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9a9dd0441df676569df85829893fe1c958632507 | 74addaa0e41490cbaf2abd313a764c96df57b05d | /Mathlib/category_theory/category/Cat.lean | 5494837c721a13db23af91263afef6307984be30 | [] | no_license | AurelienSaue/Mathlib4_auto | f538cfd0980f65a6361eadea39e6fc639e9dae14 | 590df64109b08190abe22358fabc3eae000943f2 | refs/heads/master | 1,683,906,849,776 | 1,622,564,669,000 | 1,622,564,669,000 | 371,723,747 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 2,426 | lean | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.default
import Mathlib.category_theory.concrete_category.default
import Mathlib.category_theory.discrete_category
import Mathlib.category_theory.eq_to_hom
import Mathlib.PostPort
universes v u u_1 u_2 u_3 u_4
namespace Mathlib
/-!
# Category of categories
This file contains the definition of the category `Cat` of all categories.
In this category objects are categories and
morphisms are functors between these categories.
## Implementation notes
Though `Cat` is not a concrete category, we use `bundled` to define
its carrier type.
-/
namespace category_theory
/-- Category of categories. -/
def Cat :=
bundled category
namespace Cat
protected instance inhabited : Inhabited Cat :=
{ default := bundled.mk (Type u) }
protected instance has_coe_to_sort : has_coe_to_sort Cat :=
has_coe_to_sort.mk (Type u) bundled.α
protected instance str (C : Cat) : category ↥C :=
bundled.str C
/-- Construct a bundled `Cat` from the underlying type and the typeclass. -/
def of (C : Type u) [category C] : Cat :=
bundled.of C
/-- Category structure on `Cat` -/
protected instance category : large_category Cat :=
category.mk
/-- Functor that gets the set of objects of a category. It is not
called `forget`, because it is not a faithful functor. -/
def objects : Cat ⥤ Type u :=
functor.mk (fun (C : Cat) => ↥C) fun (C D : Cat) (F : C ⟶ D) => functor.obj F
/-- Any isomorphism in `Cat` induces an equivalence of the underlying categories. -/
def equiv_of_iso {C : Cat} {D : Cat} (γ : C ≅ D) : ↥C ≌ ↥D :=
equivalence.mk' (iso.hom γ) (iso.inv γ) (eq_to_iso sorry) (eq_to_iso (iso.inv_hom_id γ))
end Cat
/--
Embedding `Type` into `Cat` as discrete categories.
This ought to be modelled as a 2-functor!
-/
@[simp] theorem Type_to_Cat_obj (X : Type u) : functor.obj Type_to_Cat X = Cat.of (discrete X) :=
Eq.refl (functor.obj Type_to_Cat X)
protected instance Type_to_Cat.faithful : faithful Type_to_Cat :=
faithful.mk
protected instance Type_to_Cat.full : full Type_to_Cat :=
full.mk
fun (X Y : Type (max (max (max u_1 u_2 u_3 u_4) u_1 u_2 u_3) (max u_1 u_2 u_3 u_4) u_1 u_2))
(F : functor.obj Type_to_Cat X ⟶ functor.obj Type_to_Cat Y) => functor.obj F
|
2035f1936fad780da98c71fbd243c8bf20b9e7f4 | 74addaa0e41490cbaf2abd313a764c96df57b05d | /Mathlib/data/polynomial/denoms_clearable_auto.lean | d8460df044a24e39f10f3b50bad3be01f1a870ad | [] | no_license | AurelienSaue/Mathlib4_auto | f538cfd0980f65a6361eadea39e6fc639e9dae14 | 590df64109b08190abe22358fabc3eae000943f2 | refs/heads/master | 1,683,906,849,776 | 1,622,564,669,000 | 1,622,564,669,000 | 371,723,747 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 2,944 | lean | /-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.default
import Mathlib.data.polynomial.erase_lead
import Mathlib.PostPort
universes u_1 u_2
namespace Mathlib
/-!
# Denominators of evaluation of polynomials at ratios
Let `i : R → K` be a homomorphism of semirings. Assume that `K` is commutative. If `a` and
`b` are elements of `R` such that `i b ∈ K` is invertible, then for any polynomial
`f ∈ polynomial R` the "mathematical" expression `b ^ f.nat_degree * f (a / b) ∈ K` is in
the image of the homomorphism `i`.
-/
-- TODO: use hypothesis (ub : is_unit (i b)) to work with localizations.
/-- `denoms_clearable` formalizes the property that `b ^ N * f (a / b)`
does not have denominators, if the inequality `f.nat_degree ≤ N` holds.
In the implementation, we also use provide an inverse in the existential.
-/
def denoms_clearable {R : Type u_1} {K : Type u_2} [semiring R] [comm_semiring K] (a : R) (b : R)
(N : ℕ) (f : polynomial R) (i : R →+* K) :=
∃ (D : R),
∃ (bi : K),
bi * coe_fn i b = 1 ∧
coe_fn i D = coe_fn i b ^ N * polynomial.eval (coe_fn i a * bi) (polynomial.map i f)
theorem denoms_clearable_zero {R : Type u_1} {K : Type u_2} [semiring R] [comm_semiring K]
{i : R →+* K} {b : R} {bi : K} (N : ℕ) (a : R) (bu : bi * coe_fn i b = 1) :
denoms_clearable a b N 0 i :=
sorry
theorem denoms_clearable_C_mul_X_pow {R : Type u_1} {K : Type u_2} [semiring R] [comm_semiring K]
{i : R →+* K} {b : R} {bi : K} {N : ℕ} (a : R) (bu : bi * coe_fn i b = 1) {n : ℕ} (r : R)
(nN : n ≤ N) : denoms_clearable a b N (coe_fn polynomial.C r * polynomial.X ^ n) i :=
sorry
theorem denoms_clearable.add {R : Type u_1} {K : Type u_2} [semiring R] [comm_semiring K]
{i : R →+* K} {a : R} {b : R} {N : ℕ} {f : polynomial R} {g : polynomial R} :
denoms_clearable a b N f i → denoms_clearable a b N g i → denoms_clearable a b N (f + g) i :=
sorry
theorem denoms_clearable_of_nat_degree_le {R : Type u_1} {K : Type u_2} [semiring R]
[comm_semiring K] {i : R →+* K} {b : R} {bi : K} (N : ℕ) (a : R) (bu : bi * coe_fn i b = 1)
(f : polynomial R) : polynomial.nat_degree f ≤ N → denoms_clearable a b N f i :=
sorry
/-- If `i : R → K` is a ring homomorphism, `f` is a polynomial with coefficients in `R`,
`a, b` are elements of `R`, with `i b` invertible, then there is a `D ∈ R` such that
`b ^ f.nat_degree * f (a / b)` equals `i D`. -/
theorem denoms_clearable_nat_degree {R : Type u_1} {K : Type u_2} [semiring R] [comm_semiring K]
{b : R} {bi : K} (i : R →+* K) (f : polynomial R) (a : R) (bu : bi * coe_fn i b = 1) :
denoms_clearable a b (polynomial.nat_degree f) f i :=
denoms_clearable_of_nat_degree_le (polynomial.nat_degree f) a bu f le_rfl
end Mathlib |
3e6859066920df241c6f3ee2b8ab889b4ec43906 | dd4e652c749fea9ac77e404005cb3470e5f75469 | /src/one_by_one_matrix.lean | 34ae6d8f894b6a19da22efb3c366e54c6effc50c | [] | no_license | skbaek/cvx | e32822ad5943541539966a37dee162b0a5495f55 | c50c790c9116f9fac8dfe742903a62bdd7292c15 | refs/heads/master | 1,623,803,010,339 | 1,618,058,958,000 | 1,618,058,958,000 | 176,293,135 | 3 | 2 | null | null | null | null | UTF-8 | Lean | false | false | 1,253 | lean | import .mat
-- TODO: move
lemma fin.eq_zero_fin1 (i : fin 1) : i = 0 :=
begin
rw fin.eq_iff_veq,
apply nat.eq_zero_of_le_zero (nat.le_of_lt_succ i.2)
end
namespace one_by_one_matrix
/-
def coe {α : Type*} : (matrix (fin 1) (fin 1) α) → α := λ A, A 0 0
instance coe_instance (α : Type*) : has_coe (matrix (fin 1) (fin 1) α) α :=
⟨coe⟩
@[simp] lemma coe_add (α : Type*) [has_add α] (A B : matrix (fin 1) (fin 1) α) :
coe (A + B) = coe A + coe B :=
by simp [coe]
@[simp] lemma coe_smul (α : Type*) [has_mul α] (a : α) (A : matrix (fin 1) (fin 1) α) :
coe (a • A) = a * coe A :=
by simp [coe, has_scalar.smul]
@[simp] lemma coe_smul' (α : Type*) [has_mul α] (a : α) (A : matrix (fin 1) (fin 1) α) :
coe (a • A) = a * coe A :=
by simp [coe, has_scalar.smul]
@[simp] lemma coe_neg (α : Type*) [has_neg α] (A : matrix (fin 1) (fin 1) α) :
coe (- A) = - coe A :=
by simp [coe]
@[simp] lemma coe_zero (α : Type*) [has_zero α] :
coe (0 : matrix (fin 1) (fin 1) α) = 0 :=
by simp [coe]
-/
@[simp] lemma transpose (α : Type*) [ring α] (A : matrix (fin 1) (fin 1) α) : Aᵀ = A :=
begin
ext i j,
unfold matrix.transpose,
rw [fin.eq_zero_fin1 i, fin.eq_zero_fin1 j]
end
end one_by_one_matrix |
cc4551fabf09eb04f47e90b5ffc5ecfb5f0cac44 | f1b175e38ffc5cc1c7c5551a72d0dbaf70786f83 | /algebra/group.lean | 790c98b66d8ef8337b2a74b2093b03dda7e6dd27 | [
"Apache-2.0"
] | permissive | mjendrusch/mathlib | df3ae884dd5ce38c7edf452bcbfd3baf4e3a6214 | 5c209edb7eb616a26f64efe3500f2b1ba95b8d55 | refs/heads/master | 1,585,663,284,800 | 1,539,062,055,000 | 1,539,062,055,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 26,782 | lean | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
Various multiplicative and additive structures.
-/
import tactic.interactive data.option
section pending_1857
/- Transport multiplicative to additive -/
section transport
open tactic
@[user_attribute]
meta def to_additive_attr : user_attribute (name_map name) name :=
{ name := `to_additive,
descr := "Transport multiplicative to additive",
cache_cfg := ⟨λ ns, ns.mfoldl (λ dict n, do
val ← to_additive_attr.get_param n,
pure $ dict.insert n val) mk_name_map, []⟩,
parser := lean.parser.ident,
after_set := some $ λ src _ _, do
env ← get_env,
dict ← to_additive_attr.get_cache,
tgt ← to_additive_attr.get_param src,
(get_decl tgt >> skip) <|>
transport_with_dict dict src tgt }
end transport
/- map operations -/
attribute [to_additive has_add.add] has_mul.mul
attribute [to_additive has_zero.zero] has_one.one
attribute [to_additive has_neg.neg] has_inv.inv
attribute [to_additive has_add] has_mul
attribute [to_additive has_zero] has_one
attribute [to_additive has_neg] has_inv
/- map constructors -/
attribute [to_additive has_add.mk] has_mul.mk
attribute [to_additive has_zero.mk] has_one.mk
attribute [to_additive has_neg.mk] has_inv.mk
/- map structures -/
attribute [to_additive add_semigroup] semigroup
attribute [to_additive add_semigroup.mk] semigroup.mk
attribute [to_additive add_semigroup.to_has_add] semigroup.to_has_mul
attribute [to_additive add_semigroup.add_assoc] semigroup.mul_assoc
attribute [to_additive add_semigroup.add] semigroup.mul
attribute [to_additive add_comm_semigroup] comm_semigroup
attribute [to_additive add_comm_semigroup.mk] comm_semigroup.mk
attribute [to_additive add_comm_semigroup.to_add_semigroup] comm_semigroup.to_semigroup
attribute [to_additive add_comm_semigroup.add_comm] comm_semigroup.mul_comm
attribute [to_additive add_left_cancel_semigroup] left_cancel_semigroup
attribute [to_additive add_left_cancel_semigroup.mk] left_cancel_semigroup.mk
attribute [to_additive add_left_cancel_semigroup.to_add_semigroup] left_cancel_semigroup.to_semigroup
attribute [to_additive add_left_cancel_semigroup.add_left_cancel] left_cancel_semigroup.mul_left_cancel
attribute [to_additive add_right_cancel_semigroup] right_cancel_semigroup
attribute [to_additive add_right_cancel_semigroup.mk] right_cancel_semigroup.mk
attribute [to_additive add_right_cancel_semigroup.to_add_semigroup] right_cancel_semigroup.to_semigroup
attribute [to_additive add_right_cancel_semigroup.add_right_cancel] right_cancel_semigroup.mul_right_cancel
attribute [to_additive add_monoid] monoid
attribute [to_additive add_monoid.mk] monoid.mk
attribute [to_additive add_monoid.to_has_zero] monoid.to_has_one
attribute [to_additive add_monoid.to_add_semigroup] monoid.to_semigroup
attribute [to_additive add_monoid.add] monoid.mul
attribute [to_additive add_monoid.add_assoc] monoid.mul_assoc
attribute [to_additive add_monoid.zero] monoid.one
attribute [to_additive add_monoid.zero_add] monoid.one_mul
attribute [to_additive add_monoid.add_zero] monoid.mul_one
attribute [to_additive add_comm_monoid] comm_monoid
attribute [to_additive add_comm_monoid.mk] comm_monoid.mk
attribute [to_additive add_comm_monoid.to_add_monoid] comm_monoid.to_monoid
attribute [to_additive add_comm_monoid.to_add_comm_semigroup] comm_monoid.to_comm_semigroup
attribute [to_additive add_group] group
attribute [to_additive add_group.mk] group.mk
attribute [to_additive add_group.to_has_neg] group.to_has_inv
attribute [to_additive add_group.to_add_monoid] group.to_monoid
attribute [to_additive add_group.add_left_neg] group.mul_left_inv
attribute [to_additive add_group.add] group.mul
attribute [to_additive add_group.add_assoc] group.mul_assoc
attribute [to_additive add_group.zero] group.one
attribute [to_additive add_group.zero_add] group.one_mul
attribute [to_additive add_group.add_zero] group.mul_one
attribute [to_additive add_group.neg] group.inv
attribute [to_additive add_comm_group] comm_group
attribute [to_additive add_comm_group.mk] comm_group.mk
attribute [to_additive add_comm_group.to_add_group] comm_group.to_group
attribute [to_additive add_comm_group.to_add_comm_monoid] comm_group.to_comm_monoid
/- map theorems -/
attribute [to_additive add_assoc] mul_assoc
attribute [to_additive add_semigroup_to_is_associative] semigroup_to_is_associative
attribute [to_additive add_comm] mul_comm
attribute [to_additive add_comm_semigroup_to_is_commutative] comm_semigroup_to_is_commutative
attribute [to_additive add_left_comm] mul_left_comm
attribute [to_additive add_right_comm] mul_right_comm
attribute [to_additive add_left_cancel] mul_left_cancel
attribute [to_additive add_right_cancel] mul_right_cancel
attribute [to_additive add_left_cancel_iff] mul_left_cancel_iff
attribute [to_additive add_right_cancel_iff] mul_right_cancel_iff
attribute [to_additive zero_add] one_mul
attribute [to_additive add_zero] mul_one
attribute [to_additive add_left_neg] mul_left_inv
attribute [to_additive neg_add_self] inv_mul_self
attribute [to_additive neg_add_cancel_left] inv_mul_cancel_left
attribute [to_additive neg_add_cancel_right] inv_mul_cancel_right
attribute [to_additive neg_eq_of_add_eq_zero] inv_eq_of_mul_eq_one
attribute [to_additive neg_zero] one_inv
attribute [to_additive neg_neg] inv_inv
attribute [to_additive add_right_neg] mul_right_inv
attribute [to_additive add_neg_self] mul_inv_self
attribute [to_additive neg_inj] inv_inj
attribute [to_additive add_group.add_left_cancel] group.mul_left_cancel
attribute [to_additive add_group.add_right_cancel] group.mul_right_cancel
attribute [to_additive add_group.to_left_cancel_add_semigroup] group.to_left_cancel_semigroup
attribute [to_additive add_group.to_right_cancel_add_semigroup] group.to_right_cancel_semigroup
attribute [to_additive add_neg_cancel_left] mul_inv_cancel_left
attribute [to_additive add_neg_cancel_right] mul_inv_cancel_right
attribute [to_additive neg_add_rev] mul_inv_rev
attribute [to_additive eq_neg_of_eq_neg] eq_inv_of_eq_inv
attribute [to_additive eq_neg_of_add_eq_zero] eq_inv_of_mul_eq_one
attribute [to_additive eq_add_neg_of_add_eq] eq_mul_inv_of_mul_eq
attribute [to_additive eq_neg_add_of_add_eq] eq_inv_mul_of_mul_eq
attribute [to_additive neg_add_eq_of_eq_add] inv_mul_eq_of_eq_mul
attribute [to_additive add_neg_eq_of_eq_add] mul_inv_eq_of_eq_mul
attribute [to_additive eq_add_of_add_neg_eq] eq_mul_of_mul_inv_eq
attribute [to_additive eq_add_of_neg_add_eq] eq_mul_of_inv_mul_eq
attribute [to_additive add_eq_of_eq_neg_add] mul_eq_of_eq_inv_mul
attribute [to_additive add_eq_of_eq_add_neg] mul_eq_of_eq_mul_inv
attribute [to_additive neg_add] mul_inv
end pending_1857
universes u v
variables {α : Type u} {β : Type v}
def additive (α : Type*) := α
def multiplicative (α : Type*) := α
instance [semigroup α] : add_semigroup (additive α) :=
{ add := ((*) : α → α → α),
add_assoc := @mul_assoc _ _ }
instance [add_semigroup α] : semigroup (multiplicative α) :=
{ mul := ((+) : α → α → α),
mul_assoc := @add_assoc _ _ }
instance [comm_semigroup α] : add_comm_semigroup (additive α) :=
{ add_comm := @mul_comm _ _,
..additive.add_semigroup }
instance [add_comm_semigroup α] : comm_semigroup (multiplicative α) :=
{ mul_comm := @add_comm _ _,
..multiplicative.semigroup }
instance [left_cancel_semigroup α] : add_left_cancel_semigroup (additive α) :=
{ add_left_cancel := @mul_left_cancel _ _,
..additive.add_semigroup }
instance [add_left_cancel_semigroup α] : left_cancel_semigroup (multiplicative α) :=
{ mul_left_cancel := @add_left_cancel _ _,
..multiplicative.semigroup }
instance [right_cancel_semigroup α] : add_right_cancel_semigroup (additive α) :=
{ add_right_cancel := @mul_right_cancel _ _,
..additive.add_semigroup }
instance [add_right_cancel_semigroup α] : right_cancel_semigroup (multiplicative α) :=
{ mul_right_cancel := @add_right_cancel _ _,
..multiplicative.semigroup }
@[simp, to_additive add_left_inj]
theorem mul_left_inj [left_cancel_semigroup α] (a : α) {b c : α} : a * b = a * c ↔ b = c :=
⟨mul_left_cancel, congr_arg _⟩
@[simp, to_additive add_right_inj]
theorem mul_right_inj [right_cancel_semigroup α] (a : α) {b c : α} : b * a = c * a ↔ b = c :=
⟨mul_right_cancel, congr_arg _⟩
structure units (α : Type u) [monoid α] :=
(val : α)
(inv : α)
(val_inv : val * inv = 1)
(inv_val : inv * val = 1)
namespace units
variables [monoid α] {a b c : units α}
instance : has_coe (units α) α := ⟨val⟩
@[extensionality] theorem ext : ∀ {a b : units α}, (a : α) = b → a = b
| ⟨v, i₁, vi₁, iv₁⟩ ⟨v', i₂, vi₂, iv₂⟩ e :=
by change v = v' at e; subst v'; congr;
simpa only [iv₂, vi₁, one_mul, mul_one] using mul_assoc i₂ v i₁
theorem ext_iff {a b : units α} : a = b ↔ (a : α) = b :=
⟨congr_arg _, ext⟩
instance [decidable_eq α] : decidable_eq (units α)
| a b := decidable_of_iff' _ ext_iff
protected def mul (u₁ u₂ : units α) : units α :=
⟨u₁.val * u₂.val, u₂.inv * u₁.inv,
have u₁.val * (u₂.val * u₂.inv) * u₁.inv = 1,
by rw [u₂.val_inv]; rw [mul_one, u₁.val_inv],
by simpa only [mul_assoc],
have u₂.inv * (u₁.inv * u₁.val) * u₂.val = 1,
by rw [u₁.inv_val]; rw [mul_one, u₂.inv_val],
by simpa only [mul_assoc]⟩
protected def inv' (u : units α) : units α :=
⟨u.inv, u.val, u.inv_val, u.val_inv⟩
instance : has_mul (units α) := ⟨units.mul⟩
instance : has_one (units α) := ⟨⟨1, 1, mul_one 1, one_mul 1⟩⟩
instance : has_inv (units α) := ⟨units.inv'⟩
variables (a b)
@[simp] lemma coe_mul : (↑(a * b) : α) = a * b := rfl
@[simp] lemma coe_one : ((1 : units α) : α) = 1 := rfl
lemma val_coe : (↑a : α) = a.val := rfl
lemma coe_inv : ((a⁻¹ : units α) : α) = a.inv := rfl
@[simp] lemma inv_mul : (↑a⁻¹ * a : α) = 1 := inv_val _
@[simp] lemma mul_inv : (a * ↑a⁻¹ : α) = 1 := val_inv _
@[simp] lemma mul_inv_cancel_left (a : units α) (b : α) : (a:α) * (↑a⁻¹ * b) = b :=
by rw [← mul_assoc, mul_inv, one_mul]
@[simp] lemma inv_mul_cancel_left (a : units α) (b : α) : (↑a⁻¹:α) * (a * b) = b :=
by rw [← mul_assoc, inv_mul, one_mul]
@[simp] lemma mul_inv_cancel_right (a : α) (b : units α) : a * b * ↑b⁻¹ = a :=
by rw [mul_assoc, mul_inv, mul_one]
@[simp] lemma inv_mul_cancel_right (a : α) (b : units α) : a * ↑b⁻¹ * b = a :=
by rw [mul_assoc, inv_mul, mul_one]
instance : group (units α) :=
by refine {mul := (*), one := 1, inv := has_inv.inv, ..};
{ intros, apply ext, simp only [coe_mul, coe_one,
mul_assoc, one_mul, mul_one, inv_mul] }
instance {α} [comm_monoid α] : comm_group (units α) :=
{ mul_comm := λ u₁ u₂, ext $ mul_comm _ _, ..units.group }
instance [has_repr α] : has_repr (units α) := ⟨repr ∘ val⟩
@[simp] theorem mul_left_inj (a : units α) {b c : α} : (a:α) * b = a * c ↔ b = c :=
⟨λ h, by simpa only [inv_mul_cancel_left] using congr_arg ((*) ↑(a⁻¹ : units α)) h, congr_arg _⟩
@[simp] theorem mul_right_inj (a : units α) {b c : α} : b * a = c * a ↔ b = c :=
⟨λ h, by simpa only [mul_inv_cancel_right] using congr_arg (* ↑(a⁻¹ : units α)) h, congr_arg _⟩
end units
theorem nat.units_eq_one (u : units ℕ) : u = 1 :=
units.ext $ nat.eq_one_of_dvd_one ⟨u.inv, u.val_inv.symm⟩
def units.mk_of_mul_eq_one [comm_monoid α] (a b : α) (hab : a * b = 1) : units α :=
⟨a, b, hab, (mul_comm b a).trans hab⟩
@[to_additive with_zero]
def with_one (α) := option α
@[to_additive with_zero.has_coe_t]
instance : has_coe_t α (with_one α) := ⟨some⟩
instance [semigroup α] : monoid (with_one α) :=
{ one := none,
mul := option.lift_or_get (*),
mul_assoc := (option.lift_or_get_assoc _).1,
one_mul := (option.lift_or_get_is_left_id _).1,
mul_one := (option.lift_or_get_is_right_id _).1 }
attribute [to_additive with_zero.add_monoid._proof_1] with_one.monoid._proof_1
attribute [to_additive with_zero.add_monoid._proof_2] with_one.monoid._proof_2
attribute [to_additive with_zero.add_monoid._proof_3] with_one.monoid._proof_3
attribute [to_additive with_zero.add_monoid] with_one.monoid
instance [semigroup α] : mul_zero_class (with_zero α) :=
{ zero := none,
mul := λ o₁ o₂, o₁.bind (λ a, o₂.map (λ b, a * b)),
zero_mul := λ a, rfl,
mul_zero := λ a, by cases a; refl,
..with_zero.add_monoid }
instance [semigroup α] : semigroup (with_zero α) :=
{ mul_assoc := λ a b c, match a, b, c with
| none, _, _ := rfl
| some a, none, _ := rfl
| some a, some b, none := rfl
| some a, some b, some c := congr_arg some (mul_assoc _ _ _)
end,
..with_zero.mul_zero_class }
instance [comm_semigroup α] : comm_semigroup (with_zero α) :=
{ mul_comm := λ a b, match a, b with
| none, _ := (mul_zero _).symm
| some a, none := rfl
| some a, some b := congr_arg some (mul_comm _ _)
end,
..with_zero.semigroup }
instance [monoid α] : monoid (with_zero α) :=
{ one := some 1,
one_mul := λ a, match a with
| none := rfl
| some a := congr_arg some $ one_mul _
end,
mul_one := λ a, match a with
| none := rfl
| some a := congr_arg some $ mul_one _
end,
..with_zero.semigroup }
instance [comm_monoid α] : comm_monoid (with_zero α) :=
{ ..with_zero.monoid, ..with_zero.comm_semigroup }
instance [monoid α] : add_monoid (additive α) :=
{ zero := (1 : α),
zero_add := @one_mul _ _,
add_zero := @mul_one _ _,
..additive.add_semigroup }
instance [add_monoid α] : monoid (multiplicative α) :=
{ one := (0 : α),
one_mul := @zero_add _ _,
mul_one := @add_zero _ _,
..multiplicative.semigroup }
section monoid
variables [monoid α] {a b c : α}
/-- Partial division. It is defined when the
second argument is invertible, and unlike the division operator
in `division_ring` it is not totalized at zero. -/
def divp (a : α) (u) : α := a * (u⁻¹ : units α)
infix ` /ₚ `:70 := divp
@[simp] theorem divp_self (u : units α) : (u : α) /ₚ u = 1 := units.mul_inv _
@[simp] theorem divp_one (a : α) : a /ₚ 1 = a := mul_one _
theorem divp_assoc (a b : α) (u : units α) : a * b /ₚ u = a * (b /ₚ u) :=
mul_assoc _ _ _
@[simp] theorem divp_mul_cancel (a : α) (u : units α) : a /ₚ u * u = a :=
(mul_assoc _ _ _).trans $ by rw [units.inv_mul, mul_one]
@[simp] theorem mul_divp_cancel (a : α) (u : units α) : (a * u) /ₚ u = a :=
(mul_assoc _ _ _).trans $ by rw [units.mul_inv, mul_one]
@[simp] theorem divp_right_inj (u : units α) {a b : α} : a /ₚ u = b /ₚ u ↔ a = b :=
units.mul_right_inj _
theorem divp_eq_one (a : α) (u : units α) : a /ₚ u = 1 ↔ a = u :=
(units.mul_right_inj u).symm.trans $ by rw [divp_mul_cancel, one_mul]
@[simp] theorem one_divp (u : units α) : 1 /ₚ u = ↑u⁻¹ :=
one_mul _
end monoid
instance [comm_semigroup α] : comm_monoid (with_one α) :=
{ mul_comm := (option.lift_or_get_comm _).1,
..with_one.monoid }
instance [add_comm_semigroup α] : add_comm_monoid (with_zero α) :=
{ add_comm := (option.lift_or_get_comm _).1,
..with_zero.add_monoid }
attribute [to_additive with_zero.add_comm_monoid] with_one.comm_monoid
instance [comm_monoid α] : add_comm_monoid (additive α) :=
{ add_comm := @mul_comm α _,
..additive.add_monoid }
instance [add_comm_monoid α] : comm_monoid (multiplicative α) :=
{ mul_comm := @add_comm α _,
..multiplicative.monoid }
instance [group α] : add_group (additive α) :=
{ neg := @has_inv.inv α _,
add_left_neg := @mul_left_inv _ _,
..additive.add_monoid }
instance [add_group α] : group (multiplicative α) :=
{ inv := @has_neg.neg α _,
mul_left_inv := @add_left_neg _ _,
..multiplicative.monoid }
section group
variables [group α] {a b c : α}
instance : has_lift α (units α) :=
⟨λ a, ⟨a, a⁻¹, mul_inv_self _, inv_mul_self _⟩⟩
@[simp, to_additive neg_inj']
theorem inv_inj' : a⁻¹ = b⁻¹ ↔ a = b :=
⟨λ h, by rw [← inv_inv a, h, inv_inv], congr_arg _⟩
@[to_additive eq_of_neg_eq_neg]
theorem eq_of_inv_eq_inv : a⁻¹ = b⁻¹ → a = b :=
inv_inj'.1
@[simp, to_additive add_self_iff_eq_zero]
theorem mul_self_iff_eq_one : a * a = a ↔ a = 1 :=
by have := @mul_left_inj _ _ a a 1; rwa mul_one at this
@[simp, to_additive neg_eq_zero]
theorem inv_eq_one : a⁻¹ = 1 ↔ a = 1 :=
by rw [← @inv_inj' _ _ a 1, one_inv]
@[simp, to_additive neg_ne_zero]
theorem inv_ne_one : a⁻¹ ≠ 1 ↔ a ≠ 1 :=
not_congr inv_eq_one
@[to_additive left_inverse_neg]
theorem left_inverse_inv (α) [group α] :
function.left_inverse (λ a : α, a⁻¹) (λ a, a⁻¹) :=
assume a, inv_inv a
attribute [simp] mul_inv_cancel_left add_neg_cancel_left
mul_inv_cancel_right add_neg_cancel_right
@[to_additive eq_neg_iff_eq_neg]
theorem eq_inv_iff_eq_inv : a = b⁻¹ ↔ b = a⁻¹ :=
⟨eq_inv_of_eq_inv, eq_inv_of_eq_inv⟩
@[to_additive neg_eq_iff_neg_eq]
theorem inv_eq_iff_inv_eq : a⁻¹ = b ↔ b⁻¹ = a :=
by rw [eq_comm, @eq_comm _ _ a, eq_inv_iff_eq_inv]
@[to_additive add_eq_zero_iff_eq_neg]
theorem mul_eq_one_iff_eq_inv : a * b = 1 ↔ a = b⁻¹ :=
by simpa [mul_left_inv, -mul_right_inj] using @mul_right_inj _ _ b a (b⁻¹)
@[to_additive add_eq_zero_iff_neg_eq]
theorem mul_eq_one_iff_inv_eq : a * b = 1 ↔ a⁻¹ = b :=
by rw [mul_eq_one_iff_eq_inv, eq_inv_iff_eq_inv, eq_comm]
@[to_additive eq_neg_iff_add_eq_zero]
theorem eq_inv_iff_mul_eq_one : a = b⁻¹ ↔ a * b = 1 :=
mul_eq_one_iff_eq_inv.symm
@[to_additive neg_eq_iff_add_eq_zero]
theorem inv_eq_iff_mul_eq_one : a⁻¹ = b ↔ a * b = 1 :=
mul_eq_one_iff_inv_eq.symm
@[to_additive eq_add_neg_iff_add_eq]
theorem eq_mul_inv_iff_mul_eq : a = b * c⁻¹ ↔ a * c = b :=
⟨λ h, by rw [h, inv_mul_cancel_right], λ h, by rw [← h, mul_inv_cancel_right]⟩
@[to_additive eq_neg_add_iff_add_eq]
theorem eq_inv_mul_iff_mul_eq : a = b⁻¹ * c ↔ b * a = c :=
⟨λ h, by rw [h, mul_inv_cancel_left], λ h, by rw [← h, inv_mul_cancel_left]⟩
@[to_additive neg_add_eq_iff_eq_add]
theorem inv_mul_eq_iff_eq_mul : a⁻¹ * b = c ↔ b = a * c :=
⟨λ h, by rw [← h, mul_inv_cancel_left], λ h, by rw [h, inv_mul_cancel_left]⟩
@[to_additive add_neg_eq_iff_eq_add]
theorem mul_inv_eq_iff_eq_mul : a * b⁻¹ = c ↔ a = c * b :=
⟨λ h, by rw [← h, inv_mul_cancel_right], λ h, by rw [h, mul_inv_cancel_right]⟩
@[to_additive add_neg_eq_zero]
theorem mul_inv_eq_one {a b : α} : a * b⁻¹ = 1 ↔ a = b :=
by rw [mul_eq_one_iff_eq_inv, inv_inv]
@[to_additive neg_comm_of_comm]
theorem inv_comm_of_comm {a b : α} (H : a * b = b * a) : a⁻¹ * b = b * a⁻¹ :=
begin
have : a⁻¹ * (b * a) * a⁻¹ = a⁻¹ * (a * b) * a⁻¹ :=
congr_arg (λ x:α, a⁻¹ * x * a⁻¹) H.symm,
rwa [inv_mul_cancel_left, mul_assoc, mul_inv_cancel_right] at this
end
end group
instance [comm_group α] : add_comm_group (additive α) :=
{ add_comm := @mul_comm α _,
..additive.add_group }
instance [add_comm_group α] : comm_group (multiplicative α) :=
{ mul_comm := @add_comm α _,
..multiplicative.group }
section add_monoid
variables [add_monoid α] {a b c : α}
@[simp] lemma bit0_zero : bit0 (0 : α) = 0 := add_zero _
@[simp] lemma bit1_zero [has_one α] : bit1 (0 : α) = 1 :=
show 0+0+1=(1:α), by rw [zero_add, zero_add]
end add_monoid
section add_group
variables [add_group α] {a b c : α}
local attribute [simp] sub_eq_add_neg
def sub_sub_cancel := @sub_sub_self
@[simp] lemma sub_left_inj : a - b = a - c ↔ b = c :=
(add_left_inj _).trans neg_inj'
@[simp] lemma sub_right_inj : b - a = c - a ↔ b = c :=
add_right_inj _
lemma sub_add_sub_cancel (a b c : α) : (a - b) + (b - c) = a - c :=
by rw [← add_sub_assoc, sub_add_cancel]
lemma sub_sub_sub_cancel_right (a b c : α) : (a - c) - (b - c) = a - b :=
by rw [← neg_sub c b, sub_neg_eq_add, sub_add_sub_cancel]
theorem sub_eq_zero : a - b = 0 ↔ a = b :=
⟨eq_of_sub_eq_zero, λ h, by rw [h, sub_self]⟩
theorem sub_ne_zero : a - b ≠ 0 ↔ a ≠ b :=
not_congr sub_eq_zero
theorem eq_sub_iff_add_eq : a = b - c ↔ a + c = b :=
eq_add_neg_iff_add_eq
theorem sub_eq_iff_eq_add : a - b = c ↔ a = c + b :=
add_neg_eq_iff_eq_add
theorem eq_iff_eq_of_sub_eq_sub {a b c d : α} (H : a - b = c - d) : a = b ↔ c = d :=
by rw [← sub_eq_zero, H, sub_eq_zero]
theorem left_inverse_sub_add_left (c : α) : function.left_inverse (λ x, x - c) (λ x, x + c) :=
assume x, add_sub_cancel x c
theorem left_inverse_add_left_sub (c : α) : function.left_inverse (λ x, x + c) (λ x, x - c) :=
assume x, sub_add_cancel x c
theorem left_inverse_add_right_neg_add (c : α) :
function.left_inverse (λ x, c + x) (λ x, - c + x) :=
assume x, add_neg_cancel_left c x
theorem left_inverse_neg_add_add_right (c : α) :
function.left_inverse (λ x, - c + x) (λ x, c + x) :=
assume x, neg_add_cancel_left c x
end add_group
section add_comm_group
variables [add_comm_group α] {a b c : α}
lemma sub_eq_neg_add (a b : α) : a - b = -b + a :=
add_comm _ _
theorem neg_add' (a b : α) : -(a + b) = -a - b := neg_add a b
lemma eq_sub_iff_add_eq' : a = b - c ↔ c + a = b :=
by rw [eq_sub_iff_add_eq, add_comm]
lemma sub_eq_iff_eq_add' : a - b = c ↔ a = b + c :=
by rw [sub_eq_iff_eq_add, add_comm]
lemma add_sub_cancel' (a b : α) : a + b - a = b :=
by rw [sub_eq_neg_add, neg_add_cancel_left]
lemma add_sub_cancel'_right (a b : α) : a + (b - a) = b :=
by rw [← add_sub_assoc, add_sub_cancel']
lemma sub_right_comm (a b c : α) : a - b - c = a - c - b :=
add_right_comm _ _ _
lemma sub_add_sub_cancel' (a b c : α) : (a - b) + (c - a) = c - b :=
by rw add_comm; apply sub_add_sub_cancel
lemma sub_sub_sub_cancel_left (a b c : α) : (c - a) - (c - b) = b - a :=
by rw [← neg_sub b c, sub_neg_eq_add, add_comm, sub_add_sub_cancel]
end add_comm_group
section is_conj
variables [group α] [group β]
def is_conj (a b : α) := ∃ c : α, c * a * c⁻¹ = b
@[refl] lemma is_conj_refl (a : α) : is_conj a a :=
⟨1, by rw [one_mul, one_inv, mul_one]⟩
@[symm] lemma is_conj_symm (a b : α) : is_conj a b → is_conj b a
| ⟨c, hc⟩ := ⟨c⁻¹, by rw [← hc, mul_assoc, mul_inv_cancel_right, inv_mul_cancel_left]⟩
@[trans] lemma is_conj_trans (a b c : α) : is_conj a b → is_conj b c → is_conj a c
| ⟨c₁, hc₁⟩ ⟨c₂, hc₂⟩ := ⟨c₂ * c₁, by rw [← hc₂, ← hc₁, mul_inv_rev]; simp only [mul_assoc]⟩
@[simp] lemma is_conj_iff_eq {α : Type*} [comm_group α] {a b : α} : is_conj a b ↔ a = b :=
⟨λ ⟨c, hc⟩, by rw [← hc, mul_right_comm, mul_inv_self, one_mul], λ h, by rw h⟩
end is_conj
class is_monoid_hom [monoid α] [monoid β] (f : α → β) : Prop :=
(map_one : f 1 = 1)
(map_mul : ∀ {x y}, f (x * y) = f x * f y)
class is_add_monoid_hom [add_monoid α] [add_monoid β] (f : α → β) : Prop :=
(map_zero : f 0 = 0)
(map_add : ∀ {x y}, f (x + y) = f x + f y)
attribute [to_additive is_add_monoid_hom] is_monoid_hom
attribute [to_additive is_add_monoid_hom.map_add] is_monoid_hom.map_mul
attribute [to_additive is_add_monoid_hom.mk] is_monoid_hom.mk
namespace is_monoid_hom
variables [monoid α] [monoid β] (f : α → β) [is_monoid_hom f]
@[to_additive is_add_monoid_hom.id]
instance id : is_monoid_hom (@id α) := by refine {..}; intros; refl
@[to_additive is_add_monoid_hom.id]
instance comp {γ} [monoid γ] (g : β → γ) [is_monoid_hom g] :
is_monoid_hom (g ∘ f) :=
{ map_mul := λ x y, show g _ = g _ * g _, by rw [map_mul f, map_mul g],
map_one := show g _ = 1, by rw [map_one f, map_one g] }
end is_monoid_hom
-- TODO rename fields of is_group_hom: mul ↝ map_mul?
/-- Predicate for group homomorphism. -/
class is_group_hom [group α] [group β] (f : α → β) : Prop :=
(mul : ∀ a b : α, f (a * b) = f a * f b)
class is_add_group_hom [add_group α] [add_group β] (f : α → β) : Prop :=
(add : ∀ a b, f (a + b) = f a + f b)
attribute [to_additive is_add_group_hom] is_group_hom
attribute [to_additive is_add_group_hom.add] is_group_hom.mul
attribute [to_additive is_add_group_hom.mk] is_group_hom.mk
namespace is_group_hom
variables [group α] [group β] (f : α → β) [is_group_hom f]
@[to_additive is_add_group_hom.zero]
theorem one : f 1 = 1 :=
mul_self_iff_eq_one.1 $ by rw [← mul f, one_mul]
@[to_additive is_add_group_hom.neg]
theorem inv (a : α) : f a⁻¹ = (f a)⁻¹ :=
eq_inv_of_mul_eq_one $ by rw [← mul f, inv_mul_self, one f]
@[to_additive is_add_group_hom.id]
instance id : is_group_hom (@id α) :=
⟨λ _ _, rfl⟩
@[to_additive is_add_group_hom.comp]
instance comp {γ} [group γ] (g : β → γ) [is_group_hom g] :
is_group_hom (g ∘ f) :=
⟨λ x y, show g _ = g _ * g _, by rw [mul f, mul g]⟩
protected lemma is_conj (f : α → β) [is_group_hom f] {a b : α} : is_conj a b → is_conj (f a) (f b)
| ⟨c, hc⟩ := ⟨f c, by rw [← is_group_hom.mul f, ← is_group_hom.inv f, ← is_group_hom.mul f, hc]⟩
end is_group_hom
/-- Predicate for group anti-homomorphism, or a homomorphism
into the opposite group. -/
class is_group_anti_hom {β : Type*} [group α] [group β] (f : α → β) : Prop :=
(mul : ∀ a b : α, f (a * b) = f b * f a)
namespace is_group_anti_hom
variables [group α] [group β] (f : α → β) [w : is_group_anti_hom f]
include w
theorem one : f 1 = 1 :=
mul_self_iff_eq_one.1 $ by rw [← mul f, one_mul]
theorem inv (a : α) : f a⁻¹ = (f a)⁻¹ :=
eq_inv_of_mul_eq_one $ by rw [← mul f, mul_inv_self, one f]
end is_group_anti_hom
theorem inv_is_group_anti_hom [group α] : is_group_anti_hom (λ x : α, x⁻¹) :=
⟨mul_inv_rev⟩
namespace is_add_group_hom
variables [add_group α] [add_group β] (f : α → β) [is_add_group_hom f]
lemma sub (a b) : f (a - b) = f a - f b :=
calc f (a - b) = f (a + -b) : rfl
... = f a + f (-b) : add f _ _
... = f a - f b : by simp[neg f]
end is_add_group_hom
namespace units
variables [monoid α] [monoid β] (f : α → β) [is_monoid_hom f]
definition map : units α → units β :=
λ u, ⟨f u.val, f u.inv,
by rw [← is_monoid_hom.map_mul f, u.val_inv, is_monoid_hom.map_one f],
by rw [← is_monoid_hom.map_mul f, u.inv_val, is_monoid_hom.map_one f] ⟩
instance : is_group_hom (units.map f) :=
⟨λ a b, by ext; exact is_monoid_hom.map_mul f ⟩
end units
|
6a520a02621fa46eab7b8c22839286c16cff09e7 | 618003631150032a5676f229d13a079ac875ff77 | /src/measure_theory/lebesgue_measure.lean | fb0127b90ff55a1721ed912d509ff4cc2c18ab63 | [
"Apache-2.0"
] | permissive | awainverse/mathlib | 939b68c8486df66cfda64d327ad3d9165248c777 | ea76bd8f3ca0a8bf0a166a06a475b10663dec44a | refs/heads/master | 1,659,592,962,036 | 1,590,987,592,000 | 1,590,987,592,000 | 268,436,019 | 1 | 0 | Apache-2.0 | 1,590,990,500,000 | 1,590,990,500,000 | null | UTF-8 | Lean | false | false | 12,254 | lean | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
Lebesgue measure on the real line
-/
import measure_theory.measure_space
import measure_theory.borel_space
noncomputable theory
open classical set filter
open nnreal (of_real)
namespace measure_theory
/-- Length of an interval. This is the largest monotonic function which correctly
measures all intervals. -/
def lebesgue_length (s : set ℝ) : ennreal := ⨅a b (h : s ⊆ Ico a b), of_real (b - a)
@[simp] lemma lebesgue_length_empty : lebesgue_length ∅ = 0 :=
le_zero_iff_eq.1 $ infi_le_of_le 0 $ infi_le_of_le 0 $ by simp
@[simp] lemma lebesgue_length_Ico (a b : ℝ) :
lebesgue_length (Ico a b) = of_real (b - a) :=
begin
refine le_antisymm (infi_le_of_le a $ infi_le_of_le b $ infi_le _ (by refl))
(le_infi $ λ a', le_infi $ λ b', le_infi $ λ h, ennreal.coe_le_coe.2 _),
cases le_or_lt b a with ab ab,
{ rw nnreal.of_real_of_nonpos (sub_nonpos.2 ab), simp },
cases (Ico_subset_Ico_iff ab).1 h with h₁ h₂,
exact nnreal.of_real_le_of_real (sub_le_sub h₂ h₁)
end
lemma lebesgue_length_mono {s₁ s₂ : set ℝ} (h : s₁ ⊆ s₂) : lebesgue_length s₁ ≤ lebesgue_length s₂ :=
infi_le_infi $ λ a, infi_le_infi $ λ b, infi_le_infi2 $ λ h', ⟨subset.trans h h', le_refl _⟩
lemma lebesgue_length_eq_infi_Ioo (s) : lebesgue_length s = ⨅a b (h : s ⊆ Ioo a b), of_real (b - a) :=
begin
refine le_antisymm
(infi_le_infi $ λ a, infi_le_infi $ λ b, infi_le_infi2 $ λ h,
⟨subset.trans h Ioo_subset_Ico_self, le_refl _⟩) _,
refine le_infi (λ a, le_infi $ λ b, le_infi $ λ h, _),
refine ennreal.le_of_forall_epsilon_le (λ ε ε0 _, _),
refine infi_le_of_le (a - ε) (infi_le_of_le b $ infi_le_of_le
(subset.trans h $ Ico_subset_Ioo_left $ (sub_lt_self_iff _).2 ε0) _),
rw [← sub_add, ← ennreal.coe_add, ennreal.coe_le_coe],
apply le_trans nnreal.of_real_add_le _,
simp,
end
@[simp] lemma lebesgue_length_Ioo (a b : ℝ) :
lebesgue_length (Ioo a b) = of_real (b - a) :=
begin
rw ← lebesgue_length_Ico,
refine le_antisymm (lebesgue_length_mono Ioo_subset_Ico_self) _,
rw lebesgue_length_eq_infi_Ioo (Ioo a b),
refine (le_infi $ λ a', le_infi $ λ b', le_infi $ λ h, _),
cases le_or_lt b a with ab ab, {simp [ab]},
cases (Ioo_subset_Ioo_iff ab).1 h with h₁ h₂,
rw [lebesgue_length_Ico, ennreal.coe_le_coe],
exact nnreal.of_real_le_of_real (sub_le_sub h₂ h₁)
end
lemma lebesgue_length_eq_infi_Icc (s) : lebesgue_length s = ⨅a b (h : s ⊆ Icc a b), of_real (b - a) :=
begin
refine le_antisymm _
(infi_le_infi $ λ a, infi_le_infi $ λ b, infi_le_infi2 $ λ h,
⟨subset.trans h Ico_subset_Icc_self, le_refl _⟩),
refine le_infi (λ a, le_infi $ λ b, le_infi $ λ h, _),
refine ennreal.le_of_forall_epsilon_le (λ ε ε0 _, _),
refine infi_le_of_le a (infi_le_of_le (b + ε) $ infi_le_of_le
(subset.trans h $ Icc_subset_Ico_right $ (lt_add_iff_pos_right _).2 ε0) _),
rw [sub_eq_add_neg, add_right_comm, ←ennreal.coe_add, ennreal.coe_le_coe],
apply le_trans nnreal.of_real_add_le,
simp [sub_eq_add_neg]
end
@[simp] lemma lebesgue_length_Icc (a b : ℝ) :
lebesgue_length (Icc a b) = of_real (b - a) :=
begin
rw ← lebesgue_length_Ico,
refine le_antisymm _ (lebesgue_length_mono Ico_subset_Icc_self),
rw lebesgue_length_eq_infi_Icc (Icc a b),
exact infi_le_of_le a (infi_le_of_le b $ infi_le_of_le (by refl) (by simp))
end
/-- The Lebesgue outer measure, as an outer measure of ℝ. -/
def lebesgue_outer : outer_measure ℝ :=
outer_measure.of_function lebesgue_length lebesgue_length_empty
lemma lebesgue_outer_le_length (s : set ℝ) : lebesgue_outer s ≤ lebesgue_length s :=
outer_measure.of_function_le _ _ _
lemma lebesgue_length_subadditive {a b : ℝ} {c d : ℕ → ℝ}
(ss : Icc a b ⊆ ⋃i, Ioo (c i) (d i)) :
(of_real (b - a) : ennreal) ≤ ∑' i, of_real (d i - c i) :=
begin
suffices : ∀ (s:finset ℕ) b
(cv : Icc a b ⊆ ⋃ i ∈ (↑s:set ℕ), Ioo (c i) (d i)),
(of_real (b - a) : ennreal) ≤ s.sum (λ i, of_real (d i - c i)),
{ rcases compact_Icc.elim_finite_subcover_image (λ (i : ℕ) (_ : i ∈ univ),
@is_open_Ioo _ _ _ _ (c i) (d i)) (by simpa using ss) with ⟨s, su, hf, hs⟩,
have e : (⋃ i ∈ (↑hf.to_finset:set ℕ),
Ioo (c i) (d i)) = (⋃ i ∈ s, Ioo (c i) (d i)), {simp [set.ext_iff]},
rw ennreal.tsum_eq_supr_sum,
refine le_trans _ (le_supr _ hf.to_finset),
exact this hf.to_finset _ (by simpa [e]) },
clear ss b,
refine λ s, finset.strong_induction_on s (λ s IH b cv, _),
cases le_total b a with ab ab,
{ rw nnreal.of_real_of_nonpos (sub_nonpos.2 ab), simp },
have := cv ⟨ab, le_refl _⟩, simp at this,
rcases this with ⟨i, is, cb, bd⟩,
rw [← finset.insert_erase is] at cv ⊢,
rw [finset.coe_insert, bUnion_insert] at cv,
rw [finset.sum_insert (finset.not_mem_erase _ _)],
refine le_trans _ (add_le_add_left' (IH _ (finset.erase_ssubset is) (c i) _)),
{ rw [← ennreal.coe_add, ennreal.coe_le_coe],
refine le_trans (nnreal.of_real_le_of_real _) nnreal.of_real_add_le,
rw sub_add_sub_cancel,
exact sub_le_sub_right (le_of_lt bd) _ },
{ rintro x ⟨h₁, h₂⟩,
refine (cv ⟨h₁, le_trans h₂ (le_of_lt cb)⟩).resolve_left
(mt and.left (not_lt_of_le h₂)) }
end
@[simp] lemma lebesgue_outer_Icc (a b : ℝ) :
lebesgue_outer (Icc a b) = of_real (b - a) :=
begin
refine le_antisymm (by rw ← lebesgue_length_Icc; apply lebesgue_outer_le_length)
(le_infi $ λ f, le_infi $ λ hf,
ennreal.le_of_forall_epsilon_le $ λ ε ε0 h, _),
rcases ennreal.exists_pos_sum_of_encodable
(ennreal.zero_lt_coe_iff.2 ε0) ℕ with ⟨ε', ε'0, hε⟩,
refine le_trans _ (add_le_add_left' (le_of_lt hε)),
rw ← ennreal.tsum_add,
choose g hg using show
∀ i, ∃ p:ℝ×ℝ, f i ⊆ Ioo p.1 p.2 ∧ (of_real (p.2 - p.1) : ennreal) < lebesgue_length (f i) + ε' i,
{ intro i,
have := (ennreal.lt_add_right (lt_of_le_of_lt (ennreal.le_tsum i) h)
(ennreal.zero_lt_coe_iff.2 (ε'0 i))),
conv at this {to_lhs, rw lebesgue_length_eq_infi_Ioo},
simpa [infi_lt_iff] },
refine le_trans _ (ennreal.tsum_le_tsum $ λ i, le_of_lt (hg i).2),
exact lebesgue_length_subadditive (subset.trans hf $
Union_subset_Union $ λ i, (hg i).1)
end
@[simp] lemma lebesgue_outer_singleton (a : ℝ) : lebesgue_outer {a} = 0 :=
by simpa using lebesgue_outer_Icc a a
@[simp] lemma lebesgue_outer_Ico (a b : ℝ) :
lebesgue_outer (Ico a b) = of_real (b - a) :=
begin
refine le_antisymm (by rw ← lebesgue_length_Ico; apply lebesgue_outer_le_length)
(ennreal.le_of_forall_epsilon_le $ λ ε ε0 h, _),
have := @nnreal.of_real_add_le (b - a - ε) ε,
rw [← ennreal.coe_le_coe, ennreal.coe_add, sub_add_cancel, sub_right_comm,
← lebesgue_outer_Icc a (b-ε), nnreal.of_real_coe] at this,
exact le_trans this (add_le_add_right' $ lebesgue_outer.mono $
Icc_subset_Ico_right $ (sub_lt_self_iff _).2 ε0)
end
@[simp] lemma lebesgue_outer_Ioo (a b : ℝ) :
lebesgue_outer (Ioo a b) = of_real (b - a) :=
begin
refine le_antisymm (by rw ← lebesgue_length_Ioo; apply lebesgue_outer_le_length)
(ennreal.le_of_forall_epsilon_le $ λ ε ε0 h, _),
have := @nnreal.of_real_add_le (b - a - ε) ε,
rw [← ennreal.coe_le_coe, ennreal.coe_add, sub_add_cancel, sub_sub,
← lebesgue_outer_Ico (a+ε) b, nnreal.of_real_coe] at this,
exact le_trans this (add_le_add_right' $ lebesgue_outer.mono $
Ico_subset_Ioo_left $ (lt_add_iff_pos_right _).2 ε0)
end
lemma is_lebesgue_measurable_Iio {c : ℝ} :
lebesgue_outer.caratheodory.is_measurable (Iio c) :=
outer_measure.caratheodory_is_measurable $ λ t,
le_infi $ λ a, le_infi $ λ b, le_infi $ λ h, begin
refine le_trans (add_le_add'
(lebesgue_length_mono $ inter_subset_inter_left _ h)
(lebesgue_length_mono $ diff_subset_diff_left h)) _,
cases le_total a c with hac hca; cases le_total b c with hbc hcb;
simp [*, -sub_eq_add_neg, sub_add_sub_cancel'];
rw [← ennreal.coe_add, ennreal.coe_le_coe],
{ simp [*, -nnreal.of_real_add, nnreal.of_real_add_of_real,
-sub_eq_add_neg, sub_add_sub_cancel'] },
{ rw nnreal.of_real_of_nonpos,
{ simp },
exact sub_nonpos.2 (le_trans hbc hca) }
end
theorem lebesgue_outer_trim : lebesgue_outer.trim = lebesgue_outer :=
begin
refine le_antisymm (λ s, _) (outer_measure.trim_ge _),
rw outer_measure.trim_eq_infi,
refine le_infi (λ f, le_infi $ λ hf,
ennreal.le_of_forall_epsilon_le $ λ ε ε0 h, _),
rcases ennreal.exists_pos_sum_of_encodable
(ennreal.zero_lt_coe_iff.2 ε0) ℕ with ⟨ε', ε'0, hε⟩,
refine le_trans _ (add_le_add_left' (le_of_lt hε)),
rw ← ennreal.tsum_add,
choose g hg using show
∀ i, ∃ s, f i ⊆ s ∧ is_measurable s ∧ lebesgue_outer s ≤ lebesgue_length (f i) + of_real (ε' i),
{ intro i,
have := (ennreal.lt_add_right (lt_of_le_of_lt (ennreal.le_tsum i) h)
(ennreal.zero_lt_coe_iff.2 (ε'0 i))),
conv at this {to_lhs, rw lebesgue_length},
simp only [infi_lt_iff] at this,
rcases this with ⟨a, b, h₁, h₂⟩,
rw ← lebesgue_outer_Ico at h₂,
exact ⟨_, h₁, is_measurable_Ico, le_of_lt $ by simpa using h₂⟩ },
simp at hg,
apply infi_le_of_le (Union g) _,
apply infi_le_of_le (subset.trans hf $ Union_subset_Union (λ i, (hg i).1)) _,
apply infi_le_of_le (is_measurable.Union (λ i, (hg i).2.1)) _,
exact le_trans (lebesgue_outer.Union _) (ennreal.tsum_le_tsum $ λ i, (hg i).2.2)
end
/-- Lebesgue measure on the Borel sets
The outer Lebesgue measure is the completion of this measure. (TODO: proof this)
-/
instance : measure_space ℝ :=
⟨{to_outer_measure := lebesgue_outer,
m_Union :=
have borel ℝ ≤ lebesgue_outer.caratheodory,
by rw real.borel_eq_generate_from_Iio_rat;
refine measurable_space.generate_from_le _;
simp [is_lebesgue_measurable_Iio] {contextual := tt},
λ f hf, lebesgue_outer.Union_eq_of_caratheodory (λ i, this _ (hf i)),
trimmed := lebesgue_outer_trim }⟩
@[simp] theorem lebesgue_to_outer_measure :
(measure_space.μ : measure ℝ).to_outer_measure = lebesgue_outer := rfl
end measure_theory
open measure_theory
section volume
open_locale interval
theorem real.volume_val (s) : volume s = lebesgue_outer s := rfl
local attribute [simp] real.volume_val
@[simp] lemma real.volume_Ico {a b : ℝ} : volume (Ico a b) = of_real (b - a) := by simp
@[simp] lemma real.volume_Icc {a b : ℝ} : volume (Icc a b) = of_real (b - a) := by simp
@[simp] lemma real.volume_Ioo {a b : ℝ} : volume (Ioo a b) = of_real (b - a) := by simp
@[simp] lemma real.volume_singleton {a : ℝ} : volume ({a} : set ℝ) = 0 := by simp
@[simp] lemma real.volume_interval {a b : ℝ} : volume [a, b] = of_real (abs (b - a)) :=
begin
rw [interval, real.volume_Icc],
congr,
exact max_sub_min_eq_abs _ _
end
open metric
lemma real.volume_lt_top_of_bounded {s : set ℝ} (h : bounded s) : volume s < ⊤ :=
begin
rw [real.bounded_iff_bdd_below_bdd_above, bdd_below_bdd_above_iff_subset_interval] at h,
rcases h with ⟨a, b, h⟩,
calc volume s ≤ volume [a, b] : volume_mono h
... < ⊤ : by { rw real.volume_interval, exact ennreal.coe_lt_top }
end
lemma real.volume_lt_top_of_compact {s : set ℝ} (h : compact s) : volume s < ⊤ :=
real.volume_lt_top_of_bounded (bounded_of_compact h)
end volume
/-
section vitali
def vitali_aux_h (x : ℝ) (h : x ∈ Icc (0:ℝ) 1) :
∃ y ∈ Icc (0:ℝ) 1, ∃ q:ℚ, ↑q = x - y :=
⟨x, h, 0, by simp⟩
def vitali_aux (x : ℝ) (h : x ∈ Icc (0:ℝ) 1) : ℝ :=
classical.some (vitali_aux_h x h)
theorem vitali_aux_mem (x : ℝ) (h : x ∈ Icc (0:ℝ) 1) : vitali_aux x h ∈ Icc (0:ℝ) 1 :=
Exists.fst (classical.some_spec (vitali_aux_h x h):_)
theorem vitali_aux_rel (x : ℝ) (h : x ∈ Icc (0:ℝ) 1) :
∃ q:ℚ, ↑q = x - vitali_aux x h :=
Exists.snd (classical.some_spec (vitali_aux_h x h):_)
def vitali : set ℝ := {x | ∃ h, x = vitali_aux x h}
theorem vitali_nonmeasurable : ¬ is_null_measurable measure_space.μ vitali :=
sorry
end vitali
-/
|
300078b1151c24158292be16ddf8157343b88fe4 | 74addaa0e41490cbaf2abd313a764c96df57b05d | /Mathlib/group_theory/archimedean_auto.lean | 0c64810b96d8c137d7bcf43ee39ba774b7d36e88 | [] | no_license | AurelienSaue/Mathlib4_auto | f538cfd0980f65a6361eadea39e6fc639e9dae14 | 590df64109b08190abe22358fabc3eae000943f2 | refs/heads/master | 1,683,906,849,776 | 1,622,564,669,000 | 1,622,564,669,000 | 371,723,747 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 2,098 | lean | /-
Copyright (c) 2020 Heather Macbeth, Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Patrick Massot
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.default
import Mathlib.group_theory.subgroup
import Mathlib.algebra.archimedean
import Mathlib.PostPort
universes u_1
namespace Mathlib
/-!
# Archimedean groups
This file proves a few facts about ordered groups which satisfy the `archimedean` property, that is:
`class archimedean (α) [ordered_add_comm_monoid α] : Prop :=`
`(arch : ∀ (x : α) {y}, 0 < y → ∃ n : ℕ, x ≤ n •ℕ y)`
They are placed here in a separate file (rather than incorporated as a continuation of
`algebra.archimedean`) because they rely on some imports from `group_theory` -- bundled subgroups
in particular.
The main result is `add_subgroup.cyclic_of_min`: a subgroup of a decidable archimedean abelian
group is cyclic, if its set of positive elements has a minimal element.
This result is used in this file to deduce `int.subgroup_cyclic`, proving that every subgroup of `ℤ`
is cyclic. (There are several other methods one could use to prove this fact, including more purely
algebraic methods, but none seem to exist in mathlib as of writing. The closest is
`subgroup.is_cyclic`, but that has not been transferred to `add_subgroup`.)
The result is also used in `topology.instances.real` as an ingredient in the classification of
subgroups of `ℝ`.
-/
/-- Given a subgroup `H` of a decidable linearly ordered archimedean abelian group `G`, if there
exists a minimal element `a` of `H ∩ G_{>0}` then `H` is generated by `a`. -/
theorem add_subgroup.cyclic_of_min {G : Type u_1} [linear_ordered_add_comm_group G] [archimedean G]
{H : add_subgroup G} {a : G} (ha : is_least (set_of fun (g : G) => g ∈ H ∧ 0 < g) a) :
H = add_subgroup.closure (singleton a) :=
sorry
/-- Every subgroup of `ℤ` is cyclic. -/
theorem int.subgroup_cyclic (H : add_subgroup ℤ) :
∃ (a : ℤ), H = add_subgroup.closure (singleton a) :=
sorry
end Mathlib |
38941336f499c32530147c715242198446a82858 | 63abd62053d479eae5abf4951554e1064a4c45b4 | /src/tactic/ring_exp.lean | 6d076b526409034a9ede963fe2b4409411e574cb | [
"Apache-2.0"
] | permissive | Lix0120/mathlib | 0020745240315ed0e517cbf32e738d8f9811dd80 | e14c37827456fc6707f31b4d1d16f1f3a3205e91 | refs/heads/master | 1,673,102,855,024 | 1,604,151,044,000 | 1,604,151,044,000 | 308,930,245 | 0 | 0 | Apache-2.0 | 1,604,164,710,000 | 1,604,163,547,000 | null | UTF-8 | Lean | false | false | 56,181 | lean | /-
Copyright (c) 2019 Tim Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Tim Baanen.
Solve equations in commutative (semi)rings with exponents.
-/
import tactic.norm_num
import control.traversable.basic
/-!
# `ring_exp` tactic
A tactic for solving equations in commutative (semi)rings,
where the exponents can also contain variables.
More precisely, expressions of the following form are supported:
- constants (non-negative integers)
- variables
- coefficients (any rational number, embedded into the (semi)ring)
- addition of expressions
- multiplication of expressions
- exponentiation of expressions (the exponent must have type `ℕ`)
- subtraction and negation of expressions (if the base is a full ring)
The motivating example is proving `2 * 2^n * b = b * 2^(n+1)`,
something that the `ring` tactic cannot do, but `ring_exp` can.
## Implementation notes
The basic approach to prove equalities is to normalise both sides and check for equality.
The normalisation is guided by building a value in the type `ex` at the meta level,
together with a proof (at the base level) that the original value is equal to
the normalised version.
The normalised version and normalisation proofs are also stored in the `ex` type.
The outline of the file:
- Define an inductive family of types `ex`, parametrised over `ex_type`,
which can represent expressions with `+`, `*`, `^` and rational numerals.
The parametrisation over `ex_type` ensures that associativity and distributivity are applied,
by restricting which kinds of subexpressions appear as arguments to the various operators.
- Represent addition, multiplication and exponentiation in the `ex` type,
thus allowing us to map expressions to `ex` (the `eval` function drives this).
We apply associativity and distributivity of the operators here (helped by `ex_type`)
and commutativity as well (by sorting the subterms; unfortunately not helped by anything).
Any expression not of the above formats is treated as an atom (the same as a variable).
There are some details we glossed over which make the plan more complicated:
- The order on atoms is not initially obvious.
We construct a list containing them in order of initial appearance in the expression,
then use the index into the list as a key to order on.
- In the tactic, a normalized expression `ps : ex` lives in the meta-world,
but the normalization proofs live in the real world.
Thus, we cannot directly say `ps.orig = ps.pretty` anywhere,
but we have to carefully construct the proof when we compute `ps`.
This was a major source of bugs in development!
- For `pow`, the exponent must be a natural number, while the base can be any semiring `α`.
We swap out operations for the base ring `α` with those for the exponent ring `ℕ`
as soon as we deal with exponents.
This is accomplished by the `in_exponent` function and is relatively painless since
we work in a `reader` monad.
- The normalized form of an expression is the one that is useful for the tactic,
but not as nice to read. To remedy this, the user-facing normalization calls `ex.simp`.
## Caveats and future work
Subtraction cancels out identical terms, but division does not.
That is: `a - a = 0 := by ring_exp` solves the goal,
but `a / a := 1 by ring_exp` doesn't.
Note that `0 / 0` is generally defined to be `0`,
so division cancelling out is not true in general.
Multiplication of powers can be simplified a little bit further:
`2 ^ n * 2 ^ n = 4 ^ n := by ring_exp` could be implemented
in a similar way that `2 * a + 2 * a = 4 * a := by ring_exp` already works.
This feature wasn't needed yet, so it's not implemented yet.
## Tags
ring, semiring, exponent, power
-/
-- The base ring `α` will have a universe level `u`.
-- We do not introduce `α` as a variable yet,
-- in order to make it explicit or implicit as required.
universes u
namespace tactic.ring_exp
open nat
/--
The `atom` structure is used to represent atomic expressions:
those which `ring_exp` cannot parse any further.
For instance, `a + (a % b)` has `a` and `(a % b)` as atoms.
The `ring_exp_eq` tactic does not normalize the subexpressions in atoms,
but `ring_exp` does if `ring_exp_eq` was not sufficient.
Atoms in fact represent equivalence classes of expressions,
modulo definitional equality.
The field `index : ℕ` should be a unique number for each class,
while `value : expr` contains a representative of this class.
The function `resolve_atom` determines the appropriate atom
for a given expression.
-/
meta structure atom : Type := (value : expr) (index : ℕ)
namespace atom
/--
The `eq` operation on `atom`s works modulo definitional equality,
ignoring their `value`s.
The invariants on `atom` ensure indices are unique per value.
Thus, `eq` indicates equality as long as the `atom`s come from the same context.
-/
meta def eq (a b : atom) : bool := a.index = b.index
/--
We order `atom`s on the order of appearance in the main expression.
-/
meta def lt (a b : atom) : bool := a.index < b.index
meta instance : has_repr atom := ⟨λ x, "(atom " ++ repr x.2 ++ ")"⟩
end atom
section expression
/-!
### `expression` section
In this section, we define the `ex` type and its basic operations.
First, we introduce the supporting types `coeff`, `ex_type` and `ex_info`.
For understanding the code, it's easier to check out `ex` itself first,
then refer back to the supporting types.
The arithmetic operations on `ex` need additional definitions,
so they are defined in a later section.
-/
/--
Coefficients in the expression are stored in a wrapper structure,
allowing for easier modification of the data structures.
The modifications might be caching of the result of `expr.of_rat`,
or using a different meta representation of numerals.
-/
@[derive decidable_eq, derive inhabited]
structure coeff : Type := (value : ℚ)
/-- The values in `ex_type` are used as parameters to `ex` to control the expression's structure. -/
@[derive decidable_eq, derive inhabited]
inductive ex_type : Type
| base : ex_type
| sum : ex_type
| prod : ex_type
| exp : ex_type
open ex_type
/--
Each `ex` stores information for its normalization proof.
The `orig` expression is the expression that was passed to `eval`.
The `pretty` expression is the normalised form that the `ex` represents.
(I didn't call this something like `norm`, because there are already
too many things called `norm` in mathematics!)
The field `proof` contains an optional proof term of type `%%orig = %%pretty`.
The value `none` for the proof indicates that everything reduces to reflexivity.
(Which saves space in quite a lot of cases.)
-/
meta structure ex_info : Type :=
(orig : expr) (pretty : expr) (proof : option expr)
/--
The `ex` type is an abstract representation of an expression with `+`, `*` and `^`.
Those operators are mapped to the `sum`, `prod` and `exp` constructors respectively.
The `zero` constructor is the base case for `ex sum`, e.g. `1 + 2` is represented
by (something along the lines of) `sum 1 (sum 2 zero)`.
The `coeff` constructor is the base case for `ex prod`, and is used for numerals.
The code maintains the invariant that the coefficient is never `0`.
The `var` constructor is the base case for `ex exp`, and is used for atoms.
The `sum_b` constructor allows for addition in the base of an exponentiation;
it serves a similar purpose as the parentheses in `(a + b)^c`.
The code maintains the invariant that the argument to `sum_b` is not `zero`
or `sum _ zero`.
All of the constructors contain an `ex_info` field,
used to carry around (arguments to) proof terms.
While the `ex_type` parameter enforces some simplification invariants,
the following ones must be manually maintained at the risk of insufficient power:
- the argument to `coeff` must be nonzero (to ensure `0 = 0 * 1`)
- the argument to `sum_b` must be of the form `sum a (sum b bs)` (to ensure `(a + 0)^n = a^n`)
- normalisation proofs of subexpressions must be `refl ps.pretty`
- if we replace `sum` with `cons` and `zero` with `nil`, the resulting list is sorted
according to the `lt` relation defined further down; similarly for `prod` and `coeff`
(to ensure `a + b = b + a`).
The first two invariants could be encoded in a subtype of `ex`,
but aren't (yet) to spare some implementation burden.
The other invariants cannot be encoded because we need the `tactic` monad to check them.
(For example, the correct equality check of `expr` is `is_def_eq : expr → expr → tactic unit`.)
-/
meta inductive ex : ex_type → Type
| zero (info : ex_info) : ex sum
| sum (info : ex_info) : ex prod → ex sum → ex sum
| coeff (info : ex_info) : coeff → ex prod
| prod (info : ex_info) : ex exp → ex prod → ex prod
| var (info : ex_info) : atom → ex base
| sum_b (info : ex_info) : ex sum → ex base
| exp (info : ex_info) : ex base → ex prod → ex exp
/--
Return the proof information associated to the `ex`.
-/
meta def ex.info : Π {et : ex_type} (ps : ex et), ex_info
| sum (ex.zero i) := i
| sum (ex.sum i _ _) := i
| prod (ex.coeff i _) := i
| prod (ex.prod i _ _) := i
| base (ex.var i _) := i
| base (ex.sum_b i _) := i
| exp (ex.exp i _ _) := i
/--
Return the original, non-normalized version of this `ex`.
Note that arguments to another `ex` are always "pre-normalized":
their `orig` and `pretty` are equal, and their `proof` is reflexivity.
-/
meta def ex.orig {et : ex_type} (ps : ex et) : expr := ps.info.orig
/--
Return the normalized version of this `ex`.
-/
meta def ex.pretty {et : ex_type} (ps : ex et) : expr := ps.info.pretty
/--
Return the normalisation proof of the given expression.
If the proof is `refl`, we give `none` instead,
which helps to control the size of proof terms.
To get an actual term, use `ex.proof_term`,
or use `mk_proof` with the correct set of arguments.
-/
meta def ex.proof {et : ex_type} (ps : ex et) : option expr := ps.info.proof
/--
Update the `orig` and `proof` fields of the `ex_info`.
Intended for use in `ex.set_info`.
-/
meta def ex_info.set (i : ex_info) (o : option expr) (pf : option expr) : ex_info :=
{orig := o.get_or_else i.pretty, proof := pf, .. i}
/--
Update the `ex_info` of the given expression.
We use this to combine intermediate normalisation proofs.
Since `pretty` only depends on the subexpressions,
which do not change, we do not set `pretty`.
-/
meta def ex.set_info : Π {et : ex_type} (ps : ex et), option expr → option expr → ex et
| sum (ex.zero i) o pf := ex.zero (i.set o pf)
| sum (ex.sum i p ps) o pf := ex.sum (i.set o pf) p ps
| prod (ex.coeff i x) o pf := ex.coeff (i.set o pf) x
| prod (ex.prod i p ps) o pf := ex.prod (i.set o pf) p ps
| base (ex.var i x) o pf := ex.var (i.set o pf) x
| base (ex.sum_b i ps) o pf := ex.sum_b (i.set o pf) ps
| exp (ex.exp i p ps) o pf := ex.exp (i.set o pf) p ps
instance coeff_has_repr : has_repr coeff := ⟨λ x, repr x.1⟩
/-- Convert an `ex` to a `string`. -/
meta def ex.repr : Π {et : ex_type}, ex et → string
| sum (ex.zero _) := "0"
| sum (ex.sum _ p ps) := ex.repr p ++ " + " ++ ex.repr ps
| prod (ex.coeff _ x) := repr x
| prod (ex.prod _ p ps) := ex.repr p ++ " * " ++ ex.repr ps
| base (ex.var _ x) := repr x
| base (ex.sum_b _ ps) := "(" ++ ex.repr ps ++ ")"
| exp (ex.exp _ p ps) := ex.repr p ++ " ^ " ++ ex.repr ps
meta instance {et : ex_type} : has_repr (ex et) := ⟨ex.repr⟩
/--
Equality test for expressions.
Since equivalence of `atom`s is not the same as equality,
we cannot make a true `(=)` operator for `ex` either.
-/
meta def ex.eq : Π {et : ex_type}, ex et → ex et → bool
| sum (ex.zero _) (ex.zero _) := tt
| sum (ex.zero _) (ex.sum _ _ _) := ff
| sum (ex.sum _ _ _) (ex.zero _) := ff
| sum (ex.sum _ p ps) (ex.sum _ q qs) := p.eq q && ps.eq qs
| prod (ex.coeff _ x) (ex.coeff _ y) := x = y
| prod (ex.coeff _ _) (ex.prod _ _ _) := ff
| prod (ex.prod _ _ _) (ex.coeff _ _) := ff
| prod (ex.prod _ p ps) (ex.prod _ q qs) := p.eq q && ps.eq qs
| base (ex.var _ x) (ex.var _ y) := x.eq y
| base (ex.var _ _) (ex.sum_b _ _) := ff
| base (ex.sum_b _ _) (ex.var _ _) := ff
| base (ex.sum_b _ ps) (ex.sum_b _ qs) := ps.eq qs
| exp (ex.exp _ p ps) (ex.exp _ q qs) := p.eq q && ps.eq qs
/--
The ordering on expressions.
As for `ex.eq`, this is a linear order only in one context.
-/
meta def ex.lt : Π {et : ex_type}, ex et → ex et → bool
| sum _ (ex.zero _) := ff
| sum (ex.zero _) _ := tt
| sum (ex.sum _ p ps) (ex.sum _ q qs) := p.lt q || (p.eq q && ps.lt qs)
| prod (ex.coeff _ x) (ex.coeff _ y) := x.1 < y.1
| prod (ex.coeff _ _) _ := tt
| prod _ (ex.coeff _ _) := ff
| prod (ex.prod _ p ps) (ex.prod _ q qs) := p.lt q || (p.eq q && ps.lt qs)
| base (ex.var _ x) (ex.var _ y) := x.lt y
| base (ex.var _ _) (ex.sum_b _ _) := tt
| base (ex.sum_b _ _) (ex.var _ _) := ff
| base (ex.sum_b _ ps) (ex.sum_b _ qs) := ps.lt qs
| exp (ex.exp _ p ps) (ex.exp _ q qs) := p.lt q || (p.eq q && ps.lt qs)
end expression
section operations
/-!
### `operations` section
This section defines the operations (on `ex`) that use tactics.
They live in the `ring_exp_m` monad,
which adds a cache and a list of encountered atoms to the `tactic` monad.
Throughout this section, we will be constructing proof terms.
The lemmas used in the construction are all defined over a commutative semiring α.
-/
variables {α : Type u} [comm_semiring α]
open tactic
open ex_type
/--
Stores the information needed in the `eval` function and its dependencies,
so they can (re)construct expressions.
The `eval_info` structure stores this information for one type,
and the `context` combines the two types, one for bases and one for exponents.
-/
meta structure eval_info :=
(α : expr) (univ : level)
-- Cache the instances for optimization and consistency
(csr_instance : expr) (ha_instance : expr) (hm_instance : expr) (hp_instance : expr)
-- Optional instances (only required for (-) and (/) respectively)
(ring_instance : option expr) (dr_instance : option expr)
-- Cache common constants.
(zero : expr) (one : expr)
/--
The `context` contains the full set of information needed for the `eval` function.
This structure has two copies of `eval_info`:
one is for the base (typically some semiring `α`) and another for the exponent (always `ℕ`).
When evaluating an exponent, we put `info_e` in `info_b`.
-/
meta structure context :=
(info_b : eval_info) (info_e : eval_info) (transp : transparency)
/--
The `ring_exp_m` monad is used instead of `tactic` to store the context.
-/
@[derive [monad, alternative]]
meta def ring_exp_m (α : Type) : Type := reader_t context (state_t (list atom) tactic) α
/--
Access the instance cache.
-/
meta def get_context : ring_exp_m context := reader_t.read
/--
Lift an operation in the `tactic` monad to the `ring_exp_m` monad.
This operation will not access the cache.
-/
meta def lift {α} (m : tactic α) : ring_exp_m α := reader_t.lift (state_t.lift m)
/--
Change the context of the given computation,
so that expressions are evaluated in the exponent ring,
instead of the base ring.
-/
meta def in_exponent {α} (mx : ring_exp_m α) : ring_exp_m α := do
ctx ← get_context,
reader_t.lift $ mx.run ⟨ctx.info_e, ctx.info_e, ctx.transp⟩
/--
Specialized version of `mk_app` where the first two arguments are `{α}` `[some_class α]`.
Should be faster because it can use the cached instances.
-/
meta def mk_app_class (f : name) (inst : expr) (args : list expr) : ring_exp_m expr := do
ctx ← get_context,
pure $ (@expr.const tt f [ctx.info_b.univ] ctx.info_b.α inst).mk_app args
/--
Specialized version of `mk_app` where the first two arguments are `{α}` `[comm_semiring α]`.
Should be faster because it can use the cached instances.
-/
meta def mk_app_csr (f : name) (args : list expr) : ring_exp_m expr := do
ctx ← get_context,
mk_app_class f (ctx.info_b.csr_instance) args
/--
Specialized version of `mk_app ``has_add.add`.
Should be faster because it can use the cached instances.
-/
meta def mk_add (args : list expr) : ring_exp_m expr := do
ctx ← get_context,
mk_app_class ``has_add.add ctx.info_b.ha_instance args
/--
Specialized version of `mk_app ``has_mul.mul`.
Should be faster because it can use the cached instances.
-/
meta def mk_mul (args : list expr) : ring_exp_m expr := do
ctx ← get_context,
mk_app_class ``has_mul.mul ctx.info_b.hm_instance args
/--
Specialized version of `mk_app ``has_pow.pow`.
Should be faster because it can use the cached instances.
-/
meta def mk_pow (args : list expr) : ring_exp_m expr := do
ctx ← get_context,
pure $ (@expr.const tt ``has_pow.pow
[ctx.info_b.univ, ctx.info_e.univ]
ctx.info_b.α ctx.info_e.α
ctx.info_b.hp_instance).mk_app args
/-- Construct a normalization proof term or return the cached one. -/
meta def ex_info.proof_term (ps : ex_info) : ring_exp_m expr :=
match ps.proof with
| none := lift $ tactic.mk_eq_refl ps.pretty
| (some p) := pure p
end
/-- Construct a normalization proof term or return the cached one. -/
meta def ex.proof_term {et : ex_type} (ps : ex et) : ring_exp_m expr := ps.info.proof_term
/--
If all `ex_info` have trivial proofs, return a trivial proof.
Otherwise, construct all proof terms.
Useful in applications where trivial proofs combine to another trivial proof,
most importantly to pass to `mk_proof_or_refl`.
-/
meta def none_or_proof_term : list ex_info → ring_exp_m (option (list expr))
| [] := pure none
| (x :: xs) := do
xs_pfs ← none_or_proof_term xs,
match (x.proof, xs_pfs) with
| (none, none) := pure none
| (some x_pf, none) := do
xs_pfs ← traverse ex_info.proof_term xs,
pure (some (x_pf :: xs_pfs))
| (_, some xs_pfs) := do
x_pf ← x.proof_term,
pure (some (x_pf :: xs_pfs))
end
/--
Use the proof terms as arguments to the given lemma.
If the lemma could reduce to reflexivity, consider using `mk_proof_or_refl.`
-/
meta def mk_proof (lem : name) (args : list expr) (hs : list ex_info) : ring_exp_m expr := do
hs' ← traverse ex_info.proof_term hs,
mk_app_csr lem (args ++ hs')
/--
Use the proof terms as arguments to the given lemma.
Often, we construct a proof term using congruence where reflexivity suffices.
To solve this, the following function tries to get away with reflexivity.
-/
meta def mk_proof_or_refl (term : expr) (lem : name) (args : list expr) (hs : list ex_info) :
ring_exp_m expr := do
hs_full ← none_or_proof_term hs,
match hs_full with
| none := lift $ mk_eq_refl term
| (some hs') := mk_app_csr lem (args ++ hs')
end
/-- A shortcut for adding the original terms of two expressions. -/
meta def add_orig {et et'} (ps : ex et) (qs : ex et') : ring_exp_m expr :=
mk_add [ps.orig, qs.orig]
/-- A shortcut for multiplying the original terms of two expressions. -/
meta def mul_orig {et et'} (ps : ex et) (qs : ex et') : ring_exp_m expr :=
mk_mul [ps.orig, qs.orig]
/-- A shortcut for exponentiating the original terms of two expressions. -/
meta def pow_orig {et et'} (ps : ex et) (qs : ex et') : ring_exp_m expr :=
mk_pow [ps.orig, qs.orig]
/-- Congruence lemma for constructing `ex.sum`. -/
lemma sum_congr {p p' ps ps' : α} : p = p' → ps = ps' → p + ps = p' + ps' := by cc
/-- Congruence lemma for constructing `ex.prod`. -/
lemma prod_congr {p p' ps ps' : α} : p = p' → ps = ps' → p * ps = p' * ps' := by cc
/-- Congruence lemma for constructing `ex.exp`. -/
lemma exp_congr {p p' : α} {ps ps' : ℕ} : p = p' → ps = ps' → p ^ ps = p' ^ ps' := by cc
/-- Constructs `ex.zero` with the correct arguments. -/
meta def ex_zero : ring_exp_m (ex sum) := do
ctx ← get_context,
pure $ ex.zero ⟨ctx.info_b.zero, ctx.info_b.zero, none⟩
/-- Constructs `ex.sum` with the correct arguments. -/
meta def ex_sum (p : ex prod) (ps : ex sum) : ring_exp_m (ex sum) := do
pps_o ← add_orig p ps,
pps_p ← mk_add [p.pretty, ps.pretty],
pps_pf ← mk_proof_or_refl pps_p ``sum_congr
[p.orig, p.pretty, ps.orig, ps.pretty]
[p.info, ps.info],
pure (ex.sum ⟨pps_o, pps_p, pps_pf⟩ (p.set_info none none) (ps.set_info none none))
/--
Constructs `ex.coeff` with the correct arguments.
There are more efficient constructors for specific numerals:
if `x = 0`, you should use `ex_zero`; if `x = 1`, use `ex_one`.
-/
meta def ex_coeff (x : rat) : ring_exp_m (ex prod) := do
ctx ← get_context,
x_p ← lift $ expr.of_rat ctx.info_b.α x,
pure (ex.coeff ⟨x_p, x_p, none⟩ ⟨x⟩)
/--
Constructs `ex.coeff 1` with the correct arguments.
This is a special case for optimization purposes.
-/
meta def ex_one : ring_exp_m (ex prod) := do
ctx ← get_context,
pure $ ex.coeff ⟨ctx.info_b.one, ctx.info_b.one, none⟩ ⟨1⟩
/-- Constructs `ex.prod` with the correct arguments. -/
meta def ex_prod (p : ex exp) (ps : ex prod) : ring_exp_m (ex prod) := do
pps_o ← mul_orig p ps,
pps_p ← mk_mul [p.pretty, ps.pretty],
pps_pf ← mk_proof_or_refl pps_p ``prod_congr
[p.orig, p.pretty, ps.orig, ps.pretty]
[p.info, ps.info],
pure (ex.prod ⟨pps_o, pps_p, pps_pf⟩ (p.set_info none none) (ps.set_info none none))
/-- Constructs `ex.var` with the correct arguments. -/
meta def ex_var (p : atom) : ring_exp_m (ex base) := pure (ex.var ⟨p.1, p.1, none⟩ p)
/-- Constructs `ex.sum_b` with the correct arguments. -/
meta def ex_sum_b (ps : ex sum) : ring_exp_m (ex base) :=
pure (ex.sum_b ps.info (ps.set_info none none))
/-- Constructs `ex.exp` with the correct arguments. -/
meta def ex_exp (p : ex base) (ps : ex prod) : ring_exp_m (ex exp) := do
ctx ← get_context,
pps_o ← pow_orig p ps,
pps_p ← mk_pow [p.pretty, ps.pretty],
pps_pf ← mk_proof_or_refl pps_p ``exp_congr
[p.orig, p.pretty, ps.orig, ps.pretty]
[p.info, ps.info],
pure (ex.exp ⟨pps_o, pps_p, pps_pf⟩ (p.set_info none none) (ps.set_info none none))
lemma base_to_exp_pf {p p' : α} : p = p' → p = p' ^ 1 := by simp
/-- Conversion from `ex base` to `ex exp`. -/
meta def base_to_exp (p : ex base) : ring_exp_m (ex exp) := do
o ← in_exponent $ ex_one,
ps ← ex_exp p o,
pf ← mk_proof ``base_to_exp_pf [p.orig, p.pretty] [p.info],
pure $ ps.set_info p.orig pf
lemma exp_to_prod_pf {p p' : α} : p = p' → p = p' * 1 := by simp
/-- Conversion from `ex exp` to `ex prod`. -/
meta def exp_to_prod (p : ex exp) : ring_exp_m (ex prod) := do
o ← ex_one,
ps ← ex_prod p o,
pf ← mk_proof ``exp_to_prod_pf [p.orig, p.pretty] [p.info],
pure $ ps.set_info p.orig pf
lemma prod_to_sum_pf {p p' : α} : p = p' → p = p' + 0 := by simp
/-- Conversion from `ex prod` to `ex sum`. -/
meta def prod_to_sum (p : ex prod) : ring_exp_m (ex sum) := do
z ← ex_zero,
ps ← ex_sum p z,
pf ← mk_proof ``prod_to_sum_pf [p.orig, p.pretty] [p.info],
pure $ ps.set_info p.orig pf
lemma atom_to_sum_pf (p : α) : p = p ^ 1 * 1 + 0 := by simp
/--
A more efficient conversion from `atom` to `ex sum`.
The result should be the same as `ex_var p >>= base_to_exp >>= exp_to_prod >>= prod_to_sum`,
except we need to calculate less intermediate steps.
-/
meta def atom_to_sum (p : atom) : ring_exp_m (ex sum) := do
p' ← ex_var p,
o ← in_exponent $ ex_one,
p' ← ex_exp p' o,
o ← ex_one,
p' ← ex_prod p' o,
z ← ex_zero,
p' ← ex_sum p' z,
pf ← mk_proof ``atom_to_sum_pf [p.1] [],
pure $ p'.set_info p.1 pf
/--
Compute the sum of two coefficients.
Note that the result might not be a valid expression:
if `p = -q`, then the result should be `ex.zero : ex sum` instead.
The caller must detect when this happens!
The returned value is of the form `ex.coeff _ (p + q)`,
with the proof of `expr.of_rat p + expr.of_rat q = expr.of_rat (p + q)`.
-/
meta def add_coeff (p_p q_p : expr) (p q : coeff) : ring_exp_m (ex prod) := do
ctx ← get_context,
pq_o ← mk_add [p_p, q_p],
(pq_p, pq_pf) ← lift $ norm_num.eval_field pq_o,
pure $ ex.coeff ⟨pq_o, pq_p, pq_pf⟩ ⟨p.1 + q.1⟩
lemma mul_coeff_pf_one_mul (q : α) : 1 * q = q := one_mul q
lemma mul_coeff_pf_mul_one (p : α) : p * 1 = p := mul_one p
/--
Compute the product of two coefficients.
The returned value is of the form `ex.coeff _ (p * q)`,
with the proof of `expr.of_rat p * expr.of_rat q = expr.of_rat (p * q)`.
-/
meta def mul_coeff (p_p q_p : expr) (p q : coeff) : ring_exp_m (ex prod) :=
match p.1, q.1 with -- Special case to speed up multiplication with 1.
| ⟨1, 1, _, _⟩, _ := do
ctx ← get_context,
pq_o ← mk_mul [p_p, q_p],
pf ← mk_app_csr ``mul_coeff_pf_one_mul [q_p],
pure $ ex.coeff ⟨pq_o, q_p, pf⟩ ⟨q.1⟩
| _, ⟨1, 1, _, _⟩ := do
ctx ← get_context,
pq_o ← mk_mul [p_p, q_p],
pf ← mk_app_csr ``mul_coeff_pf_mul_one [p_p],
pure $ ex.coeff ⟨pq_o, p_p, pf⟩ ⟨p.1⟩
| _, _ := do
ctx ← get_context,
pq' ← mk_mul [p_p, q_p],
(pq_p, pq_pf) ← lift $ norm_num.eval_field pq',
pure $ ex.coeff ⟨pq_p, pq_p, pq_pf⟩ ⟨p.1 * q.1⟩
end
/--
Represents the way in which two products are equal except coefficient.
This type is used in the function `add_overlap`.
In order to deal with equations of the form `a * 2 + a = 3 * a`,
the `add` function will add up overlapping products,
turning `a * 2 + a` into `a * 3`.
We need to distinguish `a * 2 + a` from `a * 2 + b` in order to do this,
and the `overlap` type carries the information on how it overlaps.
The case `none` corresponds to non-overlapping products, e.g. `a * 2 + b`;
the case `nonzero` to overlapping products adding to non-zero, e.g. `a * 2 + a`
(the `ex prod` field will then look like `a * 3` with a proof that `a * 2 + a = a * 3`);
the case `zero` to overlapping products adding to zero, e.g. `a * 2 + a * -2`.
We distinguish those two cases because in the second, the whole product reduces to `0`.
A potential extension to the tactic would also do this for the base of exponents,
e.g. to show `2^n * 2^n = 4^n`.
-/
meta inductive overlap : Type
| none : overlap
| nonzero : ex prod → overlap
| zero : ex sum → overlap
lemma add_overlap_pf {ps qs pq} (p : α) : ps + qs = pq → p * ps + p * qs = p * pq := λ pq_pf, calc
p * ps + p * qs = p * (ps + qs) : symm (mul_add _ _ _)
... = p * pq : by rw pq_pf
lemma add_overlap_pf_zero {ps qs} (p : α) : ps + qs = 0 → p * ps + p * qs = 0 := λ pq_pf, calc
p * ps + p * qs = p * (ps + qs) : symm (mul_add _ _ _)
... = p * 0 : by rw pq_pf
... = 0 : mul_zero _
/--
Given arguments `ps`, `qs` of the form `ps' * x` and `ps' * y` respectively
return `ps + qs = ps' * (x + y)` (with `x` and `y` arbitrary coefficients).
For other arguments, return `overlap.none`.
-/
meta def add_overlap : ex prod → ex prod → ring_exp_m overlap
| (ex.coeff x_i x) (ex.coeff y_i y) := do
xy@(ex.coeff _ xy_c) ← add_coeff x_i.pretty y_i.pretty x y
| lift $ fail "internal error: add_coeff should return ex.coeff",
if xy_c.1 = 0
then do
z ← ex_zero,
pure $ overlap.zero (z.set_info xy.orig xy.proof)
else pure $ overlap.nonzero xy
| (ex.prod _ _ _) (ex.coeff _ _) := pure overlap.none
| (ex.coeff _ _) (ex.prod _ _ _) := pure overlap.none
| pps@(ex.prod _ p ps) qqs@(ex.prod _ q qs) := if p.eq q
then do
pq_ol ← add_overlap ps qs,
pqs_o ← add_orig pps qqs,
match pq_ol with
| overlap.none := pure overlap.none
| (overlap.nonzero pq) := do
pqs ← ex_prod p pq,
pf ← mk_proof ``add_overlap_pf
[ps.pretty, qs.pretty, pq.pretty, p.pretty]
[pq.info],
pure $ overlap.nonzero (pqs.set_info pqs_o pf)
| (overlap.zero pq) := do
z ← ex_zero,
pf ← mk_proof ``add_overlap_pf_zero
[ps.pretty, qs.pretty, p.pretty]
[pq.info],
pure $ overlap.zero (z.set_info pqs_o pf)
end
else pure overlap.none
section addition
lemma add_pf_z_sum {ps qs qs' : α} : ps = 0 → qs = qs' → ps + qs = qs' := λ ps_pf qs_pf, calc
ps + qs = 0 + qs' : by rw [ps_pf, qs_pf]
... = qs' : zero_add _
lemma add_pf_sum_z {ps ps' qs : α} : ps = ps' → qs = 0 → ps + qs = ps' := λ ps_pf qs_pf, calc
ps + qs = ps' + 0 : by rw [ps_pf, qs_pf]
... = ps' : add_zero _
lemma add_pf_sum_overlap {pps p ps qqs q qs pq pqs : α} :
pps = p + ps → qqs = q + qs → p + q = pq → ps + qs = pqs → pps + qqs = pq + pqs := by cc
lemma add_pf_sum_overlap_zero {pps p ps qqs q qs pqs : α} :
pps = p + ps → qqs = q + qs → p + q = 0 → ps + qs = pqs → pps + qqs = pqs :=
λ pps_pf qqs_pf pq_pf pqs_pf, calc
pps + qqs = (p + ps) + (q + qs) : by rw [pps_pf, qqs_pf]
... = (p + q) + (ps + qs) : by cc
... = 0 + pqs : by rw [pq_pf, pqs_pf]
... = pqs : zero_add _
lemma add_pf_sum_lt {pps p ps qqs pqs : α} :
pps = p + ps → ps + qqs = pqs → pps + qqs = p + pqs := by cc
lemma add_pf_sum_gt {pps qqs q qs pqs : α} :
qqs = q + qs → pps + qs = pqs → pps + qqs = q + pqs := by cc
/--
Add two expressions.
* `0 + qs = 0`
* `ps + 0 = 0`
* `ps * x + ps * y = ps * (x + y)` (for `x`, `y` coefficients; uses `add_overlap`)
* `(p + ps) + (q + qs) = p + (ps + (q + qs))` (if `p.lt q`)
* `(p + ps) + (q + qs) = q + ((p + ps) + qs)` (if not `p.lt q`)
-/
meta def add : ex sum → ex sum → ring_exp_m (ex sum)
| ps@(ex.zero ps_i) qs := do
pf ← mk_proof ``add_pf_z_sum [ps.orig, qs.orig, qs.pretty] [ps.info, qs.info],
pqs_o ← add_orig ps qs,
pure $ qs.set_info pqs_o pf
| ps qs@(ex.zero qs_i) := do
pf ← mk_proof ``add_pf_sum_z [ps.orig, ps.pretty, qs.orig] [ps.info, qs.info],
pqs_o ← add_orig ps qs,
pure $ ps.set_info pqs_o pf
| pps@(ex.sum pps_i p ps) qqs@(ex.sum qqs_i q qs) := do
ol ← add_overlap p q,
ppqqs_o ← add_orig pps qqs,
match ol with
| (overlap.nonzero pq) := do
pqs ← add ps qs,
pqqs ← ex_sum pq pqs,
qqs_pf ← qqs.proof_term,
pf ← mk_proof ``add_pf_sum_overlap
[pps.orig, p.pretty, ps.pretty, qqs.orig, q.pretty, qs.pretty, pq.pretty, pqs.pretty]
[pps.info, qqs.info, pq.info, pqs.info],
pure $ pqqs.set_info ppqqs_o pf
| (overlap.zero pq) := do
pqs ← add ps qs,
pf ← mk_proof ``add_pf_sum_overlap_zero
[pps.orig, p.pretty, ps.pretty, qqs.orig, q.pretty, qs.pretty, pqs.pretty]
[pps.info, qqs.info, pq.info, pqs.info],
pure $ pqs.set_info ppqqs_o pf
| overlap.none := if p.lt q
then do
pqs ← add ps qqs,
ppqs ← ex_sum p pqs,
pf ← mk_proof ``add_pf_sum_lt
[pps.orig, p.pretty, ps.pretty, qqs.orig, pqs.pretty]
[pps.info, pqs.info],
pure $ ppqs.set_info ppqqs_o pf
else do
pqs ← add pps qs,
pqqs ← ex_sum q pqs,
pf ← mk_proof ``add_pf_sum_gt
[pps.orig, qqs.orig, q.pretty, qs.pretty, pqs.pretty]
[qqs.info, pqs.info],
pure $ pqqs.set_info ppqqs_o pf
end
end addition
section multiplication
lemma mul_pf_c_c {ps ps' qs qs' pq : α} :
ps = ps' → qs = qs' → ps' * qs' = pq → ps * qs = pq := by cc
lemma mul_pf_c_prod {ps qqs q qs pqs : α} :
qqs = q * qs → ps * qs = pqs → ps * qqs = q * pqs := by cc
lemma mul_pf_prod_c {pps p ps qs pqs : α} :
pps = p * ps → ps * qs = pqs → pps * qs = p * pqs := by cc
lemma mul_pp_pf_overlap {pps p_b ps qqs qs psqs : α} {p_e q_e : ℕ} :
pps = p_b ^ p_e * ps → qqs = p_b ^ q_e * qs →
p_b ^ (p_e + q_e) * (ps * qs) = psqs → pps * qqs = psqs
:= λ ps_pf qs_pf psqs_pf, by simp [symm psqs_pf, pow_add, ps_pf, qs_pf]; ac_refl
lemma mul_pp_pf_prod_lt {pps p ps qqs pqs : α} :
pps = p * ps → ps * qqs = pqs → pps * qqs = p * pqs := by cc
lemma mul_pp_pf_prod_gt {pps qqs q qs pqs : α} :
qqs = q * qs → pps * qs = pqs → pps * qqs = q * pqs := by cc
/--
Multiply two expressions.
* `x * y = (x * y)` (for `x`, `y` coefficients)
* `x * (q * qs) = q * (qs * x)` (for `x` coefficient)
* `(p * ps) * y = p * (ps * y)` (for `y` coefficient)
* `(p_b^p_e * ps) * (p_b^q_e * qs) = p_b^(p_e + q_e) * (ps * qs)`
(if `p_e` and `q_e` are identical except coefficient)
* `(p * ps) * (q * qs) = p * (ps * (q * qs))` (if `p.lt q`)
* `(p * ps) * (q * qs) = q * ((p * ps) * qs)` (if not `p.lt q`)
-/
meta def mul_pp : ex prod → ex prod → ring_exp_m (ex prod)
| ps@(ex.coeff _ x) qs@(ex.coeff _ y) := do
pq ← mul_coeff ps.pretty qs.pretty x y,
pq_o ← mul_orig ps qs,
pf ← mk_proof_or_refl pq.pretty ``mul_pf_c_c
[ps.orig, ps.pretty, qs.orig, qs.pretty, pq.pretty]
[ps.info, qs.info, pq.info],
pure $ pq.set_info pq_o pf
| ps@(ex.coeff _ x) qqs@(ex.prod _ q qs) := do
pqs ← mul_pp ps qs,
pqqs ← ex_prod q pqs,
pqqs_o ← mul_orig ps qqs,
pf ← mk_proof ``mul_pf_c_prod
[ps.orig, qqs.orig, q.pretty, qs.pretty, pqs.pretty]
[qqs.info, pqs.info],
pure $ pqqs.set_info pqqs_o pf
| pps@(ex.prod _ p ps) qs@(ex.coeff _ y) := do
pqs ← mul_pp ps qs,
ppqs ← ex_prod p pqs,
ppqs_o ← mul_orig pps qs,
pf ← mk_proof ``mul_pf_prod_c
[pps.orig, p.pretty, ps.pretty, qs.orig, pqs.pretty]
[pps.info, pqs.info],
pure $ ppqs.set_info ppqs_o pf
| pps@(ex.prod _ p@(ex.exp _ p_b p_e) ps) qqs@(ex.prod _ q@(ex.exp _ q_b q_e) qs) := do
ppqqs_o ← mul_orig pps qqs,
pq_ol ← in_exponent $ add_overlap p_e q_e,
match pq_ol, p_b.eq q_b with
| (overlap.nonzero pq_e), tt := do
psqs ← mul_pp ps qs,
pq ← ex_exp p_b pq_e,
ppsqqs ← ex_prod pq psqs,
pf ← mk_proof ``mul_pp_pf_overlap
[pps.orig, p_b.pretty, ps.pretty, qqs.orig, qs.pretty, ppsqqs.pretty, p_e.pretty, q_e.pretty]
[pps.info, qqs.info, ppsqqs.info],
pure $ ppsqqs.set_info ppqqs_o pf
| _, _ := if p.lt q then do
pqs ← mul_pp ps qqs,
ppqs ← ex_prod p pqs,
pf ← mk_proof ``mul_pp_pf_prod_lt
[pps.orig, p.pretty, ps.pretty, qqs.orig, pqs.pretty]
[pps.info, pqs.info],
pure $ ppqs.set_info ppqqs_o pf
else do
pqs ← mul_pp pps qs,
pqqs ← ex_prod q pqs,
pf ← mk_proof ``mul_pp_pf_prod_gt
[pps.orig, qqs.orig, q.pretty, qs.pretty, pqs.pretty]
[qqs.info, pqs.info],
pure $ pqqs.set_info ppqqs_o pf
end
lemma mul_p_pf_zero {ps qs : α} : ps = 0 → ps * qs = 0 :=
λ ps_pf, by rw [ps_pf, zero_mul]
lemma mul_p_pf_sum {pps p ps qs ppsqs : α} : pps = p + ps →
p * qs + ps * qs = ppsqs → pps * qs = ppsqs := λ pps_pf ppsqs_pf, calc
pps * qs = (p + ps) * qs : by rw [pps_pf]
... = p * qs + ps * qs : add_mul _ _ _
... = ppsqs : ppsqs_pf
/--
Multiply two expressions.
* `0 * qs = 0`
* `(p + ps) * qs = (p * qs) + (ps * qs)`
-/
meta def mul_p : ex sum → ex prod → ring_exp_m (ex sum)
| ps@(ex.zero ps_i) qs := do
z ← ex_zero,
z_o ← mul_orig ps qs,
pf ← mk_proof ``mul_p_pf_zero [ps.orig, qs.orig] [ps.info],
pure $ z.set_info z_o pf
| pps@(ex.sum pps_i p ps) qs := do
pqs ← mul_pp p qs >>= prod_to_sum,
psqs ← mul_p ps qs,
ppsqs ← add pqs psqs,
pps_pf ← pps.proof_term,
ppsqs_o ← mul_orig pps qs,
ppsqs_pf ← ppsqs.proof_term,
pf ← mk_proof ``mul_p_pf_sum
[pps.orig, p.pretty, ps.pretty, qs.orig, ppsqs.pretty]
[pps.info, ppsqs.info],
pure $ ppsqs.set_info ppsqs_o pf
lemma mul_pf_zero {ps qs : α} : qs = 0 → ps * qs = 0 :=
λ qs_pf, by rw [qs_pf, mul_zero]
lemma mul_pf_sum {ps qqs q qs psqqs : α} : qqs = q + qs → ps * q + ps * qs = psqqs →
ps * qqs = psqqs := λ qs_pf psqqs_pf, calc
ps * qqs = ps * (q + qs) : by rw [qs_pf]
... = ps * q + ps * qs : mul_add _ _ _
... = psqqs : psqqs_pf
/--
Multiply two expressions.
* `ps * 0 = 0`
* `ps * (q + qs) = (ps * q) + (ps * qs)`
-/
meta def mul : ex sum → ex sum → ring_exp_m (ex sum)
| ps qs@(ex.zero qs_i) := do
z ← ex_zero,
z_o ← mul_orig ps qs,
pf ← mk_proof ``mul_pf_zero [ps.orig, qs.orig] [qs.info],
pure $ z.set_info z_o pf
| ps qqs@(ex.sum qqs_i q qs) := do
psq ← mul_p ps q,
psqs ← mul ps qs,
psqqs ← add psq psqs,
psqqs_o ← mul_orig ps qqs,
pf ← mk_proof ``mul_pf_sum
[ps.orig, qqs.orig, q.orig, qs.orig, psqqs.pretty]
[qqs.info, psqqs.info],
pure $ psqqs.set_info psqqs_o pf
end multiplication
section exponentiation
lemma pow_e_pf_exp {pps p : α} {ps qs psqs : ℕ} :
pps = p ^ ps → ps * qs = psqs → pps ^ qs = p ^ psqs :=
λ pps_pf psqs_pf, calc
pps ^ qs = (p ^ ps) ^ qs : by rw [pps_pf]
... = p ^ (ps * qs) : symm (pow_mul _ _ _)
... = p ^ psqs : by rw [psqs_pf]
/--
Compute the exponentiation of two coefficients.
The returned value is of the form `ex.coeff _ (p ^ q)`,
with the proof of `expr.of_rat p ^ expr.of_rat q = expr.of_rat (p ^ q)`.
-/
meta def pow_coeff (p_p q_p : expr) (p q : coeff) : ring_exp_m (ex prod) := do
ctx ← get_context,
pq' ← mk_pow [p_p, q_p],
(pq_p, pq_pf) ← lift $ norm_num.eval_pow pq',
pure $ ex.coeff ⟨pq_p, pq_p, pq_pf⟩ ⟨p.1 * q.1⟩
/--
Exponentiate two expressions.
* `(p ^ ps) ^ qs = p ^ (ps * qs)`
-/
meta def pow_e : ex exp → ex prod → ring_exp_m (ex exp)
| pps@(ex.exp pps_i p ps) qs := do
psqs ← in_exponent $ mul_pp ps qs,
ppsqs ← ex_exp p psqs,
ppsqs_o ← pow_orig pps qs,
pf ← mk_proof ``pow_e_pf_exp
[pps.orig, p.pretty, ps.pretty, qs.orig, psqs.pretty]
[pps.info, psqs.info],
pure $ ppsqs.set_info ppsqs_o pf
lemma pow_pp_pf_one {ps : α} {qs : ℕ} : ps = 1 → ps ^ qs = 1 :=
λ ps_pf, by rw [ps_pf, one_pow]
lemma pow_pf_c_c {ps ps' pq : α} {qs qs' : ℕ} :
ps = ps' → qs = qs' → ps' ^ qs' = pq → ps ^ qs = pq := by cc
lemma pow_pp_pf_c {ps ps' pqs : α} {qs qs' : ℕ} :
ps = ps' → qs = qs' → ps' ^ qs' = pqs → ps ^ qs = pqs * 1 :=
by simp; cc
lemma pow_pp_pf_prod {pps p ps pqs psqs : α} {qs : ℕ} : pps = p * ps →
p ^ qs = pqs → ps ^ qs = psqs → pps ^ qs = pqs * psqs :=
λ pps_pf pqs_pf psqs_pf, calc
pps ^ qs = (p * ps) ^ qs : by rw [pps_pf]
... = p ^ qs * ps ^ qs : mul_pow _ _ _
... = pqs * psqs : by rw [pqs_pf, psqs_pf]
/--
Exponentiate two expressions.
* `1 ^ qs = 1`
* `x ^ qs = x ^ qs` (for `x` coefficient)
* `(p * ps) ^ qs = p ^ qs + ps ^ qs`
-/
meta def pow_pp : ex prod → ex prod → ring_exp_m (ex prod)
| ps@(ex.coeff ps_i ⟨⟨1, 1, _, _⟩⟩) qs := do
o ← ex_one,
o_o ← pow_orig ps qs,
pf ← mk_proof ``pow_pp_pf_one [ps.orig, qs.orig] [ps.info],
pure $ o.set_info o_o pf
| ps@(ex.coeff ps_i x) qs@(ex.coeff qs_i y) := do
pq ← pow_coeff ps.pretty qs.pretty x y,
pq_o ← pow_orig ps qs,
pf ← mk_proof_or_refl pq.pretty ``pow_pf_c_c
[ps.orig, ps.pretty, pq.pretty, qs.orig, qs.pretty]
[ps.info, qs.info, pq.info],
pure $ pq.set_info pq_o pf
| ps@(ex.coeff ps_i x) qs := do
ps'' ← pure ps >>= prod_to_sum >>= ex_sum_b,
pqs ← ex_exp ps'' qs,
pqs_o ← pow_orig ps qs,
pf ← mk_proof_or_refl pqs.pretty ``pow_pp_pf_c
[ps.orig, ps.pretty, pqs.pretty, qs.orig, qs.pretty]
[ps.info, qs.info, pqs.info],
pqs' ← exp_to_prod pqs,
pure $ pqs'.set_info pqs_o pf
| pps@(ex.prod pps_i p ps) qs := do
pqs ← pow_e p qs,
psqs ← pow_pp ps qs,
ppsqs ← ex_prod pqs psqs,
ppsqs_o ← pow_orig pps qs,
pf ← mk_proof ``pow_pp_pf_prod
[pps.orig, p.pretty, ps.pretty, pqs.pretty, psqs.pretty, qs.orig]
[pps.info, pqs.info, psqs.info],
pure $ ppsqs.set_info ppsqs_o pf
lemma pow_p_pf_one {ps ps' : α} {qs : ℕ} : ps = ps' → qs = succ zero → ps ^ qs = ps' :=
λ ps_pf qs_pf, calc
ps ^ qs = ps' ^ 1 : by rw [ps_pf, qs_pf]
... = ps' : pow_one _
lemma pow_p_pf_zero {ps : α} {qs qs' : ℕ} : ps = 0 → qs = succ qs' → ps ^ qs = 0 :=
λ ps_pf qs_pf, calc
ps ^ qs = 0 ^ (succ qs') : by rw [ps_pf, qs_pf]
... = 0 : zero_pow (succ_pos qs')
lemma pow_p_pf_succ {ps pqqs : α} {qs qs' : ℕ} :
qs = succ qs' → ps * ps ^ qs' = pqqs → ps ^ qs = pqqs :=
λ qs_pf pqqs_pf, calc
ps ^ qs = ps ^ succ qs' : by rw [qs_pf]
... = ps * ps ^ qs' : pow_succ _ _
... = pqqs : by rw [pqqs_pf]
lemma pow_p_pf_singleton {pps p pqs : α} {qs : ℕ} :
pps = p + 0 → p ^ qs = pqs → pps ^ qs = pqs :=
λ pps_pf pqs_pf, by rw [pps_pf, add_zero, pqs_pf]
lemma pow_p_pf_cons {ps ps' : α} {qs qs' : ℕ} :
ps = ps' → qs = qs' → ps ^ qs = ps' ^ qs' := by cc
/--
Exponentiate two expressions.
* `ps ^ 1 = ps`
* `0 ^ qs = 0` (note that this is handled *after* `ps ^ 0 = 1`)
* `(p + 0) ^ qs = p ^ qs`
* `ps ^ (qs + 1) = ps * ps ^ qs` (note that this is handled *after* `p + 0 ^ qs = p ^ qs`)
* `ps ^ qs = ps ^ qs` (otherwise)
-/
meta def pow_p : ex sum → ex prod → ring_exp_m (ex sum)
| ps qs@(ex.coeff qs_i ⟨⟨1, 1, _, _⟩⟩) := do
ps_o ← pow_orig ps qs,
pf ← mk_proof ``pow_p_pf_one [ps.orig, ps.pretty, qs.orig] [ps.info, qs.info],
pure $ ps.set_info ps_o pf
| ps@(ex.zero ps_i) qs@(ex.coeff qs_i ⟨⟨succ y, 1, _, _⟩⟩) := do
ctx ← get_context,
z ← ex_zero,
qs_pred ← lift $ expr.of_nat ctx.info_e.α y,
pf ← mk_proof ``pow_p_pf_zero [ps.orig, qs.orig, qs_pred] [ps.info, qs.info],
z_o ← pow_orig ps qs,
pure $ z.set_info z_o pf
| pps@(ex.sum pps_i p (ex.zero _)) qqs := do
pqs ← pow_pp p qqs,
pqs_o ← pow_orig pps qqs,
pf ← mk_proof ``pow_p_pf_singleton
[pps.orig, p.pretty, pqs.pretty, qqs.orig]
[pps.info, pqs.info],
prod_to_sum $ pqs.set_info pqs_o pf
| ps qs@(ex.coeff qs_i ⟨⟨int.of_nat (succ n), 1, den_pos, _⟩⟩) := do
qs' ← in_exponent $ ex_coeff ⟨int.of_nat n, 1, den_pos, coprime_one_right _⟩,
pqs ← pow_p ps qs',
pqqs ← mul ps pqs,
pqqs_o ← pow_orig ps qs,
pf ← mk_proof ``pow_p_pf_succ
[ps.orig, pqqs.pretty, qs.orig, qs'.pretty]
[qs.info, pqqs.info],
pure $ pqqs.set_info pqqs_o pf
| pps qqs := do -- fallback: treat them as atoms
pps' ← ex_sum_b pps,
psqs ← ex_exp pps' qqs,
psqs_o ← pow_orig pps qqs,
pf ← mk_proof_or_refl psqs.pretty ``pow_p_pf_cons
[pps.orig, pps.pretty, qqs.orig, qqs.pretty]
[pps.info, qqs.info],
exp_to_prod (psqs.set_info psqs_o pf) >>= prod_to_sum
lemma pow_pf_zero {ps : α} {qs : ℕ} : qs = 0 → ps ^ qs = 1 := λ qs_pf, calc
ps ^ qs = ps ^ 0 : by rw [qs_pf]
... = 1 : pow_zero _
lemma pow_pf_sum {ps psqqs : α} {qqs q qs : ℕ} : qqs = q + qs →
ps ^ q * ps ^ qs = psqqs → ps ^ qqs = psqqs := λ qqs_pf psqqs_pf, calc
ps ^ qqs = ps ^ (q + qs) : by rw [qqs_pf]
... = ps ^ q * ps ^ qs : pow_add _ _ _
... = psqqs : psqqs_pf
/--
Exponentiate two expressions.
* `ps ^ 0 = 1`
* `ps ^ (q + qs) = ps ^ q * ps ^ qs`
-/
meta def pow : ex sum → ex sum → ring_exp_m (ex sum)
| ps qs@(ex.zero qs_i) := do
o ← ex_one,
o_o ← pow_orig ps qs,
pf ← mk_proof ``pow_pf_zero [ps.orig, qs.orig] [qs.info],
prod_to_sum $ o.set_info o_o pf
| ps qqs@(ex.sum qqs_i q qs) := do
psq ← pow_p ps q,
psqs ← pow ps qs,
psqqs ← mul psq psqs,
psqqs_o ← pow_orig ps qqs,
pf ← mk_proof ``pow_pf_sum
[ps.orig, psqqs.pretty, qqs.orig, q.pretty, qs.pretty]
[qqs.info, psqqs.info],
pure $ psqqs.set_info psqqs_o pf
end exponentiation
lemma simple_pf_sum_zero {p p' : α} : p = p' → p + 0 = p' := by simp
lemma simple_pf_prod_one {p p' : α} : p = p' → p * 1 = p' := by simp
lemma simple_pf_prod_neg_one {α} [ring α] {p p' : α} : p = p' → p * -1 = - p' := by simp
lemma simple_pf_var_one (p : α) : p ^ 1 = p := by simp
lemma simple_pf_exp_one {p p' : α} : p = p' → p ^ 1 = p' := by simp
/--
Give a simpler, more human-readable representation of the normalized expression.
Normalized expressions might have the form `a^1 * 1 + 0`,
since the dummy operations reduce special cases in pattern-matching.
Humans prefer to read `a` instead.
This tactic gets rid of the dummy additions, multiplications and exponentiations.
-/
meta def ex.simple : Π {et : ex_type}, ex et → ring_exp_m (expr × expr)
| sum pps@(ex.sum pps_i p (ex.zero _)) := do
(p_p, p_pf) ← p.simple,
prod.mk p_p <$> mk_app_csr ``simple_pf_sum_zero [p.pretty, p_p, p_pf]
| sum (ex.sum pps_i p ps) := do
(p_p, p_pf) ← p.simple,
(ps_p, ps_pf) ← ps.simple,
prod.mk
<$> mk_add [p_p, ps_p]
<*> mk_app_csr ``sum_congr [p.pretty, p_p, ps.pretty, ps_p, p_pf, ps_pf]
| prod (ex.prod pps_i p (ex.coeff _ ⟨⟨1, 1, _, _⟩⟩)) := do
(p_p, p_pf) ← p.simple,
prod.mk p_p <$> mk_app_csr ``simple_pf_prod_one [p.pretty, p_p, p_pf]
| prod pps@(ex.prod pps_i p (ex.coeff _ ⟨⟨-1, 1, _, _⟩⟩)) := do
ctx ← get_context,
match ctx.info_b.ring_instance with
| none := prod.mk pps.pretty <$> pps.proof_term
| (some ringi) := do
(p_p, p_pf) ← p.simple,
prod.mk
<$> lift (mk_app ``has_neg.neg [p_p])
<*> mk_app_class ``simple_pf_prod_neg_one ringi [p.pretty, p_p, p_pf]
end
| prod (ex.prod pps_i p ps) := do
(p_p, p_pf) ← p.simple,
(ps_p, ps_pf) ← ps.simple,
prod.mk
<$> mk_mul [p_p, ps_p]
<*> mk_app_csr ``prod_congr [p.pretty, p_p, ps.pretty, ps_p, p_pf, ps_pf]
| base (ex.sum_b pps_i ps) := ps.simple
| exp (ex.exp pps_i p (ex.coeff _ ⟨⟨1, 1, _, _⟩⟩)) := do
(p_p, p_pf) ← p.simple,
prod.mk p_p <$> mk_app_csr ``simple_pf_exp_one [p.pretty, p_p, p_pf]
| exp (ex.exp pps_i p ps) := do
(p_p, p_pf) ← p.simple,
(ps_p, ps_pf) ← in_exponent $ ps.simple,
prod.mk
<$> mk_pow [p_p, ps_p]
<*> mk_app_csr ``exp_congr [p.pretty, p_p, ps.pretty, ps_p, p_pf, ps_pf]
| et ps := prod.mk ps.pretty <$> ps.proof_term
/--
Performs a lookup of the atom `a` in the list of known atoms,
or allocates a new one.
If `a` is not definitionally equal to any of the list's entries,
a new atom is appended to the list and returned.
The index of this atom is kept track of in the second inductive argument.
This function is mostly useful in `resolve_atom`,
which updates the state with the new list of atoms.
-/
meta def resolve_atom_aux (a : expr) : list atom → ℕ → ring_exp_m (atom × list atom)
| [] n := let atm : atom := ⟨a, n⟩ in pure (atm, [atm])
| bas@(b :: as) n := do
ctx ← get_context,
(lift $ is_def_eq a b.value ctx.transp >> pure (b , bas)) <|> do
(atm, as') ← resolve_atom_aux as (succ n),
pure (atm, b :: as')
/--
Convert the expression to an atom:
either look up a definitionally equal atom,
or allocate it as a new atom.
You probably want to use `eval_base` if `eval` doesn't work
instead of directly calling `resolve_atom`,
since `eval_base` can also handle numerals.
-/
meta def resolve_atom (a : expr) : ring_exp_m atom := do
atoms ← reader_t.lift $ state_t.get,
(atm, atoms') ← resolve_atom_aux a atoms 0,
reader_t.lift $ state_t.put atoms',
pure atm
/--
Treat the expression atomically: as a coefficient or atom.
Handles cases where `eval` cannot treat the expression as a known operation
because it is just a number or single variable.
-/
meta def eval_base (ps : expr) : ring_exp_m (ex sum) :=
match ps.to_rat with
| some ⟨0, 1, _, _⟩ := ex_zero
| some x := ex_coeff x >>= prod_to_sum
| none := do
a ← resolve_atom ps,
atom_to_sum a
end
lemma negate_pf {α} [ring α] {ps ps' : α} : (-1) * ps = ps' → -ps = ps' := by simp
/--
Negate an expression by multiplying with `-1`.
Only works if there is a `ring` instance; otherwise it will `fail`.
-/
meta def negate (ps : ex sum) : ring_exp_m (ex sum) := do
ctx ← get_context,
match ctx.info_b.ring_instance with
| none := lift $ fail "internal error: negate called in semiring"
| (some ring_instance) := do
minus_one ← ex_coeff (-1) >>= prod_to_sum,
ps' ← mul minus_one ps,
ps_pf ← ps'.proof_term,
pf ← mk_app_class ``negate_pf ring_instance [ps.orig, ps'.pretty, ps_pf],
ps'_o ← lift $ mk_app ``has_neg.neg [ps.orig],
pure $ ps'.set_info ps'_o pf
end
lemma inverse_pf {α} [division_ring α] {ps ps_u ps_p e' e'' : α} :
ps = ps_u → ps_u = ps_p → ps_p ⁻¹ = e' → e' = e'' → ps ⁻¹ = e'' :=
by cc
/--
Invert an expression by simplifying, applying `has_inv.inv` and treating the result as an atom.
Only works if there is a `division_ring` instance; otherwise it will `fail`.
-/
meta def inverse (ps : ex sum) : ring_exp_m (ex sum) := do
ctx ← get_context,
dri ← match ctx.info_b.dr_instance with
| none := lift $ fail "division is only supported in a division ring"
| (some dri) := pure dri
end,
(ps_simple, ps_simple_pf) ← ps.simple,
e ← lift $ mk_app ``has_inv.inv [ps_simple],
(e', e_pf) ← lift (norm_num.derive e) <|> ((λ e_pf, (e, e_pf)) <$> lift (mk_eq_refl e)),
e'' ← eval_base e',
ps_pf ← ps.proof_term,
e''_pf ← e''.proof_term,
pf ← mk_app_class ``inverse_pf dri
[ ps.orig, ps.pretty, ps_simple, e', e''.pretty,
ps_pf, ps_simple_pf, e_pf, e''_pf],
e''_o ← lift $ mk_app ``has_inv.inv [ps.orig],
pure $ e''.set_info e''_o pf
lemma sub_pf {α} [ring α] {ps qs psqs : α} : ps + -qs = psqs → ps - qs = psqs := id
lemma div_pf {α} [division_ring α] {ps qs psqs : α} : ps * qs⁻¹ = psqs → ps / qs = psqs := id
end operations
section wiring
/-!
### `wiring` section
This section deals with going from `expr` to `ex` and back.
The main attraction is `eval`, which uses `add`, `mul`, etc.
to calculate an `ex` from a given `expr`.
Other functions use `ex`es to produce `expr`s together with a proof,
or produce the context to run `ring_exp_m` from an `expr`.
-/
open tactic
open ex_type
/--
Compute a normalized form (of type `ex`) from an expression (of type `expr`).
This is the main driver of the `ring_exp` tactic,
calling out to `add`, `mul`, `pow`, etc. to parse the `expr`.
-/
meta def eval : expr → ring_exp_m (ex sum)
| e@`(%%ps + %%qs) := do
ps' ← eval ps,
qs' ← eval qs,
add ps' qs'
| e@`(%%ps - %%qs) := (do
ctx ← get_context,
ri ← match ctx.info_b.ring_instance with
| none := lift $ fail "subtraction is not directly supported in a semiring"
| (some ri) := pure ri
end,
ps' ← eval ps,
qs' ← eval qs >>= negate,
psqs ← add ps' qs',
psqs_pf ← psqs.proof_term,
pf ← mk_app_class ``sub_pf ri [ps, qs, psqs.pretty, psqs_pf],
pure (psqs.set_info e pf)) <|> eval_base e
| e@`(- %%ps) := do
ps' ← eval ps,
negate ps' <|> eval_base e
| e@`(%%ps * %%qs) := do
ps' ← eval ps,
qs' ← eval qs,
mul ps' qs'
| e@`(has_inv.inv %%ps) := do
ps' ← eval ps,
inverse ps' <|> eval_base e
| e@`(%%ps / %%qs) := do
ctx ← get_context,
dri ← match ctx.info_b.dr_instance with
| none := lift $ fail "division is only directly supported in a division ring"
| (some dri) := pure dri
end,
ps' ← eval ps,
qs' ← eval qs,
(do qs'' ← inverse qs',
psqs ← mul ps' qs'',
psqs_pf ← psqs.proof_term,
pf ← mk_app_class ``div_pf dri [ps, qs, psqs.pretty, psqs_pf],
pure (psqs.set_info e pf)) <|> eval_base e
| e@`(@has_pow.pow _ _ %%hp_instance %%ps %%qs) := do
ps' ← eval ps,
qs' ← in_exponent $ eval qs,
psqs ← pow ps' qs',
psqs_pf ← psqs.proof_term,
(do has_pow_pf ← match hp_instance with
| `(monoid.has_pow) := lift $ mk_eq_refl e
| _ := lift $ fail "has_pow instance must be nat.has_pow or monoid.has_pow"
end,
pf ← lift $ mk_eq_trans has_pow_pf psqs_pf,
pure $ psqs.set_info e pf) <|> eval_base e
| ps := eval_base ps
/--
Run `eval` on the expression and return the result together with normalization proof.
See also `eval_simple` if you want something that behaves like `norm_num`.
-/
meta def eval_with_proof (e : expr) : ring_exp_m (ex sum × expr) := do
e' ← eval e,
prod.mk e' <$> e'.proof_term
/--
Run `eval` on the expression and simplify the result.
Returns a simplified normalized expression, together with an equality proof.
See also `eval_with_proof` if you just want to check the equality of two expressions.
-/
meta def eval_simple (e : expr) : ring_exp_m (expr × expr) := do
(complicated, complicated_pf) ← eval_with_proof e,
(simple, simple_pf) ← complicated.simple,
prod.mk simple <$> lift (mk_eq_trans complicated_pf simple_pf)
/-- Compute the `eval_info` for a given type `α`. -/
meta def make_eval_info (α : expr) : tactic eval_info := do
u ← mk_meta_univ,
infer_type α >>= unify (expr.sort (level.succ u)),
u ← get_univ_assignment u,
csr_instance ← mk_app ``comm_semiring [α] >>= mk_instance,
ring_instance ← (some <$> (mk_app ``ring [α] >>= mk_instance) <|> pure none),
dr_instance ← (some <$> (mk_app ``division_ring [α] >>= mk_instance) <|> pure none),
ha_instance ← mk_app ``has_add [α] >>= mk_instance,
hm_instance ← mk_app ``has_mul [α] >>= mk_instance,
hp_instance ← mk_mapp ``monoid.has_pow [some α, none],
z ← mk_mapp ``has_zero.zero [α, none],
o ← mk_mapp ``has_one.one [α, none],
pure ⟨α, u, csr_instance, ha_instance, hm_instance, hp_instance, ring_instance, dr_instance, z, o⟩
/-- Use `e` to build the context for running `mx`. -/
meta def run_ring_exp {α} (transp : transparency) (e : expr) (mx : ring_exp_m α) : tactic α := do
info_b ← infer_type e >>= make_eval_info,
info_e ← mk_const ``nat >>= make_eval_info,
(λ x : (_ × _), x.1) <$> (state_t.run (reader_t.run mx ⟨info_b, info_e, transp⟩) [])
/-- Repeatedly apply `eval_simple` on (sub)expressions. -/
meta def normalize (transp : transparency) (e : expr) : tactic (expr × expr) := do
(_, e', pf') ← ext_simplify_core () {}
simp_lemmas.mk (λ _, failed) (λ _ _ _ _ e, do
(e'', pf) ← run_ring_exp transp e $ eval_simple e,
guard (¬ e'' =ₐ e),
return ((), e'', some pf, ff))
(λ _ _ _ _ _, failed) `eq e,
pure (e', pf')
end wiring
end tactic.ring_exp
namespace tactic.interactive
open interactive interactive.types lean.parser tactic tactic.ring_exp
local postfix `?`:9001 := optional
/--
Tactic for solving equations of *commutative* (semi)rings,
allowing variables in the exponent.
This version of `ring_exp` fails if the target is not an equality.
The variant `ring_exp_eq!` will use a more aggressive reducibility setting
to determine equality of atoms.
-/
meta def ring_exp_eq (red : parse (tk "!")?) : tactic unit := do
`(eq %%ps %%qs) ← target >>= whnf,
let transp := if red.is_some then semireducible else reducible,
((ps', ps_pf), (qs', qs_pf)) ← run_ring_exp transp ps $
prod.mk <$> eval_with_proof ps <*> eval_with_proof qs,
if ps'.eq qs'
then do
qs_pf_inv ← mk_eq_symm qs_pf,
pf ← mk_eq_trans ps_pf qs_pf_inv,
tactic.interactive.exact ``(%%pf)
else fail "ring_exp failed to prove equality"
/--
Tactic for evaluating expressions in *commutative* (semi)rings, allowing for variables in the
exponent.
This tactic extends `ring`: it should solve every goal that `ring` can solve.
Additionally, it knows how to evaluate expressions with complicated exponents
(where `ring` only understands constant exponents).
The variants `ring_exp!` and `ring_exp_eq!` use a more aggessive reducibility setting to determine
equality of atoms.
For example:
```lean
example (n : ℕ) (m : ℤ) : 2^(n+1) * m = 2 * 2^n * m := by ring_exp
example (a b : ℤ) (n : ℕ) : (a + b)^(n + 2) = (a^2 + b^2 + a * b + b * a) * (a + b)^n := by ring_exp
example (x y : ℕ) : x + id y = y + id x := by ring_exp!
```
-/
meta def ring_exp (red : parse (tk "!")?) (loc : parse location) : tactic unit :=
match loc with
| interactive.loc.ns [none] := ring_exp_eq red
| _ := failed
end <|>
do ns ← loc.get_locals,
let transp := if red.is_some then semireducible else reducible,
tt ← tactic.replace_at (normalize transp) ns loc.include_goal
| fail "ring_exp failed to simplify",
when loc.include_goal $ try tactic.reflexivity
add_tactic_doc
{ name := "ring_exp",
category := doc_category.tactic,
decl_names := [`tactic.interactive.ring_exp],
tags := ["arithmetic", "simplification", "decision procedure"] }
end tactic.interactive
namespace conv.interactive
open conv interactive
open tactic tactic.interactive (ring_exp_eq)
open tactic.ring_exp (normalize)
local postfix `?`:9001 := optional
/--
Normalises expressions in commutative (semi-)rings inside of a `conv` block using the tactic `ring_exp`.
-/
meta def ring_exp (red : parse (lean.parser.tk "!")?) : conv unit :=
let transp := if red.is_some then semireducible else reducible in
discharge_eq_lhs (ring_exp_eq red)
<|> replace_lhs (normalize transp)
<|> fail "ring_exp failed to simplify"
end conv.interactive
|
03ef9474925d9336263060eecd077530c7b176e1 | 63abd62053d479eae5abf4951554e1064a4c45b4 | /src/algebra/lie/universal_enveloping.lean | 5e393221611aa6b8c7bf1bdeebae4c59901f7628 | [
"Apache-2.0"
] | permissive | Lix0120/mathlib | 0020745240315ed0e517cbf32e738d8f9811dd80 | e14c37827456fc6707f31b4d1d16f1f3a3205e91 | refs/heads/master | 1,673,102,855,024 | 1,604,151,044,000 | 1,604,151,044,000 | 308,930,245 | 0 | 0 | Apache-2.0 | 1,604,164,710,000 | 1,604,163,547,000 | null | UTF-8 | Lean | false | false | 4,709 | lean | /-
Copyright (c) 2020 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import algebra.lie.basic
import algebra.ring_quot
import linear_algebra.tensor_algebra
/-!
# Universal enveloping algebra
Given a commutative ring `R` and a Lie algebra `L` over `R`, we construct the universal
enveloping algebra of `L`, together with its universal property.
## Main definitions
* `universal_enveloping_algebra`: the universal enveloping algebra, endowed with an
`R`-algebra structure.
* `universal_enveloping_algebra.ι`: the Lie algebra morphism from `L` to its universal
enveloping algebra.
* `universal_enveloping_algebra.lift`: given an associative algebra `A`, together with a Lie
algebra morphism `f : L →ₗ⁅R⁆ A`, `lift R L f : universal_enveloping_algebra R L →ₐ[R] A` is the
unique morphism of algebras through which `f` factors.
* `universal_enveloping_algebra.ι_comp_lift`: states that the lift of a morphism is indeed part
of a factorisation.
* `universal_enveloping_algebra.lift_unique`: states that lifts of morphisms are indeed unique.
* `universal_enveloping_algebra.hom_ext`: a restatement of `lift_unique` as an extensionality
lemma.
## Tags
lie algebra, universal enveloping algebra, tensor algebra
-/
universes u₁ u₂ u₃
variables (R : Type u₁) (L : Type u₂)
variables [comm_ring R] [lie_ring L] [lie_algebra R L]
local notation `ιₜ` := tensor_algebra.ι R
namespace universal_enveloping_algebra
/-- The quotient by the ideal generated by this relation is the universal enveloping algebra.
Note that we have avoided using the more natural expression:
| lie_compat (x y : L) : rel (ιₜ ⁅x, y⁆) ⁅ιₜ x, ιₜ y⁆
so that our construction needs only the semiring structure of the tensor algebra. -/
inductive rel : tensor_algebra R L → tensor_algebra R L → Prop
| lie_compat (x y : L) : rel (ιₜ ⁅x, y⁆ + (ιₜ y) * (ιₜ x)) ((ιₜ x) * (ιₜ y))
end universal_enveloping_algebra
/-- The universal enveloping algebra of a Lie algebra. -/
@[derive [inhabited, semiring, algebra R]]
def universal_enveloping_algebra := ring_quot (universal_enveloping_algebra.rel R L)
namespace universal_enveloping_algebra
instance : ring (universal_enveloping_algebra R L) := algebra.semiring_to_ring R
/-- The quotient map from the tensor algebra to the universal enveloping algebra as a morphism of
associative algebras. -/
def mk_alg_hom : tensor_algebra R L →ₐ[R] universal_enveloping_algebra R L :=
ring_quot.mk_alg_hom R (rel R L)
variables {L}
/-- The natural Lie algebra morphism from a Lie algebra to its universal enveloping algebra. -/
def ι : L →ₗ⁅R⁆ universal_enveloping_algebra R L :=
{ map_lie := λ x y, by
{ suffices : mk_alg_hom R L (ιₜ ⁅x, y⁆ + (ιₜ y) * (ιₜ x)) = mk_alg_hom R L ((ιₜ x) * (ιₜ y)),
{ rw alg_hom.map_mul at this, simp [lie_ring.of_associative_ring_bracket, ← this], },
exact ring_quot.mk_alg_hom_rel _ (rel.lie_compat x y), },
..(mk_alg_hom R L).to_linear_map.comp ιₜ }
variables {A : Type u₃} [ring A] [algebra R A] (f : L →ₗ⁅R⁆ A)
/-- The universal property of the universal enveloping algebra: Lie algebra morphisms into
associative algebras lift to associative algebra morphisms from the universal enveloping algebra. -/
def lift : universal_enveloping_algebra R L →ₐ[R] A :=
ring_quot.lift_alg_hom R (tensor_algebra.lift R (f : L →ₗ[R] A))
begin
intros a b h, induction h with x y,
simp [lie_ring.of_associative_ring_bracket],
end
@[simp] lemma lift_ι_apply (x : L) : lift R f (ι R x) = f x :=
begin
have : ι R x = ring_quot.mk_alg_hom R (rel R L) (ιₜ x), by refl,
simp [this, lift],
end
lemma ι_comp_lift : (lift R f) ∘ (ι R) = f :=
by { ext, simp, }
lemma lift_unique (g : universal_enveloping_algebra R L →ₐ[R] A) :
g ∘ (ι R) = f ↔ g = lift R f :=
begin
split; intros h,
{ apply ring_quot.lift_alg_hom_unique,
rw ← tensor_algebra.lift_unique,
ext x,
change _ = f x, rw ← congr h rfl,
refl, },
{ subst h, apply ι_comp_lift, },
end
@[ext] lemma hom_ext {g₁ g₂ : universal_enveloping_algebra R L →ₐ[R] A}
(h : g₁ ∘ (ι R) = g₂ ∘ (ι R)) : g₁ = g₂ :=
begin
let f₁ := (lie_algebra.of_associative_algebra_hom g₁).comp (ι R),
let f₂ := (lie_algebra.of_associative_algebra_hom g₂).comp (ι R),
have h' : f₁ = f₂, { ext, exact congr h rfl, },
have h₁ : g₁ = lift R f₁, { rw ← lift_unique, refl, },
have h₂ : g₂ = lift R f₂, { rw ← lift_unique, refl, },
rw [h₁, h₂, h'],
end
end universal_enveloping_algebra
|
7374c68f8fe9738cdc809201bc24a433b733825a | 618003631150032a5676f229d13a079ac875ff77 | /src/data/list/basic.lean | 727e564db2e978876e1ac2ef6f09ca323c8a20e5 | [
"Apache-2.0"
] | permissive | awainverse/mathlib | 939b68c8486df66cfda64d327ad3d9165248c777 | ea76bd8f3ca0a8bf0a166a06a475b10663dec44a | refs/heads/master | 1,659,592,962,036 | 1,590,987,592,000 | 1,590,987,592,000 | 268,436,019 | 1 | 0 | Apache-2.0 | 1,590,990,500,000 | 1,590,990,500,000 | null | UTF-8 | Lean | false | false | 154,468 | lean | /-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import algebra.group
import deprecated.group
import data.nat.basic
/-!
# Basic properties of lists
-/
open function nat
namespace list
universes u v w x
variables {α : Type u} {β : Type v} {γ : Type w} {δ : Type x}
attribute [inline] list.head
instance : is_left_id (list α) has_append.append [] :=
⟨ nil_append ⟩
instance : is_right_id (list α) has_append.append [] :=
⟨ append_nil ⟩
instance : is_associative (list α) has_append.append :=
⟨ append_assoc ⟩
theorem cons_ne_nil (a : α) (l : list α) : a::l ≠ [].
theorem head_eq_of_cons_eq {h₁ h₂ : α} {t₁ t₂ : list α} :
(h₁::t₁) = (h₂::t₂) → h₁ = h₂ :=
assume Peq, list.no_confusion Peq (assume Pheq Pteq, Pheq)
theorem tail_eq_of_cons_eq {h₁ h₂ : α} {t₁ t₂ : list α} :
(h₁::t₁) = (h₂::t₂) → t₁ = t₂ :=
assume Peq, list.no_confusion Peq (assume Pheq Pteq, Pteq)
theorem cons_inj {a : α} : injective (cons a) :=
assume l₁ l₂, assume Pe, tail_eq_of_cons_eq Pe
theorem cons_inj' (a : α) {l l' : list α} : a::l = a::l' ↔ l = l' :=
⟨λ e, cons_inj e, congr_arg _⟩
/-! ### mem -/
theorem mem_singleton_self (a : α) : a ∈ [a] := mem_cons_self _ _
theorem eq_of_mem_singleton {a b : α} : a ∈ [b] → a = b :=
assume : a ∈ [b], or.elim (eq_or_mem_of_mem_cons this)
(assume : a = b, this)
(assume : a ∈ [], absurd this (not_mem_nil a))
@[simp] theorem mem_singleton {a b : α} : a ∈ [b] ↔ a = b :=
⟨eq_of_mem_singleton, or.inl⟩
theorem mem_of_mem_cons_of_mem {a b : α} {l : list α} : a ∈ b::l → b ∈ l → a ∈ l :=
assume ainbl binl, or.elim (eq_or_mem_of_mem_cons ainbl)
(assume : a = b, begin subst a, exact binl end)
(assume : a ∈ l, this)
theorem eq_or_ne_mem_of_mem {a b : α} {l : list α} (h : a ∈ b :: l) : a = b ∨ (a ≠ b ∧ a ∈ l) :=
classical.by_cases or.inl $ assume : a ≠ b, h.elim or.inl $ assume h, or.inr ⟨this, h⟩
theorem not_mem_append {a : α} {s t : list α} (h₁ : a ∉ s) (h₂ : a ∉ t) : a ∉ s ++ t :=
mt mem_append.1 $ not_or_distrib.2 ⟨h₁, h₂⟩
theorem ne_nil_of_mem {a : α} {l : list α} (h : a ∈ l) : l ≠ [] :=
by intro e; rw e at h; cases h
theorem mem_split {a : α} {l : list α} (h : a ∈ l) : ∃ s t : list α, l = s ++ a :: t :=
begin
induction l with b l ih, {cases h}, rcases h with rfl | h,
{ exact ⟨[], l, rfl⟩ },
{ rcases ih h with ⟨s, t, rfl⟩,
exact ⟨b::s, t, rfl⟩ }
end
theorem mem_of_ne_of_mem {a y : α} {l : list α} (h₁ : a ≠ y) (h₂ : a ∈ y :: l) : a ∈ l :=
or.elim (eq_or_mem_of_mem_cons h₂) (λe, absurd e h₁) (λr, r)
theorem ne_of_not_mem_cons {a b : α} {l : list α} : a ∉ b::l → a ≠ b :=
assume nin aeqb, absurd (or.inl aeqb) nin
theorem not_mem_of_not_mem_cons {a b : α} {l : list α} : a ∉ b::l → a ∉ l :=
assume nin nainl, absurd (or.inr nainl) nin
theorem not_mem_cons_of_ne_of_not_mem {a y : α} {l : list α} : a ≠ y → a ∉ l → a ∉ y::l :=
assume p1 p2, not.intro (assume Pain, absurd (eq_or_mem_of_mem_cons Pain) (not_or p1 p2))
theorem ne_and_not_mem_of_not_mem_cons {a y : α} {l : list α} : a ∉ y::l → a ≠ y ∧ a ∉ l :=
assume p, and.intro (ne_of_not_mem_cons p) (not_mem_of_not_mem_cons p)
theorem mem_map_of_mem (f : α → β) {a : α} {l : list α} (h : a ∈ l) : f a ∈ map f l :=
begin
induction l with b l' ih,
{cases h},
{rcases h with rfl | h,
{exact or.inl rfl},
{exact or.inr (ih h)}}
end
theorem exists_of_mem_map {f : α → β} {b : β} {l : list α} (h : b ∈ map f l) : ∃ a, a ∈ l ∧ f a = b :=
begin
induction l with c l' ih,
{cases h},
{cases (eq_or_mem_of_mem_cons h) with h h,
{exact ⟨c, mem_cons_self _ _, h.symm⟩},
{rcases ih h with ⟨a, ha₁, ha₂⟩,
exact ⟨a, mem_cons_of_mem _ ha₁, ha₂⟩ }}
end
@[simp] theorem mem_map {f : α → β} {b : β} {l : list α} : b ∈ map f l ↔ ∃ a, a ∈ l ∧ f a = b :=
⟨exists_of_mem_map, λ ⟨a, la, h⟩, by rw [← h]; exact mem_map_of_mem f la⟩
theorem mem_map_of_inj {f : α → β} (H : injective f) {a : α} {l : list α} :
f a ∈ map f l ↔ a ∈ l :=
⟨λ m, let ⟨a', m', e⟩ := exists_of_mem_map m in H e ▸ m', mem_map_of_mem _⟩
lemma forall_mem_map_iff {f : α → β} {l : list α} {P : β → Prop} :
(∀ i ∈ l.map f, P i) ↔ ∀ j ∈ l, P (f j) :=
begin
split,
{ assume H j hj,
exact H (f j) (mem_map_of_mem f hj) },
{ assume H i hi,
rcases mem_map.1 hi with ⟨j, hj, ji⟩,
rw ← ji,
exact H j hj }
end
@[simp] lemma map_eq_nil {f : α → β} {l : list α} : list.map f l = [] ↔ l = [] :=
⟨by cases l; simp only [forall_prop_of_true, map, forall_prop_of_false, not_false_iff],
λ h, h.symm ▸ rfl⟩
@[simp] theorem mem_join {a : α} : ∀ {L : list (list α)}, a ∈ join L ↔ ∃ l, l ∈ L ∧ a ∈ l
| [] := ⟨false.elim, λ⟨_, h, _⟩, false.elim h⟩
| (c :: L) := by simp only [join, mem_append, @mem_join L, mem_cons_iff, or_and_distrib_right,
exists_or_distrib, exists_eq_left]
theorem exists_of_mem_join {a : α} {L : list (list α)} : a ∈ join L → ∃ l, l ∈ L ∧ a ∈ l :=
mem_join.1
theorem mem_join_of_mem {a : α} {L : list (list α)} {l} (lL : l ∈ L) (al : a ∈ l) : a ∈ join L :=
mem_join.2 ⟨l, lL, al⟩
@[simp]
theorem mem_bind {b : β} {l : list α} {f : α → list β} : b ∈ list.bind l f ↔ ∃ a ∈ l, b ∈ f a :=
iff.trans mem_join
⟨λ ⟨l', h1, h2⟩, let ⟨a, al, fa⟩ := exists_of_mem_map h1 in ⟨a, al, fa.symm ▸ h2⟩,
λ ⟨a, al, bfa⟩, ⟨f a, mem_map_of_mem _ al, bfa⟩⟩
theorem exists_of_mem_bind {b : β} {l : list α} {f : α → list β} :
b ∈ list.bind l f → ∃ a ∈ l, b ∈ f a :=
mem_bind.1
theorem mem_bind_of_mem {b : β} {l : list α} {f : α → list β} {a} (al : a ∈ l) (h : b ∈ f a) :
b ∈ list.bind l f :=
mem_bind.2 ⟨a, al, h⟩
lemma bind_map {g : α → list β} {f : β → γ} :
∀(l : list α), list.map f (l.bind g) = l.bind (λa, (g a).map f)
| [] := rfl
| (a::l) := by simp only [cons_bind, map_append, bind_map l]
/-! ### length -/
theorem length_eq_zero {l : list α} : length l = 0 ↔ l = [] :=
⟨eq_nil_of_length_eq_zero, λ h, h.symm ▸ rfl⟩
theorem length_pos_of_mem {a : α} : ∀ {l : list α}, a ∈ l → 0 < length l
| (b::l) _ := zero_lt_succ _
theorem exists_mem_of_length_pos : ∀ {l : list α}, 0 < length l → ∃ a, a ∈ l
| (b::l) _ := ⟨b, mem_cons_self _ _⟩
theorem length_pos_iff_exists_mem {l : list α} : 0 < length l ↔ ∃ a, a ∈ l :=
⟨exists_mem_of_length_pos, λ ⟨a, h⟩, length_pos_of_mem h⟩
theorem ne_nil_of_length_pos {l : list α} : 0 < length l → l ≠ [] :=
λ h1 h2, lt_irrefl 0 ((length_eq_zero.2 h2).subst h1)
theorem length_pos_of_ne_nil {l : list α} : l ≠ [] → 0 < length l :=
λ h, pos_iff_ne_zero.2 $ λ h0, h $ length_eq_zero.1 h0
theorem length_pos_iff_ne_nil {l : list α} : 0 < length l ↔ l ≠ [] :=
⟨ne_nil_of_length_pos, length_pos_of_ne_nil⟩
theorem length_eq_one {l : list α} : length l = 1 ↔ ∃ a, l = [a] :=
⟨match l with [a], _ := ⟨a, rfl⟩ end, λ ⟨a, e⟩, e.symm ▸ rfl⟩
lemma exists_of_length_succ {n} :
∀ l : list α, l.length = n + 1 → ∃ h t, l = h :: t
| [] H := absurd H.symm $ succ_ne_zero n
| (h :: t) H := ⟨h, t, rfl⟩
lemma injective_length_iff : injective (list.length : list α → ℕ) ↔ subsingleton α :=
begin
split,
{ intro h, refine ⟨λ x y, _⟩, suffices : [x] = [y], { simpa using this }, apply h, refl },
{ intros hα l1 l2 hl, induction l1 generalizing l2; cases l2,
{ refl }, { cases hl }, { cases hl },
congr, exactI subsingleton.elim _ _, apply l1_ih, simpa using hl }
end
lemma injective_length [subsingleton α] : injective (length : list α → ℕ) :=
injective_length_iff.mpr $ by apply_instance
/-! ### set-theoretic notation of lists -/
lemma empty_eq : (∅ : list α) = [] := by refl
lemma singleton_eq (x : α) : ({x} : list α) = [x] := rfl
lemma insert_neg [decidable_eq α] {x : α} {l : list α} (h : x ∉ l) :
has_insert.insert x l = x :: l :=
if_neg h
lemma insert_pos [decidable_eq α] {x : α} {l : list α} (h : x ∈ l) :
has_insert.insert x l = l :=
if_pos h
lemma doubleton_eq [decidable_eq α] {x y : α} (h : x ≠ y) : ({x, y} : list α) = [x, y] :=
by { rw [insert_neg, singleton_eq], rwa [singleton_eq, mem_singleton] }
/-! ### bounded quantifiers over lists -/
theorem forall_mem_nil (p : α → Prop) : ∀ x ∈ @nil α, p x.
@[simp] theorem forall_mem_cons' {p : α → Prop} {a : α} {l : list α} :
(∀ (x : α), x = a ∨ x ∈ l → p x) ↔ p a ∧ ∀ x ∈ l, p x :=
by simp only [or_imp_distrib, forall_and_distrib, forall_eq]
theorem forall_mem_cons {p : α → Prop} {a : α} {l : list α} :
(∀ x ∈ a :: l, p x) ↔ p a ∧ ∀ x ∈ l, p x :=
by simp only [mem_cons_iff, forall_mem_cons']
theorem forall_mem_of_forall_mem_cons {p : α → Prop} {a : α} {l : list α}
(h : ∀ x ∈ a :: l, p x) :
∀ x ∈ l, p x :=
(forall_mem_cons.1 h).2
theorem forall_mem_singleton {p : α → Prop} {a : α} : (∀ x ∈ [a], p x) ↔ p a :=
by simp only [mem_singleton, forall_eq]
theorem forall_mem_append {p : α → Prop} {l₁ l₂ : list α} :
(∀ x ∈ l₁ ++ l₂, p x) ↔ (∀ x ∈ l₁, p x) ∧ (∀ x ∈ l₂, p x) :=
by simp only [mem_append, or_imp_distrib, forall_and_distrib]
theorem not_exists_mem_nil (p : α → Prop) : ¬ ∃ x ∈ @nil α, p x.
theorem exists_mem_cons_of {p : α → Prop} {a : α} (l : list α) (h : p a) :
∃ x ∈ a :: l, p x :=
bex.intro a (mem_cons_self _ _) h
theorem exists_mem_cons_of_exists {p : α → Prop} {a : α} {l : list α} (h : ∃ x ∈ l, p x) :
∃ x ∈ a :: l, p x :=
bex.elim h (λ x xl px, bex.intro x (mem_cons_of_mem _ xl) px)
theorem or_exists_of_exists_mem_cons {p : α → Prop} {a : α} {l : list α} (h : ∃ x ∈ a :: l, p x) :
p a ∨ ∃ x ∈ l, p x :=
bex.elim h (λ x xal px,
or.elim (eq_or_mem_of_mem_cons xal)
(assume : x = a, begin rw ←this, left, exact px end)
(assume : x ∈ l, or.inr (bex.intro x this px)))
theorem exists_mem_cons_iff (p : α → Prop) (a : α) (l : list α) :
(∃ x ∈ a :: l, p x) ↔ p a ∨ ∃ x ∈ l, p x :=
iff.intro or_exists_of_exists_mem_cons
(assume h, or.elim h (exists_mem_cons_of l) exists_mem_cons_of_exists)
/-! ### list subset -/
theorem subset_def {l₁ l₂ : list α} : l₁ ⊆ l₂ ↔ ∀ ⦃a : α⦄, a ∈ l₁ → a ∈ l₂ := iff.rfl
theorem subset_append_of_subset_left (l l₁ l₂ : list α) : l ⊆ l₁ → l ⊆ l₁++l₂ :=
λ s, subset.trans s $ subset_append_left _ _
theorem subset_append_of_subset_right (l l₁ l₂ : list α) : l ⊆ l₂ → l ⊆ l₁++l₂ :=
λ s, subset.trans s $ subset_append_right _ _
@[simp] theorem cons_subset {a : α} {l m : list α} :
a::l ⊆ m ↔ a ∈ m ∧ l ⊆ m :=
by simp only [subset_def, mem_cons_iff, or_imp_distrib, forall_and_distrib, forall_eq]
theorem cons_subset_of_subset_of_mem {a : α} {l m : list α}
(ainm : a ∈ m) (lsubm : l ⊆ m) : a::l ⊆ m :=
cons_subset.2 ⟨ainm, lsubm⟩
theorem append_subset_of_subset_of_subset {l₁ l₂ l : list α} (l₁subl : l₁ ⊆ l) (l₂subl : l₂ ⊆ l) :
l₁ ++ l₂ ⊆ l :=
λ a h, (mem_append.1 h).elim (@l₁subl _) (@l₂subl _)
@[simp] theorem append_subset_iff {l₁ l₂ l : list α} :
l₁ ++ l₂ ⊆ l ↔ l₁ ⊆ l ∧ l₂ ⊆ l :=
begin
split,
{ intro h, simp only [subset_def] at *, split; intros; simp* },
{ rintro ⟨h1, h2⟩, apply append_subset_of_subset_of_subset h1 h2 }
end
theorem eq_nil_of_subset_nil : ∀ {l : list α}, l ⊆ [] → l = []
| [] s := rfl
| (a::l) s := false.elim $ s $ mem_cons_self a l
theorem eq_nil_iff_forall_not_mem {l : list α} : l = [] ↔ ∀ a, a ∉ l :=
show l = [] ↔ l ⊆ [], from ⟨λ e, e ▸ subset.refl _, eq_nil_of_subset_nil⟩
theorem map_subset {l₁ l₂ : list α} (f : α → β) (H : l₁ ⊆ l₂) : map f l₁ ⊆ map f l₂ :=
λ x, by simp only [mem_map, not_and, exists_imp_distrib, and_imp]; exact λ a h e, ⟨a, H h, e⟩
theorem map_subset_iff {l₁ l₂ : list α} (f : α → β) (h : injective f) : map f l₁ ⊆ map f l₂ ↔ l₁ ⊆ l₂ :=
begin
refine ⟨_, map_subset f⟩, intros h2 x hx,
rcases mem_map.1 (h2 (mem_map_of_mem f hx)) with ⟨x', hx', hxx'⟩,
cases h hxx', exact hx'
end
/-! ### append -/
lemma append_eq_has_append {L₁ L₂ : list α} : list.append L₁ L₂ = L₁ ++ L₂ := rfl
theorem append_ne_nil_of_ne_nil_left (s t : list α) : s ≠ [] → s ++ t ≠ [] :=
by induction s; intros; contradiction
theorem append_ne_nil_of_ne_nil_right (s t : list α) : t ≠ [] → s ++ t ≠ [] :=
by induction s; intros; contradiction
@[simp] lemma append_eq_nil {p q : list α} : (p ++ q) = [] ↔ p = [] ∧ q = [] :=
by cases p; simp only [nil_append, cons_append, eq_self_iff_true, true_and, false_and]
@[simp] lemma nil_eq_append_iff {a b : list α} : [] = a ++ b ↔ a = [] ∧ b = [] :=
by rw [eq_comm, append_eq_nil]
lemma append_eq_cons_iff {a b c : list α} {x : α} :
a ++ b = x :: c ↔ (a = [] ∧ b = x :: c) ∨ (∃a', a = x :: a' ∧ c = a' ++ b) :=
by cases a; simp only [and_assoc, @eq_comm _ c, nil_append, cons_append, eq_self_iff_true,
true_and, false_and, exists_false, false_or, or_false, exists_and_distrib_left, exists_eq_left']
lemma cons_eq_append_iff {a b c : list α} {x : α} :
(x :: c : list α) = a ++ b ↔ (a = [] ∧ b = x :: c) ∨ (∃a', a = x :: a' ∧ c = a' ++ b) :=
by rw [eq_comm, append_eq_cons_iff]
lemma append_eq_append_iff {a b c d : list α} :
a ++ b = c ++ d ↔ (∃a', c = a ++ a' ∧ b = a' ++ d) ∨ (∃c', a = c ++ c' ∧ d = c' ++ b) :=
begin
induction a generalizing c,
case nil { rw nil_append, split,
{ rintro rfl, left, exact ⟨_, rfl, rfl⟩ },
{ rintro (⟨a', rfl, rfl⟩ | ⟨a', H, rfl⟩), {refl}, {rw [← append_assoc, ← H], refl} } },
case cons : a as ih {
cases c,
{ simp only [cons_append, nil_append, false_and, exists_false, false_or, exists_eq_left'], exact eq_comm },
{ simp only [cons_append, @eq_comm _ a, ih, and_assoc, and_or_distrib_left, exists_and_distrib_left] } }
end
@[simp] theorem split_at_eq_take_drop : ∀ (n : ℕ) (l : list α), split_at n l = (take n l, drop n l)
| 0 a := rfl
| (succ n) [] := rfl
| (succ n) (x :: xs) := by simp only [split_at, split_at_eq_take_drop n xs, take, drop]
@[simp] theorem take_append_drop : ∀ (n : ℕ) (l : list α), take n l ++ drop n l = l
| 0 a := rfl
| (succ n) [] := rfl
| (succ n) (x :: xs) := congr_arg (cons x) $ take_append_drop n xs
-- TODO(Leo): cleanup proof after arith dec proc
theorem append_inj :
∀ {s₁ s₂ t₁ t₂ : list α}, s₁ ++ t₁ = s₂ ++ t₂ → length s₁ = length s₂ → s₁ = s₂ ∧ t₁ = t₂
| [] [] t₁ t₂ h hl := ⟨rfl, h⟩
| (a::s₁) [] t₁ t₂ h hl := list.no_confusion $ eq_nil_of_length_eq_zero hl
| [] (b::s₂) t₁ t₂ h hl := list.no_confusion $ eq_nil_of_length_eq_zero hl.symm
| (a::s₁) (b::s₂) t₁ t₂ h hl := list.no_confusion h $ λab hap,
let ⟨e1, e2⟩ := @append_inj s₁ s₂ t₁ t₂ hap (succ.inj hl) in
by rw [ab, e1, e2]; exact ⟨rfl, rfl⟩
theorem append_inj_right {s₁ s₂ t₁ t₂ : list α} (h : s₁ ++ t₁ = s₂ ++ t₂)
(hl : length s₁ = length s₂) : t₁ = t₂ :=
(append_inj h hl).right
theorem append_inj_left {s₁ s₂ t₁ t₂ : list α} (h : s₁ ++ t₁ = s₂ ++ t₂)
(hl : length s₁ = length s₂) : s₁ = s₂ :=
(append_inj h hl).left
theorem append_inj' {s₁ s₂ t₁ t₂ : list α} (h : s₁ ++ t₁ = s₂ ++ t₂) (hl : length t₁ = length t₂) :
s₁ = s₂ ∧ t₁ = t₂ :=
append_inj h $ @nat.add_right_cancel _ (length t₁) _ $
let hap := congr_arg length h in by simp only [length_append] at hap; rwa [← hl] at hap
theorem append_inj_right' {s₁ s₂ t₁ t₂ : list α} (h : s₁ ++ t₁ = s₂ ++ t₂)
(hl : length t₁ = length t₂) : t₁ = t₂ :=
(append_inj' h hl).right
theorem append_inj_left' {s₁ s₂ t₁ t₂ : list α} (h : s₁ ++ t₁ = s₂ ++ t₂)
(hl : length t₁ = length t₂) : s₁ = s₂ :=
(append_inj' h hl).left
theorem append_left_cancel {s t₁ t₂ : list α} (h : s ++ t₁ = s ++ t₂) : t₁ = t₂ :=
append_inj_right h rfl
theorem append_right_cancel {s₁ s₂ t : list α} (h : s₁ ++ t = s₂ ++ t) : s₁ = s₂ :=
append_inj_left' h rfl
theorem append_right_inj {t₁ t₂ : list α} (s) : s ++ t₁ = s ++ t₂ ↔ t₁ = t₂ :=
⟨append_left_cancel, congr_arg _⟩
theorem append_left_inj {s₁ s₂ : list α} (t) : s₁ ++ t = s₂ ++ t ↔ s₁ = s₂ :=
⟨append_right_cancel, congr_arg _⟩
theorem map_eq_append_split {f : α → β} {l : list α} {s₁ s₂ : list β}
(h : map f l = s₁ ++ s₂) : ∃ l₁ l₂, l = l₁ ++ l₂ ∧ map f l₁ = s₁ ∧ map f l₂ = s₂ :=
begin
have := h, rw [← take_append_drop (length s₁) l] at this ⊢,
rw map_append at this,
refine ⟨_, _, rfl, append_inj this _⟩,
rw [length_map, length_take, min_eq_left],
rw [← length_map f l, h, length_append],
apply nat.le_add_right
end
/-! ### repeat -/
@[simp] theorem repeat_succ (a : α) (n) : repeat a (n + 1) = a :: repeat a n := rfl
theorem eq_of_mem_repeat {a b : α} : ∀ {n}, b ∈ repeat a n → b = a
| (n+1) h := or.elim h id $ @eq_of_mem_repeat _
theorem eq_repeat_of_mem {a : α} : ∀ {l : list α}, (∀ b ∈ l, b = a) → l = repeat a l.length
| [] H := rfl
| (b::l) H := by cases forall_mem_cons.1 H with H₁ H₂;
unfold length repeat; congr; [exact H₁, exact eq_repeat_of_mem H₂]
theorem eq_repeat' {a : α} {l : list α} : l = repeat a l.length ↔ ∀ b ∈ l, b = a :=
⟨λ h, h.symm ▸ λ b, eq_of_mem_repeat, eq_repeat_of_mem⟩
theorem eq_repeat {a : α} {n} {l : list α} : l = repeat a n ↔ length l = n ∧ ∀ b ∈ l, b = a :=
⟨λ h, h.symm ▸ ⟨length_repeat _ _, λ b, eq_of_mem_repeat⟩,
λ ⟨e, al⟩, e ▸ eq_repeat_of_mem al⟩
theorem repeat_add (a : α) (m n) : repeat a (m + n) = repeat a m ++ repeat a n :=
by induction m; simp only [*, zero_add, succ_add, repeat]; split; refl
theorem repeat_subset_singleton (a : α) (n) : repeat a n ⊆ [a] :=
λ b h, mem_singleton.2 (eq_of_mem_repeat h)
@[simp] theorem map_const (l : list α) (b : β) : map (function.const α b) l = repeat b l.length :=
by induction l; [refl, simp only [*, map]]; split; refl
theorem eq_of_mem_map_const {b₁ b₂ : β} {l : list α} (h : b₁ ∈ map (function.const α b₂) l) :
b₁ = b₂ :=
by rw map_const at h; exact eq_of_mem_repeat h
@[simp] theorem map_repeat (f : α → β) (a : α) (n) : map f (repeat a n) = repeat (f a) n :=
by induction n; [refl, simp only [*, repeat, map]]; split; refl
@[simp] theorem tail_repeat (a : α) (n) : tail (repeat a n) = repeat a n.pred :=
by cases n; refl
@[simp] theorem join_repeat_nil (n : ℕ) : join (repeat [] n) = @nil α :=
by induction n; [refl, simp only [*, repeat, join, append_nil]]
/-! ### pure -/
@[simp] theorem mem_pure {α} (x y : α) :
x ∈ (pure y : list α) ↔ x = y := by simp! [pure,list.ret]
/-! ### bind -/
@[simp] theorem bind_eq_bind {α β} (f : α → list β) (l : list α) :
l >>= f = l.bind f := rfl
@[simp] theorem bind_append (f : α → list β) (l₁ l₂ : list α) :
(l₁ ++ l₂).bind f = l₁.bind f ++ l₂.bind f :=
append_bind _ _ _
/-! ### concat -/
theorem concat_nil (a : α) : concat [] a = [a] := rfl
theorem concat_cons (a b : α) (l : list α) : concat (a :: l) b = a :: concat l b := rfl
@[simp] theorem concat_eq_append (a : α) (l : list α) : concat l a = l ++ [a] :=
by induction l; simp only [*, concat]; split; refl
theorem concat_ne_nil (a : α) (l : list α) : concat l a ≠ [] :=
by simp
theorem concat_append (a : α) (l₁ l₂ : list α) : concat l₁ a ++ l₂ = l₁ ++ a :: l₂ :=
by simp
theorem length_concat (a : α) (l : list α) : length (concat l a) = succ (length l) :=
by simp only [concat_eq_append, length_append, length]
theorem append_concat (a : α) (l₁ l₂ : list α) : l₁ ++ concat l₂ a = concat (l₁ ++ l₂) a :=
by simp
/-! ### reverse -/
@[simp] theorem reverse_nil : reverse (@nil α) = [] := rfl
local attribute [simp] reverse_core
@[simp] theorem reverse_cons (a : α) (l : list α) : reverse (a::l) = reverse l ++ [a] :=
have aux : ∀ l₁ l₂, reverse_core l₁ l₂ ++ [a] = reverse_core l₁ (l₂ ++ [a]),
by intro l₁; induction l₁; intros; [refl, simp only [*, reverse_core, cons_append]],
(aux l nil).symm
theorem reverse_core_eq (l₁ l₂ : list α) : reverse_core l₁ l₂ = reverse l₁ ++ l₂ :=
by induction l₁ generalizing l₂; [refl, simp only [*, reverse_core, reverse_cons, append_assoc]];
refl
theorem reverse_cons' (a : α) (l : list α) : reverse (a::l) = concat (reverse l) a :=
by simp only [reverse_cons, concat_eq_append]
@[simp] theorem reverse_singleton (a : α) : reverse [a] = [a] := rfl
@[simp] theorem reverse_append (s t : list α) : reverse (s ++ t) = (reverse t) ++ (reverse s) :=
by induction s; [rw [nil_append, reverse_nil, append_nil],
simp only [*, cons_append, reverse_cons, append_assoc]]
@[simp] theorem reverse_reverse (l : list α) : reverse (reverse l) = l :=
by induction l; [refl, simp only [*, reverse_cons, reverse_append]]; refl
theorem reverse_injective : injective (@reverse α) :=
left_inverse.injective reverse_reverse
@[simp] theorem reverse_inj {l₁ l₂ : list α} : reverse l₁ = reverse l₂ ↔ l₁ = l₂ :=
reverse_injective.eq_iff
@[simp] theorem reverse_eq_nil {l : list α} : reverse l = [] ↔ l = [] :=
@reverse_inj _ l []
theorem concat_eq_reverse_cons (a : α) (l : list α) : concat l a = reverse (a :: reverse l) :=
by simp only [concat_eq_append, reverse_cons, reverse_reverse]
@[simp] theorem length_reverse (l : list α) : length (reverse l) = length l :=
by induction l; [refl, simp only [*, reverse_cons, length_append, length]]
@[simp] theorem map_reverse (f : α → β) (l : list α) : map f (reverse l) = reverse (map f l) :=
by induction l; [refl, simp only [*, map, reverse_cons, map_append]]
theorem map_reverse_core (f : α → β) (l₁ l₂ : list α) :
map f (reverse_core l₁ l₂) = reverse_core (map f l₁) (map f l₂) :=
by simp only [reverse_core_eq, map_append, map_reverse]
@[simp] theorem mem_reverse {a : α} {l : list α} : a ∈ reverse l ↔ a ∈ l :=
by induction l; [refl, simp only [*, reverse_cons, mem_append, mem_singleton, mem_cons_iff,
not_mem_nil, false_or, or_false, or_comm]]
@[simp] theorem reverse_repeat (a : α) (n) : reverse (repeat a n) = repeat a n :=
eq_repeat.2 ⟨by simp only [length_reverse, length_repeat],
λ b h, eq_of_mem_repeat (mem_reverse.1 h)⟩
/-- Induction principle from the right for lists: if a property holds for the empty list, and
for `l ++ [a]` if it holds for `l`, then it holds for all lists. The principle is given for
a `Sort`-valued predicate, i.e., it can also be used to construct data. -/
@[elab_as_eliminator] def reverse_rec_on {C : list α → Sort*}
(l : list α) (H0 : C [])
(H1 : ∀ (l : list α) (a : α), C l → C (l ++ [a])) : C l :=
begin
rw ← reverse_reverse l,
induction reverse l,
{ exact H0 },
{ rw reverse_cons, exact H1 _ _ ih }
end
/-! ### is_nil -/
lemma is_nil_iff_eq_nil {l : list α} : l.is_nil ↔ l = [] :=
list.cases_on l (by simp [is_nil]) (by simp [is_nil])
/-! ### last -/
@[simp] theorem last_cons {a : α} {l : list α} :
∀ (h₁ : a :: l ≠ nil) (h₂ : l ≠ nil), last (a :: l) h₁ = last l h₂ :=
by {induction l; intros, contradiction, reflexivity}
@[simp] theorem last_append {a : α} (l : list α) (h : l ++ [a] ≠ []) : last (l ++ [a]) h = a :=
by induction l;
[refl, simp only [cons_append, last_cons _ (λ H, cons_ne_nil _ _ (append_eq_nil.1 H).2), *]]
theorem last_concat {a : α} (l : list α) (h : concat l a ≠ []) : last (concat l a) h = a :=
by simp only [concat_eq_append, last_append]
@[simp] theorem last_singleton (a : α) (h : [a] ≠ []) : last [a] h = a := rfl
@[simp] theorem last_cons_cons (a₁ a₂ : α) (l : list α) (h : a₁::a₂::l ≠ []) :
last (a₁::a₂::l) h = last (a₂::l) (cons_ne_nil a₂ l) := rfl
theorem last_congr {l₁ l₂ : list α} (h₁ : l₁ ≠ []) (h₂ : l₂ ≠ []) (h₃ : l₁ = l₂) :
last l₁ h₁ = last l₂ h₂ :=
by subst l₁
theorem last_mem : ∀ {l : list α} (h : l ≠ []), last l h ∈ l
| [] h := absurd rfl h
| [a] h := or.inl rfl
| (a::b::l) h := or.inr $ by { rw [last_cons_cons], exact last_mem (cons_ne_nil b l) }
/-! ### last' -/
@[simp] theorem last'_is_none :
∀ {l : list α}, (last' l).is_none ↔ l = []
| [] := by simp
| [a] := by simp
| (a::b::l) := by simp [@last'_is_none (b::l)]
@[simp] theorem last'_is_some : ∀ {l : list α}, l.last'.is_some ↔ l ≠ []
| [] := by simp
| [a] := by simp
| (a::b::l) := by simp [@last'_is_some (b::l)]
theorem mem_last'_eq_last : ∀ {l : list α} {x : α}, x ∈ l.last' → ∃ h, x = last l h
| [] x hx := false.elim $ by simpa using hx
| [a] x hx := have a = x, by simpa using hx, this ▸ ⟨cons_ne_nil a [], rfl⟩
| (a::b::l) x hx :=
begin
rw last' at hx,
rcases mem_last'_eq_last hx with ⟨h₁, h₂⟩,
use cons_ne_nil _ _,
rwa [last_cons]
end
theorem mem_of_mem_last' {l : list α} {a : α} (ha : a ∈ l.last') : a ∈ l :=
let ⟨h₁, h₂⟩ := mem_last'_eq_last ha in h₂.symm ▸ last_mem _
theorem init_append_last' : ∀ {l : list α} (a ∈ l.last'), init l ++ [a] = l
| [] a ha := (option.not_mem_none a ha).elim
| [a] _ rfl := rfl
| (a :: b :: l) c hc := by { rw [last'] at hc, rw [init, cons_append, init_append_last' _ hc] }
theorem ilast_eq_last' [inhabited α] : ∀ l : list α, l.ilast = l.last'.iget
| [] := by simp [ilast, arbitrary]
| [a] := rfl
| [a, b] := rfl
| [a, b, c] := rfl
| (a :: b :: c :: l) := by simp [ilast, ilast_eq_last' (c :: l)]
@[simp] theorem last'_append_cons : ∀ (l₁ : list α) (a : α) (l₂ : list α),
last' (l₁ ++ a :: l₂) = last' (a :: l₂)
| [] a l₂ := rfl
| [b] a l₂ := rfl
| (b::c::l₁) a l₂ := by rw [cons_append, cons_append, last', ← cons_append, last'_append_cons]
theorem last'_append_of_ne_nil (l₁ : list α) : ∀ {l₂ : list α} (hl₂ : l₂ ≠ []),
last' (l₁ ++ l₂) = last' l₂
| [] hl₂ := by contradiction
| (b::l₂) _ := last'_append_cons l₁ b l₂
/-! ### head(') and tail -/
theorem head_eq_head' [inhabited α] (l : list α) : head l = (head' l).iget :=
by cases l; refl
theorem mem_of_mem_head' {x : α} : ∀ {l : list α}, x ∈ l.head' → x ∈ l
| [] h := (option.not_mem_none _ h).elim
| (a::l) h := by { simp only [head', option.mem_def] at h, exact h ▸ or.inl rfl }
@[simp] theorem head_cons [inhabited α] (a : α) (l : list α) : head (a::l) = a := rfl
@[simp] theorem tail_nil : tail (@nil α) = [] := rfl
@[simp] theorem tail_cons (a : α) (l : list α) : tail (a::l) = l := rfl
@[simp] theorem head_append [inhabited α] (t : list α) {s : list α} (h : s ≠ []) :
head (s ++ t) = head s :=
by {induction s, contradiction, refl}
theorem cons_head'_tail : ∀ {l : list α} {a : α} (h : a ∈ head' l), a :: tail l = l
| [] a h := by contradiction
| (b::l) a h := by { simp at h, simp [h] }
theorem head_mem_head' [inhabited α] : ∀ {l : list α} (h : l ≠ []), head l ∈ head' l
| [] h := by contradiction
| (a::l) h := rfl
theorem cons_head_tail [inhabited α] {l : list α} (h : l ≠ []) : (head l)::(tail l) = l :=
cons_head'_tail (head_mem_head' h)
@[simp] theorem head'_map (f : α → β) (l) : head' (map f l) = (head' l).map f := by cases l; refl
/-! ### sublists -/
@[simp] theorem nil_sublist : Π (l : list α), [] <+ l
| [] := sublist.slnil
| (a :: l) := sublist.cons _ _ a (nil_sublist l)
@[refl, simp] theorem sublist.refl : Π (l : list α), l <+ l
| [] := sublist.slnil
| (a :: l) := sublist.cons2 _ _ a (sublist.refl l)
@[trans] theorem sublist.trans {l₁ l₂ l₃ : list α} (h₁ : l₁ <+ l₂) (h₂ : l₂ <+ l₃) : l₁ <+ l₃ :=
sublist.rec_on h₂ (λ_ s, s)
(λl₂ l₃ a h₂ IH l₁ h₁, sublist.cons _ _ _ (IH l₁ h₁))
(λl₂ l₃ a h₂ IH l₁ h₁, @sublist.cases_on _ (λl₁ l₂', l₂' = a :: l₂ → l₁ <+ a :: l₃) _ _ h₁
(λ_, nil_sublist _)
(λl₁ l₂' a' h₁' e, match a', l₂', e, h₁' with ._, ._, rfl, h₁ :=
sublist.cons _ _ _ (IH _ h₁) end)
(λl₁ l₂' a' h₁' e, match a', l₂', e, h₁' with ._, ._, rfl, h₁ :=
sublist.cons2 _ _ _ (IH _ h₁) end) rfl)
l₁ h₁
@[simp] theorem sublist_cons (a : α) (l : list α) : l <+ a::l :=
sublist.cons _ _ _ (sublist.refl l)
theorem sublist_of_cons_sublist {a : α} {l₁ l₂ : list α} : a::l₁ <+ l₂ → l₁ <+ l₂ :=
sublist.trans (sublist_cons a l₁)
theorem cons_sublist_cons {l₁ l₂ : list α} (a : α) (s : l₁ <+ l₂) : a::l₁ <+ a::l₂ :=
sublist.cons2 _ _ _ s
@[simp] theorem sublist_append_left : Π (l₁ l₂ : list α), l₁ <+ l₁++l₂
| [] l₂ := nil_sublist _
| (a::l₁) l₂ := cons_sublist_cons _ (sublist_append_left l₁ l₂)
@[simp] theorem sublist_append_right : Π (l₁ l₂ : list α), l₂ <+ l₁++l₂
| [] l₂ := sublist.refl _
| (a::l₁) l₂ := sublist.cons _ _ _ (sublist_append_right l₁ l₂)
theorem sublist_cons_of_sublist (a : α) {l₁ l₂ : list α} : l₁ <+ l₂ → l₁ <+ a::l₂ :=
sublist.cons _ _ _
theorem sublist_append_of_sublist_left {l l₁ l₂ : list α} (s : l <+ l₁) : l <+ l₁++l₂ :=
s.trans $ sublist_append_left _ _
theorem sublist_append_of_sublist_right {l l₁ l₂ : list α} (s : l <+ l₂) : l <+ l₁++l₂ :=
s.trans $ sublist_append_right _ _
theorem sublist_of_cons_sublist_cons {l₁ l₂ : list α} : ∀ {a : α}, a::l₁ <+ a::l₂ → l₁ <+ l₂
| ._ (sublist.cons ._ ._ a s) := sublist_of_cons_sublist s
| ._ (sublist.cons2 ._ ._ a s) := s
theorem cons_sublist_cons_iff {l₁ l₂ : list α} {a : α} : a::l₁ <+ a::l₂ ↔ l₁ <+ l₂ :=
⟨sublist_of_cons_sublist_cons, cons_sublist_cons _⟩
@[simp] theorem append_sublist_append_left {l₁ l₂ : list α} : ∀ l, l++l₁ <+ l++l₂ ↔ l₁ <+ l₂
| [] := iff.rfl
| (a::l) := cons_sublist_cons_iff.trans (append_sublist_append_left l)
theorem sublist.append_right {l₁ l₂ : list α} (h : l₁ <+ l₂) (l) : l₁++l <+ l₂++l :=
begin
induction h with _ _ a _ ih _ _ a _ ih,
{ refl },
{ apply sublist_cons_of_sublist a ih },
{ apply cons_sublist_cons a ih }
end
theorem sublist_or_mem_of_sublist {l l₁ l₂ : list α} {a : α} (h : l <+ l₁ ++ a::l₂) :
l <+ l₁ ++ l₂ ∨ a ∈ l :=
begin
induction l₁ with b l₁ IH generalizing l,
{ cases h, { left, exact ‹l <+ l₂› }, { right, apply mem_cons_self } },
{ cases h with _ _ _ h _ _ _ h,
{ exact or.imp_left (sublist_cons_of_sublist _) (IH h) },
{ exact (IH h).imp (cons_sublist_cons _) (mem_cons_of_mem _) } }
end
theorem sublist.reverse {l₁ l₂ : list α} (h : l₁ <+ l₂) : l₁.reverse <+ l₂.reverse :=
begin
induction h with _ _ _ _ ih _ _ a _ ih, {refl},
{ rw reverse_cons, exact sublist_append_of_sublist_left ih },
{ rw [reverse_cons, reverse_cons], exact ih.append_right [a] }
end
@[simp] theorem reverse_sublist_iff {l₁ l₂ : list α} : l₁.reverse <+ l₂.reverse ↔ l₁ <+ l₂ :=
⟨λ h, l₁.reverse_reverse ▸ l₂.reverse_reverse ▸ h.reverse, sublist.reverse⟩
@[simp] theorem append_sublist_append_right {l₁ l₂ : list α} (l) : l₁++l <+ l₂++l ↔ l₁ <+ l₂ :=
⟨λ h, by simpa only [reverse_append, append_sublist_append_left, reverse_sublist_iff]
using h.reverse,
λ h, h.append_right l⟩
theorem sublist.append {l₁ l₂ r₁ r₂ : list α}
(hl : l₁ <+ l₂) (hr : r₁ <+ r₂) : l₁ ++ r₁ <+ l₂ ++ r₂ :=
(hl.append_right _).trans ((append_sublist_append_left _).2 hr)
theorem sublist.subset : Π {l₁ l₂ : list α}, l₁ <+ l₂ → l₁ ⊆ l₂
| ._ ._ sublist.slnil b h := h
| ._ ._ (sublist.cons l₁ l₂ a s) b h := mem_cons_of_mem _ (sublist.subset s h)
| ._ ._ (sublist.cons2 l₁ l₂ a s) b h :=
match eq_or_mem_of_mem_cons h with
| or.inl h := h ▸ mem_cons_self _ _
| or.inr h := mem_cons_of_mem _ (sublist.subset s h)
end
theorem singleton_sublist {a : α} {l} : [a] <+ l ↔ a ∈ l :=
⟨λ h, h.subset (mem_singleton_self _), λ h,
let ⟨s, t, e⟩ := mem_split h in e.symm ▸
(cons_sublist_cons _ (nil_sublist _)).trans (sublist_append_right _ _)⟩
theorem eq_nil_of_sublist_nil {l : list α} (s : l <+ []) : l = [] :=
eq_nil_of_subset_nil $ s.subset
theorem repeat_sublist_repeat (a : α) {m n} : repeat a m <+ repeat a n ↔ m ≤ n :=
⟨λ h, by simpa only [length_repeat] using length_le_of_sublist h,
λ h, by induction h; [refl, simp only [*, repeat_succ, sublist.cons]] ⟩
theorem eq_of_sublist_of_length_eq : ∀ {l₁ l₂ : list α}, l₁ <+ l₂ → length l₁ = length l₂ → l₁ = l₂
| ._ ._ sublist.slnil h := rfl
| ._ ._ (sublist.cons l₁ l₂ a s) h :=
absurd (length_le_of_sublist s) $ not_le_of_gt $ by rw h; apply lt_succ_self
| ._ ._ (sublist.cons2 l₁ l₂ a s) h :=
by rw [length, length] at h; injection h with h; rw eq_of_sublist_of_length_eq s h
theorem eq_of_sublist_of_length_le {l₁ l₂ : list α} (s : l₁ <+ l₂) (h : length l₂ ≤ length l₁) :
l₁ = l₂ :=
eq_of_sublist_of_length_eq s (le_antisymm (length_le_of_sublist s) h)
theorem sublist.antisymm {l₁ l₂ : list α} (s₁ : l₁ <+ l₂) (s₂ : l₂ <+ l₁) : l₁ = l₂ :=
eq_of_sublist_of_length_le s₁ (length_le_of_sublist s₂)
instance decidable_sublist [decidable_eq α] : ∀ (l₁ l₂ : list α), decidable (l₁ <+ l₂)
| [] l₂ := is_true $ nil_sublist _
| (a::l₁) [] := is_false $ λh, list.no_confusion $ eq_nil_of_sublist_nil h
| (a::l₁) (b::l₂) :=
if h : a = b then
decidable_of_decidable_of_iff (decidable_sublist l₁ l₂) $
by rw [← h]; exact ⟨cons_sublist_cons _, sublist_of_cons_sublist_cons⟩
else decidable_of_decidable_of_iff (decidable_sublist (a::l₁) l₂)
⟨sublist_cons_of_sublist _, λs, match a, l₁, s, h with
| a, l₁, sublist.cons ._ ._ ._ s', h := s'
| ._, ._, sublist.cons2 t ._ ._ s', h := absurd rfl h
end⟩
/-! ### index_of -/
section index_of
variable [decidable_eq α]
@[simp] theorem index_of_nil (a : α) : index_of a [] = 0 := rfl
theorem index_of_cons (a b : α) (l : list α) :
index_of a (b::l) = if a = b then 0 else succ (index_of a l) := rfl
theorem index_of_cons_eq {a b : α} (l : list α) : a = b → index_of a (b::l) = 0 :=
assume e, if_pos e
@[simp] theorem index_of_cons_self (a : α) (l : list α) : index_of a (a::l) = 0 :=
index_of_cons_eq _ rfl
@[simp, priority 990]
theorem index_of_cons_ne {a b : α} (l : list α) : a ≠ b → index_of a (b::l) = succ (index_of a l) :=
assume n, if_neg n
theorem index_of_eq_length {a : α} {l : list α} : index_of a l = length l ↔ a ∉ l :=
begin
induction l with b l ih,
{ exact iff_of_true rfl (not_mem_nil _) },
simp only [length, mem_cons_iff, index_of_cons], split_ifs,
{ exact iff_of_false (by rintro ⟨⟩) (λ H, H $ or.inl h) },
{ simp only [h, false_or], rw ← ih, exact succ_inj' }
end
@[simp, priority 980]
theorem index_of_of_not_mem {l : list α} {a : α} : a ∉ l → index_of a l = length l :=
index_of_eq_length.2
theorem index_of_le_length {a : α} {l : list α} : index_of a l ≤ length l :=
begin
induction l with b l ih, {refl},
simp only [length, index_of_cons],
by_cases h : a = b, {rw if_pos h, exact nat.zero_le _},
rw if_neg h, exact succ_le_succ ih
end
theorem index_of_lt_length {a} {l : list α} : index_of a l < length l ↔ a ∈ l :=
⟨λh, by_contradiction $ λ al, ne_of_lt h $ index_of_eq_length.2 al,
λal, lt_of_le_of_ne index_of_le_length $ λ h, index_of_eq_length.1 h al⟩
end index_of
/-! ### nth element -/
theorem nth_le_of_mem : ∀ {a} {l : list α}, a ∈ l → ∃ n h, nth_le l n h = a
| a (_ :: l) (or.inl rfl) := ⟨0, succ_pos _, rfl⟩
| a (b :: l) (or.inr m) :=
let ⟨n, h, e⟩ := nth_le_of_mem m in ⟨n+1, succ_lt_succ h, e⟩
theorem nth_le_nth : ∀ {l : list α} {n} h, nth l n = some (nth_le l n h)
| (a :: l) 0 h := rfl
| (a :: l) (n+1) h := @nth_le_nth l n _
theorem nth_len_le : ∀ {l : list α} {n}, length l ≤ n → nth l n = none
| [] n h := rfl
| (a :: l) (n+1) h := nth_len_le (le_of_succ_le_succ h)
theorem nth_eq_some {l : list α} {n a} : nth l n = some a ↔ ∃ h, nth_le l n h = a :=
⟨λ e,
have h : n < length l, from lt_of_not_ge $ λ hn,
by rw nth_len_le hn at e; contradiction,
⟨h, by rw nth_le_nth h at e;
injection e with e; apply nth_le_mem⟩,
λ ⟨h, e⟩, e ▸ nth_le_nth _⟩
theorem nth_of_mem {a} {l : list α} (h : a ∈ l) : ∃ n, nth l n = some a :=
let ⟨n, h, e⟩ := nth_le_of_mem h in ⟨n, by rw [nth_le_nth, e]⟩
theorem nth_le_mem : ∀ (l : list α) n h, nth_le l n h ∈ l
| (a :: l) 0 h := mem_cons_self _ _
| (a :: l) (n+1) h := mem_cons_of_mem _ (nth_le_mem l _ _)
theorem nth_mem {l : list α} {n a} (e : nth l n = some a) : a ∈ l :=
let ⟨h, e⟩ := nth_eq_some.1 e in e ▸ nth_le_mem _ _ _
theorem mem_iff_nth_le {a} {l : list α} : a ∈ l ↔ ∃ n h, nth_le l n h = a :=
⟨nth_le_of_mem, λ ⟨n, h, e⟩, e ▸ nth_le_mem _ _ _⟩
theorem mem_iff_nth {a} {l : list α} : a ∈ l ↔ ∃ n, nth l n = some a :=
mem_iff_nth_le.trans $ exists_congr $ λ n, nth_eq_some.symm
@[simp] theorem nth_map (f : α → β) : ∀ l n, nth (map f l) n = (nth l n).map f
| [] n := rfl
| (a :: l) 0 := rfl
| (a :: l) (n+1) := nth_map l n
theorem nth_le_map (f : α → β) {l n} (H1 H2) : nth_le (map f l) n H1 = f (nth_le l n H2) :=
option.some.inj $ by rw [← nth_le_nth, nth_map, nth_le_nth]; refl
/-- A version of `nth_le_map` that can be used for rewriting. -/
theorem nth_le_map_rev (f : α → β) {l n} (H) :
f (nth_le l n H) = nth_le (map f l) n ((length_map f l).symm ▸ H) :=
(nth_le_map f _ _).symm
@[simp] theorem nth_le_map' (f : α → β) {l n} (H) :
nth_le (map f l) n H = f (nth_le l n (length_map f l ▸ H)) :=
nth_le_map f _ _
/-- If one has `nth_le L i hi` in a formula and `h : L = L'`, one can not `rw h` in the formula as
`hi` gives `i < L.length` and not `i < L'.length`. The lemma `nth_le_of_eq` can be used to make
such a rewrite, with `rw (nth_le_of_eq h)`. -/
lemma nth_le_of_eq {L L' : list α} (h : L = L') {i : ℕ} (hi : i < L.length) :
nth_le L i hi = nth_le L' i (h ▸ hi) :=
by { congr, exact h}
@[simp] lemma nth_le_singleton (a : α) {n : ℕ} (hn : n < 1) :
nth_le [a] n hn = a :=
have hn0 : n = 0 := le_zero_iff.1 (le_of_lt_succ hn),
by subst hn0; refl
lemma nth_le_zero [inhabited α] {L : list α} (h : 0 < L.length) :
L.nth_le 0 h = L.head :=
by { cases L, cases h, simp, }
lemma nth_le_append : ∀ {l₁ l₂ : list α} {n : ℕ} (hn₁) (hn₂),
(l₁ ++ l₂).nth_le n hn₁ = l₁.nth_le n hn₂
| [] _ n hn₁ hn₂ := (not_lt_zero _ hn₂).elim
| (a::l) _ 0 hn₁ hn₂ := rfl
| (a::l) _ (n+1) hn₁ hn₂ := by simp only [nth_le, cons_append];
exact nth_le_append _ _
lemma nth_le_append_right_aux {l₁ l₂ : list α} {n : ℕ}
(h₁ : l₁.length ≤ n) (h₂ : n < (l₁ ++ l₂).length) : n - l₁.length < l₂.length :=
begin
rw list.length_append at h₂,
convert (nat.sub_lt_sub_right_iff h₁).mpr h₂,
simp,
end
lemma nth_le_append_right : ∀ {l₁ l₂ : list α} {n : ℕ} (h₁ : l₁.length ≤ n) (h₂),
(l₁ ++ l₂).nth_le n h₂ = l₂.nth_le (n - l₁.length) (nth_le_append_right_aux h₁ h₂)
| [] _ n h₁ h₂ := rfl
| (a :: l) _ (n+1) h₁ h₂ :=
begin
dsimp,
conv { to_rhs, congr, skip, rw [←nat.sub_sub, nat.sub.right_comm, nat.add_sub_cancel], },
rw nth_le_append_right (nat.lt_succ_iff.mp h₁),
end
@[simp] lemma nth_le_repeat (a : α) {n m : ℕ} (h : m < (list.repeat a n).length) :
(list.repeat a n).nth_le m h = a :=
eq_of_mem_repeat (nth_le_mem _ _ _)
lemma nth_append {l₁ l₂ : list α} {n : ℕ} (hn : n < l₁.length) :
(l₁ ++ l₂).nth n = l₁.nth n :=
have hn' : n < (l₁ ++ l₂).length := lt_of_lt_of_le hn
(by rw length_append; exact le_add_right _ _),
by rw [nth_le_nth hn, nth_le_nth hn', nth_le_append]
lemma last_eq_nth_le : ∀ (l : list α) (h : l ≠ []),
last l h = l.nth_le (l.length - 1) (sub_lt (length_pos_of_ne_nil h) one_pos)
| [] h := rfl
| [a] h := by rw [last_singleton, nth_le_singleton]
| (a :: b :: l) h := by { rw [last_cons, last_eq_nth_le (b :: l)],
refl, exact cons_ne_nil b l }
@[simp] lemma nth_concat_length : ∀ (l : list α) (a : α), (l ++ [a]).nth l.length = some a
| [] a := rfl
| (b::l) a := by rw [cons_append, length_cons, nth, nth_concat_length]
@[ext]
theorem ext : ∀ {l₁ l₂ : list α}, (∀n, nth l₁ n = nth l₂ n) → l₁ = l₂
| [] [] h := rfl
| (a::l₁) [] h := by have h0 := h 0; contradiction
| [] (a'::l₂) h := by have h0 := h 0; contradiction
| (a::l₁) (a'::l₂) h := by have h0 : some a = some a' := h 0; injection h0 with aa;
simp only [aa, ext (λn, h (n+1))]; split; refl
theorem ext_le {l₁ l₂ : list α} (hl : length l₁ = length l₂)
(h : ∀n h₁ h₂, nth_le l₁ n h₁ = nth_le l₂ n h₂) : l₁ = l₂ :=
ext $ λn, if h₁ : n < length l₁
then by rw [nth_le_nth, nth_le_nth, h n h₁ (by rwa [← hl])]
else let h₁ := le_of_not_gt h₁ in by { rw [nth_len_le h₁, nth_len_le], rwa [←hl], }
@[simp] theorem index_of_nth_le [decidable_eq α] {a : α} :
∀ {l : list α} h, nth_le l (index_of a l) h = a
| (b::l) h := by by_cases h' : a = b;
simp only [h', if_pos, if_false, index_of_cons, nth_le, @index_of_nth_le l]
@[simp] theorem index_of_nth [decidable_eq α] {a : α} {l : list α} (h : a ∈ l) :
nth l (index_of a l) = some a :=
by rw [nth_le_nth, index_of_nth_le (index_of_lt_length.2 h)]
theorem nth_le_reverse_aux1 :
∀ (l r : list α) (i h1 h2), nth_le (reverse_core l r) (i + length l) h1 = nth_le r i h2
| [] r i := λh1 h2, rfl
| (a :: l) r i :=
by rw (show i + length (a :: l) = i + 1 + length l, from add_right_comm i (length l) 1);
exact λh1 h2, nth_le_reverse_aux1 l (a :: r) (i+1) h1 (succ_lt_succ h2)
lemma index_of_inj [decidable_eq α] {l : list α} {x y : α}
(hx : x ∈ l) (hy : y ∈ l) : index_of x l = index_of y l ↔ x = y :=
⟨λ h, have nth_le l (index_of x l) (index_of_lt_length.2 hx) =
nth_le l (index_of y l) (index_of_lt_length.2 hy),
by simp only [h],
by simpa only [index_of_nth_le],
λ h, by subst h⟩
theorem nth_le_reverse_aux2 : ∀ (l r : list α) (i : nat) (h1) (h2),
nth_le (reverse_core l r) (length l - 1 - i) h1 = nth_le l i h2
| [] r i h1 h2 := absurd h2 (not_lt_zero _)
| (a :: l) r 0 h1 h2 := begin
have aux := nth_le_reverse_aux1 l (a :: r) 0,
rw zero_add at aux,
exact aux _ (zero_lt_succ _)
end
| (a :: l) r (i+1) h1 h2 := begin
have aux := nth_le_reverse_aux2 l (a :: r) i,
have heq := calc length (a :: l) - 1 - (i + 1)
= length l - (1 + i) : by rw add_comm; refl
... = length l - 1 - i : by rw nat.sub_sub,
rw [← heq] at aux,
apply aux
end
@[simp] theorem nth_le_reverse (l : list α) (i : nat) (h1 h2) :
nth_le (reverse l) (length l - 1 - i) h1 = nth_le l i h2 :=
nth_le_reverse_aux2 _ _ _ _ _
lemma eq_cons_of_length_one {l : list α} (h : l.length = 1) :
l = [l.nth_le 0 (h.symm ▸ zero_lt_one)] :=
begin
refine ext_le (by convert h) (λ n h₁ h₂, _),
simp only [nth_le_singleton],
congr,
exact eq_bot_iff.mpr (nat.lt_succ_iff.mp h₂)
end
lemma modify_nth_tail_modify_nth_tail {f g : list α → list α} (m : ℕ) :
∀n (l:list α), (l.modify_nth_tail f n).modify_nth_tail g (m + n) =
l.modify_nth_tail (λl, (f l).modify_nth_tail g m) n
| 0 l := rfl
| (n+1) [] := rfl
| (n+1) (a::l) := congr_arg (list.cons a) (modify_nth_tail_modify_nth_tail n l)
lemma modify_nth_tail_modify_nth_tail_le
{f g : list α → list α} (m n : ℕ) (l : list α) (h : n ≤ m) :
(l.modify_nth_tail f n).modify_nth_tail g m =
l.modify_nth_tail (λl, (f l).modify_nth_tail g (m - n)) n :=
begin
rcases le_iff_exists_add.1 h with ⟨m, rfl⟩,
rw [nat.add_sub_cancel_left, add_comm, modify_nth_tail_modify_nth_tail]
end
lemma modify_nth_tail_modify_nth_tail_same {f g : list α → list α} (n : ℕ) (l:list α) :
(l.modify_nth_tail f n).modify_nth_tail g n = l.modify_nth_tail (g ∘ f) n :=
by rw [modify_nth_tail_modify_nth_tail_le n n l (le_refl n), nat.sub_self]; refl
lemma modify_nth_tail_id :
∀n (l:list α), l.modify_nth_tail id n = l
| 0 l := rfl
| (n+1) [] := rfl
| (n+1) (a::l) := congr_arg (list.cons a) (modify_nth_tail_id n l)
theorem remove_nth_eq_nth_tail : ∀ n (l : list α), remove_nth l n = modify_nth_tail tail n l
| 0 l := by cases l; refl
| (n+1) [] := rfl
| (n+1) (a::l) := congr_arg (cons _) (remove_nth_eq_nth_tail _ _)
theorem update_nth_eq_modify_nth (a : α) : ∀ n (l : list α),
update_nth l n a = modify_nth (λ _, a) n l
| 0 l := by cases l; refl
| (n+1) [] := rfl
| (n+1) (b::l) := congr_arg (cons _) (update_nth_eq_modify_nth _ _)
theorem modify_nth_eq_update_nth (f : α → α) : ∀ n (l : list α),
modify_nth f n l = ((λ a, update_nth l n (f a)) <$> nth l n).get_or_else l
| 0 l := by cases l; refl
| (n+1) [] := rfl
| (n+1) (b::l) := (congr_arg (cons b)
(modify_nth_eq_update_nth n l)).trans $ by cases nth l n; refl
theorem nth_modify_nth (f : α → α) : ∀ n (l : list α) m,
nth (modify_nth f n l) m = (λ a, if n = m then f a else a) <$> nth l m
| n l 0 := by cases l; cases n; refl
| n [] (m+1) := by cases n; refl
| 0 (a::l) (m+1) := by cases nth l m; refl
| (n+1) (a::l) (m+1) := (nth_modify_nth n l m).trans $
by cases nth l m with b; by_cases n = m;
simp only [h, if_pos, if_true, if_false, option.map_none, option.map_some, mt succ_inj,
not_false_iff]
theorem modify_nth_tail_length (f : list α → list α) (H : ∀ l, length (f l) = length l) :
∀ n l, length (modify_nth_tail f n l) = length l
| 0 l := H _
| (n+1) [] := rfl
| (n+1) (a::l) := @congr_arg _ _ _ _ (+1) (modify_nth_tail_length _ _)
@[simp] theorem modify_nth_length (f : α → α) :
∀ n l, length (modify_nth f n l) = length l :=
modify_nth_tail_length _ (λ l, by cases l; refl)
@[simp] theorem update_nth_length (l : list α) (n) (a : α) :
length (update_nth l n a) = length l :=
by simp only [update_nth_eq_modify_nth, modify_nth_length]
@[simp] theorem nth_modify_nth_eq (f : α → α) (n) (l : list α) :
nth (modify_nth f n l) n = f <$> nth l n :=
by simp only [nth_modify_nth, if_pos]
@[simp] theorem nth_modify_nth_ne (f : α → α) {m n} (l : list α) (h : m ≠ n) :
nth (modify_nth f m l) n = nth l n :=
by simp only [nth_modify_nth, if_neg h, id_map']
theorem nth_update_nth_eq (a : α) (n) (l : list α) :
nth (update_nth l n a) n = (λ _, a) <$> nth l n :=
by simp only [update_nth_eq_modify_nth, nth_modify_nth_eq]
theorem nth_update_nth_of_lt (a : α) {n} {l : list α} (h : n < length l) :
nth (update_nth l n a) n = some a :=
by rw [nth_update_nth_eq, nth_le_nth h]; refl
theorem nth_update_nth_ne (a : α) {m n} (l : list α) (h : m ≠ n) :
nth (update_nth l m a) n = nth l n :=
by simp only [update_nth_eq_modify_nth, nth_modify_nth_ne _ _ h]
@[simp] lemma nth_le_update_nth_eq (l : list α) (i : ℕ) (a : α)
(h : i < (l.update_nth i a).length) : (l.update_nth i a).nth_le i h = a :=
by rw [← option.some_inj, ← nth_le_nth, nth_update_nth_eq, nth_le_nth]; simp * at *
@[simp] lemma nth_le_update_nth_of_ne {l : list α} {i j : ℕ} (h : i ≠ j) (a : α)
(hj : j < (l.update_nth i a).length) :
(l.update_nth i a).nth_le j hj = l.nth_le j (by simpa using hj) :=
by rw [← option.some_inj, ← list.nth_le_nth, list.nth_update_nth_ne _ _ h, list.nth_le_nth]
lemma mem_or_eq_of_mem_update_nth : ∀ {l : list α} {n : ℕ} {a b : α}
(h : a ∈ l.update_nth n b), a ∈ l ∨ a = b
| [] n a b h := false.elim h
| (c::l) 0 a b h := ((mem_cons_iff _ _ _).1 h).elim
or.inr (or.inl ∘ mem_cons_of_mem _)
| (c::l) (n+1) a b h := ((mem_cons_iff _ _ _).1 h).elim
(λ h, h ▸ or.inl (mem_cons_self _ _))
(λ h, (mem_or_eq_of_mem_update_nth h).elim
(or.inl ∘ mem_cons_of_mem _) or.inr)
section insert_nth
variable {a : α}
@[simp] lemma insert_nth_nil (a : α) : insert_nth 0 a [] = [a] := rfl
lemma length_insert_nth : ∀n as, n ≤ length as → length (insert_nth n a as) = length as + 1
| 0 as h := rfl
| (n+1) [] h := (nat.not_succ_le_zero _ h).elim
| (n+1) (a'::as) h := congr_arg nat.succ $ length_insert_nth n as (nat.le_of_succ_le_succ h)
lemma remove_nth_insert_nth (n:ℕ) (l : list α) : (l.insert_nth n a).remove_nth n = l :=
by rw [remove_nth_eq_nth_tail, insert_nth, modify_nth_tail_modify_nth_tail_same];
from modify_nth_tail_id _ _
lemma insert_nth_remove_nth_of_ge : ∀n m as, n < length as → m ≥ n →
insert_nth m a (as.remove_nth n) = (as.insert_nth (m + 1) a).remove_nth n
| 0 0 [] has _ := (lt_irrefl _ has).elim
| 0 0 (a::as) has hmn := by simp [remove_nth, insert_nth]
| 0 (m+1) (a::as) has hmn := rfl
| (n+1) (m+1) (a::as) has hmn :=
congr_arg (cons a) $
insert_nth_remove_nth_of_ge n m as (nat.lt_of_succ_lt_succ has) (nat.le_of_succ_le_succ hmn)
lemma insert_nth_remove_nth_of_le : ∀n m as, n < length as → m ≤ n →
insert_nth m a (as.remove_nth n) = (as.insert_nth m a).remove_nth (n + 1)
| n 0 (a :: as) has hmn := rfl
| (n + 1) (m + 1) (a :: as) has hmn :=
congr_arg (cons a) $
insert_nth_remove_nth_of_le n m as (nat.lt_of_succ_lt_succ has) (nat.le_of_succ_le_succ hmn)
lemma insert_nth_comm (a b : α) :
∀(i j : ℕ) (l : list α) (h : i ≤ j) (hj : j ≤ length l),
(l.insert_nth i a).insert_nth (j + 1) b = (l.insert_nth j b).insert_nth i a
| 0 j l := by simp [insert_nth]
| (i + 1) 0 l := assume h, (nat.not_lt_zero _ h).elim
| (i + 1) (j+1) [] := by simp
| (i + 1) (j+1) (c::l) :=
assume h₀ h₁,
by simp [insert_nth];
exact insert_nth_comm i j l (nat.le_of_succ_le_succ h₀) (nat.le_of_succ_le_succ h₁)
lemma mem_insert_nth {a b : α} : ∀ {n : ℕ} {l : list α} (hi : n ≤ l.length),
a ∈ l.insert_nth n b ↔ a = b ∨ a ∈ l
| 0 as h := iff.rfl
| (n+1) [] h := (nat.not_succ_le_zero _ h).elim
| (n+1) (a'::as) h := begin
dsimp [list.insert_nth],
erw [list.mem_cons_iff, mem_insert_nth (nat.le_of_succ_le_succ h), list.mem_cons_iff,
← or.assoc, or_comm (a = a'), or.assoc]
end
end insert_nth
/-! ### map -/
@[simp] lemma map_nil (f : α → β) : map f [] = [] := rfl
lemma map_congr {f g : α → β} : ∀ {l : list α}, (∀ x ∈ l, f x = g x) → map f l = map g l
| [] _ := rfl
| (a::l) h := let ⟨h₁, h₂⟩ := forall_mem_cons.1 h in
by rw [map, map, h₁, map_congr h₂]
lemma map_eq_map_iff {f g : α → β} {l : list α} : map f l = map g l ↔ (∀ x ∈ l, f x = g x) :=
begin
refine ⟨_, map_congr⟩, intros h x hx,
rw [mem_iff_nth_le] at hx, rcases hx with ⟨n, hn, rfl⟩,
rw [nth_le_map_rev f, nth_le_map_rev g], congr, exact h
end
theorem map_concat (f : α → β) (a : α) (l : list α) : map f (concat l a) = concat (map f l) (f a) :=
by induction l; [refl, simp only [*, concat_eq_append, cons_append, map, map_append]]; split; refl
theorem map_id' {f : α → α} (h : ∀ x, f x = x) (l : list α) : map f l = l :=
by induction l; [refl, simp only [*, map]]; split; refl
@[simp] theorem foldl_map (g : β → γ) (f : α → γ → α) (a : α) (l : list β) :
foldl f a (map g l) = foldl (λx y, f x (g y)) a l :=
by revert a; induction l; intros; [refl, simp only [*, map, foldl]]
@[simp] theorem foldr_map (g : β → γ) (f : γ → α → α) (a : α) (l : list β) :
foldr f a (map g l) = foldr (f ∘ g) a l :=
by revert a; induction l; intros; [refl, simp only [*, map, foldr]]
theorem foldl_hom (l : list γ) (f : α → β) (op : α → γ → α) (op' : β → γ → β) (a : α)
(h : ∀a x, f (op a x) = op' (f a) x) : foldl op' (f a) l = f (foldl op a l) :=
eq.symm $ by { revert a, induction l; intros; [refl, simp only [*, foldl]] }
theorem foldr_hom (l : list γ) (f : α → β) (op : γ → α → α) (op' : γ → β → β) (a : α)
(h : ∀x a, f (op x a) = op' x (f a)) : foldr op' (f a) l = f (foldr op a l) :=
by { revert a, induction l; intros; [refl, simp only [*, foldr]] }
theorem eq_nil_of_map_eq_nil {f : α → β} {l : list α} (h : map f l = nil) : l = nil :=
eq_nil_of_length_eq_zero $ by rw [← length_map f l, h]; refl
@[simp] theorem map_join (f : α → β) (L : list (list α)) :
map f (join L) = join (map (map f) L) :=
by induction L; [refl, simp only [*, join, map, map_append]]
theorem bind_ret_eq_map (f : α → β) (l : list α) :
l.bind (list.ret ∘ f) = map f l :=
by unfold list.bind; induction l; simp only [map, join, list.ret, cons_append, nil_append, *];
split; refl
@[simp] theorem map_eq_map {α β} (f : α → β) (l : list α) : f <$> l = map f l := rfl
@[simp] theorem map_tail (f : α → β) (l) : map f (tail l) = tail (map f l) :=
by cases l; refl
@[simp] theorem injective_map_iff {f : α → β} : injective (map f) ↔ injective f :=
begin
split; intros h x y hxy,
{ suffices : [x] = [y], { simpa using this }, apply h, simp [hxy] },
{ induction y generalizing x, simpa using hxy,
cases x, simpa using hxy, simp at hxy, simp [y_ih hxy.2, h hxy.1] }
end
/-! ### map₂ -/
theorem nil_map₂ (f : α → β → γ) (l : list β) : map₂ f [] l = [] :=
by cases l; refl
theorem map₂_nil (f : α → β → γ) (l : list α) : map₂ f l [] = [] :=
by cases l; refl
/-! ### take, drop -/
@[simp] theorem take_zero (l : list α) : take 0 l = [] := rfl
@[simp] theorem take_nil : ∀ n, take n [] = ([] : list α)
| 0 := rfl
| (n+1) := rfl
theorem take_cons (n) (a : α) (l : list α) : take (succ n) (a::l) = a :: take n l := rfl
@[simp] theorem take_length : ∀ (l : list α), take (length l) l = l
| [] := rfl
| (a::l) := begin change a :: (take (length l) l) = a :: l, rw take_length end
theorem take_all_of_le : ∀ {n} {l : list α}, length l ≤ n → take n l = l
| 0 [] h := rfl
| 0 (a::l) h := absurd h (not_le_of_gt (zero_lt_succ _))
| (n+1) [] h := rfl
| (n+1) (a::l) h :=
begin
change a :: take n l = a :: l,
rw [take_all_of_le (le_of_succ_le_succ h)]
end
@[simp] theorem take_left : ∀ l₁ l₂ : list α, take (length l₁) (l₁ ++ l₂) = l₁
| [] l₂ := rfl
| (a::l₁) l₂ := congr_arg (cons a) (take_left l₁ l₂)
theorem take_left' {l₁ l₂ : list α} {n} (h : length l₁ = n) :
take n (l₁ ++ l₂) = l₁ :=
by rw ← h; apply take_left
theorem take_take : ∀ (n m) (l : list α), take n (take m l) = take (min n m) l
| n 0 l := by rw [min_zero, take_zero, take_nil]
| 0 m l := by rw [zero_min, take_zero, take_zero]
| (succ n) (succ m) nil := by simp only [take_nil]
| (succ n) (succ m) (a::l) := by simp only [take, min_succ_succ, take_take n m l]; split; refl
theorem take_repeat (a : α) : ∀ (n m : ℕ), take n (repeat a m) = repeat a (min n m)
| n 0 := by simp
| 0 m := by simp
| (succ n) (succ m) := by simp [min_succ_succ, take_repeat]
lemma map_take {α β : Type*} (f : α → β) :
∀ (L : list α) (i : ℕ), (L.take i).map f = (L.map f).take i
| [] i := by simp
| L 0 := by simp
| (h :: t) (n+1) := by { dsimp, rw [map_take], }
lemma take_append_of_le_length : ∀ {l₁ l₂ : list α} {n : ℕ},
n ≤ l₁.length → (l₁ ++ l₂).take n = l₁.take n
| l₁ l₂ 0 hn := by simp
| [] l₂ (n+1) hn := absurd hn dec_trivial
| (a::l₁) l₂ (n+1) hn :=
by rw [list.take, list.cons_append, list.take, take_append_of_le_length (le_of_succ_le_succ hn)]
/-- Taking the first `l₁.length + i` elements in `l₁ ++ l₂` is the same as appending the first
`i` elements of `l₂` to `l₁`. -/
lemma take_append {l₁ l₂ : list α} (i : ℕ) :
take (l₁.length + i) (l₁ ++ l₂) = l₁ ++ (take i l₂) :=
begin
induction l₁, { simp },
have : length l₁_tl + 1 + i = (length l₁_tl + i).succ,
by { rw nat.succ_eq_add_one, exact succ_add _ _ },
simp only [cons_append, length, this, take_cons, l₁_ih, eq_self_iff_true, and_self]
end
/-- The `i`-th element of a list coincides with the `i`-th element of any of its prefixes of
length `> i`. Version designed to rewrite from the big list to the small list. -/
lemma nth_le_take (L : list α) {i j : ℕ} (hi : i < L.length) (hj : i < j) :
nth_le L i hi = nth_le (L.take j) i (by { rw length_take, exact lt_min hj hi }) :=
by { rw nth_le_of_eq (take_append_drop j L).symm hi, exact nth_le_append _ _ }
/-- The `i`-th element of a list coincides with the `i`-th element of any of its prefixes of
length `> i`. Version designed to rewrite from the small list to the big list. -/
lemma nth_le_take' (L : list α) {i j : ℕ} (hi : i < (L.take j).length) :
nth_le (L.take j) i hi = nth_le L i (lt_of_lt_of_le hi (by simp [le_refl])) :=
by { simp at hi, rw nth_le_take L _ hi.1 }
@[simp] theorem drop_nil : ∀ n, drop n [] = ([] : list α)
| 0 := rfl
| (n+1) := rfl
@[simp] theorem drop_one : ∀ l : list α, drop 1 l = tail l
| [] := rfl
| (a :: l) := rfl
theorem drop_add : ∀ m n (l : list α), drop (m + n) l = drop m (drop n l)
| m 0 l := rfl
| m (n+1) [] := (drop_nil _).symm
| m (n+1) (a::l) := drop_add m n _
@[simp] theorem drop_left : ∀ l₁ l₂ : list α, drop (length l₁) (l₁ ++ l₂) = l₂
| [] l₂ := rfl
| (a::l₁) l₂ := drop_left l₁ l₂
theorem drop_left' {l₁ l₂ : list α} {n} (h : length l₁ = n) :
drop n (l₁ ++ l₂) = l₂ :=
by rw ← h; apply drop_left
theorem drop_eq_nth_le_cons : ∀ {n} {l : list α} h,
drop n l = nth_le l n h :: drop (n+1) l
| 0 (a::l) h := rfl
| (n+1) (a::l) h := @drop_eq_nth_le_cons n _ _
@[simp] lemma drop_length (l : list α) : l.drop l.length = [] :=
calc l.drop l.length = (l ++ []).drop l.length : by simp
... = [] : drop_left _ _
lemma drop_append_of_le_length : ∀ {l₁ l₂ : list α} {n : ℕ}, n ≤ l₁.length →
(l₁ ++ l₂).drop n = l₁.drop n ++ l₂
| l₁ l₂ 0 hn := by simp
| [] l₂ (n+1) hn := absurd hn dec_trivial
| (a::l₁) l₂ (n+1) hn :=
by rw [drop, cons_append, drop, drop_append_of_le_length (le_of_succ_le_succ hn)]
/-- Dropping the elements up to `l₁.length + i` in `l₁ + l₂` is the same as dropping the elements
up to `i` in `l₂`. -/
lemma drop_append {l₁ l₂ : list α} (i : ℕ) :
drop (l₁.length + i) (l₁ ++ l₂) = drop i l₂ :=
begin
induction l₁, { simp },
have : length l₁_tl + 1 + i = (length l₁_tl + i).succ,
by { rw nat.succ_eq_add_one, exact succ_add _ _ },
simp only [cons_append, length, this, drop, l₁_ih]
end
/-- The `i + j`-th element of a list coincides with the `j`-th element of the list obtained by
dropping the first `i` elements. Version designed to rewrite from the big list to the small list. -/
lemma nth_le_drop (L : list α) {i j : ℕ} (h : i + j < L.length) :
nth_le L (i + j) h = nth_le (L.drop i) j
begin
have A : i < L.length := lt_of_le_of_lt (nat.le.intro rfl) h,
rw (take_append_drop i L).symm at h,
simpa only [le_of_lt A, min_eq_left, add_lt_add_iff_left, length_take, length_append] using h
end :=
begin
have A : length (take i L) = i, by simp [le_of_lt (lt_of_le_of_lt (nat.le.intro rfl) h)],
rw [nth_le_of_eq (take_append_drop i L).symm h, nth_le_append_right];
simp [A]
end
/-- The `i + j`-th element of a list coincides with the `j`-th element of the list obtained by
dropping the first `i` elements. Version designed to rewrite from the small list to the big list. -/
lemma nth_le_drop' (L : list α) {i j : ℕ} (h : j < (L.drop i).length) :
nth_le (L.drop i) j h = nth_le L (i + j) (nat.add_lt_of_lt_sub_left ((length_drop i L) ▸ h)) :=
by rw nth_le_drop
@[simp] theorem drop_drop (n : ℕ) : ∀ (m) (l : list α), drop n (drop m l) = drop (n + m) l
| m [] := by simp
| 0 l := by simp
| (m+1) (a::l) :=
calc drop n (drop (m + 1) (a :: l)) = drop n (drop m l) : rfl
... = drop (n + m) l : drop_drop m l
... = drop (n + (m + 1)) (a :: l) : rfl
theorem drop_take : ∀ (m : ℕ) (n : ℕ) (l : list α),
drop m (take (m + n) l) = take n (drop m l)
| 0 n _ := by simp
| (m+1) n nil := by simp
| (m+1) n (_::l) :=
have h: m + 1 + n = (m+n) + 1, by ac_refl,
by simpa [take_cons, h] using drop_take m n l
lemma map_drop {α β : Type*} (f : α → β) :
∀ (L : list α) (i : ℕ), (L.drop i).map f = (L.map f).drop i
| [] i := by simp
| L 0 := by simp
| (h :: t) (n+1) := by { dsimp, rw [map_drop], }
theorem modify_nth_tail_eq_take_drop (f : list α → list α) (H : f [] = []) :
∀ n l, modify_nth_tail f n l = take n l ++ f (drop n l)
| 0 l := rfl
| (n+1) [] := H.symm
| (n+1) (b::l) := congr_arg (cons b) (modify_nth_tail_eq_take_drop n l)
theorem modify_nth_eq_take_drop (f : α → α) :
∀ n l, modify_nth f n l = take n l ++ modify_head f (drop n l) :=
modify_nth_tail_eq_take_drop _ rfl
theorem modify_nth_eq_take_cons_drop (f : α → α) {n l} (h) :
modify_nth f n l = take n l ++ f (nth_le l n h) :: drop (n+1) l :=
by rw [modify_nth_eq_take_drop, drop_eq_nth_le_cons h]; refl
theorem update_nth_eq_take_cons_drop (a : α) {n l} (h : n < length l) :
update_nth l n a = take n l ++ a :: drop (n+1) l :=
by rw [update_nth_eq_modify_nth, modify_nth_eq_take_cons_drop _ h]
@[simp] lemma update_nth_eq_nil (l : list α) (n : ℕ) (a : α) : l.update_nth n a = [] ↔ l = [] :=
by cases l; cases n; simp only [update_nth]
section take'
variable [inhabited α]
@[simp] theorem take'_length : ∀ n l, length (@take' α _ n l) = n
| 0 l := rfl
| (n+1) l := congr_arg succ (take'_length _ _)
@[simp] theorem take'_nil : ∀ n, take' n (@nil α) = repeat (default _) n
| 0 := rfl
| (n+1) := congr_arg (cons _) (take'_nil _)
theorem take'_eq_take : ∀ {n} {l : list α},
n ≤ length l → take' n l = take n l
| 0 l h := rfl
| (n+1) (a::l) h := congr_arg (cons _) $
take'_eq_take $ le_of_succ_le_succ h
@[simp] theorem take'_left (l₁ l₂ : list α) : take' (length l₁) (l₁ ++ l₂) = l₁ :=
(take'_eq_take (by simp only [length_append, nat.le_add_right])).trans (take_left _ _)
theorem take'_left' {l₁ l₂ : list α} {n} (h : length l₁ = n) :
take' n (l₁ ++ l₂) = l₁ :=
by rw ← h; apply take'_left
end take'
/-! ### foldl, foldr -/
lemma foldl_ext (f g : α → β → α) (a : α)
{l : list β} (H : ∀ a : α, ∀ b ∈ l, f a b = g a b) :
foldl f a l = foldl g a l :=
begin
induction l with hd tl ih generalizing a, {refl},
unfold foldl,
rw [ih (λ a b bin, H a b $ mem_cons_of_mem _ bin), H a hd (mem_cons_self _ _)]
end
lemma foldr_ext (f g : α → β → β) (b : β)
{l : list α} (H : ∀ a ∈ l, ∀ b : β, f a b = g a b) :
foldr f b l = foldr g b l :=
begin
induction l with hd tl ih, {refl},
simp only [mem_cons_iff, or_imp_distrib, forall_and_distrib, forall_eq] at H,
simp only [foldr, ih H.2, H.1]
end
@[simp] theorem foldl_nil (f : α → β → α) (a : α) : foldl f a [] = a := rfl
@[simp] theorem foldl_cons (f : α → β → α) (a : α) (b : β) (l : list β) :
foldl f a (b::l) = foldl f (f a b) l := rfl
@[simp] theorem foldr_nil (f : α → β → β) (b : β) : foldr f b [] = b := rfl
@[simp] theorem foldr_cons (f : α → β → β) (b : β) (a : α) (l : list α) :
foldr f b (a::l) = f a (foldr f b l) := rfl
@[simp] theorem foldl_append (f : α → β → α) :
∀ (a : α) (l₁ l₂ : list β), foldl f a (l₁++l₂) = foldl f (foldl f a l₁) l₂
| a [] l₂ := rfl
| a (b::l₁) l₂ := by simp only [cons_append, foldl_cons, foldl_append (f a b) l₁ l₂]
@[simp] theorem foldr_append (f : α → β → β) :
∀ (b : β) (l₁ l₂ : list α), foldr f b (l₁++l₂) = foldr f (foldr f b l₂) l₁
| b [] l₂ := rfl
| b (a::l₁) l₂ := by simp only [cons_append, foldr_cons, foldr_append b l₁ l₂]
@[simp] theorem foldl_join (f : α → β → α) :
∀ (a : α) (L : list (list β)), foldl f a (join L) = foldl (foldl f) a L
| a [] := rfl
| a (l::L) := by simp only [join, foldl_append, foldl_cons, foldl_join (foldl f a l) L]
@[simp] theorem foldr_join (f : α → β → β) :
∀ (b : β) (L : list (list α)), foldr f b (join L) = foldr (λ l b, foldr f b l) b L
| a [] := rfl
| a (l::L) := by simp only [join, foldr_append, foldr_join a L, foldr_cons]
theorem foldl_reverse (f : α → β → α) (a : α) (l : list β) :
foldl f a (reverse l) = foldr (λx y, f y x) a l :=
by induction l; [refl, simp only [*, reverse_cons, foldl_append, foldl_cons, foldl_nil, foldr]]
theorem foldr_reverse (f : α → β → β) (a : β) (l : list α) :
foldr f a (reverse l) = foldl (λx y, f y x) a l :=
let t := foldl_reverse (λx y, f y x) a (reverse l) in
by rw reverse_reverse l at t; rwa t
@[simp] theorem foldr_eta : ∀ (l : list α), foldr cons [] l = l
| [] := rfl
| (x::l) := by simp only [foldr_cons, foldr_eta l]; split; refl
@[simp] theorem reverse_foldl {l : list α} : reverse (foldl (λ t h, h :: t) [] l) = l :=
by rw ←foldr_reverse; simp
/- scanl -/
lemma length_scanl {β : Type*} {f : α → β → α} :
∀ a l, length (scanl f a l) = l.length + 1
| a [] := rfl
| a (x :: l) := by erw [length_cons, length_cons, length_scanl]
/- scanr -/
@[simp] theorem scanr_nil (f : α → β → β) (b : β) : scanr f b [] = [b] := rfl
@[simp] theorem scanr_aux_cons (f : α → β → β) (b : β) : ∀ (a : α) (l : list α),
scanr_aux f b (a::l) = (foldr f b (a::l), scanr f b l)
| a [] := rfl
| a (x::l) := let t := scanr_aux_cons x l in
by simp only [scanr, scanr_aux, t, foldr_cons]
@[simp] theorem scanr_cons (f : α → β → β) (b : β) (a : α) (l : list α) :
scanr f b (a::l) = foldr f b (a::l) :: scanr f b l :=
by simp only [scanr, scanr_aux_cons, foldr_cons]; split; refl
section foldl_eq_foldr
-- foldl and foldr coincide when f is commutative and associative
variables {f : α → α → α} (hcomm : commutative f) (hassoc : associative f)
include hassoc
theorem foldl1_eq_foldr1 : ∀ a b l, foldl f a (l++[b]) = foldr f b (a::l)
| a b nil := rfl
| a b (c :: l) :=
by simp only [cons_append, foldl_cons, foldr_cons, foldl1_eq_foldr1 _ _ l]; rw hassoc
include hcomm
theorem foldl_eq_of_comm_of_assoc : ∀ a b l, foldl f a (b::l) = f b (foldl f a l)
| a b nil := hcomm a b
| a b (c::l) := by simp only [foldl_cons];
rw [← foldl_eq_of_comm_of_assoc, right_comm _ hcomm hassoc]; refl
theorem foldl_eq_foldr : ∀ a l, foldl f a l = foldr f a l
| a nil := rfl
| a (b :: l) :=
by simp only [foldr_cons, foldl_eq_of_comm_of_assoc hcomm hassoc]; rw (foldl_eq_foldr a l)
end foldl_eq_foldr
section foldl_eq_foldlr'
variables {f : α → β → α}
variables hf : ∀ a b c, f (f a b) c = f (f a c) b
include hf
theorem foldl_eq_of_comm' : ∀ a b l, foldl f a (b::l) = f (foldl f a l) b
| a b [] := rfl
| a b (c :: l) := by rw [foldl,foldl,foldl,← foldl_eq_of_comm',foldl,hf]
theorem foldl_eq_foldr' : ∀ a l, foldl f a l = foldr (flip f) a l
| a [] := rfl
| a (b :: l) := by rw [foldl_eq_of_comm' hf,foldr,foldl_eq_foldr']; refl
end foldl_eq_foldlr'
section foldl_eq_foldlr'
variables {f : α → β → β}
variables hf : ∀ a b c, f a (f b c) = f b (f a c)
include hf
theorem foldr_eq_of_comm' : ∀ a b l, foldr f a (b::l) = foldr f (f b a) l
| a b [] := rfl
| a b (c :: l) := by rw [foldr,foldr,foldr,hf,← foldr_eq_of_comm']; refl
end foldl_eq_foldlr'
section
variables {op : α → α → α} [ha : is_associative α op] [hc : is_commutative α op]
local notation a * b := op a b
local notation l <*> a := foldl op a l
include ha
lemma foldl_assoc : ∀ {l : list α} {a₁ a₂}, l <*> (a₁ * a₂) = a₁ * (l <*> a₂)
| [] a₁ a₂ := rfl
| (a :: l) a₁ a₂ :=
calc a::l <*> (a₁ * a₂) = l <*> (a₁ * (a₂ * a)) : by simp only [foldl_cons, ha.assoc]
... = a₁ * (a::l <*> a₂) : by rw [foldl_assoc, foldl_cons]
lemma foldl_op_eq_op_foldr_assoc : ∀{l : list α} {a₁ a₂}, (l <*> a₁) * a₂ = a₁ * l.foldr (*) a₂
| [] a₁ a₂ := rfl
| (a :: l) a₁ a₂ := by simp only [foldl_cons, foldr_cons, foldl_assoc, ha.assoc];
rw [foldl_op_eq_op_foldr_assoc]
include hc
lemma foldl_assoc_comm_cons {l : list α} {a₁ a₂} : (a₁ :: l) <*> a₂ = a₁ * (l <*> a₂) :=
by rw [foldl_cons, hc.comm, foldl_assoc]
end
/-! ### mfoldl, mfoldr -/
section mfoldl_mfoldr
variables {m : Type v → Type w} [monad m]
@[simp] theorem mfoldl_nil (f : β → α → m β) {b} : mfoldl f b [] = pure b := rfl
@[simp] theorem mfoldr_nil (f : α → β → m β) {b} : mfoldr f b [] = pure b := rfl
@[simp] theorem mfoldl_cons {f : β → α → m β} {b a l} :
mfoldl f b (a :: l) = f b a >>= λ b', mfoldl f b' l := rfl
@[simp] theorem mfoldr_cons {f : α → β → m β} {b a l} :
mfoldr f b (a :: l) = mfoldr f b l >>= f a := rfl
variables [is_lawful_monad m]
@[simp] theorem mfoldl_append {f : β → α → m β} : ∀ {b l₁ l₂},
mfoldl f b (l₁ ++ l₂) = mfoldl f b l₁ >>= λ x, mfoldl f x l₂
| _ [] _ := by simp only [nil_append, mfoldl_nil, pure_bind]
| _ (_::_) _ := by simp only [cons_append, mfoldl_cons, mfoldl_append, bind_assoc]
@[simp] theorem mfoldr_append {f : α → β → m β} : ∀ {b l₁ l₂},
mfoldr f b (l₁ ++ l₂) = mfoldr f b l₂ >>= λ x, mfoldr f x l₁
| _ [] _ := by simp only [nil_append, mfoldr_nil, bind_pure]
| _ (_::_) _ := by simp only [mfoldr_cons, cons_append, mfoldr_append, bind_assoc]
end mfoldl_mfoldr
/-! ### prod and sum -/
-- list.sum was already defined in defs.lean, but we couldn't tag it with `to_additive` yet.
attribute [to_additive] list.prod
section monoid
variables [monoid α] {l l₁ l₂ : list α} {a : α}
@[simp, to_additive]
theorem prod_nil : ([] : list α).prod = 1 := rfl
@[to_additive]
theorem prod_singleton : [a].prod = a := one_mul a
@[simp, to_additive]
theorem prod_cons : (a::l).prod = a * l.prod :=
calc (a::l).prod = foldl (*) (a * 1) l : by simp only [list.prod, foldl_cons, one_mul, mul_one]
... = _ : foldl_assoc
@[simp, to_additive]
theorem prod_append : (l₁ ++ l₂).prod = l₁.prod * l₂.prod :=
calc (l₁ ++ l₂).prod = foldl (*) (foldl (*) 1 l₁ * 1) l₂ : by simp [list.prod]
... = l₁.prod * l₂.prod : foldl_assoc
@[simp, to_additive]
theorem prod_join {l : list (list α)} : l.join.prod = (l.map list.prod).prod :=
by induction l; [refl, simp only [*, list.join, map, prod_append, prod_cons]]
@[to_additive]
theorem prod_eq_foldr : l.prod = foldr (*) 1 l :=
list.rec_on l rfl $ λ a l ihl, by rw [prod_cons, foldr_cons, ihl]
@[to_additive]
theorem prod_hom_rel {α β γ : Type*} [monoid β] [monoid γ] (l : list α) {r : β → γ → Prop}
{f : α → β} {g : α → γ} (h₁ : r 1 1) (h₂ : ∀⦃a b c⦄, r b c → r (f a * b) (g a * c)) :
r (l.map f).prod (l.map g).prod :=
list.rec_on l h₁ (λ a l hl, by simp only [map_cons, prod_cons, h₂ hl])
@[to_additive]
theorem prod_hom [monoid β] (l : list α) (f : α → β) [is_monoid_hom f] :
(l.map f).prod = f l.prod :=
by { simp only [prod, foldl_map, (is_monoid_hom.map_one f).symm],
exact l.foldl_hom _ _ _ 1 (is_monoid_hom.map_mul f) }
-- `to_additive` chokes on the next few lemmas, so we do them by hand below
@[simp]
lemma prod_take_mul_prod_drop :
∀ (L : list α) (i : ℕ), (L.take i).prod * (L.drop i).prod = L.prod
| [] i := by simp
| L 0 := by simp
| (h :: t) (n+1) := by { dsimp, rw [prod_cons, prod_cons, mul_assoc, prod_take_mul_prod_drop], }
@[simp]
lemma prod_take_succ :
∀ (L : list α) (i : ℕ) (p), (L.take (i + 1)).prod = (L.take i).prod * L.nth_le i p
| [] i p := by cases p
| (h :: t) 0 _ := by simp
| (h :: t) (n+1) _ := by { dsimp, rw [prod_cons, prod_cons, prod_take_succ, mul_assoc], }
/-- A list with product not one must have positive length. -/
lemma length_pos_of_prod_ne_one (L : list α) (h : L.prod ≠ 1) : 0 < L.length :=
by { cases L, { simp at h, cases h, }, { simp, }, }
end monoid
@[simp]
lemma sum_take_add_sum_drop [add_monoid α] :
∀ (L : list α) (i : ℕ), (L.take i).sum + (L.drop i).sum = L.sum
| [] i := by simp
| L 0 := by simp
| (h :: t) (n+1) := by { dsimp, rw [sum_cons, sum_cons, add_assoc, sum_take_add_sum_drop], }
@[simp]
lemma sum_take_succ [add_monoid α] :
∀ (L : list α) (i : ℕ) (p), (L.take (i + 1)).sum = (L.take i).sum + L.nth_le i p
| [] i p := by cases p
| (h :: t) 0 _ := by simp
| (h :: t) (n+1) _ := by { dsimp, rw [sum_cons, sum_cons, sum_take_succ, add_assoc], }
lemma eq_of_sum_take_eq [add_left_cancel_monoid α] {L L' : list α} (h : L.length = L'.length)
(h' : ∀ i ≤ L.length, (L.take i).sum = (L'.take i).sum) : L = L' :=
begin
apply ext_le h (λ i h₁ h₂, _),
have : (L.take (i + 1)).sum = (L'.take (i + 1)).sum := h' _ (nat.succ_le_of_lt h₁),
rw [sum_take_succ L i h₁, sum_take_succ L' i h₂, h' i (le_of_lt h₁)] at this,
exact add_left_cancel this
end
lemma monotone_sum_take [canonically_ordered_add_monoid α] (L : list α) :
monotone (λ i, (L.take i).sum) :=
begin
apply monotone_of_monotone_nat (λ n, _),
by_cases h : n < L.length,
{ rw sum_take_succ _ _ h,
exact le_add_right (le_refl _) },
{ push_neg at h,
simp [take_all_of_le h, take_all_of_le (le_trans h (nat.le_succ _))] }
end
/-- A list with sum not zero must have positive length. -/
lemma length_pos_of_sum_ne_zero [add_monoid α] (L : list α) (h : L.sum ≠ 0) : 0 < L.length :=
by { cases L, { simp at h, cases h, }, { simp, }, }
/-- If all elements in a list are bounded below by `1`, then the length of the list is bounded
by the sum of the elements. -/
lemma length_le_sum_of_one_le (L : list ℕ) (h : ∀ i ∈ L, 1 ≤ i) : L.length ≤ L.sum :=
begin
induction L with j L IH h, { simp },
rw [sum_cons, length, add_comm],
exact add_le_add (h _ (set.mem_insert _ _)) (IH (λ i hi, h i (set.mem_union_right _ hi)))
end
-- Now we tie those lemmas back to their multiplicative versions.
attribute [to_additive] prod_take_mul_prod_drop prod_take_succ length_pos_of_prod_ne_one
/-- A list with positive sum must have positive length. -/
-- This is an easy consequence of `length_pos_of_sum_ne_zero`, but often useful in applications.
lemma length_pos_of_sum_pos [ordered_cancel_add_comm_monoid α] (L : list α) (h : 0 < L.sum) :
0 < L.length :=
length_pos_of_sum_ne_zero L (ne_of_gt h)
@[simp, to_additive]
theorem prod_erase [decidable_eq α] [comm_monoid α] {a} :
Π {l : list α}, a ∈ l → a * (l.erase a).prod = l.prod
| (b::l) h :=
begin
rcases eq_or_ne_mem_of_mem h with rfl | ⟨ne, h⟩,
{ simp only [list.erase, if_pos, prod_cons] },
{ simp only [list.erase, if_neg (mt eq.symm ne), prod_cons, prod_erase h, mul_left_comm a b] }
end
lemma dvd_prod [comm_semiring α] {a} {l : list α} (ha : a ∈ l) : a ∣ l.prod :=
let ⟨s, t, h⟩ := mem_split ha in
by rw [h, prod_append, prod_cons, mul_left_comm]; exact dvd_mul_right _ _
@[simp] theorem sum_const_nat (m n : ℕ) : sum (list.repeat m n) = m * n :=
by induction n; [refl, simp only [*, repeat_succ, sum_cons, nat.mul_succ, add_comm]]
theorem dvd_sum [comm_semiring α] {a} {l : list α} (h : ∀ x ∈ l, a ∣ x) : a ∣ l.sum :=
begin
induction l with x l ih,
{ exact dvd_zero _ },
{ rw [list.sum_cons],
exact dvd_add (h _ (mem_cons_self _ _)) (ih (λ x hx, h x (mem_cons_of_mem _ hx))) }
end
@[simp] theorem length_join (L : list (list α)) : length (join L) = sum (map length L) :=
by induction L; [refl, simp only [*, join, map, sum_cons, length_append]]
@[simp] theorem length_bind (l : list α) (f : α → list β) :
length (list.bind l f) = sum (map (length ∘ f) l) :=
by rw [list.bind, length_join, map_map]
lemma exists_lt_of_sum_lt [decidable_linear_ordered_cancel_add_comm_monoid β] {l : list α}
(f g : α → β) (h : (l.map f).sum < (l.map g).sum) : ∃ x ∈ l, f x < g x :=
begin
induction l with x l,
{ exfalso, exact lt_irrefl _ h },
{ by_cases h' : f x < g x, exact ⟨x, mem_cons_self _ _, h'⟩,
rcases l_ih _ with ⟨y, h1y, h2y⟩, refine ⟨y, mem_cons_of_mem x h1y, h2y⟩, simp at h,
exact lt_of_add_lt_add_left' (lt_of_lt_of_le h $ add_le_add_right (le_of_not_gt h') _) }
end
lemma exists_le_of_sum_le [decidable_linear_ordered_cancel_add_comm_monoid β] {l : list α}
(hl : l ≠ []) (f g : α → β) (h : (l.map f).sum ≤ (l.map g).sum) : ∃ x ∈ l, f x ≤ g x :=
begin
cases l with x l,
{ contradiction },
{ by_cases h' : f x ≤ g x, exact ⟨x, mem_cons_self _ _, h'⟩,
rcases exists_lt_of_sum_lt f g _ with ⟨y, h1y, h2y⟩,
exact ⟨y, mem_cons_of_mem x h1y, le_of_lt h2y⟩, simp at h,
exact lt_of_add_lt_add_left' (lt_of_le_of_lt h $ add_lt_add_right (lt_of_not_ge h') _) }
end
-- Several lemmas about sum/head/tail for `list ℕ`.
-- These are hard to generalize well, as they rely on the fact that `default ℕ = 0`.
-- We'd like to state this as `L.head * L.tail.prod = L.prod`,
-- but because `L.head` relies on an inhabited instances and
-- returns a garbage value for the empty list, this is not possible.
-- Instead we write the statement in terms of `(L.nth 0).get_or_else 1`,
-- and below, restate the lemma just for `ℕ`.
@[to_additive]
lemma head_mul_tail_prod' [monoid α] (L : list α) :
(L.nth 0).get_or_else 1 * L.tail.prod = L.prod :=
by { cases L, { simp, refl, }, { simp, }, }
lemma head_add_tail_sum (L : list ℕ) : L.head + L.tail.sum = L.sum :=
by { cases L, { simp, refl, }, { simp, }, }
lemma head_le_sum (L : list ℕ) : L.head ≤ L.sum :=
nat.le.intro (head_add_tail_sum L)
lemma tail_sum (L : list ℕ) : L.tail.sum = L.sum - L.head :=
by rw [← head_add_tail_sum L, add_comm, nat.add_sub_cancel]
/-! ### join -/
attribute [simp] join
theorem join_eq_nil : ∀ {L : list (list α)}, join L = [] ↔ ∀ l ∈ L, l = []
| [] := iff_of_true rfl (forall_mem_nil _)
| (l::L) := by simp only [join, append_eq_nil, join_eq_nil, forall_mem_cons]
@[simp] theorem join_append (L₁ L₂ : list (list α)) : join (L₁ ++ L₂) = join L₁ ++ join L₂ :=
by induction L₁; [refl, simp only [*, join, cons_append, append_assoc]]
lemma join_join (l : list (list (list α))) : l.join.join = (l.map join).join :=
by { induction l, simp, simp [l_ih] }
/-- In a join, taking the first elements up to an index which is the sum of the lengths of the
first `i` sublists, is the same as taking the join of the first `i` sublists. -/
lemma take_sum_join (L : list (list α)) (i : ℕ) :
L.join.take ((L.map length).take i).sum = (L.take i).join :=
begin
induction L generalizing i, { simp },
cases i, { simp },
simp [take_append, L_ih]
end
/-- In a join, dropping all the elements up to an index which is the sum of the lengths of the
first `i` sublists, is the same as taking the join after dropping the first `i` sublists. -/
lemma drop_sum_join (L : list (list α)) (i : ℕ) :
L.join.drop ((L.map length).take i).sum = (L.drop i).join :=
begin
induction L generalizing i, { simp },
cases i, { simp },
simp [drop_append, L_ih],
end
/-- Taking only the first `i+1` elements in a list, and then dropping the first `i` ones, one is
left with a list of length `1` made of the `i`-th element of the original list. -/
lemma drop_take_succ_eq_cons_nth_le (L : list α) {i : ℕ} (hi : i < L.length) :
(L.take (i+1)).drop i = [nth_le L i hi] :=
begin
induction L generalizing i,
{ simp only [length] at hi, exact (nat.not_succ_le_zero i hi).elim },
cases i, { simp },
have : i < L_tl.length,
{ simp at hi,
exact nat.lt_of_succ_lt_succ hi },
simp [L_ih this],
refl
end
/-- In a join of sublists, taking the slice between the indices `A` and `B - 1` gives back the
original sublist of index `i` if `A` is the sum of the lenghts of sublists of index `< i`, and
`B` is the sum of the lengths of sublists of index `≤ i`. -/
lemma drop_take_succ_join_eq_nth_le (L : list (list α)) {i : ℕ} (hi : i < L.length) :
(L.join.take ((L.map length).take (i+1)).sum).drop ((L.map length).take i).sum = nth_le L i hi :=
begin
have : (L.map length).take i = ((L.take (i+1)).map length).take i, by simp [map_take, take_take],
simp [take_sum_join, this, drop_sum_join, drop_take_succ_eq_cons_nth_le _ hi]
end
/-- Auxiliary lemma to control elements in a join. -/
lemma sum_take_map_length_lt1 (L : list (list α)) {i j : ℕ}
(hi : i < L.length) (hj : j < (nth_le L i hi).length) :
((L.map length).take i).sum + j < ((L.map length).take (i+1)).sum :=
by simp [hi, sum_take_succ, hj]
/-- Auxiliary lemma to control elements in a join. -/
lemma sum_take_map_length_lt2 (L : list (list α)) {i j : ℕ}
(hi : i < L.length) (hj : j < (nth_le L i hi).length) :
((L.map length).take i).sum + j < L.join.length :=
begin
convert lt_of_lt_of_le (sum_take_map_length_lt1 L hi hj) (monotone_sum_take _ hi),
have : L.length = (L.map length).length, by simp,
simp [this, -length_map]
end
/-- The `n`-th element in a join of sublists is the `j`-th element of the `i`th sublist,
where `n` can be obtained in terms of `i` and `j` by adding the lengths of all the sublists
of index `< i`, and adding `j`. -/
lemma nth_le_join (L : list (list α)) {i j : ℕ}
(hi : i < L.length) (hj : j < (nth_le L i hi).length) :
nth_le L.join (((L.map length).take i).sum + j) (sum_take_map_length_lt2 L hi hj) =
nth_le (nth_le L i hi) j hj :=
by rw [nth_le_take L.join (sum_take_map_length_lt2 L hi hj) (sum_take_map_length_lt1 L hi hj),
nth_le_drop, nth_le_of_eq (drop_take_succ_join_eq_nth_le L hi)]
/-- Two lists of sublists are equal iff their joins coincide, as well as the lengths of the
sublists. -/
theorem eq_iff_join_eq (L L' : list (list α)) :
L = L' ↔ L.join = L'.join ∧ map length L = map length L' :=
begin
refine ⟨λ H, by simp [H], _⟩,
rintros ⟨join_eq, length_eq⟩,
apply ext_le,
{ have : length (map length L) = length (map length L'), by rw length_eq,
simpa using this },
{ assume n h₁ h₂,
rw [← drop_take_succ_join_eq_nth_le, ← drop_take_succ_join_eq_nth_le, join_eq, length_eq] }
end
/-! ### lexicographic ordering -/
/-- Given a strict order `<` on `α`, the lexicographic strict order on `list α`, for which
`[a0, ..., an] < [b0, ..., b_k]` if `a0 < b0` or `a0 = b0` and `[a1, ..., an] < [b1, ..., bk]`.
