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Homotopy.lean
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/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
/-!
# Chain homotopies
We define chain homotopies, and prove that homotopic chain maps induce the same map on homology.
-/
universe v u
noncomputable section
open CategoryTheory Category Limits HomologicalComplex
variable {ι : Type*}
variable {V : Type u} [Category.{v} V] [Preadditive V]
variable {c : ComplexShape ι} {C D E : HomologicalComplex V c}
variable (f g : C ⟶ D) (h k : D ⟶ E) (i : ι)
section
/-- The composition of `C.d i (c.next i) ≫ f (c.next i) i`. -/
def dNext (i : ι) : (∀ i j, C.X i ⟶ D.X j) →+ (C.X i ⟶ D.X i) :=
AddMonoidHom.mk' (fun f => C.d i (c.next i) ≫ f (c.next i) i) fun _ _ =>
Preadditive.comp_add _ _ _ _ _ _
/-- `f (c.next i) i`. -/
def fromNext (i : ι) : (∀ i j, C.X i ⟶ D.X j) →+ (C.xNext i ⟶ D.X i) :=
AddMonoidHom.mk' (fun f => f (c.next i) i) fun _ _ => rfl
@[simp]
theorem dNext_eq_dFrom_fromNext (f : ∀ i j, C.X i ⟶ D.X j) (i : ι) :
dNext i f = C.dFrom i ≫ fromNext i f :=
rfl
theorem dNext_eq (f : ∀ i j, C.X i ⟶ D.X j) {i i' : ι} (w : c.Rel i i') :
dNext i f = C.d i i' ≫ f i' i := by
obtain rfl := c.next_eq' w
rfl
lemma dNext_eq_zero (f : ∀ i j, C.X i ⟶ D.X j) (i : ι) (hi : ¬ c.Rel i (c.next i)) :
dNext i f = 0 := by
dsimp [dNext]
rw [shape _ _ _ hi, zero_comp]
-- This is not a simp lemma; the LHS already simplifies.
theorem dNext_comp_left (f : C ⟶ D) (g : ∀ i j, D.X i ⟶ E.X j) (i : ι) :
(dNext i fun i j => f.f i ≫ g i j) = f.f i ≫ dNext i g :=
(f.comm_assoc _ _ _).symm
-- This is not a simp lemma; the LHS already simplifies.
theorem dNext_comp_right (f : ∀ i j, C.X i ⟶ D.X j) (g : D ⟶ E) (i : ι) :
(dNext i fun i j => f i j ≫ g.f j) = dNext i f ≫ g.f i :=
(assoc _ _ _).symm
/-- The composition `f j (c.prev j) ≫ D.d (c.prev j) j`. -/
def prevD (j : ι) : (∀ i j, C.X i ⟶ D.X j) →+ (C.X j ⟶ D.X j) :=
AddMonoidHom.mk' (fun f => f j (c.prev j) ≫ D.d (c.prev j) j) fun _ _ =>
Preadditive.add_comp _ _ _ _ _ _
lemma prevD_eq_zero (f : ∀ i j, C.X i ⟶ D.X j) (i : ι) (hi : ¬ c.Rel (c.prev i) i) :
prevD i f = 0 := by
dsimp [prevD]
rw [shape _ _ _ hi, comp_zero]
/-- `f j (c.prev j)`. -/
def toPrev (j : ι) : (∀ i j, C.X i ⟶ D.X j) →+ (C.X j ⟶ D.xPrev j) :=
AddMonoidHom.mk' (fun f => f j (c.prev j)) fun _ _ => rfl
@[simp]
theorem prevD_eq_toPrev_dTo (f : ∀ i j, C.X i ⟶ D.X j) (j : ι) :
prevD j f = toPrev j f ≫ D.dTo j :=
rfl
theorem prevD_eq (f : ∀ i j, C.X i ⟶ D.X j) {j j' : ι} (w : c.Rel j' j) :
prevD j f = f j j' ≫ D.d j' j := by
obtain rfl := c.prev_eq' w
rfl
-- This is not a simp lemma; the LHS already simplifies.
theorem prevD_comp_left (f : C ⟶ D) (g : ∀ i j, D.X i ⟶ E.X j) (j : ι) :
(prevD j fun i j => f.f i ≫ g i j) = f.f j ≫ prevD j g :=
assoc _ _ _
-- This is not a simp lemma; the LHS already simplifies.
theorem prevD_comp_right (f : ∀ i j, C.X i ⟶ D.X j) (g : D ⟶ E) (j : ι) :
(prevD j fun i j => f i j ≫ g.f j) = prevD j f ≫ g.f j := by
dsimp [prevD]
simp only [assoc, g.comm]
theorem dNext_nat (C D : ChainComplex V ℕ) (i : ℕ) (f : ∀ i j, C.X i ⟶ D.X j) :
dNext i f = C.d i (i - 1) ≫ f (i - 1) i := by
dsimp [dNext]
cases i
· simp only [shape, ChainComplex.next_nat_zero, ComplexShape.down_Rel, not_false_iff, zero_comp,
reduceCtorEq]
· congr <;> simp
theorem prevD_nat (C D : CochainComplex V ℕ) (i : ℕ) (f : ∀ i j, C.X i ⟶ D.X j) :
prevD i f = f i (i - 1) ≫ D.d (i - 1) i := by
dsimp [prevD]
cases i
· simp only [shape, CochainComplex.prev_nat_zero, ComplexShape.up_Rel, not_false_iff, comp_zero,
reduceCtorEq]
· congr <;> simp
/-- A homotopy `h` between chain maps `f` and `g` consists of components `h i j : C.X i ⟶ D.X j`
which are zero unless `c.Rel j i`, satisfying the homotopy condition.
-/
@[ext]
structure Homotopy (f g : C ⟶ D) where
hom : ∀ i j, C.X i ⟶ D.X j
zero : ∀ i j, ¬c.Rel j i → hom i j = 0 := by cat_disch
comm : ∀ i, f.f i = dNext i hom + prevD i hom + g.f i := by cat_disch
variable {f g}
namespace Homotopy
/-- `f` is homotopic to `g` iff `f - g` is homotopic to `0`.
-/
def equivSubZero : Homotopy f g ≃ Homotopy (f - g) 0 where
toFun h :=
{ hom := fun i j => h.hom i j
zero := fun _ _ w => h.zero _ _ w
comm := fun i => by simp [h.comm] }
invFun h :=
{ hom := fun i j => h.hom i j
zero := fun _ _ w => h.zero _ _ w
comm := fun i => by simpa [sub_eq_iff_eq_add] using h.comm i }
left_inv := by cat_disch
right_inv := by cat_disch
/-- Equal chain maps are homotopic. -/
@[simps]
def ofEq (h : f = g) : Homotopy f g where
hom := 0
zero _ _ _ := rfl
/-- Every chain map is homotopic to itself. -/
@[simps!, refl]
def refl (f : C ⟶ D) : Homotopy f f :=
ofEq (rfl : f = f)
/-- `f` is homotopic to `g` iff `g` is homotopic to `f`. -/
@[simps!, symm]
def symm {f g : C ⟶ D} (h : Homotopy f g) : Homotopy g f where
hom := -h.hom
zero i j w := by rw [Pi.neg_apply, Pi.neg_apply, h.zero i j w, neg_zero]
comm i := by
rw [AddMonoidHom.map_neg, AddMonoidHom.map_neg, h.comm, ← neg_add, ← add_assoc, neg_add_cancel,
zero_add]
/-- homotopy is a transitive relation. -/
@[simps!, trans]
def trans {e f g : C ⟶ D} (h : Homotopy e f) (k : Homotopy f g) : Homotopy e g where
hom := h.hom + k.hom
zero i j w := by rw [Pi.add_apply, Pi.add_apply, h.zero i j w, k.zero i j w, zero_add]
comm i := by
rw [AddMonoidHom.map_add, AddMonoidHom.map_add, h.comm, k.comm]
abel
/-- the sum of two homotopies is a homotopy between the sum of the respective morphisms. -/
@[simps!]
def add {f₁ g₁ f₂ g₂ : C ⟶ D} (h₁ : Homotopy f₁ g₁) (h₂ : Homotopy f₂ g₂) :
Homotopy (f₁ + f₂) (g₁ + g₂) where
hom := h₁.hom + h₂.hom
zero i j hij := by rw [Pi.add_apply, Pi.add_apply, h₁.zero i j hij, h₂.zero i j hij, add_zero]
comm i := by
simp only [HomologicalComplex.add_f_apply, h₁.comm, h₂.comm, AddMonoidHom.map_add]
abel
/-- the scalar multiplication of an homotopy -/
@[simps!]
def smul {R : Type*} [Semiring R] [Linear R V] (h : Homotopy f g) (a : R) :
Homotopy (a • f) (a • g) where
hom i j := a • h.hom i j
zero i j hij := by
rw [h.zero i j hij, smul_zero]
comm i := by
dsimp
rw [h.comm]
dsimp [fromNext, toPrev]
simp only [smul_add, Linear.comp_smul, Linear.smul_comp]
/-- homotopy is closed under composition (on the right) -/
@[simps]
def compRight {e f : C ⟶ D} (h : Homotopy e f) (g : D ⟶ E) : Homotopy (e ≫ g) (f ≫ g) where
hom i j := h.hom i j ≫ g.f j
zero i j w := by rw [h.zero i j w, zero_comp]
comm i := by rw [comp_f, h.comm i, dNext_comp_right, prevD_comp_right, Preadditive.add_comp,
comp_f, Preadditive.add_comp]
/-- homotopy is closed under composition (on the left) -/
@[simps]
def compLeft {f g : D ⟶ E} (h : Homotopy f g) (e : C ⟶ D) : Homotopy (e ≫ f) (e ≫ g) where
hom i j := e.f i ≫ h.hom i j
zero i j w := by rw [h.zero i j w, comp_zero]
comm i := by rw [comp_f, h.comm i, dNext_comp_left, prevD_comp_left, comp_f,
Preadditive.comp_add, Preadditive.comp_add]
/-- homotopy is closed under composition -/
@[simps!]
def comp {C₁ C₂ C₃ : HomologicalComplex V c} {f₁ g₁ : C₁ ⟶ C₂} {f₂ g₂ : C₂ ⟶ C₃}
(h₁ : Homotopy f₁ g₁) (h₂ : Homotopy f₂ g₂) : Homotopy (f₁ ≫ f₂) (g₁ ≫ g₂) :=
(h₁.compRight _).trans (h₂.compLeft _)
/-- a variant of `Homotopy.compRight` useful for dealing with homotopy equivalences. -/
@[simps!]
def compRightId {f : C ⟶ C} (h : Homotopy f (𝟙 C)) (g : C ⟶ D) : Homotopy (f ≫ g) g :=
(h.compRight g).trans (ofEq <| id_comp _)
/-- a variant of `Homotopy.compLeft` useful for dealing with homotopy equivalences. -/
@[simps!]
def compLeftId {f : D ⟶ D} (h : Homotopy f (𝟙 D)) (g : C ⟶ D) : Homotopy (g ≫ f) g :=
(h.compLeft g).trans (ofEq <| comp_id _)
/-!
