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b5bc5529e9945aa795803585b8877fd8dd3b6a10 | 302c785c90d40ad3d6be43d33bc6a558354cc2cf | /src/data/padics/padic_integers.lean | 9aec29b62a2fc1e2a14df548c719b0815d12c2ee | [
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Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Mario Carneiro, Johan Commelin
-/
import data.int.modeq
import data.zmod.basic
import linear_algebra.adic_completion
import data.padics.padic_numbers
import ring_theory.discrete_valuation_ring
import topology.metric_space.cau_seq_filter
/-!
# p-adic integers
This file defines the p-adic integers `ℤ_p` as the subtype of `ℚ_p` with norm `≤ 1`.
We show that `ℤ_p`
* is complete
* is nonarchimedean
* is a normed ring
* is a local ring
* is a discrete valuation ring
The relation between `ℤ_[p]` and `zmod p` is established in another file.
## Important definitions
* `padic_int` : the type of p-adic numbers
## Notation
We introduce the notation `ℤ_[p]` for the p-adic integers.
## Implementation notes
Much, but not all, of this file assumes that `p` is prime. This assumption is inferred automatically
by taking `[fact (nat.prime p)] as a type class argument.
Coercions into `ℤ_p` are set up to work with the `norm_cast` tactic.
## References
* [F. Q. Gouêva, *p-adic numbers*][gouvea1997]
* [R. Y. Lewis, *A formal proof of Hensel's lemma over the p-adic integers*][lewis2019]
* <https://en.wikipedia.org/wiki/P-adic_number>
## Tags
p-adic, p adic, padic, p-adic integer
-/
open nat padic metric local_ring
noncomputable theory
open_locale classical
/-- The p-adic integers ℤ_p are the p-adic numbers with norm ≤ 1. -/
def padic_int (p : ℕ) [fact p.prime] := {x : ℚ_[p] // ∥x∥ ≤ 1}
notation `ℤ_[`p`]` := padic_int p
namespace padic_int
/-! ### Ring structure and coercion to `ℚ_[p]` -/
variables {p : ℕ} [fact p.prime]
instance : has_coe ℤ_[p] ℚ_[p] := ⟨subtype.val⟩
lemma ext {x y : ℤ_[p]} : (x : ℚ_[p]) = y → x = y := subtype.ext_iff_val.2
/-- Addition on ℤ_p is inherited from ℚ_p. -/
instance : has_add ℤ_[p] :=
⟨λ ⟨x, hx⟩ ⟨y, hy⟩, ⟨x+y,
le_trans (padic_norm_e.nonarchimedean _ _) (max_le_iff.2 ⟨hx,hy⟩)⟩⟩
/-- Multiplication on ℤ_p is inherited from ℚ_p. -/
instance : has_mul ℤ_[p] :=
⟨λ ⟨x, hx⟩ ⟨y, hy⟩, ⟨x*y,
begin rw padic_norm_e.mul, apply mul_le_one; {assumption <|> apply norm_nonneg} end⟩⟩
/-- Negation on ℤ_p is inherited from ℚ_p. -/
instance : has_neg ℤ_[p] :=
⟨λ ⟨x, hx⟩, ⟨-x, by simpa⟩⟩
/-- Subtraction on ℤ_p is inherited from ℚ_p. -/
instance : has_sub ℤ_[p] :=
⟨λ ⟨x, hx⟩ ⟨y, hy⟩, ⟨x - y,
by { rw sub_eq_add_neg, rw ← norm_neg at hy,
exact le_trans (padic_norm_e.nonarchimedean _ _) (max_le_iff.2 ⟨hx, hy⟩) }⟩⟩
/-- Zero on ℤ_p is inherited from ℚ_p. -/
instance : has_zero ℤ_[p] :=
⟨⟨0, by norm_num⟩⟩
instance : inhabited ℤ_[p] := ⟨0⟩
/-- One on ℤ_p is inherited from ℚ_p. -/
instance : has_one ℤ_[p] :=
⟨⟨1, by norm_num⟩⟩
@[simp] lemma mk_zero {h} : (⟨0, h⟩ : ℤ_[p]) = (0 : ℤ_[p]) := rfl
@[simp] lemma val_eq_coe (z : ℤ_[p]) : z.val = z := rfl
@[simp, norm_cast] lemma coe_add : ∀ (z1 z2 : ℤ_[p]), ((z1 + z2 : ℤ_[p]) : ℚ_[p]) = z1 + z2
| ⟨_, _⟩ ⟨_, _⟩ := rfl
@[simp, norm_cast] lemma coe_mul : ∀ (z1 z2 : ℤ_[p]), ((z1 * z2 : ℤ_[p]) : ℚ_[p]) = z1 * z2
| ⟨_, _⟩ ⟨_, _⟩ := rfl
@[simp, norm_cast] lemma coe_neg : ∀ (z1 : ℤ_[p]), ((-z1 : ℤ_[p]) : ℚ_[p]) = -z1
| ⟨_, _⟩ := rfl
@[simp, norm_cast] lemma coe_sub : ∀ (z1 z2 : ℤ_[p]), ((z1 - z2 : ℤ_[p]) : ℚ_[p]) = z1 - z2
| ⟨_, _⟩ ⟨_, _⟩ := rfl
@[simp, norm_cast] lemma coe_one : ((1 : ℤ_[p]) : ℚ_[p]) = 1 := rfl
@[simp, norm_cast] lemma coe_coe : ∀ n : ℕ, ((n : ℤ_[p]) : ℚ_[p]) = n
| 0 := rfl
| (k+1) := by simp [coe_coe]
@[simp, norm_cast] lemma coe_coe_int : ∀ (z : ℤ), ((z : ℤ_[p]) : ℚ_[p]) = z
| (int.of_nat n) := by simp
| -[1+n] := by simp
@[simp, norm_cast] lemma coe_zero : ((0 : ℤ_[p]) : ℚ_[p]) = 0 := rfl
instance : ring ℤ_[p] :=
begin
refine { add := (+),
mul := (*),
neg := has_neg.neg,
zero := 0,
one := 1,
sub := has_sub.sub,
sub_eq_add_neg := _,
.. };
intros; ext; simp; ring
end
/-- The coercion from ℤ[p] to ℚ[p] as a ring homomorphism. -/
def coe.ring_hom : ℤ_[p] →+* ℚ_[p] :=
{ to_fun := (coe : ℤ_[p] → ℚ_[p]),
map_zero' := rfl,
map_one' := rfl,
map_mul' := coe_mul,
map_add' := coe_add }
@[simp, norm_cast] lemma coe_pow (x : ℤ_[p]) (n : ℕ) : (↑(x^n) : ℚ_[p]) = (↑x : ℚ_[p])^n :=
coe.ring_hom.map_pow x n
@[simp] lemma mk_coe : ∀ (k : ℤ_[p]), (⟨k, k.2⟩ : ℤ_[p]) = k
| ⟨_, _⟩ := rfl
/-- The inverse of a p-adic integer with norm equal to 1 is also a p-adic integer. Otherwise, the
inverse is defined to be 0. -/
def inv : ℤ_[p] → ℤ_[p]
| ⟨k, _⟩ := if h : ∥k∥ = 1 then ⟨1/k, by simp [h]⟩ else 0
instance : char_zero ℤ_[p] :=
{ cast_injective :=
λ m n h, cast_injective $
show (m:ℚ_[p]) = n, by { rw subtype.ext_iff at h, norm_cast at h, exact h } }
@[simp, norm_cast] lemma coe_int_eq (z1 z2 : ℤ) : (z1 : ℤ_[p]) = z2 ↔ z1 = z2 :=
suffices (z1 : ℚ_[p]) = z2 ↔ z1 = z2, from iff.trans (by norm_cast) this,
by norm_cast
/--
A sequence of integers that is Cauchy with respect to the `p`-adic norm
converges to a `p`-adic integer.
-/
def of_int_seq (seq : ℕ → ℤ) (h : is_cau_seq (padic_norm p) (λ n, seq n)) : ℤ_[p] :=
⟨⟦⟨_, h⟩⟧,
show ↑(padic_seq.norm _) ≤ (1 : ℝ), begin
rw padic_seq.norm,
split_ifs with hne; norm_cast,
{ exact zero_le_one },
{ apply padic_norm.of_int }
end ⟩
end padic_int
namespace padic_int
/-!
### Instances
We now show that `ℤ_[p]` is a
* complete metric space
* normed ring
* integral domain
-/
variables (p : ℕ) [fact p.prime]
instance : metric_space ℤ_[p] := subtype.metric_space
instance complete_space : complete_space ℤ_[p] :=
begin
delta padic_int,
rw [complete_space_iff_is_complete_range uniform_embedding_subtype_coe,
subtype.range_coe_subtype],
have : is_complete (closed_ball (0 : ℚ_[p]) 1) := is_closed_ball.is_complete,
simpa [closed_ball],
end
instance : has_norm ℤ_[p] := ⟨λ z, ∥(z : ℚ_[p])∥⟩
variables {p}
protected lemma mul_comm : ∀ z1 z2 : ℤ_[p], z1*z2 = z2*z1
| ⟨q1, h1⟩ ⟨q2, h2⟩ := show (⟨q1*q2, _⟩ : ℤ_[p]) = ⟨q2*q1, _⟩, by simp [_root_.mul_comm]
protected lemma zero_ne_one : (0 : ℤ_[p]) ≠ 1 :=
show (⟨(0 : ℚ_[p]), _⟩ : ℤ_[p]) ≠ ⟨(1 : ℚ_[p]), _⟩, from mt subtype.ext_iff_val.1 zero_ne_one
protected lemma eq_zero_or_eq_zero_of_mul_eq_zero :
∀ (a b : ℤ_[p]), a * b = 0 → a = 0 ∨ b = 0
| ⟨a, ha⟩ ⟨b, hb⟩ := λ h : (⟨a * b, _⟩ : ℤ_[p]) = ⟨0, _⟩,
have a * b = 0, from subtype.ext_iff_val.1 h,
(mul_eq_zero.1 this).elim
(λ h1, or.inl (by simp [h1]; refl))
(λ h2, or.inr (by simp [h2]; refl))
lemma norm_def {z : ℤ_[p]} : ∥z∥ = ∥(z : ℚ_[p])∥ := rfl
variables (p)
instance : normed_comm_ring ℤ_[p] :=
{ dist_eq := λ ⟨_, _⟩ ⟨_, _⟩, rfl,
norm_mul := λ ⟨_, _⟩ ⟨_, _⟩, norm_mul_le _ _,
mul_comm := padic_int.mul_comm }
instance : norm_one_class ℤ_[p] := ⟨norm_def.trans norm_one⟩
instance is_absolute_value : is_absolute_value (λ z : ℤ_[p], ∥z∥) :=
{ abv_nonneg := norm_nonneg,
abv_eq_zero := λ ⟨_, _⟩, by simp [norm_eq_zero],
abv_add := λ ⟨_,_⟩ ⟨_, _⟩, norm_add_le _ _,
abv_mul := λ _ _, by simp only [norm_def, padic_norm_e.mul, padic_int.coe_mul]}
variables {p}
instance : integral_domain ℤ_[p] :=
{ eq_zero_or_eq_zero_of_mul_eq_zero := λ x y, padic_int.eq_zero_or_eq_zero_of_mul_eq_zero x y,
exists_pair_ne := ⟨0, 1, padic_int.zero_ne_one⟩,
.. padic_int.normed_comm_ring p }
end padic_int
namespace padic_int
/-! ### Norm -/
variables {p : ℕ} [fact p.prime]
lemma norm_le_one : ∀ z : ℤ_[p], ∥z∥ ≤ 1
| ⟨_, h⟩ := h
@[simp] lemma norm_mul (z1 z2 : ℤ_[p]) : ∥z1 * z2∥ = ∥z1∥ * ∥z2∥ :=
by simp [norm_def]
@[simp] lemma norm_pow (z : ℤ_[p]) : ∀ n : ℕ, ∥z^n∥ = ∥z∥^n
| 0 := by simp
| (k+1) := show ∥z*z^k∥ = ∥z∥*∥z∥^k, by {rw norm_mul, congr, apply norm_pow}
theorem nonarchimedean : ∀ (q r : ℤ_[p]), ∥q + r∥ ≤ max (∥q∥) (∥r∥)
| ⟨_, _⟩ ⟨_, _⟩ := padic_norm_e.nonarchimedean _ _
theorem norm_add_eq_max_of_ne : ∀ {q r : ℤ_[p]}, ∥q∥ ≠ ∥r∥ → ∥q+r∥ = max (∥q∥) (∥r∥)
| ⟨_, _⟩ ⟨_, _⟩ := padic_norm_e.add_eq_max_of_ne
lemma norm_eq_of_norm_add_lt_right {z1 z2 : ℤ_[p]}
(h : ∥z1 + z2∥ < ∥z2∥) : ∥z1∥ = ∥z2∥ :=
by_contradiction $ λ hne,
not_lt_of_ge (by rw norm_add_eq_max_of_ne hne; apply le_max_right) h
lemma norm_eq_of_norm_add_lt_left {z1 z2 : ℤ_[p]}
(h : ∥z1 + z2∥ < ∥z1∥) : ∥z1∥ = ∥z2∥ :=
by_contradiction $ λ hne,
not_lt_of_ge (by rw norm_add_eq_max_of_ne hne; apply le_max_left) h
@[simp] lemma padic_norm_e_of_padic_int (z : ℤ_[p]) : ∥(↑z : ℚ_[p])∥ = ∥z∥ :=
by simp [norm_def]
lemma norm_int_cast_eq_padic_norm (z : ℤ) : ∥(z : ℤ_[p])∥ = ∥(z : ℚ_[p])∥ :=
by simp [norm_def]
@[simp] lemma norm_eq_padic_norm {q : ℚ_[p]} (hq : ∥q∥ ≤ 1) :
@norm ℤ_[p] _ ⟨q, hq⟩ = ∥q∥ := rfl
@[simp] lemma norm_p : ∥(p : ℤ_[p])∥ = p⁻¹ :=
show ∥((p : ℤ_[p]) : ℚ_[p])∥ = p⁻¹, by exact_mod_cast padic_norm_e.norm_p
@[simp] lemma norm_p_pow (n : ℕ) : ∥(p : ℤ_[p])^n∥ = p^(-n:ℤ) :=
show ∥((p^n : ℤ_[p]) : ℚ_[p])∥ = p^(-n:ℤ),
by { convert padic_norm_e.norm_p_pow n, simp, }
private def cau_seq_to_rat_cau_seq (f : cau_seq ℤ_[p] norm) :
cau_seq ℚ_[p] (λ a, ∥a∥) :=
⟨ λ n, f n,
λ _ hε, by simpa [norm, norm_def] using f.cauchy hε ⟩
variables (p)
instance complete : cau_seq.is_complete ℤ_[p] norm :=
⟨ λ f,
have hqn : ∥cau_seq.lim (cau_seq_to_rat_cau_seq f)∥ ≤ 1,
from padic_norm_e_lim_le zero_lt_one (λ _, norm_le_one _),
⟨ ⟨_, hqn⟩,
λ ε, by simpa [norm, norm_def] using cau_seq.equiv_lim (cau_seq_to_rat_cau_seq f) ε⟩⟩
end padic_int
namespace padic_int
variables (p : ℕ) [hp_prime : fact p.prime]
include hp_prime
lemma exists_pow_neg_lt {ε : ℝ} (hε : 0 < ε) :
∃ (k : ℕ), ↑p ^ -((k : ℕ) : ℤ) < ε :=
begin
obtain ⟨k, hk⟩ := exists_nat_gt ε⁻¹,
use k,
rw ← inv_lt_inv hε (_root_.fpow_pos_of_pos _ _),
{ rw [fpow_neg, inv_inv', fpow_coe_nat],
apply lt_of_lt_of_le hk,
norm_cast,
apply le_of_lt,
convert nat.lt_pow_self _ _ using 1,
exact hp_prime.1.one_lt },
{ exact_mod_cast hp_prime.1.pos }
end
lemma exists_pow_neg_lt_rat {ε : ℚ} (hε : 0 < ε) :
∃ (k : ℕ), ↑p ^ -((k : ℕ) : ℤ) < ε :=
begin
obtain ⟨k, hk⟩ := @exists_pow_neg_lt p _ ε (by exact_mod_cast hε),
use k,
rw (show (p : ℝ) = (p : ℚ), by simp) at hk,
exact_mod_cast hk
end
variable {p}
lemma norm_int_lt_one_iff_dvd (k : ℤ) : ∥(k : ℤ_[p])∥ < 1 ↔ ↑p ∣ k :=
suffices ∥(k : ℚ_[p])∥ < 1 ↔ ↑p ∣ k, by rwa norm_int_cast_eq_padic_norm,
padic_norm_e.norm_int_lt_one_iff_dvd k
lemma norm_int_le_pow_iff_dvd {k : ℤ} {n : ℕ} : ∥(k : ℤ_[p])∥ ≤ ((↑p)^(-n : ℤ)) ↔ ↑p^n ∣ k :=
suffices ∥(k : ℚ_[p])∥ ≤ ((↑p)^(-n : ℤ)) ↔ ↑(p^n) ∣ k, by simpa [norm_int_cast_eq_padic_norm],
padic_norm_e.norm_int_le_pow_iff_dvd _ _
/-! ### Valuation on `ℤ_[p]` -/
/-- `padic_int.valuation` lifts the p-adic valuation on `ℚ` to `ℤ_[p]`. -/
def valuation (x : ℤ_[p]) := padic.valuation (x : ℚ_[p])
lemma norm_eq_pow_val {x : ℤ_[p]} (hx : x ≠ 0) :
∥x∥ = p^(-x.valuation) :=
begin
convert padic.norm_eq_pow_val _,
contrapose! hx,
exact subtype.val_injective hx
end
@[simp] lemma valuation_zero : valuation (0 : ℤ_[p]) = 0 :=
padic.valuation_zero
@[simp] lemma valuation_one : valuation (1 : ℤ_[p]) = 0 :=
padic.valuation_one
@[simp] lemma valuation_p : valuation (p : ℤ_[p]) = 1 :=
by simp [valuation, -cast_eq_of_rat_of_nat]
lemma valuation_nonneg (x : ℤ_[p]) : 0 ≤ x.valuation :=
begin
by_cases hx : x = 0,
{ simp [hx] },
have h : (1 : ℝ) < p := by exact_mod_cast hp_prime.1.one_lt,
rw [← neg_nonpos, ← (fpow_strict_mono h).le_iff_le],
show (p : ℝ) ^ -valuation x ≤ p ^ 0,
rw [← norm_eq_pow_val hx],
simpa using x.property,
end
@[simp] lemma valuation_p_pow_mul (n : ℕ) (c : ℤ_[p]) (hc : c ≠ 0) :
(↑p ^ n * c).valuation = n + c.valuation :=
begin
have : ∥↑p ^ n * c∥ = ∥(p ^ n : ℤ_[p])∥ * ∥c∥,
{ exact norm_mul _ _ },
have aux : ↑p ^ n * c ≠ 0,
{ contrapose! hc, rw mul_eq_zero at hc, cases hc,
{ refine (hp_prime.1.ne_zero _).elim,
exact_mod_cast (pow_eq_zero hc) },
{ exact hc } },
rwa [norm_eq_pow_val aux, norm_p_pow, norm_eq_pow_val hc,
← fpow_add, ← neg_add, fpow_inj, neg_inj] at this,
{ exact_mod_cast hp_prime.1.pos },
{ exact_mod_cast hp_prime.1.ne_one },
{ exact_mod_cast hp_prime.1.ne_zero },
end
section units
/-! ### Units of `ℤ_[p]` -/
local attribute [reducible] padic_int
lemma mul_inv : ∀ {z : ℤ_[p]}, ∥z∥ = 1 → z * z.inv = 1
| ⟨k, _⟩ h :=
begin
have hk : k ≠ 0, from λ h', @zero_ne_one ℚ_[p] _ _ (by simpa [h'] using h),
unfold padic_int.inv, split_ifs,
{ change (⟨k * (1/k), _⟩ : ℤ_[p]) = 1,
simp [hk], refl },
{ apply subtype.ext_iff_val.2, simp [mul_inv_cancel hk] }
end
lemma inv_mul {z : ℤ_[p]} (hz : ∥z∥ = 1) : z.inv * z = 1 :=
by rw [mul_comm, mul_inv hz]
lemma is_unit_iff {z : ℤ_[p]} : is_unit z ↔ ∥z∥ = 1 :=
⟨λ h, begin
rcases is_unit_iff_dvd_one.1 h with ⟨w, eq⟩,
refine le_antisymm (norm_le_one _) _,
have := mul_le_mul_of_nonneg_left (norm_le_one w) (norm_nonneg z),
rwa [mul_one, ← norm_mul, ← eq, norm_one] at this
end, λ h, ⟨⟨z, z.inv, mul_inv h, inv_mul h⟩, rfl⟩⟩
lemma norm_lt_one_add {z1 z2 : ℤ_[p]} (hz1 : ∥z1∥ < 1) (hz2 : ∥z2∥ < 1) : ∥z1 + z2∥ < 1 :=
lt_of_le_of_lt (nonarchimedean _ _) (max_lt hz1 hz2)
lemma norm_lt_one_mul {z1 z2 : ℤ_[p]} (hz2 : ∥z2∥ < 1) : ∥z1 * z2∥ < 1 :=
calc ∥z1 * z2∥ = ∥z1∥ * ∥z2∥ : by simp
... < 1 : mul_lt_one_of_nonneg_of_lt_one_right (norm_le_one _) (norm_nonneg _) hz2
@[simp] lemma mem_nonunits {z : ℤ_[p]} : z ∈ nonunits ℤ_[p] ↔ ∥z∥ < 1 :=
by rw lt_iff_le_and_ne; simp [norm_le_one z, nonunits, is_unit_iff]
/-- A `p`-adic number `u` with `∥u∥ = 1` is a unit of `ℤ_[p]`. -/
def mk_units {u : ℚ_[p]} (h : ∥u∥ = 1) : units ℤ_[p] :=
let z : ℤ_[p] := ⟨u, le_of_eq h⟩ in ⟨z, z.inv, mul_inv h, inv_mul h⟩
@[simp]
lemma mk_units_eq {u : ℚ_[p]} (h : ∥u∥ = 1) : ((mk_units h : ℤ_[p]) : ℚ_[p]) = u :=
rfl
@[simp] lemma norm_units (u : units ℤ_[p]) : ∥(u : ℤ_[p])∥ = 1 :=
is_unit_iff.mp $ by simp
/-- `unit_coeff hx` is the unit `u` in the unique representation `x = u * p ^ n`.
See `unit_coeff_spec`. -/
def unit_coeff {x : ℤ_[p]} (hx : x ≠ 0) : units ℤ_[p] :=
let u : ℚ_[p] := x*p^(-x.valuation) in
have hu : ∥u∥ = 1,
by simp [hx, nat.fpow_ne_zero_of_pos (by exact_mod_cast hp_prime.1.pos) x.valuation,
norm_eq_pow_val, fpow_neg, inv_mul_cancel, -cast_eq_of_rat_of_nat],
mk_units hu
@[simp] lemma unit_coeff_coe {x : ℤ_[p]} (hx : x ≠ 0) :
(unit_coeff hx : ℚ_[p]) = x * p ^ (-x.valuation) := rfl
lemma unit_coeff_spec {x : ℤ_[p]} (hx : x ≠ 0) :
x = (unit_coeff hx : ℤ_[p]) * p ^ int.nat_abs (valuation x) :=
begin
apply subtype.coe_injective,
push_cast,
have repr : (x : ℚ_[p]) = (unit_coeff hx) * p ^ x.valuation,
{ rw [unit_coeff_coe, mul_assoc, ← fpow_add],
{ simp },
{ exact_mod_cast hp_prime.1.ne_zero } },
convert repr using 2,
rw [← fpow_coe_nat, int.nat_abs_of_nonneg (valuation_nonneg x)],
end
end units
section norm_le_iff
/-! ### Various characterizations of open unit balls -/
lemma norm_le_pow_iff_le_valuation (x : ℤ_[p]) (hx : x ≠ 0) (n : ℕ) :
∥x∥ ≤ p ^ (-n : ℤ) ↔ ↑n ≤ x.valuation :=
begin
rw norm_eq_pow_val hx,
lift x.valuation to ℕ using x.valuation_nonneg with k hk,
simp only [int.coe_nat_le, fpow_neg, fpow_coe_nat],
have aux : ∀ n : ℕ, 0 < (p ^ n : ℝ),
{ apply pow_pos, exact_mod_cast hp_prime.1.pos },
rw [inv_le_inv (aux _) (aux _)],
have : p ^ n ≤ p ^ k ↔ n ≤ k := (pow_right_strict_mono hp_prime.1.two_le).le_iff_le,
rw [← this],
norm_cast,
end
lemma mem_span_pow_iff_le_valuation (x : ℤ_[p]) (hx : x ≠ 0) (n : ℕ) :
x ∈ (ideal.span {p ^ n} : ideal ℤ_[p]) ↔ ↑n ≤ x.valuation :=
begin
rw [ideal.mem_span_singleton],
split,
{ rintro ⟨c, rfl⟩,
suffices : c ≠ 0,
{ rw [valuation_p_pow_mul _ _ this, le_add_iff_nonneg_right], apply valuation_nonneg, },
contrapose! hx, rw [hx, mul_zero], },
{ rw [unit_coeff_spec hx] { occs := occurrences.pos [2] },
lift x.valuation to ℕ using x.valuation_nonneg with k hk,
simp only [int.nat_abs_of_nat, units.is_unit, is_unit.dvd_mul_left, int.coe_nat_le],
intro H,
obtain ⟨k, rfl⟩ := nat.exists_eq_add_of_le H,
simp only [pow_add, dvd_mul_right], }
end
lemma norm_le_pow_iff_mem_span_pow (x : ℤ_[p]) (n : ℕ) :
∥x∥ ≤ p ^ (-n : ℤ) ↔ x ∈ (ideal.span {p ^ n} : ideal ℤ_[p]) :=
begin
by_cases hx : x = 0,
{ subst hx,
simp only [norm_zero, fpow_neg, fpow_coe_nat, inv_nonneg, iff_true, submodule.zero_mem],
exact_mod_cast nat.zero_le _ },
rw [norm_le_pow_iff_le_valuation x hx, mem_span_pow_iff_le_valuation x hx],
end
lemma norm_le_pow_iff_norm_lt_pow_add_one (x : ℤ_[p]) (n : ℤ) :
∥x∥ ≤ p ^ n ↔ ∥x∥ < p ^ (n + 1) :=
begin
have aux : ∀ n : ℤ, 0 < (p ^ n : ℝ),
{ apply nat.fpow_pos_of_pos, exact hp_prime.1.pos },
by_cases hx0 : x = 0, { simp [hx0, norm_zero, aux, le_of_lt (aux _)], },
rw norm_eq_pow_val hx0,
have h1p : 1 < (p : ℝ), { exact_mod_cast hp_prime.1.one_lt },
have H := fpow_strict_mono h1p,
rw [H.le_iff_le, H.lt_iff_lt, int.lt_add_one_iff],
end
lemma norm_lt_pow_iff_norm_le_pow_sub_one (x : ℤ_[p]) (n : ℤ) :
∥x∥ < p ^ n ↔ ∥x∥ ≤ p ^ (n - 1) :=
by rw [norm_le_pow_iff_norm_lt_pow_add_one, sub_add_cancel]
lemma norm_lt_one_iff_dvd (x : ℤ_[p]) : ∥x∥ < 1 ↔ ↑p ∣ x :=
begin
have := norm_le_pow_iff_mem_span_pow x 1,
rw [ideal.mem_span_singleton, pow_one] at this,
rw [← this, norm_le_pow_iff_norm_lt_pow_add_one],
simp only [fpow_zero, int.coe_nat_zero, int.coe_nat_succ, add_left_neg, zero_add],
end
@[simp] lemma pow_p_dvd_int_iff (n : ℕ) (a : ℤ) : (p ^ n : ℤ_[p]) ∣ a ↔ ↑p ^ n ∣ a :=
by rw [← norm_int_le_pow_iff_dvd, norm_le_pow_iff_mem_span_pow, ideal.mem_span_singleton]
end norm_le_iff
section dvr
/-! ### Discrete valuation ring -/
instance : local_ring ℤ_[p] :=
local_of_nonunits_ideal zero_ne_one $ λ x y, by simp; exact norm_lt_one_add
lemma p_nonnunit : (p : ℤ_[p]) ∈ nonunits ℤ_[p] :=
have (p : ℝ)⁻¹ < 1, from inv_lt_one $ by exact_mod_cast hp_prime.1.one_lt,
by simp [this]
lemma maximal_ideal_eq_span_p : maximal_ideal ℤ_[p] = ideal.span {p} :=
begin
apply le_antisymm,
{ intros x hx,
rw ideal.mem_span_singleton,
simp only [local_ring.mem_maximal_ideal, mem_nonunits] at hx,
rwa ← norm_lt_one_iff_dvd, },
{ rw [ideal.span_le, set.singleton_subset_iff], exact p_nonnunit }
end
lemma prime_p : prime (p : ℤ_[p]) :=
begin
rw [← ideal.span_singleton_prime, ← maximal_ideal_eq_span_p],
{ apply_instance },
{ exact_mod_cast hp_prime.1.ne_zero }
end
lemma irreducible_p : irreducible (p : ℤ_[p]) :=
irreducible_of_prime prime_p
instance : discrete_valuation_ring ℤ_[p] :=
discrete_valuation_ring.of_has_unit_mul_pow_irreducible_factorization
⟨p, irreducible_p, λ x hx, ⟨x.valuation.nat_abs, unit_coeff hx,
by rw [mul_comm, ← unit_coeff_spec hx]⟩⟩
lemma ideal_eq_span_pow_p {s : ideal ℤ_[p]} (hs : s ≠ ⊥) :
∃ n : ℕ, s = ideal.span {p ^ n} :=
discrete_valuation_ring.ideal_eq_span_pow_irreducible hs irreducible_p
end dvr
end padic_int
|
871045ba10c9881bd95aaff32d6f782f908a8f59 | 6b2a480f27775cba4f3ae191b1c1387a29de586e | /group_rep1/homothetic.lean | 356355aff2e0401bf754bb5f1fe027404edb3458 | [] | no_license | Or7ando/group_representation | a681de2e19d1930a1e1be573d6735a2f0b8356cb | 9b576984f17764ebf26c8caa2a542d248f1b50d2 | refs/heads/master | 1,662,413,107,324 | 1,590,302,389,000 | 1,590,302,389,000 | 258,130,829 | 0 | 1 | null | null | null | null | UTF-8 | Lean | false | false | 1,225 | lean | import .group_representation
import .morphism
import ring_theory.algebra
open algebra
open linear_map
universe variables u v w
variables {G : Type u}[group G] {R : Type v}[comm_ring R] {M : Type w}
[add_comm_group M] [module R M]
namespace homothetic
variables (ρ : group_representation G R M)
open morphism
/-
Make an R algebra ? subalgebra of sub.type ! To check
-/
def add ( f g : ρ ⟶ ρ ) : ρ ⟶ ρ := {
ℓ := f.ℓ +g.ℓ,
commute := begin intros k,
ext,simp,
erw commute_apply,erw commute_apply ρ, exact rfl,
end
}
def spectral (t : R)(f : ρ ⟶ ρ) : ρ ⟶ ρ := { ℓ := t • f.ℓ,
commute := begin
intros g,ext,simp, erw commute_apply, exact rfl,
end }
instance has_add_morphism : has_add ( ρ ⟶ ρ ) := { add := add ρ }
instance has_scalar_morphism: has_scalar R ( ρ ⟶ ρ ) := { smul := spectral ρ }
def id := 𝟙 ρ
lemma add_ext ( f g : ρ ⟶ ρ ) : (f+g).ℓ = f.ℓ + g.ℓ := rfl
lemma one_ext : (𝟙 ρ ).ℓ =linear_map.id :=rfl
lemma smul_ext (t : R)(f : ρ ⟶ ρ) : (t • f).ℓ = t • f.ℓ := rfl
def test (t : R)(f : ρ ⟶ ρ) := f + t • 𝟙 ρ
end homothetic
#check mul_sub_left_distrib |
bf7b98ec6138500bd140400b42de8124c6ddd9cf | 96338d06deb5f54f351493a71d6ecf6c546089a2 | /priv/Lean/Cat.lean | 5d00ef5518df80d4549c17fdf972000d2f17e71e | [] | no_license | silky/exe | 5f9e4eea772d74852a1a2fac57d8d20588282d2b | e81690d6e16f2a83c105cce446011af6ae905b81 | refs/heads/master | 1,609,385,766,412 | 1,472,164,223,000 | 1,472,164,223,000 | 66,610,224 | 1 | 0 | null | 1,472,178,919,000 | 1,472,178,919,000 | null | UTF-8 | Lean | false | false | 6,132 | lean | /- Cat -/
import Setoid
set_option pp.universes true
set_option pp.metavar_args false
namespace EXE
namespace Cat
-- carrier of a category: type of morphisms
abbreviation HomType (Ob : Type) : Type := Π(Dom Cod : Ob), SetoidType
-- structure of a category
section withHom
variables {Ob : Type} (Hom : HomType Ob)
abbreviation IdType : Type :=
Π{a : Ob}, Hom a a
abbreviation MulType : Type :=
Π{a b c : Ob}, Hom b c ⥤ Hom a b ⥤ Hom a c
end withHom
-- axioms of category
section withIdMul
variables {Ob : Type} {Hom : HomType Ob}
variables (Id : IdType Hom) (Mul : MulType Hom)
abbreviation UnitLProp : Prop :=
∀{a b : Ob}, ∀(f : Hom a b), (Mul Id $ f) ≡(Hom a b)≡ f
abbreviation UnitRProp : Prop :=
∀{a b : Ob}, ∀(f : Hom a b), (Mul f $ Id) ≡(Hom a b)≡ f
abbreviation UnitCProp : Prop :=
∀{a : Ob}, (Mul Id $ Id) ≡(Hom a a)≡ Id
abbreviation UnitLInvProp : Prop :=
∀{a b : Ob}, ∀(f : Hom a b), f ≡(Hom a b)≡ (Mul Id $ f)
abbreviation UnitRInvProp : Prop :=
∀{a b : Ob}, ∀(f : Hom a b), f ≡(Hom a b)≡ (Mul f $ Id)
abbreviation UnitCInvProp : Prop :=
∀{a : Ob}, Id ≡(Hom a a)≡ (Mul Id $ Id)
abbreviation AssocProp : Prop :=
∀{a b c d : Ob}, ∀(f : Hom c d), ∀(g : Hom b c), ∀(h : Hom a b),
(Mul (Mul f $ g) $ h) ≡(Hom a d)≡ (Mul f $ (Mul g $ h))
abbreviation AssocInvProp : Prop :=
∀{a b c d : Ob}, ∀(f : Hom c d), ∀(g : Hom b c), ∀(h : Hom a b),
(Mul f $ (Mul g $ h)) ≡(Hom a d)≡ (Mul (Mul f $ g) $ h)
end withIdMul
end Cat
namespace Setoid
-- identity in the category `Setoid`
definition Id : Cat.IdType HomSet :=
λ (A : SetoidType), MkHom
/- onEl -/ ( λ (a : A), a)
/- onEqu -/ ( λ (a1 a2 : A), λ (e12 : a1 ≡_≡ a2), e12)
definition Mul : Cat.MulType HomSet :=
λ(A B C : SetoidType), MkHom2
(B⥤C) (A⥤B) (A⥤C)
/- onElEl -/ ( Setoid.Mul.onElEl )
/- onElEqu -/ ( λ(f : B⥤C), λ(g1 g2 : A⥤B), λ(g12 : g1 ≡(A⥤B)≡ g2),
λ(a : A), f (g12 a))
/- onEquEl -/ ( λ(f1 f2 : B⥤C), λ(f12 : f1 ≡(B⥤C)≡ f2), λ(g : A⥤B),
λ(a : A), f12 (g a))
definition UnitL : Cat.UnitLProp @Id @Mul :=
λ(A B : SetoidType), λ(f : A⥤B), λ(a : A), ⊜
definition UnitR : Cat.UnitRProp @Id @Mul :=
λ(A B : SetoidType), λ(f : A⥤B), λ(a : A), ⊜
definition Assoc : Cat.AssocProp @Mul :=
λ(A B C D : SetoidType), λ(f : C⥤D), λ(g : B⥤C), λ(h : A⥤B), λ(a : A), ⊜
end Setoid
-- objects of 2-category `Cat`
record CatType : Type :=
(Ob : Type)
(Hom : Cat.HomType Ob)
(Id : Cat.IdType Hom)
(Mul : Cat.MulType Hom)
(UnitL : Cat.UnitLProp @Id @Mul)
(UnitR : Cat.UnitRProp @Id @Mul)
(Assoc : Cat.AssocProp @Mul)
print CatType
abbreviation Cat.MkOb := CatType.mk
-- carrier of category
attribute CatType.Ob [coercion]
notation ` ⟦ ` C ` ⟧ ` := CatType.Ob C
notation a ` ⇒` C `⇒ ` b := CatType.Hom C a b
notation ` ① ` := CatType.Id _
notation f ` ⊙` C `⊙ ` g := CatType.Mul C f $ g
abbreviation CatType.UnitC (C : CatType)
: Cat.UnitCProp (@CatType.Id C) (@CatType.Mul C) :=
λ (X : C), CatType.UnitL C (@CatType.Id C X)
abbreviation CatType.UnitLInv (C : CatType)
: Cat.UnitLInvProp (@CatType.Id C) (@CatType.Mul C) :=
λ X Y, λ f, SetoidType.Sym _ (CatType.UnitL C f)
abbreviation CatType.UnitRInv (C : CatType)
: Cat.UnitRInvProp (@CatType.Id C) (@CatType.Mul C) :=
λ X Y, λ f, SetoidType.Sym _ (CatType.UnitR C f)
abbreviation CatType.UnitCInv (C : CatType)
: Cat.UnitCInvProp (@CatType.Id C) (@CatType.Mul C) :=
λ (X : C), SetoidType.Sym _ (@CatType.UnitC C X)
abbreviation CatType.AssocInv (C : CatType)
: Cat.AssocInvProp (@CatType.Mul C) :=
λ X Y Z T, λ f g h, SetoidType.Sym _ (CatType.Assoc C f g h)
definition CatType.MulHE (C : CatType) {X Y Z : C}
(mYZ : Y ⇒C⇒ Z)
{mXY1 mXY2 : X ⇒C⇒ Y}(e12 : mXY1 ≡(X ⇒C⇒ Y)≡ mXY2)
: (mYZ ⊙C⊙ mXY1) ≡(X ⇒C⇒ Z)≡ (mYZ ⊙C⊙ mXY2) :=
(CatType.Mul C $ mYZ) $/ e12
definition CatType.MulEH (C : CatType) {X Y Z : C}
{mYZ1 mYZ2 : Y ⇒C⇒ Z}(e12 : mYZ1 ≡(Y ⇒C⇒ Z)≡ mYZ2)
(mXY : X ⇒C⇒ Y)
: (mYZ1 ⊙C⊙ mXY) ≡(X ⇒C⇒ Z)≡ (mYZ2 ⊙C⊙ mXY) :=
(CatType.Mul C $/ e12) /$ mXY
notation f ` ⊙` C `⊙/ ` geq := CatType.MulHE C f geq
notation feq ` /⊙` C `⊙ ` g := CatType.MulEH C feq g
-- the category of `Setoid`s
definition SetoidCat : CatType :=
Cat.MkOb
SetoidType Setoid.HomSet
@Setoid.Id @Setoid.Mul
@Setoid.UnitL @Setoid.UnitR @Setoid.Assoc
namespace Cat
definition FromSet (S : SetoidType) : CatType :=
let Hom (x y : S) : SetoidType := Setoid.FromType (x ≡S≡ y) in
Cat.MkOb
/- Ob -/ S
/- Hom -/ Hom
/- Id -/ ( λ (x : S), SetoidType.Refl S)
/-Mul -/ ( λ (x y z : S), Setoid.MkHom2
(Hom y z) (Hom x y) (Hom x z)
( λ (yz : y ≡S≡ z), λ (xy : x ≡S≡ y), SetoidType.Trans S xy yz)
( λ yz xy1 xy2 xyE, true.intro)
( λ yz1 yz2 yzE xy, true.intro))
/- UnitL -/ ( λ x y, λ f, true.intro)
/- UnitR -/ ( λ x y, λ f, true.intro)
/- Assoc -/ ( λ x y z t, λ f g h, true.intro)
end Cat
namespace CatType
record IsoHom (C : CatType) (A B : C) : Type :=
(AtoB : A ⇒C⇒ B)
(BtoA : B ⇒C⇒ A)
(atA : (BtoA ⊙C⊙ AtoB) ≡(A ⇒C⇒ A)≡ ①)
(atB : (AtoB ⊙C⊙ BtoA) ≡(B ⇒C⇒ B)≡ ①)
abbreviation MkIso {C : CatType} {A B : C} := @IsoHom.mk C A B
end CatType
notation a ` ⇐` C `⇒ ` b := CatType.IsoHom C a b
infix ` ⇔ `:10 := CatType.IsoHom SetoidCat
end EXE
|
5b974c2eaa5118e2a28d6c37949bff90dfc4d887 | 367134ba5a65885e863bdc4507601606690974c1 | /src/tactic/interactive_expr.lean | 736819318d928a2ced249a04826c78c7390b4725 | [
"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 | 17,482 | lean | /-
Copyright (c) 2020 E.W.Ayers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: E.W.Ayers
-/
/-!
# Widgets used for tactic state and term-mode goal display
The vscode extension supports the display of interactive widgets.
Default implementation of these widgets are included in the core
library. We override them here using `vm_override` so that we can
change them quickly without waiting for the next Lean release.
The function `widget_override.interactive_expression.mk` renders a single
expression as a widget component. Each goal in a tactic state is rendered
using the `widget_override.tactic_view_goal` function,
a complete tactic state is rendered using
`widget_override.tactic_view_component`.
Lean itself calls the `widget_override.term_goal_widget` function to render
term-mode goals and `widget_override.tactic_state_widget` to render the
tactic state in a tactic proof.
-/
namespace widget_override
open widget
open tagged_format
open widget.html widget.attr
namespace interactive_expression
meta instance : has_mem expr.coord expr.address := list.has_mem
/-- eformat but without any of the formatting stuff like highlighting, groups etc. -/
meta inductive sf : Type
| tag_expr : expr.address → expr → sf → sf
| compose : sf → sf → sf
| of_string : string → sf
| highlight : format.color → sf → sf
| block : ℕ → sf → sf
/-- Prints a debugging representation of an `sf` object. -/
meta def sf.repr : sf → format
| (sf.tag_expr addr e a) := format.group $ format.nest 2 $
"(tag_expr " ++ to_fmt addr ++ format.line ++
"`(" ++ to_fmt e ++ ")" ++ format.line ++ a.repr ++ ")"
| (sf.compose a b) := a.repr ++ format.line ++ b.repr
| (sf.of_string s) := repr s
| (sf.block i a) := "(block " ++ to_fmt i ++ format.line ++ a.repr ++ ")"
| (sf.highlight c a) := "(highlight " ++ c.to_string ++ a.repr ++ ")"
meta instance : has_to_format sf := ⟨sf.repr⟩
meta instance : has_to_string sf := ⟨λ s, s.repr.to_string⟩
meta instance : has_repr sf := ⟨λ s, s.repr.to_string⟩
/-- Constructs an `sf` from an `eformat` by forgetting grouping, nesting, etc. -/
meta def sf.of_eformat : eformat → sf
| (tag ⟨ea,e⟩ m) := sf.tag_expr ea e $ sf.of_eformat m
| (group m) := sf.block 0 $ sf.of_eformat m
| (nest i m) := sf.block i $ sf.of_eformat m
| (highlight c m) := sf.highlight c $ sf.of_eformat m
| (of_format f) := sf.of_string $ format.to_string f
| (compose x y) := sf.compose (sf.of_eformat x) (sf.of_eformat y)
/-- Flattens an `sf`, i.e. merges adjacent `of_string` constructors. -/
meta def sf.flatten : sf → sf
| (sf.tag_expr ea e m) := (sf.tag_expr ea e $ sf.flatten m)
| (sf.compose x y) :=
match (sf.flatten x), (sf.flatten y) with
| (sf.of_string sx), (sf.of_string sy) := sf.of_string (sx ++ sy)
| (sf.of_string sx), (sf.compose (sf.of_string sy) z) := sf.compose (sf.of_string (sx ++ sy)) z
| (sf.compose x (sf.of_string sy)), (sf.of_string sz) := sf.compose x (sf.of_string (sy ++ sz))
| (sf.compose x (sf.of_string sy1)), (sf.compose (sf.of_string sy2) z) := sf.compose x (sf.compose (sf.of_string (sy1 ++ sy2)) z)
| x, y := sf.compose x y
end
| (sf.of_string s) := -- replace newline by space
sf.of_string (s.to_list.map (λ c, if c = '\n' then ' ' else c)).as_string
| (sf.block i (sf.block j a)) := (sf.block (i+j) a).flatten
| (sf.block i a) := sf.block i a.flatten
| (sf.highlight i a) := sf.highlight i a.flatten
private meta def elim_part_apps : sf → expr.address → sf
| (sf.tag_expr ea e m) acc :=
if ∀ c ∈ ea, c = expr.coord.app_fn then
elim_part_apps m (acc ++ ea)
else
sf.tag_expr (acc ++ ea) e (elim_part_apps m [])
| (sf.compose a b) acc := (elim_part_apps a acc).compose (elim_part_apps b acc)
| (sf.of_string s) _ := sf.of_string s
| (sf.block i a) acc := sf.block i $ elim_part_apps a acc
| (sf.highlight c a) acc := sf.highlight c $ elim_part_apps a acc
/--
Post-process an `sf` object to eliminate tags for partial applications by
pushing the `app_fn` as far into the expression as possible. The effect is
that clicking on a sub-expression always includes the full argument list in
the popup.
Consider the expression `id id 0`. We push the `app_fn` for the partial
application `id id` inwards and eliminate it. Before:
```
(tag_expr [app_fn]
`(id.{1} (nat -> nat) (id.{1} nat))
(tag_expr [app_fn] `(id.{1} (nat -> nat)) "id")
"\n"
(tag_expr [app_arg] `(id.{1} nat) "id"))
"\n"
(tag_expr [app_arg] `(has_zero.zero.{0} nat nat.has_zero) "0")
```
After:
```
"id"
"\n"
(tag_expr [app_fn, app_arg] `(id.{1} nat) "id")
"\n"
(tag_expr [app_arg] `(has_zero.zero.{0} nat nat.has_zero) "0")
```
-/
meta def sf.elim_part_apps (s : sf) : sf :=
elim_part_apps s []
/--
The actions accepted by an expression widget.
-/
meta inductive action (γ : Type)
| on_mouse_enter : subexpr → action
| on_mouse_leave_all : action
| on_click : subexpr → action
| on_tooltip_action : γ → action
| on_close_tooltip : action
| effect : widget.effect → action
/--
Render a 'go to definition' button for a given expression.
If there is no definition available, then returns an empty list.
-/
meta def goto_def_button {γ} : expr → tactic (list (html (action γ)))
| e := (do
(expr.const n _) ← pure $ expr.get_app_fn e,
env ← tactic.get_env,
let file := environment.decl_olean env n,
pos ← environment.decl_pos env n,
pure $ [h "button" [
cn "pointer ba br3 mr1",
on_click (λ _, action.effect $ widget.effect.reveal_position file pos),
attr.val "title" "go to definition"] ["↪"]]
) <|> pure []
/-- Due to a bug in the webview browser, we have to reduce the number of spans in the expression.
To do this, we collect the attributes from `sf.block` and `sf.highlight` after an expression boundary. -/
meta def get_block_attrs {γ}: sf → tactic (sf × list (attr γ))
| (sf.block i a) := do
let s : attr (γ) := style [
("display", "inline-block"),
("padding-left", "1ch"),
("text-indent", "-1ch"),
("white-space", "pre-wrap"),
("vertical-align", "top")
],
(a,rest) ← get_block_attrs a,
pure (a, s :: rest)
| (sf.highlight c a) := do
(a, rest) ← get_block_attrs a,
pure (a, (cn c.to_string) :: rest)
| a := pure (a,[])
/--
Renders a subexpression as a list of html elements.
-/
meta def view {γ} (tooltip_component : tc subexpr (action γ)) (click_address : option expr.address) (select_address : option expr.address) :
subexpr → sf → tactic (list (html (action γ)))
| ⟨ce, current_address⟩ (sf.tag_expr ea e m) := do
let new_address := current_address ++ ea,
let select_attrs : list (attr (action γ)) := if some new_address = select_address then [className "highlight"] else [],
click_attrs : list (attr (action γ)) ←
if some new_address = click_address then do
content ← tc.to_html tooltip_component (e, new_address),
efmt : string ← format.to_string <$> tactic.pp e,
gd_btn ← goto_def_button e,
pure [tooltip $ h "div" [] [
h "div" [cn "fr"] (gd_btn ++ [
h "button" [cn "pointer ba br3 mr1", on_click (λ _, action.effect $ widget.effect.copy_text efmt), attr.val "title" "copy expression to clipboard"] ["📋"],
h "button" [cn "pointer ba br3", on_click (λ _, action.on_close_tooltip), attr.val "title" "close"] ["×"]
]),
content
]]
else pure [],
(m, block_attrs) ← get_block_attrs m,
let as := [className "expr-boundary", key (ea)] ++ select_attrs ++ click_attrs ++ block_attrs,
inner ← view (e,new_address) m,
pure [h "span" as inner]
| ca (sf.compose x y) := pure (++) <*> view ca x <*> view ca y
| ca (sf.of_string s) := pure
[h "span" [
on_mouse_enter (λ _, action.on_mouse_enter ca),
on_click (λ _, action.on_click ca),
key s
] [html.of_string s]]
| ca b@(sf.block _ _) := do
(a, attrs) ← get_block_attrs b,
inner ← view ca a,
pure [h "span" attrs inner]
| ca b@(sf.highlight _ _) := do
(a, attrs) ← get_block_attrs b,
inner ← view ca a,
pure [h "span" attrs inner]
/-- Make an interactive expression. -/
meta def mk {γ} (tooltip : tc subexpr γ) : tc expr γ :=
let tooltip_comp :=
component.with_should_update (λ (x y : tactic_state × expr × expr.address), x.2.2 ≠ y.2.2)
$ component.map_action (action.on_tooltip_action) tooltip in
component.filter_map_action
(λ _ (a : γ ⊕ widget.effect), sum.cases_on a some (λ _, none))
$ component.with_effects (λ _ (a : γ ⊕ widget.effect),
match a with
| (sum.inl g) := []
| (sum.inr s) := [s]
end
)
$ tc.mk_simple
(action γ)
(option subexpr × option subexpr)
(λ e, pure $ (none, none))
(λ e ⟨ca, sa⟩ act, pure $
match act with
| (action.on_mouse_enter ⟨e, ea⟩) := ((ca, some (e, ea)), none)
| (action.on_mouse_leave_all) := ((ca, none), none)
| (action.on_click ⟨e, ea⟩) := if some (e,ea) = ca then ((none, sa), none) else ((some (e, ea), sa), none)
| (action.on_tooltip_action g) := ((none, sa), some $ sum.inl g)
| (action.on_close_tooltip) := ((none, sa), none)
| (action.effect e) := ((ca,sa), some $ sum.inr $ e)
end
)
(λ e ⟨ca, sa⟩, do
m ← sf.of_eformat <$> tactic.pp_tagged e,
let m := m.elim_part_apps,
let m := m.flatten,
let m := m.tag_expr [] e, -- [hack] in pp.cpp I forgot to add an expr-boundary for the root expression.
v ← view tooltip_comp (prod.snd <$> ca) (prod.snd <$> sa) ⟨e, []⟩ m,
pure $
[ h "span" [
className "expr",
key e.hash,
on_mouse_leave (λ _, action.on_mouse_leave_all) ] $ v
]
)
/-- Render the implicit arguments for an expression in fancy, little pills. -/
meta def implicit_arg_list (tooltip : tc subexpr empty) (e : expr) : tactic $ html empty := do
fn ← (mk tooltip) $ expr.get_app_fn e,
args ← list.mmap (mk tooltip) $ expr.get_app_args e,
pure $ h "div" [style [("display", "flex"), ("flexWrap", "wrap"), ("alignItems", "baseline")]]
( (h "span" [className "bg-blue br3 ma1 ph2 white"] [fn]) ::
list.map (λ a, h "span" [className "bg-gray br3 ma1 ph2 white"] [a]) args
)
/--
Component for the type tooltip.
-/
meta def type_tooltip : tc subexpr empty :=
tc.stateless (λ ⟨e,ea⟩, do
y ← tactic.infer_type e,
y_comp ← mk type_tooltip y,
implicit_args ← implicit_arg_list type_tooltip e,
pure [
h "div" [style [
("minWidth", "8rem"),
-- [note]: textIndent is inherited, and we might
-- be in an expression here where textIndent is set
("textIndent", "0")]
] [
h "div" [cn "pl1"] [y_comp],
h "hr" [] [],
implicit_args
]
]
)
end interactive_expression
/--
Supported tactic state filters.
-/
@[derive decidable_eq]
meta inductive filter_type
| none
| no_instances
| only_props
/--
Filters a local constant using the given filter.
-/
meta def filter_local : filter_type → expr → tactic bool
| (filter_type.none) e := pure tt
| (filter_type.no_instances) e := do
t ← tactic.infer_type e,
bnot <$> tactic.is_class t
| (filter_type.only_props) e := do
t ← tactic.infer_type e,
tactic.is_prop t
/--
Component for the filter dropdown.
-/
meta def filter_component : component filter_type filter_type :=
component.stateless (λ lf,
[ h "label" [] ["filter: "],
select [
⟨filter_type.none, "0", ["no filter"]⟩,
⟨filter_type.no_instances, "1", ["no instances"]⟩,
⟨filter_type.only_props, "2", ["only props"]⟩
] lf
]
)
/--
Converts a name into an html element.
-/
meta def html.of_name {α : Type} : name → html α
| n := html.of_string $ name.to_string n
open tactic
/--
Component that shows a type.
-/
meta def show_type_component : tc expr empty :=
tc.stateless (λ x, do
y ← infer_type x,
y_comp ← interactive_expression.mk interactive_expression.type_tooltip $ y,
pure y_comp
)
/-- A group of local constants in the context that should be rendered as one line. -/
@[derive decidable_eq]
meta structure local_collection :=
(key : string)
(locals : list expr)
(type : expr)
(value : option expr)
/-- Converts a single local constant into a (singleton) `local_collection` -/
meta def to_local_collection (l : expr) : tactic local_collection :=
tactic.unsafe.type_context.run $ do
lctx ← tactic.unsafe.type_context.get_local_context,
some ldecl ← pure $ lctx.get_local_decl l.local_uniq_name,
pure {
key := l.local_uniq_name.repr,
locals := [l],
type := ldecl.type,
value := ldecl.value }
/-- Groups consecutive local collections by type -/
meta def group_local_collection : list local_collection → list local_collection
| (a :: b :: rest) :=
if a.type = b.type ∧ a.value = b.value then
group_local_collection $
{ locals := a.locals ++ b.locals, ..a } :: rest
else
a :: group_local_collection (b :: rest)
| ls := ls
/-- Component that displays the main (first) goal. -/
meta def tactic_view_goal {γ} (local_c : tc local_collection γ) (target_c : tc expr γ) : tc filter_type γ :=
tc.stateless $ λ ft, do
g@(expr.mvar u_n pp_n y) ← main_goal,
t ← get_tag g,
let case_tag : list (html γ) :=
match interactive.case_tag.parse t with
| some t :=
[h "li" [key "_case"] $ [h "span" [cn "goal-case b"] ["case"]] ++
(t.case_names.bind $ λ n, [" ", n])]
| none := []
end,
lcs ← local_context,
lcs ← list.mfilter (filter_local ft) lcs,
lcs ← lcs.mmap $ to_local_collection,
let lcs := group_local_collection lcs,
lchs ← lcs.mmap (λ lc, do
lh ← local_c lc,
let ns : list (html γ) := lc.locals.map $ λ n,
h "span" [cn "goal-hyp b pr2", key n.local_uniq_name] [html.of_name n.local_pp_name],
pure $ h "li" [key lc.key] (ns ++ [": ", h "span" [cn "goal-hyp-type", key "type"] [lh]])),
t_comp ← target_c g,
pure $ h "ul" [key g.hash, className "list pl0 font-code"] $ case_tag ++ lchs ++ [
h "li" [key u_n] [
h "span" [cn "goal-vdash b"] ["⊢ "],
t_comp
]]
/--
Actions accepted by the `tactic_view_component`.
-/
meta inductive tactic_view_action (γ : Type)
| out (a:γ): tactic_view_action
| filter (f: filter_type): tactic_view_action
/--
The "goals accomplished 🎉" HTML widget. This can be overridden using:
```lean
meta def my_new_msg {α : Type} : widget.html α := "my message"
attribute [vm_override my_new_msg] widget_override.goals_accomplished_message
```
-/
meta def goals_accomplished_message {α} : html α :=
h "div" [cn "f5"] ["goals accomplished 🎉"]
/-- Component that displays all goals, together with the `$n goals` message. -/
meta def tactic_view_component {γ} (local_c : tc local_collection γ) (target_c : tc expr γ) : tc unit γ :=
tc.mk_simple
(tactic_view_action γ)
(filter_type)
(λ _, pure $ filter_type.none)
(λ ⟨⟩ ft a, match a with
| (tactic_view_action.out a) := pure (ft, some a)
| (tactic_view_action.filter ft) := pure (ft, none)
end)
(λ ⟨⟩ ft, do
gs ← get_goals,
hs ← gs.mmap (λ g, do set_goals [g], flip tc.to_html ft $ tactic_view_goal local_c target_c),
set_goals gs,
let goal_message : html γ :=
if gs.length = 0 then
goals_accomplished_message
else if gs.length = 1 then
"1 goal"
else
html.of_string $ to_string gs.length ++ " goals",
let goal_message : html γ := h "strong" [cn "goal-goals"] [goal_message],
let goals : html γ := h "ul" [className "list pl0"]
$ list.map_with_index (λ i x,
h "li" [className $ "lh-copy mt2", key i] [x])
$ (goal_message :: hs),
pure [
h "div" [className "fr"] [html.of_component ft $ component.map_action tactic_view_action.filter filter_component],
html.map_action tactic_view_action.out goals
])
/-- Component that displays the term-mode goal. -/
meta def tactic_view_term_goal {γ} (local_c : tc local_collection γ) (target_c : tc expr γ) : tc unit γ :=
tc.stateless $ λ _, do
goal ← flip tc.to_html (filter_type.none) $ tactic_view_goal local_c target_c,
pure [h "ul" [className "list pl0"] [
h "li" [className "lh-copy"] [h "strong" [cn "goal-goals"] ["expected type:"]],
h "li" [className "lh-copy"] [goal]]]
/--
Component showing a local collection.
-/
meta def show_local_collection_component : tc local_collection empty :=
tc.stateless (λ lc, do
(l::_) ← pure lc.locals,
c ← show_type_component l,
match lc.value with
| some v := do
v ← interactive_expression.mk interactive_expression.type_tooltip v,
pure [c, " := ", v]
| none := pure [c]
end)
/--
Renders the current tactic state.
-/
meta def tactic_render : tc unit empty :=
component.ignore_action $ tactic_view_component show_local_collection_component show_type_component
/--
Component showing the current tactic state.
-/
meta def tactic_state_widget : component tactic_state empty :=
tc.to_component tactic_render
/--
Widget used to display term-proof goals.
-/
meta def term_goal_widget : component tactic_state empty :=
(tactic_view_term_goal show_local_collection_component show_type_component).to_component
end widget_override
attribute [vm_override widget_override.term_goal_widget] widget.term_goal_widget
attribute [vm_override widget_override.tactic_state_widget] widget.tactic_state_widget
|
e2a6ca37ea204a594a2a9055cedc4d2526804aea | 6772a11d96d69b3f90d6eeaf7f9accddf2a7691d | /natural_transformation.lean | 4a9abe7002cbf7f4bd9534c4609ee98f4c36bf0d | [] | no_license | lbordowitz/lean-category-theory | 5397361f0f81037d65762da48de2c16ec85a5e4b | 8c59893e44af3804eba4dbc5f7fa5928ed2e0ae6 | refs/heads/master | 1,611,310,752,156 | 1,487,070,172,000 | 1,487,070,172,000 | 82,003,141 | 0 | 0 | null | 1,487,118,553,000 | 1,487,118,553,000 | null | UTF-8 | Lean | false | false | 6,413 | lean | -- Copyright (c) 2017 Scott Morrison. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Authors: Stephen Morgan, Scott Morrison
import .category
import .functor
open tqft.categories
open tqft.categories.functor
namespace tqft.categories.natural_transformation
structure NaturalTransformation { C D : Category } ( F G : Functor C D ) :=
(components: Π X : C^.Obj, D^.Hom (F X) (G X))
(naturality: ∀ { X Y : C^.Obj } (f : C^.Hom X Y),
D^.compose (F^.onMorphisms f) (components Y) = D^.compose (components X) (G^.onMorphisms f))
-- This defines a coercion so we can write `α X` for `components α X`.
instance NaturalTransformation_to_components { C D : Category } { F G : Functor C D } : has_coe_to_fun (NaturalTransformation F G) :=
{ F := λ f, Π X : C^.Obj, D^.Hom (F X) (G X),
coe := NaturalTransformation.components }
-- TODO can we simplify the next proof, by first using:
-- attribute [pointwise] funext
-- and then a bit of automation?
-- We'll want to be able to prove that two natural transformations are equal if they are componentwise equal.
@[pointwise] lemma NaturalTransformations_componentwise_equal
{ C D : Category }
{ F G : Functor C D }
( α β : NaturalTransformation F G )
( w : ∀ X : C^.Obj, α X = β X ) : α = β :=
begin
induction α with αc,
induction β with βc,
have hc : αc = βc, from funext w,
by subst hc
end
@[reducible] definition IdentityNaturalTransformation { C D : Category } (F : Functor C D) : NaturalTransformation F F :=
{
components := λ X, D^.identity (F X),
naturality := ♮
}
@[reducible] definition vertical_composition_of_NaturalTransformations
{ C D : Category }
{ F G H : Functor C D }
( α : NaturalTransformation F G )
( β : NaturalTransformation G H ) : NaturalTransformation F H :=
{
components := λ X, D^.compose (α X) (β X),
naturality := begin
blast,
/- this proof was written by a sufficient stupid process that I am confident a computer
-- could have done it! -/
rewrite D^.associativity,
rewrite - β^.naturality,
rewrite - D^.associativity,
rewrite α^.naturality,
rewrite D^.associativity
end
}
open tqft.categories.functor
@[reducible] definition horizontal_composition_of_NaturalTransformations
{ C D E : Category }
{ F G : Functor C D }
{ H I : Functor D E }
( α : NaturalTransformation F G )
( β : NaturalTransformation H I ) : NaturalTransformation (FunctorComposition F H) (FunctorComposition G I) :=
{
components := λ X : C^.Obj, E^.compose (β (F X)) (I^.onMorphisms (α X)),
naturality := begin
blast,
-- This is obscene! What is the state of automation?
-- Ideally we could just say:
-- some_tactic [ α^.naturality, β^.naturality, E^.associativity, H^.functoriality ]
-- which would search for sequences of rewrites along these equations (and their reverses)
-- or even just
-- some_other_tactic [ α, β, E, H ]
-- which would apply some_tactic using all the equations available as fields on these hypotheses,
-- or even just
-- some_other_tactic
-- which would consider all the available hypotheses.
rewrite - β^.naturality,
rewrite - β^.naturality,
rewrite - E^.associativity,
rewrite - H^.functoriality,
rewrite α^.naturality,
rewrite H^.functoriality,
rewrite E^.associativity,
rewrite E^.associativity,
rewrite β^.naturality
end
}
definition whisker_on_left
{ C D E : Category }
( F : Functor C D )
{ G H : Functor D E }
( α : NaturalTransformation G H ) :
NaturalTransformation (FunctorComposition F G) (FunctorComposition F H) :=
horizontal_composition_of_NaturalTransformations (IdentityNaturalTransformation F) α
definition whisker_on_right
{ C D E : Category }
{ F G : Functor C D }
( α : NaturalTransformation F G )
( H : Functor D E ) :
NaturalTransformation (FunctorComposition F H) (FunctorComposition G H) :=
horizontal_composition_of_NaturalTransformations α (IdentityNaturalTransformation H)
-- To define a natural isomorphism, we'll define the functor category, and ask for an isomorphism there.
-- It's then a lemma that each component is an isomorphism, and vice versa.
@[reducible] definition FunctorCategory ( C D : Category ) : Category :=
{
Obj := Functor C D,
Hom := λ F G, NaturalTransformation F G,
identity := λ F, IdentityNaturalTransformation F,
compose := @vertical_composition_of_NaturalTransformations C D,
left_identity := ♮,
right_identity := ♮,
associativity := begin
blast,
rewrite [ D^.associativity ]
end
}
definition NaturalIsomorphism { C D : Category } ( F G : Functor C D ) := Isomorphism (FunctorCategory C D) F G
-- It's a pity we need to separately define this coercion.
-- Ideally the coercion from Isomorphism along .morphism would just apply here.
-- Somehow we want the definition above to be more transparent?
instance NaturalIsomorphism_coercion_to_NaturalTransformation { C D : Category } { F G : Functor C D } : has_coe (NaturalIsomorphism F G) (NaturalTransformation F G) :=
{ coe := Isomorphism.morphism }
open NaturalTransformation
-- Getting this coercion to work is really painful. We shouldn't have to write
-- @components C D F G α X
-- below, but rather just:
-- α X
-- or at least
-- α^.components X
-- lemma components_of_NaturalIsomorphism_are_isomorphisms { C D : Category } { F G : Functor C D } { α : NaturalIsomorphism F G } { X : C^.Obj } :
-- Inverse (@components C D F G α X) :=
-- {
-- inverse := α^.inverse^.components X,
-- witness_1 := α^.witness_1, -- TODO we need to evaluate both sides of this equation at X.
-- witness_2 := sorry
-- }
end tqft.categories.natural_transformation
|
73a924dce6b41e7546600c1b37a70ce4066dbd10 | 64874bd1010548c7f5a6e3e8902efa63baaff785 | /hott/init/types/sigma.hlean | 316bb4d5eb6760e3c21b55a0c69ee17fe464e209 | [
"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 | 606 | hlean | /-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Floris van Doorn
-/
prelude
import init.num
structure sigma {A : Type} (B : A → Type) :=
mk :: (pr1 : A) (pr2 : B pr1)
notation `Σ` binders `,` r:(scoped P, sigma P) := r
namespace sigma
notation `pr₁` := pr1
notation `pr₂` := pr2
notation `⟨`:max t:(foldr `,` (e r, mk e r)) `⟩`:0 := t --input ⟨ ⟩ as \< \>
namespace ops
postfix `.1`:(max+1) := pr1
postfix `.2`:(max+1) := pr2
end ops
end sigma
|
b2ae55022ef4194bb61d7e0831f843c7eaea016a | dc253be9829b840f15d96d986e0c13520b085033 | /algebra/free_abelian_group.hlean | d86c1584e34295ef3c5c92cce4b7c4bf01431248 | [
"Apache-2.0"
] | permissive | cmu-phil/Spectral | 4ce68e5c1ef2a812ffda5260e9f09f41b85ae0ea | 3b078f5f1de251637decf04bd3fc8aa01930a6b3 | refs/heads/master | 1,685,119,195,535 | 1,684,169,772,000 | 1,684,169,772,000 | 46,450,197 | 42 | 13 | null | 1,505,516,767,000 | 1,447,883,921,000 | Lean | UTF-8 | Lean | false | false | 9,415 | hlean | /-
Copyright (c) 2015 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Egbert Rijke
Constructions with groups
-/
import algebra.group_theory hit.set_quotient types.list types.sum .free_group
open eq algebra is_trunc set_quotient relation sigma sigma.ops prod sum list trunc function equiv trunc_index
group pointed
namespace group
variables {G G' : Group} {g g' h h' k : G} {A B : AbGroup}
variables (X : Type*) {Y : Type*} [is_set X] [is_set Y] {l l' : list (X ⊎ X)}
/- Free Abelian Group on a pointed set -/
namespace free_ab_group
inductive fcg_rel : list (X ⊎ X) → list (X ⊎ X) → Type :=
| rrefl : Πl, fcg_rel l l
| cancel1 : Πx, fcg_rel [inl x, inr x] []
| cancel2 : Πx, fcg_rel [inr x, inl x] []
| cancelpt1 : fcg_rel [inl pt] []
| cancelpt2 : fcg_rel [inr pt] []
| rflip : Πx y, fcg_rel [x, y] [y, x]
| resp_append : Π{l₁ l₂ l₃ l₄}, fcg_rel l₁ l₂ → fcg_rel l₃ l₄ →
fcg_rel (l₁ ++ l₃) (l₂ ++ l₄)
| rtrans : Π{l₁ l₂ l₃}, fcg_rel l₁ l₂ → fcg_rel l₂ l₃ →
fcg_rel l₁ l₃
open fcg_rel
local abbreviation R [reducible] := fcg_rel
attribute fcg_rel.rrefl [refl]
attribute fcg_rel.rtrans [trans]
definition fcg_carrier [reducible] : Type := set_quotient (λx y, ∥R X x y∥)
local abbreviation FG := fcg_carrier
definition is_reflexive_R : is_reflexive (λx y, ∥R X x y∥) :=
begin constructor, intro s, apply tr, unfold R end
local attribute is_reflexive_R [instance]
variable {X}
theorem rel_respect_flip (r : R X l l') : R X (map sum.flip l) (map sum.flip l') :=
begin
induction r with l x x x y l₁ l₂ l₃ l₄ r₁ r₂ IH₁ IH₂ l₁ l₂ l₃ r₁ r₂ IH₁ IH₂,
{ reflexivity},
{ repeat esimp [map], exact cancel2 x},
{ repeat esimp [map], exact cancel1 x},
{ exact cancelpt2 X },
{ exact cancelpt1 X },
{ repeat esimp [map], apply fcg_rel.rflip},
{ rewrite [+map_append], exact resp_append IH₁ IH₂},
{ exact rtrans IH₁ IH₂}
end
theorem rel_respect_reverse (r : R X l l') : R X (reverse l) (reverse l') :=
begin
induction r with l x x x y l₁ l₂ l₃ l₄ r₁ r₂ IH₁ IH₂ l₁ l₂ l₃ r₁ r₂ IH₁ IH₂,
{ reflexivity},
{ repeat esimp [map], exact cancel2 x},
{ repeat esimp [map], exact cancel1 x},
{ exact cancelpt1 X },
{ exact cancelpt2 X },
{ repeat esimp [map], apply fcg_rel.rflip},
{ rewrite [+reverse_append], exact resp_append IH₂ IH₁},
{ exact rtrans IH₁ IH₂}
end
theorem rel_cons_concat (l s) : R X (s :: l) (concat s l) :=
begin
induction l with t l IH,
{ reflexivity},
{ rewrite [concat_cons], transitivity (t :: s :: l),
{ exact resp_append !rflip !rrefl},
{ exact resp_append (rrefl [t]) IH}}
end
definition fcg_one [constructor] : FG X := class_of []
definition fcg_inv [unfold 3] : FG X → FG X :=
quotient_unary_map (reverse ∘ map sum.flip)
(λl l', trunc_functor -1 (rel_respect_reverse ∘ rel_respect_flip))
definition fcg_mul [unfold 3 4] : FG X → FG X → FG X :=
quotient_binary_map append (λl l', trunc.elim (λr m m', trunc.elim (λs, tr (resp_append r s))))
section
local notation 1 := fcg_one
local postfix ⁻¹ := fcg_inv
local infix * := fcg_mul
theorem fcg_mul_assoc (g₁ g₂ g₃ : FG X) : g₁ * g₂ * g₃ = g₁ * (g₂ * g₃) :=
begin
refine set_quotient.rec_prop _ g₁,
refine set_quotient.rec_prop _ g₂,
refine set_quotient.rec_prop _ g₃,
clear g₁ g₂ g₃, intro g₁ g₂ g₃,
exact ap class_of !append.assoc
end
theorem fcg_one_mul (g : FG X) : 1 * g = g :=
begin
refine set_quotient.rec_prop _ g, clear g, intro g,
exact ap class_of !append_nil_left
end
theorem fcg_mul_one (g : FG X) : g * 1 = g :=
begin
refine set_quotient.rec_prop _ g, clear g, intro g,
exact ap class_of !append_nil_right
end
theorem fcg_mul_left_inv (g : FG X) : g⁻¹ * g = 1 :=
begin
refine set_quotient.rec_prop _ g, clear g, intro g,
apply eq_of_rel, apply tr,
induction g with s l IH,
{ reflexivity},
{ rewrite [▸*, map_cons, reverse_cons, concat_append],
refine rtrans _ IH,
apply resp_append, reflexivity,
change R X ([flip s, s] ++ l) ([] ++ l),
apply resp_append,
induction s, apply cancel2, apply cancel1,
reflexivity}
end
theorem fcg_mul_comm (g h : FG X) : g * h = h * g :=
begin
refine set_quotient.rec_prop _ g, clear g, intro g,
refine set_quotient.rec_prop _ h, clear h, intro h,
apply eq_of_rel, apply tr,
revert h, induction g with s l IH: intro h,
{ rewrite [append_nil_left, append_nil_right]},
{ rewrite [append_cons,-concat_append],
transitivity concat s (l ++ h), apply rel_cons_concat,
rewrite [-append_concat], apply IH}
end
end
end free_ab_group open free_ab_group
variables (X)
definition group_free_ab_group [constructor] : ab_group (fcg_carrier X) :=
ab_group.mk _ fcg_mul fcg_mul_assoc fcg_one fcg_one_mul fcg_mul_one
fcg_inv fcg_mul_left_inv fcg_mul_comm
definition free_ab_group [constructor] : AbGroup :=
AbGroup.mk _ (group_free_ab_group X)
/- The universal property of the free commutative group -/
variables {X A}
definition free_ab_group_inclusion [constructor] : X →* free_ab_group X :=
ppi.mk (λ x, class_of [inl x]) (eq_of_rel (tr (fcg_rel.cancelpt1 X)))
theorem fgh_helper_respect_fcg_rel (f : X →* A) (r : fcg_rel X l l')
: Π(g : A), foldl (fgh_helper f) g l = foldl (fgh_helper f) g l' :=
begin
induction r with l x x x y l₁ l₂ l₃ l₄ r₁ r₂ IH₁ IH₂ l₁ l₂ l₃ r₁ r₂ IH₁ IH₂: intro g,
{ reflexivity},
{ unfold [foldl], apply mul_inv_cancel_right},
{ unfold [foldl], apply inv_mul_cancel_right},
{ unfold [foldl], rewrite (respect_pt f), apply mul_one },
{ unfold [foldl], rewrite [respect_pt f, one_inv], apply mul_one },
{ unfold [foldl, fgh_helper], apply mul.right_comm},
{ rewrite [+foldl_append, IH₁, IH₂]},
{ exact !IH₁ ⬝ !IH₂}
end
definition free_ab_group_elim [constructor] (f : X →* A) : free_ab_group X →g A :=
begin
fapply homomorphism.mk,
{ intro g, refine set_quotient.elim _ _ g,
{ intro l, exact foldl (fgh_helper f) 1 l},
{ intro l l' r, esimp at *, refine trunc.rec _ r, clear r, intro r,
exact fgh_helper_respect_fcg_rel f r 1}},
{ refine set_quotient.rec_prop _, intro l, refine set_quotient.rec_prop _, intro l',
esimp, refine !foldl_append ⬝ _, esimp, apply fgh_helper_mul}
end
definition fn_of_free_ab_group_elim [unfold_full] (φ : free_ab_group X →g A) : X →* A :=
ppi.mk (φ ∘ free_ab_group_inclusion)
begin
refine (_ ⬝ @respect_one _ _ _ _ φ (homomorphism.p φ)),
apply ap φ, apply eq_of_rel, apply tr,
exact (fcg_rel.cancelpt1 X)
end
definition free_ab_group_elim_unique [constructor] (f : X →* A) (k : free_ab_group X →g A)
(H : k ∘ free_ab_group_inclusion ~ f) : k ~ free_ab_group_elim f :=
begin
refine set_quotient.rec_prop _, intro l, esimp,
induction l with s l IH,
{ esimp [foldl], exact to_respect_one k},
{ rewrite [foldl_cons, fgh_helper_mul],
refine to_respect_mul k (class_of [s]) (class_of l) ⬝ _,
rewrite [IH], apply ap (λx, x * _), induction s: rewrite [▸*, one_mul, -H a],
apply to_respect_inv }
end
variables (X A)
definition free_ab_group_elim_equiv_fn [constructor] : (free_ab_group X →g A) ≃ (X →* A) :=
begin
fapply equiv.MK,
{ exact fn_of_free_ab_group_elim},
{ exact free_ab_group_elim},
{ intro f, apply eq_of_phomotopy, fapply phomotopy.mk,
{ intro x, esimp, unfold [foldl], apply one_mul },
{ apply is_prop.elim } },
{ intro k, symmetry, apply homomorphism_eq, apply free_ab_group_elim_unique,
reflexivity }
end
definition free_ab_group_functor (f : X →* Y) : free_ab_group X →g free_ab_group Y :=
free_ab_group_elim (free_ab_group_inclusion ∘* f)
-- set_option pp.all true
-- definition free_ab_group.rec {P : free_ab_group X → Type} [H : Πg, is_prop (P g)]
-- (h₁ : Πx, P (free_ab_group_inclusion x))
-- (h₂ : P 0)
-- (h₃ : Πg h, P g → P h → P (g * h))
-- (h₄ : Πg, P g → P g⁻¹) :
-- Πg, P g :=
-- begin
-- refine @set_quotient.rec_prop _ _ _ H _,
-- refine @set_quotient.rec_prop _ _ _ (λx, !H) _,
-- esimp, intro l, induction l with s l ih,
-- exact h₂,
-- induction s with v v,
-- induction v with i y,
-- exact h₃ _ _ (h₁ i y) ih,
-- induction v with i y,
-- refine h₃ (gqg_map _ _ (class_of [inr ⟨i, y⟩])) _ _ ih,
-- refine transport P _ (h₁ i y⁻¹),
-- refine _ ⬝ !mul_one,
-- refine _ ⬝ ap (mul _) (to_respect_one (dirsum_incl i)),
-- apply gqg_eq_of_rel',
-- apply tr, esimp,
-- refine transport dirsum_rel _ (dirsum_rel.rmk i y⁻¹ y),
-- rewrite [mul.left_inv, mul.assoc],
-- apply ap (mul _),
-- refine _ ⬝ (mul_inv (class_of [inr ⟨i, y⟩]) (ι ⟨i, 1⟩))⁻¹ᵖ,
-- refine ap011 mul _ _,
-- end
end group
|
7b041e9b8ef3e095ad8cb5084f8600392b98601d | 36c7a18fd72e5b57229bd8ba36493daf536a19ce | /tests/lean/run/blast1.lean | 103f0f8daa5bda9b7aa75dae2bddb79dfb5d053a | [
"Apache-2.0"
] | permissive | YHVHvx/lean | 732bf0fb7a298cd7fe0f15d82f8e248c11db49e9 | 038369533e0136dd395dc252084d3c1853accbf2 | refs/heads/master | 1,610,701,080,210 | 1,449,128,595,000 | 1,449,128,595,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 55 | lean | example (a b : Prop) (Ha : a) (Hb : b) : a :=
by blast
|
d394e83e7c53724a093044622d450e50e3df328a | e61a235b8468b03aee0120bf26ec615c045005d2 | /src/Init/Lean/Elab/SyntheticMVars.lean | bb1fccdc4985cd8123aa6e92eb15a613e55b0363 | [
"Apache-2.0"
] | permissive | SCKelemen/lean4 | 140dc63a80539f7c61c8e43e1c174d8500ec3230 | e10507e6615ddbef73d67b0b6c7f1e4cecdd82bc | refs/heads/master | 1,660,973,595,917 | 1,590,278,033,000 | 1,590,278,033,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 12,303 | lean | /-
Copyright (c) 2020 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.Elab.Term
import Init.Lean.Elab.Tactic.Basic
namespace Lean
namespace Elab
namespace Term
open Tactic (TacticM evalTactic getUnsolvedGoals)
def liftTacticElabM {α} (ref : Syntax) (mvarId : MVarId) (x : TacticM α) : TermElabM α :=
withMVarContext mvarId $ fun ctx s =>
let savedSyntheticMVars := s.syntheticMVars;
match x { ctx with ref := ref, main := mvarId } { s with goals := [mvarId], syntheticMVars := [] } with
| EStateM.Result.error ex newS => EStateM.Result.error (Term.Exception.ex ex) { newS.toTermState with syntheticMVars := savedSyntheticMVars }
| EStateM.Result.ok a newS => EStateM.Result.ok a { newS.toTermState with syntheticMVars := savedSyntheticMVars }
def ensureAssignmentHasNoMVars (ref : Syntax) (mvarId : MVarId) : TermElabM Unit := do
val ← instantiateMVars ref (mkMVar mvarId);
when val.hasExprMVar $ throwError ref ("tactic failed, result still contain metavariables" ++ indentExpr val)
def runTactic (ref : Syntax) (mvarId : MVarId) (tacticCode : Syntax) : TermElabM Unit := do
modify $ fun s => { s with mctx := s.mctx.instantiateMVarDeclMVars mvarId };
remainingGoals ← liftTacticElabM ref mvarId $ do { evalTactic tacticCode; getUnsolvedGoals };
let tailRef := ref.getTailWithPos.getD ref;
unless remainingGoals.isEmpty (reportUnsolvedGoals tailRef remainingGoals);
ensureAssignmentHasNoMVars tailRef mvarId
/-- Auxiliary function used to implement `synthesizeSyntheticMVars`. -/
private def resumeElabTerm (stx : Syntax) (expectedType? : Option Expr) (errToSorry := true) : TermElabM Expr :=
-- Remark: if `ctx.errToSorry` is already false, then we don't enable it. Recall tactics disable `errToSorry`
adaptReader (fun (ctx : Context) => { ctx with errToSorry := ctx.errToSorry && errToSorry }) $
elabTerm stx expectedType? false
/--
Try to elaborate `stx` that was postponed by an elaboration method using `Expection.postpone`.
It returns `true` if it succeeded, and `false` otherwise.
It is used to implement `synthesizeSyntheticMVars`. -/
private def resumePostponed (macroStack : MacroStack) (stx : Syntax) (mvarId : MVarId) (postponeOnError : Bool) : TermElabM Bool := do
withMVarContext mvarId $ do
s ← get;
catch
(adaptReader (fun (ctx : Context) => { ctx with macroStack := macroStack }) $ do
mvarDecl ← getMVarDecl mvarId;
expectedType ← instantiateMVars stx mvarDecl.type;
result ← resumeElabTerm stx expectedType (!postponeOnError);
/- We must ensure `result` has the expected type because it is the one expected by the method that postponed stx.
That is, the method does not have an opportunity to check whether `result` has the expected type or not. -/
result ← ensureHasType stx expectedType result;
assignExprMVar mvarId result;
pure true)
(fun ex => match ex with
| Exception.postpone => do set s; pure false
| Exception.ex Elab.Exception.unsupportedSyntax => unreachable!
| Exception.ex (Elab.Exception.error msg) =>
if postponeOnError then do
set s; pure false
else do
logMessage msg; pure true)
/--
Similar to `synthesizeInstMVarCore`, but makes sure that `instMVar` local context and instances
are used. It also logs any error message produced. -/
private def synthesizePendingInstMVar (ref : Syntax) (instMVar : MVarId) : TermElabM Bool := do
withMVarContext instMVar $ catch
(synthesizeInstMVarCore ref instMVar)
(fun ex => match ex with
| Exception.ex (Elab.Exception.error errMsg) => do logMessage errMsg; pure true
| _ => unreachable!)
/--
Similar to `synthesizePendingInstMVar`, but generates type mismatch error message. -/
private def synthesizePendingCoeInstMVar (ref : Syntax) (instMVar : MVarId) (expectedType : Expr) (eType : Expr) (e : Expr) (f? : Option Expr) : TermElabM Bool := do
withMVarContext instMVar $ catch
(synthesizeInstMVarCore ref instMVar)
(fun ex => match ex with
| Exception.ex (Elab.Exception.error errMsg) => throwTypeMismatchError ref expectedType eType e f? errMsg.data
| _ => unreachable!)
/--
Return `true` iff `mvarId` is assigned to a term whose the
head is not a metavariable. We use this method to process `SyntheticMVarKind.withDefault`. -/
private def checkWithDefault (ref : Syntax) (mvarId : MVarId) : TermElabM Bool := do
val ← instantiateMVars ref (mkMVar mvarId);
pure $ !val.getAppFn.isMVar
/-- Try to synthesize the given pending synthetic metavariable. -/
private def synthesizeSyntheticMVar (mvarSyntheticDecl : SyntheticMVarDecl) (postponeOnError : Bool) (runTactics : Bool) : TermElabM Bool :=
match mvarSyntheticDecl.kind with
| SyntheticMVarKind.typeClass => synthesizePendingInstMVar mvarSyntheticDecl.ref mvarSyntheticDecl.mvarId
| SyntheticMVarKind.coe expectedType eType e f? => synthesizePendingCoeInstMVar mvarSyntheticDecl.ref mvarSyntheticDecl.mvarId expectedType eType e f?
-- NOTE: actual processing at `synthesizeSyntheticMVarsAux`
| SyntheticMVarKind.withDefault _ => checkWithDefault mvarSyntheticDecl.ref mvarSyntheticDecl.mvarId
| SyntheticMVarKind.postponed macroStack => resumePostponed macroStack mvarSyntheticDecl.ref mvarSyntheticDecl.mvarId postponeOnError
| SyntheticMVarKind.tactic tacticCode =>
if runTactics then do
runTactic mvarSyntheticDecl.ref mvarSyntheticDecl.mvarId tacticCode;
pure true
else
pure false
/--
Try to synthesize the current list of pending synthetic metavariables.
Return `true` if at least one of them was synthesized. -/
private def synthesizeSyntheticMVarsStep (postponeOnError : Bool) (runTactics : Bool) : TermElabM Bool := do
ctx ← read;
traceAtCmdPos `Elab.resuming $ fun _ =>
fmt "resuming synthetic metavariables, mayPostpone: " ++ fmt ctx.mayPostpone ++ ", postponeOnError: " ++ toString postponeOnError;
s ← get;
let syntheticMVars := s.syntheticMVars;
let numSyntheticMVars := syntheticMVars.length;
-- We reset `syntheticMVars` because new synthetic metavariables may be created by `synthesizeSyntheticMVar`.
modify $ fun s => { s with syntheticMVars := [] };
-- Recall that `syntheticMVars` is a list where head is the most recent pending synthetic metavariable.
-- We use `filterRevM` instead of `filterM` to make sure we process the synthetic metavariables using the order they were created.
-- It would not be incorrect to use `filterM`.
remainingSyntheticMVars ← syntheticMVars.filterRevM $ fun mvarDecl => do {
trace `Elab.postpone mvarDecl.ref $ fun _ => "resuming ?" ++ mvarDecl.mvarId;
succeeded ← synthesizeSyntheticMVar mvarDecl postponeOnError runTactics;
trace `Elab.postpone mvarDecl.ref $ fun _ => if succeeded then fmt "succeeded" else fmt "not ready yet";
pure $ !succeeded
};
-- Merge new synthetic metavariables with `remainingSyntheticMVars`, i.e., metavariables that still couldn't be synthesized
modify $ fun s => { s with syntheticMVars := s.syntheticMVars ++ remainingSyntheticMVars };
pure $ numSyntheticMVars != remainingSyntheticMVars.length
/-- Apply default value to any pending synthetic metavariable of kind `SyntheticMVarKind.withDefault` -/
private def synthesizeUsingDefault : TermElabM Bool := do
s ← get;
let len := s.syntheticMVars.length;
newSyntheticMVars ← s.syntheticMVars.filterM $ fun mvarDecl =>
match mvarDecl.kind with
| SyntheticMVarKind.withDefault defaultVal => withMVarContext mvarDecl.mvarId $ do
val ← instantiateMVars mvarDecl.ref (mkMVar mvarDecl.mvarId);
when val.getAppFn.isMVar $
unlessM (isDefEq mvarDecl.ref val defaultVal) $
throwError mvarDecl.ref "failed to assign default value to metavariable"; -- TODO: better error message
pure false
| _ => pure true;
modify $ fun s => { s with syntheticMVars := newSyntheticMVars };
pure $ newSyntheticMVars.length != len
/-- Report an error for each synthetic metavariable that could not be resolved. -/
private def reportStuckSyntheticMVars : TermElabM Unit := do
s ← get;
s.syntheticMVars.forM $ fun mvarSyntheticDecl =>
match mvarSyntheticDecl.kind with
| SyntheticMVarKind.typeClass =>
withMVarContext mvarSyntheticDecl.mvarId $ do
mvarDecl ← getMVarDecl mvarSyntheticDecl.mvarId;
logError mvarSyntheticDecl.ref $
"failed to create type class instance for " ++ indentExpr mvarDecl.type
| SyntheticMVarKind.coe expectedType eType e f? =>
withMVarContext mvarSyntheticDecl.mvarId $ do
mvarDecl ← getMVarDecl mvarSyntheticDecl.mvarId;
throwTypeMismatchError mvarSyntheticDecl.ref expectedType eType e f? (some ("failed to create type class instance for " ++ indentExpr mvarDecl.type))
| _ => unreachable! -- TODO handle other cases.
private def getSomeSynthethicMVarsRef : TermElabM Syntax := do
s ← get;
match s.syntheticMVars.find? $ fun (mvarDecl : SyntheticMVarDecl) => !mvarDecl.ref.getPos.isNone with
| some mvarDecl => pure mvarDecl.ref
| none => pure Syntax.missing
/--
Main loop for `synthesizeSyntheticMVars.
It keeps executing `synthesizeSyntheticMVarsStep` while progress is being made.
If `mayPostpone == false`, then it applies default values to `SyntheticMVarKind.withDefault`
metavariables that are still unresolved, and then tries to resolve metavariables
with `mayPostpone == false`. That is, we force them to produce error messages and/or commit to
a "best option". If, after that, we still haven't made progress, we report "stuck" errors. -/
private partial def synthesizeSyntheticMVarsAux (mayPostpone := true) : Unit → TermElabM Unit
| _ => do
let try (x : TermElabM Bool) (k : TermElabM Unit) : TermElabM Unit := condM x (synthesizeSyntheticMVarsAux ()) k;
ref ← getSomeSynthethicMVarsRef;
withIncRecDepth ref $ do
s ← get;
unless s.syntheticMVars.isEmpty $ do
try (synthesizeSyntheticMVarsStep false false) $
unless mayPostpone $ do
/- Resume pending metavariables with "elaboration postponement" disabled.
We postpone elaboration errors in this step by setting `postponeOnError := true`.
Example:
```
#check let x := ⟨1, 2⟩; Prod.fst x
```
The term `⟨1, 2⟩` can't be elaborated because the expected type is not know.
The `x` at `Prod.fst x` is not elaborated because the type of `x` is not known.
When we execute the following step with "elaboration postponement" disabled,
the elaborator fails at `⟨1, 2⟩` and postpones it, and succeeds at `x` and learns
that its type must be of the form `Prod ?α ?β`.
Recall that we postponed `x` at `Prod.fst x` because its type it is not known.
We the type of `x` may learn later its type and it may contain implicit and/or auto arguments.
By disabling postponement, we are essentially giving up the opportunity of learning `x`s type
and assume it does not have implict and/or auto arguments. -/
try (withoutPostponing (synthesizeSyntheticMVarsStep true false)) $
try synthesizeUsingDefault $
try (withoutPostponing (synthesizeSyntheticMVarsStep false false)) $
try (synthesizeSyntheticMVarsStep false true) $
reportStuckSyntheticMVars
/--
Try to process pending synthetic metavariables. If `mayPostpone == false`,
then `syntheticMVars` is `[]` after executing this method. -/
def synthesizeSyntheticMVars (mayPostpone := true) : TermElabM Unit :=
synthesizeSyntheticMVarsAux mayPostpone ()
/-- Elaborate `stx`, and make sure all pending synthetic metavariables created while elaborating `stx` are solved. -/
def elabTermAndSynthesize (stx : Syntax) (expectedType? : Option Expr) : TermElabM Expr := do
s ← get;
let syntheticMVars := s.syntheticMVars;
modify $ fun s => { s with syntheticMVars := [] };
finally
(do
v ← elabTerm stx expectedType?;
synthesizeSyntheticMVars false;
instantiateMVars stx v)
(modify $ fun s => { s with syntheticMVars := s.syntheticMVars ++ syntheticMVars })
end Term
end Elab
end Lean
|
b5061de0f28d0ed627a746c9cadcbac16d987ed9 | 8f0ea954c1cc4f4cda83826954d6186c68fc9536 | /library/data/hf.lean | ccdab5c17b5fc1bd080579548dde07fd103ae738 | [
"Apache-2.0"
] | permissive | piyush-kurur/lean | fb3cbd339dfa8c70c49559fbea88ac0d3102b8ca | a8db8bc61a0b00379b3d0be8ecaf0d0858dc82ee | refs/heads/master | 1,594,387,140,961 | 1,457,548,608,000 | 1,457,909,538,000 | 53,615,930 | 0 | 0 | null | 1,457,642,939,000 | 1,457,642,939,000 | null | UTF-8 | Lean | false | false | 25,371 | 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
Hereditarily finite sets: finite sets whose elements are all hereditarily finite sets.
Remark: all definitions compute, however the performace is quite poor since
we implement this module using a bijection from (finset nat) to nat, and
this bijection is implemeted using the Ackermann coding.
-/
import data.nat data.finset.equiv data.list
open nat binary
open - [notation] finset
definition hf := nat
namespace hf
local attribute hf [reducible]
protected definition prio : num := num.succ std.priority.default
protected definition is_inhabited [instance] : inhabited hf :=
nat.is_inhabited
protected definition has_decidable_eq [instance] : decidable_eq hf :=
nat.has_decidable_eq
definition of_finset (s : finset hf) : hf :=
@equiv.to_fun _ _ finset_nat_equiv_nat s
definition to_finset (h : hf) : finset hf :=
@equiv.inv _ _ finset_nat_equiv_nat h
definition to_nat (h : hf) : nat :=
h
definition of_nat (n : nat) : hf :=
n
lemma to_finset_of_finset (s : finset hf) : to_finset (of_finset s) = s :=
@equiv.left_inv _ _ finset_nat_equiv_nat s
lemma of_finset_to_finset (s : hf) : of_finset (to_finset s) = s :=
@equiv.right_inv _ _ finset_nat_equiv_nat s
lemma to_finset_inj {s₁ s₂ : hf} : to_finset s₁ = to_finset s₂ → s₁ = s₂ :=
λ h, function.injective_of_left_inverse of_finset_to_finset h
lemma of_finset_inj {s₁ s₂ : finset hf} : of_finset s₁ = of_finset s₂ → s₁ = s₂ :=
λ h, function.injective_of_left_inverse to_finset_of_finset h
/- empty -/
definition empty : hf :=
of_finset (finset.empty)
notation `∅` := hf.empty
/- insert -/
definition insert (a s : hf) : hf :=
of_finset (finset.insert a (to_finset s))
/- mem -/
definition mem (a : hf) (s : hf) : Prop :=
finset.mem a (to_finset s)
infix ∈ := mem
notation [priority finset.prio] a ∉ b := ¬ mem a b
lemma insert_lt_of_not_mem {a s : hf} : a ∉ s → s < insert a s :=
begin
unfold [insert, of_finset, equiv.to_fun, finset_nat_equiv_nat, mem, to_finset, equiv.inv],
intro h,
rewrite [finset.to_nat_insert h],
rewrite [to_nat_of_nat, -zero_add s at {1}],
apply add_lt_add_right,
apply pow_pos_of_pos _ dec_trivial
end
lemma insert_lt_insert_of_not_mem_of_not_mem_of_lt {a s₁ s₂ : hf}
: a ∉ s₁ → a ∉ s₂ → s₁ < s₂ → insert a s₁ < insert a s₂ :=
begin
unfold [insert, of_finset, equiv.to_fun, finset_nat_equiv_nat, mem, to_finset, equiv.inv],
intro h₁ h₂ h₃,
rewrite [finset.to_nat_insert h₁],
rewrite [finset.to_nat_insert h₂, *to_nat_of_nat],
apply add_lt_add_left h₃
end
open decidable
protected definition decidable_mem [instance] : ∀ a s, decidable (a ∈ s) :=
λ a s, finset.decidable_mem a (to_finset s)
lemma insert_le (a s : hf) : s ≤ insert a s :=
by_cases
(suppose a ∈ s, by rewrite [↑insert, insert_eq_of_mem this, of_finset_to_finset])
(suppose a ∉ s, le_of_lt (insert_lt_of_not_mem this))
lemma not_mem_empty (a : hf) : a ∉ ∅ :=
begin unfold [mem, empty], rewrite to_finset_of_finset, apply finset.not_mem_empty end
lemma mem_insert (a s : hf) : a ∈ insert a s :=
begin unfold [mem, insert], rewrite to_finset_of_finset, apply finset.mem_insert end
lemma mem_insert_of_mem {a s : hf} (b : hf) : a ∈ s → a ∈ insert b s :=
begin unfold [mem, insert], intros, rewrite to_finset_of_finset, apply finset.mem_insert_of_mem, assumption end
lemma eq_or_mem_of_mem_insert {a b s : hf} : a ∈ insert b s → a = b ∨ a ∈ s :=
begin unfold [mem, insert], rewrite to_finset_of_finset, intros, apply eq_or_mem_of_mem_insert, assumption end
theorem mem_of_mem_insert_of_ne {x a : hf} {s : hf} : x ∈ insert a s → x ≠ a → x ∈ s :=
begin unfold [mem, insert], rewrite to_finset_of_finset, intros, apply mem_of_mem_insert_of_ne, repeat assumption end
protected theorem ext {s₁ s₂ : hf} : (∀ a, a ∈ s₁ ↔ a ∈ s₂) → s₁ = s₂ :=
assume h,
have to_finset s₁ = to_finset s₂, from finset.ext h,
have of_finset (to_finset s₁) = of_finset (to_finset s₂), by rewrite this,
by rewrite [*of_finset_to_finset at this]; exact this
theorem insert_eq_of_mem {a : hf} {s : hf} : a ∈ s → insert a s = s :=
begin unfold mem, intro h, unfold [mem, insert], rewrite (finset.insert_eq_of_mem h), rewrite of_finset_to_finset end
protected theorem induction [recursor 4] {P : hf → Prop}
(h₁ : P empty) (h₂ : ∀ (a s : hf), a ∉ s → P s → P (insert a s)) (s : hf) : P s :=
have P (of_finset (to_finset s)), from
@finset.induction _ _ _ h₁
(λ a s nain ih,
begin
unfold [mem, insert] at h₂,
rewrite -(to_finset_of_finset s) at nain,
have P (insert a (of_finset s)), by exact h₂ a (of_finset s) nain ih,
rewrite [↑insert at this, to_finset_of_finset at this],
exact this
end)
(to_finset s),
by rewrite of_finset_to_finset at this; exact this
lemma insert_le_insert_of_le {a s₁ s₂ : hf} : a ∈ s₁ ∨ a ∉ s₂ → s₁ ≤ s₂ → insert a s₁ ≤ insert a s₂ :=
suppose a ∈ s₁ ∨ a ∉ s₂,
suppose s₁ ≤ s₂,
by_cases
(suppose s₁ = s₂, by rewrite this)
(suppose s₁ ≠ s₂,
have s₁ < s₂, from lt_of_le_of_ne `s₁ ≤ s₂` `s₁ ≠ s₂`,
by_cases
(suppose a ∈ s₁, by_cases
(suppose a ∈ s₂, by rewrite [insert_eq_of_mem `a ∈ s₁`, insert_eq_of_mem `a ∈ s₂`]; assumption)
(suppose a ∉ s₂, by rewrite [insert_eq_of_mem `a ∈ s₁`]; exact le.trans `s₁ ≤ s₂` !insert_le))
(suppose a ∉ s₁, by_cases
(suppose a ∈ s₂, or.elim `a ∈ s₁ ∨ a ∉ s₂` (by contradiction) (by contradiction))
(suppose a ∉ s₂, le_of_lt (insert_lt_insert_of_not_mem_of_not_mem_of_lt `a ∉ s₁` `a ∉ s₂` `s₁ < s₂`))))
/- union -/
definition union (s₁ s₂ : hf) : hf :=
of_finset (finset.union (to_finset s₁) (to_finset s₂))
infix [priority hf.prio] ∪ := union
theorem mem_union_left {a : hf} {s₁ : hf} (s₂ : hf) : a ∈ s₁ → a ∈ s₁ ∪ s₂ :=
begin unfold mem, intro h, unfold union, rewrite to_finset_of_finset, apply finset.mem_union_left _ h end
theorem mem_union_l {a : hf} {s₁ : hf} {s₂ : hf} : a ∈ s₁ → a ∈ s₁ ∪ s₂ :=
mem_union_left s₂
theorem mem_union_right {a : hf} {s₂ : hf} (s₁ : hf) : a ∈ s₂ → a ∈ s₁ ∪ s₂ :=
begin unfold mem, intro h, unfold union, rewrite to_finset_of_finset, apply finset.mem_union_right _ h end
theorem mem_union_r {a : hf} {s₂ : hf} {s₁ : hf} : a ∈ s₂ → a ∈ s₁ ∪ s₂ :=
mem_union_right s₁
theorem mem_or_mem_of_mem_union {a : hf} {s₁ s₂ : hf} : a ∈ s₁ ∪ s₂ → a ∈ s₁ ∨ a ∈ s₂ :=
begin unfold [mem, union], rewrite to_finset_of_finset, intro h, apply finset.mem_or_mem_of_mem_union h end
theorem mem_union_iff {a : hf} (s₁ s₂ : hf) : a ∈ s₁ ∪ s₂ ↔ a ∈ s₁ ∨ a ∈ s₂ :=
iff.intro
(λ h, mem_or_mem_of_mem_union h)
(λ d, or.elim d
(λ i, mem_union_left _ i)
(λ i, mem_union_right _ i))
theorem mem_union_eq {a : hf} (s₁ s₂ : hf) : (a ∈ s₁ ∪ s₂) = (a ∈ s₁ ∨ a ∈ s₂) :=
propext !mem_union_iff
theorem union_comm (s₁ s₂ : hf) : s₁ ∪ s₂ = s₂ ∪ s₁ :=
hf.ext (λ a, by rewrite [*mem_union_eq]; exact or.comm)
theorem union_assoc (s₁ s₂ s₃ : hf) : (s₁ ∪ s₂) ∪ s₃ = s₁ ∪ (s₂ ∪ s₃) :=
hf.ext (λ a, by rewrite [*mem_union_eq]; exact or.assoc)
theorem union_left_comm (s₁ s₂ s₃ : hf) : s₁ ∪ (s₂ ∪ s₃) = s₂ ∪ (s₁ ∪ s₃) :=
!left_comm union_comm union_assoc s₁ s₂ s₃
theorem union_right_comm (s₁ s₂ s₃ : hf) : (s₁ ∪ s₂) ∪ s₃ = (s₁ ∪ s₃) ∪ s₂ :=
!right_comm union_comm union_assoc s₁ s₂ s₃
theorem union_self (s : hf) : s ∪ s = s :=
hf.ext (λ a, iff.intro
(λ ain, or.elim (mem_or_mem_of_mem_union ain) (λ i, i) (λ i, i))
(λ i, mem_union_left _ i))
theorem union_empty (s : hf) : s ∪ ∅ = s :=
hf.ext (λ a, iff.intro
(suppose a ∈ s ∪ ∅, or.elim (mem_or_mem_of_mem_union this) (λ i, i) (λ i, absurd i !not_mem_empty))
(suppose a ∈ s, mem_union_left _ this))
theorem empty_union (s : hf) : ∅ ∪ s = s :=
calc ∅ ∪ s = s ∪ ∅ : union_comm
... = s : union_empty
/- inter -/
definition inter (s₁ s₂ : hf) : hf :=
of_finset (finset.inter (to_finset s₁) (to_finset s₂))
infix [priority hf.prio] ∩ := inter
theorem mem_of_mem_inter_left {a : hf} {s₁ s₂ : hf} : a ∈ s₁ ∩ s₂ → a ∈ s₁ :=
begin unfold mem, unfold inter, rewrite to_finset_of_finset, intro h, apply finset.mem_of_mem_inter_left h end
theorem mem_of_mem_inter_right {a : hf} {s₁ s₂ : hf} : a ∈ s₁ ∩ s₂ → a ∈ s₂ :=
begin unfold mem, unfold inter, rewrite to_finset_of_finset, intro h, apply finset.mem_of_mem_inter_right h end
theorem mem_inter {a : hf} {s₁ s₂ : hf} : a ∈ s₁ → a ∈ s₂ → a ∈ s₁ ∩ s₂ :=
begin unfold mem, intro h₁ h₂, unfold inter, rewrite to_finset_of_finset, apply finset.mem_inter h₁ h₂ end
theorem mem_inter_iff (a : hf) (s₁ s₂ : hf) : a ∈ s₁ ∩ s₂ ↔ a ∈ s₁ ∧ a ∈ s₂ :=
iff.intro
(λ h, and.intro (mem_of_mem_inter_left h) (mem_of_mem_inter_right h))
(λ h, mem_inter (and.elim_left h) (and.elim_right h))
theorem mem_inter_eq (a : hf) (s₁ s₂ : hf) : (a ∈ s₁ ∩ s₂) = (a ∈ s₁ ∧ a ∈ s₂) :=
propext !mem_inter_iff
theorem inter_comm (s₁ s₂ : hf) : s₁ ∩ s₂ = s₂ ∩ s₁ :=
hf.ext (λ a, by rewrite [*mem_inter_eq]; exact and.comm)
theorem inter_assoc (s₁ s₂ s₃ : hf) : (s₁ ∩ s₂) ∩ s₃ = s₁ ∩ (s₂ ∩ s₃) :=
hf.ext (λ a, by rewrite [*mem_inter_eq]; exact and.assoc)
theorem inter_left_comm (s₁ s₂ s₃ : hf) : s₁ ∩ (s₂ ∩ s₃) = s₂ ∩ (s₁ ∩ s₃) :=
!left_comm inter_comm inter_assoc s₁ s₂ s₃
theorem inter_right_comm (s₁ s₂ s₃ : hf) : (s₁ ∩ s₂) ∩ s₃ = (s₁ ∩ s₃) ∩ s₂ :=
!right_comm inter_comm inter_assoc s₁ s₂ s₃
theorem inter_self (s : hf) : s ∩ s = s :=
hf.ext (λ a, iff.intro
(λ h, mem_of_mem_inter_right h)
(λ h, mem_inter h h))
theorem inter_empty (s : hf) : s ∩ ∅ = ∅ :=
hf.ext (λ a, iff.intro
(suppose a ∈ s ∩ ∅, absurd (mem_of_mem_inter_right this) !not_mem_empty)
(suppose a ∈ ∅, absurd this !not_mem_empty))
theorem empty_inter (s : hf) : ∅ ∩ s = ∅ :=
calc ∅ ∩ s = s ∩ ∅ : inter_comm
... = ∅ : inter_empty
/- card -/
definition card (s : hf) : nat :=
finset.card (to_finset s)
theorem card_empty : card ∅ = 0 :=
rfl
lemma ne_empty_of_card_eq_succ {s : hf} {n : nat} : card s = succ n → s ≠ ∅ :=
by intros; substvars; contradiction
/- erase -/
definition erase (a : hf) (s : hf) : hf :=
of_finset (erase a (to_finset s))
theorem not_mem_erase (a : hf) (s : hf) : a ∉ erase a s :=
begin unfold [mem, erase], rewrite to_finset_of_finset, apply finset.not_mem_erase end
theorem card_erase_of_mem {a : hf} {s : hf} : a ∈ s → card (erase a s) = pred (card s) :=
begin unfold mem, intro h, unfold [erase, card], rewrite to_finset_of_finset, apply finset.card_erase_of_mem h end
theorem card_erase_of_not_mem {a : hf} {s : hf} : a ∉ s → card (erase a s) = card s :=
begin unfold [mem], intro h, unfold [erase, card], rewrite to_finset_of_finset, apply finset.card_erase_of_not_mem h end
theorem erase_empty (a : hf) : erase a ∅ = ∅ :=
rfl
theorem ne_of_mem_erase {a b : hf} {s : hf} : b ∈ erase a s → b ≠ a :=
by intro h beqa; subst b; exact absurd h !not_mem_erase
theorem mem_of_mem_erase {a b : hf} {s : hf} : b ∈ erase a s → b ∈ s :=
begin unfold [erase, mem], rewrite to_finset_of_finset, intro h, apply mem_of_mem_erase h end
theorem mem_erase_of_ne_of_mem {a b : hf} {s : hf} : a ≠ b → a ∈ s → a ∈ erase b s :=
begin intro h₁, unfold mem, intro h₂, unfold erase, rewrite to_finset_of_finset, apply mem_erase_of_ne_of_mem h₁ h₂ end
theorem mem_erase_iff (a b : hf) (s : hf) : a ∈ erase b s ↔ a ∈ s ∧ a ≠ b :=
iff.intro
(assume H, and.intro (mem_of_mem_erase H) (ne_of_mem_erase H))
(assume H, mem_erase_of_ne_of_mem (and.right H) (and.left H))
theorem mem_erase_eq (a b : hf) (s : hf) : a ∈ erase b s = (a ∈ s ∧ a ≠ b) :=
propext !mem_erase_iff
theorem erase_insert {a : hf} {s : hf} : a ∉ s → erase a (insert a s) = s :=
begin
unfold [mem, erase, insert],
intro h, rewrite [to_finset_of_finset, finset.erase_insert h, of_finset_to_finset]
end
theorem insert_erase {a : hf} {s : hf} : a ∈ s → insert a (erase a s) = s :=
begin
unfold mem, intro h, unfold [insert, erase],
rewrite [to_finset_of_finset, finset.insert_erase h, of_finset_to_finset]
end
/- subset -/
definition subset (s₁ s₂ : hf) : Prop :=
finset.subset (to_finset s₁) (to_finset s₂)
infix [priority hf.prio] ⊆ := subset
theorem empty_subset (s : hf) : ∅ ⊆ s :=
begin unfold [empty, subset], rewrite to_finset_of_finset, apply finset.empty_subset (to_finset s) end
theorem subset.refl (s : hf) : s ⊆ s :=
begin unfold [subset], apply finset.subset.refl (to_finset s) end
theorem subset.trans {s₁ s₂ s₃ : hf} : s₁ ⊆ s₂ → s₂ ⊆ s₃ → s₁ ⊆ s₃ :=
begin unfold [subset], intro h₁ h₂, apply finset.subset.trans h₁ h₂ end
theorem mem_of_subset_of_mem {s₁ s₂ : hf} {a : hf} : s₁ ⊆ s₂ → a ∈ s₁ → a ∈ s₂ :=
begin unfold [subset, mem], intro h₁ h₂, apply finset.mem_of_subset_of_mem h₁ h₂ end
theorem subset.antisymm {s₁ s₂ : hf} : s₁ ⊆ s₂ → s₂ ⊆ s₁ → s₁ = s₂ :=
begin unfold [subset], intro h₁ h₂, apply to_finset_inj (finset.subset.antisymm h₁ h₂) end
-- alternative name
theorem eq_of_subset_of_subset {s₁ s₂ : hf} (H₁ : s₁ ⊆ s₂) (H₂ : s₂ ⊆ s₁) : s₁ = s₂ :=
subset.antisymm H₁ H₂
theorem subset_of_forall {s₁ s₂ : hf} : (∀x, x ∈ s₁ → x ∈ s₂) → s₁ ⊆ s₂ :=
begin unfold [mem, subset], intro h, apply finset.subset_of_forall h end
theorem subset_insert (s : hf) (a : hf) : s ⊆ insert a s :=
begin unfold [subset, insert], rewrite to_finset_of_finset, apply finset.subset_insert (to_finset s) end
theorem eq_empty_of_subset_empty {x : hf} (H : x ⊆ ∅) : x = ∅ :=
subset.antisymm H (empty_subset x)
theorem subset_empty_iff (x : hf) : x ⊆ ∅ ↔ x = ∅ :=
iff.intro eq_empty_of_subset_empty (take xeq, by rewrite xeq; apply subset.refl ∅)
theorem erase_subset_erase (a : hf) {s t : hf} : s ⊆ t → erase a s ⊆ erase a t :=
begin unfold [subset, erase], intro h, rewrite *to_finset_of_finset, apply finset.erase_subset_erase a h end
theorem erase_subset (a : hf) (s : hf) : erase a s ⊆ s :=
begin unfold [subset, erase], rewrite to_finset_of_finset, apply finset.erase_subset a (to_finset s) end
theorem erase_eq_of_not_mem {a : hf} {s : hf} : a ∉ s → erase a s = s :=
begin unfold [mem, erase], intro h, rewrite [finset.erase_eq_of_not_mem h, of_finset_to_finset] end
theorem erase_insert_subset (a : hf) (s : hf) : erase a (insert a s) ⊆ s :=
begin unfold [erase, insert, subset], rewrite [*to_finset_of_finset], apply finset.erase_insert_subset a (to_finset s) end
theorem erase_subset_of_subset_insert {a : hf} {s t : hf} (H : s ⊆ insert a t) : erase a s ⊆ t :=
hf.subset.trans (!hf.erase_subset_erase H) (erase_insert_subset a t)
theorem insert_erase_subset (a : hf) (s : hf) : s ⊆ insert a (erase a s) :=
decidable.by_cases
(assume ains : a ∈ s, by rewrite [!insert_erase ains]; apply subset.refl)
(assume nains : a ∉ s,
suffices s ⊆ insert a s, by rewrite [erase_eq_of_not_mem nains]; assumption,
subset_insert s a)
theorem insert_subset_insert (a : hf) {s t : hf} : s ⊆ t → insert a s ⊆ insert a t :=
begin
unfold [subset, insert], intro h,
rewrite *to_finset_of_finset, apply finset.insert_subset_insert a h
end
theorem subset_insert_of_erase_subset {s t : hf} {a : hf} (H : erase a s ⊆ t) : s ⊆ insert a t :=
subset.trans (insert_erase_subset a s) (!insert_subset_insert H)
theorem subset_insert_iff (s t : hf) (a : hf) : s ⊆ insert a t ↔ erase a s ⊆ t :=
iff.intro !erase_subset_of_subset_insert !subset_insert_of_erase_subset
theorem le_of_subset {s₁ s₂ : hf} : s₁ ⊆ s₂ → s₁ ≤ s₂ :=
begin
revert s₂, induction s₁ with a s₁ nain ih,
take s₂, suppose ∅ ⊆ s₂, !zero_le,
take s₂, suppose insert a s₁ ⊆ s₂,
have a ∈ s₂, from mem_of_subset_of_mem this !mem_insert,
have a ∉ erase a s₂, from !not_mem_erase,
have s₁ ⊆ erase a s₂, from subset_of_forall
(take x xin, by_cases
(suppose x = a, by subst x; contradiction)
(suppose x ≠ a,
have x ∈ s₂, from mem_of_subset_of_mem `insert a s₁ ⊆ s₂` (mem_insert_of_mem _ `x ∈ s₁`),
mem_erase_of_ne_of_mem `x ≠ a` `x ∈ s₂`)),
have s₁ ≤ erase a s₂, from ih _ this,
have insert a s₁ ≤ insert a (erase a s₂), from
insert_le_insert_of_le (or.inr `a ∉ erase a s₂`) this,
by rewrite [insert_erase `a ∈ s₂` at this]; exact this
end
/- image -/
definition image (f : hf → hf) (s : hf) : hf :=
of_finset (finset.image f (to_finset s))
theorem image_empty (f : hf → hf) : image f ∅ = ∅ :=
rfl
theorem mem_image_of_mem (f : hf → hf) {s : hf} {a : hf} : a ∈ s → f a ∈ image f s :=
begin unfold [mem, image], intro h, rewrite to_finset_of_finset, apply finset.mem_image_of_mem f h end
theorem mem_image {f : hf → hf} {s : hf} {a : hf} {b : hf} (H1 : a ∈ s) (H2 : f a = b) : b ∈ image f s :=
eq.subst H2 (mem_image_of_mem f H1)
theorem exists_of_mem_image {f : hf → hf} {s : hf} {b : hf} : b ∈ image f s → ∃a, a ∈ s ∧ f a = b :=
begin unfold [mem, image], rewrite to_finset_of_finset, intro h, apply finset.exists_of_mem_image h end
theorem mem_image_iff (f : hf → hf) {s : hf} {y : hf} : y ∈ image f s ↔ ∃x, x ∈ s ∧ f x = y :=
begin unfold [mem, image], rewrite to_finset_of_finset, apply finset.mem_image_iff end
theorem mem_image_eq (f : hf → hf) {s : hf} {y : hf} : y ∈ image f s = ∃x, x ∈ s ∧ f x = y :=
propext (mem_image_iff f)
theorem mem_image_of_mem_image_of_subset {f : hf → hf} {s t : hf} {y : hf} (H1 : y ∈ image f s) (H2 : s ⊆ t) : y ∈ image f t :=
obtain x `x ∈ s` `f x = y`, from exists_of_mem_image H1,
have x ∈ t, from mem_of_subset_of_mem H2 `x ∈ s`,
show y ∈ image f t, from mem_image `x ∈ t` `f x = y`
theorem image_insert (f : hf → hf) (s : hf) (a : hf) : image f (insert a s) = insert (f a) (image f s) :=
begin unfold [image, insert], rewrite [*to_finset_of_finset, finset.image_insert] end
open function
lemma image_comp {f : hf → hf} {g : hf → hf} {s : hf} : image (f∘g) s = image f (image g s) :=
begin unfold image, rewrite [*to_finset_of_finset, finset.image_comp] end
lemma image_subset {a b : hf} (f : hf → hf) : a ⊆ b → image f a ⊆ image f b :=
begin unfold [subset, image], intro h, rewrite *to_finset_of_finset, apply finset.image_subset f h end
theorem image_union (f : hf → hf) (s t : hf) : image f (s ∪ t) = image f s ∪ image f t :=
begin unfold [image, union], rewrite [*to_finset_of_finset, finset.image_union] end
/- powerset -/
definition powerset (s : hf) : hf :=
of_finset (finset.image of_finset (finset.powerset (to_finset s)))
prefix [priority hf.prio] `𝒫`:100 := powerset
theorem powerset_empty : 𝒫 ∅ = insert ∅ ∅ :=
rfl
theorem powerset_insert {a : hf} {s : hf} : a ∉ s → 𝒫 (insert a s) = 𝒫 s ∪ image (insert a) (𝒫 s) :=
begin unfold [mem, powerset, insert, union, image], rewrite [*to_finset_of_finset], intro h,
have (λ (x : finset hf), of_finset (finset.insert a x)) = (λ (x : finset hf), of_finset (finset.insert a (to_finset (of_finset x)))), from
funext (λ x, by rewrite to_finset_of_finset),
rewrite [finset.powerset_insert h, finset.image_union, -*finset.image_comp, ↑comp, this]
end
theorem mem_powerset_iff_subset (s : hf) : ∀ x : hf, x ∈ 𝒫 s ↔ x ⊆ s :=
begin
intro x, unfold [mem, powerset, subset], rewrite [to_finset_of_finset, finset.mem_image_eq], apply iff.intro,
suppose (∃ (w : finset hf), finset.mem w (finset.powerset (to_finset s)) ∧ of_finset w = x),
obtain w h₁ h₂, from this,
begin subst x, rewrite to_finset_of_finset, exact iff.mp !finset.mem_powerset_iff_subset h₁ end,
suppose finset.subset (to_finset x) (to_finset s),
have finset.mem (to_finset x) (finset.powerset (to_finset s)), from iff.mpr !finset.mem_powerset_iff_subset this,
exists.intro (to_finset x) (and.intro this (of_finset_to_finset x))
end
theorem subset_of_mem_powerset {s t : hf} (H : s ∈ 𝒫 t) : s ⊆ t :=
iff.mp (mem_powerset_iff_subset t s) H
theorem mem_powerset_of_subset {s t : hf} (H : s ⊆ t) : s ∈ 𝒫 t :=
iff.mpr (mem_powerset_iff_subset t s) H
theorem empty_mem_powerset (s : hf) : ∅ ∈ 𝒫 s :=
mem_powerset_of_subset (empty_subset s)
/- hf as lists -/
open - [notation] list
definition of_list (s : list hf) : hf :=
@equiv.to_fun _ _ list_nat_equiv_nat s
definition to_list (h : hf) : list hf :=
@equiv.inv _ _ list_nat_equiv_nat h
lemma to_list_of_list (s : list hf) : to_list (of_list s) = s :=
@equiv.left_inv _ _ list_nat_equiv_nat s
lemma of_list_to_list (s : hf) : of_list (to_list s) = s :=
@equiv.right_inv _ _ list_nat_equiv_nat s
lemma to_list_inj {s₁ s₂ : hf} : to_list s₁ = to_list s₂ → s₁ = s₂ :=
λ h, function.injective_of_left_inverse of_list_to_list h
lemma of_list_inj {s₁ s₂ : list hf} : of_list s₁ = of_list s₂ → s₁ = s₂ :=
λ h, function.injective_of_left_inverse to_list_of_list h
definition nil : hf :=
of_list list.nil
lemma empty_eq_nil : ∅ = nil :=
rfl
definition cons (a l : hf) : hf :=
of_list (list.cons a (to_list l))
infixr :: := cons
lemma cons_ne_nil (a l : hf) : a::l ≠ nil :=
by contradiction
lemma head_eq_of_cons_eq {h₁ h₂ t₁ t₂ : hf} : (h₁::t₁) = (h₂::t₂) → h₁ = h₂ :=
begin unfold cons, intro h, apply list.head_eq_of_cons_eq (of_list_inj h) end
lemma tail_eq_of_cons_eq {h₁ h₂ t₁ t₂ : hf} : (h₁::t₁) = (h₂::t₂) → t₁ = t₂ :=
begin unfold cons, intro h, apply to_list_inj (list.tail_eq_of_cons_eq (of_list_inj h)) end
lemma cons_inj {a : hf} : injective (cons a) :=
take l₁ l₂, assume Pe, tail_eq_of_cons_eq Pe
/- append -/
definition append (l₁ l₂ : hf) : hf :=
of_list (list.append (to_list l₁) (to_list l₂))
notation l₁ ++ l₂ := append l₁ l₂
theorem append_nil_left [simp] (t : hf) : nil ++ t = t :=
begin unfold [nil, append], rewrite [to_list_of_list, list.append_nil_left, of_list_to_list] end
theorem append_cons [simp] (x s t : hf) : (x::s) ++ t = x::(s ++ t) :=
begin unfold [cons, append], rewrite [*to_list_of_list, list.append_cons] end
theorem append_nil_right [simp] (t : hf) : t ++ nil = t :=
begin unfold [nil, append], rewrite [to_list_of_list, list.append_nil_right, of_list_to_list] end
theorem append.assoc [simp] (s t u : hf) : s ++ t ++ u = s ++ (t ++ u) :=
begin unfold append, rewrite [*to_list_of_list, list.append.assoc] end
/- length -/
definition length (l : hf) : nat :=
list.length (to_list l)
theorem length_nil [simp] : length nil = 0 :=
begin unfold [length, nil] end
theorem length_cons [simp] (x t : hf) : length (x::t) = length t + 1 :=
begin unfold [length, cons], rewrite to_list_of_list end
theorem length_append [simp] (s t : hf) : length (s ++ t) = length s + length t :=
begin unfold [length, append], rewrite [to_list_of_list, list.length_append] end
theorem eq_nil_of_length_eq_zero {l : hf} : length l = 0 → l = nil :=
begin unfold [length, nil], intro h, rewrite [-list.eq_nil_of_length_eq_zero h, of_list_to_list] end
theorem ne_nil_of_length_eq_succ {l : hf} {n : nat} : length l = succ n → l ≠ nil :=
begin unfold [length, nil], intro h₁ h₂, subst l, rewrite to_list_of_list at h₁, contradiction end
/- head and tail -/
definition head (l : hf) : hf :=
list.head (to_list l)
theorem head_cons [simp] (a l : hf) : head (a::l) = a :=
begin unfold [head, cons], rewrite to_list_of_list end
private lemma to_list_ne_list_nil {s : hf} : s ≠ nil → to_list s ≠ list.nil :=
begin
unfold nil,
intro h,
suppose to_list s = list.nil,
by rewrite [-this at h, of_list_to_list at h]; exact absurd rfl h
end
theorem head_append [simp] (t : hf) {s : hf} : s ≠ nil → head (s ++ t) = head s :=
begin
unfold [nil, head, append], rewrite to_list_of_list,
suppose s ≠ of_list list.nil,
by rewrite [list.head_append _ (to_list_ne_list_nil this)]
end
definition tail (l : hf) : hf :=
of_list (list.tail (to_list l))
theorem tail_nil [simp] : tail nil = nil :=
begin unfold [tail, nil] end
theorem tail_cons [simp] (a l : hf) : tail (a::l) = l :=
begin unfold [tail, cons], rewrite [to_list_of_list, list.tail_cons, of_list_to_list] end
theorem cons_head_tail {l : hf} : l ≠ nil → (head l)::(tail l) = l :=
begin
unfold [nil, head, tail, cons],
suppose l ≠ of_list list.nil,
by rewrite [to_list_of_list, list.cons_head_tail (to_list_ne_list_nil this), of_list_to_list]
end
end hf
|
a3c2e15544404a58d569e4eb1b74507fe788af47 | 7cef822f3b952965621309e88eadf618da0c8ae9 | /src/category_theory/natural_isomorphism.lean | 1a09bed47dd8e334d21ac743ca8281a07dec0f66 | [
"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 | 5,516 | lean | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Scott Morrison, Floris van Doorn
-/
import category_theory.functor_category
import category_theory.isomorphism
open category_theory
universes v₁ v₂ v₃ v₄ u₁ u₂ u₃ u₄ -- declare the `v`'s first; see `category_theory.category` for an explanation
namespace category_theory
open nat_trans
/-- The application of a natural isomorphism to an object. We put this definition in a different namespace, so that we can use α.app -/
@[simp, reducible] def iso.app {C : Type u₁} [category.{v₁} C] {D : Type u₂} [category.{v₂} D]
{F G : C ⥤ D} (α : F ≅ G) (X : C) : F.obj X ≅ G.obj X :=
{ hom := α.hom.app X,
inv := α.inv.app X,
hom_inv_id' := begin rw [← comp_app, iso.hom_inv_id], refl end,
inv_hom_id' := begin rw [← comp_app, iso.inv_hom_id], refl end }
namespace nat_iso
open category_theory.category category_theory.functor
variables {C : Type u₁} [𝒞 : category.{v₁} C] {D : Type u₂} [𝒟 : category.{v₂} D]
{E : Type u₃} [ℰ : category.{v₃} E]
include 𝒞 𝒟
@[simp] lemma trans_app {F G H : C ⥤ D} (α : F ≅ G) (β : G ≅ H) (X : C) :
(α ≪≫ β).app X = α.app X ≪≫ β.app X := rfl
@[simp] lemma app_hom {F G : C ⥤ D} (α : F ≅ G) (X : C) : (α.app X).hom = α.hom.app X := rfl
@[simp] lemma app_inv {F G : C ⥤ D} (α : F ≅ G) (X : C) : (α.app X).inv = α.inv.app X := rfl
@[simp] lemma hom_inv_id_app {F G : C ⥤ D} (α : F ≅ G) (X : C) : α.hom.app X ≫ α.inv.app X = 𝟙 (F.obj X) :=
congr_fun (congr_arg app α.hom_inv_id) X
@[simp] lemma inv_hom_id_app {F G : C ⥤ D} (α : F ≅ G) (X : C) : α.inv.app X ≫ α.hom.app X = 𝟙 (G.obj X) :=
congr_fun (congr_arg app α.inv_hom_id) X
variables {F G : C ⥤ D}
instance hom_app_is_iso (α : F ≅ G) (X : C) : is_iso (α.hom.app X) :=
{ inv := α.inv.app X,
hom_inv_id' := begin rw [←comp_app, iso.hom_inv_id, ←id_app] end,
inv_hom_id' := begin rw [←comp_app, iso.inv_hom_id, ←id_app] end }
instance inv_app_is_iso (α : F ≅ G) (X : C) : is_iso (α.inv.app X) :=
{ inv := α.hom.app X,
hom_inv_id' := begin rw [←comp_app, iso.inv_hom_id, ←id_app] end,
inv_hom_id' := begin rw [←comp_app, iso.hom_inv_id, ←id_app] end }
@[simp] lemma hom_app_inv_app_id (α : F ≅ G) (X : C) : α.hom.app X ≫ α.inv.app X = 𝟙 _ :=
begin
rw ←comp_app, simp,
end
@[simp] lemma inv_app_hom_app_id (α : F ≅ G) (X : C) : α.inv.app X ≫ α.hom.app X = 𝟙 _ :=
begin
rw ←comp_app, simp,
end
variables {X Y : C}
@[simp] lemma naturality_1 (α : F ≅ G) (f : X ⟶ Y) :
(α.inv.app X) ≫ (F.map f) ≫ (α.hom.app Y) = G.map f :=
begin erw [naturality, ←category.assoc, is_iso.hom_inv_id, category.id_comp] end
@[simp] lemma naturality_2 (α : F ≅ G) (f : X ⟶ Y) :
(α.hom.app X) ≫ (G.map f) ≫ (α.inv.app Y) = F.map f :=
begin erw [naturality, ←category.assoc, is_iso.hom_inv_id, category.id_comp] end
def is_iso_of_is_iso_app (α : F ⟶ G) [∀ X : C, is_iso (α.app X)] : is_iso α :=
{ inv :=
{ app := λ X, inv (α.app X),
naturality' := λ X Y f,
begin
have h := congr_arg (λ f, inv (α.app X) ≫ (f ≫ inv (α.app Y))) (α.naturality f).symm,
simp only [is_iso.inv_hom_id_assoc, is_iso.hom_inv_id, assoc, comp_id, cancel_mono] at h,
exact h
end } }
instance is_iso_of_is_iso_app' (α : F ⟶ G) [H : ∀ X : C, is_iso (nat_trans.app α X)] : is_iso α :=
@nat_iso.is_iso_of_is_iso_app _ _ _ _ _ _ α H
-- TODO can we make this an instance?
def is_iso_app_of_is_iso (α : F ⟶ G) [is_iso α] (X) : is_iso (α.app X) :=
{ inv := (inv α).app X,
hom_inv_id' := congr_fun (congr_arg nat_trans.app (is_iso.hom_inv_id α)) X,
inv_hom_id' := congr_fun (congr_arg nat_trans.app (is_iso.inv_hom_id α)) X }
def of_components (app : ∀ X : C, (F.obj X) ≅ (G.obj X))
(naturality : ∀ {X Y : C} (f : X ⟶ Y), (F.map f) ≫ ((app Y).hom) = ((app X).hom) ≫ (G.map f)) :
F ≅ G :=
as_iso { app := λ X, (app X).hom }
@[simp] lemma of_components.app (app' : ∀ X : C, (F.obj X) ≅ (G.obj X)) (naturality) (X) :
(of_components app' naturality).app X = app' X :=
by tidy
@[simp] lemma of_components.hom_app (app : ∀ X : C, (F.obj X) ≅ (G.obj X)) (naturality) (X) :
(of_components app naturality).hom.app X = (app X).hom := rfl
@[simp] lemma of_components.inv_app (app : ∀ X : C, (F.obj X) ≅ (G.obj X)) (naturality) (X) :
(of_components app naturality).inv.app X = (app X).inv := rfl
include ℰ
def hcomp {F G : C ⥤ D} {H I : D ⥤ E} (α : F ≅ G) (β : H ≅ I) : F ⋙ H ≅ G ⋙ I :=
begin
refine ⟨α.hom ◫ β.hom, α.inv ◫ β.inv, _, _⟩,
{ ext, rw [←nat_trans.exchange], simp, refl },
ext, rw [←nat_trans.exchange], simp, refl
end
omit ℰ
-- declare local notation for nat_iso.hcomp
localized "infix ` ■ `:80 := category_theory.nat_iso.hcomp" in category
end nat_iso
namespace functor
variables {C : Type u₁} [𝒞 : category.{v₁} C]
include 𝒞
def ulift_down_up : ulift_down.{v₁} C ⋙ ulift_up C ≅ 𝟭 (ulift.{u₂} C) :=
{ hom := { app := λ X, @category_struct.id (ulift.{u₂} C) _ X },
inv := { app := λ X, @category_struct.id (ulift.{u₂} C) _ X } }
def ulift_up_down : ulift_up.{v₁} C ⋙ ulift_down C ≅ 𝟭 C :=
{ hom := { app := λ X, 𝟙 X },
inv := { app := λ X, 𝟙 X } }
end functor
end category_theory
|
18b6cd12e6f235a27200ee910e1cca52e870d7e0 | a9d0fb7b0e4f802bd3857b803e6c5c23d87fef91 | /library/init/string.lean | b9b536a7327222eabd7506925d0c86313091e7f5 | [
"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 | 1,288 | lean | /-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Leonardo de Moura
-/
prelude
import init.char init.list
attribute [reducible]
definition string := list char
namespace string
attribute [pattern]
definition empty : string := list.nil
attribute [pattern]
definition str : char → string → string := list.cons
definition concat (a b : string) : string :=
list.append b a
end string
attribute [instance]
definition string.has_append : has_append string :=
⟨string.concat⟩
open list
private definition utf8_length_aux : nat → nat → string → nat
| 0 r (c::s) :=
let n := char.to_nat c in
if n < 0x80 then utf8_length_aux 0 (r+1) s
else if 0xC0 ≤ n ∧ n < 0xE0 then utf8_length_aux 1 (r+1) s
else if 0xE0 ≤ n ∧ n < 0xF0 then utf8_length_aux 2 (r+1) s
else if 0xF0 ≤ n ∧ n < 0xF8 then utf8_length_aux 3 (r+1) s
else if 0xF8 ≤ n ∧ n < 0xFC then utf8_length_aux 4 (r+1) s
else if 0xFC ≤ n ∧ n < 0xFE then utf8_length_aux 5 (r+1) s
else utf8_length_aux 0 (r+1) s
| (n+1) r (c::s) := utf8_length_aux n r s
| n r [] := r
definition utf8_length : string → nat
| s := utf8_length_aux 0 0 (reverse s)
|
ec02164f778f22cd0a721a95b6f8bc48ed189069 | 75db7e3219bba2fbf41bf5b905f34fcb3c6ca3f2 | /hott/types/nat/hott.hlean | bdeaaaee93b45cf5b11ee41c964fd9b3ec05393a | [
"Apache-2.0"
] | permissive | jroesch/lean | 30ef0860fa905d35b9ad6f76de1a4f65c9af6871 | 3de4ec1a6ce9a960feb2a48eeea8b53246fa34f2 | refs/heads/master | 1,586,090,835,348 | 1,455,142,203,000 | 1,455,142,277,000 | 51,536,958 | 1 | 0 | null | 1,455,215,811,000 | 1,455,215,811,000 | null | UTF-8 | Lean | false | false | 4,645 | hlean | /-
Copyright (c) 2015 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Floris van Doorn
Theorems about the natural numbers specific to HoTT
-/
import .order
open is_trunc unit empty eq equiv algebra
namespace nat
definition is_hprop_le [instance] (n m : ℕ) : is_hprop (n ≤ m) :=
begin
assert lem : Π{n m : ℕ} (p : n ≤ m) (q : n = m), p = q ▸ le.refl n,
{ intros, cases p,
{ assert H' : q = idp, apply is_hset.elim,
cases H', reflexivity},
{ cases q, exfalso, apply not_succ_le_self a}},
apply is_hprop.mk, intro H1 H2, induction H2,
{ apply lem},
{ cases H1,
{ exfalso, apply not_succ_le_self a},
{ exact ap le.step !v_0}},
end
definition is_hprop_lt [instance] (n m : ℕ) : is_hprop (n < m) := !is_hprop_le
definition le_equiv_succ_le_succ (n m : ℕ) : (n ≤ m) ≃ (succ n ≤ succ m) :=
equiv_of_is_hprop succ_le_succ le_of_succ_le_succ
definition le_succ_equiv_pred_le (n m : ℕ) : (n ≤ succ m) ≃ (pred n ≤ m) :=
equiv_of_is_hprop pred_le_pred le_succ_of_pred_le
theorem lt_by_cases_lt {a b : ℕ} {P : Type} (H1 : a < b → P) (H2 : a = b → P)
(H3 : a > b → P) (H : a < b) : lt.by_cases H1 H2 H3 = H1 H :=
begin
unfold lt.by_cases, induction (lt.trichotomy a b) with H' H',
{ esimp, exact ap H1 !is_hprop.elim},
{ exfalso, cases H' with H' H', apply lt.irrefl, exact H' ▸ H, exact lt.asymm H H'}
end
theorem lt_by_cases_eq {a b : ℕ} {P : Type} (H1 : a < b → P) (H2 : a = b → P)
(H3 : a > b → P) (H : a = b) : lt.by_cases H1 H2 H3 = H2 H :=
begin
unfold lt.by_cases, induction (lt.trichotomy a b) with H' H',
{ exfalso, apply lt.irrefl, exact H ▸ H'},
{ cases H' with H' H', esimp, exact ap H2 !is_hprop.elim, exfalso, apply lt.irrefl, exact H ▸ H'}
end
theorem lt_by_cases_ge {a b : ℕ} {P : Type} (H1 : a < b → P) (H2 : a = b → P)
(H3 : a > b → P) (H : a > b) : lt.by_cases H1 H2 H3 = H3 H :=
begin
unfold lt.by_cases, induction (lt.trichotomy a b) with H' H',
{ exfalso, exact lt.asymm H H'},
{ cases H' with H' H', exfalso, apply lt.irrefl, exact H' ▸ H, esimp, exact ap H3 !is_hprop.elim}
end
theorem lt_ge_by_cases_lt {n m : ℕ} {P : Type} (H1 : n < m → P) (H2 : n ≥ m → P)
(H : n < m) : lt_ge_by_cases H1 H2 = H1 H :=
by apply lt_by_cases_lt
theorem lt_ge_by_cases_ge {n m : ℕ} {P : Type} (H1 : n < m → P) (H2 : n ≥ m → P)
(H : n ≥ m) : lt_ge_by_cases H1 H2 = H2 H :=
begin
unfold [lt_ge_by_cases,lt.by_cases], induction (lt.trichotomy n m) with H' H',
{ exfalso, apply lt.irrefl, exact lt_of_le_of_lt H H'},
{ cases H' with H' H'; all_goals (esimp; apply ap H2 !is_hprop.elim)}
end
theorem lt_ge_by_cases_le {n m : ℕ} {P : Type} {H1 : n ≤ m → P} {H2 : n ≥ m → P}
(H : n ≤ m) (Heq : Π(p : n = m), H1 (le_of_eq p) = H2 (le_of_eq p⁻¹))
: lt_ge_by_cases (λH', H1 (le_of_lt H')) H2 = H1 H :=
begin
unfold [lt_ge_by_cases,lt.by_cases], induction (lt.trichotomy n m) with H' H',
{ esimp, apply ap H1 !is_hprop.elim},
{ cases H' with H' H',
{ esimp, induction H', esimp, symmetry,
exact ap H1 !is_hprop.elim ⬝ Heq idp ⬝ ap H2 !is_hprop.elim},
{ exfalso, apply lt.irrefl, apply lt_of_le_of_lt H H'}}
end
protected definition code [reducible] [unfold 1 2] : ℕ → ℕ → Type₀
| code 0 0 := unit
| code 0 (succ m) := empty
| code (succ n) 0 := empty
| code (succ n) (succ m) := code n m
protected definition refl : Πn, nat.code n n
| refl 0 := star
| refl (succ n) := refl n
protected definition encode [unfold 3] {n m : ℕ} (p : n = m) : nat.code n m :=
p ▸ nat.refl n
protected definition decode : Π(n m : ℕ), nat.code n m → n = m
| decode 0 0 := λc, idp
| decode 0 (succ l) := λc, empty.elim c _
| decode (succ k) 0 := λc, empty.elim c _
| decode (succ k) (succ l) := λc, ap succ (decode k l c)
definition nat_eq_equiv (n m : ℕ) : (n = m) ≃ nat.code n m :=
equiv.MK nat.encode
!nat.decode
begin
revert m, induction n, all_goals (intro m;induction m;all_goals intro c),
all_goals try contradiction,
induction c, reflexivity,
xrewrite [↑nat.decode,-tr_compose,v_0],
end
begin
intro p, induction p, esimp, induction n,
reflexivity,
rewrite [↑nat.decode,↑nat.refl,v_0]
end
end nat
|
04cd893bb1ef6a0586e0dc4ba9c5db9ba297cbe4 | e07b1aca72e83a272dd59d24c6e0fa246034d774 | /src/surreal/ordinal/basic.lean | 43f314c4ac84c6eb4c372f5a051b7e03370af7ed | [] | no_license | pedrominicz/learn | 637a343bd4f8669d76819ac660a2d2d3e0958710 | b79b802a9846c86c21d4b6f3e17af36e7382f0ef | refs/heads/master | 1,671,746,990,402 | 1,670,778,113,000 | 1,670,778,113,000 | 265,735,177 | 1 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 6,671 | lean | import tactic
variables {α β : Type}
variables {rα : α → α → Prop} {rβ : β → β → Prop}
theorem no_infinite_descending_chain_of_wf (H : well_founded rα) :
¬ ∃ f : ℕ → α, ∀ n : ℕ, rα (f n.succ) (f n) :=
begin
rintro ⟨f, hf⟩,
cases H.min_mem _ (set.range_nonempty f) with n hn,
apply H.not_lt_min,
{ show f _ ∈ set.range f, simp },
{ rw ←hn, exact hf n }
end
namespace wf_of_no_infinite_descending_chain
noncomputable def not_acc_aux {a : α} (ha : ¬ acc rα a) :
Σ' b, rα b a ∧ ¬ acc rα b :=
begin
suffices : ∃ b, rα b a ∧ ¬ acc rα b,
{ exact ⟨classical.some this, classical.some_spec this⟩ },
by_contra h,
push_neg at h,
exact ha ⟨_, h⟩
end
noncomputable def f {a : α} (ha : ¬ acc rα a) : ℕ → Σ' b, ¬ acc rα b :=
λ n, nat.rec ⟨a, ha⟩ (λ n a, ⟨(not_acc_aux a.snd).fst, (not_acc_aux a.snd).snd.right⟩) n
theorem _root_.wf_of_no_infinite_descending_chain :
(¬ ∃ f : ℕ → α, ∀ n : ℕ, rα (f n.succ) (f n)) → well_founded rα :=
begin
by_contra h,
simp only [not_forall, exists_prop] at h,
cases h with h' h, apply h', clear h',
have : ¬ ∀ a, acc rα a := h ∘ well_founded.intro, clear h, rename this h,
push_neg at h,
cases h with a ha,
exact ⟨λ n, (f ha n).fst, λ n, (not_acc_aux (f ha n).snd).snd.left⟩
end
end wf_of_no_infinite_descending_chain
def f {A : set α} (h : Π a : α, a ∈ A → Σ' b : α, b ∈ A ∧ rα b a) {a : α} (ha : a ∈ A) :
ℕ → Σ' b, b ∈ A :=
λ n, nat.rec ⟨a, ha⟩ (λ n a, ⟨(h a.fst a.snd).fst, (h a.fst a.snd).snd.left⟩) n
example (H : well_founded rα) {A : set α} :
(Π a : α, a ∈ A → Σ' b : α, b ∈ A ∧ rα b a) → A = ∅ :=
begin
intro h,
ext a,
simp only [set.mem_empty_eq, iff_false],
intro ha,
apply no_infinite_descending_chain_of_wf H,
use λ n, (f h ha n).fst,
intro n,
suffices : (f h ha n).fst ∈ A ∧ rα (f h ha n.succ).fst (f h ha n).fst,
{ exact this.right },
induction n with n ih,
{ exact ⟨ha, (h a ha).snd.right⟩ },
{ cases ih with ih₁ ih₂,
let b := (h (f h ha n).fst ih₁).fst,
let hb : b ∈ A := (h (f h ha n).fst ih₁).snd.left,
exact ⟨hb, (h b hb).snd.right⟩ }
end
theorem set_without_min_eq_empty_of_wf (H : well_founded rα) {A : set α} :
(∀ a ∈ A, ∃ b ∈ A, rα b a) → A = ∅ :=
begin
intro h,
ext a,
rw [set.mem_empty_eq, iff_false],
intro ha,
rcases h _ (H.min_mem _ _) with ⟨b, hb, hb'⟩,
exact H.not_lt_min _ (set.nonempty_of_mem ha) hb hb'
end
theorem wf_of_set_without_min_eq_empty :
(∀ A : set α, (∀ a ∈ A, ∃ b ∈ A, rα b a) → A = ∅) → well_founded rα :=
begin
by_contra h,
simp only [exists_prop, not_forall] at h,
have := mt (@wf_of_no_infinite_descending_chain _ rα),
push_neg at this,
cases h with h₁ h₂,
specialize this h₂, clear h₂,
cases this with f hf,
let A := set.range f,
suffices : ∀ a, a ∈ A → ∃ b, b ∈ A ∧ rα b a,
{ specialize h₁ A this, finish },
clear h₁,
intros a ha,
have : ∃ n, f n = a,
{ finish },
rcases this with ⟨n, rfl⟩,
use f n.succ,
finish
end
example [iα : is_trichotomous α rα] (Hβ : well_founded rβ)
{eq : α ≃ β} (h : ∀ a₁ a₂ : α, rα a₁ a₂ → rβ (eq a₁) (eq a₂)) :
∀ b₁ b₂ : β, rβ b₁ b₂ → rα (eq.symm b₁) (eq.symm b₂) :=
begin
intros b₁ b₂ hb,
cases eq with to_fun inv_fun left_inv right_inv,
change ∀ a, inv_fun (to_fun a) = a at left_inv,
change ∀ b, to_fun (inv_fun b) = b at right_inv,
change ∀ a₁ a₂, rα a₁ a₂ → rβ (to_fun a₁) (to_fun a₂) at h,
change rα (inv_fun b₁) (inv_fun b₂),
unfreezingI { rcases iα with ⟨iα⟩ },
rcases iα (inv_fun b₁) (inv_fun b₂) with ha | ha | ha,
{ assumption },
{ exfalso,
suffices : b₁ = b₂,
{ subst this, rename b₁ b,
exact (Hβ.is_irrefl.irrefl : ∀ _, ¬ _) b hb },
rw [←right_inv b₁, ha, right_inv] },
{ exfalso,
specialize h _ _ ha,
rw [right_inv, right_inv] at h,
apply Hβ.not_gt_of_lt h hb }
end
inductive ordinal : Type 1
| mk {α : Type} (lt : α → α → Prop) [is_well_order α lt] : ordinal
namespace ordinal
def equiv : ordinal → ordinal → Prop :=
begin
rintros ⟨α, ltα, iα⟩ ⟨β, ltβ, iβ⟩,
exact ∃ eq : α ≃ β, ∀ a₁ a₂ : α, ltα a₁ a₂ ↔ ltβ (eq a₁) (eq a₂)
end
theorem equiv.refl (α : ordinal) : equiv α α :=
begin
cases α with α ltα iα,
use equiv.refl α,
simp
end
theorem equiv.rfl : ∀ {α : ordinal}, equiv α α := equiv.refl
theorem equiv.symm {α β : ordinal} : equiv α β → equiv β α :=
begin
cases α with α ltα iα,
cases β with β ltβ iβ,
rintro ⟨⟨to_fun, inv_fun, left_inv, right_inv⟩, h⟩,
change ∀ a, inv_fun (to_fun a) = a at left_inv,
change ∀ b, to_fun (inv_fun b) = b at right_inv,
change ∀ a₁ a₂, ltα a₁ a₂ ↔ ltβ (to_fun a₁) (to_fun a₂) at h,
use ⟨inv_fun, to_fun, right_inv, left_inv⟩, clear iα iβ,
change ∀ b₁ b₂, ltβ b₁ b₂ ↔ ltα (inv_fun b₁) (inv_fun b₂),
intros b₁ b₂,
specialize h (inv_fun b₁) (inv_fun b₂),
rw [right_inv, right_inv] at h,
symmetry,
assumption
end
theorem equiv.trans {α β γ : ordinal} (h₁ : equiv α β) (h₂ : equiv β γ) :
equiv α γ :=
begin
cases α with α ltα iα,
cases β with β ltβ iβ,
cases γ with γ ltγ iγ,
rcases h₁ with ⟨⟨to_fun₁, inv_fun₁, left_inv₁, right_inv₁⟩, h₁⟩,
change ∀ a, inv_fun₁ (to_fun₁ a) = a at left_inv₁,
change ∀ b, to_fun₁ (inv_fun₁ b) = b at right_inv₁,
change ∀ a₁ a₂, ltα a₁ a₂ ↔ ltβ (to_fun₁ a₁) (to_fun₁ a₂) at h₁,
rcases h₂ with ⟨⟨to_fun₂, inv_fun₂, left_inv₂, right_inv₂⟩, h₂⟩,
change ∀ b, inv_fun₂ (to_fun₂ b) = b at left_inv₂,
change ∀ c, to_fun₂ (inv_fun₂ c) = c at right_inv₂,
change ∀ b₁ b₂, ltβ b₁ b₂ ↔ ltγ (to_fun₂ b₁) (to_fun₂ b₂) at h₂,
change ∃ _, _,
use ⟨
λ a, to_fun₂ (to_fun₁ a),
λ c, inv_fun₁ (inv_fun₂ c),
function.left_inverse.comp left_inv₂ left_inv₁,
function.right_inverse.comp right_inv₂ right_inv₁⟩,
intros a₁ a₂,
change ltα a₁ a₂ ↔ ltγ (to_fun₂ (to_fun₁ a₁)) (to_fun₂ (to_fun₁ a₂)),
transitivity,
{ exact h₁ a₁ a₂ },
{ exact h₂ (to_fun₁ a₁) (to_fun₁ a₂) }
end
instance : setoid ordinal :=
⟨equiv, equiv.refl, @equiv.symm, @equiv.trans⟩
end ordinal
def Ordinal : Type 1 := quotient ordinal.setoid |
3d32715daf4d10b23d7089c3956766da74369996 | ea5678cc400c34ff95b661fa26d15024e27ea8cd | /clear_aux_decl.lean | e24fddedcc1f4d55a82aaeb4275127a2bc07e34f | [] | no_license | ChrisHughes24/leanstuff | dca0b5349c3ed893e8792ffbd98cbcadaff20411 | 9efa85f72efaccd1d540385952a6acc18fce8687 | refs/heads/master | 1,654,883,241,759 | 1,652,873,885,000 | 1,652,873,885,000 | 134,599,537 | 1 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 699 | lean | import tactic.basic tactic
open tactic
#print tactic.local_context
#print expr.is_aux_decl
#print tactic.clear
#print tactic.interactive.clear
namespace tactic.interactive
meta def clear_aux_decl_aux : list expr → tactic unit
| [] := skip
| (e::l) := do cond e.is_aux_decl (tactic.clear e) skip, clear_aux_decl_aux l
meta def clear_aux_decl : tactic unit :=
local_context >>= clear_aux_decl_aux
end tactic.interactive
meta def h : tactic unit :=
do l ← local_context, trace (l.filter (λ e : expr, e.is_aux_decl)).length
example (x y : ℕ) (h₁ : ∃ n : ℕ, n * 1 = 2) (h₂ : 1 + 1 = 2 → x * 1 = y) : x = y :=
let ⟨n, hn⟩ := h₁ in
begin
clear_aux_decl,
finish
end
|
0a8965ee4465536f8d2fbb4cf2abfdee4e254762 | 491068d2ad28831e7dade8d6dff871c3e49d9431 | /tests/lean/run/imp2.lean | e097a01063202f709e1ab7b5fa42ce769ae3a8f9 | [
"Apache-2.0"
] | permissive | davidmueller13/lean | 65a3ed141b4088cd0a268e4de80eb6778b21a0e9 | c626e2e3c6f3771e07c32e82ee5b9e030de5b050 | refs/heads/master | 1,611,278,313,401 | 1,444,021,177,000 | 1,444,021,177,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 98 | lean | import data.num
check (λ {A : Type.{1}} (a : A), a) 10
check (λ {A} a, a) 10
check (λ a, a) 10
|
403a92263530627739b06e1350ad307205193c88 | 74addaa0e41490cbaf2abd313a764c96df57b05d | /Mathlib/algebra/char_p/basic.lean | 6d2fca8033d2e3254cde77dac1cbcca667ebc010 | [] | 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 | 12,277 | lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Kenny Lau, Joey van Langen, Casper Putz
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.default
import Mathlib.data.fintype.basic
import Mathlib.data.nat.choose.default
import Mathlib.data.int.modeq
import Mathlib.algebra.module.basic
import Mathlib.algebra.iterate_hom
import Mathlib.group_theory.order_of_element
import Mathlib.algebra.group.type_tags
import Mathlib.PostPort
universes u l u_1 u_2 v
namespace Mathlib
/-!
# Characteristic of semirings
-/
/-- The generator of the kernel of the unique homomorphism ℕ → α for a semiring α -/
class char_p (α : Type u) [semiring α] (p : ℕ)
where
cast_eq_zero_iff : ∀ (x : ℕ), ↑x = 0 ↔ p ∣ x
theorem char_p.cast_eq_zero (α : Type u) [semiring α] (p : ℕ) [char_p α p] : ↑p = 0 :=
iff.mpr (char_p.cast_eq_zero_iff α p p) (dvd_refl p)
@[simp] theorem char_p.cast_card_eq_zero (R : Type u_1) [ring R] [fintype R] : ↑(fintype.card R) = 0 := sorry
theorem char_p.int_cast_eq_zero_iff (R : Type u) [ring R] (p : ℕ) [char_p R p] (a : ℤ) : ↑a = 0 ↔ ↑p ∣ a := sorry
theorem char_p.int_coe_eq_int_coe_iff (R : Type u_1) [ring R] (p : ℕ) [char_p R p] (a : ℤ) (b : ℤ) : ↑a = ↑b ↔ int.modeq (↑p) a b := sorry
theorem char_p.eq (α : Type u) [semiring α] {p : ℕ} {q : ℕ} (c1 : char_p α p) (c2 : char_p α q) : p = q :=
nat.dvd_antisymm (iff.mp (char_p.cast_eq_zero_iff α p q) (char_p.cast_eq_zero α q))
(iff.mp (char_p.cast_eq_zero_iff α q p) (char_p.cast_eq_zero α p))
protected instance char_p.of_char_zero (α : Type u) [semiring α] [char_zero α] : char_p α 0 :=
char_p.mk
fun (x : ℕ) =>
eq.mpr (id (Eq._oldrec (Eq.refl (↑x = 0 ↔ 0 ∣ x)) (propext zero_dvd_iff)))
(eq.mpr (id (Eq._oldrec (Eq.refl (↑x = 0 ↔ x = 0)) (Eq.symm nat.cast_zero)))
(eq.mpr (id (Eq._oldrec (Eq.refl (↑x = ↑0 ↔ x = 0)) (propext nat.cast_inj))) (iff.refl (x = 0))))
theorem char_p.exists (α : Type u) [semiring α] : ∃ (p : ℕ), char_p α p := sorry
theorem char_p.exists_unique (α : Type u) [semiring α] : exists_unique fun (p : ℕ) => char_p α p := sorry
theorem char_p.congr {R : Type u} [semiring R] {p : ℕ} (q : ℕ) [hq : char_p R q] (h : q = p) : char_p R p :=
h ▸ hq
/-- Noncomputable function that outputs the unique characteristic of a semiring. -/
def ring_char (α : Type u) [semiring α] : ℕ :=
classical.some (char_p.exists_unique α)
namespace ring_char
theorem spec (R : Type u) [semiring R] (x : ℕ) : ↑x = 0 ↔ ring_char R ∣ x := sorry
theorem eq (R : Type u) [semiring R] {p : ℕ} (C : char_p R p) : p = ring_char R :=
and.right (classical.some_spec (char_p.exists_unique R)) p C
protected instance char_p (R : Type u) [semiring R] : char_p R (ring_char R) :=
char_p.mk (spec R)
theorem of_eq {R : Type u} [semiring R] {p : ℕ} (h : ring_char R = p) : char_p R p :=
char_p.congr (ring_char R) h
theorem eq_iff {R : Type u} [semiring R] {p : ℕ} : ring_char R = p ↔ char_p R p :=
{ mp := of_eq, mpr := Eq.symm ∘ eq R }
theorem dvd {R : Type u} [semiring R] {x : ℕ} (hx : ↑x = 0) : ring_char R ∣ x :=
iff.mp (spec R x) hx
end ring_char
theorem add_pow_char_of_commute (R : Type u) [semiring R] {p : ℕ} [fact (nat.prime p)] [char_p R p] (x : R) (y : R) (h : commute x y) : (x + y) ^ p = x ^ p + y ^ p := sorry
theorem add_pow_char_pow_of_commute (R : Type u) [semiring R] {p : ℕ} [fact (nat.prime p)] [char_p R p] {n : ℕ} (x : R) (y : R) (h : commute x y) : (x + y) ^ p ^ n = x ^ p ^ n + y ^ p ^ n := sorry
theorem sub_pow_char_of_commute (R : Type u) [ring R] {p : ℕ} [fact (nat.prime p)] [char_p R p] (x : R) (y : R) (h : commute x y) : (x - y) ^ p = x ^ p - y ^ p := sorry
theorem sub_pow_char_pow_of_commute (R : Type u) [ring R] {p : ℕ} [fact (nat.prime p)] [char_p R p] {n : ℕ} (x : R) (y : R) (h : commute x y) : (x - y) ^ p ^ n = x ^ p ^ n - y ^ p ^ n := sorry
theorem add_pow_char (α : Type u) [comm_semiring α] {p : ℕ} [fact (nat.prime p)] [char_p α p] (x : α) (y : α) : (x + y) ^ p = x ^ p + y ^ p :=
add_pow_char_of_commute α x y (commute.all x y)
theorem add_pow_char_pow (R : Type u) [comm_semiring R] {p : ℕ} [fact (nat.prime p)] [char_p R p] {n : ℕ} (x : R) (y : R) : (x + y) ^ p ^ n = x ^ p ^ n + y ^ p ^ n :=
add_pow_char_pow_of_commute R x y (commute.all x y)
theorem sub_pow_char (α : Type u) [comm_ring α] {p : ℕ} [fact (nat.prime p)] [char_p α p] (x : α) (y : α) : (x - y) ^ p = x ^ p - y ^ p :=
sub_pow_char_of_commute α x y (commute.all x y)
theorem sub_pow_char_pow (R : Type u) [comm_ring R] {p : ℕ} [fact (nat.prime p)] [char_p R p] {n : ℕ} (x : R) (y : R) : (x - y) ^ p ^ n = x ^ p ^ n - y ^ p ^ n :=
sub_pow_char_pow_of_commute R x y (commute.all x y)
theorem eq_iff_modeq_int (R : Type u_1) [ring R] (p : ℕ) [char_p R p] (a : ℤ) (b : ℤ) : ↑a = ↑b ↔ int.modeq (↑p) a b := sorry
theorem char_p.neg_one_ne_one (R : Type u_1) [ring R] (p : ℕ) [char_p R p] [fact (bit0 1 < p)] : -1 ≠ 1 := sorry
theorem ring_hom.char_p_iff_char_p {K : Type u_1} {L : Type u_2} [field K] [field L] (f : K →+* L) (p : ℕ) : char_p K p ↔ char_p L p := sorry
/-- The frobenius map that sends x to x^p -/
def frobenius (R : Type u) [comm_semiring R] (p : ℕ) [fact (nat.prime p)] [char_p R p] : R →+* R :=
ring_hom.mk (fun (x : R) => x ^ p) sorry sorry sorry (add_pow_char R)
theorem frobenius_def {R : Type u} [comm_semiring R] (p : ℕ) [fact (nat.prime p)] [char_p R p] (x : R) : coe_fn (frobenius R p) x = x ^ p :=
rfl
theorem iterate_frobenius {R : Type u} [comm_semiring R] (p : ℕ) [fact (nat.prime p)] [char_p R p] (x : R) (n : ℕ) : nat.iterate (⇑(frobenius R p)) n x = x ^ p ^ n := sorry
theorem frobenius_mul {R : Type u} [comm_semiring R] (p : ℕ) [fact (nat.prime p)] [char_p R p] (x : R) (y : R) : coe_fn (frobenius R p) (x * y) = coe_fn (frobenius R p) x * coe_fn (frobenius R p) y :=
ring_hom.map_mul (frobenius R p) x y
theorem frobenius_one {R : Type u} [comm_semiring R] (p : ℕ) [fact (nat.prime p)] [char_p R p] : coe_fn (frobenius R p) 1 = 1 :=
one_pow p
theorem monoid_hom.map_frobenius {R : Type u} [comm_semiring R] {S : Type v} [comm_semiring S] (f : R →* S) (p : ℕ) [fact (nat.prime p)] [char_p R p] [char_p S p] (x : R) : coe_fn f (coe_fn (frobenius R p) x) = coe_fn (frobenius S p) (coe_fn f x) :=
monoid_hom.map_pow f x p
theorem ring_hom.map_frobenius {R : Type u} [comm_semiring R] {S : Type v} [comm_semiring S] (g : R →+* S) (p : ℕ) [fact (nat.prime p)] [char_p R p] [char_p S p] (x : R) : coe_fn g (coe_fn (frobenius R p) x) = coe_fn (frobenius S p) (coe_fn g x) :=
ring_hom.map_pow g x p
theorem monoid_hom.map_iterate_frobenius {R : Type u} [comm_semiring R] {S : Type v} [comm_semiring S] (f : R →* S) (p : ℕ) [fact (nat.prime p)] [char_p R p] [char_p S p] (x : R) (n : ℕ) : coe_fn f (nat.iterate (⇑(frobenius R p)) n x) = nat.iterate (⇑(frobenius S p)) n (coe_fn f x) :=
function.semiconj.iterate_right (monoid_hom.map_frobenius f p) n x
theorem ring_hom.map_iterate_frobenius {R : Type u} [comm_semiring R] {S : Type v} [comm_semiring S] (g : R →+* S) (p : ℕ) [fact (nat.prime p)] [char_p R p] [char_p S p] (x : R) (n : ℕ) : coe_fn g (nat.iterate (⇑(frobenius R p)) n x) = nat.iterate (⇑(frobenius S p)) n (coe_fn g x) :=
monoid_hom.map_iterate_frobenius (ring_hom.to_monoid_hom g) p x n
theorem monoid_hom.iterate_map_frobenius {R : Type u} [comm_semiring R] (x : R) (f : R →* R) (p : ℕ) [fact (nat.prime p)] [char_p R p] (n : ℕ) : nat.iterate (⇑f) n (coe_fn (frobenius R p) x) = coe_fn (frobenius R p) (nat.iterate (⇑f) n x) :=
monoid_hom.iterate_map_pow f x n p
theorem ring_hom.iterate_map_frobenius {R : Type u} [comm_semiring R] (x : R) (f : R →+* R) (p : ℕ) [fact (nat.prime p)] [char_p R p] (n : ℕ) : nat.iterate (⇑f) n (coe_fn (frobenius R p) x) = coe_fn (frobenius R p) (nat.iterate (⇑f) n x) :=
ring_hom.iterate_map_pow f x n p
theorem frobenius_zero (R : Type u) [comm_semiring R] (p : ℕ) [fact (nat.prime p)] [char_p R p] : coe_fn (frobenius R p) 0 = 0 :=
ring_hom.map_zero (frobenius R p)
theorem frobenius_add (R : Type u) [comm_semiring R] (p : ℕ) [fact (nat.prime p)] [char_p R p] (x : R) (y : R) : coe_fn (frobenius R p) (x + y) = coe_fn (frobenius R p) x + coe_fn (frobenius R p) y :=
ring_hom.map_add (frobenius R p) x y
theorem frobenius_nat_cast (R : Type u) [comm_semiring R] (p : ℕ) [fact (nat.prime p)] [char_p R p] (n : ℕ) : coe_fn (frobenius R p) ↑n = ↑n :=
ring_hom.map_nat_cast (frobenius R p) n
theorem frobenius_neg (R : Type u) [comm_ring R] (p : ℕ) [fact (nat.prime p)] [char_p R p] (x : R) : coe_fn (frobenius R p) (-x) = -coe_fn (frobenius R p) x :=
ring_hom.map_neg (frobenius R p) x
theorem frobenius_sub (R : Type u) [comm_ring R] (p : ℕ) [fact (nat.prime p)] [char_p R p] (x : R) (y : R) : coe_fn (frobenius R p) (x - y) = coe_fn (frobenius R p) x - coe_fn (frobenius R p) y :=
ring_hom.map_sub (frobenius R p) x y
theorem frobenius_inj (α : Type u) [comm_ring α] [no_zero_divisors α] (p : ℕ) [fact (nat.prime p)] [char_p α p] : function.injective ⇑(frobenius α p) := sorry
namespace char_p
theorem char_p_to_char_zero (α : Type u) [ring α] [char_p α 0] : char_zero α :=
char_zero_of_inj_zero fun (n : ℕ) (h0 : ↑n = 0) => eq_zero_of_zero_dvd (iff.mp (cast_eq_zero_iff α 0 n) h0)
theorem cast_eq_mod (α : Type u) [ring α] (p : ℕ) [char_p α p] (k : ℕ) : ↑k = ↑(k % p) := sorry
theorem char_ne_zero_of_fintype (α : Type u) [ring α] (p : ℕ) [hc : char_p α p] [fintype α] : p ≠ 0 :=
fun (h : p = 0) =>
(fun (this : char_zero α) => absurd nat.cast_injective (not_injective_infinite_fintype coe)) (char_p_to_char_zero α)
theorem char_ne_one (α : Type u) [integral_domain α] (p : ℕ) [hc : char_p α p] : p ≠ 1 := sorry
theorem char_is_prime_of_two_le (α : Type u) [integral_domain α] (p : ℕ) [hc : char_p α p] (hp : bit0 1 ≤ p) : nat.prime p := sorry
theorem char_is_prime_or_zero (α : Type u) [integral_domain α] (p : ℕ) [hc : char_p α p] : nat.prime p ∨ p = 0 := sorry
theorem char_is_prime_of_pos (α : Type u) [integral_domain α] (p : ℕ) [h : fact (0 < p)] [char_p α p] : fact (nat.prime p) :=
or.resolve_right (char_is_prime_or_zero α p) (iff.mp pos_iff_ne_zero h)
theorem char_is_prime (α : Type u) [integral_domain α] [fintype α] (p : ℕ) [char_p α p] : nat.prime p :=
or.resolve_right (char_is_prime_or_zero α p) (char_ne_zero_of_fintype α p)
protected instance subsingleton {R : Type u_1} [semiring R] [char_p R 1] : subsingleton R :=
subsingleton.intro
((fun (this : ∀ (r : R), r = 0) (a b : R) =>
(fun (this : a = b) => this)
(eq.mpr (id (Eq._oldrec (Eq.refl (a = b)) (this a)))
(eq.mpr (id (Eq._oldrec (Eq.refl (0 = b)) (this b))) (Eq.refl 0))))
fun (r : R) =>
Eq.trans
(Eq.trans
(Eq.trans (eq.mpr (id (Eq._oldrec (Eq.refl (r = 1 * r)) (one_mul r))) (Eq.refl r))
(eq.mpr (id (Eq._oldrec (Eq.refl (1 * r = ↑1 * r)) nat.cast_one)) (Eq.refl (1 * r))))
(eq.mpr (id (Eq._oldrec (Eq.refl (↑1 * r = 0 * r)) (cast_eq_zero R 1))) (Eq.refl (0 * r))))
(eq.mpr (id (Eq._oldrec (Eq.refl (0 * r = 0)) (zero_mul r))) (Eq.refl 0)))
theorem false_of_nontrivial_of_char_one {R : Type u_1} [semiring R] [nontrivial R] [char_p R 1] : False :=
false_of_nontrivial_of_subsingleton R
theorem ring_char_ne_one {R : Type u_1} [semiring R] [nontrivial R] : ring_char R ≠ 1 := sorry
theorem nontrivial_of_char_ne_one {v : ℕ} (hv : v ≠ 1) {R : Type u_1} [semiring R] [hr : char_p R v] : nontrivial R := sorry
end char_p
theorem char_p_of_ne_zero (n : ℕ) (R : Type u_1) [comm_ring R] [fintype R] (hn : fintype.card R = n) (hR : ∀ (i : ℕ), i < n → ↑i = 0 → i = 0) : char_p R n := sorry
theorem char_p_of_prime_pow_injective (R : Type u_1) [comm_ring R] [fintype R] (p : ℕ) [hp : fact (nat.prime p)] (n : ℕ) (hn : fintype.card R = p ^ n) (hR : ∀ (i : ℕ), i ≤ n → ↑p ^ i = 0 → i = n) : char_p R (p ^ n) := sorry
|
1ed947bf6852c8869278df27832689ed7d099443 | 437dc96105f48409c3981d46fb48e57c9ac3a3e4 | /src/group_theory/submonoid.lean | 69771d98937552d188ecc7dc852a663bc96d603f | [
"Apache-2.0"
] | permissive | dan-c-k/mathlib | 08efec79bd7481ee6da9cc44c24a653bff4fbe0d | 96efc220f6225bc7a5ed8349900391a33a38cc56 | refs/heads/master | 1,658,082,847,093 | 1,589,013,201,000 | 1,589,013,201,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 50,455 | lean | /-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import algebra.big_operators
import algebra.free_monoid
import algebra.group.prod
import data.equiv.mul_add
/-!
# Submonoids
This file defines multiplicative and additive submonoids, first in an unbundled form (deprecated)
and then in a bundled form.
We prove submonoids of a monoid form a complete lattice, and results about images and preimages of
submonoids under monoid homomorphisms. For the unbundled submonoids, these theorems use unbundled
monoid homomorphisms (also deprecated), and the bundled versions use bundled monoid homomorphisms.
There are also theorems about the submonoids generated by an element or a subset of a monoid,
defined both inductively and as the infimum of the set of submonoids containing a given
element/subset.
## Implementation notes
Unbundled submonoids will slowly be removed from mathlib.
(Bundled) submonoid inclusion is denoted `≤` rather than `⊆`, although `∈` is defined as
membership of a submonoid's underlying set.