The definition is given for any relation `r`, not only strict orders. -/
inductive lex (r : α → α → Prop) : list α → list α → Prop
| nil {a l} : lex [] (a :: l)
| cons {a l₁ l₂} (h : lex l₁ l₂) : lex (a :: l₁) (a :: l₂)
| rel {a₁ l₁ a₂ l₂} (h : r a₁ a₂) : lex (a₁ :: l₁) (a₂ :: l₂)
namespace lex
theorem cons_iff {r : α → α → Prop} [is_irrefl α r] {a l₁ l₂} :
lex r (a :: l₁) (a :: l₂) ↔ lex r l₁ l₂ :=
⟨λ h, by cases h with _ _ _ _ _ h _ _ _ _ h;
[exact h, exact (irrefl_of r a h).elim], lex.cons⟩
@[simp] theorem not_nil_right (r : α → α → Prop) (l : list α) : ¬ lex r l [].
instance is_order_connected (r : α → α → Prop)
[is_order_connected α r] [is_trichotomous α r] :
is_order_connected (list α) (lex r) :=
⟨λ l₁, match l₁ with
| _, [], c::l₃, nil := or.inr nil
| _, [], c::l₃, rel _ := or.inr nil
| _, [], c::l₃, cons _ := or.inr nil
| _, b::l₂, c::l₃, nil := or.inl nil
| a::l₁, b::l₂, c::l₃, rel h :=
(is_order_connected.conn _ b _ h).imp rel rel
| a::l₁, b::l₂, _::l₃, cons h := begin
rcases trichotomous_of r a b with ab | rfl | ab,
{ exact or.inl (rel ab) },
{ exact (_match _ l₂ _ h).imp cons cons },
{ exact or.inr (rel ab) }
end
end⟩
instance is_trichotomous (r : α → α → Prop) [is_trichotomous α r] :
is_trichotomous (list α) (lex r) :=
⟨λ l₁, match l₁ with
| [], [] := or.inr (or.inl rfl)
| [], b::l₂ := or.inl nil
| a::l₁, [] := or.inr (or.inr nil)
| a::l₁, b::l₂ := begin
rcases trichotomous_of r a b with ab | rfl | ab,
{ exact or.inl (rel ab) },
{ exact (_match l₁ l₂).imp cons
(or.imp (congr_arg _) cons) },
{ exact or.inr (or.inr (rel ab)) }
end
end⟩
instance is_asymm (r : α → α → Prop)
[is_asymm α r] : is_asymm (list α) (lex r) :=
⟨λ l₁, match l₁ with
| a::l₁, b::l₂, lex.rel h₁, lex.rel h₂ := asymm h₁ h₂
| a::l₁, b::l₂, lex.rel h₁, lex.cons h₂ := asymm h₁ h₁
| a::l₁, b::l₂, lex.cons h₁, lex.rel h₂ := asymm h₂ h₂
| a::l₁, b::l₂, lex.cons h₁, lex.cons h₂ :=
by exact _match _ _ h₁ h₂
end⟩
instance is_strict_total_order (r : α → α → Prop)
[is_strict_total_order' α r] : is_strict_total_order' (list α) (lex r) :=
{..is_strict_weak_order_of_is_order_connected}
instance decidable_rel [decidable_eq α] (r : α → α → Prop)
[decidable_rel r] : decidable_rel (lex r)
| l₁ [] := is_false $ λ h, by cases h
| [] (b::l₂) := is_true lex.nil
| (a::l₁) (b::l₂) := begin
haveI := decidable_rel l₁ l₂,
refine decidable_of_iff (r a b ∨ a = b ∧ lex r l₁ l₂) ⟨λ h, _, λ h, _⟩,
{ rcases h with h | ⟨rfl, h⟩,
{ exact lex.rel h },
{ exact lex.cons h } },
{ rcases h with _|⟨_,_,_,h⟩|⟨_,_,_,_,h⟩,
{ exact or.inr ⟨rfl, h⟩ },
{ exact or.inl h } }
end
theorem append_right (r : α → α → Prop) :
∀ {s₁ s₂} t, lex r s₁ s₂ → lex r s₁ (s₂ ++ t)
| _ _ t nil := nil
| _ _ t (cons h) := cons (append_right _ h)
| _ _ t (rel r) := rel r
theorem append_left (R : α → α → Prop) {t₁ t₂} (h : lex R t₁ t₂) :
∀ s, lex R (s ++ t₁) (s ++ t₂)
| [] := h
| (a::l) := cons (append_left l)
theorem imp {r s : α → α → Prop} (H : ∀ a b, r a b → s a b) :
∀ l₁ l₂, lex r l₁ l₂ → lex s l₁ l₂
| _ _ nil := nil
| _ _ (cons h) := cons (imp _ _ h)
| _ _ (rel r) := rel (H _ _ r)
theorem to_ne : ∀ {l₁ l₂ : list α}, lex (≠) l₁ l₂ → l₁ ≠ l₂
| _ _ (cons h) e := to_ne h (list.cons.inj e).2
| _ _ (rel r) e := r (list.cons.inj e).1
theorem ne_iff {l₁ l₂ : list α} (H : length l₁ ≤ length l₂) :
lex (≠) l₁ l₂ ↔ l₁ ≠ l₂ :=
⟨to_ne, λ h, begin
induction l₁ with a l₁ IH generalizing l₂; cases l₂ with b l₂,
{ contradiction },
{ apply nil },
{ exact (not_lt_of_ge H).elim (succ_pos _) },
{ cases classical.em (a = b) with ab ab,
{ subst b, apply cons,
exact IH (le_of_succ_le_succ H) (mt (congr_arg _) h) },
{ exact rel ab } }
end⟩
end lex
--Note: this overrides an instance in core lean
instance has_lt' [has_lt α] : has_lt (list α) := ⟨lex (<)⟩
theorem nil_lt_cons [has_lt α] (a : α) (l : list α) : [] < a :: l :=
lex.nil
instance [linear_order α] : linear_order (list α) :=
linear_order_of_STO' (lex (<))
--Note: this overrides an instance in core lean
instance has_le' [linear_order α] : has_le (list α) :=
preorder.to_has_le _
instance [decidable_linear_order α] : decidable_linear_order (list α) :=
decidable_linear_order_of_STO' (lex (<))
/-! ### all & any -/
@[simp] theorem all_nil (p : α → bool) : all [] p = tt := rfl
@[simp] theorem all_cons (p : α → bool) (a : α) (l : list α) :
all (a::l) p = (p a && all l p) := rfl
theorem all_iff_forall {p : α → bool} {l : list α} : all l p ↔ ∀ a ∈ l, p a :=
begin
induction l with a l ih,
{ exact iff_of_true rfl (forall_mem_nil _) },
simp only [all_cons, band_coe_iff, ih, forall_mem_cons]
end
theorem all_iff_forall_prop {p : α → Prop} [decidable_pred p]
{l : list α} : all l (λ a, p a) ↔ ∀ a ∈ l, p a :=
by simp only [all_iff_forall, bool.of_to_bool_iff]
@[simp] theorem any_nil (p : α → bool) : any [] p = ff := rfl
@[simp] theorem any_cons (p : α → bool) (a : α) (l : list α) :
any (a::l) p = (p a || any l p) := rfl
theorem any_iff_exists {p : α → bool} {l : list α} : any l p ↔ ∃ a ∈ l, p a :=
begin
induction l with a l ih,
{ exact iff_of_false bool.not_ff (not_exists_mem_nil _) },
simp only [any_cons, bor_coe_iff, ih, exists_mem_cons_iff]
end
theorem any_iff_exists_prop {p : α → Prop} [decidable_pred p]
{l : list α} : any l (λ a, p a) ↔ ∃ a ∈ l, p a :=
by simp [any_iff_exists]
theorem any_of_mem {p : α → bool} {a : α} {l : list α} (h₁ : a ∈ l) (h₂ : p a) : any l p :=
any_iff_exists.2 ⟨_, h₁, h₂⟩
@[priority 500] instance decidable_forall_mem {p : α → Prop} [decidable_pred p] (l : list α) :
decidable (∀ x ∈ l, p x) :=
decidable_of_iff _ all_iff_forall_prop
instance decidable_exists_mem {p : α → Prop} [decidable_pred p] (l : list α) :
decidable (∃ x ∈ l, p x) :=
decidable_of_iff _ any_iff_exists_prop
/-! ### map for partial functions -/
/-- Partial map. If `f : Π a, p a → β` is a partial function defined on
`a : α` satisfying `p`, then `pmap f l h` is essentially the same as `map f l`
but is defined only when all members of `l` satisfy `p`, using the proof
to apply `f`. -/
@[simp] def pmap {p : α → Prop} (f : Π a, p a → β) : Π l : list α, (∀ a ∈ l, p a) → list β
| [] H := []
| (a::l) H := f a (forall_mem_cons.1 H).1 :: pmap l (forall_mem_cons.1 H).2
/-- "Attach" the proof that the elements of `l` are in `l` to produce a new list
with the same elements but in the type `{x // x ∈ l}`. -/
def attach (l : list α) : list {x // x ∈ l} := pmap subtype.mk l (λ a, id)
theorem pmap_eq_map (p : α → Prop) (f : α → β) (l : list α) (H) :
@pmap _ _ p (λ a _, f a) l H = map f l :=
by induction l; [refl, simp only [*, pmap, map]]; split; refl
theorem pmap_congr {p q : α → Prop} {f : Π a, p a → β} {g : Π a, q a → β}
(l : list α) {H₁ H₂} (h : ∀ a h₁ h₂, f a h₁ = g a h₂) :
pmap f l H₁ = pmap g l H₂ :=
by induction l with _ _ ih; [refl, rw [pmap, pmap, h, ih]]
theorem map_pmap {p : α → Prop} (g : β → γ) (f : Π a, p a → β)
(l H) : map g (pmap f l H) = pmap (λ a h, g (f a h)) l H :=
by induction l; [refl, simp only [*, pmap, map]]; split; refl
theorem pmap_eq_map_attach {p : α → Prop} (f : Π a, p a → β)
(l H) : pmap f l H = l.attach.map (λ x, f x.1 (H _ x.2)) :=
by rw [attach, map_pmap]; exact pmap_congr l (λ a h₁ h₂, rfl)
theorem attach_map_val (l : list α) : l.attach.map subtype.val = l :=
by rw [attach, map_pmap]; exact (pmap_eq_map _ _ _ _).trans (map_id l)
@[simp] theorem mem_attach (l : list α) : ∀ x, x ∈ l.attach | ⟨a, h⟩ :=
by have := mem_map.1 (by rw [attach_map_val]; exact h);
{ rcases this with ⟨⟨_, _⟩, m, rfl⟩, exact m }
@[simp] theorem mem_pmap {p : α → Prop} {f : Π a, p a → β}
{l H b} : b ∈ pmap f l H ↔ ∃ a (h : a ∈ l), f a (H a h) = b :=
by simp only [pmap_eq_map_attach, mem_map, mem_attach, true_and, subtype.exists]
@[simp] theorem length_pmap {p : α → Prop} {f : Π a, p a → β}
{l H} : length (pmap f l H) = length l :=
by induction l; [refl, simp only [*, pmap, length]]
@[simp] lemma length_attach (L : list α) : L.attach.length = L.length := length_pmap
/-! ### find -/
section find
variables {p : α → Prop} [decidable_pred p] {l : list α} {a : α}
@[simp] theorem find_nil (p : α → Prop) [decidable_pred p] : find p [] = none :=
rfl
@[simp] theorem find_cons_of_pos (l) (h : p a) : find p (a::l) = some a :=
if_pos h
@[simp] theorem find_cons_of_neg (l) (h : ¬ p a) : find p (a::l) = find p l :=
if_neg h
@[simp] theorem find_eq_none : find p l = none ↔ ∀ x ∈ l, ¬ p x :=
begin
induction l with a l IH,
{ exact iff_of_true rfl (forall_mem_nil _) },
rw forall_mem_cons, by_cases h : p a,
{ simp only [find_cons_of_pos _ h, h, not_true, false_and] },
{ rwa [find_cons_of_neg _ h, iff_true_intro h, true_and] }
end
theorem find_some (H : find p l = some a) : p a :=
begin
induction l with b l IH, {contradiction},
by_cases h : p b,
{ rw find_cons_of_pos _ h at H, cases H, exact h },
{ rw find_cons_of_neg _ h at H, exact IH H }
end
@[simp] theorem find_mem (H : find p l = some a) : a ∈ l :=
begin
induction l with b l IH, {contradiction},
by_cases h : p b,
{ rw find_cons_of_pos _ h at H, cases H, apply mem_cons_self },
{ rw find_cons_of_neg _ h at H, exact mem_cons_of_mem _ (IH H) }
end
end find
/-! ### lookmap -/
section lookmap
variables (f : α → option α)
@[simp] theorem lookmap_nil : [].lookmap f = [] := rfl
@[simp] theorem lookmap_cons_none {a : α} (l : list α) (h : f a = none) :
(a :: l).lookmap f = a :: l.lookmap f :=
by simp [lookmap, h]
@[simp] theorem lookmap_cons_some {a b : α} (l : list α) (h : f a = some b) :
(a :: l).lookmap f = b :: l :=
by simp [lookmap, h]
theorem lookmap_some : ∀ l : list α, l.lookmap some = l
| [] := rfl
| (a::l) := rfl
theorem lookmap_none : ∀ l : list α, l.lookmap (λ _, none) = l
| [] := rfl
| (a::l) := congr_arg (cons a) (lookmap_none l)
theorem lookmap_congr {f g : α → option α} :
∀ {l : list α}, (∀ a ∈ l, f a = g a) → l.lookmap f = l.lookmap g
| [] H := rfl
| (a::l) H := begin
cases forall_mem_cons.1 H with H₁ H₂,
cases h : g a with b,
{ simp [h, H₁.trans h, lookmap_congr H₂] },
{ simp [lookmap_cons_some _ _ h, lookmap_cons_some _ _ (H₁.trans h)] }
end
theorem lookmap_of_forall_not {l : list α} (H : ∀ a ∈ l, f a = none) : l.lookmap f = l :=
(lookmap_congr H).trans (lookmap_none l)
theorem lookmap_map_eq (g : α → β) (h : ∀ a (b ∈ f a), g a = g b) :
∀ l : list α, map g (l.lookmap f) = map g l
| [] := rfl
| (a::l) := begin
cases h' : f a with b,
{ simp [h', lookmap_map_eq] },
{ simp [lookmap_cons_some _ _ h', h _ _ h'] }
end
theorem lookmap_id' (h : ∀ a (b ∈ f a), a = b) (l : list α) : l.lookmap f = l :=
by rw [← map_id (l.lookmap f), lookmap_map_eq, map_id]; exact h
theorem length_lookmap (l : list α) : length (l.lookmap f) = length l :=
by rw [← length_map, lookmap_map_eq _ (λ _, ()), length_map]; simp
end lookmap
/-! ### filter_map -/
@[simp] theorem filter_map_nil (f : α → option β) : filter_map f [] = [] := rfl
@[simp] theorem filter_map_cons_none {f : α → option β} (a : α) (l : list α) (h : f a = none) :
filter_map f (a :: l) = filter_map f l :=
by simp only [filter_map, h]
@[simp] theorem filter_map_cons_some (f : α → option β)
(a : α) (l : list α) {b : β} (h : f a = some b) :
filter_map f (a :: l) = b :: filter_map f l :=
by simp only [filter_map, h]; split; refl
theorem filter_map_eq_map (f : α → β) : filter_map (some ∘ f) = map f :=
begin
funext l,
induction l with a l IH, {refl},
simp only [filter_map_cons_some (some ∘ f) _ _ rfl, IH, map_cons], split; refl
end
theorem filter_map_eq_filter (p : α → Prop) [decidable_pred p] :
filter_map (option.guard p) = filter p :=
begin
funext l,
induction l with a l IH, {refl},
by_cases pa : p a,
{ simp only [filter_map, option.guard, IH, if_pos pa, filter_cons_of_pos _ pa], split; refl },
{ simp only [filter_map, option.guard, IH, if_neg pa, filter_cons_of_neg _ pa] }
end
theorem filter_map_filter_map (f : α → option β) (g : β → option γ) (l : list α) :
filter_map g (filter_map f l) = filter_map (λ x, (f x).bind g) l :=
begin
induction l with a l IH, {refl},
cases h : f a with b,
{ rw [filter_map_cons_none _ _ h, filter_map_cons_none, IH],
simp only [h, option.none_bind'] },
rw filter_map_cons_some _ _ _ h,
cases h' : g b with c;
[ rw [filter_map_cons_none _ _ h', filter_map_cons_none, IH],
rw [filter_map_cons_some _ _ _ h', filter_map_cons_some, IH] ];
simp only [h, h', option.some_bind']
end
theorem map_filter_map (f : α → option β) (g : β → γ) (l : list α) :
map g (filter_map f l) = filter_map (λ x, (f x).map g) l :=
by rw [← filter_map_eq_map, filter_map_filter_map]; refl
theorem filter_map_map (f : α → β) (g : β → option γ) (l : list α) :
filter_map g (map f l) = filter_map (g ∘ f) l :=
by rw [← filter_map_eq_map, filter_map_filter_map]; refl
theorem filter_filter_map (f : α → option β) (p : β → Prop) [decidable_pred p] (l : list α) :
filter p (filter_map f l) = filter_map (λ x, (f x).filter p) l :=
by rw [← filter_map_eq_filter, filter_map_filter_map]; refl
theorem filter_map_filter (p : α → Prop) [decidable_pred p] (f : α → option β) (l : list α) :
filter_map f (filter p l) = filter_map (λ x, if p x then f x else none) l :=
begin
rw [← filter_map_eq_filter, filter_map_filter_map], congr,
funext x,
show (option.guard p x).bind f = ite (p x) (f x) none,
by_cases h : p x,
{ simp only [option.guard, if_pos h, option.some_bind'] },
{ simp only [option.guard, if_neg h, option.none_bind'] }
end
@[simp] theorem filter_map_some (l : list α) : filter_map some l = l :=
by rw filter_map_eq_map; apply map_id
@[simp] theorem mem_filter_map (f : α → option β) (l : list α) {b : β} :
b ∈ filter_map f l ↔ ∃ a, a ∈ l ∧ f a = some b :=
begin
induction l with a l IH,
{ split, { intro H, cases H }, { rintro ⟨_, H, _⟩, cases H } },
cases h : f a with b',
{ have : f a ≠ some b, {rw h, intro, contradiction},
simp only [filter_map_cons_none _ _ h, IH, mem_cons_iff,
or_and_distrib_right, exists_or_distrib, exists_eq_left, this, false_or] },
{ have : f a = some b ↔ b = b',
{ split; intro t, {rw t at h; injection h}, {exact t.symm ▸ h} },
simp only [filter_map_cons_some _ _ _ h, IH, mem_cons_iff,
or_and_distrib_right, exists_or_distrib, this, exists_eq_left] }
end
theorem map_filter_map_of_inv (f : α → option β) (g : β → α)
(H : ∀ x : α, (f x).map g = some x) (l : list α) :
map g (filter_map f l) = l :=
by simp only [map_filter_map, H, filter_map_some]
theorem sublist.filter_map (f : α → option β) {l₁ l₂ : list α}
(s : l₁ <+ l₂) : filter_map f l₁ <+ filter_map f l₂ :=
by induction s with l₁ l₂ a s IH l₁ l₂ a s IH;
simp only [filter_map]; cases f a with b;
simp only [filter_map, IH, sublist.cons, sublist.cons2]
theorem sublist.map (f : α → β) {l₁ l₂ : list α}
(s : l₁ <+ l₂) : map f l₁ <+ map f l₂ :=
filter_map_eq_map f ▸ s.filter_map _
/-! ### filter -/
section filter
variables {p : α → Prop} [decidable_pred p]
lemma filter_congr {p q : α → Prop} [decidable_pred p] [decidable_pred q]
: ∀ {l : list α}, (∀ x ∈ l, p x ↔ q x) → filter p l = filter q l
| [] _ := rfl
| (a::l) h := by rw forall_mem_cons at h; by_cases pa : p a;
[simp only [filter_cons_of_pos _ pa, filter_cons_of_pos _ (h.1.1 pa), filter_congr h.2],
simp only [filter_cons_of_neg _ pa, filter_cons_of_neg _ (mt h.1.2 pa), filter_congr h.2]];
split; refl
@[simp] theorem filter_subset (l : list α) : filter p l ⊆ l :=
(filter_sublist l).subset
theorem of_mem_filter {a : α} : ∀ {l}, a ∈ filter p l → p a
| (b::l) ain :=
if pb : p b then
have a ∈ b :: filter p l, by simpa only [filter_cons_of_pos _ pb] using ain,
or.elim (eq_or_mem_of_mem_cons this)
(assume : a = b, begin rw [← this] at pb, exact pb end)
(assume : a ∈ filter p l, of_mem_filter this)
else
begin simp only [filter_cons_of_neg _ pb] at ain, exact (of_mem_filter ain) end
theorem mem_of_mem_filter {a : α} {l} (h : a ∈ filter p l) : a ∈ l :=
filter_subset l h
theorem mem_filter_of_mem {a : α} : ∀ {l}, a ∈ l → p a → a ∈ filter p l
| (_::l) (or.inl rfl) pa := by rw filter_cons_of_pos _ pa; apply mem_cons_self
| (b::l) (or.inr ain) pa := if pb : p b
then by rw [filter_cons_of_pos _ pb]; apply mem_cons_of_mem; apply mem_filter_of_mem ain pa
else by rw [filter_cons_of_neg _ pb]; apply mem_filter_of_mem ain pa
@[simp] theorem mem_filter {a : α} {l} : a ∈ filter p l ↔ a ∈ l ∧ p a :=
⟨λ h, ⟨mem_of_mem_filter h, of_mem_filter h⟩, λ ⟨h₁, h₂⟩, mem_filter_of_mem h₁ h₂⟩
theorem filter_eq_self {l} : filter p l = l ↔ ∀ a ∈ l, p a :=
begin
induction l with a l ih,
{ exact iff_of_true rfl (forall_mem_nil _) },
rw forall_mem_cons, by_cases p a,
{ rw [filter_cons_of_pos _ h, cons_inj', ih, and_iff_right h] },
{ rw [filter_cons_of_neg _ h],
refine iff_of_false _ (mt and.left h), intro e,
have := filter_sublist l, rw e at this,
exact not_lt_of_ge (length_le_of_sublist this) (lt_succ_self _) }
end
theorem filter_eq_nil {l} : filter p l = [] ↔ ∀ a ∈ l, ¬p a :=
by simp only [eq_nil_iff_forall_not_mem, mem_filter, not_and]
theorem filter_sublist_filter {l₁ l₂} (s : l₁ <+ l₂) : filter p l₁ <+ filter p l₂ :=
filter_map_eq_filter p ▸ s.filter_map _
theorem filter_of_map (f : β → α) (l) : filter p (map f l) = map f (filter (p ∘ f) l) :=
by rw [← filter_map_eq_map, filter_filter_map, filter_map_filter]; refl
@[simp] theorem filter_filter {q} [decidable_pred q] : ∀ l,
filter p (filter q l) = filter (λ a, p a ∧ q a) l
| [] := rfl
| (a :: l) := by by_cases hp : p a; by_cases hq : q a; simp only [hp, hq, filter, if_true, if_false,
true_and, false_and, filter_filter l, eq_self_iff_true]
@[simp] lemma filter_true {h : decidable_pred (λ a : α, true)} (l : list α) :
@filter α (λ _, true) h l = l :=
by convert filter_eq_self.2 (λ _ _, trivial)
@[simp] lemma filter_false {h : decidable_pred (λ a : α, false)} (l : list α) :
@filter α (λ _, false) h l = [] :=
by convert filter_eq_nil.2 (λ _ _, id)
@[simp] theorem span_eq_take_drop (p : α → Prop) [decidable_pred p] :
∀ (l : list α), span p l = (take_while p l, drop_while p l)
| [] := rfl
| (a::l) :=
if pa : p a then by simp only [span, if_pos pa, span_eq_take_drop l, take_while, drop_while]
else by simp only [span, take_while, drop_while, if_neg pa]
@[simp] theorem take_while_append_drop (p : α → Prop) [decidable_pred p] :
∀ (l : list α), take_while p l ++ drop_while p l = l
| [] := rfl
| (a::l) := if pa : p a then by rw [take_while, drop_while, if_pos pa, if_pos pa, cons_append,
take_while_append_drop l]
else by rw [take_while, drop_while, if_neg pa, if_neg pa, nil_append]
@[simp] theorem countp_nil (p : α → Prop) [decidable_pred p] : countp p [] = 0 := rfl
@[simp] theorem countp_cons_of_pos {a : α} (l) (pa : p a) : countp p (a::l) = countp p l + 1 :=
if_pos pa
@[simp] theorem countp_cons_of_neg {a : α} (l) (pa : ¬ p a) : countp p (a::l) = countp p l :=
if_neg pa
theorem countp_eq_length_filter (l) : countp p l = length (filter p l) :=
by induction l with x l ih; [refl, by_cases (p x)];
[simp only [filter_cons_of_pos _ h, countp, ih, if_pos h],
simp only [countp_cons_of_neg _ h, ih, filter_cons_of_neg _ h]]; refl
local attribute [simp] countp_eq_length_filter
@[simp] theorem countp_append (l₁ l₂) : countp p (l₁ ++ l₂) = countp p l₁ + countp p l₂ :=
by simp only [countp_eq_length_filter, filter_append, length_append]
theorem countp_pos {l} : 0 < countp p l ↔ ∃ a ∈ l, p a :=
by simp only [countp_eq_length_filter, length_pos_iff_exists_mem, mem_filter, exists_prop]
theorem countp_le_of_sublist {l₁ l₂} (s : l₁ <+ l₂) : countp p l₁ ≤ countp p l₂ :=
by simpa only [countp_eq_length_filter] using length_le_of_sublist (filter_sublist_filter s)
@[simp] theorem countp_filter {q} [decidable_pred q] (l : list α) :
countp p (filter q l) = countp (λ a, p a ∧ q a) l :=
by simp only [countp_eq_length_filter, filter_filter]
end filter
/-! ### count -/
section count
variable [decidable_eq α]
@[simp] theorem count_nil (a : α) : count a [] = 0 := rfl
theorem count_cons (a b : α) (l : list α) :
count a (b :: l) = if a = b then succ (count a l) else count a l := rfl
theorem count_cons' (a b : α) (l : list α) :
count a (b :: l) = count a l + (if a = b then 1 else 0) :=
begin rw count_cons, split_ifs; refl end
@[simp] theorem count_cons_self (a : α) (l : list α) : count a (a::l) = succ (count a l) :=
if_pos rfl
@[simp, priority 990]
theorem count_cons_of_ne {a b : α} (h : a ≠ b) (l : list α) : count a (b::l) = count a l :=
if_neg h
theorem count_tail : Π (l : list α) (a : α) (h : 0 < l.length),
l.tail.count a = l.count a - ite (a = list.nth_le l 0 h) 1 0
| (_ :: _) a h := by { rw [count_cons], split_ifs; simp }
theorem count_le_of_sublist (a : α) {l₁ l₂} : l₁ <+ l₂ → count a l₁ ≤ count a l₂ :=
countp_le_of_sublist
theorem count_le_count_cons (a b : α) (l : list α) : count a l ≤ count a (b :: l) :=
count_le_of_sublist _ (sublist_cons _ _)
theorem count_singleton (a : α) : count a [a] = 1 := if_pos rfl
@[simp] theorem count_append (a : α) : ∀ l₁ l₂, count a (l₁ ++ l₂) = count a l₁ + count a l₂ :=
countp_append
theorem count_concat (a : α) (l : list α) : count a (concat l a) = succ (count a l) :=
by simp [-add_comm]
theorem count_pos {a : α} {l : list α} : 0 < count a l ↔ a ∈ l :=
by simp only [count, countp_pos, exists_prop, exists_eq_right']
@[simp, priority 980]
theorem count_eq_zero_of_not_mem {a : α} {l : list α} (h : a ∉ l) : count a l = 0 :=
by_contradiction $ λ h', h $ count_pos.1 (nat.pos_of_ne_zero h')
theorem not_mem_of_count_eq_zero {a : α} {l : list α} (h : count a l = 0) : a ∉ l :=
λ h', ne_of_gt (count_pos.2 h') h
@[simp] theorem count_repeat (a : α) (n : ℕ) : count a (repeat a n) = n :=
by rw [count, countp_eq_length_filter, filter_eq_self.2, length_repeat];
exact λ b m, (eq_of_mem_repeat m).symm
theorem le_count_iff_repeat_sublist {a : α} {l : list α} {n : ℕ} :
n ≤ count a l ↔ repeat a n <+ l :=
⟨λ h, ((repeat_sublist_repeat a).2 h).trans $
have filter (eq a) l = repeat a (count a l), from eq_repeat.2
⟨by simp only [count, countp_eq_length_filter], λ b m, (of_mem_filter m).symm⟩,
by rw ← this; apply filter_sublist,
λ h, by simpa only [count_repeat] using count_le_of_sublist a h⟩
@[simp] theorem count_filter {p} [decidable_pred p]
{a} {l : list α} (h : p a) : count a (filter p l) = count a l :=
by simp only [count, countp_filter]; congr; exact
set.ext (λ b, and_iff_left_of_imp (λ e, e ▸ h))
end count
/-! ### prefix, suffix, infix -/
@[simp] theorem prefix_append (l₁ l₂ : list α) : l₁ <+: l₁ ++ l₂ := ⟨l₂, rfl⟩
@[simp] theorem suffix_append (l₁ l₂ : list α) : l₂ <:+ l₁ ++ l₂ := ⟨l₁, rfl⟩
theorem infix_append (l₁ l₂ l₃ : list α) : l₂ <:+: l₁ ++ l₂ ++ l₃ := ⟨l₁, l₃, rfl⟩
@[simp] theorem infix_append' (l₁ l₂ l₃ : list α) : l₂ <:+: l₁ ++ (l₂ ++ l₃) :=
by rw ← list.append_assoc; apply infix_append
theorem nil_prefix (l : list α) : [] <+: l := ⟨l, rfl⟩
theorem nil_suffix (l : list α) : [] <:+ l := ⟨l, append_nil _⟩
@[refl] theorem prefix_refl (l : list α) : l <+: l := ⟨[], append_nil _⟩
@[refl] theorem suffix_refl (l : list α) : l <:+ l := ⟨[], rfl⟩
@[simp] theorem suffix_cons (a : α) : ∀ l, l <:+ a :: l := suffix_append [a]
theorem prefix_concat (a : α) (l) : l <+: concat l a := by simp
theorem infix_of_prefix {l₁ l₂ : list α} : l₁ <+: l₂ → l₁ <:+: l₂ :=
λ⟨t, h⟩, ⟨[], t, h⟩
theorem infix_of_suffix {l₁ l₂ : list α} : l₁ <:+ l₂ → l₁ <:+: l₂ :=
λ⟨t, h⟩, ⟨t, [], by simp only [h, append_nil]⟩
@[refl] theorem infix_refl (l : list α) : l <:+: l := infix_of_prefix $ prefix_refl l
theorem nil_infix (l : list α) : [] <:+: l := infix_of_prefix $ nil_prefix l
theorem infix_cons {L₁ L₂ : list α} {x : α} : L₁ <:+: L₂ → L₁ <:+: x :: L₂ :=
λ⟨LP, LS, H⟩, ⟨x :: LP, LS, H ▸ rfl⟩
@[trans] theorem is_prefix.trans : ∀ {l₁ l₂ l₃ : list α}, l₁ <+: l₂ → l₂ <+: l₃ → l₁ <+: l₃
| l ._ ._ ⟨r₁, rfl⟩ ⟨r₂, rfl⟩ := ⟨r₁ ++ r₂, (append_assoc _ _ _).symm⟩
@[trans] theorem is_suffix.trans : ∀ {l₁ l₂ l₃ : list α}, l₁ <:+ l₂ → l₂ <:+ l₃ → l₁ <:+ l₃
| l ._ ._ ⟨l₁, rfl⟩ ⟨l₂, rfl⟩ := ⟨l₂ ++ l₁, append_assoc _ _ _⟩
@[trans] theorem is_infix.trans : ∀ {l₁ l₂ l₃ : list α}, l₁ <:+: l₂ → l₂ <:+: l₃ → l₁ <:+: l₃
| l ._ ._ ⟨l₁, r₁, rfl⟩ ⟨l₂, r₂, rfl⟩ := ⟨l₂ ++ l₁, r₁ ++ r₂, by simp only [append_assoc]⟩
theorem sublist_of_infix {l₁ l₂ : list α} : l₁ <:+: l₂ → l₁ <+ l₂ :=
λ⟨s, t, h⟩, by rw [← h]; exact (sublist_append_right _ _).trans (sublist_append_left _ _)
theorem sublist_of_prefix {l₁ l₂ : list α} : l₁ <+: l₂ → l₁ <+ l₂ :=
sublist_of_infix ∘ infix_of_prefix
theorem sublist_of_suffix {l₁ l₂ : list α} : l₁ <:+ l₂ → l₁ <+ l₂ :=
sublist_of_infix ∘ infix_of_suffix
theorem reverse_suffix {l₁ l₂ : list α} : reverse l₁ <:+ reverse l₂ ↔ l₁ <+: l₂ :=
⟨λ ⟨r, e⟩, ⟨reverse r,
by rw [← reverse_reverse l₁, ← reverse_append, e, reverse_reverse]⟩,
λ ⟨r, e⟩, ⟨reverse r, by rw [← reverse_append, e]⟩⟩
theorem reverse_prefix {l₁ l₂ : list α} : reverse l₁ <+: reverse l₂ ↔ l₁ <:+ l₂ :=
by rw ← reverse_suffix; simp only [reverse_reverse]
theorem length_le_of_infix {l₁ l₂ : list α} (s : l₁ <:+: l₂) : length l₁ ≤ length l₂ :=
length_le_of_sublist $ sublist_of_infix s
theorem eq_nil_of_infix_nil {l : list α} (s : l <:+: []) : l = [] :=
eq_nil_of_sublist_nil $ sublist_of_infix s
theorem eq_nil_of_prefix_nil {l : list α} (s : l <+: []) : l = [] :=
eq_nil_of_infix_nil $ infix_of_prefix s
theorem eq_nil_of_suffix_nil {l : list α} (s : l <:+ []) : l = [] :=
eq_nil_of_infix_nil $ infix_of_suffix s
theorem infix_iff_prefix_suffix (l₁ l₂ : list α) : l₁ <:+: l₂ ↔ ∃ t, l₁ <+: t ∧ t <:+ l₂ :=
⟨λ⟨s, t, e⟩, ⟨l₁ ++ t, ⟨_, rfl⟩, by rw [← e, append_assoc]; exact ⟨_, rfl⟩⟩,
λ⟨._, ⟨t, rfl⟩, ⟨s, e⟩⟩, ⟨s, t, by rw append_assoc; exact e⟩⟩
theorem eq_of_infix_of_length_eq {l₁ l₂ : list α} (s : l₁ <:+: l₂) :
length l₁ = length l₂ → l₁ = l₂ :=
eq_of_sublist_of_length_eq $ sublist_of_infix s
theorem eq_of_prefix_of_length_eq {l₁ l₂ : list α} (s : l₁ <+: l₂) :
length l₁ = length l₂ → l₁ = l₂ :=
eq_of_sublist_of_length_eq $ sublist_of_prefix s
theorem eq_of_suffix_of_length_eq {l₁ l₂ : list α} (s : l₁ <:+ l₂) :
length l₁ = length l₂ → l₁ = l₂ :=
eq_of_sublist_of_length_eq $ sublist_of_suffix s
theorem prefix_of_prefix_length_le : ∀ {l₁ l₂ l₃ : list α},
l₁ <+: l₃ → l₂ <+: l₃ → length l₁ ≤ length l₂ → l₁ <+: l₂
| [] l₂ l₃ h₁ h₂ _ := nil_prefix _
| (a::l₁) (b::l₂) _ ⟨r₁, rfl⟩ ⟨r₂, e⟩ ll := begin
injection e with _ e', subst b,
rcases prefix_of_prefix_length_le ⟨_, rfl⟩ ⟨_, e'⟩
(le_of_succ_le_succ ll) with ⟨r₃, rfl⟩,
exact ⟨r₃, rfl⟩
end
theorem prefix_or_prefix_of_prefix {l₁ l₂ l₃ : list α}
(h₁ : l₁ <+: l₃) (h₂ : l₂ <+: l₃) : l₁ <+: l₂ ∨ l₂ <+: l₁ :=
(le_total (length l₁) (length l₂)).imp
(prefix_of_prefix_length_le h₁ h₂)
(prefix_of_prefix_length_le h₂ h₁)
theorem suffix_of_suffix_length_le {l₁ l₂ l₃ : list α}
(h₁ : l₁ <:+ l₃) (h₂ : l₂ <:+ l₃) (ll : length l₁ ≤ length l₂) : l₁ <:+ l₂ :=
reverse_prefix.1 $ prefix_of_prefix_length_le
(reverse_prefix.2 h₁) (reverse_prefix.2 h₂) (by simp [ll])
theorem suffix_or_suffix_of_suffix {l₁ l₂ l₃ : list α}
(h₁ : l₁ <:+ l₃) (h₂ : l₂ <:+ l₃) : l₁ <:+ l₂ ∨ l₂ <:+ l₁ :=
(prefix_or_prefix_of_prefix (reverse_prefix.2 h₁) (reverse_prefix.2 h₂)).imp
reverse_prefix.1 reverse_prefix.1
theorem infix_of_mem_join : ∀ {L : list (list α)} {l}, l ∈ L → l <:+: join L
| (_ :: L) l (or.inl rfl) := infix_append [] _ _
| (l' :: L) l (or.inr h) :=
is_infix.trans (infix_of_mem_join h) $ infix_of_suffix $ suffix_append _ _
theorem prefix_append_right_inj {l₁ l₂ : list α} (l) : l ++ l₁ <+: l ++ l₂ ↔ l₁ <+: l₂ :=
exists_congr $ λ r, by rw [append_assoc, append_right_inj]
theorem prefix_cons_inj {l₁ l₂ : list α} (a) : a :: l₁ <+: a :: l₂ ↔ l₁ <+: l₂ :=
prefix_append_right_inj [a]
theorem take_prefix (n) (l : list α) : take n l <+: l := ⟨_, take_append_drop _ _⟩
theorem drop_suffix (n) (l : list α) : drop n l <:+ l := ⟨_, take_append_drop _ _⟩
theorem tail_suffix (l : list α) : tail l <:+ l := by rw ← drop_one; apply drop_suffix
theorem tail_subset (l : list α) : tail l ⊆ l := (sublist_of_suffix (tail_suffix l)).subset
theorem prefix_iff_eq_append {l₁ l₂ : list α} : l₁ <+: l₂ ↔ l₁ ++ drop (length l₁) l₂ = l₂ :=
⟨by rintros ⟨r, rfl⟩; rw drop_left, λ e, ⟨_, e⟩⟩
theorem suffix_iff_eq_append {l₁ l₂ : list α} :
l₁ <:+ l₂ ↔ take (length l₂ - length l₁) l₂ ++ l₁ = l₂ :=
⟨by rintros ⟨r, rfl⟩; simp only [length_append, nat.add_sub_cancel, take_left], λ e, ⟨_, e⟩⟩
theorem prefix_iff_eq_take {l₁ l₂ : list α} : l₁ <+: l₂ ↔ l₁ = take (length l₁) l₂ :=
⟨λ h, append_right_cancel $
(prefix_iff_eq_append.1 h).trans (take_append_drop _ _).symm,
λ e, e.symm ▸ take_prefix _ _⟩
theorem suffix_iff_eq_drop {l₁ l₂ : list α} : l₁ <:+ l₂ ↔ l₁ = drop (length l₂ - length l₁) l₂ :=
⟨λ h, append_left_cancel $
(suffix_iff_eq_append.1 h).trans (take_append_drop _ _).symm,
λ e, e.symm ▸ drop_suffix _ _⟩
instance decidable_prefix [decidable_eq α] : ∀ (l₁ l₂ : list α), decidable (l₁ <+: l₂)
| [] l₂ := is_true ⟨l₂, rfl⟩
| (a::l₁) [] := is_false $ λ ⟨t, te⟩, list.no_confusion te
| (a::l₁) (b::l₂) :=
if h : a = b then
@decidable_of_iff _ _ (by rw [← h, prefix_cons_inj])
(decidable_prefix l₁ l₂)
else
is_false $ λ ⟨t, te⟩, h $ by injection te
-- Alternatively, use mem_tails
instance decidable_suffix [decidable_eq α] : ∀ (l₁ l₂ : list α), decidable (l₁ <:+ l₂)
| [] l₂ := is_true ⟨l₂, append_nil _⟩
| (a::l₁) [] := is_false $ mt (length_le_of_sublist ∘ sublist_of_suffix) dec_trivial
| l₁ l₂ := let len1 := length l₁, len2 := length l₂ in
if hl : len1 ≤ len2 then
decidable_of_iff' (l₁ = drop (len2-len1) l₂) suffix_iff_eq_drop
else is_false $ λ h, hl $ length_le_of_sublist $ sublist_of_suffix h
@[simp] theorem mem_inits : ∀ (s t : list α), s ∈ inits t ↔ s <+: t
| s [] := suffices s = nil ↔ s <+: nil, by simpa only [inits, mem_singleton],
⟨λh, h.symm ▸ prefix_refl [], eq_nil_of_prefix_nil⟩
| s (a::t) :=
suffices (s = nil ∨ ∃ l ∈ inits t, a :: l = s) ↔ s <+: a :: t, by simpa,
⟨λo, match s, o with
| ._, or.inl rfl := ⟨_, rfl⟩
| s, or.inr ⟨r, hr, hs⟩ := let ⟨s, ht⟩ := (mem_inits _ _).1 hr in
by rw [← hs, ← ht]; exact ⟨s, rfl⟩
end, λmi, match s, mi with
| [], ⟨._, rfl⟩ := or.inl rfl
| (b::s), ⟨r, hr⟩ := list.no_confusion hr $ λba (st : s++r = t), or.inr $
by rw ba; exact ⟨_, (mem_inits _ _).2 ⟨_, st⟩, rfl⟩
end⟩
@[simp] theorem mem_tails : ∀ (s t : list α), s ∈ tails t ↔ s <:+ t
| s [] := by simp only [tails, mem_singleton];
exact ⟨λh, by rw h; exact suffix_refl [], eq_nil_of_suffix_nil⟩
| s (a::t) := by simp only [tails, mem_cons_iff, mem_tails s t];
exact show s = a :: t ∨ s <:+ t ↔ s <:+ a :: t, from
⟨λo, match s, t, o with
| ._, t, or.inl rfl := suffix_refl _
| s, ._, or.inr ⟨l, rfl⟩ := ⟨a::l, rfl⟩
end, λe, match s, t, e with
| ._, t, ⟨[], rfl⟩ := or.inl rfl
| s, t, ⟨b::l, he⟩ := list.no_confusion he (λab lt, or.inr ⟨l, lt⟩)
end⟩
instance decidable_infix [decidable_eq α] : ∀ (l₁ l₂ : list α), decidable (l₁ <:+: l₂)
| [] l₂ := is_true ⟨[], l₂, rfl⟩
| (a::l₁) [] := is_false $ λ⟨s, t, te⟩, absurd te $ append_ne_nil_of_ne_nil_left _ _ $
append_ne_nil_of_ne_nil_right _ _ $ λh, list.no_confusion h
| l₁ l₂ := decidable_of_decidable_of_iff (list.decidable_bex (λt, l₁ <+: t) (tails l₂)) $
by refine (exists_congr (λt, _)).trans (infix_iff_prefix_suffix _ _).symm;
exact ⟨λ⟨h1, h2⟩, ⟨h2, (mem_tails _ _).1 h1⟩, λ⟨h2, h1⟩, ⟨(mem_tails _ _).2 h1, h2⟩⟩
/-! ### sublists -/
@[simp] theorem sublists'_nil : sublists' (@nil α) = [[]] := rfl
@[simp, priority 1100] theorem sublists'_singleton (a : α) : sublists' [a] = [[], [a]] := rfl
theorem map_sublists'_aux (g : list β → list γ) (l : list α) (f r) :
map g (sublists'_aux l f r) = sublists'_aux l (g ∘ f) (map g r) :=
by induction l generalizing f r; [refl, simp only [*, sublists'_aux]]
theorem sublists'_aux_append (r' : list (list β)) (l : list α) (f r) :
sublists'_aux l f (r ++ r') = sublists'_aux l f r ++ r' :=
by induction l generalizing f r; [refl, simp only [*, sublists'_aux]]
theorem sublists'_aux_eq_sublists' (l f r) :
@sublists'_aux α β l f r = map f (sublists' l) ++ r :=
by rw [sublists', map_sublists'_aux, ← sublists'_aux_append]; refl
@[simp] theorem sublists'_cons (a : α) (l : list α) :
sublists' (a :: l) = sublists' l ++ map (cons a) (sublists' l) :=
by rw [sublists', sublists'_aux]; simp only [sublists'_aux_eq_sublists', map_id, append_nil]; refl
@[simp] theorem mem_sublists' {s t : list α} : s ∈ sublists' t ↔ s <+ t :=
begin
induction t with a t IH generalizing s,
{ simp only [sublists'_nil, mem_singleton],
exact ⟨λ h, by rw h, eq_nil_of_sublist_nil⟩ },
simp only [sublists'_cons, mem_append, IH, mem_map],
split; intro h, rcases h with h | ⟨s, h, rfl⟩,
{ exact sublist_cons_of_sublist _ h },
{ exact cons_sublist_cons _ h },
{ cases h with _ _ _ h s _ _ h,
{ exact or.inl h },
{ exact or.inr ⟨s, h, rfl⟩ } }
end
@[simp] theorem length_sublists' : ∀ l : list α, length (sublists' l) = 2 ^ length l
| [] := rfl
| (a::l) := by simp only [sublists'_cons, length_append, length_sublists' l, length_map,
length, pow_succ, mul_succ, mul_zero, zero_add]
@[simp] theorem sublists_nil : sublists (@nil α) = [[]] := rfl
@[simp] theorem sublists_singleton (a : α) : sublists [a] = [[], [a]] := rfl
theorem sublists_aux₁_eq_sublists_aux : ∀ l (f : list α → list β),
sublists_aux₁ l f = sublists_aux l (λ ys r, f ys ++ r)
| [] f := rfl
| (a::l) f := by rw [sublists_aux₁, sublists_aux]; simp only [*, append_assoc]
theorem sublists_aux_cons_eq_sublists_aux₁ (l : list α) :
sublists_aux l cons = sublists_aux₁ l (λ x, [x]) :=
by rw [sublists_aux₁_eq_sublists_aux]; refl
theorem sublists_aux_eq_foldr.aux {a : α} {l : list α}
(IH₁ : ∀ (f : list α → list β → list β), sublists_aux l f = foldr f [] (sublists_aux l cons))
(IH₂ : ∀ (f : list α → list (list α) → list (list α)),
sublists_aux l f = foldr f [] (sublists_aux l cons))
(f : list α → list β → list β) : sublists_aux (a::l) f = foldr f [] (sublists_aux (a::l) cons) :=
begin
simp only [sublists_aux, foldr_cons], rw [IH₂, IH₁], congr' 1,
induction sublists_aux l cons with _ _ ih, {refl},
simp only [ih, foldr_cons]
end
theorem sublists_aux_eq_foldr (l : list α) : ∀ (f : list α → list β → list β),
sublists_aux l f = foldr f [] (sublists_aux l cons) :=
suffices _ ∧ ∀ f : list α → list (list α) → list (list α),
sublists_aux l f = foldr f [] (sublists_aux l cons),
from this.1,
begin
induction l with a l IH, {split; intro; refl},
exact ⟨sublists_aux_eq_foldr.aux IH.1 IH.2,
sublists_aux_eq_foldr.aux IH.2 IH.2⟩
end
theorem sublists_aux_cons_cons (l : list α) (a : α) :
sublists_aux (a::l) cons = [a] :: foldr (λys r, ys :: (a :: ys) :: r) [] (sublists_aux l cons) :=
by rw [← sublists_aux_eq_foldr]; refl
theorem sublists_aux₁_append : ∀ (l₁ l₂ : list α) (f : list α → list β),
sublists_aux₁ (l₁ ++ l₂) f = sublists_aux₁ l₁ f ++
sublists_aux₁ l₂ (λ x, f x ++ sublists_aux₁ l₁ (f ∘ (++ x)))
| [] l₂ f := by simp only [sublists_aux₁, nil_append, append_nil]
| (a::l₁) l₂ f := by simp only [sublists_aux₁, cons_append, sublists_aux₁_append l₁, append_assoc];
refl
theorem sublists_aux₁_concat (l : list α) (a : α) (f : list α → list β) :
sublists_aux₁ (l ++ [a]) f = sublists_aux₁ l f ++
f [a] ++ sublists_aux₁ l (λ x, f (x ++ [a])) :=
by simp only [sublists_aux₁_append, sublists_aux₁, append_assoc, append_nil]
theorem sublists_aux₁_bind : ∀ (l : list α)
(f : list α → list β) (g : β → list γ),
(sublists_aux₁ l f).bind g = sublists_aux₁ l (λ x, (f x).bind g)
| [] f g := rfl
| (a::l) f g := by simp only [sublists_aux₁, bind_append, sublists_aux₁_bind l]
theorem sublists_aux_cons_append (l₁ l₂ : list α) :
sublists_aux (l₁ ++ l₂) cons = sublists_aux l₁ cons ++
(do x ← sublists_aux l₂ cons, (++ x) <$> sublists l₁) :=
begin
simp only [sublists, sublists_aux_cons_eq_sublists_aux₁, sublists_aux₁_append, bind_eq_bind,
sublists_aux₁_bind],
congr, funext x, apply congr_arg _,
rw [← bind_ret_eq_map, sublists_aux₁_bind], exact (append_nil _).symm
end
theorem sublists_append (l₁ l₂ : list α) :
sublists (l₁ ++ l₂) = (do x ← sublists l₂, (++ x) <$> sublists l₁) :=
by simp only [map, sublists, sublists_aux_cons_append, map_eq_map, bind_eq_bind,
cons_bind, map_id', append_nil, cons_append, map_id' (λ _, rfl)]; split; refl
@[simp] theorem sublists_concat (l : list α) (a : α) :
sublists (l ++ [a]) = sublists l ++ map (λ x, x ++ [a]) (sublists l) :=
by rw [sublists_append, sublists_singleton, bind_eq_bind, cons_bind, cons_bind, nil_bind,
map_eq_map, map_eq_map, map_id' (append_nil), append_nil]
theorem sublists_reverse (l : list α) : sublists (reverse l) = map reverse (sublists' l) :=
by induction l with hd tl ih; [refl,
simp only [reverse_cons, sublists_append, sublists'_cons, map_append, ih, sublists_singleton,
map_eq_map, bind_eq_bind, map_map, cons_bind, append_nil, nil_bind, (∘)]]
theorem sublists_eq_sublists' (l : list α) : sublists l = map reverse (sublists' (reverse l)) :=
by rw [← sublists_reverse, reverse_reverse]
theorem sublists'_reverse (l : list α) : sublists' (reverse l) = map reverse (sublists l) :=
by simp only [sublists_eq_sublists', map_map, map_id' (reverse_reverse)]
theorem sublists'_eq_sublists (l : list α) : sublists' l = map reverse (sublists (reverse l)) :=
by rw [← sublists'_reverse, reverse_reverse]
theorem sublists_aux_ne_nil : ∀ (l : list α), [] ∉ sublists_aux l cons
| [] := id
| (a::l) := begin
rw [sublists_aux_cons_cons],
refine not_mem_cons_of_ne_of_not_mem (cons_ne_nil _ _).symm _,
have := sublists_aux_ne_nil l, revert this,
induction sublists_aux l cons; intro, {rwa foldr},
simp only [foldr, mem_cons_iff, false_or, not_or_distrib],
exact ⟨ne_of_not_mem_cons this, ih (not_mem_of_not_mem_cons this)⟩
end
@[simp] theorem mem_sublists {s t : list α} : s ∈ sublists t ↔ s <+ t :=
by rw [← reverse_sublist_iff, ← mem_sublists',
sublists'_reverse, mem_map_of_inj reverse_injective]
@[simp] theorem length_sublists (l : list α) : length (sublists l) = 2 ^ length l :=
by simp only [sublists_eq_sublists', length_map, length_sublists', length_reverse]
theorem map_ret_sublist_sublists (l : list α) : map list.ret l <+ sublists l :=
reverse_rec_on l (nil_sublist _) $
λ l a IH, by simp only [map, map_append, sublists_concat]; exact
((append_sublist_append_left _).2 $ singleton_sublist.2 $
mem_map.2 ⟨[], mem_sublists.2 (nil_sublist _), by refl⟩).trans
((append_sublist_append_right _).2 IH)
/-! ### sublists_len -/
/-- Auxiliary function to construct the list of all sublists of a given length. Given an
integer `n`, a list `l`, a function `f` and an auxiliary list `L`, it returns the list made of
of `f` applied to all sublists of `l` of length `n`, concatenated with `L`. -/
def sublists_len_aux {α β : Type*} : ℕ → list α → (list α → β) → list β → list β
| 0 l f r := f [] :: r
| (n+1) [] f r := r
| (n+1) (a::l) f r := sublists_len_aux (n + 1) l f
(sublists_len_aux n l (f ∘ list.cons a) r)
/-- The list of all sublists of a list `l` that are of length `n`. For instance, for
`l = [0, 1, 2, 3]` and `n = 2`, one gets
`[[2, 3], [1, 3], [1, 2], [0, 3], [0, 2], [0, 1]]`. -/
def sublists_len {α : Type*} (n : ℕ) (l : list α) : list (list α) :=
sublists_len_aux n l id []
lemma sublists_len_aux_append {α β γ : Type*} :
∀ (n : ℕ) (l : list α) (f : list α → β) (g : β → γ) (r : list β) (s : list γ),
sublists_len_aux n l (g ∘ f) (r.map g ++ s) =
(sublists_len_aux n l f r).map g ++ s
| 0 l f g r s := rfl
| (n+1) [] f g r s := rfl
| (n+1) (a::l) f g r s := begin
unfold sublists_len_aux,
rw [show ((g ∘ f) ∘ list.cons a) = (g ∘ f ∘ list.cons a), by refl,
sublists_len_aux_append, sublists_len_aux_append]
end
lemma sublists_len_aux_eq {α β : Type*} (l : list α) (n) (f : list α → β) (r) :
sublists_len_aux n l f r = (sublists_len n l).map f ++ r :=
by rw [sublists_len, ← sublists_len_aux_append]; refl
lemma sublists_len_aux_zero {α : Type*} (l : list α) (f : list α → β) (r) :
sublists_len_aux 0 l f r = f [] :: r := by cases l; refl
@[simp] lemma sublists_len_zero {α : Type*} (l : list α) :
sublists_len 0 l = [[]] := sublists_len_aux_zero _ _ _
@[simp] lemma sublists_len_succ_nil {α : Type*} (n) :
sublists_len (n+1) (@nil α) = [] := rfl
@[simp] lemma sublists_len_succ_cons {α : Type*} (n) (a : α) (l) :
sublists_len (n + 1) (a::l) =
sublists_len (n + 1) l ++ (sublists_len n l).map (cons a) :=
by rw [sublists_len, sublists_len_aux, sublists_len_aux_eq,
sublists_len_aux_eq, map_id, append_nil]; refl
@[simp] lemma length_sublists_len {α : Type*} : ∀ n (l : list α),
length (sublists_len n l) = nat.choose (length l) n
| 0 l := by simp
| (n+1) [] := by simp
| (n+1) (a::l) := by simp [-add_comm, nat.choose, *]; apply add_comm
lemma sublists_len_sublist_sublists' {α : Type*} : ∀ n (l : list α),
sublists_len n l <+ sublists' l
| 0 l := singleton_sublist.2 (mem_sublists'.2 (nil_sublist _))
| (n+1) [] := nil_sublist _
| (n+1) (a::l) := begin
rw [sublists_len_succ_cons, sublists'_cons],
exact (sublists_len_sublist_sublists' _ _).append
((sublists_len_sublist_sublists' _ _).map _)
end
lemma sublists_len_sublist_of_sublist
{α : Type*} (n) {l₁ l₂ : list α} (h : l₁ <+ l₂) : sublists_len n l₁ <+ sublists_len n l₂ :=
begin
induction n with n IHn generalizing l₁ l₂, {simp},
induction h with l₁ l₂ a s IH l₁ l₂ a s IH, {refl},
{ refine IH.trans _,
rw sublists_len_succ_cons,
apply sublist_append_left },
{ simp [sublists_len_succ_cons],
exact IH.append ((IHn s).map _) }
end
lemma length_of_sublists_len {α : Type*} : ∀ {n} {l l' : list α},
l' ∈ sublists_len n l → length l' = n
| 0 l l' (or.inl rfl) := rfl
| (n+1) (a::l) l' h := begin
rw [sublists_len_succ_cons, mem_append, mem_map] at h,
rcases h with h | ⟨l', h, rfl⟩,
{ exact length_of_sublists_len h },
{ exact congr_arg (+1) (length_of_sublists_len h) },
end
lemma mem_sublists_len_self {α : Type*} {l l' : list α}
(h : l' <+ l) : l' ∈ sublists_len (length l') l :=
begin
induction h with l₁ l₂ a s IH l₁ l₂ a s IH,
{ exact or.inl rfl },
{ cases l₁ with b l₁,
{ exact or.inl rfl },
{ rw [length, sublists_len_succ_cons],
exact mem_append_left _ IH } },
{ rw [length, sublists_len_succ_cons],
exact mem_append_right _ (mem_map.2 ⟨_, IH, rfl⟩) }
end
@[simp] lemma mem_sublists_len {α : Type*} {n} {l l' : list α} :
l' ∈ sublists_len n l ↔ l' <+ l ∧ length l' = n :=
⟨λ h, ⟨mem_sublists'.1
((sublists_len_sublist_sublists' _ _).subset h),
length_of_sublists_len h⟩,
λ ⟨h₁, h₂⟩, h₂ ▸ mem_sublists_len_self h₁⟩
/-! ### permutations -/
section permutations
@[simp] theorem permutations_aux_nil (is : list α) : permutations_aux [] is = [] :=
by rw [permutations_aux, permutations_aux.rec]
@[simp] theorem permutations_aux_cons (t : α) (ts is : list α) :
permutations_aux (t :: ts) is = foldr (λy r, (permutations_aux2 t ts r y id).2)
(permutations_aux ts (t::is)) (permutations is) :=
by rw [permutations_aux, permutations_aux.rec]; refl
end permutations
/-! ### insert -/
section insert
variable [decidable_eq α]
@[simp] theorem insert_nil (a : α) : insert a nil = [a] := rfl
theorem insert.def (a : α) (l : list α) : insert a l = if a ∈ l then l else a :: l := rfl
@[simp, priority 980]
theorem insert_of_mem {a : α} {l : list α} (h : a ∈ l) : insert a l = l :=
by simp only [insert.def, if_pos h]
@[simp, priority 970]
theorem insert_of_not_mem {a : α} {l : list α} (h : a ∉ l) : insert a l = a :: l :=
by simp only [insert.def, if_neg h]; split; refl
@[simp] theorem mem_insert_iff {a b : α} {l : list α} : a ∈ insert b l ↔ a = b ∨ a ∈ l :=
begin
by_cases h' : b ∈ l,
{ simp only [insert_of_mem h'],
apply (or_iff_right_of_imp _).symm,
exact λ e, e.symm ▸ h' },
simp only [insert_of_not_mem h', mem_cons_iff]
end
@[simp] theorem suffix_insert (a : α) (l : list α) : l <:+ insert a l :=
by by_cases a ∈ l; [simp only [insert_of_mem h], simp only [insert_of_not_mem h, suffix_cons]]
@[simp] theorem mem_insert_self (a : α) (l : list α) : a ∈ insert a l :=
mem_insert_iff.2 (or.inl rfl)
theorem mem_insert_of_mem {a b : α} {l : list α} (h : a ∈ l) : a ∈ insert b l :=
mem_insert_iff.2 (or.inr h)
theorem eq_or_mem_of_mem_insert {a b : α} {l : list α} (h : a ∈ insert b l) : a = b ∨ a ∈ l :=
mem_insert_iff.1 h
@[simp] theorem length_insert_of_mem {a : α} {l : list α} (h : a ∈ l) :
length (insert a l) = length l :=
by rw insert_of_mem h
@[simp] theorem length_insert_of_not_mem {a : α} {l : list α} (h : a ∉ l) :
length (insert a l) = length l + 1 :=
by rw insert_of_not_mem h; refl
end insert
/-! ### erasep -/
section erasep
variables {p : α → Prop} [decidable_pred p]
@[simp] theorem erasep_nil : [].erasep p = [] := rfl
theorem erasep_cons (a : α) (l : list α) :
(a :: l).erasep p = if p a then l else a :: l.erasep p := rfl
@[simp] theorem erasep_cons_of_pos {a : α} {l : list α} (h : p a) : (a :: l).erasep p = l :=
by simp [erasep_cons, h]
@[simp] theorem erasep_cons_of_neg {a : α} {l : list α} (h : ¬ p a) :
(a::l).erasep p = a :: l.erasep p :=
by simp [erasep_cons, h]
theorem erasep_of_forall_not {l : list α}
(h : ∀ a ∈ l, ¬ p a) : l.erasep p = l :=
by induction l with _ _ ih; [refl,
simp [h _ (or.inl rfl), ih (forall_mem_of_forall_mem_cons h)]]
theorem exists_of_erasep {l : list α} {a} (al : a ∈ l) (pa : p a) :
∃ a l₁ l₂, (∀ b ∈ l₁, ¬ p b) ∧ p a ∧ l = l₁ ++ a :: l₂ ∧ l.erasep p = l₁ ++ l₂ :=
begin
induction l with b l IH, {cases al},
by_cases pb : p b,
{ exact ⟨b, [], l, forall_mem_nil _, pb, by simp [pb]⟩ },
{ rcases al with rfl | al, {exact pb.elim pa},
rcases IH al with ⟨c, l₁, l₂, h₁, h₂, h₃, h₄⟩,
exact ⟨c, b::l₁, l₂, forall_mem_cons.2 ⟨pb, h₁⟩,
h₂, by rw h₃; refl, by simp [pb, h₄]⟩ }
end
theorem exists_or_eq_self_of_erasep (p : α → Prop) [decidable_pred p] (l : list α) :
l.erasep p = l ∨ ∃ a l₁ l₂, (∀ b ∈ l₁, ¬ p b) ∧ p a ∧ l = l₁ ++ a :: l₂ ∧ l.erasep p = l₁ ++ l₂ :=
begin
by_cases h : ∃ a ∈ l, p a,
{ rcases h with ⟨a, ha, pa⟩,
exact or.inr (exists_of_erasep ha pa) },
{ simp at h, exact or.inl (erasep_of_forall_not h) }
end
@[simp] theorem length_erasep_of_mem {l : list α} {a} (al : a ∈ l) (pa : p a) :
length (l.erasep p) = pred (length l) :=
by rcases exists_of_erasep al pa with ⟨_, l₁, l₂, _, _, e₁, e₂⟩;
rw e₂; simp [-add_comm, e₁]; refl
theorem erasep_append_left {a : α} (pa : p a) :
∀ {l₁ : list α} (l₂), a ∈ l₁ → (l₁++l₂).erasep p = l₁.erasep p ++ l₂
| (x::xs) l₂ h := begin
by_cases h' : p x; simp [h'],
rw erasep_append_left l₂ (mem_of_ne_of_mem (mt _ h') h),
rintro rfl, exact pa
end
theorem erasep_append_right :
∀ {l₁ : list α} (l₂), (∀ b ∈ l₁, ¬ p b) → (l₁++l₂).erasep p = l₁ ++ l₂.erasep p
| [] l₂ h := rfl
| (x::xs) l₂ h := by simp [(forall_mem_cons.1 h).1,
erasep_append_right _ (forall_mem_cons.1 h).2]
theorem erasep_sublist (l : list α) : l.erasep p <+ l :=
by rcases exists_or_eq_self_of_erasep p l with h | ⟨c, l₁, l₂, h₁, h₂, h₃, h₄⟩;
[rw h, {rw [h₄, h₃], simp}]
theorem erasep_subset (l : list α) : l.erasep p ⊆ l :=
(erasep_sublist l).subset
theorem sublist.erasep {l₁ l₂ : list α} (s : l₁ <+ l₂) : l₁.erasep p <+ l₂.erasep p :=
begin
induction s,
case list.sublist.slnil { refl },
case list.sublist.cons : l₁ l₂ a s IH {
by_cases h : p a; simp [h],
exacts [IH.trans (erasep_sublist _), IH.cons _ _ _] },
case list.sublist.cons2 : l₁ l₂ a s IH {
by_cases h : p a; simp [h],
exacts [s, IH.cons2 _ _ _] }
end
theorem mem_of_mem_erasep {a : α} {l : list α} : a ∈ l.erasep p → a ∈ l :=
@erasep_subset _ _ _ _ _
@[simp] theorem mem_erasep_of_neg {a : α} {l : list α} (pa : ¬ p a) : a ∈ l.erasep p ↔ a ∈ l :=
⟨mem_of_mem_erasep, λ al, begin
rcases exists_or_eq_self_of_erasep p l with h | ⟨c, l₁, l₂, h₁, h₂, h₃, h₄⟩,
{ rwa h },
{ rw h₄, rw h₃ at al,
have : a ≠ c, {rintro rfl, exact pa.elim h₂},
simpa [this] using al }
end⟩
theorem erasep_map (f : β → α) :
∀ (l : list β), (map f l).erasep p = map f (l.erasep (p ∘ f))
| [] := rfl
| (b::l) := by by_cases p (f b); simp [h, erasep_map l]
@[simp] theorem extractp_eq_find_erasep :
∀ l : list α, extractp p l = (find p l, erasep p l)
| [] := rfl
| (a::l) := by by_cases pa : p a; simp [extractp, pa, extractp_eq_find_erasep l]
end erasep
/-! ### erase -/
section erase
variable [decidable_eq α]
@[simp] theorem erase_nil (a : α) : [].erase a = [] := rfl
theorem erase_cons (a b : α) (l : list α) :
(b :: l).erase a = if b = a then l else b :: l.erase a := rfl
@[simp] theorem erase_cons_head (a : α) (l : list α) : (a :: l).erase a = l :=
by simp only [erase_cons, if_pos rfl]
@[simp] theorem erase_cons_tail {a b : α} (l : list α) (h : b ≠ a) :
(b::l).erase a = b :: l.erase a :=
by simp only [erase_cons, if_neg h]; split; refl
theorem erase_eq_erasep (a : α) (l : list α) : l.erase a = l.erasep (eq a) :=
by { induction l with b l, {refl},
by_cases a = b; [simp [h], simp [h, ne.symm h, *]] }
@[simp, priority 980]
theorem erase_of_not_mem {a : α} {l : list α} (h : a ∉ l) : l.erase a = l :=
by rw [erase_eq_erasep, erasep_of_forall_not]; rintro b h' rfl; exact h h'
theorem exists_erase_eq {a : α} {l : list α} (h : a ∈ l) :
∃ l₁ l₂, a ∉ l₁ ∧ l = l₁ ++ a :: l₂ ∧ l.erase a = l₁ ++ l₂ :=
by rcases exists_of_erasep h rfl with ⟨_, l₁, l₂, h₁, rfl, h₂, h₃⟩;
rw erase_eq_erasep; exact ⟨l₁, l₂, λ h, h₁ _ h rfl, h₂, h₃⟩
@[simp] theorem length_erase_of_mem {a : α} {l : list α} (h : a ∈ l) :
length (l.erase a) = pred (length l) :=
by rw erase_eq_erasep; exact length_erasep_of_mem h rfl
theorem erase_append_left {a : α} {l₁ : list α} (l₂) (h : a ∈ l₁) :
(l₁++l₂).erase a = l₁.erase a ++ l₂ :=
by simp [erase_eq_erasep]; exact erasep_append_left (by refl) l₂ h
theorem erase_append_right {a : α} {l₁ : list α} (l₂) (h : a ∉ l₁) :
(l₁++l₂).erase a = l₁ ++ l₂.erase a :=
by rw [erase_eq_erasep, erase_eq_erasep, erasep_append_right];
rintro b h' rfl; exact h h'
theorem erase_sublist (a : α) (l : list α) : l.erase a <+ l :=
by rw erase_eq_erasep; apply erasep_sublist
theorem erase_subset (a : α) (l : list α) : l.erase a ⊆ l :=
(erase_sublist a l).subset
theorem sublist.erase (a : α) {l₁ l₂ : list α} (h : l₁ <+ l₂) : l₁.erase a <+ l₂.erase a :=
by simp [erase_eq_erasep]; exact sublist.erasep h
theorem mem_of_mem_erase {a b : α} {l : list α} : a ∈ l.erase b → a ∈ l :=
@erase_subset _ _ _ _ _
@[simp] theorem mem_erase_of_ne {a b : α} {l : list α} (ab : a ≠ b) : a ∈ l.erase b ↔ a ∈ l :=
by rw erase_eq_erasep; exact mem_erasep_of_neg ab.symm
theorem erase_comm (a b : α) (l : list α) : (l.erase a).erase b = (l.erase b).erase a :=
if ab : a = b then by rw ab else
if ha : a ∈ l then
if hb : b ∈ l then match l, l.erase a, exists_erase_eq ha, hb with
| ._, ._, ⟨l₁, l₂, ha', rfl, rfl⟩, hb :=
if h₁ : b ∈ l₁ then
by rw [erase_append_left _ h₁, erase_append_left _ h₁,
erase_append_right _ (mt mem_of_mem_erase ha'), erase_cons_head]
else
by rw [erase_append_right _ h₁, erase_append_right _ h₁, erase_append_right _ ha',
erase_cons_tail _ ab, erase_cons_head]
end
else by simp only [erase_of_not_mem hb, erase_of_not_mem (mt mem_of_mem_erase hb)]
else by simp only [erase_of_not_mem ha, erase_of_not_mem (mt mem_of_mem_erase ha)]
theorem map_erase [decidable_eq β] {f : α → β} (finj : injective f) {a : α}
(l : list α) : map f (l.erase a) = (map f l).erase (f a) :=
by rw [erase_eq_erasep, erase_eq_erasep, erasep_map]; congr;
ext b; simp [finj.eq_iff]
theorem map_foldl_erase [decidable_eq β] {f : α → β} (finj : injective f) {l₁ l₂ : list α} :
map f (foldl list.erase l₁ l₂) = foldl (λ l a, l.erase (f a)) (map f l₁) l₂ :=
by induction l₂ generalizing l₁; [refl,
simp only [foldl_cons, map_erase finj, *]]
@[simp] theorem count_erase_self (a : α) :
∀ (s : list α), count a (list.erase s a) = pred (count a s)
| [] := by simp
| (h :: t) :=
begin
rw erase_cons,
by_cases p : h = a,
{ rw [if_pos p, count_cons', if_pos p.symm], simp },
{ rw [if_neg p, count_cons', count_cons', if_neg (λ x : a = h, p x.symm), count_erase_self],
simp, }
end
@[simp] theorem count_erase_of_ne {a b : α} (ab : a ≠ b) :
∀ (s : list α), count a (list.erase s b) = count a s
| [] := by simp
| (x :: xs) :=
begin
rw erase_cons,
split_ifs with h,
{ rw [count_cons', h, if_neg ab], simp },
{ rw [count_cons', count_cons', count_erase_of_ne] }
end
end erase
/-! ### diff -/
section diff
variable [decidable_eq α]
@[simp] theorem diff_nil (l : list α) : l.diff [] = l := rfl
@[simp] theorem diff_cons (l₁ l₂ : list α) (a : α) : l₁.diff (a::l₂) = (l₁.erase a).diff l₂ :=
if h : a ∈ l₁ then by simp only [list.diff, if_pos h]
else by simp only [list.diff, if_neg h, erase_of_not_mem h]
@[simp] theorem nil_diff (l : list α) : [].diff l = [] :=
by induction l; [refl, simp only [*, diff_cons, erase_of_not_mem (not_mem_nil _)]]
theorem diff_eq_foldl : ∀ (l₁ l₂ : list α), l₁.diff l₂ = foldl list.erase l₁ l₂
| l₁ [] := rfl
| l₁ (a::l₂) := (diff_cons l₁ l₂ a).trans (diff_eq_foldl _ _)
@[simp] theorem diff_append (l₁ l₂ l₃ : list α) : l₁.diff (l₂ ++ l₃) = (l₁.diff l₂).diff l₃ :=
by simp only [diff_eq_foldl, foldl_append]
@[simp] theorem map_diff [decidable_eq β] {f : α → β} (finj : injective f) {l₁ l₂ : list α} :
map f (l₁.diff l₂) = (map f l₁).diff (map f l₂) :=
by simp only [diff_eq_foldl, foldl_map, map_foldl_erase finj]
theorem diff_sublist : ∀ l₁ l₂ : list α, l₁.diff l₂ <+ l₁
| l₁ [] := sublist.refl _
| l₁ (a::l₂) := calc l₁.diff (a :: l₂) = (l₁.erase a).diff l₂ : diff_cons _ _ _
... <+ l₁.erase a : diff_sublist _ _
... <+ l₁ : list.erase_sublist _ _
theorem diff_subset (l₁ l₂ : list α) : l₁.diff l₂ ⊆ l₁ :=
(diff_sublist _ _).subset
theorem mem_diff_of_mem {a : α} : ∀ {l₁ l₂ : list α}, a ∈ l₁ → a ∉ l₂ → a ∈ l₁.diff l₂
| l₁ [] h₁ h₂ := h₁
| l₁ (b::l₂) h₁ h₂ := by rw diff_cons; exact
mem_diff_of_mem ((mem_erase_of_ne (ne_of_not_mem_cons h₂)).2 h₁) (not_mem_of_not_mem_cons h₂)
theorem sublist.diff_right : ∀ {l₁ l₂ l₃: list α}, l₁ <+ l₂ → l₁.diff l₃ <+ l₂.diff l₃
| l₁ l₂ [] h := h
| l₁ l₂ (a::l₃) h := by simp only
[diff_cons, (h.erase _).diff_right]
theorem erase_diff_erase_sublist_of_sublist {a : α} : ∀ {l₁ l₂ : list α},
l₁ <+ l₂ → (l₂.erase a).diff (l₁.erase a) <+ l₂.diff l₁
| [] l₂ h := erase_sublist _ _
| (b::l₁) l₂ h := if heq : b = a then by simp only [heq, erase_cons_head, diff_cons]
else by simpa only [erase_cons_head, erase_cons_tail _ heq, diff_cons,
erase_comm a b l₂]
using erase_diff_erase_sublist_of_sublist (h.erase b)
end diff
/-! ### enum -/
theorem length_enum_from : ∀ n (l : list α), length (enum_from n l) = length l
| n [] := rfl
| n (a::l) := congr_arg nat.succ (length_enum_from _ _)
theorem length_enum : ∀ (l : list α), length (enum l) = length l := length_enum_from _
@[simp] theorem enum_from_nth : ∀ n (l : list α) m,
nth (enum_from n l) m = (λ a, (n + m, a)) <$> nth l m
| n [] m := rfl
| n (a :: l) 0 := rfl
| n (a :: l) (m+1) := (enum_from_nth (n+1) l m).trans $
by rw [add_right_comm]; refl
@[simp] theorem enum_nth : ∀ (l : list α) n,
nth (enum l) n = (λ a, (n, a)) <$> nth l n :=
by simp only [enum, enum_from_nth, zero_add]; intros; refl
@[simp] theorem enum_from_map_snd : ∀ n (l : list α),
map prod.snd (enum_from n l) = l
| n [] := rfl
| n (a :: l) := congr_arg (cons _) (enum_from_map_snd _ _)
@[simp] theorem enum_map_snd : ∀ (l : list α),
map prod.snd (enum l) = l := enum_from_map_snd _
theorem mem_enum_from {x : α} {i : ℕ} :
∀ {j : ℕ} (xs : list α), (i, x) ∈ xs.enum_from j → j ≤ i ∧ i < j + xs.length ∧ x ∈ xs
| j [] := by simp [enum_from]
| j (y :: ys) :=
suffices i = j ∧ x = y ∨ (i, x) ∈ enum_from (j + 1) ys →
j ≤ i ∧ i < j + (length ys + 1) ∧ (x = y ∨ x ∈ ys),
by simpa [enum_from, mem_enum_from ys],
begin
rintro (h|h),
{ refine ⟨le_of_eq h.1.symm,h.1 ▸ _,or.inl h.2⟩,
apply nat.lt_add_of_pos_right; simp },
{ obtain ⟨hji, hijlen, hmem⟩ := mem_enum_from _ h,
refine ⟨_, _, _⟩,
{ exact le_trans (nat.le_succ _) hji },
{ convert hijlen using 1, ac_refl },
{ simp [hmem] } }
end
/-! ### product -/
@[simp] theorem nil_product (l : list β) : product (@nil α) l = [] := rfl
@[simp] theorem product_cons (a : α) (l₁ : list α) (l₂ : list β)
: product (a::l₁) l₂ = map (λ b, (a, b)) l₂ ++ product l₁ l₂ := rfl
@[simp] theorem product_nil : ∀ (l : list α), product l (@nil β) = []
| [] := rfl
| (a::l) := by rw [product_cons, product_nil]; refl
@[simp] theorem mem_product {l₁ : list α} {l₂ : list β} {a : α} {b : β} :
(a, b) ∈ product l₁ l₂ ↔ a ∈ l₁ ∧ b ∈ l₂ :=
by simp only [product, mem_bind, mem_map, prod.ext_iff, exists_prop,
and.left_comm, exists_and_distrib_left, exists_eq_left, exists_eq_right]
theorem length_product (l₁ : list α) (l₂ : list β) :
length (product l₁ l₂) = length l₁ * length l₂ :=
by induction l₁ with x l₁ IH; [exact (zero_mul _).symm,
simp only [length, product_cons, length_append, IH,
right_distrib, one_mul, length_map, add_comm]]
/-! ### sigma -/
section
variable {σ : α → Type*}
@[simp] theorem nil_sigma (l : Π a, list (σ a)) : (@nil α).sigma l = [] := rfl
@[simp] theorem sigma_cons (a : α) (l₁ : list α) (l₂ : Π a, list (σ a))
: (a::l₁).sigma l₂ = map (sigma.mk a) (l₂ a) ++ l₁.sigma l₂ := rfl
@[simp] theorem sigma_nil : ∀ (l : list α), l.sigma (λ a, @nil (σ a)) = []
| [] := rfl
| (a::l) := by rw [sigma_cons, sigma_nil]; refl
@[simp] theorem mem_sigma {l₁ : list α} {l₂ : Π a, list (σ a)} {a : α} {b : σ a} :
sigma.mk a b ∈ l₁.sigma l₂ ↔ a ∈ l₁ ∧ b ∈ l₂ a :=
by simp only [list.sigma, mem_bind, mem_map, exists_prop, exists_and_distrib_left,
and.left_comm, exists_eq_left, heq_iff_eq, exists_eq_right]
theorem length_sigma (l₁ : list α) (l₂ : Π a, list (σ a)) :
length (l₁.sigma l₂) = (l₁.map (λ a, length (l₂ a))).sum :=
by induction l₁ with x l₁ IH; [refl,
simp only [map, sigma_cons, length_append, length_map, IH, sum_cons]]
end
/-! ### disjoint -/
section disjoint
theorem disjoint.symm {l₁ l₂ : list α} (d : disjoint l₁ l₂) : disjoint l₂ l₁
| a i₂ i₁ := d i₁ i₂
theorem disjoint_comm {l₁ l₂ : list α} : disjoint l₁ l₂ ↔ disjoint l₂ l₁ :=
⟨disjoint.symm, disjoint.symm⟩
theorem disjoint_left {l₁ l₂ : list α} : disjoint l₁ l₂ ↔ ∀ {a}, a ∈ l₁ → a ∉ l₂ := iff.rfl
theorem disjoint_right {l₁ l₂ : list α} : disjoint l₁ l₂ ↔ ∀ {a}, a ∈ l₂ → a ∉ l₁ :=
disjoint_comm
theorem disjoint_iff_ne {l₁ l₂ : list α} : disjoint l₁ l₂ ↔ ∀ a ∈ l₁, ∀ b ∈ l₂, a ≠ b :=
by simp only [disjoint_left, imp_not_comm, forall_eq']
theorem disjoint_of_subset_left {l₁ l₂ l : list α} (ss : l₁ ⊆ l) (d : disjoint l l₂) :
disjoint l₁ l₂
| x m₁ := d (ss m₁)
theorem disjoint_of_subset_right {l₁ l₂ l : list α} (ss : l₂ ⊆ l) (d : disjoint l₁ l) :
disjoint l₁ l₂
| x m m₁ := d m (ss m₁)
theorem disjoint_of_disjoint_cons_left {a : α} {l₁ l₂} : disjoint (a::l₁) l₂ → disjoint l₁ l₂ :=
disjoint_of_subset_left (list.subset_cons _ _)
theorem disjoint_of_disjoint_cons_right {a : α} {l₁ l₂} : disjoint l₁ (a::l₂) → disjoint l₁ l₂ :=
disjoint_of_subset_right (list.subset_cons _ _)
@[simp] theorem disjoint_nil_left (l : list α) : disjoint [] l
| a := (not_mem_nil a).elim
@[simp] theorem disjoint_nil_right (l : list α) : disjoint l [] :=
by rw disjoint_comm; exact disjoint_nil_left _
@[simp, priority 1100] theorem singleton_disjoint {l : list α} {a : α} : disjoint [a] l ↔ a ∉ l :=
by simp only [disjoint, mem_singleton, forall_eq]; refl
@[simp, priority 1100] theorem disjoint_singleton {l : list α} {a : α} : disjoint l [a] ↔ a ∉ l :=
by rw disjoint_comm; simp only [singleton_disjoint]
@[simp] theorem disjoint_append_left {l₁ l₂ l : list α} :
disjoint (l₁++l₂) l ↔ disjoint l₁ l ∧ disjoint l₂ l :=
by simp only [disjoint, mem_append, or_imp_distrib, forall_and_distrib]
@[simp] theorem disjoint_append_right {l₁ l₂ l : list α} :
disjoint l (l₁++l₂) ↔ disjoint l l₁ ∧ disjoint l l₂ :=
disjoint_comm.trans $ by simp only [disjoint_comm, disjoint_append_left]
@[simp] theorem disjoint_cons_left {a : α} {l₁ l₂ : list α} :
disjoint (a::l₁) l₂ ↔ a ∉ l₂ ∧ disjoint l₁ l₂ :=
(@disjoint_append_left _ [a] l₁ l₂).trans $ by simp only [singleton_disjoint]
@[simp] theorem disjoint_cons_right {a : α} {l₁ l₂ : list α} :
disjoint l₁ (a::l₂) ↔ a ∉ l₁ ∧ disjoint l₁ l₂ :=
disjoint_comm.trans $ by simp only [disjoint_comm, disjoint_cons_left]
theorem disjoint_of_disjoint_append_left_left {l₁ l₂ l : list α} (d : disjoint (l₁++l₂) l) :
disjoint l₁ l :=
(disjoint_append_left.1 d).1
theorem disjoint_of_disjoint_append_left_right {l₁ l₂ l : list α} (d : disjoint (l₁++l₂) l) :
disjoint l₂ l :=
(disjoint_append_left.1 d).2
theorem disjoint_of_disjoint_append_right_left {l₁ l₂ l : list α} (d : disjoint l (l₁++l₂)) :
disjoint l l₁ :=
(disjoint_append_right.1 d).1
theorem disjoint_of_disjoint_append_right_right {l₁ l₂ l : list α} (d : disjoint l (l₁++l₂)) :
disjoint l l₂ :=
(disjoint_append_right.1 d).2
end disjoint
/-! ### union -/
section union
variable [decidable_eq α]
@[simp] theorem nil_union (l : list α) : [] ∪ l = l := rfl
@[simp] theorem cons_union (l₁ l₂ : list α) (a : α) : a :: l₁ ∪ l₂ = insert a (l₁ ∪ l₂) := rfl
@[simp] theorem mem_union {l₁ l₂ : list α} {a : α} : a ∈ l₁ ∪ l₂ ↔ a ∈ l₁ ∨ a ∈ l₂ :=
by induction l₁; simp only [nil_union, not_mem_nil, false_or, cons_union, mem_insert_iff,
mem_cons_iff, or_assoc, *]
theorem mem_union_left {a : α} {l₁ : list α} (h : a ∈ l₁) (l₂ : list α) : a ∈ l₁ ∪ l₂ :=
mem_union.2 (or.inl h)
theorem mem_union_right {a : α} (l₁ : list α) {l₂ : list α} (h : a ∈ l₂) : a ∈ l₁ ∪ l₂ :=
mem_union.2 (or.inr h)
theorem sublist_suffix_of_union : ∀ l₁ l₂ : list α, ∃ t, t <+ l₁ ∧ t ++ l₂ = l₁ ∪ l₂
| [] l₂ := ⟨[], by refl, rfl⟩
| (a::l₁) l₂ := let ⟨t, s, e⟩ := sublist_suffix_of_union l₁ l₂ in
if h : a ∈ l₁ ∪ l₂
then ⟨t, sublist_cons_of_sublist _ s, by simp only [e, cons_union, insert_of_mem h]⟩
else ⟨a::t, cons_sublist_cons _ s, by simp only [cons_append, cons_union, e, insert_of_not_mem h];
split; refl⟩
theorem suffix_union_right (l₁ l₂ : list α) : l₂ <:+ l₁ ∪ l₂ :=
(sublist_suffix_of_union l₁ l₂).imp (λ a, and.right)
theorem union_sublist_append (l₁ l₂ : list α) : l₁ ∪ l₂ <+ l₁ ++ l₂ :=
let ⟨t, s, e⟩ := sublist_suffix_of_union l₁ l₂ in
e ▸ (append_sublist_append_right _).2 s
theorem forall_mem_union {p : α → Prop} {l₁ l₂ : list α} :
(∀ x ∈ l₁ ∪ l₂, p x) ↔ (∀ x ∈ l₁, p x) ∧ (∀ x ∈ l₂, p x) :=
by simp only [mem_union, or_imp_distrib, forall_and_distrib]
theorem forall_mem_of_forall_mem_union_left {p : α → Prop} {l₁ l₂ : list α}
(h : ∀ x ∈ l₁ ∪ l₂, p x) : ∀ x ∈ l₁, p x :=
(forall_mem_union.1 h).1
theorem forall_mem_of_forall_mem_union_right {p : α → Prop} {l₁ l₂ : list α}
(h : ∀ x ∈ l₁ ∪ l₂, p x) : ∀ x ∈ l₂, p x :=
(forall_mem_union.1 h).2
end union
/-! ### inter -/
section inter
variable [decidable_eq α]
@[simp] theorem inter_nil (l : list α) : [] ∩ l = [] := rfl
@[simp] theorem inter_cons_of_mem {a : α} (l₁ : list α) {l₂ : list α} (h : a ∈ l₂) :
(a::l₁) ∩ l₂ = a :: (l₁ ∩ l₂) :=
if_pos h
@[simp] theorem inter_cons_of_not_mem {a : α} (l₁ : list α) {l₂ : list α} (h : a ∉ l₂) :
(a::l₁) ∩ l₂ = l₁ ∩ l₂ :=
if_neg h
theorem mem_of_mem_inter_left {l₁ l₂ : list α} {a : α} : a ∈ l₁ ∩ l₂ → a ∈ l₁ :=
mem_of_mem_filter
theorem mem_of_mem_inter_right {l₁ l₂ : list α} {a : α} : a ∈ l₁ ∩ l₂ → a ∈ l₂ :=
of_mem_filter
theorem mem_inter_of_mem_of_mem {l₁ l₂ : list α} {a : α} : a ∈ l₁ → a ∈ l₂ → a ∈ l₁ ∩ l₂ :=
mem_filter_of_mem
@[simp] theorem mem_inter {a : α} {l₁ l₂ : list α} : a ∈ l₁ ∩ l₂ ↔ a ∈ l₁ ∧ a ∈ l₂ :=
mem_filter
theorem inter_subset_left (l₁ l₂ : list α) : l₁ ∩ l₂ ⊆ l₁ :=
filter_subset _
theorem inter_subset_right (l₁ l₂ : list α) : l₁ ∩ l₂ ⊆ l₂ :=
λ a, mem_of_mem_inter_right
theorem subset_inter {l l₁ l₂ : list α} (h₁ : l ⊆ l₁) (h₂ : l ⊆ l₂) : l ⊆ l₁ ∩ l₂ :=
λ a h, mem_inter.2 ⟨h₁ h, h₂ h⟩
theorem inter_eq_nil_iff_disjoint {l₁ l₂ : list α} : l₁ ∩ l₂ = [] ↔ disjoint l₁ l₂ :=
by simp only [eq_nil_iff_forall_not_mem, mem_inter, not_and]; refl
theorem forall_mem_inter_of_forall_left {p : α → Prop} {l₁ : list α} (h : ∀ x ∈ l₁, p x)
(l₂ : list α) :
∀ x, x ∈ l₁ ∩ l₂ → p x :=
ball.imp_left (λ x, mem_of_mem_inter_left) h
theorem forall_mem_inter_of_forall_right {p : α → Prop} (l₁ : list α) {l₂ : list α}
(h : ∀ x ∈ l₂, p x) :
∀ x, x ∈ l₁ ∩ l₂ → p x :=
ball.imp_left (λ x, mem_of_mem_inter_right) h
end inter
section choose
variables (p : α → Prop) [decidable_pred p] (l : list α)
lemma choose_spec (hp : ∃ a, a ∈ l ∧ p a) : choose p l hp ∈ l ∧ p (choose p l hp) :=
(choose_x p l hp).property
lemma choose_mem (hp : ∃ a, a ∈ l ∧ p a) : choose p l hp ∈ l := (choose_spec _ _ _).1
lemma choose_property (hp : ∃ a, a ∈ l ∧ p a) : p (choose p l hp) := (choose_spec _ _ _).2
end choose
-- A jumble of lost lemmas:
theorem ilast'_mem : ∀ a l, @ilast' α a l ∈ a :: l
| a [] := or.inl rfl
| a (b::l) := or.inr (ilast'_mem b l)
@[simp] lemma nth_le_attach (L : list α) (i) (H : i < L.attach.length) :
(L.attach.nth_le i H).1 = L.nth_le i (length_attach L ▸ H) :=
calc (L.attach.nth_le i H).1
= (L.attach.map subtype.val).nth_le i (by simpa using H) : by rw nth_le_map'
... = L.nth_le i _ : by congr; apply attach_map_val
end list
@[to_additive]
theorem monoid_hom.map_list_prod {α β : Type*} [monoid α] [monoid β] (f : α →* β) (l : list α) :
f l.prod = (l.map f).prod :=
(l.prod_hom f).symm
namespace list
@[to_additive]
theorem prod_map_hom {α β γ : Type*} [monoid β] [monoid γ] (L : list α) (f : α → β) (g : β →* γ) :
(L.map (g ∘ f)).prod = g ((L.map f).prod) :=
by {rw g.map_list_prod, exact congr_arg _ (map_map _ _ _).symm}
theorem sum_map_mul_left {α : Type*} [semiring α] {β : Type*} (L : list β)
(f : β → α) (r : α) :
(L.map (λ b, r * f b)).sum = r * (L.map f).sum :=
sum_map_hom L f $ add_monoid_hom.mul_left r
theorem sum_map_mul_right {α : Type*} [semiring α] {β : Type*} (L : list β)
(f : β → α) (r : α) :
(L.map (λ b, f b * r)).sum = (L.map f).sum * r :=
sum_map_hom L f $ add_monoid_hom.mul_right r
end list
|
836f04a2832c15ac94557382f5b2e2b374333735 | 63abd62053d479eae5abf4951554e1064a4c45b4 | /src/category_theory/limits/colimit_limit.lean | b1d32734fa1c3797a9d83a0c8de4f0693df5f306 | [
"Apache-2.0"
] | permissive | Lix0120/mathlib | 0020745240315ed0e517cbf32e738d8f9811dd80 | e14c37827456fc6707f31b4d1d16f1f3a3205e91 | refs/heads/master | 1,673,102,855,024 | 1,604,151,044,000 | 1,604,151,044,000 | 308,930,245 | 0 | 0 | Apache-2.0 | 1,604,164,710,000 | 1,604,163,547,000 | null | UTF-8 | Lean | false | false | 3,377 | lean | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import category_theory.limits.types
import category_theory.currying
/-!
# The morphism comparing a colimit of limits with the corresponding limit of colimits.
For `F : J × K ⥤ C` there is always a morphism $\colim_k \lim_j F(j,k) → \lim_j \colim_k F(j, k)$.
While it is not usually an isomorphism, with additional hypotheses on `J` and `K` it may be,
in which case we say that "colimits commute with limits".
The prototypical example, proved in `category_theory.limits.filtered_colimit_commutes_finite_limit`,
is that when `C = Type`, filtered colimits commute with finite limits.
## References
* Borceux, Handbook of categorical algebra 1, Section 2.13
* [Stacks: Filtered colimits](https://stacks.math.columbia.edu/tag/002W)
-/
universes v u
open category_theory
namespace category_theory.limits
variables {J K : Type v} [small_category J] [small_category K]
variables {C : Type u} [category.{v} C]
variables (F : J × K ⥤ C)
open category_theory.prod
lemma map_id_left_eq_curry_map {j : J} {k k' : K} {f : k ⟶ k'} :
F.map ((𝟙 j, f) : (j, k) ⟶ (j, k')) = ((curry.obj F).obj j).map f :=
rfl
lemma map_id_right_eq_curry_swap_map {j j' : J} {f : j ⟶ j'} {k : K} :
F.map ((f, 𝟙 k) : (j, k) ⟶ (j', k)) = ((curry.obj (swap K J ⋙ F)).obj k).map f :=
rfl
variables [has_limits_of_shape J C]
variables [has_colimits_of_shape K C]
/--
The universal morphism
$\colim_k \lim_j F(j,k) → \lim_j \colim_k F(j, k)$.
-/
noncomputable
def colimit_limit_to_limit_colimit :
colimit ((curry.obj (swap K J ⋙ F)) ⋙ lim) ⟶ limit ((curry.obj F) ⋙ colim) :=
limit.lift ((curry.obj F) ⋙ colim)
{ X := _,
π :=
{ app := λ j, colimit.desc ((curry.obj (swap K J ⋙ F)) ⋙ lim)
{ X := _,
ι :=
{ app := λ k,
limit.π ((curry.obj (swap K J ⋙ F)).obj k) j ≫ colimit.ι ((curry.obj F).obj j) k,
naturality' :=
begin
dsimp,
intros k k' f,
simp only [functor.comp_map, curry.obj_map_app, limits.limit.map_π_assoc, swap_map,
category.comp_id, map_id_left_eq_curry_map, colimit.w],
end }, },
naturality' :=
begin
dsimp,
intros j j' f,
ext k,
simp only [limits.colimit.ι_map, curry.obj_map_app, limits.colimit.ι_desc_assoc,
limits.colimit.ι_desc, category.id_comp, category.assoc, map_id_right_eq_curry_swap_map,
limit.w_assoc],
end } }
/--
Since `colimit_limit_to_limit_colimit` is a morphism from a colimit to a limit,
this lemma characterises it.
-/
@[simp] lemma ι_colimit_limit_to_limit_colimit_π (j) (k) :
colimit.ι _ k ≫ colimit_limit_to_limit_colimit F ≫ limit.π _ j =
limit.π ((curry.obj (swap K J ⋙ F)).obj k) j ≫ colimit.ι ((curry.obj F).obj j) k :=
by { dsimp [colimit_limit_to_limit_colimit], simp, }
@[simp] lemma ι_colimit_limit_to_limit_colimit_π_apply (F : J × K ⥤ Type v) (j) (k) (f) :
limit.π ((curry.obj F) ⋙ colim) j
(colimit_limit_to_limit_colimit F (colimit.ι ((curry.obj (swap K J ⋙ F)) ⋙ lim) k f)) =
colimit.ι ((curry.obj F).obj j) k (limit.π ((curry.obj (swap K J ⋙ F)).obj k) j f) :=
by { dsimp [colimit_limit_to_limit_colimit], simp, }
end category_theory.limits
|
9f6a731e7813147f42ea007fa27a3df8dc89cf9e | 8cae430f0a71442d02dbb1cbb14073b31048e4b0 | /src/analysis/convex/cone/basic.lean | 3feb656d347f0211c3cb34e03ee3320aef3a4aeb | [
"Apache-2.0"
] | permissive | leanprover-community/mathlib | 56a2cadd17ac88caf4ece0a775932fa26327ba0e | 442a83d738cb208d3600056c489be16900ba701d | refs/heads/master | 1,693,584,102,358 | 1,693,471,902,000 | 1,693,471,902,000 | 97,922,418 | 1,595 | 352 | Apache-2.0 | 1,694,693,445,000 | 1,500,624,130,000 | Lean | UTF-8 | Lean | false | false | 29,613 | lean | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Frédéric Dupuis
-/
import analysis.convex.hull
import data.real.basic
import linear_algebra.linear_pmap
/-!
# Convex cones
> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
> Any changes to this file require a corresponding PR to mathlib4.
In a `𝕜`-module `E`, we define a convex cone as a set `s` such that `a • x + b • y ∈ s` whenever
`x, y ∈ s` and `a, b > 0`. We prove that convex cones form a `complete_lattice`, and define their
images (`convex_cone.map`) and preimages (`convex_cone.comap`) under linear maps.
We define pointed, blunt, flat and salient cones, and prove the correspondence between
convex cones and ordered modules.
We define `convex.to_cone` to be the minimal cone that includes a given convex set.
## Main statements
We prove two extension theorems:
* `riesz_extension`:
[M. Riesz extension theorem](https://en.wikipedia.org/wiki/M._Riesz_extension_theorem) says that
if `s` is a convex cone in a real vector space `E`, `p` is a submodule of `E`
such that `p + s = E`, and `f` is a linear function `p → ℝ` which is
nonnegative on `p ∩ s`, then there exists a globally defined linear function
`g : E → ℝ` that agrees with `f` on `p`, and is nonnegative on `s`.
* `exists_extension_of_le_sublinear`:
Hahn-Banach theorem: if `N : E → ℝ` is a sublinear map, `f` is a linear map
defined on a subspace of `E`, and `f x ≤ N x` for all `x` in the domain of `f`,
then `f` can be extended to the whole space to a linear map `g` such that `g x ≤ N x`
for all `x`
We prove the following theorems:
* `convex_cone.hyperplane_separation_of_nonempty_of_is_closed_of_nmem`:
This variant of the
[hyperplane separation theorem](https://en.wikipedia.org/wiki/Hyperplane_separation_theorem)
states that given a nonempty, closed, convex cone `K` in a complete, real inner product space `H`
and a point `b` disjoint from it, there is a vector `y` which separates `b` from `K` in the sense
that for all points `x` in `K`, `0 ≤ ⟪x, y⟫_ℝ` and `⟪y, b⟫_ℝ < 0`. This is also a geometric
interpretation of the
[Farkas lemma](https://en.wikipedia.org/wiki/Farkas%27_lemma#Geometric_interpretation).
* `convex_cone.inner_dual_cone_of_inner_dual_cone_eq_self`:
## Implementation notes
While `convex 𝕜` is a predicate on sets, `convex_cone 𝕜 E` is a bundled convex cone.
## References
* https://en.wikipedia.org/wiki/Convex_cone
* [Stephen P. Boyd and Lieven Vandenberghe, *Convex Optimization*][boydVandenberghe2004]
* [Emo Welzl and Bernd Gärtner, *Cone Programming*][welzl_garter]
-/
assert_not_exists normed_space
open set linear_map
open_locale classical pointwise
variables {𝕜 E F G : Type*}
/-! ### Definition of `convex_cone` and basic properties -/
section definitions
variables (𝕜 E) [ordered_semiring 𝕜]
/-- A convex cone is a subset `s` of a `𝕜`-module such that `a • x + b • y ∈ s` whenever `a, b > 0`
and `x, y ∈ s`. -/
structure convex_cone [add_comm_monoid E] [has_smul 𝕜 E] :=
(carrier : set E)
(smul_mem' : ∀ ⦃c : 𝕜⦄, 0 < c → ∀ ⦃x : E⦄, x ∈ carrier → c • x ∈ carrier)
(add_mem' : ∀ ⦃x⦄ (hx : x ∈ carrier) ⦃y⦄ (hy : y ∈ carrier), x + y ∈ carrier)
end definitions
variables {𝕜 E}
namespace convex_cone
section ordered_semiring
variables [ordered_semiring 𝕜] [add_comm_monoid E]
section has_smul
variables [has_smul 𝕜 E] (S T : convex_cone 𝕜 E)
instance : set_like (convex_cone 𝕜 E) E :=
{ coe := carrier,
coe_injective' := λ S T h, by cases S; cases T; congr' }
@[simp] lemma coe_mk {s : set E} {h₁ h₂} : ↑(@mk 𝕜 _ _ _ _ s h₁ h₂) = s := rfl
@[simp] lemma mem_mk {s : set E} {h₁ h₂ x} : x ∈ @mk 𝕜 _ _ _ _ s h₁ h₂ ↔ x ∈ s := iff.rfl
/-- Two `convex_cone`s are equal if they have the same elements. -/
@[ext] theorem ext {S T : convex_cone 𝕜 E} (h : ∀ x, x ∈ S ↔ x ∈ T) : S = T := set_like.ext h
lemma smul_mem {c : 𝕜} {x : E} (hc : 0 < c) (hx : x ∈ S) : c • x ∈ S := S.smul_mem' hc hx
lemma add_mem ⦃x⦄ (hx : x ∈ S) ⦃y⦄ (hy : y ∈ S) : x + y ∈ S := S.add_mem' hx hy
instance : add_mem_class (convex_cone 𝕜 E) E :=
{ add_mem := λ c a b ha hb, add_mem c ha hb }
instance : has_inf (convex_cone 𝕜 E) :=
⟨λ S T, ⟨S ∩ T, λ c hc x hx, ⟨S.smul_mem hc hx.1, T.smul_mem hc hx.2⟩,
λ x hx y hy, ⟨S.add_mem hx.1 hy.1, T.add_mem hx.2 hy.2⟩⟩⟩
@[simp] lemma coe_inf : ((S ⊓ T : convex_cone 𝕜 E) : set E) = ↑S ∩ ↑T := rfl
lemma mem_inf {x} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T := iff.rfl
instance : has_Inf (convex_cone 𝕜 E) :=
⟨λ S, ⟨⋂ s ∈ S, ↑s,
λ c hc x hx, mem_bInter $ λ s hs, s.smul_mem hc $ mem_Inter₂.1 hx s hs,
λ x hx y hy, mem_bInter $ λ s hs, s.add_mem (mem_Inter₂.1 hx s hs) (mem_Inter₂.1 hy s hs)⟩⟩
@[simp] lemma coe_Inf (S : set (convex_cone 𝕜 E)) : ↑(Inf S) = ⋂ s ∈ S, (s : set E) := rfl
lemma mem_Inf {x : E} {S : set (convex_cone 𝕜 E)} : x ∈ Inf S ↔ ∀ s ∈ S, x ∈ s := mem_Inter₂
@[simp] lemma coe_infi {ι : Sort*} (f : ι → convex_cone 𝕜 E) : ↑(infi f) = ⋂ i, (f i : set E) :=
by simp [infi]
lemma mem_infi {ι : Sort*} {x : E} {f : ι → convex_cone 𝕜 E} : x ∈ infi f ↔ ∀ i, x ∈ f i :=
mem_Inter₂.trans $ by simp
variables (𝕜)
instance : has_bot (convex_cone 𝕜 E) := ⟨⟨∅, λ c hc x, false.elim, λ x, false.elim⟩⟩
lemma mem_bot (x : E) : x ∈ (⊥ : convex_cone 𝕜 E) = false := rfl
@[simp] lemma coe_bot : ↑(⊥ : convex_cone 𝕜 E) = (∅ : set E) := rfl
instance : has_top (convex_cone 𝕜 E) := ⟨⟨univ, λ c hc x hx, mem_univ _, λ x hx y hy, mem_univ _⟩⟩
lemma mem_top (x : E) : x ∈ (⊤ : convex_cone 𝕜 E) := mem_univ x
@[simp] lemma coe_top : ↑(⊤ : convex_cone 𝕜 E) = (univ : set E) := rfl
instance : complete_lattice (convex_cone 𝕜 E) :=
{ le := (≤),
lt := (<),
bot := (⊥),
bot_le := λ S x, false.elim,
top := (⊤),
le_top := λ S x hx, mem_top 𝕜 x,
inf := (⊓),
Inf := has_Inf.Inf,
sup := λ a b, Inf {x | a ≤ x ∧ b ≤ x},
Sup := λ s, Inf {T | ∀ S ∈ s, S ≤ T},
le_sup_left := λ a b, λ x hx, mem_Inf.2 $ λ s hs, hs.1 hx,
le_sup_right := λ a b, λ x hx, mem_Inf.2 $ λ s hs, hs.2 hx,
sup_le := λ a b c ha hb x hx, mem_Inf.1 hx c ⟨ha, hb⟩,
le_inf := λ a b c ha hb x hx, ⟨ha hx, hb hx⟩,
inf_le_left := λ a b x, and.left,
inf_le_right := λ a b x, and.right,
le_Sup := λ s p hs x hx, mem_Inf.2 $ λ t ht, ht p hs hx,
Sup_le := λ s p hs x hx, mem_Inf.1 hx p hs,
le_Inf := λ s a ha x hx, mem_Inf.2 $ λ t ht, ha t ht hx,
Inf_le := λ s a ha x hx, mem_Inf.1 hx _ ha,
.. set_like.partial_order }
instance : inhabited (convex_cone 𝕜 E) := ⟨⊥⟩
end has_smul
section module
variables [module 𝕜 E] (S : convex_cone 𝕜 E)
protected lemma convex : convex 𝕜 (S : set E) :=
convex_iff_forall_pos.2 $ λ x hx y hy a b ha hb _, S.add_mem (S.smul_mem ha hx) (S.smul_mem hb hy)
end module
end ordered_semiring
section linear_ordered_field
variables [linear_ordered_field 𝕜]
section add_comm_monoid
variables [add_comm_monoid E] [add_comm_monoid F] [add_comm_monoid G]
section mul_action
variables [mul_action 𝕜 E] (S : convex_cone 𝕜 E)
lemma smul_mem_iff {c : 𝕜} (hc : 0 < c) {x : E} :
c • x ∈ S ↔ x ∈ S :=
⟨λ h, inv_smul_smul₀ hc.ne' x ▸ S.smul_mem (inv_pos.2 hc) h, S.smul_mem hc⟩
end mul_action
section module
variables [module 𝕜 E] [module 𝕜 F] [module 𝕜 G]
/-- The image of a convex cone under a `𝕜`-linear map is a convex cone. -/
def map (f : E →ₗ[𝕜] F) (S : convex_cone 𝕜 E) : convex_cone 𝕜 F :=
{ carrier := f '' S,
smul_mem' := λ c hc y ⟨x, hx, hy⟩, hy ▸ f.map_smul c x ▸ mem_image_of_mem f (S.smul_mem hc hx),
add_mem' := λ y₁ ⟨x₁, hx₁, hy₁⟩ y₂ ⟨x₂, hx₂, hy₂⟩, hy₁ ▸ hy₂ ▸ f.map_add x₁ x₂ ▸
mem_image_of_mem f (S.add_mem hx₁ hx₂) }
@[simp] lemma mem_map {f : E →ₗ[𝕜] F} {S : convex_cone 𝕜 E} {y : F} :
y ∈ S.map f ↔ ∃ x ∈ S, f x = y :=
mem_image_iff_bex
lemma map_map (g : F →ₗ[𝕜] G) (f : E →ₗ[𝕜] F) (S : convex_cone 𝕜 E) :
(S.map f).map g = S.map (g.comp f) :=
set_like.coe_injective $ image_image g f S
@[simp] lemma map_id (S : convex_cone 𝕜 E) : S.map linear_map.id = S :=
set_like.coe_injective $ image_id _
/-- The preimage of a convex cone under a `𝕜`-linear map is a convex cone. -/
def comap (f : E →ₗ[𝕜] F) (S : convex_cone 𝕜 F) : convex_cone 𝕜 E :=
{ carrier := f ⁻¹' S,
smul_mem' := λ c hc x hx, by { rw [mem_preimage, f.map_smul c], exact S.smul_mem hc hx },
add_mem' := λ x hx y hy, by { rw [mem_preimage, f.map_add], exact S.add_mem hx hy } }
@[simp] lemma coe_comap (f : E →ₗ[𝕜] F) (S : convex_cone 𝕜 F) : (S.comap f : set E) = f ⁻¹' S := rfl
@[simp] lemma comap_id (S : convex_cone 𝕜 E) : S.comap linear_map.id = S :=
set_like.coe_injective preimage_id
lemma comap_comap (g : F →ₗ[𝕜] G) (f : E →ₗ[𝕜] F) (S : convex_cone 𝕜 G) :
(S.comap g).comap f = S.comap (g.comp f) :=
set_like.coe_injective $ preimage_comp.symm
@[simp] lemma mem_comap {f : E →ₗ[𝕜] F} {S : convex_cone 𝕜 F} {x : E} : x ∈ S.comap f ↔ f x ∈ S :=
iff.rfl
end module
end add_comm_monoid
section ordered_add_comm_group
variables [ordered_add_comm_group E] [module 𝕜 E]
/--
Constructs an ordered module given an `ordered_add_comm_group`, a cone, and a proof that
the order relation is the one defined by the cone.
-/
lemma to_ordered_smul (S : convex_cone 𝕜 E) (h : ∀ x y : E, x ≤ y ↔ y - x ∈ S) :
ordered_smul 𝕜 E :=
ordered_smul.mk'
begin
intros x y z xy hz,
rw [h (z • x) (z • y), ←smul_sub z y x],
exact smul_mem S hz ((h x y).mp xy.le),
end
end ordered_add_comm_group
end linear_ordered_field
/-! ### Convex cones with extra properties -/
section ordered_semiring
variables [ordered_semiring 𝕜]
section add_comm_monoid
variables [add_comm_monoid E] [has_smul 𝕜 E] (S : convex_cone 𝕜 E)
/-- A convex cone is pointed if it includes `0`. -/
def pointed (S : convex_cone 𝕜 E) : Prop := (0 : E) ∈ S
/-- A convex cone is blunt if it doesn't include `0`. -/
def blunt (S : convex_cone 𝕜 E) : Prop := (0 : E) ∉ S
lemma pointed_iff_not_blunt (S : convex_cone 𝕜 E) : S.pointed ↔ ¬S.blunt :=
⟨λ h₁ h₂, h₂ h₁, not_not.mp⟩
lemma blunt_iff_not_pointed (S : convex_cone 𝕜 E) : S.blunt ↔ ¬S.pointed :=
by rw [pointed_iff_not_blunt, not_not]
lemma pointed.mono {S T : convex_cone 𝕜 E} (h : S ≤ T) : S.pointed → T.pointed := @h _
lemma blunt.anti {S T : convex_cone 𝕜 E} (h : T ≤ S) : S.blunt → T.blunt := (∘ @@h)
end add_comm_monoid
section add_comm_group
variables [add_comm_group E] [has_smul 𝕜 E] (S : convex_cone 𝕜 E)
/-- A convex cone is flat if it contains some nonzero vector `x` and its opposite `-x`. -/
def flat : Prop := ∃ x ∈ S, x ≠ (0 : E) ∧ -x ∈ S
/-- A convex cone is salient if it doesn't include `x` and `-x` for any nonzero `x`. -/
def salient : Prop := ∀ x ∈ S, x ≠ (0 : E) → -x ∉ S
lemma salient_iff_not_flat (S : convex_cone 𝕜 E) : S.salient ↔ ¬S.flat :=
begin
split,
{ rintros h₁ ⟨x, xs, H₁, H₂⟩,
exact h₁ x xs H₁ H₂ },
{ intro h,
unfold flat at h,
push_neg at h,
exact h }
end
lemma flat.mono {S T : convex_cone 𝕜 E} (h : S ≤ T) : S.flat → T.flat
| ⟨x, hxS, hx, hnxS⟩ := ⟨x, h hxS, hx, h hnxS⟩
lemma salient.anti {S T : convex_cone 𝕜 E} (h : T ≤ S) : S.salient → T.salient :=
λ hS x hxT hx hnT, hS x (h hxT) hx (h hnT)
/-- A flat cone is always pointed (contains `0`). -/
lemma flat.pointed {S : convex_cone 𝕜 E} (hS : S.flat) : S.pointed :=
begin
obtain ⟨x, hx, _, hxneg⟩ := hS,
rw [pointed, ←add_neg_self x],
exact add_mem S hx hxneg,
end
/-- A blunt cone (one not containing `0`) is always salient. -/
lemma blunt.salient {S : convex_cone 𝕜 E} : S.blunt → S.salient :=
begin
rw [salient_iff_not_flat, blunt_iff_not_pointed],
exact mt flat.pointed,
end
/-- A pointed convex cone defines a preorder. -/
def to_preorder (h₁ : S.pointed) : preorder E :=
{ le := λ x y, y - x ∈ S,
le_refl := λ x, by change x - x ∈ S; rw [sub_self x]; exact h₁,
le_trans := λ x y z xy zy, by simpa using add_mem S zy xy }
/-- A pointed and salient cone defines a partial order. -/
def to_partial_order (h₁ : S.pointed) (h₂ : S.salient) : partial_order E :=
{ le_antisymm :=
begin
intros a b ab ba,
by_contradiction h,
have h' : b - a ≠ 0 := λ h'', h (eq_of_sub_eq_zero h'').symm,
have H := h₂ (b-a) ab h',
rw neg_sub b a at H,
exact H ba,
end,
..to_preorder S h₁ }
/-- A pointed and salient cone defines an `ordered_add_comm_group`. -/
def to_ordered_add_comm_group (h₁ : S.pointed) (h₂ : S.salient) :
ordered_add_comm_group E :=
{ add_le_add_left :=
begin
intros a b hab c,
change c + b - (c + a) ∈ S,
rw add_sub_add_left_eq_sub,
exact hab,
end,
..to_partial_order S h₁ h₂,
..show add_comm_group E, by apply_instance }
end add_comm_group
section module
variables [add_comm_monoid E] [module 𝕜 E]
instance : has_zero (convex_cone 𝕜 E) := ⟨⟨0, λ _ _, by simp, λ _, by simp⟩⟩
@[simp] lemma mem_zero (x : E) : x ∈ (0 : convex_cone 𝕜 E) ↔ x = 0 := iff.rfl
@[simp] lemma coe_zero : ((0 : convex_cone 𝕜 E) : set E) = 0 := rfl
lemma pointed_zero : (0 : convex_cone 𝕜 E).pointed := by rw [pointed, mem_zero]
instance : has_add (convex_cone 𝕜 E) := ⟨ λ K₁ K₂,
{ carrier := {z | ∃ (x y : E), x ∈ K₁ ∧ y ∈ K₂ ∧ x + y = z},
smul_mem' :=
begin
rintro c hc _ ⟨x, y, hx, hy, rfl⟩,
rw smul_add,
use [c • x, c • y, K₁.smul_mem hc hx, K₂.smul_mem hc hy],
end,
add_mem' :=
begin
rintro _ ⟨x₁, x₂, hx₁, hx₂, rfl⟩ y ⟨y₁, y₂, hy₁, hy₂, rfl⟩,
use [x₁ + y₁, x₂ + y₂, K₁.add_mem hx₁ hy₁, K₂.add_mem hx₂ hy₂],
abel,
end } ⟩
@[simp] lemma mem_add {K₁ K₂ : convex_cone 𝕜 E} {a : E} :
a ∈ K₁ + K₂ ↔ ∃ (x y : E), x ∈ K₁ ∧ y ∈ K₂ ∧ x + y = a := iff.rfl
instance : add_zero_class (convex_cone 𝕜 E) :=
⟨0, has_add.add, λ _, by {ext, simp}, λ _, by {ext, simp}⟩
instance : add_comm_semigroup (convex_cone 𝕜 E) :=
{ add := has_add.add,
add_assoc := λ _ _ _, set_like.coe_injective $ set.add_comm_semigroup.add_assoc _ _ _,
add_comm := λ _ _, set_like.coe_injective $ set.add_comm_semigroup.add_comm _ _ }
end module
end ordered_semiring
end convex_cone
namespace submodule
/-! ### Submodules are cones -/
section ordered_semiring
variables [ordered_semiring 𝕜]
section add_comm_monoid
variables [add_comm_monoid E] [module 𝕜 E]
/-- Every submodule is trivially a convex cone. -/
def to_convex_cone (S : submodule 𝕜 E) : convex_cone 𝕜 E :=
{ carrier := S,
smul_mem' := λ c hc x hx, S.smul_mem c hx,
add_mem' := λ x hx y hy, S.add_mem hx hy }
@[simp] lemma coe_to_convex_cone (S : submodule 𝕜 E) : ↑S.to_convex_cone = (S : set E) := rfl
@[simp] lemma mem_to_convex_cone {x : E} {S : submodule 𝕜 E} : x ∈ S.to_convex_cone ↔ x ∈ S :=
iff.rfl
@[simp] lemma to_convex_cone_le_iff {S T : submodule 𝕜 E} :
S.to_convex_cone ≤ T.to_convex_cone ↔ S ≤ T :=
iff.rfl
@[simp] lemma to_convex_cone_bot : (⊥ : submodule 𝕜 E).to_convex_cone = 0 := rfl
@[simp] lemma to_convex_cone_top : (⊤ : submodule 𝕜 E).to_convex_cone = ⊤ := rfl
@[simp] lemma to_convex_cone_inf (S T : submodule 𝕜 E) :
(S ⊓ T).to_convex_cone = S.to_convex_cone ⊓ T.to_convex_cone :=
rfl
@[simp] lemma pointed_to_convex_cone (S : submodule 𝕜 E) : S.to_convex_cone.pointed := S.zero_mem
end add_comm_monoid
end ordered_semiring
end submodule
namespace convex_cone
/-! ### Positive cone of an ordered module -/
section positive_cone
variables (𝕜 E) [ordered_semiring 𝕜] [ordered_add_comm_group E] [module 𝕜 E] [ordered_smul 𝕜 E]
/--
The positive cone is the convex cone formed by the set of nonnegative elements in an ordered
module.
-/
def positive : convex_cone 𝕜 E :=
{ carrier := set.Ici 0,
smul_mem' := λ c hc x (hx : _ ≤ _), smul_nonneg hc.le hx,
add_mem' := λ x (hx : _ ≤ _) y (hy : _ ≤ _), add_nonneg hx hy }
@[simp] lemma mem_positive {x : E} : x ∈ positive 𝕜 E ↔ 0 ≤ x := iff.rfl
@[simp] lemma coe_positive : ↑(positive 𝕜 E) = set.Ici (0 : E) := rfl
/-- The positive cone of an ordered module is always salient. -/
lemma salient_positive : salient (positive 𝕜 E) :=
λ x xs hx hx', lt_irrefl (0 : E)
(calc
0 < x : lt_of_le_of_ne xs hx.symm
... ≤ x + (-x) : le_add_of_nonneg_right hx'
... = 0 : add_neg_self x)
/-- The positive cone of an ordered module is always pointed. -/
lemma pointed_positive : pointed (positive 𝕜 E) := le_refl 0
/-- The cone of strictly positive elements.
Note that this naming diverges from the mathlib convention of `pos` and `nonneg` due to "positive
cone" (`convex_cone.positive`) being established terminology for the non-negative elements. -/
def strictly_positive : convex_cone 𝕜 E :=
{ carrier := set.Ioi 0,
smul_mem' := λ c hc x (hx : _ < _), smul_pos hc hx,
add_mem' := λ x hx y hy, add_pos hx hy }
@[simp] lemma mem_strictly_positive {x : E} : x ∈ strictly_positive 𝕜 E ↔ 0 < x := iff.rfl
@[simp] lemma coe_strictly_positive : ↑(strictly_positive 𝕜 E) = set.Ioi (0 : E) := rfl
lemma positive_le_strictly_positive : strictly_positive 𝕜 E ≤ positive 𝕜 E := λ x, le_of_lt
/-- The strictly positive cone of an ordered module is always salient. -/
lemma salient_strictly_positive : salient (strictly_positive 𝕜 E) :=
(salient_positive 𝕜 E).anti $ positive_le_strictly_positive 𝕜 E
/-- The strictly positive cone of an ordered module is always blunt. -/
lemma blunt_strictly_positive : blunt (strictly_positive 𝕜 E) := lt_irrefl 0
end positive_cone
end convex_cone
/-! ### Cone over a convex set -/
section cone_from_convex
variables [linear_ordered_field 𝕜] [add_comm_group E] [module 𝕜 E]
namespace convex
/-- The set of vectors proportional to those in a convex set forms a convex cone. -/
def to_cone (s : set E) (hs : convex 𝕜 s) : convex_cone 𝕜 E :=
begin
apply convex_cone.mk (⋃ (c : 𝕜) (H : 0 < c), c • s);
simp only [mem_Union, mem_smul_set],
{ rintros c c_pos _ ⟨c', c'_pos, x, hx, rfl⟩,
exact ⟨c * c', mul_pos c_pos c'_pos, x, hx, (smul_smul _ _ _).symm⟩ },
{ rintros _ ⟨cx, cx_pos, x, hx, rfl⟩ _ ⟨cy, cy_pos, y, hy, rfl⟩,
have : 0 < cx + cy, from add_pos cx_pos cy_pos,
refine ⟨_, this, _, convex_iff_div.1 hs hx hy cx_pos.le cy_pos.le this, _⟩,
simp only [smul_add, smul_smul, mul_div_assoc', mul_div_cancel_left _ this.ne'] }
end
variables {s : set E} (hs : convex 𝕜 s) {x : E}
lemma mem_to_cone : x ∈ hs.to_cone s ↔ ∃ (c : 𝕜), 0 < c ∧ ∃ y ∈ s, c • y = x :=
by simp only [to_cone, convex_cone.mem_mk, mem_Union, mem_smul_set, eq_comm, exists_prop]
lemma mem_to_cone' : x ∈ hs.to_cone s ↔ ∃ (c : 𝕜), 0 < c ∧ c • x ∈ s :=
begin
refine hs.mem_to_cone.trans ⟨_, _⟩,
{ rintros ⟨c, hc, y, hy, rfl⟩,
exact ⟨c⁻¹, inv_pos.2 hc, by rwa [smul_smul, inv_mul_cancel hc.ne', one_smul]⟩ },
{ rintros ⟨c, hc, hcx⟩,
exact ⟨c⁻¹, inv_pos.2 hc, _, hcx, by rw [smul_smul, inv_mul_cancel hc.ne', one_smul]⟩ }
end
lemma subset_to_cone : s ⊆ hs.to_cone s :=
λ x hx, hs.mem_to_cone'.2 ⟨1, zero_lt_one, by rwa one_smul⟩
/-- `hs.to_cone s` is the least cone that includes `s`. -/
lemma to_cone_is_least : is_least { t : convex_cone 𝕜 E | s ⊆ t } (hs.to_cone s) :=
begin
refine ⟨hs.subset_to_cone, λ t ht x hx, _⟩,
rcases hs.mem_to_cone.1 hx with ⟨c, hc, y, hy, rfl⟩,
exact t.smul_mem hc (ht hy)
end
lemma to_cone_eq_Inf : hs.to_cone s = Inf { t : convex_cone 𝕜 E | s ⊆ t } :=
hs.to_cone_is_least.is_glb.Inf_eq.symm
end convex
lemma convex_hull_to_cone_is_least (s : set E) :
is_least {t : convex_cone 𝕜 E | s ⊆ t} ((convex_convex_hull 𝕜 s).to_cone _) :=
begin
convert (convex_convex_hull 𝕜 s).to_cone_is_least,
ext t,
exact ⟨λ h, convex_hull_min h t.convex, (subset_convex_hull 𝕜 s).trans⟩,
end
lemma convex_hull_to_cone_eq_Inf (s : set E) :
(convex_convex_hull 𝕜 s).to_cone _ = Inf {t : convex_cone 𝕜 E | s ⊆ t} :=
eq.symm $ is_glb.Inf_eq $ is_least.is_glb $ convex_hull_to_cone_is_least s
end cone_from_convex
/-!
### M. Riesz extension theorem
Given a convex cone `s` in a vector space `E`, a submodule `p`, and a linear `f : p → ℝ`, assume
that `f` is nonnegative on `p ∩ s` and `p + s = E`. Then there exists a globally defined linear
function `g : E → ℝ` that agrees with `f` on `p`, and is nonnegative on `s`.
We prove this theorem using Zorn's lemma. `riesz_extension.step` is the main part of the proof.
It says that if the domain `p` of `f` is not the whole space, then `f` can be extended to a larger
subspace `p ⊔ span ℝ {y}` without breaking the non-negativity condition.
In `riesz_extension.exists_top` we use Zorn's lemma to prove that we can extend `f`
to a linear map `g` on `⊤ : submodule E`. Mathematically this is the same as a linear map on `E`
but in Lean `⊤ : submodule E` is isomorphic but is not equal to `E`. In `riesz_extension`
we use this isomorphism to prove the theorem.
-/
variables [add_comm_group E] [module ℝ E]
namespace riesz_extension
open submodule
variables (s : convex_cone ℝ E) (f : E →ₗ.[ℝ] ℝ)
/-- Induction step in M. Riesz extension theorem. Given a convex cone `s` in a vector space `E`,
a partially defined linear map `f : f.domain → ℝ`, assume that `f` is nonnegative on `f.domain ∩ p`
and `p + s = E`. If `f` is not defined on the whole `E`, then we can extend it to a larger
submodule without breaking the non-negativity condition. -/
lemma step (nonneg : ∀ x : f.domain, (x : E) ∈ s → 0 ≤ f x)
(dense : ∀ y, ∃ x : f.domain, (x : E) + y ∈ s) (hdom : f.domain ≠ ⊤) :
∃ g, f < g ∧ ∀ x : g.domain, (x : E) ∈ s → 0 ≤ g x :=
begin
obtain ⟨y, -, hy⟩ : ∃ (y : E) (h : y ∈ ⊤), y ∉ f.domain,
{ exact @set_like.exists_of_lt (submodule ℝ E) _ _ _ _ (lt_top_iff_ne_top.2 hdom) },
obtain ⟨c, le_c, c_le⟩ :
∃ c, (∀ x : f.domain, -(x:E) - y ∈ s → f x ≤ c) ∧ (∀ x : f.domain, (x:E) + y ∈ s → c ≤ f x),
{ set Sp := f '' {x : f.domain | (x:E) + y ∈ s},
set Sn := f '' {x : f.domain | -(x:E) - y ∈ s},
suffices : (upper_bounds Sn ∩ lower_bounds Sp).nonempty,
by simpa only [set.nonempty, upper_bounds, lower_bounds, ball_image_iff] using this,
refine exists_between_of_forall_le (nonempty.image f _) (nonempty.image f (dense y)) _,
{ rcases (dense (-y)) with ⟨x, hx⟩,
rw [← neg_neg x, add_subgroup_class.coe_neg, ← sub_eq_add_neg] at hx,
exact ⟨_, hx⟩ },
rintros a ⟨xn, hxn, rfl⟩ b ⟨xp, hxp, rfl⟩,
have := s.add_mem hxp hxn,
rw [add_assoc, add_sub_cancel'_right, ← sub_eq_add_neg, ← add_subgroup_class.coe_sub] at this,
replace := nonneg _ this,
rwa [f.map_sub, sub_nonneg] at this },
have hy' : y ≠ 0, from λ hy₀, hy (hy₀.symm ▸ zero_mem _),
refine ⟨f.sup_span_singleton y (-c) hy, _, _⟩,
{ refine lt_iff_le_not_le.2 ⟨f.left_le_sup _ _, λ H, _⟩,
replace H := linear_pmap.domain_mono.monotone H,
rw [linear_pmap.domain_sup_span_singleton, sup_le_iff, span_le, singleton_subset_iff] at H,
exact hy H.2 },
{ rintros ⟨z, hz⟩ hzs,
rcases mem_sup.1 hz with ⟨x, hx, y', hy', rfl⟩,
rcases mem_span_singleton.1 hy' with ⟨r, rfl⟩,
simp only [subtype.coe_mk] at hzs,
erw [linear_pmap.sup_span_singleton_apply_mk _ _ _ _ _ hx, smul_neg,
← sub_eq_add_neg, sub_nonneg],
rcases lt_trichotomy r 0 with hr|hr|hr,
{ have : -(r⁻¹ • x) - y ∈ s,
by rwa [← s.smul_mem_iff (neg_pos.2 hr), smul_sub, smul_neg, neg_smul, neg_neg, smul_smul,
mul_inv_cancel hr.ne, one_smul, sub_eq_add_neg, neg_smul, neg_neg],
replace := le_c (r⁻¹ • ⟨x, hx⟩) this,
rwa [← mul_le_mul_left (neg_pos.2 hr), neg_mul, neg_mul,
neg_le_neg_iff, f.map_smul, smul_eq_mul, ← mul_assoc, mul_inv_cancel hr.ne,
one_mul] at this },
{ subst r,
simp only [zero_smul, add_zero] at hzs ⊢,
apply nonneg,
exact hzs },
{ have : r⁻¹ • x + y ∈ s,
by rwa [← s.smul_mem_iff hr, smul_add, smul_smul, mul_inv_cancel hr.ne', one_smul],
replace := c_le (r⁻¹ • ⟨x, hx⟩) this,
rwa [← mul_le_mul_left hr, f.map_smul, smul_eq_mul, ← mul_assoc,
mul_inv_cancel hr.ne', one_mul] at this } }
end
theorem exists_top (p : E →ₗ.[ℝ] ℝ)
(hp_nonneg : ∀ x : p.domain, (x : E) ∈ s → 0 ≤ p x)
(hp_dense : ∀ y, ∃ x : p.domain, (x : E) + y ∈ s) :
∃ q ≥ p, q.domain = ⊤ ∧ ∀ x : q.domain, (x : E) ∈ s → 0 ≤ q x :=
begin
replace hp_nonneg : p ∈ { p | _ }, by { rw mem_set_of_eq, exact hp_nonneg },
obtain ⟨q, hqs, hpq, hq⟩ := zorn_nonempty_partial_order₀ _ _ _ hp_nonneg,
{ refine ⟨q, hpq, _, hqs⟩,
contrapose! hq,
rcases step s q hqs _ hq with ⟨r, hqr, hr⟩,
{ exact ⟨r, hr, hqr.le, hqr.ne'⟩ },
{ exact λ y, let ⟨x, hx⟩ := hp_dense y in ⟨of_le hpq.left x, hx⟩ } },
{ intros c hcs c_chain y hy,
clear hp_nonneg hp_dense p,
have cne : c.nonempty := ⟨y, hy⟩,
refine ⟨linear_pmap.Sup c c_chain.directed_on, _, λ _, linear_pmap.le_Sup c_chain.directed_on⟩,
rintros ⟨x, hx⟩ hxs,
have hdir : directed_on (≤) (linear_pmap.domain '' c),
from directed_on_image.2 (c_chain.directed_on.mono linear_pmap.domain_mono.monotone),
rcases (mem_Sup_of_directed (cne.image _) hdir).1 hx with ⟨_, ⟨f, hfc, rfl⟩, hfx⟩,
have : f ≤ linear_pmap.Sup c c_chain.directed_on, from linear_pmap.le_Sup _ hfc,
convert ← hcs hfc ⟨x, hfx⟩ hxs,
apply this.2, refl }
end
end riesz_extension
/-- M. **Riesz extension theorem**: given a convex cone `s` in a vector space `E`, a submodule `p`,
and a linear `f : p → ℝ`, assume that `f` is nonnegative on `p ∩ s` and `p + s = E`. Then
there exists a globally defined linear function `g : E → ℝ` that agrees with `f` on `p`,
and is nonnegative on `s`. -/
theorem riesz_extension (s : convex_cone ℝ E) (f : E →ₗ.[ℝ] ℝ)
(nonneg : ∀ x : f.domain, (x : E) ∈ s → 0 ≤ f x) (dense : ∀ y, ∃ x : f.domain, (x : E) + y ∈ s) :
∃ g : E →ₗ[ℝ] ℝ, (∀ x : f.domain, g x = f x) ∧ (∀ x ∈ s, 0 ≤ g x) :=
begin
rcases riesz_extension.exists_top s f nonneg dense with ⟨⟨g_dom, g⟩, ⟨hpg, hfg⟩, htop, hgs⟩,
clear hpg,
refine ⟨g ∘ₗ ↑(linear_equiv.of_top _ htop).symm, _, _⟩;
simp only [comp_apply, linear_equiv.coe_coe, linear_equiv.of_top_symm_apply],
{ exact λ x, (hfg (submodule.coe_mk _ _).symm).symm },
{ exact λ x hx, hgs ⟨x, _⟩ hx }
end
/-- **Hahn-Banach theorem**: if `N : E → ℝ` is a sublinear map, `f` is a linear map
defined on a subspace of `E`, and `f x ≤ N x` for all `x` in the domain of `f`,
then `f` can be extended to the whole space to a linear map `g` such that `g x ≤ N x`
for all `x`. -/
theorem exists_extension_of_le_sublinear (f : E →ₗ.[ℝ] ℝ) (N : E → ℝ)
(N_hom : ∀ (c : ℝ), 0 < c → ∀ x, N (c • x) = c * N x)
(N_add : ∀ x y, N (x + y) ≤ N x + N y)
(hf : ∀ x : f.domain, f x ≤ N x) :
∃ g : E →ₗ[ℝ] ℝ, (∀ x : f.domain, g x = f x) ∧ (∀ x, g x ≤ N x) :=
begin
let s : convex_cone ℝ (E × ℝ) :=
{ carrier := {p : E × ℝ | N p.1 ≤ p.2 },
smul_mem' := λ c hc p hp,
calc N (c • p.1) = c * N p.1 : N_hom c hc p.1
... ≤ c * p.2 : mul_le_mul_of_nonneg_left hp hc.le,
add_mem' := λ x hx y hy, (N_add _ _).trans (add_le_add hx hy) },
obtain ⟨g, g_eq, g_nonneg⟩ :=
riesz_extension s ((-f).coprod (linear_map.id.to_pmap ⊤)) _ _;
try { simp only [linear_pmap.coprod_apply, to_pmap_apply, id_apply,
linear_pmap.neg_apply, ← sub_eq_neg_add, sub_nonneg, subtype.coe_mk] at * },
replace g_eq : ∀ (x : f.domain) (y : ℝ), g (x, y) = y - f x,
{ intros x y,
simpa only [subtype.coe_mk, subtype.coe_eta] using g_eq ⟨(x, y), ⟨x.2, trivial⟩⟩ },
{ refine ⟨-g.comp (inl ℝ E ℝ), _, _⟩; simp only [neg_apply, inl_apply, comp_apply],
{ intro x, simp [g_eq x 0] },
{ intro x,
have A : (x, N x) = (x, 0) + (0, N x), by simp,
have B := g_nonneg ⟨x, N x⟩ (le_refl (N x)),
rw [A, map_add, ← neg_le_iff_add_nonneg'] at B,
have C := g_eq 0 (N x),
simp only [submodule.coe_zero, f.map_zero, sub_zero] at C,
rwa ← C } },
{ exact λ x hx, le_trans (hf _) hx },
{ rintros ⟨x, y⟩,
refine ⟨⟨(0, N x - y), ⟨f.domain.zero_mem, trivial⟩⟩, _⟩,
simp only [convex_cone.mem_mk, mem_set_of_eq, subtype.coe_mk, prod.fst_add, prod.snd_add,
zero_add, sub_add_cancel] }
end
|
77d36d98c5692461611eabb1fe60e2f375a740fe | 8e6cad62ec62c6c348e5faaa3c3f2079012bdd69 | /src/data/matrix/basic.lean | dbaa30f51bcb1b27044036fee9ae4ebf143c61e2 | [
"Apache-2.0"
] | permissive | benjamindavidson/mathlib | 8cc81c865aa8e7cf4462245f58d35ae9a56b150d | fad44b9f670670d87c8e25ff9cdf63af87ad731e | refs/heads/master | 1,679,545,578,362 | 1,615,343,014,000 | 1,615,343,014,000 | 312,926,983 | 0 | 0 | Apache-2.0 | 1,615,360,301,000 | 1,605,399,418,000 | Lean | UTF-8 | Lean | false | false | 42,559 | lean | /-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin
-/
import algebra.big_operators.pi
import algebra.module.pi
import algebra.module.linear_map
import algebra.big_operators.ring
import algebra.star.basic
import data.equiv.ring
import data.fintype.card
/-!
# Matrices
-/
universes u u' v w
open_locale big_operators
/-- `matrix m n` is the type of matrices whose rows are indexed by the fintype `m`
and whose columns are indexed by the fintype `n`. -/
@[nolint unused_arguments]
def matrix (m : Type u) (n : Type u') [fintype m] [fintype n] (α : Type v) : Type (max u u' v) :=
m → n → α
variables {l m n o : Type*} [fintype l] [fintype m] [fintype n] [fintype o]
variables {α : Type v}
namespace matrix
section ext
variables {M N : matrix m n α}
theorem ext_iff : (∀ i j, M i j = N i j) ↔ M = N :=
⟨λ h, funext $ λ i, funext $ h i, λ h, by simp [h]⟩
@[ext] theorem ext : (∀ i j, M i j = N i j) → M = N :=
ext_iff.mp
end ext
/-- `M.map f` is the matrix obtained by applying `f` to each entry of the matrix `M`. -/
def map (M : matrix m n α) {β : Type w} (f : α → β) : matrix m n β := λ i j, f (M i j)
@[simp]
lemma map_apply {M : matrix m n α} {β : Type w} {f : α → β} {i : m} {j : n} :
M.map f i j = f (M i j) := rfl
@[simp]
lemma map_map {M : matrix m n α} {β γ : Type*} {f : α → β} {g : β → γ} :
(M.map f).map g = M.map (g ∘ f) :=
by { ext, simp, }
/-- The transpose of a matrix. -/
def transpose (M : matrix m n α) : matrix n m α
| x y := M y x
localized "postfix `ᵀ`:1500 := matrix.transpose" in matrix
/-- `matrix.col u` is the column matrix whose entries are given by `u`. -/
def col (w : m → α) : matrix m unit α
| x y := w x
/-- `matrix.row u` is the row matrix whose entries are given by `u`. -/
def row (v : n → α) : matrix unit n α
| x y := v y
instance [inhabited α] : inhabited (matrix m n α) := pi.inhabited _
instance [has_add α] : has_add (matrix m n α) := pi.has_add
instance [add_semigroup α] : add_semigroup (matrix m n α) := pi.add_semigroup
instance [add_comm_semigroup α] : add_comm_semigroup (matrix m n α) := pi.add_comm_semigroup
instance [has_zero α] : has_zero (matrix m n α) := pi.has_zero
instance [add_monoid α] : add_monoid (matrix m n α) := pi.add_monoid
instance [add_comm_monoid α] : add_comm_monoid (matrix m n α) := pi.add_comm_monoid
instance [has_neg α] : has_neg (matrix m n α) := pi.has_neg
instance [has_sub α] : has_sub (matrix m n α) := pi.has_sub
instance [add_group α] : add_group (matrix m n α) := pi.add_group
instance [add_comm_group α] : add_comm_group (matrix m n α) := pi.add_comm_group
instance [unique α] : unique (matrix m n α) := pi.unique
instance [subsingleton α] : subsingleton (matrix m n α) := pi.subsingleton
instance [nonempty m] [nonempty n] [nontrivial α] : nontrivial (matrix m n α) :=
function.nontrivial
@[simp] theorem zero_apply [has_zero α] (i j) : (0 : matrix m n α) i j = 0 := rfl
@[simp] theorem neg_apply [has_neg α] (M : matrix m n α) (i j) : (- M) i j = - M i j := rfl
@[simp] theorem add_apply [has_add α] (M N : matrix m n α) (i j) :
(M + N) i j = M i j + N i j :=
rfl
@[simp] theorem sub_apply [has_sub α] (M N : matrix m n α) (i j) :
(M - N) i j = M i j - N i j :=
rfl
@[simp] lemma map_zero [has_zero α] {β : Type w} [has_zero β] {f : α → β} (h : f 0 = 0) :
(0 : matrix m n α).map f = 0 :=
by { ext, simp [h], }
lemma map_add [add_monoid α] {β : Type w} [add_monoid β] (f : α →+ β)
(M N : matrix m n α) : (M + N).map f = M.map f + N.map f :=
by { ext, simp, }
lemma map_sub [add_group α] {β : Type w} [add_group β] (f : α →+ β)
(M N : matrix m n α) : (M - N).map f = M.map f - N.map f :=
by { ext, simp }
lemma subsingleton_of_empty_left (hm : ¬ nonempty m) : subsingleton (matrix m n α) :=
⟨λ M N, by { ext, contrapose! hm, use i }⟩
lemma subsingleton_of_empty_right (hn : ¬ nonempty n) : subsingleton (matrix m n α) :=
⟨λ M N, by { ext, contrapose! hn, use j }⟩
end matrix
/-- The `add_monoid_hom` between spaces of matrices induced by an `add_monoid_hom` between their
coefficients. -/
def add_monoid_hom.map_matrix [add_monoid α] {β : Type w} [add_monoid β] (f : α →+ β) :
matrix m n α →+ matrix m n β :=
{ to_fun := λ M, M.map f,
map_zero' := by simp,
map_add' := matrix.map_add f, }
@[simp] lemma add_monoid_hom.map_matrix_apply [add_monoid α] {β : Type w} [add_monoid β]
(f : α →+ β) (M : matrix m n α) : f.map_matrix M = M.map f := rfl
open_locale matrix
namespace matrix
section diagonal
variables [decidable_eq n]
/-- `diagonal d` is the square matrix such that `(diagonal d) i i = d i` and `(diagonal d) i j = 0`
if `i ≠ j`. -/
def diagonal [has_zero α] (d : n → α) : matrix n n α := λ i j, if i = j then d i else 0
@[simp] theorem diagonal_apply_eq [has_zero α] {d : n → α} (i : n) : (diagonal d) i i = d i :=
by simp [diagonal]
@[simp] theorem diagonal_apply_ne [has_zero α] {d : n → α} {i j : n} (h : i ≠ j) :
(diagonal d) i j = 0 := by simp [diagonal, h]
theorem diagonal_apply_ne' [has_zero α] {d : n → α} {i j : n} (h : j ≠ i) :
(diagonal d) i j = 0 := diagonal_apply_ne h.symm
@[simp] theorem diagonal_zero [has_zero α] : (diagonal (λ _, 0) : matrix n n α) = 0 :=
by simp [diagonal]; refl
@[simp] lemma diagonal_transpose [has_zero α] (v : n → α) :
(diagonal v)ᵀ = diagonal v :=
begin
ext i j,
by_cases h : i = j,
{ simp [h, transpose] },
{ simp [h, transpose, diagonal_apply_ne' h] }
end
@[simp] theorem diagonal_add [add_monoid α] (d₁ d₂ : n → α) :
diagonal d₁ + diagonal d₂ = diagonal (λ i, d₁ i + d₂ i) :=
by ext i j; by_cases h : i = j; simp [h]
@[simp] lemma diagonal_map {β : Type w} [has_zero α] [has_zero β]
{f : α → β} (h : f 0 = 0) {d : n → α} :
(diagonal d).map f = diagonal (λ m, f (d m)) :=
by { ext, simp only [diagonal, map_apply], split_ifs; simp [h], }
section one
variables [has_zero α] [has_one α]
instance : has_one (matrix n n α) := ⟨diagonal (λ _, 1)⟩
@[simp] theorem diagonal_one : (diagonal (λ _, 1) : matrix n n α) = 1 := rfl
theorem one_apply {i j} : (1 : matrix n n α) i j = if i = j then 1 else 0 := rfl
@[simp] theorem one_apply_eq (i) : (1 : matrix n n α) i i = 1 := diagonal_apply_eq i
@[simp] theorem one_apply_ne {i j} : i ≠ j → (1 : matrix n n α) i j = 0 :=
diagonal_apply_ne
theorem one_apply_ne' {i j} : j ≠ i → (1 : matrix n n α) i j = 0 :=
diagonal_apply_ne'
@[simp] lemma one_map {β : Type w} [has_zero β] [has_one β]
{f : α → β} (h₀ : f 0 = 0) (h₁ : f 1 = 1) :
(1 : matrix n n α).map f = (1 : matrix n n β) :=
by { ext, simp only [one_apply, map_apply], split_ifs; simp [h₀, h₁], }
end one
section numeral
@[simp] lemma bit0_apply [has_add α] (M : matrix m m α) (i : m) (j : m) :
(bit0 M) i j = bit0 (M i j) := rfl
variables [add_monoid α] [has_one α]
lemma bit1_apply (M : matrix n n α) (i : n) (j : n) :
(bit1 M) i j = if i = j then bit1 (M i j) else bit0 (M i j) :=
by dsimp [bit1]; by_cases h : i = j; simp [h]
@[simp]
lemma bit1_apply_eq (M : matrix n n α) (i : n) :
(bit1 M) i i = bit1 (M i i) :=
by simp [bit1_apply]
@[simp]
lemma bit1_apply_ne (M : matrix n n α) {i j : n} (h : i ≠ j) :
(bit1 M) i j = bit0 (M i j) :=
by simp [bit1_apply, h]
end numeral
end diagonal
section dot_product
/-- `dot_product v w` is the sum of the entrywise products `v i * w i` -/
def dot_product [has_mul α] [add_comm_monoid α] (v w : m → α) : α :=
∑ i, v i * w i
lemma dot_product_assoc [semiring α] (u : m → α) (v : m → n → α) (w : n → α) :
dot_product (λ j, dot_product u (λ i, v i j)) w = dot_product u (λ i, dot_product (v i) w) :=
by simpa [dot_product, finset.mul_sum, finset.sum_mul, mul_assoc] using finset.sum_comm
lemma dot_product_comm [comm_semiring α] (v w : m → α) :
dot_product v w = dot_product w v :=
by simp_rw [dot_product, mul_comm]
@[simp] lemma dot_product_punit [add_comm_monoid α] [has_mul α] (v w : punit → α) :
dot_product v w = v ⟨⟩ * w ⟨⟩ :=
by simp [dot_product]
@[simp] lemma dot_product_zero [semiring α] (v : m → α) : dot_product v 0 = 0 :=
by simp [dot_product]
@[simp] lemma dot_product_zero' [semiring α] (v : m → α) : dot_product v (λ _, 0) = 0 :=
dot_product_zero v
@[simp] lemma zero_dot_product [semiring α] (v : m → α) : dot_product 0 v = 0 :=
by simp [dot_product]
@[simp] lemma zero_dot_product' [semiring α] (v : m → α) : dot_product (λ _, (0 : α)) v = 0 :=
zero_dot_product v
@[simp] lemma add_dot_product [semiring α] (u v w : m → α) :
dot_product (u + v) w = dot_product u w + dot_product v w :=
by simp [dot_product, add_mul, finset.sum_add_distrib]
@[simp] lemma dot_product_add [semiring α] (u v w : m → α) :
dot_product u (v + w) = dot_product u v + dot_product u w :=
by simp [dot_product, mul_add, finset.sum_add_distrib]
@[simp] lemma diagonal_dot_product [decidable_eq m] [semiring α] (v w : m → α) (i : m) :
dot_product (diagonal v i) w = v i * w i :=
have ∀ j ≠ i, diagonal v i j * w j = 0 := λ j hij, by simp [diagonal_apply_ne' hij],
by convert finset.sum_eq_single i (λ j _, this j) _ using 1; simp
@[simp] lemma dot_product_diagonal [decidable_eq m] [semiring α] (v w : m → α) (i : m) :
dot_product v (diagonal w i) = v i * w i :=
have ∀ j ≠ i, v j * diagonal w i j = 0 := λ j hij, by simp [diagonal_apply_ne' hij],
by convert finset.sum_eq_single i (λ j _, this j) _ using 1; simp
@[simp] lemma dot_product_diagonal' [decidable_eq m] [semiring α] (v w : m → α) (i : m) :
dot_product v (λ j, diagonal w j i) = v i * w i :=
have ∀ j ≠ i, v j * diagonal w j i = 0 := λ j hij, by simp [diagonal_apply_ne hij],
by convert finset.sum_eq_single i (λ j _, this j) _ using 1; simp
@[simp] lemma neg_dot_product [ring α] (v w : m → α) : dot_product (-v) w = - dot_product v w :=
by simp [dot_product]
@[simp] lemma dot_product_neg [ring α] (v w : m → α) : dot_product v (-w) = - dot_product v w :=
by simp [dot_product]
@[simp] lemma smul_dot_product [semiring α] (x : α) (v w : m → α) :
dot_product (x • v) w = x * dot_product v w :=
by simp [dot_product, finset.mul_sum, mul_assoc]
@[simp] lemma dot_product_smul [comm_semiring α] (x : α) (v w : m → α) :
dot_product v (x • w) = x * dot_product v w :=
by simp [dot_product, finset.mul_sum, mul_assoc, mul_comm, mul_left_comm]
end dot_product
/-- `M ⬝ N` is the usual product of matrices `M` and `N`, i.e. we have that
`(M ⬝ N) i k` is the dot product of the `i`-th row of `M` by the `k`-th column of `Ǹ`. -/
protected def mul [has_mul α] [add_comm_monoid α] (M : matrix l m α) (N : matrix m n α) :
matrix l n α :=
λ i k, dot_product (λ j, M i j) (λ j, N j k)
localized "infixl ` ⬝ `:75 := matrix.mul" in matrix
theorem mul_apply [has_mul α] [add_comm_monoid α] {M : matrix l m α} {N : matrix m n α} {i k} :
(M ⬝ N) i k = ∑ j, M i j * N j k := rfl
instance [has_mul α] [add_comm_monoid α] : has_mul (matrix n n α) := ⟨matrix.mul⟩
@[simp] theorem mul_eq_mul [has_mul α] [add_comm_monoid α] (M N : matrix n n α) :
M * N = M ⬝ N := rfl
theorem mul_apply' [has_mul α] [add_comm_monoid α] {M N : matrix n n α} {i k} :
(M ⬝ N) i k = dot_product (λ j, M i j) (λ j, N j k) := rfl
section semigroup
variables [semiring α]
protected theorem mul_assoc (L : matrix l m α) (M : matrix m n α) (N : matrix n o α) :
(L ⬝ M) ⬝ N = L ⬝ (M ⬝ N) :=
by { ext, apply dot_product_assoc }
instance : semigroup (matrix n n α) :=
{ mul_assoc := matrix.mul_assoc, ..matrix.has_mul }
end semigroup
@[simp] theorem diagonal_neg [decidable_eq n] [add_group α] (d : n → α) :
-diagonal d = diagonal (λ i, -d i) :=
by ext i j; by_cases i = j; simp [h]
section semiring
variables [semiring α]
@[simp] protected theorem mul_zero (M : matrix m n α) : M ⬝ (0 : matrix n o α) = 0 :=
by { ext i j, apply dot_product_zero }
@[simp] protected theorem zero_mul (M : matrix m n α) : (0 : matrix l m α) ⬝ M = 0 :=
by { ext i j, apply zero_dot_product }
protected theorem mul_add (L : matrix m n α) (M N : matrix n o α) : L ⬝ (M + N) = L ⬝ M + L ⬝ N :=
by { ext i j, apply dot_product_add }
protected theorem add_mul (L M : matrix l m α) (N : matrix m n α) : (L + M) ⬝ N = L ⬝ N + M ⬝ N :=
by { ext i j, apply add_dot_product }
@[simp] theorem diagonal_mul [decidable_eq m]
(d : m → α) (M : matrix m n α) (i j) : (diagonal d).mul M i j = d i * M i j :=
diagonal_dot_product _ _ _
@[simp] theorem mul_diagonal [decidable_eq n]
(d : n → α) (M : matrix m n α) (i j) : (M ⬝ diagonal d) i j = M i j * d j :=
by { rw ← diagonal_transpose, apply dot_product_diagonal }
@[simp] protected theorem one_mul [decidable_eq m] (M : matrix m n α) :
(1 : matrix m m α) ⬝ M = M :=
by ext i j; rw [← diagonal_one, diagonal_mul, one_mul]
@[simp] protected theorem mul_one [decidable_eq n] (M : matrix m n α) :
M ⬝ (1 : matrix n n α) = M :=
by ext i j; rw [← diagonal_one, mul_diagonal, mul_one]
instance [decidable_eq n] : monoid (matrix n n α) :=
{ one_mul := matrix.one_mul,
mul_one := matrix.mul_one,
..matrix.has_one, ..matrix.semigroup }
instance [decidable_eq n] : semiring (matrix n n α) :=
{ mul_zero := matrix.mul_zero,
zero_mul := matrix.zero_mul,
left_distrib := matrix.mul_add,
right_distrib := matrix.add_mul,
..matrix.add_comm_monoid,
..matrix.monoid }
@[simp] theorem diagonal_mul_diagonal [decidable_eq n] (d₁ d₂ : n → α) :
(diagonal d₁) ⬝ (diagonal d₂) = diagonal (λ i, d₁ i * d₂ i) :=
by ext i j; by_cases i = j; simp [h]
theorem diagonal_mul_diagonal' [decidable_eq n] (d₁ d₂ : n → α) :
diagonal d₁ * diagonal d₂ = diagonal (λ i, d₁ i * d₂ i) :=
diagonal_mul_diagonal _ _
@[simp]
lemma map_mul {L : matrix m n α} {M : matrix n o α}
{β : Type w} [semiring β] {f : α →+* β} :
(L ⬝ M).map f = L.map f ⬝ M.map f :=
by { ext, simp [mul_apply, ring_hom.map_sum], }
-- TODO: there should be a way to avoid restating these for each `foo_hom`.
/-- A version of `one_map` where `f` is a ring hom. -/
@[simp] lemma ring_hom_map_one [decidable_eq n]
{β : Type w} [semiring β] (f : α →+* β) :
(1 : matrix n n α).map f = 1 :=
one_map f.map_zero f.map_one
/-- A version of `one_map` where `f` is a `ring_equiv`. -/
@[simp] lemma ring_equiv_map_one [decidable_eq n]
{β : Type w} [semiring β] (f : α ≃+* β) :
(1 : matrix n n α).map f = 1 :=
one_map f.map_zero f.map_one
/-- A version of `map_zero` where `f` is a `zero_hom`. -/
@[simp] lemma zero_hom_map_zero
{β : Type w} [has_zero β] (f : zero_hom α β) :
(0 : matrix n n α).map f = 0 :=
map_zero f.map_zero
/-- A version of `map_zero` where `f` is a `add_monoid_hom`. -/
@[simp] lemma add_monoid_hom_map_zero
{β : Type w} [add_monoid β] (f : α →+ β) :
(0 : matrix n n α).map f = 0 :=
map_zero f.map_zero
/-- A version of `map_zero` where `f` is a `add_equiv`. -/
@[simp] lemma add_equiv_map_zero
{β : Type w} [add_monoid β] (f : α ≃+ β) :
(0 : matrix n n α).map f = 0 :=
map_zero f.map_zero
/-- A version of `map_zero` where `f` is a `linear_map`. -/
@[simp] lemma linear_map_map_zero {R : Type*} [semiring R]
{β : Type w} [add_comm_monoid β] [semimodule R α] [semimodule R β] (f : α →ₗ[R] β) :
(0 : matrix n n α).map f = 0 :=
map_zero f.map_zero
/-- A version of `map_zero` where `f` is a `linear_equiv`. -/
@[simp] lemma linear_equiv_map_zero {R : Type*} [semiring R]
{β : Type w} [add_comm_monoid β] [semimodule R α] [semimodule R β] (f : α ≃ₗ[R] β) :
(0 : matrix n n α).map f = 0 :=
map_zero f.map_zero
/-- A version of `map_zero` where `f` is a `ring_hom`. -/
@[simp] lemma ring_hom_map_zero
{β : Type w} [semiring β] (f : α →+* β) :
(0 : matrix n n α).map f = 0 :=
map_zero f.map_zero
/-- A version of `map_zero` where `f` is a `ring_equiv`. -/
@[simp] lemma ring_equiv_map_zero
{β : Type w} [semiring β] (f : α ≃+* β) :
(0 : matrix n n α).map f = 0 :=
map_zero f.map_zero
lemma is_add_monoid_hom_mul_left (M : matrix l m α) :
is_add_monoid_hom (λ x : matrix m n α, M ⬝ x) :=
{ to_is_add_hom := ⟨matrix.mul_add _⟩, map_zero := matrix.mul_zero _ }
lemma is_add_monoid_hom_mul_right (M : matrix m n α) :
is_add_monoid_hom (λ x : matrix l m α, x ⬝ M) :=
{ to_is_add_hom := ⟨λ _ _, matrix.add_mul _ _ _⟩, map_zero := matrix.zero_mul _ }
protected lemma sum_mul {β : Type*} (s : finset β) (f : β → matrix l m α)
(M : matrix m n α) : (∑ a in s, f a) ⬝ M = ∑ a in s, f a ⬝ M :=
(@finset.sum_hom _ _ _ _ _ s f (λ x, x ⬝ M)
/- This line does not type-check without `id` and `: _`. Lean did not recognize that two different
`add_monoid` instances were def-eq -/
(id (@is_add_monoid_hom_mul_right l _ _ _ _ _ _ _ M) : _)).symm
protected lemma mul_sum {β : Type*} (s : finset β) (f : β → matrix m n α)
(M : matrix l m α) : M ⬝ ∑ a in s, f a = ∑ a in s, M ⬝ f a :=
(@finset.sum_hom _ _ _ _ _ s f (λ x, M ⬝ x)
/- This line does not type-check without `id` and `: _`. Lean did not recognize that two different
`add_monoid` instances were def-eq -/
(id (@is_add_monoid_hom_mul_left _ _ n _ _ _ _ _ M) : _)).symm
@[simp]
lemma row_mul_col_apply (v w : m → α) (i j) : (row v ⬝ col w) i j = dot_product v w :=
rfl
end semiring
end matrix
/-- The `ring_hom` between spaces of square matrices induced by a `ring_hom` between their
coefficients. -/
def ring_hom.map_matrix [decidable_eq m] [semiring α] {β : Type w} [semiring β] (f : α →+* β) :
matrix m m α →+* matrix m m β :=
{ to_fun := λ M, M.map f,
map_one' := by simp,
map_mul' := λ L M, matrix.map_mul,
..(f.to_add_monoid_hom).map_matrix }
@[simp] lemma ring_hom.map_matrix_apply [decidable_eq m] [semiring α] {β : Type w} [semiring β]
(f : α →+* β) (M : matrix m m α) : f.map_matrix M = M.map f := rfl
open_locale matrix
namespace matrix
section ring
variables [ring α]
@[simp] theorem neg_mul (M : matrix m n α) (N : matrix n o α) :
(-M) ⬝ N = -(M ⬝ N) :=
by { ext, apply neg_dot_product }
@[simp] theorem mul_neg (M : matrix m n α) (N : matrix n o α) :
M ⬝ (-N) = -(M ⬝ N) :=
by { ext, apply dot_product_neg }
protected theorem sub_mul (M M' : matrix m n α) (N : matrix n o α) :
(M - M') ⬝ N = M ⬝ N - M' ⬝ N :=
by rw [sub_eq_add_neg, matrix.add_mul, neg_mul, sub_eq_add_neg]
protected theorem mul_sub (M : matrix m n α) (N N' : matrix n o α) :
M ⬝ (N - N') = M ⬝ N - M ⬝ N' :=
by rw [sub_eq_add_neg, matrix.mul_add, mul_neg, sub_eq_add_neg]
end ring
instance [decidable_eq n] [ring α] : ring (matrix n n α) :=
{ ..matrix.semiring, ..matrix.add_comm_group }
instance [semiring α] : has_scalar α (matrix m n α) := pi.has_scalar
instance {β : Type w} [semiring α] [add_comm_monoid β] [semimodule α β] :
semimodule α (matrix m n β) := pi.semimodule _ _ _
@[simp] lemma smul_apply [semiring α] (a : α) (A : matrix m n α) (i : m) (j : n) :
(a • A) i j = a * A i j := rfl
section semiring
variables [semiring α]
lemma smul_eq_diagonal_mul [decidable_eq m] (M : matrix m n α) (a : α) :
a • M = diagonal (λ _, a) ⬝ M :=
by { ext, simp }
@[simp] lemma smul_mul (M : matrix m n α) (a : α) (N : matrix n l α) : (a • M) ⬝ N = a • M ⬝ N :=
by { ext, apply smul_dot_product }
@[simp] lemma mul_mul_left (M : matrix m n α) (N : matrix n o α) (a : α) :
(λ i j, a * M i j) ⬝ N = a • (M ⬝ N) :=
begin
simp only [←smul_apply],
simp,
end
/--
The ring homomorphism `α →+* matrix n n α`
sending `a` to the diagonal matrix with `a` on the diagonal.
-/
def scalar (n : Type u) [decidable_eq n] [fintype n] : α →+* matrix n n α :=
{ to_fun := λ a, a • 1,
map_zero' := by simp,
map_add' := by { intros, ext, simp [add_mul], },
map_one' := by simp,
map_mul' := by { intros, ext, simp [mul_assoc], }, }
section scalar
variable [decidable_eq n]
@[simp] lemma coe_scalar : (scalar n : α → matrix n n α) = λ a, a • 1 := rfl
lemma scalar_apply_eq (a : α) (i : n) :
scalar n a i i = a :=
by simp only [coe_scalar, mul_one, one_apply_eq, smul_apply]
lemma scalar_apply_ne (a : α) (i j : n) (h : i ≠ j) :
scalar n a i j = 0 :=
by simp only [h, coe_scalar, one_apply_ne, ne.def, not_false_iff, smul_apply, mul_zero]
lemma scalar_inj [nonempty n] {r s : α} : scalar n r = scalar n s ↔ r = s :=
begin
split,
{ intro h,
inhabit n,
rw [← scalar_apply_eq r (arbitrary n), ← scalar_apply_eq s (arbitrary n), h] },
{ rintro rfl, refl }
end
end scalar
end semiring
section comm_semiring
variables [comm_semiring α]
lemma smul_eq_mul_diagonal [decidable_eq n] (M : matrix m n α) (a : α) :
a • M = M ⬝ diagonal (λ _, a) :=
by { ext, simp [mul_comm] }
@[simp] lemma mul_smul (M : matrix m n α) (a : α) (N : matrix n l α) : M ⬝ (a • N) = a • M ⬝ N :=
by { ext, apply dot_product_smul }
@[simp] lemma mul_mul_right (M : matrix m n α) (N : matrix n o α) (a : α) :
M ⬝ (λ i j, a * N i j) = a • (M ⬝ N) :=
begin
simp only [←smul_apply],
simp,
end
lemma scalar.commute [decidable_eq n] (r : α) (M : matrix n n α) : commute (scalar n r) M :=
by simp [commute, semiconj_by]
end comm_semiring
section semiring
variables [semiring α]
/-- For two vectors `w` and `v`, `vec_mul_vec w v i j` is defined to be `w i * v j`.
Put another way, `vec_mul_vec w v` is exactly `col w ⬝ row v`. -/
def vec_mul_vec (w : m → α) (v : n → α) : matrix m n α
| x y := w x * v y
/-- `mul_vec M v` is the matrix-vector product of `M` and `v`, where `v` is seen as a column matrix.
Put another way, `mul_vec M v` is the vector whose entries
are those of `M ⬝ col v` (see `col_mul_vec`). -/
def mul_vec (M : matrix m n α) (v : n → α) : m → α
| i := dot_product (λ j, M i j) v
/-- `vec_mul v M` is the vector-matrix product of `v` and `M`, where `v` is seen as a row matrix.
Put another way, `vec_mul v M` is the vector whose entries
are those of `row v ⬝ M` (see `row_vec_mul`). -/
def vec_mul (v : m → α) (M : matrix m n α) : n → α
| j := dot_product v (λ i, M i j)
instance mul_vec.is_add_monoid_hom_left (v : n → α) :
is_add_monoid_hom (λM:matrix m n α, mul_vec M v) :=
{ map_zero := by ext; simp [mul_vec]; refl,
map_add :=
begin
intros x y,
ext m,
apply add_dot_product
end }
lemma mul_vec_diagonal [decidable_eq m] (v w : m → α) (x : m) :
mul_vec (diagonal v) w x = v x * w x :=
diagonal_dot_product v w x
lemma vec_mul_diagonal [decidable_eq m] (v w : m → α) (x : m) :
vec_mul v (diagonal w) x = v x * w x :=
dot_product_diagonal' v w x
@[simp] lemma mul_vec_one [decidable_eq m] (v : m → α) : mul_vec 1 v = v :=
by { ext, rw [←diagonal_one, mul_vec_diagonal, one_mul] }
@[simp] lemma vec_mul_one [decidable_eq m] (v : m → α) : vec_mul v 1 = v :=
by { ext, rw [←diagonal_one, vec_mul_diagonal, mul_one] }
@[simp] lemma mul_vec_zero (A : matrix m n α) : mul_vec A 0 = 0 :=
by { ext, simp [mul_vec] }
@[simp] lemma vec_mul_zero (A : matrix m n α) : vec_mul 0 A = 0 :=
by { ext, simp [vec_mul] }
@[simp] lemma vec_mul_vec_mul (v : m → α) (M : matrix m n α) (N : matrix n o α) :
vec_mul (vec_mul v M) N = vec_mul v (M ⬝ N) :=
by { ext, apply dot_product_assoc }
@[simp] lemma mul_vec_mul_vec (v : o → α) (M : matrix m n α) (N : matrix n o α) :
mul_vec M (mul_vec N v) = mul_vec (M ⬝ N) v :=
by { ext, symmetry, apply dot_product_assoc }
lemma vec_mul_vec_eq (w : m → α) (v : n → α) :
vec_mul_vec w v = (col w) ⬝ (row v) :=
by { ext i j, simp [vec_mul_vec, mul_apply], refl }
variables [decidable_eq m] [decidable_eq n]
/--
`std_basis_matrix i j a` is the matrix with `a` in the `i`-th row, `j`-th column,
and zeroes elsewhere.
-/
def std_basis_matrix (i : m) (j : n) (a : α) : matrix m n α :=
(λ i' j', if i' = i ∧ j' = j then a else 0)
@[simp] lemma smul_std_basis_matrix (i : m) (j : n) (a b : α) :
b • std_basis_matrix i j a = std_basis_matrix i j (b • a) :=
by { unfold std_basis_matrix, ext, simp }
@[simp] lemma std_basis_matrix_zero (i : m) (j : n) :
std_basis_matrix i j (0 : α) = 0 :=
by { unfold std_basis_matrix, ext, simp }
lemma std_basis_matrix_add (i : m) (j : n) (a b : α) :
std_basis_matrix i j (a + b) = std_basis_matrix i j a + std_basis_matrix i j b :=
begin
unfold std_basis_matrix, ext,
split_ifs with h; simp [h],
end
lemma matrix_eq_sum_std_basis (x : matrix n m α) :
x = ∑ (i : n) (j : m), std_basis_matrix i j (x i j) :=
begin
ext, symmetry,
iterate 2 { rw finset.sum_apply },
convert fintype.sum_eq_single i _,
{ simp [std_basis_matrix] },
{ intros j hj,
simp [std_basis_matrix, hj.symm] }
end
-- TODO: tie this up with the `basis` machinery of linear algebra
-- this is not completely trivial because we are indexing by two types, instead of one
-- TODO: add `std_basis_vec`
lemma std_basis_eq_basis_mul_basis (i : m) (j : n) :
std_basis_matrix i j 1 = vec_mul_vec (λ i', ite (i = i') 1 0) (λ j', ite (j = j') 1 0) :=
begin
ext, norm_num [std_basis_matrix, vec_mul_vec],
split_ifs; tauto,
end
@[elab_as_eliminator] protected lemma induction_on'
{X : Type*} [semiring X] {M : matrix n n X → Prop} (m : matrix n n X)
(h_zero : M 0)
(h_add : ∀p q, M p → M q → M (p + q))
(h_std_basis : ∀ i j x, M (std_basis_matrix i j x)) :
M m :=
begin
rw [matrix_eq_sum_std_basis m, ← finset.sum_product'],
apply finset.sum_induction _ _ h_add h_zero,
{ intros, apply h_std_basis, }
end
@[elab_as_eliminator] protected lemma induction_on
[nonempty n] {X : Type*} [semiring X] {M : matrix n n X → Prop} (m : matrix n n X)
(h_add : ∀p q, M p → M q → M (p + q))
(h_std_basis : ∀ i j x, M (std_basis_matrix i j x)) :
M m :=
matrix.induction_on' m
begin
have i : n := classical.choice (by assumption),
simpa using h_std_basis i i 0,
end
h_add h_std_basis
end semiring
section ring
variables [ring α]
lemma neg_vec_mul (v : m → α) (A : matrix m n α) : vec_mul (-v) A = - vec_mul v A :=
by { ext, apply neg_dot_product }
lemma vec_mul_neg (v : m → α) (A : matrix m n α) : vec_mul v (-A) = - vec_mul v A :=
by { ext, apply dot_product_neg }
lemma neg_mul_vec (v : n → α) (A : matrix m n α) : mul_vec (-A) v = - mul_vec A v :=
by { ext, apply neg_dot_product }
lemma mul_vec_neg (v : n → α) (A : matrix m n α) : mul_vec A (-v) = - mul_vec A v :=
by { ext, apply dot_product_neg }
lemma smul_mul_vec_assoc (A : matrix n n α) (b : n → α) (a : α) :
(a • A).mul_vec b = a • (A.mul_vec b) :=
begin
ext i, change dot_product ((a • A) i) b = _,
simp only [mul_vec, smul_eq_mul, pi.smul_apply, smul_dot_product],
end
end ring
section transpose
open_locale matrix
/--
Tell `simp` what the entries are in a transposed matrix.
Compare with `mul_apply`, `diagonal_apply_eq`, etc.
-/
@[simp] lemma transpose_apply (M : matrix m n α) (i j) : M.transpose j i = M i j := rfl
@[simp] lemma transpose_transpose (M : matrix m n α) :
Mᵀᵀ = M :=
by ext; refl
@[simp] lemma transpose_zero [has_zero α] : (0 : matrix m n α)ᵀ = 0 :=
by ext i j; refl
@[simp] lemma transpose_one [decidable_eq n] [has_zero α] [has_one α] : (1 : matrix n n α)ᵀ = 1 :=
begin
ext i j,
unfold has_one.one transpose,
by_cases i = j,
{ simp only [h, diagonal_apply_eq] },
{ simp only [diagonal_apply_ne h, diagonal_apply_ne (λ p, h (symm p))] }
end
@[simp] lemma transpose_add [has_add α] (M : matrix m n α) (N : matrix m n α) :
(M + N)ᵀ = Mᵀ + Nᵀ :=
by { ext i j, simp }
@[simp] lemma transpose_sub [add_group α] (M : matrix m n α) (N : matrix m n α) :
(M - N)ᵀ = Mᵀ - Nᵀ :=
by { ext i j, simp }
@[simp] lemma transpose_mul [comm_semiring α] (M : matrix m n α) (N : matrix n l α) :
(M ⬝ N)ᵀ = Nᵀ ⬝ Mᵀ :=
begin
ext i j,
apply dot_product_comm
end
@[simp] lemma transpose_smul [semiring α] (c : α) (M : matrix m n α) :
(c • M)ᵀ = c • Mᵀ :=
by { ext i j, refl }
@[simp] lemma transpose_neg [has_neg α] (M : matrix m n α) :
(- M)ᵀ = - Mᵀ :=
by ext i j; refl
lemma transpose_map {β : Type w} {f : α → β} {M : matrix m n α} : Mᵀ.map f = (M.map f)ᵀ :=
by { ext, refl }
end transpose
section star_ring
variables [decidable_eq n] {R : Type*} [semiring R] [star_ring R]
/--
When `R` is a `*`-(semi)ring, `matrix n n R` becomes a `*`-(semi)ring with
the star operation given by taking the conjugate, and the star of each entry.
-/
instance : star_ring (matrix n n R) :=
{ star := λ M, M.transpose.map star,
star_involutive := λ M, by { ext, simp, },
star_add := λ M N, by { ext, simp, },
star_mul := λ M N, by { ext, simp [mul_apply], }, }
@[simp] lemma star_apply (M : matrix n n R) (i j) : star M i j = star (M j i) := rfl
lemma star_mul (M N : matrix n n R) : star (M ⬝ N) = star N ⬝ star M := star_mul _ _
end star_ring
/-- `M.minor row col` is the matrix obtained by reindexing the rows and the lines of
`M`, such that `M.minor row col i j = M (row i) (col j)`. Note that the total number
of row/colums doesn't have to be preserved. -/
def minor (A : matrix m n α) (row : l → m) (col : o → n) : matrix l o α :=
λ i j, A (row i) (col j)
/-- The left `n × l` part of a `n × (l+r)` matrix. -/
@[reducible]
def sub_left {m l r : nat} (A : matrix (fin m) (fin (l + r)) α) : matrix (fin m) (fin l) α :=
minor A id (fin.cast_add r)
/-- The right `n × r` part of a `n × (l+r)` matrix. -/
@[reducible]
def sub_right {m l r : nat} (A : matrix (fin m) (fin (l + r)) α) : matrix (fin m) (fin r) α :=
minor A id (fin.nat_add l)
/-- The top `u × n` part of a `(u+d) × n` matrix. -/
@[reducible]
def sub_up {d u n : nat} (A : matrix (fin (u + d)) (fin n) α) : matrix (fin u) (fin n) α :=
minor A (fin.cast_add d) id
/-- The bottom `d × n` part of a `(u+d) × n` matrix. -/
@[reducible]
def sub_down {d u n : nat} (A : matrix (fin (u + d)) (fin n) α) : matrix (fin d) (fin n) α :=
minor A (fin.nat_add u) id
/-- The top-right `u × r` part of a `(u+d) × (l+r)` matrix. -/
@[reducible]
def sub_up_right {d u l r : nat} (A: matrix (fin (u + d)) (fin (l + r)) α) :
matrix (fin u) (fin r) α :=
sub_up (sub_right A)
/-- The bottom-right `d × r` part of a `(u+d) × (l+r)` matrix. -/
@[reducible]
def sub_down_right {d u l r : nat} (A : matrix (fin (u + d)) (fin (l + r)) α) :
matrix (fin d) (fin r) α :=
sub_down (sub_right A)
/-- The top-left `u × l` part of a `(u+d) × (l+r)` matrix. -/
@[reducible]
def sub_up_left {d u l r : nat} (A : matrix (fin (u + d)) (fin (l + r)) α) :
matrix (fin u) (fin (l)) α :=
sub_up (sub_left A)
/-- The bottom-left `d × l` part of a `(u+d) × (l+r)` matrix. -/
@[reducible]
def sub_down_left {d u l r : nat} (A: matrix (fin (u + d)) (fin (l + r)) α) :
matrix (fin d) (fin (l)) α :=
sub_down (sub_left A)
section row_col
/-!
### `row_col` section
Simplification lemmas for `matrix.row` and `matrix.col`.
-/
open_locale matrix
@[simp] lemma col_add [semiring α] (v w : m → α) : col (v + w) = col v + col w := by { ext, refl }
@[simp] lemma col_smul [semiring α] (x : α) (v : m → α) : col (x • v) = x • col v :=
by { ext, refl }
@[simp] lemma row_add [semiring α] (v w : m → α) : row (v + w) = row v + row w := by { ext, refl }
@[simp] lemma row_smul [semiring α] (x : α) (v : m → α) : row (x • v) = x • row v :=
by { ext, refl }
@[simp] lemma col_apply (v : m → α) (i j) : matrix.col v i j = v i := rfl
@[simp] lemma row_apply (v : m → α) (i j) : matrix.row v i j = v j := rfl
@[simp]
lemma transpose_col (v : m → α) : (matrix.col v).transpose = matrix.row v := by {ext, refl}
@[simp]
lemma transpose_row (v : m → α) : (matrix.row v).transpose = matrix.col v := by {ext, refl}
lemma row_vec_mul [semiring α] (M : matrix m n α) (v : m → α) :
matrix.row (matrix.vec_mul v M) = matrix.row v ⬝ M := by {ext, refl}
lemma col_vec_mul [semiring α] (M : matrix m n α) (v : m → α) :
matrix.col (matrix.vec_mul v M) = (matrix.row v ⬝ M)ᵀ := by {ext, refl}
lemma col_mul_vec [semiring α] (M : matrix m n α) (v : n → α) :
matrix.col (matrix.mul_vec M v) = M ⬝ matrix.col v := by {ext, refl}
lemma row_mul_vec [semiring α] (M : matrix m n α) (v : n → α) :
matrix.row (matrix.mul_vec M v) = (M ⬝ matrix.col v)ᵀ := by {ext, refl}
end row_col
section update
/-- Update, i.e. replace the `i`th row of matrix `A` with the values in `b`. -/
def update_row [decidable_eq n] (M : matrix n m α) (i : n) (b : m → α) : matrix n m α :=
function.update M i b
/-- Update, i.e. replace the `j`th column of matrix `A` with the values in `b`. -/
def update_column [decidable_eq m] (M : matrix n m α) (j : m) (b : n → α) : matrix n m α :=
λ i, function.update (M i) j (b i)
variables {M : matrix n m α} {i : n} {j : m} {b : m → α} {c : n → α}
@[simp] lemma update_row_self [decidable_eq n] : update_row M i b i = b :=
function.update_same i b M
@[simp] lemma update_column_self [decidable_eq m] : update_column M j c i j = c i :=
function.update_same j (c i) (M i)
@[simp] lemma update_row_ne [decidable_eq n] {i' : n} (i_ne : i' ≠ i) :
update_row M i b i' = M i' := function.update_noteq i_ne b M
@[simp] lemma update_column_ne [decidable_eq m] {j' : m} (j_ne : j' ≠ j) :
update_column M j c i j' = M i j' := function.update_noteq j_ne (c i) (M i)
lemma update_row_apply [decidable_eq n] {i' : n} :
update_row M i b i' j = if i' = i then b j else M i' j :=
begin
by_cases i' = i,
{ rw [h, update_row_self, if_pos rfl] },
{ rwa [update_row_ne h, if_neg h] }
end
lemma update_column_apply [decidable_eq m] {j' : m} :
update_column M j c i j' = if j' = j then c i else M i j' :=
begin
by_cases j' = j,
{ rw [h, update_column_self, if_pos rfl] },
{ rwa [update_column_ne h, if_neg h] }
end
lemma update_row_transpose [decidable_eq m] : update_row Mᵀ j c = (update_column M j c)ᵀ :=
begin
ext i' j,
rw [transpose_apply, update_row_apply, update_column_apply],
refl
end
lemma update_column_transpose [decidable_eq n] : update_column Mᵀ i b = (update_row M i b)ᵀ :=
begin
ext i' j,
rw [transpose_apply, update_row_apply, update_column_apply],
refl
end
end update
section block_matrices
/-- We can form a single large matrix by flattening smaller 'block' matrices of compatible
dimensions. -/
def from_blocks (A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α) :
matrix (n ⊕ o) (l ⊕ m) α :=
sum.elim (λ i, sum.elim (A i) (B i))
(λ i, sum.elim (C i) (D i))
@[simp] lemma from_blocks_apply₁₁
(A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α) (i : n) (j : l) :
from_blocks A B C D (sum.inl i) (sum.inl j) = A i j :=
rfl
@[simp] lemma from_blocks_apply₁₂
(A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α) (i : n) (j : m) :
from_blocks A B C D (sum.inl i) (sum.inr j) = B i j :=
rfl
@[simp] lemma from_blocks_apply₂₁
(A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α) (i : o) (j : l) :
from_blocks A B C D (sum.inr i) (sum.inl j) = C i j :=
rfl
@[simp] lemma from_blocks_apply₂₂
(A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α) (i : o) (j : m) :
from_blocks A B C D (sum.inr i) (sum.inr j) = D i j :=
rfl
/-- Given a matrix whose row and column indexes are sum types, we can extract the correspnding
"top left" submatrix. -/
def to_blocks₁₁ (M : matrix (n ⊕ o) (l ⊕ m) α) : matrix n l α :=
λ i j, M (sum.inl i) (sum.inl j)
/-- Given a matrix whose row and column indexes are sum types, we can extract the correspnding
"top right" submatrix. -/
def to_blocks₁₂ (M : matrix (n ⊕ o) (l ⊕ m) α) : matrix n m α :=
λ i j, M (sum.inl i) (sum.inr j)
/-- Given a matrix whose row and column indexes are sum types, we can extract the correspnding
"bottom left" submatrix. -/
def to_blocks₂₁ (M : matrix (n ⊕ o) (l ⊕ m) α) : matrix o l α :=
λ i j, M (sum.inr i) (sum.inl j)
/-- Given a matrix whose row and column indexes are sum types, we can extract the correspnding
"bottom right" submatrix. -/
def to_blocks₂₂ (M : matrix (n ⊕ o) (l ⊕ m) α) : matrix o m α :=
λ i j, M (sum.inr i) (sum.inr j)
lemma from_blocks_to_blocks (M : matrix (n ⊕ o) (l ⊕ m) α) :
from_blocks M.to_blocks₁₁ M.to_blocks₁₂ M.to_blocks₂₁ M.to_blocks₂₂ = M :=
begin
ext i j, rcases i; rcases j; refl,
end
@[simp] lemma to_blocks_from_blocks₁₁
(A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α) :
(from_blocks A B C D).to_blocks₁₁ = A :=
rfl
@[simp] lemma to_blocks_from_blocks₁₂
(A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α) :
(from_blocks A B C D).to_blocks₁₂ = B :=
rfl
@[simp] lemma to_blocks_from_blocks₂₁
(A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α) :
(from_blocks A B C D).to_blocks₂₁ = C :=
rfl
@[simp] lemma to_blocks_from_blocks₂₂
(A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α) :
(from_blocks A B C D).to_blocks₂₂ = D :=
rfl
lemma from_blocks_transpose
(A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α) :
(from_blocks A B C D)ᵀ = from_blocks Aᵀ Cᵀ Bᵀ Dᵀ :=
begin
ext i j, rcases i; rcases j; simp [from_blocks],
end
variables [semiring α]
lemma from_blocks_smul
(x : α) (A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α) :
x • (from_blocks A B C D) = from_blocks (x • A) (x • B) (x • C) (x • D) :=
begin
ext i j, rcases i; rcases j; simp [from_blocks],
end
lemma from_blocks_add
(A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α)
(A' : matrix n l α) (B' : matrix n m α) (C' : matrix o l α) (D' : matrix o m α) :
(from_blocks A B C D) + (from_blocks A' B' C' D') =
from_blocks (A + A') (B + B')
(C + C') (D + D') :=
begin
ext i j, rcases i; rcases j; refl,
end
lemma from_blocks_multiply {p q : Type*} [fintype p] [fintype q]
(A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α)
(A' : matrix l p α) (B' : matrix l q α) (C' : matrix m p α) (D' : matrix m q α) :
(from_blocks A B C D) ⬝ (from_blocks A' B' C' D') =
from_blocks (A ⬝ A' + B ⬝ C') (A ⬝ B' + B ⬝ D')
(C ⬝ A' + D ⬝ C') (C ⬝ B' + D ⬝ D') :=
begin
ext i j, rcases i; rcases j;
simp only [from_blocks, mul_apply, fintype.sum_sum_type, sum.elim_inl, sum.elim_inr,
pi.add_apply],
end
variables [decidable_eq l] [decidable_eq m]
@[simp] lemma from_blocks_diagonal (d₁ : l → α) (d₂ : m → α) :
from_blocks (diagonal d₁) 0 0 (diagonal d₂) = diagonal (sum.elim d₁ d₂) :=
begin
ext i j, rcases i; rcases j; simp [diagonal],
end
@[simp] lemma from_blocks_one : from_blocks (1 : matrix l l α) 0 0 (1 : matrix m m α) = 1 :=
by { ext i j, rcases i; rcases j; simp [one_apply] }
end block_matrices
section block_diagonal
variables (M N : o → matrix m n α) [decidable_eq o]
section has_zero
variables [has_zero α]
/-- `matrix.block_diagonal M` turns `M : o → matrix m n α'` into a
`m × o`-by`n × o` block matrix which has the entries of `M` along the diagonal
and zero elsewhere. -/
def block_diagonal : matrix (m × o) (n × o) α
| ⟨i, k⟩ ⟨j, k'⟩ := if k = k' then M k i j else 0
lemma block_diagonal_apply (ik jk) :
block_diagonal M ik jk = if ik.2 = jk.2 then M ik.2 ik.1 jk.1 else 0 :=
by { cases ik, cases jk, refl }
@[simp]
lemma block_diagonal_apply_eq (i j k) :
block_diagonal M (i, k) (j, k) = M k i j :=
if_pos rfl
lemma block_diagonal_apply_ne (i j) {k k'} (h : k ≠ k') :
block_diagonal M (i, k) (j, k') = 0 :=
if_neg h
@[simp] lemma block_diagonal_transpose :
(block_diagonal M)ᵀ = (block_diagonal (λ k, (M k)ᵀ)) :=
begin
ext,
simp only [transpose_apply, block_diagonal_apply, eq_comm],
split_ifs with h,
{ rw h },
{ refl }
end
@[simp] lemma block_diagonal_zero :
block_diagonal (0 : o → matrix m n α) = 0 :=
by { ext, simp [block_diagonal_apply] }
@[simp] lemma block_diagonal_diagonal [decidable_eq m] (d : o → m → α) :
(block_diagonal (λ k, diagonal (d k))) = diagonal (λ ik, d ik.2 ik.1) :=
begin
ext ⟨i, k⟩ ⟨j, k'⟩,
simp only [block_diagonal_apply, diagonal],
split_ifs; finish
end
@[simp] lemma block_diagonal_one [decidable_eq m] [has_one α] :
(block_diagonal (1 : o → matrix m m α)) = 1 :=
show (block_diagonal (λ (_ : o), diagonal (λ (_ : m), (1 : α)))) = diagonal (λ _, 1),
by rw [block_diagonal_diagonal]
end has_zero
@[simp] lemma block_diagonal_add [add_monoid α] :
block_diagonal (M + N) = block_diagonal M + block_diagonal N :=
begin
ext,
simp only [block_diagonal_apply, add_apply],
split_ifs; simp
end
@[simp] lemma block_diagonal_neg [add_group α] :
block_diagonal (-M) = - block_diagonal M :=
begin
ext,
simp only [block_diagonal_apply, neg_apply],
split_ifs; simp
end
@[simp] lemma block_diagonal_sub [add_group α] :
block_diagonal (M - N) = block_diagonal M - block_diagonal N :=
by simp [sub_eq_add_neg]
@[simp] lemma block_diagonal_mul {p : Type*} [fintype p] [semiring α]
(N : o → matrix n p α) : block_diagonal (λ k, M k ⬝ N k) = block_diagonal M ⬝ block_diagonal N :=
begin
ext ⟨i, k⟩ ⟨j, k'⟩,
simp only [block_diagonal_apply, mul_apply, ← finset.univ_product_univ, finset.sum_product],
split_ifs with h; simp [h]
end
@[simp] lemma block_diagonal_smul {R : Type*} [semiring R] [add_comm_monoid α] [semimodule R α]
(x : R) : block_diagonal (x • M) = x • block_diagonal M :=
by { ext, simp only [block_diagonal_apply, pi.smul_apply, smul_apply], split_ifs; simp }
end block_diagonal
end matrix
namespace ring_hom
variables {β : Type*} [semiring α] [semiring β]
lemma map_matrix_mul (M : matrix m n α) (N : matrix n o α) (i : m) (j : o) (f : α →+* β) :
f (matrix.mul M N i j) = matrix.mul (λ i j, f (M i j)) (λ i j, f (N i j)) i j :=
by simp [matrix.mul_apply, ring_hom.map_sum]
end ring_hom
|
5804c3b4a66b1ec06d45dc58ad5f859edd4e25aa | f57749ca63d6416f807b770f67559503fdb21001 | /library/init/relation.lean | 672e4d0f088cac6a60afcdbba721f0cd845e6f2b | [
"Apache-2.0"
] | permissive | aliassaf/lean | bd54e85bed07b1ff6f01396551867b2677cbc6ac | f9b069b6a50756588b309b3d716c447004203152 | refs/heads/master | 1,610,982,152,948 | 1,438,916,029,000 | 1,438,916,029,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 1,659 | lean | /-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Module init.relation
Authors: Leonardo de Moura
-/
prelude
import init.logic
-- TODO(Leo): remove duplication between this file and algebra/relation.lean
-- We need some of the following definitions asap when "initializing" Lean.
variables {A B : Type} (R : B → B → Prop)
local infix `≺`:50 := R
definition reflexive := ∀x, x ≺ x
definition symmetric := ∀⦃x y⦄, x ≺ y → y ≺ x
definition transitive := ∀⦃x y z⦄, x ≺ y → y ≺ z → x ≺ z
definition equivalence := reflexive R ∧ symmetric R ∧ transitive R
definition mk_equivalence (r : reflexive R) (s : symmetric R) (t : transitive R) : equivalence R :=
and.intro r (and.intro s t)
definition irreflexive := ∀x, ¬ x ≺ x
definition anti_symmetric := ∀⦃x y⦄, x ≺ y → y ≺ x → x = y
definition empty_relation := λa₁ a₂ : A, false
definition subrelation (Q R : B → B → Prop) := ∀⦃x y⦄, Q x y → R x y
definition inv_image (f : A → B) : A → A → Prop :=
λa₁ a₂, f a₁ ≺ f a₂
theorem inv_image.trans (f : A → B) (H : transitive R) : transitive (inv_image R f) :=
λ (a₁ a₂ a₃ : A) (H₁ : inv_image R f a₁ a₂) (H₂ : inv_image R f a₂ a₃), H H₁ H₂
theorem inv_image.irreflexive (f : A → B) (H : irreflexive R) : irreflexive (inv_image R f) :=
λ (a : A) (H₁ : inv_image R f a a), H (f a) H₁
inductive tc {A : Type} (R : A → A → Prop) : A → A → Prop :=
| base : ∀a b, R a b → tc R a b
| trans : ∀a b c, tc R a b → tc R b c → tc R a c
|
17cfa2d4541110a71efad8f6394ca71b439f3559 | 8cae430f0a71442d02dbb1cbb14073b31048e4b0 | /src/ring_theory/localization/num_denom.lean | 1b1ff85f10758528578432c160c2c74d415ab07a | [
"Apache-2.0"
] | permissive | leanprover-community/mathlib | 56a2cadd17ac88caf4ece0a775932fa26327ba0e | 442a83d738cb208d3600056c489be16900ba701d | refs/heads/master | 1,693,584,102,358 | 1,693,471,902,000 | 1,693,471,902,000 | 97,922,418 | 1,595 | 352 | Apache-2.0 | 1,694,693,445,000 | 1,500,624,130,000 | Lean | UTF-8 | Lean | false | false | 4,080 | lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import ring_theory.localization.fraction_ring
import ring_theory.localization.integer
import ring_theory.unique_factorization_domain
/-!
# Numerator and denominator in a localization
> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
> Any changes to this file require a corresponding PR to mathlib4.
## Implementation notes
See `src/ring_theory/localization/basic.lean` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variables {R : Type*} [comm_ring R] (M : submonoid R) {S : Type*} [comm_ring S]
variables [algebra R S] {P : Type*} [comm_ring P]
namespace is_fraction_ring
open is_localization
section num_denom
variables (A : Type*) [comm_ring A] [is_domain A] [unique_factorization_monoid A]
variables {K : Type*} [field K] [algebra A K] [is_fraction_ring A K]
lemma exists_reduced_fraction (x : K) :
∃ (a : A) (b : non_zero_divisors A),
(∀ {d}, d ∣ a → d ∣ b → is_unit d) ∧ mk' K a b = x :=
begin
obtain ⟨⟨b, b_nonzero⟩, a, hab⟩ := exists_integer_multiple (non_zero_divisors A) x,
obtain ⟨a', b', c', no_factor, rfl, rfl⟩ :=
unique_factorization_monoid.exists_reduced_factors' a b
(mem_non_zero_divisors_iff_ne_zero.mp b_nonzero),
obtain ⟨c'_nonzero, b'_nonzero⟩ := mul_mem_non_zero_divisors.mp b_nonzero,
refine ⟨a', ⟨b', b'_nonzero⟩, @no_factor, _⟩,
refine mul_left_cancel₀
(is_fraction_ring.to_map_ne_zero_of_mem_non_zero_divisors b_nonzero) _,
simp only [subtype.coe_mk, ring_hom.map_mul, algebra.smul_def] at *,
erw [←hab, mul_assoc, mk'_spec' _ a' ⟨b', b'_nonzero⟩],
end
/-- `f.num x` is the numerator of `x : f.codomain` as a reduced fraction. -/
noncomputable def num (x : K) : A :=
classical.some (exists_reduced_fraction A x)
/-- `f.num x` is the denominator of `x : f.codomain` as a reduced fraction. -/
noncomputable def denom (x : K) : non_zero_divisors A :=
classical.some (classical.some_spec (exists_reduced_fraction A x))
lemma num_denom_reduced (x : K) {d} : d ∣ num A x → d ∣ denom A x → is_unit d :=
(classical.some_spec (classical.some_spec (exists_reduced_fraction A x))).1
@[simp] lemma mk'_num_denom (x : K) : mk' K (num A x) (denom A x) = x :=
(classical.some_spec (classical.some_spec (exists_reduced_fraction A x))).2
variables {A}
lemma num_mul_denom_eq_num_iff_eq {x y : K} :
x * algebra_map A K (denom A y) = algebra_map A K (num A y) ↔ x = y :=
⟨λ h, by simpa only [mk'_num_denom] using eq_mk'_iff_mul_eq.mpr h,
λ h, eq_mk'_iff_mul_eq.mp (by rw [h, mk'_num_denom])⟩
lemma num_mul_denom_eq_num_iff_eq' {x y : K} :
y * algebra_map A K (denom A x) = algebra_map A K (num A x) ↔ x = y :=
⟨λ h, by simpa only [eq_comm, mk'_num_denom] using eq_mk'_iff_mul_eq.mpr h,
λ h, eq_mk'_iff_mul_eq.mp (by rw [h, mk'_num_denom])⟩
lemma num_mul_denom_eq_num_mul_denom_iff_eq {x y : K} :
num A y * denom A x = num A x * denom A y ↔ x = y :=
⟨λ h, by simpa only [mk'_num_denom] using mk'_eq_of_eq' h,
λ h, by rw h⟩
lemma eq_zero_of_num_eq_zero {x : K} (h : num A x = 0) : x = 0 :=
num_mul_denom_eq_num_iff_eq'.mp (by rw [zero_mul, h, ring_hom.map_zero])
lemma is_integer_of_is_unit_denom {x : K} (h : is_unit (denom A x : A)) : is_integer A x :=
begin
cases h with d hd,
have d_ne_zero : algebra_map A K (denom A x) ≠ 0 :=
is_fraction_ring.to_map_ne_zero_of_mem_non_zero_divisors (denom A x).2,
use ↑d⁻¹ * num A x,
refine trans _ (mk'_num_denom A x),
rw [map_mul, map_units_inv, hd],
apply mul_left_cancel₀ d_ne_zero,
rw [←mul_assoc, mul_inv_cancel d_ne_zero, one_mul, mk'_spec']
end
lemma is_unit_denom_of_num_eq_zero {x : K} (h : num A x = 0) : is_unit (denom A x : A) :=
num_denom_reduced A x (h.symm ▸ dvd_zero _) dvd_rfl
end num_denom
variables (S)
end is_fraction_ring
|
97d91fe5776437a2ca15e08e4e324bc9a1f0f5b0 | c31182a012eec69da0a1f6c05f42b0f0717d212d | /src/Mbar/basic.lean | e54bc8bb2612b3e0a425be9917036ef809b1854d | [] | no_license | Ja1941/lean-liquid | fbec3ffc7fc67df1b5ca95b7ee225685ab9ffbdc | 8e80ed0cbdf5145d6814e833a674eaf05a1495c1 | refs/heads/master | 1,689,437,983,362 | 1,628,362,719,000 | 1,628,362,719,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 17,736 | lean | import data.real.basic
import algebra.big_operators.basic
import topology.algebra.infinite_sum
import data.finset.basic
import data.equiv.basic
import analysis.normed_space.basic
import analysis.specific_limits
import data.equiv.basic
import Mbar.bounded
import pseudo_normed_group.basic
import for_mathlib.tsum
/-!
## $\overline{\mathcal{M}}_{r'}(S)$
This file contains a definition of `ℳ-bar_{r'}(S) as defined at the very beginning
of section 9 of `Analytic.pdf` (p57), and its action of `T⁻¹`.
## Implementation issues
We model Tℤ[[T]] as functions ℕ → ℤ which vanish at 0.
-/
universe u
noncomputable theory
open_locale big_operators nnreal
section defs
set_option old_structure_cmd true
/-- `Mbar r' S` is the set of power series
`F_s = ∑ a_{s,n}T^n ∈ Tℤ[[T]]` such that `∑_{s,n} |a_{s,n}|r'^n` converges -/
structure Mbar (r' : ℝ≥0) (S : Type u) [fintype S] :=
(to_fun : S → ℕ → ℤ)
(coeff_zero' : ∀ s, to_fun s 0 = 0)
(summable' : ∀ s, summable (λ n, (↑(to_fun s n).nat_abs * r' ^ n)))
end defs
variables {r' : ℝ≥0} {S : Type u} [fintype S]
namespace Mbar
instance has_coe_to_fun : has_coe_to_fun (Mbar r' S) := ⟨_, Mbar.to_fun⟩
@[simp] lemma coe_mk (x h₁ h₂) : ((⟨x, h₁, h₂⟩ : Mbar r' S) : S → ℕ → ℤ) = x := rfl
@[simp] protected lemma coeff_zero (x : Mbar r' S) (s : S) : x s 0 = 0 := x.coeff_zero' s
protected lemma summable (x : Mbar r' S) (s : S) :
summable (λ n, (↑(x s n).nat_abs * r' ^ n)) := x.summable' s
lemma summable_coe_real (x : Mbar r' S) (s : S) :
summable (λ n, ∥x s n∥ * r' ^ n) :=
by simpa only [← nnreal.summable_coe, nnreal.coe_nat_cast, nnreal.coe_nat_abs,
nnreal.coe_mul, nnreal.coe_pow] using x.summable s
@[ext] lemma ext (x y : Mbar r' S) (h : ⇑x = y) : x = y :=
by { cases x, cases y, congr, exact h }
lemma ext_iff (x y : Mbar r' S) : x = y ↔ (x : S → ℕ → ℤ) = y :=
⟨congr_arg _, ext x y⟩
/-- The zero element of `Mbar r' S`, defined as the constant 0-function. -/
def zero : Mbar r' S :=
{ to_fun := 0,
coeff_zero' := λ s, rfl,
summable' := λ s, by simpa using summable_zero }
/-- Addition in `Mbar r' S`, defined as pointwise addition. -/
def add (F : Mbar r' S) (G : Mbar r' S) : Mbar r' S :=
{ to_fun := F + G,
coeff_zero' := λ s, by simp [F.coeff_zero s, G.coeff_zero s],
summable' :=
begin
intro s,
apply nnreal.summable_of_le _ ((F.summable s).add (G.summable s)),
intro n,
rw [← add_mul, ← nat.cast_add],
apply mul_le_mul_right',
rw nat.cast_le,
exact int.nat_abs_add_le _ _
end }
/-- Subtraction in `Mbar r' S`, defined as pointwise subtraction. -/
def sub (F : Mbar r' S) (G : Mbar r' S) : Mbar r' S :=
{ to_fun := F - G,
coeff_zero' := λ s, by simp [F.coeff_zero s, G.coeff_zero s],
summable' :=
begin
intro s,
apply nnreal.summable_of_le _ ((F.summable s).add (G.summable s)),
intro n,
rw [← add_mul, ← nat.cast_add],
apply mul_le_mul_right',
rw nat.cast_le,
exact int.nat_abs_sub_le _ _
end }
/-- Negation in `Mbar r' S`, defined as pointwise negation. -/
def neg (F : Mbar r' S) : Mbar r' S :=
{ to_fun := -F,
coeff_zero' := λ s, by simp [F.coeff_zero s],
summable' :=
begin
intro s,
convert F.summable s using 1,
ext n,
simp only [pi.neg_apply, int.nat_abs_neg]
end }
instance : has_zero (Mbar r' S) := ⟨zero⟩
instance : has_add (Mbar r' S) := ⟨add⟩
instance : has_sub (Mbar r' S) := ⟨sub⟩
instance : has_neg (Mbar r' S) := ⟨neg⟩
instance : inhabited (Mbar r' S) := ⟨0⟩
@[simp] lemma coe_zero : ⇑(0 : Mbar r' S) = 0 := rfl
@[simp] lemma coe_add (F G : Mbar r' S) : ⇑(F + G : Mbar r' S) = F + G := rfl
@[simp] lemma coe_sub (F G : Mbar r' S) : ⇑(F - G : Mbar r' S) = F - G := rfl
@[simp] lemma coe_neg (F : Mbar r' S) : ⇑(-F : Mbar r' S) = -F := rfl
/-- Tailored scalar multiplication by natural numbers. -/
protected def nsmul (N : ℕ) (F : Mbar r' S) : Mbar r' S :=
{ to_fun := λ s n, N • F s n,
coeff_zero' := λ s, by simp only [F.coeff_zero, smul_zero],
summable' := λ s,
begin
simp only [nsmul_eq_mul, int.nat_abs_mul, nat.cast_mul, mul_assoc, int.nat_abs_of_nat,
int.nat_cast_eq_coe_nat, nnreal.coe_nat_abs],
by_cases hN : N = 0,
{ simpa only [hN, nat.cast_zero, zero_mul] using summable_zero },
simp only [← nnreal.summable_coe, nnreal.coe_mul],
rw [← summable_mul_left_iff],
{ simp only [coe_nnnorm, nnreal.coe_nat_abs, nnreal.coe_pow], exact F.summable_coe_real s },
{ exact_mod_cast hN }
end }
/-- Tailored scalar multiplication by integers. -/
protected def gsmul (N : ℤ) (F : Mbar r' S) : Mbar r' S :=
{ to_fun := λ s n, N • F s n,
coeff_zero' := λ s, by simp only [F.coeff_zero, smul_zero],
summable' := λ s,
begin
simp only [gsmul_eq_mul, int.nat_abs_mul, int.cast_id, nat.cast_mul, mul_assoc],
by_cases hN : N.nat_abs = 0,
{ simpa only [hN, nat.cast_zero, zero_mul] using summable_zero },
simp only [← nnreal.summable_coe, nnreal.coe_mul],
rw [← summable_mul_left_iff],
{ simp only [coe_nnnorm, nnreal.coe_nat_abs, nnreal.coe_pow], exact F.summable_coe_real s },
{ exact_mod_cast hN }
end }
.
instance : add_comm_group (Mbar r' S) :=
{ zero := 0, add := (+), sub := has_sub.sub, neg := has_neg.neg,
zero_add := by { intros, ext, simp only [coe_zero, zero_add, coe_add] },
add_zero := by { intros, ext, simp only [coe_zero, add_zero, coe_add] },
add_assoc := by { intros, ext, simp only [add_assoc, coe_add] },
add_left_neg := by { intros, ext,
simp only [coe_add, coe_neg, coe_zero, pi.add_apply, pi.zero_apply, pi.neg_apply, add_left_neg] },
add_comm := by { intros, ext, simp only [coe_add, add_comm] },
sub_eq_add_neg := by { intros, ext, simp only [coe_sub, coe_add, coe_neg, sub_eq_add_neg] },
nsmul := λ n F, F.nsmul n,
nsmul_zero' := λ F, by { ext, simp only [Mbar.nsmul, zero_smul], refl },
nsmul_succ' := λ n F,
begin
ext,
simp only [Mbar.nsmul, nat.succ_eq_add_one, add_smul, one_smul, add_comm, coe_mk, coe_add,
pi.add_apply],
end,
gsmul := λ n F, F.gsmul n,
gsmul_zero' := λ F, by { ext, simp only [Mbar.gsmul, zero_smul], refl },
gsmul_succ' := λ n F,
begin
ext,
simp only [Mbar.gsmul, nat.succ_eq_add_one, algebra.id.smul_eq_mul, coe_mk, pi.add_apply,
int.coe_nat_succ, int.of_nat_eq_coe, coe_add, add_mul, one_mul, add_comm],
end,
gsmul_neg' := λ n F,
begin
ext,
simp only [Mbar.gsmul, algebra.id.smul_eq_mul, coe_mk, pi.neg_apply, int.coe_nat_succ, coe_neg,
add_mul, one_mul, neg_add_rev, int.neg_succ_of_nat_coe, neg_mul_eq_neg_mul_symm, one_mul],
end }
/-- The `coeff s n` is the additive homomorphism that sends `x : Mbar r' S`
to the coefficient `x_{s,n}`. -/
@[simps]
def coeff (s : S) (n : ℕ) : Mbar r' S →+ ℤ :=
{ to_fun := λ x, x s n,
map_zero' := rfl,
map_add' := λ x y, rfl }
/-- The norm of `F : Mbar r' S` as nonnegative real number.
It is defined as `∑ s, ∑' n, (↑(F s n).nat_abs * r' ^ n)`. -/
protected def nnnorm (F : Mbar r' S) : ℝ≥0 := ∑ s, ∑' n, (↑(F s n).nat_abs * r' ^ n)
instance : has_nnnorm (Mbar r' S) := ⟨Mbar.nnnorm⟩
lemma nnnorm_def (F : Mbar r' S) : ∥F∥₊ = ∑ s, ∑' n, (↑(F s n).nat_abs * r' ^ n) := rfl
@[simp] lemma nnnorm_zero : ∥(0 : Mbar r' S)∥₊ = 0 :=
by simp only [nnnorm_def, Mbar.coe_zero, tsum_zero, nat.cast_zero, zero_mul, pi.zero_apply,
zero_le', finset.sum_const_zero, int.nat_abs_zero]
@[simp] lemma nnnorm_neg (F : Mbar r' S) : ∥-F∥₊ = ∥F∥₊ :=
by simp only [nnnorm_def, Mbar.coe_neg, pi.neg_apply, int.nat_abs_neg, int.nat_abs,
coe_neg, set.mem_set_of_eq]
lemma nnnorm_add_le (F₁ F₂ : Mbar r' S) : ∥F₁ + F₂∥₊ ≤ ∥F₁∥₊ + ∥F₂∥₊ :=
begin
simp only [nnnorm_def, ← finset.sum_add_distrib],
refine finset.sum_le_sum _,
rintro s -,
rw ← tsum_add (F₁.summable s) (F₂.summable s),
refine tsum_le_tsum _ ((F₁ + F₂).summable _) ((F₁.summable s).add (F₂.summable s)),
intro n,
dsimp,
rw [← add_mul, ← nat.cast_add],
apply mul_le_mul_right',
rw nat.cast_le,
exact int.nat_abs_add_le _ _
end
lemma nnnorm_sum_le {ι : Type*} (s : finset ι) (F : ι → Mbar r' S) :
∥∑ i in s, F i∥₊ ≤ ∑ i in s, ∥F i∥₊ :=
begin
classical,
apply finset.induction_on s; clear s,
{ simp only [finset.sum_empty, nnnorm_zero] },
intros i s his IH,
simp only [finset.sum_insert his],
exact (nnnorm_add_le _ _).trans (add_le_add le_rfl IH)
end
instance pseudo_normed_group : pseudo_normed_group (Mbar r' S) :=
{ filtration := λ c, {F | ∥F∥₊ ≤ c},
filtration_mono := λ c₁ c₂ h F hF, le_trans (by exact hF) h, -- `by exact`, why??
zero_mem_filtration := λ c, by { dsimp, rw nnnorm_zero, apply zero_le' },
neg_mem_filtration := λ c F hF, by { dsimp, rwa nnnorm_neg },
add_mem_filtration := λ c₁ c₂ F₁ F₂ hF₁ hF₂,
by exact le_trans (nnnorm_add_le _ _) (add_le_add hF₁ hF₂) }
lemma mem_filtration_iff (x : Mbar r' S) (c : ℝ≥0) :
x ∈ pseudo_normed_group.filtration (Mbar r' S) c ↔ ∥x∥₊ ≤ c :=
iff.rfl
/--
The coercion from `Mbar r' S` to functions `S → ℕ → ℤ`.
This is an additive group homomorphism.
-/
def coe_hom : Mbar r' S →+ (S → ℕ → ℤ) :=
add_monoid_hom.mk' coe_fn $ coe_add
@[simp] lemma coe_sum {ι : Type*} (s : finset ι) (F : ι → Mbar r' S) :
⇑(∑ i in s, F i) = ∑ i in s, (F i) :=
show coe_hom (∑ i in s, F i) = ∑ i in s, coe_hom (F i), from add_monoid_hom.map_sum _ _ _
@[simp] lemma coe_gsmul (n : ℤ) (F : Mbar r' S) : ⇑(n • F) = n • F := rfl
@[simp] lemma coe_nsmul (n : ℕ) (F : Mbar r' S) : ⇑(n • F) = n • F := rfl
@[simp] lemma nnnorm_gsmul (N : ℤ) (F : Mbar r' S) : ∥N • F∥₊ = ∥N∥₊ * ∥F∥₊ :=
begin
simp only [nnnorm_def, finset.mul_sum],
apply fintype.sum_congr,
intro s,
apply nnreal.eq,
simp only [nnreal.coe_mul, nnreal.coe_tsum, ← tsum_mul_left, ← mul_assoc,
nnreal.coe_nat_abs, coe_nnnorm, coe_gsmul, pi.smul_apply, int.norm_eq_abs,
← abs_mul, ← int.cast_mul, smul_eq_mul],
end
lemma nnnorm_nsmul (x : Mbar r' S) (N : ℕ) : ∥N • x∥₊ = N • ∥x∥₊ :=
by { rw [← gsmul_coe_nat, nnnorm_gsmul, nsmul_eq_mul, ← nnreal.coe_nat_abs], refl, }
section Tinv
/--
The action of T⁻¹.
-/
def Tinv_aux {R : Type*} [has_zero R] : (ℕ → R) → ℕ → R := λ F n, if n = 0 then 0 else F (n + 1)
@[simp] lemma Tinv_aux_zero {R : Type*} [has_zero R] (f : ℕ → R) : Tinv_aux f 0 = 0 := rfl
@[simp] lemma Tinv_aux_ne_zero {R : Type*} [has_zero R] (f : ℕ → R) (i : ℕ) (hi : i ≠ 0) :
Tinv_aux f i = f (i + 1) :=
if_neg hi
@[simp] lemma Tinv_aux_succ {R : Type*} [has_zero R] (f : ℕ → R) (i : ℕ) :
Tinv_aux f (i + 1) = f (i + 2) :=
if_neg (nat.succ_ne_zero i)
lemma Tinv_aux_summable [h0r : fact (0 < r')] (F : Mbar r' S) (s : S) :
summable (λ n, (↑(Tinv_aux (F s) n).nat_abs * r' ^ n)) :=
begin
have : ∀ n:ℕ, r' ^ n = r' ^ n * r' * r'⁻¹,
{ intro, rw [mul_inv_cancel_right' h0r.out.ne'] },
conv { congr, funext, rw [this, ← mul_assoc] },
apply summable.mul_right,
refine nnreal.summable_of_le _ (nnreal.summable_nat_add _ (F.summable s) 1),
rintro ⟨i⟩,
{ simp only [nat.cast_zero, zero_mul, Tinv_aux_zero, zero_le', int.nat_abs_zero] },
{ simp only [Tinv_aux_succ, pow_add, mul_assoc, pow_one] }
end
/--
The `T⁻¹` action on `Mbar r' S`.
This is defined, essentially, as a shift in `ℕ` (accounting for the restriction at 0).
This is an additive group homomorphism.
-/
def Tinv {r : ℝ≥0} {S : Type u} [fintype S] [h0r : fact (0 < r)] :
Mbar r S →+ Mbar r S :=
add_monoid_hom.mk' (λ F,
{ to_fun := λ s, Tinv_aux (F s),
coeff_zero' := λ s, rfl,
summable' := Tinv_aux_summable F })
begin
intros F G,
ext s (_|n),
{ simp only [add_zero, pi.add_apply, Mbar.coeff_zero] },
{ simp only [coe_mk, pi.add_apply, Tinv_aux_succ, coe_add] }
end
@[simp] lemma Tinv_zero [fact (0 < r')] (F : Mbar r' S) (s : S) : Tinv F s 0 = 0 := rfl
@[simp] lemma Tinv_ne_zero [fact (0 < r')] (F : Mbar r' S) (s : S) (i : ℕ) (hi : i ≠ 0) :
Tinv F s i = F s (i + 1) :=
if_neg hi
@[simp] lemma Tinv_succ [fact (0 < r')] (F : Mbar r' S) (s : S) (i : ℕ) :
Tinv F s (i + 1) = F s (i + 2) :=
Tinv_aux_succ (F s) i
open pseudo_normed_group
lemma Tinv_mem_filtration [h0r : fact (0 < r')] :
Tinv ∈ filtration (Mbar r' S →+ Mbar r' S) (r'⁻¹) :=
begin
intros c F hF,
simp only [Tinv, add_monoid_hom.mk'_apply],
change _ ≤ _ at hF,
rw mul_comm,
apply le_mul_inv_of_mul_le h0r.out.ne',
rw [nnnorm_def, finset.sum_mul],
apply le_trans _ hF,
apply finset.sum_le_sum,
rintro s -,
rw ← nnreal.tsum_mul_right,
conv_rhs { rw [← sum_add_tsum_nat_add' 1 (F.summable s)] },
refine le_add_of_nonneg_of_le zero_le' _,
apply tsum_le_tsum,
{ rintro (_|i),
{ simp only [nat.cast_zero, zero_mul, Mbar.coeff_zero, int.nat_abs_zero], exact zero_le' },
{ simp only [Tinv_aux_succ, mul_assoc, coe_mk, pow_succ'] } },
{ exact (Tinv_aux_summable F s).mul_right _ },
{ exact (nnreal.summable_nat_add_iff 1).mpr (F.summable s) }
end
end Tinv
/-- `of_mask x mask : Mbar r' S` is `∑ a_{s,n}T^n ∈ Tℤ[[T]]`,
where `a_{s,n}` is `x_{s,n}` if `mask n s` is true and `0` otherwise. -/
@[simps]
def of_mask (x : Mbar r' S) (mask : S → ℕ → Prop) [∀ s n, decidable (mask s n)] :
Mbar r' S :=
{ to_fun := λ s n, if mask s n then x s n else 0,
coeff_zero' := λ s, by { split_ifs, { exact x.coeff_zero s }, { refl } },
summable' := λ s,
begin
apply nnreal.summable_of_le _ (x.summable s),
intro n,
refine mul_le_mul' _ le_rfl,
norm_cast,
split_ifs,
{ refl },
{ rw int.nat_abs_zero, exact zero_le' }
end }
variables (r' S)
/-- `geom r' S` is `∑ T^n ∈ Tℤ[[T]]`.
In other words, all non-constant coefficients are `1`. -/
@[simps]
def geom [hr' : fact (r' < 1)] : Mbar r' S :=
{ to_fun := λ s n, if n = 0 then 0 else 1,
coeff_zero' := λ s, if_pos rfl,
summable' := λ s,
begin
have := (normed_ring.summable_geometric_of_norm_lt_1 (r' : ℝ) _), swap,
{ rw nnreal.norm_eq, exact hr'.out },
simp only [← nnreal.coe_pow, nnreal.summable_coe] at this,
apply nnreal.summable_of_le _ this,
intro n,
refine (mul_le_mul' _ le_rfl).trans (one_mul _).le,
split_ifs; simp
end }
section map
variables {r' S} {T : Type u} [fintype T] (f : S → T)
open_locale classical
lemma has_sum_aux (F : Mbar r' S) :
has_sum (λ n : ℕ, (∑ s, ↑(F s n).nat_abs) * r' ^ n) (∑ s, ∑' n, (F s n).nat_abs * r' ^ n) :=
begin
have : (λ (n : ℕ), (∑ (s : S), ↑(F s n).nat_abs) * r' ^ n) =
(λ (n : ℕ), ∑ (s : S), (↑(F s n).nat_abs) * r' ^ n),
{ funext n,
rw finset.sum_mul },
rw this, clear this,
exact has_sum_sum (λ s _, (F.summable s).has_sum)
end
/-- Given an element of `Mbar r' S` and a function `f : S → T`, this
constructs an associated element of `Mbar r' T`. -/
@[simps]
def map : Mbar r' S → Mbar r' T := λ F,
{ to_fun := λ t n, ∑ s in finset.univ.filter (λ s', f s' = t), F s n,
coeff_zero' := λ s, by simp,
summable' := λ t, begin
have : ∀ (n : ℕ),
↑((∑ (s : S) in finset.univ.filter (λ (s' : S), f s' = t), F s n).nat_abs) * r' ^ n ≤
(∑ s : S, (F s n).nat_abs) * r' ^ n,
{ intro n,
refine mul_le_mul _ (le_refl _) zero_le' zero_le',
simp_rw ← nat.cast_sum,
apply nat.cast_le.mpr,
have : (∑ (s : S) in finset.univ.filter (λ (s' : S), f s' = t), F s n).nat_abs ≤
∑ (s : S) in finset.univ.filter (λ s', f s' = t), (F s n).nat_abs,
{ apply nat_abs_sum_le },
refine le_trans this _,
apply finset.sum_le_sum_of_subset,
{ intros t ht, simp },
{ apply_instance } },
exact nnreal.summable_of_le this (has_sum.summable (has_sum_aux _)),
end }
lemma map_id (F : Mbar r' S) : F.map id = F :=
begin
ext s n,
simp [finset.sum_filter],
end
lemma map_comp (F : Mbar r' S) {U : Type*} [fintype U] (g : T → U) :
F.map (g ∘ f) = (F.map f).map g :=
begin
ext u n,
dsimp,
rw ← finset.sum_bUnion,
{ apply finset.sum_congr,
ext s,
split,
{ intro hs,
simpa using hs },
{ intro hs,
simpa using hs },
{ tauto } },
{ rintros t1 ht1 t2 ht2 h s hs,
simp at ht1 ht2 hs ⊢,
refine h _,
rw [← hs.1, ← hs.2] }
end
lemma nnnorm_map_le_of_nnnorm_le {c : ℝ≥0} (F : Mbar r' S) (hF : ∥ F ∥₊ ≤ c) :
∥ F.map f ∥₊ ≤ c :=
begin
simp [ Mbar.nnnorm_def ] at *,
rw ← tsum_sum at hF ⊢,
{ refine le_trans (tsum_le_tsum _ _ _) hF,
{ intros n,
simp_rw ← finset.sum_mul,
refine mul_le_mul _ (le_refl _) zero_le' zero_le',
simp_rw ← nat.cast_sum,
rw nat.cast_le,
have : ∑ (x : T), (∑ (s : S) in finset.univ.filter (λ (s' : S), f s' = x), F s n).nat_abs ≤
∑ (x : T), (∑ (s : S) in finset.univ.filter (λ s', f s' = x), (F s n).nat_abs),
{ apply finset.sum_le_sum,
intros t ht,
apply nat_abs_sum_le },
refine le_trans this _,
rw ← finset.sum_bUnion,
apply finset.sum_le_sum_of_subset,
{ intros t ht, simp },
{ intros t1 ht1 t2 ht2 h s hs,
simp at ht1 ht2 hs ⊢,
apply h,
rw [← hs.1, ← hs.2] } },
{ apply summable_sum,
intros t ht,
apply (F.map f).summable },
{ apply summable_sum,
intros s hs,
apply F.summable } },
{ intros t ht,
apply (F.map f).summable },
{ intros s hs,
apply F.summable }
end
end map
end Mbar
#lint-
|
53c5f53adcce7701e041c3b64c4a5ad34b3cd333 | 9be442d9ec2fcf442516ed6e9e1660aa9071b7bd | /tests/lean/unexpander.lean | 02f385f95244066748952b79ad8ef691753dd325 | [
"Apache-2.0",
"LLVM-exception",
"NCSA",
"LGPL-3.0-only",
"LicenseRef-scancode-inner-net-2.0",
"BSD-3-Clause",
"LGPL-2.0-or-later",
"Spencer-94",
"LGPL-2.1-or-later",
"HPND",
"LicenseRef-scancode-pcre",
"ISC",
"LGPL-2.1-only",
"LicenseRef-scancode-other-permissive",
"SunPro",
"CMU-Mach"... | permissive | EdAyers/lean4 | 57ac632d6b0789cb91fab2170e8c9e40441221bd | 37ba0df5841bde51dbc2329da81ac23d4f6a4de4 | refs/heads/master | 1,676,463,245,298 | 1,660,619,433,000 | 1,660,619,433,000 | 183,433,437 | 1 | 0 | Apache-2.0 | 1,657,612,672,000 | 1,556,196,574,000 | Lean | UTF-8 | Lean | false | false | 326 | lean | open Lean PrettyPrinter
namespace ns
inductive Foo | mk: Foo
@[appUnexpander Foo.mk]
def unexpadFoo : Lean.PrettyPrinter.Unexpander | `($x) => `(unexpand)
def foo := Foo.mk
#print foo
@[appUnexpander ns.Foo.mk]
def unexpadFoo' : Lean.PrettyPrinter.Unexpander | `($x) => `(unexpand)
def bar := ns.Foo.mk
#print bar
end ns
|
64a21a8db2897c2db426135069b45baadf36b363 | 8cae430f0a71442d02dbb1cbb14073b31048e4b0 | /src/data/rat/meta_defs.lean | fbd4c5ce8cb09c1ff06455a06723bfa503a5e148 | [
"Apache-2.0"
] | permissive | leanprover-community/mathlib | 56a2cadd17ac88caf4ece0a775932fa26327ba0e | 442a83d738cb208d3600056c489be16900ba701d | refs/heads/master | 1,693,584,102,358 | 1,693,471,902,000 | 1,693,471,902,000 | 97,922,418 | 1,595 | 352 | Apache-2.0 | 1,694,693,445,000 | 1,500,624,130,000 | Lean | UTF-8 | Lean | false | false | 5,586 | lean | /-
Copyright (c) 2019 Robert Y. Lewis . All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import data.rat.defs
import tactic.core
/-!
# Meta operations on ℚ
This file defines functions for dealing with rational numbers as expressions.
They are not defined earlier in the hierarchy, in the `tactic` or `meta` folders, since we do not
want to import `data.rat.basic` there.
## Main definitions
* `rat.mk_numeral` embeds a rational `q` as a numeral expression into a type supporting the needed
operations. It does not need a tactic state.
* `rat.reflect` specializes `rat.mk_numeral` to `ℚ`.
* `expr.of_rat` behaves like `rat.mk_numeral`, but uses the tactic state to infer the needed
structure on the target type.
* `expr.to_rat` evaluates a normal numeral expression as a rat.
* `expr.eval_rat` evaluates a numeral expression with arithmetic operations as a rat.
-/
/--
`rat.mk_numeral q` embeds `q` as a numeral expression inside a type with 0, 1, +, -, and /
`type`: an expression representing the target type. This must live in Type 0.
`has_zero`, `has_one`, `has_add`: expressions of the type `has_zero %%type`, etc.
This function is similar to `expr.of_rat` but takes more hypotheses and is not tactic valued.
-/
meta def rat.mk_numeral (type has_zero has_one has_add has_neg has_div : expr) : ℚ → expr
| ⟨num, denom, _, _⟩ :=
let nume := num.mk_numeral type has_zero has_one has_add has_neg in
if denom = 1 then nume else
let dene := denom.mk_numeral type has_zero has_one has_add in
`(@has_div.div.{0} %%type %%has_div %%nume %%dene)
section
-- Note that here we are disabling the "safety" of reflected, to allow us to reuse `rat.mk_numeral`.
-- The usual way to provide the required `reflected` instance would be via rewriting to prove that
-- the expression we use here is equivalent.
local attribute [semireducible] reflected
/-- `rat.reflect q` represents the rational number `q` as a numeral expression of type `ℚ`. -/
meta instance rat.reflect : has_reflect ℚ :=
rat.mk_numeral `(ℚ) `(by apply_instance : has_zero ℚ)
`(by apply_instance : has_one ℚ) `(by apply_instance : has_add ℚ)
`(by apply_instance : has_neg ℚ) `(by apply_instance : has_div ℚ)
end
/--
`rat.to_pexpr q` creates a `pexpr` that will evaluate to `q`.
The `pexpr` does not hold any typing information:
`to_expr ``((%%(rat.to_pexpr (3/4)) : K))` will create a native `K` numeral `(3/4 : K)`.
-/
meta def rat.to_pexpr (q : ℚ) : pexpr :=
let n := q.num,
d := q.denom in
if d = 1 then n.to_pexpr
else ``(%%n.to_pexpr / %%d.to_pexpr)
/-- Evaluates an expression as a rational number,
if that expression represents a numeral or the quotient of two numerals. -/
protected meta def expr.to_nonneg_rat : expr → option ℚ
| `(%%e₁ / %%e₂) := do
m ← e₁.to_nat,
n ← e₂.to_nat,
if c : m.coprime n then if h : 1 < n then
return ⟨m, n, lt_trans zero_lt_one h, c⟩
else none else none
| e := do n ← e.to_nat, return n
/-- Evaluates an expression as a rational number,
if that expression represents a numeral, the quotient of two numerals,
the negation of a numeral, or the negation of the quotient of two numerals. -/
protected meta def expr.to_rat : expr → option ℚ
| `(has_neg.neg %%e) := do q ← e.to_nonneg_rat, some (-q)
| e := e.to_nonneg_rat
/-- Evaluates an expression into a rational number, if that expression is built up from
numerals, +, -, *, /, ⁻¹ -/
protected meta def expr.eval_rat : expr → option ℚ
| `(has_zero.zero) := some 0
| `(has_one.one) := some 1
| `(bit0 %%q) := (*) 2 <$> q.eval_rat
| `(bit1 %%q) := (+) 1 <$> (*) 2 <$> q.eval_rat
| `(%%a + %%b) := (+) <$> a.eval_rat <*> b.eval_rat
| `(%%a - %%b) := has_sub.sub <$> a.eval_rat <*> b.eval_rat
| `(%%a * %%b) := (*) <$> a.eval_rat <*> b.eval_rat
| `(%%a / %%b) := (/) <$> a.eval_rat <*> b.eval_rat
| `(-(%%a)) := has_neg.neg <$> a.eval_rat
| `((%%a)⁻¹) := has_inv.inv <$> a.eval_rat
| _ := none
/-- `expr.of_rat α q` embeds `q` as a numeral expression inside the type `α`.
Lean will try to infer the correct type classes on `α`, and the tactic will fail if it cannot.
This function is similar to `rat.mk_numeral` but it takes fewer hypotheses and is tactic valued.
-/
protected meta def expr.of_rat (α : expr) : ℚ → tactic expr
| ⟨(n:ℕ), d, h, c⟩ := do
e₁ ← expr.of_nat α n,
if d = 1 then return e₁ else
do e₂ ← expr.of_nat α d,
tactic.mk_app ``has_div.div [e₁, e₂]
| ⟨-[1+n], d, h, c⟩ := do
e₁ ← expr.of_nat α (n+1),
e ← (if d = 1 then return e₁ else do
e₂ ← expr.of_nat α d,
tactic.mk_app ``has_div.div [e₁, e₂]),
tactic.mk_app ``has_neg.neg [e]
namespace tactic
namespace instance_cache
/-- `c.of_rat q` embeds `q` as a numeral expression inside the type `α`.
Lean will try to infer the correct type classes on `c.α`, and the tactic will fail if it cannot.
This function is similar to `rat.mk_numeral` but it takes fewer hypotheses and is tactic valued.
-/
protected meta def of_rat (c : instance_cache) : ℚ → tactic (instance_cache × expr)
| ⟨(n:ℕ), d, _, _⟩ :=
if d = 1 then c.of_nat n else do
(c, e₁) ← c.of_nat n,
(c, e₂) ← c.of_nat d,
c.mk_app ``has_div.div [e₁, e₂]
| ⟨-[1+n], d, _, _⟩ := do
(c, e) ← (if d = 1 then c.of_nat (n+1) else do
(c, e₁) ← c.of_nat (n+1),
(c, e₂) ← c.of_nat d,
c.mk_app ``has_div.div [e₁, e₂]),
c.mk_app ``has_neg.neg [e]
end instance_cache
end tactic
|
33646bda4173292bde90b29ae87207776de74a3c | 4727251e0cd73359b15b664c3170e5d754078599 | /src/algebra/regular/smul.lean | b45087acf3d3afca7a7798c7f80be605f8a089ed | [
"Apache-2.0"
] | permissive | Vierkantor/mathlib | 0ea59ac32a3a43c93c44d70f441c4ee810ccceca | 83bc3b9ce9b13910b57bda6b56222495ebd31c2f | refs/heads/master | 1,658,323,012,449 | 1,652,256,003,000 | 1,652,256,003,000 | 209,296,341 | 0 | 1 | Apache-2.0 | 1,568,807,655,000 | 1,568,807,655,000 | null | UTF-8 | Lean | false | false | 7,840 | lean | /-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import algebra.smul_with_zero
import algebra.regular.basic
/-!
# Action of regular elements on a module
We introduce `M`-regular elements, in the context of an `R`-module `M`. The corresponding
predicate is called `is_smul_regular`.
There are very limited typeclass assumptions on `R` and `M`, but the "mathematical" case of interest
is a commutative ring `R` acting an a module `M`. Since the properties are "multiplicative", there
is no actual requirement of having an addition, but there is a zero in both `R` and `M`.
Smultiplications involving `0` are, of course, all trivial.
The defining property is that an element `a ∈ R` is `M`-regular if the smultiplication map
`M → M`, defined by `m ↦ a • m`, is injective.
This property is the direct generalization to modules of the property `is_left_regular` defined in
`algebra/regular`. Lemma `is_smul_regular.is_left_regular_iff` shows that indeed the two notions
coincide.
-/
variables {R S : Type*} (M : Type*) {a b : R} {s : S}
/-- An `M`-regular element is an element `c` such that multiplication on the left by `c` is an
injective map `M → M`. -/
def is_smul_regular [has_scalar R M] (c : R) := function.injective ((•) c : M → M)
lemma is_left_regular.is_smul_regular [has_mul R] {c : R} (h : is_left_regular c) :
is_smul_regular R c := h
/-- Left-regular multiplication on `R` is equivalent to `R`-regularity of `R` itself. -/
lemma is_left_regular_iff [has_mul R] {a : R} :
is_left_regular a ↔ is_smul_regular R a := iff.rfl
lemma is_right_regular.is_smul_regular [has_mul R] {c : R} (h : is_right_regular c) :
is_smul_regular R (mul_opposite.op c) := h
/-- Right-regular multiplication on `R` is equivalent to `Rᵐᵒᵖ`-regularity of `R` itself. -/
lemma is_right_regular_iff [has_mul R] {a : R} :
is_right_regular a ↔ is_smul_regular R (mul_opposite.op a) := iff.rfl
namespace is_smul_regular
variables {M}
section has_scalar
variables [has_scalar R M] [has_scalar R S] [has_scalar S M] [is_scalar_tower R S M]
/-- The product of `M`-regular elements is `M`-regular. -/
lemma smul (ra : is_smul_regular M a) (rs : is_smul_regular M s) :
is_smul_regular M (a • s) :=
λ a b ab, rs (ra ((smul_assoc _ _ _).symm.trans (ab.trans (smul_assoc _ _ _))))
/-- If an element `b` becomes `M`-regular after multiplying it on the left by an `M`-regular
element, then `b` is `M`-regular. -/
lemma of_smul (a : R) (ab : is_smul_regular M (a • s)) :
is_smul_regular M s :=
@function.injective.of_comp _ _ _ (λ m : M, a • m) _ (λ c d cd, ab
(by rwa [smul_assoc, smul_assoc]))
/-- An element is `M`-regular if and only if multiplying it on the left by an `M`-regular element
is `M`-regular. -/
@[simp] lemma smul_iff (b : S) (ha : is_smul_regular M a) :
is_smul_regular M (a • b) ↔ is_smul_regular M b :=
⟨of_smul _, ha.smul⟩
lemma is_left_regular [has_mul R] {a : R} (h : is_smul_regular R a) :
is_left_regular a := h
lemma is_right_regular [has_mul R] {a : R} (h : is_smul_regular R (mul_opposite.op a)) :
is_right_regular a := h
lemma mul [has_mul R] [is_scalar_tower R R M]
(ra : is_smul_regular M a) (rb : is_smul_regular M b) : is_smul_regular M (a * b) :=
ra.smul rb
lemma of_mul [has_mul R] [is_scalar_tower R R M] (ab : is_smul_regular M (a * b)) :
is_smul_regular M b :=
by { rw ← smul_eq_mul at ab, exact ab.of_smul _ }
@[simp] lemma mul_iff_right [has_mul R] [is_scalar_tower R R M] (ha : is_smul_regular M a) :
is_smul_regular M (a * b) ↔ is_smul_regular M b :=
⟨of_mul, ha.mul⟩
/-- Two elements `a` and `b` are `M`-regular if and only if both products `a * b` and `b * a`
are `M`-regular. -/
lemma mul_and_mul_iff [has_mul R] [is_scalar_tower R R M] :
is_smul_regular M (a * b) ∧ is_smul_regular M (b * a) ↔
is_smul_regular M a ∧ is_smul_regular M b :=
begin
refine ⟨_, _⟩,
{ rintros ⟨ab, ba⟩,
refine ⟨ba.of_mul, ab.of_mul⟩ },
{ rintros ⟨ha, hb⟩,
exact ⟨ha.mul hb, hb.mul ha⟩ }
end
end has_scalar
section monoid
variables [monoid R] [mul_action R M]
variable (M)
/-- One is `M`-regular always. -/
@[simp] lemma one : is_smul_regular M (1 : R) :=
λ a b ab, by rwa [one_smul, one_smul] at ab
variable {M}
/-- An element of `R` admitting a left inverse is `M`-regular. -/
lemma of_mul_eq_one (h : a * b = 1) : is_smul_regular M b :=
of_mul (by { rw h, exact one M })
/-- Any power of an `M`-regular element is `M`-regular. -/
lemma pow (n : ℕ) (ra : is_smul_regular M a) : is_smul_regular M (a ^ n) :=
begin
induction n with n hn,
{ simp only [one, pow_zero] },
{ rw pow_succ, exact (ra.smul_iff (a ^ n)).mpr hn }
end
/-- An element `a` is `M`-regular if and only if a positive power of `a` is `M`-regular. -/
lemma pow_iff {n : ℕ} (n0 : 0 < n) :
is_smul_regular M (a ^ n) ↔ is_smul_regular M a :=
begin
refine ⟨_, pow n⟩,
rw [← nat.succ_pred_eq_of_pos n0, pow_succ', ← smul_eq_mul],
exact of_smul _,
end
end monoid
section monoid_smul
variables [monoid S] [has_scalar R M] [has_scalar R S] [mul_action S M] [is_scalar_tower R S M]
/-- An element of `S` admitting a left inverse in `R` is `M`-regular. -/
lemma of_smul_eq_one (h : a • s = 1) : is_smul_regular M s :=
of_smul a (by { rw h, exact one M })
end monoid_smul
section monoid_with_zero
variables [monoid_with_zero R] [monoid_with_zero S] [has_zero M]
[mul_action_with_zero R M] [mul_action_with_zero R S] [mul_action_with_zero S M]
[is_scalar_tower R S M]
/-- The element `0` is `M`-regular if and only if `M` is trivial. -/
protected lemma subsingleton (h : is_smul_regular M (0 : R)) : subsingleton M :=
⟨λ a b, h (by repeat { rw mul_action_with_zero.zero_smul })⟩
/-- The element `0` is `M`-regular if and only if `M` is trivial. -/
lemma zero_iff_subsingleton : is_smul_regular M (0 : R) ↔ subsingleton M :=
⟨λ h, h.subsingleton, λ H a b h, @subsingleton.elim _ H a b⟩
/-- The `0` element is not `M`-regular, on a non-trivial module. -/
lemma not_zero_iff : ¬ is_smul_regular M (0 : R) ↔ nontrivial M :=
begin
rw [nontrivial_iff, not_iff_comm, zero_iff_subsingleton, subsingleton_iff],
push_neg,
exact iff.rfl
end
/-- The element `0` is `M`-regular when `M` is trivial. -/
lemma zero [sM : subsingleton M] : is_smul_regular M (0 : R) :=
zero_iff_subsingleton.mpr sM
/-- The `0` element is not `M`-regular, on a non-trivial module. -/
lemma not_zero [nM : nontrivial M] : ¬ is_smul_regular M (0 : R) :=
not_zero_iff.mpr nM
end monoid_with_zero
section comm_semigroup
variables [comm_semigroup R] [has_scalar R M] [is_scalar_tower R R M]
/-- A product is `M`-regular if and only if the factors are. -/
lemma mul_iff : is_smul_regular M (a * b) ↔
is_smul_regular M a ∧ is_smul_regular M b :=
begin
rw ← mul_and_mul_iff,
exact ⟨λ ab, ⟨ab, by rwa mul_comm⟩, λ rab, rab.1⟩
end
end comm_semigroup
end is_smul_regular
section group
variables {G : Type*} [group G]
/-- An element of a group acting on a Type is regular. This relies on the availability
of the inverse given by groups, since there is no `left_cancel_smul` typeclass. -/
lemma is_smul_regular_of_group [mul_action G R] (g : G) : is_smul_regular R g :=
begin
intros x y h,
convert congr_arg ((•) g⁻¹) h using 1;
simp [←smul_assoc]
end
end group
section units
variables [monoid R] [mul_action R M]
/-- Any element in `Rˣ` is `M`-regular. -/
lemma units.is_smul_regular (a : Rˣ) : is_smul_regular M (a : R) :=
is_smul_regular.of_mul_eq_one a.inv_val
/-- A unit is `M`-regular. -/
lemma is_unit.is_smul_regular (ua : is_unit a) : is_smul_regular M a :=
begin
rcases ua with ⟨a, rfl⟩,
exact a.is_smul_regular M
end
end units
|
d5a396374c39011916c921ce9041a696e1e81407 | 302c785c90d40ad3d6be43d33bc6a558354cc2cf | /src/algebra/group/units.lean | a0e1279b185574445361421f590fe6986f9a96ae | [
"Apache-2.0"
] | permissive | ilitzroth/mathlib | ea647e67f1fdfd19a0f7bdc5504e8acec6180011 | 5254ef14e3465f6504306132fe3ba9cec9ffff16 | refs/heads/master | 1,680,086,661,182 | 1,617,715,647,000 | 1,617,715,647,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 13,027 | lean | /-
Copyright (c) 2017 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johannes Hölzl, Chris Hughes, Jens Wagemaker
-/
import algebra.group.basic
import logic.nontrivial
/-!
# Units (i.e., invertible elements) of a multiplicative monoid
-/
universe u
variable {α : Type u}
/-- Units of a monoid, bundled version. An element of a `monoid` is a unit if it has a two-sided
inverse. This version bundles the inverse element so that it can be computed. For a predicate
see `is_unit`. -/
structure units (α : Type u) [monoid α] :=
(val : α)
(inv : α)
(val_inv : val * inv = 1)
(inv_val : inv * val = 1)
/-- Units of an add_monoid, bundled version. An element of an add_monoid is a unit if it has a
two-sided additive inverse. This version bundles the inverse element so that it can be
computed. For a predicate see `is_add_unit`. -/
structure add_units (α : Type u) [add_monoid α] :=
(val : α)
(neg : α)
(val_neg : val + neg = 0)
(neg_val : neg + val = 0)
attribute [to_additive add_units] units
namespace units
variables [monoid α]
@[to_additive] instance : has_coe (units α) α := ⟨val⟩
@[simp, to_additive] lemma coe_mk (a : α) (b h₁ h₂) : ↑(units.mk a b h₁ h₂) = a := rfl
@[ext, to_additive] theorem ext :
function.injective (coe : units α → α)
| ⟨v, i₁, vi₁, iv₁⟩ ⟨v', i₂, vi₂, iv₂⟩ e :=
by change v = v' at e; subst v'; congr;
simpa only [iv₂, vi₁, one_mul, mul_one] using mul_assoc i₂ v i₁
@[norm_cast, to_additive] theorem eq_iff {a b : units α} :
(a : α) = b ↔ a = b := ext.eq_iff
@[to_additive] theorem ext_iff {a b : units α} :
a = b ↔ (a : α) = b := eq_iff.symm
@[to_additive] instance [decidable_eq α] : decidable_eq (units α) :=
λ a b, decidable_of_iff' _ ext_iff
@[simp, to_additive] theorem mk_coe (u : units α) (y h₁ h₂) :
mk (u : α) y h₁ h₂ = u :=
ext rfl
/-- Units of a monoid form a group. -/
@[to_additive] instance : group (units α) :=
{ mul := λ u₁ u₂, ⟨u₁.val * u₂.val, u₂.inv * u₁.inv,
by rw [mul_assoc, ← mul_assoc u₂.val, val_inv, one_mul, val_inv],
by rw [mul_assoc, ← mul_assoc u₁.inv, inv_val, one_mul, inv_val]⟩,
one := ⟨1, 1, one_mul 1, one_mul 1⟩,
mul_one := λ u, ext $ mul_one u,
one_mul := λ u, ext $ one_mul u,
mul_assoc := λ u₁ u₂ u₃, ext $ mul_assoc u₁ u₂ u₃,
inv := λ u, ⟨u.2, u.1, u.4, u.3⟩,
mul_left_inv := λ u, ext u.inv_val }
variables (a b : units α) {c : units α}
@[simp, norm_cast, to_additive] lemma coe_mul : (↑(a * b) : α) = a * b := rfl
attribute [norm_cast] add_units.coe_add
@[simp, norm_cast, to_additive] lemma coe_one : ((1 : units α) : α) = 1 := rfl
attribute [norm_cast] add_units.coe_zero
@[simp, norm_cast, to_additive] lemma coe_eq_one {a : units α} : (a : α) = 1 ↔ a = 1 :=
by rw [←units.coe_one, eq_iff]
@[simp, to_additive] lemma inv_mk (x y : α) (h₁ h₂) : (mk x y h₁ h₂)⁻¹ = mk y x h₂ h₁ := rfl
@[to_additive] lemma val_coe : (↑a : α) = a.val := rfl
@[norm_cast, to_additive] lemma coe_inv : ((a⁻¹ : units α) : α) = a.inv := rfl
attribute [norm_cast] add_units.coe_neg
@[simp, to_additive] lemma inv_mul : (↑a⁻¹ * a : α) = 1 := inv_val _
@[simp, to_additive] lemma mul_inv : (a * ↑a⁻¹ : α) = 1 := val_inv _
@[to_additive] lemma inv_mul_of_eq {u : units α} {a : α} (h : ↑u = a) : ↑u⁻¹ * a = 1 :=
by { rw [←h, u.inv_mul], }
@[to_additive] lemma mul_inv_of_eq {u : units α} {a : α} (h : ↑u = a) : a * ↑u⁻¹ = 1 :=
by { rw [←h, u.mul_inv], }
@[simp, to_additive] lemma mul_inv_cancel_left (a : units α) (b : α) : (a:α) * (↑a⁻¹ * b) = b :=
by rw [← mul_assoc, mul_inv, one_mul]
@[simp, to_additive] lemma inv_mul_cancel_left (a : units α) (b : α) : (↑a⁻¹:α) * (a * b) = b :=
by rw [← mul_assoc, inv_mul, one_mul]
@[simp, to_additive] lemma mul_inv_cancel_right (a : α) (b : units α) : a * b * ↑b⁻¹ = a :=
by rw [mul_assoc, mul_inv, mul_one]
@[simp, to_additive] lemma inv_mul_cancel_right (a : α) (b : units α) : a * ↑b⁻¹ * b = a :=
by rw [mul_assoc, inv_mul, mul_one]
@[to_additive] instance : inhabited (units α) := ⟨1⟩
@[to_additive] instance {α} [comm_monoid α] : comm_group (units α) :=
{ mul_comm := λ u₁ u₂, ext $ mul_comm _ _, ..units.group }
@[to_additive] instance [has_repr α] : has_repr (units α) := ⟨repr ∘ val⟩
@[simp, to_additive] theorem mul_right_inj (a : units α) {b c : α} : (a:α) * b = a * c ↔ b = c :=
⟨λ h, by simpa only [inv_mul_cancel_left] using congr_arg ((*) ↑(a⁻¹ : units α)) h, congr_arg _⟩
@[simp, to_additive] theorem mul_left_inj (a : units α) {b c : α} : b * a = c * a ↔ b = c :=
⟨λ h, by simpa only [mul_inv_cancel_right] using congr_arg (* ↑(a⁻¹ : units α)) h, congr_arg _⟩
@[to_additive] theorem eq_mul_inv_iff_mul_eq {a b : α} : a = b * ↑c⁻¹ ↔ a * c = b :=
⟨λ h, by rw [h, inv_mul_cancel_right], λ h, by rw [← h, mul_inv_cancel_right]⟩
@[to_additive] theorem eq_inv_mul_iff_mul_eq {a c : α} : a = ↑b⁻¹ * c ↔ ↑b * a = c :=
⟨λ h, by rw [h, mul_inv_cancel_left], λ h, by rw [← h, inv_mul_cancel_left]⟩
@[to_additive] theorem inv_mul_eq_iff_eq_mul {b c : α} : ↑a⁻¹ * b = c ↔ b = a * c :=
⟨λ h, by rw [← h, mul_inv_cancel_left], λ h, by rw [h, inv_mul_cancel_left]⟩
@[to_additive] theorem mul_inv_eq_iff_eq_mul {a c : α} : a * ↑b⁻¹ = c ↔ a = c * b :=
⟨λ h, by rw [← h, inv_mul_cancel_right], λ h, by rw [h, mul_inv_cancel_right]⟩
lemma inv_eq_of_mul_eq_one {u : units α} {a : α} (h : ↑u * a = 1) : ↑u⁻¹ = a :=
calc ↑u⁻¹ = ↑u⁻¹ * 1 : by rw mul_one
... = ↑u⁻¹ * ↑u * a : by rw [←h, ←mul_assoc]
... = a : by rw [u.inv_mul, one_mul]
lemma inv_unique {u₁ u₂ : units α} (h : (↑u₁ : α) = ↑u₂) : (↑u₁⁻¹ : α) = ↑u₂⁻¹ :=
suffices ↑u₁ * (↑u₂⁻¹ : α) = 1, by exact inv_eq_of_mul_eq_one this, by rw [h, u₂.mul_inv]
end units
/-- For `a, b` in a `comm_monoid` such that `a * b = 1`, makes a unit out of `a`. -/
@[to_additive "For `a, b` in an `add_comm_monoid` such that `a + b = 0`, makes an add_unit
out of `a`."]
def units.mk_of_mul_eq_one [comm_monoid α] (a b : α) (hab : a * b = 1) :
units α :=
⟨a, b, hab, (mul_comm b a).trans hab⟩
@[simp, to_additive] lemma units.coe_mk_of_mul_eq_one [comm_monoid α] {a b : α} (h : a * b = 1) :
(units.mk_of_mul_eq_one a b h : α) = a := rfl
section monoid
variables [monoid α] {a b c : α}
/-- Partial division. It is defined when the
second argument is invertible, and unlike the division operator
in `division_ring` it is not totalized at zero. -/
def divp (a : α) (u) : α := a * (u⁻¹ : units α)
infix ` /ₚ `:70 := divp
@[simp] theorem divp_self (u : units α) : (u : α) /ₚ u = 1 := units.mul_inv _
@[simp] theorem divp_one (a : α) : a /ₚ 1 = a := mul_one _
theorem divp_assoc (a b : α) (u : units α) : a * b /ₚ u = a * (b /ₚ u) :=
mul_assoc _ _ _
@[simp] theorem divp_inv (u : units α) : a /ₚ u⁻¹ = a * u := rfl
@[simp] theorem divp_mul_cancel (a : α) (u : units α) : a /ₚ u * u = a :=
(mul_assoc _ _ _).trans $ by rw [units.inv_mul, mul_one]
@[simp] theorem mul_divp_cancel (a : α) (u : units α) : (a * u) /ₚ u = a :=
(mul_assoc _ _ _).trans $ by rw [units.mul_inv, mul_one]
@[simp] theorem divp_left_inj (u : units α) {a b : α} : a /ₚ u = b /ₚ u ↔ a = b :=
units.mul_left_inj _
theorem divp_divp_eq_divp_mul (x : α) (u₁ u₂ : units α) : (x /ₚ u₁) /ₚ u₂ = x /ₚ (u₂ * u₁) :=
by simp only [divp, mul_inv_rev, units.coe_mul, mul_assoc]
theorem divp_eq_iff_mul_eq {x : α} {u : units α} {y : α} : x /ₚ u = y ↔ y * u = x :=
u.mul_left_inj.symm.trans $ by rw [divp_mul_cancel]; exact ⟨eq.symm, eq.symm⟩
theorem divp_eq_one_iff_eq {a : α} {u : units α} : a /ₚ u = 1 ↔ a = u :=
(units.mul_left_inj u).symm.trans $ by rw [divp_mul_cancel, one_mul]
@[simp] theorem one_divp (u : units α) : 1 /ₚ u = ↑u⁻¹ :=
one_mul _
end monoid
section comm_monoid
variables [comm_monoid α]
theorem divp_eq_divp_iff {x y : α} {ux uy : units α} :
x /ₚ ux = y /ₚ uy ↔ x * uy = y * ux :=
by rw [divp_eq_iff_mul_eq, mul_comm, ← divp_assoc, divp_eq_iff_mul_eq, mul_comm y ux]
theorem divp_mul_divp (x y : α) (ux uy : units α) :
(x /ₚ ux) * (y /ₚ uy) = (x * y) /ₚ (ux * uy) :=
by rw [← divp_divp_eq_divp_mul, divp_assoc, mul_comm x, divp_assoc, mul_comm]
end comm_monoid
/-!
# `is_unit` predicate
In this file we define the `is_unit` predicate on a `monoid`, and
prove a few basic properties. For the bundled version see `units`. See
also `prime`, `associated`, and `irreducible` in `algebra/associated`.
-/
section is_unit
variables {M : Type*} {N : Type*}
/-- An element `a : M` of a monoid is a unit if it has a two-sided inverse.
The actual definition says that `a` is equal to some `u : units M`, where
`units M` is a bundled version of `is_unit`. -/
@[to_additive is_add_unit "An element `a : M` of an add_monoid is an `add_unit` if it has
a two-sided additive inverse. The actual definition says that `a` is equal to some
`u : add_units M`, where `add_units M` is a bundled version of `is_add_unit`."]
def is_unit [monoid M] (a : M) : Prop := ∃ u : units M, (u : M) = a
@[nontriviality] lemma is_unit_of_subsingleton [monoid M] [subsingleton M] (a : M) : is_unit a :=
⟨⟨a, a, subsingleton.elim _ _, subsingleton.elim _ _⟩, rfl⟩
@[simp, to_additive is_add_unit_add_unit]
protected lemma units.is_unit [monoid M] (u : units M) : is_unit (u : M) := ⟨u, rfl⟩
@[simp, to_additive is_add_unit_zero]
theorem is_unit_one [monoid M] : is_unit (1:M) := ⟨1, rfl⟩
@[to_additive is_add_unit_of_add_eq_zero] theorem is_unit_of_mul_eq_one [comm_monoid M]
(a b : M) (h : a * b = 1) : is_unit a :=
⟨units.mk_of_mul_eq_one a b h, rfl⟩
@[to_additive is_add_unit_iff_exists_neg] theorem is_unit_iff_exists_inv [comm_monoid M]
{a : M} : is_unit a ↔ ∃ b, a * b = 1 :=
⟨by rintro ⟨⟨a, b, hab, _⟩, rfl⟩; exact ⟨b, hab⟩,
λ ⟨b, hab⟩, is_unit_of_mul_eq_one _ b hab⟩
@[to_additive is_add_unit_iff_exists_neg'] theorem is_unit_iff_exists_inv' [comm_monoid M]
{a : M} : is_unit a ↔ ∃ b, b * a = 1 :=
by simp [is_unit_iff_exists_inv, mul_comm]
/-- Multiplication by a `u : units M` doesn't affect `is_unit`. -/
@[simp, to_additive is_add_unit_add_add_units "Addition of a `u : add_units M` doesn't affect
`is_add_unit`."]
theorem units.is_unit_mul_units [monoid M] (a : M) (u : units M) :
is_unit (a * u) ↔ is_unit a :=
iff.intro
(assume ⟨v, hv⟩,
have is_unit (a * ↑u * ↑u⁻¹), by existsi v * u⁻¹; rw [←hv, units.coe_mul],
by rwa [mul_assoc, units.mul_inv, mul_one] at this)
(assume ⟨v, hv⟩, hv ▸ ⟨v * u, (units.coe_mul v u).symm⟩)
lemma is_unit.mul [monoid M] {x y : M} : is_unit x → is_unit y → is_unit (x * y) :=
by { rintros ⟨x, rfl⟩ ⟨y, rfl⟩, exact ⟨x * y, units.coe_mul _ _⟩ }
@[to_additive is_add_unit_of_add_is_add_unit_left]
theorem is_unit_of_mul_is_unit_left [comm_monoid M] {x y : M}
(hu : is_unit (x * y)) : is_unit x :=
let ⟨z, hz⟩ := is_unit_iff_exists_inv.1 hu in
is_unit_iff_exists_inv.2 ⟨y * z, by rwa ← mul_assoc⟩
@[to_additive] theorem is_unit_of_mul_is_unit_right [comm_monoid M] {x y : M}
(hu : is_unit (x * y)) : is_unit y :=
@is_unit_of_mul_is_unit_left _ _ y x $ by rwa mul_comm
@[to_additive] theorem is_unit.mul_right_inj [monoid M] {a b c : M} (ha : is_unit a) :
a * b = a * c ↔ b = c :=
by cases ha with a ha; rw [←ha, units.mul_right_inj]
@[to_additive] theorem is_unit.mul_left_inj [monoid M] {a b c : M} (ha : is_unit a) :
b * a = c * a ↔ b = c :=
by cases ha with a ha; rw [←ha, units.mul_left_inj]
/-- The element of the group of units, corresponding to an element of a monoid which is a unit. -/
noncomputable def is_unit.unit [monoid M] {a : M} (h : is_unit a) : units M :=
classical.some h
lemma is_unit.unit_spec [monoid M] {a : M} (h : is_unit a) : ↑h.unit = a :=
classical.some_spec h
end is_unit
section noncomputable_defs
variables {M : Type*}
/-- Constructs a `group` structure on a `monoid` consisting only of units. -/
noncomputable def group_of_is_unit [hM : monoid M] (h : ∀ (a : M), is_unit a) : group M :=
{ inv := λ a, ↑((h a).unit)⁻¹,
mul_left_inv := λ a, by {
change ↑((h a).unit)⁻¹ * a = 1,
rw [units.inv_mul_eq_iff_eq_mul, (h a).unit_spec, mul_one] },
.. hM }
/-- Constructs a `comm_group` structure on a `comm_monoid` consisting only of units. -/
noncomputable def comm_group_of_is_unit [hM : comm_monoid M] (h : ∀ (a : M), is_unit a) :
comm_group M :=
{ inv := λ a, ↑((h a).unit)⁻¹,
mul_left_inv := λ a, by {
change ↑((h a).unit)⁻¹ * a = 1,
rw [units.inv_mul_eq_iff_eq_mul, (h a).unit_spec, mul_one] },
.. hM }
end noncomputable_defs
|
448e1a851987846684ef128fa76efe07fd350fef | 9dd3f3912f7321eb58ee9aa8f21778ad6221f87c | /tests/lean/1293.lean | 19c364817c3e9ce22ec92524c8a1fdf91d61d154 | [
"Apache-2.0"
] | permissive | bre7k30/lean | de893411bcfa7b3c5572e61b9e1c52951b310aa4 | 5a924699d076dab1bd5af23a8f910b433e598d7a | refs/heads/master | 1,610,900,145,817 | 1,488,006,845,000 | 1,488,006,845,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 378 | lean | open expr tactic
example : true := by whnf (var 0) >> return ()
example : true := by whnf (app (var 0) (var 0)) >> return ()
example : true := by zeta (var 0) >> return ()
example : true := by unify (var 0) (var 0) >> return ()
example : true := by is_def_eq (var 0) (var 0) >> return ()
example foo trivial := by do
t ← infer_type (var 0),
to_expr `(trivial) >>= apply
|
773bbc528eb9f1f74583477627b21059fc66ef68 | cf39355caa609c0f33405126beee2739aa3cb77e | /tests/lean/run/1623.lean | c5590950aee70c901f5f8a2f11695062936f541e | [
"Apache-2.0"
] | permissive | leanprover-community/lean | 12b87f69d92e614daea8bcc9d4de9a9ace089d0e | cce7990ea86a78bdb383e38ed7f9b5ba93c60ce0 | refs/heads/master | 1,687,508,156,644 | 1,684,951,104,000 | 1,684,951,104,000 | 169,960,991 | 457 | 107 | Apache-2.0 | 1,686,744,372,000 | 1,549,790,268,000 | C++ | UTF-8 | Lean | false | false | 158 | lean | section
parameters {A : Type} (R : list A → Prop)
structure foo (x : list A) : Prop := (bar : R x)
structure bar (x : Type)
structure baz extends bar A
end
|
65218af1342c882f2467baf9134f4c72fcfff0a0 | 74addaa0e41490cbaf2abd313a764c96df57b05d | /Mathlib/Lean3Lib/init/meta/constructor_tactic.lean | 148ff300f41eedc256798a3b9de958249c2a8408 | [] | no_license | AurelienSaue/Mathlib4_auto | f538cfd0980f65a6361eadea39e6fc639e9dae14 | 590df64109b08190abe22358fabc3eae000943f2 | refs/heads/master | 1,683,906,849,776 | 1,622,564,669,000 | 1,622,564,669,000 | 371,723,747 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 366 | lean | /-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.meta.tactic
import Mathlib.Lean3Lib.init.function
namespace Mathlib
namespace tactic
/- Return target after instantiating metavars and whnf -/
|
c931d380bd528af78d563a12743ccb9855770192 | 64874bd1010548c7f5a6e3e8902efa63baaff785 | /tests/lean/run/vector2.lean | 287e9c0cc37131e61e7fdd082cc32c3605987e30 | [
"Apache-2.0"
] | permissive | tjiaqi/lean | 4634d729795c164664d10d093f3545287c76628f | d0ce4cf62f4246b0600c07e074d86e51f2195e30 | refs/heads/master | 1,622,323,796,480 | 1,422,643,069,000 | 1,422,643,069,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 379 | lean | import logic data.nat.basic
open nat
inductive vector (A : Type) : nat → Type :=
vnil : vector A zero,
vcons : Π {n : nat}, A → vector A n → vector A (succ n)
namespace vector
theorem vcons.inj₁ {A : Type} {n : nat} (a₁ a₂ : A) (v₁ v₂ : vector A n) : vcons a₁ v₁ = vcons a₂ v₂ → a₁ = a₂ :=
assume h, no_confusion h (λ n h t, h)
end vector
|
3f0989bfccca5e0b1fdc7dc5810394cfd4349cff | 5c7fe6c4a9d4079b5457ffa5f061797d42a1cd65 | /src/exercises/src_15_negating_quantifiers.lean | 10c44cac1a856bebea3e8c605fc2b30603642b5f | [] | no_license | gihanmarasingha/mth1001_tutorial | 8e0817feeb96e7c1bb3bac49b63e3c9a3a329061 | bb277eebd5013766e1418365b91416b406275130 | refs/heads/master | 1,675,008,746,310 | 1,607,993,443,000 | 1,607,993,443,000 | 321,511,270 | 3 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 1,662 | lean | import tactic.linarith
namespace mth1001
section negating_quantifiers
/-
We *can* deal with negated quantified statements using `intro`, as we did before for other
negated statements.