Null homotopic maps can be constructed using the formula `hd+dh`. We show that
these morphisms are homotopic to `0` and provide some convenient simplification
lemmas that give a degreewise description of `hd+dh`, depending on whether we have
two differentials going to and from a certain degree, only one, or none.
-/
/-- The null homotopic map associated to a family `hom` of morphisms `C_i ⟶ D_j`.
This is the same datum as for the field `hom` in the structure `Homotopy`. For
this definition, we do not need the field `zero` of that structure
as this definition uses only the maps `C_i ⟶ C_j` when `c.Rel j i`. -/
def nullHomotopicMap (hom : ∀ i j, C.X i ⟶ D.X j) : C ⟶ D where
f i := dNext i hom + prevD i hom
comm' i j hij := by
have eq1 : prevD i hom ≫ D.d i j = 0 := by
simp only [prevD, AddMonoidHom.mk'_apply, assoc, d_comp_d, comp_zero]
have eq2 : C.d i j ≫ dNext j hom = 0 := by
simp only [dNext, AddMonoidHom.mk'_apply, d_comp_d_assoc, zero_comp]
rw [dNext_eq hom hij, prevD_eq hom hij, Preadditive.comp_add, Preadditive.add_comp, eq1, eq2,
add_zero, zero_add, assoc]
open Classical in
/-- Variant of `nullHomotopicMap` where the input consists only of the
relevant maps `C_i ⟶ D_j` such that `c.Rel j i`. -/
def nullHomotopicMap' (h : ∀ i j, c.Rel j i → (C.X i ⟶ D.X j)) : C ⟶ D :=
nullHomotopicMap fun i j => dite (c.Rel j i) (h i j) fun _ => 0
/-- Compatibility of `nullHomotopicMap` with the postcomposition by a morphism
of complexes. -/
theorem nullHomotopicMap_comp (hom : ∀ i j, C.X i ⟶ D.X j) (g : D ⟶ E) :
nullHomotopicMap hom ≫ g = nullHomotopicMap fun i j => hom i j ≫ g.f j := by
ext n
dsimp [nullHomotopicMap, fromNext, toPrev, AddMonoidHom.mk'_apply]
simp only [Preadditive.add_comp, assoc, g.comm]
/-- Compatibility of `nullHomotopicMap'` with the postcomposition by a morphism
of complexes. -/
theorem nullHomotopicMap'_comp (hom : ∀ i j, c.Rel j i → (C.X i ⟶ D.X j)) (g : D ⟶ E) :
nullHomotopicMap' hom ≫ g = nullHomotopicMap' fun i j hij => hom i j hij ≫ g.f j := by
ext n
rw [nullHomotopicMap', nullHomotopicMap_comp]
congr
ext i j
split_ifs
· rfl
· rw [zero_comp]
/-- Compatibility of `nullHomotopicMap` with the precomposition by a morphism
of complexes. -/
theorem comp_nullHomotopicMap (f : C ⟶ D) (hom : ∀ i j, D.X i ⟶ E.X j) :
f ≫ nullHomotopicMap hom = nullHomotopicMap fun i j => f.f i ≫ hom i j := by
ext n
dsimp [nullHomotopicMap, fromNext, toPrev, AddMonoidHom.mk'_apply]
simp only [Preadditive.comp_add, assoc, f.comm_assoc]
/-- Compatibility of `nullHomotopicMap'` with the precomposition by a morphism
of complexes. -/
theorem comp_nullHomotopicMap' (f : C ⟶ D) (hom : ∀ i j, c.Rel j i → (D.X i ⟶ E.X j)) :
f ≫ nullHomotopicMap' hom = nullHomotopicMap' fun i j hij => f.f i ≫ hom i j hij := by
ext n
rw [nullHomotopicMap', comp_nullHomotopicMap]
congr
ext i j
split_ifs
· rfl
· rw [comp_zero]
/-- Compatibility of `nullHomotopicMap` with the application of additive functors -/
theorem map_nullHomotopicMap {W : Type*} [Category W] [Preadditive W] (G : V ⥤ W) [G.Additive]
(hom : ∀ i j, C.X i ⟶ D.X j) :
(G.mapHomologicalComplex c).map (nullHomotopicMap hom) =
nullHomotopicMap (fun i j => by exact G.map (hom i j)) := by
ext i
dsimp [nullHomotopicMap, dNext, prevD]
simp only [G.map_comp, Functor.map_add]
/-- Compatibility of `nullHomotopicMap'` with the application of additive functors -/
theorem map_nullHomotopicMap' {W : Type*} [Category W] [Preadditive W] (G : V ⥤ W) [G.Additive]
(hom : ∀ i j, c.Rel j i → (C.X i ⟶ D.X j)) :
(G.mapHomologicalComplex c).map (nullHomotopicMap' hom) =
nullHomotopicMap' fun i j hij => by exact G.map (hom i j hij) := by
ext n
rw [nullHomotopicMap', map_nullHomotopicMap]
congr
ext i j
split_ifs
· rfl
· rw [G.map_zero]
/-- Tautological construction of the `Homotopy` to zero for maps constructed by
`nullHomotopicMap`, at least when we have the `zero` condition. -/
@[simps]
def nullHomotopy (hom : ∀ i j, C.X i ⟶ D.X j) (zero : ∀ i j, ¬c.Rel j i → hom i j = 0) :
Homotopy (nullHomotopicMap hom) 0 :=
{ hom := hom
zero := zero
comm := by
intro i
rw [HomologicalComplex.zero_f_apply, add_zero]
rfl }
open Classical in
/-- Homotopy to zero for maps constructed with `nullHomotopicMap'` -/
@[simps!]
def nullHomotopy' (h : ∀ i j, c.Rel j i → (C.X i ⟶ D.X j)) : Homotopy (nullHomotopicMap' h) 0 := by
apply nullHomotopy fun i j => dite (c.Rel j i) (h i j) fun _ => 0
grind
/-! This lemma and the following ones can be used in order to compute
the degreewise morphisms induced by the null homotopic maps constructed
with `nullHomotopicMap` or `nullHomotopicMap'` -/
-- Cannot be @[simp] because `k₀` and `k₂` can not be inferred by `simp`.
theorem nullHomotopicMap_f {k₂ k₁ k₀ : ι} (r₂₁ : c.Rel k₂ k₁) (r₁₀ : c.Rel k₁ k₀)
(hom : ∀ i j, C.X i ⟶ D.X j) :
(nullHomotopicMap hom).f k₁ = C.d k₁ k₀ ≫ hom k₀ k₁ + hom k₁ k₂ ≫ D.d k₂ k₁ := by
dsimp only [nullHomotopicMap]
rw [dNext_eq hom r₁₀, prevD_eq hom r₂₁]
-- Cannot be @[simp] because `k₀` and `k₂` can not be inferred by `simp`.
theorem nullHomotopicMap'_f {k₂ k₁ k₀ : ι} (r₂₁ : c.Rel k₂ k₁) (r₁₀ : c.Rel k₁ k₀)
(h : ∀ i j, c.Rel j i → (C.X i ⟶ D.X j)) :
(nullHomotopicMap' h).f k₁ = C.d k₁ k₀ ≫ h k₀ k₁ r₁₀ + h k₁ k₂ r₂₁ ≫ D.d k₂ k₁ := by
simp only [nullHomotopicMap']
rw [nullHomotopicMap_f r₂₁ r₁₀]
split_ifs
rfl
-- Cannot be @[simp] because `k₁` can not be inferred by `simp`.
theorem nullHomotopicMap_f_of_not_rel_left {k₁ k₀ : ι} (r₁₀ : c.Rel k₁ k₀)
(hk₀ : ∀ l : ι, ¬c.Rel k₀ l) (hom : ∀ i j, C.X i ⟶ D.X j) :
(nullHomotopicMap hom).f k₀ = hom k₀ k₁ ≫ D.d k₁ k₀ := by
dsimp only [nullHomotopicMap]
rw [prevD_eq hom r₁₀, dNext, AddMonoidHom.mk'_apply, C.shape, zero_comp, zero_add]
exact hk₀ _
-- Cannot be @[simp] because `k₁` can not be inferred by `simp`.
theorem nullHomotopicMap'_f_of_not_rel_left {k₁ k₀ : ι} (r₁₀ : c.Rel k₁ k₀)
(hk₀ : ∀ l : ι, ¬c.Rel k₀ l) (h : ∀ i j, c.Rel j i → (C.X i ⟶ D.X j)) :
(nullHomotopicMap' h).f k₀ = h k₀ k₁ r₁₀ ≫ D.d k₁ k₀ := by
simp only [nullHomotopicMap']
rw [nullHomotopicMap_f_of_not_rel_left r₁₀ hk₀]
split_ifs
rfl
-- Cannot be @[simp] because `k₀` can not be inferred by `simp`.
theorem nullHomotopicMap_f_of_not_rel_right {k₁ k₀ : ι} (r₁₀ : c.Rel k₁ k₀)
(hk₁ : ∀ l : ι, ¬c.Rel l k₁) (hom : ∀ i j, C.X i ⟶ D.X j) :
(nullHomotopicMap hom).f k₁ = C.d k₁ k₀ ≫ hom k₀ k₁ := by
dsimp only [nullHomotopicMap]
rw [dNext_eq hom r₁₀, prevD, AddMonoidHom.mk'_apply, D.shape, comp_zero, add_zero]
exact hk₁ _
-- Cannot be @[simp] because `k₀` can not be inferred by `simp`.