## Tags
submonoid, submonoids, is_submonoid
-/
variables {M : Type*} [monoid M] {s : set M}
variables {A : Type*} [add_monoid A] {t : set A}
/-- `s` is an additive submonoid: a set containing 0 and closed under addition. -/
class is_add_submonoid (s : set A) : Prop :=
(zero_mem : (0:A) ∈ s)
(add_mem {a b} : a ∈ s → b ∈ s → a + b ∈ s)
/-- `s` is a submonoid: a set containing 1 and closed under multiplication. -/
@[to_additive is_add_submonoid]
class is_submonoid (s : set M) : Prop :=
(one_mem : (1:M) ∈ s)
(mul_mem {a b} : a ∈ s → b ∈ s → a * b ∈ s)
lemma additive.is_add_submonoid
(s : set M) : ∀ [is_submonoid s], @is_add_submonoid (additive M) _ s
| ⟨h₁, h₂⟩ := ⟨h₁, @h₂⟩
theorem additive.is_add_submonoid_iff
{s : set M} : @is_add_submonoid (additive M) _ s ↔ is_submonoid s :=
⟨λ ⟨h₁, h₂⟩, ⟨h₁, @h₂⟩, λ h, by exactI additive.is_add_submonoid _⟩
lemma multiplicative.is_submonoid
(s : set A) : ∀ [is_add_submonoid s], @is_submonoid (multiplicative A) _ s
| ⟨h₁, h₂⟩ := ⟨h₁, @h₂⟩
theorem multiplicative.is_submonoid_iff
{s : set A} : @is_submonoid (multiplicative A) _ s ↔ is_add_submonoid s :=
⟨λ ⟨h₁, h₂⟩, ⟨h₁, @h₂⟩, λ h, by exactI multiplicative.is_submonoid _⟩
/-- The intersection of two submonoids of a monoid `M` is a submonoid of `M`. -/
@[to_additive "The intersection of two `add_submonoid`s of an `add_monoid` `M` is an `add_submonoid` of M."]
instance is_submonoid.inter (s₁ s₂ : set M) [is_submonoid s₁] [is_submonoid s₂] :
is_submonoid (s₁ ∩ s₂) :=
{ one_mem := ⟨is_submonoid.one_mem, is_submonoid.one_mem⟩,
mul_mem := λ x y hx hy,
⟨is_submonoid.mul_mem hx.1 hy.1, is_submonoid.mul_mem hx.2 hy.2⟩ }
/-- The intersection of an indexed set of submonoids of a monoid `M` is a submonoid of `M`. -/
@[to_additive "The intersection of an indexed set of `add_submonoid`s of an `add_monoid` `M` is an `add_submonoid` of `M`."]
instance is_submonoid.Inter {ι : Sort*} (s : ι → set M) [h : ∀ y : ι, is_submonoid (s y)] :
is_submonoid (set.Inter s) :=
{ one_mem := set.mem_Inter.2 $ λ y, is_submonoid.one_mem,
mul_mem := λ x₁ x₂ h₁ h₂, set.mem_Inter.2 $
λ y, is_submonoid.mul_mem (set.mem_Inter.1 h₁ y) (set.mem_Inter.1 h₂ y) }
/-- The union of an indexed, directed, nonempty set of submonoids of a monoid `M` is a submonoid
of `M`. -/
@[to_additive is_add_submonoid_Union_of_directed "The union of an indexed, directed, nonempty set of `add_submonoid`s of an `add_monoid` `M` is an `add_submonoid` of `M`. "]
lemma is_submonoid_Union_of_directed {ι : Type*} [hι : nonempty ι]
(s : ι → set M) [∀ i, is_submonoid (s i)]
(directed : ∀ i j, ∃ k, s i ⊆ s k ∧ s j ⊆ s k) :
is_submonoid (⋃i, s i) :=
{ one_mem := let ⟨i⟩ := hι in set.mem_Union.2 ⟨i, is_submonoid.one_mem⟩,
mul_mem := λ a b ha hb,
let ⟨i, hi⟩ := set.mem_Union.1 ha in
let ⟨j, hj⟩ := set.mem_Union.1 hb in
let ⟨k, hk⟩ := directed i j in
set.mem_Union.2 ⟨k, is_submonoid.mul_mem (hk.1 hi) (hk.2 hj)⟩ }
section powers
/-- The set of natural number powers `1, x, x², ...` of an element `x` of a monoid. -/
def powers (x : M) : set M := {y | ∃ n:ℕ, x^n = y}
/-- The set of natural number multiples `0, x, 2x, ...` of an element `x` of an `add_monoid`. -/
def multiples (x : A) : set A := {y | ∃ n:ℕ, add_monoid.smul n x = y}
attribute [to_additive multiples] powers
/-- 1 is in the set of natural number powers of an element of a monoid. -/
lemma powers.one_mem {x : M} : (1 : M) ∈ powers x := ⟨0, pow_zero _⟩
/-- 0 is in the set of natural number multiples of an element of an `add_monoid`. -/
lemma multiples.zero_mem {x : A} : (0 : A) ∈ multiples x := ⟨0, add_monoid.zero_smul _⟩
attribute [to_additive] powers.one_mem
/-- An element of a monoid is in the set of that element's natural number powers. -/
lemma powers.self_mem {x : M} : x ∈ powers x := ⟨1, pow_one _⟩
/-- An element of an `add_monoid` is in the set of that element's natural number multiples. -/
lemma multiples.self_mem {x : A} : x ∈ multiples x := ⟨1, add_monoid.one_smul _⟩
attribute [to_additive] powers.self_mem
/-- The set of natural number powers of an element of a monoid is closed under multiplication. -/
lemma powers.mul_mem {x y z : M} : (y ∈ powers x) → (z ∈ powers x) → (y * z ∈ powers x) :=
λ ⟨n₁, h₁⟩ ⟨n₂, h₂⟩, ⟨n₁ + n₂, by simp only [pow_add, *]⟩
/-- The set of natural number multiples of an element of an `add_monoid` is closed under
addition. -/
lemma multiples.add_mem {x y z : A} :
(y ∈ multiples x) → (z ∈ multiples x) → (y + z ∈ multiples x) :=
@powers.mul_mem (multiplicative A) _ _ _ _
attribute [to_additive] powers.mul_mem
/-- The set of natural number powers of an element of a monoid `M` is a submonoid of `M`. -/
@[to_additive is_add_submonoid "The set of natural number multiples of an element of an `add_monoid` `M` is an `add_submonoid` of `M`."]
instance powers.is_submonoid (x : M) : is_submonoid (powers x) :=
{ one_mem := powers.one_mem,
mul_mem := λ y z, powers.mul_mem }
/-- A monoid is a submonoid of itself. -/
@[to_additive is_add_submonoid "An `add_monoid` is an `add_submonoid` of itself."]
instance univ.is_submonoid : is_submonoid (@set.univ M) := by split; simp
/-- The preimage of a submonoid under a monoid hom is a submonoid of the domain. -/
@[to_additive is_add_submonoid "The preimage of an `add_submonoid` under an `add_monoid` hom is an `add_submonoid` of the domain."]
instance preimage.is_submonoid {N : Type*} [monoid N] (f : M → N) [is_monoid_hom f]
(s : set N) [is_submonoid s] : is_submonoid (f ⁻¹' s) :=
{ one_mem := show f 1 ∈ s, by rw is_monoid_hom.map_one f; exact is_submonoid.one_mem,
mul_mem := λ a b (ha : f a ∈ s) (hb : f b ∈ s),
show f (a * b) ∈ s, by rw is_monoid_hom.map_mul f; exact is_submonoid.mul_mem ha hb }
/-- The image of a submonoid under a monoid hom is a submonoid of the codomain. -/
@[instance, to_additive is_add_submonoid "The image of an `add_submonoid` under an `add_monoid` hom is an `add_submonoid` of the codomain."]
lemma image.is_submonoid {γ : Type*} [monoid γ] (f : M → γ) [is_monoid_hom f]
(s : set M) [is_submonoid s] : is_submonoid (f '' s) :=
{ one_mem := ⟨1, is_submonoid.one_mem, is_monoid_hom.map_one f⟩,
mul_mem := λ a b ⟨x, hx⟩ ⟨y, hy⟩, ⟨x * y, is_submonoid.mul_mem hx.1 hy.1,
by rw [is_monoid_hom.map_mul f, hx.2, hy.2]⟩ }
/-- The image of a monoid hom is a submonoid of the codomain. -/
@[to_additive is_add_submonoid "The image of an `add_monoid` hom is an `add_submonoid` of the codomain."]
instance range.is_submonoid {γ : Type*} [monoid γ] (f : M → γ) [is_monoid_hom f] :
is_submonoid (set.range f) :=
by rw ← set.image_univ; apply_instance
/-- Submonoids are closed under natural powers. -/
lemma is_submonoid.pow_mem {a : M} [is_submonoid s] (h : a ∈ s) : ∀ {n : ℕ}, a ^ n ∈ s
| 0 := is_submonoid.one_mem
| (n + 1) := is_submonoid.mul_mem h is_submonoid.pow_mem
/-- An `add_submonoid` is closed under multiplication by naturals. -/
lemma is_add_submonoid.smul_mem {a : A} [is_add_submonoid t] :
∀ (h : a ∈ t) {n : ℕ}, add_monoid.smul n a ∈ t :=
@is_submonoid.pow_mem (multiplicative A) _ _ _ (multiplicative.is_submonoid _)
attribute [to_additive smul_mem] is_submonoid.pow_mem
/-- The set of natural number powers of an element of a submonoid is a subset of the submonoid. -/
lemma is_submonoid.power_subset {a : M} [is_submonoid s] (h : a ∈ s) : powers a ⊆ s :=
assume x ⟨n, hx⟩, hx ▸ is_submonoid.pow_mem h
/-- The set of natural number multiples of an element of an `add_submonoid` is a subset of the
`add_submonoid`. -/
lemma is_add_submonoid.multiple_subset {a : A} [is_add_submonoid t] :
a ∈ t → multiples a ⊆ t :=
@is_submonoid.power_subset (multiplicative A) _ _ _ (multiplicative.is_submonoid _)
attribute [to_additive multiple_subset] is_submonoid.power_subset
end powers
namespace is_submonoid
/-- The product of a list of elements of a submonoid is an element of the submonoid. -/
@[to_additive "The sum of a list of elements of an `add_submonoid` is an element of the `add_submonoid`."]
lemma list_prod_mem [is_submonoid s] : ∀{l : list M}, (∀x∈l, x ∈ s) → l.prod ∈ s
| [] h := one_mem
| (a::l) h :=
suffices a * l.prod ∈ s, by simpa,
have a ∈ s ∧ (∀x∈l, x ∈ s), by simpa using h,
is_submonoid.mul_mem this.1 (list_prod_mem this.2)
/-- The product of a multiset of elements of a submonoid of a `comm_monoid` is an element of the submonoid. -/
@[to_additive "The sum of a multiset of elements of an `add_submonoid` of an `add_comm_monoid` is an element of the `add_submonoid`. "]
lemma multiset_prod_mem {M} [comm_monoid M] (s : set M) [is_submonoid s] (m : multiset M) :
(∀a∈m, a ∈ s) → m.prod ∈ s :=
begin
refine quotient.induction_on m (assume l hl, _),
rw [multiset.quot_mk_to_coe, multiset.coe_prod],
exact list_prod_mem hl
end
/-- The product of elements of a submonoid of a `comm_monoid` indexed by a `finset` is an element of the submonoid. -/
@[to_additive "The sum of elements of an `add_submonoid` of an `add_comm_monoid` indexed by a `finset` is an element of the `add_submonoid`."]
lemma finset_prod_mem {M A} [comm_monoid M] (s : set M) [is_submonoid s] (f : A → M) :
∀(t : finset A), (∀b∈t, f b ∈ s) → t.prod f ∈ s
| ⟨m, hm⟩ hs :=
begin
refine multiset_prod_mem s _ _,
simp,
rintros a b hb rfl,
exact hs _ hb
end
end is_submonoid
-- TODO: modify `subtype_instance` to produce this definition, then use it here
-- and for `subtype.group`
/-- Submonoids are themselves monoids. -/
@[to_additive add_monoid "An `add_submonoid` is itself an `add_monoid`."]
instance subtype.monoid {s : set M} [is_submonoid s] : monoid s :=
{ one := ⟨1, is_submonoid.one_mem⟩,
mul := λ x y, ⟨x * y, is_submonoid.mul_mem x.2 y.2⟩,
mul_one := λ x, subtype.eq $ mul_one x.1,
one_mul := λ x, subtype.eq $ one_mul x.1,
mul_assoc := λ x y z, subtype.eq $ mul_assoc x.1 y.1 z.1 }
/-- Submonoids of commutative monoids are themselves commutative monoids. -/
@[to_additive add_comm_monoid "An `add_submonoid` of a commutative `add_monoid` is itself a commutative `add_monoid`. "]
instance subtype.comm_monoid {M} [comm_monoid M] {s : set M} [is_submonoid s] : comm_monoid s :=
{ mul_comm := λ x y, subtype.eq $ mul_comm x.1 y.1,
.. subtype.monoid }
/-- Submonoids inherit the 1 of the monoid. -/
@[simp, norm_cast, to_additive "An `add_submonoid` inherits the 0 of the `add_monoid`. "]
lemma is_submonoid.coe_one [is_submonoid s] : ((1 : s) : M) = 1 := rfl
attribute [norm_cast] is_add_submonoid.coe_zero
/-- Submonoids inherit the multiplication of the monoid. -/
@[simp, norm_cast, to_additive "An `add_submonoid` inherits the addition of the `add_monoid`. "]
lemma is_submonoid.coe_mul [is_submonoid s] (a b : s) : ((a * b : s) : M) = a * b := rfl
attribute [norm_cast] is_add_submonoid.coe_add
/-- Submonoids inherit the exponentiation by naturals of the monoid. -/
@[simp, norm_cast] lemma is_submonoid.coe_pow [is_submonoid s] (a : s) (n : ℕ) :
((a ^ n : s) : M) = a ^ n :=
by induction n; simp [*, pow_succ]
/-- An `add_submonoid` inherits the multiplication by naturals of the `add_monoid`. -/
@[simp, norm_cast] lemma is_add_submonoid.smul_coe {A : Type*} [add_monoid A] {s : set A}
[is_add_submonoid s] (a : s) (n : ℕ) : ((add_monoid.smul n a : s) : A) = add_monoid.smul n a :=
by {induction n, refl, simp [*, succ_smul]}
attribute [to_additive smul_coe] is_submonoid.coe_pow
/-- The natural injection from a submonoid into the monoid is a monoid hom. -/
@[to_additive is_add_monoid_hom "The natural injection from an `add_submonoid` into the `add_monoid` is an `add_monoid` hom. "]
instance subtype_val.is_monoid_hom [is_submonoid s] : is_monoid_hom (subtype.val : s → M) :=
{ map_one := rfl, map_mul := λ _ _, rfl }
/-- The natural injection from a submonoid into the monoid is a monoid hom. -/
@[to_additive is_add_monoid_hom "The natural injection from an `add_submonoid` into the `add_monoid` is an `add_monoid` hom. "]
instance coe.is_monoid_hom [is_submonoid s] : is_monoid_hom (coe : s → M) :=
subtype_val.is_monoid_hom
/-- Given a monoid hom `f : γ → M` whose image is contained in a submonoid `s`, the induced map
from `γ` to `s` is a monoid hom. -/
@[to_additive is_add_monoid_hom "Given an `add_monoid` hom `f : γ → M` whose image is contained in an `add_submonoid` s, the induced map from `γ` to `s` is an `add_monoid` hom."]
instance subtype_mk.is_monoid_hom {γ : Type*} [monoid γ] [is_submonoid s] (f : γ → M)
[is_monoid_hom f] (h : ∀ x, f x ∈ s) : is_monoid_hom (λ x, (⟨f x, h x⟩ : s)) :=
{ map_one := subtype.eq (is_monoid_hom.map_one f),
map_mul := λ x y, subtype.eq (is_monoid_hom.map_mul f x y) }
/-- Given two submonoids `s` and `t` such that `s ⊆ t`, the natural injection from `s` into `t` is
a monoid hom. -/
@[to_additive is_add_monoid_hom "Given two `add_submonoid`s `s` and `t` such that `s ⊆ t`, the natural injection from `s` into `t` is an `add_monoid` hom."]
instance set_inclusion.is_monoid_hom (t : set M) [is_submonoid s] [is_submonoid t] (h : s ⊆ t) :
is_monoid_hom (set.inclusion h) :=
subtype_mk.is_monoid_hom _ _
namespace add_monoid
/-- The inductively defined membership predicate for the submonoid generated by a subset of a
monoid. -/
inductive in_closure (s : set A) : A → Prop
| basic {a : A} : a ∈ s → in_closure a
| zero : in_closure 0
| add {a b : A} : in_closure a → in_closure b → in_closure (a + b)
end add_monoid
namespace monoid
/-- The inductively defined membership predicate for the `add_submonoid` generated by a subset of an
add_monoid. -/
inductive in_closure (s : set M) : M → Prop
| basic {a : M} : a ∈ s → in_closure a
| one : in_closure 1
| mul {a b : M} : in_closure a → in_closure b → in_closure (a * b)
attribute [to_additive] monoid.in_closure
attribute [to_additive] monoid.in_closure.one
attribute [to_additive] monoid.in_closure.mul
/-- The inductively defined submonoid generated by a subset of a monoid. -/
@[to_additive "The inductively defined `add_submonoid` genrated by a subset of an `add_monoid`."]
def closure (s : set M) : set M := {a | in_closure s a }
@[to_additive is_add_submonoid]
instance closure.is_submonoid (s : set M) : is_submonoid (closure s) :=
{ one_mem := in_closure.one, mul_mem := assume a b, in_closure.mul }
/-- A subset of a monoid is contained in the submonoid it generates. -/
@[to_additive "A subset of an `add_monoid` is contained in the `add_submonoid` it generates."]
theorem subset_closure {s : set M} : s ⊆ closure s :=
assume a, in_closure.basic
/-- The submonoid generated by a set is contained in any submonoid that contains the set. -/
@[to_additive "The `add_submonoid` generated by a set is contained in any `add_submonoid` that contains the set."]
theorem closure_subset {s t : set M} [is_submonoid t] (h : s ⊆ t) : closure s ⊆ t :=
assume a ha, by induction ha; simp [h _, *, is_submonoid.one_mem, is_submonoid.mul_mem]
/-- Given subsets `t` and `s` of a monoid `M`, if `s ⊆ t`, the submonoid of `M` generated by `s` is
contained in the submonoid generated by `t`. -/
@[to_additive "Given subsets `t` and `s` of an `add_monoid M`, if `s ⊆ t`, the `add_submonoid` of `M` generated by `s` is contained in the `add_submonoid` generated by `t`."]
theorem closure_mono {s t : set M} (h : s ⊆ t) : closure s ⊆ closure t :=
closure_subset $ set.subset.trans h subset_closure
/-- The submonoid generated by an element of a monoid equals the set of natural number powers of
the element. -/
@[to_additive "The `add_submonoid` generated by an element of an `add_monoid` equals the set of natural number multiples of the element."]
theorem closure_singleton {x : M} : closure ({x} : set M) = powers x :=
set.eq_of_subset_of_subset (closure_subset $ set.singleton_subset_iff.2 $ powers.self_mem) $
is_submonoid.power_subset $ set.singleton_subset_iff.1 $ subset_closure
/-- The image under a monoid hom of the submonoid generated by a set equals the submonoid generated
by the image of the set under the monoid hom. -/
@[to_additive "The image under an `add_monoid` hom of the `add_submonoid` generated by a set equals the `add_submonoid` generated by the image of the set under the `add_monoid` hom."]
lemma image_closure {A : Type*} [monoid A] (f : M → A) [is_monoid_hom f] (s : set M) :
f '' closure s = closure (f '' s) :=
le_antisymm
begin
rintros _ ⟨x, hx, rfl⟩,
apply in_closure.rec_on hx; intros,
{ solve_by_elim [subset_closure, set.mem_image_of_mem] },
{ rw [is_monoid_hom.map_one f], apply is_submonoid.one_mem },
{ rw [is_monoid_hom.map_mul f], solve_by_elim [is_submonoid.mul_mem] }
end
(closure_subset $ set.image_subset _ subset_closure)
/-- Given an element `a` of the submonoid of a monoid `M` generated by a set `s`, there exists a list of
elements of `s` whose product is `a`. -/
@[to_additive "Given an element `a` of the `add_submonoid` of an `add_monoid M` generated by a set `s`, there exists a list of elements of `s` whose sum is `a`."]
theorem exists_list_of_mem_closure {s : set M} {a : M} (h : a ∈ closure s) :
(∃l:list M, (∀x∈l, x ∈ s) ∧ l.prod = a) :=
begin
induction h,
case in_closure.basic : a ha { existsi ([a]), simp [ha] },
case in_closure.one { existsi ([]), simp },
case in_closure.mul : a b _ _ ha hb {
rcases ha with ⟨la, ha, eqa⟩,
rcases hb with ⟨lb, hb, eqb⟩,
existsi (la ++ lb),
simp [eqa.symm, eqb.symm, or_imp_distrib],
exact assume a, ⟨ha a, hb a⟩
}
end
/-- Given sets `s, t` of a commutative monoid `M`, `x ∈ M` is in the submonoid of `M` generated by
`s ∪ t` iff there exists an element of the submonoid generated by `s` and an element of the
submonoid generated by `t` whose product is `x`. -/
@[to_additive "Given sets `s, t` of a commutative `add_monoid M`, `x ∈ M` is in the `add_submonoid` of `M` generated by `s ∪ t` iff there exists an element of the `add_submonoid` generated by `s` and an element of the `add_submonoid` generated by `t` whose sum is `x`."]
theorem mem_closure_union_iff {M : Type*} [comm_monoid M] {s t : set M} {x : M} :
x ∈ closure (s ∪ t) ↔ ∃ y ∈ closure s, ∃ z ∈ closure t, y * z = x :=
⟨λ hx, let ⟨L, HL1, HL2⟩ := exists_list_of_mem_closure hx in HL2 ▸
list.rec_on L (λ _, ⟨1, is_submonoid.one_mem, 1, is_submonoid.one_mem, mul_one _⟩)
(λ hd tl ih HL1, let ⟨y, hy, z, hz, hyzx⟩ := ih (list.forall_mem_of_forall_mem_cons HL1) in
or.cases_on (HL1 hd $ list.mem_cons_self _ _)
(λ hs, ⟨hd * y, is_submonoid.mul_mem (subset_closure hs) hy, z, hz, by rw [mul_assoc, list.prod_cons, ← hyzx]; refl⟩)
(λ ht, ⟨y, hy, z * hd, is_submonoid.mul_mem hz (subset_closure ht), by rw [← mul_assoc, list.prod_cons, ← hyzx, mul_comm hd]; refl⟩)) HL1,
λ ⟨y, hy, z, hz, hyzx⟩, hyzx ▸ is_submonoid.mul_mem (closure_mono (set.subset_union_left _ _) hy)
(closure_mono (set.subset_union_right _ _) hz)⟩
end monoid
/-!
### Bundled submonoids and `add_submonoid`s
-/
/-- A submonoid of a monoid `M` is a subset containing 1 and closed under multiplication. -/
structure submonoid (M : Type*) [monoid M] :=
(carrier : set M)
(one_mem' : (1 : M) ∈ carrier)
(mul_mem' {a b} : a ∈ carrier → b ∈ carrier → a * b ∈ carrier)
/-- An additive submonoid of an additive monoid `M` is a subset containing 0 and
closed under addition. -/
structure add_submonoid (M : Type*) [add_monoid M] :=
(carrier : set M)
(zero_mem' : (0 : M) ∈ carrier)
(add_mem' {a b} : a ∈ carrier → b ∈ carrier → a + b ∈ carrier)
attribute [to_additive add_submonoid] submonoid
/-- Create a bundled submonoid from a set `s` and `[is_submonoid s]`. -/
@[to_additive "Create a bundled additive submonoid from a set `s` and `[is_add_submonoid s]`."]
def submonoid.of (s : set M) [h : is_submonoid s] : submonoid M := ⟨s, h.1, h.2⟩
/-- Map from submonoids of monoid `M` to `add_submonoid`s of `additive M`. -/
def submonoid.to_add_submonoid {M : Type*} [monoid M] (S : submonoid M) :
add_submonoid (additive M) :=
{ carrier := S.carrier,
zero_mem' := S.one_mem',
add_mem' := S.mul_mem' }
/-- Map from `add_submonoid`s of `additive M` to submonoids of `M`. -/
def submonoid.of_add_submonoid {M : Type*} [monoid M] (S : add_submonoid (additive M)) :
submonoid M :=
{ carrier := S.carrier,
one_mem' := S.zero_mem',
mul_mem' := S.add_mem' }
/-- Map from `add_submonoid`s of `add_monoid M` to submonoids of `multiplicative M`. -/
def add_submonoid.to_submonoid {M : Type*} [add_monoid M] (S : add_submonoid M) :
submonoid (multiplicative M) :=
{ carrier := S.carrier,
one_mem' := S.zero_mem',
mul_mem' := S.add_mem' }
/-- Map from submonoids of `multiplicative M` to `add_submonoid`s of `add_monoid M`. -/
def add_submonoid.of_submonoid {M : Type*} [add_monoid M] (S : submonoid (multiplicative M)) :
add_submonoid M :=
{ carrier := S.carrier,
zero_mem' := S.one_mem',
add_mem' := S.mul_mem' }
/-- Submonoids of monoid `M` are isomorphic to additive submonoids of `additive M`. -/
def submonoid.add_submonoid_equiv (M : Type*) [monoid M] :
submonoid M ≃ add_submonoid (additive M) :=
{ to_fun := submonoid.to_add_submonoid,
inv_fun := submonoid.of_add_submonoid,
left_inv := λ x, by cases x; refl,
right_inv := λ x, by cases x; refl }
namespace submonoid
@[to_additive]
instance : has_coe (submonoid M) (set M) := ⟨submonoid.carrier⟩
@[to_additive]
instance : has_coe_to_sort (submonoid M) := ⟨Type*, λ S, S.carrier⟩
@[to_additive]
instance : has_mem M (submonoid M) := ⟨λ m S, m ∈ (S:set M)⟩
@[simp, norm_cast, to_additive]
lemma mem_coe {S : submonoid M} {m : M} : m ∈ (S : set M) ↔ m ∈ S := iff.rfl
@[simp, norm_cast, to_additive]
lemma coe_coe (s : submonoid M) : ↥(s : set M) = s := rfl
attribute [norm_cast] add_submonoid.mem_coe add_submonoid.coe_coe
@[to_additive]
instance is_submonoid (S : submonoid M) : is_submonoid (S : set M) := ⟨S.2, S.3⟩
end submonoid
@[to_additive]
protected lemma submonoid.exists {s : submonoid M} {p : s → Prop} :
(∃ x : s, p x) ↔ ∃ x ∈ s, p ⟨x, ‹x ∈ s›⟩ :=
set_coe.exists
namespace submonoid
variables (S : submonoid M)
/-- Two submonoids are equal if the underlying subsets are equal. -/
@[to_additive "Two `add_submonoid`s are equal if the underlying subsets are equal."]
theorem ext' {S T : submonoid M} (h : (S : set M) = T) : S = T :=
by cases S; cases T; congr'
/-- Two submonoids are equal if and only if the underlying subsets are equal. -/
@[to_additive "Two `add_submonoid`s are equal if and only if the underlying subsets are equal."]
protected theorem ext'_iff {S T : submonoid M} : S = T ↔ (S : set M) = T :=
⟨λ h, h ▸ rfl, ext'⟩
/-- Two submonoids are equal if they have the same elements. -/
@[ext, to_additive "Two `add_submonoid`s are equal if they have the same elements."]
theorem ext {S T : submonoid M}
(h : ∀ x, x ∈ S ↔ x ∈ T) : S = T := ext' $ set.ext h
attribute [ext] add_submonoid.ext
/-- A submonoid contains the monoid's 1. -/
@[to_additive "An `add_submonoid` contains the monoid's 0."]
theorem one_mem : (1 : M) ∈ S := S.one_mem'
/-- A submonoid is closed under multiplication. -/
@[to_additive "An `add_submonoid` is closed under addition."]
theorem mul_mem {x y : M} : x ∈ S → y ∈ S → x * y ∈ S := submonoid.mul_mem' S
/-- Product of a list of elements in a submonoid is in the submonoid. -/
@[to_additive "Sum of a list of elements in an `add_submonoid` is in the `add_submonoid`."]
lemma list_prod_mem : ∀ {l : list M}, (∀x ∈ l, x ∈ S) → l.prod ∈ S
| [] h := S.one_mem
| (a::l) h :=
suffices a * l.prod ∈ S, by rwa [list.prod_cons],
have a ∈ S ∧ (∀ x ∈ l, x ∈ S), from list.forall_mem_cons.1 h,
S.mul_mem this.1 (list_prod_mem this.2)
/-- Product of a multiset of elements in a submonoid of a `comm_monoid` is in the submonoid. -/
@[to_additive "Sum of a multiset of elements in an `add_submonoid` of an `add_comm_monoid` is in the `add_submonoid`."]
lemma multiset_prod_mem {M} [comm_monoid M] (S : submonoid M) (m : multiset M) :
(∀a ∈ m, a ∈ S) → m.prod ∈ S :=
begin
refine quotient.induction_on m (assume l hl, _),
rw [multiset.quot_mk_to_coe, multiset.coe_prod],
exact S.list_prod_mem hl
end
/-- Product of elements of a submonoid of a `comm_monoid` indexed by a `finset` is in the
submonoid. -/
@[to_additive "Sum of elements in an `add_submonoid` of an `add_comm_monoid` indexed by a `finset` is in the `add_submonoid`."]
lemma prod_mem {M : Type*} [comm_monoid M] (S : submonoid M)
{ι : Type*} {t : finset ι} {f : ι → M} (h : ∀c ∈ t, f c ∈ S) :
t.prod f ∈ S :=
S.multiset_prod_mem (t.1.map f) $ λ x hx, let ⟨i, hi, hix⟩ := multiset.mem_map.1 hx in hix ▸ h i hi
lemma pow_mem {x : M} (hx : x ∈ S) : ∀ n:ℕ, x^n ∈ S
| 0 := S.one_mem
| (n+1) := S.mul_mem hx (pow_mem n)
/-- A submonoid of a monoid inherits a multiplication. -/
@[to_additive "An `add_submonoid` of an `add_monoid` inherits an addition."]
instance has_mul : has_mul S := ⟨λ a b, ⟨a.1 * b.1, S.mul_mem a.2 b.2⟩⟩
/-- A submonoid of a monoid inherits a 1. -/
@[to_additive "An `add_submonoid` of an `add_monoid` inherits a zero."]
instance has_one : has_one S := ⟨⟨_, S.one_mem⟩⟩
@[simp, to_additive] lemma coe_mul (x y : S) : (↑(x * y) : M) = ↑x * ↑y := rfl
@[simp, to_additive] lemma coe_one : ((1 : S) : M) = 1 := rfl
/-- A submonoid of a monoid inherits a monoid structure. -/
@[to_additive to_add_monoid "An `add_submonoid` of an `add_monoid` inherits an `add_monoid` structure."]
instance to_monoid {M : Type*} [monoid M] (S : submonoid M) : monoid S :=
by refine { mul := (*), one := 1, ..}; by simp [mul_assoc]
/-- A submonoid of a `comm_monoid` is a `comm_monoid`. -/
@[to_additive to_add_comm_monoid "An `add_submonoid` of an `add_comm_monoid` is an `add_comm_monoid`."]
instance to_comm_monoid {M} [comm_monoid M] (S : submonoid M) : comm_monoid S :=
{ mul_comm := λ _ _, subtype.eq $ mul_comm _ _, ..S.to_monoid}
/-- The natural monoid hom from a submonoid of monoid `M` to `M`. -/
@[to_additive "The natural monoid hom from an `add_submonoid` of `add_monoid` `M` to `M`."]
def subtype : S →* M := ⟨coe, rfl, λ _ _, rfl⟩
@[simp, to_additive] theorem coe_subtype : ⇑S.subtype = coe := rfl
@[to_additive]
instance : has_le (submonoid M) := ⟨λ S T, ∀ ⦃x⦄, x ∈ S → x ∈ T⟩
@[to_additive]
instance : has_lt (submonoid M) := ⟨λ S T, S ≤ T ∧ ¬(T ≤ S)⟩
@[to_additive]
lemma le_def {S T : submonoid M} : S ≤ T ↔ ∀ ⦃x : M⦄, x ∈ S → x ∈ T := iff.rfl
@[simp, to_additive]
lemma coe_subset_coe {S T : submonoid M} : (S : set M) ⊆ T ↔ S ≤ T := iff.rfl
@[simp, to_additive]
lemma coe_ssubset_coe {S T : submonoid M} : (S : set M) ⊂ T ↔ S < T := iff.rfl
/-- The submonoid `M` of the monoid `M`. -/
@[to_additive "The `add_submonoid M` of the `add_monoid M`."]
instance : has_top (submonoid M) :=
⟨{ carrier := set.univ,
one_mem' := set.mem_univ 1,
mul_mem' := λ _ _ _ _, set.mem_univ _ }⟩
/-- The trivial submonoid `{1}` of an monoid `M`. -/
@[to_additive "The trivial `add_submonoid` `{0}` of an `add_monoid` `M`."]
instance : has_bot (submonoid M) :=
⟨{ carrier := {1},
one_mem' := set.mem_singleton 1,
mul_mem' := λ a b ha hb, by { simp only [set.mem_singleton_iff] at *, rw [ha, hb, mul_one] }}⟩
@[to_additive]
instance : inhabited (submonoid M) := ⟨⊥⟩
@[simp, to_additive] lemma mem_bot {x : M} : x ∈ (⊥ : submonoid M) ↔ x = 1 := set.mem_singleton_iff
@[simp, to_additive] lemma mem_top (x : M) : x ∈ (⊤ : submonoid M) := set.mem_univ x
@[simp, to_additive] lemma coe_top : ((⊤ : submonoid M) : set M) = set.univ := rfl
@[simp, to_additive] lemma coe_bot : ((⊥ : submonoid M) : set M) = {1} := rfl
/-- The inf of two submonoids is their intersection. -/
@[to_additive "The inf of two `add_submonoid`s is their intersection."]
instance : has_inf (submonoid M) :=
⟨λ S₁ S₂,
{ carrier := S₁ ∩ S₂,
one_mem' := ⟨S₁.one_mem, S₂.one_mem⟩,
mul_mem' := λ _ _ ⟨hx, hx'⟩ ⟨hy, hy'⟩,
⟨S₁.mul_mem hx hy, S₂.mul_mem hx' hy'⟩ }⟩
@[simp, to_additive]
lemma coe_inf (p p' : submonoid M) : ((p ⊓ p' : submonoid M) : set M) = p ∩ p' := rfl
@[simp, to_additive]
lemma mem_inf {p p' : submonoid M} {x : M} : x ∈ p ⊓ p' ↔ x ∈ p ∧ x ∈ p' := iff.rfl
@[to_additive]
instance : has_Inf (submonoid M) :=
⟨λ s, {
carrier := ⋂ t ∈ s, ↑t,
one_mem' := set.mem_bInter $ λ i h, i.one_mem,
mul_mem' := λ x y hx hy, set.mem_bInter $ λ i h,
i.mul_mem (by apply set.mem_bInter_iff.1 hx i h) (by apply set.mem_bInter_iff.1 hy i h) }⟩
@[to_additive]
lemma coe_Inf (S : set (submonoid M)) : ((Inf S : submonoid M) : set M) = ⋂ s ∈ S, ↑s := rfl
@[to_additive]
lemma mem_Inf {S : set (submonoid M)} {x : M} : x ∈ Inf S ↔ ∀ p ∈ S, x ∈ p := set.mem_bInter_iff
/-- Submonoids of a monoid form a complete lattice. -/
@[to_additive "The `add_submonoid`s of an `add_monoid` form a complete lattice."]
instance : complete_lattice (submonoid M) :=
{ le := (≤),
lt := (<),
bot := (⊥),
bot_le := λ S x hx, (mem_bot.1 hx).symm ▸ S.one_mem,
top := (⊤),
le_top := λ S x hx, mem_top 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,
.. partial_order.lift (coe : submonoid M → set M) (λ a b, ext') (by apply_instance) }
/-- The `submonoid` generated by a set. -/
@[to_additive "The `add_submonoid` generated by a set"]
def closure (s : set M) : submonoid M := Inf {S | s ⊆ S}
@[to_additive]
lemma mem_closure {x : M} : x ∈ closure s ↔ ∀ S : submonoid M, s ⊆ S → x ∈ S :=
mem_Inf
/-- The submonoid generated by a set includes the set. -/
@[simp, to_additive "The `add_submonoid` generated by a set includes the set."]
lemma subset_closure : s ⊆ closure s := λ x hx, mem_closure.2 $ λ S hS, hS hx
variable {S}
open set
/-- A submonoid `S` includes `closure s` if and only if it includes `s`. -/
@[simp, to_additive "An additive submonoid `S` includes `closure s` if and only if it includes `s`"]
lemma closure_le : closure s ≤ S ↔ s ⊆ S :=
⟨subset.trans subset_closure, λ h, Inf_le h⟩
/-- Submonoid closure of a set is monotone in its argument: if `s ⊆ t`,
then `closure s ≤ closure t`. -/
@[to_additive "Additive submonoid closure of a set is monotone in its argument: if `s ⊆ t`,
then `closure s ≤ closure t`"]
lemma closure_mono ⦃s t : set M⦄ (h : s ⊆ t) : closure s ≤ closure t :=
closure_le.2 $ subset.trans h subset_closure
@[to_additive]
lemma closure_eq_of_le (h₁ : s ⊆ S) (h₂ : S ≤ closure s) : closure s = S :=
le_antisymm (closure_le.2 h₁) h₂
variable (S)
/-- An induction principle for closure membership. If `p` holds for `1` and all elements of `s`, and
is preserved under multiplication, then `p` holds for all elements of the closure of `s`. -/
@[to_additive "An induction principle for additive closure membership. If `p` holds for `0` and all
elements of `s`, and is preserved under addition, then `p` holds for all elements
of the additive closure of `s`."]
lemma closure_induction {p : M → Prop} {x} (h : x ∈ closure s)
(Hs : ∀ x ∈ s, p x) (H1 : p 1)
(Hmul : ∀ x y, p x → p y → p (x * y)) : p x :=
(@closure_le _ _ _ ⟨p, H1, Hmul⟩).2 Hs h
attribute [elab_as_eliminator] submonoid.closure_induction add_submonoid.closure_induction
variable (M)
/-- `closure` forms a Galois insertion with the coercion to set. -/
@[to_additive "`closure` forms a Galois insertion with the coercion to set."]
protected def gi : galois_insertion (@closure M _) coe :=
{ choice := λ s _, closure s,
gc := λ s t, closure_le,
le_l_u := λ s, subset_closure,
choice_eq := λ s h, rfl }
variable {M}
/-- Closure of a submonoid `S` equals `S`. -/
@[simp, to_additive "Additive closure of an additive submonoid `S` equals `S`"]
lemma closure_eq : closure (S : set M) = S := (submonoid.gi M).l_u_eq S
@[simp, to_additive] lemma closure_empty : closure (∅ : set M) = ⊥ :=
(submonoid.gi M).gc.l_bot
@[simp, to_additive] lemma closure_univ : closure (univ : set M) = ⊤ :=
@coe_top M _ ▸ closure_eq ⊤
@[to_additive]
lemma closure_union (s t : set M) : closure (s ∪ t) = closure s ⊔ closure t :=
(submonoid.gi M).gc.l_sup
@[to_additive]
lemma closure_Union {ι} (s : ι → set M) : closure (⋃ i, s i) = ⨆ i, closure (s i) :=
(submonoid.gi M).gc.l_supr
@[to_additive]
lemma mem_supr_of_directed {ι} [hι : nonempty ι] {S : ι → submonoid M} (hS : directed (≤) S)
{x : M} :
x ∈ (supr S : submonoid M) ↔ ∃ i, x ∈ S i :=
begin
refine ⟨_, λ ⟨i, hi⟩, (le_def.1 $ le_supr S i) hi⟩,
suffices : x ∈ closure (⋃ i, (S i : set M)) → ∃ i, x ∈ S i,
by simpa only [closure_Union, closure_eq (S _)] using this,
refine (λ hx, closure_induction hx (λ _, mem_Union.1) _ _),
{ exact hι.elim (λ i, ⟨i, (S i).one_mem⟩) },
{ rintros x y ⟨i, hi⟩ ⟨j, hj⟩,
rcases hS i j with ⟨k, hki, hkj⟩,
exact ⟨k, (S k).mul_mem (hki hi) (hkj hj)⟩ }
end
@[to_additive]
lemma mem_Sup_of_directed_on {S : set (submonoid M)} (Sne : S.nonempty)
(hS : directed_on (≤) S) {x : M} :
x ∈ Sup S ↔ ∃ s ∈ S, x ∈ s :=
begin
haveI : nonempty S := Sne.to_subtype,
rw [Sup_eq_supr, supr_subtype', mem_supr_of_directed, subtype.exists],
exact (directed_on_iff_directed _).1 hS
end
variables {N : Type*} [monoid N] {P : Type*} [monoid P]
/-- The preimage of a submonoid along a monoid homomorphism is a submonoid. -/
@[to_additive "The preimage of an `add_submonoid` along an `add_monoid` homomorphism is an `add_submonoid`."]
def comap (f : M →* N) (S : submonoid N) : submonoid M :=
{ carrier := (f ⁻¹' S),
one_mem' := show f 1 ∈ S, by rw f.map_one; exact S.one_mem,
mul_mem' := λ a b ha hb,
show f (a * b) ∈ S, by rw f.map_mul; exact S.mul_mem ha hb }
@[simp, to_additive]
lemma coe_comap (S : submonoid N) (f : M →* N) : (S.comap f : set M) = f ⁻¹' S := rfl
@[simp, to_additive]
lemma mem_comap {S : submonoid N} {f : M →* N} {x : M} : x ∈ S.comap f ↔ f x ∈ S := iff.rfl
@[to_additive]
lemma comap_comap (S : submonoid P) (g : N →* P) (f : M →* N) :
(S.comap g).comap f = S.comap (g.comp f) :=
rfl
/-- The image of a submonoid along a monoid homomorphism is a submonoid. -/
@[to_additive "The image of an `add_submonoid` along an `add_monoid` homomorphism is an `add_submonoid`."]
def map (f : M →* N) (S : submonoid M) : submonoid N :=
{ carrier := (f '' S),
one_mem' := ⟨1, S.one_mem, f.map_one⟩,
mul_mem' := begin rintros _ _ ⟨x, hx, rfl⟩ ⟨y, hy, rfl⟩, exact ⟨x * y, S.mul_mem hx hy,
by rw f.map_mul; refl⟩ end }
@[simp, to_additive]
lemma coe_map (f : M →* N) (S : submonoid M) :
(S.map f : set N) = f '' S := rfl
@[simp, to_additive]
lemma mem_map {f : M →* N} {S : submonoid M} {y : N} :
y ∈ S.map f ↔ ∃ x ∈ S, f x = y :=
mem_image_iff_bex
@[to_additive]
lemma map_map (g : N →* P) (f : M →* N) : (S.map f).map g = S.map (g.comp f) :=
ext' $ image_image _ _ _
@[to_additive]
lemma map_le_iff_le_comap {f : M →* N} {S : submonoid M} {T : submonoid N} :
S.map f ≤ T ↔ S ≤ T.comap f :=
image_subset_iff
@[to_additive]
lemma gc_map_comap (f : M →* N) : galois_connection (map f) (comap f) :=
λ S T, map_le_iff_le_comap
@[to_additive]
lemma map_sup (S T : submonoid M) (f : M →* N) : (S ⊔ T).map f = S.map f ⊔ T.map f :=
(gc_map_comap f).l_sup
@[to_additive]
lemma map_supr {ι : Sort*} (f : M →* N) (s : ι → submonoid M) :
(supr s).map f = ⨆ i, (s i).map f :=
(gc_map_comap f).l_supr
@[to_additive]
lemma comap_inf (S T : submonoid N) (f : M →* N) : (S ⊓ T).comap f = S.comap f ⊓ T.comap f :=
(gc_map_comap f).u_inf
@[to_additive]
lemma comap_infi {ι : Sort*} (f : M →* N) (s : ι → submonoid N) :
(infi s).comap f = ⨅ i, (s i).comap f :=
(gc_map_comap f).u_infi
@[simp, to_additive] lemma map_bot (f : M →* N) : (⊥ : submonoid M).map f = ⊥ :=
(gc_map_comap f).l_bot
@[simp, to_additive] lemma comap_top (f : M →* N) : (⊤ : submonoid N).comap f = ⊤ :=
(gc_map_comap f).u_top
/-- Given `submonoid`s `s`, `t` of monoids `M`, `N` respectively, `s × t` as a submonoid of `M × N`. -/
@[to_additive prod "Given `add_submonoid`s `s`, `t` of `add_monoid`s `A`, `B` respectively, `s × t` as an `add_submonoid` of `A × B`."]
def prod (s : submonoid M) (t : submonoid N) : submonoid (M × N) :=
{ carrier := (s : set M).prod t,
one_mem' := ⟨s.one_mem, t.one_mem⟩,
mul_mem' := λ p q hp hq, ⟨s.mul_mem hp.1 hq.1, t.mul_mem hp.2 hq.2⟩ }
@[to_additive coe_prod]
lemma coe_prod (s : submonoid M) (t : submonoid N) :
(s.prod t : set (M × N)) = (s : set M).prod (t : set N) :=
rfl
@[to_additive mem_prod]
lemma mem_prod {s : submonoid M} {t : submonoid N} {p : M × N} :
p ∈ s.prod t ↔ p.1 ∈ s ∧ p.2 ∈ t := iff.rfl
@[to_additive prod_mono]
lemma prod_mono : ((≤) ⇒ (≤) ⇒ (≤)) (@prod M _ N _) (@prod M _ N _) :=
λ s s' hs t t' ht, set.prod_mono hs ht
@[to_additive prod_mono_right]
lemma prod_mono_right (s : submonoid M) : monotone (λ t : submonoid N, s.prod t) :=
prod_mono (le_refl s)
@[to_additive prod_mono_left]
lemma prod_mono_left (t : submonoid N) : monotone (λ s : submonoid M, s.prod t) :=
λ s₁ s₂ hs, prod_mono hs (le_refl t)
@[to_additive prod_top]
lemma prod_top (s : submonoid M) :
s.prod (⊤ : submonoid N) = s.comap (monoid_hom.fst M N) :=
ext $ λ x, by simp [mem_prod, monoid_hom.coe_fst]
@[to_additive top_prod]
lemma top_prod (s : submonoid N) :
(⊤ : submonoid M).prod s = s.comap (monoid_hom.snd M N) :=
ext $ λ x, by simp [mem_prod, monoid_hom.coe_snd]
@[simp, to_additive top_prod_top]
lemma top_prod_top : (⊤ : submonoid M).prod (⊤ : submonoid N) = ⊤ :=
(top_prod _).trans $ comap_top _
@[to_additive] lemma bot_prod_bot : (⊥ : submonoid M).prod (⊥ : submonoid N) = ⊥ :=
ext' $ by simp [coe_prod, prod.one_eq_mk]
/-- Product of submonoids is isomorphic to their product as monoids. -/
@[to_additive prod_equiv "Product of additive submonoids is isomorphic to their product
as additive monoids"]
def prod_equiv (s : submonoid M) (t : submonoid N) : s.prod t ≃* s × t :=
{ map_mul' := λ x y, rfl, .. equiv.set.prod ↑s ↑t }
end submonoid
namespace monoid_hom
variables {N : Type*} {P : Type*} [monoid N] [monoid P] (S : submonoid M)
open submonoid
/-- The range of a monoid homomorphism is a submonoid. -/
@[to_additive "The range of an `add_monoid_hom` is an `add_submonoid`."]
def mrange (f : M →* N) : submonoid N := (⊤ : submonoid M).map f
@[simp, to_additive] lemma coe_mrange (f : M →* N) :
(f.mrange : set N) = set.range f :=
set.image_univ
@[simp, to_additive] lemma mem_mrange {f : M →* N} {y : N} :
y ∈ f.mrange ↔ ∃ x, f x = y :=
by simp [mrange]
@[to_additive]
lemma map_mrange (g : N →* P) (f : M →* N) : f.mrange.map g = (g.comp f).mrange :=
(⊤ : submonoid M).map_map g f
/-- Restriction of a monoid hom to a submonoid of the domain. -/
@[to_additive "Restriction of an add_monoid hom to an `add_submonoid` of the domain."]
def restrict {N : Type*} [monoid N] (f : M →* N) (S : submonoid M) : S →* N := f.comp S.subtype
@[to_additive]
lemma restrict_apply {N : Type*} [monoid N] (f : M →* N) (x : S) : f.restrict S x = f x := rfl
@[simp, to_additive] lemma restrict_eq {N : Type} [monoid N] (f : M →* N) (x) :
f.restrict S x = f x := rfl
/-- Restriction of a monoid hom to a submonoid of the codomain. -/
@[to_additive "Restriction of an `add_monoid` hom to an `add_submonoid` of the codomain."]
def cod_restrict (f : M →* N) (S : submonoid N) (h : ∀ x, f x ∈ S) : M →* S :=
{ to_fun := λ n, ⟨f n, h n⟩,
map_one' := subtype.eq f.map_one,
map_mul' := λ x y, subtype.eq (f.map_mul x y) }
/-- Restriction of a monoid hom to its range iterpreted as a submonoid. -/
@[to_additive "Restriction of an `add_monoid` hom to its range interpreted as a submonoid."]
def range_restrict {N} [monoid N] (f : M →* N) : M →* f.mrange :=
f.cod_restrict f.mrange $ λ x, ⟨x, submonoid.mem_top x, rfl⟩
@[simp, to_additive]
lemma coe_range_restrict {N} [monoid N] (f : M →* N) (x : M) :
(f.range_restrict x : N) = f x :=
rfl
@[to_additive]
lemma mrange_top_iff_surjective {N} [monoid N] {f : M →* N} :
f.mrange = (⊤ : submonoid N) ↔ function.surjective f :=
submonoid.ext'_iff.trans $ iff.trans (by rw [coe_mrange, coe_top]) set.range_iff_surjective
/-- The range of a surjective monoid hom is the whole of the codomain. -/
@[to_additive "The range of a surjective `add_monoid` hom is the whole of the codomain."]
lemma mrange_top_of_surjective {N} [monoid N] (f : M →* N) (hf : function.surjective f) :
f.mrange = (⊤ : submonoid N) :=
mrange_top_iff_surjective.2 hf
/-- The submonoid of elements `x : M` such that `f x = g x` -/
@[to_additive "The additive submonoid of elements `x : M` such that `f x = g x`"]
def eq_mlocus (f g : M →* N) : submonoid M :=
{ carrier := {x | f x = g x},
one_mem' := by rw [set.mem_set_of_eq, f.map_one, g.map_one],
mul_mem' := λ x y (hx : _ = _) (hy : _ = _), by simp [*] }
/-- If two monoid homomorphisms are equal on a set, then they are equal on its submonoid closure. -/
@[to_additive]
lemma eq_on_mclosure {f g : M →* N} {s : set M} (h : set.eq_on f g s) :
set.eq_on f g (closure s) :=
show closure s ≤ f.eq_mlocus g, from closure_le.2 h
@[to_additive]
lemma eq_of_eq_on_mtop {f g : M →* N} (h : set.eq_on f g (⊤ : submonoid M)) :
f = g :=
ext $ λ x, h trivial
@[to_additive]
lemma eq_of_eq_on_mdense {s : set M} (hs : closure s = ⊤) {f g : M →* N} (h : s.eq_on f g) :
f = g :=
eq_of_eq_on_mtop $ hs ▸ eq_on_mclosure h
@[to_additive]
lemma closure_preimage_le (f : M →* N) (s : set N) :
closure (f ⁻¹' s) ≤ (closure s).comap f :=
closure_le.2 $ λ x hx, mem_coe.2 $ mem_comap.2 $ subset_closure hx
/-- The image under a monoid hom of the submonoid generated by a set equals the submonoid generated
by the image of the set. -/
@[to_additive "The image under an `add_monoid` hom of the `add_submonoid` generated by a set equals
the `add_submonoid` generated by the image of the set."]
lemma map_mclosure (f : M →* N) (s : set M) :
(closure s).map f = closure (f '' s) :=
le_antisymm
(map_le_iff_le_comap.2 $ le_trans (closure_mono $ set.subset_preimage_image _ _)
(closure_preimage_le _ _))
(closure_le.2 $ set.image_subset _ subset_closure)
end monoid_hom
namespace free_monoid
variables {α : Type*}
open submonoid
@[to_additive]
theorem closure_range_of : closure (set.range $ @of α) = ⊤ :=
begin
refine eq_top_iff.2 (λ x hx, _),
induction x with hd tl ih,
{ from one_mem _ },
{ rw ← of_mul_eq_cons,
exact mul_mem _ (subset_closure $ set.mem_range_self _) (ih trivial) }
end
end free_monoid
namespace submonoid
variables {N : Type*} [monoid N]
open monoid_hom lattice
/-- The monoid hom associated to an inclusion of submonoids. -/
@[to_additive "The `add_monoid` hom associated to an inclusion of submonoids."]
def inclusion {S T : submonoid M} (h : S ≤ T) : S →* T :=
S.subtype.cod_restrict _ (λ x, h x.2)
@[simp, to_additive]
lemma range_subtype (s : submonoid M) : s.subtype.mrange = s :=
ext' $ (coe_mrange _).trans $ set.range_coe_subtype s
lemma closure_singleton_eq (x : M) : closure ({x} : set M) = (powers_hom M x).mrange :=
closure_eq_of_le (set.singleton_subset_iff.2 ⟨multiplicative.of_add 1, trivial, pow_one x⟩) $
λ x ⟨n, _, hn⟩, hn ▸ pow_mem _ (subset_closure $ set.mem_singleton _) _
/-- The submonoid generated by an element of a monoid equals the set of natural number powers of
the element. -/
lemma mem_closure_singleton {x y : M} : y ∈ closure ({x} : set M) ↔ ∃ n:ℕ, x^n=y :=
by rw [closure_singleton_eq, mem_mrange]; refl
@[to_additive]
lemma closure_eq_mrange (s : set M) : closure s = (free_monoid.lift s M coe).mrange :=
by rw [mrange, ← free_monoid.closure_range_of, map_mclosure,
← set.range_comp, free_monoid.lift_comp_of, set.range_coe_subtype]
@[to_additive]
lemma exists_list_of_mem_closure {s : set M} {x : M} (hx : x ∈ closure s) :
∃ (l : list M) (hl : ∀ y ∈ l, y ∈ s), l.prod = x :=
begin
rw [closure_eq_mrange, mem_mrange] at hx,
rcases hx with ⟨l, hx⟩,
exact ⟨list.map coe l, λ y hy, let ⟨z, hz, hy⟩ := list.mem_map.1 hy in hy ▸ z.2, hx⟩
end
@[to_additive]
lemma map_inl (s : submonoid M) : s.map (inl M N) = s.prod ⊥ :=
ext $ λ p, ⟨λ ⟨x, hx, hp⟩, hp ▸ ⟨hx, set.mem_singleton 1⟩,
λ ⟨hps, hp1⟩, ⟨p.1, hps, prod.ext rfl $ (set.eq_of_mem_singleton hp1).symm⟩⟩
@[to_additive]
lemma map_inr (s : submonoid N) : s.map (inr M N) = prod ⊥ s :=
ext $ λ p, ⟨λ ⟨x, hx, hp⟩, hp ▸ ⟨set.mem_singleton 1, hx⟩,
λ ⟨hp1, hps⟩, ⟨p.2, hps, prod.ext (set.eq_of_mem_singleton hp1).symm rfl⟩⟩
@[to_additive]
lemma range_inl : (inl M N).mrange = prod ⊤ ⊥ := map_inl ⊤
@[to_additive]
lemma range_inr : (inr M N).mrange = prod ⊥ ⊤ := map_inr ⊤
@[to_additive]
lemma range_inl' : (inl M N).mrange = comap (snd M N) ⊥ := range_inl.trans (top_prod _)
@[to_additive]
lemma range_inr' : (inr M N).mrange = comap (fst M N) ⊥ := range_inr.trans (prod_top _)
@[simp, to_additive]
lemma range_fst : (fst M N).mrange = ⊤ :=
(fst M N).mrange_top_of_surjective $ @prod.fst_surjective _ _ ⟨1⟩
@[simp, to_additive]
lemma range_snd : (snd M N).mrange = ⊤ :=
(snd M N).mrange_top_of_surjective $ @prod.snd_surjective _ _ ⟨1⟩
@[simp, to_additive prod_bot_sup_bot_prod]
lemma prod_bot_sup_bot_prod (s : submonoid M) (t : submonoid N) :
(s.prod ⊥) ⊔ (prod ⊥ t) = s.prod t :=
le_antisymm (sup_le (prod_mono_right s bot_le) (prod_mono_left t bot_le)) $
assume p hp, prod.fst_mul_snd p ▸ mul_mem _
((le_sup_left : s.prod ⊥ ≤ s.prod ⊥ ⊔ prod ⊥ t) ⟨hp.1, set.mem_singleton 1⟩)
((le_sup_right : prod ⊥ t ≤ s.prod ⊥ ⊔ prod ⊥ t) ⟨set.mem_singleton 1, hp.2⟩)
@[simp, to_additive]
lemma range_inl_sup_range_inr : (inl M N).mrange ⊔ (inr M N).mrange = ⊤ :=
by simp only [range_inl, range_inr, prod_bot_sup_bot_prod, top_prod_top]
end submonoid
namespace submonoid
variables {N : Type*} [comm_monoid N]
open monoid_hom
@[to_additive]
lemma sup_eq_range (s t : submonoid N) : s ⊔ t = (s.subtype.coprod t.subtype).mrange :=
by rw [mrange, ← range_inl_sup_range_inr, map_sup, map_mrange, coprod_comp_inl,
map_mrange, coprod_comp_inr, range_subtype, range_subtype]
@[to_additive]
lemma mem_sup {s t : submonoid N} {x : N} :
x ∈ s ⊔ t ↔ ∃ (y ∈ s) (z ∈ t), y * z = x :=
by simp only [sup_eq_range, mem_mrange, coprod_apply, prod.exists, submonoid.exists,
coe_subtype, subtype.coe_mk]
end submonoid
namespace add_submonoid
open set
lemma smul_mem (S : add_submonoid A) {x : A} (hx : x ∈ S) :
∀ n : ℕ, add_monoid.smul n x ∈ S
| 0 := S.zero_mem
| (n+1) := S.add_mem hx (smul_mem n)
lemma closure_singleton_eq (x : A) : closure ({x} : set A) = (multiples_hom A x).mrange :=
closure_eq_of_le (set.singleton_subset_iff.2 ⟨1, trivial, add_monoid.one_smul x⟩) $
λ x ⟨n, _, hn⟩, hn ▸ smul_mem _ (subset_closure $ set.mem_singleton _) _
/-- The `add_submonoid` generated by an element of an `add_monoid` equals the set of
natural number multiples of the element. -/
lemma mem_closure_singleton {x y : A} : y ∈ closure ({x} : set A) ↔ ∃ n:ℕ, add_monoid.smul n x = y :=
by rw [closure_singleton_eq, add_monoid_hom.mem_mrange]; refl
end add_submonoid
namespace mul_equiv
variables {S T : submonoid M}
/-- Makes the identity isomorphism from a proof two submonoids of a multiplicative
monoid are equal. -/
@[to_additive add_submonoid_congr "Makes the identity additive isomorphism from a proof two submonoids of an additive monoid are equal."]
def submonoid_congr (h : S = T) : S ≃* T :=
{ map_mul' := λ _ _, rfl, ..equiv.set_congr $ submonoid.ext'_iff.1 h }
end mul_equiv
|
d6820b4d01b7ea3bbc19f62343238ada2e738acc | 99b36386640f67a61e85dddccbc0f82edec9bf9c | /src/pushouts.lean | 3245b402850700f5982b91e4bb4757e1d8aab0c6 | [] | no_license | aymonwuolanne/lean-theorem-proving | 1ba46fb89f446b78b7f8f8cb084aad68fadce9a3 | 701a7873f9f66f2c24962b02450ec90b20935702 | refs/heads/master | 1,586,068,982,414 | 1,575,091,205,000 | 1,575,091,205,000 | 156,647,753 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 2,083 | lean | import category_theory.instances.topological_spaces
import category_theory.limits.pullbacks
import category_theory.const
import topology.continuity
open category_theory.instances
open category_theory.limits
open category_theory.functor
open category_theory
open sum
universe u
noncomputable theory
variables {X Y Z : Top.{u}}
variables (f : X ⟶ Y) (g : X ⟶ Z)
-- in the following diagram P is the pushout we wish to construct
-- X ⟶ Y
-- ↓ ↓
-- Z ⟶ P
-- this relation identifies any x and y that are the image of the same
-- point in X
def r (x : sum Y.α Z.α) (y : sum Y.α Z.α) : Prop :=
x = y ∨
∃ a : X.α, inl (f.val a) = x ∧ inr (g.val a) = y
def P : Top := { α := quot (r f g), str := by apply_instance}
-- inclusion maps from Y and Z into the pushout P
def i_Y : Y ⟶ P f g := { val := quot.mk (r f g) ∘ inl,
property := continuous.comp continuous_inl continuous_quot_mk }
def i_Z : Z ⟶ P f g := { val := quot.mk (r f g) ∘ inr,
property := continuous.comp continuous_inr continuous_quot_mk }
lemma natural : f ≫ (i_Y f g) = g ≫ (i_Z f g) :=
subtype.eq
begin
rw [concrete_category_comp, i_Y, i_Z],
dsimp,
ext,
rw [function.comp.assoc],
apply quot.sound,
exact or.inr ⟨ x, ⟨ rfl, rfl ⟩ ⟩,
end
def Top.nat_trans_app (A : walking_span) : (span f g).obj A ⟶ ((const walking_span).obj (P f g)).obj A :=
match A with
| walking_span.zero := f ≫ (i_Y f g)
| walking_span.left := i_Y f g
| walking_span.right := i_Z f g
end
def Top.nat_trans : span f g ⟹ (const walking_span).obj (P f g) := { app := Top.nat_trans_app f g,
naturality' := sorry }
def topcat : category Top := by apply_instance
-- cosquare := λ f g, something of type cocone (span f g)
-- span f g is a functor from walking_span.{v} to Top
-- cocone (F : J ⥤ Top) is a structure taking an object X : Top and
-- a natural transformation ι taking F to (const J).obj X
-- natural transformations have an app field and a naturality' field
instance toppush : @has_pushouts Top topcat := { cosquare := sorry,
is_pushout := sorry } |
01b08b9f962e5cb3cb2b6795026380138ec2ef41 | 947b78d97130d56365ae2ec264df196ce769371a | /tests/lean/run/meta1.lean | 738be58f53e9b6f0963fe6faf1e9a284a5c1211e | [
"Apache-2.0"
] | permissive | shyamalschandra/lean4 | 27044812be8698f0c79147615b1d5090b9f4b037 | 6e7a883b21eaf62831e8111b251dc9b18f40e604 | refs/heads/master | 1,671,417,126,371 | 1,601,859,995,000 | 1,601,860,020,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 5,066 | lean | import Lean.Meta
new_frontend
open Lean
open Lean.Meta
unsafe def tstInferType (mods : List Name) (e : Expr) : IO Unit :=
withImportModules (mods.map $ fun m => {module := m}) 0 fun env => do
let (type, _, _) ← (inferType e : MetaM _).toIO {} { env := env } {} {};
IO.println (toString e ++ " : " ++ toString type)
unsafe def tstWHNF (mods : List Name) (e : Expr) (t := TransparencyMode.default) : IO Unit :=
withImportModules (mods.map $ fun m => {module := m}) 0 fun env => do
let (s, _, _) ← (whnf e : MetaM _).toIO {} { env := env };
IO.println (toString e ++ " ==> " ++ toString s)
unsafe def tstIsProp (mods : List Name) (e : Expr) : IO Unit :=
withImportModules (mods.map $ fun m => {module := m}) 0 fun env => do
let (b, _, _) ← (isProp e : MetaM _).toIO {} { env := env };
IO.println (toString e ++ ", isProp: " ++ toString b)
def t1 : Expr :=
let map := mkConst `List.map [levelOne, levelOne];
let nat := mkConst `Nat [];
let bool := mkConst `Bool [];
mkAppN map #[nat, bool]
#eval tstInferType [`Init.Data.List] t1
def t2 : Expr :=
let prop := mkSort levelZero;
mkForall `x BinderInfo.default prop prop
#eval tstInferType [`Init.Core] t2
def t3 : Expr :=
let nat := mkConst `Nat [];
let natLe := mkConst `Nat.le [];
let zero := mkLit (Literal.natVal 0);
let p := mkAppN natLe #[mkBVar 0, zero];
mkForall `x BinderInfo.default nat p
#eval tstInferType [`Init.Data.Nat] t3
def t4 : Expr :=
let nat := mkConst `Nat [];
let p := mkAppN (mkConst `Nat.succ []) #[mkBVar 0];
mkLambda `x BinderInfo.default nat p
#eval tstInferType [`Init.Core] t4
def t5 : Expr :=
let add := mkConst `Nat.add [];
mkAppN add #[mkLit (Literal.natVal 3), mkLit (Literal.natVal 5)]
#eval tstWHNF [`Init.Data.Nat] t5
#eval tstWHNF [`Init.Data.Nat] t5 TransparencyMode.reducible
set_option pp.all true
#check @List.cons Nat
def t6 : Expr :=
let map := mkConst `List.map [levelOne, levelOne];
let nat := mkConst `Nat [];
let add := mkConst `Nat.add [];
let f := mkLambda `x BinderInfo.default nat (mkAppN add #[mkBVar 0, mkLit (Literal.natVal 1)]);
let cons := mkApp (mkConst `List.cons [levelZero]) nat;
let nil := mkApp (mkConst `List.nil [levelZero]) nat;
let one := mkLit (Literal.natVal 1);
let four := mkLit (Literal.natVal 4);
let xs := mkApp (mkApp cons one) (mkApp (mkApp cons four) nil);
mkAppN map #[nat, nat, f, xs]
#eval tstInferType [`Init.Data.List] t6
#eval tstWHNF [`Init.Data.List] t6
#eval tstInferType [] $ mkSort levelZero
#eval tstInferType [`Init.Data.List] $ mkLambda `a BinderInfo.implicit (mkSort levelOne) (mkLambda `x BinderInfo.default (mkBVar 0) (mkLambda `xs BinderInfo.default (mkApp (mkConst `List [levelZero]) (mkBVar 1)) (mkBVar 0)))
def t7 : Expr :=
let nat := mkConst `Nat [];
let one := mkLit (Literal.natVal 1);
mkLet `x nat one one
#eval tstInferType [`Init.Core] $ t7
#eval tstWHNF [`Init.Core] $ t7
def t8 : Expr :=
let nat := mkConst `Nat [];
let one := mkLit (Literal.natVal 1);
let add := mkConst `Nat.add [];
mkLet `x nat one (mkAppN add #[one, mkBVar 0])
#eval tstInferType [`Init.Core] $ t8
#eval tstWHNF [`Init.Core] $ t8
def t9 : Expr :=
let nat := mkConst `Nat [];
mkLet `a (mkSort levelOne) nat (mkForall `x BinderInfo.default (mkBVar 0) (mkBVar 1))
#eval tstInferType [`Init.Core] $ t9
#eval tstWHNF [`Init.Core] $ t9
#eval tstInferType [`Init.Core] $ mkLit (Literal.natVal 10)
#eval tstInferType [`Init.Core] $ mkLit (Literal.strVal "hello")
#eval tstInferType [`Init.Core] $ mkMData {} $ mkLit (Literal.natVal 10)
#eval tstInferType [`Lean.Util.Trace] (mkProj `Inhabited 0 (mkConst `Lean.TraceState.Inhabited []))
#eval tstInferType [`Lean.Util.Trace] (mkProj `Lean.TraceState 0 (mkProj `Inhabited 0 (mkConst `Lean.TraceState.Inhabited [])))
#eval tstWHNF [`Lean.Util.Trace] (mkProj `Inhabited 0 (mkConst `Lean.TraceState.Inhabited []))
#eval tstWHNF [`Lean.Util.Trace] (mkProj `Lean.TraceState 0 (mkProj `Inhabited 0 (mkConst `Lean.TraceState.Inhabited [])))
def t10 : Expr :=
let nat := mkConst `Nat [];
let refl := mkApp (mkConst `Eq.refl [levelOne]) nat;
mkLambda `a BinderInfo.default nat (mkApp refl (mkBVar 0))
#eval tstInferType [`Init.Core] t10
#eval tstIsProp [`Init.Core] t10
#eval tstIsProp [`Init.Core] (mkAppN (mkConst `And []) #[mkConst `True [], mkConst `True []])
#eval tstIsProp [`Init.Core] (mkConst `And [])
-- Example where isPropQuick fails
#eval tstIsProp [`Init.Core] (mkAppN (mkConst `id [levelZero]) #[mkSort levelZero, mkAppN (mkConst `And []) #[mkConst `True [], mkConst
`True []]])
#eval tstIsProp [`Init.Core] (mkAppN (mkConst `Eq [levelOne]) #[mkConst `Nat [], mkLit (Literal.natVal 0), mkLit (Literal.natVal 1)])
#eval tstIsProp [`Init.Core] $
mkForall `x BinderInfo.default (mkConst `Nat [])
(mkAppN (mkConst `Eq [levelOne]) #[mkConst `Nat [], mkBVar 0, mkLit (Literal.natVal 1)])
#eval tstIsProp [`Init.Core] $
mkApp
(mkLambda `x BinderInfo.default (mkConst `Nat [])
(mkAppN (mkConst `Eq [levelOne]) #[mkConst `Nat [], mkBVar 0, mkLit (Literal.natVal 1)]))
(mkLit (Literal.natVal 0))
|
bbbcc4abf11d3d9587294f4556b8c6878247ae00 | 737dc4b96c97368cb66b925eeea3ab633ec3d702 | /stage0/src/Lean/Meta.lean | 2c23fb1ca3d0d0c56e461a8eb4893a93e3b59e26 | [
"Apache-2.0"
] | permissive | Bioye97/lean4 | 1ace34638efd9913dc5991443777b01a08983289 | bc3900cbb9adda83eed7e6affeaade7cfd07716d | refs/heads/master | 1,690,589,820,211 | 1,631,051,000,000 | 1,631,067,598,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 1,110 | 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.Meta.Basic
import Lean.Meta.LevelDefEq
import Lean.Meta.WHNF
import Lean.Meta.InferType
import Lean.Meta.FunInfo
import Lean.Meta.ExprDefEq
import Lean.Meta.DiscrTree
import Lean.Meta.Reduce
import Lean.Meta.Instances
import Lean.Meta.AbstractMVars
import Lean.Meta.SynthInstance
import Lean.Meta.AppBuilder
import Lean.Meta.Tactic
import Lean.Meta.KAbstract
import Lean.Meta.RecursorInfo
import Lean.Meta.GeneralizeTelescope
import Lean.Meta.Match
import Lean.Meta.ReduceEval
import Lean.Meta.Closure
import Lean.Meta.AbstractNestedProofs
import Lean.Meta.ForEachExpr
import Lean.Meta.Transform
import Lean.Meta.PPGoal
import Lean.Meta.UnificationHint
import Lean.Meta.Inductive
import Lean.Meta.SizeOf
import Lean.Meta.IndPredBelow
import Lean.Meta.Coe
import Lean.Meta.SortLocalDecls
import Lean.Meta.CollectFVars
import Lean.Meta.GeneralizeVars
import Lean.Meta.Injective
import Lean.Meta.Structure
import Lean.Meta.Constructions
|
e54dd440d7723f8d30a54f11ad5da830d61b6913 | a25cc44b447f2fbde61e4b848318762eb892f2bf | /src/subdiv.lean | ac4f3e0b4d12de43dcb5e68457d7b4926844820e | [] | no_license | NeilStrickland/itloc | 20f9b9bcaa4d5840f7f32e84108174b294d84b1a | 5b13b5b418766d10926b983eb3dd2ac42abf63d8 | refs/heads/master | 1,592,747,030,159 | 1,578,138,004,000 | 1,578,138,004,000 | 197,470,135 | 2 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 19,444 | lean | /-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
Given a poset `P`, we define `subdiv P` to be the poset of
finite nonempty chains `s ⊆ P`. Any such `s` has a largest
element, and the map `max : (subdiv P) → P` is a morphism
of posets. There is an approach to the homotopy theory of
finite complexes based on finite posets, and the above map
`max` plays a key role in this.
-/
import data.list.sort
import basic sort_rank
universes uP uQ uR uS
variables (P : Type uP) [partial_order P] [decidable_eq P]
variables (Q : Type uQ) [partial_order Q] [decidable_eq Q]
variables (R : Type uR) [partial_order R] [decidable_eq R]
variables (S : Type uS) [partial_order S] [decidable_eq S]
variable [decidable_rel (has_le.le : P → P → Prop)]
variable [decidable_rel (has_le.le : Q → Q → Prop)]
variable [decidable_rel (has_le.le : R → R → Prop)]
variable [decidable_rel (has_le.le : S → S → Prop)]
namespace poset
open poset
variable {P}
/-- Definition of chains
LaTeX: defn-subdiv
-/
def is_chain (s : finset P) : Prop :=
∀ (p ∈ s) (q ∈ s), (p ≤ q ∨ q ≤ p)
instance decidable_is_chain (s : finset P) : decidable (is_chain s) :=
by { unfold is_chain, apply_instance }
/-- Definition of simplices as nonempty chains
LaTeX: defn-subdiv
-/
def is_simplex (s : finset P) : Prop := s ≠ ∅ ∧ (is_chain s)
instance decidable_is_simplex (s : finset P) : decidable (is_simplex s) :=
by { unfold is_simplex, apply_instance }
variable (P)
/-- Definition of subdiv P as a type
LaTeX: defn-subdiv
-/
def subdiv := {s : finset P // is_simplex s}
/-- Vertices as 0-simplices -/
variable {P}
def vertex (p : P) : subdiv P := ⟨
finset.singleton p,
begin
split,
{exact finset.ne_empty_of_mem (finset.mem_singleton_self p)},
{intros x hx y hy,
rw [finset.mem_singleton.mp hx, finset.mem_singleton.mp hy],
left, exact le_refl p }
end⟩
variable (P)
namespace subdiv
instance : decidable_eq (subdiv P) :=
by { unfold subdiv, apply_instance }
/-- Definition of the partial order on subdiv P
LaTeX: defn-subdiv
-/
instance : partial_order (subdiv P) :=
{ le := λ s t, s.val ⊆ t.val,
le_refl := λ s, (le_refl s.val),
le_antisymm := λ s t hst hts, subtype.eq (le_antisymm hst hts),
le_trans := λ s t u (hst : s.val ⊆ t.val) (htu : t.val ⊆ u.val) p hs,
(htu (hst hs)) }
instance decidable_le : decidable_rel
(has_le.le : (subdiv P) → (subdiv P) → Prop) :=
λ s t, by { apply_instance }
variable {P}
/-- Definition of the dimension of a simplex, as one less than
the cardinality. We use the predecessor operation on
natural numbers, which sends zero to zero. Because of this,
we need a small argument to show that the cardinality is
strictly positive and thus equal to one more than the dimension.
LaTeX: defn-subdiv
-/
def dim (s : subdiv P) : ℕ := s.val.card.pred
lemma card_eq (s : subdiv P) : s.val.card = s.dim.succ :=
begin
by_cases h : s.val.card = 0,
{ exact (s.property.left (finset.card_eq_zero.mp h)).elim },
{ replace h := nat.pos_of_ne_zero h,
exact (nat.succ_pred_eq_of_pos h).symm }
end
/-- The dimension function dim : subdiv P → ℕ is monotone. -/
lemma dim_mono : monotone (dim : (subdiv P) → ℕ) :=
begin
intros s t hst,
let h := finset.card_le_of_subset hst,
rw[card_eq,card_eq] at h,
exact nat.le_of_succ_le_succ h
end
/-- If we have simplices s ≤ t with dim s ≥ dim t then s = t. -/
lemma eq_of_le_of_dim_ge (s t : subdiv P)
(hst : s ≤ t) (hd : s.dim ≥ t.dim) : s = t :=
begin
let hc := nat.succ_le_succ hd,
rw [← card_eq, ← card_eq] at hc,
exact subtype.eq (finset.eq_of_subset_of_card_le hst hc)
end
/-- This allows us to treat a simplex as a type in its own right. -/
instance : has_coe_to_sort (subdiv P) :=
⟨_,λ s, {p : P // p ∈ s.val}⟩
section els
variable (s : subdiv P)
instance els_decidable_eq : decidable_eq s := by { apply_instance }
instance els_fintype : fintype s := by { apply_instance }
lemma card_eq' : fintype.card s = s.dim.succ :=
(fintype.card_coe s.val).trans s.card_eq
/-- If s is a simplex, we can treat it as a linearly ordered set. -/
instance els_order : decidable_linear_order s :=
{ le := λ p q,p.val ≤ q.val,
le_refl := λ p, ((le_refl (p.val : P)) : p.val ≤ p.val),
le_antisymm := λ p q (hpq : p.val ≤ q.val) (hqp : q.val ≤ p.val),
subtype.eq (le_antisymm hpq hqp),
le_trans := λ p q r (hpq : p.val ≤ q.val) (hqr : q.val ≤ r.val),
le_trans hpq hqr,
le_total := λ p q,s.property.right p.val p.property q.val q.property,
lt := λ p q,p.val < q.val,
lt_iff_le_not_le := λ p q,
begin
change p.val < q.val ↔ p.val ≤ q.val ∧ ¬ q.val ≤ p.val,
apply lt_iff_le_not_le,
end,
decidable_le := λ p q, by { apply_instance } }
variable {s}
/-- The inclusion of a simplex in the full poset P. -/
def inc : s → P := λ p, p.val
/-- The inclusion map is injective. -/
lemma inc_inj (p₀ p₁ : s) : (inc p₀) = (inc p₁) → p₀ = p₁ := subtype.eq
/-- The inclusion map is monotone. -/
lemma inc_mono : monotone (inc : s → P) := λ p₀ p₁ hp, hp
variable (s)
variables {d : ℕ} (e : s.dim = d)
/-- If dim s = d, then we have a canonical bijection from s
to the set fin d.succ = { 0,1,...,d }. We define
rank_equiv s to be a package consisting of this bijection
and its inverse. We define seq s and rank s to be
auxiliary functions defined in terms of rank_equiv s,
and we prove some lemmas about the behaviour of these.
-/
def rank_equiv : s ≃ fin d.succ :=
fintype.rank_equiv (s.card_eq'.trans (congr_arg nat.succ e))
def seq : (fin d.succ) → P := λ i, inc ((rank_equiv s e).inv_fun i)
def rank (p : P) (hp : p ∈ s.val) : fin d.succ :=
(rank_equiv s e).to_fun ⟨p,hp⟩
lemma seq_mem (i : fin d.succ) : seq s e i ∈ s.val :=
((rank_equiv s e).inv_fun i).property
lemma seq_rank (p : P) (hp : p ∈ s.val) :
seq s e (rank s e p hp) = p :=
congr_arg inc ((rank_equiv s e).left_inv ⟨p,hp⟩)
lemma seq_eq (i : fin d.succ) :
((rank_equiv s e).inv_fun i) = ⟨seq s e i,seq_mem s e i⟩ :=
by { apply subtype.eq, refl }
lemma rank_seq (i : fin d.succ) :
rank s e (seq s e i) (seq_mem s e i) = i :=
begin
dsimp [rank],
rw [← seq_eq, (rank_equiv s e).right_inv]
end
lemma seq_le (i₀ i₁ : fin d.succ) :
i₀ ≤ i₁ ↔ seq s e i₀ ≤ seq s e i₁ :=
fintype.seq_le (s.card_eq'.trans (congr_arg nat.succ e)) i₀ i₁
lemma seq_lt (i₀ i₁ : fin d.succ) :
i₀ < i₁ ↔ seq s e i₀ < seq s e i₁ :=
fintype.seq_lt (s.card_eq'.trans (congr_arg nat.succ e)) i₀ i₁
variables {P Q}
def map (f : poset.hom P Q) : (poset.hom (subdiv P) (subdiv Q)) :=
begin
let sf : subdiv P → subdiv Q := λ s,
begin
let t0 : finset Q := s.val.image f.val,
have : is_simplex t0 :=
begin
split,
{ rcases finset.exists_mem_of_ne_empty s.property.1
with ⟨p,p_in_s⟩,
exact finset.ne_empty_of_mem
(finset.mem_image_of_mem f.val p_in_s) },
{ intros q₀ hq₀ q₁ hq₁,
rcases finset.mem_image.mp hq₀ with ⟨p₀,hp₀,hfp₀⟩,
rcases finset.mem_image.mp hq₁ with ⟨p₁,hp₁,hfp₁⟩,
rw [← hfp₀, ← hfp₁],
rcases s.property.2 p₀ hp₀ p₁ hp₁ with h₀₁ | h₁₀,
{ left, exact f.property h₀₁ },
{ right, exact f.property h₁₀ } }
end,
exact ⟨t0,this⟩
end,
have : monotone sf := λ s₀ s₁ hs,
begin
change s₀.val.image f.val ⊆ s₁.val.image f.val,
intros q hq,
rcases finset.mem_image.mp hq with ⟨p,hp,hfp⟩,
exact finset.mem_image.mpr ⟨p,hs hp,hfp⟩
end,
exact ⟨sf, this⟩
end
lemma map_val (f : poset.hom P Q) (s : subdiv P) :
((subdiv.map f).val s).val = s.val.image f.val := rfl
section interleave
variables [fintype P] {f g : poset.hom P Q} (hfg : f ≤ g)
include hfg
def interleave :
∀ (r : fin_ranking P) (m : ℕ), hom (subdiv P) (subdiv Q)
| ⟨n,r,r_mono⟩ m :=
⟨ λ (σ : subdiv P),
⟨ ((σ.val.filter (λ p, 2 * (r p).val < m)).image f) ∪
((σ.val.filter (λ p, 2 * (r p).val + 1 ≥ m)).image g),
begin
split,
{ rcases finset.exists_mem_of_ne_empty σ.property.1
with ⟨p, p_in_σ⟩,
by_cases h : 2 * (r p).val < m,
{ apply @finset.ne_empty_of_mem Q (f p),
apply finset.mem_union_left,
apply finset.mem_image_of_mem,
rw [finset.mem_filter],
exact ⟨p_in_σ, h⟩ },
{ replace h := le_trans (le_of_not_gt h) (nat.le_succ _),
apply @finset.ne_empty_of_mem Q (g p),
apply finset.mem_union_right,
apply finset.mem_image_of_mem,
rw [finset.mem_filter],
exact ⟨p_in_σ, h⟩ } },
{ intros q₀ h₀ q₁ h₁,
rcases finset.mem_union.mp h₀ with h₀ | h₀;
rcases finset.mem_union.mp h₁ with h₁ | h₁;
rcases finset.mem_image.mp h₀ with ⟨p₀,hf₀,he₀⟩;
rcases finset.mem_image.mp h₁ with ⟨p₁,hf₁,he₁⟩;
rw [finset.mem_filter] at hf₀ hf₁;
rw [← he₀, ← he₁];
let r₀ := (r p₀).val ; let r₁ := (r p₁).val ;
rcases σ.property.right p₀ hf₀.1 p₁ hf₁.1 with hpp | hpp;
let hfpp := f.property hpp;
let hgpp := g.property hpp,
{ left , exact hfpp },
{ right, exact hfpp },
{ left , exact le_trans hfpp (hfg p₁) },
{ by_cases hrr : r₀ = r₁,
{ rw [r.injective (fin.eq_of_veq hrr)],
left, exact hfg p₁ },
{ exfalso,
change r₀ ≠ r₁ at hrr,
replace hrr : r₁ < r₀ := lt_of_le_of_ne (r_mono hpp) hrr.symm,
change r₁ + 1 ≤ r₀ at hrr,
have := calc
2 * r₁ + 2 = 2 * (r₁ + 1) : by rw [mul_add, mul_one]
... ≤ 2 * r₀ : nat.mul_le_mul_left 2 hrr
... < m : hf₀.2
... ≤ 2 * r₁ + 1 : hf₁.2
... < 2 * r₁ + 2 : nat.lt_succ_self _,
exact lt_irrefl _ this } },
{ by_cases hrr : r₀ = r₁,
{ rw [r.injective (fin.eq_of_veq hrr)],
right, exact hfg p₁ },
{ exfalso,
change r₀ ≠ r₁ at hrr,
replace hrr : r₀ < r₁ := lt_of_le_of_ne (r_mono hpp) hrr,
change r₀ + 1 ≤ r₁ at hrr,
have := calc
2 * r₁ < m : hf₁.2
... ≤ 2 * r₀ + 1 : hf₀.2
... < 2 * r₀ + 2 : nat.lt_succ_self _
... = 2 * (r₀ + 1) : by rw [mul_add, mul_one]
... ≤ 2 * r₁ : nat.mul_le_mul_left 2 hrr,
exact lt_irrefl _ this } },
{ right, exact le_trans hfpp (hfg p₀) },
{ left , exact hgpp },
{ right, exact hgpp } }
end ⟩,
begin
intros σ₀ σ₁ h_le q hq,
rw [finset.mem_union] at hq ⊢,
rcases hq with hq | hq;
rcases finset.mem_image.mp hq with ⟨p,hm,he⟩;
rw [← he];
rw [finset.mem_filter] at hm;
replace hm := and.intro (h_le hm.1) hm.2,
{ left , apply finset.mem_image_of_mem,
rw [finset.mem_filter], exact hm },
{ right, apply finset.mem_image_of_mem,
rw [finset.mem_filter], exact hm }
end ⟩
lemma interleave_start :
∀ (r : fin_ranking P), interleave hfg r 0 = subdiv.map g
| ⟨n,r,r_mono⟩ :=
begin
ext σ,
apply subtype.eq,
change _ ∪ _ = σ.val.image g.val,
ext q,
have : σ.val.filter (λ (p : P), 2 * (r p).val < 0) = ∅ :=
begin
ext p, rw [finset.mem_filter],
simp [nat.not_lt_zero, finset.not_mem_empty]
end,
rw [this, finset.image_empty],
have : σ.val.filter (λ (p : P), 2 * (r p).val + 1 ≥ 0) = σ.val :=
begin
ext p, rw [finset.mem_filter],
have : _ ≥ 0 := nat.zero_le (2 * (r p).val + 1),
simp only [this, and_true]
end,
rw [this, finset.mem_union],
have : (g : P → Q) = g.val := rfl, rw [this],
simp [finset.not_mem_empty],
end
lemma interleave_end :
∀ (r : fin_ranking P) (m : ℕ) (hm : m ≥ 2 * r.card),
interleave hfg r m = subdiv.map f
| ⟨n,r,r_mono⟩ m hm :=
begin
change m ≥ 2 * n at hm,
ext σ,
apply subtype.eq,
change _ ∪ _ = σ.val.image f.val,
ext q,
have : σ.val.filter (λ (p : P), 2 * (r p).val < m) = σ.val :=
begin
ext p, rw [finset.mem_filter],
have := calc
2 * (r p).val < 2 * n :
nat.mul_lt_mul_of_pos_left (r p).is_lt (dec_trivial : 2 > 0)
... ≤ m : hm,
simp [this]
end,
rw [this],
have : σ.val.filter (λ (p : P), 2 * (r p).val + 1 ≥ m) = ∅ :=
begin
ext p, rw [finset.mem_filter],
have := calc
2 * (r p).val + 1 < 2 * (r p).val + 2 : nat.lt_succ_self _
... = 2 * ((r p).val + 1) : by rw [mul_add, mul_one]
... ≤ 2 * n : nat.mul_le_mul_left 2 (r p).is_lt
... ≤ m : hm,
have : ¬ (_ ≥ m) := not_le_of_gt this,
simp only [this, finset.not_mem_empty, and_false]
end,
rw [this, finset.image_empty, finset.union_empty],
have : (f : P → Q) = f.val := rfl, rw [this]
end
lemma interleave_even_step :
∀ (r : fin_ranking P) (k : ℕ),
interleave hfg r (2 * k) ≤ interleave hfg r (2 * k + 1)
| ⟨n,r,r_mono⟩ k :=
begin
intros σ q h,
change q ∈ _ ∪ _ at h,
change q ∈ _ ∪ _,
rw [finset.mem_union] at h ⊢,
rcases h with h | h;
rcases finset.mem_image.mp h with ⟨p,hf,he⟩;
rw [finset.mem_filter] at hf;
rw [← he],
{ left,
apply finset.mem_image_of_mem,
rw [finset.mem_filter],
exact ⟨hf.1, lt_trans hf.2 (nat.lt_succ_self _)⟩ },
{ right,
apply finset.mem_image_of_mem,
rw [finset.mem_filter],
rcases le_or_gt k (r p).val with hk | hk,
{ exact ⟨hf.1, nat.succ_le_succ (nat.mul_le_mul_left 2 hk)⟩ },
{ exfalso,
have := calc
2 * (r p).val + 2 = 2 * ((r p).val + 1) : by rw [mul_add, mul_one]
... ≤ 2 * k : nat.mul_le_mul_left 2 hk
... ≤ 2 * (r p).val + 1 : hf.2
... < 2 * (r p).val + 2 : nat.lt_succ_self _,
exact lt_irrefl _ this } }
end
lemma interleave_odd_step :
∀ (r : fin_ranking P) (k : ℕ),
interleave hfg r (2 * k + 2) ≤ interleave hfg r (2 * k + 1)
| ⟨n,r,r_mono⟩ k :=
begin
intros σ q h,
change q ∈ _ ∪ _ at h,
change q ∈ _ ∪ _,
rw [finset.mem_union] at h ⊢,
rcases h with h | h;
rcases finset.mem_image.mp h with ⟨p,hf,he⟩;
rw [finset.mem_filter] at hf;
rw [← he],
{ left,
apply finset.mem_image_of_mem,
rw [finset.mem_filter],
rcases le_or_gt (r p).val k with hk | hk,
{ have := calc
2 * (r p).val ≤ 2 * k : nat.mul_le_mul_left 2 hk
... < 2 * k + 1 : nat.lt_succ_self _,
exact ⟨hf.1, this⟩ },
{ exfalso,
have := calc
2 * k + 2 = 2 * (k + 1) : by rw [mul_add, mul_one]
... ≤ 2 * (r p).val : nat.mul_le_mul_left 2 hk
... < 2 * k + 2 : hf.2,
exact lt_irrefl _ this } },
{ right,
apply finset.mem_image_of_mem,
rw [finset.mem_filter],
exact ⟨hf.1, le_trans (le_of_lt (nat.lt_succ_self (2 * k + 1))) hf.2⟩ }
end
lemma interleave_component (r : fin_ranking P) (m : ℕ) :
component (interleave hfg r m) = component (interleave hfg r 0) :=
zigzag (interleave hfg r)
(interleave_even_step hfg r)
(interleave_odd_step hfg r) m
end interleave
def subdiv.mapₕ [fintype P] :
π₀ (hom P Q) → π₀ (hom (subdiv P) (subdiv Q)) :=
π₀.lift (λ f, component (subdiv.map f))
begin
intros f g hfg,
rcases exists_fin_ranking P with ⟨r⟩,
let n := r.card,
let c := interleave hfg r,
have : c 0 = subdiv.map g := interleave_start hfg r,
rw [← this],
have : c (2 * n) = subdiv.map f := interleave_end hfg r (2 * n) (le_refl _),
rw [← this],
apply interleave_component
end
/-- For a simplex s, we define max s to be the largest element. -/
def max₀ : P := seq s rfl (fin.last s.dim)
lemma max₀_mem : s.max₀ ∈ s.val := seq_mem s rfl (fin.last s.dim)
lemma le_max₀ (p : P) (hp : p ∈ s.val) : p ≤ s.max₀ :=
begin
rw [← seq_rank s rfl p hp],
apply (seq_le s rfl (rank s rfl p hp) (fin.last s.dim)).mp,
apply fin.le_last
end
end els
/-- The function max : subdiv P → P is monotone. -/
lemma max₀_mono : monotone (max₀ : subdiv P → P) :=
λ s t hst, t.le_max₀ s.max₀ (hst s.max₀_mem)
variable (P)
def max : hom (subdiv P) P := ⟨max₀,max₀_mono⟩
lemma max_mem (s : subdiv P) : max P s ∈ s.val := max₀_mem s
lemma le_max (s : subdiv P) (p : P) (hp : p ∈ s.val) : p ≤ max P s :=
le_max₀ s p hp
lemma max_cofinal : cofinalₕ (max P) :=
begin
intro p,
let C := comma (max P) p,
let c : C := ⟨vertex p,le_refl _⟩,
let T := punit.{uP + 1},
let f : hom C T := const C punit.star,
let g : hom T C := const T c,
have hfg : comp f g = id T :=
by { ext t, rcases t, refl },
let m₀ : C → C := λ x,
begin
let τ₀ := x.val.val ∪ finset.singleton p,
have h₀ : τ₀ ≠ ∅ :=
finset.ne_empty_of_mem
(finset.mem_union_right _ (finset.mem_singleton_self p)),
have h₁ : is_chain τ₀ := λ a ha b hb,
begin
rcases finset.mem_union.mp ha with hax | hap;
rcases finset.mem_union.mp hb with hbx | hbp,
{ exact x.val.property.2 a hax b hbx },
{ left,
rw [finset.mem_singleton.mp hbp],
exact le_trans (le_max P x.val a hax) x.property },
{ right,
rw [finset.mem_singleton.mp hap],
exact le_trans (le_max P x.val b hbx) x.property },
{ left,
rw [finset.mem_singleton.mp hbp],
rw [finset.mem_singleton.mp hap] }
end,
let τ : subdiv P := ⟨τ₀,⟨h₀,h₁⟩⟩,
have h₂ : max P τ ≤ p :=
begin
rcases finset.mem_union.mp (max_mem P τ) with h | h,
{ exact le_trans (le_max P x.val _ h) x.property },
{ rw [finset.mem_singleton.mp h] }
end,
exact ⟨τ,h₂⟩
end,
have m₀_mono : monotone m₀ := λ x₀ x₁ h,
begin
change x₀.val.val ⊆ x₁.val.val at h,
change x₀.val.val ∪ finset.singleton p ⊆ x₁.val.val ∪ finset.singleton p,
intros a ha,
rcases finset.mem_union.mp ha with hax | hap,
{ exact finset.mem_union_left _ (h hax) },
{ exact finset.mem_union_right _ hap }
end,
let m : hom C C := ⟨m₀,m₀_mono⟩,
have hm₀ : comp g f ≤ m := λ x a ha,
begin
change a ∈ finset.singleton p at ha,
change a ∈ x.val.val ∪ finset.singleton p,
exact finset.mem_union_right _ ha
end,
have hm₁ : id C ≤ m := λ x a ha,
begin
change a ∈ x.val.val at ha,
change a ∈ x.val.val ∪ finset.singleton p,
exact finset.mem_union_left _ ha
end,
let hgf := (π₀.sound hm₀).trans (π₀.sound hm₁).symm,
have e : equivₕ C T :=
{ to_fun := component f,
inv_fun := component g,
left_inv := hgf,
right_inv := congr_arg component hfg },
exact ⟨e⟩
end
end subdiv
end poset |
d65b3a130d0782dc4d26e0ddf7f5a461bd211a5f | 9dc8cecdf3c4634764a18254e94d43da07142918 | /src/category_theory/limits/shapes/zero_objects.lean | 040fc711649d9ea77ac7d90225b3099e3d6e0065 | [
"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 | 8,285 | lean | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Johan Commelin
-/
import category_theory.limits.shapes.products
import category_theory.limits.shapes.images
import category_theory.isomorphism_classes
/-!
# Zero objects
A category "has a zero object" if it has an object which is both initial and terminal. Having a
zero object provides zero morphisms, as the unique morphisms factoring through the zero object;
see `category_theory.limits.shapes.zero_morphisms`.
## References
* [F. Borceux, *Handbook of Categorical Algebra 2*][borceux-vol2]
-/
noncomputable theory
universes v u v' u'
open category_theory
open category_theory.category
variables {C : Type u} [category.{v} C]
variables {D : Type u'} [category.{v'} D]
namespace category_theory
namespace limits
/-- An object `X` in a category is a *zero object* if for every object `Y`
there is a unique morphism `to : X → Y` and a unique morphism `from : Y → X`.
This is a characteristic predicate for `has_zero_object`. -/
structure is_zero (X : C) : Prop :=
(unique_to : ∀ Y, nonempty (unique (X ⟶ Y)))
(unique_from : ∀ Y, nonempty (unique (Y ⟶ X)))
namespace is_zero
variables {X Y : C}
/-- If `h : is_zero X`, then `h.to Y` is a choice of unique morphism `X → Y`. -/
protected def «to» (h : is_zero X) (Y : C) : X ⟶ Y :=
@default (X ⟶ Y) $ @unique.inhabited _ $ (h.unique_to Y).some
lemma eq_to (h : is_zero X) (f : X ⟶ Y) : f = h.to Y :=
@unique.eq_default _ (id _) _
lemma to_eq (h : is_zero X) (f : X ⟶ Y) : h.to Y = f :=
(h.eq_to f).symm
/-- If `h : is_zero X`, then `h.from Y` is a choice of unique morphism `Y → X`. -/
protected def «from» (h : is_zero X) (Y : C) : Y ⟶ X :=
@default (Y ⟶ X) $ @unique.inhabited _ $ (h.unique_from Y).some
lemma eq_from (h : is_zero X) (f : Y ⟶ X) : f = h.from Y :=
@unique.eq_default _ (id _) _
lemma from_eq (h : is_zero X) (f : Y ⟶ X) : h.from Y = f :=
(h.eq_from f).symm
lemma eq_of_src (hX : is_zero X) (f g : X ⟶ Y) : f = g :=
(hX.eq_to f).trans (hX.eq_to g).symm
lemma eq_of_tgt (hX : is_zero X) (f g : Y ⟶ X) : f = g :=
(hX.eq_from f).trans (hX.eq_from g).symm
/-- Any two zero objects are isomorphic. -/
def iso (hX : is_zero X) (hY : is_zero Y) : X ≅ Y :=
{ hom := hX.to Y,
inv := hX.from Y,
hom_inv_id' := hX.eq_of_src _ _,
inv_hom_id' := hY.eq_of_src _ _, }
/-- A zero object is in particular initial. -/
protected def is_initial (hX : is_zero X) : is_initial X :=
@is_initial.of_unique _ _ X $ λ Y, (hX.unique_to Y).some
/-- A zero object is in particular terminal. -/
protected def is_terminal (hX : is_zero X) : is_terminal X :=
@is_terminal.of_unique _ _ X $ λ Y, (hX.unique_from Y).some
/-- The (unique) isomorphism between any initial object and the zero object. -/
def iso_is_initial (hX : is_zero X) (hY : is_initial Y) : X ≅ Y :=
hX.is_initial.unique_up_to_iso hY
/-- The (unique) isomorphism between any terminal object and the zero object. -/
def iso_is_terminal (hX : is_zero X) (hY : is_terminal Y) : X ≅ Y :=
hX.is_terminal.unique_up_to_iso hY
lemma of_iso (hY : is_zero Y) (e : X ≅ Y) : is_zero X :=
begin
refine ⟨λ Z, ⟨⟨⟨e.hom ≫ hY.to Z⟩, λ f, _⟩⟩, λ Z, ⟨⟨⟨hY.from Z ≫ e.inv⟩, λ f, _⟩⟩⟩,
{ rw ← cancel_epi e.inv, apply hY.eq_of_src, },
{ rw ← cancel_mono e.hom, apply hY.eq_of_tgt, },
end
end is_zero
end limits
open category_theory.limits
lemma iso.is_zero_iff {X Y : C} (e : X ≅ Y) :
is_zero X ↔ is_zero Y :=
⟨λ h, h.of_iso e.symm, λ h, h.of_iso e⟩
lemma functor.is_zero (F : C ⥤ D) (hF : ∀ X, is_zero (F.obj X)) :
is_zero F :=
begin
split; intros G; refine ⟨⟨⟨_⟩, _⟩⟩,
{ refine { app := λ X, (hF _).to _, naturality' := _ },
intros, exact (hF _).eq_of_src _ _ },
{ intro f, ext, apply (hF _).eq_of_src _ _ },
{ refine { app := λ X, (hF _).from _, naturality' := _ },
intros, exact (hF _).eq_of_tgt _ _ },
{ intro f, ext, apply (hF _).eq_of_tgt _ _ },
end
namespace limits
variables (C)
/-- A category "has a zero object" if it has an object which is both initial and terminal. -/
class has_zero_object : Prop :=
(zero : ∃ X : C, is_zero X)
instance has_zero_object_punit : has_zero_object (discrete punit) :=
{ zero := ⟨⟨⟨⟩⟩, by tidy, by tidy⟩, }
section
variables [has_zero_object C]
/--
Construct a `has_zero C` for a category with a zero object.
This can not be a global instance as it will trigger for every `has_zero C` typeclass search.
-/
protected def has_zero_object.has_zero : has_zero C :=
{ zero := has_zero_object.zero.some }
localized "attribute [instance] category_theory.limits.has_zero_object.has_zero" in zero_object
lemma is_zero_zero : is_zero (0 : C) :=
has_zero_object.zero.some_spec
end
open_locale zero_object
variables {C}
lemma is_zero.has_zero_object {X : C} (hX : is_zero X) : has_zero_object C := ⟨⟨X, hX⟩⟩
/-- Every zero object is isomorphic to *the* zero object. -/
def is_zero.iso_zero [has_zero_object C] {X : C} (hX : is_zero X) : X ≅ 0 :=
hX.iso (is_zero_zero C)
lemma is_zero.obj [has_zero_object D] {F : C ⥤ D} (hF : is_zero F) (X : C) :
is_zero (F.obj X) :=
begin
let G : C ⥤ D := (category_theory.functor.const C).obj 0,
have hG : is_zero G := functor.is_zero _ (λ X, is_zero_zero _),
let e : F ≅ G := hF.iso hG,
exact (is_zero_zero _).of_iso (e.app X),
end
namespace has_zero_object
variables [has_zero_object C]
/-- There is a unique morphism from the zero object to any object `X`. -/
protected def unique_to (X : C) : unique (0 ⟶ X) :=
((is_zero_zero C).unique_to X).some
/-- There is a unique morphism from any object `X` to the zero object. -/
protected def unique_from (X : C) : unique (X ⟶ 0) :=
((is_zero_zero C).unique_from X).some
localized "attribute [instance] category_theory.limits.has_zero_object.unique_to" in zero_object
localized "attribute [instance] category_theory.limits.has_zero_object.unique_from" in zero_object
@[ext]
lemma to_zero_ext {X : C} (f g : X ⟶ 0) : f = g :=
(is_zero_zero C).eq_of_tgt _ _
@[ext]
lemma from_zero_ext {X : C} (f g : 0 ⟶ X) : f = g :=
(is_zero_zero C).eq_of_src _ _
instance (X : C) : subsingleton (X ≅ 0) := by tidy
instance {X : C} (f : 0 ⟶ X) : mono f :=
{ right_cancellation := λ Z g h w, by ext, }
instance {X : C} (f : X ⟶ 0) : epi f :=
{ left_cancellation := λ Z g h w, by ext, }
instance zero_to_zero_is_iso (f : (0 : C) ⟶ 0) :
is_iso f :=
by convert (show is_iso (𝟙 (0 : C)), by apply_instance)
/-- A zero object is in particular initial. -/
def zero_is_initial : is_initial (0 : C) :=
(is_zero_zero C).is_initial
/-- A zero object is in particular terminal. -/
def zero_is_terminal : is_terminal (0 : C) :=
(is_zero_zero C).is_terminal
/-- A zero object is in particular initial. -/
@[priority 10]
instance has_initial : has_initial C :=
has_initial_of_unique 0
/-- A zero object is in particular terminal. -/
@[priority 10]
instance has_terminal : has_terminal C :=
has_terminal_of_unique 0
/-- The (unique) isomorphism between any initial object and the zero object. -/
def zero_iso_is_initial {X : C} (t : is_initial X) : 0 ≅ X :=
zero_is_initial.unique_up_to_iso t
/-- The (unique) isomorphism between any terminal object and the zero object. -/
def zero_iso_is_terminal {X : C} (t : is_terminal X) : 0 ≅ X :=
zero_is_terminal.unique_up_to_iso t
/-- The (unique) isomorphism between the chosen initial object and the chosen zero object. -/
def zero_iso_initial [has_initial C] : 0 ≅ ⊥_ C :=
zero_is_initial.unique_up_to_iso initial_is_initial
/-- The (unique) isomorphism between the chosen terminal object and the chosen zero object. -/
def zero_iso_terminal [has_terminal C] : 0 ≅ ⊤_ C :=
zero_is_terminal.unique_up_to_iso terminal_is_terminal
@[priority 100]
instance has_strict_initial : initial_mono_class C :=
initial_mono_class.of_is_initial zero_is_initial (λ X, category_theory.mono _)
end has_zero_object
end limits
open category_theory.limits
open_locale zero_object
lemma functor.is_zero_iff [has_zero_object D] (F : C ⥤ D) :
is_zero F ↔ ∀ X, is_zero (F.obj X) :=
⟨λ hF X, hF.obj X, functor.is_zero _⟩
end category_theory
|
2e0117f8b3fe48e5e9d87e5465cb5555a413171a | 6dc0c8ce7a76229dd81e73ed4474f15f88a9e294 | /stage0/src/Lean/Meta/Tactic/Induction.lean | e3acf657fc4984a673c556d6ef4a0e38c03c4ead | [
"Apache-2.0"
] | permissive | williamdemeo/lean4 | 72161c58fe65c3ad955d6a3050bb7d37c04c0d54 | 6d00fcf1d6d873e195f9220c668ef9c58e9c4a35 | refs/heads/master | 1,678,305,356,877 | 1,614,708,995,000 | 1,614,708,995,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 11,796 | lean | /-
Copyright (c) 2020 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Lean.Meta.RecursorInfo
import Lean.Meta.SynthInstance
import Lean.Meta.Tactic.Util
import Lean.Meta.Tactic.Revert
import Lean.Meta.Tactic.Intro
import Lean.Meta.Tactic.Clear
import Lean.Meta.Tactic.FVarSubst
namespace Lean.Meta
private partial def getTargetArity : Expr → Nat
| Expr.mdata _ b _ => getTargetArity b
| Expr.forallE _ _ b _ => getTargetArity b + 1
| e => if e.isHeadBetaTarget then getTargetArity e.headBeta else 0
private def addRecParams (mvarId : MVarId) (majorTypeArgs : Array Expr) : List (Option Nat) → Expr → MetaM Expr
| [], recursor => pure recursor
| some pos :: rest, recursor =>
if h : pos < majorTypeArgs.size then
addRecParams mvarId majorTypeArgs rest (mkApp recursor (majorTypeArgs.get ⟨pos, h⟩))
else
throwTacticEx `induction mvarId "ill-formed recursor"
| none :: rest, recursor => do
let recursorType ← inferType recursor
let recursorType ← whnfForall recursorType
match recursorType with
| Expr.forallE _ d _ _ => do
let param ← try synthInstance d catch _ => throwTacticEx `induction mvarId "failed to generate type class instance parameter"
addRecParams mvarId majorTypeArgs rest (mkApp recursor param)
| _ =>
throwTacticEx `induction mvarId "ill-formed recursor"
structure InductionSubgoal where
mvarId : MVarId
fields : Array Expr := #[]
subst : FVarSubst := {}
deriving Inhabited
private def getTypeBody (mvarId : MVarId) (type : Expr) (x : Expr) : MetaM Expr := do
let type ← whnfForall type
match type with
| Expr.forallE _ _ b _ => pure $ b.instantiate1 x
| _ => throwTacticEx `induction mvarId "ill-formed recursor"
structure AltVarNames where
explicit : Bool := false -- true if `@` modifier was used
varNames : List Name := []
deriving Inhabited
private partial def finalize
(mvarId : MVarId) (givenNames : Array AltVarNames) (recursorInfo : RecursorInfo)
(reverted : Array FVarId) (major : Expr) (indices : Array Expr) (baseSubst : FVarSubst) (recursor : Expr)
: MetaM (Array InductionSubgoal) := do
let target ← getMVarType mvarId
let initialArity := getTargetArity target
let recursorType ← inferType recursor
let numMinors := recursorInfo.produceMotive.length
let rec loop (pos : Nat) (minorIdx : Nat) (recursor recursorType : Expr) (consumedMajor : Bool) (subgoals : Array InductionSubgoal) := do
let recursorType ← whnfForall recursorType
if recursorType.isForall && pos < recursorInfo.numArgs then
if pos == recursorInfo.firstIndexPos then
let (recursor, recursorType) ← indices.foldlM (init := (recursor, recursorType)) fun (recursor, recursorType) index => do
let recursor := mkApp recursor index
let recursorType ← getTypeBody mvarId recursorType index
pure (recursor, recursorType)
let recursor := mkApp recursor major
let recursorType ← getTypeBody mvarId recursorType major
loop (pos+1+indices.size) minorIdx recursor recursorType true subgoals
else
-- consume motive
let tag ← getMVarTag mvarId
if minorIdx ≥ numMinors then throwTacticEx `induction mvarId "ill-formed recursor"
match recursorType with
| Expr.forallE n d b c =>
let d := d.headBeta
-- Remark is givenNames is not empty, then user provided explicit alternatives for each minor premise
if c.binderInfo.isInstImplicit && givenNames.isEmpty then
match (← synthInstance? d) with
| some inst =>
let recursor := mkApp recursor inst
let recursorType ← getTypeBody mvarId recursorType inst
loop (pos+1) (minorIdx+1) recursor recursorType consumedMajor subgoals
| none => do
-- Add newSubgoal if type class resolution failed
let mvar ← mkFreshExprSyntheticOpaqueMVar d (tag ++ n)
let recursor := mkApp recursor mvar
let recursorType ← getTypeBody mvarId recursorType mvar
loop (pos+1) (minorIdx+1) recursor recursorType consumedMajor (subgoals.push { mvarId := mvar.mvarId! })
else
let arity := getTargetArity d
if arity < initialArity then throwTacticEx `induction mvarId "ill-formed recursor"
let nparams := arity - initialArity -- number of fields due to minor premise
let nextra := reverted.size - indices.size - 1 -- extra dependencies that have been reverted
let minorGivenNames := if h : minorIdx < givenNames.size then givenNames.get ⟨minorIdx, h⟩ else {}
let mvar ← mkFreshExprSyntheticOpaqueMVar d (tag ++ n)
let recursor := mkApp recursor mvar
let recursorType ← getTypeBody mvarId recursorType mvar
-- Try to clear major premise from new goal
let mvarId' ← tryClear mvar.mvarId! major.fvarId!
let (fields, mvarId') ← introN mvarId' nparams minorGivenNames.varNames (useNamesForExplicitOnly := !minorGivenNames.explicit)
let (extra, mvarId') ← introNP mvarId' nextra
let subst := reverted.size.fold (init := baseSubst) fun i (subst : FVarSubst) =>
if i < indices.size + 1 then subst
else
let revertedFVarId := reverted[i]
let newFVarId := extra[i - indices.size - 1]
subst.insert revertedFVarId (mkFVar newFVarId)
let fields := fields.map mkFVar
loop (pos+1) (minorIdx+1) recursor recursorType consumedMajor (subgoals.push { mvarId := mvarId', fields := fields, subst := subst })
| _ => unreachable!
else
unless consumedMajor do throwTacticEx `induction mvarId "ill-formed recursor"
assignExprMVar mvarId recursor
pure subgoals
loop (recursorInfo.paramsPos.length + 1) 0 recursor recursorType false #[]
private def throwUnexpectedMajorType {α} (mvarId : MVarId) (majorType : Expr) : MetaM α :=
throwTacticEx `induction mvarId m!"unexpected major premise type{indentExpr majorType}"
def induction (mvarId : MVarId) (majorFVarId : FVarId) (recursorName : Name) (givenNames : Array AltVarNames := #[]) : MetaM (Array InductionSubgoal) :=
withMVarContext mvarId do
trace[Meta.Tactic.induction]! "initial\n{MessageData.ofGoal mvarId}"
checkNotAssigned mvarId `induction
let majorLocalDecl ← getLocalDecl majorFVarId
let recursorInfo ← mkRecursorInfo recursorName
let some majorType ← whnfUntil majorLocalDecl.type recursorInfo.typeName | throwUnexpectedMajorType mvarId majorLocalDecl.type
majorType.withApp fun _ majorTypeArgs => do
recursorInfo.paramsPos.forM fun paramPos? => do
match paramPos? with
| none => pure ()
| some paramPos => if paramPos ≥ majorTypeArgs.size then throwTacticEx `induction mvarId m!"major premise type is ill-formed{indentExpr majorType}"
let mctx ← getMCtx
let indices ← recursorInfo.indicesPos.toArray.mapM fun idxPos => do
if idxPos ≥ majorTypeArgs.size then throwTacticEx `induction mvarId m!"major premise type is ill-formed{indentExpr majorType}"
let idx := majorTypeArgs.get! idxPos
unless idx.isFVar do throwTacticEx `induction mvarId m!"major premise type index {idx} is not a variable{indentExpr majorType}"
majorTypeArgs.size.forM fun i => do
let arg := majorTypeArgs[i]
if i != idxPos && arg == idx then
throwTacticEx `induction mvarId m!"'{idx}' is an index in major premise, but it occurs more than once{indentExpr majorType}"
if i < idxPos && mctx.exprDependsOn arg idx.fvarId! then
throwTacticEx `induction mvarId m!"'{idx}' is an index in major premise, but it occurs in previous arguments{indentExpr majorType}"
-- If arg is also and index and a variable occurring after `idx`, we need to make sure it doesn't depend on `idx`.
-- Note that if `arg` is not a variable, we will fail anyway when we visit it.
if i > idxPos && recursorInfo.indicesPos.contains i && arg.isFVar then
let idxDecl ← getLocalDecl idx.fvarId!
if mctx.localDeclDependsOn idxDecl arg.fvarId! then
throwTacticEx `induction mvarId m!"'{idx}' is an index in major premise, but it depends on index occurring at position #{i+1}"
pure idx
let target ← getMVarType mvarId
if !recursorInfo.depElim && mctx.exprDependsOn target majorFVarId then
throwTacticEx `induction mvarId m!"recursor '{recursorName}' does not support dependent elimination, but conclusion depends on major premise"
-- Revert indices and major premise preserving variable order
let (reverted, mvarId) ← revert mvarId ((indices.map Expr.fvarId!).push majorFVarId) true
-- Re-introduce indices and major
let (indices', mvarId) ← introNP mvarId indices.size
let (majorFVarId', mvarId) ← intro1P mvarId
-- Create FVarSubst with indices
let baseSubst := do
let mut subst : FVarSubst := {}
let mut i := 0
for index in indices do
subst := subst.insert index.fvarId! (mkFVar indices'[i])
i := i + 1
pure subst
trace[Meta.Tactic.induction]! "after revert&intro\n{MessageData.ofGoal mvarId}"
-- Update indices and major
let indices := indices'.map mkFVar
let majorFVarId := majorFVarId'
let major := mkFVar majorFVarId
withMVarContext mvarId do
let target ← getMVarType mvarId
let targetLevel ← getLevel target
let targetLevel ← normalizeLevel targetLevel
let majorLocalDecl ← getLocalDecl majorFVarId
let some majorType ← whnfUntil majorLocalDecl.type recursorInfo.typeName | throwUnexpectedMajorType mvarId majorLocalDecl.type
majorType.withApp fun majorTypeFn majorTypeArgs => do
match majorTypeFn with
| Expr.const majorTypeFnName majorTypeFnLevels _ => do
let majorTypeFnLevels := majorTypeFnLevels.toArray
let (recursorLevels, foundTargetLevel) ← recursorInfo.univLevelPos.foldlM (init := (#[], false))
fun (recursorLevels, foundTargetLevel) (univPos : RecursorUnivLevelPos) => do
match univPos with
| RecursorUnivLevelPos.motive => pure (recursorLevels.push targetLevel, true)
| RecursorUnivLevelPos.majorType idx =>
if idx ≥ majorTypeFnLevels.size then throwTacticEx `induction mvarId "ill-formed recursor"
pure (recursorLevels.push (majorTypeFnLevels.get! idx), foundTargetLevel)
if !foundTargetLevel && !targetLevel.isZero then
throwTacticEx `induction mvarId m!"recursor '{recursorName}' can only eliminate into Prop"
let recursor := mkConst recursorName recursorLevels.toList
let recursor ← addRecParams mvarId majorTypeArgs recursorInfo.paramsPos recursor
-- Compute motive
let motive := target
let motive ← if recursorInfo.depElim then mkLambdaFVars #[major] motive else pure motive
let motive ← mkLambdaFVars indices motive
let recursor := mkApp recursor motive
finalize mvarId givenNames recursorInfo reverted major indices baseSubst recursor
| _ =>
throwTacticEx `induction mvarId "major premise is not of the form (C ...)"
builtin_initialize registerTraceClass `Meta.Tactic.induction
end Lean.Meta
|
ac16c4a82bea094a7e1f4378982aa42260a73e3b | 947b78d97130d56365ae2ec264df196ce769371a | /tests/lean/run/toExpr.lean | c9824ab329ff82b5994bbd07394b167f071f6924 | [
"Apache-2.0"
] | permissive | shyamalschandra/lean4 | 27044812be8698f0c79147615b1d5090b9f4b037 | 6e7a883b21eaf62831e8111b251dc9b18f40e604 | refs/heads/master | 1,671,417,126,371 | 1,601,859,995,000 | 1,601,860,020,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 797 | lean | import Lean
new_frontend
open Lean
unsafe def test {α : Type} [HasToString α] [ToExpr α] [HasBeq α] (a : α) : CoreM Unit := do
let env ← getEnv;
let auxName := `_toExpr._test;
let decl := Declaration.defnDecl {
name := auxName,
lparams := [],
value := toExpr a,
type := toTypeExpr α,
hints := ReducibilityHints.abbrev,
isUnsafe := false
};
IO.println (toExpr a);
(match env.addAndCompile {} decl with
| Except.error _ => throwError "addDecl failed"
| Except.ok env =>
match env.evalConst α auxName with
| Except.error ex => throwError ex
| Except.ok b => do
IO.println b;
«unless» (a == b) $
throwError "toExpr failed";
pure ())
#eval test #[(1, 2), (3, 4)]
#eval test ['a', 'b', 'c']
#eval test ("hello", true)
#eval test ((), 10)
|
46c11cddb82d910e68dc52f34af428e4522ce700 | a0e23cfdd129a671bf3154ee1a8a3a72bf4c7940 | /stage0/src/Lean/Server/Requests.lean | daba50109997f0ccee365d1d64ef675cd69cc0b9 | [
"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 | 6,451 | lean | /-
Copyright (c) 2021 Wojciech Nawrocki. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wojciech Nawrocki, Marc Huisinga
-/
import Lean.DeclarationRange
import Lean.Data.Json
import Lean.Data.Lsp
import Lean.Server.FileSource
import Lean.Server.FileWorker.Utils
/-! We maintain a global map of LSP request handlers. This allows user code such as plugins
to register its own handlers, for example to support ITP functionality such as goal state
visualization.
For details of how to register one, see `registerLspRequestHandler`. -/
namespace Lean.Server.Requests
structure RequestError where
code : JsonRpc.ErrorCode
message : String
namespace RequestError
open JsonRpc
def fileChanged : RequestError :=
{ code := ErrorCode.contentModified
message := "File changed." }
def methodNotFound (method : String) : RequestError :=
{ code := ErrorCode.methodNotFound
message := s!"No request handler found for '{method}'" }
instance : Coe IO.Error RequestError where
coe e := { code := ErrorCode.internalError
message := toString e }
def toLspResponseError (id : RequestID) (e : RequestError) : ResponseError Unit :=
{ id := id
code := e.code
message := e.message }
end RequestError
structure RequestContext where
srcSearchPath : SearchPath
docRef : IO.Ref FileWorker.EditableDocument
abbrev RequestTask α := Task (Except RequestError α)
/-- Workers execute request handlers in this monad. -/
abbrev RequestM := ReaderT RequestContext <| ExceptT RequestError IO
namespace RequestM
open FileWorker
open Snapshots
def readDoc : RequestM EditableDocument := fun rc =>
rc.docRef.get
def asTask (t : RequestM α) : RequestM (RequestTask α) := fun rc => do
let t ← IO.asTask <| t rc
return t.map fun
| Except.error e => throwThe RequestError e
| Except.ok v => v
def mapTask (t : Task α) (f : α → RequestM β) : RequestM (RequestTask β) := fun rc => do
let t ← (IO.mapTask · t) fun a => f a rc
return t.map fun
| Except.error e => throwThe RequestError e
| Except.ok v => v
def bindTask (t : Task α) (f : α → RequestM (RequestTask β)) : RequestM (RequestTask β) := fun rc => do
let t ← IO.bindTask t fun a => do
match (← f a rc) with
| Except.error e => return Task.pure <| Except.ok <| Except.error e
| Except.ok t => return t.map Except.ok
return t.map fun
| Except.error e => throwThe RequestError e
| Except.ok v => v
/-- Create a task which waits for a snapshot matching `p`, handles various errors,
and if a matching snapshot was found executes `x` with it. If not found, the task
executes `notFoundX`. -/
def withWaitFindSnap (doc : EditableDocument) (p : Snapshot → Bool)
(notFoundX : RequestM β)
(x : Snapshot → RequestM β)
: RequestM (RequestTask β) := do
let findTask ← doc.cmdSnaps.waitFind? p
mapTask findTask fun
/- The elaboration task that we're waiting for may be aborted if the file contents change.
In that case, we reply with the `fileChanged` error. Thanks to this, the server doesn't
get bogged down in requests for an old state of the document. -/
| Except.error FileWorker.ElabTaskError.aborted =>
throwThe RequestError RequestError.fileChanged
| Except.error (FileWorker.ElabTaskError.ioError e) =>
throwThe IO.Error e
| Except.error FileWorker.ElabTaskError.eof => notFoundX
| Except.ok none => notFoundX
| Except.ok (some snap) => x snap
end RequestM
/- The global request handlers table. -/
section HandlerTable
open Lsp
private structure RequestHandler where
fileSource : Json → Except RequestError Lsp.DocumentUri
handle : Json → RequestM (RequestTask Json)
builtin_initialize requestHandlers : IO.Ref (Std.PersistentHashMap String RequestHandler) ←
IO.mkRef {}
private def parseParams (paramType : Type) [FromJson paramType] (params : Json) : Except RequestError paramType :=
fromJson? params |>.mapError fun inner =>
{ code := JsonRpc.ErrorCode.parseError
message := s!"Cannot parse request params: {params.compress}\n{inner}" }
/-- NB: This method may only be called in `initialize`/`builtin_initialize` blocks.
A registration consists of:
- a type of JSON-parsable request data `paramType`
- a `FileSource` instance for it so the system knows where to route requests
- a type of JSON-serializable response data `respType`
- an actual `handler` which runs in the `RequestM` monad and is expected
to produce an asynchronous `RequestTask` which does any waiting/computation
A handler task may be cancelled at any time, so it should check the cancellation token when possible
to handle this cooperatively. Any exceptions thrown in a request handler will be reported to the client
as LSP error responses. -/
def registerLspRequestHandler (method : String)
paramType [FromJson paramType] [FileSource paramType]
respType [ToJson respType]
(handler : paramType → RequestM (RequestTask respType)) : IO Unit := do
if !(← IO.initializing) then
throw <| IO.userError s!"Failed to register LSP request handler for '{method}': only possible during initialization"
if (←requestHandlers.get).contains method then
throw <| IO.userError s!"Failed to register LSP request handler for '{method}': already registered"
let fileSource := fun j =>
parseParams paramType j |>.map fun p =>
Lsp.fileSource p
let handle := fun j => do
let params ← parseParams paramType j
let t ← handler params
t.map <| Except.map ToJson.toJson
requestHandlers.modify fun rhs => rhs.insert method { fileSource, handle }
private def lookupLspRequestHandler (method : String) : IO (Option RequestHandler) := do
(← requestHandlers.get).find? method
def routeLspRequest (method : String) (params : Json) : IO (Except RequestError DocumentUri) := do
match (← lookupLspRequestHandler method) with
| none => return Except.error <| RequestError.methodNotFound method
| some rh => return rh.fileSource params
def handleLspRequest (method : String) (params : Json) : RequestM (RequestTask Json) := do
match (← lookupLspRequestHandler method) with
| none => throwThe IO.Error <| IO.userError s!"internal server error: request '{method}' routed through watchdog but unknown in worker; are both using the same plugins?"
| some rh => rh.handle params
end HandlerTable
end Lean.Server.Requests |
8bf291d3b8cbedbb89a4cebc0efc24cbb5dc5c53 | c777c32c8e484e195053731103c5e52af26a25d1 | /src/analysis/convolution.lean | c46120ed87890a95a565b2953f0a893ab19a8f1e | [
"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 | 83,790 | lean | /-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import measure_theory.group.integration
import measure_theory.group.prod
import measure_theory.function.locally_integrable
import analysis.calculus.bump_function_inner
import measure_theory.integral.interval_integral
import analysis.calculus.parametric_integral
/-!
# Convolution of functions
This file defines the convolution on two functions, i.e. `x ↦ ∫ f(t)g(x - t) ∂t`.
In the general case, these functions can be vector-valued, and have an arbitrary (additive)
group as domain. We use a continuous bilinear operation `L` on these function values as
"multiplication". The domain must be equipped with a Haar measure `μ`
(though many individual results have weaker conditions on `μ`).
For many applications we can take `L = lsmul ℝ ℝ` or `L = mul ℝ ℝ`.
We also define `convolution_exists` and `convolution_exists_at` to state that the convolution is
well-defined (everywhere or at a single point). These conditions are needed for pointwise
computations (e.g. `convolution_exists_at.distrib_add`), but are generally not stong enough for any
local (or global) properties of the convolution. For this we need stronger assumptions on `f`
and/or `g`, and generally if we impose stronger conditions on one of the functions, we can impose
weaker conditions on the other.
We have proven many of the properties of the convolution assuming one of these functions
has compact support (in which case the other function only needs to be locally integrable).
We still need to prove the properties for other pairs of conditions (e.g. both functions are
rapidly decreasing)
# Design Decisions
We use a bilinear map `L` to "multiply" the two functions in the integrand.
This generality has several advantages
* This allows us to compute the total derivative of the convolution, in case the functions are
multivariate. The total derivative is again a convolution, but where the codomains of the
functions can be higher-dimensional. See `has_compact_support.has_fderiv_at_convolution_right`.
* This allows us to use `@[to_additive]` everywhere (which would not be possible if we would use
`mul`/`smul` in the integral, since `@[to_additive]` will incorrectly also try to additivize
those definitions).
* We need to support the case where at least one of the functions is vector-valued, but if we use
`smul` to multiply the functions, that would be an asymmetric definition.
# Main Definitions
* `convolution f g L μ x = (f ⋆[L, μ] g) x = ∫ t, L (f t) (g (x - t)) ∂μ` is the convolution of
`f` and `g` w.r.t. the continuous bilinear map `L` and measure `μ`.
* `convolution_exists_at f g x L μ` states that the convolution `(f ⋆[L, μ] g) x` is well-defined
(i.e. the integral exists).
* `convolution_exists f g L μ` states that the convolution `f ⋆[L, μ] g` is well-defined at each
point.
# Main Results
* `has_compact_support.has_fderiv_at_convolution_right` and
`has_compact_support.has_fderiv_at_convolution_left`: we can compute the total derivative
of the convolution as a convolution with the total derivative of the right (left) function.
* `has_compact_support.cont_diff_convolution_right` and
`has_compact_support.cont_diff_convolution_left`: the convolution is `𝒞ⁿ` if one of the functions
is `𝒞ⁿ` with compact support and the other function in locally integrable.
Versions of these statements for functions depending on a parameter are also given.
* `convolution_tendsto_right`: Given a sequence of nonnegative normalized functions whose support
tends to a small neighborhood around `0`, the convolution tends to the right argument.
This is specialized to bump functions in `cont_diff_bump.convolution_tendsto_right`.
# Notation
The following notations are localized in the locale `convolution`:
* `f ⋆[L, μ] g` for the convolution. Note: you have to use parentheses to apply the convolution
to an argument: `(f ⋆[L, μ] g) x`.
* `f ⋆[L] g := f ⋆[L, volume] g`
* `f ⋆ g := f ⋆[lsmul ℝ ℝ] g`
# To do
* Existence and (uniform) continuity of the convolution if
one of the maps is in `ℒ^p` and the other in `ℒ^q` with `1 / p + 1 / q = 1`.
This might require a generalization of `measure_theory.mem_ℒp.smul` where `smul` is generalized
to a continuous bilinear map.
(see e.g. [Fremlin, *Measure Theory* (volume 2)][fremlin_vol2], 255K)
* The convolution is a `ae_strongly_measurable` function
(see e.g. [Fremlin, *Measure Theory* (volume 2)][fremlin_vol2], 255I).
* Prove properties about the convolution if both functions are rapidly decreasing.
* Use `@[to_additive]` everywhere
-/
open set function filter measure_theory measure_theory.measure topological_space
open continuous_linear_map metric
open_locale pointwise topology nnreal filter
universes u𝕜 uG uE uE' uE'' uF uF' uF'' uP
variables {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {E'' : Type uE''}
{F : Type uF} {F' : Type uF'} {F'' : Type uF''} {P : Type uP}
variables [normed_add_comm_group E] [normed_add_comm_group E'] [normed_add_comm_group E'']
[normed_add_comm_group F]
{f f' : G → E} {g g' : G → E'} {x x' : G} {y y' : E}
section nontrivially_normed_field
variables [nontrivially_normed_field 𝕜]
variables [normed_space 𝕜 E] [normed_space 𝕜 E'] [normed_space 𝕜 E''] [normed_space 𝕜 F]
variables (L : E →L[𝕜] E' →L[𝕜] F)
section no_measurability
variables [add_group G] [topological_space G]
lemma convolution_integrand_bound_right_of_le_of_subset
{C : ℝ} (hC : ∀ i, ‖g i‖ ≤ C) {x t : G} {s u : set G} (hx : x ∈ s) (hu : - tsupport g + s ⊆ u) :
‖L (f t) (g (x - t))‖ ≤ u.indicator (λ t, ‖L‖ * ‖f t‖ * C) t :=
begin
refine le_indicator (λ t ht, _) (λ t ht, _) t,
{ refine (L.le_op_norm₂ _ _).trans _,
apply mul_le_mul_of_nonneg_left (hC _) (mul_nonneg (norm_nonneg _) (norm_nonneg _)) },
{ have : x - t ∉ support g,
{ refine mt (λ hxt, _) ht,
apply hu,
refine ⟨_, _, set.neg_mem_neg.mpr (subset_closure hxt), hx, _⟩,
rw [neg_sub, sub_add_cancel] },
rw [nmem_support.mp this, (L _).map_zero, norm_zero] }
end
lemma has_compact_support.convolution_integrand_bound_right_of_subset (hcg : has_compact_support g)
(hg : continuous g) {x t : G} {s u : set G} (hx : x ∈ s) (hu : - tsupport g + s ⊆ u) :
‖L (f t) (g (x - t))‖ ≤ u.indicator (λ t, ‖L‖ * ‖f t‖ * (⨆ i, ‖g i‖)) t :=
begin
apply convolution_integrand_bound_right_of_le_of_subset _ (λ i, _) hx hu,
exact le_csupr (hg.norm.bdd_above_range_of_has_compact_support hcg.norm) _,
end
lemma has_compact_support.convolution_integrand_bound_right (hcg : has_compact_support g)
(hg : continuous g) {x t : G} {s : set G} (hx : x ∈ s) :
‖L (f t) (g (x - t))‖ ≤ (- tsupport g + s).indicator (λ t, ‖L‖ * ‖f t‖ * (⨆ i, ‖g i‖)) t :=
hcg.convolution_integrand_bound_right_of_subset L hg hx subset.rfl
lemma continuous.convolution_integrand_fst [has_continuous_sub G] (hg : continuous g) (t : G) :
continuous (λ x, L (f t) (g (x - t))) :=
L.continuous₂.comp₂ continuous_const $ hg.comp $ continuous_id.sub continuous_const
lemma has_compact_support.convolution_integrand_bound_left (hcf : has_compact_support f)
(hf : continuous f) {x t : G} {s : set G} (hx : x ∈ s) :
‖L (f (x - t)) (g t)‖ ≤ (- tsupport f + s).indicator (λ t, ‖L‖ * (⨆ i, ‖f i‖) * ‖g t‖) t :=
by { convert hcf.convolution_integrand_bound_right L.flip hf hx,
simp_rw [L.op_norm_flip, mul_right_comm] }
end no_measurability
section measurability
variables [measurable_space G] {μ ν : measure G}
/-- The convolution of `f` and `g` exists at `x` when the function `t ↦ L (f t) (g (x - t))` is
integrable. There are various conditions on `f` and `g` to prove this. -/
def convolution_exists_at [has_sub G] (f : G → E) (g : G → E') (x : G) (L : E →L[𝕜] E' →L[𝕜] F)
(μ : measure G . volume_tac) : Prop :=
integrable (λ t, L (f t) (g (x - t))) μ
/-- The convolution of `f` and `g` exists when the function `t ↦ L (f t) (g (x - t))` is integrable
for all `x : G`. There are various conditions on `f` and `g` to prove this. -/
def convolution_exists [has_sub G] (f : G → E) (g : G → E') (L : E →L[𝕜] E' →L[𝕜] F)
(μ : measure G . volume_tac) : Prop :=
∀ x : G, convolution_exists_at f g x L μ
section convolution_exists
variables {L}
lemma convolution_exists_at.integrable [has_sub G] {x : G} (h : convolution_exists_at f g x L μ) :
integrable (λ t, L (f t) (g (x - t))) μ :=
h
variables (L)
section group
variables [add_group G]
lemma measure_theory.ae_strongly_measurable.convolution_integrand'
[has_measurable_add₂ G] [has_measurable_neg G] [sigma_finite ν]
(hf : ae_strongly_measurable f ν)
(hg : ae_strongly_measurable g $ map (λ (p : G × G), p.1 - p.2) (μ.prod ν)) :
ae_strongly_measurable (λ p : G × G, L (f p.2) (g (p.1 - p.2))) (μ.prod ν) :=
L.ae_strongly_measurable_comp₂ hf.snd $ hg.comp_measurable measurable_sub
section
variables [has_measurable_add G] [has_measurable_neg G]
lemma measure_theory.ae_strongly_measurable.convolution_integrand_snd'
(hf : ae_strongly_measurable f μ) {x : G}
(hg : ae_strongly_measurable g $ map (λ t, x - t) μ) :
ae_strongly_measurable (λ t, L (f t) (g (x - t))) μ :=
L.ae_strongly_measurable_comp₂ hf $ hg.comp_measurable $ measurable_id.const_sub x
lemma measure_theory.ae_strongly_measurable.convolution_integrand_swap_snd'
{x : G} (hf : ae_strongly_measurable f $ map (λ t, x - t) μ)
(hg : ae_strongly_measurable g μ) : ae_strongly_measurable (λ t, L (f (x - t)) (g t)) μ :=
L.ae_strongly_measurable_comp₂ (hf.comp_measurable $ measurable_id.const_sub x) hg
/-- A sufficient condition to prove that `f ⋆[L, μ] g` exists.
We assume that `f` is integrable on a set `s` and `g` is bounded and ae strongly measurable
on `x₀ - s` (note that both properties hold if `g` is continuous with compact support). -/
lemma bdd_above.convolution_exists_at' {x₀ : G}
{s : set G} (hbg : bdd_above ((λ i, ‖g i‖) '' ((λ t, - t + x₀) ⁻¹' s)))
(hs : measurable_set s) (h2s : support (λ t, L (f t) (g (x₀ - t))) ⊆ s)
(hf : integrable_on f s μ) (hmg : ae_strongly_measurable g $ map (λ t, x₀ - t) (μ.restrict s)) :
convolution_exists_at f g x₀ L μ :=
begin
rw [convolution_exists_at, ← integrable_on_iff_integrable_of_support_subset h2s],
set s' := (λ t, - t + x₀) ⁻¹' s,
have : ∀ᵐ (t : G) ∂(μ.restrict s),
‖L (f t) (g (x₀ - t))‖ ≤ s.indicator (λ t, ‖L‖ * ‖f t‖ * ⨆ i : s', ‖g i‖) t,
{ refine eventually_of_forall _,
refine le_indicator (λ t ht, _) (λ t ht, _),
{ refine (L.le_op_norm₂ _ _).trans _,
refine mul_le_mul_of_nonneg_left
(le_csupr_set hbg $ mem_preimage.mpr _)
(mul_nonneg (norm_nonneg _) (norm_nonneg _)),
rwa [neg_sub, sub_add_cancel] },
{ have : t ∉ support (λ t, L (f t) (g (x₀ - t))) := mt (λ h, h2s h) ht,
rw [nmem_support.mp this, norm_zero] } },
refine integrable.mono' _ _ this,
{ rw [integrable_indicator_iff hs], exact ((hf.norm.const_mul _).mul_const _).integrable_on },
{ exact hf.ae_strongly_measurable.convolution_integrand_snd' L hmg }
end
/-- If `‖f‖ *[μ] ‖g‖` exists, then `f *[L, μ] g` exists. -/
lemma convolution_exists_at.of_norm' {x₀ : G}
(h : convolution_exists_at (λ x, ‖f x‖) (λ x, ‖g x‖) x₀ (mul ℝ ℝ) μ)
(hmf : ae_strongly_measurable f μ)
(hmg : ae_strongly_measurable g $ map (λ t, x₀ - t) μ) :
convolution_exists_at f g x₀ L μ :=
begin
refine (h.const_mul ‖L‖).mono' (hmf.convolution_integrand_snd' L hmg)
(eventually_of_forall $ λ x, _),
rw [mul_apply', ← mul_assoc],
apply L.le_op_norm₂,
end
end
section left
variables [has_measurable_add₂ G] [has_measurable_neg G] [sigma_finite μ] [is_add_right_invariant μ]
lemma measure_theory.ae_strongly_measurable.convolution_integrand_snd
(hf : ae_strongly_measurable f μ) (hg : ae_strongly_measurable g μ)
(x : G) : ae_strongly_measurable (λ t, L (f t) (g (x - t))) μ :=
hf.convolution_integrand_snd' L $ hg.mono' $
(quasi_measure_preserving_sub_left_of_right_invariant μ x).absolutely_continuous
lemma measure_theory.ae_strongly_measurable.convolution_integrand_swap_snd
(hf : ae_strongly_measurable f μ) (hg : ae_strongly_measurable g μ)
(x : G) : ae_strongly_measurable (λ t, L (f (x - t)) (g t)) μ :=
(hf.mono' (quasi_measure_preserving_sub_left_of_right_invariant μ x).absolutely_continuous)
.convolution_integrand_swap_snd' L hg
/-- If `‖f‖ *[μ] ‖g‖` exists, then `f *[L, μ] g` exists. -/
lemma convolution_exists_at.of_norm {x₀ : G}
(h : convolution_exists_at (λ x, ‖f x‖) (λ x, ‖g x‖) x₀ (mul ℝ ℝ) μ)
(hmf : ae_strongly_measurable f μ)
(hmg : ae_strongly_measurable g μ) :
convolution_exists_at f g x₀ L μ :=
h.of_norm' L hmf $ hmg.mono'
(quasi_measure_preserving_sub_left_of_right_invariant μ x₀).absolutely_continuous
end left
section right
variables [has_measurable_add₂ G] [has_measurable_neg G]
[sigma_finite μ] [is_add_right_invariant μ] [sigma_finite ν]
lemma measure_theory.ae_strongly_measurable.convolution_integrand
(hf : ae_strongly_measurable f ν) (hg : ae_strongly_measurable g μ) :
ae_strongly_measurable (λ p : G × G, L (f p.2) (g (p.1 - p.2))) (μ.prod ν) :=
hf.convolution_integrand' L $ hg.mono'
(quasi_measure_preserving_sub_of_right_invariant μ ν).absolutely_continuous
lemma measure_theory.integrable.convolution_integrand (hf : integrable f ν) (hg : integrable g μ) :
integrable (λ p : G × G, L (f p.2) (g (p.1 - p.2))) (μ.prod ν) :=
begin
have h_meas : ae_strongly_measurable (λ (p : G × G), L (f p.2) (g (p.1 - p.2))) (μ.prod ν) :=
hf.ae_strongly_measurable.convolution_integrand L hg.ae_strongly_measurable,
have h2_meas : ae_strongly_measurable (λ (y : G), ∫ (x : G), ‖L (f y) (g (x - y))‖ ∂μ) ν :=
h_meas.prod_swap.norm.integral_prod_right',
simp_rw [integrable_prod_iff' h_meas],
refine ⟨eventually_of_forall (λ t, (L (f t)).integrable_comp (hg.comp_sub_right t)), _⟩,
refine integrable.mono' _ h2_meas (eventually_of_forall $
λ t, (_ : _ ≤ ‖L‖ * ‖f t‖ * ∫ x, ‖g (x - t)‖ ∂μ)),
{ simp_rw [integral_sub_right_eq_self (λ t, ‖ g t ‖)],
exact (hf.norm.const_mul _).mul_const _ },
{ simp_rw [← integral_mul_left],
rw [real.norm_of_nonneg],
{ exact integral_mono_of_nonneg (eventually_of_forall $ λ t, norm_nonneg _)
((hg.comp_sub_right t).norm.const_mul _) (eventually_of_forall $ λ t, L.le_op_norm₂ _ _) },
exact integral_nonneg (λ x, norm_nonneg _) }
end
lemma measure_theory.integrable.ae_convolution_exists (hf : integrable f ν) (hg : integrable g μ) :
∀ᵐ x ∂μ, convolution_exists_at f g x L ν :=
((integrable_prod_iff $ hf.ae_strongly_measurable.convolution_integrand L
hg.ae_strongly_measurable).mp $ hf.convolution_integrand L hg).1
end right
variables [topological_space G] [topological_add_group G] [borel_space G]
lemma has_compact_support.convolution_exists_at {x₀ : G}
(h : has_compact_support (λ t, L (f t) (g (x₀ - t)))) (hf : locally_integrable f μ)
(hg : continuous g) : convolution_exists_at f g x₀ L μ :=
begin
let u := (homeomorph.neg G).trans (homeomorph.add_right x₀),
let v := (homeomorph.neg G).trans (homeomorph.add_left x₀),
apply ((u.is_compact_preimage.mpr h).bdd_above_image hg.norm.continuous_on).convolution_exists_at'
L is_closed_closure.measurable_set subset_closure (hf.integrable_on_is_compact h),
have A : ae_strongly_measurable (g ∘ ⇑v)
(μ.restrict (tsupport (λ (t : G), (L (f t)) (g (x₀ - t))))),
{ apply (hg.comp v.continuous).continuous_on.ae_strongly_measurable_of_is_compact h,
exact (is_closed_tsupport _).measurable_set },
convert ((v.continuous.measurable.measure_preserving
(μ.restrict (tsupport (λ t, L (f t) (g (x₀ - t)))))).ae_strongly_measurable_comp_iff
v.to_measurable_equiv.measurable_embedding).1 A,
ext x,
simp only [homeomorph.neg, sub_eq_add_neg, coe_to_add_units, homeomorph.trans_apply,
equiv.neg_apply, equiv.to_fun_as_coe, homeomorph.homeomorph_mk_coe, equiv.coe_fn_mk,
homeomorph.coe_add_left],
end
lemma has_compact_support.convolution_exists_right
(hcg : has_compact_support g) (hf : locally_integrable f μ) (hg : continuous g) :
convolution_exists f g L μ :=
begin
intro x₀,
refine has_compact_support.convolution_exists_at L _ hf hg,
refine (hcg.comp_homeomorph (homeomorph.sub_left x₀)).mono _,
refine λ t, mt (λ ht : g (x₀ - t) = 0, _),
simp_rw [ht, (L _).map_zero]
end
lemma has_compact_support.convolution_exists_left_of_continuous_right
(hcf : has_compact_support f) (hf : locally_integrable f μ) (hg : continuous g) :
convolution_exists f g L μ :=
begin
intro x₀,
refine has_compact_support.convolution_exists_at L _ hf hg,
refine hcf.mono _,
refine λ t, mt (λ ht : f t = 0, _),
simp_rw [ht, L.map_zero₂]
end
end group
section comm_group
variables [add_comm_group G]
section measurable_group
variables [has_measurable_neg G] [is_add_left_invariant μ]
/-- A sufficient condition to prove that `f ⋆[L, μ] g` exists.
We assume that the integrand has compact support and `g` is bounded on this support (note that
both properties hold if `g` is continuous with compact support). We also require that `f` is
integrable on the support of the integrand, and that both functions are strongly measurable.
This is a variant of `bdd_above.convolution_exists_at'` in an abelian group with a left-invariant
measure. This allows us to state the boundedness and measurability of `g` in a more natural way. -/
lemma bdd_above.convolution_exists_at [has_measurable_add₂ G] [sigma_finite μ] {x₀ : G}
{s : set G} (hbg : bdd_above ((λ i, ‖g i‖) '' ((λ t, x₀ - t) ⁻¹' s)))
(hs : measurable_set s) (h2s : support (λ t, L (f t) (g (x₀ - t))) ⊆ s)
(hf : integrable_on f s μ) (hmg : ae_strongly_measurable g μ) :
convolution_exists_at f g x₀ L μ :=
begin
refine bdd_above.convolution_exists_at' L _ hs h2s hf _,
{ simp_rw [← sub_eq_neg_add, hbg] },
{ have : ae_strongly_measurable g (map (λ (t : G), x₀ - t) μ), from hmg.mono'
(quasi_measure_preserving_sub_left_of_right_invariant μ x₀).absolutely_continuous,
apply this.mono_measure,
exact map_mono_of_ae_measurable restrict_le_self
(measurable_const.sub measurable_id').ae_measurable }
end
variables {L} [has_measurable_add G] [is_neg_invariant μ]
lemma convolution_exists_at_flip :
convolution_exists_at g f x L.flip μ ↔ convolution_exists_at f g x L μ :=
by simp_rw [convolution_exists_at, ← integrable_comp_sub_left (λ t, L (f t) (g (x - t))) x,
sub_sub_cancel, flip_apply]
lemma convolution_exists_at.integrable_swap (h : convolution_exists_at f g x L μ) :
integrable (λ t, L (f (x - t)) (g t)) μ :=
by { convert h.comp_sub_left x, simp_rw [sub_sub_self] }
lemma convolution_exists_at_iff_integrable_swap :
convolution_exists_at f g x L μ ↔ integrable (λ t, L (f (x - t)) (g t)) μ :=
convolution_exists_at_flip.symm
end measurable_group
variables [topological_space G] [topological_add_group G] [borel_space G]
[is_add_left_invariant μ] [is_neg_invariant μ]
lemma has_compact_support.convolution_exists_left
(hcf : has_compact_support f) (hf : continuous f) (hg : locally_integrable g μ) :
convolution_exists f g L μ :=
λ x₀, convolution_exists_at_flip.mp $ hcf.convolution_exists_right L.flip hg hf x₀
lemma has_compact_support.convolution_exists_right_of_continuous_left
(hcg : has_compact_support g) (hf : continuous f) (hg : locally_integrable g μ) :
convolution_exists f g L μ :=
λ x₀, convolution_exists_at_flip.mp $
hcg.convolution_exists_left_of_continuous_right L.flip hg hf x₀
end comm_group
end convolution_exists
variables [normed_space ℝ F] [complete_space F]
/-- The convolution of two functions `f` and `g` with respect to a continuous bilinear map `L` and
measure `μ`. It is defined to be `(f ⋆[L, μ] g) x = ∫ t, L (f t) (g (x - t)) ∂μ`. -/
noncomputable def convolution [has_sub G] (f : G → E) (g : G → E') (L : E →L[𝕜] E' →L[𝕜] F)
(μ : measure G . volume_tac) : G → F :=
λ x, ∫ t, L (f t) (g (x - t)) ∂μ
localized "notation (name := convolution) f ` ⋆[`:67 L:67 `, ` μ:67 `] `:0 g:66 :=
convolution f g L μ" in convolution
localized "notation (name := convolution.volume) f ` ⋆[`:67 L:67 `]`:0 g:66 :=
convolution f g L measure_theory.measure_space.volume" in convolution
localized "notation (name := convolution.lsmul) f ` ⋆ `:67 g:66 :=
convolution f g (continuous_linear_map.lsmul ℝ ℝ) measure_theory.measure_space.volume"
in convolution
lemma convolution_def [has_sub G] : (f ⋆[L, μ] g) x = ∫ t, L (f t) (g (x - t)) ∂μ := rfl
/-- The definition of convolution where the bilinear operator is scalar multiplication.
Note: it often helps the elaborator to give the type of the convolution explicitly. -/
lemma convolution_lsmul [has_sub G] {f : G → 𝕜} {g : G → F} :
(f ⋆[lsmul 𝕜 𝕜, μ] g : G → F) x = ∫ t, f t • g (x - t) ∂μ := rfl
/-- The definition of convolution where the bilinear operator is multiplication. -/
lemma convolution_mul [has_sub G] [normed_space ℝ 𝕜] [complete_space 𝕜] {f : G → 𝕜} {g : G → 𝕜} :
(f ⋆[mul 𝕜 𝕜, μ] g) x = ∫ t, f t * g (x - t) ∂μ := rfl
section group
variables {L} [add_group G]
lemma smul_convolution [smul_comm_class ℝ 𝕜 F]
{y : 𝕜} : (y • f) ⋆[L, μ] g = y • (f ⋆[L, μ] g) :=
by { ext, simp only [pi.smul_apply, convolution_def, ← integral_smul, L.map_smul₂] }
lemma convolution_smul [smul_comm_class ℝ 𝕜 F]
{y : 𝕜} : f ⋆[L, μ] (y • g) = y • (f ⋆[L, μ] g) :=
by { ext, simp only [pi.smul_apply, convolution_def, ← integral_smul, (L _).map_smul] }
@[simp] lemma zero_convolution : 0 ⋆[L, μ] g = 0 :=
by { ext, simp_rw [convolution_def, pi.zero_apply, L.map_zero₂, integral_zero] }
@[simp] lemma convolution_zero : f ⋆[L, μ] 0 = 0 :=
by { ext, simp_rw [convolution_def, pi.zero_apply, (L _).map_zero, integral_zero] }
lemma convolution_exists_at.distrib_add {x : G} (hfg : convolution_exists_at f g x L μ)
(hfg' : convolution_exists_at f g' x L μ) :
(f ⋆[L, μ] (g + g')) x = (f ⋆[L, μ] g) x + (f ⋆[L, μ] g') x :=
by simp only [convolution_def, (L _).map_add, pi.add_apply, integral_add hfg hfg']
lemma convolution_exists.distrib_add (hfg : convolution_exists f g L μ)
(hfg' : convolution_exists f g' L μ) : f ⋆[L, μ] (g + g') = f ⋆[L, μ] g + f ⋆[L, μ] g' :=
by { ext, exact (hfg x).distrib_add (hfg' x) }
lemma convolution_exists_at.add_distrib {x : G} (hfg : convolution_exists_at f g x L μ)
(hfg' : convolution_exists_at f' g x L μ) :
((f + f') ⋆[L, μ] g) x = (f ⋆[L, μ] g) x + (f' ⋆[L, μ] g) x :=
by simp only [convolution_def, L.map_add₂, pi.add_apply, integral_add hfg hfg']
lemma convolution_exists.add_distrib (hfg : convolution_exists f g L μ)
(hfg' : convolution_exists f' g L μ) : (f + f') ⋆[L, μ] g = f ⋆[L, μ] g + f' ⋆[L, μ] g :=
by { ext, exact (hfg x).add_distrib (hfg' x) }
lemma convolution_mono_right {f g g' : G → ℝ}
(hfg : convolution_exists_at f g x (lsmul ℝ ℝ) μ)
(hfg' : convolution_exists_at f g' x (lsmul ℝ ℝ) μ)
(hf : ∀ x, 0 ≤ f x) (hg : ∀ x, g x ≤ g' x) :
(f ⋆[lsmul ℝ ℝ, μ] g) x ≤ (f ⋆[lsmul ℝ ℝ, μ] g') x :=
begin
apply integral_mono hfg hfg',
simp only [lsmul_apply, algebra.id.smul_eq_mul],
assume t,
apply mul_le_mul_of_nonneg_left (hg _) (hf _),
end
lemma convolution_mono_right_of_nonneg {f g g' : G → ℝ}
(hfg' : convolution_exists_at f g' x (lsmul ℝ ℝ) μ)
(hf : ∀ x, 0 ≤ f x) (hg : ∀ x, g x ≤ g' x) (hg' : ∀ x, 0 ≤ g' x) :
(f ⋆[lsmul ℝ ℝ, μ] g) x ≤ (f ⋆[lsmul ℝ ℝ, μ] g') x :=
begin
by_cases H : convolution_exists_at f g x (lsmul ℝ ℝ) μ,
{ exact convolution_mono_right H hfg' hf hg },
have : (f ⋆[lsmul ℝ ℝ, μ] g) x = 0 := integral_undef H,
rw this,
exact integral_nonneg (λ y, mul_nonneg (hf y) (hg' (x - y))),
end
variables (L)
lemma convolution_congr [has_measurable_add₂ G] [has_measurable_neg G]
[sigma_finite μ] [is_add_right_invariant μ] (h1 : f =ᵐ[μ] f') (h2 : g =ᵐ[μ] g') :
f ⋆[L, μ] g = f' ⋆[L, μ] g' :=
begin
ext x,
apply integral_congr_ae,
exact (h1.prod_mk $ h2.comp_tendsto
(quasi_measure_preserving_sub_left_of_right_invariant μ x).tendsto_ae).fun_comp ↿(λ x y, L x y)
end
lemma support_convolution_subset_swap : support (f ⋆[L, μ] g) ⊆ support g + support f :=
begin
intros x h2x,
by_contra hx,
apply h2x,
simp_rw [set.mem_add, not_exists, not_and_distrib, nmem_support] at hx,
rw [convolution_def],
convert integral_zero G F,
ext t,
rcases hx (x - t) t with h|h|h,
{ rw [h, (L _).map_zero] },
{ rw [h, L.map_zero₂] },
{ exact (h $ sub_add_cancel x t).elim }
end
section
variables [has_measurable_add₂ G] [has_measurable_neg G] [sigma_finite μ] [is_add_right_invariant μ]
lemma measure_theory.integrable.integrable_convolution (hf : integrable f μ) (hg : integrable g μ) :
integrable (f ⋆[L, μ] g) μ :=
(hf.convolution_integrand L hg).integral_prod_left
end
variables [topological_space G]
variables [topological_add_group G]
lemma has_compact_support.convolution [t2_space G] (hcf : has_compact_support f)
(hcg : has_compact_support g) : has_compact_support (f ⋆[L, μ] g) :=
is_compact_of_is_closed_subset (hcg.is_compact.add hcf) is_closed_closure $ closure_minimal
((support_convolution_subset_swap L).trans $ add_subset_add subset_closure subset_closure)
(hcg.is_compact.add hcf).is_closed
variables [borel_space G] [first_countable_topology G]
[topological_space P] [first_countable_topology P]
/-- The convolution `f * g` is continuous if `f` is locally integrable and `g` is continuous and
compactly supported. Version where `g` depends on an additional parameter in a subset `s` of
a parameter space `P` (and the compact support `k` is independent of the parameter in `s`),
not assuming `t2_space G`. -/
lemma continuous_on_convolution_right_with_param'
{g : P → G → E'} {s : set P} {k : set G} (hk : is_compact k) (h'k : is_closed k)
(hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0)
(hf : locally_integrable f μ) (hg : continuous_on (↿g) (s ×ˢ univ)) :
continuous_on (λ (q : P × G), (f ⋆[L, μ] g q.1) q.2) (s ×ˢ univ) :=
begin
assume q₀ hq₀,
replace hq₀ : q₀.1 ∈ s, by simpa only [mem_prod, mem_univ, and_true] using hq₀,
have A : ∀ p ∈ s, continuous (g p),
{ assume p hp,
apply hg.comp_continuous (continuous_const.prod_mk continuous_id') (λ x, _),
simpa only [prod_mk_mem_set_prod_eq, mem_univ, and_true] using hp },
have B : ∀ p ∈ s, tsupport (g p) ⊆ k :=
λ p hp, closure_minimal (support_subset_iff'.2 (λ z hz, hgs _ _ hp hz)) h'k,
/- We find a small neighborhood of `{q₀.1} × k` on which the function is uniformly bounded.
This follows from the continuity at all points of the compact set `k`. -/
obtain ⟨w, C, w_open, q₀w, Cnonneg, hw⟩ :
∃ w C, is_open w ∧ q₀.1 ∈ w ∧ 0 ≤ C ∧ ∀ p x, p ∈ w ∩ s → ‖g p x‖ ≤ C,
{ have A : is_compact ({q₀.1} ×ˢ k), from is_compact_singleton.prod hk,
obtain ⟨t, kt, t_open, ht⟩ :
∃ t, {q₀.1} ×ˢ k ⊆ t ∧ is_open t ∧ bounded (↿g '' (t ∩ s ×ˢ univ)),
{ apply exists_is_open_bounded_image_inter_of_is_compact_of_continuous_on A _ hg,
simp only [prod_subset_prod_iff, hq₀, singleton_subset_iff, subset_univ, and_self, true_or] },
obtain ⟨C, Cpos, hC⟩ : ∃ C, 0 < C ∧ (↿g) '' (t ∩ s ×ˢ univ) ⊆ closed_ball (0 : E') C,
from ht.subset_ball_lt 0 0,
obtain ⟨w, w_open, q₀w, hw⟩ : ∃ w, is_open w ∧ q₀.1 ∈ w ∧ w ×ˢ k ⊆ t,
{ obtain ⟨w, v, w_open, v_open, hw, hv, hvw⟩ :
∃ (w : set P) (v : set G), is_open w ∧ is_open v ∧ {q₀.fst} ⊆ w ∧ k ⊆ v ∧ w ×ˢ v ⊆ t,
from generalized_tube_lemma is_compact_singleton hk t_open kt,
exact ⟨w, w_open, singleton_subset_iff.1 hw,
subset.trans (set.prod_mono subset.rfl hv) hvw⟩ },
refine ⟨w, C, w_open, q₀w, Cpos.le, _⟩,
rintros p x ⟨hp, hps⟩,
by_cases hx : x ∈ k,
{ have H : (p, x) ∈ t,
{ apply hw,
simp only [prod_mk_mem_set_prod_eq, hp, hx, and_true], },
have H' : (p, x) ∈ (s ×ˢ univ : set (P × G)),
by simpa only [prod_mk_mem_set_prod_eq, mem_univ, and_true] using hps,
have : g p x ∈ closed_ball (0 : E') C, from hC (mem_image_of_mem _ ⟨H, H'⟩),
rwa mem_closed_ball_zero_iff at this },
{ have : g p x = 0, from hgs _ _ hps hx,
rw this,
simpa only [norm_zero] using Cpos.le } },
have I1 : ∀ᶠ (q : P × G) in 𝓝[s ×ˢ univ] q₀,
ae_strongly_measurable (λ (a : G), L (f a) (g q.1 (q.2 - a))) μ,
{ filter_upwards [self_mem_nhds_within],
rintros ⟨p, x⟩ ⟨hp, hx⟩,
refine (has_compact_support.convolution_exists_right L _ hf (A _ hp) _).1,
exact is_compact_of_is_closed_subset hk (is_closed_tsupport _) (B p hp) },
let K' := - k + {q₀.2},
have hK' : is_compact K' := hk.neg.add is_compact_singleton,
obtain ⟨U, U_open, K'U, hU⟩ : ∃ U, is_open U ∧ K' ⊆ U ∧ integrable_on f U μ,
from hf.integrable_on_nhds_is_compact hK',
let bound : G → ℝ := indicator U (λ a, ‖L‖ * ‖f a‖ * C),
have I2 : ∀ᶠ (q : P × G) in 𝓝[s ×ˢ univ] q₀, ∀ᵐ a ∂μ, ‖L (f a) (g q.1 (q.2 - a))‖ ≤ bound a,
{ obtain ⟨V, V_mem, hV⟩ : ∃ (V : set G) (H : V ∈ 𝓝 (0 : G)), K' + V ⊆ U,
from compact_open_separated_add_right hK' U_open K'U,
have : ((w ∩ s) ×ˢ ({q₀.2} + V) : set (P × G)) ∈ 𝓝[s ×ˢ univ] q₀,
{ conv_rhs { rw [← @prod.mk.eta _ _ q₀, nhds_within_prod_eq, nhds_within_univ] },
refine filter.prod_mem_prod _ (singleton_add_mem_nhds_of_nhds_zero q₀.2 V_mem),
exact mem_nhds_within_iff_exists_mem_nhds_inter.2 ⟨w, w_open.mem_nhds q₀w, subset.rfl⟩ },
filter_upwards [this],
rintros ⟨p, x⟩ hpx,
simp only [prod_mk_mem_set_prod_eq] at hpx,
apply eventually_of_forall (λ a, _),
apply convolution_integrand_bound_right_of_le_of_subset _ _ hpx.2 _,
{ assume x,
exact hw _ _ hpx.1 },
{ rw ← add_assoc,
apply subset.trans (add_subset_add_right (add_subset_add_right _)) hV,
rw neg_subset_neg,
exact B p hpx.1.2 } },
have I3 : integrable bound μ,
{ rw [integrable_indicator_iff U_open.measurable_set],
exact (hU.norm.const_mul _).mul_const _ },
have I4 : ∀ᵐ (a : G) ∂μ, continuous_within_at (λ (q : P × G), L (f a) (g q.1 (q.2 - a)))
(s ×ˢ univ) q₀,
{ apply eventually_of_forall (λ a, _),
suffices H : continuous_within_at (λ (q : P × G), (f a, g q.1 (q.2 - a))) (s ×ˢ univ) q₀,
from L.continuous₂.continuous_at.comp_continuous_within_at H,
apply continuous_within_at_const.prod,
change continuous_within_at (λ (q : P × G), ↿g (q.1, q.2 - a)) (s ×ˢ univ) q₀,
have : continuous_at (λ (q : P × G), (q.1, q.2 - a)) (q₀.1, q₀.2),
from (continuous_fst.prod_mk (continuous_snd.sub continuous_const)).continuous_at,
rw ← @prod.mk.eta _ _ q₀,
have h'q₀ : (q₀.1, q₀.2 - a) ∈ (s ×ˢ univ : set (P × G)) := ⟨hq₀, mem_univ _⟩,
refine continuous_within_at.comp (hg _ h'q₀) this.continuous_within_at _,
rintros ⟨q, x⟩ ⟨hq, hx⟩,
exact ⟨hq, mem_univ _⟩ },
exact continuous_within_at_of_dominated I1 I2 I3 I4,
end
/-- The convolution `f * g` is continuous if `f` is locally integrable and `g` is continuous and
compactly supported. Version where `g` depends on an additional parameter in a subset `s` of
a parameter space `P` (and the compact support `k` is independent of the parameter in `s`). -/
lemma continuous_on_convolution_right_with_param
[t2_space G] {g : P → G → E'} {s : set P} {k : set G} (hk : is_compact k)
(hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0)
(hf : locally_integrable f μ) (hg : continuous_on (↿g) (s ×ˢ univ)) :
continuous_on (λ (q : P × G), (f ⋆[L, μ] g q.1) q.2) (s ×ˢ univ) :=
continuous_on_convolution_right_with_param' L hk hk.is_closed hgs hf hg
/-- The convolution `f * g` is continuous if `f` is locally integrable and `g` is continuous and
compactly supported. Version where `g` depends on an additional parameter in an open subset `s` of
a parameter space `P` (and the compact support `k` is independent of the parameter in `s`),
given in terms of compositions with an additional continuous map.
Version not assuming `t2_space G`. -/
lemma continuous_on_convolution_right_with_param_comp'
{s : set P} {v : P → G} (hv : continuous_on v s)
{g : P → G → E'} {k : set G}
(hk : is_compact k) (h'k : is_closed k) (hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0)
(hf : locally_integrable f μ) (hg : continuous_on (↿g) (s ×ˢ univ)) :
continuous_on (λ x, (f ⋆[L, μ] g x) (v x)) s :=
begin
apply (continuous_on_convolution_right_with_param' L hk h'k hgs hf hg).comp
(continuous_on_id.prod hv),
assume x hx,
simp only [hx, prod_mk_mem_set_prod_eq, mem_univ, and_self, id.def],
end
/-- The convolution `f * g` is continuous if `f` is locally integrable and `g` is continuous and
compactly supported. Version where `g` depends on an additional parameter in an open subset `s` of
a parameter space `P` (and the compact support `k` is independent of the parameter in `s`),
given in terms of compositions with an additional continuous map. -/
lemma continuous_on_convolution_right_with_param_comp [t2_space G]
{s : set P} {v : P → G} (hv : continuous_on v s)
{g : P → G → E'} {k : set G}
(hk : is_compact k) (hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0)
(hf : locally_integrable f μ) (hg : continuous_on (↿g) (s ×ˢ univ)) :
continuous_on (λ x, (f ⋆[L, μ] g x) (v x)) s :=
continuous_on_convolution_right_with_param_comp' L hv hk hk.is_closed hgs hf hg
/-- The convolution is continuous if one function is locally integrable and the other has compact
support and is continuous. -/
lemma has_compact_support.continuous_convolution_right
(hcg : has_compact_support g) (hf : locally_integrable f μ)
(hg : continuous g) : continuous (f ⋆[L, μ] g) :=
begin
rw continuous_iff_continuous_on_univ,
let g' : G → G → E' := λ p q, g q,
have : continuous_on (↿g') (univ ×ˢ univ) := (hg.comp continuous_snd).continuous_on,
exact continuous_on_convolution_right_with_param_comp' L
(continuous_iff_continuous_on_univ.1 continuous_id) hcg (is_closed_tsupport _)
(λ p x hp hx, image_eq_zero_of_nmem_tsupport hx) hf this,
end
/-- The convolution is continuous if one function is integrable and the other is bounded and
continuous. -/
lemma bdd_above.continuous_convolution_right_of_integrable [second_countable_topology G]
(hbg : bdd_above (range (λ x, ‖g x‖))) (hf : integrable f μ) (hg : continuous g) :
continuous (f ⋆[L, μ] g) :=
begin
refine continuous_iff_continuous_at.mpr (λ x₀, _),
have : ∀ᶠ x in 𝓝 x₀, ∀ᵐ (t : G) ∂μ,
‖L (f t) (g (x - t))‖ ≤ ‖L‖ * ‖f t‖ * (⨆ i, ‖g i‖),
{ refine eventually_of_forall (λ x, eventually_of_forall $ λ t, _),
refine (L.le_op_norm₂ _ _).trans _,
exact mul_le_mul_of_nonneg_left (le_csupr hbg $ x - t)
(mul_nonneg (norm_nonneg _) (norm_nonneg _)) },
refine continuous_at_of_dominated _ this _ _,
{ exact eventually_of_forall
(λ x, hf.ae_strongly_measurable.convolution_integrand_snd' L hg.ae_strongly_measurable) },
{ exact (hf.norm.const_mul _).mul_const _ },
{ exact eventually_of_forall (λ t, (L.continuous₂.comp₂ continuous_const $
hg.comp $ continuous_id.sub $ by apply continuous_const).continuous_at) }
end
end group
section comm_group
variables [add_comm_group G]
lemma support_convolution_subset : support (f ⋆[L, μ] g) ⊆ support f + support g :=
(support_convolution_subset_swap L).trans (add_comm _ _).subset
variables [is_add_left_invariant μ] [is_neg_invariant μ]
section measurable
variables [has_measurable_neg G]
variables [has_measurable_add G]
variable (L)
/-- Commutativity of convolution -/
lemma convolution_flip : g ⋆[L.flip, μ] f = f ⋆[L, μ] g :=
begin
ext1 x,
simp_rw [convolution_def],
rw [← integral_sub_left_eq_self _ μ x],
simp_rw [sub_sub_self, flip_apply]
end
/-- The symmetric definition of convolution. -/
lemma convolution_eq_swap : (f ⋆[L, μ] g) x = ∫ t, L (f (x - t)) (g t) ∂μ :=
by { rw [← convolution_flip], refl }
/-- The symmetric definition of convolution where the bilinear operator is scalar multiplication. -/
lemma convolution_lsmul_swap {f : G → 𝕜} {g : G → F}:
(f ⋆[lsmul 𝕜 𝕜, μ] g : G → F) x = ∫ t, f (x - t) • g t ∂μ :=
convolution_eq_swap _
/-- The symmetric definition of convolution where the bilinear operator is multiplication. -/
lemma convolution_mul_swap [normed_space ℝ 𝕜] [complete_space 𝕜] {f : G → 𝕜} {g : G → 𝕜} :
(f ⋆[mul 𝕜 𝕜, μ] g) x = ∫ t, f (x - t) * g t ∂μ :=
convolution_eq_swap _
/-- The convolution of two even functions is also even. -/
lemma convolution_neg_of_neg_eq (h1 : ∀ᵐ x ∂μ, f (-x) = f x) (h2 : ∀ᵐ x ∂μ, g (-x) = g x) :
(f ⋆[L, μ] g) (-x) = (f ⋆[L, μ] g) x :=
calc ∫ (t : G), (L (f t)) (g (-x - t)) ∂μ
= ∫ (t : G), (L (f (-t))) (g (x + t)) ∂μ :
begin
apply integral_congr_ae,
filter_upwards [h1, (eventually_add_left_iff μ x).2 h2] with t ht h't,
simp_rw [ht, ← h't, neg_add'],
end
... = ∫ (t : G), (L (f t)) (g (x - t)) ∂μ :
by { rw ← integral_neg_eq_self, simp only [neg_neg, ← sub_eq_add_neg] }
end measurable
variables [topological_space G]
variables [topological_add_group G]
variables [borel_space G]
lemma has_compact_support.continuous_convolution_left [first_countable_topology G]
(hcf : has_compact_support f) (hf : continuous f) (hg : locally_integrable g μ) :
continuous (f ⋆[L, μ] g) :=
by { rw [← convolution_flip], exact hcf.continuous_convolution_right L.flip hg hf }
lemma bdd_above.continuous_convolution_left_of_integrable [second_countable_topology G]
(hbf : bdd_above (range (λ x, ‖f x‖))) (hf : continuous f) (hg : integrable g μ) :
continuous (f ⋆[L, μ] g) :=
by { rw [← convolution_flip], exact hbf.continuous_convolution_right_of_integrable L.flip hg hf }
end comm_group
section normed_add_comm_group
variables [seminormed_add_comm_group G]
/-- Compute `(f ⋆ g) x₀` if the support of the `f` is within `metric.ball 0 R`, and `g` is constant
on `metric.ball x₀ R`.
We can simplify the RHS further if we assume `f` is integrable, but also if `L = (•)` or more
generally if `L` has a `antilipschitz_with`-condition. -/
lemma convolution_eq_right' {x₀ : G} {R : ℝ}
(hf : support f ⊆ ball (0 : G) R)
(hg : ∀ x ∈ ball x₀ R, g x = g x₀) : (f ⋆[L, μ] g) x₀ = ∫ t, L (f t) (g x₀) ∂μ :=
begin
have h2 : ∀ t, L (f t) (g (x₀ - t)) = L (f t) (g x₀),
{ intro t, by_cases ht : t ∈ support f,
{ have h2t := hf ht,
rw [mem_ball_zero_iff] at h2t,
specialize hg (x₀ - t),
rw [sub_eq_add_neg, add_mem_ball_iff_norm, norm_neg, ← sub_eq_add_neg] at hg,
rw [hg h2t] },
{ rw [nmem_support] at ht,
simp_rw [ht, L.map_zero₂] } },
simp_rw [convolution_def, h2],
end
variables [borel_space G] [second_countable_topology G]
variables [is_add_left_invariant μ] [sigma_finite μ]
/-- Approximate `(f ⋆ g) x₀` if the support of the `f` is bounded within a ball, and `g` is near
`g x₀` on a ball with the same radius around `x₀`. See `dist_convolution_le` for a special case.
We can simplify the second argument of `dist` further if we add some extra type-classes on `E`
and `𝕜` or if `L` is scalar multiplication. -/
lemma dist_convolution_le' {x₀ : G} {R ε : ℝ} {z₀ : E'}
(hε : 0 ≤ ε)
(hif : integrable f μ)
(hf : support f ⊆ ball (0 : G) R)
(hmg : ae_strongly_measurable g μ)
(hg : ∀ x ∈ ball x₀ R, dist (g x) z₀ ≤ ε) :
dist ((f ⋆[L, μ] g : G → F) x₀) (∫ t, L (f t) z₀ ∂μ) ≤ ‖L‖ * ∫ x, ‖f x‖ ∂μ * ε :=
begin
have hfg : convolution_exists_at f g x₀ L μ,
{ refine bdd_above.convolution_exists_at L _ metric.is_open_ball.measurable_set
(subset_trans _ hf) hif.integrable_on hmg,
swap, { refine λ t, mt (λ ht : f t = 0, _), simp_rw [ht, L.map_zero₂] },
rw [bdd_above_def],
refine ⟨‖z₀‖ + ε, _⟩,
rintro _ ⟨x, hx, rfl⟩,
refine norm_le_norm_add_const_of_dist_le (hg x _),
rwa [mem_ball_iff_norm, norm_sub_rev, ← mem_ball_zero_iff] },
have h2 : ∀ t, dist (L (f t) (g (x₀ - t))) (L (f t) z₀) ≤ ‖L (f t)‖ * ε,
{ intro t, by_cases ht : t ∈ support f,
{ have h2t := hf ht,
rw [mem_ball_zero_iff] at h2t,
specialize hg (x₀ - t),
rw [sub_eq_add_neg, add_mem_ball_iff_norm, norm_neg, ← sub_eq_add_neg] at hg,
refine ((L (f t)).dist_le_op_norm _ _).trans _,
exact mul_le_mul_of_nonneg_left (hg h2t) (norm_nonneg _) },
{ rw [nmem_support] at ht,
simp_rw [ht, L.map_zero₂, L.map_zero, norm_zero, zero_mul, dist_self] } },
simp_rw [convolution_def],
simp_rw [dist_eq_norm] at h2 ⊢,
rw [← integral_sub hfg.integrable], swap, { exact (L.flip z₀).integrable_comp hif },
refine (norm_integral_le_of_norm_le ((L.integrable_comp hif).norm.mul_const ε)
(eventually_of_forall h2)).trans _,
rw [integral_mul_right],
refine mul_le_mul_of_nonneg_right _ hε,
have h3 : ∀ t, ‖L (f t)‖ ≤ ‖L‖ * ‖f t‖,
{ intros t,
exact L.le_op_norm (f t) },
refine (integral_mono (L.integrable_comp hif).norm (hif.norm.const_mul _) h3).trans_eq _,
rw [integral_mul_left]
end
variables [normed_space ℝ E] [normed_space ℝ E'] [complete_space E']
/-- Approximate `f ⋆ g` if the support of the `f` is bounded within a ball, and `g` is near `g x₀`
on a ball with the same radius around `x₀`.
This is a special case of `dist_convolution_le'` where `L` is `(•)`, `f` has integral 1 and `f` is
nonnegative. -/
lemma dist_convolution_le {f : G → ℝ} {x₀ : G} {R ε : ℝ} {z₀ : E'}
(hε : 0 ≤ ε)
(hf : support f ⊆ ball (0 : G) R)
(hnf : ∀ x, 0 ≤ f x)
(hintf : ∫ x, f x ∂μ = 1)
(hmg : ae_strongly_measurable g μ)
(hg : ∀ x ∈ ball x₀ R, dist (g x) z₀ ≤ ε) :
dist ((f ⋆[lsmul ℝ ℝ, μ] g : G → E') x₀) z₀ ≤ ε :=
begin
have hif : integrable f μ,
{ by_contra hif, exact zero_ne_one ((integral_undef hif).symm.trans hintf) },
convert (dist_convolution_le' _ hε hif hf hmg hg).trans _,
{ simp_rw [lsmul_apply, integral_smul_const, hintf, one_smul] },
{ simp_rw [real.norm_of_nonneg (hnf _), hintf, mul_one],
exact (mul_le_mul_of_nonneg_right op_norm_lsmul_le hε).trans_eq (one_mul ε) }
end
/-- `(φ i ⋆ g i) (k i)` tends to `z₀` as `i` tends to some filter `l` if
* `φ` is a sequence of nonnegative functions with integral `1` as `i` tends to `l`;
* The support of `φ` tends to small neighborhoods around `(0 : G)` as `i` tends to `l`;
* `g i` is `mu`-a.e. strongly measurable as `i` tends to `l`;
* `g i x` tends to `z₀` as `(i, x)` tends to `l ×ᶠ 𝓝 x₀`;
* `k i` tends to `x₀`.
See also `cont_diff_bump.convolution_tendsto_right`.