-/
-- Exercise 081:
-- Finish the proof below using `specialize` and `linarith`.
-- *Note*, `linarith` can prove `false` if there is a contradiction amongst the premises.
example : ¬(∀ x : ℤ, 2 * x + 6 = 0) :=
begin
intro h,
sorry
end
/-
A more natural approach is to recognise that `¬(∀ x, P x)` can always be rewritten as
`∃ x, ¬(P x)`, where `P` is any function (more accurately a *predicate*) depending on `x`.
The Lean tactic `push_neg`, which we've seen before, accomplishes this transformation.
-/
-- Exercise 082:
-- Below, `push_neg` transforms the goal into `∃ x : ℤ, 2 * x + 6 ≠ 0`.
example : ¬(∀ x : ℤ, 2 * x + 6 = 0) :=
begin
push_neg,
sorry
end
-- Exercise 083:
-- Solve the following using `push_neg`
example : ¬(∀ x : ℤ, x > 5) :=
begin
sorry
end
/-
Negating existential quantifiers follows a similar principle. We begin by using `intro`.
-/
-- Exercise 084:
-- Use `cases` and `linarith` to prove the following result.
example : ¬(∃ x : ℤ, (x + 2 ≥ 3) ∧ (2 * x + 10 ≤ 8)) :=
begin
intro h,
sorry
end
-- Exercise 085:
/-
An alternative approach is to recognise that `¬(∃ x, P x)` is equivalent to `∀ x, ¬(P x)`.
Lean offers this method via `push_neg`. Finish the proof below.
-/
example : ¬(∃ x : ℤ, (x + 2 ≥ 3) ∧ (2 * x + 10 ≤ 8)) :=
begin
push_neg,
intro x,
by_cases h : x + 2 < 3,
{ sorry, },
{ sorry, },
end
end negating_quantifiers
end mth1001
|
cf89dc0938db427aed56789e1ea94a1e64733af8 | d642a6b1261b2cbe691e53561ac777b924751b63 | /src/tactic/ext.lean | 1d6e1a62f586d7206f57bd7e929c10163a0b9a40 | [
"Apache-2.0"
] | permissive | cipher1024/mathlib | fee56b9954e969721715e45fea8bcb95f9dc03fe | d077887141000fefa5a264e30fa57520e9f03522 | refs/heads/master | 1,651,806,490,504 | 1,573,508,694,000 | 1,573,508,694,000 | 107,216,176 | 0 | 0 | Apache-2.0 | 1,647,363,136,000 | 1,508,213,014,000 | Lean | UTF-8 | Lean | false | false | 9,282 | lean | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import tactic.rcases data.list.defs data.prod data.sum
universes u₁ u₂
open interactive interactive.types
open lean.parser nat tactic
/--
`derive_struct_ext_lemma n` generates an extensionality lemma based on
the equality of all non-propositional projections.
On the following:
```
@[ext]
structure foo (α : Type*) :=
(x y : ℕ)
(z : {z // z < x})
(k : α)
(h : x < y)
```
`derive_struct_lemma` generates:
```
foo.ext : ∀ {α : Type u_1} (x y : foo α), x.x = y.x → x.y = y.y → x.z == y.z → x.k = y.k → x = y
```
-/
meta def derive_struct_ext_lemma (n : name) : tactic name :=
do e ← get_env,
fs ← e.structure_fields n,
d ← get_decl n,
n ← resolve_constant n,
let r := @expr.const tt n $ d.univ_params.map level.param,
(args,_) ← infer_type r >>= mk_local_pis,
let args := args.map expr.to_implicit,
let t := r.mk_app args,
x ← mk_local_def `x t,
y ← mk_local_def `y t,
let args_x := args ++ [x],
let args_y := args ++ [y],
bs ← fs.mmap $ λ f,
do { d ← get_decl (n ++ f),
let a := @expr.const tt (n ++ f) $ d.univ_params.map level.param,
t ← infer_type a,
s ← infer_type t,
if s ≠ `(Prop)
then do
let x := a.mk_app args_x,
let y := a.mk_app args_y,
t ← infer_type x,
t' ← infer_type y,
some <$> if t = t'
then mk_app `eq [x,y] >>= mk_local_def `h
else mk_mapp `heq [none,x,none,y] >>= mk_local_def `h
else pure none },
let bs := bs.filter_map id,
t ← mk_app `eq [x,y],
t ← pis (args ++ [x,y] ++ bs) t,
pr ← run_async $
do { (_,pr) ← solve_aux t (do
{ args ← intron args.length,
x ← intro1, y ← intro1,
cases x, cases y,
bs.mmap' (λ _,
do e ← intro1,
cases e),
reflexivity }),
instantiate_mvars pr },
let decl_n := n <.> "ext",
decl_n <$ add_decl (declaration.thm decl_n d.univ_params t pr)
meta def get_ext_subject : expr → tactic name
| (expr.pi n bi d b) :=
do v ← mk_local' n bi d,
b' ← whnf $ b.instantiate_var v,
get_ext_subject b'
| (expr.app _ e) :=
do t ← infer_type e >>= instantiate_mvars >>= head_beta,
if t.get_app_fn.is_constant then
pure $ t.get_app_fn.const_name
else if t.is_pi then
pure $ name.mk_numeral 0 name.anonymous
else if t.is_sort then
pure $ name.mk_numeral 1 name.anonymous
else do
t ← pp t,
fail format!"only constants and Pi types are supported: {t}"
| e := fail format!"Only expressions of the form `_ → _ → ... → R ... e are supported: {e}"
open native
@[reducible] def ext_param_type := option name ⊕ option name
meta def opt_minus : lean.parser (option name → ext_param_type) :=
sum.inl <$ tk "-" <|> pure sum.inr
meta def ext_param :=
opt_minus <*> ( name.mk_numeral 0 name.anonymous <$ brackets "(" ")" (tk "→" <|> tk "->") <|>
none <$ tk "*" <|>
some <$> ident )
meta def saturate_fun : name → tactic expr
| (name.mk_numeral 0 name.anonymous) :=
do v₀ ← mk_mvar,
v₁ ← mk_mvar,
return $ v₀.imp v₁
| (name.mk_numeral 1 name.anonymous) :=
do u ← mk_meta_univ,
pure $ expr.sort u
| n :=
do e ← resolve_constant n >>= mk_const,
a ← get_arity e,
e.mk_app <$> (list.iota a).mmap (λ _, mk_mvar)
meta def equiv_type_constr (n n' : name) : tactic unit :=
do e ← saturate_fun n,
e' ← saturate_fun n',
unify e e' <|> fail format!"{n} and {n'} are not definitionally equal types"
/--
Tag lemmas of the form:
```
@[ext]
lemma my_collection.ext (a b : my_collection)
(h : ∀ x, a.lookup x = b.lookup y) :
a = b := ...
```
The attribute indexes extensionality lemma using the type of the
objects (i.e. `my_collection`) which it gets from the statement of
the lemma. In some cases, the same lemma can be used to state the
extensionality of multiple types that are definitionally equivalent.
```
attribute [ext [(→),thunk,stream]] funext
```
Those parameters are cumulative. The following are equivalent:
```
attribute [ext [(→),thunk]] funext
attribute [ext [stream]] funext
```
and
```
attribute [ext [(→),thunk,stream]] funext
```
One removes type names from the list for one lemma with:
```
attribute [ext [-stream,-thunk]] funext
```
Also, the following:
```
@[ext]
lemma my_collection.ext (a b : my_collection)
(h : ∀ x, a.lookup x = b.lookup y) :
a = b := ...
```
is equivalent to
```
@[ext *]
lemma my_collection.ext (a b : my_collection)
(h : ∀ x, a.lookup x = b.lookup y) :
a = b := ...
```
This allows us specify type synonyms along with the type
that referred to in the lemma statement.
```
@[ext [*,my_type_synonym]]
lemma my_collection.ext (a b : my_collection)
(h : ∀ x, a.lookup x = b.lookup y) :
a = b := ...
```
Attribute `ext` can be applied to a structure to generate its extensionality lemma:
```
@[ext]
structure foo (α : Type*) :=
(x y : ℕ)
(z : {z // z < x})
(k : α)
(h : x < y)
```
will generate:
```
@[ext] lemma foo.ext : ∀ {α : Type u_1} (x y : foo α), x.x = y.x → x.y = y.y → x.z == y.z → x.k = y.k → x = y
```
-/
@[user_attribute]
meta def extensional_attribute : user_attribute (name_map name) (bool × list ext_param_type × list name × list (name × name)) :=
{ name := `ext,
descr := "lemmas usable by `ext` tactic",
cache_cfg := { mk_cache := λ ls,
do { attrs ← ls.mmap $ λ l,
do { ⟨_,_,ls,_⟩ ← extensional_attribute.get_param l,
pure $ prod.mk <$> ls <*> pure l },
pure $ rb_map.of_list $ attrs.join },
dependencies := [] },
parser :=
do { ls ← pure <$> ext_param <|> list_of ext_param <|> pure [],
m ← extensional_attribute.get_cache,
pure $ (ff,ls,[],m.to_list) },
after_set := some $ λ n _ b,
do (ff,ls,_,ls') ← extensional_attribute.get_param n | pure (),
e ← get_env,
n ← if (e.structure_fields n).is_some
then derive_struct_ext_lemma n
else pure n,
s ← mk_const n >>= infer_type >>= get_ext_subject,
let (rs,ls'') := if ls.empty
then ([],[s])
else ls.partition_map (sum.map (flip option.get_or_else s) (flip option.get_or_else s)),
ls''.mmap' (equiv_type_constr s),
let l := ls'' ∪ (ls'.filter $ λ l, prod.snd l = n).map prod.fst \ rs,
extensional_attribute.set n (tt,[],l,[]) b }
attribute [ext] array.ext propext prod.ext
attribute [ext [(→),thunk]] _root_.funext
namespace ulift
@[ext] lemma ext {α : Type u₁} (X Y : ulift.{u₂} α) (w : X.down = Y.down) : X = Y :=
begin
cases X, cases Y, dsimp at w, rw w,
end
end ulift
namespace plift
@[ext] lemma ext {P : Prop} (a b : plift P) : a = b :=
begin
cases a, cases b, refl
end
end plift
namespace tactic
meta def try_intros : ext_patt → tactic ext_patt
| [] := try intros $> []
| (x::xs) :=
do tgt ← target >>= whnf,
if tgt.is_pi
then rintro [x] >> try_intros xs
else pure (x :: xs)
meta def ext1 (xs : ext_patt) (cfg : apply_cfg := {}) : tactic ext_patt :=
do subject ← target >>= get_ext_subject,
m ← extensional_attribute.get_cache,
do { rule ← m.find subject,
applyc rule cfg } <|>
do { ls ← attribute.get_instances `ext,
ls.any_of (λ n, applyc n cfg) } <|>
fail format!"no applicable extensionality rule found for {subject}",
try_intros xs
meta def ext : ext_patt → option ℕ → tactic unit
| _ (some 0) := skip
| xs n := focus1 $ do
ys ← ext1 xs, try (ext ys (nat.pred <$> n))
local postfix `?`:9001 := optional
local postfix *:9001 := many
/--
`ext1 id` selects and apply one extensionality lemma (with attribute
`ext`), using `id`, if provided, to name a local constant
introduced by the lemma. If `id` is omitted, the local constant is
named automatically, as per `intro`.
-/
meta def interactive.ext1 (xs : parse ext_parse) : tactic unit :=
ext1 xs $> ()
/--
- `ext` applies as many extensionality lemmas as possible;
- `ext ids`, with `ids` a list of identifiers, finds extentionality and applies them
until it runs out of identifiers in `ids` to name the local constants.
When trying to prove:
```
α β : Type,
f g : α → set β
⊢ f = g
```
applying `ext x y` yields:
```
α β : Type,
f g : α → set β,
x : α,
y : β
⊢ y ∈ f x ↔ y ∈ f x
```
by applying functional extensionality and set extensionality.
A maximum depth can be provided with `ext x y z : 3`.
-/
meta def interactive.ext : parse ext_parse → parse (tk ":" *> small_nat)? → tactic unit
| [] (some n) := iterate_range 1 n (ext1 [] $> ())
| [] none := repeat1 (ext1 [] $> ())
| xs n := tactic.ext xs n
end tactic
|
93930b9195d055241185e1d1b92a9c9faf4adbd8 | 4d2583807a5ac6caaffd3d7a5f646d61ca85d532 | /src/linear_algebra/matrix/transvection.lean | 380e89a252cd1c253d1e95acc737f1678431b8ac | [
"Apache-2.0"
] | permissive | AntoineChambert-Loir/mathlib | 64aabb896129885f12296a799818061bc90da1ff | 07be904260ab6e36a5769680b6012f03a4727134 | refs/heads/master | 1,693,187,631,771 | 1,636,719,886,000 | 1,636,719,886,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 33,163 | lean | /-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import linear_algebra.matrix.determinant
import linear_algebra.matrix.trace
import linear_algebra.matrix.reindex
import tactic.field_simp
import data.matrix.basis
/-!
# Transvections
Transvections are matrices of the form `1 + std_basis_matrix i j c`, where `std_basis_matrix i j c`
is the basic matrix with a `c` at position `(i, j)`. Multiplying by such a transvection on the left
(resp. on the right) amounts to adding `c` times the `j`-th row to to the `i`-th row
(resp `c` times the `i`-th column to the `j`-th column). Therefore, they are useful to present
algorithms operating on rows and columns.
Transvections are a special case of *elementary matrices* (according to most references, these also
contain the matrices exchanging rows, and the matrices multiplying a row by a constant).
We show that, over a field, any matrix can be written as `L ⬝ D ⬝ L'`, where `L` and `L'` are
products of transvections and `D` is diagonal. In other words, one can reduce a matrix to diagonal
form by operations on its rows and columns, a variant of Gauss' pivot algorithm.
## Main definitions and results
* `transvection i j c` is the matrix equal to `1 + std_basis_matrix i j c`.
* `transvection_struct n R` is a structure containing the data of `i, j, c` and a proof that
`i ≠ j`. These are often easier to manipulate than straight matrices, especially in inductive
arguments.
* `exists_list_transvec_mul_diagonal_mul_list_transvec` states that any matrix `M` over a field can
be written in the form `t_1 ⬝ ... ⬝ t_k ⬝ D ⬝ t'_1 ⬝ ... ⬝ t'_l`, where `D` is diagonal and
the `t_i`, `t'_j` are transvections.
* `diagonal_transvection_induction` shows that a property which is true for diagonal matrices and
transvections, and invariant under product, is true for all matrices.
* `diagonal_transvection_induction_of_det_ne_zero` is the same statement over invertible matrices.
## Implementation details
The proof of the reduction results is done inductively on the size of the matrices, reducing an
`(r + 1) × (r + 1)` matrix to a matrix whose last row and column are zeroes, except possibly for
the last diagonal entry. This step is done as follows.
If all the coefficients on the last row and column are zero, there is nothing to do. Otherwise,
one can put a nonzero coefficient in the last diagonal entry by a row or column operation, and then
subtract this last diagonal entry from the other entries in the last row and column to make them
vanish.
This step is done in the type `fin r ⊕ unit`, where `fin r` is useful to choose arbitrarily some
order in which we cancel the coefficients, and the sum structure is useful to use the formalism of
block matrices.
To proceed with the induction, we reindex our matrices to reduce to the above situation.
-/
universes u₁ u₂
namespace matrix
open_locale matrix
variables (n p : Type*) (R : Type u₂) {𝕜 : Type*} [field 𝕜]
variables [decidable_eq n] [decidable_eq p]
variables [comm_ring R]
section transvection
variables {R n} (i j : n)
/-- The transvection matrix `transvection i j c` is equal to the identity plus `c` at position
`(i, j)`. Multiplying by it on the left (as in `transvection i j c ⬝ M`) corresponds to adding
`c` times the `j`-th line of `M` to its `i`-th line. Multiplying by it on the right corresponds
to adding `c` times the `i`-th column to the `j`-th column. -/
def transvection (c : R) : matrix n n R := 1 + matrix.std_basis_matrix i j c
@[simp] lemma transvection_zero : transvection i j (0 : R) = 1 :=
by simp [transvection]
section
variable [fintype n]
/-- A transvection matrix is obtained from the identity by adding `c` times the `j`-th row to
the `i`-th row. -/
lemma update_row_eq_transvection (c : R) :
update_row (1 : matrix n n R) i (((1 : matrix n n R)) i + c • (1 : matrix n n R) j) =
transvection i j c :=
begin
ext a b,
by_cases ha : i = a; by_cases hb : j = b,
{ simp only [update_row, transvection, ha, hb, function.update_same, std_basis_matrix.apply_same,
pi.add_apply, one_apply_eq, pi.smul_apply, mul_one, algebra.id.smul_eq_mul], },
{ simp only [update_row, transvection, ha, hb, std_basis_matrix.apply_of_ne, function.update_same,
pi.add_apply, ne.def, not_false_iff, pi.smul_apply, and_false, one_apply_ne,
algebra.id.smul_eq_mul, mul_zero] },
{ simp only [update_row, transvection, ha, ne.symm ha, std_basis_matrix.apply_of_ne, add_zero,
algebra.id.smul_eq_mul, function.update_noteq, ne.def, not_false_iff, dmatrix.add_apply,
pi.smul_apply, mul_zero, false_and] },
{ simp only [update_row, transvection, ha, hb, ne.symm ha, std_basis_matrix.apply_of_ne, add_zero,
algebra.id.smul_eq_mul, function.update_noteq, ne.def, not_false_iff, and_self,
dmatrix.add_apply, pi.smul_apply, mul_zero] }
end
lemma transvection_mul_transvection_same (h : i ≠ j) (c d : R) :
transvection i j c ⬝ transvection i j d = transvection i j (c + d) :=
by simp [transvection, matrix.add_mul, matrix.mul_add, h, h.symm, add_smul, add_assoc,
std_basis_matrix_add]
@[simp] lemma transvection_mul_apply_same (b : n) (c : R) (M : matrix n n R) :
(transvection i j c ⬝ M) i b = M i b + c * M j b :=
by simp [transvection, matrix.add_mul]
@[simp] lemma mul_transvection_apply_same (a : n) (c : R) (M : matrix n n R) :
(M ⬝ transvection i j c) a j = M a j + c * M a i :=
by simp [transvection, matrix.mul_add, mul_comm]
@[simp] lemma transvection_mul_apply_of_ne (a b : n) (ha : a ≠ i) (c : R) (M : matrix n n R) :
(transvection i j c ⬝ M) a b = M a b :=
by simp [transvection, matrix.add_mul, ha]
@[simp] lemma mul_transvection_apply_of_ne (a b : n) (hb : b ≠ j) (c : R) (M : matrix n n R) :
(M ⬝ transvection i j c) a b = M a b :=
by simp [transvection, matrix.mul_add, hb]
@[simp] lemma det_transvection_of_ne (h : i ≠ j) (c : R) : det (transvection i j c) = 1 :=
by rw [← update_row_eq_transvection i j, det_update_row_add_smul_self _ h, det_one]
end
variables (R n)
/-- A structure containing all the information from which one can build a nontrivial transvection.
This structure is easier to manipulate than transvections as one has a direct access to all the
relevant fields. -/
@[nolint has_inhabited_instance]
structure transvection_struct :=
(i j : n)
(hij : i ≠ j)
(c : R)
instance [nontrivial n] : nonempty (transvection_struct n R) :=
by { choose x y hxy using exists_pair_ne n, exact ⟨⟨x, y, hxy, 0⟩⟩ }
namespace transvection_struct
variables {R n}
/-- Associating to a `transvection_struct` the corresponding transvection matrix. -/
def to_matrix (t : transvection_struct n R) : matrix n n R :=
transvection t.i t.j t.c
@[simp] lemma to_matrix_mk (i j : n) (hij : i ≠ j) (c : R) :
transvection_struct.to_matrix ⟨i, j, hij, c⟩ = transvection i j c := rfl
@[simp] protected lemma det [fintype n] (t : transvection_struct n R) : det t.to_matrix = 1 :=
det_transvection_of_ne _ _ t.hij _
@[simp] lemma det_to_matrix_prod [fintype n] (L : list (transvection_struct n 𝕜)) :
det ((L.map to_matrix).prod) = 1 :=
begin
induction L with t L IH,
{ simp },
{ simp [IH], }
end
/-- The inverse of a `transvection_struct`, designed so that `t.inv.to_matrix` is the inverse of
`t.to_matrix`. -/
@[simps] protected def inv (t : transvection_struct n R) : transvection_struct n R :=
{ i := t.i,
j := t.j,
hij := t.hij,
c := - t.c }
section
variable [fintype n]
lemma inv_mul (t : transvection_struct n R) :
t.inv.to_matrix ⬝ t.to_matrix = 1 :=
by { rcases t, simp [to_matrix, transvection_mul_transvection_same, t_hij] }
lemma mul_inv (t : transvection_struct n R) :
t.to_matrix ⬝ t.inv.to_matrix = 1 :=
by { rcases t, simp [to_matrix, transvection_mul_transvection_same, t_hij] }
lemma reverse_inv_prod_mul_prod (L : list (transvection_struct n R)) :
(L.reverse.map (to_matrix ∘ transvection_struct.inv)).prod ⬝ (L.map to_matrix).prod = 1 :=
begin
induction L with t L IH,
{ simp },
{ suffices : (L.reverse.map (to_matrix ∘ transvection_struct.inv)).prod ⬝
(t.inv.to_matrix ⬝ t.to_matrix) ⬝ (L.map to_matrix).prod = 1, by simpa [matrix.mul_assoc],
simpa [inv_mul] using IH, }
end
lemma prod_mul_reverse_inv_prod (L : list (transvection_struct n R)) :
(L.map to_matrix).prod ⬝ (L.reverse.map (to_matrix ∘ transvection_struct.inv)).prod = 1 :=
begin
induction L with t L IH,
{ simp },
{ suffices : t.to_matrix ⬝ ((L.map to_matrix).prod ⬝
(L.reverse.map (to_matrix ∘ transvection_struct.inv)).prod) ⬝ t.inv.to_matrix = 1,
by simpa [matrix.mul_assoc],
simp_rw [IH, matrix.mul_one, t.mul_inv], }
end
end
variables (p)
open sum
/-- Given a `transvection_struct` on `n`, define the corresponding `transvection_struct` on `n ⊕ p`
using the identity on `p`. -/
def sum_inl (t : transvection_struct n R) : transvection_struct (n ⊕ p) R :=
{ i := inl t.i,
j := inl t.j,
hij := by simp [t.hij],
c := t.c }
lemma to_matrix_sum_inl (t : transvection_struct n R) :
(t.sum_inl p).to_matrix = from_blocks t.to_matrix 0 0 1 :=
begin
cases t,
ext a b,
cases a; cases b,
{ by_cases h : a = b;
simp [transvection_struct.sum_inl, transvection, h, std_basis_matrix], },
{ simp [transvection_struct.sum_inl, transvection] },
{ simp [transvection_struct.sum_inl, transvection] },
{ by_cases h : a = b;
simp [transvection_struct.sum_inl, transvection, h] },
end
@[simp] lemma sum_inl_to_matrix_prod_mul [fintype n] [fintype p]
(M : matrix n n R) (L : list (transvection_struct n R)) (N : matrix p p R) :
(L.map (to_matrix ∘ sum_inl p)).prod ⬝ from_blocks M 0 0 N
= from_blocks ((L.map to_matrix).prod ⬝ M) 0 0 N :=
begin
induction L with t L IH,
{ simp },
{ simp [matrix.mul_assoc, IH, to_matrix_sum_inl, from_blocks_multiply], },
end
@[simp] lemma mul_sum_inl_to_matrix_prod [fintype n] [fintype p]
(M : matrix n n R) (L : list (transvection_struct n R)) (N : matrix p p R) :
(from_blocks M 0 0 N) ⬝ (L.map (to_matrix ∘ sum_inl p)).prod
= from_blocks (M ⬝ (L.map to_matrix).prod) 0 0 N :=
begin
induction L with t L IH generalizing M N,
{ simp },
{ simp [IH, to_matrix_sum_inl, from_blocks_multiply], },
end
variable {p}
/-- Given a `transvection_struct` on `n` and an equivalence between `n` and `p`, define the
corresponding `transvection_struct` on `p`. -/
def reindex_equiv (e : n ≃ p) (t : transvection_struct n R) : transvection_struct p R :=
{ i := e t.i,
j := e t.j,
hij := by simp [t.hij],
c := t.c }
variables [fintype n] [fintype p]
lemma to_matrix_reindex_equiv (e : n ≃ p) (t : transvection_struct n R) :
(t.reindex_equiv e).to_matrix = reindex_alg_equiv R e t.to_matrix :=
begin
cases t,
ext a b,
simp only [reindex_equiv, transvection, mul_boole, algebra.id.smul_eq_mul, to_matrix_mk,
minor_apply, reindex_apply, dmatrix.add_apply, pi.smul_apply, reindex_alg_equiv_apply],
by_cases ha : e t_i = a; by_cases hb : e t_j = b; by_cases hab : a = b;
simp [ha, hb, hab, ← e.apply_eq_iff_eq_symm_apply, std_basis_matrix]
end
lemma to_matrix_reindex_equiv_prod (e : n ≃ p) (L : list (transvection_struct n R)) :
(L.map (to_matrix ∘ (reindex_equiv e))).prod = reindex_alg_equiv R e (L.map to_matrix).prod :=
begin
induction L with t L IH,
{ simp },
{ simp only [to_matrix_reindex_equiv, IH, function.comp_app, list.prod_cons, mul_eq_mul,
reindex_alg_equiv_apply, list.map],
exact (reindex_alg_equiv_mul _ _ _ _).symm }
end
end transvection_struct
end transvection
/-!
# Reducing matrices by left and right multiplication by transvections
In this section, we show that any matrix can be reduced to diagonal form by left and right
multiplication by transvections (or, equivalently, by elementary operations on lines and columns).
The main step is to kill the last row and column of a matrix in `fin r ⊕ unit` with nonzero last
coefficient, by subtracting this coefficient from the other ones. The list of these operations is
recorded in `list_transvec_col M` and `list_transvec_row M`. We have to analyze inductively how
these operations affect the coefficients in the last row and the last column to conclude that they
have the desired effect.
Once this is done, one concludes the reduction by induction on the size
of the matrices, through a suitable reindexing to identify any fintype with `fin r ⊕ unit`.
-/
namespace pivot
variables {R} {r : ℕ} (M : matrix (fin r ⊕ unit) (fin r ⊕ unit) 𝕜)
open sum unit fin transvection_struct
/-- A list of transvections such that multiplying on the left with these transvections will replace
the last column with zeroes. -/
def list_transvec_col : list (matrix (fin r ⊕ unit) (fin r ⊕ unit) 𝕜) :=
list.of_fn $ λ i : fin r, transvection (inl i) (inr star) $
-M (inl i) (inr star) / M (inr star) (inr star)
/-- A list of transvections such that multiplying on the right with these transvections will replace
the last row with zeroes. -/
def list_transvec_row : list (matrix (fin r ⊕ unit) (fin r ⊕ unit) 𝕜) :=
list.of_fn $ λ i : fin r, transvection (inr star) (inl i) $
-M (inr star) (inl i) / M (inr star) (inr star)
/-- Multiplying by some of the matrices in `list_transvec_col M` does not change the last row. -/
lemma list_transvec_col_mul_last_row_drop (i : fin r ⊕ unit) {k : ℕ} (hk : k ≤ r) :
(((list_transvec_col M).drop k).prod ⬝ M) (inr star) i = M (inr star) i :=
begin
apply nat.decreasing_induction' _ hk,
{ simp only [list_transvec_col, list.length_of_fn, matrix.one_mul, list.drop_eq_nil_of_le,
list.prod_nil], },
{ assume n hn hk IH,
have hn' : n < (list_transvec_col M).length, by simpa [list_transvec_col] using hn,
rw ← list.cons_nth_le_drop_succ hn',
simpa [list_transvec_col, matrix.mul_assoc] }
end
/-- Multiplying by all the matrices in `list_transvec_col M` does not change the last row. -/
lemma list_transvec_col_mul_last_row (i : fin r ⊕ unit) :
((list_transvec_col M).prod ⬝ M) (inr star) i = M (inr star) i :=
by simpa using list_transvec_col_mul_last_row_drop M i (zero_le _)
/-- Multiplying by all the matrices in `list_transvec_col M` kills all the coefficients in the
last column but the last one. -/
lemma list_transvec_col_mul_last_col (hM : M (inr star) (inr star) ≠ 0) (i : fin r) :
((list_transvec_col M).prod ⬝ M) (inl i) (inr star) = 0 :=
begin
suffices H : ∀ (k : ℕ), k ≤ r → (((list_transvec_col M).drop k).prod ⬝ M) (inl i) (inr star) =
if k ≤ i then 0 else M (inl i) (inr star),
by simpa only [if_true, list.drop.equations._eqn_1] using H 0 (zero_le _),
assume k hk,
apply nat.decreasing_induction' _ hk,
{ simp only [list_transvec_col, list.length_of_fn, matrix.one_mul, list.drop_eq_nil_of_le,
list.prod_nil],
rw if_neg,
simpa only [not_le] using i.2 },
{ assume n hn hk IH,
have hn' : n < (list_transvec_col M).length, by simpa [list_transvec_col] using hn,
let n' : fin r := ⟨n, hn⟩,
rw ← list.cons_nth_le_drop_succ hn',
have A : (list_transvec_col M).nth_le n hn' = transvection (inl n') (inr star)
(-M (inl n') (inr star) / M (inr star) (inr star)), by simp [list_transvec_col],
simp only [matrix.mul_assoc, A, matrix.mul_eq_mul, list.prod_cons],
by_cases h : n' = i,
{ have hni : n = i,
{ cases i, simp only [subtype.mk_eq_mk] at h, simp [h] },
rw [h, transvection_mul_apply_same, IH, list_transvec_col_mul_last_row_drop _ _ hn, ← hni],
field_simp [hM] },
{ have hni : n ≠ i,
{ rintros rfl, cases i, simpa using h },
simp only [transvection_mul_apply_of_ne, ne.def, not_false_iff, ne.symm h],
rw IH,
rcases le_or_lt (n+1) i with hi|hi,
{ simp only [hi, n.le_succ.trans hi, if_true] },
{ rw [if_neg, if_neg],
{ simpa only [hni.symm, not_le, or_false] using nat.lt_succ_iff_lt_or_eq.1 hi },
{ simpa only [not_le] using hi } } } }
end
/-- Multiplying by some of the matrices in `list_transvec_row M` does not change the last column. -/
lemma mul_list_transvec_row_last_col_take (i : fin r ⊕ unit) {k : ℕ} (hk : k ≤ r) :
(M ⬝ ((list_transvec_row M).take k).prod) i (inr star) = M i (inr star) :=
begin
induction k with k IH,
{ simp only [matrix.mul_one, list.take_zero, list.prod_nil], },
{ have hkr : k < r := hk,
let k' : fin r := ⟨k, hkr⟩,
have : (list_transvec_row M).nth k = ↑(transvection (inr unit.star) (inl k')
(-M (inr unit.star) (inl k') / M (inr unit.star) (inr unit.star))),
{ simp only [list_transvec_row, list.of_fn_nth_val, hkr, dif_pos, list.nth_of_fn], refl },
simp only [list.take_succ, ← matrix.mul_assoc, this, list.prod_append, matrix.mul_one,
matrix.mul_eq_mul, list.prod_cons, list.prod_nil, option.to_list_some],
rw [mul_transvection_apply_of_ne, IH hkr.le],
simp only [ne.def, not_false_iff], }
end
/-- Multiplying by all the matrices in `list_transvec_row M` does not change the last column. -/
lemma mul_list_transvec_row_last_col (i : fin r ⊕ unit) :
(M ⬝ (list_transvec_row M).prod) i (inr star) = M i (inr star) :=
begin
have A : (list_transvec_row M).length = r, by simp [list_transvec_row],
rw [← list.take_length (list_transvec_row M), A],
simpa using mul_list_transvec_row_last_col_take M i le_rfl,
end
/-- Multiplying by all the matrices in `list_transvec_row M` kills all the coefficients in the
last row but the last one. -/
lemma mul_list_transvec_row_last_row (hM : M (inr star) (inr star) ≠ 0) (i : fin r) :
(M ⬝ (list_transvec_row M).prod) (inr star) (inl i) = 0 :=
begin
suffices H : ∀ (k : ℕ), k ≤ r → (M ⬝ ((list_transvec_row M).take k).prod) (inr star) (inl i) =
if k ≤ i then M (inr star) (inl i) else 0,
{ have A : (list_transvec_row M).length = r, by simp [list_transvec_row],
rw [← list.take_length (list_transvec_row M), A],
have : ¬ (r ≤ i), by simpa using i.2,
simpa only [this, ite_eq_right_iff] using H r le_rfl },
assume k hk,
induction k with n IH,
{ simp only [if_true, matrix.mul_one, list.take_zero, zero_le', list.prod_nil] },
{ have hnr : n < r := hk,
let n' : fin r := ⟨n, hnr⟩,
have A : (list_transvec_row M).nth n = ↑(transvection (inr unit.star) (inl n')
(-M (inr unit.star) (inl n') / M (inr unit.star) (inr unit.star))),
{ simp only [list_transvec_row, list.of_fn_nth_val, hnr, dif_pos, list.nth_of_fn], refl },
simp only [list.take_succ, A, ← matrix.mul_assoc, list.prod_append, matrix.mul_one,
matrix.mul_eq_mul, list.prod_cons, list.prod_nil, option.to_list_some],
by_cases h : n' = i,
{ have hni : n = i,
{ cases i, simp only [subtype.mk_eq_mk] at h, simp only [h, coe_mk] },
have : ¬ (n.succ ≤ i), by simp only [← hni, n.lt_succ_self, not_le],
simp only [h, mul_transvection_apply_same, list.take, if_false,
mul_list_transvec_row_last_col_take _ _ hnr.le, hni.le, this, if_true, IH hnr.le],
field_simp [hM] },
{ have hni : n ≠ i,
{ rintros rfl, cases i, simpa using h },
simp only [IH hnr.le, ne.def, mul_transvection_apply_of_ne, not_false_iff, ne.symm h],
rcases le_or_lt (n+1) i with hi|hi,
{ simp [hi, n.le_succ.trans hi, if_true], },
{ rw [if_neg, if_neg],
{ simpa only [not_le] using hi },
{ simpa only [hni.symm, not_le, or_false] using nat.lt_succ_iff_lt_or_eq.1 hi } } } }
end
/-- Multiplying by all the matrices either in `list_transvec_col M` and `list_transvec_row M` kills
all the coefficients in the last row but the last one. -/
lemma list_transvec_col_mul_mul_list_transvec_row_last_col
(hM : M (inr star) (inr star) ≠ 0) (i : fin r) :
((list_transvec_col M).prod ⬝ M ⬝ (list_transvec_row M).prod) (inr star) (inl i) = 0 :=
begin
have : list_transvec_row M = list_transvec_row ((list_transvec_col M).prod ⬝ M),
by simp [list_transvec_row, list_transvec_col_mul_last_row],
rw this,
apply mul_list_transvec_row_last_row,
simpa [list_transvec_col_mul_last_row] using hM
end
/-- Multiplying by all the matrices either in `list_transvec_col M` and `list_transvec_row M` kills
all the coefficients in the last column but the last one. -/
lemma list_transvec_col_mul_mul_list_transvec_row_last_row
(hM : M (inr star) (inr star) ≠ 0) (i : fin r) :
((list_transvec_col M).prod ⬝ M ⬝ (list_transvec_row M).prod) (inl i) (inr star) = 0 :=
begin
have : list_transvec_col M = list_transvec_col (M ⬝ (list_transvec_row M).prod),
by simp [list_transvec_col, mul_list_transvec_row_last_col],
rw [this, matrix.mul_assoc],
apply list_transvec_col_mul_last_col,
simpa [mul_list_transvec_row_last_col] using hM
end
/-- Multiplying by all the matrices either in `list_transvec_col M` and `list_transvec_row M` turns
the matrix in block-diagonal form. -/
lemma is_two_block_diagonal_list_transvec_col_mul_mul_list_transvec_row
(hM : M (inr star) (inr star) ≠ 0) :
is_two_block_diagonal ((list_transvec_col M).prod ⬝ M ⬝ (list_transvec_row M).prod) :=
begin
split,
{ ext i j,
have : j = star, by simp only [eq_iff_true_of_subsingleton],
simp [to_blocks₁₂, this, list_transvec_col_mul_mul_list_transvec_row_last_row M hM] },
{ ext i j,
have : i = star, by simp only [eq_iff_true_of_subsingleton],
simp [to_blocks₂₁, this, list_transvec_col_mul_mul_list_transvec_row_last_col M hM] },
end
/-- There exist two lists of `transvection_struct` such that multiplying by them on the left and
on the right makes a matrix block-diagonal, when the last coefficient is nonzero. -/
lemma exists_is_two_block_diagonal_of_ne_zero (hM : M (inr star) (inr star) ≠ 0) :
∃ (L L' : list (transvection_struct (fin r ⊕ unit) 𝕜)),
is_two_block_diagonal ((L.map to_matrix).prod ⬝ M ⬝ (L'.map to_matrix).prod) :=
begin
let L : list (transvection_struct (fin r ⊕ unit) 𝕜) :=
list.of_fn (λ i : fin r, ⟨inl i, inr star, by simp,
-M (inl i) (inr star) / M (inr star) (inr star)⟩),
let L' : list (transvection_struct (fin r ⊕ unit) 𝕜) :=
list.of_fn (λ i : fin r, ⟨inr star, inl i, by simp,
-M (inr star) (inl i) / M (inr star) (inr star)⟩),
refine ⟨L, L', _⟩,
have A : L.map to_matrix = list_transvec_col M, by simp [L, list_transvec_col, (∘)],
have B : L'.map to_matrix = list_transvec_row M, by simp [L, list_transvec_row, (∘)],
rw [A, B],
exact is_two_block_diagonal_list_transvec_col_mul_mul_list_transvec_row M hM
end
/-- There exist two lists of `transvection_struct` such that multiplying by them on the left and
on the right makes a matrix block-diagonal. -/
lemma exists_is_two_block_diagonal_list_transvec_mul_mul_list_transvec
(M : matrix (fin r ⊕ unit) (fin r ⊕ unit) 𝕜) :
∃ (L L' : list (transvection_struct (fin r ⊕ unit) 𝕜)),
is_two_block_diagonal ((L.map to_matrix).prod ⬝ M ⬝ (L'.map to_matrix).prod) :=
begin
by_cases H : is_two_block_diagonal M, { refine ⟨list.nil, list.nil, by simpa using H⟩ },
-- we have already proved this when the last coefficient is nonzero
by_cases hM : M (inr star) (inr star) ≠ 0,
{ exact exists_is_two_block_diagonal_of_ne_zero M hM },
-- when the last coefficient is zero but there is a nonzero coefficient on the last row or the
-- last column, we will first put this nonzero coefficient in last position, and then argue as
-- above.
push_neg at hM,
simp [not_and_distrib, is_two_block_diagonal, to_blocks₁₂, to_blocks₂₁] at H,
have : ∃ (i : fin r), M (inl i) (inr star) ≠ 0 ∨ M (inr star) (inl i) ≠ 0,
{ cases H,
{ contrapose! H,
ext i j,
convert (H i).1,
simp only [eq_iff_true_of_subsingleton] },
{ contrapose! H,
ext i j,
convert (H j).2,
simp only [eq_iff_true_of_subsingleton] } },
rcases this with ⟨i, h|h⟩,
{ let M' := transvection (inr unit.star) (inl i) 1 ⬝ M,
have hM' : M' (inr star) (inr star) ≠ 0, by simpa [M', hM],
rcases exists_is_two_block_diagonal_of_ne_zero M' hM' with ⟨L, L', hLL'⟩,
rw matrix.mul_assoc at hLL',
refine ⟨L ++ [⟨inr star, inl i, by simp, 1⟩], L', _⟩,
simp only [list.map_append, list.prod_append, matrix.mul_one, to_matrix_mk, list.prod_cons,
list.prod_nil, mul_eq_mul, list.map, matrix.mul_assoc (L.map to_matrix).prod],
exact hLL' },
{ let M' := M ⬝ transvection (inl i) (inr star) 1,
have hM' : M' (inr star) (inr star) ≠ 0, by simpa [M', hM],
rcases exists_is_two_block_diagonal_of_ne_zero M' hM' with ⟨L, L', hLL'⟩,
refine ⟨L, ⟨inl i, inr star, by simp, 1⟩ :: L', _⟩,
simp only [←matrix.mul_assoc, to_matrix_mk, list.prod_cons, mul_eq_mul, list.map],
rw [matrix.mul_assoc (L.map to_matrix).prod],
exact hLL' }
end
/-- Inductive step for the reduction: if one knows that any size `r` matrix can be reduced to
diagonal form by elementary operations, then one deduces it for matrices over `fin r ⊕ unit`. -/
lemma exists_list_transvec_mul_mul_list_transvec_eq_diagonal_induction
(IH : ∀ (M : matrix (fin r) (fin r) 𝕜),
∃ (L₀ L₀' : list (transvection_struct (fin r) 𝕜)) (D₀ : (fin r) → 𝕜),
(L₀.map to_matrix).prod ⬝ M ⬝ (L₀'.map to_matrix).prod = diagonal D₀)
(M : matrix (fin r ⊕ unit) (fin r ⊕ unit) 𝕜) :
∃ (L L' : list (transvection_struct (fin r ⊕ unit) 𝕜)) (D : fin r ⊕ unit → 𝕜),
(L.map to_matrix).prod ⬝ M ⬝ (L'.map to_matrix).prod = diagonal D :=
begin
rcases exists_is_two_block_diagonal_list_transvec_mul_mul_list_transvec M with ⟨L₁, L₁', hM⟩,
let M' := (L₁.map to_matrix).prod ⬝ M ⬝ (L₁'.map to_matrix).prod,
let M'' := to_blocks₁₁ M',
rcases IH M'' with ⟨L₀, L₀', D₀, h₀⟩,
set c := M' (inr star) (inr star) with hc,
refine ⟨L₀.map (sum_inl unit) ++ L₁, L₁' ++ L₀'.map (sum_inl unit),
sum.elim D₀ (λ _, M' (inr star) (inr star)), _⟩,
suffices :
(L₀.map (to_matrix ∘ sum_inl unit)).prod ⬝ M' ⬝ (L₀'.map (to_matrix ∘ sum_inl unit)).prod =
diagonal (sum.elim D₀ (λ _, c)), by simpa [M', matrix.mul_assoc, c],
have : M' = from_blocks M'' 0 0 (diagonal (λ _, c)),
{ rw ← from_blocks_to_blocks M',
congr,
{ exact hM.1 },
{ exact hM.2 },
{ ext i j, rw [hc, to_blocks₂₂], congr } },
rw this,
simp [h₀],
end
variables {n p} [fintype n] [fintype p]
/-- Reduction to diagonal form by elementary operations is invariant under reindexing. -/
lemma reindex_exists_list_transvec_mul_mul_list_transvec_eq_diagonal (M : matrix p p 𝕜) (e : p ≃ n)
(H : ∃ (L L' : list (transvection_struct n 𝕜)) (D : n → 𝕜),
(L.map to_matrix).prod ⬝ (matrix.reindex_alg_equiv 𝕜 e M) ⬝ (L'.map to_matrix).prod
= diagonal D) :
∃ (L L' : list (transvection_struct p 𝕜)) (D : p → 𝕜),
(L.map to_matrix).prod ⬝ M ⬝ (L'.map to_matrix).prod = diagonal D :=
begin
rcases H with ⟨L₀, L₀', D₀, h₀⟩,
refine ⟨L₀.map (reindex_equiv e.symm), L₀'.map (reindex_equiv e.symm), D₀ ∘ e, _⟩,
have : M = reindex_alg_equiv 𝕜 e.symm (reindex_alg_equiv 𝕜 e M),
by simp only [equiv.symm_symm, minor_minor, reindex_apply, minor_id_id, equiv.symm_comp_self,
reindex_alg_equiv_apply],
rw this,
simp only [to_matrix_reindex_equiv_prod, list.map_map, reindex_alg_equiv_apply],
simp only [← reindex_alg_equiv_apply, ← reindex_alg_equiv_mul, h₀],
simp only [equiv.symm_symm, reindex_apply, minor_diagonal_equiv, reindex_alg_equiv_apply],
end
/-- Any matrix can be reduced to diagonal form by elementary operations. Formulated here on `Type 0`
because we will make an induction using `fin r`.
See `exists_list_transvec_mul_mul_list_transvec_eq_diagonal` for the general version (which follows
from this one and reindexing). -/
lemma exists_list_transvec_mul_mul_list_transvec_eq_diagonal_aux
(n : Type) [fintype n] [decidable_eq n]
(M : matrix n n 𝕜) : ∃ (L L' : list (transvection_struct n 𝕜)) (D : n → 𝕜),
(L.map to_matrix).prod ⬝ M ⬝ (L'.map to_matrix).prod = diagonal D :=
begin
unfreezingI { induction hn : fintype.card n with r IH generalizing n M },
{ refine ⟨list.nil, list.nil, λ _, 1, _⟩,
ext i j,
rw fintype.card_eq_zero_iff at hn,
exact hn.elim' i },
{ have e : n ≃ fin r ⊕ unit,
{ refine fintype.equiv_of_card_eq _,
rw hn,
convert (@fintype.card_sum (fin r) unit _ _).symm,
simp },
apply reindex_exists_list_transvec_mul_mul_list_transvec_eq_diagonal M e,
apply exists_list_transvec_mul_mul_list_transvec_eq_diagonal_induction
(λ N, IH (fin r) N (by simp)) }
end
/-- Any matrix can be reduced to diagonal form by elementary operations. -/
theorem exists_list_transvec_mul_mul_list_transvec_eq_diagonal
(M : matrix n n 𝕜) : ∃ (L L' : list (transvection_struct n 𝕜)) (D : n → 𝕜),
(L.map to_matrix).prod ⬝ M ⬝ (L'.map to_matrix).prod = diagonal D :=
begin
have e : n ≃ fin (fintype.card n) := fintype.equiv_of_card_eq (by simp),
apply reindex_exists_list_transvec_mul_mul_list_transvec_eq_diagonal M e,
apply exists_list_transvec_mul_mul_list_transvec_eq_diagonal_aux
end
/-- Any matrix can be written as the product of transvections, a diagonal matrix, and
transvections.-/
theorem exists_list_transvec_mul_diagonal_mul_list_transvec
(M : matrix n n 𝕜) : ∃ (L L' : list (transvection_struct n 𝕜)) (D : n → 𝕜),
M = (L.map to_matrix).prod ⬝ diagonal D ⬝ (L'.map to_matrix).prod :=
begin
rcases exists_list_transvec_mul_mul_list_transvec_eq_diagonal M with ⟨L, L', D, h⟩,
refine ⟨L.reverse.map transvection_struct.inv, L'.reverse.map transvection_struct.inv, D, _⟩,
suffices : M =
((L.reverse.map (to_matrix ∘ transvection_struct.inv)).prod ⬝ (L.map to_matrix).prod) ⬝ M ⬝
((L'.map to_matrix).prod ⬝ (L'.reverse.map (to_matrix ∘ transvection_struct.inv)).prod),
by simpa [← h, matrix.mul_assoc],
rw [reverse_inv_prod_mul_prod, prod_mul_reverse_inv_prod, matrix.one_mul, matrix.mul_one],
end
end pivot
open pivot transvection_struct
variables {n} [fintype n]
/-- Induction principle for matrices based on transvections: if a property is true for all diagonal
matrices, all transvections, and is stable under product, then it is true for all matrices. This is
the useful way to say that matrices are generated by diagonal matrices and transvections.
We state a slightly more general version: to prove a property for a matrix `M`, it suffices to
assume that the diagonal matrices we consider have the same determinant as `M`. This is useful to
obtain similar principles for `SLₙ` or `GLₙ`. -/
lemma diagonal_transvection_induction (P : matrix n n 𝕜 → Prop) (M : matrix n n 𝕜)
(hdiag : ∀ D : n → 𝕜, det (diagonal D) = det M → P (diagonal D))
(htransvec : ∀ (t : transvection_struct n 𝕜), P t.to_matrix)
(hmul : ∀ A B, P A → P B → P (A ⬝ B)) :
P M :=
begin
rcases exists_list_transvec_mul_diagonal_mul_list_transvec M with ⟨L, L', D, h⟩,
have PD : P (diagonal D) := hdiag D (by simp [h]),
suffices H : ∀ (L₁ L₂ : list (transvection_struct n 𝕜)) (E : matrix n n 𝕜),
P E → P ((L₁.map to_matrix).prod ⬝ E ⬝ (L₂.map to_matrix).prod),
by { rw h, apply H L L', exact PD },
assume L₁ L₂ E PE,
induction L₁ with t L₁ IH,
{ simp only [matrix.one_mul, list.prod_nil, list.map],
induction L₂ with t L₂ IH generalizing E,
{ simpa },
{ simp only [←matrix.mul_assoc, list.prod_cons, mul_eq_mul, list.map],
apply IH,
exact hmul _ _ PE (htransvec _) } },
{ simp only [matrix.mul_assoc, list.prod_cons, mul_eq_mul, list.map] at ⊢ IH,
exact hmul _ _ (htransvec _) IH }
end
/-- Induction principle for invertible matrices based on transvections: if a property is true for
all invertible diagonal matrices, all transvections, and is stable under product of invertible
matrices, then it is true for all invertible matrices. This is the useful way to say that
invertible matrices are generated by invertible diagonal matrices and transvections. -/
lemma diagonal_transvection_induction_of_det_ne_zero (P : matrix n n 𝕜 → Prop)
(M : matrix n n 𝕜) (hMdet : det M ≠ 0)
(hdiag : ∀ D : n → 𝕜, det (diagonal D) ≠ 0 → P (diagonal D))
(htransvec : ∀ (t : transvection_struct n 𝕜), P t.to_matrix)
(hmul : ∀ A B, det A ≠ 0 → det B ≠ 0 → P A → P B → P (A ⬝ B)) :
P M :=
begin
let Q : matrix n n 𝕜 → Prop := λ N, det N ≠ 0 ∧ P N,
have : Q M,
{ apply diagonal_transvection_induction Q M,
{ assume D hD,
have detD : det (diagonal D) ≠ 0, by { rw hD, exact hMdet },
exact ⟨detD, hdiag _ detD⟩ },
{ assume t,
exact ⟨by simp, htransvec t⟩ },
{ assume A B QA QB,
exact ⟨by simp [QA.1, QB.1], hmul A B QA.1 QB.1 QA.2 QB.2⟩ } },
exact this.2
end
end matrix
|
42a87710702adec38262b67d327d6259a7c6bf7b | dd0f5513e11c52db157d2fcc8456d9401a6cd9da | /07_Induction_and_Recursion.org.15.lean | a0b1362815440aae63b19b77cb6cd7e383e1c8cd | [] | no_license | cjmazey/lean-tutorial | ba559a49f82aa6c5848b9bf17b7389bf7f4ba645 | 381f61c9fcac56d01d959ae0fa6e376f2c4e3b34 | refs/heads/master | 1,610,286,098,832 | 1,447,124,923,000 | 1,447,124,923,000 | 43,082,433 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 156 | lean | /- page 105 -/
import standard
open nat
-- BEGIN
definition f : nat → nat → nat
| f 0 y := 1
| f x 0 := 2
| f (x+1) (y+1) := 3
-- END
|
72319a2f943bc27c1ef769fbc8babb2a9e7de952 | d406927ab5617694ec9ea7001f101b7c9e3d9702 | /src/algebra/group/with_one/basic.lean | 08b23c3badf5a48aa620090c426d5f315c3b77ad | [
"Apache-2.0"
] | permissive | alreadydone/mathlib | dc0be621c6c8208c581f5170a8216c5ba6721927 | c982179ec21091d3e102d8a5d9f5fe06c8fafb73 | refs/heads/master | 1,685,523,275,196 | 1,670,184,141,000 | 1,670,184,141,000 | 287,574,545 | 0 | 0 | Apache-2.0 | 1,670,290,714,000 | 1,597,421,623,000 | Lean | UTF-8 | Lean | false | false | 4,167 | lean | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johan Commelin
-/
import algebra.group.with_one.defs
import algebra.hom.equiv.basic
/-!
# More operations on `with_one` and `with_zero`
This file defines various bundled morphisms on `with_one` and `with_zero`
that were not available in `algebra/group/with_one/defs`.
## Main definitions
* `with_one.lift`, `with_zero.lift`
* `with_one.map`, `with_zero.map`
-/
universes u v w
variables {α : Type u} {β : Type v} {γ : Type w}
namespace with_one
section
-- workaround: we make `with_one`/`with_zero` irreducible for this definition, otherwise `simps`
-- will unfold it in the statement of the lemma it generates.
local attribute [irreducible] with_one with_zero
/-- `coe` as a bundled morphism -/
@[to_additive "`coe` as a bundled morphism", simps apply]
def coe_mul_hom [has_mul α] : α →ₙ* (with_one α) :=
{ to_fun := coe, map_mul' := λ x y, rfl }
end
section lift
local attribute [semireducible] with_one with_zero
variables [has_mul α] [mul_one_class β]
/-- Lift a semigroup homomorphism `f` to a bundled monoid homorphism. -/
@[to_additive "Lift an add_semigroup homomorphism `f` to a bundled add_monoid homorphism."]
def lift : (α →ₙ* β) ≃ (with_one α →* β) :=
{ to_fun := λ f,
{ to_fun := λ x, option.cases_on x 1 f,
map_one' := rfl,
map_mul' := λ x y,
with_one.cases_on x (by { rw one_mul, exact (one_mul _).symm }) $ λ x,
with_one.cases_on y (by { rw mul_one, exact (mul_one _).symm }) $ λ y,
f.map_mul x y },
inv_fun := λ F, F.to_mul_hom.comp coe_mul_hom,
left_inv := λ f, mul_hom.ext $ λ x, rfl,
right_inv := λ F, monoid_hom.ext $ λ x, with_one.cases_on x F.map_one.symm $ λ x, rfl }
variables (f : α →ₙ* β)
@[simp, to_additive]
lemma lift_coe (x : α) : lift f x = f x := rfl
@[simp, to_additive]
lemma lift_one : lift f 1 = 1 := rfl
@[to_additive]
theorem lift_unique (f : with_one α →* β) : f = lift (f.to_mul_hom.comp coe_mul_hom) :=
(lift.apply_symm_apply f).symm
end lift
section map
variables [has_mul α] [has_mul β] [has_mul γ]
/-- Given a multiplicative map from `α → β` returns a monoid homomorphism
from `with_one α` to `with_one β` -/
@[to_additive "Given an additive map from `α → β` returns an add_monoid homomorphism
from `with_zero α` to `with_zero β`"]
def map (f : α →ₙ* β) : with_one α →* with_one β :=
lift (coe_mul_hom.comp f)
@[simp, to_additive] lemma map_coe (f : α →ₙ* β) (a : α) : map f (a : with_one α) = f a :=
lift_coe _ _
@[simp, to_additive]
lemma map_id : map (mul_hom.id α) = monoid_hom.id (with_one α) :=
by { ext, induction x using with_one.cases_on; refl }
@[to_additive]
lemma map_map (f : α →ₙ* β) (g : β →ₙ* γ) (x) :
map g (map f x) = map (g.comp f) x :=
by { induction x using with_one.cases_on; refl }
@[simp, to_additive]
lemma map_comp (f : α →ₙ* β) (g : β →ₙ* γ) :
map (g.comp f) = (map g).comp (map f) :=
monoid_hom.ext $ λ x, (map_map f g x).symm
/-- A version of `equiv.option_congr` for `with_one`. -/
@[to_additive "A version of `equiv.option_congr` for `with_zero`.", simps apply]
def _root_.mul_equiv.with_one_congr (e : α ≃* β) : with_one α ≃* with_one β :=
{ to_fun := map e.to_mul_hom,
inv_fun := map e.symm.to_mul_hom,
left_inv := λ x, (map_map _ _ _).trans $ by induction x using with_one.cases_on; { simp },
right_inv := λ x, (map_map _ _ _).trans $ by induction x using with_one.cases_on; { simp },
.. map e.to_mul_hom }
@[simp]
lemma _root_.mul_equiv.with_one_congr_refl : (mul_equiv.refl α).with_one_congr = mul_equiv.refl _ :=
mul_equiv.to_monoid_hom_injective map_id
@[simp]
lemma _root_.mul_equiv.with_one_congr_symm (e : α ≃* β) :
e.with_one_congr.symm = e.symm.with_one_congr := rfl
@[simp]
lemma _root_.mul_equiv.with_one_congr_trans (e₁ : α ≃* β) (e₂ : β ≃* γ) :
e₁.with_one_congr.trans e₂.with_one_congr = (e₁.trans e₂).with_one_congr :=
mul_equiv.to_monoid_hom_injective (map_comp _ _).symm
end map
end with_one
|
f89759fd33ace5edaa02dd161cccf84b449b18ba | 4d2583807a5ac6caaffd3d7a5f646d61ca85d532 | /src/topology/algebra/weak_dual_topology.lean | 6c7bc8b971b69ab40df619e3493f18ec2b656faa | [
"Apache-2.0"
] | permissive | AntoineChambert-Loir/mathlib | 64aabb896129885f12296a799818061bc90da1ff | 07be904260ab6e36a5769680b6012f03a4727134 | refs/heads/master | 1,693,187,631,771 | 1,636,719,886,000 | 1,636,719,886,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 6,777 | lean | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import topology.algebra.module
/-!
# Weak dual topology
This file defines the weak-* topology on duals of suitable topological modules `E` over suitable
topological semirings `𝕜`. The (weak) dual consists of continuous linear functionals `E →L[𝕜] 𝕜`
from `E` to scalars `𝕜`. The weak-* topology is the coarsest topology on this dual
`weak_dual 𝕜 E := (E →L[𝕜] 𝕜)` w.r.t. which the evaluation maps at all `z : E` are continuous.
The weak dual is a module over `𝕜` if the semiring `𝕜` is commutative.
## Main definitions
The main definitions are the type `weak_dual 𝕜 E` and a topology instance on it.
* `weak_dual 𝕜 E` is a type synonym for `dual 𝕜 E` (when the latter is defined): both are equal to
the type `E →L[𝕜] 𝕜` of continuous linear maps from a module `E` over `𝕜` to the ring `𝕜`.
* The instance `weak_dual.topological_space` is the weak-* topology on `weak_dual 𝕜 E`, i.e., the
coarsest topology making the evaluation maps at all `z : E` continuous.
## Main results
We establish that `weak_dual 𝕜 E` has the following structure:
* `weak_dual.has_continuous_add`: The addition in `weak_dual 𝕜 E` is continuous.
* `weak_dual.module`: If the scalars `𝕜` are a commutative semiring, then `weak_dual 𝕜 E` is a
module over `𝕜`.
* `weak_dual.has_continuous_smul`: If the scalars `𝕜` are a commutative semiring, then the scalar
multiplication by `𝕜` in `weak_dual 𝕜 E` is continuous.
We prove the following results characterizing the weak-* topology:
* `weak_dual.eval_continuous`: For any `z : E`, the evaluation mapping `weak_dual 𝕜 E → 𝕜` taking
`x'`to `x' z` is continuous.
* `weak_dual.continuous_of_continuous_eval`: For a mapping to `weak_dual 𝕜 E` to be continuous,
it suffices that its compositions with evaluations at all points `z : E` are continuous.
* `weak_dual.tendsto_iff_forall_eval_tendsto`: Convergence in `weak_dual 𝕜 E` can be characterized
in terms of convergence of the evaluations at all points `z : E`.
## Notations
No new notation is introduced.
## Implementation notes
The weak-* topology is defined as the induced topology under the mapping that associates to a dual
element `x'` the functional `E → 𝕜`, when the space `E → 𝕜` of functionals is equipped with the
topology of pointwise convergence (product topology).
Typically one might assume that `𝕜` is a topological semiring in the sense of the typeclasses
`topological_space 𝕜`, `semiring 𝕜`, `has_continuous_add 𝕜`, `has_continuous_mul 𝕜`,
and that the space `E` is a topological module over `𝕜` in the sense of the typeclasses
`topological_space E`, `add_comm_monoid E`, `has_continuous_add E`, `module 𝕜 E`,
`has_continuous_smul 𝕜 E`. The definitions and results are, however, given with weaker assumptions
when possible.
## References
* https://en.wikipedia.org/wiki/Weak_topology#Weak-*_topology
## Tags
weak-star, weak dual
-/
noncomputable theory
open filter
open_locale topological_space
universes u v
section weak_star_topology
/-!
### Weak star topology on duals of topological modules
-/
variables (𝕜 : Type*) [topological_space 𝕜] [semiring 𝕜]
variables (E : Type*) [topological_space E] [add_comm_monoid E] [module 𝕜 E]
/-- The weak dual of a topological module `E` over a topological semiring `𝕜` consists of
continuous linear functionals from `E` to scalars `𝕜`. It is a type synonym with the usual dual
(when the latter is defined), but will be equipped with a different topology. -/
@[derive [inhabited, λ α, has_coe_to_fun α (λ _, E → 𝕜)]]
def weak_dual := E →L[𝕜] 𝕜
instance [has_continuous_add 𝕜] : add_comm_monoid (weak_dual 𝕜 E) :=
continuous_linear_map.add_comm_monoid
namespace weak_dual
/-- The weak-* topology instance `weak_dual.topological_space` on the dual of a topological module
`E` over a topological semiring `𝕜` is defined as the induced topology under the mapping that
associates to a dual element `x' : weak_dual 𝕜 E` the functional `E → 𝕜`, when the space `E → 𝕜`
of functionals is equipped with the topology of pointwise convergence (product topology). -/
instance : topological_space (weak_dual 𝕜 E) :=
topological_space.induced (λ x' : weak_dual 𝕜 E, λ z : E, x' z) Pi.topological_space
lemma coe_fn_continuous :
continuous (λ (x' : (weak_dual 𝕜 E)), (λ (z : E), x' z)) :=
continuous_induced_dom
lemma eval_continuous (z : E) : continuous (λ (x' : weak_dual 𝕜 E), x' z) :=
(continuous_pi_iff.mp (coe_fn_continuous 𝕜 E)) z
lemma continuous_of_continuous_eval {α : Type u} [topological_space α]
{g : α → weak_dual 𝕜 E} (h : ∀ z, continuous (λ a, g a z)) : continuous g :=
continuous_induced_rng (continuous_pi_iff.mpr h)
theorem tendsto_iff_forall_eval_tendsto {γ : Type u} {F : filter γ}
{ψs : γ → weak_dual 𝕜 E} {ψ : weak_dual 𝕜 E} :
tendsto ψs F (𝓝 ψ) ↔ ∀ z : E, tendsto (λ i, ψs i z) F (𝓝 (ψ z)) :=
begin
rw ←tendsto_pi,
split,
{ intros weak_star_conv,
exact (((coe_fn_continuous 𝕜 E).tendsto ψ).comp weak_star_conv), },
{ intro h_lim_forall,
rwa [nhds_induced, tendsto_comap_iff], },
end
/-- Addition in `weak_dual 𝕜 E` is continuous. -/
instance [has_continuous_add 𝕜] : has_continuous_add (weak_dual 𝕜 E) :=
{ continuous_add := begin
apply continuous_of_continuous_eval,
intros z,
have h : continuous (λ p : 𝕜 × 𝕜, p.1 + p.2) := continuous_add,
exact h.comp ((eval_continuous 𝕜 E z).prod_map (eval_continuous 𝕜 E z)),
end, }
/-- If the scalars `𝕜` are a commutative semiring, then `weak_dual 𝕜 E` is a module over `𝕜`. -/
instance (𝕜 : Type u) [topological_space 𝕜] [comm_semiring 𝕜]
[has_continuous_add 𝕜] [has_continuous_mul 𝕜]
(E : Type*) [topological_space E] [add_comm_group E] [module 𝕜 E] :
module 𝕜 (weak_dual 𝕜 E) :=
continuous_linear_map.module
/-- Scalar multiplication in `weak_dual 𝕜 E` is continuous (when `𝕜` is a commutative
semiring). -/
instance (𝕜 : Type u) [topological_space 𝕜] [comm_semiring 𝕜]
[has_continuous_add 𝕜] [has_continuous_mul 𝕜]
(E : Type*) [topological_space E] [add_comm_group E]
[module 𝕜 E] :
has_continuous_smul 𝕜 (weak_dual 𝕜 E) :=
{ continuous_smul := begin
apply continuous_of_continuous_eval,
intros z,
have h : continuous (λ p : 𝕜 × 𝕜, p.1 * p.2) := continuous_mul,
exact h.comp ((continuous_id').prod_map (eval_continuous 𝕜 E z)),
end, }
end weak_dual
end weak_star_topology
|
7b47c98e8c8b3cb2df50ba47417f1a0388df27b1 | d7189ea2ef694124821b033e533f18905b5e87ef | /galois/arith/ext.lean | 824e6e5a6249bff90b57ac3c9e5d9e04c5f9f065 | [
"Apache-2.0"
] | permissive | digama0/lean-protocol-support | eaa7e6f8b8e0d5bbfff1f7f52bfb79a3b11b0f59 | cabfa3abedbdd6fdca6e2da6fbbf91a13ed48dda | refs/heads/master | 1,625,421,450,627 | 1,506,035,462,000 | 1,506,035,462,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 3,530 | lean | import galois.tactic
inductive ext (A : Type) : Type
| inj {} : A → ext
| inf {} : ext
| neg_inf {} : ext
namespace ext
section
parameters (A : Type) [decidable_linear_order A]
def le : ext A → ext A → Prop
| neg_inf _ := true
| _ inf := true
| (inj _) neg_inf := false
| (inj x) (inj y) := x ≤ y
| inf (inj _) := false
| inf neg_inf := false
instance : has_le (ext A) := ⟨ le ⟩
inductive lt (x y : ext A) : Prop
| mk : ∀ (le : x ≤ y) (not_rev_le : ¬ (y ≤ x)), lt
instance : has_lt (ext A) := ⟨ lt ⟩
instance le_decidable : decidable_rel (@has_le.le (ext A) _) :=
begin
intros x y,
induction x; induction y; simp [(≤), le];
apply_instance,
end
lemma lt_equiv (a b : ext A) : a < b ↔ a ≤ b ∧ ¬b ≤ a
:= begin
split; intros H; induction H; split; assumption,
end
instance lt_decidable : decidable_rel (@has_lt.lt (ext A) _) :=
begin
intros x y, rw lt_equiv, apply_instance,
end
lemma le_refl (x : ext A) : x ≤ x :=
begin
simp only [(≤)], induction x; simp only [le],
end
lemma le_trans {x y z : ext A} (H : x ≤ y) (H' : y ≤ z)
: x ≤ z :=
begin
simp only [(≤)] at *,
induction x; induction y; induction z; simp only [le] at *;
try { contradiction },
apply le_trans; assumption
end
lemma le_antisymm (a b : ext A) : a ≤ b → b ≤ a → a = b
:= begin
simp only [(≤)], intros H H', induction a; induction b;
simp only [le] at *; try { contradiction },
f_equal, apply le_antisymm; assumption
end
lemma le_total (a b : ext A) : a ≤ b ∨ b ≤ a
:= begin
simp only [(≤)], induction a; induction b;
simp only [le]; try {cc},
apply le_total,
end
lemma neg_inf_le (x : ext A) : neg_inf ≤ x
:= begin
simp only [(≤)], induction x; simp [le],
end
lemma le_inf (x : ext A) : x ≤ inf
:= begin
simp only [(≤)], induction x; simp [le],
end
instance eq_decidable : decidable_eq (ext A) :=
by tactic.mk_dec_eq_instance
instance linear_order_decidable : decidable_linear_order (ext A) :=
begin
constructor, apply ext.le_decidable,
apply_instance, apply ext.lt_decidable,
apply le_refl, apply le_trans,
unfold auto_param, intros,
apply lt_equiv, apply le_antisymm,
apply le_total,
end
end
end ext
namespace list
def maximum_ext {α:Type _} [decidable_linear_order α] : list (ext α) → ext α
:= list.foldr max ext.neg_inf
def minimum_ext {α:Type _} [decidable_linear_order α] : list (ext α) → ext α
:= list.foldr min ext.inf
end list
lemma maximum_ext_le {α: Type _} [decidable_linear_order α]
(xs : list (ext α)) (u : ext α)
(H : ∀ x : ext α, x ∈ xs → x ≤ u)
: xs.maximum_ext ≤ u
:= begin
unfold list.maximum_ext,
induction xs; dsimp [list.foldr],
apply ext.neg_inf_le,
apply max_le, apply H, rw list.mem_cons_eq,
left, reflexivity, apply ih_1, intros,
apply H,rw list.mem_cons_eq, right, assumption
end
lemma le_minimum_ext {α: Type _} [decidable_linear_order α]
(xs : list (ext α)) (l : ext α)
(H : ∀ x : ext α, x ∈ xs → l ≤ x)
: l ≤ xs.minimum_ext
:= begin
unfold list.minimum_ext,
induction xs; dsimp [list.foldr],
apply ext.le_inf,
apply le_min, apply H, rw list.mem_cons_eq,
left, reflexivity, apply ih_1, intros,
apply H,rw list.mem_cons_eq, right, assumption
end
theorem le_ext_maximum_minimum {α: Type _} [decidable_linear_order α]
(ll ul : list (ext α))
(pr : ∀ (l u : ext α), l ∈ ll → u ∈ ul → l ≤ u)
: ll.maximum_ext ≤ ul.minimum_ext
:= begin
apply maximum_ext_le,
intros, apply le_minimum_ext, intros, apply pr; assumption
end |
80863a95567c705cb2ff03aa43494a744e905fbf | ebf7140a9ea507409ff4c994124fa36e79b4ae35 | /src/exercises_sources/thursday/groups_rings_fields.lean | 3ae3d3edba3cf809b085d3ce94297aebe2ff172b | [] | no_license | fundou/lftcm2020 | 3e88d58a92755ea5dd49f19c36239c35286ecf5e | 99d11bf3bcd71ffeaef0250caa08ecc46e69b55b | refs/heads/master | 1,685,610,799,304 | 1,624,070,416,000 | 1,624,070,416,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 9,398 | lean | import linear_algebra.finite_dimensional
import ring_theory.algebraic
import data.zmod.basic
import data.real.basic
import tactic
/-!
```
____
/ ___|_ __ ___ _ _ _ __ ___
| | _| '__/ _ \| | | | '_ \/ __|
| |_| | | | (_) | |_| | |_) \__ \_
\____|_| \___/ \__,_| .__/|___( )
|_| |/
_ __ (_) _ __ __ _ ___
| '__| | | | '_ \ / _` | / __|
| | | | | | | | | (_| | \__ \ _
|_| |_| |_| |_| \__, | |___/ ( )
|___/ |/
_ __ _ _ _
__ _ _ __ __| | / _| (_) ___ | | __| | ___
/ _` | | '_ \ / _` | | |_ | | / _ \ | | / _` | / __|
| (_| | | | | | | (_| | | _| | | | __/ | | | (_| | \__ \
\__,_| |_| |_| \__,_| |_| |_| \___| |_| \__,_| |___/
```
-/
/-! ## Reminder on updating the exercises
These instructions are now available at:
https://leanprover-community.github.io/lftcm2020/exercises.html
To get a new copy of the exercises,
run the following commands in your terminal:
```
leanproject get lftcm2020
cp -r lftcm2020/src/exercises_sources/ lftcm2020/src/my_exercises
code lftcm2020
```
To update your exercise files, run the following commands:
```
cd /path/to/lftcm2020
git pull
leanproject get-mathlib-cache
```
Don’t forget to copy the updated files to `src/my_exercises`.
-/
/-! ## What do we have?
Too much to cover in detail in 10 minutes.
Take a look at the “General algebra” section on
https://leanprover-community.github.io/mathlib-overview.html
All the basic concepts are there:
`group`, `ring`, `field`, `module`, etc...
Also versions that are compatible with an ordering, like `ordered_ring`
And versions that express compatibility with a topology: `topological_group`
Finally constructions, like `polynomial R`, or `mv_polynomial σ R`,
or `monoid_algebra K G`, or `ℤ_[p]`, or `zmod n`, or `localization R S`.
-/
/-! ## Morphisms
We are in the middle of a transition to “bundled” morphisms.
(Why? Long story... but they work better with `simp`)
* `X → Y` -- ordinary function
* `X →+ Y` -- function respects `0` and `+`
* `X →* Y` -- function respects `1` and `*`
* `X →+* Y` -- function respects `0`, `1`, `+`, `*` (surprise!)
-/
section
variables {R S : Type*} [ring R] [ring S]
-- We used to write
example (f : R → S) [is_ring_hom f] : true := trivial
-- But now we write
example (f : R →+* S) : true := trivial
/-
This heavily relies on the “coercion to function”
that we have seen a couple of times this week.
-/
end
/-! ## Where are these things in the library?
`algebra/` for basic definitions and properties; “algebraic hierarchy”
`group_theory/` ⎫
`ring_theory/` ⎬ “advanced” and “specialised” material
`field_theory/` |
`number_theory` ⎭
`data/` definitions and examples
To give an idea:
* `algebra/ordered_ring.lean`
* `ring_theory/noetherian.lean`
* `field_theory/chevalley_warning.lean`
* `data/nat/*.lean`, `data/real/*.lean`, `number_theory/padics/*.lean`
-/
/-! ## How to find things (search tools)
* `library_search` -- it often helps to carve out
the exact lemma statement that you are looking for
* online documentation: https://leanprover-community.github.io/mathlib_docs/
new search bar under construction
* Old-skool: `grep`
* Search in VS Code:
- `Ctrl - Shift - F`
-- don't forget to change settings, to search everywhere
-- click the three dots (`…`) below the search bar
-- disable the blue cogwheel
- `Ctrl - P` -- search for filenames
- `Ctrl - P`, `#` -- search for lemmas and definitions
-/
/-! ## How to find things (autocomplete)
Mathlib follows pretty strict naming conventions:
```
/-- The binomial theorem-/
theorem add_pow [comm_semiring α] (x y : α) (n : ℕ) :
(x + y) ^ n = ∑ m in range (n + 1), x ^ m * y ^ (n - m) * choose n m :=
(commute.all x y).add_pow n
```
After a while, you get the hang of this,
and you can start guessing names.
-/
open_locale big_operators -- nice notation ∑, ∏
open finset -- `finset.range n` is the finite set `{0,1,..., n-1}`
-- Demonstrate autocompletion
example (f : ℕ → ℝ) (n : ℕ) :
57 + ∑ i in range (n+1), f i = 57 + f n + ∑ i in range n, f i :=
begin
sorry
end
/-! ## How to find things (jump to definition)
Another good strategy for finding useful results about `X`,
is to “jump to the definition” and scroll through the next 3 screens of lemmas.
If you are looking for a basic fact about `X`, you will usually find it there.
-/
-- demonstrate “jump to definition”
#check polynomial.coeff
/-! ## Exercise 1
We will warm up with a well-known result:
“Subgroups of abelian groups are normal.”
Hints for proving this result:
* Notice that `normal` is a structure,
which you can see by going to the definition.
The `constructor` tactic will help you to get started.
-/
namespace add_subgroup
variables {A : Type*} [add_comm_group A]
lemma normal_of_add_comm_group (H : add_subgroup A) : normal H :=
begin
sorry
end
end add_subgroup
/-! ## Exercise 2
The following exercise will show the classical fact:
“Finite field extensions are algebraic.”
Hints for proving this result:
* Look up the definition of `finite_dimensional`.
* Search the library for useful lemmas about `is_algebraic` and `is_integral`.
-/
namespace algebra
variables {K L : Type*} [field K] [field L] [algebra K L] [finite_dimensional K L]
lemma is_algebraic_of_finite_dimensional : is_algebraic K L :=
begin
sorry
end
end algebra
/-! ## Exercise 3
In this exercise we will define the Frobenius morphism.
-/
section
variables (p : ℕ) [fact p.prime]
variables (K : Type*) [field K] [char_p K p]
/-! ### Subchallenge -/
lemma add_pow_char' (x y : K) : (x + y) ^ p = x ^ p + y ^ p :=
begin
-- Hint: `add_pow_char` already exists.
-- You can use it if you don't want to spend time on this.
/- Hints if you do want to attempt this:
* `finset.sum_range_succ`
* `finset.sum_eq_single`
* `nat.prime.ne_zero`
* `char_p.cast_eq_zero_iff`
* `nat.prime.dvd_choose_self`
* `fact.out p.prime` obtains the primality of `p` from the typeclass assumption `[fact p.prime]`
-/
sorry
end
def frobenius_hom : K →+* K :=
{ to_fun := λ x, x^p,
map_zero' :=
begin
-- Hint: `zero_pow`, search for lemmas near `nat.prime`
sorry
end,
map_one' :=
begin
sorry
end,
map_mul' :=
begin
sorry
end,
map_add' :=
begin
-- Hint: `add_pow_char` -- can you prove that one yourself?
sorry
end }
end
/-! ## Exercise 4 [challenging]
The next exercise asks to show that a monic polynomial `f ∈ ℤ[X]` is irreducible
if it is irreducible modulo a prime `p`.
This fact is also not in mathlib.
Hint: prove the helper lemma that is stated first.
Follow-up question:
Can you generalise `irreducible_of_irreducible_mod_prime`?
-/
namespace polynomial
variables {R S : Type*} [semiring R] [integral_domain S] (φ : R →+* S)
/-
Useful library lemmas (in no particular order):
- `degree_eq_zero_of_is_unit`
- `eq_C_of_degree_eq_zero`
- `is_unit.map'`
- `leading_coeff_C`
- `degree_map_eq_of_leading_coeff_ne_zero`
- `is_unit.map'`
- `is_unit.ne_zero`
-/
lemma is_unit_of_is_unit_leading_coeff_of_is_unit_map'
(f : polynomial R) (hf : is_unit (leading_coeff f)) (H : is_unit (map φ f)) :
is_unit f :=
begin
sorry
end
/-
Useful library lemmas (in no particular order):
- `is_unit.map'`
- `is_unit_of_mul_is_unit_left` (also `_right`)
- `leading_coeff_mul`
- `is_unit_of_is_unit_leading_coeff_of_is_unit_map` (the helper lemma we just proved)
- `is_unit_one`
-/
lemma irreducible_of_irreducible_mod_prime (f : polynomial ℤ) (p : ℕ) [fact p.prime]
(h_mon : monic f) (h_irr : irreducible (map (int.cast_ring_hom (zmod p)) f)) :
irreducible f :=
begin
sorry
end
end polynomial
-- SCROLL DOWN FOR THE BONUS EXERCISE
section
/-! ## Bonus exercise (wicked hard) -/
noncomputable theory -- because `polynomial` is noncomputable (implementation detail)
open polynomial -- we want to write `X`, instead of `polynomial.X`
/-
First we make some definitions
Scroll to the end for the actual exercise
-/
def partial_ramanujan_tau_polynomial (n : ℕ) : polynomial ℤ :=
X * ∏ k in finset.Ico 1 n, (1 - X^k)^24
def ramanujan_tau (n : ℕ) : ℤ :=
coeff (partial_ramanujan_tau_polynomial n) n
-- Some nice suggestive notation
prefix `τ`:300 := ramanujan_tau
/-
Some lemmas to warm up
Hint: unfold definitions, `simp`
-/
example : τ 0 = 0 :=
begin
sorry
end
example : τ 1 = 1 :=
begin
sorry
end
-- This one is nontrivial
-- Use `have : subresult,` or state helper lemmas and prove them first!
example : τ 2 = -24 :=
begin
-- Really, we ought to have a tactic that makes this easy
delta ramanujan_tau partial_ramanujan_tau_polynomial,
rw [mul_comm, coeff_mul_X],
suffices : ((1 - X) ^ 24 : polynomial ℤ).coeff 1 = -(24 : ℕ), by simpa,
generalize : (24 : ℕ) = n,
sorry
end
/-
The actual exercise. Good luck (-;
-/
theorem deligne (p : ℕ) (hp : p.prime) : (abs (τ p) : ℝ) ≤ 2 * p^(11/2) :=
begin
sorry
end
end
|
7460123428ffabf43322b84071a9cad6f5d1dad4 | 74addaa0e41490cbaf2abd313a764c96df57b05d | /Mathlib/ring_theory/integral_domain_auto.lean | c0c5922439b2e49ae5adc0ea0db38b2f4000e9c0 | [] | no_license | AurelienSaue/Mathlib4_auto | f538cfd0980f65a6361eadea39e6fc639e9dae14 | 590df64109b08190abe22358fabc3eae000943f2 | refs/heads/master | 1,683,906,849,776 | 1,622,564,669,000 | 1,622,564,669,000 | 371,723,747 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 3,546 | lean | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Chris Hughes
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.default
import Mathlib.data.fintype.card
import Mathlib.data.polynomial.ring_division
import Mathlib.group_theory.order_of_element
import Mathlib.algebra.geom_sum
import Mathlib.PostPort
universes u_1 u_2
namespace Mathlib
/-!
# Integral domains
Assorted theorems about integral domains.
## Main theorems
* `is_cyclic_of_subgroup_integral_domain` : A finite subgroup of the units of an integral domain
is cyclic.
* `field_of_integral_domain` : A finite integral domain is a field.
## Tags
integral domain, finite integral domain, finite field
-/
theorem card_nth_roots_subgroup_units {R : Type u_1} {G : Type u_2} [integral_domain R] [group G]
[fintype G] (f : G →* R) (hf : function.injective ⇑f) {n : ℕ} (hn : 0 < n) (g₀ : G) :
finset.card (has_sep.sep (fun (g : G) => g ^ n = g₀) finset.univ) ≤
coe_fn multiset.card (polynomial.nth_roots n (coe_fn f g₀)) :=
sorry
/-- A finite subgroup of the unit group of an integral domain is cyclic. -/
theorem is_cyclic_of_subgroup_integral_domain {R : Type u_1} {G : Type u_2} [integral_domain R]
[group G] [fintype G] (f : G →* R) (hf : function.injective ⇑f) : is_cyclic G :=
sorry
/-- The unit group of a finite integral domain is cyclic. -/
protected instance units.is_cyclic {R : Type u_1} [integral_domain R] [fintype R] :
is_cyclic (units R) :=
is_cyclic_of_subgroup_integral_domain (units.coe_hom R) units.ext
/-- Every finite integral domain is a field. -/
def field_of_integral_domain {R : Type u_1} [integral_domain R] [DecidableEq R] [fintype R] :
field R :=
field.mk integral_domain.add sorry integral_domain.zero sorry sorry integral_domain.neg
integral_domain.sub sorry sorry integral_domain.mul sorry integral_domain.one sorry sorry sorry
sorry sorry
(fun (a : R) => dite (a = 0) (fun (h : a = 0) => 0) fun (h : ¬a = 0) => fintype.bij_inv sorry 1)
sorry sorry sorry
/-- A finite subgroup of the units of an integral domain is cyclic. -/
protected instance subgroup_units_cyclic {R : Type u_1} [integral_domain R] (S : set (units R))
[is_subgroup S] [fintype ↥S] : is_cyclic ↥S :=
sorry
theorem card_fiber_eq_of_mem_range {G : Type u_2} [group G] [fintype G] {H : Type u_1} [group H]
[DecidableEq H] (f : G →* H) {x : H} {y : H} (hx : x ∈ set.range ⇑f) (hy : y ∈ set.range ⇑f) :
finset.card (finset.filter (fun (g : G) => coe_fn f g = x) finset.univ) =
finset.card (finset.filter (fun (g : G) => coe_fn f g = y) finset.univ) :=
sorry
/-- In an integral domain, a sum indexed by a nontrivial homomorphism from a finite group is zero. -/
theorem sum_hom_units_eq_zero {R : Type u_1} {G : Type u_2} [integral_domain R] [group G]
[fintype G] (f : G →* R) (hf : f ≠ 1) :
(finset.sum finset.univ fun (g : G) => coe_fn f g) = 0 :=
sorry
/-- In an integral domain, a sum indexed by a homomorphism from a finite group is zero,
unless the homomorphism is trivial, in which case the sum is equal to the cardinality of the group. -/
theorem sum_hom_units {R : Type u_1} {G : Type u_2} [integral_domain R] [group G] [fintype G]
(f : G →* R) [Decidable (f = 1)] :
(finset.sum finset.univ fun (g : G) => coe_fn f g) = ite (f = 1) (↑(fintype.card G)) 0 :=
sorry
end Mathlib |
dbb254c85fc74375bed1f023004af7cb98fd5203 | 556aeb81a103e9e0ac4e1fe0ce1bc6e6161c3c5e | /src/starkware/cairo/common/cairo_secp/verification/verification/signature_recover_public_key_fast_ec_add_soundness.lean | ea5a6c209f45f8a296fd880563d9f108778868e7 | [] | permissive | starkware-libs/formal-proofs | d6b731604461bf99e6ba820e68acca62a21709e8 | f5fa4ba6a471357fd171175183203d0b437f6527 | refs/heads/master | 1,691,085,444,753 | 1,690,507,386,000 | 1,690,507,386,000 | 410,476,629 | 32 | 9 | Apache-2.0 | 1,690,506,773,000 | 1,632,639,790,000 | Lean | UTF-8 | Lean | false | false | 44,345 | lean | /-
File: signature_recover_public_key_fast_ec_add_soundness.lean
Autogenerated file.
-/
import starkware.cairo.lean.semantics.soundness.hoare
import .signature_recover_public_key_code
import ..signature_recover_public_key_spec
import .signature_recover_public_key_unreduced_sqr_soundness
import .signature_recover_public_key_compute_slope_soundness
open tactic
open starkware.cairo.common.cairo_secp.ec
open starkware.cairo.common.cairo_secp.bigint
open starkware.cairo.common.cairo_secp.field
variables {F : Type} [field F] [decidable_eq F] [prelude_hyps F]
variable mem : F → F
variable σ : register_state F
/- starkware.cairo.common.cairo_secp.ec.fast_ec_add autogenerated soundness theorem -/
theorem auto_sound_fast_ec_add_block9
-- An independent ap variable.
(ap : F)
-- arguments
(range_check_ptr : F) (point0 point1 : EcPoint F)
-- code is in memory at σ.pc
(h_mem : mem_at mem code_fast_ec_add σ.pc)
-- all dependencies are in memory
(h_mem_4 : mem_at mem code_nondet_bigint3 (σ.pc - 317))
(h_mem_5 : mem_at mem code_unreduced_mul (σ.pc - 305))
(h_mem_6 : mem_at mem code_unreduced_sqr (σ.pc - 285))
(h_mem_7 : mem_at mem code_verify_zero (σ.pc - 269))
(h_mem_13 : mem_at mem code_compute_slope (σ.pc - 97))
-- input arguments on the stack
(hin_range_check_ptr : range_check_ptr = mem (σ.fp - 15))
(hin_point0 : point0 = cast_EcPoint mem (σ.fp - 14))
(hin_point1 : point1 = cast_EcPoint mem (σ.fp - 8))
(νbound : ℕ)
-- conclusion
: ensuresb_ret νbound mem
{pc := σ.pc + 28, ap := ap, fp := σ.fp}
(λ κ τ,
∃ μ ≤ κ, rc_ensures mem (rc_bound F) μ (mem (σ.fp - 15)) (mem $ τ.ap - 7)
(auto_spec_fast_ec_add_block9 mem κ range_check_ptr point0 point1 (mem (τ.ap - 7)) (cast_EcPoint mem (τ.ap - 6)))) :=
begin
have h_mem_rec := h_mem,
unpack_memory code_fast_ec_add at h_mem with ⟨hpc0, hpc1, hpc2, hpc3, hpc4, hpc5, hpc6, hpc7, hpc8, hpc9, hpc10, hpc11, hpc12, hpc13, hpc14, hpc15, hpc16, hpc17, hpc18, hpc19, hpc20, hpc21, hpc22, hpc23, hpc24, hpc25, hpc26, hpc27, hpc28, hpc29, hpc30, hpc31, hpc32, hpc33, hpc34, hpc35, hpc36, hpc37, hpc38, hpc39, hpc40, hpc41, hpc42, hpc43, hpc44, hpc45, hpc46, hpc47, hpc48, hpc49, hpc50, hpc51, hpc52, hpc53, hpc54, hpc55, hpc56, hpc57, hpc58, hpc59, hpc60, hpc61, hpc62, hpc63, hpc64, hpc65, hpc66, hpc67, hpc68, hpc69, hpc70, hpc71, hpc72, hpc73, hpc74, hpc75, hpc76, hpc77, hpc78, hpc79, hpc80, hpc81, hpc82, hpc83, hpc84, hpc85, hpc86⟩,
-- function call
step_assert_eq hpc28 with arg0,
step_assert_eq hpc29 with arg1,
step_assert_eq hpc30 with arg2,
step_assert_eq hpc31 with arg3,
step_assert_eq hpc32 with arg4,
step_assert_eq hpc33 with arg5,
step_assert_eq hpc34 with arg6,
step_assert_eq hpc35 with arg7,
step_assert_eq hpc36 with arg8,
step_assert_eq hpc37 with arg9,
step_assert_eq hpc38 with arg10,
step_assert_eq hpc39 with arg11,
step_assert_eq hpc40 with arg12,
step_sub hpc41 (auto_sound_compute_slope mem _ range_check_ptr point0 point1 _ _ _ _ _ _ _),
{ rw hpc42, norm_num2, exact h_mem_13 },
{ rw hpc42, norm_num2, exact h_mem_4 },
{ rw hpc42, norm_num2, exact h_mem_5 },
{ rw hpc42, norm_num2, exact h_mem_7 },
{ try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point0, hin_point1] },
try { dsimp [cast_EcPoint, cast_BigInt3] },
try { arith_simps }, try { simp only [arg0, arg1, arg2, arg3, arg4, arg5, arg6, arg7, arg8, arg9, arg10, arg11, arg12] },
try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } }, },
{ try { ext } ; {
try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point0, hin_point1] },
try { dsimp [cast_EcPoint, cast_BigInt3] },
try { arith_simps }, try { simp only [arg0, arg1, arg2, arg3, arg4, arg5, arg6, arg7, arg8, arg9, arg10, arg11, arg12] },
try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },}, },
{ try { ext } ; {
try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point0, hin_point1] },
try { dsimp [cast_EcPoint, cast_BigInt3] },
try { arith_simps }, try { simp only [arg0, arg1, arg2, arg3, arg4, arg5, arg6, arg7, arg8, arg9, arg10, arg11, arg12] },
try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },}, },
intros κ_call43 ap43 h_call43,
rcases h_call43 with ⟨h_call43_ap_offset, h_call43⟩,
rcases h_call43 with ⟨rc_m43, rc_mle43, hl_range_check_ptr₁, h_call43⟩,
generalize' hr_rev_range_check_ptr₁: mem (ap43 - 4) = range_check_ptr₁,
have htv_range_check_ptr₁ := hr_rev_range_check_ptr₁.symm, clear hr_rev_range_check_ptr₁,
generalize' hr_rev_slope: cast_BigInt3 mem (ap43 - 3) = slope,
simp only [hr_rev_slope] at h_call43,
have htv_slope := hr_rev_slope.symm, clear hr_rev_slope,
try { simp only [arg0 ,arg1 ,arg2 ,arg3 ,arg4 ,arg5 ,arg6 ,arg7 ,arg8 ,arg9 ,arg10 ,arg11 ,arg12] at hl_range_check_ptr₁ },
rw [←htv_range_check_ptr₁, ←hin_range_check_ptr] at hl_range_check_ptr₁,
try { simp only [arg0 ,arg1 ,arg2 ,arg3 ,arg4 ,arg5 ,arg6 ,arg7 ,arg8 ,arg9 ,arg10 ,arg11 ,arg12] at h_call43 },
rw [hin_range_check_ptr] at h_call43,
clear arg0 arg1 arg2 arg3 arg4 arg5 arg6 arg7 arg8 arg9 arg10 arg11 arg12,
-- function call
step_sub hpc43 (auto_sound_unreduced_sqr mem _ slope _ _),
{ rw hpc44, norm_num2, exact h_mem_6 },
{ try { ext } ; {
try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point0, hin_point1, htv_range_check_ptr₁, htv_slope] },
try { dsimp [cast_EcPoint, cast_BigInt3] },
try { simp only [h_call43_ap_offset] },
try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },}, },
intros κ_call45 ap45 h_call45,
rcases h_call45 with ⟨h_call45_ap_offset, h_call45⟩,
generalize' hr_rev_slope_sqr: cast_UnreducedBigInt3 mem (ap45 - 3) = slope_sqr,
simp only [hr_rev_slope_sqr] at h_call45,
have htv_slope_sqr := hr_rev_slope_sqr.symm, clear hr_rev_slope_sqr,
clear ,
-- function call
step_assert_eq hpc45 with arg0,
step_sub hpc46 (auto_sound_nondet_bigint3 mem _ range_check_ptr₁ _ _),
{ rw hpc47, norm_num2, exact h_mem_4 },
{ try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point0, hin_point1, htv_range_check_ptr₁, htv_slope, htv_slope_sqr] },
try { dsimp [cast_EcPoint, cast_BigInt3, cast_UnreducedBigInt3] },
try { arith_simps }, try { simp only [arg0] },
try { simp only [h_call43_ap_offset, h_call45_ap_offset] },
try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } }, },
intros κ_call48 ap48 h_call48,
rcases h_call48 with ⟨h_call48_ap_offset, h_call48⟩,
rcases h_call48 with ⟨rc_m48, rc_mle48, hl_range_check_ptr₂, h_call48⟩,
generalize' hr_rev_range_check_ptr₂: mem (ap48 - 4) = range_check_ptr₂,
have htv_range_check_ptr₂ := hr_rev_range_check_ptr₂.symm, clear hr_rev_range_check_ptr₂,
generalize' hr_rev_new_x: cast_BigInt3 mem (ap48 - 3) = new_x,
simp only [hr_rev_new_x] at h_call48,
have htv_new_x := hr_rev_new_x.symm, clear hr_rev_new_x,
try { simp only [arg0] at hl_range_check_ptr₂ },
try { rw [h_call45_ap_offset] at hl_range_check_ptr₂ }, try { arith_simps at hl_range_check_ptr₂ },
rw [←htv_range_check_ptr₂, ←htv_range_check_ptr₁] at hl_range_check_ptr₂,
try { simp only [arg0] at h_call48 },
try { rw [h_call45_ap_offset] at h_call48 }, try { arith_simps at h_call48 },
rw [←htv_range_check_ptr₁, hl_range_check_ptr₁, hin_range_check_ptr] at h_call48,
clear arg0,
-- function call
step_assert_eq hpc48 with arg0,
step_sub hpc49 (auto_sound_nondet_bigint3 mem _ range_check_ptr₂ _ _),
{ rw hpc50, norm_num2, exact h_mem_4 },
{ try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point0, hin_point1, htv_range_check_ptr₁, htv_slope, htv_slope_sqr, htv_range_check_ptr₂, htv_new_x] },
try { dsimp [cast_EcPoint, cast_BigInt3, cast_UnreducedBigInt3] },
try { arith_simps }, try { simp only [arg0] },
try { simp only [h_call43_ap_offset, h_call45_ap_offset, h_call48_ap_offset] },
try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } }, },
intros κ_call51 ap51 h_call51,
rcases h_call51 with ⟨h_call51_ap_offset, h_call51⟩,
rcases h_call51 with ⟨rc_m51, rc_mle51, hl_range_check_ptr₃, h_call51⟩,
generalize' hr_rev_range_check_ptr₃: mem (ap51 - 4) = range_check_ptr₃,
have htv_range_check_ptr₃ := hr_rev_range_check_ptr₃.symm, clear hr_rev_range_check_ptr₃,
generalize' hr_rev_new_y: cast_BigInt3 mem (ap51 - 3) = new_y,
simp only [hr_rev_new_y] at h_call51,
have htv_new_y := hr_rev_new_y.symm, clear hr_rev_new_y,
try { simp only [arg0] at hl_range_check_ptr₃ },
rw [←htv_range_check_ptr₃, ←htv_range_check_ptr₂] at hl_range_check_ptr₃,
try { simp only [arg0] at h_call51 },
rw [←htv_range_check_ptr₂, hl_range_check_ptr₂, hl_range_check_ptr₁, hin_range_check_ptr] at h_call51,
clear arg0,
-- function call
step_assert_eq hpc51 with arg0,
step_assert_eq hpc52 with arg1,
step_assert_eq hpc53 with arg2,
step_assert_eq hpc54 with arg3,
step_assert_eq hpc55 with arg4,
step_assert_eq hpc56 with arg5,
step_assert_eq hpc57 with arg6,
step_assert_eq hpc58 with arg7,
step_assert_eq hpc59 with arg8,
step_assert_eq hpc60 with arg9,
step_sub hpc61 (auto_sound_verify_zero mem _ range_check_ptr₃ {
d0 := slope_sqr.d0 - new_x.d0 - point0.x.d0 - point1.x.d0,
d1 := slope_sqr.d1 - new_x.d1 - point0.x.d1 - point1.x.d1,
d2 := slope_sqr.d2 - new_x.d2 - point0.x.d2 - point1.x.d2
} _ _ _),
{ rw hpc62, norm_num2, exact h_mem_7 },
{ try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point0, hin_point1, htv_range_check_ptr₁, htv_slope, htv_slope_sqr, htv_range_check_ptr₂, htv_new_x, htv_range_check_ptr₃, htv_new_y] },
try { dsimp [cast_EcPoint, cast_BigInt3, cast_UnreducedBigInt3] },
try { arith_simps }, try { simp only [(eq_sub_of_eq_add arg0), (eq_sub_of_eq_add arg1), (eq_sub_of_eq_add arg2), (eq_sub_of_eq_add arg3), (eq_sub_of_eq_add arg4), (eq_sub_of_eq_add arg5), arg6, (eq_sub_of_eq_add arg7), (eq_sub_of_eq_add arg8), (eq_sub_of_eq_add arg9)] },
try { simp only [h_call43_ap_offset, h_call45_ap_offset, h_call48_ap_offset, h_call51_ap_offset] },
try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } }, },
{ try { ext } ; {
try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point0, hin_point1, htv_range_check_ptr₁, htv_slope, htv_slope_sqr, htv_range_check_ptr₂, htv_new_x, htv_range_check_ptr₃, htv_new_y] },
try { dsimp [cast_EcPoint, cast_BigInt3, cast_UnreducedBigInt3] },
try { arith_simps }, try { simp only [(eq_sub_of_eq_add arg0), (eq_sub_of_eq_add arg1), (eq_sub_of_eq_add arg2), (eq_sub_of_eq_add arg3), (eq_sub_of_eq_add arg4), (eq_sub_of_eq_add arg5), arg6, (eq_sub_of_eq_add arg7), (eq_sub_of_eq_add arg8), (eq_sub_of_eq_add arg9)] },
try { simp only [h_call43_ap_offset, h_call45_ap_offset, h_call48_ap_offset, h_call51_ap_offset] },
try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },}, },
intros κ_call63 ap63 h_call63,
rcases h_call63 with ⟨h_call63_ap_offset, h_call63⟩,
rcases h_call63 with ⟨rc_m63, rc_mle63, hl_range_check_ptr₄, h_call63⟩,
generalize' hr_rev_range_check_ptr₄: mem (ap63 - 1) = range_check_ptr₄,
have htv_range_check_ptr₄ := hr_rev_range_check_ptr₄.symm, clear hr_rev_range_check_ptr₄,
try { simp only [arg0 ,arg1 ,arg2 ,arg3 ,arg4 ,arg5 ,arg6 ,arg7 ,arg8 ,arg9] at hl_range_check_ptr₄ },
rw [←htv_range_check_ptr₄, ←htv_range_check_ptr₃] at hl_range_check_ptr₄,
try { simp only [arg0 ,arg1 ,arg2 ,arg3 ,arg4 ,arg5 ,arg6 ,arg7 ,arg8 ,arg9] at h_call63 },
rw [←htv_range_check_ptr₃, hl_range_check_ptr₃, hl_range_check_ptr₂, hl_range_check_ptr₁, hin_range_check_ptr] at h_call63,
clear arg0 arg1 arg2 arg3 arg4 arg5 arg6 arg7 arg8 arg9,
-- function call
step_assert_eq hpc63 with arg0,
step_assert_eq hpc64 with arg1,
step_assert_eq hpc65 with arg2,
step_assert_eq hpc66 with arg3,
step_assert_eq hpc67 with arg4,
step_assert_eq hpc68 with arg5,
step_sub hpc69 (auto_sound_unreduced_mul mem _ {
d0 := point0.x.d0 - new_x.d0,
d1 := point0.x.d1 - new_x.d1,
d2 := point0.x.d2 - new_x.d2
} slope _ _ _),
{ rw hpc70, norm_num2, exact h_mem_5 },
{ try { ext } ; {
try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point0, hin_point1, htv_range_check_ptr₁, htv_slope, htv_slope_sqr, htv_range_check_ptr₂, htv_new_x, htv_range_check_ptr₃, htv_new_y, htv_range_check_ptr₄] },
try { dsimp [cast_EcPoint, cast_BigInt3, cast_UnreducedBigInt3] },
try { arith_simps }, try { simp only [(eq_sub_of_eq_add arg0), (eq_sub_of_eq_add arg1), (eq_sub_of_eq_add arg2), arg3, arg4, arg5] },
try { simp only [h_call43_ap_offset, h_call45_ap_offset, h_call48_ap_offset, h_call51_ap_offset, h_call63_ap_offset] },
try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },}, },
{ try { ext } ; {
try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point0, hin_point1, htv_range_check_ptr₁, htv_slope, htv_slope_sqr, htv_range_check_ptr₂, htv_new_x, htv_range_check_ptr₃, htv_new_y, htv_range_check_ptr₄] },
try { dsimp [cast_EcPoint, cast_BigInt3, cast_UnreducedBigInt3] },
try { arith_simps }, try { simp only [(eq_sub_of_eq_add arg0), (eq_sub_of_eq_add arg1), (eq_sub_of_eq_add arg2), arg3, arg4, arg5] },
try { simp only [h_call43_ap_offset, h_call45_ap_offset, h_call48_ap_offset, h_call51_ap_offset, h_call63_ap_offset] },
try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },}, },
intros κ_call71 ap71 h_call71,
rcases h_call71 with ⟨h_call71_ap_offset, h_call71⟩,
generalize' hr_rev_x_diff_slope: cast_UnreducedBigInt3 mem (ap71 - 3) = x_diff_slope,
simp only [hr_rev_x_diff_slope] at h_call71,
have htv_x_diff_slope := hr_rev_x_diff_slope.symm, clear hr_rev_x_diff_slope,
clear arg0 arg1 arg2 arg3 arg4 arg5,
-- function call
step_assert_eq hpc71 with arg0,
step_assert_eq hpc72 with arg1,
step_assert_eq hpc73 with arg2,
step_assert_eq hpc74 with arg3,
step_assert_eq hpc75 with arg4,
step_assert_eq hpc76 with arg5,
step_assert_eq hpc77 with arg6,
step_sub hpc78 (auto_sound_verify_zero mem _ range_check_ptr₄ {
d0 := x_diff_slope.d0 - point0.y.d0 - new_y.d0,
d1 := x_diff_slope.d1 - point0.y.d1 - new_y.d1,
d2 := x_diff_slope.d2 - point0.y.d2 - new_y.d2
} _ _ _),
{ rw hpc79, norm_num2, exact h_mem_7 },
{ try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point0, hin_point1, htv_range_check_ptr₁, htv_slope, htv_slope_sqr, htv_range_check_ptr₂, htv_new_x, htv_range_check_ptr₃, htv_new_y, htv_range_check_ptr₄, htv_x_diff_slope] },
try { dsimp [cast_EcPoint, cast_BigInt3, cast_UnreducedBigInt3] },
try { arith_simps }, try { simp only [(eq_sub_of_eq_add arg0), (eq_sub_of_eq_add arg1), (eq_sub_of_eq_add arg2), arg3, (eq_sub_of_eq_add arg4), (eq_sub_of_eq_add arg5), (eq_sub_of_eq_add arg6)] },
try { simp only [h_call43_ap_offset, h_call45_ap_offset, h_call48_ap_offset, h_call51_ap_offset, h_call63_ap_offset, h_call71_ap_offset] },
try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } }, },
{ try { ext } ; {
try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point0, hin_point1, htv_range_check_ptr₁, htv_slope, htv_slope_sqr, htv_range_check_ptr₂, htv_new_x, htv_range_check_ptr₃, htv_new_y, htv_range_check_ptr₄, htv_x_diff_slope] },
try { dsimp [cast_EcPoint, cast_BigInt3, cast_UnreducedBigInt3] },
try { arith_simps }, try { simp only [(eq_sub_of_eq_add arg0), (eq_sub_of_eq_add arg1), (eq_sub_of_eq_add arg2), arg3, (eq_sub_of_eq_add arg4), (eq_sub_of_eq_add arg5), (eq_sub_of_eq_add arg6)] },
try { simp only [h_call43_ap_offset, h_call45_ap_offset, h_call48_ap_offset, h_call51_ap_offset, h_call63_ap_offset, h_call71_ap_offset] },
try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },}, },
intros κ_call80 ap80 h_call80,
rcases h_call80 with ⟨h_call80_ap_offset, h_call80⟩,
rcases h_call80 with ⟨rc_m80, rc_mle80, hl_range_check_ptr₅, h_call80⟩,
generalize' hr_rev_range_check_ptr₅: mem (ap80 - 1) = range_check_ptr₅,
have htv_range_check_ptr₅ := hr_rev_range_check_ptr₅.symm, clear hr_rev_range_check_ptr₅,
try { simp only [arg0 ,arg1 ,arg2 ,arg3 ,arg4 ,arg5 ,arg6] at hl_range_check_ptr₅ },
try { rw [h_call71_ap_offset] at hl_range_check_ptr₅ }, try { arith_simps at hl_range_check_ptr₅ },
rw [←htv_range_check_ptr₅, ←htv_range_check_ptr₄] at hl_range_check_ptr₅,
try { simp only [arg0 ,arg1 ,arg2 ,arg3 ,arg4 ,arg5 ,arg6] at h_call80 },
try { rw [h_call71_ap_offset] at h_call80 }, try { arith_simps at h_call80 },
rw [←htv_range_check_ptr₄, hl_range_check_ptr₄, hl_range_check_ptr₃, hl_range_check_ptr₂, hl_range_check_ptr₁, hin_range_check_ptr] at h_call80,
clear arg0 arg1 arg2 arg3 arg4 arg5 arg6,
-- return
step_assert_eq hpc80 with hret0,
step_assert_eq hpc81 with hret1,
step_assert_eq hpc82 with hret2,
step_assert_eq hpc83 with hret3,
step_assert_eq hpc84 with hret4,
step_assert_eq hpc85 with hret5,
step_ret hpc86,
-- finish
step_done, use_only [rfl, rfl],
-- range check condition
use_only (rc_m43+rc_m48+rc_m51+rc_m63+rc_m80+0+0), split,
linarith [rc_mle43, rc_mle48, rc_mle51, rc_mle63, rc_mle80],
split,
{ arith_simps, try { simp only [hret0 ,hret1 ,hret2 ,hret3 ,hret4 ,hret5] },
rw [←htv_range_check_ptr₅, hl_range_check_ptr₅, hl_range_check_ptr₄, hl_range_check_ptr₃, hl_range_check_ptr₂, hl_range_check_ptr₁, hin_range_check_ptr],
try { arith_simps, refl <|> norm_cast }, try { refl } },
intro rc_h_range_check_ptr, repeat { rw [add_assoc] at rc_h_range_check_ptr },
have rc_h_range_check_ptr' := range_checked_add_right rc_h_range_check_ptr,
-- Final Proof
dsimp [auto_spec_fast_ec_add_block9],
try { norm_num1 }, try { arith_simps },
use_only [κ_call43],
use_only [range_check_ptr₁],
use_only [slope],
have rc_h_range_check_ptr₁ := range_checked_offset' rc_h_range_check_ptr,
have rc_h_range_check_ptr₁' := range_checked_add_right rc_h_range_check_ptr₁, try { norm_cast at rc_h_range_check_ptr₁' },
have spec43 := h_call43 rc_h_range_check_ptr',
rw [←hin_range_check_ptr, ←htv_range_check_ptr₁] at spec43,
try { dsimp at spec43, arith_simps at spec43 },
use_only [spec43],
use_only [κ_call45],
use_only [slope_sqr],
try { dsimp at h_call45, arith_simps at h_call45 },
try { use_only [h_call45] },
use_only [κ_call48],
use_only [range_check_ptr₂],
use_only [new_x],
have rc_h_range_check_ptr₂ := range_checked_offset' rc_h_range_check_ptr₁,
have rc_h_range_check_ptr₂' := range_checked_add_right rc_h_range_check_ptr₂, try { norm_cast at rc_h_range_check_ptr₂' },
have spec48 := h_call48 rc_h_range_check_ptr₁',
rw [←hin_range_check_ptr, ←hl_range_check_ptr₁, ←htv_range_check_ptr₂] at spec48,
try { dsimp at spec48, arith_simps at spec48 },
use_only [spec48],
use_only [κ_call51],
use_only [range_check_ptr₃],
use_only [new_y],
have rc_h_range_check_ptr₃ := range_checked_offset' rc_h_range_check_ptr₂,
have rc_h_range_check_ptr₃' := range_checked_add_right rc_h_range_check_ptr₃, try { norm_cast at rc_h_range_check_ptr₃' },
have spec51 := h_call51 rc_h_range_check_ptr₂',
rw [←hin_range_check_ptr, ←hl_range_check_ptr₁, ←hl_range_check_ptr₂, ←htv_range_check_ptr₃] at spec51,
try { dsimp at spec51, arith_simps at spec51 },
use_only [spec51],
use_only [κ_call63],
use_only [range_check_ptr₄],
have rc_h_range_check_ptr₄ := range_checked_offset' rc_h_range_check_ptr₃,
have rc_h_range_check_ptr₄' := range_checked_add_right rc_h_range_check_ptr₄, try { norm_cast at rc_h_range_check_ptr₄' },
have spec63 := h_call63 rc_h_range_check_ptr₃',
rw [←hin_range_check_ptr, ←hl_range_check_ptr₁, ←hl_range_check_ptr₂, ←hl_range_check_ptr₃, ←htv_range_check_ptr₄] at spec63,
try { dsimp at spec63, arith_simps at spec63 },
use_only [spec63],
use_only [κ_call71],
use_only [x_diff_slope],
try { dsimp at h_call71, arith_simps at h_call71 },
try { use_only [h_call71] },
use_only [κ_call80],
use_only [range_check_ptr₅],
have rc_h_range_check_ptr₅ := range_checked_offset' rc_h_range_check_ptr₄,
have rc_h_range_check_ptr₅' := range_checked_add_right rc_h_range_check_ptr₅, try { norm_cast at rc_h_range_check_ptr₅' },
have spec80 := h_call80 rc_h_range_check_ptr₄',
rw [←hin_range_check_ptr, ←hl_range_check_ptr₁, ←hl_range_check_ptr₂, ←hl_range_check_ptr₃, ←hl_range_check_ptr₄, ←htv_range_check_ptr₅] at spec80,
try { dsimp at spec80, arith_simps at spec80 },
use_only [spec80],
try { split, linarith },
try { ensures_simps; try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point0, hin_point1, htv_range_check_ptr₁, htv_slope, htv_slope_sqr, htv_range_check_ptr₂, htv_new_x, htv_range_check_ptr₃, htv_new_y, htv_range_check_ptr₄, htv_x_diff_slope, htv_range_check_ptr₅] }, },
try { dsimp [cast_EcPoint, cast_BigInt3, cast_UnreducedBigInt3] },
try { arith_simps }, try { simp only [hret0, hret1, hret2, hret3, hret4, hret5] },
try { simp only [h_call43_ap_offset, h_call45_ap_offset, h_call48_ap_offset, h_call51_ap_offset, h_call63_ap_offset, h_call71_ap_offset, h_call80_ap_offset] },
try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },
end
theorem auto_sound_fast_ec_add_block5
-- An independent ap variable.
(ap : F)
-- arguments
(range_check_ptr : F) (point0 point1 : EcPoint F)
-- code is in memory at σ.pc
(h_mem : mem_at mem code_fast_ec_add σ.pc)
-- all dependencies are in memory
(h_mem_4 : mem_at mem code_nondet_bigint3 (σ.pc - 317))
(h_mem_5 : mem_at mem code_unreduced_mul (σ.pc - 305))
(h_mem_6 : mem_at mem code_unreduced_sqr (σ.pc - 285))
(h_mem_7 : mem_at mem code_verify_zero (σ.pc - 269))
(h_mem_13 : mem_at mem code_compute_slope (σ.pc - 97))
-- input arguments on the stack
(hin_range_check_ptr : range_check_ptr = mem (σ.fp - 15))
(hin_point0 : point0 = cast_EcPoint mem (σ.fp - 14))
(hin_point1 : point1 = cast_EcPoint mem (σ.fp - 8))
(νbound : ℕ)
-- conclusion
: ensuresb_ret νbound mem
{pc := σ.pc + 14, ap := ap, fp := σ.fp}
(λ κ τ,
∃ μ ≤ κ, rc_ensures mem (rc_bound F) μ (mem (σ.fp - 15)) (mem $ τ.ap - 7)
(auto_spec_fast_ec_add_block5 mem κ range_check_ptr point0 point1 (mem (τ.ap - 7)) (cast_EcPoint mem (τ.ap - 6)))) :=
begin
have h_mem_rec := h_mem,
unpack_memory code_fast_ec_add at h_mem with ⟨hpc0, hpc1, hpc2, hpc3, hpc4, hpc5, hpc6, hpc7, hpc8, hpc9, hpc10, hpc11, hpc12, hpc13, hpc14, hpc15, hpc16, hpc17, hpc18, hpc19, hpc20, hpc21, hpc22, hpc23, hpc24, hpc25, hpc26, hpc27, hpc28, hpc29, hpc30, hpc31, hpc32, hpc33, hpc34, hpc35, hpc36, hpc37, hpc38, hpc39, hpc40, hpc41, hpc42, hpc43, hpc44, hpc45, hpc46, hpc47, hpc48, hpc49, hpc50, hpc51, hpc52, hpc53, hpc54, hpc55, hpc56, hpc57, hpc58, hpc59, hpc60, hpc61, hpc62, hpc63, hpc64, hpc65, hpc66, hpc67, hpc68, hpc69, hpc70, hpc71, hpc72, hpc73, hpc74, hpc75, hpc76, hpc77, hpc78, hpc79, hpc80, hpc81, hpc82, hpc83, hpc84, hpc85, hpc86⟩,
-- if statement
step_jnz hpc14 hpc15 with hcond hcond,
{
-- if: positive branch
have a14 : point1.x.d0 = 0, {
try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point0, hin_point1] },
try { dsimp [cast_EcPoint, cast_BigInt3] },
try { arith_simps }, try { simp only [hcond] },
try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },
},
try { dsimp at a14 }, try { arith_simps at a14 },
clear hcond,
-- if statement
step_jnz hpc16 hpc17 with hcond hcond,
{
-- if: positive branch
have a16 : point1.x.d1 = 0, {
try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point0, hin_point1] },
try { dsimp [cast_EcPoint, cast_BigInt3] },
try { arith_simps }, try { simp only [hcond] },
try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },
},
try { dsimp at a16 }, try { arith_simps at a16 },
clear hcond,
-- if statement
step_jnz hpc18 hpc19 with hcond hcond,
{
-- if: positive branch
have a18 : point1.x.d2 = 0, {
try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point0, hin_point1] },
try { dsimp [cast_EcPoint, cast_BigInt3] },
try { arith_simps }, try { simp only [hcond] },
try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },
},
try { dsimp at a18 }, try { arith_simps at a18 },
clear hcond,
-- return
step_assert_eq hpc20 with hret0,
step_assert_eq hpc21 with hret1,
step_assert_eq hpc22 with hret2,
step_assert_eq hpc23 with hret3,
step_assert_eq hpc24 with hret4,
step_assert_eq hpc25 with hret5,
step_assert_eq hpc26 with hret6,
step_ret hpc27,
-- finish
step_done, use_only [rfl, rfl],
-- range check condition
use_only (0+0), split,
linarith [],
split,
{ arith_simps, try { simp only [hret0 ,hret1 ,hret2 ,hret3 ,hret4 ,hret5 ,hret6] },
try { arith_simps, refl <|> norm_cast }, try { refl } },
intro rc_h_range_check_ptr, repeat { rw [add_assoc] at rc_h_range_check_ptr },
have rc_h_range_check_ptr' := range_checked_add_right rc_h_range_check_ptr,
-- Final Proof
dsimp [auto_spec_fast_ec_add_block5],
try { norm_num1 }, try { arith_simps },
left,
use_only [a14],
left,
use_only [a16],
left,
use_only [a18],
try { split, linarith },
try { ensures_simps; try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point0, hin_point1] }, },
try { dsimp [cast_EcPoint, cast_BigInt3] },
try { arith_simps }, try { simp only [hret0, hret1, hret2, hret3, hret4, hret5, hret6] },
try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },
},
{
-- if: negative branch
have a18 : point1.x.d2 ≠ 0, {
try { simp only [ne.def] },
try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point0, hin_point1] },
try { dsimp [cast_EcPoint, cast_BigInt3] },
try { arith_simps }, try { simp only [hcond] },
try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },
},
try { dsimp at a18 }, try { arith_simps at a18 },
clear hcond,
-- Use the block soundness theorem.
apply ensuresb_ret_trans (auto_sound_fast_ec_add_block9 mem σ _ range_check_ptr point0 point1 h_mem_rec h_mem_4 h_mem_5 h_mem_6 h_mem_7 h_mem_13 hin_range_check_ptr hin_point0 hin_point1 νbound),
intros κ_block9 τ, try { arith_simps },
intro h_block9,
rcases h_block9 with ⟨rc_m_block9, rc_m_le_block9, hblk_range_check_ptr₁, h_block9⟩,
-- range check condition
use_only (rc_m_block9+0+0), split,
linarith [rc_m_le_block9],
split,
{ arith_simps, try { simp only [hblk_range_check_ptr₁] },
try { arith_simps, refl <|> norm_cast }, try { refl } },
intro rc_h_range_check_ptr, repeat { rw [add_assoc] at rc_h_range_check_ptr },
have rc_h_range_check_ptr' := range_checked_add_right rc_h_range_check_ptr,
-- Final Proof
dsimp [auto_spec_fast_ec_add_block5],
try { norm_num1 }, try { arith_simps },
left,
use_only [a14],
left,
use_only [a16],
right,
use_only [a18],
have rc_h_range_check_ptr₁ := range_checked_offset' rc_h_range_check_ptr,
have rc_h_range_check_ptr₁' := range_checked_add_right rc_h_range_check_ptr₁, try { norm_cast at rc_h_range_check_ptr₁' },
have h_block9' := h_block9 rc_h_range_check_ptr',
try { rw [←hin_range_check_ptr] at h_block9' },
try { dsimp at h_block9, arith_simps at h_block9' },
have h_block9 := h_block9',
use_only[κ_block9],
use [h_block9],
try { linarith }
}
},
{
-- if: negative branch
have a16 : point1.x.d1 ≠ 0, {
try { simp only [ne.def] },
try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point0, hin_point1] },
try { dsimp [cast_EcPoint, cast_BigInt3] },
try { arith_simps }, try { simp only [hcond] },
try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },
},
try { dsimp at a16 }, try { arith_simps at a16 },
clear hcond,
-- Use the block soundness theorem.
apply ensuresb_ret_trans (auto_sound_fast_ec_add_block9 mem σ _ range_check_ptr point0 point1 h_mem_rec h_mem_4 h_mem_5 h_mem_6 h_mem_7 h_mem_13 hin_range_check_ptr hin_point0 hin_point1 νbound),
intros κ_block9 τ, try { arith_simps },
intro h_block9,
rcases h_block9 with ⟨rc_m_block9, rc_m_le_block9, hblk_range_check_ptr₁, h_block9⟩,
-- range check condition
use_only (rc_m_block9+0+0), split,
linarith [rc_m_le_block9],
split,
{ arith_simps, try { simp only [hblk_range_check_ptr₁] },
try { arith_simps, refl <|> norm_cast }, try { refl } },
intro rc_h_range_check_ptr, repeat { rw [add_assoc] at rc_h_range_check_ptr },
have rc_h_range_check_ptr' := range_checked_add_right rc_h_range_check_ptr,
-- Final Proof
dsimp [auto_spec_fast_ec_add_block5],
try { norm_num1 }, try { arith_simps },
left,
use_only [a14],
right,
use_only [a16],
have rc_h_range_check_ptr₁ := range_checked_offset' rc_h_range_check_ptr,
have rc_h_range_check_ptr₁' := range_checked_add_right rc_h_range_check_ptr₁, try { norm_cast at rc_h_range_check_ptr₁' },
have h_block9' := h_block9 rc_h_range_check_ptr',
try { rw [←hin_range_check_ptr] at h_block9' },
try { dsimp at h_block9, arith_simps at h_block9' },
have h_block9 := h_block9',
use_only[κ_block9],
use [h_block9],
try { linarith }
}
},
{
-- if: negative branch
have a14 : point1.x.d0 ≠ 0, {
try { simp only [ne.def] },
try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point0, hin_point1] },
try { dsimp [cast_EcPoint, cast_BigInt3] },
try { arith_simps }, try { simp only [hcond] },
try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },
},
try { dsimp at a14 }, try { arith_simps at a14 },
clear hcond,
-- Use the block soundness theorem.
apply ensuresb_ret_trans (auto_sound_fast_ec_add_block9 mem σ _ range_check_ptr point0 point1 h_mem_rec h_mem_4 h_mem_5 h_mem_6 h_mem_7 h_mem_13 hin_range_check_ptr hin_point0 hin_point1 νbound),
intros κ_block9 τ, try { arith_simps },
intro h_block9,
rcases h_block9 with ⟨rc_m_block9, rc_m_le_block9, hblk_range_check_ptr₁, h_block9⟩,
-- range check condition
use_only (rc_m_block9+0+0), split,
linarith [rc_m_le_block9],
split,
{ arith_simps, try { simp only [hblk_range_check_ptr₁] },
try { arith_simps, refl <|> norm_cast }, try { refl } },
intro rc_h_range_check_ptr, repeat { rw [add_assoc] at rc_h_range_check_ptr },
have rc_h_range_check_ptr' := range_checked_add_right rc_h_range_check_ptr,
-- Final Proof
dsimp [auto_spec_fast_ec_add_block5],
try { norm_num1 }, try { arith_simps },
right,
use_only [a14],
have rc_h_range_check_ptr₁ := range_checked_offset' rc_h_range_check_ptr,
have rc_h_range_check_ptr₁' := range_checked_add_right rc_h_range_check_ptr₁, try { norm_cast at rc_h_range_check_ptr₁' },
have h_block9' := h_block9 rc_h_range_check_ptr',
try { rw [←hin_range_check_ptr] at h_block9' },
try { dsimp at h_block9, arith_simps at h_block9' },
have h_block9 := h_block9',
use_only[κ_block9],
use [h_block9],
try { linarith }
}
end
theorem auto_sound_fast_ec_add
-- arguments
(range_check_ptr : F) (point0 point1 : EcPoint F)
-- code is in memory at σ.pc
(h_mem : mem_at mem code_fast_ec_add σ.pc)
-- all dependencies are in memory
(h_mem_4 : mem_at mem code_nondet_bigint3 (σ.pc - 317))
(h_mem_5 : mem_at mem code_unreduced_mul (σ.pc - 305))
(h_mem_6 : mem_at mem code_unreduced_sqr (σ.pc - 285))
(h_mem_7 : mem_at mem code_verify_zero (σ.pc - 269))
(h_mem_13 : mem_at mem code_compute_slope (σ.pc - 97))
-- input arguments on the stack
(hin_range_check_ptr : range_check_ptr = mem (σ.fp - 15))
(hin_point0 : point0 = cast_EcPoint mem (σ.fp - 14))
(hin_point1 : point1 = cast_EcPoint mem (σ.fp - 8))
-- conclusion
: ensures_ret mem σ (λ κ τ,
∃ μ ≤ κ, rc_ensures mem (rc_bound F) μ (mem (σ.fp - 15)) (mem $ τ.ap - 7)
(spec_fast_ec_add mem κ range_check_ptr point0 point1 (mem (τ.ap - 7)) (cast_EcPoint mem (τ.ap - 6)))) :=
begin
apply ensures_of_ensuresb, intro νbound,
have h_mem_rec := h_mem,
unpack_memory code_fast_ec_add at h_mem with ⟨hpc0, hpc1, hpc2, hpc3, hpc4, hpc5, hpc6, hpc7, hpc8, hpc9, hpc10, hpc11, hpc12, hpc13, hpc14, hpc15, hpc16, hpc17, hpc18, hpc19, hpc20, hpc21, hpc22, hpc23, hpc24, hpc25, hpc26, hpc27, hpc28, hpc29, hpc30, hpc31, hpc32, hpc33, hpc34, hpc35, hpc36, hpc37, hpc38, hpc39, hpc40, hpc41, hpc42, hpc43, hpc44, hpc45, hpc46, hpc47, hpc48, hpc49, hpc50, hpc51, hpc52, hpc53, hpc54, hpc55, hpc56, hpc57, hpc58, hpc59, hpc60, hpc61, hpc62, hpc63, hpc64, hpc65, hpc66, hpc67, hpc68, hpc69, hpc70, hpc71, hpc72, hpc73, hpc74, hpc75, hpc76, hpc77, hpc78, hpc79, hpc80, hpc81, hpc82, hpc83, hpc84, hpc85, hpc86⟩,
-- if statement
step_jnz hpc0 hpc1 with hcond hcond,
{
-- if: positive branch
have a0 : point0.x.d0 = 0, {
try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point0, hin_point1] },
try { dsimp [cast_EcPoint, cast_BigInt3] },
try { arith_simps }, try { simp only [hcond] },
try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },
},
try { dsimp at a0 }, try { arith_simps at a0 },
clear hcond,
-- if statement
step_jnz hpc2 hpc3 with hcond hcond,
{
-- if: positive branch
have a2 : point0.x.d1 = 0, {
try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point0, hin_point1] },
try { dsimp [cast_EcPoint, cast_BigInt3] },
try { arith_simps }, try { simp only [hcond] },
try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },
},
try { dsimp at a2 }, try { arith_simps at a2 },
clear hcond,
-- if statement
step_jnz hpc4 hpc5 with hcond hcond,
{
-- if: positive branch
have a4 : point0.x.d2 = 0, {
try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point0, hin_point1] },
try { dsimp [cast_EcPoint, cast_BigInt3] },
try { arith_simps }, try { simp only [hcond] },
try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },
},
try { dsimp at a4 }, try { arith_simps at a4 },
clear hcond,
-- return
step_assert_eq hpc6 with hret0,
step_assert_eq hpc7 with hret1,
step_assert_eq hpc8 with hret2,
step_assert_eq hpc9 with hret3,
step_assert_eq hpc10 with hret4,
step_assert_eq hpc11 with hret5,
step_assert_eq hpc12 with hret6,
step_ret hpc13,
-- finish
step_done, use_only [rfl, rfl],
-- range check condition
use_only (0+0), split,
linarith [],
split,
{ arith_simps, try { simp only [hret0 ,hret1 ,hret2 ,hret3 ,hret4 ,hret5 ,hret6] },
try { arith_simps, refl <|> norm_cast }, try { refl } },
intro rc_h_range_check_ptr, repeat { rw [add_assoc] at rc_h_range_check_ptr },
have rc_h_range_check_ptr' := range_checked_add_right rc_h_range_check_ptr,
-- Final Proof
-- user-provided reduction
suffices auto_spec: auto_spec_fast_ec_add mem _ range_check_ptr point0 point1 _ _,
{ apply sound_fast_ec_add, apply auto_spec },
-- prove the auto generated assertion
dsimp [auto_spec_fast_ec_add],
try { norm_num1 }, try { arith_simps },
left,
use_only [a0],
left,
use_only [a2],
left,
use_only [a4],
try { split, linarith },
try { ensures_simps; try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point0, hin_point1] }, },
try { dsimp [cast_EcPoint, cast_BigInt3] },
try { arith_simps }, try { simp only [hret0, hret1, hret2, hret3, hret4, hret5, hret6] },
try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },
},
{
-- if: negative branch
have a4 : point0.x.d2 ≠ 0, {
try { simp only [ne.def] },
try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point0, hin_point1] },
try { dsimp [cast_EcPoint, cast_BigInt3] },
try { arith_simps }, try { simp only [hcond] },
try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },
},
try { dsimp at a4 }, try { arith_simps at a4 },
clear hcond,
-- Use the block soundness theorem.
apply ensuresb_ret_trans (auto_sound_fast_ec_add_block5 mem σ _ range_check_ptr point0 point1 h_mem_rec h_mem_4 h_mem_5 h_mem_6 h_mem_7 h_mem_13 hin_range_check_ptr hin_point0 hin_point1 νbound),
intros κ_block5 τ, try { arith_simps },
intro h_block5,
rcases h_block5 with ⟨rc_m_block5, rc_m_le_block5, hblk_range_check_ptr₁, h_block5⟩,
-- range check condition
use_only (rc_m_block5+0+0), split,
linarith [rc_m_le_block5],
split,
{ arith_simps, try { simp only [hblk_range_check_ptr₁] },
try { arith_simps, refl <|> norm_cast }, try { refl } },
intro rc_h_range_check_ptr, repeat { rw [add_assoc] at rc_h_range_check_ptr },
have rc_h_range_check_ptr' := range_checked_add_right rc_h_range_check_ptr,
-- Final Proof
-- user-provided reduction
suffices auto_spec: auto_spec_fast_ec_add mem _ range_check_ptr point0 point1 _ _,
{ apply sound_fast_ec_add, apply auto_spec },
-- prove the auto generated assertion
dsimp [auto_spec_fast_ec_add],
try { norm_num1 }, try { arith_simps },
left,
use_only [a0],
left,
use_only [a2],
right,
use_only [a4],
have rc_h_range_check_ptr₁ := range_checked_offset' rc_h_range_check_ptr,
have rc_h_range_check_ptr₁' := range_checked_add_right rc_h_range_check_ptr₁, try { norm_cast at rc_h_range_check_ptr₁' },
have h_block5' := h_block5 rc_h_range_check_ptr',
try { rw [←hin_range_check_ptr] at h_block5' },
try { dsimp at h_block5, arith_simps at h_block5' },
have h_block5 := h_block5',
use_only[κ_block5],
use [h_block5],
try { linarith }
}
},
{
-- if: negative branch
have a2 : point0.x.d1 ≠ 0, {
try { simp only [ne.def] },
try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point0, hin_point1] },
try { dsimp [cast_EcPoint, cast_BigInt3] },
try { arith_simps }, try { simp only [hcond] },
try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },
},
try { dsimp at a2 }, try { arith_simps at a2 },
clear hcond,
-- Use the block soundness theorem.
apply ensuresb_ret_trans (auto_sound_fast_ec_add_block5 mem σ _ range_check_ptr point0 point1 h_mem_rec h_mem_4 h_mem_5 h_mem_6 h_mem_7 h_mem_13 hin_range_check_ptr hin_point0 hin_point1 νbound),
intros κ_block5 τ, try { arith_simps },
intro h_block5,
rcases h_block5 with ⟨rc_m_block5, rc_m_le_block5, hblk_range_check_ptr₁, h_block5⟩,
-- range check condition
use_only (rc_m_block5+0+0), split,
linarith [rc_m_le_block5],
split,
{ arith_simps, try { simp only [hblk_range_check_ptr₁] },
try { arith_simps, refl <|> norm_cast }, try { refl } },
intro rc_h_range_check_ptr, repeat { rw [add_assoc] at rc_h_range_check_ptr },
have rc_h_range_check_ptr' := range_checked_add_right rc_h_range_check_ptr,
-- Final Proof
-- user-provided reduction
suffices auto_spec: auto_spec_fast_ec_add mem _ range_check_ptr point0 point1 _ _,
{ apply sound_fast_ec_add, apply auto_spec },
-- prove the auto generated assertion
dsimp [auto_spec_fast_ec_add],
try { norm_num1 }, try { arith_simps },
left,
use_only [a0],
right,
use_only [a2],
have rc_h_range_check_ptr₁ := range_checked_offset' rc_h_range_check_ptr,
have rc_h_range_check_ptr₁' := range_checked_add_right rc_h_range_check_ptr₁, try { norm_cast at rc_h_range_check_ptr₁' },
have h_block5' := h_block5 rc_h_range_check_ptr',
try { rw [←hin_range_check_ptr] at h_block5' },
try { dsimp at h_block5, arith_simps at h_block5' },
have h_block5 := h_block5',
use_only[κ_block5],
use [h_block5],
try { linarith }
}
},
{
-- if: negative branch
have a0 : point0.x.d0 ≠ 0, {
try { simp only [ne.def] },
try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point0, hin_point1] },
try { dsimp [cast_EcPoint, cast_BigInt3] },
try { arith_simps }, try { simp only [hcond] },
try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },
},
try { dsimp at a0 }, try { arith_simps at a0 },
clear hcond,
-- Use the block soundness theorem.
apply ensuresb_ret_trans (auto_sound_fast_ec_add_block5 mem σ _ range_check_ptr point0 point1 h_mem_rec h_mem_4 h_mem_5 h_mem_6 h_mem_7 h_mem_13 hin_range_check_ptr hin_point0 hin_point1 νbound),
intros κ_block5 τ, try { arith_simps },
intro h_block5,
rcases h_block5 with ⟨rc_m_block5, rc_m_le_block5, hblk_range_check_ptr₁, h_block5⟩,
-- range check condition
use_only (rc_m_block5+0+0), split,
linarith [rc_m_le_block5],
split,
{ arith_simps, try { simp only [hblk_range_check_ptr₁] },
try { arith_simps, refl <|> norm_cast }, try { refl } },
intro rc_h_range_check_ptr, repeat { rw [add_assoc] at rc_h_range_check_ptr },
have rc_h_range_check_ptr' := range_checked_add_right rc_h_range_check_ptr,
-- Final Proof
-- user-provided reduction
suffices auto_spec: auto_spec_fast_ec_add mem _ range_check_ptr point0 point1 _ _,
{ apply sound_fast_ec_add, apply auto_spec },
-- prove the auto generated assertion
dsimp [auto_spec_fast_ec_add],
try { norm_num1 }, try { arith_simps },
right,
use_only [a0],
have rc_h_range_check_ptr₁ := range_checked_offset' rc_h_range_check_ptr,
have rc_h_range_check_ptr₁' := range_checked_add_right rc_h_range_check_ptr₁, try { norm_cast at rc_h_range_check_ptr₁' },
have h_block5' := h_block5 rc_h_range_check_ptr',
try { rw [←hin_range_check_ptr] at h_block5' },
try { dsimp at h_block5, arith_simps at h_block5' },
have h_block5 := h_block5',
use_only[κ_block5],
use [h_block5],
try { linarith }
}
end
|
4b968098e7c98f87dcbd3eaa851eb3ff07edfa7e | 9dc8cecdf3c4634764a18254e94d43da07142918 | /src/number_theory/class_number/finite.lean | 332cfe561384c1078728de2f22d64b36c7ea5720 | [
"Apache-2.0"
] | permissive | jcommelin/mathlib | d8456447c36c176e14d96d9e76f39841f69d2d9b | ee8279351a2e434c2852345c51b728d22af5a156 | refs/heads/master | 1,664,782,136,488 | 1,663,638,983,000 | 1,663,638,983,000 | 132,563,656 | 0 | 0 | Apache-2.0 | 1,663,599,929,000 | 1,525,760,539,000 | Lean | UTF-8 | Lean | false | false | 17,765 | lean | /-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import analysis.special_functions.pow
import linear_algebra.free_module.pid
import linear_algebra.matrix.absolute_value
import number_theory.class_number.admissible_absolute_value
import ring_theory.class_group
import ring_theory.dedekind_domain.integral_closure
import ring_theory.norm
/-!
# Class numbers of global fields
In this file, we use the notion of "admissible absolute value" to prove
finiteness of the class group for number fields and function fields,
and define `class_number` as the order of this group.
## Main definitions
- `class_group.fintype_of_admissible`: if `R` has an admissible absolute value,
its integral closure has a finite class group
-/
open_locale big_operators
open_locale non_zero_divisors
namespace class_group
open ring
open_locale big_operators
section euclidean_domain
variables (R S K L : Type*) [euclidean_domain R] [comm_ring S] [is_domain S]
variables [field K] [field L]
variables [algebra R K] [is_fraction_ring R K]
variables [algebra K L] [finite_dimensional K L] [is_separable K L]
variables [algRL : algebra R L] [is_scalar_tower R K L]
variables [algebra R S] [algebra S L]
variables [ist : is_scalar_tower R S L] [iic : is_integral_closure S R L]
variables {R S} (abv : absolute_value R ℤ)
variables {ι : Type*} [decidable_eq ι] [fintype ι] (bS : basis ι R S)
/-- If `b` is an `R`-basis of `S` of cardinality `n`, then `norm_bound abv b` is an integer
such that for every `R`-integral element `a : S` with coordinates `≤ y`,
we have algebra.norm a ≤ norm_bound abv b * y ^ n`. (See also `norm_le` and `norm_lt`). -/
noncomputable def norm_bound : ℤ :=
let n := fintype.card ι,
i : ι := nonempty.some bS.index_nonempty,
m : ℤ := finset.max' (finset.univ.image (λ (ijk : ι × ι × ι),
abv (algebra.left_mul_matrix bS (bS ijk.1) ijk.2.1 ijk.2.2)))
⟨_, finset.mem_image.mpr ⟨⟨i, i, i⟩, finset.mem_univ _, rfl⟩⟩
in nat.factorial n • (n • m) ^ n
lemma norm_bound_pos : 0 < norm_bound abv bS :=
begin
obtain ⟨i, j, k, hijk⟩ : ∃ i j k,
algebra.left_mul_matrix bS (bS i) j k ≠ 0,
{ by_contra' h,
obtain ⟨i⟩ := bS.index_nonempty,
apply bS.ne_zero i,
apply (injective_iff_map_eq_zero (algebra.left_mul_matrix bS)).mp
(algebra.left_mul_matrix_injective bS),
ext j k,
simp [h, dmatrix.zero_apply] },
simp only [norm_bound, algebra.smul_def, eq_nat_cast],
refine mul_pos (int.coe_nat_pos.mpr (nat.factorial_pos _)) _,
refine pow_pos (mul_pos (int.coe_nat_pos.mpr (fintype.card_pos_iff.mpr ⟨i⟩)) _) _,
refine lt_of_lt_of_le (abv.pos hijk) (finset.le_max' _ _ _),
exact finset.mem_image.mpr ⟨⟨i, j, k⟩, finset.mem_univ _, rfl⟩
end
/-- If the `R`-integral element `a : S` has coordinates `≤ y` with respect to some basis `b`,
its norm is less than `norm_bound abv b * y ^ dim S`. -/
lemma norm_le (a : S) {y : ℤ} (hy : ∀ k, abv (bS.repr a k) ≤ y) :
abv (algebra.norm R a) ≤ norm_bound abv bS * y ^ fintype.card ι :=
begin
conv_lhs { rw ← bS.sum_repr a },
rw [algebra.norm_apply, ← linear_map.det_to_matrix bS],
simp only [algebra.norm_apply, alg_hom.map_sum, alg_hom.map_smul, linear_equiv.map_sum,
linear_equiv.map_smul, algebra.to_matrix_lmul_eq, norm_bound, smul_mul_assoc, ← mul_pow],
convert matrix.det_sum_smul_le finset.univ _ hy using 3,
{ rw [finset.card_univ, smul_mul_assoc, mul_comm] },
{ intros i j k,
apply finset.le_max',
exact finset.mem_image.mpr ⟨⟨i, j, k⟩, finset.mem_univ _, rfl⟩ },
end
/-- If the `R`-integral element `a : S` has coordinates `< y` with respect to some basis `b`,
its norm is strictly less than `norm_bound abv b * y ^ dim S`. -/
lemma norm_lt {T : Type*} [linear_ordered_ring T]
(a : S) {y : T} (hy : ∀ k, (abv (bS.repr a k) : T) < y) :
(abv (algebra.norm R a) : T) < norm_bound abv bS * y ^ fintype.card ι :=
begin
obtain ⟨i⟩ := bS.index_nonempty,
have him : (finset.univ.image (λ k, abv (bS.repr a k))).nonempty :=
⟨_, finset.mem_image.mpr ⟨i, finset.mem_univ _, rfl⟩⟩,
set y' : ℤ := finset.max' _ him with y'_def,
have hy' : ∀ k, abv (bS.repr a k) ≤ y',
{ intro k,
exact finset.le_max' _ _ (finset.mem_image.mpr ⟨k, finset.mem_univ _, rfl⟩) },
have : (y' : T) < y,
{ rw [y'_def, ← finset.max'_image (show monotone (coe : ℤ → T), from λ x y h, int.cast_le.mpr h)],
apply (finset.max'_lt_iff _ (him.image _)).mpr,
simp only [finset.mem_image, exists_prop],
rintros _ ⟨x, ⟨k, -, rfl⟩, rfl⟩,
exact hy k },
have y'_nonneg : 0 ≤ y' := le_trans (abv.nonneg _) (hy' i),
apply (int.cast_le.mpr (norm_le abv bS a hy')).trans_lt,
simp only [int.cast_mul, int.cast_pow],
apply mul_lt_mul' le_rfl,
{ exact pow_lt_pow_of_lt_left this
(int.cast_nonneg.mpr y'_nonneg)
(fintype.card_pos_iff.mpr ⟨i⟩) },
{ exact pow_nonneg (int.cast_nonneg.mpr y'_nonneg) _ },
{ exact int.cast_pos.mpr (norm_bound_pos abv bS) },
{ apply_instance }
end
/-- A nonzero ideal has an element of minimal norm. -/
lemma exists_min (I : (ideal S)⁰) :
∃ b ∈ (I : ideal S), b ≠ 0 ∧
∀ c ∈ (I : ideal S), abv (algebra.norm R c) < abv (algebra.norm R b) → c = (0 : S) :=
begin
obtain ⟨_, ⟨b, b_mem, b_ne_zero, rfl⟩, min⟩ := @int.exists_least_of_bdd
(λ a, ∃ b ∈ (I : ideal S), b ≠ (0 : S) ∧ abv (algebra.norm R b) = a) _ _,
{ refine ⟨b, b_mem, b_ne_zero, _⟩,
intros c hc lt,
contrapose! lt with c_ne_zero,
exact min _ ⟨c, hc, c_ne_zero, rfl⟩ },
{ use 0,
rintros _ ⟨b, b_mem, b_ne_zero, rfl⟩,
apply abv.nonneg },
{ obtain ⟨b, b_mem, b_ne_zero⟩ := (I : ideal S).ne_bot_iff.mp (non_zero_divisors.coe_ne_zero I),
exact ⟨_, ⟨b, b_mem, b_ne_zero, rfl⟩⟩ }
end
section is_admissible
variables (L) {abv} (adm : abv.is_admissible)
include adm
/-- If we have a large enough set of elements in `R^ι`, then there will be a pair
whose remainders are close together. We'll show that all sets of cardinality
at least `cardM bS adm` elements satisfy this condition.
The value of `cardM` is not at all optimal: for specific choices of `R`,
the minimum cardinality can be exponentially smaller.
-/
noncomputable def cardM : ℕ :=
adm.card (norm_bound abv bS ^ (-1 / (fintype.card ι) : ℝ)) ^ fintype.card ι
variables [infinite R]
/-- In the following results, we need a large set of distinct elements of `R`. -/
noncomputable def distinct_elems : fin (cardM bS adm).succ ↪ R :=
fin.coe_embedding.trans (infinite.nat_embedding R)
variables [decidable_eq R]
/-- `finset_approx` is a finite set such that each fractional ideal in the integral closure
contains an element close to `finset_approx`. -/
noncomputable def finset_approx : finset R :=
(finset.univ.image $ λ xy : _ × _, distinct_elems bS adm xy.1 - distinct_elems bS adm xy.2).erase 0
lemma finset_approx.zero_not_mem : (0 : R) ∉ finset_approx bS adm :=
finset.not_mem_erase _ _
@[simp] lemma mem_finset_approx {x : R} :
x ∈ finset_approx bS adm ↔
∃ i j, i ≠ j ∧ distinct_elems bS adm i - distinct_elems bS adm j = x :=
begin
simp only [finset_approx, finset.mem_erase, finset.mem_image],
split,
{ rintros ⟨hx, ⟨i, j⟩, _, rfl⟩,
refine ⟨i, j, _, rfl⟩,
rintro rfl,
simpa using hx },
{ rintros ⟨i, j, hij, rfl⟩,
refine ⟨_, ⟨i, j⟩, finset.mem_univ _, rfl⟩,
rw [ne.def, sub_eq_zero],
exact λ h, hij ((distinct_elems bS adm).injective h) }
end
section real
open real
local attribute [-instance] real.decidable_eq
/-- We can approximate `a / b : L` with `q / r`, where `r` has finitely many options for `L`. -/
theorem exists_mem_finset_approx (a : S) {b} (hb : b ≠ (0 : R)) :
∃ (q : S) (r ∈ finset_approx bS adm),
abv (algebra.norm R (r • a - b • q)) < abv (algebra.norm R (algebra_map R S b)) :=
begin
have dim_pos := fintype.card_pos_iff.mpr bS.index_nonempty,
set ε : ℝ := norm_bound abv bS ^ (-1 / (fintype.card ι) : ℝ) with ε_eq,
have hε : 0 < ε := real.rpow_pos_of_pos (int.cast_pos.mpr (norm_bound_pos abv bS)) _,
have ε_le : (norm_bound abv bS : ℝ) * (abv b • ε) ^ fintype.card ι ≤
(abv b ^ fintype.card ι),
{ have := norm_bound_pos abv bS,
have := abv.nonneg b,
rw [ε_eq, algebra.smul_def, eq_int_cast, ← rpow_nat_cast, mul_rpow, ← rpow_mul,
div_mul_cancel, rpow_neg_one, mul_left_comm, mul_inv_cancel, mul_one, rpow_nat_cast];
try { norm_cast, linarith },
{ apply rpow_nonneg_of_nonneg,
norm_cast,
linarith } },
let μ : fin (cardM bS adm).succ ↪ R := distinct_elems bS adm,
set s := bS.repr a,
have s_eq : ∀ i, s i = bS.repr a i := λ i, rfl,
set qs := λ j i, (μ j * s i) / b,
have q_eq : ∀ j i, qs j i = (μ j * s i) / b := λ i j, rfl,
set rs := λ j i, (μ j * s i) % b with r_eq,
have r_eq : ∀ j i, rs j i = (μ j * s i) % b := λ i j, rfl,
have μ_eq : ∀ i j, μ j * s i = b * qs j i + rs j i,
{ intros i j,
rw [q_eq, r_eq, euclidean_domain.div_add_mod], },
have μ_mul_a_eq : ∀ j, μ j • a = b • ∑ i, qs j i • bS i + ∑ i, rs j i • bS i,
{ intro j,
rw ← bS.sum_repr a,
simp only [finset.smul_sum, ← finset.sum_add_distrib],
refine finset.sum_congr rfl (λ i _, _),
rw [← s_eq, ← mul_smul, μ_eq, add_smul, mul_smul] },
obtain ⟨j, k, j_ne_k, hjk⟩ := adm.exists_approx hε hb (λ j i, μ j * s i),
have hjk' : ∀ i, (abv (rs k i - rs j i) : ℝ) < abv b • ε,
{ simpa only [r_eq] using hjk },
set q := ∑ i, (qs k i - qs j i) • bS i with q_eq,
set r := μ k - μ j with r_eq,
refine ⟨q, r, (mem_finset_approx bS adm).mpr _, _⟩,
{ exact ⟨k, j, j_ne_k.symm, rfl⟩ },
have : r • a - b • q = (∑ (x : ι), (rs k x • bS x - rs j x • bS x)),
{ simp only [r_eq, sub_smul, μ_mul_a_eq, q_eq, finset.smul_sum, ← finset.sum_add_distrib,
← finset.sum_sub_distrib, smul_sub],
refine finset.sum_congr rfl (λ x _, _),
ring },
rw [this, algebra.norm_algebra_map_of_basis bS, abv.map_pow],
refine int.cast_lt.mp ((norm_lt abv bS _ (λ i, lt_of_le_of_lt _ (hjk' i))).trans_le _),
{ apply le_of_eq,
congr,
simp_rw [linear_equiv.map_sum, linear_equiv.map_sub, linear_equiv.map_smul,
finset.sum_apply', finsupp.sub_apply, finsupp.smul_apply,
finset.sum_sub_distrib, basis.repr_self_apply, smul_eq_mul, mul_boole,
finset.sum_ite_eq', finset.mem_univ, if_true] },
{ exact_mod_cast ε_le },
end
include ist iic
/-- We can approximate `a / b : L` with `q / r`, where `r` has finitely many options for `L`. -/
theorem exists_mem_finset_approx' (h : algebra.is_algebraic R L) (a : S) {b : S} (hb : b ≠ 0) :
∃ (q : S) (r ∈ finset_approx bS adm),
abv (algebra.norm R (r • a - q * b)) < abv (algebra.norm R b) :=
begin
have inj : function.injective (algebra_map R L),
{ rw is_scalar_tower.algebra_map_eq R S L,
exact (is_integral_closure.algebra_map_injective S R L).comp bS.algebra_map_injective },
obtain ⟨a', b', hb', h⟩ := is_integral_closure.exists_smul_eq_mul h inj a hb,
obtain ⟨q, r, hr, hqr⟩ := exists_mem_finset_approx bS adm a' hb',
refine ⟨q, r, hr, _⟩,
refine lt_of_mul_lt_mul_left _
(show 0 ≤ abv (algebra.norm R (algebra_map R S b')), from abv.nonneg _),
refine lt_of_le_of_lt (le_of_eq _) (mul_lt_mul hqr le_rfl
(abv.pos ((algebra.norm_ne_zero_iff_of_basis bS).mpr hb)) (abv.nonneg _)),
rw [← abv.map_mul, ← monoid_hom.map_mul, ← abv.map_mul, ← monoid_hom.map_mul, ← algebra.smul_def,
smul_sub b', sub_mul, smul_comm, h, mul_comm b a', algebra.smul_mul_assoc r a' b,
algebra.smul_mul_assoc b' q b]
end
end real
lemma prod_finset_approx_ne_zero : algebra_map R S (∏ m in finset_approx bS adm, m) ≠ 0 :=
begin
refine mt ((injective_iff_map_eq_zero _).mp bS.algebra_map_injective _) _,
simp only [finset.prod_eq_zero_iff, not_exists],
rintros x hx rfl,
exact finset_approx.zero_not_mem bS adm hx
end
lemma ne_bot_of_prod_finset_approx_mem
(J : ideal S) (h : algebra_map _ _ (∏ m in finset_approx bS adm, m) ∈ J) :
J ≠ ⊥ :=
(submodule.ne_bot_iff _).mpr ⟨_, h, prod_finset_approx_ne_zero _ _⟩
include ist iic
/-- Each class in the class group contains an ideal `J`
such that `M := Π m ∈ finset_approx` is in `J`. -/
theorem exists_mk0_eq_mk0 [is_dedekind_domain S] [is_fraction_ring S L]
(h : algebra.is_algebraic R L) (I : (ideal S)⁰) :
∃ (J : (ideal S)⁰), class_group.mk0 L I = class_group.mk0 L J ∧
algebra_map _ _ (∏ m in finset_approx bS adm, m) ∈ (J : ideal S) :=
begin
set M := ∏ m in finset_approx bS adm, m with M_eq,
have hM : algebra_map R S M ≠ 0 := prod_finset_approx_ne_zero bS adm,
obtain ⟨b, b_mem, b_ne_zero, b_min⟩ := exists_min abv I,
suffices : ideal.span {b} ∣ ideal.span {algebra_map _ _ M} * I.1,
{ obtain ⟨J, hJ⟩ := this,
refine ⟨⟨J, _⟩, _, _⟩,
{ rw mem_non_zero_divisors_iff_ne_zero,
rintro rfl,
rw [ideal.zero_eq_bot, ideal.mul_bot] at hJ,
exact hM (ideal.span_singleton_eq_bot.mp (I.2 _ hJ)) },
{ rw class_group.mk0_eq_mk0_iff,
exact ⟨algebra_map _ _ M, b, hM, b_ne_zero, hJ⟩ },
rw [← set_like.mem_coe, ← set.singleton_subset_iff, ← ideal.span_le, ← ideal.dvd_iff_le],
refine (mul_dvd_mul_iff_left _).mp _,
swap, { exact mt ideal.span_singleton_eq_bot.mp b_ne_zero },
rw [subtype.coe_mk, ideal.dvd_iff_le, ← hJ, mul_comm],
apply ideal.mul_mono le_rfl,
rw [ideal.span_le, set.singleton_subset_iff],
exact b_mem },
rw [ideal.dvd_iff_le, ideal.mul_le],
intros r' hr' a ha,
rw ideal.mem_span_singleton at ⊢ hr',
obtain ⟨q, r, r_mem, lt⟩ := exists_mem_finset_approx' L bS adm h a b_ne_zero,
apply @dvd_of_mul_left_dvd _ _ q,
simp only [algebra.smul_def] at lt,
rw ← sub_eq_zero.mp (b_min _ (I.1.sub_mem (I.1.mul_mem_left _ ha) (I.1.mul_mem_left _ b_mem)) lt),
refine mul_dvd_mul_right (dvd_trans (ring_hom.map_dvd _ _) hr') _,
exact multiset.dvd_prod (multiset.mem_map.mpr ⟨_, r_mem, rfl⟩)
end
omit iic ist
/-- `class_group.mk_M_mem` is a specialization of `class_group.mk0` to (the finite set of)
ideals that contain `M := ∏ m in finset_approx L f abs, m`.
By showing this function is surjective, we prove that the class group is finite. -/
noncomputable def mk_M_mem [is_fraction_ring S L] [is_dedekind_domain S]
(J : {J : ideal S // algebra_map _ _ (∏ m in finset_approx bS adm, m) ∈ J}) :
class_group S L :=
class_group.mk0 _ ⟨J.1, mem_non_zero_divisors_iff_ne_zero.mpr
(ne_bot_of_prod_finset_approx_mem bS adm J.1 J.2)⟩
include iic ist
lemma mk_M_mem_surjective [is_fraction_ring S L] [is_dedekind_domain S]
(h : algebra.is_algebraic R L) :
function.surjective (class_group.mk_M_mem L bS adm) :=
begin
intro I',
obtain ⟨⟨I, hI⟩, rfl⟩ := class_group.mk0_surjective I',
obtain ⟨J, mk0_eq_mk0, J_dvd⟩ := exists_mk0_eq_mk0 L bS adm h ⟨I, hI⟩,
exact ⟨⟨J, J_dvd⟩, mk0_eq_mk0.symm⟩
end
open_locale classical
/-- The main theorem: the class group of an integral closure `S` of `R` in an
algebraic extension `L` is finite if there is an admissible absolute value.
See also `class_group.fintype_of_admissible_of_finite` where `L` is a finite
extension of `K = Frac(R)`, supplying most of the required assumptions automatically.
-/
noncomputable def fintype_of_admissible_of_algebraic [is_fraction_ring S L] [is_dedekind_domain S]
(h : algebra.is_algebraic R L) : fintype (class_group S L) :=
@fintype.of_surjective _ _ _
(@fintype.of_equiv _
{J // J ∣ ideal.span ({algebra_map R S (∏ (m : R) in finset_approx bS adm, m)} : set S)}
(unique_factorization_monoid.fintype_subtype_dvd _
(by { rw [ne.def, ideal.zero_eq_bot, ideal.span_singleton_eq_bot],
exact prod_finset_approx_ne_zero bS adm }))
((equiv.refl _).subtype_equiv (λ I, ideal.dvd_iff_le.trans
(by rw [equiv.refl_apply, ideal.span_le, set.singleton_subset_iff]))))
(class_group.mk_M_mem L bS adm)
(class_group.mk_M_mem_surjective L bS adm h)
/-- The main theorem: the class group of an integral closure `S` of `R` in a
finite extension `L` of `K = Frac(R)` is finite if there is an admissible
absolute value.
See also `class_group.fintype_of_admissible_of_algebraic` where `L` is an
algebraic extension of `R`, that includes some extra assumptions.
-/
noncomputable def fintype_of_admissible_of_finite [is_dedekind_domain R] :
fintype (@class_group S L _ _ _
(is_integral_closure.is_fraction_ring_of_finite_extension R K L S)) :=
begin
letI := classical.dec_eq L,
letI := is_integral_closure.is_fraction_ring_of_finite_extension R K L S,
letI := is_integral_closure.is_dedekind_domain R K L S,
choose s b hb_int using finite_dimensional.exists_is_basis_integral R K L,
obtain ⟨n, b⟩ := submodule.basis_of_pid_of_le_span _
(is_integral_closure.range_le_span_dual_basis S b hb_int),
let bS := b.map ((linear_map.quot_ker_equiv_range _).symm ≪≫ₗ _),
refine fintype_of_admissible_of_algebraic L bS adm
(λ x, (is_fraction_ring.is_algebraic_iff R K L).mpr (algebra.is_algebraic_of_finite _ _ x)),
{ rw linear_map.ker_eq_bot.mpr,
{ exact submodule.quot_equiv_of_eq_bot _ rfl },
{ exact is_integral_closure.algebra_map_injective _ R _ } },
{ refine (basis.linear_independent _).restrict_scalars _,
simp only [algebra.smul_def, mul_one],
apply is_fraction_ring.injective }
end
end is_admissible
end euclidean_domain
end class_group
|
fb194602c008bea23574d131e0caebd0f10687fa | e030b0259b777fedcdf73dd966f3f1556d392178 | /library/init/algebra/field.lean | ccf8f200cedb69f191acb62398e8145dc43f873f | [
"Apache-2.0"
] | permissive | fgdorais/lean | 17b46a095b70b21fa0790ce74876658dc5faca06 | c3b7c54d7cca7aaa25328f0a5660b6b75fe26055 | refs/heads/master | 1,611,523,590,686 | 1,484,412,902,000 | 1,484,412,902,000 | 38,489,734 | 0 | 0 | null | 1,435,923,380,000 | 1,435,923,379,000 | null | UTF-8 | Lean | false | false | 18,244 | lean | /-
Copyright (c) 2014 Robert Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Lewis, Leonardo de Moura
Structures with multiplicative and additive components, including division rings and fields.
The development is modeled after Isabelle's library.
-/
prelude
import init.algebra.ring
universe variables u
/- Make sure instances defined in this file have lower priority than the ones
defined for concrete structures -/
set_option default_priority 100
class division_ring (α : Type u) extends ring α, has_inv α, zero_ne_one_class α :=
(mul_inv_cancel : ∀ {a : α}, a ≠ 0 → a * a⁻¹ = 1)
(inv_mul_cancel : ∀ {a : α}, a ≠ 0 → a⁻¹ * a = 1)
variable {α : Type u}
section division_ring
variables [division_ring α]
protected definition algebra.div (a b : α) : α :=
a * b⁻¹
instance division_ring_has_div [division_ring α] : has_div α :=
⟨algebra.div⟩
lemma division_def (a b : α) : a / b = a * b⁻¹ :=
rfl
lemma mul_inv_cancel {a : α} (h : a ≠ 0) : a * a⁻¹ = 1 :=
division_ring.mul_inv_cancel h
lemma inv_mul_cancel {a : α} (h : a ≠ 0) : a⁻¹ * a = 1 :=
division_ring.inv_mul_cancel h
lemma inv_eq_one_div (a : α) : a⁻¹ = 1 / a :=
eq.symm $ one_mul (a⁻¹)
local attribute [simp]
division_def mul_comm mul_assoc
mul_left_comm mul_inv_cancel inv_mul_cancel
lemma div_eq_mul_one_div (a b : α) : a / b = a * (1 / b) :=
by simp
lemma mul_one_div_cancel {a : α} (h : a ≠ 0) : a * (1 / a) = 1 :=
by simp [h]
lemma one_div_mul_cancel {a : α} (h : a ≠ 0) : (1 / a) * a = 1 :=
by simp [h]
lemma div_self {a : α} (h : a ≠ 0) : a / a = 1 :=
by simp [h]
lemma one_div_one : 1 / 1 = (1:α) :=
div_self (ne.symm zero_ne_one)
lemma mul_div_assoc (a b c : α) : (a * b) / c = a * (b / c) :=
by simp
lemma one_div_ne_zero {a : α} (h : a ≠ 0) : 1 / a ≠ 0 :=
suppose 1 / a = 0,
have 0 = (1:α), from eq.symm (by rw [-(mul_one_div_cancel h), this, mul_zero]),
absurd this zero_ne_one
lemma one_inv_eq : 1⁻¹ = (1:α) :=
suffices 1 * 1⁻¹ = (1:α), begin simp at this, assumption end,
mul_inv_cancel (ne.symm (@zero_ne_one α _))
local attribute [simp] one_inv_eq
lemma div_one (a : α) : a / 1 = a :=
by simp
lemma zero_div (a : α) : 0 / a = 0 :=
by simp
-- note: integral domain has a "mul_ne_zero". α commutative division ring is an integral
-- domain, but let's not define that class for now.
lemma division_ring.mul_ne_zero {a b : α} (ha : a ≠ 0) (hb : b ≠ 0) : a * b ≠ 0 :=
suppose a * b = 0,
have a * 1 = 0, by rw [-(mul_one_div_cancel hb), -mul_assoc, this, zero_mul],
have a = 0, by rwa mul_one at this,
absurd this ha
lemma mul_ne_zero_comm {a b : α} (h : a * b ≠ 0) : b * a ≠ 0 :=
have h₁ : a ≠ 0, from ne_zero_of_mul_ne_zero_right h,
have h₂ : b ≠ 0, from ne_zero_of_mul_ne_zero_left h,
division_ring.mul_ne_zero h₂ h₁
lemma eq_one_div_of_mul_eq_one {a b : α} (h : a * b = 1) : b = 1 / a :=
have a ≠ 0, from
suppose a = 0,
have 0 = (1:α), by rwa [this, zero_mul] at h,
absurd this zero_ne_one,
have b = (1 / a) * a * b, by rw [one_div_mul_cancel this, one_mul],
show b = 1 / a, by rwa [mul_assoc, h, mul_one] at this
lemma eq_one_div_of_mul_eq_one_left {a b : α} (h : b * a = 1) : b = 1 / a :=
have a ≠ 0, from
suppose a = 0,
have 0 = (1:α), by rwa [this, mul_zero] at h,
absurd this zero_ne_one,
by rw [-h, mul_div_assoc, div_self this, mul_one]
lemma division_ring.one_div_mul_one_div {a b : α} (ha : a ≠ 0) (hb : b ≠ 0) : (1 / a) * (1 / b) = 1 / (b * a) :=
have (b * a) * ((1 / a) * (1 / b)) = 1,
by rw [mul_assoc, -(mul_assoc a), mul_one_div_cancel ha, one_mul, mul_one_div_cancel hb],
eq_one_div_of_mul_eq_one this
lemma one_div_neg_one_eq_neg_one : (1:α) / (-1) = -1 :=
have (-1) * (-1) = (1:α), by rw [neg_mul_neg, one_mul],
eq.symm (eq_one_div_of_mul_eq_one this)
lemma division_ring.one_div_neg_eq_neg_one_div {a : α} (h : a ≠ 0) : 1 / (- a) = - (1 / a) :=
have -1 ≠ (0:α), from
(suppose -1 = 0, absurd (eq.symm (calc
1 = -(-1) : (neg_neg (1:α))^.symm
... = -0 : by rw this
... = (0:α) : neg_zero)) zero_ne_one),
calc
1 / (- a) = 1 / ((-1) * a) : by rw neg_eq_neg_one_mul
... = (1 / a) * (1 / (- 1)) : by rw (division_ring.one_div_mul_one_div h this)
... = (1 / a) * (-1) : by rw one_div_neg_one_eq_neg_one
... = - (1 / a) : by rw [mul_neg_eq_neg_mul_symm, mul_one]
lemma div_neg_eq_neg_div {a : α} (b : α) (ha : a ≠ 0) : b / (- a) = - (b / a) :=
calc
b / (- a) = b * (1 / (- a)) : by rw [-inv_eq_one_div, division_def]
... = b * -(1 / a) : by rw (division_ring.one_div_neg_eq_neg_one_div ha)
... = -(b * (1 / a)) : by rw neg_mul_eq_mul_neg
... = - (b * a⁻¹) : by rw inv_eq_one_div
lemma neg_div (a b : α) : (-b) / a = - (b / a) :=
by rw [neg_eq_neg_one_mul, mul_div_assoc, -neg_eq_neg_one_mul]
lemma division_ring.neg_div_neg_eq (a : α) {b : α} (hb : b ≠ 0) : (-a) / (-b) = a / b :=
by rw [(div_neg_eq_neg_div _ hb), neg_div, neg_neg]
lemma division_ring.one_div_one_div {a : α} (h : a ≠ 0) : 1 / (1 / a) = a :=
eq.symm (eq_one_div_of_mul_eq_one_left (mul_one_div_cancel h))
lemma division_ring.eq_of_one_div_eq_one_div {a b : α} (ha : a ≠ 0) (hb : b ≠ 0) (h : 1 / a = 1 / b) : a = b :=
by rw [-(division_ring.one_div_one_div ha), h, (division_ring.one_div_one_div hb)]
lemma mul_inv_eq {a b : α} (ha : a ≠ 0) (hb : b ≠ 0) : (b * a)⁻¹ = a⁻¹ * b⁻¹ :=
eq.symm $ calc
a⁻¹ * b⁻¹ = (1 / a) * (1 / b) : by simp
... = (1 / (b * a)) : division_ring.one_div_mul_one_div ha hb
... = (b * a)⁻¹ : by simp
lemma mul_div_cancel (a : α) {b : α} (hb : b ≠ 0) : a * b / b = a :=
by simp [hb]
lemma div_mul_cancel (a : α) {b : α} (hb : b ≠ 0) : a / b * b = a :=
by simp [hb]
lemma div_add_div_same (a b c : α) : a / c + b / c = (a + b) / c :=
eq.symm $ right_distrib a b (c⁻¹)
lemma div_sub_div_same (a b c : α) : (a / c) - (b / c) = (a - b) / c :=
by rw [sub_eq_add_neg, -neg_div, div_add_div_same, sub_eq_add_neg]
lemma one_div_mul_add_mul_one_div_eq_one_div_add_one_div {a b : α} (ha : a ≠ 0) (hb : b ≠ 0) :
(1 / a) * (a + b) * (1 / b) = 1 / a + 1 / b :=
by rw [(left_distrib (1 / a)), (one_div_mul_cancel ha), right_distrib, one_mul,
mul_assoc, (mul_one_div_cancel hb), mul_one, add_comm]
lemma one_div_mul_sub_mul_one_div_eq_one_div_add_one_div {a b : α} (ha : a ≠ 0) (hb : b ≠ 0) :
(1 / a) * (b - a) * (1 / b) = 1 / a - 1 / b :=
by rw [(mul_sub_left_distrib (1 / a)), (one_div_mul_cancel ha), mul_sub_right_distrib,
one_mul, mul_assoc, (mul_one_div_cancel hb), mul_one]
lemma div_eq_one_iff_eq (a : α) {b : α} (hb : b ≠ 0) : a / b = 1 ↔ a = b :=
iff.intro
(suppose a / b = 1, calc
a = a / b * b : by simp [hb]
... = 1 * b : by rw this
... = b : by simp)
(suppose a = b, by simp [this, hb])
lemma eq_of_div_eq_one (a : α) {b : α} (Hb : b ≠ 0) : a / b = 1 → a = b :=
iff.mp $ div_eq_one_iff_eq a Hb
lemma eq_div_iff_mul_eq (a b : α) {c : α} (hc : c ≠ 0) : a = b / c ↔ a * c = b :=
iff.intro
(suppose a = b / c, by rw [this, (div_mul_cancel _ hc)])
(suppose a * c = b, by rw [-this, mul_div_cancel _ hc])
lemma eq_div_of_mul_eq (a b : α) {c : α} (hc : c ≠ 0) : a * c = b → a = b / c :=
iff.mpr $ eq_div_iff_mul_eq a b hc
lemma mul_eq_of_eq_div (a b: α) {c : α} (hc : c ≠ 0) : a = b / c → a * c = b :=
iff.mp $ eq_div_iff_mul_eq a b hc
lemma add_div_eq_mul_add_div (a b : α) {c : α} (hc : c ≠ 0) : a + b / c = (a * c + b) / c :=
have (a + b / c) * c = a * c + b, by rw [right_distrib, (div_mul_cancel _ hc)],
(iff.mpr (eq_div_iff_mul_eq _ _ hc)) this
lemma mul_mul_div (a : α) {c : α} (hc : c ≠ 0) : a = a * c * (1 / c) :=
by simp [hc]
end division_ring
class field (α : Type u) extends division_ring α, comm_ring α
section field
variable [field α]
lemma field.one_div_mul_one_div {a b : α} (ha : a ≠ 0) (hb : b ≠ 0) : (1 / a) * (1 / b) = 1 / (a * b) :=
by rw [(division_ring.one_div_mul_one_div ha hb), mul_comm b]
lemma field.div_mul_right {a b : α} (hb : b ≠ 0) (h : a * b ≠ 0) : a / (a * b) = 1 / b :=
have a ≠ 0, from ne_zero_of_mul_ne_zero_right h,
eq.symm (calc
1 / b = a * ((1 / a) * (1 / b)) : by rw [-mul_assoc, mul_one_div_cancel this, one_mul]
... = a * (1 / (b * a)) : by rw (division_ring.one_div_mul_one_div this hb)
... = a * (a * b)⁻¹ : by rw [inv_eq_one_div, mul_comm a b])
lemma field.div_mul_left {a b : α} (ha : a ≠ 0) (h : a * b ≠ 0) : b / (a * b) = 1 / a :=
have b * a ≠ 0, from mul_ne_zero_comm h,
by rw [mul_comm a, (field.div_mul_right ha this)]
lemma mul_div_cancel_left {a : α} (b : α) (ha : a ≠ 0) : a * b / a = b :=
by rw [mul_comm a, (mul_div_cancel _ ha)]
lemma mul_div_cancel' (a : α) {b : α} (hb : b ≠ 0) : b * (a / b) = a :=
by rw [mul_comm, (div_mul_cancel _ hb)]
lemma one_div_add_one_div {a b : α} (ha : a ≠ 0) (hb : b ≠ 0) : 1 / a + 1 / b = (a + b) / (a * b) :=
have a * b ≠ 0, from (division_ring.mul_ne_zero ha hb),
by rw [add_comm, -(field.div_mul_left ha this), -(field.div_mul_right hb this),
division_def, division_def, division_def, -right_distrib]
lemma field.div_mul_div (a : α) {b : α} (c : α) {d : α} (hb : b ≠ 0) (hd : d ≠ 0) :
(a / b) * (c / d) = (a * c) / (b * d) :=
begin simp [division_def], rw [mul_inv_eq hd hb, mul_comm d⁻¹] end
lemma mul_div_mul_left (a : α) {b c : α} (hb : b ≠ 0) (hc : c ≠ 0) :
(c * a) / (c * b) = a / b :=
by rw [-(field.div_mul_div _ _ hc hb), div_self hc, one_mul]
lemma mul_div_mul_right (a : α) {b c : α} (hb : b ≠ 0) (hc : c ≠ 0) :
(a * c) / (b * c) = a / b :=
by rw [mul_comm a, mul_comm b, mul_div_mul_left _ hb hc]
lemma div_mul_eq_mul_div (a b c : α) : (b / c) * a = (b * a) / c :=
by simp [division_def]
lemma field.div_mul_eq_mul_div_comm (a b : α) {c : α} (hc : c ≠ 0) :
(b / c) * a = b * (a / c) :=
by rw [div_mul_eq_mul_div, -(one_mul c), -(field.div_mul_div _ _ (ne.symm zero_ne_one) hc),
div_one, one_mul]
lemma div_add_div (a : α) {b : α} (c : α) {d : α} (hb : b ≠ 0) (hd : d ≠ 0) :
(a / b) + (c / d) = ((a * d) + (b * c)) / (b * d) :=
by rw [-(mul_div_mul_right _ hb hd), -(mul_div_mul_left _ hd hb), div_add_div_same]
lemma div_sub_div (a : α) {b : α} (c : α) {d : α} (hb : b ≠ 0) (hd : d ≠ 0) :
(a / b) - (c / d) = ((a * d) - (b * c)) / (b * d) :=
begin
simp [sub_eq_add_neg],
rw [neg_eq_neg_one_mul, -mul_div_assoc, div_add_div _ _ hb hd,
-mul_assoc, mul_comm b, mul_assoc, -neg_eq_neg_one_mul]
end
lemma mul_eq_mul_of_div_eq_div (a : α) {b : α} (c : α) {d : α} (hb : b ≠ 0)
(hd : d ≠ 0) (h : a / b = c / d) : a * d = c * b :=
by rw [-(mul_one (a*d)), mul_assoc, (mul_comm d), -mul_assoc, -(div_self hb),
-(field.div_mul_eq_mul_div_comm _ _ hb), h, div_mul_eq_mul_div, div_mul_cancel _ hd]
lemma field.one_div_div {a b : α} (ha : a ≠ 0) (hb : b ≠ 0) : 1 / (a / b) = b / a :=
have (a / b) * (b / a) = 1, from calc
(a / b) * (b / a) = (a * b) / (b * a) : field.div_mul_div _ _ hb ha
... = (a * b) / (a * b) : by rw mul_comm
... = 1 : div_self (division_ring.mul_ne_zero ha hb),
eq.symm (eq_one_div_of_mul_eq_one this)
lemma field.div_div_eq_mul_div (a : α) {b c : α} (hb : b ≠ 0) (hc : c ≠ 0) :
a / (b / c) = (a * c) / b :=
by rw [div_eq_mul_one_div, field.one_div_div hb hc, -mul_div_assoc]
lemma field.div_div_eq_div_mul (a : α) {b c : α} (hb : b ≠ 0) (hc : c ≠ 0) :
(a / b) / c = a / (b * c) :=
by rw [div_eq_mul_one_div, field.div_mul_div _ _ hb hc, mul_one]
lemma field.div_div_div_div_eq (a : α) {b c d : α} (hb : b ≠ 0) (hc : c ≠ 0) (hd : d ≠ 0) :
(a / b) / (c / d) = (a * d) / (b * c) :=
by rw [field.div_div_eq_mul_div _ hc hd, div_mul_eq_mul_div,
field.div_div_eq_div_mul _ hb hc]
lemma field.div_mul_eq_div_mul_one_div (a : α) {b c : α} (hb : b ≠ 0) (hc : c ≠ 0) :
a / (b * c) = (a / b) * (1 / c) :=
by rw [-(field.div_div_eq_div_mul _ hb hc), -div_eq_mul_one_div]
lemma eq_of_mul_eq_mul_of_nonzero_left {a b c : α} (h : a ≠ 0) (h₂ : a * b = a * c) : b = c :=
by rw [-one_mul b, -div_self h, div_mul_eq_mul_div, h₂, mul_div_cancel_left _ h]
lemma eq_of_mul_eq_mul_of_nonzero_right {a b c : α} (h : c ≠ 0) (h2 : a * c = b * c) : a = b :=
by rw [-mul_one a, -div_self h, -mul_div_assoc, h2, mul_div_cancel _ h]
end field
class discrete_field (α : Type u) extends field α :=
(has_decidable_eq : decidable_eq α)
(inv_zero : inv zero = zero)
attribute [instance] discrete_field.has_decidable_eq
section discrete_field
variable [discrete_field α]
-- many of the lemmas in discrete_field are the same as lemmas in field or division ring,
-- but with fewer hypotheses since 0⁻¹ = 0 and equality is decidable.
lemma discrete_field.eq_zero_or_eq_zero_of_mul_eq_zero
(a b : α) (h : a * b = 0) : a = 0 ∨ b = 0 :=
decidable.by_cases
(suppose a = 0, or.inl this)
(suppose a ≠ 0,
or.inr (by rw [-(one_mul b), -(inv_mul_cancel this), mul_assoc, h, mul_zero]))
instance discrete_field.to_integral_domain [s : discrete_field α] : integral_domain α :=
{s with
eq_zero_or_eq_zero_of_mul_eq_zero := discrete_field.eq_zero_or_eq_zero_of_mul_eq_zero}
lemma inv_zero : 0⁻¹ = (0:α) :=
discrete_field.inv_zero α
lemma one_div_zero : 1 / 0 = (0:α) :=
calc
1 / 0 = (1:α) * 0⁻¹ : by rw division_def
... = 1 * 0 : by rw inv_zero
... = (0:α) : by rw mul_zero
lemma div_zero (a : α) : a / 0 = 0 :=
by rw [div_eq_mul_one_div, one_div_zero, mul_zero]
lemma ne_zero_of_one_div_ne_zero {a : α} (h : 1 / a ≠ 0) : a ≠ 0 :=
assume ha : a = 0, begin rw [ha, one_div_zero] at h, contradiction end
lemma eq_zero_of_one_div_eq_zero {a : α} (h : 1 / a = 0) : a = 0 :=
decidable.by_cases
(assume ha, ha)
(assume ha, false.elim ((one_div_ne_zero ha) h))
lemma one_div_mul_one_div' (a b : α) : (1 / a) * (1 / b) = 1 / (b * a) :=
decidable.by_cases
(suppose a = 0,
by rw [this, div_zero, zero_mul, mul_zero, div_zero])
(assume ha : a ≠ 0,
decidable.by_cases
(suppose b = 0,
by rw [this, div_zero, mul_zero, zero_mul, div_zero])
(suppose b ≠ 0, division_ring.one_div_mul_one_div ha this))
lemma one_div_neg_eq_neg_one_div (a : α) : 1 / (- a) = - (1 / a) :=
decidable.by_cases
(suppose a = 0, by rw [this, neg_zero, div_zero, neg_zero])
(suppose a ≠ 0, division_ring.one_div_neg_eq_neg_one_div this)
lemma neg_div_neg_eq (a b : α) : (-a) / (-b) = a / b :=
decidable.by_cases
(assume hb : b = 0, by rw [hb, neg_zero, div_zero, div_zero])
(assume hb : b ≠ 0, division_ring.neg_div_neg_eq _ hb)
lemma one_div_one_div (a : α) : 1 / (1 / a) = a :=
decidable.by_cases
(assume ha : a = 0, by rw [ha, div_zero, div_zero])
(assume ha : a ≠ 0, division_ring.one_div_one_div ha)
lemma eq_of_one_div_eq_one_div {a b : α} (h : 1 / a = 1 / b) : a = b :=
decidable.by_cases
(assume ha : a = 0,
have hb : b = 0, from eq_zero_of_one_div_eq_zero (by rw [-h, ha, div_zero]),
hb^.symm ▸ ha)
(assume ha : a ≠ 0,
have hb : b ≠ 0, from ne_zero_of_one_div_ne_zero (h ▸ (one_div_ne_zero ha)),
division_ring.eq_of_one_div_eq_one_div ha hb h)
lemma mul_inv' (a b : α) : (b * a)⁻¹ = a⁻¹ * b⁻¹ :=
decidable.by_cases
(assume ha : a = 0, by rw [ha, mul_zero, inv_zero, zero_mul])
(assume ha : a ≠ 0,
decidable.by_cases
(assume hb : b = 0, by rw [hb, zero_mul, inv_zero, mul_zero])
(assume hb : b ≠ 0, mul_inv_eq ha hb))
-- the following are specifically for fields
lemma one_div_mul_one_div (a b : α) : (1 / a) * (1 / b) = 1 / (a * b) :=
by rw [one_div_mul_one_div', mul_comm b]
lemma div_mul_right {a : α} (b : α) (ha : a ≠ 0) : a / (a * b) = 1 / b :=
decidable.by_cases
(assume hb : b = 0, by rw [hb, mul_zero, div_zero, div_zero])
(assume hb : b ≠ 0, field.div_mul_right hb (mul_ne_zero ha hb))
lemma div_mul_left (a : α) {b : α} (hb : b ≠ 0) : b / (a * b) = 1 / a :=
by rw [mul_comm a, div_mul_right _ hb]
lemma div_mul_div (a b c d : α) : (a / b) * (c / d) = (a * c) / (b * d) :=
decidable.by_cases
(assume hb : b = 0, by rw [hb, div_zero, zero_mul, zero_mul, div_zero])
(assume hb : b ≠ 0,
decidable.by_cases
(assume hd : d = 0, by rw [hd, div_zero, mul_zero, mul_zero, div_zero])
(assume hd : d ≠ 0, field.div_mul_div _ _ hb hd))
lemma mul_div_mul_left' (a b : α) {c : α} (hc : c ≠ 0) : (c * a) / (c * b) = a / b :=
decidable.by_cases
(assume hb : b = 0, by rw [hb, mul_zero, div_zero, div_zero])
(assume hb : b ≠ 0, mul_div_mul_left _ hb hc)
lemma mul_div_mul_right' (a b : α) {c : α} (hc : c ≠ 0) : (a * c) / (b * c) = a / b :=
by rw [mul_comm a, mul_comm b, (mul_div_mul_left' _ _ hc)]
lemma div_mul_eq_mul_div_comm (a b c : α) : (b / c) * a = b * (a / c) :=
decidable.by_cases
(assume hc : c = 0, by rw [hc, div_zero, zero_mul, div_zero, mul_zero])
(assume hc : c ≠ 0, field.div_mul_eq_mul_div_comm _ _ hc)
lemma one_div_div (a b : α) : 1 / (a / b) = b / a :=
decidable.by_cases
(assume ha : a = 0, by rw [ha, zero_div, div_zero, div_zero])
(assume ha : a ≠ 0,
decidable.by_cases
(assume hb : b = 0, by rw [hb, div_zero, zero_div, div_zero])
(assume hb : b ≠ 0, field.one_div_div ha hb))
lemma div_div_eq_mul_div (a b c : α) : a / (b / c) = (a * c) / b :=
by rw [div_eq_mul_one_div, one_div_div, -mul_div_assoc]
lemma div_div_eq_div_mul (a b c : α) : (a / b) / c = a / (b * c) :=
by rw [div_eq_mul_one_div, div_mul_div, mul_one]
lemma div_div_div_div_eq (a b c d : α) : (a / b) / (c / d) = (a * d) / (b * c) :=
by rw [div_div_eq_mul_div, div_mul_eq_mul_div, div_div_eq_div_mul]
lemma div_helper {a : α} (b : α) (h : a ≠ 0) : (1 / (a * b)) * a = 1 / b :=
by rw [div_mul_eq_mul_div, one_mul, div_mul_right _ h]
lemma div_mul_eq_div_mul_one_div (a b c : α) : a / (b * c) = (a / b) * (1 / c) :=
by rw [-div_div_eq_div_mul, -div_eq_mul_one_div]
end discrete_field
|
91e1001d610d58c051366b7e8db15415091a2578 | a0e23cfdd129a671bf3154ee1a8a3a72bf4c7940 | /tests/lean/revertlet.lean | 882e9a3e676088bf864a67049f91686fc28ad6a5 | [
"Apache-2.0"
] | permissive | WojciechKarpiel/lean4 | 7f89706b8e3c1f942b83a2c91a3a00b05da0e65b | f6e1314fa08293dea66a329e05b6c196a0189163 | refs/heads/master | 1,686,633,402,214 | 1,625,821,189,000 | 1,625,821,258,000 | 384,640,886 | 0 | 0 | Apache-2.0 | 1,625,903,617,000 | 1,625,903,026,000 | null | UTF-8 | Lean | false | false | 153 | lean | theorem ex (n : Nat) (h : n = 0) : 0 + n = 0 := by
let m := n + 1
let v := m + 1
have : v = n + 2 := rfl
traceState
subst h
traceState
rfl
|
b8b03982c0ea986b5fd551b32a9c3b8f5789cbd3 | 3f7026ea8bef0825ca0339a275c03b911baef64d | /src/tactic/default.lean | 3e8e99f272ee53c019d742f0109571bda9b0b3e0 | [
"Apache-2.0"
] | permissive | rspencer01/mathlib | b1e3afa5c121362ef0881012cc116513ab09f18c | c7d36292c6b9234dc40143c16288932ae38fdc12 | refs/heads/master | 1,595,010,346,708 | 1,567,511,503,000 | 1,567,511,503,000 | 206,071,681 | 0 | 0 | Apache-2.0 | 1,567,513,643,000 | 1,567,513,643,000 | null | UTF-8 | Lean | false | false | 832 | lean | /-
This file imports many useful tactics ("the kitchen sink").
You can use `import tactic` at the beginning of your file to get everything.
(Although you may want to strip things down when you're polishing.)
Because this file imports some complicated tactics, it has many transitive dependencies
(which of course may not use `import tactic`, and must import selectively).
As (non-exhaustive) examples, these includes things like:
* algebra.group_power
* algebra.ordered_ring
* data.rat
* data.nat.prime
* data.list.perm
* data.set.lattice
* data.equiv.encodable
* order.complete_lattice
-/
import
tactic.basic
tactic.monotonicity.interactive
tactic.finish
tactic.tauto
tactic.tidy
tactic.abel tactic.ring
tactic.linarith tactic.omega
tactic.wlog
tactic.tfae
tactic.apply_fun
tactic.localized
tactic.apply
|
975458d2dbcf1b8dbc9d7cf52634ea980839a925 | 57c233acf9386e610d99ed20ef139c5f97504ba3 | /src/order/lattice.lean | f7b1afa3db4c92c02610c1ba190459be003882ea | [
"Apache-2.0"
] | permissive | robertylewis/mathlib | 3d16e3e6daf5ddde182473e03a1b601d2810952c | 1d13f5b932f5e40a8308e3840f96fc882fae01f0 | refs/heads/master | 1,651,379,945,369 | 1,644,276,960,000 | 1,644,276,960,000 | 98,875,504 | 0 | 0 | Apache-2.0 | 1,644,253,514,000 | 1,501,495,700,000 | Lean | UTF-8 | Lean | false | false | 36,518 | lean | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import order.monotone
import order.rel_classes
import tactic.simps
import tactic.pi_instances
/-!
# (Semi-)lattices
Semilattices are partially ordered sets with join (greatest lower bound, or `sup`) or
meet (least upper bound, or `inf`) operations. Lattices are posets that are both
join-semilattices and meet-semilattices.
Distributive lattices are lattices which satisfy any of four equivalent distributivity properties,
of `sup` over `inf`, on the left or on the right.
## Main declarations
* `has_sup`: type class for the `⊔` notation
* `has_inf`: type class for the `⊓` notation
* `semilattice_sup`: a type class for join semilattices
* `semilattice_sup.mk'`: an alternative constructor for `semilattice_sup` via proofs that `⊔` is
commutative, associative and idempotent.
* `semilattice_inf`: a type class for meet semilattices
* `semilattice_sup.mk'`: an alternative constructor for `semilattice_inf` via proofs that `⊓` is
commutative, associative and idempotent.
* `lattice`: a type class for lattices
* `lattice.mk'`: an alternative constructor for `lattice` via profs that `⊔` and `⊓` are
commutative, associative and satisfy a pair of "absorption laws".
* `distrib_lattice`: a type class for distributive lattices.
## Notations
* `a ⊔ b`: the supremum or join of `a` and `b`
* `a ⊓ b`: the infimum or meet of `a` and `b`
## TODO
* (Semi-)lattice homomorphisms
* Alternative constructors for distributive lattices from the other distributive properties
## Tags
semilattice, lattice
-/
set_option old_structure_cmd true
universes u v w
variables {α : Type u} {β : Type v}
-- TODO: move this eventually, if we decide to use them
attribute [ematch] le_trans lt_of_le_of_lt lt_of_lt_of_le lt_trans
section
-- TODO: this seems crazy, but it also seems to work reasonably well
@[ematch] theorem le_antisymm' [partial_order α] : ∀ {a b : α}, (: a ≤ b :) → b ≤ a → a = b :=
@le_antisymm _ _
end
/- TODO: automatic construction of dual definitions / theorems -/
/-- Typeclass for the `⊔` (`\lub`) notation -/
@[notation_class] class has_sup (α : Type u) := (sup : α → α → α)
/-- Typeclass for the `⊓` (`\glb`) notation -/
@[notation_class] class has_inf (α : Type u) := (inf : α → α → α)
infix ⊔ := has_sup.sup
infix ⊓ := has_inf.inf
/-!
### Join-semilattices
-/
/-- A `semilattice_sup` is a join-semilattice, that is, a partial order
with a join (a.k.a. lub / least upper bound, sup / supremum) operation
`⊔` which is the least element larger than both factors. -/
class semilattice_sup (α : Type u) extends has_sup α, partial_order α :=
(le_sup_left : ∀ a b : α, a ≤ a ⊔ b)
(le_sup_right : ∀ a b : α, b ≤ a ⊔ b)
(sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c)
/--
A type with a commutative, associative and idempotent binary `sup` operation has the structure of a
join-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def semilattice_sup.mk' {α : Type*} [has_sup α]
(sup_comm : ∀ (a b : α), a ⊔ b = b ⊔ a)
(sup_assoc : ∀ (a b c : α), a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ (a : α), a ⊔ a = a) : semilattice_sup α :=
{ sup := (⊔),
le := λ a b, a ⊔ b = b,
le_refl := sup_idem,
le_trans := λ a b c hab hbc,
begin
dsimp only [(≤)] at *,
rwa [←hbc, ←sup_assoc, hab],
end,
le_antisymm := λ a b hab hba,
begin
dsimp only [(≤)] at *,
rwa [←hba, sup_comm],
end,
le_sup_left := λ a b, show a ⊔ (a ⊔ b) = (a ⊔ b), by rw [←sup_assoc, sup_idem],
le_sup_right := λ a b, show b ⊔ (a ⊔ b) = (a ⊔ b), by rw [sup_comm, sup_assoc, sup_idem],
sup_le := λ a b c hac hbc,
begin
dsimp only [(≤), preorder.le] at *,
rwa [sup_assoc, hbc],
end }
instance (α : Type*) [has_inf α] : has_sup (order_dual α) := ⟨((⊓) : α → α → α)⟩
instance (α : Type*) [has_sup α] : has_inf (order_dual α) := ⟨((⊔) : α → α → α)⟩
section semilattice_sup
variables [semilattice_sup α] {a b c d : α}
@[simp] theorem le_sup_left : a ≤ a ⊔ b :=
semilattice_sup.le_sup_left a b
@[ematch] theorem le_sup_left' : a ≤ (: a ⊔ b :) :=
le_sup_left
@[simp] theorem le_sup_right : b ≤ a ⊔ b :=
semilattice_sup.le_sup_right a b
@[ematch] theorem le_sup_right' : b ≤ (: a ⊔ b :) :=
le_sup_right
theorem le_sup_of_le_left (h : c ≤ a) : c ≤ a ⊔ b :=
le_trans h le_sup_left
theorem le_sup_of_le_right (h : c ≤ b) : c ≤ a ⊔ b :=
le_trans h le_sup_right
theorem lt_sup_of_lt_left (h : c < a) : c < a ⊔ b :=
h.trans_le le_sup_left
theorem lt_sup_of_lt_right (h : c < b) : c < a ⊔ b :=
h.trans_le le_sup_right
theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
semilattice_sup.sup_le a b c
@[simp] theorem sup_le_iff : a ⊔ b ≤ c ↔ a ≤ c ∧ b ≤ c :=
⟨assume h : a ⊔ b ≤ c, ⟨le_trans le_sup_left h, le_trans le_sup_right h⟩,
assume ⟨h₁, h₂⟩, sup_le h₁ h₂⟩
@[simp] theorem sup_eq_left : a ⊔ b = a ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_refl]
theorem sup_of_le_left (h : b ≤ a) : a ⊔ b = a :=
sup_eq_left.2 h
@[simp] theorem left_eq_sup : a = a ⊔ b ↔ b ≤ a :=
eq_comm.trans sup_eq_left
@[simp] theorem sup_eq_right : a ⊔ b = b ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_refl]
theorem sup_of_le_right (h : a ≤ b) : a ⊔ b = b :=
sup_eq_right.2 h
@[simp] theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b :=
eq_comm.trans sup_eq_right
theorem sup_le_sup (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊔ c ≤ b ⊔ d :=
sup_le (le_sup_of_le_left h₁) (le_sup_of_le_right h₂)
theorem sup_le_sup_left (h₁ : a ≤ b) (c) : c ⊔ a ≤ c ⊔ b :=
sup_le_sup le_rfl h₁
theorem sup_le_sup_right (h₁ : a ≤ b) (c) : a ⊔ c ≤ b ⊔ c :=
sup_le_sup h₁ le_rfl
theorem le_of_sup_eq (h : a ⊔ b = b) : a ≤ b :=
by { rw ← h, simp }
lemma sup_ind [is_total α (≤)] (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(is_total.total a b).elim (λ h : a ≤ b, by rwa sup_eq_right.2 h) (λ h, by rwa sup_eq_left.2 h)
@[simp] lemma sup_lt_iff [is_total α (≤)] {a b c : α} : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨λ h, ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩, λ h, sup_ind b c h.1 h.2⟩
@[simp] lemma le_sup_iff [is_total α (≤)] {a b c : α} : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c :=
⟨λ h, (total_of (≤) c b).imp
(λ bc, by rwa sup_eq_left.2 bc at h)
(λ bc, by rwa sup_eq_right.2 bc at h),
λ h, h.elim le_sup_of_le_left le_sup_of_le_right⟩
@[simp] lemma lt_sup_iff [is_total α (≤)] {a b c : α} : a < b ⊔ c ↔ a < b ∨ a < c :=
⟨λ h, (total_of (≤) c b).imp
(λ bc, by rwa sup_eq_left.2 bc at h)
(λ bc, by rwa sup_eq_right.2 bc at h),
λ h, h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
@[simp] theorem sup_idem : a ⊔ a = a :=
by apply le_antisymm; simp
instance sup_is_idempotent : is_idempotent α (⊔) := ⟨@sup_idem _ _⟩
theorem sup_comm : a ⊔ b = b ⊔ a :=
by apply le_antisymm; simp
instance sup_is_commutative : is_commutative α (⊔) := ⟨@sup_comm _ _⟩
theorem sup_assoc : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
le_antisymm
(sup_le
(sup_le le_sup_left (le_sup_of_le_right le_sup_left))
(le_sup_of_le_right le_sup_right))
(sup_le
(le_sup_of_le_left le_sup_left)
(sup_le (le_sup_of_le_left le_sup_right) le_sup_right))
instance sup_is_associative : is_associative α (⊔) := ⟨@sup_assoc _ _⟩
lemma sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a :=
by rw [sup_comm, @sup_comm _ _ a, sup_assoc]
@[simp] lemma sup_left_idem : a ⊔ (a ⊔ b) = a ⊔ b :=
by rw [← sup_assoc, sup_idem]
@[simp] lemma sup_right_idem : (a ⊔ b) ⊔ b = a ⊔ b :=
by rw [sup_assoc, sup_idem]
lemma sup_left_comm (a b c : α) : a ⊔ (b ⊔ c) = b ⊔ (a ⊔ c) :=
by rw [← sup_assoc, ← sup_assoc, @sup_comm α _ a]
lemma sup_right_comm (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ b :=
by rw [sup_assoc, sup_assoc, @sup_comm _ _ b]
lemma sup_sup_sup_comm (a b c d : α) : a ⊔ b ⊔ (c ⊔ d) = a ⊔ c ⊔ (b ⊔ d) :=
by rw [sup_assoc, sup_left_comm b, ←sup_assoc]
lemma forall_le_or_exists_lt_sup (a : α) : (∀b, b ≤ a) ∨ (∃b, a < b) :=
suffices (∃b, ¬b ≤ a) → (∃b, a < b),
by rwa [or_iff_not_imp_left, not_forall],
assume ⟨b, hb⟩,
⟨a ⊔ b, lt_of_le_of_ne le_sup_left $ mt left_eq_sup.1 hb⟩
/-- If `f` is monotone, `g` is antitone, and `f ≤ g`, then for all `a`, `b` we have `f a ≤ g b`. -/
theorem monotone.forall_le_of_antitone {β : Type*} [preorder β] {f g : α → β}
(hf : monotone f) (hg : antitone g) (h : f ≤ g) (m n : α) :
f m ≤ g n :=
calc f m ≤ f (m ⊔ n) : hf le_sup_left
... ≤ g (m ⊔ n) : h _
... ≤ g n : hg le_sup_right
theorem semilattice_sup.ext_sup {α} {A B : semilattice_sup α}
(H : ∀ x y : α, (by haveI := A; exact x ≤ y) ↔ x ≤ y)
(x y : α) : (by haveI := A; exact (x ⊔ y)) = x ⊔ y :=
eq_of_forall_ge_iff $ λ c,
by simp only [sup_le_iff]; rw [← H, @sup_le_iff α A, H, H]
theorem semilattice_sup.ext {α} {A B : semilattice_sup α}
(H : ∀ x y : α, (by haveI := A; exact x ≤ y) ↔ x ≤ y) : A = B :=
begin
have := partial_order.ext H,
have ss := funext (λ x, funext $ semilattice_sup.ext_sup H x),
casesI A, casesI B,
injection this; congr'
end
theorem exists_lt_of_sup (α : Type*) [semilattice_sup α] [nontrivial α] : ∃ a b : α, a < b :=
begin
rcases exists_pair_ne α with ⟨a, b, hne⟩,
rcases forall_le_or_exists_lt_sup b with (hb|H),
exacts [⟨a, b, (hb a).lt_of_ne hne⟩, ⟨b, H⟩]
end
lemma ite_le_sup (s s' : α) (P : Prop) [decidable P] : ite P s s' ≤ s ⊔ s' :=
if h : P then (if_pos h).trans_le le_sup_left else (if_neg h).trans_le le_sup_right
end semilattice_sup
/-!
### Meet-semilattices
-/
/-- A `semilattice_inf` is a meet-semilattice, that is, a partial order
with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation
`⊓` which is the greatest element smaller than both factors. -/
class semilattice_inf (α : Type u) extends has_inf α, partial_order α :=
(inf_le_left : ∀ a b : α, a ⊓ b ≤ a)
(inf_le_right : ∀ a b : α, a ⊓ b ≤ b)
(le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c)
instance (α) [semilattice_inf α] : semilattice_sup (order_dual α) :=
{ le_sup_left := semilattice_inf.inf_le_left,
le_sup_right := semilattice_inf.inf_le_right,
sup_le := assume a b c hca hcb, @semilattice_inf.le_inf α _ _ _ _ hca hcb,
.. order_dual.partial_order α, .. order_dual.has_sup α }
instance (α) [semilattice_sup α] : semilattice_inf (order_dual α) :=
{ inf_le_left := @le_sup_left α _,
inf_le_right := @le_sup_right α _,
le_inf := assume a b c hca hcb, @sup_le α _ _ _ _ hca hcb,
.. order_dual.partial_order α, .. order_dual.has_inf α }
theorem semilattice_sup.dual_dual (α : Type*) [H : semilattice_sup α] :
order_dual.semilattice_sup (order_dual α) = H :=
semilattice_sup.ext $ λ _ _, iff.rfl
section semilattice_inf
variables [semilattice_inf α] {a b c d : α}
@[simp] theorem inf_le_left : a ⊓ b ≤ a :=
semilattice_inf.inf_le_left a b
@[ematch] theorem inf_le_left' : (: a ⊓ b :) ≤ a :=
semilattice_inf.inf_le_left a b
@[simp] theorem inf_le_right : a ⊓ b ≤ b :=
semilattice_inf.inf_le_right a b
@[ematch] theorem inf_le_right' : (: a ⊓ b :) ≤ b :=
semilattice_inf.inf_le_right a b
theorem le_inf : a ≤ b → a ≤ c → a ≤ b ⊓ c :=
semilattice_inf.le_inf a b c
theorem inf_le_of_left_le (h : a ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_left h
theorem inf_le_of_right_le (h : b ≤ c) : a ⊓ b ≤ c :=
le_trans inf_le_right h
theorem inf_lt_of_left_lt (h : a < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_left h
theorem inf_lt_of_right_lt (h : b < c) : a ⊓ b < c :=
lt_of_le_of_lt inf_le_right h
@[simp] theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c :=
@sup_le_iff (order_dual α) _ _ _ _
@[simp] theorem inf_eq_left : a ⊓ b = a ↔ a ≤ b :=
le_antisymm_iff.trans $ by simp [le_refl]
theorem inf_of_le_left (h : a ≤ b) : a ⊓ b = a :=
inf_eq_left.2 h
@[simp] theorem left_eq_inf : a = a ⊓ b ↔ a ≤ b :=
eq_comm.trans inf_eq_left
@[simp] theorem inf_eq_right : a ⊓ b = b ↔ b ≤ a :=
le_antisymm_iff.trans $ by simp [le_refl]
theorem inf_of_le_right (h : b ≤ a) : a ⊓ b = b :=
inf_eq_right.2 h
@[simp] theorem right_eq_inf : b = a ⊓ b ↔ b ≤ a :=
eq_comm.trans inf_eq_right
theorem inf_le_inf (h₁ : a ≤ b) (h₂ : c ≤ d) : a ⊓ c ≤ b ⊓ d :=
le_inf (inf_le_of_left_le h₁) (inf_le_of_right_le h₂)
lemma inf_le_inf_right (a : α) {b c : α} (h : b ≤ c) : b ⊓ a ≤ c ⊓ a :=
inf_le_inf h le_rfl
lemma inf_le_inf_left (a : α) {b c : α} (h : b ≤ c) : a ⊓ b ≤ a ⊓ c :=
inf_le_inf le_rfl h
theorem le_of_inf_eq (h : a ⊓ b = a) : a ≤ b :=
by { rw ← h, simp }
lemma inf_ind [is_total α (≤)] (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊓ b) :=
@sup_ind (order_dual α) _ _ _ _ _ ha hb
@[simp] lemma lt_inf_iff [is_total α (≤)] {a b c : α} : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff (order_dual α) _ _ _ _ _
@[simp] lemma inf_le_iff [is_total α (≤)] {a b c : α} : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff (order_dual α) _ _ _ _ _
@[simp] theorem inf_idem : a ⊓ a = a :=
@sup_idem (order_dual α) _ _
instance inf_is_idempotent : is_idempotent α (⊓) := ⟨@inf_idem _ _⟩
theorem inf_comm : a ⊓ b = b ⊓ a :=
@sup_comm (order_dual α) _ _ _
instance inf_is_commutative : is_commutative α (⊓) := ⟨@inf_comm _ _⟩
theorem inf_assoc : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) :=
@sup_assoc (order_dual α) _ a b c
instance inf_is_associative : is_associative α (⊓) := ⟨@inf_assoc _ _⟩
lemma inf_left_right_swap (a b c : α) : a ⊓ b ⊓ c = c ⊓ b ⊓ a :=
by rw [inf_comm, @inf_comm _ _ a, inf_assoc]
@[simp] lemma inf_left_idem : a ⊓ (a ⊓ b) = a ⊓ b :=
@sup_left_idem (order_dual α) _ a b
@[simp] lemma inf_right_idem : (a ⊓ b) ⊓ b = a ⊓ b :=
@sup_right_idem (order_dual α) _ a b
lemma inf_left_comm (a b c : α) : a ⊓ (b ⊓ c) = b ⊓ (a ⊓ c) :=
@sup_left_comm (order_dual α) _ a b c
lemma inf_right_comm (a b c : α) : a ⊓ b ⊓ c = a ⊓ c ⊓ b :=
@sup_right_comm (order_dual α) _ a b c
lemma inf_inf_inf_comm (a b c d : α) : a ⊓ b ⊓ (c ⊓ d) = a ⊓ c ⊓ (b ⊓ d) :=
@sup_sup_sup_comm (order_dual α) _ _ _ _ _
lemma forall_le_or_exists_lt_inf (a : α) : (∀b, a ≤ b) ∨ (∃b, b < a) :=
@forall_le_or_exists_lt_sup (order_dual α) _ a
theorem semilattice_inf.ext_inf {α} {A B : semilattice_inf α}
(H : ∀ x y : α, (by haveI := A; exact x ≤ y) ↔ x ≤ y)
(x y : α) : (by haveI := A; exact (x ⊓ y)) = x ⊓ y :=
eq_of_forall_le_iff $ λ c,
by simp only [le_inf_iff]; rw [← H, @le_inf_iff α A, H, H]
theorem semilattice_inf.ext {α} {A B : semilattice_inf α}
(H : ∀ x y : α, (by haveI := A; exact x ≤ y) ↔ x ≤ y) : A = B :=
begin
have := partial_order.ext H,
have ss := funext (λ x, funext $ semilattice_inf.ext_inf H x),
casesI A, casesI B,
injection this; congr'
end
theorem semilattice_inf.dual_dual (α : Type*) [H : semilattice_inf α] :
order_dual.semilattice_inf (order_dual α) = H :=
semilattice_inf.ext $ λ _ _, iff.rfl
theorem exists_lt_of_inf (α : Type*) [semilattice_inf α] [nontrivial α] :
∃ a b : α, a < b :=
let ⟨a, b, h⟩ := exists_lt_of_sup (order_dual α) in ⟨b, a, h⟩
lemma inf_le_ite (s s' : α) (P : Prop) [decidable P] : s ⊓ s' ≤ ite P s s' :=
if h : P then inf_le_left.trans_eq (if_pos h).symm else inf_le_right.trans_eq (if_neg h).symm
end semilattice_inf
/--
A type with a commutative, associative and idempotent binary `inf` operation has the structure of a
meet-semilattice.
The partial order is defined so that `a ≤ b` unfolds to `b ⊓ a = a`; cf. `inf_eq_right`.
-/
def semilattice_inf.mk' {α : Type*} [has_inf α]
(inf_comm : ∀ (a b : α), a ⊓ b = b ⊓ a)
(inf_assoc : ∀ (a b c : α), a ⊓ b ⊓ c = a ⊓ (b ⊓ c))
(inf_idem : ∀ (a : α), a ⊓ a = a) : semilattice_inf α :=
begin
haveI : semilattice_sup (order_dual α) := semilattice_sup.mk' inf_comm inf_assoc inf_idem,
haveI i := order_dual.semilattice_inf (order_dual α),
exact i,
end
/-!
### Lattices
-/
/-- A lattice is a join-semilattice which is also a meet-semilattice. -/
@[protect_proj] class lattice (α : Type u) extends semilattice_sup α, semilattice_inf α
instance (α) [lattice α] : lattice (order_dual α) :=
{ .. order_dual.semilattice_sup α, .. order_dual.semilattice_inf α }
/-- The partial orders from `semilattice_sup_mk'` and `semilattice_inf_mk'` agree
if `sup` and `inf` satisfy the lattice absorption laws `sup_inf_self` (`a ⊔ a ⊓ b = a`)
and `inf_sup_self` (`a ⊓ (a ⊔ b) = a`). -/
lemma semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order {α : Type*}
[has_sup α] [has_inf α]
(sup_comm : ∀ (a b : α), a ⊔ b = b ⊔ a)
(sup_assoc : ∀ (a b c : α), a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(sup_idem : ∀ (a : α), a ⊔ a = a)
(inf_comm : ∀ (a b : α), a ⊓ b = b ⊓ a)
(inf_assoc : ∀ (a b c : α), a ⊓ b ⊓ c = a ⊓ (b ⊓ c))
(inf_idem : ∀ (a : α), a ⊓ a = a)
(sup_inf_self : ∀ (a b : α), a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ (a b : α), a ⊓ (a ⊔ b) = a) :
@semilattice_sup.to_partial_order _ (semilattice_sup.mk' sup_comm sup_assoc sup_idem) =
@semilattice_inf.to_partial_order _ (semilattice_inf.mk' inf_comm inf_assoc inf_idem) :=
partial_order.ext $ λ a b, show a ⊔ b = b ↔ b ⊓ a = a, from
⟨λ h, by rw [←h, inf_comm, inf_sup_self],
λ h, by rw [←h, sup_comm, sup_inf_self]⟩
/--
A type with a pair of commutative and associative binary operations which satisfy two absorption
laws relating the two operations has the structure of a lattice.
The partial order is defined so that `a ≤ b` unfolds to `a ⊔ b = b`; cf. `sup_eq_right`.
-/
def lattice.mk' {α : Type*} [has_sup α] [has_inf α]
(sup_comm : ∀ (a b : α), a ⊔ b = b ⊔ a)
(sup_assoc : ∀ (a b c : α), a ⊔ b ⊔ c = a ⊔ (b ⊔ c))
(inf_comm : ∀ (a b : α), a ⊓ b = b ⊓ a)
(inf_assoc : ∀ (a b c : α), a ⊓ b ⊓ c = a ⊓ (b ⊓ c))
(sup_inf_self : ∀ (a b : α), a ⊔ a ⊓ b = a)
(inf_sup_self : ∀ (a b : α), a ⊓ (a ⊔ b) = a) : lattice α :=
have sup_idem : ∀ (b : α), b ⊔ b = b := λ b,
calc b ⊔ b = b ⊔ b ⊓ (b ⊔ b) : by rw inf_sup_self
... = b : by rw sup_inf_self,
have inf_idem : ∀ (b : α), b ⊓ b = b := λ b,
calc b ⊓ b = b ⊓ (b ⊔ b ⊓ b) : by rw sup_inf_self
... = b : by rw inf_sup_self,
let semilatt_inf_inst := semilattice_inf.mk' inf_comm inf_assoc inf_idem,
semilatt_sup_inst := semilattice_sup.mk' sup_comm sup_assoc sup_idem,
-- here we help Lean to see that the two partial orders are equal
partial_order_inst := @semilattice_sup.to_partial_order _ semilatt_sup_inst in
have partial_order_eq :
partial_order_inst = @semilattice_inf.to_partial_order _ semilatt_inf_inst :=
semilattice_sup_mk'_partial_order_eq_semilattice_inf_mk'_partial_order _ _ _ _ _ _
sup_inf_self inf_sup_self,
{ inf_le_left := λ a b, by { rw partial_order_eq, apply inf_le_left },
inf_le_right := λ a b, by { rw partial_order_eq, apply inf_le_right },
le_inf := λ a b c, by { rw partial_order_eq, apply le_inf },
..partial_order_inst,
..semilatt_sup_inst,
..semilatt_inf_inst, }
section lattice
variables [lattice α] {a b c d : α}
lemma inf_le_sup : a ⊓ b ≤ a ⊔ b := inf_le_left.trans le_sup_left
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b :=
begin
split,
{ rintro H rfl, simpa using H },
{ refine λ Hne, lt_iff_le_and_ne.2 ⟨inf_le_sup, λ Heq, Hne _⟩,
refine le_antisymm _ _,
exacts [le_sup_left.trans (Heq.symm.trans_le inf_le_right),
le_sup_right.trans (Heq.symm.trans_le inf_le_left)] }
end
/-!
#### Distributivity laws
-/
/- TODO: better names? -/
theorem sup_inf_le : a ⊔ (b ⊓ c) ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _)
theorem le_inf_sup : (a ⊓ b) ⊔ (a ⊓ c) ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right)
theorem inf_sup_self : a ⊓ (a ⊔ b) = a :=
by simp
theorem sup_inf_self : a ⊔ (a ⊓ b) = a :=
by simp
theorem sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a :=
by rw [sup_eq_right, ←inf_eq_left]
theorem lattice.ext {α} {A B : lattice α}
(H : ∀ x y : α, (by haveI := A; exact x ≤ y) ↔ x ≤ y) : A = B :=
begin
have SS : @lattice.to_semilattice_sup α A =
@lattice.to_semilattice_sup α B := semilattice_sup.ext H,
have II := semilattice_inf.ext H,
casesI A, casesI B,
injection SS; injection II; congr'
end
end lattice
/-!
### Distributive lattices
-/
/-- A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`.
A classic example of a distributive lattice
is the lattice of subsets of a set, and in fact this example is
generic in the sense that every distributive lattice is realizable
as a sublattice of a powerset lattice. -/
class distrib_lattice α extends lattice α :=
(le_sup_inf : ∀x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z))
/- TODO: alternative constructors from the other distributive properties,
and perhaps a `tfae` statement -/
section distrib_lattice
variables [distrib_lattice α] {x y z : α}
theorem le_sup_inf : ∀{x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z) :=
distrib_lattice.le_sup_inf
theorem sup_inf_left : x ⊔ (y ⊓ z) = (x ⊔ y) ⊓ (x ⊔ z) :=
le_antisymm sup_inf_le le_sup_inf
theorem sup_inf_right : (y ⊓ z) ⊔ x = (y ⊔ x) ⊓ (z ⊔ x) :=
by simp only [sup_inf_left, λy:α, @sup_comm α _ y x, eq_self_iff_true]
theorem inf_sup_left : x ⊓ (y ⊔ z) = (x ⊓ y) ⊔ (x ⊓ z) :=
calc x ⊓ (y ⊔ z) = (x ⊓ (x ⊔ z)) ⊓ (y ⊔ z) : by rw [inf_sup_self]
... = x ⊓ ((x ⊓ y) ⊔ z) : by simp only [inf_assoc, sup_inf_right,
eq_self_iff_true]
... = (x ⊔ (x ⊓ y)) ⊓ ((x ⊓ y) ⊔ z) : by rw [sup_inf_self]
... = ((x ⊓ y) ⊔ x) ⊓ ((x ⊓ y) ⊔ z) : by rw [sup_comm]
... = (x ⊓ y) ⊔ (x ⊓ z) : by rw [sup_inf_left]
instance (α : Type*) [distrib_lattice α] : distrib_lattice (order_dual α) :=
{ le_sup_inf := assume x y z, le_of_eq inf_sup_left.symm,
.. order_dual.lattice α }
theorem inf_sup_right : (y ⊔ z) ⊓ x = (y ⊓ x) ⊔ (z ⊓ x) :=
by simp only [inf_sup_left, λy:α, @inf_comm α _ y x, eq_self_iff_true]
lemma le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc x ≤ (y ⊓ z) ⊔ x : le_sup_right
... = (y ⊔ x) ⊓ (x ⊔ z) : by rw [sup_inf_right, @sup_comm _ _ x]
... ≤ (y ⊔ x) ⊓ (y ⊔ z) : inf_le_inf_left _ h₂
... = y ⊔ (x ⊓ z) : sup_inf_left.symm
... ≤ y ⊔ (y ⊓ z) : sup_le_sup_left h₁ _
... ≤ _ : sup_le (le_refl y) inf_le_left
lemma eq_of_inf_eq_sup_eq {α : Type u} [distrib_lattice α] {a b c : α}
(h₁ : b ⊓ a = c ⊓ a) (h₂ : b ⊔ a = c ⊔ a) : b = c :=
le_antisymm
(le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm))
end distrib_lattice
/-!
### Lattices derived from linear orders
-/
@[priority 100] -- see Note [lower instance priority]
instance linear_order.to_lattice {α : Type u} [o : linear_order α] :
lattice α :=
{ sup := max,
le_sup_left := le_max_left,
le_sup_right := le_max_right,
sup_le := assume a b c, max_le,
inf := min,
inf_le_left := min_le_left,
inf_le_right := min_le_right,
le_inf := assume a b c, le_min,
..o }
theorem sup_eq_max [linear_order α] {x y : α} : x ⊔ y = max x y := rfl
theorem inf_eq_min [linear_order α] {x y : α} : x ⊓ y = min x y := rfl
lemma sup_eq_max_default [semilattice_sup α] [decidable_rel ((≤) : α → α → Prop)]
[is_total α (≤)] : (⊔) = (max_default : α → α → α) :=
begin
ext x y,
dunfold max_default,
split_ifs with h',
exacts [sup_of_le_left h', sup_of_le_right $ (total_of (≤) x y).resolve_right h']
end
lemma inf_eq_min_default [semilattice_inf α] [decidable_rel ((≤) : α → α → Prop)]
[is_total α (≤)] : (⊓) = (min_default : α → α → α) :=
@sup_eq_max_default (order_dual α) _ _ _
/-- A lattice with total order is a linear order.
See note [reducible non-instances]. -/
@[reducible] def lattice.to_linear_order (α : Type u) [lattice α] [decidable_eq α]
[decidable_rel ((≤) : α → α → Prop)] [decidable_rel ((<) : α → α → Prop)]
[is_total α (≤)] :
linear_order α :=
{ decidable_le := ‹_›, decidable_eq := ‹_›, decidable_lt := ‹_›,
le_total := total_of (≤),
max := (⊔),
max_def := sup_eq_max_default,
min := (⊓),
min_def := inf_eq_min_default,
..‹lattice α› }
@[priority 100] -- see Note [lower instance priority]
instance linear_order.to_distrib_lattice {α : Type u} [o : linear_order α] :
distrib_lattice α :=
{ le_sup_inf := assume a b c,
match le_total b c with
| or.inl h := inf_le_of_left_le $ sup_le_sup_left (le_inf (le_refl b) h) _
| or.inr h := inf_le_of_right_le $ sup_le_sup_left (le_inf h (le_refl c)) _
end,
..linear_order.to_lattice }
instance nat.distrib_lattice : distrib_lattice ℕ :=
by apply_instance
/-! ### Function lattices -/
namespace pi
variables {ι : Type*} {α' : ι → Type*}
instance [Π i, has_sup (α' i)] : has_sup (Π i, α' i) := ⟨λ f g i, f i ⊔ g i⟩
@[simp] lemma sup_apply [Π i, has_sup (α' i)] (f g : Π i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
lemma sup_def [Π i, has_sup (α' i)] (f g : Π i, α' i) : f ⊔ g = λ i, f i ⊔ g i := rfl
instance [Π i, has_inf (α' i)] : has_inf (Π i, α' i) := ⟨λ f g i, f i ⊓ g i⟩
@[simp] lemma inf_apply [Π i, has_inf (α' i)] (f g : Π i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl
lemma inf_def [Π i, has_inf (α' i)] (f g : Π i, α' i) : f ⊓ g = λ i, f i ⊓ g i := rfl
instance [Π i, semilattice_sup (α' i)] : semilattice_sup (Π i, α' i) :=
by refine_struct { sup := (⊔), .. pi.partial_order }; tactic.pi_instance_derive_field
instance [Π i, semilattice_inf (α' i)] : semilattice_inf (Π i, α' i) :=
by refine_struct { inf := (⊓), .. pi.partial_order }; tactic.pi_instance_derive_field
instance [Π i, lattice (α' i)] : lattice (Π i, α' i) :=
{ .. pi.semilattice_sup, .. pi.semilattice_inf }
instance [Π i, distrib_lattice (α' i)] : distrib_lattice (Π i, α' i) :=
by refine_struct { .. pi.lattice }; tactic.pi_instance_derive_field
end pi
/-!
### Monotone functions and lattices
-/
namespace monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected lemma sup [preorder α] [semilattice_sup β] {f g : α → β} (hf : monotone f)
(hg : monotone g) : monotone (f ⊔ g) :=
λ x y h, sup_le_sup (hf h) (hg h)
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected lemma inf [preorder α] [semilattice_inf β] {f g : α → β} (hf : monotone f)
(hg : monotone g) : monotone (f ⊓ g) :=
λ x y h, inf_le_inf (hf h) (hg h)
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected lemma max [preorder α] [linear_order β] {f g : α → β} (hf : monotone f)
(hg : monotone g) : monotone (λ x, max (f x) (g x)) :=
hf.sup hg
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected lemma min [preorder α] [linear_order β] {f g : α → β} (hf : monotone f)
(hg : monotone g) : monotone (λ x, min (f x) (g x)) :=
hf.inf hg
lemma le_map_sup [semilattice_sup α] [semilattice_sup β]
{f : α → β} (h : monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right)
lemma map_sup [semilattice_sup α] [is_total α (≤)] [semilattice_sup β] {f : α → β}
(hf : monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(is_total.total x y).elim
(λ h : x ≤ y, by simp only [h, hf h, sup_of_le_right])
(λ h, by simp only [h, hf h, sup_of_le_left])
lemma map_inf_le [semilattice_inf α] [semilattice_inf β]
{f : α → β} (h : monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right)
lemma map_inf [semilattice_inf α] [is_total α (≤)] [semilattice_inf β] {f : α → β}
(hf : monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
@monotone.map_sup (order_dual α) _ _ _ _ _ hf.dual x y
end monotone
namespace antitone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected lemma sup [preorder α] [semilattice_sup β] {f g : α → β} (hf : antitone f)
(hg : antitone g) : antitone (f ⊔ g) :=
λ x y h, sup_le_sup (hf h) (hg h)
/-- Pointwise infimum of two monotone functions is a monotone function. -/
protected lemma inf [preorder α] [semilattice_inf β] {f g : α → β} (hf : antitone f)
(hg : antitone g) : antitone (f ⊓ g) :=
λ x y h, inf_le_inf (hf h) (hg h)
/-- Pointwise maximum of two monotone functions is a monotone function. -/
protected lemma max [preorder α] [linear_order β] {f g : α → β} (hf : antitone f)
(hg : antitone g) : antitone (λ x, max (f x) (g x)) :=
hf.sup hg
/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected lemma min [preorder α] [linear_order β] {f g : α → β} (hf : antitone f)
(hg : antitone g) : antitone (λ x, min (f x) (g x)) :=
hf.inf hg
lemma map_sup_le [semilattice_sup α] [semilattice_inf β]
{f : α → β} (h : antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y
lemma map_sup [semilattice_sup α] [is_total α (≤)] [semilattice_inf β] {f : α → β}
(hf : antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y
lemma le_map_inf [semilattice_inf α] [semilattice_sup β]
{f : α → β} (h : antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
lemma map_inf [semilattice_inf α] [is_total α (≤)] [semilattice_sup β] {f : α → β}
(hf : antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y
end antitone
/-!
### Products of (semi-)lattices
-/
namespace prod
variables (α β)
instance [has_sup α] [has_sup β] : has_sup (α × β) := ⟨λp q, ⟨p.1 ⊔ q.1, p.2 ⊔ q.2⟩⟩
instance [has_inf α] [has_inf β] : has_inf (α × β) := ⟨λp q, ⟨p.1 ⊓ q.1, p.2 ⊓ q.2⟩⟩
instance [semilattice_sup α] [semilattice_sup β] : semilattice_sup (α × β) :=
{ sup_le := assume a b c h₁ h₂, ⟨sup_le h₁.1 h₂.1, sup_le h₁.2 h₂.2⟩,
le_sup_left := assume a b, ⟨le_sup_left, le_sup_left⟩,
le_sup_right := assume a b, ⟨le_sup_right, le_sup_right⟩,
.. prod.partial_order α β, .. prod.has_sup α β }
instance [semilattice_inf α] [semilattice_inf β] : semilattice_inf (α × β) :=
{ le_inf := assume a b c h₁ h₂, ⟨le_inf h₁.1 h₂.1, le_inf h₁.2 h₂.2⟩,
inf_le_left := assume a b, ⟨inf_le_left, inf_le_left⟩,
inf_le_right := assume a b, ⟨inf_le_right, inf_le_right⟩,
.. prod.partial_order α β, .. prod.has_inf α β }
instance [lattice α] [lattice β] : lattice (α × β) :=
{ .. prod.semilattice_inf α β, .. prod.semilattice_sup α β }
instance [distrib_lattice α] [distrib_lattice β] : distrib_lattice (α × β) :=
{ le_sup_inf := assume a b c, ⟨le_sup_inf, le_sup_inf⟩,
.. prod.lattice α β }
end prod
/-!
### Subtypes of (semi-)lattices
-/
namespace subtype
/-- A subtype forms a `⊔`-semilattice if `⊔` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilattice_sup [semilattice_sup α] {P : α → Prop}
(Psup : ∀⦃x y⦄, P x → P y → P (x ⊔ y)) : semilattice_sup {x : α // P x} :=
{ sup := λ x y, ⟨x.1 ⊔ y.1, Psup x.2 y.2⟩,
le_sup_left := λ x y, @le_sup_left _ _ (x : α) y,
le_sup_right := λ x y, @le_sup_right _ _ (x : α) y,
sup_le := λ x y z h1 h2, @sup_le α _ _ _ _ h1 h2,
..subtype.partial_order P }
/-- A subtype forms a `⊓`-semilattice if `⊓` preserves the property.
See note [reducible non-instances]. -/
@[reducible]
protected def semilattice_inf [semilattice_inf α] {P : α → Prop}
(Pinf : ∀⦃x y⦄, P x → P y → P (x ⊓ y)) : semilattice_inf {x : α // P x} :=
{ inf := λ x y, ⟨x.1 ⊓ y.1, Pinf x.2 y.2⟩,
inf_le_left := λ x y, @inf_le_left _ _ (x : α) y,
inf_le_right := λ x y, @inf_le_right _ _ (x : α) y,
le_inf := λ x y z h1 h2, @le_inf α _ _ _ _ h1 h2,
..subtype.partial_order P }
/-- A subtype forms a lattice if `⊔` and `⊓` preserve the property.
See note [reducible non-instances]. -/
@[reducible]
protected def lattice [lattice α] {P : α → Prop}
(Psup : ∀⦃x y⦄, P x → P y → P (x ⊔ y)) (Pinf : ∀⦃x y⦄, P x → P y → P (x ⊓ y)) :
lattice {x : α // P x} :=
{ ..subtype.semilattice_inf Pinf, ..subtype.semilattice_sup Psup }
end subtype
section lift
/-- A type endowed with `⊔` is a `semilattice_sup`, if it admits an injective map that
preserves `⊔` to a `semilattice_sup`.
See note [reducible non-instances]. -/
@[reducible] protected def function.injective.semilattice_sup [has_sup α] [semilattice_sup β]
(f : α → β) (hf_inj : function.injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) :
semilattice_sup α :=
{ sup := has_sup.sup,
le_sup_left := λ a b, by { change f a ≤ f (a ⊔ b), rw map_sup, exact le_sup_left, },
le_sup_right := λ a b, by { change f b ≤ f (a ⊔ b), rw map_sup, exact le_sup_right, },
sup_le := λ a b c ha hb, by { change f (a ⊔ b) ≤ f c, rw map_sup, exact sup_le ha hb, },
..partial_order.lift f hf_inj}
/-- A type endowed with `⊓` is a `semilattice_inf`, if it admits an injective map that
preserves `⊓` to a `semilattice_inf`.
See note [reducible non-instances]. -/
@[reducible] protected def function.injective.semilattice_inf [has_inf α] [semilattice_inf β]
(f : α → β) (hf_inj : function.injective f) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) :
semilattice_inf α :=
{ inf := has_inf.inf,
inf_le_left := λ a b, by { change f (a ⊓ b) ≤ f a, rw map_inf, exact inf_le_left, },
inf_le_right := λ a b, by { change f (a ⊓ b) ≤ f b, rw map_inf, exact inf_le_right, },
le_inf := λ a b c ha hb, by { change f a ≤ f (b ⊓ c), rw map_inf, exact le_inf ha hb, },
..partial_order.lift f hf_inj}
/-- A type endowed with `⊔` and `⊓` is a `lattice`, if it admits an injective map that
preserves `⊔` and `⊓` to a `lattice`.
See note [reducible non-instances]. -/
@[reducible] protected def function.injective.lattice [has_sup α] [has_inf α] [lattice β]
(f : α → β) (hf_inj : function.injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b)
(map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) :
lattice α :=
{ ..hf_inj.semilattice_sup f map_sup, ..hf_inj.semilattice_inf f map_inf}
/-- A type endowed with `⊔` and `⊓` is a `distrib_lattice`, if it admits an injective map that
preserves `⊔` and `⊓` to a `distrib_lattice`.
See note [reducible non-instances]. -/
@[reducible] protected def function.injective.distrib_lattice [has_sup α] [has_inf α]
[distrib_lattice β] (f : α → β) (hf_inj : function.injective f)
(map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) :
distrib_lattice α :=
{ le_sup_inf := λ a b c, by { change f ((a ⊔ b) ⊓ (a ⊔ c)) ≤ f (a ⊔ b ⊓ c),
rw [map_inf, map_sup, map_sup, map_sup, map_inf], exact le_sup_inf, },
..hf_inj.lattice f map_sup map_inf, }
end lift
--To avoid noncomputability poisoning from `bool.complete_boolean_algebra`
instance : distrib_lattice bool := linear_order.to_distrib_lattice
|
1ae183d4d9f250f57292e1dfd6d9bd2ec8dd99fc | 8cae430f0a71442d02dbb1cbb14073b31048e4b0 | /src/tactic/explode.lean | 4307de7d5ced615b1ed74edd0c2d43017223f1ee | [
"Apache-2.0"
] | permissive | leanprover-community/mathlib | 56a2cadd17ac88caf4ece0a775932fa26327ba0e | 442a83d738cb208d3600056c489be16900ba701d | refs/heads/master | 1,693,584,102,358 | 1,693,471,902,000 | 1,693,471,902,000 | 97,922,418 | 1,595 | 352 | Apache-2.0 | 1,694,693,445,000 | 1,500,624,130,000 | Lean | UTF-8 | Lean | false | false | 9,383 | lean | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Minchao Wu
-/
import meta.rb_map
import tactic.core
/-!
# `#explode` command
Displays a proof term in a line by line format somewhat akin to a Fitch style
proof or the Metamath proof style.
-/
open expr tactic
namespace tactic
namespace explode
@[derive inhabited]
inductive status : Type | reg | intro | lam | sintro
/--
A type to distinguish introduction or elimination rules represented as
strings from theorems referred to by their names.
-/
meta inductive thm : Type
| expr (e : expr)
| name (n : name)
| string (s : string)
/--
Turn a thm into a string.
-/
meta def thm.to_string : thm → string
| (thm.expr e) := e.to_string
| (thm.name n) := n.to_string
| (thm.string s) := s
meta structure entry : Type :=
(expr : expr)
(line : nat)
(depth : nat)
(status : status)
(thm : thm)
(deps : list nat)
meta def pad_right (l : list string) : list string :=
let n := l.foldl (λ r (s:string), max r s.length) 0 in
l.map $ λ s, nat.iterate (λ s, s.push ' ') (n - s.length) s
@[derive inhabited]
meta structure entries : Type := mk' ::
(s : expr_map entry)
(l : list entry)
meta def entries.find (es : entries) (e : expr) : option entry := es.s.find e
meta def entries.size (es : entries) : ℕ := es.s.size
meta def entries.add : entries → entry → entries
| es@⟨s, l⟩ e := if s.contains e.expr then es else ⟨s.insert e.expr e, e :: l⟩
meta def entries.head (es : entries) : option entry := es.l.head'
meta def format_aux : list string → list string → list string → list entry → tactic format
| (line :: lines) (dep :: deps) (thm :: thms) (en :: es) := do
fmt ← do
{ let margin := string.join (list.replicate en.depth " │"),
let margin := match en.status with
| status.sintro := " ├" ++ margin
| status.intro := " │" ++ margin ++ " ┌"
| status.reg := " │" ++ margin ++ ""
| status.lam := " │" ++ margin ++ ""
end,
p ← infer_type en.expr >>= pp,
let lhs := line ++ "│" ++ dep ++ "│ " ++ thm ++ margin ++ " ",
return $ format.of_string lhs ++ (p.nest lhs.length).group ++ format.line },
(++ fmt) <$> format_aux lines deps thms es
| _ _ _ _ := return format.nil
meta instance : has_to_tactic_format entries :=
⟨λ es : entries,
let lines := pad_right $ es.l.map (λ en, to_string en.line),
deps := pad_right $ es.l.map (λ en, string.intercalate "," (en.deps.map to_string)),
thms := pad_right $ es.l.map (λ en, (entry.thm en).to_string) in
format_aux lines deps thms es.l⟩
meta def append_dep (filter : expr → tactic unit)
(es : entries) (e : expr) (deps : list nat) : tactic (list nat) :=
do { ei ← es.find e,
filter ei.expr,
return (ei.line :: deps) }
<|> return deps
meta def may_be_proof (e : expr) : tactic bool :=
do expr.sort u ← infer_type e >>= infer_type,
return $ bnot u.nonzero
end explode
open explode
meta mutual def explode.core, explode.args (filter : expr → tactic unit)
with explode.core : expr → bool → nat → entries → tactic entries
| e@(lam n bi d b) si depth es := do
m ← mk_fresh_name,
let l := local_const m n bi d,
let b' := instantiate_var b l,
if si then
let en : entry := ⟨l, es.size, depth, status.sintro, thm.name n, []⟩ in do
es' ← explode.core b' si depth (es.add en),
return $ es'.add ⟨e, es'.size, depth, status.lam, thm.string "∀I", [es.size, es'.size - 1]⟩
else do
let en : entry := ⟨l, es.size, depth, status.intro, thm.name n, []⟩,
es' ← explode.core b' si (depth + 1) (es.add en),
-- in case of a "have" clause, the b' here has an annotation
deps' ← explode.append_dep filter es' b'.erase_annotations [],
deps' ← explode.append_dep filter es' l deps',
return $ es'.add ⟨e, es'.size, depth, status.lam, thm.string "∀I", deps'⟩
| e@(elet n t a b) si depth es := explode.core (reduce_lets e) si depth es
| e@(macro n l) si depth es := explode.core l.head si depth es
| e si depth es := filter e >>
match get_app_fn_args e with
| (nm@(const n _), args) :=
explode.args e args depth es (thm.expr nm) []
| (fn, []) := do
let en : entry := ⟨fn, es.size, depth, status.reg, thm.expr fn, []⟩,
return (es.add en)
| (fn, args) := do
es' ← explode.core fn ff depth es,
-- in case of a "have" clause, the fn here has an annotation
deps ← explode.append_dep filter es' fn.erase_annotations [],
explode.args e args depth es' (thm.string "∀E") deps
end
with explode.args : expr → list expr → nat → entries → thm → list nat → tactic entries
| e (arg :: args) depth es thm deps := do
es' ← explode.core arg ff depth es <|> return es,
deps' ← explode.append_dep filter es' arg deps,
explode.args e args depth es' thm deps'
| e [] depth es thm deps :=
return (es.add ⟨e, es.size, depth, status.reg, thm, deps.reverse⟩)
meta def explode_expr (e : expr) (hide_non_prop := tt) : tactic entries :=
let filter := if hide_non_prop then λ e, may_be_proof e >>= guardb else λ _, skip in
tactic.explode.core filter e tt 0 default
meta def explode (n : name) : tactic unit :=
do const n _ ← resolve_name n | fail "cannot resolve name",
d ← get_decl n,
v ← match d with
| (declaration.defn _ _ _ v _ _) := return v
| (declaration.thm _ _ _ v) := return v.get
| _ := fail "not a definition"
end,
t ← pp d.type,
explode_expr v <* trace (to_fmt n ++ " : " ++ t) >>= trace
setup_tactic_parser
/--
`#explode decl_name` displays a proof term in a line-by-line format somewhat akin to a Fitch-style
proof or the Metamath proof style.
`#explode_widget decl_name` renders a widget that displays an `#explode` proof.
`#explode iff_true_intro` produces
```lean
iff_true_intro : ∀ {a : Prop}, a → (a ↔ true)
0│ │ a ├ Prop
1│ │ h ├ a
2│ │ hl │ ┌ a
3│ │ trivial │ │ true
4│2,3│ ∀I │ a → true
5│ │ hr │ ┌ true
6│5,1│ ∀I │ true → a
7│4,6│ iff.intro │ a ↔ true
8│1,7│ ∀I │ a → (a ↔ true)
9│0,8│ ∀I │ ∀ {a : Prop}, a → (a ↔ true)
```
In more detail:
The output of `#explode` is a Fitch-style proof in a four-column diagram modeled after Metamath
proof displays like [this](http://us.metamath.org/mpeuni/ru.html). The headers of the columns are
"Step", "Hyp", "Ref", "Type" (or "Expression" in the case of Metamath):
* Step: An increasing sequence of numbers to number each step in the proof, used in the Hyp field.
* Hyp: The direct children of the current step. Most theorems are implications like `A -> B -> C`,
and so on the step proving `C` the Hyp field will refer to the steps that prove `A` and `B`.
* Ref: The name of the theorem being applied. This is well-defined in Metamath, but in Lean there
are some special steps that may have long names because the structure of proof terms doesn't
exactly match this mold.
* If the theorem is `foo (x y : Z) : A x -> B y -> C x y`:
* the Ref field will contain `foo`,
* `x` and `y` will be suppressed, because term construction is not interesting, and
* the Hyp field will reference steps proving `A x` and `B y`. This corresponds to a proof term
like `@foo x y pA pB` where `pA` and `pB` are subproofs.
* If the head of the proof term is a local constant or lambda, then in this case the Ref will
say `∀E` for forall-elimination. This happens when you have for example `h : A -> B` and
`ha : A` and prove `b` by `h ha`; we reinterpret this as if it said `∀E h ha` where `∀E` is
(n-ary) modus ponens.
* If the proof term is a lambda, we will also use `∀I` for forall-introduction, referencing the
body of the lambda. The indentation level will increase, and a bracket will surround the proof
of the body of the lambda, starting at a proof step labeled with the name of the lambda variable
and its type, and ending with the `∀I` step. Metamath doesn't have steps like this, but the
style is based on Fitch proofs in first-order logic.
* Type: This contains the type of the proof term, the theorem being proven at the current step.
This proof layout differs from `#print` in using lots of intermediate step displays so that you
can follow along and don't have to see term construction steps because they are implicitly in the
intermediate step displays.
Also, it is common for a Lean theorem to begin with a sequence of lambdas introducing local
constants of the theorem. In order to minimize the indentation level, the `∀I` steps at the end of
the proof will be introduced in a group and the indentation will stay fixed. (The indentation
brackets are only needed in order to delimit the scope of assumptions, and these assumptions
have global scope anyway so detailed tracking is not necessary.)
-/
@[user_command]
meta def explode_cmd (_ : parse $ tk "#explode") : parser unit :=
do n ← ident,
explode n
.
add_tactic_doc
{ name := "#explode / #explode_widget",
category := doc_category.cmd,
decl_names := [`tactic.explode_cmd, `tactic.explode_widget_cmd],
inherit_description_from := `tactic.explode_cmd,
tags := ["proof display", "widgets"] }
end tactic
|
f9b2e09a2e35fe4e85a5d9eb3d1c3d5bff87dc55 | 302c785c90d40ad3d6be43d33bc6a558354cc2cf | /src/measure_theory/ae_eq_fun.lean | 8d0d3fb7ba56e7412a4b2886707fa6679ce52b48 | [
"Apache-2.0"
] | permissive | ilitzroth/mathlib | ea647e67f1fdfd19a0f7bdc5504e8acec6180011 | 5254ef14e3465f6504306132fe3ba9cec9ffff16 | refs/heads/master | 1,680,086,661,182 | 1,617,715,647,000 | 1,617,715,647,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 20,401 | lean | /-
Copyright (c) 2019 Johannes Hölzl, Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Zhouhang Zhou
-/
import measure_theory.integration
import order.filter.germ
import topology.continuous_function.algebra
/-!
# Almost everywhere equal functions
Two measurable functions are treated as identical if they are almost everywhere equal. We form the
set of equivalence classes under the relation of being almost everywhere equal, which is sometimes
known as the `L⁰` space.
See `l1_space.lean` for `L¹` space.
## Notation
* `α →ₘ[μ] β` is the type of `L⁰` space, where `α` and `β` are measurable spaces and `μ`
is a measure on `α`. `f : α →ₘ β` is a "function" in `L⁰`. In comments, `[f]` is also used
to denote an `L⁰` function.
`ₘ` can be typed as `\_m`. Sometimes it is shown as a box if font is missing.
## Main statements
* The linear structure of `L⁰` :
Addition and scalar multiplication are defined on `L⁰` in the natural way, i.e.,
`[f] + [g] := [f + g]`, `c • [f] := [c • f]`. So defined, `α →ₘ β` inherits the linear structure
of `β`. For example, if `β` is a module, then `α →ₘ β` is a module over the same ring.
See `mk_add_mk`, `neg_mk`, `mk_sub_mk`, `smul_mk`,
`add_to_fun`, `neg_to_fun`, `sub_to_fun`, `smul_to_fun`
* The order structure of `L⁰` :
`≤` can be defined in a similar way: `[f] ≤ [g]` if `f a ≤ g a` for almost all `a` in domain.
And `α →ₘ β` inherits the preorder and partial order of `β`.
TODO: Define `sup` and `inf` on `L⁰` so that it forms a lattice. It seems that `β` must be a
linear order, since otherwise `f ⊔ g` may not be a measurable function.
## Implementation notes
* `f.to_fun` : To find a representative of `f : α →ₘ β`, use `f.to_fun`.
For each operation `op` in `L⁰`, there is a lemma called `op_to_fun`,
characterizing, say, `(f op g).to_fun`.
* `ae_eq_fun.mk` : To constructs an `L⁰` function `α →ₘ β` from a measurable function `f : α → β`,
use `ae_eq_fun.mk`
* `comp` : Use `comp g f` to get `[g ∘ f]` from `g : β → γ` and `[f] : α →ₘ γ`
* `comp₂` : Use `comp₂ g f₁ f₂ to get `[λa, g (f₁ a) (f₂ a)]`.
For example, `[f + g]` is `comp₂ (+)`
## Tags
function space, almost everywhere equal, `L⁰`, ae_eq_fun
-/
noncomputable theory
open_locale classical ennreal
open set filter topological_space ennreal emetric measure_theory function
variables {α β γ δ : Type*} [measurable_space α] {μ ν : measure α}
namespace measure_theory
section measurable_space
variables [measurable_space β]
variable (β)
/-- The equivalence relation of being almost everywhere equal -/
def measure.ae_eq_setoid (μ : measure α) : setoid { f : α → β // ae_measurable f μ } :=
⟨λf g, (f : α → β) =ᵐ[μ] g, λ f, ae_eq_refl f, λ f g, ae_eq_symm, λ f g h, ae_eq_trans⟩
variable (α)
/-- The space of equivalence classes of measurable functions, where two measurable functions are
equivalent if they agree almost everywhere, i.e., they differ on a set of measure `0`. -/
def ae_eq_fun (μ : measure α) : Type* := quotient (μ.ae_eq_setoid β)
variables {α β}
notation α ` →ₘ[`:25 μ `] ` β := ae_eq_fun α β μ
end measurable_space
namespace ae_eq_fun
variables [measurable_space β] [measurable_space γ] [measurable_space δ]
/-- Construct the equivalence class `[f]` of an almost everywhere measurable function `f`, based
on the equivalence relation of being almost everywhere equal. -/
def mk (f : α → β) (hf : ae_measurable f μ) : α →ₘ[μ] β := quotient.mk' ⟨f, hf⟩
/-- A measurable representative of an `ae_eq_fun` [f] -/
instance : has_coe_to_fun (α →ₘ[μ] β) :=
⟨_, λf, ae_measurable.mk _ (quotient.out' f : {f : α → β // ae_measurable f μ}).2⟩
protected lemma measurable (f : α →ₘ[μ] β) : measurable f :=
ae_measurable.measurable_mk _
protected lemma ae_measurable (f : α →ₘ[μ] β) : ae_measurable f μ :=
f.measurable.ae_measurable
@[simp] lemma quot_mk_eq_mk (f : α → β) (hf) :
(quot.mk (@setoid.r _ $ μ.ae_eq_setoid β) ⟨f, hf⟩ : α →ₘ[μ] β) = mk f hf :=
rfl
@[simp] lemma mk_eq_mk {f g : α → β} {hf hg} :
(mk f hf : α →ₘ[μ] β) = mk g hg ↔ f =ᵐ[μ] g :=
quotient.eq'
@[simp] lemma mk_coe_fn (f : α →ₘ[μ] β) : mk f f.ae_measurable = f :=
begin
conv_rhs { rw ← quotient.out_eq' f },
set g : {f : α → β // ae_measurable f μ} := quotient.out' f with hg,
have : g = ⟨g.1, g.2⟩ := subtype.eq rfl,
rw [this, ← mk, mk_eq_mk],
exact (ae_measurable.ae_eq_mk _).symm,
end
@[ext] lemma ext {f g : α →ₘ[μ] β} (h : f =ᵐ[μ] g) : f = g :=
by rwa [← f.mk_coe_fn, ← g.mk_coe_fn, mk_eq_mk]
lemma ext_iff {f g : α →ₘ[μ] β} : f = g ↔ f =ᵐ[μ] g :=
⟨λ h, by rw h, λ h, ext h⟩
lemma coe_fn_mk (f : α → β) (hf) : (mk f hf : α →ₘ[μ] β) =ᵐ[μ] f :=
begin
apply (ae_measurable.ae_eq_mk _).symm.trans,
exact @quotient.mk_out' _ (μ.ae_eq_setoid β) (⟨f, hf⟩ : {f // ae_measurable f μ})
end
@[elab_as_eliminator]
lemma induction_on (f : α →ₘ[μ] β) {p : (α →ₘ[μ] β) → Prop} (H : ∀ f hf, p (mk f hf)) : p f :=
quotient.induction_on' f $ subtype.forall.2 H
@[elab_as_eliminator]
lemma induction_on₂ {α' β' : Type*} [measurable_space α'] [measurable_space β'] {μ' : measure α'}
(f : α →ₘ[μ] β) (f' : α' →ₘ[μ'] β') {p : (α →ₘ[μ] β) → (α' →ₘ[μ'] β') → Prop}
(H : ∀ f hf f' hf', p (mk f hf) (mk f' hf')) :
p f f' :=
induction_on f $ λ f hf, induction_on f' $ H f hf
@[elab_as_eliminator]
lemma induction_on₃ {α' β' : Type*} [measurable_space α'] [measurable_space β'] {μ' : measure α'}
{α'' β'' : Type*} [measurable_space α''] [measurable_space β''] {μ'' : measure α''}
(f : α →ₘ[μ] β) (f' : α' →ₘ[μ'] β') (f'' : α'' →ₘ[μ''] β'')
{p : (α →ₘ[μ] β) → (α' →ₘ[μ'] β') → (α'' →ₘ[μ''] β'') → Prop}
(H : ∀ f hf f' hf' f'' hf'', p (mk f hf) (mk f' hf') (mk f'' hf'')) :
p f f' f'' :=
induction_on f $ λ f hf, induction_on₂ f' f'' $ H f hf
/-- Given a measurable function `g : β → γ`, and an almost everywhere equal function `[f] : α →ₘ β`,
return the equivalence class of `g ∘ f`, i.e., the almost everywhere equal function
`[g ∘ f] : α →ₘ γ`. -/
def comp (g : β → γ) (hg : measurable g) (f : α →ₘ[μ] β) : α →ₘ[μ] γ :=
quotient.lift_on' f (λ f, mk (g ∘ (f : α → β)) (hg.comp_ae_measurable f.2)) $
λ f f' H, mk_eq_mk.2 $ H.fun_comp g
@[simp] lemma comp_mk (g : β → γ) (hg : measurable g)
(f : α → β) (hf) :
comp g hg (mk f hf : α →ₘ[μ] β) = mk (g ∘ f) (hg.comp_ae_measurable hf) :=
rfl
lemma comp_eq_mk (g : β → γ) (hg : measurable g) (f : α →ₘ[μ] β) :
comp g hg f = mk (g ∘ f) (hg.comp_ae_measurable f.ae_measurable) :=
by rw [← comp_mk g hg f f.ae_measurable, mk_coe_fn]
lemma coe_fn_comp (g : β → γ) (hg : measurable g) (f : α →ₘ[μ] β) :
comp g hg f =ᵐ[μ] g ∘ f :=
by { rw [comp_eq_mk], apply coe_fn_mk }
/-- The class of `x ↦ (f x, g x)`. -/
def pair (f : α →ₘ[μ] β) (g : α →ₘ[μ] γ) : α →ₘ[μ] β × γ :=
quotient.lift_on₂' f g (λ f g, mk (λ x, (f.1 x, g.1 x)) (f.2.prod_mk g.2)) $
λ f g f' g' Hf Hg, mk_eq_mk.2 $ Hf.prod_mk Hg
@[simp] lemma pair_mk_mk (f : α → β) (hf) (g : α → γ) (hg) :
(mk f hf : α →ₘ[μ] β).pair (mk g hg) = mk (λ x, (f x, g x)) (hf.prod_mk hg) :=
rfl
lemma pair_eq_mk (f : α →ₘ[μ] β) (g : α →ₘ[μ] γ) :
f.pair g = mk (λ x, (f x, g x)) (f.ae_measurable.prod_mk g.ae_measurable) :=
by simp only [← pair_mk_mk, mk_coe_fn]
lemma coe_fn_pair (f : α →ₘ[μ] β) (g : α →ₘ[μ] γ) :
f.pair g =ᵐ[μ] (λ x, (f x, g x)) :=
by { rw pair_eq_mk, apply coe_fn_mk }
/-- Given a measurable function `g : β → γ → δ`, and almost everywhere equal functions
`[f₁] : α →ₘ β` and `[f₂] : α →ₘ γ`, return the equivalence class of the function
`λa, g (f₁ a) (f₂ a)`, i.e., the almost everywhere equal function
`[λa, g (f₁ a) (f₂ a)] : α →ₘ γ` -/
def comp₂ {γ δ : Type*} [measurable_space γ] [measurable_space δ] (g : β → γ → δ)
(hg : measurable (uncurry g)) (f₁ : α →ₘ[μ] β) (f₂ : α →ₘ[μ] γ) : α →ₘ[μ] δ :=
comp _ hg (f₁.pair f₂)
@[simp] lemma comp₂_mk_mk {γ δ : Type*} [measurable_space γ] [measurable_space δ]
(g : β → γ → δ) (hg : measurable (uncurry g)) (f₁ : α → β) (f₂ : α → γ) (hf₁ hf₂) :
comp₂ g hg (mk f₁ hf₁ : α →ₘ[μ] β) (mk f₂ hf₂) =
mk (λa, g (f₁ a) (f₂ a)) (hg.comp_ae_measurable (hf₁.prod_mk hf₂)) :=
rfl
lemma comp₂_eq_pair {γ δ : Type*} [measurable_space γ] [measurable_space δ]
(g : β → γ → δ) (hg : measurable (uncurry g)) (f₁ : α →ₘ[μ] β) (f₂ : α →ₘ[μ] γ) :
comp₂ g hg f₁ f₂ = comp _ hg (f₁.pair f₂) :=
rfl
lemma comp₂_eq_mk {γ δ : Type*} [measurable_space γ] [measurable_space δ]
(g : β → γ → δ) (hg : measurable (uncurry g)) (f₁ : α →ₘ[μ] β) (f₂ : α →ₘ[μ] γ) :
comp₂ g hg f₁ f₂ = mk (λ a, g (f₁ a) (f₂ a))
(hg.comp_ae_measurable (f₁.ae_measurable.prod_mk f₂.ae_measurable)) :=
by rw [comp₂_eq_pair, pair_eq_mk, comp_mk]; refl
lemma coe_fn_comp₂ {γ δ : Type*} [measurable_space γ] [measurable_space δ]
(g : β → γ → δ) (hg : measurable (uncurry g)) (f₁ : α →ₘ[μ] β) (f₂ : α →ₘ[μ] γ) :
comp₂ g hg f₁ f₂ =ᵐ[μ] λ a, g (f₁ a) (f₂ a) :=
by { rw comp₂_eq_mk, apply coe_fn_mk }
/-- Interpret `f : α →ₘ[μ] β` as a germ at `μ.ae` forgetting that `f` is almost everywhere
measurable. -/
def to_germ (f : α →ₘ[μ] β) : germ μ.ae β :=
quotient.lift_on' f (λ f, ((f : α → β) : germ μ.ae β)) $ λ f g H, germ.coe_eq.2 H
@[simp] lemma mk_to_germ (f : α → β) (hf) : (mk f hf : α →ₘ[μ] β).to_germ = f := rfl
lemma to_germ_eq (f : α →ₘ[μ] β) : f.to_germ = (f : α → β) :=
by rw [← mk_to_germ, mk_coe_fn]
lemma to_germ_injective : injective (to_germ : (α →ₘ[μ] β) → germ μ.ae β) :=
λ f g H, ext $ germ.coe_eq.1 $ by rwa [← to_germ_eq, ← to_germ_eq]
lemma comp_to_germ (g : β → γ) (hg : measurable g) (f : α →ₘ[μ] β) :
(comp g hg f).to_germ = f.to_germ.map g :=
induction_on f $ λ f hf, by simp
lemma comp₂_to_germ (g : β → γ → δ) (hg : measurable (uncurry g))
(f₁ : α →ₘ[μ] β) (f₂ : α →ₘ[μ] γ) :
(comp₂ g hg f₁ f₂).to_germ = f₁.to_germ.map₂ g f₂.to_germ :=
induction_on₂ f₁ f₂ $ λ f₁ hf₁ f₂ hf₂, by simp
/-- Given a predicate `p` and an equivalence class `[f]`, return true if `p` holds of `f a`
for almost all `a` -/
def lift_pred (p : β → Prop) (f : α →ₘ[μ] β) : Prop := f.to_germ.lift_pred p
/-- Given a relation `r` and equivalence class `[f]` and `[g]`, return true if `r` holds of
`(f a, g a)` for almost all `a` -/
def lift_rel (r : β → γ → Prop) (f : α →ₘ[μ] β) (g : α →ₘ[μ] γ) : Prop :=
f.to_germ.lift_rel r g.to_germ
lemma lift_rel_mk_mk {r : β → γ → Prop} {f : α → β} {g : α → γ} {hf hg} :
lift_rel r (mk f hf : α →ₘ[μ] β) (mk g hg) ↔ ∀ᵐ a ∂μ, r (f a) (g a) :=
iff.rfl
lemma lift_rel_iff_coe_fn {r : β → γ → Prop} {f : α →ₘ[μ] β} {g : α →ₘ[μ] γ} :
lift_rel r f g ↔ ∀ᵐ a ∂μ, r (f a) (g a) :=
by rw [← lift_rel_mk_mk, mk_coe_fn, mk_coe_fn]
section order
instance [preorder β] : preorder (α →ₘ[μ] β) := preorder.lift to_germ
@[simp] lemma mk_le_mk [preorder β] {f g : α → β} (hf hg) :
(mk f hf : α →ₘ[μ] β) ≤ mk g hg ↔ f ≤ᵐ[μ] g :=
iff.rfl
@[simp, norm_cast] lemma coe_fn_le [preorder β] {f g : α →ₘ[μ] β} :
(f : α → β) ≤ᵐ[μ] g ↔ f ≤ g :=
lift_rel_iff_coe_fn.symm
instance [partial_order β] : partial_order (α →ₘ[μ] β) :=
partial_order.lift to_germ to_germ_injective
/- TODO: Prove `L⁰` space is a lattice if β is linear order.