theorem nullHomotopicMap'_f_of_not_rel_right {k₁ k₀ : ι} (r₁₀ : c.Rel k₁ k₀)
(hk₁ : ∀ l : ι, ¬c.Rel l k₁) (h : ∀ i j, c.Rel j i → (C.X i ⟶ D.X j)) :
(nullHomotopicMap' h).f k₁ = C.d k₁ k₀ ≫ h k₀ k₁ r₁₀ := by
simp only [nullHomotopicMap']
rw [nullHomotopicMap_f_of_not_rel_right r₁₀ hk₁]
split_ifs
rfl
@[simp]
theorem nullHomotopicMap_f_eq_zero {k₀ : ι} (hk₀ : ∀ l : ι, ¬c.Rel k₀ l)
(hk₀' : ∀ l : ι, ¬c.Rel l k₀) (hom : ∀ i j, C.X i ⟶ D.X j) :
(nullHomotopicMap hom).f k₀ = 0 := by
dsimp [nullHomotopicMap, dNext, prevD]
rw [C.shape, D.shape, zero_comp, comp_zero, add_zero] <;> apply_assumption
@[simp]
theorem nullHomotopicMap'_f_eq_zero {k₀ : ι} (hk₀ : ∀ l : ι, ¬c.Rel k₀ l)
(hk₀' : ∀ l : ι, ¬c.Rel l k₀) (h : ∀ i j, c.Rel j i → (C.X i ⟶ D.X j)) :
(nullHomotopicMap' h).f k₀ = 0 := by
simp only [nullHomotopicMap']
apply nullHomotopicMap_f_eq_zero hk₀ hk₀'
/-!
`Homotopy.mkInductive` allows us to build a homotopy of chain complexes inductively,
so that as we construct each component, we have available the previous two components,
and the fact that they satisfy the homotopy condition.
To simplify the situation, we only construct homotopies of the form `Homotopy e 0`.
`Homotopy.equivSubZero` can provide the general case.
Notice however, that this construction does not have particularly good definitional properties:
we have to insert `eqToHom` in several places.
Hopefully this is okay in most applications, where we only need to have the existence of some
homotopy.
-/
section MkInductive
variable {P Q : ChainComplex V ℕ}
-- This is not a simp lemma; the LHS already simplifies.
theorem prevD_chainComplex (f : ∀ i j, P.X i ⟶ Q.X j) (j : ℕ) :
prevD j f = f j (j + 1) ≫ Q.d _ _ := by
dsimp [prevD]
have : (ComplexShape.down ℕ).prev j = j + 1 := ChainComplex.prev ℕ j
congr 2
-- This is not a simp lemma; the LHS already simplifies.
theorem dNext_succ_chainComplex (f : ∀ i j, P.X i ⟶ Q.X j) (i : ℕ) :
dNext (i + 1) f = P.d _ _ ≫ f i (i + 1) := by
dsimp [dNext]
have : (ComplexShape.down ℕ).next (i + 1) = i := ChainComplex.next_nat_succ _
congr 2
-- This is not a simp lemma; the LHS already simplifies.
theorem dNext_zero_chainComplex (f : ∀ i j, P.X i ⟶ Q.X j) : dNext 0 f = 0 := by
dsimp [dNext]
rw [P.shape, zero_comp]
rw [ChainComplex.next_nat_zero]; dsimp; decide
variable (e : P ⟶ Q) (zero : P.X 0 ⟶ Q.X 1) (comm_zero : e.f 0 = zero ≫ Q.d 1 0)
(one : P.X 1 ⟶ Q.X 2) (comm_one : e.f 1 = P.d 1 0 ≫ zero + one ≫ Q.d 2 1)
(succ :
∀ (n : ℕ)
(p :
Σ' (f : P.X n ⟶ Q.X (n + 1)) (f' : P.X (n + 1) ⟶ Q.X (n + 2)),
e.f (n + 1) = P.d (n + 1) n ≫ f + f' ≫ Q.d (n + 2) (n + 1)),
Σ' f'' : P.X (n + 2) ⟶ Q.X (n + 3),
e.f (n + 2) = P.d (n + 2) (n + 1) ≫ p.2.1 + f'' ≫ Q.d (n + 3) (n + 2))
/-- An auxiliary construction for `mkInductive`.
Here we build by induction a family of diagrams,
but don't require at the type level that these successive diagrams actually agree.
They do in fact agree, and we then capture that at the type level (i.e. by constructing a homotopy)
in `mkInductive`.
At this stage, we don't check the homotopy condition in degree 0,
because it "falls off the end", and is easier to treat using `xNext` and `xPrev`,
which we do in `mkInductiveAux₂`.
-/
@[simp, nolint unusedArguments]
def mkInductiveAux₁ :
∀ n,
Σ' (f : P.X n ⟶ Q.X (n + 1)) (f' : P.X (n + 1) ⟶ Q.X (n + 2)),
e.f (n + 1) = P.d (n + 1) n ≫ f + f' ≫ Q.d (n + 2) (n + 1)
| 0 => ⟨zero, one, comm_one⟩
| 1 => ⟨one, (succ 0 ⟨zero, one, comm_one⟩).1, (succ 0 ⟨zero, one, comm_one⟩).2⟩
| n + 2 =>
⟨(mkInductiveAux₁ (n + 1)).2.1, (succ (n + 1) (mkInductiveAux₁ (n + 1))).1,
(succ (n + 1) (mkInductiveAux₁ (n + 1))).2⟩
section
/-- An auxiliary construction for `mkInductive`.
-/
def mkInductiveAux₂ :
∀ n, Σ' (f : P.xNext n ⟶ Q.X n) (f' : P.X n ⟶ Q.xPrev n), e.f n = P.dFrom n ≫ f + f' ≫ Q.dTo n
| 0 => ⟨0, zero ≫ (Q.xPrevIso rfl).inv, by simpa using comm_zero⟩
| n + 1 =>
let I := mkInductiveAux₁ e zero --comm_zero
one comm_one succ n
⟨(P.xNextIso rfl).hom ≫ I.1, I.2.1 ≫ (Q.xPrevIso rfl).inv, by simpa using I.2.2⟩
-- Porting note (https://github.com/leanprover-community/mathlib4/issues/11647): during the port we marked these lemmas
-- with `@[eqns]` to emulate the old Lean 3 behaviour.
@[simp] theorem mkInductiveAux₂_zero :
mkInductiveAux₂ e zero comm_zero one comm_one succ 0 =
⟨0, zero ≫ (Q.xPrevIso rfl).inv, by simpa using comm_zero⟩ :=
rfl
@[simp] theorem mkInductiveAux₂_add_one (n) :
mkInductiveAux₂ e zero comm_zero one comm_one succ (n + 1) =
letI I := mkInductiveAux₁ e zero one comm_one succ n
⟨(P.xNextIso rfl).hom ≫ I.1, I.2.1 ≫ (Q.xPrevIso rfl).inv, by simpa using I.2.2⟩ :=
rfl
theorem mkInductiveAux₃ (i j : ℕ) (h : i + 1 = j) :
(mkInductiveAux₂ e zero comm_zero one comm_one succ i).2.1 ≫ (Q.xPrevIso h).hom =
(P.xNextIso h).inv ≫ (mkInductiveAux₂ e zero comm_zero one comm_one succ j).1 := by
subst j
rcases i with (_ | _ | i) <;> simp [mkInductiveAux₂]
/-- A constructor for a `Homotopy e 0`, for `e` a chain map between `ℕ`-indexed chain complexes,
working by induction.
You need to provide the components of the homotopy in degrees 0 and 1,
show that these satisfy the homotopy condition,
and then give a construction of each component,
and the fact that it satisfies the homotopy condition,
using as an inductive hypothesis the data and homotopy condition for the previous two components.
-/
def mkInductive : Homotopy e 0 where
hom i j :=
if h : i + 1 = j then
(mkInductiveAux₂ e zero comm_zero one comm_one succ i).2.1 ≫ (Q.xPrevIso h).hom
else 0
zero i j w := by rw [dif_neg]; exact w
comm i := by
dsimp
simp only [add_zero]
refine (mkInductiveAux₂ e zero comm_zero one comm_one succ i).2.2.trans ?_
congr
· cases i
· dsimp [fromNext, mkInductiveAux₂]
· dsimp [fromNext]
simp only [ChainComplex.next_nat_succ, dite_true]
rw [mkInductiveAux₃ e zero comm_zero one comm_one succ]
dsimp [xNextIso]
rw [id_comp]
· dsimp [toPrev]
rw [dif_pos (by simp only [ChainComplex.prev])]
simp [xPrevIso, comp_id]
end
end MkInductive
/-!
`Homotopy.mkCoinductive` allows us to build a homotopy of cochain complexes inductively,
so that as we construct each component, we have available the previous two components,
and the fact that they satisfy the homotopy condition.
-/
section MkCoinductive
variable {P Q : CochainComplex V ℕ}
-- This is not a simp lemma; the LHS already simplifies.
theorem dNext_cochainComplex (f : ∀ i j, P.X i ⟶ Q.X j) (j : ℕ) :
dNext j f = P.d _ _ ≫ f (j + 1) j := by
dsimp [dNext]
have : (ComplexShape.up ℕ).next j = j + 1 := CochainComplex.next ℕ j
congr 2
-- This is not a simp lemma; the LHS already simplifies.
theorem prevD_succ_cochainComplex (f : ∀ i j, P.X i ⟶ Q.X j) (i : ℕ) :
prevD (i + 1) f = f (i + 1) _ ≫ Q.d i (i + 1) := by
dsimp [prevD]
have : (ComplexShape.up ℕ).prev (i + 1) = i := CochainComplex.prev_nat_succ i
congr 2
-- This is not a simp lemma; the LHS already simplifies.
theorem prevD_zero_cochainComplex (f : ∀ i j, P.X i ⟶ Q.X j) : prevD 0 f = 0 := by
dsimp [prevD]
rw [Q.shape, comp_zero]
rw [CochainComplex.prev_nat_zero]; dsimp; decide
variable (e : P ⟶ Q) (zero : P.X 1 ⟶ Q.X 0) (comm_zero : e.f 0 = P.d 0 1 ≫ zero)
(one : P.X 2 ⟶ Q.X 1) (comm_one : e.f 1 = zero ≫ Q.d 0 1 + P.d 1 2 ≫ one)
(succ :
∀ (n : ℕ)
(p :
Σ' (f : P.X (n + 1) ⟶ Q.X n) (f' : P.X (n + 2) ⟶ Q.X (n + 1)),
e.f (n + 1) = f ≫ Q.d n (n + 1) + P.d (n + 1) (n + 2) ≫ f'),
Σ' f'' : P.X (n + 3) ⟶ Q.X (n + 2),
e.f (n + 2) = p.2.1 ≫ Q.d (n + 1) (n + 2) + P.d (n + 2) (n + 3) ≫ f'')
/-- An auxiliary construction for `mkCoinductive`.