-/
lemma convolution_tendsto_right
{ι} {g : ι → G → E'} {l : filter ι} {x₀ : G} {z₀ : E'}
{φ : ι → G → ℝ} {k : ι → G}
(hnφ : ∀ᶠ i in l, ∀ x, 0 ≤ φ i x)
(hiφ : ∀ᶠ i in l, ∫ x, φ i x ∂μ = 1) -- todo: we could weaken this to "the integral tends to 1"
(hφ : tendsto (λ n, support (φ n)) l (𝓝 0).small_sets)
(hmg : ∀ᶠ i in l, ae_strongly_measurable (g i) μ)
(hcg : tendsto (uncurry g) (l ×ᶠ 𝓝 x₀) (𝓝 z₀))
(hk : tendsto k l (𝓝 x₀)) :
tendsto (λ i : ι, (φ i ⋆[lsmul ℝ ℝ, μ] g i : G → E') (k i)) l (𝓝 z₀) :=
begin
simp_rw [tendsto_small_sets_iff] at hφ,
rw [metric.tendsto_nhds] at hcg ⊢,
simp_rw [metric.eventually_prod_nhds_iff] at hcg,
intros ε hε,
have h2ε : 0 < ε / 3 := div_pos hε (by norm_num),
obtain ⟨p, hp, δ, hδ, hgδ⟩ := hcg _ h2ε,
dsimp only [uncurry] at hgδ,
have h2k := hk.eventually (ball_mem_nhds x₀ $ half_pos hδ),
have h2φ := (hφ (ball (0 : G) _) $ ball_mem_nhds _ (half_pos hδ)),
filter_upwards [hp, h2k, h2φ, hnφ, hiφ, hmg] with i hpi hki hφi hnφi hiφi hmgi,
have hgi : dist (g i (k i)) z₀ < ε / 3 := hgδ hpi (hki.trans $ half_lt_self hδ),
have h1 : ∀ x' ∈ ball (k i) (δ / 2), dist (g i x') (g i (k i)) ≤ ε / 3 + ε / 3,
{ intros x' hx',
refine (dist_triangle_right _ _ _).trans (add_le_add (hgδ hpi _).le hgi.le),
exact ((dist_triangle _ _ _).trans_lt (add_lt_add hx'.out hki)).trans_eq (add_halves δ) },
have := dist_convolution_le (add_pos h2ε h2ε).le hφi hnφi hiφi hmgi h1,
refine ((dist_triangle _ _ _).trans_lt (add_lt_add_of_le_of_lt this hgi)).trans_eq _,
field_simp, ring_nf
end
end normed_add_comm_group
namespace cont_diff_bump
variables {n : ℕ∞}
variables [normed_space ℝ E']
variables [normed_add_comm_group G] [normed_space ℝ G] [has_cont_diff_bump G]
variables [complete_space E']
variables {a : G} {φ : cont_diff_bump (0 : G)}
/-- If `φ` is a bump function, compute `(φ ⋆ g) x₀` if `g` is constant on `metric.ball x₀ φ.R`. -/
lemma convolution_eq_right {x₀ : G}
(hg : ∀ x ∈ ball x₀ φ.R, g x = g x₀) : (φ ⋆[lsmul ℝ ℝ, μ] g : G → E') x₀ = integral μ φ • g x₀ :=
by simp_rw [convolution_eq_right' _ φ.support_eq.subset hg, lsmul_apply, integral_smul_const]
variables [borel_space G]
variables [is_locally_finite_measure μ] [is_open_pos_measure μ]
variables [finite_dimensional ℝ G]
/-- If `φ` is a normed bump function, compute `φ ⋆ g` if `g` is constant on `metric.ball x₀ φ.R`. -/
lemma normed_convolution_eq_right {x₀ : G}
(hg : ∀ x ∈ ball x₀ φ.R, g x = g x₀) : (φ.normed μ ⋆[lsmul ℝ ℝ, μ] g : G → E') x₀ = g x₀ :=
by { simp_rw [convolution_eq_right' _ φ.support_normed_eq.subset hg, lsmul_apply],
exact integral_normed_smul φ μ (g x₀) }
variables [is_add_left_invariant μ]
/-- If `φ` is a normed bump function, approximate `(φ ⋆ g) x₀` if `g` is near `g x₀` on a ball with
radius `φ.R` around `x₀`. -/
lemma dist_normed_convolution_le {x₀ : G} {ε : ℝ}
(hmg : ae_strongly_measurable g μ)
(hg : ∀ x ∈ ball x₀ φ.R, dist (g x) (g x₀) ≤ ε) :
dist ((φ.normed μ ⋆[lsmul ℝ ℝ, μ] g : G → E') x₀) (g x₀) ≤ ε :=
dist_convolution_le (by simp_rw [← dist_self (g x₀), hg x₀ (mem_ball_self φ.R_pos)])
φ.support_normed_eq.subset φ.nonneg_normed φ.integral_normed hmg hg
/-- `(φ i ⋆ g i) (k i)` tends to `z₀` as `i` tends to some filter `l` if
* `φ` is a sequence of normed bump functions such that `(φ i).R` tends to `0` as `i` tends to `l`;
* `g i` is `mu`-a.e. strongly measurable as `i` tends to `l`;
* `g i x` tends to `z₀` as `(i, x)` tends to `l ×ᶠ 𝓝 x₀`;
* `k i` tends to `x₀`. -/
lemma convolution_tendsto_right {ι} {φ : ι → cont_diff_bump (0 : G)}
{g : ι → G → E'} {k : ι → G} {x₀ : G} {z₀ : E'} {l : filter ι}
(hφ : tendsto (λ i, (φ i).R) l (𝓝 0))
(hig : ∀ᶠ i in l, ae_strongly_measurable (g i) μ)
(hcg : tendsto (uncurry g) (l ×ᶠ 𝓝 x₀) (𝓝 z₀))
(hk : tendsto k l (𝓝 x₀)) :
tendsto (λ i, ((λ x, (φ i).normed μ x) ⋆[lsmul ℝ ℝ, μ] g i : G → E') (k i)) l (𝓝 z₀) :=
convolution_tendsto_right (eventually_of_forall $ λ i, (φ i).nonneg_normed)
(eventually_of_forall $ λ i, (φ i).integral_normed)
(tendsto_support_normed_small_sets hφ) hig hcg hk
/-- Special case of `cont_diff_bump.convolution_tendsto_right` where `g` is continuous,
and the limit is taken only in the first function. -/
lemma convolution_tendsto_right_of_continuous {ι} {φ : ι → cont_diff_bump (0 : G)}
{l : filter ι} (hφ : tendsto (λ i, (φ i).R) l (𝓝 0))
(hg : continuous g) (x₀ : G) :
tendsto (λ i, ((λ x, (φ i).normed μ x) ⋆[lsmul ℝ ℝ, μ] g : G → E') x₀) l (𝓝 (g x₀)) :=
convolution_tendsto_right hφ (eventually_of_forall $ λ _, hg.ae_strongly_measurable)
((hg.tendsto x₀).comp tendsto_snd) tendsto_const_nhds
end cont_diff_bump
end measurability
end nontrivially_normed_field
open_locale convolution
section is_R_or_C
variables [is_R_or_C 𝕜]
variables [normed_space 𝕜 E]
variables [normed_space 𝕜 E']
variables [normed_space 𝕜 E'']
variables [normed_space ℝ F] [normed_space 𝕜 F]
variables {n : ℕ∞}
variables [complete_space F]
variables [measurable_space G] {μ ν : measure G}
variables (L : E →L[𝕜] E' →L[𝕜] F)
section assoc
variables [normed_add_comm_group F'] [normed_space ℝ F'] [normed_space 𝕜 F'] [complete_space F']
variables [normed_add_comm_group F''] [normed_space ℝ F''] [normed_space 𝕜 F''] [complete_space F'']
variables {k : G → E''}
variables (L₂ : F →L[𝕜] E'' →L[𝕜] F')
variables (L₃ : E →L[𝕜] F'' →L[𝕜] F')
variables (L₄ : E' →L[𝕜] E'' →L[𝕜] F'')
variables [add_group G]
variables [sigma_finite μ] [sigma_finite ν] [is_add_right_invariant μ]
lemma integral_convolution
[has_measurable_add₂ G] [has_measurable_neg G]
[normed_space ℝ E] [normed_space ℝ E']
[complete_space E] [complete_space E']
(hf : integrable f ν) (hg : integrable g μ) :
∫ x, (f ⋆[L, ν] g) x ∂μ = L (∫ x, f x ∂ν) (∫ x, g x ∂μ) :=
begin
refine (integral_integral_swap (by apply hf.convolution_integrand L hg)).trans _,
simp_rw [integral_comp_comm _ (hg.comp_sub_right _), integral_sub_right_eq_self],
exact (L.flip (∫ x, g x ∂μ)).integral_comp_comm hf
end
variables [has_measurable_add₂ G] [is_add_right_invariant ν] [has_measurable_neg G]
/-- Convolution is associative. This has a weak but inconvenient integrability condition.
See also `convolution_assoc`. -/
lemma convolution_assoc' (hL : ∀ (x : E) (y : E') (z : E''), L₂ (L x y) z = L₃ x (L₄ y z))
{x₀ : G}
(hfg : ∀ᵐ y ∂μ, convolution_exists_at f g y L ν)
(hgk : ∀ᵐ x ∂ν, convolution_exists_at g k x L₄ μ)
(hi : integrable (uncurry (λ x y, (L₃ (f y)) ((L₄ (g (x - y))) (k (x₀ - x))))) (μ.prod ν)) :
((f ⋆[L, ν] g) ⋆[L₂, μ] k) x₀ = (f ⋆[L₃, ν] (g ⋆[L₄, μ] k)) x₀ :=
calc ((f ⋆[L, ν] g) ⋆[L₂, μ] k) x₀
= ∫ t, L₂ (∫ s, L (f s) (g (t - s)) ∂ν) (k (x₀ - t)) ∂μ : rfl
... = ∫ t, ∫ s, L₂ (L (f s) (g (t - s))) (k (x₀ - t)) ∂ν ∂μ :
integral_congr_ae (hfg.mono $ λ t ht, ((L₂.flip (k (x₀ - t))).integral_comp_comm ht).symm)
... = ∫ t, ∫ s, L₃ (f s) (L₄ (g (t - s)) (k (x₀ - t))) ∂ν ∂μ : by simp_rw hL
... = ∫ s, ∫ t, L₃ (f s) (L₄ (g (t - s)) (k (x₀ - t))) ∂μ ∂ν : by rw [integral_integral_swap hi]
... = ∫ s, ∫ u, L₃ (f s) (L₄ (g u) (k ((x₀ - s) - u))) ∂μ ∂ν : begin
congr', ext t,
rw [eq_comm, ← integral_sub_right_eq_self _ t],
{ simp_rw [sub_sub_sub_cancel_right] },
{ apply_instance },
end
... = ∫ s, L₃ (f s) (∫ u, L₄ (g u) (k ((x₀ - s) - u)) ∂μ) ∂ν : begin
refine integral_congr_ae _,
refine ((quasi_measure_preserving_sub_left_of_right_invariant ν x₀).ae hgk).mono (λ t ht, _),
exact (L₃ (f t)).integral_comp_comm ht,
end
... = (f ⋆[L₃, ν] (g ⋆[L₄, μ] k)) x₀ : rfl
/-- Convolution is associative. This requires that
* all maps are a.e. strongly measurable w.r.t one of the measures
* `f ⋆[L, ν] g` exists almost everywhere
* `‖g‖ ⋆[μ] ‖k‖` exists almost everywhere
* `‖f‖ ⋆[ν] (‖g‖ ⋆[μ] ‖k‖)` exists at `x₀` -/
lemma convolution_assoc (hL : ∀ (x : E) (y : E') (z : E''), L₂ (L x y) z = L₃ x (L₄ y z))
{x₀ : G}
(hf : ae_strongly_measurable f ν)
(hg : ae_strongly_measurable g μ)
(hk : ae_strongly_measurable k μ)
(hfg : ∀ᵐ y ∂μ, convolution_exists_at f g y L ν)
(hgk : ∀ᵐ x ∂ν, convolution_exists_at (λ x, ‖g x‖) (λ x, ‖k x‖) x (mul ℝ ℝ) μ)
(hfgk : convolution_exists_at (λ x, ‖f x‖) ((λ x, ‖g x‖) ⋆[mul ℝ ℝ, μ] (λ x, ‖k x‖))
x₀ (mul ℝ ℝ) ν) :
((f ⋆[L, ν] g) ⋆[L₂, μ] k) x₀ = (f ⋆[L₃, ν] (g ⋆[L₄, μ] k)) x₀ :=
begin
refine convolution_assoc' L L₂ L₃ L₄ hL hfg (hgk.mono $ λ x hx, hx.of_norm L₄ hg hk) _,
/- the following is similar to `integrable.convolution_integrand` -/
have h_meas : ae_strongly_measurable
(uncurry (λ x y, L₃ (f y) (L₄ (g x) (k (x₀ - y - x))))) (μ.prod ν),
{ refine L₃.ae_strongly_measurable_comp₂ hf.snd _,
refine L₄.ae_strongly_measurable_comp₂ hg.fst _,
refine (hk.mono' _).comp_measurable ((measurable_const.sub measurable_snd).sub measurable_fst),
refine quasi_measure_preserving.absolutely_continuous _,
refine quasi_measure_preserving.prod_of_left
((measurable_const.sub measurable_snd).sub measurable_fst) (eventually_of_forall $ λ y, _),
dsimp only,
exact quasi_measure_preserving_sub_left_of_right_invariant μ _ },
have h2_meas : ae_strongly_measurable (λ y, ∫ x, ‖L₃ (f y) (L₄ (g x) (k (x₀ - y - x)))‖ ∂μ) ν :=
h_meas.prod_swap.norm.integral_prod_right',
have h3 : map (λ z : G × G, (z.1 - z.2, z.2)) (μ.prod ν) = μ.prod ν :=
(measure_preserving_sub_prod μ ν).map_eq,
suffices : integrable (uncurry (λ x y, L₃ (f y) (L₄ (g x) (k (x₀ - y - x))))) (μ.prod ν),
{ rw [← h3] at this,
convert this.comp_measurable (measurable_sub.prod_mk measurable_snd),
ext ⟨x, y⟩,
simp_rw [uncurry, function.comp_apply, sub_sub_sub_cancel_right] },
simp_rw [integrable_prod_iff' h_meas],
refine ⟨((quasi_measure_preserving_sub_left_of_right_invariant ν x₀).ae hgk).mono
(λ t ht, (L₃ (f t)).integrable_comp $ ht.of_norm L₄ hg hk), _⟩,
refine (hfgk.const_mul (‖L₃‖ * ‖L₄‖)).mono' h2_meas
(((quasi_measure_preserving_sub_left_of_right_invariant ν x₀).ae hgk).mono $ λ t ht, _),
{ simp_rw [convolution_def, mul_apply', mul_mul_mul_comm ‖L₃‖ ‖L₄‖, ← integral_mul_left],
rw [real.norm_of_nonneg],
{ refine integral_mono_of_nonneg (eventually_of_forall $ λ t, norm_nonneg _)
((ht.const_mul _).const_mul _) (eventually_of_forall $ λ s, _),
refine (L₃.le_op_norm₂ _ _).trans _,
refine mul_le_mul_of_nonneg_left _ (mul_nonneg (norm_nonneg _) (norm_nonneg _)),
rw [← mul_assoc],
apply L₄.le_op_norm₂ },
exact integral_nonneg (λ x, norm_nonneg _) }
end
end assoc
variables [normed_add_comm_group G] [borel_space G]
lemma convolution_precompR_apply {g : G → E'' →L[𝕜] E'}
(hf : locally_integrable f μ) (hcg : has_compact_support g) (hg : continuous g)
(x₀ : G) (x : E'') : (f ⋆[L.precompR E'', μ] g) x₀ x = (f ⋆[L, μ] (λ a, g a x)) x₀ :=
begin
have := hcg.convolution_exists_right (L.precompR E'' : _) hf hg x₀,
simp_rw [convolution_def, continuous_linear_map.integral_apply this],
refl,
end
variables [normed_space 𝕜 G] [sigma_finite μ] [is_add_left_invariant μ]
/-- Compute the total derivative of `f ⋆ g` if `g` is `C^1` with compact support and `f` is locally
integrable. To write down the total derivative as a convolution, we use
`continuous_linear_map.precompR`. -/
lemma has_compact_support.has_fderiv_at_convolution_right
(hcg : has_compact_support g) (hf : locally_integrable f μ) (hg : cont_diff 𝕜 1 g) (x₀ : G) :
has_fderiv_at (f ⋆[L, μ] g) ((f ⋆[L.precompR G, μ] fderiv 𝕜 g) x₀) x₀ :=
begin
rcases hcg.eq_zero_or_finite_dimensional 𝕜 hg.continuous with rfl|fin_dim,
{ have : fderiv 𝕜 (0 : G → E') = 0, from fderiv_const (0 : E'),
simp only [this, convolution_zero, pi.zero_apply],
exact has_fderiv_at_const (0 : F) x₀ },
resetI,
haveI : proper_space G, from finite_dimensional.proper_is_R_or_C 𝕜 G,
set L' := L.precompR G,
have h1 : ∀ᶠ x in 𝓝 x₀, ae_strongly_measurable (λ t, L (f t) (g (x - t))) μ :=
eventually_of_forall
(hf.ae_strongly_measurable.convolution_integrand_snd L hg.continuous.ae_strongly_measurable),
have h2 : ∀ x, ae_strongly_measurable (λ t, L' (f t) (fderiv 𝕜 g (x - t))) μ,
{ exact hf.ae_strongly_measurable.convolution_integrand_snd L'
(hg.continuous_fderiv le_rfl).ae_strongly_measurable },
have h3 : ∀ x t, has_fderiv_at (λ x, g (x - t)) (fderiv 𝕜 g (x - t)) x,
{ intros x t,
simpa using (hg.differentiable le_rfl).differentiable_at.has_fderiv_at.comp x
((has_fderiv_at_id x).sub (has_fderiv_at_const t x)) },
let K' := - tsupport (fderiv 𝕜 g) + closed_ball x₀ 1,
have hK' : is_compact K' := (hcg.fderiv 𝕜).neg.add (is_compact_closed_ball x₀ 1),
refine has_fderiv_at_integral_of_dominated_of_fderiv_le
zero_lt_one h1 _ (h2 x₀) _ _ _,
{ exact K'.indicator (λ t, ‖L'‖ * ‖f t‖ * (⨆ x, ‖fderiv 𝕜 g x‖)) },
{ exact hcg.convolution_exists_right L hf hg.continuous x₀ },
{ refine eventually_of_forall (λ t x hx, _),
exact (hcg.fderiv 𝕜).convolution_integrand_bound_right L'
(hg.continuous_fderiv le_rfl) (ball_subset_closed_ball hx) },
{ rw [integrable_indicator_iff hK'.measurable_set],
exact ((hf.integrable_on_is_compact hK').norm.const_mul _).mul_const _ },
{ exact eventually_of_forall (λ t x hx, (L _).has_fderiv_at.comp x (h3 x t)) },
end
lemma has_compact_support.has_fderiv_at_convolution_left [is_neg_invariant μ]
(hcf : has_compact_support f) (hf : cont_diff 𝕜 1 f) (hg : locally_integrable g μ) (x₀ : G) :
has_fderiv_at (f ⋆[L, μ] g) ((fderiv 𝕜 f ⋆[L.precompL G, μ] g) x₀) x₀ :=
begin
simp only [← convolution_flip] {single_pass := tt},
exact hcf.has_fderiv_at_convolution_right L.flip hg hf x₀,
end
end is_R_or_C
section real
/-! The one-variable case -/
variables [is_R_or_C 𝕜]
variables [normed_space 𝕜 E]
variables [normed_space 𝕜 E']
variables [normed_space ℝ F] [normed_space 𝕜 F]
variables {f₀ : 𝕜 → E} {g₀ : 𝕜 → E'}
variables {n : ℕ∞}
variables (L : E →L[𝕜] E' →L[𝕜] F)
variables [complete_space F]
variables {μ : measure 𝕜}
variables [is_add_left_invariant μ] [sigma_finite μ]
lemma has_compact_support.has_deriv_at_convolution_right
(hf : locally_integrable f₀ μ) (hcg : has_compact_support g₀) (hg : cont_diff 𝕜 1 g₀)
(x₀ : 𝕜) :
has_deriv_at (f₀ ⋆[L, μ] g₀) ((f₀ ⋆[L, μ] deriv g₀) x₀) x₀ :=
begin
convert (hcg.has_fderiv_at_convolution_right L hf hg x₀).has_deriv_at,
rw [convolution_precompR_apply L hf (hcg.fderiv 𝕜) (hg.continuous_fderiv le_rfl)],
refl,
end
lemma has_compact_support.has_deriv_at_convolution_left [is_neg_invariant μ]
(hcf : has_compact_support f₀) (hf : cont_diff 𝕜 1 f₀)
(hg : locally_integrable g₀ μ) (x₀ : 𝕜) :
has_deriv_at (f₀ ⋆[L, μ] g₀) ((deriv f₀ ⋆[L, μ] g₀) x₀) x₀ :=
begin
simp only [← convolution_flip] {single_pass := tt},
exact hcf.has_deriv_at_convolution_right L.flip hg hf x₀,
end
end real
section with_param
variables [is_R_or_C 𝕜] [normed_space 𝕜 E] [normed_space 𝕜 E'] [normed_space 𝕜 E'']
[normed_space ℝ F] [normed_space 𝕜 F] [complete_space F] [measurable_space G]
[normed_add_comm_group G] [borel_space G] [normed_space 𝕜 G]
[normed_add_comm_group P] [normed_space 𝕜 P]
{μ : measure G} (L : E →L[𝕜] E' →L[𝕜] F)
/-- The derivative of the convolution `f * g` is given by `f * Dg`, when `f` is locally integrable
and `g` is `C^1` and compactly supported. Version where `g` depends on an additional parameter in an
open subset `s` of a parameter space `P` (and the compact support `k` is independent of the
parameter in `s`). -/
lemma has_fderiv_at_convolution_right_with_param
{g : P → G → E'} {s : set P} {k : set G} (hs : is_open s) (hk : is_compact k)
(hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0)
(hf : locally_integrable f μ) (hg : cont_diff_on 𝕜 1 ↿g (s ×ˢ univ))
(q₀ : P × G) (hq₀ : q₀.1 ∈ s) :
has_fderiv_at (λ (q : P × G), (f ⋆[L, μ] g q.1) q.2)
((f ⋆[L.precompR (P × G), μ] (λ (x : G), fderiv 𝕜 ↿g (q₀.1, x))) q₀.2) q₀ :=
begin
let g' := fderiv 𝕜 ↿g,
have A : ∀ p ∈ s, continuous (g p),
{ assume p hp,
apply hg.continuous_on.comp_continuous (continuous_const.prod_mk continuous_id') (λ x, _),
simpa only [prod_mk_mem_set_prod_eq, mem_univ, and_true] using hp },
have A' : ∀ (q : P × G), q.1 ∈ s → s ×ˢ univ ∈ 𝓝 q,
{ assume q hq,
apply (hs.prod is_open_univ).mem_nhds,
simpa only [mem_prod, mem_univ, and_true] using hq },
/- The derivative of `g` vanishes away from `k`. -/
have g'_zero : ∀ p x, p ∈ s → x ∉ k → g' (p, x) = 0,
{ assume p x hp hx,
refine (has_fderiv_at_zero_of_eventually_const 0 _).fderiv,
have M2 : kᶜ ∈ 𝓝 x, from is_open.mem_nhds hk.is_closed.is_open_compl hx,
have M1 : s ∈ 𝓝 p, from hs.mem_nhds hp,
rw nhds_prod_eq,
filter_upwards [prod_mem_prod M1 M2],
rintros ⟨p, y⟩ ⟨hp, hy⟩,
exact hgs p y hp hy },
/- We find a small neighborhood of `{q₀.1} × k` on which the derivative is uniformly bounded. This
follows from the continuity at all points of the compact set `k`. -/
obtain ⟨ε, C, εpos, Cnonneg, h₀ε, hε⟩ :
∃ ε C, 0 < ε ∧ 0 ≤ C ∧ ball q₀.1 ε ⊆ s ∧ ∀ p x, ‖p - q₀.1‖ < ε → ‖g' (p, x)‖ ≤ C,
{ have A : is_compact ({q₀.1} ×ˢ k), from is_compact_singleton.prod hk,
obtain ⟨t, kt, t_open, ht⟩ : ∃ t, {q₀.1} ×ˢ k ⊆ t ∧ is_open t ∧ bounded (g' '' t),
{ have B : continuous_on g' (s ×ˢ univ),
from hg.continuous_on_fderiv_of_open (hs.prod is_open_univ) le_rfl,
apply exists_is_open_bounded_image_of_is_compact_of_continuous_on A
(hs.prod is_open_univ) _ B,
simp only [prod_subset_prod_iff, hq₀, singleton_subset_iff, subset_univ, and_self, true_or] },
obtain ⟨ε, εpos, hε, h'ε⟩ :
∃ (ε : ℝ), 0 < ε ∧ thickening ε ({q₀.fst} ×ˢ k) ⊆ t ∧ ball q₀.1 ε ⊆ s,
{ obtain ⟨ε, εpos, hε⟩ : ∃ (ε : ℝ), 0 < ε ∧ thickening ε ({q₀.fst} ×ˢ k) ⊆ t,
from A.exists_thickening_subset_open t_open kt,
obtain ⟨δ, δpos, hδ⟩ : ∃ (δ : ℝ) (H : 0 < δ), ball q₀.1 δ ⊆ s,
from metric.is_open_iff.1 hs _ hq₀,
refine ⟨min ε δ, lt_min εpos δpos, _, _⟩,
{ exact subset.trans (thickening_mono (min_le_left _ _) _) hε },
{ exact subset.trans (ball_subset_ball (min_le_right _ _)) hδ } },
obtain ⟨C, Cpos, hC⟩ : ∃ C, 0 < C ∧ g' '' t ⊆ closed_ball 0 C, from ht.subset_ball_lt 0 0,
refine ⟨ε, C, εpos, Cpos.le, h'ε, λ p x hp, _⟩,
have hps : p ∈ s, from h'ε (mem_ball_iff_norm.2 hp),
by_cases hx : x ∈ k,
{ have H : (p, x) ∈ t,
{ apply hε,
refine mem_thickening_iff.2 ⟨(q₀.1, x), _, _⟩,
{ simp only [hx, singleton_prod, mem_image, prod.mk.inj_iff, eq_self_iff_true, true_and,
exists_eq_right] },
{ rw ← dist_eq_norm at hp,
simpa only [prod.dist_eq, εpos, dist_self, max_lt_iff, and_true] using hp } },
have : g' (p, x) ∈ closed_ball (0 : P × G →L[𝕜] E') C, from hC (mem_image_of_mem _ H),
rwa mem_closed_ball_zero_iff at this },
{ have : g' (p, x) = 0, from g'_zero _ _ hps hx,
rw this,
simpa only [norm_zero] using Cpos.le } },
/- Now, we wish to apply a theorem on differentiation of integrals. For this, we need to check
trivial measurability or integrability assumptions (in `I1`, `I2`, `I3`), as well as a uniform
integrability assumption over the derivative (in `I4` and `I5`) and pointwise differentiability
in `I6`. -/
have I1 : ∀ᶠ (x : P × G) in 𝓝 q₀,
ae_strongly_measurable (λ (a : G), L (f a) (g x.1 (x.2 - a))) μ,
{ filter_upwards [A' q₀ hq₀],
rintros ⟨p, x⟩ ⟨hp, hx⟩,
refine (has_compact_support.convolution_exists_right L _ hf (A _ hp) _).1,
apply is_compact_of_is_closed_subset hk (is_closed_tsupport _),
exact closure_minimal (support_subset_iff'.2 (λ z hz, hgs _ _ hp hz)) hk.is_closed, },
have I2 : integrable (λ (a : G), L (f a) (g q₀.1 (q₀.2 - a))) μ,
{ have M : has_compact_support (g q₀.1),
from has_compact_support.intro hk (λ x hx, hgs q₀.1 x hq₀ hx),
apply M.convolution_exists_right L hf (A q₀.1 hq₀) q₀.2 },
have I3 : ae_strongly_measurable (λ (a : G), (L (f a)).comp (g' (q₀.fst, q₀.snd - a))) μ,
{ have T : has_compact_support (λ y, g' (q₀.1, y)),
from has_compact_support.intro hk (λ x hx, g'_zero q₀.1 x hq₀ hx),
apply (has_compact_support.convolution_exists_right (L.precompR (P × G) : _) T hf _ q₀.2).1,
have : continuous_on g' (s ×ˢ univ),
from hg.continuous_on_fderiv_of_open (hs.prod is_open_univ) le_rfl,
apply this.comp_continuous (continuous_const.prod_mk continuous_id'),
assume x,
simpa only [prod_mk_mem_set_prod_eq, mem_univ, and_true] using hq₀ },
set K' := - k + {q₀.2} with K'_def,
have hK' : is_compact K' := hk.neg.add is_compact_singleton,
obtain ⟨U, U_open, K'U, hU⟩ : ∃ U, is_open U ∧ K' ⊆ U ∧ integrable_on f U μ,
from hf.integrable_on_nhds_is_compact hK',
obtain ⟨δ, δpos, δε, hδ⟩ : ∃ δ, (0 : ℝ) < δ ∧ δ ≤ ε ∧ K' + ball 0 δ ⊆ U,
{ obtain ⟨V, V_mem, hV⟩ : ∃ (V : set G) (V_mem : V ∈ 𝓝 (0 : G)), K' + V ⊆ U,
from compact_open_separated_add_right hK' U_open K'U,
rcases metric.mem_nhds_iff.1 V_mem with ⟨δ, δpos, hδ⟩,
refine ⟨min δ ε, lt_min δpos εpos, min_le_right _ _, _⟩,
exact (add_subset_add_left ((ball_subset_ball (min_le_left _ _)).trans hδ)).trans hV },
let bound : G → ℝ := indicator U (λ a, ‖L.precompR (P × G)‖ * ‖f a‖ * C),
have I4 : ∀ᵐ (a : G) ∂μ, ∀ (x : P × G), dist x q₀ < δ →
‖L.precompR (P × G) (f a) (g' (x.fst, x.snd - a))‖ ≤ bound a,
{ apply eventually_of_forall,
assume a x hx,
rw [prod.dist_eq, dist_eq_norm, dist_eq_norm] at hx,
have : -tsupport (λ a, g' (x.1, a)) + ball q₀.2 δ ⊆ U,
{ apply subset.trans _ hδ,
rw [K'_def, add_assoc],
apply add_subset_add,
{ rw neg_subset_neg,
apply closure_minimal (support_subset_iff'.2 (λ z hz, _)) hk.is_closed,
apply g'_zero x.1 z (h₀ε _) hz,
rw mem_ball_iff_norm,
exact ((le_max_left _ _).trans_lt hx).trans_le δε },
{ simp only [add_ball, thickening_singleton, zero_vadd] } },
apply convolution_integrand_bound_right_of_le_of_subset _ _ _ this,
{ assume y,
exact hε _ _ (((le_max_left _ _).trans_lt hx).trans_le δε) },
{ rw mem_ball_iff_norm,
exact (le_max_right _ _).trans_lt hx } },
have I5 : integrable bound μ,
{ rw [integrable_indicator_iff U_open.measurable_set],
exact (hU.norm.const_mul _).mul_const _ },
have I6 : ∀ᵐ (a : G) ∂μ, ∀ (x : P × G), dist x q₀ < δ →
has_fderiv_at (λ (x : P × G), L (f a) (g x.1 (x.2 - a)))
((L (f a)).comp (g' (x.fst, x.snd - a))) x,
{ apply eventually_of_forall,
assume a x hx,
apply (L _).has_fderiv_at.comp x,
have N : s ×ˢ univ ∈ 𝓝 (x.1, x.2 - a),
{ apply A',
apply h₀ε,
rw prod.dist_eq at hx,
exact lt_of_lt_of_le (lt_of_le_of_lt (le_max_left _ _) hx) δε },
have Z := ((hg.differentiable_on le_rfl).differentiable_at N).has_fderiv_at,
have Z' : has_fderiv_at (λ (x : P × G), (x.1, x.2 - a)) (continuous_linear_map.id 𝕜 (P × G)) x,
{ have : (λ (x : P × G), (x.1, x.2 - a)) = id - (λ x, (0, a)),
{ ext x; simp only [pi.sub_apply, id.def, prod.fst_sub, sub_zero, prod.snd_sub] },
simp_rw [this],
exact (has_fderiv_at_id x).sub_const (0, a) },
exact Z.comp x Z' },
exact has_fderiv_at_integral_of_dominated_of_fderiv_le δpos I1 I2 I3 I4 I5 I6,
end
/-- The convolution `f * g` is `C^n` when `f` is locally integrable and `g` is `C^n` and compactly
supported. Version where `g` depends on an additional parameter in an open subset `s` of a
parameter space `P` (and the compact support `k` is independent of the parameter in `s`).
In this version, all the types belong to the same universe (to get an induction working in the
proof). Use instead `cont_diff_on_convolution_right_with_param`, which removes this restriction. -/
lemma cont_diff_on_convolution_right_with_param_aux
{G : Type uP} {E' : Type uP} {F : Type uP} {P : Type uP}
[normed_add_comm_group E'] [normed_add_comm_group F]
[normed_space 𝕜 E'] [normed_space ℝ F] [normed_space 𝕜 F] [complete_space F]
[measurable_space G] {μ : measure G} [normed_add_comm_group G] [borel_space G] [normed_space 𝕜 G]
[normed_add_comm_group P] [normed_space 𝕜 P]
{f : G → E} {n : ℕ∞} (L : E →L[𝕜] E' →L[𝕜] F)
{g : P → G → E'}
{s : set P} {k : set G} (hs : is_open s) (hk : is_compact k)
(hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0)
(hf : locally_integrable f μ) (hg : cont_diff_on 𝕜 n ↿g (s ×ˢ univ)) :
cont_diff_on 𝕜 n (λ (q : P × G), (f ⋆[L, μ] g q.1) q.2) (s ×ˢ univ) :=
begin
/- We have a formula for the derivation of `f * g`, which is of the same form, thanks to
`has_fderiv_at_convolution_right_with_param`. Therefore, we can prove the result by induction on
`n` (but for this we need the spaces at the different steps of the induction to live in the same
universe, which is why we make the assumption in the lemma that all the relevant spaces
come from the same universe). -/
unfreezingI { induction n using enat.nat_induction with n ih ih generalizing g E' F },
{ rw [cont_diff_on_zero] at hg ⊢,
exact continuous_on_convolution_right_with_param L hk hgs hf hg },
{ let f' : P → G → (P × G →L[𝕜] F) := λ p a,
(f ⋆[L.precompR (P × G), μ] (λ (x : G), fderiv 𝕜 (uncurry g) (p, x))) a,
have A : ∀ (q₀ : P × G), q₀.1 ∈ s →
has_fderiv_at (λ (q : P × G), (f ⋆[L, μ] g q.1) q.2) (f' q₀.1 q₀.2) q₀,
from has_fderiv_at_convolution_right_with_param L hs hk hgs hf hg.one_of_succ,
rw cont_diff_on_succ_iff_fderiv_of_open (hs.prod (@is_open_univ G _)) at ⊢ hg,
split,
{ rintros ⟨p, x⟩ ⟨hp, hx⟩,
exact (A (p, x) hp).differentiable_at.differentiable_within_at, },
{ suffices H : cont_diff_on 𝕜 n ↿f' (s ×ˢ univ),
{ apply H.congr,
rintros ⟨p, x⟩ ⟨hp, hx⟩,
exact (A (p, x) hp).fderiv },
have B : ∀ (p : P) (x : G), p ∈ s → x ∉ k → fderiv 𝕜 (uncurry g) (p, x) = 0,
{ assume p x hp hx,
apply (has_fderiv_at_zero_of_eventually_const (0 : E') _).fderiv,
have M2 : kᶜ ∈ 𝓝 x, from is_open.mem_nhds hk.is_closed.is_open_compl hx,
have M1 : s ∈ 𝓝 p, from hs.mem_nhds hp,
rw nhds_prod_eq,
filter_upwards [prod_mem_prod M1 M2],
rintros ⟨p, y⟩ ⟨hp, hy⟩,
exact hgs p y hp hy },
apply ih (L.precompR (P × G) : _) B,
convert hg.2,
apply funext,
rintros ⟨p, x⟩,
refl } },
{ rw [cont_diff_on_top] at hg ⊢,
assume n,
exact ih n L hgs (hg n) }
end
/-- The convolution `f * g` is `C^n` when `f` is locally integrable and `g` is `C^n` and compactly
supported. Version where `g` depends on an additional parameter in an open subset `s` of a
parameter space `P` (and the compact support `k` is independent of the parameter in `s`). -/
lemma cont_diff_on_convolution_right_with_param
{f : G → E} {n : ℕ∞} (L : E →L[𝕜] E' →L[𝕜] F) {g : P → G → E'}
{s : set P} {k : set G} (hs : is_open s) (hk : is_compact k)
(hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0)
(hf : locally_integrable f μ) (hg : cont_diff_on 𝕜 n ↿g (s ×ˢ univ)) :
cont_diff_on 𝕜 n (λ (q : P × G), (f ⋆[L, μ] g q.1) q.2) (s ×ˢ univ) :=
begin
/- The result is known when all the universes are the same, from
`cont_diff_on_convolution_right_with_param_aux`. We reduce to this situation by pushing
everything through `ulift` continuous linear equivalences. -/
let eG : Type (max uG uE' uF uP) := ulift G,
borelize eG,
let eE' : Type (max uE' uG uF uP) := ulift E',
let eF : Type (max uF uG uE' uP) := ulift F,
let eP : Type (max uP uG uE' uF) := ulift P,
have isoG : eG ≃L[𝕜] G := continuous_linear_equiv.ulift,
have isoE' : eE' ≃L[𝕜] E' := continuous_linear_equiv.ulift,
have isoF : eF ≃L[𝕜] F := continuous_linear_equiv.ulift,
have isoP : eP ≃L[𝕜] P := continuous_linear_equiv.ulift,
let ef := f ∘ isoG,
let eμ : measure eG := measure.map isoG.symm μ,
let eg : eP → eG → eE' := λ ep ex, isoE'.symm (g (isoP ep) (isoG ex)),
let eL := continuous_linear_map.comp
((continuous_linear_equiv.arrow_congr isoE' isoF).symm : (E' →L[𝕜] F) →L[𝕜] eE' →L[𝕜] eF) L,
let R := (λ (q : eP × eG), (ef ⋆[eL, eμ] eg q.1) q.2),
have R_contdiff : cont_diff_on 𝕜 n R ((isoP ⁻¹' s) ×ˢ univ),
{ have hek : is_compact (isoG ⁻¹' k),
from isoG.to_homeomorph.closed_embedding.is_compact_preimage hk,
have hes : is_open (isoP ⁻¹' s), from isoP.continuous.is_open_preimage _ hs,
refine cont_diff_on_convolution_right_with_param_aux eL hes hek _ _ _,
{ assume p x hp hx,
simp only [comp_app, continuous_linear_equiv.prod_apply, linear_isometry_equiv.coe_coe,
continuous_linear_equiv.map_eq_zero_iff],
exact hgs _ _ hp hx },
{ apply (locally_integrable_map_homeomorph isoG.symm.to_homeomorph).2,
convert hf,
ext1 x,
simp only [ef, continuous_linear_equiv.coe_to_homeomorph, comp_app,
continuous_linear_equiv.apply_symm_apply], },
{ apply isoE'.symm.cont_diff.comp_cont_diff_on,
apply hg.comp ((isoP.prod isoG).cont_diff).cont_diff_on,
rintros ⟨p, x⟩ ⟨hp, hx⟩,
simpa only [mem_preimage, continuous_linear_equiv.prod_apply, prod_mk_mem_set_prod_eq,
mem_univ, and_true] using hp } },
have A : cont_diff_on 𝕜 n (isoF ∘ R ∘ (isoP.prod isoG).symm) (s ×ˢ univ),
{ apply isoF.cont_diff.comp_cont_diff_on,
apply R_contdiff.comp (continuous_linear_equiv.cont_diff _).cont_diff_on,
rintros ⟨p, x⟩ ⟨hp, hx⟩,
simpa only [mem_preimage, mem_prod, mem_univ, and_true, continuous_linear_equiv.prod_symm,
continuous_linear_equiv.prod_apply, continuous_linear_equiv.apply_symm_apply] using hp },
have : isoF ∘ R ∘ (isoP.prod isoG).symm = (λ (q : P × G), (f ⋆[L, μ] g q.1) q.2),
{ apply funext,
rintros ⟨p, x⟩,
simp only [R, linear_isometry_equiv.coe_coe, comp_app, continuous_linear_equiv.prod_symm,
continuous_linear_equiv.prod_apply],
simp only [convolution, eL, coe_comp', continuous_linear_equiv.coe_coe, comp_app, eμ],
rw [closed_embedding.integral_map, ← isoF.integral_comp_comm],
swap, { exact isoG.symm.to_homeomorph.closed_embedding },
congr' 1,
ext1 a,
simp only [ef, eg, comp_app, continuous_linear_equiv.apply_symm_apply, coe_comp',
continuous_linear_equiv.prod_apply, continuous_linear_equiv.map_sub,
continuous_linear_equiv.arrow_congr, continuous_linear_equiv.arrow_congrSL_symm_apply,
continuous_linear_equiv.coe_coe, comp_app, continuous_linear_equiv.apply_symm_apply] },
simp_rw [this] at A,
exact A,
end
/-- The convolution `f * g` is `C^n` when `f` is locally integrable and `g` is `C^n` and compactly
supported. Version where `g` depends on an additional parameter in an open subset `s` of a
parameter space `P` (and the compact support `k` is independent of the parameter in `s`),
given in terms of composition with an additional smooth function. -/
lemma cont_diff_on_convolution_right_with_param_comp
{n : ℕ∞} (L : E →L[𝕜] E' →L[𝕜] F)
{s : set P} {v : P → G} (hv : cont_diff_on 𝕜 n v s)
{f : G → E} {g : P → G → E'} {k : set G} (hs : is_open s) (hk : is_compact k)
(hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0)
(hf : locally_integrable f μ) (hg : cont_diff_on 𝕜 n ↿g (s ×ˢ univ)) :
cont_diff_on 𝕜 n (λ x, (f ⋆[L, μ] g x) (v x)) s :=
begin
apply (cont_diff_on_convolution_right_with_param L hs hk hgs hf hg).comp
(cont_diff_on_id.prod hv),
assume x hx,
simp only [hx, mem_preimage, prod_mk_mem_set_prod_eq, mem_univ, and_self, id.def],
end
/-- The convolution `g * f` is `C^n` when `f` is locally integrable and `g` is `C^n` and compactly
supported. Version where `g` depends on an additional parameter in an open subset `s` of a
parameter space `P` (and the compact support `k` is independent of the parameter in `s`). -/
lemma cont_diff_on_convolution_left_with_param [μ.is_add_left_invariant] [μ.is_neg_invariant]
(L : E' →L[𝕜] E →L[𝕜] F) {f : G → E} {n : ℕ∞}
{g : P → G → E'} {s : set P} {k : set G} (hs : is_open s) (hk : is_compact k)
(hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0)
(hf : locally_integrable f μ) (hg : cont_diff_on 𝕜 n ↿g (s ×ˢ univ)) :
cont_diff_on 𝕜 n (λ (q : P × G), (g q.1 ⋆[L, μ] f) q.2) (s ×ˢ univ) :=
by simpa only [convolution_flip]
using cont_diff_on_convolution_right_with_param L.flip hs hk hgs hf hg
/-- The convolution `g * f` is `C^n` when `f` is locally integrable and `g` is `C^n` and compactly
supported. Version where `g` depends on an additional parameter in an open subset `s` of a
parameter space `P` (and the compact support `k` is independent of the parameter in `s`),
given in terms of composition with additional smooth functions. -/
lemma cont_diff_on_convolution_left_with_param_comp [μ.is_add_left_invariant] [μ.is_neg_invariant]
(L : E' →L[𝕜] E →L[𝕜] F) {s : set P} {n : ℕ∞} {v : P → G} (hv : cont_diff_on 𝕜 n v s)
{f : G → E} {g : P → G → E'} {k : set G} (hs : is_open s) (hk : is_compact k)
(hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0)
(hf : locally_integrable f μ) (hg : cont_diff_on 𝕜 n ↿g (s ×ˢ univ)) :
cont_diff_on 𝕜 n (λ x, (g x ⋆[L, μ] f) (v x)) s :=
begin
apply (cont_diff_on_convolution_left_with_param L hs hk hgs hf hg).comp (cont_diff_on_id.prod hv),
assume x hx,
simp only [hx, mem_preimage, prod_mk_mem_set_prod_eq, mem_univ, and_self, id.def],
end
lemma has_compact_support.cont_diff_convolution_right {n : ℕ∞}
(hcg : has_compact_support g) (hf : locally_integrable f μ) (hg : cont_diff 𝕜 n g) :
cont_diff 𝕜 n (f ⋆[L, μ] g) :=
begin
rcases exists_compact_iff_has_compact_support.2 hcg with ⟨k, hk, h'k⟩,
rw ← cont_diff_on_univ,
exact cont_diff_on_convolution_right_with_param_comp L cont_diff_on_id is_open_univ hk
(λ p x hp hx, h'k x hx) hf (hg.comp cont_diff_snd).cont_diff_on,
end
lemma has_compact_support.cont_diff_convolution_left [μ.is_add_left_invariant] [μ.is_neg_invariant]
{n : ℕ∞} (hcf : has_compact_support f) (hf : cont_diff 𝕜 n f) (hg : locally_integrable g μ) :
cont_diff 𝕜 n (f ⋆[L, μ] g) :=
by { rw [← convolution_flip], exact hcf.cont_diff_convolution_right L.flip hg hf }
end with_param
section nonneg
variables [normed_space ℝ E] [normed_space ℝ E'] [normed_space ℝ F] [complete_space F]
/-- The forward convolution of two functions `f` and `g` on `ℝ`, with respect to a continuous
bilinear map `L` and measure `ν`. It is defined to be the function mapping `x` to
`∫ t in 0..x, L (f t) (g (x - t)) ∂ν` if `0 < x`, and 0 otherwise. -/
noncomputable def pos_convolution
(f : ℝ → E) (g : ℝ → E') (L : E →L[ℝ] E' →L[ℝ] F) (ν : measure ℝ . volume_tac) : ℝ → F :=
indicator (Ioi (0:ℝ)) (λ x, ∫ t in 0..x, L (f t) (g (x - t)) ∂ν)
lemma pos_convolution_eq_convolution_indicator
(f : ℝ → E) (g : ℝ → E') (L : E →L[ℝ] E' →L[ℝ] F) (ν : measure ℝ . volume_tac) [has_no_atoms ν] :
pos_convolution f g L ν = convolution (indicator (Ioi 0) f) (indicator (Ioi 0) g) L ν :=
begin
ext1 x,
rw [convolution, pos_convolution, indicator],
split_ifs,
{ rw [interval_integral.integral_of_le (le_of_lt h),
integral_Ioc_eq_integral_Ioo,
←integral_indicator (measurable_set_Ioo : measurable_set (Ioo 0 x))],
congr' 1 with t : 1,
have : (t ≤ 0) ∨ (t ∈ Ioo 0 x) ∨ (x ≤ t),
{ rcases le_or_lt t 0 with h | h,
{ exact or.inl h },
{ rcases lt_or_le t x with h' | h',
exacts [or.inr (or.inl ⟨h, h'⟩), or.inr (or.inr h')] } },
rcases this with ht|ht|ht,
{ rw [indicator_of_not_mem (not_mem_Ioo_of_le ht), indicator_of_not_mem (not_mem_Ioi.mpr ht),
continuous_linear_map.map_zero, continuous_linear_map.zero_apply] },
{ rw [indicator_of_mem ht, indicator_of_mem (mem_Ioi.mpr ht.1),
indicator_of_mem (mem_Ioi.mpr $ sub_pos.mpr ht.2)] },
{ rw [indicator_of_not_mem (not_mem_Ioo_of_ge ht),
indicator_of_not_mem (not_mem_Ioi.mpr (sub_nonpos_of_le ht)),
continuous_linear_map.map_zero] } },
{ convert (integral_zero ℝ F).symm,
ext1 t,
by_cases ht : 0 < t,
{ rw [indicator_of_not_mem (_ : x - t ∉ Ioi 0), continuous_linear_map.map_zero],
rw not_mem_Ioi at h ⊢,
exact sub_nonpos.mpr (h.trans ht.le) },
{ rw [indicator_of_not_mem (mem_Ioi.not.mpr ht), continuous_linear_map.map_zero,
continuous_linear_map.zero_apply] } }
end
lemma integrable_pos_convolution {f : ℝ → E} {g : ℝ → E'} {μ ν : measure ℝ}
[sigma_finite μ] [sigma_finite ν] [is_add_right_invariant μ] [has_no_atoms ν]
(hf : integrable_on f (Ioi 0) ν) (hg : integrable_on g (Ioi 0) μ) (L : E →L[ℝ] E' →L[ℝ] F) :
integrable (pos_convolution f g L ν) μ :=
begin
rw ←integrable_indicator_iff (measurable_set_Ioi : measurable_set (Ioi (0:ℝ))) at hf hg,
rw pos_convolution_eq_convolution_indicator f g L ν,
exact (hf.convolution_integrand L hg).integral_prod_left,
end
/-- The integral over `Ioi 0` of a forward convolution of two functions is equal to the product
of their integrals over this set. (Compare `integral_convolution` for the two-sided convolution.) -/
lemma integral_pos_convolution [complete_space E] [complete_space E'] {μ ν : measure ℝ}
[sigma_finite μ] [sigma_finite ν] [is_add_right_invariant μ] [has_no_atoms ν]
{f : ℝ → E} {g : ℝ → E'} (hf : integrable_on f (Ioi 0) ν)
(hg : integrable_on g (Ioi 0) μ) (L : E →L[ℝ] E' →L[ℝ] F) :
∫ x:ℝ in Ioi 0, (∫ t:ℝ in 0..x, L (f t) (g (x - t)) ∂ν) ∂μ =
L (∫ x:ℝ in Ioi 0, f x ∂ν) (∫ x:ℝ in Ioi 0, g x ∂μ) :=
begin
rw ←integrable_indicator_iff (measurable_set_Ioi : measurable_set (Ioi (0:ℝ))) at hf hg,
simp_rw ←integral_indicator measurable_set_Ioi,
convert integral_convolution L hf hg using 2,
apply pos_convolution_eq_convolution_indicator,
end
end nonneg
|
8726e46168fcf7da8892b04f7e6a232af7f747c1 | 624f6f2ae8b3b1adc5f8f67a365c51d5126be45a | /src/Init/Lean/Meta/Tactic/Cases.lean | 42508752d796c1dd852f6ddd4121f9b58e3238c6 | [
"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 | 11,306 | lean | /-
Copyright (c) 2020 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
prelude
import Init.Lean.Meta.AppBuilder
import Init.Lean.Meta.Tactic.Induction
import Init.Lean.Meta.Tactic.Injection
import Init.Lean.Meta.Tactic.Assert
import Init.Lean.Meta.Tactic.Subst
namespace Lean
namespace Meta
private def mkEqAndProof (lhs rhs : Expr) : MetaM (Expr × Expr) := do
lhsType ← inferType lhs;
rhsType ← inferType rhs;
u ← getLevel lhsType;
condM (isDefEq lhsType rhsType)
(pure (mkApp3 (mkConst `Eq [u]) lhsType lhs rhs, mkApp2 (mkConst `Eq.refl [u]) lhsType lhs))
(pure (mkApp4 (mkConst `HEq [u]) lhsType lhs rhsType rhs, mkApp2 (mkConst `HEq.refl [u]) lhsType lhs))
private partial def withNewIndexEqsAux {α} (indices newIndices : Array Expr) (k : Array Expr → Array Expr → MetaM α) : Nat → Array Expr → Array Expr → MetaM α
| i, newEqs, newRefls =>
if h : i < indices.size then do
let index := indices.get! i;
let newIndex := newIndices.get! i;
(newEqType, newRefl) ← mkEqAndProof index newIndex;
withLocalDecl `h newEqType BinderInfo.default $ fun newEq => do
withNewIndexEqsAux (i+1) (newEqs.push newEq) (newRefls.push newRefl)
else
k newEqs newRefls
private def withNewIndexEqs {α} (indices newIndices : Array Expr) (k : Array Expr → Array Expr → MetaM α) : MetaM α :=
withNewIndexEqsAux indices newIndices k 0 #[] #[]
structure GeneralizeIndicesSubgoal :=
(mvarId : MVarId)
(indicesFVarIds : Array FVarId)
(fvarId : FVarId)
(numEqs : Nat)
/--
Given a metavariable `mvarId` representing the
```
Ctx, h : I A j, D |- T
```
where `fvarId` is `h`s id, and the type `I A j` is an inductive datatype where `A` are parameters,
and `j` the indices. Generate the goal
```
Ctx, h : I A j, D, j' : J, h' : I A j' |- j == j' -> h == h' -> T
```
Remark: `(j == j' -> h == h')` is a "telescopic" equality.
Remark: `j` is sequence of terms, and `j'` a sequence of free variables.
The result contains the fields
- `mvarId`: the new goal
- `indicesFVarIds`: `j'` ids
- `fvarId`: `h'` id
- `numEqs`: number of equations in the target -/
def generalizeIndices (mvarId : MVarId) (fvarId : FVarId) : MetaM GeneralizeIndicesSubgoal := do
withMVarContext mvarId $ do
lctx ← getLCtx;
localInsts ← getLocalInstances;
env ← getEnv;
checkNotAssigned mvarId `generalizeIndices;
fvarDecl ← getLocalDecl fvarId;
type ← whnf fvarDecl.type;
type.withApp $ fun f args => matchConst env f (fun _ => throwTacticEx `generalizeIndices mvarId "inductive type expected") $
fun cinfo _ => match cinfo with
| ConstantInfo.inductInfo val => do
unless (val.nindices > 0) $ throwTacticEx `generalizeIndices mvarId "indexed inductive type expected";
unless (args.size == val.nindices + val.nparams) $ throwTacticEx `generalizeIndices mvarId "ill-formed inductive datatype";
let indices := args.extract (args.size - val.nindices) args.size;
let IA := mkAppN f (args.extract 0 val.nparams); -- `I A`
IAType ← inferType IA;
forallTelescopeReducing IAType $ fun newIndices _ => do
let newType := mkAppN IA newIndices;
withLocalDecl fvarDecl.userName newType BinderInfo.default $ fun h' =>
withNewIndexEqs indices newIndices $ fun newEqs newRefls => do
(newEqType, newRefl) ← mkEqAndProof fvarDecl.toExpr h';
let newRefls := newRefls.push newRefl;
withLocalDecl `h newEqType BinderInfo.default $ fun newEq => do
let newEqs := newEqs.push newEq;
/- auxType `forall (j' : J) (h' : I A j'), j == j' -> h == h' -> target -/
target ← getMVarType mvarId;
tag ← getMVarTag mvarId;
auxType ← mkForall newEqs target;
auxType ← mkForall #[h'] auxType;
auxType ← mkForall newIndices auxType;
newMVar ← mkFreshExprMVarAt lctx localInsts auxType tag MetavarKind.syntheticOpaque;
/- assign mvarId := newMVar indices h refls -/
assignExprMVar mvarId (mkAppN (mkApp (mkAppN newMVar indices) fvarDecl.toExpr) newRefls);
(indicesFVarIds, newMVarId) ← introN newMVar.mvarId! newIndices.size [] false;
(fvarId, newMVarId) ← intro1 newMVarId false;
pure {
mvarId := newMVarId,
indicesFVarIds := indicesFVarIds,
fvarId := fvarId,
numEqs := newEqs.size
}
| _ => throwTacticEx `generalizeIndices mvarId "inductive type expected"
structure CasesSubgoal extends InductionSubgoal :=
(ctorName : Name)
namespace Cases
structure Context :=
(inductiveVal : InductiveVal)
(casesOnVal : DefinitionVal)
(nminors : Nat := inductiveVal.ctors.length)
(majorDecl : LocalDecl)
(majorTypeFn : Expr)
(majorTypeArgs : Array Expr)
(majorTypeIndices : Array Expr := majorTypeArgs.extract (majorTypeArgs.size - inductiveVal.nindices) majorTypeArgs.size)
private def mkCasesContext? (majorFVarId : FVarId) : MetaM (Option Context) := do
env ← getEnv;
if !env.contains `Eq || !env.contains `HEq then pure none
else do
majorDecl ← getLocalDecl majorFVarId;
majorType ← whnf majorDecl.type;
majorType.withApp $ fun f args => matchConst env f (fun _ => pure none) $ fun cinfo _ => do
match cinfo with
| ConstantInfo.inductInfo ival =>
if args.size != ival.nindices + ival.nparams then pure none
else match env.find? (mkNameStr ival.name "casesOn") with
| ConstantInfo.defnInfo cval => pure $ some {
inductiveVal := ival,
casesOnVal := cval,
majorDecl := majorDecl,
majorTypeFn := f,
majorTypeArgs := args
}
| _ => pure none
| _ => pure none
/-
We say the major premise has independent indices IF
1- its type is *not* an indexed inductive family, OR
2- its type is an indexed inductive family, but all indices are distinct free variables, and
all local declarations different from the major and its indices do not depend on the indices.
-/
private def hasIndepIndices (ctx : Context) : MetaM Bool :=
if ctx.majorTypeIndices.isEmpty then
pure true
else if ctx.majorTypeIndices.any $ fun idx => !idx.isFVar then
/- One of the indices is not a free variable. -/
pure false
else if ctx.majorTypeIndices.size.any $ fun i => i.any $ fun j => ctx.majorTypeIndices.get! i == ctx.majorTypeIndices.get! j then
/- An index ocurrs more than once -/
pure false
else do
lctx ← getLCtx;
mctx ← getMCtx;
pure $ lctx.all $ fun decl =>
decl.fvarId == ctx.majorDecl.fvarId || -- decl is the major
ctx.majorTypeIndices.any (fun index => decl.fvarId == index.fvarId!) || -- decl is one of the indices
mctx.findLocalDeclDependsOn decl (fun fvarId => ctx.majorTypeIndices.all $ fun idx => idx.fvarId! != fvarId) -- or does not depend on any index
private def elimAuxIndices (s₁ : GeneralizeIndicesSubgoal) (s₂ : Array InductionSubgoal) : MetaM (Array InductionSubgoal) :=
let indicesFVarIds := s₁.indicesFVarIds;
s₂.mapM $ fun s => do
indicesFVarIds.foldlM
(fun s indexFVarId =>
let indexFVarId' := s.subst.get indexFVarId;
(do mvarId ← clear s.mvarId indexFVarId'; pure { mvarId := mvarId, subst := s.subst.erase indexFVarId, .. s })
<|>
(pure s))
s
private def toCasesSubgoals (s : Array InductionSubgoal) (ctorNames : Array Name) : Array CasesSubgoal :=
s.mapIdx $ fun i s => { ctorName := ctorNames.get! i, toInductionSubgoal := s }
private partial def unifyEqsAux : Nat → CasesSubgoal → MetaM (Option CasesSubgoal)
| 0, s => do
trace! `Meta.cases ("unifyEqs " ++ MessageData.ofGoal s.mvarId);
pure (some s)
| n+1, s => do
trace! `Meta.cases ("unifyEqs [" ++ toString (n+1) ++ "] " ++ MessageData.ofGoal s.mvarId);
(eqFVarId, mvarId) ← intro1 s.mvarId;
withMVarContext mvarId $ do
eqDecl ← getLocalDecl eqFVarId;
match eqDecl.type.heq? with
| some (α, a, β, b) => do
prf ← mkEqOfHEq (mkFVar eqFVarId);
aEqb ← mkEq a b;
mvarId ← assert mvarId eqDecl.userName aEqb prf;
mvarId ← clear mvarId eqFVarId;
unifyEqsAux (n+1) { mvarId := mvarId, .. s }
| none => match eqDecl.type.eq? with
| some (α, a, b) =>
let skip : Unit → MetaM (Option CasesSubgoal) := fun _ => do {
mvarId ← clear mvarId eqFVarId;
unifyEqsAux n { mvarId := mvarId, .. s }
};
let substEq (symm : Bool) : MetaM (Option CasesSubgoal) := do {
(newSubst, mvarId) ← substCore mvarId eqFVarId false true;
unifyEqsAux n {
mvarId := mvarId,
subst := newSubst.compose s.subst,
fields := s.fields.map $ fun fvarId => newSubst.get fvarId,
.. s
}
};
let inj : Unit → MetaM (Option CasesSubgoal) := fun _ => do {
r ← injectionCore mvarId eqFVarId;
match r with
| InjectionResultCore.solved => pure none -- this alternative has been solved
| InjectionResultCore.subgoal mvarId numEqs => unifyEqsAux (n+numEqs) { mvarId := mvarId, .. s }
};
condM (isDefEq a b) (skip ()) $ do
a ← whnf a;
b ← whnf b;
match a, b with
| Expr.fvar aFVarId _, Expr.fvar bFVarId _ => do aDecl ← getLocalDecl aFVarId; bDecl ← getLocalDecl bFVarId; substEq (aDecl.index < bDecl.index)
| Expr.fvar _ _, _ => substEq false
| _, Expr.fvar _ _ => substEq true
| _, _ => inj ()
| none => throwTacticEx `cases mvarId "equality expected"
private def unifyEqs (numEqs : Nat) (subgoals : Array CasesSubgoal) : MetaM (Array CasesSubgoal) :=
subgoals.foldlM
(fun subgoals s => do
s? ← unifyEqsAux numEqs s;
match s? with
| none => pure $ subgoals
| some s => pure $ subgoals.push s)
#[]
def cases (mvarId : MVarId) (majorFVarId : FVarId) (givenNames : Array (List Name) := #[]) (useUnusedNames := false) : MetaM (Array CasesSubgoal) :=
withMVarContext mvarId $ do
checkNotAssigned mvarId `cases;
context? ← mkCasesContext? majorFVarId;
match context? with
| none => throwTacticEx `cases mvarId "not applicable to the given hypothesis"
| some ctx =>
let ctors := ctx.inductiveVal.ctors.toArray;
let casesOn := mkCasesOnFor ctx.inductiveVal.name;
condM (hasIndepIndices ctx)
(do
s ← induction mvarId majorFVarId casesOn givenNames useUnusedNames;
pure $ toCasesSubgoals s ctors)
(do
s₁ ← generalizeIndices mvarId majorFVarId;
trace! `Meta.cases ("after generalizeIndices" ++ Format.line ++ MessageData.ofGoal s₁.mvarId);
s₂ ← induction s₁.mvarId s₁.fvarId casesOn givenNames useUnusedNames;
s₂ ← elimAuxIndices s₁ s₂;
let s₂ := toCasesSubgoals s₂ ctors;
unifyEqs s₁.numEqs s₂)
end Cases
def cases (mvarId : MVarId) (majorFVarId : FVarId) (givenNames : Array (List Name) := #[]) (useUnusedNames := false) : MetaM (Array CasesSubgoal) :=
Cases.cases mvarId majorFVarId givenNames useUnusedNames
end Meta
end Lean
|
15b9e8b3183938bf13e234fc566171ff31ac6400 | b7f22e51856f4989b970961f794f1c435f9b8f78 | /library/data/finset/to_set.lean | d4472a5e6f954693a1cf0f36f2fb7476748d2a0d | [
"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 | 4,031 | lean | /-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Jeremy Avigad
Interactions between finset and set.
-/
import data.finset.comb data.set.function
open nat eq.ops set
namespace finset
variable {A : Type}
variable [deceq : decidable_eq A]
definition to_set [coercion] (s : finset A) : set A := λx, x ∈ s
abbreviation ts := @to_set A
variables (s t : finset A) (x y : A)
theorem mem_eq_mem_to_set : x ∈ s = (x ∈ ts s) := rfl
definition to_set.inj {s₁ s₂ : finset A} : to_set s₁ = to_set s₂ → s₁ = s₂ :=
λ h, ext (λ a, iff.of_eq (calc
(a ∈ s₁) = (a ∈ ts s₁) : mem_eq_mem_to_set
... = (a ∈ ts s₂) : h
... = (a ∈ s₂) : mem_eq_mem_to_set))
/- operations -/
theorem mem_to_set_empty : (x ∈ ts ∅) = (x ∈ ∅) := rfl
theorem to_set_empty : ts ∅ = (@set.empty A) := rfl
theorem mem_to_set_univ [h : fintype A] : (x ∈ ts univ) = (x ∈ set.univ) :=
propext (iff.intro (assume H, trivial) (assume H, !mem_univ))
theorem to_set_univ [h : fintype A] : ts univ = (set.univ : set A) := funext (λ x, !mem_to_set_univ)
theorem mem_to_set_upto (x n : ℕ) : x ∈ ts (upto n) = (x ∈ {a | a < n}) := !mem_upto_eq
theorem to_set_upto (n : ℕ) : ts (upto n) = {a | a < n} := funext (λ x, !mem_to_set_upto)
include deceq
theorem mem_to_set_insert : x ∈ insert y s = (x ∈ set.insert y s) := !mem_insert_eq
theorem to_set_insert : insert y s = set.insert y s := funext (λ x, !mem_to_set_insert)
theorem mem_to_set_union : x ∈ s ∪ t = (x ∈ ts s ∪ ts t) := !mem_union_eq
theorem to_set_union : ts (s ∪ t) = ts s ∪ ts t := funext (λ x, !mem_to_set_union)
theorem mem_to_set_inter : x ∈ s ∩ t = (x ∈ ts s ∩ ts t) := !mem_inter_eq
theorem to_set_inter : ts (s ∩ t) = ts s ∩ ts t := funext (λ x, !mem_to_set_inter)
theorem mem_to_set_diff : x ∈ s \ t = (x ∈ ts s \ ts t) := !mem_diff_eq
theorem to_set_diff : ts (s \ t) = ts s \ ts t := funext (λ x, !mem_to_set_diff)
theorem mem_to_set_sep (p : A → Prop) [h : decidable_pred p] : x ∈ sep p s = (x ∈ set.sep p s) :=
!finset.mem_sep_eq
theorem to_set_sep (p : A → Prop) [h : decidable_pred p] : sep p s = set.sep p s :=
funext (λ x, !mem_to_set_sep)
theorem mem_to_set_image {B : Type} [h : decidable_eq B] (f : A → B) {s : finset A} {y : B} :
y ∈ image f s = (y ∈ set.image f s) := !mem_image_eq
theorem to_set_image {B : Type} [h : decidable_eq B] (f : A → B) (s : finset A) :
image f s = set.image f s := funext (λ x, !mem_to_set_image)
/- relations -/
definition decidable_mem_to_set [instance] (x : A) (s : finset A) : decidable (x ∈ ts s) :=
decidable_of_decidable_of_eq _ !mem_eq_mem_to_set
theorem eq_of_to_set_eq_to_set {s t : finset A} (H : to_set s = to_set t) : s = t :=
ext (take x, by rewrite [mem_eq_mem_to_set s, H])
theorem eq_eq_to_set_eq : (s = t) = (ts s = ts t) :=
propext (iff.intro (assume H, H ▸ rfl) !eq_of_to_set_eq_to_set)
definition decidable_to_set_eq [instance] (s t : finset A) : decidable (ts s = ts t) :=
decidable_of_decidable_of_eq _ !eq_eq_to_set_eq
theorem subset_eq_to_set_subset (s t : finset A) : (s ⊆ t) = (ts s ⊆ ts t) :=
propext (iff.intro
(assume H, take x xs, mem_of_subset_of_mem H xs)
(assume H, subset_of_forall H))
definition decidable_to_set_subset (s t : finset A) : decidable (ts s ⊆ ts t) :=
decidable_of_decidable_of_eq _ !subset_eq_to_set_subset
/- bounded quantifiers -/
definition decidable_bounded_forall (s : finset A) (p : A → Prop) [h : decidable_pred p] :
decidable (∀₀ x ∈ ts s, p x) :=
decidable_of_decidable_of_iff _ !all_iff_forall
definition decidable_bounded_exists (s : finset A) (p : A → Prop) [h : decidable_pred p] :
decidable (∃₀ x ∈ ts s, p x) :=
decidable_of_decidable_of_iff _ !any_iff_exists
/- properties -/
theorem inj_on_to_set {B : Type} [h : decidable_eq B] (f : A → B) (s : finset A) :
inj_on f s = inj_on f (ts s) :=
rfl
end finset
|
b83f5a3cf96ae5085e380ab1df852c9b0af4322e | 1a61aba1b67cddccce19532a9596efe44be4285f | /library/data/finset/comb.lean | 335800a95bb22b532806626124f7d096e1452691 | [
"Apache-2.0"
] | permissive | eigengrau/lean | 07986a0f2548688c13ba36231f6cdbee82abf4c6 | f8a773be1112015e2d232661ce616d23f12874d0 | refs/heads/master | 1,610,939,198,566 | 1,441,352,386,000 | 1,441,352,494,000 | 41,903,576 | 0 | 0 | null | 1,441,352,210,000 | 1,441,352,210,000 | null | UTF-8 | Lean | false | false | 18,878 | 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
Combinators for finite sets.
-/
import data.finset.basic logic.identities
open list quot subtype decidable perm function
namespace finset
/- image (corresponds to map on list) -/
section image
variables {A B : Type}
variable [h : decidable_eq B]
include h
definition image (f : A → B) (s : finset A) : finset B :=
quot.lift_on s
(λ l, to_finset (list.map f (elt_of l)))
(λ l₁ l₂ p, quot.sound (perm_erase_dup_of_perm (perm_map _ p)))
notation [priority finset.prio] f `'[`:max a `]` := image f a
theorem image_empty (f : A → B) : image f ∅ = ∅ :=
rfl
theorem mem_image_of_mem (f : A → B) {s : finset A} {a : A} : a ∈ s → f a ∈ image f s :=
quot.induction_on s (take l, assume H : a ∈ elt_of l, mem_to_finset (mem_map f H))
theorem mem_image {f : A → B} {s : finset A} {a : A} {b : B}
(H1 : a ∈ s) (H2 : f a = b) :
b ∈ image f s :=
eq.subst H2 (mem_image_of_mem f H1)
theorem exists_of_mem_image {f : A → B} {s : finset A} {b : B} :
b ∈ image f s → ∃a, a ∈ s ∧ f a = b :=
quot.induction_on s
(take l, assume H : b ∈ erase_dup (list.map f (elt_of l)),
exists_of_mem_map (mem_of_mem_erase_dup H))
theorem mem_image_iff (f : A → B) {s : finset A} {y : B} : y ∈ image f s ↔ ∃x, x ∈ s ∧ f x = y :=
iff.intro exists_of_mem_image
(assume H,
obtain x (H₁ : x ∈ s) (H₂ : f x = y), from H,
mem_image H₁ H₂)
theorem mem_image_eq (f : A → B) {s : finset A} {y : B} : y ∈ image f s = ∃x, x ∈ s ∧ f x = y :=
propext (mem_image_iff f)
theorem mem_image_of_mem_image_of_subset {f : A → B} {s t : finset A} {y : B}
(H1 : y ∈ image f s) (H2 : s ⊆ t) : y ∈ image f t :=
obtain x (H3: x ∈ s) (H4 : f x = y), from exists_of_mem_image H1,
have H5 : x ∈ t, from mem_of_subset_of_mem H2 H3,
show y ∈ image f t, from mem_image H5 H4
theorem image_insert [h' : decidable_eq A] (f : A → B) (s : finset A) (a : A) :
image f (insert a s) = insert (f a) (image f s) :=
ext (take y, iff.intro
(assume H : y ∈ image f (insert a s),
obtain x (H1l : x ∈ insert a s) (H1r :f x = y), from exists_of_mem_image H,
have x = a ∨ x ∈ s, from eq_or_mem_of_mem_insert H1l,
or.elim this
(suppose x = a,
have f a = y, from eq.subst this H1r,
show y ∈ insert (f a) (image f s), from eq.subst this !mem_insert)
(suppose x ∈ s,
have f x ∈ image f s, from mem_image_of_mem f this,
show y ∈ insert (f a) (image f s), from eq.subst H1r (mem_insert_of_mem _ this)))
(suppose y ∈ insert (f a) (image f s),
have y = f a ∨ y ∈ image f s, from eq_or_mem_of_mem_insert this,
or.elim this
(suppose y = f a,
have f a ∈ image f (insert a s), from mem_image_of_mem f !mem_insert,
show y ∈ image f (insert a s), from eq.subst (eq.symm `y = f a`) this)
(suppose y ∈ image f s,
show y ∈ image f (insert a s), from mem_image_of_mem_image_of_subset this !subset_insert)))
lemma image_compose {C : Type} [deceqC : decidable_eq C] {f : B → C} {g : A → B} {s : finset A} :
image (f∘g) s = image f (image g s) :=
ext (take z, iff.intro
(suppose z ∈ image (f∘g) s,
obtain x (Hx : x ∈ s) (Hgfx : f (g x) = z), from exists_of_mem_image this,
by rewrite -Hgfx; apply mem_image_of_mem _ (mem_image_of_mem _ Hx))
(suppose z ∈ image f (image g s),
obtain y (Hy : y ∈ image g s) (Hfy : f y = z), from exists_of_mem_image this,
obtain x (Hx : x ∈ s) (Hgx : g x = y), from exists_of_mem_image Hy,
mem_image Hx (by esimp; rewrite [Hgx, Hfy])))
lemma image_subset {a b : finset A} (f : A → B) (H : a ⊆ b) : f '[a] ⊆ f '[b] :=
subset_of_forall
(take y, assume Hy : y ∈ f '[a],
obtain x (Hx₁ : x ∈ a) (Hx₂ : f x = y), from exists_of_mem_image Hy,
mem_image (mem_of_subset_of_mem H Hx₁) Hx₂)
theorem image_union [h' : decidable_eq A] (f : A → B) (s t : finset A) :
image f (s ∪ t) = image f s ∪ image f t :=
ext (take y, iff.intro
(assume H : y ∈ image f (s ∪ t),
obtain x [(xst : x ∈ s ∪ t) (fxy : f x = y)], from exists_of_mem_image H,
or.elim (mem_or_mem_of_mem_union xst)
(assume xs, mem_union_l (mem_image xs fxy))
(assume xt, mem_union_r (mem_image xt fxy)))
(assume H : y ∈ image f s ∪ image f t,
or.elim (mem_or_mem_of_mem_union H)
(assume yifs : y ∈ image f s,
obtain x [(xs : x ∈ s) (fxy : f x = y)], from exists_of_mem_image yifs,
mem_image (mem_union_l xs) fxy)
(assume yift : y ∈ image f t,
obtain x [(xt : x ∈ t) (fxy : f x = y)], from exists_of_mem_image yift,
mem_image (mem_union_r xt) fxy)))
end image
/- separation and set-builder notation -/
section sep
variables {A : Type} [deceq : decidable_eq A]
include deceq
variables (p : A → Prop) [decp : decidable_pred p] (s : finset A) {x : A}
include decp
definition sep : finset A :=
quot.lift_on s
(λl, to_finset_of_nodup
(list.filter p (subtype.elt_of l))
(list.nodup_filter p (subtype.has_property l)))
(λ l₁ l₂ u, quot.sound (perm.perm_filter u))
notation [priority finset.prio] `{` binder ∈ s `|` r:(scoped:1 p, sep p s) `}` := r
theorem sep_empty : sep p ∅ = ∅ := rfl
variables {p s}
theorem of_mem_sep : x ∈ sep p s → p x :=
quot.induction_on s (take l, list.of_mem_filter)
theorem mem_of_mem_sep : x ∈ sep p s → x ∈ s :=
quot.induction_on s (take l, list.mem_of_mem_filter)
theorem mem_sep_of_mem {x : A} : x ∈ s → p x → x ∈ sep p s :=
quot.induction_on s (take l, list.mem_filter_of_mem)
variables (p s)
theorem mem_sep_iff : x ∈ sep p s ↔ x ∈ s ∧ p x :=
iff.intro
(assume H, and.intro (mem_of_mem_sep H) (of_mem_sep H))
(assume H, mem_sep_of_mem (and.left H) (and.right H))
theorem mem_sep_eq : x ∈ sep p s = (x ∈ s ∧ p x) :=
propext !mem_sep_iff
variable t : finset A
theorem mem_sep_union_iff : x ∈ sep p (s ∪ t) ↔ x ∈ sep p s ∨ x ∈ sep p t :=
by rewrite [*mem_sep_iff, mem_union_iff, and.right_distrib]
end sep
section
variables {A : Type} [deceqA : decidable_eq A]
include deceqA
theorem eq_sep_of_subset {s t : finset A} (ssubt : s ⊆ t) : s = {x ∈ t | x ∈ s} :=
ext (take x, iff.intro
(suppose x ∈ s, mem_sep_of_mem (mem_of_subset_of_mem ssubt this) this)
(suppose x ∈ {x ∈ t | x ∈ s}, of_mem_sep this))
theorem mem_singleton_eq' (x a : A) : x ∈ '{a} = (x = a) :=
by rewrite [mem_insert_eq, mem_empty_eq, or_false]
end
/- set difference -/
section diff
variables {A : Type} [deceq : decidable_eq A]
include deceq
definition diff (s t : finset A) : finset A := {x ∈ s | x ∉ t}
infix [priority finset.prio] `\`:70 := diff
theorem mem_of_mem_diff {s t : finset A} {x : A} (H : x ∈ s \ t) : x ∈ s :=
mem_of_mem_sep H
theorem not_mem_of_mem_diff {s t : finset A} {x : A} (H : x ∈ s \ t) : x ∉ t :=
of_mem_sep H
theorem mem_diff {s t : finset A} {x : A} (H1 : x ∈ s) (H2 : x ∉ t) : x ∈ s \ t :=
mem_sep_of_mem H1 H2
theorem mem_diff_iff (s t : finset A) (x : A) : x ∈ s \ t ↔ x ∈ s ∧ x ∉ t :=
iff.intro
(assume H, and.intro (mem_of_mem_diff H) (not_mem_of_mem_diff H))
(assume H, mem_diff (and.left H) (and.right H))
theorem mem_diff_eq (s t : finset A) (x : A) : x ∈ s \ t = (x ∈ s ∧ x ∉ t) :=
propext !mem_diff_iff
theorem union_diff_cancel {s t : finset A} (H : s ⊆ t) : s ∪ (t \ s) = t :=
ext (take x, iff.intro
(suppose x ∈ s ∪ (t \ s),
or.elim (mem_or_mem_of_mem_union this)
(suppose x ∈ s, mem_of_subset_of_mem H this)
(suppose x ∈ t \ s, mem_of_mem_diff this))
(suppose x ∈ t,
decidable.by_cases
(suppose x ∈ s, mem_union_left _ this)
(suppose x ∉ s, mem_union_right _ (mem_diff `x ∈ t` this))))
theorem diff_union_cancel {s t : finset A} (H : s ⊆ t) : (t \ s) ∪ s = t :=
eq.subst !union.comm (!union_diff_cancel H)
end diff
/- set complement -/
section complement
variables {A : Type} [deceqA : decidable_eq A] [h : fintype A]
include deceqA h
definition complement (s : finset A) : finset A := univ \ s
prefix [priority finset.prio] - := complement
theorem mem_complement {s : finset A} {x : A} (H : x ∉ s) : x ∈ -s :=
mem_diff !mem_univ H
theorem not_mem_of_mem_complement {s : finset A} {x : A} (H : x ∈ -s) : x ∉ s :=
not_mem_of_mem_diff H
theorem mem_complement_iff (s : finset A) (x : A) : x ∈ -s ↔ x ∉ s :=
iff.intro not_mem_of_mem_complement mem_complement
section
open classical
theorem union_eq_comp_comp_inter_comp (s t : finset A) : s ∪ t = -(-s ∩ -t) :=
ext (take x, by rewrite [mem_union_iff, mem_complement_iff, mem_inter_iff, *mem_complement_iff,
or_iff_not_and_not])
theorem inter_eq_comp_comp_union_comp (s t : finset A) : s ∩ t = -(-s ∪ -t) :=
ext (take x, by rewrite [mem_inter_iff, mem_complement_iff, mem_union_iff, *mem_complement_iff,
and_iff_not_or_not])
end
end complement
/- all -/
section all
variables {A : Type}
definition all (s : finset A) (p : A → Prop) : Prop :=
quot.lift_on s
(λ l, all (elt_of l) p)
(λ l₁ l₂ p, foldr_eq_of_perm (λ a₁ a₂ q, propext !and.left_comm) p true)
theorem all_empty (p : A → Prop) : all ∅ p = true :=
rfl
theorem of_mem_of_all {p : A → Prop} {a : A} {s : finset A} : a ∈ s → all s p → p a :=
quot.induction_on s (λ l i h, list.of_mem_of_all i h)
theorem forall_of_all {p : A → Prop} {s : finset A} (H : all s p) : ∀{a}, a ∈ s → p a :=
λ a H', of_mem_of_all H' H
theorem all_of_forall {p : A → Prop} {s : finset A} : (∀a, a ∈ s → p a) → all s p :=
quot.induction_on s (λ l H, list.all_of_forall H)
theorem all_iff_forall (p : A → Prop) (s : finset A) : all s p ↔ (∀a, a ∈ s → p a) :=
iff.intro forall_of_all all_of_forall
definition decidable_all [instance] (p : A → Prop) [h : decidable_pred p] (s : finset A) :
decidable (all s p) :=
quot.rec_on_subsingleton s (λ l, list.decidable_all p (elt_of l))
theorem all_implies {p q : A → Prop} {s : finset A} : all s p → (∀ x, p x → q x) → all s q :=
quot.induction_on s (λ l h₁ h₂, list.all_implies h₁ h₂)
variable [h : decidable_eq A]
include h
theorem all_union {p : A → Prop} {s₁ s₂ : finset A} : all s₁ p → all s₂ p → all (s₁ ∪ s₂) p :=
quot.induction_on₂ s₁ s₂ (λ l₁ l₂ a₁ a₂, all_union a₁ a₂)
theorem all_of_all_union_left {p : A → Prop} {s₁ s₂ : finset A} : all (s₁ ∪ s₂) p → all s₁ p :=
quot.induction_on₂ s₁ s₂ (λ l₁ l₂ a, list.all_of_all_union_left a)
theorem all_of_all_union_right {p : A → Prop} {s₁ s₂ : finset A} : all (s₁ ∪ s₂) p → all s₂ p :=
quot.induction_on₂ s₁ s₂ (λ l₁ l₂ a, list.all_of_all_union_right a)
theorem all_insert_of_all {p : A → Prop} {a : A} {s : finset A} : p a → all s p → all (insert a s) p :=
quot.induction_on s (λ l h₁ h₂, list.all_insert_of_all h₁ h₂)
theorem all_erase_of_all {p : A → Prop} (a : A) {s : finset A}: all s p → all (erase a s) p :=
quot.induction_on s (λ l h, list.all_erase_of_all a h)
theorem all_inter_of_all_left {p : A → Prop} {s₁ : finset A} (s₂ : finset A) : all s₁ p → all (s₁ ∩ s₂) p :=
quot.induction_on₂ s₁ s₂ (λ l₁ l₂ h, list.all_inter_of_all_left _ h)
theorem all_inter_of_all_right {p : A → Prop} {s₁ : finset A} (s₂ : finset A) : all s₂ p → all (s₁ ∩ s₂) p :=
quot.induction_on₂ s₁ s₂ (λ l₁ l₂ h, list.all_inter_of_all_right _ h)
theorem subset_iff_all (s t : finset A) : s ⊆ t ↔ all s (λ x, x ∈ t) :=
iff.intro
(suppose s ⊆ t, all_of_forall (take x, suppose x ∈ s, mem_of_subset_of_mem `s ⊆ t` `x ∈ s`))
(suppose all s (λ x, x ∈ t), subset_of_forall (take x, suppose x ∈ s, of_mem_of_all `x ∈ s` `all s (λ x, x ∈ t)`))
definition decidable_subset [instance] (s t : finset A) : decidable (s ⊆ t) :=
decidable_of_decidable_of_iff _ (iff.symm !subset_iff_all)
end all
/- any -/
section any
variables {A : Type}
definition any (s : finset A) (p : A → Prop) : Prop :=
quot.lift_on s
(λ l, any (elt_of l) p)
(λ l₁ l₂ p, foldr_eq_of_perm (λ a₁ a₂ q, propext !or.left_comm) p false)
theorem any_empty (p : A → Prop) : any ∅ p = false := rfl
theorem exists_of_any {p : A → Prop} {s : finset A} : any s p → ∃a, a ∈ s ∧ p a :=
quot.induction_on s (λ l H, list.exists_of_any H)
theorem any_of_mem {p : A → Prop} {s : finset A} {a : A} : a ∈ s → p a → any s p :=
quot.induction_on s (λ l H1 H2, list.any_of_mem H1 H2)
theorem any_of_exists {p : A → Prop} {s : finset A} (H : ∃a, a ∈ s ∧ p a) : any s p :=
obtain a H₁ H₂, from H,
any_of_mem H₁ H₂
theorem any_iff_exists (p : A → Prop) (s : finset A) : any s p ↔ (∃a, a ∈ s ∧ p a) :=
iff.intro exists_of_any any_of_exists
theorem any_of_insert [h : decidable_eq A] {p : A → Prop} (s : finset A) {a : A} (H : p a) :
any (insert a s) p :=
any_of_mem (mem_insert a s) H
theorem any_of_insert_right [h : decidable_eq A] {p : A → Prop} {s : finset A} (a : A) (H : any s p) :
any (insert a s) p :=
obtain b (H₁ : b ∈ s) (H₂ : p b), from exists_of_any H,
any_of_mem (mem_insert_of_mem a H₁) H₂
definition decidable_any [instance] (p : A → Prop) [h : decidable_pred p] (s : finset A) :
decidable (any s p) :=
quot.rec_on_subsingleton s (λ l, list.decidable_any p (elt_of l))
end any
section product
variables {A B : Type}
definition product (s₁ : finset A) (s₂ : finset B) : finset (A × B) :=
quot.lift_on₂ s₁ s₂
(λ l₁ l₂,
to_finset_of_nodup (product (elt_of l₁) (elt_of l₂))
(nodup_product (has_property l₁) (has_property l₂)))
(λ v₁ v₂ w₁ w₂ p₁ p₂, quot.sound (perm_product p₁ p₂))
infix [priority finset.prio] * := product
theorem empty_product (s : finset B) : @empty A * s = ∅ :=
quot.induction_on s (λ l, rfl)
theorem mem_product {a : A} {b : B} {s₁ : finset A} {s₂ : finset B}
: a ∈ s₁ → b ∈ s₂ → (a, b) ∈ s₁ * s₂ :=
quot.induction_on₂ s₁ s₂ (λ l₁ l₂ i₁ i₂, list.mem_product i₁ i₂)
theorem mem_of_mem_product_left {a : A} {b : B} {s₁ : finset A} {s₂ : finset B}
: (a, b) ∈ s₁ * s₂ → a ∈ s₁ :=
quot.induction_on₂ s₁ s₂ (λ l₁ l₂ i, list.mem_of_mem_product_left i)
theorem mem_of_mem_product_right {a : A} {b : B} {s₁ : finset A} {s₂ : finset B}
: (a, b) ∈ s₁ * s₂ → b ∈ s₂ :=
quot.induction_on₂ s₁ s₂ (λ l₁ l₂ i, list.mem_of_mem_product_right i)
theorem product_empty (s : finset A) : s * @empty B = ∅ :=
ext (λ p,
match p with
| (a, b) := iff.intro
(λ i, absurd (mem_of_mem_product_right i) !not_mem_empty)
(λ i, absurd i !not_mem_empty)
end)
end product
/- powerset -/
section powerset
variables {A : Type} [deceqA : decidable_eq A]
include deceqA
section list_powerset
open list
definition list_powerset : list A → finset (finset A)
| [] := '{∅}
| (a :: l) := list_powerset l ∪ image (insert a) (list_powerset l)
end list_powerset
private theorem image_insert_comm (a b : A) (s : finset (finset A)) :
image (insert a) (image (insert b) s) = image (insert b) (image (insert a) s) :=
have aux' : ∀ a b : A, ∀ x : finset A,
x ∈ image (insert a) (image (insert b) s) →
x ∈ image (insert b) (image (insert a) s), from
begin
intros [a, b, x, H],
cases (exists_of_mem_image H) with [y, Hy],
cases Hy with [Hy1, Hy2],
cases (exists_of_mem_image Hy1) with [z, Hz],
cases Hz with [Hz1, Hz2],
substvars,
rewrite insert.comm,
repeat (apply mem_image_of_mem),
assumption
end,
ext (take x, iff.intro (aux' a b x) (aux' b a x))
theorem list_powerset_eq_list_powerset_of_perm {l₁ l₂ : list A} (p : l₁ ~ l₂) :
list_powerset l₁ = list_powerset l₂ :=
perm.induction_on p
rfl
(λ x l₁ l₂ p ih, by rewrite [↑list_powerset, ih])
(λ x y l, by rewrite [↑list_powerset, ↑list_powerset, *image_union, image_insert_comm,
*union.assoc, union.left_comm (finset.image (finset.insert x) _)])
(λ l₁ l₂ l₃ p₁ p₂ r₁ r₂, eq.trans r₁ r₂)
definition powerset (s : finset A) : finset (finset A) :=
quot.lift_on s
(λ l, list_powerset (elt_of l))
(λ l₁ l₂ p, list_powerset_eq_list_powerset_of_perm p)
prefix [priority finset.prio] `𝒫`:100 := powerset
theorem powerset_empty : 𝒫 (∅ : finset A) = '{∅} := rfl
theorem powerset_insert {a : A} {s : finset A} : a ∉ s → 𝒫 (insert a s) = 𝒫 s ∪ image (insert a) (𝒫 s) :=
quot.induction_on s
(λ l,
assume H : a ∉ quot.mk l,
calc
𝒫 (insert a (quot.mk l))
= list_powerset (list.insert a (elt_of l)) : rfl
... = list_powerset (#list a :: elt_of l) : by rewrite [list.insert_eq_of_not_mem H]
... = 𝒫 (quot.mk l) ∪ image (insert a) (𝒫 (quot.mk l)) : rfl)
theorem mem_powerset_iff_subset (s : finset A) : ∀ x, x ∈ 𝒫 s ↔ x ⊆ s :=
begin
induction s with a s nains ih,
intro x,
rewrite powerset_empty,
show x ∈ '{∅} ↔ x ⊆ ∅, by rewrite [mem_singleton_eq', subset_empty_iff],
intro x,
rewrite [powerset_insert nains, mem_union_iff, ih, mem_image_iff],
exact
(iff.intro
(assume H,
or.elim H
(suppose x ⊆ s, subset.trans this !subset_insert)
(suppose ∃ y, y ∈ 𝒫 s ∧ insert a y = x,
obtain y [yps iay], from this,
show x ⊆ insert a s,
begin
rewrite [-iay],
apply insert_subset_insert,
rewrite -ih,
apply yps
end))
(assume H : x ⊆ insert a s,
assert H' : erase a x ⊆ s, from erase_subset_of_subset_insert H,
decidable.by_cases
(suppose a ∈ x,
or.inr (exists.intro (erase a x)
(and.intro
(show erase a x ∈ 𝒫 s, by rewrite ih; apply H')
(show insert a (erase a x) = x, from insert_erase this))))
(suppose a ∉ x, or.inl
(show x ⊆ s, by rewrite [(erase_eq_of_not_mem this) at H']; apply H'))))
end
theorem subset_of_mem_powerset {s t : finset A} (H : s ∈ 𝒫 t) : s ⊆ t :=
iff.mp (mem_powerset_iff_subset t s) H
theorem mem_powerset_of_subset {s t : finset A} (H : s ⊆ t) : s ∈ 𝒫 t :=
iff.mpr (mem_powerset_iff_subset t s) H
theorem empty_mem_powerset (s : finset A) : ∅ ∈ 𝒫 s :=
mem_powerset_of_subset (empty_subset s)
end powerset
end finset
|
b6f4ec8259d11bc71ccfce2de243f8aff0bcfdfa | 9dc8cecdf3c4634764a18254e94d43da07142918 | /src/ring_theory/trace.lean | 4ec7c28b8cceb4165f06085b5a396d6221210929 | [
"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 | 22,540 | lean | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import linear_algebra.matrix.bilinear_form
import linear_algebra.matrix.charpoly.minpoly
import linear_algebra.determinant
import linear_algebra.finite_dimensional
import linear_algebra.vandermonde
import linear_algebra.trace
import field_theory.is_alg_closed.algebraic_closure
import field_theory.primitive_element
import field_theory.galois
import ring_theory.power_basis
/-!
# Trace for (finite) ring extensions.
Suppose we have an `R`-algebra `S` with a finite basis. For each `s : S`,
the trace of the linear map given by multiplying by `s` gives information about
the roots of the minimal polynomial of `s` over `R`.
## Main definitions
* `algebra.trace R S x`: the trace of an element `s` of an `R`-algebra `S`
* `algebra.trace_form R S`: bilinear form sending `x`, `y` to the trace of `x * y`
* `algebra.trace_matrix R b`: the matrix whose `(i j)`-th element is the trace of `b i * b j`.
* `algebra.embeddings_matrix A C b : matrix κ (B →ₐ[A] C) C` is the matrix whose
`(i, σ)` coefficient is `σ (b i)`.
* `algebra.embeddings_matrix_reindex A C b e : matrix κ κ C` is the matrix whose `(i, j)`
coefficient is `σⱼ (b i)`, where `σⱼ : B →ₐ[A] C` is the embedding corresponding to `j : κ`
given by a bijection `e : κ ≃ (B →ₐ[A] C)`.
## Main results
* `trace_algebra_map_of_basis`, `trace_algebra_map`: if `x : K`, then `Tr_{L/K} x = [L : K] x`
* `trace_trace_of_basis`, `trace_trace`: `Tr_{L/K} (Tr_{F/L} x) = Tr_{F/K} x`
* `trace_eq_sum_roots`: the trace of `x : K(x)` is the sum of all conjugate roots of `x`
* `trace_eq_sum_embeddings`: the trace of `x : K(x)` is the sum of all embeddings of `x` into an
algebraically closed field
* `trace_form_nondegenerate`: the trace form over a separable extension is a nondegenerate
bilinear form
## Implementation notes
Typically, the trace is defined specifically for finite field extensions.
The definition is as general as possible and the assumption that we have
fields or that the extension is finite is added to the lemmas as needed.
We only define the trace for left multiplication (`algebra.left_mul_matrix`,
i.e. `linear_map.mul_left`).
For now, the definitions assume `S` is commutative, so the choice doesn't matter anyway.
## References
* https://en.wikipedia.org/wiki/Field_trace
-/
universes u v w z
variables {R S T : Type*} [comm_ring R] [comm_ring S] [comm_ring T]
variables [algebra R S] [algebra R T]
variables {K L : Type*} [field K] [field L] [algebra K L]
variables {ι κ : Type w} [fintype ι]
open finite_dimensional
open linear_map
open matrix
open_locale big_operators
open_locale matrix
namespace algebra
variables (b : basis ι R S)
variables (R S)
/-- The trace of an element `s` of an `R`-algebra is the trace of `(*) s`,
as an `R`-linear map. -/
noncomputable def trace : S →ₗ[R] R :=
(linear_map.trace R S).comp (lmul R S).to_linear_map
variables {S}
-- Not a `simp` lemma since there are more interesting ways to rewrite `trace R S x`,
-- for example `trace_trace`
lemma trace_apply (x) : trace R S x = linear_map.trace R S (lmul R S x) := rfl
lemma trace_eq_zero_of_not_exists_basis
(h : ¬ ∃ (s : finset S), nonempty (basis s R S)) : trace R S = 0 :=
by { ext s, simp [trace_apply, linear_map.trace, h] }
include b
variables {R}
-- Can't be a `simp` lemma because it depends on a choice of basis
lemma trace_eq_matrix_trace [decidable_eq ι] (b : basis ι R S) (s : S) :
trace R S s = matrix.trace (algebra.left_mul_matrix b s) :=
by { rw [trace_apply, linear_map.trace_eq_matrix_trace _ b, ←to_matrix_lmul_eq], refl }
/-- If `x` is in the base field `K`, then the trace is `[L : K] * x`. -/
lemma trace_algebra_map_of_basis (x : R) :
trace R S (algebra_map R S x) = fintype.card ι • x :=
begin
haveI := classical.dec_eq ι,
rw [trace_apply, linear_map.trace_eq_matrix_trace R b, matrix.trace],
convert finset.sum_const _,
ext i,
simp [-coe_lmul_eq_mul],
end
omit b
/-- If `x` is in the base field `K`, then the trace is `[L : K] * x`.
(If `L` is not finite-dimensional over `K`, then `trace` and `finrank` return `0`.)
-/
@[simp]
lemma trace_algebra_map (x : K) : trace K L (algebra_map K L x) = finrank K L • x :=
begin
by_cases H : ∃ (s : finset L), nonempty (basis s K L),
{ rw [trace_algebra_map_of_basis H.some_spec.some, finrank_eq_card_basis H.some_spec.some] },
{ simp [trace_eq_zero_of_not_exists_basis K H, finrank_eq_zero_of_not_exists_basis_finset H] }
end
lemma trace_trace_of_basis [algebra S T] [is_scalar_tower R S T] {ι κ : Type*} [finite ι] [finite κ]
(b : basis ι R S) (c : basis κ S T) (x : T) :
trace R S (trace S T x) = trace R T x :=
begin
haveI := classical.dec_eq ι,
haveI := classical.dec_eq κ,
casesI nonempty_fintype ι,
casesI nonempty_fintype κ,
rw [trace_eq_matrix_trace (b.smul c), trace_eq_matrix_trace b, trace_eq_matrix_trace c,
matrix.trace, matrix.trace, matrix.trace,
← finset.univ_product_univ, finset.sum_product],
refine finset.sum_congr rfl (λ i _, _),
simp only [alg_hom.map_sum, smul_left_mul_matrix, finset.sum_apply, matrix.diag,
-- The unifier is not smart enough to apply this one by itself:
finset.sum_apply i _ (λ y, left_mul_matrix b (left_mul_matrix c x y y))]
end
lemma trace_comp_trace_of_basis [algebra S T] [is_scalar_tower R S T] {ι κ : Type*} [finite ι]
[fintype κ] (b : basis ι R S) (c : basis κ S T) :
(trace R S).comp ((trace S T).restrict_scalars R) = trace R T :=
by { ext, rw [linear_map.comp_apply, linear_map.restrict_scalars_apply, trace_trace_of_basis b c] }
@[simp]
lemma trace_trace [algebra K T] [algebra L T] [is_scalar_tower K L T]
[finite_dimensional K L] [finite_dimensional L T] (x : T) :
trace K L (trace L T x) = trace K T x :=
trace_trace_of_basis (basis.of_vector_space K L) (basis.of_vector_space L T) x
@[simp]
lemma trace_comp_trace [algebra K T] [algebra L T] [is_scalar_tower K L T]
[finite_dimensional K L] [finite_dimensional L T] :
(trace K L).comp ((trace L T).restrict_scalars K) = trace K T :=
by { ext, rw [linear_map.comp_apply, linear_map.restrict_scalars_apply, trace_trace] }
section trace_form
variables (R S)
/-- The `trace_form` maps `x y : S` to the trace of `x * y`.
It is a symmetric bilinear form and is nondegenerate if the extension is separable. -/
noncomputable def trace_form : bilin_form R S :=
(linear_map.compr₂ (lmul R S).to_linear_map (trace R S)).to_bilin
variables {S}
-- This is a nicer lemma than the one produced by `@[simps] def trace_form`.
@[simp] lemma trace_form_apply (x y : S) : trace_form R S x y = trace R S (x * y) := rfl
lemma trace_form_is_symm : (trace_form R S).is_symm :=
λ x y, congr_arg (trace R S) (mul_comm _ _)
lemma trace_form_to_matrix [decidable_eq ι] (i j) :
bilin_form.to_matrix b (trace_form R S) i j = trace R S (b i * b j) :=
by rw [bilin_form.to_matrix_apply, trace_form_apply]
lemma trace_form_to_matrix_power_basis (h : power_basis R S) :
bilin_form.to_matrix h.basis (trace_form R S) = of (λ i j, trace R S (h.gen ^ (↑i + ↑j : ℕ))) :=
by { ext, rw [trace_form_to_matrix, of_apply, pow_add, h.basis_eq_pow, h.basis_eq_pow] }
end trace_form
end algebra
section eq_sum_roots
open algebra polynomial
variables {F : Type*} [field F]
variables [algebra K S] [algebra K F]
/-- Given `pb : power_basis K S`, the trace of `pb.gen` is `-(minpoly K pb.gen).next_coeff`. -/
lemma power_basis.trace_gen_eq_next_coeff_minpoly [nontrivial S] (pb : power_basis K S) :
algebra.trace K S pb.gen = -(minpoly K pb.gen).next_coeff :=
begin
have d_pos : 0 < pb.dim := power_basis.dim_pos pb,
have d_pos' : 0 < (minpoly K pb.gen).nat_degree, { simpa },
haveI : nonempty (fin pb.dim) := ⟨⟨0, d_pos⟩⟩,
rw [trace_eq_matrix_trace pb.basis, trace_eq_neg_charpoly_coeff, charpoly_left_mul_matrix,
← pb.nat_degree_minpoly, fintype.card_fin, ← next_coeff_of_pos_nat_degree _ d_pos']
end
/-- Given `pb : power_basis K S`, then the trace of `pb.gen` is
`((minpoly K pb.gen).map (algebra_map K F)).roots.sum`. -/
lemma power_basis.trace_gen_eq_sum_roots [nontrivial S] (pb : power_basis K S)
(hf : (minpoly K pb.gen).splits (algebra_map K F)) :
algebra_map K F (trace K S pb.gen) =
((minpoly K pb.gen).map (algebra_map K F)).roots.sum :=
begin
rw [power_basis.trace_gen_eq_next_coeff_minpoly, ring_hom.map_neg, ← next_coeff_map
(algebra_map K F).injective, sum_roots_eq_next_coeff_of_monic_of_split
((minpoly.monic (power_basis.is_integral_gen _)).map _)
((splits_id_iff_splits _).2 hf), neg_neg]
end
namespace intermediate_field.adjoin_simple
open intermediate_field
lemma trace_gen_eq_zero {x : L} (hx : ¬ is_integral K x) :
algebra.trace K K⟮x⟯ (adjoin_simple.gen K x) = 0 :=
begin
rw [trace_eq_zero_of_not_exists_basis, linear_map.zero_apply],
contrapose! hx,
obtain ⟨s, ⟨b⟩⟩ := hx,
refine is_integral_of_mem_of_fg (K⟮x⟯).to_subalgebra _ x _,
{ exact (submodule.fg_iff_finite_dimensional _).mpr (finite_dimensional.of_fintype_basis b) },
{ exact subset_adjoin K _ (set.mem_singleton x) }
end
lemma trace_gen_eq_sum_roots (x : L)
(hf : (minpoly K x).splits (algebra_map K F)) :
algebra_map K F (trace K K⟮x⟯ (adjoin_simple.gen K x)) =
((minpoly K x).map (algebra_map K F)).roots.sum :=
begin
have injKxL := (algebra_map K⟮x⟯ L).injective,
by_cases hx : is_integral K x, swap,
{ simp [minpoly.eq_zero hx, trace_gen_eq_zero hx], },
have hx' : is_integral K (adjoin_simple.gen K x),
{ rwa [← is_integral_algebra_map_iff injKxL, adjoin_simple.algebra_map_gen],
apply_instance },
rw [← adjoin.power_basis_gen hx, (adjoin.power_basis hx).trace_gen_eq_sum_roots];
rw [adjoin.power_basis_gen hx, minpoly.eq_of_algebra_map_eq injKxL hx'];
try { simp only [adjoin_simple.algebra_map_gen _ _] },
exact hf
end
end intermediate_field.adjoin_simple
open intermediate_field
variables (K)
lemma trace_eq_trace_adjoin [finite_dimensional K L] (x : L) :
algebra.trace K L x = finrank K⟮x⟯ L • trace K K⟮x⟯ (adjoin_simple.gen K x) :=
begin
rw ← @trace_trace _ _ K K⟮x⟯ _ _ _ _ _ _ _ _ x,
conv in x { rw ← intermediate_field.adjoin_simple.algebra_map_gen K x },
rw [trace_algebra_map, linear_map.map_smul_of_tower],
end
variables {K}
lemma trace_eq_sum_roots [finite_dimensional K L]
{x : L} (hF : (minpoly K x).splits (algebra_map K F)) :
algebra_map K F (algebra.trace K L x) =
finrank K⟮x⟯ L • ((minpoly K x).map (algebra_map K _)).roots.sum :=
by rw [trace_eq_trace_adjoin K x, algebra.smul_def, ring_hom.map_mul, ← algebra.smul_def,
intermediate_field.adjoin_simple.trace_gen_eq_sum_roots _ hF, is_scalar_tower.algebra_map_smul]
end eq_sum_roots
variables {F : Type*} [field F]
variables [algebra R L] [algebra L F] [algebra R F] [is_scalar_tower R L F]
open polynomial
lemma algebra.is_integral_trace [finite_dimensional L F] {x : F} (hx : _root_.is_integral R x) :
_root_.is_integral R (algebra.trace L F x) :=
begin
have hx' : _root_.is_integral L x := is_integral_of_is_scalar_tower _ hx,
rw [← is_integral_algebra_map_iff (algebra_map L (algebraic_closure F)).injective,
trace_eq_sum_roots],
{ refine (is_integral.multiset_sum _).nsmul _,
intros y hy,
rw mem_roots_map (minpoly.ne_zero hx') at hy,
use [minpoly R x, minpoly.monic hx],
rw ← aeval_def at ⊢ hy,
exact minpoly.aeval_of_is_scalar_tower R x y hy },
{ apply is_alg_closed.splits_codomain },
{ apply_instance }
end
section eq_sum_embeddings
variables [algebra K F] [is_scalar_tower K L F]
open algebra intermediate_field
variables (F) (E : Type*) [field E] [algebra K E]
lemma trace_eq_sum_embeddings_gen
(pb : power_basis K L)
(hE : (minpoly K pb.gen).splits (algebra_map K E)) (hfx : (minpoly K pb.gen).separable) :
algebra_map K E (algebra.trace K L pb.gen) =
(@@finset.univ (power_basis.alg_hom.fintype pb)).sum (λ σ, σ pb.gen) :=
begin
letI := classical.dec_eq E,
rw [pb.trace_gen_eq_sum_roots hE, fintype.sum_equiv pb.lift_equiv', finset.sum_mem_multiset,
finset.sum_eq_multiset_sum, multiset.to_finset_val,
multiset.dedup_eq_self.mpr _, multiset.map_id],
{ exact nodup_roots ((separable_map _).mpr hfx) },
{ intro x, refl },
{ intro σ, rw [power_basis.lift_equiv'_apply_coe, id.def] }
end
variables [is_alg_closed E]
lemma sum_embeddings_eq_finrank_mul [finite_dimensional K F] [is_separable K F]
(pb : power_basis K L) :
∑ σ : F →ₐ[K] E, σ (algebra_map L F pb.gen) =
finrank L F • (@@finset.univ (power_basis.alg_hom.fintype pb)).sum
(λ σ : L →ₐ[K] E, σ pb.gen) :=
begin
haveI : finite_dimensional L F := finite_dimensional.right K L F,
haveI : is_separable L F := is_separable_tower_top_of_is_separable K L F,
letI : fintype (L →ₐ[K] E) := power_basis.alg_hom.fintype pb,
letI : ∀ (f : L →ₐ[K] E), fintype (@@alg_hom L F E _ _ _ _ f.to_ring_hom.to_algebra) :=
_, -- will be solved by unification
rw [fintype.sum_equiv alg_hom_equiv_sigma (λ (σ : F →ₐ[K] E), _) (λ σ, σ.1 pb.gen),
← finset.univ_sigma_univ, finset.sum_sigma, ← finset.sum_nsmul],
refine finset.sum_congr rfl (λ σ _, _),
{ letI : algebra L E := σ.to_ring_hom.to_algebra,
simp only [finset.sum_const, finset.card_univ],
rw alg_hom.card L F E },
{ intros σ,
simp only [alg_hom_equiv_sigma, equiv.coe_fn_mk, alg_hom.restrict_domain, alg_hom.comp_apply,
is_scalar_tower.coe_to_alg_hom'] }
end
lemma trace_eq_sum_embeddings [finite_dimensional K L] [is_separable K L]
{x : L} : algebra_map K E (algebra.trace K L x) = ∑ σ : L →ₐ[K] E, σ x :=
begin
have hx := is_separable.is_integral K x,
rw [trace_eq_trace_adjoin K x, algebra.smul_def, ring_hom.map_mul, ← adjoin.power_basis_gen hx,
trace_eq_sum_embeddings_gen E (adjoin.power_basis hx) (is_alg_closed.splits_codomain _),
← algebra.smul_def, algebra_map_smul],
{ exact (sum_embeddings_eq_finrank_mul L E (adjoin.power_basis hx)).symm },
{ haveI := is_separable_tower_bot_of_is_separable K K⟮x⟯ L,
exact is_separable.separable K _ }
end
lemma trace_eq_sum_automorphisms (x : L) [finite_dimensional K L] [is_galois K L] :
algebra_map K L (algebra.trace K L x) = ∑ (σ : L ≃ₐ[K] L), σ x :=
begin
apply no_zero_smul_divisors.algebra_map_injective L (algebraic_closure L),
rw map_sum (algebra_map L (algebraic_closure L)),
rw ← fintype.sum_equiv (normal.alg_hom_equiv_aut K (algebraic_closure L) L),
{ rw ←trace_eq_sum_embeddings (algebraic_closure L),
{ simp only [algebra_map_eq_smul_one, smul_one_smul] },
{ exact is_galois.to_is_separable } },
{ intro σ,
simp only [normal.alg_hom_equiv_aut, alg_hom.restrict_normal', equiv.coe_fn_mk,
alg_equiv.coe_of_bijective, alg_hom.restrict_normal_commutes, id.map_eq_id,
ring_hom.id_apply] },
end
end eq_sum_embeddings
section det_ne_zero
namespace algebra
variables (A : Type u) {B : Type v} (C : Type z)
variables [comm_ring A] [comm_ring B] [algebra A B] [comm_ring C] [algebra A C]
open finset
/-- Given an `A`-algebra `B` and `b`, an `κ`-indexed family of elements of `B`, we define
`trace_matrix A b` as the matrix whose `(i j)`-th element is the trace of `b i * b j`. -/
@[simp] noncomputable
def trace_matrix (b : κ → B) : matrix κ κ A
| i j := trace_form A B (b i) (b j)
lemma trace_matrix_def (b : κ → B) : trace_matrix A b = of (λ i j, trace_form A B (b i) (b j)) :=
rfl
lemma trace_matrix_reindex {κ' : Type*} (b : basis κ A B) (f : κ ≃ κ') :
trace_matrix A (b.reindex f) = reindex f f (trace_matrix A b) :=
by {ext x y, simp}
variables {A}
lemma trace_matrix_of_matrix_vec_mul [fintype κ] (b : κ → B) (P : matrix κ κ A) :
trace_matrix A ((P.map (algebra_map A B)).vec_mul b) = Pᵀ ⬝ (trace_matrix A b) ⬝ P :=
begin
ext α β,
rw [trace_matrix, vec_mul, dot_product, vec_mul, dot_product, matrix.mul_apply,
bilin_form.sum_left, fintype.sum_congr _ _ (λ (i : κ), @bilin_form.sum_right _ _ _ _ _ _ _ _
(b i * P.map (algebra_map A B) i α) (λ (y : κ), b y * P.map (algebra_map A B) y β)), sum_comm],
congr, ext x,
rw [matrix.mul_apply, sum_mul],
congr, ext y,
rw [map_apply, trace_form_apply, mul_comm (b y), ← smul_def],
simp only [id.smul_eq_mul, ring_hom.id_apply, map_apply, transpose_apply, linear_map.map_smulₛₗ,
trace_form_apply, algebra.smul_mul_assoc],
rw [mul_comm (b x), ← smul_def],
ring_nf,
simp [mul_comm],
end
lemma trace_matrix_of_matrix_mul_vec [fintype κ] (b : κ → B) (P : matrix κ κ A) :
trace_matrix A ((P.map (algebra_map A B)).mul_vec b) = P ⬝ (trace_matrix A b) ⬝ Pᵀ :=
begin
refine add_equiv.injective (transpose_add_equiv _ _ _) _,
rw [transpose_add_equiv_apply, transpose_add_equiv_apply, ← vec_mul_transpose,
← transpose_map, trace_matrix_of_matrix_vec_mul, transpose_transpose, transpose_mul,
transpose_transpose, transpose_mul]
end
lemma trace_matrix_of_basis [fintype κ] [decidable_eq κ] (b : basis κ A B) :
trace_matrix A b = bilin_form.to_matrix b (trace_form A B) :=
begin
ext i j,
rw [trace_matrix, trace_form_apply, trace_form_to_matrix]
end
lemma trace_matrix_of_basis_mul_vec (b : basis ι A B) (z : B) :
(trace_matrix A b).mul_vec (b.equiv_fun z) = (λ i, trace A B (z * (b i))) :=
begin
ext i,
rw [← col_apply ((trace_matrix A b).mul_vec (b.equiv_fun z)) i unit.star, col_mul_vec,
matrix.mul_apply, trace_matrix_def],
simp only [col_apply, trace_form_apply],
conv_lhs
{ congr, skip, funext,
rw [mul_comm _ (b.equiv_fun z _), ← smul_eq_mul, of_apply, ← linear_map.map_smul] },
rw [← linear_map.map_sum],
congr,
conv_lhs
{ congr, skip, funext,
rw [← mul_smul_comm] },
rw [← finset.mul_sum, mul_comm z],
congr,
rw [b.sum_equiv_fun ]
end
variable (A)
/-- `embeddings_matrix A C b : matrix κ (B →ₐ[A] C) C` is the matrix whose `(i, σ)` coefficient is
`σ (b i)`. It is mostly useful for fields when `fintype.card κ = finrank A B` and `C` is
algebraically closed. -/
@[simp] def embeddings_matrix (b : κ → B) : matrix κ (B →ₐ[A] C) C
| i σ := σ (b i)
/-- `embeddings_matrix_reindex A C b e : matrix κ κ C` is the matrix whose `(i, j)` coefficient
is `σⱼ (b i)`, where `σⱼ : B →ₐ[A] C` is the embedding corresponding to `j : κ` given by a
bijection `e : κ ≃ (B →ₐ[A] C)`. It is mostly useful for fields and `C` is algebraically closed.
In this case, in presence of `h : fintype.card κ = finrank A B`, one can take
`e := equiv_of_card_eq ((alg_hom.card A B C).trans h.symm)`. -/
def embeddings_matrix_reindex (b : κ → B) (e : κ ≃ (B →ₐ[A] C)) :=
reindex (equiv.refl κ) e.symm (embeddings_matrix A C b)
variable {A}
lemma embeddings_matrix_reindex_eq_vandermonde (pb : power_basis A B)
(e : fin pb.dim ≃ (B →ₐ[A] C)) :
embeddings_matrix_reindex A C pb.basis e = (vandermonde (λ i, e i pb.gen))ᵀ :=
by { ext i j, simp [embeddings_matrix_reindex, embeddings_matrix] }
section field
variables (K) {L} (E : Type z) [field E]
variables [algebra K E]
variables [module.finite K L] [is_separable K L] [is_alg_closed E]
variables (b : κ → L) (pb : power_basis K L)
lemma trace_matrix_eq_embeddings_matrix_mul_trans :
(trace_matrix K b).map (algebra_map K E) =
(embeddings_matrix K E b) ⬝ (embeddings_matrix K E b)ᵀ :=
by { ext i j, simp [trace_eq_sum_embeddings, embeddings_matrix, matrix.mul_apply] }
lemma trace_matrix_eq_embeddings_matrix_reindex_mul_trans [fintype κ]
(e : κ ≃ (L →ₐ[K] E)) : (trace_matrix K b).map (algebra_map K E) =
(embeddings_matrix_reindex K E b e) ⬝ (embeddings_matrix_reindex K E b e)ᵀ :=
by rw [trace_matrix_eq_embeddings_matrix_mul_trans, embeddings_matrix_reindex, reindex_apply,
transpose_submatrix, ← submatrix_mul_transpose_submatrix, ← equiv.coe_refl, equiv.refl_symm]
end field
end algebra
open algebra
variables (pb : power_basis K L)
lemma det_trace_matrix_ne_zero' [is_separable K L] :
det (trace_matrix K pb.basis) ≠ 0 :=
begin
suffices : algebra_map K (algebraic_closure L) (det (trace_matrix K pb.basis)) ≠ 0,
{ refine mt (λ ht, _) this,
rw [ht, ring_hom.map_zero] },
haveI : finite_dimensional K L := pb.finite_dimensional,
let e : fin pb.dim ≃ (L →ₐ[K] algebraic_closure L) := (fintype.equiv_fin_of_card_eq _).symm,
rw [ring_hom.map_det, ring_hom.map_matrix_apply,
trace_matrix_eq_embeddings_matrix_reindex_mul_trans K _ _ e,
embeddings_matrix_reindex_eq_vandermonde, det_mul, det_transpose],
refine mt mul_self_eq_zero.mp _,
{ simp only [det_vandermonde, finset.prod_eq_zero_iff, not_exists, sub_eq_zero],
intros i _ j hij h,
exact (finset.mem_Ioi.mp hij).ne' (e.injective $ pb.alg_hom_ext h) },
{ rw [alg_hom.card, pb.finrank] }
end
lemma det_trace_form_ne_zero [is_separable K L] [decidable_eq ι] (b : basis ι K L) :
det (bilin_form.to_matrix b (trace_form K L)) ≠ 0 :=
begin
haveI : finite_dimensional K L := finite_dimensional.of_fintype_basis b,
let pb : power_basis K L := field.power_basis_of_finite_of_separable _ _,
rw [← bilin_form.to_matrix_mul_basis_to_matrix pb.basis b,
← det_comm' (pb.basis.to_matrix_mul_to_matrix_flip b) _,
← matrix.mul_assoc, det_mul],
swap, { apply basis.to_matrix_mul_to_matrix_flip },
refine mul_ne_zero
(is_unit_of_mul_eq_one _ ((b.to_matrix pb.basis)ᵀ ⬝ b.to_matrix pb.basis).det _).ne_zero
_,
{ calc (pb.basis.to_matrix b ⬝ (pb.basis.to_matrix b)ᵀ).det *
((b.to_matrix pb.basis)ᵀ ⬝ b.to_matrix pb.basis).det
= (pb.basis.to_matrix b ⬝ (b.to_matrix pb.basis ⬝ pb.basis.to_matrix b)ᵀ ⬝
b.to_matrix pb.basis).det
: by simp only [← det_mul, matrix.mul_assoc, matrix.transpose_mul]
... = 1 : by simp only [basis.to_matrix_mul_to_matrix_flip, matrix.transpose_one,
matrix.mul_one, matrix.det_one] },
simpa only [trace_matrix_of_basis] using det_trace_matrix_ne_zero' pb
end
variables (K L)
theorem trace_form_nondegenerate [finite_dimensional K L] [is_separable K L] :
(trace_form K L).nondegenerate :=
bilin_form.nondegenerate_of_det_ne_zero (trace_form K L) _
(det_trace_form_ne_zero (finite_dimensional.fin_basis K L))
end det_ne_zero
|
f52339cdbd0d514a2b3abbf4c48f506149b2a6d3 | 4727251e0cd73359b15b664c3170e5d754078599 | /src/group_theory/abelianization.lean | 7bf9be76b99aebee10d21b8724b279618ce28eb4 | [
"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 | 7,202 | lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Michael Howes
-/
import group_theory.commutator
import group_theory.quotient_group
/-!
# The abelianization of a group
This file defines the commutator and the abelianization of a group. It furthermore prepares for the
result that the abelianization is left adjoint to the forgetful functor from abelian groups to
groups, which can be found in `algebra/category/Group/adjunctions`.
## Main definitions
* `commutator`: defines the commutator of a group `G` as a subgroup of `G`.
* `abelianization`: defines the abelianization of a group `G` as the quotient of a group by its
commutator subgroup.
* `abelianization.map`: lifts a group homomorphism to a homomorphism between the abelianizations
* `mul_equiv.abelianization_congr`: Equivalent groups have equivalent abelianizations
-/
universes u v w
-- Let G be a group.
variables (G : Type u) [group G]
/-- The commutator subgroup of a group G is the normal subgroup
generated by the commutators [p,q]=`p*q*p⁻¹*q⁻¹`. -/
@[derive subgroup.normal]
def commutator : subgroup G :=
⁅(⊤ : subgroup G), ⊤⁆
lemma commutator_def : commutator G = ⁅(⊤ : subgroup G), ⊤⁆ := rfl
lemma commutator_eq_closure : commutator G = subgroup.closure {g | ∃ g₁ g₂ : G, ⁅g₁, g₂⁆ = g} :=
by simp_rw [commutator, subgroup.commutator_def, subgroup.mem_top, exists_true_left]
lemma commutator_eq_normal_closure :
commutator G = subgroup.normal_closure {g | ∃ g₁ g₂ : G, ⁅g₁, g₂⁆ = g} :=
by simp_rw [commutator, subgroup.commutator_def', subgroup.mem_top, exists_true_left]
instance commutator_characteristic : (commutator G).characteristic :=
subgroup.commutator_characteristic ⊤ ⊤
lemma commutator_centralizer_commutator_le_center :
⁅(commutator G).centralizer, (commutator G).centralizer⁆ ≤ subgroup.center G :=
begin
rw [←subgroup.centralizer_top, ←subgroup.commutator_eq_bot_iff_le_centralizer],
suffices : ⁅⁅⊤, (commutator G).centralizer⁆, (commutator G).centralizer⁆ = ⊥,
{ refine subgroup.commutator_commutator_eq_bot_of_rotate _ this,
rwa subgroup.commutator_comm (commutator G).centralizer },
rw [subgroup.commutator_comm, subgroup.commutator_eq_bot_iff_le_centralizer],
exact set.centralizer_subset (subgroup.commutator_mono le_top le_top),
end
/-- The abelianization of G is the quotient of G by its commutator subgroup. -/
def abelianization : Type u :=
G ⧸ (commutator G)
namespace abelianization
local attribute [instance] quotient_group.left_rel
instance : comm_group (abelianization G) :=
{ mul_comm := λ x y, quotient.induction_on₂' x y $ λ a b, quotient.sound' $
subgroup.subset_closure ⟨b⁻¹, subgroup.mem_top b⁻¹, a⁻¹, subgroup.mem_top a⁻¹, by group⟩,
.. quotient_group.quotient.group _ }
instance : inhabited (abelianization G) := ⟨1⟩
instance [fintype G] [decidable_pred (∈ commutator G)] :
fintype (abelianization G) :=
quotient_group.fintype (commutator G)
variable {G}
/-- `of` is the canonical projection from G to its abelianization. -/
def of : G →* abelianization G :=
{ to_fun := quotient_group.mk,
map_one' := rfl,
map_mul' := λ x y, rfl }
@[simp] lemma mk_eq_of (a : G) : quot.mk _ a = of a := rfl
section lift
-- So far we have built Gᵃᵇ and proved it's an abelian group.
-- Furthremore we defined the canonical projection `of : G → Gᵃᵇ`
-- Let `A` be an abelian group and let `f` be a group homomorphism from `G` to `A`.
variables {A : Type v} [comm_group A] (f : G →* A)
lemma commutator_subset_ker : commutator G ≤ f.ker :=
begin
rw [commutator_eq_closure, subgroup.closure_le],
rintros x ⟨p, q, rfl⟩,
simp [monoid_hom.mem_ker, mul_right_comm (f p) (f q), commutator_element_def],
end
/-- If `f : G → A` is a group homomorphism to an abelian group, then `lift f` is the unique map from
the abelianization of a `G` to `A` that factors through `f`. -/
def lift : (G →* A) ≃ (abelianization G →* A) :=
{ to_fun := λ f, quotient_group.lift _ f (λ x h, f.mem_ker.2 $ commutator_subset_ker _ h),
inv_fun := λ F, F.comp of,
left_inv := λ f, monoid_hom.ext $ λ x, rfl,
right_inv := λ F, monoid_hom.ext $ λ x, quotient_group.induction_on x $ λ z, rfl }
@[simp] lemma lift.of (x : G) : lift f (of x) = f x :=
rfl
theorem lift.unique
(φ : abelianization G →* A)
-- hφ : φ agrees with f on the image of G in Gᵃᵇ
(hφ : ∀ (x : G), φ (of x) = f x)
{x : abelianization G} :
φ x = lift f x :=
quotient_group.induction_on x hφ
@[simp] lemma lift_of : lift of = monoid_hom.id (abelianization G) :=
lift.apply_symm_apply $ monoid_hom.id _
end lift
variables {A : Type v} [monoid A]
/-- See note [partially-applied ext lemmas]. -/
@[ext]
theorem hom_ext (φ ψ : abelianization G →* A)
(h : φ.comp of = ψ.comp of) : φ = ψ :=
monoid_hom.ext $ λ x, quotient_group.induction_on x $ monoid_hom.congr_fun h
section map
variables {H : Type v} [group H] (f : G →* H)
/-- The map operation of the `abelianization` functor -/
def map : abelianization G →* abelianization H := lift (of.comp f)
@[simp]
lemma map_of (x : G) : map f (of x) = of (f x) := rfl
@[simp]
lemma map_id : map (monoid_hom.id G) = monoid_hom.id (abelianization G) := hom_ext _ _ rfl
@[simp]
lemma map_comp {I : Type w} [group I] (g : H →* I) :
(map g).comp (map f) = map (g.comp f) := hom_ext _ _ rfl
@[simp]
lemma map_map_apply {I : Type w} [group I] {g : H →* I} {x : abelianization G}:
map g (map f x) = map (g.comp f) x := monoid_hom.congr_fun (map_comp _ _) x
end map
end abelianization
section abelianization_congr
variables {G} {H : Type v} [group H] (e : G ≃* H)
/-- Equivalent groups have equivalent abelianizations -/
def mul_equiv.abelianization_congr : abelianization G ≃* abelianization H :=
{ to_fun := abelianization.map e.to_monoid_hom,
inv_fun := abelianization.map e.symm.to_monoid_hom,
left_inv := by { rintros ⟨a⟩, simp },
right_inv := by { rintros ⟨a⟩, simp },
map_mul' := monoid_hom.map_mul _ }
@[simp]
lemma abelianization_congr_of (x : G) :
(e.abelianization_congr) (abelianization.of x) = abelianization.of (e x) := rfl
@[simp]
lemma abelianization_congr_refl :
(mul_equiv.refl G).abelianization_congr = mul_equiv.refl (abelianization G) :=
mul_equiv.to_monoid_hom_injective abelianization.lift_of
@[simp]
lemma abelianization_congr_symm :
e.abelianization_congr.symm = e.symm.abelianization_congr := rfl
@[simp]
lemma abelianization_congr_trans {I : Type v} [group I] (e₂ : H ≃* I) :
e.abelianization_congr.trans e₂.abelianization_congr = (e.trans e₂).abelianization_congr :=
mul_equiv.to_monoid_hom_injective (abelianization.hom_ext _ _ rfl)
end abelianization_congr
/-- An Abelian group is equivalent to its own abelianization. -/
@[simps] def abelianization.equiv_of_comm {H : Type*} [comm_group H] :
H ≃* abelianization H :=
{ to_fun := abelianization.of,
inv_fun := abelianization.lift (monoid_hom.id H),
left_inv := λ a, rfl,
right_inv := by { rintros ⟨a⟩, refl, },
.. abelianization.of }
|
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"Apache-2.0"
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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 u
/- Make sure instances defined in this file have lower priority than the ones
defined for concrete structures -/
set_option default_priority 100
set_option old_structure_cmd true
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
@[simp]
lemma mul_inv_cancel {a : α} (h : a ≠ 0) : a * a⁻¹ = 1 :=
division_ring.mul_inv_cancel h
@[simp]
lemma inv_mul_cancel {a : α} (h : a ≠ 0) : a⁻¹ * a = 1 :=
division_ring.inv_mul_cancel h
@[simp]
lemma one_div_eq_inv (a : α) : 1 / a = a⁻¹ :=
one_mul a⁻¹
lemma inv_eq_one_div (a : α) : a⁻¹ = 1 / a :=
by simp
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 :=
assume : 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 inv_ne_zero {a : α} (h : a ≠ 0) : a⁻¹ ≠ 0 :=
by rw inv_eq_one_div; exact one_div_ne_zero h
lemma one_inv_eq : 1⁻¹ = (1:α) :=
calc 1⁻¹ = 1 * 1⁻¹ : by rw [one_mul]
... = (1:α) : by simp
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 :=
assume : 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
assume : 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
assume : 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
(assume : -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.inv_inv {a : α} (h : a ≠ 0) : a⁻¹⁻¹ = a :=
by rw [inv_eq_one_div, inv_eq_one_div, division_ring.one_div_one_div 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 division_ring.one_div_div {a b : α} (ha : a ≠ 0) (hb : b ≠ 0) : 1 / (a / b) = b / a :=
by rw [one_div_eq_inv, division_def, mul_inv_eq (inv_ne_zero hb) ha,
division_ring.inv_inv hb, division_def]
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
(assume : a / b = 1, calc
a = a / b * b : by simp [hb]
... = 1 * b : by rw this
... = b : by simp)
(assume : 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
(assume : a = b / c, by rw [this, (div_mul_cancel _ hc)])
(assume : 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.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, division_ring.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
(assume : a = 0, or.inl this)
(assume : 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 :=
{ eq_zero_or_eq_zero_of_mul_eq_zero := discrete_field.eq_zero_or_eq_zero_of_mul_eq_zero,
..s }
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
(assume : a = 0,
by rw [this, div_zero, zero_mul, mul_zero, div_zero])
(assume ha : a ≠ 0,
decidable.by_cases
(assume : b = 0,
by rw [this, div_zero, mul_zero, zero_mul, div_zero])
(assume : 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
(assume : a = 0, by rw [this, neg_zero, div_zero, neg_zero])
(assume : 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, division_ring.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
|
28653adce8fa45cdab7d2bad9fac3353b87a0253 | 618003631150032a5676f229d13a079ac875ff77 | /src/ring_theory/power_series.lean | 6fa4eee5a8d9d49a8b745cfb4f17a6d6ee549442 | [
"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 | 57,816 | lean | /-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import data.mv_polynomial
import ring_theory.ideal_operations
import tactic.linarith
/-!
# Formal power series
This file defines (multivariate) formal power series
and develops the basic properties of these objects.
A formal power series is to a polynomial like an infinite sum is to a finite sum.
We provide the natural inclusion from polynomials to formal power series.
## Generalities
The file starts with setting up the (semi)ring structure on multivariate power series.
`trunc n φ` truncates a formal power series to the polynomial
that has the same coefficients as φ, for all m ≤ n, and 0 otherwise.
If the constant coefficient of a formal power series is invertible,
then this formal power series is invertible.
Formal power series over a local ring form a local ring.
## Formal power series in one variable
We prove that if the ring of coefficients is an integral domain,
then formal power series in one variable form an integral domain.
The `order` of a formal power series `φ` is the multiplicity of the variable `X` in `φ`.
If the coefficients form an integral domain, then `order` is a valuation
(`order_mul`, `order_add_ge`).
## Implementation notes
In this file we define multivariate formal power series with
variables indexed by `σ` and coefficients in `α` as
mv_power_series σ α := (σ →₀ ℕ) → α.
Unfortunately there is not yet enough API to show that they are the completion
of the ring of multivariate polynomials. However, we provide most of the infrastructure
that is needed to do this. Once I-adic completion (topological or algebraic) is available
it should not be hard to fill in the details.
Formal power series in one variable are defined as
power_series α := mv_power_series unit α.
This allows us to port a lot of proofs and properties
from the multivariate case to the single variable case.
However, it means that formal power series are indexed by (unit →₀ ℕ),
which is of course canonically isomorphic to ℕ.
We then build some glue to treat formal power series as if they are indexed by ℕ.
Occasionally this leads to proofs that are uglier than expected.
-/
noncomputable theory
open_locale classical
/-- Multivariate formal power series, where `σ` is the index set of the variables
and `α` is the coefficient ring.-/
def mv_power_series (σ : Type*) (α : Type*) := (σ →₀ ℕ) → α
namespace mv_power_series
open finsupp
variables {σ : Type*} {α : Type*}
instance [inhabited α] : inhabited (mv_power_series σ α) := ⟨λ _, default _⟩
instance [has_zero α] : has_zero (mv_power_series σ α) := pi.has_zero
instance [add_monoid α] : add_monoid (mv_power_series σ α) := pi.add_monoid
instance [add_group α] : add_group (mv_power_series σ α) := pi.add_group
instance [add_comm_monoid α] : add_comm_monoid (mv_power_series σ α) := pi.add_comm_monoid
instance [add_comm_group α] : add_comm_group (mv_power_series σ α) := pi.add_comm_group
section add_monoid
variables [add_monoid α]
variables (α)
/-- The `n`th monomial with coefficient `a` as multivariate formal power series.-/
def monomial (n : σ →₀ ℕ) : α →+ mv_power_series σ α :=
{ to_fun := λ a m, if m = n then a else 0,
map_zero' := funext $ λ m, by { split_ifs; refl },
map_add' := λ a b, funext $ λ m,
show (if m = n then a + b else 0) = (if m = n then a else 0) + (if m = n then b else 0),
from if h : m = n then by simp only [if_pos h] else by simp only [if_neg h, add_zero] }
/-- The `n`th coefficient of a multivariate formal power series.-/
def coeff (n : σ →₀ ℕ) : (mv_power_series σ α) →+ α :=
{ to_fun := λ φ, φ n,
map_zero' := rfl,
map_add' := λ _ _, rfl }
variables {α}
/-- Two multivariate formal power series are equal if all their coefficients are equal.-/
@[ext] lemma ext {φ ψ} (h : ∀ (n : σ →₀ ℕ), coeff α n φ = coeff α n ψ) :
φ = ψ :=
funext h
/-- Two multivariate formal power series are equal
if and only if all their coefficients are equal.-/
lemma ext_iff {φ ψ : mv_power_series σ α} :
φ = ψ ↔ (∀ (n : σ →₀ ℕ), coeff α n φ = coeff α n ψ) :=
⟨λ h n, congr_arg (coeff α n) h, ext⟩
lemma coeff_monomial (m n : σ →₀ ℕ) (a : α) :
coeff α m (monomial α n a) = if m = n then a else 0 := rfl
@[simp] lemma coeff_monomial' (n : σ →₀ ℕ) (a : α) :
coeff α n (monomial α n a) = a := if_pos rfl
@[simp] lemma coeff_comp_monomial (n : σ →₀ ℕ) :
(coeff α n).comp (monomial α n) = add_monoid_hom.id α :=
add_monoid_hom.ext $ coeff_monomial' n
@[simp] lemma coeff_zero (n : σ →₀ ℕ) : coeff α n (0 : mv_power_series σ α) = 0 := rfl
end add_monoid
section semiring
variables [semiring α] (n : σ →₀ ℕ) (φ ψ : mv_power_series σ α)
instance : has_one (mv_power_series σ α) := ⟨monomial α (0 : σ →₀ ℕ) 1⟩
lemma coeff_one :
coeff α n (1 : mv_power_series σ α) = if n = 0 then 1 else 0 := rfl
lemma coeff_zero_one : coeff α (0 : σ →₀ ℕ) 1 = 1 :=
coeff_monomial' 0 1
instance : has_mul (mv_power_series σ α) :=
⟨λ φ ψ n, (finsupp.antidiagonal n).support.sum (λ p, φ p.1 * ψ p.2)⟩
lemma coeff_mul : coeff α n (φ * ψ) =
(finsupp.antidiagonal n).support.sum (λ p, coeff α p.1 φ * coeff α p.2 ψ) := rfl
protected lemma zero_mul : (0 : mv_power_series σ α) * φ = 0 :=
ext $ λ n, by simp [coeff_mul]
protected lemma mul_zero : φ * 0 = 0 :=
ext $ λ n, by simp [coeff_mul]
protected lemma one_mul : (1 : mv_power_series σ α) * φ = φ :=
ext $ λ n,
begin
rw [coeff_mul, finset.sum_eq_single ((0 : σ →₀ ℕ), n)];
simp [mem_antidiagonal_support, coeff_one],
show ∀ (i j : σ →₀ ℕ), i + j = n → (i = 0 → j ≠ n) →
(if i = 0 then coeff α j φ else 0) = 0,
intros i j hij h,
rw [if_neg],
contrapose! h,
simpa [h] using hij,
end
protected lemma mul_one : φ * 1 = φ :=
ext $ λ n,
begin
rw [coeff_mul, finset.sum_eq_single (n, (0 : σ →₀ ℕ))],
rotate,
{ rintros ⟨i, j⟩ hij h,
rw [coeff_one, if_neg, mul_zero],
rw mem_antidiagonal_support at hij,
contrapose! h,
simpa [h] using hij },
all_goals { simp [mem_antidiagonal_support, coeff_one] }
end
protected lemma mul_add (φ₁ φ₂ φ₃ : mv_power_series σ α) :
φ₁ * (φ₂ + φ₃) = φ₁ * φ₂ + φ₁ * φ₃ :=
ext $ λ n, by simp only [coeff_mul, mul_add, finset.sum_add_distrib, add_monoid_hom.map_add]
protected lemma add_mul (φ₁ φ₂ φ₃ : mv_power_series σ α) :
(φ₁ + φ₂) * φ₃ = φ₁ * φ₃ + φ₂ * φ₃ :=
ext $ λ n, by simp only [coeff_mul, add_mul, finset.sum_add_distrib, add_monoid_hom.map_add]
protected lemma mul_assoc (φ₁ φ₂ φ₃ : mv_power_series σ α) :
(φ₁ * φ₂) * φ₃ = φ₁ * (φ₂ * φ₃) :=
ext $ λ n,
begin
simp only [coeff_mul],
have := @finset.sum_sigma ((σ →₀ ℕ) × (σ →₀ ℕ)) α _ _ (antidiagonal n).support
(λ p, (antidiagonal (p.1)).support) (λ x, coeff α x.2.1 φ₁ * coeff α x.2.2 φ₂ * coeff α x.1.2 φ₃),
convert this.symm using 1; clear this,
{ apply finset.sum_congr rfl,
intros p hp, exact finset.sum_mul },
have := @finset.sum_sigma ((σ →₀ ℕ) × (σ →₀ ℕ)) α _ _ (antidiagonal n).support
(λ p, (antidiagonal (p.2)).support) (λ x, coeff α x.1.1 φ₁ * (coeff α x.2.1 φ₂ * coeff α x.2.2 φ₃)),
convert this.symm using 1; clear this,
{ apply finset.sum_congr rfl, intros p hp, rw finset.mul_sum },
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_antidiagonal_support] at H ⊢, finish },
{ 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_antidiagonal_support,
and_imp, prod.mk.inj_iff, add_comm, heq_iff_eq] at H₁ H₂ ⊢,
finish },
{ rintros ⟨⟨i,j⟩, ⟨k,l⟩⟩ H, refine ⟨⟨(i+k, l), (i, k)⟩, _, _⟩;
{ simp only [finset.mem_sigma, mem_antidiagonal_support] at H ⊢, finish } }
end
instance : semiring (mv_power_series σ α) :=
{ mul_one := mv_power_series.mul_one,
one_mul := mv_power_series.one_mul,
mul_assoc := mv_power_series.mul_assoc,
mul_zero := mv_power_series.mul_zero,
zero_mul := mv_power_series.zero_mul,
left_distrib := mv_power_series.mul_add,
right_distrib := mv_power_series.add_mul,
.. mv_power_series.has_one,
.. mv_power_series.has_mul,
.. mv_power_series.add_comm_monoid }
end semiring
instance [comm_semiring α] : comm_semiring (mv_power_series σ α) :=
{ mul_comm := λ φ ψ, ext $ λ n, finset.sum_bij (λ p hp, p.swap)
(λ p hp, swap_mem_antidiagonal_support hp)
(λ p hp, mul_comm _ _)
(λ p q hp hq H, by simpa using congr_arg prod.swap H)
(λ p hp, ⟨p.swap, swap_mem_antidiagonal_support hp, p.swap_swap.symm⟩),
.. mv_power_series.semiring }
instance [ring α] : ring (mv_power_series σ α) :=
{ .. mv_power_series.semiring,
.. mv_power_series.add_comm_group }
instance [comm_ring α] : comm_ring (mv_power_series σ α) :=
{ .. mv_power_series.comm_semiring,
.. mv_power_series.add_comm_group }
section semiring
variables [semiring α]
lemma monomial_mul_monomial (m n : σ →₀ ℕ) (a b : α) :
monomial α m a * monomial α n b = monomial α (m + n) (a * b) :=
begin
ext k, rw [coeff_mul, coeff_monomial], split_ifs with h,
{ rw [h, finset.sum_eq_single (m,n)],
{ rw [coeff_monomial', coeff_monomial'] },
{ rintros ⟨i,j⟩ hij hne,
rw [ne.def, prod.mk.inj_iff, not_and] at hne,
by_cases H : i = m,
{ rw [coeff_monomial j n b, if_neg (hne H), mul_zero] },
{ rw [coeff_monomial, if_neg H, zero_mul] } },
{ intro H, rw finsupp.mem_antidiagonal_support at H,
exfalso, exact H rfl } },
{ rw [finset.sum_eq_zero], rintros ⟨i,j⟩ hij,
rw finsupp.mem_antidiagonal_support at hij,
by_cases H : i = m,
{ subst i, have : j ≠ n, { rintro rfl, exact h hij.symm },
{ rw [coeff_monomial j n b, if_neg this, mul_zero] } },
{ rw [coeff_monomial, if_neg H, zero_mul] } }
end
variables (σ) (α)
/-- The constant multivariate formal power series.-/
def C : α →+* mv_power_series σ α :=
{ map_one' := rfl,
map_mul' := λ a b, (monomial_mul_monomial 0 0 a b).symm,
.. monomial α (0 : σ →₀ ℕ) }
variables {σ} {α}
@[simp] lemma monomial_zero_eq_C : monomial α (0 : σ →₀ ℕ) = C σ α := rfl
lemma monomial_zero_eq_C_apply (a : α) : monomial α (0 : σ →₀ ℕ) a = C σ α a := rfl
lemma coeff_C (n : σ →₀ ℕ) (a : α) :
coeff α n (C σ α a) = if n = 0 then a else 0 := rfl
lemma coeff_zero_C (a : α) : coeff α (0 : σ →₀ℕ) (C σ α a) = a :=
coeff_monomial' 0 a
/-- The variables of the multivariate formal power series ring.-/
def X (s : σ) : mv_power_series σ α := monomial α (single s 1) 1
lemma coeff_X (n : σ →₀ ℕ) (s : σ) :
coeff α n (X s : mv_power_series σ α) = if n = (single s 1) then 1 else 0 := rfl
lemma coeff_index_single_X (s t : σ) :
coeff α (single t 1) (X s : mv_power_series σ α) = if t = s then 1 else 0 :=
by { simp only [coeff_X, single_left_inj one_ne_zero], split_ifs; refl }
@[simp] lemma coeff_index_single_self_X (s : σ) :
coeff α (single s 1) (X s : mv_power_series σ α) = 1 :=
if_pos rfl
lemma coeff_zero_X (s : σ) : coeff α (0 : σ →₀ ℕ) (X s : mv_power_series σ α) = 0 :=
by { rw [coeff_X, if_neg], intro h, exact one_ne_zero (single_eq_zero.mp h.symm) }
lemma X_def (s : σ) : X s = monomial α (single s 1) 1 := rfl
lemma X_pow_eq (s : σ) (n : ℕ) :
(X s : mv_power_series σ α)^n = monomial α (single s n) 1 :=
begin
induction n with n ih,
{ rw [pow_zero, finsupp.single_zero], refl },
{ rw [pow_succ', ih, nat.succ_eq_add_one, finsupp.single_add, X, monomial_mul_monomial, one_mul] }
end
lemma coeff_X_pow (m : σ →₀ ℕ) (s : σ) (n : ℕ) :
coeff α m ((X s : mv_power_series σ α)^n) = if m = single s n then 1 else 0 :=
by rw [X_pow_eq s n, coeff_monomial]
@[simp] lemma coeff_mul_C (n : σ →₀ ℕ) (φ : mv_power_series σ α) (a : α) :
coeff α n (φ * (C σ α a)) = (coeff α n φ) * a :=
begin
rw [coeff_mul n φ], rw [finset.sum_eq_single (n,(0 : σ →₀ ℕ))],
{ rw [coeff_C, if_pos rfl] },
{ rintro ⟨i,j⟩ hij hne,
rw finsupp.mem_antidiagonal_support at hij,
by_cases hj : j = 0,
{ subst hj, simp at *, contradiction },
{ rw [coeff_C, if_neg hj, mul_zero] } },
{ intro h, exfalso, apply h,
rw finsupp.mem_antidiagonal_support,
apply add_zero }
end
lemma coeff_zero_mul_X (φ : mv_power_series σ α) (s : σ) :
coeff α (0 : σ →₀ ℕ) (φ * X s) = 0 :=
begin
rw [coeff_mul _ φ, finset.sum_eq_zero],
rintro ⟨i,j⟩ hij,
obtain ⟨rfl, rfl⟩ : i = 0 ∧ j = 0,
{ rw finsupp.mem_antidiagonal_support at hij,
simpa using hij },
simp [coeff_zero_X]
end
variables (σ) (α)
/-- The constant coefficient of a formal power series.-/
def constant_coeff : (mv_power_series σ α) →+* α :=
{ to_fun := coeff α (0 : σ →₀ ℕ),
map_one' := coeff_zero_one,
map_mul' := λ φ ψ, by simp [coeff_mul, support_single_ne_zero],
.. coeff α (0 : σ →₀ ℕ) }
variables {σ} {α}
@[simp] lemma coeff_zero_eq_constant_coeff :
coeff α (0 : σ →₀ ℕ) = constant_coeff σ α := rfl
lemma coeff_zero_eq_constant_coeff_apply (φ : mv_power_series σ α) :
coeff α (0 : σ →₀ ℕ) φ = constant_coeff σ α φ := rfl
@[simp] lemma constant_coeff_C (a : α) : constant_coeff σ α (C σ α a) = a := rfl
@[simp] lemma constant_coeff_comp_C :
(constant_coeff σ α).comp (C σ α) = ring_hom.id α := rfl
@[simp] lemma constant_coeff_zero : constant_coeff σ α 0 = 0 := rfl
@[simp] lemma constant_coeff_one : constant_coeff σ α 1 = 1 := rfl
@[simp] lemma constant_coeff_X (s : σ) : constant_coeff σ α (X s) = 0 := coeff_zero_X s
/-- If a multivariate formal power series is invertible,
then so is its constant coefficient.-/
lemma is_unit_constant_coeff (φ : mv_power_series σ α) (h : is_unit φ) :
is_unit (constant_coeff σ α φ) :=
h.map' (constant_coeff σ α)
instance : semimodule α (mv_power_series σ α) :=
{ smul := λ a φ, C σ α a * φ,
one_smul := λ φ, one_mul _,
mul_smul := λ a b φ, by simp [ring_hom.map_mul, mul_assoc],
smul_add := λ a φ ψ, mul_add _ _ _,
smul_zero := λ a, mul_zero _,
add_smul := λ a b φ, by simp only [ring_hom.map_add, add_mul],
zero_smul := λ φ, by simp only [zero_mul, ring_hom.map_zero] }
end semiring
instance [ring α] : module α (mv_power_series σ α) :=
{ ..mv_power_series.semimodule }
instance [comm_ring α] : algebra α (mv_power_series σ α) :=
{ commutes' := λ _ _, mul_comm _ _,
smul_def' := λ c p, rfl,
.. C σ α, .. mv_power_series.module }
section map
variables {β : Type*} {γ : Type*} [semiring α] [semiring β] [semiring γ]
variables (f : α →+* β) (g : β →+* γ)
variable (σ)
/-- The map between multivariate formal power series induced by a map on the coefficients.-/
def map : mv_power_series σ α →+* mv_power_series σ β :=
{ to_fun := λ φ n, f $ coeff α n φ,
map_zero' := ext $ λ n, f.map_zero,
map_one' := ext $ λ n, show f ((coeff α n) 1) = (coeff β n) 1,
by { rw [coeff_one, coeff_one], split_ifs; simp [f.map_one, f.map_zero] },
map_add' := λ φ ψ, ext $ λ n,
show f ((coeff α n) (φ + ψ)) = f ((coeff α n) φ) + f ((coeff α n) ψ), by simp,
map_mul' := λ φ ψ, ext $ λ n, show f _ = _,
begin
rw [coeff_mul, ← finset.sum_hom _ f, coeff_mul, finset.sum_congr rfl],
rintros ⟨i,j⟩ hij, rw [f.map_mul], refl,
end }
variable {σ}
@[simp] lemma map_id : map σ (ring_hom.id α) = ring_hom.id _ := rfl
lemma map_comp : map σ (g.comp f) = (map σ g).comp (map σ f) := rfl
@[simp] lemma coeff_map (n : σ →₀ ℕ) (φ : mv_power_series σ α) :
coeff β n (map σ f φ) = f (coeff α n φ) := rfl
@[simp] lemma constant_coeff_map (φ : mv_power_series σ α) :
constant_coeff σ β (map σ f φ) = f (constant_coeff σ α φ) := rfl
end map
section trunc
variables [comm_semiring α] (n : σ →₀ ℕ)
-- Auxiliary definition for the truncation function.
def trunc_fun (φ : mv_power_series σ α) : mv_polynomial σ α :=
{ support := (n.antidiagonal.support.image prod.fst).filter (λ m, coeff α m φ ≠ 0),
to_fun := λ m, if m ≤ n then coeff α m φ else 0,
mem_support_to_fun := λ m,
begin
suffices : m ∈ finset.image prod.fst ((antidiagonal n).support) ↔ m ≤ n,
{ rw [finset.mem_filter, this], split,
{ intro h, rw [if_pos h.1], exact h.2 },
{ intro h, split_ifs at h with H H,
{ exact ⟨H, h⟩ },
{ exfalso, exact h rfl } } },
rw finset.mem_image, split,
{ rintros ⟨⟨i,j⟩, h, rfl⟩ s,
rw finsupp.mem_antidiagonal_support at h,
rw ← h, exact nat.le_add_right _ _ },
{ intro h, refine ⟨(m, n-m), _, rfl⟩,
rw finsupp.mem_antidiagonal_support, ext s, exact nat.add_sub_of_le (h s) }
end }
variable (α)
/-- The `n`th truncation of a multivariate formal power series to a multivariate polynomial -/
def trunc : mv_power_series σ α →+ mv_polynomial σ α :=
{ to_fun := trunc_fun n,
map_zero' := mv_polynomial.ext _ _ $ λ m, by { change ite _ _ _ = _, split_ifs; refl },
map_add' := λ φ ψ, mv_polynomial.ext _ _ $ λ m,
begin
rw mv_polynomial.coeff_add,
change ite _ _ _ = ite _ _ _ + ite _ _ _,
split_ifs with H, {refl}, {rw [zero_add]}
end }
variable {α}
lemma coeff_trunc (m : σ →₀ ℕ) (φ : mv_power_series σ α) :
mv_polynomial.coeff m (trunc α n φ) =
if m ≤ n then coeff α m φ else 0 := rfl
@[simp] lemma trunc_one : trunc α n 1 = 1 :=
mv_polynomial.ext _ _ $ λ m,
begin
rw [coeff_trunc, coeff_one],
split_ifs with H H' H',
{ subst m, erw mv_polynomial.coeff_C 0, simp },
{ symmetry, erw mv_polynomial.coeff_monomial, convert if_neg (ne.elim (ne.symm H')), },
{ symmetry, erw mv_polynomial.coeff_monomial, convert if_neg _,
intro H', apply H, subst m, intro s, exact nat.zero_le _ }
end
@[simp] lemma trunc_C (a : α) : trunc α n (C σ α a) = mv_polynomial.C a :=
mv_polynomial.ext _ _ $ λ m,
begin
rw [coeff_trunc, coeff_C, mv_polynomial.coeff_C],
split_ifs with H; refl <|> try {simp * at *},
exfalso, apply H, subst m, intro s, exact nat.zero_le _
end
end trunc
section comm_semiring
variable [comm_semiring α]
lemma X_pow_dvd_iff {s : σ} {n : ℕ} {φ : mv_power_series σ α} :
(X s : mv_power_series σ α)^n ∣ φ ↔ ∀ m : σ →₀ ℕ, m s < n → coeff α m φ = 0 :=
begin
split,
{ rintros ⟨φ, rfl⟩ m h,
rw [coeff_mul, finset.sum_eq_zero],
rintros ⟨i,j⟩ hij, rw [coeff_X_pow, if_neg, zero_mul],
contrapose! h, subst i, rw finsupp.mem_antidiagonal_support at hij,
rw [← hij, finsupp.add_apply, finsupp.single_eq_same], exact nat.le_add_right n _ },
{ intro h, refine ⟨λ m, coeff α (m + (single s n)) φ, _⟩,
ext m, by_cases H : m - single s n + single s n = m,
{ rw [coeff_mul, finset.sum_eq_single (single s n, m - single s n)],
{ rw [coeff_X_pow, if_pos rfl, one_mul],
simpa using congr_arg (λ (m : σ →₀ ℕ), coeff α m φ) H.symm },
{ rintros ⟨i,j⟩ hij hne, rw finsupp.mem_antidiagonal_support at hij,
rw coeff_X_pow, split_ifs with hi,
{ exfalso, apply hne, rw [← hij, ← hi, prod.mk.inj_iff], refine ⟨rfl, _⟩,
ext t, simp only [nat.add_sub_cancel_left, finsupp.add_apply, finsupp.nat_sub_apply] },
{ exact zero_mul _ } },
{ intro hni, exfalso, apply hni, rwa [finsupp.mem_antidiagonal_support, add_comm] } },
{ rw [h, coeff_mul, finset.sum_eq_zero],
{ rintros ⟨i,j⟩ hij, rw finsupp.mem_antidiagonal_support at hij,
rw coeff_X_pow, split_ifs with hi,
{ exfalso, apply H, rw [← hij, hi], ext, simp, cc },
{ exact zero_mul _ } },
{ classical, contrapose! H, ext t,
by_cases hst : s = t,
{ subst t, simpa using nat.sub_add_cancel H },
{ simp [finsupp.single_apply, hst] } } } }
end
lemma X_dvd_iff {s : σ} {φ : mv_power_series σ α} :
(X s : mv_power_series σ α) ∣ φ ↔ ∀ m : σ →₀ ℕ, m s = 0 → coeff α m φ = 0 :=
begin
rw [← pow_one (X s : mv_power_series σ α), X_pow_dvd_iff],
split; intros h m hm,
{ exact h m (hm.symm ▸ zero_lt_one) },
{ exact h m (nat.eq_zero_of_le_zero $ nat.le_of_succ_le_succ hm) }
end
end comm_semiring
section ring
variables [ring α]
/-
The inverse of a multivariate formal power series is defined by
well-founded recursion on the coeffients of the inverse.