What if β is only a lattice? -/
-- instance [linear_order β] : semilattice_sup (α →ₘ β) :=
-- { sup := comp₂ (⊔) (_),
-- .. ae_eq_fun.partial_order }
end order
variable (α)
/-- The equivalence class of a constant function: `[λa:α, b]`, based on the equivalence relation of
being almost everywhere equal -/
def const (b : β) : α →ₘ[μ] β := mk (λa:α, b) ae_measurable_const
lemma coe_fn_const (b : β) : (const α b : α →ₘ[μ] β) =ᵐ[μ] function.const α b :=
coe_fn_mk _ _
variable {α}
instance [inhabited β] : inhabited (α →ₘ[μ] β) := ⟨const α (default β)⟩
@[to_additive] instance [has_one β] : has_one (α →ₘ[μ] β) := ⟨const α 1⟩
@[to_additive] lemma one_def [has_one β] :
(1 : α →ₘ[μ] β) = mk (λa:α, 1) ae_measurable_const := rfl
@[to_additive] lemma coe_fn_one [has_one β] : ⇑(1 : α →ₘ[μ] β) =ᵐ[μ] 1 := coe_fn_const _ _
@[simp, to_additive] lemma one_to_germ [has_one β] : (1 : α →ₘ[μ] β).to_germ = 1 := rfl
section monoid
variables
[topological_space γ] [second_countable_topology γ] [borel_space γ]
[monoid γ] [has_continuous_mul γ]
@[to_additive]
instance : has_mul (α →ₘ[μ] γ) := ⟨comp₂ (*) measurable_mul⟩
@[simp, to_additive] lemma mk_mul_mk (f g : α → γ) (hf hg) :
(mk f hf : α →ₘ[μ] γ) * (mk g hg) = mk (f * g) (hf.mul hg) :=
rfl
@[to_additive] lemma coe_fn_mul (f g : α →ₘ[μ] γ) : ⇑(f * g) =ᵐ[μ] f * g := coe_fn_comp₂ _ _ _ _
@[simp, to_additive] lemma mul_to_germ (f g : α →ₘ[μ] γ) :
(f * g).to_germ = f.to_germ * g.to_germ :=
comp₂_to_germ _ _ _ _
@[to_additive]
instance : monoid (α →ₘ[μ] γ) :=
to_germ_injective.monoid to_germ one_to_germ mul_to_germ
end monoid
@[to_additive]
instance comm_monoid [topological_space γ] [second_countable_topology γ] [borel_space γ]
[comm_monoid γ] [has_continuous_mul γ] : comm_monoid (α →ₘ[μ] γ) :=
to_germ_injective.comm_monoid to_germ one_to_germ mul_to_germ
section group
variables [topological_space γ] [borel_space γ] [group γ] [topological_group γ]
@[to_additive] instance : has_inv (α →ₘ[μ] γ) := ⟨comp has_inv.inv measurable_inv⟩
@[simp, to_additive] lemma inv_mk (f : α → γ) (hf) : (mk f hf : α →ₘ[μ] γ)⁻¹ = mk f⁻¹ hf.inv := rfl
@[to_additive] lemma coe_fn_inv (f : α →ₘ[μ] γ) : ⇑(f⁻¹) =ᵐ[μ] f⁻¹ := coe_fn_comp _ _ _
@[to_additive] lemma inv_to_germ (f : α →ₘ[μ] γ) : (f⁻¹).to_germ = f.to_germ⁻¹ := comp_to_germ _ _ _
variables [second_countable_topology γ]
@[to_additive] instance : has_div (α →ₘ[μ] γ) := ⟨comp₂ has_div.div measurable_div⟩
@[to_additive, simp] lemma mk_div (f g : α → γ) (hf hg) :
mk (f / g) (ae_measurable.div hf hg) = (mk f hf : α →ₘ[μ] γ) / (mk g hg) :=
rfl
@[to_additive] lemma coe_fn_div (f g : α →ₘ[μ] γ) : ⇑(f / g) =ᵐ[μ] f / g := coe_fn_comp₂ _ _ _ _
@[to_additive] lemma div_to_germ (f g : α →ₘ[μ] γ) : (f / g).to_germ = f.to_germ / g.to_germ :=
comp₂_to_germ _ _ _ _
@[to_additive]
instance : group (α →ₘ[μ] γ) :=
to_germ_injective.group _ one_to_germ mul_to_germ inv_to_germ div_to_germ
end group
@[to_additive]
instance [topological_space γ] [borel_space γ] [comm_group γ] [topological_group γ]
[second_countable_topology γ] : comm_group (α →ₘ[μ] γ) :=
{ .. ae_eq_fun.group, .. ae_eq_fun.comm_monoid }
section semimodule
variables {𝕜 : Type*} [semiring 𝕜] [topological_space 𝕜] [measurable_space 𝕜]
[opens_measurable_space 𝕜]
variables [topological_space γ] [borel_space γ] [add_comm_monoid γ] [semimodule 𝕜 γ]
[has_continuous_smul 𝕜 γ]
instance : has_scalar 𝕜 (α →ₘ[μ] γ) :=
⟨λ c f, comp ((•) c) (measurable_id.const_smul c) f⟩
@[simp] lemma smul_mk (c : 𝕜) (f : α → γ) (hf) :
c • (mk f hf : α →ₘ[μ] γ) = mk (c • f) (hf.const_smul _) :=
rfl
lemma coe_fn_smul (c : 𝕜) (f : α →ₘ[μ] γ) : ⇑(c • f) =ᵐ[μ] c • f := coe_fn_comp _ _ _
lemma smul_to_germ (c : 𝕜) (f : α →ₘ[μ] γ) : (c • f).to_germ = c • f.to_germ :=
comp_to_germ _ _ _
variables [second_countable_topology γ] [has_continuous_add γ]
instance : semimodule 𝕜 (α →ₘ[μ] γ) :=
to_germ_injective.semimodule 𝕜 ⟨@to_germ α γ _ μ _, zero_to_germ, add_to_germ⟩ smul_to_germ
end semimodule
open ennreal
/-- For `f : α → ℝ≥0∞`, define `∫ [f]` to be `∫ f` -/
def lintegral (f : α →ₘ[μ] ℝ≥0∞) : ℝ≥0∞ :=
quotient.lift_on' f (λf, ∫⁻ a, (f : α → ℝ≥0∞) a ∂μ) (assume f g, lintegral_congr_ae)
@[simp] lemma lintegral_mk (f : α → ℝ≥0∞) (hf) :
(mk f hf : α →ₘ[μ] ℝ≥0∞).lintegral = ∫⁻ a, f a ∂μ := rfl
lemma lintegral_coe_fn (f : α →ₘ[μ] ℝ≥0∞) : ∫⁻ a, f a ∂μ = f.lintegral :=
by rw [← lintegral_mk, mk_coe_fn]
@[simp] lemma lintegral_zero : lintegral (0 : α →ₘ[μ] ℝ≥0∞) = 0 := lintegral_zero
@[simp] lemma lintegral_eq_zero_iff {f : α →ₘ[μ] ℝ≥0∞} : lintegral f = 0 ↔ f = 0 :=
induction_on f $ λ f hf, (lintegral_eq_zero_iff' hf).trans mk_eq_mk.symm
lemma lintegral_add (f g : α →ₘ[μ] ℝ≥0∞) : lintegral (f + g) = lintegral f + lintegral g :=
induction_on₂ f g $ λ f hf g hg, by simp [lintegral_add' hf hg]
lemma lintegral_mono {f g : α →ₘ[μ] ℝ≥0∞} : f ≤ g → lintegral f ≤ lintegral g :=
induction_on₂ f g $ λ f hf g hg hfg, lintegral_mono_ae hfg
section pos_part
variables [topological_space γ] [linear_order γ] [order_closed_topology γ]
[second_countable_topology γ] [has_zero γ] [opens_measurable_space γ]
/-- Positive part of an `ae_eq_fun`. -/
def pos_part (f : α →ₘ[μ] γ) : α →ₘ[μ] γ :=
comp (λ x, max x 0) (measurable_id.max measurable_const) f
@[simp] lemma pos_part_mk (f : α → γ) (hf) :
pos_part (mk f hf : α →ₘ[μ] γ) = mk (λ x, max (f x) 0) (hf.max ae_measurable_const) :=
rfl
lemma coe_fn_pos_part (f : α →ₘ[μ] γ) : ⇑(pos_part f) =ᵐ[μ] (λ a, max (f a) 0) :=
coe_fn_comp _ _ _
end pos_part
end ae_eq_fun
end measure_theory
namespace continuous_map
open measure_theory
variables [topological_space α] [borel_space α] (μ)
variables [topological_space β] [measurable_space β] [borel_space β]
/-- The equivalence class of `μ`-almost-everywhere measurable functions associated to a continuous
map. -/
def to_ae_eq_fun (f : C(α, β)) : α →ₘ[μ] β :=
ae_eq_fun.mk f f.continuous.measurable.ae_measurable
lemma coe_fn_to_ae_eq_fun (f : C(α, β)) : f.to_ae_eq_fun μ =ᵐ[μ] f :=
ae_eq_fun.coe_fn_mk f _
variables [group β] [topological_group β] [second_countable_topology β]
/-- The `mul_hom` from the group of continuous maps from `α` to `β` to the group of equivalence
classes of `μ`-almost-everywhere measurable functions. -/
@[to_additive "The `add_hom` from the group of continuous maps from `α` to `β` to the group of
equivalence classes of `μ`-almost-everywhere measurable functions."]
def to_ae_eq_fun_mul_hom : C(α, β) →* α →ₘ[μ] β :=
{ to_fun := continuous_map.to_ae_eq_fun μ,
map_one' := rfl,
map_mul' := λ f g, ae_eq_fun.mk_mul_mk f g f.continuous.measurable.ae_measurable
g.continuous.measurable.ae_measurable }
variables {𝕜 : Type*} [semiring 𝕜] [topological_space 𝕜] [measurable_space 𝕜]
[opens_measurable_space 𝕜]
variables [topological_space γ] [measurable_space γ] [borel_space γ] [add_comm_group γ]
[semimodule 𝕜 γ] [topological_add_group γ] [has_continuous_smul 𝕜 γ]
[second_countable_topology γ]
/-- The linear map from the group of continuous maps from `α` to `β` to the group of equivalence
classes of `μ`-almost-everywhere measurable functions. -/
def to_ae_eq_fun_linear_map : C(α, γ) →ₗ[𝕜] α →ₘ[μ] γ :=
{ map_smul' := λ c f, ae_eq_fun.smul_mk c f f.continuous.measurable.ae_measurable,
.. to_ae_eq_fun_add_hom μ }
end continuous_map
|
864f4137a67995061899c4930d69065b28d33aac | 59b654f4ee2fef898a3487dc03554a569051b63a | /src/old/generic_cube_graph.lean | 72058cd72d17dfccccadd2b4cb423ea665f4b672 | [] | no_license | gunpinyo/twisted_cube_formalisation | 180c9157478b66ec2b11ca47c8ff998a3e978a88 | f78206ac495e84bd43a9b820fa10b6c94722e0ec | refs/heads/master | 1,624,501,222,992 | 1,607,081,624,000 | 1,607,081,624,000 | 166,885,106 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 19,707 | lean | import graph
section
variable tw : bool
variable G : graph
def prism_graph.nodes := bool × G.nodes
def prism_graph.edges := (bool × G.edges) ⊕ G.nodes
def prism_graph.srctrg (b : bool) : prism_graph.edges G → prism_graph.nodes G
| (sum.inl (b', e)) := (b', G.srctrg (bxor (band b' tw) b) e)
| (sum.inr v) := (b, v)
lemma prism_graph.edge_ext :
∀ (tw : bool) (G : graph) (e e' : prism_graph.edges G),
(∀ b, prism_graph.srctrg tw G b e = prism_graph.srctrg tw G b e') →
e = e'
| tw G (sum.inl (ff, e)) (sum.inl (ff, e')) p :=
begin congr, apply G.edge_ext e e',
cases tw, all_goals { simp [prism_graph.srctrg] at p, assumption,},
end
| tw G (sum.inl (ff, e)) (sum.inl (tt, e')) p :=
begin simp [prism_graph.srctrg] at p, contradiction, end
| tw G (sum.inl (tt, e)) (sum.inl (ff, e')) p :=
begin simp [prism_graph.srctrg] at p, contradiction, end
| tw G (sum.inl (tt, e)) (sum.inl (tt, e')) p :=
begin
congr, apply G.edge_ext e e',
cases tw, { simp [prism_graph.srctrg] at p, assumption,},
simp [prism_graph.srctrg] at p,
intro b, cases b, exact p tt, exact p ff,
end
| tw G (sum.inl (b, e)) (sum.inr v') p :=
begin let pnb := p (not b), cases b,
all_goals { simp [prism_graph.srctrg] at pnb, contradiction}, end
| tw G (sum.inr v) (sum.inl (b', e')) p :=
begin let pnb := p (not b'), cases b',
all_goals { simp [prism_graph.srctrg] at pnb, contradiction}, end
| tw G (sum.inr v) (sum.inr v') p :=
begin simp [prism_graph.srctrg] at p, congr, assumption, end
def prism_graph : graph :=
{ nodes := prism_graph.nodes G
, edges := prism_graph.edges G
, srctrg := prism_graph.srctrg tw G
, edge_ext := prism_graph.edge_ext tw G
}
end
def cube_graph (tw : bool) : ℕ → graph
| 0 := graph.singleton_reflexive
| (n +1) := prism_graph tw (cube_graph n)
section
variable tw : bool
variable n : ℕ
def cube_graph_alt.nodes := bitvec n
def cube_graph_alt.edges := bitvec n ⊕ (fin n × bitvec (nat.pred n))
def cube_graph_alt.num_zeros_is_odd :
Π {n : ℕ} (i : fin (n +1)) (v : bitvec n), bool
| 0 _ _ := ff
| (n +1) i v := fin.maybe_pred_rec ff (λ i',
bxor (v.head) (cube_graph_alt.num_zeros_is_odd i' v.tail)) i
def cube_graph_alt.srctrg (b : bool) :
cube_graph_alt.edges n → cube_graph_alt.nodes n
| (sum.inl v) := v
| (sum.inr (i, v)) := match n, i, v with
| 0, i, v := i.elim0
| (_ +1), i, v :=
let tw_bit := cube_graph_alt.num_zeros_is_odd i v
in vector.insert_nth (bxor (band tw tw_bit) b) i v
end
private lemma cube_graph_alt.edge_ext_helper :
∀ (n : ℕ) (i : fin (n +1)) (v : bitvec n) (v' : bitvec (n +1)),
(∀ b, v.insert_nth b i = v') → false
| 0 i v v' p :=
begin
replace p := p (bnot v'.head),
cases i, cases i_val, swap,
{ exact nat.not_succ_le_zero i_val (nat.le_of_lt_succ i_is_lt),},
rw [fin.to_zero, vector.insert_nth_zero] at p,
replace p := (vector.cons_injection p).1, simp at p,
exact bnot_self p,
end
| (n +1) ⟨0, pz⟩ v v' p :=
begin
replace p := p (bnot v'.head),
simp [fin.to_zero, vector.insert_nth_zero] at p,
replace p := (vector.cons_injection p).1, simp at p,
exact bnot_self p,
end
| (n +1) ⟨i +1, psi⟩ v v' p :=
begin
let fin_i := fin.mk_from_succ i psi,
refine cube_graph_alt.edge_ext_helper n fin_i v.tail v'.tail _,
intro b, replace p := congr_arg vector.tail (p b),
rwa [fin.to_succ, vector.insert_nth_succ] at p,
end
lemma cube_graph_alt.edge_ext :
∀ (tw : bool) (n : ℕ) (e e' : cube_graph_alt.edges n),
(∀ b, cube_graph_alt.srctrg tw n b e
= cube_graph_alt.srctrg tw n b e') →
e = e'
| tw n (sum.inl v) (sum.inl v') p :=
begin simp [cube_graph_alt.srctrg] at p, congr, assumption, end
| tw (n +1) (sum.inl v) (sum.inr (i', v')) p :=
begin
exfalso, simp [cube_graph_alt.srctrg] at p,
refine cube_graph_alt.edge_ext_helper n i' v' v _, intro b, symmetry,
replace p := p (bxor(band tw (cube_graph_alt.num_zeros_is_odd i' v'))b),
rwa [bxor_comm, bxor_bxor_id] at p,
end
| tw (n +1) (sum.inr (i, v)) (sum.inl v') p :=
begin
exfalso, simp [cube_graph_alt.srctrg] at p,
refine cube_graph_alt.edge_ext_helper n i v v' _, intro b,
replace p := p (bxor (band tw (cube_graph_alt.num_zeros_is_odd i v)) b),
rwa [bxor_comm, bxor_bxor_id] at p,
end
| tw (n +1) (sum.inr (⟨0, pi⟩, v)) (sum.inr (⟨0, pi'⟩, v')) p :=
begin
simp [cube_graph_alt.srctrg, vector.insert_nth, list.insert_nth] at p,
congr, exact subtype.eq (p tt).2
end
| tw 1 (sum.inr (⟨0, pz⟩, v)) (sum.inr (⟨i' +1, psi'⟩, v')) p :=
begin exfalso, exact nat.not_lt_zero i' (nat.pred_le_pred psi') end
| tw 1 (sum.inr (⟨i +1, psi⟩, v)) (sum.inr (⟨0, pz'⟩, v')) p :=
begin exfalso, exact nat.not_lt_zero i (nat.pred_le_pred psi) end
| tw 1 (sum.inr (⟨i +1, psi⟩, v)) (sum.inr (⟨i' +1, psi'⟩, v')) p :=
begin exfalso, exact nat.not_lt_zero i (nat.pred_le_pred psi) end
| tw (n +2) (sum.inr (⟨0, pz⟩, v)) (sum.inr (⟨i' +1, psi'⟩, v')) p :=
begin
exfalso, replace p := p (bnot v'.head),
simp [cube_graph_alt.srctrg, fin.to_succ, fin.to_zero] at p,
rw [vector.insert_nth_succ, vector.insert_nth_zero] at p,
simp [vector.insert_nth, list.insert_nth] at p,
replace p := (vector.cons_injection p).1,
simp [fin.zero, cube_graph_alt.num_zeros_is_odd] at p,
exact bnot_self p,
end
| tw (n +2) (sum.inr (⟨i +1, psi⟩, v)) (sum.inr (⟨0, pz'⟩, v')) p :=
begin
exfalso, replace p := p (bnot v.head),
simp [cube_graph_alt.srctrg, fin.to_succ, fin.to_zero] at p,
rw [vector.insert_nth_succ, vector.insert_nth_zero] at p,
simp [vector.insert_nth, list.insert_nth] at p,
replace p := (vector.cons_injection p).1,
simp [fin.zero, cube_graph_alt.num_zeros_is_odd] at p,
exact bnot_self (eq.symm p),
end
| tw (n +2) (sum.inr (⟨i +1, psi⟩, v)) (sum.inr (⟨i' +1, psi'⟩, v')) p :=
begin
let fin_i := fin.mk_from_succ i psi,
let fin_i' := fin.mk_from_succ i' psi',
simp [fin.to_succ] at p,
suffices : ∀ b, v.head = v'.head ∧
cube_graph_alt.srctrg tw (n + 1) b (sum.inr (fin_i, v.tail)) =
cube_graph_alt.srctrg tw (n + 1) b (sum.inr (fin_i', v'.tail)),
{ let e_eq := cube_graph_alt.edge_ext tw (n + 1)
(sum.inr (fin_i, v.tail)) (sum.inr (fin_i', v'.tail))
(λ b, (this b).2),
injection e_eq with e_eq',
injection e_eq' with i_i'_fin_eq v_v'_tail_eq,
congr, {injection i_i'_fin_eq,},
rw [←v.cons_head_tail, ←v'.cons_head_tail],
rw [(this tt).1, v_v'_tail_eq],},
suffices : ∀ (fin_i : fin (n +1)) (v : bitvec (n +1)) (b : bool),
cube_graph_alt.srctrg tw (n + 2) b
(sum.inr (fin_i.succ, v)) =
v.head :: cube_graph_alt.srctrg tw (n + 1) (bxor (band tw v.head) b)
(sum.inr (fin_i, v.tail)),
{ intro b,
replace p := p (bxor (band tw v.head) b),
rw [this, this] at p,
let p' := vector.cons_injection p, simp at p',
rw [←p'.1, bxor_bxor_id] at p, rw [←p'.1] at |-,
exact vector.cons_injection p,},
clear p fin_i fin_i' psi psi' i i' v v',
intros fin_i v b,
simp [cube_graph_alt.srctrg],
rw ←vector.insert_nth_succ,
cases tw, {simp,}, simp,
suffices : cube_graph_alt.num_zeros_is_odd fin_i.succ v
= bxor (vector.head v)
(cube_graph_alt.num_zeros_is_odd fin_i (vector.tail v)),
{rw this,},
rw ←vector.cons_head_tail v,
cases fin_i with i pi,
simp [fin.succ, cube_graph_alt.num_zeros_is_odd],
refl,
end
def cube_graph_alt : graph :=
{ nodes := cube_graph_alt.nodes n
, edges := cube_graph_alt.edges n
, srctrg := cube_graph_alt.srctrg tw n
, edge_ext := cube_graph_alt.edge_ext tw n
}
end
namespace cg_to_cg'
section
variable tw : bool
def nodes_map : Π (n : ℕ), (cube_graph tw n).nodes
→ (cube_graph_alt tw n).nodes
| 0 _ := vector.nil
| (n +1) (b, v) := b :: (nodes_map n v)
def edges_map : Π (n : ℕ), (cube_graph tw n).edges
→ (cube_graph_alt tw n).edges
| 0 _ := sum.inl vector.nil
| (n +1) (sum.inl (b, e)) :=
match edges_map n e with
| sum.inl v := sum.inl (b :: v)
| sum.inr (i, v) := match n, i, v with
| 0, i, v := i.elim0
| (_ +1), i, v := sum.inr (i.succ, b :: v)
end
end
| (n +1) (sum.inr v) := sum.inr (fin.zero, nodes_map tw n v)
lemma srctrg_map (n : ℕ) (b : bool) (e : (cube_graph tw n).edges)
: nodes_map tw n ((cube_graph tw n).srctrg b e)
= (cube_graph_alt tw n).srctrg b (edges_map tw n e) :=
begin
revert b, induction n with n IH, { intro, refl,}, intro b,
cases e with be v,
{ cases be with b' e,
dsimp [cube_graph, prism_graph, prism_graph.srctrg,
nodes_map, edges_map],
rw [IH], dsimp [cube_graph_alt],
cases edges_map tw n e with v iv,
{ dsimp [edges_map._match_1, cube_graph_alt.srctrg], refl,},
{ cases n, { cases iv with i, exact i.elim0,},
cases iv with i v,
dsimp [edges_map._match_1, edges_map._match_2, cube_graph_alt.srctrg],
rw vector.insert_nth_succ,
cases tw, {simp,}, simp,
suffices : cube_graph_alt.num_zeros_is_odd i.succ (b' :: v)
= bxor b' (cube_graph_alt.num_zeros_is_odd i v), rw this,
cases i with i_val i_p, cases v with v_l v_p,
simp [fin.succ, vector.cons, cube_graph_alt.num_zeros_is_odd],
refl,},},
{ transitivity (b :: nodes_map tw n v), refl,
have : (edges_map tw (n +1) (sum.inr v)
: (cube_graph_alt tw (n +1)).edges)
= (sum.inr (fin.zero, nodes_map tw n v)), refl, rw this,
transitivity cube_graph_alt.srctrg tw (n +1) b
(sum.inr (fin.zero, nodes_map tw n v)),
swap, {refl,},
simp [cube_graph_alt.srctrg], rw [vector.insert_nth_zero], congr,
induction n with n IH, {simp [cube_graph_alt.num_zeros_is_odd],},
simp [fin.zero, cube_graph_alt.num_zeros_is_odd],},
end
end
end cg_to_cg'
def cg_to_cg' (tw : bool) (n : ℕ)
: graph_cat.hom (cube_graph tw n) (cube_graph_alt tw n) :=
{ nodes_map := cg_to_cg'.nodes_map tw n
, edges_map := cg_to_cg'.edges_map tw n
, srctrg_map := cg_to_cg'.srctrg_map tw n
}
namespace cg'_to_cg
section
variable tw : bool
def nodes_map : Π (n : ℕ), (cube_graph_alt tw n).nodes
→ (cube_graph tw n).nodes
| 0 _ := unit.star
| (n +1) v := (v.head, nodes_map n v.tail)
def edges_map : Π (n : ℕ), (cube_graph_alt tw n).edges
→ (cube_graph tw n).edges
| 0 _ := unit.star
| (n +1) (sum.inl v) := sum.inl (v.head, edges_map n (sum.inl v.tail))
| (n +1) (sum.inr (i, v)) := fin.maybe_pred_rec (sum.inr (nodes_map tw n v))
(λ i' : fin n, sum.inl (match n, i', v with
| 0, i', _ := i'.elim0
| (_ +1), _, v := v.head
end, edges_map n (sum.inr (i', v.tail)))) i
lemma srctrg_map : ∀ (n : ℕ) (b : bool) (e : (cube_graph_alt tw n).edges),
nodes_map tw n ((cube_graph_alt tw n).srctrg b e) =
(cube_graph tw n).srctrg b (edges_map tw n e)
| 0 b e := rfl
| (n +1) b (sum.inl v) :=
begin
dsimp [cube_graph_alt, cube_graph_alt.srctrg],
cases v with b'l p, cases b'l with b' l, {simp at p, contradiction,},
simp [nodes_map, edges_map, vector.head, vector.tail],
dunfold cube_graph, dunfold prism_graph, simp [prism_graph.srctrg],
rw ←srctrg_map, congr,
end
| 1 b (sum.inr (i, v)) :=
begin
cases i with i_val i_is_lt, cases i_val with i'_val,
swap, { exfalso,
exact nat.not_succ_le_zero i'_val (nat.le_of_lt_succ i_is_lt),},
dsimp [cube_graph_alt, cube_graph_alt.srctrg],
rw [vector.eq_nil v, fin.to_zero, vector.insert_nth_zero],
simp [nodes_map, fin.zero, cube_graph_alt.num_zeros_is_odd],
simp [edges_map, fin.zero],
dsimp [cube_graph, prism_graph, prism_graph.srctrg, nodes_map], refl,
end
| (n +2) b (sum.inr (⟨0, pz⟩, v)) :=
begin
dsimp [cube_graph_alt, cube_graph_alt.srctrg],
rw [fin.to_zero, vector.insert_nth_zero],
simp [nodes_map, fin.zero, cube_graph_alt.num_zeros_is_odd],
unfold1 edges_map, simp [fin.zero, edges_map._match_1],
dsimp [cube_graph, prism_graph, prism_graph.srctrg, nodes_map], refl,
end
| (n +2) b (sum.inr (⟨i +1, psi⟩, v)) :=
begin
dsimp [cube_graph_alt, cube_graph_alt.srctrg],
rw [fin.to_succ, vector.insert_nth_succ],
unfold1 nodes_map, simp [vector.head, fin.succ],
simp [cube_graph_alt.num_zeros_is_odd], symmetry,
unfold1 edges_map, simp [edges_map._match_1],
dsimp [cube_graph, prism_graph, prism_graph.srctrg], congr,
transitivity (cube_graph tw (n +1)).srctrg (bxor (vector.head v && tw) b)
(edges_map tw (n + 1) (sum.inr (⟨i, _⟩, vector.tail v))), refl,
rw ←srctrg_map, dsimp [cube_graph_alt, cube_graph_alt.srctrg], simp,
have : ∀ a b c, bxor (band a b) (band a c) = band a (bxor b c),
{ intros a b c, cases a, refl, simp,}, rw this,
end
end
end cg'_to_cg
def cg'_to_cg (tw : bool) (n : ℕ)
: graph_cat.hom (cube_graph_alt tw n) (cube_graph tw n) :=
{ nodes_map := cg'_to_cg.nodes_map tw n
, edges_map := cg'_to_cg.edges_map tw n
, srctrg_map := cg'_to_cg.srctrg_map tw n
}
section
variable tw : bool
lemma cg_cg'_cg_eq_id.nodes_map_eq :
∀ (n : ℕ) (v : (cube_graph tw n).nodes),
cg'_to_cg.nodes_map tw n (cg_to_cg'.nodes_map tw n v) = v
| 0 v := begin cases v, refl end
| (n +1) (b, v) := begin simp [cg_to_cg'.nodes_map, cg'_to_cg.nodes_map],
exact cg_cg'_cg_eq_id.nodes_map_eq n v end
lemma cg'_cg_cg'_eq_id.nodes_map_eq :
∀ (n : ℕ)(v : (cube_graph_alt tw n).nodes),
cg_to_cg'.nodes_map tw n (cg'_to_cg.nodes_map tw n v) = v
| 0 v := begin change vector bool 0 at v, rw vector.eq_nil v,
simp [cg'_to_cg.nodes_map, cg_to_cg'.nodes_map], end
| (n +1) v := begin simp [cg'_to_cg.nodes_map, cg_to_cg'.nodes_map],
rw cg'_cg_cg'_eq_id.nodes_map_eq, simp, end
lemma cg_cg'_cg_eq_id.edges_map_eq : ∀ (n : ℕ) (e : (cube_graph tw n).edges),
cg'_to_cg.edges_map tw n (cg_to_cg'.edges_map tw n e) = e
| 0 e := begin cases e, refl end
| (n +1) (sum.inl (b, e)) :=
begin
transitivity sum.inl (b, cg'_to_cg.edges_map tw n
(cg_to_cg'.edges_map tw n e)), swap,
{ congr, exact cg_cg'_cg_eq_id.edges_map_eq n e,},
set e' := cg_to_cg'.edges_map tw n e with ←e'_prop,
unfold cg_to_cg'.edges_map,
rcases cg_to_cg'.edges_map tw n e with ⟨v⟩ | ⟨i, v⟩,
{ intro e'_prop,
unfold cg_to_cg'.edges_map._match_1, unfold cg'_to_cg.edges_map,
rw [vector.head_cons, vector.tail_cons], congr, assumption,},
intro e'_prop, unfold cg_to_cg'.edges_map._match_1,
cases n with n, {exact i.elim0,}, unfold cg_to_cg'.edges_map._match_2,
cases i, unfold fin.succ, unfold1 cg'_to_cg.edges_map,
unfold fin.maybe_pred_rec,
congr, { unfold cg'_to_cg.edges_map._match_1, simp,}, simpa,
end
| (n +1) (sum.inr v) :=
begin
simp [cg_to_cg'.edges_map, cg'_to_cg.edges_map, fin.zero],
exact cg_cg'_cg_eq_id.nodes_map_eq tw n v,
end
lemma cg'_cg_cg'_eq_id.edges_map_eq (n : ℕ) (e : (cube_graph_alt tw n).edges) :
cg_to_cg'.edges_map tw n (cg'_to_cg.edges_map tw n e) = e :=
begin
induction n with n IH,
{ dsimp [cg'_to_cg.edges_map, cg_to_cg'.edges_map],
rcases e with ⟨⟩ | ⟨i, v⟩, {congr,}, exact i.elim0,},
rcases e with ⟨v⟩ | ⟨i, v⟩,
{ rcases v with ⟨bl, p⟩, rcases bl with ⟨⟩ | ⟨b, l⟩, {contradiction,},
dsimp [cg'_to_cg.edges_map, vector.head, vector.tail,
cg_to_cg'.edges_map],
rw IH, unfold cg_to_cg'.edges_map._match_1, refl,},
cases i, rcases i_val with ⟨⟩ | ⟨i'_val⟩,
{ dsimp [cg'_to_cg.edges_map, fin.maybe_pred_rec, cg_to_cg'.edges_map],
congr, apply cg'_cg_cg'_eq_id.nodes_map_eq,},
cases n,
{ exfalso, exact nat.not_succ_le_zero i'_val (nat.le_of_lt_succ i_is_lt)},
rcases v with ⟨bl, p⟩, rcases bl with ⟨⟩ | ⟨b, l⟩, {contradiction,},
unfold1 cg'_to_cg.edges_map, unfold1 cg'_to_cg.edges_map._match_1,
unfold1 fin.maybe_pred_rec,
simp [vector.head, vector.tail, cg_to_cg'.edges_map], rw IH,
simp [cg_to_cg'.edges_map._match_1, cg_to_cg'.edges_map._match_2],
split, {refl,}, refl,
end
lemma cg_cg'_cg_eq_id (n : ℕ) :
graph_cat.comp (cg_to_cg' tw n) (cg'_to_cg tw n) =
graph_cat.id (cube_graph tw n) :=
graph_cat.hom.eq (funext (cg_cg'_cg_eq_id.nodes_map_eq tw n))
(funext (cg_cg'_cg_eq_id.edges_map_eq tw n))
lemma cg'_cg_cg'_eq_id (n : ℕ) :
graph_cat.comp (cg'_to_cg tw n) (cg_to_cg' tw n) =
graph_cat.id (cube_graph_alt tw n) :=
graph_cat.hom.eq (funext (cg'_cg_cg'_eq_id.nodes_map_eq tw n))
(funext (cg'_cg_cg'_eq_id.edges_map_eq tw n))
def cg_iso_cg' (n : ℕ) :
graph_cat.iso (cube_graph tw n) (cube_graph_alt tw n) :=
{ dir := cg_to_cg' tw n
, inv := cg'_to_cg tw n
, dir_inv := cg_cg'_cg_eq_id tw n
, inv_dir := cg'_cg_cg'_eq_id tw n
}
def cube_graph_cat : category :=
{ obj := ℕ
, hom := λ m n, graph_cat.hom (cube_graph tw m) (cube_graph tw n)
, id := λ n, graph_cat.id (cube_graph tw n)
, comp := λ {n n' n''}, @graph_cat.comp
(cube_graph tw n) (cube_graph tw n') (cube_graph tw n'')
, id_l := λ {m n}, @graph_cat.id_l (cube_graph tw m) (cube_graph tw n)
, id_r := λ {m n}, @graph_cat.id_r (cube_graph tw m) (cube_graph tw n)
, assoc := λ {n n' n'' n'''}, @graph_cat.assoc (cube_graph tw n)
(cube_graph tw n') (cube_graph tw n'') (cube_graph tw n''')
}
end
|
9ed5d7f0a5fab8e519a7e73a6a7d3875a68ace58 | 94637389e03c919023691dcd05bd4411b1034aa5 | /src/inClassNotes/final/imp.lean | caeef0e3215083e71f7d127a5e678132120f8441 | [] | no_license | kevinsullivan/complogic-s21 | 7c4eef2105abad899e46502270d9829d913e8afc | 99039501b770248c8ceb39890be5dfe129dc1082 | refs/heads/master | 1,682,985,669,944 | 1,621,126,241,000 | 1,621,126,241,000 | 335,706,272 | 0 | 38 | null | 1,618,325,669,000 | 1,612,374,118,000 | Lean | UTF-8 | Lean | false | false | 3,005 | lean | import .arith_expr
import .bool_expr
/-
Syntax of our little language (OLL).
OLL supports mutable variables and
values of types arithmetic (bool) and
arithmetic (nat). It has an assignment
command for each of these two types.
It also supports sequential composition
of smaller programs into larger ones.
-/
inductive cmd : Type
| skip
| a_assn (v : var nat) (e : arith_expr)
| b_assn (v : var bool) (e : bool_expr)
| seq (c1 c2 : cmd) : cmd
| ifelse (b : bool_expr) (c1 c2 : cmd)
| while (b : bool_expr) (c : cmd)
open cmd
notation v = e := b_assn v e
notation v = a := a_assn v a
notation c1 ; c2 := seq c1 c2
notation `IF ` b ` THEN ` c1 ` ELSE ` c2 := ifelse b c1 c2
notation `WHILE ` b ` DO ` c ` END`:= while b c
/-
Computational semantics
-/
meta def c_eval : cmd → env → env
| skip st := st
| (b_assn v e) st := override_bool st v e
| (a_assn v e) st := override_nat st v e
| (c1 ; c2) st := c_eval c2 (c_eval c1 st)
| (IF b THEN c1 ELSE c2) st :=
if (bool_eval b st)
then c_eval c1 st
else c_eval c2 st
| (WHILE b DO c END) st :=
if (bool_eval b st)
then c_eval (c; WHILE b DO c END) st
else st
/-
-- run c; repeat loop.
-- Unroll loop by one run of c
-- Running c can but needn't make b false
-- side effects via big shared memory (BSM)
-/
/-
Logical semantics
The rules are the axioms you can then
use to prove c_sem relations.
-/
inductive c_sem : cmd → env → env → Prop
-- c_sem skip pre pre
| c_sem_skip : ∀ (st : env),
c_sem skip st st
-- c_sem (v =a e) pre post -- arith assignment
| c_sem_arith_assn :
∀ (pre post : env) (v : var nat) (e : arith_expr),
(override_nat pre v e = post) →
c_sem (a_assn v e) pre post
-- c_sem (v =b e) pre post -- bool assignment
| c_sem_bool_assn :
∀ (pre post : env) (v : var bool) (e : bool_expr),
(override_bool pre v e = post) →
c_sem (b_assn v e) pre post
-- c_sem (c1 ; c2) pre post
| c_sem_seq :
∀ (pre is post : env) (c1 c2 : cmd),
c_sem c1 pre is →
c_sem c2 is post →
c_sem (c1 ; c2) pre post
-- c_sem if false c1 c2
| c_sem_if_false :
∀ (pre is post : env) (b : bool_expr) (c1 c2 : cmd),
bool_eval b pre = ff →
c_sem c2 pre post →
c_sem (IF b THEN c1 ELSE c2) pre post
-- c_sem if true c1 c2
| c_sem_if_true :
∀ (pre is post : env)
(b : bool_expr)
(c1 c2 : cmd),
bool_eval b pre = tt →
c_sem c1 pre post →
c_sem (IF b THEN c1 ELSE c2) pre post
-- c_sem (while false do c) pre post
| c_sem_while_false :
∀ (pre : env)
(b : bool_expr)
(c : cmd),
bool_eval b pre = ff →
c_sem (WHILE b DO c END) pre pre
-- c_sem (while true do c) pre post
| c_sem_while_true :
∀ (pre is post : env)
(b : bool_expr)
(c : cmd),
bool_eval b pre = tt →
c_sem c pre is → -- one iteration of c: pre->is
c_sem (WHILE b DO c END) is post →
c_sem (WHILE b DO c END) pre post |
919c70ca6a77e2a2eb7d02c59210c1b948e41034 | 55c7fc2bf55d496ace18cd6f3376e12bb14c8cc5 | /src/ring_theory/valuation/basic.lean | c2e61a27fa121806a7dd9b0a568babe8cb72c60b | [
"Apache-2.0"
] | permissive | dupuisf/mathlib | 62de4ec6544bf3b79086afd27b6529acfaf2c1bb | 8582b06b0a5d06c33ee07d0bdf7c646cae22cf36 | refs/heads/master | 1,669,494,854,016 | 1,595,692,409,000 | 1,595,692,409,000 | 272,046,630 | 0 | 0 | Apache-2.0 | 1,592,066,143,000 | 1,592,066,142,000 | null | UTF-8 | Lean | false | false | 14,493 | lean | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Patrick Massot
-/
import algebra.linear_ordered_comm_group_with_zero
import algebra.group_power
import ring_theory.ideal_operations
import ring_theory.subring
import algebra.punit_instances
/-!
# The basics of valuation theory.
The basic theory of valuations (non-archimedean norms) on a commutative ring,
following T. Wedhorn's unpublished notes “Adic Spaces” ([wedhorn_adic]).
The definition of a valuation we use here is Definition 1.22 of [wedhorn_adic].
A valuation on a ring `R` is a monoid homomorphism `v` to a linearly ordered
commutative group with zero, that in addition satisfies the following two axioms:
* `v 0 = 0`
* `∀ x y, v (x + y) ≤ max (v x) (v y)`
`valuation R Γ₀`is the type of valuations `R → Γ₀`, with a coercion to the underlying
function. If `v` is a valuation from `R` to `Γ₀` then the induced group
homomorphism `units(R) → Γ₀` is called `unit_map v`.
The equivalence "relation" `is_equiv v₁ v₂ : Prop` defined in 1.27 of [wedhorn_adic] is not strictly
speaking a relation, because `v₁ : valuation R Γ₁` and `v₂ : valuation R Γ₂` might
not have the same type. This corresponds in ZFC to the set-theoretic difficulty
that the class of all valuations (as `Γ₀` varies) on a ring `R` is not a set.
The "relation" is however reflexive, symmetric and transitive in the obvious
sense. Note that we use 1.27(iii) of [wedhorn_adic] as the definition of equivalence.
The support of a valuation `v : valuation R Γ₀` is `supp v`. If `J` is an ideal of `R`
with `h : J ⊆ supp v` then the induced valuation
on R / J = `ideal.quotient J` is `on_quot v h`.
## Main definitions
* `valuation R Γ₀`, the type of valuations on `R` with values in `Γ₀`
* `valuation.is_equiv`, the heterogeneous equivalence relation on valuations
* `valuation.supp`, the support of a valuation
-/
open_locale classical
noncomputable theory
local attribute [instance, priority 0] classical.DLO
open function ideal
-- universes u u₀ u₁ u₂ -- v is used for valuations
variables {R : Type*} -- This will be a ring, assumed commutative in some sections
variables {Γ₀ : Type*} [linear_ordered_comm_group_with_zero Γ₀]
variables {Γ'₀ : Type*} [linear_ordered_comm_group_with_zero Γ'₀]
variables {Γ''₀ : Type*} [linear_ordered_comm_group_with_zero Γ''₀]
set_option old_structure_cmd true
section
variables (R) (Γ₀) [ring R]
/-- The type of Γ₀-valued valuations on R. -/
@[nolint has_inhabited_instance]
structure valuation extends R →* Γ₀ :=
(map_zero' : to_fun 0 = 0)
(map_add' : ∀ x y, to_fun (x + y) ≤ max (to_fun x) (to_fun y))
run_cmd tactic.add_doc_string `valuation.to_monoid_hom "The `monoid_hom` underlying a valuation."
end
namespace valuation
section basic
variables (R) (Γ₀) [ring R]
/-- A valuation is coerced to the underlying function R → Γ₀. -/
instance : has_coe_to_fun (valuation R Γ₀) := { F := λ _, R → Γ₀, coe := valuation.to_fun }
/-- A valuation is coerced to a monoid morphism R → Γ₀. -/
instance : has_coe (valuation R Γ₀) (R →* Γ₀) := ⟨valuation.to_monoid_hom⟩
variables {R} {Γ₀} (v : valuation R Γ₀) {x y z : R}
@[simp, norm_cast] lemma coe_coe : ((v : R →* Γ₀) : R → Γ₀) = v := rfl
@[simp] lemma map_zero : v 0 = 0 := v.map_zero'
@[simp] lemma map_one : v 1 = 1 := v.map_one'
@[simp] lemma map_mul : ∀ x y, v (x * y) = v x * v y := v.map_mul'
@[simp] lemma map_add : ∀ x y, v (x + y) ≤ max (v x) (v y) := v.map_add'
@[simp] lemma map_pow : ∀ x (n:ℕ), v (x^n) = (v x)^n :=
v.to_monoid_hom.map_pow
@[ext] lemma ext {v₁ v₂ : valuation R Γ₀} (h : ∀ r, v₁ r = v₂ r) : v₁ = v₂ :=
by { cases v₁, cases v₂, congr, funext r, exact h r }
lemma ext_iff {v₁ v₂ : valuation R Γ₀} : v₁ = v₂ ↔ ∀ r, v₁ r = v₂ r :=
⟨λ h r, congr_arg _ h, ext⟩
-- The following definition is not an instance, because we have more than one `v` on a given `R`.
-- In addition, type class inference would not be able to infer `v`.
/-- A valuation gives a preorder on the underlying ring. -/
def to_preorder : preorder R := preorder.lift v
/-- If `v` is a valuation on a division ring then `v(x) = 0` iff `x = 0`. -/
@[simp] lemma zero_iff {K : Type*} [division_ring K]
(v : valuation K Γ₀) {x : K} : v x = 0 ↔ x = 0 :=
begin
split ; intro h,
{ contrapose! h,
exact ((is_unit.mk0 _ h).map (v : K →* Γ₀)).ne_zero },
{ exact h.symm ▸ v.map_zero },
end
lemma ne_zero_iff {K : Type*} [division_ring K]
(v : valuation K Γ₀) {x : K} : v x ≠ 0 ↔ x ≠ 0 :=
not_iff_not_of_iff v.zero_iff
@[simp] lemma map_inv {K : Type*} [division_ring K]
(v : valuation K Γ₀) {x : K} : v x⁻¹ = (v x)⁻¹ :=
begin
by_cases h : x = 0,
{ subst h, rw [inv_zero, v.map_zero, inv_zero] },
{ apply eq_inv_of_mul_right_eq_one,
rw [← v.map_mul, mul_inv_cancel h, v.map_one] }
end
lemma map_units_inv (x : units R) : v (x⁻¹ : units R) = (v x)⁻¹ :=
eq_inv_of_mul_right_eq_one $ by rw [← v.map_mul, units.mul_inv, v.map_one]
@[simp] theorem unit_map_eq (u : units R) :
(units.map (v : R →* Γ₀) u : Γ₀) = v u := rfl
theorem map_neg_one : v (-1) = 1 :=
begin
apply eq_one_of_pow_eq_one (nat.succ_ne_zero 1) (_ : _ ^ 2 = _),
rw [pow_two, ← v.map_mul, neg_one_mul, neg_neg, v.map_one],
end
@[simp] lemma map_neg (x : R) : v (-x) = v x :=
calc v (-x) = v (-1 * x) : by rw [neg_one_mul]
... = v (-1) * v x : map_mul _ _ _
... = v x : by rw [v.map_neg_one, one_mul]
lemma map_sub_swap (x y : R) : v (x - y) = v (y - x) :=
calc v (x - y) = v (-(y - x)) : by rw show x - y = -(y-x), by abel
... = _ : map_neg _ _
lemma map_sub_le_max (x y : R) : v (x - y) ≤ max (v x) (v y) :=
calc v (x - y) = v (x + -y) : by rw [sub_eq_add_neg]
... ≤ max (v x) (v $ -y) : v.map_add _ _
... = max (v x) (v y) : by rw map_neg
lemma map_add_of_distinct_val (h : v x ≠ v y) : v (x + y) = max (v x) (v y) :=
begin
suffices : ¬v (x + y) < max (v x) (v y),
from or_iff_not_imp_right.1 (le_iff_eq_or_lt.1 (v.map_add x y)) this,
intro h',
wlog vyx : v y < v x using x y,
{ apply lt_or_gt_of_ne h.symm },
{ rw max_eq_left_of_lt vyx at h',
apply lt_irrefl (v x),
calc v x = v ((x+y) - y) : by simp
... ≤ max (v $ x + y) (v y) : map_sub_le_max _ _ _
... < v x : max_lt h' vyx },
{ apply this h.symm,
rwa [add_comm, max_comm] at h' }
end
lemma map_eq_of_sub_lt (h : v (y - x) < v x) : v y = v x :=
begin
have := valuation.map_add_of_distinct_val v (ne_of_gt h).symm,
rw max_eq_right (le_of_lt h) at this,
simpa using this
end
/-- A ring homomorphism S → R induces a map valuation R Γ₀ → valuation S Γ₀ -/
def comap {S : Type*} [ring S] (f : S →+* R) (v : valuation R Γ₀) :
valuation S Γ₀ :=
by refine_struct { to_fun := v ∘ f, .. }; intros;
simp only [comp_app, map_one, map_mul, map_zero, map_add,
f.map_one, f.map_mul, f.map_zero, f.map_add]
@[simp] lemma comap_id : v.comap (ring_hom.id R) = v := ext $ λ r, rfl
lemma comap_comp {S₁ : Type*} {S₂ : Type*} [ring S₁] [ring S₂] (f : S₁ →+* S₂) (g : S₂ →+* R) :
v.comap (g.comp f) = (v.comap g).comap f :=
ext $ λ r, rfl
/-- A ≤-preserving group homomorphism Γ₀ → Γ'₀ induces a map valuation R Γ₀ → valuation R Γ'₀. -/
def map (f : Γ₀ →* Γ'₀) (h₀ : f 0 = 0) (hf : monotone f) (v : valuation R Γ₀) : valuation R Γ'₀ :=
{ to_fun := f ∘ v,
map_zero' := show f (v 0) = 0, by rw [v.map_zero, h₀],
map_add' := λ r s,
calc f (v (r + s)) ≤ f (max (v r) (v s)) : hf (v.map_add r s)
... = max (f (v r)) (f (v s)) : hf.map_max,
.. monoid_hom.comp f (v : R →* Γ₀) }
/-- Two valuations on R are defined to be equivalent if they induce the same preorder on R. -/
def is_equiv (v₁ : valuation R Γ₀) (v₂ : valuation R Γ'₀) : Prop :=
∀ r s, v₁ r ≤ v₁ s ↔ v₂ r ≤ v₂ s
end basic -- end of section
namespace is_equiv
variables [ring R]
variables {v : valuation R Γ₀}
variables {v₁ : valuation R Γ₀} {v₂ : valuation R Γ'₀} {v₃ : valuation R Γ''₀}
@[refl] lemma refl : v.is_equiv v :=
λ _ _, iff.refl _
@[symm] lemma symm (h : v₁.is_equiv v₂) : v₂.is_equiv v₁ :=
λ _ _, iff.symm (h _ _)
@[trans] lemma trans (h₁₂ : v₁.is_equiv v₂) (h₂₃ : v₂.is_equiv v₃) : v₁.is_equiv v₃ :=
λ _ _, iff.trans (h₁₂ _ _) (h₂₃ _ _)
lemma of_eq {v' : valuation R Γ₀} (h : v = v') : v.is_equiv v' :=
by { subst h }
lemma map {v' : valuation R Γ₀} (f : Γ₀ →* Γ'₀) (h₀ : f 0 = 0) (hf : monotone f) (inf : injective f)
(h : v.is_equiv v') :
(v.map f h₀ hf).is_equiv (v'.map f h₀ hf) :=
let H : strict_mono f := strict_mono_of_monotone_of_injective hf inf in
λ r s,
calc f (v r) ≤ f (v s) ↔ v r ≤ v s : by rw H.le_iff_le
... ↔ v' r ≤ v' s : h r s
... ↔ f (v' r) ≤ f (v' s) : by rw H.le_iff_le
/-- `comap` preserves equivalence. -/
lemma comap {S : Type*} [ring S] (f : S →+* R) (h : v₁.is_equiv v₂) :
(v₁.comap f).is_equiv (v₂.comap f) :=
λ r s, h (f r) (f s)
lemma val_eq (h : v₁.is_equiv v₂) {r s : R} :
v₁ r = v₁ s ↔ v₂ r = v₂ s :=
by simpa only [le_antisymm_iff] using and_congr (h r s) (h s r)
lemma ne_zero (h : v₁.is_equiv v₂) {r : R} :
v₁ r ≠ 0 ↔ v₂ r ≠ 0 :=
begin
have : v₁ r ≠ v₁ 0 ↔ v₂ r ≠ v₂ 0 := not_iff_not_of_iff h.val_eq,
rwa [v₁.map_zero, v₂.map_zero] at this,
end
end is_equiv -- end of namespace
lemma is_equiv_of_map_strict_mono [ring R] {v : valuation R Γ₀}
(f : Γ₀ →* Γ'₀) (h₀ : f 0 = 0) (H : strict_mono f) :
is_equiv (v.map f h₀ (H.monotone)) v :=
λ x y, ⟨H.le_iff_le.mp, λ h, H.monotone h⟩
lemma is_equiv_of_val_le_one {K : Type*} [division_ring K]
(v : valuation K Γ₀) (v' : valuation K Γ'₀) (h : ∀ {x:K}, v x ≤ 1 ↔ v' x ≤ 1) :
v.is_equiv v' :=
begin
intros x y,
by_cases hy : y = 0, { simp [hy, zero_iff], },
rw show y = 1 * y, by rw one_mul,
rw [← (inv_mul_cancel_right' hy x)],
iterate 2 {rw [v.map_mul _ y, v'.map_mul _ y]},
rw [v.map_one, v'.map_one],
split; intro H,
{ apply mul_le_mul_right',
replace hy := v.ne_zero_iff.mpr hy,
replace H := le_of_le_mul_right hy H,
rwa h at H, },
{ apply mul_le_mul_right',
replace hy := v'.ne_zero_iff.mpr hy,
replace H := le_of_le_mul_right hy H,
rwa h, },
end
section supp
variables [comm_ring R]
variables (v : valuation R Γ₀)
/-- The support of a valuation `v : R → Γ₀` is the ideal of `R` where `v` vanishes. -/
def supp : ideal R :=
{ carrier := {x | v x = 0},
zero_mem' := map_zero v,
add_mem' := λ x y hx hy, le_zero_iff.mp $
calc v (x + y) ≤ max (v x) (v y) : v.map_add x y
... ≤ 0 : max_le (le_zero_iff.mpr hx) (le_zero_iff.mpr hy),
smul_mem' := λ c x hx, calc v (c * x)
= v c * v x : map_mul v c x
... = v c * 0 : congr_arg _ hx
... = 0 : mul_zero _ }
@[simp] lemma mem_supp_iff (x : R) : x ∈ supp v ↔ v x = 0 := iff.rfl
-- @[simp] lemma mem_supp_iff' (x : R) : x ∈ (supp v : set R) ↔ v x = 0 := iff.rfl
/-- The support of a valuation is a prime ideal. -/
instance : ideal.is_prime (supp v) :=
⟨λ (h : v.supp = ⊤), one_ne_zero $ show (1 : Γ₀) = 0,
from calc 1 = v 1 : v.map_one.symm
... = 0 : show (1:R) ∈ supp v, by { rw h, trivial },
λ x y hxy, begin
show v x = 0 ∨ v y = 0,
change v (x * y) = 0 at hxy,
rw [v.map_mul x y] at hxy,
exact eq_zero_or_eq_zero_of_mul_eq_zero hxy
end⟩
lemma map_add_supp (a : R) {s : R} (h : s ∈ supp v) : v (a + s) = v a :=
begin
have aux : ∀ a s, v s = 0 → v (a + s) ≤ v a,
{ intros a' s' h', refine le_trans (v.map_add a' s') (max_le (le_refl _) _), simp [h'], },
apply le_antisymm (aux a s h),
calc v a = v (a + s + -s) : by simp
... ≤ v (a + s) : aux (a + s) (-s) (by rwa ←ideal.neg_mem_iff at h)
end
/-- If `hJ : J ⊆ supp v` then `on_quot_val hJ` is the induced function on R/J as a function.
Note: it's just the function; the valuation is `on_quot hJ`. -/
def on_quot_val {J : ideal R} (hJ : J ≤ supp v) :
J.quotient → Γ₀ :=
λ q, quotient.lift_on' q v $ λ a b h,
calc v a = v (b + (a - b)) : by simp
... = v b : v.map_add_supp b (hJ h)
/-- The extension of valuation v on R to valuation on R/J if J ⊆ supp v -/
def on_quot {J : ideal R} (hJ : J ≤ supp v) :
valuation J.quotient Γ₀ :=
{ to_fun := v.on_quot_val hJ,
map_zero' := v.map_zero,
map_one' := v.map_one,
map_mul' := λ xbar ybar, quotient.ind₂' v.map_mul xbar ybar,
map_add' := λ xbar ybar, quotient.ind₂' v.map_add xbar ybar }
@[simp] lemma on_quot_comap_eq {J : ideal R} (hJ : J ≤ supp v) :
(v.on_quot hJ).comap (ideal.quotient.mk J) = v :=
ext $ λ r,
begin
refine @quotient.lift_on_beta _ _ (J.quotient_rel) v (λ a b h, _) _,
calc v a = v (b + (a - b)) : by simp
... = v b : v.map_add_supp b (hJ h)
end
lemma comap_supp {S : Type*} [comm_ring S] (f : S →+* R) :
supp (v.comap f) = ideal.comap f v.supp :=
ideal.ext $ λ x,
begin
rw [mem_supp_iff, ideal.mem_comap, mem_supp_iff],
refl,
end
lemma self_le_supp_comap (J : ideal R) (v : valuation (quotient J) Γ₀) :
J ≤ (v.comap (ideal.quotient.mk J)).supp :=
by { rw [comap_supp, ← ideal.map_le_iff_le_comap], simp }
@[simp] lemma comap_on_quot_eq (J : ideal R) (v : valuation J.quotient Γ₀) :
(v.comap (ideal.quotient.mk J)).on_quot (v.self_le_supp_comap J) = v :=
ext $ by { rintro ⟨x⟩, refl }
/-- The quotient valuation on R/J has support supp(v)/J if J ⊆ supp v. -/
lemma supp_quot {J : ideal R} (hJ : J ≤ supp v) :
supp (v.on_quot hJ) = (supp v).map (ideal.quotient.mk J) :=
begin
apply le_antisymm,
{ rintro ⟨x⟩ hx,
apply ideal.subset_span,
exact ⟨x, hx, rfl⟩ },
{ rw ideal.map_le_iff_le_comap,
intros x hx, exact hx }
end
lemma supp_quot_supp : supp (v.on_quot (le_refl _)) = 0 :=
by { rw supp_quot, exact ideal.map_quotient_self _ }
end supp -- end of section
end valuation
|
205a813e7dbdc48022fa2e25a7d4c4c47adaf22e | 69d4931b605e11ca61881fc4f66db50a0a875e39 | /src/algebra/group_power/lemmas.lean | 34d7bfe326083946cd2e0e82c14044f41710ebd6 | [
"Apache-2.0"
] | permissive | abentkamp/mathlib | d9a75d291ec09f4637b0f30cc3880ffb07549ee5 | 5360e476391508e092b5a1e5210bd0ed22dc0755 | refs/heads/master | 1,682,382,954,948 | 1,622,106,077,000 | 1,622,106,077,000 | 149,285,665 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 32,819 | lean | /-
Copyright (c) 2015 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis
-/
import algebra.group_power.basic
import algebra.invertible
import algebra.opposites
import data.list.basic
import data.int.cast
import data.equiv.basic
import data.equiv.mul_add
import deprecated.group
/-!
# Lemmas about power operations on monoids and groups
This file contains lemmas about `monoid.pow`, `group.pow`, `nsmul`, `gsmul`
which require additional imports besides those available in `.basic`.
-/
universes u v w x y z u₁ u₂
variables {M : Type u} {N : Type v} {G : Type w} {H : Type x} {A : Type y} {B : Type z}
{R : Type u₁} {S : Type u₂}
/-!
### (Additive) monoid
-/
section monoid
variables [monoid M] [monoid N] [add_monoid A] [add_monoid B]
@[simp] theorem nsmul_one [has_one A] : ∀ n : ℕ, n • (1 : A) = n :=
add_monoid_hom.eq_nat_cast
⟨λ n, n • (1 : A), zero_nsmul _, λ _ _, add_nsmul _ _ _⟩
(one_nsmul _)
@[simp, priority 500]
theorem list.prod_repeat (a : M) (n : ℕ) : (list.repeat a n).prod = a ^ n :=
begin
induction n with n ih,
{ rw pow_zero, refl },
{ rw [list.repeat_succ, list.prod_cons, ih, pow_succ] }
end
@[simp, priority 500]
theorem list.sum_repeat : ∀ (a : A) (n : ℕ), (list.repeat a n).sum = n • a :=
@list.prod_repeat (multiplicative A) _
@[simp, norm_cast] lemma units.coe_pow (u : units M) (n : ℕ) : ((u ^ n : units M) : M) = u ^ n :=
(units.coe_hom M).map_pow u n
instance invertible_pow (m : M) [invertible m] (n : ℕ) : invertible (m ^ n) :=
{ inv_of := ⅟ m ^ n,
inv_of_mul_self := by rw [← (commute_inv_of m).symm.mul_pow, inv_of_mul_self, one_pow],
mul_inv_of_self := by rw [← (commute_inv_of m).mul_pow, mul_inv_of_self, one_pow] }
lemma inv_of_pow (m : M) [invertible m] (n : ℕ) [invertible (m ^ n)] :
⅟(m ^ n) = ⅟m ^ n :=
@invertible_unique M _ (m ^ n) (m ^ n) rfl ‹_› (invertible_pow m n)
lemma is_unit.pow {m : M} (n : ℕ) : is_unit m → is_unit (m ^ n) :=
λ ⟨u, hu⟩, ⟨u ^ n, by simp *⟩
/-- If `x ^ n.succ = 1` then `x` has an inverse, `x^n`. -/
def invertible_of_pow_succ_eq_one (x : M) (n : ℕ) (hx : x ^ n.succ = 1) :
invertible x :=
⟨x ^ n, (pow_succ' x n).symm.trans hx, (pow_succ x n).symm.trans hx⟩
/-- If `x ^ n = 1` then `x` has an inverse, `x^(n - 1)`. -/
def invertible_of_pow_eq_one (x : M) (n : ℕ) (hx : x ^ n = 1) (hn : 0 < n) :
invertible x :=
begin
apply invertible_of_pow_succ_eq_one x (n - 1),
convert hx,
exact nat.sub_add_cancel (nat.succ_le_of_lt hn),
end
lemma is_unit_of_pow_eq_one (x : M) (n : ℕ) (hx : x ^ n = 1) (hn : 0 < n) :
is_unit x :=
begin
haveI := invertible_of_pow_eq_one x n hx hn,
exact is_unit_of_invertible x
end
end monoid
theorem nat.nsmul_eq_mul (m n : ℕ) : m • n = m * n :=
by induction m with m ih; [rw [zero_nsmul, zero_mul],
rw [succ_nsmul', ih, nat.succ_mul]]
section group
variables [group G] [group H] [add_group A] [add_group B]
open int
local attribute [ematch] le_of_lt
open nat
theorem gsmul_one [has_one A] (n : ℤ) : n • (1 : A) = n :=
by cases n; simp
lemma gpow_add_one (a : G) : ∀ n : ℤ, a ^ (n + 1) = a ^ n * a
| (of_nat n) := by simp [← int.coe_nat_succ, pow_succ']
| -[1+0] := by simp [int.neg_succ_of_nat_eq]
| -[1+(n+1)] := by rw [int.neg_succ_of_nat_eq, gpow_neg, neg_add, neg_add_cancel_right, gpow_neg,
← int.coe_nat_succ, gpow_coe_nat, gpow_coe_nat, pow_succ _ (n + 1), mul_inv_rev,
inv_mul_cancel_right]
theorem add_one_gsmul : ∀ (a : A) (i : ℤ), (i + 1) • a = i • a + a :=
@gpow_add_one (multiplicative A) _
lemma gpow_sub_one (a : G) (n : ℤ) : a ^ (n - 1) = a ^ n * a⁻¹ :=
calc a ^ (n - 1) = a ^ (n - 1) * a * a⁻¹ : (mul_inv_cancel_right _ _).symm
... = a^n * a⁻¹ : by rw [← gpow_add_one, sub_add_cancel]
lemma gpow_add (a : G) (m n : ℤ) : a ^ (m + n) = a ^ m * a ^ n :=
begin
induction n using int.induction_on with n ihn n ihn,
case hz : { simp },
{ simp only [← add_assoc, gpow_add_one, ihn, mul_assoc] },
{ rw [gpow_sub_one, ← mul_assoc, ← ihn, ← gpow_sub_one, add_sub_assoc] }
end
lemma mul_self_gpow (b : G) (m : ℤ) : b*b^m = b^(m+1) :=
by { conv_lhs {congr, rw ← gpow_one b }, rw [← gpow_add, add_comm] }
lemma mul_gpow_self (b : G) (m : ℤ) : b^m*b = b^(m+1) :=
by { conv_lhs {congr, skip, rw ← gpow_one b }, rw [← gpow_add, add_comm] }
theorem add_gsmul : ∀ (a : A) (i j : ℤ), (i + j) • a = i • a + j • a :=
@gpow_add (multiplicative A) _
lemma gpow_sub (a : G) (m n : ℤ) : a ^ (m - n) = a ^ m * (a ^ n)⁻¹ :=
by rw [sub_eq_add_neg, gpow_add, gpow_neg]
lemma sub_gsmul (m n : ℤ) (a : A) : (m - n) • a = m • a - n • a :=
by simpa only [sub_eq_add_neg] using @gpow_sub (multiplicative A) _ _ _ _
theorem gpow_one_add (a : G) (i : ℤ) : a ^ (1 + i) = a * a ^ i :=
by rw [gpow_add, gpow_one]
theorem one_add_gsmul : ∀ (a : A) (i : ℤ), (1 + i) • a = a + i • a :=
@gpow_one_add (multiplicative A) _
theorem gpow_mul_comm (a : G) (i j : ℤ) : a ^ i * a ^ j = a ^ j * a ^ i :=
by rw [← gpow_add, ← gpow_add, add_comm]
theorem gsmul_add_comm : ∀ (a : A) (i j : ℤ), i • a + j • a = j • a + i • a :=
@gpow_mul_comm (multiplicative A) _
theorem gpow_mul (a : G) (m n : ℤ) : a ^ (m * n) = (a ^ m) ^ n :=
int.induction_on n (by simp) (λ n ihn, by simp [mul_add, gpow_add, ihn])
(λ n ihn, by simp only [mul_sub, gpow_sub, ihn, mul_one, gpow_one])
theorem gsmul_mul' : ∀ (a : A) (m n : ℤ), (m * n) • a = n • (m • a) :=
@gpow_mul (multiplicative A) _
theorem gpow_mul' (a : G) (m n : ℤ) : a ^ (m * n) = (a ^ n) ^ m :=
by rw [mul_comm, gpow_mul]
theorem mul_gsmul (a : A) (m n : ℤ) : (m * n) • a = m • (n • a) :=
by rw [mul_comm, gsmul_mul']
theorem gpow_bit0 (a : G) (n : ℤ) : a ^ bit0 n = a ^ n * a ^ n := gpow_add _ _ _
theorem bit0_gsmul (a : A) (n : ℤ) : bit0 n • a = n • a + n • a :=
@gpow_bit0 (multiplicative A) _ _ _
theorem gpow_bit1 (a : G) (n : ℤ) : a ^ bit1 n = a ^ n * a ^ n * a :=
by rw [bit1, gpow_add, gpow_bit0, gpow_one]
theorem bit1_gsmul : ∀ (a : A) (n : ℤ), bit1 n • a = n • a + n • a + a :=
@gpow_bit1 (multiplicative A) _
@[simp] theorem monoid_hom.map_gpow (f : G →* H) (a : G) (n : ℤ) : f (a ^ n) = f a ^ n :=
by cases n; simp
@[simp] theorem add_monoid_hom.map_gsmul (f : A →+ B) (a : A) (n : ℤ) : f (n • a) = n • f a :=
f.to_multiplicative.map_gpow a n
@[simp, norm_cast] lemma units.coe_gpow (u : units G) (n : ℤ) : ((u ^ n : units G) : G) = u ^ n :=
(units.coe_hom G).map_gpow u n
end group
section ordered_add_comm_group
variables [ordered_add_comm_group A]
/-! Lemmas about `gsmul` under ordering, placed here (rather than in `algebra.group_power.order`
with their friends) because they require facts from `data.int.basic`-/
open int
lemma gsmul_pos {a : A} (ha : 0 < a) {k : ℤ} (hk : (0:ℤ) < k) : 0 < k • a :=
begin
lift k to ℕ using int.le_of_lt hk,
rw gsmul_coe_nat,
apply nsmul_pos ha,
exact coe_nat_pos.mp hk,
end
theorem gsmul_le_gsmul {a : A} {n m : ℤ} (ha : 0 ≤ a) (h : n ≤ m) : n • a ≤ m • a :=
calc n • a = n • a + 0 : (add_zero _).symm
... ≤ n • a + (m - n) • a : add_le_add_left (gsmul_nonneg ha (sub_nonneg.mpr h)) _
... = m • a : by { rw [← add_gsmul], simp }
theorem gsmul_lt_gsmul {a : A} {n m : ℤ} (ha : 0 < a) (h : n < m) : n • a < m • a :=
calc n • a = n • a + 0 : (add_zero _).symm
... < n • a + (m - n) • a : add_lt_add_left (gsmul_pos ha (sub_pos.mpr h)) _
... = m • a : by { rw [← add_gsmul], simp }
lemma abs_nsmul {α : Type*} [linear_ordered_add_comm_group α] (n : ℕ) (a : α) :
abs (n • a) = n • abs a :=
begin
cases le_total a 0 with hneg hpos,
{ rw [abs_of_nonpos hneg, ← abs_neg, ← neg_nsmul, abs_of_nonneg],
exact nsmul_nonneg (neg_nonneg.mpr hneg) n },
{ rw [abs_of_nonneg hpos, abs_of_nonneg],
exact nsmul_nonneg hpos n }
end
lemma abs_gsmul {α : Type*} [linear_ordered_add_comm_group α] (n : ℤ) (a : α) :
abs (n • a) = (abs n) • abs a :=
begin
by_cases n0 : 0 ≤ n,
{ lift n to ℕ using n0,
simp only [abs_nsmul, coe_nat_abs, gsmul_coe_nat] },
{ lift (- n) to ℕ using int.le_of_lt (neg_pos.mpr (not_le.mp n0)) with m h,
rw [← abs_neg (n • a), ← neg_gsmul, ← abs_neg n, ← h, gsmul_coe_nat, coe_nat_abs,
gsmul_coe_nat],
exact abs_nsmul m _ },
end
lemma abs_add_eq_add_abs_le {α : Type*} [linear_ordered_add_comm_group α] {a b : α} (hle : a ≤ b) :
abs (a + b) = abs a + abs b ↔ (0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0) :=
begin
by_cases a0 : 0 ≤ a; by_cases b0 : 0 ≤ b,
{ simp [a0, b0, abs_of_nonneg, add_nonneg a0 b0] },
{ exact (lt_irrefl (0 : α) (a0.trans_lt (hle.trans_lt (not_le.mp b0)))).elim },
any_goals { simp [(not_le.mp a0).le, (not_le.mp b0).le, abs_of_nonpos, add_nonpos, add_comm] },
obtain F := (not_le.mp a0),
have : (abs (a + b) = -a + b ↔ b ≤ 0) ↔ (abs (a + b) =
abs a + abs b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0),
{ simp [a0, b0, abs_of_neg, abs_of_nonneg, F, F.le] },
refine this.mp ⟨λ h, _, λ h, by simp only [le_antisymm h b0, abs_of_neg F, add_zero]⟩,
by_cases ba : a + b ≤ 0,
{ refine le_of_eq (eq_zero_of_neg_eq _),
rwa [abs_of_nonpos ba, neg_add_rev, add_comm, add_right_inj] at h },
{ refine (lt_irrefl (0 : α) _).elim,
rw [abs_of_pos (not_le.mp ba), add_left_inj] at h,
rwa eq_zero_of_neg_eq h.symm at F }
end
lemma abs_add_eq_add_abs_iff {α : Type*} [linear_ordered_add_comm_group α] (a b : α) :
abs (a + b) = abs a + abs b ↔ (0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0) :=
begin
by_cases ab : a ≤ b,
{ exact abs_add_eq_add_abs_le ab },
{ rw [add_comm a, add_comm (abs _), abs_add_eq_add_abs_le ((not_le.mp ab).le), and.comm,
@and.comm (b ≤ 0 ) _] }
end
end ordered_add_comm_group
section linear_ordered_add_comm_group
variable [linear_ordered_add_comm_group A]
theorem gsmul_le_gsmul_iff {a : A} {n m : ℤ} (ha : 0 < a) : n • a ≤ m • a ↔ n ≤ m :=
begin
refine ⟨λ h, _, gsmul_le_gsmul $ le_of_lt ha⟩,
by_contra H,
exact lt_irrefl _ (lt_of_lt_of_le (gsmul_lt_gsmul ha (not_le.mp H)) h)
end
theorem gsmul_lt_gsmul_iff {a : A} {n m : ℤ} (ha : 0 < a) : n • a < m • a ↔ n < m :=
begin
refine ⟨λ h, _, gsmul_lt_gsmul ha⟩,
by_contra H,
exact lt_irrefl _ (lt_of_le_of_lt (gsmul_le_gsmul (le_of_lt ha) $ not_lt.mp H) h)
end
theorem nsmul_le_nsmul_iff {a : A} {n m : ℕ} (ha : 0 < a) : n • a ≤ m • a ↔ n ≤ m :=
begin
refine ⟨λ h, _, nsmul_le_nsmul $ le_of_lt ha⟩,
by_contra H,
exact lt_irrefl _ (lt_of_lt_of_le (nsmul_lt_nsmul ha (not_le.mp H)) h)
end
theorem nsmul_lt_nsmul_iff {a : A} {n m : ℕ} (ha : 0 < a) : n • a < m • a ↔ n < m :=
begin
refine ⟨λ h, _, nsmul_lt_nsmul ha⟩,
by_contra H,
exact lt_irrefl _ (lt_of_le_of_lt (nsmul_le_nsmul (le_of_lt ha) $ not_lt.mp H) h)
end
end linear_ordered_add_comm_group
@[simp] lemma with_bot.coe_nsmul [add_monoid A] (a : A) (n : ℕ) :
((n • a : A) : with_bot A) = n • a :=
add_monoid_hom.map_nsmul ⟨(coe : A → with_bot A), with_bot.coe_zero, with_bot.coe_add⟩ a n
theorem nsmul_eq_mul' [semiring R] (a : R) (n : ℕ) : n • a = a * n :=
by induction n with n ih; [rw [zero_nsmul, nat.cast_zero, mul_zero],
rw [succ_nsmul', ih, nat.cast_succ, mul_add, mul_one]]
@[simp] theorem nsmul_eq_mul [semiring R] (n : ℕ) (a : R) : n • a = n * a :=
by rw [nsmul_eq_mul', (n.cast_commute a).eq]
theorem mul_nsmul_left [semiring R] (a b : R) (n : ℕ) : n • (a * b) = a * (n • b) :=
by rw [nsmul_eq_mul', nsmul_eq_mul', mul_assoc]
theorem mul_nsmul_assoc [semiring R] (a b : R) (n : ℕ) : n • (a * b) = n • a * b :=
by rw [nsmul_eq_mul, nsmul_eq_mul, mul_assoc]
@[simp, norm_cast] theorem nat.cast_pow [semiring R] (n m : ℕ) : (↑(n ^ m) : R) = ↑n ^ m :=
begin
induction m with m ih,
{ rw [pow_zero, pow_zero], exact nat.cast_one },
{ rw [pow_succ', pow_succ', nat.cast_mul, ih] }
end
@[simp, norm_cast] theorem int.coe_nat_pow (n m : ℕ) : ((n ^ m : ℕ) : ℤ) = n ^ m :=
by induction m with m ih; [exact int.coe_nat_one, rw [pow_succ', pow_succ', int.coe_nat_mul, ih]]
theorem int.nat_abs_pow (n : ℤ) (k : ℕ) : int.nat_abs (n ^ k) = (int.nat_abs n) ^ k :=
by induction k with k ih; [refl, rw [pow_succ', int.nat_abs_mul, pow_succ', ih]]
-- The next four lemmas allow us to replace multiplication by a numeral with a `gsmul` expression.
-- They are used by the `noncomm_ring` tactic, to normalise expressions before passing to `abel`.
lemma bit0_mul [ring R] {n r : R} : bit0 n * r = (2 : ℤ) • (n * r) :=
by { dsimp [bit0], rw [add_mul, add_gsmul, one_gsmul], }
lemma mul_bit0 [ring R] {n r : R} : r * bit0 n = (2 : ℤ) • (r * n) :=
by { dsimp [bit0], rw [mul_add, add_gsmul, one_gsmul], }
lemma bit1_mul [ring R] {n r : R} : bit1 n * r = (2 : ℤ) • (n * r) + r :=
by { dsimp [bit1], rw [add_mul, bit0_mul, one_mul], }
lemma mul_bit1 [ring R] {n r : R} : r * bit1 n = (2 : ℤ) • (r * n) + r :=
by { dsimp [bit1], rw [mul_add, mul_bit0, mul_one], }
@[simp] theorem gsmul_eq_mul [ring R] (a : R) : ∀ (n : ℤ), n • a = n * a
| (n : ℕ) := by { rw [gsmul_coe_nat, nsmul_eq_mul], refl }
| -[1+ n] := by simp [nat.cast_succ, neg_add_rev, int.cast_neg_succ_of_nat, add_mul]
theorem gsmul_eq_mul' [ring R] (a : R) (n : ℤ) : n • a = a * n :=
by rw [gsmul_eq_mul, (n.cast_commute a).eq]
theorem mul_gsmul_left [ring R] (a b : R) (n : ℤ) : n • (a * b) = a * (n • b) :=
by rw [gsmul_eq_mul', gsmul_eq_mul', mul_assoc]
theorem mul_gsmul_assoc [ring R] (a b : R) (n : ℤ) : n • (a * b) = n • a * b :=
by rw [gsmul_eq_mul, gsmul_eq_mul, mul_assoc]
lemma gsmul_int_int (a b : ℤ) : a • b = a * b := by simp
lemma gsmul_int_one (n : ℤ) : n • 1 = n := by simp
@[simp, norm_cast] theorem int.cast_pow [ring R] (n : ℤ) (m : ℕ) : (↑(n ^ m) : R) = ↑n ^ m :=
begin
induction m with m ih,
{ rw [pow_zero, pow_zero, int.cast_one] },
{ rw [pow_succ, pow_succ, int.cast_mul, ih] }
end
lemma neg_one_pow_eq_pow_mod_two [ring R] {n : ℕ} : (-1 : R) ^ n = (-1) ^ (n % 2) :=
by rw [← nat.mod_add_div n 2, pow_add, pow_mul]; simp [sq]
section ordered_semiring
variable [ordered_semiring R]
/-- Bernoulli's inequality. This version works for semirings but requires
additional hypotheses `0 ≤ a * a` and `0 ≤ (1 + a) * (1 + a)`. -/
theorem one_add_mul_le_pow' {a : R} (Hsq : 0 ≤ a * a) (Hsq' : 0 ≤ (1 + a) * (1 + a))
(H : 0 ≤ 2 + a) :
∀ (n : ℕ), 1 + (n : R) * a ≤ (1 + a) ^ n
| 0 := by simp
| 1 := by simp
| (n+2) :=
have 0 ≤ (n : R) * (a * a * (2 + a)) + a * a,
from add_nonneg (mul_nonneg n.cast_nonneg (mul_nonneg Hsq H)) Hsq,
calc 1 + (↑(n + 2) : R) * a ≤ 1 + ↑(n + 2) * a + (n * (a * a * (2 + a)) + a * a) :
(le_add_iff_nonneg_right _).2 this
... = (1 + a) * (1 + a) * (1 + n * a) :
by { simp [add_mul, mul_add, bit0, mul_assoc, (n.cast_commute (_ : R)).left_comm],
ac_refl }
... ≤ (1 + a) * (1 + a) * (1 + a)^n :
mul_le_mul_of_nonneg_left (one_add_mul_le_pow' n) Hsq'
... = (1 + a)^(n + 2) : by simp only [pow_succ, mul_assoc]
private lemma pow_lt_pow_of_lt_one_aux {a : R} (h : 0 < a) (ha : a < 1) (i : ℕ) :
∀ k : ℕ, a ^ (i + k + 1) < a ^ i
| 0 :=
begin
rw [←one_mul (a^i), add_zero, pow_succ],
exact mul_lt_mul ha (le_refl _) (pow_pos h _) zero_le_one
end
| (k+1) :=
begin
rw [←one_mul (a^i), pow_succ],
apply mul_lt_mul ha _ _ zero_le_one,
{ apply le_of_lt, apply pow_lt_pow_of_lt_one_aux },
{ show 0 < a ^ (i + (k + 1) + 0), apply pow_pos h }
end
private lemma pow_le_pow_of_le_one_aux {a : R} (h : 0 ≤ a) (ha : a ≤ 1) (i : ℕ) :
∀ k : ℕ, a ^ (i + k) ≤ a ^ i
| 0 := by simp
| (k+1) := by { rw [←add_assoc, ←one_mul (a^i), pow_succ],
exact mul_le_mul ha (pow_le_pow_of_le_one_aux _) (pow_nonneg h _) zero_le_one }
lemma pow_lt_pow_of_lt_one {a : R} (h : 0 < a) (ha : a < 1)
{i j : ℕ} (hij : i < j) : a ^ j < a ^ i :=
let ⟨k, hk⟩ := nat.exists_eq_add_of_lt hij in
by rw hk; exact pow_lt_pow_of_lt_one_aux h ha _ _
lemma pow_lt_pow_iff_of_lt_one {a : R} {n m : ℕ} (hpos : 0 < a) (h : a < 1) :
a ^ m < a ^ n ↔ n < m :=
begin
have : strict_mono (λ (n : order_dual ℕ), a ^ (id n : ℕ)) := λ m n, pow_lt_pow_of_lt_one hpos h,
exact this.lt_iff_lt
end
lemma pow_le_pow_of_le_one {a : R} (h : 0 ≤ a) (ha : a ≤ 1)
{i j : ℕ} (hij : i ≤ j) : a ^ j ≤ a ^ i :=
let ⟨k, hk⟩ := nat.exists_eq_add_of_le hij in
by rw hk; exact pow_le_pow_of_le_one_aux h ha _ _
lemma pow_le_one {x : R} : ∀ (n : ℕ) (h0 : 0 ≤ x) (h1 : x ≤ 1), x ^ n ≤ 1
| 0 h0 h1 := by rw [pow_zero]
| (n+1) h0 h1 := by { rw [pow_succ], exact mul_le_one h1 (pow_nonneg h0 _) (pow_le_one n h0 h1) }
end ordered_semiring
section linear_ordered_semiring
variables [linear_ordered_semiring R]
lemma sign_cases_of_C_mul_pow_nonneg {C r : R} (h : ∀ n : ℕ, 0 ≤ C * r ^ n) :
C = 0 ∨ (0 < C ∧ 0 ≤ r) :=
begin
have : 0 ≤ C, by simpa only [pow_zero, mul_one] using h 0,
refine this.eq_or_lt.elim (λ h, or.inl h.symm) (λ hC, or.inr ⟨hC, _⟩),
refine nonneg_of_mul_nonneg_left _ hC,
simpa only [pow_one] using h 1
end
end linear_ordered_semiring
section linear_ordered_ring
variables [linear_ordered_ring R] {a : R} {n : ℕ}
@[simp] lemma abs_pow (a : R) (n : ℕ) : abs (a ^ n) = abs a ^ n :=
(pow_abs a n).symm
@[simp] theorem pow_bit1_neg_iff : a ^ bit1 n < 0 ↔ a < 0 :=
⟨λ h, not_le.1 $ λ h', not_le.2 h $ pow_nonneg h' _,
λ h, by { rw [bit1, pow_succ], exact mul_neg_of_neg_of_pos h (pow_bit0_pos h.ne _)}⟩
@[simp] theorem pow_bit1_nonneg_iff : 0 ≤ a ^ bit1 n ↔ 0 ≤ a :=
le_iff_le_iff_lt_iff_lt.2 pow_bit1_neg_iff
@[simp] theorem pow_bit1_nonpos_iff : a ^ bit1 n ≤ 0 ↔ a ≤ 0 :=
by simp only [le_iff_lt_or_eq, pow_bit1_neg_iff, pow_eq_zero_iff (bit1_pos (zero_le n))]
@[simp] theorem pow_bit1_pos_iff : 0 < a ^ bit1 n ↔ 0 < a :=
lt_iff_lt_of_le_iff_le pow_bit1_nonpos_iff
theorem pow_even_nonneg (a : R) (hn : even n) : 0 ≤ a ^ n :=
by cases hn with k hk; simpa only [hk, two_mul] using pow_bit0_nonneg a k
theorem pow_even_pos (ha : a ≠ 0) (hn : even n) : 0 < a ^ n :=
by cases hn with k hk; simpa only [hk, two_mul] using pow_bit0_pos ha k
theorem pow_odd_nonneg (ha : 0 ≤ a) (hn : odd n) : 0 ≤ a ^ n :=
by cases hn with k hk; simpa only [hk, two_mul] using pow_bit1_nonneg_iff.mpr ha
theorem pow_odd_pos (ha : 0 < a) (hn : odd n) : 0 < a ^ n :=
by cases hn with k hk; simpa only [hk, two_mul] using pow_bit1_pos_iff.mpr ha
theorem pow_odd_nonpos (ha : a ≤ 0) (hn : odd n) : a ^ n ≤ 0:=
by cases hn with k hk; simpa only [hk, two_mul] using pow_bit1_nonpos_iff.mpr ha
theorem pow_odd_neg (ha : a < 0) (hn : odd n) : a ^ n < 0:=
by cases hn with k hk; simpa only [hk, two_mul] using pow_bit1_neg_iff.mpr ha
lemma pow_even_abs (a : R) {p : ℕ} (hp : even p) :
abs a ^ p = a ^ p :=
begin
rw [←abs_pow, abs_eq_self],
exact pow_even_nonneg _ hp
end
@[simp] lemma pow_bit0_abs (a : R) (p : ℕ) :
abs a ^ bit0 p = a ^ bit0 p :=
pow_even_abs _ (even_bit0 _)
lemma strict_mono_pow_bit1 (n : ℕ) : strict_mono (λ a : R, a ^ bit1 n) :=
begin
intros a b hab,
cases le_total a 0 with ha ha,
{ cases le_or_lt b 0 with hb hb,
{ rw [← neg_lt_neg_iff, ← neg_pow_bit1, ← neg_pow_bit1],
exact pow_lt_pow_of_lt_left (neg_lt_neg hab) (neg_nonneg.2 hb) (bit1_pos (zero_le n)) },
{ exact (pow_bit1_nonpos_iff.2 ha).trans_lt (pow_bit1_pos_iff.2 hb) } },
{ exact pow_lt_pow_of_lt_left hab ha (bit1_pos (zero_le n)) }
end
/-- Bernoulli's inequality for `n : ℕ`, `-2 ≤ a`. -/
theorem one_add_mul_le_pow (H : -2 ≤ a) (n : ℕ) : 1 + (n : R) * a ≤ (1 + a) ^ n :=
one_add_mul_le_pow' (mul_self_nonneg _) (mul_self_nonneg _) (neg_le_iff_add_nonneg'.1 H) _
/-- Bernoulli's inequality reformulated to estimate `a^n`. -/
theorem one_add_mul_sub_le_pow (H : -1 ≤ a) (n : ℕ) : 1 + (n : R) * (a - 1) ≤ a ^ n :=
have -2 ≤ a - 1, by rwa [bit0, neg_add, ← sub_eq_add_neg, sub_le_sub_iff_right],
by simpa only [add_sub_cancel'_right] using one_add_mul_le_pow this n
end linear_ordered_ring
/-- Bernoulli's inequality reformulated to estimate `(n : K)`. -/
theorem nat.cast_le_pow_sub_div_sub {K : Type*} [linear_ordered_field K] {a : K} (H : 1 < a)
(n : ℕ) :
(n : K) ≤ (a ^ n - 1) / (a - 1) :=
(le_div_iff (sub_pos.2 H)).2 $ le_sub_left_of_add_le $
one_add_mul_sub_le_pow ((neg_le_self $ @zero_le_one K _).trans H.le) _
/-- For any `a > 1` and a natural `n` we have `n ≤ a ^ n / (a - 1)`. See also
`nat.cast_le_pow_sub_div_sub` for a stronger inequality with `a ^ n - 1` in the numerator. -/
theorem nat.cast_le_pow_div_sub {K : Type*} [linear_ordered_field K] {a : K} (H : 1 < a) (n : ℕ) :
(n : K) ≤ a ^ n / (a - 1) :=
(n.cast_le_pow_sub_div_sub H).trans $ div_le_div_of_le (sub_nonneg.2 H.le)
(sub_le_self _ zero_le_one)
namespace int
lemma units_sq (u : units ℤ) : u ^ 2 = 1 :=
(sq u).symm ▸ units_mul_self u
alias int.units_sq ← int.units_pow_two
lemma units_pow_eq_pow_mod_two (u : units ℤ) (n : ℕ) : u ^ n = u ^ (n % 2) :=
by conv {to_lhs, rw ← nat.mod_add_div n 2}; rw [pow_add, pow_mul, units_sq, one_pow, mul_one]
@[simp] lemma nat_abs_sq (x : ℤ) : (x.nat_abs ^ 2 : ℤ) = x ^ 2 :=
by rw [sq, int.nat_abs_mul_self', sq]
alias int.nat_abs_sq ← int.nat_abs_pow_two
lemma abs_le_self_sq (a : ℤ) : (int.nat_abs a : ℤ) ≤ a ^ 2 :=
by { rw [← int.nat_abs_sq a, sq], norm_cast, apply nat.le_mul_self }
alias int.abs_le_self_sq ← int.abs_le_self_pow_two
lemma le_self_sq (b : ℤ) : b ≤ b ^ 2 := le_trans (le_nat_abs) (abs_le_self_sq _)
alias int.le_self_sq ← int.le_self_pow_two
end int
variables (M G A)
/-- Monoid homomorphisms from `multiplicative ℕ` are defined by the image
of `multiplicative.of_add 1`. -/
def powers_hom [monoid M] : M ≃ (multiplicative ℕ →* M) :=
{ to_fun := λ x, ⟨λ n, x ^ n.to_add, by { convert pow_zero x, exact to_add_one },
λ m n, pow_add x m n⟩,
inv_fun := λ f, f (multiplicative.of_add 1),
left_inv := pow_one,
right_inv := λ f, monoid_hom.ext $ λ n, by { simp [← f.map_pow, ← of_add_nsmul] } }
/-- Monoid homomorphisms from `multiplicative ℤ` are defined by the image
of `multiplicative.of_add 1`. -/
def gpowers_hom [group G] : G ≃ (multiplicative ℤ →* G) :=
{ to_fun := λ x, ⟨λ n, x ^ n.to_add, gpow_zero x, λ m n, gpow_add x m n⟩,
inv_fun := λ f, f (multiplicative.of_add 1),
left_inv := gpow_one,
right_inv := λ f, monoid_hom.ext $ λ n, by { simp [← f.map_gpow, ← of_add_gsmul ] } }
/-- Additive homomorphisms from `ℕ` are defined by the image of `1`. -/
def multiples_hom [add_monoid A] : A ≃ (ℕ →+ A) :=
{ to_fun := λ x, ⟨λ n, n • x, zero_nsmul x, λ m n, add_nsmul _ _ _⟩,
inv_fun := λ f, f 1,
left_inv := one_nsmul,
right_inv := λ f, add_monoid_hom.ext_nat $ one_nsmul (f 1) }
/-- Additive homomorphisms from `ℤ` are defined by the image of `1`. -/
def gmultiples_hom [add_group A] : A ≃ (ℤ →+ A) :=
{ to_fun := λ x, ⟨λ n, n • x, zero_gsmul x, λ m n, add_gsmul _ _ _⟩,
inv_fun := λ f, f 1,
left_inv := one_gsmul,
right_inv := λ f, add_monoid_hom.ext_int $ one_gsmul (f 1) }
variables {M G A}
@[simp] lemma powers_hom_apply [monoid M] (x : M) (n : multiplicative ℕ) :
powers_hom M x n = x ^ n.to_add := rfl
@[simp] lemma powers_hom_symm_apply [monoid M] (f : multiplicative ℕ →* M) :
(powers_hom M).symm f = f (multiplicative.of_add 1) := rfl
@[simp] lemma gpowers_hom_apply [group G] (x : G) (n : multiplicative ℤ) :
gpowers_hom G x n = x ^ n.to_add := rfl
@[simp] lemma gpowers_hom_symm_apply [group G] (f : multiplicative ℤ →* G) :
(gpowers_hom G).symm f = f (multiplicative.of_add 1) := rfl
@[simp] lemma multiples_hom_apply [add_monoid A] (x : A) (n : ℕ) :
multiples_hom A x n = n • x := rfl
@[simp] lemma multiples_hom_symm_apply [add_monoid A] (f : ℕ →+ A) :
(multiples_hom A).symm f = f 1 := rfl
@[simp] lemma gmultiples_hom_apply [add_group A] (x : A) (n : ℤ) :
gmultiples_hom A x n = n • x := rfl
@[simp] lemma gmultiples_hom_symm_apply [add_group A] (f : ℤ →+ A) :
(gmultiples_hom A).symm f = f 1 := rfl
lemma monoid_hom.apply_mnat [monoid M] (f : multiplicative ℕ →* M) (n : multiplicative ℕ) :
f n = (f (multiplicative.of_add 1)) ^ n.to_add :=
by rw [← powers_hom_symm_apply, ← powers_hom_apply, equiv.apply_symm_apply]
@[ext] lemma monoid_hom.ext_mnat [monoid M] ⦃f g : multiplicative ℕ →* M⦄
(h : f (multiplicative.of_add 1) = g (multiplicative.of_add 1)) : f = g :=
monoid_hom.ext $ λ n, by rw [f.apply_mnat, g.apply_mnat, h]
lemma monoid_hom.apply_mint [group M] (f : multiplicative ℤ →* M) (n : multiplicative ℤ) :
f n = (f (multiplicative.of_add 1)) ^ n.to_add :=
by rw [← gpowers_hom_symm_apply, ← gpowers_hom_apply, equiv.apply_symm_apply]
/-! `monoid_hom.ext_mint` is defined in `data.int.cast` -/
lemma add_monoid_hom.apply_nat [add_monoid M] (f : ℕ →+ M) (n : ℕ) :
f n = n • (f 1) :=
by rw [← multiples_hom_symm_apply, ← multiples_hom_apply, equiv.apply_symm_apply]
/-! `add_monoid_hom.ext_nat` is defined in `data.nat.cast` -/
lemma add_monoid_hom.apply_int [add_group M] (f : ℤ →+ M) (n : ℤ) :
f n = n • (f 1) :=
by rw [← gmultiples_hom_symm_apply, ← gmultiples_hom_apply, equiv.apply_symm_apply]
/-! `add_monoid_hom.ext_int` is defined in `data.int.cast` -/
variables (M G A)
/-- If `M` is commutative, `powers_hom` is a multiplicative equivalence. -/
def powers_mul_hom [comm_monoid M] : M ≃* (multiplicative ℕ →* M) :=
{ map_mul' := λ a b, monoid_hom.ext $ by simp [mul_pow],
..powers_hom M}
/-- If `M` is commutative, `gpowers_hom` is a multiplicative equivalence. -/
def gpowers_mul_hom [comm_group G] : G ≃* (multiplicative ℤ →* G) :=
{ map_mul' := λ a b, monoid_hom.ext $ by simp [mul_gpow],
..gpowers_hom G}
/-- If `M` is commutative, `multiples_hom` is an additive equivalence. -/
def multiples_add_hom [add_comm_monoid A] : A ≃+ (ℕ →+ A) :=
{ map_add' := λ a b, add_monoid_hom.ext $ by simp [nsmul_add],
..multiples_hom A}
/-- If `M` is commutative, `gmultiples_hom` is an additive equivalence. -/
def gmultiples_add_hom [add_comm_group A] : A ≃+ (ℤ →+ A) :=
{ map_add' := λ a b, add_monoid_hom.ext $ by simp [gsmul_add],
..gmultiples_hom A}
variables {M G A}
@[simp] lemma powers_mul_hom_apply [comm_monoid M] (x : M) (n : multiplicative ℕ) :
powers_mul_hom M x n = x ^ n.to_add := rfl
@[simp] lemma powers_mul_hom_symm_apply [comm_monoid M] (f : multiplicative ℕ →* M) :
(powers_mul_hom M).symm f = f (multiplicative.of_add 1) := rfl
@[simp] lemma gpowers_mul_hom_apply [comm_group G] (x : G) (n : multiplicative ℤ) :
gpowers_mul_hom G x n = x ^ n.to_add := rfl
@[simp] lemma gpowers_mul_hom_symm_apply [comm_group G] (f : multiplicative ℤ →* G) :
(gpowers_mul_hom G).symm f = f (multiplicative.of_add 1) := rfl
@[simp] lemma multiples_add_hom_apply [add_comm_monoid A] (x : A) (n : ℕ) :
multiples_add_hom A x n = n • x := rfl
@[simp] lemma multiples_add_hom_symm_apply [add_comm_monoid A] (f : ℕ →+ A) :
(multiples_add_hom A).symm f = f 1 := rfl
@[simp] lemma gmultiples_add_hom_apply [add_comm_group A] (x : A) (n : ℤ) :
gmultiples_add_hom A x n = n • x := rfl
@[simp] lemma gmultiples_add_hom_symm_apply [add_comm_group A] (f : ℤ →+ A) :
(gmultiples_add_hom A).symm f = f 1 := rfl
/-!
### Commutativity (again)
Facts about `semiconj_by` and `commute` that require `gpow` or `gsmul`, or the fact that integer
multiplication equals semiring multiplication.
-/
namespace semiconj_by
section
variables [semiring R] {a x y : R}
@[simp] lemma cast_nat_mul_right (h : semiconj_by a x y) (n : ℕ) :
semiconj_by a ((n : R) * x) (n * y) :=
semiconj_by.mul_right (nat.commute_cast _ _) h
@[simp] lemma cast_nat_mul_left (h : semiconj_by a x y) (n : ℕ) : semiconj_by ((n : R) * a) x y :=
semiconj_by.mul_left (nat.cast_commute _ _) h
@[simp] lemma cast_nat_mul_cast_nat_mul (h : semiconj_by a x y) (m n : ℕ) :
semiconj_by ((m : R) * a) (n * x) (n * y) :=
(h.cast_nat_mul_left m).cast_nat_mul_right n
end
variables [monoid M] [group G] [ring R]
@[simp] lemma units_gpow_right {a : M} {x y : units M} (h : semiconj_by a x y) :
∀ m : ℤ, semiconj_by a (↑(x^m)) (↑(y^m))
| (n : ℕ) := by simp only [gpow_coe_nat, units.coe_pow, h, pow_right]
| -[1+n] := by simp only [gpow_neg_succ_of_nat, units.coe_pow, units_inv_right, h, pow_right]
variables {a b x y x' y' : R}
@[simp] lemma cast_int_mul_right (h : semiconj_by a x y) (m : ℤ) :
semiconj_by a ((m : ℤ) * x) (m * y) :=
semiconj_by.mul_right (int.commute_cast _ _) h
@[simp] lemma cast_int_mul_left (h : semiconj_by a x y) (m : ℤ) : semiconj_by ((m : R) * a) x y :=
semiconj_by.mul_left (int.cast_commute _ _) h
@[simp] lemma cast_int_mul_cast_int_mul (h : semiconj_by a x y) (m n : ℤ) :
semiconj_by ((m : R) * a) (n * x) (n * y) :=
(h.cast_int_mul_left m).cast_int_mul_right n
end semiconj_by
namespace commute
section
variables [semiring R] {a b : R}
@[simp] theorem cast_nat_mul_right (h : commute a b) (n : ℕ) : commute a ((n : R) * b) :=
h.cast_nat_mul_right n
@[simp] theorem cast_nat_mul_left (h : commute a b) (n : ℕ) : commute ((n : R) * a) b :=
h.cast_nat_mul_left n
@[simp] theorem cast_nat_mul_cast_nat_mul (h : commute a b) (m n : ℕ) :
commute ((m : R) * a) (n * b) :=
h.cast_nat_mul_cast_nat_mul m n
@[simp] theorem self_cast_nat_mul (n : ℕ) : commute a (n * a) :=
(commute.refl a).cast_nat_mul_right n
@[simp] theorem cast_nat_mul_self (n : ℕ) : commute ((n : R) * a) a :=
(commute.refl a).cast_nat_mul_left n
@[simp] theorem self_cast_nat_mul_cast_nat_mul (m n : ℕ) : commute ((m : R) * a) (n * a) :=
(commute.refl a).cast_nat_mul_cast_nat_mul m n
end
variables [monoid M] [group G] [ring R]
@[simp] lemma units_gpow_right {a : M} {u : units M} (h : commute a u) (m : ℤ) :
commute a (↑(u^m)) :=
h.units_gpow_right m
@[simp] lemma units_gpow_left {u : units M} {a : M} (h : commute ↑u a) (m : ℤ) :
commute (↑(u^m)) a :=
(h.symm.units_gpow_right m).symm
variables {a b : R}
@[simp] lemma cast_int_mul_right (h : commute a b) (m : ℤ) : commute a (m * b) :=
h.cast_int_mul_right m
@[simp] lemma cast_int_mul_left (h : commute a b) (m : ℤ) : commute ((m : R) * a) b :=
h.cast_int_mul_left m
lemma cast_int_mul_cast_int_mul (h : commute a b) (m n : ℤ) : commute ((m : R) * a) (n * b) :=
h.cast_int_mul_cast_int_mul m n
variables (a) (m n : ℤ)
@[simp] theorem self_cast_int_mul : commute a (n * a) := (commute.refl a).cast_int_mul_right n
@[simp] theorem cast_int_mul_self : commute ((n : R) * a) a := (commute.refl a).cast_int_mul_left n
theorem self_cast_int_mul_cast_int_mul : commute ((m : R) * a) (n * a) :=
(commute.refl a).cast_int_mul_cast_int_mul m n
end commute
section multiplicative
open multiplicative
@[simp] lemma nat.to_add_pow (a : multiplicative ℕ) (b : ℕ) : to_add (a ^ b) = to_add a * b :=
begin
induction b with b ih,
{ erw [pow_zero, to_add_one, mul_zero] },
{ simp [*, pow_succ, add_comm, nat.mul_succ] }
end
@[simp] lemma nat.of_add_mul (a b : ℕ) : of_add (a * b) = of_add a ^ b :=
(nat.to_add_pow _ _).symm
@[simp] lemma int.to_add_pow (a : multiplicative ℤ) (b : ℕ) : to_add (a ^ b) = to_add a * b :=
by induction b; simp [*, mul_add, pow_succ, add_comm]
@[simp] lemma int.to_add_gpow (a : multiplicative ℤ) (b : ℤ) : to_add (a ^ b) = to_add a * b :=
int.induction_on b (by simp)
(by simp [gpow_add, mul_add] {contextual := tt})
(by simp [gpow_add, mul_add, sub_eq_add_neg, -int.add_neg_one] {contextual := tt})
@[simp] lemma int.of_add_mul (a b : ℤ) : of_add (a * b) = of_add a ^ b :=
(int.to_add_gpow _ _).symm
end multiplicative
namespace units
variables [monoid M]
lemma conj_pow (u : units M) (x : M) (n : ℕ) : (↑u * x * ↑(u⁻¹))^n = u * x^n * ↑(u⁻¹) :=
(divp_eq_iff_mul_eq.2 ((u.mk_semiconj_by x).pow_right n).eq.symm).symm
lemma conj_pow' (u : units M) (x : M) (n : ℕ) : (↑(u⁻¹) * x * u)^n = ↑(u⁻¹) * x^n * u:=
(u⁻¹).conj_pow x n
open opposite
/-- Moving to the opposite monoid commutes with taking powers. -/
@[simp] lemma op_pow (x : M) (n : ℕ) : op (x ^ n) = (op x) ^ n :=
begin
induction n with n h,
{ simp },
{ rw [pow_succ', op_mul, h, pow_succ] }
end
@[simp] lemma unop_pow (x : Mᵒᵖ) (n : ℕ) : unop (x ^ n) = (unop x) ^ n :=
begin
induction n with n h,
{ simp },
{ rw [pow_succ', unop_mul, h, pow_succ] }
end
end units
|
1471d6c20d8f1803536121adc5094005a7e6e2e7 | 88fb7558b0636ec6b181f2a548ac11ad3919f8a5 | /library/tools/super/clause.lean | a7180232884c910c0f1725cefa8f59685a612501 | [
"Apache-2.0"
] | permissive | moritayasuaki/lean | 9f666c323cb6fa1f31ac597d777914aed41e3b7a | ae96ebf6ee953088c235ff7ae0e8c95066ba8001 | refs/heads/master | 1,611,135,440,814 | 1,493,852,869,000 | 1,493,852,869,000 | 90,269,903 | 0 | 0 | null | 1,493,906,291,000 | 1,493,906,291,000 | null | UTF-8 | Lean | false | false | 8,662 | lean | /-
Copyright (c) 2016 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner
-/
import init.meta.tactic .utils .trim
open expr list tactic monad decidable
namespace super
meta def is_local_not (local_false : expr) (e : expr) : option expr :=
match e with
| (pi _ _ a b) := if b = local_false then some a else none
| _ := if local_false = ```(false) then is_not e else none
end
meta structure clause :=
(num_quants : ℕ)
(num_lits : ℕ)
(proof : expr)
(type : expr)
(local_false : expr)
namespace clause
meta def num_binders (c : clause) : ℕ := num_quants c + num_lits c
meta def inst (c : clause) (e : expr) : clause :=
(if num_quants c > 0
then mk (num_quants c - 1) (num_lits c)
else mk 0 (num_lits c - 1))
(app (proof c) e) (instantiate_var (binding_body (type c)) e) c.local_false
meta def instn (c : clause) (es : list expr) : clause :=
foldr (λe c', inst c' e) c es
meta def open_const (c : clause) : tactic (clause × expr) := do
n ← mk_fresh_name,
b ← return $ local_const n (binding_name (type c)) (binding_info (type c)) (binding_domain (type c)),
return (inst c b, b)
meta def open_meta (c : clause) : tactic (clause × expr) := do
b ← mk_meta_var (binding_domain (type c)),
return (inst c b, b)
meta def close_const (c : clause) (e : expr) : clause :=
match e with
| local_const uniq pp info t :=
let abst_type' := abstract_local (type c) (local_uniq_name e) in
let type' := pi pp binder_info.default t (abstract_local (type c) uniq) in
let abs_prf := abstract_local (proof c) uniq in
let proof' := lambdas [e] c.proof in
if num_quants c > 0 ∨ has_var abst_type' then
{ c with num_quants := c.num_quants + 1, proof := proof', type := type' }
else
{ c with num_lits := c.num_lits + 1, proof := proof', type := type' }
| _ := ⟨0, 0, default expr, default expr, default expr⟩
end
meta def open_constn : clause → ℕ → tactic (clause × list expr)
| c 0 := return (c, nil)
| c (n+1) := do
(c', b) ← open_const c,
(c'', bs) ← open_constn c' n,
return (c'', b::bs)
meta def open_metan : clause → ℕ → tactic (clause × list expr)
| c 0 := return (c, nil)
| c (n+1) := do
(c', b) ← open_meta c,
(c'', bs) ← open_metan c' n,
return (c'', b::bs)
meta def close_constn : clause → list expr → clause
| c [] := c
| c (b::bs') := close_const (close_constn c bs') b
private meta def parse_clause (local_false : expr) : expr → expr → tactic clause
| proof (pi n bi d b) := do
lc_n ← mk_fresh_name,
lc ← return $ local_const lc_n n bi d,
c ← parse_clause (app proof lc) (instantiate_var b lc),
return $ c.close_const $ local_const lc_n n binder_info.default d
| proof ```(not %%formula) := parse_clause proof ```(%%formula → false)
| proof type :=
if type = local_false then do
return { num_quants := 0, num_lits := 0, proof := proof, type := type, local_false := local_false }
else do
univ ← infer_univ type,
not_type ← return $ imp type local_false,
parse_clause (lam `H binder_info.default not_type $ app (mk_var 0) proof) (imp not_type local_false)
meta def of_proof_and_type (local_false proof type : expr) : tactic clause :=
parse_clause local_false proof type
meta def of_proof (local_false proof : expr) : tactic clause := do
type ← infer_type proof,
of_proof_and_type local_false proof type
meta def of_classical_proof (proof : expr) : tactic clause :=
of_proof ```(false) proof
meta def inst_mvars (c : clause) : tactic clause := do
proof' ← instantiate_mvars (proof c),
type' ← instantiate_mvars (type c),
return { c with proof := proof', type := type' }
meta inductive literal
| left : expr → literal
| right : expr → literal
namespace literal
meta instance : decidable_eq literal := by mk_dec_eq_instance
meta def formula : literal → expr
| (left f) := f
| (right f) := f
meta def is_neg : literal → bool
| (left _) := tt
| (right _) := ff
meta def is_pos (l : literal) : bool := bnot l.is_neg
meta def to_formula (l : literal) : expr :=
if l.is_neg
then app (const ``not []) l.formula
else formula l
meta def type_str : literal → string
| (literal.left _) := "left"
| (literal.right _) := "right"
meta instance : has_to_tactic_format literal :=
⟨λl, do
pp_f ← pp l.formula,
return $ to_fmt l.type_str ++ " (" ++ pp_f ++ ")"⟩
end literal
private meta def get_binding_body : expr → ℕ → expr
| e 0 := e
| e (i+1) := get_binding_body e.binding_body i
meta def get_binder (e : expr) (i : nat) :=
binding_domain (get_binding_body e i)
meta def validate (c : clause) : tactic unit := do
concl ← return $ get_binding_body c.type c.num_binders,
unify concl c.local_false
<|> (do pp_concl ← pp concl, pp_lf ← pp c.local_false,
fail $ to_fmt "wrong local false: " ++ pp_concl ++ " =!= " ++ pp_lf),
type' ← infer_type c.proof,
unify c.type type' <|> (do pp_ty ← pp c.type, pp_ty' ← pp type',
fail (to_fmt "wrong type: " ++ pp_ty ++ " =!= " ++ pp_ty'))
meta def get_lit (c : clause) (i : nat) : literal :=
let bind := get_binder (type c) (num_quants c + i) in
match is_local_not c.local_false bind with
| some formula := literal.right formula
| none := literal.left bind
end
meta def lits_where (c : clause) (p : literal → bool) : list nat :=
list.filter (λl, p (get_lit c l)) (range (num_lits c))
meta def get_lits (c : clause) : list literal :=
list.map (get_lit c) (range c.num_lits)
private meta def tactic_format (c : clause) : tactic format := do
c ← c.open_metan c.num_quants,
pp (do l ← c.1.get_lits, [l.to_formula])
meta instance : has_to_tactic_format clause := ⟨tactic_format⟩
meta def is_maximal (gt : expr → expr → bool) (c : clause) (i : nat) : bool :=
list.empty (list.filter (λj, gt (get_lit c j).formula (get_lit c i).formula) (range c.num_lits))
meta def normalize (c : clause) : tactic clause := do
opened ← open_constn c (num_binders c),
lconsts_in_types ← return $ contained_lconsts_list (list.map local_type opened.2),
quants' ← return $ list.filter (λlc, rb_map.contains lconsts_in_types (local_uniq_name lc))
opened.2,
lits' ← return $ list.filter (λlc, ¬rb_map.contains lconsts_in_types (local_uniq_name lc))
opened.2,
return $ close_constn opened.1 (quants' ++ lits')
meta def whnf_head_lit (c : clause) : tactic clause := do
atom' ← whnf $ literal.formula $ get_lit c 0,
return $
if literal.is_neg (get_lit c 0) then
{ c with type := ```(%%atom' → %%(binding_body c.type)) }
else
{ c with type := ```(not %%atom' → %%(c.type.binding_body)) }
end clause
meta def unify_lit (l1 l2 : clause.literal) : tactic unit :=
if clause.literal.is_pos l1 = clause.literal.is_pos l2 then
unify (clause.literal.formula l1) (clause.literal.formula l2) transparency.none
else
fail "cannot unify literals"
-- FIXME: this is most definitely broken with meta-variables that were already in the goal
meta def sort_and_constify_metas : list expr → tactic (list expr)
| exprs_with_metas := do
inst_exprs ← mapm instantiate_mvars exprs_with_metas,
metas ← return $ inst_exprs >>= get_metas,
match list.filter (λm, ¬has_meta_var (get_meta_type m)) metas with
| [] :=
if metas.empty then
return []
else do
for' metas (λm, do trace (expr.to_string m), t ← infer_type m, trace (expr.to_string t)),
fail "could not sort metas"
| ((mvar n t) :: _) := do
c ← infer_type (mvar n t) >>= mk_local_def `x,
unify c (mvar n t),
rest ← sort_and_constify_metas metas,
c ← instantiate_mvars c,
return ([c] ++ rest)
| _ := failed
end
namespace clause
meta def meta_closure (metas : list expr) (qf : clause) : tactic clause := do
bs ← sort_and_constify_metas metas,
qf' ← clause.inst_mvars qf,
clause.inst_mvars $ clause.close_constn qf' bs
private meta def distinct' (local_false : expr) : list expr → expr → clause
| [] proof := ⟨ 0, 0, proof, local_false, local_false ⟩
| (h::hs) proof :=
let (dups, rest) := partition (λh' : expr, h.local_type = h'.local_type) hs,
proof_wo_dups := foldl (λproof (h' : expr),
instantiate_var (abstract_local proof h'.local_uniq_name) h)
proof dups in
(distinct' rest proof_wo_dups).close_const h
meta def distinct (c : clause) : tactic clause := do
(qf, vs) ← c.open_constn c.num_quants,
(fls, hs) ← qf.open_constn qf.num_lits,
return $ (distinct' c.local_false hs fls.proof).close_constn vs
end clause
end super
|
a79ff831cf3a672e62fdb388d9b148f0edb7f753 | 7282d49021d38dacd06c4ce45a48d09627687fe0 | /tests/lean/eq3.lean | a8c4741ae415676ca3d529047742f048a8ff7109 | [
"Apache-2.0"
] | permissive | steveluc/lean | 5a0b4431acefaf77f15b25bbb49294c2449923ad | 92ba4e8b2d040a799eda7deb8d2a7cdd3e69c496 | refs/heads/master | 1,611,332,256,930 | 1,391,013,244,000 | 1,391,013,244,000 | 16,361,079 | 1 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 199 | lean | import heq
variable Vector : Nat -> Type
variable n : Nat
variable v1 : Vector n
variable v2 : Vector (n + 0)
variable v3 : Vector (0 + n)
axiom H1 : v1 == v2
axiom H2 : v2 == v3
check htrans H1 H2
|
bc4f924e5b06c12cccd9664de641f7b430578c0f | 77c5b91fae1b966ddd1db969ba37b6f0e4901e88 | /src/number_theory/arithmetic_function.lean | 0019a5e08fb337dd6ef6ff1226c05d5bfe2a7db9 | [
"Apache-2.0"
] | permissive | dexmagic/mathlib | ff48eefc56e2412429b31d4fddd41a976eb287ce | 7a5d15a955a92a90e1d398b2281916b9c41270b2 | refs/heads/master | 1,693,481,322,046 | 1,633,360,193,000 | 1,633,360,193,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 32,509 | lean | /-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import algebra.big_operators.ring
import number_theory.divisors
import algebra.squarefree
import algebra.invertible
/-!
# Arithmetic Functions and Dirichlet Convolution
This file defines arithmetic functions, which are functions from `ℕ` to a specified type that map 0
to 0. In the literature, they are often instead defined as functions from `ℕ+`. These arithmetic
functions are endowed with a multiplication, given by Dirichlet convolution, and pointwise addition,
to form the Dirichlet ring.
## Main Definitions
* `arithmetic_function R` consists of functions `f : ℕ → R` such that `f 0 = 0`.
* An arithmetic function `f` `is_multiplicative` when `x.coprime y → f (x * y) = f x * f y`.
* The pointwise operations `pmul` and `ppow` differ from the multiplication
and power instances on `arithmetic_function R`, which use Dirichlet multiplication.
* `ζ` is the arithmetic function such that `ζ x = 1` for `0 < x`.
* `σ k` is the arithmetic function such that `σ k x = ∑ y in divisors x, y ^ k` for `0 < x`.
* `pow k` is the arithmetic function such that `pow k x = x ^ k` for `0 < x`.
* `id` is the identity arithmetic function on `ℕ`.
* `ω n` is the number of distinct prime factors of `n`.
* `Ω n` is the number of prime factors of `n` counted with multiplicity.
* `μ` is the Möbius function.
## Main Results
* Several forms of Möbius inversion:
* `sum_eq_iff_sum_mul_moebius_eq` for functions to a `comm_ring`
* `sum_eq_iff_sum_smul_moebius_eq` for functions to an `add_comm_group`
* `prod_eq_iff_prod_pow_moebius_eq` for functions to a `comm_group`
* `prod_eq_iff_prod_pow_moebius_eq_of_nonzero` for functions to a `comm_group_with_zero`
## Notation
The arithmetic functions `ζ` and `σ` have Greek letter names, which are localized notation in
the namespace `arithmetic_function`.
## Tags
arithmetic functions, dirichlet convolution, divisors
-/
open finset
open_locale big_operators
namespace nat
variable (R : Type*)
/-- An arithmetic function is a function from `ℕ` that maps 0 to 0. In the literature, they are
often instead defined as functions from `ℕ+`. Multiplication on `arithmetic_functions` is by
Dirichlet convolution. -/
@[derive [has_coe_to_fun, has_zero, inhabited]]
def arithmetic_function [has_zero R] := zero_hom ℕ R
variable {R}
namespace arithmetic_function
section has_zero
variable [has_zero R]
@[simp] lemma to_fun_eq (f : arithmetic_function R) : f.to_fun = f := rfl
@[simp]
lemma map_zero {f : arithmetic_function R} : f 0 = 0 :=
zero_hom.map_zero' f
theorem coe_inj {f g : arithmetic_function R} : (f : ℕ → R) = g ↔ f = g :=
⟨λ h, zero_hom.coe_inj h, λ h, h ▸ rfl⟩
@[simp]
lemma zero_apply {x : ℕ} : (0 : arithmetic_function R) x = 0 :=
zero_hom.zero_apply x
@[ext] theorem ext ⦃f g : arithmetic_function R⦄ (h : ∀ x, f x = g x) : f = g :=
zero_hom.ext h
theorem ext_iff {f g : arithmetic_function R} : f = g ↔ ∀ x, f x = g x :=
zero_hom.ext_iff
section has_one
variable [has_one R]
instance : has_one (arithmetic_function R) := ⟨⟨λ x, ite (x = 1) 1 0, rfl⟩⟩
@[simp]
lemma one_one : (1 : arithmetic_function R) 1 = 1 := rfl
@[simp]
lemma one_apply_ne {x : ℕ} (h : x ≠ 1) : (1 : arithmetic_function R) x = 0 := if_neg h
end has_one
end has_zero
instance nat_coe [has_zero R] [has_one R] [has_add R] :
has_coe (arithmetic_function ℕ) (arithmetic_function R) :=
⟨λ f, ⟨↑(f : ℕ → ℕ), by { transitivity ↑(f 0), refl, simp }⟩⟩
@[simp]
lemma nat_coe_nat (f : arithmetic_function ℕ) :
(↑f : arithmetic_function ℕ) = f :=
ext $ λ _, cast_id _
@[simp]
lemma nat_coe_apply [has_zero R] [has_one R] [has_add R] {f : arithmetic_function ℕ} {x : ℕ} :
(f : arithmetic_function R) x = f x := rfl
instance int_coe [has_zero R] [has_one R] [has_add R] [has_neg R] :
has_coe (arithmetic_function ℤ) (arithmetic_function R) :=
⟨λ f, ⟨↑(f : ℕ → ℤ), by { transitivity ↑(f 0), refl, simp }⟩⟩
@[simp]
lemma int_coe_int (f : arithmetic_function ℤ) :
(↑f : arithmetic_function ℤ) = f :=
ext $ λ _, int.cast_id _
@[simp]
lemma int_coe_apply [has_zero R] [has_one R] [has_add R] [has_neg R]
{f : arithmetic_function ℤ} {x : ℕ} :
(f : arithmetic_function R) x = f x := rfl
@[simp]
lemma coe_coe [has_zero R] [has_one R] [has_add R] [has_neg R] {f : arithmetic_function ℕ} :
((f : arithmetic_function ℤ) : arithmetic_function R) = f :=
by { ext, simp, }
section add_monoid
variable [add_monoid R]
instance : has_add (arithmetic_function R) := ⟨λ f g, ⟨λ n, f n + g n, by simp⟩⟩
@[simp]
lemma add_apply {f g : arithmetic_function R} {n : ℕ} : (f + g) n = f n + g n := rfl
instance : add_monoid (arithmetic_function R) :=
{ add_assoc := λ _ _ _, ext (λ _, add_assoc _ _ _),
zero_add := λ _, ext (λ _, zero_add _),
add_zero := λ _, ext (λ _, add_zero _),
.. arithmetic_function.has_zero R,
.. arithmetic_function.has_add }
end add_monoid
instance [add_comm_monoid R] : add_comm_monoid (arithmetic_function R) :=
{ add_comm := λ _ _, ext (λ _, add_comm _ _),
.. arithmetic_function.add_monoid }
instance [add_group R] : add_group (arithmetic_function R) :=
{ neg := λ f, ⟨λ n, - f n, by simp⟩,
add_left_neg := λ _, ext (λ _, add_left_neg _),
.. arithmetic_function.add_monoid }
instance [add_comm_group R] : add_comm_group (arithmetic_function R) :=
{ .. arithmetic_function.add_comm_monoid,
.. arithmetic_function.add_group }
section has_scalar
variables {M : Type*} [has_zero R] [add_comm_monoid M] [has_scalar R M]
/-- The Dirichlet convolution of two arithmetic functions `f` and `g` is another arithmetic function
such that `(f * g) n` is the sum of `f x * g y` over all `(x,y)` such that `x * y = n`. -/
instance : has_scalar (arithmetic_function R) (arithmetic_function M) :=
⟨λ f g, ⟨λ n, ∑ x in divisors_antidiagonal n, f x.fst • g x.snd, by simp⟩⟩
@[simp]
lemma smul_apply {f : arithmetic_function R} {g : arithmetic_function M} {n : ℕ} :
(f • g) n = ∑ x in divisors_antidiagonal n, f x.fst • g x.snd := rfl
end has_scalar
/-- The Dirichlet convolution of two arithmetic functions `f` and `g` is another arithmetic function
such that `(f * g) n` is the sum of `f x * g y` over all `(x,y)` such that `x * y = n`. -/
instance [semiring R] : has_mul (arithmetic_function R) := ⟨(•)⟩
@[simp]
lemma mul_apply [semiring R] {f g : arithmetic_function R} {n : ℕ} :
(f * g) n = ∑ x in divisors_antidiagonal n, f x.fst * g x.snd := rfl
section module
variables {M : Type*} [semiring R] [add_comm_monoid M] [module R M]
lemma mul_smul' (f g : arithmetic_function R) (h : arithmetic_function M) :
(f * g) • h = f • g • h :=
begin
ext n,
simp only [mul_apply, smul_apply, sum_smul, mul_smul, smul_sum, finset.sum_sigma'],
apply finset.sum_bij,
swap 5,
{ rintros ⟨⟨i,j⟩, ⟨k,l⟩⟩ H, exact ⟨(k, l*j), (l, j)⟩ },
{ rintros ⟨⟨i,j⟩, ⟨k,l⟩⟩ H,
simp only [finset.mem_sigma, mem_divisors_antidiagonal] at H ⊢,
rcases H with ⟨⟨rfl, n0⟩, rfl, i0⟩,
refine ⟨⟨(mul_assoc _ _ _).symm, n0⟩, rfl, _⟩,
rw mul_ne_zero_iff at *,
exact ⟨i0.2, n0.2⟩, },
{ rintros ⟨⟨i,j⟩, ⟨k,l⟩⟩ H, simp only [mul_assoc] },
{ rintros ⟨⟨a,b⟩, ⟨c,d⟩⟩ ⟨⟨i,j⟩, ⟨k,l⟩⟩ H₁ H₂,
simp only [finset.mem_sigma, mem_divisors_antidiagonal,
and_imp, prod.mk.inj_iff, add_comm, heq_iff_eq] at H₁ H₂ ⊢,
rintros rfl h2 rfl rfl,
exact ⟨⟨eq.trans H₁.2.1.symm H₂.2.1, rfl⟩, rfl, rfl⟩ },
{ rintros ⟨⟨i,j⟩, ⟨k,l⟩⟩ H, refine ⟨⟨(i*k, l), (i, k)⟩, _, _⟩,
{ simp only [finset.mem_sigma, mem_divisors_antidiagonal] at H ⊢,
rcases H with ⟨⟨rfl, n0⟩, rfl, j0⟩,
refine ⟨⟨mul_assoc _ _ _, n0⟩, rfl, _⟩,
rw mul_ne_zero_iff at *,
exact ⟨n0.1, j0.1⟩ },
{ simp only [true_and, mem_divisors_antidiagonal, and_true, prod.mk.inj_iff, eq_self_iff_true,
ne.def, mem_sigma, heq_iff_eq] at H ⊢,
rw H.2.1 } }
end
lemma one_smul' (b : arithmetic_function M) :
(1 : arithmetic_function R) • b = b :=
begin
ext,
rw smul_apply,
by_cases x0 : x = 0, {simp [x0]},
have h : {(1,x)} ⊆ divisors_antidiagonal x := by simp [x0],
rw ← sum_subset h, {simp},
intros y ymem ynmem,
have y1ne : y.fst ≠ 1,
{ intro con,
simp only [con, mem_divisors_antidiagonal, one_mul, ne.def] at ymem,
simp only [mem_singleton, prod.ext_iff] at ynmem,
tauto },
simp [y1ne],
end
end module
section semiring
variable [semiring R]
instance : monoid (arithmetic_function R) :=
{ one_mul := one_smul',
mul_one := λ f,
begin
ext,
rw mul_apply,
by_cases x0 : x = 0, {simp [x0]},
have h : {(x,1)} ⊆ divisors_antidiagonal x := by simp [x0],
rw ← sum_subset h, {simp},
intros y ymem ynmem,
have y2ne : y.snd ≠ 1,
{ intro con,
simp only [con, mem_divisors_antidiagonal, mul_one, ne.def] at ymem,
simp only [mem_singleton, prod.ext_iff] at ynmem,
tauto },
simp [y2ne],
end,
mul_assoc := mul_smul',
.. arithmetic_function.has_one,
.. arithmetic_function.has_mul }
instance : semiring (arithmetic_function R) :=
{ zero_mul := λ f, by { ext, simp only [mul_apply, zero_mul, sum_const_zero, zero_apply] },
mul_zero := λ f, by { ext, simp only [mul_apply, sum_const_zero, mul_zero, zero_apply] },
left_distrib := λ a b c, by { ext, simp only [←sum_add_distrib, mul_add, mul_apply, add_apply] },
right_distrib := λ a b c, by { ext, simp only [←sum_add_distrib, add_mul, mul_apply, add_apply] },
.. arithmetic_function.has_zero R,
.. arithmetic_function.has_mul,
.. arithmetic_function.has_add,
.. arithmetic_function.add_comm_monoid,
.. arithmetic_function.monoid }
end semiring
instance [comm_semiring R] : comm_semiring (arithmetic_function R) :=
{ mul_comm := λ f g, by { ext,
rw [mul_apply, ← map_swap_divisors_antidiagonal, sum_map],
simp [mul_comm] },
.. arithmetic_function.semiring }
instance [comm_ring R] : comm_ring (arithmetic_function R) :=
{ .. arithmetic_function.add_comm_group,
.. arithmetic_function.comm_semiring }
instance {M : Type*} [semiring R] [add_comm_monoid M] [module R M] :
module (arithmetic_function R) (arithmetic_function M) :=
{ one_smul := one_smul',
mul_smul := mul_smul',
smul_add := λ r x y, by { ext, simp only [sum_add_distrib, smul_add, smul_apply, add_apply] },
smul_zero := λ r, by { ext, simp only [smul_apply, sum_const_zero, smul_zero, zero_apply] },
add_smul := λ r s x, by { ext, simp only [add_smul, sum_add_distrib, smul_apply, add_apply] },
zero_smul := λ r, by { ext, simp only [smul_apply, sum_const_zero, zero_smul, zero_apply] }, }
section zeta
/-- `ζ 0 = 0`, otherwise `ζ x = 1`. The Dirichlet Series is the Riemann ζ. -/
def zeta : arithmetic_function ℕ :=
⟨λ x, ite (x = 0) 0 1, rfl⟩
localized "notation `ζ` := zeta" in arithmetic_function
@[simp]
lemma zeta_apply {x : ℕ} : ζ x = if (x = 0) then 0 else 1 := rfl
lemma zeta_apply_ne {x : ℕ} (h : x ≠ 0) : ζ x = 1 := if_neg h
@[simp]
theorem coe_zeta_mul_apply [semiring R] {f : arithmetic_function R} {x : ℕ} :
(↑ζ * f) x = ∑ i in divisors x, f i :=
begin
rw mul_apply,
transitivity ∑ i in divisors_antidiagonal x, f i.snd,
{ apply sum_congr rfl,
intros i hi,
rcases mem_divisors_antidiagonal.1 hi with ⟨rfl, h⟩,
rw [nat_coe_apply, zeta_apply_ne (left_ne_zero_of_mul h), cast_one, one_mul] },
{ apply sum_bij (λ i h, prod.snd i),
{ rintros ⟨a, b⟩ h, simp [snd_mem_divisors_of_mem_antidiagonal h] },
{ rintros ⟨a, b⟩ h, refl },
{ rintros ⟨a1, b1⟩ ⟨a2, b2⟩ h1 h2 h,
dsimp at h,
rw h at *,
rw mem_divisors_antidiagonal at *,
ext, swap, {refl},
simp only [prod.fst, prod.snd] at *,
apply nat.eq_of_mul_eq_mul_right _ (eq.trans h1.1 h2.1.symm),
rcases h1 with ⟨rfl, h⟩,
apply nat.pos_of_ne_zero (right_ne_zero_of_mul h) },
{ intros a ha,
rcases mem_divisors.1 ha with ⟨⟨b, rfl⟩, ne0⟩,
use (b, a),
simp [ne0, mul_comm] } }
end
theorem coe_zeta_smul_apply {M : Type*} [comm_ring R] [add_comm_group M] [module R M]
{f : arithmetic_function M} {x : ℕ} :
((↑ζ : arithmetic_function R) • f) x = ∑ i in divisors x, f i :=
begin
rw smul_apply,
transitivity ∑ i in divisors_antidiagonal x, f i.snd,
{ apply sum_congr rfl,
intros i hi,
rcases mem_divisors_antidiagonal.1 hi with ⟨rfl, h⟩,
rw [nat_coe_apply, zeta_apply_ne (left_ne_zero_of_mul h), cast_one, one_smul] },
{ apply sum_bij (λ i h, prod.snd i),
{ rintros ⟨a, b⟩ h, simp [snd_mem_divisors_of_mem_antidiagonal h] },
{ rintros ⟨a, b⟩ h, refl },
{ rintros ⟨a1, b1⟩ ⟨a2, b2⟩ h1 h2 h,
dsimp at h,
rw h at *,
rw mem_divisors_antidiagonal at *,
ext, swap, {refl},
simp only [prod.fst, prod.snd] at *,
apply nat.eq_of_mul_eq_mul_right _ (eq.trans h1.1 h2.1.symm),
rcases h1 with ⟨rfl, h⟩,
apply nat.pos_of_ne_zero (right_ne_zero_of_mul h) },
{ intros a ha,
rcases mem_divisors.1 ha with ⟨⟨b, rfl⟩, ne0⟩,
use (b, a),
simp [ne0, mul_comm] } }
end
@[simp]
theorem coe_mul_zeta_apply [semiring R] {f : arithmetic_function R} {x : ℕ} :
(f * ζ) x = ∑ i in divisors x, f i :=
begin
apply opposite.op_injective,
rw [op_sum],
convert @coe_zeta_mul_apply Rᵒᵖ _ { to_fun := opposite.op ∘ f, map_zero' := by simp} x,
rw [mul_apply, mul_apply, op_sum],
conv_lhs { rw ← map_swap_divisors_antidiagonal, },
rw sum_map,
apply sum_congr rfl,
intros y hy,
by_cases h1 : y.fst = 0,
{ simp [function.comp_apply, h1] },
{ simp only [h1, mul_one, one_mul, prod.fst_swap, function.embedding.coe_fn_mk, prod.snd_swap,
if_false, zeta_apply, zero_hom.coe_mk, nat_coe_apply, cast_one] }
end
theorem zeta_mul_apply {f : arithmetic_function ℕ} {x : ℕ} :
(ζ * f) x = ∑ i in divisors x, f i :=
by rw [← nat_coe_nat ζ, coe_zeta_mul_apply]
theorem mul_zeta_apply {f : arithmetic_function ℕ} {x : ℕ} :
(f * ζ) x = ∑ i in divisors x, f i :=
by rw [← nat_coe_nat ζ, coe_mul_zeta_apply]
end zeta
open_locale arithmetic_function
section pmul
/-- This is the pointwise product of `arithmetic_function`s. -/
def pmul [mul_zero_class R] (f g : arithmetic_function R) :
arithmetic_function R :=
⟨λ x, f x * g x, by simp⟩
@[simp]
lemma pmul_apply [mul_zero_class R] {f g : arithmetic_function R} {x : ℕ} :
f.pmul g x = f x * g x := rfl
lemma pmul_comm [comm_monoid_with_zero R] (f g : arithmetic_function R) :
f.pmul g = g.pmul f :=
by { ext, simp [mul_comm] }
variable [semiring R]
@[simp]
lemma pmul_zeta (f : arithmetic_function R) : f.pmul ↑ζ = f :=
begin
ext x,
cases x;
simp [nat.succ_ne_zero],
end
@[simp]
lemma zeta_pmul (f : arithmetic_function R) : (ζ : arithmetic_function R).pmul f = f :=
begin
ext x,
cases x;
simp [nat.succ_ne_zero],
end
/-- This is the pointwise power of `arithmetic_function`s. -/
def ppow (f : arithmetic_function R) (k : ℕ) :
arithmetic_function R :=
if h0 : k = 0 then ζ else ⟨λ x, (f x) ^ k,
by { rw [map_zero], exact zero_pow (nat.pos_of_ne_zero h0) }⟩
@[simp]
lemma ppow_zero {f : arithmetic_function R} : f.ppow 0 = ζ :=
by rw [ppow, dif_pos rfl]
@[simp]
lemma ppow_apply {f : arithmetic_function R} {k x : ℕ} (kpos : 0 < k) :
f.ppow k x = (f x) ^ k :=
by { rw [ppow, dif_neg (ne_of_gt kpos)], refl }
lemma ppow_succ {f : arithmetic_function R} {k : ℕ} :
f.ppow (k + 1) = f.pmul (f.ppow k) :=
begin
ext x,
rw [ppow_apply (nat.succ_pos k), pow_succ],
induction k; simp,
end
lemma ppow_succ' {f : arithmetic_function R} {k : ℕ} {kpos : 0 < k} :
f.ppow (k + 1) = (f.ppow k).pmul f :=
begin
ext x,
rw [ppow_apply (nat.succ_pos k), pow_succ'],
induction k; simp,
end
end pmul
/-- Multiplicative functions -/
def is_multiplicative [monoid_with_zero R] (f : arithmetic_function R) : Prop :=
f 1 = 1 ∧ (∀ {m n : ℕ}, m.coprime n → f (m * n) = f m * f n)
namespace is_multiplicative
section monoid_with_zero
variable [monoid_with_zero R]
@[simp]
lemma map_one {f : arithmetic_function R} (h : f.is_multiplicative) : f 1 = 1 :=
h.1
@[simp]
lemma map_mul_of_coprime {f : arithmetic_function R} (hf : f.is_multiplicative)
{m n : ℕ} (h : m.coprime n) : f (m * n) = f m * f n :=
hf.2 h
end monoid_with_zero
lemma nat_cast {f : arithmetic_function ℕ} [semiring R] (h : f.is_multiplicative) :
is_multiplicative (f : arithmetic_function R) :=
⟨by simp [h], λ m n cop, by simp [cop, h]⟩
lemma int_cast {f : arithmetic_function ℤ} [ring R] (h : f.is_multiplicative) :
is_multiplicative (f : arithmetic_function R) :=
⟨by simp [h], λ m n cop, by simp [cop, h]⟩
lemma mul [comm_semiring R] {f g : arithmetic_function R}
(hf : f.is_multiplicative) (hg : g.is_multiplicative) :
is_multiplicative (f * g) :=
⟨by { simp [hf, hg], }, begin
simp only [mul_apply],
intros m n cop,
rw sum_mul_sum,
symmetry,
apply sum_bij (λ (x : (ℕ × ℕ) × ℕ × ℕ) h, (x.1.1 * x.2.1, x.1.2 * x.2.2)),
{ rintros ⟨⟨a1, a2⟩, ⟨b1, b2⟩⟩ h,
simp only [mem_divisors_antidiagonal, ne.def, mem_product] at h,
rcases h with ⟨⟨rfl, ha⟩, ⟨rfl, hb⟩⟩,
simp only [mem_divisors_antidiagonal, nat.mul_eq_zero, ne.def],
split, {ring},
rw nat.mul_eq_zero at *,
apply not_or ha hb },
{ rintros ⟨⟨a1, a2⟩, ⟨b1, b2⟩⟩ h,
simp only [mem_divisors_antidiagonal, ne.def, mem_product] at h,
rcases h with ⟨⟨rfl, ha⟩, ⟨rfl, hb⟩⟩,
dsimp only,
rw [hf.map_mul_of_coprime cop.coprime_mul_right.coprime_mul_right_right,
hg.map_mul_of_coprime cop.coprime_mul_left.coprime_mul_left_right],
ring, },
{ rintros ⟨⟨a1, a2⟩, ⟨b1, b2⟩⟩ ⟨⟨c1, c2⟩, ⟨d1, d2⟩⟩ hab hcd h,
simp only [mem_divisors_antidiagonal, ne.def, mem_product] at hab,
rcases hab with ⟨⟨rfl, ha⟩, ⟨rfl, hb⟩⟩,
simp only [mem_divisors_antidiagonal, ne.def, mem_product] at hcd,
simp only [prod.mk.inj_iff] at h,
ext; dsimp only,
{ transitivity nat.gcd (a1 * a2) (a1 * b1),
{ rw [nat.gcd_mul_left, cop.coprime_mul_left.coprime_mul_right_right.gcd_eq_one, mul_one] },
{ rw [← hcd.1.1, ← hcd.2.1] at cop,
rw [← hcd.1.1, h.1, nat.gcd_mul_left,
cop.coprime_mul_left.coprime_mul_right_right.gcd_eq_one, mul_one] } },
{ transitivity nat.gcd (a1 * a2) (a2 * b2),
{ rw [mul_comm, nat.gcd_mul_left, cop.coprime_mul_right.coprime_mul_left_right.gcd_eq_one,
mul_one] },
{ rw [← hcd.1.1, ← hcd.2.1] at cop,
rw [← hcd.1.1, h.2, mul_comm, nat.gcd_mul_left,
cop.coprime_mul_right.coprime_mul_left_right.gcd_eq_one, mul_one] } },
{ transitivity nat.gcd (b1 * b2) (a1 * b1),
{ rw [mul_comm, nat.gcd_mul_right,
cop.coprime_mul_right.coprime_mul_left_right.symm.gcd_eq_one, one_mul] },
{ rw [← hcd.1.1, ← hcd.2.1] at cop,
rw [← hcd.2.1, h.1, mul_comm c1 d1, nat.gcd_mul_left,
cop.coprime_mul_right.coprime_mul_left_right.symm.gcd_eq_one, mul_one] } },
{ transitivity nat.gcd (b1 * b2) (a2 * b2),
{ rw [nat.gcd_mul_right,
cop.coprime_mul_left.coprime_mul_right_right.symm.gcd_eq_one, one_mul] },
{ rw [← hcd.1.1, ← hcd.2.1] at cop,
rw [← hcd.2.1, h.2, nat.gcd_mul_right,
cop.coprime_mul_left.coprime_mul_right_right.symm.gcd_eq_one, one_mul] } } },
{ rintros ⟨b1, b2⟩ h,
simp only [mem_divisors_antidiagonal, ne.def, mem_product] at h,
use ((b1.gcd m, b2.gcd m), (b1.gcd n, b2.gcd n)),
simp only [exists_prop, prod.mk.inj_iff, ne.def, mem_product, mem_divisors_antidiagonal],
rw [← cop.gcd_mul _, ← cop.gcd_mul _, ← h.1, nat.gcd_mul_gcd_of_coprime_of_mul_eq_mul cop h.1,
nat.gcd_mul_gcd_of_coprime_of_mul_eq_mul cop.symm _],
{ rw [nat.mul_eq_zero, decidable.not_or_iff_and_not] at h, simp [h.2.1, h.2.2] },
rw [mul_comm n m, h.1] }
end⟩
lemma pmul [comm_semiring R] {f g : arithmetic_function R}
(hf : f.is_multiplicative) (hg : g.is_multiplicative) :
is_multiplicative (f.pmul g) :=
⟨by { simp [hf, hg], }, λ m n cop, begin
simp only [pmul_apply, hf.map_mul_of_coprime cop, hg.map_mul_of_coprime cop],
ring,
end⟩
end is_multiplicative
section special_functions
/-- The identity on `ℕ` as an `arithmetic_function`. -/
def id : arithmetic_function ℕ := ⟨id, rfl⟩
@[simp]
lemma id_apply {x : ℕ} : id x = x := rfl
/-- `pow k n = n ^ k`, except `pow 0 0 = 0`. -/
def pow (k : ℕ) : arithmetic_function ℕ := id.ppow k
@[simp]
lemma pow_apply {k n : ℕ} : pow k n = if (k = 0 ∧ n = 0) then 0 else n ^ k :=
begin
cases k,
{ simp [pow] },
simp [pow, (ne_of_lt (nat.succ_pos k)).symm],
end
/-- `σ k n` is the sum of the `k`th powers of the divisors of `n` -/
def sigma (k : ℕ) : arithmetic_function ℕ :=
⟨λ n, ∑ d in divisors n, d ^ k, by simp⟩
localized "notation `σ` := sigma" in arithmetic_function
@[simp]
lemma sigma_apply {k n : ℕ} : σ k n = ∑ d in divisors n, d ^ k := rfl
lemma sigma_one_apply {n : ℕ} : σ 1 n = ∑ d in divisors n, d := by simp
lemma zeta_mul_pow_eq_sigma {k : ℕ} : ζ * pow k = σ k :=
begin
ext,
rw [sigma, zeta_mul_apply],
apply sum_congr rfl,
intros x hx,
rw [pow_apply, if_neg (not_and_of_not_right _ _)],
contrapose! hx,
simp [hx],
end
lemma is_multiplicative_zeta : is_multiplicative ζ :=
⟨by simp, λ m n cop, begin
cases m, {simp},
cases n, {simp},
simp [nat.succ_ne_zero]
end⟩
lemma is_multiplicative_id : is_multiplicative arithmetic_function.id :=
⟨rfl, λ _ _ _, rfl⟩
lemma is_multiplicative.ppow [comm_semiring R] {f : arithmetic_function R}
(hf : f.is_multiplicative) {k : ℕ} :
is_multiplicative (f.ppow k) :=
begin
induction k with k hi,
{ exact is_multiplicative_zeta.nat_cast },
{ rw ppow_succ,
apply hf.pmul hi },
end
lemma is_multiplicative_pow {k : ℕ} : is_multiplicative (pow k) :=
is_multiplicative_id.ppow
lemma is_multiplicative_sigma {k : ℕ} :
is_multiplicative (sigma k) :=
begin
rw [← zeta_mul_pow_eq_sigma],
apply ((is_multiplicative_zeta).mul is_multiplicative_pow)
end
/-- `Ω n` is the number of prime factors of `n`. -/
def card_factors : arithmetic_function ℕ :=
⟨λ n, n.factors.length, by simp⟩
localized "notation `Ω` := card_factors" in arithmetic_function
lemma card_factors_apply {n : ℕ} :
Ω n = n.factors.length := rfl
@[simp]
lemma card_factors_one : Ω 1 = 0 := by simp [card_factors]
lemma card_factors_eq_one_iff_prime {n : ℕ} :
Ω n = 1 ↔ n.prime :=
begin
refine ⟨λ h, _, λ h, list.length_eq_one.2 ⟨n, factors_prime h⟩⟩,
cases n,
{ contrapose! h,
simp },
rcases list.length_eq_one.1 h with ⟨x, hx⟩,
rw [← prod_factors n.succ_pos, hx, list.prod_singleton],
apply prime_of_mem_factors,
rw [hx, list.mem_singleton]
end
lemma card_factors_mul {m n : ℕ} (m0 : m ≠ 0) (n0 : n ≠ 0) :
Ω (m * n) = Ω m + Ω n :=
by rw [card_factors_apply, card_factors_apply, card_factors_apply, ← multiset.coe_card,
← factors_eq, unique_factorization_monoid.normalized_factors_mul m0 n0, factors_eq, factors_eq,
multiset.card_add, multiset.coe_card, multiset.coe_card]
lemma card_factors_multiset_prod {s : multiset ℕ} (h0 : s.prod ≠ 0) :
Ω s.prod = (multiset.map Ω s).sum :=
begin
revert h0,
apply s.induction_on, by simp,
intros a t h h0,
rw [multiset.prod_cons, mul_ne_zero_iff] at h0,
simp [h0, card_factors_mul, h],
end
/-- `ω n` is the number of distinct prime factors of `n`. -/
def card_distinct_factors : arithmetic_function ℕ :=
⟨λ n, n.factors.erase_dup.length, by simp⟩
localized "notation `ω` := card_distinct_factors" in arithmetic_function
lemma card_distinct_factors_zero : ω 0 = 0 := by simp
lemma card_distinct_factors_apply {n : ℕ} :
ω n = n.factors.erase_dup.length := rfl
lemma card_distinct_factors_eq_card_factors_iff_squarefree {n : ℕ} (h0 : n ≠ 0) :
ω n = Ω n ↔ squarefree n :=
begin
rw [squarefree_iff_nodup_factors h0, card_distinct_factors_apply],
split; intro h,
{ rw ← list.eq_of_sublist_of_length_eq n.factors.erase_dup_sublist h,
apply list.nodup_erase_dup },
{ rw list.erase_dup_eq_self.2 h,
refl }
end
/-- `μ` is the Möbius function. If `n` is squarefree with an even number of distinct prime factors,
`μ n = 1`. If `n` is squarefree with an odd number of distinct prime factors, `μ n = -1`.
If `n` is not squarefree, `μ n = 0`. -/
def moebius : arithmetic_function ℤ :=
⟨λ n, if squarefree n then (-1) ^ (card_factors n) else 0, by simp⟩
localized "notation `μ` := moebius" in arithmetic_function
@[simp]
lemma moebius_apply_of_squarefree {n : ℕ} (h : squarefree n): μ n = (-1) ^ (card_factors n) :=
if_pos h
@[simp]
lemma moebius_eq_zero_of_not_squarefree {n : ℕ} (h : ¬ squarefree n): μ n = 0 := if_neg h
lemma moebius_ne_zero_iff_squarefree {n : ℕ} : μ n ≠ 0 ↔ squarefree n :=
begin
split; intro h,
{ contrapose! h,
simp [h] },
{ simp [h, pow_ne_zero] }
end
lemma moebius_ne_zero_iff_eq_or {n : ℕ} : μ n ≠ 0 ↔ μ n = 1 ∨ μ n = -1 :=
begin
split; intro h,
{ rw moebius_ne_zero_iff_squarefree at h,
rw moebius_apply_of_squarefree h,
apply neg_one_pow_eq_or },
{ rcases h with h | h; simp [h] }
end
open unique_factorization_monoid
@[simp] lemma coe_moebius_mul_coe_zeta [comm_ring R] : (μ * ζ : arithmetic_function R) = 1 :=
begin
ext x,
cases x,
{ simp only [divisors_zero, sum_empty, ne.def, not_false_iff, coe_mul_zeta_apply,
zero_ne_one, one_apply_ne] },
cases x,
{ simp only [moebius_apply_of_squarefree, card_factors_one, squarefree_one, divisors_one,
int.cast_one, sum_singleton, coe_mul_zeta_apply, one_one, int_coe_apply, pow_zero] },
rw [coe_mul_zeta_apply, one_apply_ne (ne_of_gt (succ_lt_succ (nat.succ_pos _)))],
simp_rw [int_coe_apply],
rw [←int.cast_sum, ← sum_filter_ne_zero],
convert int.cast_zero,
simp only [moebius_ne_zero_iff_squarefree],
suffices :
∑ (y : finset ℕ) in
(unique_factorization_monoid.normalized_factors x.succ.succ).to_finset.powerset,
ite (squarefree y.val.prod) ((-1:ℤ) ^ Ω y.val.prod) 0 = 0,
{ have h : ∑ i in _, ite (squarefree i) ((-1:ℤ) ^ Ω i) 0 = _ :=
(sum_divisors_filter_squarefree (nat.succ_ne_zero _)),
exact (eq.trans (by congr') h).trans this },
apply eq.trans (sum_congr rfl _) (sum_powerset_neg_one_pow_card_of_nonempty _),
{ intros y hy,
rw [finset.mem_powerset, ← finset.val_le_iff, multiset.to_finset_val] at hy,
have h : unique_factorization_monoid.normalized_factors y.val.prod = y.val,
{ apply factors_multiset_prod_of_irreducible,
intros z hz,
apply irreducible_of_normalized_factor _ (multiset.subset_of_le
(le_trans hy (multiset.erase_dup_le _)) hz) },
rw [if_pos],
{ rw [card_factors_apply, ← multiset.coe_card, ← factors_eq, h, finset.card] },
rw [unique_factorization_monoid.squarefree_iff_nodup_normalized_factors, h],
{ apply y.nodup },
rw [ne.def, multiset.prod_eq_zero_iff],
intro con,
rw ← h at con,
exact not_irreducible_zero (irreducible_of_normalized_factor 0 con) },
{ rw finset.nonempty,
rcases wf_dvd_monoid.exists_irreducible_factor _ (nat.succ_ne_zero _) with ⟨i, hi⟩,
{ rcases exists_mem_normalized_factors_of_dvd (nat.succ_ne_zero _) hi.1 hi.2 with ⟨j, hj, hj2⟩,
use j,
apply multiset.mem_to_finset.2 hj },
rw nat.is_unit_iff,
norm_num },
end
@[simp] lemma coe_zeta_mul_coe_moebius [comm_ring R] : (ζ * μ : arithmetic_function R) = 1 :=
by rw [mul_comm, coe_moebius_mul_coe_zeta]
@[simp] lemma moebius_mul_coe_zeta : (μ * ζ : arithmetic_function ℤ) = 1 :=
by rw [← int_coe_int μ, coe_moebius_mul_coe_zeta]
@[simp] lemma coe_zeta_mul_moebius : (ζ * μ : arithmetic_function ℤ) = 1 :=
by rw [← int_coe_int μ, coe_zeta_mul_coe_moebius]
section comm_ring
variable [comm_ring R]
instance : invertible (ζ : arithmetic_function R) :=
{ inv_of := μ,
inv_of_mul_self := coe_moebius_mul_coe_zeta,
mul_inv_of_self := coe_zeta_mul_coe_moebius}
/-- A unit in `arithmetic_function R` that evaluates to `ζ`, with inverse `μ`. -/
def zeta_unit : units (arithmetic_function R) :=
⟨ζ, μ, coe_zeta_mul_coe_moebius, coe_moebius_mul_coe_zeta⟩
@[simp]
lemma coe_zeta_unit :
((zeta_unit : units (arithmetic_function R)) : arithmetic_function R) = ζ := rfl
@[simp]
lemma inv_zeta_unit :
((zeta_unit⁻¹ : units (arithmetic_function R)) : arithmetic_function R) = μ := rfl
end comm_ring
/-- Möbius inversion for functions to an `add_comm_group`. -/
theorem sum_eq_iff_sum_smul_moebius_eq
[add_comm_group R] {f g : ℕ → R} :
(∀ (n : ℕ), 0 < n → ∑ i in (n.divisors), f i = g n) ↔
∀ (n : ℕ), 0 < n → ∑ (x : ℕ × ℕ) in n.divisors_antidiagonal, μ x.fst • g x.snd = f n :=
begin
let f' : arithmetic_function R := ⟨λ x, if x = 0 then 0 else f x, if_pos rfl⟩,
let g' : arithmetic_function R := ⟨λ x, if x = 0 then 0 else g x, if_pos rfl⟩,
transitivity (ζ : arithmetic_function ℤ) • f' = g',
{ rw ext_iff,
apply forall_congr,
intro n,
cases n, { simp },
rw coe_zeta_smul_apply,
simp only [n.succ_ne_zero, forall_prop_of_true, succ_pos', if_false, zero_hom.coe_mk],
rw sum_congr rfl (λ x hx, _),
rw (if_neg (ne_of_gt (nat.pos_of_mem_divisors hx))) },
transitivity μ • g' = f',
{ split; intro h,
{ rw [← h, ← mul_smul, moebius_mul_coe_zeta, one_smul] },
{ rw [← h, ← mul_smul, coe_zeta_mul_moebius, one_smul] } },
{ rw ext_iff,
apply forall_congr,
intro n,
cases n, { simp },
simp only [n.succ_ne_zero, forall_prop_of_true, succ_pos', smul_apply,
if_false, zero_hom.coe_mk],
rw sum_congr rfl (λ x hx, _),
rw (if_neg (ne_of_gt (nat.pos_of_mem_divisors (snd_mem_divisors_of_mem_antidiagonal hx)))) },
end
/-- Möbius inversion for functions to a `comm_ring`. -/
theorem sum_eq_iff_sum_mul_moebius_eq [comm_ring R] {f g : ℕ → R} :
(∀ (n : ℕ), 0 < n → ∑ i in (n.divisors), f i = g n) ↔
∀ (n : ℕ), 0 < n → ∑ (x : ℕ × ℕ) in n.divisors_antidiagonal, (μ x.fst : R) * g x.snd = f n :=
begin
rw sum_eq_iff_sum_smul_moebius_eq,
apply forall_congr,
intro a,
apply imp_congr (iff.refl _) (eq.congr_left (sum_congr rfl (λ x hx, _))),
rw [gsmul_eq_mul],
end
/-- Möbius inversion for functions to a `comm_group`. -/
theorem prod_eq_iff_prod_pow_moebius_eq [comm_group R] {f g : ℕ → R} :
(∀ (n : ℕ), 0 < n → ∏ i in (n.divisors), f i = g n) ↔
∀ (n : ℕ), 0 < n → ∏ (x : ℕ × ℕ) in n.divisors_antidiagonal, g x.snd ^ (μ x.fst) = f n :=
@sum_eq_iff_sum_smul_moebius_eq (additive R) _ _ _
/-- Möbius inversion for functions to a `comm_group_with_zero`. -/
theorem prod_eq_iff_prod_pow_moebius_eq_of_nonzero [comm_group_with_zero R] {f g : ℕ → R}
(hf : ∀ (n : ℕ), 0 < n → f n ≠ 0) (hg : ∀ (n : ℕ), 0 < n → g n ≠ 0) :
(∀ (n : ℕ), 0 < n → ∏ i in (n.divisors), f i = g n) ↔
∀ (n : ℕ), 0 < n → ∏ (x : ℕ × ℕ) in n.divisors_antidiagonal, g x.snd ^ (μ x.fst) = f n :=
begin
refine iff.trans (iff.trans (forall_congr (λ n, _)) (@prod_eq_iff_prod_pow_moebius_eq (units R) _
(λ n, if h : 0 < n then units.mk0 (f n) (hf n h) else 1)
(λ n, if h : 0 < n then units.mk0 (g n) (hg n h) else 1))) (forall_congr (λ n, _));
refine imp_congr_right (λ hn, _),
{ dsimp,
rw [dif_pos hn, ← units.eq_iff, ← units.coe_hom_apply, monoid_hom.map_prod, units.coe_mk0,
prod_congr rfl _],
intros x hx,
rw [dif_pos (nat.pos_of_mem_divisors hx), units.coe_hom_apply, units.coe_mk0] },
{ dsimp,
rw [dif_pos hn, ← units.eq_iff, ← units.coe_hom_apply, monoid_hom.map_prod, units.coe_mk0,
prod_congr rfl _],
intros x hx,
rw [dif_pos (nat.pos_of_mem_divisors (nat.snd_mem_divisors_of_mem_antidiagonal hx)),
units.coe_hom_apply, units.coe_gpow₀, units.coe_mk0] }
end
end special_functions
end arithmetic_function
end nat
|
eb84c9ef7d5438f25de4d5ad0d864bfc7bce86af | f618aea02cb4104ad34ecf3b9713065cc0d06103 | /src/analysis/normed_space/basic.lean | af715c62849e4f71ee23c197d3275313057b0c8f | [
"Apache-2.0"
] | permissive | joehendrix/mathlib | 84b6603f6be88a7e4d62f5b1b0cbb523bb82b9a5 | c15eab34ad754f9ecd738525cb8b5a870e834ddc | refs/heads/master | 1,589,606,591,630 | 1,555,946,393,000 | 1,555,946,393,000 | 182,813,854 | 0 | 0 | null | 1,555,946,309,000 | 1,555,946,308,000 | null | UTF-8 | Lean | false | false | 24,570 | lean | /-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Normed spaces.
Authors: Patrick Massot, Johannes Hölzl
-/
import algebra.pi_instances
import linear_algebra.basic
import topology.instances.nnreal topology.instances.complex
variables {α : Type*} {β : Type*} {γ : Type*} {ι : Type*}
local attribute [instance, priority 0] nat.cast_coe
local attribute [instance, priority 0] int.cast_coe
local attribute [instance, priority 0] rat.cast_coe
noncomputable theory
open filter metric
local notation f `→_{`:50 a `}`:0 b := tendsto f (nhds a) (nhds b)
class has_norm (α : Type*) := (norm : α → ℝ)
export has_norm (norm)
notation `∥`:1024 e:1 `∥`:1 := norm e
class normed_group (α : Type*) extends has_norm α, add_comm_group α, metric_space α :=
(dist_eq : ∀ x y, dist x y = norm (x - y))
/-- Construct a normed group from a translation invariant distance -/
def normed_group.of_add_dist [has_norm α] [add_comm_group α] [metric_space α]
(H1 : ∀ x:α, ∥x∥ = dist x 0)
(H2 : ∀ x y z : α, dist x y ≤ dist (x + z) (y + z)) : normed_group α :=
{ dist_eq := λ x y, begin
rw H1, apply le_antisymm,
{ rw [sub_eq_add_neg, ← add_right_neg y], apply H2 },
{ have := H2 (x-y) 0 y, rwa [sub_add_cancel, zero_add] at this }
end }
/-- Construct a normed group from a translation invariant distance -/
def normed_group.of_add_dist' [has_norm α] [add_comm_group α] [metric_space α]
(H1 : ∀ x:α, ∥x∥ = dist x 0)
(H2 : ∀ x y z : α, dist (x + z) (y + z) ≤ dist x y) : normed_group α :=
{ dist_eq := λ x y, begin
rw H1, apply le_antisymm,
{ have := H2 (x-y) 0 y, rwa [sub_add_cancel, zero_add] at this },
{ rw [sub_eq_add_neg, ← add_right_neg y], apply H2 }
end }
section normed_group
variables [normed_group α] [normed_group β]
lemma dist_eq_norm (g h : α) : dist g h = ∥g - h∥ :=
normed_group.dist_eq _ _
@[simp] lemma dist_zero_right (g : α) : dist g 0 = ∥g∥ :=
by { rw[dist_eq_norm], simp }
lemma norm_triangle (g h : α) : ∥g + h∥ ≤ ∥g∥ + ∥h∥ :=
calc ∥g + h∥ = ∥g - (-h)∥ : by simp
... = dist g (-h) : by simp[dist_eq_norm]
... ≤ dist g 0 + dist 0 (-h) : by apply dist_triangle
... = ∥g∥ + ∥h∥ : by simp[dist_eq_norm]
@[simp] lemma norm_nonneg (g : α) : 0 ≤ ∥g∥ :=
by { rw[←dist_zero_right], exact dist_nonneg }
lemma norm_eq_zero (g : α) : ∥g∥ = 0 ↔ g = 0 :=
by { rw[←dist_zero_right], exact dist_eq_zero }
@[simp] lemma norm_zero : ∥(0:α)∥ = 0 := (norm_eq_zero _).2 (by simp)
lemma norm_triangle_sum {β} : ∀(s : finset β) (f : β → α), ∥s.sum f∥ ≤ s.sum (λa, ∥ f a ∥) :=
finset.le_sum_of_subadditive norm norm_zero norm_triangle
lemma norm_pos_iff (g : α) : 0 < ∥ g ∥ ↔ g ≠ 0 :=
begin
split ; intro h ; rw[←dist_zero_right] at *,
{ exact dist_pos.1 h },
{ exact dist_pos.2 h }
end
lemma norm_le_zero_iff (g : α) : ∥g∥ ≤ 0 ↔ g = 0 :=
by { rw[←dist_zero_right], exact dist_le_zero }
@[simp] lemma norm_neg (g : α) : ∥-g∥ = ∥g∥ :=
calc ∥-g∥ = ∥0 - g∥ : by simp
... = dist 0 g : (dist_eq_norm 0 g).symm
... = dist g 0 : dist_comm _ _
... = ∥g - 0∥ : (dist_eq_norm g 0)
... = ∥g∥ : by simp
lemma norm_reverse_triangle' (a b : α) : ∥a∥ - ∥b∥ ≤ ∥a - b∥ :=
by simpa using add_le_add (norm_triangle (a - b) (b)) (le_refl (-∥b∥))
lemma norm_reverse_triangle (a b : α) : abs(∥a∥ - ∥b∥) ≤ ∥a - b∥ :=
suffices -(∥a∥ - ∥b∥) ≤ ∥a - b∥, from abs_le_of_le_of_neg_le (norm_reverse_triangle' a b) this,
calc -(∥a∥ - ∥b∥) = ∥b∥ - ∥a∥ : by abel
... ≤ ∥b - a∥ : norm_reverse_triangle' b a
... = ∥a - b∥ : by rw ← norm_neg (a - b); simp
lemma norm_triangle_sub {a b : α} : ∥a - b∥ ≤ ∥a∥ + ∥b∥ :=
by simpa only [sub_eq_add_neg, norm_neg] using norm_triangle a (-b)
lemma abs_norm_sub_norm_le (g h : α) : abs(∥g∥ - ∥h∥) ≤ ∥g - h∥ :=
abs_le.2 $ and.intro
(suffices -∥g - h∥ ≤ -(∥h∥ - ∥g∥), by simpa,
neg_le_neg $ sub_right_le_of_le_add $
calc ∥h∥ = ∥h - g + g∥ : by simp
... ≤ ∥h - g∥ + ∥g∥ : norm_triangle _ _
... = ∥-(g - h)∥ + ∥g∥ : by simp
... = ∥g - h∥ + ∥g∥ : by { rw [norm_neg (g-h)] })
(sub_right_le_of_le_add $ calc ∥g∥ = ∥g - h + h∥ : by simp ... ≤ ∥g-h∥ + ∥h∥ : norm_triangle _ _)
lemma dist_norm_norm_le (g h : α) : dist ∥g∥ ∥h∥ ≤ ∥g - h∥ :=
abs_norm_sub_norm_le g h
lemma norm_sub_rev (g h : α) : ∥g - h∥ = ∥h - g∥ :=
by rw ←norm_neg; simp
lemma ball_0_eq (ε : ℝ) : ball (0:α) ε = {x | ∥x∥ < ε} :=
set.ext $ assume a, by simp
theorem normed_space.tendsto_nhds_zero {f : γ → α} {l : filter γ} :
tendsto f l (nhds 0) ↔ ∀ ε > 0, { x | ∥ f x ∥ < ε } ∈ l :=
begin
rw [metric.tendsto_nhds], simp only [normed_group.dist_eq, sub_zero],
split,
{ intros h ε εgt0,
rcases h ε εgt0 with ⟨s, ssets, hs⟩,
exact mem_sets_of_superset ssets hs },
intros h ε εgt0,
exact ⟨_, h ε εgt0, set.subset.refl _⟩
end
section nnnorm
def nnnorm (a : α) : nnreal := ⟨norm a, norm_nonneg a⟩
@[simp] lemma coe_nnnorm (a : α) : (nnnorm a : ℝ) = norm a := rfl
lemma nndist_eq_nnnorm (a b : α) : nndist a b = nnnorm (a - b) := nnreal.eq $ dist_eq_norm _ _
lemma nnnorm_eq_zero (a : α) : nnnorm a = 0 ↔ a = 0 :=
by simp only [nnreal.eq_iff.symm, nnreal.coe_zero, coe_nnnorm, norm_eq_zero]
@[simp] lemma nnnorm_zero : nnnorm (0 : α) = 0 :=
nnreal.eq norm_zero
lemma nnnorm_triangle (g h : α) : nnnorm (g + h) ≤ nnnorm g + nnnorm h :=
by simpa [nnreal.coe_le] using norm_triangle g h
@[simp] lemma nnnorm_neg (g : α) : nnnorm (-g) = nnnorm g :=
nnreal.eq $ norm_neg g
lemma nndist_nnnorm_nnnorm_le (g h : α) : nndist (nnnorm g) (nnnorm h) ≤ nnnorm (g - h) :=
nnreal.coe_le.2 $ dist_norm_norm_le g h
end nnnorm
instance prod.normed_group [normed_group β] : normed_group (α × β) :=
{ norm := λx, max ∥x.1∥ ∥x.2∥,
dist_eq := assume (x y : α × β),
show max (dist x.1 y.1) (dist x.2 y.2) = (max ∥(x - y).1∥ ∥(x - y).2∥), by simp [dist_eq_norm] }
lemma norm_fst_le (x : α × β) : ∥x.1∥ ≤ ∥x∥ :=
begin have : ∥x∥ = max (∥x.fst∥) (∥x.snd∥) := rfl, rw this, simp[le_max_left] end
lemma norm_snd_le (x : α × β) : ∥x.2∥ ≤ ∥x∥ :=
begin have : ∥x∥ = max (∥x.fst∥) (∥x.snd∥) := rfl, rw this, simp[le_max_right] end
instance fintype.normed_group {π : α → Type*} [fintype α] [∀i, normed_group (π i)] :
normed_group (Πb, π b) :=
{ norm := λf, ((finset.sup finset.univ (λ b, nnnorm (f b)) : nnreal) : ℝ),
dist_eq := assume x y,
congr_arg (coe : nnreal → ℝ) $ congr_arg (finset.sup finset.univ) $ funext $ assume a,
show nndist (x a) (y a) = nnnorm (x a - y a), from nndist_eq_nnnorm _ _ }
lemma tendsto_iff_norm_tendsto_zero {f : ι → β} {a : filter ι} {b : β} :
tendsto f a (nhds b) ↔ tendsto (λ e, ∥ f e - b ∥) a (nhds 0) :=
by rw tendsto_iff_dist_tendsto_zero ; simp only [(dist_eq_norm _ _).symm]
lemma tendsto_zero_iff_norm_tendsto_zero [normed_group α] [normed_group β]
{f : γ → β} {a : filter γ} :
tendsto f a (nhds 0) ↔ tendsto (λ e, ∥ f e ∥) a (nhds 0) :=
have tendsto f a (nhds 0) ↔ tendsto (λ e, ∥ f e - 0 ∥) a (nhds 0) :=
tendsto_iff_norm_tendsto_zero,
by simpa
lemma lim_norm (x : α) : (λg:α, ∥g - x∥) →_{x} 0 :=
tendsto_iff_norm_tendsto_zero.1 (continuous_iff_continuous_at.1 continuous_id x)
lemma lim_norm_zero : (λg:α, ∥g∥) →_{0} 0 :=
by simpa using lim_norm (0:α)
lemma continuous_norm : continuous (λg:α, ∥g∥) :=
begin
rw continuous_iff_continuous_at,
intro x,
rw [continuous_at, tendsto_iff_dist_tendsto_zero],
exact squeeze_zero (λ t, abs_nonneg _) (λ t, abs_norm_sub_norm_le _ _) (lim_norm x)
end
lemma continuous_nnnorm : continuous (nnnorm : α → nnreal) :=
continuous_subtype_mk _ continuous_norm
instance normed_uniform_group : uniform_add_group α :=
begin
refine ⟨metric.uniform_continuous_iff.2 $ assume ε hε, ⟨ε / 2, half_pos hε, assume a b h, _⟩⟩,
rw [prod.dist_eq, max_lt_iff, dist_eq_norm, dist_eq_norm] at h,
calc dist (a.1 - a.2) (b.1 - b.2) = ∥(a.1 - b.1) - (a.2 - b.2)∥ : by simp [dist_eq_norm]
... ≤ ∥a.1 - b.1∥ + ∥a.2 - b.2∥ : norm_triangle_sub
... < ε / 2 + ε / 2 : add_lt_add h.1 h.2
... = ε : add_halves _
end
instance normed_top_monoid : topological_add_monoid α := by apply_instance
instance normed_top_group : topological_add_group α := by apply_instance
end normed_group
section normed_ring
class normed_ring (α : Type*) extends has_norm α, ring α, metric_space α :=
(dist_eq : ∀ x y, dist x y = norm (x - y))
(norm_mul : ∀ a b, norm (a * b) ≤ norm a * norm b)
instance normed_ring.to_normed_group [β : normed_ring α] : normed_group α := { ..β }
lemma norm_mul_le {α : Type*} [normed_ring α] (a b : α) : (∥a*b∥) ≤ (∥a∥) * (∥b∥) :=
normed_ring.norm_mul _ _
lemma norm_pow_le {α : Type*} [normed_ring α] (a : α) : ∀ {n : ℕ}, n > 0 → ∥a^n∥ ≤ ∥a∥^n
| 1 h := by simp
| (n+2) h :=
le_trans (norm_mul_le a (a^(n+1)))
(mul_le_mul (le_refl _)
(norm_pow_le (nat.succ_pos _)) (norm_nonneg _) (norm_nonneg _))
instance prod.normed_ring [normed_ring α] [normed_ring β] : normed_ring (α × β) :=
{ norm_mul := assume x y,
calc
∥x * y∥ = ∥(x.1*y.1, x.2*y.2)∥ : rfl
... = (max ∥x.1*y.1∥ ∥x.2*y.2∥) : rfl
... ≤ (max (∥x.1∥*∥y.1∥) (∥x.2∥*∥y.2∥)) :
max_le_max (norm_mul_le (x.1) (y.1)) (norm_mul_le (x.2) (y.2))
... = (max (∥x.1∥*∥y.1∥) (∥y.2∥*∥x.2∥)) : by simp[mul_comm]
... ≤ (max (∥x.1∥) (∥x.2∥)) * (max (∥y.2∥) (∥y.1∥)) : by { apply max_mul_mul_le_max_mul_max; simp [norm_nonneg] }
... = (max (∥x.1∥) (∥x.2∥)) * (max (∥y.1∥) (∥y.2∥)) : by simp[max_comm]
... = (∥x∥*∥y∥) : rfl,
..prod.normed_group }
end normed_ring
instance normed_ring_top_monoid [normed_ring α] : topological_monoid α :=
⟨ continuous_iff_continuous_at.2 $ λ x, tendsto_iff_norm_tendsto_zero.2 $
have ∀ e : α × α, e.fst * e.snd - x.fst * x.snd =
e.fst * e.snd - e.fst * x.snd + (e.fst * x.snd - x.fst * x.snd), by intro; rw sub_add_sub_cancel,
begin
apply squeeze_zero,
{ intro, apply norm_nonneg },
{ simp only [this], intro, apply norm_triangle },
{ rw ←zero_add (0 : ℝ), apply tendsto_add,
{ apply squeeze_zero,
{ intro, apply norm_nonneg },
{ intro t, show ∥t.fst * t.snd - t.fst * x.snd∥ ≤ ∥t.fst∥ * ∥t.snd - x.snd∥,
rw ←mul_sub, apply norm_mul_le },
{ rw ←mul_zero (∥x.fst∥), apply tendsto_mul,
{ apply continuous_iff_continuous_at.1,
apply continuous.comp,
{ apply continuous_fst },
{ apply continuous_norm }},
{ apply tendsto_iff_norm_tendsto_zero.1,
apply continuous_iff_continuous_at.1,
apply continuous_snd }}},
{ apply squeeze_zero,
{ intro, apply norm_nonneg },
{ intro t, show ∥t.fst * x.snd - x.fst * x.snd∥ ≤ ∥t.fst - x.fst∥ * ∥x.snd∥,
rw ←sub_mul, apply norm_mul_le },
{ rw ←zero_mul (∥x.snd∥), apply tendsto_mul,
{ apply tendsto_iff_norm_tendsto_zero.1,
apply continuous_iff_continuous_at.1,
apply continuous_fst },
{ apply tendsto_const_nhds }}}}
end ⟩
instance normed_top_ring [normed_ring α] : topological_ring α :=
⟨ continuous_iff_continuous_at.2 $ λ x, tendsto_iff_norm_tendsto_zero.2 $
have ∀ e : α, -e - -x = -(e - x), by intro; simp,
by simp only [this, norm_neg]; apply lim_norm ⟩
section normed_field
class normed_field (α : Type*) extends has_norm α, discrete_field α, metric_space α :=
(dist_eq : ∀ x y, dist x y = norm (x - y))
(norm_mul : ∀ a b, norm (a * b) = norm a * norm b)
class nondiscrete_normed_field (α : Type*) extends normed_field α :=
(non_trivial : ∃x:α, 1<∥x∥)
instance normed_field.to_normed_ring [i : normed_field α] : normed_ring α :=
{ norm_mul := by finish [i.norm_mul], ..i }
@[simp] lemma norm_one {α : Type*} [normed_field α] : ∥(1 : α)∥ = 1 :=
have ∥(1 : α)∥ * ∥(1 : α)∥ = ∥(1 : α)∥ * 1, by calc
∥(1 : α)∥ * ∥(1 : α)∥ = ∥(1 : α) * (1 : α)∥ : by rw normed_field.norm_mul
... = ∥(1 : α)∥ * 1 : by simp,
eq_of_mul_eq_mul_left (ne_of_gt ((norm_pos_iff _).2 (by simp))) this
@[simp] lemma norm_mul [normed_field α] (a b : α) : ∥a * b∥ = ∥a∥ * ∥b∥ :=
normed_field.norm_mul a b
instance normed_field.is_monoid_hom_norm [normed_field α] : is_monoid_hom (norm : α → ℝ) :=
{ map_one := norm_one, map_mul := norm_mul }
@[simp] lemma norm_pow [normed_field α] (a : α) : ∀ (n : ℕ), ∥a^n∥ = ∥a∥^n :=
is_monoid_hom.map_pow norm a
@[simp] lemma norm_prod {β : Type*} [normed_field α] (s : finset β) (f : β → α) :
∥s.prod f∥ = s.prod (λb, ∥f b∥) :=
eq.symm (finset.prod_hom norm)
@[simp] lemma norm_div {α : Type*} [normed_field α] (a b : α) : ∥a/b∥ = ∥a∥/∥b∥ :=
if hb : b = 0 then by simp [hb] else
begin
apply eq_div_of_mul_eq,
{ apply ne_of_gt, apply (norm_pos_iff _).mpr hb },
{ rw [←normed_field.norm_mul, div_mul_cancel _ hb] }
end
@[simp] lemma norm_inv {α : Type*} [normed_field α] (a : α) : ∥a⁻¹∥ = ∥a∥⁻¹ :=
by simp only [inv_eq_one_div, norm_div, norm_one]
@[simp] lemma norm_fpow {α : Type*} [normed_field α] (a : α) : ∀n : ℤ,
∥a^n∥ = ∥a∥^n
| (n : ℕ) := norm_pow a n
| -[1+ n] := by simp [fpow_neg_succ_of_nat]
lemma exists_one_lt_norm (α : Type*) [i : nondiscrete_normed_field α] : ∃x : α, 1 < ∥x∥ :=
i.non_trivial
lemma exists_norm_lt_one (α : Type*) [nondiscrete_normed_field α] : ∃x : α, 0 < ∥x∥ ∧ ∥x∥ < 1 :=
begin
rcases exists_one_lt_norm α with ⟨y, hy⟩,
refine ⟨y⁻¹, _, _⟩,
{ simp only [inv_eq_zero, ne.def, norm_pos_iff],
assume h,
rw ← norm_eq_zero at h,
rw h at hy,
exact lt_irrefl _ (lt_trans zero_lt_one hy) },
{ simp [inv_lt_one hy] }
end
instance : normed_field ℝ :=
{ norm := λ x, abs x,
dist_eq := assume x y, rfl,
norm_mul := abs_mul }
instance : nondiscrete_normed_field ℝ :=
{ non_trivial := ⟨2, by { unfold norm, rw abs_of_nonneg; norm_num }⟩ }
lemma real.norm_eq_abs (r : ℝ): norm r = abs r := rfl
end normed_field
@[simp] lemma norm_norm [normed_group α] (x : α) : ∥∥x∥∥ = ∥x∥ :=
by rw [real.norm_eq_abs, abs_of_nonneg (norm_nonneg _)]
section normed_space
class normed_space (α : Type*) (β : Type*) [normed_field α]
extends normed_group β, vector_space α β :=
(norm_smul : ∀ (a:α) b, norm (a • b) = has_norm.norm a * norm b)
variables [normed_field α]
instance normed_field.to_normed_space : normed_space α α :=
{ dist_eq := normed_field.dist_eq,
norm_smul := normed_field.norm_mul }
set_option class.instance_max_depth 43
lemma norm_smul [normed_space α β] (s : α) (x : β) : ∥s • x∥ = ∥s∥ * ∥x∥ :=
normed_space.norm_smul s x
lemma nnnorm_smul [normed_space α β] (s : α) (x : β) : nnnorm (s • x) = nnnorm s * nnnorm x :=
nnreal.eq $ norm_smul s x
variables {E : Type*} {F : Type*} [normed_space α E] [normed_space α F]
lemma tendsto_smul {f : γ → α} { g : γ → F} {e : filter γ} {s : α} {b : F} :
(tendsto f e (nhds s)) → (tendsto g e (nhds b)) → tendsto (λ x, (f x) • (g x)) e (nhds (s • b)) :=
begin
intros limf limg,
rw tendsto_iff_norm_tendsto_zero,
have ineq := λ x : γ, calc
∥f x • g x - s • b∥ = ∥(f x • g x - s • g x) + (s • g x - s • b)∥ : by simp[add_assoc]
... ≤ ∥f x • g x - s • g x∥ + ∥s • g x - s • b∥ : norm_triangle (f x • g x - s • g x) (s • g x - s • b)
... ≤ ∥f x - s∥*∥g x∥ + ∥s∥*∥g x - b∥ : by { rw [←smul_sub, ←sub_smul, norm_smul, norm_smul] },
apply squeeze_zero,
{ intro t, exact norm_nonneg _ },
{ exact ineq },
{ clear ineq,
have limf': tendsto (λ x, ∥f x - s∥) e (nhds 0) := tendsto_iff_norm_tendsto_zero.1 limf,
have limg' : tendsto (λ x, ∥g x∥) e (nhds ∥b∥) := filter.tendsto.comp limg (continuous_iff_continuous_at.1 continuous_norm _),
have lim1 := tendsto_mul limf' limg',
simp only [zero_mul, sub_eq_add_neg] at lim1,
have limg3 := tendsto_iff_norm_tendsto_zero.1 limg,
have lim2 := tendsto_mul (tendsto_const_nhds : tendsto _ _ (nhds ∥ s ∥)) limg3,
simp only [sub_eq_add_neg, mul_zero] at lim2,
rw [show (0:ℝ) = 0 + 0, by simp],
exact tendsto_add lim1 lim2 }
end
lemma tendsto_smul_const {g : γ → F} {e : filter γ} (s : α) {b : F} :
(tendsto g e (nhds b)) → tendsto (λ x, s • (g x)) e (nhds (s • b)) :=
tendsto_smul tendsto_const_nhds
lemma continuous_smul [topological_space γ] {f : γ → α} {g : γ → E}
(hf : continuous f) (hg : continuous g) : continuous (λc, f c • g c) :=
continuous_iff_continuous_at.2 $ assume c,
tendsto_smul (continuous_iff_continuous_at.1 hf _) (continuous_iff_continuous_at.1 hg _)
/-- If there is a scalar `c` with `∥c∥>1`, then any element can be moved by scalar multiplication to
any shell of width `∥c∥`. Also recap information on the norm of the rescaling element that shows
up in applications. -/
lemma rescale_to_shell {c : α} (hc : 1 < ∥c∥) {ε : ℝ} (εpos : 0 < ε) {x : E} (hx : x ≠ 0) :
∃d:α, d ≠ 0 ∧ ∥d • x∥ ≤ ε ∧ (ε/∥c∥ ≤ ∥d • x∥) ∧ (∥d∥⁻¹ ≤ ε⁻¹ * ∥c∥ * ∥x∥) :=
begin
have xεpos : 0 < ∥x∥/ε := div_pos_of_pos_of_pos ((norm_pos_iff _).2 hx) εpos,
rcases exists_int_pow_near xεpos hc with ⟨n, hn⟩,
have cpos : 0 < ∥c∥ := lt_trans (zero_lt_one : (0 :ℝ) < 1) hc,
have cnpos : 0 < ∥c^(n+1)∥ := by { rw norm_fpow, exact lt_trans xεpos hn.2 },
refine ⟨(c^(n+1))⁻¹, _, _, _, _⟩,
show (c ^ (n + 1))⁻¹ ≠ 0,
by rwa [ne.def, inv_eq_zero, ← ne.def, ← norm_pos_iff],
show ∥(c ^ (n + 1))⁻¹ • x∥ ≤ ε,
{ rw [norm_smul, norm_inv, ← div_eq_inv_mul, div_le_iff cnpos, mul_comm, norm_fpow],
exact (div_le_iff εpos).1 (le_of_lt (hn.2)) },
show ε / ∥c∥ ≤ ∥(c ^ (n + 1))⁻¹ • x∥,
{ rw [div_le_iff cpos, norm_smul, norm_inv, norm_fpow, fpow_add (ne_of_gt cpos),
fpow_one, mul_inv', mul_comm, ← mul_assoc, ← mul_assoc, mul_inv_cancel (ne_of_gt cpos),
one_mul, ← div_eq_inv_mul, le_div_iff (fpow_pos_of_pos cpos _), mul_comm],
exact (le_div_iff εpos).1 hn.1 },
show ∥(c ^ (n + 1))⁻¹∥⁻¹ ≤ ε⁻¹ * ∥c∥ * ∥x∥,
{ have : ε⁻¹ * ∥c∥ * ∥x∥ = ε⁻¹ * ∥x∥ * ∥c∥, by ring,
rw [norm_inv, inv_inv', norm_fpow, fpow_add (ne_of_gt cpos), fpow_one, this, ← div_eq_inv_mul],
exact mul_le_mul_of_nonneg_right hn.1 (norm_nonneg _) }
end
instance : normed_space α (E × F) :=
{ norm_smul :=
begin
intros s x,
cases x with x₁ x₂,
change max (∥s • x₁∥) (∥s • x₂∥) = ∥s∥ * max (∥x₁∥) (∥x₂∥),
rw [norm_smul, norm_smul, ← mul_max_of_nonneg _ _ (norm_nonneg _)]
end,
add_smul := λ r x y, prod.ext (add_smul _ _ _) (add_smul _ _ _),
smul_add := λ r x y, prod.ext (smul_add _ _ _) (smul_add _ _ _),
..prod.normed_group,
..prod.vector_space }
instance fintype.normed_space {ι : Type*} {E : ι → Type*} [fintype ι] [∀i, normed_space α (E i)] :
normed_space α (Πi, E i) :=
{ norm := λf, ((finset.univ.sup (λb, nnnorm (f b)) : nnreal) : ℝ),
dist_eq :=
assume f g, congr_arg coe $ congr_arg (finset.sup finset.univ) $
by funext i; exact nndist_eq_nnnorm _ _,
norm_smul := λ a f,
show (↑(finset.sup finset.univ (λ (b : ι), nnnorm (a • f b))) : ℝ) =
nnnorm a * ↑(finset.sup finset.univ (λ (b : ι), nnnorm (f b))),
by simp only [(nnreal.coe_mul _ _).symm, nnreal.mul_finset_sup, nnnorm_smul],
..metric_space_pi,
..pi.vector_space α }
/-- A normed space can be built from a norm that satisfies algebraic properties. This is
formalised in this structure. -/
structure normed_space.core (α : Type*) (β : Type*)
[normed_field α] [add_comm_group β] [has_scalar α β] [has_norm β] :=
(norm_eq_zero_iff : ∀ x : β, ∥x∥ = 0 ↔ x = 0)
(norm_smul : ∀ c : α, ∀ x : β, ∥c • x∥ = ∥c∥ * ∥x∥)
(triangle : ∀ x y : β, ∥x + y∥ ≤ ∥x∥ + ∥y∥)
noncomputable def normed_space.of_core (α : Type*) (β : Type*)
[normed_field α] [add_comm_group β] [vector_space α β] [has_norm β]
(C : normed_space.core α β) : normed_space α β :=
{ dist := λ x y, ∥x - y∥,
dist_eq := assume x y, by refl,
dist_self := assume x, (C.norm_eq_zero_iff (x - x)).mpr (show x - x = 0, by simp),
eq_of_dist_eq_zero := assume x y h, show (x = y), from sub_eq_zero.mp $ (C.norm_eq_zero_iff (x - y)).mp h,
dist_triangle := assume x y z,
calc ∥x - z∥ = ∥x - y + (y - z)∥ : by simp
... ≤ ∥x - y∥ + ∥y - z∥ : C.triangle _ _,
dist_comm := assume x y,
calc ∥x - y∥ = ∥ -(1 : α) • (y - x)∥ : by simp
... = ∥y - x∥ : begin rw[C.norm_smul], simp end,
norm_smul := C.norm_smul }
end normed_space
section summable
local attribute [instance] classical.prop_decidable
open finset filter
variables [normed_group α] [complete_space α]
lemma summable_iff_vanishing_norm {f : ι → α} :
summable f ↔ ∀ε>0, (∃s:finset ι, ∀t, disjoint t s → ∥ t.sum f ∥ < ε) :=
begin
simp only [summable_iff_vanishing, metric.mem_nhds_iff, exists_imp_distrib],
split,
{ assume h ε hε, refine h {x | ∥x∥ < ε} ε hε _, rw [ball_0_eq ε] },
{ assume h s ε hε hs,
rcases h ε hε with ⟨t, ht⟩,
refine ⟨t, assume u hu, hs _⟩,
rw [ball_0_eq],
exact ht u hu }
end
lemma summable_of_norm_bounded {f : ι → α} (g : ι → ℝ) (hf : summable g) (h : ∀i, ∥f i∥ ≤ g i) :
summable f :=
summable_iff_vanishing_norm.2 $ assume ε hε,
let ⟨s, hs⟩ := summable_iff_vanishing_norm.1 hf ε hε in
⟨s, assume t ht,
have ∥t.sum g∥ < ε := hs t ht,
have nn : 0 ≤ t.sum g := finset.zero_le_sum (assume a _, le_trans (norm_nonneg _) (h a)),
lt_of_le_of_lt (norm_triangle_sum t f) $ lt_of_le_of_lt (finset.sum_le_sum $ assume i _, h i) $
by rwa [real.norm_eq_abs, abs_of_nonneg nn] at this⟩
lemma summable_of_summable_norm {f : ι → α} (hf : summable (λa, ∥f a∥)) : summable f :=
summable_of_norm_bounded _ hf (assume i, le_refl _)
lemma norm_tsum_le_tsum_norm {f : ι → α} (hf : summable (λi, ∥f i∥)) : ∥(∑i, f i)∥ ≤ (∑ i, ∥f i∥) :=
have h₁ : tendsto (λs:finset ι, ∥s.sum f∥) at_top (nhds ∥(∑ i, f i)∥) :=
(has_sum_tsum $ summable_of_summable_norm hf).comp (continuous_norm.tendsto _),
have h₂ : tendsto (λs:finset ι, s.sum (λi, ∥f i∥)) at_top (nhds (∑ i, ∥f i∥)) :=
has_sum_tsum hf,
le_of_tendsto_of_tendsto at_top_ne_bot h₁ h₂ $ univ_mem_sets' $ assume s, norm_triangle_sum _ _
end summable
namespace complex
instance : normed_field ℂ :=
{ norm := complex.abs,
dist_eq := λ _ _, rfl,
norm_mul := complex.abs_mul,
.. complex.discrete_field }
instance : nondiscrete_normed_field ℂ :=
{ non_trivial := ⟨2, by simp [norm]; norm_num⟩ }
@[simp] lemma norm_real (r : ℝ) : ∥(r : ℂ)∥ = ∥r∥ := complex.abs_of_real _
@[simp] lemma norm_rat (r : ℚ) : ∥(r : ℂ)∥ = _root_.abs (r : ℝ) :=
suffices ∥((r : ℝ) : ℂ)∥ = _root_.abs r, by simpa,
by rw [norm_real, real.norm_eq_abs]
@[simp] lemma norm_nat (n : ℕ) : ∥(n : ℂ)∥ = n := complex.abs_of_nat _
@[simp] lemma norm_int {n : ℤ} : ∥(n : ℂ)∥ = _root_.abs n :=
suffices ∥((n : ℝ) : ℂ)∥ = _root_.abs n, by simpa,
by rw [norm_real, real.norm_eq_abs]
lemma norm_int_of_nonneg {n : ℤ} (hn : n ≥ 0) : ∥(n : ℂ)∥ = n :=
by rw [norm_int, _root_.abs_of_nonneg]; exact int.cast_nonneg.2 hn
end complex
|
60da68fecd5d19d6398a0087dc735a9933d655cc | d406927ab5617694ec9ea7001f101b7c9e3d9702 | /src/category_theory/triangulated/rotate.lean | 4e5f57120189f02c8972bc7f84ec054edcbe4a45 | [
"Apache-2.0"
] | permissive | alreadydone/mathlib | dc0be621c6c8208c581f5170a8216c5ba6721927 | c982179ec21091d3e102d8a5d9f5fe06c8fafb73 | refs/heads/master | 1,685,523,275,196 | 1,670,184,141,000 | 1,670,184,141,000 | 287,574,545 | 0 | 0 | Apache-2.0 | 1,670,290,714,000 | 1,597,421,623,000 | Lean | UTF-8 | Lean | false | false | 12,157 | lean | /-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw
-/
import category_theory.preadditive.additive_functor
import category_theory.triangulated.basic
/-!
# Rotate
This file adds the ability to rotate triangles and triangle morphisms.
It also shows that rotation gives an equivalence on the category of triangles.
-/
noncomputable theory
open category_theory
open category_theory.preadditive
open category_theory.limits
universes v v₀ v₁ v₂ u u₀ u₁ u₂
namespace category_theory.pretriangulated
open category_theory.category
/--
We work in an preadditive category `C` equipped with an additive shift.
-/
variables {C : Type u} [category.{v} C] [preadditive C]
variables [has_shift C ℤ]
variables (X : C)
/--
If you rotate a triangle, you get another triangle.
Given a triangle of the form:
```
f g h
X ───> Y ───> Z ───> X⟦1⟧
```
applying `rotate` gives a triangle of the form:
```
g h -f⟦1⟧'
Y ───> Z ───> X⟦1⟧ ───> Y⟦1⟧
```
-/
@[simps]
def triangle.rotate (T : triangle C) : triangle C := triangle.mk T.mor₂ T.mor₃ (-T.mor₁⟦1⟧')
section
local attribute [semireducible] shift_shift_neg shift_neg_shift
/--
Given a triangle of the form:
```
f g h
X ───> Y ───> Z ───> X⟦1⟧
```
applying `inv_rotate` gives a triangle that can be thought of as:
```
-h⟦-1⟧' f g
Z⟦-1⟧ ───> X ───> Y ───> Z
```
(note that this diagram doesn't technically fit the definition of triangle, as `Z⟦-1⟧⟦1⟧` is
not necessarily equal to `Z`, but it is isomorphic, by the `counit_iso` of `shift C`)
-/
@[simps]
def triangle.inv_rotate (T : triangle C) : triangle C :=
triangle.mk (-T.mor₃⟦(-1:ℤ)⟧' ≫ (shift_shift_neg _ _).hom) T.mor₁
(T.mor₂ ≫ (shift_neg_shift _ _).inv)
end
namespace triangle_morphism
variables {T₁ T₂ T₃ T₄: triangle C}
open triangle
/--
You can also rotate a triangle morphism to get a morphism between the two rotated triangles.
Given a triangle morphism of the form:
```
f g h
X ───> Y ───> Z ───> X⟦1⟧
│ │ │ │
│a │b │c │a⟦1⟧
V V V V
X' ───> Y' ───> Z' ───> X'⟦1⟧
f' g' h'
```
applying `rotate` gives a triangle morphism of the form:
```
g h -f⟦1⟧
Y ───> Z ───> X⟦1⟧ ───> Y⟦1⟧
│ │ │ │
│b │c │a⟦1⟧ │b⟦1⟧'
V V V V
Y' ───> Z' ───> X'⟦1⟧ ───> Y'⟦1⟧
g' h' -f'⟦1⟧
```
-/
@[simps]
def rotate (f : triangle_morphism T₁ T₂) :
triangle_morphism (T₁.rotate) (T₂.rotate):=
{ hom₁ := f.hom₂,
hom₂ := f.hom₃,
hom₃ := f.hom₁⟦1⟧',
comm₃' := begin
dsimp,
simp only [rotate_mor₃, comp_neg, neg_comp, ← functor.map_comp, f.comm₁]
end }
/--
Given a triangle morphism of the form:
```
f g h
X ───> Y ───> Z ───> X⟦1⟧
│ │ │ │
│a │b │c │a⟦1⟧
V V V V
X' ───> Y' ───> Z' ───> X'⟦1⟧
f' g' h'
```
applying `inv_rotate` gives a triangle morphism that can be thought of as:
```
-h⟦-1⟧ f g
Z⟦-1⟧ ───> X ───> Y ───> Z
│ │ │ │
│c⟦-1⟧' │a │b │c
V V V V
Z'⟦-1⟧ ───> X' ───> Y' ───> Z'
-h'⟦-1⟧ f' g'
```
(note that this diagram doesn't technically fit the definition of triangle morphism,
as `Z⟦-1⟧⟦1⟧` is not necessarily equal to `Z`, and `Z'⟦-1⟧⟦1⟧` is not necessarily equal to `Z'`,
but they are isomorphic, by the `counit_iso` of `shift C`)
-/
@[simps]
def inv_rotate (f : triangle_morphism T₁ T₂) :
triangle_morphism (T₁.inv_rotate) (T₂.inv_rotate) :=
{ hom₁ := f.hom₃⟦-1⟧',
hom₂ := f.hom₁,
hom₃ := f.hom₂,
comm₁' := begin
dsimp [inv_rotate_mor₁],
simp only [discrete.functor_map_id, id_comp, preadditive.comp_neg, assoc,
neg_inj, nat_trans.id_app, preadditive.neg_comp],
rw [← functor.map_comp_assoc, ← f.comm₃, functor.map_comp_assoc, μ_naturality_assoc,
nat_trans.naturality, functor.id_map],
end,
comm₃' := begin
dsimp,
simp only [discrete.functor_map_id, id_comp, μ_inv_naturality,
category.assoc, nat_trans.id_app, unit_of_tensor_iso_unit_inv_app],
erw ε_naturality_assoc,
rw comm₂_assoc
end }
end triangle_morphism
variables (C)
/--
Rotating triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def rotate : triangle C ⥤ triangle C :=
{ obj := triangle.rotate,
map := λ _ _ f, f.rotate }
/--
The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def inv_rotate : triangle C ⥤ triangle C :=
{ obj := triangle.inv_rotate,
map := λ _ _ f, f.inv_rotate }
variables {C}
variables [∀ n : ℤ, functor.additive (shift_functor C n)]
/-- There is a natural map from a triangle to the `inv_rotate` of its `rotate`. -/
@[simps]
def to_inv_rotate_rotate (T : triangle C) : T ⟶ (inv_rotate C).obj ((rotate C).obj T) :=
{ hom₁ := (shift_shift_neg _ _).inv,
hom₂ := 𝟙 T.obj₂,
hom₃ := 𝟙 T.obj₃,
comm₃' := begin
dsimp,
simp only [ε_app_obj, eq_to_iso.hom, discrete.functor_map_id, id_comp, eq_to_iso.inv,
category.assoc, obj_μ_inv_app, functor.map_comp, nat_trans.id_app, obj_ε_app,
unit_of_tensor_iso_unit_inv_app],
erw μ_inv_hom_app_assoc,
refl
end }
/--
There is a natural transformation between the identity functor on triangles in `C`,
and the composition of a rotation with an inverse rotation.
-/
@[simps]
def rot_comp_inv_rot_hom : 𝟭 (triangle C) ⟶ rotate C ⋙ inv_rotate C :=
{ app := to_inv_rotate_rotate,
naturality' := begin
introv, ext,
{ dsimp,
simp only [nat_iso.cancel_nat_iso_inv_right_assoc, discrete.functor_map_id, id_comp,
μ_inv_naturality, assoc, nat_trans.id_app, unit_of_tensor_iso_unit_inv_app],
erw ε_naturality },
{ dsimp, rw [comp_id, id_comp] },
{ dsimp, rw [comp_id, id_comp] },
end }
/-- There is a natural map from the `inv_rotate` of the `rotate` of a triangle to itself. -/
@[simps]
def from_inv_rotate_rotate (T : triangle C) : (inv_rotate C).obj ((rotate C).obj T) ⟶ T :=
{ hom₁ := (shift_equiv C 1).unit_inv.app T.obj₁,
hom₂ := 𝟙 T.obj₂,
hom₃ := 𝟙 T.obj₃,
comm₃' := begin
dsimp,
rw [unit_of_tensor_iso_unit_inv_app, ε_app_obj],
simp only [discrete.functor_map_id, nat_trans.id_app, id_comp, assoc, functor.map_comp,
obj_μ_app, obj_ε_inv_app, comp_id, μ_inv_hom_app_assoc],
erw [μ_inv_hom_app, μ_inv_hom_app_assoc, category.comp_id]
end }
/--
There is a natural transformation between the composition of a rotation with an inverse rotation
on triangles in `C`, and the identity functor.
-/
@[simps]
def rot_comp_inv_rot_inv : rotate C ⋙ inv_rotate C ⟶ 𝟭 (triangle C) :=
{ app := from_inv_rotate_rotate }
/--
The natural transformations between the identity functor on triangles in `C` and the composition
of a rotation with an inverse rotation are natural isomorphisms (they are isomorphisms in the
category of functors).
-/
@[simps]
def rot_comp_inv_rot : 𝟭 (triangle C) ≅ rotate C ⋙ inv_rotate C :=
{ hom := rot_comp_inv_rot_hom,
inv := rot_comp_inv_rot_inv }
/-- There is a natural map from the `rotate` of the `inv_rotate` of a triangle to itself. -/
@[simps]
def from_rotate_inv_rotate (T : triangle C) : (rotate C).obj ((inv_rotate C).obj T) ⟶ T :=
{ hom₁ := 𝟙 T.obj₁,
hom₂ := 𝟙 T.obj₂,
hom₃ := (shift_equiv C 1).counit.app T.obj₃,
comm₂' := begin
dsimp,
rw unit_of_tensor_iso_unit_inv_app,
simp only [discrete.functor_map_id, nat_trans.id_app,
id_comp, add_neg_equiv_counit_iso_hom, eq_to_hom_refl, nat_trans.comp_app, assoc,
μ_inv_hom_app_assoc, ε_hom_inv_app],
exact category.comp_id _,
end,
comm₃' := begin
dsimp,
simp only [discrete.functor_map_id, nat_trans.id_app, id_comp, functor.map_neg,
functor.map_comp, obj_μ_app, obj_ε_inv_app, comp_id, assoc, μ_naturality_assoc, neg_neg,
category_theory.functor.map_id, add_neg_equiv_counit_iso_hom, eq_to_hom_refl,
nat_trans.comp_app],
erw [μ_inv_hom_app, category.comp_id, obj_zero_map_μ_app],
rw [discrete.functor_map_id, nat_trans.id_app, comp_id],
end }
/--
There is a natural transformation between the composition of an inverse rotation with a rotation
on triangles in `C`, and the identity functor.
-/
@[simps]
def inv_rot_comp_rot_hom : inv_rotate C ⋙ rotate C ⟶ 𝟭 (triangle C) :=
{ app := from_rotate_inv_rotate }
/-- There is a natural map from a triangle to the `rotate` of its `inv_rotate`. -/
@[simps]
def to_rotate_inv_rotate (T : triangle C) : T ⟶ (rotate C).obj ((inv_rotate C).obj T) :=
{ hom₁ := 𝟙 T.obj₁,
hom₂ := 𝟙 T.obj₂,
hom₃ := (shift_equiv C 1).counit_inv.app T.obj₃,
comm₃' := begin
dsimp,
rw category_theory.functor.map_id,
simp only [comp_id, add_neg_equiv_counit_iso_inv, eq_to_hom_refl, id_comp, nat_trans.comp_app,
discrete.functor_map_id, nat_trans.id_app, functor.map_neg, functor.map_comp, obj_μ_app,
obj_ε_inv_app, assoc, μ_naturality_assoc, neg_neg, μ_inv_hom_app_assoc],
erw [μ_inv_hom_app, category.comp_id, obj_zero_map_μ_app],
simp only [discrete.functor_map_id, nat_trans.id_app, comp_id, ε_hom_inv_app_assoc],
end }
/--
There is a natural transformation between the identity functor on triangles in `C`,
and the composition of an inverse rotation with a rotation.
-/
@[simps]
def inv_rot_comp_rot_inv : 𝟭 (triangle C) ⟶ inv_rotate C ⋙ rotate C :=
{ app := to_rotate_inv_rotate,
naturality' := begin
introv, ext,
{ dsimp, rw [comp_id, id_comp] },
{ dsimp, rw [comp_id, id_comp] },
{ dsimp,
rw [add_neg_equiv_counit_iso_inv, eq_to_hom_map, eq_to_hom_refl, id_comp],
simp only [nat_trans.comp_app, assoc],
erw [μ_inv_naturality, ε_naturality_assoc] },
end }
/--
The natural transformations between the composition of a rotation with an inverse rotation
on triangles in `C`, and the identity functor on triangles are natural isomorphisms
(they are isomorphisms in the category of functors).
-/
@[simps]
def inv_rot_comp_rot : inv_rotate C ⋙ rotate C ≅ 𝟭 (triangle C) :=
{ hom := inv_rot_comp_rot_hom,
inv := inv_rot_comp_rot_inv }
variables (C)
/--
Rotating triangles gives an auto-equivalence on the category of triangles in `C`.
-/
@[simps]
def triangle_rotation : equivalence (triangle C) (triangle C) :=
{ functor := rotate C,
inverse := inv_rotate C,
unit_iso := rot_comp_inv_rot,
counit_iso := inv_rot_comp_rot,
functor_unit_iso_comp' := begin
introv, ext,
{ dsimp, rw comp_id },
{ dsimp, rw comp_id },
{ dsimp,
rw unit_of_tensor_iso_unit_inv_app,
simp only [discrete.functor_map_id, nat_trans.id_app, id_comp, functor.map_comp, obj_ε_app,
obj_μ_inv_app, assoc, add_neg_equiv_counit_iso_hom, eq_to_hom_refl, nat_trans.comp_app,
ε_inv_app_obj, comp_id, μ_inv_hom_app_assoc],
erw [μ_inv_hom_app_assoc, μ_inv_hom_app],
refl }
end }
variables {C}
instance : is_equivalence (rotate C) :=
by { change is_equivalence (triangle_rotation C).functor, apply_instance, }
instance : is_equivalence (inv_rotate C) :=
by { change is_equivalence (triangle_rotation C).inverse, apply_instance, }
end category_theory.pretriangulated
|
61b52020b753fa7283760824292452ea01bcb61d | 9dc8cecdf3c4634764a18254e94d43da07142918 | /src/algebra/algebra/tower.lean | 7416808106c253283dd70a459d7f3f7dfd5d00ae | [
"Apache-2.0"
] | permissive | jcommelin/mathlib | d8456447c36c176e14d96d9e76f39841f69d2d9b | ee8279351a2e434c2852345c51b728d22af5a156 | refs/heads/master | 1,664,782,136,488 | 1,663,638,983,000 | 1,663,638,983,000 | 132,563,656 | 0 | 0 | Apache-2.0 | 1,663,599,929,000 | 1,525,760,539,000 | Lean | UTF-8 | Lean | false | false | 14,049 | lean | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import algebra.algebra.subalgebra.basic
import algebra.algebra.bilinear
/-!
# Towers of algebras
In this file we prove basic facts about towers of algebra.
An algebra tower A/S/R is expressed by having instances of `algebra A S`,
`algebra R S`, `algebra R A` and `is_scalar_tower R S A`, the later asserting the
compatibility condition `(r • s) • a = r • (s • a)`.
An important definition is `to_alg_hom R S A`, the canonical `R`-algebra homomorphism `S →ₐ[R] A`.
-/
open_locale pointwise
universes u v w u₁ v₁
variables (R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)
namespace algebra
variables [comm_semiring R] [semiring A] [algebra R A]
variables [add_comm_monoid M] [module R M] [module A M] [is_scalar_tower R A M]
variables {A}
/-- The `R`-algebra morphism `A → End (M)` corresponding to the representation of the algebra `A`
on the `R`-module `M`.
This is a stronger version of `distrib_mul_action.to_linear_map`, and could also have been
called `algebra.to_module_End`. -/
def lsmul : A →ₐ[R] module.End R M :=
{ to_fun := distrib_mul_action.to_linear_map R M,
map_one' := linear_map.ext $ λ _, one_smul A _,
map_mul' := λ a b, linear_map.ext $ smul_assoc a b,
map_zero' := linear_map.ext $ λ _, zero_smul A _,
map_add' := λ a b, linear_map.ext $ λ _, add_smul _ _ _,
commutes' := λ r, linear_map.ext $ algebra_map_smul A r, }
@[simp] lemma lsmul_coe (a : A) : (lsmul R M a : M → M) = (•) a := rfl
lemma lmul_algebra_map (x : R) :
algebra.lmul R A (algebra_map R A x) = algebra.lsmul R A x :=
eq.symm $ linear_map.ext $ smul_def x
end algebra
namespace is_scalar_tower
section module
variables [comm_semiring R] [semiring A] [algebra R A]
variables [has_smul R M] [mul_action A M] [is_scalar_tower R A M]
variables {R} (A) {M}
theorem algebra_map_smul (r : R) (x : M) : algebra_map R A r • x = r • x :=
by rw [algebra.algebra_map_eq_smul_one, smul_assoc, one_smul]
end module
section semiring
variables [comm_semiring R] [comm_semiring S] [semiring A] [semiring B]
variables [algebra R S] [algebra S A] [algebra S B]
variables {R S A}
theorem of_algebra_map_eq [algebra R A]
(h : ∀ x, algebra_map R A x = algebra_map S A (algebra_map R S x)) :
is_scalar_tower R S A :=
⟨λ x y z, by simp_rw [algebra.smul_def, ring_hom.map_mul, mul_assoc, h]⟩
/-- See note [partially-applied ext lemmas]. -/
theorem of_algebra_map_eq' [algebra R A]
(h : algebra_map R A = (algebra_map S A).comp (algebra_map R S)) :
is_scalar_tower R S A :=
of_algebra_map_eq $ ring_hom.ext_iff.1 h
variables (R S A)
instance subalgebra (S₀ : subalgebra R S) : is_scalar_tower S₀ S A :=
of_algebra_map_eq $ λ x, rfl
variables [algebra R A] [algebra R B]
variables [is_scalar_tower R S A] [is_scalar_tower R S B]
theorem algebra_map_eq :
algebra_map R A = (algebra_map S A).comp (algebra_map R S) :=
ring_hom.ext $ λ x, by simp_rw [ring_hom.comp_apply, algebra.algebra_map_eq_smul_one,
smul_assoc, one_smul]
theorem algebra_map_apply (x : R) : algebra_map R A x = algebra_map S A (algebra_map R S x) :=
by rw [algebra_map_eq R S A, ring_hom.comp_apply]
instance subalgebra' (S₀ : subalgebra R S) : is_scalar_tower R S₀ A :=
@is_scalar_tower.of_algebra_map_eq R S₀ A _ _ _ _ _ _ $ λ _,
(is_scalar_tower.algebra_map_apply R S A _ : _)
@[ext] lemma algebra.ext {S : Type u} {A : Type v} [comm_semiring S] [semiring A]
(h1 h2 : algebra S A) (h : ∀ (r : S) (x : A), (by haveI := h1; exact r • x) = r • x) : h1 = h2 :=
algebra.algebra_ext _ _ $ λ r, by
simpa only [@algebra.smul_def _ _ _ _ h1, @algebra.smul_def _ _ _ _ h2, mul_one] using h r 1
/-- In a tower, the canonical map from the middle element to the top element is an
algebra homomorphism over the bottom element. -/
def to_alg_hom : S →ₐ[R] A :=
{ commutes' := λ _, (algebra_map_apply _ _ _ _).symm,
.. algebra_map S A }
lemma to_alg_hom_apply (y : S) : to_alg_hom R S A y = algebra_map S A y := rfl
@[simp] lemma coe_to_alg_hom : ↑(to_alg_hom R S A) = algebra_map S A :=
ring_hom.ext $ λ _, rfl
@[simp] lemma coe_to_alg_hom' : (to_alg_hom R S A : S → A) = algebra_map S A :=
rfl
variables {R S A B}
@[simp, priority 900] lemma _root_.alg_hom.map_algebra_map (f : A →ₐ[S] B) (r : R) :
f (algebra_map R A r) = algebra_map R B r :=
by rw [algebra_map_apply R S A r, f.commutes, ← algebra_map_apply R S B]
variables (R)
@[simp, priority 900] lemma _root_.alg_hom.comp_algebra_map_of_tower (f : A →ₐ[S] B) :
(f : A →+* B).comp (algebra_map R A) = algebra_map R B :=
ring_hom.ext f.map_algebra_map
variables (R) {S A B}
-- conflicts with is_scalar_tower.subalgebra
@[priority 999] instance subsemiring (U : subsemiring S) : is_scalar_tower U S A :=
of_algebra_map_eq $ λ x, rfl
@[nolint instance_priority]
instance of_ring_hom {R A B : Type*} [comm_semiring R] [comm_semiring A] [comm_semiring B]
[algebra R A] [algebra R B] (f : A →ₐ[R] B) :
@is_scalar_tower R A B _ (f.to_ring_hom.to_algebra.to_has_smul) _ :=
by { letI := (f : A →+* B).to_algebra, exact of_algebra_map_eq (λ x, (f.commutes x).symm) }
end semiring
end is_scalar_tower
section homs
variables [comm_semiring R] [comm_semiring S] [semiring A] [semiring B]
variables [algebra R S] [algebra S A] [algebra S B]
variables [algebra R A] [algebra R B]
variables [is_scalar_tower R S A] [is_scalar_tower R S B]
variables (R) {A S B}
open is_scalar_tower
namespace alg_hom
/-- R ⟶ S induces S-Alg ⥤ R-Alg -/
def restrict_scalars (f : A →ₐ[S] B) : A →ₐ[R] B :=
{ commutes' := λ r, by { rw [algebra_map_apply R S A, algebra_map_apply R S B],
exact f.commutes (algebra_map R S r) },
.. (f : A →+* B) }
lemma restrict_scalars_apply (f : A →ₐ[S] B) (x : A) : f.restrict_scalars R x = f x := rfl
@[simp] lemma coe_restrict_scalars (f : A →ₐ[S] B) : (f.restrict_scalars R : A →+* B) = f := rfl
@[simp] lemma coe_restrict_scalars' (f : A →ₐ[S] B) : (restrict_scalars R f : A → B) = f := rfl
lemma restrict_scalars_injective :
function.injective (restrict_scalars R : (A →ₐ[S] B) → (A →ₐ[R] B)) :=
λ f g h, alg_hom.ext (alg_hom.congr_fun h : _)
end alg_hom
namespace alg_equiv
/-- R ⟶ S induces S-Alg ⥤ R-Alg -/
def restrict_scalars (f : A ≃ₐ[S] B) : A ≃ₐ[R] B :=
{ commutes' := λ r, by { rw [algebra_map_apply R S A, algebra_map_apply R S B],
exact f.commutes (algebra_map R S r) },
.. (f : A ≃+* B) }
lemma restrict_scalars_apply (f : A ≃ₐ[S] B) (x : A) : f.restrict_scalars R x = f x := rfl
@[simp] lemma coe_restrict_scalars (f : A ≃ₐ[S] B) : (f.restrict_scalars R : A ≃+* B) = f := rfl
@[simp] lemma coe_restrict_scalars' (f : A ≃ₐ[S] B) : (restrict_scalars R f : A → B) = f := rfl
lemma restrict_scalars_injective :
function.injective (restrict_scalars R : (A ≃ₐ[S] B) → (A ≃ₐ[R] B)) :=
λ f g h, alg_equiv.ext (alg_equiv.congr_fun h : _)
end alg_equiv
end homs
namespace subalgebra
open is_scalar_tower
section semiring
variables (R) {S A B} [comm_semiring R] [comm_semiring S] [semiring A] [semiring B]
variables [algebra R S] [algebra S A] [algebra R A] [algebra S B] [algebra R B]
variables [is_scalar_tower R S A] [is_scalar_tower R S B]
/-- Given a tower `A / ↥U / S / R` of algebras, where `U` is an `S`-subalgebra of `A`, reinterpret
`U` as an `R`-subalgebra of `A`. -/
def restrict_scalars (U : subalgebra S A) : subalgebra R A :=
{ algebra_map_mem' := λ x, by { rw algebra_map_apply R S A, exact U.algebra_map_mem _ },
.. U }
@[simp] lemma coe_restrict_scalars {U : subalgebra S A} :
(restrict_scalars R U : set A) = (U : set A) := rfl
@[simp] lemma restrict_scalars_top : restrict_scalars R (⊤ : subalgebra S A) = ⊤ :=
set_like.coe_injective rfl
@[simp] lemma restrict_scalars_to_submodule {U : subalgebra S A} :
(U.restrict_scalars R).to_submodule = U.to_submodule.restrict_scalars R :=
set_like.coe_injective rfl
@[simp] lemma mem_restrict_scalars {U : subalgebra S A} {x : A} :
x ∈ restrict_scalars R U ↔ x ∈ U := iff.rfl
lemma restrict_scalars_injective :
function.injective (restrict_scalars R : subalgebra S A → subalgebra R A) :=
λ U V H, ext $ λ x, by rw [← mem_restrict_scalars R, H, mem_restrict_scalars]
/-- Produces an `R`-algebra map from `U.restrict_scalars R` given an `S`-algebra map from `U`.
This is a special case of `alg_hom.restrict_scalars` that can be helpful in elaboration. -/
@[simp]
def of_restrict_scalars (U : subalgebra S A) (f : U →ₐ[S] B) : U.restrict_scalars R →ₐ[R] B :=
f.restrict_scalars R
end semiring
end subalgebra
namespace algebra
variables {R A} [comm_semiring R] [semiring A] [algebra R A]
variables {M} [add_comm_monoid M] [module A M] [module R M] [is_scalar_tower R A M]
lemma span_restrict_scalars_eq_span_of_surjective
(h : function.surjective (algebra_map R A)) (s : set M) :
(submodule.span A s).restrict_scalars R = submodule.span R s :=
begin
refine le_antisymm (λ x hx, _) (submodule.span_subset_span _ _ _),
refine submodule.span_induction hx _ _ _ _,
{ exact λ x hx, submodule.subset_span hx },
{ exact submodule.zero_mem _ },
{ exact λ x y, submodule.add_mem _ },
{ intros c x hx,
obtain ⟨c', rfl⟩ := h c,
rw is_scalar_tower.algebra_map_smul,
exact submodule.smul_mem _ _ hx },
end
lemma coe_span_eq_span_of_surjective
(h : function.surjective (algebra_map R A)) (s : set M) :
(submodule.span A s : set M) = submodule.span R s :=
congr_arg coe (algebra.span_restrict_scalars_eq_span_of_surjective h s)
end algebra
namespace is_scalar_tower
open subalgebra
variables [comm_semiring R] [comm_semiring S] [comm_semiring A]
variables [algebra R S] [algebra S A] [algebra R A] [is_scalar_tower R S A]
theorem adjoin_range_to_alg_hom (t : set A) :
(algebra.adjoin (to_alg_hom R S A).range t).restrict_scalars R =
(algebra.adjoin S t).restrict_scalars R :=
subalgebra.ext $ λ z,
show z ∈ subsemiring.closure (set.range (algebra_map (to_alg_hom R S A).range A) ∪ t : set A) ↔
z ∈ subsemiring.closure (set.range (algebra_map S A) ∪ t : set A),
from suffices set.range (algebra_map (to_alg_hom R S A).range A) = set.range (algebra_map S A),
by rw this,
by { ext z, exact ⟨λ ⟨⟨x, y, h1⟩, h2⟩, ⟨y, h2 ▸ h1⟩, λ ⟨y, hy⟩, ⟨⟨z, y, hy⟩, rfl⟩⟩ }
end is_scalar_tower
section semiring
variables {R S A}
namespace submodule
section module
variables [semiring R] [semiring S] [add_comm_monoid A]
variables [module R S] [module S A] [module R A] [is_scalar_tower R S A]
open is_scalar_tower
theorem smul_mem_span_smul_of_mem {s : set S} {t : set A} {k : S} (hks : k ∈ span R s)
{x : A} (hx : x ∈ t) : k • x ∈ span R (s • t) :=
span_induction hks (λ c hc, subset_span $ set.mem_smul.2 ⟨c, x, hc, hx, rfl⟩)
(by { rw zero_smul, exact zero_mem _ })
(λ c₁ c₂ ih₁ ih₂, by { rw add_smul, exact add_mem ih₁ ih₂ })
(λ b c hc, by { rw is_scalar_tower.smul_assoc, exact smul_mem _ _ hc })
variables [smul_comm_class R S A]
theorem smul_mem_span_smul {s : set S} (hs : span R s = ⊤) {t : set A} {k : S}
{x : A} (hx : x ∈ span R t) :
k • x ∈ span R (s • t) :=
span_induction hx (λ x hx, smul_mem_span_smul_of_mem (hs.symm ▸ mem_top) hx)
(by { rw smul_zero, exact zero_mem _ })
(λ x y ihx ihy, by { rw smul_add, exact add_mem ihx ihy })
(λ c x hx, smul_comm c k x ▸ smul_mem _ _ hx)
theorem smul_mem_span_smul' {s : set S} (hs : span R s = ⊤) {t : set A} {k : S}
{x : A} (hx : x ∈ span R (s • t)) :
k • x ∈ span R (s • t) :=
span_induction hx (λ x hx, let ⟨p, q, hp, hq, hpq⟩ := set.mem_smul.1 hx in
by { rw [← hpq, smul_smul], exact smul_mem_span_smul_of_mem (hs.symm ▸ mem_top) hq })
(by { rw smul_zero, exact zero_mem _ })
(λ x y ihx ihy, by { rw smul_add, exact add_mem ihx ihy })
(λ c x hx, smul_comm c k x ▸ smul_mem _ _ hx)
theorem span_smul {s : set S} (hs : span R s = ⊤) (t : set A) :
span R (s • t) = (span S t).restrict_scalars R :=
le_antisymm (span_le.2 $ λ x hx, let ⟨p, q, hps, hqt, hpqx⟩ := set.mem_smul.1 hx in
hpqx ▸ (span S t).smul_mem p (subset_span hqt)) $
λ p hp, span_induction hp (λ x hx, one_smul S x ▸ smul_mem_span_smul hs (subset_span hx))
(zero_mem _)
(λ _ _, add_mem)
(λ k x hx, smul_mem_span_smul' hs hx)
end module
section algebra
variables [comm_semiring R] [semiring S] [add_comm_monoid A]
variables [algebra R S] [module S A] [module R A] [is_scalar_tower R S A]
/-- A variant of `submodule.span_image` for `algebra_map`. -/
lemma span_algebra_map_image (a : set R) :
submodule.span R (algebra_map R S '' a) =
(submodule.span R a).map (algebra.linear_map R S) :=
(submodule.span_image $ algebra.linear_map R S).trans rfl
lemma span_algebra_map_image_of_tower {S T : Type*} [comm_semiring S] [semiring T]
[module R S] [is_scalar_tower R S S] [algebra R T] [algebra S T] [is_scalar_tower R S T]
(a : set S) :
submodule.span R (algebra_map S T '' a) =
(submodule.span R a).map ((algebra.linear_map S T).restrict_scalars R) :=
(submodule.span_image $ (algebra.linear_map S T).restrict_scalars R).trans rfl
lemma map_mem_span_algebra_map_image {S T : Type*} [comm_semiring S] [semiring T]
[algebra R S] [algebra R T] [algebra S T] [is_scalar_tower R S T]
(x : S) (a : set S) (hx : x ∈ submodule.span R a) :
algebra_map S T x ∈ submodule.span R (algebra_map S T '' a) :=
by { rw [span_algebra_map_image_of_tower, mem_map], exact ⟨x, hx, rfl⟩ }
end algebra
end submodule
end semiring
section ring
namespace algebra
variables [comm_semiring R] [semiring A] [algebra R A]
variables [add_comm_group M] [module A M] [module R M] [is_scalar_tower R A M]
lemma lsmul_injective [no_zero_smul_divisors A M] {x : A} (hx : x ≠ 0) :
function.injective (lsmul R M x) :=
smul_right_injective _ hx
end algebra
end ring
|
5c420b4cd5950df06f20ae36386cfea5da0bb8cb | a9d0fb7b0e4f802bd3857b803e6c5c23d87fef91 | /tests/lean/univ_vars.lean | 92f4c7848d4772775fb05c78ee9e7d845d320dc6 | [
"Apache-2.0"
] | permissive | soonhokong/lean-osx | 4a954262c780e404c1369d6c06516161d07fcb40 | 3670278342d2f4faa49d95b46d86642d7875b47c | refs/heads/master | 1,611,410,334,552 | 1,474,425,686,000 | 1,474,425,686,000 | 12,043,103 | 5 | 1 | null | null | null | null | UTF-8 | Lean | false | false | 417 | lean | --
set_option pp.universes true
universe u
variable A : Type.{u}
definition id1 (a : A) : A := a
check @id1
variable B : Type
definition id2 (a : B) : B := a
check @id2
universe variable k
variable C : Type.{k}
definition id3 (a : C) := a
check @id3
universe variables l m
variable A₁ : Type.{l}
variable A₂ : Type.{l}
definition foo (a₁ : A₁) (a₂ : A₂) := a₁ == a₂
check @foo
check Type.{m}
|
7807d566378b4ac62403746a47365885208ce2f4 | 9b9a16fa2cb737daee6b2785474678b6fa91d6d4 | /src/analysis/exponential.lean | 812ff22b73e39f7de2fee4b8df9b3a38b1c6ce55 | [
"Apache-2.0"
] | permissive | johoelzl/mathlib | 253f46daa30b644d011e8e119025b01ad69735c4 | 592e3c7a2dfbd5826919b4605559d35d4d75938f | refs/heads/master | 1,625,657,216,488 | 1,551,374,946,000 | 1,551,374,946,000 | 98,915,829 | 0 | 0 | Apache-2.0 | 1,522,917,267,000 | 1,501,524,499,000 | Lean | UTF-8 | Lean | false | false | 57,799 | lean | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo
-/
import topology.instances.complex tactic.linarith data.complex.exponential group_theory.quotient_group
open finset filter metric
namespace complex
lemma tendsto_exp_zero_one : tendsto exp (nhds 0) (nhds 1) :=
tendsto_nhds_nhds.2 $ λ ε ε0,
⟨min (ε / 2) 1, lt_min (div_pos ε0 (by norm_num)) (by norm_num),
λ x h, have h : abs x < min (ε / 2) 1, by simpa [dist_eq] using h,
calc abs (exp x - 1) ≤ 2 * abs x : abs_exp_sub_one_le
(le_trans (le_of_lt h) (min_le_right _ _))
... = abs x + abs x : two_mul (abs x)
... < ε / 2 + ε / 2 : add_lt_add
(lt_of_lt_of_le h (min_le_left _ _)) (lt_of_lt_of_le h (min_le_left _ _))
... = ε : by rw add_halves⟩
lemma continuous_exp : continuous exp :=
continuous_iff_continuous_at.2 (λ x,
have H1 : tendsto (λ h, exp (x + h)) (nhds 0) (nhds (exp x)),
by simpa [exp_add] using tendsto_mul tendsto_const_nhds tendsto_exp_zero_one,
have H2 : tendsto (λ y, y - x) (nhds x) (nhds (x - x)) :=
tendsto_sub tendsto_id (@tendsto_const_nhds _ _ _ x _),
suffices tendsto ((λ h, exp (x + h)) ∘
(λ y, id y - (λ z, x) y)) (nhds x) (nhds (exp x)),
by simp only [function.comp, add_sub_cancel'_right, id.def] at this;
exact this,
tendsto.comp (by rw [sub_self] at H2; exact H2) H1)
lemma continuous_sin : continuous sin :=
continuous_mul
(continuous_mul
(continuous_sub
((continuous_mul continuous_neg' continuous_const).comp continuous_exp)
((continuous_mul continuous_id continuous_const).comp continuous_exp))
continuous_const)
continuous_const
lemma continuous_cos : continuous cos :=
continuous_mul
(continuous_add
((continuous_mul continuous_id continuous_const).comp continuous_exp)
((continuous_mul continuous_neg' continuous_const).comp continuous_exp))
continuous_const
lemma continuous_tan : continuous (λ x : {x // cos x ≠ 0}, tan x) :=
continuous_mul
(continuous_subtype_val.comp continuous_sin)
(continuous_inv subtype.property
(continuous_subtype_val.comp continuous_cos))
lemma continuous_sinh : continuous sinh :=
continuous_mul
(continuous_sub
continuous_exp
(continuous_neg'.comp continuous_exp))
continuous_const
lemma continuous_cosh : continuous cosh :=
continuous_mul
(continuous_add
continuous_exp
(continuous_neg'.comp continuous_exp))
continuous_const
end complex
namespace real
lemma continuous_exp : continuous exp :=
(complex.continuous_of_real.comp complex.continuous_exp).comp
complex.continuous_re
lemma continuous_sin : continuous sin :=
(complex.continuous_of_real.comp complex.continuous_sin).comp
complex.continuous_re
lemma continuous_cos : continuous cos :=
(complex.continuous_of_real.comp complex.continuous_cos).comp
complex.continuous_re
lemma continuous_tan : continuous (λ x : {x // cos x ≠ 0}, tan x) :=
by simp only [tan_eq_sin_div_cos]; exact
continuous_mul
(continuous_subtype_val.comp continuous_sin)
(continuous_inv subtype.property
(continuous_subtype_val.comp continuous_cos))
lemma continuous_sinh : continuous sinh :=
(complex.continuous_of_real.comp complex.continuous_sinh).comp
complex.continuous_re
lemma continuous_cosh : continuous cosh :=
(complex.continuous_of_real.comp complex.continuous_cosh).comp
complex.continuous_re
private lemma exists_exp_eq_of_one_le {x : ℝ} (hx : 1 ≤ x) : ∃ y, exp y = x :=
let ⟨y, hy⟩ := @intermediate_value real.exp 0 (x - 1) x
(λ _ _ _, continuous_iff_continuous_at.1 continuous_exp _) (by simpa)
(by simpa using add_one_le_exp_of_nonneg (sub_nonneg.2 hx)) (sub_nonneg.2 hx) in
⟨y, hy.2.2⟩
lemma exists_exp_eq_of_pos {x : ℝ} (hx : 0 < x) : ∃ y, exp y = x :=
match le_total x 1 with
| (or.inl hx1) := let ⟨y, hy⟩ := exists_exp_eq_of_one_le (one_le_inv hx hx1) in
⟨-y, by rw [exp_neg, hy, inv_inv']⟩
| (or.inr hx1) := exists_exp_eq_of_one_le hx1
end
noncomputable def log (x : ℝ) : ℝ :=
if hx : 0 < x then classical.some (exists_exp_eq_of_pos hx) else 0
lemma exp_log {x : ℝ} (hx : 0 < x) : exp (log x) = x :=
by rw [log, dif_pos hx]; exact classical.some_spec (exists_exp_eq_of_pos hx)
@[simp] lemma log_exp (x : ℝ) : log (exp x) = x :=
exp_injective $ exp_log (exp_pos x)
@[simp] lemma log_zero : log 0 = 0 :=
by simp [log, lt_irrefl]
@[simp] lemma log_one : log 1 = 0 :=
exp_injective $ by rw [exp_log zero_lt_one, exp_zero]
lemma log_mul {x y : ℝ} (hx : 0 < x) (hy : 0 < y) : log (x * y) = log x + log y :=
exp_injective $ by rw [exp_log (mul_pos hx hy), exp_add, exp_log hx, exp_log hy]
lemma exists_cos_eq_zero : ∃ x, 1 ≤ x ∧ x ≤ 2 ∧ cos x = 0 :=
real.intermediate_value'
(λ x _ _, continuous_iff_continuous_at.1 continuous_cos _)
(le_of_lt cos_one_pos)
(le_of_lt cos_two_neg) (by norm_num)
noncomputable def pi : ℝ := 2 * classical.some exists_cos_eq_zero
local notation `π` := pi
@[simp] lemma cos_pi_div_two : cos (π / 2) = 0 :=
by rw [pi, mul_div_cancel_left _ (@two_ne_zero' ℝ _ _ _)];
exact (classical.some_spec exists_cos_eq_zero).2.2
lemma one_le_pi_div_two : (1 : ℝ) ≤ π / 2 :=
by rw [pi, mul_div_cancel_left _ (@two_ne_zero' ℝ _ _ _)];
exact (classical.some_spec exists_cos_eq_zero).1
lemma pi_div_two_le_two : π / 2 ≤ 2 :=
by rw [pi, mul_div_cancel_left _ (@two_ne_zero' ℝ _ _ _)];
exact (classical.some_spec exists_cos_eq_zero).2.1
lemma two_le_pi : (2 : ℝ) ≤ π :=
(div_le_div_right (show (0 : ℝ) < 2, by norm_num)).1
(by rw div_self (@two_ne_zero' ℝ _ _ _); exact one_le_pi_div_two)
lemma pi_le_four : π ≤ 4 :=
(div_le_div_right (show (0 : ℝ) < 2, by norm_num)).1
(calc π / 2 ≤ 2 : pi_div_two_le_two
... = 4 / 2 : by norm_num)
lemma pi_pos : 0 < π :=
lt_of_lt_of_le (by norm_num) two_le_pi
lemma pi_div_two_pos : 0 < π / 2 :=
half_pos pi_pos
lemma two_pi_pos : 0 < 2 * π :=
by linarith using [pi_pos]
@[simp] lemma sin_pi : sin π = 0 :=
by rw [← mul_div_cancel_left pi (@two_ne_zero ℝ _), two_mul, add_div,
sin_add, cos_pi_div_two]; simp
@[simp] lemma cos_pi : cos π = -1 :=
by rw [← mul_div_cancel_left pi (@two_ne_zero ℝ _), mul_div_assoc,
cos_two_mul, cos_pi_div_two];
simp [bit0, pow_add]
@[simp] lemma sin_two_pi : sin (2 * π) = 0 :=
by simp [two_mul, sin_add]
@[simp] lemma cos_two_pi : cos (2 * π) = 1 :=
by simp [two_mul, cos_add]
lemma sin_add_pi (x : ℝ) : sin (x + π) = -sin x :=
by simp [sin_add]
lemma sin_add_two_pi (x : ℝ) : sin (x + 2 * π) = sin x :=
by simp [sin_add_pi, sin_add, sin_two_pi, cos_two_pi]
lemma cos_add_two_pi (x : ℝ) : cos (x + 2 * π) = cos x :=
by simp [cos_add, cos_two_pi, sin_two_pi]
lemma sin_pi_sub (x : ℝ) : sin (π - x) = sin x :=
by simp [sin_add]
lemma cos_add_pi (x : ℝ) : cos (x + π) = -cos x :=
by simp [cos_add]
lemma cos_pi_sub (x : ℝ) : cos (π - x) = -cos x :=
by simp [cos_add]
lemma sin_pos_of_pos_of_lt_pi {x : ℝ} (h0x : 0 < x) (hxp : x < π) : 0 < sin x :=
if hx2 : x ≤ 2 then sin_pos_of_pos_of_le_two h0x hx2
else
have (2 : ℝ) + 2 = 4, from rfl,
have π - x ≤ 2, from sub_le_iff_le_add.2
(le_trans pi_le_four (this ▸ add_le_add_left (le_of_not_ge hx2) _)),
sin_pi_sub x ▸ sin_pos_of_pos_of_le_two (sub_pos.2 hxp) this
lemma sin_nonneg_of_nonneg_of_le_pi {x : ℝ} (h0x : 0 ≤ x) (hxp : x ≤ π) : 0 ≤ sin x :=
match lt_or_eq_of_le h0x with
| or.inl h0x := (lt_or_eq_of_le hxp).elim
(le_of_lt ∘ sin_pos_of_pos_of_lt_pi h0x)
(λ hpx, by simp [hpx])
| or.inr h0x := by simp [h0x.symm]
end
lemma sin_neg_of_neg_of_neg_pi_lt {x : ℝ} (hx0 : x < 0) (hpx : -π < x) : sin x < 0 :=
neg_pos.1 $ sin_neg x ▸ sin_pos_of_pos_of_lt_pi (neg_pos.2 hx0) (neg_lt.1 hpx)
lemma sin_nonpos_of_nonnpos_of_neg_pi_le {x : ℝ} (hx0 : x ≤ 0) (hpx : -π ≤ x) : sin x ≤ 0 :=
neg_nonneg.1 $ sin_neg x ▸ sin_nonneg_of_nonneg_of_le_pi (neg_nonneg.2 hx0) (neg_le.1 hpx)
@[simp] lemma sin_pi_div_two : sin (π / 2) = 1 :=
have sin (π / 2) = 1 ∨ sin (π / 2) = -1 :=
by simpa [pow_two, mul_self_eq_one_iff] using sin_pow_two_add_cos_pow_two (π / 2),
this.resolve_right
(λ h, (show ¬(0 : ℝ) < -1, by norm_num) $
h ▸ sin_pos_of_pos_of_lt_pi pi_div_two_pos (half_lt_self pi_pos))
lemma sin_add_pi_div_two (x : ℝ) : sin (x + π / 2) = cos x :=
by simp [sin_add]
lemma sin_sub_pi_div_two (x : ℝ) : sin (x - π / 2) = -cos x :=
by simp [sin_add]
lemma sin_pi_div_two_sub (x : ℝ) : sin (π / 2 - x) = cos x :=
by simp [sin_add]
lemma cos_add_pi_div_two (x : ℝ) : cos (x + π / 2) = -sin x :=
by simp [cos_add]
lemma cos_sub_pi_div_two (x : ℝ) : cos (x - π / 2) = sin x :=
by simp [cos_add]
lemma cos_pi_div_two_sub (x : ℝ) : cos (π / 2 - x) = sin x :=
by rw [← cos_neg, neg_sub, cos_sub_pi_div_two]
lemma cos_pos_of_neg_pi_div_two_lt_of_lt_pi_div_two
{x : ℝ} (hx₁ : -(π / 2) < x) (hx₂ : x < π / 2) : 0 < cos x :=
sin_add_pi_div_two x ▸ sin_pos_of_pos_of_lt_pi (by linarith) (by linarith)
lemma cos_nonneg_of_neg_pi_div_two_le_of_le_pi_div_two
{x : ℝ} (hx₁ : -(π / 2) ≤ x) (hx₂ : x ≤ π / 2) : 0 ≤ cos x :=
match lt_or_eq_of_le hx₁, lt_or_eq_of_le hx₂ with
| or.inl hx₁, or.inl hx₂ := le_of_lt (cos_pos_of_neg_pi_div_two_lt_of_lt_pi_div_two hx₁ hx₂)
| or.inl hx₁, or.inr hx₂ := by simp [hx₂]
| or.inr hx₁, _ := by simp [hx₁.symm]
end
lemma cos_neg_of_pi_div_two_lt_of_lt {x : ℝ} (hx₁ : π / 2 < x) (hx₂ : x < π + π / 2) : cos x < 0 :=
neg_pos.1 $ cos_pi_sub x ▸
cos_pos_of_neg_pi_div_two_lt_of_lt_pi_div_two (by linarith) (by linarith)
lemma cos_nonpos_of_pi_div_two_le_of_le {x : ℝ} (hx₁ : π / 2 ≤ x) (hx₂ : x ≤ π + π / 2) : cos x ≤ 0 :=
neg_nonneg.1 $ cos_pi_sub x ▸
cos_nonneg_of_neg_pi_div_two_le_of_le_pi_div_two (by linarith) (by linarith)
lemma sin_nat_mul_pi (n : ℕ) : sin (n * π) = 0 :=
by induction n; simp [add_mul, sin_add, *]
lemma sin_int_mul_pi (n : ℤ) : sin (n * π) = 0 :=
by cases n; simp [add_mul, sin_add, *, sin_nat_mul_pi]
lemma cos_nat_mul_two_pi (n : ℕ) : cos (n * (2 * π)) = 1 :=
by induction n; simp [*, mul_add, cos_add, add_mul, cos_two_pi, sin_two_pi]
lemma cos_int_mul_two_pi (n : ℤ) : cos (n * (2 * π)) = 1 :=
by cases n; simp only [cos_nat_mul_two_pi, int.of_nat_eq_coe,
int.neg_succ_of_nat_coe, int.cast_coe_nat, int.cast_neg,
(neg_mul_eq_neg_mul _ _).symm, cos_neg]
lemma cos_int_mul_two_pi_add_pi (n : ℤ) : cos (n * (2 * π) + π) = -1 :=
by simp [cos_add, sin_add, cos_int_mul_two_pi]
lemma sin_eq_zero_iff_of_lt_of_lt {x : ℝ} (hx₁ : -π < x) (hx₂ : x < π) :
sin x = 0 ↔ x = 0 :=
⟨λ h, le_antisymm
(le_of_not_gt (λ h0, lt_irrefl (0 : ℝ) $
calc 0 < sin x : sin_pos_of_pos_of_lt_pi h0 hx₂
... = 0 : h))
(le_of_not_gt (λ h0, lt_irrefl (0 : ℝ) $
calc 0 = sin x : h.symm
... < 0 : sin_neg_of_neg_of_neg_pi_lt h0 hx₁)),
λ h, by simp [h]⟩
lemma sin_eq_zero_iff {x : ℝ} : sin x = 0 ↔ ∃ n : ℤ, (n : ℝ) * π = x :=
⟨λ h, ⟨⌊x / π⌋, le_antisymm (sub_nonneg.1 (sub_floor_div_mul_nonneg _ pi_pos))
(sub_nonpos.1 $ le_of_not_gt $ λ h₃, ne_of_lt (sin_pos_of_pos_of_lt_pi h₃ (sub_floor_div_mul_lt _ pi_pos))
(by simp [sin_add, h, sin_int_mul_pi]))⟩,
λ ⟨n, hn⟩, hn ▸ sin_int_mul_pi _⟩
lemma sin_eq_zero_iff_cos_eq {x : ℝ} : sin x = 0 ↔ cos x = 1 ∨ cos x = -1 :=
by rw [← mul_self_eq_one_iff (cos x), ← sin_pow_two_add_cos_pow_two x,
pow_two, pow_two, ← sub_eq_iff_eq_add, sub_self];
exact ⟨λ h, by rw [h, mul_zero], eq_zero_of_mul_self_eq_zero ∘ eq.symm⟩
theorem sin_sub_sin (θ ψ : ℝ) : sin θ - sin ψ = 2 * sin((θ - ψ)/2) * cos((θ + ψ)/2) :=
begin
have s1 := sin_add ((θ + ψ) / 2) ((θ - ψ) / 2),
have s2 := sin_sub ((θ + ψ) / 2) ((θ - ψ) / 2),
rw [div_add_div_same, add_sub, add_right_comm, add_sub_cancel, add_self_div_two] at s1,
rw [div_sub_div_same, ←sub_add, add_sub_cancel', add_self_div_two] at s2,
rw [s1, s2, ←sub_add, add_sub_cancel', ← two_mul, ← mul_assoc, mul_right_comm]
end
lemma cos_eq_one_iff (x : ℝ) : cos x = 1 ↔ ∃ n : ℤ, (n : ℝ) * (2 * π) = x :=
⟨λ h, let ⟨n, hn⟩ := sin_eq_zero_iff.1 (sin_eq_zero_iff_cos_eq.2 (or.inl h)) in
⟨n / 2, (int.mod_two_eq_zero_or_one n).elim
(λ hn0, by rwa [← mul_assoc, ← @int.cast_two ℝ, ← int.cast_mul, int.div_mul_cancel
((int.dvd_iff_mod_eq_zero _ _).2 hn0)])
(λ hn1, by rw [← int.mod_add_div n 2, hn1, int.cast_add, int.cast_one, add_mul,
one_mul, add_comm, mul_comm (2 : ℤ), int.cast_mul, mul_assoc, int.cast_two] at hn;
rw [← hn, cos_int_mul_two_pi_add_pi] at h;
exact absurd h (by norm_num))⟩,
λ ⟨n, hn⟩, hn ▸ cos_int_mul_two_pi _⟩
theorem cos_eq_zero_iff {θ : ℝ} : cos θ = 0 ↔ ∃ k : ℤ, θ = (2 * k + 1) * pi / 2 :=
begin
rw [←real.sin_pi_div_two_sub, sin_eq_zero_iff],
split,
{ rintro ⟨n, hn⟩, existsi -n,
rw [int.cast_neg, add_mul, add_div, mul_assoc, mul_div_cancel_left _ two_ne_zero,
one_mul, ←neg_mul_eq_neg_mul, hn, neg_sub, sub_add_cancel] },
{ rintro ⟨n, hn⟩, existsi -n,
rw [hn, add_mul, one_mul, add_div, mul_assoc, mul_div_cancel_left _ two_ne_zero,
sub_add_eq_sub_sub_swap, sub_self, zero_sub, neg_mul_eq_neg_mul, int.cast_neg] }
end
lemma cos_eq_one_iff_of_lt_of_lt {x : ℝ} (hx₁ : -(2 * π) < x) (hx₂ : x < 2 * π) : cos x = 1 ↔ x = 0 :=
⟨λ h, let ⟨n, hn⟩ := (cos_eq_one_iff x).1 h in
begin
clear _let_match,
subst hn,
rw [mul_lt_iff_lt_one_left two_pi_pos, ← int.cast_one, int.cast_lt, ← int.le_sub_one_iff, sub_self] at hx₂,
rw [neg_lt, neg_mul_eq_neg_mul, mul_lt_iff_lt_one_left two_pi_pos, neg_lt,
← int.cast_one, ← int.cast_neg, int.cast_lt, ← int.add_one_le_iff, neg_add_self] at hx₁,
exact mul_eq_zero.2 (or.inl (int.cast_eq_zero.2 (le_antisymm hx₂ hx₁))),
end,
λ h, by simp [h]⟩
theorem cos_sub_cos (θ ψ : ℝ) : cos θ - cos ψ = -2 * sin((θ + ψ)/2) * sin((θ - ψ)/2) :=
by rw [← sin_pi_div_two_sub, ← sin_pi_div_two_sub, sin_sub_sin, sub_sub_sub_cancel_left,
add_sub, sub_add_eq_add_sub, add_halves, sub_sub, sub_div π, cos_pi_div_two_sub,
← neg_sub, neg_div, sin_neg, ← neg_mul_eq_mul_neg, neg_mul_eq_neg_mul, mul_right_comm]
lemma cos_lt_cos_of_nonneg_of_le_pi_div_two {x y : ℝ} (hx₁ : 0 ≤ x) (hx₂ : x ≤ π / 2)
(hy₁ : 0 ≤ y) (hy₂ : y ≤ π / 2) (hxy : x < y) : cos y < cos x :=
calc cos y = cos x * cos (y - x) - sin x * sin (y - x) :
by rw [← cos_add, add_sub_cancel'_right]
... < (cos x * 1) - sin x * sin (y - x) :
sub_lt_sub_right ((mul_lt_mul_left
(cos_pos_of_neg_pi_div_two_lt_of_lt_pi_div_two (lt_of_lt_of_le (neg_neg_of_pos pi_div_two_pos) hx₁)
(lt_of_lt_of_le hxy hy₂))).2
(lt_of_le_of_ne (cos_le_one _) (mt (cos_eq_one_iff_of_lt_of_lt
(show -(2 * π) < y - x, by linarith) (show y - x < 2 * π, by linarith)).1
(sub_ne_zero.2 (ne_of_lt hxy).symm)))) _
... ≤ _ : by rw mul_one;
exact sub_le_self _ (mul_nonneg (sin_nonneg_of_nonneg_of_le_pi hx₁ (by linarith))
(sin_nonneg_of_nonneg_of_le_pi (by linarith) (by linarith)))
lemma cos_lt_cos_of_nonneg_of_le_pi {x y : ℝ} (hx₁ : 0 ≤ x) (hx₂ : x ≤ π)
(hy₁ : 0 ≤ y) (hy₂ : y ≤ π) (hxy : x < y) : cos y < cos x :=
match (le_total x (π / 2) : x ≤ π / 2 ∨ π / 2 ≤ x), le_total y (π / 2) with
| or.inl hx, or.inl hy := cos_lt_cos_of_nonneg_of_le_pi_div_two hx₁ hx hy₁ hy hxy
| or.inl hx, or.inr hy := (lt_or_eq_of_le hx).elim
(λ hx, calc cos y ≤ 0 : cos_nonpos_of_pi_div_two_le_of_le hy (by linarith using [pi_pos])
... < cos x : cos_pos_of_neg_pi_div_two_lt_of_lt_pi_div_two (by linarith) hx)
(λ hx, calc cos y < 0 : cos_neg_of_pi_div_two_lt_of_lt (by linarith) (by linarith using [pi_pos])
... = cos x : by rw [hx, cos_pi_div_two])
| or.inr hx, or.inl hy := by linarith
| or.inr hx, or.inr hy := neg_lt_neg_iff.1 (by rw [← cos_pi_sub, ← cos_pi_sub];
apply cos_lt_cos_of_nonneg_of_le_pi_div_two; linarith)
end
lemma cos_le_cos_of_nonneg_of_le_pi {x y : ℝ} (hx₁ : 0 ≤ x) (hx₂ : x ≤ π)
(hy₁ : 0 ≤ y) (hy₂ : y ≤ π) (hxy : x ≤ y) : cos y ≤ cos x :=
(lt_or_eq_of_le hxy).elim
(le_of_lt ∘ cos_lt_cos_of_nonneg_of_le_pi hx₁ hx₂ hy₁ hy₂)
(λ h, h ▸ le_refl _)
lemma sin_lt_sin_of_le_of_le_pi_div_two {x y : ℝ} (hx₁ : -(π / 2) ≤ x) (hx₂ : x ≤ π / 2) (hy₁ : -(π / 2) ≤ y)
(hy₂ : y ≤ π / 2) (hxy : x < y) : sin x < sin y :=
by rw [← cos_sub_pi_div_two, ← cos_sub_pi_div_two, ← cos_neg (x - _), ← cos_neg (y - _)];
apply cos_lt_cos_of_nonneg_of_le_pi; linarith
lemma sin_le_sin_of_le_of_le_pi_div_two {x y : ℝ} (hx₁ : -(π / 2) ≤ x) (hx₂ : x ≤ π / 2) (hy₁ : -(π / 2) ≤ y)
(hy₂ : y ≤ π / 2) (hxy : x ≤ y) : sin x ≤ sin y :=
(lt_or_eq_of_le hxy).elim
(le_of_lt ∘ sin_lt_sin_of_le_of_le_pi_div_two hx₁ hx₂ hy₁ hy₂)
(λ h, h ▸ le_refl _)
lemma sin_inj_of_le_of_le_pi_div_two {x y : ℝ} (hx₁ : -(π / 2) ≤ x) (hx₂ : x ≤ π / 2) (hy₁ : -(π / 2) ≤ y)
(hy₂ : y ≤ π / 2) (hxy : sin x = sin y) : x = y :=
match lt_trichotomy x y with
| or.inl h := absurd (sin_lt_sin_of_le_of_le_pi_div_two hx₁ hx₂ hy₁ hy₂ h) (by rw hxy; exact lt_irrefl _)
| or.inr (or.inl h) := h
| or.inr (or.inr h) := absurd (sin_lt_sin_of_le_of_le_pi_div_two hy₁ hy₂ hx₁ hx₂ h) (by rw hxy; exact lt_irrefl _)
end
lemma cos_inj_of_nonneg_of_le_pi {x y : ℝ} (hx₁ : 0 ≤ x) (hx₂ : x ≤ π) (hy₁ : 0 ≤ y) (hy₂ : y ≤ π)
(hxy : cos x = cos y) : x = y :=
begin
rw [← sin_pi_div_two_sub, ← sin_pi_div_two_sub] at hxy,
refine (sub_left_inj).1 (sin_inj_of_le_of_le_pi_div_two _ _ _ _ hxy);
linarith
end
lemma exists_sin_eq {x : ℝ} (hx₁ : -1 ≤ x) (hx₂ : x ≤ 1) : ∃ y, -(π / 2) ≤ y ∧ y ≤ π / 2 ∧ sin y = x :=
@real.intermediate_value sin (-(π / 2)) (π / 2) x
(λ _ _ _, continuous_iff_continuous_at.1 continuous_sin _)
(by rwa [sin_neg, sin_pi_div_two]) (by rwa sin_pi_div_two)
(le_trans (neg_nonpos.2 (le_of_lt pi_div_two_pos)) (le_of_lt pi_div_two_pos))
namespace angle
/-- The type of angles -/
def angle : Type :=
quotient_add_group.quotient (gmultiples (2 * π))
instance angle.add_comm_group : add_comm_group angle :=
quotient_add_group.add_comm_group _
instance angle.has_coe : has_coe ℝ angle :=
⟨quotient.mk'⟩
instance angle.is_add_group_hom : is_add_group_hom (coe : ℝ → angle) :=
@quotient_add_group.is_add_group_hom _ _ _ (normal_add_subgroup_of_add_comm_group _)
@[simp] lemma coe_zero : ↑(0 : ℝ) = (0 : angle) := rfl
@[simp] lemma coe_add (x y : ℝ) : ↑(x + y : ℝ) = (↑x + ↑y : angle) := rfl
@[simp] lemma coe_neg (x : ℝ) : ↑(-x : ℝ) = -(↑x : angle) := rfl
@[simp] lemma coe_sub (x y : ℝ) : ↑(x - y : ℝ) = (↑x - ↑y : angle) := rfl
@[simp] lemma coe_gsmul (x : ℝ) (n : ℤ) : ↑(gsmul n x : ℝ) = gsmul n (↑x : angle) := is_add_group_hom.gsmul _ _ _
@[simp] lemma coe_two_pi : ↑(2 * π : ℝ) = (0 : angle) :=
quotient.sound' ⟨-1, by dsimp only; rw [neg_one_gsmul, add_zero]⟩
lemma angle_eq_iff_two_pi_dvd_sub {ψ θ : ℝ} : (θ : angle) = ψ ↔ ∃ k : ℤ, θ - ψ = 2 * π * k :=
by simp only [quotient_add_group.eq, gmultiples, set.mem_range, gsmul_eq_mul', (sub_eq_neg_add _ _).symm, eq_comm]
theorem cos_eq_iff_eq_or_eq_neg {θ ψ : ℝ} : cos θ = cos ψ ↔ (θ : angle) = ψ ∨ (θ : angle) = -ψ :=
begin
split,
{ intro Hcos,
rw [←sub_eq_zero, cos_sub_cos, mul_eq_zero, mul_eq_zero, neg_eq_zero, eq_false_intro two_ne_zero,
false_or, sin_eq_zero_iff, sin_eq_zero_iff] at Hcos,
rcases Hcos with ⟨n, hn⟩ | ⟨n, hn⟩,
{ right,
rw [eq_div_iff_mul_eq _ _ two_ne_zero, ← sub_eq_iff_eq_add] at hn,
rw [← hn, coe_sub, eq_neg_iff_add_eq_zero, sub_add_cancel, mul_assoc,
← gsmul_eq_mul, coe_gsmul, mul_comm, coe_two_pi, gsmul_zero] },
{ left,
rw [eq_div_iff_mul_eq _ _ two_ne_zero, eq_sub_iff_add_eq] at hn,
rw [← hn, coe_add, mul_assoc,
← gsmul_eq_mul, coe_gsmul, mul_comm, coe_two_pi, gsmul_zero, zero_add] } },
{ rw [angle_eq_iff_two_pi_dvd_sub, ← coe_neg, angle_eq_iff_two_pi_dvd_sub],
rintro (⟨k, H⟩ | ⟨k, H⟩),
rw [← sub_eq_zero_iff_eq, cos_sub_cos, H, mul_assoc 2 π k, mul_div_cancel_left _ two_ne_zero,
mul_comm π _, sin_int_mul_pi, mul_zero],
rw [←sub_eq_zero_iff_eq, cos_sub_cos, ← sub_neg_eq_add, H, mul_assoc 2 π k,
mul_div_cancel_left _ two_ne_zero, mul_comm π _, sin_int_mul_pi, mul_zero, zero_mul] }
end
theorem sin_eq_iff_eq_or_add_eq_pi {θ ψ : ℝ} : sin θ = sin ψ ↔ (θ : angle) = ψ ∨ (θ : angle) + ψ = π :=
begin
split,
{ intro Hsin, rw [← cos_pi_div_two_sub, ← cos_pi_div_two_sub] at Hsin,
cases cos_eq_iff_eq_or_eq_neg.mp Hsin with h h,
{ left, rw coe_sub at h, exact sub_left_inj.1 h },
right, rw [coe_sub, coe_sub, eq_neg_iff_add_eq_zero, add_sub,
sub_add_eq_add_sub, ← coe_add, add_halves, sub_sub, sub_eq_zero] at h,
exact h.symm },
{ rw [angle_eq_iff_two_pi_dvd_sub, ←eq_sub_iff_add_eq, ←coe_sub, angle_eq_iff_two_pi_dvd_sub],
rintro (⟨k, H⟩ | ⟨k, H⟩),
rw [← sub_eq_zero_iff_eq, sin_sub_sin, H, mul_assoc 2 π k, mul_div_cancel_left _ two_ne_zero,
mul_comm π _, sin_int_mul_pi, mul_zero, zero_mul],
have H' : θ + ψ = (2 * k) * π + π := by rwa [←sub_add, sub_add_eq_add_sub, sub_eq_iff_eq_add,
mul_assoc, mul_comm π _, ←mul_assoc] at H,
rw [← sub_eq_zero_iff_eq, sin_sub_sin, H', add_div, mul_assoc 2 _ π, mul_div_cancel_left _ two_ne_zero,
cos_add_pi_div_two, sin_int_mul_pi, neg_zero, mul_zero] }
end
theorem cos_sin_inj {θ ψ : ℝ} (Hcos : cos θ = cos ψ) (Hsin : sin θ = sin ψ) : (θ : angle) = ψ :=
begin
cases cos_eq_iff_eq_or_eq_neg.mp Hcos with hc hc, { exact hc },
cases sin_eq_iff_eq_or_add_eq_pi.mp Hsin with hs hs, { exact hs },
rw [eq_neg_iff_add_eq_zero, hs] at hc,
cases quotient.exact' hc with n hn, dsimp only at hn,
rw [← neg_one_mul, add_zero, ← sub_eq_zero_iff_eq, gsmul_eq_mul, ← mul_assoc, ← sub_mul,
mul_eq_zero, eq_false_intro (ne_of_gt pi_pos), or_false, sub_neg_eq_add,
← int.cast_zero, ← int.cast_one, ← int.cast_bit0, ← int.cast_mul, ← int.cast_add, int.cast_inj] at hn,
have : (n * 2 + 1) % (2:ℤ) = 0 % (2:ℤ) := congr_arg (%(2:ℤ)) hn,
rw [add_comm, int.add_mul_mod_self] at this,
exact absurd this one_ne_zero
end
end angle
/-- Inverse of the `sin` function, returns values in the range `-π / 2 ≤ arcsin x` and `arcsin x ≤ π / 2`.