Here we build by induction a family of diagrams,
but don't require at the type level that these successive diagrams actually agree.
They do in fact agree, and we then capture that at the type level (i.e. by constructing a homotopy)
in `mkCoinductive`.
At this stage, we don't check the homotopy condition in degree 0,
because it "falls off the end", and is easier to treat using `xNext` and `xPrev`,
which we do in `mkInductiveAux₂`.
-/
@[simp]
def mkCoinductiveAux₁ :
∀ n,
Σ' (f : P.X (n + 1) ⟶ Q.X n) (f' : P.X (n + 2) ⟶ Q.X (n + 1)),
e.f (n + 1) = f ≫ Q.d n (n + 1) + P.d (n + 1) (n + 2) ≫ f'
| 0 => ⟨zero, one, comm_one⟩
| 1 => ⟨one, (succ 0 ⟨zero, one, comm_one⟩).1, (succ 0 ⟨zero, one, comm_one⟩).2⟩
| n + 2 =>
⟨(mkCoinductiveAux₁ (n + 1)).2.1, (succ (n + 1) (mkCoinductiveAux₁ (n + 1))).1,
(succ (n + 1) (mkCoinductiveAux₁ (n + 1))).2⟩
section
/-- An auxiliary construction for `mkInductive`.
-/
def mkCoinductiveAux₂ :
∀ n, Σ' (f : P.X n ⟶ Q.xPrev n) (f' : P.xNext n ⟶ Q.X n), e.f n = f ≫ Q.dTo n + P.dFrom n ≫ f'
| 0 => ⟨0, (P.xNextIso rfl).hom ≫ zero, by simpa using comm_zero⟩
| n + 1 =>
let I := mkCoinductiveAux₁ e zero one comm_one succ n
⟨I.1 ≫ (Q.xPrevIso rfl).inv, (P.xNextIso rfl).hom ≫ I.2.1, by simpa using I.2.2⟩
-- Porting note (https://github.com/leanprover-community/mathlib4/issues/11647): during the port we marked these lemmas with `@[eqns]`
-- to emulate the old Lean 3 behaviour.
@[simp] theorem mkCoinductiveAux₂_zero :
mkCoinductiveAux₂ e zero comm_zero one comm_one succ 0 =
⟨0, (P.xNextIso rfl).hom ≫ zero, by simpa using comm_zero⟩ :=
rfl
@[simp] theorem mkCoinductiveAux₂_add_one (n) :
mkCoinductiveAux₂ e zero comm_zero one comm_one succ (n + 1) =
letI I := mkCoinductiveAux₁ e zero one comm_one succ n
⟨I.1 ≫ (Q.xPrevIso rfl).inv, (P.xNextIso rfl).hom ≫ I.2.1, by simpa using I.2.2⟩ :=
rfl
theorem mkCoinductiveAux₃ (i j : ℕ) (h : i + 1 = j) :
(P.xNextIso h).inv ≫ (mkCoinductiveAux₂ e zero comm_zero one comm_one succ i).2.1 =
(mkCoinductiveAux₂ e zero comm_zero one comm_one succ j).1 ≫ (Q.xPrevIso h).hom := by
subst j
rcases i with (_ | _ | i) <;> simp [mkCoinductiveAux₂]
/-- A constructor for a `Homotopy e 0`, for `e` a chain map between `ℕ`-indexed cochain complexes,
working by induction.
You need to provide the components of the homotopy in degrees 0 and 1,
show that these satisfy the homotopy condition,
and then give a construction of each component,
and the fact that it satisfies the homotopy condition,
using as an inductive hypothesis the data and homotopy condition for the previous two components.
-/
def mkCoinductive : Homotopy e 0 where
hom i j :=
if h : j + 1 = i then
(P.xNextIso h).inv ≫ (mkCoinductiveAux₂ e zero comm_zero one comm_one succ j).2.1
else 0
zero i j w := by rw [dif_neg]; exact w
comm i := by
dsimp
simp only [add_zero]
rw [add_comm]
refine (mkCoinductiveAux₂ e zero comm_zero one comm_one succ i).2.2.trans ?_
congr
· cases i
· dsimp [toPrev, mkCoinductiveAux₂]
· dsimp [toPrev]
simp only [CochainComplex.prev_nat_succ, dite_true]
rw [mkCoinductiveAux₃ e zero comm_zero one comm_one succ]
dsimp [xPrevIso]
rw [comp_id]
· dsimp [fromNext]
rw [dif_pos (by simp only [CochainComplex.next])]
simp [xNextIso, id_comp]
end
end MkCoinductive
end Homotopy
/-- A homotopy equivalence between two chain complexes consists of a chain map each way,
and homotopies from the compositions to the identity chain maps.
Note that this contains data;
arguably it might be more useful for many applications if we truncated it to a Prop.
-/
structure HomotopyEquiv (C D : HomologicalComplex V c) where
/-- The forward chain map -/
hom : C ⟶ D
/-- The backward chain map -/
inv : D ⟶ C
/-- A homotopy showing that composing the forward and backward maps is homotopic to the identity
on C -/
homotopyHomInvId : Homotopy (hom ≫ inv) (𝟙 C)
/-- A homotopy showing that composing the backward and forward maps is homotopic to the identity
on D -/
homotopyInvHomId : Homotopy (inv ≫ hom) (𝟙 D)
variable (V c) in
/-- The morphism property on `HomologicalComplex V c` given by homotopy equivalences. -/
def HomologicalComplex.homotopyEquivalences :
MorphismProperty (HomologicalComplex V c) :=
fun X Y f => ∃ (e : HomotopyEquiv X Y), e.hom = f
namespace HomotopyEquiv
/-- Any complex is homotopy equivalent to itself. -/
@[refl]
def refl (C : HomologicalComplex V c) : HomotopyEquiv C C where
hom := 𝟙 C
inv := 𝟙 C
homotopyHomInvId := Homotopy.ofEq (by simp)
homotopyInvHomId := Homotopy.ofEq (by simp)
instance : Inhabited (HomotopyEquiv C C) :=
⟨refl C⟩
/-- Being homotopy equivalent is a symmetric relation. -/
@[symm]
def symm {C D : HomologicalComplex V c} (f : HomotopyEquiv C D) : HomotopyEquiv D C where
hom := f.inv
inv := f.hom
homotopyHomInvId := f.homotopyInvHomId
homotopyInvHomId := f.homotopyHomInvId
/-- Homotopy equivalence is a transitive relation. -/
@[trans]
def trans {C D E : HomologicalComplex V c} (f : HomotopyEquiv C D) (g : HomotopyEquiv D E) :
HomotopyEquiv C E where
hom := f.hom ≫ g.hom
inv := g.inv ≫ f.inv
homotopyHomInvId := by simpa using
((g.homotopyHomInvId.compRightId f.inv).compLeft f.hom).trans f.homotopyHomInvId
homotopyInvHomId := by simpa using
((f.homotopyInvHomId.compRightId g.hom).compLeft g.inv).trans g.homotopyInvHomId
/-- An isomorphism of complexes induces a homotopy equivalence. -/
def ofIso {ι : Type*} {V : Type u} [Category.{v} V] [Preadditive V] {c : ComplexShape ι}
{C D : HomologicalComplex V c} (f : C ≅ D) : HomotopyEquiv C D :=
⟨f.hom, f.inv, Homotopy.ofEq f.3, Homotopy.ofEq f.4⟩
end HomotopyEquiv
end
namespace CategoryTheory
variable {W : Type*} [Category W] [Preadditive W]
/-- An additive functor takes homotopies to homotopies. -/
@[simps]
def Functor.mapHomotopy (F : V ⥤ W) [F.Additive] {f g : C ⟶ D} (h : Homotopy f g) :
Homotopy ((F.mapHomologicalComplex c).map f) ((F.mapHomologicalComplex c).map g) where
hom i j := F.map (h.hom i j)
zero i j w := by dsimp; rw [h.zero i j w, F.map_zero]
comm i := by
have H := h.comm i
dsimp [dNext, prevD] at H ⊢
simp [H]
/-- An additive functor preserves homotopy equivalences. -/
@[simps]
def Functor.mapHomotopyEquiv (F : V ⥤ W) [F.Additive] (h : HomotopyEquiv C D) :
HomotopyEquiv ((F.mapHomologicalComplex c).obj C) ((F.mapHomologicalComplex c).obj D) where
hom := (F.mapHomologicalComplex c).map h.hom
inv := (F.mapHomologicalComplex c).map h.inv
homotopyHomInvId := by
rw [← (F.mapHomologicalComplex c).map_comp, ← (F.mapHomologicalComplex c).map_id]
exact F.mapHomotopy h.homotopyHomInvId
homotopyInvHomId := by
rw [← (F.mapHomologicalComplex c).map_comp, ← (F.mapHomologicalComplex c).map_id]
exact F.mapHomotopy h.homotopyInvHomId
end CategoryTheory
section
open HomologicalComplex CategoryTheory
variable {C : Type*} [Category C] [Preadditive C] {ι : Type _} {c : ComplexShape ι}
[DecidableRel c.Rel] {K L : HomologicalComplex C c} {f g : K ⟶ L}
/-- A homotopy between morphisms of homological complexes `K ⟶ L` induces a homotopy
between morphisms of short complexes `K.sc i ⟶ L.sc i`. -/
noncomputable def Homotopy.toShortComplex (ho : Homotopy f g) (i : ι) :
ShortComplex.Homotopy ((shortComplexFunctor C c i).map f)
((shortComplexFunctor C c i).map g) where
h₀ :=
if c.Rel (c.prev i) i
then ho.hom _ (c.prev (c.prev i)) ≫ L.d _ _
else f.f _ - g.f _ - K.d _ i ≫ ho.hom i _
h₁ := ho.hom _ _
h₂ := ho.hom _ _
h₃ :=
if c.Rel i (c.next i)
then K.d _ _ ≫ ho.hom (c.next (c.next i)) _
else f.f _ - g.f _ - ho.hom _ i ≫ L.d _ _
h₀_f := by
split_ifs with h
· dsimp
simp only [assoc, d_comp_d, comp_zero]
· dsimp
rw [L.shape _ _ h, comp_zero]
g_h₃ := by
split_ifs with h
· simp
· dsimp
rw [K.shape _ _ h, zero_comp]
comm₁ := by
dsimp
split_ifs with h
· rw [ho.comm (c.prev i)]
dsimp [dFrom, dTo, fromNext, toPrev]
rw [congr_arg (fun j => d K (c.prev i) j ≫ ho.hom j (c.prev i)) (c.next_eq' h)]
· abel
comm₂ := ho.comm i
comm₃ := by
dsimp
split_ifs with h
· rw [ho.comm (c.next i)]
dsimp [dFrom, dTo, fromNext, toPrev]
rw [congr_arg (fun j => ho.hom (c.next i) j ≫ L.d j (c.next i)) (c.prev_eq' h)]
· abel
lemma Homotopy.homologyMap_eq (ho : Homotopy f g) (i : ι) [K.HasHomology i] [L.HasHomology i] :
homologyMap f i = homologyMap g i :=
ShortComplex.Homotopy.homologyMap_congr (ho.toShortComplex i)
/-- The isomorphism in homology induced by an homotopy equivalence. -/
noncomputable def HomotopyEquiv.toHomologyIso (h : HomotopyEquiv K L) (i : ι)
[K.HasHomology i] [L.HasHomology i] : K.homology i ≅ L.homology i where
hom := homologyMap h.hom i
inv := homologyMap h.inv i
hom_inv_id := by rw [← homologyMap_comp, h.homotopyHomInvId.homologyMap_eq, homologyMap_id]
inv_hom_id := by rw [← homologyMap_comp, h.homotopyInvHomId.homologyMap_eq, homologyMap_id]
end
|
div.v
|
(* (c) Copyright 2006-2016 Microsoft Corporation and Inria. *)
(* Distributed under the terms of CeCILL-B. *)
From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq.