-/
/-- Auxiliary definition that unifies
the totalised inverse formal power series `(_)⁻¹` and
the inverse formal power series that depends on
an inverse of the constant coefficient `inv_of_unit`.-/
protected noncomputable def inv.aux (a : α) (φ : mv_power_series σ α) : mv_power_series σ α
| n := if n = 0 then a else
- a * n.antidiagonal.support.sum (λ (x : (σ →₀ ℕ) × (σ →₀ ℕ)),
if h : x.2 < n then coeff α x.1 φ * inv.aux x.2 else 0)
using_well_founded
{ rel_tac := λ _ _, `[exact ⟨_, finsupp.lt_wf σ⟩],
dec_tac := tactic.assumption }
lemma coeff_inv_aux (n : σ →₀ ℕ) (a : α) (φ : mv_power_series σ α) :
coeff α n (inv.aux a φ) = if n = 0 then a else
- a * n.antidiagonal.support.sum (λ (x : (σ →₀ ℕ) × (σ →₀ ℕ)),
if x.2 < n then coeff α x.1 φ * coeff α x.2 (inv.aux a φ) else 0) :=
show inv.aux a φ n = _, by { rw inv.aux, refl }
/-- A multivariate formal power series is invertible if the constant coefficient is invertible.-/
def inv_of_unit (φ : mv_power_series σ α) (u : units α) : mv_power_series σ α :=
inv.aux (↑u⁻¹) φ
lemma coeff_inv_of_unit (n : σ →₀ ℕ) (φ : mv_power_series σ α) (u : units α) :
coeff α n (inv_of_unit φ u) = if n = 0 then ↑u⁻¹ else
- ↑u⁻¹ * n.antidiagonal.support.sum (λ (x : (σ →₀ ℕ) × (σ →₀ ℕ)),
if x.2 < n then coeff α x.1 φ * coeff α x.2 (inv_of_unit φ u) else 0) :=
coeff_inv_aux n (↑u⁻¹) φ
@[simp] lemma constant_coeff_inv_of_unit (φ : mv_power_series σ α) (u : units α) :
constant_coeff σ α (inv_of_unit φ u) = ↑u⁻¹ :=
by rw [← coeff_zero_eq_constant_coeff_apply, coeff_inv_of_unit, if_pos rfl]
lemma mul_inv_of_unit (φ : mv_power_series σ α) (u : units α) (h : constant_coeff σ α φ = u) :
φ * inv_of_unit φ u = 1 :=
ext $ λ n, if H : n = 0 then by { rw H, simp [coeff_mul, support_single_ne_zero, h], }
else
begin
have : ((0 : σ →₀ ℕ), n) ∈ n.antidiagonal.support,
{ rw [finsupp.mem_antidiagonal_support, zero_add] },
rw [coeff_one, if_neg H, coeff_mul,
← finset.insert_erase this, finset.sum_insert (finset.not_mem_erase _ _),
coeff_zero_eq_constant_coeff_apply, h, coeff_inv_of_unit, if_neg H,
neg_mul_eq_neg_mul_symm, mul_neg_eq_neg_mul_symm, units.mul_inv_cancel_left,
← finset.insert_erase this, finset.sum_insert (finset.not_mem_erase _ _),
finset.insert_erase this, if_neg (not_lt_of_ge $ le_refl _), zero_add, add_comm,
← sub_eq_add_neg, sub_eq_zero, finset.sum_congr rfl],
rintros ⟨i,j⟩ hij, rw [finset.mem_erase, finsupp.mem_antidiagonal_support] at hij,
cases hij with h₁ h₂,
subst n, rw if_pos,
suffices : (0 : _) + j < i + j, {simpa},
apply add_lt_add_right,
split,
{ intro s, exact nat.zero_le _ },
{ intro H, apply h₁,
suffices : i = 0, {simp [this]},
ext1 s, exact nat.eq_zero_of_le_zero (H s) }
end
end ring
section comm_ring
variable [comm_ring α]
/-- Multivariate formal power series over a local ring form a local ring.-/
lemma is_local_ring (h : is_local_ring α) : is_local_ring (mv_power_series σ α) :=
begin
split,
{ have H : (0:α) ≠ 1 := ‹is_local_ring α›.1, contrapose! H,
simpa using congr_arg (constant_coeff σ α) H },
{ intro φ, rcases ‹is_local_ring α›.2 (constant_coeff σ α φ) with ⟨u,h⟩|⟨u,h⟩; [left, right];
{ refine is_unit_of_mul_eq_one _ _ (mul_inv_of_unit _ u _),
simpa using h.symm } }
end
-- TODO(jmc): once adic topology lands, show that this is complete
end comm_ring
section nonzero
variables [semiring α] [nonzero α]
instance : nonzero (mv_power_series σ α) :=
{ zero_ne_one := assume h, zero_ne_one $ show (0:α) = 1, from congr_arg (constant_coeff σ α) h }
lemma X_inj {s t : σ} : (X s : mv_power_series σ α) = X t ↔ s = t :=
⟨begin
intro h, replace h := congr_arg (coeff α (single s 1)) h, rw [coeff_X, if_pos rfl, coeff_X] at h,
split_ifs at h with H,
{ rw finsupp.single_eq_single_iff at H,
cases H, { exact H.1 }, { exfalso, exact one_ne_zero H.1 } },
{ exfalso, exact one_ne_zero h }
end, congr_arg X⟩
end nonzero
section local_ring
variables {β : Type*} [local_ring α] [local_ring β] (f : α →+* β) [is_local_ring_hom f]
instance : local_ring (mv_power_series σ α) :=
local_of_is_local_ring $ is_local_ring ⟨zero_ne_one, local_ring.is_local⟩
instance map.is_local_ring_hom : is_local_ring_hom (map σ f) :=
⟨begin
rintros φ ⟨ψ, h⟩,
replace h := congr_arg (constant_coeff σ β) h,
rw constant_coeff_map at h,
have : is_unit (constant_coeff σ β ↑ψ) := @is_unit_constant_coeff σ β _ (↑ψ) (is_unit_unit ψ),
rw h at this,
rcases is_unit_of_map_unit f _ this with ⟨c, hc⟩,
exact is_unit_of_mul_eq_one φ (inv_of_unit φ c) (mul_inv_of_unit φ c hc.symm)
end⟩
end local_ring
section field
variables [field α]
protected def inv (φ : mv_power_series σ α) : mv_power_series σ α :=
inv.aux (constant_coeff σ α φ)⁻¹ φ
instance : has_inv (mv_power_series σ α) := ⟨mv_power_series.inv⟩
lemma coeff_inv (n : σ →₀ ℕ) (φ : mv_power_series σ α) :
coeff α n (φ⁻¹) = if n = 0 then (constant_coeff σ α φ)⁻¹ else
- (constant_coeff σ α φ)⁻¹ * n.antidiagonal.support.sum (λ (x : (σ →₀ ℕ) × (σ →₀ ℕ)),
if x.2 < n then coeff α x.1 φ * coeff α x.2 (φ⁻¹) else 0) :=
coeff_inv_aux n _ φ
@[simp] lemma constant_coeff_inv (φ : mv_power_series σ α) :
constant_coeff σ α (φ⁻¹) = (constant_coeff σ α φ)⁻¹ :=
by rw [← coeff_zero_eq_constant_coeff_apply, coeff_inv, if_pos rfl]
lemma inv_eq_zero {φ : mv_power_series σ α} :
φ⁻¹ = 0 ↔ constant_coeff σ α φ = 0 :=
⟨λ h, by simpa using congr_arg (constant_coeff σ α) h,
λ h, ext $ λ n, by { rw coeff_inv, split_ifs;
simp only [h, mv_power_series.coeff_zero, zero_mul, inv_zero, neg_zero] }⟩
@[simp, priority 1100] lemma inv_of_unit_eq (φ : mv_power_series σ α) (h : constant_coeff σ α φ ≠ 0) :
inv_of_unit φ (units.mk0 _ h) = φ⁻¹ := rfl
@[simp] lemma inv_of_unit_eq' (φ : mv_power_series σ α) (u : units α) (h : constant_coeff σ α φ = u) :
inv_of_unit φ u = φ⁻¹ :=
begin
rw ← inv_of_unit_eq φ (h.symm ▸ u.coe_ne_zero),
congr' 1, rw [units.ext_iff], exact h.symm,
end
@[simp] protected lemma mul_inv (φ : mv_power_series σ α) (h : constant_coeff σ α φ ≠ 0) :
φ * φ⁻¹ = 1 :=
by rw [← inv_of_unit_eq φ h, mul_inv_of_unit φ (units.mk0 _ h) rfl]
@[simp] protected lemma inv_mul (φ : mv_power_series σ α) (h : constant_coeff σ α φ ≠ 0) :
φ⁻¹ * φ = 1 :=
by rw [mul_comm, φ.mul_inv h]
end field
end mv_power_series
namespace mv_polynomial
open finsupp
variables {σ : Type*} {α : Type*} [comm_semiring α]
/-- The natural inclusion from multivariate polynomials into multivariate formal power series.-/
instance coe_to_mv_power_series : has_coe (mv_polynomial σ α) (mv_power_series σ α) :=
⟨λ φ n, coeff n φ⟩
@[simp, norm_cast] lemma coeff_coe (φ : mv_polynomial σ α) (n : σ →₀ ℕ) :
mv_power_series.coeff α n ↑φ = coeff n φ := rfl
@[simp, norm_cast] lemma coe_monomial (n : σ →₀ ℕ) (a : α) :
(monomial n a : mv_power_series σ α) = mv_power_series.monomial α n a :=
mv_power_series.ext $ λ m,
begin
rw [coeff_coe, coeff_monomial, mv_power_series.coeff_monomial],
split_ifs with h₁ h₂; refl <|> subst m; contradiction
end
@[simp, norm_cast] lemma coe_zero : ((0 : mv_polynomial σ α) : mv_power_series σ α) = 0 := rfl
@[simp, norm_cast] lemma coe_one : ((1 : mv_polynomial σ α) : mv_power_series σ α) = 1 :=
coe_monomial _ _
@[simp, norm_cast] lemma coe_add (φ ψ : mv_polynomial σ α) :
((φ + ψ : mv_polynomial σ α) : mv_power_series σ α) = φ + ψ := rfl
@[simp, norm_cast] lemma coe_mul (φ ψ : mv_polynomial σ α) :
((φ * ψ : mv_polynomial σ α) : mv_power_series σ α) = φ * ψ :=
mv_power_series.ext $ λ n,
by simp only [coeff_coe, mv_power_series.coeff_mul, coeff_mul]
@[simp, norm_cast] lemma coe_C (a : α) :
((C a : mv_polynomial σ α) : mv_power_series σ α) = mv_power_series.C σ α a :=
coe_monomial _ _
@[simp, norm_cast] lemma coe_X (s : σ) :
((X s : mv_polynomial σ α) : mv_power_series σ α) = mv_power_series.X s :=
coe_monomial _ _
namespace coe_to_mv_power_series
instance : is_semiring_hom (coe : mv_polynomial σ α → mv_power_series σ α) :=
{ map_zero := coe_zero,
map_one := coe_one,
map_add := coe_add,
map_mul := coe_mul }
end coe_to_mv_power_series
end mv_polynomial
/-- Formal power series over the coefficient ring `α`.-/
def power_series (α : Type*) := mv_power_series unit α
namespace power_series
open finsupp (single)
variable {α : Type*}
instance [inhabited α] : inhabited (power_series α) := by delta power_series; apply_instance
instance [add_monoid α] : add_monoid (power_series α) := by delta power_series; apply_instance
instance [add_group α] : add_group (power_series α) := by delta power_series; apply_instance
instance [add_comm_monoid α] : add_comm_monoid (power_series α) := by delta power_series; apply_instance
instance [add_comm_group α] : add_comm_group (power_series α) := by delta power_series; apply_instance
instance [semiring α] : semiring (power_series α) := by delta power_series; apply_instance
instance [comm_semiring α] : comm_semiring (power_series α) := by delta power_series; apply_instance
instance [ring α] : ring (power_series α) := by delta power_series; apply_instance
instance [comm_ring α] : comm_ring (power_series α) := by delta power_series; apply_instance
instance [semiring α] [nonzero α] : nonzero (power_series α) := by delta power_series; apply_instance
instance [semiring α] : semimodule α (power_series α) := by delta power_series; apply_instance
instance [ring α] : module α (power_series α) := by delta power_series; apply_instance
instance [comm_ring α] : algebra α (power_series α) := by delta power_series; apply_instance
section add_monoid
variables (α) [add_monoid α]
/-- The `n`th coefficient of a formal power series.-/
def coeff (n : ℕ) : power_series α →+ α := mv_power_series.coeff α (single () n)
/-- The `n`th monomial with coefficient `a` as formal power series.-/
def monomial (n : ℕ) : α →+ power_series α := mv_power_series.monomial α (single () n)
variables {α}
lemma coeff_def {s : unit →₀ ℕ} {n : ℕ} (h : s () = n) :
coeff α n = mv_power_series.coeff α s :=
by erw [coeff, ← h, ← finsupp.unique_single s]
/-- Two formal power series are equal if all their coefficients are equal.-/
@[ext] lemma ext {φ ψ : power_series α} (h : ∀ n, coeff α n φ = coeff α n ψ) :
φ = ψ :=
mv_power_series.ext $ λ n,
by { rw ← coeff_def, { apply h }, refl }
/-- Two formal power series are equal if all their coefficients are equal.-/
lemma ext_iff {φ ψ : power_series α} : φ = ψ ↔ (∀ n, coeff α n φ = coeff α n ψ) :=
⟨λ h n, congr_arg (coeff α n) h, ext⟩
/-- Constructor for formal power series.-/
def mk {α} (f : ℕ → α) : power_series α := λ s, f (s ())
@[simp] lemma coeff_mk (n : ℕ) (f : ℕ → α) : coeff α n (mk f) = f n :=
congr_arg f finsupp.single_eq_same
lemma coeff_monomial (m n : ℕ) (a : α) :
coeff α m (monomial α n a) = if m = n then a else 0 :=
calc coeff α m (monomial α n a) = _ : mv_power_series.coeff_monomial _ _ _
... = if m = n then a else 0 :
by { simp only [finsupp.unique_single_eq_iff], split_ifs; refl }
lemma monomial_eq_mk (n : ℕ) (a : α) :
monomial α n a = mk (λ m, if m = n then a else 0) :=
ext $ λ m, by { rw [coeff_monomial, coeff_mk] }
@[simp] lemma coeff_monomial' (n : ℕ) (a : α) :
coeff α n (monomial α n a) = a :=
by convert if_pos rfl
@[simp] lemma coeff_comp_monomial (n : ℕ) :
(coeff α n).comp (monomial α n) = add_monoid_hom.id α :=
add_monoid_hom.ext $ coeff_monomial' n
end add_monoid
section semiring
variable [semiring α]
variable (α)
/--The constant coefficient of a formal power series. -/
def constant_coeff : power_series α →+* α := mv_power_series.constant_coeff unit α
/-- The constant formal power series.-/
def C : α →+* power_series α := mv_power_series.C unit α
variable {α}
/-- The variable of the formal power series ring.-/
def X : power_series α := mv_power_series.X ()
@[simp] lemma coeff_zero_eq_constant_coeff :
coeff α 0 = constant_coeff α :=
begin
rw [constant_coeff, ← mv_power_series.coeff_zero_eq_constant_coeff, coeff_def], refl
end
lemma coeff_zero_eq_constant_coeff_apply (φ : power_series α) :
coeff α 0 φ = constant_coeff α φ :=
by rw [coeff_zero_eq_constant_coeff]; refl
@[simp] lemma monomial_zero_eq_C : monomial α 0 = C α :=
by rw [monomial, finsupp.single_zero, mv_power_series.monomial_zero_eq_C, C]
lemma monomial_zero_eq_C_apply (a : α) : monomial α 0 a = C α a :=
by simp
lemma coeff_C (n : ℕ) (a : α) :
coeff α n (C α a : power_series α) = if n = 0 then a else 0 :=
by rw [← monomial_zero_eq_C_apply, coeff_monomial]
lemma coeff_zero_C (a : α) : coeff α 0 (C α a) = a :=
by rw [← monomial_zero_eq_C_apply, coeff_monomial' 0 a]
lemma X_eq : (X : power_series α) = monomial α 1 1 := rfl
lemma coeff_X (n : ℕ) :
coeff α n (X : power_series α) = if n = 1 then 1 else 0 :=
by rw [X_eq, coeff_monomial]
lemma coeff_zero_X : coeff α 0 (X : power_series α) = 0 :=
by rw [coeff, finsupp.single_zero, X, mv_power_series.coeff_zero_X]
@[simp] lemma coeff_one_X : coeff α 1 (X : power_series α) = 1 :=
by rw [coeff_X, if_pos rfl]
lemma X_pow_eq (n : ℕ) : (X : power_series α)^n = monomial α n 1 :=
mv_power_series.X_pow_eq _ n
lemma coeff_X_pow (m n : ℕ) :
coeff α m ((X : power_series α)^n) = if m = n then 1 else 0 :=
by rw [X_pow_eq, coeff_monomial]
@[simp] lemma coeff_X_pow_self (n : ℕ) :
coeff α n ((X : power_series α)^n) = 1 :=
by rw [coeff_X_pow, if_pos rfl]
@[simp] lemma coeff_one (n : ℕ) :
coeff α n (1 : power_series α) = if n = 0 then 1 else 0 :=
calc coeff α n (1 : power_series α) = _ : mv_power_series.coeff_one _
... = if n = 0 then 1 else 0 :
by { simp only [finsupp.single_eq_zero], split_ifs; refl }
lemma coeff_zero_one : coeff α 0 (1 : power_series α) = 1 :=
coeff_zero_C 1
lemma coeff_mul (n : ℕ) (φ ψ : power_series α) :
coeff α n (φ * ψ) = (finset.nat.antidiagonal n).sum (λ p, coeff α p.1 φ * coeff α p.2 ψ) :=
begin
symmetry,
apply finset.sum_bij (λ (p : ℕ × ℕ) h, (single () p.1, single () p.2)),
{ rintros ⟨i,j⟩ hij, rw finset.nat.mem_antidiagonal at hij,
rw [finsupp.mem_antidiagonal_support, ← finsupp.single_add, hij], },
{ rintros ⟨i,j⟩ hij, refl },
{ rintros ⟨i,j⟩ ⟨k,l⟩ hij hkl,
simpa only [prod.mk.inj_iff, finsupp.unique_single_eq_iff] using id },
{ rintros ⟨f,g⟩ hfg,
refine ⟨(f (), g ()), _, _⟩,
{ rw finsupp.mem_antidiagonal_support at hfg,
rw [finset.nat.mem_antidiagonal, ← finsupp.add_apply, hfg, finsupp.single_eq_same] },
{ rw prod.mk.inj_iff, dsimp,
exact ⟨finsupp.unique_single f, finsupp.unique_single g⟩ } }
end
@[simp] lemma coeff_mul_C (n : ℕ) (φ : power_series α) (a : α) :
coeff α n (φ * (C α a)) = (coeff α n φ) * a :=
mv_power_series.coeff_mul_C _ φ a
@[simp] lemma coeff_succ_mul_X (n : ℕ) (φ : power_series α) :
coeff α (n+1) (φ * X) = coeff α n φ :=
begin
rw [coeff_mul _ φ, finset.sum_eq_single (n,1)],
{ rw [coeff_X, if_pos rfl, mul_one] },
{ rintro ⟨i,j⟩ hij hne,
by_cases hj : j = 1,
{ subst hj, simp at *, contradiction },
{ simp [coeff_X, hj] } },
{ intro h, exfalso, apply h, simp },
end
@[simp] lemma constant_coeff_C (a : α) : constant_coeff α (C α a) = a := rfl
@[simp] lemma constant_coeff_comp_C :
(constant_coeff α).comp (C α) = ring_hom.id α := rfl
@[simp] lemma constant_coeff_zero : constant_coeff α 0 = 0 := rfl
@[simp] lemma constant_coeff_one : constant_coeff α 1 = 1 := rfl
@[simp] lemma constant_coeff_X : constant_coeff α X = 0 := mv_power_series.coeff_zero_X _
lemma coeff_zero_mul_X (φ : power_series α) : coeff α 0 (φ * X) = 0 := by simp
/-- If a formal power series is invertible, then so is its constant coefficient.-/
lemma is_unit_constant_coeff (φ : power_series α) (h : is_unit φ) :
is_unit (constant_coeff α φ) :=
mv_power_series.is_unit_constant_coeff φ h
section map
variables {β : Type*} {γ : Type*} [semiring β] [semiring γ]
variables (f : α →+* β) (g : β →+* γ)
/-- The map between formal power series induced by a map on the coefficients.-/
def map : power_series α →+* power_series β :=
mv_power_series.map _ f
@[simp] lemma map_id : (map (ring_hom.id α) :
power_series α → power_series α) = id := rfl
lemma map_comp : map (g.comp f) = (map g).comp (map f) := rfl
@[simp] lemma coeff_map (n : ℕ) (φ : power_series α) :
coeff β n (map f φ) = f (coeff α n φ) := rfl
end map
end semiring
section comm_semiring
variables [comm_semiring α]
lemma X_pow_dvd_iff {n : ℕ} {φ : power_series α} :
(X : power_series α)^n ∣ φ ↔ ∀ m, m < n → coeff α m φ = 0 :=
begin
convert @mv_power_series.X_pow_dvd_iff unit α _ () n φ, apply propext,
classical, split; intros h m hm,
{ rw finsupp.unique_single m, convert h _ hm },
{ apply h, simpa only [finsupp.single_eq_same] using hm }
end
lemma X_dvd_iff {φ : power_series α} :
(X : power_series α) ∣ φ ↔ constant_coeff α φ = 0 :=
begin
rw [← pow_one (X : power_series α), X_pow_dvd_iff, ← coeff_zero_eq_constant_coeff_apply],
split; intro h,
{ exact h 0 zero_lt_one },
{ intros m hm, rwa nat.eq_zero_of_le_zero (nat.le_of_succ_le_succ hm) }
end
section trunc
/-- The `n`th truncation of a formal power series to a polynomial -/
def trunc (n : ℕ) (φ : power_series α) : polynomial α :=
{ support := ((finset.nat.antidiagonal n).image prod.fst).filter (λ m, coeff α m φ ≠ 0),
to_fun := λ m, if m ≤ n then coeff α m φ else 0,
mem_support_to_fun := λ m,
begin
suffices : m ∈ ((finset.nat.antidiagonal n).image prod.fst) ↔ m ≤ n,
{ rw [finset.mem_filter, this], split,
{ intro h, rw [if_pos h.1], exact h.2 },
{ intro h, split_ifs at h with H H,
{ exact ⟨H, h⟩ },
{ exfalso, exact h rfl } } },
rw finset.mem_image, split,
{ rintros ⟨⟨i,j⟩, h, rfl⟩,
rw finset.nat.mem_antidiagonal at h,
rw ← h, exact nat.le_add_right _ _ },
{ intro h, refine ⟨(m, n-m), _, rfl⟩,
rw finset.nat.mem_antidiagonal, exact nat.add_sub_of_le h }
end }
lemma coeff_trunc (m) (n) (φ : power_series α) :
polynomial.coeff (trunc n φ) m = if m ≤ n then coeff α m φ else 0 := rfl
@[simp] lemma trunc_zero (n) : trunc n (0 : power_series α) = 0 :=
polynomial.ext $ λ m,
begin
rw [coeff_trunc, add_monoid_hom.map_zero, polynomial.coeff_zero],
split_ifs; refl
end
@[simp] lemma trunc_one (n) : trunc n (1 : power_series α) = 1 :=
polynomial.ext $ λ m,
begin
rw [coeff_trunc, coeff_one],
split_ifs with H H' H'; rw [polynomial.coeff_one],
{ subst m, rw [if_pos rfl] },
{ symmetry, exact if_neg (ne.elim (ne.symm H')) },
{ symmetry, refine if_neg _,
intro H', apply H, subst m, exact nat.zero_le _ }
end
@[simp] lemma trunc_C (n) (a : α) : trunc n (C α a) = polynomial.C a :=
polynomial.ext $ λ m,
begin
rw [coeff_trunc, coeff_C, polynomial.coeff_C],
split_ifs with H; refl <|> try {simp * at *}
end
@[simp] lemma trunc_add (n) (φ ψ : power_series α) :
trunc n (φ + ψ) = trunc n φ + trunc n ψ :=
polynomial.ext $ λ m,
begin
simp only [coeff_trunc, add_monoid_hom.map_add, polynomial.coeff_add],
split_ifs with H, {refl}, {rw [zero_add]}
end
end trunc
end comm_semiring
section ring
variables [ring α]
protected def inv.aux : α → power_series α → power_series α :=
mv_power_series.inv.aux
lemma coeff_inv_aux (n : ℕ) (a : α) (φ : power_series α) :
coeff α n (inv.aux a φ) = if n = 0 then a else
- a * (finset.nat.antidiagonal n).sum (λ (x : ℕ × ℕ),
if x.2 < n then coeff α x.1 φ * coeff α x.2 (inv.aux a φ) else 0) :=
begin
rw [coeff, inv.aux, mv_power_series.coeff_inv_aux],
simp only [finsupp.single_eq_zero],
split_ifs, {refl},
congr' 1,
symmetry,
apply finset.sum_bij (λ (p : ℕ × ℕ) h, (single () p.1, single () p.2)),
{ rintros ⟨i,j⟩ hij, rw finset.nat.mem_antidiagonal at hij,
rw [finsupp.mem_antidiagonal_support, ← finsupp.single_add, hij], },
{ rintros ⟨i,j⟩ hij,
by_cases H : j < n,
{ rw [if_pos H, if_pos], {refl},
split,
{ rintro ⟨⟩, simpa [finsupp.single_eq_same] using le_of_lt H },
{ intro hh, rw lt_iff_not_ge at H, apply H,
simpa [finsupp.single_eq_same] using hh () } },
{ rw [if_neg H, if_neg], rintro ⟨h₁, h₂⟩, apply h₂, rintro ⟨⟩,
simpa [finsupp.single_eq_same] using not_lt.1 H } },
{ rintros ⟨i,j⟩ ⟨k,l⟩ hij hkl,
simpa only [prod.mk.inj_iff, finsupp.unique_single_eq_iff] using id },
{ rintros ⟨f,g⟩ hfg,
refine ⟨(f (), g ()), _, _⟩,
{ rw finsupp.mem_antidiagonal_support at hfg,
rw [finset.nat.mem_antidiagonal, ← finsupp.add_apply, hfg, finsupp.single_eq_same] },
{ rw prod.mk.inj_iff, dsimp,
exact ⟨finsupp.unique_single f, finsupp.unique_single g⟩ } }
end
/-- A formal power series is invertible if the constant coefficient is invertible.-/
def inv_of_unit (φ : power_series α) (u : units α) : power_series α :=
mv_power_series.inv_of_unit φ u
lemma coeff_inv_of_unit (n : ℕ) (φ : power_series α) (u : units α) :
coeff α n (inv_of_unit φ u) = if n = 0 then ↑u⁻¹ else
- ↑u⁻¹ * (finset.nat.antidiagonal n).sum (λ (x : ℕ × ℕ),
if x.2 < n then coeff α x.1 φ * coeff α x.2 (inv_of_unit φ u) else 0) :=
coeff_inv_aux n ↑u⁻¹ φ
@[simp] lemma constant_coeff_inv_of_unit (φ : power_series α) (u : units α) :
constant_coeff α (inv_of_unit φ u) = ↑u⁻¹ :=
by rw [← coeff_zero_eq_constant_coeff_apply, coeff_inv_of_unit, if_pos rfl]
lemma mul_inv_of_unit (φ : power_series α) (u : units α) (h : constant_coeff α φ = u) :
φ * inv_of_unit φ u = 1 :=
mv_power_series.mul_inv_of_unit φ u $ h
end ring
section integral_domain
variable [integral_domain α]
lemma eq_zero_or_eq_zero_of_mul_eq_zero (φ ψ : power_series α) (h : φ * ψ = 0) :
φ = 0 ∨ ψ = 0 :=
begin
rw classical.or_iff_not_imp_left, intro H,
have ex : ∃ m, coeff α m φ ≠ 0, { contrapose! H, exact ext H },
let P : ℕ → Prop := λ k, coeff α k φ ≠ 0,
let m := nat.find ex,
have hm₁ : coeff α m φ ≠ 0 := nat.find_spec ex,
have hm₂ : ∀ k < m, ¬coeff α k φ ≠ 0 := λ k, nat.find_min ex,
ext n, rw (coeff α n).map_zero, apply nat.strong_induction_on n,
clear n, intros n ih,
replace h := congr_arg (coeff α (m + n)) h,
rw [add_monoid_hom.map_zero, coeff_mul, finset.sum_eq_single (m,n)] at h,
{ replace h := eq_zero_or_eq_zero_of_mul_eq_zero h,
rw or_iff_not_imp_left at h, exact h hm₁ },
{ rintro ⟨i,j⟩ hij hne,
by_cases hj : j < n, { rw [ih j hj, mul_zero] },
by_cases hi : i < m,
{ specialize hm₂ _ hi, push_neg at hm₂, rw [hm₂, zero_mul] },
rw finset.nat.mem_antidiagonal at hij,
push_neg at hi hj,
suffices : m < i,
{ have : m + n < i + j := add_lt_add_of_lt_of_le this hj,
exfalso, exact ne_of_lt this hij.symm },
contrapose! hne, have : i = m := le_antisymm hne hi, subst i, clear hi hne,
simpa [ne.def, prod.mk.inj_iff] using (add_right_inj m).mp hij },
{ contrapose!, intro h, rw finset.nat.mem_antidiagonal }
end
instance : integral_domain (power_series α) :=
{ eq_zero_or_eq_zero_of_mul_eq_zero := eq_zero_or_eq_zero_of_mul_eq_zero,
.. power_series.comm_ring,
.. power_series.nonzero }
/-- The ideal spanned by the variable in the power series ring
over an integral domain is a prime ideal.-/
lemma span_X_is_prime : (ideal.span ({X} : set (power_series α))).is_prime :=
begin
suffices : ideal.span ({X} : set (power_series α)) = (constant_coeff α).ker,
{ rw this, exact ring_hom.ker_is_prime _ },
apply ideal.ext, intro φ,
rw [ring_hom.mem_ker, ideal.mem_span_singleton, X_dvd_iff]
end
/-- The variable of the power series ring over an integral domain is prime.-/
lemma X_prime : prime (X : power_series α) :=
begin
rw ← ideal.span_singleton_prime,
{ exact span_X_is_prime },
{ intro h, simpa using congr_arg (coeff α 1) h }
end
end integral_domain
section local_ring
variables [comm_ring α]
lemma is_local_ring (h : is_local_ring α) :
is_local_ring (power_series α) :=
mv_power_series.is_local_ring h
end local_ring
section local_ring
variables {β : Type*} [local_ring α] [local_ring β] (f : α →+* β) [is_local_ring_hom f]
instance : local_ring (power_series α) :=
mv_power_series.local_ring
instance map.is_local_ring_hom :
is_local_ring_hom (map f) :=
mv_power_series.map.is_local_ring_hom f
end local_ring
section field
variables [field α]
protected def inv : power_series α → power_series α :=
mv_power_series.inv
instance : has_inv (power_series α) := ⟨power_series.inv⟩
lemma inv_eq_inv_aux (φ : power_series α) :
φ⁻¹ = inv.aux (constant_coeff α φ)⁻¹ φ := rfl
lemma coeff_inv (n) (φ : power_series α) :
coeff α n (φ⁻¹) = if n = 0 then (constant_coeff α φ)⁻¹ else
- (constant_coeff α φ)⁻¹ * (finset.nat.antidiagonal n).sum (λ (x : ℕ × ℕ),
if x.2 < n then coeff α x.1 φ * coeff α x.2 (φ⁻¹) else 0) :=
by rw [inv_eq_inv_aux, coeff_inv_aux n (constant_coeff α φ)⁻¹ φ]
@[simp] lemma constant_coeff_inv (φ : power_series α) :
constant_coeff α (φ⁻¹) = (constant_coeff α φ)⁻¹ :=
mv_power_series.constant_coeff_inv φ
lemma inv_eq_zero {φ : power_series α} :
φ⁻¹ = 0 ↔ constant_coeff α φ = 0 :=
mv_power_series.inv_eq_zero
@[simp, priority 1100] lemma inv_of_unit_eq (φ : power_series α) (h : constant_coeff α φ ≠ 0) :
inv_of_unit φ (units.mk0 _ h) = φ⁻¹ :=
mv_power_series.inv_of_unit_eq _ _
@[simp] lemma inv_of_unit_eq' (φ : power_series α) (u : units α) (h : constant_coeff α φ = u) :
inv_of_unit φ u = φ⁻¹ :=
mv_power_series.inv_of_unit_eq' φ _ h
@[simp] protected lemma mul_inv (φ : power_series α) (h : constant_coeff α φ ≠ 0) :
φ * φ⁻¹ = 1 :=
mv_power_series.mul_inv φ h
@[simp] protected lemma inv_mul (φ : power_series α) (h : constant_coeff α φ ≠ 0) :
φ⁻¹ * φ = 1 :=
mv_power_series.inv_mul φ h
end field
end power_series
namespace power_series
variable {α : Type*}
local attribute [instance, priority 1] classical.prop_decidable
noncomputable theory
section order_basic
open multiplicity
variables [comm_semiring α]
/-- The order of a formal power series `φ` is the smallest `n : enat`
such that `X^n` divides `φ`. The order is `⊤` if and only if `φ = 0`. -/
@[reducible] def order (φ : power_series α) : enat :=
multiplicity X φ
lemma order_finite_of_coeff_ne_zero (φ : power_series α) (h : ∃ n, coeff α n φ ≠ 0) :
(order φ).dom :=
begin
cases h with n h, refine ⟨n, _⟩,
rw X_pow_dvd_iff, push_neg, exact ⟨n, lt_add_one n, h⟩
end
/-- If the order of a formal power series is finite,
then the coefficient indexed by the order is nonzero.-/
lemma coeff_order (φ : power_series α) (h : (order φ).dom) :
coeff α (φ.order.get h) φ ≠ 0 :=
begin
have H := nat.find_spec h, contrapose! H, rw X_pow_dvd_iff,
intros m hm, by_cases Hm : m < nat.find h,
{ have := nat.find_min h Hm, push_neg at this,
rw X_pow_dvd_iff at this, exact this m (lt_add_one m) },
have : m = nat.find h, {linarith}, {rwa this}
end
/-- If the `n`th coefficient of a formal power series is nonzero,
then the order of the power series is less than or equal to `n`.-/
lemma order_le (φ : power_series α) (n : ℕ) (h : coeff α n φ ≠ 0) :
order φ ≤ n :=
begin
have h : ¬ X^(n+1) ∣ φ,
{ rw X_pow_dvd_iff, push_neg, exact ⟨n, lt_add_one n, h⟩ },
have : (order φ).dom := ⟨n, h⟩,
rw [← enat.coe_get this, enat.coe_le_coe],
refine nat.find_min' this h
end
/-- The `n`th coefficient of a formal power series is `0` if `n` is strictly
smaller than the order of the power series.-/
lemma coeff_of_lt_order (φ : power_series α) (n : ℕ) (h: ↑n < order φ) :
coeff α n φ = 0 :=
by { contrapose! h, exact order_le _ _ h }
/-- The `0` power series is the unique power series with infinite order.-/
lemma order_eq_top {φ : power_series α} :
φ.order = ⊤ ↔ φ = 0 :=
begin
rw multiplicity.eq_top_iff,
split,
{ intro h, ext n, specialize h (n+1), rw X_pow_dvd_iff at h, exact h n (lt_add_one _) },
{ rintros rfl n, exact dvd_zero _ }
end
/-- The order of the `0` power series is infinite.-/
@[simp] lemma order_zero : order (0 : power_series α) = ⊤ :=
multiplicity.zero _
/-- The order of a formal power series is at least `n` if
the `i`th coefficient is `0` for all `i < n`.-/
lemma order_ge_nat (φ : power_series α) (n : ℕ) (h : ∀ i < n, coeff α i φ = 0) :
order φ ≥ n :=
begin
by_contra H, rw not_le at H,
have : (order φ).dom := enat.dom_of_le_some (le_of_lt H),
rw [← enat.coe_get this, enat.coe_lt_coe] at H,
exact coeff_order _ this (h _ H)
end
/-- The order of a formal power series is at least `n` if
the `i`th coefficient is `0` for all `i < n`.-/
lemma order_ge (φ : power_series α) (n : enat) (h : ∀ i : ℕ, ↑i < n → coeff α i φ = 0) :
order φ ≥ n :=
begin
induction n using enat.cases_on,
{ show _ ≤ _, rw [top_le_iff, order_eq_top],
ext i, exact h _ (enat.coe_lt_top i) },
{ apply order_ge_nat, simpa only [enat.coe_lt_coe] using h }
end
/-- The order of a formal power series is exactly `n` if the `n`th coefficient is nonzero,
and the `i`th coefficient is `0` for all `i < n`.-/
lemma order_eq_nat {φ : power_series α} {n : ℕ} :
order φ = n ↔ (coeff α n φ ≠ 0) ∧ (∀ i, i < n → coeff α i φ = 0) :=
begin
simp only [eq_some_iff, X_pow_dvd_iff], push_neg,
split,
{ rintros ⟨h₁, m, hm₁, hm₂⟩, refine ⟨_, h₁⟩,
suffices : n = m, { rwa this },
suffices : m ≥ n, { linarith },
contrapose! hm₂, exact h₁ _ hm₂ },
{ rintros ⟨h₁, h₂⟩, exact ⟨h₂, n, lt_add_one n, h₁⟩ }
end
/-- The order of a formal power series is exactly `n` if the `n`th coefficient is nonzero,
and the `i`th coefficient is `0` for all `i < n`.-/
lemma order_eq {φ : power_series α} {n : enat} :
order φ = n ↔ (∀ i:ℕ, ↑i = n → coeff α i φ ≠ 0) ∧ (∀ i:ℕ, ↑i < n → coeff α i φ = 0) :=
begin
induction n using enat.cases_on,
{ rw order_eq_top, split,
{ rintro rfl, split; intros,
{ exfalso, exact enat.coe_ne_top ‹_› ‹_› },
{ exact (coeff _ _).map_zero } },
{ rintro ⟨h₁, h₂⟩, ext i, exact h₂ i (enat.coe_lt_top i) } },
{ simpa [enat.coe_inj] using order_eq_nat }
end
/-- The order of the sum of two formal power series
is at least the minimum of their orders.-/
lemma order_add_ge (φ ψ : power_series α) :
order (φ + ψ) ≥ min (order φ) (order ψ) :=
multiplicity.min_le_multiplicity_add
private lemma order_add_of_order_eq.aux (φ ψ : power_series α)
(h : order φ ≠ order ψ) (H : order φ < order ψ) :
order (φ + ψ) ≤ order φ ⊓ order ψ :=
begin
suffices : order (φ + ψ) = order φ,
{ rw [le_inf_iff, this], exact ⟨le_refl _, le_of_lt H⟩ },
{ rw order_eq, split,
{ intros i hi, rw [(coeff _ _).map_add, coeff_of_lt_order ψ i (hi.symm ▸ H), add_zero],
exact (order_eq_nat.1 hi.symm).1 },
{ intros i hi,
rw [(coeff _ _).map_add, coeff_of_lt_order φ i hi,
coeff_of_lt_order ψ i (lt_trans hi H), zero_add] } }
end
/-- The order of the sum of two formal power series
is the minimum of their orders if their orders differ.-/
lemma order_add_of_order_eq (φ ψ : power_series α) (h : order φ ≠ order ψ) :
order (φ + ψ) = order φ ⊓ order ψ :=
begin
refine le_antisymm _ (order_add_ge _ _),
by_cases H₁ : order φ < order ψ,
{ apply order_add_of_order_eq.aux _ _ h H₁ },
by_cases H₂ : order ψ < order φ,
{ simpa only [add_comm, inf_comm] using order_add_of_order_eq.aux _ _ h.symm H₂ },
exfalso, exact h (le_antisymm (not_lt.1 H₂) (not_lt.1 H₁))
end
/-- The order of the product of two formal power series
is at least the sum of their orders.-/
lemma order_mul_ge (φ ψ : power_series α) :
order (φ * ψ) ≥ order φ + order ψ :=
begin
apply order_ge,
intros n hn, rw [coeff_mul, finset.sum_eq_zero],
rintros ⟨i,j⟩ hij,
by_cases hi : ↑i < order φ,
{ rw [coeff_of_lt_order φ i hi, zero_mul] },
by_cases hj : ↑j < order ψ,
{ rw [coeff_of_lt_order ψ j hj, mul_zero] },
rw not_lt at hi hj, rw finset.nat.mem_antidiagonal at hij,
exfalso,
apply ne_of_lt (lt_of_lt_of_le hn $ add_le_add' hi hj),
rw [← enat.coe_add, hij]
end
/-- The order of the monomial `a*X^n` is infinite if `a = 0` and `n` otherwise.-/
lemma order_monomial (n : ℕ) (a : α) :
order (monomial α n a) = if a = 0 then ⊤ else n :=
begin
split_ifs with h,
{ rw [h, order_eq_top, add_monoid_hom.map_zero] },
{ rw [order_eq], split; intros i hi,
{ rw [enat.coe_inj] at hi, rwa [hi, coeff_monomial'] },
{ rw [enat.coe_lt_coe] at hi, rw [coeff_monomial, if_neg], exact ne_of_lt hi } }
end
/-- The order of the monomial `a*X^n` is `n` if `a ≠ 0`.-/
lemma order_monomial_of_ne_zero (n : ℕ) (a : α) (h : a ≠ 0) :
order (monomial α n a) = n :=
by rw [order_monomial, if_neg h]
end order_basic
section order_zero_ne_one
variables [comm_semiring α] [nonzero α]
/-- The order of the formal power series `1` is `0`.-/
@[simp] lemma order_one : order (1 : power_series α) = 0 :=
by simpa using order_monomial_of_ne_zero 0 (1:α) one_ne_zero
/-- The order of the formal power series `X` is `1`.-/
@[simp] lemma order_X : order (X : power_series α) = 1 :=
order_monomial_of_ne_zero 1 (1:α) one_ne_zero
/-- The order of the formal power series `X^n` is `n`.-/
@[simp] lemma order_X_pow (n : ℕ) : order ((X : power_series α)^n) = n :=
by { rw [X_pow_eq, order_monomial_of_ne_zero], exact one_ne_zero }
end order_zero_ne_one
section order_integral_domain
variables [integral_domain α]
/-- The order of the product of two formal power series over an integral domain
is the sum of their orders.-/
lemma order_mul (φ ψ : power_series α) :
order (φ * ψ) = order φ + order ψ :=
multiplicity.mul (X_prime)
end order_integral_domain
end power_series
namespace polynomial
open finsupp
variables {σ : Type*} {α : Type*} [comm_semiring α]
/-- The natural inclusion from polynomials into formal power series.-/
instance coe_to_power_series : has_coe (polynomial α) (power_series α) :=
⟨λ φ, power_series.mk $ λ n, coeff φ n⟩
@[simp, norm_cast] lemma coeff_coe (φ : polynomial α) (n) :
power_series.coeff α n φ = coeff φ n :=
congr_arg (coeff φ) (finsupp.single_eq_same)
@[simp, norm_cast] lemma coe_monomial (n : ℕ) (a : α) :
(monomial n a : power_series α) = power_series.monomial α n a :=
power_series.ext $ λ m,
begin
rw [coeff_coe, power_series.coeff_monomial],
simp only [@eq_comm _ m n],
convert finsupp.single_apply,
end
@[simp, norm_cast] lemma coe_zero : ((0 : polynomial α) : power_series α) = 0 := rfl
@[simp, norm_cast] lemma coe_one : ((1 : polynomial α) : power_series α) = 1 :=
begin
have := coe_monomial 0 (1:α),
rwa power_series.monomial_zero_eq_C_apply at this,
end
@[simp, norm_cast] lemma coe_add (φ ψ : polynomial α) :
((φ + ψ : polynomial α) : power_series α) = φ + ψ := rfl
@[simp, norm_cast] lemma coe_mul (φ ψ : polynomial α) :
((φ * ψ : polynomial α) : power_series α) = φ * ψ :=
power_series.ext $ λ n,
by simp only [coeff_coe, power_series.coeff_mul, coeff_mul]
@[simp, norm_cast] lemma coe_C (a : α) :
((C a : polynomial α) : power_series α) = power_series.C α a :=
begin
have := coe_monomial 0 a,
rwa power_series.monomial_zero_eq_C_apply at this,
end
@[simp, norm_cast] lemma coe_X :
((X : polynomial α) : power_series α) = power_series.X :=
coe_monomial _ _
namespace coe_to_mv_power_series
instance : is_semiring_hom (coe : polynomial α → power_series α) :=
{ map_zero := coe_zero,
map_one := coe_one,
map_add := coe_add,
map_mul := coe_mul }
end coe_to_mv_power_series
end polynomial
|
144e1c57e7933bf241140d9321e59b5ae31f953a | 624f6f2ae8b3b1adc5f8f67a365c51d5126be45a | /tests/lean/run/kernel1.lean | 06681400648cc4560fc01ba30e8b636c88176da6 | [
"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 | 779 | lean | import Init.Lean
open Lean
def checkDefEq (a b : Name) : MetaIO Unit := do
env ← MetaIO.getEnv;
let a := mkConst a;
let b := mkConst b;
let r := Kernel.isDefEq env {} a b;
IO.println (toString a ++ " =?= " ++ toString b ++ " := " ++ toString r)
def a1 := 100 + 100
def a2 := 200
def a3 := 20
#eval checkDefEq `a1 `a2
#eval checkDefEq `a1 `a3
def v1 := 100000000000 + 100000000000
def v2 := 200000000000
def v3 := 200000000001
def v4 : Bool := 20000000000 > 200000000001
def v5 := 100000000000 - 100000000000
def c1 := reduceNat v1
def c2 := reduceNat v2
def c3 := reduceNat v3
def c4 := reduceBool v4
def c5 := reduceNat v5
#eval checkDefEq `c1 `c2
#eval checkDefEq `c1 `c3
#eval checkDefEq `c5 `Nat.zero
#eval checkDefEq `Nat.zero `c5
#eval checkDefEq `c4 `Bool.true
|
59c963423f153d0441d58573ddc029577f72e0f9 | b7f22e51856f4989b970961f794f1c435f9b8f78 | /hott/init/nat.hlean | be8aff41a246efdf8995e80f94011cb8ae70b173 | [
"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 | 9,955 | hlean | /-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Leonardo de Moura
-/
prelude
import init.tactic init.num init.types init.path
open eq eq.ops decidable
open algebra sum
set_option class.force_new true
notation `ℕ` := nat
namespace nat
protected definition rec_on [reducible] [recursor] [unfold 2]
{C : ℕ → Type} (n : ℕ) (H₁ : C 0) (H₂ : Π (a : ℕ), C a → C (succ a)) : C n :=
nat.rec H₁ H₂ n
protected definition cases [reducible] [unfold 4] {M : ℕ → Type} (mz : M zero)
(ms : Πn, M (succ n)) : Πn, M n :=
nat.rec mz (λn dummy, ms n)
protected definition cases_on [reducible] [recursor] [unfold 2]
{C : ℕ → Type} (n : ℕ) (H₁ : C 0) (H₂ : Π (a : ℕ), C (succ a)) : C n :=
nat.rec H₁ (λ a ih, H₂ a) n
protected definition no_confusion_type.{u} [reducible] (P : Type.{u}) (v₁ v₂ : ℕ) : Type.{u} :=
nat.rec
(nat.rec
(P → lift P)
(λ a₂ ih, lift P)
v₂)
(λ a₁ ih, nat.rec
(lift P)
(λ a₂ ih, (a₁ = a₂ → P) → lift P)
v₂)
v₁
protected definition no_confusion [reducible] [unfold 4]
{P : Type} {v₁ v₂ : ℕ} (H : v₁ = v₂) : nat.no_confusion_type P v₁ v₂ :=
eq.rec (λ H₁ : v₁ = v₁, nat.rec (λ h, lift.up h) (λ a ih h, lift.up (h (eq.refl a))) v₁) H H
/- basic definitions on natural numbers -/
inductive le (a : ℕ) : ℕ → Type :=
| nat_refl : le a a -- use nat_refl to avoid overloading le.refl
| step : Π {b}, le a b → le a (succ b)
definition nat_has_le [instance] [priority nat.prio]: has_le nat := has_le.mk nat.le
protected definition le_refl [refl] : Π a : nat, a ≤ a :=
le.nat_refl
protected definition lt [reducible] (n m : ℕ) := succ n ≤ m
definition nat_has_lt [instance] [priority nat.prio] : has_lt nat := has_lt.mk nat.lt
definition pred [unfold 1] (a : nat) : nat :=
nat.cases_on a zero (λ a₁, a₁)
-- add is defined in init.reserved_notation
protected definition sub (a b : nat) : nat :=
nat.rec_on b a (λ b₁, pred)
protected definition mul (a b : nat) : nat :=
nat.rec_on b zero (λ b₁ r, r + a)
definition nat_has_sub [instance] [priority nat.prio] : has_sub nat :=
has_sub.mk nat.sub
definition nat_has_mul [instance] [priority nat.prio] : has_mul nat :=
has_mul.mk nat.mul
/- properties of ℕ -/
protected definition is_inhabited [instance] : inhabited nat :=
inhabited.mk zero
protected definition has_decidable_eq [instance] [priority nat.prio] : Π x y : nat, decidable (x = y)
| has_decidable_eq zero zero := inl rfl
| has_decidable_eq (succ x) zero := inr (by contradiction)
| has_decidable_eq zero (succ y) := inr (by contradiction)
| has_decidable_eq (succ x) (succ y) :=
match has_decidable_eq x y with
| inl xeqy := inl (by rewrite xeqy)
| inr xney := inr (λ h : succ x = succ y, by injection h with xeqy; exact absurd xeqy xney)
end
/- properties of inequality -/
protected definition le_of_eq {n m : ℕ} (p : n = m) : n ≤ m := p ▸ !nat.le_refl
definition le_succ (n : ℕ) : n ≤ succ n := le.step !nat.le_refl
definition pred_le (n : ℕ) : pred n ≤ n := by cases n;repeat constructor
definition le_succ_iff_unit [simp] (n : ℕ) : n ≤ succ n ↔ unit :=
iff_unit_intro (le_succ n)
definition pred_le_iff_unit [simp] (n : ℕ) : pred n ≤ n ↔ unit :=
iff_unit_intro (pred_le n)
protected definition le_trans {n m k : ℕ} (H1 : n ≤ m) : m ≤ k → n ≤ k :=
le.rec H1 (λp H2, le.step)
definition le_succ_of_le {n m : ℕ} (H : n ≤ m) : n ≤ succ m := nat.le_trans H !le_succ
definition le_of_succ_le {n m : ℕ} (H : succ n ≤ m) : n ≤ m := nat.le_trans !le_succ H
protected definition le_of_lt {n m : ℕ} (H : n < m) : n ≤ m := le_of_succ_le H
definition succ_le_succ {n m : ℕ} : n ≤ m → succ n ≤ succ m :=
le.rec !nat.le_refl (λa b, le.step)
theorem pred_le_pred {n m : ℕ} : n ≤ m → pred n ≤ pred m :=
le.rec !nat.le_refl (nat.rec (λa b, b) (λa b c, le.step))
theorem le_of_succ_le_succ {n m : ℕ} : succ n ≤ succ m → n ≤ m :=
pred_le_pred
theorem le_succ_of_pred_le {n m : ℕ} : pred n ≤ m → n ≤ succ m :=
nat.cases_on n le.step (λa, succ_le_succ)
theorem not_succ_le_zero (n : ℕ) : ¬succ n ≤ 0 :=
by intro H; cases H
theorem succ_le_zero_iff_empty (n : ℕ) : succ n ≤ 0 ↔ empty :=
iff_empty_intro !not_succ_le_zero
theorem not_succ_le_self : Π {n : ℕ}, ¬succ n ≤ n :=
nat.rec !not_succ_le_zero (λa b c, b (le_of_succ_le_succ c))
theorem succ_le_self_iff_empty [simp] (n : ℕ) : succ n ≤ n ↔ empty :=
iff_empty_intro not_succ_le_self
definition zero_le : Π (n : ℕ), 0 ≤ n :=
nat.rec !nat.le_refl (λa, le.step)
theorem zero_le_iff_unit [simp] (n : ℕ) : 0 ≤ n ↔ unit :=
iff_unit_intro !zero_le
theorem lt.step {n m : ℕ} : n < m → n < succ m := le.step
theorem zero_lt_succ (n : ℕ) : 0 < succ n :=
succ_le_succ !zero_le
theorem zero_lt_succ_iff_unit [simp] (n : ℕ) : 0 < succ n ↔ unit :=
iff_unit_intro (zero_lt_succ n)
protected theorem lt_trans {n m k : ℕ} (H1 : n < m) : m < k → n < k :=
nat.le_trans (le.step H1)
protected theorem lt_of_le_of_lt {n m k : ℕ} (H1 : n ≤ m) : m < k → n < k :=
nat.le_trans (succ_le_succ H1)
protected theorem lt_of_lt_of_le {n m k : ℕ} : n < m → m ≤ k → n < k := nat.le_trans
protected theorem lt_irrefl (n : ℕ) : ¬n < n := not_succ_le_self
theorem lt_self_iff_empty (n : ℕ) : n < n ↔ empty :=
iff_empty_intro (λ H, absurd H (nat.lt_irrefl n))
theorem self_lt_succ (n : ℕ) : n < succ n := !nat.le_refl
theorem self_lt_succ_iff_unit [simp] (n : ℕ) : n < succ n ↔ unit :=
iff_unit_intro (self_lt_succ n)
theorem lt.base (n : ℕ) : n < succ n := !nat.le_refl
theorem le_lt_antisymm {n m : ℕ} (H1 : n ≤ m) (H2 : m < n) : empty :=
!nat.lt_irrefl (nat.lt_of_le_of_lt H1 H2)
protected theorem le_antisymm {n m : ℕ} (H1 : n ≤ m) : m ≤ n → n = m :=
le.cases_on H1 (λa, rfl) (λa b c, absurd (nat.lt_of_le_of_lt b c) !nat.lt_irrefl)
theorem lt_le_antisymm {n m : ℕ} (H1 : n < m) (H2 : m ≤ n) : empty :=
le_lt_antisymm H2 H1
protected theorem nat.lt_asymm {n m : ℕ} (H1 : n < m) : ¬ m < n :=
le_lt_antisymm (nat.le_of_lt H1)
theorem not_lt_zero (a : ℕ) : ¬ a < 0 := !not_succ_le_zero
theorem lt_zero_iff_empty [simp] (a : ℕ) : a < 0 ↔ empty :=
iff_empty_intro (not_lt_zero a)
protected theorem eq_sum_lt_of_le {a b : ℕ} (H : a ≤ b) : a = b ⊎ a < b :=
le.cases_on H (inl rfl) (λn h, inr (succ_le_succ h))
protected theorem le_of_eq_sum_lt {a b : ℕ} (H : a = b ⊎ a < b) : a ≤ b :=
sum.rec_on H !nat.le_of_eq !nat.le_of_lt
theorem succ_lt_succ {a b : ℕ} : a < b → succ a < succ b :=
succ_le_succ
theorem lt_of_succ_lt {a b : ℕ} : succ a < b → a < b :=
le_of_succ_le
theorem lt_of_succ_lt_succ {a b : ℕ} : succ a < succ b → a < b :=
le_of_succ_le_succ
definition decidable_le [instance] [priority nat.prio] : Π a b : nat, decidable (a ≤ b) :=
nat.rec (λm, (decidable.inl !zero_le))
(λn IH m, !nat.cases_on (decidable.inr (not_succ_le_zero n))
(λm, decidable.rec (λH, inl (succ_le_succ H))
(λH, inr (λa, H (le_of_succ_le_succ a))) (IH m)))
definition decidable_lt [instance] [priority nat.prio] : Π a b : nat, decidable (a < b) :=
λ a b, decidable_le (succ a) b
protected theorem lt_sum_ge (a b : ℕ) : a < b ⊎ a ≥ b :=
nat.rec (inr !zero_le) (λn, sum.rec
(λh, inl (le_succ_of_le h))
(λh, sum.rec_on (nat.eq_sum_lt_of_le h) (λe, inl (eq.subst e !nat.le_refl)) inr)) b
protected definition lt_ge_by_cases {a b : ℕ} {P : Type} (H1 : a < b → P) (H2 : a ≥ b → P) : P :=
by_cases H1 (λh, H2 (sum.rec_on !nat.lt_sum_ge (λa, absurd a h) (λa, a)))
protected definition lt_by_cases {a b : ℕ} {P : Type} (H1 : a < b → P) (H2 : a = b → P)
(H3 : b < a → P) : P :=
nat.lt_ge_by_cases H1 (λh₁,
nat.lt_ge_by_cases H3 (λh₂, H2 (nat.le_antisymm h₂ h₁)))
protected theorem lt_trichotomy (a b : ℕ) : a < b ⊎ a = b ⊎ b < a :=
nat.lt_by_cases (λH, inl H) (λH, inr (inl H)) (λH, inr (inr H))
protected theorem eq_sum_lt_of_not_lt {a b : ℕ} (hnlt : ¬ a < b) : a = b ⊎ b < a :=
sum.rec_on (nat.lt_trichotomy a b)
(λ hlt, absurd hlt hnlt)
(λ h, h)
theorem lt_succ_of_le {a b : ℕ} : a ≤ b → a < succ b :=
succ_le_succ
theorem lt_of_succ_le {a b : ℕ} (h : succ a ≤ b) : a < b := h
theorem succ_le_of_lt {a b : ℕ} (h : a < b) : succ a ≤ b := h
theorem succ_sub_succ_eq_sub [simp] (a b : ℕ) : succ a - succ b = a - b :=
nat.rec (by esimp) (λ b, ap pred) b
theorem sub_eq_succ_sub_succ (a b : ℕ) : a - b = succ a - succ b :=
inverse !succ_sub_succ_eq_sub
theorem zero_sub_eq_zero [simp] (a : ℕ) : 0 - a = 0 :=
nat.rec rfl (λ a, ap pred) a
theorem zero_eq_zero_sub (a : ℕ) : 0 = 0 - a :=
inverse !zero_sub_eq_zero
theorem sub_le (a b : ℕ) : a - b ≤ a :=
nat.rec_on b !nat.le_refl (λ b₁, nat.le_trans !pred_le)
theorem sub_le_iff_unit [simp] (a b : ℕ) : a - b ≤ a ↔ unit :=
iff_unit_intro (sub_le a b)
theorem sub_lt {a b : ℕ} (H1 : 0 < a) (H2 : 0 < b) : a - b < a :=
!nat.cases_on (λh, absurd h !nat.lt_irrefl)
(λa h, succ_le_succ (!nat.cases_on (λh, absurd h !nat.lt_irrefl)
(λb c, tr_rev _ !succ_sub_succ_eq_sub !sub_le) H2)) H1
theorem sub_lt_succ (a b : ℕ) : a - b < succ a :=
lt_succ_of_le !sub_le
theorem sub_lt_succ_iff_unit [simp] (a b : ℕ) : a - b < succ a ↔ unit :=
iff_unit_intro !sub_lt_succ
end nat
|
9ecadecbec0969986a07d16afa80f26a29c83142 | ba4794a0deca1d2aaa68914cd285d77880907b5c | /src/game/world8/level1.lean | 308d0b589c93182654f908839dac52a754e0459e | [
"Apache-2.0"
] | permissive | ChrisHughes24/natural_number_game | c7c00aa1f6a95004286fd456ed13cf6e113159ce | 9d09925424da9f6275e6cfe427c8bcf12bb0944f | refs/heads/master | 1,600,715,773,528 | 1,573,910,462,000 | 1,573,910,462,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 1,682 | lean | import mynat.definition -- hide
import mynat.add -- hide
import game.world2.level6 -- hide
namespace mynat -- hide
/- Axiom : succ_inj (a b : mynat) :
succ(a) = succ(b) → a = b
-/
/-
# Advanced Addition World
## Level 1: `succ_inj`. A function.
Peano's original collection of axioms for the natural numbers contained two further
assumptions, which have not yet been mentioned in the game:
```
succ_inj (a b : mynat) :
succ(a) = succ(b) → a = b
zero_ne_succ (a : mynat) :
zero ≠ succ(a)
```
The reason they have not been used yet is that they are both implications,
that is,
of the form $P\implies Q$. This is clear for `succ_inj a b`, which
says that for all $a$ and $b$ we have $succ(a)=succ(b)\implies a=b$.
For `zero_ne_succ` the trick is that `X ≠ Y` is *defined to mean*
$X = Y\implies{\tt false}$. If you have played through Proposition world,
you now have the required Lean skills (i.e., you know the required
tactics) to work with these implications.
Let's finally learn how to use `succ_inj`. You should know a couple
of ways to prove the below -- one directly using an `exact`, and one which uses an
`apply` first.
-/
/- Theorem : no-side-bar
For all naturals $a$ and $b$, if we assume $succ(a)=succ(b)$, then we can
deduce $a=b$.
-/
theorem succ_inj' {a b : mynat} (hs : succ(a) = succ(b)) : a = b :=
begin [less_leaky]
exact succ_inj(hs),
end
/-
## Important thing.
You can rewrite proofs of *equalities*. If `h : A = B` then `rw h` changes `A`s to `B`s.
But you *cannot rewrite proofs of implications*. `rw succ_inj` will *never work*
because `succ_inj` isn't of the form $A = B$, it's of the form $A\implies B$.
-/
end mynat -- hide |
0ec897b2cee8b84453339f8f5c3c2c34f97bc3e8 | 64874bd1010548c7f5a6e3e8902efa63baaff785 | /library/logic/eq.lean | 53110825b7de5677cde7e40086b08b8cbc6801a6 | [
"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 | 4,461 | lean | /-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Module: logic.eq
Authors: Leonardo de Moura, Jeremy Avigad, Floris van Doorn
Additional declarations/theorems about equality. See also init.datatypes and init.logic.
-/
open eq.ops
namespace eq
variables {A B : Type} {a a' a₁ a₂ a₃ a₄ : A}
theorem irrel (H₁ H₂ : a = a') : H₁ = H₂ :=
!proof_irrel
theorem id_refl (H₁ : a = a) : H₁ = (eq.refl a) :=
rfl
definition drec_on {B : Πa' : A, a = a' → Type} (H₁ : a = a') (H₂ : B a (refl a)) : B a' H₁ :=
eq.rec (λH₁ : a = a, show B a H₁, from H₂) H₁ H₁
theorem rec_on_id {B : A → Type} (H : a = a) (b : B a) : rec_on H b = b :=
rfl
theorem rec_on_constant (H : a = a') {B : Type} (b : B) : rec_on H b = b :=
drec_on H rfl
theorem rec_on_constant2 (H₁ : a₁ = a₂) (H₂ : a₃ = a₄) (b : B) : rec_on H₁ b = rec_on H₂ b :=
rec_on_constant H₁ b ⬝ (rec_on_constant H₂ b)⁻¹
theorem rec_on_irrel_arg {f : A → B} {D : B → Type} (H : a = a') (H' : f a = f a') (b : D (f a)) :
rec_on H b = rec_on H' b :=
drec_on H (λ(H' : f a = f a), !rec_on_id⁻¹) H'
theorem rec_on_irrel {a a' : A} {D : A → Type} (H H' : a = a') (b : D a) :
drec_on H b = drec_on H' b :=
proof_irrel H H' ▸ rfl
theorem rec_on_compose {a b c : A} {P : A → Type} (H₁ : a = b) (H₂ : b = c)
(u : P a) : rec_on H₂ (rec_on H₁ u) = rec_on (trans H₁ H₂) u :=
(show ∀ H₂ : b = c, rec_on H₂ (rec_on H₁ u) = rec_on (trans H₁ H₂) u,
from drec_on H₂ (take (H₂ : b = b), rec_on_id H₂ _))
H₂
end eq
open eq
section
variables {A B C D E F : Type}
variables {a a' : A} {b b' : B} {c c' : C} {d d' : D} {e e' : E}
theorem congr_fun {B : A → Type} {f g : Π x, B x} (H : f = g) (a : A) : f a = g a :=
H ▸ rfl
theorem congr_arg (f : A → B) (H : a = a') : f a = f a' :=
H ▸ rfl
theorem congr {f g : A → B} (H₁ : f = g) (H₂ : a = a') : f a = g a' :=
H₁ ▸ H₂ ▸ rfl
theorem congr_arg2 (f : A → B → C) (Ha : a = a') (Hb : b = b') : f a b = f a' b' :=
congr (congr_arg f Ha) Hb
theorem congr_arg3 (f : A → B → C → D) (Ha : a = a') (Hb : b = b') (Hc : c = c')
: f a b c = f a' b' c' :=
congr (congr_arg2 f Ha Hb) Hc
theorem congr_arg4 (f : A → B → C → D → E) (Ha : a = a') (Hb : b = b') (Hc : c = c') (Hd : d = d')
: f a b c d = f a' b' c' d' :=
congr (congr_arg3 f Ha Hb Hc) Hd
theorem congr_arg5 (f : A → B → C → D → E → F)
(Ha : a = a') (Hb : b = b') (Hc : c = c') (Hd : d = d') (He : e = e')
: f a b c d e = f a' b' c' d' e' :=
congr (congr_arg4 f Ha Hb Hc Hd) He
theorem congr2 (f f' : A → B → C) (Hf : f = f') (Ha : a = a') (Hb : b = b') : f a b = f' a' b' :=
Hf ▸ congr_arg2 f Ha Hb
theorem congr3 (f f' : A → B → C → D) (Hf : f = f') (Ha : a = a') (Hb : b = b') (Hc : c = c')
: f a b c = f' a' b' c' :=
Hf ▸ congr_arg3 f Ha Hb Hc
theorem congr4 (f f' : A → B → C → D → E)
(Hf : f = f') (Ha : a = a') (Hb : b = b') (Hc : c = c') (Hd : d = d')
: f a b c d = f' a' b' c' d' :=
Hf ▸ congr_arg4 f Ha Hb Hc Hd
theorem congr5 (f f' : A → B → C → D → E → F)
(Hf : f = f') (Ha : a = a') (Hb : b = b') (Hc : c = c') (Hd : d = d') (He : e = e')
: f a b c d e = f' a' b' c' d' e' :=
Hf ▸ congr_arg5 f Ha Hb Hc Hd He
end
theorem equal_f {A : Type} {B : A → Type} {f g : Π x, B x} (H : f = g) : ∀x, f x = g x :=
take x, congr_fun H x
section
variables {a b c : Prop}
theorem eqmp (H₁ : a = b) (H₂ : a) : b :=
H₁ ▸ H₂
theorem eqmpr (H₁ : a = b) (H₂ : b) : a :=
H₁⁻¹ ▸ H₂
theorem imp_trans (H₁ : a → b) (H₂ : b → c) : a → c :=
assume Ha, H₂ (H₁ Ha)
theorem imp_eq_trans (H₁ : a → b) (H₂ : b = c) : a → c :=
assume Ha, H₂ ▸ (H₁ Ha)
theorem eq_imp_trans (H₁ : a = b) (H₂ : b → c) : a → c :=
assume Ha, H₂ (H₁ ▸ Ha)
end
section
variables {p : Prop}
theorem p_ne_false : p → p ≠ false :=
assume (Hp : p) (Heq : p = false), Heq ▸ Hp
theorem p_ne_true : ¬p → p ≠ true :=
assume (Hnp : ¬p) (Heq : p = true), absurd trivial (Heq ▸ Hnp)
end
theorem true_ne_false : ¬true = false :=
assume H : true = false,
H ▸ trivial
|
59c1b456a8c4ac78d3fcc438cba648a211743a2d | 74addaa0e41490cbaf2abd313a764c96df57b05d | /Mathlib/data/polynomial/ring_division.lean | 314bc7573d26107037bb7a05fb5bd4ecd7b5dbc4 | [] | 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 | 14,012 | lean | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker, Johan Commelin
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.default
import Mathlib.data.polynomial.basic
import Mathlib.data.polynomial.div
import Mathlib.data.polynomial.algebra_map
import Mathlib.data.set.finite
import Mathlib.PostPort
universes u v u_1
namespace Mathlib
/-!
# Theory of univariate polynomials
This file starts looking like the ring theory of $ R[X] $
-/
namespace polynomial
theorem nat_degree_pos_of_aeval_root {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] [algebra R S] {p : polynomial R} (hp : p ≠ 0) {z : S} (hz : coe_fn (aeval z) p = 0) (inj : ∀ (x : R), coe_fn (algebra_map R S) x = 0 → x = 0) : 0 < nat_degree p :=
nat_degree_pos_of_eval₂_root hp (algebra_map R S) hz inj
theorem degree_pos_of_aeval_root {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] [algebra R S] {p : polynomial R} (hp : p ≠ 0) {z : S} (hz : coe_fn (aeval z) p = 0) (inj : ∀ (x : R), coe_fn (algebra_map R S) x = 0 → x = 0) : 0 < degree p :=
iff.mp nat_degree_pos_iff_degree_pos (nat_degree_pos_of_aeval_root hp hz inj)
theorem aeval_mod_by_monic_eq_self_of_root {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] [algebra R S] {p : polynomial R} {q : polynomial R} (hq : monic q) {x : S} (hx : coe_fn (aeval x) q = 0) : coe_fn (aeval x) (p %ₘ q) = coe_fn (aeval x) p :=
eval₂_mod_by_monic_eq_self_of_root hq hx
protected instance integral_domain {R : Type u} [integral_domain R] : integral_domain (polynomial R) :=
integral_domain.mk comm_ring.add sorry comm_ring.zero sorry sorry comm_ring.neg comm_ring.sub sorry sorry comm_ring.mul
sorry comm_ring.one sorry sorry sorry sorry sorry sorry sorry
theorem nat_degree_mul {R : Type u} [integral_domain R] {p : polynomial R} {q : polynomial R} (hp : p ≠ 0) (hq : q ≠ 0) : nat_degree (p * q) = nat_degree p + nat_degree q := sorry
@[simp] theorem nat_degree_pow {R : Type u} [integral_domain R] (p : polynomial R) (n : ℕ) : nat_degree (p ^ n) = n * nat_degree p := sorry
theorem root_mul {R : Type u} {a : R} [integral_domain R] {p : polynomial R} {q : polynomial R} : is_root (p * q) a ↔ is_root p a ∨ is_root q a := sorry
theorem root_or_root_of_root_mul {R : Type u} {a : R} [integral_domain R] {p : polynomial R} {q : polynomial R} (h : is_root (p * q) a) : is_root p a ∨ is_root q a :=
iff.mp root_mul h
theorem degree_le_mul_left {R : Type u} [integral_domain R] {q : polynomial R} (p : polynomial R) (hq : q ≠ 0) : degree p ≤ degree (p * q) := sorry
theorem nat_degree_le_of_dvd {R : Type u} [integral_domain R] {p : polynomial R} {q : polynomial R} (h1 : p ∣ q) (h2 : q ≠ 0) : nat_degree p ≤ nat_degree q := sorry
theorem degree_eq_zero_of_is_unit {R : Type u} [integral_domain R] {p : polynomial R} (h : is_unit p) : degree p = 0 := sorry
@[simp] theorem degree_coe_units {R : Type u} [integral_domain R] (u : units (polynomial R)) : degree ↑u = 0 :=
degree_eq_zero_of_is_unit (Exists.intro u rfl)
theorem prime_X_sub_C {R : Type u} [integral_domain R] {r : R} : prime (X - coe_fn C r) := sorry
theorem prime_X {R : Type u} [integral_domain R] : prime X := sorry
theorem prime_of_degree_eq_one_of_monic {R : Type u} [integral_domain R] {p : polynomial R} (hp1 : degree p = 1) (hm : monic p) : prime p := sorry
theorem irreducible_X_sub_C {R : Type u} [integral_domain R] (r : R) : irreducible (X - coe_fn C r) :=
irreducible_of_prime prime_X_sub_C
theorem irreducible_X {R : Type u} [integral_domain R] : irreducible X :=
irreducible_of_prime prime_X
theorem irreducible_of_degree_eq_one_of_monic {R : Type u} [integral_domain R] {p : polynomial R} (hp1 : degree p = 1) (hm : monic p) : irreducible p :=
irreducible_of_prime (prime_of_degree_eq_one_of_monic hp1 hm)
theorem eq_of_monic_of_associated {R : Type u} [integral_domain R] {p : polynomial R} {q : polynomial R} (hp : monic p) (hq : monic q) (hpq : associated p q) : p = q := sorry
@[simp] theorem root_multiplicity_zero {R : Type u} [integral_domain R] {x : R} : root_multiplicity x 0 = 0 :=
dif_pos rfl
theorem root_multiplicity_eq_zero {R : Type u} [integral_domain R] {p : polynomial R} {x : R} (h : ¬is_root p x) : root_multiplicity x p = 0 := sorry
theorem root_multiplicity_pos {R : Type u} [integral_domain R] {p : polynomial R} (hp : p ≠ 0) {x : R} : 0 < root_multiplicity x p ↔ is_root p x := sorry
theorem root_multiplicity_mul {R : Type u} [integral_domain R] {p : polynomial R} {q : polynomial R} {x : R} (hpq : p * q ≠ 0) : root_multiplicity x (p * q) = root_multiplicity x p + root_multiplicity x q := sorry
theorem root_multiplicity_X_sub_C_self {R : Type u} [integral_domain R] {x : R} : root_multiplicity x (X - coe_fn C x) = 1 := sorry
theorem root_multiplicity_X_sub_C {R : Type u} [integral_domain R] {x : R} {y : R} : root_multiplicity x (X - coe_fn C y) = ite (x = y) 1 0 := sorry
/-- The multiplicity of `a` as root of `(X - a) ^ n` is `n`. -/
theorem root_multiplicity_X_sub_C_pow {R : Type u} [integral_domain R] (a : R) (n : ℕ) : root_multiplicity a ((X - coe_fn C a) ^ n) = n := sorry
/-- If `(X - a) ^ n` divides a polynomial `p` then the multiplicity of `a` as root of `p` is at
least `n`. -/
theorem root_multiplicity_of_dvd {R : Type u} [integral_domain R] {p : polynomial R} {a : R} {n : ℕ} (hzero : p ≠ 0) (h : (X - coe_fn C a) ^ n ∣ p) : n ≤ root_multiplicity a p := sorry
/-- The multiplicity of `p + q` is at least the minimum of the multiplicities. -/
theorem root_multiplicity_add {R : Type u} [integral_domain R] {p : polynomial R} {q : polynomial R} (a : R) (hzero : p + q ≠ 0) : min (root_multiplicity a p) (root_multiplicity a q) ≤ root_multiplicity a (p + q) :=
root_multiplicity_of_dvd hzero (min_pow_dvd_add (pow_root_multiplicity_dvd p a) (pow_root_multiplicity_dvd q a))
theorem exists_multiset_roots {R : Type u} [integral_domain R] {p : polynomial R} (hp : p ≠ 0) : ∃ (s : multiset R), ↑(coe_fn multiset.card s) ≤ degree p ∧ ∀ (a : R), multiset.count a s = root_multiplicity a p := sorry
/-- `roots p` noncomputably gives a multiset containing all the roots of `p`,
including their multiplicities. -/
def roots {R : Type u} [integral_domain R] (p : polynomial R) : multiset R :=
dite (p = 0) (fun (h : p = 0) => ∅) fun (h : ¬p = 0) => classical.some (exists_multiset_roots h)
@[simp] theorem roots_zero {R : Type u} [integral_domain R] : roots 0 = 0 :=
dif_pos rfl
theorem card_roots {R : Type u} [integral_domain R] {p : polynomial R} (hp0 : p ≠ 0) : ↑(coe_fn multiset.card (roots p)) ≤ degree p := sorry
theorem card_roots' {R : Type u} [integral_domain R] {p : polynomial R} (hp0 : p ≠ 0) : coe_fn multiset.card (roots p) ≤ nat_degree p :=
iff.mp with_bot.coe_le_coe (le_trans (card_roots hp0) (le_of_eq (degree_eq_nat_degree hp0)))
theorem card_roots_sub_C {R : Type u} [integral_domain R] {p : polynomial R} {a : R} (hp0 : 0 < degree p) : ↑(coe_fn multiset.card (roots (p - coe_fn C a))) ≤ degree p := sorry
theorem card_roots_sub_C' {R : Type u} [integral_domain R] {p : polynomial R} {a : R} (hp0 : 0 < degree p) : coe_fn multiset.card (roots (p - coe_fn C a)) ≤ nat_degree p := sorry
@[simp] theorem count_roots {R : Type u} {a : R} [integral_domain R] {p : polynomial R} (hp : p ≠ 0) : multiset.count a (roots p) = root_multiplicity a p := sorry
@[simp] theorem mem_roots {R : Type u} {a : R} [integral_domain R] {p : polynomial R} (hp : p ≠ 0) : a ∈ roots p ↔ is_root p a := sorry
theorem eq_zero_of_infinite_is_root {R : Type u} [integral_domain R] (p : polynomial R) (h : set.infinite (set_of fun (x : R) => is_root p x)) : p = 0 := sorry
theorem eq_of_infinite_eval_eq {R : Type u_1} [integral_domain R] (p : polynomial R) (q : polynomial R) (h : set.infinite (set_of fun (x : R) => eval x p = eval x q)) : p = q := sorry
theorem roots_mul {R : Type u} [integral_domain R] {p : polynomial R} {q : polynomial R} (hpq : p * q ≠ 0) : roots (p * q) = roots p + roots q := sorry
@[simp] theorem mem_roots_sub_C {R : Type u} [integral_domain R] {p : polynomial R} {a : R} {x : R} (hp0 : 0 < degree p) : x ∈ roots (p - coe_fn C a) ↔ eval x p = a := sorry
@[simp] theorem roots_X_sub_C {R : Type u} [integral_domain R] (r : R) : roots (X - coe_fn C r) = r ::ₘ 0 := sorry
@[simp] theorem roots_C {R : Type u} [integral_domain R] (x : R) : roots (coe_fn C x) = 0 := sorry
@[simp] theorem roots_one {R : Type u} [integral_domain R] : roots 1 = ∅ :=
roots_C 1
theorem roots_list_prod {R : Type u} [integral_domain R] (L : List (polynomial R)) : (∀ (p : polynomial R), p ∈ L → p ≠ 0) → roots (list.prod L) = multiset.bind (↑L) roots := sorry
theorem roots_multiset_prod {R : Type u} [integral_domain R] (m : multiset (polynomial R)) : (∀ (p : polynomial R), p ∈ m → p ≠ 0) → roots (multiset.prod m) = multiset.bind m roots := sorry
theorem roots_prod {R : Type u} [integral_domain R] {ι : Type u_1} (f : ι → polynomial R) (s : finset ι) : finset.prod s f ≠ 0 → roots (finset.prod s f) = multiset.bind (finset.val s) fun (i : ι) => roots (f i) := sorry
theorem roots_prod_X_sub_C {R : Type u} [integral_domain R] (s : finset R) : roots (finset.prod s fun (a : R) => X - coe_fn C a) = finset.val s := sorry
theorem card_roots_X_pow_sub_C {R : Type u} [integral_domain R] {n : ℕ} (hn : 0 < n) (a : R) : coe_fn multiset.card (roots (X ^ n - coe_fn C a)) ≤ n :=
iff.mp with_bot.coe_le_coe (trans_rel_left LessEq (card_roots (X_pow_sub_C_ne_zero hn a)) (degree_X_pow_sub_C hn a))
/-- `nth_roots n a` noncomputably returns the solutions to `x ^ n = a`-/
def nth_roots {R : Type u} [integral_domain R] (n : ℕ) (a : R) : multiset R :=
roots (X ^ n - coe_fn C a)
@[simp] theorem mem_nth_roots {R : Type u} [integral_domain R] {n : ℕ} (hn : 0 < n) {a : R} {x : R} : x ∈ nth_roots n a ↔ x ^ n = a := sorry
@[simp] theorem nth_roots_zero {R : Type u} [integral_domain R] (r : R) : nth_roots 0 r = 0 := sorry
theorem card_nth_roots {R : Type u} [integral_domain R] (n : ℕ) (a : R) : coe_fn multiset.card (nth_roots n a) ≤ n := sorry
/-- The multiset `nth_roots ↑n (1 : R)` as a finset. -/
def nth_roots_finset (n : ℕ) (R : Type u_1) [integral_domain R] : finset R :=
multiset.to_finset (nth_roots n 1)
@[simp] theorem mem_nth_roots_finset {R : Type u} [integral_domain R] {n : ℕ} (h : 0 < n) {x : R} : x ∈ nth_roots_finset n R ↔ x ^ n = 1 := sorry
theorem coeff_comp_degree_mul_degree {R : Type u} [integral_domain R] {p : polynomial R} {q : polynomial R} (hqd0 : nat_degree q ≠ 0) : coeff (comp p q) (nat_degree p * nat_degree q) = leading_coeff p * leading_coeff q ^ nat_degree p := sorry
theorem nat_degree_comp {R : Type u} [integral_domain R] {p : polynomial R} {q : polynomial R} : nat_degree (comp p q) = nat_degree p * nat_degree q := sorry
theorem leading_coeff_comp {R : Type u} [integral_domain R] {p : polynomial R} {q : polynomial R} (hq : nat_degree q ≠ 0) : leading_coeff (comp p q) = leading_coeff p * leading_coeff q ^ nat_degree p := sorry
theorem units_coeff_zero_smul {R : Type u} [integral_domain R] (c : units (polynomial R)) (p : polynomial R) : coeff (↑c) 0 • p = ↑c * p := sorry
@[simp] theorem nat_degree_coe_units {R : Type u} [integral_domain R] (u : units (polynomial R)) : nat_degree ↑u = 0 :=
nat_degree_eq_of_degree_eq_some (degree_coe_units u)
theorem zero_of_eval_zero {R : Type u} [integral_domain R] [infinite R] (p : polynomial R) (h : ∀ (x : R), eval x p = 0) : p = 0 := sorry
theorem funext {R : Type u} [integral_domain R] [infinite R] {p : polynomial R} {q : polynomial R} (ext : ∀ (r : R), eval r p = eval r q) : p = q := sorry
theorem is_unit_iff {R : Type u} [integral_domain R] {f : polynomial R} : is_unit f ↔ ∃ (r : R), is_unit r ∧ coe_fn C r = f := sorry
theorem coeff_coe_units_zero_ne_zero {R : Type u} [integral_domain R] (u : units (polynomial R)) : coeff (↑u) 0 ≠ 0 := sorry
theorem degree_eq_degree_of_associated {R : Type u} [integral_domain R] {p : polynomial R} {q : polynomial R} (h : associated p q) : degree p = degree q := sorry
theorem degree_eq_one_of_irreducible_of_root {R : Type u} [integral_domain R] {p : polynomial R} (hi : irreducible p) {x : R} (hx : is_root p x) : degree p = 1 := sorry
/-- Division by a monic polynomial doesn't change the leading coefficient. -/
theorem leading_coeff_div_by_monic_of_monic {R : Type u} [integral_domain R] {p : polynomial R} {q : polynomial R} (hmonic : monic q) (hdegree : degree q ≤ degree p) : leading_coeff (p /ₘ q) = leading_coeff p := sorry
theorem eq_of_monic_of_dvd_of_nat_degree_le {R : Type u} [integral_domain R] {p : polynomial R} {q : polynomial R} (hp : monic p) (hq : monic q) (hdiv : p ∣ q) (hdeg : nat_degree q ≤ nat_degree p) : q = p := sorry
theorem is_unit_of_is_unit_leading_coeff_of_is_unit_map {R : Type u} {S : Type v} [semiring R] [integral_domain S] (φ : R →+* S) (f : polynomial R) (hf : is_unit (leading_coeff f)) (H : is_unit (map φ f)) : is_unit f := sorry
/--
A polynomial over an integral domain `R` is irreducible if it is monic and
irreducible after mapping into an integral domain `S`.
A special case of this lemma is that a polynomial over `ℤ` is irreducible if
it is monic and irreducible over `ℤ/pℤ` for some prime `p`.
-/
theorem monic.irreducible_of_irreducible_map {R : Type u} {S : Type v} [integral_domain R] [integral_domain S] (φ : R →+* S) (f : polynomial R) (h_mon : monic f) (h_irr : irreducible (map φ f)) : irreducible f := sorry
end polynomial
namespace is_integral_domain
/-- Lift evidence that `is_integral_domain R` to `is_integral_domain (polynomial R)`. -/
theorem polynomial {R : Type u_1} [comm_ring R] (h : is_integral_domain R) : is_integral_domain (polynomial R) :=
integral_domain.to_is_integral_domain (polynomial R)
|
01a73d2a77dc5b98b13be1d40d30810fd8c93c51 | 6432ea7a083ff6ba21ea17af9ee47b9c371760f7 | /src/lake/examples/ffi/app/Test.lean | 5131ef7b8ab968ed0ef2569f6c41b3cf270ca406 | [
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"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 | 28 | lean | import FFI
#eval myAdd 3 4
|
899f9a23c3f859e97f462f74313e6a049a1f62c5 | 367134ba5a65885e863bdc4507601606690974c1 | /src/order/ord_continuous.lean | e6bf6789aa02597d7b6a1da5f05ea793b99b6c14 | [
"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 | 8,022 | lean | /-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Johannes Hölzl
-/
import order.conditionally_complete_lattice
import logic.function.iterate
import order.rel_iso
/-!
# Order continuity
We say that a function is *left order continuous* if it sends all least upper bounds
to least upper bounds. The order dual notion is called *right order continuity*.
For monotone functions `ℝ → ℝ` these notions correspond to the usual left and right continuity.
We prove some basic lemmas (`map_sup`, `map_Sup` etc) and prove that an `rel_iso` is both left
and right order continuous.
-/
universes u v w x
variables {α : Type u} {β : Type v} {γ : Type w} {ι : Sort x}
open set function
/-!
### Definitions
-/
/-- A function `f` between preorders is left order continuous if it preserves all suprema. We
define it using `is_lub` instead of `Sup` so that the proof works both for complete lattices and
conditionally complete lattices. -/
def left_ord_continuous [preorder α] [preorder β] (f : α → β) :=
∀ ⦃s : set α⦄ ⦃x⦄, is_lub s x → is_lub (f '' s) (f x)
/-- A function `f` between preorders is right order continuous if it preserves all infima. We
define it using `is_glb` instead of `Inf` so that the proof works both for complete lattices and
conditionally complete lattices. -/
def right_ord_continuous [preorder α] [preorder β] (f : α → β) :=
∀ ⦃s : set α⦄ ⦃x⦄, is_glb s x → is_glb (f '' s) (f x)
namespace left_ord_continuous
section preorder
variables (α) [preorder α] [preorder β] [preorder γ] {g : β → γ} {f : α → β}
protected lemma id : left_ord_continuous (id : α → α) := λ s x h, by simpa only [image_id] using h
variable {α}
protected lemma order_dual (hf : left_ord_continuous f) :
@right_ord_continuous (order_dual α) (order_dual β) _ _ f := hf
lemma map_is_greatest (hf : left_ord_continuous f) {s : set α} {x : α} (h : is_greatest s x):
is_greatest (f '' s) (f x) :=
⟨mem_image_of_mem f h.1, (hf h.is_lub).1⟩
lemma mono (hf : left_ord_continuous f) : monotone f :=
λ a₁ a₂ h,
have is_greatest {a₁, a₂} a₂ := ⟨or.inr rfl, by simp [*]⟩,
(hf.map_is_greatest this).2 $ mem_image_of_mem _ (or.inl rfl)
lemma comp (hg : left_ord_continuous g) (hf : left_ord_continuous f) :
left_ord_continuous (g ∘ f) :=
λ s x h, by simpa only [image_image] using hg (hf h)
protected lemma iterate {f : α → α} (hf : left_ord_continuous f) (n : ℕ) :
left_ord_continuous (f^[n]) :=
nat.rec_on n (left_ord_continuous.id α) $ λ n ihn, ihn.comp hf
end preorder
section semilattice_sup
variables [semilattice_sup α] [semilattice_sup β] {f : α → β}
lemma map_sup (hf : left_ord_continuous f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(hf is_lub_pair).unique $ by simp only [image_pair, is_lub_pair]
lemma le_iff (hf : left_ord_continuous f) (h : injective f) {x y} :
f x ≤ f y ↔ x ≤ y :=
by simp only [← sup_eq_right, ← hf.map_sup, h.eq_iff]
lemma lt_iff (hf : left_ord_continuous f) (h : injective f) {x y} :
f x < f y ↔ x < y :=
by simp only [lt_iff_le_not_le, hf.le_iff h]
variable (f)
/-- Convert an injective left order continuous function to an order embedding. -/
def to_order_embedding (hf : left_ord_continuous f) (h : injective f) :
α ↪o β :=
⟨⟨f, h⟩, λ x y, hf.le_iff h⟩
variable {f}
@[simp] lemma coe_to_order_embedding (hf : left_ord_continuous f) (h : injective f) :
⇑(hf.to_order_embedding f h) = f := rfl
end semilattice_sup
section complete_lattice
variables [complete_lattice α] [complete_lattice β] {f : α → β}
lemma map_Sup' (hf : left_ord_continuous f) (s : set α) :
f (Sup s) = Sup (f '' s) :=
(hf $ is_lub_Sup s).Sup_eq.symm
lemma map_Sup (hf : left_ord_continuous f) (s : set α) :
f (Sup s) = ⨆ x ∈ s, f x :=
by rw [hf.map_Sup', Sup_image]
lemma map_supr (hf : left_ord_continuous f) (g : ι → α) :
f (⨆ i, g i) = ⨆ i, f (g i) :=
by simp only [supr, hf.map_Sup', ← range_comp]
end complete_lattice
section conditionally_complete_lattice
variables [conditionally_complete_lattice α] [conditionally_complete_lattice β] [nonempty ι]
{f : α → β}
lemma map_cSup (hf : left_ord_continuous f) {s : set α} (sne : s.nonempty) (sbdd : bdd_above s) :
f (Sup s) = Sup (f '' s) :=
((hf $ is_lub_cSup sne sbdd).cSup_eq $ sne.image f).symm
lemma map_csupr (hf : left_ord_continuous f) {g : ι → α} (hg : bdd_above (range g)) :
f (⨆ i, g i) = ⨆ i, f (g i) :=
by simp only [supr, hf.map_cSup (range_nonempty _) hg, ← range_comp]
end conditionally_complete_lattice
end left_ord_continuous
namespace right_ord_continuous
section preorder
variables (α) [preorder α] [preorder β] [preorder γ] {g : β → γ} {f : α → β}
protected lemma id : right_ord_continuous (id : α → α) := λ s x h, by simpa only [image_id] using h
variable {α}
protected lemma order_dual (hf : right_ord_continuous f) :
@left_ord_continuous (order_dual α) (order_dual β) _ _ f := hf
lemma map_is_least (hf : right_ord_continuous f) {s : set α} {x : α} (h : is_least s x):
is_least (f '' s) (f x) :=
hf.order_dual.map_is_greatest h
lemma mono (hf : right_ord_continuous f) : monotone f :=
hf.order_dual.mono.order_dual
lemma comp (hg : right_ord_continuous g) (hf : right_ord_continuous f) :
right_ord_continuous (g ∘ f) :=
hg.order_dual.comp hf.order_dual
protected lemma iterate {f : α → α} (hf : right_ord_continuous f) (n : ℕ) :
right_ord_continuous (f^[n]) :=
hf.order_dual.iterate n
end preorder
section semilattice_inf
variables [semilattice_inf α] [semilattice_inf β] {f : α → β}
lemma map_inf (hf : right_ord_continuous f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.order_dual.map_sup x y
lemma le_iff (hf : right_ord_continuous f) (h : injective f) {x y} :
f x ≤ f y ↔ x ≤ y :=
hf.order_dual.le_iff h
lemma lt_iff (hf : right_ord_continuous f) (h : injective f) {x y} :
f x < f y ↔ x < y :=
hf.order_dual.lt_iff h
variable (f)
/-- Convert an injective left order continuous function to a `order_embedding`. -/
def to_order_embedding (hf : right_ord_continuous f) (h : injective f) : α ↪o β :=
⟨⟨f, h⟩, λ x y, hf.le_iff h⟩
variable {f}
@[simp] lemma coe_to_order_embedding (hf : right_ord_continuous f) (h : injective f) :
⇑(hf.to_order_embedding f h) = f := rfl
end semilattice_inf
section complete_lattice
variables [complete_lattice α] [complete_lattice β] {f : α → β}
lemma map_Inf' (hf : right_ord_continuous f) (s : set α) :
f (Inf s) = Inf (f '' s) :=
hf.order_dual.map_Sup' s
lemma map_Inf (hf : right_ord_continuous f) (s : set α) :
f (Inf s) = ⨅ x ∈ s, f x :=
hf.order_dual.map_Sup s
lemma map_infi (hf : right_ord_continuous f) (g : ι → α) :
f (⨅ i, g i) = ⨅ i, f (g i) :=
hf.order_dual.map_supr g
end complete_lattice
section conditionally_complete_lattice
variables [conditionally_complete_lattice α] [conditionally_complete_lattice β] [nonempty ι]
{f : α → β}
lemma map_cInf (hf : right_ord_continuous f) {s : set α} (sne : s.nonempty) (sbdd : bdd_below s) :
f (Inf s) = Inf (f '' s) :=
hf.order_dual.map_cSup sne sbdd
lemma map_cinfi (hf : right_ord_continuous f) {g : ι → α} (hg : bdd_below (range g)) :
f (⨅ i, g i) = ⨅ i, f (g i) :=
hf.order_dual.map_csupr hg
end conditionally_complete_lattice
end right_ord_continuous
namespace order_iso
section preorder
variables [preorder α] [preorder β] (e : α ≃o β)
{s : set α} {x : α}
protected lemma left_ord_continuous : left_ord_continuous e :=
λ s x hx,
⟨monotone.mem_upper_bounds_image (λ x y, e.map_rel_iff.2) hx.1,
λ y hy, e.rel_symm_apply.1 $ (is_lub_le_iff hx).2 $ λ x' hx', e.rel_symm_apply.2 $ hy $
mem_image_of_mem _ hx'⟩
protected lemma right_ord_continuous : right_ord_continuous e := order_iso.left_ord_continuous e.dual
end preorder
end order_iso
|
39f3989a41b03331c1a65648fd34700efa7b5b99 | 8eeb99d0fdf8125f5d39a0ce8631653f588ee817 | /src/data/quot.lean | 67d080b227e116ca897a57614b5271b8df4b77c2 | [
"Apache-2.0"
] | permissive | jesse-michael-han/mathlib | a15c58378846011b003669354cbab7062b893cfe | fa6312e4dc971985e6b7708d99a5bc3062485c89 | refs/heads/master | 1,625,200,760,912 | 1,602,081,753,000 | 1,602,081,753,000 | 181,787,230 | 0 | 0 | null | 1,555,460,682,000 | 1,555,460,682,000 | null | UTF-8 | Lean | false | false | 19,578 | 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 logic.relator
/-!
# Quotient types
This module extensd the core library's treatment of quotient types (`init.data.quot`).
## Tags
quotient
-/
variables {α : Sort*} {β : Sort*}
namespace setoid
lemma ext {α : Sort*} :
∀{s t : setoid α}, (∀a b, @setoid.r α s a b ↔ @setoid.r α t a b) → s = t
| ⟨r, _⟩ ⟨p, _⟩ eq :=
have r = p, from funext $ assume a, funext $ assume b, propext $ eq a b,
by subst this
end setoid
namespace quot
variables {ra : α → α → Prop} {rb : β → β → Prop} {φ : quot ra → quot rb → Sort*}
local notation `⟦`:max a `⟧` := quot.mk _ a
instance [inhabited α] : inhabited (quot ra) := ⟨⟦default _⟧⟩
/-- Recursion on two `quotient` arguments `a` and `b`, result type depends on `⟦a⟧` and `⟦b⟧`. -/
protected def hrec_on₂ (qa : quot ra) (qb : quot rb) (f : Π a b, φ ⟦a⟧ ⟦b⟧)
(ca : ∀ {b a₁ a₂}, ra a₁ a₂ → f a₁ b == f a₂ b)
(cb : ∀ {a b₁ b₂}, rb b₁ b₂ → f a b₁ == f a b₂) : φ qa qb :=
quot.hrec_on qa (λ a, quot.hrec_on qb (f a) (λ b₁ b₂ pb, cb pb)) $ λ a₁ a₂ pa,
quot.induction_on qb $ λ b,
calc @quot.hrec_on _ _ (φ _) ⟦b⟧ (f a₁) (@cb _)
== f a₁ b : by simp
... == f a₂ b : ca pa
... == @quot.hrec_on _ _ (φ _) ⟦b⟧ (f a₂) (@cb _) : by simp
/-- Map a function `f : α → β` such that `ra x y` implies `rb (f x) (f y)`
to a map `quot ra → quot rb`. -/
protected def map (f : α → β) (h : (ra ⇒ rb) f f) : quot ra → quot rb :=
quot.lift (λ x, ⟦f x⟧) $ assume x y (h₁ : ra x y), quot.sound $ h h₁
/-- If `ra` is a subrelation of `ra'`, then we have a natural map `quot ra → quot ra'`. -/
protected def map_right {ra' : α → α → Prop} (h : ∀a₁ a₂, ra a₁ a₂ → ra' a₁ a₂) :
quot ra → quot ra' :=
quot.map id h
/-- weaken the relation of a quotient -/
def factor {α : Type*} (r s : α → α → Prop) (h : ∀ x y, r x y → s x y) :
quot r → quot s :=
quot.lift (quot.mk s) (λ x y rxy, quot.sound (h x y rxy))
lemma factor_mk_eq {α : Type*} (r s : α → α → Prop) (h : ∀ x y, r x y → s x y) :
factor r s h ∘ quot.mk _ = quot.mk _ := rfl
variables {γ : Sort*} {r : α → α → Prop} {s : β → β → Prop}
/-- Descends a function `f : α → β → γ` to quotients of `α` and `β`. -/
attribute [reducible, elab_as_eliminator]
protected def lift₂
(f : α → β → γ)
(hr : ∀ a b₁ b₂, s b₁ b₂ → f a b₁ = f a b₂)
(hs : ∀ a₁ a₂ b, r a₁ a₂ → f a₁ b = f a₂ b)
(q₁ : quot r) (q₂ : quot s) : γ :=
quot.lift (λ a, quot.lift (f a) (hr a))
(λ a₁ a₂ ha, funext (λ q, quot.induction_on q (λ b, hs a₁ a₂ b ha)))
q₁ q₂
@[simp]
lemma lift₂_mk (a : α) (b : β) (f : α → β → γ)
(hr : ∀ a b₁ b₂, s b₁ b₂ → f a b₁ = f a b₂)
(hs : ∀ a₁ a₂ b, r a₁ a₂ → f a₁ b = f a₂ b) :
quot.lift₂ f hr hs (quot.mk r a) (quot.mk s b) = f a b := rfl
/-- Descends a function `f : α → β → γ` to quotients of `α` and `β` and applies it. -/
attribute [reducible, elab_as_eliminator]
protected def lift_on₂ (p : quot r) (q : quot s) (f : α → β → γ)
(hr : ∀ a b₁ b₂, s b₁ b₂ → f a b₁ = f a b₂)
(hs : ∀ a₁ a₂ b, r a₁ a₂ → f a₁ b = f a₂ b) : γ := quot.lift₂ f hr hs p q
@[simp]
lemma lift_on₂_mk (a : α) (b : β) (f : α → β → γ)
(hr : ∀ a b₁ b₂, s b₁ b₂ → f a b₁ = f a b₂)
(hs : ∀ a₁ a₂ b, r a₁ a₂ → f a₁ b = f a₂ b) :
quot.lift_on₂ (quot.mk r a) (quot.mk s b) f hr hs = f a b := rfl
variables {t : γ → γ → Prop}
/-- Descends a function `f : α → β → γ` to quotients of `α` and `β` wih values in a quotient of
`γ`. -/
protected def map₂ (f : α → β → γ)
(hr : ∀ a b₁ b₂, s b₁ b₂ → t (f a b₁) (f a b₂))
(hs : ∀ a₁ a₂ b, r a₁ a₂ → t (f a₁ b) (f a₂ b))
(q₁ : quot r) (q₂ : quot s) : quot t :=
quot.lift₂ (λ a b, quot.mk t $ f a b) (λ a b₁ b₂ hb, quot.sound (hr a b₁ b₂ hb))
(λ a₁ a₂ b ha, quot.sound (hs a₁ a₂ b ha)) q₁ q₂
@[simp]
lemma map₂_mk (f : α → β → γ)
(hr : ∀ a b₁ b₂, s b₁ b₂ → t (f a b₁) (f a b₂))
(hs : ∀ a₁ a₂ b, r a₁ a₂ → t (f a₁ b) (f a₂ b))
(a : α) (b : β) : quot.map₂ f hr hs (quot.mk r a) (quot.mk s b) = quot.mk t (f a b) := rfl
attribute [elab_as_eliminator]
protected lemma induction_on₂
{δ : quot r → quot s → Prop} (q₁ : quot r) (q₂ : quot s)
(h : ∀ a b, δ (quot.mk r a) (quot.mk s b)) : δ q₁ q₂ :=
quot.ind (λ a₁, quot.ind (λ a₂, h a₁ a₂) q₂) q₁
attribute [elab_as_eliminator]
protected lemma induction_on₃
{δ : quot r → quot s → quot t → Prop} (q₁ : quot r) (q₂ : quot s) (q₃ : quot t)
(h : ∀ a b c, δ (quot.mk r a) (quot.mk s b) (quot.mk t c)) : δ q₁ q₂ q₃ :=
quot.ind (λ a₁, quot.ind (λ a₂, quot.ind (λ a₃, h a₁ a₂ a₃) q₃) q₂) q₁
end quot
namespace quotient
variables [sa : setoid α] [sb : setoid β]
variables {φ : quotient sa → quotient sb → Sort*}
instance [inhabited α] : inhabited (quotient sa) := ⟨⟦default _⟧⟩
/-- Induction on two `quotient` arguments `a` and `b`, result type depends on `⟦a⟧` and `⟦b⟧`. -/
protected def hrec_on₂ (qa : quotient sa) (qb : quotient sb) (f : Π a b, φ ⟦a⟧ ⟦b⟧)
(c : ∀ a₁ b₁ a₂ b₂, a₁ ≈ a₂ → b₁ ≈ b₂ → f a₁ b₁ == f a₂ b₂) : φ qa qb :=
quot.hrec_on₂ qa qb f
(λ _ _ _ p, c _ _ _ _ p (setoid.refl _))
(λ _ _ _ p, c _ _ _ _ (setoid.refl _) p)
/-- Map a function `f : α → β` that sends equivalent elements to equivalent elements
to a function `quotient sa → quotient sb`. Useful to define unary operations on quotients. -/
protected def map (f : α → β) (h : ((≈) ⇒ (≈)) f f) : quotient sa → quotient sb :=
quot.map f h
@[simp] lemma map_mk (f : α → β) (h : ((≈) ⇒ (≈)) f f) (x : α) :
quotient.map f h (⟦x⟧ : quotient sa) = (⟦f x⟧ : quotient sb) :=
rfl
variables {γ : Sort*} [sc : setoid γ]
/-- Map a function `f : α → β → γ` that sends equivalent elements to equivalent elements
to a function `f : quotient sa → quotient sb → quotient sc`.
Useful to define binary operations on quotients. -/
protected def map₂ (f : α → β → γ) (h : ((≈) ⇒ (≈) ⇒ (≈)) f f) :
quotient sa → quotient sb → quotient sc :=
quotient.lift₂ (λ x y, ⟦f x y⟧) (λ x₁ y₁ x₂ y₂ h₁ h₂, quot.sound $ h h₁ h₂)
end quotient
lemma quot.eq {α : Type*} {r : α → α → Prop} {x y : α} :
quot.mk r x = quot.mk r y ↔ eqv_gen r x y :=
⟨quot.exact r, quot.eqv_gen_sound⟩
@[simp] theorem quotient.eq [r : setoid α] {x y : α} : ⟦x⟧ = ⟦y⟧ ↔ x ≈ y :=
⟨quotient.exact, quotient.sound⟩
theorem forall_quotient_iff {α : Type*} [r : setoid α] {p : quotient r → Prop} :
(∀a:quotient r, p a) ↔ (∀a:α, p ⟦a⟧) :=
⟨assume h x, h _, assume h a, a.induction_on h⟩
@[simp] lemma quotient.lift_beta [s : setoid α] (f : α → β) (h : ∀ (a b : α), a ≈ b → f a = f b)
(x : α) :
quotient.lift f h (quotient.mk x) = f x := rfl
@[simp] lemma quotient.lift_on_beta [s : setoid α] (f : α → β) (h : ∀ (a b : α), a ≈ b → f a = f b)
(x : α) :
quotient.lift_on (quotient.mk x) f h = f x := rfl
@[simp] theorem quotient.lift_on_beta₂ {α : Type} {β : Type} [setoid α] (f : α → α → β)
(h : ∀ (a₁ a₂ b₁ b₂ : α), a₁ ≈ b₁ → a₂ ≈ b₂ → f a₁ a₂ = f b₁ b₂) (x y : α) :
quotient.lift_on₂ (quotient.mk x) (quotient.mk y) f h = f x y := rfl
/-- Choose an element of the equivalence class using the axiom of choice.
Sound but noncomputable. -/
noncomputable def quot.out {r : α → α → Prop} (q : quot r) : α :=
classical.some (quot.exists_rep q)
/-- Unwrap the VM representation of a quotient to obtain an element of the equivalence class.
Computable but unsound. -/
meta def quot.unquot {r : α → α → Prop} : quot r → α := unchecked_cast
@[simp] theorem quot.out_eq {r : α → α → Prop} (q : quot r) : quot.mk r q.out = q :=
classical.some_spec (quot.exists_rep q)
/-- Choose an element of the equivalence class using the axiom of choice.
Sound but noncomputable. -/
noncomputable def quotient.out [s : setoid α] : quotient s → α := quot.out
@[simp] theorem quotient.out_eq [s : setoid α] (q : quotient s) : ⟦q.out⟧ = q := q.out_eq
theorem quotient.mk_out [s : setoid α] (a : α) : ⟦a⟧.out ≈ a :=
quotient.exact (quotient.out_eq _)
instance pi_setoid {ι : Sort*} {α : ι → Sort*} [∀ i, setoid (α i)] : setoid (Π i, α i) :=
{ r := λ a b, ∀ i, a i ≈ b i,
iseqv := ⟨
λ a i, setoid.refl _,
λ a b h i, setoid.symm (h _),
λ a b c h₁ h₂ i, setoid.trans (h₁ _) (h₂ _)⟩ }
/-- Given a function `f : Π i, quotient (S i)`, returns the class of functions `Π i, α i` sending
each `i` to an element of the class `f i`. -/
noncomputable def quotient.choice {ι : Type*} {α : ι → Type*} [S : Π i, setoid (α i)]
(f : Π i, quotient (S i)) : @quotient (Π i, α i) (by apply_instance) :=
⟦λ i, (f i).out⟧
theorem quotient.choice_eq {ι : Type*} {α : ι → Type*} [Π i, setoid (α i)]
(f : Π i, α i) : quotient.choice (λ i, ⟦f i⟧) = ⟦f⟧ :=
quotient.sound $ λ i, quotient.mk_out _
lemma nonempty_quotient_iff (s : setoid α) : nonempty (quotient s) ↔ nonempty α :=
⟨assume ⟨a⟩, quotient.induction_on a nonempty.intro, assume ⟨a⟩, ⟨⟦a⟧⟩⟩
/-- `trunc α` is the quotient of `α` by the always-true relation. This
is related to the propositional truncation in HoTT, and is similar
in effect to `nonempty α`, but unlike `nonempty α`, `trunc α` is data,
so the VM representation is the same as `α`, and so this can be used to
maintain computability. -/
def {u} trunc (α : Sort u) : Sort u := @quot α (λ _ _, true)
theorem true_equivalence : @equivalence α (λ _ _, true) :=
⟨λ _, trivial, λ _ _ _, trivial, λ _ _ _ _ _, trivial⟩
namespace trunc
/-- Constructor for `trunc α` -/
def mk (a : α) : trunc α := quot.mk _ a
instance [inhabited α] : inhabited (trunc α) := ⟨mk (default _)⟩
/-- Any constant function lifts to a function out of the truncation -/
def lift (f : α → β) (c : ∀ a b : α, f a = f b) : trunc α → β :=
quot.lift f (λ a b _, c a b)
theorem ind {β : trunc α → Prop} : (∀ a : α, β (mk a)) → ∀ q : trunc α, β q := quot.ind
protected theorem lift_beta (f : α → β) (c) (a : α) : lift f c (mk a) = f a := rfl
/-- Lift a constant function on `q : trunc α`. -/
@[reducible, elab_as_eliminator]
protected def lift_on (q : trunc α) (f : α → β)
(c : ∀ a b : α, f a = f b) : β := lift f c q
@[elab_as_eliminator]
protected theorem induction_on {β : trunc α → Prop} (q : trunc α)
(h : ∀ a, β (mk a)) : β q := ind h q
theorem exists_rep (q : trunc α) : ∃ a : α, mk a = q := quot.exists_rep q
attribute [elab_as_eliminator]
protected theorem induction_on₂ {C : trunc α → trunc β → Prop} (q₁ : trunc α) (q₂ : trunc β)
(h : ∀ a b, C (mk a) (mk b)) : C q₁ q₂ :=
trunc.induction_on q₁ $ λ a₁, trunc.induction_on q₂ (h a₁)
protected theorem eq (a b : trunc α) : a = b :=
trunc.induction_on₂ a b (λ x y, quot.sound trivial)
instance : subsingleton (trunc α) := ⟨trunc.eq⟩
/-- The `bind` operator for the `trunc` monad. -/
def bind (q : trunc α) (f : α → trunc β) : trunc β :=
trunc.lift_on q f (λ a b, trunc.eq _ _)
/-- A function `f : α → β` defines a function `map f : trunc α → trunc β`. -/
def map (f : α → β) (q : trunc α) : trunc β := bind q (trunc.mk ∘ f)
instance : monad trunc :=
{ pure := @trunc.mk,
bind := @trunc.bind }
instance : is_lawful_monad trunc :=
{ id_map := λ α q, trunc.eq _ _,
pure_bind := λ α β q f, rfl,
bind_assoc := λ α β γ x f g, trunc.eq _ _ }
variable {C : trunc α → Sort*}
/-- Recursion/induction principle for `trunc`. -/
@[reducible, elab_as_eliminator]
protected def rec
(f : Π a, C (mk a)) (h : ∀ (a b : α), (eq.rec (f a) (trunc.eq (mk a) (mk b)) : C (mk b)) = f b)
(q : trunc α) : C q :=
quot.rec f (λ a b _, h a b) q
/-- A version of `trunc.rec` taking `q : trunc α` as the first argument. -/
@[reducible, elab_as_eliminator]
protected def rec_on (q : trunc α) (f : Π a, C (mk a))
(h : ∀ (a b : α), (eq.rec (f a) (trunc.eq (mk a) (mk b)) : C (mk b)) = f b) : C q :=
trunc.rec f h q
/-- A version of `trunc.rec_on` assuming the codomain is a `subsingleton`. -/
@[reducible, elab_as_eliminator]
protected def rec_on_subsingleton
[∀ a, subsingleton (C (mk a))] (q : trunc α) (f : Π a, C (mk a)) : C q :=
trunc.rec f (λ a b, subsingleton.elim _ (f b)) q
/-- Noncomputably extract a representative of `trunc α` (using the axiom of choice). -/
noncomputable def out : trunc α → α := quot.out
@[simp] theorem out_eq (q : trunc α) : mk q.out = q := trunc.eq _ _
end trunc
theorem nonempty_of_trunc (q : trunc α) : nonempty α :=
let ⟨a, _⟩ := q.exists_rep in ⟨a⟩
namespace quotient
variables {γ : Sort*} {φ : Sort*}
{s₁ : setoid α} {s₂ : setoid β} {s₃ : setoid γ}
/- Versions of quotient definitions and lemmas ending in `'` use unification instead
of typeclass inference for inferring the `setoid` argument. This is useful when there are
several different quotient relations on a type, for example quotient groups, rings and modules -/
/-- A version of `quotient.mk` taking `{s : setoid α}` as an implicit argument instead of an
instance argument. -/
protected def mk' (a : α) : quotient s₁ := quot.mk s₁.1 a
/-- A version of `quotient.lift_on` taking `{s : setoid α}` as an implicit argument instead of an
instance argument. -/
@[elab_as_eliminator, reducible]
protected def lift_on' (q : quotient s₁) (f : α → φ)
(h : ∀ a b, @setoid.r α s₁ a b → f a = f b) : φ := quotient.lift_on q f h
/-- A version of `quotient.lift_on₂` taking `{s₁ : setoid α} {s₂ : setoid β}` as implicit arguments
instead of instance arguments. -/
@[elab_as_eliminator, reducible]
protected def lift_on₂' (q₁ : quotient s₁) (q₂ : quotient s₂) (f : α → β → γ)
(h : ∀ a₁ a₂ b₁ b₂, @setoid.r α s₁ a₁ b₁ → @setoid.r β s₂ a₂ b₂ → f a₁ a₂ = f b₁ b₂) : γ :=
quotient.lift_on₂ q₁ q₂ f h
/-- A version of `quotient.ind` taking `{s : setoid α}` as an implicit argument instead of an
instance argument. -/
@[elab_as_eliminator]
protected lemma ind' {p : quotient s₁ → Prop}
(h : ∀ a, p (quotient.mk' a)) (q : quotient s₁) : p q :=
quotient.ind h q
/-- A version of `quotient.ind₂` taking `{s₁ : setoid α} {s₂ : setoid β}` as implicit arguments
instead of instance arguments. -/
@[elab_as_eliminator]
protected lemma ind₂' {p : quotient s₁ → quotient s₂ → Prop}
(h : ∀ a₁ a₂, p (quotient.mk' a₁) (quotient.mk' a₂))
(q₁ : quotient s₁) (q₂ : quotient s₂) : p q₁ q₂ :=
quotient.ind₂ h q₁ q₂
/-- A version of `quotient.induction_on` taking `{s : setoid α}` as an implicit argument instead
of an instance argument. -/
@[elab_as_eliminator]
protected lemma induction_on' {p : quotient s₁ → Prop} (q : quotient s₁)
(h : ∀ a, p (quotient.mk' a)) : p q := quotient.induction_on q h
/-- A version of `quotient.induction_on₂` taking `{s₁ : setoid α} {s₂ : setoid β}` as implicit
arguments instead of instance arguments. -/
@[elab_as_eliminator]
protected lemma induction_on₂' {p : quotient s₁ → quotient s₂ → Prop} (q₁ : quotient s₁)
(q₂ : quotient s₂) (h : ∀ a₁ a₂, p (quotient.mk' a₁) (quotient.mk' a₂)) : p q₁ q₂ :=
quotient.induction_on₂ q₁ q₂ h
/-- A version of `quotient.induction_on₃` taking `{s₁ : setoid α} {s₂ : setoid β} {s₃ : setoid γ}`
as implicit arguments instead of instance arguments. -/
@[elab_as_eliminator]
protected lemma induction_on₃' {p : quotient s₁ → quotient s₂ → quotient s₃ → Prop}
(q₁ : quotient s₁) (q₂ : quotient s₂) (q₃ : quotient s₃)
(h : ∀ a₁ a₂ a₃, p (quotient.mk' a₁) (quotient.mk' a₂) (quotient.mk' a₃)) : p q₁ q₂ q₃ :=
quotient.induction_on₃ q₁ q₂ q₃ h
/-- Recursion on a `quotient` argument `a`, result type depends on `⟦a⟧`. -/
protected def hrec_on' {φ : quotient s₁ → Sort*} (qa : quotient s₁) (f : Π a, φ (quotient.mk' a))
(c : ∀ a₁ a₂, a₁ ≈ a₂ → f a₁ == f a₂) : φ qa :=
quot.hrec_on qa f c
@[simp] lemma hrec_on'_mk' {φ : quotient s₁ → Sort*} (f : Π a, φ (quotient.mk' a))
(c : ∀ a₁ a₂, a₁ ≈ a₂ → f a₁ == f a₂) (x : α) :
(quotient.mk' x).hrec_on' f c = f x :=
rfl
/-- Recursion on two `quotient` arguments `a` and `b`, result type depends on `⟦a⟧` and `⟦b⟧`. -/
protected def hrec_on₂' {φ : quotient s₁ → quotient s₂ → Sort*} (qa : quotient s₁)
(qb : quotient s₂) (f : ∀ a b, φ (quotient.mk' a) (quotient.mk' b))
(c : ∀ a₁ b₁ a₂ b₂, a₁ ≈ a₂ → b₁ ≈ b₂ → f a₁ b₁ == f a₂ b₂) : φ qa qb :=
quotient.hrec_on₂ qa qb f c
@[simp] lemma hrec_on₂'_mk' {φ : quotient s₁ → quotient s₂ → Sort*}
(f : ∀ a b, φ (quotient.mk' a) (quotient.mk' b))
(c : ∀ a₁ b₁ a₂ b₂, a₁ ≈ a₂ → b₁ ≈ b₂ → f a₁ b₁ == f a₂ b₂) (x : α) (qb : quotient s₂) :
(quotient.mk' x).hrec_on₂' qb f c = qb.hrec_on' (f x) (λ b₁ b₂, c _ _ _ _ (setoid.refl _)) :=
rfl
/-- Map a function `f : α → β` that sends equivalent elements to equivalent elements
to a function `quotient sa → quotient sb`. Useful to define unary operations on quotients. -/
protected def map' (f : α → β) (h : ((≈) ⇒ (≈)) f f) :
quotient s₁ → quotient s₂ :=
quot.map f h
@[simp] lemma map'_mk' (f : α → β) (h) (x : α) :
(quotient.mk' x : quotient s₁).map' f h = (quotient.mk' (f x) : quotient s₂) :=
rfl
/-- A version of `quotient.map₂` using curly braces and unification. -/
protected def map₂' (f : α → β → γ) (h : ((≈) ⇒ (≈) ⇒ (≈)) f f) :
quotient s₁ → quotient s₂ → quotient s₃ :=
quotient.map₂ f h
@[simp] lemma map₂'_mk' (f : α → β → γ) (h) (x : α) :
(quotient.mk' x : quotient s₁).map₂' f h =
(quotient.map' (f x) (h (setoid.refl x)) : quotient s₂ → quotient s₃) :=
rfl
lemma exact' {a b : α} :
(quotient.mk' a : quotient s₁) = quotient.mk' b → @setoid.r _ s₁ a b :=
quotient.exact
lemma sound' {a b : α} : @setoid.r _ s₁ a b → @quotient.mk' α s₁ a = quotient.mk' b :=
quotient.sound
@[simp]
protected lemma eq' {a b : α} : @quotient.mk' α s₁ a = quotient.mk' b ↔ @setoid.r _ s₁ a b :=
quotient.eq
/-- A version of `quotient.out` taking `{s₁ : setoid α}` as an implicit argument instead of an
instance argument. -/
noncomputable def out' (a : quotient s₁) : α := quotient.out a
@[simp] theorem out_eq' (q : quotient s₁) : quotient.mk' q.out' = q := q.out_eq
theorem mk_out' (a : α) : @setoid.r α s₁ (quotient.mk' a : quotient s₁).out' a :=
quotient.exact (quotient.out_eq _)
end quotient
|
9806c468e37eb063b9ead7bfac4e34bf2b4c1c72 | 57fdc8de88f5ea3bfde4325e6ecd13f93a274ab5 | /group_theory/subgroup.lean | c43704f2ecd0badda8418893b9bcfccb68c0d231 | [
"Apache-2.0"
] | permissive | louisanu/mathlib | 11f56f2d40dc792bc05ee2f78ea37d73e98ecbfe | 2bd5e2159d20a8f20d04fc4d382e65eea775ed39 | refs/heads/master | 1,617,706,993,439 | 1,523,163,654,000 | 1,523,163,654,000 | 124,519,997 | 0 | 0 | Apache-2.0 | 1,520,588,283,000 | 1,520,588,283,000 | null | UTF-8 | Lean | false | false | 9,705 | lean | /-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import data.finset algebra.big_operators data.equiv data.set data.nat.basic set_theory.cardinal
open set function finset
universes u v w
variables {α : Type u} {β : Type v} {s : set α} {a a₁ a₂ : α}
namespace finset
open finset
lemma mem_range_iff_mem_finset_range_of_mod_eq [decidable_eq α] {f : ℤ → α} {a : α} {n : ℕ}
(hn : 0 < n) (h : ∀i, f (i % n) = f i) :
a ∈ set.range f ↔ a ∈ (finset.range n).image (λi, f i) :=
suffices (∃i, f (i % n) = a) ↔ ∃i, i < n ∧ f ↑i = a, by simpa [h],
have hn' : 0 < (n : ℤ), from int.coe_nat_lt.mpr hn,
iff.intro
(assume ⟨i, hi⟩,
have 0 ≤ i % ↑n, from int.mod_nonneg _ (ne_of_gt hn'),
⟨int.to_nat (i % n),
by rw [←int.coe_nat_lt, int.to_nat_of_nonneg this]; exact ⟨int.mod_lt_of_pos i hn', hi⟩⟩)
(assume ⟨i, hi, ha⟩,
⟨i, by rw [int.mod_eq_of_lt (int.coe_zero_le _) (int.coe_nat_lt_coe_nat_of_lt hi), ha]⟩)
end finset
variables [group α]
/-- A subset of a group closed under the group operations. -/
structure is_subgroup (s : set α) : Prop :=
(one_mem : (1:α) ∈ s)
(mul_inv_mem : ∀a∈s, ∀b∈s, a * b⁻¹ ∈ s)
def cosets (s : set α) : set (set α) := range (λa, (*) a '' s)
namespace is_subgroup
lemma inv_mem (hs : is_subgroup s) (h : a ∈ s) : a⁻¹ ∈ s :=
have 1 * a⁻¹ ∈ s, from hs.mul_inv_mem _ hs.one_mem _ h,
by simpa
lemma inv_mem_iff (hs : is_subgroup s) : a⁻¹ ∈ s ↔ a ∈ s :=
iff.intro (assume h, have a⁻¹⁻¹ ∈ s, from hs.inv_mem h, by simpa) hs.inv_mem
lemma mul_mem (hs : is_subgroup s) (h₁ : a₁ ∈ s) (h₂ : a₂ ∈ s) : a₁ * a₂ ∈ s :=
have a₁ * a₂⁻¹⁻¹ ∈ s, from hs.mul_inv_mem _ h₁ _ (hs.inv_mem h₂),
by simpa
lemma mul_image (hs : is_subgroup s) (a : α) (ha : a ∈ s) :
(*) a '' s = s :=
ext $ assume a', iff.intro
(assume ⟨a'', ha'', eq⟩, eq ▸ hs.mul_mem ha ha'')
(assume ha', ⟨a⁻¹ * a', hs.mul_mem (hs.inv_mem ha) ha', by simp⟩)
lemma injective_mul {a : α} : injective ((*) a) :=
assume a₁ a₂ h,
have a⁻¹ * a * a₁ = a⁻¹ * a * a₂, by rw [mul_assoc, mul_assoc, h],
by rwa [inv_mul_self, one_mul, one_mul] at this
lemma subgroup_mem_cosets (hs : is_subgroup s) : s ∈ cosets s :=
⟨1, hs.mul_image _ hs.one_mem⟩
lemma cosets_disjoint (hs : is_subgroup s) :
∀{s₁ s₂ : set α}, s₁ ∈ cosets s → s₂ ∈ cosets s → ∀{a}, a ∈ s₁ → a ∈ s₂ → s₁ = s₂
| _ _ ⟨b₁, rfl⟩ ⟨b₂, rfl⟩ a ⟨c₁, hc₁, eq₁⟩ ⟨c₂, hc₂, eq₂⟩ :=
have b_eq : b₁ = b₂ * c₂ * c₁⁻¹, by rw [eq_mul_inv_iff_mul_eq, eq₁, eq₂],
have hc : c₂ * c₁⁻¹ ∈ s, from hs.mul_mem hc₂ (hs.inv_mem hc₁),
calc (*) b₁ '' s = (*) b₂ '' ((*) (c₂ * c₁⁻¹) '' s) :
by rw [←image_comp, (∘), b_eq]; apply image_congr _; simp [mul_assoc]
... = (*) b₂ '' s :
by rw [hs.mul_image _ hc]
lemma pairwise_cosets_disjoint (hs : is_subgroup s) : pairwise_on (cosets s) disjoint :=
assume s₁ h₁ s₂ h₂ ne, eq_empty_iff_forall_not_mem.mpr $ assume a ⟨ha₁, ha₂⟩,
ne $ hs.cosets_disjoint h₁ h₂ ha₁ ha₂
lemma cosets_equiv_subgroup (hs : is_subgroup s) : ∀{t : set α}, t ∈ cosets s → nonempty (t ≃ s)
| _ ⟨a, rfl⟩ := ⟨(equiv.set.image ((*) a) s injective_mul).symm⟩
lemma Union_cosets_eq_univ (hs : is_subgroup s) : ⋃₀ cosets s = univ :=
eq_univ_of_forall $ assume a, ⟨(*) a '' s, mem_range_self _, ⟨1, hs.one_mem, mul_one _⟩⟩
lemma group_equiv_cosets_times_subgroup (hs : is_subgroup s) : nonempty (α ≃ (cosets s × s)) :=
⟨calc α ≃ (@set.univ α) :
(equiv.set.univ α).symm
... ≃ (⋃t∈cosets s, t) :
by rw [←hs.Union_cosets_eq_univ]; simp
... ≃ (Σt:cosets s, t) :
equiv.set.bUnion_eq_sigma_of_disjoint hs.pairwise_cosets_disjoint
... ≃ (Σt:cosets s, s) :
equiv.sigma_congr_right $ λ⟨t, ht⟩, classical.choice $ hs.cosets_equiv_subgroup ht
... ≃ (cosets s × s) :
equiv.sigma_equiv_prod _ _⟩
end is_subgroup
lemma is_subgroup_range_gpow : is_subgroup (range ((^) a : ℤ → α)) :=
⟨⟨0, rfl⟩, assume a ⟨i, ha⟩ b ⟨j, hb⟩, ⟨i - j, by simp [gpow_add, gpow_neg, ha.symm, hb.symm]⟩⟩
section finite_group
variables [fintype α] [decidable_eq α]
lemma exists_gpow_eq_one (a : α) : ∃i≠0, a ^ (i:ℤ) = 1 :=
have ¬ injective (λi, a ^ i),
from not_injective_int_fintype,
let ⟨i, j, a_eq, ne⟩ := show ∃(i j : ℤ), a ^ i = a ^ j ∧ i ≠ j,
by rw [injective] at this; simpa [classical.not_forall] in
have a ^ (i - j) = 1,
by simp [gpow_add, gpow_neg, a_eq],
⟨i - j, sub_ne_zero.mpr ne, this⟩
lemma exists_pow_eq_one (a : α) : ∃i≠0, a ^ i = 1 :=
let ⟨i, hi, eq⟩ := exists_gpow_eq_one a in
begin
cases i,
{ exact ⟨i, by simp [int.of_nat_eq_coe, *] at *, eq⟩ },
{ exact ⟨i + 1, dec_trivial, inv_eq_one.1 eq⟩ }
end
/-- `order_of a` is the order of the element, i.e. the `n ≥ 1`, s.t. `a ^ n = 1` -/
def order_of (a : α) : ℕ := nat.find (exists_pow_eq_one a)
lemma pow_order_of_eq_one (a : α) : a ^ order_of a = 1 :=
let ⟨h₁, h₂⟩ := nat.find_spec (exists_pow_eq_one a) in h₂
lemma order_of_ne_zero (a : α) : order_of a ≠ 0 :=
let ⟨h₁, h₂⟩ := nat.find_spec (exists_pow_eq_one a) in h₁
private lemma pow_injective_aux {n m : ℕ} (a : α) (h : n ≤ m)
(hn : n < order_of a) (hm : m < order_of a) (eq : a ^ n = a ^ m) : n = m :=
decidable.by_contradiction $ assume ne : n ≠ m,
have h₁ : m - n ≠ 0, by simp [nat.sub_eq_iff_eq_add h, ne.symm],
have h₂ : a ^ (m - n) = 1, by simp [pow_sub _ h, eq],
have le : order_of a ≤ m - n, from nat.find_min' (exists_pow_eq_one a) ⟨h₁, h₂⟩,
have lt : m - n < order_of a,
from (nat.sub_lt_left_iff_lt_add h).mpr $ nat.lt_add_left _ _ _ hm,
lt_irrefl _ (lt_of_le_of_lt le lt)
lemma pow_injective_of_lt_order_of {n m : ℕ} (a : α)
(hn : n < order_of a) (hm : m < order_of a) (eq : a ^ n = a ^ m) : n = m :=
(le_total n m).elim
(assume h, pow_injective_aux a h hn hm eq)
(assume h, (pow_injective_aux a h hm hn eq.symm).symm)
lemma order_of_le_card_univ : order_of a ≤ fintype.card α :=
finset.card_le_of_inj_on ((^) a)
(assume n _, fintype.complete _)
(assume i j, pow_injective_of_lt_order_of a)
lemma pow_eq_mod_order_of {n : ℕ} : a ^ n = a ^ (n % order_of a) :=
calc a ^ n = a ^ (n % order_of a + order_of a * (n / order_of a)) :
by rw [nat.mod_add_div]
... = a ^ (n % order_of a) :
by simp [pow_add, pow_mul, pow_order_of_eq_one]
lemma gpow_eq_mod_order_of {i : ℤ} : a ^ i = a ^ (i % order_of a) :=
calc a ^ i = a ^ (i % order_of a + order_of a * (i / order_of a)) :
by rw [int.mod_add_div]
... = a ^ (i % order_of a) :
by simp [gpow_add, gpow_mul, pow_order_of_eq_one]
lemma mem_range_gpow_iff_mem_range_order_of {a a' : α} :
a' ∈ range ((^) a : ℤ → α) ↔ a' ∈ (finset.range (order_of a)).image ((^) a : ℕ → α) :=
finset.mem_range_iff_mem_finset_range_of_mod_eq
(nat.pos_iff_ne_zero.mpr (order_of_ne_zero a))
(assume i, gpow_eq_mod_order_of.symm)
instance decidable_range_gpow : decidable_pred (range ((^) a : ℤ → α)) :=
assume a', decidable_of_iff'
(a' ∈ (finset.range (order_of a)).image ((^) a))
mem_range_gpow_iff_mem_range_order_of
section
local attribute [instance] set_fintype
lemma order_eq_card_range_gpow : order_of a = fintype.card (range ((^) a : ℤ → α)) :=
begin
refine (finset.card_eq_of_bijective _ _ _ _).symm,
{ exact λn hn, ⟨gpow a n, mem_range_self n⟩ },
{ exact assume ⟨_, i, rfl⟩ _,
have pos: (0:int) < order_of a,
from int.coe_nat_lt.mpr $ nat.pos_iff_ne_zero.mpr $ order_of_ne_zero a,
have 0 ≤ i % (order_of a),
from int.mod_nonneg _ $ ne_of_gt pos,
⟨int.to_nat (i % order_of a),
by rw [← int.coe_nat_lt, int.to_nat_of_nonneg this];
exact ⟨int.mod_lt_of_pos _ pos, subtype.eq gpow_eq_mod_order_of.symm⟩⟩ },
{ intros, exact finset.mem_univ _ },
{ exact assume i j hi hj eq, pow_injective_of_lt_order_of a hi hj $ by simpa using eq }
end
section classical
local attribute [instance] classical.prop_decidable
-- TODO: use cardinal theory, or introduce `card : set α → ℕ`
lemma order_of_dvd_card_univ : order_of a ∣ fintype.card α :=
let s := range $ gpow a in
have hs : is_subgroup s, from is_subgroup_range_gpow,
let ⟨equiv⟩ := hs.group_equiv_cosets_times_subgroup in
have ft_prod : fintype (cosets s × s),
from fintype.of_equiv α equiv,
have ft_s : fintype s,
from @fintype.fintype_prod_right _ _ _ ft_prod ⟨⟨s, hs.subgroup_mem_cosets⟩⟩,
have ft_cosets : fintype (cosets s),
from @fintype.fintype_prod_left _ _ _ ft_prod ⟨⟨1, hs.one_mem⟩⟩,
have ft : fintype (cosets s × s),
from @prod.fintype _ _ ft_cosets ft_s,
have eq₁ : fintype.card α = @fintype.card _ ft_cosets * @fintype.card _ ft_s,
from calc fintype.card α = @fintype.card _ ft_prod :
(@fintype.card_eq _ _ _ ft_prod).2 hs.group_equiv_cosets_times_subgroup
... = @fintype.card _ (@prod.fintype _ _ ft_cosets ft_s) :
congr_arg (@fintype.card _) $ subsingleton.elim _ _
... = @fintype.card _ ft_cosets * @fintype.card _ ft_s :
@fintype.card_prod _ _ ft_cosets ft_s,
have eq₂ : order_of a = @fintype.card _ ft_s,
from calc order_of a = _ : order_eq_card_range_gpow
... = _ : congr_arg (@fintype.card _) $ subsingleton.elim _ _,
dvd.intro (@fintype.card (cosets s) ft_cosets) $
by rw [eq₁, eq₂, mul_comm]
end classical
end
end finite_group
|
c0f86debded8004e1130d982e3e406657689ef31 | 40ad357bbd0d327dd1e3e7f7beb868bd4e5b0a9d | /src/temporal_logic/pair.lean | 4ed3cc99fa97f51c2dbc87e8d406c77f4dc53148 | [] | no_license | unitb/temporal-logic | 9966424f015976d5997a9ffa30cbd77cc3a9cb1c | accec04d1b09ca841be065511c9e206b725b16e9 | refs/heads/master | 1,633,868,382,769 | 1,541,072,223,000 | 1,541,072,223,000 | 114,790,987 | 5 | 3 | null | null | null | null | UTF-8 | Lean | false | false | 2,357 | lean | import temporal_logic.basic
universe variables u u₀ u₁ u₂
namespace temporal
open predicate
section pair
variables {α : Type u} {β : Type u₀} {γ : Type u₁} {ω : Type u₂}
variables (x : tvar α) (y : tvar β) (z : tvar γ)
@[tl_simp, simp]
lemma pair_model (i : ℕ) :
i ⊨ ⦃x,y⦄ = (i ⊨ x,i ⊨ y) :=
by { cases x, cases y, refl }
@[reducible]
def pair.fst : var (α × β) α :=
⟨ @prod.fst α β ⟩
@[reducible]
def pair.snd : var (α × β) β :=
⟨ @prod.snd α β ⟩
abbreviation tvar.fst (p : var γ (α × β)) : var γ α :=
pair.fst!p
abbreviation tvar.snd (p : var γ (α × β)) : var γ β :=
pair.snd!p
@[tl_simp, simp]
def pair.fst_mk (x : tvar α) (y : tvar β)
: pair.fst ! ⦃x,y⦄ = x :=
by lifted_pred
@[tl_simp, simp]
def pair.fst_mk' (x : tvar α) (y : tvar β)
: ⟨ @prod.fst α β ⟩ ! ⦃x,y⦄ = x :=
pair.fst_mk _ _
@[tl_simp, simp]
def pair.snd_mk (x : tvar α) (y : tvar β)
: pair.snd ! ⦃x,y⦄ = y :=
by lifted_pred
@[tl_simp, simp]
def pair.snd_mk' (x : tvar α) (y : tvar β)
: ⟨ @prod.snd α β ⟩ ! ⦃x,y⦄ = y :=
by lifted_pred
@[tl_simp, simp]
def prod.map_left {α β γ} (f : α → β) : α × γ → β × γ
| (x,y) := (f x, y)
@[tl_simp, simp]
def prod.map_right {α β γ} (f : β → γ) : α × β → α × γ
| (x,y) := (x,f y)
open temporal.prod
@[tl_simp, simp]
lemma map_right_proj_pair (f : β → γ)
: ⟨map_right f⟩ ! ⦃x,y⦄ = ⦃x, ⟨f⟩ ! y⦄ :=
by ext i ; simp [map_right] with tl_simp
@[tl_simp, simp]
lemma map_left_proj_pair (f : α → γ)
: ⟨map_left f⟩ ! ⦃x,y⦄ = ⦃⟨f⟩ ! x, y⦄ :=
by ext i ; simp [map_left] with tl_simp
@[tl_simp, simp]
lemma map_proj_pair (f : α → γ) (g : β → ω)
: ⟨prod.map f g⟩ ! ⦃x,y⦄ = ⦃⟨f⟩ ! x,⟨g⟩ ! y⦄ :=
by ext i ; simp [prod.map] with tl_simp
@[tl_simp, simp]
lemma eta_pair (w : tvar (α × β))
: ⦃w.fst, w.snd⦄ = w :=
by ext i ; simp [prod.map] with tl_simp
@[tl_simp, simp]
lemma next_pair (v₀ : tvar α) (v₁ : tvar β)
: ⊙⦃v₀,v₁⦄ = ⦃⊙v₀,⊙v₁⦄ :=
by lifted_pred [next]
@[tl_simp, simp, predicate]
lemma to_fun_var_pair (f : tvar α → tvar β) (g : tvar α → tvar γ) (w : tvar α)
: to_fun_var (λ w : tvar α, ⦃f w,g w⦄) w = ⦃to_fun_var f w,to_fun_var g w⦄ :=
by { lifted_pred, simp! }
end pair
end temporal
|
55ba4fc90998dadd372ee251b0d55b0fb7f9248c | d406927ab5617694ec9ea7001f101b7c9e3d9702 | /src/category_theory/Fintype.lean | 49fd297aeb31061f243a4efd30895072b290e382 | [
"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 | 4,825 | lean | /-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Adam Topaz
-/
import category_theory.concrete_category.basic
import category_theory.full_subcategory
import category_theory.skeletal
import category_theory.elementwise
import data.fintype.card
/-!
# The category of finite types.
We define the category of finite types, denoted `Fintype` as
(bundled) types with a `fintype` instance.
We also define `Fintype.skeleton`, the standard skeleton of `Fintype` whose objects are `fin n`
for `n : ℕ`. We prove that the obvious inclusion functor `Fintype.skeleton ⥤ Fintype` is an
equivalence of categories in `Fintype.skeleton.equivalence`.
We prove that `Fintype.skeleton` is a skeleton of `Fintype` in `Fintype.is_skeleton`.
-/
open_locale classical
open category_theory
/-- The category of finite types. -/
def Fintype := bundled fintype
namespace Fintype
instance : has_coe_to_sort Fintype Type* := bundled.has_coe_to_sort
/-- Construct a bundled `Fintype` from the underlying type and typeclass. -/
def of (X : Type*) [fintype X] : Fintype := bundled.of X
instance : inhabited Fintype := ⟨⟨pempty⟩⟩
instance {X : Fintype} : fintype X := X.2
instance : category Fintype := induced_category.category bundled.α
/-- The fully faithful embedding of `Fintype` into the category of types. -/
@[derive [full, faithful], simps]
def incl : Fintype ⥤ Type* := induced_functor _
instance concrete_category_Fintype : concrete_category Fintype := ⟨incl⟩
@[simp] lemma id_apply (X : Fintype) (x : X) : (𝟙 X : X → X) x = x := rfl
@[simp] lemma comp_apply {X Y Z : Fintype} (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) :
(f ≫ g) x = g (f x) := rfl
/-- Equivalences between finite types are the same as isomorphisms in `Fintype`. -/
-- See `equiv_equiv_iso` in the root namespace for the analogue in `Type`.
@[simps]
def equiv_equiv_iso {A B : Fintype} : (A ≃ B) ≃ (A ≅ B) :=
{ to_fun := λ e,
{ hom := e,
inv := e.symm, },
inv_fun := λ i,
{ to_fun := i.hom,
inv_fun := i.inv,
left_inv := iso.hom_inv_id_apply i,
right_inv := iso.inv_hom_id_apply i, },
left_inv := by tidy,
right_inv := by tidy, }
universe u
/--
The "standard" skeleton for `Fintype`. This is the full subcategory of `Fintype` spanned by objects
of the form `ulift (fin n)` for `n : ℕ`. We parameterize the objects of `Fintype.skeleton`
directly as `ulift ℕ`, as the type `ulift (fin m) ≃ ulift (fin n)` is
nonempty if and only if `n = m`. Specifying universes, `skeleton : Type u` is a small
skeletal category equivalent to `Fintype.{u}`.
-/
def skeleton : Type u := ulift ℕ
namespace skeleton
/-- Given any natural number `n`, this creates the associated object of `Fintype.skeleton`. -/
def mk : ℕ → skeleton := ulift.up
instance : inhabited skeleton := ⟨mk 0⟩
/-- Given any object of `Fintype.skeleton`, this returns the associated natural number. -/
def len : skeleton → ℕ := ulift.down
@[ext]
lemma ext (X Y : skeleton) : X.len = Y.len → X = Y := ulift.ext _ _
instance : small_category skeleton.{u} :=
{ hom := λ X Y, ulift.{u} (fin X.len) → ulift.{u} (fin Y.len),
id := λ _, id,
comp := λ _ _ _ f g, g ∘ f }
lemma is_skeletal : skeletal skeleton.{u} := λ X Y ⟨h⟩, ext _ _ $ fin.equiv_iff_eq.mp $
nonempty.intro $
{ to_fun := λ x, (h.hom ⟨x⟩).down,
inv_fun := λ x, (h.inv ⟨x⟩).down,
left_inv := begin
intro a,
change ulift.down _ = _,
rw ulift.up_down,
change ((h.hom ≫ h.inv) _).down = _,
simpa,
end,
right_inv := begin
intro a,
change ulift.down _ = _,
rw ulift.up_down,
change ((h.inv ≫ h.hom) _).down = _,
simpa,
end }
/-- The canonical fully faithful embedding of `Fintype.skeleton` into `Fintype`. -/
def incl : skeleton.{u} ⥤ Fintype.{u} :=
{ obj := λ X, Fintype.of (ulift (fin X.len)),
map := λ _ _ f, f }
instance : full incl := { preimage := λ _ _ f, f }
instance : faithful incl := {}
instance : ess_surj incl :=
ess_surj.mk $ λ X, let F := fintype.equiv_fin X in ⟨mk (fintype.card X), nonempty.intro
{ hom := F.symm ∘ ulift.down,
inv := ulift.up ∘ F }⟩
noncomputable instance : is_equivalence incl :=
equivalence.of_fully_faithfully_ess_surj _
/-- The equivalence between `Fintype.skeleton` and `Fintype`. -/
noncomputable def equivalence : skeleton ≌ Fintype := incl.as_equivalence
@[simp] lemma incl_mk_nat_card (n : ℕ) : fintype.card (incl.obj (mk n)) = n :=
begin
convert finset.card_fin n,
apply fintype.of_equiv_card,
end
end skeleton
/-- `Fintype.skeleton` is a skeleton of `Fintype`. -/
noncomputable def is_skeleton : is_skeleton_of Fintype skeleton skeleton.incl :=
{ skel := skeleton.is_skeletal,
eqv := by apply_instance }
end Fintype
|
e173cd25dd8c405936c37bc72eeede57c5948e93 | 2c096fdfecf64e46ea7bc6ce5521f142b5926864 | /tests/lean/derivingRpcEncoding.lean | de5eb9233ff42009359beb9f9aa10636fa581087 | [
"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 | Kha/lean4 | 1005785d2c8797ae266a303968848e5f6ce2fe87 | b99e11346948023cd6c29d248cd8f3e3fb3474cf | refs/heads/master | 1,693,355,498,027 | 1,669,080,461,000 | 1,669,113,138,000 | 184,748,176 | 0 | 0 | Apache-2.0 | 1,665,995,520,000 | 1,556,884,930,000 | Lean | UTF-8 | Lean | false | false | 2,486 | lean | import Lean.Server.Rpc.Basic
open Lean Server
abbrev M := StateM RpcObjectStore
def M.run (x : ExceptT String M α) : Except String α :=
x.run' {}
def test (α : Type) [RpcEncodable α] (a : α) := M.run do
let json ← rpcEncode a
let _a : α ← ofExcept (rpcDecode json (← get))
return json
structure FooRef where
a : Array Nat
deriving Inhabited, TypeName
#check instTypeNameFooRef
#eval test (WithRpcRef FooRef) default
structure FooJson where
s : String
deriving FromJson, ToJson, Inhabited
structure Bar where
fooRef : WithRpcRef FooRef
fooJson : FooJson
deriving RpcEncodable, Inhabited
#check instRpcEncodableBar
#eval test Bar default
structure BarTrans where
bar : Bar
deriving RpcEncodable, Inhabited
#check instRpcEncodableBarTrans
#eval test BarTrans default
structure Baz where
arr : Array String -- non-constant field
deriving RpcEncodable, Inhabited
#check instRpcEncodableBaz
#eval test Baz default
structure FooGeneric (α : Type) where
a : α
b? : Option α
deriving RpcEncodable, Inhabited
#check @instRpcEncodableFooGeneric
#eval test (FooGeneric Nat) default
#eval test (FooGeneric Nat) { a := 3, b? := some 42 }
inductive BazInductive
| baz (arr : Array Bar)
deriving RpcEncodable, Inhabited
#check @instRpcEncodableBazInductive
#eval test BazInductive ⟨#[default, default]⟩
inductive FooInductive (α : Type) where
| a : α → WithRpcRef FooRef → FooInductive α
| b : (n : Nat) → (a : α) → (m : Nat) → FooInductive α
deriving RpcEncodable, Inhabited
#check @instRpcEncodableFooInductive
#eval test (FooInductive BazInductive) (.a default default)
#eval test (FooInductive BazInductive) (.b 42 default default)
inductive FooNested (α : Type) where
| a : α → Array (FooNested α) → FooNested α
deriving RpcEncodable, Inhabited
#eval test (FooNested BazInductive) (.a default #[default])
inductive FooParam (n : Nat) where
| a : Nat → FooParam n
deriving RpcEncodable, Inhabited
#check @instRpcEncodableFooParam
#eval test (FooParam 10) (.a 42)
inductive Unused (α : Type) | a
deriving RpcEncodable, Inhabited
#check @instRpcEncodableUnused
structure NoRpcEncodable
#eval test (Unused NoRpcEncodable) default
structure UnusedStruct (α : Type)
deriving RpcEncodable, Inhabited
#check @instRpcEncodableUnusedStruct
#eval test (UnusedStruct NoRpcEncodable) default
deriving instance Repr, RpcEncodable for Empty
#eval rpcDecode (α := Empty) .null {}
|
65d8d74e7d27e606f6f20080a12f1d2ff74c7af7 | 02fbe05a45fda5abde7583464416db4366eedfbf | /library/init/data/nat/bitwise.lean | e45188a80df9896c0076914a88d0d084939a43e3 | [
"Apache-2.0"
] | permissive | jasonrute/lean | cc12807e11f9ac6b01b8951a8bfb9c2eb35a0154 | 4be962c167ca442a0ec5e84472d7ff9f5302788f | refs/heads/master | 1,672,036,664,637 | 1,601,642,826,000 | 1,601,642,826,000 | 260,777,966 | 0 | 0 | Apache-2.0 | 1,588,454,819,000 | 1,588,454,818,000 | null | UTF-8 | Lean | false | false | 10,112 | lean | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Mario Carneiro
-/
prelude
import init.data.nat.lemmas init.meta.well_founded_tactics
universe u
namespace nat
def bodd_div2 : ℕ → bool × ℕ
| 0 := (ff, 0)
| (succ n) :=
match bodd_div2 n with
| (ff, m) := (tt, m)
| (tt, m) := (ff, succ m)
end
def div2 (n : ℕ) : ℕ := (bodd_div2 n).2
def bodd (n : ℕ) : bool := (bodd_div2 n).1
@[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_succ (n : ℕ) : bodd (succ n) = bnot (bodd n) :=
by unfold bodd bodd_div2; cases bodd_div2 n; cases fst; refl
@[simp] lemma bodd_add (m n : ℕ) : bodd (m + n) = bxor (bodd m) (bodd n) :=
begin
induction n with n IH,
{ simp, cases bodd m; refl },
{ simp [IH], cases bodd m; cases bodd n; refl }
end
@[simp] lemma bodd_mul (m n : ℕ) : bodd (m * n) = bodd m && bodd n :=
begin
induction n with n IH,
{ simp, cases bodd m; refl },
{ simp [mul_succ, IH], cases bodd m; cases bodd n; refl }
end
lemma mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 :=
begin
have := congr_arg bodd (mod_add_div n 2),
simp [bnot] at this,
rw [show ∀ b, ff && b = ff, by intros; cases b; refl,
show ∀ b, bxor b ff = b, by intros; cases b; refl] at this,
rw [← this],
cases mod_two_eq_zero_or_one n with h h; rw h; refl
end
@[simp] lemma div2_zero : div2 0 = 0 := rfl
@[simp] lemma div2_one : div2 1 = 0 := rfl
@[simp] lemma div2_two : div2 2 = 1 := rfl
@[simp] lemma div2_succ (n : ℕ) : div2 (succ n) = cond (bodd n) (succ (div2 n)) (div2 n) :=
by unfold bodd div2 bodd_div2; cases bodd_div2 n; cases fst; refl
local attribute [simp] nat.add_comm nat.add_assoc nat.add_left_comm nat.mul_comm nat.mul_assoc
theorem bodd_add_div2 : ∀ n, cond (bodd n) 1 0 + 2 * div2 n = n
| 0 := rfl
| (succ n) := begin
simp,
refine eq.trans _ (congr_arg succ (bodd_add_div2 n)),
cases bodd n; simp [cond, bnot],
{ rw [nat.add_comm, nat.zero_add], },
{ rw [succ_mul, nat.add_comm 1, nat.zero_add] }
end
theorem div2_val (n) : div2 n = n / 2 :=
begin
refine eq_of_mul_eq_mul_left dec_trivial
(nat.add_left_cancel (eq.trans _ (nat.mod_add_div n 2).symm)),
rw [mod_two_of_bodd, bodd_add_div2]
end
def bit (b : bool) : ℕ → ℕ := cond b bit1 bit0
lemma bit0_val (n : nat) : bit0 n = 2 * n :=
calc n + n = 0 + n + n : by rw nat.zero_add
... = n * 2 : rfl
... = 2 * n : nat.mul_comm _ _
lemma bit1_val (n : nat) : bit1 n = 2 * n + 1 := congr_arg succ (bit0_val _)
lemma bit_val (b n) : bit b n = 2 * n + cond b 1 0 :=
by { cases b, apply bit0_val, apply bit1_val }
lemma bit_decomp (n : nat) : bit (bodd n) (div2 n) = n :=
(bit_val _ _).trans $ (nat.add_comm _ _).trans $ bodd_add_div2 _
def bit_cases_on {C : nat → 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
def shiftl' (b : bool) (m : ℕ) : ℕ → ℕ
| 0 := m
| (n+1) := bit b (shiftl' n)
def shiftl : ℕ → ℕ → ℕ := shiftl' ff
@[simp] theorem shiftl_zero (m) : shiftl m 0 = m := rfl
@[simp] theorem shiftl_succ (m n) : shiftl m (n + 1) = bit0 (shiftl m n) := rfl
def shiftr : ℕ → ℕ → ℕ
| m 0 := m
| m (n+1) := div2 (shiftr m n)
def test_bit (m n : ℕ) : bool := bodd (shiftr m n)
def binary_rec {C : nat → Sort u} (z : C 0) (f : ∀ b n, C n → C (bit b n)) : Π n, C n
| n := if n0 : n = 0 then by rw n0; exact z else let n' := div2 n in
have n' < n, begin
change div2 n < n, rw div2_val,
apply (div_lt_iff_lt_mul _ _ (succ_pos 1)).2,
have := nat.mul_lt_mul_of_pos_left (lt_succ_self 1)
(lt_of_le_of_ne (zero_le _) (ne.symm n0)),
rwa nat.mul_one at this
end,
by rw [← show bit (bodd n) n' = n, from bit_decomp n]; exact
f (bodd n) n' (binary_rec n')
def size : ℕ → ℕ := binary_rec 0 (λ_ _, succ)
def bits : ℕ → list bool := binary_rec [] (λb _ IH, b :: IH)
def bitwise (f : bool → bool → bool) : ℕ → ℕ → ℕ :=
binary_rec
(λn, cond (f ff tt) n 0)
(λa m Ia, binary_rec
(cond (f tt ff) (bit a m) 0)
(λb n _, bit (f a b) (Ia n)))
def lor : ℕ → ℕ → ℕ := bitwise bor
def land : ℕ → ℕ → ℕ := bitwise band
def ldiff : ℕ → ℕ → ℕ := bitwise (λ a b, a && bnot b)
def lxor : ℕ → ℕ → ℕ := bitwise bxor
@[simp] lemma binary_rec_zero {C : nat → Sort u} (z : C 0) (f : ∀ b n, C n → C (bit b n)) :
binary_rec z f 0 = z :=
by {rw [binary_rec], refl}
/- bitwise ops -/
lemma bodd_bit (b n) : bodd (bit b n) = b :=
by rw bit_val; simp; cases b; cases bodd n; refl
lemma div2_bit (b n) : div2 (bit b n) = n :=
by rw [bit_val, div2_val, nat.add_comm, add_mul_div_left, div_eq_of_lt, nat.zero_add];
cases b; exact dec_trivial
lemma shiftl'_add (b m n) : ∀ k, shiftl' b m (n + k) = shiftl' b (shiftl' b m n) k
| 0 := rfl
| (k+1) := congr_arg (bit b) (shiftl'_add k)
lemma shiftl_add : ∀ m n k, shiftl m (n + k) = shiftl (shiftl m n) k := shiftl'_add _
lemma shiftr_add (m n) : ∀ k, shiftr m (n + k) = shiftr (shiftr m n) k
| 0 := rfl
| (k+1) := congr_arg div2 (shiftr_add k)
lemma shiftl'_sub (b m) : ∀ {n k}, k ≤ n → shiftl' b m (n - k) = shiftr (shiftl' b m n) k
| n 0 h := rfl
| (n+1) (k+1) h := begin
simp [shiftl'], rw [nat.add_comm, shiftr_add],
simp [shiftr, div2_bit],
apply shiftl'_sub (nat.le_of_succ_le_succ h)
end
lemma shiftl_sub : ∀ m {n k}, k ≤ n → shiftl m (n - k) = shiftr (shiftl m n) k := shiftl'_sub _
@[simp] lemma test_bit_zero (b n) : test_bit (bit b n) 0 = b := bodd_bit _ _
lemma test_bit_succ (m b n) : test_bit (bit b n) (succ m) = test_bit n m :=
have bodd (shiftr (shiftr (bit b n) 1) m) = bodd (shiftr n m),
by dsimp [shiftr]; rw div2_bit,
by rw [← shiftr_add, nat.add_comm] at this; exact this
lemma binary_rec_eq {C : nat → Sort u} {z : C 0} {f : ∀ b n, C n → C (bit b n)}
(h : f ff 0 z = z) (b n) :
binary_rec z f (bit b n) = f b n (binary_rec z f n) :=
begin
rw [binary_rec],
with_cases { by_cases bit b n = 0 },
case pos : h' {
simp [dif_pos h'],
generalize : binary_rec._main._pack._proof_1 (bit b n) h' = e,
revert e,
have bf := bodd_bit b n,
have n0 := div2_bit b n,
rw h' at bf n0,
simp at bf n0,
rw [← bf, ← n0, binary_rec_zero],
intros, exact h.symm },
case neg : h' {
simp [dif_neg h'],
generalize : binary_rec._main._pack._proof_2 (bit b n) = e,
revert e,
rw [bodd_bit, div2_bit],
intros, refl}
end
lemma bitwise_bit_aux {f : bool → bool → bool} (h : f ff ff = ff) :
@binary_rec (λ_, ℕ)
(cond (f tt ff) (bit ff 0) 0)
(λ b n _, bit (f ff b) (cond (f ff tt) n 0)) =
λ (n : ℕ), cond (f ff tt) n 0 :=
begin
funext n,
apply bit_cases_on n, intros b n, rw [binary_rec_eq],
{ cases b; try {rw h}; induction fft : f ff tt; simp [cond]; refl },
{ rw [h, show cond (f ff tt) 0 0 = 0, by cases f ff tt; refl,
show cond (f tt ff) (bit ff 0) 0 = 0, by cases f tt ff; refl]; refl }
end
@[simp] lemma bitwise_zero_left (f : bool → bool → bool) (n) :
bitwise f 0 n = cond (f ff tt) n 0 :=
by unfold bitwise; rw [binary_rec_zero]
@[simp] lemma bitwise_zero_right (f : bool → bool → bool) (h : f ff ff = ff) (m) :
bitwise f m 0 = cond (f tt ff) m 0 :=
by unfold bitwise; apply bit_cases_on m; intros;
rw [binary_rec_eq, binary_rec_zero]; exact bitwise_bit_aux h
@[simp] lemma bitwise_zero (f : bool → bool → bool) :
bitwise f 0 0 = 0 :=
by rw bitwise_zero_left; cases f ff tt; refl
@[simp] lemma bitwise_bit {f : bool → bool → bool} (h : f ff ff = ff) (a m b n) :
bitwise f (bit a m) (bit b n) = bit (f a b) (bitwise f m n) :=
begin
unfold bitwise,
rw [binary_rec_eq, binary_rec_eq],
{ induction ftf : f tt ff; dsimp [cond],
rw [show f a ff = ff, by cases a; assumption],
apply @congr_arg _ _ _ 0 (bit ff), tactic.swap,
rw [show f a ff = a, by cases a; assumption],
apply congr_arg (bit a),
all_goals {
apply bit_cases_on m, intros a m,
rw [binary_rec_eq, binary_rec_zero],
rw [← bitwise_bit_aux h, ftf], refl } },
{ exact bitwise_bit_aux h }
end
theorem bitwise_swap {f : bool → bool → bool} (h : f ff ff = ff) :
bitwise (function.swap f) = function.swap (bitwise f) :=
begin
funext m n, revert n,
dsimp [function.swap],
apply binary_rec _ (λ a m' IH, _) m; intro n,
{ rw [bitwise_zero_left, bitwise_zero_right], exact h },
apply bit_cases_on n; intros b n',
rw [bitwise_bit, bitwise_bit, IH]; exact h
end
@[simp] lemma lor_bit : ∀ (a m b n),
lor (bit a m) (bit b n) = bit (a || b) (lor m n) := bitwise_bit rfl
@[simp] lemma land_bit : ∀ (a m b n),
land (bit a m) (bit b n) = bit (a && b) (land m n) := bitwise_bit rfl
@[simp] lemma ldiff_bit : ∀ (a m b n),
ldiff (bit a m) (bit b n) = bit (a && bnot b) (ldiff m n) := bitwise_bit rfl
@[simp] lemma lxor_bit : ∀ (a m b n),
lxor (bit a m) (bit b n) = bit (bxor a b) (lxor m n) := bitwise_bit rfl
@[simp] lemma test_bit_bitwise {f : bool → bool → bool} (h : f ff ff = ff) (m n k) :
test_bit (bitwise f m n) k = f (test_bit m k) (test_bit n k) :=
begin
revert m n; induction k with k IH; intros m n;
apply bit_cases_on m; intros a m';
apply bit_cases_on n; intros b n';
rw bitwise_bit h,
{ 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 := test_bit_bitwise rfl
@[simp] lemma test_bit_land : ∀ (m n k),
test_bit (land m n) k = test_bit m k && test_bit n k := test_bit_bitwise rfl
@[simp] lemma test_bit_ldiff : ∀ (m n k),
test_bit (ldiff m n) k = test_bit m k && bnot (test_bit n k) := test_bit_bitwise rfl
@[simp] lemma test_bit_lxor : ∀ (m n k),
test_bit (lxor m n) k = bxor (test_bit m k) (test_bit n k) := test_bit_bitwise rfl
end nat
|
a55878a5cb77e3880e6784d799e56c3433e18210 | 08a8ee10652ba4f8592710ceb654b37e951d9082 | /src/hott/init/trunc.lean | db4f04ec0f164e03de60527f45f96f9d0885fc9c | [
"Apache-2.0"
] | permissive | felixwellen/hott3 | e9f299c84d30a782a741c40d38741ec024d391fb | 8ac87a2699ab94c23ea7984b4a5fbd5a7052575c | refs/heads/master | 1,619,972,899,098 | 1,509,047,351,000 | 1,518,040,986,000 | 120,676,559 | 0 | 0 | null | 1,518,040,503,000 | 1,518,040,503,000 | null | UTF-8 | Lean | false | false | 13,263 | lean | /-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Floris van Doorn
Definition of is_trunc (n-truncatedness)
Ported from Coq HoTT.
-/
import .equiv .pathover .logic
universes u v l
hott_theory
namespace hott
open hott.eq nat sigma unit
/- Truncation levels -/
inductive trunc_index : Type
| minus_two : trunc_index
| succ : trunc_index → trunc_index
open trunc_index
/-
notation for trunc_index is -2, -1, 0, 1, ...
from 0 and up this comes from the way numerals are parsed in Lean.
Any structure with a 0, a 1, and a + have numerals defined in them.
-/
notation `ℕ₋₂` := trunc_index -- input using \N-2
instance has_zero_trunc_index : has_zero ℕ₋₂ :=
has_zero.mk (succ (succ minus_two))
instance has_one_trunc_index : has_one ℕ₋₂ :=
has_one.mk (succ (succ (succ minus_two)))
namespace trunc_index
notation `-1` := trunc_index.succ trunc_index.minus_two -- ISSUE: -1 gets printed as -2.+1?
notation `-2` := trunc_index.minus_two
postfix `.+1`:(max+1) := trunc_index.succ
postfix `.+2`:(max+1) := λn, (n .+1 .+1)
--addition, where we add two to the result
@[hott] def add_plus_two (n m : ℕ₋₂) : ℕ₋₂ :=
trunc_index.rec_on m n (λ k l, l .+1)
infix ` +2+ `:65 := trunc_index.add_plus_two
-- addition of trunc_indices, where results smaller than -2 are changed to -2
@[hott] protected def add (n m : ℕ₋₂) : ℕ₋₂ :=
trunc_index.cases_on m
(trunc_index.cases_on n -2 (λn', (trunc_index.cases_on n' -2 id)))
(λm', trunc_index.cases_on m'
(trunc_index.cases_on n -2 id)
(add_plus_two n))
/- we give a weird name to the reflexivity step to avoid overloading le.refl
(which can be used if types.trunc is imported) -/
inductive le (a : ℕ₋₂) : ℕ₋₂ → Type
| tr_refl : le a
| step : Π {b}, le b → le (b.+1)
attribute [refl] le.tr_refl
end trunc_index
local infix ` ≤ ` := trunc_index.le -- TODO(gabriel)
instance has_add_trunc_index : has_add ℕ₋₂ :=
has_add.mk trunc_index.add
namespace trunc_index
@[hott] def sub_two (n : ℕ) : ℕ₋₂ :=
nat.rec_on n -2 (λ n k, k.+1)
@[hott] def add_two (n : ℕ₋₂) : ℕ :=
trunc_index.rec_on n nat.zero (λ n k, nat.succ k)
postfix `.-2`:(max+1) := sub_two
postfix `.-1`:(max+1) := λn, (n .-2 .+1)
@[hott] def of_nat (n : ℕ) : ℕ₋₂ :=
n.-2.+2
instance: has_coe ℕ ℕ₋₂ := ⟨of_nat⟩
@[hott] def succ_le_succ {n m : ℕ₋₂} (H : n ≤ m) : n.+1 ≤ m.+1 :=
begin induction H, refl, apply le.step; assumption end
@[hott] def minus_two_le (n : ℕ₋₂) : -2 ≤ n :=
begin induction n, refl, apply le.step; assumption end
end trunc_index open trunc_index
namespace is_trunc
-- export [notation] [coercion] trunc_index
/- truncated types -/
/-
Just as in Coq HoTT we define an internal version of contractibility and is_trunc, but we only
use `is_trunc` and `is_contr`
-/
structure contr_internal (A : Type u) :=
(center : A)
(center_eq : Π(a : A), center = a)
@[hott] def is_trunc_internal (n : ℕ₋₂) : Type u → Type u :=
trunc_index.rec_on n
(λA, contr_internal A)
(λn trunc_n A, (Π(x y : A), trunc_n (x = y)))
end is_trunc open is_trunc
class is_trunc (n : ℕ₋₂) (A : Type u) :=
(to_internal : is_trunc_internal n A)
open nat trunc_index
namespace is_trunc
notation `is_contr` := is_trunc -2
notation `is_prop` := is_trunc -1
notation `is_set` := is_trunc 0
variables {A : Type u} {B : Type v}
@[hott] def is_trunc_succ_intro {A : Type _} {n : ℕ₋₂} (H : ∀x y : A, is_trunc n (x = y))
: is_trunc n.+1 A :=
is_trunc.mk (λ x y, is_trunc.to_internal _ _)
@[hott, instance] def is_trunc_eq
(n : ℕ₋₂) [H : is_trunc (n.+1) A] (x y : A) : is_trunc n (x = y) :=
is_trunc.mk (is_trunc.to_internal (n.+1) A x y)
/- contractibility -/
@[hott] def is_contr.mk (center : A) (center_eq : Π(a : A), center = a) : is_contr A :=
is_trunc.mk (contr_internal.mk center center_eq)
@[hott] def center (A : Type _) [H : is_contr A] : A :=
contr_internal.center (is_trunc.to_internal -2 A)
@[hott] def center' {A : Type _} (H : is_contr A) : A := center A
@[hott] def center_eq [H : is_contr A] (a : A) : center _ = a :=
contr_internal.center_eq (is_trunc.to_internal -2 A) a
@[hott] def eq_of_is_contr [H : is_contr A] (x y : A) : x = y :=
(center_eq x)⁻¹ ⬝ (center_eq y)
@[hott] def prop_eq_of_is_contr {A : Type _} [H : is_contr A] {x y : A} (p q : x = y) : p = q :=
have K : ∀ (r : x = y), eq_of_is_contr x y = r, from (λ r, by induction r; apply con.left_inv),
(K p)⁻¹ ⬝ K q
@[hott] def is_contr_eq {A : Type _} [H : is_contr A] (x y : A) : is_contr (x = y) :=
is_contr.mk (eq_of_is_contr _ _) (λ p, prop_eq_of_is_contr _ _)
/- truncation is upward close -/
-- n-types are also (n+1)-types
@[hott, instance] def is_trunc_succ (A : Type _) (n : ℕ₋₂)
[H : is_trunc n A] : is_trunc (n.+1) A :=
begin
induction n with n IH generalizing A; apply is_trunc_succ_intro; intros x y,
{ exact @is_contr_eq A H x y },
{ refine @IH _ (@is_trunc_eq _ _ H _ _) , }
end
--in the proof the type of H is given explicitly to make it available for class inference
@[hott] def is_trunc_of_le (A : Type.{l}) {n m : ℕ₋₂} (Hnm : n ≤ m)
[Hn : is_trunc n A] : is_trunc m A :=
begin
induction Hnm with m Hnm IH,
{ exact Hn },
{ exact @is_trunc_succ _ _ IH }
end
@[hott] def is_trunc_of_imp_is_trunc {n : ℕ₋₂} (H : A → is_trunc (n.+1) A)
: is_trunc (n.+1) A :=
@is_trunc_succ_intro _ _ (λx y, @is_trunc_eq _ _ (H x) x y)
@[hott] def is_trunc_of_imp_is_trunc_of_le {n : ℕ₋₂} (Hn : -1 ≤ n) (H : A → is_trunc n A)
: is_trunc n A :=
begin
induction Hn with n' Hn'; apply is_trunc_of_imp_is_trunc H
end
-- these must be definitions, because we need them to compute sometimes
@[hott] def is_trunc_of_is_contr (A : Type _) (n : ℕ₋₂) [H : is_contr A] : is_trunc n A :=
trunc_index.rec_on n H (λn H, by apply_instance)
@[hott] def is_trunc_succ_of_is_prop (A : Type _) (n : ℕ₋₂) [H : is_prop A]
: is_trunc (n.+1) A :=
is_trunc_of_le A (show -1 ≤ n.+1, from succ_le_succ (minus_two_le n))
@[hott] def is_trunc_succ_succ_of_is_set (A : Type _) (n : ℕ₋₂) [H : is_set A]
: is_trunc (n.+2) A :=
is_trunc_of_le A (show 0 ≤ n.+2, from succ_le_succ (succ_le_succ (minus_two_le n)))
/- props -/
@[hott] def is_prop.elim [H : is_prop A] (x y : A) : x = y :=
by apply center
@[hott] def is_prop.elim' (x y : A) (H : is_prop A) : x = y :=
is_prop.elim x y
@[hott] def is_contr_of_inhabited_prop {A : Type _} [H : is_prop A] (x : A) : is_contr A :=
is_contr.mk x (λy, by apply is_prop.elim)
@[hott] def is_prop_of_imp_is_contr {A : Type _} (H : A → is_contr A) : is_prop A :=
@is_trunc_succ_intro A -2
(λx y,
have H2 : is_contr A, from H x,
by apply is_contr_eq)
@[hott] def is_prop.mk {A : Type _} (H : ∀x y : A, x = y) : is_prop A :=
is_prop_of_imp_is_contr (λ x, is_contr.mk x (H x))
@[hott] def is_prop_elim_self {A : Type _} {H : is_prop A} (x : A) : is_prop.elim x x = idp :=
by apply is_prop.elim
/- sets -/
@[hott] def is_set.mk (A : Type _) (H : ∀(x y : A) (p q : x = y), p = q) : is_set A :=
@is_trunc_succ_intro _ _ (λ x y, is_prop.mk (H x y))
@[hott] def is_set.elim [H : is_set A] ⦃x y : A⦄ (p q : x = y) : p = q :=
by apply is_prop.elim
@[hott] def is_set.elim' ⦃x y : A⦄ (p q : x = y) (H : is_set A) : p = q :=
is_set.elim p q
/- instances -/
@[instance, hott] def is_contr_sigma_eq {A : Type u} (a : A)
: is_contr (Σ(x : A), a = x) :=
is_contr.mk (sigma.mk a idp) (λ ⟨b,q⟩, by induction q; refl)
@[instance, hott] def is_contr_sigma_eq' {A : Type u} (a : A)
: is_contr (Σ(x : A), x = a) :=
is_contr.mk (sigma.mk a idp) (λ ⟨b,q⟩, by clear _fun_match; induction q; refl)
@[hott] def ap_fst_center_eq_sigma_eq {A : Type _} {a x : A} (p : a = x)
: ap sigma.fst (center_eq ⟨x, p⟩) = p :=
by induction p; reflexivity
@[hott] def ap_fst_center_eq_sigma_eq' {A : Type _} {a x : A} (p : x = a)
: ap sigma.fst (center_eq ⟨x, p⟩) = p⁻¹ :=
by induction p; reflexivity
@[hott] def is_contr_unit : is_contr unit :=
is_contr.mk star (λp, unit.rec_on p idp)
@[hott] def is_prop_empty : is_prop empty :=
is_prop.mk (λx, by induction x)
local attribute [instance] is_contr_unit is_prop_empty
@[hott,instance] def is_trunc_unit (n : ℕ₋₂) : is_trunc n unit :=
by apply is_trunc_of_is_contr
@[hott,instance] def is_trunc_empty (n : ℕ₋₂) : is_trunc (n.+1) empty :=
by apply is_trunc_succ_of_is_prop
/- interaction with equivalences -/
section
open hott.is_equiv hott.equiv
@[hott] def is_contr_is_equiv_closed (f : A → B) [Hf : is_equiv f] [HA: is_contr A]
: (is_contr B) :=
is_contr.mk (f (center A)) (λp, eq_of_eq_inv (center_eq _))
@[hott] def is_contr_equiv_closed (H : A ≃ B) [HA: is_contr A] : is_contr B :=
is_contr_is_equiv_closed H
@[hott] def equiv_of_is_contr_of_is_contr [HA : is_contr A] [HB : is_contr B] : A ≃ B :=
equiv.mk
(λa, center B)
(is_equiv.adjointify (λa, center B) (λb, center A) center_eq center_eq)
@[hott] def is_trunc_is_equiv_closed (n : ℕ₋₂) (f : A → B) [H : is_equiv f]
(HA : is_trunc n A) : is_trunc n B :=
begin
revert A B f H HA, induction n with n IH; intros,
{ exact is_contr_is_equiv_closed f },
{ apply is_trunc_succ_intro, intros, apply IH (ap f⁻¹ᶠ)⁻¹ᶠ, all_goals {apply_instance} }
end
@[hott] def is_trunc_is_equiv_closed_rev (n : ℕ₋₂) (f : A → B) [H : is_equiv f]
(HA : is_trunc n B) : is_trunc n A :=
is_trunc_is_equiv_closed n f⁻¹ᶠ HA
@[hott] def is_trunc_equiv_closed (n : ℕ₋₂) (f : A ≃ B) (HA : is_trunc n A)
: is_trunc n B :=
is_trunc_is_equiv_closed n f HA
@[hott] def is_trunc_equiv_closed_rev (n : ℕ₋₂) (f : A ≃ B) (HA : is_trunc n B)
: is_trunc n A :=
is_trunc_is_equiv_closed n f⁻¹ᶠ HA
@[hott] def is_equiv_of_is_prop [HA : is_prop A] [HB : is_prop B]
(f : A → B) (g : B → A) : is_equiv f :=
by fapply is_equiv.mk f g; intros; {apply is_prop.elim <|> apply is_set.elim}
@[hott] def is_equiv_of_is_contr [HA : is_contr A] [HB : is_contr B]
(f : A → B) : is_equiv f :=
by fapply is_equiv.mk f; intros; {apply center <|> apply is_prop.elim <|> apply is_set.elim}
@[hott] def equiv_of_is_prop [HA : is_prop A] [HB : is_prop B]
(f : A → B) (g : B → A) : A ≃ B :=
equiv.mk f (is_equiv_of_is_prop f g)
@[hott] def equiv_of_iff_of_is_prop [HA : is_prop A] [HB : is_prop B] (H : A ↔ B) : A ≃ B :=
equiv_of_is_prop (iff.elim_left H) (iff.elim_right H)
/- truncatedness of lift -/
@[hott,instance] def is_trunc_lift (A : Type _) (n : ℕ₋₂)
[H : is_trunc n A] : is_trunc n (ulift A) :=
is_trunc_equiv_closed _ (equiv_lift _) H
end
/- interaction with the Unit type -/
open equiv
/- A contractible type is equivalent to unit. -/
variable (A)
@[hott] def equiv_unit_of_is_contr [H : is_contr A] : A ≃ unit :=
equiv.MK (λ (x : A), ())
(λ (u : unit), center A)
(λ (u : unit), unit.rec_on u idp)
(λ (x : A), center_eq x)
/- interaction with pathovers -/
variable {A}
variables {C : A → Type _}
{a a₂ : A} (p : a = a₂)
(c : C a) (c₂ : C a₂)
instance is_trunc_pathover
(n : ℕ₋₂) [H : is_trunc (n.+1) (C a)] : is_trunc n (c =[p] c₂) :=
begin
refine @is_trunc_equiv_closed_rev _ _ n _ _, tactic.swap,
apply pathover_equiv_eq_tr,
apply is_trunc_eq,
end
@[hott] def is_prop.elimo [H : is_prop (C a)] : c =[p] c₂ :=
pathover_of_eq_tr (by apply is_prop.elim)
@[hott] def is_prop.elimo' (H : is_prop (C a)) : c =[p] c₂ :=
is_prop.elimo p c c₂
@[hott] def is_prop_elimo_self {A : Type _} (B : A → Type _) {a : A} (b : B a) {H : is_prop (B a)} :
@is_prop.elimo A B a a idp b b H = idpo :=
by apply is_prop.elim
variables {p c c₂}
@[hott] def is_set.elimo (q q' : c =[p] c₂) [H : is_set (C a)] : q = q' :=
by apply is_prop.elim
-- TODO: port "Truncated morphisms"
/- truncated universe -/
end is_trunc
structure trunctype (n : ℕ₋₂) :=
(carrier : Type u)
(struct : is_trunc n carrier)
@[hott] instance (n): has_coe_to_sort (trunctype n) := {
S := Type u,
coe := trunctype.carrier
}
notation n `-Type` := trunctype n
hott_theory_cmd "local notation Prop := -1-Type"
notation `Set` := 0-Type
-- attribute trunctype.carrier [coercion]
attribute [instance] trunctype.struct
notation `Prop.mk` := @trunctype.mk -1
notation `Set.mk` := @trunctype.mk (-1.+1)
@[hott] protected def trunctype.mk' (n : ℕ₋₂) (A : Type _) [H : is_trunc n A]
: n-Type :=
trunctype.mk A H
namespace is_trunc
@[hott] def tlift {n : ℕ₋₂} (A : trunctype.{u} n)
: trunctype.{max u v} n :=
trunctype.mk (ulift A) (is_trunc_lift _ _)
end is_trunc
end hott
|
5f9916deca64524d82797e7f64fff32c0ac22632 | 302c785c90d40ad3d6be43d33bc6a558354cc2cf | /src/data/sym2.lean | 5c41a2e062a6de92b83e95aa015f60f29b0b7d83 | [
"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 | 13,744 | lean | /-
Copyright (c) 2020 Kyle Miller All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import tactic.linarith
import data.sym
open function
open sym
/-!
# The symmetric square
This file defines the symmetric square, which is `α × α` modulo
swapping. This is also known as the type of unordered pairs.
More generally, the symmetric square is the second symmetric power
(see `data.sym`). The equivalence is `sym2.equiv_sym`.
From the point of view that an unordered pair is equivalent to a
multiset of cardinality two (see `sym2.equiv_multiset`), there is a
`has_mem` instance `sym2.mem`, which is a `Prop`-valued membership
test. Given `h : a ∈ z` for `z : sym2 α`, then `h.other` is the other
element of the pair, defined using `classical.choice`. If `α` has
decidable equality, then `h.other'` computably gives the other element.
Recall that an undirected graph (allowing self loops, but no multiple
edges) is equivalent to a symmetric relation on the vertex type `α`.
Given a symmetric relation on `α`, the corresponding edge set is
constructed by `sym2.from_rel`.
## Notation
The symmetric square has a setoid instance, so `⟦(a, b)⟧` denotes a
term of the symmetric square.
## Tags
symmetric square, unordered pairs, symmetric powers
-/
universe u
variables {α : Type u}
namespace sym2
/--
This is the relation capturing the notion of pairs equivalent up to permutations.
-/
inductive rel (α : Type u) : (α × α) → (α × α) → Prop
| refl (x y : α) : rel (x, y) (x, y)
| swap (x y : α) : rel (x, y) (y, x)
attribute [refl] rel.refl
@[symm] lemma rel.symm {x y : α × α} : rel α x y → rel α y x :=
by rintro ⟨_, _⟩; constructor
@[trans] lemma rel.trans {x y z : α × α} : rel α x y → rel α y z → rel α x z :=
by { intros a b, cases_matching* rel _ _ _; apply rel.refl <|> apply rel.swap }
lemma rel.is_equivalence : equivalence (rel α) := by tidy; apply rel.trans; assumption
instance rel.setoid (α : Type u) : setoid (α × α) := ⟨rel α, rel.is_equivalence⟩
end sym2
/--
`sym2 α` is the symmetric square of `α`, which, in other words, is the
type of unordered pairs.
It is equivalent in a natural way to multisets of cardinality 2 (see
`sym2.equiv_multiset`).
-/
@[reducible]
def sym2 (α : Type u) := quotient (sym2.rel.setoid α)
namespace sym2
lemma eq_swap {a b : α} : ⟦(a, b)⟧ = ⟦(b, a)⟧ :=
by { rw quotient.eq, apply rel.swap }
lemma congr_right {a b c : α} : ⟦(a, b)⟧ = ⟦(a, c)⟧ ↔ b = c :=
by { split; intro h, { rw quotient.eq at h, cases h; refl }, rw h }
lemma congr_left {a b c : α} : ⟦(b, a)⟧ = ⟦(c, a)⟧ ↔ b = c :=
by { split; intro h, { rw quotient.eq at h, cases h; refl }, rw h }
/--
The functor `sym2` is functorial, and this function constructs the induced maps.
-/
def map {α β : Type*} (f : α → β) : sym2 α → sym2 β :=
quotient.map (prod.map f f)
(by { rintros _ _ h, cases h, { refl }, apply rel.swap })
@[simp]
lemma map_id : sym2.map (@id α) = id := by tidy
lemma map_comp {α β γ : Type*} {g : β → γ} {f : α → β} :
sym2.map (g ∘ f) = sym2.map g ∘ sym2.map f := by tidy
@[simp]
lemma map_pair_eq {α β : Type*} (f : α → β) (x y : α) : map f ⟦(x, y)⟧ = ⟦(f x, f y)⟧ :=
by simp [map]
section membership
/-! ### Declarations about membership -/
/--
This is a predicate that determines whether a given term is a member of a term of the
symmetric square. From this point of view, the symmetric square is the subtype of
cardinality-two multisets on `α`.
-/
def mem (x : α) (z : sym2 α) : Prop :=
∃ (y : α), z = ⟦(x, y)⟧
instance : has_mem α (sym2 α) := ⟨mem⟩
lemma mk_has_mem (x y : α) : x ∈ ⟦(x, y)⟧ := ⟨y, rfl⟩
lemma mk_has_mem_right (x y : α) : y ∈ ⟦(x, y)⟧ := by { rw eq_swap, apply mk_has_mem }
/--
Given an element of the unordered pair, give the other element using `classical.some`.
See also `mem.other'` for the computable version.
-/
noncomputable def mem.other {a : α} {z : sym2 α} (h : a ∈ z) : α :=
classical.some h
@[simp]
lemma mem_other_spec {a : α} {z : sym2 α} (h : a ∈ z) : ⟦(a, h.other)⟧ = z :=
by erw ← classical.some_spec h
lemma eq_iff {x y z w : α} :
⟦(x, y)⟧ = ⟦(z, w)⟧ ↔ (x = z ∧ y = w) ∨ (x = w ∧ y = z) :=
begin
split; intro h,
{ rw quotient.eq at h, cases h; tidy },
{ cases h; rw [h.1, h.2], rw eq_swap }
end
@[simp] lemma mem_iff {a b c : α} : a ∈ ⟦(b, c)⟧ ↔ a = b ∨ a = c :=
{ mp := by { rintro ⟨_, h⟩, rw eq_iff at h, tidy },
mpr := by { rintro ⟨_⟩; subst a, { apply mk_has_mem }, apply mk_has_mem_right } }
lemma mem_other_mem {a : α} {z : sym2 α} (h : a ∈ z) :
h.other ∈ z :=
by { convert mk_has_mem_right a h.other, rw mem_other_spec h }
lemma elems_iff_eq {x y : α} {z : sym2 α} (hne : x ≠ y) :
x ∈ z ∧ y ∈ z ↔ z = ⟦(x, y)⟧ :=
begin
split,
{ refine quotient.rec_on_subsingleton z _,
rintros ⟨z₁, z₂⟩ ⟨hx, hy⟩,
rw eq_iff,
cases mem_iff.mp hx with hx hx; cases mem_iff.mp hy with hy hy; cc },
{ rintro rfl, simp },
end
@[ext]
lemma sym2_ext (z z' : sym2 α) (h : ∀ x, x ∈ z ↔ x ∈ z') : z = z' :=
begin
refine quotient.rec_on_subsingleton z (λ w, _) h,
refine quotient.rec_on_subsingleton z' (λ w', _),
intro h,
cases w with x y, cases w' with x' y',
simp only [mem_iff] at h,
apply eq_iff.mpr,
have hx := h x, have hy := h y, have hx' := h x', have hy' := h y',
simp only [true_iff, true_or, eq_self_iff_true, iff_true, or_true] at hx hy hx' hy',
cases hx; subst x; cases hy; subst y; cases hx'; try { subst x' }; cases hy'; try { subst y' };
cc,
end
instance mem.decidable [decidable_eq α] (x : α) (z : sym2 α) : decidable (x ∈ z) :=
quotient.rec_on_subsingleton z (λ ⟨y₁, y₂⟩, decidable_of_iff' _ mem_iff)
end membership
/--
A type `α` is naturally included in the diagonal of `α × α`, and this function gives the image
of this diagonal in `sym2 α`.
-/
def diag (x : α) : sym2 α := ⟦(x, x)⟧
/--
A predicate for testing whether an element of `sym2 α` is on the diagonal.
-/
def is_diag (z : sym2 α) : Prop := z ∈ set.range (@diag α)
@[simp]
lemma diag_is_diag (a : α) : is_diag (diag a) :=
by use a
@[simp]
lemma is_diag_iff_proj_eq (z : α × α) : is_diag ⟦z⟧ ↔ z.1 = z.2 :=
begin
cases z with a, split,
{ rintro ⟨_, h⟩, dsimp only, erw eq_iff at h, rcases h; cc },
{ rintro ⟨⟩, use a, refl },
end
instance is_diag.decidable_pred (α : Type u) [decidable_eq α] : decidable_pred (@is_diag α) :=
by { refine λ z, quotient.rec_on_subsingleton z (λ a, _), erw is_diag_iff_proj_eq, apply_instance }
lemma mem_other_ne {a : α} {z : sym2 α} (hd : ¬is_diag z) (h : a ∈ z) : h.other ≠ a :=
begin
intro hn, apply hd,
have h' := sym2.mem_other_spec h,
rw hn at h',
rw ←h',
simp,
end
section relations
/-! ### Declarations about symmetric relations -/
variables {r : α → α → Prop}
/--
Symmetric relations define a set on `sym2 α` by taking all those pairs
of elements that are related.
-/
def from_rel (sym : symmetric r) : set (sym2 α) :=
quotient.lift (uncurry r) (by { rintros _ _ ⟨_, _⟩, tidy })
@[simp]
lemma from_rel_proj_prop {sym : symmetric r} {z : α × α} :
⟦z⟧ ∈ from_rel sym ↔ r z.1 z.2 := iff.rfl
@[simp]
lemma from_rel_prop {sym : symmetric r} {a b : α} :
⟦(a, b)⟧ ∈ from_rel sym ↔ r a b := by simp only [from_rel_proj_prop]
lemma from_rel_irreflexive {sym : symmetric r} :
irreflexive r ↔ ∀ {z}, z ∈ from_rel sym → ¬is_diag z :=
{ mp := by { intros h z hr hd, induction z,
erw is_diag_iff_proj_eq at hd, erw from_rel_proj_prop at hr, tidy },
mpr := by { intros h x hr, rw ← @from_rel_prop _ _ sym at hr, exact h hr ⟨x, rfl⟩ }}
lemma mem_from_rel_irrefl_other_ne {sym : symmetric r} (irrefl : irreflexive r)
{a : α} {z : sym2 α} (hz : z ∈ from_rel sym) (h : a ∈ z) : h.other ≠ a :=
mem_other_ne (from_rel_irreflexive.mp irrefl hz) h
instance from_rel.decidable_pred (sym : symmetric r) [h : decidable_rel r] :
decidable_pred (sym2.from_rel sym) :=
λ z, quotient.rec_on_subsingleton z (λ x, h _ _)
end relations
section sym_equiv
/-! ### Equivalence to the second symmetric power -/
local attribute [instance] vector.perm.is_setoid
private def from_vector {α : Type*} : vector α 2 → α × α
| ⟨[a, b], h⟩ := (a, b)
private lemma perm_card_two_iff {α : Type*} {a₁ b₁ a₂ b₂ : α} :
[a₁, b₁].perm [a₂, b₂] ↔ (a₁ = a₂ ∧ b₁ = b₂) ∨ (a₁ = b₂ ∧ b₁ = a₂) :=
{ mp := by { simp [← multiset.coe_eq_coe, ← multiset.cons_coe, multiset.cons_eq_cons]; tidy },
mpr := by { intro h, cases h; rw [h.1, h.2], apply list.perm.swap', refl } }
/--
The symmetric square is equivalent to length-2 vectors up to permutations.
-/
def sym2_equiv_sym' {α : Type*} : equiv (sym2 α) (sym' α 2) :=
{ to_fun := quotient.map
(λ (x : α × α), ⟨[x.1, x.2], rfl⟩)
(by { rintros _ _ ⟨_⟩, { refl }, apply list.perm.swap', refl }),
inv_fun := quotient.map from_vector (begin
rintros ⟨x, hx⟩ ⟨y, hy⟩ h,
cases x with _ x, { simpa using hx, },
cases x with _ x, { simpa using hx, },
cases x with _ x, swap, { exfalso, simp at hx, linarith [hx] },
cases y with _ y, { simpa using hy, },
cases y with _ y, { simpa using hy, },
cases y with _ y, swap, { exfalso, simp at hy, linarith [hy] },
rcases perm_card_two_iff.mp h with ⟨rfl,rfl⟩|⟨rfl,rfl⟩, { refl },
apply sym2.rel.swap,
end),
left_inv := by tidy,
right_inv := λ x, begin
refine quotient.rec_on_subsingleton x (λ x, _),
{ cases x with x hx,
cases x with _ x, { simpa using hx, },
cases x with _ x, { simpa using hx, },
cases x with _ x, swap, { exfalso, simp at hx, linarith [hx] },
refl },
end }
/--
The symmetric square is equivalent to the second symmetric power.
-/
def equiv_sym (α : Type*) : sym2 α ≃ sym α 2 :=
equiv.trans sym2_equiv_sym' sym_equiv_sym'.symm
/--
The symmetric square is equivalent to multisets of cardinality
two. (This is currently a synonym for `equiv_sym`, but it's provided
in case the definition for `sym` changes.)
-/
def equiv_multiset (α : Type*) : sym2 α ≃ {s : multiset α // s.card = 2} :=
equiv_sym α
end sym_equiv
section decidable
/--
An algorithm for computing `sym2.rel`.
-/
def rel_bool [decidable_eq α] (x y : α × α) : bool :=
if x.1 = y.1 then x.2 = y.2 else
if x.1 = y.2 then x.2 = y.1 else ff
lemma rel_bool_spec [decidable_eq α] (x y : α × α) :
↥(rel_bool x y) ↔ rel α x y :=
begin
cases x with x₁ x₂, cases y with y₁ y₂,
dsimp [rel_bool], split_ifs;
simp only [false_iff, bool.coe_sort_ff, bool.of_to_bool_iff],
rotate 2, { contrapose! h, cases h; cc },
all_goals { subst x₁, split; intro h1,
{ subst h1; apply sym2.rel.swap },
{ cases h1; cc } }
end
/--
Given `[decidable_eq α]` and `[fintype α]`, the following instance gives `fintype (sym2 α)`.
-/
instance (α : Type*) [decidable_eq α] : decidable_rel (sym2.rel α) :=
λ x y, decidable_of_bool (rel_bool x y) (rel_bool_spec x y)
/--
A function that gives the other element of a pair given one of the elements. Used in `mem.other'`.
-/
private def pair_other [decidable_eq α] (a : α) (z : α × α) : α := if a = z.1 then z.2 else z.1
/--
Get the other element of the unordered pair using the decidable equality.
This is the computable version of `mem.other`.
-/
def mem.other' [decidable_eq α] {a : α} {z : sym2 α} (h : a ∈ z) : α :=
quot.rec (λ x h', pair_other a x) (begin
clear h z,
intros x y h,
ext hy,
convert_to pair_other a x = _,
{ have h' : ∀ {c e h}, @eq.rec _ ⟦x⟧ (λ s, a ∈ s → α)
(λ _, pair_other a x) c e h = pair_other a x,
{ intros _ e _, subst e },
apply h', },
have h' := (rel_bool_spec x y).mpr h,
cases x with x₁ x₂, cases y with y₁ y₂,
cases mem_iff.mp hy with hy'; subst a; dsimp [rel_bool] at h';
split_ifs at h'; try { rw bool.of_to_bool_iff at h', subst x₁, subst x₂ }; dsimp [pair_other],
simp only [ne.symm h_1, if_true, eq_self_iff_true, if_false],
exfalso, exact bool.not_ff h',
simp only [h_1, if_true, eq_self_iff_true, if_false],
exfalso, exact bool.not_ff h',
end) z h
@[simp]
lemma mem_other_spec' [decidable_eq α] {a : α} {z : sym2 α} (h : a ∈ z) :
⟦(a, h.other')⟧ = z :=
begin
induction z, cases z with x y,
have h' := mem_iff.mp h,
dsimp [mem.other', quot.rec, pair_other],
cases h'; subst a,
{ simp only [if_true, eq_self_iff_true], refl, },
{ split_ifs, subst h_1, refl, rw eq_swap, refl, },
refl,
end
@[simp]
lemma other_eq_other' [decidable_eq α] {a : α} {z : sym2 α} (h : a ∈ z) : h.other = h.other' :=
by rw [←congr_right, mem_other_spec' h, mem_other_spec]
lemma mem_other_mem' [decidable_eq α] {a : α} {z : sym2 α} (h : a ∈ z) :
h.other' ∈ z :=
by { rw ←other_eq_other', exact mem_other_mem h }
lemma other_invol' [decidable_eq α] {a : α} {z : sym2 α} (ha : a ∈ z) (hb : ha.other' ∈ z):
hb.other' = a :=
begin
induction z, cases z with x y,
dsimp [mem.other', quot.rec, pair_other] at hb,
split_ifs at hb; dsimp [mem.other', quot.rec, pair_other],
simp only [h, if_true, eq_self_iff_true],
split_ifs, assumption, refl,
simp only [h, if_false, if_true, eq_self_iff_true],
cases mem_iff.mp ha; cc,
refl,
end
lemma other_invol {a : α} {z : sym2 α} (ha : a ∈ z) (hb : ha.other ∈ z):
hb.other = a :=
begin
classical,
rw other_eq_other' at hb ⊢,
convert other_invol' ha hb,
rw other_eq_other',
end
end decidable
end sym2
|
76db5f8f8ceda6543580bf178bdfccde3093334c | 5749d8999a76f3a8fddceca1f6941981e33aaa96 | /src/ring_theory/adjoin_root.lean | 6eec11df42f42f9f08e7b84359c8457adb73e1aa | [
"Apache-2.0"
] | permissive | jdsalchow/mathlib | 13ab43ef0d0515a17e550b16d09bd14b76125276 | 497e692b946d93906900bb33a51fd243e7649406 | refs/heads/master | 1,585,819,143,348 | 1,580,072,892,000 | 1,580,072,892,000 | 154,287,128 | 0 | 0 | Apache-2.0 | 1,540,281,610,000 | 1,540,281,609,000 | null | UTF-8 | Lean | false | false | 3,421 | lean | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
Adjoining roots of polynomials
-/
import data.polynomial ring_theory.principal_ideal_domain
noncomputable theory
universes u v w
variables {α : Type u} {β : Type v} {γ : Type w}
open polynomial ideal
def adjoin_root [comm_ring α] (f : polynomial α) : Type u :=
ideal.quotient (span {f} : ideal (polynomial α))
namespace adjoin_root
section comm_ring
variables [comm_ring α] (f : polynomial α)
instance : comm_ring (adjoin_root f) := ideal.quotient.comm_ring _
instance : decidable_eq (adjoin_root f) := classical.dec_eq _
variable {f}
def mk : polynomial α → adjoin_root f := ideal.quotient.mk _
def root : adjoin_root f := mk X
def of (x : α) : adjoin_root f := mk (C x)
instance adjoin_root.has_coe_t : has_coe_t α (adjoin_root f) := ⟨of⟩
instance mk.is_ring_hom : is_ring_hom (mk : polynomial α → adjoin_root f) :=
ideal.quotient.is_ring_hom_mk _
@[simp] lemma mk_self : (mk f : adjoin_root f) = 0 :=
quotient.sound' (mem_span_singleton.2 $ by simp)
instance : is_ring_hom (coe : α → adjoin_root f) :=
@is_ring_hom.comp _ _ _ _ C _ _ _ mk mk.is_ring_hom
lemma eval₂_root (f : polynomial α) : f.eval₂ coe (root : adjoin_root f) = 0 :=
quotient.induction_on' (root : adjoin_root f)
(λ (g : polynomial α) (hg : mk g = mk X),
show finsupp.sum f (λ (e : ℕ) (a : α), mk (C a) * mk g ^ e) = 0,
by simp only [hg, (is_semiring_hom.map_pow (mk : polynomial α → adjoin_root f) _ _).symm,
(is_ring_hom.map_mul (mk : polynomial α → adjoin_root f)).symm];
rw [finsupp.sum, f.support.sum_hom (mk : polynomial α → adjoin_root f),
show finset.sum _ _ = _, from sum_C_mul_X_eq _, mk_self])
(show (root : adjoin_root f) = mk X, from rfl)
lemma is_root_root (f : polynomial α) : is_root (f.map coe) (root : adjoin_root f) :=
by rw [is_root, eval_map, eval₂_root]
variables [comm_ring β]
def lift (i : α → β) [is_ring_hom i] (x : β) (h : f.eval₂ i x = 0) : (adjoin_root f) → β :=
ideal.quotient.lift _ (eval₂ i x) $ λ g H,
begin
simp [mem_span_singleton] at H,
cases H with y H,
rw [H, eval₂_mul],
simp [h]
end
variables {i : α → β} [is_ring_hom i] {a : β} {h : f.eval₂ i a = 0}
@[simp] lemma lift_mk {g : polynomial α} : lift i a h (mk g) = g.eval₂ i a :=
ideal.quotient.lift_mk
@[simp] lemma lift_root : lift i a h root = a := by simp [root, h]
@[simp] lemma lift_of {x : α} : lift i a h x = i x :=
by show lift i a h (ideal.quotient.mk _ (C x)) = i x;
convert ideal.quotient.lift_mk; simp
instance is_ring_hom_lift : is_ring_hom (lift i a h) :=
by unfold lift; apply_instance
end comm_ring
variables [discrete_field α] {f : polynomial α} [irreducible f]
instance is_maximal_span : is_maximal (span {f} : ideal (polynomial α)) :=
principal_ideal_domain.is_maximal_of_irreducible ‹irreducible f›
noncomputable instance field : discrete_field (adjoin_root f) :=
ideal.quotient.field (span {f} : ideal (polynomial α))
lemma coe_injective : function.injective (coe : α → adjoin_root f) :=
is_ring_hom.injective _
lemma mul_div_root_cancel (f : polynomial α) [irreducible f] :
(X - C (root : adjoin_root f)) * (f.map coe / (X - C root)) = f.map coe :=
mul_div_eq_iff_is_root.2 $ is_root_root _
end adjoin_root
|
96e6b05c07cd24c8d45aaa94dc4f12fba46fa0af | a7eef317ddec01b9fc6cfbb876fe7ac00f205ac7 | /src/category_theory/limits/shapes/finite_products.lean | eb3d92275e50f20a5f66ec9aec3dc59c33a6b936 | [
"Apache-2.0"
] | permissive | kmill/mathlib | ea5a007b67ae4e9e18dd50d31d8aa60f650425ee | 1a419a9fea7b959317eddd556e1bb9639f4dcc05 | refs/heads/master | 1,668,578,197,719 | 1,593,629,163,000 | 1,593,629,163,000 | 276,482,939 | 0 | 0 | null | 1,593,637,960,000 | 1,593,637,959,000 | null | UTF-8 | Lean | false | false | 1,579 | lean | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import category_theory.limits.shapes.finite_limits
universes v u
open category_theory
namespace category_theory.limits
variables (C : Type u) [category.{v} C]
class has_finite_products :=
(has_limits_of_shape : Π (J : Type v) [fintype J] [decidable_eq J], has_limits_of_shape (discrete J) C)
class has_finite_coproducts :=
(has_colimits_of_shape : Π (J : Type v) [fintype J] [decidable_eq J], has_colimits_of_shape (discrete J) C)
attribute [instance] has_finite_products.has_limits_of_shape has_finite_coproducts.has_colimits_of_shape
@[priority 100] -- see Note [lower instance priority]
instance has_finite_products_of_has_products [has_products C] : has_finite_products C :=
{ has_limits_of_shape := λ J _, by apply_instance }
@[priority 100] -- see Note [lower instance priority]
instance has_finite_coproducts_of_has_coproducts [has_coproducts C] : has_finite_coproducts C :=
{ has_colimits_of_shape := λ J _, by apply_instance }
@[priority 100] -- see Note [lower instance priority]
instance has_finite_products_of_has_finite_limits [has_finite_limits C] : has_finite_products C :=
{ has_limits_of_shape := λ J _ _, by { resetI, apply_instance } }
@[priority 100] -- see Note [lower instance priority]
instance has_finite_coproducts_of_has_finite_colimits [has_finite_colimits C] : has_finite_coproducts C :=
{ has_colimits_of_shape := λ J _ _, by { resetI, apply_instance } }
end category_theory.limits
|
9c91efa5d4724a5a24df26c15bb7eb20d3b73fc9 | 32da3d0f92cab08875472ef6cacc1931c2b3eafa | /src/data/real/ennreal.lean | a1262b7d9f1d50d744a1cbadf49500d0da17cc1f | [
"Apache-2.0"
] | permissive | karthiknadig/mathlib | b6073c3748860bfc9a3e55da86afcddba62dc913 | 33a86cfff12d7f200d0010cd03b95e9b69a6c1a5 | refs/heads/master | 1,676,389,371,851 | 1,610,061,127,000 | 1,610,061,127,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 55,722 | lean | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Johannes Hölzl, Yury Kudryashov
-/
import data.real.nnreal
import data.set.intervals
/-!