If the argument is not between `-1` and `1` it defaults to `0` -/
noncomputable def arcsin (x : ℝ) : ℝ :=
if hx : -1 ≤ x ∧ x ≤ 1 then classical.some (exists_sin_eq hx.1 hx.2) else 0
lemma arcsin_le_pi_div_two (x : ℝ) : arcsin x ≤ π / 2 :=
if hx : -1 ≤ x ∧ x ≤ 1
then by rw [arcsin, dif_pos hx]; exact (classical.some_spec (exists_sin_eq hx.1 hx.2)).2.1
else by rw [arcsin, dif_neg hx]; exact le_of_lt pi_div_two_pos
lemma neg_pi_div_two_le_arcsin (x : ℝ) : -(π / 2) ≤ arcsin x :=
if hx : -1 ≤ x ∧ x ≤ 1
then by rw [arcsin, dif_pos hx]; exact (classical.some_spec (exists_sin_eq hx.1 hx.2)).1
else by rw [arcsin, dif_neg hx]; exact neg_nonpos.2 (le_of_lt pi_div_two_pos)
lemma sin_arcsin {x : ℝ} (hx₁ : -1 ≤ x) (hx₂ : x ≤ 1) : sin (arcsin x) = x :=
by rw [arcsin, dif_pos (and.intro hx₁ hx₂)];
exact (classical.some_spec (exists_sin_eq hx₁ hx₂)).2.2
lemma arcsin_sin {x : ℝ} (hx₁ : -(π / 2) ≤ x) (hx₂ : x ≤ π / 2) : arcsin (sin x) = x :=
sin_inj_of_le_of_le_pi_div_two (neg_pi_div_two_le_arcsin _) (arcsin_le_pi_div_two _) hx₁ hx₂
(by rw sin_arcsin (neg_one_le_sin _) (sin_le_one _))
lemma arcsin_inj {x y : ℝ} (hx₁ : -1 ≤ x) (hx₂ : x ≤ 1) (hy₁ : -1 ≤ y) (hy₂ : y ≤ 1)
(hxy : arcsin x = arcsin y) : x = y :=
by rw [← sin_arcsin hx₁ hx₂, ← sin_arcsin hy₁ hy₂, hxy]
@[simp] lemma arcsin_zero : arcsin 0 = 0 :=
sin_inj_of_le_of_le_pi_div_two
(neg_pi_div_two_le_arcsin _)
(arcsin_le_pi_div_two _)
(neg_nonpos.2 (le_of_lt pi_div_two_pos))
(le_of_lt pi_div_two_pos)
(by rw [sin_arcsin, sin_zero]; norm_num)
@[simp] lemma arcsin_one : arcsin 1 = π / 2 :=
sin_inj_of_le_of_le_pi_div_two
(neg_pi_div_two_le_arcsin _)
(arcsin_le_pi_div_two _)
(by linarith using [pi_pos])
(le_refl _)
(by rw [sin_arcsin, sin_pi_div_two]; norm_num)
@[simp] lemma arcsin_neg (x : ℝ) : arcsin (-x) = -arcsin x :=
if h : -1 ≤ x ∧ x ≤ 1 then
have -1 ≤ -x ∧ -x ≤ 1, by rwa [neg_le_neg_iff, neg_le, and.comm],
sin_inj_of_le_of_le_pi_div_two
(neg_pi_div_two_le_arcsin _)
(arcsin_le_pi_div_two _)
(neg_le_neg (arcsin_le_pi_div_two _))
(neg_le.1 (neg_pi_div_two_le_arcsin _))
(by rw [sin_arcsin this.1 this.2, sin_neg, sin_arcsin h.1 h.2])
else
have ¬(-1 ≤ -x ∧ -x ≤ 1) := by rwa [neg_le_neg_iff, neg_le, and.comm],
by rw [arcsin, arcsin, dif_neg h, dif_neg this, neg_zero]
@[simp] lemma arcsin_neg_one : arcsin (-1) = -(π / 2) := by simp
lemma arcsin_nonneg {x : ℝ} (hx : 0 ≤ x) : 0 ≤ arcsin x :=
if hx₁ : x ≤ 1 then
not_lt.1 (λ h, not_lt.2 hx begin
have := sin_lt_sin_of_le_of_le_pi_div_two
(neg_pi_div_two_le_arcsin _) (arcsin_le_pi_div_two _)
(neg_nonpos.2 (le_of_lt pi_div_two_pos)) (le_of_lt pi_div_two_pos) h,
rw [real.sin_arcsin, sin_zero] at this; linarith
end)
else by rw [arcsin, dif_neg]; simp [hx₁]
lemma arcsin_eq_zero_iff {x : ℝ} (hx₁ : -1 ≤ x) (hx₂ : x ≤ 1) : arcsin x = 0 ↔ x = 0 :=
⟨λ h, have sin (arcsin x) = 0, by simp [h],
by rwa [sin_arcsin hx₁ hx₂] at this,
λ h, by simp [h]⟩
lemma arcsin_pos {x : ℝ} (hx₁ : 0 < x) (hx₂ : x ≤ 1) : 0 < arcsin x :=
lt_of_le_of_ne (arcsin_nonneg (le_of_lt hx₁))
(ne.symm (mt (arcsin_eq_zero_iff (by linarith) hx₂).1 (ne_of_lt hx₁).symm))
lemma arcsin_nonpos {x : ℝ} (hx : x ≤ 0) : arcsin x ≤ 0 :=
neg_nonneg.1 (arcsin_neg x ▸ arcsin_nonneg (neg_nonneg.2 hx))
/-- Inverse of the `cos` function, returns values in the range `0 ≤ arccos x` and `arccos x ≤ π`.
If the argument is not between `-1` and `1` it defaults to `π / 2` -/
noncomputable def arccos (x : ℝ) : ℝ :=
π / 2 - arcsin x
lemma arccos_eq_pi_div_two_sub_arcsin (x : ℝ) : arccos x = π / 2 - arcsin x := rfl
lemma arcsin_eq_pi_div_two_sub_arccos (x : ℝ) : arcsin x = π / 2 - arccos x := by simp [arccos]
lemma arccos_le_pi (x : ℝ) : arccos x ≤ π :=
by unfold arccos; linarith using [neg_pi_div_two_le_arcsin x]
lemma arccos_nonneg (x : ℝ) : 0 ≤ arccos x :=
by unfold arccos; linarith using [arcsin_le_pi_div_two x]
lemma cos_arccos {x : ℝ} (hx₁ : -1 ≤ x) (hx₂ : x ≤ 1) : cos (arccos x) = x :=
by rw [arccos, cos_pi_div_two_sub, sin_arcsin hx₁ hx₂]
lemma arccos_cos {x : ℝ} (hx₁ : 0 ≤ x) (hx₂ : x ≤ π) : arccos (cos x) = x :=
by rw [arccos, ← sin_pi_div_two_sub, arcsin_sin]; simp; linarith
lemma arccos_inj {x y : ℝ} (hx₁ : -1 ≤ x) (hx₂ : x ≤ 1) (hy₁ : -1 ≤ y) (hy₂ : y ≤ 1)
(hxy : arccos x = arccos y) : x = y :=
arcsin_inj hx₁ hx₂ hy₁ hy₂ $ by simp [arccos, *] at *
@[simp] lemma arccos_zero : arccos 0 = π / 2 := by simp [arccos]
@[simp] lemma arccos_one : arccos 1 = 0 := by simp [arccos]
@[simp] lemma arccos_neg_one : arccos (-1) = π := by simp [arccos, add_halves]
lemma arccos_neg (x : ℝ) : arccos (-x) = π - arccos x :=
by rw [← add_halves π, arccos, arcsin_neg, arccos, add_sub_assoc, sub_sub_self]; simp
lemma cos_arcsin_nonneg (x : ℝ) : 0 ≤ cos (arcsin x) :=
cos_nonneg_of_neg_pi_div_two_le_of_le_pi_div_two
(neg_pi_div_two_le_arcsin _) (arcsin_le_pi_div_two _)
lemma cos_arcsin {x : ℝ} (hx₁ : -1 ≤ x) (hx₂ : x ≤ 1) : cos (arcsin x) = sqrt (1 - x ^ 2) :=
have sin (arcsin x) ^ 2 + cos (arcsin x) ^ 2 = 1 := sin_pow_two_add_cos_pow_two (arcsin x),
begin
rw [← eq_sub_iff_add_eq', ← sqrt_inj (pow_two_nonneg _) (sub_nonneg.2 (sin_pow_two_le_one (arcsin x))),
pow_two, sqrt_mul_self (cos_arcsin_nonneg _)] at this,
rw [this, sin_arcsin hx₁ hx₂],
end
lemma sin_arccos {x : ℝ} (hx₁ : -1 ≤ x) (hx₂ : x ≤ 1) : sin (arccos x) = sqrt (1 - x ^ 2) :=
by rw [arccos_eq_pi_div_two_sub_arcsin, sin_pi_div_two_sub, cos_arcsin hx₁ hx₂]
lemma abs_div_sqrt_one_add_lt (x : ℝ) : abs (x / sqrt (1 + x ^ 2)) < 1 :=
have h₁ : 0 < 1 + x ^ 2, from add_pos_of_pos_of_nonneg zero_lt_one (pow_two_nonneg _),
have h₂ : 0 < sqrt (1 + x ^ 2), from sqrt_pos.2 h₁,
by rw [abs_div, div_lt_iff (abs_pos_of_pos h₂), one_mul,
mul_self_lt_mul_self_iff (abs_nonneg x) (abs_nonneg _),
← abs_mul, ← abs_mul, mul_self_sqrt (add_nonneg zero_le_one (pow_two_nonneg _)),
abs_of_nonneg (mul_self_nonneg x), abs_of_nonneg (le_of_lt h₁), pow_two, add_comm];
exact lt_add_one _
lemma div_sqrt_one_add_lt_one (x : ℝ) : x / sqrt (1 + x ^ 2) < 1 :=
(abs_lt.1 (abs_div_sqrt_one_add_lt _)).2
lemma neg_one_lt_div_sqrt_one_add (x : ℝ) : -1 < x / sqrt (1 + x ^ 2) :=
(abs_lt.1 (abs_div_sqrt_one_add_lt _)).1
lemma tan_pos_of_pos_of_lt_pi_div_two {x : ℝ} (h0x : 0 < x) (hxp : x < π / 2) : 0 < tan x :=
by rw tan_eq_sin_div_cos; exact div_pos (sin_pos_of_pos_of_lt_pi h0x (by linarith))
(cos_pos_of_neg_pi_div_two_lt_of_lt_pi_div_two (by linarith) hxp)
lemma tan_nonneg_of_nonneg_of_le_pi_div_two {x : ℝ} (h0x : 0 ≤ x) (hxp : x ≤ π / 2) : 0 ≤ tan x :=
match lt_or_eq_of_le h0x, lt_or_eq_of_le hxp with
| or.inl hx0, or.inl hxp := le_of_lt (tan_pos_of_pos_of_lt_pi_div_two hx0 hxp)
| or.inl hx0, or.inr hxp := by simp [hxp, tan_eq_sin_div_cos]
| or.inr hx0, _ := by simp [hx0.symm]
end
lemma tan_neg_of_neg_of_pi_div_two_lt {x : ℝ} (hx0 : x < 0) (hpx : -(π / 2) < x) : tan x < 0 :=
neg_pos.1 (tan_neg x ▸ tan_pos_of_pos_of_lt_pi_div_two (by linarith) (by linarith using [pi_pos]))
lemma tan_nonpos_of_nonpos_of_neg_pi_div_two_le {x : ℝ} (hx0 : x ≤ 0) (hpx : -(π / 2) ≤ x) : tan x ≤ 0 :=
neg_nonneg.1 (tan_neg x ▸ tan_nonneg_of_nonneg_of_le_pi_div_two (by linarith) (by linarith using [pi_pos]))
lemma tan_lt_tan_of_nonneg_of_lt_pi_div_two {x y : ℝ} (hx₁ : 0 ≤ x) (hx₂ : x < π / 2) (hy₁ : 0 ≤ y)
(hy₂ : y < π / 2) (hxy : x < y) : tan x < tan y :=
begin
rw [tan_eq_sin_div_cos, tan_eq_sin_div_cos],
exact div_lt_div
(sin_lt_sin_of_le_of_le_pi_div_two (by linarith) (le_of_lt hx₂)
(by linarith) (le_of_lt hy₂) hxy)
(cos_le_cos_of_nonneg_of_le_pi hx₁ (by linarith) hy₁ (by linarith) (le_of_lt hxy))
(sin_nonneg_of_nonneg_of_le_pi hy₁ (by linarith))
(cos_pos_of_neg_pi_div_two_lt_of_lt_pi_div_two (by linarith) hy₂)
end
lemma tan_lt_tan_of_lt_of_lt_pi_div_two {x y : ℝ} (hx₁ : -(π / 2) < x) (hx₂ : x < π / 2)
(hy₁ : -(π / 2) < y) (hy₂ : y < π / 2) (hxy : x < y) : tan x < tan y :=
match le_total x 0, le_total y 0 with
| or.inl hx0, or.inl hy0 := neg_lt_neg_iff.1 $ by rw [← tan_neg, ← tan_neg]; exact
tan_lt_tan_of_nonneg_of_lt_pi_div_two (neg_nonneg.2 hy0) (neg_lt.2 hy₁)
(neg_nonneg.2 hx0) (neg_lt.2 hx₁) (neg_lt_neg hxy)
| or.inl hx0, or.inr hy0 := (lt_or_eq_of_le hy0).elim
(λ hy0, calc tan x ≤ 0 : tan_nonpos_of_nonpos_of_neg_pi_div_two_le hx0 (le_of_lt hx₁)
... < tan y : tan_pos_of_pos_of_lt_pi_div_two hy0 hy₂)
(λ hy0, by rw [← hy0, tan_zero]; exact
tan_neg_of_neg_of_pi_div_two_lt (hy0.symm ▸ hxy) hx₁)
| or.inr hx0, or.inl hy0 := by linarith
| or.inr hx0, or.inr hy0 := tan_lt_tan_of_nonneg_of_lt_pi_div_two hx0 hx₂ hy0 hy₂ hxy
end
lemma tan_inj_of_lt_of_lt_pi_div_two {x y : ℝ} (hx₁ : -(π / 2) < x) (hx₂ : x < π / 2)
(hy₁ : -(π / 2) < y) (hy₂ : y < π / 2) (hxy : tan x = tan y) : x = y :=
match lt_trichotomy x y with
| or.inl h := absurd (tan_lt_tan_of_lt_of_lt_pi_div_two hx₁ hx₂ hy₁ hy₂ h) (by rw hxy; exact lt_irrefl _)
| or.inr (or.inl h) := h
| or.inr (or.inr h) := absurd (tan_lt_tan_of_lt_of_lt_pi_div_two hy₁ hy₂ hx₁ hx₂ h) (by rw hxy; exact lt_irrefl _)
end
/-- Inverse of the `tan` function, returns values in the range `-π / 2 < arctan x` and `arctan x < π / 2` -/
noncomputable def arctan (x : ℝ) : ℝ :=
arcsin (x / sqrt (1 + x ^ 2))
lemma sin_arctan (x : ℝ) : sin (arctan x) = x / sqrt (1 + x ^ 2) :=
sin_arcsin (le_of_lt (neg_one_lt_div_sqrt_one_add _)) (le_of_lt (div_sqrt_one_add_lt_one _))
lemma cos_arctan (x : ℝ) : cos (arctan x) = 1 / sqrt (1 + x ^ 2) :=
have h₁ : (0 : ℝ) < 1 + x ^ 2,
from add_pos_of_pos_of_nonneg zero_lt_one (pow_two_nonneg _),
have h₂ : (x / sqrt (1 + x ^ 2)) ^ 2 < 1,
by rw [pow_two, ← abs_mul_self, _root_.abs_mul];
exact mul_lt_one_of_nonneg_of_lt_one_left (abs_nonneg _)
(abs_div_sqrt_one_add_lt _) (le_of_lt (abs_div_sqrt_one_add_lt _)),
by rw [arctan, cos_arcsin (le_of_lt (neg_one_lt_div_sqrt_one_add _)) (le_of_lt (div_sqrt_one_add_lt_one _)),
one_div_eq_inv, ← sqrt_inv, sqrt_inj (sub_nonneg.2 (le_of_lt h₂)) (inv_nonneg.2 (le_of_lt h₁)),
div_pow _ (mt sqrt_eq_zero'.1 (not_le.2 h₁)), pow_two (sqrt _), mul_self_sqrt (le_of_lt h₁),
← domain.mul_left_inj (ne.symm (ne_of_lt h₁)), mul_sub,
mul_div_cancel' _ (ne.symm (ne_of_lt h₁)), mul_inv_cancel (ne.symm (ne_of_lt h₁))];
simp
lemma tan_arctan (x : ℝ) : tan (arctan x) = x :=
by rw [tan_eq_sin_div_cos, sin_arctan, cos_arctan, div_div_div_div_eq, mul_one,
mul_div_assoc,
div_self (mt sqrt_eq_zero'.1 (not_le_of_gt (add_pos_of_pos_of_nonneg zero_lt_one (pow_two_nonneg x)))),
mul_one]
lemma arctan_lt_pi_div_two (x : ℝ) : arctan x < π / 2 :=
lt_of_le_of_ne (arcsin_le_pi_div_two _)
(λ h, ne_of_lt (div_sqrt_one_add_lt_one x) $
by rw [← sin_arcsin (le_of_lt (neg_one_lt_div_sqrt_one_add _))
(le_of_lt (div_sqrt_one_add_lt_one _)), ← arctan, h, sin_pi_div_two])
lemma neg_pi_div_two_lt_arctan (x : ℝ) : -(π / 2) < arctan x :=
lt_of_le_of_ne (neg_pi_div_two_le_arcsin _)
(λ h, ne_of_lt (neg_one_lt_div_sqrt_one_add x) $
by rw [← sin_arcsin (le_of_lt (neg_one_lt_div_sqrt_one_add _))
(le_of_lt (div_sqrt_one_add_lt_one _)), ← arctan, ← h, sin_neg, sin_pi_div_two])
lemma tan_surjective : function.surjective tan :=
function.surjective_of_has_right_inverse ⟨_, tan_arctan⟩
lemma arctan_tan {x : ℝ} (hx₁ : -(π / 2) < x) (hx₂ : x < π / 2) : arctan (tan x) = x :=
tan_inj_of_lt_of_lt_pi_div_two (neg_pi_div_two_lt_arctan _)
(arctan_lt_pi_div_two _) hx₁ hx₂ (by rw tan_arctan)
@[simp] lemma arctan_zero : arctan 0 = 0 :=
by simp [arctan]
@[simp] lemma arctan_neg (x : ℝ) : arctan (-x) = - arctan x :=
by simp [arctan, neg_div]
end real
namespace complex
local notation `π` := real.pi
/-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x,abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` -/
noncomputable def arg (x : ℂ) : ℝ :=
if 0 ≤ x.re
then real.arcsin (x.im / x.abs)
else if 0 ≤ x.im
then real.arcsin ((-x).im / x.abs) + π
else real.arcsin ((-x).im / x.abs) - π
lemma arg_le_pi (x : ℂ) : arg x ≤ π :=
if hx₁ : 0 ≤ x.re
then by rw [arg, if_pos hx₁];
exact le_trans (real.arcsin_le_pi_div_two _) (le_of_lt (half_lt_self real.pi_pos))
else
have hx : x ≠ 0, from λ h, by simpa [h, lt_irrefl] using hx₁,
if hx₂ : 0 ≤ x.im
then by rw [arg, if_neg hx₁, if_pos hx₂];
exact le_sub_iff_add_le.1 (by rw sub_self;
exact real.arcsin_nonpos (by rw [neg_im, neg_div, neg_nonpos]; exact div_nonneg hx₂ (abs_pos.2 hx)))
else by rw [arg, if_neg hx₁, if_neg hx₂];
exact sub_le_iff_le_add.2 (le_trans (real.arcsin_le_pi_div_two _)
(by linarith using [real.pi_pos]))
lemma neg_pi_lt_arg (x : ℂ) : -π < arg x :=
if hx₁ : 0 ≤ x.re
then by rw [arg, if_pos hx₁];
exact lt_of_lt_of_le (neg_lt_neg (half_lt_self real.pi_pos)) (real.neg_pi_div_two_le_arcsin _)
else
have hx : x ≠ 0, from λ h, by simpa [h, lt_irrefl] using hx₁,
if hx₂ : 0 ≤ x.im
then by rw [arg, if_neg hx₁, if_pos hx₂];
exact sub_lt_iff_lt_add.1
(lt_of_lt_of_le (by linarith using [real.pi_pos]) (real.neg_pi_div_two_le_arcsin _))
else by rw [arg, if_neg hx₁, if_neg hx₂];
exact lt_sub_iff_add_lt.2 (by rw neg_add_self;
exact real.arcsin_pos (by rw [neg_im]; exact div_pos (neg_pos.2 (lt_of_not_ge hx₂))
(abs_pos.2 hx)) (by rw [← abs_neg x]; exact (abs_le.1 (abs_im_div_abs_le_one _)).2))
lemma arg_eq_arg_neg_add_pi_of_im_nonneg_of_re_neg {x : ℂ} (hxr : x.re < 0) (hxi : 0 ≤ x.im) :
arg x = arg (-x) + π :=
have 0 ≤ (-x).re, from le_of_lt $ by simpa [neg_pos],
by rw [arg, arg, if_neg (not_le.2 hxr), if_pos this, if_pos hxi, abs_neg]
lemma arg_eq_arg_neg_sub_pi_of_im_neg_of_re_neg {x : ℂ} (hxr : x.re < 0) (hxi : x.im < 0) :
arg x = arg (-x) - π :=
have 0 ≤ (-x).re, from le_of_lt $ by simpa [neg_pos],
by rw [arg, arg, if_neg (not_le.2 hxr), if_neg (not_le.2 hxi), if_pos this, abs_neg]
@[simp] lemma arg_zero : arg 0 = 0 :=
by simp [arg, le_refl]
@[simp] lemma arg_one : arg 1 = 0 :=
by simp [arg, zero_le_one]
@[simp] lemma arg_neg_one : arg (-1) = π :=
by simp [arg, le_refl, not_le.2 (@zero_lt_one ℝ _)]
@[simp] lemma arg_I : arg I = π / 2 :=
by simp [arg, le_refl]
@[simp] lemma arg_neg_I : arg (-I) = -(π / 2) :=
by simp [arg, le_refl]
lemma sin_arg (x : ℂ) : real.sin (arg x) = x.im / x.abs :=
by unfold arg; split_ifs;
simp [arg, real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1
(abs_le.1 (abs_im_div_abs_le_one x)).2, real.sin_add, neg_div, real.arcsin_neg,
real.sin_neg]
private lemma cos_arg_of_re_nonneg {x : ℂ} (hx : x ≠ 0) (hxr : 0 ≤ x.re) : real.cos (arg x) = x.re / x.abs :=
have 0 ≤ 1 - (x.im / abs x) ^ 2,
from sub_nonneg.2 $ by rw [pow_two, ← _root_.abs_mul_self, _root_.abs_mul, ← pow_two];
exact pow_le_one _ (_root_.abs_nonneg _) (abs_im_div_abs_le_one _),
by rw [eq_div_iff_mul_eq _ _ (mt abs_eq_zero.1 hx), ← real.mul_self_sqrt (abs_nonneg x),
arg, if_pos hxr, real.cos_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1
(abs_le.1 (abs_im_div_abs_le_one x)).2, ← real.sqrt_mul (abs_nonneg _), ← real.sqrt_mul this,
sub_mul, div_pow _ (mt abs_eq_zero.1 hx), ← pow_two, div_mul_cancel _ (pow_ne_zero 2 (mt abs_eq_zero.1 hx)),
one_mul, pow_two, mul_self_abs, norm_sq, pow_two, add_sub_cancel, real.sqrt_mul_self hxr]
lemma cos_arg {x : ℂ} (hx : x ≠ 0) : real.cos (arg x) = x.re / x.abs :=
if hxr : 0 ≤ x.re then cos_arg_of_re_nonneg hx hxr
else
have 0 ≤ (-x).re, from le_of_lt $ by simpa [neg_pos] using hxr,
if hxi : 0 ≤ x.im
then have 0 ≤ (-x).re, from le_of_lt $ by simpa [neg_pos] using hxr,
by rw [arg_eq_arg_neg_add_pi_of_im_nonneg_of_re_neg (not_le.1 hxr) hxi, real.cos_add_pi,
cos_arg_of_re_nonneg (neg_ne_zero.2 hx) this];
simp [neg_div]
else by rw [arg_eq_arg_neg_sub_pi_of_im_neg_of_re_neg (not_le.1 hxr) (not_le.1 hxi)];
simp [real.cos_add, neg_div, cos_arg_of_re_nonneg (neg_ne_zero.2 hx) this]
lemma tan_arg {x : ℂ} : real.tan (arg x) = x.im / x.re :=
if hx : x = 0 then by simp [hx]
else by rw [real.tan_eq_sin_div_cos, sin_arg, cos_arg hx,
div_div_div_cancel_right _ _ (mt abs_eq_zero.1 hx)]
lemma arg_cos_add_sin_mul_I {x : ℝ} (hx₁ : -π < x) (hx₂ : x ≤ π) :
arg (cos x + sin x * I) = x :=
if hx₃ : -(π / 2) ≤ x ∧ x ≤ π / 2
then
have hx₄ : 0 ≤ (cos x + sin x * I).re,
by simp; exact real.cos_nonneg_of_neg_pi_div_two_le_of_le_pi_div_two hx₃.1 hx₃.2,
by rw [arg, if_pos hx₄];
simp [abs_cos_add_sin_mul_I, sin_of_real_re, real.arcsin_sin hx₃.1 hx₃.2]
else if hx₄ : x < -(π / 2)
then
have hx₅ : ¬0 ≤ (cos x + sin x * I).re :=
suffices ¬ 0 ≤ real.cos x, by simpa,
not_le.2 $ by rw ← real.cos_neg;
apply real.cos_neg_of_pi_div_two_lt_of_lt; linarith,
have hx₆ : ¬0 ≤ (cos ↑x + sin ↑x * I).im :=
suffices real.sin x < 0, by simpa,
by apply real.sin_neg_of_neg_of_neg_pi_lt; linarith,
suffices -π + -real.arcsin (real.sin x) = x,
by rw [arg, if_neg hx₅, if_neg hx₆];
simpa [abs_cos_add_sin_mul_I, sin_of_real_re],
by rw [← real.arcsin_neg, ← real.sin_add_pi, real.arcsin_sin]; simp; linarith
else
have hx₅ : π / 2 < x, by cases not_and_distrib.1 hx₃; linarith,
have hx₆ : ¬0 ≤ (cos x + sin x * I).re :=
suffices ¬0 ≤ real.cos x, by simpa,
not_le.2 $ by apply real.cos_neg_of_pi_div_two_lt_of_lt; linarith,
have hx₇ : 0 ≤ (cos x + sin x * I).im :=
suffices 0 ≤ real.sin x, by simpa,
by apply real.sin_nonneg_of_nonneg_of_le_pi; linarith,
suffices π - real.arcsin (real.sin x) = x,
by rw [arg, if_neg hx₆, if_pos hx₇];
simpa [abs_cos_add_sin_mul_I, sin_of_real_re],
by rw [← real.sin_pi_sub, real.arcsin_sin]; simp; linarith
lemma arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) :
arg x = arg y ↔ (abs y / abs x : ℂ) * x = y :=
have hax : abs x ≠ 0, from (mt abs_eq_zero.1 hx),
have hay : abs y ≠ 0, from (mt abs_eq_zero.1 hy),
⟨λ h,
begin
have hcos := congr_arg real.cos h,
rw [cos_arg hx, cos_arg hy, div_eq_div_iff hax hay] at hcos,
have hsin := congr_arg real.sin h,
rw [sin_arg, sin_arg, div_eq_div_iff hax hay] at hsin,
apply complex.ext,
{ rw [mul_re, ← of_real_div, of_real_re, of_real_im, zero_mul, sub_zero, mul_comm,
← mul_div_assoc, hcos, mul_div_cancel _ hax] },
{ rw [mul_im, ← of_real_div, of_real_re, of_real_im, zero_mul, add_zero,
mul_comm, ← mul_div_assoc, hsin, mul_div_cancel _ hax] }
end,
λ h,
have hre : abs (y / x) * x.re = y.re,
by rw ← of_real_div at h;
simpa [-of_real_div] using congr_arg re h,
have hre' : abs (x / y) * y.re = x.re,
by rw [← hre, abs_div, abs_div, ← mul_assoc, div_mul_div,
mul_comm (abs _), div_self (mul_ne_zero hay hax), one_mul],
have him : abs (y / x) * x.im = y.im,
by rw ← of_real_div at h;
simpa [-of_real_div] using congr_arg im h,
have him' : abs (x / y) * y.im = x.im,
by rw [← him, abs_div, abs_div, ← mul_assoc, div_mul_div,
mul_comm (abs _), div_self (mul_ne_zero hay hax), one_mul],
have hxya : x.im / abs x = y.im / abs y,
by rw [← him, abs_div, mul_comm, ← mul_div_comm, mul_div_cancel_left _ hay],
have hnxya : (-x).im / abs x = (-y).im / abs y,
by rw [neg_im, neg_im, neg_div, neg_div, hxya],
if hxr : 0 ≤ x.re
then
have hyr : 0 ≤ y.re, from hre ▸ mul_nonneg (abs_nonneg _) hxr,
by simp [arg, *] at *
else
have hyr : ¬ 0 ≤ y.re, from λ hyr, hxr $ hre' ▸ mul_nonneg (abs_nonneg _) hyr,
if hxi : 0 ≤ x.im
then
have hyi : 0 ≤ y.im, from him ▸ mul_nonneg (abs_nonneg _) hxi,
by simp [arg, *] at *
else
have hyi : ¬ 0 ≤ y.im, from λ hyi, hxi $ him' ▸ mul_nonneg (abs_nonneg _) hyi,
by simp [arg, *] at *⟩
lemma arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x :=
if hx : x = 0 then by simp [hx]
else (arg_eq_arg_iff (mul_ne_zero (of_real_ne_zero.2 (ne_of_lt hr).symm) hx) hx).2 $
by rw [abs_mul, abs_of_nonneg (le_of_lt hr), ← mul_assoc,
of_real_mul, mul_comm (r : ℂ), ← div_div_eq_div_mul,
div_mul_cancel _ (of_real_ne_zero.2 (ne_of_lt hr).symm),
div_self (of_real_ne_zero.2 (mt abs_eq_zero.1 hx)), one_mul]
lemma ext_abs_arg {x y : ℂ} (h₁ : x.abs = y.abs) (h₂ : x.arg = y.arg) : x = y :=
if hy : y = 0 then by simp * at *
else have hx : x ≠ 0, from λ hx, by simp [*, eq_comm] at *,
by rwa [arg_eq_arg_iff hx hy, h₁, div_self (of_real_ne_zero.2 (mt abs_eq_zero.1 hy)), one_mul] at h₂
lemma arg_of_real_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 :=
by simp [arg, hx]
lemma arg_of_real_of_neg {x : ℝ} (hx : x < 0) : arg x = π :=
by rw [arg_eq_arg_neg_add_pi_of_im_nonneg_of_re_neg, ← of_real_neg, arg_of_real_of_nonneg];
simp [*, le_iff_eq_or_lt, lt_neg]
/-- Inverse of the `exp` function. Returns values such that `(log x).im > - π` and `(log x).im ≤ π`.
`log 0 = 0`-/
noncomputable def log (x : ℂ) : ℂ := x.abs.log + arg x * I
lemma log_re (x : ℂ) : x.log.re = x.abs.log := by simp [log]
lemma log_im (x : ℂ) : x.log.im = x.arg := by simp [log]
lemma exp_log {x : ℂ} (hx : x ≠ 0) : exp (log x) = x :=
by rw [log, exp_add_mul_I, ← of_real_sin, sin_arg, ← of_real_cos, cos_arg hx,
← of_real_exp, real.exp_log (abs_pos.2 hx), mul_add, of_real_div, of_real_div,
mul_div_cancel' _ (of_real_ne_zero.2 (mt abs_eq_zero.1 hx)), ← mul_assoc,
mul_div_cancel' _ (of_real_ne_zero.2 (mt abs_eq_zero.1 hx)), re_add_im]
lemma exp_inj_of_neg_pi_lt_of_le_pi {x y : ℂ} (hx₁ : -π < x.im) (hx₂ : x.im ≤ π)
(hy₁ : - π < y.im) (hy₂ : y.im ≤ π) (hxy : exp x = exp y) : x = y :=
by rw [exp_eq_exp_re_mul_sin_add_cos, exp_eq_exp_re_mul_sin_add_cos y] at hxy;
exact complex.ext
(real.exp_injective $
by simpa [abs_mul, abs_cos_add_sin_mul_I] using congr_arg complex.abs hxy)
(by simpa [(of_real_exp _).symm, - of_real_exp, arg_real_mul _ (real.exp_pos _),
arg_cos_add_sin_mul_I hx₁ hx₂, arg_cos_add_sin_mul_I hy₁ hy₂] using congr_arg arg hxy)
lemma log_exp {x : ℂ} (hx₁ : -π < x.im) (hx₂: x.im ≤ π) : log (exp x) = x :=
exp_inj_of_neg_pi_lt_of_le_pi
(by rw log_im; exact neg_pi_lt_arg _)
(by rw log_im; exact arg_le_pi _)
hx₁ hx₂ (by rw [exp_log (exp_ne_zero _)])
lemma of_real_log {x : ℝ} (hx : 0 ≤ x) : (x.log : ℂ) = log x :=
complex.ext
(by rw [log_re, of_real_re, abs_of_nonneg hx])
(by rw [of_real_im, log_im, arg_of_real_of_nonneg hx])
@[simp] lemma log_zero : log 0 = 0 := by simp [log]
@[simp] lemma log_one : log 1 = 0 := by simp [log]
lemma log_neg_one : log (-1) = π * I := by simp [log]
lemma log_I : log I = π / 2 * I := by simp [log]
lemma log_neg_I : log (-I) = -(π / 2) * I := by simp [log]
lemma exp_eq_one_iff {x : ℂ} : exp x = 1 ↔ ∃ n : ℤ, x = n * ((2 * π) * I) :=
have real.exp (x.re) * real.cos (x.im) = 1 → real.cos x.im ≠ -1,
from λ h₁ h₂, begin
rw [h₂, mul_neg_eq_neg_mul_symm, mul_one, neg_eq_iff_neg_eq] at h₁,
have := real.exp_pos x.re,
rw ← h₁ at this,
exact absurd this (by norm_num)
end,
calc exp x = 1 ↔ (exp x).re = 1 ∧ (exp x).im = 0 : by simp [complex.ext_iff]
... ↔ real.cos x.im = 1 ∧ real.sin x.im = 0 ∧ x.re = 0 :
begin
rw exp_eq_exp_re_mul_sin_add_cos,
simp [complex.ext_iff, cos_of_real_re, sin_of_real_re, exp_of_real_re,
real.exp_ne_zero],
split; finish [real.sin_eq_zero_iff_cos_eq]
end
... ↔ (∃ n : ℤ, ↑n * (2 * π) = x.im) ∧ (∃ n : ℤ, ↑n * π = x.im) ∧ x.re = 0 :
by rw [real.sin_eq_zero_iff, real.cos_eq_one_iff]
... ↔ ∃ n : ℤ, x = n * ((2 * π) * I) :
⟨λ ⟨⟨n, hn⟩, ⟨m, hm⟩, h⟩, ⟨n, by simp [complex.ext_iff, hn.symm, h]⟩,
λ ⟨n, hn⟩, ⟨⟨n, by simp [hn]⟩, ⟨2 * n, by simp [hn, mul_comm, mul_assoc, mul_left_comm]⟩,
by simp [hn]⟩⟩
lemma exp_eq_exp_iff_exp_sub_eq_one {x y : ℂ} : exp x = exp y ↔ exp (x - y) = 1 :=
by rw [exp_sub, div_eq_one_iff_eq _ (exp_ne_zero _)]
lemma exp_eq_exp_iff_exists_int {x y : ℂ} : exp x = exp y ↔ ∃ n : ℤ, x = y + n * ((2 * π) * I) :=
by simp only [exp_eq_exp_iff_exp_sub_eq_one, exp_eq_one_iff, sub_eq_iff_eq_add']
@[simp] lemma cos_pi_div_two : cos (π / 2) = 0 :=
calc cos (π / 2) = real.cos (π / 2) : by rw [of_real_cos]; simp
... = 0 : by simp
@[simp] lemma sin_pi_div_two : sin (π / 2) = 1 :=
calc sin (π / 2) = real.sin (π / 2) : by rw [of_real_sin]; simp
... = 1 : by simp
@[simp] lemma sin_pi : sin π = 0 :=
by rw [← of_real_sin, real.sin_pi]; simp
@[simp] lemma cos_pi : cos π = -1 :=
by rw [← of_real_cos, real.cos_pi]; simp
@[simp] lemma sin_two_pi : sin (2 * π) = 0 :=
by simp [two_mul, sin_add]
@[simp] lemma cos_two_pi : cos (2 * π) = 1 :=
by simp [two_mul, cos_add]
lemma sin_add_pi (x : ℝ) : sin (x + π) = -sin x :=
by simp [sin_add]
lemma sin_add_two_pi (x : ℝ) : sin (x + 2 * π) = sin x :=
by simp [sin_add_pi, sin_add, sin_two_pi, cos_two_pi]
lemma cos_add_two_pi (x : ℝ) : cos (x + 2 * π) = cos x :=
by simp [cos_add, cos_two_pi, sin_two_pi]
lemma sin_pi_sub (x : ℝ) : sin (π - x) = sin x :=
by simp [sin_add]
lemma cos_add_pi (x : ℝ) : cos (x + π) = -cos x :=
by simp [cos_add]
lemma cos_pi_sub (x : ℝ) : cos (π - x) = -cos x :=
by simp [cos_add]
lemma sin_add_pi_div_two (x : ℝ) : sin (x + π / 2) = cos x :=
by simp [sin_add]
lemma sin_sub_pi_div_two (x : ℝ) : sin (x - π / 2) = -cos x :=
by simp [sin_add]
lemma sin_pi_div_two_sub (x : ℝ) : sin (π / 2 - x) = cos x :=
by simp [sin_add]
lemma cos_add_pi_div_two (x : ℝ) : cos (x + π / 2) = -sin x :=
by simp [cos_add]
lemma cos_sub_pi_div_two (x : ℝ) : cos (x - π / 2) = sin x :=
by simp [cos_add]
lemma cos_pi_div_two_sub (x : ℝ) : cos (π / 2 - x) = sin x :=
by rw [← cos_neg, neg_sub, cos_sub_pi_div_two]
lemma sin_nat_mul_pi (n : ℕ) : sin (n * π) = 0 :=
by induction n; simp [add_mul, sin_add, *]
lemma sin_int_mul_pi (n : ℤ) : sin (n * π) = 0 :=
by cases n; simp [add_mul, sin_add, *, sin_nat_mul_pi]
lemma cos_nat_mul_two_pi (n : ℕ) : cos (n * (2 * π)) = 1 :=
by induction n; simp [*, mul_add, cos_add, add_mul, cos_two_pi, sin_two_pi]
lemma cos_int_mul_two_pi (n : ℤ) : cos (n * (2 * π)) = 1 :=
by cases n; simp only [cos_nat_mul_two_pi, int.of_nat_eq_coe,
int.neg_succ_of_nat_coe, int.cast_coe_nat, int.cast_neg,
(neg_mul_eq_neg_mul _ _).symm, cos_neg]
lemma cos_int_mul_two_pi_add_pi (n : ℤ) : cos (n * (2 * π) + π) = -1 :=
by simp [cos_add, sin_add, cos_int_mul_two_pi]
section pow
noncomputable def cpow (x y : ℂ) : ℂ :=
if x = 0
then if y = 0
then 1
else 0
else exp (log x * y)
noncomputable instance : has_pow ℂ ℂ := ⟨cpow⟩
lemma cpow_def (x y : ℂ) : x ^ y =
if x = 0
then if y = 0
then 1
else 0
else exp (log x * y) := rfl
@[simp] lemma cpow_zero (x : ℂ) : x ^ (0 : ℂ) = 1 := by simp [cpow_def]
@[simp] lemma zero_cpow {x : ℂ} (h : x ≠ 0) : (0 : ℂ) ^ x = 0 :=
by simp [cpow_def, *]
@[simp] lemma cpow_one (x : ℂ) : x ^ (1 : ℂ) = x :=
if hx : x = 0 then by simp [hx, cpow_def]
else by rw [cpow_def, if_neg (@one_ne_zero ℂ _), if_neg hx, mul_one, exp_log hx]
@[simp] lemma one_cpow (x : ℂ) : (1 : ℂ) ^ x = 1 :=
by rw cpow_def; split_ifs; simp [one_ne_zero, *] at *
lemma cpow_add {x : ℂ} (y z : ℂ) (hx : x ≠ 0) : x ^ (y + z) = x ^ y * x ^ z :=
by simp [cpow_def]; split_ifs; simp [*, exp_add, mul_add] at *
lemma cpow_mul {x y : ℂ} (z : ℂ) (h₁ : -π < (log x * y).im) (h₂ : (log x * y).im ≤ π) :
x ^ (y * z) = (x ^ y) ^ z :=
begin
simp [cpow_def],
split_ifs;
simp [*, exp_ne_zero, log_exp h₁ h₂, mul_assoc] at *
end
lemma cpow_neg (x y : ℂ) : x ^ -y = (x ^ y)⁻¹ :=
by simp [cpow_def]; split_ifs; simp [exp_neg]
@[simp] lemma cpow_nat_cast (x : ℂ) : ∀ (n : ℕ), x ^ (n : ℂ) = x ^ n
| 0 := by simp
| (n + 1) := if hx : x = 0 then by simp only [hx, pow_succ,
complex.zero_cpow (nat.cast_ne_zero.2 (nat.succ_ne_zero _)), zero_mul]
else by simp [cpow_def, hx, mul_add, exp_add, pow_succ, (cpow_nat_cast n).symm, exp_log hx]
@[simp] lemma cpow_int_cast (x : ℂ) : ∀ (n : ℤ), x ^ (n : ℂ) = x ^ n
| (n : ℕ) := by simp; refl
| -[1+ n] := by rw fpow_neg_succ_of_nat;
simp only [int.neg_succ_of_nat_coe, int.cast_neg, complex.cpow_neg, inv_eq_one_div,
int.cast_coe_nat, cpow_nat_cast]
lemma cpow_nat_inv_pow (x : ℂ) {n : ℕ} (hn : 0 < n) : (x ^ (n⁻¹ : ℂ)) ^ n = x :=
have (log x * (↑n)⁻¹).im = (log x).im / n,
by rw [div_eq_mul_inv, ← of_real_nat_cast, ← of_real_inv, mul_im,
of_real_re, of_real_im]; simp,
have h : -π < (log x * (↑n)⁻¹).im ∧ (log x * (↑n)⁻¹).im ≤ π,
from (le_total (log x).im 0).elim
(λ h, ⟨calc -π < (log x).im : by simp [log, neg_pi_lt_arg]
... ≤ ((log x).im * 1) / n : le_div_of_mul_le (nat.cast_pos.2 hn)
(mul_le_mul_of_nonpos_left (by rw ← nat.cast_one; exact nat.cast_le.2 hn) h)
... = (log x * (↑n)⁻¹).im : by simp [this],
this.symm ▸ le_trans (div_nonpos_of_nonpos_of_pos h (nat.cast_pos.2 hn))
(le_of_lt real.pi_pos)⟩)
(λ h, ⟨this.symm ▸ lt_of_lt_of_le (neg_neg_of_pos real.pi_pos)
(div_nonneg h (nat.cast_pos.2 hn)),
calc (log x * (↑n)⁻¹).im = (1 * (log x).im) / n : by simp [this]
... ≤ (log x).im : (div_le_of_le_mul (nat.cast_pos.2 hn)
(mul_le_mul_of_nonneg_right (by rw ← nat.cast_one; exact nat.cast_le.2 hn) h))
... ≤ _ : by simp [log, arg_le_pi]⟩),
by rw [← cpow_nat_cast, ← cpow_mul _ h.1 h.2,
inv_mul_cancel (show (n : ℂ) ≠ 0, from nat.cast_ne_zero.2 (nat.pos_iff_ne_zero.1 hn)),
cpow_one]
end pow
end complex
namespace real
noncomputable def rpow (x y : ℝ) := ((x : ℂ) ^ (y : ℂ)).re
noncomputable instance : has_pow ℝ ℝ := ⟨rpow⟩
lemma rpow_def (x y : ℝ) : x ^ y = ((x : ℂ) ^ (y : ℂ)).re := rfl
lemma rpow_def_of_nonneg {x : ℝ} (hx : 0 ≤ x) (y : ℝ) : x ^ y =
if x = 0
then if y = 0
then 1
else 0
else exp (log x * y) :=
by simp only [rpow_def, complex.cpow_def];
split_ifs;
simp [*, (complex.of_real_log hx).symm, -complex.of_real_mul,
(complex.of_real_mul _ _).symm, complex.exp_of_real_re] at *
end real
namespace complex
lemma of_real_cpow {x : ℝ} (hx : 0 ≤ x) (y : ℝ) : ((x ^ y : ℝ) : ℂ) = (x : ℂ) ^ (y : ℂ) :=
by simp [real.rpow_def_of_nonneg hx, complex.cpow_def]; split_ifs; simp [complex.of_real_log hx]
@[simp] lemma abs_cpow_real (x : ℂ) (y : ℝ) : abs (x ^ (y : ℂ)) = x.abs ^ y :=
begin
rw [real.rpow_def_of_nonneg (abs_nonneg _), complex.cpow_def],
split_ifs;
simp [*, abs_of_nonneg (le_of_lt (real.exp_pos _)), complex.log, complex.exp_add,
add_mul, mul_right_comm _ I, exp_mul_I, abs_cos_add_sin_mul_I,
(complex.of_real_mul _ _).symm, -complex.of_real_mul] at *
end
@[simp] lemma abs_cpow_inv_nat (x : ℂ) (n : ℕ) : abs (x ^ (n⁻¹ : ℂ)) = x.abs ^ (n⁻¹ : ℝ) :=
by rw ← abs_cpow_real; simp [-abs_cpow_real]
end complex
namespace real
@[simp] lemma rpow_zero (x : ℝ) : x ^ (0 : ℝ) = 1 := by simp [rpow_def]
@[simp] lemma zero_rpow {x : ℝ} (h : x ≠ 0) : (0 : ℝ) ^ x = 0 :=
by simp [rpow_def, *]
@[simp] lemma rpow_one (x : ℝ) : x ^ (1 : ℝ) = x := by simp [rpow_def]
@[simp] lemma one_rpow (x : ℝ) : (1 : ℝ) ^ x = 1 := by simp [rpow_def]
lemma rpow_nonneg_of_nonneg {x : ℝ} (hx : 0 ≤ x) (y : ℝ) : 0 ≤ x ^ y :=
by rw [rpow_def_of_nonneg hx];
split_ifs; simp only [zero_le_one, le_refl, le_of_lt (exp_pos _)]
lemma rpow_add {x : ℝ} (y z : ℝ) (hx : 0 < x) : x ^ (y + z) = x ^ y * x ^ z :=
by simp only [rpow_def_of_nonneg (le_of_lt hx), if_neg (ne_of_lt hx).symm, mul_add, exp_add]
lemma rpow_mul {x : ℝ} (hx : 0 ≤ x) (y z : ℝ) : x ^ (y * z) = (x ^ y) ^ z :=
by rw [← complex.of_real_inj, complex.of_real_cpow (rpow_nonneg_of_nonneg hx _),
complex.of_real_cpow hx, complex.of_real_mul, complex.cpow_mul, complex.of_real_cpow hx];
simp only [(complex.of_real_mul _ _).symm, (complex.of_real_log hx).symm,
complex.of_real_im, neg_lt_zero, pi_pos, le_of_lt pi_pos]
lemma rpow_neg {x : ℝ} (hx : 0 ≤ x) (y : ℝ) : x ^ -y = (x ^ y)⁻¹ :=
by simp only [rpow_def_of_nonneg hx]; split_ifs; simp [*, exp_neg] at *
@[simp] lemma rpow_nat_cast (x : ℝ) (n : ℕ) : x ^ (n : ℝ) = x ^ n :=
by simp only [rpow_def, (complex.of_real_pow _ _).symm, complex.cpow_nat_cast,
complex.of_real_nat_cast, complex.of_real_re]
@[simp] lemma rpow_int_cast (x : ℝ) (n : ℤ) : x ^ (n : ℝ) = x ^ n :=
by simp only [rpow_def, (complex.of_real_fpow _ _).symm, complex.cpow_int_cast,
complex.of_real_int_cast, complex.of_real_re]
lemma pow_nat_rpow_nat_inv {x : ℝ} (hx : 0 ≤ x) {n : ℕ} (hn : 0 < n) :
(x ^ n) ^ (n⁻¹ : ℝ) = x :=
have hn0 : (n : ℝ) ≠ 0, by simpa [nat.pos_iff_ne_zero'] using hn,
by rw [← rpow_nat_cast, ← rpow_mul hx, mul_inv_cancel hn0, rpow_one]
end real
|
1c3753f233768c3d5e0e720ea9f46cdb0d558ad0 | a9d0fb7b0e4f802bd3857b803e6c5c23d87fef91 | /tests/lean/defeq_simp_lemmas2.lean | 99f04490efe485b29c052fd7a2459c34c2ea2376 | [
"Apache-2.0"
] | permissive | soonhokong/lean-osx | 4a954262c780e404c1369d6c06516161d07fcb40 | 3670278342d2f4faa49d95b46d86642d7875b47c | refs/heads/master | 1,611,410,334,552 | 1,474,425,686,000 | 1,474,425,686,000 | 12,043,103 | 5 | 1 | null | null | null | null | UTF-8 | Lean | false | false | 1,436 | lean | namespace foo
universe l
constants (A : Type.{l})
noncomputable definition q (x : A) := x
noncomputable definition h (x : A) : A := q x
noncomputable definition g (x y : A) := h y
noncomputable definition f (x y z : A) := g (g x y) z
noncomputable definition d (x y z w : A) := f (f x y z) (f y z w) (f x w z)
attribute [defeq]
definition h.rfl (x : A) : h x = x := rfl
attribute [defeq]
definition g.rfl (x y : A) : g x y = y := rfl
attribute [defeq]
definition f.rfl (x y z : A) : f x y z = z := rfl
attribute [defeq]
definition d.rfl (x y z w : A) : d x y z w = z := rfl
attribute [defeq]
definition h.def (x : A) : h x = q x := rfl
attribute [defeq]
definition g.def (x y : A) : g x y = h y := rfl
attribute [defeq]
definition f.def (x y z : A) : f x y z = g (g x y) z := rfl
attribute [defeq]
definition d.def (x y z w : A) : d x y z w = f (f x y z) (f y z w) (f x w z) := rfl
-- Confirm that more recent annotations get priority
print [defeq]
attribute h.rfl [defeq, priority 1001]
attribute g.rfl [defeq, priority 1001]
attribute f.rfl [defeq, priority 1001]
attribute d.rfl [defeq, priority 1001]
-- Confirm that priority annotations override
print [defeq]
attribute h.def [defeq, priority 1001]
attribute g.def [defeq, priority 1001]
attribute f.def [defeq, priority 1001]
attribute d.def [defeq, priority 1001]
-- Confirm that most recent annotations get priority to break explicit priority ties
print [defeq]
end foo
|
2246d3338f1ae1bca2bb3868731008d41851d5fd | b7f22e51856f4989b970961f794f1c435f9b8f78 | /tests/lean/run/blast_simp3.lean | cbbb6b549ea785754bd2a7bee14137d6990c9fcd | [
"Apache-2.0"
] | permissive | soonhokong/lean | cb8aa01055ffe2af0fb99a16b4cda8463b882cd1 | 38607e3eb57f57f77c0ac114ad169e9e4262e24f | refs/heads/master | 1,611,187,284,081 | 1,450,766,737,000 | 1,476,122,547,000 | 11,513,992 | 2 | 0 | null | 1,401,763,102,000 | 1,374,182,235,000 | C++ | UTF-8 | Lean | false | false | 152 | lean | example (A : Type₁) (a₁ a₂ : A) : a₁ = a₂ →
(λ (B : Type₁) (f : A → B), f a₁) = (λ (B : Type₁) (f : A → B), f a₂) :=
by simp
|
5c902de533afbf13815e118117d5f3c0ada38649 | 84c325c9a58de83324b777adc78ac188ecdbceae | /src/util/io.lean | 08fd78324f372bbf87b5a84a30ea04a227814e63 | [
"Apache-2.0"
] | permissive | brando90/lean-gym | 0c3b433dac9c1a1079623721cdb3307a0981d86b | f4a94fcf54f8729be416ebdca82097d95a6f1c39 | refs/heads/main | 1,683,583,589,195 | 1,622,558,076,000 | 1,622,558,076,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 1,146 | lean | /-
Copyright (c) 2021 OpenAI. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author(s): Stanislas Polu
Helper functions to work with the io monad.
-/
import system.io
import tactic.gptf.utils.util
section io
open interaction_monad interaction_monad.result
namespace io
/-- verion of io.run_tactic' which does not suppress the exception msg -/
meta def run_tactic'' {α} (tac :tactic α) : io α := do {
io.run_tactic $ do {
result ← tactic.capture tac,
match result with
| (success val _) := pure val
| (exception m_fmt pos _) := do {
let fmt_msg := (m_fmt.get_or_else (λ _, format!"n/a")) (),
let msg := format!"run_tactic_failed: failed: pos={pos} msg={fmt_msg}",
tactic.trace format!"TRACE RUN_TACTIC FAILURE: {msg}",
tactic.fail msg
}
end
}
}
end io
end io
-- convenience function for command-line argument parsing
meta def list.nth_except {α} : list α → ℕ → string → io α := λ xs pos msg,
match (xs.nth pos) with
| (some result) := pure result
| none := do
io.fail' format!"must supply {msg} as argument {pos}"
end |
9d994db1d9b32698e419d119ed303467bbfcc657 | 9dc8cecdf3c4634764a18254e94d43da07142918 | /src/geometry/manifold/cont_mdiff.lean | 4b7650eb57e3d001f015e3658516759fe710487e | [
"Apache-2.0"
] | permissive | jcommelin/mathlib | d8456447c36c176e14d96d9e76f39841f69d2d9b | ee8279351a2e434c2852345c51b728d22af5a156 | refs/heads/master | 1,664,782,136,488 | 1,663,638,983,000 | 1,663,638,983,000 | 132,563,656 | 0 | 0 | Apache-2.0 | 1,663,599,929,000 | 1,525,760,539,000 | Lean | UTF-8 | Lean | false | false | 93,542 | lean | /-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import geometry.manifold.mfderiv
import geometry.manifold.local_invariant_properties
/-!
# Smooth functions between smooth manifolds
We define `Cⁿ` functions between smooth manifolds, as functions which are `Cⁿ` in charts, and prove
basic properties of these notions.
## Main definitions and statements
Let `M ` and `M'` be two smooth manifolds, with respect to model with corners `I` and `I'`. Let
`f : M → M'`.
* `cont_mdiff_within_at I I' n f s x` states that the function `f` is `Cⁿ` within the set `s`
around the point `x`.
* `cont_mdiff_at I I' n f x` states that the function `f` is `Cⁿ` around `x`.
* `cont_mdiff_on I I' n f s` states that the function `f` is `Cⁿ` on the set `s`
* `cont_mdiff I I' n f` states that the function `f` is `Cⁿ`.
* `cont_mdiff_on.comp` gives the invariance of the `Cⁿ` property under composition
* `cont_mdiff_on.cont_mdiff_on_tangent_map_within` states that the bundled derivative
of a `Cⁿ` function in a domain is `Cᵐ` when `m + 1 ≤ n`.
* `cont_mdiff.cont_mdiff_tangent_map` states that the bundled derivative
of a `Cⁿ` function is `Cᵐ` when `m + 1 ≤ n`.
* `cont_mdiff_iff_cont_diff` states that, for functions between vector spaces,
manifold-smoothness is equivalent to usual smoothness.
We also give many basic properties of smooth functions between manifolds, following the API of
smooth functions between vector spaces.
## Implementation details
Many properties follow for free from the corresponding properties of functions in vector spaces,
as being `Cⁿ` is a local property invariant under the smooth groupoid. We take advantage of the
general machinery developed in `local_invariant_properties.lean` to get these properties
automatically. For instance, the fact that being `Cⁿ` does not depend on the chart one considers
is given by `lift_prop_within_at_indep_chart`.
For this to work, the definition of `cont_mdiff_within_at` and friends has to
follow definitionally the setup of local invariant properties. Still, we recast the definition
in terms of extended charts in `cont_mdiff_on_iff` and `cont_mdiff_iff`.
-/
open set function filter charted_space smooth_manifold_with_corners
open_locale topological_space manifold
/-! ### Definition of smooth functions between manifolds -/
variables {𝕜 : Type*} [nontrivially_normed_field 𝕜]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*} [normed_add_comm_group E] [normed_space 𝕜 E]
{H : Type*} [topological_space H] (I : model_with_corners 𝕜 E H)
{M : Type*} [topological_space M] [charted_space H M] [Is : smooth_manifold_with_corners I M]
-- declare a smooth manifold `M'` over the pair `(E', H')`.
{E' : Type*} [normed_add_comm_group E'] [normed_space 𝕜 E']
{H' : Type*} [topological_space H'] (I' : model_with_corners 𝕜 E' H')
{M' : Type*} [topological_space M'] [charted_space H' M'] [I's : smooth_manifold_with_corners I' M']
-- declare a smooth manifold `N` over the pair `(F, G)`.
{F : Type*} [normed_add_comm_group F] [normed_space 𝕜 F]
{G : Type*} [topological_space G] {J : model_with_corners 𝕜 F G}
{N : Type*} [topological_space N] [charted_space G N] [Js : smooth_manifold_with_corners J N]
-- declare a smooth manifold `N'` over the pair `(F', G')`.
{F' : Type*} [normed_add_comm_group F'] [normed_space 𝕜 F']
{G' : Type*} [topological_space G'] {J' : model_with_corners 𝕜 F' G'}
{N' : Type*} [topological_space N'] [charted_space G' N'] [J's : smooth_manifold_with_corners J' N']
-- declare functions, sets, points and smoothness indices
{f f₁ : M → M'} {s s₁ t : set M} {x : M} {m n : ℕ∞}
/-- Property in the model space of a model with corners of being `C^n` within at set at a point,
when read in the model vector space. This property will be lifted to manifolds to define smooth
functions between manifolds. -/
def cont_diff_within_at_prop (n : ℕ∞) (f : H → H') (s : set H) (x : H) : Prop :=
cont_diff_within_at 𝕜 n (I' ∘ f ∘ I.symm) (I.symm ⁻¹' s ∩ range I) (I x)
lemma cont_diff_within_at_prop_self_source {f : E → H'} {s : set E} {x : E} :
cont_diff_within_at_prop 𝓘(𝕜, E) I' n f s x ↔ cont_diff_within_at 𝕜 n (I' ∘ f) s x :=
begin
simp_rw [cont_diff_within_at_prop, model_with_corners_self_coe, range_id, inter_univ],
refl
end
lemma cont_diff_within_at_prop_self {f : E → E'} {s : set E} {x : E} :
cont_diff_within_at_prop 𝓘(𝕜, E) 𝓘(𝕜, E') n f s x ↔ cont_diff_within_at 𝕜 n f s x :=
cont_diff_within_at_prop_self_source 𝓘(𝕜, E')
lemma cont_diff_within_at_prop_self_target {f : H → E'} {s : set H} {x : H} :
cont_diff_within_at_prop I 𝓘(𝕜, E') n f s x ↔
cont_diff_within_at 𝕜 n (f ∘ I.symm) (I.symm ⁻¹' s ∩ range I) (I x) :=
iff.rfl
/-- Being `Cⁿ` in the model space is a local property, invariant under smooth maps. Therefore,
it will lift nicely to manifolds. -/
lemma cont_diff_within_at_local_invariant_prop (n : ℕ∞) :
(cont_diff_groupoid ∞ I).local_invariant_prop (cont_diff_groupoid ∞ I')
(cont_diff_within_at_prop I I' n) :=
{ is_local :=
begin
assume s x u f u_open xu,
have : I.symm ⁻¹' (s ∩ u) ∩ range I = (I.symm ⁻¹' s ∩ range I) ∩ I.symm ⁻¹' u,
by simp only [inter_right_comm, preimage_inter],
rw [cont_diff_within_at_prop, cont_diff_within_at_prop, this],
symmetry,
apply cont_diff_within_at_inter,
have : u ∈ 𝓝 (I.symm (I x)),
by { rw [model_with_corners.left_inv], exact is_open.mem_nhds u_open xu },
apply continuous_at.preimage_mem_nhds I.continuous_symm.continuous_at this,
end,
right_invariance' :=
begin
assume s x f e he hx h,
rw cont_diff_within_at_prop at h ⊢,
have : I x = (I ∘ e.symm ∘ I.symm) (I (e x)), by simp only [hx] with mfld_simps,
rw this at h,
have : I (e x) ∈ (I.symm) ⁻¹' e.target ∩ range I, by simp only [hx] with mfld_simps,
have := ((mem_groupoid_of_pregroupoid.2 he).2.cont_diff_within_at this).of_le le_top,
convert (h.comp' _ this).mono_of_mem _ using 1,
{ ext y, simp only with mfld_simps },
refine mem_nhds_within.mpr ⟨I.symm ⁻¹' e.target, e.open_target.preimage I.continuous_symm,
by simp_rw [mem_preimage, I.left_inv, e.maps_to hx], _⟩,
mfld_set_tac
end,
congr_of_forall :=
begin
assume s x f g h hx hf,
apply hf.congr,
{ assume y hy,
simp only with mfld_simps at hy,
simp only [h, hy] with mfld_simps },
{ simp only [hx] with mfld_simps }
end,
left_invariance' :=
begin
assume s x f e' he' hs hx h,
rw cont_diff_within_at_prop at h ⊢,
have A : (I' ∘ f ∘ I.symm) (I x) ∈ (I'.symm ⁻¹' e'.source ∩ range I'),
by simp only [hx] with mfld_simps,
have := ((mem_groupoid_of_pregroupoid.2 he').1.cont_diff_within_at A).of_le le_top,
convert this.comp _ h _,
{ ext y, simp only with mfld_simps },
{ assume y hy, simp only with mfld_simps at hy, simpa only [hy] with mfld_simps using hs hy.1 }
end }
lemma cont_diff_within_at_prop_mono (n : ℕ∞)
⦃s x t⦄ ⦃f : H → H'⦄ (hts : t ⊆ s) (h : cont_diff_within_at_prop I I' n f s x) :
cont_diff_within_at_prop I I' n f t x :=
begin
apply h.mono (λ y hy, _),
simp only with mfld_simps at hy,
simp only [hy, hts _] with mfld_simps
end
lemma cont_diff_within_at_prop_id (x : H) :
cont_diff_within_at_prop I I ∞ id univ x :=
begin
simp [cont_diff_within_at_prop],
have : cont_diff_within_at 𝕜 ∞ id (range I) (I x) :=
cont_diff_id.cont_diff_at.cont_diff_within_at,
apply this.congr (λ y hy, _),
{ simp only with mfld_simps },
{ simp only [model_with_corners.right_inv I hy] with mfld_simps }
end
/-- A function is `n` times continuously differentiable within a set at a point in a manifold if
it is continuous and it is `n` times continuously differentiable in this set around this point, when
read in the preferred chart at this point. -/
def cont_mdiff_within_at (n : ℕ∞) (f : M → M') (s : set M) (x : M) :=
lift_prop_within_at (cont_diff_within_at_prop I I' n) f s x
/-- Abbreviation for `cont_mdiff_within_at I I' ⊤ f s x`. See also documentation for `smooth`.
-/
@[reducible] def smooth_within_at (f : M → M') (s : set M) (x : M) :=
cont_mdiff_within_at I I' ⊤ f s x
/-- A function is `n` times continuously differentiable at a point in a manifold if
it is continuous and it is `n` times continuously differentiable around this point, when
read in the preferred chart at this point. -/
def cont_mdiff_at (n : ℕ∞) (f : M → M') (x : M) :=
cont_mdiff_within_at I I' n f univ x
lemma cont_mdiff_at_iff {n : ℕ∞} {f : M → M'} {x : M} :
cont_mdiff_at I I' n f x ↔ continuous_at f x ∧ cont_diff_within_at 𝕜 n
(ext_chart_at I' (f x) ∘ f ∘ (ext_chart_at I x).symm) (range I) (ext_chart_at I x x) :=
lift_prop_at_iff.trans $ by { rw [cont_diff_within_at_prop, preimage_univ, univ_inter], refl }
/-- Abbreviation for `cont_mdiff_at I I' ⊤ f x`. See also documentation for `smooth`. -/
@[reducible] def smooth_at (f : M → M') (x : M) := cont_mdiff_at I I' ⊤ f x
/-- A function is `n` times continuously differentiable in a set of a manifold if it is continuous
and, for any pair of points, it is `n` times continuously differentiable on this set in the charts
around these points. -/
def cont_mdiff_on (n : ℕ∞) (f : M → M') (s : set M) :=
∀ x ∈ s, cont_mdiff_within_at I I' n f s x
/-- Abbreviation for `cont_mdiff_on I I' ⊤ f s`. See also documentation for `smooth`. -/
@[reducible] def smooth_on (f : M → M') (s : set M) := cont_mdiff_on I I' ⊤ f s
/-- A function is `n` times continuously differentiable in a manifold if it is continuous
and, for any pair of points, it is `n` times continuously differentiable in the charts
around these points. -/
def cont_mdiff (n : ℕ∞) (f : M → M') :=
∀ x, cont_mdiff_at I I' n f x
/-- Abbreviation for `cont_mdiff I I' ⊤ f`.
Short note to work with these abbreviations: a lemma of the form `cont_mdiff_foo.bar` will
apply fine to an assumption `smooth_foo` using dot notation or normal notation.
If the consequence `bar` of the lemma involves `cont_diff`, it is still better to restate
the lemma replacing `cont_diff` with `smooth` both in the assumption and in the conclusion,
to make it possible to use `smooth` consistently.
This also applies to `smooth_at`, `smooth_on` and `smooth_within_at`.-/
@[reducible] def smooth (f : M → M') := cont_mdiff I I' ⊤ f
/-! ### Basic properties of smooth functions between manifolds -/
variables {I I'}
lemma cont_mdiff.smooth (h : cont_mdiff I I' ⊤ f) : smooth I I' f := h
lemma smooth.cont_mdiff (h : smooth I I' f) : cont_mdiff I I' ⊤ f := h
lemma cont_mdiff_on.smooth_on (h : cont_mdiff_on I I' ⊤ f s) : smooth_on I I' f s := h
lemma smooth_on.cont_mdiff_on (h : smooth_on I I' f s) : cont_mdiff_on I I' ⊤ f s := h
lemma cont_mdiff_at.smooth_at (h : cont_mdiff_at I I' ⊤ f x) : smooth_at I I' f x := h
lemma smooth_at.cont_mdiff_at (h : smooth_at I I' f x) : cont_mdiff_at I I' ⊤ f x := h
lemma cont_mdiff_within_at.smooth_within_at (h : cont_mdiff_within_at I I' ⊤ f s x) :
smooth_within_at I I' f s x := h
lemma smooth_within_at.cont_mdiff_within_at (h : smooth_within_at I I' f s x) :
cont_mdiff_within_at I I' ⊤ f s x := h
lemma cont_mdiff.cont_mdiff_at (h : cont_mdiff I I' n f) :
cont_mdiff_at I I' n f x :=
h x
lemma smooth.smooth_at (h : smooth I I' f) :
smooth_at I I' f x := cont_mdiff.cont_mdiff_at h
lemma cont_mdiff_within_at_univ :
cont_mdiff_within_at I I' n f univ x ↔ cont_mdiff_at I I' n f x :=
iff.rfl
lemma smooth_within_at_univ :
smooth_within_at I I' f univ x ↔ smooth_at I I' f x := cont_mdiff_within_at_univ
lemma cont_mdiff_on_univ :
cont_mdiff_on I I' n f univ ↔ cont_mdiff I I' n f :=
by simp only [cont_mdiff_on, cont_mdiff, cont_mdiff_within_at_univ,
forall_prop_of_true, mem_univ]
lemma smooth_on_univ : smooth_on I I' f univ ↔ smooth I I' f := cont_mdiff_on_univ
/-- One can reformulate smoothness within a set at a point as continuity within this set at this
point, and smoothness in the corresponding extended chart. -/
lemma cont_mdiff_within_at_iff :
cont_mdiff_within_at I I' n f s x ↔ continuous_within_at f s x ∧
cont_diff_within_at 𝕜 n ((ext_chart_at I' (f x)) ∘ f ∘ (ext_chart_at I x).symm)
((ext_chart_at I x).symm ⁻¹' s ∩ range I)
(ext_chart_at I x x) :=
iff.rfl
/-- One can reformulate smoothness within a set at a point as continuity within this set at this
point, and smoothness in the corresponding extended chart. This form states smoothness of `f`
written in such a way that the set is restricted to lie within the domain/codomain of the
corresponding charts.
Even though this expression is more complicated than the one in `cont_mdiff_within_at_iff`, it is
a smaller set, but their germs at `ext_chart_at I x x` are equal. It is sometimes useful to rewrite
using this in the goal.
-/
lemma cont_mdiff_within_at_iff' :
cont_mdiff_within_at I I' n f s x ↔ continuous_within_at f s x ∧
cont_diff_within_at 𝕜 n ((ext_chart_at I' (f x)) ∘ f ∘ (ext_chart_at I x).symm)
((ext_chart_at I x).target ∩
(ext_chart_at I x).symm ⁻¹' (s ∩ f ⁻¹' (ext_chart_at I' (f x)).source))
(ext_chart_at I x x) :=
begin
rw [cont_mdiff_within_at_iff, and.congr_right_iff],
set e := ext_chart_at I x, set e' := ext_chart_at I' (f x),
refine λ hc, cont_diff_within_at_congr_nhds _,
rw [← e.image_source_inter_eq', ← ext_chart_at_map_nhds_within_eq_image,
← ext_chart_at_map_nhds_within, inter_comm, nhds_within_inter_of_mem],
exact hc (ext_chart_at_source_mem_nhds _ _)
end
/-- One can reformulate smoothness within a set at a point as continuity within this set at this
point, and smoothness in the corresponding extended chart in the target. -/
lemma cont_mdiff_within_at_iff_target :
cont_mdiff_within_at I I' n f s x ↔ continuous_within_at f s x ∧
cont_mdiff_within_at I 𝓘(𝕜, E') n (ext_chart_at I' (f x) ∘ f) s x :=
begin
simp_rw [cont_mdiff_within_at, lift_prop_within_at, ← and_assoc],
have cont : (continuous_within_at f s x ∧
continuous_within_at (ext_chart_at I' (f x) ∘ f) s x) ↔
continuous_within_at f s x,
{ refine ⟨λ h, h.1, λ h, ⟨h, _⟩⟩,
have h₂ := (chart_at H' (f x)).continuous_to_fun.continuous_within_at (mem_chart_source _ _),
refine ((I'.continuous_at.comp_continuous_within_at h₂).comp' h).mono_of_mem _,
exact inter_mem self_mem_nhds_within (h.preimage_mem_nhds_within $
(chart_at _ _).open_source.mem_nhds $ mem_chart_source _ _) },
simp_rw [cont, cont_diff_within_at_prop, ext_chart_at, local_equiv.coe_trans,
model_with_corners.to_local_equiv_coe, local_homeomorph.coe_coe, model_with_corners_self_coe,
chart_at_self_eq, local_homeomorph.refl_apply, comp.left_id]
end
lemma smooth_within_at_iff :
smooth_within_at I I' f s x ↔ continuous_within_at f s x ∧
cont_diff_within_at 𝕜 ∞ (ext_chart_at I' (f x) ∘ f ∘ (ext_chart_at I x).symm)
((ext_chart_at I x).symm ⁻¹' s ∩ range I)
(ext_chart_at I x x) :=
cont_mdiff_within_at_iff
lemma smooth_within_at_iff_target :
smooth_within_at I I' f s x ↔ continuous_within_at f s x ∧
smooth_within_at I 𝓘(𝕜, E') (ext_chart_at I' (f x) ∘ f) s x :=
cont_mdiff_within_at_iff_target
lemma cont_mdiff_at_iff_target {x : M} :
cont_mdiff_at I I' n f x ↔
continuous_at f x ∧ cont_mdiff_at I 𝓘(𝕜, E') n (ext_chart_at I' (f x) ∘ f) x :=
by rw [cont_mdiff_at, cont_mdiff_at, cont_mdiff_within_at_iff_target, continuous_within_at_univ]
lemma smooth_at_iff_target {x : M} :
smooth_at I I' f x ↔ continuous_at f x ∧ smooth_at I 𝓘(𝕜, E') (ext_chart_at I' (f x) ∘ f) x :=
cont_mdiff_at_iff_target
include Is I's
/-- One can reformulate smoothness within a set at a point as continuity within this set at this
point, and smoothness in any chart containing that point. -/
lemma cont_mdiff_within_at_iff_of_mem_source {x' : M} {y : M'} (hx : x' ∈ (chart_at H x).source)
(hy : f x' ∈ (chart_at H' y).source) :
cont_mdiff_within_at I I' n f s x' ↔ continuous_within_at f s x' ∧
cont_diff_within_at 𝕜 n (ext_chart_at I' y ∘ f ∘ (ext_chart_at I x).symm)
((ext_chart_at I x).symm ⁻¹' s ∩ range I)
(ext_chart_at I x x') :=
(cont_diff_within_at_local_invariant_prop I I' n).lift_prop_within_at_indep_chart
(structure_groupoid.chart_mem_maximal_atlas _ x) hx
(structure_groupoid.chart_mem_maximal_atlas _ y) hy
lemma cont_mdiff_within_at_iff_of_mem_source' {x' : M} {y : M'} (hx : x' ∈ (chart_at H x).source)
(hy : f x' ∈ (chart_at H' y).source) :
cont_mdiff_within_at I I' n f s x' ↔ continuous_within_at f s x' ∧
cont_diff_within_at 𝕜 n ((ext_chart_at I' y) ∘ f ∘ (ext_chart_at I x).symm)
((ext_chart_at I x).target ∩ (ext_chart_at I x).symm ⁻¹' (s ∩ f ⁻¹' (ext_chart_at I' y).source))
(ext_chart_at I x x') :=
begin
refine (cont_mdiff_within_at_iff_of_mem_source hx hy).trans _,
rw [← ext_chart_at_source I] at hx,
rw [← ext_chart_at_source I'] at hy,
rw [and.congr_right_iff],
set e := ext_chart_at I x, set e' := ext_chart_at I' (f x),
refine λ hc, cont_diff_within_at_congr_nhds _,
rw [← e.image_source_inter_eq', ← ext_chart_at_map_nhds_within_eq_image' I x hx,
← ext_chart_at_map_nhds_within' I x hx, inter_comm, nhds_within_inter_of_mem],
exact hc ((ext_chart_at_open_source _ _).mem_nhds hy)
end
lemma cont_mdiff_at_iff_of_mem_source {x' : M} {y : M'} (hx : x' ∈ (chart_at H x).source)
(hy : f x' ∈ (chart_at H' y).source) :
cont_mdiff_at I I' n f x' ↔ continuous_at f x' ∧
cont_diff_within_at 𝕜 n (ext_chart_at I' y ∘ f ∘ (ext_chart_at I x).symm)
(range I)
(ext_chart_at I x x') :=
(cont_mdiff_within_at_iff_of_mem_source hx hy).trans $
by rw [continuous_within_at_univ, preimage_univ, univ_inter]
omit Is
lemma cont_mdiff_within_at_iff_target_of_mem_source
{x : M} {y : M'} (hy : f x ∈ (chart_at H' y).source) :
cont_mdiff_within_at I I' n f s x ↔ continuous_within_at f s x ∧
cont_mdiff_within_at I 𝓘(𝕜, E') n (ext_chart_at I' y ∘ f) s x :=
begin
simp_rw [cont_mdiff_within_at],
rw [(cont_diff_within_at_local_invariant_prop I I' n).lift_prop_within_at_indep_chart_target
(chart_mem_maximal_atlas I' y) hy, and_congr_right],
intro hf,
simp_rw [structure_groupoid.lift_prop_within_at_self_target],
simp_rw [((chart_at H' y).continuous_at hy).comp_continuous_within_at hf],
rw [← ext_chart_at_source I'] at hy,
simp_rw [(ext_chart_at_continuous_at' I' _ hy).comp_continuous_within_at hf],
refl,
end
lemma cont_mdiff_at_iff_target_of_mem_source
{x : M} {y : M'} (hy : f x ∈ (chart_at H' y).source) :
cont_mdiff_at I I' n f x ↔ continuous_at f x ∧
cont_mdiff_at I 𝓘(𝕜, E') n (ext_chart_at I' y ∘ f) x :=
begin
rw [cont_mdiff_at, cont_mdiff_within_at_iff_target_of_mem_source hy,
continuous_within_at_univ, cont_mdiff_at],
apply_instance
end
omit I's
variable (I)
lemma model_with_corners.symm_continuous_within_at_comp_right_iff {X} [topological_space X]
{f : H → X} {s : set H} {x : H} :
continuous_within_at (f ∘ I.symm) (I.symm ⁻¹' s ∩ range I) (I x) ↔ continuous_within_at f s x :=
begin
refine ⟨λ h, _, λ h, _⟩,
{ have := h.comp I.continuous_within_at (maps_to_preimage _ _),
simp_rw [preimage_inter, preimage_preimage, I.left_inv, preimage_id', preimage_range,
inter_univ] at this,
rwa [function.comp.assoc, I.symm_comp_self] at this },
{ rw [← I.left_inv x] at h, exact h.comp I.continuous_within_at_symm (inter_subset_left _ _) }
end
variable {I}
lemma ext_chart_at_symm_continuous_within_at_comp_right_iff {X} [topological_space X] {f : M → X}
{s : set M} {x x' : M} :
continuous_within_at (f ∘ (ext_chart_at I x).symm) ((ext_chart_at I x).symm ⁻¹' s ∩ range I)
(ext_chart_at I x x') ↔
continuous_within_at (f ∘ (chart_at H x).symm) ((chart_at H x).symm ⁻¹' s) (chart_at H x x') :=
by convert I.symm_continuous_within_at_comp_right_iff; refl
include Is
lemma cont_mdiff_within_at_iff_source_of_mem_source
{x' : M} (hx' : x' ∈ (chart_at H x).source) :
cont_mdiff_within_at I I' n f s x' ↔
cont_mdiff_within_at 𝓘(𝕜, E) I' n (f ∘ (ext_chart_at I x).symm)
((ext_chart_at I x).symm ⁻¹' s ∩ range I) (ext_chart_at I x x') :=
begin
have h2x' := hx', rw [← ext_chart_at_source I] at h2x',
simp_rw [cont_mdiff_within_at,
(cont_diff_within_at_local_invariant_prop I I' n).lift_prop_within_at_indep_chart_source
(chart_mem_maximal_atlas I x) hx', structure_groupoid.lift_prop_within_at_self_source,
ext_chart_at_symm_continuous_within_at_comp_right_iff, cont_diff_within_at_prop_self_source,
cont_diff_within_at_prop, function.comp, (chart_at H x).left_inv hx',
(ext_chart_at I x).left_inv h2x'],
refl,
end
lemma cont_mdiff_at_iff_source_of_mem_source
{x' : M} (hx' : x' ∈ (chart_at H x).source) :
cont_mdiff_at I I' n f x' ↔ cont_mdiff_within_at 𝓘(𝕜, E) I' n (f ∘ (ext_chart_at I x).symm)
(range I) (ext_chart_at I x x') :=
by simp_rw [cont_mdiff_at, cont_mdiff_within_at_iff_source_of_mem_source hx', preimage_univ,
univ_inter]
lemma cont_mdiff_at_ext_chart_at' {x' : M} (h : x' ∈ (chart_at H x).source) :
cont_mdiff_at I 𝓘(𝕜, E) n (ext_chart_at I x) x' :=
begin
refine (cont_mdiff_at_iff_of_mem_source h (mem_chart_source _ _)).mpr _,
rw [← ext_chart_at_source I] at h,
refine ⟨ext_chart_at_continuous_at' _ _ h, _⟩,
refine cont_diff_within_at_id.congr_of_eventually_eq _ _,
{ refine eventually_eq_of_mem (ext_chart_at_target_mem_nhds_within' I x h) (λ x₂ hx₂, _),
simp_rw [function.comp_apply, (ext_chart_at I x).right_inv hx₂], refl },
simp_rw [function.comp_apply, (ext_chart_at I x).right_inv ((ext_chart_at I x).maps_to h)], refl
end
lemma cont_mdiff_at_ext_chart_at : cont_mdiff_at I 𝓘(𝕜, E) n (ext_chart_at I x) x :=
cont_mdiff_at_ext_chart_at' $ mem_chart_source H x
include I's
/-- If the set where you want `f` to be smooth lies entirely in a single chart, and `f` maps it
into a single chart, the smoothness of `f` on that set can be expressed by purely looking in
these charts.
Note: this lemma uses `ext_chart_at I x '' s` instead of `(ext_chart_at I x).symm ⁻¹' s` to ensure
that this set lies in `(ext_chart_at I x).target`. -/
lemma cont_mdiff_on_iff_of_subset_source {x : M} {y : M'}
(hs : s ⊆ (chart_at H x).source)
(h2s : maps_to f s (chart_at H' y).source) :
cont_mdiff_on I I' n f s ↔ continuous_on f s ∧
cont_diff_on 𝕜 n (ext_chart_at I' y ∘ f ∘ (ext_chart_at I x).symm)
(ext_chart_at I x '' s) :=
begin
split,
{ refine λ H, ⟨λ x hx, (H x hx).1, _⟩,
rintro _ ⟨x', hx', rfl⟩,
exact ((cont_mdiff_within_at_iff_of_mem_source (hs hx')
(h2s.image_subset $ mem_image_of_mem f hx')).mp (H _ hx')).2.mono
(maps_to_ext_chart_at I x hs).image_subset },
{ rintro ⟨h1, h2⟩ x' hx',
refine (cont_mdiff_within_at_iff_of_mem_source (hs hx')
(h2s.image_subset $ mem_image_of_mem f hx')).mpr
⟨h1.continuous_within_at hx', _⟩,
refine (h2 _ $ mem_image_of_mem _ hx').mono_of_mem _,
rw [← ext_chart_at_source I] at hs,
rw [(ext_chart_at I x).image_eq_target_inter_inv_preimage hs],
refine inter_mem _ (ext_chart_preimage_mem_nhds_within' I x (hs hx') self_mem_nhds_within),
have := ext_chart_at_target_mem_nhds_within' I x (hs hx'),
refine nhds_within_mono _ (inter_subset_right _ _) this }
end
/-- One can reformulate smoothness on a set as continuity on this set, and smoothness in any
extended chart. -/
lemma cont_mdiff_on_iff :
cont_mdiff_on I I' n f s ↔ continuous_on f s ∧
∀ (x : M) (y : M'), cont_diff_on 𝕜 n (ext_chart_at I' y ∘ f ∘ (ext_chart_at I x).symm)
((ext_chart_at I x).target ∩
(ext_chart_at I x).symm ⁻¹' (s ∩ f ⁻¹' (ext_chart_at I' y).source)) :=
begin
split,
{ assume h,
refine ⟨λ x hx, (h x hx).1, λ x y z hz, _⟩,
simp only with mfld_simps at hz,
let w := (ext_chart_at I x).symm z,
have : w ∈ s, by simp only [w, hz] with mfld_simps,
specialize h w this,
have w1 : w ∈ (chart_at H x).source, by simp only [w, hz] with mfld_simps,
have w2 : f w ∈ (chart_at H' y).source, by simp only [w, hz] with mfld_simps,
convert ((cont_mdiff_within_at_iff_of_mem_source w1 w2).mp h).2.mono _,
{ simp only [w, hz] with mfld_simps },
{ mfld_set_tac } },
{ rintros ⟨hcont, hdiff⟩ x hx,
refine ((cont_diff_within_at_local_invariant_prop I I' n).lift_prop_within_at_iff $
hcont x hx).mpr _,
dsimp [cont_diff_within_at_prop],
convert hdiff x (f x) (ext_chart_at I x x) (by simp only [hx] with mfld_simps) using 1,
mfld_set_tac }
end
/-- One can reformulate smoothness on a set as continuity on this set, and smoothness in any
extended chart in the target. -/
lemma cont_mdiff_on_iff_target :
cont_mdiff_on I I' n f s ↔ continuous_on f s ∧ ∀ (y : M'),
cont_mdiff_on I 𝓘(𝕜, E') n (ext_chart_at I' y ∘ f)
(s ∩ f ⁻¹' (ext_chart_at I' y).source) :=
begin
inhabit E',
simp only [cont_mdiff_on_iff, model_with_corners.source_eq, chart_at_self_eq,
local_homeomorph.refl_local_equiv, local_equiv.refl_trans, ext_chart_at.equations._eqn_1,
set.preimage_univ, set.inter_univ, and.congr_right_iff],
intros h,
split,
{ refine λ h' y, ⟨_, λ x _, h' x y⟩,
have h'' : continuous_on _ univ := (model_with_corners.continuous I').continuous_on,
convert (h''.comp' (chart_at H' y).continuous_to_fun).comp' h,
simp },
{ exact λ h' x y, (h' y).2 x default }
end
lemma smooth_on_iff :
smooth_on I I' f s ↔ continuous_on f s ∧
∀ (x : M) (y : M'), cont_diff_on 𝕜 ⊤ (ext_chart_at I' y ∘ f ∘ (ext_chart_at I x).symm)
((ext_chart_at I x).target ∩
(ext_chart_at I x).symm ⁻¹' (s ∩ f ⁻¹' (ext_chart_at I' y).source)) :=
cont_mdiff_on_iff
lemma smooth_on_iff_target :
smooth_on I I' f s ↔ continuous_on f s ∧ ∀ (y : M'),
smooth_on I 𝓘(𝕜, E') (ext_chart_at I' y ∘ f)
(s ∩ f ⁻¹' (ext_chart_at I' y).source) :=
cont_mdiff_on_iff_target
/-- One can reformulate smoothness as continuity and smoothness in any extended chart. -/
lemma cont_mdiff_iff :
cont_mdiff I I' n f ↔ continuous f ∧
∀ (x : M) (y : M'), cont_diff_on 𝕜 n (ext_chart_at I' y ∘ f ∘ (ext_chart_at I x).symm)
((ext_chart_at I x).target ∩ (ext_chart_at I x).symm ⁻¹' (f ⁻¹' (ext_chart_at I' y).source)) :=
by simp [← cont_mdiff_on_univ, cont_mdiff_on_iff, continuous_iff_continuous_on_univ]
/-- One can reformulate smoothness as continuity and smoothness in any extended chart in the
target. -/
lemma cont_mdiff_iff_target :
cont_mdiff I I' n f ↔ continuous f ∧
∀ (y : M'), cont_mdiff_on I 𝓘(𝕜, E') n (ext_chart_at I' y ∘ f)
(f ⁻¹' (ext_chart_at I' y).source) :=
begin
rw [← cont_mdiff_on_univ, cont_mdiff_on_iff_target],
simp [continuous_iff_continuous_on_univ]
end
lemma smooth_iff :
smooth I I' f ↔ continuous f ∧
∀ (x : M) (y : M'), cont_diff_on 𝕜 ⊤ (ext_chart_at I' y ∘ f ∘ (ext_chart_at I x).symm)
((ext_chart_at I x).target ∩ (ext_chart_at I x).symm ⁻¹' (f ⁻¹' (ext_chart_at I' y).source)) :=
cont_mdiff_iff
lemma smooth_iff_target :
smooth I I' f ↔ continuous f ∧ ∀ (y : M'), smooth_on I 𝓘(𝕜, E') (ext_chart_at I' y ∘ f)
(f ⁻¹' (ext_chart_at I' y).source) :=
cont_mdiff_iff_target
omit Is I's
/-! ### Deducing smoothness from higher smoothness -/
lemma cont_mdiff_within_at.of_le (hf : cont_mdiff_within_at I I' n f s x) (le : m ≤ n) :
cont_mdiff_within_at I I' m f s x :=
⟨hf.1, hf.2.of_le le⟩
lemma cont_mdiff_at.of_le (hf : cont_mdiff_at I I' n f x) (le : m ≤ n) :
cont_mdiff_at I I' m f x :=
cont_mdiff_within_at.of_le hf le
lemma cont_mdiff_on.of_le (hf : cont_mdiff_on I I' n f s) (le : m ≤ n) :
cont_mdiff_on I I' m f s :=
λ x hx, (hf x hx).of_le le
lemma cont_mdiff.of_le (hf : cont_mdiff I I' n f) (le : m ≤ n) :
cont_mdiff I I' m f :=
λ x, (hf x).of_le le
/-! ### Deducing smoothness from smoothness one step beyond -/
lemma cont_mdiff_within_at.of_succ {n : ℕ}
(h : cont_mdiff_within_at I I' n.succ f s x) :
cont_mdiff_within_at I I' n f s x :=
h.of_le (with_top.coe_le_coe.2 (nat.le_succ n))
lemma cont_mdiff_at.of_succ {n : ℕ} (h : cont_mdiff_at I I' n.succ f x) :
cont_mdiff_at I I' n f x :=
cont_mdiff_within_at.of_succ h
lemma cont_mdiff_on.of_succ {n : ℕ} (h : cont_mdiff_on I I' n.succ f s) :
cont_mdiff_on I I' n f s :=
λ x hx, (h x hx).of_succ
lemma cont_mdiff.of_succ {n : ℕ} (h : cont_mdiff I I' n.succ f) :
cont_mdiff I I' n f :=
λ x, (h x).of_succ
/-! ### Deducing continuity from smoothness -/
lemma cont_mdiff_within_at.continuous_within_at
(hf : cont_mdiff_within_at I I' n f s x) : continuous_within_at f s x :=
hf.1
lemma cont_mdiff_at.continuous_at
(hf : cont_mdiff_at I I' n f x) : continuous_at f x :=
(continuous_within_at_univ _ _ ).1 $ cont_mdiff_within_at.continuous_within_at hf
lemma cont_mdiff_on.continuous_on
(hf : cont_mdiff_on I I' n f s) : continuous_on f s :=
λ x hx, (hf x hx).continuous_within_at
lemma cont_mdiff.continuous (hf : cont_mdiff I I' n f) :
continuous f :=
continuous_iff_continuous_at.2 $ λ x, (hf x).continuous_at
/-! ### Deducing differentiability from smoothness -/
lemma cont_mdiff_within_at.mdifferentiable_within_at
(hf : cont_mdiff_within_at I I' n f s x) (hn : 1 ≤ n) :
mdifferentiable_within_at I I' f s x :=
begin
suffices h : mdifferentiable_within_at I I' f (s ∩ (f ⁻¹' (ext_chart_at I' (f x)).source)) x,
{ rwa mdifferentiable_within_at_inter' at h,
apply (hf.1).preimage_mem_nhds_within,
exact is_open.mem_nhds (ext_chart_at_open_source I' (f x)) (mem_ext_chart_source I' (f x)) },
rw mdifferentiable_within_at_iff,
exact ⟨hf.1.mono (inter_subset_left _ _),
(hf.2.differentiable_within_at hn).mono (by mfld_set_tac)⟩,
end
lemma cont_mdiff_at.mdifferentiable_at (hf : cont_mdiff_at I I' n f x) (hn : 1 ≤ n) :
mdifferentiable_at I I' f x :=
mdifferentiable_within_at_univ.1 $ cont_mdiff_within_at.mdifferentiable_within_at hf hn
lemma cont_mdiff_on.mdifferentiable_on (hf : cont_mdiff_on I I' n f s) (hn : 1 ≤ n) :
mdifferentiable_on I I' f s :=
λ x hx, (hf x hx).mdifferentiable_within_at hn
lemma cont_mdiff.mdifferentiable (hf : cont_mdiff I I' n f) (hn : 1 ≤ n) :
mdifferentiable I I' f :=
λ x, (hf x).mdifferentiable_at hn
lemma smooth.mdifferentiable (hf : smooth I I' f) : mdifferentiable I I' f :=
cont_mdiff.mdifferentiable hf le_top
lemma smooth.mdifferentiable_at (hf : smooth I I' f) : mdifferentiable_at I I' f x :=
hf.mdifferentiable x
lemma smooth.mdifferentiable_within_at (hf : smooth I I' f) :
mdifferentiable_within_at I I' f s x :=
hf.mdifferentiable_at.mdifferentiable_within_at
/-! ### `C^∞` smoothness -/
lemma cont_mdiff_within_at_top :
smooth_within_at I I' f s x ↔ (∀n:ℕ, cont_mdiff_within_at I I' n f s x) :=
⟨λ h n, ⟨h.1, cont_diff_within_at_top.1 h.2 n⟩,
λ H, ⟨(H 0).1, cont_diff_within_at_top.2 (λ n, (H n).2)⟩⟩
lemma cont_mdiff_at_top :
smooth_at I I' f x ↔ (∀n:ℕ, cont_mdiff_at I I' n f x) :=
cont_mdiff_within_at_top
lemma cont_mdiff_on_top :
smooth_on I I' f s ↔ (∀n:ℕ, cont_mdiff_on I I' n f s) :=
⟨λ h n, h.of_le le_top, λ h x hx, cont_mdiff_within_at_top.2 (λ n, h n x hx)⟩
lemma cont_mdiff_top :
smooth I I' f ↔ (∀n:ℕ, cont_mdiff I I' n f) :=
⟨λ h n, h.of_le le_top, λ h x, cont_mdiff_within_at_top.2 (λ n, h n x)⟩
lemma cont_mdiff_within_at_iff_nat :
cont_mdiff_within_at I I' n f s x ↔
(∀m:ℕ, (m : ℕ∞) ≤ n → cont_mdiff_within_at I I' m f s x) :=
begin
refine ⟨λ h m hm, h.of_le hm, λ h, _⟩,
cases n,
{ exact cont_mdiff_within_at_top.2 (λ n, h n le_top) },
{ exact h n le_rfl }
end
/-! ### Restriction to a smaller set -/
lemma cont_mdiff_within_at.mono (hf : cont_mdiff_within_at I I' n f s x) (hts : t ⊆ s) :
cont_mdiff_within_at I I' n f t x :=
structure_groupoid.local_invariant_prop.lift_prop_within_at_mono
(cont_diff_within_at_prop_mono I I' n) hf hts
lemma cont_mdiff_at.cont_mdiff_within_at (hf : cont_mdiff_at I I' n f x) :
cont_mdiff_within_at I I' n f s x :=
cont_mdiff_within_at.mono hf (subset_univ _)
lemma smooth_at.smooth_within_at (hf : smooth_at I I' f x) :
smooth_within_at I I' f s x :=
cont_mdiff_at.cont_mdiff_within_at hf
lemma cont_mdiff_on.mono (hf : cont_mdiff_on I I' n f s) (hts : t ⊆ s) :
cont_mdiff_on I I' n f t :=
λ x hx, (hf x (hts hx)).mono hts
lemma cont_mdiff.cont_mdiff_on (hf : cont_mdiff I I' n f) :
cont_mdiff_on I I' n f s :=
λ x hx, (hf x).cont_mdiff_within_at
lemma smooth.smooth_on (hf : smooth I I' f) :
smooth_on I I' f s :=
cont_mdiff.cont_mdiff_on hf
lemma cont_mdiff_within_at_inter' (ht : t ∈ 𝓝[s] x) :
cont_mdiff_within_at I I' n f (s ∩ t) x ↔ cont_mdiff_within_at I I' n f s x :=
(cont_diff_within_at_local_invariant_prop I I' n).lift_prop_within_at_inter' ht
lemma cont_mdiff_within_at_inter (ht : t ∈ 𝓝 x) :
cont_mdiff_within_at I I' n f (s ∩ t) x ↔ cont_mdiff_within_at I I' n f s x :=
(cont_diff_within_at_local_invariant_prop I I' n).lift_prop_within_at_inter ht
lemma cont_mdiff_within_at.cont_mdiff_at
(h : cont_mdiff_within_at I I' n f s x) (ht : s ∈ 𝓝 x) :
cont_mdiff_at I I' n f x :=
(cont_diff_within_at_local_invariant_prop I I' n).lift_prop_at_of_lift_prop_within_at h ht
lemma smooth_within_at.smooth_at
(h : smooth_within_at I I' f s x) (ht : s ∈ 𝓝 x) :
smooth_at I I' f x :=
cont_mdiff_within_at.cont_mdiff_at h ht
lemma cont_mdiff_on.cont_mdiff_at (h : cont_mdiff_on I I' n f s) (hx : s ∈ 𝓝 x) :
cont_mdiff_at I I' n f x :=
(h x (mem_of_mem_nhds hx)).cont_mdiff_at hx
lemma smooth_on.smooth_at (h : smooth_on I I' f s) (hx : s ∈ 𝓝 x) : smooth_at I I' f x :=
h.cont_mdiff_at hx
include Is
lemma cont_mdiff_on_ext_chart_at :
cont_mdiff_on I 𝓘(𝕜, E) n (ext_chart_at I x) (chart_at H x).source :=
λ x' hx', (cont_mdiff_at_ext_chart_at' hx').cont_mdiff_within_at
include I's
/-- A function is `C^n` within a set at a point, for `n : ℕ`, if and only if it is `C^n` on
a neighborhood of this point. -/
lemma cont_mdiff_within_at_iff_cont_mdiff_on_nhds {n : ℕ} :
cont_mdiff_within_at I I' n f s x ↔
∃ u ∈ 𝓝[insert x s] x, cont_mdiff_on I I' n f u :=
begin
split,
{ assume h,
-- the property is true in charts. We will pull such a good neighborhood in the chart to the
-- manifold. For this, we need to restrict to a small enough set where everything makes sense
obtain ⟨o, o_open, xo, ho, h'o⟩ : ∃ (o : set M),
is_open o ∧ x ∈ o ∧ o ⊆ (chart_at H x).source ∧ o ∩ s ⊆ f ⁻¹' (chart_at H' (f x)).source,
{ have : (chart_at H' (f x)).source ∈ 𝓝 (f x) :=
is_open.mem_nhds (local_homeomorph.open_source _) (mem_chart_source H' (f x)),
rcases mem_nhds_within.1 (h.1.preimage_mem_nhds_within this) with ⟨u, u_open, xu, hu⟩,
refine ⟨u ∩ (chart_at H x).source, _, ⟨xu, mem_chart_source _ _⟩, _, _⟩,
{ exact is_open.inter u_open (local_homeomorph.open_source _) },
{ assume y hy, exact hy.2 },
{ assume y hy, exact hu ⟨hy.1.1, hy.2⟩ } },
have h' : cont_mdiff_within_at I I' n f (s ∩ o) x := h.mono (inter_subset_left _ _),
simp only [cont_mdiff_within_at, lift_prop_within_at, cont_diff_within_at_prop] at h',
-- let `u` be a good neighborhood in the chart where the function is smooth
rcases h.2.cont_diff_on le_rfl with ⟨u, u_nhds, u_subset, hu⟩,
-- pull it back to the manifold, and intersect with a suitable neighborhood of `x`, to get the
-- desired good neighborhood `v`.
let v := ((insert x s) ∩ o) ∩ (ext_chart_at I x) ⁻¹' u,
have v_incl : v ⊆ (chart_at H x).source := λ y hy, ho hy.1.2,
have v_incl' : ∀ y ∈ v, f y ∈ (chart_at H' (f x)).source,
{ assume y hy,
rcases hy.1.1 with rfl|h',
{ simp only with mfld_simps },
{ apply h'o ⟨hy.1.2, h'⟩ } },
refine ⟨v, _, _⟩,
show v ∈ 𝓝[insert x s] x,
{ rw nhds_within_restrict _ xo o_open,
refine filter.inter_mem self_mem_nhds_within _,
suffices : u ∈ 𝓝[(ext_chart_at I x) '' (insert x s ∩ o)] (ext_chart_at I x x),
from (ext_chart_at_continuous_at I x).continuous_within_at.preimage_mem_nhds_within' this,
apply nhds_within_mono _ _ u_nhds,
rw image_subset_iff,
assume y hy,
rcases hy.1 with rfl|h',
{ simp only [mem_insert_iff] with mfld_simps },
{ simp only [mem_insert_iff, ho hy.2, h', h'o ⟨hy.2, h'⟩] with mfld_simps } },
show cont_mdiff_on I I' n f v,
{ assume y hy,
have : continuous_within_at f v y,
{ apply (((ext_chart_at_continuous_on_symm I' (f x) _ _).comp'
(hu _ hy.2).continuous_within_at).comp' (ext_chart_at_continuous_on I x _ _)).congr_mono,
{ assume z hz,
simp only [v_incl hz, v_incl' z hz] with mfld_simps },
{ assume z hz,
simp only [v_incl hz, v_incl' z hz] with mfld_simps,
exact hz.2 },
{ simp only [v_incl hy, v_incl' y hy] with mfld_simps },
{ simp only [v_incl hy, v_incl' y hy] with mfld_simps },
{ simp only [v_incl hy] with mfld_simps } },
refine (cont_mdiff_within_at_iff_of_mem_source' (v_incl hy) (v_incl' y hy)).mpr ⟨this, _⟩,
{ apply hu.mono,
{ assume z hz,
simp only [v] with mfld_simps at hz,
have : I ((chart_at H x) (((chart_at H x).symm) (I.symm z))) ∈ u, by simp only [hz],
simpa only [hz] with mfld_simps using this },
{ have exty : I (chart_at H x y) ∈ u := hy.2,
simp only [v_incl hy, v_incl' y hy, exty, hy.1.1, hy.1.2] with mfld_simps } } } },
{ rintros ⟨u, u_nhds, hu⟩,
have : cont_mdiff_within_at I I' ↑n f (insert x s ∩ u) x,
{ have : x ∈ insert x s := mem_insert x s,
exact hu.mono (inter_subset_right _ _) _ ⟨this, mem_of_mem_nhds_within this u_nhds⟩ },
rw cont_mdiff_within_at_inter' u_nhds at this,
exact this.mono (subset_insert x s) }
end
/-- A function is `C^n` at a point, for `n : ℕ`, if and only if it is `C^n` on
a neighborhood of this point. -/
lemma cont_mdiff_at_iff_cont_mdiff_on_nhds {n : ℕ} :
cont_mdiff_at I I' n f x ↔ ∃ u ∈ 𝓝 x, cont_mdiff_on I I' n f u :=
by simp [← cont_mdiff_within_at_univ, cont_mdiff_within_at_iff_cont_mdiff_on_nhds,
nhds_within_univ]
/-- Note: This does not hold for `n = ∞`. `f` being `C^∞` at `x` means that for every `n`, `f` is
`C^n` on some neighborhood of `x`, but this neighborhood can depend on `n`. -/
lemma cont_mdiff_at_iff_cont_mdiff_at_nhds {n : ℕ} :
cont_mdiff_at I I' n f x ↔ ∀ᶠ x' in 𝓝 x, cont_mdiff_at I I' n f x' :=
begin
refine ⟨_, λ h, h.self_of_nhds⟩,
rw [cont_mdiff_at_iff_cont_mdiff_on_nhds],
rintro ⟨u, hu, h⟩,
refine (eventually_mem_nhds.mpr hu).mono (λ x' hx', _),
exact (h x' $ mem_of_mem_nhds hx').cont_mdiff_at hx'
end
omit Is I's
/-! ### Congruence lemmas -/
lemma cont_mdiff_within_at.congr
(h : cont_mdiff_within_at I I' n f s x) (h₁ : ∀ y ∈ s, f₁ y = f y)
(hx : f₁ x = f x) : cont_mdiff_within_at I I' n f₁ s x :=
(cont_diff_within_at_local_invariant_prop I I' n).lift_prop_within_at_congr h h₁ hx
lemma cont_mdiff_within_at_congr (h₁ : ∀ y ∈ s, f₁ y = f y) (hx : f₁ x = f x) :
cont_mdiff_within_at I I' n f₁ s x ↔ cont_mdiff_within_at I I' n f s x :=
(cont_diff_within_at_local_invariant_prop I I' n).lift_prop_within_at_congr_iff h₁ hx
lemma cont_mdiff_within_at.congr_of_eventually_eq
(h : cont_mdiff_within_at I I' n f s x) (h₁ : f₁ =ᶠ[𝓝[s] x] f)
(hx : f₁ x = f x) : cont_mdiff_within_at I I' n f₁ s x :=
(cont_diff_within_at_local_invariant_prop I I' n).lift_prop_within_at_congr_of_eventually_eq
h h₁ hx
lemma filter.eventually_eq.cont_mdiff_within_at_iff
(h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) :
cont_mdiff_within_at I I' n f₁ s x ↔ cont_mdiff_within_at I I' n f s x :=
(cont_diff_within_at_local_invariant_prop I I' n)
.lift_prop_within_at_congr_iff_of_eventually_eq h₁ hx
lemma cont_mdiff_at.congr_of_eventually_eq
(h : cont_mdiff_at I I' n f x) (h₁ : f₁ =ᶠ[𝓝 x] f) :
cont_mdiff_at I I' n f₁ x :=
(cont_diff_within_at_local_invariant_prop I I' n).lift_prop_at_congr_of_eventually_eq h h₁
lemma filter.eventually_eq.cont_mdiff_at_iff (h₁ : f₁ =ᶠ[𝓝 x] f) :
cont_mdiff_at I I' n f₁ x ↔ cont_mdiff_at I I' n f x :=
(cont_diff_within_at_local_invariant_prop I I' n).lift_prop_at_congr_iff_of_eventually_eq h₁
lemma cont_mdiff_on.congr (h : cont_mdiff_on I I' n f s) (h₁ : ∀ y ∈ s, f₁ y = f y) :
cont_mdiff_on I I' n f₁ s :=
(cont_diff_within_at_local_invariant_prop I I' n).lift_prop_on_congr h h₁
lemma cont_mdiff_on_congr (h₁ : ∀ y ∈ s, f₁ y = f y) :
cont_mdiff_on I I' n f₁ s ↔ cont_mdiff_on I I' n f s :=
(cont_diff_within_at_local_invariant_prop I I' n).lift_prop_on_congr_iff h₁
/-! ### Locality -/
/-- Being `C^n` is a local property. -/
lemma cont_mdiff_on_of_locally_cont_mdiff_on
(h : ∀x∈s, ∃u, is_open u ∧ x ∈ u ∧ cont_mdiff_on I I' n f (s ∩ u)) :
cont_mdiff_on I I' n f s :=
(cont_diff_within_at_local_invariant_prop I I' n).lift_prop_on_of_locally_lift_prop_on h
lemma cont_mdiff_of_locally_cont_mdiff_on
(h : ∀x, ∃u, is_open u ∧ x ∈ u ∧ cont_mdiff_on I I' n f u) :
cont_mdiff I I' n f :=
(cont_diff_within_at_local_invariant_prop I I' n).lift_prop_of_locally_lift_prop_on h
/-! ### Smoothness of the composition of smooth functions between manifolds -/
section composition
variables {E'' : Type*} [normed_add_comm_group E''] [normed_space 𝕜 E'']
{H'' : Type*} [topological_space H''] {I'' : model_with_corners 𝕜 E'' H''}
{M'' : Type*} [topological_space M''] [charted_space H'' M'']
/-- The composition of `C^n` functions within domains at points is `C^n`. -/
lemma cont_mdiff_within_at.comp {t : set M'} {g : M' → M''} (x : M)
(hg : cont_mdiff_within_at I' I'' n g t (f x))
(hf : cont_mdiff_within_at I I' n f s x)
(st : maps_to f s t) : cont_mdiff_within_at I I'' n (g ∘ f) s x :=
begin
rw cont_mdiff_within_at_iff at hg hf ⊢,
refine ⟨hg.1.comp hf.1 st, _⟩,
set e := ext_chart_at I x,
set e' := ext_chart_at I' (f x),
set e'' := ext_chart_at I'' (g (f x)),
have : e' (f x) = (written_in_ext_chart_at I I' x f) (e x),
by simp only [e, e'] with mfld_simps,
rw this at hg,
have A : ∀ᶠ y in 𝓝[e.symm ⁻¹' s ∩ range I] e x,
y ∈ e.target ∧ f (e.symm y) ∈ t ∧ f (e.symm y) ∈ e'.source ∧ g (f (e.symm y)) ∈ e''.source,
{ simp only [← ext_chart_at_map_nhds_within, eventually_map],
filter_upwards [hf.1.tendsto (ext_chart_at_source_mem_nhds I' (f x)),
(hg.1.comp hf.1 st).tendsto (ext_chart_at_source_mem_nhds I'' (g (f x))),
(inter_mem_nhds_within s (ext_chart_at_source_mem_nhds I x))],
rintros x' (hfx' : f x' ∈ _) (hgfx' : g (f x') ∈ _) ⟨hx's, hx'⟩,
simp only [e.map_source hx', true_and, e.left_inv hx', st hx's, *] },
refine ((hg.2.comp _ (hf.2.mono (inter_subset_right _ _)) (inter_subset_left _ _)).mono_of_mem
(inter_mem _ self_mem_nhds_within)).congr_of_eventually_eq _ _,
{ filter_upwards [A],
rintro x' ⟨hx', ht, hfx', hgfx'⟩,
simp only [*, mem_preimage, written_in_ext_chart_at, (∘), mem_inter_eq, e'.left_inv, true_and],
exact mem_range_self _ },
{ filter_upwards [A],
rintro x' ⟨hx', ht, hfx', hgfx'⟩,
simp only [*, (∘), written_in_ext_chart_at, e'.left_inv] },
{ simp only [written_in_ext_chart_at, (∘), mem_ext_chart_source, e.left_inv, e'.left_inv] }
end
/-- The composition of `C^n` functions on domains is `C^n`. -/
lemma cont_mdiff_on.comp {t : set M'} {g : M' → M''}
(hg : cont_mdiff_on I' I'' n g t) (hf : cont_mdiff_on I I' n f s)
(st : s ⊆ f ⁻¹' t) : cont_mdiff_on I I'' n (g ∘ f) s :=
λ x hx, (hg _ (st hx)).comp x (hf x hx) st
/-- The composition of `C^n` functions on domains is `C^n`. -/
lemma cont_mdiff_on.comp' {t : set M'} {g : M' → M''}
(hg : cont_mdiff_on I' I'' n g t) (hf : cont_mdiff_on I I' n f s) :
cont_mdiff_on I I'' n (g ∘ f) (s ∩ f ⁻¹' t) :=
hg.comp (hf.mono (inter_subset_left _ _)) (inter_subset_right _ _)
/-- The composition of `C^n` functions is `C^n`. -/
lemma cont_mdiff.comp {g : M' → M''}
(hg : cont_mdiff I' I'' n g) (hf : cont_mdiff I I' n f) :
cont_mdiff I I'' n (g ∘ f) :=
begin
rw ← cont_mdiff_on_univ at hf hg ⊢,
exact hg.comp hf subset_preimage_univ,
end
/-- The composition of `C^n` functions within domains at points is `C^n`. -/
lemma cont_mdiff_within_at.comp' {t : set M'} {g : M' → M''} (x : M)
(hg : cont_mdiff_within_at I' I'' n g t (f x))
(hf : cont_mdiff_within_at I I' n f s x) :
cont_mdiff_within_at I I'' n (g ∘ f) (s ∩ f⁻¹' t) x :=
hg.comp x (hf.mono (inter_subset_left _ _)) (inter_subset_right _ _)
/-- `g ∘ f` is `C^n` within `s` at `x` if `g` is `C^n` at `f x` and
`f` is `C^n` within `s` at `x`. -/
lemma cont_mdiff_at.comp_cont_mdiff_within_at {g : M' → M''} (x : M)
(hg : cont_mdiff_at I' I'' n g (f x)) (hf : cont_mdiff_within_at I I' n f s x) :
cont_mdiff_within_at I I'' n (g ∘ f) s x :=
hg.comp x hf (maps_to_univ _ _)
/-- The composition of `C^n` functions at points is `C^n`. -/
lemma cont_mdiff_at.comp {g : M' → M''} (x : M)
(hg : cont_mdiff_at I' I'' n g (f x)) (hf : cont_mdiff_at I I' n f x) :
cont_mdiff_at I I'' n (g ∘ f) x :=
hg.comp x hf (maps_to_univ _ _)
lemma cont_mdiff.comp_cont_mdiff_on {f : M → M'} {g : M' → M''} {s : set M}
(hg : cont_mdiff I' I'' n g) (hf : cont_mdiff_on I I' n f s) :
cont_mdiff_on I I'' n (g ∘ f) s :=
hg.cont_mdiff_on.comp hf set.subset_preimage_univ
lemma smooth.comp_smooth_on {f : M → M'} {g : M' → M''} {s : set M}
(hg : smooth I' I'' g) (hf : smooth_on I I' f s) :
smooth_on I I'' (g ∘ f) s :=
hg.smooth_on.comp hf set.subset_preimage_univ
lemma cont_mdiff_on.comp_cont_mdiff {t : set M'} {g : M' → M''}
(hg : cont_mdiff_on I' I'' n g t) (hf : cont_mdiff I I' n f)
(ht : ∀ x, f x ∈ t) : cont_mdiff I I'' n (g ∘ f) :=
cont_mdiff_on_univ.mp $ hg.comp hf.cont_mdiff_on (λ x _, ht x)
lemma smooth_on.comp_smooth {t : set M'} {g : M' → M''}
(hg : smooth_on I' I'' g t) (hf : smooth I I' f)
(ht : ∀ x, f x ∈ t) : smooth I I'' (g ∘ f) :=
hg.comp_cont_mdiff hf ht
end composition
/-! ### Atlas members are smooth -/
section atlas
variables {e : local_homeomorph M H}
include Is
/-- An atlas member is `C^n` for any `n`. -/
lemma cont_mdiff_on_of_mem_maximal_atlas
(h : e ∈ maximal_atlas I M) : cont_mdiff_on I I n e e.source :=
cont_mdiff_on.of_le
((cont_diff_within_at_local_invariant_prop I I ∞).lift_prop_on_of_mem_maximal_atlas
(cont_diff_within_at_prop_id I) h) le_top
/-- The inverse of an atlas member is `C^n` for any `n`. -/
lemma cont_mdiff_on_symm_of_mem_maximal_atlas
(h : e ∈ maximal_atlas I M) : cont_mdiff_on I I n e.symm e.target :=
cont_mdiff_on.of_le
((cont_diff_within_at_local_invariant_prop I I ∞).lift_prop_on_symm_of_mem_maximal_atlas
(cont_diff_within_at_prop_id I) h) le_top
lemma cont_mdiff_on_chart :
cont_mdiff_on I I n (chart_at H x) (chart_at H x).source :=
cont_mdiff_on_of_mem_maximal_atlas
((cont_diff_groupoid ⊤ I).chart_mem_maximal_atlas x)
lemma cont_mdiff_on_chart_symm :
cont_mdiff_on I I n (chart_at H x).symm (chart_at H x).target :=
cont_mdiff_on_symm_of_mem_maximal_atlas
((cont_diff_groupoid ⊤ I).chart_mem_maximal_atlas x)
end atlas
/-! ### The identity is smooth -/
section id
lemma cont_mdiff_id : cont_mdiff I I n (id : M → M) :=
cont_mdiff.of_le ((cont_diff_within_at_local_invariant_prop I I ∞).lift_prop_id
(cont_diff_within_at_prop_id I)) le_top
lemma smooth_id : smooth I I (id : M → M) := cont_mdiff_id
lemma cont_mdiff_on_id : cont_mdiff_on I I n (id : M → M) s :=
cont_mdiff_id.cont_mdiff_on
lemma smooth_on_id : smooth_on I I (id : M → M) s := cont_mdiff_on_id
lemma cont_mdiff_at_id : cont_mdiff_at I I n (id : M → M) x :=
cont_mdiff_id.cont_mdiff_at
lemma smooth_at_id : smooth_at I I (id : M → M) x := cont_mdiff_at_id
lemma cont_mdiff_within_at_id : cont_mdiff_within_at I I n (id : M → M) s x :=
cont_mdiff_at_id.cont_mdiff_within_at
lemma smooth_within_at_id : smooth_within_at I I (id : M → M) s x := cont_mdiff_within_at_id
end id
/-! ### Constants are smooth -/
section id
variable {c : M'}
lemma cont_mdiff_const : cont_mdiff I I' n (λ (x : M), c) :=
begin
assume x,
refine ⟨continuous_within_at_const, _⟩,
simp only [cont_diff_within_at_prop, (∘)],
exact cont_diff_within_at_const,
end
@[to_additive]
lemma cont_mdiff_one [has_one M'] : cont_mdiff I I' n (1 : M → M') :=
by simp only [pi.one_def, cont_mdiff_const]
lemma smooth_const : smooth I I' (λ (x : M), c) := cont_mdiff_const
@[to_additive]
lemma smooth_one [has_one M'] : smooth I I' (1 : M → M') :=
by simp only [pi.one_def, smooth_const]
lemma cont_mdiff_on_const : cont_mdiff_on I I' n (λ (x : M), c) s :=
cont_mdiff_const.cont_mdiff_on
@[to_additive]
lemma cont_mdiff_on_one [has_one M'] : cont_mdiff_on I I' n (1 : M → M') s :=
cont_mdiff_one.cont_mdiff_on
lemma smooth_on_const : smooth_on I I' (λ (x : M), c) s :=
cont_mdiff_on_const
@[to_additive]
lemma smooth_on_one [has_one M'] : smooth_on I I' (1 : M → M') s :=
cont_mdiff_on_one
lemma cont_mdiff_at_const : cont_mdiff_at I I' n (λ (x : M), c) x :=
cont_mdiff_const.cont_mdiff_at
@[to_additive]
lemma cont_mdiff_at_one [has_one M'] : cont_mdiff_at I I' n (1 : M → M') x :=
cont_mdiff_one.cont_mdiff_at
lemma smooth_at_const : smooth_at I I' (λ (x : M), c) x :=
cont_mdiff_at_const
@[to_additive]
lemma smooth_at_one [has_one M'] : smooth_at I I' (1 : M → M') x :=
cont_mdiff_at_one
lemma cont_mdiff_within_at_const : cont_mdiff_within_at I I' n (λ (x : M), c) s x :=
cont_mdiff_at_const.cont_mdiff_within_at
@[to_additive]
lemma cont_mdiff_within_at_one [has_one M'] :
cont_mdiff_within_at I I' n (1 : M → M') s x :=
cont_mdiff_at_const.cont_mdiff_within_at
lemma smooth_within_at_const : smooth_within_at I I' (λ (x : M), c) s x :=
cont_mdiff_within_at_const
@[to_additive]
lemma smooth_within_at_one [has_one M'] : smooth_within_at I I' (1 : M → M') s x :=
cont_mdiff_within_at_one
end id
lemma cont_mdiff_of_support {f : M → F}
(hf : ∀ x ∈ tsupport f, cont_mdiff_at I 𝓘(𝕜, F) n f x) :
cont_mdiff I 𝓘(𝕜, F) n f :=
begin
intro x,
by_cases hx : x ∈ tsupport f,
{ exact hf x hx },
{ refine cont_mdiff_at.congr_of_eventually_eq _ (eventually_eq_zero_nhds.2 hx),
exact cont_mdiff_at_const }
end
/-! ### Equivalence with the basic definition for functions between vector spaces -/
section module
lemma cont_mdiff_within_at_iff_cont_diff_within_at {f : E → E'} {s : set E} {x : E} :
cont_mdiff_within_at 𝓘(𝕜, E) 𝓘(𝕜, E') n f s x
↔ cont_diff_within_at 𝕜 n f s x :=
begin
simp only [cont_mdiff_within_at, lift_prop_within_at, cont_diff_within_at_prop,
iff_def] with mfld_simps {contextual := tt},
exact cont_diff_within_at.continuous_within_at
end
alias cont_mdiff_within_at_iff_cont_diff_within_at ↔
cont_mdiff_within_at.cont_diff_within_at
cont_diff_within_at.cont_mdiff_within_at
lemma cont_mdiff_at_iff_cont_diff_at {f : E → E'} {x : E} :
cont_mdiff_at 𝓘(𝕜, E) 𝓘(𝕜, E') n f x
↔ cont_diff_at 𝕜 n f x :=
by rw [← cont_mdiff_within_at_univ,
cont_mdiff_within_at_iff_cont_diff_within_at, cont_diff_within_at_univ]
alias cont_mdiff_at_iff_cont_diff_at ↔
cont_mdiff_at.cont_diff_at cont_diff_at.cont_mdiff_at
lemma cont_mdiff_on_iff_cont_diff_on {f : E → E'} {s : set E} :
cont_mdiff_on 𝓘(𝕜, E) 𝓘(𝕜, E') n f s
↔ cont_diff_on 𝕜 n f s :=
forall_congr $ by simp [cont_mdiff_within_at_iff_cont_diff_within_at]
alias cont_mdiff_on_iff_cont_diff_on ↔
cont_mdiff_on.cont_diff_on cont_diff_on.cont_mdiff_on
lemma cont_mdiff_iff_cont_diff {f : E → E'} :
cont_mdiff 𝓘(𝕜, E) 𝓘(𝕜, E') n f
↔ cont_diff 𝕜 n f :=
by rw [← cont_diff_on_univ, ← cont_mdiff_on_univ,
cont_mdiff_on_iff_cont_diff_on]
alias cont_mdiff_iff_cont_diff ↔
cont_mdiff.cont_diff cont_diff.cont_mdiff
end module
/-! ### The tangent map of a smooth function is smooth -/
section tangent_map
/-- If a function is `C^n` with `1 ≤ n` on a domain with unique derivatives, then its bundled
derivative is continuous. In this auxiliary lemma, we prove this fact when the source and target
space are model spaces in models with corners. The general fact is proved in
`cont_mdiff_on.continuous_on_tangent_map_within`-/
lemma cont_mdiff_on.continuous_on_tangent_map_within_aux
{f : H → H'} {s : set H}
(hf : cont_mdiff_on I I' n f s) (hn : 1 ≤ n) (hs : unique_mdiff_on I s) :
continuous_on (tangent_map_within I I' f s) ((tangent_bundle.proj I H) ⁻¹' s) :=
begin
suffices h : continuous_on (λ (p : H × E), (f p.fst,
(fderiv_within 𝕜 (written_in_ext_chart_at I I' p.fst f) (I.symm ⁻¹' s ∩ range I)
((ext_chart_at I p.fst) p.fst) : E →L[𝕜] E') p.snd)) (prod.fst ⁻¹' s),
{ have A := (tangent_bundle_model_space_homeomorph H I).continuous,
rw continuous_iff_continuous_on_univ at A,
have B := ((tangent_bundle_model_space_homeomorph H' I').symm.continuous.comp_continuous_on h)
.comp' A,
have : (univ ∩ ⇑(tangent_bundle_model_space_homeomorph H I) ⁻¹' (prod.fst ⁻¹' s)) =
tangent_bundle.proj I H ⁻¹' s,
by { ext ⟨x, v⟩, simp only with mfld_simps },
rw this at B,
apply B.congr,
rintros ⟨x, v⟩ hx,
dsimp [tangent_map_within],
ext, { refl },
simp only with mfld_simps,
apply congr_fun,
apply congr_arg,
rw mdifferentiable_within_at.mfderiv_within (hf.mdifferentiable_on hn x hx),
refl },
suffices h : continuous_on (λ (p : H × E), (fderiv_within 𝕜 (I' ∘ f ∘ I.symm)
(I.symm ⁻¹' s ∩ range I) (I p.fst) : E →L[𝕜] E') p.snd) (prod.fst ⁻¹' s),
{ dsimp [written_in_ext_chart_at, ext_chart_at],
apply continuous_on.prod
(continuous_on.comp hf.continuous_on continuous_fst.continuous_on (subset.refl _)),
apply h.congr,
assume p hp,
refl },
suffices h : continuous_on (fderiv_within 𝕜 (I' ∘ f ∘ I.symm)
(I.symm ⁻¹' s ∩ range I)) (I '' s),
{ have C := continuous_on.comp h I.continuous_to_fun.continuous_on (subset.refl _),
have A : continuous (λq : (E →L[𝕜] E') × E, q.1 q.2) :=
is_bounded_bilinear_map_apply.continuous,
have B : continuous_on (λp : H × E,
(fderiv_within 𝕜 (I' ∘ f ∘ I.symm) (I.symm ⁻¹' s ∩ range I)
(I p.1), p.2)) (prod.fst ⁻¹' s),
{ apply continuous_on.prod _ continuous_snd.continuous_on,
refine (continuous_on.comp C continuous_fst.continuous_on _ : _),
exact preimage_mono (subset_preimage_image _ _) },
exact A.comp_continuous_on B },
rw cont_mdiff_on_iff at hf,
let x : H := I.symm (0 : E),
let y : H' := I'.symm (0 : E'),
have A := hf.2 x y,
simp only [I.image_eq, inter_comm] with mfld_simps at A ⊢,
apply A.continuous_on_fderiv_within _ hn,
convert hs.unique_diff_on_target_inter x using 1,
simp only [inter_comm] with mfld_simps
end
/-- If a function is `C^n` on a domain with unique derivatives, then its bundled derivative is
`C^m` when `m+1 ≤ n`. In this auxiliary lemma, we prove this fact when the source and target space
are model spaces in models with corners. The general fact is proved in
`cont_mdiff_on.cont_mdiff_on_tangent_map_within` -/
lemma cont_mdiff_on.cont_mdiff_on_tangent_map_within_aux
{f : H → H'} {s : set H}
(hf : cont_mdiff_on I I' n f s) (hmn : m + 1 ≤ n) (hs : unique_mdiff_on I s) :
cont_mdiff_on I.tangent I'.tangent m (tangent_map_within I I' f s)
((tangent_bundle.proj I H) ⁻¹' s) :=
begin
have m_le_n : m ≤ n,
{ apply le_trans _ hmn,
have : m + 0 ≤ m + 1 := add_le_add_left (zero_le _) _,
simpa only [add_zero] using this },
have one_le_n : 1 ≤ n,
{ apply le_trans _ hmn,
change 0 + 1 ≤ m + 1,
exact add_le_add_right (zero_le _) _ },
have U': unique_diff_on 𝕜 (range I ∩ I.symm ⁻¹' s),
{ assume y hy,
simpa only [unique_mdiff_on, unique_mdiff_within_at, hy.1, inter_comm] with mfld_simps
using hs (I.symm y) hy.2 },
rw cont_mdiff_on_iff,
refine ⟨hf.continuous_on_tangent_map_within_aux one_le_n hs, λp q, _⟩,
have A : range I ×ˢ univ ∩
((equiv.sigma_equiv_prod H E).symm ∘ λ (p : E × E), ((I.symm) p.fst, p.snd)) ⁻¹'
(tangent_bundle.proj I H ⁻¹' s)
= (range I ∩ I.symm ⁻¹' s) ×ˢ univ,
by { ext ⟨x, v⟩, simp only with mfld_simps },
suffices h : cont_diff_on 𝕜 m (((λ (p : H' × E'), (I' p.fst, p.snd)) ∘
(equiv.sigma_equiv_prod H' E')) ∘ tangent_map_within I I' f s ∘
((equiv.sigma_equiv_prod H E).symm) ∘ λ (p : E × E), (I.symm p.fst, p.snd))
((range ⇑I ∩ ⇑(I.symm) ⁻¹' s) ×ˢ univ),
by simpa [A] using h,
change cont_diff_on 𝕜 m (λ (p : E × E),
((I' (f (I.symm p.fst)), ((mfderiv_within I I' f s (I.symm p.fst)) : E → E') p.snd) : E' × E'))
((range I ∩ I.symm ⁻¹' s) ×ˢ univ),
-- check that all bits in this formula are `C^n`
have hf' := cont_mdiff_on_iff.1 hf,
have A : cont_diff_on 𝕜 m (I' ∘ f ∘ I.symm) (range I ∩ I.symm ⁻¹' s) :=
by simpa only with mfld_simps using (hf'.2 (I.symm 0) (I'.symm 0)).of_le m_le_n,
have B : cont_diff_on 𝕜 m ((I' ∘ f ∘ I.symm) ∘ prod.fst)
((range I ∩ I.symm ⁻¹' s) ×ˢ univ) :=
A.comp (cont_diff_fst.cont_diff_on) (prod_subset_preimage_fst _ _),
suffices C : cont_diff_on 𝕜 m (λ (p : E × E),
((fderiv_within 𝕜 (I' ∘ f ∘ I.symm) (I.symm ⁻¹' s ∩ range I) p.1 : _) p.2))
((range I ∩ I.symm ⁻¹' s) ×ˢ univ),
{ apply cont_diff_on.prod B _,
apply C.congr (λp hp, _),
simp only with mfld_simps at hp,
simp only [mfderiv_within, hf.mdifferentiable_on one_le_n _ hp.2, hp.1, dif_pos]
with mfld_simps },
have D : cont_diff_on 𝕜 m (λ x,
(fderiv_within 𝕜 (I' ∘ f ∘ I.symm) (I.symm ⁻¹' s ∩ range I) x))
(range I ∩ I.symm ⁻¹' s),
{ have : cont_diff_on 𝕜 n (I' ∘ f ∘ I.symm) (range I ∩ I.symm ⁻¹' s) :=
by simpa only with mfld_simps using (hf'.2 (I.symm 0) (I'.symm 0)),
simpa only [inter_comm] using this.fderiv_within U' hmn },
have := D.comp (cont_diff_fst.cont_diff_on) (prod_subset_preimage_fst _ _),
have := cont_diff_on.prod this (cont_diff_snd.cont_diff_on),
exact is_bounded_bilinear_map_apply.cont_diff.comp_cont_diff_on this,
end
include Is I's
/-- If a function is `C^n` on a domain with unique derivatives, then its bundled derivative
is `C^m` when `m+1 ≤ n`. -/
theorem cont_mdiff_on.cont_mdiff_on_tangent_map_within
(hf : cont_mdiff_on I I' n f s) (hmn : m + 1 ≤ n) (hs : unique_mdiff_on I s) :
cont_mdiff_on I.tangent I'.tangent m (tangent_map_within I I' f s)
((tangent_bundle.proj I M) ⁻¹' s) :=
begin
/- The strategy of the proof is to avoid unfolding the definitions, and reduce by functoriality
to the case of functions on the model spaces, where we have already proved the result.
Let `l` and `r` be the charts to the left and to the right, so that we have
```
l^{-1} f r
H --------> M ---> M' ---> H'
```
Then the tangent map `T(r ∘ f ∘ l)` is smooth by a previous result. Consider the composition
```
Tl T(r ∘ f ∘ l^{-1}) Tr^{-1}
TM -----> TH -------------------> TH' ---------> TM'
```
where `Tr^{-1}` and `Tl` are the tangent maps of `r^{-1}` and `l`. Writing `Tl` and `Tr^{-1}` as
composition of charts (called `Dl` and `il` for `l` and `Dr` and `ir` in the proof below), it
follows that they are smooth. The composition of all these maps is `Tf`, and is therefore smooth
as a composition of smooth maps.
-/
have m_le_n : m ≤ n,
{ apply le_trans _ hmn,
have : m + 0 ≤ m + 1 := add_le_add_left (zero_le _) _,
simpa only [add_zero] },
have one_le_n : 1 ≤ n,
{ apply le_trans _ hmn,
change 0 + 1 ≤ m + 1,
exact add_le_add_right (zero_le _) _ },
/- First step: local reduction on the space, to a set `s'` which is contained in chart domains. -/
refine cont_mdiff_on_of_locally_cont_mdiff_on (λp hp, _),
have hf' := cont_mdiff_on_iff.1 hf,
simp [tangent_bundle.proj] at hp,
let l := chart_at H p.1,
set Dl := chart_at (model_prod H E) p with hDl,
let r := chart_at H' (f p.1),
let Dr := chart_at (model_prod H' E') (tangent_map_within I I' f s p),
let il := chart_at (model_prod H E) (tangent_map I I l p),
let ir := chart_at (model_prod H' E') (tangent_map I I' (r ∘ f) p),
let s' := f ⁻¹' r.source ∩ s ∩ l.source,
let s'_lift := (tangent_bundle.proj I M)⁻¹' s',
let s'l := l.target ∩ l.symm ⁻¹' s',
let s'l_lift := (tangent_bundle.proj I H) ⁻¹' s'l,
rcases continuous_on_iff'.1 hf'.1 r.source r.open_source with ⟨o, o_open, ho⟩,
suffices h : cont_mdiff_on I.tangent I'.tangent m (tangent_map_within I I' f s) s'_lift,
{ refine ⟨(tangent_bundle.proj I M)⁻¹' (o ∩ l.source), _, _, _⟩,
show is_open ((tangent_bundle.proj I M)⁻¹' (o ∩ l.source)), from
(is_open.inter o_open l.open_source).preimage (tangent_bundle_proj_continuous _ _) ,
show p ∈ tangent_bundle.proj I M ⁻¹' (o ∩ l.source),
{ simp [tangent_bundle.proj] at ⊢,
have : p.1 ∈ f ⁻¹' r.source ∩ s, by simp [hp],
rw ho at this,
exact this.1 },
{ have : tangent_bundle.proj I M ⁻¹' s ∩ tangent_bundle.proj I M ⁻¹' (o ∩ l.source) = s'_lift,
{ dsimp only [s'_lift, s'], rw [ho], mfld_set_tac },
rw this,
exact h } },
/- Second step: check that all functions are smooth, and use the chain rule to write the bundled
derivative as a composition of a function between model spaces and of charts.
Convention: statements about the differentiability of `a ∘ b ∘ c` are named `diff_abc`. Statements
about differentiability in the bundle have a `_lift` suffix. -/
have U' : unique_mdiff_on I s',
{ apply unique_mdiff_on.inter _ l.open_source,
rw [ho, inter_comm],
exact hs.inter o_open },
have U'l : unique_mdiff_on I s'l :=
U'.unique_mdiff_on_preimage (mdifferentiable_chart _ _),
have diff_f : cont_mdiff_on I I' n f s' :=
hf.mono (by mfld_set_tac),
have diff_r : cont_mdiff_on I' I' n r r.source :=
cont_mdiff_on_chart,
have diff_rf : cont_mdiff_on I I' n (r ∘ f) s',
{ apply cont_mdiff_on.comp diff_r diff_f (λx hx, _),
simp only [s'] with mfld_simps at hx, simp only [hx] with mfld_simps },
have diff_l : cont_mdiff_on I I n l.symm s'l,
{ have A : cont_mdiff_on I I n l.symm l.target :=
cont_mdiff_on_chart_symm,
exact A.mono (by mfld_set_tac) },
have diff_rfl : cont_mdiff_on I I' n (r ∘ f ∘ l.symm) s'l,
{ apply cont_mdiff_on.comp diff_rf diff_l,
mfld_set_tac },
have diff_rfl_lift : cont_mdiff_on I.tangent I'.tangent m
(tangent_map_within I I' (r ∘ f ∘ l.symm) s'l) s'l_lift :=
diff_rfl.cont_mdiff_on_tangent_map_within_aux hmn U'l,
have diff_irrfl_lift : cont_mdiff_on I.tangent I'.tangent m
(ir ∘ (tangent_map_within I I' (r ∘ f ∘ l.symm) s'l)) s'l_lift,
{ have A : cont_mdiff_on I'.tangent I'.tangent m ir ir.source := cont_mdiff_on_chart,
exact cont_mdiff_on.comp A diff_rfl_lift (λp hp, by simp only [ir] with mfld_simps) },
have diff_Drirrfl_lift : cont_mdiff_on I.tangent I'.tangent m
(Dr.symm ∘ (ir ∘ (tangent_map_within I I' (r ∘ f ∘ l.symm) s'l))) s'l_lift,
{ have A : cont_mdiff_on I'.tangent I'.tangent m Dr.symm Dr.target :=
cont_mdiff_on_chart_symm,
apply cont_mdiff_on.comp A diff_irrfl_lift (λp hp, _),
simp only [s'l_lift, tangent_bundle.proj] with mfld_simps at hp,
simp only [ir, @local_equiv.refl_coe (model_prod H' E'), hp] with mfld_simps },
-- conclusion of this step: the composition of all the maps above is smooth
have diff_DrirrflilDl : cont_mdiff_on I.tangent I'.tangent m
(Dr.symm ∘ (ir ∘ (tangent_map_within I I' (r ∘ f ∘ l.symm) s'l)) ∘
(il.symm ∘ Dl)) s'_lift,
{ have A : cont_mdiff_on I.tangent I.tangent m Dl Dl.source := cont_mdiff_on_chart,
have A' : cont_mdiff_on I.tangent I.tangent m Dl s'_lift,
{ apply A.mono (λp hp, _),
simp only [s'_lift, tangent_bundle.proj] with mfld_simps at hp,
simp only [Dl, hp] with mfld_simps },
have B : cont_mdiff_on I.tangent I.tangent m il.symm il.target :=
cont_mdiff_on_chart_symm,
have C : cont_mdiff_on I.tangent I.tangent m (il.symm ∘ Dl) s'_lift :=
cont_mdiff_on.comp B A' (λp hp, by simp only [il] with mfld_simps),
apply cont_mdiff_on.comp diff_Drirrfl_lift C (λp hp, _),
simp only [s'_lift, tangent_bundle.proj] with mfld_simps at hp,
simp only [il, s'l_lift, hp, tangent_bundle.proj] with mfld_simps },
/- Third step: check that the composition of all the maps indeed coincides with the derivative we
are looking for -/
have eq_comp : ∀q ∈ s'_lift, tangent_map_within I I' f s q =
(Dr.symm ∘ ir ∘ (tangent_map_within I I' (r ∘ f ∘ l.symm) s'l) ∘
(il.symm ∘ Dl)) q,
{ assume q hq,
simp only [s'_lift, tangent_bundle.proj] with mfld_simps at hq,
have U'q : unique_mdiff_within_at I s' q.1,
by { apply U', simp only [hq, s'] with mfld_simps },
have U'lq : unique_mdiff_within_at I s'l (Dl q).1,
by { apply U'l, simp only [hq, s'l] with mfld_simps },
have A : tangent_map_within I I' ((r ∘ f) ∘ l.symm) s'l (il.symm (Dl q)) =
tangent_map_within I I' (r ∘ f) s' (tangent_map_within I I l.symm s'l (il.symm (Dl q))),
{ refine tangent_map_within_comp_at (il.symm (Dl q)) _ _ (λp hp, _) U'lq,
{ apply diff_rf.mdifferentiable_on one_le_n,
simp only [hq] with mfld_simps },
{ apply diff_l.mdifferentiable_on one_le_n,
simp only [s'l, hq] with mfld_simps },
{ simp only with mfld_simps at hp, simp only [hp] with mfld_simps } },
have B : tangent_map_within I I l.symm s'l (il.symm (Dl q)) = q,
{ have : tangent_map_within I I l.symm s'l (il.symm (Dl q))
= tangent_map I I l.symm (il.symm (Dl q)),
{ refine tangent_map_within_eq_tangent_map U'lq _,
refine mdifferentiable_at_atlas_symm _ (chart_mem_atlas _ _) _,
simp only [hq] with mfld_simps },
rw [this, tangent_map_chart_symm, hDl],
{ simp only [hq] with mfld_simps,
have : q ∈ (chart_at (model_prod H E) p).source, by simp only [hq] with mfld_simps,
exact (chart_at (model_prod H E) p).left_inv this },
{ simp only [hq] with mfld_simps } },
have C : tangent_map_within I I' (r ∘ f) s' q
= tangent_map_within I' I' r r.source (tangent_map_within I I' f s' q),
{ refine tangent_map_within_comp_at q _ _ (λr hr, _) U'q,
{ apply diff_r.mdifferentiable_on one_le_n,
simp only [hq] with mfld_simps },
{ apply diff_f.mdifferentiable_on one_le_n,
simp only [hq] with mfld_simps },
{ simp only [s'] with mfld_simps at hr,
simp only [hr] with mfld_simps } },
have D : Dr.symm (ir (tangent_map_within I' I' r r.source (tangent_map_within I I' f s' q)))
= tangent_map_within I I' f s' q,
{ have A : tangent_map_within I' I' r r.source (tangent_map_within I I' f s' q) =
tangent_map I' I' r (tangent_map_within I I' f s' q),
{ apply tangent_map_within_eq_tangent_map,
{ apply is_open.unique_mdiff_within_at _ r.open_source, simp [hq] },
{ refine mdifferentiable_at_atlas _ (chart_mem_atlas _ _) _,
simp only [hq] with mfld_simps } },
have : f p.1 = (tangent_map_within I I' f s p).1 := rfl,
rw [A],
dsimp [r, Dr],
rw [this, tangent_map_chart],
{ simp only [hq] with mfld_simps,
have : tangent_map_within I I' f s' q ∈
(chart_at (model_prod H' E') (tangent_map_within I I' f s p)).source,
by simp only [hq] with mfld_simps,
exact (chart_at (model_prod H' E') (tangent_map_within I I' f s p)).left_inv this },
{ simp only [hq] with mfld_simps } },
have E : tangent_map_within I I' f s' q = tangent_map_within I I' f s q,
{ refine tangent_map_within_subset (by mfld_set_tac) U'q _,
apply hf.mdifferentiable_on one_le_n,
simp only [hq] with mfld_simps },
simp only [(∘), A, B, C, D, E.symm] },
exact diff_DrirrflilDl.congr eq_comp,
end
/-- If a function is `C^n` on a domain with unique derivatives, with `1 ≤ n`, then its bundled
derivative is continuous there. -/
theorem cont_mdiff_on.continuous_on_tangent_map_within
(hf : cont_mdiff_on I I' n f s) (hmn : 1 ≤ n) (hs : unique_mdiff_on I s) :
continuous_on (tangent_map_within I I' f s) ((tangent_bundle.proj I M) ⁻¹' s) :=
begin
have : cont_mdiff_on I.tangent I'.tangent 0 (tangent_map_within I I' f s)
((tangent_bundle.proj I M) ⁻¹' s) :=
hf.cont_mdiff_on_tangent_map_within hmn hs,
exact this.continuous_on
end
/-- If a function is `C^n`, then its bundled derivative is `C^m` when `m+1 ≤ n`. -/
theorem cont_mdiff.cont_mdiff_tangent_map
(hf : cont_mdiff I I' n f) (hmn : m + 1 ≤ n) :
cont_mdiff I.tangent I'.tangent m (tangent_map I I' f) :=
begin
rw ← cont_mdiff_on_univ at hf ⊢,
convert hf.cont_mdiff_on_tangent_map_within hmn unique_mdiff_on_univ,
rw tangent_map_within_univ
end
/-- If a function is `C^n`, with `1 ≤ n`, then its bundled derivative is continuous. -/
theorem cont_mdiff.continuous_tangent_map
(hf : cont_mdiff I I' n f) (hmn : 1 ≤ n) :
continuous (tangent_map I I' f) :=
begin
rw ← cont_mdiff_on_univ at hf,
rw continuous_iff_continuous_on_univ,
convert hf.continuous_on_tangent_map_within hmn unique_mdiff_on_univ,
rw tangent_map_within_univ
end
end tangent_map
/-! ### Smoothness of the projection in a basic smooth bundle -/
namespace basic_smooth_vector_bundle_core
variables (Z : basic_smooth_vector_bundle_core I M E')
/-- A version of `cont_mdiff_at_iff_target` when the codomain is the total space of
a `basic_smooth_vector_bundle_core`. The continuity condition in the RHS is weaker. -/
lemma cont_mdiff_at_iff_target {f : N → Z.to_topological_vector_bundle_core.total_space}
{x : N} {n : ℕ∞} :
cont_mdiff_at J (I.prod 𝓘(𝕜, E')) n f x ↔ continuous_at (bundle.total_space.proj ∘ f) x ∧
cont_mdiff_at J 𝓘(𝕜, E × E') n (ext_chart_at (I.prod 𝓘(𝕜, E')) (f x) ∘ f) x :=
begin
let Z' := Z.to_topological_vector_bundle_core,
rw [cont_mdiff_at_iff_target, and.congr_left_iff],
refine λ hf, ⟨λ h, Z'.continuous_proj.continuous_at.comp h, λ h, _⟩,
exact (Z'.local_triv ⟨chart_at _ (f x).1, chart_mem_atlas _ _⟩).to_fiber_bundle_trivialization
.continuous_at_of_comp_left h (mem_chart_source _ _) (h.prod hf.continuous_at.snd)
end
lemma cont_mdiff_proj :
cont_mdiff (I.prod 𝓘(𝕜, E')) I n Z.to_topological_vector_bundle_core.proj :=
begin
assume x,
rw [cont_mdiff_at, cont_mdiff_within_at_iff'],
refine ⟨Z.to_topological_vector_bundle_core.continuous_proj.continuous_within_at, _⟩,
simp only [(∘), chart_at, chart] with mfld_simps,
apply cont_diff_within_at_fst.congr,
{ rintros ⟨a, b⟩ hab,
simp only with mfld_simps at hab,
simp only [hab] with mfld_simps },
{ simp only with mfld_simps }
end
lemma smooth_proj :
smooth (I.prod 𝓘(𝕜, E')) I Z.to_topological_vector_bundle_core.proj :=
cont_mdiff_proj Z
lemma cont_mdiff_on_proj {s : set (Z.to_topological_vector_bundle_core.total_space)} :
cont_mdiff_on (I.prod 𝓘(𝕜, E')) I n
Z.to_topological_vector_bundle_core.proj s :=
Z.cont_mdiff_proj.cont_mdiff_on
lemma smooth_on_proj {s : set (Z.to_topological_vector_bundle_core.total_space)} :
smooth_on (I.prod 𝓘(𝕜, E')) I Z.to_topological_vector_bundle_core.proj s :=
cont_mdiff_on_proj Z
lemma cont_mdiff_at_proj {p : Z.to_topological_vector_bundle_core.total_space} :
cont_mdiff_at (I.prod 𝓘(𝕜, E')) I n
Z.to_topological_vector_bundle_core.proj p :=
Z.cont_mdiff_proj.cont_mdiff_at
lemma smooth_at_proj {p : Z.to_topological_vector_bundle_core.total_space} :
smooth_at (I.prod 𝓘(𝕜, E')) I Z.to_topological_vector_bundle_core.proj p :=
Z.cont_mdiff_at_proj
lemma cont_mdiff_within_at_proj
{s : set (Z.to_topological_vector_bundle_core.total_space)}
{p : Z.to_topological_vector_bundle_core.total_space} :
cont_mdiff_within_at (I.prod 𝓘(𝕜, E')) I n
Z.to_topological_vector_bundle_core.proj s p :=
Z.cont_mdiff_at_proj.cont_mdiff_within_at
lemma smooth_within_at_proj
{s : set (Z.to_topological_vector_bundle_core.total_space)}
{p : Z.to_topological_vector_bundle_core.total_space} :
smooth_within_at (I.prod 𝓘(𝕜, E')) I
Z.to_topological_vector_bundle_core.proj s p :=
Z.cont_mdiff_within_at_proj
/-- If an element of `E'` is invariant under all coordinate changes, then one can define a
corresponding section of the fiber bundle, which is smooth. This applies in particular to the
zero section of a vector bundle. Another example (not yet defined) would be the identity
section of the endomorphism bundle of a vector bundle. -/
lemma smooth_const_section (v : E')
(h : ∀ (i j : atlas H M), ∀ x ∈ i.1.source ∩ j.1.source, Z.coord_change i j (i.1 x) v = v) :
smooth I (I.prod 𝓘(𝕜, E'))
(show M → Z.to_topological_vector_bundle_core.total_space, from λ x, ⟨x, v⟩) :=
begin
assume x,
rw [cont_mdiff_at, cont_mdiff_within_at_iff'],
split,
{ apply continuous.continuous_within_at,
apply topological_fiber_bundle_core.continuous_const_section,
assume i j y hy,
exact h _ _ _ hy },
{ have : cont_diff 𝕜 ⊤ (λ (y : E), (y, v)) := cont_diff_id.prod cont_diff_const,
apply this.cont_diff_within_at.congr,
{ assume y hy,
simp only with mfld_simps at hy,
simp only [chart, hy, chart_at, prod.mk.inj_iff, to_topological_vector_bundle_core]
with mfld_simps,
apply h,
simp only [hy, subtype.val_eq_coe] with mfld_simps,
exact mem_chart_source H (((chart_at H x).symm) ((model_with_corners.symm I) y)) },
{ simp only [chart, chart_at, prod.mk.inj_iff, to_topological_vector_bundle_core]
with mfld_simps,
apply h,
simp only [subtype.val_eq_coe] with mfld_simps,
exact mem_chart_source H x, } }
end
end basic_smooth_vector_bundle_core
/-! ### Smoothness of the tangent bundle projection -/
namespace tangent_bundle
include Is
lemma cont_mdiff_proj :
cont_mdiff I.tangent I n (proj I M) :=
basic_smooth_vector_bundle_core.cont_mdiff_proj _
lemma smooth_proj : smooth I.tangent I (proj I M) :=
basic_smooth_vector_bundle_core.smooth_proj _
lemma cont_mdiff_on_proj {s : set (tangent_bundle I M)} :
cont_mdiff_on I.tangent I n (proj I M) s :=
basic_smooth_vector_bundle_core.cont_mdiff_on_proj _
lemma smooth_on_proj {s : set (tangent_bundle I M)} :
smooth_on I.tangent I (proj I M) s :=
basic_smooth_vector_bundle_core.smooth_on_proj _
lemma cont_mdiff_at_proj {p : tangent_bundle I M} :
cont_mdiff_at I.tangent I n
(proj I M) p :=
basic_smooth_vector_bundle_core.cont_mdiff_at_proj _
lemma smooth_at_proj {p : tangent_bundle I M} :
smooth_at I.tangent I (proj I M) p :=
basic_smooth_vector_bundle_core.smooth_at_proj _
lemma cont_mdiff_within_at_proj
{s : set (tangent_bundle I M)} {p : tangent_bundle I M} :
cont_mdiff_within_at I.tangent I n
(proj I M) s p :=
basic_smooth_vector_bundle_core.cont_mdiff_within_at_proj _
lemma smooth_within_at_proj
{s : set (tangent_bundle I M)} {p : tangent_bundle I M} :
smooth_within_at I.tangent I
(proj I M) s p :=
basic_smooth_vector_bundle_core.smooth_within_at_proj _
variables (I M)
/-- The zero section of the tangent bundle -/
def zero_section : M → tangent_bundle I M := λ x, ⟨x, 0⟩
variables {I M}
lemma smooth_zero_section : smooth I I.tangent (zero_section I M) :=
begin
apply basic_smooth_vector_bundle_core.smooth_const_section (tangent_bundle_core I M) 0,
assume i j x hx,
simp only [tangent_bundle_core, continuous_linear_map.map_zero, continuous_linear_map.coe_coe]
with mfld_simps,
end
open bundle
/-- The derivative of the zero section of the tangent bundle maps `⟨x, v⟩` to `⟨⟨x, 0⟩, ⟨v, 0⟩⟩`.
Note that, as currently framed, this is a statement in coordinates, thus reliant on the choice
of the coordinate system we use on the tangent bundle.
However, the result itself is coordinate-dependent only to the extent that the coordinates
determine a splitting of the tangent bundle. Moreover, there is a canonical splitting at each
point of the zero section (since there is a canonical horizontal space there, the tangent space
to the zero section, in addition to the canonical vertical space which is the kernel of the
derivative of the projection), and this canonical splitting is also the one that comes from the
coordinates on the tangent bundle in our definitions. So this statement is not as crazy as it
may seem.
TODO define splittings of vector bundles; state this result invariantly. -/
lemma tangent_map_tangent_bundle_pure (p : tangent_bundle I M) :
tangent_map I I.tangent (tangent_bundle.zero_section I M) p = ⟨⟨p.1, 0⟩, ⟨p.2, 0⟩⟩ :=
begin
rcases p with ⟨x, v⟩,
have N : I.symm ⁻¹' (chart_at H x).target ∈ 𝓝 (I ((chart_at H x) x)),
{ apply is_open.mem_nhds,
apply (local_homeomorph.open_target _).preimage I.continuous_inv_fun,
simp only with mfld_simps },
have A : mdifferentiable_at I I.tangent (λ x, @total_space_mk M (tangent_space I) x 0) x :=
tangent_bundle.smooth_zero_section.mdifferentiable_at,
have B : fderiv_within 𝕜 (λ (x_1 : E), (x_1, (0 : E))) (set.range ⇑I) (I ((chart_at H x) x)) v
= (v, 0),
{ rw [fderiv_within_eq_fderiv, differentiable_at.fderiv_prod],
{ simp },
{ exact differentiable_at_id' },
{ exact differentiable_at_const _ },
{ exact model_with_corners.unique_diff_at_image I },
{ exact differentiable_at_id'.prod (differentiable_at_const _) } },
simp only [tangent_bundle.zero_section, tangent_map, mfderiv,
A, dif_pos, chart_at, basic_smooth_vector_bundle_core.chart,
basic_smooth_vector_bundle_core.to_topological_vector_bundle_core, tangent_bundle_core,
function.comp, continuous_linear_map.map_zero] with mfld_simps,
rw ← fderiv_within_inter N (I.unique_diff (I ((chart_at H x) x)) (set.mem_range_self _)) at B,
rw [← fderiv_within_inter N (I.unique_diff (I ((chart_at H x) x)) (set.mem_range_self _)), ← B],
congr' 2,
apply fderiv_within_congr _ (λ y hy, _),
{ simp only [prod.mk.inj_iff] with mfld_simps,
exact ((tangent_bundle_core I M).to_topological_vector_bundle_core.coord_change
((tangent_bundle_core I M).to_topological_vector_bundle_core.index_at (((chart_at H x).symm)
(I.symm (I ((chart_at H x) x))))) ⟨chart_at H x, _⟩ (((chart_at H x).symm)
(I.symm (I ((chart_at H x) x))))).map_zero, },
{ apply unique_diff_within_at.inter (I.unique_diff _ _) N,
simp only with mfld_simps },
{ simp only with mfld_simps at hy,
simp only [hy, prod.mk.inj_iff] with mfld_simps,
exact ((tangent_bundle_core I M).to_topological_vector_bundle_core.coord_change
((tangent_bundle_core I M).to_topological_vector_bundle_core.index_at (((chart_at H x).symm)
(I.symm y))) ⟨chart_at H x, _⟩ (((chart_at H x).symm) (I.symm y))).map_zero, },
end
end tangent_bundle
/-! ### Smoothness of standard maps associated to the product of manifolds -/
section prod_mk
lemma cont_mdiff_within_at.prod_mk {f : M → M'} {g : M → N'}
(hf : cont_mdiff_within_at I I' n f s x) (hg : cont_mdiff_within_at I J' n g s x) :
cont_mdiff_within_at I (I'.prod J') n (λ x, (f x, g x)) s x :=
begin
rw cont_mdiff_within_at_iff at *,
exact ⟨hf.1.prod hg.1, hf.2.prod hg.2⟩,
end
lemma cont_mdiff_within_at.prod_mk_space {f : M → E'} {g : M → F'}
(hf : cont_mdiff_within_at I 𝓘(𝕜, E') n f s x)
(hg : cont_mdiff_within_at I 𝓘(𝕜, F') n g s x) :
cont_mdiff_within_at I 𝓘(𝕜, E' × F') n (λ x, (f x, g x)) s x :=
begin
rw cont_mdiff_within_at_iff at *,
exact ⟨hf.1.prod hg.1, hf.2.prod hg.2⟩,
end
lemma cont_mdiff_at.prod_mk {f : M → M'} {g : M → N'}
(hf : cont_mdiff_at I I' n f x) (hg : cont_mdiff_at I J' n g x) :
cont_mdiff_at I (I'.prod J') n (λ x, (f x, g x)) x :=
hf.prod_mk hg
lemma cont_mdiff_at.prod_mk_space {f : M → E'} {g : M → F'}
(hf : cont_mdiff_at I 𝓘(𝕜, E') n f x) (hg : cont_mdiff_at I 𝓘(𝕜, F') n g x) :
cont_mdiff_at I 𝓘(𝕜, E' × F') n (λ x, (f x, g x)) x :=
hf.prod_mk_space hg
lemma cont_mdiff_on.prod_mk {f : M → M'} {g : M → N'}
(hf : cont_mdiff_on I I' n f s) (hg : cont_mdiff_on I J' n g s) :
cont_mdiff_on I (I'.prod J') n (λ x, (f x, g x)) s :=
λ x hx, (hf x hx).prod_mk (hg x hx)
lemma cont_mdiff_on.prod_mk_space {f : M → E'} {g : M → F'}
(hf : cont_mdiff_on I 𝓘(𝕜, E') n f s) (hg : cont_mdiff_on I 𝓘(𝕜, F') n g s) :
cont_mdiff_on I 𝓘(𝕜, E' × F') n (λ x, (f x, g x)) s :=
λ x hx, (hf x hx).prod_mk_space (hg x hx)
lemma cont_mdiff.prod_mk {f : M → M'} {g : M → N'}
(hf : cont_mdiff I I' n f) (hg : cont_mdiff I J' n g) :
cont_mdiff I (I'.prod J') n (λ x, (f x, g x)) :=
λ x, (hf x).prod_mk (hg x)
lemma cont_mdiff.prod_mk_space {f : M → E'} {g : M → F'}
(hf : cont_mdiff I 𝓘(𝕜, E') n f) (hg : cont_mdiff I 𝓘(𝕜, F') n g) :
cont_mdiff I 𝓘(𝕜, E' × F') n (λ x, (f x, g x)) :=
λ x, (hf x).prod_mk_space (hg x)
lemma smooth_within_at.prod_mk {f : M → M'} {g : M → N'}
(hf : smooth_within_at I I' f s x) (hg : smooth_within_at I J' g s x) :
smooth_within_at I (I'.prod J') (λ x, (f x, g x)) s x :=
hf.prod_mk hg
lemma smooth_within_at.prod_mk_space {f : M → E'} {g : M → F'}
(hf : smooth_within_at I 𝓘(𝕜, E') f s x) (hg : smooth_within_at I 𝓘(𝕜, F') g s x) :
smooth_within_at I 𝓘(𝕜, E' × F') (λ x, (f x, g x)) s x :=
hf.prod_mk_space hg
lemma smooth_at.prod_mk {f : M → M'} {g : M → N'}
(hf : smooth_at I I' f x) (hg : smooth_at I J' g x) :
smooth_at I (I'.prod J') (λ x, (f x, g x)) x :=
hf.prod_mk hg
lemma smooth_at.prod_mk_space {f : M → E'} {g : M → F'}
(hf : smooth_at I 𝓘(𝕜, E') f x) (hg : smooth_at I 𝓘(𝕜, F') g x) :
smooth_at I 𝓘(𝕜, E' × F') (λ x, (f x, g x)) x :=
hf.prod_mk_space hg
lemma smooth_on.prod_mk {f : M → M'} {g : M → N'}
(hf : smooth_on I I' f s) (hg : smooth_on I J' g s) :
smooth_on I (I'.prod J') (λ x, (f x, g x)) s :=
hf.prod_mk hg
lemma smooth_on.prod_mk_space {f : M → E'} {g : M → F'}
(hf : smooth_on I 𝓘(𝕜, E') f s) (hg : smooth_on I 𝓘(𝕜, F') g s) :
smooth_on I 𝓘(𝕜, E' × F') (λ x, (f x, g x)) s :=
hf.prod_mk_space hg
lemma smooth.prod_mk {f : M → M'} {g : M → N'}
(hf : smooth I I' f) (hg : smooth I J' g) :
smooth I (I'.prod J') (λ x, (f x, g x)) :=
hf.prod_mk hg
lemma smooth.prod_mk_space {f : M → E'} {g : M → F'}
(hf : smooth I 𝓘(𝕜, E') f) (hg : smooth I 𝓘(𝕜, F') g) :
smooth I 𝓘(𝕜, E' × F') (λ x, (f x, g x)) :=
hf.prod_mk_space hg
end prod_mk
section projections
lemma cont_mdiff_within_at_fst {s : set (M × N)} {p : M × N} :
cont_mdiff_within_at (I.prod J) I n prod.fst s p :=
begin
rw cont_mdiff_within_at_iff',
refine ⟨continuous_within_at_fst, _⟩,
refine cont_diff_within_at_fst.congr (λ y hy, _) _,
{ simp only with mfld_simps at hy,
simp only [hy] with mfld_simps },
{ simp only with mfld_simps }
end
lemma cont_mdiff_at_fst {p : M × N} :
cont_mdiff_at (I.prod J) I n prod.fst p :=
cont_mdiff_within_at_fst
lemma cont_mdiff_on_fst {s : set (M × N)} :
cont_mdiff_on (I.prod J) I n prod.fst s :=
λ x hx, cont_mdiff_within_at_fst
lemma cont_mdiff_fst :
cont_mdiff (I.prod J) I n (@prod.fst M N) :=
λ x, cont_mdiff_at_fst
lemma smooth_within_at_fst {s : set (M × N)} {p : M × N} :
smooth_within_at (I.prod J) I prod.fst s p :=
cont_mdiff_within_at_fst
lemma smooth_at_fst {p : M × N} :
smooth_at (I.prod J) I prod.fst p :=
cont_mdiff_at_fst
lemma smooth_on_fst {s : set (M × N)} :
smooth_on (I.prod J) I prod.fst s :=
cont_mdiff_on_fst
lemma smooth_fst :
smooth (I.prod J) I (@prod.fst M N) :=
cont_mdiff_fst
lemma cont_mdiff_within_at_snd {s : set (M × N)} {p : M × N} :
cont_mdiff_within_at (I.prod J) J n prod.snd s p :=
begin
rw cont_mdiff_within_at_iff',
refine ⟨continuous_within_at_snd, _⟩,
refine cont_diff_within_at_snd.congr (λ y hy, _) _,
{ simp only with mfld_simps at hy,
simp only [hy] with mfld_simps },
{ simp only with mfld_simps }
end
lemma cont_mdiff_at_snd {p : M × N} :
cont_mdiff_at (I.prod J) J n prod.snd p :=
cont_mdiff_within_at_snd
lemma cont_mdiff_on_snd {s : set (M × N)} :
cont_mdiff_on (I.prod J) J n prod.snd s :=
λ x hx, cont_mdiff_within_at_snd
lemma cont_mdiff_snd :
cont_mdiff (I.prod J) J n (@prod.snd M N) :=
λ x, cont_mdiff_at_snd
lemma smooth_within_at_snd {s : set (M × N)} {p : M × N} :
smooth_within_at (I.prod J) J prod.snd s p :=
cont_mdiff_within_at_snd
lemma smooth_at_snd {p : M × N} :
smooth_at (I.prod J) J prod.snd p :=
cont_mdiff_at_snd
lemma smooth_on_snd {s : set (M × N)} :
smooth_on (I.prod J) J prod.snd s :=
cont_mdiff_on_snd
lemma smooth_snd :
smooth (I.prod J) J (@prod.snd M N) :=
cont_mdiff_snd
lemma smooth_iff_proj_smooth {f : M → M' × N'} :
(smooth I (I'.prod J') f) ↔ (smooth I I' (prod.fst ∘ f)) ∧ (smooth I J' (prod.snd ∘ f)) :=
begin
split,
{ intro h, exact ⟨smooth_fst.comp h, smooth_snd.comp h⟩ },
{ rintro ⟨h_fst, h_snd⟩, simpa only [prod.mk.eta] using h_fst.prod_mk h_snd, }
end
end projections
section prod_map
variables {g : N → N'} {r : set N} {y : N}
/-- The product map of two `C^n` functions within a set at a point is `C^n`
within the product set at the product point. -/
lemma cont_mdiff_within_at.prod_map' {p : M × N}
(hf : cont_mdiff_within_at I I' n f s p.1)
(hg : cont_mdiff_within_at J J' n g r p.2) :
cont_mdiff_within_at (I.prod J) (I'.prod J') n (prod.map f g) (s ×ˢ r) p :=
(hf.comp p cont_mdiff_within_at_fst (prod_subset_preimage_fst _ _)).prod_mk $
hg.comp p cont_mdiff_within_at_snd (prod_subset_preimage_snd _ _)
lemma cont_mdiff_within_at.prod_map
(hf : cont_mdiff_within_at I I' n f s x) (hg : cont_mdiff_within_at J J' n g r y) :
cont_mdiff_within_at (I.prod J) (I'.prod J') n (prod.map f g) (s ×ˢ r) (x, y) :=
cont_mdiff_within_at.prod_map' hf hg
lemma cont_mdiff_at.prod_map
(hf : cont_mdiff_at I I' n f x) (hg : cont_mdiff_at J J' n g y) :
cont_mdiff_at (I.prod J) (I'.prod J') n (prod.map f g) (x, y) :=
begin
rw ← cont_mdiff_within_at_univ at *,
convert hf.prod_map hg,
exact univ_prod_univ.symm
end
lemma cont_mdiff_at.prod_map' {p : M × N}
(hf : cont_mdiff_at I I' n f p.1) (hg : cont_mdiff_at J J' n g p.2) :
cont_mdiff_at (I.prod J) (I'.prod J') n (prod.map f g) p :=
begin
rcases p,
exact hf.prod_map hg
end
lemma cont_mdiff_on.prod_map
(hf : cont_mdiff_on I I' n f s) (hg : cont_mdiff_on J J' n g r) :
cont_mdiff_on (I.prod J) (I'.prod J') n (prod.map f g) (s ×ˢ r) :=
(hf.comp cont_mdiff_on_fst (prod_subset_preimage_fst _ _)).prod_mk $
hg.comp (cont_mdiff_on_snd) (prod_subset_preimage_snd _ _)
lemma cont_mdiff.prod_map
(hf : cont_mdiff I I' n f) (hg : cont_mdiff J J' n g) :
cont_mdiff (I.prod J) (I'.prod J') n (prod.map f g) :=
begin
assume p,
exact (hf p.1).prod_map' (hg p.2)
end
lemma smooth_within_at.prod_map
(hf : smooth_within_at I I' f s x) (hg : smooth_within_at J J' g r y) :
smooth_within_at (I.prod J) (I'.prod J') (prod.map f g) (s ×ˢ r) (x, y) :=
hf.prod_map hg
lemma smooth_at.prod_map
(hf : smooth_at I I' f x) (hg : smooth_at J J' g y) :
smooth_at (I.prod J) (I'.prod J') (prod.map f g) (x, y) :=
hf.prod_map hg
lemma smooth_on.prod_map
(hf : smooth_on I I' f s) (hg : smooth_on J J' g r) :
smooth_on (I.prod J) (I'.prod J') (prod.map f g) (s ×ˢ r) :=
hf.prod_map hg
lemma smooth.prod_map
(hf : smooth I I' f) (hg : smooth J J' g) :
smooth (I.prod J) (I'.prod J') (prod.map f g) :=
hf.prod_map hg
end prod_map
section pi_space
/-!
### Smoothness of functions with codomain `Π i, F i`
We have no `model_with_corners.pi` yet, so we prove lemmas about functions `f : M → Π i, F i` and
use `𝓘(𝕜, Π i, F i)` as the model space.
-/
variables {ι : Type*} [fintype ι] {Fi : ι → Type*} [Π i, normed_add_comm_group (Fi i)]
[Π i, normed_space 𝕜 (Fi i)] {φ : M → Π i, Fi i}
lemma cont_mdiff_within_at_pi_space :
cont_mdiff_within_at I (𝓘(𝕜, Π i, Fi i)) n φ s x ↔
∀ i, cont_mdiff_within_at I (𝓘(𝕜, Fi i)) n (λ x, φ x i) s x :=
by simp only [cont_mdiff_within_at_iff, continuous_within_at_pi,
cont_diff_within_at_pi, forall_and_distrib, written_in_ext_chart_at,
ext_chart_model_space_eq_id, (∘), local_equiv.refl_coe, id]
lemma cont_mdiff_on_pi_space :
cont_mdiff_on I (𝓘(𝕜, Π i, Fi i)) n φ s ↔
∀ i, cont_mdiff_on I (𝓘(𝕜, Fi i)) n (λ x, φ x i) s :=
⟨λ h i x hx, cont_mdiff_within_at_pi_space.1 (h x hx) i,
λ h x hx, cont_mdiff_within_at_pi_space.2 (λ i, h i x hx)⟩
lemma cont_mdiff_at_pi_space :
cont_mdiff_at I (𝓘(𝕜, Π i, Fi i)) n φ x ↔
∀ i, cont_mdiff_at I (𝓘(𝕜, Fi i)) n (λ x, φ x i) x :=
cont_mdiff_within_at_pi_space
lemma cont_mdiff_pi_space :
cont_mdiff I (𝓘(𝕜, Π i, Fi i)) n φ ↔
∀ i, cont_mdiff I (𝓘(𝕜, Fi i)) n (λ x, φ x i) :=
⟨λ h i x, cont_mdiff_at_pi_space.1 (h x) i,
λ h x, cont_mdiff_at_pi_space.2 (λ i, h i x)⟩
lemma smooth_within_at_pi_space :
smooth_within_at I (𝓘(𝕜, Π i, Fi i)) φ s x ↔
∀ i, smooth_within_at I (𝓘(𝕜, Fi i)) (λ x, φ x i) s x :=
cont_mdiff_within_at_pi_space
lemma smooth_on_pi_space :
smooth_on I (𝓘(𝕜, Π i, Fi i)) φ s ↔ ∀ i, smooth_on I (𝓘(𝕜, Fi i)) (λ x, φ x i) s :=
cont_mdiff_on_pi_space
lemma smooth_at_pi_space :
smooth_at I (𝓘(𝕜, Π i, Fi i)) φ x ↔ ∀ i, smooth_at I (𝓘(𝕜, Fi i)) (λ x, φ x i) x :=
cont_mdiff_at_pi_space
lemma smooth_pi_space :
smooth I (𝓘(𝕜, Π i, Fi i)) φ ↔ ∀ i, smooth I (𝓘(𝕜, Fi i)) (λ x, φ x i) :=
cont_mdiff_pi_space
end pi_space
/-! ### Linear maps between normed spaces are smooth -/
lemma continuous_linear_map.cont_mdiff (L : E →L[𝕜] F) :
cont_mdiff 𝓘(𝕜, E) 𝓘(𝕜, F) n L :=
L.cont_diff.cont_mdiff
/-! ### Smoothness of standard operations -/
variables {V : Type*} [normed_add_comm_group V] [normed_space 𝕜 V]
/-- On any vector space, multiplication by a scalar is a smooth operation. -/
lemma smooth_smul : smooth (𝓘(𝕜).prod 𝓘(𝕜, V)) 𝓘(𝕜, V) (λp : 𝕜 × V, p.1 • p.2) :=
smooth_iff.2 ⟨continuous_smul, λ x y, cont_diff_smul.cont_diff_on⟩
lemma cont_mdiff_within_at.smul {f : M → 𝕜} {g : M → V} (hf : cont_mdiff_within_at I 𝓘(𝕜) n f s x)
(hg : cont_mdiff_within_at I 𝓘(𝕜, V) n g s x) :
cont_mdiff_within_at I 𝓘(𝕜, V) n (λ p, f p • g p) s x :=
(smooth_smul.of_le le_top).cont_mdiff_at.comp_cont_mdiff_within_at x (hf.prod_mk hg)
lemma cont_mdiff_at.smul {f : M → 𝕜} {g : M → V} (hf : cont_mdiff_at I 𝓘(𝕜) n f x)
(hg : cont_mdiff_at I 𝓘(𝕜, V) n g x) :
cont_mdiff_at I 𝓘(𝕜, V) n (λ p, f p • g p) x :=
hf.smul hg
lemma cont_mdiff_on.smul {f : M → 𝕜} {g : M → V} (hf : cont_mdiff_on I 𝓘(𝕜) n f s)
(hg : cont_mdiff_on I 𝓘(𝕜, V) n g s) :
cont_mdiff_on I 𝓘(𝕜, V) n (λ p, f p • g p) s :=
λ x hx, (hf x hx).smul (hg x hx)
lemma cont_mdiff.smul {f : M → 𝕜} {g : M → V} (hf : cont_mdiff I 𝓘(𝕜) n f)
(hg : cont_mdiff I 𝓘(𝕜, V) n g) :
cont_mdiff I 𝓘(𝕜, V) n (λ p, f p • g p) :=
λ x, (hf x).smul (hg x)
lemma smooth_within_at.smul {f : M → 𝕜} {g : M → V} (hf : smooth_within_at I 𝓘(𝕜) f s x)
(hg : smooth_within_at I 𝓘(𝕜, V) g s x) :
smooth_within_at I 𝓘(𝕜, V) (λ p, f p • g p) s x :=
hf.smul hg
lemma smooth_at.smul {f : M → 𝕜} {g : M → V} (hf : smooth_at I 𝓘(𝕜) f x)
(hg : smooth_at I 𝓘(𝕜, V) g x) :
smooth_at I 𝓘(𝕜, V) (λ p, f p • g p) x :=
hf.smul hg
lemma smooth_on.smul {f : M → 𝕜} {g : M → V} (hf : smooth_on I 𝓘(𝕜) f s)
(hg : smooth_on I 𝓘(𝕜, V) g s) :
smooth_on I 𝓘(𝕜, V) (λ p, f p • g p) s :=
hf.smul hg
lemma smooth.smul {f : M → 𝕜} {g : M → V} (hf : smooth I 𝓘(𝕜) f) (hg : smooth I 𝓘(𝕜, V) g) :
smooth I 𝓘(𝕜, V) (λ p, f p • g p) :=
hf.smul hg
|
6d68220135e630aa35821453eb4387d8361bb5f8 | 57c233acf9386e610d99ed20ef139c5f97504ba3 | /src/linear_algebra/orientation.lean | ebc1425acc5d6dc7554b75860afeac820e11038d | [
"Apache-2.0"
] | permissive | robertylewis/mathlib | 3d16e3e6daf5ddde182473e03a1b601d2810952c | 1d13f5b932f5e40a8308e3840f96fc882fae01f0 | refs/heads/master | 1,651,379,945,369 | 1,644,276,960,000 | 1,644,276,960,000 | 98,875,504 | 0 | 0 | Apache-2.0 | 1,644,253,514,000 | 1,501,495,700,000 | Lean | UTF-8 | Lean | false | false | 28,617 | lean | /-
Copyright (c) 2021 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import linear_algebra.determinant
/-!
# Orientations of modules and rays in modules
This file defines rays in modules and orientations of modules.
## Main definitions
* `module.ray` is a type for the equivalence class of nonzero vectors in a module with some
common positive multiple.
* `orientation` is a type synonym for `module.ray` for the case where the module is that of
alternating maps from a module to its underlying ring. An orientation may be associated with an
alternating map or with a basis.
* `module.oriented` is a type class for a choice of orientation of a module that is considered
the positive orientation.
## Implementation notes
`orientation` is defined for an arbitrary index type, but the main intended use case is when
that index type is a `fintype` and there exists a basis of the same cardinality.
## References
* https://en.wikipedia.org/wiki/Orientation_(vector_space)
-/
noncomputable theory
open_locale big_operators
section ordered_comm_semiring
variables (R : Type*) [ordered_comm_semiring R]
variables {M : Type*} [add_comm_monoid M] [module R M]
variables {N : Type*} [add_comm_monoid N] [module R N]
variables (ι : Type*) [decidable_eq ι]
/-- Two vectors are in the same ray if some positive multiples of them are equal (in the typical
case over a field, this means each is a positive multiple of the other). Over a field, this
is equivalent to `mul_action.orbit_rel`. -/
def same_ray (v₁ v₂ : M) : Prop :=
∃ (r₁ r₂ : R), 0 < r₁ ∧ 0 < r₂ ∧ r₁ • v₁ = r₂ • v₂
variables {R}
/-- `same_ray` is reflexive. -/
@[refl] lemma same_ray.refl [nontrivial R] (x : M) : same_ray R x x :=
⟨1, 1, zero_lt_one, zero_lt_one, rfl⟩
/-- `same_ray` is symmetric. -/
@[symm] lemma same_ray.symm {x y : M} : same_ray R x y → same_ray R y x :=
λ ⟨r₁, r₂, hr₁, hr₂, h⟩, ⟨r₂, r₁, hr₂, hr₁, h.symm⟩
/-- `same_ray` is transitive. -/
@[trans] lemma same_ray.trans {x y z : M} : same_ray R x y → same_ray R y z → same_ray R x z :=
λ ⟨r₁, r₂, hr₁, hr₂, h₁⟩ ⟨r₃, r₄, hr₃, hr₄, h₂⟩,
⟨r₃ * r₁, r₂ * r₄, mul_pos hr₃ hr₁, mul_pos hr₂ hr₄,
by rw [mul_smul, mul_smul, h₁, ←h₂, smul_comm]⟩
lemma same_ray_comm {x y : M} : same_ray R x y ↔ same_ray R y x :=
⟨same_ray.symm, same_ray.symm⟩
variables (R M)
/-- `same_ray` is an equivalence relation. -/
lemma equivalence_same_ray [nontrivial R] : equivalence (same_ray R : M → M → Prop) :=
⟨same_ray.refl, λ _ _, same_ray.symm, λ _ _ _, same_ray.trans⟩
variables {R M}
/-- A vector is in the same ray as a positive multiple of itself. -/
lemma same_ray_pos_smul_right (v : M) {r : R} (h : 0 < r) : same_ray R v (r • v) :=
⟨r, 1, h, let f := nontrivial_of_lt _ _ h in by exactI zero_lt_one, (one_smul _ _).symm⟩
/-- A vector is in the same ray as a positive multiple of one it is in the same ray as. -/
lemma same_ray.pos_smul_right {v₁ v₂ : M} {r : R} (h : same_ray R v₁ v₂) (hr : 0 < r) :
same_ray R v₁ (r • v₂) :=
h.trans (same_ray_pos_smul_right v₂ hr)
/-- A positive multiple of a vector is in the same ray as that vector. -/
lemma same_ray_pos_smul_left (v : M) {r : R} (h : 0 < r) : same_ray R (r • v) v :=
⟨1, r, let f := nontrivial_of_lt _ _ h in by exactI zero_lt_one, h, one_smul _ _⟩
/-- A positive multiple of a vector is in the same ray as one it is in the same ray as. -/
lemma same_ray.pos_smul_left {v₁ v₂ : M} {r : R} (h : same_ray R v₁ v₂) (hr : 0 < r) :
same_ray R (r • v₁) v₂ :=
(same_ray_pos_smul_left v₁ hr).trans h
/-- If two vectors are on the same ray then they remain so after appling a linear map. -/
lemma same_ray.map {v₁ v₂ : M} (f : M →ₗ[R] N)
(h : same_ray R v₁ v₂) : same_ray R (f v₁) (f v₂) :=
let ⟨r₁, r₂, hr₁, hr₂, h⟩ := h in
⟨r₁, r₂, hr₁, hr₂, by rw [←f.map_smul, ←f.map_smul, h]⟩
/-- If two vectors are on the same ray then they remain so after appling a linear equivalence. -/
@[simp] lemma same_ray_map_iff {v₁ v₂ : M} (e : M ≃ₗ[R] N) :
same_ray R (e v₁) (e v₂) ↔ same_ray R v₁ v₂ :=
⟨λ h, by simpa using same_ray.map e.symm.to_linear_map h, same_ray.map e.to_linear_map⟩
/-- If two vectors are on the same ray then both scaled by the same action are also on the same
ray. -/
lemma same_ray.smul {S : Type*} [has_scalar S M] [smul_comm_class R S M] {v₁ v₂ : M} (s : S)
(h : same_ray R v₁ v₂) : same_ray R (s • v₁) (s • v₂) :=
let ⟨r₁, r₂, hr₁, hr₂, h⟩ := h in
⟨r₁, r₂, hr₁, hr₂, by rw [smul_comm r₁ s v₁, smul_comm r₂ s v₂, h]⟩
variables (R M)
/-- The setoid of the `same_ray` relation for elements of a module. -/
def same_ray_setoid [nontrivial R] : setoid M :=
{ r := same_ray R, iseqv := equivalence_same_ray R M }
/-- Nonzero vectors, as used to define rays. -/
@[reducible] def ray_vector := {v : M // v ≠ 0}
/-- The setoid of the `same_ray` relation for the subtype of nonzero vectors. -/
def ray_vector.same_ray_setoid [nontrivial R] : setoid (ray_vector M) :=
(same_ray_setoid R M).comap coe
local attribute [instance] ray_vector.same_ray_setoid
variables {R M}
/-- Equivalence of nonzero vectors, in terms of same_ray. -/
lemma equiv_iff_same_ray [nontrivial R] (v₁ v₂ : ray_vector M) :
v₁ ≈ v₂ ↔ same_ray R (v₁ : M) v₂ :=
iff.rfl
variables (R M)
/-- A ray (equivalence class of nonzero vectors with common positive multiples) in a module. -/
@[nolint has_inhabited_instance]
def module.ray [nontrivial R] := quotient (ray_vector.same_ray_setoid R M)
/-- An orientation of a module, intended to be used when `ι` is a `fintype` with the same
cardinality as a basis. -/
abbreviation orientation [nontrivial R] := module.ray R (alternating_map R M R ι)
/-- A type class fixing an orientation of a module. -/
class module.oriented [nontrivial R] :=
(positive_orientation : orientation R M ι)
variables {M}
/-- The ray given by a nonzero vector. -/
protected def ray_of_ne_zero [nontrivial R] (v : M) (h : v ≠ 0) : module.ray R M :=
⟦⟨v, h⟩⟧
/-- An induction principle for `module.ray`, used as `induction x using module.ray.ind`. -/
lemma module.ray.ind [nontrivial R] {C : module.ray R M → Prop}
(h : Π v (hv : v ≠ 0), C (ray_of_ne_zero R v hv)) (x : module.ray R M) : C x :=
quotient.ind (subtype.rec $ by exact h) x
/-- The rays given by two nonzero vectors are equal if and only if those vectors
satisfy `same_ray`. -/
lemma ray_eq_iff [nontrivial R] {v₁ v₂ : M} (hv₁ : v₁ ≠ 0) (hv₂ : v₂ ≠ 0) :
ray_of_ne_zero R _ hv₁ = ray_of_ne_zero R _ hv₂ ↔ same_ray R v₁ v₂ :=
quotient.eq
variables {R}
/-- The ray given by a positive multiple of a nonzero vector. -/
@[simp] lemma ray_pos_smul [nontrivial R] {v : M} (h : v ≠ 0) {r : R} (hr : 0 < r)
(hrv : r • v ≠ 0) : ray_of_ne_zero R _ hrv = ray_of_ne_zero R _ h :=
begin
rw ray_eq_iff,
exact same_ray_pos_smul_left v hr
end
/-- An equivalence between modules implies an equivalence between ray vectors. -/
def ray_vector.map_linear_equiv (e : M ≃ₗ[R] N) : ray_vector M ≃ ray_vector N :=
equiv.subtype_equiv e.to_equiv $ λ _, e.map_ne_zero_iff.symm
/-- An equivalence between modules implies an equivalence between rays. -/
def module.ray.map [nontrivial R] (e : M ≃ₗ[R] N) : module.ray R M ≃ module.ray R N :=
quotient.congr (ray_vector.map_linear_equiv e) $ λ ⟨a, ha⟩ ⟨b, hb⟩, (same_ray_map_iff _).symm
@[simp] lemma module.ray.map_apply [nontrivial R] (e : M ≃ₗ[R] N) (v : M) (hv : v ≠ 0) :
module.ray.map e (ray_of_ne_zero _ v hv) = ray_of_ne_zero _ (e v) (e.map_ne_zero_iff.2 hv) := rfl
@[simp] lemma module.ray.map_refl [nontrivial R] :
(module.ray.map $ linear_equiv.refl R M) = equiv.refl _ :=
equiv.ext $ module.ray.ind R $ λ _ _, rfl
@[simp] lemma module.ray.map_symm [nontrivial R] (e : M ≃ₗ[R] N) :
(module.ray.map e).symm = module.ray.map e.symm := rfl
/-- An equivalence between modules implies an equivalence between orientations. -/
def orientation.map [nontrivial R] (e : M ≃ₗ[R] N) : orientation R M ι ≃ orientation R N ι :=
module.ray.map $ alternating_map.dom_lcongr R R ι R e
@[simp] lemma orientation.map_apply [nontrivial R] (e : M ≃ₗ[R] N) (v : alternating_map R M R ι)
(hv : v ≠ 0) :
orientation.map ι e (ray_of_ne_zero _ v hv) = ray_of_ne_zero _ (v.comp_linear_map e.symm)
(mt (v.comp_linear_equiv_eq_zero_iff e.symm).mp hv) := rfl
@[simp] lemma orientation.map_refl [nontrivial R] :
(orientation.map ι $ linear_equiv.refl R M) = equiv.refl _ :=
by rw [orientation.map, alternating_map.dom_lcongr_refl, module.ray.map_refl]
@[simp] lemma orientation.map_symm [nontrivial R] (e : M ≃ₗ[R] N) :
(orientation.map ι e).symm = orientation.map ι e.symm := rfl
section action
variables {G : Type*} [group G] [nontrivial R] [distrib_mul_action G M] [smul_comm_class R G M]
/-- Any invertible action preserves the non-zeroness of ray vectors. This is primarily of interest
when `G = Rˣ` -/
instance : mul_action G (ray_vector M) :=
{ smul := λ r, (subtype.map ((•) r) $ λ a, (smul_ne_zero_iff_ne _).2),
mul_smul := λ a b m, subtype.ext $ mul_smul a b _,
one_smul := λ m, subtype.ext $ one_smul _ _ }
/-- Any invertible action preserves the non-zeroness of rays. This is primarily of interest when
`G = Rˣ` -/
instance : mul_action G (module.ray R M) :=
{ smul := λ r, quotient.map ((•) r) (λ a b, same_ray.smul _),
mul_smul := λ a b, quotient.ind $ by exact(λ m, congr_arg quotient.mk $ mul_smul a b _),
one_smul := quotient.ind $ by exact (λ m, congr_arg quotient.mk $ one_smul _ _), }
/-- The action via `linear_equiv.apply_distrib_mul_action` corresponds to `module.ray.map`. -/
@[simp] lemma module.ray.linear_equiv_smul_eq_map (e : M ≃ₗ[R] M) (v : module.ray R M) :
e • v = module.ray.map e v := rfl
@[simp] lemma smul_ray_of_ne_zero (g : G) (v : M) (hv) :
g • ray_of_ne_zero R v hv = ray_of_ne_zero R (g • v) ((smul_ne_zero_iff_ne _).2 hv) := rfl
end action
namespace module.ray
/-- Scaling by a positive unit is a no-op. -/
lemma units_smul_of_pos [nontrivial R] (u : Rˣ) (hu : 0 < (u : R)) (v : module.ray R M) :
u • v = v :=
begin
induction v using module.ray.ind,
rw [smul_ray_of_ne_zero, ray_eq_iff],
exact same_ray_pos_smul_left _ hu,
end
/-- An arbitrary `ray_vector` giving a ray. -/
def some_ray_vector [nontrivial R] (x : module.ray R M) : ray_vector M :=
quotient.out x
/-- The ray of `some_ray_vector`. -/
@[simp] lemma some_ray_vector_ray [nontrivial R] (x : module.ray R M) :
(⟦x.some_ray_vector⟧ : module.ray R M) = x :=
quotient.out_eq _
/-- An arbitrary nonzero vector giving a ray. -/
def some_vector [nontrivial R] (x : module.ray R M) : M :=
x.some_ray_vector
/-- `some_vector` is nonzero. -/
@[simp] lemma some_vector_ne_zero [nontrivial R] (x : module.ray R M) : x.some_vector ≠ 0 :=
x.some_ray_vector.property
/-- The ray of `some_vector`. -/
@[simp] lemma some_vector_ray [nontrivial R] (x : module.ray R M) :
ray_of_ne_zero R _ x.some_vector_ne_zero = x :=
(congr_arg _ (subtype.coe_eta _ _) : _).trans x.out_eq
end module.ray
end ordered_comm_semiring
section ordered_comm_ring
local attribute [instance] ray_vector.same_ray_setoid
variables {R : Type*} [ordered_comm_ring R]
variables {M N : Type*} [add_comm_group M] [add_comm_group N] [module R M] [module R N]
/-- If two vectors are in the same ray, so are their negations. -/
lemma same_ray.neg {v₁ v₂ : M} : same_ray R v₁ v₂ → same_ray R (-v₁) (-v₂) :=
λ ⟨r₁, r₂, hr₁, hr₂, h⟩, ⟨r₁, r₂, hr₁, hr₂, by rwa [smul_neg, smul_neg, neg_inj]⟩
/-- `same_ray.neg` as an `iff`. -/
@[simp] lemma same_ray_neg_iff {v₁ v₂ : M} : same_ray R (-v₁) (-v₂) ↔ same_ray R v₁ v₂ :=
⟨λ h, by simpa only [neg_neg] using h.neg, same_ray.neg⟩
lemma same_ray_neg_swap {v₁ v₂ : M} : same_ray R (-v₁) v₂ ↔ same_ray R v₁ (-v₂) :=
⟨λ h, by simpa only [neg_neg] using h.neg, λ h, by simpa only [neg_neg] using h.neg⟩
/-- If a vector is in the same ray as its negation, that vector is zero. -/
lemma eq_zero_of_same_ray_self_neg [no_zero_smul_divisors R M] {v₁ : M} (h : same_ray R v₁ (-v₁)) :
v₁ = 0 :=
begin
rcases h with ⟨r₁, r₂, hr₁, hr₂, h⟩,
rw [smul_neg, ←neg_smul, ←sub_eq_zero, ←sub_smul, sub_neg_eq_add, smul_eq_zero] at h,
exact h.resolve_left (add_pos hr₁ hr₂).ne',
end
namespace ray_vector
variables {R}
/-- Negating a nonzero vector. -/
instance : has_neg (ray_vector M) := ⟨λ v, ⟨-v, neg_ne_zero.2 v.prop⟩⟩
/-- Negating a nonzero vector commutes with coercion to the underlying module. -/
@[simp, norm_cast] lemma coe_neg (v : ray_vector M) : ↑(-v) = -(v : M) := rfl
/-- Negating a nonzero vector twice produces the original vector. -/
@[simp] protected lemma neg_neg (v : ray_vector M) : -(-v) = v :=
by rw [subtype.ext_iff, coe_neg, coe_neg, neg_neg]
variables (R)
/-- If two nonzero vectors are equivalent, so are their negations. -/
@[simp] lemma equiv_neg_iff [nontrivial R] (v₁ v₂ : ray_vector M) : -v₁ ≈ -v₂ ↔ v₁ ≈ v₂ :=
by rw [equiv_iff_same_ray, equiv_iff_same_ray, coe_neg, coe_neg, same_ray_neg_iff]
end ray_vector
variables (R)
/-- Negating a ray. -/
instance [nontrivial R] : has_neg (module.ray R M) :=
⟨quotient.map (λ v, -v) (λ v₁ v₂, (ray_vector.equiv_neg_iff R v₁ v₂).2)⟩
/-- The ray given by the negation of a nonzero vector. -/
lemma ray_neg [nontrivial R] (v : M) (h : v ≠ 0) :
ray_of_ne_zero R _ (show -v ≠ 0, by rw neg_ne_zero; exact h) = -(ray_of_ne_zero R _ h) :=
rfl
namespace module.ray
variables {R}
/-- Negating a ray twice produces the original ray. -/
@[simp] protected lemma neg_neg [nontrivial R] (x : module.ray R M) : -(-x) = x :=
quotient.ind (λ a, congr_arg quotient.mk $ ray_vector.neg_neg _) x
variables (R M)
/-- Negating a ray is involutive. -/
lemma neg_involutive [nontrivial R] : function.involutive (λ x : module.ray R M, -x) :=
λ x, module.ray.neg_neg x
variables {R M}
protected lemma eq_neg_iff_eq_neg [nontrivial R] (x y : module.ray R M) : x = -y ↔ y = -x :=
by rw [←module.ray.neg_neg x, (neg_involutive R M).injective.eq_iff, module.ray.neg_neg x, eq_comm]
/-- A ray does not equal its own negation. -/
lemma ne_neg_self [nontrivial R] [no_zero_smul_divisors R M] (x : module.ray R M) : x ≠ -x :=
begin
intro h,
induction x using module.ray.ind,
rw [←ray_neg, ray_eq_iff] at h,
exact x_hv (eq_zero_of_same_ray_self_neg h)
end
/-- Scaling by a negative unit is negation. -/
lemma units_smul_of_neg [nontrivial R] (u : Rˣ) (hu : (u : R) < 0) (v : module.ray R M) :
u • v = -v :=
begin
induction v using module.ray.ind,
rw [smul_ray_of_ne_zero, ←ray_neg, ray_eq_iff, ←same_ray_neg_swap, units.smul_def, ←neg_smul],
exact same_ray_pos_smul_left _ (neg_pos_of_neg hu),
end
end module.ray
namespace basis
variables {R} {ι : Type*} [fintype ι] [decidable_eq ι]
/-- The orientation given by a basis. -/
protected def orientation [nontrivial R] (e : basis ι R M) : orientation R M ι :=
ray_of_ne_zero R _ e.det_ne_zero
lemma orientation_map [nontrivial R] (e : basis ι R M)
(f : M ≃ₗ[R] N) : (e.map f).orientation = orientation.map ι f e.orientation :=
by simp_rw [basis.orientation, orientation.map_apply, basis.det_map']
/-- The value of `orientation.map` when the index type has the cardinality of a basis, in terms
of `f.det`. -/
lemma map_orientation_eq_det_inv_smul [nontrivial R] [is_domain R] (e : basis ι R M)
(x : orientation R M ι) (f : M ≃ₗ[R] M) : orientation.map ι f x = (f.det)⁻¹ • x :=
begin
induction x using module.ray.ind with g hg,
rw [orientation.map_apply, smul_ray_of_ne_zero, ray_eq_iff, units.smul_def,
(g.comp_linear_map ↑f.symm).eq_smul_basis_det e, g.eq_smul_basis_det e,
alternating_map.comp_linear_map_apply, alternating_map.smul_apply, basis.det_comp,
basis.det_self, mul_one, smul_eq_mul, mul_comm, mul_smul, linear_equiv.coe_inv_det],
end
/-- The orientation given by a basis derived using `units_smul`, in terms of the product of those
units. -/
lemma orientation_units_smul [nontrivial R] (e : basis ι R M) (w : ι → units R) :
(e.units_smul w).orientation = (∏ i, w i)⁻¹ • e.orientation :=
begin
rw [basis.orientation, basis.orientation, smul_ray_of_ne_zero, ray_eq_iff,
e.det.eq_smul_basis_det (e.units_smul w), det_units_smul, units.smul_def, smul_smul],
norm_cast,
simp
end
end basis
end ordered_comm_ring
section linear_ordered_comm_ring
variables {R : Type*} [linear_ordered_comm_ring R]
variables {M : Type*} [add_comm_group M] [module R M]
variables {ι : Type*} [decidable_eq ι]
/-- `same_ray` follows from membership of `mul_action.orbit` for the `units.pos_subgroup`. -/
lemma same_ray_of_mem_orbit {v₁ v₂ : M} (h : v₁ ∈ mul_action.orbit (units.pos_subgroup R) v₂) :
same_ray R v₁ v₂ :=
begin
rcases h with ⟨⟨r, hr⟩, (rfl : r • v₂ = v₁)⟩,
exact same_ray_pos_smul_left _ hr,
end
/-- Scaling by an inverse unit is the same as scaling by itself. -/
@[simp] lemma units_inv_smul (u : Rˣ) (v : module.ray R M) :
u⁻¹ • v = u • v :=
begin
induction v using module.ray.ind with v hv,
rw [smul_ray_of_ne_zero, smul_ray_of_ne_zero, ray_eq_iff],
have : ∀ {u : Rˣ}, 0 < (u : R) → same_ray R (u⁻¹ • v) (u • v) :=
λ u h, ((same_ray.refl v).pos_smul_left $ units.inv_pos.mpr h).pos_smul_right h,
cases lt_or_lt_iff_ne.2 u.ne_zero,
{ rw [←units.neg_neg u, units.neg_inv, (- u).neg_smul, units.neg_smul],
refine (this _).neg,
exact neg_pos_of_neg h },
{ exact this h, },
end
section
variables [no_zero_smul_divisors R M]
/-- A nonzero vector is in the same ray as a multiple of itself if and only if that multiple
is positive. -/
@[simp] lemma same_ray_smul_right_iff {v : M} (hv : v ≠ 0) (r : R) :
same_ray R v (r • v) ↔ 0 < r :=
begin
split,
{ rintros ⟨r₁, r₂, hr₁, hr₂, h⟩,
rw [smul_smul, ←sub_eq_zero, ←sub_smul, sub_eq_add_neg, neg_mul_eq_mul_neg] at h,
by_contradiction hr,
rw [not_lt, ←neg_le_neg_iff, neg_zero] at hr,
have hzzz := ne_of_gt (add_pos_of_pos_of_nonneg hr₁ (mul_nonneg hr₂.le hr)),
simpa [ne_of_gt (add_pos_of_pos_of_nonneg hr₁ (mul_nonneg hr₂.le hr)),
-mul_neg_eq_neg_mul_symm] using h },
{ exact λ h, same_ray_pos_smul_right v h }
end
/-- A multiple of a nonzero vector is in the same ray as that vector if and only if that multiple
is positive. -/
@[simp] lemma same_ray_smul_left_iff {v : M} (hv : v ≠ 0) (r : R) :
same_ray R (r • v) v ↔ 0 < r :=
begin
rw same_ray_comm,
exact same_ray_smul_right_iff hv r
end
/-- The negation of a nonzero vector is in the same ray as a multiple of that vector if and
only if that multiple is negative. -/
@[simp] lemma same_ray_neg_smul_right_iff {v : M} (hv : v ≠ 0) (r : R) :
same_ray R (-v) (r • v) ↔ r < 0 :=
begin
rw [←same_ray_neg_iff, neg_neg, ←neg_smul, same_ray_smul_right_iff hv (-r)],
exact right.neg_pos_iff
end
/-- A multiple of a nonzero vector is in the same ray as the negation of that vector if and
only if that multiple is negative. -/
@[simp] lemma same_ray_neg_smul_left_iff {v : M} (hv : v ≠ 0) (r : R) :
same_ray R (r • v) (-v) ↔ r < 0 :=
begin
rw [←same_ray_neg_iff, neg_neg, ←neg_smul, same_ray_smul_left_iff hv (-r)],
exact left.neg_pos_iff
end
/-- A nonzero vector is in the same ray as a multiple of itself if and only if that multiple
is positive. -/
@[simp] lemma units_smul_eq_self_iff {u : Rˣ} {v : module.ray R M} :
u • v = v ↔ (0 : R) < u :=
begin
induction v using module.ray.ind with v hv,
rw [smul_ray_of_ne_zero, ray_eq_iff, units.smul_def],
exact same_ray_smul_left_iff hv _,
end
/-- A nonzero vector is in the same ray as a multiple of itself if and only if that multiple
is positive. -/
@[simp] lemma units_smul_eq_neg_iff {u : Rˣ} {v : module.ray R M} :
u • v = -v ↔ ↑u < (0 : R) :=
begin
induction v using module.ray.ind with v hv,
rw [smul_ray_of_ne_zero, ←ray_neg, ray_eq_iff, units.smul_def],
exact same_ray_neg_smul_left_iff hv _,
end
end
namespace basis
variables [fintype ι]
/-- The orientations given by two bases are equal if and only if the determinant of one basis
with respect to the other is positive. -/
lemma orientation_eq_iff_det_pos (e₁ e₂ : basis ι R M) :
e₁.orientation = e₂.orientation ↔ 0 < e₁.det e₂ :=
by rw [basis.orientation, basis.orientation, ray_eq_iff,
e₁.det.eq_smul_basis_det e₂, alternating_map.smul_apply, basis.det_self, smul_eq_mul,
mul_one, same_ray_smul_left_iff e₂.det_ne_zero (_ : R)]
/-- Given a basis, any orientation equals the orientation given by that basis or its negation. -/
lemma orientation_eq_or_eq_neg (e : basis ι R M) (x : orientation R M ι) :
x = e.orientation ∨ x = -e.orientation :=
begin
induction x using module.ray.ind with x hx,
rw [basis.orientation, ray_eq_iff, ←ray_neg, ray_eq_iff, x.eq_smul_basis_det e,
same_ray_neg_smul_left_iff e.det_ne_zero (_ : R), same_ray_smul_left_iff e.det_ne_zero (_ : R),
lt_or_lt_iff_ne, ne_comm, alternating_map.map_basis_ne_zero_iff],
exact hx
end
/-- Given a basis, an orientation equals the negation of that given by that basis if and only
if it does not equal that given by that basis. -/
lemma orientation_ne_iff_eq_neg (e : basis ι R M) (x : orientation R M ι) :
x ≠ e.orientation ↔ x = -e.orientation :=
⟨λ h, (e.orientation_eq_or_eq_neg x).resolve_left h,
λ h, h.symm ▸ (module.ray.ne_neg_self e.orientation).symm⟩
/-- Composing a basis with a linear equiv gives the same orientation if and only if the
determinant is positive. -/
lemma orientation_comp_linear_equiv_eq_iff_det_pos (e : basis ι R M) (f : M ≃ₗ[R] M) :
(e.map f).orientation = e.orientation ↔ 0 < (f : M →ₗ[R] M).det :=
by rw [orientation_map, e.map_orientation_eq_det_inv_smul, units_inv_smul, units_smul_eq_self_iff,
linear_equiv.coe_det]
/-- Composing a basis with a linear equiv gives the negation of that orientation if and only if
the determinant is negative. -/
lemma orientation_comp_linear_equiv_eq_neg_iff_det_neg (e : basis ι R M) (f : M ≃ₗ[R] M) :
(e.map f).orientation = -e.orientation ↔ (f : M →ₗ[R] M).det < 0 :=
by rw [orientation_map, e.map_orientation_eq_det_inv_smul, units_inv_smul, units_smul_eq_neg_iff,
linear_equiv.coe_det]
/-- Negating a single basis vector (represented using `units_smul`) negates the corresponding
orientation. -/
@[simp] lemma orientation_neg_single [nontrivial R] (e : basis ι R M) (i : ι) :
(e.units_smul (function.update 1 i (-1))).orientation = -e.orientation :=
begin
rw [orientation_units_smul, finset.prod_update_of_mem (finset.mem_univ _)],
simp
end
/-- Given a basis and an orientation, return a basis giving that orientation: either the original
basis, or one constructed by negating a single (arbitrary) basis vector. -/
def adjust_to_orientation [nontrivial R] [nonempty ι] (e : basis ι R M) (x : orientation R M ι) :
basis ι R M :=
by haveI := classical.dec_eq (orientation R M ι); exact if e.orientation = x then e else
(e.units_smul (function.update 1 (classical.arbitrary ι) (-1)))
/-- `adjust_to_orientation` gives a basis with the required orientation. -/
@[simp] lemma orientation_adjust_to_orientation [nontrivial R] [nonempty ι] (e : basis ι R M)
(x : orientation R M ι) : (e.adjust_to_orientation x).orientation = x :=
begin
rw adjust_to_orientation,
split_ifs with h,
{ exact h },
{ rw [orientation_neg_single, eq_comm, ←orientation_ne_iff_eq_neg, ne_comm],
exact h }
end
/-- Every basis vector from `adjust_to_orientation` is either that from the original basis or its
negation. -/
lemma adjust_to_orientation_apply_eq_or_eq_neg [nontrivial R] [nonempty ι] (e : basis ι R M)
(x : orientation R M ι) (i : ι) :
e.adjust_to_orientation x i = e i ∨ e.adjust_to_orientation x i = -(e i) :=
begin
rw adjust_to_orientation,
split_ifs with h,
{ simp },
{ by_cases hi : i = classical.arbitrary ι;
simp [units_smul_apply, hi] }
end
end basis
end linear_ordered_comm_ring
section linear_ordered_field
variables (R : Type*) [linear_ordered_field R]
variables {M : Type*} [add_comm_group M] [module R M]
variables {ι : Type*} [decidable_eq ι]
/-- `same_ray` is equivalent to membership of `mul_action.orbit` for the `units.pos_subgroup`. -/
lemma same_ray_iff_mem_orbit (v₁ v₂ : M) :
same_ray R v₁ v₂ ↔ v₁ ∈ mul_action.orbit (units.pos_subgroup R) v₂ :=
begin
split,
{ rintros ⟨r₁, r₂, hr₁, hr₂, h⟩,
rw mul_action.mem_orbit_iff,
have h' : (r₁⁻¹ * r₂) • v₂ = v₁,
{ rw [mul_smul, ←h, ←mul_smul, inv_mul_cancel (ne_of_lt hr₁).symm, one_smul] },
have hr' : 0 < (r₁⁻¹ * r₂) := mul_pos (inv_pos.2 hr₁) hr₂,
change (⟨units.mk0 (r₁⁻¹ * r₂) (ne_of_lt hr').symm, hr'⟩ : units.pos_subgroup R) • v₂ = v₁
at h',
exact ⟨_, h'⟩ },
{ exact same_ray_of_mem_orbit }
end
/-- `same_ray_setoid` equals `mul_action.orbit_rel` for the `units.pos_subgroup`. -/
lemma same_ray_setoid_eq_orbit_rel :
same_ray_setoid R M = mul_action.orbit_rel (units.pos_subgroup R) M :=
setoid.ext' $ same_ray_iff_mem_orbit R
variables {R}
namespace orientation
variables [fintype ι] [finite_dimensional R M]
open finite_dimensional
/-- If the index type has cardinality equal to the finite dimension, any two orientations are
equal or negations. -/
lemma eq_or_eq_neg (x₁ x₂ : orientation R M ι) (h : fintype.card ι = finrank R M) :
x₁ = x₂ ∨ x₁ = -x₂ :=
begin
have e := (fin_basis R M).reindex (fintype.equiv_fin_of_card_eq h).symm,
rcases e.orientation_eq_or_eq_neg x₁ with h₁|h₁;
rcases e.orientation_eq_or_eq_neg x₂ with h₂|h₂;
simp [h₁, h₂]
end
/-- If the index type has cardinality equal to the finite dimension, an orientation equals the
negation of another orientation if and only if they are not equal. -/
lemma ne_iff_eq_neg (x₁ x₂ : orientation R M ι) (h : fintype.card ι = finrank R M) :
x₁ ≠ x₂ ↔ x₁ = -x₂ :=
⟨λ hn, (eq_or_eq_neg x₁ x₂ h).resolve_left hn, λ he, he.symm ▸ (module.ray.ne_neg_self x₂).symm⟩
/-- The value of `orientation.map` when the index type has cardinality equal to the finite
dimension, in terms of `f.det`. -/
lemma map_eq_det_inv_smul (x : orientation R M ι) (f : M ≃ₗ[R] M)
(h : fintype.card ι = finrank R M) :
orientation.map ι f x = (f.det)⁻¹ • x :=
begin
have e := (fin_basis R M).reindex (fintype.equiv_fin_of_card_eq h).symm,
exact e.map_orientation_eq_det_inv_smul x f
end
/-- If the index type has cardinality equal to the finite dimension, composing an alternating
map with the same linear equiv on each argument gives the same orientation if and only if the
determinant is positive. -/
lemma map_eq_iff_det_pos (x : orientation R M ι) (f : M ≃ₗ[R] M)
(h : fintype.card ι = finrank R M) :
orientation.map ι f x = x ↔ 0 < (f : M →ₗ[R] M).det :=
by rw [map_eq_det_inv_smul _ _ h, units_inv_smul, units_smul_eq_self_iff, linear_equiv.coe_det]
/-- If the index type has cardinality equal to the finite dimension, composing an alternating
map with the same linear equiv on each argument gives the negation of that orientation if and
only if the determinant is negative. -/
lemma map_eq_neg_iff_det_neg (x : orientation R M ι) (f : M ≃ₗ[R] M)
(h : fintype.card ι = finrank R M) :
orientation.map ι f x = -x ↔ (f : M →ₗ[R] M).det < 0 :=
by rw [map_eq_det_inv_smul _ _ h, units_inv_smul, units_smul_eq_neg_iff, linear_equiv.coe_det]
/-- If the index type has cardinality equal to the finite dimension, a basis with the given
orientation. -/
def some_basis [nonempty ι] (x : orientation R M ι) (h : fintype.card ι = finrank R M) :
basis ι R M :=
((fin_basis R M).reindex (fintype.equiv_fin_of_card_eq h).symm).adjust_to_orientation x
/-- `some_basis` gives a basis with the required orientation. -/
@[simp] lemma some_basis_orientation [nonempty ι] (x : orientation R M ι)
(h : fintype.card ι = finrank R M) : (x.some_basis h).orientation = x :=
basis.orientation_adjust_to_orientation _ _
end orientation
end linear_ordered_field
|
367f42e9b91474ba2ff5d661bc560a83df572c5e | 853df553b1d6ca524e3f0a79aedd32dde5d27ec3 | /src/order/filter/ultrafilter.lean | 8e860e8031ae709b692706d978f172aff87be1e4 | [
"Apache-2.0"
] | permissive | DanielFabian/mathlib | efc3a50b5dde303c59eeb6353ef4c35a345d7112 | f520d07eba0c852e96fe26da71d85bf6d40fcc2a | refs/heads/master | 1,668,739,922,971 | 1,595,201,756,000 | 1,595,201,756,000 | 279,469,476 | 0 | 0 | null | 1,594,696,604,000 | 1,594,696,604,000 | null | UTF-8 | Lean | false | false | 12,716 | lean | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import order.filter.cofinite
/-!
# Ultrafilters
An ultrafilter is a minimal (maximal in the set order) proper filter.
In this file we define
* `is_ultrafilter`: a predicate stating that a given filter is an ultrafiler;
* `ultrafilter_of`: an ultrafilter that is less than or equal to a given filter;
* `ultrafilter`: subtype of ultrafilters;
* `ultrafilter.pure`: `pure x` as an `ultrafiler`;
* `ultrafilter.map`, `ultrafilter.bind` : operations on ultrafilters;
* `hyperfilter`: the ultra-filter extending the cofinite filter.
-/
universes u v
variables {α : Type u} {β : Type v}
namespace filter
open set zorn
open_locale classical filter
variables {f g : filter α}
/-- An ultrafilter is a minimal (maximal in the set order) proper filter. -/
def is_ultrafilter (f : filter α) := f ≠ ⊥ ∧ ∀g, g ≠ ⊥ → g ≤ f → f ≤ g
lemma ultrafilter_unique (hg : is_ultrafilter g) (hf : f ≠ ⊥) (h : f ≤ g) : f = g :=
le_antisymm h (hg.right _ hf h)
lemma le_of_ultrafilter {g : filter α} (hf : is_ultrafilter f) (h : g ⊓ f ≠ ⊥) :
f ≤ g :=
by { rw inf_comm at h, exact le_of_inf_eq (ultrafilter_unique hf h inf_le_left) }
/-- Equivalent characterization of ultrafilters:
A filter f is an ultrafilter if and only if for each set s,
-s belongs to f if and only if s does not belong to f. -/
lemma ultrafilter_iff_compl_mem_iff_not_mem :
is_ultrafilter f ↔ (∀ s, sᶜ ∈ f ↔ s ∉ f) :=
⟨assume hf s,
⟨assume hns hs,
hf.1 $ empty_in_sets_eq_bot.mp $ by convert f.inter_sets hs hns; rw [inter_compl_self],
assume hs,
have f ≤ 𝓟 sᶜ, from
le_of_ultrafilter hf $ assume h, hs $ mem_sets_of_eq_bot $
by rwa inf_comm,
by simp only [le_principal_iff] at this; assumption⟩,
assume hf,
⟨mt empty_in_sets_eq_bot.mpr ((hf ∅).mp (by convert f.univ_sets; rw [compl_empty])),
assume g hg g_le s hs, classical.by_contradiction $ mt (hf s).mpr $
assume : sᶜ ∈ f,
have s ∩ sᶜ ∈ g, from inter_mem_sets hs (g_le this),
by simp only [empty_in_sets_eq_bot, hg, inter_compl_self] at this; contradiction⟩⟩
lemma mem_or_compl_mem_of_ultrafilter (hf : is_ultrafilter f) (s : set α) :
s ∈ f ∨ sᶜ ∈ f :=
classical.or_iff_not_imp_left.2 (ultrafilter_iff_compl_mem_iff_not_mem.mp hf s).mpr
lemma mem_or_mem_of_ultrafilter {s t : set α} (hf : is_ultrafilter f) (h : s ∪ t ∈ f) :
s ∈ f ∨ t ∈ f :=
(mem_or_compl_mem_of_ultrafilter hf s).imp_right
(assume : sᶜ ∈ f, by filter_upwards [this, h] assume x hnx hx, hx.resolve_left hnx)
lemma is_ultrafilter.em (hf : is_ultrafilter f) (p : α → Prop) :
(∀ᶠ x in f, p x) ∨ ∀ᶠ x in f, ¬p x :=
mem_or_compl_mem_of_ultrafilter hf {x | p x}
lemma is_ultrafilter.eventually_or (hf : is_ultrafilter f) {p q : α → Prop} :
(∀ᶠ x in f, p x ∨ q x) ↔ (∀ᶠ x in f, p x) ∨ ∀ᶠ x in f, q x :=
⟨mem_or_mem_of_ultrafilter hf, λ H, H.elim (λ hp, hp.mono $ λ x, or.inl)
(λ hp, hp.mono $ λ x, or.inr)⟩
lemma is_ultrafilter.eventually_not (hf : is_ultrafilter f) {p : α → Prop} :
(∀ᶠ x in f, ¬p x) ↔ ¬∀ᶠ x in f, p x :=
ultrafilter_iff_compl_mem_iff_not_mem.1 hf {x | p x}
lemma is_ultrafilter.eventually_imp (hf : is_ultrafilter f) {p q : α → Prop} :
(∀ᶠ x in f, p x → q x) ↔ (∀ᶠ x in f, p x) → ∀ᶠ x in f, q x :=
by simp only [imp_iff_not_or, hf.eventually_or, hf.eventually_not]
lemma mem_of_finite_sUnion_ultrafilter {s : set (set α)} (hf : is_ultrafilter f) (hs : finite s)
: ⋃₀ s ∈ f → ∃t∈s, t ∈ f :=
finite.induction_on hs (by simp only [empty_in_sets_eq_bot, hf.left, mem_empty_eq, sUnion_empty,
forall_prop_of_false, exists_false, not_false_iff, exists_prop_of_false]) $
λ t s' ht' hs' ih, by simp only [exists_prop, mem_insert_iff, set.sUnion_insert]; exact
assume h, (mem_or_mem_of_ultrafilter hf h).elim
(assume : t ∈ f, ⟨t, or.inl rfl, this⟩)
(assume h, let ⟨t, hts', ht⟩ := ih h in ⟨t, or.inr hts', ht⟩)
lemma mem_of_finite_Union_ultrafilter {is : set β} {s : β → set α}
(hf : is_ultrafilter f) (his : finite is) (h : (⋃i∈is, s i) ∈ f) : ∃i∈is, s i ∈ f :=
have his : finite (image s is), from his.image s,
have h : (⋃₀ image s is) ∈ f, from by simp only [sUnion_image, set.sUnion_image]; assumption,
let ⟨t, ⟨i, hi, h_eq⟩, (ht : t ∈ f)⟩ := mem_of_finite_sUnion_ultrafilter hf his h in
⟨i, hi, h_eq.symm ▸ ht⟩
lemma ultrafilter_map {f : filter α} {m : α → β} (h : is_ultrafilter f) :
is_ultrafilter (map m f) :=
by rw ultrafilter_iff_compl_mem_iff_not_mem at ⊢ h; exact assume s, h (m ⁻¹' s)
lemma ultrafilter_pure {a : α} : is_ultrafilter (pure a) :=
begin
rw ultrafilter_iff_compl_mem_iff_not_mem, intro s,
rw [mem_pure_sets, mem_pure_sets], exact iff.rfl
end
lemma ultrafilter_bind {f : filter α} (hf : is_ultrafilter f) {m : α → filter β}
(hm : ∀ a, is_ultrafilter (m a)) : is_ultrafilter (f.bind m) :=
begin
simp only [ultrafilter_iff_compl_mem_iff_not_mem] at ⊢ hf hm, intro s,
dsimp [bind, join, map, preimage],
simp only [hm], apply hf
end
/-- The ultrafilter lemma: Any proper filter is contained in an ultrafilter. -/
lemma exists_ultrafilter (h : f ≠ ⊥) : ∃u, u ≤ f ∧ is_ultrafilter u :=
let
τ := {f' // f' ≠ ⊥ ∧ f' ≤ f},
r : τ → τ → Prop := λt₁ t₂, t₂.val ≤ t₁.val,
⟨a, ha⟩ := nonempty_of_mem_sets h univ_mem_sets,
top : τ := ⟨f, h, le_refl f⟩,
sup : Π(c:set τ), chain r c → τ :=
λc hc, ⟨⨅a:{a:τ // a ∈ insert top c}, a.val.val,
infi_ne_bot_of_directed ⟨a⟩
(directed_of_chain $ chain_insert hc $ assume ⟨b, _, hb⟩ _ _, or.inl hb)
(assume ⟨⟨a, ha, _⟩, _⟩, ha),
infi_le_of_le ⟨top, mem_insert _ _⟩ (le_refl _)⟩
in
have ∀c (hc: chain r c) a (ha : a ∈ c), r a (sup c hc),
from assume c hc a ha, infi_le_of_le ⟨a, mem_insert_of_mem _ ha⟩ (le_refl _),
have (∃ (u : τ), ∀ (a : τ), r u a → r a u),
from exists_maximal_of_chains_bounded (assume c hc, ⟨sup c hc, this c hc⟩)
(assume f₁ f₂ f₃ h₁ h₂, le_trans h₂ h₁),
let ⟨uτ, hmin⟩ := this in
⟨uτ.val, uτ.property.right, uτ.property.left, assume g hg₁ hg₂,
hmin ⟨g, hg₁, le_trans hg₂ uτ.property.right⟩ hg₂⟩
/-- Construct an ultrafilter extending a given filter.
The ultrafilter lemma is the assertion that such a filter exists;
we use the axiom of choice to pick one. -/
noncomputable def ultrafilter_of (f : filter α) : filter α :=
if h : f = ⊥ then ⊥ else classical.epsilon (λu, u ≤ f ∧ is_ultrafilter u)
lemma ultrafilter_of_spec (h : f ≠ ⊥) : ultrafilter_of f ≤ f ∧ is_ultrafilter (ultrafilter_of f) :=
begin
have h' := classical.epsilon_spec (exists_ultrafilter h),
simp only [ultrafilter_of, dif_neg, h, dif_neg, not_false_iff],
simp only at h',
assumption
end
lemma ultrafilter_of_le : ultrafilter_of f ≤ f :=
if h : f = ⊥ then by simp only [ultrafilter_of, h, dif_pos, le_bot_iff]
else (ultrafilter_of_spec h).left
lemma ultrafilter_ultrafilter_of (h : f ≠ ⊥) : is_ultrafilter (ultrafilter_of f) :=
(ultrafilter_of_spec h).right
lemma ultrafilter_of_ultrafilter (h : is_ultrafilter f) : ultrafilter_of f = f :=
ultrafilter_unique h (ultrafilter_ultrafilter_of h.left).left ultrafilter_of_le
/-- A filter equals the intersection of all the ultrafilters which contain it. -/
lemma sup_of_ultrafilters (f : filter α) : f = ⨆ (g) (u : is_ultrafilter g) (H : g ≤ f), g :=
begin
refine le_antisymm _ (supr_le $ λ g, supr_le $ λ u, supr_le $ λ H, H),
intros s hs,
-- If s ∉ f.sets, we'll apply the ultrafilter lemma to the restriction of f to -s.
by_contradiction hs',
let j : sᶜ → α := subtype.val,
have j_inv_s : j ⁻¹' s = ∅, by
erw [←preimage_inter_range, subtype.range_coe, inter_compl_self, preimage_empty],
let f' := comap j f,
have : f' ≠ ⊥,
{ apply mt empty_in_sets_eq_bot.mpr,
rintro ⟨t, htf, ht⟩,
suffices : t ⊆ s, from absurd (f.sets_of_superset htf this) hs',
rw [subset_empty_iff] at ht,
have : j '' (j ⁻¹' t) = ∅, by rw [ht, image_empty],
erw [image_preimage_eq_inter_range, subtype.range_coe, ←subset_compl_iff_disjoint,
set.compl_compl] at this,
exact this },
rcases exists_ultrafilter this with ⟨g', g'f', u'⟩,
simp only [supr_sets_eq, mem_Inter] at hs,
have := hs (g'.map subtype.val) (ultrafilter_map u') (map_le_iff_le_comap.mpr g'f'),
rw [←le_principal_iff, map_le_iff_le_comap, comap_principal, j_inv_s, principal_empty,
le_bot_iff] at this,
exact absurd this u'.1
end
/-- The `tendsto` relation can be checked on ultrafilters. -/
lemma tendsto_iff_ultrafilter (f : α → β) (l₁ : filter α) (l₂ : filter β) :
tendsto f l₁ l₂ ↔ ∀ g, is_ultrafilter g → g ≤ l₁ → g.map f ≤ l₂ :=
⟨assume h g u gx, le_trans (map_mono gx) h,
assume h, by rw [sup_of_ultrafilters l₁]; simpa only [tendsto, map_supr, supr_le_iff]⟩
/-- The ultrafilter monad. The monad structure on ultrafilters is the
restriction of the one on filters. -/
def ultrafilter (α : Type u) : Type u := {f : filter α // is_ultrafilter f}
/-- Push-forward for ultra-filters. -/
def ultrafilter.map (m : α → β) (u : ultrafilter α) : ultrafilter β :=
⟨u.val.map m, ultrafilter_map u.property⟩
/-- The principal ultra-filter associated to a point `x`. -/
def ultrafilter.pure (x : α) : ultrafilter α := ⟨pure x, ultrafilter_pure⟩
/-- Monadic bind for ultra-filters, coming from the one on filters
defined in terms of map and join.-/
def ultrafilter.bind (u : ultrafilter α) (m : α → ultrafilter β) : ultrafilter β :=
⟨u.val.bind (λ a, (m a).val), ultrafilter_bind u.property (λ a, (m a).property)⟩
instance ultrafilter.has_pure : has_pure ultrafilter := ⟨@ultrafilter.pure⟩
instance ultrafilter.has_bind : has_bind ultrafilter := ⟨@ultrafilter.bind⟩
instance ultrafilter.functor : functor ultrafilter := { map := @ultrafilter.map }
instance ultrafilter.monad : monad ultrafilter := { map := @ultrafilter.map }
instance ultrafilter.inhabited [inhabited α] : inhabited (ultrafilter α) := ⟨pure (default _)⟩
/-- The ultra-filter extending the cofinite filter. -/
noncomputable def hyperfilter : filter α := ultrafilter_of cofinite
lemma hyperfilter_le_cofinite : @hyperfilter α ≤ cofinite :=
ultrafilter_of_le
lemma is_ultrafilter_hyperfilter [infinite α] : is_ultrafilter (@hyperfilter α) :=
(ultrafilter_of_spec cofinite_ne_bot).2
@[simp] lemma hyperfilter_ne_bot [infinite α] : @hyperfilter α ≠ ⊥ :=
is_ultrafilter_hyperfilter.1
@[simp] lemma bot_ne_hyperfilter [infinite α] : ⊥ ≠ @hyperfilter α :=
is_ultrafilter_hyperfilter.1.symm
theorem nmem_hyperfilter_of_finite [infinite α] {s : set α} (hf : s.finite) :
s ∉ @hyperfilter α :=
λ hy,
have hx : sᶜ ∉ hyperfilter :=
λ hs, (ultrafilter_iff_compl_mem_iff_not_mem.mp is_ultrafilter_hyperfilter s).mp hs hy,
have ht : sᶜ ∈ cofinite.sets := by show sᶜ ∈ {s | _}; rwa [set.mem_set_of_eq, compl_compl],
hx $ hyperfilter_le_cofinite ht
theorem compl_mem_hyperfilter_of_finite [infinite α] {s : set α} (hf : set.finite s) :
sᶜ ∈ @hyperfilter α :=
(ultrafilter_iff_compl_mem_iff_not_mem.mp is_ultrafilter_hyperfilter s).mpr $
nmem_hyperfilter_of_finite hf
theorem mem_hyperfilter_of_finite_compl [infinite α] {s : set α} (hf : set.finite sᶜ) :
s ∈ @hyperfilter α :=
s.compl_compl ▸ compl_mem_hyperfilter_of_finite hf
section
local attribute [instance] filter.monad filter.is_lawful_monad
instance ultrafilter.is_lawful_monad : is_lawful_monad ultrafilter :=
{ id_map := assume α f, subtype.eq (id_map f.val),
pure_bind := assume α β a f, subtype.eq (pure_bind a (subtype.val ∘ f)),
bind_assoc := assume α β γ f m₁ m₂, subtype.eq (filter_eq rfl),
bind_pure_comp_eq_map := assume α β f x, subtype.eq (bind_pure_comp_eq_map f x.val) }
end
lemma ultrafilter.eq_iff_val_le_val {u v : ultrafilter α} : u = v ↔ u.val ≤ v.val :=
⟨assume h, by rw h; exact le_refl _,
assume h, by rw subtype.ext_iff_val; apply ultrafilter_unique v.property u.property.1 h⟩
lemma exists_ultrafilter_iff (f : filter α) : (∃ (u : ultrafilter α), u.val ≤ f) ↔ f ≠ ⊥ :=
⟨assume ⟨u, uf⟩, ne_bot_of_le_ne_bot u.property.1 uf,
assume h, let ⟨u, uf, hu⟩ := exists_ultrafilter h in ⟨⟨u, hu⟩, uf⟩⟩
end filter
|
4e78a92a97f11c548c6ac242dc02ed710655c4cb | 02005f45e00c7ecf2c8ca5db60251bd1e9c860b5 | /src/data/nat/choose/dvd.lean | 6278e650fd7ca51760a33b736465ff9bc538f751 | [
"Apache-2.0"
] | permissive | anthony2698/mathlib | 03cd69fe5c280b0916f6df2d07c614c8e1efe890 | 407615e05814e98b24b2ff322b14e8e3eb5e5d67 | refs/heads/master | 1,678,792,774,873 | 1,614,371,563,000 | 1,614,371,563,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 1,957 | lean | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Patrick Stevens
-/
import data.nat.choose.basic
import data.nat.prime
import data.rat.floor
/-!
# Divisibility properties of binomial coefficients
-/
namespace nat
namespace prime
open_locale nat
lemma dvd_choose_add {p a b : ℕ} (hap : a < p) (hbp : b < p) (h : p ≤ a + b)
(hp : prime p) : p ∣ choose (a + b) a :=
have h₁ : p ∣ (a + b)!, from hp.dvd_factorial.2 h,
have h₂ : ¬p ∣ a!, from mt hp.dvd_factorial.1 (not_le_of_gt hap),
have h₃ : ¬p ∣ b!, from mt hp.dvd_factorial.1 (not_le_of_gt hbp),
by
rw [← choose_mul_factorial_mul_factorial (le.intro rfl), mul_assoc, hp.dvd_mul, hp.dvd_mul,
nat.add_sub_cancel_left a b] at h₁;
exact h₁.resolve_right (not_or_distrib.2 ⟨h₂, h₃⟩)
lemma dvd_choose_self {p k : ℕ} (hk : 0 < k) (hkp : k < p) (hp : prime p) :
p ∣ choose p k :=
begin
have r : k + (p - k) = p,
by rw [← nat.add_sub_assoc (nat.le_of_lt hkp) k, nat.add_sub_cancel_left],
have e : p ∣ choose (k + (p - k)) k,
by exact dvd_choose_add hkp (sub_lt (hk.trans hkp) hk) (by rw r) hp,
rwa r at e,
end
lemma choose_eq_factorial_div_factorial' {a b : ℕ}
(hab : a ≤ b) : (b.choose a : ℚ) = b! / (a! * (b - a)!) :=
begin
field_simp [mul_ne_zero, factorial_ne_zero], norm_cast,
rw ← choose_mul_factorial_mul_factorial hab, ring,
end
lemma choose_mul {n k s : ℕ} (hn : k ≤ n) (hs : s ≤ k) :
(n.choose k : ℚ) * k.choose s = n.choose s * (n - s).choose (k - s) :=
begin
rw [choose_eq_factorial_div_factorial' hn, choose_eq_factorial_div_factorial' hs,
choose_eq_factorial_div_factorial' (le_trans hs hn), choose_eq_factorial_div_factorial' ],
swap,
{ exact nat.sub_le_sub_right hn s, },
{ field_simp [mul_ne_zero, factorial_ne_zero],
rw sub_sub_sub_cancel_right hs, ring, },
end
end prime
end nat
|
fbaa8a7f1b27b645523f45c24eccccb3b32e115c | d406927ab5617694ec9ea7001f101b7c9e3d9702 | /src/order/symm_diff.lean | 401c9105dc451cab8b3e0d801ca21d6d68011cc1 | [
"Apache-2.0"
] | permissive | alreadydone/mathlib | dc0be621c6c8208c581f5170a8216c5ba6721927 | c982179ec21091d3e102d8a5d9f5fe06c8fafb73 | refs/heads/master | 1,685,523,275,196 | 1,670,184,141,000 | 1,670,184,141,000 | 287,574,545 | 0 | 0 | Apache-2.0 | 1,670,290,714,000 | 1,597,421,623,000 | Lean | UTF-8 | Lean | false | false | 23,517 | lean | /-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaël Dillies
-/
import order.boolean_algebra
import logic.equiv.basic
/-!
# Symmetric difference and bi-implication
This file defines the symmetric difference and bi-implication operators in (co-)Heyting algebras.
## Examples
Some examples are
* The symmetric difference of two sets is the set of elements that are in either but not both.
* The symmetric difference on propositions is `xor`.
* The symmetric difference on `bool` is `bxor`.
* The equivalence of propositions. Two propositions are equivalent if they imply each other.
* The symmetric difference translates to addition when considering a Boolean algebra as a Boolean
ring.
## Main declarations
* `symm_diff`: The symmetric difference operator, defined as `(a \ b) ⊔ (b \ a)`
* `bihimp`: The bi-implication operator, defined as `(b ⇨ a) ⊓ (a ⇨ b)`
In generalized Boolean algebras, the symmetric difference operator is:
* `symm_diff_comm`: commutative, and
* `symm_diff_assoc`: associative.