(******************************************************************************)
(* This file deals with divisibility for natural numbers. *)
(* It contains the definitions of: *)
(* edivn m d == the pair composed of the quotient and remainder *)
(* of the Euclidean division of m by d. *)
(* m %/ d == quotient of the Euclidean division of m by d. *)
(* m %% d == remainder of the Euclidean division of m by d. *)
(* m = n %[mod d] <-> m equals n modulo d. *)
(* m == n %[mod d] <=> m equals n modulo d (boolean version). *)
(* m <> n %[mod d] <-> m differs from n modulo d. *)
(* m != n %[mod d] <=> m differs from n modulo d (boolean version). *)
(* d %| m <=> d divides m. *)
(* gcdn m n == the GCD of m and n. *)
(* egcdn m n == the extended GCD (Bezout coefficient pair) of m and n. *)
(* If egcdn m n = (u, v), then gcdn m n = m * u - n * v. *)
(* lcmn m n == the LCM of m and n. *)
(* coprime m n <=> m and n are coprime (:= gcdn m n == 1). *)
(* chinese m n r s == witness of the chinese remainder theorem. *)
(* We adjoin an m to operator suffixes to indicate a nested %% (modn), as in *)
(* modnDml : m %% d + n = m + n %[mod d]. *)
(******************************************************************************)
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
(** Euclidean division *)
Definition edivn_rec d :=
fix loop m q := if m - d is m'.+1 then loop m' q.+1 else (q, m).
Definition edivn m d := if d > 0 then edivn_rec d.-1 m 0 else (0, m).
Variant edivn_spec m d : nat * nat -> Type :=
EdivnSpec q r of m = q * d + r & (d > 0) ==> (r < d) : edivn_spec m d (q, r).
Lemma edivnP m d : edivn_spec m d (edivn m d).
Proof.
rewrite -[m in edivn_spec m]/(0 * d + m) /edivn; case: d => //= d.
elim/ltn_ind: m 0 => -[|m] IHm q //=; rewrite subn_if_gt.
case: ltnP => // le_dm; rewrite -[in m.+1](subnKC le_dm) -addSn.
by rewrite addnA -mulSnr; apply/IHm/leq_subr.
Qed.
Lemma edivn_eq d q r : r < d -> edivn (q * d + r) d = (q, r).
Proof.
move=> lt_rd; have d_gt0: 0 < d by apply: leq_trans lt_rd.
case: edivnP lt_rd => q' r'; rewrite d_gt0 /=.
wlog: q q' r r' / q <= q' by case/orP: (leq_total q q'); last symmetry; eauto.
have [||-> _ /addnI ->] //= := ltngtP q q'.
rewrite -(leq_pmul2r d_gt0) => /leq_add lt_qr _ eq_qr _ /lt_qr {lt_qr}.
by rewrite addnS ltnNge mulSn -addnA eq_qr addnCA addnA leq_addr.
Qed.
Definition divn m d := (edivn m d).1.
Notation "m %/ d" := (divn m d) : nat_scope.
(* We redefine modn so that it is structurally decreasing. *)
Definition modn_rec d := fix loop m := if m - d is m'.+1 then loop m' else m.
Definition modn m d := if d > 0 then modn_rec d.-1 m else m.
Notation "m %% d" := (modn m d) : nat_scope.
Notation "m = n %[mod d ]" := (m %% d = n %% d) : nat_scope.
Notation "m == n %[mod d ]" := (m %% d == n %% d) : nat_scope.
Notation "m <> n %[mod d ]" := (m %% d <> n %% d) : nat_scope.
Notation "m != n %[mod d ]" := (m %% d != n %% d) : nat_scope.
Lemma modn_def m d : m %% d = (edivn m d).2.
Proof.
case: d => //= d; rewrite /modn /edivn /=; elim/ltn_ind: m 0 => -[|m] IHm q //=.
by rewrite !subn_if_gt; case: (d <= m) => //; apply/IHm/leq_subr.
Qed.
Lemma edivn_def m d : edivn m d = (m %/ d, m %% d).
Proof. by rewrite /divn modn_def; case: (edivn m d). Qed.
Lemma divn_eq m d : m = m %/ d * d + m %% d.
Proof. by rewrite /divn modn_def; case: edivnP. Qed.
Lemma div0n d : 0 %/ d = 0. Proof. by case: d. Qed.
Lemma divn0 m : m %/ 0 = 0. Proof. by []. Qed.
Lemma mod0n d : 0 %% d = 0. Proof. by case: d. Qed.
Lemma modn0 m : m %% 0 = m. Proof. by []. Qed.
Lemma divn_small m d : m < d -> m %/ d = 0.
Proof. by move=> lt_md; rewrite /divn (edivn_eq 0). Qed.
Lemma divnMDl q m d : 0 < d -> (q * d + m) %/ d = q + m %/ d.
Proof.
move=> d_gt0; rewrite [in LHS](divn_eq m d) addnA -mulnDl.
by rewrite /divn edivn_eq // modn_def; case: edivnP; rewrite d_gt0.
Qed.
Lemma mulnK m d : 0 < d -> m * d %/ d = m.
Proof. by move=> d_gt0; rewrite -[m * d]addn0 divnMDl // div0n addn0. Qed.
Lemma mulKn m d : 0 < d -> d * m %/ d = m.
Proof. by move=> d_gt0; rewrite mulnC mulnK. Qed.
Lemma expnB p m n : p > 0 -> m >= n -> p ^ (m - n) = p ^ m %/ p ^ n.
Proof.
by move=> p_gt0 /subnK-Dm; rewrite -[in RHS]Dm expnD mulnK // expn_gt0 p_gt0.
Qed.
Lemma modn1 m : m %% 1 = 0.
Proof. by rewrite modn_def; case: edivnP => ? []. Qed.
Lemma divn1 m : m %/ 1 = m.
Proof. by rewrite [RHS](@divn_eq m 1) // modn1 addn0 muln1. Qed.
Lemma divnn d : d %/ d = (0 < d).
Proof. by case: d => // d; rewrite -[n in n %/ _]muln1 mulKn. Qed.
Lemma divnMl p m d : p > 0 -> p * m %/ (p * d) = m %/ d.
Proof.
move=> p_gt0; have [->|d_gt0] := posnP d; first by rewrite muln0.
rewrite [RHS]/divn; case: edivnP; rewrite d_gt0 /= => q r ->{m} lt_rd.
rewrite mulnDr mulnCA divnMDl; last by rewrite muln_gt0 p_gt0.
by rewrite addnC divn_small // ltn_pmul2l.
Qed.
Arguments divnMl [p m d].
Lemma divnMr p m d : p > 0 -> m * p %/ (d * p) = m %/ d.
Proof. by move=> p_gt0; rewrite -!(mulnC p) divnMl. Qed.
Arguments divnMr [p m d].
Lemma ltn_mod m d : (m %% d < d) = (0 < d).
Proof. by case: d => // d; rewrite modn_def; case: edivnP. Qed.
Lemma ltn_pmod m d : 0 < d -> m %% d < d.
Proof. by rewrite ltn_mod. Qed.
Lemma leq_divM m d : m %/ d * d <= m.
Proof. by rewrite [leqRHS](divn_eq m d) leq_addr. Qed.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to leq_divM.")]
Notation leq_trunc_div := leq_divM.
Lemma leq_mod m d : m %% d <= m.
Proof. by rewrite [leqRHS](divn_eq m d) leq_addl. Qed.
Lemma leq_div m d : m %/ d <= m.
Proof.
by case: d => // d; apply: leq_trans (leq_pmulr _ _) (leq_divM _ _).
Qed.
Lemma ltn_ceil m d : 0 < d -> m < (m %/ d).+1 * d.