# Extended non-negative reals
We define `ennreal := with_no ℝ≥0` to be the type of extended nonnegative real numbers, i.e., the
interval `[0, +∞]`. This type is used as the codomain of a `measure_theory.measure`, and of the
extended distance `edist` in a `emetric_space`. In this file we define some algebraic operations and
a linear order on `ennreal` and prove basic properties of these operations, order, and conversions
to/from `ℝ`, `ℝ≥0`, and `ℕ`.
## Main definitions
* `ennreal`: the extended nonnegative real numbers `[0, ∞]`; defined as `with_top ℝ≥0`; it is
equipped with the following structures:
- coercion from `ℝ≥0` defined in the natural way;
- the natural structure of a complete dense linear order: `↑p ≤ ↑q ↔ p ≤ q` and `∀ a, a ≤ ∞`;
- `a + b` is defined so that `↑p + ↑q = ↑(p + q)` for `(p q : ℝ≥0)` and `a + ∞ = ∞ + a = ∞`;
- `a * b` is defined so that `↑p * ↑q = ↑(p * q)` for `(p q : ℝ≥0)`, `0 * ∞ = ∞ * 0 = 0`, and `a *
∞ = ∞ * a = ∞` for `a ≠ 0`;
- `a - b` is defined as the minimal `d` such that `a ≤ d + b`; this way we have
`↑p - ↑q = ↑(p - q)`, `∞ - ↑p = ∞`, `↑p - ∞ = ∞ - ∞ = 0`; note that there is no negation, only
subtraction;
- `a⁻¹` is defined as `Inf {b | 1 ≤ a * b}`. This way we have `(↑p)⁻¹ = ↑(p⁻¹)` for
`p : ℝ≥0`, `p ≠ 0`, `0⁻¹ = ∞`, and `∞⁻¹ = 0`.
- `a / b` is defined as `a * b⁻¹`.
The addition and multiplication defined this way together with `0 = ↑0` and `1 = ↑1` turn
`ennreal` into a canonically ordered commutative semiring of characteristic zero.
* Coercions to/from other types:
- coercion `ℝ≥0 → ennreal` is defined as `has_coe`, so one can use `(p : ℝ≥0)` in a context that
expects `a : ennreal`, and Lean will apply `coe` automatically;
- `ennreal.to_nnreal` sends `↑p` to `p` and `∞` to `0`;
- `ennreal.to_real := coe ∘ ennreal.to_nnreal` sends `↑p`, `p : ℝ≥0` to `(↑p : ℝ)` and `∞` to `0`;
- `ennreal.of_real := coe ∘ nnreal.of_real` sends `x : ℝ` to `↑⟨max x 0, _⟩`
- `ennreal.ne_top_equiv_nnreal` is an equivalence between `{a : ennreal // a ≠ 0}` and `ℝ≥0`.
## Implementation notes
We define a `can_lift ennreal ℝ≥0` instance, so one of the ways to prove theorems about an `ennreal`
number `a` is to consider the cases `a = ∞` and `a ≠ ∞`, and use the tactic `lift a to ℝ≥0 using ha`
in the second case. This instance is even more useful if one already has `ha : a ≠ ∞` in the
context, or if we have `(f : α → ennreal) (hf : ∀ x, f x ≠ ∞)`.
## Notations
* `ℝ≥0`: type of nonnegative real numbers `[0, ∞)`; defined in `data.real.nnreal`;
* `∞`: a localized notation in `ennreal` for `⊤ : ennreal`.
-/
noncomputable theory
open classical set
open_locale classical big_operators nnreal
variables {α : Type*} {β : Type*}
/-- The extended nonnegative real numbers. This is usually denoted [0, ∞],
and is relevant as the codomain of a measure. -/
@[derive canonically_ordered_comm_semiring, derive complete_linear_order, derive densely_ordered,
derive nontrivial]
def ennreal := with_top ℝ≥0
localized "notation `∞` := (⊤ : ennreal)" in ennreal
namespace ennreal
variables {a b c d : ennreal} {r p q : ℝ≥0}
instance : inhabited ennreal := ⟨0⟩
instance : has_coe ℝ≥0 ennreal := ⟨ option.some ⟩
instance : can_lift ennreal ℝ≥0 :=
{ coe := coe,
cond := λ r, r ≠ ∞,
prf := λ x hx, ⟨option.get $ option.ne_none_iff_is_some.1 hx, option.some_get _⟩ }
@[simp] lemma none_eq_top : (none : ennreal) = ∞ := rfl
@[simp] lemma some_eq_coe (a : ℝ≥0) : (some a : ennreal) = (↑a : ennreal) := rfl
/-- `to_nnreal x` returns `x` if it is real, otherwise 0. -/
protected def to_nnreal : ennreal → ℝ≥0
| (some r) := r
| none := 0
/-- `to_real x` returns `x` if it is real, `0` otherwise. -/
protected def to_real (a : ennreal) : real := coe (a.to_nnreal)
/-- `of_real x` returns `x` if it is nonnegative, `0` otherwise. -/
protected def of_real (r : real) : ennreal := coe (nnreal.of_real r)
@[simp, norm_cast] lemma to_nnreal_coe : (r : ennreal).to_nnreal = r := rfl
@[simp] lemma coe_to_nnreal : ∀{a:ennreal}, a ≠ ∞ → ↑(a.to_nnreal) = a
| (some r) h := rfl
| none h := (h rfl).elim
@[simp] lemma of_real_to_real {a : ennreal} (h : a ≠ ∞) : ennreal.of_real (a.to_real) = a :=
by simp [ennreal.to_real, ennreal.of_real, h]
@[simp] lemma to_real_of_real {r : ℝ} (h : 0 ≤ r) : ennreal.to_real (ennreal.of_real r) = r :=
by simp [ennreal.to_real, ennreal.of_real, nnreal.coe_of_real _ h]
lemma to_real_of_real' {r : ℝ} : ennreal.to_real (ennreal.of_real r) = max r 0 := rfl
lemma coe_to_nnreal_le_self : ∀{a:ennreal}, ↑(a.to_nnreal) ≤ a
| (some r) := by rw [some_eq_coe, to_nnreal_coe]; exact le_refl _
| none := le_top
lemma coe_nnreal_eq (r : ℝ≥0) : (r : ennreal) = ennreal.of_real r :=
by { rw [ennreal.of_real, nnreal.of_real], cases r with r h, congr, dsimp, rw max_eq_left h }
lemma of_real_eq_coe_nnreal {x : ℝ} (h : 0 ≤ x) :
ennreal.of_real x = @coe ℝ≥0 ennreal _ (⟨x, h⟩ : ℝ≥0) :=
by { rw [coe_nnreal_eq], refl }
@[simp] lemma of_real_coe_nnreal : ennreal.of_real p = p := (coe_nnreal_eq p).symm
@[simp, norm_cast] lemma coe_zero : ↑(0 : ℝ≥0) = (0 : ennreal) := rfl
@[simp, norm_cast] lemma coe_one : ↑(1 : ℝ≥0) = (1 : ennreal) := rfl
@[simp] lemma to_real_nonneg {a : ennreal} : 0 ≤ a.to_real := by simp [ennreal.to_real]
@[simp] lemma top_to_nnreal : ∞.to_nnreal = 0 := rfl
@[simp] lemma top_to_real : ∞.to_real = 0 := rfl
@[simp] lemma one_to_real : (1 : ennreal).to_real = 1 := rfl
@[simp] lemma one_to_nnreal : (1 : ennreal).to_nnreal = 1 := rfl
@[simp] lemma coe_to_real (r : ℝ≥0) : (r : ennreal).to_real = r := rfl
@[simp] lemma zero_to_nnreal : (0 : ennreal).to_nnreal = 0 := rfl
@[simp] lemma zero_to_real : (0 : ennreal).to_real = 0 := rfl
@[simp] lemma of_real_zero : ennreal.of_real (0 : ℝ) = 0 :=
by simp [ennreal.of_real]; refl
@[simp] lemma of_real_one : ennreal.of_real (1 : ℝ) = (1 : ennreal) :=
by simp [ennreal.of_real]
lemma of_real_to_real_le {a : ennreal} : ennreal.of_real (a.to_real) ≤ a :=
if ha : a = ∞ then ha.symm ▸ le_top else le_of_eq (of_real_to_real ha)
lemma forall_ennreal {p : ennreal → Prop} : (∀a, p a) ↔ (∀r:ℝ≥0, p r) ∧ p ∞ :=
⟨assume h, ⟨assume r, h _, h _⟩,
assume ⟨h₁, h₂⟩ a, match a with some r := h₁ _ | none := h₂ end⟩
lemma forall_ne_top {p : ennreal → Prop} : (∀ a ≠ ∞, p a) ↔ ∀ r : ℝ≥0, p r :=
option.ball_ne_none
lemma exists_ne_top {p : ennreal → Prop} : (∃ a ≠ ∞, p a) ↔ ∃ r : ℝ≥0, p r :=
option.bex_ne_none
lemma to_nnreal_eq_zero_iff (x : ennreal) : x.to_nnreal = 0 ↔ x = 0 ∨ x = ∞ :=
⟨begin
cases x,
{ simp [none_eq_top] },
{ have A : some (0:ℝ≥0) = (0:ennreal) := rfl,
simp [ennreal.to_nnreal, A] {contextual := tt} }
end,
by intro h; cases h; simp [h]⟩
lemma to_real_eq_zero_iff (x : ennreal) : x.to_real = 0 ↔ x = 0 ∨ x = ∞ :=
by simp [ennreal.to_real, to_nnreal_eq_zero_iff]
@[simp] lemma coe_ne_top : (r : ennreal) ≠ ∞ := with_top.coe_ne_top
@[simp] lemma top_ne_coe : ∞ ≠ (r : ennreal) := with_top.top_ne_coe
@[simp] lemma of_real_ne_top {r : ℝ} : ennreal.of_real r ≠ ∞ := by simp [ennreal.of_real]
@[simp] lemma of_real_lt_top {r : ℝ} : ennreal.of_real r < ∞ := lt_top_iff_ne_top.2 of_real_ne_top
@[simp] lemma top_ne_of_real {r : ℝ} : ∞ ≠ ennreal.of_real r := by simp [ennreal.of_real]
@[simp] lemma zero_ne_top : 0 ≠ ∞ := coe_ne_top
@[simp] lemma top_ne_zero : ∞ ≠ 0 := top_ne_coe
@[simp] lemma one_ne_top : 1 ≠ ∞ := coe_ne_top
@[simp] lemma top_ne_one : ∞ ≠ 1 := top_ne_coe
@[simp, norm_cast] lemma coe_eq_coe : (↑r : ennreal) = ↑q ↔ r = q := with_top.coe_eq_coe
@[simp, norm_cast] lemma coe_le_coe : (↑r : ennreal) ≤ ↑q ↔ r ≤ q := with_top.coe_le_coe
@[simp, norm_cast] lemma coe_lt_coe : (↑r : ennreal) < ↑q ↔ r < q := with_top.coe_lt_coe
lemma coe_mono : monotone (coe : ℝ≥0 → ennreal) := λ _ _, coe_le_coe.2
@[simp, norm_cast] lemma coe_eq_zero : (↑r : ennreal) = 0 ↔ r = 0 := coe_eq_coe
@[simp, norm_cast] lemma zero_eq_coe : 0 = (↑r : ennreal) ↔ 0 = r := coe_eq_coe
@[simp, norm_cast] lemma coe_eq_one : (↑r : ennreal) = 1 ↔ r = 1 := coe_eq_coe
@[simp, norm_cast] lemma one_eq_coe : 1 = (↑r : ennreal) ↔ 1 = r := coe_eq_coe
@[simp, norm_cast] lemma coe_nonneg : 0 ≤ (↑r : ennreal) ↔ 0 ≤ r := coe_le_coe
@[simp, norm_cast] lemma coe_pos : 0 < (↑r : ennreal) ↔ 0 < r := coe_lt_coe
@[simp, norm_cast] lemma coe_add : ↑(r + p) = (r + p : ennreal) := with_top.coe_add
@[simp, norm_cast] lemma coe_mul : ↑(r * p) = (r * p : ennreal) := with_top.coe_mul
@[simp, norm_cast] lemma coe_bit0 : (↑(bit0 r) : ennreal) = bit0 r := coe_add
@[simp, norm_cast] lemma coe_bit1 : (↑(bit1 r) : ennreal) = bit1 r := by simp [bit1]
lemma coe_two : ((2:ℝ≥0) : ennreal) = 2 := by norm_cast
protected lemma zero_lt_one : 0 < (1 : ennreal) :=
canonically_ordered_semiring.zero_lt_one
@[simp] lemma one_lt_two : (1 : ennreal) < 2 :=
coe_one ▸ coe_two ▸ by exact_mod_cast (@one_lt_two ℕ _ _)
@[simp] lemma zero_lt_two : (0:ennreal) < 2 := lt_trans ennreal.zero_lt_one one_lt_two
lemma two_ne_zero : (2:ennreal) ≠ 0 := (ne_of_lt zero_lt_two).symm
lemma two_ne_top : (2:ennreal) ≠ ∞ := coe_two ▸ coe_ne_top
/-- The set of `ennreal` numbers that are not equal to `∞` is equivalent to `ℝ≥0`. -/
def ne_top_equiv_nnreal : {a | a ≠ ∞} ≃ ℝ≥0 :=
{ to_fun := λ x, ennreal.to_nnreal x,
inv_fun := λ x, ⟨x, coe_ne_top⟩,
left_inv := λ ⟨x, hx⟩, subtype.eq $ coe_to_nnreal hx,
right_inv := λ x, to_nnreal_coe }
lemma cinfi_ne_top [has_Inf α] (f : ennreal → α) : (⨅ x : {x // x ≠ ∞}, f x) = ⨅ x : ℝ≥0, f x :=
eq.symm $ infi_congr _ ne_top_equiv_nnreal.symm.surjective $ λ x, rfl
lemma infi_ne_top [complete_lattice α] (f : ennreal → α) : (⨅ x ≠ ∞, f x) = ⨅ x : ℝ≥0, f x :=
by rw [infi_subtype', cinfi_ne_top]
lemma csupr_ne_top [has_Sup α] (f : ennreal → α) : (⨆ x : {x // x ≠ ∞}, f x) = ⨆ x : ℝ≥0, f x :=
@cinfi_ne_top (order_dual α) _ _
lemma supr_ne_top [complete_lattice α] (f : ennreal → α) : (⨆ x ≠ ∞, f x) = ⨆ x : ℝ≥0, f x :=
@infi_ne_top (order_dual α) _ _
lemma infi_ennreal {α : Type*} [complete_lattice α] {f : ennreal → α} :
(⨅ n, f n) = (⨅ n : ℝ≥0, f n) ⊓ f ∞ :=
le_antisymm
(le_inf (le_infi $ assume i, infi_le _ _) (infi_le _ _))
(le_infi $ forall_ennreal.2 ⟨assume r, inf_le_left_of_le $ infi_le _ _, inf_le_right⟩)
lemma supr_ennreal {α : Type*} [complete_lattice α] {f : ennreal → α} :
(⨆ n, f n) = (⨆ n : ℝ≥0, f n) ⊔ f ∞ :=
@infi_ennreal (order_dual α) _ _
@[simp] lemma add_top : a + ∞ = ∞ := with_top.add_top
@[simp] lemma top_add : ∞ + a = ∞ := with_top.top_add
/-- Coercion `ℝ≥0 → ennreal` as a `ring_hom`. -/
def of_nnreal_hom : ℝ≥0 →+* ennreal :=
⟨coe, coe_one, λ _ _, coe_mul, coe_zero, λ _ _, coe_add⟩
@[simp] lemma coe_of_nnreal_hom : ⇑of_nnreal_hom = coe := rfl
@[simp, norm_cast] lemma coe_indicator {α} (s : set α) (f : α → ℝ≥0) (a : α) :
((s.indicator f a : ℝ≥0) : ennreal) = s.indicator (λ x, f x) a :=
(of_nnreal_hom : ℝ≥0 →+ ennreal).map_indicator _ _ _
@[simp, norm_cast] lemma coe_pow (n : ℕ) : (↑(r^n) : ennreal) = r^n :=
of_nnreal_hom.map_pow r n
@[simp] lemma add_eq_top : a + b = ∞ ↔ a = ∞ ∨ b = ∞ := with_top.add_eq_top
@[simp] lemma add_lt_top : a + b < ∞ ↔ a < ∞ ∧ b < ∞ := with_top.add_lt_top
lemma to_nnreal_add {r₁ r₂ : ennreal} (h₁ : r₁ < ∞) (h₂ : r₂ < ∞) :
(r₁ + r₂).to_nnreal = r₁.to_nnreal + r₂.to_nnreal :=
begin
rw [← coe_eq_coe, coe_add, coe_to_nnreal, coe_to_nnreal, coe_to_nnreal];
apply @ne_top_of_lt ennreal _ _ ∞,
exact h₂,
exact h₁,
exact add_lt_top.2 ⟨h₁, h₂⟩
end
/- rw has trouble with the generic lt_top_iff_ne_top and bot_lt_iff_ne_bot
(contrary to erw). This is solved with the next lemmas -/
protected lemma lt_top_iff_ne_top : a < ∞ ↔ a ≠ ∞ := lt_top_iff_ne_top
protected lemma bot_lt_iff_ne_bot : 0 < a ↔ a ≠ 0 := bot_lt_iff_ne_bot
lemma not_lt_top {x : ennreal} : ¬ x < ∞ ↔ x = ∞ := by rw [lt_top_iff_ne_top, not_not]
lemma add_ne_top : a + b ≠ ∞ ↔ a ≠ ∞ ∧ b ≠ ∞ :=
by simpa only [lt_top_iff_ne_top] using add_lt_top
lemma mul_top : a * ∞ = (if a = 0 then 0 else ∞) :=
begin split_ifs, { simp [h] }, { exact with_top.mul_top h } end
lemma top_mul : ∞ * a = (if a = 0 then 0 else ∞) :=
begin split_ifs, { simp [h] }, { exact with_top.top_mul h } end
@[simp] lemma top_mul_top : ∞ * ∞ = ∞ := with_top.top_mul_top
lemma top_pow {n:ℕ} (h : 0 < n) : ∞^n = ∞ :=
nat.le_induction (pow_one _) (λ m hm hm', by rw [pow_succ, hm', top_mul_top])
_ (nat.succ_le_of_lt h)
lemma mul_eq_top : a * b = ∞ ↔ (a ≠ 0 ∧ b = ∞) ∨ (a = ∞ ∧ b ≠ 0) :=
with_top.mul_eq_top_iff
lemma mul_lt_top : a < ∞ → b < ∞ → a * b < ∞ :=
with_top.mul_lt_top
lemma mul_ne_top : a ≠ ∞ → b ≠ ∞ → a * b ≠ ∞ :=
by simpa only [lt_top_iff_ne_top] using mul_lt_top
lemma ne_top_of_mul_ne_top_left (h : a * b ≠ ∞) (hb : b ≠ 0) : a ≠ ∞ :=
by { simp [mul_eq_top, hb, not_or_distrib] at h ⊢, exact h.2 }
lemma ne_top_of_mul_ne_top_right (h : a * b ≠ ∞) (ha : a ≠ 0) : b ≠ ∞ :=
ne_top_of_mul_ne_top_left (by rwa [mul_comm]) ha
lemma lt_top_of_mul_lt_top_left (h : a * b < ∞) (hb : b ≠ 0) : a < ∞ :=
by { rw [ennreal.lt_top_iff_ne_top] at h ⊢, exact ne_top_of_mul_ne_top_left h hb }
lemma lt_top_of_mul_lt_top_right (h : a * b < ∞) (ha : a ≠ 0) : b < ∞ :=
lt_top_of_mul_lt_top_left (by rwa [mul_comm]) ha
lemma mul_lt_top_iff {a b : ennreal} : a * b < ∞ ↔ (a < ∞ ∧ b < ∞) ∨ a = 0 ∨ b = 0 :=
begin
split,
{ intro h, rw [← or_assoc, or_iff_not_imp_right, or_iff_not_imp_right], intros hb ha,
exact ⟨lt_top_of_mul_lt_top_left h hb, lt_top_of_mul_lt_top_right h ha⟩ },
{ rintro (⟨ha, hb⟩|rfl|rfl); [exact mul_lt_top ha hb, simp, simp] }
end
@[simp] lemma mul_pos : 0 < a * b ↔ 0 < a ∧ 0 < b :=
by simp only [zero_lt_iff_ne_zero, ne.def, mul_eq_zero, not_or_distrib]
lemma pow_eq_top : ∀ n:ℕ, a^n=∞ → a=∞
| 0 := by simp
| (n+1) := λ o, (mul_eq_top.1 o).elim (λ h, pow_eq_top n h.2) and.left
lemma pow_ne_top (h : a ≠ ∞) {n:ℕ} : a^n ≠ ∞ :=
mt (pow_eq_top n) h
lemma pow_lt_top : a < ∞ → ∀ n:ℕ, a^n < ∞ :=
by simpa only [lt_top_iff_ne_top] using pow_ne_top
@[simp, norm_cast] lemma coe_finset_sum {s : finset α} {f : α → ℝ≥0} :
↑(∑ a in s, f a) = (∑ a in s, f a : ennreal) :=
of_nnreal_hom.map_sum f s
@[simp, norm_cast] lemma coe_finset_prod {s : finset α} {f : α → ℝ≥0} :
↑(∏ a in s, f a) = ((∏ a in s, f a) : ennreal) :=
of_nnreal_hom.map_prod f s
section order
@[simp] lemma bot_eq_zero : (⊥ : ennreal) = 0 := rfl
@[simp] lemma coe_lt_top : coe r < ∞ := with_top.coe_lt_top r
@[simp] lemma not_top_le_coe : ¬ ∞ ≤ ↑r := with_top.not_top_le_coe r
lemma zero_lt_coe_iff : 0 < (↑p : ennreal) ↔ 0 < p := coe_lt_coe
@[simp, norm_cast] lemma one_le_coe_iff : (1:ennreal) ≤ ↑r ↔ 1 ≤ r := coe_le_coe
@[simp, norm_cast] lemma coe_le_one_iff : ↑r ≤ (1:ennreal) ↔ r ≤ 1 := coe_le_coe
@[simp, norm_cast] lemma coe_lt_one_iff : (↑p : ennreal) < 1 ↔ p < 1 := coe_lt_coe
@[simp, norm_cast] lemma one_lt_coe_iff : 1 < (↑p : ennreal) ↔ 1 < p := coe_lt_coe
@[simp, norm_cast] lemma coe_nat (n : nat) : ((n : ℝ≥0) : ennreal) = n := with_top.coe_nat n
@[simp] lemma nat_ne_top (n : nat) : (n : ennreal) ≠ ∞ := with_top.nat_ne_top n
@[simp] lemma top_ne_nat (n : nat) : ∞ ≠ n := with_top.top_ne_nat n
@[simp] lemma one_lt_top : 1 < ∞ := coe_lt_top
lemma le_coe_iff : a ≤ ↑r ↔ (∃p:ℝ≥0, a = p ∧ p ≤ r) := with_top.le_coe_iff
lemma coe_le_iff : ↑r ≤ a ↔ (∀p:ℝ≥0, a = p → r ≤ p) := with_top.coe_le_iff
lemma lt_iff_exists_coe : a < b ↔ (∃p:ℝ≥0, a = p ∧ ↑p < b) := with_top.lt_iff_exists_coe
@[simp, norm_cast] lemma coe_finset_sup {s : finset α} {f : α → ℝ≥0} :
↑(s.sup f) = s.sup (λ x, (f x : ennreal)) :=
finset.comp_sup_eq_sup_comp_of_is_total _ coe_mono rfl
lemma pow_le_pow {n m : ℕ} (ha : 1 ≤ a) (h : n ≤ m) : a ^ n ≤ a ^ m :=
begin
cases a,
{ cases m,
{ rw eq_bot_iff.mpr h,
exact le_refl _ },
{ rw [none_eq_top, top_pow (nat.succ_pos m)],
exact le_top } },
{ rw [some_eq_coe, ← coe_pow, ← coe_pow, coe_le_coe],
exact pow_le_pow (by simpa using ha) h }
end
@[simp] lemma max_eq_zero_iff : max a b = 0 ↔ a = 0 ∧ b = 0 :=
by simp only [le_zero_iff_eq.symm, max_le_iff]
@[simp] lemma max_zero_left : max 0 a = a := max_eq_right (zero_le a)
@[simp] lemma max_zero_right : max a 0 = a := max_eq_left (zero_le a)
-- TODO: why this is not a `rfl`? There is some hidden diamond here.
@[simp] lemma sup_eq_max : a ⊔ b = max a b :=
eq_of_forall_ge_iff $ λ c, sup_le_iff.trans max_le_iff.symm
protected lemma pow_pos : 0 < a → ∀ n : ℕ, 0 < a^n :=
canonically_ordered_semiring.pow_pos
protected lemma pow_ne_zero : a ≠ 0 → ∀ n : ℕ, a^n ≠ 0 :=
by simpa only [zero_lt_iff_ne_zero] using ennreal.pow_pos
@[simp] lemma not_lt_zero : ¬ a < 0 := by simp
lemma add_lt_add_iff_left : a < ∞ → (a + c < a + b ↔ c < b) :=
with_top.add_lt_add_iff_left
lemma add_lt_add_iff_right : a < ∞ → (c + a < b + a ↔ c < b) :=
with_top.add_lt_add_iff_right
lemma lt_add_right (ha : a < ∞) (hb : 0 < b) : a < a + b :=
by rwa [← add_lt_add_iff_left ha, add_zero] at hb
lemma le_of_forall_epsilon_le : ∀{a b : ennreal}, (∀ε:ℝ≥0, 0 < ε → b < ∞ → a ≤ b + ε) → a ≤ b
| a none h := le_top
| none (some a) h :=
have ∞ ≤ ↑a + ↑(1:ℝ≥0), from h 1 zero_lt_one coe_lt_top,
by rw [← coe_add] at this; exact (not_top_le_coe this).elim
| (some a) (some b) h :=
by simp only [none_eq_top, some_eq_coe, coe_add.symm, coe_le_coe, coe_lt_top, true_implies_iff]
at *; exact nnreal.le_of_forall_epsilon_le h
lemma lt_iff_exists_rat_btwn :
a < b ↔ (∃q:ℚ, 0 ≤ q ∧ a < nnreal.of_real q ∧ (nnreal.of_real q:ennreal) < b) :=
⟨λ h,
begin
rcases lt_iff_exists_coe.1 h with ⟨p, rfl, _⟩,
rcases exists_between h with ⟨c, pc, cb⟩,
rcases lt_iff_exists_coe.1 cb with ⟨r, rfl, _⟩,
rcases (nnreal.lt_iff_exists_rat_btwn _ _).1 (coe_lt_coe.1 pc) with ⟨q, hq0, pq, qr⟩,
exact ⟨q, hq0, coe_lt_coe.2 pq, lt_trans (coe_lt_coe.2 qr) cb⟩
end,
λ ⟨q, q0, qa, qb⟩, lt_trans qa qb⟩
lemma lt_iff_exists_real_btwn :
a < b ↔ (∃r:ℝ, 0 ≤ r ∧ a < ennreal.of_real r ∧ (ennreal.of_real r:ennreal) < b) :=
⟨λ h, let ⟨q, q0, aq, qb⟩ := ennreal.lt_iff_exists_rat_btwn.1 h in
⟨q, rat.cast_nonneg.2 q0, aq, qb⟩,
λ ⟨q, q0, qa, qb⟩, lt_trans qa qb⟩
lemma lt_iff_exists_nnreal_btwn :
a < b ↔ (∃r:ℝ≥0, a < r ∧ (r : ennreal) < b) :=
with_top.lt_iff_exists_coe_btwn
lemma lt_iff_exists_add_pos_lt : a < b ↔ (∃ r : ℝ≥0, 0 < r ∧ a + r < b) :=
begin
refine ⟨λ hab, _, λ ⟨r, rpos, hr⟩, lt_of_le_of_lt (le_add_right (le_refl _)) hr⟩,
cases a, { simpa using hab },
rcases lt_iff_exists_real_btwn.1 hab with ⟨c, c_nonneg, ac, cb⟩,
let d : ℝ≥0 := ⟨c, c_nonneg⟩,
have ad : a < d,
{ rw of_real_eq_coe_nnreal c_nonneg at ac,
exact coe_lt_coe.1 ac },
refine ⟨d-a, nnreal.sub_pos.2 ad, _⟩,
rw [some_eq_coe, ← coe_add],
convert cb,
have : nnreal.of_real c = d,
by { rw [← nnreal.coe_eq, nnreal.coe_of_real _ c_nonneg], refl },
rw [add_comm, this],
exact nnreal.sub_add_cancel_of_le (le_of_lt ad)
end
lemma coe_nat_lt_coe {n : ℕ} : (n : ennreal) < r ↔ ↑n < r := ennreal.coe_nat n ▸ coe_lt_coe
lemma coe_lt_coe_nat {n : ℕ} : (r : ennreal) < n ↔ r < n := ennreal.coe_nat n ▸ coe_lt_coe
@[norm_cast] lemma coe_nat_lt_coe_nat {m n : ℕ} : (m : ennreal) < n ↔ m < n :=
ennreal.coe_nat n ▸ coe_nat_lt_coe.trans nat.cast_lt
lemma coe_nat_ne_top {n : ℕ} : (n : ennreal) ≠ ∞ := ennreal.coe_nat n ▸ coe_ne_top
lemma coe_nat_mono : strict_mono (coe : ℕ → ennreal) := λ _ _, coe_nat_lt_coe_nat.2
@[norm_cast] lemma coe_nat_le_coe_nat {m n : ℕ} : (m : ennreal) ≤ n ↔ m ≤ n :=
coe_nat_mono.le_iff_le
instance : char_zero ennreal := ⟨coe_nat_mono.injective⟩
protected lemma exists_nat_gt {r : ennreal} (h : r ≠ ∞) : ∃n:ℕ, r < n :=
begin
lift r to ℝ≥0 using h,
rcases exists_nat_gt r with ⟨n, hn⟩,
exact ⟨n, coe_lt_coe_nat.2 hn⟩,
end
lemma add_lt_add (ac : a < c) (bd : b < d) : a + b < c + d :=
begin
lift a to ℝ≥0 using ne_top_of_lt ac,
lift b to ℝ≥0 using ne_top_of_lt bd,
cases c, { simp }, cases d, { simp },
simp only [← coe_add, some_eq_coe, coe_lt_coe] at *,
exact add_lt_add ac bd
end
@[norm_cast] lemma coe_min : ((min r p:ℝ≥0):ennreal) = min r p :=
coe_mono.map_min
@[norm_cast] lemma coe_max : ((max r p:ℝ≥0):ennreal) = max r p :=
coe_mono.map_max
lemma le_of_top_imp_top_of_to_nnreal_le {a b : ennreal} (h : a = ⊤ → b = ⊤)
(h_nnreal : a ≠ ⊤ → b ≠ ⊤ → a.to_nnreal ≤ b.to_nnreal) :
a ≤ b :=
begin
by_cases ha : a = ⊤,
{ rw h ha,
exact le_top, },
by_cases hb : b = ⊤,
{ rw hb,
exact le_top, },
rw [←coe_to_nnreal hb, ←coe_to_nnreal ha, coe_le_coe],
exact h_nnreal ha hb,
end
end order
section complete_lattice
lemma coe_Sup {s : set ℝ≥0} : bdd_above s → (↑(Sup s) : ennreal) = (⨆a∈s, ↑a) := with_top.coe_Sup
lemma coe_Inf {s : set ℝ≥0} : s.nonempty → (↑(Inf s) : ennreal) = (⨅a∈s, ↑a) := with_top.coe_Inf
@[simp] lemma top_mem_upper_bounds {s : set ennreal} : ∞ ∈ upper_bounds s :=
assume x hx, le_top
lemma coe_mem_upper_bounds {s : set ℝ≥0} :
↑r ∈ upper_bounds ((coe : ℝ≥0 → ennreal) '' s) ↔ r ∈ upper_bounds s :=
by simp [upper_bounds, ball_image_iff, -mem_image, *] {contextual := tt}
end complete_lattice
section mul
lemma mul_le_mul : a ≤ b → c ≤ d → a * c ≤ b * d :=
canonically_ordered_semiring.mul_le_mul
lemma mul_lt_mul (ac : a < c) (bd : b < d) : a * b < c * d :=
begin
rcases lt_iff_exists_nnreal_btwn.1 ac with ⟨a', aa', a'c⟩,
lift a to ℝ≥0 using ne_top_of_lt aa',
rcases lt_iff_exists_nnreal_btwn.1 bd with ⟨b', bb', b'd⟩,
lift b to ℝ≥0 using ne_top_of_lt bb',
norm_cast at *,
calc ↑(a * b) < ↑(a' * b') :
coe_lt_coe.2 (mul_lt_mul' aa'.le bb' (zero_le _) ((zero_le a).trans_lt aa'))
... = ↑a' * ↑b' : coe_mul
... ≤ c * d : mul_le_mul a'c.le b'd.le
end
lemma mul_left_mono : monotone ((*) a) := λ b c, mul_le_mul (le_refl a)
lemma mul_right_mono : monotone (λ x, x * a) := λ b c h, mul_le_mul h (le_refl a)
lemma max_mul : max a b * c = max (a * c) (b * c) :=
mul_right_mono.map_max
lemma mul_max : a * max b c = max (a * b) (a * c) :=
mul_left_mono.map_max
lemma mul_eq_mul_left : a ≠ 0 → a ≠ ∞ → (a * b = a * c ↔ b = c) :=
begin
cases a; cases b; cases c;
simp [none_eq_top, some_eq_coe, mul_top, top_mul, -coe_mul, coe_mul.symm,
nnreal.mul_eq_mul_left] {contextual := tt},
end
lemma mul_eq_mul_right : c ≠ 0 → c ≠ ∞ → (a * c = b * c ↔ a = b) :=
mul_comm c a ▸ mul_comm c b ▸ mul_eq_mul_left
lemma mul_le_mul_left : a ≠ 0 → a ≠ ∞ → (a * b ≤ a * c ↔ b ≤ c) :=
begin
cases a; cases b; cases c;
simp [none_eq_top, some_eq_coe, mul_top, top_mul, -coe_mul, coe_mul.symm] {contextual := tt},
assume h, exact mul_le_mul_left (zero_lt_iff_ne_zero.2 h)
end
lemma mul_le_mul_right : c ≠ 0 → c ≠ ∞ → (a * c ≤ b * c ↔ a ≤ b) :=
mul_comm c a ▸ mul_comm c b ▸ mul_le_mul_left
lemma mul_lt_mul_left : a ≠ 0 → a ≠ ∞ → (a * b < a * c ↔ b < c) :=
λ h0 ht, by simp only [mul_le_mul_left h0 ht, lt_iff_le_not_le]
lemma mul_lt_mul_right : c ≠ 0 → c ≠ ∞ → (a * c < b * c ↔ a < b) :=
mul_comm c a ▸ mul_comm c b ▸ mul_lt_mul_left
end mul
section sub
instance : has_sub ennreal := ⟨λa b, Inf {d | a ≤ d + b}⟩
@[norm_cast] lemma coe_sub : ↑(p - r) = (↑p:ennreal) - r :=
le_antisymm
(le_Inf $ assume b (hb : ↑p ≤ b + r), coe_le_iff.2 $
by rintros d rfl; rwa [← coe_add, coe_le_coe, ← nnreal.sub_le_iff_le_add] at hb)
(Inf_le $ show (↑p : ennreal) ≤ ↑(p - r) + ↑r,
by rw [← coe_add, coe_le_coe, ← nnreal.sub_le_iff_le_add])
@[simp] lemma top_sub_coe : ∞ - ↑r = ∞ :=
top_unique $ le_Inf $ by simp [add_eq_top]
@[simp] lemma sub_eq_zero_of_le (h : a ≤ b) : a - b = 0 :=
le_antisymm (Inf_le $ le_add_left h) (zero_le _)
@[simp] lemma sub_self : a - a = 0 := sub_eq_zero_of_le $ le_refl _
@[simp] lemma zero_sub : 0 - a = 0 :=
le_antisymm (Inf_le $ zero_le $ 0 + a) (zero_le _)
@[simp] lemma sub_infty : a - ∞ = 0 :=
le_antisymm (Inf_le $ by simp) (zero_le _)
lemma sub_le_sub (h₁ : a ≤ b) (h₂ : d ≤ c) : a - c ≤ b - d :=
Inf_le_Inf $ assume e (h : b ≤ e + d),
calc a ≤ b : h₁
... ≤ e + d : h
... ≤ e + c : add_le_add (le_refl _) h₂
@[simp] lemma add_sub_self : ∀{a b : ennreal}, b < ∞ → (a + b) - b = a
| a none := by simp [none_eq_top]
| none (some b) := by simp [none_eq_top, some_eq_coe]
| (some a) (some b) :=
by simp [some_eq_coe]; rw [← coe_add, ← coe_sub, coe_eq_coe, nnreal.add_sub_cancel]
@[simp] lemma add_sub_self' (h : a < ∞) : (a + b) - a = b :=
by rw [add_comm, add_sub_self h]
lemma add_right_inj (h : a < ∞) : a + b = a + c ↔ b = c :=
⟨λ e, by simpa [h] using congr_arg (λ x, x - a) e, congr_arg _⟩
lemma add_left_inj (h : a < ∞) : b + a = c + a ↔ b = c :=
by rw [add_comm, add_comm c, add_right_inj h]
@[simp] lemma sub_add_cancel_of_le : ∀{a b : ennreal}, b ≤ a → (a - b) + b = a :=
begin
simp [forall_ennreal, le_coe_iff, -add_comm] {contextual := tt},
rintros r p x rfl h,
rw [← coe_sub, ← coe_add, nnreal.sub_add_cancel_of_le h]
end
@[simp] lemma add_sub_cancel_of_le (h : b ≤ a) : b + (a - b) = a :=
by rwa [add_comm, sub_add_cancel_of_le]
lemma sub_add_self_eq_max : (a - b) + b = max a b :=
match le_total a b with
| or.inl h := by simp [h, max_eq_right]
| or.inr h := by simp [h, max_eq_left]
end
lemma le_sub_add_self : a ≤ (a - b) + b :=
by { rw sub_add_self_eq_max, exact le_max_left a b }
@[simp] protected lemma sub_le_iff_le_add : a - b ≤ c ↔ a ≤ c + b :=
iff.intro
(assume h : a - b ≤ c,
calc a ≤ (a - b) + b : le_sub_add_self
... ≤ c + b : add_le_add_right h _)
(assume h : a ≤ c + b,
calc a - b ≤ (c + b) - b : sub_le_sub h (le_refl _)
... ≤ c : Inf_le (le_refl (c + b)))
protected lemma sub_le_iff_le_add' : a - b ≤ c ↔ a ≤ b + c :=
add_comm c b ▸ ennreal.sub_le_iff_le_add
lemma sub_eq_of_add_eq : b ≠ ∞ → a + b = c → c - b = a :=
λ hb hc, hc ▸ add_sub_self (lt_top_iff_ne_top.2 hb)
protected lemma sub_le_of_sub_le (h : a - b ≤ c) : a - c ≤ b :=
ennreal.sub_le_iff_le_add.2 $ by { rw add_comm, exact ennreal.sub_le_iff_le_add.1 h }
protected lemma sub_lt_sub_self : a ≠ ∞ → a ≠ 0 → 0 < b → a - b < a :=
match a, b with
| none, _ := by { have := none_eq_top, assume h, contradiction }
| (some a), none := by {intros, simp only [none_eq_top, sub_infty, zero_lt_iff_ne_zero], assumption}
| (some a), (some b) :=
begin
simp only [some_eq_coe, coe_sub.symm, coe_pos, coe_eq_zero, coe_lt_coe, ne.def],
assume h₁ h₂, apply nnreal.sub_lt_self, exact zero_lt_iff_ne_zero.2 h₂
end
end
@[simp] lemma sub_eq_zero_iff_le : a - b = 0 ↔ a ≤ b :=
by simpa [-ennreal.sub_le_iff_le_add] using @ennreal.sub_le_iff_le_add a b 0
@[simp] lemma zero_lt_sub_iff_lt : 0 < a - b ↔ b < a :=
by simpa [ennreal.bot_lt_iff_ne_bot, -sub_eq_zero_iff_le]
using not_iff_not.2 (@sub_eq_zero_iff_le a b)
lemma lt_sub_iff_add_lt : a < b - c ↔ a + c < b :=
begin
cases a, { simp },
cases c, { simp },
cases b, { simp only [true_iff, coe_lt_top, some_eq_coe, top_sub_coe, none_eq_top, ← coe_add] },
simp only [some_eq_coe],
rw [← coe_add, ← coe_sub, coe_lt_coe, coe_lt_coe, nnreal.lt_sub_iff_add_lt],
end
lemma sub_le_self (a b : ennreal) : a - b ≤ a :=
ennreal.sub_le_iff_le_add.2 $ le_add_right (le_refl a)
@[simp] lemma sub_zero : a - 0 = a :=
eq.trans (add_zero (a - 0)).symm $ by simp
/-- A version of triangle inequality for difference as a "distance". -/
lemma sub_le_sub_add_sub : a - c ≤ a - b + (b - c) :=
ennreal.sub_le_iff_le_add.2 $
calc a ≤ a - b + b : le_sub_add_self
... ≤ a - b + ((b - c) + c) : add_le_add_left le_sub_add_self _
... = a - b + (b - c) + c : (add_assoc _ _ _).symm
lemma sub_sub_cancel (h : a < ∞) (h2 : b ≤ a) : a - (a - b) = b :=
by rw [← add_left_inj (lt_of_le_of_lt (sub_le_self _ _) h),
sub_add_cancel_of_le (sub_le_self _ _), add_sub_cancel_of_le h2]
lemma sub_right_inj {a b c : ennreal} (ha : a < ∞) (hb : b ≤ a) (hc : c ≤ a) :
a - b = a - c ↔ b = c :=
iff.intro
begin
assume h, have : a - (a - b) = a - (a - c), rw h,
rw [sub_sub_cancel ha hb, sub_sub_cancel ha hc] at this, exact this
end
(λ h, by rw h)
lemma sub_mul (h : 0 < b → b < a → c ≠ ∞) : (a - b) * c = a * c - b * c :=
begin
cases le_or_lt a b with hab hab,
{ simp [hab, mul_right_mono hab] },
symmetry,
cases eq_or_lt_of_le (zero_le b) with hb hb,
{ subst b, simp },
apply sub_eq_of_add_eq,
{ exact mul_ne_top (ne_top_of_lt hab) (h hb hab) },
rw [← add_mul, sub_add_cancel_of_le (le_of_lt hab)]
end
lemma mul_sub (h : 0 < c → c < b → a ≠ ∞) :
a * (b - c) = a * b - a * c :=
by { simp only [mul_comm a], exact sub_mul h }
lemma sub_mul_ge : a * c - b * c ≤ (a - b) * c :=
begin
-- with `0 < b → b < a → c ≠ ∞` Lean names the first variable `a`
by_cases h : ∀ (hb : 0 < b), b < a → c ≠ ∞,
{ rw [sub_mul h],
exact le_refl _ },
{ push_neg at h,
rcases h with ⟨hb, hba, hc⟩,
subst c,
simp only [mul_top, if_neg (ne_of_gt hb), if_neg (ne_of_gt $ lt_trans hb hba), sub_self,
zero_le] }
end
end sub
section sum
open finset
/-- A product of finite numbers is still finite -/
lemma prod_lt_top {s : finset α} {f : α → ennreal} (h : ∀a∈s, f a < ∞) : (∏ a in s, f a) < ∞ :=
with_top.prod_lt_top h
/-- A sum of finite numbers is still finite -/
lemma sum_lt_top {s : finset α} {f : α → ennreal} :
(∀a∈s, f a < ∞) → ∑ a in s, f a < ∞ :=
with_top.sum_lt_top
/-- A sum of finite numbers is still finite -/
lemma sum_lt_top_iff {s : finset α} {f : α → ennreal} :
∑ a in s, f a < ∞ ↔ (∀a∈s, f a < ∞) :=
with_top.sum_lt_top_iff
/-- A sum of numbers is infinite iff one of them is infinite -/
lemma sum_eq_top_iff {s : finset α} {f : α → ennreal} :
(∑ x in s, f x) = ∞ ↔ (∃a∈s, f a = ∞) :=
with_top.sum_eq_top_iff
/-- seeing `ennreal` as `ℝ≥0` does not change their sum, unless one of the `ennreal` is
infinity -/
lemma to_nnreal_sum {s : finset α} {f : α → ennreal} (hf : ∀a∈s, f a < ∞) :
ennreal.to_nnreal (∑ a in s, f a) = ∑ a in s, ennreal.to_nnreal (f a) :=
begin
rw [← coe_eq_coe, coe_to_nnreal, coe_finset_sum, sum_congr],
{ refl },
{ intros x hx, exact (coe_to_nnreal (hf x hx).ne).symm },
{ exact (sum_lt_top hf).ne }
end
/-- seeing `ennreal` as `real` does not change their sum, unless one of the `ennreal` is infinity -/
lemma to_real_sum {s : finset α} {f : α → ennreal} (hf : ∀a∈s, f a < ∞) :
ennreal.to_real (∑ a in s, f a) = ∑ a in s, ennreal.to_real (f a) :=
by { rw [ennreal.to_real, to_nnreal_sum hf, nnreal.coe_sum], refl }
end sum
section interval
variables {x y z : ennreal} {ε ε₁ ε₂ : ennreal} {s : set ennreal}
protected lemma Ico_eq_Iio : (Ico 0 y) = (Iio y) :=
ext $ assume a, iff.intro
(assume ⟨_, hx⟩, hx)
(assume hx, ⟨zero_le _, hx⟩)
lemma mem_Iio_self_add : x ≠ ∞ → 0 < ε → x ∈ Iio (x + ε) :=
assume xt ε0, lt_add_right (by rwa lt_top_iff_ne_top) ε0
lemma not_mem_Ioo_self_sub : x = 0 → x ∉ Ioo (x - ε) y :=
assume x0, by simp [x0]
lemma mem_Ioo_self_sub_add : x ≠ ∞ → x ≠ 0 → 0 < ε₁ → 0 < ε₂ → x ∈ Ioo (x - ε₁) (x + ε₂) :=
assume xt x0 ε0 ε0',
⟨ennreal.sub_lt_sub_self xt x0 ε0, lt_add_right (by rwa [lt_top_iff_ne_top]) ε0'⟩
end interval
section bit
@[simp] lemma bit0_inj : bit0 a = bit0 b ↔ a = b :=
⟨λh, begin
rcases (lt_trichotomy a b) with h₁| h₂| h₃,
{ exact (absurd h (ne_of_lt (add_lt_add h₁ h₁))) },
{ exact h₂ },
{ exact (absurd h.symm (ne_of_lt (add_lt_add h₃ h₃))) }
end,
λh, congr_arg _ h⟩
@[simp] lemma bit0_eq_zero_iff : bit0 a = 0 ↔ a = 0 :=
by simpa only [bit0_zero] using @bit0_inj a 0
@[simp] lemma bit0_eq_top_iff : bit0 a = ∞ ↔ a = ∞ :=
by rw [bit0, add_eq_top, or_self]
@[simp] lemma bit1_inj : bit1 a = bit1 b ↔ a = b :=
⟨λh, begin
unfold bit1 at h,
rwa [add_left_inj, bit0_inj] at h,
simp [lt_top_iff_ne_top]
end,
λh, congr_arg _ h⟩
@[simp] lemma bit1_ne_zero : bit1 a ≠ 0 :=
by unfold bit1; simp
@[simp] lemma bit1_eq_one_iff : bit1 a = 1 ↔ a = 0 :=
by simpa only [bit1_zero] using @bit1_inj a 0
@[simp] lemma bit1_eq_top_iff : bit1 a = ∞ ↔ a = ∞ :=
by unfold bit1; rw add_eq_top; simp
end bit
section inv
instance : has_inv ennreal := ⟨λa, Inf {b | 1 ≤ a * b}⟩
instance : has_div ennreal := ⟨λa b, a * b⁻¹⟩
lemma div_def : a / b = a * b⁻¹ := rfl
lemma mul_div_assoc : (a * b) / c = a * (b / c) :=
mul_assoc _ _ _
@[simp] lemma inv_zero : (0 : ennreal)⁻¹ = ∞ :=
show Inf {b : ennreal | 1 ≤ 0 * b} = ∞, by simp; refl
@[simp] lemma inv_top : ∞⁻¹ = 0 :=
bot_unique $ le_of_forall_le_of_dense $ λ a (h : a > 0), Inf_le $ by simp [*, ne_of_gt h, top_mul]
@[simp, norm_cast] lemma coe_inv (hr : r ≠ 0) : (↑r⁻¹ : ennreal) = (↑r)⁻¹ :=
le_antisymm
(le_Inf $ assume b (hb : 1 ≤ ↑r * b), coe_le_iff.2 $
by rintros b rfl; rwa [← coe_mul, ← coe_one, coe_le_coe, ← nnreal.inv_le hr] at hb)
(Inf_le $ by simp; rw [← coe_mul, nnreal.mul_inv_cancel hr]; exact le_refl 1)
lemma coe_inv_le : (↑r⁻¹ : ennreal) ≤ (↑r)⁻¹ :=
if hr : r = 0 then by simp only [hr, nnreal.inv_zero, inv_zero, coe_zero, zero_le]
else by simp only [coe_inv hr, le_refl]
@[norm_cast] lemma coe_inv_two : ((2⁻¹:ℝ≥0):ennreal) = 2⁻¹ :=
by rw [coe_inv (ne_of_gt _root_.zero_lt_two), coe_two]
@[simp, norm_cast] lemma coe_div (hr : r ≠ 0) : (↑(p / r) : ennreal) = p / r :=
show ↑(p * r⁻¹) = ↑p * (↑r)⁻¹, by rw [coe_mul, coe_inv hr]
@[simp] lemma inv_one : (1:ennreal)⁻¹ = 1 :=
by simpa only [coe_inv one_ne_zero, coe_one] using coe_eq_coe.2 nnreal.inv_one
@[simp] lemma div_one {a : ennreal} : a / 1 = a :=
by simp [ennreal.div_def]
protected lemma inv_pow {n : ℕ} : (a^n)⁻¹ = (a⁻¹)^n :=
begin
by_cases a = 0; cases a; cases n; simp [*, none_eq_top, some_eq_coe,
zero_pow, top_pow, nat.zero_lt_succ] at *,
rw [← coe_inv h, ← coe_pow, ← coe_inv, nnreal.inv_pow, coe_pow],
rw [← ne.def] at h,
rw [← zero_lt_iff_ne_zero] at *,
apply pow_pos h
end
@[simp] lemma inv_inv : (a⁻¹)⁻¹ = a :=
by by_cases a = 0; cases a; simp [*, none_eq_top, some_eq_coe,
-coe_inv, (coe_inv _).symm] at *
lemma inv_involutive : function.involutive (λ a:ennreal, a⁻¹) :=
λ a, ennreal.inv_inv
lemma inv_bijective : function.bijective (λ a:ennreal, a⁻¹) :=
ennreal.inv_involutive.bijective
@[simp] lemma inv_eq_inv : a⁻¹ = b⁻¹ ↔ a = b := inv_bijective.1.eq_iff
@[simp] lemma inv_eq_top : a⁻¹ = ∞ ↔ a = 0 :=
inv_zero ▸ inv_eq_inv
lemma inv_ne_top : a⁻¹ ≠ ∞ ↔ a ≠ 0 := by simp
@[simp] lemma inv_lt_top {x : ennreal} : x⁻¹ < ∞ ↔ 0 < x :=
by { simp only [lt_top_iff_ne_top, inv_ne_top, zero_lt_iff_ne_zero] }
lemma div_lt_top {x y : ennreal} (h1 : x < ∞) (h2 : 0 < y) : x / y < ∞ :=
mul_lt_top h1 (inv_lt_top.mpr h2)
@[simp] lemma inv_eq_zero : a⁻¹ = 0 ↔ a = ∞ :=
inv_top ▸ inv_eq_inv
lemma inv_ne_zero : a⁻¹ ≠ 0 ↔ a ≠ ∞ := by simp
@[simp] lemma inv_pos : 0 < a⁻¹ ↔ a ≠ ∞ :=
zero_lt_iff_ne_zero.trans inv_ne_zero
@[simp] lemma inv_lt_inv : a⁻¹ < b⁻¹ ↔ b < a :=
begin
cases a; cases b; simp only [some_eq_coe, none_eq_top, inv_top],
{ simp only [lt_irrefl] },
{ exact inv_pos.trans lt_top_iff_ne_top.symm },
{ simp only [not_lt_zero, not_top_lt] },
{ cases eq_or_lt_of_le (zero_le a) with ha ha;
cases eq_or_lt_of_le (zero_le b) with hb hb,
{ subst a, subst b, simp },
{ subst a, simp },
{ subst b, simp [zero_lt_iff_ne_zero, lt_top_iff_ne_top, inv_ne_top] },
{ rw [← coe_inv (ne_of_gt ha), ← coe_inv (ne_of_gt hb), coe_lt_coe, coe_lt_coe],
simp only [nnreal.coe_lt_coe.symm] at *,
exact inv_lt_inv ha hb } }
end
lemma inv_lt_iff_inv_lt : a⁻¹ < b ↔ b⁻¹ < a :=
by simpa only [inv_inv] using @inv_lt_inv a b⁻¹
lemma lt_inv_iff_lt_inv : a < b⁻¹ ↔ b < a⁻¹ :=
by simpa only [inv_inv] using @inv_lt_inv a⁻¹ b
@[simp, priority 1100] -- higher than le_inv_iff_mul_le
lemma inv_le_inv : a⁻¹ ≤ b⁻¹ ↔ b ≤ a :=
by simp only [le_iff_lt_or_eq, inv_lt_inv, inv_eq_inv, eq_comm]
lemma inv_le_iff_inv_le : a⁻¹ ≤ b ↔ b⁻¹ ≤ a :=
by simpa only [inv_inv] using @inv_le_inv a b⁻¹
lemma le_inv_iff_le_inv : a ≤ b⁻¹ ↔ b ≤ a⁻¹ :=
by simpa only [inv_inv] using @inv_le_inv a⁻¹ b
@[simp] lemma inv_lt_one : a⁻¹ < 1 ↔ 1 < a :=
inv_lt_iff_inv_lt.trans $ by rw [inv_one]
@[simp] lemma div_top : a / ∞ = 0 := by simp only [div_def, inv_top, mul_zero]
@[simp] lemma top_div_coe : ∞ / p = ∞ := by simp [div_def, top_mul]
lemma top_div_of_ne_top (h : a ≠ ∞) : ∞ / a = ∞ :=
by { lift a to ℝ≥0 using h, exact top_div_coe }
lemma top_div_of_lt_top (h : a < ∞) : ∞ / a = ∞ :=
top_div_of_ne_top h.ne
lemma top_div : ∞ / a = if a = ∞ then 0 else ∞ :=
by by_cases a = ∞; simp [top_div_of_ne_top, *]
@[simp] lemma zero_div : 0 / a = 0 := zero_mul a⁻¹
lemma div_eq_top : a / b = ∞ ↔ (a ≠ 0 ∧ b = 0) ∨ (a = ∞ ∧ b ≠ ∞) :=
by simp [ennreal.div_def, ennreal.mul_eq_top]
lemma le_div_iff_mul_le (h0 : b ≠ 0 ∨ c ≠ 0) (ht : b ≠ ∞ ∨ c ≠ ∞) :
a ≤ c / b ↔ a * b ≤ c :=
begin
cases b,
{ simp at ht,
split,
{ assume ha, simp at ha, simp [ha] },
{ contrapose,
assume ha,
simp at ha,
have : a * ∞ = ∞, by simp [ennreal.mul_eq_top, ha],
simp [this, ht] } },
by_cases hb : b ≠ 0,
{ have : (b : ennreal) ≠ 0, by simp [hb],
rw [← ennreal.mul_le_mul_left this coe_ne_top],
suffices : ↑b * a ≤ (↑b * ↑b⁻¹) * c ↔ a * ↑b ≤ c,
{ simpa [some_eq_coe, div_def, hb, mul_left_comm, mul_comm, mul_assoc] },
rw [← coe_mul, nnreal.mul_inv_cancel hb, coe_one, one_mul, mul_comm] },
{ simp at hb,
simp [hb] at h0,
have : c / 0 = ∞, by simp [div_eq_top, h0],
simp [hb, this] }
end
lemma div_le_iff_le_mul (hb0 : b ≠ 0 ∨ c ≠ ∞) (hbt : b ≠ ∞ ∨ c ≠ 0) : a / b ≤ c ↔ a ≤ c * b :=
begin
suffices : a * b⁻¹ ≤ c ↔ a ≤ c / b⁻¹, by simpa [div_def],
apply (le_div_iff_mul_le _ _).symm,
simpa [inv_ne_zero] using hbt,
simpa [inv_ne_zero] using hb0
end
lemma div_le_of_le_mul (h : a ≤ b * c) : a / c ≤ b :=
begin
by_cases h0 : c = 0,
{ have : a = 0, by simpa [h0] using h, simp [*] },
by_cases hinf : c = ∞, by simp [hinf],
exact (div_le_iff_le_mul (or.inl h0) (or.inl hinf)).2 h
end
protected lemma div_lt_iff (h0 : b ≠ 0 ∨ c ≠ 0) (ht : b ≠ ∞ ∨ c ≠ ∞) :
c / b < a ↔ c < a * b :=
lt_iff_lt_of_le_iff_le $ le_div_iff_mul_le h0 ht
lemma mul_lt_of_lt_div (h : a < b / c) : a * c < b :=
by { contrapose! h, exact ennreal.div_le_of_le_mul h }
lemma inv_le_iff_le_mul : (b = ∞ → a ≠ 0) → (a = ∞ → b ≠ 0) → (a⁻¹ ≤ b ↔ 1 ≤ a * b) :=
begin
cases a; cases b; simp [none_eq_top, some_eq_coe, mul_top, top_mul] {contextual := tt},
by_cases a = 0; simp [*, -coe_mul, coe_mul.symm, -coe_inv, (coe_inv _).symm, nnreal.inv_le]
end
@[simp] lemma le_inv_iff_mul_le : a ≤ b⁻¹ ↔ a * b ≤ 1 :=
begin
cases b, { by_cases a = 0; simp [*, none_eq_top, mul_top] },
by_cases b = 0; simp [*, some_eq_coe, le_div_iff_mul_le],
suffices : a ≤ 1 / b ↔ a * b ≤ 1, { simpa [div_def, h] },
exact le_div_iff_mul_le (or.inl (mt coe_eq_coe.1 h)) (or.inl coe_ne_top)
end
lemma mul_inv_cancel (h0 : a ≠ 0) (ht : a ≠ ∞) : a * a⁻¹ = 1 :=
begin
lift a to ℝ≥0 using ht,
norm_cast at h0,
norm_cast,
exact nnreal.mul_inv_cancel h0
end
lemma inv_mul_cancel (h0 : a ≠ 0) (ht : a ≠ ∞) : a⁻¹ * a = 1 :=
mul_comm a a⁻¹ ▸ mul_inv_cancel h0 ht
lemma mul_le_iff_le_inv {a b r : ennreal} (hr₀ : r ≠ 0) (hr₁ : r ≠ ∞) : (r * a ≤ b ↔ a ≤ r⁻¹ * b) :=
by rw [← @ennreal.mul_le_mul_left _ a _ hr₀ hr₁, ← mul_assoc, mul_inv_cancel hr₀ hr₁, one_mul]
lemma le_of_forall_nnreal_lt {x y : ennreal} (h : ∀ r : ℝ≥0, ↑r < x → ↑r ≤ y) : x ≤ y :=
begin
refine le_of_forall_ge_of_dense (λ r hr, _),
lift r to ℝ≥0 using ne_top_of_lt hr,
exact h r hr
end
lemma div_add_div_same {a b c : ennreal} : a / c + b / c = (a + b) / c :=
eq.symm $ right_distrib a b (c⁻¹)
lemma div_self (h0 : a ≠ 0) (hI : a ≠ ∞) : a / a = 1 :=
mul_inv_cancel h0 hI
lemma mul_div_cancel (h0 : a ≠ 0) (hI : a ≠ ∞) : (b / a) * a = b :=
by rw [div_def, mul_assoc, inv_mul_cancel h0 hI, mul_one]
lemma mul_div_cancel' (h0 : a ≠ 0) (hI : a ≠ ∞) : a * (b / a) = b :=
by rw [mul_comm, mul_div_cancel h0 hI]
lemma inv_two_add_inv_two : (2:ennreal)⁻¹ + 2⁻¹ = 1 :=
by rw [← two_mul, ← div_def, div_self two_ne_zero two_ne_top]
lemma add_halves (a : ennreal) : a / 2 + a / 2 = a :=
by rw [div_def, ← mul_add, inv_two_add_inv_two, mul_one]
@[simp] lemma div_zero_iff : a / b = 0 ↔ a = 0 ∨ b = ∞ :=
by simp [div_def, mul_eq_zero]
@[simp] lemma div_pos_iff : 0 < a / b ↔ a ≠ 0 ∧ b ≠ ∞ :=
by simp [zero_lt_iff_ne_zero, not_or_distrib]
lemma half_pos {a : ennreal} (h : 0 < a) : 0 < a / 2 :=
by simp [ne_of_gt h]
lemma one_half_lt_one : (2⁻¹:ennreal) < 1 := inv_lt_one.2 $ one_lt_two
lemma half_lt_self {a : ennreal} (hz : a ≠ 0) (ht : a ≠ ∞) : a / 2 < a :=
begin
lift a to ℝ≥0 using ht,
have h : (2 : ennreal) = ((2 : ℝ≥0) : ennreal), from rfl,
have h' : (2 : ℝ≥0) ≠ 0, from _root_.two_ne_zero',
rw [h, ← coe_div h', coe_lt_coe], -- `norm_cast` fails to apply `coe_div`
norm_cast at hz,
exact nnreal.half_lt_self hz
end
lemma sub_half (h : a ≠ ∞) : a - a / 2 = a / 2 :=
begin
lift a to ℝ≥0 using h,
exact sub_eq_of_add_eq (mul_ne_top coe_ne_top $ by simp) (add_halves a)
end
lemma one_sub_inv_two : (1:ennreal) - 2⁻¹ = 2⁻¹ :=
by simpa only [div_def, one_mul] using sub_half one_ne_top
lemma exists_inv_nat_lt {a : ennreal} (h : a ≠ 0) :
∃n:ℕ, (n:ennreal)⁻¹ < a :=
@inv_inv a ▸ by simp only [inv_lt_inv, ennreal.exists_nat_gt (inv_ne_top.2 h)]
lemma exists_nat_pos_mul_gt (ha : a ≠ 0) (hb : b ≠ ∞) :
∃ n > 0, b < (n : ℕ) * a :=
begin
have : b / a ≠ ∞, from mul_ne_top hb (inv_ne_top.2 ha),
refine (ennreal.exists_nat_gt this).imp (λ n hn, _),
have : 0 < (n : ennreal), from (zero_le _).trans_lt hn,
refine ⟨coe_nat_lt_coe_nat.1 this, _⟩,
rwa [← ennreal.div_lt_iff (or.inl ha) (or.inr hb)]
end
lemma exists_nat_mul_gt (ha : a ≠ 0) (hb : b ≠ ∞) :
∃ n : ℕ, b < n * a :=
(exists_nat_pos_mul_gt ha hb).imp $ λ n, Exists.snd
lemma exists_nat_pos_inv_mul_lt (ha : a ≠ ∞) (hb : b ≠ 0) :
∃ n > 0, ((n : ℕ) : ennreal)⁻¹ * a < b :=
begin
rcases exists_nat_pos_mul_gt hb ha with ⟨n, npos, hn⟩,
have : (n : ennreal) ≠ 0 := nat.cast_ne_zero.2 npos.lt.ne',
use [n, npos],
rwa [← one_mul b, ← inv_mul_cancel this coe_nat_ne_top,
mul_assoc, mul_lt_mul_left (inv_ne_zero.2 coe_nat_ne_top) (inv_ne_top.2 this)]
end
lemma exists_nnreal_pos_mul_lt (ha : a ≠ ∞) (hb : b ≠ 0) :
∃ n > 0, ↑(n : ℝ≥0) * a < b :=
begin
rcases exists_nat_pos_inv_mul_lt ha hb with ⟨n, npos : 0 < n, hn⟩,
use (n : ℝ≥0)⁻¹,
simp [*, npos.ne', zero_lt_one]
end
end inv
section real
lemma to_real_add (ha : a ≠ ∞) (hb : b ≠ ∞) : (a+b).to_real = a.to_real + b.to_real :=
begin
lift a to ℝ≥0 using ha,
lift b to ℝ≥0 using hb,
refl
end
lemma to_real_add_le : (a+b).to_real ≤ a.to_real + b.to_real :=
if ha : a = ∞ then by simp only [ha, top_add, top_to_real, zero_add, to_real_nonneg]
else if hb : b = ∞ then by simp only [hb, add_top, top_to_real, add_zero, to_real_nonneg]
else le_of_eq (to_real_add ha hb)
lemma of_real_add {p q : ℝ} (hp : 0 ≤ p) (hq : 0 ≤ q) :
ennreal.of_real (p + q) = ennreal.of_real p + ennreal.of_real q :=
by rw [ennreal.of_real, ennreal.of_real, ennreal.of_real, ← coe_add,
coe_eq_coe, nnreal.of_real_add hp hq]
lemma of_real_add_le {p q : ℝ} : ennreal.of_real (p + q) ≤ ennreal.of_real p + ennreal.of_real q :=
coe_le_coe.2 nnreal.of_real_add_le
@[simp] lemma to_real_le_to_real (ha : a ≠ ∞) (hb : b ≠ ∞) : a.to_real ≤ b.to_real ↔ a ≤ b :=
begin
lift a to ℝ≥0 using ha,
lift b to ℝ≥0 using hb,
norm_cast
end
@[simp] lemma to_real_lt_to_real (ha : a ≠ ∞) (hb : b ≠ ∞) : a.to_real < b.to_real ↔ a < b :=
begin
lift a to ℝ≥0 using ha,
lift b to ℝ≥0 using hb,
norm_cast
end
lemma to_real_max (hr : a ≠ ∞) (hp : b ≠ ∞) :
ennreal.to_real (max a b) = max (ennreal.to_real a) (ennreal.to_real b) :=
(le_total a b).elim
(λ h, by simp only [h, (ennreal.to_real_le_to_real hr hp).2 h, max_eq_right])
(λ h, by simp only [h, (ennreal.to_real_le_to_real hp hr).2 h, max_eq_left])
lemma to_nnreal_pos_iff : 0 < a.to_nnreal ↔ (0 < a ∧ a ≠ ∞) :=
begin
cases a,
{ simp [none_eq_top] },
{ simp [some_eq_coe] }
end
lemma to_real_pos_iff : 0 < a.to_real ↔ (0 < a ∧ a ≠ ∞):=
(nnreal.coe_pos).trans to_nnreal_pos_iff
lemma of_real_le_of_real {p q : ℝ} (h : p ≤ q) : ennreal.of_real p ≤ ennreal.of_real q :=
by simp [ennreal.of_real, nnreal.of_real_le_of_real h]
lemma of_real_le_of_le_to_real {a : ℝ} {b : ennreal} (h : a ≤ ennreal.to_real b) :
ennreal.of_real a ≤ b :=
(of_real_le_of_real h).trans of_real_to_real_le
@[simp] lemma of_real_le_of_real_iff {p q : ℝ} (h : 0 ≤ q) :
ennreal.of_real p ≤ ennreal.of_real q ↔ p ≤ q :=
by rw [ennreal.of_real, ennreal.of_real, coe_le_coe, nnreal.of_real_le_of_real_iff h]
@[simp] lemma of_real_lt_of_real_iff {p q : ℝ} (h : 0 < q) :
ennreal.of_real p < ennreal.of_real q ↔ p < q :=
by rw [ennreal.of_real, ennreal.of_real, coe_lt_coe, nnreal.of_real_lt_of_real_iff h]
lemma of_real_lt_of_real_iff_of_nonneg {p q : ℝ} (hp : 0 ≤ p) :
ennreal.of_real p < ennreal.of_real q ↔ p < q :=
by rw [ennreal.of_real, ennreal.of_real, coe_lt_coe, nnreal.of_real_lt_of_real_iff_of_nonneg hp]
@[simp] lemma of_real_pos {p : ℝ} : 0 < ennreal.of_real p ↔ 0 < p :=
by simp [ennreal.of_real]
@[simp] lemma of_real_eq_zero {p : ℝ} : ennreal.of_real p = 0 ↔ p ≤ 0 :=
by simp [ennreal.of_real]
lemma of_real_le_iff_le_to_real {a : ℝ} {b : ennreal} (hb : b ≠ ∞) :
ennreal.of_real a ≤ b ↔ a ≤ ennreal.to_real b :=
begin
lift b to ℝ≥0 using hb,
simpa [ennreal.of_real, ennreal.to_real] using nnreal.of_real_le_iff_le_coe
end
lemma of_real_lt_iff_lt_to_real {a : ℝ} {b : ennreal} (ha : 0 ≤ a) (hb : b ≠ ∞) :
ennreal.of_real a < b ↔ a < ennreal.to_real b :=
begin
lift b to ℝ≥0 using hb,
simpa [ennreal.of_real, ennreal.to_real] using nnreal.of_real_lt_iff_lt_coe ha
end
lemma le_of_real_iff_to_real_le {a : ennreal} {b : ℝ} (ha : a ≠ ∞) (hb : 0 ≤ b) :
a ≤ ennreal.of_real b ↔ ennreal.to_real a ≤ b :=
begin
lift a to ℝ≥0 using ha,
simpa [ennreal.of_real, ennreal.to_real] using nnreal.le_of_real_iff_coe_le hb
end
lemma to_real_le_of_le_of_real {a : ennreal} {b : ℝ} (hb : 0 ≤ b) (h : a ≤ ennreal.of_real b) :
ennreal.to_real a ≤ b :=
have ha : a ≠ ∞, from ne_top_of_le_ne_top of_real_ne_top h,
(le_of_real_iff_to_real_le ha hb).1 h
lemma lt_of_real_iff_to_real_lt {a : ennreal} {b : ℝ} (ha : a ≠ ∞) :
a < ennreal.of_real b ↔ ennreal.to_real a < b :=
begin
lift a to ℝ≥0 using ha,
simpa [ennreal.of_real, ennreal.to_real] using nnreal.lt_of_real_iff_coe_lt
end
lemma of_real_mul {p q : ℝ} (hp : 0 ≤ p) :
ennreal.of_real (p * q) = (ennreal.of_real p) * (ennreal.of_real q) :=
by { simp only [ennreal.of_real, coe_mul.symm, coe_eq_coe], exact nnreal.of_real_mul hp }
lemma of_real_inv_of_pos {x : ℝ} (hx : 0 < x) :
(ennreal.of_real x)⁻¹ = ennreal.of_real x⁻¹ :=
by rw [ennreal.of_real, ennreal.of_real, ←@coe_inv (nnreal.of_real x) (by simp [hx]), coe_eq_coe,
nnreal.of_real_inv.symm]
lemma of_real_div_of_pos {x y : ℝ} (hy : 0 < y) :
ennreal.of_real (x / y) = ennreal.of_real x / ennreal.of_real y :=
by rw [div_def, of_real_inv_of_pos hy, mul_comm, ←of_real_mul (inv_nonneg.mpr (le_of_lt hy)),
div_eq_mul_inv, mul_comm]
lemma to_real_of_real_mul (c : ℝ) (a : ennreal) (h : 0 ≤ c) :
ennreal.to_real ((ennreal.of_real c) * a) = c * ennreal.to_real a :=
begin
cases a,
{ simp only [none_eq_top, ennreal.to_real, top_to_nnreal, nnreal.coe_zero, mul_zero, mul_top],
by_cases h' : c ≤ 0,
{ rw [if_pos], { simp }, { convert of_real_zero, exact le_antisymm h' h } },
{ rw [if_neg], refl, rw [of_real_eq_zero], assumption } },
{ simp only [ennreal.to_real, ennreal.to_nnreal],
simp only [some_eq_coe, ennreal.of_real, coe_mul.symm, to_nnreal_coe, nnreal.coe_mul],
congr, apply nnreal.coe_of_real, exact h }
end
@[simp] lemma to_nnreal_mul_top (a : ennreal) : ennreal.to_nnreal (a * ∞) = 0 :=
begin
by_cases h : a = 0,
{ rw [h, zero_mul, zero_to_nnreal] },
{ rw [mul_top, if_neg h, top_to_nnreal] }
end
@[simp] lemma to_nnreal_top_mul (a : ennreal) : ennreal.to_nnreal (∞ * a) = 0 :=
by rw [mul_comm, to_nnreal_mul_top]
@[simp] lemma to_real_mul_top (a : ennreal) : ennreal.to_real (a * ∞) = 0 :=
by rw [ennreal.to_real, to_nnreal_mul_top, nnreal.coe_zero]
@[simp] lemma to_real_top_mul (a : ennreal) : ennreal.to_real (∞ * a) = 0 :=
by { rw mul_comm, exact to_real_mul_top _ }
lemma to_real_eq_to_real (ha : a < ∞) (hb : b < ∞) :
ennreal.to_real a = ennreal.to_real b ↔ a = b :=
begin
lift a to ℝ≥0 using ha.ne,
lift b to ℝ≥0 using hb.ne,
simp only [coe_eq_coe, nnreal.coe_eq, coe_to_real],
end
/-- `ennreal.to_nnreal` as a `monoid_hom`. -/
def to_nnreal_hom : ennreal →* ℝ≥0 :=
{ to_fun := ennreal.to_nnreal,
map_one' := to_nnreal_coe,
map_mul' := by rintro (_|x) (_|y); simp only [← coe_mul, none_eq_top, some_eq_coe,
to_nnreal_top_mul, to_nnreal_mul_top, top_to_nnreal, mul_zero, zero_mul, to_nnreal_coe] }
lemma to_nnreal_mul {a b : ennreal}: (a * b).to_nnreal = a.to_nnreal * b.to_nnreal :=
to_nnreal_hom.map_mul a b
lemma to_nnreal_pow (a : ennreal) (n : ℕ) : (a ^ n).to_nnreal = a.to_nnreal ^ n :=
to_nnreal_hom.map_pow a n
lemma to_nnreal_prod {ι : Type*} {s : finset ι} {f : ι → ennreal} :
(∏ i in s, f i).to_nnreal = ∏ i in s, (f i).to_nnreal :=
to_nnreal_hom.map_prod _ _
/-- `ennreal.to_real` as a `monoid_hom`. -/
def to_real_hom : ennreal →* ℝ :=
(nnreal.to_real_hom : ℝ≥0 →* ℝ).comp to_nnreal_hom
lemma to_real_mul : (a * b).to_real = a.to_real * b.to_real :=
to_real_hom.map_mul a b
lemma to_real_pow (a : ennreal) (n : ℕ) : (a ^ n).to_real = a.to_real ^ n :=
to_real_hom.map_pow a n
lemma to_real_prod {ι : Type*} {s : finset ι} {f : ι → ennreal} :
(∏ i in s, f i).to_real = ∏ i in s, (f i).to_real :=
to_real_hom.map_prod _ _
end real
section infi
variables {ι : Sort*} {f g : ι → ennreal}
lemma infi_add : infi f + a = ⨅i, f i + a :=
le_antisymm
(le_infi $ assume i, add_le_add (infi_le _ _) $ le_refl _)
(ennreal.sub_le_iff_le_add.1 $ le_infi $ assume i, ennreal.sub_le_iff_le_add.2 $ infi_le _ _)
lemma supr_sub : (⨆i, f i) - a = (⨆i, f i - a) :=
le_antisymm
(ennreal.sub_le_iff_le_add.2 $ supr_le $ assume i, ennreal.sub_le_iff_le_add.1 $ le_supr _ i)
(supr_le $ assume i, ennreal.sub_le_sub (le_supr _ _) (le_refl a))
lemma sub_infi : a - (⨅i, f i) = (⨆i, a - f i) :=
begin
refine (eq_of_forall_ge_iff $ λ c, _),
rw [ennreal.sub_le_iff_le_add, add_comm, infi_add],
simp [ennreal.sub_le_iff_le_add, sub_eq_add_neg, add_comm],
end
lemma Inf_add {s : set ennreal} : Inf s + a = ⨅b∈s, b + a :=
by simp [Inf_eq_infi, infi_add]
lemma add_infi {a : ennreal} : a + infi f = ⨅b, a + f b :=
by rw [add_comm, infi_add]; simp [add_comm]
lemma infi_add_infi (h : ∀i j, ∃k, f k + g k ≤ f i + g j) : infi f + infi g = (⨅a, f a + g a) :=
suffices (⨅a, f a + g a) ≤ infi f + infi g,
from le_antisymm (le_infi $ assume a, add_le_add (infi_le _ _) (infi_le _ _)) this,
calc (⨅a, f a + g a) ≤ (⨅ a a', f a + g a') :
le_infi $ assume a, le_infi $ assume a',
let ⟨k, h⟩ := h a a' in infi_le_of_le k h
... ≤ infi f + infi g :
by simp [add_infi, infi_add, -add_comm, -le_infi_iff]; exact le_refl _
lemma infi_sum {f : ι → α → ennreal} {s : finset α} [nonempty ι]
(h : ∀(t : finset α) (i j : ι), ∃k, ∀a∈t, f k a ≤ f i a ∧ f k a ≤ f j a) :
(⨅i, ∑ a in s, f i a) = ∑ a in s, ⨅i, f i a :=
finset.induction_on s (by simp) $ assume a s ha ih,
have ∀ (i j : ι), ∃ (k : ι), f k a + ∑ b in s, f k b ≤ f i a + ∑ b in s, f j b,
from assume i j,
let ⟨k, hk⟩ := h (insert a s) i j in
⟨k, add_le_add (hk a (finset.mem_insert_self _ _)).left $ finset.sum_le_sum $
assume a ha, (hk _ $ finset.mem_insert_of_mem ha).right⟩,
by simp [ha, ih.symm, infi_add_infi this]
lemma infi_mul {ι} [nonempty ι] {f : ι → ennreal} {x : ennreal} (h : x ≠ ∞) :
infi f * x = ⨅i, f i * x :=
begin
by_cases h2 : x = 0, simp only [h2, mul_zero, infi_const],
refine le_antisymm
(le_infi $ λ i, mul_right_mono $ infi_le _ _)
((div_le_iff_le_mul (or.inl h2) $ or.inl h).mp $ le_infi $
λ i, (div_le_iff_le_mul (or.inl h2) $ or.inl h).mpr $ infi_le _ _)
end
lemma mul_infi {ι} [nonempty ι] {f : ι → ennreal} {x : ennreal} (h : x ≠ ∞) :
x * infi f = ⨅i, x * f i :=
by { rw [mul_comm, infi_mul h], simp only [mul_comm], assumption }
/-! `supr_mul`, `mul_supr` and variants are in `topology.instances.ennreal`. -/
end infi
section supr
lemma supr_coe_nat : (⨆n:ℕ, (n : ennreal)) = ∞ :=
(supr_eq_top _).2 $ assume b hb, ennreal.exists_nat_gt (lt_top_iff_ne_top.1 hb)
end supr
/-- `le_of_add_le_add_left` is normally applicable to `ordered_cancel_add_comm_monoid`,
but it holds in `ennreal` with the additional assumption that `a < ∞`. -/
lemma le_of_add_le_add_left {a b c : ennreal} : a < ∞ →
a + b ≤ a + c → b ≤ c :=
by cases a; cases b; cases c; simp [← ennreal.coe_add, ennreal.coe_le_coe]
end ennreal
|
812e81c13d22e3f193eca85adf6685d08c4ed1c8 | 77c5b91fae1b966ddd1db969ba37b6f0e4901e88 | /src/tactic/lint/misc.lean | 80ec3ca59662eb2dbc9c283f2d7f9f70c529029b | [
"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 | 16,417 | lean | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Robert Y. Lewis
-/
import tactic.lint.basic
/-!
# Various linters
This file defines several small linters:
- `ge_or_gt` checks that `>` and `≥` do not occur in the statement of theorems.
- `dup_namespace` checks that no declaration has a duplicated namespace such as `list.list.monad`.
- `unused_arguments` checks that definitions and theorems do not have unused arguments.
- `doc_blame` checks that every definition has a documentation string.
- `doc_blame_thm` checks that every theorem has a documentation string (not enabled by default).
- `def_lemma` checks that a declaration is a lemma iff its type is a proposition.
- `check_type` checks that the statement of a declaration is well-typed.
- `check_univs` checks that there are no bad `max u v` universe levels.
- `syn_taut` checks that declarations are not syntactic tautologies.
-/
open tactic expr
/-!
## Linter against use of `>`/`≥`
-/
/-- The names of `≥` and `>`, mostly disallowed in lemma statements -/
private meta def illegal_ge_gt : list name := [`gt, `ge]
set_option eqn_compiler.max_steps 20000
/--
Checks whether `≥` and `>` occurs in an illegal way in the expression.
The main ways we legally use these orderings are:
- `f (≥)`
- `∃ x ≥ t, b`. This corresponds to the expression
`@Exists α (fun (x : α), (@Exists (x > t) (λ (H : x > t), b)))`
This function returns `tt` when it finds `ge`/`gt`, except in the following patterns
(which are the same for `gt`):
- `f (@ge _ _)`
- `f (&0 ≥ y) (λ x : t, b)`
- `λ H : &0 ≥ t, b`
Here `&0` is the 0-th de Bruijn variable.
-/
private meta def contains_illegal_ge_gt : expr → bool
| (const nm us) := if nm ∈ illegal_ge_gt then tt else ff
| (app f e@(app (app (const nm us) tp) tc)) :=
contains_illegal_ge_gt f || if nm ∈ illegal_ge_gt then ff else contains_illegal_ge_gt e
| (app (app custom_binder (app (app (app (app (const nm us) tp) tc) (var 0)) t))
e@(lam var_name bi var_type body)) :=
contains_illegal_ge_gt e || if nm ∈ illegal_ge_gt then ff else contains_illegal_ge_gt e
| (app f x) := contains_illegal_ge_gt f || contains_illegal_ge_gt x
| (lam `H bi type@(app (app (app (app (const nm us) tp) tc) (var 0)) t) body) :=
contains_illegal_ge_gt body || if nm ∈ illegal_ge_gt then ff else contains_illegal_ge_gt type
| (lam var_name bi var_type body) := contains_illegal_ge_gt var_type || contains_illegal_ge_gt body
| (pi `H bi type@(app (app (app (app (const nm us) tp) tc) (var 0)) t) body) :=
contains_illegal_ge_gt body || if nm ∈ illegal_ge_gt then ff else contains_illegal_ge_gt type
| (pi var_name bi var_type body) := contains_illegal_ge_gt var_type || contains_illegal_ge_gt body
| (elet var_name type assignment body) :=
contains_illegal_ge_gt type || contains_illegal_ge_gt assignment || contains_illegal_ge_gt body
| _ := ff
/-- Checks whether a `>`/`≥` is used in the statement of `d`.
It first does a quick check to see if there is any `≥` or `>` in the statement, and then does a
slower check whether the occurrences of `≥` and `>` are allowed.
Currently it checks only the conclusion of the declaration, to eliminate false positive from
binders such as `∀ ε > 0, ...` -/
private meta def ge_or_gt_in_statement (d : declaration) : tactic (option string) :=
return $ if d.type.contains_constant (λ n, n ∈ illegal_ge_gt) &&
contains_illegal_ge_gt d.type
then some "the type contains ≥/>. Use ≤/< instead."
else none
-- TODO: the commented out code also checks for classicality in statements, but needs fixing
-- TODO: this probably needs to also check whether the argument is a variable or @eq <var> _ _
-- meta def illegal_constants_in_statement (d : declaration) : tactic (option string) :=
-- return $ if d.type.contains_constant (λ n, (n.get_prefix = `classical ∧
-- n.last ∈ ["prop_decidable", "dec", "dec_rel", "dec_eq"]) ∨ n ∈ [`gt, `ge])
-- then
-- let illegal1 := [`classical.prop_decidable, `classical.dec, `classical.dec_rel,
-- `classical.dec_eq],
-- illegal2 := [`gt, `ge],
-- occur1 := illegal1.filter (λ n, d.type.contains_constant (eq n)),
-- occur2 := illegal2.filter (λ n, d.type.contains_constant (eq n)) in
-- some $ sformat!"the type contains the following declarations: {occur1 ++ occur2}." ++
-- (if occur1 = [] then "" else " Add decidability type-class arguments instead.") ++
-- (if occur2 = [] then "" else " Use ≤/< instead.")
-- else none
/-- A linter for checking whether illegal constants (≥, >) appear in a declaration's type. -/
@[linter] meta def linter.ge_or_gt : linter :=
{ test := ge_or_gt_in_statement,
auto_decls := ff,
no_errors_found := "Not using ≥/> in declarations.",
errors_found := "The following declarations use ≥/>, probably in a way where we would prefer
to use ≤/< instead. See note [nolint_ge] for more information.",
is_fast := ff }
/--
Currently, the linter forbids the use of `>` and `≥` in definitions and
statements, as they cause problems in rewrites.
They are still allowed in statements such as `bounded (≥)` or `∀ ε > 0` or `⨆ n ≥ m`,
and the linter allows that.
If you write a pattern where you bind two or more variables, like `∃ n m > 0`, the linter will
flag this as illegal, but it is also allowed. In this case, add the line
```
@[nolint ge_or_gt] -- see Note [nolint_ge]
```
-/
library_note "nolint_ge"
/-!
## Linter for duplicate namespaces
-/
/-- Checks whether a declaration has a namespace twice consecutively in its name -/
private meta def dup_namespace (d : declaration) : tactic (option string) :=
is_instance d.to_name >>= λ is_inst,
return $ let nm := d.to_name.components in if nm.chain' (≠) ∨ is_inst then none
else let s := (nm.find $ λ n, nm.count n ≥ 2).iget.to_string in
some $ "The namespace `" ++ s ++ "` is duplicated in the name"
/-- A linter for checking whether a declaration has a namespace twice consecutively in its name. -/
@[linter] meta def linter.dup_namespace : linter :=
{ test := dup_namespace,
auto_decls := ff,
no_errors_found := "No declarations have a duplicate namespace.",
errors_found := "DUPLICATED NAMESPACES IN NAME:" }
/-!
## Linter for unused arguments
-/
/-- Auxiliary definition for `check_unused_arguments` -/
private meta def check_unused_arguments_aux : list ℕ → ℕ → ℕ → expr → list ℕ | l n n_max e :=
if n > n_max then l else
if ¬ is_lambda e ∧ ¬ is_pi e then l else
let b := e.binding_body in
let l' := if b.has_var_idx 0 then l else n :: l in check_unused_arguments_aux l' (n+1) n_max b
/-- Check which arguments of a declaration are not used.
Prints a list of natural numbers corresponding to which arguments are not used (e.g.
this outputs [1, 4] if the first and fourth arguments are unused).
Checks both the type and the value of `d` for whether the argument is used
(in rare cases an argument is used in the type but not in the value).
We return [] if the declaration was automatically generated.
We print arguments that are larger than the arity of the type of the declaration
(without unfolding definitions). -/
meta def check_unused_arguments (d : declaration) : option (list ℕ) :=
let l := check_unused_arguments_aux [] 1 d.type.pi_arity d.value in
if l = [] then none else
let l2 := check_unused_arguments_aux [] 1 d.type.pi_arity d.type in
(l.filter $ λ n, n ∈ l2).reverse
/-- Check for unused arguments, and print them with their position, variable name, type and whether
the argument is a duplicate.
See also `check_unused_arguments`.
This tactic additionally filters out all unused arguments of type `parse _`. -/
private meta def unused_arguments (d : declaration) : tactic (option string) := do
let ns := check_unused_arguments d,
if ¬ ns.is_some then return none else do
let ns := ns.iget,
(ds, _) ← get_pi_binders d.type,
let ns := ns.map (λ n, (n, (ds.nth $ n - 1).iget)),
let ns := ns.filter (λ x, x.2.type.get_app_fn ≠ const `interactive.parse []),
if ns = [] then return none else do
ds' ← ds.mmap pp,
ns ← ns.mmap (λ ⟨n, b⟩, (λ s, to_fmt "argument " ++ to_fmt n ++ ": " ++ s ++
(if ds.countp (λ b', b.type = b'.type) ≥ 2 then " (duplicate)" else "")) <$> pp b),
return $ some $ ns.to_string_aux tt
/-- A linter object for checking for unused arguments. This is in the default linter set. -/
@[linter] meta def linter.unused_arguments : linter :=
{ test := unused_arguments,
auto_decls := ff,
no_errors_found := "No unused arguments.",
errors_found := "UNUSED ARGUMENTS." }
attribute [nolint unused_arguments] imp_intro
/-!
## Linter for documentation strings
-/
/-- Reports definitions and constants that are missing doc strings -/
private meta def doc_blame_report_defn : declaration → tactic (option string)
| (declaration.defn n _ _ _ _ _) := doc_string n >> return none <|> return "def missing doc string"
| (declaration.cnst n _ _ _) := doc_string n >> return none <|> return "constant missing doc string"
| _ := return none
/-- Reports definitions and constants that are missing doc strings -/
private meta def doc_blame_report_thm : declaration → tactic (option string)
| (declaration.thm n _ _ _) := doc_string n >> return none <|> return "theorem missing doc string"
| _ := return none
/-- A linter for checking definition doc strings -/
@[linter] meta def linter.doc_blame : linter :=
{ test := λ d, mcond (bnot <$> has_attribute' `instance d.to_name)
(doc_blame_report_defn d) (return none),
auto_decls := ff,
no_errors_found := "No definitions are missing documentation.",
errors_found := "DEFINITIONS ARE MISSING DOCUMENTATION STRINGS:" }
/-- A linter for checking theorem doc strings. This is not in the default linter set. -/
meta def linter.doc_blame_thm : linter :=
{ test := doc_blame_report_thm,
auto_decls := ff,
no_errors_found := "No theorems are missing documentation.",
errors_found := "THEOREMS ARE MISSING DOCUMENTATION STRINGS:",
is_fast := ff }
/-!
## Linter for correct usage of `lemma`/`def`
-/
/--
Checks whether the correct declaration constructor (definition or theorem) by
comparing it to its sort. Instances will not be printed.
This test is not very quick: maybe we can speed-up testing that something is a proposition?
This takes almost all of the execution time.
-/
private meta def incorrect_def_lemma (d : declaration) : tactic (option string) :=
if d.is_constant ∨ d.is_axiom
then return none else do
is_instance_d ← is_instance d.to_name,
if is_instance_d then return none else do
-- the following seems to be a little quicker than `is_prop d.type`.
expr.sort n ← infer_type d.type,
is_pattern ← has_attribute' `pattern d.to_name,
return $
if d.is_theorem ↔ n = level.zero then none
else if d.is_theorem then "is a lemma/theorem, should be a def"
else if is_pattern then none -- declarations with `@[pattern]` are allowed to be a `def`.
else "is a def, should be a lemma/theorem"
/-- A linter for checking whether the correct declaration constructor (definition or theorem)
has been used. -/
@[linter] meta def linter.def_lemma : linter :=
{ test := incorrect_def_lemma,
auto_decls := ff,
no_errors_found := "All declarations correctly marked as def/lemma.",
errors_found := "INCORRECT DEF/LEMMA:" }
attribute [nolint def_lemma] classical.dec classical.dec_pred classical.dec_rel classical.dec_eq
/-!
## Linter that checks whether declarations are well-typed
-/
/-- Checks whether the statement of a declaration is well-typed. -/
meta def check_type (d : declaration) : tactic (option string) :=
(type_check d.type >> return none) <|> return "The statement doesn't type-check"
/-- A linter for missing checking whether statements of declarations are well-typed. -/
@[linter]
meta def linter.check_type : linter :=
{ test := check_type,
auto_decls := ff,
no_errors_found :=
"The statements of all declarations type-check with default reducibility settings.",
errors_found := "THE STATEMENTS OF THE FOLLOWING DECLARATIONS DO NOT TYPE-CHECK.
Some definitions in the statement are marked `@[irreducible]`, which means that the statement " ++
"is now ill-formed. It is likely that these definitions were locally marked as `@[reducible]` " ++
"or `@[semireducible]`. This can especially cause problems with type class inference or " ++
"`@[simps]`.",
is_fast := tt }
/-!
## Linter for universe parameters
-/
open native
/--
The good parameters are the parameters that occur somewhere in the `rb_set` as a singleton or
(recursively) with only other good parameters.
All other parameters in the `rb_set` are bad.
-/
meta def bad_params : rb_set (list name) → list name | l :=
let good_levels : name_set :=
l.fold mk_name_set $ λ us prev, if us.length = 1 then prev.insert us.head else prev in
if good_levels.empty then
l.fold [] list.union
else bad_params $ rb_set.of_list $ l.to_list.map $ λ us, us.filter $ λ nm, !good_levels.contains nm
/--
Checks whether all universe levels `u` in `d` are "good".
This means that `u` either occurs in a `level` of `d` by itself, or (recursively)
with only other good levels.
When this fails, usually this means that there is a level `max u v`, where neither `u` nor `v`
occur by themselves in a level. It is ok if *one* of `u` or `v` never occurs alone. For example,
`(α : Type u) (β : Type (max u v))` is a occasionally useful method of saying that `β` lives in
a higher universe level than `α`.
-/
meta def check_univs (d : declaration) : tactic (option string) := do
let l := d.type.univ_params_grouped.union d.value.univ_params_grouped,
let bad := bad_params l,
if bad.empty then return none else return $ some $ "universes " ++ to_string bad ++
" only occur together."
/-- A linter for checking that there are no bad `max u v` universe levels. -/
@[linter]
meta def linter.check_univs : linter :=
{ test := check_univs,
auto_decls := tt,
no_errors_found :=
"All declarations have good universe levels.",
errors_found := "THE STATEMENTS OF THE FOLLOWING DECLARATIONS HAVE BAD UNIVERSE LEVELS. " ++
"This usually means that there is a `max u v` in the declaration where neither `u` nor `v` " ++
"occur by themselves. Solution: Find the type (or type bundled with data) that has this " ++
"universe argument and provide the universe level explicitly. If this happens in an implicit " ++
"argument of the declaration, a better solution is to move this argument to a `variables` " ++
"command (where the universe level can be kept implicit).
Note: if the linter flags an automatically generated declaration `xyz._proof_i`, it means that
the universe problem is with `xyz` itself (even if the linter doesn't flag `xyz`)",
is_fast := tt }
/-!
## Linter for syntactic tautologies
-/
/--
Checks whether a lemma is a declaration of the form `∀ a b ... z, e₁ = e₂`
where `e₁` and `e₂` are identical exprs.
We call declarations of this form syntactic tautologies.
Such lemmas are (mostly) useless and sometimes introduced unintentionally when proving basic facts
with rfl when elaboration results in a different term than the user intended.
-/
meta def syn_taut (d : declaration) : tactic (option string) :=
(do (el, er) ← d.type.pi_codomain.is_eq,
guardb (el =ₐ er),
return $ some "LHS equals RHS syntactically") <|>
return none
/-- A linter for checking that declarations aren't syntactic tautologies. -/
@[linter]
meta def linter.syn_taut : linter :=
{ test := syn_taut,
auto_decls := ff, -- many false positives with this enabled
no_errors_found :=
"No declarations are syntactic tautologies.",
errors_found := "THE FOLLOWING DECLARATIONS ARE SYNTACTIC TAUTOLOGIES. " ++
"This usually means that they are of the form `∀ a b ... z, e₁ = e₂` where `e₁` and `e₂` are " ++
"identical expressions. We call declarations of this form syntactic tautologies. " ++
"Such lemmas are (mostly) useless and sometimes introduced unintentionally when proving " ++
"basic facts using `rfl`, when elaboration results in a different term than the user intended. " ++
"You should check that the declaration really says what you think it does.",
is_fast := tt }
attribute [nolint syn_taut] rfl
|
1c39e61f9b77b95976e1f4487c990484f621c28f | 302c785c90d40ad3d6be43d33bc6a558354cc2cf | /src/combinatorics/pigeonhole.lean | c2c782c521927e3ea51ffb31baa8a2dbe5c554a7 | [
"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 | 19,412 | lean | /-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Yury Kudryashov
-/
import data.set.finite
import data.nat.modeq
import algebra.big_operators.order
/-!
# Pigeonhole principles
Given pigeons (possibly infinitely many) in pigeonholes, the
pigeonhole principle states that, if there are more pigeons than
pigeonholes, then there is a pigeonhole with two or more pigeons.
There are a few variations on this statement, and the conclusion can
be made stronger depending on how many pigeons you know you might
have.
The basic statements of the pigeonhole principle appear in the
following locations:
* `data.finset.basic` has `finset.exists_ne_map_eq_of_card_lt_of_maps_to`
* `data.fintype.basic` has `fintype.exists_ne_map_eq_of_card_lt`
* `data.fintype.basic` has `fintype.exists_ne_map_eq_of_infinite`
* `data.fintype.basic` has `fintype.exists_infinite_fiber`
* `data.set.finite` has `set.infinite.exists_ne_map_eq_of_maps_to`
This module gives access to these pigeonhole principles along with 20 more.
The versions vary by:
* using a function between `fintype`s or a function between possibly infinite types restricted to
`finset`s;
* counting pigeons by a general weight function (`∑ x in s, w x`) or by heads (`finset.card s`);
* using strict or non-strict inequalities;
* establishing upper or lower estimate on the number (or the total weight) of the pigeons in one
pigeonhole;
* in case when we count pigeons by some weight function `w` and consider a function `f` between
`finset`s `s` and `t`, we can either assume that each pigeon is in one of the pigeonholes
(`∀ x ∈ s, f x ∈ t`), or assume that for `y ∉ t`, the total weight of the pigeons in this
pigeonhole `∑ x in s.filter (λ x, f x = y), w x` is nonpositive or nonnegative depending on
the inequality we are proving.
Lemma names follow `mathlib` convention (e.g.,
`finset.exists_lt_sum_fiber_of_maps_to_of_nsmul_lt_sum`); "pigeonhole principle" is mentioned in the
docstrings instead of the names.
## See also
* `ordinal.infinite_pigeonhole`: pigeonhole principle for cardinals, formulated using cofinality;
* `measure_theory.exists_nonempty_inter_of_measure_univ_lt_tsum_measure`,
`measure_theory.exists_nonempty_inter_of_measure_univ_lt_sum_measure`: pigeonhole principle in a
measure space.
## Tags
pigeonhole principle
-/
universes u v w
variables {α : Type u} {β : Type v} {M : Type w} [linear_ordered_cancel_add_comm_monoid M]
[decidable_eq β]
open_locale big_operators
namespace finset
variables {s : finset α} {t : finset β} {f : α → β} {w : α → M} {b : M} {n : ℕ}
/-!
### The pigeonhole principles on `finset`s, pigeons counted by weight
In this section we prove the following version of the pigeonhole principle: if the total weight of a
finite set of pigeons is greater than `n •ℕ b`, and they are sorted into `n` pigeonholes, then for
some pigeonhole, the total weight of the pigeons in this pigeonhole is greateer than `b`, and a few
variations of this theorem.
The principle is formalized in the following way, see
`finset.exists_lt_sum_fiber_of_maps_to_of_nsmul_lt_sum`: if `f : α → β` is a function which maps all
elements of `s : finset α` to `t : finset β` and `card t •ℕ b < ∑ x in s, w x`, where `w : α → M` is
a weight function taking values in a `linear_ordered_cancel_add_comm_monoid`, then for
some `y ∈ t`, the sum of the weights of all `x ∈ s` such that `f x = y` is greater than `b`.
There are a few bits we can change in this theorem:
* reverse all inequalities, with obvious adjustments to the name;
* replace the assumption `∀ a ∈ s, f a ∈ t` with
`∀ y ∉ t, (∑ x in s.filter (λ x, f x = y), w x) ≤ 0`,
and replace `of_maps_to` with `of_sum_fiber_nonpos` in the name;
* use non-strict inequalities assuming `t` is nonempty.
We can do all these variations independently, so we have eight versions of the theorem.
-/
/-!
#### Strict inequality versions
-/
/-- The pigeonhole principle for finitely many pigeons counted by weight, strict inequality version:
if the total weight of a finite set of pigeons is greater than `n •ℕ b`, and they are sorted into
`n` pigeonholes, then for some pigeonhole, the total weight of the pigeons in this pigeonhole is
greater than `b`. -/
lemma exists_lt_sum_fiber_of_maps_to_of_nsmul_lt_sum (hf : ∀ a ∈ s, f a ∈ t)
(hb : t.card •ℕ b < ∑ x in s, w x) :
∃ y ∈ t, b < ∑ x in s.filter (λ x, f x = y), w x :=
exists_lt_of_sum_lt $ by simpa only [sum_fiberwise_of_maps_to hf, sum_const]
/-- The pigeonhole principle for finitely many pigeons counted by weight, strict inequality version:
if the total weight of a finite set of pigeons is less than `n •ℕ b`, and they are sorted into `n`
pigeonholes, then for some pigeonhole, the total weight of the pigeons in this pigeonhole is less
than `b`. -/
lemma exists_sum_fiber_lt_of_maps_to_of_sum_lt_nsmul (hf : ∀ a ∈ s, f a ∈ t)
(hb : (∑ x in s, w x) < t.card •ℕ b) :
∃ y ∈ t, (∑ x in s.filter (λ x, f x = y), w x) < b :=
@exists_lt_sum_fiber_of_maps_to_of_nsmul_lt_sum α β (order_dual M) _ _ _ _ _ _ _ hf hb
/-- The pigeonhole principle for finitely many pigeons counted by weight, strict inequality version:
if the total weight of a finite set of pigeons is greater than `n •ℕ b`, they are sorted into some
pigeonholes, and for all but `n` pigeonholes the total weight of the pigeons there is nonpositive,
then for at least one of these `n` pigeonholes, the total weight of the pigeons in this pigeonhole
is greater than `b`. -/
lemma exists_lt_sum_fiber_of_sum_fiber_nonpos_of_nsmul_lt_sum
(ht : ∀ y ∉ t, (∑ x in s.filter (λ x, f x = y), w x) ≤ 0) (hb : t.card •ℕ b < ∑ x in s, w x) :
∃ y ∈ t, b < ∑ x in s.filter (λ x, f x = y), w x :=
exists_lt_of_sum_lt $
calc (∑ y in t, b) < ∑ x in s, w x : by simpa
... ≤ ∑ y in t, ∑ x in s.filter (λ x, f x = y), w x :
sum_le_sum_fiberwise_of_sum_fiber_nonpos ht
/-- The pigeonhole principle for finitely many pigeons counted by weight, strict inequality version:
if the total weight of a finite set of pigeons is less than `n •ℕ b`, they are sorted into some
pigeonholes, and for all but `n` pigeonholes the total weight of the pigeons there is nonnegative,
then for at least one of these `n` pigeonholes, the total weight of the pigeons in this pigeonhole
is less than `b`. -/
lemma exists_sum_fiber_lt_of_sum_fiber_nonneg_of_sum_lt_nsmul
(ht : ∀ y ∉ t, (0:M) ≤ ∑ x in s.filter (λ x, f x = y), w x) (hb : (∑ x in s, w x) < t.card •ℕ b) :
∃ y ∈ t, (∑ x in s.filter (λ x, f x = y), w x) < b :=
@exists_lt_sum_fiber_of_sum_fiber_nonpos_of_nsmul_lt_sum α β (order_dual M) _ _ _ _ _ _ _ ht hb
/-!
#### Non-strict inequality versions
-/
/-- The pigeonhole principle for finitely many pigeons counted by weight, non-strict inequality
version: if the total weight of a finite set of pigeons is greater than or equal to `n •ℕ b`, and
they are sorted into `n > 0` pigeonholes, then for some pigeonhole, the total weight of the pigeons
in this pigeonhole is greater than or equal to `b`. -/
lemma exists_le_sum_fiber_of_maps_to_of_nsmul_le_sum (hf : ∀ a ∈ s, f a ∈ t) (ht : t.nonempty)
(hb : t.card •ℕ b ≤ ∑ x in s, w x) :
∃ y ∈ t, b ≤ ∑ x in s.filter (λ x, f x = y), w x :=
exists_le_of_sum_le ht $ by simpa only [sum_fiberwise_of_maps_to hf, sum_const]
/-- The pigeonhole principle for finitely many pigeons counted by weight, non-strict inequality
version: if the total weight of a finite set of pigeons is less than or equal to `n •ℕ b`, and they
are sorted into `n > 0` pigeonholes, then for some pigeonhole, the total weight of the pigeons in
this pigeonhole is less than or equal to `b`. -/
lemma exists_sum_fiber_le_of_maps_to_of_sum_le_nsmul (hf : ∀ a ∈ s, f a ∈ t) (ht : t.nonempty)
(hb : (∑ x in s, w x) ≤ t.card •ℕ b) :
∃ y ∈ t, (∑ x in s.filter (λ x, f x = y), w x) ≤ b :=
@exists_le_sum_fiber_of_maps_to_of_nsmul_le_sum α β (order_dual M) _ _ _ _ _ _ _ hf ht hb
/-- The pigeonhole principle for finitely many pigeons counted by weight, non-strict inequality
version: if the total weight of a finite set of pigeons is greater than or equal to `n •ℕ b`, they
are sorted into some pigeonholes, and for all but `n > 0` pigeonholes the total weight of the
pigeons there is nonpositive, then for at least one of these `n` pigeonholes, the total weight of
the pigeons in this pigeonhole is greater than or equal to `b`. -/
lemma exists_le_sum_fiber_of_sum_fiber_nonpos_of_nsmul_le_sum
(hf : ∀ y ∉ t, (∑ x in s.filter (λ x, f x = y), w x) ≤ 0) (ht : t.nonempty)
(hb : t.card •ℕ b ≤ ∑ x in s, w x) :
∃ y ∈ t, b ≤ ∑ x in s.filter (λ x, f x = y), w x :=
exists_le_of_sum_le ht $
calc (∑ y in t, b) ≤ ∑ x in s, w x : by simpa
... ≤ ∑ y in t, ∑ x in s.filter (λ x, f x = y), w x :
sum_le_sum_fiberwise_of_sum_fiber_nonpos hf
/-- The pigeonhole principle for finitely many pigeons counted by weight, non-strict inequality
version: if the total weight of a finite set of pigeons is less than or equal to `n •ℕ b`, they are
sorted into some pigeonholes, and for all but `n > 0` pigeonholes the total weight of the pigeons
there is nonnegative, then for at least one of these `n` pigeonholes, the total weight of the
pigeons in this pigeonhole is less than or equal to `b`. -/
lemma exists_sum_fiber_le_of_sum_fiber_nonneg_of_sum_le_nsmul
(hf : ∀ y ∉ t, (0:M) ≤ ∑ x in s.filter (λ x, f x = y), w x) (ht : t.nonempty)
(hb : (∑ x in s, w x) ≤ t.card •ℕ b) :
∃ y ∈ t, (∑ x in s.filter (λ x, f x = y), w x) ≤ b :=
@exists_le_sum_fiber_of_sum_fiber_nonpos_of_nsmul_le_sum α β (order_dual M) _ _ _ _ _ _ _ hf ht hb
/-!
### The pigeonhole principles on `finset`s, pigeons counted by heads
In this section we formalize a few versions of the following pigeonhole principle: there is a
pigeonhole with at least as many pigeons as the ceiling of the average number of pigeons across all
pigeonholes.
First, we can use strict or non-strict inequalities. While the versions with non-strict inequalities
are weaker than those with strict inequalities, sometimes it might be more convenient to apply the
weaker version. Second, we can either state that there exists a pigeonhole with at least `n`
pigeons, or state that there exists a pigeonhole with at most `n` pigeons. In the latter case we do
not need the assumption `∀ a ∈ s, f a ∈ t`.
So, we prove four theorems: `finset.exists_lt_card_fiber_of_maps_to_of_mul_lt_card`,
`finset.exists_le_card_fiber_of_maps_to_of_mul_le_card`,
`finset.exists_card_fiber_lt_of_card_lt_mul`, and `finset.exists_card_fiber_le_of_card_le_mul`. -/
/-- The pigeonhole principle for finitely many pigeons counted by heads: there is a pigeonhole with
at least as many pigeons as the ceiling of the average number of pigeons across all pigeonholes.
("The maximum is at least the mean" specialized to integers.)
More formally, given a function between finite sets `s` and `t` and a natural number `n` such that
`card t * n < card s`, there exists `y ∈ t` such that its preimage in `s` has more than `n`
elements. -/
lemma exists_lt_card_fiber_of_mul_lt_card_of_maps_to (hf : ∀ a ∈ s, f a ∈ t)
(hn : t.card * n < s.card) :
∃ y ∈ t, n < (s.filter (λ x, f x = y)).card :=
begin
simp only [card_eq_sum_ones],
apply exists_lt_sum_fiber_of_maps_to_of_nsmul_lt_sum hf,
simpa
end
/-- The pigeonhole principle for finitely many pigeons counted by heads: there is a pigeonhole with
at most as many pigeons as the floor of the average number of pigeons across all pigeonholes. ("The
minimum is at most the mean" specialized to integers.)
More formally, given a function `f`, a finite sets `s` in its domain, a finite set `t` in its
codomain, and a natural number `n` such that `card s < card t * n`, there exists `y ∈ t` such that
its preimage in `s` has less than `n` elements. -/
lemma exists_card_fiber_lt_of_card_lt_mul (hn : s.card < t.card * n) :
∃ y ∈ t, (s.filter (λ x, f x = y)).card < n:=
begin
simp only [card_eq_sum_ones],
apply exists_sum_fiber_lt_of_sum_fiber_nonneg_of_sum_lt_nsmul (λ _ _, nat.zero_le _),
simpa
end
/-- The pigeonhole principle for finitely many pigeons counted by heads: given a function between
finite sets `s` and `t` and a natural number `n` such that `card t * n ≤ card s`, there exists `y ∈
t` such that its preimage in `s` has at least `n` elements. See also
`finset.exists_lt_card_fiber_of_mul_lt_card_of_maps_to` for a stronger statement. -/
lemma exists_le_card_fiber_of_mul_le_card_of_maps_to (hf : ∀ a ∈ s, f a ∈ t) (ht : t.nonempty)
(hn : t.card * n ≤ s.card) :
∃ y ∈ t, n ≤ (s.filter (λ x, f x = y)).card :=
begin
simp only [card_eq_sum_ones],
apply exists_le_sum_fiber_of_maps_to_of_nsmul_le_sum hf ht,
simpa
end
/-- The pigeonhole principle for finitely many pigeons counted by heads: given a function `f`, a
finite sets `s` in its domain, a finite set `t` in its codomain, and a natural number `n` such that
`card s ≤ card t * n`, there exists `y ∈ t` such that its preimage in `s` has no more than `n`
elements. See also `finset.exists_card_fiber_lt_of_card_lt_mul` for a stronger statement. -/
lemma exists_card_fiber_le_of_card_le_mul (ht : t.nonempty) (hn : s.card ≤ t.card * n) :
∃ y ∈ t, (s.filter (λ x, f x = y)).card ≤ n:=
begin
simp only [card_eq_sum_ones],
apply exists_sum_fiber_le_of_sum_fiber_nonneg_of_sum_le_nsmul (λ _ _, nat.zero_le _) ht,
simpa
end
end finset
namespace fintype
open finset
variables [fintype α] [fintype β] (f : α → β) {w : α → M} {b : M} {n : ℕ}
/-!
### The pigeonhole principles on `fintypes`s, pigeons counted by weight
In this section we specialize theorems from the previous section to the special case of functions
between `fintype`s and `s = univ`, `t = univ`. In this case the assumption `∀ x ∈ s, f x ∈ t` always
holds, so we have four theorems instead of eight. -/
/-- The pigeonhole principle for finitely many pigeons of different weights, strict inequality
version: there is a pigeonhole with the total weight of pigeons in it greater than `b` provided that
the total number of pigeonholes times `b` is less than the total weight of all pigeons. -/
lemma exists_lt_sum_fiber_of_nsmul_lt_sum (hb : card β •ℕ b < ∑ x, w x) :
∃ y, b < ∑ x in univ.filter (λ x, f x = y), w x :=
let ⟨y, _, hy⟩ := exists_lt_sum_fiber_of_maps_to_of_nsmul_lt_sum (λ _ _, mem_univ _) hb in ⟨y, hy⟩
/-- The pigeonhole principle for finitely many pigeons of different weights, non-strict inequality
version: there is a pigeonhole with the total weight of pigeons in it greater than or equal to `b`
provided that the total number of pigeonholes times `b` is less than or equal to the total weight of
all pigeons. -/
lemma exists_le_sum_fiber_of_nsmul_le_sum [nonempty β] (hb : card β •ℕ b ≤ ∑ x, w x) :
∃ y, b ≤ ∑ x in univ.filter (λ x, f x = y), w x :=
let ⟨y, _, hy⟩ :=
exists_le_sum_fiber_of_maps_to_of_nsmul_le_sum (λ _ _, mem_univ _) univ_nonempty hb
in ⟨y, hy⟩
/-- The pigeonhole principle for finitely many pigeons of different weights, strict inequality
version: there is a pigeonhole with the total weight of pigeons in it less than `b` provided that
the total number of pigeonholes times `b` is greater than the total weight of all pigeons. -/
lemma exists_sum_fiber_lt_of_sum_lt_nsmul (hb : (∑ x, w x) < card β •ℕ b) :
∃ y, (∑ x in univ.filter (λ x, f x = y), w x) < b :=
@exists_lt_sum_fiber_of_nsmul_lt_sum α β (order_dual M) _ _ _ _ _ w b hb
/-- The pigeonhole principle for finitely many pigeons of different weights, non-strict inequality
version: there is a pigeonhole with the total weight of pigeons in it less than or equal to `b`
provided that the total number of pigeonholes times `b` is greater than or equal to the total weight
of all pigeons. -/
lemma exists_sum_fiber_le_of_sum_le_nsmul [nonempty β] (hb : (∑ x, w x) ≤ card β •ℕ b) :
∃ y, (∑ x in univ.filter (λ x, f x = y), w x) ≤ b :=
@exists_le_sum_fiber_of_nsmul_le_sum α β (order_dual M) _ _ _ _ _ w b _ hb
/--
The strong pigeonhole principle for finitely many pigeons and pigeonholes.
There is a pigeonhole with at least as many pigeons as
the ceiling of the average number of pigeons across all pigeonholes.
("The maximum is at least the mean" specialized to integers.)
More formally, given a function `f` between finite types `α` and `β` and a number `n` such that
`card β * n < card α`, there exists an element `y : β` such that its preimage has more than `n`
elements. -/
lemma exists_lt_card_fiber_of_mul_lt_card (hn : card β * n < card α) :
∃ y : β, n < (univ.filter (λ x, f x = y)).card :=
let ⟨y, _, h⟩ := exists_lt_card_fiber_of_mul_lt_card_of_maps_to (λ _ _, mem_univ _) hn in ⟨y, h⟩
/--
The strong pigeonhole principle for finitely many pigeons and pigeonholes.
There is a pigeonhole with at most as many pigeons as
the floor of the average number of pigeons across all pigeonholes.
("The minimum is at most the mean" specialized to integers.)
More formally, given a function `f` between finite types `α` and `β` and a number `n` such that
`card α < card β * n`, there exists an element `y : β` such that its preimage has less than `n`
elements. -/
lemma exists_card_fiber_lt_of_card_lt_mul (hn : card α < card β * n) :
∃ y : β, (univ.filter (λ x, f x = y)).card < n :=
let ⟨y, _, h⟩ := exists_card_fiber_lt_of_card_lt_mul hn in ⟨y, h⟩
/-- The strong pigeonhole principle for finitely many pigeons and pigeonholes. Given a function `f`
between finite types `α` and `β` and a number `n` such that `card β * n ≤ card α`, there exists an
element `y : β` such that its preimage has at least `n` elements. See also
`fintype.exists_lt_card_fiber_of_mul_lt_card` for a stronger statement. -/
lemma exists_le_card_fiber_of_mul_le_card [nonempty β] (hn : card β * n ≤ card α) :
∃ y : β, n ≤ (univ.filter (λ x, f x = y)).card :=
let ⟨y, _, h⟩ := exists_le_card_fiber_of_mul_le_card_of_maps_to (λ _ _, mem_univ _) univ_nonempty hn
in ⟨y, h⟩
/-- The strong pigeonhole principle for finitely many pigeons and pigeonholes. Given a function `f`
between finite types `α` and `β` and a number `n` such that `card α ≤ card β * n`, there exists an
element `y : β` such that its preimage has at most `n` elements. See also
`fintype.exists_card_fiber_lt_of_card_lt_mul` for a stronger statement. -/
lemma exists_card_fiber_le_of_card_le_mul [nonempty β] (hn : card α ≤ card β * n) :
∃ y : β, (univ.filter (λ x, f x = y)).card ≤ n :=
let ⟨y, _, h⟩ := exists_card_fiber_le_of_card_le_mul univ_nonempty hn in ⟨y, h⟩
end fintype
namespace nat
open set
/-- If `s` is an infinite set of natural numbers and `k > 0`, then `s` contains two elements `m < n`
that are equal mod `k`. -/
theorem exists_lt_modeq_of_infinite {s : set ℕ} (hs : s.infinite) {k : ℕ} (hk : 0 < k) :
∃ (m ∈ s) (n ∈ s), m < n ∧ m ≡ n [MOD k] :=
hs.exists_lt_map_eq_of_maps_to (λ n _, show n % k ∈ Iio k, from nat.mod_lt n hk) $
finite_lt_nat k
end nat
|
608f03849f5e2517c42fcd240c766fa86e3592b8 | 55c7fc2bf55d496ace18cd6f3376e12bb14c8cc5 | /src/algebra/quadratic_discriminant.lean | 4d6ca65afec5dd4a72a26c41eb022fb120122384 | [
"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 | 6,848 | lean | /-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou
-/
import tactic.linarith
/-!
# Quadratic discriminants and roots of a quadratic
This file defines the discriminant of a quadratic and gives the solution to a quadratic equation.
## Main definition
The discriminant of a quadratic `a*x*x + b*x + c` is `b*b - 4*a*c`.
## Main statements
• Roots of a quadratic can be written as `(-b + s) / (2 * a)` or `(-b - s) / (2 * a)`,
where `s` is the square root of the discriminant.
• If the discriminant has no square root, then the corresponding quadratic has no root.
• If a quadratic is always non-negative, then its discriminant is non-positive.
## Tags
polynomial, quadratic, discriminant, root
-/
variables {α : Type*}
section lemmas
variables [linear_ordered_field α] {a b c : α}
lemma exists_le_mul_self : ∀ a : α, ∃ x : α, a ≤ x * x :=
begin
classical, -- TODO: otherwise linarith performance sucks
assume a, cases le_total 1 a with ha ha,
{ use a, exact le_mul_of_ge_one_left (by linarith) ha },
{ use 1, linarith }
end
lemma exists_lt_mul_self : ∀ a : α, ∃ x : α, a < x * x :=
begin
classical, -- todo: otherwise linarith performance sucks
assume a, rcases (exists_le_mul_self a) with ⟨x, hx⟩,
cases le_total 0 x with hx' hx',
{ use (x + 1),
have : (x+1)*(x+1) = x*x + 2*x + 1, {ring},
exact lt_of_le_of_lt hx (by rw this; linarith) },
{ use (x - 1),
have : (x-1)*(x-1) = x*x - 2*x + 1, {ring},
exact lt_of_le_of_lt hx (by rw this; linarith) }
end
end lemmas
variables [linear_ordered_field α] {a b c x : α}
/-- Discriminant of a quadratic -/
def discrim [ring α] (a b c : α) : α := b^2 - 4 * a * c
/--
A quadratic has roots if and only if its discriminant equals some square.
-/
lemma quadratic_eq_zero_iff_discrim_eq_square (ha : a ≠ 0) :
∀ x : α, a * x * x + b * x + c = 0 ↔ discrim a b c = (2 * a * x + b)^2 :=
by classical; exact -- TODO: otherwise linarith performance sucks
assume x, iff.intro
(assume h, calc
discrim a b c = 4*a*(a*x*x + b*x + c) + b*b - 4*a*c : by rw [h, discrim]; ring
... = (2*a*x + b)^2 : by ring)
(assume h,
have ha : 2*2*a ≠ 0 := mul_ne_zero (mul_ne_zero two_ne_zero two_ne_zero) ha,
mul_left_cancel' ha $
calc
2 * 2 * a * (a * x * x + b * x + c) = (2*a*x + b)^2 - (b^2 - 4*a*c) : by ring
... = 0 : by { rw [← h, discrim], ring }
... = 2*2*a*0 : by ring)
/-- Roots of a quadratic -/
lemma quadratic_eq_zero_iff (ha : a ≠ 0) {s : α} (h : discrim a b c = s * s) :
∀ x : α, a * x * x + b * x + c = 0 ↔ x = (-b + s) / (2 * a) ∨ x = (-b - s) / (2 * a) := assume x,
begin
classical, -- TODO: otherwise linarith performance sucks
rw [quadratic_eq_zero_iff_discrim_eq_square ha, h, pow_two, mul_self_eq_mul_self_iff],
have ne : 2 * a ≠ 0 := mul_ne_zero two_ne_zero ha,
have : x = 2 * a * x / (2 * a) := (mul_div_cancel_left x ne).symm,
have h₁ : 2 * a * ((-b + s) / (2 * a)) = -b + s := mul_div_cancel' _ ne,
have h₂ : 2 * a * ((-b - s) / (2 * a)) = -b - s := mul_div_cancel' _ ne,
split,
{ intro h', rcases h',
{ left, rw h', simpa [add_comm] },
{ right, rw h', simpa [add_comm, sub_eq_add_neg] } },
{ intro h', rcases h', { left, rw [h', h₁], ring }, { right, rw [h', h₂], ring } }
end
/-- A quadratic has roots if its discriminant has square roots -/
lemma exist_quadratic_eq_zero (ha : a ≠ 0) (h : ∃ s, discrim a b c = s * s) :
∃ x, a * x * x + b * x + c = 0 :=
begin
rcases h with ⟨s, hs⟩,
use (-b + s) / (2 * a),
rw quadratic_eq_zero_iff ha hs,
simp
end
/-- Root of a quadratic when its discriminant equals zero -/
lemma quadratic_eq_zero_iff_of_discrim_eq_zero (ha : a ≠ 0) (h : discrim a b c = 0) :
∀ x : α, a * x * x + b * x + c = 0 ↔ x = -b / (2 * a) := assume x,
begin
classical, -- TODO: otherwise linarith performance sucks
have : discrim a b c = 0 * 0 := eq.trans h (by ring),
rw quadratic_eq_zero_iff ha this,
simp
end
/-- A quadratic has no root if its discriminant has no square root. -/
lemma quadratic_ne_zero_of_discrim_ne_square (ha : a ≠ 0) (h : ∀ s : α, discrim a b c ≠ s * s) :
∀ (x : α), a * x * x + b * x + c ≠ 0 :=
begin
assume x h',
rw [quadratic_eq_zero_iff_discrim_eq_square ha, pow_two] at h',
have := h _,
contradiction
end
/-- If a polynomial of degree 2 is always nonnegative, then its discriminant is nonpositive -/
lemma discriminant_le_zero {a b c : α} (h : ∀ x : α, 0 ≤ a*x*x + b*x + c) : discrim a b c ≤ 0 :=
by classical; exact -- TODO: otherwise linarith performance sucks
have hc : 0 ≤ c, by { have := h 0, linarith },
begin
rw [discrim, pow_two],
cases lt_trichotomy a 0 with ha ha,
-- if a < 0
cases classical.em (b = 0) with hb hb,
{ rw hb at *,
rcases exists_lt_mul_self (-c/a) with ⟨x, hx⟩,
have := mul_lt_mul_of_neg_left hx ha,
rw [mul_div_cancel' _ (ne_of_lt ha), ← mul_assoc] at this,
have h₂ := h x, linarith },
{ cases classical.em (c = 0) with hc' hc',
{ rw hc' at *,
have : -(a*-b*-b + b*-b + 0) = (1-a)*(b*b), {ring},
have h := h (-b), rw [← neg_nonpos, this] at h,
have : b * b ≤ 0 := nonpos_of_mul_nonpos_left h (by linarith),
linarith },
{ have h := h (-c/b),
have : a*(-c/b)*(-c/b) + b*(-c/b) + c = a*((c/b)*(c/b)),
{ rw mul_div_cancel' _ hb, ring },
rw this at h,
have : 0 ≤ a := nonneg_of_mul_nonneg_right h (mul_self_pos $ div_ne_zero hc' hb),
linarith [ha] } },
cases ha with ha ha,
-- if a = 0
cases classical.em (b = 0) with hb hb,
{ rw [ha, hb], linarith },
{ have := h ((-c-1)/b), rw [ha, mul_div_cancel' _ hb] at this, linarith },
-- if a > 0
have := calc
4*a* (a*(-(b/a)*(1/2))*(-(b/a)*(1/2)) + b*(-(b/a)*(1/2)) + c)
= (a*(b/a)) * (a*(b/a)) - 2*(a*(b/a))*b + 4*a*c : by ring
... = -(b*b - 4*a*c) : by { simp only [mul_div_cancel' b (ne_of_gt ha)], ring },
have ha' : 0 ≤ 4*a, {linarith},
have h := (mul_nonneg ha' (h (-(b/a) * (1/2)))),
rw this at h, rwa ← neg_nonneg
end
/--
If a polynomial of degree 2 is always positive, then its discriminant is negative,
at least when the coefficient of the quadratic term is nonzero.
-/
lemma discriminant_lt_zero {a b c : α} (ha : a ≠ 0) (h : ∀ x : α, 0 < a*x*x + b*x + c) :
discrim a b c < 0 :=
begin
classical, -- TODO: otherwise linarith performance sucks
have : ∀ x : α, 0 ≤ a*x*x + b*x + c := assume x, le_of_lt (h x),
refine lt_of_le_of_ne (discriminant_le_zero this) _,
assume h',
have := h (-b / (2 * a)),
have : a * (-b / (2 * a)) * (-b / (2 * a)) + b * (-b / (2 * a)) + c = 0,
{ rw [quadratic_eq_zero_iff_of_discrim_eq_zero ha h' (-b / (2 * a))] },
linarith
end
|
d97974d73aa15adcbb0725f855e3d996c9af3491 | 6ae186a0c6ab366b39397ec9250541c9d5aeb023 | /src/category_theory/functor/isomorphism.lean | 8fa2af6b07d6c15699f7d8e8effe308ddcab20a9 | [] | no_license | ThanhPhamPhuong/lean-category-theory | 0d5c4fe1137866b4fe29ec2753d99aa0d0667881 | 968a29fe7c0b20e10d8a27e120aca8ddc184e1ea | refs/heads/master | 1,587,206,682,489 | 1,544,045,056,000 | 1,544,045,056,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 603 | 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
import category_theory.isomorphism
open category_theory
namespace category_theory
universes u₁ v₁ u₂ v₂
variables {C : Type u₁} [category.{u₁ v₁} C] {D : Type u₂} [category.{u₂ v₂} D]
class reflects_isos (F : C ⥤ D) :=
(reflects : Π {X Y : C} (f : X ⟶ Y) (w : is_iso (F.map f)), is_iso f)
-- TODO
-- instance (F : C ⥤ D) [faithful F] : reflects_isos F := sorry
end category_theory |
33b23e84c64489d776933d6825e83dddfe99fc23 | 8cae430f0a71442d02dbb1cbb14073b31048e4b0 | /src/order/max.lean | 01aa6e0e063d57d4d98beaa438791b9a83eb921c | [
"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 | 12,628 | lean | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Yury Kudryashov, Yaël Dillies
-/
import order.synonym
/-!
# Minimal/maximal and bottom/top elements
> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
> Any changes to this file require a corresponding PR to mathlib4.
This file defines predicates for elements to be minimal/maximal or bottom/top and typeclasses
saying that there are no such elements.
## Predicates
* `is_bot`: An element is *bottom* if all elements are greater than it.
* `is_top`: An element is *top* if all elements are less than it.
* `is_min`: An element is *minimal* if no element is strictly less than it.
* `is_max`: An element is *maximal* if no element is strictly greater than it.
See also `is_bot_iff_is_min` and `is_top_iff_is_max` for the equivalences in a (co)directed order.
## Typeclasses
* `no_bot_order`: An order without bottom elements.
* `no_top_order`: An order without top elements.
* `no_min_order`: An order without minimal elements.
* `no_max_order`: An order without maximal elements.
-/
open order_dual
variables {ι α β : Type*} {π : ι → Type*}
/-- Order without bottom elements. -/
class no_bot_order (α : Type*) [has_le α] : Prop :=
(exists_not_ge (a : α) : ∃ b, ¬ a ≤ b)
/-- Order without top elements. -/
class no_top_order (α : Type*) [has_le α] : Prop :=
(exists_not_le (a : α) : ∃ b, ¬ b ≤ a)
/-- Order without minimal elements. Sometimes called coinitial or dense. -/
class no_min_order (α : Type*) [has_lt α] : Prop :=
(exists_lt (a : α) : ∃ b, b < a)
/-- Order without maximal elements. Sometimes called cofinal. -/
class no_max_order (α : Type*) [has_lt α] : Prop :=
(exists_gt (a : α) : ∃ b, a < b)
export no_bot_order (exists_not_ge)
export no_top_order (exists_not_le)
export no_min_order (exists_lt)
export no_max_order (exists_gt)
instance nonempty_lt [has_lt α] [no_min_order α] (a : α) : nonempty {x // x < a} :=
nonempty_subtype.2 (exists_lt a)
instance nonempty_gt [has_lt α] [no_max_order α] (a : α) : nonempty {x // a < x} :=
nonempty_subtype.2 (exists_gt a)
instance order_dual.no_bot_order (α : Type*) [has_le α] [no_top_order α] : no_bot_order αᵒᵈ :=
⟨λ a, @exists_not_le α _ _ a⟩
instance order_dual.no_top_order (α : Type*) [has_le α] [no_bot_order α] : no_top_order αᵒᵈ :=
⟨λ a, @exists_not_ge α _ _ a⟩
instance order_dual.no_min_order (α : Type*) [has_lt α] [no_max_order α] : no_min_order αᵒᵈ :=
⟨λ a, @exists_gt α _ _ a⟩
instance order_dual.no_max_order (α : Type*) [has_lt α] [no_min_order α] : no_max_order αᵒᵈ :=
⟨λ a, @exists_lt α _ _ a⟩
instance no_max_order_of_left [preorder α] [preorder β] [no_max_order α] : no_max_order (α × β) :=
⟨λ ⟨a, b⟩, by { obtain ⟨c, h⟩ := exists_gt a, exact ⟨(c, b), prod.mk_lt_mk_iff_left.2 h⟩ }⟩
instance no_max_order_of_right [preorder α] [preorder β] [no_max_order β] : no_max_order (α × β) :=
⟨λ ⟨a, b⟩, by { obtain ⟨c, h⟩ := exists_gt b, exact ⟨(a, c), prod.mk_lt_mk_iff_right.2 h⟩ }⟩
instance no_min_order_of_left [preorder α] [preorder β] [no_min_order α] : no_min_order (α × β) :=
⟨λ ⟨a, b⟩, by { obtain ⟨c, h⟩ := exists_lt a, exact ⟨(c, b), prod.mk_lt_mk_iff_left.2 h⟩ }⟩
instance no_min_order_of_right [preorder α] [preorder β] [no_min_order β] : no_min_order (α × β) :=
⟨λ ⟨a, b⟩, by { obtain ⟨c, h⟩ := exists_lt b, exact ⟨(a, c), prod.mk_lt_mk_iff_right.2 h⟩ }⟩
instance [nonempty ι] [Π i, preorder (π i)] [Π i, no_max_order (π i)] : no_max_order (Π i, π i) :=
⟨λ a, begin
classical,
obtain ⟨b, hb⟩ := exists_gt (a $ classical.arbitrary _),
exact ⟨_, lt_update_self_iff.2 hb⟩,
end⟩
instance [nonempty ι] [Π i, preorder (π i)] [Π i, no_min_order (π i)] : no_min_order (Π i, π i) :=
⟨λ a, begin
classical,
obtain ⟨b, hb⟩ := exists_lt (a $ classical.arbitrary _),
exact ⟨_, update_lt_self_iff.2 hb⟩,
end⟩
@[priority 100] -- See note [lower instance priority]
instance no_min_order.to_no_bot_order (α : Type*) [preorder α] [no_min_order α] : no_bot_order α :=
⟨λ a, (exists_lt a).imp $ λ _, not_le_of_lt⟩
@[priority 100] -- See note [lower instance priority]
instance no_max_order.to_no_top_order (α : Type*) [preorder α] [no_max_order α] : no_top_order α :=
⟨λ a, (exists_gt a).imp $ λ _, not_le_of_lt⟩
lemma no_bot_order.to_no_min_order (α : Type*) [linear_order α] [no_bot_order α] : no_min_order α :=
{ exists_lt := by { convert λ a : α, exists_not_ge a, simp_rw not_le, } }
lemma no_top_order.to_no_max_order (α : Type*) [linear_order α] [no_top_order α] : no_max_order α :=
{ exists_gt := by { convert λ a : α, exists_not_le a, simp_rw not_le, } }
lemma no_bot_order_iff_no_min_order (α : Type*) [linear_order α] :
no_bot_order α ↔ no_min_order α :=
⟨λ h, by { haveI := h, exact no_bot_order.to_no_min_order α },
λ h, by { haveI := h, exact no_min_order.to_no_bot_order α }⟩
lemma no_top_order_iff_no_max_order (α : Type*) [linear_order α] :
no_top_order α ↔ no_max_order α :=
⟨λ h, by { haveI := h, exact no_top_order.to_no_max_order α },
λ h, by { haveI := h, exact no_max_order.to_no_top_order α }⟩
theorem no_min_order.not_acc [has_lt α] [no_min_order α] (a : α) : ¬ acc (<) a :=
λ h, acc.rec_on h $ λ x _, (exists_lt x).rec_on
theorem no_max_order.not_acc [has_lt α] [no_max_order α] (a : α) : ¬ acc (>) a :=
λ h, acc.rec_on h $ λ x _, (exists_gt x).rec_on
section has_le
variables [has_le α] {a b : α}
/-- `a : α` is a bottom element of `α` if it is less than or equal to any other element of `α`.
This predicate is roughly an unbundled version of `order_bot`, except that a preorder may have
several bottom elements. When `α` is linear, this is useful to make a case disjunction on
`no_min_order α` within a proof. -/
def is_bot (a : α) : Prop := ∀ b, a ≤ b
/-- `a : α` is a top element of `α` if it is greater than or equal to any other element of `α`.
This predicate is roughly an unbundled version of `order_bot`, except that a preorder may have
several top elements. When `α` is linear, this is useful to make a case disjunction on
`no_max_order α` within a proof. -/
def is_top (a : α) : Prop := ∀ b, b ≤ a
/-- `a` is a minimal element of `α` if no element is strictly less than it. We spell it without `<`
to avoid having to convert between `≤` and `<`. Instead, `is_min_iff_forall_not_lt` does the
conversion. -/
def is_min (a : α) : Prop := ∀ ⦃b⦄, b ≤ a → a ≤ b
/-- `a` is a maximal element of `α` if no element is strictly greater than it. We spell it without
`<` to avoid having to convert between `≤` and `<`. Instead, `is_max_iff_forall_not_lt` does the
conversion. -/
def is_max (a : α) : Prop := ∀ ⦃b⦄, a ≤ b → b ≤ a
@[simp] lemma not_is_bot [no_bot_order α] (a : α) : ¬is_bot a :=
λ h, let ⟨b, hb⟩ := exists_not_ge a in hb $ h _
@[simp] lemma not_is_top [no_top_order α] (a : α) : ¬is_top a :=
λ h, let ⟨b, hb⟩ := exists_not_le a in hb $ h _
protected lemma is_bot.is_min (h : is_bot a) : is_min a := λ b _, h b
protected lemma is_top.is_max (h : is_top a) : is_max a := λ b _, h b
@[simp] lemma is_bot_to_dual_iff : is_bot (to_dual a) ↔ is_top a := iff.rfl
@[simp] lemma is_top_to_dual_iff : is_top (to_dual a) ↔ is_bot a := iff.rfl
@[simp] lemma is_min_to_dual_iff : is_min (to_dual a) ↔ is_max a := iff.rfl
@[simp] lemma is_max_to_dual_iff : is_max (to_dual a) ↔ is_min a := iff.rfl
@[simp] lemma is_bot_of_dual_iff {a : αᵒᵈ} : is_bot (of_dual a) ↔ is_top a := iff.rfl
@[simp] lemma is_top_of_dual_iff {a : αᵒᵈ} : is_top (of_dual a) ↔ is_bot a := iff.rfl
@[simp] lemma is_min_of_dual_iff {a : αᵒᵈ} : is_min (of_dual a) ↔ is_max a := iff.rfl
@[simp] lemma is_max_of_dual_iff {a : αᵒᵈ} : is_max (of_dual a) ↔ is_min a := iff.rfl
alias is_bot_to_dual_iff ↔ _ is_top.to_dual
alias is_top_to_dual_iff ↔ _ is_bot.to_dual
alias is_min_to_dual_iff ↔ _ is_max.to_dual
alias is_max_to_dual_iff ↔ _ is_min.to_dual
alias is_bot_of_dual_iff ↔ _ is_top.of_dual
alias is_top_of_dual_iff ↔ _ is_bot.of_dual
alias is_min_of_dual_iff ↔ _ is_max.of_dual
alias is_max_of_dual_iff ↔ _ is_min.of_dual
end has_le
section preorder
variables [preorder α] {a b : α}
lemma is_bot.mono (ha : is_bot a) (h : b ≤ a) : is_bot b := λ c, h.trans $ ha _
lemma is_top.mono (ha : is_top a) (h : a ≤ b) : is_top b := λ c, (ha _).trans h
lemma is_min.mono (ha : is_min a) (h : b ≤ a) : is_min b := λ c hc, h.trans $ ha $ hc.trans h
lemma is_max.mono (ha : is_max a) (h : a ≤ b) : is_max b := λ c hc, (ha $ h.trans hc).trans h
lemma is_min.not_lt (h : is_min a) : ¬ b < a := λ hb, hb.not_le $ h hb.le
lemma is_max.not_lt (h : is_max a) : ¬ a < b := λ hb, hb.not_le $ h hb.le
@[simp] lemma not_is_min_of_lt (h : b < a) : ¬ is_min a := λ ha, ha.not_lt h
@[simp] lemma not_is_max_of_lt (h : a < b) : ¬ is_max a := λ ha, ha.not_lt h
alias not_is_min_of_lt ← has_lt.lt.not_is_min
alias not_is_max_of_lt ← has_lt.lt.not_is_max
lemma is_min_iff_forall_not_lt : is_min a ↔ ∀ b, ¬ b < a :=
⟨λ h _, h.not_lt, λ h b hba, of_not_not $ λ hab, h _ $ hba.lt_of_not_le hab⟩
lemma is_max_iff_forall_not_lt : is_max a ↔ ∀ b, ¬ a < b :=
⟨λ h _, h.not_lt, λ h b hba, of_not_not $ λ hab, h _ $ hba.lt_of_not_le hab⟩
@[simp] lemma not_is_min_iff : ¬ is_min a ↔ ∃ b, b < a :=
by simp_rw [lt_iff_le_not_le, is_min, not_forall, exists_prop]
@[simp] lemma not_is_max_iff : ¬ is_max a ↔ ∃ b, a < b :=
by simp_rw [lt_iff_le_not_le, is_max, not_forall, exists_prop]
@[simp] lemma not_is_min [no_min_order α] (a : α) : ¬ is_min a := not_is_min_iff.2 $ exists_lt a
@[simp] lemma not_is_max [no_max_order α] (a : α) : ¬ is_max a := not_is_max_iff.2 $ exists_gt a
namespace subsingleton
variable [subsingleton α]
protected lemma is_bot (a : α) : is_bot a := λ _, (subsingleton.elim _ _).le
protected lemma is_top (a : α) : is_top a := λ _, (subsingleton.elim _ _).le
protected lemma is_min (a : α) : is_min a := (subsingleton.is_bot _).is_min
protected lemma is_max (a : α) : is_max a := (subsingleton.is_top _).is_max
end subsingleton
end preorder
section partial_order
variables [partial_order α] {a b : α}
protected lemma is_min.eq_of_le (ha : is_min a) (h : b ≤ a) : b = a := h.antisymm $ ha h
protected lemma is_min.eq_of_ge (ha : is_min a) (h : b ≤ a) : a = b := h.antisymm' $ ha h
protected lemma is_max.eq_of_le (ha : is_max a) (h : a ≤ b) : a = b := h.antisymm $ ha h
protected lemma is_max.eq_of_ge (ha : is_max a) (h : a ≤ b) : b = a := h.antisymm' $ ha h
end partial_order
section prod
variables [preorder α] [preorder β] {a a₁ a₂ : α} {b b₁ b₂ : β} {x y : α × β}
lemma is_bot.prod_mk (ha : is_bot a) (hb : is_bot b) : is_bot (a, b) := λ c, ⟨ha _, hb _⟩
lemma is_top.prod_mk (ha : is_top a) (hb : is_top b) : is_top (a, b) := λ c, ⟨ha _, hb _⟩
lemma is_min.prod_mk (ha : is_min a) (hb : is_min b) : is_min (a, b) := λ c hc, ⟨ha hc.1, hb hc.2⟩
lemma is_max.prod_mk (ha : is_max a) (hb : is_max b) : is_max (a, b) := λ c hc, ⟨ha hc.1, hb hc.2⟩
lemma is_bot.fst (hx : is_bot x) : is_bot x.1 := λ c, (hx (c, x.2)).1
lemma is_bot.snd (hx : is_bot x) : is_bot x.2 := λ c, (hx (x.1, c)).2
lemma is_top.fst (hx : is_top x) : is_top x.1 := λ c, (hx (c, x.2)).1
lemma is_top.snd (hx : is_top x) : is_top x.2 := λ c, (hx (x.1, c)).2
lemma is_min.fst (hx : is_min x) : is_min x.1 :=
λ c hc, (hx $ show (c, x.2) ≤ x, from (and_iff_left le_rfl).2 hc).1
lemma is_min.snd (hx : is_min x) : is_min x.2 :=
λ c hc, (hx $ show (x.1, c) ≤ x, from (and_iff_right le_rfl).2 hc).2
lemma is_max.fst (hx : is_max x) : is_max x.1 :=
λ c hc, (hx $ show x ≤ (c, x.2), from (and_iff_left le_rfl).2 hc).1
lemma is_max.snd (hx : is_max x) : is_max x.2 :=
λ c hc, (hx $ show x ≤ (x.1, c), from (and_iff_right le_rfl).2 hc).2
lemma prod.is_bot_iff : is_bot x ↔ is_bot x.1 ∧ is_bot x.2 :=
⟨λ hx, ⟨hx.fst, hx.snd⟩, λ h, h.1.prod_mk h.2⟩
lemma prod.is_top_iff : is_top x ↔ is_top x.1 ∧ is_top x.2 :=
⟨λ hx, ⟨hx.fst, hx.snd⟩, λ h, h.1.prod_mk h.2⟩
lemma prod.is_min_iff : is_min x ↔ is_min x.1 ∧ is_min x.2 :=
⟨λ hx, ⟨hx.fst, hx.snd⟩, λ h, h.1.prod_mk h.2⟩
lemma prod.is_max_iff : is_max x ↔ is_max x.1 ∧ is_max x.2 :=
⟨λ hx, ⟨hx.fst, hx.snd⟩, λ h, h.1.prod_mk h.2⟩
end prod
|
b0ddfbb40cc0cde0578066c5a37bb05e659cabd1 | 35960c5b117752aca7e3e7767c0b393e4dbd72a7 | /src/tactics.lean | 7d1c4b27b7de486c8cc5ef7f8a6d4fc131b0247d | [
"Apache-2.0"
] | permissive | spl/tts | 461dc76b83df8db47e4660d0941dc97e6d4fd7d1 | b65298fea68ce47c8ed3ba3dbce71c1a20dd3481 | refs/heads/master | 1,541,049,198,347 | 1,537,967,023,000 | 1,537,967,029,000 | 119,653,145 | 1 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 1,020 | lean | import tactic.basic
meta def name.update_suffix : name → (string → string) → name
| name.anonymous _ := name.anonymous
| (name.mk_string s p) f := name.mk_string (f s) p
| (name.mk_numeral n p) _ := name.mk_numeral n p
namespace tactic ---------------------------------------------------------------
namespace interactive ----------------------------------------------------------
open interactive
open interactive.types
meta def note_all_applied (p : parse texpr) : tactic unit :=
do e ← to_expr p,
et ← infer_type e,
guard et.is_pi <|> fail format!"'note_all_applied' expects a function, got '{et}'",
ctx ← local_context,
ctx.for_each $ λ h, do
ht ← infer_type h,
tactic.try $ do
unify ht et.binding_domain,
note (h.local_pp_name.update_suffix (++ "'")) none (e.mk_app [h])
end /- namespace -/ interactive ------------------------------------------------
end /- namespace -/ tactic -----------------------------------------------------
|
d0ff6ce570a7bb2e9c7abc7ffc06d5d415a17244 | fe208a542cea7b2d6d7ff79f94d535f6d11d814a | /src/Logic/Q6a_term_solution.lean | 5353c94cecbd378fdafd9f6b77f570ffc2d82f0a | [] | no_license | ImperialCollegeLondon/M1F_room_342_questions | c4b98b14113fe900a7f388762269305faff73e63 | 63de9a6ab9c27a433039dd5530bc9b10b1d227f7 | refs/heads/master | 1,585,807,312,561 | 1,545,232,972,000 | 1,545,232,972,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 712 | lean | inductive fml
| atom (i : ℕ)
| imp (a b : fml)
| not (a : fml)
open fml
infixr ` →' `:50 := imp -- right associative
prefix `¬' `:40 := fml.not
inductive prf : fml → Type
| axk (p q) : prf (p →' q →' p)
| axs (p q r) : prf $ (p →' q →' r) →' (p →' q) →' (p →' r)
| axX (p q) : prf $ ((¬' q) →' (¬' p)) →' p →' q
| mp {p q} : prf (p →' q) → prf p → prf q -- bracket change
open prf
-- example usage:
lemma p_of_p_of_p_of_q (p q : fml) : prf $ (p →' q) →' (p →' p) :=
begin
apply mp (axs p q p),
exact (axk p q)
end
lemma Q6a (p : fml) : prf $ p →' p :=
mp (p_of_p_of_p_of_q _ (p →' p)) $ axk p p
theorem Q6b (p : fml) : prf $ p →' ¬' ¬' p := sorry |
3ceac997e2f8e8bec5e232dfcc8119ead51007db | 6432ea7a083ff6ba21ea17af9ee47b9c371760f7 | /src/Lean/Compiler/LCNF/BaseTypes.lean | 662b98853d9fa0e691f3244bcc14359847aacf69 | [
"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 | 1,216 | lean | /-
Copyright (c) 2022 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Lean.Compiler.LCNF.CompilerM
import Lean.Compiler.LCNF.Types
namespace Lean.Compiler.LCNF
/--
State for the environment extension used to save the LCNF base phase type for declarations
that do not have code associated with them.
Example: constructors, inductive types, foreign functions.
-/
structure BaseTypeExtState where
/-- The LCNF type for the `base` phase. -/
base : PHashMap Name Expr := {}
deriving Inhabited
builtin_initialize baseTypeExt : EnvExtension BaseTypeExtState ←
registerEnvExtension (pure {})
def getOtherDeclBaseType (declName : Name) (us : List Level) : CoreM Expr := do
let info ← getConstInfo declName
let type ← match baseTypeExt.getState (← getEnv) |>.base.find? declName with
| some type => pure type
| none =>
let type ← Meta.MetaM.run' <| toLCNFType info.type
modifyEnv fun env => baseTypeExt.modifyState env fun s => { s with base := s.base.insert declName type }
pure type
return type.instantiateLevelParamsNoCache info.levelParams us
end Lean.Compiler.LCNF
|
5c9ff06b621e3a0e410cff4611f81ddc19ffbfb6 | dd0f5513e11c52db157d2fcc8456d9401a6cd9da | /08_Building_Theories_and_Proofs.org.28.lean | f9e3382949dfab74f726ab0f02b47eb3780f92f1 | [] | 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 | 34 | lean | import standard
print metaclasses
|
597c6a7125c0913c3fabfbde5f955b0b07012ac3 | d4df7c538bbc0b06771d7b7a6f46fbadebb8c5eb | /src/identity_t.lean | 278619cb84769568ee4be8d89264614a38390b9e | [] | no_license | breakerzirconia/LeanTransformers | 11bb924b59ac589d0c6b39586d27cccc8f9e939c | 3c52441b8990bab7d8cf1817df6e04f657a14e1c | refs/heads/main | 1,682,610,526,027 | 1,620,477,129,000 | 1,620,477,129,000 | 365,507,838 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 1,883 | lean | import comonad_transformers.comonad
import comonad_transformers.comonad_trans
import monad_transformers.monad_trans
----------------------------
-- Definition + instances --
----------------------------
structure IdentityT (m : Type → Type) (a : Type) := (run : m a)
open IdentityT
instance {m : Type → Type} [functor m] : functor (IdentityT m) := {
map := λ α β f idntt, mk $ f <$> idntt.run
}
instance {m : Type → Type} [has_pure m] : has_pure (IdentityT m) := {
pure := λ α a, mk $ pure a
}
instance {m : Type → Type} [has_seq m] : has_seq (IdentityT m) := {
seq := λ α β idnttab idntta, mk $ idnttab.run <*> idntta.run
}
instance {m : Type → Type} [applicative m] : applicative (IdentityT m) := {}
instance {m : Type → Type} [has_bind m] : has_bind (IdentityT m) := {
bind := λ α β idntt f, mk $ do a ← idntt.run, (f a).run
}
instance {m : Type → Type} [monad m] : monad (IdentityT m) := {}
instance {m : Type → Type} [has_orelse m] : has_orelse (IdentityT m) := {
orelse := λ α idntt1 idntt2, mk $ idntt1.run <|> idntt2.run
}
instance {m : Type → Type} [alternative m] : alternative (IdentityT m) := {
failure := λ α, mk failure
}
protected def IdentityT_lift {m : Type → Type} [monad m] {a : Type} : m a → IdentityT m a :=
λ ma, mk $ ma
instance : monad_trans IdentityT := ⟨@IdentityT_lift⟩
instance {w : Type → Type} [has_extract w] : has_extract (IdentityT w) := {
extract := λ α idntt, *idntt.run
}
instance {w : Type → Type} [has_extend w] : has_extend (IdentityT w) := {
extend := λ α β f idntt, mk $ (<<=) _ _ (f ∘ mk) idntt.run
}
instance {w : Type → Type} [comonad w] : comonad (IdentityT w) := {}
protected def IdentityT_lower {w : Type → Type} [comonad w] {a : Type} : IdentityT w a → w a :=
λ idntt, idntt.run
instance : comonad_trans IdentityT := ⟨@IdentityT_lower⟩ |
db036391c2bd86693fca8981a78d91c64d8bf9be | 097294e9b80f0d9893ac160b9c7219aa135b51b9 | /answers/exam1_prep_key.lean | f56eb86cb6bc3551aeaaf3d177c44598013929bb | [] | no_license | AbigailCastro17/CS2102-Discrete-Math | cf296251be9418ce90206f5e66bde9163e21abf9 | d741e4d2d6a9b2e0c8380e51706218b8f608cee4 | refs/heads/main | 1,682,891,087,358 | 1,621,401,341,000 | 1,621,401,341,000 | 368,749,959 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 7,186 | lean | /-
Note: on this exam, you are meant to use
built-in Lean types on all problems. You
won't need to use our dm_ versions. Those
we developed just so you understand how
these types are defined and how they work.
-/
/-
1. Product types
A. Use Lean's "structure" mechanism to define an
inductive product type, called employee, with a
constructor, mk, that takes four arguments:
(name : string), (id : nat), (salary : nat),
and (active : bool).
-/
structure employee : Type :=
mk :: (name : string)
(id : nat)
(salary : nat)
(active : bool)
/-
B. Define e1 to be an active employee named
John Smith with id 123 and a salary of 50.
-/
def e1 := employee.mk "John Smith" 123 50 tt
#eval e1.salary
/-
C. Write a function called update_salary that
takes an value of the employee type and a natural
number representing a new salary for that employee
and that returns a new employee object just like
the given one but with the salary field updated
to the given value.
-/
def update_salary : employee → ℕ → employee
| (employee.mk n i _ a) r := (employee.mk n i r a)
/-
D. Use #reduce or #eval to evaluate an expression
in which you use this function to "raise Mr. Smith's
salary to 60."
-/
#eval update_salary e1 60 -- Lean has issues here
/-
2. Sum types
Define an inductive type called rock_paper_scissors
with three values, called "rock", "paper", and
"scissors". Then writen a function called next, that
takes a value of this type and returns a value of
this type according to the following rules: given rock,
return paper; given paper return scissors; given
scissors, return rock.
-/
inductive rock_paper_scissors : Type
| rock : rock_paper_scissors
| paper
| scissors
open rock_paper_scissors -- not required
def next : rock_paper_scissors → rock_paper_scissors
| rock := paper
| paper := scissors
| scissors := rock
/-
3. Inductive definitions
Define a type called nested_doll, where a value of
this type is either "solid_doll" or "shell" applied
to a smaller value of this same type. Then write a
function, depth, that takes an object of this type and
that returns the number of shells it has. The solid
doll has zero shells around it, of course.
-/
inductive nested_doll : Type
| solid_doll
| shell_doll (n : nested_doll)
open nested_doll
def d0 := nested_doll.solid_doll
def d1 := shell_doll d0
def d2 := shell_doll d1
-- ad infinitum
def depth : nested_doll → ℕ
| solid_doll := 0
| (shell_doll d) := 1 + depth d
#eval depth d2
/-
4. Polymorphic types and functions
We have represented ordered pairs as values of
type prod S T, where S and T are type parameters.
An object of this type can be said to have two
fields, fst of type S and snd of type T. Define
an analogous type, called three_tuple S T U,
where S, T, and U are type arguments, and where
an object of this type has three fields, fst: S,
snd : T, and thd : U. Then write a function,
rotate_left, that takes a value of this type and
returns a new value of this type in which the
field values are all rotated on position to the
left. So, the first element becomes the third
in the result, the second becomes the first, and
the third becomes the second. Test your function.
-/
structure three_tuple (S T U : Type) : Type :=
mk :: (fst : S) (snd : T) (thd : U)
-- implicit args best here but not required
def rotate_left {S T U : Type}:
three_tuple S T U → three_tuple T U S
| x := three_tuple.mk x.snd x.thd x.fst
-- destructuring in lieu of projection also works
def rotate_left' {S T U : Type}:
three_tuple S T U → three_tuple T U S
| (three_tuple.mk s t u) := three_tuple.mk t u s
def rotate_left'' {S T U : Type} (x : three_tuple S T U) :=
match x with
| (three_tuple.mk s t u) := three_tuple.mk t u s
end
#reduce rotate_left (three_tuple.mk ff 5 tt)
#reduce rotate_left' (three_tuple.mk ff 5 tt)
/-
5. Sequences (lists) / recursion
A. Write a function, contains_zero, that takes
a value of type "list nat" and that returns true
if any value in the list is zero and otherwise
false . An empty list has no zero values so the
function should return false in this case. Do
not use if..then..else; use pattern matching
within one of the cases to solve this problem.
-/
def contains_zero : list ℕ → bool
| [] := ff
| (h :: t) := match h with
| 0 := tt
| _ := contains_zero t
end
def contains_zero'' : list ℕ → bool
| list.nil := ff
| (list.cons h t) := match h with
| 0 := tt
| _ := contains_zero t
end
-- we'll also take this answer
def contains_zero' : list ℕ → bool
| [] := ff
| (0 :: t) := tt
| (_ :: t) := contains_zero t
-- This answer show Lean's list notations
/-
B. Write a function, inc_values, that takes a
value of type "list nat" and that returns a
list just like the given one but which each value
incremented by one. So for example, given a list,
[1, 2, 3], inc_values will return [2, 3, 4].
-/
def inc_values : list ℕ → list ℕ
| [] := []
| (h :: t) := ((h + 1) :: (inc_values t))
/-
inc_values [1, 2, 3]
2 :: (inc_values [2, 3])
2 :: (3 :: (inc_values [3]))
2 :: 3 :: 4 :: inc_values []
2 :: 3 :: 4 :: nil
[2, 3, 4]
-/
#eval inc_values [1, 2, 3]
/-
6. Binary trees / recursion
A. Define a type called binary_tree_nat. Its values
will represent binary trees of natural numbers. Such
a tree is either "empty" or it is a "node" containing
a natural number and two smaller binary_tree_nat values.
-/
inductive binary_tree_nat : Type
| empty
| node (n : ℕ)
(left : binary_tree_nat)
(right : binary_tree_nat)
/-
B. Then define a function that takes a binary_tree_nat
value as an argument and that returns the sum of all of
the natural numbers in the tree. An empty tree contains
no natural numbers, so the result must be zero in this
case.
-/
def tree_sum : binary_tree_nat → ℕ
| binary_tree_nat.empty := 0
| (binary_tree_nat.node n l r) := n + tree_sum l + tree_sum r
/-
7. Partial functions.
A partial function is a function that is not defined
for all values of its argument type. Yet all function
definitions in Lean must be total. To represent a
partial function as a total funtion in Lean, we use the
option type. We define a function that always returns
an option, and so is total, but the result is in one
of two forms: either "some result" when the result of
the underlying partial function is defined, or "none"
when it's not.
We have implemented a predecessor function, pred : ℕ → ℕ,
that's a little "funky": when applied to zero, it returns
zero, even though it could be argued that the predecessor
of zero (in the natural numbers) should be defined to be
undefined.
Implement a version of pred, let's call it pred_partial,
that for any non-zero argument, n, returns "some result"
where "result" is the predecessor of n, but that when
applied to zero returns "none", to flag that the function
is not defined for zero. Write the function pred_partial
and test it for zero and non-zero argument values.
-/
def pred_partial : ℕ → option ℕ
| 0 := none
| (nat.succ n') := some n' |
839ca231f158de9d2a215e511bcf7bf7240d7cca | ee8cdbabf07f77e7be63a449b8483ce308d37218 | /lean/src/valid/mathd-algebra-149.lean | 77adcd850b5d73c43a998e84c0fa9221e5b97db4 | [
"MIT",
"Apache-2.0"
] | permissive | zeta1999/miniF2F | 6d66c75d1c18152e224d07d5eed57624f731d4b7 | c1ba9629559c5273c92ec226894baa0c1ce27861 | refs/heads/main | 1,681,897,460,642 | 1,620,646,361,000 | 1,620,646,361,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 417 | lean | /-
Copyright (c) 2021 OpenAI. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kunhao Zheng
-/
import data.real.basic
import data.finset.basic
open_locale big_operators
example (f : ℝ → ℝ) (h₀ : ∀ x < -5, f x = x ^ 2 + 5) (h₁ : ∀ x ≥ -5, f x = 3 * x -8) (h₂ : fintype (f⁻¹' {10})): ∑ k in (f⁻¹' {10}).to_finset, k = 6 :=
begin
sorry
end
|
5d183ce78a37bde6dd64f85977108c408044b9d7 | a45212b1526d532e6e83c44ddca6a05795113ddc | /src/group_theory/free_group.lean | ddfd29d8738f3491403507f5354f6366ef74b568 | [
"Apache-2.0"
] | permissive | fpvandoorn/mathlib | b21ab4068db079cbb8590b58fda9cc4bc1f35df4 | b3433a51ea8bc07c4159c1073838fc0ee9b8f227 | refs/heads/master | 1,624,791,089,608 | 1,556,715,231,000 | 1,556,715,231,000 | 165,722,980 | 5 | 0 | Apache-2.0 | 1,552,657,455,000 | 1,547,494,646,000 | Lean | UTF-8 | Lean | false | false | 30,833 | lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
Free groups as a quotient over the reduction relation `a * x * x⁻¹ * b = a * b`.
First we introduce the one step reduction relation
`free_group.red.step`: w * x * x⁻¹ * v ~> w * v
its reflexive transitive closure:
`free_group.red.trans`
and proof that its join is an equivalence relation.
Then we introduce `free_group α` as a quotient over `free_group.red.step`.