## Notations
* `a ∆ b`: `symm_diff a b`
* `a ⇔ b`: `bihimp a b`
## References
The proof of associativity follows the note "Associativity of the Symmetric Difference of Sets: A
Proof from the Book" by John McCuan:
* <https://people.math.gatech.edu/~mccuan/courses/4317/symmetricdifference.pdf>
## Tags
boolean ring, generalized boolean algebra, boolean algebra, symmetric difference, bi-implication,
Heyting
-/
open function order_dual
variables {ι α β : Type*} {π : ι → Type*}
/-- The symmetric difference operator on a type with `⊔` and `\` is `(A \ B) ⊔ (B \ A)`. -/
def symm_diff [has_sup α] [has_sdiff α] (a b : α) : α := a \ b ⊔ b \ a
/-- The Heyting bi-implication is `(b ⇨ a) ⊓ (a ⇨ b)`. This generalizes equivalence of
propositions. -/
def bihimp [has_inf α] [has_himp α] (a b : α) : α := (b ⇨ a) ⊓ (a ⇨ b)
/- This notation might conflict with the Laplacian once we have it. Feel free to put it in locale
`order` or `symm_diff` if that happens. -/
infix ` ∆ `:100 := symm_diff
infix ` ⇔ `:100 := bihimp
lemma symm_diff_def [has_sup α] [has_sdiff α] (a b : α) : a ∆ b = (a \ b) ⊔ (b \ a) := rfl
lemma bihimp_def [has_inf α] [has_himp α] (a b : α) : a ⇔ b = (b ⇨ a) ⊓ (a ⇨ b):= rfl
lemma symm_diff_eq_xor (p q : Prop) : p ∆ q = xor p q := rfl
@[simp] lemma bihimp_iff_iff {p q : Prop} : p ⇔ q ↔ (p ↔ q) :=
(iff_iff_implies_and_implies _ _).symm.trans iff.comm
@[simp] lemma bool.symm_diff_eq_bxor : ∀ p q : bool, p ∆ q = bxor p q := dec_trivial
section generalized_coheyting_algebra
variables [generalized_coheyting_algebra α] (a b c d : α)
@[simp] lemma to_dual_symm_diff : to_dual (a ∆ b) = to_dual a ⇔ to_dual b := rfl
@[simp] lemma of_dual_bihimp (a b : αᵒᵈ) : of_dual (a ⇔ b) = of_dual a ∆ of_dual b := rfl
lemma symm_diff_comm : a ∆ b = b ∆ a := by simp only [(∆), sup_comm]
instance symm_diff_is_comm : is_commutative α (∆) := ⟨symm_diff_comm⟩
@[simp] lemma symm_diff_self : a ∆ a = ⊥ := by rw [(∆), sup_idem, sdiff_self]
@[simp] lemma symm_diff_bot : a ∆ ⊥ = a := by rw [(∆), sdiff_bot, bot_sdiff, sup_bot_eq]
@[simp] lemma bot_symm_diff : ⊥ ∆ a = a := by rw [symm_diff_comm, symm_diff_bot]
@[simp] lemma symm_diff_eq_bot {a b : α} : a ∆ b = ⊥ ↔ a = b :=
by simp_rw [symm_diff, sup_eq_bot_iff, sdiff_eq_bot_iff, le_antisymm_iff]
lemma symm_diff_of_le {a b : α} (h : a ≤ b) : a ∆ b = b \ a :=
by rw [symm_diff, sdiff_eq_bot_iff.2 h, bot_sup_eq]
lemma symm_diff_of_ge {a b : α} (h : b ≤ a) : a ∆ b = a \ b :=
by rw [symm_diff, sdiff_eq_bot_iff.2 h, sup_bot_eq]
lemma symm_diff_le {a b c : α} (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ∆ b ≤ c :=
sup_le (sdiff_le_iff.2 ha) $ sdiff_le_iff.2 hb
lemma symm_diff_le_iff {a b c : α} : a ∆ b ≤ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c :=
by simp_rw [symm_diff, sup_le_iff, sdiff_le_iff]
@[simp] lemma symm_diff_le_sup {a b : α} : a ∆ b ≤ a ⊔ b := sup_le_sup sdiff_le sdiff_le
lemma symm_diff_eq_sup_sdiff_inf : a ∆ b = (a ⊔ b) \ (a ⊓ b) := by simp [sup_sdiff, symm_diff]
lemma disjoint.symm_diff_eq_sup {a b : α} (h : disjoint a b) : a ∆ b = a ⊔ b :=
by rw [(∆), h.sdiff_eq_left, h.sdiff_eq_right]
lemma symm_diff_sdiff : (a ∆ b) \ c = a \ (b ⊔ c) ⊔ b \ (a ⊔ c) :=
by rw [symm_diff, sup_sdiff_distrib, sdiff_sdiff_left, sdiff_sdiff_left]
@[simp] lemma symm_diff_sdiff_inf : a ∆ b \ (a ⊓ b) = a ∆ b :=
by { rw symm_diff_sdiff, simp [symm_diff] }
@[simp] lemma symm_diff_sdiff_eq_sup : a ∆ (b \ a) = a ⊔ b :=
begin
rw [symm_diff, sdiff_idem],
exact le_antisymm (sup_le_sup sdiff_le sdiff_le)
(sup_le le_sdiff_sup $ le_sdiff_sup.trans $ sup_le le_sup_right le_sdiff_sup),
end
@[simp] lemma sdiff_symm_diff_eq_sup : (a \ b) ∆ b = a ⊔ b :=
by rw [symm_diff_comm, symm_diff_sdiff_eq_sup, sup_comm]
@[simp] lemma symm_diff_sup_inf : a ∆ b ⊔ a ⊓ b = a ⊔ b :=
begin
refine le_antisymm (sup_le symm_diff_le_sup inf_le_sup) _,
rw [sup_inf_left, symm_diff],
refine sup_le (le_inf le_sup_right _) (le_inf _ le_sup_right),
{ rw sup_right_comm,
exact le_sup_of_le_left le_sdiff_sup },
{ rw sup_assoc,
exact le_sup_of_le_right le_sdiff_sup }
end
@[simp] lemma inf_sup_symm_diff : a ⊓ b ⊔ a ∆ b = a ⊔ b := by rw [sup_comm, symm_diff_sup_inf]
@[simp] lemma symm_diff_symm_diff_inf : a ∆ b ∆ (a ⊓ b) = a ⊔ b :=
by rw [←symm_diff_sdiff_inf a, sdiff_symm_diff_eq_sup, symm_diff_sup_inf]
@[simp] lemma inf_symm_diff_symm_diff : (a ⊓ b) ∆ (a ∆ b) = a ⊔ b :=
by rw [symm_diff_comm, symm_diff_symm_diff_inf]
lemma symm_diff_triangle : a ∆ c ≤ a ∆ b ⊔ b ∆ c :=
begin
refine (sup_le_sup (sdiff_triangle a b c) $ sdiff_triangle _ b _).trans_eq _,
rw [@sup_comm _ _ (c \ b), sup_sup_sup_comm, symm_diff, symm_diff],
end
end generalized_coheyting_algebra
section generalized_heyting_algebra
variables [generalized_heyting_algebra α] (a b c d : α)
@[simp] lemma to_dual_bihimp : to_dual (a ⇔ b) = to_dual a ∆ to_dual b := rfl
@[simp] lemma of_dual_symm_diff (a b : αᵒᵈ) : of_dual (a ∆ b) = of_dual a ⇔ of_dual b := rfl
lemma bihimp_comm : a ⇔ b = b ⇔ a := by simp only [(⇔), inf_comm]
instance bihimp_is_comm : is_commutative α (⇔) := ⟨bihimp_comm⟩
@[simp] lemma bihimp_self : a ⇔ a = ⊤ := by rw [(⇔), inf_idem, himp_self]
@[simp] lemma bihimp_top : a ⇔ ⊤ = a := by rw [(⇔), himp_top, top_himp, inf_top_eq]
@[simp] lemma top_bihimp : ⊤ ⇔ a = a := by rw [bihimp_comm, bihimp_top]
@[simp] lemma bihimp_eq_top {a b : α} : a ⇔ b = ⊤ ↔ a = b := @symm_diff_eq_bot αᵒᵈ _ _ _
lemma bihimp_of_le {a b : α} (h : a ≤ b) : a ⇔ b = b ⇨ a :=
by rw [bihimp, himp_eq_top_iff.2 h, inf_top_eq]
lemma bihimp_of_ge {a b : α} (h : b ≤ a) : a ⇔ b = a ⇨ b :=
by rw [bihimp, himp_eq_top_iff.2 h, top_inf_eq]
lemma le_bihimp {a b c : α} (hb : a ⊓ b ≤ c) (hc : a ⊓ c ≤ b) : a ≤ b ⇔ c :=
le_inf (le_himp_iff.2 hc) $ le_himp_iff.2 hb
lemma le_bihimp_iff {a b c : α} : a ≤ b ⇔ c ↔ a ⊓ b ≤ c ∧ a ⊓ c ≤ b :=
by simp_rw [bihimp, le_inf_iff, le_himp_iff, and.comm]
@[simp] lemma inf_le_bihimp {a b : α} : a ⊓ b ≤ a ⇔ b := inf_le_inf le_himp le_himp
lemma bihimp_eq_inf_himp_inf : a ⇔ b = (a ⊔ b) ⇨ (a ⊓ b) := by simp [himp_inf_distrib, bihimp]
lemma codisjoint.bihimp_eq_inf {a b : α} (h : codisjoint a b) : a ⇔ b = a ⊓ b :=
by rw [(⇔), h.himp_eq_left, h.himp_eq_right]
lemma himp_bihimp : a ⇨ (b ⇔ c) = ((a ⊓ c) ⇨ b) ⊓ ((a ⊓ b) ⇨ c) :=
by rw [bihimp, himp_inf_distrib, himp_himp, himp_himp]
@[simp] lemma sup_himp_bihimp : (a ⊔ b) ⇨ (a ⇔ b) = a ⇔ b :=
by { rw himp_bihimp, simp [bihimp] }
@[simp] lemma bihimp_himp_eq_inf : a ⇔ (a ⇨ b) = a ⊓ b := @symm_diff_sdiff_eq_sup αᵒᵈ _ _ _
@[simp] lemma himp_bihimp_eq_inf : (b ⇨ a) ⇔ b = a ⊓ b := @sdiff_symm_diff_eq_sup αᵒᵈ _ _ _
@[simp] lemma bihimp_inf_sup : a ⇔ b ⊓ (a ⊔ b) = a ⊓ b := @symm_diff_sup_inf αᵒᵈ _ _ _
@[simp] lemma sup_inf_bihimp : (a ⊔ b) ⊓ a ⇔ b = a ⊓ b := @inf_sup_symm_diff αᵒᵈ _ _ _
@[simp] lemma bihimp_bihimp_sup : a ⇔ b ⇔ (a ⊔ b) = a ⊓ b := @symm_diff_symm_diff_inf αᵒᵈ _ _ _
@[simp] lemma sup_bihimp_bihimp : (a ⊔ b) ⇔ (a ⇔ b) = a ⊓ b := @inf_symm_diff_symm_diff αᵒᵈ _ _ _
lemma bihimp_triangle : a ⇔ b ⊓ b ⇔ c ≤ a ⇔ c := @symm_diff_triangle αᵒᵈ _ _ _ _
end generalized_heyting_algebra
section coheyting_algebra
variables [coheyting_algebra α] (a : α)
@[simp] lemma symm_diff_top' : a ∆ ⊤ = ￢a := by simp [symm_diff]
@[simp] lemma top_symm_diff' : ⊤ ∆ a = ￢a := by simp [symm_diff]
@[simp] lemma hnot_symm_diff_self : (￢a) ∆ a = ⊤ :=
begin
rw [eq_top_iff, symm_diff, hnot_sdiff, sup_sdiff_self],
exact codisjoint.top_le codisjoint_hnot_left
end
@[simp] lemma symm_diff_hnot_self : a ∆ ￢a = ⊤ := by rw [symm_diff_comm, hnot_symm_diff_self]
lemma is_compl.symm_diff_eq_top {a b : α} (h : is_compl a b) : a ∆ b = ⊤ :=
by rw [h.eq_hnot, hnot_symm_diff_self]
end coheyting_algebra
section heyting_algebra
variables [heyting_algebra α] (a : α)
@[simp] lemma bihimp_bot : a ⇔ ⊥ = aᶜ := by simp [bihimp]
@[simp] lemma bot_bihimp : ⊥ ⇔ a = aᶜ := by simp [bihimp]
@[simp] lemma compl_bihimp_self : aᶜ ⇔ a = ⊥ := @hnot_symm_diff_self αᵒᵈ _ _
@[simp] lemma bihimp_hnot_self : a ⇔ aᶜ = ⊥ := @symm_diff_hnot_self αᵒᵈ _ _
lemma is_compl.bihimp_eq_bot {a b : α} (h : is_compl a b) : a ⇔ b = ⊥ :=
by rw [h.eq_compl, compl_bihimp_self]
end heyting_algebra
section generalized_boolean_algebra
variables [generalized_boolean_algebra α] (a b c d : α)
@[simp] lemma sup_sdiff_symm_diff : (a ⊔ b) \ (a ∆ b) = a ⊓ b :=
sdiff_eq_symm inf_le_sup (by rw symm_diff_eq_sup_sdiff_inf)
lemma disjoint_symm_diff_inf : disjoint (a ∆ b) (a ⊓ b) :=
begin
rw [symm_diff_eq_sup_sdiff_inf],
exact disjoint_sdiff_self_left,
end
lemma inf_symm_diff_distrib_left : a ⊓ (b ∆ c) = (a ⊓ b) ∆ (a ⊓ c) :=
by rw [symm_diff_eq_sup_sdiff_inf, inf_sdiff_distrib_left, inf_sup_left, inf_inf_distrib_left,
symm_diff_eq_sup_sdiff_inf]
lemma inf_symm_diff_distrib_right : (a ∆ b) ⊓ c = (a ⊓ c) ∆ (b ⊓ c) :=
by simp_rw [@inf_comm _ _ _ c, inf_symm_diff_distrib_left]
lemma sdiff_symm_diff : c \ (a ∆ b) = (c ⊓ a ⊓ b) ⊔ ((c \ a) ⊓ (c \ b)) :=
by simp only [(∆), sdiff_sdiff_sup_sdiff']
lemma sdiff_symm_diff' : c \ (a ∆ b) = (c ⊓ a ⊓ b) ⊔ (c \ (a ⊔ b)) :=
by rw [sdiff_symm_diff, sdiff_sup, sup_comm]
@[simp] lemma symm_diff_sdiff_left : (a ∆ b) \ a = b \ a :=
by rw [symm_diff_def, sup_sdiff, sdiff_idem, sdiff_sdiff_self, bot_sup_eq]
@[simp] lemma symm_diff_sdiff_right : (a ∆ b) \ b = a \ b :=
by rw [symm_diff_comm, symm_diff_sdiff_left]
@[simp] lemma sdiff_symm_diff_left : a \ (a ∆ b) = a ⊓ b := by simp [sdiff_symm_diff]
@[simp] lemma sdiff_symm_diff_right : b \ (a ∆ b) = a ⊓ b :=
by rw [symm_diff_comm, inf_comm, sdiff_symm_diff_left]
lemma symm_diff_eq_sup : a ∆ b = a ⊔ b ↔ disjoint a b :=
begin
refine ⟨λ h, _, disjoint.symm_diff_eq_sup⟩,
rw [symm_diff_eq_sup_sdiff_inf, sdiff_eq_self_iff_disjoint] at h,
exact h.of_disjoint_inf_of_le le_sup_left,
end
@[simp] lemma le_symm_diff_iff_left : a ≤ a ∆ b ↔ disjoint a b :=
begin
refine ⟨λ h, _, λ h, h.symm_diff_eq_sup.symm ▸ le_sup_left⟩,
rw symm_diff_eq_sup_sdiff_inf at h,
exact disjoint_iff_inf_le.mpr (le_sdiff_iff.1 $ inf_le_of_left_le h).le,
end
@[simp] lemma le_symm_diff_iff_right : b ≤ a ∆ b ↔ disjoint a b :=
by rw [symm_diff_comm, le_symm_diff_iff_left, disjoint.comm]
lemma symm_diff_symm_diff_left :
a ∆ b ∆ c = (a \ (b ⊔ c)) ⊔ (b \ (a ⊔ c)) ⊔ (c \ (a ⊔ b)) ⊔ (a ⊓ b ⊓ c) :=
calc a ∆ b ∆ c = ((a ∆ b) \ c) ⊔ (c \ (a ∆ b)) : symm_diff_def _ _
... = (a \ (b ⊔ c)) ⊔ (b \ (a ⊔ c)) ⊔
((c \ (a ⊔ b)) ⊔ (c ⊓ a ⊓ b)) :
by rw [sdiff_symm_diff', @sup_comm _ _ (c ⊓ a ⊓ b), symm_diff_sdiff]
... = (a \ (b ⊔ c)) ⊔ (b \ (a ⊔ c)) ⊔
(c \ (a ⊔ b)) ⊔ (a ⊓ b ⊓ c) : by ac_refl
lemma symm_diff_symm_diff_right :
a ∆ (b ∆ c) = (a \ (b ⊔ c)) ⊔ (b \ (a ⊔ c)) ⊔ (c \ (a ⊔ b)) ⊔ (a ⊓ b ⊓ c) :=
calc a ∆ (b ∆ c) = (a \ (b ∆ c)) ⊔ ((b ∆ c) \ a) : symm_diff_def _ _
... = (a \ (b ⊔ c)) ⊔ (a ⊓ b ⊓ c) ⊔
(b \ (c ⊔ a) ⊔ c \ (b ⊔ a)) :
by rw [sdiff_symm_diff', @sup_comm _ _ (a ⊓ b ⊓ c), symm_diff_sdiff]
... = (a \ (b ⊔ c)) ⊔ (b \ (a ⊔ c)) ⊔
(c \ (a ⊔ b)) ⊔ (a ⊓ b ⊓ c) : by ac_refl
lemma symm_diff_assoc : a ∆ b ∆ c = a ∆ (b ∆ c) :=
by rw [symm_diff_symm_diff_left, symm_diff_symm_diff_right]
instance symm_diff_is_assoc : is_associative α (∆) := ⟨symm_diff_assoc⟩
lemma symm_diff_left_comm : a ∆ (b ∆ c) = b ∆ (a ∆ c) :=
by simp_rw [←symm_diff_assoc, symm_diff_comm]
lemma symm_diff_right_comm : a ∆ b ∆ c = a ∆ c ∆ b := by simp_rw [symm_diff_assoc, symm_diff_comm]
lemma symm_diff_symm_diff_symm_diff_comm : (a ∆ b) ∆ (c ∆ d) = (a ∆ c) ∆ (b ∆ d) :=
by simp_rw [symm_diff_assoc, symm_diff_left_comm]
@[simp] lemma symm_diff_symm_diff_cancel_left : a ∆ (a ∆ b) = b := by simp [←symm_diff_assoc]
@[simp] lemma symm_diff_symm_diff_cancel_right : b ∆ a ∆ a = b := by simp [symm_diff_assoc]
@[simp] lemma symm_diff_symm_diff_self' : a ∆ b ∆ a = b :=
by rw [symm_diff_comm,symm_diff_symm_diff_cancel_left]
lemma symm_diff_left_involutive (a : α) : involutive (∆ a) := symm_diff_symm_diff_cancel_right _
lemma symm_diff_right_involutive (a : α) : involutive ((∆) a) := symm_diff_symm_diff_cancel_left _
lemma symm_diff_left_injective (a : α) : injective (∆ a) := (symm_diff_left_involutive _).injective
lemma symm_diff_right_injective (a : α) : injective ((∆) a) :=
(symm_diff_right_involutive _).injective
lemma symm_diff_left_surjective (a : α) : surjective (∆ a) :=
(symm_diff_left_involutive _).surjective
lemma symm_diff_right_surjective (a : α) : surjective ((∆) a) :=
(symm_diff_right_involutive _).surjective
variables {a b c}
@[simp] lemma symm_diff_left_inj : a ∆ b = c ∆ b ↔ a = c := (symm_diff_left_injective _).eq_iff
@[simp] lemma symm_diff_right_inj : a ∆ b = a ∆ c ↔ b = c := (symm_diff_right_injective _).eq_iff
@[simp] lemma symm_diff_eq_left : a ∆ b = a ↔ b = ⊥ :=
calc a ∆ b = a ↔ a ∆ b = a ∆ ⊥ : by rw symm_diff_bot
... ↔ b = ⊥ : by rw symm_diff_right_inj
@[simp] lemma symm_diff_eq_right : a ∆ b = b ↔ a = ⊥ := by rw [symm_diff_comm, symm_diff_eq_left]
protected lemma disjoint.symm_diff_left (ha : disjoint a c) (hb : disjoint b c) :
disjoint (a ∆ b) c :=
by { rw symm_diff_eq_sup_sdiff_inf, exact (ha.sup_left hb).disjoint_sdiff_left }
protected lemma disjoint.symm_diff_right (ha : disjoint a b) (hb : disjoint a c) :
disjoint a (b ∆ c) :=
(ha.symm.symm_diff_left hb.symm).symm
lemma symm_diff_eq_iff_sdiff_eq (ha : a ≤ c) : a ∆ b = c ↔ c \ a = b :=
begin
rw ←symm_diff_of_le ha,
exact ((symm_diff_right_involutive a).to_perm _).apply_eq_iff_eq_symm_apply.trans eq_comm,
end
end generalized_boolean_algebra
section boolean_algebra
variables [boolean_algebra α] (a b c d : α)
/- `cogeneralized_boolean_algebra` isn't actually a typeclass, but the lemmas in here are dual to
the `generalized_boolean_algebra` ones -/
section cogeneralized_boolean_algebra
@[simp] lemma inf_himp_bihimp : (a ⇔ b) ⇨ (a ⊓ b) = a ⊔ b := @sup_sdiff_symm_diff αᵒᵈ _ _ _
lemma codisjoint_bihimp_sup : codisjoint (a ⇔ b) (a ⊔ b) := @disjoint_symm_diff_inf αᵒᵈ _ _ _
@[simp] lemma himp_bihimp_left : a ⇨ (a ⇔ b) = a ⇨ b := @symm_diff_sdiff_left αᵒᵈ _ _ _
@[simp] lemma himp_bihimp_right : b ⇨ (a ⇔ b) = b ⇨ a := @symm_diff_sdiff_right αᵒᵈ _ _ _
@[simp] lemma bihimp_himp_left : (a ⇔ b) ⇨ a = a ⊔ b := @sdiff_symm_diff_left αᵒᵈ _ _ _
@[simp] lemma bihimp_himp_right : (a ⇔ b) ⇨ b = a ⊔ b := @sdiff_symm_diff_right αᵒᵈ _ _ _
@[simp] lemma bihimp_eq_inf : a ⇔ b = a ⊓ b ↔ codisjoint a b := @symm_diff_eq_sup αᵒᵈ _ _ _
@[simp] lemma bihimp_le_iff_left : a ⇔ b ≤ a ↔ codisjoint a b := @le_symm_diff_iff_left αᵒᵈ _ _ _
@[simp] lemma bihimp_le_iff_right : a ⇔ b ≤ b ↔ codisjoint a b := @le_symm_diff_iff_right αᵒᵈ _ _ _
lemma bihimp_assoc : a ⇔ b ⇔ c = a ⇔ (b ⇔ c) := @symm_diff_assoc αᵒᵈ _ _ _ _
instance bihimp_is_assoc : is_associative α (⇔) := ⟨bihimp_assoc⟩
lemma bihimp_left_comm : a ⇔ (b ⇔ c) = b ⇔ (a ⇔ c) := by simp_rw [←bihimp_assoc, bihimp_comm]
lemma bihimp_right_comm : a ⇔ b ⇔ c = a ⇔ c ⇔ b := by simp_rw [bihimp_assoc, bihimp_comm]
lemma bihimp_bihimp_bihimp_comm : (a ⇔ b) ⇔ (c ⇔ d) = (a ⇔ c) ⇔ (b ⇔ d) :=
by simp_rw [bihimp_assoc, bihimp_left_comm]
@[simp] lemma bihimp_bihimp_cancel_left : a ⇔ (a ⇔ b) = b := by simp [←bihimp_assoc]
@[simp] lemma bihimp_bihimp_cancel_right : b ⇔ a ⇔ a = b := by simp [bihimp_assoc]
@[simp] lemma bihimp_bihimp_self : a ⇔ b ⇔ a = b := by rw [bihimp_comm, bihimp_bihimp_cancel_left]
lemma bihimp_left_involutive (a : α) : involutive (⇔ a) := bihimp_bihimp_cancel_right _
lemma bihimp_right_involutive (a : α) : involutive ((⇔) a) := bihimp_bihimp_cancel_left _
lemma bihimp_left_injective (a : α) : injective (⇔ a) := @symm_diff_left_injective αᵒᵈ _ _
lemma bihimp_right_injective (a : α) : injective ((⇔) a) := @symm_diff_right_injective αᵒᵈ _ _
lemma bihimp_left_surjective (a : α) : surjective (⇔ a) := @symm_diff_left_surjective αᵒᵈ _ _
lemma bihimp_right_surjective (a : α) : surjective ((⇔) a) := @symm_diff_right_surjective αᵒᵈ _ _
variables {a b c}
@[simp] lemma bihimp_left_inj : a ⇔ b = c ⇔ b ↔ a = c := (bihimp_left_injective _).eq_iff
@[simp] lemma bihimp_right_inj : a ⇔ b = a ⇔ c ↔ b = c := (bihimp_right_injective _).eq_iff
@[simp] lemma bihimp_eq_left : a ⇔ b = a ↔ b = ⊤ := @symm_diff_eq_left αᵒᵈ _ _ _
@[simp] lemma bihimp_eq_right : a ⇔ b = b ↔ a = ⊤ := @symm_diff_eq_right αᵒᵈ _ _ _
protected lemma codisjoint.bihimp_left (ha : codisjoint a c) (hb : codisjoint b c) :
codisjoint (a ⇔ b) c :=
(ha.inf_left hb).mono_left inf_le_bihimp
protected lemma codisjoint.bihimp_right (ha : codisjoint a b) (hb : codisjoint a c) :
codisjoint a (b ⇔ c) :=
(ha.inf_right hb).mono_right inf_le_bihimp
end cogeneralized_boolean_algebra
lemma symm_diff_eq : a ∆ b = (a ⊓ bᶜ) ⊔ (b ⊓ aᶜ) := by simp only [(∆), sdiff_eq]
lemma bihimp_eq : a ⇔ b = (a ⊔ bᶜ) ⊓ (b ⊔ aᶜ) := by simp only [(⇔), himp_eq]
lemma symm_diff_eq' : a ∆ b = (a ⊔ b) ⊓ (aᶜ ⊔ bᶜ) :=
by rw [symm_diff_eq_sup_sdiff_inf, sdiff_eq, compl_inf]
lemma bihimp_eq' : a ⇔ b = (a ⊓ b) ⊔ (aᶜ ⊓ bᶜ) := @symm_diff_eq' αᵒᵈ _ _ _
lemma symm_diff_top : a ∆ ⊤ = aᶜ := symm_diff_top' _
lemma top_symm_diff : ⊤ ∆ a = aᶜ := top_symm_diff' _
@[simp] lemma compl_symm_diff : (a ∆ b)ᶜ = a ⇔ b :=
by simp_rw [symm_diff, compl_sup_distrib, compl_sdiff, bihimp, inf_comm]
@[simp] lemma compl_bihimp : (a ⇔ b)ᶜ = a ∆ b := @compl_symm_diff αᵒᵈ _ _ _
@[simp] lemma compl_symm_diff_compl : aᶜ ∆ bᶜ = a ∆ b :=
sup_comm.trans $ by simp_rw [compl_sdiff_compl, sdiff_eq, symm_diff_eq]
@[simp] lemma compl_bihimp_compl : aᶜ ⇔ bᶜ = a ⇔ b := @compl_symm_diff_compl αᵒᵈ _ _ _
@[simp] lemma symm_diff_eq_top : a ∆ b = ⊤ ↔ is_compl a b :=
by rw [symm_diff_eq', ←compl_inf, inf_eq_top_iff, compl_eq_top, is_compl_iff, disjoint_iff,
codisjoint_iff, and.comm]
@[simp] lemma bihimp_eq_bot : a ⇔ b = ⊥ ↔ is_compl a b :=
by rw [bihimp_eq', ←compl_sup, sup_eq_bot_iff, compl_eq_bot, is_compl_iff, disjoint_iff,
codisjoint_iff]
@[simp] lemma compl_symm_diff_self : aᶜ ∆ a = ⊤ := hnot_symm_diff_self _
@[simp] lemma symm_diff_compl_self : a ∆ aᶜ = ⊤ := symm_diff_hnot_self _
lemma symm_diff_symm_diff_right' :
a ∆ (b ∆ c) = (a ⊓ b ⊓ c) ⊔ (a ⊓ bᶜ ⊓ cᶜ) ⊔ (aᶜ ⊓ b ⊓ cᶜ) ⊔ (aᶜ ⊓ bᶜ ⊓ c) :=
calc a ∆ (b ∆ c) = (a ⊓ ((b ⊓ c) ⊔ (bᶜ ⊓ cᶜ))) ⊔
(((b ⊓ cᶜ) ⊔ (c ⊓ bᶜ)) ⊓ aᶜ) : by rw [symm_diff_eq, compl_symm_diff,
bihimp_eq', symm_diff_eq]
... = (a ⊓ b ⊓ c) ⊔ (a ⊓ bᶜ ⊓ cᶜ) ⊔
(b ⊓ cᶜ ⊓ aᶜ) ⊔ (c ⊓ bᶜ ⊓ aᶜ) : by rw [inf_sup_left, inf_sup_right,
←sup_assoc, ←inf_assoc, ←inf_assoc]
... = (a ⊓ b ⊓ c) ⊔ (a ⊓ bᶜ ⊓ cᶜ) ⊔
(aᶜ ⊓ b ⊓ cᶜ) ⊔ (aᶜ ⊓ bᶜ ⊓ c) : begin
congr' 1,
{ congr' 1,
rw [inf_comm, inf_assoc], },
{ apply inf_left_right_swap }
end
variables {a b c}
lemma disjoint.le_symm_diff_sup_symm_diff_left (h : disjoint a b) : c ≤ a ∆ c ⊔ b ∆ c :=
begin
transitivity c \ (a ⊓ b),
{ rw [h.eq_bot, sdiff_bot] },
{ rw sdiff_inf,
exact sup_le_sup le_sup_right le_sup_right }
end
lemma disjoint.le_symm_diff_sup_symm_diff_right (h : disjoint b c) : a ≤ a ∆ b ⊔ a ∆ c :=
by { simp_rw symm_diff_comm a, exact h.le_symm_diff_sup_symm_diff_left }
lemma codisjoint.bihimp_inf_bihimp_le_left (h : codisjoint a b) : a ⇔ c ⊓ b ⇔ c ≤ c :=
h.dual.le_symm_diff_sup_symm_diff_left
lemma codisjoint.bihimp_inf_bihimp_le_right (h : codisjoint b c) : a ⇔ b ⊓ a ⇔ c ≤ a :=
h.dual.le_symm_diff_sup_symm_diff_right
end boolean_algebra
/-! ### Prod -/
section prod
@[simp] lemma symm_diff_fst [generalized_coheyting_algebra α] [generalized_coheyting_algebra β]
(a b : α × β) :
(a ∆ b).1 = a.1 ∆ b.1 := rfl
@[simp] lemma symm_diff_snd [generalized_coheyting_algebra α] [generalized_coheyting_algebra β]
(a b : α × β) :
(a ∆ b).2 = a.2 ∆ b.2 := rfl
@[simp] lemma bihimp_fst [generalized_heyting_algebra α] [generalized_heyting_algebra β]
(a b : α × β) :
(a ⇔ b).1 = a.1 ⇔ b.1 := rfl
@[simp] lemma bihimp_snd [generalized_heyting_algebra α] [generalized_heyting_algebra β]
(a b : α × β) :
(a ⇔ b).2 = a.2 ⇔ b.2 := rfl
end prod
/-! ### Pi -/
namespace pi
lemma symm_diff_def [Π i, generalized_coheyting_algebra (π i)] (a b : Π i, π i) :
a ∆ b = λ i, a i ∆ b i := rfl
lemma bihimp_def [Π i, generalized_heyting_algebra (π i)] (a b : Π i, π i) :
a ⇔ b = λ i, a i ⇔ b i := rfl
@[simp] lemma symm_diff_apply [Π i, generalized_coheyting_algebra (π i)] (a b : Π i, π i) (i : ι) :
(a ∆ b) i = a i ∆ b i := rfl
@[simp] lemma bihimp_apply [Π i, generalized_heyting_algebra (π i)] (a b : Π i, π i) (i : ι) :
(a ⇔ b) i = a i ⇔ b i := rfl
end pi
|
98a6501f985d14ad754e5c4f9a9368ff1573c288 | 367134ba5a65885e863bdc4507601606690974c1 | /src/topology/algebra/continuous_functions.lean | f34ccf6863545fac2756fbae7f18ad3d8207cc43 | [
"Apache-2.0"
] | permissive | kodyvajjha/mathlib | 9bead00e90f68269a313f45f5561766cfd8d5cad | b98af5dd79e13a38d84438b850a2e8858ec21284 | refs/heads/master | 1,624,350,366,310 | 1,615,563,062,000 | 1,615,563,062,000 | 162,666,963 | 0 | 0 | Apache-2.0 | 1,545,367,651,000 | 1,545,367,651,000 | null | UTF-8 | Lean | false | false | 14,533 | lean | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Nicolò Cavalleri
-/
import topology.algebra.module
import topology.continuous_map
/-!
# Algebraic structures over continuous functions
In this file we define instances of algebraic structures over continuous functions. Instances are
present both in the case of the subtype of continuous functions and the type of continuous bundled
functions. Both implementations have advantages and disadvantages, but many experienced people in
Zulip have expressed a preference towards continuous bundled maps, so when there is no particular
reason to use the subtype, continuous bundled functions should be used for the sake of uniformity.
-/
local attribute [elab_simple] continuous.comp
namespace continuous_functions
variables {α : Type*} {β : Type*} [topological_space α] [topological_space β]
variables {f g : {f : α → β | continuous f }}
instance : has_coe_to_fun {f : α → β | continuous f} := ⟨_, subtype.val⟩
end continuous_functions
namespace continuous_map
@[to_additive]
instance has_mul {α : Type*} {β : Type*} [topological_space α] [topological_space β]
[has_mul β] [has_continuous_mul β] : has_mul C(α, β) :=
⟨λ f g, ⟨f * g, continuous_mul.comp (f.continuous.prod_mk g.continuous : _)⟩⟩
@[to_additive]
instance {α : Type*} {β : Type*} [topological_space α] [topological_space β]
[has_one β] : has_one C(α, β) := ⟨const (1 : β)⟩
end continuous_map
section group_structure
/-!
### Group stucture
In this section we show that continuous functions valued in a topological group inherit
the structure of a group.
-/
section subtype
@[to_additive]
instance continuous_submonoid (α : Type*) (β : Type*) [topological_space α] [topological_space β]
[monoid β] [has_continuous_mul β] : is_submonoid { f : α → β | continuous f } :=
{ one_mem := @continuous_const _ _ _ _ 1,
mul_mem := λ f g fc gc, continuous.comp
has_continuous_mul.continuous_mul (continuous.prod_mk fc gc : _) }
@[to_additive]
instance continuous_subgroup (α : Type*) (β : Type*) [topological_space α] [topological_space β]
[group β] [topological_group β] : is_subgroup { f : α → β | continuous f } :=
{ inv_mem := λ f fc, continuous.comp (@topological_group.continuous_inv β _ _ _) fc,
..continuous_submonoid α β, }.
@[to_additive]
instance continuous_monoid {α : Type*} {β : Type*} [topological_space α] [topological_space β]
[monoid β] [has_continuous_mul β] : monoid { f : α → β | continuous f } :=
subtype.monoid
@[to_additive]
instance continuous_group {α : Type*} {β : Type*} [topological_space α] [topological_space β]
[group β] [topological_group β] : group { f : α → β | continuous f } :=
subtype.group
@[to_additive]
instance continuous_comm_group {α : Type*} {β : Type*} [topological_space α] [topological_space β]
[comm_group β] [topological_group β] : comm_group { f : α → β | continuous f } :=
@subtype.comm_group _ _ _ (continuous_subgroup α β) -- infer_instance doesn't work?!
end subtype
section continuous_map
@[to_additive]
instance continuous_map_semigroup {α : Type*} {β : Type*} [topological_space α]
[topological_space β] [semigroup β] [has_continuous_mul β] : semigroup C(α, β) :=
{ mul_assoc := λ a b c, by ext; exact mul_assoc _ _ _,
..continuous_map.has_mul}
@[to_additive]
instance continuous_map_monoid {α : Type*} {β : Type*} [topological_space α] [topological_space β]
[monoid β] [has_continuous_mul β] : monoid C(α, β) :=
{ one_mul := λ a, by ext; exact one_mul _,
mul_one := λ a, by ext; exact mul_one _,
..continuous_map_semigroup,
..continuous_map.has_one }
@[to_additive]
instance continuous_map_comm_monoid {α : Type*} {β : Type*} [topological_space α]
[topological_space β] [comm_monoid β] [has_continuous_mul β] : comm_monoid C(α, β) :=
{ one_mul := λ a, by ext; exact one_mul _,
mul_one := λ a, by ext; exact mul_one _,
mul_comm := λ a b, by ext; exact mul_comm _ _,
..continuous_map_semigroup,
..continuous_map.has_one }
@[to_additive]
instance continuous_map_group {α : Type*} {β : Type*} [topological_space α] [topological_space β]
[group β] [topological_group β] : group C(α, β) :=
{ inv := λ f, ⟨λ x, (f x)⁻¹, continuous_inv.comp f.continuous⟩,
mul_left_inv := λ a, by ext; exact mul_left_inv _,
..continuous_map_monoid }
@[to_additive]
instance continuous_map_comm_group {α : Type*} {β : Type*} [topological_space α]
[topological_space β] [comm_group β] [topological_group β] : comm_group C(α, β) :=
{ ..continuous_map_group,
..continuous_map_comm_monoid }
end continuous_map
end group_structure
section ring_structure
/-!
### Ring stucture
In this section we show that continuous functions valued in a topological ring `R` inherit
the structure of a ring.
-/
section subtype
instance continuous_subring (α : Type*) (R : Type*) [topological_space α] [topological_space R]
[ring R] [topological_ring R] : is_subring { f : α → R | continuous f } :=
{ ..continuous_add_subgroup α R,
..continuous_submonoid α R }.
instance continuous_ring {α : Type*} {R : Type*} [topological_space α] [topological_space R]
[ring R] [topological_ring R] : ring { f : α → R | continuous f } :=
@subtype.ring _ _ _ (continuous_subring α R) -- infer_instance doesn't work?!
instance continuous_comm_ring {α : Type*} {R : Type*} [topological_space α] [topological_space R]
[comm_ring R] [topological_ring R] : comm_ring { f : α → R | continuous f } :=
@subtype.comm_ring _ _ _ (continuous_subring α R) -- infer_instance doesn't work?!
end subtype
section continuous_map
instance continuous_map_semiring {α : Type*} {β : Type*} [topological_space α] [topological_space β]
[semiring β] [topological_semiring β] : semiring C(α, β) :=
{ left_distrib := λ a b c, by ext; exact left_distrib _ _ _,
right_distrib := λ a b c, by ext; exact right_distrib _ _ _,
zero_mul := λ a, by ext; exact zero_mul _,
mul_zero := λ a, by ext; exact mul_zero _,
..continuous_map_add_comm_monoid,
..continuous_map_monoid }
instance continuous_map_ring {α : Type*} {β : Type*} [topological_space α] [topological_space β]
[ring β] [topological_ring β] : ring C(α, β) :=
{ ..continuous_map_semiring,
..continuous_map_add_comm_group, }
instance continuous_map_comm_ring {α : Type*} {β : Type*} [topological_space α]
[topological_space β] [comm_ring β] [topological_ring β] : comm_ring C(α, β) :=
{ ..continuous_map_semiring,
..continuous_map_add_comm_group,
..continuous_map_comm_monoid,}
end continuous_map
end ring_structure
local attribute [ext] subtype.eq
section semimodule_structure
/-!
### Semiodule stucture
In this section we show that continuous functions valued in a topological semimodule `M` over a
topological semiring `R` inherit the structure of a semimodule.
-/
section subtype
instance continuous_has_scalar {α : Type*} [topological_space α]
{R : Type*} [semiring R] [topological_space R]
{M : Type*} [topological_space M] [add_comm_group M]
[semimodule R M] [topological_semimodule R M] :
has_scalar R { f : α → M | continuous f } :=
⟨λ r f, ⟨r • f, continuous_const.smul f.property⟩⟩
instance continuous_semimodule {α : Type*} [topological_space α]
{R : Type*} [semiring R] [topological_space R]
{M : Type*} [topological_space M] [add_comm_group M] [topological_add_group M]
[semimodule R M] [topological_semimodule R M] :
semimodule R { f : α → M | continuous f } :=
semimodule.of_core $
{ smul := (•),
smul_add := λ c f g, by ext x; exact smul_add c (f x) (g x),
add_smul := λ c₁ c₂ f, by ext x; exact add_smul c₁ c₂ (f x),
mul_smul := λ c₁ c₂ f, by ext x; exact mul_smul c₁ c₂ (f x),
one_smul := λ f, by ext x; exact one_smul R (f x) }
end subtype
section continuous_map
instance continuous_map_has_scalar {α : Type*} [topological_space α]
{R : Type*} [semiring R] [topological_space R]
{M : Type*} [topological_space M] [add_comm_monoid M]
[semimodule R M] [topological_semimodule R M] :
has_scalar R C(α, M) :=
⟨λ r f, ⟨r • f, continuous_const.smul f.continuous⟩⟩
instance continuous_map_semimodule {α : Type*} [topological_space α]
{R : Type*} [semiring R] [topological_space R]
{M : Type*} [topological_space M] [add_comm_monoid M] [has_continuous_add M]
[semimodule R M] [topological_semimodule R M] :
semimodule R C(α, M) :=
{ smul := (•),
smul_add := λ c f g, by { ext, exact smul_add c (f x) (g x) },
add_smul := λ c₁ c₂ f, by { ext, exact add_smul c₁ c₂ (f x) },
mul_smul := λ c₁ c₂ f, by { ext, exact mul_smul c₁ c₂ (f x) },
one_smul := λ f, by { ext, exact one_smul R (f x) },
zero_smul := λ f, by { ext, exact zero_smul _ _ },
smul_zero := λ r, by { ext, exact smul_zero _ } }
end continuous_map
end semimodule_structure
section algebra_structure
/-!
### Algebra structure
In this section we show that continuous functions valued in a topological algebra `A` over a ring
`R` inherit the structure of an algebra. Note that the hypothesis that `A` is a topological algebra
is obtained by requiring that `A` be both a `topological_semimodule` and a `topological_semiring`
(by now we require `topological_ring`: see TODO below).-/
section subtype
variables {α : Type*} [topological_space α]
{R : Type*} [comm_semiring R]
{A : Type*} [topological_space A] [ring A]
[algebra R A] [topological_ring A]
/-- Continuous constant functions as a `ring_hom`. -/
def continuous.C : R →+* { f : α → A | continuous f } :=
{ to_fun := λ c : R, ⟨λ x: α, ((algebra_map R A) c), continuous_const⟩,
map_one' := by ext x; exact (algebra_map R A).map_one,
map_mul' := λ c₁ c₂, by ext x; exact (algebra_map R A).map_mul _ _,
map_zero' := by ext x; exact (algebra_map R A).map_zero,
map_add' := λ c₁ c₂, by ext x; exact (algebra_map R A).map_add _ _ }
variables [topological_space R] [topological_semimodule R A]
instance : algebra R { f : α → A | continuous f } :=
{ to_ring_hom := continuous.C,
commutes' := λ c f, by ext x; exact algebra.commutes' _ _,
smul_def' := λ c f, by ext x; exact algebra.smul_def' _ _,
..continuous_semimodule,
..continuous_ring }
/- TODO: We are assuming `A` to be a ring and not a semiring just because there is not yet an
instance of semiring. In turn, we do not want to define yet an instance of semiring because there is
no `is_subsemiring` but only `subsemiring`, and it will make sense to change this when the whole
file will have no more `is_subobject`s but only `subobject`s. It does not make sense to change
it yet in this direction as `subring` does not exist yet, so everything is being blocked by
`subring`: afterwards everything will need to be updated to the new conventions of Mathlib.
Then the instance of `topological_ring` can also be removed, as it is below for `continuous_map`. -/
end subtype
section continuous_map
variables {α : Type*} [topological_space α]
{R : Type*} [comm_semiring R]
{A : Type*} [topological_space A] [semiring A]
[algebra R A] [topological_semiring A]
/-- Continuous constant functions as a `ring_hom`. -/
def continuous_map.C : R →+* C(α, A) :=
{ to_fun := λ c : R, ⟨λ x: α, ((algebra_map R A) c), continuous_const⟩,
map_one' := by ext x; exact (algebra_map R A).map_one,
map_mul' := λ c₁ c₂, by ext x; exact (algebra_map R A).map_mul _ _,
map_zero' := by ext x; exact (algebra_map R A).map_zero,
map_add' := λ c₁ c₂, by ext x; exact (algebra_map R A).map_add _ _ }
variables [topological_space R] [topological_semimodule R A]
instance continuous_map_algebra : algebra R C(α, A) :=
{ to_ring_hom := continuous_map.C,
commutes' := λ c f, by ext x; exact algebra.commutes' _ _,
smul_def' := λ c f, by ext x; exact algebra.smul_def' _ _,
..continuous_map_semiring }
end continuous_map
end algebra_structure
section module_over_continuous_functions
/-!
### Structure as module over scalar functions
If `M` is a module over `R`, then we show that the space of continuous functions from `α` to `M`
is naturally a module over the ring of continuous functions from `α` to `M`. -/
section subtype
instance continuous_has_scalar' {α : Type*} [topological_space α]
{R : Type*} [semiring R] [topological_space R]
{M : Type*} [topological_space M] [add_comm_group M]
[semimodule R M] [topological_semimodule R M] :
has_scalar { f : α → R | continuous f } { f : α → M | continuous f } :=
⟨λ f g, ⟨λ x, (f x) • (g x), (continuous.smul f.2 g.2)⟩⟩
instance continuous_module' {α : Type*} [topological_space α]
(R : Type*) [ring R] [topological_space R] [topological_ring R]
(M : Type*) [topological_space M] [add_comm_group M] [topological_add_group M]
[module R M] [topological_module R M]
: module { f : α → R | continuous f } { f : α → M | continuous f } :=
semimodule.of_core $
{ smul := (•),
smul_add := λ c f g, by ext x; exact smul_add (c x) (f x) (g x),
add_smul := λ c₁ c₂ f, by ext x; exact add_smul (c₁ x) (c₂ x) (f x),
mul_smul := λ c₁ c₂ f, by ext x; exact mul_smul (c₁ x) (c₂ x) (f x),
one_smul := λ f, by ext x; exact one_smul R (f x) }
end subtype
section continuous_map
instance continuous_map_has_scalar' {α : Type*} [topological_space α]
{R : Type*} [semiring R] [topological_space R]
{M : Type*} [topological_space M] [add_comm_monoid M]
[semimodule R M] [topological_semimodule R M] :
has_scalar C(α, R) C(α, M) :=
⟨λ f g, ⟨λ x, (f x) • (g x), (continuous.smul f.2 g.2)⟩⟩
instance continuous_map_module' {α : Type*} [topological_space α]
(R : Type*) [ring R] [topological_space R] [topological_ring R]
(M : Type*) [topological_space M] [add_comm_monoid M] [has_continuous_add M]
[semimodule R M] [topological_semimodule R M] :
semimodule C(α, R) C(α, M) :=
{ smul := (•),
smul_add := λ c f g, by ext x; exact smul_add (c x) (f x) (g x),
add_smul := λ c₁ c₂ f, by ext x; exact add_smul (c₁ x) (c₂ x) (f x),
mul_smul := λ c₁ c₂ f, by ext x; exact mul_smul (c₁ x) (c₂ x) (f x),
one_smul := λ f, by ext x; exact one_smul R (f x),
zero_smul := λ f, by ext x; exact zero_smul _ _,
smul_zero := λ r, by ext x; exact smul_zero _, }
end continuous_map
end module_over_continuous_functions
|
aba15b6761c5b6e2b371b83122fa461f1e49f035 | 137c667471a40116a7afd7261f030b30180468c2 | /src/analysis/calculus/parametric_integral.lean | 5db27ab9e8b4c20c1a918bfe17c0b94dcd38822f | [
"Apache-2.0"
] | permissive | bragadeesh153/mathlib | 46bf814cfb1eecb34b5d1549b9117dc60f657792 | b577bb2cd1f96eb47031878256856020b76f73cd | refs/heads/master | 1,687,435,188,334 | 1,626,384,207,000 | 1,626,384,207,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 14,753 | lean | /-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import measure_theory.set_integral
import analysis.calculus.mean_value
/-!
# Derivatives of integrals depending on parameters
A parametric integral is a function with shape `f = λ x : H, ∫ a : α, F x a ∂μ` for some
`F : H → α → E`, where `H` and `E` are normed spaces and `α` is a measured space with measure `μ`.
We already know from `continuous_of_dominated` in `measure_theory.bochner_integral` how to
guarantee that `f` is continuous using the dominated convergence theorem. In this file,
we want to express the derivative of `f` as the integral of the derivative of `F` with respect
to `x`.
## Main results
As explained above, all results express the derivative of a parametric integral as the integral of
a derivative. The variations come from the assumptions and from the different ways of expressing
derivative, especially Fréchet derivatives vs elementary derivative of function of one real
variable.
* `has_fderiv_at_of_dominated_loc_of_lip`: this version assumes
`F x` is ae-measurable for x near `x₀`, `F x₀` is integrable,
`λ x, F x a` has derivative `F' a : H →L[ℝ] E` at `x₀` which is ae-measurable,
`λ x, F x a` is locally Lipschitz near `x₀` for almost every `a`, with a Lipschitz bound which
is integrable with respect to `a`. A subtle point is that the "near x₀" in the last condition
has to be uniform in `a`. This is controlled by a positive number `ε`.
* `has_fderiv_at_of_dominated_of_fderiv_le`: this version assume `λ x, F x a` has derivative
`F' x a` for `x` near `x₀` and `F' x` is bounded by an integrable function independent from
`x` near `x₀`.
`has_deriv_at_of_dominated_loc_of_lip` and `has_deriv_at_of_dominated_loc_of_deriv_le ` are versions
of the above two results that assume `H = ℝ` and use the high-school derivative `deriv` instead of
Fréchet derivative `fderiv`.
-/
noncomputable theory
open topological_space measure_theory filter metric
open_locale topological_space filter
variables {α : Type*} [measurable_space α] {μ : measure α}
{E : Type*} [normed_group E] [normed_space ℝ E]
[complete_space E] [second_countable_topology E]
[measurable_space E] [borel_space E]
{H : Type*} [normed_group H] [normed_space ℝ H]
[second_countable_topology $ H →L[ℝ] E]
/-- Differentiation under integral of `x ↦ ∫ F x a` at a given point `x₀`, assuming
`F x₀` is integrable, `x ↦ F x a` is locally Lipschitz on a ball around `x₀` for ae `a` with
integrable Lipschitz bound (with a ball radius independent of `a`), and `F x` is
ae-measurable for `x` in the same ball. See `has_fderiv_at_of_dominated_loc_of_lip` for a
slightly more general version. -/
lemma has_fderiv_at_of_dominated_loc_of_lip' {F : H → α → E} {F' : α → (H →L[ℝ] E)} {x₀ : H}
{bound : α → ℝ}
{ε : ℝ} (ε_pos : 0 < ε)
(hF_meas : ∀ x ∈ ball x₀ ε, ae_measurable (F x) μ)
(hF_int : integrable (F x₀) μ)
(hF'_meas : ae_measurable F' μ)
(h_lipsch : ∀ᵐ a ∂μ, lipschitz_on_with (real.nnabs $ bound a) (λ x, F x a) (ball x₀ ε))
(bound_integrable : integrable (bound : α → ℝ) μ)
(h_diff : ∀ᵐ a ∂μ, has_fderiv_at (λ x, F x a) (F' a) x₀) :
integrable F' μ ∧ has_fderiv_at (λ x, ∫ a, F x a ∂μ) (∫ a, F' a ∂μ) x₀ :=
begin
have x₀_in : x₀ ∈ ball x₀ ε := mem_ball_self ε_pos,
have nneg : ∀ x, 0 ≤ ∥x - x₀∥⁻¹ := λ x, inv_nonneg.mpr (norm_nonneg _) ,
set b : α → ℝ := λ a, abs (bound a),
have b_int : integrable b μ := bound_integrable.norm,
have b_nonneg : ∀ a, 0 ≤ b a := λ a, abs_nonneg _,
have hF_int' : ∀ x ∈ ball x₀ ε, integrable (F x) μ,
{ intros x x_in,
have : ∀ᵐ a ∂μ, ∥F x₀ a - F x a∥ ≤ ε * ∥(bound a : ℝ)∥,
{ apply h_lipsch.mono,
intros a ha,
rw lipschitz_on_with_iff_norm_sub_le at ha,
apply (ha x₀ x₀_in x x_in).trans,
rw [mul_comm, nnreal.coe_nnabs, real.norm_eq_abs],
rw [mem_ball, dist_eq_norm, norm_sub_rev] at x_in,
exact mul_le_mul_of_nonneg_right (le_of_lt x_in) (abs_nonneg _) },
exact integrable_of_norm_sub_le (hF_meas x x_in) hF_int
(integrable.const_mul bound_integrable.norm ε) this },
have hF'_int : integrable F' μ,
{ have : ∀ᵐ a ∂μ, ∥F' a∥ ≤ b a,
{ apply (h_diff.and h_lipsch).mono,
rintros a ⟨ha_diff, ha_lip⟩,
exact ha_diff.le_of_lip (ball_mem_nhds _ ε_pos) ha_lip },
exact b_int.mono' hF'_meas this },
refine ⟨hF'_int, _⟩,
have h_ball: ball x₀ ε ∈ 𝓝 x₀ := ball_mem_nhds x₀ ε_pos,
have : ∀ᶠ x in 𝓝 x₀,
∥x - x₀∥⁻¹ * ∥∫ a, F x a ∂μ - ∫ a, F x₀ a ∂μ - (∫ a, F' a ∂μ) (x - x₀)∥ =
∥∫ a, ∥x - x₀∥⁻¹ • (F x a - F x₀ a - F' a (x - x₀)) ∂μ∥,
{ apply mem_sets_of_superset (ball_mem_nhds _ ε_pos),
intros x x_in,
rw [set.mem_set_of_eq, ← norm_smul_of_nonneg (nneg _), integral_smul,
integral_sub, integral_sub, ← continuous_linear_map.integral_apply hF'_int],
exacts [hF_int' x x_in, hF_int, (hF_int' x x_in).sub hF_int,
hF'_int.apply_continuous_linear_map _] },
rw [has_fderiv_at_iff_tendsto, tendsto_congr' this, ← tendsto_zero_iff_norm_tendsto_zero,
← show ∫ (a : α), ∥x₀ - x₀∥⁻¹ • (F x₀ a - F x₀ a - (F' a) (x₀ - x₀)) ∂μ = 0, by simp],
apply tendsto_integral_filter_of_dominated_convergence,
{ apply is_countably_generated_nhds },
{ filter_upwards [h_ball],
intros x x_in,
apply ae_measurable.const_smul,
exact ((hF_meas _ x_in).sub (hF_meas _ x₀_in)).sub (hF'_meas.apply_continuous_linear_map _) },
{ simp [measurable_const] },
{ apply mem_sets_of_superset h_ball,
intros x hx,
apply (h_diff.and h_lipsch).mono,
rintros a ⟨ha_deriv, ha_bound⟩,
show ∥∥x - x₀∥⁻¹ • (F x a - F x₀ a - F' a (x - x₀))∥ ≤ b a + ∥F' a∥,
replace ha_bound : ∥F x a - F x₀ a∥ ≤ b a * ∥x - x₀∥,
{ rw lipschitz_on_with_iff_norm_sub_le at ha_bound,
exact ha_bound _ hx _ x₀_in },
calc ∥∥x - x₀∥⁻¹ • (F x a - F x₀ a - F' a (x - x₀))∥
= ∥∥x - x₀∥⁻¹ • (F x a - F x₀ a) - ∥x - x₀∥⁻¹ • F' a (x - x₀)∥ : by rw smul_sub
... ≤ ∥∥x - x₀∥⁻¹ • (F x a - F x₀ a)∥ + ∥∥x - x₀∥⁻¹ • F' a (x - x₀)∥ : norm_sub_le _ _
... = ∥x - x₀∥⁻¹ * ∥F x a - F x₀ a∥ + ∥x - x₀∥⁻¹ * ∥F' a (x - x₀)∥ :
by { rw [norm_smul_of_nonneg, norm_smul_of_nonneg] ; exact nneg _}
... ≤ ∥x - x₀∥⁻¹ * (b a * ∥x - x₀∥) + ∥x - x₀∥⁻¹ * (∥F' a∥ * ∥x - x₀∥) : add_le_add _ _
... ≤ b a + ∥F' a∥ : _,
exact mul_le_mul_of_nonneg_left ha_bound (nneg _),
apply mul_le_mul_of_nonneg_left ((F' a).le_op_norm _) (nneg _),
by_cases h : ∥x - x₀∥ = 0,
{ simpa [h] using add_nonneg (b_nonneg a) (norm_nonneg (F' a)) },
{ field_simp [h] } },
{ exact b_int.add hF'_int.norm },
{ apply h_diff.mono,
intros a ha,
suffices : tendsto (λ x, ∥x - x₀∥⁻¹ • (F x a - F x₀ a - F' a (x - x₀))) (𝓝 x₀) (𝓝 0),
by simpa,
rw tendsto_zero_iff_norm_tendsto_zero,
have : (λ x, ∥x - x₀∥⁻¹ * ∥F x a - F x₀ a - F' a (x - x₀)∥) =
λ x, ∥∥x - x₀∥⁻¹ • (F x a - F x₀ a - F' a (x - x₀))∥,
{ ext x,
rw norm_smul_of_nonneg (nneg _) },
rwa [has_fderiv_at_iff_tendsto, this] at ha },
end
/-- Differentiation under integral of `x ↦ ∫ F x a` at a given point `x₀`, assuming
`F x₀` is integrable, `x ↦ F x a` is locally Lipschitz on a ball around `x₀` for ae `a`
(with a ball radius independent of `a`) with integrable Lipschitz bound, and `F x` is ae-measurable
for `x` in a possibly smaller neighborhood of `x₀`. -/
lemma has_fderiv_at_of_dominated_loc_of_lip {F : H → α → E} {F' : α → (H →L[ℝ] E)} {x₀ : H}
{bound : α → ℝ}
{ε : ℝ} (ε_pos : 0 < ε)
(hF_meas : ∀ᶠ x in 𝓝 x₀, ae_measurable (F x) μ)
(hF_int : integrable (F x₀) μ)
(hF'_meas : ae_measurable F' μ)
(h_lip : ∀ᵐ a ∂μ, lipschitz_on_with (real.nnabs $ bound a) (λ x, F x a) (ball x₀ ε))
(bound_integrable : integrable (bound : α → ℝ) μ)
(h_diff : ∀ᵐ a ∂μ, has_fderiv_at (λ x, F x a) (F' a) x₀) :
integrable F' μ ∧ has_fderiv_at (λ x, ∫ a, F x a ∂μ) (∫ a, F' a ∂μ) x₀ :=
begin
obtain ⟨ε', ε'_pos, h'⟩ : ∃ ε' > 0, ∀ x ∈ ball x₀ ε', ae_measurable (F x) μ,
by simpa using nhds_basis_ball.eventually_iff.mp hF_meas,
set δ := min ε ε',
have δ_pos : 0 < δ := lt_min ε_pos ε'_pos,
replace h' : ∀ x, x ∈ ball x₀ δ → ae_measurable (F x) μ,
{ intros x x_in,
exact h' _ (ball_subset_ball (min_le_right ε ε') x_in) },
replace h_lip : ∀ᵐ (a : α) ∂μ, lipschitz_on_with (real.nnabs $ bound a) (λ x, F x a) (ball x₀ δ),
{ apply h_lip.mono,
intros a lip,
exact lip.mono (ball_subset_ball $ min_le_left ε ε') },
apply has_fderiv_at_of_dominated_loc_of_lip' δ_pos ; assumption
end
/-- Differentiation under integral of `x ↦ ∫ F x a` at a given point `x₀`, assuming
`F x₀` is integrable, `x ↦ F x a` is differentiable on a ball around `x₀` for ae `a` with
derivative norm uniformly bounded by an integrable function (the ball radius is independent of `a`),
and `F x` is ae-measurable for `x` in a possibly smaller neighborhood of `x₀`. -/
lemma has_fderiv_at_of_dominated_of_fderiv_le {F : H → α → E} {F' : H → α → (H →L[ℝ] E)} {x₀ : H}
{bound : α → ℝ}
{ε : ℝ} (ε_pos : 0 < ε)
(hF_meas : ∀ᶠ x in 𝓝 x₀, ae_measurable (F x) μ)
(hF_int : integrable (F x₀) μ)
(hF'_meas : ae_measurable (F' x₀) μ)
(h_bound : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, ∥F' x a∥ ≤ bound a)
(bound_integrable : integrable (bound : α → ℝ) μ)
(h_diff : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, has_fderiv_at (λ x, F x a) (F' x a) x) :
has_fderiv_at (λ x, ∫ a, F x a ∂μ) (∫ a, F' x₀ a ∂μ) x₀ :=
begin
have x₀_in : x₀ ∈ ball x₀ ε := mem_ball_self ε_pos,
have diff_x₀ : ∀ᵐ a ∂μ, has_fderiv_at (λ x, F x a) (F' x₀ a) x₀ :=
h_diff.mono (λ a ha, ha x₀ x₀_in),
have : ∀ᵐ a ∂μ, lipschitz_on_with (real.nnabs (bound a)) (λ x, F x a) (ball x₀ ε),
{ apply (h_diff.and h_bound).mono,
rintros a ⟨ha_deriv, ha_bound⟩,
have bound_nonneg : 0 ≤ bound a := (norm_nonneg (F' x₀ a)).trans (ha_bound x₀ x₀_in),
rw show real.nnabs (bound a) = real.to_nnreal (bound a), by simp [bound_nonneg],
apply convex.lipschitz_on_with_of_norm_has_fderiv_within_le _ ha_bound (convex_ball _ _),
intros x x_in,
exact (ha_deriv x x_in).has_fderiv_within_at, },
exact (has_fderiv_at_of_dominated_loc_of_lip ε_pos hF_meas hF_int
hF'_meas this bound_integrable diff_x₀).2
end
/-- Derivative under integral of `x ↦ ∫ F x a` at a given point `x₀ : ℝ`, assuming
`F x₀` is integrable, `x ↦ F x a` is locally Lipschitz on an interval around `x₀` for ae `a`
(with interval radius independent of `a`) with integrable Lipschitz bound, and `F x` is
ae-measurable for `x` in a possibly smaller neighborhood of `x₀`. -/
lemma has_deriv_at_of_dominated_loc_of_lip {F : ℝ → α → E} {F' : α → E} {x₀ : ℝ}
{ε : ℝ} (ε_pos : 0 < ε)
(hF_meas : ∀ᶠ x in 𝓝 x₀, ae_measurable (F x) μ)
(hF_int : integrable (F x₀) μ)
(hF'_meas : ae_measurable F' μ) {bound : α → ℝ}
(h_lipsch : ∀ᵐ a ∂μ, lipschitz_on_with (real.nnabs $ bound a) (λ x, F x a) (ball x₀ ε))
(bound_integrable : integrable (bound : α → ℝ) μ)
(h_diff : ∀ᵐ a ∂μ, has_deriv_at (λ x, F x a) (F' a) x₀) :
(integrable F' μ) ∧ has_deriv_at (λ x, ∫ a, F x a ∂μ) (∫ a, F' a ∂μ) x₀ :=
begin
have hm := (continuous_linear_map.smul_rightL ℝ ℝ E 1).continuous.measurable.comp_ae_measurable
hF'_meas,
cases has_fderiv_at_of_dominated_loc_of_lip ε_pos hF_meas hF_int hm h_lipsch bound_integrable
h_diff with hF'_int key,
replace hF'_int : integrable F' μ,
{ rw [← integrable_norm_iff hm] at hF'_int,
simpa only [integrable_norm_iff, hF'_meas, one_mul, continuous_linear_map.norm_id_field',
continuous_linear_map.norm_smul_rightL_apply] using hF'_int},
refine ⟨hF'_int, _⟩,
simp_rw has_deriv_at_iff_has_fderiv_at at h_diff ⊢,
rwa continuous_linear_map.integral_comp_comm _ hF'_int at key,
all_goals { apply_instance, },
end
/-- Derivative under integral of `x ↦ ∫ F x a` at a given point `x₀ : ℝ`, assuming
`F x₀` is integrable, `x ↦ F x a` is differentiable on an interval around `x₀` for ae `a`
(with interval radius independent of `a`) with derivative uniformly bounded by an integrable
function, and `F x` is ae-measurable for `x` in a possibly smaller neighborhood of `x₀`. -/
lemma has_deriv_at_of_dominated_loc_of_deriv_le {F : ℝ → α → E} {F' : ℝ → α → E} {x₀ : ℝ}
{ε : ℝ} (ε_pos : 0 < ε)
(hF_meas : ∀ᶠ x in 𝓝 x₀, ae_measurable (F x) μ)
(hF_int : integrable (F x₀) μ)
(hF'_meas : ae_measurable (F' x₀) μ)
{bound : α → ℝ}
(h_bound : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, ∥F' x a∥ ≤ bound a)
(bound_integrable : integrable bound μ)
(h_diff : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, has_deriv_at (λ x, F x a) (F' x a) x) :
(integrable (F' x₀) μ) ∧ has_deriv_at (λn, ∫ a, F n a ∂μ) (∫ a, F' x₀ a ∂μ) x₀ :=
begin
have x₀_in : x₀ ∈ ball x₀ ε := mem_ball_self ε_pos,
have diff_x₀ : ∀ᵐ a ∂μ, has_deriv_at (λ x, F x a) (F' x₀ a) x₀ :=
h_diff.mono (λ a ha, ha x₀ x₀_in),
have : ∀ᵐ a ∂μ, lipschitz_on_with (real.nnabs (bound a)) (λ (x : ℝ), F x a) (ball x₀ ε),
{ apply (h_diff.and h_bound).mono,
rintros a ⟨ha_deriv, ha_bound⟩,
have bound_nonneg : 0 ≤ bound a := (norm_nonneg (F' x₀ a)).trans (ha_bound x₀ x₀_in),
rw show real.nnabs (bound a) = real.to_nnreal (bound a), by simp [bound_nonneg],
apply convex.lipschitz_on_with_of_norm_has_deriv_within_le (convex_ball _ _)
(λ x x_in, (ha_deriv x x_in).has_deriv_within_at) ha_bound },
exact has_deriv_at_of_dominated_loc_of_lip ε_pos hF_meas hF_int hF'_meas this
bound_integrable diff_x₀
end
|
857c645646756f477b6fc798e412424a132f009f | 4727251e0cd73359b15b664c3170e5d754078599 | /src/control/equiv_functor.lean | 5305c7e0136e6eeea1926d4cb72e975b9167c9e7 | [
"Apache-2.0"
] | permissive | Vierkantor/mathlib | 0ea59ac32a3a43c93c44d70f441c4ee810ccceca | 83bc3b9ce9b13910b57bda6b56222495ebd31c2f | refs/heads/master | 1,658,323,012,449 | 1,652,256,003,000 | 1,652,256,003,000 | 209,296,341 | 0 | 1 | Apache-2.0 | 1,568,807,655,000 | 1,568,807,655,000 | null | UTF-8 | Lean | false | false | 3,278 | lean | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import logic.equiv.basic
/-!
# Functions functorial with respect to equivalences
An `equiv_functor` is a function from `Type → Type` equipped with the additional data of
coherently mapping equivalences to equivalences.
In categorical language, it is an endofunctor of the "core" of the category `Type`.
-/
universes u₀ u₁ u₂ v₀ v₁ v₂
open function
/--
An `equiv_functor` is only functorial with respect to equivalences.
To construct an `equiv_functor`, it suffices to supply just the function `f α → f β` from
an equivalence `α ≃ β`, and then prove the functor laws. It's then a consequence that
this function is part of an equivalence, provided by `equiv_functor.map_equiv`.
-/
class equiv_functor (f : Type u₀ → Type u₁) :=
(map : Π {α β}, (α ≃ β) → (f α → f β))
(map_refl' : Π α, map (equiv.refl α) = @id (f α) . obviously)
(map_trans' : Π {α β γ} (k : α ≃ β) (h : β ≃ γ),
map (k.trans h) = (map h) ∘ (map k) . obviously)
restate_axiom equiv_functor.map_refl'
restate_axiom equiv_functor.map_trans'
attribute [simp] equiv_functor.map_refl
namespace equiv_functor
section
variables (f : Type u₀ → Type u₁) [equiv_functor f] {α β : Type u₀} (e : α ≃ β)
/-- An `equiv_functor` in fact takes every equiv to an equiv. -/
def map_equiv :
f α ≃ f β :=
{ to_fun := equiv_functor.map e,
inv_fun := equiv_functor.map e.symm,
left_inv := λ x, by { convert (congr_fun (equiv_functor.map_trans e e.symm) x).symm, simp, },
right_inv := λ y, by { convert (congr_fun (equiv_functor.map_trans e.symm e) y).symm, simp, }, }
@[simp] lemma map_equiv_apply (x : f α) :
map_equiv f e x = equiv_functor.map e x := rfl
lemma map_equiv_symm_apply (y : f β) :
(map_equiv f e).symm y = equiv_functor.map e.symm y := rfl
@[simp] lemma map_equiv_refl (α) :
map_equiv f (equiv.refl α) = equiv.refl (f α) :=
by simpa [equiv_functor.map_equiv]
@[simp] lemma map_equiv_symm :
(map_equiv f e).symm = map_equiv f e.symm :=
equiv.ext $ map_equiv_symm_apply f e
/--
The composition of `map_equiv`s is carried over the `equiv_functor`.
For plain `functor`s, this lemma is named `map_map` when applied
or `map_comp_map` when not applied.
-/
@[simp] lemma map_equiv_trans {γ : Type u₀} (ab : α ≃ β) (bc : β ≃ γ) :
(map_equiv f ab).trans (map_equiv f bc) = map_equiv f (ab.trans bc) :=
equiv.ext $ λ x, by simp [map_equiv, map_trans']
end
@[priority 100]
instance of_is_lawful_functor
(f : Type u₀ → Type u₁) [functor f] [is_lawful_functor f] : equiv_functor f :=
{ map := λ α β e, functor.map e,
map_refl' := λ α, by { ext, apply is_lawful_functor.id_map, },
map_trans' := λ α β γ k h, by { ext x, apply (is_lawful_functor.comp_map k h x), } }
lemma map_equiv.injective
(f : Type u₀ → Type u₁) [applicative f] [is_lawful_applicative f] {α β : Type u₀}
(h : ∀ γ, function.injective (pure : γ → f γ)) :
function.injective (@equiv_functor.map_equiv f _ α β) :=
λ e₁ e₂ H, equiv.ext $ λ x, h β (by simpa [equiv_functor.map] using equiv.congr_fun H (pure x))
end equiv_functor
|
a699b1f3224a99d922115306c3ce73d521982f5e | 6432ea7a083ff6ba21ea17af9ee47b9c371760f7 | /tests/lean/run/229.lean | 3653a727c507b02b3f4a5c7113ade338868f68bc | [
"Apache-2.0",
"LLVM-exception",
"NCSA",
"LGPL-3.0-only",
"LicenseRef-scancode-inner-net-2.0",
"BSD-3-Clause",
"LGPL-2.0-or-later",
"Spencer-94",
"LGPL-2.1-or-later",
"HPND",
"LicenseRef-scancode-pcre",
"ISC",
"LGPL-2.1-only",
"LicenseRef-scancode-other-permissive",
"SunPro",
"CMU-Mach"... | permissive | leanprover/lean4 | 4bdf9790294964627eb9be79f5e8f6157780b4cc | f1f9dc0f2f531af3312398999d8b8303fa5f096b | refs/heads/master | 1,693,360,665,786 | 1,693,350,868,000 | 1,693,350,868,000 | 129,571,436 | 2,827 | 311 | Apache-2.0 | 1,694,716,156,000 | 1,523,760,560,000 | Lean | UTF-8 | Lean | false | false | 693 | lean | def f1 (xs : Option (Array Nat)) : Nat :=
match xs with
| some #[x, y] => x
| _ => 0
def f2 (xs : Option (Array Nat)) : Nat :=
match xs with
| some #[0, y] => y
| some #[x+1, y] => x
| _ => 0
theorem ex1 : f2 (some #[0, 2]) = 2 := rfl
theorem ex2 : f2 (some #[10, 2]) = 9 := rfl
def f3 (xs : Option (Array (Array Nat))) : Nat :=
match xs with
| some #[#[0], #[x]] => x
| some #[#[x+1], #[y]] => x
| _ => 0
theorem ex3 : f3 (some #[#[10], #[5]]) = 9 := rfl
theorem ex4 : f3 (some #[#[0], #[5]]) = 5 := rfl
theorem ex5 : f3 (some #[#[0], #[5], #[4]]) = 0 := rfl
#check match some #[1, 2] with
| some #[x, y] => x
| _ => 0
|
44ea99810bb9747211d773ac91b0797bd54893b9 | 8b9f17008684d796c8022dab552e42f0cb6fb347 | /tests/lean/run/class2.lean | 434df42a9b4a33670f4ea3c1d0a9d172f893bcfc | [
"Apache-2.0"
] | permissive | chubbymaggie/lean | 0d06ae25f9dd396306fb02190e89422ea94afd7b | d2c7b5c31928c98f545b16420d37842c43b4ae9a | refs/heads/master | 1,611,313,622,901 | 1,430,266,839,000 | 1,430,267,083,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 190 | lean | import logic data.prod data.num
open prod nonempty inhabited
theorem H {A B : Type} (H1 : inhabited A) : inhabited (Prop × A × (B → num))
:= _
(*
print(get_env():find("H"):value())
*)
|
5c618172708ab285589bc17ae5877fc7648334bf | d406927ab5617694ec9ea7001f101b7c9e3d9702 | /src/order/extension/well.lean | 76c290e56f5d060990c7e3228c661a6967b4f236 | [
"Apache-2.0"
] | permissive | alreadydone/mathlib | dc0be621c6c8208c581f5170a8216c5ba6721927 | c982179ec21091d3e102d8a5d9f5fe06c8fafb73 | refs/heads/master | 1,685,523,275,196 | 1,670,184,141,000 | 1,670,184,141,000 | 287,574,545 | 0 | 0 | Apache-2.0 | 1,670,290,714,000 | 1,597,421,623,000 | Lean | UTF-8 | Lean | false | false | 3,271 | lean | /-
Copyright (c) 2022 Yaël Dillies, Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Junyan Xu
-/
import data.prod.lex
import set_theory.ordinal.arithmetic
/-!
# Extend a well-founded order to a well-order
This file constructs a well-order (linear well-founded order) which is an extension of a given
well-founded order.
## Proof idea
We can map our order into two well-orders:
* the first map respects the order but isn't necessarily injective. Namely, this is the *rank*
function `rank : α → ordinal`.
* the second map is injective but doesn't necessarily respect the order. This is an arbitrary
well-order on `α`.
Then their lexicographic product is a well-founded linear order which our original order injects in.
-/
universe u
variables {α : Type u} {r : α → α → Prop}
namespace well_founded
variable (hwf : well_founded r)
include hwf
/-- An arbitrary well order on `α` that extends `r`.
The construction maps `r` into two well-orders: the first map is `well_founded.rank`, which is not
necessarily injective but respects the order `r`; the other map is the identity (with an arbitrarily
chosen well-order on `α`), which is injective but doesn't respect `r`.
By taking the lexicographic product of the two, we get both properties, so we can pull it back and
get an well-order that extend our original order `r`. Another way to view this is that we choose an
arbitrary well-order to serve as a tiebreak between two elements of same rank.
-/
noncomputable def well_order_extension : linear_order α :=
let l : linear_order α := is_well_order.linear_order well_ordering_rel in by exactI
@linear_order.lift' α (ordinal ×ₗ α) _
(λ a : α, (well_founded.rank.{u u} hwf a, a)) (λ _ _, congr_arg prod.snd)
instance well_order_extension.is_well_founded_lt : is_well_founded α hwf.well_order_extension.lt :=
⟨inv_image.wf _ $ prod.lex_wf ordinal.well_founded_lt.wf well_ordering_rel.is_well_order.wf⟩
/-- Any well-founded relation can be extended to a well-ordering on that type. -/
lemma exists_well_order_ge : ∃ s, r ≤ s ∧ is_well_order α s :=
⟨hwf.well_order_extension.lt, λ a b h, prod.lex.left _ _ (hwf.rank_lt_of_rel h), by split⟩
end well_founded
/-- A type alias for `α`, intended to extend a well-founded order on `α` to a well-order. -/
def well_order_extension (α) : Type* := α
instance [inhabited α] : inhabited (well_order_extension α) := ‹inhabited (well_order_extension α)›
/-- "Identity" equivalence between a well-founded order and its well-order extension. -/
def to_well_order_extension : α ≃ well_order_extension α := equiv.refl _
noncomputable instance [has_lt α] [well_founded_lt α] : linear_order (well_order_extension α) :=
(is_well_founded.wf : @well_founded α (<)).well_order_extension
instance well_order_extension.well_founded_lt [has_lt α] [well_founded_lt α] :
well_founded_lt (well_order_extension α) :=
well_founded.well_order_extension.is_well_founded_lt _
lemma to_well_order_extension_strict_mono [preorder α] [well_founded_lt α] :
strict_mono (to_well_order_extension : α → well_order_extension α) :=
λ a b h, prod.lex.left _ _ $ well_founded.rank_lt_of_rel _ h
|
f2845a6a773adcdf0308533a214b3aadb121e7f8 | fe25de614feb5587799621c41487aaee0d083b08 | /src/Lean/Elab/App.lean | 920273258df72f12ec9c055b952eed2d3d34e5d4 | [
"Apache-2.0"
] | permissive | pollend/lean4 | e8469c2f5fb8779b773618c3267883cf21fb9fac | c913886938c4b3b83238a3f99673c6c5a9cec270 | refs/heads/master | 1,687,973,251,481 | 1,628,039,739,000 | 1,628,039,739,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 43,019 | lean | /-
Copyright (c) 2019 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Lean.Util.FindMVar
import Lean.Parser.Term
import Lean.Elab.Term
import Lean.Elab.Binders
import Lean.Elab.SyntheticMVars
import Lean.Elab.Arg
namespace Lean.Elab.Term
open Meta
builtin_initialize elabWithoutExpectedTypeAttr : TagAttribute ←
registerTagAttribute `elabWithoutExpectedType "mark that applications of the given declaration should be elaborated without the expected type"
def hasElabWithoutExpectedType (env : Environment) (declName : Name) : Bool :=
elabWithoutExpectedTypeAttr.hasTag env declName
instance : ToString Arg := ⟨fun
| Arg.stx val => toString val
| Arg.expr val => toString val⟩
instance : ToString NamedArg where
toString s := "(" ++ toString s.name ++ " := " ++ toString s.val ++ ")"
def throwInvalidNamedArg {α} (namedArg : NamedArg) (fn? : Option Name) : TermElabM α :=
withRef namedArg.ref <| match fn? with
| some fn => throwError "invalid argument name '{namedArg.name}' for function '{fn}'"
| none => throwError "invalid argument name '{namedArg.name}' for function"
private def ensureArgType (f : Expr) (arg : Expr) (expectedType : Expr) : TermElabM Expr := do
let argType ← inferType arg
ensureHasTypeAux expectedType argType arg f
/-
Relevant definitions:
```
class CoeFun (α : Sort u) (γ : α → outParam (Sort v))
abbrev coeFun {α : Sort u} {γ : α → Sort v} (a : α) [CoeFun α γ] : γ a
```
-/
private def tryCoeFun? (α : Expr) (a : Expr) : TermElabM (Option Expr) := do
let v ← mkFreshLevelMVar
let type ← mkArrow α (mkSort v)
let γ ← mkFreshExprMVar type
let u ← getLevel α
let coeFunInstType := mkAppN (Lean.mkConst ``CoeFun [u, v]) #[α, γ]
let mvar ← mkFreshExprMVar coeFunInstType MetavarKind.synthetic
let mvarId := mvar.mvarId!
try
if (← synthesizeCoeInstMVarCore mvarId) then
expandCoe <| mkAppN (Lean.mkConst ``coeFun [u, v]) #[α, γ, a, mvar]
else
return none
catch _ =>
return none
def synthesizeAppInstMVars (instMVars : Array MVarId) : TermElabM Unit :=
for mvarId in instMVars do
unless (← synthesizeInstMVarCore mvarId) do
registerSyntheticMVarWithCurrRef mvarId SyntheticMVarKind.typeClass
namespace ElabAppArgs
/- Auxiliary structure for elaborating the application `f args namedArgs`. -/
structure State where
explicit : Bool -- true if `@` modifier was used
f : Expr
fType : Expr
args : List Arg -- remaining regular arguments
namedArgs : List NamedArg -- remaining named arguments to be processed
ellipsis : Bool := false
expectedType? : Option Expr
etaArgs : Array Expr := #[]
toSetErrorCtx : Array MVarId := #[] -- metavariables that we need the set the error context using the application being built
instMVars : Array MVarId := #[] -- metavariables for the instance implicit arguments that have already been processed
-- The following field is used to implement the `propagateExpectedType` heuristic.
propagateExpected : Bool -- true when expectedType has not been propagated yet
abbrev M := StateRefT State TermElabM
/- Add the given metavariable to the collection of metavariables associated with instance-implicit arguments. -/
private def addInstMVar (mvarId : MVarId) : M Unit :=
modify fun s => { s with instMVars := s.instMVars.push mvarId }
/-
Try to synthesize metavariables are `instMVars` using type class resolution.
The ones that cannot be synthesized yet are registered.
Remark: we use this method before trying to apply coercions to function. -/
def synthesizeAppInstMVars : M Unit := do
let s ← get
let instMVars := s.instMVars
modify fun s => { s with instMVars := #[] }
Lean.Elab.Term.synthesizeAppInstMVars instMVars
/- fType may become a forallE after we synthesize pending metavariables. -/
private def synthesizePendingAndNormalizeFunType : M Unit := do
synthesizeAppInstMVars
synthesizeSyntheticMVars
let s ← get
let fType ← whnfForall s.fType
if fType.isForall then
modify fun s => { s with fType := fType }
else
match (← tryCoeFun? fType s.f) with
| some f =>
let fType ← inferType f
modify fun s => { s with f := f, fType := fType }
| none =>
for namedArg in s.namedArgs do
let f := s.f.getAppFn
if f.isConst then
throwInvalidNamedArg namedArg f.constName!
else
throwInvalidNamedArg namedArg none
throwError "function expected at{indentExpr s.f}\nterm has type{indentExpr fType}"
/- Normalize and return the function type. -/
private def normalizeFunType : M Expr := do
let s ← get
let fType ← whnfForall s.fType
modify fun s => { s with fType := fType }
pure fType
/- Return the binder name at `fType`. This method assumes `fType` is a function type. -/
private def getBindingName : M Name := return (← get).fType.bindingName!
/- Return the next argument expected type. This method assumes `fType` is a function type. -/
private def getArgExpectedType : M Expr := return (← get).fType.bindingDomain!
def eraseNamedArgCore (namedArgs : List NamedArg) (binderName : Name) : List NamedArg :=
namedArgs.filter (·.name != binderName)
/- Remove named argument with name `binderName` from `namedArgs`. -/
def eraseNamedArg (binderName : Name) : M Unit :=
modify fun s => { s with namedArgs := eraseNamedArgCore s.namedArgs binderName }
/-
Add a new argument to the result. That is, `f := f arg`, update `fType`.
This method assumes `fType` is a function type. -/
private def addNewArg (arg : Expr) : M Unit :=
modify fun s => { s with f := mkApp s.f arg, fType := s.fType.bindingBody!.instantiate1 arg }
/-
Elaborate the given `Arg` and add it to the result. See `addNewArg`.
Recall that, `Arg` may be wrapping an already elaborated `Expr`. -/
private def elabAndAddNewArg (arg : Arg) : M Unit := do
let s ← get
let expectedType ← getArgExpectedType
match arg with
| Arg.expr val =>
let arg ← ensureArgType s.f val expectedType
addNewArg arg
| Arg.stx val =>
let val ← elabTerm val expectedType
let arg ← ensureArgType s.f val expectedType
addNewArg arg
/- Return true if the given type contains `OptParam` or `AutoParams` -/
private def hasOptAutoParams (type : Expr) : M Bool := do
forallTelescopeReducing type fun xs type =>
xs.anyM fun x => do
let xType ← inferType x
return xType.getOptParamDefault?.isSome || xType.getAutoParamTactic?.isSome
/- Return true if `fType` contains `OptParam` or `AutoParams` -/
private def fTypeHasOptAutoParams : M Bool := do
hasOptAutoParams (← get).fType
/- Auxiliary function for retrieving the resulting type of a function application.
See `propagateExpectedType`.
Remark: `(explicit : Bool) == true` when `@` modifier is used. -/
private partial def getForallBody (explicit : Bool) : Nat → List NamedArg → Expr → Option Expr
| i, namedArgs, type@(Expr.forallE n d b c) =>
match namedArgs.find? fun (namedArg : NamedArg) => namedArg.name == n with
| some _ => getForallBody explicit i (eraseNamedArgCore namedArgs n) b
| none =>
if !explicit && !c.binderInfo.isExplicit then
getForallBody explicit i namedArgs b
else if i > 0 then
getForallBody explicit (i-1) namedArgs b
else if d.isAutoParam || d.isOptParam then
getForallBody explicit i namedArgs b
else
some type
| 0, [], type => some type
| _, _, _ => none
private def shouldPropagateExpectedTypeFor (nextArg : Arg) : Bool :=
match nextArg with
| Arg.expr _ => false -- it has already been elaborated
| Arg.stx stx =>
-- TODO: make this configurable?
stx.getKind != ``Lean.Parser.Term.hole &&
stx.getKind != ``Lean.Parser.Term.syntheticHole &&
stx.getKind != ``Lean.Parser.Term.byTactic
/-
Auxiliary method for propagating the expected type. We call it as soon as we find the first explict
argument. The goal is to propagate the expected type in applications of functions such as
```lean
Add.add {α : Type u} : α → α → α
List.cons {α : Type u} : α → List α → List α
```
This is particularly useful when there applicable coercions. For example,
assume we have a coercion from `Nat` to `Int`, and we have
`(x : Nat)` and the expected type is `List Int`. Then, if we don't use this function,
the elaborator will fail to elaborate
```
List.cons x []
```
First, the elaborator creates a new metavariable `?α` for the implicit argument `{α : Type u}`.
Then, when it processes `x`, it assigns `?α := Nat`, and then obtain the
resultant type `List Nat` which is **not** definitionally equal to `List Int`.
We solve the problem by executing this method before we elaborate the first explicit argument (`x` in this example).
This method infers that the resultant type is `List ?α` and unifies it with `List Int`.
Then, when we elaborate `x`, the elaborate realizes the coercion from `Nat` to `Int` must be used, and the
term
```
@List.cons Int (coe x) (@List.nil Int)
```
is produced.
The method will do nothing if
1- The resultant type depends on the remaining arguments (i.e., `!eTypeBody.hasLooseBVars`).
2- The resultant type contains optional/auto params.
We have considered adding the following extra conditions
a) The resultant type does not contain any type metavariable.
b) The resultant type contains a nontype metavariable.
These two conditions would restrict the method to simple functions that are "morally" in
the Hindley&Milner fragment.
If users need to disable expected type propagation, we can add an attribute `[elabWithoutExpectedType]`.
-/
private def propagateExpectedType (arg : Arg) : M Unit := do
if shouldPropagateExpectedTypeFor arg then
let s ← get
-- TODO: handle s.etaArgs.size > 0
unless !s.etaArgs.isEmpty || !s.propagateExpected do
match s.expectedType? with
| none => pure ()
| some expectedType =>
/- We don't propagate `Prop` because we often use `Prop` as a more general "Bool" (e.g., `if-then-else`).
If we propagate `expectedType == Prop` in the following examples, the elaborator would fail
```
def f1 (s : Nat × Bool) : Bool := if s.2 then false else true
def f2 (s : List Bool) : Bool := if s.head! then false else true
def f3 (s : List Bool) : Bool := if List.head! (s.map not) then false else true
```
They would all fail for the same reason. So, let's focus on the first one.
We would elaborate `s.2` with `expectedType == Prop`.
Before we elaborate `s`, this method would be invoked, and `s.fType` is `?α × ?β → ?β` and after
propagation we would have `?α × Prop → Prop`. Then, when we would try to elaborate `s`, and
get a type error because `?α × Prop` cannot be unified with `Nat × Bool`
Most users would have a hard time trying to understand why these examples failed.
Here is a possible alternative workarounds. We give up the idea of using `Prop` at `if-then-else`.
Drawback: users use `if-then-else` with conditions that are not Decidable.
So, users would have to embrace `propDecidable` and `choice`.
This may not be that bad since the developers and users don't seem to care about constructivism.
We currently use a different workaround, we just don't propagate the expected type when it is `Prop`. -/
if expectedType.isProp then
modify fun s => { s with propagateExpected := false }
else
let numRemainingArgs := s.args.length
trace[Elab.app.propagateExpectedType] "etaArgs.size: {s.etaArgs.size}, numRemainingArgs: {numRemainingArgs}, fType: {s.fType}"
match getForallBody s.explicit numRemainingArgs s.namedArgs s.fType with
| none => pure ()
| some fTypeBody =>
unless fTypeBody.hasLooseBVars do
unless (← hasOptAutoParams fTypeBody) do
trace[Elab.app.propagateExpectedType] "{expectedType} =?= {fTypeBody}"
if (← isDefEq expectedType fTypeBody) then
/- Note that we only set `propagateExpected := false` when propagation has succeeded. -/
modify fun s => { s with propagateExpected := false }
/-
Create a fresh local variable with the current binder name and argument type, add it to `etaArgs` and `f`,
and then execute the continuation `k`.-/
private def addEtaArg (k : M Expr) : M Expr := do
let n ← getBindingName
let type ← getArgExpectedType
withLocalDeclD n type fun x => do
modify fun s => { s with etaArgs := s.etaArgs.push x }
addNewArg x
k
/- This method execute after all application arguments have been processed. -/
private def finalize : M Expr := do
let s ← get
let mut e := s.f
-- all user explicit arguments have been consumed
trace[Elab.app.finalize] e
let ref ← getRef
-- Register the error context of implicits
for mvarId in s.toSetErrorCtx do
registerMVarErrorImplicitArgInfo mvarId ref e
if !s.etaArgs.isEmpty then
e ← mkLambdaFVars s.etaArgs e
/-
Remark: we should not use `s.fType` as `eType` even when
`s.etaArgs.isEmpty`. Reason: it may have been unfolded.
-/
let eType ← inferType e
trace[Elab.app.finalize] "after etaArgs, {e} : {eType}"
match s.expectedType? with
| none => pure ()
| some expectedType =>
trace[Elab.app.finalize] "expected type: {expectedType}"
-- Try to propagate expected type. Ignore if types are not definitionally equal, caller must handle it.
discard <| isDefEq expectedType eType
synthesizeAppInstMVars
pure e
private def addImplicitArg (k : M Expr) : M Expr := do
let argType ← getArgExpectedType
let arg ← mkFreshExprMVar argType
modify fun s => { s with toSetErrorCtx := s.toSetErrorCtx.push arg.mvarId! }
addNewArg arg
k
/- Return true if there is a named argument that depends on the next argument. -/
private def anyNamedArgDependsOnCurrent : M Bool := do
let s ← get
if s.namedArgs.isEmpty then
return false
else
forallTelescopeReducing s.fType fun xs _ => do
let curr := xs[0]
for i in [1:xs.size] do
let xDecl ← getLocalDecl xs[i].fvarId!
if s.namedArgs.any fun arg => arg.name == xDecl.userName then
if (← getMCtx).localDeclDependsOn xDecl curr.fvarId! then
return true
return false
/-
Process a `fType` of the form `(x : A) → B x`.
This method assume `fType` is a function type -/
private def processExplictArg (k : M Expr) : M Expr := do
let s ← get
match s.args with
| arg::args =>
propagateExpectedType arg
modify fun s => { s with args := args }
elabAndAddNewArg arg
k
| _ =>
let argType ← getArgExpectedType
match s.explicit, argType.getOptParamDefault?, argType.getAutoParamTactic? with
| false, some defVal, _ => addNewArg defVal; k
| false, _, some (Expr.const tacticDecl _ _) =>
let env ← getEnv
let opts ← getOptions
match evalSyntaxConstant env opts tacticDecl with
| Except.error err => throwError err
| Except.ok tacticSyntax =>
-- TODO(Leo): does this work correctly for tactic sequences?
let tacticBlock ← `(by $tacticSyntax)
let argType := argType.getArg! 0 -- `autoParam type := by tactic` ==> `type`
let argNew := Arg.stx tacticBlock
propagateExpectedType argNew
elabAndAddNewArg argNew
k
| false, _, some _ =>
throwError "invalid autoParam, argument must be a constant"
| _, _, _ =>
if !s.namedArgs.isEmpty then
if (← anyNamedArgDependsOnCurrent) then
addImplicitArg k
else
addEtaArg k
else if !s.explicit then
if (← fTypeHasOptAutoParams) then
addEtaArg k
else if (← get).ellipsis then
addImplicitArg k
else
finalize
else
finalize
/-
Process a `fType` of the form `{x : A} → B x`.
This method assume `fType` is a function type -/
private def processImplicitArg (k : M Expr) : M Expr := do
if (← get).explicit then
processExplictArg k
else
addImplicitArg k
/- Return true if the next argument at `args` is of the form `_` -/
private def isNextArgHole : M Bool := do
match (← get).args with
| Arg.stx (Syntax.node ``Lean.Parser.Term.hole _) :: _ => pure true
| _ => pure false
/-
Process a `fType` of the form `[x : A] → B x`.
This method assume `fType` is a function type -/
private def processInstImplicitArg (k : M Expr) : M Expr := do
if (← get).explicit then
if (← isNextArgHole) then
/- Recall that if '@' has been used, and the argument is '_', then we still use type class resolution -/
let arg ← mkFreshExprMVar (← getArgExpectedType) MetavarKind.synthetic
modify fun s => { s with args := s.args.tail! }
addInstMVar arg.mvarId!
addNewArg arg
k
else
processExplictArg k
else
let arg ← mkFreshExprMVar (← getArgExpectedType) MetavarKind.synthetic
addInstMVar arg.mvarId!
addNewArg arg
k
/- Return true if there are regular or named arguments to be processed. -/
private def hasArgsToProcess : M Bool := do
let s ← get
pure $ !s.args.isEmpty || !s.namedArgs.isEmpty
/- Elaborate function application arguments. -/
partial def main : M Expr := do
let s ← get
let fType ← normalizeFunType
if fType.isForall then
let binderName := fType.bindingName!
let binfo := fType.bindingInfo!
let s ← get
match s.namedArgs.find? fun (namedArg : NamedArg) => namedArg.name == binderName with
| some namedArg =>
propagateExpectedType namedArg.val
eraseNamedArg binderName
elabAndAddNewArg namedArg.val
main
| none =>
match binfo with
| BinderInfo.implicit => processImplicitArg main
| BinderInfo.instImplicit => processInstImplicitArg main
| _ => processExplictArg main
else if (← hasArgsToProcess) then
synthesizePendingAndNormalizeFunType
main
else
finalize
end ElabAppArgs
private def propagateExpectedTypeFor (f : Expr) : TermElabM Bool :=
match f.getAppFn.constName? with
| some declName => return !hasElabWithoutExpectedType (← getEnv) declName
| _ => return true
def elabAppArgs (f : Expr) (namedArgs : Array NamedArg) (args : Array Arg)
(expectedType? : Option Expr) (explicit ellipsis : Bool) : TermElabM Expr := do
let fType ← inferType f
let fType ← instantiateMVars fType
trace[Elab.app.args] "explicit: {explicit}, {f} : {fType}"
unless namedArgs.isEmpty && args.isEmpty do
tryPostponeIfMVar fType
ElabAppArgs.main.run' {
args := args.toList,
expectedType? := expectedType?,
explicit := explicit,
ellipsis := ellipsis,
namedArgs := namedArgs.toList,
f := f,
fType := fType
propagateExpected := (← propagateExpectedTypeFor f)
}
/-- Auxiliary inductive datatype that represents the resolution of an `LVal`. -/
inductive LValResolution where
| projFn (baseStructName : Name) (structName : Name) (fieldName : Name)
| projIdx (structName : Name) (idx : Nat)
| const (baseStructName : Name) (structName : Name) (constName : Name)
| localRec (baseName : Name) (fullName : Name) (fvar : Expr)
| getOp (fullName : Name) (idx : Syntax)
private def throwLValError {α} (e : Expr) (eType : Expr) (msg : MessageData) : TermElabM α :=
throwError "{msg}{indentExpr e}\nhas type{indentExpr eType}"
/-- `findMethod? env S fName`.
1- If `env` contains `S ++ fName`, return `(S, S++fName)`
2- Otherwise if `env` contains private name `prv` for `S ++ fName`, return `(S, prv)`, o
3- Otherwise for each parent structure `S'` of `S`, we try `findMethod? env S' fname` -/
private partial def findMethod? (env : Environment) (structName fieldName : Name) : Option (Name × Name) :=
let fullName := structName ++ fieldName
match env.find? fullName with
| some _ => some (structName, fullName)
| none =>
let fullNamePrv := mkPrivateName env fullName
match env.find? fullNamePrv with
| some _ => some (structName, fullNamePrv)
| none =>
if isStructure env structName then
(getParentStructures env structName).findSome? fun parentStructName => findMethod? env parentStructName fieldName
else
none
private def resolveLValAux (e : Expr) (eType : Expr) (lval : LVal) : TermElabM LValResolution := do
match eType.getAppFn.constName?, lval with
| some structName, LVal.fieldIdx _ idx =>
if idx == 0 then
throwError "invalid projection, index must be greater than 0"
let env ← getEnv
unless isStructureLike env structName do
throwLValError e eType "invalid projection, structure expected"
let numFields := getStructureLikeNumFields env structName
if idx - 1 < numFields then
if isStructure env structName then
let fieldNames := getStructureFields env structName
return LValResolution.projFn structName structName fieldNames[idx - 1]
else
/- `structName` was declared using `inductive` command.
So, we don't projection functions for it. Thus, we use `Expr.proj` -/
return LValResolution.projIdx structName (idx - 1)
else
throwLValError e eType m!"invalid projection, structure has only {numFields} field(s)"
| some structName, LVal.fieldName _ fieldName _ _ =>
let env ← getEnv
let searchEnv : Unit → TermElabM LValResolution := fun _ => do
match findMethod? env structName (Name.mkSimple fieldName) with
| some (baseStructName, fullName) => pure $ LValResolution.const baseStructName structName fullName
| none =>
throwLValError e eType
m!"invalid field '{fieldName}', the environment does not contain '{Name.mkStr structName fieldName}'"
-- search local context first, then environment
let searchCtx : Unit → TermElabM LValResolution := fun _ => do
let fullName := Name.mkStr structName fieldName
let currNamespace ← getCurrNamespace
let localName := fullName.replacePrefix currNamespace Name.anonymous
let lctx ← getLCtx
match lctx.findFromUserName? localName with
| some localDecl =>
if localDecl.binderInfo == BinderInfo.auxDecl then
/- LVal notation is being used to make a "local" recursive call. -/
pure $ LValResolution.localRec structName fullName localDecl.toExpr
else
searchEnv ()
| none => searchEnv ()
if isStructure env structName then
match findField? env structName (Name.mkSimple fieldName) with
| some baseStructName => pure $ LValResolution.projFn baseStructName structName (Name.mkSimple fieldName)
| none => searchCtx ()
else
searchCtx ()
| some structName, LVal.getOp _ idx =>
let env ← getEnv
let fullName := Name.mkStr structName "getOp"
match env.find? fullName with
| some _ => pure $ LValResolution.getOp fullName idx
| none => throwLValError e eType m!"invalid [..] notation because environment does not contain '{fullName}'"
| none, LVal.fieldName _ _ (some suffix) _ =>
if e.isConst then
throwUnknownConstant (e.constName! ++ suffix)
else
throwLValError e eType "invalid field notation, type is not of the form (C ...) where C is a constant"
| _, LVal.getOp _ idx =>
throwLValError e eType "invalid [..] notation, type is not of the form (C ...) where C is a constant"
| _, _ =>
throwLValError e eType "invalid field notation, type is not of the form (C ...) where C is a constant"
/- whnfCore + implicit consumption.
Example: given `e` with `eType := {α : Type} → (fun β => List β) α `, it produces `(e ?m, List ?m)` where `?m` is fresh metavariable. -/
private partial def consumeImplicits (stx : Syntax) (e eType : Expr) : TermElabM (Expr × Expr) := do
let eType ← whnfCore eType
match eType with
| Expr.forallE n d b c =>
if c.binderInfo.isImplicit then
let mvar ← mkFreshExprMVar d
registerMVarErrorHoleInfo mvar.mvarId! stx
consumeImplicits stx (mkApp e mvar) (b.instantiate1 mvar)
else if c.binderInfo.isInstImplicit then
let mvar ← mkInstMVar d
consumeImplicits stx (mkApp e mvar) (b.instantiate1 mvar)
else match d.getOptParamDefault? with
| some defVal => consumeImplicits stx (mkApp e defVal) (b.instantiate1 defVal)
-- TODO: we do not handle autoParams here.
| _ => pure (e, eType)
| _ => pure (e, eType)
private partial def resolveLValLoop (lval : LVal) (e eType : Expr) (previousExceptions : Array Exception) : TermElabM (Expr × LValResolution) := do
let (e, eType) ← consumeImplicits lval.getRef e eType
tryPostponeIfMVar eType
try
let lvalRes ← resolveLValAux e eType lval
pure (e, lvalRes)
catch
| ex@(Exception.error _ _) =>
let eType? ← unfoldDefinition? eType
match eType? with
| some eType => resolveLValLoop lval e eType (previousExceptions.push ex)
| none =>
previousExceptions.forM fun ex => logException ex
throw ex
| ex@(Exception.internal _ _) => throw ex
private def resolveLVal (e : Expr) (lval : LVal) : TermElabM (Expr × LValResolution) := do
let eType ← inferType e
resolveLValLoop lval e eType #[]
private partial def mkBaseProjections (baseStructName : Name) (structName : Name) (e : Expr) : TermElabM Expr := do
let env ← getEnv
match getPathToBaseStructure? env baseStructName structName with
| none => throwError "failed to access field in parent structure"
| some path =>
let mut e := e
for projFunName in path do
let projFn ← mkConst projFunName
e ← elabAppArgs projFn #[{ name := `self, val := Arg.expr e }] (args := #[]) (expectedType? := none) (explicit := false) (ellipsis := false)
return e
/- Auxiliary method for field notation. It tries to add `e` as a new argument to `args` or `namedArgs`.
This method first finds the parameter with a type of the form `(baseName ...)`.
When the parameter is found, if it an explicit one and `args` is big enough, we add `e` to `args`.
Otherwise, if there isn't another parameter with the same name, we add `e` to `namedArgs`.
Remark: `fullName` is the name of the resolved "field" access function. It is used for reporting errors -/
private def addLValArg (baseName : Name) (fullName : Name) (e : Expr) (args : Array Arg) (namedArgs : Array NamedArg) (fType : Expr)
: TermElabM (Array Arg × Array NamedArg) :=
forallTelescopeReducing fType fun xs _ => do
let mut argIdx := 0 -- position of the next explicit argument
let mut remainingNamedArgs := namedArgs
for i in [:xs.size] do
let x := xs[i]
let xDecl ← getLocalDecl x.fvarId!
/- If there is named argument with name `xDecl.userName`, then we skip it. -/
match remainingNamedArgs.findIdx? (fun namedArg => namedArg.name == xDecl.userName) with
| some idx =>
remainingNamedArgs := remainingNamedArgs.eraseIdx idx
| none =>
let mut foundIt := false
let type := xDecl.type
if type.consumeMData.isAppOf baseName then
foundIt := true
if !foundIt then
/- Normalize type and try again -/
let type ← withReducible $ whnf type
if type.consumeMData.isAppOf baseName then
foundIt := true
if foundIt then
/- We found a type of the form (baseName ...).
First, we check if the current argument is an explicit one,
and the current explicit position "fits" at `args` (i.e., it must be ≤ arg.size) -/
if argIdx ≤ args.size && xDecl.binderInfo.isExplicit then
/- We insert `e` as an explicit argument -/
return (args.insertAt argIdx (Arg.expr e), namedArgs)
/- If we can't add `e` to `args`, we try to add it using a named argument, but this is only possible
if there isn't an argument with the same name occurring before it. -/
for j in [:i] do
let prev := xs[j]
let prevDecl ← getLocalDecl prev.fvarId!
if prevDecl.userName == xDecl.userName then
throwError "invalid field notation, function '{fullName}' has argument with the expected type{indentExpr type}\nbut it cannot be used"
return (args, namedArgs.push { name := xDecl.userName, val := Arg.expr e })
if xDecl.binderInfo.isExplicit then
-- advance explicit argument position
argIdx := argIdx + 1
throwError "invalid field notation, function '{fullName}' does not have argument with type ({baseName} ...) that can be used, it must be explicit or implicit with an unique name"
private def elabAppLValsAux (namedArgs : Array NamedArg) (args : Array Arg) (expectedType? : Option Expr) (explicit ellipsis : Bool)
(f : Expr) (lvals : List LVal) : TermElabM Expr :=
let rec loop : Expr → List LVal → TermElabM Expr
| f, [] => elabAppArgs f namedArgs args expectedType? explicit ellipsis
| f, lval::lvals => do
if let LVal.fieldName (ref := fieldStx) (targetStx := targetStx) .. := lval then
addDotCompletionInfo targetStx f expectedType? fieldStx
let (f, lvalRes) ← resolveLVal f lval
match lvalRes with
| LValResolution.projIdx structName idx =>
let f := mkProj structName idx f
addTermInfo lval.getRef f
loop f lvals
| LValResolution.projFn baseStructName structName fieldName =>
let f ← mkBaseProjections baseStructName structName f
if let some info := getFieldInfo? (← getEnv) baseStructName fieldName then
if isPrivateNameFromImportedModule (← getEnv) info.projFn then
throwError "field '{fieldName}' from structure '{structName}' is private"
let projFn ← mkConst info.projFn
addTermInfo lval.getRef projFn
if lvals.isEmpty then
let namedArgs ← addNamedArg namedArgs { name := `self, val := Arg.expr f }
elabAppArgs projFn namedArgs args expectedType? explicit ellipsis
else
let f ← elabAppArgs projFn #[{ name := `self, val := Arg.expr f }] #[] (expectedType? := none) (explicit := false) (ellipsis := false)
loop f lvals
else
unreachable!
| LValResolution.const baseStructName structName constName =>
let f ← if baseStructName != structName then mkBaseProjections baseStructName structName f else pure f
let projFn ← mkConst constName
addTermInfo lval.getRef projFn
if lvals.isEmpty then
let projFnType ← inferType projFn
let (args, namedArgs) ← addLValArg baseStructName constName f args namedArgs projFnType
elabAppArgs projFn namedArgs args expectedType? explicit ellipsis
else
let f ← elabAppArgs projFn #[] #[Arg.expr f] (expectedType? := none) (explicit := false) (ellipsis := false)
loop f lvals
| LValResolution.localRec baseName fullName fvar =>
addTermInfo lval.getRef fvar
if lvals.isEmpty then
let fvarType ← inferType fvar
let (args, namedArgs) ← addLValArg baseName fullName f args namedArgs fvarType
elabAppArgs fvar namedArgs args expectedType? explicit ellipsis
else
let f ← elabAppArgs fvar #[] #[Arg.expr f] (expectedType? := none) (explicit := false) (ellipsis := false)
loop f lvals
| LValResolution.getOp fullName idx =>
let getOpFn ← mkConst fullName
addTermInfo lval.getRef getOpFn
if lvals.isEmpty then
let namedArgs ← addNamedArg namedArgs { name := `self, val := Arg.expr f }
let namedArgs ← addNamedArg namedArgs { name := `idx, val := Arg.stx idx }
elabAppArgs getOpFn namedArgs args expectedType? explicit ellipsis
else
let f ← elabAppArgs getOpFn #[{ name := `self, val := Arg.expr f }, { name := `idx, val := Arg.stx idx }]
#[] (expectedType? := none) (explicit := false) (ellipsis := false)
loop f lvals
loop f lvals
private def elabAppLVals (f : Expr) (lvals : List LVal) (namedArgs : Array NamedArg) (args : Array Arg)
(expectedType? : Option Expr) (explicit ellipsis : Bool) : TermElabM Expr := do
if !lvals.isEmpty && explicit then
throwError "invalid use of field notation with `@` modifier"
elabAppLValsAux namedArgs args expectedType? explicit ellipsis f lvals
def elabExplicitUnivs (lvls : Array Syntax) : TermElabM (List Level) := do
lvls.foldrM (fun stx lvls => do pure ((← elabLevel stx)::lvls)) []
/-
Interaction between `errToSorry` and `observing`.
- The method `elabTerm` catches exceptions, log them, and returns a synthetic sorry (IF `ctx.errToSorry` == true).
- When we elaborate choice nodes (and overloaded identifiers), we track multiple results using the `observing x` combinator.
The `observing x` executes `x` and returns a `TermElabResult`.
`observing `x does not check for synthetic sorry's, just an exception. Thus, it may think `x` worked when it didn't
if a synthetic sorry was introduced. We decided that checking for synthetic sorrys at `observing` is not a good solution
because it would not be clear to decide what the "main" error message for the alternative is. When the result contains
a synthetic `sorry`, it is not clear which error message corresponds to the `sorry`. Moreover, while executing `x`, many
error messages may have been logged. Recall that we need an error per alternative at `mergeFailures`.
Thus, we decided to set `errToSorry` to `false` whenever processing choice nodes and overloaded symbols.
Important: we rely on the property that after `errToSorry` is set to
false, no elaboration function executed by `x` will reset it to
`true`.
-/
private partial def elabAppFnId (fIdent : Syntax) (fExplicitUnivs : List Level) (lvals : List LVal)
(namedArgs : Array NamedArg) (args : Array Arg) (expectedType? : Option Expr) (explicit ellipsis overloaded : Bool) (acc : Array (TermElabResult Expr))
: TermElabM (Array (TermElabResult Expr)) := do
let funLVals ← withRef fIdent <| resolveName' fIdent fExplicitUnivs expectedType?
let overloaded := overloaded || funLVals.length > 1
-- Set `errToSorry` to `false` if `funLVals` > 1. See comment above about the interaction between `errToSorry` and `observing`.
withReader (fun ctx => { ctx with errToSorry := funLVals.length == 1 && ctx.errToSorry }) do
funLVals.foldlM (init := acc) fun acc (f, fIdent, fields) => do
let lvals' := toLVals fields (first := true)
let s ← observing do
addTermInfo fIdent f expectedType?
let e ← elabAppLVals f (lvals' ++ lvals) namedArgs args expectedType? explicit ellipsis
if overloaded then ensureHasType expectedType? e else pure e
return acc.push s
where
toName : List Syntax → Name
| [] => Name.anonymous
| field :: fields => Name.mkStr (toName fields) field.getId.toString
toLVals : List Syntax → (first : Bool) → List LVal
| [], _ => []
| field::fields, true => LVal.fieldName field field.getId.toString (toName (field::fields)) fIdent :: toLVals fields false
| field::fields, false => LVal.fieldName field field.getId.toString none fIdent :: toLVals fields false
private partial def elabAppFn (f : Syntax) (lvals : List LVal) (namedArgs : Array NamedArg) (args : Array Arg)
(expectedType? : Option Expr) (explicit ellipsis overloaded : Bool) (acc : Array (TermElabResult Expr)) : TermElabM (Array (TermElabResult Expr)) :=
if f.getKind == choiceKind then
-- Set `errToSorry` to `false` when processing choice nodes. See comment above about the interaction between `errToSorry` and `observing`.
withReader (fun ctx => { ctx with errToSorry := false }) do
f.getArgs.foldlM (fun acc f => elabAppFn f lvals namedArgs args expectedType? explicit ellipsis true acc) acc
else
let elabFieldName (e field : Syntax) := do
let newLVals := field.identComponents.map fun comp =>
-- We use `none` in `suffix?` since `field` can't be part of a composite name
LVal.fieldName comp (toString comp.getId) none e
elabAppFn e (newLVals ++ lvals) namedArgs args expectedType? explicit ellipsis overloaded acc
let elabFieldIdx (e idxStx : Syntax) := do
let idx := idxStx.isFieldIdx?.get!
elabAppFn e (LVal.fieldIdx idxStx idx :: lvals) namedArgs args expectedType? explicit ellipsis overloaded acc
match f with
| `($(e).$idx:fieldIdx) => elabFieldIdx e idx
| `($e |>.$idx:fieldIdx) => elabFieldIdx e idx
| `($(e).$field:ident) => elabFieldName e field
| `($e |>.$field:ident) => elabFieldName e field
| `($e[%$bracket $idx]) => elabAppFn e (LVal.getOp bracket idx :: lvals) namedArgs args expectedType? explicit ellipsis overloaded acc
| `($id:ident@$t:term) =>
throwError "unexpected occurrence of named pattern"
| `($id:ident) => do
elabAppFnId id [] lvals namedArgs args expectedType? explicit ellipsis overloaded acc
| `($id:ident.{$us,*}) => do
let us ← elabExplicitUnivs us
elabAppFnId id us lvals namedArgs args expectedType? explicit ellipsis overloaded acc
| `(@$id:ident) =>
elabAppFn id lvals namedArgs args expectedType? (explicit := true) ellipsis overloaded acc
| `(@$id:ident.{$us,*}) =>
elabAppFn (f.getArg 1) lvals namedArgs args expectedType? (explicit := true) ellipsis overloaded acc
| `(@$t) => throwUnsupportedSyntax -- invalid occurrence of `@`
| `(_) => throwError "placeholders '_' cannot be used where a function is expected"
| _ => do
let catchPostpone := !overloaded
/- If we are processing a choice node, then we should use `catchPostpone == false` when elaborating terms.
Recall that `observing` does not catch `postponeExceptionId`. -/
if lvals.isEmpty && namedArgs.isEmpty && args.isEmpty then
/- Recall that elabAppFn is used for elaborating atomics terms **and** choice nodes that may contain
arbitrary terms. If they are not being used as a function, we should elaborate using the expectedType. -/
let s ←
if overloaded then
observing <| elabTermEnsuringType f expectedType? catchPostpone
else
observing <| elabTerm f expectedType?
return acc.push s
else
let s ← observing do
let f ← elabTerm f none catchPostpone
let e ← elabAppLVals f lvals namedArgs args expectedType? explicit ellipsis
if overloaded then ensureHasType expectedType? e else pure e
return acc.push s
private def isSuccess (candidate : TermElabResult Expr) : Bool :=
match candidate with
| EStateM.Result.ok _ _ => true
| _ => false
private def getSuccess (candidates : Array (TermElabResult Expr)) : Array (TermElabResult Expr) :=
candidates.filter isSuccess
private def toMessageData (ex : Exception) : TermElabM MessageData := do
let pos ← getRefPos
match ex.getRef.getPos? with
| none => return ex.toMessageData
| some exPos =>
if pos == exPos then
return ex.toMessageData
else
let exPosition := (← getFileMap).toPosition exPos
return m!"{exPosition.line}:{exPosition.column} {ex.toMessageData}"
private def toMessageList (msgs : Array MessageData) : MessageData :=
indentD (MessageData.joinSep msgs.toList m!"\n\n")
private def mergeFailures {α} (failures : Array (TermElabResult Expr)) : TermElabM α := do
let msgs ← failures.mapM fun failure =>
match failure with
| EStateM.Result.ok _ _ => unreachable!
| EStateM.Result.error ex _ => toMessageData ex
throwError "overloaded, errors {toMessageList msgs}"
private def elabAppAux (f : Syntax) (namedArgs : Array NamedArg) (args : Array Arg) (ellipsis : Bool) (expectedType? : Option Expr) : TermElabM Expr := do
let candidates ← elabAppFn f [] namedArgs args expectedType? (explicit := false) (ellipsis := ellipsis) (overloaded := false) #[]
if candidates.size == 1 then
applyResult candidates[0]
else
let successes := getSuccess candidates
if successes.size == 1 then
applyResult successes[0]
else if successes.size > 1 then
let lctx ← getLCtx
let opts ← getOptions
let msgs : Array MessageData := successes.map fun success => match success with
| EStateM.Result.ok e s => MessageData.withContext { env := s.meta.core.env, mctx := s.meta.meta.mctx, lctx := lctx, opts := opts } e
| _ => unreachable!
throwErrorAt f "ambiguous, possible interpretations {toMessageList msgs}"
else
withRef f <| mergeFailures candidates
@[builtinTermElab app] def elabApp : TermElab := fun stx expectedType? =>
withoutPostponingUniverseConstraints do
let (f, namedArgs, args, ellipsis) ← expandApp stx
elabAppAux f namedArgs args (ellipsis := ellipsis) expectedType?
private def elabAtom : TermElab := fun stx expectedType? =>
elabAppAux stx #[] #[] (ellipsis := false) expectedType?
@[builtinTermElab ident] def elabIdent : TermElab := elabAtom
@[builtinTermElab namedPattern] def elabNamedPattern : TermElab := elabAtom
@[builtinTermElab explicitUniv] def elabExplicitUniv : TermElab := elabAtom
@[builtinTermElab pipeProj] def elabPipeProj : TermElab
| `($e |>.$f $args*), expectedType? =>
withoutPostponingUniverseConstraints do
let (namedArgs, args, ellipsis) ← expandArgs args
elabAppAux (← `($e |>.$f)) namedArgs args (ellipsis := ellipsis) expectedType?
| _, _ => throwUnsupportedSyntax
@[builtinTermElab explicit] def elabExplicit : TermElab := fun stx expectedType? =>
match stx with
| `(@$id:ident) => elabAtom stx expectedType? -- Recall that `elabApp` also has support for `@`
| `(@$id:ident.{$us,*}) => elabAtom stx expectedType?
| `(@($t)) => elabTerm t expectedType? (implicitLambda := false) -- `@` is being used just to disable implicit lambdas
| `(@$t) => elabTerm t expectedType? (implicitLambda := false) -- `@` is being used just to disable implicit lambdas
| _ => throwUnsupportedSyntax
@[builtinTermElab choice] def elabChoice : TermElab := elabAtom
@[builtinTermElab proj] def elabProj : TermElab := elabAtom
@[builtinTermElab arrayRef] def elabArrayRef : TermElab := elabAtom
builtin_initialize
registerTraceClass `Elab.app
end Lean.Elab.Term
|
e9cb7a8ed1ff03d2b0aded6ed26648137264d91b | 8cae430f0a71442d02dbb1cbb14073b31048e4b0 | /src/linear_algebra/projective_space/basic.lean | 78593079ab34d81fa6826a13271c1891dce3dd90 | [
"Apache-2.0"
] | permissive | leanprover-community/mathlib | 56a2cadd17ac88caf4ece0a775932fa26327ba0e | 442a83d738cb208d3600056c489be16900ba701d | refs/heads/master | 1,693,584,102,358 | 1,693,471,902,000 | 1,693,471,902,000 | 97,922,418 | 1,595 | 352 | Apache-2.0 | 1,694,693,445,000 | 1,500,624,130,000 | Lean | UTF-8 | Lean | false | false | 8,164 | lean | /-
Copyright (c) 2022 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import linear_algebra.finite_dimensional
/-!
# Projective Spaces
> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
> Any changes to this file require a corresponding PR to mathlib4.
This file contains the definition of the projectivization of a vector space over a field,
as well as the bijection between said projectivization and the collection of all one
dimensional subspaces of the vector space.
## Notation
`ℙ K V` is notation for `projectivization K V`, the projectivization of a `K`-vector space `V`.
## Constructing terms of `ℙ K V`.
We have three ways to construct terms of `ℙ K V`:
- `projectivization.mk K v hv` where `v : V` and `hv : v ≠ 0`.
- `projectivization.mk' K v` where `v : { w : V // w ≠ 0 }`.
- `projectivization.mk'' H h` where `H : submodule K V` and `h : finrank H = 1`.
## Other definitions
- For `v : ℙ K V`, `v.submodule` gives the corresponding submodule of `V`.
- `projectivization.equiv_submodule` is the equivalence between `ℙ K V`
and `{ H : submodule K V // finrank H = 1 }`.
- For `v : ℙ K V`, `v.rep : V` is a representative of `v`.
-/
variables (K V : Type*) [division_ring K] [add_comm_group V] [module K V]
/-- The setoid whose quotient is the projectivization of `V`. -/
def projectivization_setoid : setoid { v : V // v ≠ 0 } :=
(mul_action.orbit_rel Kˣ V).comap coe
/-- The projectivization of the `K`-vector space `V`.
The notation `ℙ K V` is preferred. -/
@[nolint has_nonempty_instance]
def projectivization := quotient (projectivization_setoid K V)
notation `ℙ` := projectivization
namespace projectivization
variables {V}
/-- Construct an element of the projectivization from a nonzero vector. -/
def mk (v : V) (hv : v ≠ 0) : ℙ K V := quotient.mk' ⟨v,hv⟩
/-- A variant of `projectivization.mk` in terms of a subtype. `mk` is preferred. -/
def mk' (v : { v : V // v ≠ 0 }) : ℙ K V := quotient.mk' v
@[simp] lemma mk'_eq_mk (v : { v : V // v ≠ 0}) :
mk' K v = mk K v v.2 :=
by { dsimp [mk, mk'], congr' 1, simp }
instance [nontrivial V] : nonempty (ℙ K V) :=
let ⟨v, hv⟩ := exists_ne (0 : V) in ⟨mk K v hv⟩
variable {K}
/-- Choose a representative of `v : projectivization K V` in `V`. -/
protected noncomputable def rep (v : ℙ K V) : V := v.out'
lemma rep_nonzero (v : ℙ K V) : v.rep ≠ 0 := v.out'.2
@[simp]
lemma mk_rep (v : ℙ K V) :
mk K v.rep v.rep_nonzero = v :=
by { dsimp [mk, projectivization.rep], simp }
open finite_dimensional
/-- Consider an element of the projectivization as a submodule of `V`. -/
protected def submodule (v : ℙ K V) : submodule K V :=
quotient.lift_on' v (λ v, K ∙ (v : V)) $ begin
rintro ⟨a, ha⟩ ⟨b, hb⟩ ⟨x, (rfl : x • b = a)⟩,
exact (submodule.span_singleton_group_smul_eq _ x _),
end
variable (K)
lemma mk_eq_mk_iff (v w : V) (hv : v ≠ 0) (hw : w ≠ 0) :
mk K v hv = mk K w hw ↔ ∃ (a : Kˣ), a • w = v :=
quotient.eq'
/-- Two nonzero vectors go to the same point in projective space if and only if one is
a scalar multiple of the other. -/
lemma mk_eq_mk_iff' (v w : V) (hv : v ≠ 0) (hw : w ≠ 0) : mk K v hv = mk K w hw ↔
∃ (a : K), a • w = v :=
begin
rw mk_eq_mk_iff K v w hv hw,
split,
{ rintro ⟨a, ha⟩, exact ⟨a, ha⟩ },
{ rintro ⟨a, ha⟩, refine ⟨units.mk0 a (λ c, hv.symm _), ha⟩, rwa [c, zero_smul] at ha }
end
lemma exists_smul_eq_mk_rep
(v : V) (hv : v ≠ 0) : ∃ (a : Kˣ), a • v = (mk K v hv).rep :=
show (projectivization_setoid K V).rel _ _, from quotient.mk_out' ⟨v, hv⟩
variable {K}
/-- An induction principle for `projectivization`.
Use as `induction v using projectivization.ind`. -/
@[elab_as_eliminator]
lemma ind {P : ℙ K V → Prop} (h : ∀ (v : V) (h : v ≠ 0), P (mk K v h)) :
∀ p, P p :=
quotient.ind' $ subtype.rec $ by exact h
@[simp]
lemma submodule_mk (v : V) (hv : v ≠ 0) : (mk K v hv).submodule = K ∙ v := rfl
lemma submodule_eq (v : ℙ K V) : v.submodule = K ∙ v.rep :=
by { conv_lhs { rw ← v.mk_rep }, refl }
lemma finrank_submodule (v : ℙ K V) : finrank K v.submodule = 1 :=
begin
rw submodule_eq,
exact finrank_span_singleton v.rep_nonzero,
end
instance (v : ℙ K V) : finite_dimensional K v.submodule :=
by { rw ← v.mk_rep, change finite_dimensional K (K ∙ v.rep), apply_instance }
lemma submodule_injective : function.injective
(projectivization.submodule : ℙ K V → submodule K V) :=
begin
intros u v h, replace h := le_of_eq h,
simp only [submodule_eq] at h,
rw submodule.le_span_singleton_iff at h,
rw [← mk_rep v, ← mk_rep u],
apply quotient.sound',
obtain ⟨a,ha⟩ := h u.rep (submodule.mem_span_singleton_self _),
have : a ≠ 0 := λ c, u.rep_nonzero (by simpa [c] using ha.symm),
use [units.mk0 a this, ha],
end
variables (K V)
/-- The equivalence between the projectivization and the
collection of subspaces of dimension 1. -/
noncomputable
def equiv_submodule : ℙ K V ≃ { H : submodule K V // finrank K H = 1 } :=
equiv.of_bijective (λ v, ⟨v.submodule, v.finrank_submodule⟩)
begin
split,
{ intros u v h, apply_fun (λ e, e.val) at h,
apply submodule_injective h },
{ rintros ⟨H, h⟩,
rw finrank_eq_one_iff' at h,
obtain ⟨v, hv, h⟩ := h,
have : (v : V) ≠ 0 := λ c, hv (subtype.coe_injective c),
use mk K v this,
symmetry,
ext x, revert x, erw ← set.ext_iff, ext x,
dsimp [-set_like.mem_coe],
rw [submodule.span_singleton_eq_range],
refine ⟨λ hh, _, _⟩,
{ obtain ⟨c,hc⟩ := h ⟨x,hh⟩,
exact ⟨c, congr_arg coe hc⟩ },
{ rintros ⟨c,rfl⟩,
refine submodule.smul_mem _ _ v.2 } }
end
variables {K V}
/-- Construct an element of the projectivization from a subspace of dimension 1. -/
noncomputable
def mk'' (H : _root_.submodule K V) (h : finrank K H = 1) : ℙ K V :=
(equiv_submodule K V).symm ⟨H,h⟩
@[simp]
lemma submodule_mk'' (H : _root_.submodule K V) (h : finrank K H = 1) :
(mk'' H h).submodule = H :=
begin
suffices : (equiv_submodule K V) (mk'' H h) = ⟨H,h⟩, by exact congr_arg coe this,
dsimp [mk''],
simp
end
@[simp]
lemma mk''_submodule (v : ℙ K V) : mk'' v.submodule v.finrank_submodule = v :=
show (equiv_submodule K V).symm (equiv_submodule K V _) = _, by simp
section map
variables {L W : Type*} [division_ring L] [add_comm_group W] [module L W]
/-- An injective semilinear map of vector spaces induces a map on projective spaces. -/
def map {σ : K →+* L} (f : V →ₛₗ[σ] W) (hf : function.injective f) :
ℙ K V → ℙ L W :=
quotient.map' (λ v, ⟨f v, λ c, v.2 (hf (by simp [c]))⟩)
begin
rintros ⟨u,hu⟩ ⟨v,hv⟩ ⟨a,ha⟩,
use units.map σ.to_monoid_hom a,
dsimp at ⊢ ha,
erw [← f.map_smulₛₗ, ha],
end
/-- Mapping with respect to a semilinear map over an isomorphism of fields yields
an injective map on projective spaces. -/
lemma map_injective {σ : K →+* L} {τ : L →+* K} [ring_hom_inv_pair σ τ]
(f : V →ₛₗ[σ] W) (hf : function.injective f) :
function.injective (map f hf) :=
begin
intros u v h,
rw [← u.mk_rep, ← v.mk_rep] at *,
apply quotient.sound',
dsimp [map, mk] at h,
simp only [quotient.eq'] at h,
obtain ⟨a,ha⟩ := h,
use units.map τ.to_monoid_hom a,
dsimp at ⊢ ha,
have : (a : L) = σ (τ a), by rw ring_hom_inv_pair.comp_apply_eq₂,
change (a : L) • f v.rep = f u.rep at ha,
rw [this, ← f.map_smulₛₗ] at ha,
exact hf ha,
end
@[simp]
lemma map_id : map
(linear_map.id : V →ₗ[K] V)
(linear_equiv.refl K V).injective = id :=
by { ext v, induction v using projectivization.ind, refl }
@[simp]
lemma map_comp {F U : Type*} [field F] [add_comm_group U] [module F U]
{σ : K →+* L} {τ : L →+* F} {γ : K →+* F} [ring_hom_comp_triple σ τ γ]
(f : V →ₛₗ[σ] W) (hf : function.injective f)
(g : W →ₛₗ[τ] U) (hg : function.injective g) :
map (g.comp f) (hg.comp hf) = map g hg ∘ map f hf :=
by { ext v, induction v using projectivization.ind, refl }
end map
end projectivization
|
8b7c1b9f199e893bc2b2546398815fe9d75b0c2a | 7cef822f3b952965621309e88eadf618da0c8ae9 | /src/category_theory/groupoid.lean | ef96cfba620bd4eacb370bc8c616354474528f61 | [
"Apache-2.0"
] | permissive | rmitta/mathlib | 8d90aee30b4db2b013e01f62c33f297d7e64a43d | 883d974b608845bad30ae19e27e33c285200bf84 | refs/heads/master | 1,585,776,832,544 | 1,576,874,096,000 | 1,576,874,096,000 | 153,663,165 | 0 | 2 | Apache-2.0 | 1,544,806,490,000 | 1,539,884,365,000 | Lean | UTF-8 | Lean | false | false | 1,573 | lean | /-
Copyright (c) 2018 Reid Barton All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton
-/
import category_theory.category
import category_theory.isomorphism
import data.equiv.basic
namespace category_theory
universes v u -- declare the `v`'s first; see `category_theory.category` for an explanation
section prio
set_option default_priority 100 -- see Note [default priority]
/-- A `groupoid` is a category such that all morphisms are isomorphisms. -/
class groupoid (obj : Type u) extends category.{v} obj : Type (max u (v+1)) :=
(inv : Π {X Y : obj}, (X ⟶ Y) → (Y ⟶ X))
(inv_comp' : ∀ {X Y : obj} (f : X ⟶ Y), comp (inv f) f = id Y . obviously)
(comp_inv' : ∀ {X Y : obj} (f : X ⟶ Y), comp f (inv f) = id X . obviously)
end prio
restate_axiom groupoid.inv_comp'
restate_axiom groupoid.comp_inv'
attribute [simp] groupoid.inv_comp groupoid.comp_inv
abbreviation large_groupoid (C : Type (u+1)) : Type (u+1) := groupoid.{u} C
abbreviation small_groupoid (C : Type u) : Type (u+1) := groupoid.{u} C
section
variables {C : Type u} [𝒞 : groupoid.{v} C] {X Y : C}
include 𝒞
@[priority 100] -- see Note [lower instance priority]
instance is_iso.of_groupoid (f : X ⟶ Y) : is_iso f := { inv := groupoid.inv f }
variables (X Y)
/-- In a groupoid, isomorphisms are equivalent to morphisms. -/
def groupoid.iso_equiv_hom : (X ≅ Y) ≃ (X ⟶ Y) :=
{ to_fun := iso.hom,
inv_fun := λ f, as_iso f,
left_inv := λ i, iso.ext rfl,
right_inv := λ f, rfl }
end
end category_theory
|
1d555ab0b70a0beeba080c60cd116c5bc2e0d7cc | cf39355caa609c0f33405126beee2739aa3cb77e | /tests/lean/sec_param_pp2.lean | 770b29f4d9459291704ed3025cd0cbfab7b24d35 | [
"Apache-2.0"
] | permissive | leanprover-community/lean | 12b87f69d92e614daea8bcc9d4de9a9ace089d0e | cce7990ea86a78bdb383e38ed7f9b5ba93c60ce0 | refs/heads/master | 1,687,508,156,644 | 1,684,951,104,000 | 1,684,951,104,000 | 169,960,991 | 457 | 107 | Apache-2.0 | 1,686,744,372,000 | 1,549,790,268,000 | C++ | UTF-8 | Lean | false | false | 234 | lean | section
parameters {A : Type} (a : A)
section
parameters {B : Type} (b : B)
variable f : A → B → A
definition id2 := f a b
#check id2
set_option pp.universes true
#check id2
end
#check id2
end
#check id2
|
70928715ed1667410f7cf3c054071b0161c179f9 | cf39355caa609c0f33405126beee2739aa3cb77e | /tests/lean/widget/widget5.lean | b89c06dd821ddf3e07b0dd26dbbd2c2ef5116550 | [
"Apache-2.0"
] | permissive | leanprover-community/lean | 12b87f69d92e614daea8bcc9d4de9a9ace089d0e | cce7990ea86a78bdb383e38ed7f9b5ba93c60ce0 | refs/heads/master | 1,687,508,156,644 | 1,684,951,104,000 | 1,684,951,104,000 | 169,960,991 | 457 | 107 | Apache-2.0 | 1,686,744,372,000 | 1,549,790,268,000 | C++ | UTF-8 | Lean | false | false | 99 | lean | open tactic widget
#html (component.pure $ λ _, [h "p" [attr.style [("color", "blue")]] ["hi"]])
|
2f43701c23c6f01eb5b9b54bc7c2959890c1d6b5 | 82e44445c70db0f03e30d7be725775f122d72f3e | /src/algebra/homology/exact.lean | 086f111614d16a078455a57155f0e88de83991a3 | [
"Apache-2.0"
] | permissive | stjordanis/mathlib | 51e286d19140e3788ef2c470bc7b953e4991f0c9 | 2568d41bca08f5d6bf39d915434c8447e21f42ee | refs/heads/master | 1,631,748,053,501 | 1,627,938,886,000 | 1,627,938,886,000 | 228,728,358 | 0 | 0 | Apache-2.0 | 1,576,630,588,000 | 1,576,630,587,000 | null | UTF-8 | Lean | false | false | 9,617 | lean | /-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import algebra.homology.image_to_kernel
/-!
# Exact sequences
In a category with zero morphisms, images, and equalizers we say that `f : A ⟶ B` and `g : B ⟶ C`
are exact if `f ≫ g = 0` and the natural map `image f ⟶ kernel g` is an epimorphism.
In any preadditive category this is equivalent to the homology at `B` vanishing.
However in general it is weaker than other reasonable definitions of exactness,
particularly that
1. the inclusion map `image.ι f` is a kernel of `g` or
2. `image f ⟶ kernel g` is an isomorphism or
3. `image_subobject f = kernel_subobject f`.
However when the category is abelian, these all become equivalent;
these results are found in `category_theory/abelian/exact.lean`.
# Main results
* Suppose that cokernels exist and that `f` and `g` are exact.
If `s` is any kernel fork over `g` and `t` is any cokernel cofork over `f`,
then `fork.ι s ≫ cofork.π t = 0`.
* Precomposing the first morphism with an epimorphism retains exactness.
Postcomposing the second morphism with a monomorphism retains exactness.
* If `f` and `g` are exact and `i` is an isomorphism,
then `f ≫ i.hom` and `i.inv ≫ g` are also exact.
# Future work
* Short exact sequences, split exact sequences, the splitting lemma (maybe only for abelian
categories?)
* Two adjacent maps in a chain complex are exact iff the homology vanishes
-/
universes v u
open category_theory
open category_theory.limits
variables {V : Type u} [category.{v} V]
variables [has_images V]
namespace category_theory
/--
Two morphisms `f : A ⟶ B`, `g : B ⟶ C` are called exact if `w : f ≫ g = 0` and the natural map
`image_to_kernel f g w : image_subobject f ⟶ kernel_subobject g` is an epimorphism.
In any preadditive category, this is equivalent to `w : f ≫ g = 0` and `homology f g w ≅ 0`.
In an abelian category, this is equivalent to `image_to_kernel f g w` being an isomorphism,
and hence equivalent to the usual definition,
`image_subobject f = kernel_subobject g`.
-/
-- One nice feature of this definition is that we have
-- `epi f → exact g h → exact (f ≫ g) h` and `exact f g → mono h → exact f (g ≫ h)`,
-- which do not necessarily hold in a non-abelian category with the usual definition of `exact`.
class exact [has_zero_morphisms V] [has_kernels V] {A B C : V} (f : A ⟶ B) (g : B ⟶ C) : Prop :=
(w : f ≫ g = 0)
(epi : epi (image_to_kernel f g w))
attribute [instance] exact.epi
attribute [simp, reassoc] exact.w
section
variables [has_zero_object V] [preadditive V] [has_kernels V] [has_cokernels V]
open_locale zero_object
/--
In any preadditive category,
composable morphisms `f g` are exact iff they compose to zero and the homology vanishes.
-/
lemma preadditive.exact_iff_homology_zero {A B C : V} (f : A ⟶ B) (g : B ⟶ C) :
exact f g ↔ ∃ w : f ≫ g = 0, nonempty (homology f g w ≅ 0) :=
⟨λ h, ⟨h.w, ⟨cokernel.of_epi _⟩⟩,
λ h, begin
obtain ⟨w, ⟨i⟩⟩ := h,
exact ⟨w, preadditive.epi_of_cokernel_zero ((cancel_mono i.hom).mp (by ext))⟩,
end⟩
end
section
variables [has_zero_morphisms V] [has_kernels V]
lemma comp_eq_zero_of_image_eq_kernel {A B C : V} (f : A ⟶ B) (g : B ⟶ C)
(p : image_subobject f = kernel_subobject g) : f ≫ g = 0 :=
begin
rw [←image_subobject_arrow_comp f, category.assoc],
convert comp_zero,
rw p,
simp,
end
lemma image_to_kernel_is_iso_of_image_eq_kernel {A B C : V} (f : A ⟶ B) (g : B ⟶ C)
(p : image_subobject f = kernel_subobject g) :
is_iso (image_to_kernel f g (comp_eq_zero_of_image_eq_kernel f g p)) :=
begin
refine ⟨⟨subobject.of_le _ _ p.ge, _⟩⟩,
dsimp [image_to_kernel],
simp only [subobject.of_le_comp_of_le, subobject.of_le_refl],
simp,
end
-- We'll prove the converse later, when `V` is abelian.
lemma exact_of_image_eq_kernel {A B C : V} (f : A ⟶ B) (g : B ⟶ C)
(p : image_subobject f = kernel_subobject g) : exact f g :=
{ w := comp_eq_zero_of_image_eq_kernel f g p,
epi := begin
haveI := image_to_kernel_is_iso_of_image_eq_kernel f g p,
apply_instance,
end }
end
variables {A B C D : V} {f : A ⟶ B} {g : B ⟶ C} {h : C ⟶ D}
local attribute [instance] epi_comp
section
variables [has_zero_morphisms V] [has_equalizers V]
instance exact_comp_hom_inv_comp [exact f g] (i : B ≅ D) : exact (f ≫ i.hom) (i.inv ≫ g) :=
begin
refine ⟨by simp, _⟩,
rw image_to_kernel_comp_hom_inv_comp,
apply_instance,
end
instance exact_comp_inv_hom_comp [exact f g] (i : D ≅ B) : exact (f ≫ i.inv) (i.hom ≫ g) :=
category_theory.exact_comp_hom_inv_comp i.symm
lemma exact_comp_hom_inv_comp_iff (i : B ≅ D) : exact (f ≫ i.hom) (i.inv ≫ g) ↔ exact f g :=
begin
refine ⟨_, by { introI, apply_instance }⟩,
introI,
have : exact ((f ≫ i.hom) ≫ i.inv) (i.hom ≫ i.inv ≫ g) := infer_instance,
simpa using this
end
lemma exact_epi_comp [exact g h] [epi f] : exact (f ≫ g) h :=
begin
refine ⟨by simp, _⟩,
rw image_to_kernel_comp_left,
apply_instance,
end
@[simp]
lemma exact_iso_comp [is_iso f] : exact (f ≫ g) h ↔ exact g h :=
⟨λ w, by { rw ←is_iso.inv_hom_id_assoc f g, exactI exact_epi_comp, }, λ w, by exactI exact_epi_comp⟩
lemma exact_comp_mono [exact f g] [mono h] : exact f (g ≫ h) :=
begin
refine ⟨by simp, _⟩,
rw image_to_kernel_comp_right f g h exact.w,
apply_instance,
end
@[simp]
lemma exact_comp_iso [is_iso h] : exact f (g ≫ h) ↔ exact f g :=
⟨λ w, begin
rw [←category.comp_id g, ←is_iso.hom_inv_id h, ←category.assoc],
exactI exact_comp_mono,
end,
λ w, by exactI exact_comp_mono⟩
lemma exact_kernel_subobject_arrow : exact (kernel_subobject f).arrow f :=
begin
refine ⟨by simp, _⟩,
apply @is_iso.epi_of_iso _ _ _ _ _ _,
exact ⟨⟨factor_thru_image_subobject _, by { ext, simp, }, by { ext, simp, }⟩⟩,
end
lemma exact_kernel_ι : exact (kernel.ι f) f :=
by { rw [←kernel_subobject_arrow', exact_iso_comp], exact exact_kernel_subobject_arrow }
instance [exact f g] : epi (factor_thru_kernel_subobject g f (by simp)) :=
begin
rw ←factor_thru_image_subobject_comp_image_to_kernel,
apply epi_comp,
end
instance [exact f g] : epi (kernel.lift g f (by simp)) :=
begin
rw ←factor_thru_kernel_subobject_comp_kernel_subobject_iso,
apply epi_comp
end
variables (A)
lemma kernel_subobject_arrow_eq_zero_of_exact_zero_left [exact (0 : A ⟶ B) g] :
(kernel_subobject g).arrow = 0 :=
begin
rw [←cancel_epi (image_to_kernel (0 : A ⟶ B) g exact.w),
←cancel_epi (factor_thru_image_subobject (0 : A ⟶ B))],
simp
end
lemma kernel_ι_eq_zero_of_exact_zero_left [exact (0 : A ⟶ B) g] :
kernel.ι g = 0 :=
by { rw ←kernel_subobject_arrow', simp [kernel_subobject_arrow_eq_zero_of_exact_zero_left A], }
lemma exact_zero_left_of_mono [has_zero_object V] [mono g] : exact (0 : A ⟶ B) g :=
⟨by simp, image_to_kernel_epi_of_zero_of_mono _⟩
end
section has_cokernels
variables [has_zero_morphisms V] [has_equalizers V] [has_cokernels V] (f g)
@[simp, reassoc] lemma kernel_comp_cokernel [exact f g] : kernel.ι g ≫ cokernel.π f = 0 :=
begin
rw [←kernel_subobject_arrow', category.assoc],
convert comp_zero,
apply zero_of_epi_comp (image_to_kernel f g exact.w) _,
rw [image_to_kernel_arrow_assoc, ←image_subobject_arrow, category.assoc, ←iso.eq_inv_comp],
ext,
simp,
end
lemma comp_eq_zero_of_exact [exact f g] {X Y : V} {ι : X ⟶ B} (hι : ι ≫ g = 0) {π : B ⟶ Y}
(hπ : f ≫ π = 0) : ι ≫ π = 0 :=
by rw [←kernel.lift_ι _ _ hι, ←cokernel.π_desc _ _ hπ, category.assoc, kernel_comp_cokernel_assoc,
zero_comp, comp_zero]
@[simp, reassoc] lemma fork_ι_comp_cofork_π [exact f g] (s : kernel_fork g)
(t : cokernel_cofork f) : fork.ι s ≫ cofork.π t = 0 :=
comp_eq_zero_of_exact f g (kernel_fork.condition s) (cokernel_cofork.condition t)
end has_cokernels
section
variables [has_zero_object V]
open_locale zero_object
section
variables [has_zero_morphisms V] [has_kernels V]
instance exact_of_zero {A C : V} (f : A ⟶ 0) (g : 0 ⟶ C) : exact f g :=
begin
obtain rfl : f = 0 := by ext,
obtain rfl : g = 0 := by ext,
fsplit,
{ simp, },
{ exact image_to_kernel_epi_of_zero_of_mono 0, },
end
instance exact_zero_mono {B C : V} (f : B ⟶ C) [mono f] : exact (0 : (0 ⟶ B)) f :=
⟨by simp, infer_instance⟩
instance exact_epi_zero {A B : V} (f : A ⟶ B) [epi f] : exact f (0 : (B ⟶ 0)) :=
⟨by simp, infer_instance⟩
end
section
variables [preadditive V]
lemma mono_iff_exact_zero_left [has_kernels V]{B C : V} (f : B ⟶ C) :
mono f ↔ exact (0 : (0 ⟶ B)) f :=
⟨λ h, by { resetI, apply_instance, },
λ h, preadditive.mono_of_kernel_iso_zero
((kernel_subobject_iso f).symm ≪≫ iso_zero_of_epi_zero (by simpa using h.epi))⟩
lemma epi_iff_exact_zero_right [has_equalizers V] {A B : V} (f : A ⟶ B) :
epi f ↔ exact f (0 : (B ⟶ 0)) :=
⟨λ h, by { resetI, apply_instance, },
λ h, begin
have e₁ := h.epi,
rw image_to_kernel_zero_right at e₁,
have e₂ : epi (((image_subobject f).arrow ≫ inv (kernel_subobject 0).arrow) ≫
(kernel_subobject 0).arrow) := @epi_comp _ _ _ _ _ _ e₁ _ _,
rw [category.assoc, is_iso.inv_hom_id, category.comp_id] at e₂,
rw [←image_subobject_arrow] at e₂,
resetI,
haveI : epi (image.ι f) := epi_of_epi (image_subobject_iso f).hom (image.ι f),
apply epi_of_epi_image,
end⟩
end
end
end category_theory
|
48bf3d3a4657e6f8d975f7f97f3664b189c4f2d2 | 453dcd7c0d1ef170b0843a81d7d8caedc9741dce | /data/int/basic.lean | 2b2bd2ca4acd78c8bdcb7804e5528776f5c2f1b5 | [
"Apache-2.0"
] | permissive | amswerdlow/mathlib | 9af77a1f08486d8fa059448ae2d97795bd12ec0c | 27f96e30b9c9bf518341705c99d641c38638dfd0 | refs/heads/master | 1,585,200,953,598 | 1,534,275,532,000 | 1,534,275,532,000 | 144,564,700 | 0 | 0 | null | 1,534,156,197,000 | 1,534,156,197,000 | null | UTF-8 | Lean | false | false | 43,279 | lean | /-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
The integers, with addition, multiplication, and subtraction.
-/
import data.nat.basic algebra.char_zero algebra.order_functions
open nat
namespace int
instance : inhabited ℤ := ⟨0⟩
meta instance : has_to_format ℤ := ⟨λ z, int.rec_on z (λ k, ↑k) (λ k, "-("++↑k++"+1)")⟩
attribute [simp] int.coe_nat_add int.coe_nat_mul int.coe_nat_zero int.coe_nat_one int.coe_nat_succ
@[simp] theorem coe_nat_mul_neg_succ (m n : ℕ) : (m : ℤ) * -[1+ n] = -(m * succ n) := rfl
@[simp] theorem neg_succ_mul_coe_nat (m n : ℕ) : -[1+ m] * n = -(succ m * n) := rfl
@[simp] theorem neg_succ_mul_neg_succ (m n : ℕ) : -[1+ m] * -[1+ n] = succ m * succ n := rfl
theorem coe_nat_le {m n : ℕ} : (↑m : ℤ) ≤ ↑n ↔ m ≤ n := coe_nat_le_coe_nat_iff m n
theorem coe_nat_lt {m n : ℕ} : (↑m : ℤ) < ↑n ↔ m < n := coe_nat_lt_coe_nat_iff m n
theorem coe_nat_inj' {m n : ℕ} : (↑m : ℤ) = ↑n ↔ m = n := int.coe_nat_eq_coe_nat_iff m n
@[simp] theorem coe_nat_pos {n : ℕ} : (0 : ℤ) < n ↔ 0 < n :=
by rw [← int.coe_nat_zero, coe_nat_lt]
@[simp] theorem coe_nat_eq_zero {n : ℕ} : (n : ℤ) = 0 ↔ n = 0 :=
by rw [← int.coe_nat_zero, coe_nat_inj']
@[simp] theorem coe_nat_ne_zero {n : ℕ} : (n : ℤ) ≠ 0 ↔ n ≠ 0 :=
not_congr coe_nat_eq_zero
lemma coe_nat_nonneg (n : ℕ) : 0 ≤ (n : ℤ) := coe_nat_le.2 (zero_le _)
lemma coe_nat_ne_zero_iff_pos {n : ℕ} : (n : ℤ) ≠ 0 ↔ 0 < n :=
⟨λ h, nat.pos_of_ne_zero (coe_nat_ne_zero.1 h),
λ h, (ne_of_lt (coe_nat_lt.2 h)).symm⟩
/- succ and pred -/
/-- Immediate successor of an integer: `succ n = n + 1` -/
def succ (a : ℤ) := a + 1
/-- Immediate predecessor of an integer: `pred n = n - 1` -/
def pred (a : ℤ) := a - 1
theorem nat_succ_eq_int_succ (n : ℕ) : (nat.succ n : ℤ) = int.succ n := rfl
theorem pred_succ (a : ℤ) : pred (succ a) = a := add_sub_cancel _ _
theorem succ_pred (a : ℤ) : succ (pred a) = a := sub_add_cancel _ _
theorem neg_succ (a : ℤ) : -succ a = pred (-a) := neg_add _ _
theorem succ_neg_succ (a : ℤ) : succ (-succ a) = -a :=
by rw [neg_succ, succ_pred]
theorem neg_pred (a : ℤ) : -pred a = succ (-a) :=
by rw [eq_neg_of_eq_neg (neg_succ (-a)).symm, neg_neg]
theorem pred_neg_pred (a : ℤ) : pred (-pred a) = -a :=
by rw [neg_pred, pred_succ]
theorem pred_nat_succ (n : ℕ) : pred (nat.succ n) = n := pred_succ n
theorem neg_nat_succ (n : ℕ) : -(nat.succ n : ℤ) = pred (-n) := neg_succ n
theorem succ_neg_nat_succ (n : ℕ) : succ (-nat.succ n) = -n := succ_neg_succ n
theorem lt_succ_self (a : ℤ) : a < succ a :=
lt_add_of_pos_right _ zero_lt_one
theorem pred_self_lt (a : ℤ) : pred a < a :=
sub_lt_self _ zero_lt_one
theorem add_one_le_iff {a b : ℤ} : a + 1 ≤ b ↔ a < b := iff.rfl
theorem lt_add_one_iff {a b : ℤ} : a < b + 1 ↔ a ≤ b :=
@add_le_add_iff_right _ _ a b 1
theorem sub_one_le_iff {a b : ℤ} : a - 1 < b ↔ a ≤ b :=
sub_lt_iff_lt_add.trans lt_add_one_iff
theorem le_sub_one_iff {a b : ℤ} : a ≤ b - 1 ↔ a < b :=
le_sub_iff_add_le
@[elab_as_eliminator] protected lemma induction_on {p : ℤ → Prop}
(i : ℤ) (hz : p 0) (hp : ∀i, p i → p (i + 1)) (hn : ∀i, p i → p (i - 1)) : p i :=
begin
induction i,
{ induction i,
{ exact hz },
{ exact hp _ i_ih } },
{ have : ∀n:ℕ, p (- n),
{ intro n, induction n,
{ simp [hz] },
{ have := hn _ n_ih, simpa } },
exact this (i + 1) }
end
/- / -/
@[simp] theorem of_nat_div (m n : ℕ) : of_nat (m / n) = (of_nat m) / (of_nat n) := rfl
@[simp] theorem coe_nat_div (m n : ℕ) : ((m / n : ℕ) : ℤ) = m / n := rfl
theorem neg_succ_of_nat_div (m : ℕ) {b : ℤ} (H : b > 0) :
-[1+m] / b = -(m / b + 1) :=
match b, eq_succ_of_zero_lt H with ._, ⟨n, rfl⟩ := rfl end
@[simp] protected theorem div_neg : ∀ (a b : ℤ), a / -b = -(a / b)
| (m : ℕ) 0 := show of_nat (m / 0) = -(m / 0 : ℕ), by rw nat.div_zero; refl
| (m : ℕ) (n+1:ℕ) := rfl
| 0 -[1+ n] := rfl
| (m+1:ℕ) -[1+ n] := (neg_neg _).symm
| -[1+ m] 0 := rfl
| -[1+ m] (n+1:ℕ) := rfl
| -[1+ m] -[1+ n] := rfl
theorem div_of_neg_of_pos {a b : ℤ} (Ha : a < 0) (Hb : b > 0) : a / b = -((-a - 1) / b + 1) :=
match a, b, eq_neg_succ_of_lt_zero Ha, eq_succ_of_zero_lt Hb with
| ._, ._, ⟨m, rfl⟩, ⟨n, rfl⟩ :=
by change (- -[1+ m] : ℤ) with (m+1 : ℤ); rw add_sub_cancel; refl
end
protected theorem div_nonneg {a b : ℤ} (Ha : a ≥ 0) (Hb : b ≥ 0) : a / b ≥ 0 :=
match a, b, eq_coe_of_zero_le Ha, eq_coe_of_zero_le Hb with
| ._, ._, ⟨m, rfl⟩, ⟨n, rfl⟩ := coe_zero_le _
end
protected theorem div_nonpos {a b : ℤ} (Ha : a ≥ 0) (Hb : b ≤ 0) : a / b ≤ 0 :=
nonpos_of_neg_nonneg $ by rw [← int.div_neg]; exact int.div_nonneg Ha (neg_nonneg_of_nonpos Hb)
theorem div_neg' {a b : ℤ} (Ha : a < 0) (Hb : b > 0) : a / b < 0 :=
match a, b, eq_neg_succ_of_lt_zero Ha, eq_succ_of_zero_lt Hb with
| ._, ._, ⟨m, rfl⟩, ⟨n, rfl⟩ := neg_succ_lt_zero _
end
@[simp] protected theorem zero_div : ∀ (b : ℤ), 0 / b = 0
| 0 := rfl
| (n+1:ℕ) := rfl
| -[1+ n] := rfl
@[simp] protected theorem div_zero : ∀ (a : ℤ), a / 0 = 0
| 0 := rfl
| (n+1:ℕ) := rfl
| -[1+ n] := rfl
@[simp] protected theorem div_one : ∀ (a : ℤ), a / 1 = a
| 0 := rfl
| (n+1:ℕ) := congr_arg of_nat (nat.div_one _)
| -[1+ n] := congr_arg neg_succ_of_nat (nat.div_one _)
theorem div_eq_zero_of_lt {a b : ℤ} (H1 : 0 ≤ a) (H2 : a < b) : a / b = 0 :=
match a, b, eq_coe_of_zero_le H1, eq_succ_of_zero_lt (lt_of_le_of_lt H1 H2), H2 with
| ._, ._, ⟨m, rfl⟩, ⟨n, rfl⟩, H2 :=
congr_arg of_nat $ nat.div_eq_of_lt $ lt_of_coe_nat_lt_coe_nat H2
end
theorem div_eq_zero_of_lt_abs {a b : ℤ} (H1 : 0 ≤ a) (H2 : a < abs b) : a / b = 0 :=
match b, abs b, abs_eq_nat_abs b, H2 with
| (n : ℕ), ._, rfl, H2 := div_eq_zero_of_lt H1 H2
| -[1+ n], ._, rfl, H2 := neg_inj $ by rw [← int.div_neg]; exact div_eq_zero_of_lt H1 H2
end
protected theorem add_mul_div_right (a b : ℤ) {c : ℤ} (H : c ≠ 0) :
(a + b * c) / c = a / c + b :=
have ∀ {k n : ℕ} {a : ℤ}, (a + n * k.succ) / k.succ = a / k.succ + n, from
λ k n a, match a with
| (m : ℕ) := congr_arg of_nat $ nat.add_mul_div_right _ _ k.succ_pos
| -[1+ m] := show ((n * k.succ:ℕ) - m.succ : ℤ) / k.succ =
n - (m / k.succ + 1 : ℕ), begin
cases lt_or_ge m (n*k.succ) with h h,
{ rw [← int.coe_nat_sub h,
← int.coe_nat_sub ((nat.div_lt_iff_lt_mul _ _ k.succ_pos).2 h)],
apply congr_arg of_nat,
rw [mul_comm, nat.mul_sub_div], rwa mul_comm },
{ change (↑(n * nat.succ k) - (m + 1) : ℤ) / ↑(nat.succ k) =
↑n - ((m / nat.succ k : ℕ) + 1),
rw [← sub_sub, ← sub_sub, ← neg_sub (m:ℤ), ← neg_sub _ (n:ℤ),
← int.coe_nat_sub h,
← int.coe_nat_sub ((nat.le_div_iff_mul_le _ _ k.succ_pos).2 h),
← neg_succ_of_nat_coe', ← neg_succ_of_nat_coe'],
{ apply congr_arg neg_succ_of_nat,
rw [mul_comm, nat.sub_mul_div], rwa mul_comm } }
end
end,
have ∀ {a b c : ℤ}, c > 0 → (a + b * c) / c = a / c + b, from
λ a b c H, match c, eq_succ_of_zero_lt H, b with
| ._, ⟨k, rfl⟩, (n : ℕ) := this
| ._, ⟨k, rfl⟩, -[1+ n] :=
show (a - n.succ * k.succ) / k.succ = (a / k.succ) - n.succ, from
eq_sub_of_add_eq $ by rw [← this, sub_add_cancel]
end,
match lt_trichotomy c 0 with
| or.inl hlt := neg_inj $ by rw [← int.div_neg, neg_add, ← int.div_neg, ← neg_mul_neg];
apply this (neg_pos_of_neg hlt)
| or.inr (or.inl heq) := absurd heq H
| or.inr (or.inr hgt) := this hgt
end
protected theorem add_mul_div_left (a : ℤ) {b : ℤ} (c : ℤ) (H : b ≠ 0) :
(a + b * c) / b = a / b + c :=
by rw [mul_comm, int.add_mul_div_right _ _ H]
@[simp] protected theorem mul_div_cancel (a : ℤ) {b : ℤ} (H : b ≠ 0) : a * b / b = a :=
by have := int.add_mul_div_right 0 a H;
rwa [zero_add, int.zero_div, zero_add] at this
@[simp] protected theorem mul_div_cancel_left {a : ℤ} (b : ℤ) (H : a ≠ 0) : a * b / a = b :=
by rw [mul_comm, int.mul_div_cancel _ H]
@[simp] protected theorem div_self {a : ℤ} (H : a ≠ 0) : a / a = 1 :=
by have := int.mul_div_cancel 1 H; rwa one_mul at this
/- mod -/
theorem of_nat_mod (m n : nat) : (m % n : ℤ) = of_nat (m % n) := rfl
@[simp] theorem coe_nat_mod (m n : ℕ) : (↑(m % n) : ℤ) = ↑m % ↑n := rfl
theorem neg_succ_of_nat_mod (m : ℕ) {b : ℤ} (bpos : b > 0) :
-[1+m] % b = b - 1 - m % b :=
by rw [sub_sub, add_comm]; exact
match b, eq_succ_of_zero_lt bpos with ._, ⟨n, rfl⟩ := rfl end
@[simp] theorem mod_neg : ∀ (a b : ℤ), a % -b = a % b
| (m : ℕ) n := @congr_arg ℕ ℤ _ _ (λ i, ↑(m % i)) (nat_abs_neg _)
| -[1+ m] n := @congr_arg ℕ ℤ _ _ (λ i, sub_nat_nat i (nat.succ (m % i))) (nat_abs_neg _)
@[simp] theorem mod_abs (a b : ℤ) : a % (abs b) = a % b :=
abs_by_cases (λ i, a % i = a % b) rfl (mod_neg _ _)
@[simp] theorem zero_mod (b : ℤ) : 0 % b = 0 :=
congr_arg of_nat $ nat.zero_mod _
@[simp] theorem mod_zero : ∀ (a : ℤ), a % 0 = a
| (m : ℕ) := congr_arg of_nat $ nat.mod_zero _
| -[1+ m] := congr_arg neg_succ_of_nat $ nat.mod_zero _
@[simp] theorem mod_one : ∀ (a : ℤ), a % 1 = 0
| (m : ℕ) := congr_arg of_nat $ nat.mod_one _
| -[1+ m] := show (1 - (m % 1).succ : ℤ) = 0, by rw nat.mod_one; refl
theorem mod_eq_of_lt {a b : ℤ} (H1 : 0 ≤ a) (H2 : a < b) : a % b = a :=
match a, b, eq_coe_of_zero_le H1, eq_coe_of_zero_le (le_trans H1 (le_of_lt H2)), H2 with
| ._, ._, ⟨m, rfl⟩, ⟨n, rfl⟩, H2 :=
congr_arg of_nat $ nat.mod_eq_of_lt (lt_of_coe_nat_lt_coe_nat H2)
end
theorem mod_nonneg : ∀ (a : ℤ) {b : ℤ}, b ≠ 0 → a % b ≥ 0
| (m : ℕ) n H := coe_zero_le _
| -[1+ m] n H :=
sub_nonneg_of_le $ coe_nat_le_coe_nat_of_le $ nat.mod_lt _ (nat_abs_pos_of_ne_zero H)
theorem mod_lt_of_pos (a : ℤ) {b : ℤ} (H : b > 0) : a % b < b :=
match a, b, eq_succ_of_zero_lt H with
| (m : ℕ), ._, ⟨n, rfl⟩ := coe_nat_lt_coe_nat_of_lt (nat.mod_lt _ (nat.succ_pos _))
| -[1+ m], ._, ⟨n, rfl⟩ := sub_lt_self _ (coe_nat_lt_coe_nat_of_lt $ nat.succ_pos _)
end
theorem mod_lt (a : ℤ) {b : ℤ} (H : b ≠ 0) : a % b < abs b :=
by rw [← mod_abs]; exact mod_lt_of_pos _ (abs_pos_of_ne_zero H)
theorem mod_add_div_aux (m n : ℕ) : (n - (m % n + 1) - (n * (m / n) + n) : ℤ) = -[1+ m] :=
begin
rw [← sub_sub, neg_succ_of_nat_coe, sub_sub (n:ℤ)],
apply eq_neg_of_eq_neg,
rw [neg_sub, sub_sub_self, add_right_comm],
exact @congr_arg ℕ ℤ _ _ (λi, (i + 1 : ℤ)) (nat.mod_add_div _ _).symm
end
theorem mod_add_div : ∀ (a b : ℤ), a % b + b * (a / b) = a
| (m : ℕ) 0 := congr_arg of_nat (nat.mod_add_div _ _)
| (m : ℕ) (n+1:ℕ) := congr_arg of_nat (nat.mod_add_div _ _)
| 0 -[1+ n] := rfl
| (m+1:ℕ) -[1+ n] := show (_ + -(n+1) * -((m + 1) / (n + 1) : ℕ) : ℤ) = _,
by rw [neg_mul_neg]; exact congr_arg of_nat (nat.mod_add_div _ _)
| -[1+ m] 0 := by rw [mod_zero, int.div_zero]; refl
| -[1+ m] (n+1:ℕ) := mod_add_div_aux m n.succ
| -[1+ m] -[1+ n] := mod_add_div_aux m n.succ
theorem mod_def (a b : ℤ) : a % b = a - b * (a / b) :=
eq_sub_of_add_eq (mod_add_div _ _)
@[simp] theorem add_mul_mod_self {a b c : ℤ} : (a + b * c) % c = a % c :=
if cz : c = 0 then by rw [cz, mul_zero, add_zero] else
by rw [mod_def, mod_def, int.add_mul_div_right _ _ cz,
mul_add, mul_comm, add_sub_add_right_eq_sub]
@[simp] theorem add_mul_mod_self_left (a b c : ℤ) : (a + b * c) % b = a % b :=
by rw [mul_comm, add_mul_mod_self]
@[simp] theorem add_mod_self {a b : ℤ} : (a + b) % b = a % b :=
by have := add_mul_mod_self_left a b 1; rwa mul_one at this
@[simp] theorem add_mod_self_left {a b : ℤ} : (a + b) % a = b % a :=
by rw [add_comm, add_mod_self]
@[simp] theorem mod_add_mod (m n k : ℤ) : (m % n + k) % n = (m + k) % n :=
by have := (add_mul_mod_self_left (m % n + k) n (m / n)).symm;
rwa [add_right_comm, mod_add_div] at this
@[simp] theorem add_mod_mod (m n k : ℤ) : (m + n % k) % k = (m + n) % k :=
by rw [add_comm, mod_add_mod, add_comm]
theorem add_mod_eq_add_mod_right {m n k : ℤ} (i : ℤ) (H : m % n = k % n) :
(m + i) % n = (k + i) % n :=
by rw [← mod_add_mod, ← mod_add_mod k, H]
theorem add_mod_eq_add_mod_left {m n k : ℤ} (i : ℤ) (H : m % n = k % n) :
(i + m) % n = (i + k) % n :=
by rw [add_comm, add_mod_eq_add_mod_right _ H, add_comm]
theorem mod_add_cancel_right {m n k : ℤ} (i) : (m + i) % n = (k + i) % n ↔
m % n = k % n :=
⟨λ H, by have := add_mod_eq_add_mod_right (-i) H;
rwa [add_neg_cancel_right, add_neg_cancel_right] at this,
add_mod_eq_add_mod_right _⟩
theorem mod_add_cancel_left {m n k i : ℤ} :
(i + m) % n = (i + k) % n ↔ m % n = k % n :=
by rw [add_comm, add_comm i, mod_add_cancel_right]
theorem mod_sub_cancel_right {m n k : ℤ} (i) : (m - i) % n = (k - i) % n ↔
m % n = k % n :=
mod_add_cancel_right _
theorem mod_eq_mod_iff_mod_sub_eq_zero {m n k : ℤ} : m % n = k % n ↔ (m - k) % n = 0 :=
(mod_sub_cancel_right k).symm.trans $ by simp
@[simp] theorem mul_mod_left (a b : ℤ) : (a * b) % b = 0 :=
by rw [← zero_add (a * b), add_mul_mod_self, zero_mod]
@[simp] theorem mul_mod_right (a b : ℤ) : (a * b) % a = 0 :=
by rw [mul_comm, mul_mod_left]
@[simp] theorem mod_self {a : ℤ} : a % a = 0 :=
by have := mul_mod_left 1 a; rwa one_mul at this
@[simp] lemma mod_mod (a b : ℤ) : a % b % b = a % b :=
by conv {to_rhs, rw [← mod_add_div a b, add_mul_mod_self_left]}
/- properties of / and % -/
@[simp] theorem mul_div_mul_of_pos {a : ℤ} (b c : ℤ) (H : a > 0) : a * b / (a * c) = b / c :=
suffices ∀ (m k : ℕ) (b : ℤ), (m.succ * b / (m.succ * k) : ℤ) = b / k, from
match a, eq_succ_of_zero_lt H, c, eq_coe_or_neg c with
| ._, ⟨m, rfl⟩, ._, ⟨k, or.inl rfl⟩ := this _ _ _
| ._, ⟨m, rfl⟩, ._, ⟨k, or.inr rfl⟩ :=
by rw [← neg_mul_eq_mul_neg, int.div_neg, int.div_neg];
apply congr_arg has_neg.neg; apply this
end,
λ m k b, match b, k with
| (n : ℕ), k := congr_arg of_nat (nat.mul_div_mul _ _ m.succ_pos)
| -[1+ n], 0 := by rw [int.coe_nat_zero, mul_zero, int.div_zero, int.div_zero]
| -[1+ n], k+1 := congr_arg neg_succ_of_nat $
show (m.succ * n + m) / (m.succ * k.succ) = n / k.succ, begin
apply nat.div_eq_of_lt_le,
{ refine le_trans _ (nat.le_add_right _ _),
rw [← nat.mul_div_mul _ _ m.succ_pos],
apply nat.div_mul_le_self },
{ change m.succ * n.succ ≤ _,
rw [mul_left_comm],
apply nat.mul_le_mul_left,
apply (nat.div_lt_iff_lt_mul _ _ k.succ_pos).1,
apply nat.lt_succ_self }
end
end
@[simp] theorem mul_div_mul_of_pos_left (a : ℤ) {b : ℤ} (c : ℤ) (H : b > 0) :
a * b / (c * b) = a / c :=
by rw [mul_comm, mul_comm c, mul_div_mul_of_pos _ _ H]
@[simp] theorem mul_mod_mul_of_pos {a : ℤ} (b c : ℤ) (H : a > 0) : a * b % (a * c) = a * (b % c) :=
by rw [mod_def, mod_def, mul_div_mul_of_pos _ _ H, mul_sub_left_distrib, mul_assoc]
theorem lt_div_add_one_mul_self (a : ℤ) {b : ℤ} (H : b > 0) : a < (a / b + 1) * b :=
by rw [add_mul, one_mul, mul_comm]; apply lt_add_of_sub_left_lt;
rw [← mod_def]; apply mod_lt_of_pos _ H
theorem abs_div_le_abs : ∀ (a b : ℤ), abs (a / b) ≤ abs a :=
suffices ∀ (a : ℤ) (n : ℕ), abs (a / n) ≤ abs a, from
λ a b, match b, eq_coe_or_neg b with
| ._, ⟨n, or.inl rfl⟩ := this _ _
| ._, ⟨n, or.inr rfl⟩ := by rw [int.div_neg, abs_neg]; apply this
end,
λ a n, by rw [abs_eq_nat_abs, abs_eq_nat_abs]; exact
coe_nat_le_coe_nat_of_le (match a, n with
| (m : ℕ), n := nat.div_le_self _ _
| -[1+ m], 0 := nat.zero_le _
| -[1+ m], n+1 := nat.succ_le_succ (nat.div_le_self _ _)
end)
theorem div_le_self {a : ℤ} (b : ℤ) (Ha : a ≥ 0) : a / b ≤ a :=
by have := le_trans (le_abs_self _) (abs_div_le_abs a b);
rwa [abs_of_nonneg Ha] at this
theorem mul_div_cancel_of_mod_eq_zero {a b : ℤ} (H : a % b = 0) : b * (a / b) = a :=
by have := mod_add_div a b; rwa [H, zero_add] at this
theorem div_mul_cancel_of_mod_eq_zero {a b : ℤ} (H : a % b = 0) : a / b * b = a :=
by rw [mul_comm, mul_div_cancel_of_mod_eq_zero H]
/- dvd -/
theorem coe_nat_dvd {m n : ℕ} : (↑m : ℤ) ∣ ↑n ↔ m ∣ n :=
⟨λ ⟨a, ae⟩, m.eq_zero_or_pos.elim
(λm0, by simp [m0] at ae; simp [ae, m0])
(λm0l, by {
cases eq_coe_of_zero_le (@nonneg_of_mul_nonneg_left ℤ _ m a
(by simp [ae.symm]) (by simpa using m0l)) with k e,
subst a, exact ⟨k, int.coe_nat_inj ae⟩ }),
λ ⟨k, e⟩, dvd.intro k $ by rw [e, int.coe_nat_mul]⟩
theorem dvd_antisymm {a b : ℤ} (H1 : a ≥ 0) (H2 : b ≥ 0) : a ∣ b → b ∣ a → a = b :=
begin
rw [← abs_of_nonneg H1, ← abs_of_nonneg H2, abs_eq_nat_abs, abs_eq_nat_abs],
rw [coe_nat_dvd, coe_nat_dvd, coe_nat_inj'],
apply nat.dvd_antisymm
end
theorem dvd_of_mod_eq_zero {a b : ℤ} (H : b % a = 0) : a ∣ b :=
⟨b / a, (mul_div_cancel_of_mod_eq_zero H).symm⟩
theorem mod_eq_zero_of_dvd : ∀ {a b : ℤ}, a ∣ b → b % a = 0
| a ._ ⟨c, rfl⟩ := mul_mod_right _ _
theorem dvd_iff_mod_eq_zero (a b : ℤ) : a ∣ b ↔ b % a = 0 :=
⟨mod_eq_zero_of_dvd, dvd_of_mod_eq_zero⟩
theorem nat_abs_dvd {a b : ℤ} : (a.nat_abs : ℤ) ∣ b ↔ a ∣ b :=
(nat_abs_eq a).elim (λ e, by rw ← e) (λ e, by rw [← neg_dvd_iff_dvd, ← e])
theorem dvd_nat_abs {a b : ℤ} : a ∣ b.nat_abs ↔ a ∣ b :=
(nat_abs_eq b).elim (λ e, by rw ← e) (λ e, by rw [← dvd_neg_iff_dvd, ← e])
instance decidable_dvd : @decidable_rel ℤ (∣) :=
assume a n, decidable_of_decidable_of_iff (by apply_instance) (dvd_iff_mod_eq_zero _ _).symm
protected theorem div_mul_cancel {a b : ℤ} (H : b ∣ a) : a / b * b = a :=
div_mul_cancel_of_mod_eq_zero (mod_eq_zero_of_dvd H)
protected theorem mul_div_cancel' {a b : ℤ} (H : a ∣ b) : a * (b / a) = b :=
by rw [mul_comm, int.div_mul_cancel H]
protected theorem mul_div_assoc (a : ℤ) : ∀ {b c : ℤ}, c ∣ b → (a * b) / c = a * (b / c)
| ._ c ⟨d, rfl⟩ := if cz : c = 0 then by simp [cz] else
by rw [mul_left_comm, int.mul_div_cancel_left _ cz, int.mul_div_cancel_left _ cz]
theorem div_dvd_div : ∀ {a b c : ℤ} (H1 : a ∣ b) (H2 : b ∣ c), b / a ∣ c / a
| a ._ ._ ⟨b, rfl⟩ ⟨c, rfl⟩ := if az : a = 0 then by simp [az] else
by rw [int.mul_div_cancel_left _ az, mul_assoc, int.mul_div_cancel_left _ az];
apply dvd_mul_right
protected theorem eq_mul_of_div_eq_right {a b c : ℤ} (H1 : b ∣ a) (H2 : a / b = c) :
a = b * c :=
by rw [← H2, int.mul_div_cancel' H1]
protected theorem div_eq_of_eq_mul_right {a b c : ℤ} (H1 : b ≠ 0) (H2 : a = b * c) :
a / b = c :=
by rw [H2, int.mul_div_cancel_left _ H1]
protected theorem div_eq_iff_eq_mul_right {a b c : ℤ} (H : b ≠ 0) (H' : b ∣ a) :
a / b = c ↔ a = b * c :=
⟨int.eq_mul_of_div_eq_right H', int.div_eq_of_eq_mul_right H⟩
protected theorem div_eq_iff_eq_mul_left {a b c : ℤ} (H : b ≠ 0) (H' : b ∣ a) :
a / b = c ↔ a = c * b :=
by rw mul_comm; exact int.div_eq_iff_eq_mul_right H H'
protected theorem eq_mul_of_div_eq_left {a b c : ℤ} (H1 : b ∣ a) (H2 : a / b = c) :
a = c * b :=
by rw [mul_comm, int.eq_mul_of_div_eq_right H1 H2]
protected theorem div_eq_of_eq_mul_left {a b c : ℤ} (H1 : b ≠ 0) (H2 : a = c * b) :
a / b = c :=
int.div_eq_of_eq_mul_right H1 (by rw [mul_comm, H2])
theorem neg_div_of_dvd : ∀ {a b : ℤ} (H : b ∣ a), -a / b = -(a / b)
| ._ b ⟨c, rfl⟩ := if bz : b = 0 then by simp [bz] else
by rw [neg_mul_eq_mul_neg, int.mul_div_cancel_left _ bz, int.mul_div_cancel_left _ bz]
theorem div_sign : ∀ a b, a / sign b = a * sign b
| a (n+1:ℕ) := by unfold sign; simp
| a 0 := by simp [sign]
| a -[1+ n] := by simp [sign]
@[simp] theorem sign_mul : ∀ a b, sign (a * b) = sign a * sign b
| a 0 := by simp
| 0 b := by simp
| (m+1:ℕ) (n+1:ℕ) := rfl
| (m+1:ℕ) -[1+ n] := rfl
| -[1+ m] (n+1:ℕ) := rfl
| -[1+ m] -[1+ n] := rfl
protected theorem sign_eq_div_abs (a : ℤ) : sign a = a / (abs a) :=
if az : a = 0 then by simp [az] else
(int.div_eq_of_eq_mul_left (mt eq_zero_of_abs_eq_zero az)
(sign_mul_abs _).symm).symm
theorem mul_sign : ∀ (i : ℤ), i * sign i = nat_abs i
| (n+1:ℕ) := mul_one _
| 0 := mul_zero _
| -[1+ n] := mul_neg_one _
theorem le_of_dvd {a b : ℤ} (bpos : b > 0) (H : a ∣ b) : a ≤ b :=
match a, b, eq_succ_of_zero_lt bpos, H with
| (m : ℕ), ._, ⟨n, rfl⟩, H := coe_nat_le_coe_nat_of_le $
nat.le_of_dvd n.succ_pos $ coe_nat_dvd.1 H
| -[1+ m], ._, ⟨n, rfl⟩, _ :=
le_trans (le_of_lt $ neg_succ_lt_zero _) (coe_zero_le _)
end
theorem eq_one_of_dvd_one {a : ℤ} (H : a ≥ 0) (H' : a ∣ 1) : a = 1 :=
match a, eq_coe_of_zero_le H, H' with
| ._, ⟨n, rfl⟩, H' := congr_arg coe $
nat.eq_one_of_dvd_one $ coe_nat_dvd.1 H'
end
theorem eq_one_of_mul_eq_one_right {a b : ℤ} (H : a ≥ 0) (H' : a * b = 1) : a = 1 :=
eq_one_of_dvd_one H ⟨b, H'.symm⟩
theorem eq_one_of_mul_eq_one_left {a b : ℤ} (H : b ≥ 0) (H' : a * b = 1) : b = 1 :=
eq_one_of_mul_eq_one_right H (by rw [mul_comm, H'])
/- / and ordering -/
protected theorem div_mul_le (a : ℤ) {b : ℤ} (H : b ≠ 0) : a / b * b ≤ a :=
le_of_sub_nonneg $ by rw [mul_comm, ← mod_def]; apply mod_nonneg _ H
protected theorem div_le_of_le_mul {a b c : ℤ} (H : c > 0) (H' : a ≤ b * c) : a / c ≤ b :=
le_of_mul_le_mul_right (le_trans (int.div_mul_le _ (ne_of_gt H)) H') H
protected theorem mul_lt_of_lt_div {a b c : ℤ} (H : c > 0) (H3 : a < b / c) : a * c < b :=
lt_of_not_ge $ mt (int.div_le_of_le_mul H) (not_le_of_gt H3)
protected theorem mul_le_of_le_div {a b c : ℤ} (H1 : c > 0) (H2 : a ≤ b / c) : a * c ≤ b :=
le_trans (mul_le_mul_of_nonneg_right H2 (le_of_lt H1)) (int.div_mul_le _ (ne_of_gt H1))
protected theorem le_div_of_mul_le {a b c : ℤ} (H1 : c > 0) (H2 : a * c ≤ b) : a ≤ b / c :=
le_of_lt_add_one $ lt_of_mul_lt_mul_right
(lt_of_le_of_lt H2 (lt_div_add_one_mul_self _ H1)) (le_of_lt H1)
protected theorem le_div_iff_mul_le {a b c : ℤ} (H : c > 0) : a ≤ b / c ↔ a * c ≤ b :=
⟨int.mul_le_of_le_div H, int.le_div_of_mul_le H⟩
protected theorem div_le_div {a b c : ℤ} (H : c > 0) (H' : a ≤ b) : a / c ≤ b / c :=
int.le_div_of_mul_le H (le_trans (int.div_mul_le _ (ne_of_gt H)) H')
protected theorem div_lt_of_lt_mul {a b c : ℤ} (H : c > 0) (H' : a < b * c) : a / c < b :=
lt_of_not_ge $ mt (int.mul_le_of_le_div H) (not_le_of_gt H')
protected theorem lt_mul_of_div_lt {a b c : ℤ} (H1 : c > 0) (H2 : a / c < b) : a < b * c :=
lt_of_not_ge $ mt (int.le_div_of_mul_le H1) (not_le_of_gt H2)
protected theorem div_lt_iff_lt_mul {a b c : ℤ} (H : c > 0) : a / c < b ↔ a < b * c :=
⟨int.lt_mul_of_div_lt H, int.div_lt_of_lt_mul H⟩
protected theorem le_mul_of_div_le {a b c : ℤ} (H1 : b ≥ 0) (H2 : b ∣ a) (H3 : a / b ≤ c) :
a ≤ c * b :=
by rw [← int.div_mul_cancel H2]; exact mul_le_mul_of_nonneg_right H3 H1
protected theorem lt_div_of_mul_lt {a b c : ℤ} (H1 : b ≥ 0) (H2 : b ∣ c) (H3 : a * b < c) :
a < c / b :=
lt_of_not_ge $ mt (int.le_mul_of_div_le H1 H2) (not_le_of_gt H3)
protected theorem lt_div_iff_mul_lt {a b : ℤ} (c : ℤ) (H : c > 0) (H' : c ∣ b) :
a < b / c ↔ a * c < b :=
⟨int.mul_lt_of_lt_div H, int.lt_div_of_mul_lt (le_of_lt H) H'⟩
theorem div_pos_of_pos_of_dvd {a b : ℤ} (H1 : a > 0) (H2 : b ≥ 0) (H3 : b ∣ a) : a / b > 0 :=
int.lt_div_of_mul_lt H2 H3 (by rwa zero_mul)
theorem div_eq_div_of_mul_eq_mul {a b c d : ℤ} (H1 : b ∣ a) (H2 : d ∣ c) (H3 : b ≠ 0)
(H4 : d ≠ 0) (H5 : a * d = b * c) :
a / b = c / d :=
int.div_eq_of_eq_mul_right H3 $
by rw [← int.mul_div_assoc _ H2]; exact
(int.div_eq_of_eq_mul_left H4 H5.symm).symm
theorem of_nat_add_neg_succ_of_nat_of_lt {m n : ℕ}
(h : m < n.succ) : of_nat m + -[1+n] = -[1+ n - m] :=
begin
change sub_nat_nat _ _ = _,
have h' : n.succ - m = (n - m).succ,
apply succ_sub,
apply le_of_lt_succ h,
simp [*, sub_nat_nat]
end
theorem of_nat_add_neg_succ_of_nat_of_ge {m n : ℕ}
(h : m ≥ n.succ) : of_nat m + -[1+n] = of_nat (m - n.succ) :=
begin
change sub_nat_nat _ _ = _,
have h' : n.succ - m = 0,
apply sub_eq_zero_of_le h,
simp [*, sub_nat_nat]
end
@[simp] theorem neg_add_neg (m n : ℕ) : -[1+m] + -[1+n] = -[1+nat.succ(m+n)] := rfl
/- nat abs -/
attribute [simp] nat_abs nat_abs_of_nat nat_abs_zero nat_abs_one
theorem nat_abs_add_le (a b : ℤ) : nat_abs (a + b) ≤ nat_abs a + nat_abs b :=
begin
have, {
refine (λ a b : ℕ, sub_nat_nat_elim a b.succ
(λ m n i, n = b.succ → nat_abs i ≤ (m + b).succ) _ _ rfl);
intros i n e,
{ subst e, rw [add_comm _ i, add_assoc],
exact nat.le_add_right i (b.succ + b).succ },
{ apply succ_le_succ,
rw [← succ_inj e, ← add_assoc, add_comm],
apply nat.le_add_right } },
cases a; cases b with b b; simp [nat_abs, nat.succ_add];
try {refl}; [skip, rw add_comm a b]; apply this
end
theorem nat_abs_neg_of_nat (n : nat) : nat_abs (neg_of_nat n) = n :=
by cases n; refl
theorem nat_abs_mul (a b : ℤ) : nat_abs (a * b) = (nat_abs a) * (nat_abs b) :=
by cases a; cases b; simp [(*), int.mul, nat_abs_neg_of_nat]
theorem neg_succ_of_nat_eq' (m : ℕ) : -[1+ m] = -m - 1 :=
by simp [neg_succ_of_nat_eq]
/- to_nat -/
theorem to_nat_eq_max : ∀ (a : ℤ), (to_nat a : ℤ) = max a 0
| (n : ℕ) := (max_eq_left (coe_zero_le n)).symm
| -[1+ n] := (max_eq_right (le_of_lt (neg_succ_lt_zero n))).symm
@[simp] theorem to_nat_of_nonneg {a : ℤ} (h : 0 ≤ a) : (to_nat a : ℤ) = a :=
by rw [to_nat_eq_max, max_eq_left h]
@[simp] theorem to_nat_coe_nat (n : ℕ) : to_nat ↑n = n := rfl
theorem le_to_nat (a : ℤ) : a ≤ to_nat a :=
by rw [to_nat_eq_max]; apply le_max_left
@[simp] theorem to_nat_le (a : ℤ) (n : ℕ) : to_nat a ≤ n ↔ a ≤ n :=
by rw [(coe_nat_le_coe_nat_iff _ _).symm, to_nat_eq_max, max_le_iff];
exact and_iff_left (coe_zero_le _)
def to_nat' : ℤ → option ℕ
| (n : ℕ) := some n
| -[1+ n] := none
theorem mem_to_nat' : ∀ (a : ℤ) (n : ℕ), n ∈ to_nat' a ↔ a = n
| (m : ℕ) n := option.some_inj.trans coe_nat_inj'.symm
| -[1+ m] n := by split; intro h; cases h
/- units -/
@[simp] theorem units_nat_abs (u : units ℤ) : nat_abs u = 1 :=
units.ext_iff.1 $ nat.units_eq_one ⟨nat_abs u, nat_abs ↑u⁻¹,
by rw [← nat_abs_mul, units.mul_inv]; refl,
by rw [← nat_abs_mul, units.inv_mul]; refl⟩
theorem units_eq_one_or (u : units ℤ) : u = 1 ∨ u = -1 :=
by simpa [units.ext_iff, units_nat_abs] using nat_abs_eq u
/- bitwise ops -/
@[simp] lemma bodd_zero : bodd 0 = ff := rfl
@[simp] lemma bodd_one : bodd 1 = tt := rfl
@[simp] lemma bodd_two : bodd 2 = ff := rfl
@[simp] lemma bodd_sub_nat_nat (m n : ℕ) : bodd (sub_nat_nat m n) = bxor m.bodd n.bodd :=
by apply sub_nat_nat_elim m n (λ m n i, bodd i = bxor m.bodd n.bodd);
intros i m; simp [bodd]; cases i.bodd; cases m.bodd; refl
@[simp] lemma bodd_neg_of_nat (n : ℕ) : bodd (neg_of_nat n) = n.bodd :=
by cases n; simp; refl
@[simp] lemma bodd_neg (n : ℤ) : bodd (-n) = bodd n :=
by cases n; unfold has_neg.neg; simp [int.coe_nat_eq, int.neg, bodd]
@[simp] lemma bodd_add (m n : ℤ) : bodd (m + n) = bxor (bodd m) (bodd n) :=
by cases m with m m; cases n with n n; unfold has_add.add; simp [int.add, bodd];
cases m.bodd; cases n.bodd; refl
@[simp] lemma bodd_mul (m n : ℤ) : bodd (m * n) = bodd m && bodd n :=
by cases m with m m; cases n with n n; unfold has_mul.mul; simp [int.mul, bodd];
cases m.bodd; cases n.bodd; refl
theorem bodd_add_div2 : ∀ n, cond (bodd n) 1 0 + 2 * div2 n = n
| (n : ℕ) :=
by rw [show (cond (bodd n) 1 0 : ℤ) = (cond (bodd n) 1 0 : ℕ),
by cases bodd n; refl]; exact congr_arg of_nat n.bodd_add_div2
| -[1+ n] := begin
refine eq.trans _ (congr_arg neg_succ_of_nat n.bodd_add_div2),
dsimp [bodd], cases nat.bodd n; dsimp [cond, bnot, div2, int.mul],
{ change -[1+ 2 * nat.div2 n] = _, rw zero_add },
{ rw [zero_add, add_comm], refl }
end
theorem div2_val : ∀ n, div2 n = n / 2
| (n : ℕ) := congr_arg of_nat n.div2_val
| -[1+ n] := congr_arg neg_succ_of_nat n.div2_val
lemma bit0_val (n : ℤ) : bit0 n = 2 * n := (two_mul _).symm
lemma bit1_val (n : ℤ) : bit1 n = 2 * n + 1 := congr_arg (+(1:ℤ)) (bit0_val _)
lemma bit_val (b n) : bit b n = 2 * n + cond b 1 0 :=
by { cases b, apply (bit0_val n).trans (add_zero _).symm, apply bit1_val }
lemma bit_decomp (n : ℤ) : bit (bodd n) (div2 n) = n :=
(bit_val _ _).trans $ (add_comm _ _).trans $ bodd_add_div2 _
def {u} bit_cases_on {C : ℤ → Sort u} (n) (h : ∀ b n, C (bit b n)) : C n :=
by rw [← bit_decomp n]; apply h
@[simp] lemma bit_zero : bit ff 0 = 0 := rfl
@[simp] lemma bit_coe_nat (b) (n : ℕ) : bit b n = nat.bit b n :=
by rw [bit_val, nat.bit_val]; cases b; refl
@[simp] lemma bit_neg_succ (b) (n : ℕ) : bit b -[1+ n] = -[1+ nat.bit (bnot b) n] :=
by rw [bit_val, nat.bit_val]; cases b; refl
@[simp] lemma bodd_bit (b n) : bodd (bit b n) = b :=
by rw bit_val; simp; cases b; cases bodd n; refl
@[simp] lemma div2_bit (b n) : div2 (bit b n) = n :=
begin
rw [bit_val, div2_val, add_comm, int.add_mul_div_left, (_ : (_/2:ℤ) = 0), zero_add],
cases b, all_goals {exact dec_trivial}
end
@[simp] lemma test_bit_zero (b) : ∀ n, test_bit (bit b n) 0 = b
| (n : ℕ) := by rw [bit_coe_nat]; apply nat.test_bit_zero
| -[1+ n] := by rw [bit_neg_succ]; dsimp [test_bit]; rw [nat.test_bit_zero];
clear test_bit_zero; cases b; refl
@[simp] lemma test_bit_succ (m b) : ∀ n, test_bit (bit b n) (nat.succ m) = test_bit n m
| (n : ℕ) := by rw [bit_coe_nat]; apply nat.test_bit_succ
| -[1+ n] := by rw [bit_neg_succ]; dsimp [test_bit]; rw [nat.test_bit_succ]
private meta def bitwise_tac : tactic unit := `[
funext m,
funext n,
cases m with m m; cases n with n n; try {refl},
all_goals {
apply congr_arg of_nat <|> apply congr_arg neg_succ_of_nat,
try {dsimp [nat.land, nat.ldiff, nat.lor]},
try {rw [
show nat.bitwise (λ a b, a && bnot b) n m =
nat.bitwise (λ a b, b && bnot a) m n, from
congr_fun (congr_fun (@nat.bitwise_swap (λ a b, b && bnot a) rfl) n) m]},
apply congr_arg (λ f, nat.bitwise f m n),
funext a,
funext b,
cases a; cases b; refl
},
all_goals {unfold nat.land nat.ldiff nat.lor}
]
theorem bitwise_or : bitwise bor = lor := by bitwise_tac
theorem bitwise_and : bitwise band = land := by bitwise_tac
theorem bitwise_diff : bitwise (λ a b, a && bnot b) = ldiff := by bitwise_tac
theorem bitwise_xor : bitwise bxor = lxor := by bitwise_tac
@[simp] lemma bitwise_bit (f : bool → bool → bool) (a m b n) :
bitwise f (bit a m) (bit b n) = bit (f a b) (bitwise f m n) :=
begin
cases m with m m; cases n with n n;
repeat { rw [← int.coe_nat_eq] <|> rw bit_coe_nat <|> rw bit_neg_succ };
unfold bitwise nat_bitwise bnot;
[ induction h : f ff ff,
induction h : f ff tt,
induction h : f tt ff,
induction h : f tt tt ],
all_goals {
unfold cond, rw nat.bitwise_bit,
repeat { rw bit_coe_nat <|> rw bit_neg_succ <|> rw bnot_bnot } },
all_goals { unfold bnot {fail_if_unchanged := ff}; rw h; refl }
end
@[simp] lemma lor_bit (a m b n) : lor (bit a m) (bit b n) = bit (a || b) (lor m n) :=
by rw [← bitwise_or, bitwise_bit]
@[simp] lemma land_bit (a m b n) : land (bit a m) (bit b n) = bit (a && b) (land m n) :=
by rw [← bitwise_and, bitwise_bit]
@[simp] lemma ldiff_bit (a m b n) : ldiff (bit a m) (bit b n) = bit (a && bnot b) (ldiff m n) :=
by rw [← bitwise_diff, bitwise_bit]
@[simp] lemma lxor_bit (a m b n) : lxor (bit a m) (bit b n) = bit (bxor a b) (lxor m n) :=
by rw [← bitwise_xor, bitwise_bit]
@[simp] lemma lnot_bit (b) : ∀ n, lnot (bit b n) = bit (bnot b) (lnot n)
| (n : ℕ) := by simp [lnot]
| -[1+ n] := by simp [lnot]
@[simp] lemma test_bit_bitwise (f : bool → bool → bool) (m n k) :
test_bit (bitwise f m n) k = f (test_bit m k) (test_bit n k) :=
begin
induction k with k IH generalizing m n;
apply bit_cases_on m; intros a m';
apply bit_cases_on n; intros b n';
rw bitwise_bit,
{ simp [test_bit_zero] },
{ simp [test_bit_succ, IH] }
end
@[simp] lemma test_bit_lor (m n k) : test_bit (lor m n) k = test_bit m k || test_bit n k :=
by rw [← bitwise_or, test_bit_bitwise]
@[simp] lemma test_bit_land (m n k) : test_bit (land m n) k = test_bit m k && test_bit n k :=
by rw [← bitwise_and, test_bit_bitwise]
@[simp] lemma test_bit_ldiff (m n k) : test_bit (ldiff m n) k = test_bit m k && bnot (test_bit n k) :=
by rw [← bitwise_diff, test_bit_bitwise]
@[simp] lemma test_bit_lxor (m n k) : test_bit (lxor m n) k = bxor (test_bit m k) (test_bit n k) :=
by rw [← bitwise_xor, test_bit_bitwise]
@[simp] lemma test_bit_lnot : ∀ n k, test_bit (lnot n) k = bnot (test_bit n k)
| (n : ℕ) k := by simp [lnot, test_bit]
| -[1+ n] k := by simp [lnot, test_bit]
lemma shiftl_add : ∀ (m : ℤ) (n : ℕ) (k : ℤ), shiftl m (n + k) = shiftl (shiftl m n) k
| (m : ℕ) n (k:ℕ) := congr_arg of_nat (nat.shiftl_add _ _ _)
| -[1+ m] n (k:ℕ) := congr_arg neg_succ_of_nat (nat.shiftl'_add _ _ _ _)
| (m : ℕ) n -[1+k] := sub_nat_nat_elim n k.succ
(λ n k i, shiftl ↑m i = nat.shiftr (nat.shiftl m n) k)
(λ i n, congr_arg coe $
by rw [← nat.shiftl_sub, nat.add_sub_cancel_left]; apply nat.le_add_right)
(λ i n, congr_arg coe $
by rw [add_assoc, nat.shiftr_add, ← nat.shiftl_sub, nat.sub_self]; refl)
| -[1+ m] n -[1+k] := sub_nat_nat_elim n k.succ
(λ n k i, shiftl -[1+ m] i = -[1+ nat.shiftr (nat.shiftl' tt m n) k])
(λ i n, congr_arg neg_succ_of_nat $
by rw [← nat.shiftl'_sub, nat.add_sub_cancel_left]; apply nat.le_add_right)
(λ i n, congr_arg neg_succ_of_nat $
by rw [add_assoc, nat.shiftr_add, ← nat.shiftl'_sub, nat.sub_self]; refl)
lemma shiftl_sub (m : ℤ) (n : ℕ) (k : ℤ) : shiftl m (n - k) = shiftr (shiftl m n) k :=
shiftl_add _ _ _
@[simp] lemma shiftl_neg (m n : ℤ) : shiftl m (-n) = shiftr m n := rfl
@[simp] lemma shiftr_neg (m n : ℤ) : shiftr m (-n) = shiftl m n := by rw [← shiftl_neg, neg_neg]
@[simp] lemma shiftl_coe_nat (m n : ℕ) : shiftl m n = nat.shiftl m n := rfl
@[simp] lemma shiftr_coe_nat (m n : ℕ) : shiftr m n = nat.shiftr m n := by cases n; refl
@[simp] lemma shiftl_neg_succ (m n : ℕ) : shiftl -[1+ m] n = -[1+ nat.shiftl' tt m n] := rfl
@[simp] lemma shiftr_neg_succ (m n : ℕ) : shiftr -[1+ m] n = -[1+ nat.shiftr m n] := by cases n; refl
lemma shiftr_add : ∀ (m : ℤ) (n k : ℕ), shiftr m (n + k) = shiftr (shiftr m n) k
| (m : ℕ) n k := by rw [shiftr_coe_nat, shiftr_coe_nat,
← int.coe_nat_add, shiftr_coe_nat, nat.shiftr_add]
| -[1+ m] n k := by rw [shiftr_neg_succ, shiftr_neg_succ,
← int.coe_nat_add, shiftr_neg_succ, nat.shiftr_add]
lemma shiftl_eq_mul_pow : ∀ (m : ℤ) (n : ℕ), shiftl m n = m * ↑(2 ^ n)
| (m : ℕ) n := congr_arg coe (nat.shiftl_eq_mul_pow _ _)
| -[1+ m] n := @congr_arg ℕ ℤ _ _ (λi, -i) (nat.shiftl'_tt_eq_mul_pow _ _)
lemma shiftr_eq_div_pow : ∀ (m : ℤ) (n : ℕ), shiftr m n = m / ↑(2 ^ n)
| (m : ℕ) n := by rw shiftr_coe_nat; exact congr_arg coe (nat.shiftr_eq_div_pow _ _)
| -[1+ m] n := begin
rw [shiftr_neg_succ, neg_succ_of_nat_div, nat.shiftr_eq_div_pow], refl,
exact coe_nat_lt_coe_nat_of_lt (nat.pos_pow_of_pos _ dec_trivial)
end
lemma one_shiftl (n : ℕ) : shiftl 1 n = (2 ^ n : ℕ) :=
congr_arg coe (nat.one_shiftl _)
@[simp] lemma zero_shiftl : ∀ n : ℤ, shiftl 0 n = 0
| (n : ℕ) := congr_arg coe (nat.zero_shiftl _)
| -[1+ n] := congr_arg coe (nat.zero_shiftr _)
@[simp] lemma zero_shiftr (n) : shiftr 0 n = 0 := zero_shiftl _
/- Least upper bound property for integers -/
theorem exists_least_of_bdd {P : ℤ → Prop} [HP : decidable_pred P]
(Hbdd : ∃ b : ℤ, ∀ z : ℤ, P z → b ≤ z)
(Hinh : ∃ z : ℤ, P z) : ∃ lb : ℤ, P lb ∧ (∀ z : ℤ, P z → lb ≤ z) :=
let ⟨b, Hb⟩ := Hbdd in
have EX : ∃ n : ℕ, P (b + n), from
let ⟨elt, Helt⟩ := Hinh in
match elt, le.dest (Hb _ Helt), Helt with
| ._, ⟨n, rfl⟩, Hn := ⟨n, Hn⟩
end,
⟨b + (nat.find EX : ℤ), nat.find_spec EX, λ z h,
match z, le.dest (Hb _ h), h with
| ._, ⟨n, rfl⟩, h := add_le_add_left
(int.coe_nat_le.2 $ nat.find_min' _ h) _
end⟩
theorem exists_greatest_of_bdd {P : ℤ → Prop} [HP : decidable_pred P]
(Hbdd : ∃ b : ℤ, ∀ z : ℤ, P z → z ≤ b)
(Hinh : ∃ z : ℤ, P z) : ∃ ub : ℤ, P ub ∧ (∀ z : ℤ, P z → z ≤ ub) :=
have Hbdd' : ∃ (b : ℤ), ∀ (z : ℤ), P (-z) → b ≤ z, from
let ⟨b, Hb⟩ := Hbdd in ⟨-b, λ z h, neg_le.1 (Hb _ h)⟩,
have Hinh' : ∃ z : ℤ, P (-z), from
let ⟨elt, Helt⟩ := Hinh in ⟨-elt, by rw [neg_neg]; exact Helt⟩,
let ⟨lb, Plb, al⟩ := exists_least_of_bdd Hbdd' Hinh' in
⟨-lb, Plb, λ z h, le_neg.1 $ al _ $ by rwa neg_neg⟩
/- cast (injection into groups with one) -/
@[simp] theorem nat_cast_eq_coe_nat : ∀ n,
@coe ℕ ℤ (@coe_to_lift _ _ (@coe_base _ _ nat.cast_coe)) n =
@coe ℕ ℤ (@coe_to_lift _ _ (@coe_base _ _ int.has_coe)) n
| 0 := rfl
| (n+1) := congr_arg (+(1:ℤ)) (nat_cast_eq_coe_nat n)
section cast
variables {α : Type*}
section
variables [has_zero α] [has_one α] [has_add α] [has_neg α]
/-- Canonical homomorphism from the integers to any ring(-like) structure `α` -/
protected def cast : ℤ → α
| (n : ℕ) := n
| -[1+ n] := -(n+1)
@[priority 0] instance cast_coe : has_coe ℤ α := ⟨int.cast⟩
@[simp] theorem cast_zero : ((0 : ℤ) : α) = 0 := rfl
@[simp] theorem cast_of_nat (n : ℕ) : (of_nat n : α) = n := rfl
@[simp] theorem cast_coe_nat (n : ℕ) : ((n : ℤ) : α) = n := rfl
@[simp] theorem cast_coe_nat' (n : ℕ) :
(@coe ℕ ℤ (@coe_to_lift _ _ (@coe_base _ _ nat.cast_coe)) n : α) = n :=
by simp
@[simp] theorem cast_neg_succ_of_nat (n : ℕ) : (-[1+ n] : α) = -(n + 1) := rfl
end
@[simp] theorem cast_one [add_monoid α] [has_one α] [has_neg α] : ((1 : ℤ) : α) = 1 := nat.cast_one
@[simp] theorem cast_sub_nat_nat [add_group α] [has_one α] (m n) : ((int.sub_nat_nat m n : ℤ) : α) = m - n :=
begin
unfold sub_nat_nat, cases e : n - m,
{ simp [sub_nat_nat, e, nat.le_of_sub_eq_zero e] },
{ rw [sub_nat_nat, cast_neg_succ_of_nat, ← nat.cast_succ, ← e,
nat.cast_sub $ _root_.le_of_lt $ nat.lt_of_sub_eq_succ e, neg_sub] },
end
@[simp] theorem cast_neg_of_nat [add_group α] [has_one α] : ∀ n, ((neg_of_nat n : ℤ) : α) = -n
| 0 := neg_zero.symm
| (n+1) := rfl
@[simp] theorem cast_add [add_group α] [has_one α] : ∀ m n, ((m + n : ℤ) : α) = m + n
| (m : ℕ) (n : ℕ) := nat.cast_add _ _
| (m : ℕ) -[1+ n] := cast_sub_nat_nat _ _
| -[1+ m] (n : ℕ) := (cast_sub_nat_nat _ _).trans $ sub_eq_of_eq_add $
show (n:α) = -(m+1) + n + (m+1),
by rw [add_assoc, ← cast_succ, ← nat.cast_add, add_comm,
nat.cast_add, cast_succ, neg_add_cancel_left]
| -[1+ m] -[1+ n] := show -((m + n + 1 + 1 : ℕ) : α) = -(m + 1) + -(n + 1),
by rw [← neg_add_rev, ← nat.cast_add_one, ← nat.cast_add_one, ← nat.cast_add];
apply congr_arg (λ x:ℕ, -(x:α)); simp
@[simp] theorem cast_neg [add_group α] [has_one α] : ∀ n, ((-n : ℤ) : α) = -n
| (n : ℕ) := cast_neg_of_nat _
| -[1+ n] := (neg_neg _).symm
theorem cast_sub [add_group α] [has_one α] (m n) : ((m - n : ℤ) : α) = m - n :=
by simp
@[simp] theorem cast_eq_zero [add_group α] [has_one α] [char_zero α] {n : ℤ} : (n : α) = 0 ↔ n = 0 :=
⟨λ h, begin cases n,
{ exact congr_arg coe (nat.cast_eq_zero.1 h) },
{ rw [cast_neg_succ_of_nat, neg_eq_zero, ← cast_succ, nat.cast_eq_zero] at h,
contradiction }
end, λ h, by rw [h, cast_zero]⟩
@[simp] theorem cast_inj [add_group α] [has_one α] [char_zero α] {m n : ℤ} : (m : α) = n ↔ m = n :=
by rw [← sub_eq_zero, ← cast_sub, cast_eq_zero, sub_eq_zero]
theorem cast_injective [add_group α] [has_one α] [char_zero α] : function.injective (coe : ℤ → α)
| m n := cast_inj.1
@[simp] theorem cast_ne_zero [add_group α] [has_one α] [char_zero α] {n : ℤ} : (n : α) ≠ 0 ↔ n ≠ 0 :=
not_congr cast_eq_zero
@[simp] theorem cast_mul [ring α] : ∀ m n, ((m * n : ℤ) : α) = m * n
| (m : ℕ) (n : ℕ) := nat.cast_mul _ _
| (m : ℕ) -[1+ n] := (cast_neg_of_nat _).trans $
show (-(m * (n + 1) : ℕ) : α) = m * -(n + 1),
by rw [nat.cast_mul, nat.cast_add_one, neg_mul_eq_mul_neg]
| -[1+ m] (n : ℕ) := (cast_neg_of_nat _).trans $
show (-((m + 1) * n : ℕ) : α) = -(m + 1) * n,
by rw [nat.cast_mul, nat.cast_add_one, neg_mul_eq_neg_mul]
| -[1+ m] -[1+ n] := show (((m + 1) * (n + 1) : ℕ) : α) = -(m + 1) * -(n + 1),
by rw [nat.cast_mul, nat.cast_add_one, nat.cast_add_one, neg_mul_neg]
theorem mul_cast_comm [ring α] (a : α) (n : ℤ) : a * n = n * a :=
by cases n; simp [nat.mul_cast_comm, left_distrib, right_distrib, *]
@[simp] theorem cast_bit0 [ring α] (n : ℤ) : ((bit0 n : ℤ) : α) = bit0 n := cast_add _ _
@[simp] theorem cast_bit1 [ring α] (n : ℤ) : ((bit1 n : ℤ) : α) = bit1 n :=
by rw [bit1, cast_add, cast_one, cast_bit0]; refl
theorem cast_nonneg [linear_ordered_ring α] : ∀ {n : ℤ}, (0 : α) ≤ n ↔ 0 ≤ n
| (n : ℕ) := by simp
| -[1+ n] := by simpa [not_le_of_gt (neg_succ_lt_zero n)] using
show -(n:α) < 1, from lt_of_le_of_lt (by simp) zero_lt_one
@[simp] theorem cast_le [linear_ordered_ring α] {m n : ℤ} : (m : α) ≤ n ↔ m ≤ n :=
by rw [← sub_nonneg, ← cast_sub, cast_nonneg, sub_nonneg]
@[simp] theorem cast_lt [linear_ordered_ring α] {m n : ℤ} : (m : α) < n ↔ m < n :=
by simpa [-cast_le] using not_congr (@cast_le α _ n m)
@[simp] theorem cast_nonpos [linear_ordered_ring α] {n : ℤ} : (n : α) ≤ 0 ↔ n ≤ 0 :=
by rw [← cast_zero, cast_le]
@[simp] theorem cast_pos [linear_ordered_ring α] {n : ℤ} : (0 : α) < n ↔ 0 < n :=
by rw [← cast_zero, cast_lt]
@[simp] theorem cast_lt_zero [linear_ordered_ring α] {n : ℤ} : (n : α) < 0 ↔ n < 0 :=
by rw [← cast_zero, cast_lt]
theorem eq_cast [add_group α] [has_one α] (f : ℤ → α)
(H1 : f 1 = 1) (Hadd : ∀ x y, f (x + y) = f x + f y) (n : ℤ) : f n = n :=
begin
have H : ∀ (n : ℕ), f n = n :=
nat.eq_cast' (λ n, f n) H1 (λ x y, Hadd x y),
cases n, {apply H},
apply eq_neg_of_add_eq_zero,
rw [← nat.cast_zero, ← H 0, int.coe_nat_zero,
← show -[1+ n] + (↑n + 1) = 0, from neg_add_self (↑n+1),
Hadd, show f (n+1) = n+1, from H (n+1)]
end
@[simp] theorem cast_id (n : ℤ) : ↑n = n :=
(eq_cast id rfl (λ _ _, rfl) n).symm
@[simp] theorem cast_min [decidable_linear_ordered_comm_ring α] {a b : ℤ} : (↑(min a b) : α) = min a b :=
by by_cases a ≤ b; simp [h, min]
@[simp] theorem cast_max [decidable_linear_ordered_comm_ring α] {a b : ℤ} : (↑(max a b) : α) = max a b :=
by by_cases a ≤ b; simp [h, max]
@[simp] theorem cast_abs [decidable_linear_ordered_comm_ring α] {q : ℤ} : ((abs q : ℤ) : α) = abs q :=
by simp [abs]
end cast
end int
|
7ca935bc2134d2cd0bd27633dc47510f8d701ec5 | 77c5b91fae1b966ddd1db969ba37b6f0e4901e88 | /src/algebra/order/ring.lean | cb5de550c1e54a068ca0e95d9e980c16be56ab77 | [
"Apache-2.0"
] | permissive | dexmagic/mathlib | ff48eefc56e2412429b31d4fddd41a976eb287ce | 7a5d15a955a92a90e1d398b2281916b9c41270b2 | refs/heads/master | 1,693,481,322,046 | 1,633,360,193,000 | 1,633,360,193,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 68,748 | lean | /-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro
-/
import algebra.invertible
import algebra.order.group
import algebra.order.sub
import data.set.intervals.basic
/-!
# Ordered rings and semirings
This file develops the basics of ordered (semi)rings.
Each typeclass here comprises
* an algebraic class (`semiring`, `comm_semiring`, `ring`, `comm_ring`)
* an order class (`partial_order`, `linear_order`)
* assumptions on how both interact ((strict) monotonicity, canonicity)
For short,
* "`+` respects `≤`" means "monotonicity of addition"
* "`*` respects `<`" means "strict monotonicity of multiplication by a positive number".
## Typeclasses
* `ordered_semiring`: Semiring with a partial order such that `+` respects `≤` and `*` respects `<`.
* `ordered_comm_semiring`: Commutative semiring with a partial order such that `+` respects `≤` and
`*` respects `<`.
* `ordered_ring`: Ring with a partial order such that `+` respects `≤` and `*` respects `<`.
* `ordered_comm_ring`: Commutative ring with a partial order such that `+` respects `≤` and
`*` respects `<`.
* `linear_ordered_semiring`: Semiring with a linear order such that `+` respects `≤` and
`*` respects `<`.
* `linear_ordered_ring`: Ring with a linear order such that `+` respects `≤` and `*` respects `<`.
* `linear_ordered_comm_ring`: Commutative ring with a linear order such that `+` respects `≤` and
`*` respects `<`.
* `canonically_ordered_comm_semiring`: Commutative semiring with a partial order such that `+`
respects `≤`, `*` respects `<`, and `a ≤ b ↔ ∃ c, b = a + c`.
and some typeclasses to define ordered rings by specifying their nonegative elements:
* `nonneg_ring`: To define `ordered_ring`s.
* `linear_nonneg_ring`: To define `linear_ordered_ring`s.
## Hierarchy
The hardest part of proving order lemmas might be to figure out the correct generality and its
corresponding typeclass. Here's an attempt at demystifying it. For each typeclass, we list its
immediate predecessors and what conditions are added to each of them.
* `ordered_semiring`
- `ordered_cancel_add_comm_monoid` & multiplication & `*` respects `<`
- `semiring` & partial order structure & `+` respects `≤` & `*` respects `<`
* `ordered_comm_semiring`
- `ordered_semiring` & commutativity of multiplication
- `comm_semiring` & partial order structure & `+` respects `≤` & `*` respects `<`
* `ordered_ring`
- `ordered_semiring` & additive inverses
- `ordered_add_comm_group` & multiplication & `*` respects `<`
- `ring` & partial order structure & `+` respects `≤` & `*` respects `<`
* `ordered_comm_ring`
- `ordered_ring` & commutativity of multiplication
- `ordered_comm_semiring` & additive inverses
- `comm_ring` & partial order structure & `+` respects `≤` & `*` respects `<`
* `linear_ordered_semiring`
- `ordered_semiring` & totality of the order & nontriviality
- `linear_ordered_add_comm_monoid` & multiplication & nontriviality & `*` respects `<`
* `linear_ordered_ring`
- `ordered_ring` & totality of the order & nontriviality
- `linear_ordered_semiring` & additive inverses
- `linear_ordered_add_comm_group` & multiplication & `*` respects `<`
- `domain` & linear order structure
* `linear_ordered_comm_ring`
- `ordered_comm_ring` & totality of the order & nontriviality
- `linear_ordered_ring` & commutativity of multiplication
- `integral_domain` & linear order structure
* `canonically_ordered_comm_semiring`
- `canonically_ordered_add_monoid` & multiplication & `*` respects `<` & no zero divisors
- `comm_semiring` & `a ≤ b ↔ ∃ c, b = a + c` & no zero divisors
## TODO
We're still missing some typeclasses, like
* `linear_ordered_comm_semiring`
* `canonically_ordered_semiring`
They have yet to come up in practice.
-/
set_option old_structure_cmd true
universe u
variable {α : Type u}
lemma add_one_le_two_mul [preorder α] [semiring α] [covariant_class α α (+) (≤)]
{a : α} (a1 : 1 ≤ a) :
a + 1 ≤ 2 * a :=
calc a + 1 ≤ a + a : add_le_add_left a1 a
... = 2 * a : (two_mul _).symm
/-- An `ordered_semiring α` is a semiring `α` with a partial order such that
addition is monotone and multiplication by a positive number is strictly monotone. -/
@[protect_proj]
class ordered_semiring (α : Type u) extends semiring α, ordered_cancel_add_comm_monoid α :=
(zero_le_one : 0 ≤ (1 : α))
(mul_lt_mul_of_pos_left : ∀ a b c : α, a < b → 0 < c → c * a < c * b)
(mul_lt_mul_of_pos_right : ∀ a b c : α, a < b → 0 < c → a * c < b * c)
section ordered_semiring
variables [ordered_semiring α] {a b c d : α}
@[simp] lemma zero_le_one : 0 ≤ (1:α) :=
ordered_semiring.zero_le_one
lemma zero_le_two : 0 ≤ (2:α) :=
add_nonneg zero_le_one zero_le_one
lemma one_le_two : 1 ≤ (2:α) :=
calc (1:α) = 0 + 1 : (zero_add _).symm
... ≤ 1 + 1 : add_le_add_right zero_le_one _
section nontrivial
variables [nontrivial α]
@[simp] lemma zero_lt_one : 0 < (1 : α) :=
lt_of_le_of_ne zero_le_one zero_ne_one
lemma zero_lt_two : 0 < (2:α) := add_pos zero_lt_one zero_lt_one
@[field_simps] lemma two_ne_zero : (2:α) ≠ 0 :=
ne.symm (ne_of_lt zero_lt_two)
lemma one_lt_two : 1 < (2:α) :=
calc (2:α) = 1+1 : one_add_one_eq_two
... > 1+0 : add_lt_add_left zero_lt_one _
... = 1 : add_zero 1
lemma zero_lt_three : 0 < (3:α) := add_pos zero_lt_two zero_lt_one
lemma zero_lt_four : 0 < (4:α) := add_pos zero_lt_two zero_lt_two
end nontrivial
lemma mul_lt_mul_of_pos_left (h₁ : a < b) (h₂ : 0 < c) : c * a < c * b :=
ordered_semiring.mul_lt_mul_of_pos_left a b c h₁ h₂
lemma mul_lt_mul_of_pos_right (h₁ : a < b) (h₂ : 0 < c) : a * c < b * c :=
ordered_semiring.mul_lt_mul_of_pos_right a b c h₁ h₂
-- See Note [decidable namespace]
protected lemma decidable.mul_le_mul_of_nonneg_left [@decidable_rel α (≤)]
(h₁ : a ≤ b) (h₂ : 0 ≤ c) : c * a ≤ c * b :=
begin
by_cases ba : b ≤ a, { simp [ba.antisymm h₁] },
by_cases c0 : c ≤ 0, { simp [c0.antisymm h₂] },
exact (mul_lt_mul_of_pos_left (h₁.lt_of_not_le ba) (h₂.lt_of_not_le c0)).le,
end
lemma mul_le_mul_of_nonneg_left : a ≤ b → 0 ≤ c → c * a ≤ c * b :=
by classical; exact decidable.mul_le_mul_of_nonneg_left
-- See Note [decidable namespace]
protected lemma decidable.mul_le_mul_of_nonneg_right [@decidable_rel α (≤)]
(h₁ : a ≤ b) (h₂ : 0 ≤ c) : a * c ≤ b * c :=
begin
by_cases ba : b ≤ a, { simp [ba.antisymm h₁] },
by_cases c0 : c ≤ 0, { simp [c0.antisymm h₂] },
exact (mul_lt_mul_of_pos_right (h₁.lt_of_not_le ba) (h₂.lt_of_not_le c0)).le,
end
lemma mul_le_mul_of_nonneg_right : a ≤ b → 0 ≤ c → a * c ≤ b * c :=
by classical; exact decidable.mul_le_mul_of_nonneg_right
-- TODO: there are four variations, depending on which variables we assume to be nonneg
-- See Note [decidable namespace]
protected lemma decidable.mul_le_mul [@decidable_rel α (≤)]
(hac : a ≤ c) (hbd : b ≤ d) (nn_b : 0 ≤ b) (nn_c : 0 ≤ c) : a * b ≤ c * d :=
calc
a * b ≤ c * b : decidable.mul_le_mul_of_nonneg_right hac nn_b
... ≤ c * d : decidable.mul_le_mul_of_nonneg_left hbd nn_c
lemma mul_le_mul : a ≤ c → b ≤ d → 0 ≤ b → 0 ≤ c → a * b ≤ c * d :=
by classical; exact decidable.mul_le_mul
-- See Note [decidable namespace]
protected lemma decidable.mul_nonneg_le_one_le {α : Type*} [ordered_semiring α]
[@decidable_rel α (≤)] {a b c : α}
(h₁ : 0 ≤ c) (h₂ : a ≤ c) (h₃ : 0 ≤ b) (h₄ : b ≤ 1) : a * b ≤ c :=
by simpa only [mul_one] using decidable.mul_le_mul h₂ h₄ h₃ h₁
lemma mul_nonneg_le_one_le {α : Type*} [ordered_semiring α] {a b c : α} :
0 ≤ c → a ≤ c → 0 ≤ b → b ≤ 1 → a * b ≤ c :=
by classical; exact decidable.mul_nonneg_le_one_le
-- See Note [decidable namespace]
protected lemma decidable.mul_nonneg [@decidable_rel α (≤)]
(ha : 0 ≤ a) (hb : 0 ≤ b) : 0 ≤ a * b :=
have h : 0 * b ≤ a * b, from decidable.mul_le_mul_of_nonneg_right ha hb,
by rwa [zero_mul] at h
lemma mul_nonneg : 0 ≤ a → 0 ≤ b → 0 ≤ a * b := by classical; exact decidable.mul_nonneg
-- See Note [decidable namespace]
protected lemma decidable.mul_nonpos_of_nonneg_of_nonpos [@decidable_rel α (≤)]
(ha : 0 ≤ a) (hb : b ≤ 0) : a * b ≤ 0 :=
have h : a * b ≤ a * 0, from decidable.mul_le_mul_of_nonneg_left hb ha,
by rwa mul_zero at h
lemma mul_nonpos_of_nonneg_of_nonpos : 0 ≤ a → b ≤ 0 → a * b ≤ 0 :=
by classical; exact decidable.mul_nonpos_of_nonneg_of_nonpos
-- See Note [decidable namespace]
protected lemma decidable.mul_nonpos_of_nonpos_of_nonneg [@decidable_rel α (≤)]
(ha : a ≤ 0) (hb : 0 ≤ b) : a * b ≤ 0 :=
have h : a * b ≤ 0 * b, from decidable.mul_le_mul_of_nonneg_right ha hb,
by rwa zero_mul at h
lemma mul_nonpos_of_nonpos_of_nonneg : a ≤ 0 → 0 ≤ b → a * b ≤ 0 :=
by classical; exact decidable.mul_nonpos_of_nonpos_of_nonneg
-- See Note [decidable namespace]
protected lemma decidable.mul_lt_mul [@decidable_rel α (≤)]
(hac : a < c) (hbd : b ≤ d) (pos_b : 0 < b) (nn_c : 0 ≤ c) : a * b < c * d :=
calc
a * b < c * b : mul_lt_mul_of_pos_right hac pos_b
... ≤ c * d : decidable.mul_le_mul_of_nonneg_left hbd nn_c
lemma mul_lt_mul : a < c → b ≤ d → 0 < b → 0 ≤ c → a * b < c * d :=
by classical; exact decidable.mul_lt_mul
-- See Note [decidable namespace]
protected lemma decidable.mul_lt_mul' [@decidable_rel α (≤)]
(h1 : a ≤ c) (h2 : b < d) (h3 : 0 ≤ b) (h4 : 0 < c) : a * b < c * d :=
calc
a * b ≤ c * b : decidable.mul_le_mul_of_nonneg_right h1 h3
... < c * d : mul_lt_mul_of_pos_left h2 h4
lemma mul_lt_mul' : a ≤ c → b < d → 0 ≤ b → 0 < c → a * b < c * d :=
by classical; exact decidable.mul_lt_mul'
lemma mul_pos (ha : 0 < a) (hb : 0 < b) : 0 < a * b :=
have h : 0 * b < a * b, from mul_lt_mul_of_pos_right ha hb,
by rwa zero_mul at h
lemma mul_neg_of_pos_of_neg (ha : 0 < a) (hb : b < 0) : a * b < 0 :=
have h : a * b < a * 0, from mul_lt_mul_of_pos_left hb ha,
by rwa mul_zero at h
lemma mul_neg_of_neg_of_pos (ha : a < 0) (hb : 0 < b) : a * b < 0 :=
have h : a * b < 0 * b, from mul_lt_mul_of_pos_right ha hb,
by rwa zero_mul at h
-- See Note [decidable namespace]
protected lemma decidable.mul_self_lt_mul_self [@decidable_rel α (≤)]
(h1 : 0 ≤ a) (h2 : a < b) : a * a < b * b :=
decidable.mul_lt_mul' h2.le h2 h1 $ h1.trans_lt h2
lemma mul_self_lt_mul_self (h1 : 0 ≤ a) (h2 : a < b) : a * a < b * b :=
mul_lt_mul' h2.le h2 h1 $ h1.trans_lt h2
-- See Note [decidable namespace]
protected lemma decidable.strict_mono_on_mul_self [@decidable_rel α (≤)] :
strict_mono_on (λ x : α, x * x) (set.Ici 0) :=
λ x hx y hy hxy, decidable.mul_self_lt_mul_self hx hxy
lemma strict_mono_on_mul_self : strict_mono_on (λ x : α, x * x) (set.Ici 0) :=
λ x hx y hy hxy, mul_self_lt_mul_self hx hxy
-- See Note [decidable namespace]
protected lemma decidable.mul_self_le_mul_self [@decidable_rel α (≤)]
(h1 : 0 ≤ a) (h2 : a ≤ b) : a * a ≤ b * b :=
decidable.mul_le_mul h2 h2 h1 $ h1.trans h2
lemma mul_self_le_mul_self (h1 : 0 ≤ a) (h2 : a ≤ b) : a * a ≤ b * b :=
mul_le_mul h2 h2 h1 $ h1.trans h2
-- See Note [decidable namespace]
protected lemma decidable.mul_lt_mul'' [@decidable_rel α (≤)]
(h1 : a < c) (h2 : b < d) (h3 : 0 ≤ a) (h4 : 0 ≤ b) : a * b < c * d :=
h4.lt_or_eq_dec.elim
(λ b0, decidable.mul_lt_mul h1 h2.le b0 $ h3.trans h1.le)
(λ b0, by rw [← b0, mul_zero]; exact
mul_pos (h3.trans_lt h1) (h4.trans_lt h2))
lemma mul_lt_mul'' : a < c → b < d → 0 ≤ a → 0 ≤ b → a * b < c * d :=
by classical; exact decidable.mul_lt_mul''
-- See Note [decidable namespace]
protected lemma decidable.le_mul_of_one_le_right [@decidable_rel α (≤)]
(hb : 0 ≤ b) (h : 1 ≤ a) : b ≤ b * a :=
suffices b * 1 ≤ b * a, by rwa mul_one at this,
decidable.mul_le_mul_of_nonneg_left h hb
lemma le_mul_of_one_le_right : 0 ≤ b → 1 ≤ a → b ≤ b * a :=
by classical; exact decidable.le_mul_of_one_le_right
-- See Note [decidable namespace]
protected lemma decidable.le_mul_of_one_le_left [@decidable_rel α (≤)]
(hb : 0 ≤ b) (h : 1 ≤ a) : b ≤ a * b :=
suffices 1 * b ≤ a * b, by rwa one_mul at this,
decidable.mul_le_mul_of_nonneg_right h hb
lemma le_mul_of_one_le_left : 0 ≤ b → 1 ≤ a → b ≤ a * b :=
by classical; exact decidable.le_mul_of_one_le_left
-- See Note [decidable namespace]
protected lemma decidable.lt_mul_of_one_lt_right [@decidable_rel α (≤)]
(hb : 0 < b) (h : 1 < a) : b < b * a :=
suffices b * 1 < b * a, by rwa mul_one at this,
decidable.mul_lt_mul' (le_refl _) h zero_le_one hb
lemma lt_mul_of_one_lt_right : 0 < b → 1 < a → b < b * a :=
by classical; exact decidable.lt_mul_of_one_lt_right
-- See Note [decidable namespace]
protected lemma decidable.lt_mul_of_one_lt_left [@decidable_rel α (≤)]
(hb : 0 < b) (h : 1 < a) : b < a * b :=
suffices 1 * b < a * b, by rwa one_mul at this,
decidable.mul_lt_mul h (le_refl _) hb (zero_le_one.trans h.le)
lemma lt_mul_of_one_lt_left : 0 < b → 1 < a → b < a * b :=
by classical; exact decidable.lt_mul_of_one_lt_left
-- See Note [decidable namespace]
protected lemma decidable.add_le_mul_two_add [@decidable_rel α (≤)] {a b : α}
(a2 : 2 ≤ a) (b0 : 0 ≤ b) : a + (2 + b) ≤ a * (2 + b) :=
calc a + (2 + b) ≤ a + (a + a * b) :
add_le_add_left (add_le_add a2 (decidable.le_mul_of_one_le_left b0 (one_le_two.trans a2))) a
... ≤ a * (2 + b) : by rw [mul_add, mul_two, add_assoc]
lemma add_le_mul_two_add {a b : α} : 2 ≤ a → 0 ≤ b → a + (2 + b) ≤ a * (2 + b) :=
by classical; exact decidable.add_le_mul_two_add
-- See Note [decidable namespace]
protected lemma decidable.one_le_mul_of_one_le_of_one_le [@decidable_rel α (≤)]
{a b : α} (a1 : 1 ≤ a) (b1 : 1 ≤ b) : (1 : α) ≤ a * b :=
(mul_one (1 : α)).symm.le.trans (decidable.mul_le_mul a1 b1 zero_le_one (zero_le_one.trans a1))
lemma one_le_mul_of_one_le_of_one_le {a b : α} : 1 ≤ a → 1 ≤ b → (1 : α) ≤ a * b :=
by classical; exact decidable.one_le_mul_of_one_le_of_one_le
/-- Pullback an `ordered_semiring` under an injective map.
See note [reducible non-instances]. -/
@[reducible]
def function.injective.ordered_semiring {β : Type*}
[has_zero β] [has_one β] [has_add β] [has_mul β]
(f : β → α) (hf : function.injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y) :
ordered_semiring β :=
{ zero_le_one := show f 0 ≤ f 1, by simp only [zero, one, zero_le_one],
mul_lt_mul_of_pos_left := λ a b c ab c0, show f (c * a) < f (c * b),
begin
rw [mul, mul],
refine mul_lt_mul_of_pos_left ab _,
rwa ← zero,
end,
mul_lt_mul_of_pos_right := λ a b c ab c0, show f (a * c) < f (b * c),
begin
rw [mul, mul],
refine mul_lt_mul_of_pos_right ab _,
rwa ← zero,
end,
..hf.ordered_cancel_add_comm_monoid f zero add,
..hf.semiring f zero one add mul }
section
variable [nontrivial α]
lemma bit1_pos (h : 0 ≤ a) : 0 < bit1 a :=
lt_add_of_le_of_pos (add_nonneg h h) zero_lt_one
lemma lt_add_one (a : α) : a < a + 1 :=
lt_add_of_le_of_pos le_rfl zero_lt_one
lemma lt_one_add (a : α) : a < 1 + a :=
by { rw [add_comm], apply lt_add_one }
end
lemma bit1_pos' (h : 0 < a) : 0 < bit1 a :=
begin
nontriviality,
exact bit1_pos h.le,
end
-- See Note [decidable namespace]
protected lemma decidable.one_lt_mul [@decidable_rel α (≤)]
(ha : 1 ≤ a) (hb : 1 < b) : 1 < a * b :=
begin
nontriviality,
exact (one_mul (1 : α)) ▸ decidable.mul_lt_mul' ha hb zero_le_one (zero_lt_one.trans_le ha)
end
lemma one_lt_mul : 1 ≤ a → 1 < b → 1 < a * b :=
by classical; exact decidable.one_lt_mul
-- See Note [decidable namespace]
protected lemma decidable.mul_le_one [@decidable_rel α (≤)]
(ha : a ≤ 1) (hb' : 0 ≤ b) (hb : b ≤ 1) : a * b ≤ 1 :=
begin rw ← one_mul (1 : α), apply decidable.mul_le_mul; {assumption <|> apply zero_le_one} end
lemma mul_le_one : a ≤ 1 → 0 ≤ b → b ≤ 1 → a * b ≤ 1 :=
by classical; exact decidable.mul_le_one
-- See Note [decidable namespace]
protected lemma decidable.one_lt_mul_of_le_of_lt [@decidable_rel α (≤)]
(ha : 1 ≤ a) (hb : 1 < b) : 1 < a * b :=
begin
nontriviality,
calc 1 = 1 * 1 : by rw one_mul
... < a * b : decidable.mul_lt_mul' ha hb zero_le_one (zero_lt_one.trans_le ha)
end
lemma one_lt_mul_of_le_of_lt : 1 ≤ a → 1 < b → 1 < a * b :=
by classical; exact decidable.one_lt_mul_of_le_of_lt
-- See Note [decidable namespace]
protected lemma decidable.one_lt_mul_of_lt_of_le [@decidable_rel α (≤)]
(ha : 1 < a) (hb : 1 ≤ b) : 1 < a * b :=
begin
nontriviality,
calc 1 = 1 * 1 : by rw one_mul
... < a * b : decidable.mul_lt_mul ha hb zero_lt_one $ zero_le_one.trans ha.le
end
lemma one_lt_mul_of_lt_of_le : 1 < a → 1 ≤ b → 1 < a * b :=
by classical; exact decidable.one_lt_mul_of_lt_of_le
-- See Note [decidable namespace]
protected lemma decidable.mul_le_of_le_one_right [@decidable_rel α (≤)]
(ha : 0 ≤ a) (hb1 : b ≤ 1) : a * b ≤ a :=
calc a * b ≤ a * 1 : decidable.mul_le_mul_of_nonneg_left hb1 ha
... = a : mul_one a
lemma mul_le_of_le_one_right : 0 ≤ a → b ≤ 1 → a * b ≤ a :=
by classical; exact decidable.mul_le_of_le_one_right
-- See Note [decidable namespace]
protected lemma decidable.mul_le_of_le_one_left [@decidable_rel α (≤)]
(hb : 0 ≤ b) (ha1 : a ≤ 1) : a * b ≤ b :=
calc a * b ≤ 1 * b : decidable.mul_le_mul ha1 le_rfl hb zero_le_one
... = b : one_mul b
lemma mul_le_of_le_one_left : 0 ≤ b → a ≤ 1 → a * b ≤ b :=
by classical; exact decidable.mul_le_of_le_one_left
-- See Note [decidable namespace]
protected lemma decidable.mul_lt_one_of_nonneg_of_lt_one_left [@decidable_rel α (≤)]
(ha0 : 0 ≤ a) (ha : a < 1) (hb : b ≤ 1) : a * b < 1 :=
calc a * b ≤ a : decidable.mul_le_of_le_one_right ha0 hb
... < 1 : ha
lemma mul_lt_one_of_nonneg_of_lt_one_left : 0 ≤ a → a < 1 → b ≤ 1 → a * b < 1 :=
by classical; exact decidable.mul_lt_one_of_nonneg_of_lt_one_left
-- See Note [decidable namespace]
protected lemma decidable.mul_lt_one_of_nonneg_of_lt_one_right [@decidable_rel α (≤)]
(ha : a ≤ 1) (hb0 : 0 ≤ b) (hb : b < 1) : a * b < 1 :=
calc a * b ≤ b : decidable.mul_le_of_le_one_left hb0 ha
... < 1 : hb
lemma mul_lt_one_of_nonneg_of_lt_one_right : a ≤ 1 → 0 ≤ b → b < 1 → a * b < 1 :=
by classical; exact decidable.mul_lt_one_of_nonneg_of_lt_one_right
end ordered_semiring
section ordered_comm_semiring
/-- An `ordered_comm_semiring α` is a commutative semiring `α` with a partial order such that
addition is monotone and multiplication by a positive number is strictly monotone. -/
@[protect_proj]
class ordered_comm_semiring (α : Type u) extends ordered_semiring α, comm_semiring α
/-- Pullback an `ordered_comm_semiring` under an injective map.
See note [reducible non-instances]. -/
@[reducible]
def function.injective.ordered_comm_semiring [ordered_comm_semiring α] {β : Type*}
[has_zero β] [has_one β] [has_add β] [has_mul β]
(f : β → α) (hf : function.injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y) :
ordered_comm_semiring β :=
{ ..hf.comm_semiring f zero one add mul,
..hf.ordered_semiring f zero one add mul }
end ordered_comm_semiring
/--
A `linear_ordered_semiring α` is a nontrivial semiring `α` with a linear order
such that addition is monotone and multiplication by a positive number is strictly monotone.
-/
-- It's not entirely clear we should assume `nontrivial` at this point;
-- it would be reasonable to explore changing this,
-- but be warned that the instances involving `domain` may cause
-- typeclass search loops.
@[protect_proj]
class linear_ordered_semiring (α : Type u)
extends ordered_semiring α, linear_ordered_add_comm_monoid α, nontrivial α
section linear_ordered_semiring
variables [linear_ordered_semiring α] {a b c d : α}
-- `norm_num` expects the lemma stating `0 < 1` to have a single typeclass argument
-- (see `norm_num.prove_pos_nat`).
-- Rather than working out how to relax that assumption,
-- we provide a synonym for `zero_lt_one` (which needs both `ordered_semiring α` and `nontrivial α`)
-- with only a `linear_ordered_semiring` typeclass argument.
lemma zero_lt_one' : 0 < (1 : α) := zero_lt_one
lemma lt_of_mul_lt_mul_left (h : c * a < c * b) (hc : 0 ≤ c) : a < b :=
by haveI := @linear_order.decidable_le α _; exact lt_of_not_ge
(assume h1 : b ≤ a,
have h2 : c * b ≤ c * a, from decidable.mul_le_mul_of_nonneg_left h1 hc,
h2.not_lt h)
lemma lt_of_mul_lt_mul_right (h : a * c < b * c) (hc : 0 ≤ c) : a < b :=
by haveI := @linear_order.decidable_le α _; exact lt_of_not_ge
(assume h1 : b ≤ a,
have h2 : b * c ≤ a * c, from decidable.mul_le_mul_of_nonneg_right h1 hc,
h2.not_lt h)
lemma le_of_mul_le_mul_left (h : c * a ≤ c * b) (hc : 0 < c) : a ≤ b :=
le_of_not_gt
(assume h1 : b < a,
have h2 : c * b < c * a, from mul_lt_mul_of_pos_left h1 hc,
h2.not_le h)
lemma le_of_mul_le_mul_right (h : a * c ≤ b * c) (hc : 0 < c) : a ≤ b :=
le_of_not_gt
(assume h1 : b < a,
have h2 : b * c < a * c, from mul_lt_mul_of_pos_right h1 hc,
h2.not_le h)
lemma pos_and_pos_or_neg_and_neg_of_mul_pos (hab : 0 < a * b) :
(0 < a ∧ 0 < b) ∨ (a < 0 ∧ b < 0) :=
begin
haveI := @linear_order.decidable_le α _,
rcases lt_trichotomy 0 a with (ha|rfl|ha),
{ refine or.inl ⟨ha, lt_imp_lt_of_le_imp_le (λ hb, _) hab⟩,
exact decidable.mul_nonpos_of_nonneg_of_nonpos ha.le hb },
{ rw [zero_mul] at hab, exact hab.false.elim },
{ refine or.inr ⟨ha, lt_imp_lt_of_le_imp_le (λ hb, _) hab⟩,
exact decidable.mul_nonpos_of_nonpos_of_nonneg ha.le hb }
end
lemma nonneg_and_nonneg_or_nonpos_and_nonpos_of_mul_nnonneg (hab : 0 ≤ a * b) :
(0 ≤ a ∧ 0 ≤ b) ∨ (a ≤ 0 ∧ b ≤ 0) :=
begin
haveI := @linear_order.decidable_le α _,
refine decidable.or_iff_not_and_not.2 _,
simp only [not_and, not_le], intros ab nab, apply not_lt_of_le hab _,
rcases lt_trichotomy 0 a with (ha|rfl|ha),
exacts [mul_neg_of_pos_of_neg ha (ab ha.le), ((ab le_rfl).asymm (nab le_rfl)).elim,
mul_neg_of_neg_of_pos ha (nab ha.le)]
end
lemma pos_of_mul_pos_left (h : 0 < a * b) (ha : 0 ≤ a) : 0 < b :=
((pos_and_pos_or_neg_and_neg_of_mul_pos h).resolve_right $ λ h, h.1.not_le ha).2
lemma pos_of_mul_pos_right (h : 0 < a * b) (hb : 0 ≤ b) : 0 < a :=
((pos_and_pos_or_neg_and_neg_of_mul_pos h).resolve_right $ λ h, h.2.not_le hb).1
@[simp] lemma inv_of_pos [invertible a] : 0 < ⅟a ↔ 0 < a :=
begin
have : 0 < a * ⅟a, by simp only [mul_inv_of_self, zero_lt_one],
exact ⟨λ h, pos_of_mul_pos_right this h.le, λ h, pos_of_mul_pos_left this h.le⟩
end
@[simp] lemma inv_of_nonpos [invertible a] : ⅟a ≤ 0 ↔ a ≤ 0 :=
by simp only [← not_lt, inv_of_pos]
lemma nonneg_of_mul_nonneg_left (h : 0 ≤ a * b) (h1 : 0 < a) : 0 ≤ b :=
le_of_not_gt (assume h2 : b < 0, (mul_neg_of_pos_of_neg h1 h2).not_le h)
lemma nonneg_of_mul_nonneg_right (h : 0 ≤ a * b) (h1 : 0 < b) : 0 ≤ a :=
le_of_not_gt (assume h2 : a < 0, (mul_neg_of_neg_of_pos h2 h1).not_le h)
@[simp] lemma inv_of_nonneg [invertible a] : 0 ≤ ⅟a ↔ 0 ≤ a :=
begin
have : 0 < a * ⅟a, by simp only [mul_inv_of_self, zero_lt_one],
exact ⟨λ h, (pos_of_mul_pos_right this h).le, λ h, (pos_of_mul_pos_left this h).le⟩
end
@[simp] lemma inv_of_lt_zero [invertible a] : ⅟a < 0 ↔ a < 0 :=
by simp only [← not_le, inv_of_nonneg]
@[simp] lemma inv_of_le_one [invertible a] (h : 1 ≤ a) : ⅟a ≤ 1 :=
by haveI := @linear_order.decidable_le α _; exact
mul_inv_of_self a ▸ decidable.le_mul_of_one_le_left (inv_of_nonneg.2 $ zero_le_one.trans h) h
lemma neg_of_mul_neg_left (h : a * b < 0) (h1 : 0 ≤ a) : b < 0 :=
by haveI := @linear_order.decidable_le α _; exact
lt_of_not_ge (assume h2 : b ≥ 0, (decidable.mul_nonneg h1 h2).not_lt h)
lemma neg_of_mul_neg_right (h : a * b < 0) (h1 : 0 ≤ b) : a < 0 :=
by haveI := @linear_order.decidable_le α _; exact
lt_of_not_ge (assume h2 : a ≥ 0, (decidable.mul_nonneg h2 h1).not_lt h)
lemma nonpos_of_mul_nonpos_left (h : a * b ≤ 0) (h1 : 0 < a) : b ≤ 0 :=
le_of_not_gt (assume h2 : b > 0, (mul_pos h1 h2).not_le h)
lemma nonpos_of_mul_nonpos_right (h : a * b ≤ 0) (h1 : 0 < b) : a ≤ 0 :=
le_of_not_gt (assume h2 : a > 0, (mul_pos h2 h1).not_le h)
@[simp] lemma mul_le_mul_left (h : 0 < c) : c * a ≤ c * b ↔ a ≤ b :=
by haveI := @linear_order.decidable_le α _; exact
⟨λ h', le_of_mul_le_mul_left h' h, λ h', decidable.mul_le_mul_of_nonneg_left h' h.le⟩
@[simp] lemma mul_le_mul_right (h : 0 < c) : a * c ≤ b * c ↔ a ≤ b :=
by haveI := @linear_order.decidable_le α _; exact
⟨λ h', le_of_mul_le_mul_right h' h, λ h', decidable.mul_le_mul_of_nonneg_right h' h.le⟩
@[simp] lemma mul_lt_mul_left (h : 0 < c) : c * a < c * b ↔ a < b :=
by haveI := @linear_order.decidable_le α _; exact
⟨lt_imp_lt_of_le_imp_le $ λ h', decidable.mul_le_mul_of_nonneg_left h' h.le,
λ h', mul_lt_mul_of_pos_left h' h⟩
@[simp] lemma mul_lt_mul_right (h : 0 < c) : a * c < b * c ↔ a < b :=
by haveI := @linear_order.decidable_le α _; exact
⟨lt_imp_lt_of_le_imp_le $ λ h', decidable.mul_le_mul_of_nonneg_right h' h.le,
λ h', mul_lt_mul_of_pos_right h' h⟩
@[simp] lemma zero_le_mul_left (h : 0 < c) : 0 ≤ c * b ↔ 0 ≤ b :=
by { convert mul_le_mul_left h, simp }
@[simp] lemma zero_le_mul_right (h : 0 < c) : 0 ≤ b * c ↔ 0 ≤ b :=
by { convert mul_le_mul_right h, simp }
@[simp] lemma zero_lt_mul_left (h : 0 < c) : 0 < c * b ↔ 0 < b :=
by { convert mul_lt_mul_left h, simp }
@[simp] lemma zero_lt_mul_right (h : 0 < c) : 0 < b * c ↔ 0 < b :=
by { convert mul_lt_mul_right h, simp }
lemma add_le_mul_of_left_le_right (a2 : 2 ≤ a) (ab : a ≤ b) : a + b ≤ a * b :=
have 0 < b, from
calc 0 < 2 : zero_lt_two
... ≤ a : a2
... ≤ b : ab,
calc a + b ≤ b + b : add_le_add_right ab b
... = 2 * b : (two_mul b).symm
... ≤ a * b : (mul_le_mul_right this).mpr a2
lemma add_le_mul_of_right_le_left (b2 : 2 ≤ b) (ba : b ≤ a) : a + b ≤ a * b :=
have 0 < a, from
calc 0 < 2 : zero_lt_two
... ≤ b : b2
... ≤ a : ba,
calc a + b ≤ a + a : add_le_add_left ba a
... = a * 2 : (mul_two a).symm
... ≤ a * b : (mul_le_mul_left this).mpr b2
lemma add_le_mul (a2 : 2 ≤ a) (b2 : 2 ≤ b) : a + b ≤ a * b :=
if hab : a ≤ b then add_le_mul_of_left_le_right a2 hab
else add_le_mul_of_right_le_left b2 (le_of_not_le hab)
lemma add_le_mul' (a2 : 2 ≤ a) (b2 : 2 ≤ b) : a + b ≤ b * a :=
(le_of_eq (add_comm _ _)).trans (add_le_mul b2 a2)
section
variables [nontrivial α]
@[simp] lemma bit0_le_bit0 : bit0 a ≤ bit0 b ↔ a ≤ b :=
by rw [bit0, bit0, ← two_mul, ← two_mul, mul_le_mul_left (zero_lt_two : 0 < (2:α))]
@[simp] lemma bit0_lt_bit0 : bit0 a < bit0 b ↔ a < b :=
by rw [bit0, bit0, ← two_mul, ← two_mul, mul_lt_mul_left (zero_lt_two : 0 < (2:α))]
@[simp] lemma bit1_le_bit1 : bit1 a ≤ bit1 b ↔ a ≤ b :=
(add_le_add_iff_right 1).trans bit0_le_bit0
@[simp] lemma bit1_lt_bit1 : bit1 a < bit1 b ↔ a < b :=
(add_lt_add_iff_right 1).trans bit0_lt_bit0
@[simp] lemma one_le_bit1 : (1 : α) ≤ bit1 a ↔ 0 ≤ a :=
by rw [bit1, le_add_iff_nonneg_left, bit0, ← two_mul, zero_le_mul_left (zero_lt_two : 0 < (2:α))]
@[simp] lemma one_lt_bit1 : (1 : α) < bit1 a ↔ 0 < a :=
by rw [bit1, lt_add_iff_pos_left, bit0, ← two_mul, zero_lt_mul_left (zero_lt_two : 0 < (2:α))]
@[simp] lemma zero_le_bit0 : (0 : α) ≤ bit0 a ↔ 0 ≤ a :=
by rw [bit0, ← two_mul, zero_le_mul_left (zero_lt_two : 0 < (2:α))]
@[simp] lemma zero_lt_bit0 : (0 : α) < bit0 a ↔ 0 < a :=
by rw [bit0, ← two_mul, zero_lt_mul_left (zero_lt_two : 0 < (2:α))]
end
lemma le_mul_iff_one_le_left (hb : 0 < b) : b ≤ a * b ↔ 1 ≤ a :=
suffices 1 * b ≤ a * b ↔ 1 ≤ a, by rwa one_mul at this,
mul_le_mul_right hb
lemma lt_mul_iff_one_lt_left (hb : 0 < b) : b < a * b ↔ 1 < a :=
suffices 1 * b < a * b ↔ 1 < a, by rwa one_mul at this,
mul_lt_mul_right hb
lemma le_mul_iff_one_le_right (hb : 0 < b) : b ≤ b * a ↔ 1 ≤ a :=
suffices b * 1 ≤ b * a ↔ 1 ≤ a, by rwa mul_one at this,
mul_le_mul_left hb
lemma lt_mul_iff_one_lt_right (hb : 0 < b) : b < b * a ↔ 1 < a :=
suffices b * 1 < b * a ↔ 1 < a, by rwa mul_one at this,
mul_lt_mul_left hb
theorem mul_nonneg_iff_right_nonneg_of_pos (ha : 0 < a) : 0 ≤ b * a ↔ 0 ≤ b :=
by haveI := @linear_order.decidable_le α _; exact
⟨λ h, nonneg_of_mul_nonneg_right h ha, λ h, decidable.mul_nonneg h ha.le⟩
lemma mul_le_iff_le_one_left (hb : 0 < b) : a * b ≤ b ↔ a ≤ 1 :=
⟨ λ h, le_of_not_lt (mt (lt_mul_iff_one_lt_left hb).2 h.not_lt),
λ h, le_of_not_lt (mt (lt_mul_iff_one_lt_left hb).1 h.not_lt) ⟩
lemma mul_lt_iff_lt_one_left (hb : 0 < b) : a * b < b ↔ a < 1 :=
⟨ λ h, lt_of_not_ge (mt (le_mul_iff_one_le_left hb).2 h.not_le),
λ h, lt_of_not_ge (mt (le_mul_iff_one_le_left hb).1 h.not_le) ⟩
lemma mul_le_iff_le_one_right (hb : 0 < b) : b * a ≤ b ↔ a ≤ 1 :=
⟨ λ h, le_of_not_lt (mt (lt_mul_iff_one_lt_right hb).2 h.not_lt),
λ h, le_of_not_lt (mt (lt_mul_iff_one_lt_right hb).1 h.not_lt) ⟩
lemma mul_lt_iff_lt_one_right (hb : 0 < b) : b * a < b ↔ a < 1 :=
⟨ λ h, lt_of_not_ge (mt (le_mul_iff_one_le_right hb).2 h.not_le),
λ h, lt_of_not_ge (mt (le_mul_iff_one_le_right hb).1 h.not_le) ⟩
lemma nonpos_of_mul_nonneg_left (h : 0 ≤ a * b) (hb : b < 0) : a ≤ 0 :=
le_of_not_gt (λ ha, absurd h (mul_neg_of_pos_of_neg ha hb).not_le)
lemma nonpos_of_mul_nonneg_right (h : 0 ≤ a * b) (ha : a < 0) : b ≤ 0 :=
le_of_not_gt (λ hb, absurd h (mul_neg_of_neg_of_pos ha hb).not_le)
lemma neg_of_mul_pos_left (h : 0 < a * b) (hb : b ≤ 0) : a < 0 :=
by haveI := @linear_order.decidable_le α _; exact
lt_of_not_ge (λ ha, absurd h (decidable.mul_nonpos_of_nonneg_of_nonpos ha hb).not_lt)
lemma neg_of_mul_pos_right (h : 0 < a * b) (ha : a ≤ 0) : b < 0 :=
by haveI := @linear_order.decidable_le α _; exact
lt_of_not_ge (λ hb, absurd h (decidable.mul_nonpos_of_nonpos_of_nonneg ha hb).not_lt)
@[priority 100] -- see Note [lower instance priority]
instance linear_ordered_semiring.to_no_top_order {α : Type*} [linear_ordered_semiring α] :
no_top_order α :=
⟨assume a, ⟨a + 1, lt_add_of_pos_right _ zero_lt_one⟩⟩
/-- Pullback a `linear_ordered_semiring` under an injective map.
See note [reducible non-instances]. -/
@[reducible]
def function.injective.linear_ordered_semiring {β : Type*}
[has_zero β] [has_one β] [has_add β] [has_mul β] [nontrivial β]
(f : β → α) (hf : function.injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y) :
linear_ordered_semiring β :=
{ ..linear_order.lift f hf,
..‹nontrivial β›,
..hf.ordered_semiring f zero one add mul }
end linear_ordered_semiring
section mono
variables {β : Type*} [linear_ordered_semiring α] [preorder β] {f g : β → α} {a : α}
lemma monotone_mul_left_of_nonneg (ha : 0 ≤ a) : monotone (λ x, a*x) :=
by haveI := @linear_order.decidable_le α _; exact
assume b c b_le_c, decidable.mul_le_mul_of_nonneg_left b_le_c ha
lemma monotone_mul_right_of_nonneg (ha : 0 ≤ a) : monotone (λ x, x*a) :=
by haveI := @linear_order.decidable_le α _; exact
assume b c b_le_c, decidable.mul_le_mul_of_nonneg_right b_le_c ha
lemma monotone.mul_const (hf : monotone f) (ha : 0 ≤ a) :
monotone (λ x, (f x) * a) :=
(monotone_mul_right_of_nonneg ha).comp hf
lemma monotone.const_mul (hf : monotone f) (ha : 0 ≤ a) :
monotone (λ x, a * (f x)) :=
(monotone_mul_left_of_nonneg ha).comp hf
lemma monotone.mul (hf : monotone f) (hg : monotone g) (hf0 : ∀ x, 0 ≤ f x) (hg0 : ∀ x, 0 ≤ g x) :
monotone (λ x, f x * g x) :=
by haveI := @linear_order.decidable_le α _; exact
λ x y h, decidable.mul_le_mul (hf h) (hg h) (hg0 x) (hf0 y)
lemma strict_mono_mul_left_of_pos (ha : 0 < a) : strict_mono (λ x, a * x) :=
assume b c b_lt_c, (mul_lt_mul_left ha).2 b_lt_c
lemma strict_mono_mul_right_of_pos (ha : 0 < a) : strict_mono (λ x, x * a) :=
assume b c b_lt_c, (mul_lt_mul_right ha).2 b_lt_c
lemma strict_mono.mul_const (hf : strict_mono f) (ha : 0 < a) :
strict_mono (λ x, (f x) * a) :=
(strict_mono_mul_right_of_pos ha).comp hf
lemma strict_mono.const_mul (hf : strict_mono f) (ha : 0 < a) :
strict_mono (λ x, a * (f x)) :=
(strict_mono_mul_left_of_pos ha).comp hf
lemma strict_mono.mul_monotone (hf : strict_mono f) (hg : monotone g) (hf0 : ∀ x, 0 ≤ f x)
(hg0 : ∀ x, 0 < g x) :
strict_mono (λ x, f x * g x) :=
by haveI := @linear_order.decidable_le α _; exact
λ x y h, decidable.mul_lt_mul (hf h) (hg h.le) (hg0 x) (hf0 y)
lemma monotone.mul_strict_mono (hf : monotone f) (hg : strict_mono g) (hf0 : ∀ x, 0 < f x)
(hg0 : ∀ x, 0 ≤ g x) :
strict_mono (λ x, f x * g x) :=
by haveI := @linear_order.decidable_le α _; exact
λ x y h, decidable.mul_lt_mul' (hf h.le) (hg h) (hg0 x) (hf0 y)
lemma strict_mono.mul (hf : strict_mono f) (hg : strict_mono g) (hf0 : ∀ x, 0 ≤ f x)
(hg0 : ∀ x, 0 ≤ g x) :
strict_mono (λ x, f x * g x) :=
by haveI := @linear_order.decidable_le α _; exact
λ x y h, decidable.mul_lt_mul'' (hf h) (hg h) (hf0 x) (hg0 x)
end mono
section linear_ordered_semiring
variables [linear_ordered_semiring α] {a b c : α}
lemma mul_max_of_nonneg (b c : α) (ha : 0 ≤ a) : a * max b c = max (a * b) (a * c) :=
(monotone_mul_left_of_nonneg ha).map_max
lemma mul_min_of_nonneg (b c : α) (ha : 0 ≤ a) : a * min b c = min (a * b) (a * c) :=
(monotone_mul_left_of_nonneg ha).map_min
lemma max_mul_of_nonneg (a b : α) (hc : 0 ≤ c) : max a b * c = max (a * c) (b * c) :=
(monotone_mul_right_of_nonneg hc).map_max
lemma min_mul_of_nonneg (a b : α) (hc : 0 ≤ c) : min a b * c = min (a * c) (b * c) :=
(monotone_mul_right_of_nonneg hc).map_min
end linear_ordered_semiring
/-- An `ordered_ring α` is a ring `α` with a partial order such that
addition is monotone and multiplication by a positive number is strictly monotone. -/
@[protect_proj]
class ordered_ring (α : Type u) extends ring α, ordered_add_comm_group α :=
(zero_le_one : 0 ≤ (1 : α))
(mul_pos : ∀ a b : α, 0 < a → 0 < b → 0 < a * b)
section ordered_ring
variables [ordered_ring α] {a b c : α}
-- See Note [decidable namespace]
protected lemma decidable.ordered_ring.mul_nonneg [@decidable_rel α (≤)]
{a b : α} (h₁ : 0 ≤ a) (h₂ : 0 ≤ b) : 0 ≤ a * b :=
begin
by_cases ha : a ≤ 0, { simp [le_antisymm ha h₁] },
by_cases hb : b ≤ 0, { simp [le_antisymm hb h₂] },
exact (le_not_le_of_lt (ordered_ring.mul_pos a b (h₁.lt_of_not_le ha) (h₂.lt_of_not_le hb))).1,
end
lemma ordered_ring.mul_nonneg : 0 ≤ a → 0 ≤ b → 0 ≤ a * b :=
by classical; exact decidable.ordered_ring.mul_nonneg
-- See Note [decidable namespace]
protected lemma decidable.ordered_ring.mul_le_mul_of_nonneg_left
[@decidable_rel α (≤)] (h₁ : a ≤ b) (h₂ : 0 ≤ c) : c * a ≤ c * b :=
begin
rw [← sub_nonneg, ← mul_sub],
exact decidable.ordered_ring.mul_nonneg h₂ (sub_nonneg.2 h₁),
end
lemma ordered_ring.mul_le_mul_of_nonneg_left : a ≤ b → 0 ≤ c → c * a ≤ c * b :=
by classical; exact decidable.ordered_ring.mul_le_mul_of_nonneg_left
-- See Note [decidable namespace]
protected lemma decidable.ordered_ring.mul_le_mul_of_nonneg_right
[@decidable_rel α (≤)] (h₁ : a ≤ b) (h₂ : 0 ≤ c) : a * c ≤ b * c :=
begin
rw [← sub_nonneg, ← sub_mul],
exact decidable.ordered_ring.mul_nonneg (sub_nonneg.2 h₁) h₂,
end
lemma ordered_ring.mul_le_mul_of_nonneg_right : a ≤ b → 0 ≤ c → a * c ≤ b * c :=
by classical; exact decidable.ordered_ring.mul_le_mul_of_nonneg_right
lemma ordered_ring.mul_lt_mul_of_pos_left (h₁ : a < b) (h₂ : 0 < c) : c * a < c * b :=
begin
rw [← sub_pos, ← mul_sub],
exact ordered_ring.mul_pos _ _ h₂ (sub_pos.2 h₁),
end
lemma ordered_ring.mul_lt_mul_of_pos_right (h₁ : a < b) (h₂ : 0 < c) : a * c < b * c :=
begin
rw [← sub_pos, ← sub_mul],
exact ordered_ring.mul_pos _ _ (sub_pos.2 h₁) h₂,
end
@[priority 100] -- see Note [lower instance priority]
instance ordered_ring.to_ordered_semiring : ordered_semiring α :=
{ mul_zero := mul_zero,
zero_mul := zero_mul,
add_left_cancel := @add_left_cancel α _,
le_of_add_le_add_left := @le_of_add_le_add_left α _ _ _,
mul_lt_mul_of_pos_left := @ordered_ring.mul_lt_mul_of_pos_left α _,
mul_lt_mul_of_pos_right := @ordered_ring.mul_lt_mul_of_pos_right α _,
..‹ordered_ring α› }
-- See Note [decidable namespace]
protected lemma decidable.mul_le_mul_of_nonpos_left [@decidable_rel α (≤)]
{a b c : α} (h : b ≤ a) (hc : c ≤ 0) : c * a ≤ c * b :=
have -c ≥ 0, from neg_nonneg_of_nonpos hc,
have -c * b ≤ -c * a, from decidable.mul_le_mul_of_nonneg_left h this,
have -(c * b) ≤ -(c * a), by rwa [← neg_mul_eq_neg_mul, ← neg_mul_eq_neg_mul] at this,
le_of_neg_le_neg this
lemma mul_le_mul_of_nonpos_left {a b c : α} : b ≤ a → c ≤ 0 → c * a ≤ c * b :=
by classical; exact decidable.mul_le_mul_of_nonpos_left
-- See Note [decidable namespace]
protected lemma decidable.mul_le_mul_of_nonpos_right [@decidable_rel α (≤)]
{a b c : α} (h : b ≤ a) (hc : c ≤ 0) : a * c ≤ b * c :=
have -c ≥ 0, from neg_nonneg_of_nonpos hc,
have b * -c ≤ a * -c, from decidable.mul_le_mul_of_nonneg_right h this,
have -(b * c) ≤ -(a * c), by rwa [← neg_mul_eq_mul_neg, ← neg_mul_eq_mul_neg] at this,
le_of_neg_le_neg this
lemma mul_le_mul_of_nonpos_right {a b c : α} : b ≤ a → c ≤ 0 → a * c ≤ b * c :=
by classical; exact decidable.mul_le_mul_of_nonpos_right
-- See Note [decidable namespace]
protected lemma decidable.mul_nonneg_of_nonpos_of_nonpos [@decidable_rel α (≤)]
{a b : α} (ha : a ≤ 0) (hb : b ≤ 0) : 0 ≤ a * b :=
have 0 * b ≤ a * b, from decidable.mul_le_mul_of_nonpos_right ha hb,
by rwa zero_mul at this
lemma mul_nonneg_of_nonpos_of_nonpos {a b : α} : a ≤ 0 → b ≤ 0 → 0 ≤ a * b :=
by classical; exact decidable.mul_nonneg_of_nonpos_of_nonpos
lemma mul_lt_mul_of_neg_left {a b c : α} (h : b < a) (hc : c < 0) : c * a < c * b :=
have -c > 0, from neg_pos_of_neg hc,
have -c * b < -c * a, from mul_lt_mul_of_pos_left h this,
have -(c * b) < -(c * a), by rwa [← neg_mul_eq_neg_mul, ← neg_mul_eq_neg_mul] at this,
lt_of_neg_lt_neg this
lemma mul_lt_mul_of_neg_right {a b c : α} (h : b < a) (hc : c < 0) : a * c < b * c :=
have -c > 0, from neg_pos_of_neg hc,
have b * -c < a * -c, from mul_lt_mul_of_pos_right h this,
have -(b * c) < -(a * c), by rwa [← neg_mul_eq_mul_neg, ← neg_mul_eq_mul_neg] at this,
lt_of_neg_lt_neg this
lemma mul_pos_of_neg_of_neg {a b : α} (ha : a < 0) (hb : b < 0) : 0 < a * b :=
have 0 * b < a * b, from mul_lt_mul_of_neg_right ha hb,
by rwa zero_mul at this
/-- Pullback an `ordered_ring` under an injective map.
See note [reducible non-instances]. -/
@[reducible]
def function.injective.ordered_ring {β : Type*}
[has_zero β] [has_one β] [has_add β] [has_mul β] [has_neg β] [has_sub β]
(f : β → α) (hf : function.injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(neg : ∀ x, f (- x) = - f x) (sub : ∀ x y, f (x - y) = f x - f y) :
ordered_ring β :=
{ mul_pos := λ a b a0 b0, show f 0 < f (a * b), by { rw [zero, mul], apply mul_pos; rwa ← zero },
..hf.ordered_semiring f zero one add mul,
..hf.ring f zero one add mul neg sub }
end ordered_ring
section ordered_comm_ring
/-- An `ordered_comm_ring α` is a commutative ring `α` with a partial order such that
addition is monotone and multiplication by a positive number is strictly monotone. -/
@[protect_proj]
class ordered_comm_ring (α : Type u) extends ordered_ring α, comm_ring α
@[priority 100] -- See note [lower instance priority]
instance ordered_comm_ring.to_ordered_comm_semiring {α : Type u} [ordered_comm_ring α] :
ordered_comm_semiring α :=
{ .. (by apply_instance : ordered_semiring α),
.. ‹ordered_comm_ring α› }
/-- Pullback an `ordered_comm_ring` under an injective map.