Proof.
by move=> d_gt0; rewrite [in m.+1](divn_eq m d) -addnS mulSnr leq_add2l ltn_mod.
Qed.
Lemma ltn_divLR m n d : d > 0 -> (m %/ d < n) = (m < n * d).
Proof.
move=> d_gt0; apply/idP/idP.
by rewrite -(leq_pmul2r d_gt0); apply: leq_trans (ltn_ceil _ _).
rewrite !ltnNge -(@leq_pmul2r d n) //; apply: contra => le_nd_floor.
exact: leq_trans le_nd_floor (leq_divM _ _).
Qed.
Lemma leq_divRL m n d : d > 0 -> (m <= n %/ d) = (m * d <= n).
Proof. by move=> d_gt0; rewrite leqNgt ltn_divLR // -leqNgt. Qed.
Lemma ltn_Pdiv m d : 1 < d -> 0 < m -> m %/ d < m.
Proof. by move=> d_gt1 m_gt0; rewrite ltn_divLR ?ltn_Pmulr // ltnW. Qed.
Lemma divn_gt0 d m : 0 < d -> (0 < m %/ d) = (d <= m).
Proof. by move=> d_gt0; rewrite leq_divRL ?mul1n. Qed.
Lemma leq_div2r d m n : m <= n -> m %/ d <= n %/ d.
Proof.
have [-> //| d_gt0 le_mn] := posnP d.
by rewrite leq_divRL // (leq_trans _ le_mn) -?leq_divRL.
Qed.
Lemma leq_div2l m d e : 0 < d -> d <= e -> m %/ e <= m %/ d.
Proof.
move/leq_divRL=> -> le_de.
by apply: leq_trans (leq_divM m e); apply: leq_mul.
Qed.
Lemma edivnD m n d (offset := m %% d + n %% d >= d) : 0 < d ->
edivn (m + n) d = (m %/ d + n %/ d + offset, m %% d + n %% d - offset * d).
Proof.
rewrite {}/offset; case: d => // d _; rewrite /divn !modn_def.
case: (edivnP m d.+1) (edivnP n d.+1) => [/= q r -> r_lt] [/= p s -> s_lt].
rewrite addnACA -mulnDl; have [r_le s_le] := (ltnW r_lt, ltnW s_lt).
have [d_ge|d_lt] := leqP; first by rewrite addn0 mul0n subn0 edivn_eq.
rewrite addn1 mul1n -[in LHS](subnKC d_lt) addnA -mulSnr edivn_eq//.
by rewrite ltn_subLR// -addnS leq_add.
Qed.
Lemma divnD m n d : 0 < d ->
(m + n) %/ d = (m %/ d) + (n %/ d) + (m %% d + n %% d >= d).
Proof. by move=> /(@edivnD m n); rewrite edivn_def => -[]. Qed.
Lemma modnD m n d : 0 < d ->
(m + n) %% d = m %% d + n %% d - (m %% d + n %% d >= d) * d.
Proof. by move=> /(@edivnD m n); rewrite edivn_def => -[]. Qed.
Lemma leqDmod m n d : 0 < d ->
(d <= m %% d + n %% d) = ((m + n) %% d < n %% d).
Proof.
move=> d_gt0; rewrite modnD//.
have [d_le|_] := leqP d; last by rewrite subn0 ltnNge leq_addl.
by rewrite -(ltn_add2r d) mul1n (subnK d_le) addnC ltn_add2l ltn_pmod.
Qed.
Lemma divnB n m d : 0 < d ->
(m - n) %/ d = (m %/ d) - (n %/ d) - (m %% d < n %% d).
Proof.
move=> d_gt0; have [mn|/ltnW nm] := leqP m n.
by rewrite (eqP mn) (eqP (leq_div2r _ _)) ?div0n.
by rewrite -[in m %/ d](subnK nm) divnD// addnAC addnK leqDmod ?subnK ?addnK.
Qed.
Lemma modnB m n d : 0 < d -> n <= m ->
(m - n) %% d = (m %% d < n %% d) * d + m %% d - n %% d.
Proof.
move=> d_gt0 nm; rewrite -[in m %% _](subnK nm) -leqDmod// modnD//.
have [d_le|_] := leqP d; last by rewrite mul0n add0n subn0 addnK.
by rewrite mul1n addnBA// addnC !addnK.
Qed.
Lemma edivnB m n d (offset := m %% d < n %% d) : 0 < d -> n <= m ->
edivn (m - n) d = (m %/ d - n %/ d - offset, offset * d + m %% d - n %% d).
Proof. by move=> d_gt0 le_nm; rewrite edivn_def divnB// modnB. Qed.
Lemma leq_divDl p m n : (m + n) %/ p <= m %/ p + n %/ p + 1.
Proof. by have [->//|p_gt0] := posnP p; rewrite divnD// !leq_add// leq_b1. Qed.
Lemma geq_divBl k m p : k %/ p - m %/ p <= (k - m) %/ p + 1.
Proof.
rewrite leq_subLR addnA; apply: leq_trans (leq_divDl _ _ _).
by rewrite -maxnE leq_div2r ?leq_maxr.
Qed.
Lemma divnMA m n p : m %/ (n * p) = m %/ n %/ p.
Proof.
case: n p => [|n] [|p]; rewrite ?muln0 ?div0n //.
rewrite [in RHS](divn_eq m (n.+1 * p.+1)) mulnA mulnAC !divnMDl //.
by rewrite [_ %/ p.+1]divn_small ?addn0 // ltn_divLR // mulnC ltn_mod.
Qed.
Lemma divnAC m n p : m %/ n %/ p = m %/ p %/ n.
Proof. by rewrite -!divnMA mulnC. Qed.
Lemma modn_small m d : m < d -> m %% d = m.
Proof. by move=> lt_md; rewrite [RHS](divn_eq m d) divn_small. Qed.
Lemma modn_mod m d : m %% d = m %[mod d].
Proof. by case: d => // d; apply: modn_small; rewrite ltn_mod. Qed.
Lemma modnMDl p m d : p * d + m = m %[mod d].
Proof.
have [->|d_gt0] := posnP d; first by rewrite muln0.
by rewrite [in LHS](divn_eq m d) addnA -mulnDl modn_def edivn_eq // ltn_mod.
Qed.
Lemma muln_modr p m d : p * (m %% d) = (p * m) %% (p * d).
Proof.
have [->//|p_gt0] := posnP p; apply: (@addnI (p * (m %/ d * d))).
by rewrite -mulnDr -divn_eq mulnCA -(divnMl p_gt0) -divn_eq.
Qed.
Lemma muln_modl p m d : (m %% d) * p = (m * p) %% (d * p).
Proof. by rewrite -!(mulnC p); apply: muln_modr. Qed.
Lemma modn_divl m n d : (m %/ d) %% n = m %% (n * d) %/ d.
Proof.
case: d n => [|d] [|n] //; rewrite [in LHS]/divn [in LHS]modn_def.
case: (edivnP m d.+1) edivnP => [/= _ r -> le_rd] [/= q s -> le_sn].
rewrite mulnDl -mulnA -addnA modnMDl modn_small ?divnMDl ?divn_small ?addn0//.
by rewrite mulSnr -addnS leq_add ?leq_mul2r.
Qed.
Lemma modnDl m d : d + m = m %[mod d].
Proof. by rewrite -[m %% _](modnMDl 1) mul1n. Qed.
Lemma modnDr m d : m + d = m %[mod d]. Proof. by rewrite addnC modnDl. Qed.
Lemma modnn d : d %% d = 0. Proof. by rewrite [d %% d](modnDr 0) mod0n. Qed.
Lemma modnMl p d : p * d %% d = 0.
Proof. by rewrite -[p * d]addn0 modnMDl mod0n. Qed.
Lemma modnMr p d : d * p %% d = 0. Proof. by rewrite mulnC modnMl. Qed.
Lemma modnDml m n d : m %% d + n = m + n %[mod d].
Proof. by rewrite [in RHS](divn_eq m d) -addnA modnMDl. Qed.
Lemma modnDmr m n d : m + n %% d = m + n %[mod d].
Proof. by rewrite !(addnC m) modnDml. Qed.
Lemma modnDm m n d : m %% d + n %% d = m + n %[mod d].
Proof. by rewrite modnDml modnDmr. Qed.
Lemma eqn_modDl p m n d : (p + m == p + n %[mod d]) = (m == n %[mod d]).
Proof.
case: d => [|d]; first by rewrite !modn0 eqn_add2l.
apply/eqP/eqP=> eq_mn; last by rewrite -modnDmr eq_mn modnDmr.
rewrite -(modnMDl p m) -(modnMDl p n) !mulnSr -!addnA.
by rewrite -modnDmr eq_mn modnDmr.
Qed.
Lemma eqn_modDr p m n d : (m + p == n + p %[mod d]) = (m == n %[mod d]).
Proof. by rewrite -!(addnC p) eqn_modDl. Qed.
Lemma modnMml m n d : m %% d * n = m * n %[mod d].
Proof. by rewrite [in RHS](divn_eq m d) mulnDl mulnAC modnMDl. Qed.
Lemma modnMmr m n d : m * (n %% d) = m * n %[mod d].
Proof. by rewrite !(mulnC m) modnMml. Qed.
Lemma modnMm m n d : m %% d * (n %% d) = m * n %[mod d].
Proof. by rewrite modnMml modnMmr. Qed.
Lemma modn2 m : m %% 2 = odd m.
Proof. by elim: m => //= m IHm; rewrite -addn1 -modnDml IHm; case odd. Qed.
Lemma divn2 m : m %/ 2 = m./2.
Proof. by rewrite [in RHS](divn_eq m 2) modn2 muln2 addnC half_bit_double. Qed.
Lemma odd_mod m d : odd d = false -> odd (m %% d) = odd m.
Proof.
by move=> d_even; rewrite [in RHS](divn_eq m d) oddD oddM d_even andbF.
Qed.
Lemma modnXm m n a : (a %% n) ^ m = a ^ m %[mod n].
Proof. by elim: m => // m IHm; rewrite !expnS -modnMmr IHm modnMml modnMmr. Qed.
Lemma modnMDXl p m n d : (p * d + m) ^ n = m ^ n %[mod d].