-/
import logic.relation
import algebra.group algebra.group_power
import data.fintype data.list.basic
import group_theory.subgroup
open relation
universes u v w
variables {α : Type u}
local attribute [simp] list.append_eq_has_append
namespace free_group
variables {L L₁ L₂ L₃ L₄ : list (α × bool)}
/-- Reduction step: `w * x * x⁻¹ * v ~> w * v` -/
inductive red.step : list (α × bool) → list (α × bool) → Prop
| bnot {L₁ L₂ x b} : red.step (L₁ ++ (x, b) :: (x, bnot b) :: L₂) (L₁ ++ L₂)
attribute [simp] red.step.bnot
/-- Reflexive-transitive closure of red.step -/
def red : list (α × bool) → list (α × bool) → Prop := refl_trans_gen red.step
@[refl] lemma red.refl : red L L := refl_trans_gen.refl
@[trans] lemma red.trans : red L₁ L₂ → red L₂ L₃ → red L₁ L₃ := refl_trans_gen.trans
namespace red
/-- Predicate asserting that word `w₁` can be reduced to `w₂` in one step, i.e. there are words
`w₃ w₄` and letter `x` such that `w₁ = w₃xx⁻¹w₄` and `w₂ = w₃w₄` -/
theorem step.length : ∀ {L₁ L₂ : list (α × bool)}, step L₁ L₂ → L₂.length + 2 = L₁.length
| _ _ (@red.step.bnot _ L1 L2 x b) := by rw [list.length_append, list.length_append]; refl
@[simp] lemma step.bnot_rev {x b} : step (L₁ ++ (x, bnot b) :: (x, b) :: L₂) (L₁ ++ L₂) :=
by cases b; from step.bnot
@[simp] lemma step.cons_bnot {x b} : red.step ((x, b) :: (x, bnot b) :: L) L :=
@step.bnot _ [] _ _ _
@[simp] lemma step.cons_bnot_rev {x b} : red.step ((x, bnot b) :: (x, b) :: L) L :=
@red.step.bnot_rev _ [] _ _ _
theorem step.append_left : ∀ {L₁ L₂ L₃ : list (α × bool)}, step L₂ L₃ → step (L₁ ++ L₂) (L₁ ++ L₃)
| _ _ _ red.step.bnot := by rw [← list.append_assoc, ← list.append_assoc]; constructor
theorem step.cons {x} (H : red.step L₁ L₂) : red.step (x :: L₁) (x :: L₂) :=
@step.append_left _ [x] _ _ H
theorem step.append_right : ∀ {L₁ L₂ L₃ : list (α × bool)}, step L₁ L₂ → step (L₁ ++ L₃) (L₂ ++ L₃)
| _ _ _ red.step.bnot := by simp
lemma not_step_nil : ¬ step [] L :=
begin
generalize h' : [] = L',
assume h,
cases h with L₁ L₂,
simp [list.nil_eq_append_iff] at h',
contradiction
end
lemma step.cons_left_iff {a : α} {b : bool} :
step ((a, b) :: L₁) L₂ ↔ (∃L, step L₁ L ∧ L₂ = (a, b) :: L) ∨ (L₁ = (a, bnot b)::L₂) :=
begin
split,
{ generalize hL : ((a, b) :: L₁ : list _) = L,
assume h,
rcases h with ⟨_ | ⟨p, s'⟩, e, a', b'⟩,
{ simp at hL, simp [*] },
{ simp at hL,
rcases hL with ⟨rfl, rfl⟩,
refine or.inl ⟨s' ++ e, step.bnot, _⟩,
simp } },
{ assume h,
rcases h with ⟨L, h, rfl⟩ | rfl,
{ exact step.cons h },
{ exact step.cons_bnot } }
end
lemma not_step_singleton : ∀ {p : α × bool}, ¬ step [p] L
| (a, b) := by simp [step.cons_left_iff, not_step_nil]
lemma step.cons_cons_iff : ∀{p : α × bool}, step (p :: L₁) (p :: L₂) ↔ step L₁ L₂ :=
by simp [step.cons_left_iff, iff_def, or_imp_distrib] {contextual := tt}
lemma step.append_left_iff : ∀L, step (L ++ L₁) (L ++ L₂) ↔ step L₁ L₂
| [] := by simp
| (p :: l) := by simp [step.append_left_iff l, step.cons_cons_iff]
private theorem step.diamond_aux : ∀ {L₁ L₂ L₃ L₄ : list (α × bool)} {x1 b1 x2 b2},
L₁ ++ (x1, b1) :: (x1, bnot b1) :: L₂ = L₃ ++ (x2, b2) :: (x2, bnot b2) :: L₄ →
L₁ ++ L₂ = L₃ ++ L₄ ∨ ∃ L₅, red.step (L₁ ++ L₂) L₅ ∧ red.step (L₃ ++ L₄) L₅
| [] _ [] _ _ _ _ _ H := by injections; subst_vars; simp
| [] _ [(x3,b3)] _ _ _ _ _ H := by injections; subst_vars; simp
| [(x3,b3)] _ [] _ _ _ _ _ H := by injections; subst_vars; simp
| [] _ ((x3,b3)::(x4,b4)::tl) _ _ _ _ _ H :=
by injections; subst_vars; simp; right; exact ⟨_, red.step.bnot, red.step.cons_bnot⟩
| ((x3,b3)::(x4,b4)::tl) _ [] _ _ _ _ _ H :=
by injections; subst_vars; simp; right; exact ⟨_, red.step.cons_bnot, red.step.bnot⟩
| ((x3,b3)::tl) _ ((x4,b4)::tl2) _ _ _ _ _ H :=
let ⟨H1, H2⟩ := list.cons.inj H in
match step.diamond_aux H2 with
| or.inl H3 := or.inl $ by simp [H1, H3]
| or.inr ⟨L₅, H3, H4⟩ := or.inr
⟨_, step.cons H3, by simpa [H1] using step.cons H4⟩
end
theorem step.diamond : ∀ {L₁ L₂ L₃ L₄ : list (α × bool)},
red.step L₁ L₃ → red.step L₂ L₄ → L₁ = L₂ →
L₃ = L₄ ∨ ∃ L₅, red.step L₃ L₅ ∧ red.step L₄ L₅
| _ _ _ _ red.step.bnot red.step.bnot H := step.diamond_aux H
lemma step.to_red : step L₁ L₂ → red L₁ L₂ :=
refl_trans_gen.single
/-- Church-Rosser theorem for word reduction: If `w1 w2 w3` are words such that `w1` reduces to `w2`
and `w3` respectively, then there is a word `w4` such that `w2` and `w3` reduce to `w4` respectively. -/
theorem church_rosser : red L₁ L₂ → red L₁ L₃ → join red L₂ L₃ :=
relation.church_rosser (assume a b c hab hac,
match b, c, red.step.diamond hab hac rfl with
| b, _, or.inl rfl := ⟨b, by refl, by refl⟩
| b, c, or.inr ⟨d, hbd, hcd⟩ := ⟨d, refl_gen.single hbd, hcd.to_red⟩
end)
lemma cons_cons {p} : red L₁ L₂ → red (p :: L₁) (p :: L₂) :=
refl_trans_gen_lift (list.cons p) (assume a b, step.cons)
lemma cons_cons_iff (p) : red (p :: L₁) (p :: L₂) ↔ red L₁ L₂ :=
iff.intro
begin
generalize eq₁ : (p :: L₁ : list _) = LL₁,
generalize eq₂ : (p :: L₂ : list _) = LL₂,
assume h,
induction h using relation.refl_trans_gen.head_induction_on
with L₁ L₂ h₁₂ h ih
generalizing L₁ L₂,
{ subst_vars, cases eq₂, constructor },
{ subst_vars,
cases p with a b,
rw [step.cons_left_iff] at h₁₂,
rcases h₁₂ with ⟨L, h₁₂, rfl⟩ | rfl,
{ exact (ih rfl rfl).head h₁₂ },
{ exact (cons_cons h).tail step.cons_bnot_rev } }
end
cons_cons
lemma append_append_left_iff : ∀L, red (L ++ L₁) (L ++ L₂) ↔ red L₁ L₂
| [] := iff.refl _
| (p :: L) := by simp [append_append_left_iff L, cons_cons_iff]
lemma append_append (h₁ : red L₁ L₃) (h₂ : red L₂ L₄) : red (L₁ ++ L₂) (L₃ ++ L₄) :=
(refl_trans_gen_lift (λL, L ++ L₂) (assume a b, step.append_right) h₁).trans
((append_append_left_iff _).2 h₂)
lemma to_append_iff : red L (L₁ ++ L₂) ↔ (∃L₃ L₄, L = L₃ ++ L₄ ∧ red L₃ L₁ ∧ red L₄ L₂) :=
iff.intro
begin
generalize eq : L₁ ++ L₂ = L₁₂,
assume h,
induction h with L' L₁₂ hLL' h ih generalizing L₁ L₂,
{ exact ⟨_, _, eq.symm, by refl, by refl⟩ },
{ cases h with s e a b,
rcases list.append_eq_append_iff.1 eq with ⟨s', rfl, rfl⟩ | ⟨e', rfl, rfl⟩,
{ have : L₁ ++ (s' ++ ((a, b) :: (a, bnot b) :: e)) = (L₁ ++ s') ++ ((a, b) :: (a, bnot b) :: e),
{ simp },
rcases ih this with ⟨w₁, w₂, rfl, h₁, h₂⟩,
exact ⟨w₁, w₂, rfl, h₁, h₂.tail step.bnot⟩ },
{ have : (s ++ ((a, b) :: (a, bnot b) :: e')) ++ L₂ = s ++ ((a, b) :: (a, bnot b) :: (e' ++ L₂)),
{ simp },
rcases ih this with ⟨w₁, w₂, rfl, h₁, h₂⟩,
exact ⟨w₁, w₂, rfl, h₁.tail step.bnot, h₂⟩ }, }
end
(assume ⟨L₃, L₄, eq, h₃, h₄⟩, eq.symm ▸ append_append h₃ h₄)
/-- The empty word `[]` only reduces to itself. -/
theorem nil_iff : red [] L ↔ L = [] :=
refl_trans_gen_iff_eq (assume l, red.not_step_nil)
/-- A letter only reduces to itself. -/
theorem singleton_iff {x} : red [x] L₁ ↔ L₁ = [x] :=
refl_trans_gen_iff_eq (assume l, not_step_singleton)
/-- If `x` is a letter and `w` is a word such that `xw` reduces to the empty word, then `w` reduces
to `x⁻¹` -/
theorem cons_nil_iff_singleton {x b} : red ((x, b) :: L) [] ↔ red L [(x, bnot b)] :=
iff.intro
(assume h,
have h₁ : red ((x, bnot b) :: (x, b) :: L) [(x, bnot b)], from cons_cons h,
have h₂ : red ((x, bnot b) :: (x, b) :: L) L, from refl_trans_gen.single step.cons_bnot_rev,
let ⟨L', h₁, h₂⟩ := church_rosser h₁ h₂ in
by rw [singleton_iff] at h₁; subst L'; assumption)
(assume h, (cons_cons h).tail step.cons_bnot)
theorem red_iff_irreducible {x1 b1 x2 b2} (h : (x1, b1) ≠ (x2, b2)) :
red [(x1, bnot b1), (x2, b2)] L ↔ L = [(x1, bnot b1), (x2, b2)] :=
begin
apply refl_trans_gen_iff_eq,
generalize eq : [(x1, bnot b1), (x2, b2)] = L',
assume L h',
cases h',
simp [list.cons_eq_append_iff, list.nil_eq_append_iff] at eq,
rcases eq with ⟨rfl, ⟨rfl, rfl⟩, ⟨rfl, rfl⟩, rfl⟩, subst_vars,
simp at h,
contradiction
end
/-- If `x` and `y` are distinct letters and `w₁ w₂` are words such that `xw₁` reduces to `yw₂`, then
`w₁` reduces to `x⁻¹yw₂`. -/
theorem inv_of_red_of_ne {x1 b1 x2 b2}
(H1 : (x1, b1) ≠ (x2, b2))
(H2 : red ((x1, b1) :: L₁) ((x2, b2) :: L₂)) :
red L₁ ((x1, bnot b1) :: (x2, b2) :: L₂) :=
begin
have : red ((x1, b1) :: L₁) ([(x2, b2)] ++ L₂), from H2,
rcases to_append_iff.1 this with ⟨_ | ⟨p, L₃⟩, L₄, eq, h₁, h₂⟩,
{ simp [nil_iff] at h₁, contradiction },
{ cases eq,
show red (L₃ ++ L₄) ([(x1, bnot b1), (x2, b2)] ++ L₂),
apply append_append _ h₂,
have h₁ : red ((x1, bnot b1) :: (x1, b1) :: L₃) [(x1, bnot b1), (x2, b2)],
{ exact cons_cons h₁ },
have h₂ : red ((x1, bnot b1) :: (x1, b1) :: L₃) L₃,
{ exact step.cons_bnot_rev.to_red },
rcases church_rosser h₁ h₂ with ⟨L', h₁, h₂⟩,
rw [red_iff_irreducible H1] at h₁,
rwa [h₁] at h₂ }
end
theorem step.sublist (H : red.step L₁ L₂) : L₂ <+ L₁ :=
by cases H; simp; constructor; constructor; refl
/-- If `w₁ w₂` are words such that `w₁` reduces to `w₂`, then `w₂` is a sublist of `w₁`. -/
theorem sublist : red L₁ L₂ → L₂ <+ L₁ :=
refl_trans_gen_of_transitive_reflexive
(λl, list.sublist.refl l) (λa b c hab hbc, list.sublist.trans hbc hab) (λa b, red.step.sublist)
theorem sizeof_of_step : ∀ {L₁ L₂ : list (α × bool)}, step L₁ L₂ → L₂.sizeof < L₁.sizeof
| _ _ (@step.bnot _ L1 L2 x b) :=
begin
induction L1 with hd tl ih,
case list.nil
{ dsimp [list.sizeof],
have H : 1 + sizeof (x, b) + (1 + sizeof (x, bnot b) + list.sizeof L2)
= (list.sizeof L2 + 1) + (sizeof (x, b) + sizeof (x, bnot b) + 1),
{ ac_refl },
rw H,
exact nat.le_add_right _ _ },
case list.cons
{ dsimp [list.sizeof],
exact nat.add_lt_add_left ih _ }
end
theorem length (h : red L₁ L₂) : ∃ n, L₁.length = L₂.length + 2 * n :=
begin
induction h with L₂ L₃ h₁₂ h₂₃ ih,
{ exact ⟨0, rfl⟩ },
{ rcases ih with ⟨n, eq⟩,
existsi (1 + n),
simp [mul_add, eq, (step.length h₂₃).symm] }
end
theorem antisymm (h₁₂ : red L₁ L₂) : red L₂ L₁ → L₁ = L₂ :=
match L₁, h₁₂.cases_head with
| _, or.inl rfl := assume h, rfl
| L₁, or.inr ⟨L₃, h₁₃, h₃₂⟩ := assume h₂₁,
let ⟨n, eq⟩ := length (h₃₂.trans h₂₁) in
have list.length L₃ + 0 = list.length L₃ + (2 * n + 2),
by simpa [(step.length h₁₃).symm, add_comm, add_assoc] using eq,
(nat.no_confusion $ nat.add_left_cancel this)
end
end red
theorem equivalence_join_red : equivalence (join (@red α)) :=
equivalence_join_refl_trans_gen $ assume a b c hab hac,
(match b, c, red.step.diamond hab hac rfl with
| b, _, or.inl rfl := ⟨b, by refl, by refl⟩
| b, c, or.inr ⟨d, hbd, hcd⟩ := ⟨d, refl_gen.single hbd, refl_trans_gen.single hcd⟩
end)
theorem join_red_of_step (h : red.step L₁ L₂) : join red L₁ L₂ :=
join_of_single reflexive_refl_trans_gen h.to_red
theorem eqv_gen_step_iff_join_red : eqv_gen red.step L₁ L₂ ↔ join red L₁ L₂ :=
iff.intro
(assume h,
have eqv_gen (join red) L₁ L₂ := eqv_gen_mono (assume a b, join_red_of_step) h,
(eqv_gen_iff_of_equivalence $ equivalence_join_red).1 this)
(join_of_equivalence (eqv_gen.is_equivalence _) $ assume a b,
refl_trans_gen_of_equivalence (eqv_gen.is_equivalence _) eqv_gen.rel)
end free_group
/-- The free group over a type, i.e. the words formed by the elements of the type and their formal
inverses, quotient by one step reduction. -/
def free_group (α : Type u) : Type u :=
quot $ @free_group.red.step α
namespace free_group
variables {α} {L L₁ L₂ L₃ L₄ : list (α × bool)}
def mk (L) : free_group α := quot.mk red.step L
@[simp] lemma quot_mk_eq_mk : quot.mk red.step L = mk L := rfl
@[simp] lemma quot_lift_mk (β : Type v) (f : list (α × bool) → β)
(H : ∀ L₁ L₂, red.step L₁ L₂ → f L₁ = f L₂) :
quot.lift f H (mk L) = f L := rfl
@[simp] lemma quot_lift_on_mk (β : Type v) (f : list (α × bool) → β)
(H : ∀ L₁ L₂, red.step L₁ L₂ → f L₁ = f L₂) :
quot.lift_on (mk L) f H = f L := rfl
instance : has_one (free_group α) := ⟨mk []⟩
lemma one_eq_mk : (1 : free_group α) = mk [] := rfl
instance : has_mul (free_group α) :=
⟨λ x y, quot.lift_on x
(λ L₁, quot.lift_on y (λ L₂, mk $ L₁ ++ L₂) (λ L₂ L₃ H, quot.sound $ red.step.append_left H))
(λ L₁ L₂ H, quot.induction_on y $ λ L₃, quot.sound $ red.step.append_right H)⟩
@[simp] lemma mul_mk : mk L₁ * mk L₂ = mk (L₁ ++ L₂) := rfl
instance : has_inv (free_group α) :=
⟨λx, quot.lift_on x (λ L, mk (L.map $ λ x : α × bool, (x.1, bnot x.2)).reverse)
(assume a b h, quot.sound $ by cases h; simp)⟩
@[simp] lemma inv_mk : (mk L)⁻¹ = mk (L.map $ λ x : α × bool, (x.1, bnot x.2)).reverse := rfl
instance : group (free_group α) :=
{ mul := (*),
one := 1,
inv := has_inv.inv,
mul_assoc := by rintros ⟨L₁⟩ ⟨L₂⟩ ⟨L₃⟩; simp,
one_mul := by rintros ⟨L⟩; refl,
mul_one := by rintros ⟨L⟩; simp [one_eq_mk],
mul_left_inv := by rintros ⟨L⟩; exact (list.rec_on L rfl $
λ ⟨x, b⟩ tl ih, eq.trans (quot.sound $ by simp [one_eq_mk]) ih) }
/-- `of x` is the canonical injection from the type to the free group over that type by sending each
element to the equivalence class of the letter that is the element. -/
def of (x : α) : free_group α :=
mk [(x, tt)]
theorem red.exact : mk L₁ = mk L₂ ↔ join red L₁ L₂ :=
calc (mk L₁ = mk L₂) ↔ eqv_gen red.step L₁ L₂ : iff.intro (quot.exact _) quot.eqv_gen_sound
... ↔ join red L₁ L₂ : eqv_gen_step_iff_join_red
/-- The canonical injection from the type to the free group is an injection. -/
theorem of.inj {x y : α} (H : of x = of y) : x = y :=
let ⟨L₁, hx, hy⟩ := red.exact.1 H in
by simp [red.singleton_iff] at hx hy; cc
section to_group
variables {β : Type v} [group β] (f : α → β) {x y : free_group α}
def to_group.aux : list (α × bool) → β :=
λ L, list.prod $ L.map $ λ x, cond x.2 (f x.1) (f x.1)⁻¹
theorem red.step.to_group {f : α → β} (H : red.step L₁ L₂) :
to_group.aux f L₁ = to_group.aux f L₂ :=
by cases H with _ _ _ b; cases b; simp [to_group.aux]
/-- If `β` is a group, then any function from `α` to `β`
extends uniquely to a group homomorphism from
the free group over `α` to `β` -/
def to_group : free_group α → β :=
quot.lift (to_group.aux f) $ λ L₁ L₂ H, red.step.to_group H
variable {f}
@[simp] lemma to_group.mk : to_group f (mk L) =
list.prod (L.map $ λ x, cond x.2 (f x.1) (f x.1)⁻¹) :=
rfl
@[simp] lemma to_group.of {x} : to_group f (of x) = f x :=
one_mul _
instance to_group.is_group_hom : is_group_hom (to_group f) :=
⟨by rintros ⟨L₁⟩ ⟨L₂⟩; simp⟩
@[simp] lemma to_group.mul : to_group f (x * y) = to_group f x * to_group f y :=
is_group_hom.map_mul _ _ _
@[simp] lemma to_group.one : to_group f 1 = 1 :=
is_group_hom.map_one _
@[simp] lemma to_group.inv : to_group f x⁻¹ = (to_group f x)⁻¹ :=
is_group_hom.map_inv _ _
theorem to_group.unique (g : free_group α → β) [is_group_hom g]
(hg : ∀ x, g (of x) = f x) : ∀{x}, g x = to_group f x :=
by rintros ⟨L⟩; exact list.rec_on L (is_group_hom.map_one g)
(λ ⟨x, b⟩ t (ih : g (mk t) = _), bool.rec_on b
(show g ((of x)⁻¹ * mk t) = to_group f (mk ((x, ff) :: t)),
by simp [is_group_hom.map_mul g, is_group_hom.map_inv g, hg, ih, to_group, to_group.aux])
(show g (of x * mk t) = to_group f (mk ((x, tt) :: t)),
by simp [is_group_hom.map_mul g, is_group_hom.map_inv g, hg, ih, to_group, to_group.aux]))
theorem to_group.of_eq (x : free_group α) : to_group of x = x :=
eq.symm $ to_group.unique id (λ x, rfl)
theorem to_group.range_subset {s : set β} [is_subgroup s] (H : set.range f ⊆ s) :
set.range (to_group f) ⊆ s :=
by rintros _ ⟨⟨L⟩, rfl⟩; exact list.rec_on L (is_submonoid.one_mem s)
(λ ⟨x, b⟩ tl ih, bool.rec_on b
(by simp at ih ⊢; from is_submonoid.mul_mem
(is_subgroup.inv_mem $ H ⟨x, rfl⟩) ih)
(by simp at ih ⊢; from is_submonoid.mul_mem (H ⟨x, rfl⟩) ih))
theorem to_group.range_eq_closure :
set.range (to_group f) = group.closure (set.range f) :=
set.subset.antisymm
(to_group.range_subset group.subset_closure)
(group.closure_subset $ λ y ⟨x, hx⟩, ⟨of x, by simpa⟩)
end to_group
section map
variables {β : Type v} (f : α → β) {x y : free_group α}
def map.aux (L : list (α × bool)) : list (β × bool) :=
L.map $ λ x, (f x.1, x.2)
/-- Any function from `α` to `β` extends uniquely
to a group homomorphism from the free group
ver `α` to the free group over `β`. -/
def map (x : free_group α) : free_group β :=
x.lift_on (λ L, mk $ map.aux f L) $
λ L₁ L₂ H, quot.sound $ by cases H; simp [map.aux]
instance map.is_group_hom : is_group_hom (map f) :=
⟨by rintros ⟨L₁⟩ ⟨L₂⟩; simp [map, map.aux]⟩
variable {f}
@[simp] lemma map.mk : map f (mk L) = mk (L.map (λ x, (f x.1, x.2))) :=
rfl
@[simp] lemma map.id : map id x = x :=
have H1 : (λ (x : α × bool), x) = id := rfl,
by rcases x with ⟨L⟩; simp [H1]
@[simp] lemma map.id' : map (λ z, z) x = x := map.id
theorem map.comp {γ : Type w} {f : α → β} {g : β → γ} {x} :
map g (map f x) = map (g ∘ f) x :=
by rcases x with ⟨L⟩; simp
@[simp] lemma map.of {x} : map f (of x) = of (f x) := rfl
@[simp] lemma map.mul : map f (x * y) = map f x * map f y :=
is_group_hom.map_mul _ x y
@[simp] lemma map.one : map f 1 = 1 :=
is_group_hom.map_one _
@[simp] lemma map.inv : map f x⁻¹ = (map f x)⁻¹ :=
is_group_hom.map_inv _ x
theorem map.unique (g : free_group α → free_group β) [is_group_hom g]
(hg : ∀ x, g (of x) = of (f x)) : ∀{x}, g x = map f x :=
by rintros ⟨L⟩; exact list.rec_on L (is_group_hom.map_one g)
(λ ⟨x, b⟩ t (ih : g (mk t) = map f (mk t)), bool.rec_on b
(show g ((of x)⁻¹ * mk t) = map f ((of x)⁻¹ * mk t),
by simp [is_group_hom.map_mul g, is_group_hom.map_inv g, hg, ih])
(show g (of x * mk t) = map f (of x * mk t),
by simp [is_group_hom.map_mul g, hg, ih]))
/-- Equivalent types give rise to equivalent free groups. -/
def free_group_congr {α β} (e : α ≃ β) : free_group α ≃ free_group β :=
⟨map e, map e.symm,
λ x, by simp [function.comp, map.comp],
λ x, by simp [function.comp, map.comp]⟩
theorem map_eq_to_group : map f x = to_group (of ∘ f) x :=
eq.symm $ map.unique _ $ λ x, by simp
end map
section prod
variables [group α] (x y : free_group α)
/-- If `α` is a group, then any function from `α` to `α`
extends uniquely to a homomorphism from the
free group over `α` to `α`. This is the multiplicative
version of `sum`. -/
def prod : α :=
to_group id x
variables {x y}
@[simp] lemma prod_mk :
prod (mk L) = list.prod (L.map $ λ x, cond x.2 x.1 x.1⁻¹) :=
rfl
@[simp] lemma prod.of {x : α} : prod (of x) = x :=
to_group.of
instance prod.is_group_hom : is_group_hom (@prod α _) :=
to_group.is_group_hom
@[simp] lemma prod.mul : prod (x * y) = prod x * prod y :=
to_group.mul
@[simp] lemma prod.one : prod (1:free_group α) = 1 :=
to_group.one
@[simp] lemma prod.inv : prod x⁻¹ = (prod x)⁻¹ :=
to_group.inv
lemma prod.unique (g : free_group α → α) [is_group_hom g]
(hg : ∀ x, g (of x) = x) {x} :
g x = prod x :=
to_group.unique g hg
end prod
theorem to_group_eq_prod_map {β : Type v} [group β] {f : α → β} {x} :
to_group f x = prod (map f x) :=
eq.symm $ to_group.unique (prod ∘ map f) $ λ _, by simp
section sum
variables [add_group α] (x y : free_group α)
/-- If `α` is a group, then any function from `α` to `α`
extends uniquely to a homomorphism from the
free group over `α` to `α`. This is the additive
version of `prod`. -/
def sum : α :=
@prod (multiplicative _) _ x
variables {x y}
@[simp] lemma sum_mk :
sum (mk L) = list.sum (L.map $ λ x, cond x.2 x.1 (-x.1)) :=
rfl
@[simp] lemma sum.of {x : α} : sum (of x) = x :=
prod.of
instance sum.is_group_hom : is_group_hom (@sum α _) :=
prod.is_group_hom
@[simp] lemma sum.sum : sum (x * y) = sum x + sum y :=
prod.mul
@[simp] lemma sum.one : sum (1:free_group α) = 0 :=
prod.one
@[simp] lemma sum.inv : sum x⁻¹ = -sum x :=
prod.inv
end sum
def free_group_empty_equiv_unit : free_group empty ≃ unit :=
{ to_fun := λ _, (),
inv_fun := λ _, 1,
left_inv := by rintros ⟨_ | ⟨⟨⟨⟩, _⟩, _⟩⟩; refl,
right_inv := λ ⟨⟩, rfl }
def free_group_unit_equiv_int : free_group unit ≃ int :=
{ to_fun := λ x, sum $ map (λ _, 1) x,
inv_fun := λ x, of () ^ x,
left_inv := by rintros ⟨L⟩; exact list.rec_on L rfl
(λ ⟨⟨⟩, b⟩ tl ih, by cases b; simp [gpow_add] at ih ⊢; rw ih; refl),
right_inv := λ x, int.induction_on x (by simp)
(λ i ih, by simp at ih; simp [gpow_add, ih])
(λ i ih, by simp at ih; simp [gpow_add, ih]) }
section category
variables {β : Type u}
instance : monad free_group.{u} :=
{ pure := λ α, of,
map := λ α β, map,
bind := λ α β x f, to_group f x }
@[elab_as_eliminator]
protected theorem induction_on
{C : free_group α → Prop}
(z : free_group α)
(C1 : C 1)
(Cp : ∀ x, C $ pure x)
(Ci : ∀ x, C (pure x) → C (pure x)⁻¹)
(Cm : ∀ x y, C x → C y → C (x * y)) : C z :=
quot.induction_on z $ λ L, list.rec_on L C1 $ λ ⟨x, b⟩ tl ih,
bool.rec_on b (Cm _ _ (Ci _ $ Cp x) ih) (Cm _ _ (Cp x) ih)
@[simp] lemma map_pure (f : α → β) (x : α) : f <$> (pure x : free_group α) = pure (f x) :=
map.of
@[simp] lemma map_one (f : α → β) : f <$> (1 : free_group α) = 1 :=
map.one
@[simp] lemma map_mul (f : α → β) (x y : free_group α) : f <$> (x * y) = f <$> x * f <$> y :=
map.mul
@[simp] lemma map_inv (f : α → β) (x : free_group α) : f <$> (x⁻¹) = (f <$> x)⁻¹ :=
map.inv
@[simp] lemma pure_bind (f : α → free_group β) (x) : pure x >>= f = f x :=
to_group.of
@[simp] lemma one_bind (f : α → free_group β) : 1 >>= f = 1 :=
@@to_group.one _ f
@[simp] lemma mul_bind (f : α → free_group β) (x y : free_group α) : x * y >>= f = (x >>= f) * (y >>= f) :=
to_group.mul
@[simp] lemma inv_bind (f : α → free_group β) (x : free_group α) : x⁻¹ >>= f = (x >>= f)⁻¹ :=
to_group.inv
instance : is_lawful_monad free_group.{u} :=
{ id_map := λ α x, free_group.induction_on x (map_one id) (λ x, map_pure id x)
(λ x ih, by rw [map_inv, ih]) (λ x y ihx ihy, by rw [map_mul, ihx, ihy]),
pure_bind := λ α β x f, pure_bind f x,
bind_assoc := λ α β γ x f g, free_group.induction_on x
(by iterate 3 { rw one_bind }) (λ x, by iterate 2 { rw pure_bind })
(λ x ih, by iterate 3 { rw inv_bind }; rw ih)
(λ x y ihx ihy, by iterate 3 { rw mul_bind }; rw [ihx, ihy]),
bind_pure_comp_eq_map := λ α β f x, free_group.induction_on x
(by rw [one_bind, map_one]) (λ x, by rw [pure_bind, map_pure])
(λ x ih, by rw [inv_bind, map_inv, ih]) (λ x y ihx ihy, by rw [mul_bind, map_mul, ihx, ihy]) }
end category
section reduce
variable [decidable_eq α]
/-- The maximal reduction of a word. It is computable
iff `α` has decidable equality. -/
def reduce (L : list (α × bool)) : list (α × bool) :=
list.rec_on L [] $ λ hd1 tl1 ih,
list.cases_on ih [hd1] $ λ hd2 tl2,
if hd1.1 = hd2.1 ∧ hd1.2 = bnot hd2.2 then tl2
else hd1 :: hd2 :: tl2
@[simp] lemma reduce.cons (x) : reduce (x :: L) =
list.cases_on (reduce L) [x] (λ hd tl,
if x.1 = hd.1 ∧ x.2 = bnot hd.2 then tl
else x :: hd :: tl) := rfl
/-- The first theorem that characterises the function
`reduce`: a word reduces to its maximal reduction. -/
theorem reduce.red : red L (reduce L) :=
begin
induction L with hd1 tl1 ih,
case list.nil
{ constructor },
case list.cons
{ dsimp,
revert ih,
generalize htl : reduce tl1 = TL,
intro ih,
cases TL with hd2 tl2,
case list.nil
{ exact red.cons_cons ih },
case list.cons
{ dsimp,
by_cases h : hd1.fst = hd2.fst ∧ hd1.snd = bnot (hd2.snd),
{ rw [if_pos h],
transitivity,
{ exact red.cons_cons ih },
{ cases hd1, cases hd2, cases h,
dsimp at *, subst_vars,
exact red.step.cons_bnot_rev.to_red } },
{ rw [if_neg h],
exact red.cons_cons ih } } }
end
theorem reduce.not {p : Prop} : ∀ {L₁ L₂ L₃ : list (α × bool)} {x b}, reduce L₁ = L₂ ++ (x, b) :: (x, bnot b) :: L₃ → p
| [] L2 L3 _ _ := λ h, by cases L2; injections
| ((x,b)::L1) L2 L3 x' b' := begin
dsimp,
cases r : reduce L1,
{ dsimp, intro h,
have := congr_arg list.length h,
simp [-add_comm] at this,
exact absurd this dec_trivial },
cases hd with y c,
by_cases x = y ∧ b = bnot c; simp [h]; intro H,
{ rw H at r,
exact @reduce.not L1 ((y,c)::L2) L3 x' b' r },
rcases L2 with _|⟨a, L2⟩,
{ injections, subst_vars,
simp at h, cc },
{ refine @reduce.not L1 L2 L3 x' b' _,
injection H with _ H,
rw [r, H], refl }
end
/-- The second theorem that characterises the
function `reduce`: the maximal reduction of a word
only reduces to itself. -/
theorem reduce.min (H : red (reduce L₁) L₂) : reduce L₁ = L₂ :=
begin
induction H with L1 L' L2 H1 H2 ih,
{ refl },
{ cases H1 with L4 L5 x b,
exact reduce.not H2 }
end
/-- `reduce` is idempotent, i.e. the maximal reduction
of the maximal reduction of a word is the maximal
reduction of the word. -/
theorem reduce.idem : reduce (reduce L) = reduce L :=
eq.symm $ reduce.min reduce.red
theorem reduce.step.eq (H : red.step L₁ L₂) : reduce L₁ = reduce L₂ :=
let ⟨L₃, HR13, HR23⟩ := red.church_rosser reduce.red (reduce.red.head H) in
(reduce.min HR13).trans (reduce.min HR23).symm
/-- If a word reduces to another word, then they have
a common maximal reduction. -/
theorem reduce.eq_of_red (H : red L₁ L₂) : reduce L₁ = reduce L₂ :=
let ⟨L₃, HR13, HR23⟩ := red.church_rosser reduce.red (red.trans H reduce.red) in
(reduce.min HR13).trans (reduce.min HR23).symm
/-- If two words correspond to the same element in
the free group, then they have a common maximal
reduction. This is the proof that the function that
sends an element of the free group to its maximal
reduction is well-defined. -/
theorem reduce.sound (H : mk L₁ = mk L₂) : reduce L₁ = reduce L₂ :=
let ⟨L₃, H13, H23⟩ := red.exact.1 H in
(reduce.eq_of_red H13).trans (reduce.eq_of_red H23).symm
/-- If two words have a common maximal reduction,
then they correspond to the same element in the free group. -/
theorem reduce.exact (H : reduce L₁ = reduce L₂) : mk L₁ = mk L₂ :=
red.exact.2 ⟨reduce L₂, H ▸ reduce.red, reduce.red⟩
/-- A word and its maximal reduction correspond to
the same element of the free group. -/
theorem reduce.self : mk (reduce L) = mk L :=
reduce.exact reduce.idem
/-- If words `w₁ w₂` are such that `w₁` reduces to `w₂`,
then `w₂` reduces to the maximal reduction of `w₁`. -/
theorem reduce.rev (H : red L₁ L₂) : red L₂ (reduce L₁) :=
(reduce.eq_of_red H).symm ▸ reduce.red
/-- The function that sends an element of the free
group to its maximal reduction. -/
def to_word : free_group α → list (α × bool) :=
quot.lift reduce $ λ L₁ L₂ H, reduce.step.eq H
def to_word.mk : ∀{x : free_group α}, mk (to_word x) = x :=
by rintros ⟨L⟩; exact reduce.self
def to_word.inj : ∀(x y : free_group α), to_word x = to_word y → x = y :=
by rintros ⟨L₁⟩ ⟨L₂⟩; exact reduce.exact
/-- Constructive Church-Rosser theorem (compare `church_rosser`). -/
def reduce.church_rosser (H12 : red L₁ L₂) (H13 : red L₁ L₃) :
{ L₄ // red L₂ L₄ ∧ red L₃ L₄ } :=
⟨reduce L₁, reduce.rev H12, reduce.rev H13⟩
instance : decidable_eq (free_group α) :=
function.injective.decidable_eq to_word.inj
instance red.decidable_rel : decidable_rel (@red α)
| [] [] := is_true red.refl
| [] (hd2::tl2) := is_false $ λ H, list.no_confusion (red.nil_iff.1 H)
| ((x,b)::tl) [] := match red.decidable_rel tl [(x, bnot b)] with
| is_true H := is_true $ red.trans (red.cons_cons H) $
(@red.step.bnot _ [] [] _ _).to_red
| is_false H := is_false $ λ H2, H $ red.cons_nil_iff_singleton.1 H2
end
| ((x1,b1)::tl1) ((x2,b2)::tl2) := if h : (x1, b1) = (x2, b2)
then match red.decidable_rel tl1 tl2 with
| is_true H := is_true $ h ▸ red.cons_cons H
| is_false H := is_false $ λ H2, H $ h ▸ (red.cons_cons_iff _).1 $ H2
end
else match red.decidable_rel tl1 ((x1,bnot b1)::(x2,b2)::tl2) with
| is_true H := is_true $ (red.cons_cons H).tail red.step.cons_bnot
| is_false H := is_false $ λ H2, H $ red.inv_of_red_of_ne h H2
end
/-- A list containing every word that `w₁` reduces to. -/
def red.enum (L₁ : list (α × bool)) : list (list (α × bool)) :=
list.filter (λ L₂, red L₁ L₂) (list.sublists L₁)
theorem red.enum.sound (H : L₂ ∈ red.enum L₁) : red L₁ L₂ :=
list.of_mem_filter H
theorem red.enum.complete (H : red L₁ L₂) : L₂ ∈ red.enum L₁ :=
list.mem_filter_of_mem (list.mem_sublists.2 $ red.sublist H) H
instance : fintype { L₂ // red L₁ L₂ } :=
fintype.subtype (list.to_finset $ red.enum L₁) $
λ L₂, ⟨λ H, red.enum.sound $ list.mem_to_finset.1 H,
λ H, list.mem_to_finset.2 $ red.enum.complete H⟩
end reduce
end free_group
|
aa9b6655c2a881dc3c58ea0fd4bad2006bb6862f | 2c096fdfecf64e46ea7bc6ce5521f142b5926864 | /src/Lean/MetavarContext.lean | 37ad643c03e8dd9a66b14ce2a419fb9f647e0766 | [
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"BSD-3-Clause",
"LGPL-2.0-or-later",
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"HPND",
"LicenseRef-scancode-pcre",
"ISC",
"LGPL-2.1-only",
"LicenseRef-scancode-other-permissive",
"SunPro",
"CMU-Mach"... | permissive | Kha/lean4 | 1005785d2c8797ae266a303968848e5f6ce2fe87 | b99e11346948023cd6c29d248cd8f3e3fb3474cf | refs/heads/master | 1,693,355,498,027 | 1,669,080,461,000 | 1,669,113,138,000 | 184,748,176 | 0 | 0 | Apache-2.0 | 1,665,995,520,000 | 1,556,884,930,000 | Lean | UTF-8 | Lean | false | false | 63,836 | 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.MonadCache
import Lean.LocalContext
namespace Lean
/-!
The metavariable context stores metavariable declarations and their
assignments. It is used in the elaborator, tactic framework, unifier
(aka `isDefEq`), and type class resolution (TC). First, we list all
the requirements imposed by these modules.
- We may invoke TC while executing `isDefEq`. We need this feature to
be able to solve unification problems such as:
```
f ?a (ringAdd ?s) ?x ?y =?= f Int intAdd n m
```
where `(?a : Type) (?s : Ring ?a) (?x ?y : ?a)`
During `isDefEq` (i.e., unification), it will need to solve the constrain
```
ringAdd ?s =?= intAdd
```
We say `ringAdd ?s` is stuck because it cannot be reduced until we
synthesize the term `?s : Ring ?a` using TC. This can be done since we
have assigned `?a := Int` when solving `?a =?= Int`.
- TC uses `isDefEq`, and `isDefEq` may create TC problems as shown
above. Thus, we may have nested TC problems.
- `isDefEq` extends the local context when going inside binders. Thus,
the local context for nested TC may be an extension of the local
context for outer TC.
- TC should not assign metavariables created by the elaborator, simp,
tactic framework, and outer TC problems. Reason: TC commits to the
first solution it finds. Consider the TC problem `Coe Nat ?x`,
where `?x` is a metavariable created by the caller. There are many
solutions to this problem (e.g., `?x := Int`, `?x := Real`, ...),
and it doesn’t make sense to commit to the first one since TC does
not know the constraints the caller may impose on `?x` after the
TC problem is solved.
Remark: we claim it is not feasible to make the whole system backtrackable,
and allow the caller to backtrack back to TC and ask it for another solution
if the first one found did not work. We claim it would be too inefficient.
- TC metavariables should not leak outside of TC. Reason: we want to
get rid of them after we synthesize the instance.
- `simp` invokes `isDefEq` for matching the left-hand-side of
equations to terms in our goal. Thus, it may invoke TC indirectly.
- In Lean3, we didn’t have to create a fresh pattern for trying to
match the left-hand-side of equations when executing `simp`. We had a
mechanism called "tmp" metavariables. It avoided this overhead, but it
created many problems since `simp` may indirectly call TC which may
recursively call TC. Moreover, we may want to allow TC to invoke
tactics in the future. Thus, when `simp` invokes `isDefEq`, it may indirectly invoke
a tactic and `simp` itself. The Lean3 approach assumed that
metavariables were short-lived, this is not true in Lean4, and to some
extent was also not true in Lean3 since `simp`, in principle, could
trigger an arbitrary number of nested TC problems.
- Here are some possible call stack traces we could have in Lean3 (and Lean4).
```
Elaborator (-> TC -> isDefEq)+
Elaborator -> isDefEq (-> TC -> isDefEq)*
Elaborator -> simp -> isDefEq (-> TC -> isDefEq)*
```
In Lean4, TC may also invoke tactics in the future.
- In Lean3 and Lean4, TC metavariables are not really short-lived. We
solve an arbitrary number of unification problems, and we may have
nested TC invocations.
- TC metavariables do not share the same local context even in the
same invocation. In the C++ and Lean implementations we use a trick to
ensure they do:
https://github.com/leanprover/lean/blob/92826917a252a6092cffaf5fc5f1acb1f8cef379/src/library/type_context.cpp#L3583-L3594
- Metavariables may be natural, synthetic or syntheticOpaque.
a) Natural metavariables may be assigned by unification (i.e., `isDefEq`).
b) Synthetic metavariables may still be assigned by unification,
but whenever possible `isDefEq` will avoid the assignment. For example,
if we have the unification constraint `?m =?= ?n`, where `?m` is synthetic,
but `?n` is not, `isDefEq` solves it by using the assignment `?n := ?m`.
We use synthetic metavariables for type class resolution.
Any module that creates synthetic metavariables, must also check
whether they have been assigned by `isDefEq`, and then still synthesize
them, and check whether the synthesized result is compatible with the one
assigned by `isDefEq`.
c) SyntheticOpaque metavariables are never assigned by `isDefEq`.
That is, the constraint `?n =?= Nat.succ Nat.zero` always fail
if `?n` is a syntheticOpaque metavariable. This kind of metavariable
is created by tactics such as `intro`. Reason: in the tactic framework,
subgoals as represented as metavariables, and a subgoal `?n` is considered
as solved whenever the metavariable is assigned.
This distinction was not precise in Lean3 and produced
counterintuitive behavior. For example, the following hack was added
in Lean3 to work around one of these issues:
https://github.com/leanprover/lean/blob/92826917a252a6092cffaf5fc5f1acb1f8cef379/src/library/type_context.cpp#L2751
- When creating lambda/forall expressions, we need to convert/abstract
free variables and convert them to bound variables. Now, suppose we a
trying to create a lambda/forall expression by abstracting free
variable `xs` and a term `t[?m]` which contains a metavariable `?m`,
and the local context of `?m` contains `xs`. The term
```
fun xs => t[?m]
```
will be ill-formed if we later assign a term `s` to `?m`, and
`s` contains free variables in `xs`. We address this issue by changing
the free variable abstraction procedure. We consider two cases: `?m`
is natural, `?m` is synthetic. Assume the type of `?m` is
`A[xs]`. Then, in both cases we create an auxiliary metavariable `?n` with
type `forall xs => A[xs]`, and local context := local context of `?m` - `xs`.
In both cases, we produce the term `fun xs => t[?n xs]`
1- If `?m` is natural or synthetic, then we assign `?m := ?n xs`, and we produce
the term `fun xs => t[?n xs]`
2- If `?m` is syntheticOpaque, then we mark `?n` as a syntheticOpaque variable.
However, `?n` is managed by the metavariable context itself.
We say we have a "delayed assignment" `?n xs := ?m`.
That is, after a term `s` is assigned to `?m`, and `s`
does not contain metavariables, we replace any occurrence
`?n ts` with `s[xs := ts]`.
Gruesome details:
- When we create the type `forall xs => A` for `?n`, we may
encounter the same issue if `A` contains metavariables. So, the
process above is recursive. We claim it terminates because we keep
creating new metavariables with smaller local contexts.
- Suppose, we have `t[?m]` and we want to create a let-expression by
abstracting a let-decl free variable `x`, and the local context of
`?m` contains `x`. Similarly to the previous case
```
let x : T := v; t[?m]
```
will be ill-formed if we later assign a term `s` to `?m`, and
`s` contains free variable `x`. Again, assume the type of `?m` is `A[x]`.
1- If `?m` is natural or synthetic, then we create `?n : (let x : T := v; A[x])` with
and local context := local context of `?m` - `x`, we assign `?m := ?n`,
and produce the term `let x : T := v; t[?n]`. That is, we are just making
sure `?n` must never be assigned to a term containing `x`.
2- If `?m` is syntheticOpaque, we create a fresh syntheticOpaque `?n`
with type `?n : T -> (let x : T := v; A[x])` and local context := local context of `?m` - `x`,
create the delayed assignment `?n #[x] := ?m`, and produce the term `let x : T := v; t[?n x]`.
Now suppose we assign `s` to `?m`. We do not assign the term `fun (x : T) => s` to `?n`, since
`fun (x : T) => s` may not even be type correct. Instead, we just replace applications `?n r`
with `s[x/r]`. The term `r` may not necessarily be a bound variable. For example, a tactic
may have reduced `let x : T := v; t[?n x]` into `t[?n v]`.
We are essentially using the pair "delayed assignment + application" to implement a delayed
substitution.
- We use TC for implementing coercions. Both Joe Hendrix and Reid Barton
reported a nasty limitation. In Lean3, TC will not be used if there are
metavariables in the TC problem. For example, the elaborator will not try
to synthesize `Coe Nat ?x`. This is good, but this constraint is too
strict for problems such as `Coe (Vector Bool ?n) (BV ?n)`. The coercion
exists independently of `?n`. Thus, during TC, we want `isDefEq` to throw
an exception instead of return `false` whenever it tries to assign
a metavariable owned by its caller. The idea is to sign to the caller that
it cannot solve the TC problem at this point, and more information is needed.
That is, the caller must make progress an assign its metavariables before
trying to invoke TC again.
In Lean4, we are using a simpler design for the `MetavarContext`.
- No distinction between temporary and regular metavariables.
- Metavariables have a `depth` Nat field.
- MetavarContext also has a `depth` field.
- We bump the `MetavarContext` depth when we create a nested problem.
Example: Elaborator (depth = 0) -> Simplifier matcher (depth = 1) -> TC (level = 2) -> TC (level = 3) -> ...
- When `MetavarContext` is at depth N, `isDefEq` does not assign variables from `depth < N`.
- Metavariables from depth N+1 must be fully assigned before we return to level N.
- New design even allows us to invoke tactics from TC.
* Main concern
We don't have tmp metavariables anymore in Lean4. Thus, before trying to match
the left-hand-side of an equation in `simp`. We first must bump the level of the `MetavarContext`,
create fresh metavariables, then create a new pattern by replacing the free variable on the left-hand-side with
these metavariables. We are hoping to minimize this overhead by
- Using better indexing data structures in `simp`. They should reduce the number of time `simp` must invoke `isDefEq`.
- Implementing `isDefEqApprox` which ignores metavariables and returns only `false` or `undef`.
It is a quick filter that allows us to fail quickly and avoid the creation of new fresh metavariables,
and a new pattern.
- Adding built-in support for arithmetic, Logical connectives, etc. Thus, we avoid a bunch of lemmas in the simp set.
- Adding support for AC-rewriting. In Lean3, users use AC lemmas as
rewriting rules for "sorting" terms. This is inefficient, requires
a quadratic number of rewrite steps, and does not preserve the
structure of the goal.
The temporary metavariables were also used in the "app builder" module used in Lean3. The app builder uses
`isDefEq`. So, it could, in principle, invoke an arbitrary number of nested TC problems. However, in Lean3,
all app builder uses are controlled. That is, it is mainly used to synthesize implicit arguments using
very simple unification and/or non-nested TC. So, if the "app builder" becomes a bottleneck without tmp metavars,
we may solve the issue by implementing `isDefEqCheap` that never invokes TC and uses tmp metavars.
-/
/--
`LocalInstance` represents a local typeclass instance registered by and for
the elaborator. It stores the name of the typeclass in `className`, and the
concrete typeclass instance in `fvar`. Note that the kernel does not care about
this information, since typeclasses are entirely eliminated during elaboration.
-/
structure LocalInstance where
className : Name
fvar : Expr
deriving Inhabited
abbrev LocalInstances := Array LocalInstance
instance : BEq LocalInstance where
beq i₁ i₂ := i₁.fvar == i₂.fvar
/-- Remove local instance with the given `fvarId`. Do nothing if `localInsts` does not contain any free variable with id `fvarId`. -/
def LocalInstances.erase (localInsts : LocalInstances) (fvarId : FVarId) : LocalInstances :=
match localInsts.findIdx? (fun inst => inst.fvar.fvarId! == fvarId) with
| some idx => localInsts.eraseIdx idx
| _ => localInsts
/-- A kind for the metavariable that determines its unification behaviour.
For more information see the large comment at the beginning of this file. -/
inductive MetavarKind where
/-- Normal unification behaviour -/
| natural
/-- `isDefEq` avoids assignment -/
| synthetic
/-- Never assigned by isDefEq -/
| syntheticOpaque
deriving Inhabited, Repr
def MetavarKind.isSyntheticOpaque : MetavarKind → Bool
| MetavarKind.syntheticOpaque => true
| _ => false
def MetavarKind.isNatural : MetavarKind → Bool
| MetavarKind.natural => true
| _ => false
/-- Information about a metavariable. -/
structure MetavarDecl where
/-- A user-friendly name for the metavariable. If anonymous then there is no such name. -/
userName : Name := Name.anonymous
/-- The local context containing the free variables that the mvar is permitted to depend upon. -/
lctx : LocalContext
/-- The type of the metavarible, in the given `lctx`. -/
type : Expr
/--
The nesting depth of this metavariable. We do not want
unification subproblems to influence the results of parent
problems. The depth keeps track of this information and ensures
that unification subproblems cannot leak information out, by unifying
based on depth.
-/
depth : Nat
localInstances : LocalInstances
kind : MetavarKind
/-- See comment at `CheckAssignment` `Meta/ExprDefEq.lean` -/
numScopeArgs : Nat := 0
/-- We use this field to track how old a metavariable is. It is set using a counter at `MetavarContext` -/
index : Nat
deriving Inhabited
/--
A delayed assignment for a metavariable `?m`. It represents an assignment of the form `?m := (fun fvars => (mkMVar mvarIdPending))`.
`mvarIdPending` is a `syntheticOpaque` metavariable that has not been synthesized yet. The delayed assignment becomes a real one
as soon as `mvarIdPending` has been fully synthesized.
`fvars` are variables in the `mvarIdPending` local context.
See the comment below `assignDelayedMVar ` for the rationale of delayed assignments.
Recall that we use a locally nameless approach when dealing with binders. Suppose we are
trying to synthesize `?n` in the expression `e`, in the context of `(fun x => e)`.
The metavariable `?n` might depend on the bound variable `x`. However, since we are locally nameless,
the bound variable `x` is in fact represented by some free variable `fvar_x`. Thus, when we exit
the scope, we must rebind the value of `fvar_x` in `?n` to the de-bruijn index of the bound variable `x`.
-/
structure DelayedMetavarAssignment where
fvars : Array Expr
mvarIdPending : MVarId
/-- The metavariable context is a set of metavariable declarations and their assignments.
For more information on specifics see the comment in the file that `MetavarContext` is defined in.
-/
structure MetavarContext where
/-- Depth is used to control whether an mvar can be assigned in unification. -/
depth : Nat := 0
/-- Counter for setting the field `index` at `MetavarDecl` -/
mvarCounter : Nat := 0
lDepth : PersistentHashMap LMVarId Nat := {}
/-- Metavariable declarations. -/
decls : PersistentHashMap MVarId MetavarDecl := {}
/-- Index mapping user-friendly names to ids. -/
userNames : PersistentHashMap Name MVarId := {}
/-- Assignment table for universe level metavariables.-/
lAssignment : PersistentHashMap LMVarId Level := {}
/-- Assignment table for expression metavariables.-/
eAssignment : PersistentHashMap MVarId Expr := {}
/-- Assignment table for delayed abstraction metavariables.
For more information about delayed abstraction, see the docstring for `DelayedMetavarAssignment`. -/
dAssignment : PersistentHashMap MVarId DelayedMetavarAssignment := {}
/-- A monad with a stateful metavariable context, defining `getMCtx` and `modifyMCtx`. -/
class MonadMCtx (m : Type → Type) where
getMCtx : m MetavarContext
modifyMCtx : (MetavarContext → MetavarContext) → m Unit
export MonadMCtx (getMCtx modifyMCtx)
@[always_inline]
instance (m n) [MonadLift m n] [MonadMCtx m] : MonadMCtx n where
getMCtx := liftM (getMCtx : m _)
modifyMCtx := fun f => liftM (modifyMCtx f : m _)
abbrev setMCtx [MonadMCtx m] (mctx : MetavarContext) : m Unit :=
modifyMCtx fun _ => mctx
abbrev getLevelMVarAssignment? [Monad m] [MonadMCtx m] (mvarId : LMVarId) : m (Option Level) :=
return (← getMCtx).lAssignment.find? mvarId
def MetavarContext.getExprAssignmentCore? (m : MetavarContext) (mvarId : MVarId) : Option Expr :=
m.eAssignment.find? mvarId
def getExprMVarAssignment? [Monad m] [MonadMCtx m] (mvarId : MVarId) : m (Option Expr) :=
return (← getMCtx).getExprAssignmentCore? mvarId
def getDelayedMVarAssignment? [Monad m] [MonadMCtx m] (mvarId : MVarId) : m (Option DelayedMetavarAssignment) :=
return (← getMCtx).dAssignment.find? mvarId
/-- Given a sequence of delayed assignments
```
mvarId₁ := mvarId₂ ...;
...
mvarIdₙ := mvarId_root ... -- where `mvarId_root` is not delayed assigned
```
in `mctx`, `getDelayedRoot mctx mvarId₁` return `mvarId_root`.
If `mvarId₁` is not delayed assigned then return `mvarId₁` -/
partial def getDelayedMVarRoot [Monad m] [MonadMCtx m] (mvarId : MVarId) : m MVarId := do
match (← getDelayedMVarAssignment? mvarId) with
| some d => getDelayedMVarRoot d.mvarIdPending
| none => return mvarId
def isLevelMVarAssigned [Monad m] [MonadMCtx m] (mvarId : LMVarId) : m Bool :=
return (← getMCtx).lAssignment.contains mvarId
/-- Return `true` if the give metavariable is already assigned. -/
def _root_.Lean.MVarId.isAssigned [Monad m] [MonadMCtx m] (mvarId : MVarId) : m Bool :=
return (← getMCtx).eAssignment.contains mvarId
@[deprecated MVarId.isAssigned]
def isExprMVarAssigned [Monad m] [MonadMCtx m] (mvarId : MVarId) : m Bool := do
mvarId.isAssigned
def _root_.Lean.MVarId.isDelayedAssigned [Monad m] [MonadMCtx m] (mvarId : MVarId) : m Bool :=
return (← getMCtx).dAssignment.contains mvarId
@[deprecated MVarId.isDelayedAssigned]
def isMVarDelayedAssigned [Monad m] [MonadMCtx m] (mvarId : MVarId) : m Bool := do
mvarId.isDelayedAssigned
def isLevelMVarAssignable [Monad m] [MonadMCtx m] (mvarId : LMVarId) : m Bool := do
let mctx ← getMCtx
match mctx.lDepth.find? mvarId with
| some d => return d == mctx.depth
| _ => panic! "unknown universe metavariable"
def MetavarContext.getDecl (mctx : MetavarContext) (mvarId : MVarId) : MetavarDecl :=
match mctx.decls.find? mvarId with
| some decl => decl
| none => panic! "unknown metavariable"
def _root_.Lean.MVarId.isAssignable [Monad m] [MonadMCtx m] (mvarId : MVarId) : m Bool := do
let mctx ← getMCtx
let decl := mctx.getDecl mvarId
return decl.depth == mctx.depth
@[deprecated MVarId.isAssignable]
def isExprMVarAssignable [Monad m] [MonadMCtx m] (mvarId : MVarId) : m Bool := do
mvarId.isAssignable
/-- Return true iff the given level contains an assigned metavariable. -/
def hasAssignedLevelMVar [Monad m] [MonadMCtx m] : Level → m Bool
| .succ lvl => pure lvl.hasMVar <&&> hasAssignedLevelMVar lvl
| .max lvl₁ lvl₂ => (pure lvl₁.hasMVar <&&> hasAssignedLevelMVar lvl₁) <||> (pure lvl₂.hasMVar <&&> hasAssignedLevelMVar lvl₂)
| .imax lvl₁ lvl₂ => (pure lvl₁.hasMVar <&&> hasAssignedLevelMVar lvl₁) <||> (pure lvl₂.hasMVar <&&> hasAssignedLevelMVar lvl₂)
| .mvar mvarId => isLevelMVarAssigned mvarId
| .zero => pure false
| .param _ => pure false
/-- Return `true` iff expression contains assigned (level/expr) metavariables or delayed assigned mvars -/
def hasAssignedMVar [Monad m] [MonadMCtx m] : Expr → m Bool
| .const _ lvls => lvls.anyM hasAssignedLevelMVar
| .sort lvl => hasAssignedLevelMVar lvl
| .app f a => (pure f.hasMVar <&&> hasAssignedMVar f) <||> (pure a.hasMVar <&&> hasAssignedMVar a)
| .letE _ t v b _ => (pure t.hasMVar <&&> hasAssignedMVar t) <||> (pure v.hasMVar <&&> hasAssignedMVar v) <||> (pure b.hasMVar <&&> hasAssignedMVar b)
| .forallE _ d b _ => (pure d.hasMVar <&&> hasAssignedMVar d) <||> (pure b.hasMVar <&&> hasAssignedMVar b)
| .lam _ d b _ => (pure d.hasMVar <&&> hasAssignedMVar d) <||> (pure b.hasMVar <&&> hasAssignedMVar b)
| .fvar _ => return false
| .bvar _ => return false
| .lit _ => return false
| .mdata _ e => pure e.hasMVar <&&> hasAssignedMVar e
| .proj _ _ e => pure e.hasMVar <&&> hasAssignedMVar e
| .mvar mvarId => mvarId.isAssigned <||> mvarId.isDelayedAssigned
/-- Return true iff the given level contains a metavariable that can be assigned. -/
def hasAssignableLevelMVar [Monad m] [MonadMCtx m] : Level → m Bool
| .succ lvl => pure lvl.hasMVar <&&> hasAssignableLevelMVar lvl
| .max lvl₁ lvl₂ => (pure lvl₁.hasMVar <&&> hasAssignableLevelMVar lvl₁) <||> (pure lvl₂.hasMVar <&&> hasAssignableLevelMVar lvl₂)
| .imax lvl₁ lvl₂ => (pure lvl₁.hasMVar <&&> hasAssignableLevelMVar lvl₁) <||> (pure lvl₂.hasMVar <&&> hasAssignableLevelMVar lvl₂)
| .mvar mvarId => isLevelMVarAssignable mvarId
| .zero => return false
| .param _ => return false
/-- Return `true` iff expression contains a metavariable that can be assigned. -/
def hasAssignableMVar [Monad m] [MonadMCtx m] : Expr → m Bool
| .const _ lvls => lvls.anyM hasAssignableLevelMVar
| .sort lvl => hasAssignableLevelMVar lvl
| .app f a => (pure f.hasMVar <&&> hasAssignableMVar f) <||> (pure a.hasMVar <&&> hasAssignableMVar a)
| .letE _ t v b _ => (pure t.hasMVar <&&> hasAssignableMVar t) <||> (pure v.hasMVar <&&> hasAssignableMVar v) <||> (pure b.hasMVar <&&> hasAssignableMVar b)
| .forallE _ d b _ => (pure d.hasMVar <&&> hasAssignableMVar d) <||> (pure b.hasMVar <&&> hasAssignableMVar b)
| .lam _ d b _ => (pure d.hasMVar <&&> hasAssignableMVar d) <||> (pure b.hasMVar <&&> hasAssignableMVar b)
| .fvar _ => return false
| .bvar _ => return false
| .lit _ => return false
| .mdata _ e => pure e.hasMVar <&&> hasAssignableMVar e
| .proj _ _ e => pure e.hasMVar <&&> hasAssignableMVar e
| .mvar mvarId => mvarId.isAssignable
/--
Add `mvarId := u` to the universe metavariable assignment.
This method does not check whether `mvarId` is already assigned, nor it checks whether
a cycle is being introduced.
This is a low-level API, and it is safer to use `isLevelDefEq (mkLevelMVar mvarId) u`.
-/
def assignLevelMVar [MonadMCtx m] (mvarId : LMVarId) (val : Level) : m Unit :=
modifyMCtx fun m => { m with lAssignment := m.lAssignment.insert mvarId val }
/--
Add `mvarId := x` to the metavariable assignment.
This method does not check whether `mvarId` is already assigned, nor it checks whether
a cycle is being introduced, or whether the expression has the right type.
This is a low-level API, and it is safer to use `isDefEq (mkMVar mvarId) x`.
-/
def _root_.Lean.MVarId.assign [MonadMCtx m] (mvarId : MVarId) (val : Expr) : m Unit :=
modifyMCtx fun m => { m with eAssignment := m.eAssignment.insert mvarId val }
@[deprecated MVarId.assign]
def assignExprMVar [MonadMCtx m] (mvarId : MVarId) (val : Expr) : m Unit :=
mvarId.assign val
def assignDelayedMVar [MonadMCtx m] (mvarId : MVarId) (fvars : Array Expr) (mvarIdPending : MVarId) : m Unit :=
modifyMCtx fun m => { m with dAssignment := m.dAssignment.insert mvarId { fvars, mvarIdPending } }
/-!
Notes on artificial eta-expanded terms due to metavariables.
We try avoid synthetic terms such as `((fun x y => t) a b)` in the output produced by the elaborator.
This kind of term may be generated when instantiating metavariable assignments.
This module tries to avoid their generation because they often introduce unnecessary dependencies and
may affect automation.
When elaborating terms, we use metavariables to represent "holes". Each hole has a context which includes
all free variables that may be used to "fill" the hole. Suppose, we create a metavariable (hole) `?m : Nat` in a context
containing `(x : Nat) (y : Nat) (b : Bool)`, then we can assign terms such as `x + y` to `?m` since `x` and `y`
are in the context used to create `?m`. Now, suppose we have the term `?m + 1` and we want to create the lambda expression
`fun x => ?m + 1`. This term is not correct since we may assign to `?m` a term containing `x`.
We address this issue by create a synthetic metavariable `?n : Nat → Nat` and adding the delayed assignment
`?n #[x] := ?m`, and the term `fun x => ?n x + 1`. When we later assign a term `t[x]` to `?m`, `fun x => t[x]` is assigned to
`?n`, and if we substitute it at `fun x => ?n x + 1`, we produce `fun x => ((fun x => t[x]) x) + 1`.
To avoid this term eta-expanded term, we apply beta-reduction when instantiating metavariable assignments in this module.
This operation is performed at `instantiateExprMVars`, `elimMVarDeps`, and `levelMVarToParam`.
-/
partial def instantiateLevelMVars [Monad m] [MonadMCtx m] : Level → m Level
| lvl@(Level.succ lvl₁) => return Level.updateSucc! lvl (← instantiateLevelMVars lvl₁)
| lvl@(Level.max lvl₁ lvl₂) => return Level.updateMax! lvl (← instantiateLevelMVars lvl₁) (← instantiateLevelMVars lvl₂)
| lvl@(Level.imax lvl₁ lvl₂) => return Level.updateIMax! lvl (← instantiateLevelMVars lvl₁) (← instantiateLevelMVars lvl₂)
| lvl@(Level.mvar mvarId) => do
match (← getLevelMVarAssignment? mvarId) with
| some newLvl =>
if !newLvl.hasMVar then pure newLvl
else do
let newLvl' ← instantiateLevelMVars newLvl
assignLevelMVar mvarId newLvl'
pure newLvl'
| none => pure lvl
| lvl => pure lvl
/-- instantiateExprMVars main function -/
partial def instantiateExprMVars [Monad m] [MonadMCtx m] [STWorld ω m] [MonadLiftT (ST ω) m] (e : Expr) : MonadCacheT ExprStructEq Expr m Expr :=
if !e.hasMVar then
pure e
else checkCache { val := e : ExprStructEq } fun _ => do match e with
| .proj _ _ s => return e.updateProj! (← instantiateExprMVars s)
| .forallE _ d b _ => return e.updateForallE! (← instantiateExprMVars d) (← instantiateExprMVars b)
| .lam _ d b _ => return e.updateLambdaE! (← instantiateExprMVars d) (← instantiateExprMVars b)
| .letE _ t v b _ => return e.updateLet! (← instantiateExprMVars t) (← instantiateExprMVars v) (← instantiateExprMVars b)
| .const _ lvls => return e.updateConst! (← lvls.mapM instantiateLevelMVars)
| .sort lvl => return e.updateSort! (← instantiateLevelMVars lvl)
| .mdata _ b => return e.updateMData! (← instantiateExprMVars b)
| .app .. => e.withApp fun f args => do
let instArgs (f : Expr) : MonadCacheT ExprStructEq Expr m Expr := do
let args ← args.mapM instantiateExprMVars
pure (mkAppN f args)
let instApp : MonadCacheT ExprStructEq Expr m Expr := do
let wasMVar := f.isMVar
let f ← instantiateExprMVars f
if wasMVar && f.isLambda then
/- Some of the arguments in args are irrelevant after we beta reduce. -/
instantiateExprMVars (f.betaRev args.reverse)
else
instArgs f
match f with
| .mvar mvarId =>
match (← getDelayedMVarAssignment? mvarId) with
| none => instApp
| some { fvars, mvarIdPending } =>
/-
Apply "delayed substitution" (i.e., delayed assignment + application).
That is, `f` is some metavariable `?m`, that is delayed assigned to `val`.
If after instantiating `val`, we obtain `newVal`, and `newVal` does not contain
metavariables, we replace the free variables `fvars` in `newVal` with the first
`fvars.size` elements of `args`. -/
if fvars.size > args.size then
/- We don't have sufficient arguments for instantiating the free variables `fvars`.
This can only happy if a tactic or elaboration function is not implemented correctly.
We decided to not use `panic!` here and report it as an error in the frontend
when we are checking for unassigned metavariables in an elaborated term. -/
instArgs f
else
let newVal ← instantiateExprMVars (mkMVar mvarIdPending)
if newVal.hasExprMVar then
instArgs f
else do
let args ← args.mapM instantiateExprMVars
/-
Example: suppose we have
`?m t1 t2 t3`
That is, `f := ?m` and `args := #[t1, t2, t3]`
Morever, `?m` is delayed assigned
`?m #[x, y] := f x y`
where, `fvars := #[x, y]` and `newVal := f x y`.
After abstracting `newVal`, we have `f (Expr.bvar 0) (Expr.bvar 1)`.
After `instantiaterRevRange 0 2 args`, we have `f t1 t2`.
After `mkAppRange 2 3`, we have `f t1 t2 t3` -/
let newVal := newVal.abstract fvars
let result := newVal.instantiateRevRange 0 fvars.size args
let result := mkAppRange result fvars.size args.size args
pure result
| _ => instApp
| e@(.mvar mvarId) => checkCache { val := e : ExprStructEq } fun _ => do
match (← getExprMVarAssignment? mvarId) with
| some newE => do
let newE' ← instantiateExprMVars newE
mvarId.assign newE'
pure newE'
| none => pure e
| e => pure e
instance : MonadMCtx (StateRefT MetavarContext (ST ω)) where
getMCtx := get
modifyMCtx := modify
def instantiateMVarsCore (mctx : MetavarContext) (e : Expr) : Expr × MetavarContext :=
let instantiate {ω} (e : Expr) : (MonadCacheT ExprStructEq Expr <| StateRefT MetavarContext (ST ω)) Expr :=
instantiateExprMVars e
runST fun _ => instantiate e |>.run |>.run mctx
def instantiateMVars [Monad m] [MonadMCtx m] (e : Expr) : m Expr := do
if !e.hasMVar then
return e
else
let (r, mctx) := instantiateMVarsCore (← getMCtx) e
modifyMCtx fun _ => mctx
return r
def instantiateLCtxMVars [Monad m] [MonadMCtx m] (lctx : LocalContext) : m LocalContext :=
lctx.foldlM (init := {}) fun lctx ldecl => do
match ldecl with
| .cdecl _ fvarId userName type bi k =>
let type ← instantiateMVars type
return lctx.mkLocalDecl fvarId userName type bi k
| .ldecl _ fvarId userName type value nonDep k =>
let type ← instantiateMVars type
let value ← instantiateMVars value
return lctx.mkLetDecl fvarId userName type value nonDep k
def instantiateMVarDeclMVars [Monad m] [MonadMCtx m] (mvarId : MVarId) : m Unit := do
let mvarDecl := (← getMCtx).getDecl mvarId
let lctx ← instantiateLCtxMVars mvarDecl.lctx
let type ← instantiateMVars mvarDecl.type
modifyMCtx fun mctx => { mctx with decls := mctx.decls.insert mvarId { mvarDecl with lctx, type } }
def instantiateLocalDeclMVars [Monad m] [MonadMCtx m] (localDecl : LocalDecl) : m LocalDecl := do
match localDecl with
| .cdecl idx id n type bi k =>
return .cdecl idx id n (← instantiateMVars type) bi k
| .ldecl idx id n type val nonDep k =>
return .ldecl idx id n (← instantiateMVars type) (← instantiateMVars val) nonDep k
namespace DependsOn
structure State where
visited : ExprSet := {}
mctx : MetavarContext
private abbrev M := StateM State
instance : MonadMCtx M where
getMCtx := return (← get).mctx
modifyMCtx f := modify fun s => { s with mctx := f s.mctx }
private def shouldVisit (e : Expr) : M Bool := do
if !e.hasMVar && !e.hasFVar then
return false
else if (← get).visited.contains e then
return false
else
modify fun s => { s with visited := s.visited.insert e }
return true
@[specialize] private partial def dep (pf : FVarId → Bool) (pm : MVarId → Bool) (e : Expr) : M Bool :=
let rec
visit (e : Expr) : M Bool := do
if !(← shouldVisit e) then
pure false
else
visitMain e,
visitApp : Expr → M Bool
| .app f a .. => visitApp f <||> visit a
| e => visit e,
visitMain : Expr → M Bool
| .proj _ _ s => visit s
| .forallE _ d b _ => visit d <||> visit b
| .lam _ d b _ => visit d <||> visit b
| .letE _ t v b _ => visit t <||> visit v <||> visit b
| .mdata _ b => visit b
| e@(.app ..) => do
let f := e.getAppFn
if f.isMVar then
let e' ← instantiateMVars e
if e'.getAppFn != f then
visitMain e'
else if pm f.mvarId! then
return true
else
visitApp e
else
visitApp e
| .mvar mvarId => do
match (← getExprMVarAssignment? mvarId) with
| some a => visit a
| none =>
if pm mvarId then
return true
else
let lctx := (← getMCtx).getDecl mvarId |>.lctx
return lctx.any fun decl => pf decl.fvarId
| .fvar fvarId => return pf fvarId
| _ => pure false
visit e
@[inline] partial def main (pf : FVarId → Bool) (pm : MVarId → Bool) (e : Expr) : M Bool :=
if !e.hasFVar && !e.hasMVar then pure false else dep pf pm e
end DependsOn
/--
Return `true` iff `e` depends on a free variable `x` s.t. `pf x` is `true`, or an unassigned metavariable `?m` s.t. `pm ?m` is true.
For each metavariable `?m` (that does not satisfy `pm` occurring in `x`
1- If `?m := t`, then we visit `t` looking for `x`
2- If `?m` is unassigned, then we consider the worst case and check whether `x` is in the local context of `?m`.
This case is a "may dependency". That is, we may assign a term `t` to `?m` s.t. `t` contains `x`. -/
@[inline] def findExprDependsOn [Monad m] [MonadMCtx m] (e : Expr) (pf : FVarId → Bool := fun _ => false) (pm : MVarId → Bool := fun _ => false) : m Bool := do
let (result, { mctx, .. }) := DependsOn.main pf pm e |>.run { mctx := (← getMCtx) }
setMCtx mctx
return result
/--
Similar to `findExprDependsOn`, but checks the expressions in the given local declaration
depends on a free variable `x` s.t. `pf x` is `true` or an unassigned metavariable `?m` s.t. `pm ?m` is true. -/
@[inline] def findLocalDeclDependsOn [Monad m] [MonadMCtx m] (localDecl : LocalDecl) (pf : FVarId → Bool := fun _ => false) (pm : MVarId → Bool := fun _ => false) : m Bool := do
match localDecl with
| .cdecl (type := t) .. => findExprDependsOn t pf pm
| .ldecl (type := t) (value := v) .. =>
let (result, { mctx, .. }) := (DependsOn.main pf pm t <||> DependsOn.main pf pm v).run { mctx := (← getMCtx) }
setMCtx mctx
return result
def exprDependsOn [Monad m] [MonadMCtx m] (e : Expr) (fvarId : FVarId) : m Bool :=
findExprDependsOn e (fvarId == ·)
/-- Return true iff `e` depends on the free variable `fvarId` -/
def dependsOn [Monad m] [MonadMCtx m] (e : Expr) (fvarId : FVarId) : m Bool :=
exprDependsOn e fvarId
/-- Return true iff `e` depends on the free variable `fvarId` -/
def localDeclDependsOn [Monad m] [MonadMCtx m] (localDecl : LocalDecl) (fvarId : FVarId) : m Bool :=
findLocalDeclDependsOn localDecl (fvarId == ·)
/-- Similar to `exprDependsOn`, but `x` can be a free variable or an unassigned metavariable. -/
def exprDependsOn' [Monad m] [MonadMCtx m] (e : Expr) (x : Expr) : m Bool :=
if x.isFVar then
findExprDependsOn e (x.fvarId! == ·)
else if x.isMVar then
findExprDependsOn e (pm := (x.mvarId! == ·))
else
return false
/-- Similar to `localDeclDependsOn`, but `x` can be a free variable or an unassigned metavariable. -/
def localDeclDependsOn' [Monad m] [MonadMCtx m] (localDecl : LocalDecl) (x : Expr) : m Bool :=
if x.isFVar then
findLocalDeclDependsOn localDecl (x.fvarId! == ·)
else if x.isMVar then
findLocalDeclDependsOn localDecl (pm := (x.mvarId! == ·))
else
return false
/-- Return true iff `e` depends on a free variable `x` s.t. `pf x`, or an unassigned metavariable `?m` s.t. `pm ?m` is true. -/
def dependsOnPred [Monad m] [MonadMCtx m] (e : Expr) (pf : FVarId → Bool := fun _ => false) (pm : MVarId → Bool := fun _ => false) : m Bool :=
findExprDependsOn e pf pm
/-- Return true iff the local declaration `localDecl` depends on a free variable `x` s.t. `pf x`, an unassigned metavariable `?m` s.t. `pm ?m` is true. -/
def localDeclDependsOnPred [Monad m] [MonadMCtx m] (localDecl : LocalDecl) (pf : FVarId → Bool := fun _ => false) (pm : MVarId → Bool := fun _ => false) : m Bool := do
findLocalDeclDependsOn localDecl pf pm
namespace MetavarContext
instance : Inhabited MetavarContext := ⟨{}⟩
@[export lean_mk_metavar_ctx]
def mkMetavarContext : Unit → MetavarContext := fun _ => {}
/-- Low level API for adding/declaring metavariable declarations.
It is used to implement actions in the monads `MetaM`, `ElabM` and `TacticM`.
It should not be used directly since the argument `(mvarId : MVarId)` is assumed to be "unique". -/
def addExprMVarDecl (mctx : MetavarContext)
(mvarId : MVarId)
(userName : Name)
(lctx : LocalContext)
(localInstances : LocalInstances)
(type : Expr)
(kind : MetavarKind := MetavarKind.natural)
(numScopeArgs : Nat := 0) : MetavarContext :=
{ mctx with
mvarCounter := mctx.mvarCounter + 1
decls := mctx.decls.insert mvarId {
depth := mctx.depth
index := mctx.mvarCounter
userName
lctx
localInstances
type
kind
numScopeArgs }
userNames := if userName.isAnonymous then mctx.userNames else mctx.userNames.insert userName mvarId }
def addExprMVarDeclExp (mctx : MetavarContext) (mvarId : MVarId) (userName : Name) (lctx : LocalContext) (localInstances : LocalInstances)
(type : Expr) (kind : MetavarKind) : MetavarContext :=
addExprMVarDecl mctx mvarId userName lctx localInstances type kind
/-- Low level API for adding/declaring universe level metavariable declarations.
It is used to implement actions in the monads `MetaM`, `ElabM` and `TacticM`.
It should not be used directly since the argument `(mvarId : MVarId)` is assumed to be "unique". -/
def addLevelMVarDecl (mctx : MetavarContext) (mvarId : LMVarId) : MetavarContext :=
{ mctx with lDepth := mctx.lDepth.insert mvarId mctx.depth }
def findDecl? (mctx : MetavarContext) (mvarId : MVarId) : Option MetavarDecl :=
mctx.decls.find? mvarId
def findUserName? (mctx : MetavarContext) (userName : Name) : Option MVarId :=
mctx.userNames.find? userName
def setMVarKind (mctx : MetavarContext) (mvarId : MVarId) (kind : MetavarKind) : MetavarContext :=
let decl := mctx.getDecl mvarId
{ mctx with decls := mctx.decls.insert mvarId { decl with kind := kind } }
/--
Set the metavariable user facing name.
-/
def setMVarUserName (mctx : MetavarContext) (mvarId : MVarId) (userName : Name) : MetavarContext :=
let decl := mctx.getDecl mvarId
{ mctx with
decls := mctx.decls.insert mvarId { decl with userName := userName }
userNames :=
let userNames := mctx.userNames.erase decl.userName
if userName.isAnonymous then userNames else userNames.insert userName mvarId }
/--
Low-level version of `setMVarUserName`.
It does not update the table `userNames`. Thus, `findUserName?` cannot see the modification.
It is meant for `mkForallFVars'` where we temporarily set the user facing name of metavariables to get more
meaningful binder names.
-/
def setMVarUserNameTemporarily (mctx : MetavarContext) (mvarId : MVarId) (userName : Name) : MetavarContext :=
let decl := mctx.getDecl mvarId
{ mctx with decls := mctx.decls.insert mvarId { decl with userName := userName } }
/-- Update the type of the given metavariable. This function assumes the new type is
definitionally equal to the current one -/
def setMVarType (mctx : MetavarContext) (mvarId : MVarId) (type : Expr) : MetavarContext :=
let decl := mctx.getDecl mvarId
{ mctx with decls := mctx.decls.insert mvarId { decl with type := type } }
def findLevelDepth? (mctx : MetavarContext) (mvarId : LMVarId) : Option Nat :=
mctx.lDepth.find? mvarId
def getLevelDepth (mctx : MetavarContext) (mvarId : LMVarId) : Nat :=
match mctx.findLevelDepth? mvarId with
| some d => d
| none => panic! "unknown metavariable"
def isAnonymousMVar (mctx : MetavarContext) (mvarId : MVarId) : Bool :=
match mctx.findDecl? mvarId with
| none => false
| some mvarDecl => mvarDecl.userName.isAnonymous
def incDepth (mctx : MetavarContext) : MetavarContext :=
{ mctx with depth := mctx.depth + 1 }
instance : MonadMCtx (StateRefT MetavarContext (ST ω)) where
getMCtx := get
modifyMCtx := modify
namespace MkBinding
inductive Exception where
| revertFailure (mctx : MetavarContext) (lctx : LocalContext) (toRevert : Array Expr) (varName : String)
instance : ToString Exception where
toString
| Exception.revertFailure _ lctx toRevert varName =>
"failed to revert "
++ toString (toRevert.map (fun x => "'" ++ toString (lctx.getFVar! x).userName ++ "'"))
++ ", '" ++ toString varName ++ "' depends on them, and it is an auxiliary declaration created by the elaborator"
++ " (possible solution: use tactic 'clear' to remove '" ++ toString varName ++ "' from local context)"
/--
`MkBinding` and `elimMVarDepsAux` are mutually recursive, but `cache` is only used at `elimMVarDepsAux`.
We use a single state object for convenience.
We have a `NameGenerator` because we need to generate fresh auxiliary metavariables. -/
structure State where
mctx : MetavarContext
nextMacroScope : MacroScope
ngen : NameGenerator
cache : HashMap ExprStructEq Expr := {}
structure Context where
mainModule : Name
preserveOrder : Bool
/-- When creating binders for abstracted metavariables, we use the following `BinderInfo`. -/
binderInfoForMVars : BinderInfo := BinderInfo.implicit
/-- Set of unassigned metavariables being abstracted. -/
mvarIdsToAbstract : MVarIdSet := {}
abbrev MCore := EStateM Exception State
abbrev M := ReaderT Context MCore
instance : MonadMCtx M where
getMCtx := return (← get).mctx
modifyMCtx f := modify fun s => { s with mctx := f s.mctx }
private def mkFreshBinderName (n : Name := `x) : M Name := do
let fresh ← modifyGet fun s => (s.nextMacroScope, { s with nextMacroScope := s.nextMacroScope + 1 })
return addMacroScope (← read).mainModule n fresh
def preserveOrder : M Bool :=
return (← read).preserveOrder
instance : MonadHashMapCacheAdapter ExprStructEq Expr M where
getCache := do let s ← get; pure s.cache
modifyCache := fun f => modify fun s => { s with cache := f s.cache }
/-- Return the local declaration of the free variable `x` in `xs` with the smallest index -/
private def getLocalDeclWithSmallestIdx (lctx : LocalContext) (xs : Array Expr) : LocalDecl := Id.run do
let mut d : LocalDecl := lctx.getFVar! xs[0]!
for x in xs[1:] do
if x.isFVar then
let curr := lctx.getFVar! x
if curr.index < d.index then
d := curr
return d
/--
Given `toRevert` an array of free variables s.t. `lctx` contains their declarations,
return a new array of free variables that contains `toRevert` and all free variables
in `lctx` that may depend on `toRevert`.
Remark: the result is sorted by `LocalDecl` indices.
Remark: We used to throw an `Exception.revertFailure` exception when an auxiliary declaration
had to be reversed. Recall that auxiliary declarations are created when compiling (mutually)
recursive definitions. The `revertFailure` due to auxiliary declaration dependency was originally
introduced in Lean3 to address issue https://github.com/leanprover/lean/issues/1258.
In Lean4, this solution is not satisfactory because all definitions/theorems are potentially
recursive. So, even an simple (incomplete) definition such as
```
variables {α : Type} in
def f (a : α) : List α :=
_
```
would trigger the `Exception.revertFailure` exception. In the definition above,
the elaborator creates the auxiliary definition `f : {α : Type} → List α`.
The `_` is elaborated as a new fresh variable `?m` that contains `α : Type`, `a : α`, and `f : α → List α` in its context,
When we try to create the lambda `fun {α : Type} (a : α) => ?m`, we first need to create
an auxiliary `?n` which do not contain `α` and `a` in its context. That is,
we create the metavariable `?n : {α : Type} → (a : α) → (f : α → List α) → List α`,
add the delayed assignment `?n #[α, a, f] := ?m α a f`, and create the lambda
`fun {α : Type} (a : α) => ?n α a f`.
See `elimMVarDeps` for more information.
If we kept using the Lean3 approach, we would get the `Exception.revertFailure` exception because we are
reverting the auxiliary definition `f`.
Note that https://github.com/leanprover/lean/issues/1258 is not an issue in Lean4 because
we have changed how we compile recursive definitions.
-/
def collectForwardDeps (lctx : LocalContext) (toRevert : Array Expr) : M (Array Expr) := do
if toRevert.size == 0 then
pure toRevert
else
if (← preserveOrder) then
-- Make sure toRevert[j] does not depend on toRevert[i] for j > i
toRevert.size.forM fun i => do
let fvar := toRevert[i]!
i.forM fun j => do
let prevFVar := toRevert[j]!
let prevDecl := lctx.getFVar! prevFVar
if (← localDeclDependsOn prevDecl fvar.fvarId!) then
throw (Exception.revertFailure (← getMCtx) lctx toRevert prevDecl.userName.toString)
let newToRevert := if (← preserveOrder) then toRevert else Array.mkEmpty toRevert.size
let firstDeclToVisit := getLocalDeclWithSmallestIdx lctx toRevert
let initSize := newToRevert.size
lctx.foldlM (init := newToRevert) (start := firstDeclToVisit.index) fun (newToRevert : Array Expr) decl => do
if initSize.any fun i => decl.fvarId == newToRevert[i]!.fvarId! then
return newToRevert
else if toRevert.any fun x => decl.fvarId == x.fvarId! then
return newToRevert.push decl.toExpr
else if (← findLocalDeclDependsOn decl (newToRevert.any fun x => x.fvarId! == ·)) then
return newToRevert.push decl.toExpr
else
return newToRevert
/-- Create a new `LocalContext` by removing the free variables in `toRevert` from `lctx`.
We use this function when we create auxiliary metavariables at `elimMVarDepsAux`. -/
def reduceLocalContext (lctx : LocalContext) (toRevert : Array Expr) : LocalContext :=
toRevert.foldr (init := lctx) fun x lctx =>
if x.isFVar then lctx.erase x.fvarId! else lctx
/-- Return free variables in `xs` that are in the local context `lctx` -/
private def getInScope (lctx : LocalContext) (xs : Array Expr) : Array Expr :=
xs.foldl (init := #[]) fun scope x =>
if !x.isFVar then
scope
else if lctx.contains x.fvarId! then
scope.push x
else
scope
/-- Execute `x` with an empty cache, and then restore the original cache. -/
@[inline] private def withFreshCache (x : M α) : M α := do
let cache ← modifyGet fun s => (s.cache, { s with cache := {} })
let a ← x
modify fun s => { s with cache := cache }
pure a
/--
Create an application `mvar ys` where `ys` are the free variables.
See "Gruesome details" in the beginning of the file for understanding
how let-decl free variables are handled. -/
private def mkMVarApp (lctx : LocalContext) (mvar : Expr) (xs : Array Expr) (kind : MetavarKind) : Expr :=
xs.foldl (init := mvar) fun e x =>
if !x.isFVar then
e
else
match kind with
| MetavarKind.syntheticOpaque => mkApp e x
| _ => if (lctx.getFVar! x).isLet then e else mkApp e x
mutual
private partial def visit (xs : Array Expr) (e : Expr) : M Expr :=
if !e.hasMVar then pure e else checkCache { val := e : ExprStructEq } fun _ => elim xs e
private partial def elim (xs : Array Expr) (e : Expr) : M Expr :=
match e with
| .proj _ _ s => return e.updateProj! (← visit xs s)
| .forallE _ d b _ => return e.updateForallE! (← visit xs d) (← visit xs b)
| .lam _ d b _ => return e.updateLambdaE! (← visit xs d) (← visit xs b)
| .letE _ t v b _ => return e.updateLet! (← visit xs t) (← visit xs v) (← visit xs b)
| .mdata _ b => return e.updateMData! (← visit xs b)
| .app .. => e.withApp fun f args => elimApp xs f args
| .mvar _ => elimApp xs e #[]
| e => return e
private partial def mkAuxMVarType (lctx : LocalContext) (xs : Array Expr) (kind : MetavarKind) (e : Expr) : M Expr := do
let e ← abstractRangeAux xs xs.size e
xs.size.foldRevM (init := e) fun i e => do
let x := xs[i]!
if x.isFVar then
match lctx.getFVar! x with
| LocalDecl.cdecl _ _ n type bi _ =>
let type := type.headBeta
let type ← abstractRangeAux xs i type
return Lean.mkForall n bi type e
| LocalDecl.ldecl _ _ n type value nonDep _ =>
let type := type.headBeta
let type ← abstractRangeAux xs i type
let value ← abstractRangeAux xs i value
let e := mkLet n type value e nonDep
match kind with
| MetavarKind.syntheticOpaque =>
-- See "Gruesome details" section in the beginning of the file
let e := e.liftLooseBVars 0 1
return mkForall n BinderInfo.default type e
| _ => pure e
else
let mvarDecl := (← get).mctx.getDecl x.mvarId!
let type := mvarDecl.type.headBeta
let type ← abstractRangeAux xs i type
let id ← if mvarDecl.userName.isAnonymous then mkFreshBinderName else pure mvarDecl.userName
return Lean.mkForall id (← read).binderInfoForMVars type e
where
abstractRangeAux (xs : Array Expr) (i : Nat) (e : Expr) : M Expr := do
let e ← elim xs e
pure (e.abstractRange i xs)
private partial def elimMVar (xs : Array Expr) (mvarId : MVarId) (args : Array Expr) : M (Expr × Array Expr) := do
let mvarDecl := (← getMCtx).getDecl mvarId
let mvarLCtx := mvarDecl.lctx
let toRevert := getInScope mvarLCtx xs
if toRevert.size == 0 then
let args ← args.mapM (visit xs)
return (mkAppN (mkMVar mvarId) args, #[])
else
/- `newMVarKind` is the kind for the new auxiliary metavariable.
There is an alternative approach where we use
```
let newMVarKind := if !mctx.isExprAssignable mvarId || mvarDecl.isSyntheticOpaque then MetavarKind.syntheticOpaque else MetavarKind.natural
```
In this approach, we use the natural kind for the new auxiliary metavariable if the original metavariable is synthetic and assignable.
Since we mainly use synthetic metavariables for pending type class (TC) resolution problems,
this approach may minimize the number of TC resolution problems that may need to be resolved.
A potential disadvantage is that `isDefEq` will not eagerly use `synthPending` for natural metavariables.
That being said, we should try this approach as soon as we have an extensive test suite.
-/
let newMVarKind := if !(← mvarId.isAssignable) then MetavarKind.syntheticOpaque else mvarDecl.kind
let args ← args.mapM (visit xs)
let toRevert ← collectForwardDeps mvarLCtx toRevert
let newMVarLCtx := reduceLocalContext mvarLCtx toRevert
-- Note that `toRevert` only contains free variables since it is the result of `getInScope`
let newLocalInsts := mvarDecl.localInstances.filter fun inst => toRevert.all fun x => inst.fvar != x
-- Remark: we must reset the before processing `mkAuxMVarType` because `toRevert` may not be equal to `xs`
let newMVarType ← withFreshCache do mkAuxMVarType mvarLCtx toRevert newMVarKind mvarDecl.type
let newMVarId := { name := (← get).ngen.curr }
let newMVar := mkMVar newMVarId
let result := mkMVarApp mvarLCtx newMVar toRevert newMVarKind
let numScopeArgs := mvarDecl.numScopeArgs + result.getAppNumArgs
modify fun s => { s with
mctx := s.mctx.addExprMVarDecl newMVarId Name.anonymous newMVarLCtx newLocalInsts newMVarType newMVarKind numScopeArgs,
ngen := s.ngen.next
}
if !mvarDecl.kind.isSyntheticOpaque then
mvarId.assign result
else
/- If `mvarId` is the lhs of a delayed assignment `?m #[x_1, ... x_n] := ?mvarPending`,
then `nestedFVars` is `#[x_1, ..., x_n]`.
In this case, `newMVarId` is also `syntheticOpaque` and we add the delayed assignment delayed assignment
```
?newMVar #[y_1, ..., y_m, x_1, ... x_n] := ?m
```
where `#[y_1, ..., y_m]` is `toRevert` after `collectForwardDeps`.
-/
let (mvarIdPending, nestedFVars) ← match (← getDelayedMVarAssignment? mvarId) with
| none => pure (mvarId, #[])
| some { fvars, mvarIdPending } => pure (mvarIdPending, fvars)
assignDelayedMVar newMVarId (toRevert ++ nestedFVars) mvarIdPending
return (mkAppN result args, toRevert)
private partial def elimApp (xs : Array Expr) (f : Expr) (args : Array Expr) : M Expr := do
match f with
| Expr.mvar mvarId =>
match (← getExprMVarAssignment? mvarId) with
| some newF =>
if newF.isLambda then
let args ← args.mapM (visit xs)
elim xs <| newF.betaRev args.reverse
else
elimApp xs newF args
| none =>
if (← read).mvarIdsToAbstract.contains mvarId then
return mkAppN f (← args.mapM (visit xs))
else
return (← elimMVar xs mvarId args).1
| _ =>
return mkAppN (← visit xs f) (← args.mapM (visit xs))
end
partial def elimMVarDeps (xs : Array Expr) (e : Expr) : M Expr :=
if !e.hasMVar then
return e
else
withFreshCache do
elim xs e
partial def revert (xs : Array Expr) (mvarId : MVarId) : M (Expr × Array Expr) :=
withFreshCache do
elimMVar xs mvarId #[]
/--
Similar to `Expr.abstractRange`, but handles metavariables correctly.
It uses `elimMVarDeps` to ensure `e` and the type of the free variables `xs` do not
contain a metavariable `?m` s.t. local context of `?m` contains a free variable in `xs`.
`elimMVarDeps` is defined later in this file. -/
@[inline] def abstractRange (xs : Array Expr) (i : Nat) (e : Expr) : M Expr := do
let e ← elimMVarDeps xs e
pure (e.abstractRange i xs)
/--
Similar to `LocalContext.mkBinding`, but handles metavariables correctly.
If `usedOnly == false` then `forall` and `lambda` expressions are created only for used variables.
If `usedLetOnly == false` then `let` expressions are created only for used (let-) variables. -/
@[specialize] def mkBinding (isLambda : Bool) (lctx : LocalContext) (xs : Array Expr) (e : Expr) (usedOnly : Bool) (usedLetOnly : Bool) : M (Expr × Nat) := do
let e ← abstractRange xs xs.size e
xs.size.foldRevM (init := (e, 0)) fun i (e, num) => do
let x := xs[i]!
if x.isFVar then
match lctx.getFVar! x with
| LocalDecl.cdecl _ _ n type bi _ =>
if !usedOnly || e.hasLooseBVar 0 then
let type := type.headBeta;
let type ← abstractRange xs i type
if isLambda then
return (Lean.mkLambda n bi type e, num + 1)
else
return (Lean.mkForall n bi type e, num + 1)
else
return (e.lowerLooseBVars 1 1, num)
| LocalDecl.ldecl _ _ n type value nonDep _ =>
if !usedLetOnly || e.hasLooseBVar 0 then
let type ← abstractRange xs i type
let value ← abstractRange xs i value
return (mkLet n type value e nonDep, num + 1)
else
return (e.lowerLooseBVars 1 1, num)
else
let mvarDecl := (← get).mctx.getDecl x.mvarId!
let type := mvarDecl.type.headBeta
let type ← abstractRange xs i type
let id ← if mvarDecl.userName.isAnonymous then mkFreshBinderName else pure mvarDecl.userName
if isLambda then
return (Lean.mkLambda id (← read).binderInfoForMVars type e, num + 1)
else
return (Lean.mkForall id (← read).binderInfoForMVars type e, num + 1)
end MkBinding
structure MkBindingM.Context where
mainModule : Name
lctx : LocalContext
abbrev MkBindingM := ReaderT MkBindingM.Context MkBinding.MCore
def elimMVarDeps (xs : Array Expr) (e : Expr) (preserveOrder : Bool) : MkBindingM Expr := fun ctx =>
MkBinding.elimMVarDeps xs e { preserveOrder, mainModule := ctx.mainModule }
def revert (xs : Array Expr) (mvarId : MVarId) (preserveOrder : Bool) : MkBindingM (Expr × Array Expr) := fun ctx =>
MkBinding.revert xs mvarId { preserveOrder, mainModule := ctx.mainModule }
def mkBinding (isLambda : Bool) (xs : Array Expr) (e : Expr) (usedOnly : Bool := false) (usedLetOnly : Bool := true) (binderInfoForMVars := BinderInfo.implicit) : MkBindingM (Expr × Nat) := fun ctx =>
let mvarIdsToAbstract := xs.foldl (init := {}) fun s x => if x.isMVar then s.insert x.mvarId! else s
MkBinding.mkBinding isLambda ctx.lctx xs e usedOnly usedLetOnly { preserveOrder := false, binderInfoForMVars, mvarIdsToAbstract, mainModule := ctx.mainModule }
@[inline] def mkLambda (xs : Array Expr) (e : Expr) (usedOnly : Bool := false) (usedLetOnly : Bool := true) (binderInfoForMVars := BinderInfo.implicit) : MkBindingM Expr :=
return (← mkBinding (isLambda := true) xs e usedOnly usedLetOnly binderInfoForMVars).1
@[inline] def mkForall (xs : Array Expr) (e : Expr) (usedOnly : Bool := false) (usedLetOnly : Bool := true) (binderInfoForMVars := BinderInfo.implicit) : MkBindingM Expr :=
return (← mkBinding (isLambda := false) xs e usedOnly usedLetOnly binderInfoForMVars).1
@[inline] def abstractRange (e : Expr) (n : Nat) (xs : Array Expr) : MkBindingM Expr := fun ctx =>
MkBinding.abstractRange xs n e { preserveOrder := false, mainModule := ctx.mainModule }
@[inline] def collectForwardDeps (toRevert : Array Expr) (preserveOrder : Bool) : MkBindingM (Array Expr) := fun ctx =>
MkBinding.collectForwardDeps ctx.lctx toRevert { preserveOrder, mainModule := ctx.mainModule }
/--
`isWellFormed mctx lctx e` return true if
- All locals in `e` are declared in `lctx`
- All metavariables `?m` in `e` have a local context which is a subprefix of `lctx` or are assigned, and the assignment is well-formed. -/
partial def isWellFormed [Monad m] [MonadMCtx m] (lctx : LocalContext) : Expr → m Bool
| .mdata _ e => isWellFormed lctx e
| .proj _ _ e => isWellFormed lctx e
| e@(.app f a) => pure (!e.hasExprMVar && !e.hasFVar) <||> (isWellFormed lctx f <&&> isWellFormed lctx a)
| e@(.lam _ d b _) => pure (!e.hasExprMVar && !e.hasFVar) <||> (isWellFormed lctx d <&&> isWellFormed lctx b)
| e@(.forallE _ d b _) => pure (!e.hasExprMVar && !e.hasFVar) <||> (isWellFormed lctx d <&&> isWellFormed lctx b)
| e@(.letE _ t v b _) => pure (!e.hasExprMVar && !e.hasFVar) <||> (isWellFormed lctx t <&&> isWellFormed lctx v <&&> isWellFormed lctx b)
| .const .. => return true
| .bvar .. => return true
| .sort .. => return true
| .lit .. => return true
| .mvar mvarId => do
let mvarDecl := (← getMCtx).getDecl mvarId;
if mvarDecl.lctx.isSubPrefixOf lctx then
return true
else match (← getExprMVarAssignment? mvarId) with
| none => return false
| some v => isWellFormed lctx v
| .fvar fvarId => return lctx.contains fvarId
namespace LevelMVarToParam
structure Context where
paramNamePrefix : Name
alreadyUsedPred : Name → Bool
except : LMVarId → Bool
structure State where
mctx : MetavarContext
paramNames : Array Name := #[]
nextParamIdx : Nat
cache : HashMap ExprStructEq Expr := {}
abbrev M := ReaderT Context <| StateM State
instance : MonadMCtx M where
getMCtx := return (← get).mctx
modifyMCtx f := modify fun s => { s with mctx := f s.mctx }
instance : MonadCache ExprStructEq Expr M where
findCached? e := return (← get).cache.find? e
cache e v := modify fun s => { s with cache := s.cache.insert e v }
partial def mkParamName : M Name := do
let ctx ← read
let s ← get
let newParamName := ctx.paramNamePrefix.appendIndexAfter s.nextParamIdx
if ctx.alreadyUsedPred newParamName then
modify fun s => { s with nextParamIdx := s.nextParamIdx + 1 }
mkParamName
else do
modify fun s => { s with nextParamIdx := s.nextParamIdx + 1, paramNames := s.paramNames.push newParamName }
pure newParamName
partial def visitLevel (u : Level) : M Level := do
match u with
| .succ v => return u.updateSucc! (← visitLevel v)
| .max v₁ v₂ => return u.updateMax! (← visitLevel v₁) (← visitLevel v₂)
| .imax v₁ v₂ => return u.updateIMax! (← visitLevel v₁) (← visitLevel v₂)
| .zero => return u
| .param .. => return u
| .mvar mvarId =>
match (← getLevelMVarAssignment? mvarId) with
| some v => visitLevel v
| none =>
if (← read).except mvarId then
return u
else
let p ← mkParamName
let p := mkLevelParam p
assignLevelMVar mvarId p
return p
partial def main (e : Expr) : M Expr :=
if !e.hasMVar then
return e
else
checkCache { val := e : ExprStructEq } fun _ => do
match e with
| .proj _ _ s => return e.updateProj! (← main s)
| .forallE _ d b _ => return e.updateForallE! (← main d) (← main b)
| .lam _ d b _ => return e.updateLambdaE! (← main d) (← main b)
| .letE _ t v b _ => return e.updateLet! (← main t) (← main v) (← main b)
| .app .. => e.withApp fun f args => visitApp f args
| .mdata _ b => return e.updateMData! (← main b)
| .const _ us => return e.updateConst! (← us.mapM visitLevel)
| .sort u => return e.updateSort! (← visitLevel u)
| .mvar .. => visitApp e #[]
| e => return e
where
visitApp (f : Expr) (args : Array Expr) : M Expr := do
match f with
| .mvar mvarId .. =>
match (← getExprMVarAssignment? mvarId) with
| some v => return (← visitApp v args).headBeta
| none => return mkAppN f (← args.mapM main)
| _ => return mkAppN (← main f) (← args.mapM main)
end LevelMVarToParam
structure UnivMVarParamResult where
mctx : MetavarContext
newParamNames : Array Name
nextParamIdx : Nat
expr : Expr
def levelMVarToParam (mctx : MetavarContext) (alreadyUsedPred : Name → Bool) (except : LMVarId → Bool) (e : Expr) (paramNamePrefix : Name := `u) (nextParamIdx : Nat := 1)
: UnivMVarParamResult :=
let (e, s) := LevelMVarToParam.main e { except, paramNamePrefix, alreadyUsedPred } { mctx, nextParamIdx }
{ mctx := s.mctx
newParamNames := s.paramNames
nextParamIdx := s.nextParamIdx
expr := e }
def getExprAssignmentDomain (mctx : MetavarContext) : Array MVarId :=
mctx.eAssignment.foldl (init := #[]) fun a mvarId _ => Array.push a mvarId
end MetavarContext
end Lean
|
4a47818eac8985dcb7736a4970ed888a08b92c44 | 4727251e0cd73359b15b664c3170e5d754078599 | /src/linear_algebra/matrix/trace.lean | f4953a34463cce5cfac26adc500f94c25385f360 | [
"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 | 4,475 | 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 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 matrix
namespace matrix
variables {ι m n p : Type*} {α R S : Type*}
variables [fintype m] [fintype n] [fintype p]
section add_comm_monoid
variables [add_comm_monoid R]
/-- The trace of a square matrix. For more bundled versions, see:
* `matrix.trace_add_monoid_hom`
* `matrix.trace_linear_map`
-/
def trace (A : matrix n n R) : R := ∑ i, diag A i
variables (n R)
@[simp] lemma trace_zero : trace (0 : matrix n n R) = 0 :=
(finset.sum_const (0 : R)).trans $ smul_zero _
variables {n R}
@[simp] lemma trace_add (A B : matrix n n R) : trace (A + B) = trace A + trace B :=
finset.sum_add_distrib
@[simp] lemma trace_smul [monoid α] [distrib_mul_action α R] (r : α) (A : matrix n n R) :
trace (r • A) = r • trace A :=
finset.smul_sum.symm
@[simp] lemma trace_transpose (A : matrix n n R) : trace Aᵀ = trace A := rfl
@[simp] lemma trace_conj_transpose [star_add_monoid R] (A : matrix n n R) :
trace Aᴴ = star (trace A) :=
(star_sum _ _).symm
variables (n α R)
/-- `matrix.trace` as an `add_monoid_hom` -/
@[simps]
def trace_add_monoid_hom : matrix n n R →+ R :=
{ to_fun := trace, map_zero' := trace_zero n R, map_add' := trace_add }
/-- `matrix.trace` as a `linear_map` -/
@[simps]
def trace_linear_map [semiring α] [module α R] : matrix n n R →ₗ[α] R :=
{ to_fun := trace, map_add' := trace_add, map_smul' := trace_smul }
variables {n α R}
@[simp] lemma trace_list_sum (l : list (matrix n n R)) : trace l.sum = (l.map trace).sum :=
map_list_sum (trace_add_monoid_hom n R) l
@[simp] lemma trace_multiset_sum (s : multiset (matrix n n R)) : trace s.sum = (s.map trace).sum :=
map_multiset_sum (trace_add_monoid_hom n R) s
@[simp] lemma trace_sum (s : finset ι) (f : ι → matrix n n R) :
trace (∑ i in s, f i) = ∑ i in s, trace (f i) :=
map_sum (trace_add_monoid_hom n R) f s
end add_comm_monoid
section add_comm_group
variables [add_comm_group R]
@[simp] lemma trace_sub (A B : matrix n n R) : trace (A - B) = trace A - trace B :=
finset.sum_sub_distrib
@[simp] lemma trace_neg (A : matrix n n R) : trace (-A) = -trace A :=
finset.sum_neg_distrib
end add_comm_group
section one
variables [decidable_eq n] [add_comm_monoid R] [has_one R]
@[simp] lemma trace_one : trace (1 : matrix n n R) = fintype.card n :=
by simp_rw [trace, diag_one, pi.one_def, finset.sum_const, nsmul_one, finset.card_univ]
end one
section mul
@[simp] lemma trace_transpose_mul [add_comm_monoid R] [has_mul R]
(A : matrix m n R) (B : matrix n m R) : trace (Aᵀ ⬝ Bᵀ) = trace (A ⬝ B) := finset.sum_comm
lemma trace_mul_comm [add_comm_monoid R] [comm_semigroup R] (A : matrix m n R) (B : matrix n m R) :
trace (A ⬝ B) = trace (B ⬝ A) :=
by rw [←trace_transpose, ←trace_transpose_mul, transpose_mul]
lemma trace_mul_cycle [non_unital_comm_semiring R]
(A : matrix m n R) (B : matrix n p R) (C : matrix p m R) :
trace (A ⬝ B ⬝ C) = trace (C ⬝ A ⬝ B) :=
by rw [trace_mul_comm, matrix.mul_assoc]
lemma trace_mul_cycle' [non_unital_comm_semiring R]
(A : matrix m n R) (B : matrix n p R) (C : matrix p m R) :
trace (A ⬝ (B ⬝ C)) = trace (C ⬝ (A ⬝ B)) :=
by rw [←matrix.mul_assoc, trace_mul_comm]
@[simp] lemma trace_col_mul_row [non_unital_non_assoc_semiring R] (a b : n → R) :
trace (col a ⬝ row b) = dot_product a b :=
by simp [dot_product, trace]
end mul
section fin
variables [add_comm_monoid R]
/-! ### Special cases for `fin n`
While `simp [fin.sum_univ_succ]` can prove these, we include them for convenience and consistency
with `matrix.det_fin_two` etc.
-/
@[simp] lemma trace_fin_zero (A : matrix (fin 0) (fin 0) R) : trace A = 0 :=
rfl
lemma trace_fin_one (A : matrix (fin 1) (fin 1) R) : trace A = A 0 0 :=
add_zero _
lemma trace_fin_two (A : matrix (fin 2) (fin 2) R) : trace A = A 0 0 + A 1 1 :=
congr_arg ((+) _) (add_zero (A 1 1))
lemma trace_fin_three (A : matrix (fin 3) (fin 3) R) : trace A = A 0 0 + A 1 1 + A 2 2 :=
by { rw [← add_zero (A 2 2), add_assoc], refl }
end fin
end matrix
|
188e952491cefad787fe72e0dae378752a57a75e | 74addaa0e41490cbaf2abd313a764c96df57b05d | /Mathlib/data/mv_polynomial/monad.lean | a1c339505d3ece9c54760088a0733ddfb9f54c7c | [] | 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 | 17,949 | lean | /-
Copyright (c) 2020 Johan Commelin and Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin and Robert Y. Lewis
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.default
import Mathlib.data.mv_polynomial.rename
import Mathlib.PostPort
universes u_1 u_2 u_3 u_4 u_5
namespace Mathlib
/-!
# Monad operations on `mv_polynomial`
This file defines two monadic operations on `mv_polynomial`. Given `p : mv_polynomial σ R`,
* `mv_polynomial.bind₁` and `mv_polynomial.join₁` operate on the variable type `σ`.
* `mv_polynomial.bind₂` and `mv_polynomial.join₂` operate on the coefficient type `R`.
- `mv_polynomial.bind₁ f φ` with `f : σ → mv_polynomial τ R` and `φ : mv_polynomial σ R`,
is the polynomial `φ(f 1, ..., f i, ...) : mv_polynomial τ R`.
- `mv_polynomial.join₁ φ` with `φ : mv_polynomial (mv_polynomial σ R) R` collapses `φ` to
a `mv_polynomial σ R`, by evaluating `φ` under the map `X f ↦ f` for `f : mv_polynomial σ R`.
In other words, if you have a polynomial `φ` in a set of variables indexed by a polynomial ring,
you evaluate the polynomial in these indexing polynomials.
- `mv_polynomial.bind₂ f φ` with `f : R →+* mv_polynomial σ S` and `φ : mv_polynomial σ R`
is the `mv_polynomial σ S` obtained from `φ` by mapping the coefficients of `φ` through `f`
and considering the resulting polynomial as polynomial expression in `mv_polynomial σ R`.
- `mv_polynomial.join₂ φ` with `φ : mv_polynomial σ (mv_polynomial σ R)` collapses `φ` to
a `mv_polynomial σ R`, by considering `φ` as polynomial expression in `mv_polynomial σ R`.
These operations themselves have algebraic structure: `mv_polynomial.bind₁`
and `mv_polynomial.join₁` are algebra homs and
`mv_polynomial.bind₂` and `mv_polynomial.join₂` are ring homs.
They interact in convenient ways with `mv_polynomial.rename`, `mv_polynomial.map`,
`mv_polynomial.vars`, and other polynomial operations.
Indeed, `mv_polynomial.rename` is the "map" operation for the (`bind₁`, `join₁`) pair,
whereas `mv_polynomial.map` is the "map" operation for the other pair.
## Implementation notes
We add an `is_lawful_monad` instance for the (`bind₁`, `join₁`) pair.
The second pair cannot be instantiated as a `monad`,
since it is not a monad in `Type` but in `CommRing` (or rather `CommSemiRing`).
-/
namespace mv_polynomial
/--
`bind₁` is the "left hand side" bind operation on `mv_polynomial`, operating on the variable type.
Given a polynomial `p : mv_polynomial σ R` and a map `f : σ → mv_polynomial τ R` taking variables
in `p` to polynomials in the variable type `τ`, `bind₁ f p` replaces each variable in `p` with
its value under `f`, producing a new polynomial in `τ`. The coefficient type remains the same.
This operation is an algebra hom.
-/
def bind₁ {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [comm_semiring R] (f : σ → mv_polynomial τ R) : alg_hom R (mv_polynomial σ R) (mv_polynomial τ R) :=
aeval f
/--
`bind₂` is the "right hand side" bind operation on `mv_polynomial`, operating on the coefficient type.
Given a polynomial `p : mv_polynomial σ R` and a map `f : R → mv_polynomial σ S` taking coefficients
in `p` to polynomials over a new ring `S`, `bind₂ f p` replaces each coefficient in `p` with its
value under `f`, producing a new polynomial over `S`. The variable type remains the same.
This operation is a ring hom.
-/
def bind₂ {σ : Type u_1} {R : Type u_3} {S : Type u_4} [comm_semiring R] [comm_semiring S] (f : R →+* mv_polynomial σ S) : mv_polynomial σ R →+* mv_polynomial σ S :=
eval₂_hom f X
/--
`join₁` is the monadic join operation corresponding to `mv_polynomial.bind₁`. Given a polynomial `p`
with coefficients in `R` whose variables are polynomials in `σ` with coefficients in `R`,
`join₁ p` collapses `p` to a polynomial with variables in `σ` and coefficients in `R`.
This operation is an algebra hom.
-/
def join₁ {σ : Type u_1} {R : Type u_3} [comm_semiring R] : alg_hom R (mv_polynomial (mv_polynomial σ R) R) (mv_polynomial σ R) :=
aeval id
/--
`join₂` is the monadic join operation corresponding to `mv_polynomial.bind₂`. Given a polynomial `p`
with variables in `σ` whose coefficients are polynomials in `σ` with coefficients in `R`,
`join₂ p` collapses `p` to a polynomial with variables in `σ` and coefficients in `R`.
This operation is a ring hom.
-/
def join₂ {σ : Type u_1} {R : Type u_3} [comm_semiring R] : mv_polynomial σ (mv_polynomial σ R) →+* mv_polynomial σ R :=
eval₂_hom (ring_hom.id (mv_polynomial σ R)) X
@[simp] theorem aeval_eq_bind₁ {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [comm_semiring R] (f : σ → mv_polynomial τ R) : aeval f = bind₁ f :=
rfl
@[simp] theorem eval₂_hom_C_eq_bind₁ {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [comm_semiring R] (f : σ → mv_polynomial τ R) : eval₂_hom C f = ↑(bind₁ f) :=
rfl
@[simp] theorem eval₂_hom_eq_bind₂ {σ : Type u_1} {R : Type u_3} {S : Type u_4} [comm_semiring R] [comm_semiring S] (f : R →+* mv_polynomial σ S) : eval₂_hom f X = bind₂ f :=
rfl
@[simp] theorem aeval_id_eq_join₁ (σ : Type u_1) (R : Type u_3) [comm_semiring R] : aeval id = join₁ :=
rfl
theorem eval₂_hom_C_id_eq_join₁ (σ : Type u_1) (R : Type u_3) [comm_semiring R] (φ : mv_polynomial (mv_polynomial σ R) R) : coe_fn (eval₂_hom C id) φ = coe_fn join₁ φ :=
rfl
@[simp] theorem eval₂_hom_id_X_eq_join₂ (σ : Type u_1) (R : Type u_3) [comm_semiring R] : eval₂_hom (ring_hom.id (mv_polynomial σ R)) X = join₂ :=
rfl
-- In this file, we don't want to use these simp lemmas,
-- because we first need to show how these new definitions interact
-- and the proofs fall back on unfolding the definitions and call simp afterwards
@[simp] theorem bind₁_X_right {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [comm_semiring R] (f : σ → mv_polynomial τ R) (i : σ) : coe_fn (bind₁ f) (X i) = f i :=
aeval_X f i
@[simp] theorem bind₂_X_right {σ : Type u_1} {R : Type u_3} {S : Type u_4} [comm_semiring R] [comm_semiring S] (f : R →+* mv_polynomial σ S) (i : σ) : coe_fn (bind₂ f) (X i) = X i :=
eval₂_hom_X' f X i
@[simp] theorem bind₁_X_left {σ : Type u_1} {R : Type u_3} [comm_semiring R] : bind₁ X = alg_hom.id R (mv_polynomial σ R) := sorry
theorem aeval_X_left {σ : Type u_1} {R : Type u_3} [comm_semiring R] : aeval X = alg_hom.id R (mv_polynomial σ R) :=
eq.mpr (id (Eq._oldrec (Eq.refl (aeval X = alg_hom.id R (mv_polynomial σ R))) (aeval_eq_bind₁ X)))
(eq.mpr (id (Eq._oldrec (Eq.refl (bind₁ X = alg_hom.id R (mv_polynomial σ R))) bind₁_X_left))
(Eq.refl (alg_hom.id R (mv_polynomial σ R))))
theorem aeval_X_left_apply {σ : Type u_1} {R : Type u_3} [comm_semiring R] (φ : mv_polynomial σ R) : coe_fn (aeval X) φ = φ := sorry
@[simp] theorem bind₁_C_right {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [comm_semiring R] (f : σ → mv_polynomial τ R) (x : R) : coe_fn (bind₁ f) (coe_fn C x) = coe_fn C x := sorry
@[simp] theorem bind₂_C_right {σ : Type u_1} {R : Type u_3} {S : Type u_4} [comm_semiring R] [comm_semiring S] (f : R →+* mv_polynomial σ S) (r : R) : coe_fn (bind₂ f) (coe_fn C r) = coe_fn f r :=
eval₂_hom_C f X r
@[simp] theorem bind₂_C_left {σ : Type u_1} {R : Type u_3} [comm_semiring R] : bind₂ C = ring_hom.id (mv_polynomial σ R) := sorry
@[simp] theorem bind₂_comp_C {σ : Type u_1} {R : Type u_3} {S : Type u_4} [comm_semiring R] [comm_semiring S] (f : R →+* mv_polynomial σ S) : ring_hom.comp (bind₂ f) C = f :=
ring_hom.ext (bind₂_C_right f)
@[simp] theorem join₂_map {σ : Type u_1} {R : Type u_3} {S : Type u_4} [comm_semiring R] [comm_semiring S] (f : R →+* mv_polynomial σ S) (φ : mv_polynomial σ R) : coe_fn join₂ (coe_fn (map f) φ) = coe_fn (bind₂ f) φ := sorry
@[simp] theorem join₂_comp_map {σ : Type u_1} {R : Type u_3} {S : Type u_4} [comm_semiring R] [comm_semiring S] (f : R →+* mv_polynomial σ S) : ring_hom.comp join₂ (map f) = bind₂ f :=
ring_hom.ext (join₂_map f)
theorem aeval_id_rename {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [comm_semiring R] (f : σ → mv_polynomial τ R) (p : mv_polynomial σ R) : coe_fn (aeval id) (coe_fn (rename f) p) = coe_fn (aeval f) p :=
eq.mpr (id (Eq._oldrec (Eq.refl (coe_fn (aeval id) (coe_fn (rename f) p) = coe_fn (aeval f) p)) (aeval_rename f id p)))
(eq.mpr (id (Eq._oldrec (Eq.refl (coe_fn (aeval (id ∘ f)) p = coe_fn (aeval f) p)) (function.comp.left_id f)))
(Eq.refl (coe_fn (aeval f) p)))
@[simp] theorem join₁_rename {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [comm_semiring R] (f : σ → mv_polynomial τ R) (φ : mv_polynomial σ R) : coe_fn join₁ (coe_fn (rename f) φ) = coe_fn (bind₁ f) φ :=
aeval_id_rename f φ
@[simp] theorem bind₁_id {σ : Type u_1} {R : Type u_3} [comm_semiring R] : bind₁ id = join₁ :=
rfl
@[simp] theorem bind₂_id {σ : Type u_1} {R : Type u_3} [comm_semiring R] : bind₂ (ring_hom.id (mv_polynomial σ R)) = join₂ :=
rfl
theorem bind₁_bind₁ {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [comm_semiring R] {υ : Type u_4} (f : σ → mv_polynomial τ R) (g : τ → mv_polynomial υ R) (φ : mv_polynomial σ R) : coe_fn (bind₁ g) (coe_fn (bind₁ f) φ) = coe_fn (bind₁ fun (i : σ) => coe_fn (bind₁ g) (f i)) φ := sorry
theorem bind₁_comp_bind₁ {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [comm_semiring R] {υ : Type u_4} (f : σ → mv_polynomial τ R) (g : τ → mv_polynomial υ R) : alg_hom.comp (bind₁ g) (bind₁ f) = bind₁ fun (i : σ) => coe_fn (bind₁ g) (f i) :=
alg_hom_ext fun (i : σ) => bind₁_bind₁ (fun (n : σ) => f n) (fun (n : τ) => g n) (X i)
theorem bind₂_comp_bind₂ {σ : Type u_1} {R : Type u_3} {S : Type u_4} {T : Type u_5} [comm_semiring R] [comm_semiring S] [comm_semiring T] (f : R →+* mv_polynomial σ S) (g : S →+* mv_polynomial σ T) : ring_hom.comp (bind₂ g) (bind₂ f) = bind₂ (ring_hom.comp (bind₂ g) f) := sorry
theorem bind₂_bind₂ {σ : Type u_1} {R : Type u_3} {S : Type u_4} {T : Type u_5} [comm_semiring R] [comm_semiring S] [comm_semiring T] (f : R →+* mv_polynomial σ S) (g : S →+* mv_polynomial σ T) (φ : mv_polynomial σ R) : coe_fn (bind₂ g) (coe_fn (bind₂ f) φ) = coe_fn (bind₂ (ring_hom.comp (bind₂ g) f)) φ :=
ring_hom.congr_fun (bind₂_comp_bind₂ f g) φ
theorem rename_comp_bind₁ {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [comm_semiring R] {υ : Type u_4} (f : σ → mv_polynomial τ R) (g : τ → υ) : alg_hom.comp (rename g) (bind₁ f) = bind₁ fun (i : σ) => coe_fn (rename g) (f i) := sorry
theorem rename_bind₁ {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [comm_semiring R] {υ : Type u_4} (f : σ → mv_polynomial τ R) (g : τ → υ) (φ : mv_polynomial σ R) : coe_fn (rename g) (coe_fn (bind₁ f) φ) = coe_fn (bind₁ fun (i : σ) => coe_fn (rename g) (f i)) φ :=
alg_hom.congr_fun (rename_comp_bind₁ f g) φ
theorem map_bind₂ {σ : Type u_1} {R : Type u_3} {S : Type u_4} {T : Type u_5} [comm_semiring R] [comm_semiring S] [comm_semiring T] (f : R →+* mv_polynomial σ S) (g : S →+* T) (φ : mv_polynomial σ R) : coe_fn (map g) (coe_fn (bind₂ f) φ) = coe_fn (bind₂ (ring_hom.comp (map g) f)) φ := sorry
theorem bind₁_comp_rename {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [comm_semiring R] {υ : Type u_4} (f : τ → mv_polynomial υ R) (g : σ → τ) : alg_hom.comp (bind₁ f) (rename g) = bind₁ (f ∘ g) := sorry
theorem bind₁_rename {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [comm_semiring R] {υ : Type u_4} (f : τ → mv_polynomial υ R) (g : σ → τ) (φ : mv_polynomial σ R) : coe_fn (bind₁ f) (coe_fn (rename g) φ) = coe_fn (bind₁ (f ∘ g)) φ :=
alg_hom.congr_fun (bind₁_comp_rename f g) φ
theorem bind₂_map {σ : Type u_1} {R : Type u_3} {S : Type u_4} {T : Type u_5} [comm_semiring R] [comm_semiring S] [comm_semiring T] (f : S →+* mv_polynomial σ T) (g : R →+* S) (φ : mv_polynomial σ R) : coe_fn (bind₂ f) (coe_fn (map g) φ) = coe_fn (bind₂ (ring_hom.comp f g)) φ := sorry
@[simp] theorem map_comp_C {σ : Type u_1} {R : Type u_3} {S : Type u_4} [comm_semiring R] [comm_semiring S] (f : R →+* S) : ring_hom.comp (map f) C = ring_hom.comp C f :=
ring_hom.ext fun (x : R) => map_C f x
-- mixing the two monad structures
theorem hom_bind₁ {σ : Type u_1} {τ : Type u_2} {R : Type u_3} {S : Type u_4} [comm_semiring R] [comm_semiring S] (f : mv_polynomial τ R →+* S) (g : σ → mv_polynomial τ R) (φ : mv_polynomial σ R) : coe_fn f (coe_fn (bind₁ g) φ) = coe_fn (eval₂_hom (ring_hom.comp f C) fun (i : σ) => coe_fn f (g i)) φ := sorry
theorem map_bind₁ {σ : Type u_1} {τ : Type u_2} {R : Type u_3} {S : Type u_4} [comm_semiring R] [comm_semiring S] (f : R →+* S) (g : σ → mv_polynomial τ R) (φ : mv_polynomial σ R) : coe_fn (map f) (coe_fn (bind₁ g) φ) = coe_fn (bind₁ fun (i : σ) => coe_fn (map f) (g i)) (coe_fn (map f) φ) := sorry
@[simp] theorem eval₂_hom_comp_C {σ : Type u_1} {R : Type u_3} {S : Type u_4} [comm_semiring R] [comm_semiring S] (f : R →+* S) (g : σ → S) : ring_hom.comp (eval₂_hom f g) C = f :=
ring_hom.ext fun (r : R) => eval₂_C f g r
theorem eval₂_hom_bind₁ {σ : Type u_1} {τ : Type u_2} {R : Type u_3} {S : Type u_4} [comm_semiring R] [comm_semiring S] (f : R →+* S) (g : τ → S) (h : σ → mv_polynomial τ R) (φ : mv_polynomial σ R) : coe_fn (eval₂_hom f g) (coe_fn (bind₁ h) φ) = coe_fn (eval₂_hom f fun (i : σ) => coe_fn (eval₂_hom f g) (h i)) φ := sorry
theorem aeval_bind₁ {σ : Type u_1} {τ : Type u_2} {R : Type u_3} {S : Type u_4} [comm_semiring R] [comm_semiring S] [algebra R S] (f : τ → S) (g : σ → mv_polynomial τ R) (φ : mv_polynomial σ R) : coe_fn (aeval f) (coe_fn (bind₁ g) φ) = coe_fn (aeval fun (i : σ) => coe_fn (aeval f) (g i)) φ :=
eval₂_hom_bind₁ (algebra_map R S) (fun (n : τ) => f n) g φ
theorem aeval_comp_bind₁ {σ : Type u_1} {τ : Type u_2} {R : Type u_3} {S : Type u_4} [comm_semiring R] [comm_semiring S] [algebra R S] (f : τ → S) (g : σ → mv_polynomial τ R) : alg_hom.comp (aeval f) (bind₁ g) = aeval fun (i : σ) => coe_fn (aeval f) (g i) :=
alg_hom_ext fun (i : σ) => aeval_bind₁ (fun (n : τ) => f n) (fun (n : σ) => g n) (X i)
theorem eval₂_hom_comp_bind₂ {σ : Type u_1} {R : Type u_3} {S : Type u_4} {T : Type u_5} [comm_semiring R] [comm_semiring S] [comm_semiring T] (f : S →+* T) (g : σ → T) (h : R →+* mv_polynomial σ S) : ring_hom.comp (eval₂_hom f g) (bind₂ h) = eval₂_hom (ring_hom.comp (eval₂_hom f g) h) g := sorry
theorem eval₂_hom_bind₂ {σ : Type u_1} {R : Type u_3} {S : Type u_4} {T : Type u_5} [comm_semiring R] [comm_semiring S] [comm_semiring T] (f : S →+* T) (g : σ → T) (h : R →+* mv_polynomial σ S) (φ : mv_polynomial σ R) : coe_fn (eval₂_hom f g) (coe_fn (bind₂ h) φ) = coe_fn (eval₂_hom (ring_hom.comp (eval₂_hom f g) h) g) φ :=
ring_hom.congr_fun (eval₂_hom_comp_bind₂ f g h) φ
theorem aeval_bind₂ {σ : Type u_1} {R : Type u_3} {S : Type u_4} {T : Type u_5} [comm_semiring R] [comm_semiring S] [comm_semiring T] [algebra S T] (f : σ → T) (g : R →+* mv_polynomial σ S) (φ : mv_polynomial σ R) : coe_fn (aeval f) (coe_fn (bind₂ g) φ) = coe_fn (eval₂_hom (ring_hom.comp (↑(aeval f)) g) f) φ :=
eval₂_hom_bind₂ (algebra_map S T) (fun (n : σ) => f n) g φ
theorem eval₂_hom_C_left {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [comm_semiring R] (f : σ → mv_polynomial τ R) : eval₂_hom C f = ↑(bind₁ f) :=
rfl
theorem bind₁_monomial {σ : Type u_1} {τ : Type u_2} {R : Type u_3} [comm_semiring R] (f : σ → mv_polynomial τ R) (d : σ →₀ ℕ) (r : R) : coe_fn (bind₁ f) (monomial d r) = coe_fn C r * finset.prod (finsupp.support d) fun (i : σ) => f i ^ coe_fn d i := sorry
theorem bind₂_monomial {σ : Type u_1} {R : Type u_3} {S : Type u_4} [comm_semiring R] [comm_semiring S] (f : R →+* mv_polynomial σ S) (d : σ →₀ ℕ) (r : R) : coe_fn (bind₂ f) (monomial d r) = coe_fn f r * monomial d 1 := sorry
@[simp] theorem bind₂_monomial_one {σ : Type u_1} {R : Type u_3} {S : Type u_4} [comm_semiring R] [comm_semiring S] (f : R →+* mv_polynomial σ S) (d : σ →₀ ℕ) : coe_fn (bind₂ f) (monomial d 1) = monomial d 1 := sorry
protected instance monad {R : Type u_3} [comm_semiring R] : Monad fun (σ : Type u_1) => mv_polynomial σ R := sorry
protected instance is_lawful_functor {R : Type u_3} [comm_semiring R] : is_lawful_functor fun (σ : Type u_1) => mv_polynomial σ R :=
is_lawful_functor.mk
(fun (α : Type u_1) (x : mv_polynomial α R) =>
eq.mpr
(id
(Eq.trans
((fun (a a_1 : mv_polynomial α R) (e_1 : a = a_1) (ᾰ ᾰ_1 : mv_polynomial α R) (e_2 : ᾰ = ᾰ_1) =>
congr (congr_arg Eq e_1) e_2)
(id <$> x) x (rename_id x) x x (Eq.refl x))
(propext (eq_self_iff_true x))))
trivial)
fun (α β γ : Type u_1) (g : α → β) (h : β → γ) (x : mv_polynomial α R) =>
eq.mpr
(id
(Eq.trans
((fun (a a_1 : mv_polynomial γ R) (e_1 : a = a_1) (ᾰ ᾰ_1 : mv_polynomial γ R) (e_2 : ᾰ = ᾰ_1) =>
congr (congr_arg Eq e_1) e_2)
((h ∘ g) <$> x) (coe_fn (rename (h ∘ g)) x) (Eq.refl (coe_fn (rename (h ∘ g)) x)) (h <$> g <$> x)
(coe_fn (rename (h ∘ g)) x) (rename_rename g h x))
(propext (eq_self_iff_true (coe_fn (rename (h ∘ g)) x)))))
trivial
protected instance is_lawful_monad {R : Type u_3} [comm_semiring R] : is_lawful_monad fun (σ : Type u_1) => mv_polynomial σ R := sorry
|
1b1d4b66b2a9d536c1258a1820c65edec53a83df | e9dbaaae490bc072444e3021634bf73664003760 | /src/Problems/2003/IMO_2003_P3.lean | 02cdd8e38d2ad9299b0b3be0e37b75dcbb5740cf | [
"Apache-2.0"
] | permissive | liaofei1128/geometry | 566d8bfe095ce0c0113d36df90635306c60e975b | 3dd128e4eec8008764bb94e18b932f9ffd66e6b3 | refs/heads/master | 1,678,996,510,399 | 1,581,454,543,000 | 1,583,337,839,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 335 | lean | import Geo.Geo.Core
namespace Geo
open Analytic Polygon
def IMO_2003_P3 : Prop :=
∀ (hex : Polygon 6),
convex hex →
(hex.oppoSides rfl).allP (λ ⟨s₁, s₂⟩ =>
let midpDist := dist s₁.midp s₂.midp;
let sumLengths := ulen s₁ + ulen s₂;
midpDist = (Real.sqrt 3) / 2 * sumLengths
) →
hex.angles.allEq
end Geo
|
5d43ea2c42ad22bb28f13391a76d5ec4915ee871 | 5e3548e65f2c037cb94cd5524c90c623fbd6d46a | /AMC_2021_A_25.lean | 86b54df9c7e2428ada2b41049d43f43844168ec0 | [] | no_license | ahayat16/lean_exos | d4f08c30adb601a06511a71b5ffb4d22d12ef77f | 682f2552d5b04a8c8eb9e4ab15f875a91b03845c | refs/heads/main | 1,693,101,073,585 | 1,636,479,336,000 | 1,636,479,336,000 | 415,000,441 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 229 | lean | import data.complex.basic
import number_theory.divisors
theorem AMC_2021_A_25 (N:ℕ)(f:ℕ → ℝ)(h:∀(n:ℕ), f n = (finset.card (nat.divisors n))/(n^(1/3)))(hN:∀(n:ℕ),n≠N → (f N > f n)):
N=2520
:=
begin
sorry
end |
6670375493f7950e69d81025d7c94a097fa13aab | 9dd3f3912f7321eb58ee9aa8f21778ad6221f87c | /tests/lean/run/simplifier_custom_relations.lean | 0590d0234db571f5ec0a315ec116df911f478b18 | [
"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 | 987 | lean | open tactic
constants (A : Type) (rel : A → A → Prop)
(rel.refl : ∀ a, rel a a)
(rel.symm : ∀ a b, rel a b → rel b a)
(rel.trans : ∀ a b c, rel a b → rel b c → rel a c)
attribute [refl] rel.refl
attribute [symm] rel.symm
attribute [trans] rel.trans
constants (x y z : A) (f g h : A → A)
(H₁ : rel (f x) (g y))
(H₂ : rel (h (g y)) z)
(Hf : ∀ (a b : A), rel a b → rel (f a) (f b))
(Hg : ∀ (a b : A), rel a b → rel (g a) (g b))
(Hh : ∀ (a b : A), rel a b → rel (h a) (h b))
attribute [simp] H₁ H₂
attribute [congr] Hf Hg Hh
print [simp] default
print [congr] default
meta definition relsimp_core (e : expr) : tactic (expr × expr) :=
do S ← simp_lemmas.mk_default,
e_type ← infer_type e >>= whnf,
simplify_core {} S `rel e
example : rel (h (f x)) z :=
by do e₁ ← to_expr `(h (f x)),
(e₁', pf) ← relsimp_core e₁,
exact pf
|
339213f5ce4b698cbc7742cbb8c67d8335d298df | 367134ba5a65885e863bdc4507601606690974c1 | /src/group_theory/sylow.lean | 5a4a5bd49790d3837e4ec978640002334098fead | [
"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 | 13,251 | lean | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import group_theory.group_action
import group_theory.quotient_group
import group_theory.order_of_element
import data.zmod.basic
import data.fintype.card
import data.list.rotate
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
In this file, currently only the first of these results is proven.
## Main statements
* `exists_prime_order_of_dvd_card`: For every prime `p` dividing the order of `G` there exists an
element of order `p` in `G`. This is known as Cauchy`s theorem.
* `exists_subgroup_card_pow_prime`: A generalisation of the first of the Sylow theorems: For every
prime power `pⁿ` dividing `G`, there exists a subgroup of `G` of order `pⁿ`.
## TODO
* Prove the second and third of the Sylow theorems.
* Sylow theorems for infinite groups
-/
open equiv fintype finset mul_action function
open equiv.perm subgroup list quotient_group
open_locale big_operators
universes u v w
variables {G : Type u} {α : Type v} {β : Type w} [group G]
local attribute [instance, priority 10] subtype.fintype set_fintype classical.prop_decidable
namespace mul_action
variables [mul_action G α]
lemma mem_fixed_points_iff_card_orbit_eq_one {a : α}
[fintype (orbit G a)] : a ∈ fixed_points G α ↔ card (orbit G a) = 1 :=
begin
rw [fintype.card_eq_one_iff, mem_fixed_points],
split,
{ exact λ h, ⟨⟨a, mem_orbit_self _⟩, λ ⟨b, ⟨x, hx⟩⟩, subtype.eq $ by simp [h x, hx.symm]⟩ },
{ assume h x,
rcases h with ⟨⟨z, hz⟩, hz₁⟩,
exact calc x • a = z : subtype.mk.inj (hz₁ ⟨x • a, mem_orbit _ _⟩)
... = a : (subtype.mk.inj (hz₁ ⟨a, mem_orbit_self _⟩)).symm }
end
lemma card_modeq_card_fixed_points [fintype α] [fintype G] [fintype (fixed_points G α)]
(p : ℕ) {n : ℕ} [hp : fact p.prime] (h : card G = p ^ n) :
card α ≡ card (fixed_points G α) [MOD p] :=
calc card α = card (Σ y : quotient (orbit_rel G α), {x // quotient.mk' x = y}) :
card_congr (sigma_preimage_equiv (@quotient.mk' _ (orbit_rel G α))).symm
... = ∑ a : quotient (orbit_rel G α), card {x // quotient.mk' x = a} : card_sigma _
... ≡ ∑ a : fixed_points G α, 1 [MOD p] :
begin
rw [← zmod.eq_iff_modeq_nat p, sum_nat_cast, sum_nat_cast],
refine eq.symm (sum_bij_ne_zero (λ a _ _, quotient.mk' a.1)
(λ _ _ _, mem_univ _)
(λ a₁ a₂ _ _ _ _ h,
subtype.eq ((mem_fixed_points' α).1 a₂.2 a₁.1 (quotient.exact' h)))
(λ b, _)
(λ a ha _, by rw [← mem_fixed_points_iff_card_orbit_eq_one.1 a.2];
simp only [quotient.eq']; congr)),
{ refine quotient.induction_on' b (λ b _ hb, _),
have : card (orbit G b) ∣ p ^ n,
{ rw [← h, fintype.card_congr (orbit_equiv_quotient_stabilizer G b)],
exact card_quotient_dvd_card _ },
rcases (nat.dvd_prime_pow hp).1 this with ⟨k, _, hk⟩,
have hb' :¬ p ^ 1 ∣ p ^ k,
{ rw [pow_one, ← hk, ← nat.modeq.modeq_zero_iff, ← zmod.eq_iff_modeq_nat,
nat.cast_zero, ← ne.def],
exact eq.mpr (by simp only [quotient.eq']; congr) hb },
have : k = 0 := nat.le_zero_iff.1 (nat.le_of_lt_succ (lt_of_not_ge (mt (pow_dvd_pow p) hb'))),
refine ⟨⟨b, mem_fixed_points_iff_card_orbit_eq_one.2 $ by rw [hk, this, pow_zero]⟩,
mem_univ _, _, rfl⟩,
rw [nat.cast_one], exact one_ne_zero }
end
... = _ : by simp; refl
end mul_action
lemma quotient_group.card_preimage_mk [fintype G] (s : subgroup G)
(t : set (quotient s)) : fintype.card (quotient_group.mk ⁻¹' t) =
fintype.card s * fintype.card t :=
by rw [← fintype.card_prod, fintype.card_congr
(preimage_mk_equiv_subgroup_times_set _ _)]
namespace sylow
/-- Given a vector `v` of length `n`, make a vector of length `n+1` whose product is `1`,
by consing the the inverse of the product of `v`. -/
def mk_vector_prod_eq_one (n : ℕ) (v : vector G n) : vector G (n+1) :=
v.to_list.prod⁻¹ ::ᵥ v
lemma mk_vector_prod_eq_one_injective (n : ℕ) : injective (@mk_vector_prod_eq_one G _ n) :=
λ ⟨v, _⟩ ⟨w, _⟩ h, subtype.eq (show v = w, by injection h with h; injection h)
/-- The type of vectors with terms from `G`, length `n`, and product equal to `1:G`. -/
def vectors_prod_eq_one (G : Type*) [group G] (n : ℕ) : set (vector G n) :=
{v | v.to_list.prod = 1}
lemma mem_vectors_prod_eq_one {n : ℕ} (v : vector G n) :
v ∈ vectors_prod_eq_one G n ↔ v.to_list.prod = 1 := iff.rfl
lemma mem_vectors_prod_eq_one_iff {n : ℕ} (v : vector G (n + 1)) :
v ∈ vectors_prod_eq_one G (n + 1) ↔ v ∈ set.range (@mk_vector_prod_eq_one G _ n) :=
⟨λ (h : v.to_list.prod = 1), ⟨v.tail,
begin
unfold mk_vector_prod_eq_one,
conv {to_rhs, rw ← vector.cons_head_tail v},
suffices : (v.tail.to_list.prod)⁻¹ = v.head,
{ rw this },
rw [← mul_left_inj v.tail.to_list.prod, inv_mul_self, ← list.prod_cons,
← vector.to_list_cons, vector.cons_head_tail, h]
end⟩,
λ ⟨w, hw⟩, by rw [mem_vectors_prod_eq_one, ← hw, mk_vector_prod_eq_one,
vector.to_list_cons, list.prod_cons, inv_mul_self]⟩
/-- The rotation action of `zmod n` (viewed as multiplicative group) on
`vectors_prod_eq_one G n`, where `G` is a multiplicative group. -/
def rotate_vectors_prod_eq_one (G : Type*) [group G] (n : ℕ)
(m : multiplicative (zmod n)) (v : vectors_prod_eq_one G n) : vectors_prod_eq_one G n :=
⟨⟨v.1.to_list.rotate m.val, by simp⟩, prod_rotate_eq_one_of_prod_eq_one v.2 _⟩
instance rotate_vectors_prod_eq_one.mul_action (n : ℕ) [fact (0 < n)] :
mul_action (multiplicative (zmod n)) (vectors_prod_eq_one G n) :=
{ smul := (rotate_vectors_prod_eq_one G n),
one_smul :=
begin
intro v, apply subtype.eq, apply vector.eq _ _,
show rotate _ (0 : zmod n).val = _, rw zmod.val_zero,
exact rotate_zero v.1.to_list
end,
mul_smul := λ a b ⟨⟨v, hv₁⟩, hv₂⟩, subtype.eq $ vector.eq _ _ $
show v.rotate ((a + b : zmod n).val) = list.rotate (list.rotate v (b.val)) (a.val),
by rw [zmod.val_add, rotate_rotate, ← rotate_mod _ (b.val + a.val), add_comm, hv₁] }
lemma one_mem_vectors_prod_eq_one (n : ℕ) : vector.repeat (1 : G) n ∈ vectors_prod_eq_one G n :=
by simp [vector.repeat, vectors_prod_eq_one]
lemma one_mem_fixed_points_rotate (n : ℕ) [fact (0 < n)] :
(⟨vector.repeat (1 : G) n, one_mem_vectors_prod_eq_one n⟩ : vectors_prod_eq_one G n) ∈
fixed_points (multiplicative (zmod n)) (vectors_prod_eq_one G n) :=
λ m, subtype.eq $ vector.eq _ _ $
rotate_eq_self_iff_eq_repeat.2 ⟨(1 : G),
show list.repeat (1 : G) n = list.repeat 1 (list.repeat (1 : G) n).length, by simp⟩ _
/-- Cauchy's theorem -/
lemma exists_prime_order_of_dvd_card [fintype G] (p : ℕ) [hp : fact p.prime]
(hdvd : p ∣ card G) : ∃ x : G, order_of x = p :=
let n : ℕ+ := ⟨p - 1, nat.sub_pos_of_lt hp.one_lt⟩ in
have hn : p = n + 1 := nat.succ_sub hp.pos,
have hcard : card (vectors_prod_eq_one G (n + 1)) = card G ^ (n : ℕ),
by rw [set.ext mem_vectors_prod_eq_one_iff,
set.card_range_of_injective (mk_vector_prod_eq_one_injective _), card_vector],
have hzmod : fintype.card (multiplicative (zmod p)) = p ^ 1,
by { rw pow_one p, exact zmod.card p },
have hmodeq : _ = _ := @mul_action.card_modeq_card_fixed_points
(multiplicative (zmod p)) (vectors_prod_eq_one G p) _ _ _ _ _ _ 1 hp hzmod,
have hdvdcard : p ∣ fintype.card (vectors_prod_eq_one G (n + 1)) :=
calc p ∣ card G ^ 1 : by rwa pow_one
... ∣ card G ^ (n : ℕ) : pow_dvd_pow _ n.2
... = card (vectors_prod_eq_one G (n + 1)) : hcard.symm,
have hdvdcard₂ : p ∣ card (fixed_points (multiplicative (zmod p)) (vectors_prod_eq_one G p)),
by { rw nat.dvd_iff_mod_eq_zero at hdvdcard ⊢, rwa [← hn, hmodeq] at hdvdcard },
have hcard_pos : 0 < card (fixed_points (multiplicative (zmod p)) (vectors_prod_eq_one G p)) :=
fintype.card_pos_iff.2 ⟨⟨⟨vector.repeat 1 p, one_mem_vectors_prod_eq_one _⟩,
one_mem_fixed_points_rotate _⟩⟩,
have hlt : 1 < card (fixed_points (multiplicative (zmod p)) (vectors_prod_eq_one G p)) :=
calc (1 : ℕ) < p : hp.one_lt
... ≤ _ : nat.le_of_dvd hcard_pos hdvdcard₂,
let ⟨⟨⟨⟨x, hx₁⟩, hx₂⟩, hx₃⟩, hx₄⟩ := fintype.exists_ne_of_one_lt_card hlt
⟨_, one_mem_fixed_points_rotate p⟩ in
have hx : x ≠ list.repeat (1 : G) p, from λ h, by simpa [h, vector.repeat] using hx₄,
have ∃ a, x = list.repeat a x.length := by exactI rotate_eq_self_iff_eq_repeat.1 (λ n,
have list.rotate x (n : zmod p).val = x :=
subtype.mk.inj (subtype.mk.inj (hx₃ (n : zmod p))),
by rwa [zmod.val_nat_cast, ← hx₁, rotate_mod] at this),
let ⟨a, ha⟩ := this in
⟨a, have hx1 : x.prod = 1 := hx₂,
have ha1: a ≠ 1, from λ h, hx (ha.symm ▸ h ▸ hx₁ ▸ rfl),
have a ^ p = 1, by rwa [ha, list.prod_repeat, hx₁] at hx1,
(hp.2 _ (order_of_dvd_of_pow_eq_one this)).resolve_left
(λ h, ha1 (order_of_eq_one_iff.1 h))⟩
open subgroup submonoid is_group_hom mul_action
lemma mem_fixed_points_mul_left_cosets_iff_mem_normalizer {H : subgroup G}
[fintype ((H : set G) : Type u)] {x : G} :
(x : quotient H) ∈ fixed_points H (quotient H) ↔ x ∈ normalizer H :=
⟨λ hx, have ha : ∀ {y : quotient H}, y ∈ orbit H (x : quotient H) → y = x,
from λ _, ((mem_fixed_points' _).1 hx _),
(inv_mem_iff _).1 (@mem_normalizer_fintype _ _ _ _inst_2 _ (λ n (hn : n ∈ H),
have (n⁻¹ * x)⁻¹ * x ∈ H := quotient_group.eq.1 (ha (mem_orbit _ ⟨n⁻¹, H.inv_mem hn⟩)),
show _ ∈ H, by {rw [mul_inv_rev, inv_inv] at this, convert this, rw inv_inv}
)),
λ (hx : ∀ (n : G), n ∈ H ↔ x * n * x⁻¹ ∈ H),
(mem_fixed_points' _).2 $ λ y, quotient.induction_on' y $ λ y hy, quotient_group.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy in
have hb₂ : (b * x)⁻¹ * y ∈ H := quotient_group.eq.1 hb₂,
(inv_mem_iff H).1 $ (hx _).2 $ (mul_mem_cancel_left H (H.inv_mem hb₁)).1
$ by rw hx at hb₂;
simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
def fixed_points_mul_left_cosets_equiv_quotient (H : subgroup G) [fintype (H : set G)] :
fixed_points H (quotient H) ≃
quotient (subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
@subtype_quotient_equiv_quotient_subtype G (normalizer H : set G) (id _) (id _) (fixed_points _ _)
(λ a, (@mem_fixed_points_mul_left_cosets_iff_mem_normalizer _ _ _ _inst_2 _).symm)
(by intros; refl)
/-- The first of the Sylow theorems. -/
theorem exists_subgroup_card_pow_prime [fintype G] (p : ℕ) : ∀ {n : ℕ} [hp : fact p.prime]
(hdvd : p ^ n ∣ card G), ∃ H : subgroup G, fintype.card H = p ^ n
| 0 := λ _ _, ⟨(⊥ : subgroup G), by convert card_trivial⟩
| (n+1) := λ hp hdvd,
let ⟨H, hH2⟩ := @exists_subgroup_card_pow_prime _ hp
(dvd.trans (pow_dvd_pow _ (nat.le_succ _)) hdvd) in
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd in
have hcard : card (quotient H) = s * p :=
(nat.mul_left_inj (show card H > 0, from fintype.card_pos_iff.2
⟨⟨1, H.one_mem⟩⟩)).1
(by rwa [← card_eq_card_quotient_mul_card_subgroup H, hH2, hs,
pow_succ', mul_assoc, mul_comm p]),
have hm : s * p % p =
card (quotient (subgroup.comap ((normalizer H).subtype : normalizer H →* G) H)) % p :=
card_congr (fixed_points_mul_left_cosets_equiv_quotient H) ▸ hcard ▸
@card_modeq_card_fixed_points _ _ _ _ _ _ _ p _ hp hH2,
have hm' : p ∣ card (quotient (subgroup.comap ((normalizer H).subtype : normalizer H →* G) H)) :=
nat.dvd_of_mod_eq_zero
(by rwa [nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm),
let ⟨x, hx⟩ := @exists_prime_order_of_dvd_card _ (quotient_group.quotient.group _) _ _ hp hm' in
have hequiv : H ≃ (subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
⟨λ a, ⟨⟨a.1, le_normalizer a.2⟩, a.2⟩, λ a, ⟨a.1.1, a.2⟩,
λ ⟨_, _⟩, rfl, λ ⟨⟨_, _⟩, _⟩, rfl⟩,
-- begin proof of ∃ H : subgroup G, fintype.card H = p ^ n
⟨subgroup.map ((normalizer H).subtype) (subgroup.comap
(quotient_group.mk' (comap H.normalizer.subtype H)) (gpowers x)),
begin
show card ↥(map H.normalizer.subtype
(comap (mk' (comap H.normalizer.subtype H)) (subgroup.gpowers x))) = p ^ (n + 1),
suffices : card ↥(subtype.val '' ((subgroup.comap (mk' (comap H.normalizer.subtype H))
(gpowers x)) : set (↥(H.normalizer)))) = p^(n+1),
{ convert this using 2 },
rw [set.card_image_of_injective
(subgroup.comap (mk' (comap H.normalizer.subtype H)) (gpowers x) : set (H.normalizer))
subtype.val_injective,
pow_succ', ← hH2, fintype.card_congr hequiv, ← hx, order_eq_card_gpowers,
← fintype.card_prod],
exact @fintype.card_congr _ _ (id _) (id _) (preimage_mk_equiv_subgroup_times_set _ _)
end⟩
end sylow
|
5f4e499a5803db03abac24d28b74707f2b94c7a2 | 63abd62053d479eae5abf4951554e1064a4c45b4 | /src/data/seq/computation.lean | 0b8a0754940ddff2b4d3889c8543ce97dfdfe27c | [
"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 | 37,543 | lean | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Mario Carneiro
Coinductive formalization of unbounded computations.
-/
import data.stream
import tactic.basic
universes u v w
/-
coinductive computation (α : Type u) : Type u
| return : α → computation α
| think : computation α → computation α
-/
/-- `computation α` is the type of unbounded computations returning `α`.
An element of `computation α` is an infinite sequence of `option α` such
that if `f n = some a` for some `n` then it is constantly `some a` after that. -/
def computation (α : Type u) : Type u :=
{ f : stream (option α) // ∀ {n a}, f n = some a → f (n+1) = some a }
namespace computation
variables {α : Type u} {β : Type v} {γ : Type w}
-- constructors
/-- `return a` is the computation that immediately terminates with result `a`. -/
def return (a : α) : computation α := ⟨stream.const (some a), λn a', id⟩
instance : has_coe_t α (computation α) := ⟨return⟩ -- note [use has_coe_t]
/-- `think c` is the computation that delays for one "tick" and then performs
computation `c`. -/
def think (c : computation α) : computation α :=
⟨none :: c.1, λn a h, by {cases n with n, contradiction, exact c.2 h}⟩
/-- `thinkN c n` is the computation that delays for `n` ticks and then performs
computation `c`. -/
def thinkN (c : computation α) : ℕ → computation α
| 0 := c
| (n+1) := think (thinkN n)
-- check for immediate result
/-- `head c` is the first step of computation, either `some a` if `c = return a`
or `none` if `c = think c'`. -/
def head (c : computation α) : option α := c.1.head
-- one step of computation
/-- `tail c` is the remainder of computation, either `c` if `c = return a`
or `c'` if `c = think c'`. -/
def tail (c : computation α) : computation α :=
⟨c.1.tail, λ n a, let t := c.2 in t⟩
/-- `empty α` is the computation that never returns, an infinite sequence of
`think`s. -/
def empty (α) : computation α := ⟨stream.const none, λn a', id⟩
instance : inhabited (computation α) := ⟨empty _⟩
/-- `run_for c n` evaluates `c` for `n` steps and returns the result, or `none`
if it did not terminate after `n` steps. -/
def run_for : computation α → ℕ → option α := subtype.val
/-- `destruct c` is the destructor for `computation α` as a coinductive type.
It returns `inl a` if `c = return a` and `inr c'` if `c = think c'`. -/
def destruct (c : computation α) : α ⊕ computation α :=
match c.1 0 with
| none := sum.inr (tail c)
| some a := sum.inl a
end
/-- `run c` is an unsound meta function that runs `c` to completion, possibly
resulting in an infinite loop in the VM. -/
meta def run : computation α → α | c :=
match destruct c with
| sum.inl a := a
| sum.inr ca := run ca
end
theorem destruct_eq_ret {s : computation α} {a : α} :
destruct s = sum.inl a → s = return a :=
begin
dsimp [destruct],
induction f0 : s.1 0; intro h,
{ contradiction },
{ apply subtype.eq, funext n,
induction n with n IH,
{ injection h with h', rwa h' at f0 },
{ exact s.2 IH } }
end
theorem destruct_eq_think {s : computation α} {s'} :
destruct s = sum.inr s' → s = think s' :=
begin
dsimp [destruct],
induction f0 : s.1 0 with a'; intro h,
{ injection h with h', rw ←h',
cases s with f al,
apply subtype.eq, dsimp [think, tail],
rw ←f0, exact (stream.eta f).symm },
{ contradiction }
end
@[simp] theorem destruct_ret (a : α) : destruct (return a) = sum.inl a := rfl
@[simp] theorem destruct_think : ∀ s : computation α, destruct (think s) = sum.inr s
| ⟨f, al⟩ := rfl
@[simp] theorem destruct_empty : destruct (empty α) = sum.inr (empty α) := rfl
@[simp] theorem head_ret (a : α) : head (return a) = some a := rfl
@[simp] theorem head_think (s : computation α) : head (think s) = none := rfl
@[simp] theorem head_empty : head (empty α) = none := rfl
@[simp] theorem tail_ret (a : α) : tail (return a) = return a := rfl
@[simp] theorem tail_think (s : computation α) : tail (think s) = s :=
by cases s with f al; apply subtype.eq; dsimp [tail, think]; rw [stream.tail_cons]
@[simp] theorem tail_empty : tail (empty α) = empty α := rfl
theorem think_empty : empty α = think (empty α) :=
destruct_eq_think destruct_empty
def cases_on {C : computation α → Sort v} (s : computation α)
(h1 : ∀ a, C (return a)) (h2 : ∀ s, C (think s)) : C s := begin
induction H : destruct s with v v,
{ rw destruct_eq_ret H, apply h1 },
{ cases v with a s', rw destruct_eq_think H, apply h2 }
end
def corec.F (f : β → α ⊕ β) : α ⊕ β → option α × (α ⊕ β)
| (sum.inl a) := (some a, sum.inl a)
| (sum.inr b) := (match f b with
| sum.inl a := some a
| sum.inr b' := none
end, f b)
/-- `corec f b` is the corecursor for `computation α` as a coinductive type.
If `f b = inl a` then `corec f b = return a`, and if `f b = inl b'` then
`corec f b = think (corec f b')`. -/
def corec (f : β → α ⊕ β) (b : β) : computation α :=
begin
refine ⟨stream.corec' (corec.F f) (sum.inr b), λn a' h, _⟩,
rw stream.corec'_eq,
change stream.corec' (corec.F f) (corec.F f (sum.inr b)).2 n = some a',
revert h, generalize : sum.inr b = o, revert o,
induction n with n IH; intro o,
{ change (corec.F f o).1 = some a' → (corec.F f (corec.F f o).2).1 = some a',
cases o with a b; intro h, { exact h },
dsimp [corec.F] at h, dsimp [corec.F],
cases f b with a b', { exact h },
{ contradiction } },
{ rw [stream.corec'_eq (corec.F f) (corec.F f o).2,
stream.corec'_eq (corec.F f) o],
exact IH (corec.F f o).2 }
end
/-- left map of `⊕` -/
def lmap (f : α → β) : α ⊕ γ → β ⊕ γ
| (sum.inl a) := sum.inl (f a)
| (sum.inr b) := sum.inr b
/-- right map of `⊕` -/
def rmap (f : β → γ) : α ⊕ β → α ⊕ γ
| (sum.inl a) := sum.inl a
| (sum.inr b) := sum.inr (f b)
attribute [simp] lmap rmap
@[simp] lemma corec_eq (f : β → α ⊕ β) (b : β) :
destruct (corec f b) = rmap (corec f) (f b) :=
begin
dsimp [corec, destruct],
change stream.corec' (corec.F f) (sum.inr b) 0 with corec.F._match_1 (f b),
induction h : f b with a b', { refl },
dsimp [corec.F, destruct],
apply congr_arg, apply subtype.eq,
dsimp [corec, tail],
rw [stream.corec'_eq, stream.tail_cons],
dsimp [corec.F], rw h
end
section bisim
variable (R : computation α → computation α → Prop)
local infix ~ := R
def bisim_o : α ⊕ computation α → α ⊕ computation α → Prop
| (sum.inl a) (sum.inl a') := a = a'
| (sum.inr s) (sum.inr s') := R s s'
| _ _ := false
attribute [simp] bisim_o
def is_bisimulation := ∀ ⦃s₁ s₂⦄, s₁ ~ s₂ → bisim_o R (destruct s₁) (destruct s₂)
-- If two computations are bisimilar, then they are equal
theorem eq_of_bisim (bisim : is_bisimulation R) {s₁ s₂} (r : s₁ ~ s₂) : s₁ = s₂ :=
begin
apply subtype.eq,
apply stream.eq_of_bisim (λx y, ∃ s s' : computation α, s.1 = x ∧ s'.1 = y ∧ R s s'),
dsimp [stream.is_bisimulation],
intros t₁ t₂ e,
exact match t₁, t₂, e with ._, ._, ⟨s, s', rfl, rfl, r⟩ :=
suffices head s = head s' ∧ R (tail s) (tail s'), from
and.imp id (λr, ⟨tail s, tail s',
by cases s; refl, by cases s'; refl, r⟩) this,
begin
have := bisim r, revert r this,
apply cases_on s _ _; intros; apply cases_on s' _ _; intros; intros r this,
{ constructor, dsimp at this, rw this, assumption },
{ rw [destruct_ret, destruct_think] at this,
exact false.elim this },
{ rw [destruct_ret, destruct_think] at this,
exact false.elim this },
{ simp at this, simp [*] }
end
end,
exact ⟨s₁, s₂, rfl, rfl, r⟩
end
end bisim
-- It's more of a stretch to use ∈ for this relation, but it
-- asserts that the computation limits to the given value.
protected def mem (a : α) (s : computation α) := some a ∈ s.1
instance : has_mem α (computation α) := ⟨computation.mem⟩
theorem le_stable (s : computation α) {a m n} (h : m ≤ n) :
s.1 m = some a → s.1 n = some a :=
by {cases s with f al, induction h with n h IH, exacts [id, λ h2, al (IH h2)]}
theorem mem_unique :
relator.left_unique ((∈) : α → computation α → Prop) :=
λa s b ⟨m, ha⟩ ⟨n, hb⟩, by injection
(le_stable s (le_max_left m n) ha.symm).symm.trans
(le_stable s (le_max_right m n) hb.symm)
/-- `terminates s` asserts that the computation `s` eventually terminates with some value. -/
@[class] def terminates (s : computation α) : Prop := ∃ a, a ∈ s
theorem terminates_of_mem {s : computation α} {a : α} : a ∈ s → terminates s :=
exists.intro a
theorem terminates_def (s : computation α) : terminates s ↔ ∃ n, (s.1 n).is_some :=
⟨λ⟨a, n, h⟩, ⟨n, by {dsimp [stream.nth] at h, rw ←h, exact rfl}⟩,
λ⟨n, h⟩, ⟨option.get h, n, (option.eq_some_of_is_some h).symm⟩⟩
theorem ret_mem (a : α) : a ∈ return a :=
exists.intro 0 rfl
theorem eq_of_ret_mem {a a' : α} (h : a' ∈ return a) : a' = a :=
mem_unique h (ret_mem _)
instance ret_terminates (a : α) : terminates (return a) :=
terminates_of_mem (ret_mem _)
theorem think_mem {s : computation α} {a} : a ∈ s → a ∈ think s
| ⟨n, h⟩ := ⟨n+1, h⟩
instance think_terminates (s : computation α) :
∀ [terminates s], terminates (think s)
| ⟨a, n, h⟩ := ⟨a, n+1, h⟩
theorem of_think_mem {s : computation α} {a} : a ∈ think s → a ∈ s
| ⟨n, h⟩ := by {cases n with n', contradiction, exact ⟨n', h⟩}
theorem of_think_terminates {s : computation α} :
terminates (think s) → terminates s
| ⟨a, h⟩ := ⟨a, of_think_mem h⟩
theorem not_mem_empty (a : α) : a ∉ empty α :=
λ ⟨n, h⟩, by clear _fun_match; contradiction
theorem not_terminates_empty : ¬ terminates (empty α) :=
λ ⟨a, h⟩, not_mem_empty a h
theorem eq_empty_of_not_terminates {s} (H : ¬ terminates s) : s = empty α :=
begin
apply subtype.eq, funext n,
induction h : s.val n, {refl},
refine absurd _ H, exact ⟨_, _, h.symm⟩
end
theorem thinkN_mem {s : computation α} {a} : ∀ n, a ∈ thinkN s n ↔ a ∈ s
| 0 := iff.rfl
| (n+1) := iff.trans ⟨of_think_mem, think_mem⟩ (thinkN_mem n)
instance thinkN_terminates (s : computation α) :
∀ [terminates s] n, terminates (thinkN s n)
| ⟨a, h⟩ n := ⟨a, (thinkN_mem n).2 h⟩
theorem of_thinkN_terminates (s : computation α) (n) :
terminates (thinkN s n) → terminates s
| ⟨a, h⟩ := ⟨a, (thinkN_mem _).1 h⟩
/-- `promises s a`, or `s ~> a`, asserts that although the computation `s`
may not terminate, if it does, then the result is `a`. -/
def promises (s : computation α) (a : α) : Prop := ∀ ⦃a'⦄, a' ∈ s → a = a'
infix ` ~> `:50 := promises
theorem mem_promises {s : computation α} {a : α} : a ∈ s → s ~> a :=
λ h a', mem_unique h
theorem empty_promises (a : α) : empty α ~> a :=
λ a' h, absurd h (not_mem_empty _)
section get
variables (s : computation α) [h : terminates s]
include s h
/-- `length s` gets the number of steps of a terminating computation -/
def length : ℕ := nat.find ((terminates_def _).1 h)
/-- `get s` returns the result of a terminating computation -/
def get : α := option.get (nat.find_spec $ (terminates_def _).1 h)
theorem get_mem : get s ∈ s :=
exists.intro (length s) (option.eq_some_of_is_some _).symm
theorem get_eq_of_mem {a} : a ∈ s → get s = a :=
mem_unique (get_mem _)
theorem mem_of_get_eq {a} : get s = a → a ∈ s :=
by intro h; rw ←h; apply get_mem
@[simp] theorem get_think : get (think s) = get s :=
get_eq_of_mem _ $ let ⟨n, h⟩ := get_mem s in ⟨n+1, h⟩
@[simp] theorem get_thinkN (n) : get (thinkN s n) = get s :=
get_eq_of_mem _ $ (thinkN_mem _).2 (get_mem _)
theorem get_promises : s ~> get s := λ a, get_eq_of_mem _
theorem mem_of_promises {a} (p : s ~> a) : a ∈ s :=
by { casesI h with a' h, rw p h, exact h }
theorem get_eq_of_promises {a} : s ~> a → get s = a :=
get_eq_of_mem _ ∘ mem_of_promises _
end get
/-- `results s a n` completely characterizes a terminating computation:
it asserts that `s` terminates after exactly `n` steps, with result `a`. -/
def results (s : computation α) (a : α) (n : ℕ) :=
∃ (h : a ∈ s), @length _ s (terminates_of_mem h) = n
theorem results_of_terminates (s : computation α) [T : terminates s] :
results s (get s) (length s) :=
⟨get_mem _, rfl⟩
theorem results_of_terminates' (s : computation α) [T : terminates s] {a} (h : a ∈ s) :
results s a (length s) :=
by rw ←get_eq_of_mem _ h; apply results_of_terminates
theorem results.mem {s : computation α} {a n} : results s a n → a ∈ s
| ⟨m, _⟩ := m
theorem results.terminates {s : computation α} {a n} (h : results s a n) : terminates s :=
terminates_of_mem h.mem
theorem results.length {s : computation α} {a n} [T : terminates s] :
results s a n → length s = n
| ⟨_, h⟩ := h
theorem results.val_unique {s : computation α} {a b m n}
(h1 : results s a m) (h2 : results s b n) : a = b :=
mem_unique h1.mem h2.mem
theorem results.len_unique {s : computation α} {a b m n}
(h1 : results s a m) (h2 : results s b n) : m = n :=
by haveI := h1.terminates; haveI := h2.terminates; rw [←h1.length, h2.length]
theorem exists_results_of_mem {s : computation α} {a} (h : a ∈ s) : ∃ n, results s a n :=
by haveI := terminates_of_mem h; exact ⟨_, results_of_terminates' s h⟩
@[simp] theorem get_ret (a : α) : get (return a) = a :=
get_eq_of_mem _ ⟨0, rfl⟩
@[simp] theorem length_ret (a : α) : length (return a) = 0 :=
let h := computation.ret_terminates a in
nat.eq_zero_of_le_zero $ nat.find_min' ((terminates_def (return a)).1 h) rfl
theorem results_ret (a : α) : results (return a) a 0 :=
⟨_, length_ret _⟩
@[simp] theorem length_think (s : computation α) [h : terminates s] :
length (think s) = length s + 1 :=
begin
apply le_antisymm,
{ exact nat.find_min' _ (nat.find_spec ((terminates_def _).1 h)) },
{ have : (option.is_some ((think s).val (length (think s))) : Prop) :=
nat.find_spec ((terminates_def _).1 s.think_terminates),
cases length (think s) with n,
{ contradiction },
{ apply nat.succ_le_succ, apply nat.find_min', apply this } }
end
theorem results_think {s : computation α} {a n}
(h : results s a n) : results (think s) a (n + 1) :=
by haveI := h.terminates; exact ⟨think_mem h.mem, by rw [length_think, h.length]⟩
theorem of_results_think {s : computation α} {a n}
(h : results (think s) a n) : ∃ m, results s a m ∧ n = m + 1 :=
begin
haveI := of_think_terminates h.terminates,
have := results_of_terminates' _ (of_think_mem h.mem),
exact ⟨_, this, results.len_unique h (results_think this)⟩,
end
@[simp] theorem results_think_iff {s : computation α} {a n} :
results (think s) a (n + 1) ↔ results s a n :=
⟨λ h, let ⟨n', r, e⟩ := of_results_think h in by injection e with h'; rwa h',
results_think⟩
theorem results_thinkN {s : computation α} {a m} :
∀ n, results s a m → results (thinkN s n) a (m + n)
| 0 h := h
| (n+1) h := results_think (results_thinkN n h)
theorem results_thinkN_ret (a : α) (n) : results (thinkN (return a) n) a n :=
by have := results_thinkN n (results_ret a); rwa nat.zero_add at this
@[simp] theorem length_thinkN (s : computation α) [h : terminates s] (n) :
length (thinkN s n) = length s + n :=
(results_thinkN n (results_of_terminates _)).length
theorem eq_thinkN {s : computation α} {a n} (h : results s a n) :
s = thinkN (return a) n :=
begin
revert s,
induction n with n IH; intro s;
apply cases_on s (λ a', _) (λ s, _); intro h,
{ rw ←eq_of_ret_mem h.mem, refl },
{ cases of_results_think h with n h, cases h, contradiction },
{ have := h.len_unique (results_ret _), contradiction },
{ rw IH (results_think_iff.1 h), refl }
end
theorem eq_thinkN' (s : computation α) [h : terminates s] :
s = thinkN (return (get s)) (length s) :=
eq_thinkN (results_of_terminates _)
def mem_rec_on {C : computation α → Sort v} {a s} (M : a ∈ s)
(h1 : C (return a)) (h2 : ∀ s, C s → C (think s)) : C s :=
begin
haveI T := terminates_of_mem M,
rw [eq_thinkN' s, get_eq_of_mem s M],
generalize : length s = n,
induction n with n IH, exacts [h1, h2 _ IH]
end
def terminates_rec_on {C : computation α → Sort v} (s) [terminates s]
(h1 : ∀ a, C (return a)) (h2 : ∀ s, C s → C (think s)) : C s :=
mem_rec_on (get_mem s) (h1 _) h2
/-- Map a function on the result of a computation. -/
def map (f : α → β) : computation α → computation β
| ⟨s, al⟩ := ⟨s.map (λo, option.cases_on o none (some ∘ f)),
λn b, begin
dsimp [stream.map, stream.nth],
induction e : s n with a; intro h,
{ contradiction }, { rw [al e, ←h] }
end⟩
def bind.G : β ⊕ computation β → β ⊕ computation α ⊕ computation β
| (sum.inl b) := sum.inl b
| (sum.inr cb') := sum.inr $ sum.inr cb'
def bind.F (f : α → computation β) :
computation α ⊕ computation β → β ⊕ computation α ⊕ computation β
| (sum.inl ca) :=
match destruct ca with
| sum.inl a := bind.G $ destruct (f a)
| sum.inr ca' := sum.inr $ sum.inl ca'
end
| (sum.inr cb) := bind.G $ destruct cb
/-- Compose two computations into a monadic `bind` operation. -/
def bind (c : computation α) (f : α → computation β) : computation β :=
corec (bind.F f) (sum.inl c)
instance : has_bind computation := ⟨@bind⟩
theorem has_bind_eq_bind {β} (c : computation α) (f : α → computation β) :
c >>= f = bind c f := rfl
/-- Flatten a computation of computations into a single computation. -/
def join (c : computation (computation α)) : computation α := c >>= id
@[simp] theorem map_ret (f : α → β) (a) : map f (return a) = return (f a) := rfl
@[simp] theorem map_think (f : α → β) : ∀ s, map f (think s) = think (map f s)
| ⟨s, al⟩ := by apply subtype.eq; dsimp [think, map]; rw stream.map_cons
@[simp]
theorem destruct_map (f : α → β) (s) : destruct (map f s) = lmap f (rmap (map f) (destruct s)) :=
by apply s.cases_on; intro; simp
@[simp] theorem map_id : ∀ (s : computation α), map id s = s
| ⟨f, al⟩ := begin
apply subtype.eq; simp [map, function.comp],
have e : (@option.rec α (λ_, option α) none some) = id,
{ ext ⟨⟩; refl },
simp [e, stream.map_id]
end
theorem map_comp (f : α → β) (g : β → γ) :
∀ (s : computation α), map (g ∘ f) s = map g (map f s)
| ⟨s, al⟩ := begin
apply subtype.eq; dsimp [map],
rw stream.map_map,
apply congr_arg (λ f : _ → option γ, stream.map f s),
ext ⟨⟩; refl
end
@[simp] theorem ret_bind (a) (f : α → computation β) :
bind (return a) f = f a :=
begin
apply eq_of_bisim (λc₁ c₂,
c₁ = bind (return a) f ∧ c₂ = f a ∨
c₁ = corec (bind.F f) (sum.inr c₂)),
{ intros c₁ c₂ h,
exact match c₁, c₂, h with
| ._, ._, or.inl ⟨rfl, rfl⟩ := begin
simp [bind, bind.F],
cases destruct (f a) with b cb; simp [bind.G]
end
| ._, c, or.inr rfl := begin
simp [bind.F],
cases destruct c with b cb; simp [bind.G]
end end },
{ simp }
end
@[simp] theorem think_bind (c) (f : α → computation β) :
bind (think c) f = think (bind c f) :=
destruct_eq_think $ by simp [bind, bind.F]
@[simp] theorem bind_ret (f : α → β) (s) : bind s (return ∘ f) = map f s :=
begin
apply eq_of_bisim (λc₁ c₂, c₁ = c₂ ∨
∃ s, c₁ = bind s (return ∘ f) ∧ c₂ = map f s),
{ intros c₁ c₂ h,
exact match c₁, c₂, h with
| _, _, or.inl (eq.refl c) := begin cases destruct c with b cb; simp end
| _, _, or.inr ⟨s, rfl, rfl⟩ := begin
apply cases_on s; intros s; simp,
exact or.inr ⟨s, rfl, rfl⟩
end end },
{ exact or.inr ⟨s, rfl, rfl⟩ }
end
@[simp] theorem bind_ret' (s : computation α) : bind s return = s :=
by rw bind_ret; change (λ x : α, x) with @id α; rw map_id
@[simp] theorem bind_assoc (s : computation α) (f : α → computation β) (g : β → computation γ) :
bind (bind s f) g = bind s (λ (x : α), bind (f x) g) :=
begin
apply eq_of_bisim (λc₁ c₂, c₁ = c₂ ∨
∃ s, c₁ = bind (bind s f) g ∧ c₂ = bind s (λ (x : α), bind (f x) g)),
{ intros c₁ c₂ h,
exact match c₁, c₂, h with
| _, _, or.inl (eq.refl c) := by cases destruct c with b cb; simp
| ._, ._, or.inr ⟨s, rfl, rfl⟩ := begin
apply cases_on s; intros s; simp,
{ generalize : f s = fs,
apply cases_on fs; intros t; simp,
{ cases destruct (g t) with b cb; simp } },
{ exact or.inr ⟨s, rfl, rfl⟩ }
end end },
{ exact or.inr ⟨s, rfl, rfl⟩ }
end
theorem results_bind {s : computation α} {f : α → computation β} {a b m n}
(h1 : results s a m) (h2 : results (f a) b n) : results (bind s f) b (n + m) :=
begin
have := h1.mem, revert m,
apply mem_rec_on this _ (λ s IH, _); intros m h1,
{ rw [ret_bind], rw h1.len_unique (results_ret _), exact h2 },
{ rw [think_bind], cases of_results_think h1 with m' h, cases h with h1 e,
rw e, exact results_think (IH h1) }
end
theorem mem_bind {s : computation α} {f : α → computation β} {a b}
(h1 : a ∈ s) (h2 : b ∈ f a) : b ∈ bind s f :=
let ⟨m, h1⟩ := exists_results_of_mem h1,
⟨n, h2⟩ := exists_results_of_mem h2 in (results_bind h1 h2).mem
instance terminates_bind (s : computation α) (f : α → computation β)
[terminates s] [terminates (f (get s))] :
terminates (bind s f) :=
terminates_of_mem (mem_bind (get_mem s) (get_mem (f (get s))))
@[simp] theorem get_bind (s : computation α) (f : α → computation β)
[terminates s] [terminates (f (get s))] :
get (bind s f) = get (f (get s)) :=
get_eq_of_mem _ (mem_bind (get_mem s) (get_mem (f (get s))))
@[simp] theorem length_bind (s : computation α) (f : α → computation β)
[T1 : terminates s] [T2 : terminates (f (get s))] :
length (bind s f) = length (f (get s)) + length s :=
(results_of_terminates _).len_unique $
results_bind (results_of_terminates _) (results_of_terminates _)
theorem of_results_bind {s : computation α} {f : α → computation β} {b k} :
results (bind s f) b k →
∃ a m n, results s a m ∧ results (f a) b n ∧ k = n + m :=
begin
induction k with n IH generalizing s;
apply cases_on s (λ a, _) (λ s', _); intro e,
{ simp [thinkN] at e, refine ⟨a, _, _, results_ret _, e, rfl⟩ },
{ have := congr_arg head (eq_thinkN e), contradiction },
{ simp at e, refine ⟨a, _, n+1, results_ret _, e, rfl⟩ },
{ simp at e, exact let ⟨a, m, n', h1, h2, e'⟩ := IH e in
by rw e'; exact ⟨a, m.succ, n', results_think h1, h2, rfl⟩ }
end
theorem exists_of_mem_bind {s : computation α} {f : α → computation β} {b}
(h : b ∈ bind s f) : ∃ a ∈ s, b ∈ f a :=
let ⟨k, h⟩ := exists_results_of_mem h,
⟨a, m, n, h1, h2, e⟩ := of_results_bind h in ⟨a, h1.mem, h2.mem⟩
theorem bind_promises {s : computation α} {f : α → computation β} {a b}
(h1 : s ~> a) (h2 : f a ~> b) : bind s f ~> b :=
λ b' bB, begin
rcases exists_of_mem_bind bB with ⟨a', a's, ba'⟩,
rw ←h1 a's at ba', exact h2 ba'
end
instance : monad computation :=
{ map := @map,
pure := @return,
bind := @bind }
instance : is_lawful_monad computation :=
{ id_map := @map_id,
bind_pure_comp_eq_map := @bind_ret,
pure_bind := @ret_bind,
bind_assoc := @bind_assoc }
theorem has_map_eq_map {β} (f : α → β) (c : computation α) : f <$> c = map f c := rfl
@[simp] theorem return_def (a) : (_root_.return a : computation α) = return a := rfl
@[simp] theorem map_ret' {α β} : ∀ (f : α → β) (a), f <$> return a = return (f a) := map_ret
@[simp] theorem map_think' {α β} : ∀ (f : α → β) s, f <$> think s = think (f <$> s) := map_think
theorem mem_map (f : α → β) {a} {s : computation α} (m : a ∈ s) : f a ∈ map f s :=
by rw ←bind_ret; apply mem_bind m; apply ret_mem
theorem exists_of_mem_map {f : α → β} {b : β} {s : computation α} (h : b ∈ map f s) :
∃ a, a ∈ s ∧ f a = b :=
by rw ←bind_ret at h; exact
let ⟨a, as, fb⟩ := exists_of_mem_bind h in ⟨a, as, mem_unique (ret_mem _) fb⟩
instance terminates_map (f : α → β) (s : computation α) [terminates s] : terminates (map f s) :=
by rw ←bind_ret; apply_instance
theorem terminates_map_iff (f : α → β) (s : computation α) :
terminates (map f s) ↔ terminates s :=
⟨λ⟨a, h⟩, let ⟨b, h1, _⟩ := exists_of_mem_map h in ⟨_, h1⟩, @computation.terminates_map _ _ _ _⟩
-- Parallel computation
/-- `c₁ <|> c₂` calculates `c₁` and `c₂` simultaneously, returning
the first one that gives a result. -/
def orelse (c₁ c₂ : computation α) : computation α :=
@computation.corec α (computation α × computation α)
(λ⟨c₁, c₂⟩, match destruct c₁ with
| sum.inl a := sum.inl a
| sum.inr c₁' := match destruct c₂ with
| sum.inl a := sum.inl a
| sum.inr c₂' := sum.inr (c₁', c₂')
end
end) (c₁, c₂)
instance : alternative computation :=
{ orelse := @orelse, failure := @empty, ..computation.monad }
@[simp] theorem ret_orelse (a : α) (c₂ : computation α) :
(return a <|> c₂) = return a :=
destruct_eq_ret $ by unfold has_orelse.orelse; simp [orelse]
@[simp] theorem orelse_ret (c₁ : computation α) (a : α) :
(think c₁ <|> return a) = return a :=
destruct_eq_ret $ by unfold has_orelse.orelse; simp [orelse]
@[simp] theorem orelse_think (c₁ c₂ : computation α) :
(think c₁ <|> think c₂) = think (c₁ <|> c₂) :=
destruct_eq_think $ by unfold has_orelse.orelse; simp [orelse]
@[simp] theorem empty_orelse (c) : (empty α <|> c) = c :=
begin
apply eq_of_bisim (λc₁ c₂, (empty α <|> c₂) = c₁) _ rfl,
intros s' s h, rw ←h,
apply cases_on s; intros s; rw think_empty; simp,
rw ←think_empty,
end
@[simp] theorem orelse_empty (c : computation α) : (c <|> empty α) = c :=
begin
apply eq_of_bisim (λc₁ c₂, (c₂ <|> empty α) = c₁) _ rfl,
intros s' s h, rw ←h,
apply cases_on s; intros s; rw think_empty; simp,
rw←think_empty,
end
/-- `c₁ ~ c₂` asserts that `c₁` and `c₂` either both terminate with the same result,
or both loop forever. -/
def equiv (c₁ c₂ : computation α) : Prop := ∀ a, a ∈ c₁ ↔ a ∈ c₂
infix ~ := equiv
@[refl] theorem equiv.refl (s : computation α) : s ~ s := λ_, iff.rfl
@[symm] theorem equiv.symm {s t : computation α} : s ~ t → t ~ s :=
λh a, (h a).symm
@[trans] theorem equiv.trans {s t u : computation α} : s ~ t → t ~ u → s ~ u :=
λh1 h2 a, (h1 a).trans (h2 a)
theorem equiv.equivalence : equivalence (@equiv α) :=
⟨@equiv.refl _, @equiv.symm _, @equiv.trans _⟩
theorem equiv_of_mem {s t : computation α} {a} (h1 : a ∈ s) (h2 : a ∈ t) : s ~ t :=
λa', ⟨λma, by rw mem_unique ma h1; exact h2,
λma, by rw mem_unique ma h2; exact h1⟩
theorem terminates_congr {c₁ c₂ : computation α}
(h : c₁ ~ c₂) : terminates c₁ ↔ terminates c₂ :=
exists_congr h
theorem promises_congr {c₁ c₂ : computation α}
(h : c₁ ~ c₂) (a) : c₁ ~> a ↔ c₂ ~> a :=
forall_congr (λa', imp_congr (h a') iff.rfl)
theorem get_equiv {c₁ c₂ : computation α} (h : c₁ ~ c₂)
[terminates c₁] [terminates c₂] : get c₁ = get c₂ :=
get_eq_of_mem _ $ (h _).2 $ get_mem _
theorem think_equiv (s : computation α) : think s ~ s :=
λ a, ⟨of_think_mem, think_mem⟩
theorem thinkN_equiv (s : computation α) (n) : thinkN s n ~ s :=
λ a, thinkN_mem n
theorem bind_congr {s1 s2 : computation α} {f1 f2 : α → computation β}
(h1 : s1 ~ s2) (h2 : ∀ a, f1 a ~ f2 a) : bind s1 f1 ~ bind s2 f2 :=
λ b, ⟨λh, let ⟨a, ha, hb⟩ := exists_of_mem_bind h in
mem_bind ((h1 a).1 ha) ((h2 a b).1 hb),
λh, let ⟨a, ha, hb⟩ := exists_of_mem_bind h in
mem_bind ((h1 a).2 ha) ((h2 a b).2 hb)⟩
theorem equiv_ret_of_mem {s : computation α} {a} (h : a ∈ s) : s ~ return a :=
equiv_of_mem h (ret_mem _)
/-- `lift_rel R ca cb` is a generalization of `equiv` to relations other than
equality. It asserts that if `ca` terminates with `a`, then `cb` terminates with
some `b` such that `R a b`, and if `cb` terminates with `b` then `ca` terminates
with some `a` such that `R a b`. -/
def lift_rel (R : α → β → Prop) (ca : computation α) (cb : computation β) : Prop :=
(∀ {a}, a ∈ ca → ∃ {b}, b ∈ cb ∧ R a b) ∧
∀ {b}, b ∈ cb → ∃ {a}, a ∈ ca ∧ R a b
theorem lift_rel.swap (R : α → β → Prop) (ca : computation α) (cb : computation β) :
lift_rel (function.swap R) cb ca ↔ lift_rel R ca cb :=
and_comm _ _
theorem lift_eq_iff_equiv (c₁ c₂ : computation α) : lift_rel (=) c₁ c₂ ↔ c₁ ~ c₂ :=
⟨λ⟨h1, h2⟩ a,
⟨λ a1, let ⟨b, b2, ab⟩ := h1 a1 in by rwa ab,
λ a2, let ⟨b, b1, ab⟩ := h2 a2 in by rwa ←ab⟩,
λe, ⟨λ a a1, ⟨a, (e _).1 a1, rfl⟩, λ a a2, ⟨a, (e _).2 a2, rfl⟩⟩⟩
theorem lift_rel.refl (R : α → α → Prop) (H : reflexive R) : reflexive (lift_rel R) :=
λ s, ⟨λ a as, ⟨a, as, H a⟩, λ b bs, ⟨b, bs, H b⟩⟩
theorem lift_rel.symm (R : α → α → Prop) (H : symmetric R) : symmetric (lift_rel R) :=
λ s1 s2 ⟨l, r⟩,
⟨λ a a2, let ⟨b, b1, ab⟩ := r a2 in ⟨b, b1, H ab⟩,
λ a a1, let ⟨b, b2, ab⟩ := l a1 in ⟨b, b2, H ab⟩⟩
theorem lift_rel.trans (R : α → α → Prop) (H : transitive R) : transitive (lift_rel R) :=
λ s1 s2 s3 ⟨l1, r1⟩ ⟨l2, r2⟩,
⟨λ a a1, let ⟨b, b2, ab⟩ := l1 a1, ⟨c, c3, bc⟩ := l2 b2 in ⟨c, c3, H ab bc⟩,
λ c c3, let ⟨b, b2, bc⟩ := r2 c3, ⟨a, a1, ab⟩ := r1 b2 in ⟨a, a1, H ab bc⟩⟩
theorem lift_rel.equiv (R : α → α → Prop) : equivalence R → equivalence (lift_rel R)
| ⟨refl, symm, trans⟩ :=
⟨lift_rel.refl R refl, lift_rel.symm R symm, lift_rel.trans R trans⟩
theorem lift_rel.imp {R S : α → β → Prop} (H : ∀ {a b}, R a b → S a b) (s t) :
lift_rel R s t → lift_rel S s t | ⟨l, r⟩ :=
⟨λ a as, let ⟨b, bt, ab⟩ := l as in ⟨b, bt, H ab⟩,
λ b bt, let ⟨a, as, ab⟩ := r bt in ⟨a, as, H ab⟩⟩
theorem terminates_of_lift_rel {R : α → β → Prop} {s t} :
lift_rel R s t → (terminates s ↔ terminates t) | ⟨l, r⟩ :=
⟨λ ⟨a, as⟩, let ⟨b, bt, ab⟩ := l as in ⟨b, bt⟩,
λ ⟨b, bt⟩, let ⟨a, as, ab⟩ := r bt in ⟨a, as⟩⟩
theorem rel_of_lift_rel {R : α → β → Prop} {ca cb} :
lift_rel R ca cb → ∀ {a b}, a ∈ ca → b ∈ cb → R a b
| ⟨l, r⟩ a b ma mb :=
let ⟨b', mb', ab'⟩ := l ma in by rw mem_unique mb mb'; exact ab'
theorem lift_rel_of_mem {R : α → β → Prop} {a b ca cb}
(ma : a ∈ ca) (mb : b ∈ cb) (ab : R a b) : lift_rel R ca cb :=
⟨λ a' ma', by rw mem_unique ma' ma; exact ⟨b, mb, ab⟩,
λ b' mb', by rw mem_unique mb' mb; exact ⟨a, ma, ab⟩⟩
theorem exists_of_lift_rel_left {R : α → β → Prop} {ca cb}
(H : lift_rel R ca cb) {a} (h : a ∈ ca) : ∃ {b}, b ∈ cb ∧ R a b :=
H.left h
theorem exists_of_lift_rel_right {R : α → β → Prop} {ca cb}
(H : lift_rel R ca cb) {b} (h : b ∈ cb) : ∃ {a}, a ∈ ca ∧ R a b :=
H.right h
theorem lift_rel_def {R : α → β → Prop} {ca cb} : lift_rel R ca cb ↔
(terminates ca ↔ terminates cb) ∧ ∀ {a b}, a ∈ ca → b ∈ cb → R a b :=
⟨λh, ⟨terminates_of_lift_rel h, λ a b ma mb,
let ⟨b', mb', ab⟩ := h.left ma in by rwa mem_unique mb mb'⟩,
λ⟨l, r⟩,
⟨λ a ma, let ⟨b, mb⟩ := l.1 ⟨_, ma⟩ in ⟨b, mb, r ma mb⟩,
λ b mb, let ⟨a, ma⟩ := l.2 ⟨_, mb⟩ in ⟨a, ma, r ma mb⟩⟩⟩
theorem lift_rel_bind {δ} (R : α → β → Prop) (S : γ → δ → Prop)
{s1 : computation α} {s2 : computation β}
{f1 : α → computation γ} {f2 : β → computation δ}
(h1 : lift_rel R s1 s2) (h2 : ∀ {a b}, R a b → lift_rel S (f1 a) (f2 b))
: lift_rel S (bind s1 f1) (bind s2 f2) :=
let ⟨l1, r1⟩ := h1 in
⟨λ c cB,
let ⟨a, a1, c₁⟩ := exists_of_mem_bind cB,
⟨b, b2, ab⟩ := l1 a1,
⟨l2, r2⟩ := h2 ab,
⟨d, d2, cd⟩ := l2 c₁ in
⟨_, mem_bind b2 d2, cd⟩,
λ d dB,
let ⟨b, b1, d1⟩ := exists_of_mem_bind dB,
⟨a, a2, ab⟩ := r1 b1,
⟨l2, r2⟩ := h2 ab,
⟨c, c₂, cd⟩ := r2 d1 in
⟨_, mem_bind a2 c₂, cd⟩⟩
@[simp] theorem lift_rel_return_left (R : α → β → Prop) (a : α) (cb : computation β) :
lift_rel R (return a) cb ↔ ∃ {b}, b ∈ cb ∧ R a b :=
⟨λ⟨l, r⟩, l (ret_mem _),
λ⟨b, mb, ab⟩,
⟨λ a' ma', by rw eq_of_ret_mem ma'; exact ⟨b, mb, ab⟩,
λ b' mb', ⟨_, ret_mem _, by rw mem_unique mb' mb; exact ab⟩⟩⟩
@[simp] theorem lift_rel_return_right (R : α → β → Prop) (ca : computation α) (b : β) :
lift_rel R ca (return b) ↔ ∃ {a}, a ∈ ca ∧ R a b :=
by rw [lift_rel.swap, lift_rel_return_left]
@[simp] theorem lift_rel_return (R : α → β → Prop) (a : α) (b : β) :
lift_rel R (return a) (return b) ↔ R a b :=
by rw [lift_rel_return_left]; exact
⟨λ⟨b', mb', ab'⟩, by rwa eq_of_ret_mem mb' at ab',
λab, ⟨_, ret_mem _, ab⟩⟩
@[simp] theorem lift_rel_think_left (R : α → β → Prop) (ca : computation α) (cb : computation β) :
lift_rel R (think ca) cb ↔ lift_rel R ca cb :=
and_congr (forall_congr $ λb, imp_congr ⟨of_think_mem, think_mem⟩ iff.rfl)
(forall_congr $ λb, imp_congr iff.rfl $
exists_congr $ λ b, and_congr ⟨of_think_mem, think_mem⟩ iff.rfl)
@[simp] theorem lift_rel_think_right (R : α → β → Prop) (ca : computation α) (cb : computation β) :
lift_rel R ca (think cb) ↔ lift_rel R ca cb :=
by rw [←lift_rel.swap R, ←lift_rel.swap R]; apply lift_rel_think_left
theorem lift_rel_mem_cases {R : α → β → Prop} {ca cb}
(Ha : ∀ a ∈ ca, lift_rel R ca cb)
(Hb : ∀ b ∈ cb, lift_rel R ca cb) : lift_rel R ca cb :=
⟨λ a ma, (Ha _ ma).left ma, λ b mb, (Hb _ mb).right mb⟩
theorem lift_rel_congr {R : α → β → Prop} {ca ca' : computation α} {cb cb' : computation β}
(ha : ca ~ ca') (hb : cb ~ cb') : lift_rel R ca cb ↔ lift_rel R ca' cb' :=
and_congr
(forall_congr $ λ a, imp_congr (ha _) $ exists_congr $ λ b, and_congr (hb _) iff.rfl)
(forall_congr $ λ b, imp_congr (hb _) $ exists_congr $ λ a, and_congr (ha _) iff.rfl)
theorem lift_rel_map {δ} (R : α → β → Prop) (S : γ → δ → Prop)
{s1 : computation α} {s2 : computation β}
{f1 : α → γ} {f2 : β → δ}
(h1 : lift_rel R s1 s2) (h2 : ∀ {a b}, R a b → S (f1 a) (f2 b))
: lift_rel S (map f1 s1) (map f2 s2) :=
by rw [←bind_ret, ←bind_ret]; apply lift_rel_bind _ _ h1; simp; exact @h2
theorem map_congr (R : α → α → Prop) (S : β → β → Prop)
{s1 s2 : computation α} {f : α → β}
(h1 : s1 ~ s2) : map f s1 ~ map f s2 :=
by rw [←lift_eq_iff_equiv];
exact lift_rel_map eq _ ((lift_eq_iff_equiv _ _).2 h1) (λ a b, congr_arg _)
def lift_rel_aux (R : α → β → Prop)
(C : computation α → computation β → Prop) :
α ⊕ computation α → β ⊕ computation β → Prop
| (sum.inl a) (sum.inl b) := R a b
| (sum.inl a) (sum.inr cb) := ∃ {b}, b ∈ cb ∧ R a b
| (sum.inr ca) (sum.inl b) := ∃ {a}, a ∈ ca ∧ R a b
| (sum.inr ca) (sum.inr cb) := C ca cb
attribute [simp] lift_rel_aux
@[simp] lemma lift_rel_aux.ret_left (R : α → β → Prop)
(C : computation α → computation β → Prop) (a cb) :
lift_rel_aux R C (sum.inl a) (destruct cb) ↔ ∃ {b}, b ∈ cb ∧ R a b :=
begin
apply cb.cases_on (λ b, _) (λ cb, _),
{ exact ⟨λ h, ⟨_, ret_mem _, h⟩, λ ⟨b', mb, h⟩,
by rw [mem_unique (ret_mem _) mb]; exact h⟩ },
{ rw [destruct_think],
exact ⟨λ ⟨b, h, r⟩, ⟨b, think_mem h, r⟩,
λ ⟨b, h, r⟩, ⟨b, of_think_mem h, r⟩⟩ }
end
theorem lift_rel_aux.swap (R : α → β → Prop) (C) (a b) :
lift_rel_aux (function.swap R) (function.swap C) b a = lift_rel_aux R C a b :=
by cases a with a ca; cases b with b cb; simp only [lift_rel_aux]
@[simp] lemma lift_rel_aux.ret_right (R : α → β → Prop)
(C : computation α → computation β → Prop) (b ca) :
lift_rel_aux R C (destruct ca) (sum.inl b) ↔ ∃ {a}, a ∈ ca ∧ R a b :=
by rw [←lift_rel_aux.swap, lift_rel_aux.ret_left]
theorem lift_rel_rec.lem {R : α → β → Prop} (C : computation α → computation β → Prop)
(H : ∀ {ca cb}, C ca cb → lift_rel_aux R C (destruct ca) (destruct cb))
(ca cb) (Hc : C ca cb) (a) (ha : a ∈ ca) : lift_rel R ca cb :=
begin
revert cb, refine mem_rec_on ha _ (λ ca' IH, _);
intros cb Hc; have h := H Hc,
{ simp at h, simp [h] },
{ have h := H Hc, simp, revert h, apply cb.cases_on (λ b, _) (λ cb', _);
intro h; simp at h; simp [h], exact IH _ h }
end
theorem lift_rel_rec {R : α → β → Prop} (C : computation α → computation β → Prop)
(H : ∀ {ca cb}, C ca cb → lift_rel_aux R C (destruct ca) (destruct cb))
(ca cb) (Hc : C ca cb) : lift_rel R ca cb :=
lift_rel_mem_cases (lift_rel_rec.lem C @H ca cb Hc) (λ b hb,
(lift_rel.swap _ _ _).2 $
lift_rel_rec.lem (function.swap C)
(λ cb ca h, cast (lift_rel_aux.swap _ _ _ _).symm $ H h)
cb ca Hc b hb)
end computation
|
1d37d8c2fe30273ddc1256f0dcde3187eeab2d41 | 5749d8999a76f3a8fddceca1f6941981e33aaa96 | /src/data/finsupp.lean | 76d8b513e3d8a1ab7900564cfd1d03ea7fdf95f1 | [
"Apache-2.0"
] | permissive | jdsalchow/mathlib | 13ab43ef0d0515a17e550b16d09bd14b76125276 | 497e692b946d93906900bb33a51fd243e7649406 | refs/heads/master | 1,585,819,143,348 | 1,580,072,892,000 | 1,580,072,892,000 | 154,287,128 | 0 | 0 | Apache-2.0 | 1,540,281,610,000 | 1,540,281,609,000 | null | UTF-8 | Lean | false | false | 61,700 | lean | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
Type of functions with finite support.