See note [reducible non-instances]. -/
@[reducible]
def function.injective.ordered_comm_ring [ordered_comm_ring α] {β : Type*}
[has_zero β] [has_one β] [has_add β] [has_mul β] [has_neg β] [has_sub β]
(f : β → α) (hf : function.injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(neg : ∀ x, f (- x) = - f x) (sub : ∀ x y, f (x - y) = f x - f y) :
ordered_comm_ring β :=
{ ..hf.ordered_ring f zero one add mul neg sub,
..hf.comm_ring f zero one add mul neg sub }
end ordered_comm_ring
/-- A `linear_ordered_ring α` is a ring `α` with a linear order such that
addition is monotone and multiplication by a positive number is strictly monotone. -/
@[protect_proj] class linear_ordered_ring (α : Type u)
extends ordered_ring α, linear_order α, nontrivial α
@[priority 100] -- see Note [lower instance priority]
instance linear_ordered_ring.to_linear_ordered_add_comm_group [s : linear_ordered_ring α] :
linear_ordered_add_comm_group α :=
{ .. s }
section linear_ordered_ring
variables [linear_ordered_ring α] {a b c : α}
@[priority 100] -- see Note [lower instance priority]
instance linear_ordered_ring.to_linear_ordered_semiring : linear_ordered_semiring α :=
{ mul_zero := mul_zero,
zero_mul := zero_mul,
add_left_cancel := @add_left_cancel α _,
le_of_add_le_add_left := @le_of_add_le_add_left α _ _ _,
mul_lt_mul_of_pos_left := @mul_lt_mul_of_pos_left α _,
mul_lt_mul_of_pos_right := @mul_lt_mul_of_pos_right α _,
le_total := linear_ordered_ring.le_total,
..‹linear_ordered_ring α› }
@[priority 100] -- see Note [lower instance priority]
instance linear_ordered_ring.to_domain : domain α :=
{ eq_zero_or_eq_zero_of_mul_eq_zero :=
begin
intros a b hab,
refine decidable.or_iff_not_and_not.2 (λ h, _), revert hab,
cases lt_or_gt_of_ne h.1 with ha ha; cases lt_or_gt_of_ne h.2 with hb hb,
exacts [(mul_pos_of_neg_of_neg ha hb).ne.symm, (mul_neg_of_neg_of_pos ha hb).ne,
(mul_neg_of_pos_of_neg ha hb).ne, (mul_pos ha hb).ne.symm]
end,
.. ‹linear_ordered_ring α› }
@[simp] lemma abs_one : |(1 : α)| = 1 := abs_of_pos zero_lt_one
@[simp] lemma abs_two : |(2 : α)| = 2 := abs_of_pos zero_lt_two
lemma abs_mul (a b : α) : |a * b| = |a| * |b| :=
begin
haveI := @linear_order.decidable_le α _,
rw [abs_eq (decidable.mul_nonneg (abs_nonneg a) (abs_nonneg b))],
cases le_total a 0 with ha ha; cases le_total b 0 with hb hb;
simp only [abs_of_nonpos, abs_of_nonneg, true_or, or_true, eq_self_iff_true,
neg_mul_eq_neg_mul_symm, mul_neg_eq_neg_mul_symm, neg_neg, *]
end
/-- `abs` as a `monoid_with_zero_hom`. -/
def abs_hom : monoid_with_zero_hom α α := ⟨abs, abs_zero, abs_one, abs_mul⟩
@[simp] lemma abs_mul_abs_self (a : α) : |a| * |a| = a * a :=
abs_by_cases (λ x, x * x = a * a) rfl (neg_mul_neg a a)
@[simp] lemma abs_mul_self (a : α) : |a * a| = a * a :=
by rw [abs_mul, abs_mul_abs_self]
lemma mul_pos_iff : 0 < a * b ↔ 0 < a ∧ 0 < b ∨ a < 0 ∧ b < 0 :=
⟨pos_and_pos_or_neg_and_neg_of_mul_pos,
λ h, h.elim (and_imp.2 mul_pos) (and_imp.2 mul_pos_of_neg_of_neg)⟩
lemma mul_neg_iff : a * b < 0 ↔ 0 < a ∧ b < 0 ∨ a < 0 ∧ 0 < b :=
by rw [← neg_pos, neg_mul_eq_mul_neg, mul_pos_iff, neg_pos, neg_lt_zero]
lemma mul_nonneg_iff : 0 ≤ a * b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0 :=
by haveI := @linear_order.decidable_le α _; exact
⟨nonneg_and_nonneg_or_nonpos_and_nonpos_of_mul_nnonneg,
λ h, h.elim (and_imp.2 decidable.mul_nonneg) (and_imp.2 decidable.mul_nonneg_of_nonpos_of_nonpos)⟩
/-- Out of three elements of a `linear_ordered_ring`, two must have the same sign. -/
lemma mul_nonneg_of_three (a b c : α) :
0 ≤ a * b ∨ 0 ≤ b * c ∨ 0 ≤ c * a :=
by iterate 3 { rw mul_nonneg_iff };
have := le_total 0 a; have := le_total 0 b; have := le_total 0 c; itauto
lemma mul_nonpos_iff : a * b ≤ 0 ↔ 0 ≤ a ∧ b ≤ 0 ∨ a ≤ 0 ∧ 0 ≤ b :=
by rw [← neg_nonneg, neg_mul_eq_mul_neg, mul_nonneg_iff, neg_nonneg, neg_nonpos]
lemma mul_self_nonneg (a : α) : 0 ≤ a * a :=
abs_mul_self a ▸ abs_nonneg _
@[simp] lemma neg_le_self_iff : -a ≤ a ↔ 0 ≤ a :=
by simp [neg_le_iff_add_nonneg, ← two_mul, mul_nonneg_iff, zero_le_one, (@zero_lt_two α _ _).not_le]
@[simp] lemma neg_lt_self_iff : -a < a ↔ 0 < a :=
by simp [neg_lt_iff_pos_add, ← two_mul, mul_pos_iff, zero_lt_one, (@zero_lt_two α _ _).not_lt]
@[simp] lemma le_neg_self_iff : a ≤ -a ↔ a ≤ 0 :=
calc a ≤ -a ↔ -(-a) ≤ -a : by rw neg_neg
... ↔ 0 ≤ -a : neg_le_self_iff
... ↔ a ≤ 0 : neg_nonneg
@[simp] lemma lt_neg_self_iff : a < -a ↔ a < 0 :=
calc a < -a ↔ -(-a) < -a : by rw neg_neg
... ↔ 0 < -a : neg_lt_self_iff
... ↔ a < 0 : neg_pos
@[simp] lemma abs_eq_self : |a| = a ↔ 0 ≤ a := by simp [abs_eq_max_neg]
@[simp] lemma abs_eq_neg_self : |a| = -a ↔ a ≤ 0 := by simp [abs_eq_max_neg]
/-- For an element `a` of a linear ordered ring, either `abs a = a` and `0 ≤ a`,
or `abs a = -a` and `a < 0`.
Use cases on this lemma to automate linarith in inequalities -/
lemma abs_cases (a : α) : (|a| = a ∧ 0 ≤ a) ∨ (|a| = -a ∧ a < 0) :=
begin
by_cases 0 ≤ a,
{ left,
exact ⟨abs_eq_self.mpr h, h⟩ },
{ right,
push_neg at h,
exact ⟨abs_eq_neg_self.mpr (le_of_lt h), h⟩ }
end
lemma gt_of_mul_lt_mul_neg_left (h : c * a < c * b) (hc : c ≤ 0) : b < a :=
have nhc : 0 ≤ -c, from neg_nonneg_of_nonpos hc,
have h2 : -(c * b) < -(c * a), from neg_lt_neg h,
have h3 : (-c) * b < (-c) * a, from calc
(-c) * b = - (c * b) : by rewrite neg_mul_eq_neg_mul
... < -(c * a) : h2
... = (-c) * a : by rewrite neg_mul_eq_neg_mul,
lt_of_mul_lt_mul_left h3 nhc
lemma neg_one_lt_zero : -1 < (0:α) := neg_lt_zero.2 zero_lt_one
lemma le_of_mul_le_of_one_le {a b c : α} (h : a * c ≤ b) (hb : 0 ≤ b) (hc : 1 ≤ c) : a ≤ b :=
by haveI := @linear_order.decidable_le α _; exact
have h' : a * c ≤ b * c, from calc
a * c ≤ b : h
... = b * 1 : by rewrite mul_one
... ≤ b * c : decidable.mul_le_mul_of_nonneg_left hc hb,
le_of_mul_le_mul_right h' (zero_lt_one.trans_le hc)
lemma nonneg_le_nonneg_of_sq_le_sq {a b : α} (hb : 0 ≤ b) (h : a * a ≤ b * b) : a ≤ b :=
by haveI := @linear_order.decidable_le α _; exact
le_of_not_gt (λhab, (decidable.mul_self_lt_mul_self hb hab).not_le h)
lemma mul_self_le_mul_self_iff {a b : α} (h1 : 0 ≤ a) (h2 : 0 ≤ b) : a ≤ b ↔ a * a ≤ b * b :=
by haveI := @linear_order.decidable_le α _; exact
⟨decidable.mul_self_le_mul_self h1, nonneg_le_nonneg_of_sq_le_sq h2⟩
lemma mul_self_lt_mul_self_iff {a b : α} (h1 : 0 ≤ a) (h2 : 0 ≤ b) : a < b ↔ a * a < b * b :=
by haveI := @linear_order.decidable_le α _; exact
((@decidable.strict_mono_on_mul_self α _ _).lt_iff_lt h1 h2).symm
lemma mul_self_inj {a b : α} (h1 : 0 ≤ a) (h2 : 0 ≤ b) : a * a = b * b ↔ a = b :=
by haveI := @linear_order.decidable_le α _; exact
(@decidable.strict_mono_on_mul_self α _ _).inj_on.eq_iff h1 h2
@[simp] lemma mul_le_mul_left_of_neg {a b c : α} (h : c < 0) : c * a ≤ c * b ↔ b ≤ a :=
by haveI := @linear_order.decidable_le α _; exact
⟨le_imp_le_of_lt_imp_lt $ λ h', mul_lt_mul_of_neg_left h' h,
λ h', decidable.mul_le_mul_of_nonpos_left h' h.le⟩
@[simp] lemma mul_le_mul_right_of_neg {a b c : α} (h : c < 0) : a * c ≤ b * c ↔ b ≤ a :=
by haveI := @linear_order.decidable_le α _; exact
⟨le_imp_le_of_lt_imp_lt $ λ h', mul_lt_mul_of_neg_right h' h,
λ h', decidable.mul_le_mul_of_nonpos_right h' h.le⟩
@[simp] lemma mul_lt_mul_left_of_neg {a b c : α} (h : c < 0) : c * a < c * b ↔ b < a :=
lt_iff_lt_of_le_iff_le (mul_le_mul_left_of_neg h)
@[simp] lemma mul_lt_mul_right_of_neg {a b c : α} (h : c < 0) : a * c < b * c ↔ b < a :=
lt_iff_lt_of_le_iff_le (mul_le_mul_right_of_neg h)
lemma sub_one_lt (a : α) : a - 1 < a :=
sub_lt_iff_lt_add.2 (lt_add_one a)
lemma mul_self_pos {a : α} (ha : a ≠ 0) : 0 < a * a :=
by rcases lt_trichotomy a 0 with h|h|h;
[exact mul_pos_of_neg_of_neg h h, exact (ha h).elim, exact mul_pos h h]
lemma mul_self_le_mul_self_of_le_of_neg_le {x y : α} (h₁ : x ≤ y) (h₂ : -x ≤ y) : x * x ≤ y * y :=
begin
haveI := @linear_order.decidable_le α _,
rw [← abs_mul_abs_self x],
exact decidable.mul_self_le_mul_self (abs_nonneg x) (abs_le.2 ⟨neg_le.2 h₂, h₁⟩)
end
lemma nonneg_of_mul_nonpos_left {a b : α} (h : a * b ≤ 0) (hb : b < 0) : 0 ≤ a :=
le_of_not_gt (λ ha, absurd h (mul_pos_of_neg_of_neg ha hb).not_le)
lemma nonneg_of_mul_nonpos_right {a b : α} (h : a * b ≤ 0) (ha : a < 0) : 0 ≤ b :=
le_of_not_gt (λ hb, absurd h (mul_pos_of_neg_of_neg ha hb).not_le)
lemma pos_of_mul_neg_left {a b : α} (h : a * b < 0) (hb : b ≤ 0) : 0 < a :=
by haveI := @linear_order.decidable_le α _; exact
lt_of_not_ge (λ ha, absurd h (decidable.mul_nonneg_of_nonpos_of_nonpos ha hb).not_lt)
lemma pos_of_mul_neg_right {a b : α} (h : a * b < 0) (ha : a ≤ 0) : 0 < b :=
by haveI := @linear_order.decidable_le α _; exact
lt_of_not_ge (λ hb, absurd h (decidable.mul_nonneg_of_nonpos_of_nonpos ha hb).not_lt)
/-- The sum of two squares is zero iff both elements are zero. -/
lemma mul_self_add_mul_self_eq_zero {x y : α} : x * x + y * y = 0 ↔ x = 0 ∧ y = 0 :=
by rw [add_eq_zero_iff', mul_self_eq_zero, mul_self_eq_zero]; apply mul_self_nonneg
lemma eq_zero_of_mul_self_add_mul_self_eq_zero (h : a * a + b * b = 0) : a = 0 :=
(mul_self_add_mul_self_eq_zero.mp h).left
lemma abs_eq_iff_mul_self_eq : |a| = |b| ↔ a * a = b * b :=
begin
rw [← abs_mul_abs_self, ← abs_mul_abs_self b],
exact (mul_self_inj (abs_nonneg a) (abs_nonneg b)).symm,
end
lemma abs_lt_iff_mul_self_lt : |a| < |b| ↔ a * a < b * b :=
begin
rw [← abs_mul_abs_self, ← abs_mul_abs_self b],
exact mul_self_lt_mul_self_iff (abs_nonneg a) (abs_nonneg b)
end
lemma abs_le_iff_mul_self_le : |a| ≤ |b| ↔ a * a ≤ b * b :=
begin
rw [← abs_mul_abs_self, ← abs_mul_abs_self b],
exact mul_self_le_mul_self_iff (abs_nonneg a) (abs_nonneg b)
end
lemma abs_le_one_iff_mul_self_le_one : |a| ≤ 1 ↔ a * a ≤ 1 :=
by simpa only [abs_one, one_mul] using @abs_le_iff_mul_self_le α _ a 1
/-- Pullback a `linear_ordered_ring` under an injective map.
See note [reducible non-instances]. -/
@[reducible]
def function.injective.linear_ordered_ring {β : Type*}
[has_zero β] [has_one β] [has_add β] [has_mul β] [has_neg β] [has_sub β] [nontrivial β]
(f : β → α) (hf : function.injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(neg : ∀ x, f (-x) = -f x) (sub : ∀ x y, f (x - y) = f x - f y) :
linear_ordered_ring β :=
{ ..linear_order.lift f hf,
..‹nontrivial β›,
..hf.ordered_ring f zero one add mul neg sub }
end linear_ordered_ring
/-- A `linear_ordered_comm_ring α` is a commutative ring `α` with a linear order
such that addition is monotone and multiplication by a positive number is strictly monotone. -/
@[protect_proj]
class linear_ordered_comm_ring (α : Type u) extends linear_ordered_ring α, comm_monoid α
@[priority 100] -- see Note [lower instance priority]
instance linear_ordered_comm_ring.to_ordered_comm_ring [d : linear_ordered_comm_ring α] :
ordered_comm_ring α :=
{ ..d }
@[priority 100] -- see Note [lower instance priority]
instance linear_ordered_comm_ring.to_integral_domain [s : linear_ordered_comm_ring α] :
integral_domain α :=
{ ..linear_ordered_ring.to_domain, ..s }
@[priority 100] -- see Note [lower instance priority]
instance linear_ordered_comm_ring.to_linear_ordered_semiring [d : linear_ordered_comm_ring α] :
linear_ordered_semiring α :=
{ .. d, ..linear_ordered_ring.to_linear_ordered_semiring }
section linear_ordered_comm_ring
variables [linear_ordered_comm_ring α] {a b c d : α}
lemma max_mul_mul_le_max_mul_max (b c : α) (ha : 0 ≤ a) (hd: 0 ≤ d) :
max (a * b) (d * c) ≤ max a c * max d b :=
by haveI := @linear_order.decidable_le α _; exact
have ba : b * a ≤ max d b * max c a, from
decidable.mul_le_mul (le_max_right d b) (le_max_right c a) ha (le_trans hd (le_max_left d b)),
have cd : c * d ≤ max a c * max b d, from
decidable.mul_le_mul (le_max_right a c) (le_max_right b d) hd (le_trans ha (le_max_left a c)),
max_le
(by simpa [mul_comm, max_comm] using ba)
(by simpa [mul_comm, max_comm] using cd)
lemma abs_sub_sq (a b : α) : |a - b| * |a - b| = a * a + b * b - (1 + 1) * a * b :=
begin
rw abs_mul_abs_self,
simp only [mul_add, add_comm, add_left_comm, mul_comm, sub_eq_add_neg,
mul_one, mul_neg_eq_neg_mul_symm, neg_add_rev, neg_neg],
end
end linear_ordered_comm_ring
section
variables [ring α] [linear_order α] {a b : α}
@[simp] lemma abs_dvd (a b : α) : |a| ∣ b ↔ a ∣ b :=
by { cases abs_choice a with h h; simp only [h, neg_dvd] }
lemma abs_dvd_self (a : α) : |a| ∣ a :=
(abs_dvd a a).mpr (dvd_refl a)
@[simp] lemma dvd_abs (a b : α) : a ∣ |b| ↔ a ∣ b :=
by { cases abs_choice b with h h; simp only [h, dvd_neg] }
lemma self_dvd_abs (a : α) : a ∣ |a| :=
(dvd_abs a a).mpr (dvd_refl a)
lemma abs_dvd_abs (a b : α) : |a| ∣ |b| ↔ a ∣ b :=
(abs_dvd _ _).trans (dvd_abs _ _)
lemma even_abs {a : α} : even (|a|) ↔ even a :=
dvd_abs _ _
lemma odd_abs {a : α} : odd (abs a) ↔ odd a :=
by { cases abs_choice a with h h; simp only [h, odd_neg] }
end
section linear_ordered_comm_ring
variables [linear_ordered_comm_ring α]
/-- Pullback a `linear_ordered_comm_ring` under an injective map.
See note [reducible non-instances]. -/
@[reducible]
def function.injective.linear_ordered_comm_ring {β : Type*}
[has_zero β] [has_one β] [has_add β] [has_mul β] [has_neg β] [has_sub β] [nontrivial β]
(f : β → α) (hf : function.injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(neg : ∀ x, f (-x) = -f x) (sub : ∀ x y, f (x - y) = f x - f y) :
linear_ordered_comm_ring β :=
{ ..linear_order.lift f hf,
..‹nontrivial β›,
..hf.ordered_comm_ring f zero one add mul neg sub }
end linear_ordered_comm_ring
/-- Extend `nonneg_add_comm_group` to support ordered rings
specified by their nonnegative elements -/
class nonneg_ring (α : Type*) extends ring α, nonneg_add_comm_group α :=
(one_nonneg : nonneg 1)
(mul_nonneg : ∀ {a b}, nonneg a → nonneg b → nonneg (a * b))
(mul_pos : ∀ {a b}, pos a → pos b → pos (a * b))
/-- Extend `nonneg_add_comm_group` to support linearly ordered rings
specified by their nonnegative elements -/
class linear_nonneg_ring (α : Type*) extends domain α, nonneg_add_comm_group α :=
(one_pos : pos 1)
(mul_nonneg : ∀ {a b}, nonneg a → nonneg b → nonneg (a * b))
(nonneg_total : ∀ a, nonneg a ∨ nonneg (-a))
[dec_nonneg : decidable_pred nonneg]
namespace nonneg_ring
open nonneg_add_comm_group
variable [nonneg_ring α]
/-- `to_linear_nonneg_ring` shows that a `nonneg_ring` with a linear order is a `domain`,
hence a `linear_nonneg_ring`. -/
def to_linear_nonneg_ring [nontrivial α] [decidable_pred (@nonneg α _)]
(nonneg_total : ∀ a : α, nonneg a ∨ nonneg (-a))
: linear_nonneg_ring α :=
{ one_pos := (pos_iff 1).mpr ⟨one_nonneg, λ h, zero_ne_one (nonneg_antisymm one_nonneg h).symm⟩,
nonneg_total := nonneg_total,
eq_zero_or_eq_zero_of_mul_eq_zero :=
suffices ∀ {a} b : α, nonneg a → a * b = 0 → a = 0 ∨ b = 0,
from λ a b, (nonneg_total a).elim (this b)
(λ na, by simpa using this b na),
suffices ∀ {a b : α}, nonneg a → nonneg b → a * b = 0 → a = 0 ∨ b = 0,
from λ a b na, (nonneg_total b).elim (this na)
(λ nb, by simpa using this na nb),
λ a b na nb z, decidable.by_cases
(λ nna : nonneg (-a), or.inl (nonneg_antisymm na nna))
(λ pa, decidable.by_cases
(λ nnb : nonneg (-b), or.inr (nonneg_antisymm nb nnb))
(λ pb, absurd z $ ne_of_gt $ pos_def.1 $ mul_pos
((pos_iff _).2 ⟨na, pa⟩)
((pos_iff _).2 ⟨nb, pb⟩))),
..‹nontrivial α›,
..‹nonneg_ring α› }
end nonneg_ring
namespace linear_nonneg_ring
open nonneg_add_comm_group
variable [linear_nonneg_ring α]
@[priority 100] -- see Note [lower instance priority]
instance to_nonneg_ring : nonneg_ring α :=
{ one_nonneg := ((pos_iff _).mp one_pos).1,
mul_pos := λ a b pa pb,
let ⟨a1, a2⟩ := (pos_iff a).1 pa,
⟨b1, b2⟩ := (pos_iff b).1 pb in
have ab : nonneg (a * b), from mul_nonneg a1 b1,
(pos_iff _).2 ⟨ab, λ hn,
have a * b = 0, from nonneg_antisymm ab hn,
(eq_zero_or_eq_zero_of_mul_eq_zero _ _ this).elim
(ne_of_gt (pos_def.1 pa))
(ne_of_gt (pos_def.1 pb))⟩,
..‹linear_nonneg_ring α› }
/-- Construct `linear_order` from `linear_nonneg_ring`. This is not an instance
because we don't use it in `mathlib`. -/
local attribute [instance]
def to_linear_order [decidable_pred (nonneg : α → Prop)] : linear_order α :=
{ le_total := nonneg_total_iff.1 nonneg_total,
decidable_le := by apply_instance,
decidable_lt := by apply_instance,
..‹linear_nonneg_ring α›, ..(infer_instance : ordered_add_comm_group α) }
/-- Construct `linear_ordered_ring` from `linear_nonneg_ring`.
This is not an instance because we don't use it in `mathlib`. -/
local attribute [instance]
def to_linear_ordered_ring [decidable_pred (nonneg : α → Prop)] : linear_ordered_ring α :=
{ mul_pos := by simp [pos_def.symm]; exact @nonneg_ring.mul_pos _ _,
zero_le_one := le_of_lt $ lt_of_not_ge $ λ (h : nonneg (0 - 1)), begin
rw [zero_sub] at h,
have := mul_nonneg h h, simp at this,
exact zero_ne_one (nonneg_antisymm this h).symm
end,
..‹linear_nonneg_ring α›, ..(infer_instance : ordered_add_comm_group α),
..(infer_instance : linear_order α) }
/-- Convert a `linear_nonneg_ring` with a commutative multiplication and
decidable non-negativity into a `linear_ordered_comm_ring` -/
def to_linear_ordered_comm_ring
[decidable_pred (@nonneg α _)]
[comm : @is_commutative α (*)]
: linear_ordered_comm_ring α :=
{ mul_comm := is_commutative.comm,
..@linear_nonneg_ring.to_linear_ordered_ring _ _ _ }
end linear_nonneg_ring
/-- A canonically ordered commutative semiring is an ordered, commutative semiring
in which `a ≤ b` iff there exists `c` with `b = a + c`. This is satisfied by the
natural numbers, for example, but not the integers or other ordered groups. -/
@[protect_proj]
class canonically_ordered_comm_semiring (α : Type*) extends
canonically_ordered_add_monoid α, comm_semiring α :=
(eq_zero_or_eq_zero_of_mul_eq_zero : ∀ a b : α, a * b = 0 → a = 0 ∨ b = 0)
namespace canonically_ordered_comm_semiring
variables [canonically_ordered_comm_semiring α] {a b : α}
@[priority 100] -- see Note [lower instance priority]
instance to_no_zero_divisors : no_zero_divisors α :=
⟨canonically_ordered_comm_semiring.eq_zero_or_eq_zero_of_mul_eq_zero⟩
@[priority 100] -- see Note [lower instance priority]
instance to_covariant_mul_le : covariant_class α α (*) (≤) :=
begin
refine ⟨λ a b c h, _⟩,
rcases le_iff_exists_add.1 h with ⟨c, rfl⟩,
rw mul_add,
apply self_le_add_right
end
/-- A version of `zero_lt_one : 0 < 1` for a `canonically_ordered_comm_semiring`. -/
lemma zero_lt_one [nontrivial α] : (0:α) < 1 := (zero_le 1).lt_of_ne zero_ne_one
@[simp] lemma mul_pos : 0 < a * b ↔ (0 < a) ∧ (0 < b) :=
by simp only [pos_iff_ne_zero, ne.def, mul_eq_zero, not_or_distrib]
variables [has_sub α] [has_ordered_sub α] [contravariant_class α α (+) (≤)] [is_total α (≤)]
lemma _root_.mul_sub' (a b c : α) : a * (b - c) = a * b - a * c :=
begin
cases total_of (≤) b c with hbc hcb,
{ rw [sub_eq_zero_iff_le.2 hbc, mul_zero, sub_eq_zero_iff_le.2 (mul_le_mul_left' hbc a)] },
{ apply eq_sub_of_add_eq'',
rw [← mul_add, sub_add_cancel_of_le hcb] }
end
lemma _root_.sub_mul' (a b c : α) : (a - b) * c = a * c - b * c :=
by simp only [← mul_comm c, mul_sub']
end canonically_ordered_comm_semiring
/-! ### Structures involving `*` and `0` on `with_top` and `with_bot`
The main results of this section are `with_top.canonically_ordered_comm_semiring` and
`with_bot.comm_monoid_with_zero`.
-/
namespace with_top
instance [nonempty α] : nontrivial (with_top α) :=
option.nontrivial
variable [decidable_eq α]
section has_mul
variables [has_zero α] [has_mul α]
instance : mul_zero_class (with_top α) :=
{ zero := 0,
mul := λm n, if m = 0 ∨ n = 0 then 0 else m.bind (λa, n.bind $ λb, ↑(a * b)),
zero_mul := assume a, if_pos $ or.inl rfl,
mul_zero := assume a, if_pos $ or.inr rfl }
lemma mul_def {a b : with_top α} :
a * b = if a = 0 ∨ b = 0 then 0 else a.bind (λa, b.bind $ λb, ↑(a * b)) := rfl
@[simp] lemma mul_top {a : with_top α} (h : a ≠ 0) : a * ⊤ = ⊤ :=
by cases a; simp [mul_def, h]; refl
@[simp] lemma top_mul {a : with_top α} (h : a ≠ 0) : ⊤ * a = ⊤ :=
by cases a; simp [mul_def, h]; refl
@[simp] lemma top_mul_top : (⊤ * ⊤ : with_top α) = ⊤ :=
top_mul top_ne_zero
end has_mul
section mul_zero_class
variables [mul_zero_class α]
@[norm_cast] lemma coe_mul {a b : α} : (↑(a * b) : with_top α) = a * b :=
decidable.by_cases (assume : a = 0, by simp [this]) $ assume ha,
decidable.by_cases (assume : b = 0, by simp [this]) $ assume hb,
by { simp [*, mul_def], refl }
lemma mul_coe {b : α} (hb : b ≠ 0) : ∀{a : with_top α}, a * b = a.bind (λa:α, ↑(a * b))
| none := show (if (⊤:with_top α) = 0 ∨ (b:with_top α) = 0 then 0 else ⊤ : with_top α) = ⊤,
by simp [hb]
| (some a) := show ↑a * ↑b = ↑(a * b), from coe_mul.symm
@[simp] lemma mul_eq_top_iff {a b : with_top α} : a * b = ⊤ ↔ (a ≠ 0 ∧ b = ⊤) ∨ (a = ⊤ ∧ b ≠ 0) :=
begin
cases a; cases b; simp only [none_eq_top, some_eq_coe],
{ simp [← coe_mul] },
{ suffices : ⊤ * (b : with_top α) = ⊤ ↔ b ≠ 0, by simpa,
by_cases hb : b = 0; simp [hb] },
{ suffices : (a : with_top α) * ⊤ = ⊤ ↔ a ≠ 0, by simpa,
by_cases ha : a = 0; simp [ha] },
{ simp [← coe_mul] }
end
lemma mul_lt_top [partial_order α] {a b : with_top α} (ha : a ≠ ⊤) (hb : b ≠ ⊤) : a * b < ⊤ :=
begin
lift a to α using ha,
lift b to α using hb,
simp only [← coe_mul, coe_lt_top]
end
end mul_zero_class
/-- `nontrivial α` is needed here as otherwise we have `1 * ⊤ = ⊤` but also `= 0 * ⊤ = 0`. -/
instance [mul_zero_one_class α] [nontrivial α] : mul_zero_one_class (with_top α) :=
{ mul := (*),
one := 1,
zero := 0,
one_mul := λ a, match a with
| none := show ((1:α) : with_top α) * ⊤ = ⊤, by simp [-with_top.coe_one]
| (some a) := show ((1:α) : with_top α) * a = a, by simp [coe_mul.symm, -with_top.coe_one]
end,
mul_one := λ a, match a with
| none := show ⊤ * ((1:α) : with_top α) = ⊤, by simp [-with_top.coe_one]
| (some a) := show ↑a * ((1:α) : with_top α) = a, by simp [coe_mul.symm, -with_top.coe_one]
end,
.. with_top.mul_zero_class }
instance [mul_zero_class α] [no_zero_divisors α] : no_zero_divisors (with_top α) :=
⟨λ a b, by cases a; cases b; dsimp [mul_def]; split_ifs;
simp [*, none_eq_top, some_eq_coe, mul_eq_zero] at *⟩
instance [semigroup_with_zero α] [no_zero_divisors α] : semigroup_with_zero (with_top α) :=
{ mul := (*),
zero := 0,
mul_assoc := λ a b c, begin
cases a,
{ by_cases hb : b = 0; by_cases hc : c = 0;
simp [*, none_eq_top] },
cases b,
{ by_cases ha : a = 0; by_cases hc : c = 0;
simp [*, none_eq_top, some_eq_coe] },
cases c,
{ by_cases ha : a = 0; by_cases hb : b = 0;
simp [*, none_eq_top, some_eq_coe] },
simp [some_eq_coe, coe_mul.symm, mul_assoc]
end,
.. with_top.mul_zero_class }
instance [monoid_with_zero α] [no_zero_divisors α] [nontrivial α] : monoid_with_zero (with_top α) :=
{ .. with_top.mul_zero_one_class, .. with_top.semigroup_with_zero }
instance [comm_monoid_with_zero α] [no_zero_divisors α] [nontrivial α] :
comm_monoid_with_zero (with_top α) :=
{ mul := (*),
zero := 0,
mul_comm := λ a b, begin
by_cases ha : a = 0, { simp [ha] },
by_cases hb : b = 0, { simp [hb] },
simp [ha, hb, mul_def, option.bind_comm a b, mul_comm]
end,
.. with_top.monoid_with_zero }
variables [canonically_ordered_comm_semiring α]
private lemma distrib' (a b c : with_top α) : (a + b) * c = a * c + b * c :=
begin
cases c,
{ show (a + b) * ⊤ = a * ⊤ + b * ⊤,
by_cases ha : a = 0; simp [ha] },
{ show (a + b) * c = a * c + b * c,
by_cases hc : c = 0, { simp [hc] },
simp [mul_coe hc], cases a; cases b,
repeat { refl <|> exact congr_arg some (add_mul _ _ _) } }
end
/-- This instance requires `canonically_ordered_comm_semiring` as it is the smallest class
that derives from both `non_assoc_non_unital_semiring` and `canonically_ordered_add_monoid`, both
of which are required for distributivity. -/
instance [nontrivial α] : comm_semiring (with_top α) :=
{ right_distrib := distrib',
left_distrib := assume a b c, by rw [mul_comm, distrib', mul_comm b, mul_comm c]; refl,
.. with_top.add_comm_monoid, .. with_top.comm_monoid_with_zero,}
instance [nontrivial α] : canonically_ordered_comm_semiring (with_top α) :=
{ .. with_top.comm_semiring,
.. with_top.canonically_ordered_add_monoid,
.. with_top.no_zero_divisors, }
end with_top
namespace with_bot
instance [nonempty α] : nontrivial (with_bot α) :=
option.nontrivial
variable [decidable_eq α]
section has_mul
variables [has_zero α] [has_mul α]
instance : mul_zero_class (with_bot α) :=
with_top.mul_zero_class
lemma mul_def {a b : with_bot α} :
a * b = if a = 0 ∨ b = 0 then 0 else a.bind (λa, b.bind $ λb, ↑(a * b)) := rfl
@[simp] lemma mul_bot {a : with_bot α} (h : a ≠ 0) : a * ⊥ = ⊥ :=
with_top.mul_top h
@[simp] lemma bot_mul {a : with_bot α} (h : a ≠ 0) : ⊥ * a = ⊥ :=
with_top.top_mul h
@[simp] lemma bot_mul_bot : (⊥ * ⊥ : with_bot α) = ⊥ :=
with_top.top_mul_top
end has_mul
section mul_zero_class
variables [mul_zero_class α]
@[norm_cast] lemma coe_mul {a b : α} : (↑(a * b) : with_bot α) = a * b :=
decidable.by_cases (assume : a = 0, by simp [this]) $ assume ha,
decidable.by_cases (assume : b = 0, by simp [this]) $ assume hb,
by { simp [*, mul_def], refl }
lemma mul_coe {b : α} (hb : b ≠ 0) {a : with_bot α} : a * b = a.bind (λa:α, ↑(a * b)) :=
with_top.mul_coe hb
@[simp] lemma mul_eq_bot_iff {a b : with_bot α} : a * b = ⊥ ↔ (a ≠ 0 ∧ b = ⊥) ∨ (a = ⊥ ∧ b ≠ 0) :=
with_top.mul_eq_top_iff
lemma bot_lt_mul [partial_order α] {a b : with_bot α} (ha : ⊥ < a) (hb : ⊥ < b) : ⊥ < a * b :=
begin
lift a to α using ne_bot_of_gt ha,
lift b to α using ne_bot_of_gt hb,
simp only [← coe_mul, bot_lt_coe],
end
end mul_zero_class
/-- `nontrivial α` is needed here as otherwise we have `1 * ⊥ = ⊥` but also `= 0 * ⊥ = 0`. -/
instance [mul_zero_one_class α] [nontrivial α] : mul_zero_one_class (with_bot α) :=
with_top.mul_zero_one_class
instance [mul_zero_class α] [no_zero_divisors α] : no_zero_divisors (with_bot α) :=
with_top.no_zero_divisors
instance [semigroup_with_zero α] [no_zero_divisors α] : semigroup_with_zero (with_bot α) :=
with_top.semigroup_with_zero
instance [monoid_with_zero α] [no_zero_divisors α] [nontrivial α] : monoid_with_zero (with_bot α) :=
with_top.monoid_with_zero
instance [comm_monoid_with_zero α] [no_zero_divisors α] [nontrivial α] :
comm_monoid_with_zero (with_bot α) :=
with_top.comm_monoid_with_zero
instance [canonically_ordered_comm_semiring α] [nontrivial α] : comm_semiring (with_bot α) :=
with_top.comm_semiring
end with_bot
|
040850485a0ec07caa91f2c8207172874de61f9c | acc85b4be2c618b11fc7cb3005521ae6858a8d07 | /data/finset/basic.lean | 0b7c23172100571daecd5a62108425d23ae822c8 | [
"Apache-2.0"
] | permissive | linpingchuan/mathlib | d49990b236574df2a45d9919ba43c923f693d341 | 5ad8020f67eb13896a41cc7691d072c9331b1f76 | refs/heads/master | 1,626,019,377,808 | 1,508,048,784,000 | 1,508,048,784,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 28,120 | lean | /-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
Finite sets.
-/
import data.list.basic data.list.perm order.boolean_algebra algebra.functions
open list subtype nat lattice
variables {α : Type*} {β : Type*} {γ : Type*}
def nodup_list (α : Type*) := {l : list α // nodup l}
def to_nodup_list_of_nodup {l : list α} (n : nodup l) : nodup_list α :=
⟨l, n⟩
def to_nodup_list [decidable_eq α] (l : list α) : nodup_list α :=
⟨erase_dup l, nodup_erase_dup l⟩
private def eqv (l₁ l₂ : nodup_list α) :=
perm l₁.1 l₂.1
local infix ~ := eqv
private def eqv.refl (l : nodup_list α) : l ~ l :=
perm.refl _
private def eqv.symm {l₁ l₂ : nodup_list α} : l₁ ~ l₂ → l₂ ~ l₁ :=
perm.symm
private def eqv.trans {l₁ l₂ l₃ : nodup_list α} : l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ :=
perm.trans
instance finset.nodup_list_setoid (α : Type*) : setoid (nodup_list α) :=
setoid.mk (@eqv α) (mk_equivalence (@eqv α) (@eqv.refl α) (@eqv.symm α) (@eqv.trans α))
def {u} finset (α : Type u) : Type u :=
quotient (finset.nodup_list_setoid α)
namespace finset
def to_finset_of_nodup (l : list α) (n : nodup l) : finset α :=
⟦to_nodup_list_of_nodup n⟧
@[elab_as_eliminator]
protected theorem induction_on_to_finset {α : Type*} {p : finset α → Prop} (s : finset α)
(h : ∀ (l : list α) (h : nodup l), p (to_finset_of_nodup l h)) : p s :=
quot.induction_on s $ assume ⟨l, hl⟩, h l hl
def to_finset [decidable_eq α] (l : list α) : finset α :=
⟦to_nodup_list l⟧
lemma to_finset_eq_of_nodup [decidable_eq α] {l : list α} (n : nodup l) :
to_finset_of_nodup l n = to_finset l :=
quotient.sound $ show l.perm (erase_dup l), by rw erase_dup_eq_self.2 n
instance has_decidable_eq [decidable_eq α] : decidable_eq (finset α)
| s₁ s₂ := quotient.rec_on_subsingleton₂ s₁ s₂ $ λ l₁ l₂,
decidable_of_iff (l₁.1.perm l₂.1) ⟨λ e, quot.sound e, quotient.exact⟩
section mem
def mem (a : α) (s : finset α) : Prop :=
quot.lift_on s (λ l, a ∈ l.1) (λ l₁ l₂ (e : l₁ ~ l₂), propext $ mem_iff_mem_of_perm e)
instance : has_mem α (finset α) := ⟨mem⟩
@[simp] lemma mem_to_finset_of_nodup_eq {a : α} {l : list α} (n : nodup l) :
(a ∈ to_finset_of_nodup l n) = (a ∈ l) :=
rfl
theorem mem_of_mem_list {a : α} {l : nodup_list α} (h : a ∈ l.1) : a ∈ @id (finset α) ⟦l⟧ := h
theorem mem_list_of_mem {a : α} {l : nodup_list α} (h : a ∈ @id (finset α) ⟦l⟧) : a ∈ l.1 := h
theorem mem_to_finset_of_mem [decidable_eq α] {a : α} {l : list α} : a ∈ l → a ∈ to_finset l :=
mem_erase_dup.2
@[simp] theorem mem_to_finset [decidable_eq α] {a : α} {l : list α} : a ∈ to_finset l ↔ a ∈ l :=
mem_erase_dup
instance decidable_mem [h : decidable_eq α] (a : α) (s : finset α) : decidable (a ∈ s) :=
quot.rec_on_subsingleton s $ λ l, list.decidable_mem a l.1
/- extensionality -/
theorem ext {s₁ s₂ : finset α} : (∀ a, a ∈ s₁ ↔ a ∈ s₂) → s₁ = s₂ :=
quotient.induction_on₂ s₁ s₂ $ λ l₁ l₂ e, quot.sound $ (perm_ext l₁.2 l₂.2).2 e
end mem
/- subset -/
section subset
protected def subset (s₁ s₂ : finset α) : Prop :=
quotient.lift_on₂ s₁ s₂ (λ l₁ l₂, l₁.1 ⊆ l₂.1)
(λ v₁ v₂ w₁ w₂ p₁ p₂, propext (iff.intro
(λ s₁ a i, mem_of_perm p₂ (s₁ (mem_of_perm p₁.symm i)))
(λ s₂ a i, mem_of_perm p₂.symm (s₂ (mem_of_perm p₁ i)))))
instance : has_subset (finset α) := ⟨finset.subset⟩
-- theorem subset_univ [h : fintype α] (s : finset α) : s ⊆ univ :=
-- quot.induction_on s (λ l a i, fintype.complete a)
@[simp] theorem subset.refl (s : finset α) : s ⊆ s :=
quot.induction_on s (λ l, by refl)
theorem subset.trans {s₁ s₂ s₃ : finset α} : s₁ ⊆ s₂ → s₂ ⊆ s₃ → s₁ ⊆ s₃ :=
quotient.induction_on₃ s₁ s₂ s₃ (λ l₁ l₂ l₃, list.subset.trans)
theorem mem_of_subset_of_mem {s₁ s₂ : finset α} {a : α} : s₁ ⊆ s₂ → a ∈ s₁ → a ∈ s₂ :=
quotient.induction_on₂ s₁ s₂ (λ l₁ l₂ h₁ h₂, h₁ h₂)
theorem subset.antisymm {s₁ s₂ : finset α} (H₁ : s₁ ⊆ s₂) (H₂ : s₂ ⊆ s₁) : s₁ = s₂ :=
ext (λ x, ⟨mem_of_subset_of_mem H₁, mem_of_subset_of_mem H₂⟩)
theorem subset_iff {s₁ s₂ : finset α} : s₁ ⊆ s₂ ↔ (∀x, x ∈ s₁ → x ∈ s₂) :=
⟨λ s a, mem_of_subset_of_mem s, quotient.induction_on₂ s₁ s₂ (λ l₁ l₂ H, H)⟩
end subset
section empty
variables {s : finset α} {a b : α}
/- empty -/
protected def empty : finset α :=
to_finset_of_nodup [] nodup_nil
instance : has_emptyc (finset α) := ⟨finset.empty⟩
instance : inhabited (finset α) := ⟨∅⟩
@[simp] theorem not_mem_empty : a ∉ (∅ : finset α) := id
theorem empty_subset : ∅ ⊆ s := quot.induction_on s (λ l, list.nil_subset l.1)
theorem eq_empty_of_forall_not_mem (H : ∀x, x ∉ s) : s = ∅ := ext (λ x, iff_false_intro (H x))
theorem eq_empty_of_subset_empty (h : s ⊆ ∅) : s = ∅ := subset.antisymm h empty_subset
theorem subset_empty_iff (x : finset α) : x ⊆ ∅ ↔ x = ∅ :=
iff.intro eq_empty_of_subset_empty (λ xeq, by rw xeq; apply subset.refl ∅)
theorem exists_mem_of_ne_empty : s ≠ ∅ → ∃ a : α, a ∈ s :=
finset.induction_on_to_finset s $ assume l hl,
match l, hl with
| [] := assume _ h, false.elim $ h rfl
| (a :: l) := assume _ _, ⟨a, by simp⟩
end
end empty
-- /- universe -/
-- def univ [h : fintype A] : finset A :=
-- to_finset_of_nodup (@fintype.elems A h) (@fintype.unique A h)
-- theorem mem_univ [fintype A] (x : A) : x ∈ univ :=
-- fintype.complete x
-- theorem mem_univ_eq [fintype A] (x : A) : x ∈ univ = true := propext (iff_true_intro !mem_univ)
/- insert -/
section insert
variables [decidable_eq α] {s t : finset α} {a b : α}
protected def insert (a : α) (s : finset α) : finset α :=
quot.lift_on s
(λ l, to_finset_of_nodup (insert a l.1) (nodup_insert l.2))
(λ (l₁ l₂ : nodup_list α) (p : l₁ ~ l₂), quot.sound (perm_insert a p))
instance : has_insert α (finset α) := ⟨finset.insert⟩
@[simp] theorem mem_insert : a ∈ insert b s ↔ (a = b ∨ a ∈ s) :=
finset.induction_on_to_finset s $ assume l hl, show a ∈ insert b l ↔ (a = b ∨ a ∈ l), by simp
theorem mem_insert_self : a ∈ insert a s := by simp
theorem mem_insert_of_mem : a ∈ s → a ∈ insert b s := by simp {contextual := tt}
theorem mem_of_mem_insert_of_ne (h : b ∈ insert a s) : b ≠ a → b ∈ s :=
(mem_insert.1 h).resolve_left
@[simp] theorem insert_eq_of_mem (h : a ∈ s) : insert a s = s :=
ext (λ x, by rw mem_insert; apply or_iff_right_of_imp; intro eq; rw eq; assumption)
@[simp] theorem insert.comm : insert a (insert b s) = insert b (insert a s) :=
ext $ by simp
@[simp] theorem insert_idem : insert a (insert a s) = insert a s :=
ext $ by simp [mem_insert]
@[simp] theorem insert_ne_empty : insert a s ≠ ∅ :=
assume h, @not_mem_empty α a $ h ▸ by simp
theorem subset_insert [h : decidable_eq α] : s ⊆ insert a s :=
subset_iff.2 (λ x h, mem_insert_of_mem h)
theorem insert_subset_insert (h : s ⊆ t) : insert a s ⊆ insert a t :=
subset_iff.2 $ assume x, by simp [-forall_or_distrib_left]; exact or.imp_right (subset_iff.1 h _)
@[recursor 6] protected theorem induction {p : finset α → Prop}
(h₁ : p ∅) (h₂ : ∀ ⦃a : α⦄ {s : finset α}, a ∉ s → p s → p (insert a s)) (s) : p s :=
finset.induction_on_to_finset s $ λl, list.rec_on l
(assume _, h₁)
(assume a l ih hal,
let l' := to_finset_of_nodup l $ nodup_of_nodup_cons hal in
have insert a l' = to_finset_of_nodup (a :: l) hal,
from ext $ by simp,
this ▸ @h₂ a l' (not_mem_of_nodup_cons hal) (ih _))
protected theorem induction_on {p : finset α → Prop} (s : finset α)
(h₁ : p ∅) (h₂ : ∀ ⦃a : α⦄ {s : finset α}, a ∉ s → p s → p (insert a s)) : p s :=
finset.induction h₁ h₂ s
-- useful in proofs by induction
theorem forall_of_forall_insert {p : α → Prop} (H : ∀ x, x ∈ insert a s → p x) :
∀ x, x ∈ s → p x :=
λ x xs, H x (mem_insert_of_mem xs)
end insert
section singleton
variables [decidable_eq α] {a b : α} {s : finset α}
@[simp] theorem mem_singleton : b ∈ ({a} : finset α) ↔ (b = a) :=
show b ∈ insert a ∅ ↔ b = a, by simp
theorem mem_singleton_self : a ∈ ({a} : finset α) := mem_insert_self
theorem singleton_inj (h : {a} = ({b}:finset α)) : a = b :=
have a ∈ ({b} : finset α), by rw ← h; apply mem_singleton_self,
mem_singleton.1 this
@[simp] theorem singleton_ne_empty : ({a} : finset α) ≠ ∅ := insert_ne_empty
@[simp] theorem insert_singelton_self_eq : ({a, a} : finset α) = {a} :=
show insert a {a} = ({a} : finset α), by rw [insert_eq_of_mem]; apply mem_singleton_self
end singleton
/-
section sep
variables [decidable_eq α]
instance : has_sep α (finset α) :=
⟨λp s, quotient.lift_on s (λl, to_finset_of_nodup (l.val.filter p) _) _⟩
end sep
-/
/- union -/
section union
variable [decidable_eq α]
protected def union (s₁ s₂ : finset α) : finset α :=
quotient.lift_on₂ s₁ s₂
(λ l₁ l₂,
to_finset_of_nodup (list.union l₁.1 l₂.1)
(nodup_union l₁.1 l₂.2))
(λ v₁ v₂ w₁ w₂ p₁ p₂, quot.sound (perm_union p₁ p₂))
instance : has_union (finset α) := ⟨finset.union⟩
@[simp] theorem mem_union {a : α} {s₁ s₂ : finset α} : a ∈ s₁ ∪ s₂ ↔ a ∈ s₁ ∨ a ∈ s₂ :=
quotient.induction_on₂ s₁ s₂ (λ l₁ l₂, list.mem_union)
theorem mem_union_left {a : α} {s₁ : finset α} (s₂ : finset α) : a ∈ s₁ → a ∈ s₁ ∪ s₂ :=
by simp {contextual := tt}
theorem mem_union_right {a : α} {s₂ : finset α} (s₁ : finset α) : a ∈ s₂ → a ∈ s₁ ∪ s₂ :=
by simp {contextual := tt}
theorem mem_or_mem_of_mem_union {a : α} {s₁ s₂ : finset α} : a ∈ s₁ ∪ s₂ → a ∈ s₁ ∨ a ∈ s₂ :=
mem_union.1
theorem union_subset {s₁ s₂ s₃ : finset α} : s₁ ⊆ s₃ → s₂ ⊆ s₃ → s₁ ∪ s₂ ⊆ s₃ :=
by simp [subset_iff] {contextual:=tt}; finish
theorem subset_union_left {s₁ s₂ : finset α} : s₁ ⊆ s₁ ∪ s₂ :=
subset_iff.mpr $ assume x, mem_union_left _
theorem subset_union_right {s₁ s₂ : finset α} : s₂ ⊆ s₁ ∪ s₂ :=
subset_iff.mpr $ assume x, mem_union_right _
@[simp] theorem union_comm (s₁ s₂ : finset α) : s₁ ∪ s₂ = s₂ ∪ s₁ :=
ext $ by simp
instance : is_commutative (finset α) (∪) := ⟨union_comm⟩
@[simp] theorem union_assoc (s₁ s₂ s₃ : finset α) : (s₁ ∪ s₂) ∪ s₃ = s₁ ∪ (s₂ ∪ s₃) :=
ext $ by simp
instance : is_associative (finset α) (∪) := ⟨union_assoc⟩
@[simp] theorem union_idempotent (s : finset α) : (s ∪ s) = s :=
ext $ by simp
instance : is_idempotent (finset α) (∪) := ⟨union_idempotent⟩
theorem union_left_comm (s₁ s₂ s₃ : finset α) : s₁ ∪ (s₂ ∪ s₃) = s₂ ∪ (s₁ ∪ s₃) :=
ext $ by simp
theorem union_right_comm (s₁ s₂ s₃ : finset α) : (s₁ ∪ s₂) ∪ s₃ = (s₁ ∪ s₃) ∪ s₂ :=
ext $ by simp
@[simp] theorem union_self (s : finset α) : s ∪ s = s :=
ext $ by simp
@[simp] theorem union_empty (s : finset α) : s ∪ ∅ = s :=
ext $ by simp
@[simp] theorem empty_union (s : finset α) : ∅ ∪ s = s :=
ext $ by simp
theorem insert_eq (a : α) (s : finset α) : insert a s = {a} ∪ s :=
ext $ by simp
@[simp] theorem insert_union (a : α) (s t : finset α) : insert a s ∪ t = insert a (s ∪ t) :=
ext $ by simp
@[simp] theorem union_insert (a : α) (s t : finset α) : s ∪ insert a t = insert a (s ∪ t) :=
ext $ by simp
end union
/- inter -/
section inter
variable [decidable_eq α]
protected def inter (s₁ s₂ : finset α) : finset α :=
quotient.lift_on₂ s₁ s₂
(λ l₁ l₂, to_finset_of_nodup (list.inter l₁.1 l₂.1) (nodup_inter_of_nodup _ l₁.2))
(λ v₁ v₂ w₁ w₂ p₁ p₂, quot.sound (perm_inter p₁ p₂))
instance : has_inter (finset α) := ⟨finset.inter⟩
@[simp] theorem mem_inter (a : α) (s₁ s₂ : finset α) : a ∈ s₁ ∩ s₂ ↔ a ∈ s₁ ∧ a ∈ s₂ :=
quotient.induction_on₂ s₁ s₂ (λ l₁ l₂, mem_inter)
theorem mem_of_mem_inter_left {a : α} {s₁ s₂ : finset α} : a ∈ s₁ ∩ s₂ → a ∈ s₁ :=
by simp {contextual := tt}
theorem mem_of_mem_inter_right {a : α} {s₁ s₂ : finset α} : a ∈ s₁ ∩ s₂ → a ∈ s₂ :=
by simp {contextual := tt}
theorem mem_inter_of_mem {a : α} {s₁ s₂ : finset α} : a ∈ s₁ → a ∈ s₂ → a ∈ s₁ ∩ s₂ :=
by simp {contextual := tt}
theorem inter_subset_left {s₁ s₂ : finset α} : s₁ ∩ s₂ ⊆ s₁ :=
by simp [subset_iff] {contextual:=tt}; finish
theorem inter_subset_right {s₁ s₂ : finset α} : s₁ ∩ s₂ ⊆ s₂ :=
by simp [subset_iff] {contextual:=tt}; finish
theorem subset_inter {s₁ s₂ s₃ : finset α} : s₁ ⊆ s₂ → s₁ ⊆ s₃ → s₁ ⊆ s₂ ∩ s₃ :=
by simp [subset_iff] {contextual:=tt}; finish
@[simp] theorem inter_comm (s₁ s₂ : finset α) : s₁ ∩ s₂ = s₂ ∩ s₁ :=
ext $ by simp
@[simp] theorem inter_assoc (s₁ s₂ s₃ : finset α) : (s₁ ∩ s₂) ∩ s₃ = s₁ ∩ (s₂ ∩ s₃) :=
ext $ by simp
@[simp] theorem inter_left_comm (s₁ s₂ s₃ : finset α) : s₁ ∩ (s₂ ∩ s₃) = s₂ ∩ (s₁ ∩ s₃) :=
ext $ by simp
@[simp] theorem inter_right_comm (s₁ s₂ s₃ : finset α) : (s₁ ∩ s₂) ∩ s₃ = (s₁ ∩ s₃) ∩ s₂ :=
ext $ by simp
@[simp] theorem inter_self (s : finset α) : s ∩ s = s :=
ext $ by simp
@[simp] theorem inter_empty (s : finset α) : s ∩ ∅ = ∅ :=
ext $ by simp
@[simp] theorem empty_inter (s : finset α) : ∅ ∩ s = ∅ :=
ext $ by simp
@[simp] theorem insert_inter_of_mem {s₁ s₂ : finset α} {a : α} (h : a ∈ s₂) :
insert a s₁ ∩ s₂ = insert a (s₁ ∩ s₂) :=
ext $ by simp; intro x; constructor; finish
@[simp] theorem inter_insert_of_mem {s₁ s₂ : finset α} {a : α} (h : a ∈ s₁) :
s₁ ∩ insert a s₂ = insert a (s₁ ∩ s₂) :=
by rw [inter_comm, insert_inter_of_mem h, inter_comm]
@[simp] theorem insert_inter_of_not_mem {s₁ s₂ : finset α} {a : α} (h : a ∉ s₂) :
insert a s₁ ∩ s₂ = s₁ ∩ s₂ :=
ext $ assume a', by by_cases a' = a with h'; simp [mem_inter, mem_insert, h, h']
@[simp] theorem inter_insert_of_not_mem {s₁ s₂ : finset α} {a : α} (h : a ∉ s₁) :
s₁ ∩ insert a s₂ = s₁ ∩ s₂ :=
by rw [inter_comm, insert_inter_of_not_mem h, inter_comm]
@[simp] theorem singleton_inter_of_mem {a : α} {s : finset α} : a ∈ s → {a} ∩ s = {a} :=
show a ∈ s → insert a ∅ ∩ s = insert a ∅, by simp {contextual := tt}
@[simp] theorem singleton_inter_of_not_mem {a : α} {s : finset α} : a ∉ s → {a} ∩ s = ∅ :=
show a ∉ s → insert a ∅ ∩ s = ∅, by simp {contextual := tt}
@[simp] theorem inter_singleton_of_mem {a : α} {s : finset α} (h : a ∈ s) : s ∩ {a} = {a} :=
by rw [inter_comm, singleton_inter_of_mem h]
@[simp] theorem inter_singleton_of_not_mem {a : α} {s : finset α} (h : a ∉ s) : s ∩ {a} = ∅ :=
by rw [inter_comm, singleton_inter_of_not_mem h]
end inter
section lattice
instance [decidable_eq α] : lattice (finset α) :=
{ le := (⊆),
le_refl := subset.refl,
le_trans := assume a b c, subset.trans,
le_antisymm := assume a b, subset.antisymm,
sup := (∪),
sup_le := assume a b c, union_subset,
le_sup_left := assume a b, subset_union_left,
le_sup_right := assume a b, subset_union_right,
inf := (∩),
le_inf := assume a b c, subset_inter,
inf_le_left := assume a b, inter_subset_left,
inf_le_right := assume a b, inter_subset_right }
instance [decidable_eq α] : semilattice_inf_bot (finset α) :=
{ finset.lattice.lattice with
bot := ∅,
bot_le := assume a, empty_subset }
instance [decidable_eq α] : distrib_lattice (finset α) :=
{ finset.lattice.lattice with
le_sup_inf := assume a b c, show (a ∪ b) ∩ (a ∪ c) ⊆ a ∪ b ∩ c,
by simp [subset_iff, and_imp, or_imp_distrib] {contextual:=tt} }
end lattice
/- distributivity laws -/
section inter
variable [decidable_eq α]
theorem inter_distrib_left (s t u : finset α) : s ∩ (t ∪ u) = (s ∩ t) ∪ (s ∩ u) :=
ext $ by simp [mem_inter, mem_union]; intro; split; finish
theorem inter_distrib_right (s t u : finset α) : (s ∪ t) ∩ u = (s ∩ u) ∪ (t ∩ u) :=
ext $ by simp [mem_inter, mem_union]; intro; split; finish
theorem union_distrib_left (s t u : finset α) : s ∪ (t ∩ u) = (s ∪ t) ∩ (s ∪ u) :=
ext $ by simp [mem_inter, mem_union]; intro; split; finish
theorem union_distrib_right (s t u : finset α) : (s ∩ t) ∪ u = (s ∪ u) ∩ (t ∪ u) :=
ext $ by simp [mem_inter, mem_union]; intro; split; finish
end inter
/- erase -/
section erase
variables [decidable_eq α] {a b c : α} {s t : finset α}
def erase (a : α) (s : finset α) : finset α :=
quot.lift_on s
(λ l, to_finset_of_nodup (l.1.erase a) (nodup_erase_of_nodup a l.2))
(λ (l₁ l₂ : nodup_list α) (p : l₁ ~ l₂), quot.sound (erase_perm_erase a p))
@[simp] theorem mem_erase : a ∈ erase b s ↔ a ≠ b ∧ a ∈ s :=
finset.induction_on_to_finset s $ λ l hl, mem_erase_iff_of_nodup hl
theorem not_mem_erase : a ∉ erase a s := by simp
@[simp] theorem erase_empty : erase a ∅ = ∅ := rfl
theorem ne_of_mem_erase : b ∈ erase a s → b ≠ a := by simp {contextual:=tt}
theorem mem_of_mem_erase : b ∈ erase a s → b ∈ s := by simp {contextual:=tt}
theorem mem_erase_of_ne_of_mem : a ≠ b → a ∈ s → a ∈ erase b s := by simp {contextual:=tt}
theorem erase_insert (h : a ∉ s) : erase a (insert a s) = s :=
ext $ assume x, by simp; constructor; finish
theorem insert_erase (h : a ∈ s) : insert a (erase a s) = s :=
ext $ assume x, by simp; constructor; finish
theorem erase_subset_erase (h : s ⊆ t) : erase a s ⊆ erase a t :=
subset_iff.2 $ assume x, by simp [-and_imp]; exact and.imp_right (mem_of_subset_of_mem h)
theorem erase_subset : erase a s ⊆ s :=
subset_iff.2 $ assume x, by simp {contextual:=tt}
theorem erase_eq_of_not_mem (h : a ∉ s) : erase a s = s :=
ext $ assume b, by by_cases b = a; simp *
theorem erase_insert_subset : erase a (insert a s) ⊆ s :=
by by_cases a ∈ s; simp [h, erase_subset, erase_insert, subset.refl]
theorem erase_subset_of_subset_insert (h : s ⊆ insert a t) : erase a s ⊆ t :=
subset.trans (erase_subset_erase h) erase_insert_subset
theorem insert_erase_subset : s ⊆ insert a (erase a s) :=
decidable.by_cases
(λ ains : a ∈ s, by rw [insert_erase ains]; apply subset.refl)
(λ nains : a ∉ s, by rw[erase_eq_of_not_mem nains]; apply subset_insert)
theorem subset_insert_of_erase_subset (h : erase a s ⊆ t) : s ⊆ insert a t :=
subset.trans insert_erase_subset (insert_subset_insert h)
theorem subset_insert_iff (s t : finset α) (a : α) : s ⊆ insert a t ↔ erase a s ⊆ t :=
iff.intro erase_subset_of_subset_insert subset_insert_of_erase_subset
end erase
section filter
variables [decidable_eq α] {p : α → Prop} [decidable_pred p] {s s₁ s₂ : finset α} {a : α}
protected def filter (p : α → Prop) [decidable_pred p] (s : finset α) : finset α :=
quotient.lift_on s
(λl, to_finset_of_nodup (l.val.filter p) $ nodup_filter _ l.property)
(assume l₁ l₂ (h : perm l₁.val l₂.val), quot.sound $ perm_filter _ h)
@[simp] theorem mem_filter : a ∈ s.filter p ↔ (a ∈ s ∧ p a) :=
finset.induction_on_to_finset s $ λl₁ h₁, list.mem_filter
theorem filter_subset : s.filter p ⊆ s := by simp [subset_iff] {contextual := tt}
theorem filter_union : (s₁ ∪ s₂).filter p = s₁.filter p ∪ s₂.filter p :=
ext $ assume a, by simp [and_or_distrib_left]
theorem filter_filter {p' : α → Prop} [decidable_pred p'] :
(s.filter p).filter p' = s.filter (λa, p a ∧ p' a) :=
ext $ assume a, by simp
@[simp] theorem filter_false {h : decidable_pred $ λa:α, false} :
@finset.filter α _ (λa:α, false) h s = ∅ :=
ext $ assume a, by simp
theorem filter_union_filter_neg_eq : s.filter p ∪ s.filter (λa, ¬ p a) = s :=
ext $ assume a, by by_cases p a; simp [iff_iff_implies_and_implies, *] {contextual := tt}
theorem filter_inter_filter_neg_eq : s.filter p ∩ s.filter (λa, ¬ p a) = ∅ :=
ext $ assume a, by by_cases p a; simp * {contextual := tt}
end filter
section sdiff
variables [decidable_eq α] {a : α} {s₁ s₂ t₁ t₂ : finset α}
instance : has_sdiff (finset α) := ⟨λs₁ s₂, s₁.filter (λa, a ∉ s₂)⟩
@[simp] theorem mem_sdiff_iff : a ∈ s₁ \ s₂ ↔ (a ∈ s₁ ∧ a ∉ s₂) := mem_filter
lemma sdiff_union_of_subset (h : s₁ ⊆ s₂) : (s₂ \ s₁) ∪ s₁ = s₂ :=
ext $ assume a,
by rw [subset_iff] at h;
by_cases a ∈ s₁; simp [iff_iff_implies_and_implies, *] {contextual := tt}
lemma sdiff_inter_self : (s₂ \ s₁) ∩ s₁ = ∅ :=
ext $ by simp {contextual := tt}
lemma sdiff_subset_sdiff : t₁ ⊆ t₂ → s₂ ⊆ s₁ → t₁ \ s₁ ⊆ t₂ \ s₂ :=
begin
simp [subset_iff, mem_sdiff_iff] {contextual := tt},
exact assume h₁ h₂ a _ ha₁ ha₂, ha₁ $ h₂ _ ha₂
end
end sdiff
/- range -/
section range
variables {n m l : ℕ}
def range (n : ℕ) : finset ℕ :=
to_finset_of_nodup (list.range n) (nodup_range n)
@[simp] theorem mem_range : m ∈ range n ↔ m < n :=
list.mem_range
@[simp] theorem range_zero : range 0 = ∅ := rfl
@[simp] theorem range_succ : range (succ n) = insert n (range n) :=
ext $ by simp [mem_insert, le_iff_lt_or_eq, lt_succ_iff]
@[simp] theorem not_mem_range_self : n ∉ range n :=
not_mem_range_self
@[simp] theorem range_subset {n m} : range n ⊆ range m ↔ n ≤ m :=
range_subset
theorem exists_nat_subset_range {s : finset ℕ} : ∃n : ℕ, s ⊆ range n :=
s.induction_on ⟨0, by simp⟩ $
assume a s ha ⟨n, hn⟩,
⟨max (a + 1) n, subset_iff.mpr $ assume x,
begin
simp [subset_iff, mem_range] at hn,
simp [or_imp_distrib, mem_range, lt_max_iff, hn] {contextual := tt},
exact assume _, or.inr (lt_succ_self a)
end⟩
end range
/- useful rules for calculations with quantifiers -/
theorem exists_mem_empty_iff (p : α → Prop) : (∃ x, x ∈ (∅ : finset α) ∧ p x) ↔ false :=
⟨λ⟨x, hx⟩, not_mem_empty (hx.left), false.elim⟩
theorem exists_mem_insert [d : decidable_eq α]
(a : α) (s : finset α) (p : α → Prop) :
(∃ x, x ∈ insert a s ∧ p x) ↔ p a ∨ (∃ x, x ∈ s ∧ p x) :=
iff.intro
(λ H,
let ⟨x,H1,H2⟩ := H in
or.elim (mem_insert.mp H1)
(λ l, or.inl (eq.subst l H2))
(λ r, or.inr ⟨x, ⟨r, H2⟩⟩))
(λ H,
or.elim H
(λ l, ⟨a, ⟨mem_insert_self, l⟩⟩)
(λ r, let ⟨x,H2,H3⟩ := r in ⟨x, ⟨mem_insert_of_mem H2, H3⟩⟩))
theorem forall_mem_empty_iff (p : α → Prop) : (∀ x, x ∈ (∅ : finset α) → p x) ↔ true :=
iff.intro (λ H, trivial) (λ H x H', absurd H' not_mem_empty)
theorem forall_mem_insert [d : decidable_eq α]
(a : α) (s : finset α) (p : α → Prop) :
(∀ x, x ∈ insert a s → p x) ↔ p a ∧ (∀ x, x ∈ s → p x) :=
by simp [or_imp_distrib, forall_and_distrib]
section image
variables (f : α → β) (s s₁ s₂ : finset α) [decidable_eq β]
protected def image : finset β :=
quot.lift_on s (λl, to_finset $ l.val.map f) $ assume ⟨l₁, hl₁⟩ ⟨l₂, hl₂⟩ (h : perm l₁ l₂),
quotient.sound $ perm_erase_dup_of_perm $ perm_map _ $ h
@[simp] lemma image_empty {f : α → β} : (∅ : finset α).image f = ∅ := rfl
variables {f s s₁ s₂} {b : β} [decidable_eq α]
lemma mem_image_iff : b ∈ s.image f ↔ (∃a∈s, f a = b) :=
finset.induction_on_to_finset s $ assume l h,
show b ∈ to_finset (l.map f) ↔ _, by simp [mem_to_finset]
lemma erase_dup_map_erase_dup_eq {f : α → β} :
∀{l : list α}, erase_dup (map f (erase_dup l)) = (erase_dup $ map f l)
| [] := by simp
| (x :: xs) :=
have f x ∈ map f (erase_dup xs) ↔ f x ∈ map f xs, by simp,
by by_cases x ∈ xs; by_cases f x ∈ map f xs; simp * at *
lemma image_to_finset {l : list α} : (to_finset l).image f = to_finset (l.map f) :=
quot.sound $ show perm (erase_dup $ map f $ erase_dup l) (erase_dup $ map f l),
by rw [erase_dup_map_erase_dup_eq]
lemma image_to_finset_of_nodup {l : list α} (hl : nodup l) (h : ∀x∈l, ∀y∈l, f x = f y → x = y) :
(to_finset_of_nodup l hl).image f = to_finset_of_nodup (l.map f) (nodup_map_on h hl) :=
quot.sound $ show perm (erase_dup (map f l)) (l.map f),
by rw [erase_dup_eq_self.2 (nodup_map_on h hl)]
lemma image_id : s.image id = s :=
quot.induction_on s $ assume ⟨l, hl⟩, show to_finset (l.map id) = to_finset_of_nodup l hl,
by rw [map_id, to_finset_eq_of_nodup]
lemma image_image [decidable_eq γ] {g : β → γ} : (s.image f).image g = s.image (g ∘ f) :=
quot.induction_on s $ assume ⟨l, hl⟩,
show ((to_finset_of_nodup l hl).image f).image g = (to_finset_of_nodup l hl).image (g ∘ f),
by simp [to_finset_eq_of_nodup, image_to_finset]
lemma image_subset_image : s₁ ⊆ s₂ → s₁.image f ⊆ s₂.image f :=
by simp [subset_iff, mem_image_iff] {contextual:=tt};
from assume h b a h_eq ha, ⟨a, h_eq, h _ ha⟩
lemma image_filter {p : β → Prop} [decidable_pred p] :
(s.image f).filter p = (s.filter (p ∘ f)).image f :=
ext $ assume b, by simp [mem_image_iff]; exact iff.intro
(assume ⟨hb, ⟨a, h_eq, ha⟩⟩, ⟨a, h_eq, show p (f a), from h_eq.symm ▸ hb, ha⟩)
(assume ⟨a, h_eq, ha, has⟩, ⟨h_eq ▸ ha, a, h_eq, has⟩)
lemma image_union {f : α → β} : (s₁ ∪ s₂).image f = s₁.image f ∪ s₂.image f :=
ext $ assume b, by simp [mem_image_iff, and_or_distrib_left, exists_or_distrib]
lemma image_inter (hf : ∀x y, f x = f y → x = y) : (s₁ ∩ s₂).image f = s₁.image f ∩ s₂.image f :=
ext $ assume b,
by simp [mem_image_iff, iff_def];
from and.intro
(assume x x_eq hx y y_eq hy,
have y = x, from hf _ _ $ by simp *,
⟨x, x_eq, hx, this ▸ hy⟩)
(assume x x_eq h₁ h₂, ⟨⟨x, x_eq, h₁⟩, ⟨x, x_eq, h₂⟩⟩)
@[simp] lemma image_singleton {f : α → β} {a : α} : ({a}: finset α).image f = {f a} :=
ext $ by simp [mem_image_iff, and_or_distrib_left, exists_or_distrib, iff_def] {contextual := tt}
@[simp] lemma image_insert {f : α → β} {s : finset α} {a : α} :
(insert a s).image f = insert (f a) (s.image f) :=
by simp [insert_eq, image_union]
@[simp] lemma image_eq_empty_iff : s.image f = ∅ ↔ s = ∅ :=
iff.intro
(assume h,
have ∀a, a ∉ s,
from assume a ha,
have f a ∈ s.image f, from mem_image_iff.mpr ⟨a, ha, rfl⟩,
by simp * at *,
ext $ by simp *)
(assume h, by simp *)
end image
section card
variables [decidable_eq α] {a : α} {s : finset α} {n : ℕ}
/- card -/
def card (s : finset α) : nat :=
quot.lift_on s (λ l, length l.1) (λ l₁ l₂, perm_length)
@[simp] theorem card_empty : card (∅ : finset α) = 0 := rfl
lemma ne_empty_of_card_eq_succ : card s = succ n → s ≠ ∅ :=
λ h hn, by rw hn at h; contradiction
@[simp] theorem card_insert_of_not_mem : a ∉ s → card (insert a s) = card s + 1 :=
quot.induction_on s (λ (l : nodup_list α) (nainl : a ∉ ⟦l⟧), list.length_insert_of_not_mem nainl)
theorem card_insert_le : card (insert a s) ≤ card s + 1 :=
if h : a ∈ s then by simp [h, le_add_left] else by rw [card_insert_of_not_mem h]
theorem eq_empty_of_card_eq_zero : card s = 0 → s = ∅ :=
s.induction_on
(assume _, rfl)
(assume a s ha ih, by rw [card_insert_of_not_mem ha]; exact assume h, nat.no_confusion h)
theorem card_erase_of_mem : a ∈ s → card (erase a s) = pred (card s) :=
quot.induction_on s (λ l ainl, list.length_erase_of_mem ainl)
theorem card_range : card (range n) = n :=
list.length_range n
end card
end finset
|
e77f92f0c7e6b1898a4c2631107366e2a4ee7d46 | 624f6f2ae8b3b1adc5f8f67a365c51d5126be45a | /src/Init/Lean/Parser/Module.lean | d8924f4180b8271e4f4bf5f3fdae6fbfe5883de6 | [
"Apache-2.0"
] | permissive | mhuisi/lean4 | 28d35a4febc2e251c7f05492e13f3b05d6f9b7af | dda44bc47f3e5d024508060dac2bcb59fd12e4c0 | refs/heads/master | 1,621,225,489,283 | 1,585,142,689,000 | 1,585,142,689,000 | 250,590,438 | 0 | 2 | Apache-2.0 | 1,602,443,220,000 | 1,585,327,814,000 | C | UTF-8 | Lean | false | false | 4,770 | lean | /-
Copyright (c) 2019 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Sebastian Ullrich
-/
prelude
import Init.Lean.Message
import Init.Lean.Parser.Command
namespace Lean
namespace Parser
namespace Module
def «prelude» := parser! "prelude"
def «import» := parser! "import " >> optional "runtime" >> ident
def header := parser! optional «prelude» >> many «import»
def updateTokens (c : ParserContext) : ParserContext :=
{ tokens := match addParserTokens c.tokens header.info with
| Except.ok tables => tables
| Except.error _ => unreachable!,
.. c }
end Module
structure ModuleParserState :=
(pos : String.Pos := 0)
(recovering : Bool := false)
instance ModuleParserState.inhabited : Inhabited ModuleParserState :=
⟨{}⟩
private def mkErrorMessage (c : ParserContext) (pos : String.Pos) (errorMsg : String) : Message :=
let pos := c.fileMap.toPosition pos;
{ fileName := c.fileName, pos := pos, data := errorMsg }
def parseHeader (env : Environment) (inputCtx : InputContext) : Syntax × ModuleParserState × MessageLog :=
let ctx := mkParserContext env inputCtx;
let ctx := Module.updateTokens ctx;
let s := mkParserState ctx.input;
let s := whitespace ctx s;
let s := Module.header.fn ctx s;
let stx := s.stxStack.back;
match s.errorMsg with
| some errorMsg =>
let msg := mkErrorMessage ctx s.pos (toString errorMsg);
(stx, { pos := s.pos, recovering := true }, { MessageLog . }.add msg)
| none =>
(stx, { pos := s.pos }, {})
private def mkEOI (pos : String.Pos) : Syntax :=
let atom := mkAtom { pos := pos, trailing := "".toSubstring, leading := "".toSubstring } "";
Syntax.node `Lean.Parser.Module.eoi #[atom]
def isEOI (s : Syntax) : Bool :=
s.isOfKind `Lean.Parser.Module.eoi
def isExitCommand (s : Syntax) : Bool :=
s.isOfKind `Lean.Parser.Command.exit
private def consumeInput (c : ParserContext) (pos : String.Pos) : String.Pos :=
let s : ParserState := { cache := initCacheForInput c.input, pos := pos };
let s := tokenFn c s;
match s.errorMsg with
| some _ => pos + 1
| none => s.pos
partial def parseCommand (env : Environment) (inputCtx : InputContext) : ModuleParserState → MessageLog → Syntax × ModuleParserState × MessageLog
| s@{ pos := pos, recovering := recovering }, messages =>
if inputCtx.input.atEnd pos then
(mkEOI pos, s, messages)
else
let c := mkParserContext env inputCtx;
let s := { ParserState . cache := initCacheForInput c.input, pos := pos };
let s := whitespace c s;
let s := (commandParser : Parser).fn c s;
match s.errorMsg with
| none =>
let stx := s.stxStack.back;
(stx, { pos := s.pos }, messages)
| some errorMsg =>
if recovering then
parseCommand { pos := consumeInput c s.pos, recovering := true } messages
else
let msg := mkErrorMessage c s.pos (toString errorMsg);
let messages := messages.add msg;
parseCommand { pos := consumeInput c s.pos, recovering := true } messages
private partial def testModuleParserAux (env : Environment) (inputCtx : InputContext) (displayStx : Bool) : ModuleParserState → MessageLog → IO Bool
| s, messages =>
match parseCommand env inputCtx s messages with
| (stx, s, messages) =>
if isEOI stx || isExitCommand stx then do
messages.forM $ fun msg => IO.println msg;
pure (!messages.hasErrors)
else do
when displayStx (IO.println stx);
testModuleParserAux s messages
@[export lean_test_module_parser]
def testModuleParser (env : Environment) (input : String) (fileName := "<input>") (displayStx := false) : IO Bool :=
timeit (fileName ++ " parser") $ do
let inputCtx := mkInputContext input fileName;
let (stx, s, messages) := parseHeader env inputCtx;
when displayStx (IO.println stx);
testModuleParserAux env inputCtx displayStx s messages
partial def parseFileAux (env : Environment) (inputCtx : InputContext) : ModuleParserState → MessageLog → Array Syntax → IO Syntax
| state, msgs, stxs =>
match parseCommand env inputCtx state msgs with
| (stx, state, msgs) =>
if isEOI stx then
if msgs.isEmpty then
let stx := mkListNode stxs;
pure stx.updateLeading
else do
msgs.forM $ fun msg => IO.println msg;
throw (IO.userError "failed to parse file")
else
parseFileAux state msgs (stxs.push stx)
def parseFile (env : Environment) (fname : String) : IO Syntax := do
fname ← IO.realPath fname;
contents ← IO.FS.readFile fname;
let inputCtx := mkInputContext contents fname;
let (stx, state, messages) := parseHeader env inputCtx;
parseFileAux env inputCtx state messages #[stx]
end Parser
end Lean
|
0f778e86141250a2dabfc9ca0b8fb3509b8928d2 | 92e157ec9825b5e4597a6d715a8928703bc8e3b2 | /src/mywork/lecture_5.lean | ac4ff3e32781912ce3c4d2fad961d23062f1cab2 | [] | no_license | exb3dg/cs2120f21 | 9e566bc508762573c023d3e70f83cb839c199ec8 | 319b8bf0d63bf96437bf17970ce0198d0b3525cd | refs/heads/main | 1,692,970,909,568 | 1,634,584,540,000 | 1,634,584,540,000 | 399,947,025 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 2,420 | lean | /-
INTRODUCTION and ELIMINATION RULES
-/
/-
For ∀ x, P x (every x has property P)
- introduction rule: assume arbitrary x, then show P x
- elimination rule: *apply* a proof of ∀ x, P x, as a
kind of function to a specific value of x, say k, to
produce a proof of P k.
-/
theorem silly : ∀ (n : ℕ), true :=
begin
assume (n : ℕ),
exact true.intro,
end
/-
The proposition true is unconditionally true,
as proven by an always available proof called
(in Lean) true.intro.
-/
#check silly 7
/-
The check command will tell you the type of any
expression (aka term) in Lean. Here we can see
that silly is like a function, and that when we
apply it to the specific argument, 7, we get back
a proof of the resulting proposition (which is
just, "true"). We'll soon be equipped to deal
with more interesting "return types".
-/
/-
For P → Q (if P is true then Q must also be true)
- introduction rule: assume arbitrary P, then show Q
- elimination rule: *apply* a proof of P → Q, as a
- kind of function, to any proof of P to derive a proof of Q!
-/
def foo : ∀ (x : ℕ), x = 0 → x + 1 = 1 :=
begin
assume x h,
rw h,
end
/-
Wow! ∀ and → sure do seem similar. Indeed they're the same!
They define function types. We construct a proof of ∀ or →
by assuming the premise and showing that in that context we
can derive a result of the conclusion type. We can then use
a proof of a ∀ or → by treating it as a function that can
be applied to a specific value to derive a proof *for that
specific value. Indeed, in Lean, → is really just another
notation for forall!
-/
/-
Introduction rule for and ∧
Given a proof of P and a proof of Q
get back a proof of (P ∧ Q).
-/
axioms (P Q : Prop)
#check P
#check (P ∧ Q)
axioms (p : P) (q : Q)
example : P ∧ Q := and.intro p q
/-
Prive that if arbitrary porpositions P and Q
are trie (which is to say that we have a proof
of each of them), that the proposition P ∧ Q
is also true
Proof: The conjecture that P ∧ Q is true
is proved by application of the introduction
rule for and.)
-/
theorem bar : 0 = 0 ∧ 1 = 1 :=
begin
apply and.intro (eq.refl 0) (eq.refl 1)
end
#check bracket
#check and.elim_left bar
#check and.elim_right bar
theorem and_commutative : ∀ (P Q : Prop), P ∧ Q → Q ∧ P :=
begin
assume P Q h,
apply and.intro _ _,
apply and.elim_right h,
apply and.elim_left h,
end
|
8b15866f655e2f325a80d6d7d5219fe01cd55937 | 9dc8cecdf3c4634764a18254e94d43da07142918 | /src/linear_algebra/affine_space/affine_map.lean | 878eb6c3a187e0ccd20357257d1bcec7908aa273 | [
"Apache-2.0"
] | permissive | jcommelin/mathlib | d8456447c36c176e14d96d9e76f39841f69d2d9b | ee8279351a2e434c2852345c51b728d22af5a156 | refs/heads/master | 1,664,782,136,488 | 1,663,638,983,000 | 1,663,638,983,000 | 132,563,656 | 0 | 0 | Apache-2.0 | 1,663,599,929,000 | 1,525,760,539,000 | Lean | UTF-8 | Lean | false | false | 24,067 | lean | /-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import algebra.add_torsor
import data.set.intervals.unordered_interval
import linear_algebra.affine_space.basic
import linear_algebra.bilinear_map
import linear_algebra.pi
import linear_algebra.prod
import tactic.abel
/-!
# Affine maps
This file defines affine maps.
## Main definitions
* `affine_map` is the type of affine maps between two affine spaces with the same ring `k`. Various
basic examples of affine maps are defined, including `const`, `id`, `line_map` and `homothety`.
## Notations
* `P1 →ᵃ[k] P2` is a notation for `affine_map k P1 P2`;
* `affine_space V P`: a localized notation for `add_torsor V P` defined in
`linear_algebra.affine_space.basic`.
## Implementation notes
`out_param` is used in the definition of `[add_torsor V P]` to make `V` an implicit argument
(deduced from `P`) in most cases; `include V` is needed in many cases for `V`, and type classes
using it, to be added as implicit arguments to individual lemmas. As for modules, `k` is an
explicit argument rather than implied by `P` or `V`.
This file only provides purely algebraic definitions and results. Those depending on analysis or
topology are defined elsewhere; see `analysis.normed_space.add_torsor` and
`topology.algebra.affine`.
## References
* https://en.wikipedia.org/wiki/Affine_space
* https://en.wikipedia.org/wiki/Principal_homogeneous_space
-/
open_locale affine
/-- An `affine_map k P1 P2` (notation: `P1 →ᵃ[k] P2`) is a map from `P1` to `P2` that
induces a corresponding linear map from `V1` to `V2`. -/
structure affine_map (k : Type*) {V1 : Type*} (P1 : Type*) {V2 : Type*} (P2 : Type*)
[ring k]
[add_comm_group V1] [module k V1] [affine_space V1 P1]
[add_comm_group V2] [module k V2] [affine_space V2 P2] :=
(to_fun : P1 → P2)
(linear : V1 →ₗ[k] V2)
(map_vadd' : ∀ (p : P1) (v : V1), to_fun (v +ᵥ p) = linear v +ᵥ to_fun p)
notation P1 ` →ᵃ[`:25 k:25 `] `:0 P2:0 := affine_map k P1 P2
instance (k : Type*) {V1 : Type*} (P1 : Type*) {V2 : Type*} (P2 : Type*)
[ring k]
[add_comm_group V1] [module k V1] [affine_space V1 P1]
[add_comm_group V2] [module k V2] [affine_space V2 P2]:
has_coe_to_fun (P1 →ᵃ[k] P2) (λ _, P1 → P2) := ⟨affine_map.to_fun⟩
namespace linear_map
variables {k : Type*} {V₁ : Type*} {V₂ : Type*} [ring k] [add_comm_group V₁] [module k V₁]
[add_comm_group V₂] [module k V₂] (f : V₁ →ₗ[k] V₂)
/-- Reinterpret a linear map as an affine map. -/
def to_affine_map : V₁ →ᵃ[k] V₂ :=
{ to_fun := f,
linear := f,
map_vadd' := λ p v, f.map_add v p }
@[simp] lemma coe_to_affine_map : ⇑f.to_affine_map = f := rfl
@[simp] lemma to_affine_map_linear : f.to_affine_map.linear = f := rfl
end linear_map
namespace affine_map
variables {k : Type*} {V1 : Type*} {P1 : Type*} {V2 : Type*} {P2 : Type*}
{V3 : Type*} {P3 : Type*} {V4 : Type*} {P4 : Type*} [ring k]
[add_comm_group V1] [module k V1] [affine_space V1 P1]
[add_comm_group V2] [module k V2] [affine_space V2 P2]
[add_comm_group V3] [module k V3] [affine_space V3 P3]
[add_comm_group V4] [module k V4] [affine_space V4 P4]
include V1 V2
/-- Constructing an affine map and coercing back to a function
produces the same map. -/
@[simp] lemma coe_mk (f : P1 → P2) (linear add) :
((mk f linear add : P1 →ᵃ[k] P2) : P1 → P2) = f := rfl
/-- `to_fun` is the same as the result of coercing to a function. -/
@[simp] lemma to_fun_eq_coe (f : P1 →ᵃ[k] P2) : f.to_fun = ⇑f := rfl
/-- An affine map on the result of adding a vector to a point produces
the same result as the linear map applied to that vector, added to the
affine map applied to that point. -/
@[simp] lemma map_vadd (f : P1 →ᵃ[k] P2) (p : P1) (v : V1) :
f (v +ᵥ p) = f.linear v +ᵥ f p := f.map_vadd' p v
/-- The linear map on the result of subtracting two points is the
result of subtracting the result of the affine map on those two
points. -/
@[simp] lemma linear_map_vsub (f : P1 →ᵃ[k] P2) (p1 p2 : P1) :
f.linear (p1 -ᵥ p2) = f p1 -ᵥ f p2 :=
by conv_rhs { rw [←vsub_vadd p1 p2, map_vadd, vadd_vsub] }
/-- Two affine maps are equal if they coerce to the same function. -/
@[ext] lemma ext {f g : P1 →ᵃ[k] P2} (h : ∀ p, f p = g p) : f = g :=
begin
rcases f with ⟨f, f_linear, f_add⟩,
rcases g with ⟨g, g_linear, g_add⟩,
obtain rfl : f = g := funext h,
congr' with v,
cases (add_torsor.nonempty : nonempty P1) with p,
apply vadd_right_cancel (f p),
erw [← f_add, ← g_add]
end
lemma ext_iff {f g : P1 →ᵃ[k] P2} : f = g ↔ ∀ p, f p = g p := ⟨λ h p, h ▸ rfl, ext⟩
lemma coe_fn_injective : @function.injective (P1 →ᵃ[k] P2) (P1 → P2) coe_fn :=
λ f g H, ext $ congr_fun H
protected lemma congr_arg (f : P1 →ᵃ[k] P2) {x y : P1} (h : x = y) : f x = f y :=
congr_arg _ h
protected lemma congr_fun {f g : P1 →ᵃ[k] P2} (h : f = g) (x : P1) : f x = g x :=
h ▸ rfl
variables (k P1)
/-- Constant function as an `affine_map`. -/
def const (p : P2) : P1 →ᵃ[k] P2 :=
{ to_fun := function.const P1 p,
linear := 0,
map_vadd' := λ p v, by simp }
@[simp] lemma coe_const (p : P2) : ⇑(const k P1 p) = function.const P1 p := rfl
@[simp] lemma const_linear (p : P2) : (const k P1 p).linear = 0 := rfl
variables {k P1}
lemma linear_eq_zero_iff_exists_const (f : P1 →ᵃ[k] P2) :
f.linear = 0 ↔ ∃ q, f = const k P1 q :=
begin
refine ⟨λ h, _, λ h, _⟩,
{ use f (classical.arbitrary P1),
ext,
rw [coe_const, function.const_apply, ← @vsub_eq_zero_iff_eq V2, ← f.linear_map_vsub, h,
linear_map.zero_apply], },
{ rcases h with ⟨q, rfl⟩,
exact const_linear k P1 q, },
end
instance nonempty : nonempty (P1 →ᵃ[k] P2) :=
(add_torsor.nonempty : nonempty P2).elim $ λ p, ⟨const k P1 p⟩
/-- Construct an affine map by verifying the relation between the map and its linear part at one
base point. Namely, this function takes a map `f : P₁ → P₂`, a linear map `f' : V₁ →ₗ[k] V₂`, and
a point `p` such that for any other point `p'` we have `f p' = f' (p' -ᵥ p) +ᵥ f p`. -/
def mk' (f : P1 → P2) (f' : V1 →ₗ[k] V2) (p : P1) (h : ∀ p' : P1, f p' = f' (p' -ᵥ p) +ᵥ f p) :
P1 →ᵃ[k] P2 :=
{ to_fun := f,
linear := f',
map_vadd' := λ p' v, by rw [h, h p', vadd_vsub_assoc, f'.map_add, vadd_vadd] }
@[simp] lemma coe_mk' (f : P1 → P2) (f' : V1 →ₗ[k] V2) (p h) : ⇑(mk' f f' p h) = f := rfl
@[simp] lemma mk'_linear (f : P1 → P2) (f' : V1 →ₗ[k] V2) (p h) : (mk' f f' p h).linear = f' := rfl
section has_smul
variables {R : Type*} [monoid R] [distrib_mul_action R V2] [smul_comm_class k R V2]
/-- The space of affine maps to a module inherits an `R`-action from the action on its codomain. -/
instance : mul_action R (P1 →ᵃ[k] V2) :=
{ smul := λ c f, ⟨c • f, c • f.linear, λ p v, by simp [smul_add]⟩,
one_smul := λ f, ext $ λ p, one_smul _ _,
mul_smul := λ c₁ c₂ f, ext $ λ p, mul_smul _ _ _ }
@[simp, norm_cast] lemma coe_smul (c : R) (f : P1 →ᵃ[k] V2) : ⇑(c • f) = c • f := rfl
@[simp] lemma smul_linear (t : R) (f : P1 →ᵃ[k] V2) : (t • f).linear = t • f.linear := rfl
instance [distrib_mul_action Rᵐᵒᵖ V2] [is_central_scalar R V2] :
is_central_scalar R (P1 →ᵃ[k] V2) :=
{ op_smul_eq_smul := λ r x, ext $ λ _, op_smul_eq_smul _ _ }
end has_smul
instance : has_zero (P1 →ᵃ[k] V2) := { zero := ⟨0, 0, λ p v, (zero_vadd _ _).symm⟩ }
instance : has_add (P1 →ᵃ[k] V2) :=
{ add := λ f g, ⟨f + g, f.linear + g.linear, λ p v, by simp [add_add_add_comm]⟩ }
instance : has_sub (P1 →ᵃ[k] V2) :=
{ sub := λ f g, ⟨f - g, f.linear - g.linear, λ p v, by simp [sub_add_sub_comm]⟩ }
instance : has_neg (P1 →ᵃ[k] V2) := { neg := λ f, ⟨-f, -f.linear, λ p v, by simp [add_comm]⟩ }
@[simp, norm_cast] lemma coe_zero : ⇑(0 : P1 →ᵃ[k] V2) = 0 := rfl
@[simp, norm_cast] lemma coe_add (f g : P1 →ᵃ[k] V2) : ⇑(f + g) = f + g := rfl
@[simp, norm_cast] lemma coe_neg (f : P1 →ᵃ[k] V2) : ⇑(-f) = -f := rfl
@[simp, norm_cast] lemma coe_sub (f g : P1 →ᵃ[k] V2) : ⇑(f - g) = f - g := rfl
@[simp] lemma zero_linear : (0 : P1 →ᵃ[k] V2).linear = 0 := rfl
@[simp] lemma add_linear (f g : P1 →ᵃ[k] V2) : (f + g).linear = f.linear + g.linear := rfl
@[simp] lemma sub_linear (f g : P1 →ᵃ[k] V2) : (f - g).linear = f.linear - g.linear := rfl
@[simp] lemma neg_linear (f : P1 →ᵃ[k] V2) : (-f).linear = -f.linear := rfl
/-- The set of affine maps to a vector space is an additive commutative group. -/
instance : add_comm_group (P1 →ᵃ[k] V2) :=
coe_fn_injective.add_comm_group _
coe_zero coe_add coe_neg coe_sub (λ _ _, coe_smul _ _) (λ _ _, coe_smul _ _)
/-- The space of affine maps from `P1` to `P2` is an affine space over the space of affine maps
from `P1` to the vector space `V2` corresponding to `P2`. -/
instance : affine_space (P1 →ᵃ[k] V2) (P1 →ᵃ[k] P2) :=
{ vadd := λ f g, ⟨λ p, f p +ᵥ g p, f.linear + g.linear, λ p v,
by simp [vadd_vadd, add_right_comm]⟩,
zero_vadd := λ f, ext $ λ p, zero_vadd _ (f p),
add_vadd := λ f₁ f₂ f₃, ext $ λ p, add_vadd (f₁ p) (f₂ p) (f₃ p),
vsub := λ f g, ⟨λ p, f p -ᵥ g p, f.linear - g.linear, λ p v,
by simp [vsub_vadd_eq_vsub_sub, vadd_vsub_assoc, add_sub, sub_add_eq_add_sub]⟩,
vsub_vadd' := λ f g, ext $ λ p, vsub_vadd (f p) (g p),
vadd_vsub' := λ f g, ext $ λ p, vadd_vsub (f p) (g p) }
@[simp] lemma vadd_apply (f : P1 →ᵃ[k] V2) (g : P1 →ᵃ[k] P2) (p : P1) :
(f +ᵥ g) p = f p +ᵥ g p :=
rfl
@[simp] lemma vsub_apply (f g : P1 →ᵃ[k] P2) (p : P1) :
(f -ᵥ g : P1 →ᵃ[k] V2) p = f p -ᵥ g p :=
rfl
/-- `prod.fst` as an `affine_map`. -/
def fst : (P1 × P2) →ᵃ[k] P1 :=
{ to_fun := prod.fst,
linear := linear_map.fst k V1 V2,
map_vadd' := λ _ _, rfl }
@[simp] lemma coe_fst : ⇑(fst : (P1 × P2) →ᵃ[k] P1) = prod.fst := rfl
@[simp] lemma fst_linear : (fst : (P1 × P2) →ᵃ[k] P1).linear = linear_map.fst k V1 V2 := rfl
/-- `prod.snd` as an `affine_map`. -/
def snd : (P1 × P2) →ᵃ[k] P2 :=
{ to_fun := prod.snd,
linear := linear_map.snd k V1 V2,
map_vadd' := λ _ _, rfl }
@[simp] lemma coe_snd : ⇑(snd : (P1 × P2) →ᵃ[k] P2) = prod.snd := rfl
@[simp] lemma snd_linear : (snd : (P1 × P2) →ᵃ[k] P2).linear = linear_map.snd k V1 V2 := rfl
variables (k P1)
omit V2
/-- Identity map as an affine map. -/
def id : P1 →ᵃ[k] P1 :=
{ to_fun := id,
linear := linear_map.id,
map_vadd' := λ p v, rfl }
/-- The identity affine map acts as the identity. -/
@[simp] lemma coe_id : ⇑(id k P1) = _root_.id := rfl
@[simp] lemma id_linear : (id k P1).linear = linear_map.id := rfl
variable {P1}
/-- The identity affine map acts as the identity. -/
lemma id_apply (p : P1) : id k P1 p = p := rfl
variables {k P1}
instance : inhabited (P1 →ᵃ[k] P1) := ⟨id k P1⟩
include V2 V3
/-- Composition of affine maps. -/
def comp (f : P2 →ᵃ[k] P3) (g : P1 →ᵃ[k] P2) : P1 →ᵃ[k] P3 :=
{ to_fun := f ∘ g,
linear := f.linear.comp g.linear,
map_vadd' := begin
intros p v,
rw [function.comp_app, g.map_vadd, f.map_vadd],
refl
end }
/-- Composition of affine maps acts as applying the two functions. -/
@[simp] lemma coe_comp (f : P2 →ᵃ[k] P3) (g : P1 →ᵃ[k] P2) :
⇑(f.comp g) = f ∘ g := rfl
/-- Composition of affine maps acts as applying the two functions. -/
lemma comp_apply (f : P2 →ᵃ[k] P3) (g : P1 →ᵃ[k] P2) (p : P1) :
f.comp g p = f (g p) := rfl
omit V3
@[simp] lemma comp_id (f : P1 →ᵃ[k] P2) : f.comp (id k P1) = f := ext $ λ p, rfl
@[simp] lemma id_comp (f : P1 →ᵃ[k] P2) : (id k P2).comp f = f := ext $ λ p, rfl
include V3 V4
lemma comp_assoc (f₃₄ : P3 →ᵃ[k] P4) (f₂₃ : P2 →ᵃ[k] P3) (f₁₂ : P1 →ᵃ[k] P2) :
(f₃₄.comp f₂₃).comp f₁₂ = f₃₄.comp (f₂₃.comp f₁₂) :=
rfl
omit V2 V3 V4
instance : monoid (P1 →ᵃ[k] P1) :=
{ one := id k P1,
mul := comp,
one_mul := id_comp,
mul_one := comp_id,
mul_assoc := comp_assoc }
@[simp] lemma coe_mul (f g : P1 →ᵃ[k] P1) : ⇑(f * g) = f ∘ g := rfl
@[simp] lemma coe_one : ⇑(1 : P1 →ᵃ[k] P1) = _root_.id := rfl
/-- `affine_map.linear` on endomorphisms is a `monoid_hom`. -/
@[simps] def linear_hom : (P1 →ᵃ[k] P1) →* (V1 →ₗ[k] V1) :=
{ to_fun := linear,
map_one' := rfl,
map_mul' := λ _ _, rfl }
include V2
@[simp] lemma injective_iff_linear_injective (f : P1 →ᵃ[k] P2) :
function.injective f.linear ↔ function.injective f :=
begin
obtain ⟨p⟩ := (infer_instance : nonempty P1),
have h : ⇑f.linear = (equiv.vadd_const (f p)).symm ∘ f ∘ (equiv.vadd_const p),
{ ext v, simp [f.map_vadd, vadd_vsub_assoc], },
rw [h, equiv.comp_injective, equiv.injective_comp],
end
@[simp] lemma surjective_iff_linear_surjective (f : P1 →ᵃ[k] P2) :
function.surjective f.linear ↔ function.surjective f :=
begin
obtain ⟨p⟩ := (infer_instance : nonempty P1),
have h : ⇑f.linear = (equiv.vadd_const (f p)).symm ∘ f ∘ (equiv.vadd_const p),
{ ext v, simp [f.map_vadd, vadd_vsub_assoc], },
rw [h, equiv.comp_surjective, equiv.surjective_comp],
end
lemma image_vsub_image {s t : set P1} (f : P1 →ᵃ[k] P2) :
(f '' s) -ᵥ (f '' t) = f.linear '' (s -ᵥ t) :=
begin
ext v,
simp only [set.mem_vsub, set.mem_image, exists_exists_and_eq_and, exists_and_distrib_left,
← f.linear_map_vsub],
split,
{ rintros ⟨x, hx, y, hy, hv⟩,
exact ⟨x -ᵥ y, ⟨x, hx, y, hy, rfl⟩, hv⟩, },
{ rintros ⟨-, ⟨x, hx, y, hy, rfl⟩, rfl⟩,
exact ⟨x, hx, y, hy, rfl⟩, },
end
omit V2
/-! ### Definition of `affine_map.line_map` and lemmas about it -/
/-- The affine map from `k` to `P1` sending `0` to `p₀` and `1` to `p₁`. -/
def line_map (p₀ p₁ : P1) : k →ᵃ[k] P1 :=
((linear_map.id : k →ₗ[k] k).smul_right (p₁ -ᵥ p₀)).to_affine_map +ᵥ const k k p₀
lemma coe_line_map (p₀ p₁ : P1) : (line_map p₀ p₁ : k → P1) = λ c, c • (p₁ -ᵥ p₀) +ᵥ p₀ := rfl
lemma line_map_apply (p₀ p₁ : P1) (c : k) : line_map p₀ p₁ c = c • (p₁ -ᵥ p₀) +ᵥ p₀ := rfl
lemma line_map_apply_module' (p₀ p₁ : V1) (c : k) : line_map p₀ p₁ c = c • (p₁ - p₀) + p₀ := rfl
lemma line_map_apply_module (p₀ p₁ : V1) (c : k) : line_map p₀ p₁ c = (1 - c) • p₀ + c • p₁ :=
by simp [line_map_apply_module', smul_sub, sub_smul]; abel
omit V1
lemma line_map_apply_ring' (a b c : k) : line_map a b c = c * (b - a) + a :=
rfl
lemma line_map_apply_ring (a b c : k) : line_map a b c = (1 - c) * a + c * b :=
line_map_apply_module a b c
include V1
lemma line_map_vadd_apply (p : P1) (v : V1) (c : k) :
line_map p (v +ᵥ p) c = c • v +ᵥ p :=
by rw [line_map_apply, vadd_vsub]
@[simp] lemma line_map_linear (p₀ p₁ : P1) :
(line_map p₀ p₁ : k →ᵃ[k] P1).linear = linear_map.id.smul_right (p₁ -ᵥ p₀) :=
add_zero _
lemma line_map_same_apply (p : P1) (c : k) : line_map p p c = p := by simp [line_map_apply]
@[simp] lemma line_map_same (p : P1) : line_map p p = const k k p :=
ext $ line_map_same_apply p
@[simp] lemma line_map_apply_zero (p₀ p₁ : P1) : line_map p₀ p₁ (0:k) = p₀ :=
by simp [line_map_apply]
@[simp] lemma line_map_apply_one (p₀ p₁ : P1) : line_map p₀ p₁ (1:k) = p₁ :=
by simp [line_map_apply]
include V2
@[simp] lemma apply_line_map (f : P1 →ᵃ[k] P2) (p₀ p₁ : P1) (c : k) :
f (line_map p₀ p₁ c) = line_map (f p₀) (f p₁) c :=
by simp [line_map_apply]
@[simp] lemma comp_line_map (f : P1 →ᵃ[k] P2) (p₀ p₁ : P1) :
f.comp (line_map p₀ p₁) = line_map (f p₀) (f p₁) :=
ext $ f.apply_line_map p₀ p₁
@[simp] lemma fst_line_map (p₀ p₁ : P1 × P2) (c : k) :
(line_map p₀ p₁ c).1 = line_map p₀.1 p₁.1 c :=
fst.apply_line_map p₀ p₁ c
@[simp] lemma snd_line_map (p₀ p₁ : P1 × P2) (c : k) :
(line_map p₀ p₁ c).2 = line_map p₀.2 p₁.2 c :=
snd.apply_line_map p₀ p₁ c
omit V2
lemma line_map_symm (p₀ p₁ : P1) :
line_map p₀ p₁ = (line_map p₁ p₀).comp (line_map (1:k) (0:k)) :=
by { rw [comp_line_map], simp }
lemma line_map_apply_one_sub (p₀ p₁ : P1) (c : k) :
line_map p₀ p₁ (1 - c) = line_map p₁ p₀ c :=
by { rw [line_map_symm p₀, comp_apply], congr, simp [line_map_apply] }
@[simp] lemma line_map_vsub_left (p₀ p₁ : P1) (c : k) :
line_map p₀ p₁ c -ᵥ p₀ = c • (p₁ -ᵥ p₀) :=
vadd_vsub _ _
@[simp] lemma left_vsub_line_map (p₀ p₁ : P1) (c : k) :
p₀ -ᵥ line_map p₀ p₁ c = c • (p₀ -ᵥ p₁) :=
by rw [← neg_vsub_eq_vsub_rev, line_map_vsub_left, ← smul_neg, neg_vsub_eq_vsub_rev]
@[simp] lemma line_map_vsub_right (p₀ p₁ : P1) (c : k) :
line_map p₀ p₁ c -ᵥ p₁ = (1 - c) • (p₀ -ᵥ p₁) :=
by rw [← line_map_apply_one_sub, line_map_vsub_left]
@[simp] lemma right_vsub_line_map (p₀ p₁ : P1) (c : k) :
p₁ -ᵥ line_map p₀ p₁ c = (1 - c) • (p₁ -ᵥ p₀) :=
by rw [← line_map_apply_one_sub, left_vsub_line_map]
lemma line_map_vadd_line_map (v₁ v₂ : V1) (p₁ p₂ : P1) (c : k) :
line_map v₁ v₂ c +ᵥ line_map p₁ p₂ c = line_map (v₁ +ᵥ p₁) (v₂ +ᵥ p₂) c :=
((fst : V1 × P1 →ᵃ[k] V1) +ᵥ snd).apply_line_map (v₁, p₁) (v₂, p₂) c
lemma line_map_vsub_line_map (p₁ p₂ p₃ p₄ : P1) (c : k) :
line_map p₁ p₂ c -ᵥ line_map p₃ p₄ c = line_map (p₁ -ᵥ p₃) (p₂ -ᵥ p₄) c :=
-- Why Lean fails to find this instance without a hint?
by letI : affine_space (V1 × V1) (P1 × P1) := prod.add_torsor; exact
((fst : P1 × P1 →ᵃ[k] P1) -ᵥ (snd : P1 × P1 →ᵃ[k] P1)).apply_line_map (_, _) (_, _) c
/-- Decomposition of an affine map in the special case when the point space and vector space
are the same. -/
lemma decomp (f : V1 →ᵃ[k] V2) : (f : V1 → V2) = f.linear + (λ z, f 0) :=
begin
ext x,
calc
f x = f.linear x +ᵥ f 0 : by simp [← f.map_vadd]
... = (f.linear.to_fun + λ (z : V1), f 0) x : by simp
end
/-- Decomposition of an affine map in the special case when the point space and vector space
are the same. -/
lemma decomp' (f : V1 →ᵃ[k] V2) : (f.linear : V1 → V2) = f - (λ z, f 0) :=
by rw decomp ; simp only [linear_map.map_zero, pi.add_apply, add_sub_cancel, zero_add]
omit V1
lemma image_interval {k : Type*} [linear_ordered_field k] (f : k →ᵃ[k] k)
(a b : k) :
f '' set.interval a b = set.interval (f a) (f b) :=
begin
have : ⇑f = (λ x, x + f 0) ∘ λ x, x * (f 1 - f 0),
{ ext x,
change f x = x • (f 1 -ᵥ f 0) +ᵥ f 0,
rw [← f.linear_map_vsub, ← f.linear.map_smul, ← f.map_vadd],
simp only [vsub_eq_sub, add_zero, mul_one, vadd_eq_add, sub_zero, smul_eq_mul] },
rw [this, set.image_comp],
simp only [set.image_add_const_interval, set.image_mul_const_interval]
end
section
variables {ι : Type*} {V : Π i : ι, Type*} {P : Π i : ι, Type*} [Π i, add_comm_group (V i)]
[Π i, module k (V i)] [Π i, add_torsor (V i) (P i)]
include V
/-- Evaluation at a point as an affine map. -/
def proj (i : ι) : (Π i : ι, P i) →ᵃ[k] P i :=
{ to_fun := λ f, f i,
linear := @linear_map.proj k ι _ V _ _ i,
map_vadd' := λ p v, rfl }
@[simp] lemma proj_apply (i : ι) (f : Π i, P i) : @proj k _ ι V P _ _ _ i f = f i := rfl
@[simp] lemma proj_linear (i : ι) :
(@proj k _ ι V P _ _ _ i).linear = @linear_map.proj k ι _ V _ _ i := rfl
lemma pi_line_map_apply (f g : Π i, P i) (c : k) (i : ι) :
line_map f g c i = line_map (f i) (g i) c :=
(proj i : (Π i, P i) →ᵃ[k] P i).apply_line_map f g c
end
end affine_map
namespace affine_map
variables {R k V1 P1 V2 : Type*}
section ring
variables [ring k] [add_comm_group V1] [affine_space V1 P1] [add_comm_group V2]
variables [module k V1] [module k V2]
include V1
section distrib_mul_action
variables [monoid R] [distrib_mul_action R V2] [smul_comm_class k R V2]
/-- The space of affine maps to a module inherits an `R`-action from the action on its codomain. -/
instance : distrib_mul_action R (P1 →ᵃ[k] V2) :=
{ smul_add := λ c f g, ext $ λ p, smul_add _ _ _,
smul_zero := λ c, ext $ λ p, smul_zero _ }
end distrib_mul_action
section module
variables [semiring R] [module R V2] [smul_comm_class k R V2]
/-- The space of affine maps taking values in an `R`-module is an `R`-module. -/
instance : module R (P1 →ᵃ[k] V2) :=
{ smul := (•),
add_smul := λ c₁ c₂ f, ext $ λ p, add_smul _ _ _,
zero_smul := λ f, ext $ λ p, zero_smul _ _,
.. affine_map.distrib_mul_action }
variables (R)
/-- The space of affine maps between two modules is linearly equivalent to the product of the
domain with the space of linear maps, by taking the value of the affine map at `(0 : V1)` and the
linear part.
See note [bundled maps over different rings]-/
@[simps] def to_const_prod_linear_map : (V1 →ᵃ[k] V2) ≃ₗ[R] V2 × (V1 →ₗ[k] V2) :=
{ to_fun := λ f, ⟨f 0, f.linear⟩,
inv_fun := λ p, p.2.to_affine_map + const k V1 p.1,
left_inv := λ f, by { ext, rw f.decomp, simp, },
right_inv := by { rintros ⟨v, f⟩, ext; simp, },
map_add' := by simp,
map_smul' := by simp, }
end module
end ring
section comm_ring
variables [comm_ring k] [add_comm_group V1] [affine_space V1 P1] [add_comm_group V2]
variables [module k V1] [module k V2]
include V1
/-- `homothety c r` is the homothety (also known as dilation) about `c` with scale factor `r`. -/
def homothety (c : P1) (r : k) : P1 →ᵃ[k] P1 :=
r • (id k P1 -ᵥ const k P1 c) +ᵥ const k P1 c
lemma homothety_def (c : P1) (r : k) :
homothety c r = r • (id k P1 -ᵥ const k P1 c) +ᵥ const k P1 c :=
rfl
lemma homothety_apply (c : P1) (r : k) (p : P1) : homothety c r p = r • (p -ᵥ c : V1) +ᵥ c := rfl
lemma homothety_eq_line_map (c : P1) (r : k) (p : P1) : homothety c r p = line_map c p r := rfl
@[simp] lemma homothety_one (c : P1) : homothety c (1:k) = id k P1 :=
by { ext p, simp [homothety_apply] }
@[simp] lemma homothety_apply_same (c : P1) (r : k) : homothety c r c = c := line_map_same_apply c r
lemma homothety_mul_apply (c : P1) (r₁ r₂ : k) (p : P1) :
homothety c (r₁ * r₂) p = homothety c r₁ (homothety c r₂ p) :=
by simp [homothety_apply, mul_smul]
lemma homothety_mul (c : P1) (r₁ r₂ : k) :
homothety c (r₁ * r₂) = (homothety c r₁).comp (homothety c r₂) :=
ext $ homothety_mul_apply c r₁ r₂
@[simp] lemma homothety_zero (c : P1) : homothety c (0:k) = const k P1 c :=
by { ext p, simp [homothety_apply] }
@[simp] lemma homothety_add (c : P1) (r₁ r₂ : k) :
homothety c (r₁ + r₂) = r₁ • (id k P1 -ᵥ const k P1 c) +ᵥ homothety c r₂ :=
by simp only [homothety_def, add_smul, vadd_vadd]
/-- `homothety` as a multiplicative monoid homomorphism. -/
def homothety_hom (c : P1) : k →* P1 →ᵃ[k] P1 :=
⟨homothety c, homothety_one c, homothety_mul c⟩
@[simp] lemma coe_homothety_hom (c : P1) : ⇑(homothety_hom c : k →* _) = homothety c := rfl
/-- `homothety` as an affine map. -/
def homothety_affine (c : P1) : k →ᵃ[k] (P1 →ᵃ[k] P1) :=
⟨homothety c, (linear_map.lsmul k _).flip (id k P1 -ᵥ const k P1 c),
function.swap (homothety_add c)⟩
@[simp] lemma coe_homothety_affine (c : P1) :
⇑(homothety_affine c : k →ᵃ[k] _) = homothety c :=
rfl
end comm_ring
end affine_map
section
variables {𝕜 E F : Type*} [ring 𝕜] [add_comm_group E] [add_comm_group F] [module 𝕜 E] [module 𝕜 F]
/-- Applying an affine map to an affine combination of two points yields an affine combination of
the images. -/
lemma convex.combo_affine_apply {x y : E} {a b : 𝕜} {f : E →ᵃ[𝕜] F} (h : a + b = 1) :
f (a • x + b • y) = a • f x + b • f y :=
by { simp only [convex.combo_eq_smul_sub_add h, ←vsub_eq_sub], exact f.apply_line_map _ _ _ }
end
|
07991f9c895f38914321fa3a183d0895bec88abe | c777c32c8e484e195053731103c5e52af26a25d1 | /src/analysis/special_functions/trigonometric/arctan.lean | 10ce7f424c9fcaaa5bc4ffd2455bfa5c9aeb4b22 | [
"Apache-2.0"
] | permissive | kbuzzard/mathlib | 2ff9e85dfe2a46f4b291927f983afec17e946eb8 | 58537299e922f9c77df76cb613910914a479c1f7 | refs/heads/master | 1,685,313,702,744 | 1,683,974,212,000 | 1,683,974,212,000 | 128,185,277 | 1 | 0 | null | 1,522,920,600,000 | 1,522,920,600,000 | null | UTF-8 | Lean | false | false | 7,968 | lean | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import analysis.special_functions.trigonometric.complex
/-!
# The `arctan` function.
Inequalities, derivatives,
and `real.tan` as a `local_homeomorph` between `(-(π / 2), π / 2)` and the whole line.
-/
noncomputable theory
namespace real
open set filter
open_locale topology real
lemma tan_add {x y : ℝ}
(h : ((∀ k : ℤ, x ≠ (2 * k + 1) * π / 2) ∧ ∀ l : ℤ, y ≠ (2 * l + 1) * π / 2)
∨ ((∃ k : ℤ, x = (2 * k + 1) * π / 2) ∧ ∃ l : ℤ, y = (2 * l + 1) * π / 2)) :
tan (x + y) = (tan x + tan y) / (1 - tan x * tan y) :=
by simpa only [← complex.of_real_inj, complex.of_real_sub, complex.of_real_add, complex.of_real_div,
complex.of_real_mul, complex.of_real_tan]
using @complex.tan_add (x:ℂ) (y:ℂ) (by convert h; norm_cast)
lemma tan_add' {x y : ℝ}
(h : ((∀ k : ℤ, x ≠ (2 * k + 1) * π / 2) ∧ ∀ l : ℤ, y ≠ (2 * l + 1) * π / 2)) :
tan (x + y) = (tan x + tan y) / (1 - tan x * tan y) :=
tan_add (or.inl h)
lemma tan_two_mul {x:ℝ} : tan (2 * x) = 2 * tan x / (1 - tan x ^ 2) :=
by simpa only [← complex.of_real_inj, complex.of_real_sub, complex.of_real_div, complex.of_real_pow,
complex.of_real_mul, complex.of_real_tan, complex.of_real_bit0, complex.of_real_one]
using complex.tan_two_mul
lemma tan_ne_zero_iff {θ : ℝ} : tan θ ≠ 0 ↔ ∀ k : ℤ, θ ≠ k * π / 2 :=
by rw [← complex.of_real_ne_zero, complex.of_real_tan, complex.tan_ne_zero_iff]; norm_cast
lemma tan_eq_zero_iff {θ : ℝ} : tan θ = 0 ↔ ∃ k : ℤ, θ = k * π / 2 :=
by rw [← not_iff_not, not_exists, ← ne, tan_ne_zero_iff]
lemma tan_int_mul_pi_div_two (n : ℤ) : tan (n * π/2) = 0 :=
tan_eq_zero_iff.mpr (by use n)
lemma continuous_on_tan : continuous_on tan {x | cos x ≠ 0} :=
begin
suffices : continuous_on (λ x, sin x / cos x) {x | cos x ≠ 0},
{ have h_eq : (λ x, sin x / cos x) = tan, by {ext1 x, rw tan_eq_sin_div_cos, },
rwa h_eq at this, },
exact continuous_on_sin.div continuous_on_cos (λ x, id),
end
@[continuity]
lemma continuous_tan : continuous (λ x : {x | cos x ≠ 0}, tan x) :=
continuous_on_iff_continuous_restrict.1 continuous_on_tan
lemma continuous_on_tan_Ioo : continuous_on tan (Ioo (-(π/2)) (π/2)) :=
begin
refine continuous_on.mono continuous_on_tan (λ x, _),
simp only [and_imp, mem_Ioo, mem_set_of_eq, ne.def],
rw cos_eq_zero_iff,
rintros hx_gt hx_lt ⟨r, hxr_eq⟩,
cases le_or_lt 0 r,
{ rw lt_iff_not_ge at hx_lt,
refine hx_lt _,
rw [hxr_eq, ← one_mul (π / 2), mul_div_assoc, ge_iff_le, mul_le_mul_right (half_pos pi_pos)],
simp [h], },
{ rw lt_iff_not_ge at hx_gt,
refine hx_gt _,
rw [hxr_eq, ← one_mul (π / 2), mul_div_assoc, ge_iff_le, neg_mul_eq_neg_mul,
mul_le_mul_right (half_pos pi_pos)],
have hr_le : r ≤ -1, by rwa [int.lt_iff_add_one_le, ← le_neg_iff_add_nonpos_right] at h,
rw [← le_sub_iff_add_le, mul_comm, ← le_div_iff],
{ norm_num, rw [← int.cast_one, ← int.cast_neg], norm_cast, exact hr_le, },
{ exact zero_lt_two, }, },
end
lemma surj_on_tan : surj_on tan (Ioo (-(π / 2)) (π / 2)) univ :=
have _ := neg_lt_self pi_div_two_pos,
continuous_on_tan_Ioo.surj_on_of_tendsto (nonempty_Ioo.2 this)
(by simp [tendsto_tan_neg_pi_div_two, this]) (by simp [tendsto_tan_pi_div_two, this])
lemma tan_surjective : function.surjective tan :=
λ x, surj_on_tan.subset_range trivial
lemma image_tan_Ioo : tan '' (Ioo (-(π / 2)) (π / 2)) = univ :=
univ_subset_iff.1 surj_on_tan
/-- `real.tan` as an `order_iso` between `(-(π / 2), π / 2)` and `ℝ`. -/
def tan_order_iso : Ioo (-(π / 2)) (π / 2) ≃o ℝ :=
(strict_mono_on_tan.order_iso _ _).trans $ (order_iso.set_congr _ _ image_tan_Ioo).trans
order_iso.set.univ
/-- Inverse of the `tan` function, returns values in the range `-π / 2 < arctan x` and
`arctan x < π / 2` -/
@[pp_nodot] noncomputable def arctan (x : ℝ) : ℝ :=
tan_order_iso.symm x
@[simp] lemma tan_arctan (x : ℝ) : tan (arctan x) = x :=
tan_order_iso.apply_symm_apply x
lemma arctan_mem_Ioo (x : ℝ) : arctan x ∈ Ioo (-(π / 2)) (π / 2) :=
subtype.coe_prop _
@[simp] lemma range_arctan : range arctan = Ioo (-(π / 2)) (π / 2) :=
((equiv_like.surjective _).range_comp _).trans subtype.range_coe
lemma arctan_tan {x : ℝ} (hx₁ : -(π / 2) < x) (hx₂ : x < π / 2) : arctan (tan x) = x :=
subtype.ext_iff.1 $ tan_order_iso.symm_apply_apply ⟨x, hx₁, hx₂⟩
lemma cos_arctan_pos (x : ℝ) : 0 < cos (arctan x) :=
cos_pos_of_mem_Ioo $ arctan_mem_Ioo x
lemma cos_sq_arctan (x : ℝ) : cos (arctan x) ^ 2 = 1 / (1 + x ^ 2) :=
by rw [one_div, ← inv_one_add_tan_sq (cos_arctan_pos x).ne', tan_arctan]
lemma sin_arctan (x : ℝ) : sin (arctan x) = x / sqrt (1 + x ^ 2) :=
by rw [← tan_div_sqrt_one_add_tan_sq (cos_arctan_pos x), tan_arctan]
lemma cos_arctan (x : ℝ) : cos (arctan x) = 1 / sqrt (1 + x ^ 2) :=
by rw [one_div, ← inv_sqrt_one_add_tan_sq (cos_arctan_pos x), tan_arctan]
lemma arctan_lt_pi_div_two (x : ℝ) : arctan x < π / 2 :=
(arctan_mem_Ioo x).2
lemma neg_pi_div_two_lt_arctan (x : ℝ) : -(π / 2) < arctan x :=
(arctan_mem_Ioo x).1
lemma arctan_eq_arcsin (x : ℝ) : arctan x = arcsin (x / sqrt (1 + x ^ 2)) :=
eq.symm $ arcsin_eq_of_sin_eq (sin_arctan x) (mem_Icc_of_Ioo $ arctan_mem_Ioo x)
lemma arcsin_eq_arctan {x : ℝ} (h : x ∈ Ioo (-(1:ℝ)) 1) :
arcsin x = arctan (x / sqrt (1 - x ^ 2)) :=
begin
rw [arctan_eq_arcsin, div_pow, sq_sqrt, one_add_div, div_div,
← sqrt_mul, mul_div_cancel', sub_add_cancel, sqrt_one, div_one];
nlinarith [h.1, h.2],
end
@[simp] lemma arctan_zero : arctan 0 = 0 :=
by simp [arctan_eq_arcsin]
lemma arctan_eq_of_tan_eq {x y : ℝ} (h : tan x = y) (hx : x ∈ Ioo (-(π / 2)) (π / 2)) :
arctan y = x :=
inj_on_tan (arctan_mem_Ioo _) hx (by rw [tan_arctan, h])
@[simp] lemma arctan_one : arctan 1 = π / 4 :=
arctan_eq_of_tan_eq tan_pi_div_four $ by split; linarith [pi_pos]
@[simp] lemma arctan_neg (x : ℝ) : arctan (-x) = - arctan x :=
by simp [arctan_eq_arcsin, neg_div]
lemma arctan_eq_arccos {x : ℝ} (h : 0 ≤ x) : arctan x = arccos ((sqrt (1 + x ^ 2))⁻¹) :=
begin
rw [arctan_eq_arcsin, arccos_eq_arcsin], swap, { exact inv_nonneg.2 (sqrt_nonneg _) },
congr' 1,
rw [←sqrt_inv, sq_sqrt, ←one_div, one_sub_div, add_sub_cancel', sqrt_div, sqrt_sq h],
all_goals { positivity }
end
-- The junk values for `arccos` and `sqrt` make this true even for `1 < x`.
lemma arccos_eq_arctan {x : ℝ} (h : 0 < x) :
arccos x = arctan (sqrt (1 - x ^ 2) / x) :=
begin
rw [arccos, eq_comm],
refine arctan_eq_of_tan_eq _ ⟨_, _⟩,
{ rw [tan_pi_div_two_sub, tan_arcsin, inv_div] },
{ linarith only [arcsin_le_pi_div_two x, pi_pos] },
{ linarith only [arcsin_pos.2 h] }
end
@[continuity]
lemma continuous_arctan : continuous arctan :=
continuous_subtype_coe.comp tan_order_iso.to_homeomorph.continuous_inv_fun
lemma continuous_at_arctan {x : ℝ} : continuous_at arctan x := continuous_arctan.continuous_at
/-- `real.tan` as a `local_homeomorph` between `(-(π / 2), π / 2)` and the whole line. -/
def tan_local_homeomorph : local_homeomorph ℝ ℝ :=
{ to_fun := tan,
inv_fun := arctan,
source := Ioo (-(π / 2)) (π / 2),
target := univ,
map_source' := maps_to_univ _ _,
map_target' := λ y hy, arctan_mem_Ioo y,
left_inv' := λ x hx, arctan_tan hx.1 hx.2,
right_inv' := λ y hy, tan_arctan y,
open_source := is_open_Ioo,
open_target := is_open_univ,
continuous_to_fun := continuous_on_tan_Ioo,
continuous_inv_fun := continuous_arctan.continuous_on }
@[simp] lemma coe_tan_local_homeomorph : ⇑tan_local_homeomorph = tan := rfl
@[simp] lemma coe_tan_local_homeomorph_symm : ⇑tan_local_homeomorph.symm = arctan := rfl
end real
|
75bd964a78bf8bc6737d92984e2628f8510e03ed | 367134ba5a65885e863bdc4507601606690974c1 | /src/group_theory/perm/subgroup.lean | ef0d70cae526760db29b4db65c8ddb704815f9fa | [
"Apache-2.0"
] | permissive | kodyvajjha/mathlib | 9bead00e90f68269a313f45f5561766cfd8d5cad | b98af5dd79e13a38d84438b850a2e8858ec21284 | refs/heads/master | 1,624,350,366,310 | 1,615,563,062,000 | 1,615,563,062,000 | 162,666,963 | 0 | 0 | Apache-2.0 | 1,545,367,651,000 | 1,545,367,651,000 | null | UTF-8 | Lean | false | false | 2,617 | lean | /-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import group_theory.perm.basic
import data.fintype.basic
import group_theory.subgroup
/-!
# Lemmas about subgroups within the permutations (self-equivalences) of a type `α`
This file provides extra lemmas about some `subgroup`s that exist within `equiv.perm α`.
`group_theory.subgroup` depends on `group_theory.perm.basic`, so these need to be in a separate
file.
It also provides decidable instances on membership in these subgroups, since
`monoid_hom.decidable_mem_range` cannot be inferred without the help of a lambda.
The presence of these instances induces a `fintype` instance on the `quotient_group.quotient` of
these subgroups.
-/
namespace equiv
namespace perm
universes u
instance sum_congr_hom.decidable_mem_range {α β : Type*}
[decidable_eq α] [decidable_eq β] [fintype α] [fintype β] :
decidable_pred (λ x, x ∈ (sum_congr_hom α β).range) :=
λ x, infer_instance
@[simp]
lemma sum_congr_hom.card_range {α β : Type*}
[fintype (sum_congr_hom α β).range] [fintype (perm α × perm β)] :
fintype.card (sum_congr_hom α β).range = fintype.card (perm α × perm β) :=
fintype.card_eq.mpr ⟨(set.range (sum_congr_hom α β) sum_congr_hom_injective).symm⟩
instance sigma_congr_right_hom.decidable_mem_range {α : Type*} {β : α → Type*}
[decidable_eq α] [∀ a, decidable_eq (β a)] [fintype α] [∀ a, fintype (β a)] :
decidable_pred (λ x, x ∈ (sigma_congr_right_hom β).range) :=
λ x, infer_instance
@[simp]
lemma sigma_congr_right_hom.card_range {α : Type*} {β : α → Type*}
[fintype (sigma_congr_right_hom β).range] [fintype (Π a, perm (β a))] :
fintype.card (sigma_congr_right_hom β).range = fintype.card (Π a, perm (β a)) :=
fintype.card_eq.mpr ⟨(set.range (sigma_congr_right_hom β) sigma_congr_right_hom_injective).symm⟩
instance subtype_congr_hom.decidable_mem_range {α : Type*} (p : α → Prop) [decidable_pred p]
[fintype (perm {a // p a} × perm {a // ¬ p a})] [decidable_eq (perm α)] :
decidable_pred (λ x, x ∈ (subtype_congr_hom p).range) :=
λ x, infer_instance
@[simp]
lemma subtype_congr_hom.card_range {α : Type*} (p : α → Prop) [decidable_pred p]
[fintype (subtype_congr_hom p).range] [fintype (perm {a // p a} × perm {a // ¬ p a})] :
fintype.card (subtype_congr_hom p).range = fintype.card (perm {a // p a} × perm {a // ¬ p a}) :=
fintype.card_eq.mpr ⟨(set.range (subtype_congr_hom p) (subtype_congr_hom_injective p)).symm⟩
end perm
end equiv
|
672cbab43759b09537d9f8dea2071d689fcccc70 | e898bfefd5cb60a60220830c5eba68cab8d02c79 | /uexp/src/uexp/rules/decorrelateExists.lean | 8869aaec24fbc9c112f3e523c13efce8f1e08c8d | [
"BSD-2-Clause"
] | permissive | kkpapa/Cosette | 9ed09e2dc4c1ecdef815c30b5501f64a7383a2ce | fda8fdbbf0de6c1be9b4104b87bbb06cede46329 | refs/heads/master | 1,584,573,128,049 | 1,526,370,422,000 | 1,526,370,422,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 1,626 | lean | import ..sql
import ..tactics
import ..u_semiring
import ..cosette_lemmas
import ..extra_constants
import ..TDP
import ..cosette_tactics
open Expr
open Proj
open Pred
open SQL
variable i : const datatypes.int
theorem rule :
forall (Γ scm_emp : Schema)
(rel_emp : relation scm_emp)
(emp_empno : Column datatypes.int scm_emp)
(emp_ename : Column datatypes.int scm_emp)
(emp_job : Column datatypes.int scm_emp)
(emp_mgr : Column datatypes.int scm_emp)
(emp_hiredate : Column datatypes.int scm_emp)
(emp_comm : Column datatypes.int scm_emp)
(emp_sal : Column datatypes.int scm_emp)
(emp_deptno : Column datatypes.int scm_emp)
(emp_slacker : Column datatypes.int scm_emp),
denoteSQL
(SELECT *
FROM1 (table rel_emp
WHERE EXISTS (SELECT *
FROM1 table rel_emp
WHERE (equal (uvariable (left⋅right⋅emp_deptno)) (uvariable (right⋅emp_deptno)))))
: SQL Γ _) =
denoteSQL
(SELECT1 (right⋅left⋅star)
FROM1 (product (table rel_emp)
(DISTINCT (SELECT1 (combine (right⋅emp_deptno)
(e2p (constantExpr i)))
FROM1 table rel_emp))
WHERE (equal (uvariable (right⋅left⋅emp_deptno))
(uvariable (right⋅right⋅left)))) : SQL Γ _) :=
begin
intros,
unfold_all_denotations,
funext,
simp,
print_size,
apply ueq_symm,
-- remove_dup_sigs_lhs,
sorry
end |
99ae0c5793f222d08a205070be0ef24af52006b1 | df7bb3acd9623e489e95e85d0bc55590ab0bc393 | /lean/love11_logical_foundations_of_mathematics_exercise_solution.lean | 474f39fb2c8d19969f14b1bce5fd146e5106d450 | [] | no_license | MaschavanderMarel/logical_verification_2020 | a41c210b9237c56cb35f6cd399e3ac2fe42e775d | 7d562ef174cc6578ca6013f74db336480470b708 | refs/heads/master | 1,692,144,223,196 | 1,634,661,675,000 | 1,634,661,675,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 2,860 | lean | import .love11_logical_foundations_of_mathematics_demo
/- # LoVe Exercise 11: Logical Foundations of Mathematics -/
set_option pp.beta true
set_option pp.generalized_field_notation false
namespace LoVe
/- ## Question 1: Vectors as Subtypes
Recall the definition of vectors from the demo: -/
#check vector
/- The following function adds two lists of integers elementwise. If one
function is longer than the other, the tail of the longer function is
truncated. -/
def list.add : list ℤ → list ℤ → list ℤ
| [] [] := []
| (x :: xs) (y :: ys) := (x + y) :: list.add xs ys
| [] (y :: ys) := []
| (x :: xs) [] := []
/- 1.1. Show that if the lists have the same length, the resulting list also
has that length. -/
lemma list.length_add :
∀(xs : list ℤ) (ys : list ℤ) (h : list.length xs = list.length ys),
list.length (list.add xs ys) = list.length xs
| [] [] :=
by simp [list.add]
| (x :: xs) (y :: ys) :=
begin
simp [list.add, length],
intro h,
rw list.length_add xs ys h
end
| [] (y :: ys) :=
begin
intro h,
cases' h
end
| (x :: xs) [] :=
begin
intro h,
cases' h
end
/- 1.2. Define componentwise addition on vectors using `list.add` and
`list.length_add`. -/
def vector.add {n : ℕ} : vector ℤ n → vector ℤ n → vector ℤ n
| u v := subtype.mk (list.add (subtype.val u) (subtype.val v))
begin
rw list.length_add,
{ exact subtype.property u },
{ rw subtype.property u, rw subtype.property v }
end
/- 1.3. Show that `list.add` and `vector.add` are commutative. -/
lemma list.add.comm :
∀(xs : list ℤ) (ys : list ℤ), list.add xs ys = list.add ys xs
| [] [] := by refl
| (x :: xs) (y :: ys) := by simp [list.add, add_comm]; exact list.add.comm xs ys
| [] (y :: ys) := by refl
| (x :: xs) [] := by refl
lemma vector.add.comm {n : ℕ} (x y : vector ℤ n) :
vector.add x y = vector.add y x :=
begin
apply subtype.eq,
simp [vector.add],
apply list.add.comm
end
/- ## Question 2: Integers as Quotients
Recall the construction of integers from the lecture, not to be confused with
Lean's predefined type `int` (= `ℤ`): -/
#check int.rel
#check int.rel_iff
#check int
/- 2.1. Define negation on these integers. -/
def int.neg : int → int :=
quotient.lift (λpn : ℕ × ℕ, ⟦(prod.snd pn, prod.fst pn)⟧)
begin
intros a a' ha,
cases' a with p n,
cases' a' with p' n',
apply quotient.sound,
simp at *,
rw int.rel_iff at *,
cc
end
/- 2.2. Prove the following lemmas about negation. -/
lemma int.neg_eq (p n : ℕ) :
int.neg ⟦(p, n)⟧ = ⟦(n, p)⟧ :=
by refl
lemma int.neg_neg (a : int) :
int.neg (int.neg a) = a :=
begin
apply quotient.induction_on a,
intro a,
cases' a,
simp [int.neg_eq]
end
end LoVe
|
ddac99be788ef3a6b8fa4e0de98736c46d6e44b4 | 856e2e1615a12f95b551ed48fa5b03b245abba44 | /src/linear_algebra/direct_sum/tensor_product.lean | 060e60bace54a08435cfdf293d418c69390da1f1 | [
"Apache-2.0"
] | permissive | pimsp/mathlib | 8b77e1ccfab21703ba8fbe65988c7de7765aa0e5 | 913318ca9d6979686996e8d9b5ebf7e74aae1c63 | refs/heads/master | 1,669,812,465,182 | 1,597,133,610,000 | 1,597,133,610,000 | 281,890,685 | 1 | 0 | null | 1,595,491,577,000 | 1,595,491,576,000 | null | UTF-8 | Lean | false | false | 2,262 | lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro
-/
import linear_algebra.tensor_product
import linear_algebra.direct_sum_module
section ring
namespace tensor_product
open_locale tensor_product
open linear_map
variables (R : Type*) [comm_ring R]
variables (ι₁ : Type*) (ι₂ : Type*)
variables [decidable_eq ι₁] [decidable_eq ι₂]
variables (M₁ : ι₁ → Type*) (M₂ : ι₂ → Type*)
variables [Π i₁, add_comm_group (M₁ i₁)] [Π i₂, add_comm_group (M₂ i₂)]
variables [Π i₁, module R (M₁ i₁)] [Π i₂, module R (M₂ i₂)]
/-- The linear equivalence `(⊕ i₁, M₁ i₁) ⊗ (⊕ i₂, M₂ i₂) ≃ (⊕ i₁, ⊕ i₂, M₁ i₁ ⊗ M₂ i₂)`, i.e.
"tensor product distributes over direct sum". -/
def direct_sum :
direct_sum ι₁ M₁ ⊗[R] direct_sum ι₂ M₂ ≃ₗ[R] direct_sum (ι₁ × ι₂) (λ i, M₁ i.1 ⊗[R] M₂ i.2) :=
begin
refine linear_equiv.of_linear
(lift $ direct_sum.to_module R _ _ $ λ i₁, flip $ direct_sum.to_module R _ _ $ λ i₂,
flip $ curry $ direct_sum.lof R (ι₁ × ι₂) (λ i, M₁ i.1 ⊗[R] M₂ i.2) (i₁, i₂))
(direct_sum.to_module R _ _ $ λ i, map (direct_sum.lof R _ _ _) (direct_sum.lof R _ _ _))
(linear_map.ext $ direct_sum.to_module.ext _ $ λ i, mk_compr₂_inj $
linear_map.ext $ λ x₁, linear_map.ext $ λ x₂, _)
(mk_compr₂_inj $ linear_map.ext $ direct_sum.to_module.ext _ $ λ i₁, linear_map.ext $ λ x₁,
linear_map.ext $ direct_sum.to_module.ext _ $ λ i₂, linear_map.ext $ λ x₂, _),
repeat { rw compr₂_apply <|> rw comp_apply <|> rw id_apply <|> rw mk_apply <|>
rw direct_sum.to_module_lof <|> rw map_tmul <|> rw lift.tmul <|> rw flip_apply <|>
rw curry_apply },
cases i; refl
end
@[simp] theorem direct_sum_lof_tmul_lof (i₁ : ι₁) (m₁ : M₁ i₁) (i₂ : ι₂) (m₂ : M₂ i₂) :
direct_sum R ι₁ ι₂ M₁ M₂ (direct_sum.lof R ι₁ M₁ i₁ m₁ ⊗ₜ direct_sum.lof R ι₂ M₂ i₂ m₂) =
direct_sum.lof R (ι₁ × ι₂) (λ i, M₁ i.1 ⊗[R] M₂ i.2) (i₁, i₂) (m₁ ⊗ₜ m₂) :=
by simp [direct_sum]
end tensor_product
end ring
|
b00461034c4e7b0661b5eb00c8dc540aa7dfdc38 | 42610cc2e5db9c90269470365e6056df0122eaa0 | /library/theories/group_theory/basic.lean | b1bfe3d165f8c7a0dc7a52b06c16fd9600731d3d | [
"Apache-2.0"
] | permissive | tomsib2001/lean | 2ab59bfaebd24a62109f800dcf4a7139ebd73858 | eb639a7d53fb40175bea5c8da86b51d14bb91f76 | refs/heads/master | 1,586,128,387,740 | 1,468,968,950,000 | 1,468,968,950,000 | 61,027,234 | 0 | 0 | null | 1,465,813,585,000 | 1,465,813,585,000 | null | UTF-8 | Lean | false | false | 39,831 | lean | /-
Copyright (c) 2016 Andrew Zipperer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Zipperer, Jeremy Avigad
Basic group theory: subgroups, homomorphisms on a set, homomorphic images, cosets,
normal cosets and the normalizer, the kernel of a homomorphism, the centralizer, etc.
For notation a * S and S * a for cosets, open the namespace "coset_notation".
For notation a^b and S^a, open the namespace "conj_notation".
TODO: homomorphisms on sets should be refactored and moved to algebra.
-/
import data.set algebra.homomorphism theories.topology.basic theories.move
open eq.ops set function
namespace group_theory
variables {A B C : Type}
/- subgroups -/
structure is_one_closed [class] [has_one A] (S : set A) : Prop :=
(one_mem : one ∈ S)
proposition one_mem [has_one A] {S : set A} [is_one_closed S] : 1 ∈ S :=
is_one_closed.one_mem _ S
structure is_mul_closed [class] [has_mul A] (S : set A) : Prop :=
(mul_mem : ∀₀ a ∈ S, ∀₀ b ∈ S, a * b ∈ S)
proposition mul_mem [has_mul A] {S : set A} [is_mul_closed S] {a b : A} (aS : a ∈ S) (bS : b ∈ S) :
a * b ∈ S :=
is_mul_closed.mul_mem _ S aS bS
structure is_inv_closed [class] [has_inv A] (S : set A) : Prop :=
(inv_mem : ∀₀ a ∈ S, a⁻¹ ∈ S)
proposition inv_mem [has_inv A] {S : set A} [is_inv_closed S] {a : A} (aS : a ∈ S) : a⁻¹ ∈ S :=
is_inv_closed.inv_mem _ S aS
structure is_subgroup [class] [group A] (S : set A)
extends is_one_closed S, is_mul_closed S, is_inv_closed S : Prop
section groupA
variable [group A]
proposition mem_of_inv_mem {a : A} {S : set A} [is_subgroup S] (H : a⁻¹ ∈ S) : a ∈ S :=
have (a⁻¹)⁻¹ ∈ S, from inv_mem H,
by rewrite inv_inv at this; apply this
proposition inv_mem_iff (a : A) (S : set A) [is_subgroup S] : a⁻¹ ∈ S ↔ a ∈ S :=
iff.intro mem_of_inv_mem inv_mem
proposition is_subgroup_univ [instance] : is_subgroup (@univ A) :=
⦃ is_subgroup,
one_mem := trivial,
mul_mem := λ a au b bu, trivial,
inv_mem := λ a au, trivial ⦄
proposition is_subgroup_inter [instance] (G H : set A) [is_subgroup G] [is_subgroup H] :
is_subgroup (G ∩ H) :=
⦃ is_subgroup,
one_mem := and.intro one_mem one_mem,
mul_mem := λ a ai b bi, and.intro (mul_mem (and.left ai) (and.left bi))
(mul_mem (and.right ai) (and.right bi)),
inv_mem := λ a ai, and.intro (inv_mem (and.left ai)) (inv_mem (and.right ai)) ⦄
end groupA
/- homomorphisms on sets -/
section has_mulABC
variables [has_mul A] [has_mul B] [has_mul C]
-- in group theory, we can use is_hom for is_mul_hom
abbreviation is_hom := @is_mul_hom
definition is_hom_on [class] (f : A → B) (S : set A) : Prop :=
∀₀ a₁ ∈ S, ∀₀ a₂ ∈ S, f (a₁ * a₂) = f a₁ * f a₂
proposition hom_on_mul (f : A → B) {S : set A} [H : is_hom_on f S] {a₁ a₂ : A}
(a₁S : a₁ ∈ S) (a₂S : a₂ ∈ S) : f (a₁ * a₂) = (f a₁) * (f a₂) :=
H a₁S a₂S
proposition is_hom_on_of_is_hom (f : A → B) (S : set A) [H : is_hom f] : is_hom_on f S :=
forallb_of_forall₂ S S (hom_mul f)
proposition is_hom_of_is_hom_on_univ (f : A → B) [H : is_hom_on f univ] : is_hom f :=
is_mul_hom.mk (forall_of_forallb_univ₂ H)
proposition is_hom_on_univ_iff (f : A → B) : is_hom_on f univ ↔ is_hom f :=
iff.intro (λH, is_hom_of_is_hom_on_univ f) (λ H, is_hom_on_of_is_hom f univ)
proposition is_hom_on_of_subset (f : A → B) {S T : set A} (ssubt : S ⊆ T) [H : is_hom_on f T] :
is_hom_on f S :=
forallb_of_subset₂ ssubt ssubt H
proposition is_hom_on_id (S : set A) : is_hom_on id S :=
have H : is_hom (@id A), from is_mul_hom_id,
is_hom_on_of_is_hom id S
proposition is_hom_on_comp {S : set A} {T : set B} {g : B → C} {f : A → B}
(H₁ : is_hom_on f S) (H₂ : is_hom_on g T) (H₃ : maps_to f S T) : is_hom_on (g ∘ f) S :=
take a₁, assume a₁S, take a₂, assume a₂S,
have f a₁ ∈ T, from H₃ a₁S,
have f a₂ ∈ T, from H₃ a₂S,
show g (f (a₁ * a₂)) = g (f a₁) * g (f a₂), by rewrite [H₁ a₁S a₂S, H₂ `f a₁ ∈ T` `f a₂ ∈ T`]
end has_mulABC
section groupAB
variables [group A] [group B]
proposition hom_on_one (f : A → B) (G : set A) [is_subgroup G] [H : is_hom_on f G] : f 1 = 1 :=
have f 1 * f 1 = f 1 * 1, by rewrite [-H one_mem one_mem, *mul_one],
eq_of_mul_eq_mul_left' this
proposition hom_on_inv (f : A → B) {G : set A} [is_subgroup G] [H : is_hom_on f G]
{a : A} (aG : a ∈ G) :
f a⁻¹ = (f a)⁻¹ :=
have f a⁻¹ * f a = 1, by rewrite [-H (inv_mem aG) aG, mul.left_inv, hom_on_one f G],
eq_inv_of_mul_eq_one this
proposition is_subgroup_image [instance] (f : A → B) (G : set A)
[is_subgroup G] [is_hom_on f G] :
is_subgroup (f ' G) :=
⦃ is_subgroup,
one_mem := mem_image one_mem (hom_on_one f G),
mul_mem := λ a afG b bfG,
obtain c (cG : c ∈ G)(Hc : f c = a), from afG,
obtain d (dG : d ∈ G)(Hd : f d = b), from bfG,
show a * b ∈ f ' G, from mem_image (mul_mem cG dG) (by rewrite [hom_on_mul f cG dG, Hc, Hd]),
inv_mem := λ a afG,
obtain c (cG : c ∈ G)(Hc : f c = a), from afG,
show a⁻¹ ∈ f ' G, from mem_image (inv_mem cG) (by rewrite [hom_on_inv f cG, Hc]) ⦄
end groupAB
/- cosets -/
definition lcoset [has_mul A] (a : A) (N : set A) : set A := (mul a) 'N
definition rcoset [has_mul A] (N : set A) (a : A) : set A := (mul^~ a) 'N
-- overload multiplication
namespace coset_notation
infix * := lcoset
infix * := rcoset
end coset_notation
open coset_notation
section has_mulA
variable [has_mul A]
proposition mul_mem_lcoset {S : set A} {x : A} (a : A) (xS : x ∈ S) : a * x ∈ a * S :=
mem_image_of_mem (mul a) xS
proposition mul_mem_rcoset [has_mul A] {S : set A} {x : A} (xS : x ∈ S) (a : A) :
x * a ∈ S * a :=
mem_image_of_mem (mul^~ a) xS
definition lcoset_equiv (S : set A) (a b : A) : Prop := a * S = b * S
proposition equivalence_lcoset_equiv (S : set A) : equivalence (lcoset_equiv S) :=
mk_equivalence (lcoset_equiv S) (λ a, rfl) (λ a b, !eq.symm) (λ a b c, !eq.trans)
proposition lcoset_subset_lcoset {S T : set A} (a : A) (H : S ⊆ T) : a * S ⊆ a * T :=
image_subset _ H
proposition rcoset_subset_rcoset {S T : set A} (H : S ⊆ T) (a : A) : S * a ⊆ T * a :=
image_subset _ H
proposition image_lcoset_of_is_hom_on {B : Type} [has_mul B] {f : A → B} {S : set A} {a : A}
{G : set A} (SsubG : S ⊆ G) (aG : a ∈ G) [is_hom_on f G] :
f ' (a * S) = f a * f ' S :=
ext (take x, iff.intro
(assume fas : x ∈ f ' (a * S),
obtain t [s (sS : s ∈ S) (seq : a * s = t)] (teq : f t = x), from fas,
have x = f a * f s, by rewrite [-teq, -seq, hom_on_mul f aG (SsubG sS)],
show x ∈ f a * f ' S, by rewrite this; apply mul_mem_lcoset _ (mem_image_of_mem _ sS))
(assume fafs : x ∈ f a * f ' S,
obtain t [s (sS : s ∈ S) (seq : f s = t)] (teq : f a * t = x), from fafs,
have x = f (a * s), by rewrite [-teq, -seq, hom_on_mul f aG (SsubG sS)],
show x ∈ f ' (a * S), by rewrite this; exact mem_image_of_mem _ (mul_mem_lcoset _ sS)))
proposition image_rcoset_of_is_hom_on {B : Type} [has_mul B] {f : A → B} {S : set A} {a : A}
{G : set A} (SsubG : S ⊆ G) (aG : a ∈ G) [is_hom_on f G] :
f ' (S * a) = f ' S * f a :=
ext (take x, iff.intro
(assume fas : x ∈ f ' (S * a),
obtain t [s (sS : s ∈ S) (seq : s * a = t)] (teq : f t = x), from fas,
have x = f s * f a, by rewrite [-teq, -seq, hom_on_mul f (SsubG sS) aG],
show x ∈ f ' S * f a, by rewrite this; exact mul_mem_rcoset (mem_image_of_mem _ sS) _)
(assume fafs : x ∈ f ' S * f a,
obtain t [s (sS : s ∈ S) (seq : f s = t)] (teq : t * f a = x), from fafs,
have x = f (s * a), by rewrite [-teq, -seq, hom_on_mul f (SsubG sS) aG],
show x ∈ f ' (S * a), by rewrite this; exact mem_image_of_mem _ (mul_mem_rcoset sS _)))
proposition image_lcoset_of_is_hom {B : Type} [has_mul B] (f : A → B) (a : A) (S : set A)
[is_hom f] :
f ' (a * S) = f a * f ' S :=
have is_hom_on f univ, from is_hom_on_of_is_hom f univ,
image_lcoset_of_is_hom_on (subset_univ S) !mem_univ
proposition image_rcoset_of_is_hom {B : Type} [has_mul B] (f : A → B) (S : set A) (a : A)
[is_hom f] :
f ' (S * a) = f ' S * f a :=
have is_hom_on f univ, from is_hom_on_of_is_hom f univ,
image_rcoset_of_is_hom_on (subset_univ S) !mem_univ
end has_mulA
section semigroupA
variable [semigroup A]
proposition rcoset_rcoset (S : set A) (a b : A) : S * a * b = S * (a * b) :=
have H : (mul^~ b) ∘ (mul^~ a) = mul^~ (a * b), from funext (take x, !mul.assoc),
calc
S * a * b = ((mul^~ b) ∘ (mul^~ a)) 'S : image_comp
... = S * (a * b) : by rewrite [↑rcoset, H]
proposition lcoset_lcoset (S : set A) (a b : A) : a * (b * S) = (a * b) * S :=
have H : (mul a) ∘ (mul b) = mul (a * b), from funext (take x, !mul.assoc⁻¹),
calc
a * (b * S) = ((mul a) ∘ (mul b)) 'S : image_comp
... = (a * b) * S : by rewrite [↑lcoset, H]
proposition lcoset_rcoset [semigroup A] (S : set A) (a b : A) : a * S * b = a * (S * b) :=
have H : (mul^~ b) ∘ (mul a) = (mul a) ∘ (mul^~ b), from funext (take x, !mul.assoc),
calc
a * S * b = ((mul^~ b) ∘ (mul a)) 'S : image_comp
... = ((mul a) ∘ (mul^~ b)) 'S : H
... = a * (S * b) : image_comp
end semigroupA
section monoidA
variable [monoid A]
proposition one_lcoset (S : set A) : 1 * S = S :=
ext (take x, iff.intro
(suppose x ∈ 1 * S,
obtain s (sS : s ∈ S) (eqx : 1 * s = x), from this,
show x ∈ S, by rewrite [-eqx, one_mul]; apply sS)
(suppose x ∈ S,
have 1 * x ∈ 1 * S, from mem_image_of_mem (mul 1) this,
show x ∈ 1 * S, by rewrite one_mul at this; apply this))
proposition rcoset_one (S : set A) : S * 1 = S :=
ext (take x, iff.intro
(suppose x ∈ S * 1,
obtain s (sS : s ∈ S) (eqx : s * 1 = x), from this,
show x ∈ S, by rewrite [-eqx, mul_one]; apply sS)
(suppose x ∈ S,
have x * 1 ∈ S * 1, from mem_image_of_mem (mul^~ 1) this,
show x ∈ S * 1, by rewrite mul_one at this; apply this))
end monoidA
section groupA
variable [group A]
proposition lcoset_inv_lcoset (a : A) (S : set A) : a * (a⁻¹ * S) = S :=
by rewrite [lcoset_lcoset, mul.right_inv, one_lcoset]
proposition inv_lcoset_lcoset (a : A) (S : set A) : a⁻¹ * (a * S) = S :=
by rewrite [lcoset_lcoset, mul.left_inv, one_lcoset]
proposition rcoset_inv_rcoset (S : set A) (a : A) : (S * a⁻¹) * a = S :=
by rewrite [rcoset_rcoset, mul.left_inv, rcoset_one]
proposition rcoset_rcoset_inv (S : set A) (a : A) : (S * a) * a⁻¹ = S :=
by rewrite [rcoset_rcoset, mul.right_inv, rcoset_one]
proposition eq_of_lcoset_eq_lcoset {a : A} {S T : set A} (H : a * S = a * T) : S = T :=
by rewrite [-inv_lcoset_lcoset a S, -inv_lcoset_lcoset a T, H]
proposition eq_of_rcoset_eq_rcoset {a : A} {S T : set A} (H : S * a = T * a) : S = T :=
by rewrite [-rcoset_rcoset_inv S a, -rcoset_rcoset_inv T a, H]
proposition mem_of_mul_mem_lcoset {a b : A} {S : set A} (abaS : a * b ∈ a * S) : b ∈ S :=
have a⁻¹ * (a * b) ∈ a⁻¹ * (a * S), from mul_mem_lcoset _ abaS,
by rewrite [inv_mul_cancel_left at this, inv_lcoset_lcoset at this]; apply this
proposition mul_mem_lcoset_iff (a b : A) (S : set A) : a * b ∈ a * S ↔ b ∈ S :=
iff.intro !mem_of_mul_mem_lcoset !mul_mem_lcoset
proposition mem_of_mul_mem_rcoset {a b : A} {S : set A} (abSb : a * b ∈ S * b) : a ∈ S :=
have (a * b) * b⁻¹ ∈ (S * b) * b⁻¹, from mul_mem_rcoset abSb _,
by rewrite [mul_inv_cancel_right at this, rcoset_rcoset_inv at this]; apply this
proposition mul_mem_rcoset_iff (a b : A) (S : set A) : a * b ∈ S * b ↔ a ∈ S :=
iff.intro !mem_of_mul_mem_rcoset (λ H, mul_mem_rcoset H _)
proposition inv_mul_mem_of_mem_lcoset {a b : A} {S : set A} (abS : a ∈ b * S) : b⁻¹ * a ∈ S :=
have b⁻¹ * a ∈ b⁻¹ * (b * S), from mul_mem_lcoset b⁻¹ abS,
by rewrite inv_lcoset_lcoset at this; apply this
proposition mem_lcoset_of_inv_mul_mem {a b : A} {S : set A} (H : b⁻¹ * a ∈ S) : a ∈ b * S :=
have b * (b⁻¹ * a) ∈ b * S, from mul_mem_lcoset b H,
by rewrite mul_inv_cancel_left at this; apply this
proposition mem_lcoset_iff (a b : A) (S : set A) : a ∈ b * S ↔ b⁻¹ * a ∈ S :=
iff.intro inv_mul_mem_of_mem_lcoset mem_lcoset_of_inv_mul_mem
proposition mul_inv_mem_of_mem_rcoset {a b : A} {S : set A} (aSb : a ∈ S * b) : a * b⁻¹ ∈ S :=
have a * b⁻¹ ∈ (S * b) * b⁻¹, from mul_mem_rcoset aSb b⁻¹,
by rewrite rcoset_rcoset_inv at this; apply this
proposition mem_rcoset_of_mul_inv_mem {a b : A} {S : set A} (H : a * b⁻¹ ∈ S) : a ∈ S * b :=
have a * b⁻¹ * b ∈ S * b, from mul_mem_rcoset H b,
by rewrite inv_mul_cancel_right at this; apply this
proposition mem_rcoset_iff (a b : A) (S : set A) : a ∈ S * b ↔ a * b⁻¹ ∈ S :=
iff.intro mul_inv_mem_of_mem_rcoset mem_rcoset_of_mul_inv_mem
proposition lcoset_eq_iff_eq_inv_lcoset (a : A) (S T : set A) : (a * S = T) ↔ (S = a⁻¹ * T) :=
iff.intro (assume H, by rewrite [-H, inv_lcoset_lcoset])
(assume H, by rewrite [H, lcoset_inv_lcoset])
proposition rcoset_eq_iff_eq_rcoset_inv (a : A) (S T : set A) : (S * a = T) ↔ (S = T * a⁻¹) :=
iff.intro (assume H, by rewrite [-H, rcoset_rcoset_inv])
(assume H, by rewrite [H, rcoset_inv_rcoset])
proposition lcoset_inter (a : A) (S T : set A) [is_subgroup S] [is_subgroup T] :
a * (S ∩ T) = (a * S) ∩ (a * T) :=
eq_of_subset_of_subset
(image_inter_subset _ S T)
(take b, suppose b ∈ (a * S) ∩ (a * T),
obtain [s [smem (seq : a * s = b)]] [t [tmem (teq : a * t = b)]], from this,
have s = t, from eq_of_mul_eq_mul_left' (eq.trans seq (eq.symm teq)),
show b ∈ a * (S ∩ T),
begin
rewrite -seq,
apply mul_mem_lcoset,
apply and.intro smem,
rewrite this, apply tmem
end)
proposition inter_rcoset (a : A) (S T : set A) [is_subgroup S] [is_subgroup T] :
(S ∩ T) * a = (S * a) ∩ (T * a) :=
eq_of_subset_of_subset
(image_inter_subset _ S T)
(take b, suppose b ∈ (S * a) ∩ (T * a),
obtain [s [smem (seq : s * a = b)]] [t [tmem (teq : t * a = b)]], from this,
have s = t, from eq_of_mul_eq_mul_right' (eq.trans seq (eq.symm teq)),
show b ∈ (S ∩ T) * a,
begin
rewrite -seq,
apply mul_mem_rcoset,
apply and.intro smem,
rewrite this, apply tmem
end)
end groupA
section subgroupG
variables [group A] {G : set A} [is_subgroup G]
proposition lcoset_eq_self_of_mem {a : A} (aG : a ∈ G) : a * G = G :=
ext (take x, iff.intro
(assume xaG, obtain g [gG xeq], from xaG,
show x ∈ G, by rewrite -xeq; exact (mul_mem aG gG))
(assume xG, show x ∈ a * G, from mem_image
(show a⁻¹ * x ∈ G, from (mul_mem (inv_mem aG) xG)) !mul_inv_cancel_left))
proposition rcoset_eq_self_of_mem {a : A} (aG : a ∈ G) : G * a = G :=
ext (take x, iff.intro
(assume xGa, obtain g [gG xeq], from xGa,
show x ∈ G, by rewrite -xeq; exact (mul_mem gG aG))
(assume xG, show x ∈ G * a, from mem_image
(show x * a⁻¹ ∈ G, from (mul_mem xG (inv_mem aG))) !inv_mul_cancel_right))
proposition mem_lcoset_self (a : A) : a ∈ a * G :=
by rewrite [-mul_one a at {1}]; exact mul_mem_lcoset a one_mem
proposition mem_rcoset_self (a : A) : a ∈ G * a :=
by rewrite [-one_mul a at {1}]; exact mul_mem_rcoset one_mem a
proposition mem_of_lcoset_eq_self {a : A} (H : a * G = G) : a ∈ G :=
by rewrite [-H]; exact mem_lcoset_self a
proposition mem_of_rcoset_eq_self {a : A} (H : G * a = G) : a ∈ G :=
by rewrite [-H]; exact mem_rcoset_self a
variable (G)
proposition lcoset_eq_self_iff (a : A) : a * G = G ↔ a ∈ G :=
iff.intro mem_of_lcoset_eq_self lcoset_eq_self_of_mem
proposition rcoset_eq_self_iff (a : A) : G * a = G ↔ a ∈ G :=
iff.intro mem_of_rcoset_eq_self rcoset_eq_self_of_mem
variable {G}
proposition lcoset_eq_lcoset {a b : A} (H : b⁻¹ * a ∈ G) : a * G = b * G :=
have b⁻¹ * (a * G) = b⁻¹ * (b * G),
by rewrite [inv_lcoset_lcoset, lcoset_lcoset, lcoset_eq_self_of_mem H],
eq_of_lcoset_eq_lcoset this
proposition inv_mul_mem_of_lcoset_eq_lcoset {a b : A} (H : a * G = b * G) : b⁻¹ * a ∈ G :=
mem_of_lcoset_eq_self (by rewrite [-lcoset_lcoset, H, inv_lcoset_lcoset])
proposition lcoset_eq_lcoset_iff (a b : A) : a * G = b * G ↔ b⁻¹ * a ∈ G :=
iff.intro inv_mul_mem_of_lcoset_eq_lcoset lcoset_eq_lcoset
proposition rcoset_eq_rcoset {a b : A} (H : a * b⁻¹ ∈ G) : G * a = G * b :=
have G * a * b⁻¹ = G * b * b⁻¹,
by rewrite [rcoset_rcoset_inv, rcoset_rcoset, rcoset_eq_self_of_mem H],
eq_of_rcoset_eq_rcoset this
proposition mul_inv_mem_of_rcoset_eq_rcoset {a b : A} (H : G * a = G * b) : a * b⁻¹ ∈ G :=
mem_of_rcoset_eq_self (by rewrite [-rcoset_rcoset, H, rcoset_rcoset_inv])
proposition rcoset_eq_rcoset_iff (a b : A) : G * a = G * b ↔ a * b⁻¹ ∈ G :=
iff.intro mul_inv_mem_of_rcoset_eq_rcoset rcoset_eq_rcoset
end subgroupG
/- normal cosets and the normalizer -/
section has_mulA
variable [has_mul A]
abbreviation normalizes [reducible] (a : A) (S : set A) : Prop := a * S = S * a
definition is_normal [class] (S : set A) : Prop := ∀ a, normalizes a S
definition normalizer (S : set A) : set A := { a : A | normalizes a S }
definition is_normal_in [class] (S T : set A) : Prop := T ⊆ normalizer S
abbreviation normalizer_in [reducible] (S T : set A) : set A := T ∩ normalizer S
proposition lcoset_eq_rcoset (a : A) (S : set A) [H : is_normal S] : a * S = S * a := H a
proposition subset_normalizer (S T : set A) [H : is_normal_in T S] : S ⊆ normalizer T := H
proposition lcoset_eq_rcoset_of_mem {a : A} (S : set A) {T : set A} [H : is_normal_in S T]
(amemT : a ∈ T) :
a * S = S * a := H amemT
proposition is_normal_in_of_is_normal (S T : set A) [H : is_normal S] : is_normal_in S T :=
forallb_of_forall T H
proposition is_normal_of_is_normal_in_univ {S : set A} (H : is_normal_in S univ) :
is_normal S :=
forall_of_forallb_univ H
proposition is_normal_in_univ_iff_is_normal (S : set A) : is_normal_in S univ ↔ is_normal S :=
forallb_univ_iff_forall _
proposition is_normal_in_of_subset {S T U : set A} (H : T ⊆ U) (H' : is_normal_in S U) :
is_normal_in S T :=
forallb_of_subset H H'
proposition normalizes_of_mem_normalizer {a : A} {S : set A} (H : a ∈ normalizer S) :
normalizes a S := H
proposition mem_normalizer_iff_normalizes (a : A) (S : set A) :
a ∈ normalizer S ↔ normalizes a S := iff.refl _
proposition is_normal_in_normalizer [instance] (S : set A) : is_normal_in S (normalizer S) :=
subset.refl (normalizer S)
end has_mulA
section groupA
variable [group A]
proposition is_normal_in_of_forall_subset {S G : set A} [is_subgroup G]
(H : ∀₀ x ∈ G, x * S ⊆ S * x) :
is_normal_in S G :=
take x, assume xG,
show x * S = S * x, from eq_of_subset_of_subset (H xG)
(have x * (x⁻¹ * S) * x ⊆ x * (S * x⁻¹) * x,
from rcoset_subset_rcoset (lcoset_subset_lcoset x (H (inv_mem xG))) x,
show S * x ⊆ x * S,
begin
rewrite [lcoset_inv_lcoset at this, lcoset_rcoset at this, rcoset_inv_rcoset at this],
exact this
end)
proposition is_normal_of_forall_subset {S : set A} (H : ∀ x, x * S ⊆ S * x) : is_normal S :=
begin
rewrite [-is_normal_in_univ_iff_is_normal],
apply is_normal_in_of_forall_subset,
intro x xuniv, exact H x
end
proposition subset_normalizer_self (G : set A) [is_subgroup G] : G ⊆ normalizer G :=
take a, assume aG, show a * G = G * a,
by rewrite [lcoset_eq_self_of_mem aG, rcoset_eq_self_of_mem aG]
end groupA
section normalG
variables [group A] (G : set A) [is_normal G]
proposition lcoset_equiv_mul {a₁ a₂ b₁ b₂ : A}
(H₁ : lcoset_equiv G a₁ a₂) (H₂ : lcoset_equiv G b₁ b₂) : lcoset_equiv G (a₁ * b₁) (a₂ * b₂) :=
begin
unfold lcoset_equiv at *,
rewrite [-lcoset_lcoset, H₂, lcoset_eq_rcoset, -lcoset_rcoset, H₁, lcoset_rcoset,
-lcoset_eq_rcoset, lcoset_lcoset]
end
proposition lcoset_equiv_inv {a₁ a₂ : A} (H : lcoset_equiv G a₁ a₂) : lcoset_equiv G a₁⁻¹ a₂⁻¹ :=
begin
unfold lcoset_equiv at *,
have a₁⁻¹ * G = a₂⁻¹ * (a₂ * G) * a₁⁻¹, by rewrite [inv_lcoset_lcoset, lcoset_eq_rcoset],
rewrite [this, -H, lcoset_rcoset, lcoset_eq_rcoset, rcoset_rcoset_inv]
end
end normalG
/- the normalizer is a subgroup -/
section semigroupA
variable [semigroup A]
proposition mul_mem_normalizer {S : set A} {a b : A}
(Ha : a ∈ normalizer S) (Hb : b ∈ normalizer S) : a * b ∈ normalizer S :=
show a * b * S = S * (a * b),
by rewrite [-lcoset_lcoset, normalizes_of_mem_normalizer Hb, -lcoset_rcoset,
normalizes_of_mem_normalizer Ha, rcoset_rcoset]
end semigroupA
section monoidA
variable [monoid A]
proposition one_mem_normalizer (S : set A) : 1 ∈ normalizer S :=
by rewrite [↑normalizer, mem_set_of_iff, one_lcoset, rcoset_one]
end monoidA
section groupA
variable [group A]
proposition inv_mem_normalizer {S : set A} {a : A} (H : a ∈ normalizer S) : a⁻¹ ∈ normalizer S :=
have a⁻¹ * S = S * a⁻¹,
begin
apply iff.mp (rcoset_eq_iff_eq_rcoset_inv _ _ _),
rewrite [lcoset_rcoset, -normalizes_of_mem_normalizer H, inv_lcoset_lcoset]
end,
by rewrite [↑normalizer, mem_set_of_iff, this]
proposition is_subgroup_normalizer [instance] (S : set A) : is_subgroup (normalizer S) :=
⦃ is_subgroup,
one_mem := one_mem_normalizer S,
mul_mem := λ a Ha b Hb, mul_mem_normalizer Ha Hb,
inv_mem := λ a H, inv_mem_normalizer H⦄
end groupA
section subgroupG
variables [group A] {G : set A} [is_subgroup G]
proposition normalizes_image_of_is_hom_on [group B] {a : A} (aG : a ∈ G) {S : set A}
(SsubG : S ⊆ G) (H : normalizes a S) (f : A → B) [is_hom_on f G] :
normalizes (f a) (f ' S) :=
by rewrite [-image_lcoset_of_is_hom_on SsubG aG, -image_rcoset_of_is_hom_on SsubG aG,
↑normalizes at H, H]
proposition is_normal_in_image_image [group B] {S T : set A} (SsubT : S ⊆ T)
[H : is_normal_in S T] (f : A → B) [is_subgroup T] [is_hom_on f T] :
is_normal_in (f ' S) (f ' T) :=
take a, assume afT,
obtain b [bT (beq : f b = a)], from afT,
show normalizes a (f ' S),
begin rewrite -beq, apply (normalizes_image_of_is_hom_on bT SsubT (H bT)) end
proposition normalizes_image_of_is_hom [group B] {a : A} {S : set A}
(H : normalizes a S) (f : A → B) [is_hom f] :
normalizes (f a) (f ' S) :=
by rewrite [-image_lcoset_of_is_hom f a S, -image_rcoset_of_is_hom f S a,
↑normalizes at H, H]
proposition is_normal_in_image_image_univ [group B] {S : set A}
[H : is_normal S] (f : A → B) [is_hom f] :
is_normal_in (f ' S) (f ' univ) :=
take a, assume afT,
obtain b [buniv (beq : f b = a)], from afT,
show normalizes a (f ' S),
begin rewrite -beq, apply (normalizes_image_of_is_hom (H b) f) end
end subgroupG
/- conjugation -/
definition conj [reducible] [group A] (a b : A) : A := b⁻¹ * a * b
definition set_conj [reducible] [group A] (S : set A)(a : A) : set A := a⁻¹ * S * a
-- conj^~ a ' S
namespace conj_notation
infix `^` := conj
infix `^` := set_conj
end conj_notation
open conj_notation
section groupA
variables [group A]
proposition set_conj_eq_image_conj (S : set A) (a : A) : S^a = conj^~ a 'S :=
eq.symm !image_comp
proposition set_conj_eq_self_of_normalizes {S : set A} {a : A} (H : normalizes a S) : S^a = S :=
by rewrite [lcoset_rcoset, ↑normalizes at H, -H, inv_lcoset_lcoset]
proposition normalizes_of_set_conj_eq_self {S : set A} {a : A} (H : S^a = S) : normalizes a S :=
by rewrite [-H at {1}, ↑set_conj, lcoset_rcoset, lcoset_inv_lcoset]
proposition set_conj_eq_self_iff_normalizes (S : set A) (a : A) : S^a = S ↔ normalizes a S :=
iff.intro normalizes_of_set_conj_eq_self set_conj_eq_self_of_normalizes
proposition set_conj_eq_self_of_mem_normalizer {S : set A} {a : A} (H : a ∈ normalizer S) :
S^a = S := set_conj_eq_self_of_normalizes H
proposition mem_normalizer_of_set_conj_eq_self {S : set A} {a : A} (H : S^a = S) :
a ∈ normalizer S := normalizes_of_set_conj_eq_self H
proposition set_conj_eq_self_iff_mem_normalizer (S : set A) (a : A) :
S^a = S ↔ a ∈ normalizer S :=
iff.intro mem_normalizer_of_set_conj_eq_self set_conj_eq_self_of_mem_normalizer
proposition conj_one (a : A) : a ^ (1 : A) = a :=
by rewrite [↑conj, one_inv, one_mul, mul_one]
proposition conj_conj (a b c : A) : (a^b)^c = a^(b * c) :=
by rewrite [↑conj, mul_inv, *mul.assoc]
proposition conj_inv (a b : A) : (a^b)⁻¹ = (a⁻¹)^b :=
by rewrite[mul_inv, mul_inv, inv_inv, mul.assoc]
proposition mul_conj (a b c : A) : (a * b)^c = a^c * b^c :=
by rewrite[↑conj, *mul.assoc, mul_inv_cancel_left]
end groupA
/- the kernel -/
definition ker [has_one B] (f : A → B) : set A := { x | f x = 1 }
section hasoneB
variable [has_one B]
proposition eq_one_of_mem_ker {f : A → B} {a : A} (H : a ∈ ker f) : f a = 1 := H
proposition mem_ker_iff (f : A → B) (a : A) : a ∈ ker f ↔ f a = 1 := iff.rfl
proposition ker_eq_preimage_one (f : A → B) : ker f = f '- '{1} :=
ext (take x, by rewrite [mem_ker_iff, -mem_preimage_iff, mem_singleton_iff])
definition ker_in (f : A → B) (S : set A) : set A := ker f ∩ S
proposition ker_in_univ (f : A → B) : ker_in f univ = ker f :=
!inter_univ
end hasoneB
section groupAB
variables [group A] [group B]
variable {f : A → B}
proposition eq_of_mul_inv_mem_ker [is_hom f] {a₁ a₂ : A} (H : a₁ * a₂⁻¹ ∈ ker f) :
f a₁ = f a₂ :=
eq_of_mul_inv_eq_one (by rewrite [-hom_inv f, -hom_mul f]; exact H)
proposition mul_inv_mem_ker_of_eq [is_hom f] {a₁ a₂ : A} (H : f a₁ = f a₂) :
a₁ * a₂⁻¹ ∈ ker f :=
show f (a₁ * a₂⁻¹) = 1, by rewrite [hom_mul f, hom_inv f, H, mul.right_inv]
proposition eq_iff_mul_inv_mem_ker [is_hom f] (a₁ a₂ : A) : f a₁ = f a₂ ↔ a₁ * a₂⁻¹ ∈ ker f :=
iff.intro mul_inv_mem_ker_of_eq eq_of_mul_inv_mem_ker
proposition eq_of_mul_inv_mem_ker_in {G : set A} [is_subgroup G] [is_hom_on f G]
{a₁ a₂ : A} (a₁G : a₁ ∈ G) (a₂G : a₂ ∈ G) (H : a₁ * a₂⁻¹ ∈ ker_in f G) :
f a₁ = f a₂ :=
eq_of_mul_inv_eq_one (by rewrite [-hom_on_inv f a₂G, -hom_on_mul f a₁G (inv_mem a₂G)];
exact and.left H)
proposition mul_inv_mem_ker_in_of_eq {G : set A} [is_subgroup G] [is_hom_on f G]
{a₁ a₂ : A} (a₁G : a₁ ∈ G) (a₂G : a₂ ∈ G) (H : f a₁ = f a₂) :
a₁ * a₂⁻¹ ∈ ker_in f G :=
and.intro
(show f (a₁ * a₂⁻¹) = 1,
by rewrite [hom_on_mul f a₁G (inv_mem a₂G), hom_on_inv f a₂G, H, mul.right_inv])
(mul_mem a₁G (inv_mem a₂G))
proposition eq_iff_mul_inv_mem_ker_in {G : set A} [is_subgroup G] [is_hom_on f G]
{a₁ a₂ : A} (a₁G : a₁ ∈ G) (a₂G : a₂ ∈ G) :
f a₁ = f a₂ ↔ a₁ * a₂⁻¹ ∈ ker_in f G :=
iff.intro (mul_inv_mem_ker_in_of_eq a₁G a₂G) (eq_of_mul_inv_mem_ker_in a₁G a₂G)
-- Ouch! These versions are not equivalent to the ones before.
proposition eq_of_inv_mul_mem_ker [is_hom f] {a₁ a₂ : A} (H : a₁⁻¹ * a₂ ∈ ker f) :
f a₁ = f a₂ :=
eq.symm (eq_of_inv_mul_eq_one (by rewrite [-hom_inv f, -hom_mul f]; exact H))
proposition inv_mul_mem_ker_of_eq [is_hom f] {a₁ a₂ : A} (H : f a₁ = f a₂) :
a₁⁻¹ * a₂ ∈ ker f :=
show f (a₁⁻¹ * a₂) = 1, by rewrite [hom_mul f, hom_inv f, H, mul.left_inv]
proposition eq_iff_inv_mul_mem_ker [is_hom f] (a₁ a₂ : A) : f a₁ = f a₂ ↔ a₁⁻¹ * a₂ ∈ ker f :=
iff.intro inv_mul_mem_ker_of_eq eq_of_inv_mul_mem_ker
proposition eq_of_inv_mul_mem_ker_in {G : set A} [is_subgroup G] [is_hom_on f G]
{a₁ a₂ : A} (a₁G : a₁ ∈ G) (a₂G : a₂ ∈ G) (H : a₁⁻¹ * a₂ ∈ ker_in f G) :
f a₁ = f a₂ :=
eq.symm (eq_of_inv_mul_eq_one (by rewrite [-hom_on_inv f a₁G, -hom_on_mul f (inv_mem a₁G) a₂G];
exact and.left H))
proposition inv_mul_mem_ker_in_of_eq {G : set A} [is_subgroup G] [is_hom_on f G]
{a₁ a₂ : A} (a₁G : a₁ ∈ G) (a₂G : a₂ ∈ G) (H : f a₁ = f a₂) :
a₁⁻¹ * a₂ ∈ ker_in f G :=
and.intro
(show f (a₁⁻¹ * a₂) = 1,
by rewrite [hom_on_mul f (inv_mem a₁G) a₂G, hom_on_inv f a₁G, H, mul.left_inv])
(mul_mem (inv_mem a₁G) a₂G)
proposition eq_iff_inv_mul_mem_ker_in {G : set A} [is_subgroup G] [is_hom_on f G]
{a₁ a₂ : A} (a₁G : a₁ ∈ G) (a₂G : a₂ ∈ G) :
f a₁ = f a₂ ↔ a₁⁻¹ * a₂ ∈ ker_in f G :=
iff.intro (inv_mul_mem_ker_in_of_eq a₁G a₂G) (eq_of_inv_mul_mem_ker_in a₁G a₂G)
proposition eq_one_of_eq_one_of_injective [is_hom f] (H : injective f) {x : A}
(H' : f x = 1) :
x = 1 :=
H (by rewrite [H', hom_one f])
proposition eq_one_iff_eq_one_of_injective [is_hom f] (H : injective f) (x : A) :
f x = 1 ↔ x = 1 :=
iff.intro (eq_one_of_eq_one_of_injective H) (λ H', by rewrite [H', hom_one f])
proposition injective_of_forall_eq_one [is_hom f] (H : ∀ x, f x = 1 → x = 1) : injective f :=
take a₁ a₂, assume Heq,
have f (a₁ * a₂⁻¹) = 1, by rewrite [hom_mul f, hom_inv f, Heq, mul.right_inv],
eq_of_mul_inv_eq_one (H _ this)
proposition injective_of_ker_eq_singleton_one [is_hom f] (H : ker f = '{1}) : injective f :=
injective_of_forall_eq_one
(take x, suppose x ∈ ker f, by rewrite [H at this]; exact eq_of_mem_singleton this)
proposition ker_eq_singleton_one_of_injective [is_hom f] (H : injective f) : ker f = '{1} :=
ext (take x, by rewrite [mem_ker_iff, mem_singleton_iff, eq_one_iff_eq_one_of_injective H])
variable (f)
proposition injective_iff_ker_eq_singleton_one [is_hom f] : injective f ↔ ker f = '{1} :=
iff.intro ker_eq_singleton_one_of_injective injective_of_ker_eq_singleton_one
variable {f}
proposition eq_one_of_eq_one_of_inj_on {G : set A} [is_subgroup G] [is_hom_on f G]
(H : inj_on f G) {x : A} (xG : x ∈ G) (H' : f x = 1) :
x = 1 :=
H xG one_mem (by rewrite [H', hom_on_one f G])
proposition eq_one_iff_eq_one_of_inj_on {G : set A} [is_subgroup G] [is_hom_on f G]
(H : inj_on f G) {x : A} (xG : x ∈ G) [is_hom_on f G] :
f x = 1 ↔ x = 1 :=
iff.intro (eq_one_of_eq_one_of_inj_on H xG) (λ H', by rewrite [H', hom_on_one f G])
proposition inj_on_of_forall_eq_one {G : set A} [is_subgroup G] [is_hom_on f G]
(H : ∀₀ x ∈ G, f x = 1 → x = 1) : inj_on f G :=
take a₁ a₂, assume a₁G a₂G Heq,
have f (a₁ * a₂⁻¹) = 1,
by rewrite [hom_on_mul f a₁G (inv_mem a₂G), hom_on_inv f a₂G, Heq, mul.right_inv],
eq_of_mul_inv_eq_one (H (mul_mem a₁G (inv_mem a₂G)) this)
proposition inj_on_of_ker_in_eq_singleton_one {G : set A} [is_subgroup G] [is_hom_on f G]
(H : ker_in f G = '{1}) : inj_on f G :=
inj_on_of_forall_eq_one
(take x, assume xG fxone,
have x ∈ ker_in f G, from and.intro fxone xG,
by rewrite [H at this]; exact eq_of_mem_singleton this)
proposition ker_in_eq_singleton_one_of_inj_on {G : set A} [is_subgroup G] [is_hom_on f G]
(H : inj_on f G) : ker_in f G = '{1} :=
ext (take x,
begin
rewrite [↑ker_in, mem_inter_iff, mem_ker_iff, mem_singleton_iff],
apply iff.intro,
{intro H', cases H' with fxone xG, exact eq_one_of_eq_one_of_inj_on H xG fxone},
intro xone, rewrite xone, split, exact hom_on_one f G, exact one_mem
end)
variable (f)
proposition inj_on_iff_ker_in_eq_singleton_one (G : set A) [is_subgroup G] [is_hom_on f G] :
inj_on f G ↔ ker_in f G = '{1} :=
iff.intro ker_in_eq_singleton_one_of_inj_on inj_on_of_ker_in_eq_singleton_one
variable {f}
proposition conj_mem_ker [is_hom f] {a₁ : A} (a₂ : A) (H : a₁ ∈ ker f) : a₁^a₂ ∈ ker f :=
show f (a₁^a₂) = 1,
by rewrite [↑conj, *(hom_mul f), hom_inv f, eq_one_of_mem_ker H, mul_one, mul.left_inv]
variable (f)
proposition is_subgroup_ker_in [instance] (S : set A) [is_subgroup S] [is_hom_on f S] :
is_subgroup (ker_in f S) :=
⦃ is_subgroup,
one_mem := and.intro (hom_on_one f S) one_mem,
mul_mem := λ a aker b bker,
obtain (fa : f a = 1) (aS : a ∈ S), from aker,
obtain (fb : f b = 1) (bS : b ∈ S), from bker,
and.intro (show f (a * b) = 1, by rewrite [hom_on_mul f aS bS, fa, fb, one_mul])
(mul_mem aS bS),
inv_mem := λ a aker,
obtain (fa : f a = 1) (aS : a ∈ S), from aker,
and.intro (show f (a⁻¹) = 1, by rewrite [hom_on_inv f aS, fa, one_inv])
(inv_mem aS)
⦄
proposition is_subgroup_ker [instance] [is_hom f] : is_subgroup (ker f) :=
begin
rewrite [-ker_in_univ f],
have is_hom_on f univ, from is_hom_on_of_is_hom f univ,
apply is_subgroup_ker_in f univ
end
proposition is_normal_in_ker_in [instance] (G : set A) [is_subgroup G] [is_hom_on f G] :
is_normal_in (ker_in f G) G :=
is_normal_in_of_forall_subset
(take x, assume xG, take y, assume yker,
obtain z [[(fz : f z = 1) zG] (yeq : x * z = y)], from yker,
have y = x * z * x⁻¹ * x, by rewrite [yeq, inv_mul_cancel_right],
show y ∈ ker_in f G * x,
begin
rewrite this,
apply mul_mem_rcoset,
apply and.intro,
show f (x * z * x⁻¹) = 1,
by rewrite [hom_on_mul f (mul_mem xG zG) (inv_mem xG), hom_on_mul f xG zG, fz,
hom_on_inv f xG, mul_one, mul.right_inv],
show x * z * x⁻¹ ∈ G, from mul_mem (mul_mem xG zG) (inv_mem xG)
end)
proposition is_normal_ker [instance] [H : is_hom f] : is_normal (ker f) :=
begin
rewrite [-ker_in_univ, -is_normal_in_univ_iff_is_normal],
apply is_normal_in_ker_in,
exact is_hom_on_of_is_hom f univ
end
end groupAB
section subgroupH
variables [group A] [group B] {H : set A} [is_subgroup H]
variables {f : A → B} [is_hom f]
proposition subset_ker_of_forall (hyp : ∀ x y, x * H = y * H → f x = f y) : H ⊆ ker f :=
take h, assume hH,
have h * H = 1 * H, by rewrite [lcoset_eq_self_of_mem hH, one_lcoset],
have f h = f 1, from hyp h 1 this,
show f h = 1, by rewrite [this, hom_one f]
proposition eq_of_lcoset_eq_lcoset_of_subset_ker {x y : A} (hyp₀ : x * H = y * H) (hyp₁ : H ⊆ ker f) :
f x = f y :=
have y⁻¹ * x ∈ H, from inv_mul_mem_of_lcoset_eq_lcoset hyp₀,
eq.symm (eq_of_inv_mul_mem_ker (hyp₁ this))
variables (H f)
proposition subset_ker_iff : H ⊆ ker f ↔ ∀ x y, x * H = y * H → f x = f y :=
iff.intro (λ h₁ x y h₀, eq_of_lcoset_eq_lcoset_of_subset_ker h₀ h₁) subset_ker_of_forall
end subgroupH
section subgroupGH
variables [group A] [group B] {G H : set A} [is_subgroup G] [is_subgroup H]
variables {f : A → B} [is_hom_on f G]
proposition subset_ker_in_of_forall (hyp₀ : ∀₀ x ∈ G, ∀₀ y ∈ G, x * H = y * H → f x = f y)
(hyp₁ : H ⊆ G) :
H ⊆ ker_in f G :=
take h, assume hH,
have hG : h ∈ G, from hyp₁ hH,
and.intro
(have h * H = 1 * H, by rewrite [lcoset_eq_self_of_mem hH, one_lcoset],
have f h = f 1, from hyp₀ hG one_mem this,
show f h = 1, by rewrite [this, hom_on_one f G])
hG
proposition eq_of_lcoset_eq_lcoset_of_subset_ker_in {x : A} (xG : x ∈ G) {y : A} (yG : y ∈ G)
(hyp₀ : x * H = y * H) (hyp₁ : H ⊆ ker_in f G) :
f x = f y :=
have y⁻¹ * x ∈ H, from inv_mul_mem_of_lcoset_eq_lcoset hyp₀,
eq.symm (eq_of_inv_mul_mem_ker_in yG xG (hyp₁ this))
variables (H f)
proposition subset_ker_in_iff :
H ⊆ ker_in f G ↔ (H ⊆ G ∧ ∀₀ x ∈ G, ∀₀ y ∈ G, x * H = y * H → f x = f y) :=
iff.intro
(λ h₁, and.intro
(subset.trans h₁ (inter_subset_right _ _))
(λ x xG y yG h₀, eq_of_lcoset_eq_lcoset_of_subset_ker_in xG yG h₀ h₁))
(λ h, subset_ker_in_of_forall (and.right h) (and.left h))
end subgroupGH
/- the centralizer -/
section has_mulA
variable [has_mul A]
abbreviation centralizes [reducible] (a : A) (S : set A) : Prop := ∀₀ b ∈ S, a * b = b * a
definition centralizer (S : set A) : set A := { a : A | centralizes a S }
abbreviation is_centralized_by (S T : set A) : Prop := T ⊆ centralizer S
abbreviation centralizer_in (S T : set A) : set A := T ∩ centralizer S
proposition mem_centralizer_iff_centralizes (a : A) (S : set A) :
a ∈ centralizer S ↔ centralizes a S := iff.refl _
proposition normalizes_of_centralizes {a : A} {S : set A} (H : centralizes a S) :
normalizes a S :=
ext (take b, iff.intro
(suppose b ∈ a * S,
obtain s [ains (beq : a * s = b)], from this,
show b ∈ S * a, by rewrite[-beq, H ains]; apply mem_image_of_mem _ ains)
(suppose b ∈ S * a,
obtain s [ains (beq : s * a = b)], from this,
show b ∈ a * S, by rewrite[-beq, -H ains]; apply mem_image_of_mem _ ains))
proposition centralizer_subset_normalizer (S : set A) : centralizer S ⊆ normalizer S :=
λ a acent, normalizes_of_centralizes acent
proposition centralizer_subset_centralizer {S T : set A} (ssubt : S ⊆ T) :
centralizer T ⊆ centralizer S :=
λ x xCentT s sS, xCentT _ (ssubt sS)
end has_mulA
section groupA
variable [group A]
proposition is_subgroup_centralizer [instance] [group A] (S : set A) :
is_subgroup (centralizer S) :=
⦃ is_subgroup,
one_mem := λ b bS, by rewrite [one_mul, mul_one],
mul_mem := λ a acent b bcent c cS, by rewrite [mul.assoc, bcent cS, -*mul.assoc, acent cS],
inv_mem := λ a acent c cS, eq_mul_inv_of_mul_eq
(by rewrite [mul.assoc, -acent cS, inv_mul_cancel_left])⦄
end groupA
/- the subgroup generated by a set -/
section groupA
variable [group A]
inductive subgroup_generated_by (S : set A) : A → Prop :=
| generators_mem : ∀ x, x ∈ S → subgroup_generated_by S x
| one_mem : subgroup_generated_by S 1
| mul_mem : ∀ x y, subgroup_generated_by S x → subgroup_generated_by S y →
subgroup_generated_by S (x * y)
| inv_mem : ∀ x, subgroup_generated_by S x → subgroup_generated_by S (x⁻¹)
theorem generators_subset_subgroup_generated_by (S : set A) : S ⊆ subgroup_generated_by S :=
subgroup_generated_by.generators_mem
theorem is_subgroup_subgroup_generated_by [instance] (S : set A) :
is_subgroup (subgroup_generated_by S) :=
⦃ is_subgroup,
one_mem := subgroup_generated_by.one_mem S,
mul_mem := λ a amem b bmem, subgroup_generated_by.mul_mem a b amem bmem,
inv_mem := λ a amem, subgroup_generated_by.inv_mem a amem ⦄
theorem subgroup_generated_by_subset {S G : set A} [is_subgroup G] (H : S ⊆ G) :
subgroup_generated_by S ⊆ G :=
begin
intro x xgenS,
induction xgenS with a aS a b agen bgen aG bG a agen aG,
{exact H aS},
{exact one_mem},
{exact mul_mem aG bG},
exact inv_mem aG
end
end groupA
end group_theory
|
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"LicenseRef-scancode-other-permissive",
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"CMU-Mach"... | permissive | EdAyers/lean4 | 57ac632d6b0789cb91fab2170e8c9e40441221bd | 37ba0df5841bde51dbc2329da81ac23d4f6a4de4 | refs/heads/master | 1,676,463,245,298 | 1,660,619,433,000 | 1,660,619,433,000 | 183,433,437 | 1 | 0 | Apache-2.0 | 1,657,612,672,000 | 1,556,196,574,000 | Lean | UTF-8 | Lean | false | false | 10,900 | lean | /-!
# Binary Search Trees
If the type of keys can be totally ordered -- that is, it supports a well-behaved `≤` comparison --
then maps can be implemented with binary search trees (BSTs). Insert and lookup operations on BSTs take time
proportional to the height of the tree. If the tree is balanced, the operations therefore take logarithmic time.
This example is based on a similar example found in the ["Sofware Foundations"](https://softwarefoundations.cis.upenn.edu/vfa-current/SearchTree.html)
book (volume 3).
-/
/-!
We use `Nat` as the key type in our implementation of BSTs,
since it has a convenient total order with lots of theorems and automation available.
We leave as an exercise to the reader the generalization to arbitrary types.
-/
inductive Tree (β : Type v) where
| leaf
| node (left : Tree β) (key : Nat) (value : β) (right : Tree β)
deriving Repr
/-!
The function `contains` returns `true` iff the given tree contains the key `k`.
-/
def Tree.contains (t : Tree β) (k : Nat) : Bool :=
match t with
| leaf => false
| node left key value right =>
if k < key then
left.contains k
else if key < k then
right.contains k
else
true
/-!
`t.find? k` returns `some v` if `v` is the value bound to key `k` in the tree `t`. It returns `none` otherwise.
-/
def Tree.find? (t : Tree β) (k : Nat) : Option β :=
match t with
| leaf => none
| node left key value right =>
if k < key then
left.find? k
else if key < k then
right.find? k
else
some value
/-!
`t.insert k v` is the map containing all the bindings of `t` along with a binding of `k` to `v`.
-/
def Tree.insert (t : Tree β) (k : Nat) (v : β) : Tree β :=
match t with
| leaf => node leaf k v leaf
| node left key value right =>
if k < key then
node (left.insert k v) key value right
else if key < k then
node left key value (right.insert k v)
else
node left k v right
/-!
Let's add a new operation to our tree: converting it to an association list that contains the key--value bindings from the tree stored as pairs.
If that list is sorted by the keys, then any two trees that represent the same map would be converted to the same list.
Here's a function that does so with an in-order traversal of the tree.
-/
def Tree.toList (t : Tree β) : List (Nat × β) :=
match t with
| leaf => []
| node l k v r => l.toList ++ [(k, v)] ++ r.toList
#eval Tree.leaf.insert 2 "two"
|>.insert 3 "three"
|>.insert 1 "one"
#eval Tree.leaf.insert 2 "two"
|>.insert 3 "three"
|>.insert 1 "one"
|>.toList
/-!
The implemention of `Tree.toList` is inefficient because of how it uses the `++` operator.
On a balanced tree its running time is linearithmic, because it does a linear number of
concatentations at each level of the tree. On an unbalanced tree it's quadratic time.
Here's a tail-recursive implementation than runs in linear time, regardless of whether the tree is balanced:
-/
def Tree.toListTR (t : Tree β) : List (Nat × β) :=
go t []
where
go (t : Tree β) (acc : List (Nat × β)) : List (Nat × β) :=
match t with
| leaf => acc
| node l k v r => go l ((k, v) :: go r acc)
/-!
We now prove that `t.toList` and `t.toListTR` return the same list.
The proof is on induction, and as we used the auxiliary function `go`
to define `Tree.toListTR`, we use the auxiliary theorem `go` to prove the theorem.
The proof of the auxiliary theorem is by induction on `t`.
The `generalizing acc` modifier instructs Lean to revert `acc`, apply the
induction theorem for `Tree`s, and then reintroduce `acc` in each case.
By using `generalizing`, we obtain the more general induction hypotheses
- `left_ih : ∀ acc, toListTR.go left acc = toList left ++ acc`
- `right_ih : ∀ acc, toListTR.go right acc = toList right ++ acc`
Recall that the combinator `tac <;> tac'` runs `tac` on the main goal and `tac'` on each produced goal,
concatenating all goals produced by `tac'`. In this theorem, we use it to apply
`simp` and close each subgoal produced by the `induction` tactic.
The `simp` parameters `toListTR.go` and `toList` instruct the simplifier to try to reduce
and/or apply auto generated equation theorems for these two functions.
The parameter `*` intructs the simplifier to use any equation in a goal as rewriting rules.
In this particular case, `simp` uses the induction hypotheses as rewriting rules.
Finally, the parameter `List.append_assoc` intructs the simplifier to use the
`List.append_assoc` theorem as a rewriting rule.
-/
theorem Tree.toList_eq_toListTR (t : Tree β)
: t.toList = t.toListTR := by
simp [toListTR, go t []]
where
go (t : Tree β) (acc : List (Nat × β))
: toListTR.go t acc = t.toList ++ acc := by
induction t generalizing acc <;>
simp [toListTR.go, toList, *, List.append_assoc]
/-!
The `[csimp]` annotation instructs the Lean code generator to replace
any `Tree.toList` with `Tree.toListTR` when generating code.
-/
@[csimp] theorem Tree.toList_eq_toListTR_csimp
: @Tree.toList = @Tree.toListTR := by
funext β t
apply toList_eq_toListTR
/-!
The implementations of `Tree.find?` and `Tree.insert` assume that values of type tree obey the BST invariant:
for any non-empty node with key `k`, all the values of the `left` subtree are less than `k` and all the values
of the right subtree are greater than `k`. But that invariant is not part of the definition of tree.
So, let's formalize the BST invariant. Here's one way to do so. First, we define a helper `ForallTree`
to express that idea that a predicate holds at every node of a tree:
-/
inductive ForallTree (p : Nat → β → Prop) : Tree β → Prop
| leaf : ForallTree p .leaf
| node :
ForallTree p left →
p key value →
ForallTree p right →
ForallTree p (.node left key value right)
/-!
Second, we define the BST invariant:
An empty tree is a BST.
A non-empty tree is a BST if all its left nodes have a lesser key, its right nodes have a greater key, and the left and right subtrees are themselves BSTs.
-/
inductive BST : Tree β → Prop
| leaf : BST .leaf
| node :
ForallTree (fun k v => k < key) left →
ForallTree (fun k v => key < k) right →
BST left → BST right →
BST (.node left key value right)
/-!
We can use the `macro` command to create helper tactics for organizing our proofs.
The macro `have_eq x y` tries to prove `x = y` using linear arithmetic, and then
immediately uses the new equality to substitute `x` with `y` everywhere in the goal.
The modifier `local` specifies the scope of the macro.
-/
/-- The `have_eq lhs rhs` tactic (tries to) prove that `lhs = rhs`,
and then replaces `lhs` with `rhs`. -/
local macro "have_eq " lhs:term:max rhs:term:max : tactic =>
`((have h : $lhs = $rhs :=
-- TODO: replace with linarith
by simp_arith at *; apply Nat.le_antisymm <;> assumption
try subst $lhs))
/-!
The `by_cases' e` is just the regular `by_cases` followed by `simp` using all
hypotheses in the current goal as rewriting rules.
Recall that the `by_cases` tactic creates two goals. One where we have `h : e` and
another one containing `h : ¬ e`. The simplier uses the `h` to rewrite `e` to `True`
in the first subgoal, and `e` to `False` in the second. This is particularly
useful if `e` is the condition of an `if`-statement.
-/
/-- `by_cases' e` is a shorthand form `by_cases e <;> simp[*]` -/
local macro "by_cases' " e:term : tactic =>
`(by_cases $e <;> simp [*])
/-!
We can use the attribute `[simp]` to instruct the simplifier to reduce given definitions or
apply rewrite theorems. The `local` modifier limits the scope of this modification to this file.
-/
attribute [local simp] Tree.insert
/-!
We now prove that `Tree.insert` preserves the BST invariant using induction and case analysis.
Recall that the tactic `. tac` focuses on the main goal and tries to solve it using `tac`, or else fails.
It is used to structure proofs in Lean.
The notation `‹e›` is just syntax sugar for `(by assumption : e)`. That is, it tries to find a hypothesis `h : e`.
It is useful to access hypothesis that have auto generated names (aka "inaccessible") names.
-/
theorem Tree.forall_insert_of_forall
(h₁ : ForallTree p t) (h₂ : p key value)
: ForallTree p (t.insert key value) := by
induction h₁ with
| leaf => exact .node .leaf h₂ .leaf
| node hl hp hr ihl ihr =>
rename Nat => k
by_cases' key < k
. exact .node ihl hp hr
. by_cases' k < key
. exact .node hl hp ihr
. have_eq key k
exact .node hl h₂ hr
theorem Tree.bst_insert_of_bst
{t : Tree β} (h : BST t) (key : Nat) (value : β)
: BST (t.insert key value) := by
induction h with
| leaf => exact .node .leaf .leaf .leaf .leaf
| node h₁ h₂ b₁ b₂ ih₁ ih₂ =>
rename Nat => k
simp
by_cases' key < k
. exact .node (forall_insert_of_forall h₁ ‹key < k›) h₂ ih₁ b₂
. by_cases' k < key
. exact .node h₁ (forall_insert_of_forall h₂ ‹k < key›) b₁ ih₂
. have_eq key k
exact .node h₁ h₂ b₁ b₂
/-!
Now, we define the type `BinTree` using a `Subtype` that states that only trees satisfying the BST invariant are `BinTree`s.
-/
def BinTree (β : Type u) := { t : Tree β // BST t }
def BinTree.mk : BinTree β :=
⟨.leaf, .leaf⟩
def BinTree.contains (b : BinTree β) (k : Nat) : Bool :=
b.val.contains k
def BinTree.find? (b : BinTree β) (k : Nat) : Option β :=
b.val.find? k
def BinTree.insert (b : BinTree β) (k : Nat) (v : β) : BinTree β :=
⟨b.val.insert k v, b.val.bst_insert_of_bst b.property k v⟩
/-!
Finally, we prove that `BinTree.find?` and `BinTree.insert` satisfy the map properties.
-/
attribute [local simp]
BinTree.mk BinTree.contains BinTree.find?
BinTree.insert Tree.find? Tree.contains Tree.insert
theorem BinTree.find_mk (k : Nat)
: BinTree.mk.find? k = (none : Option β) := by
simp
theorem BinTree.find_insert (b : BinTree β) (k : Nat) (v : β)
: (b.insert k v).find? k = some v := by
let ⟨t, h⟩ := b; simp
induction t with simp
| node left key value right ihl ihr =>
by_cases' k < key
. cases h; apply ihl; assumption
. by_cases' key < k
cases h; apply ihr; assumption
theorem BinTree.find_insert_of_ne (b : BinTree β) (h : k ≠ k') (v : β)
: (b.insert k v).find? k' = b.find? k' := by
let ⟨t, h⟩ := b; simp
induction t with simp
| leaf =>
split <;> simp <;> split <;> simp
have_eq k k'
contradiction
| node left key value right ihl ihr =>
let .node hl hr bl br := h
specialize ihl bl
specialize ihr br
by_cases' k < key; by_cases' key < k
have_eq key k
by_cases' k' < k; by_cases' k < k'
have_eq k k'
contradiction
|
cdee1f0e3566594dac29a44c9cc224dcbe059573 | 680b0d1592ce164979dab866b232f6fa743f2cc8 | /hott/types/pointed.hlean | 02caf0818275c4a84849fcf86c856be74ca55105 | [
"Apache-2.0"
] | permissive | syohex/lean | 657428ab520f8277fc18cf04bea2ad200dbae782 | 081ad1212b686780f3ff8a6d0e5f8a1d29a7d8bc | refs/heads/master | 1,611,274,838,635 | 1,452,668,188,000 | 1,452,668,188,000 | 49,562,028 | 0 | 0 | null | 1,452,675,604,000 | 1,452,675,602,000 | null | UTF-8 | Lean | false | false | 12,082 | hlean | /-
Copyright (c) 2014 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer, Floris van Doorn
Ported from Coq HoTT
-/
import arity .eq .bool .unit .sigma .nat.basic
open is_trunc eq prod sigma nat equiv option is_equiv bool unit algebra
structure pointed [class] (A : Type) :=
(point : A)
structure Pointed :=
{carrier : Type}
(Point : carrier)
open Pointed
notation `Type*` := Pointed
namespace pointed
attribute Pointed.carrier [coercion]
variables {A B : Type}
definition pt [unfold 2] [H : pointed A] := point A
protected definition Mk [constructor] := @Pointed.mk
protected definition MK [constructor] (A : Type) (a : A) := Pointed.mk a
protected definition mk' [constructor] (A : Type) [H : pointed A] : Type* :=
Pointed.mk (point A)
definition pointed_carrier [instance] [constructor] (A : Type*) : pointed A :=
pointed.mk (Point A)
-- Any contractible type is pointed
definition pointed_of_is_contr [instance] [priority 800] [constructor]
(A : Type) [H : is_contr A] : pointed A :=
pointed.mk !center
-- A pi type with a pointed target is pointed
definition pointed_pi [instance] [constructor] (P : A → Type) [H : Πx, pointed (P x)]
: pointed (Πx, P x) :=
pointed.mk (λx, pt)
-- A sigma type of pointed components is pointed
definition pointed_sigma [instance] [constructor] (P : A → Type) [G : pointed A]
[H : pointed (P pt)] : pointed (Σx, P x) :=
pointed.mk ⟨pt,pt⟩
definition pointed_prod [instance] [constructor] (A B : Type) [H1 : pointed A] [H2 : pointed B]
: pointed (A × B) :=
pointed.mk (pt,pt)
definition pointed_loop [instance] [constructor] (a : A) : pointed (a = a) :=
pointed.mk idp
definition pointed_bool [instance] [constructor] : pointed bool :=
pointed.mk ff
definition Bool [constructor] : Type* :=
pointed.mk' bool
definition Unit [constructor] : Type* :=
Pointed.mk unit.star
definition pointed_fun_closed [constructor] (f : A → B) [H : pointed A] : pointed B :=
pointed.mk (f pt)
definition Loop_space [reducible] [constructor] (A : Type*) : Type* :=
pointed.mk' (point A = point A)
definition Iterated_loop_space [unfold 1] [reducible] : ℕ → Type* → Type*
| Iterated_loop_space 0 X := X
| Iterated_loop_space (n+1) X := Loop_space (Iterated_loop_space n X)
prefix `Ω`:(max+5) := Loop_space
notation `Ω[`:95 n:0 `] `:0 A:95 := Iterated_loop_space n A
definition rfln [constructor] [reducible] {A : Type*} {n : ℕ} : Ω[n] A := pt
definition refln [constructor] [reducible] (A : Type*) (n : ℕ) : Ω[n] A := pt
definition refln_eq_refl (A : Type*) (n : ℕ) : rfln = rfl :> Ω[succ n] A := rfl
definition iterated_loop_space [unfold 3] (A : Type) [H : pointed A] (n : ℕ) : Type :=
Ω[n] (pointed.mk' A)
open equiv.ops
definition Pointed_eq {A B : Type*} (f : A ≃ B) (p : f pt = pt) : A = B :=
begin
cases A with A a, cases B with B b, esimp at *,
fapply apd011 @Pointed.mk,
{ apply ua f},
{ rewrite [cast_ua,p]},
end
protected definition Pointed.sigma_char.{u} : Pointed.{u} ≃ Σ(X : Type.{u}), X :=
begin
fapply equiv.MK,
{ intro x, induction x with X x, exact ⟨X, x⟩},
{ intro x, induction x with X x, exact pointed.MK X x},
{ intro x, induction x with X x, reflexivity},
{ intro x, induction x with X x, reflexivity},
end
definition add_point [constructor] (A : Type) : Type* :=
Pointed.mk (none : option A)
postfix `₊`:(max+1) := add_point
-- the inclusion A → A₊ is called "some", the extra point "pt" or "none" ("@none A")
end pointed
open pointed
structure pmap (A B : Type*) :=
(map : A → B)
(resp_pt : map (Point A) = Point B)
open pmap
namespace pointed
abbreviation respect_pt [unfold 3] := @pmap.resp_pt
notation `map₊` := pmap
infix ` →* `:30 := pmap
attribute pmap.map [coercion]
variables {A B C D : Type*} {f g h : A →* B}
definition pmap_eq (r : Πa, f a = g a) (s : respect_pt f = (r pt) ⬝ respect_pt g) : f = g :=
begin
cases f with f p, cases g with g q,
esimp at *,
fapply apo011 pmap.mk,
{ exact eq_of_homotopy r},
{ apply concato_eq, apply pathover_eq_Fl, apply inv_con_eq_of_eq_con,
rewrite [ap_eq_ap10,↑ap10,apd10_eq_of_homotopy,s]}
end
definition pid [constructor] (A : Type*) : A →* A :=
pmap.mk id idp
definition pcompose [constructor] (g : B →* C) (f : A →* B) : A →* C :=
pmap.mk (λa, g (f a)) (ap g (respect_pt f) ⬝ respect_pt g)
infixr ` ∘* `:60 := pcompose
structure phomotopy (f g : A →* B) :=
(homotopy : f ~ g)
(homotopy_pt : homotopy pt ⬝ respect_pt g = respect_pt f)
infix ` ~* `:50 := phomotopy
abbreviation to_homotopy_pt [unfold 5] := @phomotopy.homotopy_pt
abbreviation to_homotopy [coercion] [unfold 5] (p : f ~* g) : Πa, f a = g a :=
phomotopy.homotopy p
definition passoc (h : C →* D) (g : B →* C) (f : A →* B) : (h ∘* g) ∘* f ~* h ∘* (g ∘* f) :=
begin
fconstructor, intro a, reflexivity,
cases A, cases B, cases C, cases D, cases f with f pf, cases g with g pg, cases h with h ph,
esimp at *,
induction pf, induction pg, induction ph, reflexivity
end
definition pid_comp (f : A →* B) : pid B ∘* f ~* f :=
begin
fconstructor,
{ intro a, reflexivity},
{ esimp, exact !idp_con ⬝ !ap_id⁻¹}
end
definition comp_pid (f : A →* B) : f ∘* pid A ~* f :=
begin
fconstructor,
{ intro a, reflexivity},
{ reflexivity}
end
definition pmap_equiv_left (A : Type) (B : Type*) : A₊ →* B ≃ (A → B) :=
begin
fapply equiv.MK,
{ intro f a, cases f with f p, exact f (some a)},
{ intro f, fconstructor,
intro a, cases a, exact pt, exact f a,
reflexivity},
{ intro f, reflexivity},
{ intro f, cases f with f p, esimp, fapply pmap_eq,
{ intro a, cases a; all_goals (esimp at *), exact p⁻¹},
{ esimp, exact !con.left_inv⁻¹}},
end
-- set_option pp.notation false
-- definition pmap_equiv_right (A : Type*) (B : Type)
-- : (Σ(b : B), map₊ A (pointed.Mk b)) ≃ (A → B) :=
-- begin
-- fapply equiv.MK,
-- { intro u a, cases u with b f, cases f with f p, esimp at f, exact f a},
-- { intro f, refine ⟨f pt, _⟩, fapply pmap.mk,
-- intro a, esimp, exact f a,
-- reflexivity},
-- { intro f, reflexivity},
-- { intro u, cases u with b f, cases f with f p, esimp at *, apply sigma_eq p,
-- esimp, apply sorry
-- }
-- end
definition pmap_bool_equiv (B : Type*) : map₊ Bool B ≃ B :=
begin
fapply equiv.MK,
{ intro f, cases f with f p, exact f tt},
{ intro b, fconstructor,
intro u, cases u, exact pt, exact b,
reflexivity},
{ intro b, reflexivity},
{ intro f, cases f with f p, esimp, fapply pmap_eq,
{ intro a, cases a; all_goals (esimp at *), exact p⁻¹},
{ esimp, exact !con.left_inv⁻¹}},
end
definition ap1 [constructor] (f : A →* B) : Ω A →* Ω B :=
begin
fconstructor,
{ intro p, exact !respect_pt⁻¹ ⬝ ap f p ⬝ !respect_pt},
{ esimp, apply con.left_inv}
end
definition apn [unfold 3] (n : ℕ) (f : map₊ A B) : Ω[n] A →* Ω[n] B :=
begin
induction n with n IH,
{ exact f},
{ esimp [Iterated_loop_space], exact ap1 IH}
end
variable (A)
definition loop_space_succ_eq_in (n : ℕ) : Ω[succ n] A = Ω[n] (Ω A) :=
begin
induction n with n IH,
{ reflexivity},
{ exact ap Loop_space IH}
end
definition loop_space_add (n m : ℕ) : Ω[n] (Ω[m] A) = Ω[m+n] (A) :=
begin
induction n with n IH,
{ reflexivity},
{ exact ap Loop_space IH}
end
definition loop_space_succ_eq_out (n : ℕ) : Ω[succ n] A = Ω(Ω[n] A) :=
idp
variable {A}
/- the equality [loop_space_succ_eq_in] preserves concatenation -/
theorem loop_space_succ_eq_in_concat {n : ℕ} (p q : Ω[succ (succ n)] A) :
transport carrier (ap Loop_space (loop_space_succ_eq_in A n)) (p ⬝ q)
= transport carrier (ap Loop_space (loop_space_succ_eq_in A n)) p
⬝ transport carrier (ap Loop_space (loop_space_succ_eq_in A n)) q :=
begin
rewrite [-+tr_compose, ↑function.compose],
rewrite [+@transport_eq_FlFr_D _ _ _ _ Point Point, +con.assoc], apply whisker_left,
rewrite [-+con.assoc], apply whisker_right, rewrite [con_inv_cancel_right, ▸*, -ap_con]
end
definition loop_space_loop_irrel (p : point A = point A) : Ω(Pointed.mk p) = Ω[2] A :=
begin
intros, fapply Pointed_eq,
{ esimp, transitivity _,
apply eq_equiv_fn_eq_of_equiv (equiv_eq_closed_right _ p⁻¹),
esimp, apply eq_equiv_eq_closed, apply con.right_inv, apply con.right_inv},
{ esimp, apply con.left_inv}
end
definition iterated_loop_space_loop_irrel (n : ℕ) (p : point A = point A)
: Ω[succ n](Pointed.mk p) = Ω[succ (succ n)] A :> Pointed :=
calc
Ω[succ n](Pointed.mk p) = Ω[n](Ω (Pointed.mk p)) : loop_space_succ_eq_in
... = Ω[n] (Ω[2] A) : loop_space_loop_irrel
... = Ω[2+n] A : loop_space_add
... = Ω[n+2] A : by rewrite [algebra.add.comm]
-- TODO:
-- definition apn_compose (n : ℕ) (g : B →* C) (f : A →* B) : apn n (g ∘* f) ~* apn n g ∘* apn n f :=
-- _
definition ap1_compose (g : B →* C) (f : A →* B) : ap1 (g ∘* f) ~* ap1 g ∘* ap1 f :=
begin
induction B, induction C, induction g with g pg, induction f with f pf, esimp at *,
induction pg, induction pf,
fconstructor,
{ intro p, esimp, apply whisker_left, exact ap_compose g f p ⬝ ap (ap g) !idp_con⁻¹},
{ reflexivity}
end
protected definition phomotopy.refl [refl] (f : A →* B) : f ~* f :=
begin
fconstructor,
{ intro a, exact idp},
{ apply idp_con}
end
protected definition phomotopy.trans [trans] (p : f ~* g) (q : g ~* h)
: f ~* h :=
begin
fconstructor,
{ intro a, exact p a ⬝ q a},
{ induction f, induction g, induction p with p p', induction q with q q', esimp at *,
induction p', induction q', esimp, apply con.assoc}
end
protected definition phomotopy.symm [symm] (p : f ~* g) : g ~* f :=
begin
fconstructor,
{ intro a, exact (p a)⁻¹},
{ induction f, induction p with p p', esimp at *,
induction p', esimp, apply inv_con_cancel_left}
end
infix ` ⬝* `:75 := phomotopy.trans
postfix `⁻¹*`:(max+1) := phomotopy.symm
definition eq_of_phomotopy (p : f ~* g) : f = g :=
begin
fapply pmap_eq,
{ intro a, exact p a},
{ exact !to_homotopy_pt⁻¹}
end
definition pwhisker_left (h : B →* C) (p : f ~* g) : h ∘* f ~* h ∘* g :=
begin
fconstructor,
{ intro a, exact ap h (p a)},
{ induction A, induction B, induction C,
induction f with f pf, induction g with g pg, induction h with h ph,
induction p with p p', esimp at *, induction ph, induction pg, induction p', reflexivity}
end
definition pwhisker_right (h : C →* A) (p : f ~* g) : f ∘* h ~* g ∘* h :=
begin
fconstructor,
{ intro a, exact p (h a)},
{ induction A, induction B, induction C,
induction f with f pf, induction g with g pg, induction h with h ph,
induction p with p p', esimp at *, induction ph, induction pg, induction p', esimp,
exact !idp_con⁻¹}
end
structure pequiv (A B : Type*) :=
(to_pmap : A →* B)
(is_equiv_to_pmap : is_equiv to_pmap)
infix ` ≃* `:25 := pequiv
attribute pequiv.to_pmap [coercion]
attribute pequiv.is_equiv_to_pmap [instance]
definition equiv_of_pequiv [constructor] (f : A ≃* B) : A ≃ B :=
equiv.mk f _
definition to_pinv [constructor] (f : A ≃* B) : B →* A :=
pmap.mk f⁻¹ (ap f⁻¹ (respect_pt f)⁻¹ ⬝ !left_inv)
definition pua {A B : Type*} (f : A ≃* B) : A = B :=
Pointed_eq (equiv_of_pequiv f) !respect_pt
end pointed
|
98780f9bfc742b3780493082f4c9ed8e8d475dae | 4727251e0cd73359b15b664c3170e5d754078599 | /src/ring_theory/fractional_ideal.lean | 630cce4cc9454786d275027ba56efbad7771efac | [
"Apache-2.0"
] | permissive | Vierkantor/mathlib | 0ea59ac32a3a43c93c44d70f441c4ee810ccceca | 83bc3b9ce9b13910b57bda6b56222495ebd31c2f | refs/heads/master | 1,658,323,012,449 | 1,652,256,003,000 | 1,652,256,003,000 | 209,296,341 | 0 | 1 | Apache-2.0 | 1,568,807,655,000 | 1,568,807,655,000 | null | UTF-8 | Lean | false | false | 51,881 | lean | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import algebra.big_operators.finprod
import ring_theory.integral_closure
import ring_theory.localization.integer
import ring_theory.localization.submodule
import ring_theory.noetherian
import ring_theory.principal_ideal_domain
import tactic.field_simp
/-!
# Fractional ideals
This file defines fractional ideals of an integral domain and proves basic facts about them.
## Main definitions
Let `S` be a submonoid of an integral domain `R`, `P` the localization of `R` at `S`, and `f` the
natural ring hom from `R` to `P`.
* `is_fractional` defines which `R`-submodules of `P` are fractional ideals
* `fractional_ideal S P` is the type of fractional ideals in `P`
* `has_coe_t (ideal R) (fractional_ideal S P)` instance
* `comm_semiring (fractional_ideal S P)` instance:
the typical ideal operations generalized to fractional ideals
* `lattice (fractional_ideal S P)` instance
* `map` is the pushforward of a fractional ideal along an algebra morphism
Let `K` be the localization of `R` at `R⁰ = R \ {0}` (i.e. the field of fractions).
* `fractional_ideal R⁰ K` is the type of fractional ideals in the field of fractions
* `has_div (fractional_ideal R⁰ K)` instance:
the ideal quotient `I / J` (typically written $I : J$, but a `:` operator cannot be defined)
## Main statements
* `mul_left_mono` and `mul_right_mono` state that ideal multiplication is monotone
* `prod_one_self_div_eq` states that `1 / I` is the inverse of `I` if one exists
* `is_noetherian` states that every fractional ideal of a noetherian integral domain is noetherian
## Implementation notes
Fractional ideals are considered equal when they contain the same elements,
independent of the denominator `a : R` such that `a I ⊆ R`.
Thus, we define `fractional_ideal` to be the subtype of the predicate `is_fractional`,
instead of having `fractional_ideal` be a structure of which `a` is a field.
Most definitions in this file specialize operations from submodules to fractional ideals,
proving that the result of this operation is fractional if the input is fractional.
Exceptions to this rule are defining `(+) := (⊔)` and `⊥ := 0`,
in order to re-use their respective proof terms.
We can still use `simp` to show `↑I + ↑J = ↑(I + J)` and `↑⊥ = ↑0`.
Many results in fact do not need that `P` is a localization, only that `P` is an
`R`-algebra. We omit the `is_localization` parameter whenever this is practical.
Similarly, we don't assume that the localization is a field until we need it to
define ideal quotients. When this assumption is needed, we replace `S` with `R⁰`,
making the localization a field.
## References
* https://en.wikipedia.org/wiki/Fractional_ideal
## Tags
fractional ideal, fractional ideals, invertible ideal
-/
open is_localization
open_locale pointwise
open_locale non_zero_divisors
section defs
variables {R : Type*} [comm_ring R] {S : submonoid R} {P : Type*} [comm_ring P]
variables [algebra R P]
variables (S)
/-- A submodule `I` is a fractional ideal if `a I ⊆ R` for some `a ≠ 0`. -/
def is_fractional (I : submodule R P) :=
∃ a ∈ S, ∀ b ∈ I, is_integer R (a • b)
variables (S P)
/-- The fractional ideals of a domain `R` are ideals of `R` divided by some `a ∈ R`.
More precisely, let `P` be a localization of `R` at some submonoid `S`,
then a fractional ideal `I ⊆ P` is an `R`-submodule of `P`,
such that there is a nonzero `a : R` with `a I ⊆ R`.
-/
def fractional_ideal :=
{I : submodule R P // is_fractional S I}
end defs
namespace fractional_ideal
open set
open submodule
variables {R : Type*} [comm_ring R] {S : submonoid R} {P : Type*} [comm_ring P]
variables [algebra R P] [loc : is_localization S P]
/-- Map a fractional ideal `I` to a submodule by forgetting that `∃ a, a I ⊆ R`.
This coercion is typically called `coe_to_submodule` in lemma names
(or `coe` when the coercion is clear from the context),
not to be confused with `is_localization.coe_submodule : ideal R → submodule R P`
(which we use to define `coe : ideal R → fractional_ideal S P`,
referred to as `coe_ideal` in theorem names).
-/
instance : has_coe (fractional_ideal S P) (submodule R P) := ⟨λ I, I.val⟩
protected lemma is_fractional (I : fractional_ideal S P) :
is_fractional S (I : submodule R P) :=
I.prop
section set_like
instance : set_like (fractional_ideal S P) P :=
{ coe := λ I, ↑(I : submodule R P),
coe_injective' := set_like.coe_injective.comp subtype.coe_injective }
@[simp] lemma mem_coe {I : fractional_ideal S P} {x : P} :
x ∈ (I : submodule R P) ↔ x ∈ I :=
iff.rfl
@[ext] lemma ext {I J : fractional_ideal S P} : (∀ x, x ∈ I ↔ x ∈ J) → I = J := set_like.ext
/-- Copy of a `fractional_ideal` with a new underlying set equal to the old one.
Useful to fix definitional equalities. -/
protected def copy (p : fractional_ideal S P) (s : set P) (hs : s = ↑p) : fractional_ideal S P :=
⟨submodule.copy p s hs, by { convert p.is_fractional, ext, simp only [hs], refl }⟩
end set_like
@[simp] lemma val_eq_coe (I : fractional_ideal S P) : I.val = I := rfl
@[simp, norm_cast] lemma coe_mk (I : submodule R P) (hI : is_fractional S I) :
(subtype.mk I hI : submodule R P) = I := rfl
lemma coe_to_submodule_injective :
function.injective (coe : fractional_ideal S P → submodule R P) :=
subtype.coe_injective
lemma is_fractional_of_le_one (I : submodule R P) (h : I ≤ 1) :
is_fractional S I :=
begin
use [1, S.one_mem],
intros b hb,
rw one_smul,
obtain ⟨b', b'_mem, rfl⟩ := h hb,
exact set.mem_range_self b',
end
lemma is_fractional_of_le {I : submodule R P} {J : fractional_ideal S P}
(hIJ : I ≤ J) : is_fractional S I :=
begin
obtain ⟨a, a_mem, ha⟩ := J.is_fractional,
use [a, a_mem],
intros b b_mem,
exact ha b (hIJ b_mem)
end
/-- Map an ideal `I` to a fractional ideal by forgetting `I` is integral.
This is a bundled version of `is_localization.coe_submodule : ideal R → submodule R P`,
which is not to be confused with the `coe : fractional_ideal S P → submodule R P`,
also called `coe_to_submodule` in theorem names.
This map is available as a ring hom, called `fractional_ideal.coe_ideal_hom`.
-/
-- Is a `coe_t` rather than `coe` to speed up failing inference, see library note [use has_coe_t]
instance coe_to_fractional_ideal : has_coe_t (ideal R) (fractional_ideal S P) :=
⟨λ I, ⟨coe_submodule P I, is_fractional_of_le_one _
(by simpa using coe_submodule_mono P (le_top : I ≤ ⊤))⟩⟩
@[simp, norm_cast] lemma coe_coe_ideal (I : ideal R) :
((I : fractional_ideal S P) : submodule R P) = coe_submodule P I := rfl
variables (S)
@[simp] lemma mem_coe_ideal {x : P} {I : ideal R} :
x ∈ (I : fractional_ideal S P) ↔ ∃ x', x' ∈ I ∧ algebra_map R P x' = x :=
mem_coe_submodule _ _
lemma mem_coe_ideal_of_mem {x : R} {I : ideal R} (hx : x ∈ I) :
algebra_map R P x ∈ (I : fractional_ideal S P) :=
(mem_coe_ideal S).mpr ⟨x, hx, rfl⟩
lemma coe_ideal_le_coe_ideal' [is_localization S P] (h : S ≤ non_zero_divisors R)
{I J : ideal R} : (I : fractional_ideal S P) ≤ J ↔ I ≤ J :=
coe_submodule_le_coe_submodule h
@[simp] lemma coe_ideal_le_coe_ideal (K : Type*) [comm_ring K] [algebra R K] [is_fraction_ring R K]
{I J : ideal R} : (I : fractional_ideal R⁰ K) ≤ J ↔ I ≤ J :=
is_fraction_ring.coe_submodule_le_coe_submodule
instance : has_zero (fractional_ideal S P) := ⟨(0 : ideal R)⟩
@[simp] lemma mem_zero_iff {x : P} : x ∈ (0 : fractional_ideal S P) ↔ x = 0 :=
⟨(λ ⟨x', x'_mem_zero, x'_eq_x⟩,
have x'_eq_zero : x' = 0 := x'_mem_zero,
by simp [x'_eq_x.symm, x'_eq_zero]),
(λ hx, ⟨0, rfl, by simp [hx]⟩)⟩
variables {S}
@[simp, norm_cast] lemma coe_zero : ↑(0 : fractional_ideal S P) = (⊥ : submodule R P) :=
submodule.ext $ λ _, mem_zero_iff S
@[simp, norm_cast] lemma coe_to_fractional_ideal_bot : ((⊥ : ideal R) : fractional_ideal S P) = 0 :=
rfl
variables (P)
include loc
@[simp] lemma exists_mem_to_map_eq {x : R} {I : ideal R} (h : S ≤ non_zero_divisors R) :
(∃ x', x' ∈ I ∧ algebra_map R P x' = algebra_map R P x) ↔ x ∈ I :=
⟨λ ⟨x', hx', eq⟩, is_localization.injective _ h eq ▸ hx', λ h, ⟨x, h, rfl⟩⟩
variables {P}
lemma coe_to_fractional_ideal_injective (h : S ≤ non_zero_divisors R) :
function.injective (coe : ideal R → fractional_ideal S P) :=
λ I J heq, have
∀ (x : R), algebra_map R P x ∈ (I : fractional_ideal S P) ↔
algebra_map R P x ∈ (J : fractional_ideal S P) :=
λ x, heq ▸ iff.rfl,
ideal.ext (by simpa only [mem_coe_ideal, exists_prop, exists_mem_to_map_eq P h] using this)
lemma coe_to_fractional_ideal_eq_zero {I : ideal R} (hS : S ≤ non_zero_divisors R) :
(I : fractional_ideal S P) = 0 ↔ I = (⊥ : ideal R) :=
⟨λ h, coe_to_fractional_ideal_injective hS h,
λ h, by rw [h, coe_to_fractional_ideal_bot]⟩
lemma coe_to_fractional_ideal_ne_zero {I : ideal R} (hS : S ≤ non_zero_divisors R) :
(I : fractional_ideal S P) ≠ 0 ↔ I ≠ (⊥ : ideal R) :=
not_iff_not.mpr (coe_to_fractional_ideal_eq_zero hS)
omit loc
lemma coe_to_submodule_eq_bot {I : fractional_ideal S P} :
(I : submodule R P) = ⊥ ↔ I = 0 :=
⟨λ h, coe_to_submodule_injective (by simp [h]),
λ h, by simp [h]⟩
lemma coe_to_submodule_ne_bot {I : fractional_ideal S P} :
↑I ≠ (⊥ : submodule R P) ↔ I ≠ 0 :=
not_iff_not.mpr coe_to_submodule_eq_bot
instance : inhabited (fractional_ideal S P) := ⟨0⟩
instance : has_one (fractional_ideal S P) :=
⟨(⊤ : ideal R)⟩
variables (S)
@[simp, norm_cast] lemma coe_ideal_top : ((⊤ : ideal R) : fractional_ideal S P) = 1 :=
rfl
lemma mem_one_iff {x : P} : x ∈ (1 : fractional_ideal S P) ↔ ∃ x' : R, algebra_map R P x' = x :=
iff.intro (λ ⟨x', _, h⟩, ⟨x', h⟩) (λ ⟨x', h⟩, ⟨x', ⟨⟩, h⟩)
lemma coe_mem_one (x : R) : algebra_map R P x ∈ (1 : fractional_ideal S P) :=
(mem_one_iff S).mpr ⟨x, rfl⟩
lemma one_mem_one : (1 : P) ∈ (1 : fractional_ideal S P) :=
(mem_one_iff S).mpr ⟨1, ring_hom.map_one _⟩
variables {S}
/-- `(1 : fractional_ideal S P)` is defined as the R-submodule `f(R) ≤ P`.
However, this is not definitionally equal to `1 : submodule R P`,
which is proved in the actual `simp` lemma `coe_one`. -/
lemma coe_one_eq_coe_submodule_top :
↑(1 : fractional_ideal S P) = coe_submodule P (⊤ : ideal R) :=
rfl
@[simp, norm_cast] lemma coe_one :
(↑(1 : fractional_ideal S P) : submodule R P) = 1 :=
by rw [coe_one_eq_coe_submodule_top, coe_submodule_top]
section lattice
/-!
### `lattice` section
Defines the order on fractional ideals as inclusion of their underlying sets,
and ports the lattice structure on submodules to fractional ideals.
-/
@[simp] lemma coe_le_coe {I J : fractional_ideal S P} :
(I : submodule R P) ≤ (J : submodule R P) ↔ I ≤ J :=
iff.rfl
lemma zero_le (I : fractional_ideal S P) : 0 ≤ I :=
begin
intros x hx,
convert submodule.zero_mem _,
simpa using hx
end
instance order_bot : order_bot (fractional_ideal S P) :=
{ bot := 0,
bot_le := zero_le }
@[simp] lemma bot_eq_zero : (⊥ : fractional_ideal S P) = 0 :=
rfl
@[simp] lemma le_zero_iff {I : fractional_ideal S P} : I ≤ 0 ↔ I = 0 :=
le_bot_iff
lemma eq_zero_iff {I : fractional_ideal S P} : I = 0 ↔ (∀ x ∈ I, x = (0 : P)) :=
⟨ (λ h x hx, by simpa [h, mem_zero_iff] using hx),
(λ h, le_bot_iff.mp (λ x hx, (mem_zero_iff S).mpr (h x hx))) ⟩
lemma _root_.is_fractional.sup {I J : submodule R P} :
is_fractional S I → is_fractional S J → is_fractional S (I ⊔ J)
| ⟨aI, haI, hI⟩ ⟨aJ, haJ, hJ⟩ := ⟨aI * aJ, S.mul_mem haI haJ, λ b hb, begin
rcases mem_sup.mp hb with ⟨bI, hbI, bJ, hbJ, rfl⟩,
rw smul_add,
apply is_integer_add,
{ rw [mul_smul, smul_comm],
exact is_integer_smul (hI bI hbI), },
{ rw mul_smul,
exact is_integer_smul (hJ bJ hbJ) }
end⟩
lemma _root_.is_fractional.inf_right {I : submodule R P} :
is_fractional S I → ∀ J, is_fractional S (I ⊓ J)
| ⟨aI, haI, hI⟩ J := ⟨aI, haI, λ b hb, begin
rcases mem_inf.mp hb with ⟨hbI, hbJ⟩,
exact hI b hbI
end⟩
instance : has_inf (fractional_ideal S P) := ⟨λ I J, ⟨I ⊓ J, I.is_fractional.inf_right J⟩⟩
@[simp, norm_cast]
lemma coe_inf (I J : fractional_ideal S P) : ↑(I ⊓ J) = (I ⊓ J : submodule R P) := rfl
instance : has_sup (fractional_ideal S P) := ⟨λ I J, ⟨I ⊔ J, I.is_fractional.sup J.is_fractional⟩⟩
@[norm_cast]
lemma coe_sup (I J : fractional_ideal S P) : ↑(I ⊔ J) = (I ⊔ J : submodule R P) := rfl
instance lattice : lattice (fractional_ideal S P) :=
function.injective.lattice _ subtype.coe_injective coe_sup coe_inf
instance : semilattice_sup (fractional_ideal S P) :=
{ ..fractional_ideal.lattice }
end lattice
section semiring
instance : has_add (fractional_ideal S P) := ⟨(⊔)⟩
@[simp]
lemma sup_eq_add (I J : fractional_ideal S P) : I ⊔ J = I + J := rfl
@[simp, norm_cast]
lemma coe_add (I J : fractional_ideal S P) : (↑(I + J) : submodule R P) = I + J := rfl
@[simp, norm_cast]
lemma coe_ideal_sup (I J : ideal R) : ↑(I ⊔ J) = (I + J : fractional_ideal S P) :=
coe_to_submodule_injective $ coe_submodule_sup _ _ _
lemma _root_.is_fractional.nsmul {I : submodule R P} :
Π n : ℕ, is_fractional S I → is_fractional S (n • I : submodule R P)
| 0 _ := begin
rw [zero_smul],
convert ((0 : ideal R) : fractional_ideal S P).is_fractional,
simp,
end
| (n + 1) h := begin
rw succ_nsmul,
exact h.sup (_root_.is_fractional.nsmul n h)
end
instance : has_scalar ℕ (fractional_ideal S P) :=
{ smul := λ n I, ⟨n • I, I.is_fractional.nsmul n⟩}
@[norm_cast]
lemma coe_nsmul (n : ℕ) (I : fractional_ideal S P) : (↑(n • I) : submodule R P) = n • I := rfl
lemma _root_.is_fractional.mul {I J : submodule R P} :
is_fractional S I → is_fractional S J → is_fractional S (I * J : submodule R P)
| ⟨aI, haI, hI⟩ ⟨aJ, haJ, hJ⟩ := ⟨aI * aJ, S.mul_mem haI haJ, λ b hb, begin
apply submodule.mul_induction_on hb,
{ intros m hm n hn,
obtain ⟨n', hn'⟩ := hJ n hn,
rw [mul_smul, mul_comm m, ← smul_mul_assoc, ← hn', ← algebra.smul_def],
apply hI,
exact submodule.smul_mem _ _ hm },
{ intros x y hx hy,
rw smul_add,
apply is_integer_add hx hy },
end⟩
lemma _root_.is_fractional.pow {I : submodule R P} (h : is_fractional S I) :
∀ n : ℕ, is_fractional S (I ^ n : submodule R P)
| 0 := is_fractional_of_le_one _ (pow_zero _).le
| (n + 1) := (pow_succ I n).symm ▸ h.mul (_root_.is_fractional.pow n)
/-- `fractional_ideal.mul` is the product of two fractional ideals,
used to define the `has_mul` instance.
This is only an auxiliary definition: the preferred way of writing `I.mul J` is `I * J`.
Elaborated terms involving `fractional_ideal` tend to grow quite large,
so by making definitions irreducible, we hope to avoid deep unfolds.
-/
@[irreducible]
def mul (I J : fractional_ideal S P) : fractional_ideal S P :=
⟨I * J, I.is_fractional.mul J.is_fractional⟩
local attribute [semireducible] mul
instance : has_mul (fractional_ideal S P) := ⟨λ I J, mul I J⟩
@[simp] lemma mul_eq_mul (I J : fractional_ideal S P) : mul I J = I * J := rfl
@[simp, norm_cast]
lemma coe_mul (I J : fractional_ideal S P) : (↑(I * J) : submodule R P) = I * J := rfl
@[simp, norm_cast]
lemma coe_ideal_mul (I J : ideal R) : (↑(I * J) : fractional_ideal S P) = I * J :=
coe_to_submodule_injective $ coe_submodule_mul _ _ _
lemma mul_left_mono (I : fractional_ideal S P) : monotone ((*) I) :=
λ J J' h, mul_le.mpr (λ x hx y hy, mul_mem_mul hx (h hy))
lemma mul_right_mono (I : fractional_ideal S P) : monotone (λ J, J * I) :=
λ J J' h, mul_le.mpr (λ x hx y hy, mul_mem_mul (h hx) hy)
lemma mul_mem_mul {I J : fractional_ideal S P} {i j : P} (hi : i ∈ I) (hj : j ∈ J) :
i * j ∈ I * J := submodule.mul_mem_mul hi hj
lemma mul_le {I J K : fractional_ideal S P} :
I * J ≤ K ↔ (∀ (i ∈ I) (j ∈ J), i * j ∈ K) :=
submodule.mul_le
instance : has_pow (fractional_ideal S P) ℕ := ⟨λ I n, ⟨I^n, I.is_fractional.pow n⟩⟩
@[simp, norm_cast]
lemma coe_pow (I : fractional_ideal S P) (n : ℕ) : ↑(I ^ n) = (I ^ n : submodule R P) := rfl
@[elab_as_eliminator] protected theorem mul_induction_on
{I J : fractional_ideal S P}
{C : P → Prop} {r : P} (hr : r ∈ I * J)
(hm : ∀ (i ∈ I) (j ∈ J), C (i * j))
(ha : ∀ x y, C x → C y → C (x + y)) : C r :=
submodule.mul_induction_on hr hm ha
instance : comm_semiring (fractional_ideal S P) :=
function.injective.comm_semiring _ subtype.coe_injective
coe_zero coe_one coe_add coe_mul (λ _ _, coe_nsmul _ _) coe_pow
section order
lemma add_le_add_left {I J : fractional_ideal S P} (hIJ : I ≤ J) (J' : fractional_ideal S P) :
J' + I ≤ J' + J :=
sup_le_sup_left hIJ J'
lemma mul_le_mul_left {I J : fractional_ideal S P} (hIJ : I ≤ J) (J' : fractional_ideal S P) :
J' * I ≤ J' * J :=
mul_le.mpr (λ k hk j hj, mul_mem_mul hk (hIJ hj))
lemma le_self_mul_self {I : fractional_ideal S P} (hI: 1 ≤ I) : I ≤ I * I :=
begin
convert mul_left_mono I hI,
exact (mul_one I).symm
end
lemma mul_self_le_self {I : fractional_ideal S P} (hI: I ≤ 1) : I * I ≤ I :=
begin
convert mul_left_mono I hI,
exact (mul_one I).symm
end
lemma coe_ideal_le_one {I : ideal R} : (I : fractional_ideal S P) ≤ 1 :=
λ x hx, let ⟨y, _, hy⟩ := (fractional_ideal.mem_coe_ideal S).mp hx
in (fractional_ideal.mem_one_iff S).mpr ⟨y, hy⟩
lemma le_one_iff_exists_coe_ideal {J : fractional_ideal S P} :
J ≤ (1 : fractional_ideal S P) ↔ ∃ (I : ideal R), ↑I = J :=
begin
split,
{ intro hJ,
refine ⟨⟨{x : R | algebra_map R P x ∈ J}, _, _, _⟩, _⟩,
{ intros a b ha hb,
rw [mem_set_of_eq, ring_hom.map_add],
exact J.val.add_mem ha hb },
{ rw [mem_set_of_eq, ring_hom.map_zero],
exact J.val.zero_mem },
{ intros c x hx,
rw [smul_eq_mul, mem_set_of_eq, ring_hom.map_mul, ← algebra.smul_def],
exact J.val.smul_mem c hx },
{ ext x,
split,
{ rintros ⟨y, hy, eq_y⟩,
rwa ← eq_y },
{ intro hx,
obtain ⟨y, eq_x⟩ := (fractional_ideal.mem_one_iff S).mp (hJ hx),
rw ← eq_x at *,
exact ⟨y, hx, rfl⟩ } } },
{ rintro ⟨I, hI⟩,
rw ← hI,
apply coe_ideal_le_one },
end
variables (S P)
/-- `coe_ideal_hom (S : submonoid R) P` is `coe : ideal R → fractional_ideal S P` as a ring hom -/
@[simps]
def coe_ideal_hom : ideal R →+* fractional_ideal S P :=
{ to_fun := coe,
map_add' := coe_ideal_sup,
map_mul' := coe_ideal_mul,
map_one' := by rw [ideal.one_eq_top, coe_ideal_top],
map_zero' := coe_to_fractional_ideal_bot }
lemma coe_ideal_pow (I : ideal R) (n : ℕ) : (↑(I^n) : fractional_ideal S P) = I^n :=
(fractional_ideal.coe_ideal_hom S P).map_pow _ n
open_locale big_operators
lemma coe_ideal_finprod [is_localization S P] {α : Sort*} {f : α → ideal R}
(hS : S ≤ non_zero_divisors R) :
((∏ᶠ a : α, f a : ideal R) : fractional_ideal S P) = ∏ᶠ a : α, (f a : fractional_ideal S P) :=
monoid_hom.map_finprod_of_injective (fractional_ideal.coe_ideal_hom S P).to_monoid_hom
(fractional_ideal.coe_to_fractional_ideal_injective hS) f
end order
variables {P' : Type*} [comm_ring P'] [algebra R P'] [loc' : is_localization S P']
variables {P'' : Type*} [comm_ring P''] [algebra R P''] [loc'' : is_localization S P'']
lemma _root_.is_fractional.map (g : P →ₐ[R] P') {I : submodule R P} :
is_fractional S I → is_fractional S (submodule.map g.to_linear_map I)
| ⟨a, a_nonzero, hI⟩ := ⟨a, a_nonzero, λ b hb, begin
obtain ⟨b', b'_mem, hb'⟩ := submodule.mem_map.mp hb,
obtain ⟨x, hx⟩ := hI b' b'_mem,
use x,
erw [←g.commutes, hx, g.map_smul, hb']
end⟩
/-- `I.map g` is the pushforward of the fractional ideal `I` along the algebra morphism `g` -/
def map (g : P →ₐ[R] P') :
fractional_ideal S P → fractional_ideal S P' :=
λ I, ⟨submodule.map g.to_linear_map I, I.is_fractional.map g⟩
@[simp, norm_cast] lemma coe_map (g : P →ₐ[R] P') (I : fractional_ideal S P) :
↑(map g I) = submodule.map g.to_linear_map I := rfl
@[simp] lemma mem_map {I : fractional_ideal S P} {g : P →ₐ[R] P'}
{y : P'} : y ∈ I.map g ↔ ∃ x, x ∈ I ∧ g x = y :=
submodule.mem_map
variables (I J : fractional_ideal S P) (g : P →ₐ[R] P')
@[simp] lemma map_id : I.map (alg_hom.id _ _) = I :=
coe_to_submodule_injective (submodule.map_id I)
@[simp] lemma map_comp (g' : P' →ₐ[R] P'') :
I.map (g'.comp g) = (I.map g).map g' :=
coe_to_submodule_injective (submodule.map_comp g.to_linear_map g'.to_linear_map I)
@[simp, norm_cast] lemma map_coe_ideal (I : ideal R) :
(I : fractional_ideal S P).map g = I :=
begin
ext x,
simp only [mem_coe_ideal],
split,
{ rintro ⟨_, ⟨y, hy, rfl⟩, rfl⟩,
exact ⟨y, hy, (g.commutes y).symm⟩ },
{ rintro ⟨y, hy, rfl⟩,
exact ⟨_, ⟨y, hy, rfl⟩, g.commutes y⟩ },
end
@[simp] lemma map_one :
(1 : fractional_ideal S P).map g = 1 :=
map_coe_ideal g ⊤
@[simp] lemma map_zero :
(0 : fractional_ideal S P).map g = 0 :=
map_coe_ideal g 0
@[simp] lemma map_add : (I + J).map g = I.map g + J.map g :=
coe_to_submodule_injective (submodule.map_sup _ _ _)
@[simp] lemma map_mul : (I * J).map g = I.map g * J.map g :=
coe_to_submodule_injective (submodule.map_mul _ _ _)
@[simp] lemma map_map_symm (g : P ≃ₐ[R] P') :
(I.map (g : P →ₐ[R] P')).map (g.symm : P' →ₐ[R] P) = I :=
by rw [←map_comp, g.symm_comp, map_id]
@[simp] lemma map_symm_map (I : fractional_ideal S P') (g : P ≃ₐ[R] P') :
(I.map (g.symm : P' →ₐ[R] P)).map (g : P →ₐ[R] P') = I :=
by rw [←map_comp, g.comp_symm, map_id]
lemma map_mem_map {f : P →ₐ[R] P'} (h : function.injective f) {x : P} {I : fractional_ideal S P} :
f x ∈ map f I ↔ x ∈ I :=
mem_map.trans ⟨λ ⟨x', hx', x'_eq⟩, h x'_eq ▸ hx', λ h, ⟨x, h, rfl⟩⟩
lemma map_injective (f : P →ₐ[R] P') (h : function.injective f) :
function.injective (map f : fractional_ideal S P → fractional_ideal S P') :=
λ I J hIJ, fractional_ideal.ext (λ x, (fractional_ideal.map_mem_map h).symm.trans
(hIJ.symm ▸ fractional_ideal.map_mem_map h))
/-- If `g` is an equivalence, `map g` is an isomorphism -/
def map_equiv (g : P ≃ₐ[R] P') :
fractional_ideal S P ≃+* fractional_ideal S P' :=
{ to_fun := map g,
inv_fun := map g.symm,
map_add' := λ I J, map_add I J _,
map_mul' := λ I J, map_mul I J _,
left_inv := λ I, by { rw [←map_comp, alg_equiv.symm_comp, map_id] },
right_inv := λ I, by { rw [←map_comp, alg_equiv.comp_symm, map_id] } }
@[simp] lemma coe_fun_map_equiv (g : P ≃ₐ[R] P') :
(map_equiv g : fractional_ideal S P → fractional_ideal S P') = map g :=
rfl
@[simp] lemma map_equiv_apply (g : P ≃ₐ[R] P') (I : fractional_ideal S P) :
map_equiv g I = map ↑g I := rfl
@[simp] lemma map_equiv_symm (g : P ≃ₐ[R] P') :
((map_equiv g).symm : fractional_ideal S P' ≃+* _) = map_equiv g.symm := rfl
@[simp] lemma map_equiv_refl :
map_equiv alg_equiv.refl = ring_equiv.refl (fractional_ideal S P) :=
ring_equiv.ext (λ x, by simp)
lemma is_fractional_span_iff {s : set P} :
is_fractional S (span R s) ↔ ∃ a ∈ S, ∀ (b : P), b ∈ s → is_integer R (a • b) :=
⟨λ ⟨a, a_mem, h⟩, ⟨a, a_mem, λ b hb, h b (subset_span hb)⟩,
λ ⟨a, a_mem, h⟩, ⟨a, a_mem, λ b hb, span_induction hb
h
(by { rw smul_zero, exact is_integer_zero })
(λ x y hx hy, by { rw smul_add, exact is_integer_add hx hy })
(λ s x hx, by { rw smul_comm, exact is_integer_smul hx })⟩⟩
include loc
lemma is_fractional_of_fg {I : submodule R P} (hI : I.fg) :
is_fractional S I :=
begin
rcases hI with ⟨I, rfl⟩,
rcases exist_integer_multiples_of_finset S I with ⟨⟨s, hs1⟩, hs⟩,
rw is_fractional_span_iff,
exact ⟨s, hs1, hs⟩,
end
omit loc
lemma mem_span_mul_finite_of_mem_mul {I J : fractional_ideal S P} {x : P} (hx : x ∈ I * J) :
∃ (T T' : finset P), (T : set P) ⊆ I ∧ (T' : set P) ⊆ J ∧ x ∈ span R (T * T' : set P) :=
submodule.mem_span_mul_finite_of_mem_mul (by simpa using mem_coe.mpr hx)
variables (S)
lemma coe_ideal_fg (inj : function.injective (algebra_map R P)) (I : ideal R) :
fg ((I : fractional_ideal S P) : submodule R P) ↔ I.fg :=
coe_submodule_fg _ inj _
variables {S}
lemma fg_unit (I : (fractional_ideal S P)ˣ) :
fg (I : submodule R P) :=
begin
have : (1 : P) ∈ (I * ↑I⁻¹ : fractional_ideal S P),
{ rw units.mul_inv, exact one_mem_one _ },
obtain ⟨T, T', hT, hT', one_mem⟩ := mem_span_mul_finite_of_mem_mul this,
refine ⟨T, submodule.span_eq_of_le _ hT _⟩,
rw [← one_mul ↑I, ← mul_one (span R ↑T)],
conv_rhs { rw [← fractional_ideal.coe_one, ← units.mul_inv I, fractional_ideal.coe_mul,
mul_comm ↑↑I, ← mul_assoc] },
refine submodule.mul_le_mul_left
(le_trans _ (submodule.mul_le_mul_right (submodule.span_le.mpr hT'))),
rwa [submodule.one_le, submodule.span_mul_span]
end
lemma fg_of_is_unit (I : fractional_ideal S P) (h : is_unit I) :
fg (I : submodule R P) :=
by { rcases h with ⟨I, rfl⟩, exact fg_unit I }
lemma _root_.ideal.fg_of_is_unit (inj : function.injective (algebra_map R P))
(I : ideal R) (h : is_unit (I : fractional_ideal S P)) :
I.fg :=
by { rw ← coe_ideal_fg S inj I, exact fg_of_is_unit I h }
variables (S P P')
include loc loc'
/-- `canonical_equiv f f'` is the canonical equivalence between the fractional
ideals in `P` and in `P'` -/
@[irreducible]
noncomputable def canonical_equiv :
fractional_ideal S P ≃+* fractional_ideal S P' :=
map_equiv
{ commutes' := λ r, ring_equiv_of_ring_equiv_eq _ _,
..ring_equiv_of_ring_equiv P P' (ring_equiv.refl R)
(show S.map _ = S, by rw [ring_equiv.to_monoid_hom_refl, submonoid.map_id]) }
@[simp] lemma mem_canonical_equiv_apply {I : fractional_ideal S P} {x : P'} :
x ∈ canonical_equiv S P P' I ↔
∃ y ∈ I, is_localization.map P' (ring_hom.id R)
(λ y (hy : y ∈ S), show ring_hom.id R y ∈ S, from hy) (y : P) = x :=
begin
rw [canonical_equiv, map_equiv_apply, mem_map],
exact ⟨λ ⟨y, mem, eq⟩, ⟨y, mem, eq⟩, λ ⟨y, mem, eq⟩, ⟨y, mem, eq⟩⟩
end
@[simp] lemma canonical_equiv_symm :
(canonical_equiv S P P').symm = canonical_equiv S P' P :=
ring_equiv.ext $ λ I, set_like.ext_iff.mpr $ λ x,
by { rw [mem_canonical_equiv_apply, canonical_equiv, map_equiv_symm, map_equiv,
ring_equiv.coe_mk, mem_map],
exact ⟨λ ⟨y, mem, eq⟩, ⟨y, mem, eq⟩, λ ⟨y, mem, eq⟩, ⟨y, mem, eq⟩⟩ }
@[simp] lemma canonical_equiv_flip (I) :
canonical_equiv S P P' (canonical_equiv S P' P I) = I :=
by rw [←canonical_equiv_symm, ring_equiv.symm_apply_apply]
end semiring
section is_fraction_ring
/-!
### `is_fraction_ring` section
This section concerns fractional ideals in the field of fractions,
i.e. the type `fractional_ideal R⁰ K` where `is_fraction_ring R K`.
-/
variables {K K' : Type*} [field K] [field K']
variables [algebra R K] [is_fraction_ring R K] [algebra R K'] [is_fraction_ring R K']
variables {I J : fractional_ideal R⁰ K} (h : K →ₐ[R] K')
/-- Nonzero fractional ideals contain a nonzero integer. -/
lemma exists_ne_zero_mem_is_integer [nontrivial R] (hI : I ≠ 0) :
∃ x ≠ (0 : R), algebra_map R K x ∈ I :=
begin
obtain ⟨y, y_mem, y_not_mem⟩ := set_like.exists_of_lt
(by simpa only using bot_lt_iff_ne_bot.mpr hI),
have y_ne_zero : y ≠ 0 := by simpa using y_not_mem,
obtain ⟨z, ⟨x, hx⟩⟩ := exists_integer_multiple R⁰ y,
refine ⟨x, _, _⟩,
{ rw [ne.def, ← @is_fraction_ring.to_map_eq_zero_iff R _ K, hx, algebra.smul_def],
exact mul_ne_zero (is_fraction_ring.to_map_ne_zero_of_mem_non_zero_divisors z.2) y_ne_zero },
{ rw hx,
exact smul_mem _ _ y_mem }
end
lemma map_ne_zero [nontrivial R] (hI : I ≠ 0) : I.map h ≠ 0 :=
begin
obtain ⟨x, x_ne_zero, hx⟩ := exists_ne_zero_mem_is_integer hI,
contrapose! x_ne_zero with map_eq_zero,
refine is_fraction_ring.to_map_eq_zero_iff.mp (eq_zero_iff.mp map_eq_zero _ (mem_map.mpr _)),
exact ⟨algebra_map R K x, hx, h.commutes x⟩,
end
@[simp] lemma map_eq_zero_iff [nontrivial R] : I.map h = 0 ↔ I = 0 :=
⟨imp_of_not_imp_not _ _ (map_ne_zero _), λ hI, hI.symm ▸ map_zero h⟩
lemma coe_ideal_injective : function.injective (coe : ideal R → fractional_ideal R⁰ K) :=
injective_of_le_imp_le _ (λ _ _, (coe_ideal_le_coe_ideal _).mp)
@[simp] lemma coe_ideal_eq_zero_iff {I : ideal R} : (I : fractional_ideal R⁰ K) = 0 ↔ I = ⊥ :=
coe_ideal_injective.eq_iff
lemma coe_ideal_ne_zero_iff {I : ideal R} : (I : fractional_ideal R⁰ K) ≠ 0 ↔ I ≠ ⊥ :=
not_iff_not.mpr coe_ideal_eq_zero_iff
lemma coe_ideal_ne_zero {I : ideal R} (hI : I ≠ ⊥) : (I : fractional_ideal R⁰ K) ≠ 0 :=
coe_ideal_ne_zero_iff.mpr hI
@[simp] lemma coe_ideal_eq_one_iff {I : ideal R} : (I : fractional_ideal R⁰ K) = 1 ↔ I = 1 :=
by simpa only [ideal.one_eq_top] using coe_ideal_injective.eq_iff
end is_fraction_ring
section quotient
/-!