Proof. by elim: n => // n IH; rewrite !expnS -modnMm IH modnMDl modnMm. Qed.
(** Divisibility **)
Definition dvdn d m := m %% d == 0.
Notation "m %| d" := (dvdn m d) : nat_scope.
Lemma dvdnP d m : reflect (exists k, m = k * d) (d %| m).
Proof.
apply: (iffP eqP) => [md0 | [k ->]]; last by rewrite modnMl.
by exists (m %/ d); rewrite [LHS](divn_eq m d) md0 addn0.
Qed.
Arguments dvdnP {d m}.
Lemma dvdn0 d : d %| 0.
Proof. by case: d. Qed.
Lemma dvd0n n : (0 %| n) = (n == 0).
Proof. by case: n. Qed.
Lemma dvdn1 d : (d %| 1) = (d == 1).
Proof. by case: d => [|[|d]] //; rewrite /dvdn modn_small. Qed.
Lemma dvd1n m : 1 %| m.
Proof. by rewrite /dvdn modn1. Qed.
Lemma dvdn_gt0 d m : m > 0 -> d %| m -> d > 0.
Proof. by case: d => // /prednK <-. Qed.
Lemma dvdnn m : m %| m.
Proof. by rewrite /dvdn modnn. Qed.
Lemma dvdn_mull d m n : d %| n -> d %| m * n.
Proof. by case/dvdnP=> n' ->; rewrite /dvdn mulnA modnMl. Qed.
Lemma dvdn_mulr d m n : d %| m -> d %| m * n.
Proof. by move=> d_m; rewrite mulnC dvdn_mull. Qed.
#[global] Hint Resolve dvdn0 dvd1n dvdnn dvdn_mull dvdn_mulr : core.
Lemma dvdn_mul d1 d2 m1 m2 : d1 %| m1 -> d2 %| m2 -> d1 * d2 %| m1 * m2.
Proof.
by move=> /dvdnP[q1 ->] /dvdnP[q2 ->]; rewrite mulnCA -mulnA 2?dvdn_mull.
Qed.
Lemma dvdn_trans n d m : d %| n -> n %| m -> d %| m.
Proof. by move=> d_dv_n /dvdnP[n1 ->]; apply: dvdn_mull. Qed.
Lemma dvdn_eq d m : (d %| m) = (m %/ d * d == m).
Proof.
apply/eqP/eqP=> [modm0 | <-]; last exact: modnMl.
by rewrite [RHS](divn_eq m d) modm0 addn0.
Qed.
Lemma dvdn2 n : (2 %| n) = ~~ odd n.
Proof. by rewrite /dvdn modn2; case (odd n). Qed.
Lemma dvdn_odd m n : m %| n -> odd n -> odd m.
Proof. by move=> m_dv_n; apply: contraTT; rewrite -!dvdn2 => /dvdn_trans->. Qed.
Lemma divnK d m : d %| m -> m %/ d * d = m.
Proof. by rewrite dvdn_eq; move/eqP. Qed.
Lemma leq_divLR d m n : d %| m -> (m %/ d <= n) = (m <= n * d).
Proof. by case: d m => [|d] [|m] ///divnK=> {2}<-; rewrite leq_pmul2r. Qed.
Lemma ltn_divRL d m n : d %| m -> (n < m %/ d) = (n * d < m).
Proof. by move=> dv_d_m; rewrite !ltnNge leq_divLR. Qed.
Lemma eqn_div d m n : d > 0 -> d %| m -> (n == m %/ d) = (n * d == m).
Proof. by move=> d_gt0 dv_d_m; rewrite -(eqn_pmul2r d_gt0) divnK. Qed.
Lemma eqn_mul d m n : d > 0 -> d %| m -> (m == n * d) = (m %/ d == n).
Proof. by move=> d_gt0 dv_d_m; rewrite eq_sym -eqn_div // eq_sym. Qed.
Lemma divn_mulAC d m n : d %| m -> m %/ d * n = m * n %/ d.
Proof.
case: d m => [[] //| d m] dv_d_m; apply/eqP.
by rewrite eqn_div ?dvdn_mulr // mulnAC divnK.
Qed.
Lemma muln_divA d m n : d %| n -> m * (n %/ d) = m * n %/ d.
Proof. by move=> dv_d_m; rewrite !(mulnC m) divn_mulAC. Qed.
Lemma muln_divCA d m n : d %| m -> d %| n -> m * (n %/ d) = n * (m %/ d).
Proof. by move=> dv_d_m dv_d_n; rewrite mulnC divn_mulAC ?muln_divA. Qed.
Lemma divnA m n p : p %| n -> m %/ (n %/ p) = m * p %/ n.
Proof. by case: p => [|p] dv_n; rewrite -[in RHS](divnK dv_n) // divnMr. Qed.
Lemma modn_dvdm m n d : d %| m -> n %% m = n %[mod d].
Proof.
by case/dvdnP=> q def_m; rewrite [in RHS](divn_eq n m) def_m mulnA modnMDl.
Qed.
Lemma dvdn_leq d m : 0 < m -> d %| m -> d <= m.
Proof. by move=> m_gt0 /dvdnP[[|k] Dm]; rewrite Dm // leq_addr in m_gt0 *. Qed.
Lemma gtnNdvd n d : 0 < n -> n < d -> (d %| n) = false.
Proof. by move=> n_gt0 lt_nd; rewrite /dvdn eqn0Ngt modn_small ?n_gt0. Qed.
Lemma eqn_dvd m n : (m == n) = (m %| n) && (n %| m).
Proof.
case: m n => [|m] [|n] //; apply/idP/andP => [/eqP -> //| []].
by rewrite eqn_leq => Hmn Hnm; do 2 rewrite dvdn_leq //.
Qed.
Lemma dvdn_pmul2l p d m : 0 < p -> (p * d %| p * m) = (d %| m).
Proof. by case: p => // p _; rewrite /dvdn -muln_modr // muln_eq0. Qed.
Arguments dvdn_pmul2l [p d m].
Lemma dvdn_pmul2r p d m : 0 < p -> (d * p %| m * p) = (d %| m).
Proof. by move=> p_gt0; rewrite -!(mulnC p) dvdn_pmul2l. Qed.
Arguments dvdn_pmul2r [p d m].
Lemma dvdn_divLR p d m : 0 < p -> p %| d -> (d %/ p %| m) = (d %| m * p).
Proof. by move=> /(@dvdn_pmul2r p _ m) <- /divnK->. Qed.
Lemma dvdn_divRL p d m : p %| m -> (d %| m %/ p) = (d * p %| m).
Proof.
have [-> | /(@dvdn_pmul2r p d) <- /divnK-> //] := posnP p.
by rewrite divn0 muln0 dvdn0.
Qed.
Lemma dvdn_div d m : d %| m -> m %/ d %| m.
Proof. by move/divnK=> {2}<-; apply: dvdn_mulr. Qed.
Lemma dvdn_exp2l p m n : m <= n -> p ^ m %| p ^ n.
Proof. by move/subnK <-; rewrite expnD dvdn_mull. Qed.
Lemma dvdn_Pexp2l p m n : p > 1 -> (p ^ m %| p ^ n) = (m <= n).
Proof.
move=> p_gt1; case: leqP => [|gt_n_m]; first exact: dvdn_exp2l.
by rewrite gtnNdvd ?ltn_exp2l ?expn_gt0 // ltnW.
Qed.
Lemma dvdn_exp2r m n k : m %| n -> m ^ k %| n ^ k.
Proof. by case/dvdnP=> q ->; rewrite expnMn dvdn_mull. Qed.
Lemma divn_modl m n d : d %| n -> (m %% n) %/ d = (m %/ d) %% (n %/ d).
Proof. by move=> dvd_dn; rewrite modn_divl divnK. Qed.
Lemma dvdn_addr m d n : d %| m -> (d %| m + n) = (d %| n).
Proof. by case/dvdnP=> q ->; rewrite /dvdn modnMDl. Qed.
Lemma dvdn_addl n d m : d %| n -> (d %| m + n) = (d %| m).
Proof. by rewrite addnC; apply: dvdn_addr. Qed.
Lemma dvdn_add d m n : d %| m -> d %| n -> d %| m + n.
Proof. by move/dvdn_addr->. Qed.
Lemma dvdn_add_eq d m n : d %| m + n -> (d %| m) = (d %| n).
Proof. by move=> dv_d_mn; apply/idP/idP => [/dvdn_addr | /dvdn_addl] <-. Qed.
Lemma dvdn_subr d m n : n <= m -> d %| m -> (d %| m - n) = (d %| n).
Proof. by move=> le_n_m dv_d_m; apply: dvdn_add_eq; rewrite subnK. Qed.
Lemma dvdn_subl d m n : n <= m -> d %| n -> (d %| m - n) = (d %| m).
Proof. by move=> le_n_m dv_d_m; rewrite -(dvdn_addl _ dv_d_m) subnK. Qed.
Lemma dvdn_sub d m n : d %| m -> d %| n -> d %| m - n.
Proof.
by case: (leqP n m) => [le_nm /dvdn_subr <- // | /ltnW/eqnP ->]; rewrite dvdn0.
Qed.
Lemma dvdn_exp k d m : 0 < k -> d %| m -> d %| (m ^ k).
Proof. by case: k => // k _ d_dv_m; rewrite expnS dvdn_mulr. Qed.
Lemma dvdn_fact m n : 0 < m <= n -> m %| n`!.
Proof.
case: m => //= m; elim: n => //= n IHn; rewrite ltnS.
have [/IHn/dvdn_mull->||-> _] // := ltngtP m n; exact: dvdn_mulr.
Qed.
#[global] Hint Resolve dvdn_add dvdn_sub dvdn_exp : core.
Lemma eqn_mod_dvd d m n : n <= m -> (m == n %[mod d]) = (d %| m - n).
Proof.
by move/subnK=> Dm; rewrite -[n in LHS]add0n -[in LHS]Dm eqn_modDr mod0n.
Qed.
Lemma divnDMl q m d : 0 < d -> (m + q * d) %/ d = (m %/ d) + q.
Proof. by move=> d_gt0; rewrite addnC divnMDl// addnC. Qed.