Functions with finite support provide the basis for the following concrete instances:
* ℕ →₀ α: Polynomials (where α is a ring)
* (σ →₀ ℕ) →₀ α: Multivariate Polynomials (again α is a ring, and σ are variable names)
* α →₀ ℕ: Multisets
* α →₀ ℤ: Abelian groups freely generated by α
* β →₀ α: Linear combinations over β where α is the scalar ring
Most of the theory assumes that the range is a commutative monoid. This gives us the big sum
operator as a powerful way to construct `finsupp` elements.
A general advice is to not use α →₀ β directly, as the type class setup might not be fitting.
The best is to define a copy and select the instances best suited.
-/
import data.finset data.set.finite algebra.big_operators algebra.module
noncomputable theory
open_locale classical
open finset
variables {α : Type*} {β : Type*} {γ : Type*} {δ : Type*} {ι : Type*}
{α₁ : Type*} {α₂ : Type*} {β₁ : Type*} {β₂ : Type*}
/-- `finsupp α β`, denoted `α →₀ β`, is the type of functions `f : α → β` such that
`f x = 0` for all but finitely many `x`. -/
structure finsupp (α : Type*) (β : Type*) [has_zero β] :=
(support : finset α)
(to_fun : α → β)
(mem_support_to_fun : ∀a, a ∈ support ↔ to_fun a ≠ 0)
infixr ` →₀ `:25 := finsupp
namespace finsupp
section basic
variable [has_zero β]
instance : has_coe_to_fun (α →₀ β) := ⟨λ_, α → β, finsupp.to_fun⟩
instance : has_zero (α →₀ β) := ⟨⟨∅, (λ_, 0), λ _, ⟨false.elim, λ H, H rfl⟩⟩⟩
@[simp] lemma zero_apply {a : α} : (0 : α →₀ β) a = 0 := rfl
@[simp] lemma support_zero : (0 : α →₀ β).support = ∅ := rfl
instance : inhabited (α →₀ β) := ⟨0⟩
@[simp] lemma mem_support_iff {f : α →₀ β} : ∀{a:α}, a ∈ f.support ↔ f a ≠ 0 :=
f.mem_support_to_fun
lemma not_mem_support_iff {f : α →₀ β} {a} : a ∉ f.support ↔ f a = 0 :=
not_iff_comm.1 mem_support_iff.symm
@[ext]
lemma ext : ∀{f g : α →₀ β}, (∀a, f a = g a) → f = g
| ⟨s, f, hf⟩ ⟨t, g, hg⟩ h :=
begin
have : f = g, { funext a, exact h a },
subst this,
have : s = t, { ext a, exact (hf a).trans (hg a).symm },
subst this
end
lemma ext_iff {f g : α →₀ β} : f = g ↔ (∀a:α, f a = g a) :=
⟨by rintros rfl a; refl, ext⟩
@[simp] lemma support_eq_empty {f : α →₀ β} : f.support = ∅ ↔ f = 0 :=
⟨assume h, ext $ assume a, by_contradiction $ λ H, (finset.ext.1 h a).1 $
mem_support_iff.2 H, by rintro rfl; refl⟩
instance finsupp.decidable_eq [decidable_eq α] [decidable_eq β] : decidable_eq (α →₀ β) :=
assume f g, decidable_of_iff (f.support = g.support ∧ (∀a∈f.support, f a = g a))
⟨assume ⟨h₁, h₂⟩, ext $ assume a,
if h : a ∈ f.support then h₂ a h else
have hf : f a = 0, by rwa [mem_support_iff, not_not] at h,
have hg : g a = 0, by rwa [h₁, mem_support_iff, not_not] at h,
by rw [hf, hg],
by rintro rfl; exact ⟨rfl, λ _ _, rfl⟩⟩
lemma finite_supp (f : α →₀ β) : set.finite {a | f a ≠ 0} :=
⟨fintype.of_finset f.support (λ _, mem_support_iff)⟩
lemma support_subset_iff {s : set α} {f : α →₀ β} :
↑f.support ⊆ s ↔ (∀a∉s, f a = 0) :=
by simp only [set.subset_def, mem_coe, mem_support_iff];
exact forall_congr (assume a, @not_imp_comm _ _ (classical.dec _) (classical.dec _))
def equiv_fun_on_fintype [fintype α] : (α →₀ β) ≃ (α → β) :=
⟨λf a, f a, λf, mk (finset.univ.filter $ λa, f a ≠ 0) f (by simp),
begin intro f, ext a, refl end,
begin intro f, ext a, refl end⟩
end basic
section single
variables [has_zero β] {a a' : α} {b : β}
/-- `single a b` is the finitely supported function which has
value `b` at `a` and zero otherwise. -/
def single (a : α) (b : β) : α →₀ β :=
⟨if b = 0 then ∅ else finset.singleton a, λ a', if a = a' then b else 0, λ a', begin
by_cases hb : b = 0; by_cases a = a';
simp only [hb, h, if_pos, if_false, mem_singleton],
{ exact ⟨false.elim, λ H, H rfl⟩ },
{ exact ⟨false.elim, λ H, H rfl⟩ },
{ exact ⟨λ _, hb, λ _, rfl⟩ },
{ exact ⟨λ H _, h H.symm, λ H, (H rfl).elim⟩ }
end⟩
lemma single_apply : (single a b : α →₀ β) a' = if a = a' then b else 0 := rfl
@[simp] lemma single_eq_same : (single a b : α →₀ β) a = b := if_pos rfl
@[simp] lemma single_eq_of_ne (h : a ≠ a') : (single a b : α →₀ β) a' = 0 := if_neg h
@[simp] lemma single_zero : (single a 0 : α →₀ β) = 0 :=
ext $ assume a',
begin
by_cases h : a = a',
{ rw [h, single_eq_same, zero_apply] },
{ rw [single_eq_of_ne h, zero_apply] }
end
lemma support_single_ne_zero (hb : b ≠ 0) : (single a b).support = {a} :=
if_neg hb
lemma support_single_subset : (single a b).support ⊆ {a} :=
show ite _ _ _ ⊆ _, by split_ifs; [exact empty_subset _, exact subset.refl _]
lemma injective_single (a : α) : function.injective (single a : β → α →₀ β) :=
assume b₁ b₂ eq,
have (single a b₁ : α →₀ β) a = (single a b₂ : α →₀ β) a, by rw eq,
by rwa [single_eq_same, single_eq_same] at this
lemma single_eq_single_iff (a₁ a₂ : α) (b₁ b₂ : β) :
single a₁ b₁ = single a₂ b₂ ↔ ((a₁ = a₂ ∧ b₁ = b₂) ∨ (b₁ = 0 ∧ b₂ = 0)) :=
begin
split,
{ assume eq,
by_cases a₁ = a₂,
{ refine or.inl ⟨h, _⟩,
rwa [h, (injective_single a₂).eq_iff] at eq },
{ rw [finsupp.ext_iff] at eq,
have h₁ := eq a₁,
have h₂ := eq a₂,
simp only [single_eq_same, single_eq_of_ne h, single_eq_of_ne (ne.symm h)] at h₁ h₂,
exact or.inr ⟨h₁, h₂.symm⟩ } },
{ rintros (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩),
{ refl },
{ rw [single_zero, single_zero] } }
end
lemma single_right_inj (h : b ≠ 0) :
single a b = single a' b ↔ a = a' :=
⟨λ H, by simpa [h, single_eq_single_iff] using H, λ H, by rw [H]⟩
lemma single_eq_zero : single a b = 0 ↔ b = 0 :=
⟨λ h, by { rw ext_iff at h, simpa only [finsupp.single_eq_same, finsupp.zero_apply] using h a },
λ h, by rw [h, single_zero]⟩
lemma single_swap {α β : Type*} [has_zero β] (a₁ a₂ : α) (b : β) :
(single a₁ b : α → β) a₂ = (single a₂ b : α → β) a₁ :=
by simp [single_apply]; ac_refl
lemma unique_single [unique α] (x : α →₀ β) : x = single (default α) (x (default α)) :=
by ext i; simp [unique.eq_default i]
@[simp] lemma unique_single_eq_iff [unique α] {b' : β} :
single a b = single a' b' ↔ b = b' :=
begin
rw [single_eq_single_iff],
split,
{ rintros (⟨_, rfl⟩ | ⟨rfl, rfl⟩); refl },
{ intro h, left, exact ⟨subsingleton.elim _ _, h⟩ }
end
end single
section on_finset
variables [has_zero β]
/-- `on_finset s f hf` is the finsupp function representing `f` restricted to the set `s`.
The function needs to be 0 outside of `s`. Use this when the set needs filtered anyway, otherwise
often better set representation is available. -/
def on_finset (s : finset α) (f : α → β) (hf : ∀a, f a ≠ 0 → a ∈ s) : α →₀ β :=
⟨s.filter (λa, f a ≠ 0), f,
assume a, classical.by_cases
(assume h : f a = 0, by rw mem_filter; exact ⟨and.right, λ H, (H h).elim⟩)
(assume h : f a ≠ 0, by rw mem_filter; simp only [iff_true_intro h, hf a h, true_and])⟩
@[simp] lemma on_finset_apply {s : finset α} {f : α → β} {hf a} :
(on_finset s f hf : α →₀ β) a = f a :=
rfl
@[simp] lemma support_on_finset_subset {s : finset α} {f : α → β} {hf} :
(on_finset s f hf).support ⊆ s := filter_subset _
end on_finset
section map_range
variables [has_zero β₁] [has_zero β₂]
/-- The composition of `f : β₁ → β₂` and `g : α →₀ β₁` is
`map_range f hf g : α →₀ β₂`, well defined when `f 0 = 0`. -/
def map_range (f : β₁ → β₂) (hf : f 0 = 0) (g : α →₀ β₁) : α →₀ β₂ :=
on_finset g.support (f ∘ g) $
assume a, by rw [mem_support_iff, not_imp_not]; exact λ H, (congr_arg f H).trans hf
@[simp] lemma map_range_apply {f : β₁ → β₂} {hf : f 0 = 0} {g : α →₀ β₁} {a : α} :
map_range f hf g a = f (g a) :=
rfl
@[simp] lemma map_range_zero {f : β₁ → β₂} {hf : f 0 = 0} : map_range f hf (0 : α →₀ β₁) = 0 :=
finsupp.ext $ λ a, by simp [hf]
lemma support_map_range {f : β₁ → β₂} {hf : f 0 = 0} {g : α →₀ β₁} :
(map_range f hf g).support ⊆ g.support :=
support_on_finset_subset
@[simp] lemma map_range_single {f : β₁ → β₂} {hf : f 0 = 0} {a : α} {b : β₁} :
map_range f hf (single a b) = single a (f b) :=
finsupp.ext $ λ a', show f (ite _ _ _) = ite _ _ _, by split_ifs; [refl, exact hf]
end map_range
section emb_domain
variables [has_zero β]
/-- Given `f : α₁ ↪ α₂` and `v : α₁ →₀ β`, `emb_domain f v : α₂ →₀ β` is the finitely supported
function whose value at `f a : α₂` is `v a`. For a `b : α₂` outside the range of `f` it is zero. -/
def emb_domain (f : α₁ ↪ α₂) (v : α₁ →₀ β) : α₂ →₀ β :=
begin
refine ⟨v.support.map f, λa₂,
if h : a₂ ∈ v.support.map f then v (v.support.choose (λa₁, f a₁ = a₂) _) else 0, _⟩,
{ rcases finset.mem_map.1 h with ⟨a, ha, rfl⟩,
exact exists_unique.intro a ⟨ha, rfl⟩ (assume b ⟨_, hb⟩, f.inj hb) },
{ assume a₂,
split_ifs,
{ simp [h],
rw [← finsupp.not_mem_support_iff, not_not],
apply finset.choose_mem },
{ simp [h] } }
end
lemma support_emb_domain (f : α₁ ↪ α₂) (v : α₁ →₀ β) :
(emb_domain f v).support = v.support.map f :=
rfl
lemma emb_domain_zero (f : α₁ ↪ α₂) : (emb_domain f 0 : α₂ →₀ β) = 0 :=
rfl
lemma emb_domain_apply (f : α₁ ↪ α₂) (v : α₁ →₀ β) (a : α₁) :
emb_domain f v (f a) = v a :=
begin
change dite _ _ _ = _,
split_ifs; rw [finset.mem_map' f] at h,
{ refine congr_arg (v : α₁ → β) (f.inj' _),
exact finset.choose_property (λa₁, f a₁ = f a) _ _ },
{ exact (finsupp.not_mem_support_iff.1 h).symm }
end
lemma emb_domain_notin_range (f : α₁ ↪ α₂) (v : α₁ →₀ β) (a : α₂) (h : a ∉ set.range f) :
emb_domain f v a = 0 :=
begin
refine dif_neg (mt (assume h, _) h),
rcases finset.mem_map.1 h with ⟨a, h, rfl⟩,
exact set.mem_range_self a
end
lemma emb_domain_inj {f : α₁ ↪ α₂} {l₁ l₂ : α₁ →₀ β} :
emb_domain f l₁ = emb_domain f l₂ ↔ l₁ = l₂ :=
⟨λ h, finsupp.ext $ λ a, by simpa [emb_domain_apply] using finsupp.ext_iff.1 h (f a),
λ h, by rw h⟩
lemma emb_domain_map_range
{β₁ β₂ : Type*} [has_zero β₁] [has_zero β₂]
(f : α₁ ↪ α₂) (g : β₁ → β₂) (p : α₁ →₀ β₁) (hg : g 0 = 0) :
emb_domain f (map_range g hg p) = map_range g hg (emb_domain f p) :=
begin
ext a,
by_cases a ∈ set.range f,
{ rcases h with ⟨a', rfl⟩,
rw [map_range_apply, emb_domain_apply, emb_domain_apply, map_range_apply] },
{ rw [map_range_apply, emb_domain_notin_range, emb_domain_notin_range, ← hg]; assumption }
end
lemma single_of_emb_domain_single
(l : α₁ →₀ β) (f : α₁ ↪ α₂) (a : α₂) (b : β) (hb : b ≠ 0)
(h : l.emb_domain f = finsupp.single a b) :
∃ x, l = finsupp.single x b ∧ f x = a :=
begin
have h_map_support : finset.map f (l.support) = finset.singleton a,
by rw [←finsupp.support_emb_domain, h, finsupp.support_single_ne_zero hb]; refl,
have ha : a ∈ finset.map f (l.support),
by simp [h_map_support],
rcases finset.mem_map.1 ha with ⟨c, hc₁, hc₂⟩,
use c,
split,
{ ext d,
rw [← finsupp.emb_domain_apply f l, h],
by_cases h_cases : c = d,
{ simp [h_cases.symm, hc₂] },
{ rw [finsupp.single_apply, finsupp.single_apply, if_neg, if_neg h_cases],
by_contra hfd,
exact h_cases (f.inj (hc₂.trans hfd)) } },
{ exact hc₂ }
end
end emb_domain
section zip_with
variables [has_zero β] [has_zero β₁] [has_zero β₂]
/-- `zip_with f hf g₁ g₂` is the finitely supported function satisfying
`zip_with f hf g₁ g₂ a = f (g₁ a) (g₂ a)`, and well defined when `f 0 0 = 0`. -/
def zip_with (f : β₁ → β₂ → β) (hf : f 0 0 = 0) (g₁ : α →₀ β₁) (g₂ : α →₀ β₂) : (α →₀ β) :=
on_finset (g₁.support ∪ g₂.support) (λa, f (g₁ a) (g₂ a)) $ λ a H, begin
simp only [mem_union, mem_support_iff, ne], rw [← not_and_distrib],
rintro ⟨h₁, h₂⟩, rw [h₁, h₂] at H, exact H hf
end
@[simp] lemma zip_with_apply
{f : β₁ → β₂ → β} {hf : f 0 0 = 0} {g₁ : α →₀ β₁} {g₂ : α →₀ β₂} {a : α} :
zip_with f hf g₁ g₂ a = f (g₁ a) (g₂ a) := rfl
lemma support_zip_with {f : β₁ → β₂ → β} {hf : f 0 0 = 0} {g₁ : α →₀ β₁} {g₂ : α →₀ β₂} :
(zip_with f hf g₁ g₂).support ⊆ g₁.support ∪ g₂.support :=
support_on_finset_subset
end zip_with
section erase
def erase [has_zero β] (a : α) (f : α →₀ β) : α →₀ β :=
⟨f.support.erase a, (λa', if a' = a then 0 else f a'),
assume a', by rw [mem_erase, mem_support_iff]; split_ifs;
[exact ⟨λ H _, H.1 h, λ H, (H rfl).elim⟩,
exact and_iff_right h]⟩
@[simp] lemma support_erase [has_zero β] {a : α} {f : α →₀ β} :
(f.erase a).support = f.support.erase a :=
rfl
@[simp] lemma erase_same [has_zero β] {a : α} {f : α →₀ β} : (f.erase a) a = 0 :=
if_pos rfl
@[simp] lemma erase_ne [has_zero β] {a a' : α} {f : α →₀ β} (h : a' ≠ a) : (f.erase a) a' = f a' :=
if_neg h
end erase
-- [to_additive sum] for finsupp.prod doesn't work, the equation lemmas are not generated
/-- `sum f g` is the sum of `g a (f a)` over the support of `f`. -/
def sum [has_zero β] [add_comm_monoid γ] (f : α →₀ β) (g : α → β → γ) : γ :=
f.support.sum (λa, g a (f a))
/-- `prod f g` is the product of `g a (f a)` over the support of `f`. -/
@[to_additive]
def prod [has_zero β] [comm_monoid γ] (f : α →₀ β) (g : α → β → γ) : γ :=
f.support.prod (λa, g a (f a))
@[to_additive]
lemma prod_map_range_index [has_zero β₁] [has_zero β₂] [comm_monoid γ]
{f : β₁ → β₂} {hf : f 0 = 0} {g : α →₀ β₁} {h : α → β₂ → γ} (h0 : ∀a, h a 0 = 1) :
(map_range f hf g).prod h = g.prod (λa b, h a (f b)) :=
finset.prod_subset support_map_range $ λ _ _ H,
by rw [not_mem_support_iff.1 H, h0]
@[to_additive]
lemma prod_zero_index [add_comm_monoid β] [comm_monoid γ] {h : α → β → γ} :
(0 : α →₀ β).prod h = 1 := rfl
section nat_sub
instance nat_sub : has_sub (α →₀ ℕ) := ⟨zip_with (λ m n, m - n) (nat.sub_zero 0)⟩
@[simp] lemma nat_sub_apply {g₁ g₂ : α →₀ ℕ} {a : α} :
(g₁ - g₂) a = g₁ a - g₂ a := rfl
end nat_sub
section add_monoid
variables [add_monoid β]
@[to_additive]
lemma prod_single_index [comm_monoid γ] {a : α} {b : β} {h : α → β → γ} (h_zero : h a 0 = 1) :
(single a b).prod h = h a b :=
begin
by_cases h : b = 0,
{ simp only [h, h_zero, single_zero]; refl },
{ simp only [finsupp.prod, support_single_ne_zero h, insert_empty_eq_singleton,
prod_singleton, single_eq_same] }
end
instance : has_add (α →₀ β) := ⟨zip_with (+) (add_zero 0)⟩
@[simp] lemma add_apply {g₁ g₂ : α →₀ β} {a : α} : (g₁ + g₂) a = g₁ a + g₂ a :=
rfl
lemma support_add {g₁ g₂ : α →₀ β} : (g₁ + g₂).support ⊆ g₁.support ∪ g₂.support :=
support_zip_with
lemma support_add_eq {g₁ g₂ : α →₀ β} (h : disjoint g₁.support g₂.support) :
(g₁ + g₂).support = g₁.support ∪ g₂.support :=
le_antisymm support_zip_with $ assume a ha,
(finset.mem_union.1 ha).elim
(assume ha, have a ∉ g₂.support, from disjoint_left.1 h ha,
by simp only [mem_support_iff, not_not] at *;
simpa only [add_apply, this, add_zero])
(assume ha, have a ∉ g₁.support, from disjoint_right.1 h ha,
by simp only [mem_support_iff, not_not] at *;
simpa only [add_apply, this, zero_add])
@[simp] lemma single_add {a : α} {b₁ b₂ : β} : single a (b₁ + b₂) = single a b₁ + single a b₂ :=
ext $ assume a',
begin
by_cases h : a = a',
{ rw [h, add_apply, single_eq_same, single_eq_same, single_eq_same] },
{ rw [add_apply, single_eq_of_ne h, single_eq_of_ne h, single_eq_of_ne h, zero_add] }
end
instance : add_monoid (α →₀ β) :=
{ add_monoid .
zero := 0,
add := (+),
add_assoc := assume ⟨s, f, hf⟩ ⟨t, g, hg⟩ ⟨u, h, hh⟩, ext $ assume a, add_assoc _ _ _,
zero_add := assume ⟨s, f, hf⟩, ext $ assume a, zero_add _,
add_zero := assume ⟨s, f, hf⟩, ext $ assume a, add_zero _ }
instance (a : α) : is_add_monoid_hom (λ g : α →₀ β, g a) :=
{ map_add := λ _ _, add_apply, map_zero := zero_apply }
lemma single_add_erase {a : α} {f : α →₀ β} : single a (f a) + f.erase a = f :=
ext $ λ a',
if h : a = a' then by subst h; simp only [add_apply, single_eq_same, erase_same, add_zero]
else by simp only [add_apply, single_eq_of_ne h, zero_add, erase_ne (ne.symm h)]
lemma erase_add_single {a : α} {f : α →₀ β} : f.erase a + single a (f a) = f :=
ext $ λ a',
if h : a = a' then by subst h; simp only [add_apply, single_eq_same, erase_same, zero_add]
else by simp only [add_apply, single_eq_of_ne h, add_zero, erase_ne (ne.symm h)]
@[elab_as_eliminator]
protected theorem induction {p : (α →₀ β) → Prop} (f : α →₀ β)
(h0 : p 0) (ha : ∀a b (f : α →₀ β), a ∉ f.support → b ≠ 0 → p f → p (single a b + f)) :
p f :=
suffices ∀s (f : α →₀ β), f.support = s → p f, from this _ _ rfl,
assume s, finset.induction_on s (λ f hf, by rwa [support_eq_empty.1 hf]) $
assume a s has ih f hf,
suffices p (single a (f a) + f.erase a), by rwa [single_add_erase] at this,
begin
apply ha,
{ rw [support_erase, mem_erase], exact λ H, H.1 rfl },
{ rw [← mem_support_iff, hf], exact mem_insert_self _ _ },
{ apply ih _ _,
rw [support_erase, hf, finset.erase_insert has] }
end
lemma induction₂ {p : (α →₀ β) → Prop} (f : α →₀ β)
(h0 : p 0) (ha : ∀a b (f : α →₀ β), a ∉ f.support → b ≠ 0 → p f → p (f + single a b)) :
p f :=
suffices ∀s (f : α →₀ β), f.support = s → p f, from this _ _ rfl,
assume s, finset.induction_on s (λ f hf, by rwa [support_eq_empty.1 hf]) $
assume a s has ih f hf,
suffices p (f.erase a + single a (f a)), by rwa [erase_add_single] at this,
begin
apply ha,
{ rw [support_erase, mem_erase], exact λ H, H.1 rfl },
{ rw [← mem_support_iff, hf], exact mem_insert_self _ _ },
{ apply ih _ _,
rw [support_erase, hf, finset.erase_insert has] }
end
lemma map_range_add [add_monoid β₁] [add_monoid β₂]
{f : β₁ → β₂} {hf : f 0 = 0} (hf' : ∀ x y, f (x + y) = f x + f y) (v₁ v₂ : α →₀ β₁) :
map_range f hf (v₁ + v₂) = map_range f hf v₁ + map_range f hf v₂ :=
finsupp.ext $ λ a, by simp [hf']
end add_monoid
instance [add_comm_monoid β] : add_comm_monoid (α →₀ β) :=
{ add_comm := assume ⟨s, f, _⟩ ⟨t, g, _⟩, ext $ assume a, add_comm _ _,
.. finsupp.add_monoid }
instance [add_group β] : add_group (α →₀ β) :=
{ neg := map_range (has_neg.neg) neg_zero,
add_left_neg := assume ⟨s, f, _⟩, ext $ assume x, add_left_neg _,
.. finsupp.add_monoid }
lemma single_multiset_sum [add_comm_monoid β] (s : multiset β) (a : α) :
single a s.sum = (s.map (single a)).sum :=
multiset.induction_on s single_zero $ λ a s ih,
by rw [multiset.sum_cons, single_add, ih, multiset.map_cons, multiset.sum_cons]
lemma single_finset_sum [add_comm_monoid β] (s : finset γ) (f : γ → β) (a : α) :
single a (s.sum f) = s.sum (λb, single a (f b)) :=
begin
transitivity,
apply single_multiset_sum,
rw [multiset.map_map],
refl
end
lemma single_sum [has_zero γ] [add_comm_monoid β] (s : δ →₀ γ) (f : δ → γ → β) (a : α) :
single a (s.sum f) = s.sum (λd c, single a (f d c)) :=
single_finset_sum _ _ _
@[to_additive]
lemma prod_neg_index [add_group β] [comm_monoid γ]
{g : α →₀ β} {h : α → β → γ} (h0 : ∀a, h a 0 = 1) :
(-g).prod h = g.prod (λa b, h a (- b)) :=
prod_map_range_index h0
@[simp] lemma neg_apply [add_group β] {g : α →₀ β} {a : α} : (- g) a = - g a := rfl
@[simp] lemma sub_apply [add_group β] {g₁ g₂ : α →₀ β} {a : α} : (g₁ - g₂) a = g₁ a - g₂ a := rfl
@[simp] lemma support_neg [add_group β] {f : α →₀ β} : support (-f) = support f :=
finset.subset.antisymm
support_map_range
(calc support f = support (- (- f)) : congr_arg support (neg_neg _).symm
... ⊆ support (- f) : support_map_range)
instance [add_comm_group β] : add_comm_group (α →₀ β) :=
{ add_comm := add_comm, ..finsupp.add_group }
@[simp] lemma sum_apply [has_zero β₁] [add_comm_monoid β]
{f : α₁ →₀ β₁} {g : α₁ → β₁ → α →₀ β} {a₂ : α} :
(f.sum g) a₂ = f.sum (λa₁ b, g a₁ b a₂) :=
(f.support.sum_hom (λf : α →₀ β, f a₂)).symm
lemma support_sum [has_zero β₁] [add_comm_monoid β]
{f : α₁ →₀ β₁} {g : α₁ → β₁ → (α →₀ β)} :
(f.sum g).support ⊆ f.support.bind (λa, (g a (f a)).support) :=
have ∀a₁ : α, f.sum (λ (a : α₁) (b : β₁), (g a b) a₁) ≠ 0 →
(∃ (a : α₁), f a ≠ 0 ∧ ¬ (g a (f a)) a₁ = 0),
from assume a₁ h,
let ⟨a, ha, ne⟩ := finset.exists_ne_zero_of_sum_ne_zero h in
⟨a, mem_support_iff.mp ha, ne⟩,
by simpa only [finset.subset_iff, mem_support_iff, finset.mem_bind, sum_apply, exists_prop] using this
@[simp] lemma sum_zero [add_comm_monoid β] [add_comm_monoid γ] {f : α →₀ β} :
f.sum (λa b, (0 : γ)) = 0 :=
finset.sum_const_zero
@[simp] lemma sum_add [add_comm_monoid β] [add_comm_monoid γ] {f : α →₀ β}
{h₁ h₂ : α → β → γ} :
f.sum (λa b, h₁ a b + h₂ a b) = f.sum h₁ + f.sum h₂ :=
finset.sum_add_distrib
@[simp] lemma sum_neg [add_comm_monoid β] [add_comm_group γ] {f : α →₀ β}
{h : α → β → γ} : f.sum (λa b, - h a b) = - f.sum h :=
f.support.sum_hom (@has_neg.neg γ _)
@[simp] lemma sum_sub [add_comm_monoid β] [add_comm_group γ] {f : α →₀ β}
{h₁ h₂ : α → β → γ} :
f.sum (λa b, h₁ a b - h₂ a b) = f.sum h₁ - f.sum h₂ :=
by rw [sub_eq_add_neg, ←sum_neg, ←sum_add]; refl
@[simp] lemma sum_single [add_comm_monoid β] (f : α →₀ β) :
f.sum single = f :=
have ∀a:α, f.sum (λa' b, ite (a' = a) b 0) =
({a} : finset α).sum (λa', ite (a' = a) (f a') 0),
begin
intro a,
by_cases h : a ∈ f.support,
{ have : (finset.singleton a : finset α) ⊆ f.support,
{ simpa only [finset.subset_iff, mem_singleton, forall_eq] },
refine (finset.sum_subset this (λ _ _ H, _)).symm,
exact if_neg (mt mem_singleton.2 H) },
{ transitivity (f.support.sum (λa, (0 : β))),
{ refine (finset.sum_congr rfl $ λ a' ha', if_neg _),
rintro rfl, exact h ha' },
{ rw [sum_const_zero, insert_empty_eq_singleton, sum_singleton,
if_pos rfl, not_mem_support_iff.1 h] } }
end,
ext $ assume a, by simp only [sum_apply, single_apply, this,
insert_empty_eq_singleton, sum_singleton, if_pos]
@[to_additive]
lemma prod_add_index [add_comm_monoid β] [comm_monoid γ] {f g : α →₀ β}
{h : α → β → γ} (h_zero : ∀a, h a 0 = 1) (h_add : ∀a b₁ b₂, h a (b₁ + b₂) = h a b₁ * h a b₂) :
(f + g).prod h = f.prod h * g.prod h :=
have f_eq : (f.support ∪ g.support).prod (λa, h a (f a)) = f.prod h,
from (finset.prod_subset (finset.subset_union_left _ _) $
by intros _ _ H; rw [not_mem_support_iff.1 H, h_zero]).symm,
have g_eq : (f.support ∪ g.support).prod (λa, h a (g a)) = g.prod h,
from (finset.prod_subset (finset.subset_union_right _ _) $
by intros _ _ H; rw [not_mem_support_iff.1 H, h_zero]).symm,
calc (f + g).support.prod (λa, h a ((f + g) a)) =
(f.support ∪ g.support).prod (λa, h a ((f + g) a)) :
finset.prod_subset support_add $
by intros _ _ H; rw [not_mem_support_iff.1 H, h_zero]
... = (f.support ∪ g.support).prod (λa, h a (f a)) *
(f.support ∪ g.support).prod (λa, h a (g a)) :
by simp only [add_apply, h_add, finset.prod_mul_distrib]
... = _ : by rw [f_eq, g_eq]
lemma sum_sub_index [add_comm_group β] [add_comm_group γ] {f g : α →₀ β}
{h : α → β → γ} (h_sub : ∀a b₁ b₂, h a (b₁ - b₂) = h a b₁ - h a b₂) :
(f - g).sum h = f.sum h - g.sum h :=
have h_zero : ∀a, h a 0 = 0,
from assume a,
have h a (0 - 0) = h a 0 - h a 0, from h_sub a 0 0,
by simpa only [sub_self] using this,
have h_neg : ∀a b, h a (- b) = - h a b,
from assume a b,
have h a (0 - b) = h a 0 - h a b, from h_sub a 0 b,
by simpa only [h_zero, zero_sub] using this,
have h_add : ∀a b₁ b₂, h a (b₁ + b₂) = h a b₁ + h a b₂,
from assume a b₁ b₂,
have h a (b₁ - (- b₂)) = h a b₁ - h a (- b₂), from h_sub a b₁ (-b₂),
by simpa only [h_neg, sub_neg_eq_add] using this,
calc (f - g).sum h = (f + - g).sum h : rfl
... = f.sum h + - g.sum h : by simp only [sum_add_index h_zero h_add, sum_neg_index h_zero, h_neg, sum_neg]
... = f.sum h - g.sum h : rfl
@[to_additive]
lemma prod_finset_sum_index [add_comm_monoid β] [comm_monoid γ]
{s : finset ι} {g : ι → α →₀ β}
{h : α → β → γ} (h_zero : ∀a, h a 0 = 1) (h_add : ∀a b₁ b₂, h a (b₁ + b₂) = h a b₁ * h a b₂) :
s.prod (λi, (g i).prod h) = (s.sum g).prod h :=
finset.induction_on s rfl $ λ a s has ih,
by rw [prod_insert has, ih, sum_insert has, prod_add_index h_zero h_add]
@[to_additive]
lemma prod_sum_index
[add_comm_monoid β₁] [add_comm_monoid β] [comm_monoid γ]
{f : α₁ →₀ β₁} {g : α₁ → β₁ → α →₀ β}
{h : α → β → γ} (h_zero : ∀a, h a 0 = 1) (h_add : ∀a b₁ b₂, h a (b₁ + b₂) = h a b₁ * h a b₂) :
(f.sum g).prod h = f.prod (λa b, (g a b).prod h) :=
(prod_finset_sum_index h_zero h_add).symm
lemma multiset_sum_sum_index
[add_comm_monoid β] [add_comm_monoid γ]
(f : multiset (α →₀ β)) (h : α → β → γ)
(h₀ : ∀a, h a 0 = 0) (h₁ : ∀ (a : α) (b₁ b₂ : β), h a (b₁ + b₂) = h a b₁ + h a b₂) :
(f.sum.sum h) = (f.map $ λg:α →₀ β, g.sum h).sum :=
multiset.induction_on f rfl $ assume a s ih,
by rw [multiset.sum_cons, multiset.map_cons, multiset.sum_cons, sum_add_index h₀ h₁, ih]
lemma multiset_map_sum [has_zero β] {f : α →₀ β} {m : γ → δ} {h : α → β → multiset γ} :
multiset.map m (f.sum h) = f.sum (λa b, (h a b).map m) :=
(f.support.sum_hom _).symm
lemma multiset_sum_sum [has_zero β] [add_comm_monoid γ] {f : α →₀ β} {h : α → β → multiset γ} :
multiset.sum (f.sum h) = f.sum (λa b, multiset.sum (h a b)) :=
(f.support.sum_hom multiset.sum).symm
section map_range
variables
[add_comm_monoid β₁] [add_comm_monoid β₂]
(f : β₁ → β₂) [hf : is_add_monoid_hom f]
instance is_add_monoid_hom_map_range :
is_add_monoid_hom (map_range f hf.map_zero : (α →₀ β₁) → (α →₀ β₂)) :=
{ map_zero := map_range_zero, map_add := λ a b, map_range_add hf.map_add _ _ }
lemma map_range_multiset_sum (m : multiset (α →₀ β₁)) :
map_range f hf.map_zero m.sum = (m.map $ λx, map_range f hf.map_zero x).sum :=
(m.sum_hom (map_range f hf.map_zero)).symm
lemma map_range_finset_sum {ι : Type*} (s : finset ι) (g : ι → (α →₀ β₁)) :
map_range f hf.map_zero (s.sum g) = s.sum (λx, map_range f hf.map_zero (g x)) :=
by rw [finset.sum.equations._eqn_1, map_range_multiset_sum, multiset.map_map]; refl
end map_range
section map_domain
variables [add_comm_monoid β] {v v₁ v₂ : α →₀ β}
/-- Given `f : α₁ → α₂` and `v : α₁ →₀ β`, `map_domain f v : α₂ →₀ β`
is the finitely supported function whose value at `a : α₂` is the sum
of `v x` over all `x` such that `f x = a`. -/
def map_domain (f : α₁ → α₂) (v : α₁ →₀ β) : α₂ →₀ β :=
v.sum $ λa, single (f a)
lemma map_domain_apply {f : α₁ → α₂} (hf : function.injective f) (x : α₁ →₀ β) (a : α₁) :
map_domain f x (f a) = x a :=
begin
rw [map_domain, sum_apply, sum, finset.sum_eq_single a, single_eq_same],
{ assume b _ hba, exact single_eq_of_ne (hf.ne hba) },
{ simp only [(∉), (≠), not_not, mem_support_iff],
assume h,
rw [h, single_zero],
refl }
end
lemma map_domain_notin_range {f : α₁ → α₂} (x : α₁ →₀ β) (a : α₂) (h : a ∉ set.range f) :
map_domain f x a = 0 :=
begin
rw [map_domain, sum_apply, sum],
exact finset.sum_eq_zero
(assume a' h', single_eq_of_ne $ assume eq, h $ eq ▸ set.mem_range_self _)
end
lemma map_domain_id : map_domain id v = v := sum_single _
lemma map_domain_comp {f : α → α₁} {g : α₁ → α₂} :
map_domain (g ∘ f) v = map_domain g (map_domain f v) :=
begin
refine ((sum_sum_index _ _).trans _).symm,
{ intros, exact single_zero },
{ intros, exact single_add },
refine sum_congr rfl (λ _ _, sum_single_index _),
{ exact single_zero }
end
lemma map_domain_single {f : α → α₁} {a : α} {b : β} : map_domain f (single a b) = single (f a) b :=
sum_single_index single_zero
@[simp] lemma map_domain_zero {f : α → α₂} : map_domain f 0 = (0 : α₂ →₀ β) :=
sum_zero_index
lemma map_domain_congr {f g : α → α₂} (h : ∀x∈v.support, f x = g x) :
v.map_domain f = v.map_domain g :=
finset.sum_congr rfl $ λ _ H, by simp only [h _ H]
lemma map_domain_add {f : α → α₂} : map_domain f (v₁ + v₂) = map_domain f v₁ + map_domain f v₂ :=
sum_add_index (λ _, single_zero) (λ _ _ _, single_add)
lemma map_domain_finset_sum {f : α → α₂} {s : finset ι} {v : ι → α →₀ β} :
map_domain f (s.sum v) = s.sum (λi, map_domain f (v i)) :=
eq.symm $ sum_finset_sum_index (λ _, single_zero) (λ _ _ _, single_add)
lemma map_domain_sum [has_zero β₁] {f : α → α₂} {s : α →₀ β₁} {v : α → β₁ → α →₀ β} :
map_domain f (s.sum v) = s.sum (λa b, map_domain f (v a b)) :=
eq.symm $ sum_finset_sum_index (λ _, single_zero) (λ _ _ _, single_add)
lemma map_domain_support {f : α → α₂} {s : α →₀ β} :
(s.map_domain f).support ⊆ s.support.image f :=
finset.subset.trans support_sum $
finset.subset.trans (finset.bind_mono $ assume a ha, support_single_subset) $
by rw [finset.bind_singleton]; exact subset.refl _
@[to_additive]
lemma prod_map_domain_index [comm_monoid γ] {f : α → α₂} {s : α →₀ β}
{h : α₂ → β → γ} (h_zero : ∀a, h a 0 = 1) (h_add : ∀a b₁ b₂, h a (b₁ + b₂) = h a b₁ * h a b₂) :
(s.map_domain f).prod h = s.prod (λa b, h (f a) b) :=
(prod_sum_index h_zero h_add).trans $ prod_congr rfl $ λ _ _, prod_single_index (h_zero _)
lemma emb_domain_eq_map_domain (f : α₁ ↪ α₂) (v : α₁ →₀ β) :
emb_domain f v = map_domain f v :=
begin
ext a,
by_cases a ∈ set.range f,
{ rcases h with ⟨a, rfl⟩,
rw [map_domain_apply (function.embedding.inj' _), emb_domain_apply] },
{ rw [map_domain_notin_range, emb_domain_notin_range]; assumption }
end
lemma injective_map_domain {f : α₁ → α₂} (hf : function.injective f) :
function.injective (map_domain f : (α₁ →₀ β) → (α₂ →₀ β)) :=
begin
assume v₁ v₂ eq, ext a,
have : map_domain f v₁ (f a) = map_domain f v₂ (f a), { rw eq },
rwa [map_domain_apply hf, map_domain_apply hf] at this,
end
end map_domain
section comap_domain
noncomputable def comap_domain {α₁ α₂ γ : Type*} [has_zero γ]
(f : α₁ → α₂) (l : α₂ →₀ γ) (hf : set.inj_on f (f ⁻¹' l.support.to_set)) : α₁ →₀ γ :=
{ support := l.support.preimage hf,
to_fun := (λ a, l (f a)),
mem_support_to_fun :=
begin
intros a,
simp only [finset.mem_def.symm, finset.mem_preimage],
exact l.mem_support_to_fun (f a),
end }
lemma comap_domain_apply {α₁ α₂ γ : Type*} [has_zero γ]
(f : α₁ → α₂) (l : α₂ →₀ γ) (hf : set.inj_on f (f ⁻¹' l.support.to_set)) (a : α₁) :
comap_domain f l hf a = l (f a) :=
begin
unfold_coes,
unfold comap_domain,
simp,
refl
end
lemma sum_comap_domain {α₁ α₂ β γ : Type*} [has_zero β] [add_comm_monoid γ]
(f : α₁ → α₂) (l : α₂ →₀ β) (g : α₂ → β → γ) (hf : set.bij_on f (f ⁻¹' l.support.to_set) l.support.to_set):
(comap_domain f l (set.inj_on_of_bij_on hf)).sum (g ∘ f) = l.sum g :=
begin
unfold sum,
haveI := classical.dec_eq α₂,
simp only [comap_domain, comap_domain_apply, finset.sum_preimage f _ _ (λ (x : α₂), g x (l x))],
end
lemma eq_zero_of_comap_domain_eq_zero {α₁ α₂ γ : Type*} [add_comm_monoid γ]
(f : α₁ → α₂) (l : α₂ →₀ γ) (hf : set.bij_on f (f ⁻¹' l.support.to_set) l.support.to_set) :
comap_domain f l (set.inj_on_of_bij_on hf) = 0 → l = 0 :=
begin
rw [← support_eq_empty, ← support_eq_empty, comap_domain],
simp only [finset.ext, finset.not_mem_empty, iff_false, mem_preimage],
assume h a ha,
cases hf.2.2 ha with b hb,
exact h b (hb.2.symm ▸ ha)
end
lemma map_domain_comap_domain {α₁ α₂ γ : Type*} [add_comm_monoid γ]
(f : α₁ → α₂) (l : α₂ →₀ γ)
(hf : function.injective f) (hl : ↑l.support ⊆ set.range f):
map_domain f (comap_domain f l (set.inj_on_of_injective _ hf)) = l :=
begin
ext a,
haveI := classical.dec (a ∈ set.range f),
by_cases h_cases: a ∈ set.range f,
{ rcases set.mem_range.1 h_cases with ⟨b, hb⟩,
rw [hb.symm, map_domain_apply hf, comap_domain_apply] },
{ rw map_domain_notin_range _ _ h_cases,
by_contra h_contr,
apply h_cases (hl (finset.mem_coe.2 (mem_support_iff.2 (λ h, h_contr h.symm)))) }
end
end comap_domain
/-- The product of `f g : α →₀ β` is the finitely supported function
whose value at `a` is the sum of `f x * g y` over all pairs `x, y`
such that `x + y = a`. (Think of the product of multivariate
polynomials where `α` is the monoid of monomial exponents.) -/
instance [has_add α] [semiring β] : has_mul (α →₀ β) :=
⟨λf g, f.sum $ λa₁ b₁, g.sum $ λa₂ b₂, single (a₁ + a₂) (b₁ * b₂)⟩
lemma mul_def [has_add α] [semiring β] {f g : α →₀ β} :
f * g = (f.sum $ λa₁ b₁, g.sum $ λa₂ b₂, single (a₁ + a₂) (b₁ * b₂)) := rfl
lemma support_mul [has_add α] [semiring β] (a b : α →₀ β) :
(a * b).support ⊆ a.support.bind (λa₁, b.support.bind $ λa₂, {a₁ + a₂}) :=
subset.trans support_sum $ bind_mono $ assume a₁ _,
subset.trans support_sum $ bind_mono $ assume a₂ _, support_single_subset
/-- The unit of the multiplication is `single 0 1`, i.e. the function
that is 1 at 0 and zero elsewhere. -/
instance [has_zero α] [has_zero β] [has_one β] : has_one (α →₀ β) :=
⟨single 0 1⟩
lemma one_def [has_zero α] [has_zero β] [has_one β] : 1 = (single 0 1 : α →₀ β) := rfl
section filter
section has_zero
variables [has_zero β] (p : α → Prop) (f : α →₀ β)
/-- `filter p f` is the function which is `f a` if `p a` is true and 0 otherwise. -/
def filter (p : α → Prop) (f : α →₀ β) : α →₀ β :=
on_finset f.support (λa, if p a then f a else 0) $ λ a H,
mem_support_iff.2 $ λ h, by rw [h, if_t_t] at H; exact H rfl
@[simp] lemma filter_apply_pos {a : α} (h : p a) : f.filter p a = f a :=
if_pos h
@[simp] lemma filter_apply_neg {a : α} (h : ¬ p a) : f.filter p a = 0 :=
if_neg h
@[simp] lemma support_filter : (f.filter p).support = f.support.filter p :=
finset.ext.mpr $ assume a, if H : p a
then by simp only [mem_support_iff, filter_apply_pos _ _ H, mem_filter, H, and_true]
else by simp only [mem_support_iff, filter_apply_neg _ _ H, mem_filter, H, and_false, ne.def, ne_self_iff_false]
lemma filter_zero : (0 : α →₀ β).filter p = 0 :=
by rw [← support_eq_empty, support_filter, support_zero, finset.filter_empty]
@[simp] lemma filter_single_of_pos
{a : α} {b : β} (h : p a) : (single a b).filter p = single a b :=
finsupp.ext $ λ x, begin
by_cases h' : p x; simp [h'],
rw single_eq_of_ne, rintro rfl, exact h' h
end
@[simp] lemma filter_single_of_neg
{a : α} {b : β} (h : ¬ p a) : (single a b).filter p = 0 :=
finsupp.ext $ λ x, begin
by_cases h' : p x; simp [h'],
rw single_eq_of_ne, rintro rfl, exact h h'
end
end has_zero
lemma filter_pos_add_filter_neg [add_monoid β] (f : α →₀ β) (p : α → Prop) :
f.filter p + f.filter (λa, ¬ p a) = f :=
finsupp.ext $ assume a, if H : p a
then by simp only [add_apply, filter_apply_pos, filter_apply_neg, H, not_not, add_zero]
else by simp only [add_apply, filter_apply_pos, filter_apply_neg, H, not_false_iff, zero_add]
end filter
section frange
variables [has_zero β]
def frange (f : α →₀ β) : finset β :=
finset.image f f.support
theorem mem_frange {f : α →₀ β} {y : β} :
y ∈ f.frange ↔ y ≠ 0 ∧ ∃ x, f x = y :=
finset.mem_image.trans
⟨λ ⟨x, hx1, hx2⟩, ⟨hx2 ▸ mem_support_iff.1 hx1, x, hx2⟩,
λ ⟨hy, x, hx⟩, ⟨x, mem_support_iff.2 (hx.symm ▸ hy), hx⟩⟩
theorem zero_not_mem_frange {f : α →₀ β} : (0:β) ∉ f.frange :=
λ H, (mem_frange.1 H).1 rfl
theorem frange_single {x : α} {y : β} : frange (single x y) ⊆ {y} :=
λ r hr, let ⟨t, ht1, ht2⟩ := mem_frange.1 hr in ht2 ▸
(by rw single_apply at ht2 ⊢; split_ifs at ht2 ⊢; [exact finset.mem_singleton_self _, cc])
end frange
section subtype_domain
variables {α' : Type*} [has_zero δ] {p : α → Prop}
section zero
variables [has_zero β] {v v' : α' →₀ β}
/-- `subtype_domain p f` is the restriction of the finitely supported function
`f` to the subtype `p`. -/
def subtype_domain (p : α → Prop) (f : α →₀ β) : (subtype p →₀ β) :=
⟨f.support.subtype p, f ∘ subtype.val, λ a, by simp only [mem_subtype, mem_support_iff]⟩
@[simp] lemma support_subtype_domain {f : α →₀ β} :
(subtype_domain p f).support = f.support.subtype p :=
rfl
@[simp] lemma subtype_domain_apply {a : subtype p} {v : α →₀ β} :
(subtype_domain p v) a = v (a.val) :=
rfl
@[simp] lemma subtype_domain_zero : subtype_domain p (0 : α →₀ β) = 0 :=
rfl
@[to_additive]
lemma prod_subtype_domain_index [comm_monoid γ] {v : α →₀ β}
{h : α → β → γ} (hp : ∀x∈v.support, p x) :
(v.subtype_domain p).prod (λa b, h a.1 b) = v.prod h :=
prod_bij (λp _, p.val)
(λ _, mem_subtype.1)
(λ _ _, rfl)
(λ _ _ _ _, subtype.eq)
(λ b hb, ⟨⟨b, hp b hb⟩, mem_subtype.2 hb, rfl⟩)
end zero
section monoid
variables [add_monoid β] {v v' : α' →₀ β}
@[simp] lemma subtype_domain_add {v v' : α →₀ β} :
(v + v').subtype_domain p = v.subtype_domain p + v'.subtype_domain p :=
ext $ λ _, rfl
instance subtype_domain.is_add_monoid_hom :
is_add_monoid_hom (subtype_domain p : (α →₀ β) → subtype p →₀ β) :=
{ map_add := λ _ _, subtype_domain_add, map_zero := subtype_domain_zero }
@[simp] lemma filter_add {v v' : α →₀ β} :
(v + v').filter p = v.filter p + v'.filter p :=
ext $ λ a, by by_cases p a; simp [h]
instance filter.is_add_monoid_hom (p : α → Prop) :
is_add_monoid_hom (filter p : (α →₀ β) → (α →₀ β)) :=
{ map_zero := filter_zero p, map_add := λ x y, filter_add }
end monoid
section comm_monoid
variables [add_comm_monoid β]
lemma subtype_domain_sum {s : finset γ} {h : γ → α →₀ β} :
(s.sum h).subtype_domain p = s.sum (λc, (h c).subtype_domain p) :=
eq.symm (s.sum_hom _)
lemma subtype_domain_finsupp_sum {s : γ →₀ δ} {h : γ → δ → α →₀ β} :
(s.sum h).subtype_domain p = s.sum (λc d, (h c d).subtype_domain p) :=
subtype_domain_sum
lemma filter_sum (s : finset γ) (f : γ → α →₀ β) :
(s.sum f).filter p = s.sum (λa, filter p (f a)) :=
(s.sum_hom (filter p)).symm
end comm_monoid
section group
variables [add_group β] {v v' : α' →₀ β}
@[simp] lemma subtype_domain_neg {v : α →₀ β} :
(- v).subtype_domain p = - v.subtype_domain p :=
ext $ λ _, rfl
@[simp] lemma subtype_domain_sub {v v' : α →₀ β} :
(v - v').subtype_domain p = v.subtype_domain p - v'.subtype_domain p :=
ext $ λ _, rfl
end group
end subtype_domain
section multiset
def to_multiset (f : α →₀ ℕ) : multiset α :=
f.sum (λa n, add_monoid.smul n {a})
lemma to_multiset_zero : (0 : α →₀ ℕ).to_multiset = 0 :=
rfl
lemma to_multiset_add (m n : α →₀ ℕ) :
(m + n).to_multiset = m.to_multiset + n.to_multiset :=
sum_add_index (assume a, add_monoid.zero_smul _) (assume a b₁ b₂, add_monoid.add_smul _ _ _)
lemma to_multiset_single (a : α) (n : ℕ) : to_multiset (single a n) = add_monoid.smul n {a} :=
by rw [to_multiset, sum_single_index]; apply add_monoid.zero_smul
instance is_add_monoid_hom.to_multiset : is_add_monoid_hom (to_multiset : _ → multiset α) :=
{ map_zero := to_multiset_zero, map_add := to_multiset_add }
lemma card_to_multiset (f : α →₀ ℕ) : f.to_multiset.card = f.sum (λa, id) :=
begin
refine f.induction _ _,
{ rw [to_multiset_zero, multiset.card_zero, sum_zero_index] },
{ assume a n f _ _ ih,
rw [to_multiset_add, multiset.card_add, ih, sum_add_index, to_multiset_single,
sum_single_index, multiset.card_smul, multiset.singleton_eq_singleton,
multiset.card_singleton, mul_one]; intros; refl }
end
lemma to_multiset_map (f : α →₀ ℕ) (g : α → β) :
f.to_multiset.map g = (f.map_domain g).to_multiset :=
begin
refine f.induction _ _,
{ rw [to_multiset_zero, multiset.map_zero, map_domain_zero, to_multiset_zero] },
{ assume a n f _ _ ih,
rw [to_multiset_add, multiset.map_add, ih, map_domain_add, map_domain_single,
to_multiset_single, to_multiset_add, to_multiset_single,
is_add_monoid_hom.map_smul (multiset.map g)],
refl }
end
lemma prod_to_multiset [comm_monoid α] (f : α →₀ ℕ) :
f.to_multiset.prod = f.prod (λa n, a ^ n) :=
begin
refine f.induction _ _,
{ rw [to_multiset_zero, multiset.prod_zero, finsupp.prod_zero_index] },
{ assume a n f _ _ ih,
rw [to_multiset_add, multiset.prod_add, ih, to_multiset_single, finsupp.prod_add_index,
finsupp.prod_single_index, multiset.prod_smul, multiset.singleton_eq_singleton,
multiset.prod_singleton],
{ exact pow_zero a },
{ exact pow_zero },
{ exact pow_add } }
end
lemma to_finset_to_multiset (f : α →₀ ℕ) : f.to_multiset.to_finset = f.support :=
begin
refine f.induction _ _,
{ rw [to_multiset_zero, multiset.to_finset_zero, support_zero] },
{ assume a n f ha hn ih,
rw [to_multiset_add, multiset.to_finset_add, ih, to_multiset_single, support_add_eq,
support_single_ne_zero hn, multiset.to_finset_smul _ _ hn,
multiset.singleton_eq_singleton, multiset.to_finset_cons, multiset.to_finset_zero],
refl,
refine disjoint_mono support_single_subset (subset.refl _) _,
rwa [finset.singleton_eq_singleton, finset.singleton_disjoint] }
end
@[simp] lemma count_to_multiset (f : α →₀ ℕ) (a : α) :
f.to_multiset.count a = f a :=
calc f.to_multiset.count a = f.sum (λx n, (add_monoid.smul n {x} : multiset α).count a) :
(f.support.sum_hom $ multiset.count a).symm
... = f.sum (λx n, n * ({x} : multiset α).count a) : by simp only [multiset.count_smul]
... = f.sum (λx n, n * (x :: 0 : multiset α).count a) : rfl
... = f a * (a :: 0 : multiset α).count a : sum_eq_single _
(λ a' _ H, by simp only [multiset.count_cons_of_ne (ne.symm H), multiset.count_zero, mul_zero])
(λ H, by simp only [not_mem_support_iff.1 H, zero_mul])
... = f a : by simp only [multiset.count_singleton, mul_one]
def of_multiset (m : multiset α) : α →₀ ℕ :=
on_finset m.to_finset (λa, m.count a) $ λ a H, multiset.mem_to_finset.2 $
by_contradiction (mt multiset.count_eq_zero.2 H)
@[simp] lemma of_multiset_apply (m : multiset α) (a : α) :
of_multiset m a = m.count a :=
rfl
def equiv_multiset : (α →₀ ℕ) ≃ (multiset α) :=
⟨ to_multiset, of_multiset,
assume f, finsupp.ext $ λ a, by rw [of_multiset_apply, count_to_multiset],
assume m, multiset.ext.2 $ λ a, by rw [count_to_multiset, of_multiset_apply] ⟩
lemma mem_support_multiset_sum [add_comm_monoid β]
{s : multiset (α →₀ β)} (a : α) :
a ∈ s.sum.support → ∃f∈s, a ∈ (f : α →₀ β).support :=
multiset.induction_on s false.elim
begin
assume f s ih ha,
by_cases a ∈ f.support,
{ exact ⟨f, multiset.mem_cons_self _ _, h⟩ },
{ simp only [multiset.sum_cons, mem_support_iff, add_apply,
not_mem_support_iff.1 h, zero_add] at ha,
rcases ih (mem_support_iff.2 ha) with ⟨f', h₀, h₁⟩,
exact ⟨f', multiset.mem_cons_of_mem h₀, h₁⟩ }
end
lemma mem_support_finset_sum [add_comm_monoid β]
{s : finset γ} {h : γ → α →₀ β} (a : α) (ha : a ∈ (s.sum h).support) : ∃c∈s, a ∈ (h c).support :=
let ⟨f, hf, hfa⟩ := mem_support_multiset_sum a ha in
let ⟨c, hc, eq⟩ := multiset.mem_map.1 hf in
⟨c, hc, eq.symm ▸ hfa⟩
lemma mem_support_single [has_zero β] (a a' : α) (b : β) :
a ∈ (single a' b).support ↔ a = a' ∧ b ≠ 0 :=
⟨λ H : (a ∈ ite _ _ _), if h : b = 0
then by rw if_pos h at H; exact H.elim
else ⟨by rw if_neg h at H; exact mem_singleton.1 H, h⟩,
λ ⟨h1, h2⟩, show a ∈ ite _ _ _, by rw [if_neg h2]; exact mem_singleton.2 h1⟩
end multiset
section curry_uncurry
protected def curry [add_comm_monoid γ]
(f : (α × β) →₀ γ) : α →₀ (β →₀ γ) :=
f.sum $ λp c, single p.1 (single p.2 c)
lemma sum_curry_index
[add_comm_monoid γ] [add_comm_monoid δ]
(f : (α × β) →₀ γ) (g : α → β → γ → δ)
(hg₀ : ∀ a b, g a b 0 = 0) (hg₁ : ∀a b c₀ c₁, g a b (c₀ + c₁) = g a b c₀ + g a b c₁) :
f.curry.sum (λa f, f.sum (g a)) = f.sum (λp c, g p.1 p.2 c) :=
begin
rw [finsupp.curry],
transitivity,
{ exact sum_sum_index (assume a, sum_zero_index)
(assume a b₀ b₁, sum_add_index (assume a, hg₀ _ _) (assume c d₀ d₁, hg₁ _ _ _ _)) },
congr, funext p c,
transitivity,
{ exact sum_single_index sum_zero_index },
exact sum_single_index (hg₀ _ _)
end
protected def uncurry [add_comm_monoid γ] (f : α →₀ (β →₀ γ)) : (α × β) →₀ γ :=
f.sum $ λa g, g.sum $ λb c, single (a, b) c
def finsupp_prod_equiv [add_comm_monoid γ] : ((α × β) →₀ γ) ≃ (α →₀ (β →₀ γ)) :=
by refine ⟨finsupp.curry, finsupp.uncurry, λ f, _, λ f, _⟩; simp only [
finsupp.curry, finsupp.uncurry, sum_sum_index, sum_zero_index, sum_add_index,
sum_single_index, single_zero, single_add, eq_self_iff_true, forall_true_iff,
forall_3_true_iff, prod.mk.eta, (single_sum _ _ _).symm, sum_single]
lemma filter_curry [add_comm_monoid β] (f : α₁ × α₂ →₀ β) (p : α₁ → Prop) :
(f.filter (λa:α₁×α₂, p a.1)).curry = f.curry.filter p :=
begin
rw [finsupp.curry, finsupp.curry, finsupp.sum, finsupp.sum,
@filter_sum _ (α₂ →₀ β) _ p _ f.support _],
rw [support_filter, sum_filter],
refine finset.sum_congr rfl _,
rintros ⟨a₁, a₂⟩ ha,
dsimp only,
split_ifs,
{ rw [filter_apply_pos, filter_single_of_pos]; exact h },
{ rwa [filter_single_of_neg] }
end
lemma support_curry [add_comm_monoid β] (f : α₁ × α₂ →₀ β) :
f.curry.support ⊆ f.support.image prod.fst :=
begin
rw ← finset.bind_singleton,
refine finset.subset.trans support_sum _,
refine finset.bind_mono (assume a _, support_single_subset)
end
end curry_uncurry
section
variables [add_monoid α] [semiring β]
-- TODO: the simplifier unfolds 0 in the instance proof!
private lemma zero_mul (f : α →₀ β) : 0 * f = 0 := by simp only [mul_def, sum_zero_index]
private lemma mul_zero (f : α →₀ β) : f * 0 = 0 := by simp only [mul_def, sum_zero_index, sum_zero]
private lemma left_distrib (a b c : α →₀ β) : a * (b + c) = a * b + a * c :=
by simp only [mul_def, sum_add_index, mul_add, _root_.mul_zero, single_zero, single_add,
eq_self_iff_true, forall_true_iff, forall_3_true_iff, sum_add]
private lemma right_distrib (a b c : α →₀ β) : (a + b) * c = a * c + b * c :=
by simp only [mul_def, sum_add_index, add_mul, _root_.mul_zero, _root_.zero_mul, single_zero, single_add,
eq_self_iff_true, forall_true_iff, forall_3_true_iff, sum_zero, sum_add]
instance : semiring (α →₀ β) :=
{ one := 1,
mul := (*),
one_mul := assume f, by simp only [mul_def, one_def, sum_single_index, _root_.zero_mul, single_zero, sum_zero,
zero_add, one_mul, sum_single],
mul_one := assume f, by simp only [mul_def, one_def, sum_single_index, _root_.mul_zero, single_zero, sum_zero,
add_zero, mul_one, sum_single],
zero_mul := zero_mul,
mul_zero := mul_zero,
mul_assoc := assume f g h, by simp only [mul_def, sum_sum_index, sum_zero_index, sum_add_index, sum_single_index,
single_zero, single_add, eq_self_iff_true, forall_true_iff, forall_3_true_iff,
add_mul, mul_add, add_assoc, mul_assoc, _root_.zero_mul, _root_.mul_zero, sum_zero, sum_add],
left_distrib := left_distrib,
right_distrib := right_distrib,
.. finsupp.add_comm_monoid }
end
instance [add_comm_monoid α] [comm_semiring β] : comm_semiring (α →₀ β) :=
{ mul_comm := assume f g,
begin
simp only [mul_def, finsupp.sum, mul_comm],
rw [finset.sum_comm],
simp only [add_comm]
end,
.. finsupp.semiring }
instance [add_monoid α] [ring β] : ring (α →₀ β) :=
{ neg := has_neg.neg,
add_left_neg := add_left_neg,
.. finsupp.semiring }
instance [add_comm_monoid α] [comm_ring β] : comm_ring (α →₀ β) :=
{ mul_comm := mul_comm, .. finsupp.ring}
lemma single_mul_single [has_add α] [semiring β] {a₁ a₂ : α} {b₁ b₂ : β}:
single a₁ b₁ * single a₂ b₂ = single (a₁ + a₂) (b₁ * b₂) :=
(sum_single_index (by simp only [_root_.zero_mul, single_zero, sum_zero])).trans
(sum_single_index (by rw [_root_.mul_zero, single_zero]))
lemma prod_single [add_comm_monoid α] [comm_semiring β]
{s : finset ι} {a : ι → α} {b : ι → β} :
s.prod (λi, single (a i) (b i)) = single (s.sum a) (s.prod b) :=
finset.induction_on s rfl $ λ a s has ih, by rw [prod_insert has, ih,
single_mul_single, sum_insert has, prod_insert has]
section
instance [semiring γ] [add_comm_monoid β] [semimodule γ β] : has_scalar γ (α →₀ β) := ⟨λa v, v.map_range ((•) a) (smul_zero _)⟩
variables (α β)
@[simp] lemma smul_apply' {R:semiring γ} [add_comm_monoid β] [semimodule γ β] {a : α} {b : γ} {v : α →₀ β} :
(b • v) a = b • (v a) := rfl
instance [semiring γ] [add_comm_monoid β] [semimodule γ β] : semimodule γ (α →₀ β) :=
{ smul := (•),
smul_add := λ a x y, finsupp.ext $ λ _, smul_add _ _ _,
add_smul := λ a x y, finsupp.ext $ λ _, add_smul _ _ _,
one_smul := λ x, finsupp.ext $ λ _, one_smul _ _,
mul_smul := λ r s x, finsupp.ext $ λ _, mul_smul _ _ _,
zero_smul := λ x, finsupp.ext $ λ _, zero_smul _ _,
smul_zero := λ x, finsupp.ext $ λ _, smul_zero _ }
instance [ring γ] [add_comm_group β] [module γ β] : module γ (α →₀ β) :=
{ ..finsupp.semimodule α β }
instance [discrete_field γ] [add_comm_group β] [vector_space γ β] : vector_space γ (α →₀ β) :=
{ ..finsupp.module α β }
variables {α β}
lemma support_smul {R:semiring γ} [add_comm_monoid β] [semimodule γ β] {b : γ} {g : α →₀ β} :
(b • g).support ⊆ g.support :=
λ a, by simp; exact mt (λ h, h.symm ▸ smul_zero _)
section
variables {α' : Type*} [has_zero δ] {p : α → Prop}
@[simp] lemma filter_smul {R : semiring γ} [add_comm_monoid β] [semimodule γ β]
{b : γ} {v : α →₀ β} : (b • v).filter p = b • v.filter p :=
ext $ λ a, by by_cases p a; simp [h]
end
lemma map_domain_smul {α'} {R : semiring γ} [add_comm_monoid β] [semimodule γ β]
{f : α → α'} (b : γ) (v : α →₀ β) : map_domain f (b • v) = b • map_domain f v :=
begin
change map_domain f (map_range _ _ _) = map_range _ _ _,
apply finsupp.induction v, {simp},
intros a b v' hv₁ hv₂ IH,
rw [map_range_add, map_domain_add, IH, map_domain_add, map_range_add,
map_range_single, map_domain_single, map_domain_single, map_range_single];
apply smul_add
end
@[simp] lemma smul_single {R : semiring γ} [add_comm_monoid β] [semimodule γ β]
(c : γ) (a : α) (b : β) : c • finsupp.single a b = finsupp.single a (c • b) :=
ext $ λ a', by by_cases a = a'; [{subst h, simp}, simp [h]]
end
@[simp] lemma smul_apply [ring β] {a : α} {b : β} {v : α →₀ β} :
(b • v) a = b • (v a) := rfl
lemma sum_smul_index [ring β] [add_comm_monoid γ] {g : α →₀ β} {b : β} {h : α → β → γ}
(h0 : ∀i, h i 0 = 0) : (b • g).sum h = g.sum (λi a, h i (b * a)) :=
finsupp.sum_map_range_index h0
section
variables [semiring β] [semiring γ]
lemma sum_mul (b : γ) (s : α →₀ β) {f : α → β → γ} :
(s.sum f) * b = s.sum (λ a c, (f a (s a)) * b) :=
by simp only [finsupp.sum, finset.sum_mul]
lemma mul_sum (b : γ) (s : α →₀ β) {f : α → β → γ} :
b * (s.sum f) = s.sum (λ a c, b * (f a (s a))) :=
by simp only [finsupp.sum, finset.mul_sum]
protected lemma eq_zero_of_zero_eq_one
(zero_eq_one : (0 : β) = 1) (l : α →₀ β) : l = 0 :=
by ext i; simp [eq_zero_of_zero_eq_one β zero_eq_one (l i)]
end
def restrict_support_equiv [add_comm_monoid β] (s : set α) :
{f : α →₀ β // ↑f.support ⊆ s } ≃ (s →₀ β):=
begin
refine ⟨λf, subtype_domain (λx, x ∈ s) f.1, λ f, ⟨f.map_domain subtype.val, _⟩, _, _⟩,
{ refine set.subset.trans (finset.coe_subset.2 map_domain_support) _,
rw [finset.coe_image, set.image_subset_iff],
exact assume x hx, x.2 },
{ rintros ⟨f, hf⟩,
apply subtype.eq,
ext a,
dsimp only,
refine classical.by_cases (assume h : a ∈ set.range (subtype.val : s → α), _) (assume h, _),
{ rcases h with ⟨x, rfl⟩,
rw [map_domain_apply subtype.val_injective, subtype_domain_apply] },
{ convert map_domain_notin_range _ _ h,
rw [← not_mem_support_iff],
refine mt _ h,
exact assume ha, ⟨⟨a, hf ha⟩, rfl⟩ } },
{ assume f,
ext ⟨a, ha⟩,
dsimp only,
rw [subtype_domain_apply, map_domain_apply subtype.val_injective] }
end
protected def dom_congr [add_comm_monoid β] (e : α₁ ≃ α₂) : (α₁ →₀ β) ≃ (α₂ →₀ β) :=
⟨map_domain e, map_domain e.symm,
begin
assume v,
simp only [map_domain_comp.symm, (∘), equiv.symm_apply_apply],
exact map_domain_id
end,
begin
assume v,
simp only [map_domain_comp.symm, (∘), equiv.apply_symm_apply],
exact map_domain_id
end⟩
section sigma
variables {αs : ι → Type*} [has_zero β] (l : (Σ i, αs i) →₀ β)
noncomputable def split (i : ι) : αs i →₀ β :=
l.comap_domain (sigma.mk i) (λ x1 x2 _ _ hx, heq_iff_eq.1 (sigma.mk.inj hx).2)
lemma split_apply (i : ι) (x : αs i) : split l i x = l ⟨i, x⟩ :=
begin
dunfold split,
rw comap_domain_apply
end
def split_support : finset ι := l.support.image sigma.fst
lemma mem_split_support_iff_nonzero (i : ι) :
i ∈ split_support l ↔ split l i ≠ 0 :=
begin
classical,
rw [split_support, mem_image, ne.def, ← support_eq_empty, ← ne.def,
← finset.nonempty_iff_ne_empty, split, comap_domain, finset.nonempty],
simp
end
noncomputable def split_comp [has_zero γ] (g : Π i, (αs i →₀ β) → γ)
(hg : ∀ i x, x = 0 ↔ g i x = 0) : ι →₀ γ :=
{ support := split_support l,
to_fun := λ i, g i (split l i),
mem_support_to_fun :=
begin
intros i,
rw mem_split_support_iff_nonzero,
haveI := classical.dec,
rwa not_iff_not,
exact hg _ _,
end }
lemma sigma_support : l.support = l.split_support.sigma (λ i, (l.split i).support) :=
by simp [finset.ext, split_support, split, comap_domain]; tauto
lemma sigma_sum [add_comm_monoid γ] (f : (Σ (i : ι), αs i) → β → γ) :
l.sum f = (split_support l).sum (λ (i : ι), (split l i).sum (λ (a : αs i) b, f ⟨i, a⟩ b)) :=
by simp [sum, sigma_support, sum_sigma,split_apply]
end sigma
end finsupp
namespace multiset
def to_finsupp (s : multiset α) : α →₀ ℕ :=
{ support := s.to_finset,
to_fun := λ a, s.count a,
mem_support_to_fun := λ a,
begin
rw mem_to_finset,
convert not_iff_not_of_iff (count_eq_zero.symm),
rw not_not
end }
@[simp] lemma to_finsupp_support (s : multiset α) :
s.to_finsupp.support = s.to_finset := rfl
@[simp] lemma to_finsupp_apply (s : multiset α) (a : α) :
s.to_finsupp a = s.count a := rfl
@[simp] lemma to_finsupp_zero :
to_finsupp (0 : multiset α) = 0 :=
finsupp.ext $ λ a, count_zero a
@[simp] lemma to_finsupp_add (s t : multiset α) :
to_finsupp (s + t) = to_finsupp s + to_finsupp t :=
finsupp.ext $ λ a, count_add a s t
lemma to_finsupp_singleton (a : α) :
to_finsupp {a} = finsupp.single a 1 :=
finsupp.ext $ λ b,
if h : a = b then by simp [finsupp.single_apply, h] else
begin
rw [to_finsupp_apply, finsupp.single_apply, if_neg h, count_eq_zero,
singleton_eq_singleton, mem_singleton],
rintro rfl, exact h rfl
end
namespace to_finsupp
instance : is_add_monoid_hom (to_finsupp : multiset α → α →₀ ℕ) :=
{ map_zero := to_finsupp_zero,
map_add := to_finsupp_add }
end to_finsupp
@[simp] lemma to_finsupp_to_multiset (s : multiset α) :
s.to_finsupp.to_multiset = s :=
ext.2 $ λ a, by rw [finsupp.count_to_multiset, to_finsupp_apply]
end multiset
namespace finsupp
variables {σ : Type*}
instance [preorder α] [has_zero α] : preorder (σ →₀ α) :=
{ le := λ f g, ∀ s, f s ≤ g s,
le_refl := λ f s, le_refl _,
le_trans := λ f g h Hfg Hgh s, le_trans (Hfg s) (Hgh s) }
instance [partial_order α] [has_zero α] : partial_order (σ →₀ α) :=
{ le_antisymm := λ f g hfg hgf, finsupp.ext $ λ s, le_antisymm (hfg s) (hgf s),
.. finsupp.preorder }
instance [ordered_cancel_comm_monoid α] :
add_left_cancel_semigroup (σ →₀ α) :=
{ add_left_cancel := λ a b c h, finsupp.ext $ λ s,
by { rw finsupp.ext_iff at h, exact add_left_cancel (h s) },
.. finsupp.add_monoid }
instance [ordered_cancel_comm_monoid α] :
add_right_cancel_semigroup (σ →₀ α) :=
{ add_right_cancel := λ a b c h, finsupp.ext $ λ s,
by { rw finsupp.ext_iff at h, exact add_right_cancel (h s) },
.. finsupp.add_monoid }
instance [ordered_cancel_comm_monoid α] :
ordered_cancel_comm_monoid (σ →₀ α) :=
{ add_le_add_left := λ a b h c s, add_le_add_left (h s) (c s),
le_of_add_le_add_left := λ a b c h s, le_of_add_le_add_left (h s),
.. finsupp.add_comm_monoid, .. finsupp.partial_order,
.. finsupp.add_left_cancel_semigroup, .. finsupp.add_right_cancel_semigroup }
lemma le_iff [canonically_ordered_monoid α] (f g : σ →₀ α) :
f ≤ g ↔ ∀ s ∈ f.support, f s ≤ g s :=
⟨λ h s hs, h s,
λ h s, if H : s ∈ f.support then h s H else (not_mem_support_iff.1 H).symm ▸ zero_le (g s)⟩
attribute [simp] to_multiset_zero to_multiset_add
@[simp] lemma to_multiset_to_finsupp (f : σ →₀ ℕ) :
f.to_multiset.to_finsupp = f :=
ext $ λ s, by rw [multiset.to_finsupp_apply, count_to_multiset]
lemma to_multiset_strict_mono : strict_mono (@to_multiset σ) :=
λ m n h,
begin
rw lt_iff_le_and_ne at h ⊢, cases h with h₁ h₂,
split,
{ rw multiset.le_iff_count, intro s, erw [count_to_multiset m s, count_to_multiset], exact h₁ s },
{ intro H, apply h₂, replace H := congr_arg multiset.to_finsupp H, simpa using H }
end
lemma sum_id_lt_of_lt (m n : σ →₀ ℕ) (h : m < n) :
m.sum (λ _, id) < n.sum (λ _, id) :=
begin
rw [← card_to_multiset, ← card_to_multiset],
apply multiset.card_lt_of_lt,
exact to_multiset_strict_mono h
end
variable (σ)
/-- The order on σ →₀ ℕ is well-founded.-/
lemma lt_wf : well_founded (@has_lt.lt (σ →₀ ℕ) _) :=
subrelation.wf (sum_id_lt_of_lt) $ inv_image.wf _ nat.lt_wf
instance decidable_le : decidable_rel (@has_le.le (σ →₀ ℕ) _) :=
λ m n, by rw le_iff; apply_instance
variable {σ}
def antidiagonal (f : σ →₀ ℕ) : ((σ →₀ ℕ) × (σ →₀ ℕ)) →₀ ℕ :=
(f.to_multiset.antidiagonal.map (prod.map multiset.to_finsupp multiset.to_finsupp)).to_finsupp
lemma mem_antidiagonal_support {f : σ →₀ ℕ} {p : (σ →₀ ℕ) × (σ →₀ ℕ)} :
p ∈ (antidiagonal f).support ↔ p.1 + p.2 = f :=
begin
erw [multiset.mem_to_finset, multiset.mem_map],
split,
{ rintros ⟨⟨a, b⟩, h, rfl⟩,
rw multiset.mem_antidiagonal at h,
simpa using congr_arg multiset.to_finsupp h },
{ intro h,
refine ⟨⟨p.1.to_multiset, p.2.to_multiset⟩, _, _⟩,
{ simpa using congr_arg to_multiset h },
{ rw [prod.map, to_multiset_to_finsupp, to_multiset_to_finsupp, prod.mk.eta] } }
end
@[simp] lemma antidiagonal_zero : antidiagonal (0 : σ →₀ ℕ) = single (0,0) 1 :=
by rw [← multiset.to_finsupp_singleton]; refl
lemma swap_mem_antidiagonal_support {n : σ →₀ ℕ} {f} (hf : f ∈ (antidiagonal n).support) :
f.swap ∈ (antidiagonal n).support :=
by simpa [mem_antidiagonal_support, add_comm] using hf
end finsupp
|
641104ca9dbb5d5446081badc6780fa7bd9fa006 | 618003631150032a5676f229d13a079ac875ff77 | /src/data/set/intervals/basic.lean | 49bdee9c4ed1fcf394ccab7cff6bc44bd28cb328 | [
"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 | 28,577 | lean | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov
-/
import tactic.tauto
import algebra.order_functions
import algebra.ordered_field
/-!
# Intervals
In any preorder `α`, we define intervals (which on each side can be either infinite, open, or
closed) using the following naming conventions:
- `i`: infinite
- `o`: open
- `c`: closed
Each interval has the name `I` + letter for left side + letter for right side. For instance,
`Ioc a b` denotes the inverval `(a, b]`.
This file contains these definitions, and basic facts on inclusion, intersection, difference of
intervals (where the precise statements may depend on the properties of the order, in particular
for some statements it should be `linear_order` or `densely_ordered`).
This file also contains statements on lower and upper bounds of intervals.
TODO: This is just the beginning; a lot of rules are missing
-/
universe u
namespace set
open set
section intervals
variables {α : Type u} [preorder α] {a a₁ a₂ b b₁ b₂ x : α}
/-- Left-open right-open interval -/
def Ioo (a b : α) := {x | a < x ∧ x < b}
/-- Left-closed right-open interval -/
def Ico (a b : α) := {x | a ≤ x ∧ x < b}
/-- Left-infinite right-open interval -/
def Iio (a : α) := {x | x < a}
/-- Left-closed right-closed interval -/
def Icc (a b : α) := {x | a ≤ x ∧ x ≤ b}
/-- Left-infinite right-closed interval -/
def Iic (b : α) := {x | x ≤ b}
/-- Left-open right-closed interval -/
def Ioc (a b : α) := {x | a < x ∧ x ≤ b}
/-- Left-closed right-infinite interval -/
def Ici (a : α) := {x | a ≤ x}
/-- Left-open right-infinite interval -/
def Ioi (a : α) := {x | a < x}
@[simp] lemma mem_Ioo : x ∈ Ioo a b ↔ a < x ∧ x < b := iff.rfl
@[simp] lemma mem_Ico : x ∈ Ico a b ↔ a ≤ x ∧ x < b := iff.rfl
@[simp] lemma mem_Iio : x ∈ Iio b ↔ x < b := iff.rfl
@[simp] lemma mem_Icc : x ∈ Icc a b ↔ a ≤ x ∧ x ≤ b := iff.rfl
@[simp] lemma mem_Iic : x ∈ Iic b ↔ x ≤ b := iff.rfl
@[simp] lemma mem_Ioc : x ∈ Ioc a b ↔ a < x ∧ x ≤ b := iff.rfl
@[simp] lemma mem_Ici : x ∈ Ici a ↔ a ≤ x := iff.rfl
@[simp] lemma mem_Ioi : x ∈ Ioi a ↔ a < x := iff.rfl
@[simp] lemma left_mem_Ioo : a ∈ Ioo a b ↔ false := by simp [lt_irrefl]
@[simp] lemma left_mem_Ico : a ∈ Ico a b ↔ a < b := by simp [le_refl]
@[simp] lemma left_mem_Icc : a ∈ Icc a b ↔ a ≤ b := by simp [le_refl]
@[simp] lemma left_mem_Ioc : a ∈ Ioc a b ↔ false := by simp [lt_irrefl]
lemma left_mem_Ici : a ∈ Ici a := by simp
@[simp] lemma right_mem_Ioo : b ∈ Ioo a b ↔ false := by simp [lt_irrefl]
@[simp] lemma right_mem_Ico : b ∈ Ico a b ↔ false := by simp [lt_irrefl]
@[simp] lemma right_mem_Icc : b ∈ Icc a b ↔ a ≤ b := by simp [le_refl]
@[simp] lemma right_mem_Ioc : b ∈ Ioc a b ↔ a < b := by simp [le_refl]
lemma right_mem_Iic : a ∈ Iic a := by simp
@[simp] lemma dual_Ici : @Ici (order_dual α) _ a = @Iic α _ a := rfl
@[simp] lemma dual_Iic : @Iic (order_dual α) _ a = @Ici α _ a := rfl
@[simp] lemma dual_Ioi : @Ioi (order_dual α) _ a = @Iio α _ a := rfl
@[simp] lemma dual_Iio : @Iio (order_dual α) _ a = @Ioi α _ a := rfl
@[simp] lemma dual_Icc : @Icc (order_dual α) _ a b = @Icc α _ b a :=
set.ext $ λ x, and_comm _ _
@[simp] lemma dual_Ioc : @Ioc (order_dual α) _ a b = @Ico α _ b a :=
set.ext $ λ x, and_comm _ _
@[simp] lemma dual_Ico : @Ico (order_dual α) _ a b = @Ioc α _ b a :=
set.ext $ λ x, and_comm _ _
@[simp] lemma dual_Ioo : @Ioo (order_dual α) _ a b = @Ioo α _ b a :=
set.ext $ λ x, and_comm _ _
@[simp] lemma nonempty_Icc : (Icc a b).nonempty ↔ a ≤ b :=
⟨λ ⟨x, hx⟩, le_trans hx.1 hx.2, λ h, ⟨a, left_mem_Icc.2 h⟩⟩
@[simp] lemma nonempty_Ico : (Ico a b).nonempty ↔ a < b :=
⟨λ ⟨x, hx⟩, lt_of_le_of_lt hx.1 hx.2, λ h, ⟨a, left_mem_Ico.2 h⟩⟩
@[simp] lemma nonempty_Ioc : (Ioc a b).nonempty ↔ a < b :=
⟨λ ⟨x, hx⟩, lt_of_lt_of_le hx.1 hx.2, λ h, ⟨b, right_mem_Ioc.2 h⟩⟩
@[simp] lemma nonempty_Ici : (Ici a).nonempty := ⟨a, left_mem_Ici⟩
@[simp] lemma nonempty_Iic : (Iic a).nonempty := ⟨a, right_mem_Iic⟩
@[simp] lemma nonempty_Ioo [densely_ordered α] : (Ioo a b).nonempty ↔ a < b :=
⟨λ ⟨x, ha, hb⟩, lt_trans ha hb, dense⟩
@[simp] lemma nonempty_Ioi [no_top_order α] : (Ioi a).nonempty := no_top a
@[simp] lemma nonempty_Iio [no_bot_order α] : (Iio a).nonempty := no_bot a
@[simp] lemma Ioo_eq_empty (h : b ≤ a) : Ioo a b = ∅ :=
eq_empty_iff_forall_not_mem.2 $ λ x ⟨h₁, h₂⟩, not_le_of_lt (lt_trans h₁ h₂) h
@[simp] lemma Ico_eq_empty (h : b ≤ a) : Ico a b = ∅ :=
eq_empty_iff_forall_not_mem.2 $ λ x ⟨h₁, h₂⟩, not_le_of_lt (lt_of_le_of_lt h₁ h₂) h
@[simp] lemma Icc_eq_empty (h : b < a) : Icc a b = ∅ :=
eq_empty_iff_forall_not_mem.2 $ λ x ⟨h₁, h₂⟩, not_lt_of_le (le_trans h₁ h₂) h
@[simp] lemma Ioc_eq_empty (h : b ≤ a) : Ioc a b = ∅ :=
eq_empty_iff_forall_not_mem.2 $ λ x ⟨h₁, h₂⟩, not_lt_of_le (le_trans h₂ h) h₁
@[simp] lemma Ioo_self (a : α) : Ioo a a = ∅ := Ioo_eq_empty $ le_refl _
@[simp] lemma Ico_self (a : α) : Ico a a = ∅ := Ico_eq_empty $ le_refl _
@[simp] lemma Ioc_self (a : α) : Ioc a a = ∅ := Ioc_eq_empty $ le_refl _
lemma Ici_subset_Ici : Ici a ⊆ Ici b ↔ b ≤ a :=
⟨λ h, h $ left_mem_Ici, λ h x hx, le_trans h hx⟩
lemma Iic_subset_Iic : Iic a ⊆ Iic b ↔ a ≤ b :=
@Ici_subset_Ici (order_dual α) _ _ _
lemma Ici_subset_Ioi : Ici a ⊆ Ioi b ↔ b < a :=
⟨λ h, h left_mem_Ici, λ h x hx, lt_of_lt_of_le h hx⟩
lemma Iic_subset_Iio : Iic a ⊆ Iio b ↔ a < b :=
⟨λ h, h right_mem_Iic, λ h x hx, lt_of_le_of_lt hx h⟩
lemma Ioo_subset_Ioo (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂) :
Ioo a₁ b₁ ⊆ Ioo a₂ b₂ :=
λ x ⟨hx₁, hx₂⟩, ⟨lt_of_le_of_lt h₁ hx₁, lt_of_lt_of_le hx₂ h₂⟩
lemma Ioo_subset_Ioo_left (h : a₁ ≤ a₂) : Ioo a₂ b ⊆ Ioo a₁ b :=
Ioo_subset_Ioo h (le_refl _)
lemma Ioo_subset_Ioo_right (h : b₁ ≤ b₂) : Ioo a b₁ ⊆ Ioo a b₂ :=
Ioo_subset_Ioo (le_refl _) h
lemma Ico_subset_Ico (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂) :
Ico a₁ b₁ ⊆ Ico a₂ b₂ :=
λ x ⟨hx₁, hx₂⟩, ⟨le_trans h₁ hx₁, lt_of_lt_of_le hx₂ h₂⟩
lemma Ico_subset_Ico_left (h : a₁ ≤ a₂) : Ico a₂ b ⊆ Ico a₁ b :=
Ico_subset_Ico h (le_refl _)
lemma Ico_subset_Ico_right (h : b₁ ≤ b₂) : Ico a b₁ ⊆ Ico a b₂ :=
Ico_subset_Ico (le_refl _) h
lemma Icc_subset_Icc (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂) :
Icc a₁ b₁ ⊆ Icc a₂ b₂ :=
λ x ⟨hx₁, hx₂⟩, ⟨le_trans h₁ hx₁, le_trans hx₂ h₂⟩
lemma Icc_subset_Icc_left (h : a₁ ≤ a₂) : Icc a₂ b ⊆ Icc a₁ b :=
Icc_subset_Icc h (le_refl _)
lemma Icc_subset_Icc_right (h : b₁ ≤ b₂) : Icc a b₁ ⊆ Icc a b₂ :=
Icc_subset_Icc (le_refl _) h
lemma Ioc_subset_Ioc (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂) :
Ioc a₁ b₁ ⊆ Ioc a₂ b₂ :=
λ x ⟨hx₁, hx₂⟩, ⟨lt_of_le_of_lt h₁ hx₁, le_trans hx₂ h₂⟩
lemma Ioc_subset_Ioc_left (h : a₁ ≤ a₂) : Ioc a₂ b ⊆ Ioc a₁ b :=
Ioc_subset_Ioc h (le_refl _)
lemma Ioc_subset_Ioc_right (h : b₁ ≤ b₂) : Ioc a b₁ ⊆ Ioc a b₂ :=
Ioc_subset_Ioc (le_refl _) h
lemma Ico_subset_Ioo_left (h₁ : a₁ < a₂) : Ico a₂ b ⊆ Ioo a₁ b :=
λ x, and.imp_left $ lt_of_lt_of_le h₁
lemma Icc_subset_Ico_right (h₁ : b₁ < b₂) : Icc a b₁ ⊆ Ico a b₂ :=
λ x, and.imp_right $ λ h₂, lt_of_le_of_lt h₂ h₁
lemma Ioo_subset_Ico_self : Ioo a b ⊆ Ico a b := λ x, and.imp_left le_of_lt
lemma Ioo_subset_Ioc_self : Ioo a b ⊆ Ioc a b := λ x, and.imp_right le_of_lt
lemma Ico_subset_Icc_self : Ico a b ⊆ Icc a b := λ x, and.imp_right le_of_lt
lemma Ioc_subset_Icc_self : Ioc a b ⊆ Icc a b := λ x, and.imp_left le_of_lt
lemma Ioo_subset_Icc_self : Ioo a b ⊆ Icc a b :=
subset.trans Ioo_subset_Ico_self Ico_subset_Icc_self
lemma Ico_subset_Iio_self : Ico a b ⊆ Iio b := λ x, and.right
lemma Ioo_subset_Iio_self : Ioo a b ⊆ Iio b := λ x, and.right
lemma Ioc_subset_Ioi_self : Ioc a b ⊆ Ioi a := λ x, and.left
lemma Ioo_subset_Ioi_self : Ioo a b ⊆ Ioi a := λ x, and.left
lemma Ioi_subset_Ici_self : Ioi a ⊆ Ici a := λx hx, le_of_lt hx
lemma Iio_subset_Iic_self : Iio a ⊆ Iic a := λx hx, le_of_lt hx
lemma Icc_subset_Icc_iff (h₁ : a₁ ≤ b₁) :
Icc a₁ b₁ ⊆ Icc a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ ≤ b₂ :=
⟨λ h, ⟨(h ⟨le_refl _, h₁⟩).1, (h ⟨h₁, le_refl _⟩).2⟩,
λ ⟨h, h'⟩ x ⟨hx, hx'⟩, ⟨le_trans h hx, le_trans hx' h'⟩⟩
lemma Icc_subset_Ioo_iff (h₁ : a₁ ≤ b₁) :
Icc a₁ b₁ ⊆ Ioo a₂ b₂ ↔ a₂ < a₁ ∧ b₁ < b₂ :=
⟨λ h, ⟨(h ⟨le_refl _, h₁⟩).1, (h ⟨h₁, le_refl _⟩).2⟩,
λ ⟨h, h'⟩ x ⟨hx, hx'⟩, ⟨lt_of_lt_of_le h hx, lt_of_le_of_lt hx' h'⟩⟩
lemma Icc_subset_Ico_iff (h₁ : a₁ ≤ b₁) :
Icc a₁ b₁ ⊆ Ico a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ < b₂ :=
⟨λ h, ⟨(h ⟨le_refl _, h₁⟩).1, (h ⟨h₁, le_refl _⟩).2⟩,
λ ⟨h, h'⟩ x ⟨hx, hx'⟩, ⟨le_trans h hx, lt_of_le_of_lt hx' h'⟩⟩
lemma Icc_subset_Ioc_iff (h₁ : a₁ ≤ b₁) :
Icc a₁ b₁ ⊆ Ioc a₂ b₂ ↔ a₂ < a₁ ∧ b₁ ≤ b₂ :=
⟨λ h, ⟨(h ⟨le_refl _, h₁⟩).1, (h ⟨h₁, le_refl _⟩).2⟩,
λ ⟨h, h'⟩ x ⟨hx, hx'⟩, ⟨lt_of_lt_of_le h hx, le_trans hx' h'⟩⟩
lemma Icc_subset_Iio_iff (h₁ : a₁ ≤ b₁) :
Icc a₁ b₁ ⊆ Iio b₂ ↔ b₁ < b₂ :=
⟨λ h, h ⟨h₁, le_refl _⟩, λ h x ⟨hx, hx'⟩, lt_of_le_of_lt hx' h⟩
lemma Icc_subset_Ioi_iff (h₁ : a₁ ≤ b₁) :
Icc a₁ b₁ ⊆ Ioi a₂ ↔ a₂ < a₁ :=
⟨λ h, h ⟨le_refl _, h₁⟩, λ h x ⟨hx, hx'⟩, lt_of_lt_of_le h hx⟩
lemma Icc_subset_Iic_iff (h₁ : a₁ ≤ b₁) :
Icc a₁ b₁ ⊆ Iic b₂ ↔ b₁ ≤ b₂ :=
⟨λ h, h ⟨h₁, le_refl _⟩, λ h x ⟨hx, hx'⟩, le_trans hx' h⟩
lemma Icc_subset_Ici_iff (h₁ : a₁ ≤ b₁) :
Icc a₁ b₁ ⊆ Ici a₂ ↔ a₂ ≤ a₁ :=
⟨λ h, h ⟨le_refl _, h₁⟩, λ h x ⟨hx, hx'⟩, le_trans h hx⟩
/-- If `a ≤ b`, then `(b, +∞) ⊆ (a, +∞)`. In preorders, this is just an implication. If you need
the equivalence in linear orders, use `Ioi_subset_Ioi_iff`. -/
lemma Ioi_subset_Ioi (h : a ≤ b) : Ioi b ⊆ Ioi a :=
λx hx, lt_of_le_of_lt h hx
/-- If `a ≤ b`, then `(b, +∞) ⊆ [a, +∞)`. In preorders, this is just an implication. If you need
the equivalence in dense linear orders, use `Ioi_subset_Ici_iff`. -/
lemma Ioi_subset_Ici (h : a ≤ b) : Ioi b ⊆ Ici a :=
subset.trans (Ioi_subset_Ioi h) Ioi_subset_Ici_self
/-- If `a ≤ b`, then `(-∞, a) ⊆ (-∞, b)`. In preorders, this is just an implication. If you need
the equivalence in linear orders, use `Iio_subset_Iio_iff`. -/
lemma Iio_subset_Iio (h : a ≤ b) : Iio a ⊆ Iio b :=
λx hx, lt_of_lt_of_le hx h
/-- If `a ≤ b`, then `(-∞, a) ⊆ (-∞, b]`. In preorders, this is just an implication. If you need
the equivalence in dense linear orders, use `Iio_subset_Iic_iff`. -/
lemma Iio_subset_Iic (h : a ≤ b) : Iio a ⊆ Iic b :=
subset.trans (Iio_subset_Iio h) Iio_subset_Iic_self
lemma Ici_inter_Iic : Ici a ∩ Iic b = Icc a b := rfl
lemma Ici_inter_Iio : Ici a ∩ Iio b = Ico a b := rfl
lemma Ioi_inter_Iic : Ioi a ∩ Iic b = Ioc a b := rfl
lemma Ioi_inter_Iio : Ioi a ∩ Iio b = Ioo a b := rfl
end intervals
section partial_order
variables {α : Type u} [partial_order α] {a b : α}
@[simp] lemma Icc_self (a : α) : Icc a a = {a} :=
set.ext $ by simp [Icc, le_antisymm_iff, and_comm]
lemma Ico_diff_Ioo_eq_singleton (h : a < b) : Ico a b \ Ioo a b = {a} :=
set.ext $ λ x, begin
simp [not_and'], split,
{ rintro ⟨⟨ax, xb⟩, hne⟩,
exact (eq_or_lt_of_le ax).elim eq.symm (λ h', absurd h' (hne xb)) },
{ rintro rfl, exact ⟨⟨le_refl _, h⟩, λ _, lt_irrefl x⟩ }
end
lemma Ioc_diff_Ioo_eq_singleton (h : a < b) : Ioc a b \ Ioo a b = {b} :=
set.ext $ λ x, begin
simp, split,
{ rintro ⟨⟨ax, xb⟩, hne⟩,
exact (eq_or_lt_of_le xb).elim id (λ h', absurd h' (hne ax)) },
{ rintro rfl, exact ⟨⟨h, le_refl _⟩, λ _, lt_irrefl x⟩ }
end
lemma Icc_diff_Ico_eq_singleton (h : a ≤ b) : Icc a b \ Ico a b = {b} :=
set.ext $ λ x, begin
simp, split,
{ rintro ⟨⟨ax, xb⟩, h⟩,
exact classical.by_contradiction
(λ ne, h ax (lt_of_le_of_ne xb ne)) },
{ rintro rfl, exact ⟨⟨h, le_refl _⟩, λ _, lt_irrefl x⟩ }
end
lemma Icc_diff_Ioc_eq_singleton (h : a ≤ b) : Icc a b \ Ioc a b = {a} :=
set.ext $ λ x, begin
simp [not_and'], split,
{ rintro ⟨⟨ax, xb⟩, h⟩,
exact classical.by_contradiction
(λ hne, h xb (lt_of_le_of_ne ax (ne.symm hne))) },
{ rintro rfl, exact ⟨⟨le_refl _, h⟩, λ _, lt_irrefl x⟩ }
end
end partial_order
section linear_order
variables {α : Type u} [linear_order α] {a a₁ a₂ b b₁ b₂ : α}
lemma compl_Iic : -(Iic a) = Ioi a := ext $ λ _, not_le
lemma compl_Ici : -(Ici a) = Iio a := ext $ λ _, not_le
lemma compl_Iio : -(Iio a) = Ici a := ext $ λ _, not_lt
lemma compl_Ioi : -(Ioi a) = Iic a := ext $ λ _, not_lt
lemma Ioo_eq_empty_iff [densely_ordered α] : Ioo a b = ∅ ↔ b ≤ a :=
⟨λ eq, le_of_not_lt $ λ h,
let ⟨x, h₁, h₂⟩ := dense h in
eq_empty_iff_forall_not_mem.1 eq x ⟨h₁, h₂⟩,
Ioo_eq_empty⟩
lemma Ico_eq_empty_iff : Ico a b = ∅ ↔ b ≤ a :=
⟨λ eq, le_of_not_lt $ λ h, eq_empty_iff_forall_not_mem.1 eq a ⟨le_refl _, h⟩,
Ico_eq_empty⟩
lemma Icc_eq_empty_iff : Icc a b = ∅ ↔ b < a :=
⟨λ eq, lt_of_not_ge $ λ h, eq_empty_iff_forall_not_mem.1 eq a ⟨le_refl _, h⟩,
Icc_eq_empty⟩
lemma Ico_subset_Ico_iff (h₁ : a₁ < b₁) :
Ico a₁ b₁ ⊆ Ico a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ ≤ b₂ :=
⟨λ h, have a₂ ≤ a₁ ∧ a₁ < b₂ := h ⟨le_refl _, h₁⟩,
⟨this.1, le_of_not_lt $ λ h', lt_irrefl b₂ (h ⟨le_of_lt this.2, h'⟩).2⟩,
λ ⟨h₁, h₂⟩, Ico_subset_Ico h₁ h₂⟩
lemma Ioo_subset_Ioo_iff [densely_ordered α] (h₁ : a₁ < b₁) :
Ioo a₁ b₁ ⊆ Ioo a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ ≤ b₂ :=
⟨λ h, begin
rcases dense h₁ with ⟨x, xa, xb⟩,
split; refine le_of_not_lt (λ h', _),
{ have ab := lt_trans (h ⟨xa, xb⟩).1 xb,
exact lt_irrefl _ (h ⟨h', ab⟩).1 },
{ have ab := lt_trans xa (h ⟨xa, xb⟩).2,
exact lt_irrefl _ (h ⟨ab, h'⟩).2 }
end, λ ⟨h₁, h₂⟩, Ioo_subset_Ioo h₁ h₂⟩
lemma Ico_eq_Ico_iff (h : a₁ < b₁ ∨ a₂ < b₂) : Ico a₁ b₁ = Ico a₂ b₂ ↔ a₁ = a₂ ∧ b₁ = b₂ :=
⟨λ e, begin
simp [subset.antisymm_iff] at e, simp [le_antisymm_iff],
cases h; simp [Ico_subset_Ico_iff h] at e;
[ rcases e with ⟨⟨h₁, h₂⟩, e'⟩, rcases e with ⟨e', ⟨h₁, h₂⟩⟩ ];
have := (Ico_subset_Ico_iff (lt_of_le_of_lt h₁ $ lt_of_lt_of_le h h₂)).1 e';
tauto
end, λ ⟨h₁, h₂⟩, by rw [h₁, h₂]⟩
open_locale classical
@[simp] lemma Ioi_subset_Ioi_iff : Ioi b ⊆ Ioi a ↔ a ≤ b :=
begin
refine ⟨λh, _, λh, Ioi_subset_Ioi h⟩,
by_contradiction ba,
exact lt_irrefl _ (h (not_le.mp ba))
end
@[simp] lemma Ioi_subset_Ici_iff [densely_ordered α] : Ioi b ⊆ Ici a ↔ a ≤ b :=
begin
refine ⟨λh, _, λh, Ioi_subset_Ici h⟩,
by_contradiction ba,
obtain ⟨c, bc, ca⟩ : ∃c, b < c ∧ c < a := dense (not_le.mp ba),
exact lt_irrefl _ (lt_of_lt_of_le ca (h bc))
end
@[simp] lemma Iio_subset_Iio_iff : Iio a ⊆ Iio b ↔ a ≤ b :=
begin
refine ⟨λh, _, λh, Iio_subset_Iio h⟩,
by_contradiction ab,
exact lt_irrefl _ (h (not_le.mp ab))
end
@[simp] lemma Iio_subset_Iic_iff [densely_ordered α] : Iio a ⊆ Iic b ↔ a ≤ b :=
begin
refine ⟨λh, _, λh, Iio_subset_Iic h⟩,
by_contradiction ba,
obtain ⟨c, bc, ca⟩ : ∃c, b < c ∧ c < a := dense (not_le.mp ba),
exact lt_irrefl _ (lt_of_lt_of_le bc (h ca))
end
/-! ### Unions of adjacent intervals -/
/-! #### Two infinite intervals -/
@[simp] lemma Iic_union_Ici : Iic a ∪ Ici a = univ := eq_univ_of_forall (λ x, le_total x a)
@[simp] lemma Iio_union_Ici : Iio a ∪ Ici a = univ := eq_univ_of_forall (λ x, lt_or_le x a)
@[simp] lemma Iic_union_Ioi : Iic a ∪ Ioi a = univ := eq_univ_of_forall (λ x, le_or_lt x a)
/-! #### A finite and an infinite interval -/
@[simp] lemma Ioc_union_Ici_eq_Ioi (h : a < b) : Ioc a b ∪ Ici b = Ioi a :=
ext $ λ x, ⟨λ hx, hx.elim and.left (lt_of_lt_of_le h),
λ hx, (le_total x b).elim (λ hxb, or.inl ⟨hx, hxb⟩) (λ hxb, or.inr hxb)⟩
@[simp] lemma Icc_union_Ici_eq_Ioi (h : a ≤ b) : Icc a b ∪ Ici b = Ici a :=
ext $ λ x, ⟨λ hx, hx.elim and.left (le_trans h),
λ hx, (le_total x b).elim (λ hxb, or.inl ⟨hx, hxb⟩) (λ hxb, or.inr hxb)⟩
@[simp] lemma Ioo_union_Ici_eq_Ioi (h : a < b) : Ioo a b ∪ Ici b = Ioi a :=
ext $ λ x, ⟨λ hx, hx.elim and.left (lt_of_lt_of_le h),
λ hx, (lt_or_le x b).elim (λ hxb, or.inl ⟨hx, hxb⟩) (λ hxb, or.inr hxb)⟩
@[simp] lemma Ico_union_Ici_eq_Ioi (h : a ≤ b) : Ico a b ∪ Ici b = Ici a :=
ext $ λ x, ⟨λ hx, hx.elim and.left (le_trans h),
λ hx, (lt_or_le x b).elim (λ hxb, or.inl ⟨hx, hxb⟩) (λ hxb, or.inr hxb)⟩
@[simp] lemma Ioc_union_Ioi_eq_Ioi (h : a ≤ b) : Ioc a b ∪ Ioi b = Ioi a :=
ext $ λ x, ⟨λ hx, hx.elim and.left (lt_of_le_of_lt h),
λ hx, (le_or_lt x b).elim (λ hxb, or.inl ⟨hx, hxb⟩) (λ hxb, or.inr hxb)⟩
@[simp] lemma Icc_union_Ioi_eq_Ioi (h : a ≤ b) : Icc a b ∪ Ioi b = Ici a :=
ext $ λ x, ⟨λ hx, hx.elim and.left (λ hx, le_trans h (le_of_lt hx)),
λ hx, (le_or_lt x b).elim (λ hxb, or.inl ⟨hx, hxb⟩) (λ hxb, or.inr hxb)⟩
/-! #### An infinite and a finite interval -/
@[simp] lemma Iic_union_Icc_eq_Iic (h : a ≤ b) : Iic a ∪ Icc a b = Iic b :=
ext $ λ x, ⟨λ hx, hx.elim (λ hx, le_trans hx h) and.right,
λ hx, (le_total x a).elim (λ hxa, or.inl hxa) (λ hxa, or.inr ⟨hxa, hx⟩)⟩
@[simp] lemma Iic_union_Ico_eq_Iio (h : a < b) : Iic a ∪ Ico a b = Iio b :=
ext $ λ x, ⟨λ hx, hx.elim (λ hx, lt_of_le_of_lt hx h) and.right,
λ hx, (le_total x a).elim (λ hxa, or.inl hxa) (λ hxa, or.inr ⟨hxa, hx⟩)⟩
@[simp] lemma Iio_union_Icc_eq_Iic (h : a ≤ b) : Iio a ∪ Icc a b = Iic b :=
ext $ λ x, ⟨λ hx, hx.elim (λ hx, le_trans (le_of_lt hx) h) and.right,
λ hx, (lt_or_le x a).elim (λ hxa, or.inl hxa) (λ hxa, or.inr ⟨hxa, hx⟩)⟩
@[simp] lemma Iio_union_Ico_eq_Iio (h : a ≤ b) : Iio a ∪ Ico a b = Iio b :=
ext $ λ x, ⟨λ hx, hx.elim (λ hx, lt_of_lt_of_le hx h) and.right,
λ hx, (lt_or_le x a).elim (λ hxa, or.inl hxa) (λ hxa, or.inr ⟨hxa, hx⟩)⟩
@[simp] lemma Iic_union_Ioc_eq_Iic (h : a ≤ b) : Iic a ∪ Ioc a b = Iic b :=
ext $ λ x, ⟨λ hx, hx.elim (λ hx, le_trans hx h) and.right,
λ hx, (le_or_lt x a).elim (λ hxa, or.inl hxa) (λ hxa, or.inr ⟨hxa, hx⟩)⟩
@[simp] lemma Iic_union_Ioo_eq_Iio (h : a < b) : Iic a ∪ Ioo a b = Iio b :=
ext $ λ x, ⟨λ hx, hx.elim (λ hx, lt_of_le_of_lt hx h) and.right,
λ hx, (le_or_lt x a).elim (λ hxa, or.inl hxa) (λ hxa, or.inr ⟨hxa, hx⟩)⟩
/-! #### Two finite intervals with a common point -/
@[simp] lemma Ioc_union_Ico_eq_Ioo {c} (h₁ : a < b) (h₂ : b < c) : Ioc a b ∪ Ico b c = Ioo a c :=
ext $ λ x,
⟨λ hx, hx.elim (λ hx, ⟨hx.1, lt_of_le_of_lt hx.2 h₂⟩) (λ hx, ⟨lt_of_lt_of_le h₁ hx.1, hx.2⟩),
λ hx, (le_total x b).elim (λ hxb, or.inl ⟨hx.1, hxb⟩) (λ hxb, or.inr ⟨hxb, hx.2⟩)⟩
@[simp] lemma Icc_union_Ico_eq_Ico {c} (h₁ : a ≤ b) (h₂ : b < c) : Icc a b ∪ Ico b c = Ico a c :=
ext $ λ x,
⟨λ hx, hx.elim (λ hx, ⟨hx.1, lt_of_le_of_lt hx.2 h₂⟩) (λ hx, ⟨le_trans h₁ hx.1, hx.2⟩),
λ hx, (le_total x b).elim (λ hxb, or.inl ⟨hx.1, hxb⟩) (λ hxb, or.inr ⟨hxb, hx.2⟩)⟩
@[simp] lemma Icc_union_Icc_eq_Icc {c} (h₁ : a ≤ b) (h₂ : b ≤ c) : Icc a b ∪ Icc b c = Icc a c :=
ext $ λ x,
⟨λ hx, hx.elim (λ hx, ⟨hx.1, le_trans hx.2 h₂⟩) (λ hx, ⟨le_trans h₁ hx.1, hx.2⟩),
λ hx, (le_total x b).elim (λ hxb, or.inl ⟨hx.1, hxb⟩) (λ hxb, or.inr ⟨hxb, hx.2⟩)⟩
@[simp] lemma Ioc_union_Icc_eq_Ioc {c} (h₁ : a < b) (h₂ : b ≤ c) : Ioc a b ∪ Icc b c = Ioc a c :=
ext $ λ x,
⟨λ hx, hx.elim (λ hx, ⟨hx.1, le_trans hx.2 h₂⟩) (λ hx, ⟨lt_of_lt_of_le h₁ hx.1, hx.2⟩),
λ hx, (le_total x b).elim (λ hxb, or.inl ⟨hx.1, hxb⟩) (λ hxb, or.inr ⟨hxb, hx.2⟩)⟩
/-! #### Two finite intervals, `I?o` and `Ic?` -/
@[simp] lemma Ioo_union_Ico_eq_Ioo {c} (h₁ : a < b) (h₂ : b ≤ c) : Ioo a b ∪ Ico b c = Ioo a c :=
ext $ λ x,
⟨λ hx, hx.elim (λ hx, ⟨hx.1, lt_of_lt_of_le hx.2 h₂⟩) (λ hx, ⟨lt_of_lt_of_le h₁ hx.1, hx.2⟩),
λ hx, (lt_or_le x b).elim (λ hxb, or.inl ⟨hx.1, hxb⟩) (λ hxb, or.inr ⟨hxb, hx.2⟩)⟩
@[simp] lemma Ico_union_Ico_eq_Ico {c} (h₁ : a ≤ b) (h₂ : b ≤ c) : Ico a b ∪ Ico b c = Ico a c :=
ext $ λ x,
⟨λ hx, hx.elim (λ hx, ⟨hx.1, lt_of_lt_of_le hx.2 h₂⟩) (λ hx, ⟨le_trans h₁ hx.1, hx.2⟩),
λ hx, (lt_or_le x b).elim (λ hxb, or.inl ⟨hx.1, hxb⟩) (λ hxb, or.inr ⟨hxb, hx.2⟩)⟩
@[simp] lemma Ico_union_Icc_eq_Icc {c} (h₁ : a ≤ b) (h₂ : b ≤ c) : Ico a b ∪ Icc b c = Icc a c :=
ext $ λ x,
⟨λ hx, hx.elim (λ hx, ⟨hx.1, le_trans (le_of_lt hx.2) h₂⟩) (λ hx, ⟨le_trans h₁ hx.1, hx.2⟩),
λ hx, (lt_or_le x b).elim (λ hxb, or.inl ⟨hx.1, hxb⟩) (λ hxb, or.inr ⟨hxb, hx.2⟩)⟩
@[simp] lemma Ioo_union_Icc_eq_Ioc {c} (h₁ : a < b) (h₂ : b ≤ c) : Ioo a b ∪ Icc b c = Ioc a c :=
ext $ λ x,
⟨λ hx, hx.elim (λ hx, ⟨hx.1, le_trans (le_of_lt hx.2) h₂⟩) (λ hx, ⟨lt_of_lt_of_le h₁ hx.1, hx.2⟩),
λ hx, (lt_or_le x b).elim (λ hxb, or.inl ⟨hx.1, hxb⟩) (λ hxb, or.inr ⟨hxb, hx.2⟩)⟩
/-! #### Two finite intervals, `I?c` and `Io?` -/
@[simp] lemma Ioc_union_Ioo_eq_Ioo {c} (h₁ : a ≤ b) (h₂ : b < c) : Ioc a b ∪ Ioo b c = Ioo a c :=
ext $ λ x,
⟨λ hx, hx.elim (λ hx, ⟨hx.1, lt_of_le_of_lt hx.2 h₂⟩) (λ hx, ⟨lt_of_le_of_lt h₁ hx.1, hx.2⟩),
λ hx, (le_or_lt x b).elim (λ hxb, or.inl ⟨hx.1, hxb⟩) (λ hxb, or.inr ⟨hxb, hx.2⟩)⟩
@[simp] lemma Icc_union_Ioo_eq_Ico {c} (h₁ : a ≤ b) (h₂ : b < c) : Icc a b ∪ Ioo b c = Ico a c :=
ext $ λ x,
⟨λ hx, hx.elim (λ hx, ⟨hx.1, lt_of_le_of_lt hx.2 h₂⟩) (λ hx, ⟨le_trans h₁ (le_of_lt hx.1), hx.2⟩),
λ hx, (le_or_lt x b).elim (λ hxb, or.inl ⟨hx.1, hxb⟩) (λ hxb, or.inr ⟨hxb, hx.2⟩)⟩
@[simp] lemma Icc_union_Ioc_eq_Icc {c} (h₁ : a ≤ b) (h₂ : b ≤ c) : Icc a b ∪ Ioc b c = Icc a c :=
ext $ λ x,
⟨λ hx, hx.elim (λ hx, ⟨hx.1, le_trans hx.2 h₂⟩) (λ hx, ⟨le_trans h₁ (le_of_lt hx.1), hx.2⟩),
λ hx, (le_or_lt x b).elim (λ hxb, or.inl ⟨hx.1, hxb⟩) (λ hxb, or.inr ⟨hxb, hx.2⟩)⟩
@[simp] lemma Ioc_union_Ioc_eq_Ioc {c} (h₁ : a ≤ b) (h₂ : b ≤ c) : Ioc a b ∪ Ioc b c = Ioc a c :=
ext $ λ x,
⟨λ hx, hx.elim (λ hx, ⟨hx.1, le_trans hx.2 h₂⟩) (λ hx, ⟨lt_of_le_of_lt h₁ hx.1, hx.2⟩),
λ hx, (le_or_lt x b).elim (λ hxb, or.inl ⟨hx.1, hxb⟩) (λ hxb, or.inr ⟨hxb, hx.2⟩)⟩
end linear_order
section lattice
section inf
variables {α : Type u} [semilattice_inf α]
@[simp] lemma Iic_inter_Iic {a b : α} : Iic a ∩ Iic b = Iic (a ⊓ b) :=
by { ext x, simp [Iic] }
@[simp] lemma Iio_inter_Iio [is_total α (≤)] {a b : α} : Iio a ∩ Iio b = Iio (a ⊓ b) :=
by { ext x, simp [Iio] }
end inf
section sup
variables {α : Type u} [semilattice_sup α]
@[simp] lemma Ici_inter_Ici {a b : α} : Ici a ∩ Ici b = Ici (a ⊔ b) :=
by { ext x, simp [Ici] }
@[simp] lemma Ioi_inter_Ioi [is_total α (≤)] {a b : α} : Ioi a ∩ Ioi b = Ioi (a ⊔ b) :=
by { ext x, simp [Ioi] }
end sup
section both
variables {α : Type u} [lattice α] [ht : is_total α (≤)] {a b c a₁ a₂ b₁ b₂ : α}
lemma Icc_inter_Icc : Icc a₁ b₁ ∩ Icc a₂ b₂ = Icc (a₁ ⊔ a₂) (b₁ ⊓ b₂) :=
by simp only [Ici_inter_Iic.symm, Ici_inter_Ici.symm, Iic_inter_Iic.symm]; ac_refl
@[simp] lemma Icc_inter_Icc_eq_singleton (hab : a ≤ b) (hbc : b ≤ c) :
Icc a b ∩ Icc b c = {b} :=
by rw [Icc_inter_Icc, sup_of_le_right hab, inf_of_le_left hbc, Icc_self]
include ht
lemma Ico_inter_Ico : Ico a₁ b₁ ∩ Ico a₂ b₂ = Ico (a₁ ⊔ a₂) (b₁ ⊓ b₂) :=
by simp only [Ici_inter_Iio.symm, Ici_inter_Ici.symm, Iio_inter_Iio.symm]; ac_refl
lemma Ioc_inter_Ioc : Ioc a₁ b₁ ∩ Ioc a₂ b₂ = Ioc (a₁ ⊔ a₂) (b₁ ⊓ b₂) :=
by simp only [Ioi_inter_Iic.symm, Ioi_inter_Ioi.symm, Iic_inter_Iic.symm]; ac_refl
lemma Ioo_inter_Ioo : Ioo a₁ b₁ ∩ Ioo a₂ b₂ = Ioo (a₁ ⊔ a₂) (b₁ ⊓ b₂) :=
by simp only [Ioi_inter_Iio.symm, Ioi_inter_Ioi.symm, Iio_inter_Iio.symm]; ac_refl
end both
end lattice
section decidable_linear_order
variables {α : Type u} [decidable_linear_order α] {a a₁ a₂ b b₁ b₂ : α}
@[simp] lemma Ico_diff_Iio {a b c : α} : Ico a b \ Iio c = Ico (max a c) b :=
set.ext $ by simp [Ico, Iio, iff_def, max_le_iff] {contextual:=tt}
@[simp] lemma Ico_inter_Iio {a b c : α} : Ico a b ∩ Iio c = Ico a (min b c) :=
set.ext $ by simp [Ico, Iio, iff_def, lt_min_iff] {contextual:=tt}
end decidable_linear_order
section ordered_add_comm_group
variables {α : Type u} [ordered_add_comm_group α]
lemma image_add_left_Icc (a b c : α) : ((+) a) '' Icc b c = Icc (a + b) (a + c) :=
begin
ext x,
split,
{ rintros ⟨x, hx, rfl⟩,
exact ⟨add_le_add_left hx.1 a, add_le_add_left hx.2 a⟩},
{ intro hx,
refine ⟨-a + x, _, add_neg_cancel_left _ _⟩,
exact ⟨le_neg_add_iff_add_le.2 hx.1, neg_add_le_iff_le_add.2 hx.2⟩ }
end
lemma image_add_right_Icc (a b c : α) : (λ x, x + c) '' Icc a b = Icc (a + c) (b + c) :=
by convert image_add_left_Icc c a b using 1; simp only [add_comm _ c]
lemma image_neg_Iio (r : α) : image (λz, -z) (Iio r) = Ioi (-r) :=
begin
ext z,
apply iff.intro,
{ intros hz,
apply exists.elim hz,
intros z' hz',
rw [←hz'.2],
simp only [mem_Ioi, neg_lt_neg_iff],
exact hz'.1 },
{ intros hz,
simp only [mem_image, mem_Iio],
use -z,
simp [hz],
exact neg_lt.1 hz }
end
lemma image_neg_Iic (r : α) : image (λz, -z) (Iic r) = Ici (-r) :=
begin
apply set.ext,
intros z,
apply iff.intro,
{ intros hz,
apply exists.elim hz,
intros z' hz',
rw [←hz'.2],
simp only [neg_le_neg_iff, mem_Ici],
exact hz'.1 },
{ intros hz,
simp only [mem_image, mem_Iic],
use -z,
simp [hz],
exact neg_le.1 hz }
end
end ordered_add_comm_group
section decidable_linear_ordered_add_comm_group
variables {α : Type u} [decidable_linear_ordered_add_comm_group α]
/-- If we remove a smaller interval from a larger, the result is nonempty -/
lemma nonempty_Ico_sdiff {x dx y dy : α} (h : dy < dx) (hx : 0 < dx) :
nonempty ↥(Ico x (x + dx) \ Ico y (y + dy)) :=
begin
cases lt_or_le x y with h' h',
{ use x, simp* },
{ use max x (x + dy), simp [*, le_refl] }
end
end decidable_linear_ordered_add_comm_group
section linear_ordered_field
variables {α : Type u} [linear_ordered_field α]
lemma image_mul_right_Icc' (a b : α) {c : α} (h : 0 < c) :
(λ x, x * c) '' Icc a b = Icc (a * c) (b * c) :=
begin
ext x,
split,
{ rintros ⟨x, hx, rfl⟩,
exact ⟨mul_le_mul_of_nonneg_right hx.1 (le_of_lt h),
mul_le_mul_of_nonneg_right hx.2 (le_of_lt h)⟩ },
{ intro hx,
refine ⟨x / c, _, div_mul_cancel x (ne_of_gt h)⟩,
exact ⟨le_div_of_mul_le h hx.1, div_le_of_le_mul h (mul_comm b c ▸ hx.2)⟩ }
end
lemma image_mul_right_Icc {a b c : α} (hab : a ≤ b) (hc : 0 ≤ c) :
(λ x, x * c) '' Icc a b = Icc (a * c) (b * c) :=
begin
cases eq_or_lt_of_le hc,
{ subst c,
simp [(nonempty_Icc.2 hab).image_const] },
exact image_mul_right_Icc' a b ‹0 < c›
end
lemma image_mul_left_Icc' {a : α} (h : 0 < a) (b c : α) :
((*) a) '' Icc b c = Icc (a * b) (a * c) :=
by { convert image_mul_right_Icc' b c h using 1; simp only [mul_comm _ a] }
lemma image_mul_left_Icc {a b c : α} (ha : 0 ≤ a) (hbc : b ≤ c) :
((*) a) '' Icc b c = Icc (a * b) (a * c) :=
by { convert image_mul_right_Icc hbc ha using 1; simp only [mul_comm _ a] }
end linear_ordered_field
end set
|
ac4758453afce9fa64efb499ff59cc7468f998bc | bb31430994044506fa42fd667e2d556327e18dfe | /src/ring_theory/ideal/basic.lean | fd1ffa938c20c51f856cdf3e3bf92faf864fb713 | [
"Apache-2.0"
] | permissive | sgouezel/mathlib | 0cb4e5335a2ba189fa7af96d83a377f83270e503 | 00638177efd1b2534fc5269363ebf42a7871df9a | refs/heads/master | 1,674,527,483,042 | 1,673,665,568,000 | 1,673,665,568,000 | 119,598,202 | 0 | 0 | null | 1,517,348,647,000 | 1,517,348,646,000 | null | UTF-8 | Lean | false | false | 25,691 | lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro
-/
import algebra.associated
import linear_algebra.basic
import order.atoms
import order.compactly_generated
import tactic.abel
import data.nat.choose.sum
import linear_algebra.finsupp
/-!
# Ideals over a ring
This file defines `ideal R`, the type of (left) ideals over a ring `R`.
Note that over commutative rings, left ideals and two-sided ideals are equivalent.
## Implementation notes
`ideal R` is implemented using `submodule R R`, where `•` is interpreted as `*`.
## TODO
Support right ideals, and two-sided ideals over non-commutative rings.
-/
universes u v w
variables {α : Type u} {β : Type v}
open set function
open_locale classical big_operators pointwise
/-- A (left) ideal in a semiring `R` is an additive submonoid `s` such that
`a * b ∈ s` whenever `b ∈ s`. If `R` is a ring, then `s` is an additive subgroup. -/
@[reducible] def ideal (R : Type u) [semiring R] := submodule R R
section semiring
namespace ideal
variables [semiring α] (I : ideal α) {a b : α}
protected lemma zero_mem : (0 : α) ∈ I := I.zero_mem
protected lemma add_mem : a ∈ I → b ∈ I → a + b ∈ I := I.add_mem
variables (a)
lemma mul_mem_left : b ∈ I → a * b ∈ I := I.smul_mem a
variables {a}
@[ext] lemma ext {I J : ideal α} (h : ∀ x, x ∈ I ↔ x ∈ J) : I = J :=
submodule.ext h
lemma sum_mem (I : ideal α) {ι : Type*} {t : finset ι} {f : ι → α} :
(∀c∈t, f c ∈ I) → (∑ i in t, f i) ∈ I := submodule.sum_mem I
theorem eq_top_of_unit_mem
(x y : α) (hx : x ∈ I) (h : y * x = 1) : I = ⊤ :=
eq_top_iff.2 $ λ z _, calc
z = z * (y * x) : by simp [h]
... = (z * y) * x : eq.symm $ mul_assoc z y x
... ∈ I : I.mul_mem_left _ hx
theorem eq_top_of_is_unit_mem {x} (hx : x ∈ I) (h : is_unit x) : I = ⊤ :=
let ⟨y, hy⟩ := h.exists_left_inv in eq_top_of_unit_mem I x y hx hy
theorem eq_top_iff_one : I = ⊤ ↔ (1:α) ∈ I :=
⟨by rintro rfl; trivial,
λ h, eq_top_of_unit_mem _ _ 1 h (by simp)⟩
theorem ne_top_iff_one : I ≠ ⊤ ↔ (1:α) ∉ I :=
not_congr I.eq_top_iff_one
@[simp]
theorem unit_mul_mem_iff_mem {x y : α} (hy : is_unit y) : y * x ∈ I ↔ x ∈ I :=
begin
refine ⟨λ h, _, λ h, I.mul_mem_left y h⟩,
obtain ⟨y', hy'⟩ := hy.exists_left_inv,
have := I.mul_mem_left y' h,
rwa [← mul_assoc, hy', one_mul] at this,
end
/-- The ideal generated by a subset of a ring -/
def span (s : set α) : ideal α := submodule.span α s
@[simp] lemma submodule_span_eq {s : set α} :
submodule.span α s = ideal.span s :=
rfl
@[simp] lemma span_empty : span (∅ : set α) = ⊥ := submodule.span_empty
@[simp] lemma span_univ : span (set.univ : set α) = ⊤ := submodule.span_univ
lemma span_union (s t : set α) : span (s ∪ t) = span s ⊔ span t :=
submodule.span_union _ _
lemma span_Union {ι} (s : ι → set α) : span (⋃ i, s i) = ⨆ i, span (s i) :=
submodule.span_Union _
lemma mem_span {s : set α} (x) : x ∈ span s ↔ ∀ p : ideal α, s ⊆ p → x ∈ p :=
mem_Inter₂
lemma subset_span {s : set α} : s ⊆ span s := submodule.subset_span
lemma span_le {s : set α} {I} : span s ≤ I ↔ s ⊆ I := submodule.span_le
lemma span_mono {s t : set α} : s ⊆ t → span s ≤ span t := submodule.span_mono
@[simp] lemma span_eq : span (I : set α) = I := submodule.span_eq _
@[simp] lemma span_singleton_one : span ({1} : set α) = ⊤ :=
(eq_top_iff_one _).2 $ subset_span $ mem_singleton _
lemma mem_span_insert {s : set α} {x y} :
x ∈ span (insert y s) ↔ ∃ a (z ∈ span s), x = a * y + z := submodule.mem_span_insert
lemma mem_span_singleton' {x y : α} :
x ∈ span ({y} : set α) ↔ ∃ a, a * y = x := submodule.mem_span_singleton
lemma span_singleton_le_iff_mem {x : α} : span {x} ≤ I ↔ x ∈ I :=
submodule.span_singleton_le_iff_mem _ _
lemma span_singleton_mul_left_unit {a : α} (h2 : is_unit a) (x : α) :
span ({a * x} : set α) = span {x} :=
begin
apply le_antisymm; rw [span_singleton_le_iff_mem, mem_span_singleton'],
exacts [⟨a, rfl⟩, ⟨_, h2.unit.inv_mul_cancel_left x⟩],
end
lemma span_insert (x) (s : set α) : span (insert x s) = span ({x} : set α) ⊔ span s :=
submodule.span_insert x s
lemma span_eq_bot {s : set α} : span s = ⊥ ↔ ∀ x ∈ s, (x:α) = 0 := submodule.span_eq_bot
@[simp] lemma span_singleton_eq_bot {x} : span ({x} : set α) = ⊥ ↔ x = 0 :=
submodule.span_singleton_eq_bot
lemma span_singleton_ne_top {α : Type*} [comm_semiring α] {x : α} (hx : ¬ is_unit x) :
ideal.span ({x} : set α) ≠ ⊤ :=
(ideal.ne_top_iff_one _).mpr $ λ h1, let ⟨y, hy⟩ := ideal.mem_span_singleton'.mp h1 in
hx ⟨⟨x, y, mul_comm y x ▸ hy, hy⟩, rfl⟩
@[simp] lemma span_zero : span (0 : set α) = ⊥ := by rw [←set.singleton_zero, span_singleton_eq_bot]
@[simp] lemma span_one : span (1 : set α) = ⊤ := by rw [←set.singleton_one, span_singleton_one]
lemma span_eq_top_iff_finite (s : set α) :
span s = ⊤ ↔ ∃ s' : finset α, ↑s' ⊆ s ∧ span (s' : set α) = ⊤ :=
begin
simp_rw eq_top_iff_one,
exact ⟨submodule.mem_span_finite_of_mem_span, λ ⟨s', h₁, h₂⟩, span_mono h₁ h₂⟩
end
lemma mem_span_singleton_sup {S : Type*} [comm_semiring S] {x y : S} {I : ideal S} :
x ∈ ideal.span {y} ⊔ I ↔ ∃ (a : S) (b ∈ I), a * y + b = x :=
begin
rw submodule.mem_sup,
split,
{ rintro ⟨ya, hya, b, hb, rfl⟩,
obtain ⟨a, rfl⟩ := mem_span_singleton'.mp hya,
exact ⟨a, b, hb, rfl⟩ },
{ rintro ⟨a, b, hb, rfl⟩,
exact ⟨a * y, ideal.mem_span_singleton'.mpr ⟨a, rfl⟩, b, hb, rfl⟩ }
end
/--
The ideal generated by an arbitrary binary relation.
-/
def of_rel (r : α → α → Prop) : ideal α :=
submodule.span α { x | ∃ (a b) (h : r a b), x + b = a }
/-- An ideal `P` of a ring `R` is prime if `P ≠ R` and `xy ∈ P → x ∈ P ∨ y ∈ P` -/
class is_prime (I : ideal α) : Prop :=
(ne_top' : I ≠ ⊤)
(mem_or_mem' : ∀ {x y : α}, x * y ∈ I → x ∈ I ∨ y ∈ I)
theorem is_prime_iff {I : ideal α} :
is_prime I ↔ I ≠ ⊤ ∧ ∀ {x y : α}, x * y ∈ I → x ∈ I ∨ y ∈ I :=
⟨λ h, ⟨h.1, λ _ _, h.2⟩, λ h, ⟨h.1, λ _ _, h.2⟩⟩
theorem is_prime.ne_top {I : ideal α} (hI : I.is_prime) : I ≠ ⊤ := hI.1
theorem is_prime.mem_or_mem {I : ideal α} (hI : I.is_prime) {x y : α} :
x * y ∈ I → x ∈ I ∨ y ∈ I := hI.2
theorem is_prime.mem_or_mem_of_mul_eq_zero {I : ideal α} (hI : I.is_prime)
{x y : α} (h : x * y = 0) : x ∈ I ∨ y ∈ I :=
hI.mem_or_mem (h.symm ▸ I.zero_mem)
theorem is_prime.mem_of_pow_mem {I : ideal α} (hI : I.is_prime)
{r : α} (n : ℕ) (H : r^n ∈ I) : r ∈ I :=
begin
induction n with n ih,
{ rw pow_zero at H, exact (mt (eq_top_iff_one _).2 hI.1).elim H },
{ rw pow_succ at H, exact or.cases_on (hI.mem_or_mem H) id ih }
end
lemma not_is_prime_iff {I : ideal α} : ¬ I.is_prime ↔ I = ⊤ ∨ ∃ (x ∉ I) (y ∉ I), x * y ∈ I :=
begin
simp_rw [ideal.is_prime_iff, not_and_distrib, ne.def, not_not, not_forall, not_or_distrib],
exact or_congr iff.rfl
⟨λ ⟨x, y, hxy, hx, hy⟩, ⟨x, hx, y, hy, hxy⟩, λ ⟨x, hx, y, hy, hxy⟩, ⟨x, y, hxy, hx, hy⟩⟩
end
theorem zero_ne_one_of_proper {I : ideal α} (h : I ≠ ⊤) : (0:α) ≠ 1 :=
λ hz, I.ne_top_iff_one.1 h $ hz ▸ I.zero_mem
lemma bot_prime {R : Type*} [ring R] [is_domain R] : (⊥ : ideal R).is_prime :=
⟨λ h, one_ne_zero (by rwa [ideal.eq_top_iff_one, submodule.mem_bot] at h),
λ x y h, mul_eq_zero.mp (by simpa only [submodule.mem_bot] using h)⟩
/-- An ideal is maximal if it is maximal in the collection of proper ideals. -/
class is_maximal (I : ideal α) : Prop := (out : is_coatom I)
theorem is_maximal_def {I : ideal α} : I.is_maximal ↔ is_coatom I := ⟨λ h, h.1, λ h, ⟨h⟩⟩
theorem is_maximal.ne_top {I : ideal α} (h : I.is_maximal) : I ≠ ⊤ := (is_maximal_def.1 h).1
theorem is_maximal_iff {I : ideal α} : I.is_maximal ↔
(1:α) ∉ I ∧ ∀ (J : ideal α) x, I ≤ J → x ∉ I → x ∈ J → (1:α) ∈ J :=
is_maximal_def.trans $ and_congr I.ne_top_iff_one $ forall_congr $ λ J,
by rw [lt_iff_le_not_le]; exact
⟨λ H x h hx₁ hx₂, J.eq_top_iff_one.1 $
H ⟨h, not_subset.2 ⟨_, hx₂, hx₁⟩⟩,
λ H ⟨h₁, h₂⟩, let ⟨x, xJ, xI⟩ := not_subset.1 h₂ in
J.eq_top_iff_one.2 $ H x h₁ xI xJ⟩
theorem is_maximal.eq_of_le {I J : ideal α}
(hI : I.is_maximal) (hJ : J ≠ ⊤) (IJ : I ≤ J) : I = J :=
eq_iff_le_not_lt.2 ⟨IJ, λ h, hJ (hI.1.2 _ h)⟩
instance : is_coatomic (ideal α) :=
begin
apply complete_lattice.coatomic_of_top_compact,
rw ←span_singleton_one,
exact submodule.singleton_span_is_compact_element 1,
end
/-- **Krull's theorem**: if `I` is an ideal that is not the whole ring, then it is included in some
maximal ideal. -/
theorem exists_le_maximal (I : ideal α) (hI : I ≠ ⊤) :
∃ M : ideal α, M.is_maximal ∧ I ≤ M :=
let ⟨m, hm⟩ := (eq_top_or_exists_le_coatom I).resolve_left hI in ⟨m, ⟨⟨hm.1⟩, hm.2⟩⟩
variables (α)
/-- Krull's theorem: a nontrivial ring has a maximal ideal. -/
theorem exists_maximal [nontrivial α] : ∃ M : ideal α, M.is_maximal :=
let ⟨I, ⟨hI, _⟩⟩ := exists_le_maximal (⊥ : ideal α) bot_ne_top in ⟨I, hI⟩
variables {α}
instance [nontrivial α] : nontrivial (ideal α) :=
begin
rcases @exists_maximal α _ _ with ⟨M, hM, _⟩,
exact nontrivial_of_ne M ⊤ hM
end
/-- If P is not properly contained in any maximal ideal then it is not properly contained
in any proper ideal -/
lemma maximal_of_no_maximal {R : Type u} [semiring R] {P : ideal R}
(hmax : ∀ m : ideal R, P < m → ¬is_maximal m) (J : ideal R) (hPJ : P < J) : J = ⊤ :=
begin
by_contradiction hnonmax,
rcases exists_le_maximal J hnonmax with ⟨M, hM1, hM2⟩,
exact hmax M (lt_of_lt_of_le hPJ hM2) hM1,
end
lemma span_pair_comm {x y : α} : (span {x, y} : ideal α) = span {y, x} :=
by simp only [span_insert, sup_comm]
theorem mem_span_pair {x y z : α} :
z ∈ span ({x, y} : set α) ↔ ∃ a b, a * x + b * y = z :=
submodule.mem_span_pair
@[simp] lemma span_pair_add_mul_left {R : Type u} [comm_ring R] {x y : R} (z : R) :
(span {x + y * z, y} : ideal R) = span {x, y} :=
begin
ext,
rw [mem_span_pair, mem_span_pair],
exact ⟨λ ⟨a, b, h⟩, ⟨a, b + a * z, by { rw [← h], ring1 }⟩,
λ ⟨a, b, h⟩, ⟨a, b - a * z, by { rw [← h], ring1 }⟩⟩
end
@[simp] lemma span_pair_add_mul_right {R : Type u} [comm_ring R] {x y : R} (z : R) :
(span {x, y + x * z} : ideal R) = span {x, y} :=
by rw [span_pair_comm, span_pair_add_mul_left, span_pair_comm]
theorem is_maximal.exists_inv {I : ideal α}
(hI : I.is_maximal) {x} (hx : x ∉ I) : ∃ y, ∃ i ∈ I, y * x + i = 1 :=
begin
cases is_maximal_iff.1 hI with H₁ H₂,
rcases mem_span_insert.1 (H₂ (span (insert x I)) x
(set.subset.trans (subset_insert _ _) subset_span)
hx (subset_span (mem_insert _ _))) with ⟨y, z, hz, hy⟩,
refine ⟨y, z, _, hy.symm⟩,
rwa ← span_eq I,
end
section lattice
variables {R : Type u} [semiring R]
lemma mem_sup_left {S T : ideal R} : ∀ {x : R}, x ∈ S → x ∈ S ⊔ T :=
show S ≤ S ⊔ T, from le_sup_left
lemma mem_sup_right {S T : ideal R} : ∀ {x : R}, x ∈ T → x ∈ S ⊔ T :=
show T ≤ S ⊔ T, from le_sup_right
lemma mem_supr_of_mem {ι : Sort*} {S : ι → ideal R} (i : ι) :
∀ {x : R}, x ∈ S i → x ∈ supr S :=
show S i ≤ supr S, from le_supr _ _
lemma mem_Sup_of_mem {S : set (ideal R)} {s : ideal R}
(hs : s ∈ S) : ∀ {x : R}, x ∈ s → x ∈ Sup S :=
show s ≤ Sup S, from le_Sup hs
theorem mem_Inf {s : set (ideal R)} {x : R} :
x ∈ Inf s ↔ ∀ ⦃I⦄, I ∈ s → x ∈ I :=
⟨λ hx I his, hx I ⟨I, infi_pos his⟩, λ H I ⟨J, hij⟩, hij ▸ λ S ⟨hj, hS⟩, hS ▸ H hj⟩
@[simp] lemma mem_inf {I J : ideal R} {x : R} : x ∈ I ⊓ J ↔ x ∈ I ∧ x ∈ J := iff.rfl
@[simp] lemma mem_infi {ι : Sort*} {I : ι → ideal R} {x : R} : x ∈ infi I ↔ ∀ i, x ∈ I i :=
submodule.mem_infi _
@[simp] lemma mem_bot {x : R} : x ∈ (⊥ : ideal R) ↔ x = 0 :=
submodule.mem_bot _
end lattice
section pi
variables (ι : Type v)
/-- `I^n` as an ideal of `R^n`. -/
def pi : ideal (ι → α) :=
{ carrier := { x | ∀ i, x i ∈ I },
zero_mem' := λ i, I.zero_mem,
add_mem' := λ a b ha hb i, I.add_mem (ha i) (hb i),
smul_mem' := λ a b hb i, I.mul_mem_left (a i) (hb i) }
lemma mem_pi (x : ι → α) : x ∈ I.pi ι ↔ ∀ i, x i ∈ I := iff.rfl
end pi
lemma Inf_is_prime_of_is_chain {s : set (ideal α)} (hs : s.nonempty) (hs' : is_chain (≤) s)
(H : ∀ p ∈ s, ideal.is_prime p) :
(Inf s).is_prime :=
⟨λ e, let ⟨x, hx⟩ := hs in (H x hx).ne_top (eq_top_iff.mpr (e.symm.trans_le (Inf_le hx))),
λ x y e, or_iff_not_imp_left.mpr $ λ hx, begin
rw ideal.mem_Inf at hx ⊢ e,
push_neg at hx,
obtain ⟨I, hI, hI'⟩ := hx,
intros J hJ,
cases hs'.total hI hJ,
{ exact h (((H I hI).mem_or_mem (e hI)).resolve_left hI') },
{ exact ((H J hJ).mem_or_mem (e hJ)).resolve_left (λ x, hI' $ h x) },
end⟩
end ideal
end semiring
section comm_semiring
variables {a b : α}
-- A separate namespace definition is needed because the variables were historically in a different
-- order.
namespace ideal
variables [comm_semiring α] (I : ideal α)
@[simp]
theorem mul_unit_mem_iff_mem {x y : α} (hy : is_unit y) : x * y ∈ I ↔ x ∈ I :=
mul_comm y x ▸ unit_mul_mem_iff_mem I hy
lemma mem_span_singleton {x y : α} : x ∈ span ({y} : set α) ↔ y ∣ x :=
mem_span_singleton'.trans $ exists_congr $ λ _, by rw [eq_comm, mul_comm]
lemma mem_span_singleton_self (x : α) : x ∈ span ({x} : set α) := mem_span_singleton.mpr dvd_rfl
lemma span_singleton_le_span_singleton {x y : α} :
span ({x} : set α) ≤ span ({y} : set α) ↔ y ∣ x :=
span_le.trans $ singleton_subset_iff.trans mem_span_singleton
lemma span_singleton_eq_span_singleton {α : Type u} [comm_ring α] [is_domain α] {x y : α} :
span ({x} : set α) = span ({y} : set α) ↔ associated x y :=
begin
rw [←dvd_dvd_iff_associated, le_antisymm_iff, and_comm],
apply and_congr;
rw span_singleton_le_span_singleton,
end
lemma span_singleton_mul_right_unit {a : α} (h2 : is_unit a) (x : α) :
span ({x * a} : set α) = span {x} := by rw [mul_comm, span_singleton_mul_left_unit h2]
lemma span_singleton_eq_top {x} : span ({x} : set α) = ⊤ ↔ is_unit x :=
by rw [is_unit_iff_dvd_one, ← span_singleton_le_span_singleton, span_singleton_one,
eq_top_iff]
theorem span_singleton_prime {p : α} (hp : p ≠ 0) :
is_prime (span ({p} : set α)) ↔ prime p :=
by simp [is_prime_iff, prime, span_singleton_eq_top, hp, mem_span_singleton]
theorem is_maximal.is_prime {I : ideal α} (H : I.is_maximal) : I.is_prime :=
⟨H.1.1, λ x y hxy, or_iff_not_imp_left.2 $ λ hx, begin
let J : ideal α := submodule.span α (insert x ↑I),
have IJ : I ≤ J := (set.subset.trans (subset_insert _ _) subset_span),
have xJ : x ∈ J := ideal.subset_span (set.mem_insert x I),
cases is_maximal_iff.1 H with _ oJ,
specialize oJ J x IJ hx xJ,
rcases submodule.mem_span_insert.mp oJ with ⟨a, b, h, oe⟩,
obtain (F : y * 1 = y * (a • x + b)) := congr_arg (λ g : α, y * g) oe,
rw [← mul_one y, F, mul_add, mul_comm, smul_eq_mul, mul_assoc],
refine submodule.add_mem I (I.mul_mem_left a hxy) (submodule.smul_mem I y _),
rwa submodule.span_eq at h,
end⟩
@[priority 100] -- see Note [lower instance priority]
instance is_maximal.is_prime' (I : ideal α) : ∀ [H : I.is_maximal], I.is_prime :=
is_maximal.is_prime
lemma span_singleton_lt_span_singleton [comm_ring β] [is_domain β] {x y : β} :
span ({x} : set β) < span ({y} : set β) ↔ dvd_not_unit y x :=
by rw [lt_iff_le_not_le, span_singleton_le_span_singleton, span_singleton_le_span_singleton,
dvd_and_not_dvd_iff]
lemma factors_decreasing [comm_ring β] [is_domain β]
(b₁ b₂ : β) (h₁ : b₁ ≠ 0) (h₂ : ¬ is_unit b₂) :
span ({b₁ * b₂} : set β) < span {b₁} :=
lt_of_le_not_le (ideal.span_le.2 $ singleton_subset_iff.2 $
ideal.mem_span_singleton.2 ⟨b₂, rfl⟩) $ λ h,
h₂ $ is_unit_of_dvd_one _ $ (mul_dvd_mul_iff_left h₁).1 $
by rwa [mul_one, ← ideal.span_singleton_le_span_singleton]
variables (b)
lemma mul_mem_right (h : a ∈ I) : a * b ∈ I := mul_comm b a ▸ I.mul_mem_left b h
variables {b}
lemma pow_mem_of_mem (ha : a ∈ I) (n : ℕ) (hn : 0 < n) : a ^ n ∈ I :=
nat.cases_on n (not.elim dec_trivial) (λ m hm, (pow_succ a m).symm ▸ I.mul_mem_right (a^m) ha) hn
theorem is_prime.mul_mem_iff_mem_or_mem {I : ideal α} (hI : I.is_prime) :
∀ {x y : α}, x * y ∈ I ↔ x ∈ I ∨ y ∈ I :=
λ x y, ⟨hI.mem_or_mem, by { rintro (h | h), exacts [I.mul_mem_right y h, I.mul_mem_left x h] }⟩
theorem is_prime.pow_mem_iff_mem {I : ideal α} (hI : I.is_prime)
{r : α} (n : ℕ) (hn : 0 < n) : r ^ n ∈ I ↔ r ∈ I :=
⟨hI.mem_of_pow_mem n, (λ hr, I.pow_mem_of_mem hr n hn)⟩
theorem pow_multiset_sum_mem_span_pow (s : multiset α) (n : ℕ) :
s.sum ^ (s.card * n + 1) ∈ span ((s.map (λ x, x ^ (n + 1))).to_finset : set α) :=
begin
induction s using multiset.induction_on with a s hs,
{ simp },
simp only [finset.coe_insert, multiset.map_cons, multiset.to_finset_cons, multiset.sum_cons,
multiset.card_cons, add_pow],
refine submodule.sum_mem _ _,
intros c hc,
rw mem_span_insert,
by_cases h : n+1 ≤ c,
{ refine ⟨a ^ (c - (n + 1)) * s.sum ^ ((s.card + 1) * n + 1 - c) *
(((s.card + 1) * n + 1).choose c), 0, submodule.zero_mem _, _⟩,
rw mul_comm _ (a ^ (n + 1)),
simp_rw ← mul_assoc,
rw [← pow_add, add_zero, add_tsub_cancel_of_le h], },
{ use 0,
simp_rw [zero_mul, zero_add],
refine ⟨_,_,rfl⟩,
replace h : c ≤ n := nat.lt_succ_iff.mp (not_le.mp h),
have : (s.card + 1) * n + 1 - c = s.card * n + 1 + (n - c),
{ rw [add_mul, one_mul, add_assoc, add_comm n 1, ← add_assoc, add_tsub_assoc_of_le h] },
rw [this, pow_add],
simp_rw [mul_assoc, mul_comm (s.sum ^ (s.card * n + 1)), ← mul_assoc],
exact mul_mem_left _ _ hs }
end
theorem sum_pow_mem_span_pow {ι} (s : finset ι) (f : ι → α) (n : ℕ) :
(∑ i in s, f i) ^ (s.card * n + 1) ∈ span ((λ i, f i ^ (n + 1)) '' s) :=
begin
convert pow_multiset_sum_mem_span_pow (s.1.map f) n,
{ rw multiset.card_map, refl },
rw [multiset.map_map, multiset.to_finset_map, finset.val_to_finset, finset.coe_image]
end
theorem span_pow_eq_top (s : set α)
(hs : span s = ⊤) (n : ℕ) : span ((λ x, x ^ n) '' s) = ⊤ :=
begin
rw eq_top_iff_one,
cases n,
{ obtain rfl | ⟨x, hx⟩ := eq_empty_or_nonempty s,
{ rw [set.image_empty, hs],
trivial },
{ exact subset_span ⟨_, hx, pow_zero _⟩ } },
rw [eq_top_iff_one, span, finsupp.mem_span_iff_total] at hs,
rcases hs with ⟨f, hf⟩,
change f.support.sum (λ a, f a * a) = 1 at hf,
have := sum_pow_mem_span_pow f.support (λ a, f a * a) n,
rw [hf, one_pow] at this,
refine (span_le).mpr _ this,
rintros _ hx,
simp_rw [finset.mem_coe, set.mem_image] at hx,
rcases hx with ⟨x, hx, rfl⟩,
have : span ({x ^ (n + 1)} : set α) ≤ span ((λ (x : α), x ^ (n + 1)) '' s),
{ rw [span_le, set.singleton_subset_iff],
exact subset_span ⟨x, x.prop, rfl⟩ },
refine this _,
rw [mul_pow, mem_span_singleton],
exact ⟨f x ^ (n + 1), mul_comm _ _⟩
end
end ideal
end comm_semiring
section ring
namespace ideal
variables [ring α] (I : ideal α) {a b : α}
protected lemma neg_mem_iff : -a ∈ I ↔ a ∈ I := neg_mem_iff
protected lemma add_mem_iff_left : b ∈ I → (a + b ∈ I ↔ a ∈ I) := I.add_mem_iff_left
protected lemma add_mem_iff_right : a ∈ I → (a + b ∈ I ↔ b ∈ I) := I.add_mem_iff_right
protected lemma sub_mem : a ∈ I → b ∈ I → a - b ∈ I := sub_mem
lemma mem_span_insert' {s : set α} {x y} :
x ∈ span (insert y s) ↔ ∃a, x + a * y ∈ span s := submodule.mem_span_insert'
@[simp] lemma span_singleton_neg (x : α) : (span {-x} : ideal α) = span {x} :=
by { ext, simp only [mem_span_singleton'],
exact ⟨λ ⟨y, h⟩, ⟨-y, h ▸ neg_mul_comm y x⟩, λ ⟨y, h⟩, ⟨-y, h ▸ neg_mul_neg y x⟩⟩ }
end ideal
end ring
section division_ring
variables {K : Type u} [division_ring K] (I : ideal K)
namespace ideal
/-- All ideals in a division ring are trivial. -/
lemma eq_bot_or_top : I = ⊥ ∨ I = ⊤ :=
begin
rw or_iff_not_imp_right,
change _ ≠ _ → _,
rw ideal.ne_top_iff_one,
intro h1,
rw eq_bot_iff,
intros r hr,
by_cases H : r = 0, {simpa},
simpa [H, h1] using I.mul_mem_left r⁻¹ hr,
end
/-- Ideals of a `division_ring` are a simple order. Thanks to the way abbreviations work, this
automatically gives a `is_simple_module K` instance. -/
instance : is_simple_order (ideal K) := ⟨eq_bot_or_top⟩
lemma eq_bot_of_prime [h : I.is_prime] : I = ⊥ :=
or_iff_not_imp_right.mp I.eq_bot_or_top h.1
lemma bot_is_maximal : is_maximal (⊥ : ideal K) :=
⟨⟨λ h, absurd ((eq_top_iff_one (⊤ : ideal K)).mp rfl) (by rw ← h; simp),
λ I hI, or_iff_not_imp_left.mp (eq_bot_or_top I) (ne_of_gt hI)⟩⟩
end ideal
end division_ring
section comm_ring
namespace ideal
theorem mul_sub_mul_mem {R : Type*} [comm_ring R] (I : ideal R) {a b c d : R}
(h1 : a - b ∈ I) (h2 : c - d ∈ I) : a * c - b * d ∈ I :=
begin
rw (show a * c - b * d = (a - b) * c + b * (c - d), by {rw [sub_mul, mul_sub], abel}),
exact I.add_mem (I.mul_mem_right _ h1) (I.mul_mem_left _ h2),
end
end ideal
end comm_ring
namespace ring
variables {R : Type*} [comm_ring R]
lemma not_is_field_of_subsingleton {R : Type*} [ring R] [subsingleton R] : ¬ is_field R :=
λ ⟨⟨x, y, hxy⟩, _, _⟩, hxy (subsingleton.elim x y)
lemma exists_not_is_unit_of_not_is_field [nontrivial R] (hf : ¬ is_field R) :
∃ x ≠ (0 : R), ¬ is_unit x :=
begin
have : ¬ _ := λ h, hf ⟨exists_pair_ne R, mul_comm, h⟩,
simp_rw is_unit_iff_exists_inv,
push_neg at ⊢ this,
obtain ⟨x, hx, not_unit⟩ := this,
exact ⟨x, hx, not_unit⟩
end
lemma not_is_field_iff_exists_ideal_bot_lt_and_lt_top [nontrivial R] :
¬ is_field R ↔ ∃ I : ideal R, ⊥ < I ∧ I < ⊤ :=
begin
split,
{ intro h,
obtain ⟨x, nz, nu⟩ := exists_not_is_unit_of_not_is_field h,
use ideal.span {x},
rw [bot_lt_iff_ne_bot, lt_top_iff_ne_top],
exact ⟨mt ideal.span_singleton_eq_bot.mp nz, mt ideal.span_singleton_eq_top.mp nu⟩ },
{ rintros ⟨I, bot_lt, lt_top⟩ hf,
obtain ⟨x, mem, ne_zero⟩ := set_like.exists_of_lt bot_lt,
rw submodule.mem_bot at ne_zero,
obtain ⟨y, hy⟩ := hf.mul_inv_cancel ne_zero,
rw [lt_top_iff_ne_top, ne.def, ideal.eq_top_iff_one, ← hy] at lt_top,
exact lt_top (I.mul_mem_right _ mem), }
end
lemma not_is_field_iff_exists_prime [nontrivial R] :
¬ is_field R ↔ ∃ p : ideal R, p ≠ ⊥ ∧ p.is_prime :=
not_is_field_iff_exists_ideal_bot_lt_and_lt_top.trans
⟨λ ⟨I, bot_lt, lt_top⟩, let ⟨p, hp, le_p⟩ := I.exists_le_maximal (lt_top_iff_ne_top.mp lt_top) in
⟨p, bot_lt_iff_ne_bot.mp (lt_of_lt_of_le bot_lt le_p), hp.is_prime⟩,
λ ⟨p, ne_bot, prime⟩, ⟨p, bot_lt_iff_ne_bot.mpr ne_bot, lt_top_iff_ne_top.mpr prime.1⟩⟩
/-- When a ring is not a field, the maximal ideals are nontrivial. -/
lemma ne_bot_of_is_maximal_of_not_is_field [nontrivial R] {M : ideal R} (max : M.is_maximal)
(not_field : ¬ is_field R) : M ≠ ⊥ :=
begin
rintros h,
rw h at max,
rcases max with ⟨⟨h1, h2⟩⟩,
obtain ⟨I, hIbot, hItop⟩ := not_is_field_iff_exists_ideal_bot_lt_and_lt_top.mp not_field,
exact ne_of_lt hItop (h2 I hIbot),
end
end ring
namespace ideal
/-- Maximal ideals in a non-field are nontrivial. -/
variables {R : Type u} [comm_ring R] [nontrivial R]
lemma bot_lt_of_maximal (M : ideal R) [hm : M.is_maximal] (non_field : ¬ is_field R) : ⊥ < M :=
begin
rcases (ring.not_is_field_iff_exists_ideal_bot_lt_and_lt_top.1 non_field)
with ⟨I, Ibot, Itop⟩,
split, { simp },
intro mle,
apply @irrefl _ (<) _ (⊤ : ideal R),
have : M = ⊥ := eq_bot_iff.mpr mle,
rw this at *,
rwa hm.1.2 I Ibot at Itop,
end
end ideal
variables {a b : α}
/-- The set of non-invertible elements of a monoid. -/
def nonunits (α : Type u) [monoid α] : set α := { a | ¬is_unit a }
@[simp] theorem mem_nonunits_iff [monoid α] : a ∈ nonunits α ↔ ¬ is_unit a := iff.rfl
theorem mul_mem_nonunits_right [comm_monoid α] :
b ∈ nonunits α → a * b ∈ nonunits α :=
mt is_unit_of_mul_is_unit_right
theorem mul_mem_nonunits_left [comm_monoid α] :
a ∈ nonunits α → a * b ∈ nonunits α :=
mt is_unit_of_mul_is_unit_left
theorem zero_mem_nonunits [semiring α] : 0 ∈ nonunits α ↔ (0:α) ≠ 1 :=
not_congr is_unit_zero_iff
@[simp] theorem one_not_mem_nonunits [monoid α] : (1:α) ∉ nonunits α :=
not_not_intro is_unit_one
theorem coe_subset_nonunits [semiring α] {I : ideal α} (h : I ≠ ⊤) :
(I : set α) ⊆ nonunits α :=
λ x hx hu, h $ I.eq_top_of_is_unit_mem hx hu
lemma exists_max_ideal_of_mem_nonunits [comm_semiring α] (h : a ∈ nonunits α) :
∃ I : ideal α, I.is_maximal ∧ a ∈ I :=
begin
have : ideal.span ({a} : set α) ≠ ⊤,
{ intro H, rw ideal.span_singleton_eq_top at H, contradiction },
rcases ideal.exists_le_maximal _ this with ⟨I, Imax, H⟩,
use [I, Imax], apply H, apply ideal.subset_span, exact set.mem_singleton a
end
|
897b41c80189411a8f176e4468386a065397557f | 9dd3f3912f7321eb58ee9aa8f21778ad6221f87c | /tests/lean/run/fingerprint.lean | 7d613f90d5701b0105ed9a1fc4d5f084fb578eec | [
"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 | 775 | lean | open tactic
meta def nat.to_expr (n : nat) : tactic expr := to_expr (quote n)
run_command attribute.fingerprint `reducible >>= trace
definition ex0 : nat :=
by attribute.fingerprint `reducible >>= nat.to_expr >>= exact
attribute [reducible]
definition f : nat := 10
run_command attribute.fingerprint `reducible >>= trace
definition ex1 : nat :=
by attribute.fingerprint `reducible >>= nat.to_expr >>= exact
vm_eval ex1
definition g : nat := 20
run_command attribute.fingerprint `reducible >>= trace
definition ex2 : nat :=
by attribute.fingerprint `reducible >>= nat.to_expr >>= exact
vm_eval ex2
example : ex1 = ex2 :=
rfl
definition h : nat := 20
definition ex3 : nat :=
by attribute.fingerprint `reducible >>= nat.to_expr >>= exact
example : ex1 = ex3 :=
rfl
|
a5561f7e166fc6babce7bd0e84c0329dded496d3 | 3dc4623269159d02a444fe898d33e8c7e7e9461b | /.github/workflows/project_1_a_decrire/lean-scheme-submission/src/sheaves/sheaf_on_standard_basis.lean | 2180b8a8035e4d9eaa6a586505d389e5b8bc5729 | [] | no_license | Or7ando/lean | cc003e6c41048eae7c34aa6bada51c9e9add9e66 | d41169cf4e416a0d42092fb6bdc14131cee9dd15 | refs/heads/master | 1,650,600,589,722 | 1,587,262,906,000 | 1,587,262,906,000 | 255,387,160 | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 6,063 | lean | /-
Sheaf (of types) on standard basis.
https://stacks.math.columbia.edu/tag/009L
-/
import topology.basic
import to_mathlib.opens
import sheaves.covering.covering_on_standard_basis
import sheaves.presheaf_of_rings_on_basis
universe u
open topological_space
namespace sheaf_on_standard_basis
variables {α : Type u} [topological_space α]
variables {B : set (opens α)} {HB : opens.is_basis B}
variables (Bstd : opens.univ ∈ B ∧ ∀ {U V}, U ∈ B → V ∈ B → U ∩ V ∈ B)
include Bstd
section properties
-- Compactness definition.
-- Basis elementsfinite subcover
def basis_is_compact : Prop :=
∀ {U} (BU : U ∈ B) (OC : covering_standard_basis B U),
∃ (γ : Type u) (Hfin : fintype γ) (f : γ → OC.γ), (⋃ (OC.Uis ∘ f)) = U
end properties
section sheaf_condition
-- Restriction map from U to U ∩ V.
def res_to_inter_left (F : presheaf_on_basis α HB) {U V} (BU : U ∈ B) (BV : V ∈ B)
: (F BU) → (F (Bstd.2 BU BV)) :=
F.res BU (Bstd.2 BU BV) (set.inter_subset_left U V)
@[simp] lemma res_to_inter_left'
(F : presheaf_on_basis α HB) {U V} (BU : U ∈ B) (BV : V ∈ B)
: sheaf_on_standard_basis.res_to_inter_left Bstd F BU BV =
F.res BU (Bstd.2 BU BV) (set.inter_subset_left U V)
:= rfl
-- Restriction map from V to U ∩ V.
def res_to_inter_right (F : presheaf_on_basis α HB) {U V} (BU : U ∈ B) (BV : V ∈ B)
: (F BV) → (F (Bstd.2 BU BV)) :=
F.res BV (Bstd.2 BU BV) (set.inter_subset_right U V)
@[simp] lemma res_to_inter_right'
(F : presheaf_on_basis α HB) {U V} (BU : U ∈ B) (BV : V ∈ B)
: sheaf_on_standard_basis.res_to_inter_right Bstd F BU BV =
F.res BV (Bstd.2 BU BV) (set.inter_subset_right U V)
:= rfl
-- Sheaf condition.
def locality
(F : presheaf_on_basis α HB) {U} (BU : U ∈ B) (OC : covering_standard_basis B U) :=
∀ (s t : F BU),
(∀ i, F.res BU (OC.BUis i) (subset_covering i) s =
F.res BU (OC.BUis i) (subset_covering i) t) →
s = t
def gluing
(F : presheaf_on_basis α HB) {U} (BU : U ∈ B) (OC : covering_standard_basis B U) :=
∀ (s : Π i, F (OC.BUis i)),
(∀ j k, sheaf_on_standard_basis.res_to_inter_left Bstd F (OC.BUis j) (OC.BUis k) (s j) =
sheaf_on_standard_basis.res_to_inter_right Bstd F (OC.BUis j) (OC.BUis k) (s k)) →
∃ S, ∀ i, F.res BU (OC.BUis i) (subset_covering i) S = s i
def is_sheaf_on_standard_basis (F : presheaf_on_basis α HB) :=
∀ {U} (BU : U ∈ B) (OC : covering_standard_basis B U),
locality Bstd F BU OC ∧ gluing Bstd F BU OC
end sheaf_condition
section cofinal_system
-- Suffices to prove the sheaf condition for finite covers.
def is_sheaf_on_standard_basis_cofinal_system (F : presheaf_on_basis α HB) : Prop :=
∀ {U} (BU : U ∈ B) (OC : covering_standard_basis B U) (Hfin : fintype OC.γ),
locality Bstd F BU OC ∧ gluing Bstd F BU OC
theorem cofinal_systems_coverings_standard_case
(F : presheaf_on_basis α HB)
(Hcompact : basis_is_compact Bstd)
: is_sheaf_on_standard_basis_cofinal_system Bstd F → is_sheaf_on_standard_basis Bstd F :=
begin
intros Hcofinal,
have Hcofinal' := @Hcofinal,
intros U BU OC,
rcases (Hcompact BU OC) with ⟨γ, Hfinγ, fγ, Hcov⟩,
have BUisfin : ∀ i, ((OC.Uis ∘ fγ) i) ∈ B := λ i, OC.BUis (fγ i),
let OCfin : covering_standard_basis B U
:= {γ := γ, Uis := OC.Uis ∘ fγ, Hcov := Hcov, BUis := BUisfin},
replace Hcofinal := Hcofinal BU OCfin Hfinγ,
rcases Hcofinal with ⟨Hloc, Hglue⟩,
split,
-- Locality.
{ intros s t Hres,
apply Hloc,
intros i,
apply Hres, },
-- Gluing.
{ intros s Hinter,
have Hinterfin
: ∀ (j k),
sheaf_on_standard_basis.res_to_inter_left Bstd F _ _ (s (fγ j)) =
sheaf_on_standard_basis.res_to_inter_right Bstd F _ _ (s (fγ k)) :=
λ j k, Hinter (fγ j) (fγ k),
have Hglobal := Hglue (λ i, s (fγ i)) Hinterfin,
rcases Hglobal with ⟨S, HS⟩,
existsi S,
intros i,
let Vjs := λ j : γ, (OCfin.Uis j) ∩ (OC.Uis i),
have BVjs : ∀ j, (Vjs j) ∈ B := λ j, Bstd.2 (OCfin.BUis j) (OC.BUis i),
have HVjscov : (⋃ Vjs) = OC.Uis i,
apply opens.ext,
apply set.ext,
intros x,
split,
{ intros HxUVjs,
rcases HxUVjs with ⟨Vj, HVj, HxVj⟩,
rcases HVj with ⟨VjO, ⟨⟨j, HVjO⟩, HVjeq⟩⟩,
rw [←HVjeq, ←HVjO] at HxVj,
exact HxVj.2, },
{ intros HxUi,
have HxU : x ∈ U := OC.Hcov ▸ (opens_supr_mem OC.Uis i x HxUi),
rw ←Hcov at HxU,
rcases HxU with ⟨Uj, HUj, HxUj⟩,
rcases HUj with ⟨UjO, ⟨⟨j, HUjO⟩, HUjeq⟩⟩,
use [(Vjs j).1, ⟨(Vjs j), ⟨⟨j, rfl⟩, rfl⟩⟩],
rw [←HUjeq, ←HUjO] at HxUj,
exact ⟨HxUj, HxUi⟩, },
have HVjUVjs : ∀ j, Vjs j ⊆ OC.Uis i
:= HVjscov ▸ λ j x Hx, opens_supr_mem Vjs j x Hx,
have HVjU : ∀ j, Vjs j ⊆ U
:= λ j, set.subset.trans (HVjUVjs j) (subset_covering i),
let Ui : opens α := OC.Uis i,
let BUi : Ui ∈ B := OC.BUis i,
let OCfin' : covering_standard_basis B Ui
:= {γ := γ, Uis := Vjs, Hcov := HVjscov, BUis := BVjs},
have Hloc' := (@Hcofinal' Ui BUi OCfin' Hfinγ).1,
have Hglue' := (@Hcofinal' Ui BUi OCfin' Hfinγ).2,
have Hglue'' := Hglue' (λ j, F.res BU (BVjs j) (HVjU j) S),
have Hres' : ∀ (j k),
sheaf_on_standard_basis.res_to_inter_left Bstd
F (BVjs j) (BVjs k) (F.res BU (BVjs j) (HVjU j) S) =
sheaf_on_standard_basis.res_to_inter_right Bstd
F (BVjs j) (BVjs k) (F.res BU (BVjs k) (HVjU k) S),
intros j k,
simp,
rw ←F.Hcomp',
rw ←F.Hcomp',
rcases (Hglue'' Hres') with ⟨S', HS'⟩,
refine @eq.trans _ _ S' _ _ _,
{ apply Hloc',
intro j,
rw HS',
rw ←F.Hcomp',
refl, },
{ apply eq.symm,
apply Hloc',
intro j,
simp at Hinter,
erw ←Hinter,
rw HS',
have Hsfj : s (fγ j) = _ := (HS j).symm,
rw Hsfj,
rw ←F.Hcomp',
refl, }, },
end
end cofinal_system
end sheaf_on_standard_basis
|
34ee267e309cc7d7b323cd9a19e530db46308d91 | 74addaa0e41490cbaf2abd313a764c96df57b05d | /Mathlib/data/multiset/nodup_auto.lean | 22ef4a5acff35809130a0f31ff5bc1fd97f6ee1b | [] | 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 | 9,537 | lean | /-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Mario Carneiro
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.default
import Mathlib.data.multiset.powerset
import Mathlib.data.multiset.range
import Mathlib.PostPort
universes u_1 u_2 u_3
namespace Mathlib
/-!