### `quotient` section
This section defines the ideal quotient of fractional ideals.
In this section we need that each non-zero `y : R` has an inverse in
the localization, i.e. that the localization is a field. We satisfy this
assumption by taking `S = non_zero_divisors R`, `R`'s localization at which
is a field because `R` is a domain.
-/
open_locale classical
variables {R₁ : Type*} [comm_ring R₁] {K : Type*} [field K]
variables [algebra R₁ K] [frac : is_fraction_ring R₁ K]
instance : nontrivial (fractional_ideal R₁⁰ K) :=
⟨⟨0, 1, λ h,
have this : (1 : K) ∈ (0 : fractional_ideal R₁⁰ K) :=
by { rw ← (algebra_map R₁ K).map_one, simpa only [h] using coe_mem_one R₁⁰ 1 },
one_ne_zero ((mem_zero_iff _).mp this)⟩⟩
lemma ne_zero_of_mul_eq_one (I J : fractional_ideal R₁⁰ K) (h : I * J = 1) : I ≠ 0 :=
λ hI, @zero_ne_one (fractional_ideal R₁⁰ K) _ _ (by { convert h, simp [hI], })
variables [is_domain R₁]
include frac
lemma _root_.is_fractional.div_of_nonzero {I J : submodule R₁ K} :
is_fractional R₁⁰ I → is_fractional R₁⁰ J → J ≠ 0 → is_fractional R₁⁰ (I / J)
| ⟨aI, haI, hI⟩ ⟨aJ, haJ, hJ⟩ h := begin
obtain ⟨y, mem_J, not_mem_zero⟩ := set_like.exists_of_lt
(by simpa only using bot_lt_iff_ne_bot.mpr h),
obtain ⟨y', hy'⟩ := hJ y mem_J,
use (aI * y'),
split,
{ apply (non_zero_divisors R₁).mul_mem haI (mem_non_zero_divisors_iff_ne_zero.mpr _),
intro y'_eq_zero,
have : algebra_map R₁ K aJ * y = 0,
{ rw [← algebra.smul_def, ←hy', y'_eq_zero, ring_hom.map_zero] },
have y_zero := (mul_eq_zero.mp this).resolve_left
(mt ((injective_iff_map_eq_zero (algebra_map R₁ K)).1 (is_fraction_ring.injective _ _) _)
(mem_non_zero_divisors_iff_ne_zero.mp haJ)),
apply not_mem_zero,
simpa only using (mem_zero_iff R₁⁰).mpr y_zero, },
intros b hb,
convert hI _ (hb _ (submodule.smul_mem _ aJ mem_J)) using 1,
rw [← hy', mul_comm b, ← algebra.smul_def, mul_smul]
end
lemma fractional_div_of_nonzero {I J : fractional_ideal R₁⁰ K} (h : J ≠ 0) :
is_fractional R₁⁰ (I / J : submodule R₁ K) :=
I.is_fractional.div_of_nonzero J.is_fractional $ λ H, h $
coe_to_submodule_injective $ H.trans coe_zero.symm
noncomputable instance fractional_ideal_has_div :
has_div (fractional_ideal R₁⁰ K) :=
⟨ λ I J, if h : J = 0 then 0 else ⟨I / J, fractional_div_of_nonzero h⟩ ⟩
variables {I J : fractional_ideal R₁⁰ K} [ J ≠ 0 ]
@[simp] lemma div_zero {I : fractional_ideal R₁⁰ K} :
I / 0 = 0 :=
dif_pos rfl
lemma div_nonzero {I J : fractional_ideal R₁⁰ K} (h : J ≠ 0) :
(I / J) = ⟨I / J, fractional_div_of_nonzero h⟩ :=
dif_neg h
@[simp] lemma coe_div {I J : fractional_ideal R₁⁰ K} (hJ : J ≠ 0) :
(↑(I / J) : submodule R₁ K) = ↑I / (↑J : submodule R₁ K) :=
congr_arg _ (dif_neg hJ)
lemma mem_div_iff_of_nonzero {I J : fractional_ideal R₁⁰ K} (h : J ≠ 0) {x} :
x ∈ I / J ↔ ∀ y ∈ J, x * y ∈ I :=
by { rw div_nonzero h, exact submodule.mem_div_iff_forall_mul_mem }
lemma mul_one_div_le_one {I : fractional_ideal R₁⁰ K} : I * (1 / I) ≤ 1 :=
begin
by_cases hI : I = 0,
{ rw [hI, div_zero, mul_zero],
exact zero_le 1 },
{ rw [← coe_le_coe, coe_mul, coe_div hI, coe_one],
apply submodule.mul_one_div_le_one },
end
lemma le_self_mul_one_div {I : fractional_ideal R₁⁰ K} (hI : I ≤ (1 : fractional_ideal R₁⁰ K)) :
I ≤ I * (1 / I) :=
begin
by_cases hI_nz : I = 0,
{ rw [hI_nz, div_zero, mul_zero], exact zero_le 0 },
{ rw [← coe_le_coe, coe_mul, coe_div hI_nz, coe_one],
rw [← coe_le_coe, coe_one] at hI,
exact submodule.le_self_mul_one_div hI },
end
lemma le_div_iff_of_nonzero {I J J' : fractional_ideal R₁⁰ K} (hJ' : J' ≠ 0) :
I ≤ J / J' ↔ ∀ (x ∈ I) (y ∈ J'), x * y ∈ J :=
⟨ λ h x hx, (mem_div_iff_of_nonzero hJ').mp (h hx),
λ h x hx, (mem_div_iff_of_nonzero hJ').mpr (h x hx) ⟩
lemma le_div_iff_mul_le {I J J' : fractional_ideal R₁⁰ K} (hJ' : J' ≠ 0) :
I ≤ J / J' ↔ I * J' ≤ J :=
begin
rw div_nonzero hJ',
convert submodule.le_div_iff_mul_le using 1,
rw [← coe_mul, coe_le_coe]
end
@[simp] lemma div_one {I : fractional_ideal R₁⁰ K} : I / 1 = I :=
begin
rw [div_nonzero (@one_ne_zero (fractional_ideal R₁⁰ K) _ _)],
ext,
split; intro h,
{ simpa using mem_div_iff_forall_mul_mem.mp h 1
((algebra_map R₁ K).map_one ▸ coe_mem_one R₁⁰ 1) },
{ apply mem_div_iff_forall_mul_mem.mpr,
rintros y ⟨y', _, rfl⟩,
rw mul_comm,
convert submodule.smul_mem _ y' h,
exact (algebra.smul_def _ _).symm }
end
theorem eq_one_div_of_mul_eq_one_right (I J : fractional_ideal R₁⁰ K) (h : I * J = 1) :
J = 1 / I :=
begin
have hI : I ≠ 0 := ne_zero_of_mul_eq_one I J h,
suffices h' : I * (1 / I) = 1,
{ exact (congr_arg units.inv $
@units.ext _ _ (units.mk_of_mul_eq_one _ _ h) (units.mk_of_mul_eq_one _ _ h') rfl) },
apply le_antisymm,
{ apply mul_le.mpr _,
intros x hx y hy,
rw mul_comm,
exact (mem_div_iff_of_nonzero hI).mp hy x hx },
rw ← h,
apply mul_left_mono I,
apply (le_div_iff_of_nonzero hI).mpr _,
intros y hy x hx,
rw mul_comm,
exact mul_mem_mul hx hy,
end
theorem mul_div_self_cancel_iff {I : fractional_ideal R₁⁰ K} :
I * (1 / I) = 1 ↔ ∃ J, I * J = 1 :=
⟨λ h, ⟨(1 / I), h⟩, λ ⟨J, hJ⟩, by rwa [← eq_one_div_of_mul_eq_one_right I J hJ]⟩
variables {K' : Type*} [field K'] [algebra R₁ K'] [is_fraction_ring R₁ K']
@[simp] lemma map_div (I J : fractional_ideal R₁⁰ K) (h : K ≃ₐ[R₁] K') :
(I / J).map (h : K →ₐ[R₁] K') = I.map h / J.map h :=
begin
by_cases H : J = 0,
{ rw [H, div_zero, map_zero, div_zero] },
{ apply coe_to_submodule_injective,
simp [div_nonzero H, div_nonzero (map_ne_zero _ H), submodule.map_div] }
end
@[simp] lemma map_one_div (I : fractional_ideal R₁⁰ K) (h : K ≃ₐ[R₁] K') :
(1 / I).map (h : K →ₐ[R₁] K') = 1 / I.map h :=
by rw [map_div, map_one]
end quotient
section field
variables {R₁ K L : Type*} [comm_ring R₁] [is_domain R₁] [field K] [field L]
variables [algebra R₁ K] [is_fraction_ring R₁ K] [algebra K L] [is_fraction_ring K L]
lemma eq_zero_or_one (I : fractional_ideal K⁰ L) : I = 0 ∨ I = 1 :=
begin
rw or_iff_not_imp_left,
intro hI,
simp_rw [@set_like.ext_iff _ _ _ I 1, fractional_ideal.mem_one_iff],
intro x,
split,
{ intro x_mem,
obtain ⟨n, d, rfl⟩ := is_localization.mk'_surjective K⁰ x,
refine ⟨n / d, _⟩,
rw [ring_hom.map_div, is_fraction_ring.mk'_eq_div] },
{ rintro ⟨x, rfl⟩,
obtain ⟨y, y_ne, y_mem⟩ := fractional_ideal.exists_ne_zero_mem_is_integer hI,
rw [← div_mul_cancel x y_ne, ring_hom.map_mul, ← algebra.smul_def],
exact submodule.smul_mem I _ y_mem }
end
lemma eq_zero_or_one_of_is_field (hF : is_field R₁) (I : fractional_ideal R₁⁰ K) : I = 0 ∨ I = 1 :=
begin
letI : field R₁ := hF.to_field,
-- TODO can this be less ugly?
exact @eq_zero_or_one R₁ K _ _ _ (by { unfreezingI {cases _inst_4}, convert _inst_9 }) I
end
end field
section principal_ideal_ring
variables {R₁ : Type*} [comm_ring R₁] {K : Type*} [field K]
variables [algebra R₁ K] [is_fraction_ring R₁ K]
open_locale classical
variables (R₁)
/-- `fractional_ideal.span_finset R₁ s f` is the fractional ideal of `R₁` generated by `f '' s`. -/
@[simps] def span_finset {ι : Type*} (s : finset ι) (f : ι → K) :
fractional_ideal R₁⁰ K :=
⟨submodule.span R₁ (f '' s), begin
obtain ⟨a', ha'⟩ := is_localization.exist_integer_multiples R₁⁰ s f,
refine ⟨a', a'.2, λ x hx, submodule.span_induction hx _ _ _ _⟩,
{ rintro _ ⟨i, hi, rfl⟩, exact ha' i hi },
{ rw smul_zero, exact is_localization.is_integer_zero },
{ intros x y hx hy, rw smul_add, exact is_localization.is_integer_add hx hy },
{ intros c x hx, rw smul_comm, exact is_localization.is_integer_smul hx }
end⟩
variables {R₁}
@[simp] lemma span_finset_eq_zero {ι : Type*} {s : finset ι} {f : ι → K} :
span_finset R₁ s f = 0 ↔ ∀ j ∈ s, f j = 0 :=
by simp only [← coe_to_submodule_injective.eq_iff, span_finset_coe, coe_zero, submodule.span_eq_bot,
set.mem_image, finset.mem_coe, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂]
lemma span_finset_ne_zero {ι : Type*} {s : finset ι} {f : ι → K} :
span_finset R₁ s f ≠ 0 ↔ ∃ j ∈ s, f j ≠ 0 :=
by simp
open submodule.is_principal
include loc
lemma is_fractional_span_singleton (x : P) : is_fractional S (span R {x} : submodule R P) :=
let ⟨a, ha⟩ := exists_integer_multiple S x in
is_fractional_span_iff.mpr ⟨a, a.2, λ x' hx', (set.mem_singleton_iff.mp hx').symm ▸ ha⟩
variables (S)
/-- `span_singleton x` is the fractional ideal generated by `x` if `0 ∉ S` -/
@[irreducible]
def span_singleton (x : P) : fractional_ideal S P :=
⟨span R {x}, is_fractional_span_singleton x⟩
local attribute [semireducible] span_singleton
@[simp] lemma coe_span_singleton (x : P) :
(span_singleton S x : submodule R P) = span R {x} := rfl
@[simp] lemma mem_span_singleton {x y : P} :
x ∈ span_singleton S y ↔ ∃ (z : R), z • y = x :=
submodule.mem_span_singleton
lemma mem_span_singleton_self (x : P) :
x ∈ span_singleton S x :=
(mem_span_singleton S).mpr ⟨1, one_smul _ _⟩
variables {S}
lemma span_singleton_eq_span_singleton [no_zero_smul_divisors R P] {x y : P} :
span_singleton S x = span_singleton S y ↔ ∃ z : Rˣ, z • x = y :=
by { rw [← submodule.span_singleton_eq_span_singleton], exact subtype.mk_eq_mk }
lemma eq_span_singleton_of_principal (I : fractional_ideal S P)
[is_principal (I : submodule R P)] :
I = span_singleton S (generator (I : submodule R P)) :=
coe_to_submodule_injective (span_singleton_generator ↑I).symm
lemma is_principal_iff (I : fractional_ideal S P) :
is_principal (I : submodule R P) ↔ ∃ x, I = span_singleton S x :=
⟨λ h, ⟨@generator _ _ _ _ _ ↑I h, @eq_span_singleton_of_principal _ _ _ _ _ _ _ I h⟩,
λ ⟨x, hx⟩, { principal := ⟨x, trans (congr_arg _ hx) (coe_span_singleton _ x)⟩ } ⟩
@[simp] lemma span_singleton_zero : span_singleton S (0 : P) = 0 :=
by { ext, simp [submodule.mem_span_singleton, eq_comm] }
lemma span_singleton_eq_zero_iff {y : P} : span_singleton S y = 0 ↔ y = 0 :=
⟨λ h, span_eq_bot.mp (by simpa using congr_arg subtype.val h : span R {y} = ⊥) y (mem_singleton y),
λ h, by simp [h] ⟩
lemma span_singleton_ne_zero_iff {y : P} : span_singleton S y ≠ 0 ↔ y ≠ 0 :=
not_congr span_singleton_eq_zero_iff
@[simp] lemma span_singleton_one : span_singleton S (1 : P) = 1 :=
begin
ext,
refine (mem_span_singleton S).trans ((exists_congr _).trans (mem_one_iff S).symm),
intro x',
rw [algebra.smul_def, mul_one]
end
@[simp]
lemma span_singleton_mul_span_singleton (x y : P) :
span_singleton S x * span_singleton S y = span_singleton S (x * y) :=
begin
apply coe_to_submodule_injective,
simp only [coe_mul, coe_span_singleton, span_mul_span, singleton_mul_singleton],
end
@[simp]
lemma span_singleton_pow (x : P) (n : ℕ) : span_singleton S x ^ n = span_singleton S (x ^ n) :=
begin
induction n with n hn,
{ rw [pow_zero, pow_zero, span_singleton_one] },
{ rw [pow_succ, hn, span_singleton_mul_span_singleton, pow_succ] }
end
@[simp]
lemma coe_ideal_span_singleton (x : R) :
(↑(ideal.span {x} : ideal R) : fractional_ideal S P) = span_singleton S (algebra_map R P x) :=
begin
ext y,
refine (mem_coe_ideal S).trans (iff.trans _ (mem_span_singleton S).symm),
split,
{ rintros ⟨y', hy', rfl⟩,
obtain ⟨x', rfl⟩ := submodule.mem_span_singleton.mp hy',
use x',
rw [smul_eq_mul, ring_hom.map_mul, algebra.smul_def] },
{ rintros ⟨y', rfl⟩,
refine ⟨y' * x, submodule.mem_span_singleton.mpr ⟨y', rfl⟩, _⟩,
rw [ring_hom.map_mul, algebra.smul_def] }
end
@[simp]
lemma canonical_equiv_span_singleton {P'} [comm_ring P'] [algebra R P'] [is_localization S P']
(x : P) :
canonical_equiv S P P' (span_singleton S x) =
span_singleton S (is_localization.map P' (ring_hom.id R)
(λ y (hy : y ∈ S), show ring_hom.id R y ∈ S, from hy) x) :=
begin
apply set_like.ext_iff.mpr,
intro y,
split; intro h,
{ rw mem_span_singleton,
obtain ⟨x', hx', rfl⟩ := (mem_canonical_equiv_apply _ _ _).mp h,
obtain ⟨z, rfl⟩ := (mem_span_singleton _).mp hx',
use z,
rw is_localization.map_smul,
refl },
{ rw mem_canonical_equiv_apply,
obtain ⟨z, rfl⟩ := (mem_span_singleton _).mp h,
use z • x,
use (mem_span_singleton _).mpr ⟨z, rfl⟩,
simp [is_localization.map_smul] }
end
lemma mem_singleton_mul {x y : P} {I : fractional_ideal S P} :
y ∈ span_singleton S x * I ↔ ∃ y' ∈ I, y = x * y' :=
begin
split,
{ intro h,
apply fractional_ideal.mul_induction_on h,
{ intros x' hx' y' hy',
obtain ⟨a, ha⟩ := (mem_span_singleton S).mp hx',
use [a • y', submodule.smul_mem I a hy'],
rw [←ha, algebra.mul_smul_comm, algebra.smul_mul_assoc] },
{ rintros _ _ ⟨y, hy, rfl⟩ ⟨y', hy', rfl⟩,
exact ⟨y + y', submodule.add_mem I hy hy', (mul_add _ _ _).symm⟩ } },
{ rintros ⟨y', hy', rfl⟩,
exact mul_mem_mul ((mem_span_singleton S).mpr ⟨1, one_smul _ _⟩) hy' }
end
omit loc
variables (K)
lemma mk'_mul_coe_ideal_eq_coe_ideal {I J : ideal R₁} {x y : R₁} (hy : y ∈ R₁⁰) :
span_singleton R₁⁰ (is_localization.mk' K x ⟨y, hy⟩) * I = (J : fractional_ideal R₁⁰ K) ↔
ideal.span {x} * I = ideal.span {y} * J :=
begin
have inj : function.injective (coe : ideal R₁ → fractional_ideal R₁⁰ K) :=
fractional_ideal.coe_ideal_injective,
have : span_singleton R₁⁰ (is_localization.mk' _ (1 : R₁) ⟨y, hy⟩) *
span_singleton R₁⁰ (algebra_map R₁ K y) = 1,
{ rw [span_singleton_mul_span_singleton, mul_comm, ← is_localization.mk'_eq_mul_mk'_one,
is_localization.mk'_self, span_singleton_one] },
let y' : (fractional_ideal R₁⁰ K)ˣ := units.mk_of_mul_eq_one _ _ this,
have coe_y' : ↑y' = span_singleton R₁⁰ (is_localization.mk' K (1 : R₁) ⟨y, hy⟩) := rfl,
refine iff.trans _ (y'.mul_right_inj.trans inj.eq_iff),
rw [coe_y', coe_ideal_mul, coe_ideal_span_singleton, coe_ideal_mul, coe_ideal_span_singleton,
←mul_assoc, span_singleton_mul_span_singleton, ←mul_assoc, span_singleton_mul_span_singleton,
mul_comm (mk' _ _ _), ← is_localization.mk'_eq_mul_mk'_one,
mul_comm (mk' _ _ _), ← is_localization.mk'_eq_mul_mk'_one,
is_localization.mk'_self, span_singleton_one, one_mul],
end
variables {K}
lemma span_singleton_mul_coe_ideal_eq_coe_ideal {I J : ideal R₁} {z : K} :
span_singleton R₁⁰ z * (I : fractional_ideal R₁⁰ K) = J ↔
ideal.span {((is_localization.sec R₁⁰ z).1 : R₁)} * I =
ideal.span {(is_localization.sec R₁⁰ z).2} * J :=
-- `erw` to deal with the distinction between `y` and `⟨y.1, y.2⟩`
by erw [← mk'_mul_coe_ideal_eq_coe_ideal K (is_localization.sec R₁⁰ z).2.prop,
is_localization.mk'_sec K z]
variables [is_domain R₁]
lemma one_div_span_singleton (x : K) :
1 / span_singleton R₁⁰ x = span_singleton R₁⁰ (x⁻¹) :=
if h : x = 0 then by simp [h] else (eq_one_div_of_mul_eq_one_right _ _ (by simp [h])).symm
@[simp] lemma div_span_singleton (J : fractional_ideal R₁⁰ K) (d : K) :
J / span_singleton R₁⁰ d = span_singleton R₁⁰ (d⁻¹) * J :=
begin
rw ← one_div_span_singleton,
by_cases hd : d = 0,
{ simp only [hd, span_singleton_zero, div_zero, zero_mul] },
have h_spand : span_singleton R₁⁰ d ≠ 0 := mt span_singleton_eq_zero_iff.mp hd,
apply le_antisymm,
{ intros x hx,
rw [← mem_coe, coe_div h_spand, submodule.mem_div_iff_forall_mul_mem] at hx,
specialize hx d (mem_span_singleton_self R₁⁰ d),
have h_xd : x = d⁻¹ * (x * d), { field_simp },
rw [← mem_coe, coe_mul, one_div_span_singleton, h_xd],
exact submodule.mul_mem_mul (mem_span_singleton_self R₁⁰ _) hx },
{ rw [le_div_iff_mul_le h_spand, mul_assoc, mul_left_comm, one_div_span_singleton,
span_singleton_mul_span_singleton, inv_mul_cancel hd, span_singleton_one, mul_one],
exact le_refl J },
end
lemma exists_eq_span_singleton_mul (I : fractional_ideal R₁⁰ K) :
∃ (a : R₁) (aI : ideal R₁), a ≠ 0 ∧ I = span_singleton R₁⁰ (algebra_map R₁ K a)⁻¹ * aI :=
begin
obtain ⟨a_inv, nonzero, ha⟩ := I.is_fractional,
have nonzero := mem_non_zero_divisors_iff_ne_zero.mp nonzero,
have map_a_nonzero : algebra_map R₁ K a_inv ≠ 0 :=
mt is_fraction_ring.to_map_eq_zero_iff.mp nonzero,
refine ⟨a_inv,
submodule.comap (algebra.linear_map R₁ K)
↑(span_singleton R₁⁰ (algebra_map R₁ K a_inv) * I),
nonzero,
ext (λ x, iff.trans ⟨_, _⟩ mem_singleton_mul.symm)⟩,
{ intro hx,
obtain ⟨x', hx'⟩ := ha x hx,
rw algebra.smul_def at hx',
refine ⟨algebra_map R₁ K x', (mem_coe_ideal _).mpr ⟨x', mem_singleton_mul.mpr _, rfl⟩, _⟩,
{ exact ⟨x, hx, hx'⟩ },
{ rw [hx', ← mul_assoc, inv_mul_cancel map_a_nonzero, one_mul] } },
{ rintros ⟨y, hy, rfl⟩,
obtain ⟨x', hx', rfl⟩ := (mem_coe_ideal _).mp hy,
obtain ⟨y', hy', hx'⟩ := mem_singleton_mul.mp hx',
rw algebra.linear_map_apply at hx',
rwa [hx', ←mul_assoc, inv_mul_cancel map_a_nonzero, one_mul] }
end
instance is_principal {R} [comm_ring R] [is_domain R] [is_principal_ideal_ring R]
[algebra R K] [is_fraction_ring R K]
(I : fractional_ideal R⁰ K) : (I : submodule R K).is_principal :=
begin
obtain ⟨a, aI, -, ha⟩ := exists_eq_span_singleton_mul I,
use (algebra_map R K a)⁻¹ * algebra_map R K (generator aI),
suffices : I = span_singleton R⁰ ((algebra_map R K a)⁻¹ * algebra_map R K (generator aI)),
{ exact congr_arg subtype.val this },
conv_lhs { rw [ha, ←span_singleton_generator aI] },
rw [ideal.submodule_span_eq, coe_ideal_span_singleton (generator aI),
span_singleton_mul_span_singleton]
end
include loc
lemma le_span_singleton_mul_iff {x : P} {I J : fractional_ideal S P} :
I ≤ span_singleton S x * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI :=
show (∀ {zI} (hzI : zI ∈ I), zI ∈ span_singleton _ x * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI,
by simp only [fractional_ideal.mem_singleton_mul, eq_comm]
lemma span_singleton_mul_le_iff {x : P} {I J : fractional_ideal S P} :
span_singleton _ x * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J :=
begin
simp only [fractional_ideal.mul_le, fractional_ideal.mem_singleton_mul,
fractional_ideal.mem_span_singleton],
split,
{ intros h zI hzI,
exact h x ⟨1, one_smul _ _⟩ zI hzI },
{ rintros h _ ⟨z, rfl⟩ zI hzI,
rw [algebra.smul_mul_assoc],
exact submodule.smul_mem J.1 _ (h zI hzI) },
end
lemma eq_span_singleton_mul {x : P} {I J : fractional_ideal S P} :
I = span_singleton _ x * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I :=
by simp only [le_antisymm_iff, fractional_ideal.le_span_singleton_mul_iff,
fractional_ideal.span_singleton_mul_le_iff]
end principal_ideal_ring
variables {R₁ : Type*} [comm_ring R₁]
variables {K : Type*} [field K] [algebra R₁ K] [frac : is_fraction_ring R₁ K]
local attribute [instance] classical.prop_decidable
lemma is_noetherian_zero : is_noetherian R₁ (0 : fractional_ideal R₁⁰ K) :=
is_noetherian_submodule.mpr (λ I (hI : I ≤ (0 : fractional_ideal R₁⁰ K)),
by { rw coe_zero at hI, rw le_bot_iff.mp hI, exact fg_bot })
lemma is_noetherian_iff {I : fractional_ideal R₁⁰ K} :
is_noetherian R₁ I ↔ ∀ J ≤ I, (J : submodule R₁ K).fg :=
is_noetherian_submodule.trans ⟨λ h J hJ, h _ hJ, λ h J hJ, h ⟨J, is_fractional_of_le hJ⟩ hJ⟩
lemma is_noetherian_coe_to_fractional_ideal [_root_.is_noetherian_ring R₁] (I : ideal R₁) :
is_noetherian R₁ (I : fractional_ideal R₁⁰ K) :=
begin
rw is_noetherian_iff,
intros J hJ,
obtain ⟨J, rfl⟩ := le_one_iff_exists_coe_ideal.mp (le_trans hJ coe_ideal_le_one),
exact (is_noetherian.noetherian J).map _,
end
include frac
variables [is_domain R₁]
lemma is_noetherian_span_singleton_inv_to_map_mul (x : R₁) {I : fractional_ideal R₁⁰ K}
(hI : is_noetherian R₁ I) :
is_noetherian R₁ (span_singleton R₁⁰ (algebra_map R₁ K x)⁻¹ * I : fractional_ideal R₁⁰ K) :=
begin
by_cases hx : x = 0,
{ rw [hx, ring_hom.map_zero, _root_.inv_zero, span_singleton_zero, zero_mul],
exact is_noetherian_zero },
have h_gx : algebra_map R₁ K x ≠ 0,
from mt ((injective_iff_map_eq_zero (algebra_map R₁ K)).mp
(is_fraction_ring.injective _ _) x) hx,
have h_spanx : span_singleton R₁⁰ (algebra_map R₁ K x) ≠ 0,
from span_singleton_ne_zero_iff.mpr h_gx,
rw is_noetherian_iff at ⊢ hI,
intros J hJ,
rw [← div_span_singleton, le_div_iff_mul_le h_spanx] at hJ,
obtain ⟨s, hs⟩ := hI _ hJ,
use s * {(algebra_map R₁ K x)⁻¹},
rw [finset.coe_mul, finset.coe_singleton, ← span_mul_span, hs, ← coe_span_singleton R₁⁰,
← coe_mul, mul_assoc, span_singleton_mul_span_singleton, mul_inv_cancel h_gx,
span_singleton_one, mul_one],
end
/-- Every fractional ideal of a noetherian integral domain is noetherian. -/
theorem is_noetherian [_root_.is_noetherian_ring R₁] (I : fractional_ideal R₁⁰ K) :
is_noetherian R₁ I :=
begin
obtain ⟨d, J, h_nzd, rfl⟩ := exists_eq_span_singleton_mul I,
apply is_noetherian_span_singleton_inv_to_map_mul,
apply is_noetherian_coe_to_fractional_ideal,
end
section adjoin
include loc
omit frac
variables {R P} (S) (x : P) (hx : is_integral R x)
/-- `A[x]` is a fractional ideal for every integral `x`. -/
lemma is_fractional_adjoin_integral :
is_fractional S (algebra.adjoin R ({x} : set P)).to_submodule :=
is_fractional_of_fg (fg_adjoin_singleton_of_integral x hx)
/-- `fractional_ideal.adjoin_integral (S : submonoid R) x hx` is `R[x]` as a fractional ideal,
where `hx` is a proof that `x : P` is integral over `R`. -/
@[simps]
def adjoin_integral : fractional_ideal S P :=
⟨_, is_fractional_adjoin_integral S x hx⟩
lemma mem_adjoin_integral_self :
x ∈ adjoin_integral S x hx :=
algebra.subset_adjoin (set.mem_singleton x)
end adjoin
end fractional_ideal
|
a42e09701a7ffa92b7acc52bc1572d9af1625576 | 59a4b050600ed7b3d5826a8478db0a9bdc190252 | /src/category_theory/functor_categories/whiskering.lean | 40cbd4fd5baea3eeb25e324d4f267b14cd403744 | [] | no_license | rwbarton/lean-category-theory | f720268d800b62a25d69842ca7b5d27822f00652 | 00df814d463406b7a13a56f5dcda67758ba1b419 | refs/heads/master | 1,585,366,296,767 | 1,536,151,349,000 | 1,536,151,349,000 | 147,652,096 | 0 | 0 | null | 1,536,226,960,000 | 1,536,226,960,000 | null | UTF-8 | Lean | false | false | 2,069 | lean | -- Copyright (c) 2018 Scott Morrison. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Authors: Scott Morrison
import category_theory.functor_category
import category_theory.tactics.obviously
namespace category_theory
universes u₁ v₁ u₂ v₂ u₃ v₃
section
variables (C : Type u₁) [𝒞 : category.{u₁ v₁} C] (D : Type u₂) [𝒟 : category.{u₂ v₂} D] (E : Type u₃) [ℰ : category.{u₃ v₃} E]
include 𝒞 𝒟 ℰ
-- set_option trace.tidy true
def whiskering_on_left : (C ⥤ D) ⥤ ((D ⥤ E) ⥤ (C ⥤ E)) :=
{ obj := λ F, { obj := λ G, F ⋙ G,
map' := λ _ _ α, (nat_trans.id _) ◫ α },
map' := λ F G τ, { app := λ H, { app := λ c, H.map (τ c) } } }
def whiskering_on_right : (D ⥤ E) ⥤ ((C ⥤ D) ⥤ (C ⥤ E)) :=
{ obj := λ H, { obj := λ F, F ⋙ H,
map' := λ _ _ α, α ◫ (nat_trans.id _) },
map' := λ G H τ, { app := λ F, { app := λ c, τ (F c) } } }
end
def whisker_on_left_functor {C : Type u₁} [𝒞 : category.{u₁ v₁} C] {D : Type u₂} [𝒟 : category.{u₂ v₂} D] (F : C ⥤ D) (E : Type u₃) [ℰ : category.{u₃ v₃} E] : (D ⥤ E) ⥤ (C ⥤ E) :=
(whiskering_on_left C D E) F
def whisker_on_right_functor (C : Type u₁) [𝒞 : category.{u₁ v₁} C] {D : Type u₂} [𝒟 : category.{u₂ v₂} D] {E : Type u₃} [ℰ : category.{u₃ v₃} E] (H : D ⥤ E) : (C ⥤ D) ⥤ (C ⥤ E) :=
(whiskering_on_right C D E) H
def whisker_on_left {C : Type u₁} [𝒞 : category.{u₁ v₁} C] {D : Type u₂} [𝒟 : category.{u₂ v₂} D] {E : Type u₃} [ℰ : category.{u₃ v₃} E] (F : C ⥤ D) {G H : D ⥤ E} (α : G ⟹ H) : (F ⋙ G) ⟹ (F ⋙ H) :=
(whisker_on_left_functor F E).map α
def whisker_on_right {C : Type u₁} [𝒞 : category.{u₁ v₁} C] {D : Type u₂} [𝒟 : category.{u₂ v₂} D] {E : Type u₃} [ℰ : category.{u₃ v₃} E] {G H : C ⥤ D} (α : G ⟹ H) (F : D ⥤ E) : (G ⋙ F) ⟹ (H ⋙ F) :=
(whisker_on_right_functor C F).map α
end category_theory |
616837991e583cd1a43b66da80f4049a48af1874 | bbecf0f1968d1fba4124103e4f6b55251d08e9c4 | /src/linear_algebra/matrix/trace.lean | fa68bafdb405f71d4b86b9557c57812579516a68 | [
"Apache-2.0"
] | permissive | waynemunro/mathlib | e3fd4ff49f4cb43d4a8ded59d17be407bc5ee552 | 065a70810b5480d584033f7bbf8e0409480c2118 | refs/heads/master | 1,693,417,182,397 | 1,634,644,781,000 | 1,634,644,781,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 2,350 | lean | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import data.matrix.basic
/-!
# Trace of a matrix
This file defines the trace of a matrix, the linear map
sending a matrix to the sum of its diagonal entries.
See also `linear_algebra.trace` for the trace of an endomorphism.
## Tags
matrix, trace, diagonal
-/
open_locale big_operators
open_locale matrix
namespace matrix
section trace
universes u v w
variables {m : Type*} (n : Type*)
variables (R : Type*) (M : Type*) [semiring R] [add_comm_monoid M] [module R M]
/--
The diagonal of a square matrix.
-/
def diag : (matrix n n M) →ₗ[R] n → M :=
{ to_fun := λ A i, A i i,
map_add' := by { intros, ext, refl, },
map_smul' := by { intros, ext, refl, } }
variables {n} {R} {M}
@[simp] lemma diag_apply (A : matrix n n M) (i : n) : diag n R M A i = A i i := rfl
@[simp] lemma diag_one [decidable_eq n] :
diag n R R 1 = λ i, 1 := by { dunfold diag, ext, simp [one_apply_eq] }
@[simp] lemma diag_transpose (A : matrix n n M) : diag n R M Aᵀ = diag n R M A := rfl
variables (n) (R) (M)
/--
The trace of a square matrix.
-/
def trace [fintype n] : (matrix n n M) →ₗ[R] M :=
{ to_fun := λ A, ∑ i, diag n R M A i,
map_add' := by { intros, apply finset.sum_add_distrib, },
map_smul' := by { intros, simp [finset.smul_sum], } }
variables {n} {R} {M} [fintype n] [fintype m]
@[simp] lemma trace_diag (A : matrix n n M) : trace n R M A = ∑ i, diag n R M A i := rfl
lemma trace_apply (A : matrix n n M) : trace n R M A = ∑ i, A i i := rfl
@[simp] lemma trace_one [decidable_eq n] :
trace n R R 1 = fintype.card n :=
have h : trace n R R 1 = ∑ i, diag n R R 1 i := rfl,
by simp_rw [h, diag_one, finset.sum_const, nsmul_one]; refl
@[simp] lemma trace_transpose (A : matrix n n M) : trace n R M Aᵀ = trace n R M A := rfl
@[simp] lemma trace_transpose_mul (A : matrix m n R) (B : matrix n m R) :
trace n R R (Aᵀ ⬝ Bᵀ) = trace m R R (A ⬝ B) := finset.sum_comm
lemma trace_mul_comm {S : Type v} [comm_semiring S] (A : matrix m n S) (B : matrix n m S) :
trace n S S (B ⬝ A) = trace m S S (A ⬝ B) :=
by rw [←trace_transpose, ←trace_transpose_mul, transpose_mul]
end trace
end matrix
|
2738462435f96905d853485ebdc200f6c2b3e9af | 9d2e3d5a2e2342a283affd97eead310c3b528a24 | /src/solutions/friday/manifolds.lean | 216ffc21d0b4f46307b1f7c54537e618ca8d2672 | [] | permissive | Vtec234/lftcm2020 | ad2610ab614beefe44acc5622bb4a7fff9a5ea46 | bbbd4c8162f8c2ef602300ab8fdeca231886375d | refs/heads/master | 1,668,808,098,623 | 1,594,989,081,000 | 1,594,990,079,000 | 280,423,039 | 0 | 0 | MIT | 1,594,990,209,000 | 1,594,990,209,000 | null | UTF-8 | Lean | false | false | 41,871 | lean | import for_mathlib.manifolds
noncomputable theory
open_locale manifold classical big_operators
open set
universe u
/-!
## Reminder on updating the exercises
These instructions are now available at:
https://leanprover-community.github.io/lftcm2020/exercises.html
To get a new copy of the exercises,
run the following commands in your terminal:
```
leanproject get lftcm2020
cp -r lftcm2020/src/exercises_sources/ lftcm2020/src/my_exercises
code lftcm2020
```
To update your exercise files, run the following commands:
```
cd /path/to/lftcm2020
git pull
leanproject get-mathlib-cache
```
Don’t forget to copy the updated files to `src/my_exercises`.
-/
/-!
## An overview of manifolds in Lean, discussing design decisions
Warning: there are sorries in this section, they are not supposed to be filled! The exercises section
start later, and there you will have plenty of sorries to fill.
What is a manifold?
1) allow field other than `ℝ` or `ℂ`?
2) allow infinite dimension?
3) allow boundary?
4) allow model space depending on the point of the manifold?
Bourbaki: 2, 4 (and just definitions and statements, no proofs!)
Lean: 1, 2, 3
Perelman geometrization theorem : any compact connected irreducible 3-manifold can
be cut along tori into finitely many pieces, each of which has a _geometric structure_ of
finite volume, i.e., it is locally like a model space, with changes of coordinates given
locally by the action of a Lie group
Typical dynamics theorem : let `M` be a compact manifold, and `f : M → M` a map with property
such and such. Then ...
Or : Consider a hyperbolic surface of genus `g`, and a random geodesic of length `T`. How many
times does it typically self-intersect?
Manifold in Lean:
* charted space structure, i.e., set of local homeos to a model space. This is data, fixed
once and for all (and a typeclass)
* compatibility condition, i.e., the change of coordinates should belong to some subgroup
of the group of local homeos of the model space. This is Prop (and a typeclass). The same
manifold can be at the same time an analytic manifold, a smooth manifold and a topological
manifold (with the same fixed atlas).
* A charted space is a smooth manifold (with corners) if it is compatible with the smooth
groupoid on the model space. To cover uniformly both situations with and without boundary,
the smooth groupoid is with respect to a map `I : H → E` (think of `H` as the half-space and
`E` the full space), which is the identity in the boundaryless situation, the inclusion in
the half-space situation. This map `I` is called a _model with corners_. The most standard ones
(identity in `ℝ^n` and inclusion of half-space in `ℝ^n`) have dedicated notations:
`𝓡 n` and `𝓡∂ n`.
-/
#check charted_space (euclidean_half_space 1) (Icc (0 : ℝ) 1)
#check has_groupoid (Icc (0 : ℝ) 1) (times_cont_diff_groupoid ∞ (𝓡∂ 1))
#check smooth_manifold_with_corners (𝓡∂ 1) (Icc (0 : ℝ) 1)
-- atlases are not maximal in general
#check (times_cont_diff_groupoid ∞ (𝓡∂ 1)).maximal_atlas (Icc (0 : ℝ) 1)
-- let's try to put a smooth manifold structure on the sphere
-- (we don't have submanifolds yet, but it's coming in the near future)
@[derive topological_space]
definition sphere (n : ℕ) : Type := metric.sphere (0 : euclidean_space (fin (n+1))) 1
instance (n : ℕ) : has_coe (sphere n) (euclidean_space (fin (n+1))) := ⟨subtype.val⟩
instance (n : ℕ) : charted_space (euclidean_space (fin n)) (sphere n) :=
{ atlas := begin sorry end,
chart_at := begin sorry end,
mem_chart_source := begin sorry end,
chart_mem_atlas := begin sorry end }
instance (n : ℕ) : smooth_manifold_with_corners (𝓡 n) (sphere n) :=
{ compatible := begin
assume e e' he he',
sorry
end }
-- smooth functions
def inc (n : ℕ) : sphere n → euclidean_space (fin (n+1)) :=
λ p : sphere n, (p : euclidean_space (fin (n+1)))
lemma inc_smooth (n : ℕ) : times_cont_mdiff (𝓡 n) (𝓡 (n+1)) ∞ (inc n) :=
begin
rw times_cont_mdiff_iff,
split,
{ exact continuous_subtype_coe, },
{ assume x y,
sorry }
end
lemma inc_continuous (n : ℕ) : continuous (inc n) :=
(inc_smooth n).continuous
lemma inc_mdifferentiable (n : ℕ) : mdifferentiable (𝓡 n) (𝓡 (n+1)) (inc n) :=
(inc_smooth n).mdifferentiable le_top
-- tangent space and tangent bundles
example (n : ℕ) (p : sphere n) (v : tangent_space (𝓡 n) p) :
tangent_bundle (𝓡 n) (sphere n) :=
⟨p, v⟩
-- tangent map, derivatives
example (n : ℕ) : times_cont_mdiff ((𝓡 n).prod (𝓡 n)) ((𝓡 (n+1)).prod (𝓡 (n+1))) ∞
(tangent_map (𝓡 n) (𝓡 (n+1)) (inc n)) :=
(inc_smooth n).times_cont_mdiff_tangent_map le_top
example (n : ℕ) (f : sphere n → sphere (n^2)) (p : sphere n) (v : tangent_space (𝓡 n) p) :
mfderiv (𝓡 n) (𝓡 (n^2)) f p v = (tangent_map (𝓡 n) (𝓡 (n^2)) f ⟨p, v⟩).2 :=
rfl
/- Can you express the sphere eversion theorem, i.e., the fact that there is a smooth isotopy
of immersions between the canonical embedding of the sphere `S^2` and `ℝ^3`, and the antipodal
embedding?
Note that we haven't defined immersions in mathlib, but you can jut require that the fiber
derivative is injective everywhere, which is easy to express if you know that the derivative
of a function `f` from a manifold of dimension `2` to a manifold of dimension `3` at a point `x` is
`mfderiv (𝓡 2) (𝓡 3) f x`.
Don't forget to require the global smoothness of the map! You may need to know that the interval
`[0,1]`, called `Icc (0 : ℝ) 1` in Lean, already has a manifold (with boundary!) structure,
where the corresponding model with corners is called `𝓡∂ 1`.
-/
theorem sphere_eversion :
∃ f : (Icc (0 : ℝ) 1) × sphere 2 → euclidean_space (fin 3),
times_cont_mdiff ((𝓡∂ 1).prod (𝓡 2)) (𝓡 3) ∞ f
∧ ∀ (t : (Icc (0 : ℝ) 1)), ∀ (p : sphere 2),
function.injective (mfderiv (𝓡 2) (𝓡 3) (f ∘ λ y, (t, y)) p)
∧ ∀ (p : sphere 2), f (0, p) = p
∧ ∀ (p : sphere 2), f (1, p) = - p :=
sorry
/- Dicussing three (controversial?) design decisions
#### Local homeos
What is a local homeo `f` between an open subset of `E` and an open subset of `F`?
1) a map defined on a subtype: `f x` only makes sense for `x : f.source`
2) a map defined on the whole space `E`, but taking values in `option F = F ∪ {junk}`, with
`f x = junk` when `x ∉ f.source`
3) a map defined on the whole space `E`, taking values in `F`, and we don't care about its values
outside of `f.source`.
Just like division by zero! But worse:
* issue with 1): you keep intersecting chart domains. But the subtype `u ∩ v` is not the same as
the subtype `v ∩ u`, so you keep adding casts everywhere
* issue with 2): if you want to say that a chart is smooth, then you define to define smooth functions
between `option E` and `option F` when `E` and `F` are vector spaces. All notions need to be
redefined with `option`.
* issue with 3): it works perfectly well, but it makes mathematicians unhappy/uneasy (and it is *not*
equivalent to 1) or 2) when one of the spaces is empty)
I picked 3)
#### Tangent vectors
What is a tangent vector (for a manifold `M` modelled on a vector space `E`)?
1) An equivalence class of germs of curves
2) A derivation
3) Physicist point of view: I don't know what a tangent vector is, but I know in charts.
Mathematician's interpretation: equivalence class of `(e, v)` where `e` is a chart at `x`, `v` a vector
in the vector space, and `(e, v) ∼ (e', v')` if `D(e' ∘ e ⁻¹) v = v'`
4) ...
Issues:
1) Pictures are pretty, but this doesn't bring anything compared to 3) when you go down to details.
And what about boundaries, where you can only have a half-curve
2) Need partitions of unity to show that this is local and coincides with the usual point of view.
Doesn't work well in finite smoothness, nor in complex manifolds
3) Fine, works in all situations, but requires a lot of work to define the equivalence classes,
the topology, check that the topology is compatible with the vector space structure, and so on.
In a vector space, the tangent space is not defeq to the vector space itself
4) Pick one favorite chart at `x`, say `e_x`, and *define* the tangent space at `x` to be `E`,
but "seen" in the chart `e_x` (this will show up in the definition of the derivative : the
derivative of `f : M → M'` at `x` is defined to be the derivative of the map
`e_{f x} ∘ f ∘ e_x⁻¹`). Works perfectly fine, but makes mathematicians unhappy/uneasy.
(Axiom of choice? In fact we put the choice of `e_x` in the *definition* of charted spaces,
so not further choice)
I picked 4)
#### Smooth functions in manifolds with boundary
Usual definition of smooth functions in a half space: extend to a smooth function a little bit
beyond the boundary, so one only really needs to speak of smooth functions in open subsets of
vector spaces.
When you define the derivative, you will need to check that it does not depend on the choice
of the extension. Even worse when you want to define the tangent bundle: choose an open extension
of your manifold with boundary, and then check that the restriction of the tangent bundle does
not depend on the choice of the extension. Very easy when handwaving, nightmare to formalize.
(What is the extension of the manifold with boundary? Another type?)
Instead, if you define derivatives in (non-open) domains, you can talk of smooth functions in
domains, and do everything without extending. Need to know this early enough: when starting to
define derivatives, you should already think of manifolds with boundaries! That's what we did
in mathlib.
Difficulty: if a domain `s` is too small (think `s = ℝ ⊆ ℝ^2`), the values of `f` on `s` do not
prescribe uniquely a derivative, so `fderiv_within_at ℝ f s x` may behave badly: derivative of
a sum might be different from sum of derivatives, as there is an arbitrary choice to be made.
This does not happen with the half-space, as it is large enough: derivatives within domains only
work well if the tangent directions span the whole space. Predicate `unique_diff_on` for sets
in vector spaces. You won't find this in books!
-/
/-! ## Exercises -/
/-! ### Local homeomorphisms
Local homeomorphisms are globally defined maps with a globally defined "inverse", but the only
relevant set is the *source*, which should be mapped homeomorphically to the *target*.
-/
/- Define a local homeomorphism from `ℝ` to `ℝ` which is just `x ↦ -x`, but on `(-1, 1)`. In
Lean, the interval `(-1, 1)` is denoted by `Ioo (-1 : ℝ) 1` (where `o` stands for _open_). -/
-- set up a simple helper simp lemma to simplify our life later.
@[simp] lemma neg_mem_Ioo_minus_one_one (x : ℝ) : -x ∈ Ioo (-1 : ℝ) 1 ↔ x ∈ Ioo (-1 : ℝ) 1 :=
begin
-- sorry
simp [neg_lt, and_comm],
-- sorry
end
def my_first_local_homeo : local_homeomorph ℝ ℝ :=
{ to_fun := λ x, -x,
inv_fun := λ x, -x,
source := Ioo (-1) 1,
target := /- inline sorry -/Ioo (-1) 1/- inline sorry -/,
map_source' :=
begin
-- sorry
assume x hx,
simp [hx],
-- sorry
end,
map_target' :=
begin
-- sorry
assume x hx,
simp [hx],
-- sorry
end,
left_inv' :=
begin
-- sorry
simp,
-- sorry
end,
right_inv' :=
begin
-- sorry
simp,
-- sorry
end,
open_source := /- inline sorry -/is_open_Ioo/- inline sorry -/,
open_target := /- inline sorry -/is_open_Ioo/- inline sorry -/,
continuous_to_fun := /- inline sorry -/continuous_neg.continuous_on/- inline sorry -/,
continuous_inv_fun := /- inline sorry -/continuous_neg.continuous_on/- inline sorry -/ }
/- Two simple lemmas that will prove useful below. You can leave them sorried if you like. -/
lemma ne_3_of_mem_Ioo {x : ℝ} (h : x ∈ Ioo (-1 : ℝ) 1) : x ≠ 3 :=
begin
-- sorry
exact ne_of_lt (lt_trans h.2 (by norm_num))
-- sorry
end
lemma neg_ne_3_of_mem_Ioo {x : ℝ} (h : x ∈ Ioo (-1 : ℝ) 1) : -x ≠ 3 :=
begin
-- sorry
assume h',
simp at h,
linarith,
-- sorry
end
/- Now, define a second local homeomorphism which is almost like the previous one. You may find the
following lemma useful for `continuous_to_fun`: -/
#check continuous_on.congr
def my_second_local_homeo : local_homeomorph ℝ ℝ :=
{ to_fun := λ x, if x = 3 then 0 else - x,
inv_fun := λ x, -x,
source := Ioo (-1) 1,
target := /- inline sorry -/Ioo (-1) 1/- inline sorry -/,
map_source' := /- inline sorry -/λ x hx, by simp [hx, ne_3_of_mem_Ioo hx]/- inline sorry -/,
map_target' := /- inline sorry -/λ x hx, by simp [hx]/- inline sorry -/,
left_inv' := /- inline sorry -/λ x hx, by simp [hx, ne_3_of_mem_Ioo hx]/- inline sorry -/,
right_inv' := /- inline sorry -/λ x hx, by simp [hx, neg_ne_3_of_mem_Ioo hx]/- inline sorry -/,
open_source := /- inline sorry -/is_open_Ioo/- inline sorry -/,
open_target := /- inline sorry -/is_open_Ioo/- inline sorry -/,
continuous_to_fun :=
begin
-- sorry
refine continuous_neg.continuous_on.congr (λ x hx, _),
simp [hx, ne_3_of_mem_Ioo hx],
-- sorry
end,
continuous_inv_fun := /- inline sorry -/continuous_neg.continuous_on/- inline sorry -/ }
/- Although the two above local homeos are the same for all practical purposes as they coincide
where relevant, they are not *equal*: -/
lemma my_first_local_homeo_ne_my_second_local_homeo :
my_first_local_homeo ≠ my_second_local_homeo :=
begin
-- sorry
assume h,
have : my_first_local_homeo 3 = my_second_local_homeo 3, by rw h,
simp [my_first_local_homeo, my_second_local_homeo] at this,
linarith,
-- sorry
end
/- The right equivalence relation for local homeos is not equality, but `eq_on_source`.
Indeed, the two local homeos we have defined above coincide from this point of view. -/
#check local_homeomorph.eq_on_source
lemma eq_on_source_my_first_local_homeo_my_second_local_homeo :
local_homeomorph.eq_on_source my_first_local_homeo my_second_local_homeo :=
begin
-- sorry
refine ⟨rfl, λ x hx, _⟩,
simp [my_first_local_homeo, my_second_local_homeo, ne_3_of_mem_Ioo hx],
-- sorry
end
/-! ### An example of a charted space structure on `ℝ`
A charted space is a topological space together with a set of local homeomorphisms to a model space,
whose sources cover the whole space. For instance, `ℝ` is already endowed with a charted space
structure with model space `ℝ`, where the unique chart is the identity:
-/
#check charted_space_self ℝ
/- For educational purposes only, we will put another charted space structure on `ℝ` using the
local homeomorphisms we have constructed above. To avoid using too much structure of `ℝ` (and to
avoid confusing Lean), we will work with a copy of `ℝ`, on which we will only register the
topology. -/
@[derive topological_space]
def myℝ : Type := ℝ
instance : charted_space ℝ myℝ :=
{ atlas := { local_homeomorph.refl ℝ, my_first_local_homeo },
chart_at := λ x, if x ∈ Ioo (-1 : ℝ) 1 then my_first_local_homeo else local_homeomorph.refl ℝ,
mem_chart_source :=
begin
-- sorry
assume x,
split_ifs,
{ exact h },
{ exact mem_univ _ }
-- sorry
end,
chart_mem_atlas :=
begin
-- sorry
assume x,
split_ifs;
simp,
-- sorry
end }
/- Now come more interesting bits. We have endowed `myℝ` with a charted space structure, with charts
taking values in `ℝ`. We want to say that this is a smooth structure, i.e., the changes of
coordinates are smooth. In Lean, this is written with `has_groupoid`. A groupoid is a set
of local homeomorphisms of the model space (for example, local homeos that are smooth on their
domain). A charted space admits the groupoid as a structure groupoid if all the changes of
coordinates belong to the groupoid.
There is a difficulty that the definitions are set up to be able to also speak of smooth manifolds
with boundary or with corners, so the name of the smooth groupoid on `ℝ` has the slightly strange
name `times_cont_diff_groupoid ∞ (model_with_corners_self ℝ ℝ)`. To avoid typing again and again
`model_with_corners_self ℝ ℝ`, let us introduce a shortcut
-/
abbreviation 𝓡1 := model_with_corners_self ℝ ℝ
/- In the library, there are such shortcuts for manifolds modelled on `ℝ^n`, denoted with `𝓡 n`,
but for `n = 1` this does not coincide with the above one, as `ℝ^1` (a.k.a. `fin 1 → ℝ`) is not
the same as `ℝ`! Still, since they are of the same nature, the notation we have just introduced
is very close, compare `𝓡1` with `𝓡 1` (and try not to get confused): -/
instance : has_groupoid myℝ (times_cont_diff_groupoid ∞ 𝓡1) :=
begin
-- in theory, we should prove that all compositions of charts are diffeos, i.e., they are smooth
-- and their inverse are smooth. For symmetry reasons, it suffices to check one direction
apply has_groupoid_of_pregroupoid,
-- take two charts `e` and `e'`
assume e e' he he',
-- if next line is a little bit slow for your taste, you can replace `simp` with `squeeze_simp`
-- and then follow the advice
simp [atlas] at he he',
dsimp,
-- to continue, some hints:
-- (1) don't hesitate to use the fact that the restriction of a smooth function to a
-- subset is still smooth there (`times_cont_diff.times_cont_diff_on`)
-- (2) hopefully, there is a theorem saying that the negation function is smooth.
-- you can either try to guess its name, or hope that `suggest` will help you there.
-- sorry
rcases he with rfl|rfl; rcases he' with rfl|rfl,
{ exact times_cont_diff_id.times_cont_diff_on },
{ exact times_cont_diff_id.neg.times_cont_diff_on },
{ exact times_cont_diff_id.neg.times_cont_diff_on },
{ convert times_cont_diff_id.times_cont_diff_on,
ext x,
simp [my_first_local_homeo], },
-- sorry
end
/- The statement of the previous instance is not very readable. There is a shortcut notation: -/
instance : smooth_manifold_with_corners 𝓡1 myℝ := {}
/- We will now study a very simple map from `myℝ` to `ℝ`, the identity. -/
def my_map : myℝ → ℝ := λ x, x
/- The map `my_map` is a map going from the type `myℝ` to the type `ℝ`. From the point of view of
the kernel of Lean, it is just the identity, but from the point of view of structures on `myℝ`
and `ℝ` it might not be trivial, as we have registered different instances on these two types. -/
/- The continuity should be trivial, as the topologies on `myℝ` and `ℝ` are definitionally the
same. So `continuous_id` might help. -/
lemma continuous_my_map : continuous my_map :=
-- sorry
continuous_id
-- sorry
/- Smoothness should not be obvious, though, as the manifold structures are not the same: the atlas
on `myℝ` has two elements, while the atlas on `ℝ` has one single element.
Note that `myℝ` is not a vector space, nor a normed space, so one can not ask whether `my_map`
is smooth in the usual sense (as a map between vector spaces): -/
-- lemma times_cont_diff_my_map : times_cont_diff ℝ ∞ my_map := sorry
/- does not make sense (try uncommenting it!) However, we can ask whether `my_map` is a smooth
map between manifolds, i.e., whether it is smooth when read in the charts. When we mention the
smoothness of a map, we should always specify explicitly the model with corners we are using,
because there might be several around (think of a complex manifold that you may want to consider
as a real manifold, to talk about functions which are real-smooth but not holomorphic) -/
lemma times_cont_mdiff_my_map : times_cont_mdiff 𝓡1 𝓡1 ∞ my_map :=
begin
-- put things in a nicer form. The simpset `mfld_simps` registers many simplification rules for
-- manifolds. `simp` is used heavily in manifold files to bring everything into manageable form.
rw times_cont_mdiff_iff,
simp only [continuous_my_map] with mfld_simps,
-- simp has erased the chart in the target, as it knows that the only chart in the manifold `ℝ`
-- is the identity.
assume x y,
-- sorry
simp [my_map, (∘), chart_at],
split_ifs,
{ exact times_cont_diff_id.neg.times_cont_diff_on },
{ exact times_cont_diff_id.times_cont_diff_on },
-- sorry
end
/- Now, let's go to tangent bundles. We have a smooth manifold, so its tangent bundle should also
be a smooth manifold. -/
-- the type `tangent_bundle I myℝ` makes sense
#check tangent_bundle 𝓡1 myℝ
/- The tangent space above a point of `myℝ` is just a one-dimensional vector space (identified with `ℝ`).
So, one can prescribe an element of the tangent bundle as a pair (more on this below) -/
example : tangent_bundle 𝓡1 myℝ := ((4 : ℝ), 0)
/- Construct the smooth manifold structure on the tangent bundle. Hint: the answer is a one-liner,
and this instance is not really needed. -/
instance tangent_bundle_myℝ : smooth_manifold_with_corners (𝓡1.prod 𝓡1) (tangent_bundle 𝓡1 myℝ) :=
-- sorry
by apply_instance
-- sorry
/-
NB: the model space for the tangent bundle to a product manifold or a tangent space is not
`ℝ × ℝ`, but a copy called `model_prod ℝ ℝ`. Otherwise, `ℝ × ℝ` would have two charted space
structures with model `ℝ × ℝ`, the identity one and the product one, which are not definitionally
equal. And this would be bad.
-/
#check tangent_bundle.charted_space 𝓡1 myℝ
/- A smooth map between manifolds induces a map between their tangent bundles. In `mathlib` this is
called the `tangent_map` (you might instead know it as the "differential" or "pushforward" of the
map). Let us check that the `tangent_map` of `my_map` is smooth. -/
lemma times_cont_mdiff_tangent_map_my_map :
times_cont_mdiff (𝓡1.prod 𝓡1) (𝓡1.prod 𝓡1) ∞ (tangent_map 𝓡1 𝓡1 my_map) :=
begin
-- hopefully, there is a theorem providing the general result, i.e. the tangent map to a smooth
-- map is smooth.
-- you can either try to guess its name, or hope that `suggest` will help you there.
-- sorry
exact times_cont_mdiff_my_map.times_cont_mdiff_tangent_map le_top,
-- sorry
end
/- (Harder question) Can you show that this tangent bundle is homeomorphic to `ℝ × ℝ`? You could
try to build the homeomorphism by hand, using `tangent_map I I my_map` in one direction and a
similar map in the other direction, but it is probably more efficient to use one of the charts of
the tangent bundle.
Remember, the model space for `tangent_bundle I myℝ` is `model_prod ℝ ℝ`, not `ℝ × ℝ`. But the
topologies on `model_prod ℝ ℝ` and `ℝ × ℝ` are the same, so it is by definition good enough to
construct a homeomorphism with `model_prod ℝ ℝ`.
-/
def my_homeo : tangent_bundle 𝓡1 myℝ ≃ₜ (ℝ × ℝ) :=
begin
-- sorry
let p : tangent_bundle 𝓡1 myℝ := ((4 : ℝ), 0),
let F := chart_at (model_prod ℝ ℝ) p,
have A : ¬ ((4 : ℝ) < 1), by norm_num,
have S : F.source = univ, by simp [F, chart_at, A, @local_homeomorph.refl_source ℝ _],
have T : F.target = univ, by simp [F, chart_at, A, @local_homeomorph.refl_target ℝ _],
exact F.to_homeomorph_of_source_eq_univ_target_eq_univ S T,
-- sorry
end
/- Up to now, we have never used the definition of the tangent bundle, and this corresponds to
the usual mathematical practice: one doesn't care if the tangent space is defined using germs of
curves, or spaces of derivations, or whatever equivalent definition. Instead, one relies all the
time on functoriality (i.e., a smooth map has a well defined derivative, and they compose well,
together with the fact that the tangent bundle to a vector space is the product).
If you want to know more about the internals of the tangent bundle in mathlib, you can browse
through the next section, but it is maybe wiser to skip it on first reading, as it is not needed
to use the library
-/
section you_should_probably_skip_this
/- If `M` is a manifold modelled on a vector space `E`, then the underlying type for the tangent
bundle is just `M × E` -/
lemma tangent_bundle_myℝ_is_prod : tangent_bundle 𝓡1 myℝ = (myℝ × ℝ) :=
/- inline sorry -/rfl/- inline sorry -/
/- This means that you can specify a point in the tangent bundle as a pair `(x, y)`.
However, in general, a tangent bundle is not trivial: the topology on `tangent_bundle I myℝ` is *not*
the product topology. Instead, the tangent space at a point `x` is identified with `ℝ` through some
preferred chart at `x`, called `chart_at ℝ x`, but the way they are glued together depends on the
manifold and the charts.
In vector spaces, the tangent space is canonically the product space, with the same topology, as
there is only one chart so there is no strange gluing at play. The equality of the topologies
is given in `tangent_bundle_model_space_topology_eq_prod`, but they are not definitionally equal
so one can get strange behavior if abusing identifications.
Let us register the identification explicitly, as a homeomorphism:
-/
def tangent_bundle_vector_space_triv (E : Type u) [normed_group E] [normed_space ℝ E] :
tangent_bundle (model_with_corners_self ℝ E) E ≃ₜ E × E :=
{ to_fun := id,
inv_fun := id,
left_inv := /- inline sorry -/λ x, rfl/- inline sorry -/,
right_inv := /- inline sorry -/λ x, rfl/- inline sorry -/,
continuous_to_fun := begin
-- if you think that `continuous_id` should work but `exact continuous_id` fails, you
-- can try `convert continuous_id`: it might show you what doesn't match and let you
-- fix it afterwards.
-- sorry
convert continuous_id,
exact (tangent_bundle_model_space_topology_eq_prod _ _).symm
-- sorry
end,
continuous_inv_fun :=
begin
-- sorry
convert continuous_id,
exact (tangent_bundle_model_space_topology_eq_prod _ _)
-- sorry
end }
/- Even though the tangent bundle to `myℝ` is trivial abstractly, with this construction the
tangent bundle is *not* the product space with the product topology, as we have used various charts
so the gluing is not trivial. The following exercise unfolds the definition to see what is going on.
It is not a reasonable exercise, in the sense that one should never ever do this when working
with a manifold! -/
lemma crazy_formula_after_identifications (x : ℝ) (v : ℝ) :
let p : tangent_bundle 𝓡1 myℝ := ((3 : ℝ), 0) in
chart_at (model_prod ℝ ℝ) p (x, v) = if x ∈ Ioo (-1 : ℝ) 1 then (x, -v) else (x, v) :=
begin
-- this exercise is not easy (and shouldn't be: you are not supposed to use the library like this!)
-- if you really want to do this, you should unfold as much as you can using simp and dsimp, until you
-- are left with a statement speaking of derivatives of real functions, without any manifold code left.
-- sorry
have : ¬ ((3 : ℝ) < 1), by norm_num,
simp only [chart_at, this, mem_Ioo, if_false, and_false],
dsimp [tangent_bundle_core, basic_smooth_bundle_core.chart,
topological_fiber_bundle_core.local_triv, topological_fiber_bundle_core.local_triv',
topological_fiber_bundle_core.index_at,
basic_smooth_bundle_core.to_topological_fiber_bundle_core],
split_ifs,
{ simp only [chart_at, h, my_first_local_homeo, if_true, fderiv_within_univ, prod.mk.inj_iff, mem_Ioo,
fderiv_neg differentiable_at_id', fderiv_id', id.def, continuous_linear_map.coe_id',
continuous_linear_map.neg_apply] with mfld_simps },
{ simp only [chart_at, h, fderiv_within_univ, mem_Ioo, if_false, @local_homeomorph.refl_symm ℝ,
fderiv_id, continuous_linear_map.coe_id'] with mfld_simps }
-- sorry
end
end you_should_probably_skip_this
/-!
### The language of manifolds
In this paragraph, we will try to write down interesting statements of theorems, without proving them. The
goal here is that Lean should not complain on the statement, but the proof should be sorried.
-/
/- Here is a first example, already filled up, to show you how diffeomorphisms are currently named
(we will probably introduce an abbreviation, but this hasn't been done yet).
Don't try to fill the sorried proof! -/
/-- Two zero-dimensional connected manifolds are diffeomorphic. -/
theorem diffeomorph_of_zero_dim_connected
(M M' : Type*) [topological_space M] [topological_space M']
[charted_space (euclidean_space (fin 0)) M] [charted_space (euclidean_space (fin 0)) M']
[connected_space M] [connected_space M'] :
nonempty (structomorph (times_cont_diff_groupoid ∞ (𝓡 0)) M M') :=
sorry
/- Do you think that this statement is correct? (note that we have not assumed that our manifolds
are smooth, nor that they are separated, but this is maybe automatic in zero dimension).
Now, write down a version of this theorem in dimension 1, replacing the first sorry with meaningful content
(and adding what is needed before the colon).
Don't try to fill the sorried proof! -/
/-- Two one-dimensional smooth compact connected manifolds are diffeomorphic. -/
theorem diffeomorph_of_one_dim_compact_connected
-- omit
(M M' : Type*) [topological_space M] [topological_space M']
[charted_space (euclidean_space (fin 1)) M] [charted_space (euclidean_space (fin 1)) M']
[connected_space M] [connected_space M'] [compact_space M] [compact_space M']
[t2_space M] [t2_space M']
[smooth_manifold_with_corners (𝓡 1) M] [smooth_manifold_with_corners (𝓡 1) M']
-- omit
:
-- sorry
nonempty (structomorph (times_cont_diff_groupoid ∞ (𝓡 1)) M M')
-- sorry
:= sorry
/- You will definitely need to require smoothness and separation in this case, as it is wrong otherwise.
Note that Lean won't complain if you don't put these assumptions, as the theorem would still make
sense, but it would just turn out to be wrong.
The previous statement is not really satisfactory: we would instead like to express that any such
manifold is diffeomorphic to the circle. The trouble is that we don't have the circle as a smooth
manifold yet. Since we have cheated and introduced it (with sorries) at the beginning of the tutorial,
let's cheat again and use it to reformulate the previous statement.
-/
-- the next result is not trivial, leave it sorried (but you can work on it if you don't like
-- manifolds and prefer topology -- then please PR it to mathlib!).
instance connected_sphere (n : ℕ) : connected_space (sphere (n+1)) := sorry
/- The next two instances are easier to prove, you can prove them or leave them sorried
as you like. For the second one, you may need to use facts of the library such as -/
#check compact_iff_compact_space
#check metric.compact_iff_closed_bounded
instance (n : ℕ) : t2_space (sphere n) :=
begin
-- sorry
dunfold sphere,
apply_instance
-- sorry
end
instance (n : ℕ) : compact_space (sphere n) :=
begin
-- sorry
dunfold sphere,
apply compact_iff_compact_space.1,
rw metric.compact_iff_closed_bounded,
split,
{ exact metric.is_closed_sphere },
{ rw metric.bounded_iff_subset_ball (0 : euclidean_space (fin (n+1))),
exact ⟨1, metric.sphere_subset_closed_ball⟩ }
-- sorry
end
/- Now, you can prove that any one-dimensional compact connected manifold is diffeomorphic to
the circle. Here, you should fill the `sorry` (but luckily you may use
`diffeomorph_of_one_dim_compact_connected`). -/
theorem diffeomorph_circle_of_one_dim_compact_connected
(M : Type*) [topological_space M] [charted_space (euclidean_space (fin 1)) M]
[connected_space M] [compact_space M] [t2_space M] [smooth_manifold_with_corners (𝓡 1) M] :
nonempty (structomorph (times_cont_diff_groupoid ∞ (𝓡 1)) M (sphere 1)) :=
-- sorry
diffeomorph_of_one_dim_compact_connected M (sphere 1)
-- sorry
/- What about trying to say that there are uncountably many different smooth structures on `ℝ⁴`?
(see https://en.wikipedia.org/wiki/Exotic_R4). The library is not really designed with this in mind,
as in general we only work with one differentiable structure on a space, but it is perfectly
capable of expressing this fact if one uses the `@` version of some definitions.
Don't try to fill the sorried proof!
-/
theorem exotic_ℝ4 :
-- sorry
let E := (euclidean_space (fin 4)) in
∃ f : ℝ → charted_space E E,
∀ i, @has_groupoid E _ E _ (f i) (times_cont_diff_groupoid ∞ (𝓡 4))
∧ ∀ i j, nonempty (@structomorph _ _ (times_cont_diff_groupoid ∞ (𝓡 4)) E E _ _ (f i) (f j)) →
i = j
-- sorry
:=
sorry
/-!
### Smooth functions on `[0, 1]`
In this paragraph, you will prove several (math-trivial but Lean-nontrivial) statements on the smooth
structure of `[0,1]`. These facts should be Lean-trivial, but they are not (yet) since there is essentially
nothing in this direction for now in the library.
The goal is as much to be able to write the statements as to prove them. Most of the necessary vocabulary
has been introduced above, so don't hesitate to browse the file if you are stuck. Additionally, you will
need the notion of a smooth function on a subset: it is `times_cont_diff_on` for functions between vector
spaces and `times_cont_mdiff_on` for functions between manifolds.
Lemma times_cont_mdiff_g : the inclusion `g` of `[0, 1]` in `ℝ` is smooth.
Lemma msmooth_of_smooth : Consider a function `f : ℝ → [0, 1]`, which is smooth in the usual sense as a function
from `ℝ` to `ℝ` on a set `s`. Then it is manifold-smooth on `s`.
Definition : construct a function `f` from `ℝ` to `[0,1]` which is the identity on `[0, 1]`.
Theorem : the tangent bundle to `[0, 1]` is homeomorphic to `[0, 1] × ℝ`
Hint for Theorem 4: don't try to unfold the definition of the tangent bundle, it will only get you
into trouble. Instead, use the derivatives of the maps `f` and `g`, and rely on functoriality
to check that they are inverse to each other. (This advice is slightly misleading as these derivatives
do not go between the right spaces, so you will need to massage them a little bit).
A global advice: don't hesitate to use and abuse `simp`, it is the main workhorse in this
area of mathlib.
-/
/- After doing the exercise myself, I realized it was (way!) too hard. So I will give at least the statements
of the lemmas, to guide you a little bit more. To let you try the original version if you want,
I have left a big blank space to avoid spoilers. -/
def g : Icc (0 : ℝ) 1 → ℝ := subtype.val
-- smoothness results for `euclidean_space` are expressed for general `L^p` spaces
-- (as `euclidean_space` has the `L^2` norm), in:
#check pi_Lp.times_cont_diff_coord
#check pi_Lp.times_cont_diff_on_iff_coord
lemma times_cont_mdiff_g : times_cont_mdiff (𝓡∂ 1) 𝓡1 ∞ g :=
begin
-- sorry
rw times_cont_mdiff_iff,
refine ⟨continuous_subtype_val, λ x y, _⟩,
by_cases h : (x : ℝ) < 1,
{ simp only [g, chart_at, h, Icc_left_chart, function.comp, model_with_corners_euclidean_half_space,
add_zero, dif_pos, if_true, max_lt_iff, preimage_set_of_eq, sub_zero, subtype.range_coe_subtype,
subtype.coe_mk, subtype.val_eq_coe] with mfld_simps,
refine (pi_Lp.times_cont_diff_coord 0).times_cont_diff_on.congr (λ x hx, _),
simp only [mem_inter_eq, mem_set_of_eq] at hx,
simp only [hx, le_of_lt hx.right.left, min_eq_left, max_eq_left] },
{ simp only [chart_at, h, Icc_right_chart, function.comp, model_with_corners_euclidean_half_space, dif_pos,
max_lt_iff, preimage_set_of_eq, sub_zero, subtype.range_coe_subtype, if_false, subtype.coe_mk,
subtype.val_eq_coe, g] with mfld_simps,
have : times_cont_diff ℝ ⊤ (λ (x : euclidean_space (fin 1)), 1 - x 0) :=
times_cont_diff_const.sub (pi_Lp.times_cont_diff_coord 0),
apply this.times_cont_diff_on.congr (λ x hx, _),
simp only [mem_inter_eq, mem_set_of_eq] at hx,
have : 0 ≤ 1 - x 0, by linarith,
simp only [hx, this, max_eq_left] }
-- sorry
end
lemma msmooth_of_smooth {f : ℝ → Icc (0 : ℝ) 1} {s : set ℝ} (h : times_cont_diff_on ℝ ∞ (λ x, (f x : ℝ)) s) :
times_cont_mdiff_on 𝓡1 (𝓡∂ 1) ∞ f s :=
begin
-- sorry
rw times_cont_mdiff_on_iff,
split,
{ have : embedding (subtype.val : Icc (0 : ℝ) 1 → ℝ) := embedding_subtype_coe,
exact (embedding.continuous_on_iff this).2 h.continuous_on },
simp only with mfld_simps,
assume y,
by_cases hy : (y : ℝ) < 1,
{ simp [chart_at, model_with_corners_euclidean_half_space, (∘), hy, Icc_left_chart,
pi_Lp.times_cont_diff_on_iff_coord],
apply h.mono (inter_subset_left _ _) },
{ simp [chart_at, model_with_corners_euclidean_half_space, (∘), hy, Icc_right_chart,
pi_Lp.times_cont_diff_on_iff_coord],
assume i,
apply (times_cont_diff_on_const.sub h).mono (inter_subset_left _ _) }
-- sorry
end
/- A function from `ℝ` to `[0,1]` which is the identity on `[0,1]`. -/
def f : ℝ → Icc (0 : ℝ) 1 :=
λ x, ⟨max (min x 1) 0, by simp [le_refl, zero_le_one]⟩
lemma times_cont_mdiff_on_f : times_cont_mdiff_on 𝓡1 (𝓡∂ 1) ∞ f (Icc 0 1) :=
begin
-- sorry
apply msmooth_of_smooth,
apply times_cont_diff_id.times_cont_diff_on.congr,
assume x hx,
simp at hx,
simp [f, hx],
-- sorry
end
lemma fog : f ∘ g = id :=
begin
-- sorry
ext x,
rcases x with ⟨x', h'⟩,
simp at h',
simp [f, g, h'],
-- sorry
end
lemma gof : ∀ x ∈ Icc (0 : ℝ) 1, g (f x) = x :=
begin
-- sorry
assume x hx,
simp at hx,
simp [g, f],
simp [hx],
-- sorry
end
def G : tangent_bundle (𝓡∂ 1) (Icc (0 : ℝ) 1) → (Icc (0 : ℝ) 1) × ℝ :=
λ p, (p.1, (tangent_map (𝓡∂ 1) 𝓡1 g p).2)
lemma continuous_G : continuous G :=
begin
-- sorry
apply continuous.prod_mk (tangent_bundle_proj_continuous _ _),
refine continuous_snd.comp _,
have Z := times_cont_mdiff_g.continuous_tangent_map le_top,
convert Z,
exact (tangent_bundle_model_space_topology_eq_prod ℝ 𝓡1).symm
-- sorry
end
/- in the definition of `F`, we use the map `tangent_bundle_vector_space_triv`
(which is just the identity pointwise) to make sure that Lean is not lost
between the different topologies. -/
def F : (Icc (0 : ℝ) 1) × ℝ → tangent_bundle (𝓡∂ 1) (Icc (0 : ℝ) 1) :=
λ p, tangent_map_within 𝓡1 (𝓡∂ 1) f (Icc 0 1)
((tangent_bundle_vector_space_triv ℝ).symm (p.1, p.2))
lemma continuous_F : continuous F :=
begin
-- sorry
rw continuous_iff_continuous_on_univ,
apply (times_cont_mdiff_on_f.continuous_on_tangent_map_within le_top _).comp,
{ apply ((tangent_bundle_vector_space_triv ℝ).symm.continuous.comp _).continuous_on,
apply (continuous_subtype_coe.comp continuous_fst).prod_mk continuous_snd },
{ rintros ⟨⟨x, hx⟩, v⟩ _,
simp [tangent_bundle_vector_space_triv],
exact hx },
{ rw unique_mdiff_on_iff_unique_diff_on,
exact unique_diff_on_Icc_zero_one }
-- sorry
end
lemma FoG : F ∘ G = id :=
begin
-- sorry
ext1 p,
rcases p with ⟨x, v⟩,
simp [F, G, tangent_map_within, tangent_bundle_vector_space_triv, f],
dsimp,
split,
{ rcases x with ⟨x', h'⟩,
simp at h',
simp [h'] },
{ change (tangent_map_within 𝓡1 (𝓡∂ 1) f (Icc 0 1) (tangent_map (𝓡∂ 1) 𝓡1 g (x, v))).snd = v,
rw [← tangent_map_within_univ, ← tangent_map_within_comp_at, fog, tangent_map_within_univ, tangent_map_id],
{ refl },
{ apply times_cont_mdiff_on_f.mdifferentiable_on le_top,
simpa [g] using x.2 },
{ apply (times_cont_mdiff_g.times_cont_mdiff_at.mdifferentiable_at le_top).mdifferentiable_within_at },
{ assume z hz,
simpa [g] using z.2 },
{ apply unique_mdiff_on_univ _ (mem_univ _) } }
-- sorry
end
lemma GoF : G ∘ F = id :=
begin
-- sorry
ext1 p,
rcases p with ⟨x, v⟩,
simp [F, G, tangent_map_within, tangent_bundle_vector_space_triv, f],
dsimp,
split,
{ rcases x with ⟨x', h'⟩,
simp at h',
simp [h'] },
{ have A : unique_mdiff_within_at 𝓡1 (Icc 0 1) ((x : ℝ), v).fst,
{ rw unique_mdiff_within_at_iff_unique_diff_within_at,
apply unique_diff_on_Icc_zero_one _ x.2 },
change (tangent_map (𝓡∂ 1) 𝓡1 g (tangent_map_within 𝓡1 (𝓡∂ 1) f (Icc 0 1) (x, v))).snd = v,
rw [← tangent_map_within_univ, ← tangent_map_within_comp_at _ _ _ _ A],
{ have : tangent_map_within 𝓡1 𝓡1 (g ∘ f) (Icc 0 1) (x, v)
= tangent_map_within 𝓡1 𝓡1 id (Icc 0 1) (x, v) :=
tangent_map_within_congr gof _ x.2 A,
rw [this, tangent_map_within_id A] },
{ apply times_cont_mdiff_g.times_cont_mdiff_on.mdifferentiable_on le_top _ (mem_univ _) },
{ apply times_cont_mdiff_on_f.mdifferentiable_on le_top _ x.2 },
{ simp only [preimage_univ, subset_univ], } }
-- sorry
end
def my_tangent_homeo : tangent_bundle (𝓡∂ 1) (Icc (0 : ℝ) 1) ≃ₜ (Icc (0 : ℝ) 1) × ℝ :=
-- sorry
{ to_fun := G,
inv_fun := F,
continuous_to_fun := continuous_G,
continuous_inv_fun := continuous_F,
left_inv := λ p, show (F ∘ G) p = id p, by rw FoG,
right_inv := λ p, show (G ∘ F) p = id p, by rw GoF }
-- sorry
/-!
### Further things to do
1) can you prove `diffeomorph_of_zero_dim_connected` or `connected_sphere`?
2) Try to express and then prove the local inverse theorem in real manifolds: if a map between
real manifolds (without boundary, modelled on a complete vector space) is smooth, then it is
a local homeomorphism around each point. We already have versions of this statement in mathlib
for functions between vector spaces, but this is very much a work in progress.
3) What about trying to prove `diffeomorph_of_one_dim_compact_connected`? (I am not sure mathlib
is ready for this, as the proofs I am thinking of are currently a little bit too high-powered.
If you manage to do it, you should absolutely PR it!)
4) Why not contribute to the proof of `sphere_eversion`? You can have a look at
https://leanprover-community.github.io/sphere-eversion/ to learn more about this project
by Patrick Massot.
-/
|
d534899f9a62fc998c3efb871d49019daafc85ff | fa02ed5a3c9c0adee3c26887a16855e7841c668b | /src/category_theory/limits/punit.lean | 95732e70d9e95a0ba30cdce54a8b8cd1d4a240c5 | [
"Apache-2.0"
] | permissive | jjgarzella/mathlib | 96a345378c4e0bf26cf604aed84f90329e4896a2 | 395d8716c3ad03747059d482090e2bb97db612c8 | refs/heads/master | 1,686,480,124,379 | 1,625,163,323,000 | 1,625,163,323,000 | 281,190,421 | 2 | 0 | Apache-2.0 | 1,595,268,170,000 | 1,595,268,169,000 | null | UTF-8 | Lean | false | false | 1,365 | lean | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import category_theory.punit
import category_theory.limits.has_limits
/-!
# `discrete punit` has limits and colimits
Mostly for the sake of constructing trivial examples, we show all (co)cones into `discrete punit`
are (co)limit (co)cones. We also show that such (co)cones exist, and that `discrete punit` has all
(co)limits.
-/
universe v
open category_theory
namespace category_theory.limits
variables {J : Type v} [small_category J] {F : J ⥤ discrete punit.{v+1}}
/-- A trivial cone for a functor into `punit`. `punit_cone_is_limit` shows it is a limit. -/
def punit_cone : cone F :=
⟨punit.star, (functor.punit_ext _ _).hom⟩
/-- A trivial cocone for a functor into `punit`. `punit_cocone_is_limit` shows it is a colimit. -/
def punit_cocone : cocone F :=
⟨punit.star, (functor.punit_ext _ _).hom⟩
/--
Any cone over a functor into `punit` is a limit cone.
-/
def punit_cone_is_limit {c : cone F} : is_limit c :=
by tidy
/--
Any cocone over a functor into `punit` is a colimit cocone.
-/
def punit_cocone_is_colimit {c : cocone F} : is_colimit c :=
by tidy
instance : has_limits (discrete punit) :=
by tidy
instance : has_colimits (discrete punit) :=
by tidy
end category_theory.limits
|
a4e80a7d0873203812091d84d87bf6438f95d791 | 55c7fc2bf55d496ace18cd6f3376e12bb14c8cc5 | /src/category_theory/adjunction/limits.lean | 3a9b1825092a83f65660c79559d7256d0f1af35c | [
"Apache-2.0"
] | permissive | dupuisf/mathlib | 62de4ec6544bf3b79086afd27b6529acfaf2c1bb | 8582b06b0a5d06c33ee07d0bdf7c646cae22cf36 | refs/heads/master | 1,669,494,854,016 | 1,595,692,409,000 | 1,595,692,409,000 | 272,046,630 | 0 | 0 | Apache-2.0 | 1,592,066,143,000 | 1,592,066,142,000 | null | UTF-8 | Lean | false | false | 8,528 | lean | /-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Johan Commelin
-/
import category_theory.adjunction.basic
import category_theory.limits.creates
open opposite
namespace category_theory.adjunction
open category_theory
open category_theory.functor
open category_theory.limits
universes u₁ u₂ v
variables {C : Type u₁} [category.{v} C] {D : Type u₂} [category.{v} D]
variables {F : C ⥤ D} {G : D ⥤ C} (adj : F ⊣ G)
include adj
section preservation_colimits
variables {J : Type v} [small_category J] (K : J ⥤ C)
def functoriality_right_adjoint : cocone (K ⋙ F) ⥤ cocone K :=
(cocones.functoriality _ G) ⋙
(cocones.precompose (K.right_unitor.inv ≫ (whisker_left K adj.unit) ≫ (associator _ _ _).inv))
local attribute [reducible] functoriality_right_adjoint
@[simps] def functoriality_unit : 𝟭 (cocone K) ⟶ cocones.functoriality _ F ⋙ functoriality_right_adjoint adj K :=
{ app := λ c, { hom := adj.unit.app c.X } }
@[simps] def functoriality_counit : functoriality_right_adjoint adj K ⋙ cocones.functoriality _ F ⟶ 𝟭 (cocone (K ⋙ F)) :=
{ app := λ c, { hom := adj.counit.app c.X } }
def functoriality_is_left_adjoint :
is_left_adjoint (cocones.functoriality K F) :=
{ right := functoriality_right_adjoint adj K,
adj := mk_of_unit_counit
{ unit := functoriality_unit adj K,
counit := functoriality_counit adj K } }
/-- A left adjoint preserves colimits. -/
def left_adjoint_preserves_colimits : preserves_colimits F :=
{ preserves_colimits_of_shape := λ J 𝒥,
{ preserves_colimit := λ F,
by exactI
{ preserves := λ c hc, is_colimit.iso_unique_cocone_morphism.inv
(λ s, @equiv.unique _ _ (is_colimit.iso_unique_cocone_morphism.hom hc _)
(((adj.functoriality_is_left_adjoint _).adj).hom_equiv _ _)) } } }.
omit adj
@[priority 100] -- see Note [lower instance priority]
instance is_equivalence_preserves_colimits (E : C ⥤ D) [is_equivalence E] : preserves_colimits E :=
left_adjoint_preserves_colimits E.adjunction
-- verify the preserve_colimits instance works as expected:
example (E : C ⥤ D) [is_equivalence E]
(c : cocone K) (h : is_colimit c) : is_colimit (E.map_cocone c) :=
preserves_colimit.preserves h
instance has_colimit_comp_equivalence (E : C ⥤ D) [is_equivalence E] [has_colimit K] :
has_colimit (K ⋙ E) :=
{ cocone := E.map_cocone (colimit.cocone K),
is_colimit := preserves_colimit.preserves (colimit.is_colimit K) }
def has_colimit_of_comp_equivalence (E : C ⥤ D) [is_equivalence E] [has_colimit (K ⋙ E)] :
has_colimit K :=
@has_colimit_of_iso _ _ _ _ (K ⋙ E ⋙ inv E) K
(@adjunction.has_colimit_comp_equivalence _ _ _ _ _ _ (K ⋙ E) (inv E) _ _)
((functor.right_unitor _).symm ≪≫ (iso_whisker_left K (fun_inv_id E)).symm)
end preservation_colimits
section preservation_limits
variables {J : Type v} [small_category J] (K : J ⥤ D)
def functoriality_left_adjoint : cone (K ⋙ G) ⥤ cone K :=
(cones.functoriality _ F) ⋙ (cones.postcompose
((associator _ _ _).hom ≫ (whisker_left K adj.counit) ≫ K.right_unitor.hom))
local attribute [reducible] functoriality_left_adjoint
@[simps] def functoriality_unit' : 𝟭 (cone (K ⋙ G)) ⟶ functoriality_left_adjoint adj K ⋙ cones.functoriality _ G :=
{ app := λ c, { hom := adj.unit.app c.X, } }
@[simps] def functoriality_counit' : cones.functoriality _ G ⋙ functoriality_left_adjoint adj K ⟶ 𝟭 (cone K) :=
{ app := λ c, { hom := adj.counit.app c.X, } }
def functoriality_is_right_adjoint :
is_right_adjoint (cones.functoriality K G) :=
{ left := functoriality_left_adjoint adj K,
adj := mk_of_unit_counit
{ unit := functoriality_unit' adj K,
counit := functoriality_counit' adj K } }
/-- A right adjoint preserves limits. -/
def right_adjoint_preserves_limits : preserves_limits G :=
{ preserves_limits_of_shape := λ J 𝒥,
{ preserves_limit := λ K,
by exactI
{ preserves := λ c hc, is_limit.iso_unique_cone_morphism.inv
(λ s, @equiv.unique _ _ (is_limit.iso_unique_cone_morphism.hom hc _)
(((adj.functoriality_is_right_adjoint _).adj).hom_equiv _ _).symm) } } }.
omit adj
@[priority 100] -- see Note [lower instance priority]
instance is_equivalence_preserves_limits (E : D ⥤ C) [is_equivalence E] : preserves_limits E :=
right_adjoint_preserves_limits E.inv.adjunction
@[priority 100] -- see Note [lower instance priority]
instance is_equivalence_reflects_limits (E : D ⥤ C) [is_equivalence E] : reflects_limits E :=
{ reflects_limits_of_shape := λ J 𝒥, by exactI
{ reflects_limit := λ K,
{ reflects := λ c t,
begin
have l: is_limit (E.inv.map_cone (E.map_cone c)) := preserves_limit.preserves t,
convert is_limit.map_cone_equiv E.fun_inv_id l,
{ rw functor.comp_id },
{ cases c,
cases c_π,
congr; rw functor.comp_id }
end } } }
@[priority 100] -- see Note [lower instance priority]
instance is_equivalence_creates_limits (H : D ⥤ C) [is_equivalence H] : creates_limits H :=
{ creates_limits_of_shape := λ J 𝒥, by exactI
{ creates_limit := λ F,
{ lifts := λ c t,
{ lifted_cone := H.map_cone_inv c,
valid_lift := H.map_cone_map_cone_inv c } } } }
-- verify the preserve_limits instance works as expected:
example (E : D ⥤ C) [is_equivalence E]
(c : cone K) [h : is_limit c] : is_limit (E.map_cone c) :=
preserves_limit.preserves h
instance has_limit_comp_equivalence (E : D ⥤ C) [is_equivalence E] [has_limit K] :
has_limit (K ⋙ E) :=
{ cone := E.map_cone (limit.cone K),
is_limit := preserves_limit.preserves (limit.is_limit K) }
def has_limit_of_comp_equivalence (E : D ⥤ C) [is_equivalence E] [has_limit (K ⋙ E)] :
has_limit K :=
@has_limit_of_iso _ _ _ _ (K ⋙ E ⋙ inv E) K
(@adjunction.has_limit_comp_equivalence _ _ _ _ _ _ (K ⋙ E) (inv E) _ _)
((iso_whisker_left K (fun_inv_id E)) ≪≫ (functor.right_unitor _))
end preservation_limits
/-- auxilliary construction for `cocones_iso` -/
@[simps]
def cocones_iso_component_hom {J : Type v} [small_category J] {K : J ⥤ C}
(Y : D) (t : ((cocones J D).obj (op (K ⋙ F))).obj Y) :
(G ⋙ (cocones J C).obj (op K)).obj Y :=
{ app := λ j, (adj.hom_equiv (K.obj j) Y) (t.app j),
naturality' := λ j j' f, by erw [← adj.hom_equiv_naturality_left, t.naturality]; dsimp; simp }
/-- auxilliary construction for `cocones_iso` -/
@[simps]
def cocones_iso_component_inv {J : Type v} [small_category J] {K : J ⥤ C}
(Y : D) (t : (G ⋙ (cocones J C).obj (op K)).obj Y) :
((cocones J D).obj (op (K ⋙ F))).obj Y :=
{ app := λ j, (adj.hom_equiv (K.obj j) Y).symm (t.app j),
naturality' := λ j j' f,
begin
erw [← adj.hom_equiv_naturality_left_symm, ← adj.hom_equiv_naturality_right_symm, t.naturality],
dsimp, simp
end }
-- Note: this is natural in K, but we do not yet have the tools to formulate that.
def cocones_iso {J : Type v} [small_category J] {K : J ⥤ C} :
(cocones J D).obj (op (K ⋙ F)) ≅ G ⋙ ((cocones J C).obj (op K)) :=
nat_iso.of_components (λ Y,
{ hom := cocones_iso_component_hom adj Y,
inv := cocones_iso_component_inv adj Y, })
(by tidy)
/-- auxilliary construction for `cones_iso` -/
@[simps]
def cones_iso_component_hom {J : Type v} [small_category J] {K : J ⥤ D}
(X : Cᵒᵖ) (t : (functor.op F ⋙ (cones J D).obj K).obj X) :
((cones J C).obj (K ⋙ G)).obj X :=
{ app := λ j, (adj.hom_equiv (unop X) (K.obj j)) (t.app j),
naturality' := λ j j' f,
begin
erw [← adj.hom_equiv_naturality_right, ← t.naturality, category.id_comp, category.id_comp],
refl
end }
/-- auxilliary construction for `cones_iso` -/
@[simps]
def cones_iso_component_inv {J : Type v} [small_category J] {K : J ⥤ D}
(X : Cᵒᵖ) (t : ((cones J C).obj (K ⋙ G)).obj X) :
(functor.op F ⋙ (cones J D).obj K).obj X :=
{ app := λ j, (adj.hom_equiv (unop X) (K.obj j)).symm (t.app j),
naturality' := λ j j' f,
begin
erw [← adj.hom_equiv_naturality_right_symm, ← t.naturality, category.id_comp, category.id_comp]
end }
-- Note: this is natural in K, but we do not yet have the tools to formulate that.
def cones_iso {J : Type v} [small_category J] {K : J ⥤ D} :
F.op ⋙ ((cones J D).obj K) ≅ (cones J C).obj (K ⋙ G) :=
nat_iso.of_components (λ X,
{ hom := cones_iso_component_hom adj X,
inv := cones_iso_component_inv adj X, } )
(by tidy)
end category_theory.adjunction
|
9856670910c0312fb37447c2f07e6abaefc1b417 | 1446f520c1db37e157b631385707cc28a17a595e | /tests/compiler/closure_bug6.lean | a3f8ac43f2e85b31fde056c3207cf2d2c215299c | [
"Apache-2.0"
] | permissive | bdbabiak/lean4 | cab06b8a2606d99a168dd279efdd404edb4e825a | 3f4d0d78b2ce3ef541cb643bbe21496bd6b057ac | refs/heads/master | 1,615,045,275,530 | 1,583,793,696,000 | 1,583,793,696,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 509 | lean | def f (x : Nat) : Nat × (Nat → String) :=
let x1 := x + 1;
let x2 := x + 2;
let x3 := x + 3;
let x4 := x + 4;
let x5 := x + 5;
let x6 := x + 6;
let x7 := x + 7;
let x8 := x + 8;
let x9 := x + 9;
let x10 := x + 10;
let x11 := x + 11;
let x12 := x + 12;
let x13 := x + 13;
let x14 := x + 14;
let x15 := x + 15;
let x16 := x + 16;
let x17 := x + 17;
(x, fun y => toString [x1, x2, x3, x4, x5, x6, x7, x8])
def main (xs : List String) : IO Unit :=
IO.println ((f (xs.headD "0").toNat).2 (xs.headD "0").toNat)
|
77790b6a1c3e047c999ee1b6e482295bca3ed8f8 | 6432ea7a083ff6ba21ea17af9ee47b9c371760f7 | /tests/lean/run/909.lean | f2955ed52c67504d5e3401ddf6775f3735b445ec | [
"Apache-2.0",
"LLVM-exception",
"NCSA",
"LGPL-3.0-only",
"LicenseRef-scancode-inner-net-2.0",
"BSD-3-Clause",
"LGPL-2.0-or-later",
"Spencer-94",
"LGPL-2.1-or-later",
"HPND",
"LicenseRef-scancode-pcre",
"ISC",
"LGPL-2.1-only",
"LicenseRef-scancode-other-permissive",
"SunPro",
"CMU-Mach"... | permissive | leanprover/lean4 | 4bdf9790294964627eb9be79f5e8f6157780b4cc | f1f9dc0f2f531af3312398999d8b8303fa5f096b | refs/heads/master | 1,693,360,665,786 | 1,693,350,868,000 | 1,693,350,868,000 | 129,571,436 | 2,827 | 311 | Apache-2.0 | 1,694,716,156,000 | 1,523,760,560,000 | Lean | UTF-8 | Lean | false | false | 942 | lean | structure Date where
val : Nat
deriving Repr
instance : LE Date := ⟨InvImage (Nat.le) Date.val⟩
instance bad (a b : Date) : Decidable (a <= b) :=
if h0 : (a.val <= b.val) then isTrue h0 else isFalse (fun hf => False.elim (h0 hf))
instance : Min Date := minOfLe
/-
This implementation also fails:
instance (a b : Date) : Decidable (a <= b) :=
inferInstanceAs (Decidable (a.val <= b.val))
-/
/-
This implemenation evaluates successfully:
instance (a b : Date) : Decidable (a <= b) :=
dite (a.val <= b.val) isTrue (fun nle => isFalse (fun hf => False.elim (nle hf)))
-/
instance : ToString Date where
toString d := s!"D{d.val}"
structure DateRange where
start_ : Date
finish_ : Date
h : start_ <= finish_
def r1 := (DateRange.mk (Date.mk 0) (Date.mk 10) (by decide))
-- If you change the start_ value here to `0`, it works.
def r2 := (DateRange.mk (Date.mk 1) (Date.mk 10) (by decide))
#eval min r1.start_ r2.start_
|
2dfe4d73409deed5cda1ff207ffba4c3be7afbf0 | 7cef822f3b952965621309e88eadf618da0c8ae9 | /src/data/fin.lean | dcd83e481c1a5264d62f09d592dff43f7c63c930 | [
"Apache-2.0"
] | permissive | rmitta/mathlib | 8d90aee30b4db2b013e01f62c33f297d7e64a43d | 883d974b608845bad30ae19e27e33c285200bf84 | refs/heads/master | 1,585,776,832,544 | 1,576,874,096,000 | 1,576,874,096,000 | 153,663,165 | 0 | 2 | Apache-2.0 | 1,544,806,490,000 | 1,539,884,365,000 | Lean | UTF-8 | Lean | false | false | 14,243 | lean | /-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
More about finite numbers.
-/
import data.nat.basic
open fin nat function
/-- `fin 0` is empty -/
def fin_zero_elim {C : Sort*} : fin 0 → C :=
λ x, false.elim $ nat.not_lt_zero x.1 x.2
def {u} fin_zero_elim' {α : fin 0 → Sort u} : ∀(x : fin 0), α x
| ⟨n, hn⟩ := false.elim (nat.not_lt_zero n hn)
namespace fin
variables {n m : ℕ} {a b : fin n}
@[simp] protected lemma eta (a : fin n) (h : a.1 < n) : (⟨a.1, h⟩ : fin n) = a :=
by cases a; refl
protected lemma ext_iff (a b : fin n) : a = b ↔ a.val = b.val :=
iff.intro (congr_arg _) fin.eq_of_veq
lemma injective_val {n : ℕ} : injective (val : fin n → ℕ) := λ _ _, fin.eq_of_veq
lemma eq_iff_veq (a b : fin n) : a = b ↔ a.1 = b.1 :=
⟨veq_of_eq, eq_of_veq⟩
@[simp] protected lemma mk.inj_iff {n a b : ℕ} {ha : a < n} {hb : b < n} :
fin.mk a ha = fin.mk b hb ↔ a = b :=
⟨fin.mk.inj, λ h, by subst h⟩
instance fin_to_nat (n : ℕ) : has_coe (fin n) nat := ⟨fin.val⟩
@[simp] lemma mk_val {m n : ℕ} (h : m < n) : (⟨m, h⟩ : fin n).val = m := rfl
@[simp] lemma coe_mk {m n : ℕ} (h : m < n) : ((⟨m, h⟩ : fin n) : ℕ) = m := rfl
lemma coe_eq_val (a : fin n) : (a : ℕ) = a.val := rfl
@[simp] lemma val_one {n : ℕ} : (1 : fin (n+2)).val = 1 := rfl
@[simp] lemma val_two {n : ℕ} : (2 : fin (n+3)).val = 2 := rfl
@[simp] lemma coe_zero {n : ℕ} : ((0 : fin (n+1)) : ℕ) = 0 := rfl
@[simp] lemma coe_one {n : ℕ} : ((1 : fin (n+2)) : ℕ) = 1 := rfl
@[simp] lemma coe_two {n : ℕ} : ((2 : fin (n+3)) : ℕ) = 2 := rfl
instance {n : ℕ} : decidable_linear_order (fin n) :=
decidable_linear_order.lift fin.val (@fin.eq_of_veq _) (by apply_instance)
lemma exists_iff {p : fin n → Prop} : (∃ i, p i) ↔ ∃ i h, p ⟨i, h⟩ :=
⟨λ h, exists.elim h (λ ⟨i, hi⟩ hpi, ⟨i, hi, hpi⟩),
λ h, exists.elim h (λ i hi, ⟨⟨i, hi.fst⟩, hi.snd⟩)⟩
lemma forall_iff {p : fin n → Prop} : (∀ i, p i) ↔ ∀ i h, p ⟨i, h⟩ :=
⟨λ h i hi, h ⟨i, hi⟩, λ h ⟨i, hi⟩, h i hi⟩
lemma zero_le (a : fin (n + 1)) : 0 ≤ a := zero_le a.1
lemma lt_iff_val_lt_val : a < b ↔ a.val < b.val := iff.rfl
lemma le_iff_val_le_val : a ≤ b ↔ a.val ≤ b.val := iff.rfl
@[simp] lemma succ_val (j : fin n) : j.succ.val = j.val.succ :=
by cases j; simp [fin.succ]
protected theorem succ.inj (p : fin.succ a = fin.succ b) : a = b :=
by cases a; cases b; exact eq_of_veq (nat.succ.inj (veq_of_eq p))
lemma succ_ne_zero {n} : ∀ k : fin n, fin.succ k ≠ 0
| ⟨k, hk⟩ heq := nat.succ_ne_zero k $ (fin.ext_iff _ _).1 heq
@[simp] lemma pred_val (j : fin (n+1)) (h : j ≠ 0) : (j.pred h).val = j.val.pred :=
by cases j; simp [fin.pred]
@[simp] lemma succ_pred : ∀(i : fin (n+1)) (h : i ≠ 0), (i.pred h).succ = i
| ⟨0, h⟩ hi := by contradiction
| ⟨n + 1, h⟩ hi := rfl
@[simp] lemma pred_succ (i : fin n) {h : i.succ ≠ 0} : i.succ.pred h = i :=
by cases i; refl
@[simp] lemma pred_inj :
∀ {a b : fin (n + 1)} {ha : a ≠ 0} {hb : b ≠ 0}, a.pred ha = b.pred hb ↔ a = b
| ⟨0, _⟩ b ha hb := by contradiction
| ⟨i+1, _⟩ ⟨0, _⟩ ha hb := by contradiction
| ⟨i+1, hi⟩ ⟨j+1, hj⟩ ha hb := by simp [fin.eq_iff_veq]
/-- The greatest value of `fin (n+1)` -/
def last (n : ℕ) : fin (n+1) := ⟨_, n.lt_succ_self⟩
/-- `cast_lt i h` embeds `i` into a `fin` where `h` proves it belongs into. -/
def cast_lt (i : fin m) (h : i.1 < n) : fin n := ⟨i.1, h⟩
/-- `cast_le h i` embeds `i` into a larger `fin` type. -/
def cast_le (h : n ≤ m) (a : fin n) : fin m := cast_lt a (lt_of_lt_of_le a.2 h)
/-- `cast eq i` embeds `i` into a equal `fin` type. -/
def cast (eq : n = m) : fin n → fin m := cast_le $ le_of_eq eq
/-- `cast_add m i` embedds `i` in `fin (n+m)`. -/
def cast_add (m) : fin n → fin (n + m) := cast_le $ le_add_right n m
/-- `cast_succ i` embedds `i` in `fin (n+1)`. -/
def cast_succ : fin n → fin (n + 1) := cast_add 1
/-- `succ_above p i` embeds into `fin (n + 1)` with a hole around `p`. -/
def succ_above (p : fin (n+1)) (i : fin n) : fin (n+1) :=
if i.1 < p.1 then i.cast_succ else i.succ
/-- `pred_above p i h` embeds `i` into `fin n` by ignoring `p`. -/
def pred_above (p : fin (n+1)) (i : fin (n+1)) (hi : i ≠ p) : fin n :=
if h : i < p
then i.cast_lt (lt_of_lt_of_le h $ nat.le_of_lt_succ p.2)
else i.pred $
have p < i, from lt_of_le_of_ne (le_of_not_gt h) hi.symm,
ne_of_gt (lt_of_le_of_lt (zero_le p) this)
/-- `sub_nat i h` subtracts `m` from `i`, generalizes `fin.pred`. -/
def sub_nat (m) (i : fin (n + m)) (h : m ≤ i.val) : fin n :=
⟨i.val - m, by simp [nat.sub_lt_right_iff_lt_add h, i.is_lt]⟩
/-- `add_nat i h` adds `m` on `i`, generalizes `fin.succ`. -/
def add_nat (m) (i : fin n) : fin (n + m) :=
⟨i.1 + m, add_lt_add_right i.2 _⟩
/-- `nat_add i h` adds `n` on `i` -/
def nat_add (n) {m} (i : fin m) : fin (n + m) :=
⟨n + i.1, add_lt_add_left i.2 _⟩
theorem le_last (i : fin (n+1)) : i ≤ last n :=
le_of_lt_succ i.is_lt
@[simp] lemma cast_val (k : fin n) (h : n = m) : (fin.cast h k).val = k.val := rfl
@[simp] lemma cast_succ_val (k : fin n) : k.cast_succ.val = k.val := rfl
@[simp] lemma cast_lt_val (k : fin m) (h : k.1 < n) : (k.cast_lt h).val = k.val := rfl
@[simp] lemma cast_le_val (k : fin m) (h : m ≤ n) : (k.cast_le h).val = k.val := rfl
@[simp] lemma cast_add_val (k : fin m) : (k.cast_add n).val = k.val := rfl
@[simp] lemma last_val (n : ℕ) : (last n).val = n := rfl
@[simp] lemma cast_succ_cast_lt (i : fin (n + 1)) (h : i.val < n) : cast_succ (cast_lt i h) = i :=
fin.eq_of_veq rfl
@[simp] lemma cast_lt_cast_succ {n : ℕ} (a : fin n) (h : a.1 < n) : cast_lt (cast_succ a) h = a :=
by cases a; refl
@[simp] lemma sub_nat_val (i : fin (n + m)) (h : m ≤ i.val) : (i.sub_nat m h).val = i.val - m :=
rfl
@[simp] lemma add_nat_val (i : fin (n + m)) (h : m ≤ i.val) : (i.add_nat m).val = i.val + m :=
rfl
@[simp] lemma cast_succ_inj {a b : fin n} : a.cast_succ = b.cast_succ ↔ a = b :=
by simp [eq_iff_veq]
def clamp (n m : ℕ) : fin (m + 1) := fin.of_nat $ min n m
@[simp] lemma clamp_val (n m : ℕ) : (clamp n m).val = min n m :=
nat.mod_eq_of_lt $ nat.lt_succ_iff.mpr $ min_le_right _ _
lemma injective_cast_le {n₁ n₂ : ℕ} (h : n₁ ≤ n₂) : injective (fin.cast_le h)
| ⟨i₁, h₁⟩ ⟨i₂, h₂⟩ eq := fin.eq_of_veq $ show i₁ = i₂, from fin.veq_of_eq eq
theorem succ_above_ne (p : fin (n+1)) (i : fin n) : p.succ_above i ≠ p :=
begin
assume eq,
unfold fin.succ_above at eq,
split_ifs at eq with h;
simpa [lt_irrefl, nat.lt_succ_self, eq.symm] using h
end
@[simp] lemma succ_above_descend : ∀(p i : fin (n+1)) (h : i ≠ p), p.succ_above (p.pred_above i h) = i
| ⟨p, hp⟩ ⟨0, hi⟩ h := fin.eq_of_veq $ by simp [succ_above, pred_above]; split_ifs; simp * at *
| ⟨p, hp⟩ ⟨i+1, hi⟩ h := fin.eq_of_veq
begin
have : i + 1 ≠ p, by rwa [(≠), fin.ext_iff] at h,
unfold succ_above pred_above,
split_ifs with h1 h2;
simp only [fin.cast_succ_cast_lt, add_right_inj, pred_val, ne.def, cast_succ_val,
nat.pred_succ, fin.succ_pred, add_right_inj] at *,
exact (this (le_antisymm h2 (le_of_not_gt h1))).elim
end
@[simp] lemma pred_above_succ_above (p : fin (n+1)) (i : fin n) (h : p.succ_above i ≠ p) :
p.pred_above (p.succ_above i) h = i :=
begin
unfold fin.succ_above,
apply eq_of_veq,
split_ifs with h₀,
{ simp [pred_above, h₀, lt_iff_val_lt_val], },
{ unfold pred_above,
split_ifs with h₁,
{ exfalso,
rw [lt_iff_val_lt_val, succ_val] at h₁,
exact h₀ (lt_trans (nat.lt_succ_self _) h₁) },
{ rw [pred_succ] } }
end
section rec
@[elab_as_eliminator] def succ_rec
{C : ∀ n, fin n → Sort*}
(H0 : ∀ n, C (succ n) 0)
(Hs : ∀ n i, C n i → C (succ n) i.succ) : ∀ {n : ℕ} (i : fin n), C n i
| 0 i := i.elim0
| (succ n) ⟨0, _⟩ := H0 _
| (succ n) ⟨succ i, h⟩ := Hs _ _ (succ_rec ⟨i, lt_of_succ_lt_succ h⟩)
@[elab_as_eliminator] def succ_rec_on {n : ℕ} (i : fin n)
{C : ∀ n, fin n → Sort*}
(H0 : ∀ n, C (succ n) 0)
(Hs : ∀ n i, C n i → C (succ n) i.succ) : C n i :=
i.succ_rec H0 Hs
@[simp] theorem succ_rec_on_zero {C : ∀ n, fin n → Sort*} {H0 Hs} (n) :
@fin.succ_rec_on (succ n) 0 C H0 Hs = H0 n :=
rfl
@[simp] theorem succ_rec_on_succ {C : ∀ n, fin n → Sort*} {H0 Hs} {n} (i : fin n) :
@fin.succ_rec_on (succ n) i.succ C H0 Hs = Hs n i (fin.succ_rec_on i H0 Hs) :=
by cases i; refl
@[elab_as_eliminator] def cases
{C : fin (succ n) → Sort*} (H0 : C 0) (Hs : ∀ i : fin n, C (i.succ)) :
∀ (i : fin (succ n)), C i
| ⟨0, h⟩ := H0
| ⟨succ i, h⟩ := Hs ⟨i, lt_of_succ_lt_succ h⟩
@[simp] theorem cases_zero {n} {C : fin (succ n) → Sort*} {H0 Hs} : @fin.cases n C H0 Hs 0 = H0 :=
rfl
@[simp] theorem cases_succ {n} {C : fin (succ n) → Sort*} {H0 Hs} (i : fin n) :
@fin.cases n C H0 Hs i.succ = Hs i :=
by cases i; refl
lemma forall_fin_succ {P : fin (n+1) → Prop} :
(∀ i, P i) ↔ P 0 ∧ (∀ i:fin n, P i.succ) :=
⟨λ H, ⟨H 0, λ i, H _⟩, λ ⟨H0, H1⟩ i, fin.cases H0 H1 i⟩
lemma exists_fin_succ {P : fin (n+1) → Prop} :
(∃ i, P i) ↔ P 0 ∨ (∃i:fin n, P i.succ) :=
⟨λ ⟨i, h⟩, fin.cases or.inl (λ i hi, or.inr ⟨i, hi⟩) i h,
λ h, or.elim h (λ h, ⟨0, h⟩) $ λ⟨i, hi⟩, ⟨i.succ, hi⟩⟩
end rec
section tuple
/- We can think of the type `fin n → α` as `n`-tuples in `α`. Here are some relevant operations. -/
def tail {α} (p : fin (n+1) → α) : fin n → α := λ i, p i.succ
def cons {α} (x : α) (v : fin n → α) : fin (n+1) → α :=
λ j, fin.cases x v j
@[simp] lemma tail_cons {α} (x : α) (p : fin n → α) : tail (cons x p) = p :=
by simp [tail, cons]
@[simp] lemma cons_succ {α} (x : α) (p : fin n → α) (i : fin n) : cons x p i.succ = p i :=
by simp [cons]
@[simp] lemma cons_zero {α} (x : α) (p : fin n → α) : cons x p 0 = x :=
by simp [cons]
end tuple
section find
def find : Π {n : ℕ} (p : fin n → Prop) [decidable_pred p], option (fin n)
| 0 p _ := none
| (n+1) p _ := by resetI; exact option.cases_on
(@find n (λ i, p (i.cast_lt (nat.lt_succ_of_lt i.2))) _)
(if h : p (fin.last n) then some (fin.last n) else none)
(λ i, some (i.cast_lt (nat.lt_succ_of_lt i.2)))
lemma find_spec : Π {n : ℕ} (p : fin n → Prop) [decidable_pred p] {i : fin n}
(hi : i ∈ by exactI fin.find p), p i
| 0 p I i hi := option.no_confusion hi
| (n+1) p I i hi := begin
dsimp [find] at hi,
resetI,
cases h : find (λ i : fin n, (p (i.cast_lt (nat.lt_succ_of_lt i.2)))) with j,
{ rw h at hi,
dsimp at hi,
split_ifs at hi with hl hl,
{ exact option.some_inj.1 hi ▸ hl },
{ exact option.no_confusion hi } },
{ rw h at hi,
rw [← option.some_inj.1 hi],
exact find_spec _ h }
end
lemma is_some_find_iff : Π {n : ℕ} {p : fin n → Prop} [decidable_pred p],
by exactI (find p).is_some ↔ ∃ i, p i
| 0 p _ := iff_of_false (λ h, bool.no_confusion h) (λ ⟨i, _⟩, fin.elim0 i)
| (n+1) p _ := ⟨λ h, begin
resetI,
rw [option.is_some_iff_exists] at h,
cases h with i hi,
exact ⟨i, find_spec _ hi⟩
end, λ ⟨⟨i, hin⟩, hi⟩,
begin
resetI,
dsimp [find],
cases h : find (λ i : fin n, (p (i.cast_lt (nat.lt_succ_of_lt i.2)))) with j,
{ split_ifs with hl hl,
{ exact option.is_some_some },
{ have := (@is_some_find_iff n (λ x, p (x.cast_lt (nat.lt_succ_of_lt x.2))) _).2
⟨⟨i, lt_of_le_of_ne (nat.le_of_lt_succ hin)
(λ h, by clear_aux_decl; subst h; exact hl hi)⟩, hi⟩,
rw h at this,
exact this } },
{ simp }
end⟩
lemma find_eq_none_iff {n : ℕ} {p : fin n → Prop} [decidable_pred p] :
find p = none ↔ ∀ i, ¬ p i :=
by rw [← not_exists, ← is_some_find_iff]; cases (find p); simp
lemma find_min : Π {n : ℕ} {p : fin n → Prop} [decidable_pred p] {i : fin n}
(hi : i ∈ by exactI fin.find p) {j : fin n} (hj : j < i), ¬ p j
| 0 p _ i hi j hj hpj := option.no_confusion hi
| (n+1) p _ i hi ⟨j, hjn⟩ hj hpj := begin
resetI,
dsimp [find] at hi,
cases h : find (λ i : fin n, (p (i.cast_lt (nat.lt_succ_of_lt i.2)))) with k,
{ rw [h] at hi,
split_ifs at hi with hl hl,
{ have := option.some_inj.1 hi,
subst this,
rw [find_eq_none_iff] at h,
exact h ⟨j, hj⟩ hpj },
{ exact option.no_confusion hi } },
{ rw h at hi,
dsimp at hi,
have := option.some_inj.1 hi,
subst this,
exact find_min h (show (⟨j, lt_trans hj k.2⟩ : fin n) < k, from hj) hpj }
end
lemma find_min' {p : fin n → Prop} [decidable_pred p] {i : fin n}
(h : i ∈ fin.find p) {j : fin n} (hj : p j) : i ≤ j :=
le_of_not_gt (λ hij, find_min h hij hj)
lemma nat_find_mem_find {p : fin n → Prop} [decidable_pred p]
(h : ∃ i, ∃ hin : i < n, p ⟨i, hin⟩) :
(⟨nat.find h, (nat.find_spec h).fst⟩ : fin n) ∈ find p :=
let ⟨i, hin, hi⟩ := h in
begin
cases hf : find p with f,
{ rw [find_eq_none_iff] at hf,
exact (hf ⟨i, hin⟩ hi).elim },
{ refine option.some_inj.2 (le_antisymm _ _),
{ exact find_min' hf (nat.find_spec h).snd },
{ exact nat.find_min' _ ⟨f.2, by convert find_spec p hf;
exact fin.eta _ _⟩ } }
end
lemma mem_find_iff {p : fin n → Prop} [decidable_pred p] {i : fin n} :
i ∈ fin.find p ↔ p i ∧ ∀ j, p j → i ≤ j :=
⟨λ hi, ⟨find_spec _ hi, λ _, find_min' hi⟩,
begin
rintros ⟨hpi, hj⟩,
cases hfp : fin.find p,
{ rw [find_eq_none_iff] at hfp,
exact (hfp _ hpi).elim },
{ exact option.some_inj.2 (le_antisymm (find_min' hfp hpi) (hj _ (find_spec _ hfp))) }
end⟩
lemma find_eq_some_iff {p : fin n → Prop} [decidable_pred p] {i : fin n} :
fin.find p = some i ↔ p i ∧ ∀ j, p j → i ≤ j :=
mem_find_iff
lemma mem_find_of_unique {p : fin n → Prop} [decidable_pred p]
(h : ∀ i j, p i → p j → i = j) {i : fin n} (hi : p i) : i ∈ fin.find p :=
mem_find_iff.2 ⟨hi, λ j hj, le_of_eq $ h i j hi hj⟩
end find
end fin
|
38b8630c7a0078b12d535a8a1e51f43ca0314b6b | 86f6f4f8d827a196a32bfc646234b73328aeb306 | /examples/basics/unnamed_1989.lean | bfde4eaf49fc9102e8dad5986e460ceefbda72a9 | [] | no_license | jamescheuk91/mathematics_in_lean | 09f1f87d2b0dce53464ff0cbe592c568ff59cf5e | 4452499264e2975bca2f42565c0925506ba5dda3 | refs/heads/master | 1,679,716,410,967 | 1,613,957,947,000 | 1,613,957,947,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 162 | lean | import algebra.ordered_ring
variables {R : Type*} [ordered_ring R]
variables a b c : R
-- BEGIN
#check (mul_nonneg : 0 ≤ a → 0 ≤ b → 0 ≤ a * b)
-- END |
a769fd1752ed4c42bbedced0999ded24928de69a | b5d813b41740060da28e55b77c69760b7c52760d | /lean_stuff_4.lean | 37d6c77a4bd5090cb7e4e165cf62826225731712 | [] | no_license | ImperialCollegeLondon/SF-solns-zak | 4e85518c5cd4093b995a7593eee8295887e8727e | 7edc4424938e01a6ef4bb518fa8ec1757a2226b1 | refs/heads/master | 1,584,849,829,155 | 1,533,216,269,000 | 1,533,216,269,000 | 139,470,368 | 3 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 5,278 | lean |
namespace z3
inductive List (α : Type)
| emp {} : List
| ap : α → List → List
open List
notation {} := emp
notation x :: y := ap x y
definition append : Π {α:Type}, List α → List α → List α
| α emp L := L
| α (x::L₁) L₂ := x::(append L₁ L₂)
definition repeat : Π {α:Type}, α → ℕ → List α
| α _ 0 := emp
| α x (nat.succ n) := append (x::{}) (repeat x (n))
definition len : Π {α:Type}, List α → ℕ
| α emp := 0
| α (_::L) := len L + 1
--todo : {} for append
theorem list_app_eq : ∀ α : Type, ∀ L₁ L₂ : List α,
len (append L₁ L₂) = len L₁ + len L₂ := begin
intro α,
intros L₁ L₂,
induction L₁ with new L_shorter H,{unfold append, unfold len, simp},
{
unfold append,
unfold len,
rw H,
simp
}end
-------------- products -----------------
inductive prodd (α β : Type)
| pair : α → β → prodd
open prodd
def fst : Π {α β : Type}, prodd α β → α
| _ _ (pair x y) := x
def snd : Π {α β : Type}, prodd α β → β
| _ _ (pair x y) := y
def zip : Π {α β : Type}, List α → List β → List (prodd α β)
| α β {} {} := {}
| α β (A::L₁) (B::L₂) := ap (pair A B) (zip L₁ L₂)
-- if different lengths, give up
| _ _ _ _ := {}
def unzip : Π {α β : Type}, List (prodd α β) → prodd (List α) (List β)
| α β {} := pair {} {}
| α β (x::L) := pair (ap (fst x) (fst (unzip L)) ) (ap (snd x) (snd (unzip L)) )
theorem A : ∀ α β:Type, ∀ L₁ L₂:List α, len L₁ = len L₁ → pair L₁ L₂ = unzip (zip L₁ L₂) := begin
intros α β,
intros L₁ L₂,
intro H,
admit
end
inductive option (α : Type)
| None : option
| Some : α → option
def get_nth {α : Type}: List α → ℕ → option α
| {} _ := option.None α
| (x::_) 0 := option.Some x
| (_::y) (nat.succ n) := get_nth y n
def filter : Π {α : Type}, List α → (α → bool) → List α
| α {} _ := {}
| α (x::L) F := if F x = tt then ap x (filter L F) else filter L F
def nat_even : ℕ → bool
| 0 := tt
| 1 := ff
| (nat.succ (nat.succ n)) := nat_even n
def bool_and : bool → bool → bool
| tt tt := tt
| _ _ := ff
def bool_inv : bool → bool
| tt := ff
| _ := tt
def gt (a b : ℕ) : bool := if a > b then tt else ff
def task (L₁ : List ℕ) : List ℕ := filter L₁ (λ n, bool_and (nat_even n) (gt n 7) )
def inverse_func : Π {α : Type}, (α → bool) → (α → bool) :=
--λ α F, λ b, bool_inv (F b)
λ α F, bool_inv ∘ F
-- f ∘ g = λ b, f (g b)
def partition : Π {α : Type}, (α → bool) → List α → prodd (List α) (List α) :=
λ α F L, pair (filter L F) (filter L (inverse_func F))
def fold {α β : Type} : (α → β → β) → List α → β → β
| _ {} st := st
| F (a::L) st := F a (fold F L st)
def map : Π {α β : Type}, (α → β) → List α → List β
| α β _ {} := {}
| α β F (a::L) := ap (F a) (map F L)
definition fold_map : Π{α β:Type}, (α → β) → List α → List β :=
λ α β F L, fold (λ a b, ap (F a) b) L {}
-- ∀ n : (α → β), ∀ L : List α, fold_map n L = map n L
def reverse : Π {α : Type}, List α → List α
| α {} := {}
| α (ap n L) := append (reverse L) (n::{})
def flatten_list : Π {α : Type}, List (List α) → List α
| α {} := {}
| α (a::L) := append a (flatten_list L)
def flat_map : Π {α β : Type}, (α → List β) → List α → List β := λ α β F L, flatten_list (map F L)
definition fold_length : Π {α : Type}, List α → ℕ :=
λ α L, fold (λ _ n, nat.succ n) L 0
definition prod_curry {X Y Z : Type} (f : prodd X Y → Z) (x : X) (y : Y) : Z :=
f (pair x y)
definition prod_uncurry {X Y Z : Type} : (X → Y → Z) → (prodd X Y) → Z
| f p := f (fst p) (snd p)
#check (λ a b, a+b) 5 6
#check prod_curry (prod_uncurry (λ a b, a+b) (pair 1 2))
#reduce prod_curry (λ pair (a:ℕ) (b:ℕ), a+1) 50 6
@[simp]theorem map_app : Π {α β : Type}, ∀ F : α → β, ∀ L : List α, ∀ a : α,
map F (append L (a::{})) = append (map F L) ((F a)::{}) := begin
intros α β F L a,
induction L with new L_sh H,{refl},
{
unfold map,
unfold append,
rw ←H,
unfold map
}
end
theorem map_rev : Π {α β : Type}, ∀ F : α → β, ∀ L : List α,
map F (reverse L) = reverse (map F L) := begin
intros α β,
intro F,
intro L,
induction L with new L_sh H₁,
{
unfold reverse,
unfold map,
refl
},
{
unfold map reverse,
simp,
rw H₁
--apply map_app F (reverse L_sh) new,
}
end
theorem fold_length_eq : Π {α : Type}, ∀ L : List α, fold_length L = len L := begin
intros α L,
induction L with new L_sh prev, {refl},
{
unfold len,
rw ←prev,
unfold fold_length,
refl
}
end
theorem curry {α β γ : Type} : ∀ F : (α → β → γ),
F = prod_curry (prod_uncurry F) := begin
intro F,
apply funext,
intro a,
apply funext,
intro b,
unfold prod_curry,
unfold prod_uncurry,
refl
end
theorem a : ∀ X n L, len L = n → @get_nth X L n = option.None X :=begin
intros X n L,
revert n,
induction L with new Lh H₂,
{
unfold get_nth,
intros a b,
refl
},
{
unfold len get_nth,
have H₃ := H₂ (len Lh) rfl,
intro n,
intro H₄,
rw ←H₄,
clear H₂,
unfold get_nth,
rw H₃
}
end
end z3
|
2b7542f9d6b81a1d3cbc60118c45df7f9fde59b5 | 88fb7558b0636ec6b181f2a548ac11ad3919f8a5 | /tests/lean/pp_shadowed_const.lean | 2acd96fe6ae72f528f9c70274d6649bbf8280912 | [
"Apache-2.0"
] | permissive | moritayasuaki/lean | 9f666c323cb6fa1f31ac597d777914aed41e3b7a | ae96ebf6ee953088c235ff7ae0e8c95066ba8001 | refs/heads/master | 1,611,135,440,814 | 1,493,852,869,000 | 1,493,852,869,000 | 90,269,903 | 0 | 0 | null | 1,493,906,291,000 | 1,493,906,291,000 | null | UTF-8 | Lean | false | false | 102 | lean | def f : Π (x : nat) (b : bool), bool = bool := λ bool, _
def g (heq : 1 == 1) := heq.symm
#print g
|
1c6047ef4334e960da94be5b240ce68e7110c0cf | 4efff1f47634ff19e2f786deadd394270a59ecd2 | /src/data/fin.lean | 4d0420556ce9ba7aefba3020b1748b9975d087bb | [
"Apache-2.0"
] | permissive | agjftucker/mathlib | d634cd0d5256b6325e3c55bb7fb2403548371707 | 87fe50de17b00af533f72a102d0adefe4a2285e8 | refs/heads/master | 1,625,378,131,941 | 1,599,166,526,000 | 1,599,166,526,000 | 160,748,509 | 0 | 0 | Apache-2.0 | 1,544,141,789,000 | 1,544,141,789,000 | null | UTF-8 | Lean | false | false | 39,333 | lean | /-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import data.nat.cast
import tactic.localized
import logic.embedding
/-!
# The finite type with `n` elements
`fin n` is the type whose elements are natural numbers smaller than `n`.
This file expands on the development in the core library.
## Main definitions
### Induction principles
* `fin_zero_elim` : Elimination principle for the empty set `fin 0`, generalizes `fin.elim0`.
* `fin.succ_rec` : Define `C n i` by induction on `i : fin n` interpreted
as `(0 : fin (n - i)).succ.succ…`. This function has two arguments: `H0 n` defines
`0`-th element `C (n+1) 0` of an `(n+1)`-tuple, and `Hs n i` defines `(i+1)`-st element
of `(n+1)`-tuple based on `n`, `i`, and `i`-th element of `n`-tuple.
* `fin.succ_rec_on` : same as `fin.succ_rec` but `i : fin n` is the first argument;
### Casts
* `cast_lt i h` : embed `i` into a `fin` where `h` proves it belongs into;
* `cast_le h` : embed `fin n` into `fin m`, `h : n ≤ m`;
* `cast eq` : embed `fin n` into `fin m`, `eq : n = m`;
* `cast_add m` : embed `fin n` into `fin (n+m)`;
* `cast_succ` : embed `fin n` into `fin (n+1)`;
* `succ_above p` : embed `fin n` into `fin (n + 1)` with a hole around `p`;
* `pred_above p i h` : embed `i : fin (n+1)` into `fin n` by ignoring `p`;
* `sub_nat i h` : subtract `m` from `i ≥ m`, generalizes `fin.pred`;
* `add_nat i h` : add `m` on `i` on the right, generalizes `fin.succ`;
* `nat_add i h` adds `n` on `i` on the left;
* `clamp n m` : `min n m` as an element of `fin (m + 1)`;
### Operation on tuples
We interpret maps `Π i : fin n, α i` as tuples `(α 0, …, α (n-1))`.
If `α i` is a constant map, then tuples are isomorphic (but not definitionally equal)
to `vector`s.
We define the following operations:
* `tail` : the tail of an `n+1` tuple, i.e., its last `n` entries;
* `cons` : adding an element at the beginning of an `n`-tuple, to get an `n+1`-tuple;
* `init` : the beginning of an `n+1` tuple, i.e., its first `n` entries;
* `snoc` : adding an element at the end of an `n`-tuple, to get an `n+1`-tuple. The name `snoc`
comes from `cons` (i.e., adding an element to the left of a tuple) read in reverse order.
* `find p` : returns the first index `n` where `p n` is satisfied, and `none` if it is never
satisfied.
### Misc definitions
* `fin.last n` : The greatest value of `fin (n+1)`.
-/
universe u
open fin nat function
/-- Elimination principle for the empty set `fin 0`, dependent version. -/
def fin_zero_elim {α : fin 0 → Sort u} (x : fin 0) : α x := x.elim0
lemma fact.succ.pos {n} : fact (0 < succ n) := zero_lt_succ _
lemma fact.bit0.pos {n} [h : fact (0 < n)] : fact (0 < bit0 n) :=
nat.zero_lt_bit0 $ ne_of_gt h
lemma fact.bit1.pos {n} : fact (0 < bit1 n) :=
nat.zero_lt_bit1 _
lemma fact.pow.pos {p n : ℕ} [h : fact $ 0 < p] : fact (0 < p ^ n) :=
pow_pos h _
localized "attribute [instance] fact.succ.pos" in fin_fact
localized "attribute [instance] fact.bit0.pos" in fin_fact
localized "attribute [instance] fact.bit1.pos" in fin_fact
localized "attribute [instance] fact.pow.pos" in fin_fact
namespace fin
variables {n m : ℕ} {a b : fin n}
instance fin_to_nat (n : ℕ) : has_coe (fin n) nat := ⟨subtype.val⟩
lemma is_lt (i : fin n) : (i : ℕ) < n := i.2
/-- convert a `ℕ` to `fin n`, provided `n` is positive -/
def of_nat' [h : fact (0 < n)] (i : ℕ) : fin n := ⟨i%n, mod_lt _ h⟩
@[simp] protected lemma eta (a : fin n) (h : (a : ℕ) < n) : (⟨(a : ℕ), h⟩ : fin n) = a :=
by cases a; refl
@[ext]
lemma ext {a b : fin n} (h : (a : ℕ) = b) : a = b := eq_of_veq h
lemma ext_iff (a b : fin n) : a = b ↔ (a : ℕ) = b :=
iff.intro (congr_arg _) fin.eq_of_veq
lemma coe_injective {n : ℕ} : injective (coe : fin n → ℕ) := subtype.coe_injective
lemma eq_iff_veq (a b : fin n) : a = b ↔ a.1 = b.1 :=
⟨veq_of_eq, eq_of_veq⟩
lemma ne_iff_vne (a b : fin n) : a ≠ b ↔ a.1 ≠ b.1 :=
⟨vne_of_ne, ne_of_vne⟩
@[simp] lemma mk_eq_subtype_mk (a : ℕ) (h : a < n) : mk a h = ⟨a, h⟩ := rfl
protected lemma mk.inj_iff {n a b : ℕ} {ha : a < n} {hb : b < n} :
(⟨a, ha⟩ : fin n) = ⟨b, hb⟩ ↔ a = b :=
⟨subtype.mk.inj, λ h, by subst h⟩
lemma mk_val {m n : ℕ} (h : m < n) : (⟨m, h⟩ : fin n).val = m := rfl
lemma eq_mk_iff_coe_eq {k : ℕ} {hk : k < n} : a = ⟨k, hk⟩ ↔ (a : ℕ) = k :=
fin.eq_iff_veq a ⟨k, hk⟩
@[simp, norm_cast] lemma coe_mk {m n : ℕ} (h : m < n) : ((⟨m, h⟩ : fin n) : ℕ) = m := rfl
lemma mk_coe (i : fin n) : (⟨i, i.is_lt⟩ : fin n) = i :=
fin.eta _ _
lemma coe_eq_val (a : fin n) : (a : ℕ) = a.val := rfl
@[simp] lemma val_eq_coe (a : fin n) : a.val = a := rfl
attribute [simp] val_zero
@[simp] lemma val_one {n : ℕ} : (1 : fin (n+2)).val = 1 := rfl
@[simp] lemma val_two {n : ℕ} : (2 : fin (n+3)).val = 2 := rfl
@[simp] lemma coe_zero {n : ℕ} : ((0 : fin (n+1)) : ℕ) = 0 := rfl
@[simp] lemma coe_one {n : ℕ} : ((1 : fin (n+2)) : ℕ) = 1 := rfl
@[simp] lemma coe_two {n : ℕ} : ((2 : fin (n+3)) : ℕ) = 2 := rfl
/-- `a < b` as natural numbers if and only if `a < b` in `fin n`. -/
@[norm_cast, simp] lemma coe_fin_lt {n : ℕ} {a b : fin n} : (a : ℕ) < (b : ℕ) ↔ a < b :=
iff.rfl
/-- `a ≤ b` as natural numbers if and only if `a ≤ b` in `fin n`. -/
@[norm_cast, simp] lemma coe_fin_le {n : ℕ} {a b : fin n} : (a : ℕ) ≤ (b : ℕ) ↔ a ≤ b :=
iff.rfl
lemma val_add {n : ℕ} : ∀ a b : fin n, (a + b).val = (a.val + b.val) % n
| ⟨_, _⟩ ⟨_, _⟩ := rfl
lemma coe_add {n : ℕ} : ∀ a b : fin n, ((a + b : fin n) : ℕ) = (a + b) % n
| ⟨_, _⟩ ⟨_, _⟩ := rfl
lemma val_mul {n : ℕ} : ∀ a b : fin n, (a * b).val = (a.val * b.val) % n
| ⟨_, _⟩ ⟨_, _⟩ := rfl
lemma coe_mul {n : ℕ} : ∀ a b : fin n, ((a * b : fin n) : ℕ) = (a * b) % n
| ⟨_, _⟩ ⟨_, _⟩ := rfl
lemma one_val {n : ℕ} : (1 : fin (n+1)).val = 1 % (n+1) := rfl
lemma coe_one' {n : ℕ} : ((1 : fin (n+1)) : ℕ) = 1 % (n+1) := rfl
@[simp] lemma val_zero' (n) : (0 : fin (n+1)).val = 0 := rfl
@[simp] lemma mk_zero : (⟨0, nat.succ_pos'⟩ : fin (n + 1)) = (0 : fin _) := rfl
@[simp] lemma mk_one : (⟨1, nat.succ_lt_succ (nat.succ_pos n)⟩ : fin (n + 2)) = (1 : fin _) := rfl
section bit
@[simp] lemma mk_bit0 {m n : ℕ} (h : bit0 m < n) :
(⟨bit0 m, h⟩ : fin n) = (bit0 ⟨m, (nat.le_add_right m m).trans_lt h⟩ : fin _) :=
eq_of_veq (nat.mod_eq_of_lt h).symm
@[simp] lemma mk_bit1 {m n : ℕ} (h : bit1 m < n + 1) :
(⟨bit1 m, h⟩ : fin (n + 1)) = (bit1 ⟨m, (nat.le_add_right m m).trans_lt
((m + m).lt_succ_self.trans h)⟩ : fin _) :=
begin
ext,
simp only [bit1, bit0] at h,
simp only [bit1, bit0, coe_add, coe_one', coe_mk, ←nat.add_mod, nat.mod_eq_of_lt h],
end
end bit
@[simp]
lemma of_nat_eq_coe (n : ℕ) (a : ℕ) : (of_nat a : fin (n+1)) = a :=
begin
induction a with a ih, { refl },
ext, show (a+1) % (n+1) = subtype.val (a+1 : fin (n+1)),
{ rw [val_add, ← ih, of_nat],
exact add_mod _ _ _ }
end
/-- Converting an in-range number to `fin (n + 1)` produces a result
whose value is the original number. -/
lemma coe_val_of_lt {n : ℕ} {a : ℕ} (h : a < n + 1) :
((a : fin (n + 1)).val) = a :=
begin
rw ←of_nat_eq_coe,
exact nat.mod_eq_of_lt h
end
/-- Converting the value of a `fin (n + 1)` to `fin (n + 1)` results
in the same value. -/
lemma coe_val_eq_self {n : ℕ} (a : fin (n + 1)) : (a.val : fin (n + 1)) = a :=
begin
rw fin.eq_iff_veq,
exact coe_val_of_lt a.property
end
/-- Coercing an in-range number to `fin (n + 1)`, and converting back
to `ℕ`, results in that number. -/
lemma coe_coe_of_lt {n : ℕ} {a : ℕ} (h : a < n + 1) :
((a : fin (n + 1)) : ℕ) = a :=
coe_val_of_lt h
/-- Converting a `fin (n + 1)` to `ℕ` and back results in the same
value. -/
@[simp] lemma coe_coe_eq_self {n : ℕ} (a : fin (n + 1)) : ((a : ℕ) : fin (n + 1)) = a :=
coe_val_eq_self a
/-- Assume `k = l`. If two functions defined on `fin k` and `fin l` are equal on each element,
then they coincide (in the heq sense). -/
protected lemma heq_fun_iff {α : Type*} {k l : ℕ} (h : k = l) {f : fin k → α} {g : fin l → α} :
f == g ↔ (∀ (i : fin k), f i = g ⟨(i : ℕ), h ▸ i.2⟩) :=
by { induction h, simp [heq_iff_eq, function.funext_iff] }
protected lemma heq_ext_iff {k l : ℕ} (h : k = l) {i : fin k} {j : fin l} :
i == j ↔ (i : ℕ) = (j : ℕ) :=
by { induction h, simp [ext_iff] }
instance {n : ℕ} : nontrivial (fin (n + 2)) := ⟨⟨0, 1, dec_trivial⟩⟩
instance {n : ℕ} : linear_order (fin n) :=
{ le := (≤), lt := (<), ..linear_order.lift (coe : fin n → ℕ) (@fin.eq_of_veq _) }
instance {n : ℕ} : decidable_linear_order (fin n) :=
{ decidable_le := fin.decidable_le,
decidable_lt := fin.decidable_lt,
decidable_eq := fin.decidable_eq _,
..fin.linear_order }
lemma exists_iff {p : fin n → Prop} : (∃ i, p i) ↔ ∃ i h, p ⟨i, h⟩ :=
⟨λ h, exists.elim h (λ ⟨i, hi⟩ hpi, ⟨i, hi, hpi⟩),
λ h, exists.elim h (λ i hi, ⟨⟨i, hi.fst⟩, hi.snd⟩)⟩
lemma forall_iff {p : fin n → Prop} : (∀ i, p i) ↔ ∀ i h, p ⟨i, h⟩ :=
⟨λ h i hi, h ⟨i, hi⟩, λ h ⟨i, hi⟩, h i hi⟩
lemma lt_iff_coe_lt_coe : a < b ↔ (a : ℕ) < b := iff.rfl
lemma le_iff_coe_le_coe : a ≤ b ↔ (a : ℕ) ≤ b := iff.rfl
lemma zero_le (a : fin (n + 1)) : 0 ≤ a := zero_le a.1
@[simp] lemma coe_succ (j : fin n) : (j.succ : ℕ) = j + 1 :=
by cases j; simp [fin.succ]
lemma succ_pos (a : fin n) : (0 : fin (n + 1)) < a.succ := by simp [lt_iff_coe_lt_coe]
protected theorem succ.inj (p : fin.succ a = fin.succ b) : a = b :=
by cases a; cases b; exact eq_of_veq (nat.succ.inj (veq_of_eq p))
@[simp] lemma succ_inj {a b : fin n} : a.succ = b.succ ↔ a = b :=
⟨λh, succ.inj h, λh, by rw h⟩
@[simp] lemma succ_le_succ_iff : a.succ ≤ b.succ ↔ a ≤ b :=
by { simp only [le_iff_coe_le_coe, coe_succ], exact ⟨le_of_succ_le_succ, succ_le_succ⟩ }
@[simp] lemma succ_lt_succ_iff : a.succ < b.succ ↔ a < b :=
by { simp only [lt_iff_coe_lt_coe, coe_succ], exact ⟨lt_of_succ_lt_succ, succ_lt_succ⟩ }
lemma succ_injective (n : ℕ) : injective (@fin.succ n) :=
λa b, succ.inj
lemma succ_ne_zero {n} : ∀ k : fin n, fin.succ k ≠ 0
| ⟨k, hk⟩ heq := nat.succ_ne_zero k $ (ext_iff _ _).1 heq
@[simp] lemma succ_zero_eq_one : fin.succ (0 : fin (n + 1)) = 1 := rfl
lemma mk_succ_pos (i : ℕ) (h : i < n) : (0 : fin (n + 1)) < ⟨i.succ, add_lt_add_right h 1⟩ :=
by { rw [lt_iff_coe_lt_coe, coe_zero], exact nat.succ_pos i }
lemma one_lt_succ_succ (a : fin n) : (1 : fin (n + 2)) < a.succ.succ :=
by { cases n, { exact fin_zero_elim a }, { rw [←succ_zero_eq_one, succ_lt_succ_iff], exact succ_pos a } }
lemma succ_succ_ne_one (a : fin n) : fin.succ (fin.succ a) ≠ 1 := ne_of_gt (one_lt_succ_succ a)
@[simp] lemma coe_pred (j : fin (n+1)) (h : j ≠ 0) : (j.pred h : ℕ) = j - 1 :=
by { cases j, refl }
@[simp] lemma succ_pred : ∀(i : fin (n+1)) (h : i ≠ 0), (i.pred h).succ = i
| ⟨0, h⟩ hi := by contradiction
| ⟨n + 1, h⟩ hi := rfl
@[simp] lemma pred_succ (i : fin n) {h : i.succ ≠ 0} : i.succ.pred h = i :=
by { cases i, refl }
@[simp] lemma pred_mk_succ (i : ℕ) (h : i < n + 1) :
fin.pred ⟨i + 1, add_lt_add_right h 1⟩ (ne_of_vne (ne_of_gt (mk_succ_pos i h))) = ⟨i, h⟩ :=
by simp only [ext_iff, coe_pred, coe_mk, nat.add_sub_cancel]
@[simp] lemma pred_inj :
∀ {a b : fin (n + 1)} {ha : a ≠ 0} {hb : b ≠ 0}, a.pred ha = b.pred hb ↔ a = b
| ⟨0, _⟩ b ha hb := by contradiction
| ⟨i+1, _⟩ ⟨0, _⟩ ha hb := by contradiction
| ⟨i+1, hi⟩ ⟨j+1, hj⟩ ha hb := by simp [fin.eq_iff_veq]
/-- The greatest value of `fin (n+1)` -/
def last (n : ℕ) : fin (n+1) := ⟨_, n.lt_succ_self⟩
/-- `cast_lt i h` embeds `i` into a `fin` where `h` proves it belongs into. -/
def cast_lt (i : fin m) (h : i.1 < n) : fin n := ⟨i.1, h⟩
/-- `cast_le h i` embeds `i` into a larger `fin` type. -/
def cast_le (h : n ≤ m) (a : fin n) : fin m := cast_lt a (lt_of_lt_of_le a.2 h)
/-- `cast eq i` embeds `i` into a equal `fin` type. -/
def cast (eq : n = m) : fin n → fin m := cast_le $ le_of_eq eq
/-- `cast_add m i` embeds `i : fin n` in `fin (n+m)`. -/
def cast_add (m) : fin n → fin (n + m) := cast_le $ le_add_right n m
/-- `cast_succ i` embeds `i : fin n` in `fin (n+1)`. -/
def cast_succ : fin n → fin (n + 1) := cast_add 1
/-- `succ_above p i` embeds `fin n` into `fin (n + 1)` with a hole around `p`. -/
def succ_above (p : fin (n + 1)) (i : fin n) : fin (n + 1) :=
if i.cast_succ < p then i.cast_succ else i.succ
/-- `pred_above p i h` embeds `i : fin (n+1)` into `fin n` by ignoring `p`. -/
def pred_above (p : fin (n+1)) (i : fin (n+1)) (hi : i ≠ p) : fin n :=
if h : i < p
then i.cast_lt (lt_of_lt_of_le h $ nat.le_of_lt_succ p.2)
else i.pred $
have p < i, from lt_of_le_of_ne (le_of_not_gt h) hi.symm,
ne_of_gt (lt_of_le_of_lt (zero_le p) this)
/-- `sub_nat i h` subtracts `m` from `i`, generalizes `fin.pred`. -/
def sub_nat (m) (i : fin (n + m)) (h : m ≤ (i : ℕ)) : fin n :=
⟨(i : ℕ) - m, by { rw [nat.sub_lt_right_iff_lt_add h], exact i.is_lt }⟩
/-- `add_nat i h` adds `m` on `i`, generalizes `fin.succ`. -/
def add_nat (m) (i : fin n) : fin (n + m) :=
⟨(i : ℕ) + m, add_lt_add_right i.2 _⟩
/-- `nat_add i h` adds `n` on `i` -/
def nat_add (n) {m} (i : fin m) : fin (n + m) :=
⟨n + (i : ℕ), add_lt_add_left i.2 _⟩
theorem le_last (i : fin (n+1)) : i ≤ last n :=
le_of_lt_succ i.is_lt
@[simp] lemma coe_cast (k : fin n) (h : n = m) : (fin.cast h k : ℕ) = k := rfl
@[simp] lemma coe_cast_succ (k : fin n) : (k.cast_succ : ℕ) = k := rfl
@[simp] lemma coe_cast_lt (k : fin m) (h : (k : ℕ) < n) : (k.cast_lt h : ℕ) = k := rfl
@[simp] lemma coe_cast_le (k : fin m) (h : m ≤ n) : (k.cast_le h : ℕ) = k := rfl
@[simp] lemma coe_cast_add (k : fin m) : (k.cast_add n : ℕ) = k := rfl
lemma last_val (n : ℕ) : (last n).val = n := rfl
@[simp, norm_cast] lemma coe_last {n : ℕ} : (last n : ℕ) = n := rfl
@[simp] lemma succ_last (n : ℕ) : (last n).succ = last (n.succ) := rfl
@[simp] lemma cast_succ_cast_lt (i : fin (n + 1)) (h : (i : ℕ) < n) : cast_succ (cast_lt i h) = i :=
fin.eq_of_veq rfl
@[simp] lemma cast_lt_cast_succ {n : ℕ} (a : fin n) (h : (a : ℕ) < n) : cast_lt (cast_succ a) h = a :=
by cases a; refl
@[simp] lemma coe_sub_nat (i : fin (n + m)) (h : m ≤ i) : (i.sub_nat m h : ℕ) = i - m :=
rfl
@[simp] lemma coe_add_nat (i : fin (n + m)) : (i.add_nat m : ℕ) = i + m :=
rfl
@[simp] lemma cast_succ_inj {a b : fin n} : a.cast_succ = b.cast_succ ↔ a = b :=
by simp [eq_iff_veq]
lemma cast_succ_lt_last (a : fin n) : cast_succ a < last n := lt_iff_coe_lt_coe.mpr a.is_lt
@[simp] lemma cast_succ_zero : cast_succ (0 : fin (n + 1)) = 0 := rfl
/-- `cast_succ i` is positive when `i` is positive -/
lemma cast_succ_pos (i : fin (n + 1)) (h : 0 < i) : 0 < cast_succ i :=
by simpa [lt_iff_coe_lt_coe] using h
lemma last_pos : (0 : fin (n + 2)) < last (n + 1) :=
by simp [lt_iff_coe_lt_coe]
lemma coe_nat_eq_last (n) : (n : fin (n + 1)) = fin.last n :=
by { rw [←fin.of_nat_eq_coe, fin.of_nat, fin.last], simp only [nat.mod_eq_of_lt n.lt_succ_self] }
lemma le_coe_last (i : fin (n + 1)) : i ≤ n :=
by { rw fin.coe_nat_eq_last, exact fin.le_last i }
lemma eq_last_of_not_lt {i : fin (n+1)} (h : ¬ (i : ℕ) < n) : i = last n :=
le_antisymm (le_last i) (not_lt.1 h)
lemma add_one_pos (i : fin (n + 1)) (h : i < fin.last n) : (0 : fin (n + 1)) < i + 1 :=
begin
cases n,
{ exact absurd h (nat.not_lt_zero _) },
{ rw [lt_iff_coe_lt_coe, coe_last, ←add_lt_add_iff_right 1] at h,
rw [lt_iff_coe_lt_coe, coe_add, coe_zero, coe_one, nat.mod_eq_of_lt h],
exact nat.zero_lt_succ _ }
end
lemma one_pos : (0 : fin (n + 2)) < 1 := succ_pos 0
lemma zero_ne_one : (0 : fin (n + 2)) ≠ 1 := ne_of_lt one_pos
lemma cast_succ_fin_succ (n : ℕ) (j : fin n) :
cast_succ (fin.succ j) = fin.succ (cast_succ j) :=
by { simp [fin.ext_iff], }
lemma cast_succ_lt_succ (i : fin n) : i.cast_succ < i.succ :=
by { rw [lt_iff_coe_lt_coe, cast_succ, coe_cast_add, coe_succ], exact lt_add_one _ }
@[norm_cast, simp] lemma coe_eq_cast_succ : (a : fin (n + 1)) = a.cast_succ :=
begin
rw [cast_succ, cast_add, cast_le, cast_lt, eq_iff_veq],
exact coe_val_of_lt (nat.lt.step a.is_lt),
end
@[simp] lemma coe_succ_eq_succ : a.cast_succ + 1 = a.succ :=
begin
cases n,
{ exact fin_zero_elim a },
{ simp [a.is_lt, eq_iff_veq, add_def, nat.mod_eq_of_lt] }
end
lemma lt_succ : a.cast_succ < a.succ :=
by { rw [cast_succ, lt_iff_coe_lt_coe, coe_cast_add, coe_succ], exact lt_add_one a.val }
@[simp] lemma pred_one {n : ℕ} : fin.pred (1 : fin (n + 2)) (ne.symm (ne_of_lt one_pos)) = 0 := rfl
lemma pred_add_one (i : fin (n + 2)) (h : (i : ℕ) < n + 1) :
pred (i + 1) (ne_of_gt (add_one_pos _ (lt_iff_coe_lt_coe.mpr h))) = cast_lt i h :=
begin
rw [ext_iff, coe_pred, coe_cast_lt, coe_add, coe_one, mod_eq_of_lt, nat.add_sub_cancel],
exact add_lt_add_right h 1,
end
/-- `min n m` as an element of `fin (m + 1)` -/
def clamp (n m : ℕ) : fin (m + 1) := of_nat $ min n m
@[simp] lemma coe_clamp (n m : ℕ) : (clamp n m : ℕ) = min n m :=
nat.mod_eq_of_lt $ nat.lt_succ_iff.mpr $ min_le_right _ _
lemma cast_le_injective {n₁ n₂ : ℕ} (h : n₁ ≤ n₂) : injective (fin.cast_le h)
| ⟨i₁, h₁⟩ ⟨i₂, h₂⟩ eq := fin.eq_of_veq $ show i₁ = i₂, from fin.veq_of_eq eq
lemma cast_succ_injective (n : ℕ) : injective (@fin.cast_succ n) :=
cast_le_injective (le_add_right n 1)
/-- Embedding `i : fin n` into `fin (n + 1)` with a hole around `p : fin (n + 1)`
embeds `i` by `cast_succ` when the resulting `i.cast_succ < p` -/
lemma succ_above_below (p : fin (n + 1)) (i : fin n) (h : i.cast_succ < p) :
p.succ_above i = i.cast_succ :=
by { rw [succ_above], exact if_pos h }
/-- Embedding `fin n` into `fin (n + 1)` with a hole around zero embeds by `succ` -/
@[simp] lemma succ_above_zero : succ_above (0 : fin (n + 1)) = fin.succ := rfl
/-- Embedding `fin n` into `fin (n + 1)` with a whole around `last n` embeds by `cast_succ` -/
@[simp] lemma succ_above_last : succ_above (fin.last n) = cast_succ :=
by { ext, simp only [succ_above, cast_succ_lt_last, if_true] }
/-- Embedding `i : fin n` into `fin (n + 1)` with a hole around `p : fin (n + 1)`
embeds `i` by `succ` when the resulting `p < i.succ` -/
lemma succ_above_above (p : fin (n + 1)) (i : fin n) (h : p ≤ i.cast_succ) :
p.succ_above i = i.succ :=
by { rw [succ_above], exact if_neg (not_lt_of_le h) }
/-- Embedding `i : fin n` into `fin (n + 1)` is always about some hole `p` -/
lemma succ_above_lt_ge (p : fin (n + 1)) (i : fin n) : i.cast_succ < p ∨ p ≤ i.cast_succ :=
lt_or_ge (cast_succ i) p
/-- Embedding `i : fin n` into `fin (n + 1)` is always about some hole `p` -/
lemma succ_above_lt_gt (p : fin (n + 1)) (i : fin n) : i.cast_succ < p ∨ p < i.succ :=
or.cases_on (succ_above_lt_ge p i)
(λ h, or.inl h) (λ h, or.inr (lt_of_le_of_lt h (cast_succ_lt_succ i)))
/-- Embedding `i : fin n` into `fin (n + 1)` with a hole around `p : fin (n + 1)`
never results in `p` itself -/
theorem succ_above_ne (p : fin (n + 1)) (i : fin n) : p.succ_above i ≠ p :=
begin
intro eq,
by_cases H : i.cast_succ < p,
{ simpa [lt_irrefl, ←succ_above_below _ _ H, eq] using H },
{ simpa [←succ_above_above _ _ (le_of_not_lt H), eq] using cast_succ_lt_succ i }
end
/-- Embedding a positive `fin n` results in a positive fin (n + 1)` -/
lemma succ_above_pos (p : fin (n + 2)) (i : fin (n + 1)) (h : 0 < i) : 0 < p.succ_above i :=
begin
by_cases H : i.cast_succ < p,
{ simpa [succ_above_below _ _ H] using cast_succ_pos _ h },
{ simpa [succ_above_above _ _ (le_of_not_lt H)] using succ_pos _ },
end
/-- Given a fixed pivot `x : fin (n + 1)`, `x.succ_above` is injective -/
lemma succ_above_right_inj {x : fin (n + 1)} :
x.succ_above a = x.succ_above b ↔ a = b :=
begin
refine iff.intro _ (λ h, by rw h),
intro h,
cases succ_above_lt_ge x a with ha ha;
cases succ_above_lt_ge x b with hb hb,
{ simpa only [succ_above_below, ha, hb, cast_succ_inj] using h },
{ simp only [succ_above_below, succ_above_above, ha, hb] at h,
rw h at ha,
exact absurd (lt_of_le_of_lt hb (cast_succ_lt_succ _)) (asymm ha) },
{ simp only [succ_above_below, succ_above_above, ha, hb] at h,
rw ←h at hb,
exact absurd (lt_of_le_of_lt ha (cast_succ_lt_succ _)) (asymm hb) },
{ simpa only [succ_above_above, ha, hb, succ_inj] using h },
end
/-- Given a fixed pivot `x : fin (n + 1)`, `x.succ_above` is injective -/
lemma succ_above_right_injective {x : fin (n + 1)} : injective (succ_above x) :=
λ _ _, succ_above_right_inj.mp
/-- Embedding a `fin (n + 1)` into `fin n` and embedding it back around the same hole
gives the starting `fin (n + 1)` -/
@[simp] lemma succ_above_descend (p i : fin (n + 1)) (h : i ≠ p) :
p.succ_above (p.pred_above i h) = i :=
begin
rw pred_above,
cases lt_or_le i p with H H,
{ simp only [succ_above_below, cast_succ_cast_lt, H, dif_pos]},
{ rw le_iff_coe_le_coe at H,
rw succ_above_above,
{ simp only [le_iff_coe_le_coe, H, not_lt, dif_neg, succ_pred] },
{ simp only [le_iff_coe_le_coe, H, coe_pred, not_lt, dif_neg, coe_cast_succ],
exact le_pred_of_lt (lt_of_le_of_ne H (vne_of_ne h.symm)) } }
end
/-- Embedding a `fin n` into `fin (n + 1)` and embedding it back around the same hole
gives the starting `fin n` -/
@[simp] lemma pred_above_succ_above (p : fin (n + 1)) (i : fin n) :
p.pred_above (p.succ_above i) (succ_above_ne _ _) = i :=
begin
rw pred_above,
by_cases H : i.cast_succ < p,
{ simp [succ_above_below _ _ H, H] },
{ cases succ_above_lt_gt p i with h h,
{ exact absurd h H },
{ simp [succ_above_above _ _ (le_of_not_lt H), pred_succ, dif_neg (asymm h)] } }
end
/-- `succ_above` is injective at the pivot -/
lemma succ_above_left_inj {x y : fin (n + 1)} :
x.succ_above = y.succ_above ↔ x = y :=
begin
refine iff.intro _ (λ h, by rw h),
contrapose!,
intros H h,
have key := congr_fun h (y.pred_above x H),
rw [succ_above_descend] at key,
exact absurd key (succ_above_ne x _)
end
/-- `succ_above` is injective at the pivot -/
lemma succ_above_left_injective : injective (@succ_above n) :=
λ _ _, succ_above_left_inj.mp
/-- A function `f` on `fin n` is strictly monotone if and only if `f i < f (i+1)` for all `i`. -/
lemma strict_mono_iff_lt_succ {α : Type*} [preorder α] {f : fin n → α} :
strict_mono f ↔ ∀ i (h : i + 1 < n), f ⟨i, lt_of_le_of_lt (nat.le_succ i) h⟩ < f ⟨i+1, h⟩ :=
begin
split,
{ assume H i hi,
apply H,
exact nat.lt_succ_self _ },
{ assume H,
have A : ∀ i j (h : i < j) (h' : j < n), f ⟨i, lt_trans h h'⟩ < f ⟨j, h'⟩,
{ assume i j h h',
induction h with k h IH,
{ exact H _ _ },
{ exact lt_trans (IH (nat.lt_of_succ_lt h')) (H _ _) } },
assume i j hij,
convert A (i : ℕ) (j : ℕ) hij j.2; ext; simp only [subtype.coe_eta] }
end
section rec
/-- Define `C n i` by induction on `i : fin n` interpreted as `(0 : fin (n - i)).succ.succ…`.
This function has two arguments: `H0 n` defines `0`-th element `C (n+1) 0` of an `(n+1)`-tuple,
and `Hs n i` defines `(i+1)`-st element of `(n+1)`-tuple based on `n`, `i`, and `i`-th element
of `n`-tuple. -/
@[elab_as_eliminator] def succ_rec
{C : Π n, fin n → Sort*}
(H0 : Π n, C (succ n) 0)
(Hs : Π n i, C n i → C (succ n) i.succ) : Π {n : ℕ} (i : fin n), C n i
| 0 i := i.elim0
| (succ n) ⟨0, _⟩ := H0 _
| (succ n) ⟨succ i, h⟩ := Hs _ _ (succ_rec ⟨i, lt_of_succ_lt_succ h⟩)
/-- Define `C n i` by induction on `i : fin n` interpreted as `(0 : fin (n - i)).succ.succ…`.
This function has two arguments: `H0 n` defines `0`-th element `C (n+1) 0` of an `(n+1)`-tuple,
and `Hs n i` defines `(i+1)`-st element of `(n+1)`-tuple based on `n`, `i`, and `i`-th element
of `n`-tuple.
A version of `fin.succ_rec` taking `i : fin n` as the first argument. -/
@[elab_as_eliminator] def succ_rec_on {n : ℕ} (i : fin n)
{C : Π n, fin n → Sort*}
(H0 : Π n, C (succ n) 0)
(Hs : Π n i, C n i → C (succ n) i.succ) : C n i :=
i.succ_rec H0 Hs
@[simp] theorem succ_rec_on_zero {C : ∀ n, fin n → Sort*} {H0 Hs} (n) :
@fin.succ_rec_on (succ n) 0 C H0 Hs = H0 n :=
rfl
@[simp] theorem succ_rec_on_succ {C : ∀ n, fin n → Sort*} {H0 Hs} {n} (i : fin n) :
@fin.succ_rec_on (succ n) i.succ C H0 Hs = Hs n i (fin.succ_rec_on i H0 Hs) :=
by cases i; refl
/-- Define `f : Π i : fin n.succ, C i` by separately handling the cases `i = 0` and
`i = j.succ`, `j : fin n`. -/
@[elab_as_eliminator] def cases
{C : fin (succ n) → Sort*} (H0 : C 0) (Hs : Π i : fin n, C (i.succ)) :
Π (i : fin (succ n)), C i
| ⟨0, h⟩ := H0
| ⟨succ i, h⟩ := Hs ⟨i, lt_of_succ_lt_succ h⟩
@[simp] theorem cases_zero {n} {C : fin (succ n) → Sort*} {H0 Hs} : @fin.cases n C H0 Hs 0 = H0 :=
rfl
@[simp] theorem cases_succ {n} {C : fin (succ n) → Sort*} {H0 Hs} (i : fin n) :
@fin.cases n C H0 Hs i.succ = Hs i :=
by cases i; refl
lemma forall_fin_succ {P : fin (n+1) → Prop} :
(∀ i, P i) ↔ P 0 ∧ (∀ i:fin n, P i.succ) :=
⟨λ H, ⟨H 0, λ i, H _⟩, λ ⟨H0, H1⟩ i, fin.cases H0 H1 i⟩
lemma exists_fin_succ {P : fin (n+1) → Prop} :
(∃ i, P i) ↔ P 0 ∨ (∃i:fin n, P i.succ) :=
⟨λ ⟨i, h⟩, fin.cases or.inl (λ i hi, or.inr ⟨i, hi⟩) i h,
λ h, or.elim h (λ h, ⟨0, h⟩) $ λ⟨i, hi⟩, ⟨i.succ, hi⟩⟩
end rec
section tuple
/-!
### Tuples
We can think of the type `Π(i : fin n), α i` as `n`-tuples of elements of possibly varying type
`α i`. A particular case is `fin n → α` of elements with all the same type. Here are some relevant
operations, first about adding or removing elements at the beginning of a tuple.
-/
/-- There is exactly one tuple of size zero. -/
instance tuple0_unique (α : fin 0 → Type u) : unique (Π i : fin 0, α i) :=
{ default := fin_zero_elim, uniq := λ x, funext fin_zero_elim }
variables {α : fin (n+1) → Type u} (x : α 0) (q : Πi, α i) (p : Π(i : fin n), α (i.succ))
(i : fin n) (y : α i.succ) (z : α 0)
/-- The tail of an `n+1` tuple, i.e., its last `n` entries -/
def tail (q : Πi, α i) : (Π(i : fin n), α (i.succ)) := λ i, q i.succ
/-- Adding an element at the beginning of an `n`-tuple, to get an `n+1`-tuple -/
def cons (x : α 0) (p : Π(i : fin n), α (i.succ)) : Πi, α i :=
λ j, fin.cases x p j
@[simp] lemma tail_cons : tail (cons x p) = p :=
by simp [tail, cons]
@[simp] lemma cons_succ : cons x p i.succ = p i :=
by simp [cons]
@[simp] lemma cons_zero : cons x p 0 = x :=
by simp [cons]
/-- Updating a tuple and adding an element at the beginning commute. -/
@[simp] lemma cons_update : cons x (update p i y) = update (cons x p) i.succ y :=
begin
ext j,
by_cases h : j = 0,
{ rw h, simp [ne.symm (succ_ne_zero i)] },
{ let j' := pred j h,
have : j'.succ = j := succ_pred j h,
rw [← this, cons_succ],
by_cases h' : j' = i,
{ rw h', simp },
{ have : j'.succ ≠ i.succ, by rwa [ne.def, succ_inj],
rw [update_noteq h', update_noteq this, cons_succ] } }
end
/-- Adding an element at the beginning of a tuple and then updating it amounts to adding it
directly. -/
lemma update_cons_zero : update (cons x p) 0 z = cons z p :=
begin
ext j,
by_cases h : j = 0,
{ rw h, simp },
{ simp only [h, update_noteq, ne.def, not_false_iff],
let j' := pred j h,
have : j'.succ = j := succ_pred j h,
rw [← this, cons_succ, cons_succ] }
end
/-- Concatenating the first element of a tuple with its tail gives back the original tuple -/
@[simp] lemma cons_self_tail : cons (q 0) (tail q) = q :=
begin
ext j,
by_cases h : j = 0,
{ rw h, simp },
{ let j' := pred j h,
have : j'.succ = j := succ_pred j h,
rw [← this, tail, cons_succ] }
end
/-- Updating the first element of a tuple does not change the tail. -/
@[simp] lemma tail_update_zero : tail (update q 0 z) = tail q :=
by { ext j, simp [tail, fin.succ_ne_zero] }
/-- Updating a nonzero element and taking the tail commute. -/
@[simp] lemma tail_update_succ :
tail (update q i.succ y) = update (tail q) i y :=
begin
ext j,
by_cases h : j = i,
{ rw h, simp [tail] },
{ simp [tail, (fin.succ_injective n).ne h, h] }
end
lemma comp_cons {α : Type*} {β : Type*} (g : α → β) (y : α) (q : fin n → α) :
g ∘ (cons y q) = cons (g y) (g ∘ q) :=
begin
ext j,
by_cases h : j = 0,
{ rw h, refl },
{ let j' := pred j h,
have : j'.succ = j := succ_pred j h,
rw [← this, cons_succ, comp_app, cons_succ] }
end
lemma comp_tail {α : Type*} {β : Type*} (g : α → β) (q : fin n.succ → α) :
g ∘ (tail q) = tail (g ∘ q) :=
by { ext j, simp [tail] }
end tuple
section tuple_right
/-! In the previous section, we have discussed inserting or removing elements on the left of a
tuple. In this section, we do the same on the right. A difference is that `fin (n+1)` is constructed
inductively from `fin n` starting from the left, not from the right. This implies that Lean needs
more help to realize that elements belong to the right types, i.e., we need to insert casts at
several places. -/
variables {α : fin (n+1) → Type u} (x : α (last n)) (q : Πi, α i) (p : Π(i : fin n), α i.cast_succ)
(i : fin n) (y : α i.cast_succ) (z : α (last n))
/-- The beginning of an `n+1` tuple, i.e., its first `n` entries -/
def init (q : Πi, α i) (i : fin n) : α i.cast_succ :=
q i.cast_succ
/-- Adding an element at the end of an `n`-tuple, to get an `n+1`-tuple. The name `snoc` comes from
`cons` (i.e., adding an element to the left of a tuple) read in reverse order. -/
def snoc (p : Π(i : fin n), α i.cast_succ) (x : α (last n)) (i : fin (n+1)) : α i :=
if h : i.val < n
then _root_.cast (by rw fin.cast_succ_cast_lt i h) (p (cast_lt i h))
else _root_.cast (by rw eq_last_of_not_lt h) x
@[simp] lemma init_snoc : init (snoc p x) = p :=
begin
ext i,
have h' := fin.cast_lt_cast_succ i i.is_lt,
simp [init, snoc, i.is_lt, h'],
convert cast_eq rfl (p i)
end
@[simp] lemma snoc_cast_succ : snoc p x i.cast_succ = p i :=
begin
have : i.cast_succ.val < n := i.is_lt,
have h' := fin.cast_lt_cast_succ i i.is_lt,
simp [snoc, this, h'],
convert cast_eq rfl (p i)
end
@[simp] lemma snoc_last : snoc p x (last n) = x :=
by { simp [snoc] }
/-- Updating a tuple and adding an element at the end commute. -/
@[simp] lemma snoc_update : snoc (update p i y) x = update (snoc p x) i.cast_succ y :=
begin
ext j,
by_cases h : j.val < n,
{ simp only [snoc, h, dif_pos],
by_cases h' : j = cast_succ i,
{ have C1 : α i.cast_succ = α j, by rw h',
have E1 : update (snoc p x) i.cast_succ y j = _root_.cast C1 y,
{ have : update (snoc p x) j (_root_.cast C1 y) j = _root_.cast C1 y, by simp,
convert this,
{ exact h'.symm },
{ exact heq_of_eq_mp (congr_arg α (eq.symm h')) rfl } },
have C2 : α i.cast_succ = α (cast_succ (cast_lt j h)),
by rw [cast_succ_cast_lt, h'],
have E2 : update p i y (cast_lt j h) = _root_.cast C2 y,
{ have : update p (cast_lt j h) (_root_.cast C2 y) (cast_lt j h) = _root_.cast C2 y,
by simp,
convert this,
{ simp [h, h'] },
{ exact heq_of_eq_mp C2 rfl } },
rw [E1, E2],
exact eq_rec_compose _ _ _ },
{ have : ¬(cast_lt j h = i),
by { assume E, apply h', rw [← E, cast_succ_cast_lt] },
simp [h', this, snoc, h] } },
{ rw eq_last_of_not_lt h,
simp [ne.symm (ne_of_lt (cast_succ_lt_last i))] }
end
/-- Adding an element at the beginning of a tuple and then updating it amounts to adding it
directly. -/
lemma update_snoc_last : update (snoc p x) (last n) z = snoc p z :=
begin
ext j,
by_cases h : j.val < n,
{ have : j ≠ last n := ne_of_lt h,
simp [h, update_noteq, this, snoc] },
{ rw eq_last_of_not_lt h,
simp }
end
/-- Concatenating the first element of a tuple with its tail gives back the original tuple -/
@[simp] lemma snoc_init_self : snoc (init q) (q (last n)) = q :=
begin
ext j,
by_cases h : j.val < n,
{ have : j ≠ last n := ne_of_lt h,
simp [h, update_noteq, this, snoc, init, cast_succ_cast_lt],
have A : cast_succ (cast_lt j h) = j := cast_succ_cast_lt _ _,
rw ← cast_eq rfl (q j),
congr' 1; rw A },
{ rw eq_last_of_not_lt h,
simp }
end
/-- Updating the last element of a tuple does not change the beginning. -/
@[simp] lemma init_update_last : init (update q (last n) z) = init q :=
by { ext j, simp [init, ne_of_lt, cast_succ_lt_last] }
/-- Updating an element and taking the beginning commute. -/
@[simp] lemma init_update_cast_succ :
init (update q i.cast_succ y) = update (init q) i y :=
begin
ext j,
by_cases h : j = i,
{ rw h, simp [init] },
{ simp [init, h] }
end
/-- `tail` and `init` commute. We state this lemma in a non-dependent setting, as otherwise it
would involve a cast to convince Lean that the two types are equal, making it harder to use. -/
lemma tail_init_eq_init_tail {β : Type*} (q : fin (n+2) → β) :
tail (init q) = init (tail q) :=
by { ext i, simp [tail, init, cast_succ_fin_succ] }
/-- `cons` and `snoc` commute. We state this lemma in a non-dependent setting, as otherwise it
would involve a cast to convince Lean that the two types are equal, making it harder to use. -/
lemma cons_snoc_eq_snoc_cons {β : Type*} (a : β) (q : fin n → β) (b : β) :
@cons n.succ (λ i, β) a (snoc q b) = snoc (cons a q) b :=
begin
ext i,
by_cases h : i = 0,
{ rw h, refl },
set j := pred i h with ji,
have : i = j.succ, by rw [ji, succ_pred],
rw [this, cons_succ],
by_cases h' : j.val < n,
{ set k := cast_lt j h' with jk,
have : j = k.cast_succ, by rw [jk, cast_succ_cast_lt],
rw [this, ← cast_succ_fin_succ],
simp },
rw [eq_last_of_not_lt h', succ_last],
simp
end
lemma comp_snoc {α : Type*} {β : Type*} (g : α → β) (q : fin n → α) (y : α) :
g ∘ (snoc q y) = snoc (g ∘ q) (g y) :=
begin
ext j,
by_cases h : j.val < n,
{ have : j ≠ last n := ne_of_lt h,
simp [h, this, snoc, cast_succ_cast_lt] },
{ rw eq_last_of_not_lt h,
simp }
end
lemma comp_init {α : Type*} {β : Type*} (g : α → β) (q : fin n.succ → α) :
g ∘ (init q) = init (g ∘ q) :=
by { ext j, simp [init] }
end tuple_right
section find
/-- `find p` returns the first index `n` where `p n` is satisfied, and `none` if it is never
satisfied. -/
def find : Π {n : ℕ} (p : fin n → Prop) [decidable_pred p], option (fin n)
| 0 p _ := none
| (n+1) p _ := by resetI; exact option.cases_on
(@find n (λ i, p (i.cast_lt (nat.lt_succ_of_lt i.2))) _)
(if h : p (fin.last n) then some (fin.last n) else none)
(λ i, some (i.cast_lt (nat.lt_succ_of_lt i.2)))
/-- If `find p = some i`, then `p i` holds -/
lemma find_spec : Π {n : ℕ} (p : fin n → Prop) [decidable_pred p] {i : fin n}
(hi : i ∈ by exactI fin.find p), p i
| 0 p I i hi := option.no_confusion hi
| (n+1) p I i hi := begin
dsimp [find] at hi,
resetI,
cases h : find (λ i : fin n, (p (i.cast_lt (nat.lt_succ_of_lt i.2)))) with j,
{ rw h at hi,
dsimp at hi,
split_ifs at hi with hl hl,
{ exact option.some_inj.1 hi ▸ hl },
{ exact option.no_confusion hi } },
{ rw h at hi,
rw [← option.some_inj.1 hi],
exact find_spec _ h }
end
/-- `find p` does not return `none` if and only if `p i` holds at some index `i`. -/
lemma is_some_find_iff : Π {n : ℕ} {p : fin n → Prop} [decidable_pred p],
by exactI (find p).is_some ↔ ∃ i, p i
| 0 p _ := iff_of_false (λ h, bool.no_confusion h) (λ ⟨i, _⟩, fin_zero_elim i)
| (n+1) p _ := ⟨λ h, begin
rw [option.is_some_iff_exists] at h,
cases h with i hi,
exactI ⟨i, find_spec _ hi⟩
end, λ ⟨⟨i, hin⟩, hi⟩,
begin
resetI,
dsimp [find],
cases h : find (λ i : fin n, (p (i.cast_lt (nat.lt_succ_of_lt i.2)))) with j,
{ split_ifs with hl hl,
{ exact option.is_some_some },
{ have := (@is_some_find_iff n (λ x, p (x.cast_lt (nat.lt_succ_of_lt x.2))) _).2
⟨⟨i, lt_of_le_of_ne (nat.le_of_lt_succ hin)
(λ h, by clear_aux_decl; cases h; exact hl hi)⟩, hi⟩,
rw h at this,
exact this } },
{ simp }
end⟩
/-- `find p` returns `none` if and only if `p i` never holds. -/
lemma find_eq_none_iff {n : ℕ} {p : fin n → Prop} [decidable_pred p] :
find p = none ↔ ∀ i, ¬ p i :=
by rw [← not_exists, ← is_some_find_iff]; cases (find p); simp
/-- If `find p` returns `some i`, then `p j` does not hold for `j < i`, i.e., `i` is minimal among
the indices where `p` holds. -/
lemma find_min : Π {n : ℕ} {p : fin n → Prop} [decidable_pred p] {i : fin n}
(hi : i ∈ by exactI fin.find p) {j : fin n} (hj : j < i), ¬ p j
| 0 p _ i hi j hj hpj := option.no_confusion hi
| (n+1) p _ i hi ⟨j, hjn⟩ hj hpj := begin
resetI,
dsimp [find] at hi,
cases h : find (λ i : fin n, (p (i.cast_lt (nat.lt_succ_of_lt i.2)))) with k,
{ rw [h] at hi,
split_ifs at hi with hl hl,
{ have := option.some_inj.1 hi,
subst this,
rw [find_eq_none_iff] at h,
exact h ⟨j, hj⟩ hpj },
{ exact option.no_confusion hi } },
{ rw h at hi,
dsimp at hi,
have := option.some_inj.1 hi,
subst this,
exact find_min h (show (⟨j, lt_trans hj k.2⟩ : fin n) < k, from hj) hpj }
end
lemma find_min' {p : fin n → Prop} [decidable_pred p] {i : fin n}
(h : i ∈ fin.find p) {j : fin n} (hj : p j) : i ≤ j :=
le_of_not_gt (λ hij, find_min h hij hj)
lemma nat_find_mem_find {p : fin n → Prop} [decidable_pred p]
(h : ∃ i, ∃ hin : i < n, p ⟨i, hin⟩) :
(⟨nat.find h, (nat.find_spec h).fst⟩ : fin n) ∈ find p :=
let ⟨i, hin, hi⟩ := h in
begin
cases hf : find p with f,
{ rw [find_eq_none_iff] at hf,
exact (hf ⟨i, hin⟩ hi).elim },
{ refine option.some_inj.2 (le_antisymm _ _),
{ exact find_min' hf (nat.find_spec h).snd },
{ exact nat.find_min' _ ⟨f.2, by convert find_spec p hf;
exact fin.eta _ _⟩ } }
end
lemma mem_find_iff {p : fin n → Prop} [decidable_pred p] {i : fin n} :
i ∈ fin.find p ↔ p i ∧ ∀ j, p j → i ≤ j :=
⟨λ hi, ⟨find_spec _ hi, λ _, find_min' hi⟩,
begin
rintros ⟨hpi, hj⟩,
cases hfp : fin.find p,
{ rw [find_eq_none_iff] at hfp,
exact (hfp _ hpi).elim },
{ exact option.some_inj.2 (le_antisymm (find_min' hfp hpi) (hj _ (find_spec _ hfp))) }
end⟩
lemma find_eq_some_iff {p : fin n → Prop} [decidable_pred p] {i : fin n} :
fin.find p = some i ↔ p i ∧ ∀ j, p j → i ≤ j :=
mem_find_iff
lemma mem_find_of_unique {p : fin n → Prop} [decidable_pred p]
(h : ∀ i j, p i → p j → i = j) {i : fin n} (hi : p i) : i ∈ fin.find p :=
mem_find_iff.2 ⟨hi, λ j hj, le_of_eq $ h i j hi hj⟩
end find
@[simp]
lemma coe_of_nat_eq_mod (m n : ℕ) :
((n : fin (succ m)) : ℕ) = n % succ m :=
by rw [← of_nat_eq_coe]; refl
@[simp] lemma coe_of_nat_eq_mod' (m n : ℕ) [I : fact (0 < m)] :
(@fin.of_nat' _ I n : ℕ) = n % m :=
rfl
end fin
|
5387a69ed59c40e92f5915073dc564e35501d9d8 | fa02ed5a3c9c0adee3c26887a16855e7841c668b | /src/set_theory/ordinal.lean | 1f3503a7c2fbfa66928d1cff39c286506cc4eb5a | [
"Apache-2.0"
] | permissive | jjgarzella/mathlib | 96a345378c4e0bf26cf604aed84f90329e4896a2 | 395d8716c3ad03747059d482090e2bb97db612c8 | refs/heads/master | 1,686,480,124,379 | 1,625,163,323,000 | 1,625,163,323,000 | 281,190,421 | 2 | 0 | Apache-2.0 | 1,595,268,170,000 | 1,595,268,169,000 | null | UTF-8 | Lean | false | false | 56,660 | lean | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn
-/
import set_theory.cardinal
/-!
# Ordinals
Ordinals are defined as equivalences of well-ordered sets under order isomorphism. They are endowed
with a total order, where an ordinal is smaller than another one if it embeds into it as an
initial segment (or, equivalently, in any way). This total order is well founded.
## Main definitions
* `initial_seg r s`: type of order embeddings of `r` into `s` for which the range is an initial
segment (i.e., if `b` belongs to the range, then any `b' < b` also belongs to the range).
It is denoted by `r ≼i s`.
* `principal_seg r s`: Type of order embeddings of `r` into `s` for which the range is a principal
segment, i.e., an interval of the form `(-∞, top)` for some element `top`. It is denoted by
`r ≺i s`.
* `ordinal`: the type of ordinals (in a given universe)
* `ordinal.type r`: given a well-founded order `r`, this is the corresponding ordinal
* `ordinal.typein r a`: given a well-founded order `r` on a type `α`, and `a : α`, the ordinal
corresponding to all elements smaller than `a`.
* `enum r o h`: given a well-order `r` on a type `α`, and an ordinal `o` strictly smaller than
the ordinal corresponding to `r` (this is the assumption `h`), returns the `o`-th element of `α`.
In other words, the elements of `α` can be enumerated using ordinals up to `type r`.
* `ordinal.card o`: the cardinality of an ordinal `o`.
* `ordinal.lift` lifts an ordinal in universe `u` to an ordinal in universe `max u v`.
For a version registering additionally that this is an initial segment embedding, see
`ordinal.lift.initial_seg`.
For a version regiserting that it is a principal segment embedding if `u < v`, see
`ordinal.lift.principal_seg`.
* `ordinal.omega` is the first infinite ordinal. It is the order type of `ℕ`.
* `o₁ + o₂` is the order on the disjoint union of `o₁` and `o₂` obtained by declaring that
every element of `o₁` is smaller than every element of `o₂`. The main properties of addition
(and the other operations on ordinals) are stated and proved in `ordinal_arithmetic.lean`. Here,
we only introduce it and prove its basic properties to deduce the fact that the order on ordinals
is total (and well founded).
* `succ o` is the successor of the ordinal `o`.
* `ordinal.min`: the minimal element of a nonempty indexed family of ordinals
* `ordinal.omin` : the minimal element of a nonempty set of ordinals
* `cardinal.ord c`: when `c` is a cardinal, `ord c` is the smallest ordinal with this cardinality.
It is the canonical way to represent a cardinal with an ordinal.
## Notations
* `r ≼i s`: the type of initial segment embeddings of `r` into `s`.
* `r ≺i s`: the type of principal segment embeddings of `r` into `s`.
* `ω` is a notation for the first infinite ordinal in the locale `ordinal`.
-/
noncomputable theory
open function cardinal set equiv
open_locale classical cardinal
universes u v w
variables {α : Type*} {β : Type*} {γ : Type*}
{r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop}
/-!
### Initial segments
Order embeddings whose range is an initial segment of `s` (i.e., if `b` belongs to the range, then
any `b' < b` also belongs to the range). The type of these embeddings from `r` to `s` is called
`initial_seg r s`, and denoted by `r ≼i s`.
-/
/-- If `r` is a relation on `α` and `s` in a relation on `β`, then `f : r ≼i s` is an order
embedding whose range is an initial segment. That is, whenever `b < f a` in `β` then `b` is in the
range of `f`. -/
@[nolint has_inhabited_instance]
structure initial_seg {α β : Type*} (r : α → α → Prop) (s : β → β → Prop) extends r ↪r s :=
(init : ∀ a b, s b (to_rel_embedding a) → ∃ a', to_rel_embedding a' = b)
local infix ` ≼i `:25 := initial_seg
namespace initial_seg
instance : has_coe (r ≼i s) (r ↪r s) := ⟨initial_seg.to_rel_embedding⟩
instance : has_coe_to_fun (r ≼i s) := ⟨λ _, α → β, λ f x, (f : r ↪r s) x⟩
@[simp] theorem coe_fn_mk (f : r ↪r s) (o) :
(@initial_seg.mk _ _ r s f o : α → β) = f := rfl
@[simp] theorem coe_fn_to_rel_embedding (f : r ≼i s) : (f.to_rel_embedding : α → β) = f := rfl
@[simp] theorem coe_coe_fn (f : r ≼i s) : ((f : r ↪r s) : α → β) = f := rfl
theorem init' (f : r ≼i s) {a : α} {b : β} : s b (f a) → ∃ a', f a' = b :=
f.init _ _
theorem init_iff (f : r ≼i s) {a : α} {b : β} : s b (f a) ↔ ∃ a', f a' = b ∧ r a' a :=
⟨λ h, let ⟨a', e⟩ := f.init' h in ⟨a', e, (f : r ↪r s).map_rel_iff.1 (e.symm ▸ h)⟩,
λ ⟨a', e, h⟩, e ▸ (f : r ↪r s).map_rel_iff.2 h⟩
/-- An order isomorphism is an initial segment -/
def of_iso (f : r ≃r s) : r ≼i s :=
⟨f, λ a b h, ⟨f.symm b, rel_iso.apply_symm_apply f _⟩⟩
/-- The identity function shows that `≼i` is reflexive -/
@[refl] protected def refl (r : α → α → Prop) : r ≼i r :=
⟨rel_embedding.refl _, λ a b h, ⟨_, rfl⟩⟩
/-- Composition of functions shows that `≼i` is transitive -/
@[trans] protected def trans (f : r ≼i s) (g : s ≼i t) : r ≼i t :=
⟨f.1.trans g.1, λ a c h, begin
simp at h ⊢,
rcases g.2 _ _ h with ⟨b, rfl⟩, have h := g.1.map_rel_iff.1 h,
rcases f.2 _ _ h with ⟨a', rfl⟩, exact ⟨a', rfl⟩
end⟩
@[simp] theorem refl_apply (x : α) : initial_seg.refl r x = x := rfl
@[simp] theorem trans_apply (f : r ≼i s) (g : s ≼i t) (a : α) : (f.trans g) a = g (f a) := rfl
theorem unique_of_extensional [is_extensional β s] :
well_founded r → subsingleton (r ≼i s) | ⟨h⟩ :=
⟨λ f g, begin
suffices : (f : α → β) = g, { cases f, cases g,
congr, exact rel_embedding.coe_fn_inj this },
funext a, have := h a, induction this with a H IH,
refine @is_extensional.ext _ s _ _ _ (λ x, ⟨λ h, _, λ h, _⟩),
{ rcases f.init_iff.1 h with ⟨y, rfl, h'⟩,
rw IH _ h', exact (g : r ↪r s).map_rel_iff.2 h' },
{ rcases g.init_iff.1 h with ⟨y, rfl, h'⟩,
rw ← IH _ h', exact (f : r ↪r s).map_rel_iff.2 h' }
end⟩
instance [is_well_order β s] : subsingleton (r ≼i s) :=
⟨λ a, @subsingleton.elim _ (unique_of_extensional
(@rel_embedding.well_founded _ _ r s a is_well_order.wf)) a⟩
protected theorem eq [is_well_order β s] (f g : r ≼i s) (a) : f a = g a :=
by rw subsingleton.elim f g
theorem antisymm.aux [is_well_order α r] (f : r ≼i s) (g : s ≼i r) : left_inverse g f :=
initial_seg.eq (f.trans g) (initial_seg.refl _)
/-- If we have order embeddings between `α` and `β` whose images are initial segments, and `β`
is a well-order then `α` and `β` are order-isomorphic. -/
def antisymm [is_well_order β s] (f : r ≼i s) (g : s ≼i r) : r ≃r s :=
by haveI := f.to_rel_embedding.is_well_order; exact
⟨⟨f, g, antisymm.aux f g, antisymm.aux g f⟩, f.map_rel_iff'⟩
@[simp] theorem antisymm_to_fun [is_well_order β s]
(f : r ≼i s) (g : s ≼i r) : (antisymm f g : α → β) = f := rfl
@[simp] theorem antisymm_symm [is_well_order α r] [is_well_order β s]
(f : r ≼i s) (g : s ≼i r) : (antisymm f g).symm = antisymm g f :=
rel_iso.coe_fn_injective rfl
theorem eq_or_principal [is_well_order β s] (f : r ≼i s) :
surjective f ∨ ∃ b, ∀ x, s x b ↔ ∃ y, f y = x :=
or_iff_not_imp_right.2 $ λ h b,
acc.rec_on (is_well_order.wf.apply b : acc s b) $ λ x H IH,
not_forall_not.1 $ λ hn,
h ⟨x, λ y, ⟨(IH _), λ ⟨a, e⟩, by rw ← e; exact
(trichotomous _ _).resolve_right
(not_or (hn a) (λ hl, not_exists.2 hn (f.init' hl)))⟩⟩
/-- Restrict the codomain of an initial segment -/
def cod_restrict (p : set β) (f : r ≼i s) (H : ∀ a, f a ∈ p) : r ≼i subrel s p :=
⟨rel_embedding.cod_restrict p f H, λ a ⟨b, m⟩ (h : s b (f a)),
let ⟨a', e⟩ := f.init' h in ⟨a', by clear _let_match; subst e; refl⟩⟩
@[simp] theorem cod_restrict_apply (p) (f : r ≼i s) (H a) : cod_restrict p f H a = ⟨f a, H a⟩ := rfl
/-- Initial segment embedding of an order `r` into the disjoint union of `r` and `s`. -/
def le_add (r : α → α → Prop) (s : β → β → Prop) : r ≼i sum.lex r s :=
⟨⟨⟨sum.inl, λ _ _, sum.inl.inj⟩, λ a b, sum.lex_inl_inl⟩,
λ a b, by cases b; [exact λ _, ⟨_, rfl⟩, exact false.elim ∘ sum.lex_inr_inl]⟩
@[simp] theorem le_add_apply (r : α → α → Prop) (s : β → β → Prop)
(a) : le_add r s a = sum.inl a := rfl
end initial_seg
/-!
### Principal segments
Order embeddings whose range is a principal segment of `s` (i.e., an interval of the form
`(-∞, top)` for some element `top` of `β`). The type of these embeddings from `r` to `s` is called
`principal_seg r s`, and denoted by `r ≺i s`. Principal segments are in particular initial
segments.
-/
/-- If `r` is a relation on `α` and `s` in a relation on `β`, then `f : r ≺i s` is an order
embedding whose range is an open interval `(-∞, top)` for some element `top` of `β`. Such order
embeddings are called principal segments -/
@[nolint has_inhabited_instance]
structure principal_seg {α β : Type*} (r : α → α → Prop) (s : β → β → Prop) extends r ↪r s :=
(top : β)
(down : ∀ b, s b top ↔ ∃ a, to_rel_embedding a = b)
local infix ` ≺i `:25 := principal_seg
namespace principal_seg
instance : has_coe (r ≺i s) (r ↪r s) := ⟨principal_seg.to_rel_embedding⟩
instance : has_coe_to_fun (r ≺i s) := ⟨λ _, α → β, λ f, f⟩
@[simp] theorem coe_fn_mk (f : r ↪r s) (t o) :
(@principal_seg.mk _ _ r s f t o : α → β) = f := rfl
@[simp] theorem coe_fn_to_rel_embedding (f : r ≺i s) : (f.to_rel_embedding : α → β) = f := rfl
@[simp] theorem coe_coe_fn (f : r ≺i s) : ((f : r ↪r s) : α → β) = f := rfl
theorem down' (f : r ≺i s) {b : β} : s b f.top ↔ ∃ a, f a = b :=
f.down _
theorem lt_top (f : r ≺i s) (a : α) : s (f a) f.top :=
f.down'.2 ⟨_, rfl⟩
theorem init [is_trans β s] (f : r ≺i s) {a : α} {b : β} (h : s b (f a)) : ∃ a', f a' = b :=
f.down'.1 $ trans h $ f.lt_top _
/-- A principal segment is in particular an initial segment. -/
instance has_coe_initial_seg [is_trans β s] : has_coe (r ≺i s) (r ≼i s) :=
⟨λ f, ⟨f.to_rel_embedding, λ a b, f.init⟩⟩
theorem coe_coe_fn' [is_trans β s] (f : r ≺i s) : ((f : r ≼i s) : α → β) = f := rfl
theorem init_iff [is_trans β s] (f : r ≺i s) {a : α} {b : β} :
s b (f a) ↔ ∃ a', f a' = b ∧ r a' a :=
@initial_seg.init_iff α β r s f a b
theorem irrefl (r : α → α → Prop) [is_well_order α r] (f : r ≺i r) : false :=
begin
have := f.lt_top f.top,
rw [show f f.top = f.top, from
initial_seg.eq ↑f (initial_seg.refl r) f.top] at this,
exact irrefl _ this
end
/-- Composition of a principal segment with an initial segment, as a principal segment -/
def lt_le (f : r ≺i s) (g : s ≼i t) : r ≺i t :=
⟨@rel_embedding.trans _ _ _ r s t f g, g f.top, λ a,
by simp only [g.init_iff, f.down', exists_and_distrib_left.symm,
exists_swap, rel_embedding.trans_apply, exists_eq_right']; refl⟩
@[simp] theorem lt_le_apply (f : r ≺i s) (g : s ≼i t) (a : α) : (f.lt_le g) a = g (f a) :=
rel_embedding.trans_apply _ _ _
@[simp] theorem lt_le_top (f : r ≺i s) (g : s ≼i t) : (f.lt_le g).top = g f.top := rfl
/-- Composition of two principal segments as a principal segment -/
@[trans] protected def trans [is_trans γ t] (f : r ≺i s) (g : s ≺i t) : r ≺i t :=
lt_le f g
@[simp] theorem trans_apply [is_trans γ t] (f : r ≺i s) (g : s ≺i t) (a : α) :
(f.trans g) a = g (f a) :=
lt_le_apply _ _ _
@[simp] theorem trans_top [is_trans γ t] (f : r ≺i s) (g : s ≺i t) :
(f.trans g).top = g f.top := rfl
/-- Composition of an order isomorphism with a principal segment, as a principal segment -/
def equiv_lt (f : r ≃r s) (g : s ≺i t) : r ≺i t :=
⟨@rel_embedding.trans _ _ _ r s t f g, g.top, λ c,
suffices (∃ (a : β), g a = c) ↔ ∃ (a : α), g (f a) = c, by simpa [g.down],
⟨λ ⟨b, h⟩, ⟨f.symm b, by simp only [h, rel_iso.apply_symm_apply, rel_iso.coe_coe_fn]⟩,
λ ⟨a, h⟩, ⟨f a, h⟩⟩⟩
/-- Composition of a principal segment with an order isomorphism, as a principal segment -/
def lt_equiv {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop}
(f : principal_seg r s) (g : s ≃r t) : principal_seg r t :=
⟨@rel_embedding.trans _ _ _ r s t f g, g f.top,
begin
intro x,
rw [← g.apply_symm_apply x, g.map_rel_iff, f.down', exists_congr],
intro y, exact ⟨congr_arg g, λ h, g.to_equiv.bijective.1 h⟩
end⟩
@[simp] theorem equiv_lt_apply (f : r ≃r s) (g : s ≺i t) (a : α) : (equiv_lt f g) a = g (f a) :=
rel_embedding.trans_apply _ _ _
@[simp] theorem equiv_lt_top (f : r ≃r s) (g : s ≺i t) : (equiv_lt f g).top = g.top := rfl
/-- Given a well order `s`, there is a most one principal segment embedding of `r` into `s`. -/
instance [is_well_order β s] : subsingleton (r ≺i s) :=
⟨λ f g, begin
have ef : (f : α → β) = g,
{ show ((f : r ≼i s) : α → β) = g,
rw @subsingleton.elim _ _ (f : r ≼i s) g, refl },
have et : f.top = g.top,
{ refine @is_extensional.ext _ s _ _ _ (λ x, _),
simp only [f.down, g.down, ef, coe_fn_to_rel_embedding] },
cases f, cases g,
have := rel_embedding.coe_fn_inj ef; congr'
end⟩
theorem top_eq [is_well_order γ t]
(e : r ≃r s) (f : r ≺i t) (g : s ≺i t) : f.top = g.top :=
by rw subsingleton.elim f (principal_seg.equiv_lt e g); refl
lemma top_lt_top {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop}
[is_well_order γ t]
(f : principal_seg r s) (g : principal_seg s t) (h : principal_seg r t) : t h.top g.top :=
by { rw [subsingleton.elim h (f.trans g)], apply principal_seg.lt_top }
/-- Any element of a well order yields a principal segment -/
def of_element {α : Type*} (r : α → α → Prop) (a : α) : subrel r {b | r b a} ≺i r :=
⟨subrel.rel_embedding _ _, a, λ b,
⟨λ h, ⟨⟨_, h⟩, rfl⟩, λ ⟨⟨_, h⟩, rfl⟩, h⟩⟩
@[simp] theorem of_element_apply {α : Type*} (r : α → α → Prop) (a : α) (b) :
of_element r a b = b.1 := rfl
@[simp] theorem of_element_top {α : Type*} (r : α → α → Prop) (a : α) :
(of_element r a).top = a := rfl
/-- Restrict the codomain of a principal segment -/
def cod_restrict (p : set β) (f : r ≺i s)
(H : ∀ a, f a ∈ p) (H₂ : f.top ∈ p) : r ≺i subrel s p :=
⟨rel_embedding.cod_restrict p f H, ⟨f.top, H₂⟩, λ ⟨b, h⟩,
f.down'.trans $ exists_congr $ λ a,
show (⟨f a, H a⟩ : p).1 = _ ↔ _, from ⟨subtype.eq, congr_arg _⟩⟩
@[simp]
theorem cod_restrict_apply (p) (f : r ≺i s) (H H₂ a) : cod_restrict p f H H₂ a = ⟨f a, H a⟩ := rfl
@[simp]
theorem cod_restrict_top (p) (f : r ≺i s) (H H₂) : (cod_restrict p f H H₂).top = ⟨f.top, H₂⟩ := rfl
end principal_seg
/-! ### Properties of initial and principal segments -/
/-- To an initial segment taking values in a well order, one can associate either a principal
segment (if the range is not everything, hence one can take as top the minimum of the complement
of the range) or an order isomorphism (if the range is everything). -/
def initial_seg.lt_or_eq [is_well_order β s] (f : r ≼i s) :
(r ≺i s) ⊕ (r ≃r s) :=
if h : surjective f then sum.inr (rel_iso.of_surjective f h) else
have h' : _, from (initial_seg.eq_or_principal f).resolve_left h,
sum.inl ⟨f, classical.some h', classical.some_spec h'⟩
theorem initial_seg.lt_or_eq_apply_left [is_well_order β s]
(f : r ≼i s) (g : r ≺i s) (a : α) : g a = f a :=
@initial_seg.eq α β r s _ g f a
theorem initial_seg.lt_or_eq_apply_right [is_well_order β s]
(f : r ≼i s) (g : r ≃r s) (a : α) : g a = f a :=
initial_seg.eq (initial_seg.of_iso g) f a
/-- Composition of an initial segment taking values in a well order and a principal segment. -/
def initial_seg.le_lt [is_well_order β s] [is_trans γ t] (f : r ≼i s) (g : s ≺i t) : r ≺i t :=
match f.lt_or_eq with
| sum.inl f' := f'.trans g
| sum.inr f' := principal_seg.equiv_lt f' g
end
@[simp] theorem initial_seg.le_lt_apply [is_well_order β s] [is_trans γ t]
(f : r ≼i s) (g : s ≺i t) (a : α) : (f.le_lt g) a = g (f a) :=
begin
delta initial_seg.le_lt, cases h : f.lt_or_eq with f' f',
{ simp only [principal_seg.trans_apply, f.lt_or_eq_apply_left] },
{ simp only [principal_seg.equiv_lt_apply, f.lt_or_eq_apply_right] }
end
namespace rel_embedding
/-- Given an order embedding into a well order, collapse the order embedding by filling the
gaps, to obtain an initial segment. Here, we construct the collapsed order embedding pointwise,
but the proof of the fact that it is an initial segment will be given in `collapse`. -/
def collapse_F [is_well_order β s] (f : r ↪r s) : Π a, {b // ¬ s (f a) b} :=
(rel_embedding.well_founded f $ is_well_order.wf).fix $ λ a IH, begin
let S := {b | ∀ a h, s (IH a h).1 b},
have : f a ∈ S, from λ a' h, ((trichotomous _ _)
.resolve_left $ λ h', (IH a' h).2 $ trans (f.map_rel_iff.2 h) h')
.resolve_left $ λ h', (IH a' h).2 $ h' ▸ f.map_rel_iff.2 h,
exact ⟨is_well_order.wf.min S ⟨_, this⟩,
is_well_order.wf.not_lt_min _ _ this⟩
end
theorem collapse_F.lt [is_well_order β s] (f : r ↪r s) {a : α}
: ∀ {a'}, r a' a → s (collapse_F f a').1 (collapse_F f a).1 :=
show (collapse_F f a).1 ∈ {b | ∀ a' (h : r a' a), s (collapse_F f a').1 b}, begin
unfold collapse_F, rw well_founded.fix_eq,
apply well_founded.min_mem _ _
end
theorem collapse_F.not_lt [is_well_order β s] (f : r ↪r s) (a : α)
{b} (h : ∀ a' (h : r a' a), s (collapse_F f a').1 b) : ¬ s b (collapse_F f a).1 :=
begin
unfold collapse_F, rw well_founded.fix_eq,
exact well_founded.not_lt_min _ _ _
(show b ∈ {b | ∀ a' (h : r a' a), s (collapse_F f a').1 b}, from h)
end
/-- Construct an initial segment from an order embedding into a well order, by collapsing it
to fill the gaps. -/
def collapse [is_well_order β s] (f : r ↪r s) : r ≼i s :=
by haveI := rel_embedding.is_well_order f; exact
⟨rel_embedding.of_monotone
(λ a, (collapse_F f a).1) (λ a b, collapse_F.lt f),
λ a b, acc.rec_on (is_well_order.wf.apply b : acc s b) (λ b H IH a h, begin
let S := {a | ¬ s (collapse_F f a).1 b},
have : S.nonempty := ⟨_, asymm h⟩,
existsi (is_well_order.wf : well_founded r).min S this,
refine ((@trichotomous _ s _ _ _).resolve_left _).resolve_right _,
{ exact (is_well_order.wf : well_founded r).min_mem S this },
{ refine collapse_F.not_lt f _ (λ a' h', _),
by_contradiction hn,
exact is_well_order.wf.not_lt_min S this hn h' }
end) a⟩
theorem collapse_apply [is_well_order β s] (f : r ↪r s)
(a) : collapse f a = (collapse_F f a).1 := rfl
end rel_embedding
/-! ### Well order on an arbitrary type -/
section well_ordering_thm
parameter {σ : Type u}
open function
theorem nonempty_embedding_to_cardinal : nonempty (σ ↪ cardinal.{u}) :=
embedding.total.resolve_left $ λ ⟨⟨f, hf⟩⟩,
let g : σ → cardinal.{u} := inv_fun f in
let ⟨x, (hx : g x = 2 ^ sum g)⟩ := inv_fun_surjective hf (2 ^ sum g) in
have g x ≤ sum g, from le_sum.{u u} g x,
not_le_of_gt (by rw hx; exact cantor _) this
/-- An embedding of any type to the set of cardinals. -/
def embedding_to_cardinal : σ ↪ cardinal.{u} := classical.choice nonempty_embedding_to_cardinal
/-- Any type can be endowed with a well order, obtained by pulling back the well order over
cardinals by some embedding. -/
def well_ordering_rel : σ → σ → Prop := embedding_to_cardinal ⁻¹'o (<)
instance well_ordering_rel.is_well_order : is_well_order σ well_ordering_rel :=
(rel_embedding.preimage _ _).is_well_order
end well_ordering_thm
/-! ### Definition of ordinals -/
/-- Bundled structure registering a well order on a type. Ordinals will be defined as a quotient
of this type. -/
structure Well_order : Type (u+1) :=
(α : Type u)
(r : α → α → Prop)
(wo : is_well_order α r)
attribute [instance] Well_order.wo
namespace Well_order
instance : inhabited Well_order := ⟨⟨pempty, _, empty_relation.is_well_order⟩⟩
end Well_order
/-- Equivalence relation on well orders on arbitrary types in universe `u`, given by order
isomorphism. -/
instance ordinal.is_equivalent : setoid Well_order :=
{ r := λ ⟨α, r, wo⟩ ⟨β, s, wo'⟩, nonempty (r ≃r s),
iseqv := ⟨λ⟨α, r, _⟩, ⟨rel_iso.refl _⟩,
λ⟨α, r, _⟩ ⟨β, s, _⟩ ⟨e⟩, ⟨e.symm⟩,
λ⟨α, r, _⟩ ⟨β, s, _⟩ ⟨γ, t, _⟩ ⟨e₁⟩ ⟨e₂⟩, ⟨e₁.trans e₂⟩⟩ }
/-- `ordinal.{u}` is the type of well orders in `Type u`, up to order isomorphism. -/
def ordinal : Type (u + 1) := quotient ordinal.is_equivalent
namespace ordinal
/-- The order type of a well order is an ordinal. -/
def type (r : α → α → Prop) [wo : is_well_order α r] : ordinal :=
⟦⟨α, r, wo⟩⟧
/-- The order type of an element inside a well order. For the embedding as a principal segment, see
`typein.principal_seg`. -/
def typein (r : α → α → Prop) [is_well_order α r] (a : α) : ordinal :=
type (subrel r {b | r b a})
theorem type_def (r : α → α → Prop) [wo : is_well_order α r] :
@eq ordinal ⟦⟨α, r, wo⟩⟧ (type r) := rfl
@[simp] theorem type_def' (r : α → α → Prop) [is_well_order α r] {wo} :
@eq ordinal ⟦⟨α, r, wo⟩⟧ (type r) := rfl
theorem type_eq {α β} {r : α → α → Prop} {s : β → β → Prop}
[is_well_order α r] [is_well_order β s] :
type r = type s ↔ nonempty (r ≃r s) := quotient.eq
@[simp] lemma type_out (o : ordinal) : type o.out.r = o :=
by { refine eq.trans _ (by rw [←quotient.out_eq o]), cases quotient.out o, refl }
@[elab_as_eliminator] theorem induction_on {C : ordinal → Prop}
(o : ordinal) (H : ∀ α r [is_well_order α r], by exactI C (type r)) : C o :=
quot.induction_on o $ λ ⟨α, r, wo⟩, @H α r wo
/-! ### The order on ordinals -/
/-- Ordinal less-equal is defined such that
well orders `r` and `s` satisfy `type r ≤ type s` if there exists
a function embedding `r` as an initial segment of `s`. -/
protected def le (a b : ordinal) : Prop :=
quotient.lift_on₂ a b (λ ⟨α, r, wo⟩ ⟨β, s, wo'⟩, nonempty (r ≼i s)) $
λ ⟨α₁, r₁, o₁⟩ ⟨α₂, r₂, o₂⟩ ⟨β₁, s₁, p₁⟩ ⟨β₂, s₂, p₂⟩ ⟨f⟩ ⟨g⟩,
propext ⟨
λ ⟨h⟩, ⟨(initial_seg.of_iso f.symm).trans $
h.trans (initial_seg.of_iso g)⟩,
λ ⟨h⟩, ⟨(initial_seg.of_iso f).trans $
h.trans (initial_seg.of_iso g.symm)⟩⟩
instance : has_le ordinal := ⟨ordinal.le⟩
theorem type_le {α β} {r : α → α → Prop} {s : β → β → Prop}
[is_well_order α r] [is_well_order β s] :
type r ≤ type s ↔ nonempty (r ≼i s) := iff.rfl
theorem type_le' {α β} {r : α → α → Prop} {s : β → β → Prop}
[is_well_order α r] [is_well_order β s] : type r ≤ type s ↔ nonempty (r ↪r s) :=
⟨λ ⟨f⟩, ⟨f⟩, λ ⟨f⟩, ⟨f.collapse⟩⟩
/-- Ordinal less-than is defined such that
well orders `r` and `s` satisfy `type r < type s` if there exists
a function embedding `r` as a principal segment of `s`. -/
def lt (a b : ordinal) : Prop :=
quotient.lift_on₂ a b (λ ⟨α, r, wo⟩ ⟨β, s, wo'⟩, nonempty (r ≺i s)) $
λ ⟨α₁, r₁, o₁⟩ ⟨α₂, r₂, o₂⟩ ⟨β₁, s₁, p₁⟩ ⟨β₂, s₂, p₂⟩ ⟨f⟩ ⟨g⟩,
by exactI propext ⟨
λ ⟨h⟩, ⟨principal_seg.equiv_lt f.symm $
h.lt_le (initial_seg.of_iso g)⟩,
λ ⟨h⟩, ⟨principal_seg.equiv_lt f $
h.lt_le (initial_seg.of_iso g.symm)⟩⟩
instance : has_lt ordinal := ⟨ordinal.lt⟩
@[simp] theorem type_lt {α β} {r : α → α → Prop} {s : β → β → Prop}
[is_well_order α r] [is_well_order β s] :
type r < type s ↔ nonempty (r ≺i s) := iff.rfl
instance : partial_order ordinal :=
{ le := (≤),
lt := (<),
le_refl := quot.ind $ by exact λ ⟨α, r, wo⟩, ⟨initial_seg.refl _⟩,
le_trans := λ a b c, quotient.induction_on₃ a b c $
λ ⟨α, r, _⟩ ⟨β, s, _⟩ ⟨γ, t, _⟩ ⟨f⟩ ⟨g⟩, ⟨f.trans g⟩,
lt_iff_le_not_le := λ a b, quotient.induction_on₂ a b $
λ ⟨α, r, _⟩ ⟨β, s, _⟩, by exactI
⟨λ ⟨f⟩, ⟨⟨f⟩, λ ⟨g⟩, (f.lt_le g).irrefl _⟩,
λ ⟨⟨f⟩, h⟩, sum.rec_on f.lt_or_eq (λ g, ⟨g⟩)
(λ g, (h ⟨initial_seg.of_iso g.symm⟩).elim)⟩,
le_antisymm := λ x b, show x ≤ b → b ≤ x → x = b, from
quotient.induction_on₂ x b $ λ ⟨α, r, _⟩ ⟨β, s, _⟩ ⟨h₁⟩ ⟨h₂⟩,
by exactI quot.sound ⟨initial_seg.antisymm h₁ h₂⟩ }
/-- Given two ordinals `α ≤ β`, then `initial_seg_out α β` is the initial segment embedding
of `α` to `β`, as map from a model type for `α` to a model type for `β`. -/
def initial_seg_out {α β : ordinal} (h : α ≤ β) : initial_seg α.out.r β.out.r :=
begin
rw [←quotient.out_eq α, ←quotient.out_eq β] at h, revert h,
cases quotient.out α, cases quotient.out β, exact classical.choice
end
/-- Given two ordinals `α < β`, then `principal_seg_out α β` is the principal segment embedding
of `α` to `β`, as map from a model type for `α` to a model type for `β`. -/
def principal_seg_out {α β : ordinal} (h : α < β) : principal_seg α.out.r β.out.r :=
begin
rw [←quotient.out_eq α, ←quotient.out_eq β] at h, revert h,
cases quotient.out α, cases quotient.out β, exact classical.choice
end
/-- Given two ordinals `α = β`, then `rel_iso_out α β` is the order isomorphism between two
model types for `α` and `β`. -/
def rel_iso_out {α β : ordinal} (h : α = β) : α.out.r ≃r β.out.r :=
begin
rw [←quotient.out_eq α, ←quotient.out_eq β] at h, revert h,
cases quotient.out α, cases quotient.out β, exact classical.choice ∘ quotient.exact
end
theorem typein_lt_type (r : α → α → Prop) [is_well_order α r]
(a : α) : typein r a < type r :=
⟨principal_seg.of_element _ _⟩
@[simp] theorem typein_top {α β} {r : α → α → Prop} {s : β → β → Prop}
[is_well_order α r] [is_well_order β s] (f : r ≺i s) :
typein s f.top = type r :=
eq.symm $ quot.sound ⟨rel_iso.of_surjective
(rel_embedding.cod_restrict _ f f.lt_top)
(λ ⟨a, h⟩, by rcases f.down'.1 h with ⟨b, rfl⟩; exact ⟨b, rfl⟩)⟩
@[simp] theorem typein_apply {α β} {r : α → α → Prop} {s : β → β → Prop}
[is_well_order α r] [is_well_order β s] (f : r ≼i s) (a : α) :
ordinal.typein s (f a) = ordinal.typein r a :=
eq.symm $ quotient.sound ⟨rel_iso.of_surjective
(rel_embedding.cod_restrict _
((subrel.rel_embedding _ _).trans f)
(λ ⟨x, h⟩, by rw [rel_embedding.trans_apply]; exact f.to_rel_embedding.map_rel_iff.2 h))
(λ ⟨y, h⟩, by rcases f.init' h with ⟨a, rfl⟩;
exact ⟨⟨a, f.to_rel_embedding.map_rel_iff.1 h⟩, subtype.eq $ rel_embedding.trans_apply _ _ _⟩)⟩
@[simp] theorem typein_lt_typein (r : α → α → Prop) [is_well_order α r]
{a b : α} : typein r a < typein r b ↔ r a b :=
⟨λ ⟨f⟩, begin
have : f.top.1 = a,
{ let f' := principal_seg.of_element r a,
let g' := f.trans (principal_seg.of_element r b),
have : g'.top = f'.top, {rw subsingleton.elim f' g'},
exact this },
rw ← this, exact f.top.2
end, λ h, ⟨principal_seg.cod_restrict _
(principal_seg.of_element r a)
(λ x, @trans _ r _ _ _ _ x.2 h) h⟩⟩
theorem typein_surj (r : α → α → Prop) [is_well_order α r]
{o} (h : o < type r) : ∃ a, typein r a = o :=
induction_on o (λ β s _ ⟨f⟩, by exactI ⟨f.top, typein_top _⟩) h
lemma typein_injective (r : α → α → Prop) [is_well_order α r] : injective (typein r) :=
injective_of_increasing r (<) (typein r) (λ x y, (typein_lt_typein r).2)
theorem typein_inj (r : α → α → Prop) [is_well_order α r]
{a b} : typein r a = typein r b ↔ a = b :=
injective.eq_iff (typein_injective r)
/-! ### Enumerating elements in a well-order with ordinals. -/
/-- `enum r o h` is the `o`-th element of `α` ordered by `r`.
That is, `enum` maps an initial segment of the ordinals, those
less than the order type of `r`, to the elements of `α`. -/
def enum (r : α → α → Prop) [is_well_order α r] (o) : o < type r → α :=
quot.rec_on o (λ ⟨β, s, _⟩ h, (classical.choice h).top) $
λ ⟨β, s, _⟩ ⟨γ, t, _⟩ ⟨h⟩, begin
resetI, refine funext (λ (H₂ : type t < type r), _),
have H₁ : type s < type r, {rwa type_eq.2 ⟨h⟩},
have : ∀ {o e} (H : o < type r), @@eq.rec
(λ (o : ordinal), o < type r → α)
(λ (h : type s < type r), (classical.choice h).top)
e H = (classical.choice H₁).top, {intros, subst e},
exact (this H₂).trans (principal_seg.top_eq h
(classical.choice H₁) (classical.choice H₂))
end
theorem enum_type {α β} {r : α → α → Prop} {s : β → β → Prop}
[is_well_order α r] [is_well_order β s] (f : s ≺i r)
{h : type s < type r} : enum r (type s) h = f.top :=
principal_seg.top_eq (rel_iso.refl _) _ _
@[simp] theorem enum_typein (r : α → α → Prop) [is_well_order α r] (a : α)
{h : typein r a < type r} : enum r (typein r a) h = a :=
enum_type (principal_seg.of_element r a)
@[simp] theorem typein_enum (r : α → α → Prop) [is_well_order α r]
{o} (h : o < type r) : typein r (enum r o h) = o :=
let ⟨a, e⟩ := typein_surj r h in
by clear _let_match; subst e; rw enum_typein
/-- A well order `r` is order isomorphic to the set of ordinals strictly smaller than the
ordinal version of `r`. -/
def typein_iso (r : α → α → Prop) [is_well_order α r] : r ≃r subrel (<) (< type r) :=
⟨⟨λ x, ⟨typein r x, typein_lt_type r x⟩, λ x, enum r x.1 x.2, λ y, enum_typein r y,
λ ⟨y, hy⟩, subtype.eq (typein_enum r hy)⟩,
λ a b, (typein_lt_typein r)⟩
theorem enum_lt {r : α → α → Prop} [is_well_order α r]
{o₁ o₂ : ordinal} (h₁ : o₁ < type r) (h₂ : o₂ < type r) :
r (enum r o₁ h₁) (enum r o₂ h₂) ↔ o₁ < o₂ :=
by rw [← typein_lt_typein r, typein_enum, typein_enum]
lemma rel_iso_enum' {α β : Type u} {r : α → α → Prop} {s : β → β → Prop}
[is_well_order α r] [is_well_order β s]
(f : r ≃r s) (o : ordinal) : ∀(hr : o < type r) (hs : o < type s),
f (enum r o hr) = enum s o hs :=
begin
refine induction_on o _, rintros γ t wo ⟨g⟩ ⟨h⟩,
resetI, rw [enum_type g, enum_type (principal_seg.lt_equiv g f)], refl
end
lemma rel_iso_enum {α β : Type u} {r : α → α → Prop} {s : β → β → Prop}
[is_well_order α r] [is_well_order β s]
(f : r ≃r s) (o : ordinal) (hr : o < type r) :
f (enum r o hr) =
enum s o (by {convert hr using 1, apply quotient.sound, exact ⟨f.symm⟩ }) :=
rel_iso_enum' _ _ _ _
theorem wf : @well_founded ordinal (<) :=
⟨λ a, induction_on a $ λ α r wo, by exactI
suffices ∀ a, acc (<) (typein r a), from
⟨_, λ o h, let ⟨a, e⟩ := typein_surj r h in e ▸ this a⟩,
λ a, acc.rec_on (wo.wf.apply a) $ λ x H IH, ⟨_, λ o h, begin
rcases typein_surj r (lt_trans h (typein_lt_type r _)) with ⟨b, rfl⟩,
exact IH _ ((typein_lt_typein r).1 h)
end⟩⟩
instance : has_well_founded ordinal := ⟨(<), wf⟩
/-- Principal segment version of the `typein` function, embedding a well order into
ordinals as a principal segment. -/
def typein.principal_seg {α : Type u} (r : α → α → Prop) [is_well_order α r] :
@principal_seg α ordinal.{u} r (<) :=
⟨rel_embedding.of_monotone (typein r)
(λ a b, (typein_lt_typein r).2), type r, λ b,
⟨λ h, ⟨enum r _ h, typein_enum r h⟩,
λ ⟨a, e⟩, e ▸ typein_lt_type _ _⟩⟩
@[simp] theorem typein.principal_seg_coe (r : α → α → Prop) [is_well_order α r] :
(typein.principal_seg r : α → ordinal) = typein r := rfl
/-! ### Cardinality of ordinals -/
/-- The cardinal of an ordinal is the cardinal of any
set with that order type. -/
def card (o : ordinal) : cardinal :=
quot.lift_on o (λ ⟨α, r, _⟩, mk α) $
λ ⟨α, r, _⟩ ⟨β, s, _⟩ ⟨e⟩, quotient.sound ⟨e.to_equiv⟩
@[simp] theorem card_type (r : α → α → Prop) [is_well_order α r] :
card (type r) = mk α := rfl
lemma card_typein {r : α → α → Prop} [wo : is_well_order α r] (x : α) :
mk {y // r y x} = (typein r x).card := rfl
theorem card_le_card {o₁ o₂ : ordinal} : o₁ ≤ o₂ → card o₁ ≤ card o₂ :=
induction_on o₁ $ λ α r _, induction_on o₂ $ λ β s _ ⟨⟨⟨f, _⟩, _⟩⟩, ⟨f⟩
instance : has_zero ordinal :=
⟨⟦⟨pempty, empty_relation, by apply_instance⟩⟧⟩
instance : inhabited ordinal := ⟨0⟩
theorem zero_eq_type_empty : 0 = @type empty empty_relation _ :=
quotient.sound ⟨⟨empty_equiv_pempty.symm, λ _ _, iff.rfl⟩⟩
@[simp] theorem card_zero : card 0 = 0 := rfl
protected theorem zero_le (o : ordinal) : 0 ≤ o :=
induction_on o $ λ α r _, ⟨⟨⟨embedding.of_is_empty, is_empty_elim⟩, is_empty_elim⟩⟩
@[simp] protected theorem le_zero {o : ordinal} : o ≤ 0 ↔ o = 0 :=
by simp only [le_antisymm_iff, ordinal.zero_le, and_true]
protected theorem pos_iff_ne_zero {o : ordinal} : 0 < o ↔ o ≠ 0 :=
by simp only [lt_iff_le_and_ne, ordinal.zero_le, true_and, ne.def, eq_comm]
instance : has_one ordinal :=
⟨⟦⟨punit, empty_relation, by apply_instance⟩⟧⟩
theorem one_eq_type_unit : 1 = @type unit empty_relation _ :=
quotient.sound ⟨⟨punit_equiv_punit, λ _ _, iff.rfl⟩⟩
@[simp] theorem card_one : card 1 = 1 := rfl
/-! ### Lifting ordinals to a higher universe -/
/-- The universe lift operation for ordinals, which embeds `ordinal.{u}` as
a proper initial segment of `ordinal.{v}` for `v > u`. For the initial segment version,
see `lift.initial_seg`. -/
def lift (o : ordinal.{u}) : ordinal.{max u v} :=
quotient.lift_on o (λ ⟨α, r, wo⟩,
@type _ _ (@rel_embedding.is_well_order _ _ (@equiv.ulift.{v} α ⁻¹'o r) r
(rel_iso.preimage equiv.ulift.{v} r) wo)) $
λ ⟨α, r, _⟩ ⟨β, s, _⟩ ⟨f⟩,
quot.sound ⟨(rel_iso.preimage equiv.ulift r).trans $
f.trans (rel_iso.preimage equiv.ulift s).symm⟩
theorem lift_type {α} (r : α → α → Prop) [is_well_order α r] :
∃ wo', lift (type r) = @type _ (@equiv.ulift.{v} α ⁻¹'o r) wo' :=
⟨_, rfl⟩
theorem lift_umax : lift.{u (max u v)} = lift.{u v} :=
funext $ λ a, induction_on a $ λ α r _,
quotient.sound ⟨(rel_iso.preimage equiv.ulift r).trans (rel_iso.preimage equiv.ulift r).symm⟩
theorem lift_id' (a : ordinal) : lift a = a :=
induction_on a $ λ α r _,
quotient.sound ⟨rel_iso.preimage equiv.ulift r⟩
@[simp] theorem lift_id : ∀ a, lift.{u u} a = a := lift_id'.{u u}
@[simp]
theorem lift_lift (a : ordinal) : lift.{(max u v) w} (lift.{u v} a) = lift.{u (max v w)} a :=
induction_on a $ λ α r _,
quotient.sound ⟨(rel_iso.preimage equiv.ulift _).trans $
(rel_iso.preimage equiv.ulift _).trans (rel_iso.preimage equiv.ulift _).symm⟩
theorem lift_type_le {α : Type u} {β : Type v} {r s} [is_well_order α r] [is_well_order β s] :
lift.{u (max v w)} (type r) ≤ lift.{v (max u w)} (type s) ↔ nonempty (r ≼i s) :=
⟨λ ⟨f⟩, ⟨(initial_seg.of_iso (rel_iso.preimage equiv.ulift r).symm).trans $
f.trans (initial_seg.of_iso (rel_iso.preimage equiv.ulift s))⟩,
λ ⟨f⟩, ⟨(initial_seg.of_iso (rel_iso.preimage equiv.ulift r)).trans $
f.trans (initial_seg.of_iso (rel_iso.preimage equiv.ulift s).symm)⟩⟩
theorem lift_type_eq {α : Type u} {β : Type v} {r s} [is_well_order α r] [is_well_order β s] :
lift.{u (max v w)} (type r) = lift.{v (max u w)} (type s) ↔ nonempty (r ≃r s) :=
quotient.eq.trans
⟨λ ⟨f⟩, ⟨(rel_iso.preimage equiv.ulift r).symm.trans $
f.trans (rel_iso.preimage equiv.ulift s)⟩,
λ ⟨f⟩, ⟨(rel_iso.preimage equiv.ulift r).trans $
f.trans (rel_iso.preimage equiv.ulift s).symm⟩⟩
theorem lift_type_lt {α : Type u} {β : Type v} {r s} [is_well_order α r] [is_well_order β s] :
lift.{u (max v w)} (type r) < lift.{v (max u w)} (type s) ↔ nonempty (r ≺i s) :=
by haveI := @rel_embedding.is_well_order _ _ (@equiv.ulift.{(max v w)} α ⁻¹'o r)
r (rel_iso.preimage equiv.ulift.{(max v w)} r) _;
haveI := @rel_embedding.is_well_order _ _ (@equiv.ulift.{(max u w)} β ⁻¹'o s)
s (rel_iso.preimage equiv.ulift.{(max u w)} s) _; exact
⟨λ ⟨f⟩, ⟨(f.equiv_lt (rel_iso.preimage equiv.ulift r).symm).lt_le
(initial_seg.of_iso (rel_iso.preimage equiv.ulift s))⟩,
λ ⟨f⟩, ⟨(f.equiv_lt (rel_iso.preimage equiv.ulift r)).lt_le
(initial_seg.of_iso (rel_iso.preimage equiv.ulift s).symm)⟩⟩
@[simp] theorem lift_le {a b : ordinal} : lift.{u v} a ≤ lift b ↔ a ≤ b :=
induction_on a $ λ α r _, induction_on b $ λ β s _,
by rw ← lift_umax; exactI lift_type_le
@[simp] theorem lift_inj {a b : ordinal} : lift a = lift b ↔ a = b :=
by simp only [le_antisymm_iff, lift_le]
@[simp] theorem lift_lt {a b : ordinal} : lift a < lift b ↔ a < b :=
by simp only [lt_iff_le_not_le, lift_le]
@[simp] theorem lift_zero : lift 0 = 0 :=
quotient.sound ⟨(rel_iso.preimage equiv.ulift _).trans
⟨pempty_equiv_pempty, λ a b, iff.rfl⟩⟩
theorem zero_eq_lift_type_empty : 0 = lift.{0 u} (@type empty empty_relation _) :=
by rw [← zero_eq_type_empty, lift_zero]
@[simp] theorem lift_one : lift 1 = 1 :=
quotient.sound ⟨(rel_iso.preimage equiv.ulift _).trans
⟨punit_equiv_punit, λ a b, iff.rfl⟩⟩
theorem one_eq_lift_type_unit : 1 = lift.{0 u} (@type unit empty_relation _) :=
by rw [← one_eq_type_unit, lift_one]
@[simp] theorem lift_card (a) : (card a).lift = card (lift a) :=
induction_on a $ λ α r _, rfl
theorem lift_down' {a : cardinal.{u}} {b : ordinal.{max u v}}
(h : card b ≤ a.lift) : ∃ a', lift a' = b :=
let ⟨c, e⟩ := cardinal.lift_down h in
quotient.induction_on c (λ α, induction_on b $ λ β s _ e', begin
resetI,
rw [mk_def, card_type, ← cardinal.lift_id'.{(max u v) u} (mk β),
← cardinal.lift_umax.{u v}, lift_mk_eq.{u (max u v) (max u v)}] at e',
cases e' with f,
have g := rel_iso.preimage f s,
haveI := (g : ⇑f ⁻¹'o s ↪r s).is_well_order,
have := lift_type_eq.{u (max u v) (max u v)}.2 ⟨g⟩,
rw [lift_id, lift_umax.{u v}] at this,
exact ⟨_, this⟩
end) e
theorem lift_down {a : ordinal.{u}} {b : ordinal.{max u v}}
(h : b ≤ lift a) : ∃ a', lift a' = b :=
@lift_down' (card a) _ (by rw lift_card; exact card_le_card h)
theorem le_lift_iff {a : ordinal.{u}} {b : ordinal.{max u v}} :
b ≤ lift a ↔ ∃ a', lift a' = b ∧ a' ≤ a :=
⟨λ h, let ⟨a', e⟩ := lift_down h in ⟨a', e, lift_le.1 $ e.symm ▸ h⟩,
λ ⟨a', e, h⟩, e ▸ lift_le.2 h⟩
theorem lt_lift_iff {a : ordinal.{u}} {b : ordinal.{max u v}} :
b < lift a ↔ ∃ a', lift a' = b ∧ a' < a :=
⟨λ h, let ⟨a', e⟩ := lift_down (le_of_lt h) in
⟨a', e, lift_lt.1 $ e.symm ▸ h⟩,
λ ⟨a', e, h⟩, e ▸ lift_lt.2 h⟩
/-- Initial segment version of the lift operation on ordinals, embedding `ordinal.{u}` in
`ordinal.{v}` as an initial segment when `u ≤ v`. -/
def lift.initial_seg : @initial_seg ordinal.{u} ordinal.{max u v} (<) (<) :=
⟨⟨⟨lift.{u v}, λ a b, lift_inj.1⟩, λ a b, lift_lt⟩,
λ a b h, lift_down (le_of_lt h)⟩
@[simp] theorem lift.initial_seg_coe : (lift.initial_seg : ordinal → ordinal) = lift := rfl
/-! ### The first infinite ordinal `omega` -/
/-- `ω` is the first infinite ordinal, defined as the order type of `ℕ`. -/
def omega : ordinal.{u} := lift $ @type ℕ (<) _
localized "notation `ω` := ordinal.omega.{0}" in ordinal
theorem card_omega : card omega = cardinal.omega := rfl
@[simp] theorem lift_omega : lift omega = omega := lift_lift _
/-!
### Definition and first properties of addition on ordinals
In this paragraph, we introduce the addition on ordinals, and prove just enough properties to
deduce that the order on ordinals is total (and therefore well-founded). Further properties of
the addition, together with properties of the other operations, are proved in
`ordinal_arithmetic.lean`.
-/
/-- `o₁ + o₂` is the order on the disjoint union of `o₁` and `o₂` obtained by declaring that
every element of `o₁` is smaller than every element of `o₂`. -/
instance : has_add ordinal.{u} :=
⟨λo₁ o₂, quotient.lift_on₂ o₁ o₂
(λ ⟨α, r, wo⟩ ⟨β, s, wo'⟩, ⟦⟨α ⊕ β, sum.lex r s, by exactI sum.lex.is_well_order⟩⟧
: Well_order → Well_order → ordinal) $
λ ⟨α₁, r₁, o₁⟩ ⟨α₂, r₂, o₂⟩ ⟨β₁, s₁, p₁⟩ ⟨β₂, s₂, p₂⟩ ⟨f⟩ ⟨g⟩,
quot.sound ⟨rel_iso.sum_lex_congr f g⟩⟩
@[simp] theorem card_add (o₁ o₂ : ordinal) : card (o₁ + o₂) = card o₁ + card o₂ :=
induction_on o₁ $ λ α r _, induction_on o₂ $ λ β s _, rfl
@[simp] theorem card_nat (n : ℕ) : card.{u} n = n :=
by induction n; [refl, simp only [card_add, card_one, nat.cast_succ, *]]
@[simp] theorem type_add {α β : Type u} (r : α → α → Prop) (s : β → β → Prop)
[is_well_order α r] [is_well_order β s] : type r + type s = type (sum.lex r s) := rfl
/-- The ordinal successor is the smallest ordinal larger than `o`.
It is defined as `o + 1`. -/
def succ (o : ordinal) : ordinal := o + 1
theorem succ_eq_add_one (o) : succ o = o + 1 := rfl
instance : add_monoid ordinal.{u} :=
{ add := (+),
zero := 0,
zero_add := λ o, induction_on o $ λ α r _, eq.symm $ quotient.sound
⟨⟨(empty_sum pempty α).symm, λ a b, sum.lex_inr_inr⟩⟩,
add_zero := λ o, induction_on o $ λ α r _, eq.symm $ quotient.sound
⟨⟨(sum_empty α pempty).symm, λ a b, sum.lex_inl_inl⟩⟩,
add_assoc := λ o₁ o₂ o₃, quotient.induction_on₃ o₁ o₂ o₃ $
λ ⟨α, r, _⟩ ⟨β, s, _⟩ ⟨γ, t, _⟩, quot.sound
⟨⟨sum_assoc _ _ _, λ a b,
begin rcases a with ⟨a|a⟩|a; rcases b with ⟨b|b⟩|b;
simp only [sum_assoc_apply_in1, sum_assoc_apply_in2, sum_assoc_apply_in3,
sum.lex_inl_inl, sum.lex_inr_inr, sum.lex.sep, sum.lex_inr_inl] end⟩⟩ }
theorem add_le_add_left {a b : ordinal} : a ≤ b → ∀ c, c + a ≤ c + b :=
induction_on a $ λ α₁ r₁ _, induction_on b $ λ α₂ r₂ _ ⟨⟨⟨f, fo⟩, fi⟩⟩ c,
induction_on c $ λ β s _,
⟨⟨⟨(embedding.refl _).sum_map f,
λ a b, match a, b with
| sum.inl a, sum.inl b := sum.lex_inl_inl.trans sum.lex_inl_inl.symm
| sum.inl a, sum.inr b := by apply iff_of_true; apply sum.lex.sep
| sum.inr a, sum.inl b := by apply iff_of_false; exact sum.lex_inr_inl
| sum.inr a, sum.inr b := sum.lex_inr_inr.trans $ fo.trans sum.lex_inr_inr.symm
end⟩,
λ a b H, match a, b, H with
| _, sum.inl b, _ := ⟨sum.inl b, rfl⟩
| sum.inl a, sum.inr b, H := (sum.lex_inr_inl H).elim
| sum.inr a, sum.inr b, H := let ⟨w, h⟩ := fi _ _ (sum.lex_inr_inr.1 H) in
⟨sum.inr w, congr_arg sum.inr h⟩
end⟩⟩
theorem le_add_right (a b : ordinal) : a ≤ a + b :=
by simpa only [add_zero] using add_le_add_left (ordinal.zero_le b) a
theorem add_le_add_right {a b : ordinal} : a ≤ b → ∀ c, a + c ≤ b + c :=
induction_on a $ λ α₁ r₁ hr₁, induction_on b $ λ α₂ r₂ hr₂ ⟨⟨⟨f, fo⟩, fi⟩⟩ c,
induction_on c $ λ β s hs, (@type_le' _ _ _ _
(@sum.lex.is_well_order _ _ _ _ hr₁ hs)
(@sum.lex.is_well_order _ _ _ _ hr₂ hs)).2
⟨⟨f.sum_map (embedding.refl _), λ a b, begin
split; intro H,
{ cases a with a a; cases b with b b; cases H; constructor; [rwa ← fo, assumption] },
{ cases H; constructor; [rwa fo, assumption] }
end⟩⟩
theorem le_add_left (a b : ordinal) : a ≤ b + a :=
by simpa only [zero_add] using add_le_add_right (ordinal.zero_le b) a
theorem lt_succ_self (o : ordinal.{u}) : o < succ o :=
induction_on o $ λ α r _, ⟨⟨⟨⟨λ x, sum.inl x, λ _ _, sum.inl.inj⟩,
λ _ _, sum.lex_inl_inl⟩,
sum.inr punit.star, λ b, sum.rec_on b
(λ x, ⟨λ _, ⟨x, rfl⟩, λ _, sum.lex.sep _ _⟩)
(λ x, sum.lex_inr_inr.trans ⟨false.elim, λ ⟨x, H⟩, sum.inl_ne_inr H⟩)⟩⟩
theorem succ_le {a b : ordinal} : succ a ≤ b ↔ a < b :=
⟨lt_of_lt_of_le (lt_succ_self _),
induction_on a $ λ α r hr, induction_on b $ λ β s hs ⟨⟨f, t, hf⟩⟩, begin
refine ⟨⟨@rel_embedding.of_monotone (α ⊕ punit) β _ _
(@sum.lex.is_well_order _ _ _ _ hr _).1.1
(@is_asymm_of_is_trans_of_is_irrefl _ _ hs.1.2.2 hs.1.2.1)
(sum.rec _ _) (λ a b, _), λ a b, _⟩⟩,
{ exact f }, { exact λ _, t },
{ rcases a with a|_; rcases b with b|_,
{ simpa only [sum.lex_inl_inl] using f.map_rel_iff.2 },
{ intro _, rw hf, exact ⟨_, rfl⟩ },
{ exact false.elim ∘ sum.lex_inr_inl },
{ exact false.elim ∘ sum.lex_inr_inr.1 } },
{ rcases a with a|_,
{ intro h, have := @principal_seg.init _ _ _ _ hs.1.2.2 ⟨f, t, hf⟩ _ _ h,
cases this with w h, exact ⟨sum.inl w, h⟩ },
{ intro h, cases (hf b).1 h with w h, exact ⟨sum.inl w, h⟩ } }
end⟩
theorem le_total (a b : ordinal) : a ≤ b ∨ b ≤ a :=
match lt_or_eq_of_le (le_add_left b a), lt_or_eq_of_le (le_add_right a b) with
| or.inr h, _ := by rw h; exact or.inl (le_add_right _ _)
| _, or.inr h := by rw h; exact or.inr (le_add_left _ _)
| or.inl h₁, or.inl h₂ := induction_on a (λ α₁ r₁ _,
induction_on b $ λ α₂ r₂ _ ⟨f⟩ ⟨g⟩, begin
resetI,
rw [← typein_top f, ← typein_top g, le_iff_lt_or_eq,
le_iff_lt_or_eq, typein_lt_typein, typein_lt_typein],
rcases trichotomous_of (sum.lex r₁ r₂) g.top f.top with h|h|h;
[exact or.inl (or.inl h), {left, right, rw h}, exact or.inr (or.inl h)]
end) h₁ h₂
end
instance : linear_order ordinal :=
{ le_total := le_total,
decidable_le := classical.dec_rel _,
..ordinal.partial_order }
instance : is_well_order ordinal (<) := ⟨wf⟩
@[simp] lemma typein_le_typein (r : α → α → Prop) [is_well_order α r] {x x' : α} :
typein r x ≤ typein r x' ↔ ¬r x' x :=
by rw [←not_lt, typein_lt_typein]
lemma enum_le_enum (r : α → α → Prop) [is_well_order α r] {o o' : ordinal}
(ho : o < type r) (ho' : o' < type r) : ¬r (enum r o' ho') (enum r o ho) ↔ o ≤ o' :=
by rw [←@not_lt _ _ o' o, enum_lt ho']
/-- `univ.{u v}` is the order type of the ordinals of `Type u` as a member
of `ordinal.{v}` (when `u < v`). It is an inaccessible cardinal. -/
def univ := lift.{(u+1) v} (@type ordinal.{u} (<) _)
theorem univ_id : univ.{u (u+1)} = @type ordinal.{u} (<) _ := lift_id _
@[simp] theorem lift_univ : lift.{_ w} univ.{u v} = univ.{u (max v w)} := lift_lift _
theorem univ_umax : univ.{u (max (u+1) v)} = univ.{u v} := congr_fun lift_umax _
/-- Principal segment version of the lift operation on ordinals, embedding `ordinal.{u}` in
`ordinal.{v}` as a principal segment when `u < v`. -/
def lift.principal_seg : @principal_seg ordinal.{u} ordinal.{max (u+1) v} (<) (<) :=
⟨↑lift.initial_seg.{u (max (u+1) v)}, univ.{u v}, begin
refine λ b, induction_on b _, introsI β s _,
rw [univ, ← lift_umax], split; intro h,
{ rw ← lift_id (type s) at h ⊢,
cases lift_type_lt.1 h with f, cases f with f a hf,
existsi a, revert hf,
apply induction_on a, introsI α r _ hf,
refine lift_type_eq.{u (max (u+1) v) (max (u+1) v)}.2
⟨(rel_iso.of_surjective (rel_embedding.of_monotone _ _) _).symm⟩,
{ exact λ b, enum r (f b) ((hf _).2 ⟨_, rfl⟩) },
{ refine λ a b h, (typein_lt_typein r).1 _,
rw [typein_enum, typein_enum],
exact f.map_rel_iff.2 h },
{ intro a', cases (hf _).1 (typein_lt_type _ a') with b e,
existsi b, simp, simp [e] } },
{ cases h with a e, rw [← e],
apply induction_on a, introsI α r _,
exact lift_type_lt.{u (u+1) (max (u+1) v)}.2
⟨typein.principal_seg r⟩ }
end⟩
@[simp] theorem lift.principal_seg_coe :
(lift.principal_seg.{u v} : ordinal → ordinal) = lift.{u (max (u+1) v)} := rfl
@[simp] theorem lift.principal_seg_top : lift.principal_seg.top = univ := rfl
theorem lift.principal_seg_top' :
lift.principal_seg.{u (u+1)}.top = @type ordinal.{u} (<) _ :=
by simp only [lift.principal_seg_top, univ_id]
/-! ### Minimum -/
/-- The minimal element of a nonempty family of ordinals -/
def min {ι} (I : nonempty ι) (f : ι → ordinal) : ordinal :=
wf.min (set.range f) (let ⟨i⟩ := I in ⟨_, set.mem_range_self i⟩)
theorem min_eq {ι} (I) (f : ι → ordinal) : ∃ i, min I f = f i :=
let ⟨i, e⟩ := wf.min_mem (set.range f) _ in ⟨i, e.symm⟩
theorem min_le {ι I} (f : ι → ordinal) (i) : min I f ≤ f i :=
le_of_not_gt $ wf.not_lt_min (set.range f) _ (set.mem_range_self i)
theorem le_min {ι I} {f : ι → ordinal} {a} : a ≤ min I f ↔ ∀ i, a ≤ f i :=
⟨λ h i, le_trans h (min_le _ _),
λ h, let ⟨i, e⟩ := min_eq I f in e.symm ▸ h i⟩
/-- The minimal element of a nonempty set of ordinals -/
def omin (S : set ordinal.{u}) (H : ∃ x, x ∈ S) : ordinal.{u} :=
@min.{(u+2) u} S (let ⟨x, px⟩ := H in ⟨⟨x, px⟩⟩) subtype.val
theorem omin_mem (S H) : omin S H ∈ S :=
let ⟨⟨i, h⟩, e⟩ := @min_eq S _ _ in
(show omin S H = i, from e).symm ▸ h
theorem le_omin {S H a} : a ≤ omin S H ↔ ∀ i ∈ S, a ≤ i :=
le_min.trans set_coe.forall
theorem omin_le {S H i} (h : i ∈ S) : omin S H ≤ i :=
le_omin.1 (le_refl _) _ h
@[simp] theorem lift_min {ι} (I) (f : ι → ordinal) : lift (min I f) = min I (lift ∘ f) :=
le_antisymm (le_min.2 $ λ a, lift_le.2 $ min_le _ a) $
let ⟨i, e⟩ := min_eq I (lift ∘ f) in
by rw e; exact lift_le.2 (le_min.2 $ λ j, lift_le.1 $
by have := min_le (lift ∘ f) j; rwa e at this)
end ordinal
/-! ### Representing a cardinal with an ordinal -/
namespace cardinal
open ordinal
/-- The ordinal corresponding to a cardinal `c` is the least ordinal
whose cardinal is `c`. For the order-embedding version, see `ord.order_embedding`. -/
def ord (c : cardinal) : ordinal :=
begin
let ι := λ α, {r // is_well_order α r},
have : Π α, ι α := λ α, ⟨well_ordering_rel, by apply_instance⟩,
let F := λ α, ordinal.min ⟨this _⟩ (λ i:ι α, ⟦⟨α, i.1, i.2⟩⟧),
refine quot.lift_on c F _,
suffices : ∀ {α β}, α ≈ β → F α ≤ F β,
from λ α β h, le_antisymm (this h) (this (setoid.symm h)),
intros α β h, cases h with f, refine ordinal.le_min.2 (λ i, _),
haveI := @rel_embedding.is_well_order _ _
(f ⁻¹'o i.1) _ ↑(rel_iso.preimage f i.1) i.2,
rw ← show type (f ⁻¹'o i.1) = ⟦⟨β, i.1, i.2⟩⟧, from
quot.sound ⟨rel_iso.preimage f i.1⟩,
exact ordinal.min_le (λ i:ι α, ⟦⟨α, i.1, i.2⟩⟧) ⟨_, _⟩
end
lemma ord_eq_min (α : Type u) : ord (mk α) =
@ordinal.min {r // is_well_order α r} ⟨⟨well_ordering_rel, by apply_instance⟩⟩
(λ i, ⟦⟨α, i.1, i.2⟩⟧) := rfl
theorem ord_eq (α) : ∃ (r : α → α → Prop) [wo : is_well_order α r],
ord (mk α) = @type α r wo :=
let ⟨⟨r, wo⟩, h⟩ := @ordinal.min_eq {r // is_well_order α r}
⟨⟨well_ordering_rel, by apply_instance⟩⟩
(λ i:{r // is_well_order α r}, ⟦⟨α, i.1, i.2⟩⟧) in
⟨r, wo, h⟩
theorem ord_le_type (r : α → α → Prop) [is_well_order α r] : ord (mk α) ≤ ordinal.type r :=
@ordinal.min_le {r // is_well_order α r}
⟨⟨well_ordering_rel, by apply_instance⟩⟩
(λ i:{r // is_well_order α r}, ⟦⟨α, i.1, i.2⟩⟧) ⟨r, _⟩
theorem ord_le {c o} : ord c ≤ o ↔ c ≤ o.card :=
quotient.induction_on c $ λ α, induction_on o $ λ β s _,
let ⟨r, _, e⟩ := ord_eq α in begin
resetI, simp only [mk_def, card_type], split; intro h,
{ rw e at h, exact let ⟨f⟩ := h in ⟨f.to_embedding⟩ },
{ cases h with f,
have g := rel_embedding.preimage f s,
haveI := rel_embedding.is_well_order g,
exact le_trans (ord_le_type _) (type_le'.2 ⟨g⟩) }
end
theorem lt_ord {c o} : o < ord c ↔ o.card < c :=
by rw [← not_le, ← not_le, ord_le]
@[simp] theorem card_ord (c) : (ord c).card = c :=
quotient.induction_on c $ λ α,
let ⟨r, _, e⟩ := ord_eq α in by simp only [mk_def, e, card_type]
theorem ord_card_le (o : ordinal) : o.card.ord ≤ o :=
ord_le.2 (le_refl _)
lemma lt_ord_succ_card (o : ordinal) : o < o.card.succ.ord :=
by { rw [lt_ord], apply cardinal.lt_succ_self }
@[simp] theorem ord_le_ord {c₁ c₂} : ord c₁ ≤ ord c₂ ↔ c₁ ≤ c₂ :=
by simp only [ord_le, card_ord]
@[simp] theorem ord_lt_ord {c₁ c₂} : ord c₁ < ord c₂ ↔ c₁ < c₂ :=
by simp only [lt_ord, card_ord]
@[simp] theorem ord_zero : ord 0 = 0 :=
le_antisymm (ord_le.2 $ zero_le _) (ordinal.zero_le _)
@[simp] theorem ord_nat (n : ℕ) : ord n = n :=
le_antisymm (ord_le.2 $ by simp only [card_nat]) $ begin
induction n with n IH,
{ apply ordinal.zero_le },
{ exact (@ordinal.succ_le n _).2 (lt_of_le_of_lt IH $
ord_lt_ord.2 $ nat_cast_lt.2 (nat.lt_succ_self n)) }
end
@[simp] theorem lift_ord (c) : (ord c).lift = ord (lift c) :=
eq_of_forall_ge_iff $ λ o, le_iff_le_iff_lt_iff_lt.2 $ begin
split; intro h,
{ rcases ordinal.lt_lift_iff.1 h with ⟨a, e, h⟩,
rwa [← e, lt_ord, ← lift_card, lift_lt, ← lt_ord] },
{ rw lt_ord at h,
rcases lift_down' (le_of_lt h) with ⟨o, rfl⟩,
rw [← lift_card, lift_lt] at h,
rwa [ordinal.lift_lt, lt_ord] }
end
lemma mk_ord_out (c : cardinal) : mk c.ord.out.α = c :=
by rw [←card_type c.ord.out.r, type_out, card_ord]
lemma card_typein_lt (r : α → α → Prop) [is_well_order α r] (x : α)
(h : ord (mk α) = type r) : card (typein r x) < mk α :=
by { rw [←ord_lt_ord, h], refine lt_of_le_of_lt (ord_card_le _) (typein_lt_type r x) }
lemma card_typein_out_lt (c : cardinal) (x : c.ord.out.α) : card (typein c.ord.out.r x) < c :=
by { convert card_typein_lt c.ord.out.r x _, rw [mk_ord_out], rw [type_out, mk_ord_out] }
lemma ord_injective : injective ord :=
by { intros c c' h, rw [←card_ord c, ←card_ord c', h] }
/-- The ordinal corresponding to a cardinal `c` is the least ordinal
whose cardinal is `c`. This is the order-embedding version. For the regular function, see `ord`.
-/
def ord.order_embedding : cardinal ↪o ordinal :=
rel_embedding.order_embedding_of_lt_embedding
(rel_embedding.of_monotone cardinal.ord $ λ a b, cardinal.ord_lt_ord.2)
@[simp] theorem ord.order_embedding_coe :
(ord.order_embedding : cardinal → ordinal) = ord := rfl
/-- The cardinal `univ` is the cardinality of ordinal `univ`, or
equivalently the cardinal of `ordinal.{u}`, or `cardinal.{u}`,
as an element of `cardinal.{v}` (when `u < v`). -/
def univ := lift.{(u+1) v} (mk ordinal)
theorem univ_id : univ.{u (u+1)} = mk ordinal := lift_id _
@[simp] theorem lift_univ : lift.{_ w} univ.{u v} = univ.{u (max v w)} := lift_lift _
theorem univ_umax : univ.{u (max (u+1) v)} = univ.{u v} := congr_fun lift_umax _
theorem lift_lt_univ (c : cardinal) : lift.{u (u+1)} c < univ.{u (u+1)} :=
by simpa only [lift.principal_seg_coe, lift_ord, lift_succ, ord_le, succ_le] using le_of_lt
(lift.principal_seg.{u (u+1)}.lt_top (succ c).ord)
theorem lift_lt_univ' (c : cardinal) : lift.{u (max (u+1) v)} c < univ.{u v} :=
by simpa only [lift_lift, lift_univ, univ_umax] using
lift_lt.{_ (max (u+1) v)}.2 (lift_lt_univ c)
@[simp] theorem ord_univ : ord univ.{u v} = ordinal.univ.{u v} :=
le_antisymm (ord_card_le _) $ le_of_forall_lt $ λ o h,
lt_ord.2 begin
rcases lift.principal_seg.{u v}.down'.1
(by simpa only [lift.principal_seg_coe] using h) with ⟨o', rfl⟩,
simp only [lift.principal_seg_coe], rw [← lift_card],
apply lift_lt_univ'
end
theorem lt_univ {c} : c < univ.{u (u+1)} ↔ ∃ c', c = lift.{u (u+1)} c' :=
⟨λ h, begin
have := ord_lt_ord.2 h,
rw ord_univ at this,
cases lift.principal_seg.{u (u+1)}.down'.1
(by simpa only [lift.principal_seg_top]) with o e,
have := card_ord c,
rw [← e, lift.principal_seg_coe, ← lift_card] at this,
exact ⟨_, this.symm⟩
end, λ ⟨c', e⟩, e.symm ▸ lift_lt_univ _⟩
theorem lt_univ' {c} : c < univ.{u v} ↔ ∃ c', c = lift.{u (max (u+1) v)} c' :=
⟨λ h, let ⟨a, e, h'⟩ := lt_lift_iff.1 h in begin
rw [← univ_id] at h',
rcases lt_univ.{u}.1 h' with ⟨c', rfl⟩,
exact ⟨c', by simp only [e.symm, lift_lift]⟩
end, λ ⟨c', e⟩, e.symm ▸ lift_lt_univ' _⟩
end cardinal
namespace ordinal
@[simp] theorem card_univ : card univ = cardinal.univ := rfl
end ordinal
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