# The `nodup` predicate for multisets without duplicate elements.
-/
namespace multiset
/- nodup -/
/-- `nodup s` means that `s` has no duplicates, i.e. the multiplicity of
any element is at most 1. -/
def nodup {α : Type u_1} (s : multiset α) := quot.lift_on s list.nodup sorry
@[simp] theorem coe_nodup {α : Type u_1} {l : List α} : nodup ↑l ↔ list.nodup l := iff.rfl
@[simp] theorem nodup_zero {α : Type u_1} : nodup 0 := list.pairwise.nil
@[simp] theorem nodup_cons {α : Type u_1} {a : α} {s : multiset α} :
nodup (a ::ₘ s) ↔ ¬a ∈ s ∧ nodup s :=
quot.induction_on s fun (l : List α) => list.nodup_cons
theorem nodup_cons_of_nodup {α : Type u_1} {a : α} {s : multiset α} (m : ¬a ∈ s) (n : nodup s) :
nodup (a ::ₘ s) :=
iff.mpr nodup_cons { left := m, right := n }
theorem nodup_singleton {α : Type u_1} (a : α) : nodup (a ::ₘ 0) := list.nodup_singleton
theorem nodup_of_nodup_cons {α : Type u_1} {a : α} {s : multiset α} (h : nodup (a ::ₘ s)) :
nodup s :=
and.right (iff.mp nodup_cons h)
theorem not_mem_of_nodup_cons {α : Type u_1} {a : α} {s : multiset α} (h : nodup (a ::ₘ s)) :
¬a ∈ s :=
and.left (iff.mp nodup_cons h)
theorem nodup_of_le {α : Type u_1} {s : multiset α} {t : multiset α} (h : s ≤ t) :
nodup t → nodup s :=
le_induction_on h fun (l₁ l₂ : List α) => list.nodup_of_sublist
theorem not_nodup_pair {α : Type u_1} (a : α) : ¬nodup (a ::ₘ a ::ₘ 0) := list.not_nodup_pair
theorem nodup_iff_le {α : Type u_1} {s : multiset α} : nodup s ↔ ∀ (a : α), ¬a ::ₘ a ::ₘ 0 ≤ s :=
quot.induction_on s
fun (l : List α) =>
iff.trans list.nodup_iff_sublist
(forall_congr fun (a : α) => not_congr (iff.symm repeat_le_coe))
theorem nodup_iff_ne_cons_cons {α : Type u_1} {s : multiset α} :
nodup s ↔ ∀ (a : α) (t : multiset α), s ≠ a ::ₘ a ::ₘ t :=
sorry
theorem nodup_iff_count_le_one {α : Type u_1} [DecidableEq α] {s : multiset α} :
nodup s ↔ ∀ (a : α), count a s ≤ 1 :=
quot.induction_on s fun (l : List α) => list.nodup_iff_count_le_one
@[simp] theorem count_eq_one_of_mem {α : Type u_1} [DecidableEq α] {a : α} {s : multiset α}
(d : nodup s) (h : a ∈ s) : count a s = 1 :=
le_antisymm (iff.mp nodup_iff_count_le_one d a) (iff.mpr count_pos h)
theorem nodup_iff_pairwise {α : Type u_1} {s : multiset α} : nodup s ↔ pairwise ne s :=
quotient.induction_on s
fun (l : List α) => iff.symm (pairwise_coe_iff_pairwise fun (a b : α) => ne.symm)
theorem pairwise_of_nodup {α : Type u_1} {r : α → α → Prop} {s : multiset α} :
(∀ (a : α), a ∈ s → ∀ (b : α), b ∈ s → a ≠ b → r a b) → nodup s → pairwise r s :=
sorry
theorem forall_of_pairwise {α : Type u_1} {r : α → α → Prop} (H : symmetric r) {s : multiset α}
(hs : pairwise r s) (a : α) : a ∈ s → ∀ (b : α), b ∈ s → a ≠ b → r a b :=
sorry
theorem nodup_add {α : Type u_1} {s : multiset α} {t : multiset α} :
nodup (s + t) ↔ nodup s ∧ nodup t ∧ disjoint s t :=
quotient.induction_on₂ s t fun (l₁ l₂ : List α) => list.nodup_append
theorem disjoint_of_nodup_add {α : Type u_1} {s : multiset α} {t : multiset α} (d : nodup (s + t)) :
disjoint s t :=
and.right (and.right (iff.mp nodup_add d))
theorem nodup_add_of_nodup {α : Type u_1} {s : multiset α} {t : multiset α} (d₁ : nodup s)
(d₂ : nodup t) : nodup (s + t) ↔ disjoint s t :=
sorry
theorem nodup_of_nodup_map {α : Type u_1} {β : Type u_2} (f : α → β) {s : multiset α} :
nodup (map f s) → nodup s :=
quot.induction_on s fun (l : List α) => list.nodup_of_nodup_map f
theorem nodup_map_on {α : Type u_1} {β : Type u_2} {f : α → β} {s : multiset α} :
(∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → f x = f y → x = y) → nodup s → nodup (map f s) :=
quot.induction_on s fun (l : List α) => list.nodup_map_on
theorem nodup_map {α : Type u_1} {β : Type u_2} {f : α → β} {s : multiset α}
(hf : function.injective f) : nodup s → nodup (map f s) :=
nodup_map_on fun (x : α) (_x : x ∈ s) (y : α) (_x : y ∈ s) (h : f x = f y) => hf h
theorem nodup_filter {α : Type u_1} (p : α → Prop) [decidable_pred p] {s : multiset α} :
nodup s → nodup (filter p s) :=
quot.induction_on s fun (l : List α) => list.nodup_filter p
@[simp] theorem nodup_attach {α : Type u_1} {s : multiset α} : nodup (attach s) ↔ nodup s :=
quot.induction_on s fun (l : List α) => list.nodup_attach
theorem nodup_pmap {α : Type u_1} {β : Type u_2} {p : α → Prop} {f : (a : α) → p a → β}
{s : multiset α} {H : ∀ (a : α), a ∈ s → p a}
(hf : ∀ (a : α) (ha : p a) (b : α) (hb : p b), f a ha = f b hb → a = b) :
nodup s → nodup (pmap f s H) :=
quot.induction_on s
(fun (l : List α) (H : ∀ (a : α), a ∈ Quot.mk setoid.r l → p a) => list.nodup_pmap hf) H
protected instance nodup_decidable {α : Type u_1} [DecidableEq α] (s : multiset α) :
Decidable (nodup s) :=
quotient.rec_on_subsingleton s fun (l : List α) => list.nodup_decidable l
theorem nodup_erase_eq_filter {α : Type u_1} [DecidableEq α] (a : α) {s : multiset α} :
nodup s → erase s a = filter (fun (_x : α) => _x ≠ a) s :=
quot.induction_on s
fun (l : List α) (d : nodup (Quot.mk setoid.r l)) =>
congr_arg coe (list.nodup_erase_eq_filter a d)
theorem nodup_erase_of_nodup {α : Type u_1} [DecidableEq α] (a : α) {l : multiset α} :
nodup l → nodup (erase l a) :=
nodup_of_le (erase_le a l)
theorem mem_erase_iff_of_nodup {α : Type u_1} [DecidableEq α] {a : α} {b : α} {l : multiset α}
(d : nodup l) : a ∈ erase l b ↔ a ≠ b ∧ a ∈ l :=
sorry
theorem mem_erase_of_nodup {α : Type u_1} [DecidableEq α] {a : α} {l : multiset α} (h : nodup l) :
¬a ∈ erase l a :=
sorry
theorem nodup_product {α : Type u_1} {β : Type u_2} {s : multiset α} {t : multiset β} :
nodup s → nodup t → nodup (product s t) :=
sorry
theorem nodup_sigma {α : Type u_1} {σ : α → Type u_2} {s : multiset α}
{t : (a : α) → multiset (σ a)} :
nodup s → (∀ (a : α), nodup (t a)) → nodup (multiset.sigma s t) :=
sorry
theorem nodup_filter_map {α : Type u_1} {β : Type u_2} (f : α → Option β) {s : multiset α}
(H : ∀ (a a' : α) (b : β), b ∈ f a → b ∈ f a' → a = a') : nodup s → nodup (filter_map f s) :=
quot.induction_on s fun (l : List α) => list.nodup_filter_map H
theorem nodup_range (n : ℕ) : nodup (range n) := list.nodup_range n
theorem nodup_inter_left {α : Type u_1} [DecidableEq α] {s : multiset α} (t : multiset α) :
nodup s → nodup (s ∩ t) :=
nodup_of_le (inter_le_left s t)
theorem nodup_inter_right {α : Type u_1} [DecidableEq α] (s : multiset α) {t : multiset α} :
nodup t → nodup (s ∩ t) :=
nodup_of_le (inter_le_right s t)
@[simp] theorem nodup_union {α : Type u_1} [DecidableEq α] {s : multiset α} {t : multiset α} :
nodup (s ∪ t) ↔ nodup s ∧ nodup t :=
sorry
@[simp] theorem nodup_powerset {α : Type u_1} {s : multiset α} : nodup (powerset s) ↔ nodup s :=
sorry
theorem nodup_powerset_len {α : Type u_1} {n : ℕ} {s : multiset α} (h : nodup s) :
nodup (powerset_len n s) :=
nodup_of_le (powerset_len_le_powerset n s) (iff.mpr nodup_powerset h)
@[simp] theorem nodup_bind {α : Type u_1} {β : Type u_2} {s : multiset α} {t : α → multiset β} :
nodup (bind s t) ↔
(∀ (a : α), a ∈ s → nodup (t a)) ∧ pairwise (fun (a b : α) => disjoint (t a) (t b)) s :=
sorry
theorem nodup_ext {α : Type u_1} {s : multiset α} {t : multiset α} :
nodup s → nodup t → (s = t ↔ ∀ (a : α), a ∈ s ↔ a ∈ t) :=
quotient.induction_on₂ s t
fun (l₁ l₂ : List α) (d₁ : nodup (quotient.mk l₁)) (d₂ : nodup (quotient.mk l₂)) =>
iff.trans quotient.eq (list.perm_ext d₁ d₂)
theorem le_iff_subset {α : Type u_1} {s : multiset α} {t : multiset α} :
nodup s → (s ≤ t ↔ s ⊆ t) :=
quotient.induction_on₂ s t
fun (l₁ l₂ : List α) (d : nodup (quotient.mk l₁)) =>
{ mp := subset_of_le, mpr := list.subperm_of_subset_nodup d }
theorem range_le {m : ℕ} {n : ℕ} : range m ≤ range n ↔ m ≤ n :=
iff.trans (le_iff_subset (nodup_range m)) range_subset
theorem mem_sub_of_nodup {α : Type u_1} [DecidableEq α] {a : α} {s : multiset α} {t : multiset α}
(d : nodup s) : a ∈ s - t ↔ a ∈ s ∧ ¬a ∈ t :=
sorry
theorem map_eq_map_of_bij_of_nodup {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → γ)
(g : β → γ) {s : multiset α} {t : multiset β} (hs : nodup s) (ht : nodup t)
(i : (a : α) → a ∈ s → β) (hi : ∀ (a : α) (ha : a ∈ s), i a ha ∈ t)
(h : ∀ (a : α) (ha : a ∈ s), f a = g (i a ha))
(i_inj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), i a₁ ha₁ = i a₂ ha₂ → a₁ = a₂)
(i_surj : ∀ (b : β), b ∈ t → ∃ (a : α), ∃ (ha : a ∈ s), b = i a ha) : map f s = map g t :=
sorry
end Mathlib |
592a37595149dd798a55d233778a3ddaa6623a05 | ce89339993655da64b6ccb555c837ce6c10f9ef4 | /ukikagi/top39.lean | 4dde198c9197d93ce524e7c90d970d5fc404c63c | [] | no_license | zeptometer/LearnLean | ef32dc36a22119f18d843f548d0bb42f907bff5d | bb84d5dbe521127ba134d4dbf9559b294a80b9f7 | refs/heads/master | 1,625,710,824,322 | 1,601,382,570,000 | 1,601,382,570,000 | 195,228,870 | 2 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 583 | lean | theorem pnp {P: Prop}: ¬(P ↔ ¬P)
:=
begin
intro h,
have hnp: ¬P, {
intro hp,
exact h.1 hp hp
},
exact hnp (h.2 hnp)
end
example:
(∀ P Q R, (P ↔ Q) ∨ (Q ↔ R) ∨ (R ↔ P))
→ ∀ P, P ∨ ¬P
:=
begin
intros h P,
cases h (P ∨ ¬P) ¬(P ∨ ¬P) ¬¬(P ∨ ¬P) with h1 h23, {
exfalso,
exact pnp h1,
},
cases h23 with h2 h3, {
exfalso,
exact pnp h2,
}, {
apply h3.1,
intro H,
have hnp: ¬P, {
intro hp,
apply H,
left,
assumption
},
apply H,
right,
assumption,
}
end
|
fee456a46dd77456c797b5a59da89dd9c8d66b0c | 74addaa0e41490cbaf2abd313a764c96df57b05d | /Mathlib/tactic/localized.lean | 6fa977f7238cde7270a2584f50a39de88e20a77f | [] | 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 | 1,066 | lean | /-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.default
import Mathlib.tactic.core
import Mathlib.PostPort
namespace Mathlib
/-!
# Localized notation
This consists of two user-commands which allow you to declare notation and commands localized to a
locale.
See the tactic doc entry below for more information.
The code is inspired by code from Gabriel Ebner from the
[hott3 repository](https://github.com/gebner/hott3).
-/
/-- Get all commands in the given locale and return them as a list of strings -/
/-- Execute all commands in the given locale -/
/-- Add a new command to a locale and execute it right now.
The new command is added as a declaration to the environment with name `_localized_decl.<number>`.
This declaration has attribute `_localized` and as value a name-string pair. -/
/--
This consists of two user-commands which allow you to declare notation and commands localized to a
|
eff3c7cee5ce1d7ad7f3be4806dc1e62fe415c89 | 94e33a31faa76775069b071adea97e86e218a8ee | /src/linear_algebra/prod.lean | cfb0e6a6e6183b02a6bcced48d551f81eb7da55f | [
"Apache-2.0"
] | permissive | urkud/mathlib | eab80095e1b9f1513bfb7f25b4fa82fa4fd02989 | 6379d39e6b5b279df9715f8011369a301b634e41 | refs/heads/master | 1,658,425,342,662 | 1,658,078,703,000 | 1,658,078,703,000 | 186,910,338 | 0 | 0 | Apache-2.0 | 1,568,512,083,000 | 1,557,958,709,000 | Lean | UTF-8 | Lean | false | false | 31,170 | lean | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Eric Wieser
-/
import linear_algebra.span
import order.partial_sups
import algebra.algebra.basic
/-! ### Products of modules
This file defines constructors for linear maps whose domains or codomains are products.
It contains theorems relating these to each other, as well as to `submodule.prod`, `submodule.map`,
`submodule.comap`, `linear_map.range`, and `linear_map.ker`.
## Main definitions
- products in the domain:
- `linear_map.fst`
- `linear_map.snd`
- `linear_map.coprod`
- `linear_map.prod_ext`
- products in the codomain:
- `linear_map.inl`
- `linear_map.inr`
- `linear_map.prod`
- products in both domain and codomain:
- `linear_map.prod_map`
- `linear_equiv.prod_map`
- `linear_equiv.skew_prod`
-/
universes u v w x y z u' v' w' y'
variables {R : Type u} {K : Type u'} {M : Type v} {V : Type v'} {M₂ : Type w} {V₂ : Type w'}
variables {M₃ : Type y} {V₃ : Type y'} {M₄ : Type z} {ι : Type x}
variables {M₅ M₆ : Type*}
section prod
namespace linear_map
variables (S : Type*) [semiring R] [semiring S]
variables [add_comm_monoid M] [add_comm_monoid M₂] [add_comm_monoid M₃] [add_comm_monoid M₄]
variables [add_comm_monoid M₅] [add_comm_monoid M₆]
variables [module R M] [module R M₂] [module R M₃] [module R M₄]
variables [module R M₅] [module R M₆]
variables (f : M →ₗ[R] M₂)
section
variables (R M M₂)
/-- The first projection of a product is a linear map. -/
def fst : M × M₂ →ₗ[R] M := { to_fun := prod.fst, map_add' := λ x y, rfl, map_smul' := λ x y, rfl }
/-- The second projection of a product is a linear map. -/
def snd : M × M₂ →ₗ[R] M₂ := { to_fun := prod.snd, map_add' := λ x y, rfl, map_smul' := λ x y, rfl }
end
@[simp] theorem fst_apply (x : M × M₂) : fst R M M₂ x = x.1 := rfl
@[simp] theorem snd_apply (x : M × M₂) : snd R M M₂ x = x.2 := rfl
theorem fst_surjective : function.surjective (fst R M M₂) := λ x, ⟨(x, 0), rfl⟩
theorem snd_surjective : function.surjective (snd R M M₂) := λ x, ⟨(0, x), rfl⟩
/-- The prod of two linear maps is a linear map. -/
@[simps] def prod (f : M →ₗ[R] M₂) (g : M →ₗ[R] M₃) : (M →ₗ[R] M₂ × M₃) :=
{ to_fun := pi.prod f g,
map_add' := λ x y, by simp only [pi.prod, prod.mk_add_mk, map_add],
map_smul' := λ c x, by simp only [pi.prod, prod.smul_mk, map_smul, ring_hom.id_apply] }
lemma coe_prod (f : M →ₗ[R] M₂) (g : M →ₗ[R] M₃) : ⇑(f.prod g) = pi.prod f g := rfl
@[simp] theorem fst_prod (f : M →ₗ[R] M₂) (g : M →ₗ[R] M₃) :
(fst R M₂ M₃).comp (prod f g) = f := by ext; refl
@[simp] theorem snd_prod (f : M →ₗ[R] M₂) (g : M →ₗ[R] M₃) :
(snd R M₂ M₃).comp (prod f g) = g := by ext; refl
@[simp] theorem pair_fst_snd : prod (fst R M M₂) (snd R M M₂) = linear_map.id :=
fun_like.coe_injective pi.prod_fst_snd
/-- Taking the product of two maps with the same domain is equivalent to taking the product of
their codomains.
See note [bundled maps over different rings] for why separate `R` and `S` semirings are used. -/
@[simps] def prod_equiv
[module S M₂] [module S M₃] [smul_comm_class R S M₂] [smul_comm_class R S M₃] :
((M →ₗ[R] M₂) × (M →ₗ[R] M₃)) ≃ₗ[S] (M →ₗ[R] M₂ × M₃) :=
{ to_fun := λ f, f.1.prod f.2,
inv_fun := λ f, ((fst _ _ _).comp f, (snd _ _ _).comp f),
left_inv := λ f, by ext; refl,
right_inv := λ f, by ext; refl,
map_add' := λ a b, rfl,
map_smul' := λ r a, rfl }
section
variables (R M M₂)
/-- The left injection into a product is a linear map. -/
def inl : M →ₗ[R] M × M₂ := prod linear_map.id 0
/-- The right injection into a product is a linear map. -/
def inr : M₂ →ₗ[R] M × M₂ := prod 0 linear_map.id
theorem range_inl : range (inl R M M₂) = ker (snd R M M₂) :=
begin
ext x,
simp only [mem_ker, mem_range],
split,
{ rintros ⟨y, rfl⟩, refl },
{ intro h, exact ⟨x.fst, prod.ext rfl h.symm⟩ }
end
theorem ker_snd : ker (snd R M M₂) = range (inl R M M₂) :=
eq.symm $ range_inl R M M₂
theorem range_inr : range (inr R M M₂) = ker (fst R M M₂) :=
begin
ext x,
simp only [mem_ker, mem_range],
split,
{ rintros ⟨y, rfl⟩, refl },
{ intro h, exact ⟨x.snd, prod.ext h.symm rfl⟩ }
end
theorem ker_fst : ker (fst R M M₂) = range (inr R M M₂) :=
eq.symm $ range_inr R M M₂
end
@[simp] theorem coe_inl : (inl R M M₂ : M → M × M₂) = λ x, (x, 0) := rfl
theorem inl_apply (x : M) : inl R M M₂ x = (x, 0) := rfl
@[simp] theorem coe_inr : (inr R M M₂ : M₂ → M × M₂) = prod.mk 0 := rfl
theorem inr_apply (x : M₂) : inr R M M₂ x = (0, x) := rfl
theorem inl_eq_prod : inl R M M₂ = prod linear_map.id 0 := rfl
theorem inr_eq_prod : inr R M M₂ = prod 0 linear_map.id := rfl
theorem inl_injective : function.injective (inl R M M₂) :=
λ _, by simp
theorem inr_injective : function.injective (inr R M M₂) :=
λ _, by simp
/-- The coprod function `λ x : M × M₂, f x.1 + g x.2` is a linear map. -/
def coprod (f : M →ₗ[R] M₃) (g : M₂ →ₗ[R] M₃) : M × M₂ →ₗ[R] M₃ :=
f.comp (fst _ _ _) + g.comp (snd _ _ _)
@[simp] theorem coprod_apply (f : M →ₗ[R] M₃) (g : M₂ →ₗ[R] M₃) (x : M × M₂) :
coprod f g x = f x.1 + g x.2 := rfl
@[simp] theorem coprod_inl (f : M →ₗ[R] M₃) (g : M₂ →ₗ[R] M₃) :
(coprod f g).comp (inl R M M₂) = f :=
by ext; simp only [map_zero, add_zero, coprod_apply, inl_apply, comp_apply]
@[simp] theorem coprod_inr (f : M →ₗ[R] M₃) (g : M₂ →ₗ[R] M₃) :
(coprod f g).comp (inr R M M₂) = g :=
by ext; simp only [map_zero, coprod_apply, inr_apply, zero_add, comp_apply]
@[simp] theorem coprod_inl_inr : coprod (inl R M M₂) (inr R M M₂) = linear_map.id :=
by ext; simp only [prod.mk_add_mk, add_zero, id_apply, coprod_apply,
inl_apply, inr_apply, zero_add]
theorem comp_coprod (f : M₃ →ₗ[R] M₄) (g₁ : M →ₗ[R] M₃) (g₂ : M₂ →ₗ[R] M₃) :
f.comp (g₁.coprod g₂) = (f.comp g₁).coprod (f.comp g₂) :=
ext $ λ x, f.map_add (g₁ x.1) (g₂ x.2)
theorem fst_eq_coprod : fst R M M₂ = coprod linear_map.id 0 := by ext; simp
theorem snd_eq_coprod : snd R M M₂ = coprod 0 linear_map.id := by ext; simp
@[simp] theorem coprod_comp_prod (f : M₂ →ₗ[R] M₄) (g : M₃ →ₗ[R] M₄)
(f' : M →ₗ[R] M₂) (g' : M →ₗ[R] M₃) :
(f.coprod g).comp (f'.prod g') = f.comp f' + g.comp g' :=
rfl
@[simp]
lemma coprod_map_prod (f : M →ₗ[R] M₃) (g : M₂ →ₗ[R] M₃) (S : submodule R M)
(S' : submodule R M₂) :
(submodule.prod S S').map (linear_map.coprod f g) = S.map f ⊔ S'.map g :=
set_like.coe_injective $ begin
simp only [linear_map.coprod_apply, submodule.coe_sup, submodule.map_coe],
rw [←set.image2_add, set.image2_image_left, set.image2_image_right],
exact set.image_prod (λ m m₂, f m + g m₂),
end
/-- Taking the product of two maps with the same codomain is equivalent to taking the product of
their domains.
See note [bundled maps over different rings] for why separate `R` and `S` semirings are used. -/
@[simps] def coprod_equiv [module S M₃] [smul_comm_class R S M₃] :
((M →ₗ[R] M₃) × (M₂ →ₗ[R] M₃)) ≃ₗ[S] (M × M₂ →ₗ[R] M₃) :=
{ to_fun := λ f, f.1.coprod f.2,
inv_fun := λ f, (f.comp (inl _ _ _), f.comp (inr _ _ _)),
left_inv := λ f, by simp only [prod.mk.eta, coprod_inl, coprod_inr],
right_inv := λ f, by simp only [←comp_coprod, comp_id, coprod_inl_inr],
map_add' := λ a b,
by { ext, simp only [prod.snd_add, add_apply, coprod_apply, prod.fst_add, add_add_add_comm] },
map_smul' := λ r a,
by { dsimp, ext, simp only [smul_add, smul_apply, prod.smul_snd, prod.smul_fst,
coprod_apply] } }
theorem prod_ext_iff {f g : M × M₂ →ₗ[R] M₃} :
f = g ↔ f.comp (inl _ _ _) = g.comp (inl _ _ _) ∧ f.comp (inr _ _ _) = g.comp (inr _ _ _) :=
(coprod_equiv ℕ).symm.injective.eq_iff.symm.trans prod.ext_iff
/--
Split equality of linear maps from a product into linear maps over each component, to allow `ext`
to apply lemmas specific to `M →ₗ M₃` and `M₂ →ₗ M₃`.
See note [partially-applied ext lemmas]. -/
@[ext] theorem prod_ext {f g : M × M₂ →ₗ[R] M₃}
(hl : f.comp (inl _ _ _) = g.comp (inl _ _ _))
(hr : f.comp (inr _ _ _) = g.comp (inr _ _ _)) :
f = g :=
prod_ext_iff.2 ⟨hl, hr⟩
/-- `prod.map` of two linear maps. -/
def prod_map (f : M →ₗ[R] M₃) (g : M₂ →ₗ[R] M₄) : (M × M₂) →ₗ[R] (M₃ × M₄) :=
(f.comp (fst R M M₂)).prod (g.comp (snd R M M₂))
@[simp] theorem prod_map_apply (f : M →ₗ[R] M₃) (g : M₂ →ₗ[R] M₄) (x) :
f.prod_map g x = (f x.1, g x.2) := rfl
lemma prod_map_comap_prod (f : M →ₗ[R] M₂) (g : M₃ →ₗ[R] M₄) (S : submodule R M₂)
(S' : submodule R M₄) :
(submodule.prod S S').comap (linear_map.prod_map f g) = (S.comap f).prod (S'.comap g) :=
set_like.coe_injective $ set.preimage_prod_map_prod f g _ _
lemma ker_prod_map (f : M →ₗ[R] M₂) (g : M₃ →ₗ[R] M₄) :
(linear_map.prod_map f g).ker = submodule.prod f.ker g.ker :=
begin
dsimp only [ker],
rw [←prod_map_comap_prod, submodule.prod_bot],
end
@[simp]
lemma prod_map_id : (id : M →ₗ[R] M).prod_map (id : M₂ →ₗ[R] M₂) = id :=
linear_map.ext $ λ _, prod.mk.eta
@[simp]
lemma prod_map_one : (1 : M →ₗ[R] M).prod_map (1 : M₂ →ₗ[R] M₂) = 1 :=
linear_map.ext $ λ _, prod.mk.eta
lemma prod_map_comp (f₁₂ : M →ₗ[R] M₂) (f₂₃ : M₂ →ₗ[R] M₃) (g₁₂ : M₄ →ₗ[R] M₅) (g₂₃ : M₅ →ₗ[R] M₆) :
f₂₃.prod_map g₂₃ ∘ₗ f₁₂.prod_map g₁₂ = (f₂₃ ∘ₗ f₁₂).prod_map (g₂₃ ∘ₗ g₁₂) := rfl
lemma prod_map_mul (f₁₂ : M →ₗ[R] M) (f₂₃ : M →ₗ[R] M) (g₁₂ : M₂ →ₗ[R] M₂) (g₂₃ : M₂ →ₗ[R] M₂) :
f₂₃.prod_map g₂₃ * f₁₂.prod_map g₁₂ = (f₂₃ * f₁₂).prod_map (g₂₃ * g₁₂) := rfl
lemma prod_map_add (f₁ : M →ₗ[R] M₃) (f₂ : M →ₗ[R] M₃) (g₁ : M₂ →ₗ[R] M₄) (g₂ : M₂ →ₗ[R] M₄) :
(f₁ + f₂).prod_map (g₁ + g₂) = f₁.prod_map g₁ + f₂.prod_map g₂ := rfl
@[simp] lemma prod_map_zero :
(0 : M →ₗ[R] M₂).prod_map (0 : M₃ →ₗ[R] M₄) = 0 := rfl
@[simp] lemma prod_map_smul
[module S M₃] [module S M₄] [smul_comm_class R S M₃] [smul_comm_class R S M₄]
(s : S) (f : M →ₗ[R] M₃) (g : M₂ →ₗ[R] M₄) : prod_map (s • f) (s • g) = s • prod_map f g := rfl
variables (R M M₂ M₃ M₄)
/-- `linear_map.prod_map` as a `linear_map` -/
@[simps]
def prod_map_linear
[module S M₃] [module S M₄] [smul_comm_class R S M₃] [smul_comm_class R S M₄] :
((M →ₗ[R] M₃) × (M₂ →ₗ[R] M₄)) →ₗ[S] ((M × M₂) →ₗ[R] (M₃ × M₄)) :=
{ to_fun := λ f, prod_map f.1 f.2,
map_add' := λ _ _, rfl,
map_smul' := λ _ _, rfl}
/-- `linear_map.prod_map` as a `ring_hom` -/
@[simps]
def prod_map_ring_hom : (M →ₗ[R] M) × (M₂ →ₗ[R] M₂) →+* ((M × M₂) →ₗ[R] (M × M₂)) :=
{ to_fun := λ f, prod_map f.1 f.2,
map_one' := prod_map_one,
map_zero' := rfl,
map_add' := λ _ _, rfl,
map_mul' := λ _ _, rfl }
variables {R M M₂ M₃ M₄}
section map_mul
variables {A : Type*} [non_unital_non_assoc_semiring A] [module R A]
variables {B : Type*} [non_unital_non_assoc_semiring B] [module R B]
lemma inl_map_mul (a₁ a₂ : A) : linear_map.inl R A B (a₁ * a₂) =
linear_map.inl R A B a₁ * linear_map.inl R A B a₂ :=
prod.ext rfl (by simp)
lemma inr_map_mul (b₁ b₂ : B) : linear_map.inr R A B (b₁ * b₂) =
linear_map.inr R A B b₁ * linear_map.inr R A B b₂ :=
prod.ext (by simp) rfl
end map_mul
end linear_map
end prod
namespace linear_map
variables (R M M₂)
variables [comm_semiring R]
variables [add_comm_monoid M] [add_comm_monoid M₂]
variables [module R M] [module R M₂]
/-- `linear_map.prod_map` as an `algebra_hom` -/
@[simps]
def prod_map_alg_hom : (module.End R M) × (module.End R M₂) →ₐ[R] module.End R (M × M₂) :=
{ commutes' := λ _, rfl, ..prod_map_ring_hom R M M₂ }
end linear_map
namespace linear_map
open submodule
variables [semiring R]
[add_comm_monoid M] [add_comm_monoid M₂] [add_comm_monoid M₃] [add_comm_monoid M₄]
[module R M] [module R M₂] [module R M₃] [module R M₄]
lemma range_coprod (f : M →ₗ[R] M₃) (g : M₂ →ₗ[R] M₃) :
(f.coprod g).range = f.range ⊔ g.range :=
submodule.ext $ λ x, by simp [mem_sup]
lemma is_compl_range_inl_inr : is_compl (inl R M M₂).range (inr R M M₂).range :=
begin
split,
{ rintros ⟨_, _⟩ ⟨⟨x, hx⟩, ⟨y, hy⟩⟩,
simp only [prod.ext_iff, inl_apply, inr_apply, mem_bot] at hx hy ⊢,
exact ⟨hy.1.symm, hx.2.symm⟩ },
{ rintros ⟨x, y⟩ -,
simp only [mem_sup, mem_range, exists_prop],
refine ⟨(x, 0), ⟨x, rfl⟩, (0, y), ⟨y, rfl⟩, _⟩,
simp }
end
lemma sup_range_inl_inr : (inl R M M₂).range ⊔ (inr R M M₂).range = ⊤ :=
is_compl_range_inl_inr.sup_eq_top
lemma disjoint_inl_inr : disjoint (inl R M M₂).range (inr R M M₂).range :=
by simp [disjoint_def, @eq_comm M 0, @eq_comm M₂ 0] {contextual := tt}; intros; refl
theorem map_coprod_prod (f : M →ₗ[R] M₃) (g : M₂ →ₗ[R] M₃)
(p : submodule R M) (q : submodule R M₂) :
map (coprod f g) (p.prod q) = map f p ⊔ map g q :=
begin
refine le_antisymm _ (sup_le (map_le_iff_le_comap.2 _) (map_le_iff_le_comap.2 _)),
{ rw set_like.le_def, rintro _ ⟨x, ⟨h₁, h₂⟩, rfl⟩,
exact mem_sup.2 ⟨_, ⟨_, h₁, rfl⟩, _, ⟨_, h₂, rfl⟩, rfl⟩ },
{ exact λ x hx, ⟨(x, 0), by simp [hx]⟩ },
{ exact λ x hx, ⟨(0, x), by simp [hx]⟩ }
end
theorem comap_prod_prod (f : M →ₗ[R] M₂) (g : M →ₗ[R] M₃)
(p : submodule R M₂) (q : submodule R M₃) :
comap (prod f g) (p.prod q) = comap f p ⊓ comap g q :=
submodule.ext $ λ x, iff.rfl
theorem prod_eq_inf_comap (p : submodule R M) (q : submodule R M₂) :
p.prod q = p.comap (linear_map.fst R M M₂) ⊓ q.comap (linear_map.snd R M M₂) :=
submodule.ext $ λ x, iff.rfl
theorem prod_eq_sup_map (p : submodule R M) (q : submodule R M₂) :
p.prod q = p.map (linear_map.inl R M M₂) ⊔ q.map (linear_map.inr R M M₂) :=
by rw [← map_coprod_prod, coprod_inl_inr, map_id]
lemma span_inl_union_inr {s : set M} {t : set M₂} :
span R (inl R M M₂ '' s ∪ inr R M M₂ '' t) = (span R s).prod (span R t) :=
by rw [span_union, prod_eq_sup_map, ← span_image, ← span_image]
@[simp] lemma ker_prod (f : M →ₗ[R] M₂) (g : M →ₗ[R] M₃) :
ker (prod f g) = ker f ⊓ ker g :=
by rw [ker, ← prod_bot, comap_prod_prod]; refl
lemma range_prod_le (f : M →ₗ[R] M₂) (g : M →ₗ[R] M₃) :
range (prod f g) ≤ (range f).prod (range g) :=
begin
simp only [set_like.le_def, prod_apply, mem_range, set_like.mem_coe, mem_prod,
exists_imp_distrib],
rintro _ x rfl,
exact ⟨⟨x, rfl⟩, ⟨x, rfl⟩⟩
end
lemma ker_prod_ker_le_ker_coprod {M₂ : Type*} [add_comm_group M₂] [module R M₂]
{M₃ : Type*} [add_comm_group M₃] [module R M₃]
(f : M →ₗ[R] M₃) (g : M₂ →ₗ[R] M₃) :
(ker f).prod (ker g) ≤ ker (f.coprod g) :=
by { rintros ⟨y, z⟩, simp {contextual := tt} }
lemma ker_coprod_of_disjoint_range {M₂ : Type*} [add_comm_group M₂] [module R M₂]
{M₃ : Type*} [add_comm_group M₃] [module R M₃]
(f : M →ₗ[R] M₃) (g : M₂ →ₗ[R] M₃) (hd : disjoint f.range g.range) :
ker (f.coprod g) = (ker f).prod (ker g) :=
begin
apply le_antisymm _ (ker_prod_ker_le_ker_coprod f g),
rintros ⟨y, z⟩ h,
simp only [mem_ker, mem_prod, coprod_apply] at h ⊢,
have : f y ∈ f.range ⊓ g.range,
{ simp only [true_and, mem_range, mem_inf, exists_apply_eq_apply],
use -z,
rwa [eq_comm, map_neg, ← sub_eq_zero, sub_neg_eq_add] },
rw [hd.eq_bot, mem_bot] at this,
rw [this] at h,
simpa [this] using h,
end
end linear_map
namespace submodule
open linear_map
variables [semiring R]
variables [add_comm_monoid M] [add_comm_monoid M₂]
variables [module R M] [module R M₂]
lemma sup_eq_range (p q : submodule R M) : p ⊔ q = (p.subtype.coprod q.subtype).range :=
submodule.ext $ λ x, by simp [submodule.mem_sup, set_like.exists]
variables (p : submodule R M) (q : submodule R M₂)
@[simp] theorem map_inl : p.map (inl R M M₂) = prod p ⊥ :=
by { ext ⟨x, y⟩, simp only [and.left_comm, eq_comm, mem_map, prod.mk.inj_iff, inl_apply, mem_bot,
exists_eq_left', mem_prod] }
@[simp] theorem map_inr : q.map (inr R M M₂) = prod ⊥ q :=
by ext ⟨x, y⟩; simp [and.left_comm, eq_comm]
@[simp] theorem comap_fst : p.comap (fst R M M₂) = prod p ⊤ :=
by ext ⟨x, y⟩; simp
@[simp] theorem comap_snd : q.comap (snd R M M₂) = prod ⊤ q :=
by ext ⟨x, y⟩; simp
@[simp] theorem prod_comap_inl : (prod p q).comap (inl R M M₂) = p := by ext; simp
@[simp] theorem prod_comap_inr : (prod p q).comap (inr R M M₂) = q := by ext; simp
@[simp] theorem prod_map_fst : (prod p q).map (fst R M M₂) = p :=
by ext x; simp [(⟨0, zero_mem _⟩ : ∃ x, x ∈ q)]
@[simp] theorem prod_map_snd : (prod p q).map (snd R M M₂) = q :=
by ext x; simp [(⟨0, zero_mem _⟩ : ∃ x, x ∈ p)]
@[simp] theorem ker_inl : (inl R M M₂).ker = ⊥ :=
by rw [ker, ← prod_bot, prod_comap_inl]
@[simp] theorem ker_inr : (inr R M M₂).ker = ⊥ :=
by rw [ker, ← prod_bot, prod_comap_inr]
@[simp] theorem range_fst : (fst R M M₂).range = ⊤ :=
by rw [range_eq_map, ← prod_top, prod_map_fst]
@[simp] theorem range_snd : (snd R M M₂).range = ⊤ :=
by rw [range_eq_map, ← prod_top, prod_map_snd]
variables (R M M₂)
/-- `M` as a submodule of `M × N`. -/
def fst : submodule R (M × M₂) := (⊥ : submodule R M₂).comap (linear_map.snd R M M₂)
/-- `M` as a submodule of `M × N` is isomorphic to `M`. -/
@[simps] def fst_equiv : submodule.fst R M M₂ ≃ₗ[R] M :=
{ to_fun := λ x, x.1.1,
inv_fun := λ m, ⟨⟨m, 0⟩, by tidy⟩,
map_add' := by simp,
map_smul' := by simp,
left_inv := by tidy,
right_inv := by tidy, }
lemma fst_map_fst : (submodule.fst R M M₂).map (linear_map.fst R M M₂) = ⊤ :=
by tidy
lemma fst_map_snd : (submodule.fst R M M₂).map (linear_map.snd R M M₂) = ⊥ :=
by { tidy, exact 0, }
/-- `N` as a submodule of `M × N`. -/
def snd : submodule R (M × M₂) := (⊥ : submodule R M).comap (linear_map.fst R M M₂)
/-- `N` as a submodule of `M × N` is isomorphic to `N`. -/
@[simps] def snd_equiv : submodule.snd R M M₂ ≃ₗ[R] M₂ :=
{ to_fun := λ x, x.1.2,
inv_fun := λ n, ⟨⟨0, n⟩, by tidy⟩,
map_add' := by simp,
map_smul' := by simp,
left_inv := by tidy,
right_inv := by tidy, }
lemma snd_map_fst : (submodule.snd R M M₂).map (linear_map.fst R M M₂) = ⊥ :=
by { tidy, exact 0, }
lemma snd_map_snd : (submodule.snd R M M₂).map (linear_map.snd R M M₂) = ⊤ :=
by tidy
lemma fst_sup_snd : submodule.fst R M M₂ ⊔ submodule.snd R M M₂ = ⊤ :=
begin
rw eq_top_iff,
rintro ⟨m, n⟩ -,
rw [show (m, n) = (m, 0) + (0, n), by simp],
apply submodule.add_mem (submodule.fst R M M₂ ⊔ submodule.snd R M M₂),
{ exact submodule.mem_sup_left (submodule.mem_comap.mpr (by simp)), },
{ exact submodule.mem_sup_right (submodule.mem_comap.mpr (by simp)), },
end
lemma fst_inf_snd : submodule.fst R M M₂ ⊓ submodule.snd R M M₂ = ⊥ := by tidy
lemma le_prod_iff {p₁ : submodule R M} {p₂ : submodule R M₂} {q : submodule R (M × M₂)} :
q ≤ p₁.prod p₂ ↔ map (linear_map.fst R M M₂) q ≤ p₁ ∧ map (linear_map.snd R M M₂) q ≤ p₂ :=
begin
split,
{ intros h,
split,
{ rintros x ⟨⟨y1,y2⟩, ⟨hy1,rfl⟩⟩, exact (h hy1).1 },
{ rintros x ⟨⟨y1,y2⟩, ⟨hy1,rfl⟩⟩, exact (h hy1).2 }, },
{ rintros ⟨hH, hK⟩ ⟨x1, x2⟩ h, exact ⟨hH ⟨_ , h, rfl⟩, hK ⟨ _, h, rfl⟩⟩, }
end
lemma prod_le_iff {p₁ : submodule R M} {p₂ : submodule R M₂} {q : submodule R (M × M₂)} :
p₁.prod p₂ ≤ q ↔ map (linear_map.inl R M M₂) p₁ ≤ q ∧ map (linear_map.inr R M M₂) p₂ ≤ q :=
begin
split,
{ intros h,
split,
{ rintros _ ⟨x, hx, rfl⟩, apply h, exact ⟨hx, zero_mem p₂⟩, },
{ rintros _ ⟨x, hx, rfl⟩, apply h, exact ⟨zero_mem p₁, hx⟩, }, },
{ rintros ⟨hH, hK⟩ ⟨x1, x2⟩ ⟨h1, h2⟩,
have h1' : (linear_map.inl R _ _) x1 ∈ q, { apply hH, simpa using h1, },
have h2' : (linear_map.inr R _ _) x2 ∈ q, { apply hK, simpa using h2, },
simpa using add_mem h1' h2', }
end
lemma prod_eq_bot_iff {p₁ : submodule R M} {p₂ : submodule R M₂} :
p₁.prod p₂ = ⊥ ↔ p₁ = ⊥ ∧ p₂ = ⊥ :=
by simp only [eq_bot_iff, prod_le_iff, (gc_map_comap _).le_iff_le, comap_bot, ker_inl, ker_inr]
lemma prod_eq_top_iff {p₁ : submodule R M} {p₂ : submodule R M₂} :
p₁.prod p₂ = ⊤ ↔ p₁ = ⊤ ∧ p₂ = ⊤ :=
by simp only [eq_top_iff, le_prod_iff, ← (gc_map_comap _).le_iff_le, map_top, range_fst, range_snd]
end submodule
namespace linear_equiv
/-- Product of modules is commutative up to linear isomorphism. -/
@[simps apply]
def prod_comm (R M N : Type*) [semiring R] [add_comm_monoid M] [add_comm_monoid N]
[module R M] [module R N] : (M × N) ≃ₗ[R] (N × M) :=
{ to_fun := prod.swap,
map_smul' := λ r ⟨m, n⟩, rfl,
..add_equiv.prod_comm }
section
variables [semiring R]
variables [add_comm_monoid M] [add_comm_monoid M₂] [add_comm_monoid M₃] [add_comm_monoid M₄]
variables {module_M : module R M} {module_M₂ : module R M₂}
variables {module_M₃ : module R M₃} {module_M₄ : module R M₄}
variables (e₁ : M ≃ₗ[R] M₂) (e₂ : M₃ ≃ₗ[R] M₄)
/-- Product of linear equivalences; the maps come from `equiv.prod_congr`. -/
protected def prod :
(M × M₃) ≃ₗ[R] (M₂ × M₄) :=
{ map_smul' := λ c x, prod.ext (e₁.map_smulₛₗ c _) (e₂.map_smulₛₗ c _),
.. e₁.to_add_equiv.prod_congr e₂.to_add_equiv }
lemma prod_symm : (e₁.prod e₂).symm = e₁.symm.prod e₂.symm := rfl
@[simp] lemma prod_apply (p) :
e₁.prod e₂ p = (e₁ p.1, e₂ p.2) := rfl
@[simp, norm_cast] lemma coe_prod :
(e₁.prod e₂ : (M × M₃) →ₗ[R] (M₂ × M₄)) = (e₁ : M →ₗ[R] M₂).prod_map (e₂ : M₃ →ₗ[R] M₄) := rfl
end
section
variables [semiring R]
variables [add_comm_monoid M] [add_comm_monoid M₂] [add_comm_monoid M₃] [add_comm_group M₄]
variables {module_M : module R M} {module_M₂ : module R M₂}
variables {module_M₃ : module R M₃} {module_M₄ : module R M₄}
variables (e₁ : M ≃ₗ[R] M₂) (e₂ : M₃ ≃ₗ[R] M₄)
/-- Equivalence given by a block lower diagonal matrix. `e₁` and `e₂` are diagonal square blocks,
and `f` is a rectangular block below the diagonal. -/
protected def skew_prod (f : M →ₗ[R] M₄) :
(M × M₃) ≃ₗ[R] M₂ × M₄ :=
{ inv_fun := λ p : M₂ × M₄, (e₁.symm p.1, e₂.symm (p.2 - f (e₁.symm p.1))),
left_inv := λ p, by simp,
right_inv := λ p, by simp,
.. ((e₁ : M →ₗ[R] M₂).comp (linear_map.fst R M M₃)).prod
((e₂ : M₃ →ₗ[R] M₄).comp (linear_map.snd R M M₃) +
f.comp (linear_map.fst R M M₃)) }
@[simp] lemma skew_prod_apply (f : M →ₗ[R] M₄) (x) :
e₁.skew_prod e₂ f x = (e₁ x.1, e₂ x.2 + f x.1) := rfl
@[simp] lemma skew_prod_symm_apply (f : M →ₗ[R] M₄) (x) :
(e₁.skew_prod e₂ f).symm x = (e₁.symm x.1, e₂.symm (x.2 - f (e₁.symm x.1))) := rfl
end
end linear_equiv
namespace linear_map
open submodule
variables [ring R]
variables [add_comm_group M] [add_comm_group M₂] [add_comm_group M₃]
variables [module R M] [module R M₂] [module R M₃]
/-- If the union of the kernels `ker f` and `ker g` spans the domain, then the range of
`prod f g` is equal to the product of `range f` and `range g`. -/
lemma range_prod_eq {f : M →ₗ[R] M₂} {g : M →ₗ[R] M₃} (h : ker f ⊔ ker g = ⊤) :
range (prod f g) = (range f).prod (range g) :=
begin
refine le_antisymm (f.range_prod_le g) _,
simp only [set_like.le_def, prod_apply, mem_range, set_like.mem_coe, mem_prod, exists_imp_distrib,
and_imp, prod.forall, pi.prod],
rintros _ _ x rfl y rfl,
simp only [prod.mk.inj_iff, ← sub_mem_ker_iff],
have : y - x ∈ ker f ⊔ ker g, { simp only [h, mem_top] },
rcases mem_sup.1 this with ⟨x', hx', y', hy', H⟩,
refine ⟨x' + x, _, _⟩,
{ rwa add_sub_cancel },
{ rwa [← eq_sub_iff_add_eq.1 H, add_sub_add_right_eq_sub, ← neg_mem_iff, neg_sub,
add_sub_cancel'] }
end
end linear_map
namespace linear_map
/-!
## Tunnels and tailings
Some preliminary work for establishing the strong rank condition for noetherian rings.
Given a morphism `f : M × N →ₗ[R] M` which is `i : injective f`,
we can find an infinite decreasing `tunnel f i n` of copies of `M` inside `M`,
and sitting beside these, an infinite sequence of copies of `N`.
We picturesquely name these as `tailing f i n` for each individual copy of `N`,
and `tailings f i n` for the supremum of the first `n+1` copies:
they are the pieces left behind, sitting inside the tunnel.
By construction, each `tailing f i (n+1)` is disjoint from `tailings f i n`;
later, when we assume `M` is noetherian, this implies that `N` must be trivial,
and establishes the strong rank condition for any left-noetherian ring.
-/
section tunnel
-- (This doesn't work over a semiring: we need to use that `submodule R M` is a modular lattice,
-- which requires cancellation.)
variables [ring R]
variables {N : Type*} [add_comm_group M] [module R M] [add_comm_group N] [module R N]
open function
/-- An auxiliary construction for `tunnel`.
The composition of `f`, followed by the isomorphism back to `K`,
followed by the inclusion of this submodule back into `M`. -/
def tunnel_aux (f : M × N →ₗ[R] M) (Kφ : Σ K : submodule R M, K ≃ₗ[R] M) :
M × N →ₗ[R] M :=
(Kφ.1.subtype.comp Kφ.2.symm.to_linear_map).comp f
lemma tunnel_aux_injective
(f : M × N →ₗ[R] M) (i : injective f) (Kφ : Σ K : submodule R M, K ≃ₗ[R] M) :
injective (tunnel_aux f Kφ) :=
(subtype.val_injective.comp Kφ.2.symm.injective).comp i
noncomputable theory
/-- Auxiliary definition for `tunnel`. -/
-- Even though we have `noncomputable theory`,
-- we get an error without another `noncomputable` here.
noncomputable def tunnel' (f : M × N →ₗ[R] M) (i : injective f) :
ℕ → Σ (K : submodule R M), K ≃ₗ[R] M
| 0 := ⟨⊤, linear_equiv.of_top ⊤ rfl⟩
| (n+1) :=
⟨(submodule.fst R M N).map (tunnel_aux f (tunnel' n)),
((submodule.fst R M N).equiv_map_of_injective _ (tunnel_aux_injective f i (tunnel' n))).symm.trans
(submodule.fst_equiv R M N)⟩
/--
Give an injective map `f : M × N →ₗ[R] M` we can find a nested sequence of submodules
all isomorphic to `M`.
-/
def tunnel (f : M × N →ₗ[R] M) (i : injective f) : ℕ →o (submodule R M)ᵒᵈ :=
⟨λ n, (tunnel' f i n).1, monotone_nat_of_le_succ (λ n, begin
dsimp [tunnel', tunnel_aux],
rw [submodule.map_comp, submodule.map_comp],
apply submodule.map_subtype_le,
end)⟩
/--
Give an injective map `f : M × N →ₗ[R] M` we can find a sequence of submodules
all isomorphic to `N`.
-/
def tailing (f : M × N →ₗ[R] M) (i : injective f) (n : ℕ) : submodule R M :=
(submodule.snd R M N).map (tunnel_aux f (tunnel' f i n))
/-- Each `tailing f i n` is a copy of `N`. -/
def tailing_linear_equiv (f : M × N →ₗ[R] M) (i : injective f) (n : ℕ) : tailing f i n ≃ₗ[R] N :=
((submodule.snd R M N).equiv_map_of_injective _
(tunnel_aux_injective f i (tunnel' f i n))).symm.trans (submodule.snd_equiv R M N)
lemma tailing_le_tunnel (f : M × N →ₗ[R] M) (i : injective f) (n : ℕ) :
tailing f i n ≤ tunnel f i n :=
begin
dsimp [tailing, tunnel_aux],
rw [submodule.map_comp, submodule.map_comp],
apply submodule.map_subtype_le,
end
lemma tailing_disjoint_tunnel_succ (f : M × N →ₗ[R] M) (i : injective f) (n : ℕ) :
disjoint (tailing f i n) (tunnel f i (n+1)) :=
begin
rw disjoint_iff,
dsimp [tailing, tunnel, tunnel'],
rw [submodule.map_inf_eq_map_inf_comap,
submodule.comap_map_eq_of_injective (tunnel_aux_injective _ i _), inf_comm,
submodule.fst_inf_snd, submodule.map_bot],
end
lemma tailing_sup_tunnel_succ_le_tunnel (f : M × N →ₗ[R] M) (i : injective f) (n : ℕ) :
tailing f i n ⊔ tunnel f i (n+1) ≤ tunnel f i n :=
begin
dsimp [tailing, tunnel, tunnel', tunnel_aux],
rw [←submodule.map_sup, sup_comm, submodule.fst_sup_snd, submodule.map_comp, submodule.map_comp],
apply submodule.map_subtype_le,
end
/-- The supremum of all the copies of `N` found inside the tunnel. -/
def tailings (f : M × N →ₗ[R] M) (i : injective f) : ℕ → submodule R M :=
partial_sups (tailing f i)
@[simp] lemma tailings_zero (f : M × N →ₗ[R] M) (i : injective f) :
tailings f i 0 = tailing f i 0 :=
by simp [tailings]
@[simp] lemma tailings_succ (f : M × N →ₗ[R] M) (i : injective f) (n : ℕ) :
tailings f i (n+1) = tailings f i n ⊔ tailing f i (n+1) :=
by simp [tailings]
lemma tailings_disjoint_tunnel (f : M × N →ₗ[R] M) (i : injective f) (n : ℕ) :
disjoint (tailings f i n) (tunnel f i (n+1)) :=
begin
induction n with n ih,
{ simp only [tailings_zero],
apply tailing_disjoint_tunnel_succ, },
{ simp only [tailings_succ],
refine disjoint.disjoint_sup_left_of_disjoint_sup_right _ _,
apply tailing_disjoint_tunnel_succ,
apply disjoint.mono_right _ ih,
apply tailing_sup_tunnel_succ_le_tunnel, },
end
lemma tailings_disjoint_tailing (f : M × N →ₗ[R] M) (i : injective f) (n : ℕ) :
disjoint (tailings f i n) (tailing f i (n+1)) :=
disjoint.mono_right (tailing_le_tunnel f i _) (tailings_disjoint_tunnel f i _)
end tunnel
section graph
variables [semiring R] [add_comm_monoid M] [add_comm_monoid M₂]
[add_comm_group M₃] [add_comm_group M₄] [module R M] [module R M₂]
[module R M₃] [module R M₄] (f : M →ₗ[R] M₂) (g : M₃ →ₗ[R] M₄)
/-- Graph of a linear map. -/
def graph : submodule R (M × M₂) :=
{ carrier := {p | p.2 = f p.1},
add_mem' := λ a b (ha : _ = _) (hb : _ = _),
begin
change _ + _ = f (_ + _),
rw [map_add, ha, hb]
end,
zero_mem' := eq.symm (map_zero f),
smul_mem' := λ c x (hx : _ = _),
begin
change _ • _ = f (_ • _),
rw [map_smul, hx]
end }
@[simp] lemma mem_graph_iff (x : M × M₂) : x ∈ f.graph ↔ x.2 = f x.1 := iff.rfl
lemma graph_eq_ker_coprod : g.graph = ((-g).coprod linear_map.id).ker :=
begin
ext x,
change _ = _ ↔ -(g x.1) + x.2 = _,
rw [add_comm, add_neg_eq_zero]
end
lemma graph_eq_range_prod : f.graph = (linear_map.id.prod f).range :=
begin
ext x,
exact ⟨λ hx, ⟨x.1, prod.ext rfl hx.symm⟩, λ ⟨u, hu⟩, hu ▸ rfl⟩
end
end graph
end linear_map
|
e8b1f3e96597ada7aa1f2b8ae3f0c97e97e3d923 | 1a61aba1b67cddccce19532a9596efe44be4285f | /library/init/setoid.lean | a253d90ee4dde59314bbbd433ab84c76a1eb8e12 | [
"Apache-2.0"
] | permissive | eigengrau/lean | 07986a0f2548688c13ba36231f6cdbee82abf4c6 | f8a773be1112015e2d232661ce616d23f12874d0 | refs/heads/master | 1,610,939,198,566 | 1,441,352,386,000 | 1,441,352,494,000 | 41,903,576 | 0 | 0 | null | 1,441,352,210,000 | 1,441,352,210,000 | null | UTF-8 | Lean | false | false | 738 | lean | /-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
prelude
import init.relation
structure setoid [class] (A : Type) :=
(r : A → A → Prop) (iseqv : equivalence r)
namespace setoid
infix `≈` := setoid.r
variable {A : Type}
variable [s : setoid A]
include s
theorem refl (a : A) : a ≈ a :=
and.elim_left (@setoid.iseqv A s) a
theorem symm {a b : A} : a ≈ b → b ≈ a :=
λ H, and.elim_left (and.elim_right (@setoid.iseqv A s)) a b H
theorem trans {a b c : A} : a ≈ b → b ≈ c → a ≈ c :=
λ H₁ H₂, and.elim_right (and.elim_right (@setoid.iseqv A s)) a b c H₁ H₂
end setoid
|
a040a68fceece0cbf4450c9026879e1a39e4bc5e | 957a80ea22c5abb4f4670b250d55534d9db99108 | /tests/lean/run/struct_value.lean | 6046a0ab200cf1b0010281094bee36a3a226bb13 | [
"Apache-2.0"
] | permissive | GaloisInc/lean | aa1e64d604051e602fcf4610061314b9a37ab8cd | f1ec117a24459b59c6ff9e56a1d09d9e9e60a6c0 | refs/heads/master | 1,592,202,909,807 | 1,504,624,387,000 | 1,504,624,387,000 | 75,319,626 | 2 | 1 | Apache-2.0 | 1,539,290,164,000 | 1,480,616,104,000 | C++ | UTF-8 | Lean | false | false | 756 | lean | #check { prod . fst := 10, snd := 20 }
#check ({ fst := 10, snd := 20 } : nat × nat)
definition p : nat × nat :=
{ snd := 20, fst := 10 }
#print p
meta def attr : user_attribute :=
{ name := `foo, descr := "hello world" }
#print attr
definition p2 :=
{ p with fst := 20 }
#print p2
structure point :=
(x : nat) (y : nat)
structure point3d extends point :=
(z : nat)
definition p1 : point := { x := 1, y := 1 }
definition p3 : point3d := { p1 with z := 10 }
#print p3
#check { point3d . x := 1, y := 2, z := 3 }
#check (⟨10, rfl⟩ : Σ' x : nat, x = x)
#check ((| 10, rfl |) : Σ' x : nat, x = x)
#check ({ fst := 10, snd := rfl } : Σ' x : nat, x = x)
definition f (a : nat) : Σ' x : nat, x = x :=
{ fst := a, snd := rfl }
#print f
|
a2ec13638e34acd2523c39d8634d3c66d6144669 | 8cae430f0a71442d02dbb1cbb14073b31048e4b0 | /src/algebra/divisibility/units.lean | 6cd70984f1a41bf8e1370a40e493182a49207ec1 | [
"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 | 3,864 | 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, Floris van Doorn, Amelia Livingston, Yury Kudryashov,
Neil Strickland, Aaron Anderson
-/
import algebra.divisibility.basic
import algebra.group.units
/-!
# Lemmas about divisibility and units
> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
> Any changes to this file require a corresponding PR to mathlib4.
-/
variables {α : Type*}
namespace units
section monoid
variables [monoid α] {a b : α} {u : αˣ}
/-- Elements of the unit group of a monoid represented as elements of the monoid
divide any element of the monoid. -/
lemma coe_dvd : ↑u ∣ a := ⟨↑u⁻¹ * a, by simp⟩
/-- In a monoid, an element `a` divides an element `b` iff `a` divides all
associates of `b`. -/
lemma dvd_mul_right : a ∣ b * u ↔ a ∣ b :=
iff.intro
(assume ⟨c, eq⟩, ⟨c * ↑u⁻¹, by rw [← mul_assoc, ← eq, units.mul_inv_cancel_right]⟩)
(assume ⟨c, eq⟩, eq.symm ▸ (dvd_mul_right _ _).mul_right _)
/-- In a monoid, an element `a` divides an element `b` iff all associates of `a` divide `b`. -/
lemma mul_right_dvd : a * u ∣ b ↔ a ∣ b :=
iff.intro
(λ ⟨c, eq⟩, ⟨↑u * c, eq.trans (mul_assoc _ _ _)⟩)
(λ h, dvd_trans (dvd.intro ↑u⁻¹ (by rw [mul_assoc, u.mul_inv, mul_one])) h)
end monoid
section comm_monoid
variables [comm_monoid α] {a b : α} {u : αˣ}
/-- In a commutative monoid, an element `a` divides an element `b` iff `a` divides all left
associates of `b`. -/
lemma dvd_mul_left : a ∣ u * b ↔ a ∣ b := by { rw mul_comm, apply dvd_mul_right }
/-- In a commutative monoid, an element `a` divides an element `b` iff all
left associates of `a` divide `b`.-/
lemma mul_left_dvd : ↑u * a ∣ b ↔ a ∣ b :=
by { rw mul_comm, apply mul_right_dvd }
end comm_monoid
end units
namespace is_unit
section monoid
variables [monoid α] {a b u : α} (hu : is_unit u)
include hu
/-- Units of a monoid divide any element of the monoid. -/
@[simp] lemma dvd : u ∣ a := by { rcases hu with ⟨u, rfl⟩, apply units.coe_dvd, }
@[simp] lemma dvd_mul_right : a ∣ b * u ↔ a ∣ b :=
by { rcases hu with ⟨u, rfl⟩, apply units.dvd_mul_right, }
/-- In a monoid, an element a divides an element b iff all associates of `a` divide `b`.-/
@[simp] lemma mul_right_dvd : a * u ∣ b ↔ a ∣ b :=
by { rcases hu with ⟨u, rfl⟩, apply units.mul_right_dvd, }
end monoid
section comm_monoid
variables [comm_monoid α] (a b u : α) (hu : is_unit u)
include hu
/-- In a commutative monoid, an element `a` divides an element `b` iff `a` divides all left
associates of `b`. -/
@[simp] lemma dvd_mul_left : a ∣ u * b ↔ a ∣ b :=
by { rcases hu with ⟨u, rfl⟩, apply units.dvd_mul_left, }
/-- In a commutative monoid, an element `a` divides an element `b` iff all
left associates of `a` divide `b`.-/
@[simp] lemma mul_left_dvd : u * a ∣ b ↔ a ∣ b :=
by { rcases hu with ⟨u, rfl⟩, apply units.mul_left_dvd, }
end comm_monoid
end is_unit
section comm_monoid
variables [comm_monoid α]
theorem is_unit_iff_dvd_one {x : α} : is_unit x ↔ x ∣ 1 :=
⟨is_unit.dvd, λ ⟨y, h⟩, ⟨⟨x, y, h.symm, by rw [h, mul_comm]⟩, rfl⟩⟩
theorem is_unit_iff_forall_dvd {x : α} :
is_unit x ↔ ∀ y, x ∣ y :=
is_unit_iff_dvd_one.trans ⟨λ h y, h.trans (one_dvd _), λ h, h _⟩
theorem is_unit_of_dvd_unit {x y : α}
(xy : x ∣ y) (hu : is_unit y) : is_unit x :=
is_unit_iff_dvd_one.2 $ xy.trans $ is_unit_iff_dvd_one.1 hu
lemma is_unit_of_dvd_one : ∀a ∣ 1, is_unit (a:α)
| a ⟨b, eq⟩ := ⟨units.mk_of_mul_eq_one a b eq.symm, rfl⟩
lemma not_is_unit_of_not_is_unit_dvd {a b : α} (ha : ¬is_unit a) (hb : a ∣ b) :
¬ is_unit b :=
mt (is_unit_of_dvd_unit hb) ha
end comm_monoid
|
f87c9a3f2ac7e2307285c9ce873f767edd23b6e5 | 74addaa0e41490cbaf2abd313a764c96df57b05d | /Mathlib/data/polynomial/degree/lemmas.lean | 658ae828537f2c1590795ad13e4f2f03358f3ce8 | [] | 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 | 3,734 | lean | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.default
import Mathlib.data.polynomial.eval
import Mathlib.tactic.interval_cases
import Mathlib.PostPort
universes u v
namespace Mathlib
/-!
# Theory of degrees of polynomials
Some of the main results include
- `nat_degree_comp_le` : The degree of the composition is at most the product of degrees
-/
namespace polynomial
theorem nat_degree_comp_le {R : Type u} [semiring R] {p : polynomial R} {q : polynomial R} : nat_degree (comp p q) ≤ nat_degree p * nat_degree q := sorry
theorem degree_map_le {R : Type u} {S : Type v} [semiring R] {p : polynomial R} [semiring S] (f : R →+* S) : degree (map f p) ≤ degree p := sorry
theorem nat_degree_map_le {R : Type u} {S : Type v} [semiring R] {p : polynomial R} [semiring S] (f : R →+* S) : nat_degree (map f p) ≤ nat_degree p := sorry
theorem degree_map_eq_of_leading_coeff_ne_zero {R : Type u} {S : Type v} [semiring R] {p : polynomial R} [semiring S] (f : R →+* S) (hf : coe_fn f (leading_coeff p) ≠ 0) : degree (map f p) = degree p := sorry
theorem nat_degree_map_of_leading_coeff_ne_zero {R : Type u} {S : Type v} [semiring R] {p : polynomial R} [semiring S] (f : R →+* S) (hf : coe_fn f (leading_coeff p) ≠ 0) : nat_degree (map f p) = nat_degree p :=
nat_degree_eq_of_degree_eq (degree_map_eq_of_leading_coeff_ne_zero f hf)
theorem leading_coeff_map_of_leading_coeff_ne_zero {R : Type u} {S : Type v} [semiring R] {p : polynomial R} [semiring S] (f : R →+* S) (hf : coe_fn f (leading_coeff p) ≠ 0) : leading_coeff (map f p) = coe_fn f (leading_coeff p) := sorry
theorem degree_pos_of_root {R : Type u} {a : R} [semiring R] {p : polynomial R} (hp : p ≠ 0) (h : is_root p a) : 0 < degree p := sorry
theorem nat_degree_pos_of_eval₂_root {R : Type u} {S : Type v} [semiring R] [semiring S] {p : polynomial R} (hp : p ≠ 0) (f : R →+* S) {z : S} (hz : eval₂ f z p = 0) (inj : ∀ (x : R), coe_fn f x = 0 → x = 0) : 0 < nat_degree p := sorry
theorem degree_pos_of_eval₂_root {R : Type u} {S : Type v} [semiring R] [semiring S] {p : polynomial R} (hp : p ≠ 0) (f : R →+* S) {z : S} (hz : eval₂ f z p = 0) (inj : ∀ (x : R), coe_fn f x = 0 → x = 0) : 0 < degree p :=
iff.mp nat_degree_pos_iff_degree_pos (nat_degree_pos_of_eval₂_root hp f hz inj)
theorem degree_map_eq_of_injective {R : Type u} {S : Type v} [semiring R] [semiring S] {f : R →+* S} (hf : function.injective ⇑f) (p : polynomial R) : degree (map f p) = degree p := sorry
theorem degree_map' {R : Type u} {S : Type v} [semiring R] [semiring S] {f : R →+* S} (hf : function.injective ⇑f) (p : polynomial R) : degree (map f p) = degree p :=
degree_map_eq_of_injective hf p
theorem nat_degree_map' {R : Type u} {S : Type v} [semiring R] [semiring S] {f : R →+* S} (hf : function.injective ⇑f) (p : polynomial R) : nat_degree (map f p) = nat_degree p :=
nat_degree_eq_of_degree_eq (degree_map' hf p)
theorem leading_coeff_map' {R : Type u} {S : Type v} [semiring R] [semiring S] {f : R →+* S} (hf : function.injective ⇑f) (p : polynomial R) : leading_coeff (map f p) = coe_fn f (leading_coeff p) := sorry
theorem monomial_nat_degree_leading_coeff_eq_self {R : Type u} [semiring R] {f : polynomial R} (h : finset.card (finsupp.support f) ≤ 1) : coe_fn (monomial (nat_degree f)) (leading_coeff f) = f := sorry
theorem C_mul_X_pow_eq_self {R : Type u} [semiring R] {f : polynomial R} (h : finset.card (finsupp.support f) ≤ 1) : coe_fn C (leading_coeff f) * X ^ nat_degree f = f := sorry
|
4a1fbd76d8cdc7b39e78d4f0a45289cf985de6ad | 4f643cce24b2d005aeeb5004c2316a8d6cc7f3b1 | /src/for_mathlib/finvec.lean | 0aee8894a77eea9c58ff9e15d27be2e5c56557dd | [] | no_license | rwbarton/lean-omin | da209ed061d64db65a8f7f71f198064986f30eb9 | fd733c6d95ef6f4743aae97de5e15df79877c00e | refs/heads/master | 1,674,408,673,325 | 1,607,343,535,000 | 1,607,343,535,000 | 285,150,399 | 9 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 12,848 | lean | import data.equiv.fin
import for_mathlib.fin
import tactic.abel
import tactic.fin_cases
/-
Finite vectors, implemented as functions on `fin n`.
The main advantage of `finvec` over alternatives like `vector` is that
extracting coordinates (components) of a `finvec` is just function application,
so it's easy to reason about things like permuting coordinates.
-/
/-- A (homogeneous) "vector" of `n` `α`s, implemented as a function from `fin n`.
The `j`th component of such a vector `x` (where `j : fin n`) is simply `x j`. -/
def finvec (n : ℕ) (α : Type*) : Type* := fin n → α
namespace finvec
variables {α : Type*}
/-- Transport a vector across an equality of dimensions.
This is implemented without using `eq.rec` so that it will reduce
when evaluated at a constructor `⟨j, h : j < n'⟩` of `fin n'`. -/
protected def cast {n n' : ℕ} (h : n = n') : finvec n α → finvec n' α :=
λ x, x ∘ fin.cast h.symm
@[simp] lemma cast_app {n n' : ℕ} {h : n = n'} {x : finvec n α} {j : fin n'} :
(x.cast h) j = x (j.cast h.symm) :=
rfl
@[simp] lemma cast_id {n : ℕ} {h : n = n} {x : finvec n α} : x.cast h = x :=
by { ext ⟨i, _⟩, refl }
section set
/-- Transport a set of vectors across an equality of dimensions. -/
protected def set.cast {n n' : ℕ} (h : n = n') : set (finvec n α) → set (finvec n' α) :=
λ s, s ∘ finvec.cast h.symm
lemma set.cast_id {n : ℕ} : finvec.set.cast rfl = (id : set (finvec n α) → set (finvec n α)) :=
begin
ext s x,
change x.cast rfl ∈ s ↔ x ∈ s,
rw cast_id
end
lemma set.heq_iff_cast_eq {n n' : ℕ} (h : n = n') {s : set (finvec n α)} {s' : set (finvec n' α)} :
s == s' ↔ s = finvec.set.cast h.symm s' :=
begin
cases h,
rw [heq_iff_eq, set.cast_id],
refl
end
end set
instance : unique (finvec 0 α) :=
⟨⟨fin_zero_elim⟩, by { intro x, ext i, fin_cases i }⟩
section prod
/-! ### Products
The isomorphism αⁿ⁺ᵐ ≅ αⁿ × αᵐ is ubiquitous in mathematics, and usually
invoked implicitly. Lean maintains a distinction between the two sides of course.
Here, we provide an API which presents `finvec (n+m) α` as though it were
a structure with two fields `left : finvec n α` and `right : finvec m α`,
with constructor `append : finvec n α → finvec m α → finvec (n+m) α`.
The arguments `n m : ℕ` are implicit throughout this API, to be inferred
from the argument or result types; this may seem problematic, since a number
can be written in the form `n+m` in many ways, but actually works well
in practice. -/
variables {n m : ℕ}
/-- Project `αⁿ⁺ᵐ ≅ αⁿ × αᵐ` to the first factor. -/
protected def left : finvec (n + m) α → finvec n α :=
λ x, x ∘ fin.cast_add m
/-- Project `αⁿ⁺ᵐ ≅ αⁿ × αᵐ` to the second factor. -/
protected def right : finvec (n + m) α → finvec m α :=
λ x, x ∘ fin.nat_add n
/-- The canonical equivalence `αⁿ⁺ᵐ ≅ αⁿ × αᵐ`. -/
def append_equiv : finvec (n + m) α ≃ finvec n α × finvec m α :=
calc (fin (n + m) → α)
≃ (fin n ⊕ fin m → α) : equiv.arrow_congr sum_fin_sum_equiv.symm (equiv.refl _)
... ≃ (fin n → α) × (fin m → α) : (equiv.sum_arrow_equiv_prod_arrow (fin n) (fin m) α)
/-- The concatenation of two vectors. -/
def append (x : finvec n α) (y : finvec m α) : finvec (n + m) α :=
append_equiv.symm (x, y)
lemma append_equiv_app {z : finvec (n + m) α} : append_equiv z = (z.left, z.right) := rfl
localized "infixr ` ++ ` := finvec.append" in finvec
lemma append_equiv_symm_app {x : finvec n α} {y : finvec m α} :
append_equiv.symm (x, y) = x ++ y :=
rfl
@[simp] lemma left_append_right (x : finvec (n + m) α) : x.left ++ x.right = x :=
append_equiv.symm_apply_apply x
/-- Induction principle for `finvec (n + m) α`,
imagined as a structure containing its left and right parts. -/
@[elab_as_eliminator]
protected lemma rec {C : finvec (n + m) α → Prop}
(h : Π (x : finvec n α) (y : finvec m α), C (x ++ y)) (z : finvec (n + m) α) : C z :=
begin
rw ←left_append_right z,
apply h
end
@[simp] lemma left_append {x : finvec n α} {y : finvec m α} : (x ++ y).left = x :=
congr_arg prod.fst (append_equiv.apply_symm_apply (x, y) : _)
@[simp] lemma right_append {x : finvec n α} {y : finvec m α} : (x ++ y).right = y :=
congr_arg prod.snd (append_equiv.apply_symm_apply (x, y) : _)
lemma prod_ext {z z' : finvec (n + m) α} :
z = z' ↔ z.left = z'.left ∧ z.right = z'.right :=
append_equiv.apply_eq_iff_eq.symm.trans prod.mk.inj_iff
lemma append.inj_iff {x x' : finvec n α} {y y' : finvec m α} :
x ++ y = x' ++ y' ↔ x = x' ∧ y = y' :=
by simp only [append, equiv.apply_eq_iff_eq, prod.mk.inj_iff]
lemma left_zero {x : finvec (n + 0) α} : x.left = x :=
by { ext ⟨_, _⟩, refl }
lemma right_zero {x : finvec (0 + n) α} : x.right = x.cast (zero_add n) :=
funext $ λ _, congr_arg x (by rw fin.nat_add_zero)
@[simp] lemma append_zero {x : finvec n α} {y : finvec 0 α} : x ++ y = x :=
begin
rw [prod_ext, left_append, left_zero, right_append],
simp
end
/-- The product (in the sense of `set.prod`) of sets of vectors,
under the isomorphism `αⁿ × αᵐ ≅ αⁿ⁺ᵐ`. -/
def prod (s : set (finvec n α)) (t : set (finvec m α)) :
set (finvec (n + m) α) :=
{z | z.left ∈ s ∧ z.right ∈ t}
localized "infixr ` ⊠ `:70 := finvec.prod" in finvec
lemma mem_prod_iff {s : set (finvec n α)} {t : set (finvec m α)}
(z : finvec (n + m) α) : z ∈ s ⊠ t ↔ z.left ∈ s ∧ z.right ∈ t :=
iff.rfl
lemma append_mem_prod_iff {s : set (finvec n α)} {t : set (finvec m α)}
{x : finvec n α} {y : finvec m α} : x ++ y ∈ s ⊠ t ↔ x ∈ s ∧ y ∈ t :=
begin
convert mem_prod_iff (x ++ y),
{ rw left_append }, { rw right_append }
end
def prod_univ (s : set (finvec n α)) (m : ℕ) : set (finvec (n + m) α) :=
{z | z.left ∈ s}
section
variables (n)
def univ_prod (n : ℕ) (t : set (finvec m α)) : set (finvec (n + m) α) :=
{z | z.right ∈ t}
end
lemma prod_univ_eq {s : set (finvec n α)} :
prod_univ s m = s ⊠ (set.univ : set (finvec m α)) :=
by { ext, exact (and_true _).symm }
lemma univ_prod_eq {t : set (finvec m α)} :
univ_prod n t = (set.univ : set (finvec n α)) ⊠ t :=
by { ext, exact (true_and _).symm }
lemma prod_eq_prod_univ_inter_univ_prod {s : set (finvec n α)} {t : set (finvec m α)} :
s ⊠ t = prod_univ s m ∩ univ_prod n t :=
rfl
@[simp] lemma append_mem_prod_univ_iff {s : set (finvec n α)}
{x : finvec n α} {y : finvec m α} : x ++ y ∈ prod_univ s m ↔ x ∈ s :=
begin
change (x ++ y).left ∈ s ↔ x ∈ s,
rw left_append
end
@[simp] lemma append_mem_univ_prod_iff {t : set (finvec m α)}
{x : finvec n α} {y : finvec m α} : x ++ y ∈ univ_prod n t ↔ y ∈ t :=
begin
change (x ++ y).right ∈ t ↔ y ∈ t,
rw right_append
end
-- Lemmas for inductively adding coordinates on the right.
-- These lemma statements (and proof) take advantage of
-- the definitional equalities `n + 0 = n`, `n + (m + 1) = (n + m) + 1`.
lemma prod_univ_zero_eq {s : set (finvec n α)} :
prod_univ s 0 = s :=
begin
ext x,
change x.left ∈ s ↔ x ∈ s,
rw left_zero
end
lemma prod_univ_plus_one_eq {s : set (finvec n α)} :
prod_univ s (m + 1) = (prod_univ (prod_univ s m) 1 : _) :=
begin
ext x,
change _ ∈ s ↔ _ ∈ s,
refl
end
-- Lemmas for inductively adding coordinates on the left.
-- For these lemmas we do not have corresponding definitional equalities
-- so TODO
lemma univ_prod_zero_like {ι : Sort*} (C : Π {l : ℕ}, set (finvec l α) → ι)
{t : set (finvec m α)} : C (univ_prod 0 t) = C t :=
begin
have : 0 + m = m := zero_add m,
congr',
rw set.heq_iff_cast_eq this,
ext x,
change x.right ∈ t ↔ x.cast _ ∈ t,
rw right_zero
end
lemma univ_prod_plus_one_like {ι : Sort*} (C : Π {l : ℕ}, set (finvec l α) → ι)
{t : set (finvec m α)} :
C (univ_prod (n + 1) t) = C (univ_prod 1 (univ_prod n t)) :=
begin
have : (n + 1) + m = 1 + (n + m) := by abel,
congr' 1, { exact this },
rw set.heq_iff_cast_eq this,
ext x,
change x.right ∈ t ↔ (x.cast _).right.right ∈ t,
congr' 2,
ext ⟨i, h⟩,
change x _ = x _,
congr,
ext,
change (n + 1) + i = 1 + (n + i),
abel
end
end prod
/-
-- TODO: is it useful to introduce this definition?
-- use in `tame_of_def`, `tame_of_constrained` etc.
def finvec_one_equiv_self {α : Type*} : finvec 1 α ≃ α :=
equiv.fun_unique (fin 1) α
-/
section snoc
-- TODO: for append notation; this is a bit ugly
open_locale finvec
/-! ### Appending a single element (snoc)
This section is parallel to the section on products
but based on the isomorphism αⁿ⁺¹ ≅ αⁿ × α.
We name the projections `init : finvec (n+1) α → finvec n α`
(which equals `left` specialized to `m = 1`) and `last : finvec (n+1) α → α`,
with constructor `snoc : finvec n α → α → finvec (n+1) α`. -/
variables {n : ℕ}
protected def init : finvec (n + 1) α → finvec n α :=
λ x, x ∘ fin.cast_succ
lemma left_eq_init {x : finvec (n + 1) α} : x.left = x.init := rfl
-- TODO: simp lemmas? see uses of `simp [finvec.last]`
protected def last : finvec (n + 1) α → α :=
λ x, x (fin.last n)
lemma right_eq_last {x : finvec (n + 1) α} : x.right = function.const (fin 1) x.last :=
begin
ext i,
fin_cases i,
refl
end
/-- The canonical equivalence `αⁿ⁺¹ ≅ αⁿ × α`. -/
def snoc_equiv : finvec (n + 1) α ≃ finvec n α × α :=
calc (fin (n + 1) → α)
≃ (fin n → α) × (fin 1 → α) : append_equiv
... ≃ (fin n → α) × α : equiv.prod_congr (equiv.refl _) (equiv.fun_unique (fin 1) α)
def snoc (x : finvec n α) (a : α) : finvec (n + 1) α :=
snoc_equiv.symm (x, a)
lemma snoc_equiv_app {z : finvec (n + 1) α} : snoc_equiv z = (z.init, z.last) := rfl
lemma snoc_equiv_symm_app {x : finvec n α} {a : α} :
snoc_equiv.symm (x, a) = x.snoc a :=
rfl
lemma init_snoc_last (x : finvec (n + 1) α) : x.init.snoc x.last = x :=
snoc_equiv.symm_apply_apply x
/-- Induction principle for `finvec (n + 1) α`,
imagined as a structure containing its initial part and last element. -/
@[elab_as_eliminator]
protected lemma snoc_rec {C : finvec (n + 1) α → Prop}
(h : Π (x : finvec n α) (a : α), C (snoc x a)) (z : finvec (n + 1) α) : C z :=
begin
rw ←init_snoc_last z,
apply h
end
@[simp] lemma init_snoc {x : finvec n α} {a : α} : (x.snoc a).init = x :=
congr_arg prod.fst (snoc_equiv.apply_symm_apply (x, a) : _)
@[simp] lemma last_snoc {x : finvec n α} {a : α} : (x.snoc a).last = a :=
congr_arg prod.snd (snoc_equiv.apply_symm_apply (x, a) : _)
lemma snoc.inj_iff {x x' : finvec n α} {a a' : α} :
x.snoc a = x'.snoc a' ↔ x = x' ∧ a = a' :=
by simp only [snoc, equiv.apply_eq_iff_eq, prod.mk.inj_iff]
-- TODO: rather than `function.const (fin 1)`
-- should we use a specialized `finvec_one_equiv_self`?
lemma snoc_eq_append {x : finvec n α} {a : α} :
x.snoc a = x ++ function.const (fin 1) a :=
begin
-- TODO: think about whether this proof is still logical
refine (left_append_right (x.snoc a : fin (n+1) → α)).symm.trans _,
rw append.inj_iff,
split,
{ rw [left_eq_init, init_snoc] },
{ rw [right_eq_last, last_snoc] }
end
-- TODO: add versions for `left`, `right`?
lemma mem_image_init {s : set (finvec (n + 1) α)} {y : finvec n α} :
y ∈ finvec.init '' s ↔ ∃ z : α, finvec.snoc y z ∈ s :=
begin
split; intro h,
{ obtain ⟨x, hx, rfl⟩ := h,
refine ⟨x.last, _⟩,
rw finvec.init_snoc_last,
exact hx },
{ obtain ⟨z, hz⟩ := h,
refine ⟨finvec.snoc y z, hz, _⟩,
simp }
end
-- TODO: comparison to prod, fin.snoc.
end snoc
section tail
-- Same as `finvec.right` but with the "wrong" type:
-- `finvec (n + 1) α → finvec n α` rather than `finvec (1 + n) α → finvec n α`.
variables {n : ℕ}
def tail (x : finvec (n + 1) α) : finvec n α :=
x ∘ fin.succ
lemma univ_prod_one_like_preimage_tail {ι : Sort*} (C : Π {l : ℕ}, set (finvec l α) → ι)
{t : set (finvec n α)} :
C (univ_prod 1 t) = C (tail ⁻¹' t) :=
begin
have : 1 + n = n + 1 := add_comm 1 n,
congr',
rw set.heq_iff_cast_eq this,
ext x,
change x.right ∈ t ↔ (x.cast _).tail ∈ t,
congr',
ext i,
change x _ = x _,
congr,
ext,
rw [fin.coe_cast, fin.coe_succ],
apply add_comm
end
end tail
/-- The canonical isomorphism `α² ≅ α × α`. -/
-- TODO: This most likely needs some lemmas. See its use in `o_minimal.mk'`.
def finvec_two_equiv_prod_self {α : Type*} : finvec 2 α ≃ α × α :=
{ to_fun := λ f, (f 0, f 1),
inv_fun := λ p, finvec.append (λ _, p.1) (λ _, p.2),
left_inv := λ f, by { ext i, fin_cases i; refl },
right_inv := λ p, by { cases p, refl } }
-- TODO: instances? e.g. topological_space, add_comm_group, etc.?
end finvec
|
b4f46b63f5e98223b545af43f9e7007ceb70706d | 624f6f2ae8b3b1adc5f8f67a365c51d5126be45a | /tests/bench/rbmap_checkpoint.lean | 46a5a0f672225e481b95ba8546a8146457b13e66 | [
"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 | 3,417 | lean | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
prelude
import Init.Data.Option.Basic
import Init.Data.List.BasicAux
import Init.Data.String
import Init.System.IO
universes u v w w'
inductive color
| Red | Black
inductive Tree
| Leaf {} : Tree
| Node (color : color) (lchild : Tree) (key : Nat) (val : Bool) (rchild : Tree) : Tree
instance : Inhabited Tree := ⟨Tree.Leaf⟩
variables {σ : Type w}
open color Nat Tree
def fold (f : Nat → Bool → σ → σ) : Tree → σ → σ
| Leaf, b => b
| Node _ l k v r, b => fold r (f k v (fold l b))
@[inline]
def balance1 : Nat → Bool → Tree → Tree → Tree
| kv, vv, t, Node _ (Node Red l kx vx r₁) ky vy r₂ => Node Red (Node Black l kx vx r₁) ky vy (Node Black r₂ kv vv t)
| kv, vv, t, Node _ l₁ ky vy (Node Red l₂ kx vx r) => Node Red (Node Black l₁ ky vy l₂) kx vx (Node Black r kv vv t)
| kv, vv, t, Node _ l ky vy r => Node Black (Node Red l ky vy r) kv vv t
| _, _, _, _ => Leaf
@[inline]
def balance2 : Tree → Nat → Bool → Tree → Tree
| t, kv, vv, Node _ (Node Red l kx₁ vx₁ r₁) ky vy r₂ => Node Red (Node Black t kv vv l) kx₁ vx₁ (Node Black r₁ ky vy r₂)
| t, kv, vv, Node _ l₁ ky vy (Node Red l₂ kx₂ vx₂ r₂) => Node Red (Node Black t kv vv l₁) ky vy (Node Black l₂ kx₂ vx₂ r₂)
| t, kv, vv, Node _ l ky vy r => Node Black t kv vv (Node Red l ky vy r)
| _, _, _, _ => Leaf
def isRed : Tree → Bool
| Node Red _ _ _ _ => true
| _ => false
def ins : Tree → Nat → Bool → Tree
| Leaf, kx, vx => Node Red Leaf kx vx Leaf
| Node Red a ky vy b, kx, vx =>
(if kx < ky then Node Red (ins a kx vx) ky vy b
else if kx = ky then Node Red a kx vx b
else Node Red a ky vy (ins b kx vx))
| Node Black a ky vy b, kx, vx =>
if kx < ky then
(if isRed a then balance1 ky vy b (ins a kx vx)
else Node Black (ins a kx vx) ky vy b)
else if kx = ky then Node Black a kx vx b
else if isRed b then balance2 a ky vy (ins b kx vx)
else Node Black a ky vy (ins b kx vx)
def setBlack : Tree → Tree
| Node _ l k v r => Node Black l k v r
| e => e
def insert (t : Tree) (k : Nat) (v : Bool) : Tree :=
if isRed t then setBlack (ins t k v)
else ins t k v
def mkMapAux (freq : Nat) : Nat → Tree → List Tree → List Tree
| 0, m, r => m::r
| n+1, m, r =>
let m := insert m n (n % 10 = 0);
let r := if n % freq == 0 then m::r else r;
mkMapAux n m r
def mkMap (n : Nat) (freq : Nat) : List Tree :=
mkMapAux freq n Leaf []
def myLen : List Tree → Nat → Nat
| Node _ _ _ _ _ :: xs, r => myLen xs (r + 1)
| _ :: xs, r => myLen xs r
| [], r => r
def main (xs : List String) : IO UInt32 := do
[n, freq] ← pure xs | throw $ IO.userError "invalid input";
let n := n.toNat!;
let freq := freq.toNat!;
let freq := if freq == 0 then 1 else freq;
let mList := mkMap n freq;
let v := fold (fun (k : Nat) (v : Bool) (r : Nat) => if v then r + 1 else r) mList.head! 0;
IO.println (toString (myLen mList 0) ++ " " ++ toString v) *>
pure 0
|
c382fa1536764b6f4e87f89bed124e204d45182a | 32025d5c2d6e33ad3b6dd8a3c91e1e838066a7f7 | /stage0/src/Lean/Meta/Tactic/LocalDecl.lean | c297fa467ca45542f225f502064f64b07a5c2b50 | [
"Apache-2.0"
] | permissive | walterhu1015/lean4 | b2c71b688975177402758924eaa513475ed6ce72 | 2214d81e84646a905d0b20b032c89caf89c737ad | refs/heads/master | 1,671,342,096,906 | 1,599,695,985,000 | 1,599,695,985,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 799 | lean | /-
Copyright (c) 2020 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Lean.Meta.AppBuilder
import Lean.Meta.Tactic.Util
import Lean.Meta.Tactic.Assert
import Lean.Meta.Tactic.Intro
import Lean.Meta.Tactic.Clear
namespace Lean
namespace Meta
def replaceLocalDecl (mvarId : MVarId) (fvarId : FVarId) (newType : Expr) (eqProof : Expr) : MetaM (FVarId × MVarId) := do
withMVarContext mvarId $ do
localDecl ← getLocalDecl fvarId;
newTypePr ← mkEqMP eqProof (mkFVar fvarId);
mvarId ← assert mvarId localDecl.userName newType newTypePr;
(fvarIdNew, mvarId) ← intro1 mvarId;
(do mvarId ← clear mvarId fvarId; pure (fvarIdNew, mvarId)) <|> pure (fvarIdNew, mvarId)
end Meta
end Lean
|
e244cfd31fea92a1062f50e8d08318b533dea78e | 3dd1b66af77106badae6edb1c4dea91a146ead30 | /tests/lean/run/tactic25.lean | 5fa7c72058777155fe0e13b0d4574b65c8a2e9b1 | [
"Apache-2.0"
] | permissive | silky/lean | 79c20c15c93feef47bb659a2cc139b26f3614642 | df8b88dca2f8da1a422cb618cd476ef5be730546 | refs/heads/master | 1,610,737,587,697 | 1,406,574,534,000 | 1,406,574,534,000 | 22,362,176 | 1 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 538 | lean | import standard
using tactic
definition my_tac1 := apply @refl
definition my_tac2 := repeat (apply @and_intro; assumption)
tactic_hint my_tac1
tactic_hint my_tac2
theorem T1 {A : Type.{2}} (a : A) : a = a
theorem T2 {a b c : Prop} (Ha : a) (Hb : b) (Hc : c) : a ∧ b ∧ c
definition my_tac3 := fixpoint (λ f, [apply @or_intro_left; f |
apply @or_intro_right; f |
assumption])
tactic_hint [or] my_tac3
theorem T3 {a b c : Prop} (Hb : b) : a ∨ b ∨ c
|
8e330d8d7530cb52d52fc1978bee575b4c86661b | cf39355caa609c0f33405126beee2739aa3cb77e | /tests/lean/interactive/hole3.lean | c7f6faca58467846f5ad63479c897ad82b492d12 | [
"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 | 145 | lean | def x : nat → nat :=
λ a, {!
0 = a
--^ "command": "hole", "action": "Infer"
!}
--^ "command": "hole", "action": "Infer"
|
f52b5f97e0ffe34899abef7dc40013ddfb3e9e1b | 9dc8cecdf3c4634764a18254e94d43da07142918 | /src/category_theory/preadditive/eilenberg_moore.lean | 4ac0e531313ea5667ab3c65cf5d9554d1d9dfb38 | [
"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 | 4,625 | lean | /-
Copyright (c) 2022 Julian Kuelshammer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Kuelshammer
-/
import category_theory.preadditive.default
import category_theory.monad.algebra
import category_theory.preadditive.additive_functor
/-!
# Preadditive structure on algebras over a monad
If `C` is a preadditive categories and `T` is an additive monad on `C` then `algebra T` is also
preadditive. Dually, if `U` is an additive comonad on `C` then `coalgebra U` is preadditive as well.
-/
universes v₁ u₁ -- morphism levels before object levels. See note [category_theory universes].
namespace category_theory
variables (C : Type u₁) [category.{v₁} C] [preadditive C] (T : monad C)
[functor.additive (T : C ⥤ C)]
open category_theory.limits preadditive
/-- The category of algebras over an additive monad on a preadditive category is preadditive. -/
@[simps]
instance monad.algebra_preadditive : preadditive (monad.algebra T) :=
{ hom_group := λ F G,
{ add := λ α β,
{ f := α.f + β.f,
h' := by simp only [functor.map_add, add_comp, monad.algebra.hom.h, comp_add] },
zero :=
{ f := 0,
h' := by simp only [functor.map_zero, zero_comp, comp_zero] },
nsmul := λ n α,
{ f := n • α.f,
h' := by rw [functor.map_nsmul, nsmul_comp, monad.algebra.hom.h, comp_nsmul] },
neg := λ α,
{ f := -α.f,
h' := by simp only [functor.map_neg, neg_comp, monad.algebra.hom.h, comp_neg] },
sub := λ α β,
{ f := α.f - β.f,
h' := by simp only [functor.map_sub, sub_comp, monad.algebra.hom.h, comp_sub] },
zsmul := λ r α,
{ f := r • α.f,
h' := by rw [functor.map_zsmul, zsmul_comp, monad.algebra.hom.h, comp_zsmul] },
add_assoc := by { intros, ext, apply add_assoc },
zero_add := by { intros, ext, apply zero_add },
add_zero := by { intros, ext, apply add_zero },
nsmul_zero' := by { intros, ext, apply zero_smul },
nsmul_succ' := by { intros, ext, apply succ_nsmul },
sub_eq_add_neg := by { intros, ext, apply sub_eq_add_neg },
zsmul_zero' := by { intros, ext, apply zero_smul },
zsmul_succ' := by { intros, ext, dsimp, simp only [coe_nat_zsmul, succ_nsmul], refl, },
zsmul_neg' := by { intros, ext, simp only [zsmul_neg_succ_of_nat, neg_inj,
nsmul_eq_smul_cast ℤ] },
add_left_neg := by { intros, ext, apply add_left_neg },
add_comm := by { intros, ext, apply add_comm } },
add_comp' := by { intros, ext, apply add_comp },
comp_add' := by { intros, ext, apply comp_add } }
instance monad.forget_additive : (monad.forget T).additive := {}
variables (U : comonad C) [functor.additive (U : C ⥤ C)]
/-- The category of coalgebras over an additive comonad on a preadditive category is preadditive. -/
@[simps]
instance comonad.coalgebra_preadditive : preadditive (comonad.coalgebra U) :=
{ hom_group := λ F G,
{ add := λ α β,
{ f := α.f + β.f,
h' := by simp only [functor.map_add, comp_add, comonad.coalgebra.hom.h, add_comp] },
zero :=
{ f := 0,
h' := by simp only [functor.map_zero, comp_zero, zero_comp] },
nsmul := λ n α,
{ f := n • α.f,
h' := by rw [functor.map_nsmul, comp_nsmul, comonad.coalgebra.hom.h, nsmul_comp] },
neg := λ α,
{ f := -α.f,
h' := by simp only [functor.map_neg, comp_neg, comonad.coalgebra.hom.h, neg_comp] },
sub := λ α β,
{ f := α.f - β.f,
h' := by simp only [functor.map_sub, comp_sub, comonad.coalgebra.hom.h, sub_comp] },
zsmul := λ r α,
{ f := r • α.f,
h' := by rw [functor.map_zsmul, comp_zsmul, comonad.coalgebra.hom.h, zsmul_comp] },
add_assoc := by { intros, ext, apply add_assoc },
zero_add := by { intros, ext, apply zero_add },
add_zero := by { intros, ext, apply add_zero },
nsmul_zero' := by { intros, ext, apply zero_smul },
nsmul_succ' := by { intros, ext, apply succ_nsmul },
sub_eq_add_neg := by { intros, ext, apply sub_eq_add_neg },
zsmul_zero' := by { intros, ext, apply zero_smul },
zsmul_succ' := by { intros, ext, dsimp, simp only [coe_nat_zsmul, succ_nsmul], refl, },
zsmul_neg' := by { intros, ext, simp only [zsmul_neg_succ_of_nat, neg_inj,
nsmul_eq_smul_cast ℤ] },
add_left_neg := by { intros, ext, apply add_left_neg },
add_comm := by { intros, ext, apply add_comm } },
add_comp' := by { intros, ext, apply add_comp },
comp_add' := by { intros, ext, apply comp_add } }
instance comonad.forget_additive : (comonad.forget U).additive := {}
end category_theory
|
d5f81a2ad0d2e9d4b46e94e80c2bef6941943487 | ac2987d8c7832fb4a87edb6bee26141facbb6fa0 | /Mathlib/Tactic/Core.lean | d7c9378d25a6665120408ae24c4e21173dfaecce | [
"Apache-2.0"
] | permissive | AurelienSaue/mathlib4 | 52204b9bd9d207c922fe0cf3397166728bb6c2e2 | 84271fe0875bafdaa88ac41f1b5a7c18151bd0d5 | refs/heads/master | 1,689,156,096,545 | 1,629,378,840,000 | 1,629,378,840,000 | 389,648,603 | 0 | 0 | Apache-2.0 | 1,627,307,284,000 | 1,627,307,284,000 | null | UTF-8 | Lean | false | false | 710 | lean | /-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Aurélien Saue
-/
import Lean.Expr
namespace Lean.Expr
private def getAppFnArgsAux : Expr → Array Expr → Nat → Name × Array Expr
| app f a _, as, i => getAppFnArgsAux f (as.set! i a) (i-1)
| const n _ _, as, i => (n, as)
| _, as, _ => (Name.anonymous, as)
def getAppFnArgs (e : Expr) : Name × Array Expr :=
let nargs := e.getAppNumArgs
getAppFnArgsAux e (mkArray nargs arbitrary) (nargs-1)
def natLit! : Expr → Nat
| lit (Literal.natVal v) _ => v
| _ => panic! "nat literal expected"
end Expr
|
050e6c8c113ae239a04384e7e3031f8a3bd969cc | c777c32c8e484e195053731103c5e52af26a25d1 | /src/number_theory/modular_forms/jacobi_theta.lean | dd6578c6a37ae4c22e5595651b96dc4e2f34e508 | [
"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 | 9,139 | lean | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import analysis.special_functions.gaussian
import analysis.complex.locally_uniform_limit
import analysis.complex.upper_half_plane.functions_bounded_at_infty
import analysis.complex.upper_half_plane.topology
/-! # Jacobi's theta function
This file defines the Jacobi theta function
$$\theta(\tau) = \sum_{n \in \mathbb{Z}} \exp (i \pi n ^ 2 \tau),$$
and proves the modular transformation properties `θ (τ + 2) = θ τ` and
`θ (-1 / τ) = (-I * τ) ^ (1 / 2) * θ τ`, using Poisson's summation formula for the latter. We also
show that `θ` is differentiable on `ℍ`, and `θ(τ) - 1` has exponential decay as `im τ → ∞`.
-/
open complex real asymptotics
open_locale real big_operators upper_half_plane manifold
/-- Jacobi's theta function `∑' (n : ℤ), exp (π * I * n ^ 2 * τ)`. -/
noncomputable def jacobi_theta (τ : ℍ) : ℂ := ∑' (n : ℤ), cexp (π * I * n ^ 2 * τ)
lemma norm_exp_mul_sq_le {z : ℂ} (hz : 0 < z.im) (n : ℤ) :
‖cexp (π * I * n ^ 2 * z)‖ ≤ exp (-π * z.im) ^ n.nat_abs :=
begin
let y := rexp (-π * z.im),
have h : y < 1, from exp_lt_one_iff.mpr (mul_neg_of_neg_of_pos (neg_lt_zero.mpr pi_pos) hz),
refine (le_of_eq _).trans (_ : y ^ (n ^ 2) ≤ _),
{ rw [complex.norm_eq_abs, complex.abs_exp],
have : (↑π * I * n ^ 2 * z).re = (-π * z.im) * n ^ 2,
{ rw [(by { push_cast, ring } : ↑π * I * n ^ 2 * z = ↑(π * n ^ 2) * (z * I)),
of_real_mul_re, mul_I_re],
ring },
obtain ⟨m, hm⟩ := int.eq_coe_of_zero_le (sq_nonneg n),
rw [this, exp_mul, ←int.cast_pow, rpow_int_cast, hm, zpow_coe_nat] },
{ have : n ^ 2 = ↑(n.nat_abs ^ 2), by rw [nat.cast_pow, int.nat_abs_sq],
rw [this, zpow_coe_nat],
exact pow_le_pow_of_le_one (exp_pos _).le h.le ((sq n.nat_abs).symm ▸ n.nat_abs.le_mul_self) },
end
lemma exists_summable_bound_exp_mul_sq {R : ℝ} (hR : 0 < R) :
∃ (bd : ℤ → ℝ), (summable bd) ∧
(∀ {τ : ℂ} (hτ : R ≤ τ.im) (n : ℤ), ‖cexp (π * I * n ^ 2 * τ)‖ ≤ bd n) :=
begin
let y := rexp (-π * R),
have h : y < 1, from exp_lt_one_iff.mpr (mul_neg_of_neg_of_pos (neg_lt_zero.mpr pi_pos) hR),
refine ⟨λ n, y ^ n.nat_abs, summable_int_of_summable_nat _ _, λ τ hτ n, _⟩, swap 3,
{ refine (norm_exp_mul_sq_le (hR.trans_le hτ) n).trans _,
refine pow_le_pow_of_le_left (exp_pos _).le (real.exp_le_exp.mpr _) _,
rwa [mul_le_mul_left_of_neg (neg_lt_zero.mpr pi_pos)] },
all_goals { simpa only [int.nat_abs_neg, int.nat_abs_of_nat]
using summable_geometric_of_lt_1 (real.exp_pos _).le h },
end
lemma summable_exp_mul_sq {z : ℂ} (hz : 0 < z.im) :
summable (λ n : ℤ, cexp (π * I * n ^ 2 * z)) :=
let ⟨bd, h, h'⟩ := exists_summable_bound_exp_mul_sq hz in
summable_norm_iff.mp (summable_of_nonneg_of_le (λ n, norm_nonneg _) (h' $ le_refl _) h)
lemma jacobi_theta_two_vadd (τ : ℍ) : jacobi_theta ((2 : ℝ) +ᵥ τ) = jacobi_theta τ :=
begin
refine tsum_congr (λ n, _),
rw [upper_half_plane.coe_vadd, of_real_bit0, of_real_one],
suffices : cexp (↑π * I * ↑n ^ 2 * 2) = 1, by rw [mul_add, complex.exp_add, this, one_mul],
rw [(by { push_cast, ring } : ↑π * I * ↑n ^ 2 * 2 = ↑(n ^ 2) * (2 * π * I)),
complex.exp_int_mul, complex.exp_two_pi_mul_I, one_zpow],
end
lemma jacobi_theta_T_sq_smul (τ : ℍ) : jacobi_theta (modular_group.T ^ 2 • τ) = jacobi_theta τ :=
begin
suffices : (2 : ℝ) +ᵥ τ = modular_group.T ^ (2 : ℤ) • τ, from this ▸ (jacobi_theta_two_vadd τ),
simp only [←subtype.coe_inj, upper_half_plane.modular_T_zpow_smul, int.cast_two],
end
lemma jacobi_theta_S_smul (τ : ℍ) :
jacobi_theta (modular_group.S • τ) = (-I * τ) ^ (1 / 2 : ℂ) * jacobi_theta τ :=
begin
unfold jacobi_theta,
rw [upper_half_plane.modular_S_smul, upper_half_plane.coe_mk],
have ha : 0 < (-I * τ).re,
{ rw [neg_mul, neg_re, mul_re, I_re, I_im, zero_mul, one_mul, zero_sub, neg_neg],
exact τ.im_pos },
have ha' : (-I * τ) ^ (1 / 2 : ℂ) ≠ 0,
{ rw [ne.def, cpow_eq_zero_iff],
contrapose! ha,
rw [ha.1, zero_re] },
have hτ : (τ : ℂ) ≠ 0, from τ.ne_zero,
have := complex.tsum_exp_neg_mul_int_sq ha,
rw [mul_comm ((1:ℂ) / _) _, mul_one_div, eq_div_iff ha', mul_comm _ (_ ^ _), eq_comm] at this,
convert this using 3,
{ ext1 n,
congr' 1,
field_simp [hτ, I_ne_zero],
ring_nf,
rw [I_sq, mul_neg, mul_one, neg_mul, neg_neg] },
{ ext1 n,
congr' 1,
ring_nf }
end
lemma has_sum_nat_jacobi_theta (τ : ℍ) :
has_sum (λ (n : ℕ), cexp (π * I * (n + 1) ^ 2 * τ)) ((jacobi_theta τ - 1) / 2) :=
begin
have := (summable_exp_mul_sq τ.im_pos).has_sum.sum_nat_of_sum_int,
rw ←@has_sum_nat_add_iff' ℂ _ _ _ _ 1 at this,
simp_rw [finset.sum_range_one, int.cast_neg, int.cast_coe_nat, nat.cast_zero, neg_zero,
int.cast_zero, sq (0:ℂ), mul_zero, zero_mul, neg_sq, ←mul_two, complex.exp_zero,
add_sub_assoc, (by norm_num : (1 : ℂ) - 1 * 2 = -1), ←sub_eq_add_neg,
nat.cast_add, nat.cast_one] at this,
convert this.div_const 2,
simp_rw mul_div_cancel _ two_ne_zero,
end
lemma jacobi_theta_eq_tsum_nat (τ : ℍ) :
jacobi_theta τ = 1 + 2 * ∑' (n : ℕ), cexp (π * I * (n + 1) ^ 2 * τ) :=
by rw [(has_sum_nat_jacobi_theta τ).tsum_eq, mul_div_cancel' _ (two_ne_zero' ℂ), ←add_sub_assoc,
add_sub_cancel']
/-- An explicit upper bound for `‖jacobi_theta τ - 1‖`. -/
lemma norm_jacobi_theta_sub_one_le (τ : ℍ) :
‖jacobi_theta τ - 1‖ ≤ 2 / (1 - exp (-π * τ.im)) * exp (-π * τ.im) :=
begin
suffices : ‖∑' (n : ℕ), cexp (π * I * (n + 1) ^ 2 * τ)‖ ≤ exp (-π * τ.im) / (1 - exp (-π * τ.im)),
{ calc ‖jacobi_theta τ - 1‖ = 2 * ‖∑' (n : ℕ), cexp (π * I * (n + 1) ^ 2 * τ)‖ :
by rw [sub_eq_iff_eq_add'.mpr (jacobi_theta_eq_tsum_nat τ), norm_mul, complex.norm_eq_abs,
complex.abs_two]
... ≤ 2 * (rexp (-π * τ.im) / (1 - rexp (-π * τ.im))) :
by rwa [mul_le_mul_left (zero_lt_two' ℝ)]
... = 2 / (1 - rexp (-π * τ.im)) * rexp (-π * τ.im) : by rw [div_mul_comm, mul_comm] },
have : ∀ (n : ℕ), ‖cexp (π * I * (n + 1) ^ 2 * τ)‖ ≤ exp (-π * τ.im) ^ (n + 1),
{ intro n,
simpa only [int.cast_add, int.cast_one] using norm_exp_mul_sq_le τ.im_pos (n + 1) },
have s : has_sum (λ n : ℕ, rexp (-π * τ.im) ^ (n + 1)) (exp (-π * τ.im) / (1 - exp (-π * τ.im))),
{ simp_rw [pow_succ, div_eq_mul_inv, has_sum_mul_left_iff (real.exp_ne_zero _)],
exact has_sum_geometric_of_lt_1 (exp_pos (-π * τ.im)).le
(exp_lt_one_iff.mpr $ (mul_neg_of_neg_of_pos (neg_lt_zero.mpr pi_pos) τ.im_pos)) },
have aux : summable (λ (n : ℕ), ‖cexp (↑π * I * (↑n + 1) ^ 2 * ↑τ)‖),
from summable_of_nonneg_of_le (λ n, norm_nonneg _) this s.summable,
exact (norm_tsum_le_tsum_norm aux).trans
((tsum_mono aux s.summable this).trans (le_of_eq s.tsum_eq)),
end
/-- The norm of `jacobi_theta τ - 1` decays exponentially as `im τ → ∞`. -/
lemma is_O_at_im_infty_jacobi_theta_sub_one :
is_O upper_half_plane.at_im_infty (λ τ, jacobi_theta τ - 1) (λ τ, rexp (-π * τ.im)) :=
begin
simp_rw [is_O, is_O_with, filter.eventually, upper_half_plane.at_im_infty_mem],
refine ⟨2 / (1 - rexp (-π)), 1, (λ τ hτ, (norm_jacobi_theta_sub_one_le τ).trans _)⟩,
rw [real.norm_eq_abs, real.abs_exp],
refine mul_le_mul_of_nonneg_right _ (exp_pos _).le,
rw [div_le_div_left (zero_lt_two' ℝ), sub_le_sub_iff_left, exp_le_exp, neg_mul, neg_le_neg_iff],
{ exact le_mul_of_one_le_right pi_pos.le hτ },
{ rw [sub_pos, exp_lt_one_iff, neg_mul, neg_lt_zero], exact mul_pos pi_pos τ.im_pos },
{ rw [sub_pos, exp_lt_one_iff, neg_lt_zero], exact pi_pos }
end
lemma differentiable_at_tsum_exp_mul_sq (τ : ℍ) :
differentiable_at ℂ (λ z, ∑' (n : ℤ), cexp (π * I * n ^ 2 * z)) ↑τ :=
begin
suffices : ∀ (y : ℝ) (hy : 0 < y),
differentiable_on ℂ (λ z, ∑' (n : ℤ), cexp (π * I * n ^ 2 * z)) {w : ℂ | y < im w},
from let ⟨y, hy, hy'⟩ := exists_between τ.im_pos in (this y hy).differentiable_at
((complex.continuous_im.is_open_preimage _ is_open_Ioi).mem_nhds (τ.coe_im ▸ hy')),
intros y hy,
have h1 : ∀ (n : ℤ) (w : ℂ) (hw : y < im w), differentiable_within_at ℂ
(λ (v : ℂ), cexp (↑π * I * ↑n ^ 2 * v)) {z : ℂ | y < im z} w,
from λ n w hw, (differentiable_at_id.const_mul _).cexp.differentiable_within_at,
have h2 : is_open {w : ℂ | y < im w}, from continuous_im.is_open_preimage _ is_open_Ioi,
obtain ⟨bd, bd_s, le_bd⟩ := exists_summable_bound_exp_mul_sq hy,
exact differentiable_on_tsum_of_summable_norm bd_s h1 h2 (λ i w hw, le_bd (le_of_lt hw) i),
end
lemma mdifferentiable_jacobi_theta : mdifferentiable 𝓘(ℂ) 𝓘(ℂ) jacobi_theta :=
λ τ, (differentiable_at_tsum_exp_mul_sq τ).mdifferentiable_at.comp τ τ.mdifferentiable_coe
lemma continuous_jacobi_theta : continuous jacobi_theta := mdifferentiable_jacobi_theta.continuous
|
afe63675ce5e088310edf25dc43c3e078351bcf4 | 6dc0c8ce7a76229dd81e73ed4474f15f88a9e294 | /tests/lean/run/induction1.lean | 3fd38124fe8af3f4d425c74a7c62bc2d31f20da0 | [
"Apache-2.0"
] | permissive | williamdemeo/lean4 | 72161c58fe65c3ad955d6a3050bb7d37c04c0d54 | 6d00fcf1d6d873e195f9220c668ef9c58e9c4a35 | refs/heads/master | 1,678,305,356,877 | 1,614,708,995,000 | 1,614,708,995,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 3,479 | lean | @[recursor 4]
def Or.elim2 {p q r : Prop} (major : p ∨ q) (left : p → r) (right : q → r) : r :=
match major with
| Or.inl h => left h
| Or.inr h => right h
theorem tst0 {p q : Prop } (h : p ∨ q) : q ∨ p :=
by {
induction h;
{ apply Or.inr; assumption };
{ apply Or.inl; assumption }
}
theorem tst0' {p q : Prop } (h : p ∨ q) : q ∨ p := by
induction h
focus
apply Or.inr
assumption
focus
apply Or.inl
assumption
theorem tst1 {p q : Prop } (h : p ∨ q) : q ∨ p := by
induction h with
| inr h2 => exact Or.inl h2
| inl h1 => exact Or.inr h1
theorem tst2 {p q : Prop } (h : p ∨ q) : q ∨ p := by
induction h using elim2 with
| left _ => apply Or.inr; assumption
| right _ => apply Or.inl; assumption
theorem tst2b {p q : Prop } (h : p ∨ q) : q ∨ p := by
induction h using elim2 with
| left => apply Or.inr; assumption
| _ => apply Or.inl; assumption
theorem tst3 {p q : Prop } (h : p ∨ q) : q ∨ p := by
induction h using elim2 with
| right h => exact Or.inl h
| left h => exact Or.inr h
theorem tst4 {p q : Prop } (h : p ∨ q) : q ∨ p := by
induction h using elim2 with
| right h => ?myright
| left h => ?myleft
case myleft => exact Or.inr h
case myright => exact Or.inl h
theorem tst5 {p q : Prop } (h : p ∨ q) : q ∨ p := by
induction h using elim2 with
| right h => _
| left h =>
refine Or.inr ?_
exact h
case right => exact Or.inl h
theorem tst6 {p q : Prop } (h : p ∨ q) : q ∨ p :=
by {
cases h with
| inr h2 => exact Or.inl h2
| inl h1 => exact Or.inr h1
}
theorem tst7 {α : Type} (xs : List α) (h : (a : α) → (as : List α) → xs ≠ a :: as) : xs = [] :=
by {
induction xs with
| nil => exact rfl
| cons z zs ih => exact absurd rfl (h z zs)
}
theorem tst8 {α : Type} (xs : List α) (h : (a : α) → (as : List α) → xs ≠ a :: as) : xs = [] := by {
induction xs;
exact rfl;
exact absurd rfl $ h _ _
}
theorem tst9 {α : Type} (xs : List α) (h : (a : α) → (as : List α) → xs ≠ a :: as) : xs = [] := by
cases xs with
| nil => exact rfl
| cons z zs => exact absurd rfl (h z zs)
theorem tst10 {p q : Prop } (h₁ : p ↔ q) (h₂ : p) : q := by
induction h₁ with
| intro h _ => exact h h₂
def Iff2 (m p q : Prop) := p ↔ q
theorem tst11 {p q r : Prop } (h₁ : Iff2 r p q) (h₂ : p) : q := by
induction h₁ using Iff.rec with
| intro h _ => exact h h₂
theorem tst12 {p q : Prop } (h₁ : p ∨ q) (h₂ : p ↔ q) (h₃ : p) : q := by
failIfSuccess induction h₁ using Iff.casesOn
induction h₂ using Iff.casesOn with
| intro h _ =>
exact h h₃
inductive Tree
| leaf₁
| leaf₂
| node : Tree → Tree → Tree
def Tree.isLeaf₁ : Tree → Bool
| leaf₁ => true
| _ => false
theorem tst13 (x : Tree) (h : x = Tree.leaf₁) : x.isLeaf₁ = true := by
cases x with
| leaf₁ => rfl
| _ => injection h
theorem tst14 (x : Tree) (h : x = Tree.leaf₁) : x.isLeaf₁ = true := by
induction x with
| leaf₁ => rfl
| _ => injection h
inductive Vec (α : Type) : Nat → Type
| nil : Vec α 0
| cons : (a : α) → {n : Nat} → (as : Vec α n) → Vec α (n+1)
def getHeads {α β} {n} (xs : Vec α (n+1)) (ys : Vec β (n+1)) : α × β := by
cases xs
cases ys
apply Prod.mk
repeat
traceState
assumption
done
theorem ex1 (n m o : Nat) : n = m + 0 → m = o → m = o := by
intro (h₁ : n = m) h₂
rw [← h₁, ← h₂]
assumption
|
29f3b8d0b82cbfcf438e62a5929b23ba53aa27a3 | 63abd62053d479eae5abf4951554e1064a4c45b4 | /src/algebraic_geometry/stalks.lean | 6f2de5987d13b3ea01b5f682759c52596dc0dbbc | [
"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 | 2,850 | lean | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import algebraic_geometry.presheafed_space
import topology.sheaves.stalks
/-!
# Stalks for presheaved spaces
This file lifts constructions of stalks and pushforwards of stalks to work with
the category of presheafed spaces.
-/
noncomputable theory
universes v u v' u'
open category_theory
open category_theory.limits category_theory.category category_theory.functor
open algebraic_geometry
open topological_space
variables {C : Type u} [category.{v} C] [has_colimits C]
local attribute [tidy] tactic.op_induction'
open Top.presheaf
namespace algebraic_geometry.PresheafedSpace
/--
The stalk at `x` of a `PresheafedSpace`.
-/
def stalk (X : PresheafedSpace C) (x : X) : C := X.presheaf.stalk x
/--
A morphism of presheafed spaces induces a morphism of stalks.
-/
def stalk_map {X Y : PresheafedSpace C} (α : X ⟶ Y) (x : X) : Y.stalk (α.base x) ⟶ X.stalk x :=
(stalk_functor C (α.base x)).map (α.c) ≫ X.presheaf.stalk_pushforward C α.base x
section restrict
-- PROJECT: restriction preserves stalks.
-- We'll want to define cofinal functors, show precomposing with a cofinal functor preserves colimits,
-- and (easily) verify that "open neighbourhoods of x within U" is cofinal in "open neighbourhoods of x".
/-
def restrict_stalk_iso {U : Top} (X : PresheafedSpace C)
(f : U ⟶ (X : Top.{v})) (h : open_embedding f) (x : U) :
(X.restrict f h).stalk x ≅ X.stalk (f x) :=
begin
dsimp only [stalk, Top.presheaf.stalk, stalk_functor],
dsimp [colim],
sorry
end
-- TODO `restrict_stalk_iso` is compatible with `germ`.
-/
end restrict
namespace stalk_map
@[simp] lemma id (X : PresheafedSpace C) (x : X) : stalk_map (𝟙 X) x = 𝟙 (X.stalk x) :=
begin
dsimp [stalk_map],
simp only [stalk_pushforward.id],
rw [←map_comp],
convert (stalk_functor C x).map_id X.presheaf,
tidy,
end
-- TODO understand why this proof is still gross (i.e. requires using `erw`)
@[simp] lemma comp {X Y Z : PresheafedSpace C} (α : X ⟶ Y) (β : Y ⟶ Z) (x : X) :
stalk_map (α ≫ β) x =
(stalk_map β (α.base x) : Z.stalk (β.base (α.base x)) ⟶ Y.stalk (α.base x)) ≫
(stalk_map α x : Y.stalk (α.base x) ⟶ X.stalk x) :=
begin
dsimp [stalk_map, stalk_functor, stalk_pushforward],
ext U,
op_induction U,
cases U,
simp only [colimit.ι_map_assoc, colimit.ι_pre_assoc, colimit.ι_pre,
whisker_left_app, whisker_right_app,
assoc, id_comp, map_id, map_comp],
dsimp,
simp only [map_id, assoc, pushforward.comp_inv_app],
-- FIXME Why doesn't simp do this:
erw [category_theory.functor.map_id],
erw [category_theory.functor.map_id],
erw [id_comp, id_comp, id_comp],
end
end stalk_map
end algebraic_geometry.PresheafedSpace
|
b54024fd7444f7c345e9e34485ba5498b53f3145 | 26b8b0964ca8e1c2e203585ba5940f83fe05e48a | /src/tidy/applicable.lean | 10983948705e1309fc1a12ff17b6902826f0919f | [] | no_license | jcommelin/lean-tidy | ef3cd32a3804221d93f0dff9e180bb2c52f4b143 | 9cecf497e90db64b5ea140ad6ae1603976dcd402 | refs/heads/master | 1,585,129,919,276 | 1,533,512,680,000 | 1,533,512,680,000 | 143,677,361 | 0 | 0 | null | 1,616,803,481,000 | 1,533,530,576,000 | Lean | UTF-8 | Lean | false | false | 3,434 | lean | -- Copyright (c) 2017 Scott Morrison. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Authors: Scott Morrison
import .pempty
open tactic
meta def applicable_attribute : user_attribute := {
name := `applicable,
descr := "A lemma that should be applied to a goal whenever possible."
}
run_cmd attribute.register `applicable_attribute
/- Try to apply one of the given lemmas, it succeeds if one of them succeeds. -/
meta def any_apply : list name → tactic name
| [] := failed
| (c::cs) := (mk_const c >>= apply >> pure c) <|> any_apply cs
/- Try to apply any lemma marked with the attribute @[applicable] -/
meta def applicable : tactic name :=
do cs ← attribute.get_instances `applicable,
(any_apply cs) <|> fail "no @[applicable] lemmas could be applied"
meta def semiapplicable_attribute : user_attribute := {
name := `semiapplicable,
descr := "A lemma that should be applied to a goal whenever possible, as long as it create no new goals."
}
run_cmd attribute.register `semiapplicable_attribute
/- Try to apply one of the given lemmas, fulfilling all new goals using existing hypotheses. It succeeds if one of them succeeds. -/
meta def any_apply_no_new_goals : list name → tactic name
| [] := failed
| (c::cs) := (do n ← num_goals,
t ← mk_const c,
r ← seq (apply t >> skip) assumption,
n' ← num_goals,
guard (n = n' + 1),
pure c) <|> any_apply_no_new_goals cs
/- Try to apply any lemma marked with the attribute @[semiapplicable], as long as all hypotheses of that lemma can be satisfied immediately from hypotheses. -/
meta def semiapplicable : tactic name :=
do cs ← attribute.get_instances `semiapplicable,
(any_apply_no_new_goals cs) <|> fail "no @[semiapplicable] lemmas could be applied"
attribute [applicable] subsingleton.elim
attribute [applicable] funext
attribute [applicable] set.ext -- Order matters here: putting the attribute on set.ext after funext makes applicable prefer set.ext
attribute [applicable] propext
attribute [applicable] subtype.eq
universes u₁ u₂
@[applicable] definition empty_exfalso (x : false) : empty := begin exfalso, trivial end
@[applicable] definition pempty_exfalso (x : false) : pempty := begin exfalso, trivial end
@[applicable] lemma punit.ext (a b : punit.{u₁}): a = b := by induction a; induction b; refl
@[applicable] lemma ulift.ext {α : Type u₁} (X Y : ulift.{u₂} α) (w : X.down = Y.down) : X = Y :=
begin
induction X, induction Y, dsimp at w, rw w,
end
@[applicable] lemma sigma.ext {α : Type u₁} (Z : α → Type u₂) (X Y : Σ a : α, Z a) (w1 : X.1 = Y.1) (w2 : @eq.rec_on _ X.1 _ _ w1 X.2 = Y.2) : X = Y :=
begin
induction X, induction Y, dsimp at w1, dsimp at w2, induction w1, induction w2, refl,
end
@[applicable] lemma pair.ext {α : Type u₁} {β : Type u₂} {X Y : α × β} (p1 : X.1 = Y.1) (p2 : X.2 = Y.2) : X = Y :=
begin
induction X, induction Y, dsimp at *, rw p1, rw p2,
end
-- TODO should `apply_instance` be in tidy? If so, these shouldn't be needed.
@[applicable] definition decidable_true : decidable true := is_true dec_trivial
@[applicable] definition decidable_false : decidable false := is_false dec_trivial
attribute [applicable] quotient.mk quotient.sound
attribute [semiapplicable] eqv_gen.rel
attribute [semiapplicable] Exists.intro
|
e0ea617ba10a5d31e10671898d029dcc84dacd6a | 200b12985a863d01fbbde6abfc9326bb82424a8b | /src/propLogic/Ex009.lean | 815b8b62412a53cdf2b0e06b80d0a86a33f0e1e0 | [] | no_license | SvenWille/LeanLogicExercises | 38eacd36733ac48e5a7aacf863c681c9a9a48271 | 2dbc920feadd63bbc50f87e69646c0081db26eba | refs/heads/master | 1,629,676,667,365 | 1,512,161,459,000 | 1,512,161,459,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 802 | lean | open classical
theorem Ex009(a b c: Prop): ¬c → a ∨ ((a ∨ c) → b):=
assume H1:¬c ,
have A:¬(a ∨ ((a ∨ c) → b)) → false, from
(
assume H2:¬(a ∨ ((a ∨ c) → b)),
have B:(a ∨ c) → b, from
(
assume H3:a ∨ c,
show b , from or.elim H3
(
assume H4:a,
have D:a ∨ ((a ∨ c) → b), from or.inl H4,
have false, from H2 D,
show b,from false.elim this
)
(
assume H5:c,
have E:false, from H1 H5,
show b,from false.elim E
)
),
have a ∨ ((a ∨ c) → b), from or.inr B,
show false, from H2 this
),
have F:¬¬(a ∨ ((a ∨ c) → b)), from not.intro A,
by_contradiction
(
assume H6:¬(a ∨ ((a ∨ c) → b)),
show false, from F H6
)
|
14226a1e5aa4e4c51436c92c913b33c0a1c1b218 | 9dc8cecdf3c4634764a18254e94d43da07142918 | /src/number_theory/legendre_symbol/mul_character.lean | 117f6c465394c541b669cde174df16cd047c2426 | [
"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 | 17,497 | lean | /-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import ring_theory.integral_domain
/-!
# Multiplicative characters of finite rings and fields
Let `R` and `R'` be a commutative rings.
A *multiplicative character* of `R` with values in `R'` is a morphism of
monoids from the multiplicative monoid of `R` into that of `R'`
that sends non-units to zero.
We use the namespace `mul_char` for the definitions and results.
## Main results
We show that the multiplicative characters form a group (if `R'` is commutative);
see `mul_char.comm_group`. We also provide an equivalence with the
homomorphisms `Rˣ →* R'ˣ`; see `mul_char.equiv_to_unit_hom`.
We define a multiplicative character to be *quadratic* if its values
are among `0`, `1` and `-1`, and we prove some properties of quadratic characters.
Finally, we show that the sum of all values of a nontrivial multiplicative
character vanishes; see `mul_char.is_nontrivial.sum_eq_zero`.
## Tags
multiplicative character
-/
section definition_and_group
/-!
### Definitions related to multiplicative characters
Even though the intended use is when domain and target of the characters
are commutative rings, we define them in the more general setting when
the domain is a commutative monoid and the target is a commutative monoid
with zero. (We need a zero in the target, since non-units are supposed
to map to zero.)
In this setting, there is an equivalence between multiplicative characters
`R → R'` and group homomorphisms `Rˣ → R'ˣ`, and the multiplicative characters
have a natural structure as a commutative group.
-/
universes u v
section defi
-- The domain of our multiplicative characters
variables (R : Type u) [comm_monoid R]
-- The target
variables (R' : Type v) [comm_monoid_with_zero R']
/-- Define a structure for multiplicative characters.
A multiplicative character from a commutative monoid `R` to a commutative monoid with zero `R'`
is a homomorphism of (multiplicative) monoids that sends non-units to zero. -/
structure mul_char extends monoid_hom R R' :=
(map_nonunit' : ∀ a : R, ¬ is_unit a → to_fun a = 0)
/-- This is the corresponding extension of `monoid_hom_class`. -/
class mul_char_class (F : Type*) (R R' : out_param $ Type*) [comm_monoid R]
[comm_monoid_with_zero R']
extends monoid_hom_class F R R' :=
(map_nonunit : ∀ (χ : F) {a : R} (ha : ¬ is_unit a), χ a = 0)
attribute [simp] mul_char_class.map_nonunit
end defi
section group
namespace mul_char
-- The domain of our multiplicative characters
variables {R : Type u} [comm_monoid R]
-- The target
variables {R' : Type v} [comm_monoid_with_zero R']
instance coe_to_fun : has_coe_to_fun (mul_char R R') (λ _, R → R') :=
⟨λ χ, χ.to_fun⟩
/-- See note [custom simps projection] -/
protected def simps.apply (χ : mul_char R R') : R → R' := χ
initialize_simps_projections mul_char (to_monoid_hom_to_fun → apply, -to_monoid_hom)
section trivial
variables (R R')
/-- The trivial multiplicative character. It takes the value `0` on non-units and
the value `1` on units. -/
@[simps]
noncomputable
def trivial : mul_char R R' :=
{ to_fun := by { classical, exact λ x, if is_unit x then 1 else 0 },
map_nonunit' := by { intros a ha, simp only [ha, if_false], },
map_one' := by simp only [is_unit_one, if_true],
map_mul' := by { intros x y,
simp only [is_unit.mul_iff, boole_mul],
split_ifs; tauto, } }
end trivial
@[simp]
lemma coe_coe (χ : mul_char R R') : (χ.to_monoid_hom : R → R') = χ := rfl
@[simp]
lemma to_fun_eq_coe (χ : mul_char R R') : χ.to_fun = χ := rfl
@[simp]
lemma coe_mk (f : R →* R') (hf) : (mul_char.mk f hf : R → R') = f := rfl
/-- Extensionality. See `ext` below for the version that will actually be used. -/
lemma ext' {χ χ' : mul_char R R'} (h : ∀ a, χ a = χ' a) : χ = χ' :=
begin
cases χ,
cases χ',
congr,
exact monoid_hom.ext h,
end
instance : mul_char_class (mul_char R R') R R' :=
{ coe := λ χ, χ.to_monoid_hom.to_fun,
coe_injective' := λ f g h, ext' (λ a, congr_fun h a),
map_mul := λ χ, χ.map_mul',
map_one := λ χ, χ.map_one',
map_nonunit := λ χ, χ.map_nonunit', }
lemma map_nonunit (χ : mul_char R R') {a : R} (ha : ¬ is_unit a) : χ a = 0 :=
χ.map_nonunit' a ha
/-- Extensionality. Since `mul_char`s always take the value zero on non-units, it is sufficient
to compare the values on units. -/
@[ext]
lemma ext {χ χ' : mul_char R R'} (h : ∀ a : Rˣ, χ a = χ' a) : χ = χ' :=
begin
apply ext',
intro a,
by_cases ha : is_unit a,
{ exact h ha.unit, },
{ rw [map_nonunit χ ha, map_nonunit χ' ha], },
end
lemma ext_iff {χ χ' : mul_char R R'} : χ = χ' ↔ ∀ a : Rˣ, χ a = χ' a :=
⟨by { rintro rfl a, refl }, ext⟩
/-!
### Equivalence of multiplicative characters with homomorphisms on units
We show that restriction / extension by zero gives an equivalence
between `mul_char R R'` and `Rˣ →* R'ˣ`.
-/
/-- Turn a `mul_char` into a homomorphism between the unit groups. -/
def to_unit_hom (χ : mul_char R R') : Rˣ →* R'ˣ := units.map χ
lemma coe_to_unit_hom (χ : mul_char R R') (a : Rˣ) :
↑(χ.to_unit_hom a) = χ a :=
rfl
/-- Turn a homomorphism between unit groups into a `mul_char`. -/
noncomputable
def of_unit_hom (f : Rˣ →* R'ˣ) : mul_char R R' :=
{ to_fun := by { classical, exact λ x, if hx : is_unit x then f hx.unit else 0 },
map_one' := by { have h1 : (is_unit_one.unit : Rˣ) = 1 := units.eq_iff.mp rfl,
simp only [h1, dif_pos, units.coe_eq_one, map_one, is_unit_one], },
map_mul' :=
begin
intros x y,
by_cases hx : is_unit x,
{ simp only [hx, is_unit.mul_iff, true_and, dif_pos],
by_cases hy : is_unit y,
{ simp only [hy, dif_pos],
have hm : (is_unit.mul_iff.mpr ⟨hx, hy⟩).unit = hx.unit * hy.unit := units.eq_iff.mp rfl,
rw [hm, map_mul],
norm_cast, },
{ simp only [hy, not_false_iff, dif_neg, mul_zero], }, },
{ simp only [hx, is_unit.mul_iff, false_and, not_false_iff, dif_neg, zero_mul], },
end ,
map_nonunit' := by { intros a ha, simp only [ha, not_false_iff, dif_neg], }, }
lemma of_unit_hom_coe (f : Rˣ →* R'ˣ) (a : Rˣ) :
of_unit_hom f ↑a = f a :=
by simp [of_unit_hom]
/-- The equivalence between multiplicative characters and homomorphisms of unit groups. -/
noncomputable
def equiv_to_unit_hom : mul_char R R' ≃ (Rˣ →* R'ˣ) :=
{ to_fun := to_unit_hom,
inv_fun := of_unit_hom,
left_inv :=
by { intro χ, ext x, rw [of_unit_hom_coe, coe_to_unit_hom] },
right_inv :=
by { intro f, ext x, rw [coe_to_unit_hom, of_unit_hom_coe], } }
@[simp]
lemma to_unit_hom_eq (χ : mul_char R R') : to_unit_hom χ = equiv_to_unit_hom χ := rfl
@[simp]
lemma of_unit_hom_eq (χ : Rˣ →* R'ˣ) : of_unit_hom χ = equiv_to_unit_hom.symm χ := rfl
@[simp]
lemma coe_equiv_to_unit_hom (χ : mul_char R R') (a : Rˣ) :
↑(equiv_to_unit_hom χ a) = χ a :=
coe_to_unit_hom χ a
@[simp]
lemma equiv_unit_hom_symm_coe (f : Rˣ →* R'ˣ) (a : Rˣ) :
equiv_to_unit_hom.symm f ↑a = f a :=
of_unit_hom_coe f a
/-!
### Commutative group structure on multiplicative characters
The multiplicative characters `R → R'` form a commutative group.
-/
protected
lemma map_one (χ : mul_char R R') : χ (1 : R) = 1 :=
χ.map_one'
/-- If the domain has a zero (and is nontrivial), then `χ 0 = 0`. -/
protected
lemma map_zero {R : Type u} [comm_monoid_with_zero R] [nontrivial R] (χ : mul_char R R') :
χ (0 : R) = 0 :=
by rw [map_nonunit χ not_is_unit_zero]
/-- If the domain is a ring `R`, then `χ (ring_char R) = 0`. -/
lemma map_ring_char {R : Type u} [comm_ring R] [nontrivial R] (χ : mul_char R R') :
χ (ring_char R) = 0 :=
by rw [ring_char.nat.cast_ring_char, χ.map_zero]
noncomputable
instance has_one : has_one (mul_char R R') := ⟨trivial R R'⟩
noncomputable
instance inhabited : inhabited (mul_char R R') := ⟨1⟩
/-- Evaluation of the trivial character -/
@[simp]
lemma one_apply_coe (a : Rˣ) : (1 : mul_char R R') a = 1 :=
dif_pos a.is_unit
/-- Multiplication of multiplicative characters. (This needs the target to be commutative.) -/
def mul (χ χ' : mul_char R R') : mul_char R R' :=
{ to_fun := χ * χ',
map_nonunit' := λ a ha, by simp [map_nonunit χ ha],
..χ.to_monoid_hom * χ'.to_monoid_hom }
instance has_mul : has_mul (mul_char R R') := ⟨mul⟩
lemma mul_apply (χ χ' : mul_char R R') (a : R) : (χ * χ') a = χ a * χ' a := rfl
@[simp]
lemma coe_to_fun_mul (χ χ' : mul_char R R') : ⇑(χ * χ') = χ * χ' := rfl
protected
lemma one_mul (χ : mul_char R R') : (1 : mul_char R R') * χ = χ := by { ext, simp }
protected
lemma mul_one (χ : mul_char R R') : χ * 1 = χ := by { ext, simp }
/-- The inverse of a multiplicative character. We define it as `inverse ∘ χ`. -/
noncomputable
def inv (χ : mul_char R R') : mul_char R R' :=
{ to_fun := λ a, monoid_with_zero.inverse (χ a),
map_nonunit' := λ a ha, by simp [map_nonunit _ ha],
..monoid_with_zero.inverse.to_monoid_hom.comp χ.to_monoid_hom }
noncomputable
instance has_inv : has_inv (mul_char R R') := ⟨inv⟩
/-- The inverse of a multiplicative character `χ`, applied to `a`, is the inverse of `χ a`. -/
lemma inv_apply_eq_inv (χ : mul_char R R') (a : R) :
χ⁻¹ a = ring.inverse (χ a) :=
eq.refl $ inv χ a
/-- The inverse of a multiplicative character `χ`, applied to `a`, is the inverse of `χ a`.
Variant when the target is a field -/
lemma inv_apply_eq_inv' {R' : Type v} [field R'] (χ : mul_char R R') (a : R) :
χ⁻¹ a = (χ a)⁻¹ :=
(inv_apply_eq_inv χ a).trans $ ring.inverse_eq_inv (χ a)
/-- When the domain has a zero, then the inverse of a multiplicative character `χ`,
applied to `a`, is `χ` applied to the inverse of `a`. -/
lemma inv_apply {R : Type u} [comm_monoid_with_zero R] (χ : mul_char R R') (a : R) :
χ⁻¹ a = χ (ring.inverse a) :=
begin
by_cases ha : is_unit a,
{ rw [inv_apply_eq_inv],
have h := is_unit.map χ ha,
apply_fun ((*) (χ a)) using is_unit.mul_right_injective h,
rw [ring.mul_inverse_cancel _ h, ← map_mul, ring.mul_inverse_cancel _ ha, mul_char.map_one], },
{ revert ha, nontriviality R, intro ha, -- `nontriviality R` by itself doesn't do it
rw [map_nonunit _ ha, ring.inverse_non_unit a ha, mul_char.map_zero χ], },
end
/-- When the domain has a zero, then the inverse of a multiplicative character `χ`,
applied to `a`, is `χ` applied to the inverse of `a`. -/
lemma inv_apply' {R : Type u} [field R] (χ : mul_char R R') (a : R) : χ⁻¹ a = χ a⁻¹ :=
(inv_apply χ a).trans $ congr_arg _ (ring.inverse_eq_inv a)
/-- The product of a character with its inverse is the trivial character. -/
@[simp]
lemma inv_mul (χ : mul_char R R') : χ⁻¹ * χ = 1 :=
begin
ext x,
rw [coe_to_fun_mul, pi.mul_apply, inv_apply_eq_inv,
ring.inverse_mul_cancel _ (is_unit.map _ x.is_unit), one_apply_coe],
end
/-- The commutative group structure on `mul_char R R'`. -/
noncomputable
instance comm_group : comm_group (mul_char R R') :=
{ one := 1,
mul := (*),
inv := has_inv.inv,
mul_left_inv := inv_mul,
mul_assoc := by { intros χ₁ χ₂ χ₃, ext a, simp [mul_assoc], },
mul_comm := by { intros χ₁ χ₂, ext a, simp [mul_comm], },
one_mul := one_mul,
mul_one := mul_one, }
/-- If `a` is a unit and `n : ℕ`, then `(χ ^ n) a = (χ a) ^ n`. -/
lemma pow_apply_coe (χ : mul_char R R') (n : ℕ) (a : Rˣ) :
(χ ^ n) a = (χ a) ^ n :=
begin
induction n with n ih,
{ rw [pow_zero, pow_zero, one_apply_coe], },
{ rw [pow_succ, pow_succ, mul_apply, ih], },
end
/-- If `n` is positive, then `(χ ^ n) a = (χ a) ^ n`. -/
lemma pow_apply' (χ : mul_char R R') {n : ℕ} (hn : 0 < n) (a : R) :
(χ ^ n) a = (χ a) ^ n :=
begin
by_cases ha : is_unit a,
{ exact pow_apply_coe χ n ha.unit, },
{ rw [map_nonunit (χ ^ n) ha, map_nonunit χ ha, zero_pow hn], },
end
end mul_char
end group
end definition_and_group
/-!
### Properties of multiplicative characters
We introduce the properties of being nontrivial or quadratic and prove
some basic facts about them.
We now assume that domain and target are commutative rings.
-/
section properties
namespace mul_char
universes u v w
variables {R : Type u} [comm_ring R] {R' : Type v} [comm_ring R'] {R'' : Type w} [comm_ring R'']
/-- A multiplicative character is *nontrivial* if it takes a value `≠ 1` on a unit. -/
def is_nontrivial (χ : mul_char R R') : Prop := ∃ a : Rˣ, χ a ≠ 1
/-- A multiplicative character is nontrivial iff it is not the trivial character. -/
lemma is_nontrivial_iff (χ : mul_char R R') : χ.is_nontrivial ↔ χ ≠ 1 :=
by simp only [is_nontrivial, ne.def, ext_iff, not_forall, one_apply_coe]
/-- A multiplicative character is *quadratic* if it takes only the values `0`, `1`, `-1`. -/
def is_quadratic (χ : mul_char R R') : Prop := ∀ a, χ a = 0 ∨ χ a = 1 ∨ χ a = -1
/-- If two values of quadratic characters with target `ℤ` agree after coercion into a ring
of characteristic not `2`, then they agree in `ℤ`. -/
lemma is_quadratic.eq_of_eq_coe {χ : mul_char R ℤ} (hχ : is_quadratic χ)
{χ' : mul_char R' ℤ} (hχ' : is_quadratic χ') [nontrivial R''] (hR'' : ring_char R'' ≠ 2)
{a : R} {a' : R'} (h : (χ a : R'') = χ' a') :
χ a = χ' a' :=
int.cast_inj_on_of_ring_char_ne_two hR'' (hχ a) (hχ' a') h
/-- We can post-compose a multiplicative character with a ring homomorphism. -/
@[simps]
def ring_hom_comp (χ : mul_char R R') (f : R' →+* R'') : mul_char R R'' :=
{ to_fun := λ a, f (χ a),
map_nonunit' := λ a ha, by simp only [map_nonunit χ ha, map_zero],
..f.to_monoid_hom.comp χ.to_monoid_hom }
/-- Composition with an injective ring homomorphism preserves nontriviality. -/
lemma is_nontrivial.comp {χ : mul_char R R'} (hχ : χ.is_nontrivial)
{f : R' →+* R''} (hf : function.injective f) :
(χ.ring_hom_comp f).is_nontrivial :=
begin
obtain ⟨a, ha⟩ := hχ,
use a,
rw [ring_hom_comp_apply, ← ring_hom.map_one f],
exact λ h, ha (hf h),
end
/-- Composition with a ring homomorphism preserves the property of being a quadratic character. -/
lemma is_quadratic.comp {χ : mul_char R R'} (hχ : χ.is_quadratic) (f : R' →+* R'') :
(χ.ring_hom_comp f).is_quadratic :=
begin
intro a,
rcases hχ a with (ha | ha | ha);
simp [ha],
end
/-- The inverse of a quadratic character is itself. → -/
lemma is_quadratic.inv {χ : mul_char R R'} (hχ : χ.is_quadratic) : χ⁻¹ = χ :=
begin
ext x,
rw [inv_apply_eq_inv],
rcases hχ x with h₀ | h₁ | h₂,
{ rw [h₀, ring.inverse_zero], },
{ rw [h₁, ring.inverse_one], },
{ rw [h₂, (by norm_cast : (-1 : R') = (-1 : R'ˣ)), ring.inverse_unit (-1 : R'ˣ)],
refl, },
end
/-- The square of a quadratic character is the trivial character. -/
lemma is_quadratic.sq_eq_one {χ : mul_char R R'} (hχ : χ.is_quadratic) : χ ^ 2 = 1 :=
begin
convert mul_left_inv _,
rw [pow_two, hχ.inv],
end
/-- The `p`th power of a quadratic character is itself, when `p` is the (prime) characteristic
of the target ring. -/
lemma is_quadratic.pow_char {χ : mul_char R R'} (hχ : χ.is_quadratic)
(p : ℕ) [hp : fact p.prime] [char_p R' p] :
χ ^ p = χ :=
begin
ext x,
rw [pow_apply_coe],
rcases hχ x with (hx | hx | hx); rw hx,
{ rw [zero_pow (fact.out p.prime).pos], },
{ rw [one_pow], },
{ exact char_p.neg_one_pow_char R' p, },
end
/-- The `n`th power of a quadratic character is the trivial character, when `n` is even. -/
lemma is_quadratic.pow_even {χ : mul_char R R'} (hχ : χ.is_quadratic) {n : ℕ} (hn : even n) :
χ ^ n = 1 :=
begin
obtain ⟨n, rfl⟩ := even_iff_two_dvd.mp hn,
rw [pow_mul, hχ.sq_eq_one, one_pow]
end
/-- The `n`th power of a quadratic character is itself, when `n` is odd. -/
lemma is_quadratic.pow_odd {χ : mul_char R R'} (hχ : χ.is_quadratic) {n : ℕ} (hn : odd n) :
χ ^ n = χ :=
begin
obtain ⟨n, rfl⟩ := hn,
rw [pow_add, pow_one, hχ.pow_even (even_two_mul _), one_mul]
end
open_locale big_operators
/-- The sum over all values of a nontrivial multiplicative character on a finite ring is zero
(when the target is a domain). -/
lemma is_nontrivial.sum_eq_zero [fintype R] [is_domain R'] {χ : mul_char R R'}
(hχ : χ.is_nontrivial) :
∑ a, χ a = 0 :=
begin
rcases hχ with ⟨b, hb⟩,
refine eq_zero_of_mul_eq_self_left hb _,
simp only [finset.mul_sum, ← map_mul],
exact fintype.sum_bijective _ (units.mul_left_bijective b) _ _ (λ x, rfl)
end
/-- The sum over all values of the trivial multiplicative character on a finite ring is
the cardinality of its unit group. -/
lemma sum_one_eq_card_units [fintype R] [decidable_eq R] :
∑ a, (1 : mul_char R R') a = fintype.card Rˣ :=
begin
calc ∑ a, (1 : mul_char R R') a
= ∑ a : R, if is_unit a then 1 else 0 : finset.sum_congr rfl (λ a _, _)
... = ((finset.univ : finset R).filter is_unit).card : finset.sum_boole
... = (finset.univ.map (⟨(coe : Rˣ → R), units.ext⟩)).card : _
... = fintype.card Rˣ : congr_arg _ (finset.card_map _),
{ split_ifs with h h,
{ exact one_apply_coe h.unit },
{ exact map_nonunit _ h } },
{ congr,
ext a,
simp only [finset.mem_filter, finset.mem_univ, true_and, finset.mem_map,
function.embedding.coe_fn_mk, exists_true_left, is_unit], },
end
end mul_char
end properties
|
6e3e75b4083036b421665cea468b3910b0800177 | 19cc34575500ee2e3d4586c15544632aa07a8e66 | /src/data/complex/basic.lean | cab431a0ca0597611c3cfc0c480ccb2d2ce49476 | [
"Apache-2.0"
] | permissive | LibertasSpZ/mathlib | b9fcd46625eb940611adb5e719a4b554138dade6 | 33f7870a49d7cc06d2f3036e22543e6ec5046e68 | refs/heads/master | 1,672,066,539,347 | 1,602,429,158,000 | 1,602,429,158,000 | null | 0 | 0 | null | null | null | null | UTF-8 | Lean | false | false | 20,427 | lean | /-
Copyright (c) 2017 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Mario Carneiro
-/
import data.real.basic
open_locale big_operators
/-!
# The complex numbers
The complex numbers are modelled as ℝ^2 in the obvious way.
-/
/-! ### Definition and basic arithmmetic -/
/-- Complex numbers consist of two `real`s: a real part `re` and an imaginary part `im`. -/
structure complex : Type :=
(re : ℝ) (im : ℝ)
notation `ℂ` := complex
namespace complex
noncomputable instance : decidable_eq ℂ := classical.dec_eq _
/-- The equivalence between the complex numbers and `ℝ × ℝ`. -/
def equiv_real_prod : ℂ ≃ (ℝ × ℝ) :=
{ to_fun := λ z, ⟨z.re, z.im⟩,
inv_fun := λ p, ⟨p.1, p.2⟩,
left_inv := λ ⟨x, y⟩, rfl,
right_inv := λ ⟨x, y⟩, rfl }
@[simp] theorem equiv_real_prod_apply (z : ℂ) : equiv_real_prod z = (z.re, z.im) := rfl
theorem equiv_real_prod_symm_re (x y : ℝ) : (equiv_real_prod.symm (x, y)).re = x := rfl
theorem equiv_real_prod_symm_im (x y : ℝ) : (equiv_real_prod.symm (x, y)).im = y := rfl
@[simp] theorem eta : ∀ z : ℂ, complex.mk z.re z.im = z
| ⟨a, b⟩ := rfl
@[ext]
theorem ext : ∀ {z w : ℂ}, z.re = w.re → z.im = w.im → z = w
| ⟨zr, zi⟩ ⟨_, _⟩ rfl rfl := rfl
theorem ext_iff {z w : ℂ} : z = w ↔ z.re = w.re ∧ z.im = w.im :=
⟨λ H, by simp [H], and.rec ext⟩
instance : has_coe ℝ ℂ := ⟨λ r, ⟨r, 0⟩⟩
@[simp, norm_cast] lemma of_real_re (r : ℝ) : (r : ℂ).re = r := rfl
@[simp, norm_cast] lemma of_real_im (r : ℝ) : (r : ℂ).im = 0 := rfl
@[simp, norm_cast] theorem of_real_inj {z w : ℝ} : (z : ℂ) = w ↔ z = w :=
⟨congr_arg re, congr_arg _⟩
instance : has_zero ℂ := ⟨(0 : ℝ)⟩
instance : inhabited ℂ := ⟨0⟩
@[simp] lemma zero_re : (0 : ℂ).re = 0 := rfl
@[simp] lemma zero_im : (0 : ℂ).im = 0 := rfl
@[simp, norm_cast] lemma of_real_zero : ((0 : ℝ) : ℂ) = 0 := rfl
@[simp] theorem of_real_eq_zero {z : ℝ} : (z : ℂ) = 0 ↔ z = 0 := of_real_inj
theorem of_real_ne_zero {z : ℝ} : (z : ℂ) ≠ 0 ↔ z ≠ 0 := not_congr of_real_eq_zero
instance : has_one ℂ := ⟨(1 : ℝ)⟩
@[simp] lemma one_re : (1 : ℂ).re = 1 := rfl
@[simp] lemma one_im : (1 : ℂ).im = 0 := rfl
@[simp, norm_cast] lemma of_real_one : ((1 : ℝ) : ℂ) = 1 := rfl
instance : has_add ℂ := ⟨λ z w, ⟨z.re + w.re, z.im + w.im⟩⟩
@[simp] lemma add_re (z w : ℂ) : (z + w).re = z.re + w.re := rfl
@[simp] lemma add_im (z w : ℂ) : (z + w).im = z.im + w.im := rfl
@[simp] lemma bit0_re (z : ℂ) : (bit0 z).re = bit0 z.re := rfl
@[simp] lemma bit1_re (z : ℂ) : (bit1 z).re = bit1 z.re := rfl
@[simp] lemma bit0_im (z : ℂ) : (bit0 z).im = bit0 z.im := eq.refl _
@[simp] lemma bit1_im (z : ℂ) : (bit1 z).im = bit0 z.im := add_zero _
@[simp, norm_cast] lemma of_real_add (r s : ℝ) : ((r + s : ℝ) : ℂ) = r + s :=
ext_iff.2 $ by simp
@[simp, norm_cast] lemma of_real_bit0 (r : ℝ) : ((bit0 r : ℝ) : ℂ) = bit0 r :=
ext_iff.2 $ by simp [bit0]
@[simp, norm_cast] lemma of_real_bit1 (r : ℝ) : ((bit1 r : ℝ) : ℂ) = bit1 r :=
ext_iff.2 $ by simp [bit1]
instance : has_neg ℂ := ⟨λ z, ⟨-z.re, -z.im⟩⟩
@[simp] lemma neg_re (z : ℂ) : (-z).re = -z.re := rfl
@[simp] lemma neg_im (z : ℂ) : (-z).im = -z.im := rfl
@[simp, norm_cast] lemma of_real_neg (r : ℝ) : ((-r : ℝ) : ℂ) = -r := ext_iff.2 $ by simp
instance : has_mul ℂ := ⟨λ z w, ⟨z.re * w.re - z.im * w.im, z.re * w.im + z.im * w.re⟩⟩
@[simp] lemma mul_re (z w : ℂ) : (z * w).re = z.re * w.re - z.im * w.im := rfl
@[simp] lemma mul_im (z w : ℂ) : (z * w).im = z.re * w.im + z.im * w.re := rfl
@[simp, norm_cast] lemma of_real_mul (r s : ℝ) : ((r * s : ℝ) : ℂ) = r * s := ext_iff.2 $ by simp
lemma smul_re (r : ℝ) (z : ℂ) : (↑r * z).re = r * z.re := by simp
lemma smul_im (r : ℝ) (z : ℂ) : (↑r * z).im = r * z.im := by simp
lemma of_real_smul (r : ℝ) (z : ℂ) : (↑r * z) = ⟨r * z.re, r * z.im⟩ := ext (smul_re _ _) (smul_im _ _)
/-! ### The imaginary unit, `I` -/
/-- The imaginary unit. -/
def I : ℂ := ⟨0, 1⟩
@[simp] lemma I_re : I.re = 0 := rfl
@[simp] lemma I_im : I.im = 1 := rfl
@[simp] lemma I_mul_I : I * I = -1 := ext_iff.2 $ by simp
lemma I_mul (z : ℂ) : I * z = ⟨-z.im, z.re⟩ :=
ext_iff.2 $ by simp
lemma I_ne_zero : (I : ℂ) ≠ 0 := mt (congr_arg im) zero_ne_one.symm
lemma mk_eq_add_mul_I (a b : ℝ) : complex.mk a b = a + b * I :=
ext_iff.2 $ by simp
@[simp] lemma re_add_im (z : ℂ) : (z.re : ℂ) + z.im * I = z :=
ext_iff.2 $ by simp
/-! ### Commutative ring instance and lemmas -/
instance : comm_ring ℂ :=
by refine { zero := 0, add := (+), neg := has_neg.neg, one := 1, mul := (*), ..};
{ intros, apply ext_iff.2; split; simp; ring }
instance re.is_add_group_hom : is_add_group_hom complex.re :=
{ map_add := complex.add_re }
instance im.is_add_group_hom : is_add_group_hom complex.im :=
{ map_add := complex.add_im }
/-! ### Complex conjugation -/
/-- The complex conjugate. -/
def conj : ℂ →+* ℂ :=
begin
refine_struct { to_fun := λ z : ℂ, (⟨z.re, -z.im⟩ : ℂ), .. };
{ intros, ext; simp [add_comm], },
end
@[simp] lemma conj_re (z : ℂ) : (conj z).re = z.re := rfl
@[simp] lemma conj_im (z : ℂ) : (conj z).im = -z.im := rfl
@[simp] lemma conj_of_real (r : ℝ) : conj r = r := ext_iff.2 $ by simp [conj]
@[simp] lemma conj_I : conj I = -I := ext_iff.2 $ by simp
@[simp] lemma conj_bit0 (z : ℂ) : conj (bit0 z) = bit0 (conj z) := ext_iff.2 $ by simp [bit0]
@[simp] lemma conj_bit1 (z : ℂ) : conj (bit1 z) = bit1 (conj z) := ext_iff.2 $ by simp [bit0]
@[simp] lemma conj_neg_I : conj (-I) = I := ext_iff.2 $ by simp
@[simp] lemma conj_conj (z : ℂ) : conj (conj z) = z :=
ext_iff.2 $ by simp
lemma conj_involutive : function.involutive conj := conj_conj
lemma conj_bijective : function.bijective conj := conj_involutive.bijective
lemma conj_inj {z w : ℂ} : conj z = conj w ↔ z = w :=
conj_bijective.1.eq_iff
@[simp] lemma conj_eq_zero {z : ℂ} : conj z = 0 ↔ z = 0 :=
by simpa using @conj_inj z 0
lemma eq_conj_iff_real {z : ℂ} : conj z = z ↔ ∃ r : ℝ, z = r :=
⟨λ h, ⟨z.re, ext rfl $ eq_zero_of_neg_eq (congr_arg im h)⟩,
λ ⟨h, e⟩, by rw [e, conj_of_real]⟩
lemma eq_conj_iff_re {z : ℂ} : conj z = z ↔ (z.re : ℂ) = z :=
eq_conj_iff_real.trans ⟨by rintro ⟨r, rfl⟩; simp, λ h, ⟨_, h.symm⟩⟩
/-! ### Norm squared -/
/-- The norm squared function. -/
@[pp_nodot] def norm_sq (z : ℂ) : ℝ := z.re * z.re + z.im * z.im
@[simp] lemma norm_sq_of_real (r : ℝ) : norm_sq r = r * r :=
by simp [norm_sq]
@[simp] lemma norm_sq_zero : norm_sq 0 = 0 := by simp [norm_sq]
@[simp] lemma norm_sq_one : norm_sq 1 = 1 := by simp [norm_sq]
@[simp] lemma norm_sq_I : norm_sq I = 1 := by simp [norm_sq]
lemma norm_sq_nonneg (z : ℂ) : 0 ≤ norm_sq z :=
add_nonneg (mul_self_nonneg _) (mul_self_nonneg _)
@[simp] lemma norm_sq_eq_zero {z : ℂ} : norm_sq z = 0 ↔ z = 0 :=
⟨λ h, ext
(eq_zero_of_mul_self_add_mul_self_eq_zero h)
(eq_zero_of_mul_self_add_mul_self_eq_zero $ (add_comm _ _).trans h),
λ h, h.symm ▸ norm_sq_zero⟩
@[simp] lemma norm_sq_pos {z : ℂ} : 0 < norm_sq z ↔ z ≠ 0 :=
by rw [lt_iff_le_and_ne, ne, eq_comm]; simp [norm_sq_nonneg]
@[simp] lemma norm_sq_neg (z : ℂ) : norm_sq (-z) = norm_sq z :=
by simp [norm_sq]
@[simp] lemma norm_sq_conj (z : ℂ) : norm_sq (conj z) = norm_sq z :=
by simp [norm_sq]
@[simp] lemma norm_sq_mul (z w : ℂ) : norm_sq (z * w) = norm_sq z * norm_sq w :=
by dsimp [norm_sq]; ring
lemma norm_sq_add (z w : ℂ) : norm_sq (z + w) =
norm_sq z + norm_sq w + 2 * (z * conj w).re :=
by dsimp [norm_sq]; ring
lemma re_sq_le_norm_sq (z : ℂ) : z.re * z.re ≤ norm_sq z :=
le_add_of_nonneg_right (mul_self_nonneg _)
lemma im_sq_le_norm_sq (z : ℂ) : z.im * z.im ≤ norm_sq z :=
le_add_of_nonneg_left (mul_self_nonneg _)
theorem mul_conj (z : ℂ) : z * conj z = norm_sq z :=
ext_iff.2 $ by simp [norm_sq, mul_comm, sub_eq_neg_add, add_comm]
theorem add_conj (z : ℂ) : z + conj z = (2 * z.re : ℝ) :=
ext_iff.2 $ by simp [two_mul]
/-- The coercion `ℝ → ℂ` as a `ring_hom`. -/
def of_real : ℝ →+* ℂ := ⟨coe, of_real_one, of_real_mul, of_real_zero, of_real_add⟩
@[simp] lemma of_real_eq_coe (r : ℝ) : of_real r = r := rfl
@[simp] lemma I_sq : I ^ 2 = -1 := by rw [pow_two, I_mul_I]
@[simp] lemma sub_re (z w : ℂ) : (z - w).re = z.re - w.re := rfl
@[simp] lemma sub_im (z w : ℂ) : (z - w).im = z.im - w.im := rfl
@[simp, norm_cast] lemma of_real_sub (r s : ℝ) : ((r - s : ℝ) : ℂ) = r - s := ext_iff.2 $ by simp
@[simp, norm_cast] lemma of_real_pow (r : ℝ) (n : ℕ) : ((r ^ n : ℝ) : ℂ) = r ^ n :=
by induction n; simp [*, of_real_mul, pow_succ]
theorem sub_conj (z : ℂ) : z - conj z = (2 * z.im : ℝ) * I :=
ext_iff.2 $ by simp [two_mul, sub_eq_add_neg]
lemma norm_sq_sub (z w : ℂ) : norm_sq (z - w) =
norm_sq z + norm_sq w - 2 * (z * conj w).re :=
by rw [sub_eq_add_neg, norm_sq_add]; simp [-mul_re, add_comm, add_left_comm, sub_eq_add_neg]
/-! ### Inversion -/
noncomputable instance : has_inv ℂ := ⟨λ z, conj z * ((norm_sq z)⁻¹:ℝ)⟩
theorem inv_def (z : ℂ) : z⁻¹ = conj z * ((norm_sq z)⁻¹:ℝ) := rfl
@[simp] lemma inv_re (z : ℂ) : (z⁻¹).re = z.re / norm_sq z := by simp [inv_def, division_def]
@[simp] lemma inv_im (z : ℂ) : (z⁻¹).im = -z.im / norm_sq z := by simp [inv_def, division_def]
@[simp, norm_cast] lemma of_real_inv (r : ℝ) : ((r⁻¹ : ℝ) : ℂ) = r⁻¹ :=
ext_iff.2 $ begin
simp,
by_cases r = 0, { simp [h] },
{ rw [← div_div_eq_div_mul, div_self h, one_div] },
end
protected lemma inv_zero : (0⁻¹ : ℂ) = 0 :=
by rw [← of_real_zero, ← of_real_inv, inv_zero]
protected theorem mul_inv_cancel {z : ℂ} (h : z ≠ 0) : z * z⁻¹ = 1 :=
by rw [inv_def, ← mul_assoc, mul_conj, ← of_real_mul,
mul_inv_cancel (mt norm_sq_eq_zero.1 h), of_real_one]
/-! ### Field instance and lemmas -/
noncomputable instance : field ℂ :=
{ inv := has_inv.inv,
exists_pair_ne := ⟨0, 1, mt (congr_arg re) zero_ne_one⟩,
mul_inv_cancel := @complex.mul_inv_cancel,
inv_zero := complex.inv_zero,
..complex.comm_ring }
lemma div_re (z w : ℂ) : (z / w).re = z.re * w.re / norm_sq w + z.im * w.im / norm_sq w :=
by simp [div_eq_mul_inv, mul_assoc, sub_eq_add_neg]
lemma div_im (z w : ℂ) : (z / w).im = z.im * w.re / norm_sq w - z.re * w.im / norm_sq w :=
by simp [div_eq_mul_inv, mul_assoc, sub_eq_add_neg, add_comm]
@[simp, norm_cast] lemma of_real_div (r s : ℝ) : ((r / s : ℝ) : ℂ) = r / s :=
of_real.map_div r s
@[simp, norm_cast] lemma of_real_fpow (r : ℝ) (n : ℤ) : ((r ^ n : ℝ) : ℂ) = (r : ℂ) ^ n :=
of_real.map_fpow r n
@[simp] lemma div_I (z : ℂ) : z / I = -(z * I) :=
(div_eq_iff_mul_eq I_ne_zero).2 $ by simp [mul_assoc]
@[simp] lemma inv_I : I⁻¹ = -I :=
by simp [inv_eq_one_div]
@[simp] lemma norm_sq_inv (z : ℂ) : norm_sq z⁻¹ = (norm_sq z)⁻¹ :=
if h : z = 0 then by simp [h] else
mul_right_cancel' (mt norm_sq_eq_zero.1 h) $
by rw [← norm_sq_mul]; simp [h, -norm_sq_mul]
@[simp] lemma norm_sq_div (z w : ℂ) : norm_sq (z / w) = norm_sq z / norm_sq w :=
by rw [division_def, norm_sq_mul, norm_sq_inv]; refl
/-! ### Cast lemmas -/
@[simp, norm_cast] theorem of_real_nat_cast (n : ℕ) : ((n : ℝ) : ℂ) = n :=
of_real.map_nat_cast n
@[simp, norm_cast] lemma nat_cast_re (n : ℕ) : (n : ℂ).re = n :=
by rw [← of_real_nat_cast, of_real_re]
@[simp, norm_cast] lemma nat_cast_im (n : ℕ) : (n : ℂ).im = 0 :=
by rw [← of_real_nat_cast, of_real_im]
@[simp, norm_cast] theorem of_real_int_cast (n : ℤ) : ((n : ℝ) : ℂ) = n :=
of_real.map_int_cast n
@[simp, norm_cast] lemma int_cast_re (n : ℤ) : (n : ℂ).re = n :=
by rw [← of_real_int_cast, of_real_re]
@[simp, norm_cast] lemma int_cast_im (n : ℤ) : (n : ℂ).im = 0 :=
by rw [← of_real_int_cast, of_real_im]
@[simp, norm_cast] theorem of_real_rat_cast (n : ℚ) : ((n : ℝ) : ℂ) = n :=
of_real.map_rat_cast n
@[simp, norm_cast] lemma rat_cast_re (q : ℚ) : (q : ℂ).re = q :=
by rw [← of_real_rat_cast, of_real_re]
@[simp, norm_cast] lemma rat_cast_im (q : ℚ) : (q : ℂ).im = 0 :=
by rw [← of_real_rat_cast, of_real_im]
/-! ### Characteristic zero -/
instance char_zero_complex : char_zero ℂ :=
char_zero_of_inj_zero $ λ n h,
by rwa [← of_real_nat_cast, of_real_eq_zero, nat.cast_eq_zero] at h
theorem re_eq_add_conj (z : ℂ) : (z.re : ℂ) = (z + conj z) / 2 :=
by rw [add_conj]; simp; rw [mul_div_cancel_left (z.re:ℂ) two_ne_zero']
/-! ### Absolute value -/
/-- The complex absolute value function, defined as the square root of the norm squared. -/
@[pp_nodot] noncomputable def abs (z : ℂ) : ℝ := (norm_sq z).sqrt
local notation `abs'` := _root_.abs
@[simp, norm_cast] lemma abs_of_real (r : ℝ) : abs r = abs' r :=
by simp [abs, norm_sq_of_real, real.sqrt_mul_self_eq_abs]
lemma abs_of_nonneg {r : ℝ} (h : 0 ≤ r) : abs r = r :=
(abs_of_real _).trans (abs_of_nonneg h)
lemma abs_of_nat (n : ℕ) : complex.abs n = n :=
calc complex.abs n = complex.abs (n:ℝ) : by rw [of_real_nat_cast]
... = _ : abs_of_nonneg (nat.cast_nonneg n)
lemma mul_self_abs (z : ℂ) : abs z * abs z = norm_sq z :=
real.mul_self_sqrt (norm_sq_nonneg _)
@[simp] lemma abs_zero : abs 0 = 0 := by simp [abs]
@[simp] lemma abs_one : abs 1 = 1 := by simp [abs]
@[simp] lemma abs_I : abs I = 1 := by simp [abs]
@[simp] lemma abs_two : abs 2 = 2 :=
calc abs 2 = abs (2 : ℝ) : by rw [of_real_bit0, of_real_one]
... = (2 : ℝ) : abs_of_nonneg (by norm_num)
lemma abs_nonneg (z : ℂ) : 0 ≤ abs z :=
real.sqrt_nonneg _
@[simp] lemma abs_eq_zero {z : ℂ} : abs z = 0 ↔ z = 0 :=
(real.sqrt_eq_zero $ norm_sq_nonneg _).trans norm_sq_eq_zero
lemma abs_ne_zero {z : ℂ} : abs z ≠ 0 ↔ z ≠ 0 :=
not_congr abs_eq_zero
@[simp] lemma abs_conj (z : ℂ) : abs (conj z) = abs z :=
by simp [abs]
@[simp] lemma abs_mul (z w : ℂ) : abs (z * w) = abs z * abs w :=
by rw [abs, norm_sq_mul, real.sqrt_mul (norm_sq_nonneg _)]; refl
lemma abs_re_le_abs (z : ℂ) : abs' z.re ≤ abs z :=
by rw [mul_self_le_mul_self_iff (_root_.abs_nonneg z.re) (abs_nonneg _),
abs_mul_abs_self, mul_self_abs];
apply re_sq_le_norm_sq
lemma abs_im_le_abs (z : ℂ) : abs' z.im ≤ abs z :=
by rw [mul_self_le_mul_self_iff (_root_.abs_nonneg z.im) (abs_nonneg _),
abs_mul_abs_self, mul_self_abs];
apply im_sq_le_norm_sq
lemma re_le_abs (z : ℂ) : z.re ≤ abs z :=
(abs_le.1 (abs_re_le_abs _)).2
lemma im_le_abs (z : ℂ) : z.im ≤ abs z :=
(abs_le.1 (abs_im_le_abs _)).2
lemma abs_add (z w : ℂ) : abs (z + w) ≤ abs z + abs w :=
(mul_self_le_mul_self_iff (abs_nonneg _)
(add_nonneg (abs_nonneg _) (abs_nonneg _))).2 $
begin
rw [mul_self_abs, add_mul_self_eq, mul_self_abs, mul_self_abs,
add_right_comm, norm_sq_add, add_le_add_iff_left,
mul_assoc, mul_le_mul_left (@zero_lt_two ℝ _)],
simpa [-mul_re] using re_le_abs (z * conj w)
end
instance : is_absolute_value abs :=
{ abv_nonneg := abs_nonneg,
abv_eq_zero := λ _, abs_eq_zero,
abv_add := abs_add,
abv_mul := abs_mul }
open is_absolute_value
@[simp] lemma abs_abs (z : ℂ) : abs' (abs z) = abs z :=
_root_.abs_of_nonneg (abs_nonneg _)
@[simp] lemma abs_pos {z : ℂ} : 0 < abs z ↔ z ≠ 0 := abv_pos abs
@[simp] lemma abs_neg : ∀ z, abs (-z) = abs z := abv_neg abs
lemma abs_sub : ∀ z w, abs (z - w) = abs (w - z) := abv_sub abs
lemma abs_sub_le : ∀ a b c, abs (a - c) ≤ abs (a - b) + abs (b - c) := abv_sub_le abs
@[simp] theorem abs_inv : ∀ z, abs z⁻¹ = (abs z)⁻¹ := abv_inv abs
@[simp] theorem abs_div : ∀ z w, abs (z / w) = abs z / abs w := abv_div abs
lemma abs_abs_sub_le_abs_sub : ∀ z w, abs' (abs z - abs w) ≤ abs (z - w) :=
abs_abv_sub_le_abv_sub abs
lemma abs_le_abs_re_add_abs_im (z : ℂ) : abs z ≤ abs' z.re + abs' z.im :=
by simpa [re_add_im] using abs_add z.re (z.im * I)
lemma abs_re_div_abs_le_one (z : ℂ) : abs' (z.re / z.abs) ≤ 1 :=
if hz : z = 0 then by simp [hz, zero_le_one]
else by { simp_rw [_root_.abs_div, abs_abs, div_le_iff (abs_pos.2 hz), one_mul, abs_re_le_abs] }
lemma abs_im_div_abs_le_one (z : ℂ) : abs' (z.im / z.abs) ≤ 1 :=
if hz : z = 0 then by simp [hz, zero_le_one]
else by { simp_rw [_root_.abs_div, abs_abs, div_le_iff (abs_pos.2 hz), one_mul, abs_im_le_abs] }
@[simp, norm_cast] lemma abs_cast_nat (n : ℕ) : abs (n : ℂ) = n :=
by rw [← of_real_nat_cast, abs_of_nonneg (nat.cast_nonneg n)]
lemma norm_sq_eq_abs (x : ℂ) : norm_sq x = abs x ^ 2 :=
by rw [abs, pow_two, real.mul_self_sqrt (norm_sq_nonneg _)]
/-! ### Cauchy sequences -/
theorem is_cau_seq_re (f : cau_seq ℂ abs) : is_cau_seq abs' (λ n, (f n).re) :=
λ ε ε0, (f.cauchy ε0).imp $ λ i H j ij,
lt_of_le_of_lt (by simpa using abs_re_le_abs (f j - f i)) (H _ ij)
theorem is_cau_seq_im (f : cau_seq ℂ abs) : is_cau_seq abs' (λ n, (f n).im) :=
λ ε ε0, (f.cauchy ε0).imp $ λ i H j ij,
lt_of_le_of_lt (by simpa using abs_im_le_abs (f j - f i)) (H _ ij)
/-- The real part of a complex Cauchy sequence, as a real Cauchy sequence. -/
noncomputable def cau_seq_re (f : cau_seq ℂ abs) : cau_seq ℝ abs' :=
⟨_, is_cau_seq_re f⟩
/-- The imaginary part of a complex Cauchy sequence, as a real Cauchy sequence. -/
noncomputable def cau_seq_im (f : cau_seq ℂ abs) : cau_seq ℝ abs' :=
⟨_, is_cau_seq_im f⟩
lemma is_cau_seq_abs {f : ℕ → ℂ} (hf : is_cau_seq abs f) :
is_cau_seq abs' (abs ∘ f) :=
λ ε ε0, let ⟨i, hi⟩ := hf ε ε0 in
⟨i, λ j hj, lt_of_le_of_lt (abs_abs_sub_le_abs_sub _ _) (hi j hj)⟩
/-- The limit of a Cauchy sequence of complex numbers. -/
noncomputable def lim_aux (f : cau_seq ℂ abs) : ℂ :=
⟨cau_seq.lim (cau_seq_re f), cau_seq.lim (cau_seq_im f)⟩
theorem equiv_lim_aux (f : cau_seq ℂ abs) : f ≈ cau_seq.const abs (lim_aux f) :=
λ ε ε0, (exists_forall_ge_and
(cau_seq.equiv_lim ⟨_, is_cau_seq_re f⟩ _ (half_pos ε0))
(cau_seq.equiv_lim ⟨_, is_cau_seq_im f⟩ _ (half_pos ε0))).imp $
λ i H j ij, begin
cases H _ ij with H₁ H₂,
apply lt_of_le_of_lt (abs_le_abs_re_add_abs_im _),
dsimp [lim_aux] at *,
have := add_lt_add H₁ H₂,
rwa add_halves at this,
end
noncomputable instance : cau_seq.is_complete ℂ abs :=
⟨λ f, ⟨lim_aux f, equiv_lim_aux f⟩⟩
open cau_seq
lemma lim_eq_lim_im_add_lim_re (f : cau_seq ℂ abs) : lim f =
↑(lim (cau_seq_re f)) + ↑(lim (cau_seq_im f)) * I :=
lim_eq_of_equiv_const $
calc f ≈ _ : equiv_lim_aux f
... = cau_seq.const abs (↑(lim (cau_seq_re f)) + ↑(lim (cau_seq_im f)) * I) :
cau_seq.ext (λ _, complex.ext (by simp [lim_aux, cau_seq_re]) (by simp [lim_aux, cau_seq_im]))
lemma lim_re (f : cau_seq ℂ abs) : lim (cau_seq_re f) = (lim f).re :=
by rw [lim_eq_lim_im_add_lim_re]; simp
lemma lim_im (f : cau_seq ℂ abs) : lim (cau_seq_im f) = (lim f).im :=
by rw [lim_eq_lim_im_add_lim_re]; simp
lemma is_cau_seq_conj (f : cau_seq ℂ abs) : is_cau_seq abs (λ n, conj (f n)) :=
λ ε ε0, let ⟨i, hi⟩ := f.2 ε ε0 in
⟨i, λ j hj, by rw [← conj.map_sub, abs_conj]; exact hi j hj⟩
/-- The complex conjugate of a complex Cauchy sequence, as a complex Cauchy sequence. -/
noncomputable def cau_seq_conj (f : cau_seq ℂ abs) : cau_seq ℂ abs :=
⟨_, is_cau_seq_conj f⟩
lemma lim_conj (f : cau_seq ℂ abs) : lim (cau_seq_conj f) = conj (lim f) :=
complex.ext (by simp [cau_seq_conj, (lim_re _).symm, cau_seq_re])
(by simp [cau_seq_conj, (lim_im _).symm, cau_seq_im, (lim_neg _).symm]; refl)
/-- The absolute value of a complex Cauchy sequence, as a real Cauchy sequence. -/
noncomputable def cau_seq_abs (f : cau_seq ℂ abs) : cau_seq ℝ abs' :=
⟨_, is_cau_seq_abs f.2⟩
lemma lim_abs (f : cau_seq ℂ abs) : lim (cau_seq_abs f) = abs (lim f) :=
lim_eq_of_equiv_const (λ ε ε0,
let ⟨i, hi⟩ := equiv_lim f ε ε0 in
⟨i, λ j hj, lt_of_le_of_lt (abs_abs_sub_le_abs_sub _ _) (hi j hj)⟩)
@[simp, norm_cast] lemma of_real_prod {α : Type*} (s : finset α) (f : α → ℝ) :
((∏ i in s, f i : ℝ) : ℂ) = ∏ i in s, (f i : ℂ) :=
ring_hom.map_prod of_real _ _
@[simp, norm_cast] lemma of_real_sum {α : Type*} (s : finset α) (f : α → ℝ) :
((∑ i in s, f i : ℝ) : ℂ) = ∑ i in s, (f i : ℂ) :=
ring_hom.map_sum of_real _ _
end complex
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