Lemma divnMBl q m d : 0 < d -> (q * d - m) %/ d = q - (m %/ d) - (~~ (d %| m)).
Proof. by move=> d_gt0; rewrite divnB// mulnK// modnMl lt0n. Qed.
Lemma divnBMl q m d : (m - q * d) %/ d = (m %/ d) - q.
Proof. by case: d => [|d]//=; rewrite divnB// mulnK// modnMl ltn0 subn0. Qed.
Lemma divnDl m n d : d %| m -> (m + n) %/ d = m %/ d + n %/ d.
Proof. by case: d => // d /divnK-Dm; rewrite -[in LHS]Dm divnMDl. Qed.
Lemma divnDr m n d : d %| n -> (m + n) %/ d = m %/ d + n %/ d.
Proof. by move=> dv_n; rewrite addnC divnDl // addnC. Qed.
Lemma divnBl m n d : d %| m -> (m - n) %/ d = m %/ d - (n %/ d) - (~~ (d %| n)).
Proof. by case: d => [|d] // /divnK-Dm; rewrite -[in LHS]Dm divnMBl. Qed.
Lemma divnBr m n d : d %| n -> (m - n) %/ d = m %/ d - n %/ d.
Proof. by case: d => [|d]// /divnK-Dm; rewrite -[in LHS]Dm divnBMl. Qed.
Lemma edivnS m d : 0 < d -> edivn m.+1 d =
if d %| m.+1 then ((m %/ d).+1, 0) else (m %/ d, (m %% d).+1).
Proof.
case: d => [|[|d]] //= _; first by rewrite edivn_def modn1 dvd1n !divn1.
rewrite -addn1 /dvdn modn_def edivnD//= (@modn_small 1)// (@divn_small 1)//.
rewrite addn1 addn0 ltnS; have [||<-] := ltngtP d.+1.
- by rewrite ltnNge -ltnS ltn_pmod.
- by rewrite addn0 mul0n subn0.
- by rewrite addn1 mul1n subnn.
Qed.
Lemma modnS m d : m.+1 %% d = if d %| m.+1 then 0 else (m %% d).+1.
Proof. by case: d => [|d]//; rewrite modn_def edivnS//; case: ifP. Qed.
Lemma divnS m d : 0 < d -> m.+1 %/ d = (d %| m.+1) + m %/ d.
Proof. by move=> d_gt0; rewrite /divn edivnS//; case: ifP. Qed.
Lemma divn_pred m d : m.-1 %/ d = (m %/ d) - (d %| m).
Proof.
by case: d m => [|d] [|m]; rewrite ?divn1 ?dvd1n ?subn1//= divnS// addnC addnK.
Qed.
Lemma modn_pred m d : d != 1 -> 0 < m ->
m.-1 %% d = if d %| m then d.-1 else (m %% d).-1.
Proof.
rewrite -subn1; case: d m => [|[|d]] [|m]//= _ _.
by rewrite ?modn1 ?dvd1n ?modn0 ?subn1.
rewrite modnB// (@modn_small 1)// [_ < _]leqn0 /dvdn mulnbl/= subn1.
by case: eqP => // ->; rewrite addn0.
Qed.
Lemma edivn_pred m d : d != 1 -> 0 < m ->
edivn m.-1 d = if d %| m then ((m %/ d).-1, d.-1) else (m %/ d, (m %% d).-1).
Proof.
move=> d_neq1 m_gt0; rewrite edivn_def divn_pred modn_pred//.
by case: ifP; rewrite ?subn0 ?subn1.
Qed.
(***********************************************************************)
(* A function that computes the gcd of 2 numbers *)
(***********************************************************************)
Fixpoint gcdn m n :=
let n' := n %% m in if n' is 0 then m else
if m - n'.-1 is m'.+1 then gcdn (m' %% n') n' else n'.
Arguments gcdn : simpl never.
Lemma gcdnE m n : gcdn m n = if m == 0 then n else gcdn (n %% m) m.
Proof.
elim/ltn_ind: m n => -[|m] IHm [|n] //=; rewrite /gcdn -/gcdn.
case def_p: (_ %% _) => // [p].
have{def_p} lt_pm: p.+1 < m.+1 by rewrite -def_p ltn_pmod.
rewrite {}IHm // subn_if_gt ltnW //=; congr gcdn.
by rewrite -(subnK (ltnW lt_pm)) modnDr.
Qed.
Lemma gcdnn : idempotent_op gcdn.
Proof. by case=> // n; rewrite gcdnE modnn. Qed.
Lemma gcdnC : commutative gcdn.
Proof.
move=> m n; wlog lt_nm: m n / n < m by have [? ->|? <-|-> //] := ltngtP n m.
by rewrite gcdnE -[in m == 0](ltn_predK lt_nm) modn_small.
Qed.
Lemma gcd0n : left_id 0 gcdn. Proof. by case. Qed.
Lemma gcdn0 : right_id 0 gcdn. Proof. by case. Qed.
Lemma gcd1n : left_zero 1 gcdn.
Proof. by move=> n; rewrite gcdnE modn1. Qed.
Lemma gcdn1 : right_zero 1 gcdn.
Proof. by move=> n; rewrite gcdnC gcd1n. Qed.
Lemma dvdn_gcdr m n : gcdn m n %| n.
Proof.
elim/ltn_ind: m n => -[|m] IHm [|n] //=.
rewrite gcdnE; case def_p: (_ %% _) => [|p]; first by rewrite /dvdn def_p.
have lt_pm: p < m by rewrite -ltnS -def_p ltn_pmod.
rewrite /= (divn_eq n.+1 m.+1) def_p dvdn_addr ?dvdn_mull //; last exact: IHm.
by rewrite gcdnE /= IHm // (ltn_trans (ltn_pmod _ _)).
Qed.
Lemma dvdn_gcdl m n : gcdn m n %| m.
Proof. by rewrite gcdnC dvdn_gcdr. Qed.
Lemma gcdn_gt0 m n : (0 < gcdn m n) = (0 < m) || (0 < n).
Proof.
by case: m n => [|m] [|n] //; apply: (@dvdn_gt0 _ m.+1) => //; apply: dvdn_gcdl.
Qed.
Lemma gcdnMDl k m n : gcdn m (k * m + n) = gcdn m n.
Proof. by rewrite !(gcdnE m) modnMDl mulnC; case: m. Qed.
Lemma gcdnDl m n : gcdn m (m + n) = gcdn m n.
Proof. by rewrite -[m in m + n]mul1n gcdnMDl. Qed.
Lemma gcdnDr m n : gcdn m (n + m) = gcdn m n.
Proof. by rewrite addnC gcdnDl. Qed.
Lemma gcdnMl n m : gcdn n (m * n) = n.
Proof. by case: n => [|n]; rewrite gcdnE modnMl // muln0. Qed.
Lemma gcdnMr n m : gcdn n (n * m) = n.
Proof. by rewrite mulnC gcdnMl. Qed.
Lemma gcdn_idPl {m n} : reflect (gcdn m n = m) (m %| n).
Proof.
by apply: (iffP idP) => [/dvdnP[q ->] | <-]; rewrite (gcdnMl, dvdn_gcdr).
Qed.
Lemma gcdn_idPr {m n} : reflect (gcdn m n = n) (n %| m).
Proof. by rewrite gcdnC; apply: gcdn_idPl. Qed.
Lemma expn_min e m n : e ^ minn m n = gcdn (e ^ m) (e ^ n).
Proof. by case: leqP => [|/ltnW] /(dvdn_exp2l e) /gcdn_idPl; rewrite gcdnC. Qed.
Lemma gcdn_modr m n : gcdn m (n %% m) = gcdn m n.
Proof. by rewrite [in RHS](divn_eq n m) gcdnMDl. Qed.
Lemma gcdn_modl m n : gcdn (m %% n) n = gcdn m n.
Proof. by rewrite !(gcdnC _ n) gcdn_modr. Qed.
(* Extended gcd, which computes Bezout coefficients. *)
Fixpoint Bezout_rec km kn qs :=
if qs is q :: qs' then Bezout_rec kn (NatTrec.add_mul q kn km) qs'
else (km, kn).
Fixpoint egcdn_rec m n s qs :=
if s is s'.+1 then
let: (q, r) := edivn m n in
if r > 0 then egcdn_rec n r s' (q :: qs) else
if odd (size qs) then qs else q.-1 :: qs
else [::0].
Definition egcdn m n := Bezout_rec 0 1 (egcdn_rec m n n [::]).
Variant egcdn_spec m n : nat * nat -> Type :=
EgcdnSpec km kn of km * m = kn * n + gcdn m n & kn * gcdn m n < m :
egcdn_spec m n (km, kn).
Lemma egcd0n n : egcdn 0 n = (1, 0).
Proof. by case: n. Qed.
Lemma egcdnP m n : m > 0 -> egcdn_spec m n (egcdn m n).
Proof.
have [-> /= | n_gt0 m_gt0] := posnP n; first by split; rewrite // mul1n gcdn0.
rewrite /egcdn; set s := (s in egcdn_rec _ _ s); pose bz := Bezout_rec n m [::].
have: n < s.+1 by []; move defSpec: (egcdn_spec bz.2 bz.1) s => Spec s.
elim: s => [[]|s IHs] //= in n m (qs := [::]) bz defSpec n_gt0 m_gt0 *.
case: edivnP => q r def_m; rewrite n_gt0 ltnS /= => lt_rn le_ns1.
case: posnP => [r0 {s le_ns1 IHs lt_rn}|r_gt0]; last first.
by apply: IHs => //=; [rewrite natTrecE -def_m | rewrite (leq_trans lt_rn)].
rewrite {r}r0 addn0 in def_m; set b := odd _; pose d := gcdn m n.
pose km := ~~ b : nat; pose kn := if b then 1 else q.-1.
rewrite [bz in Spec bz](_ : _ = Bezout_rec km kn qs); last first.
by rewrite /kn /km; case: (b) => //=; rewrite natTrecE addn0 muln1.
have def_d: d = n by rewrite /d def_m gcdnC gcdnE modnMl gcd0n -[n]prednK.
have: km * m + 2 * b * d = kn * n + d.
rewrite {}/kn {}/km def_m def_d -mulSnr; case: b; rewrite //= addn0 mul1n.
by rewrite prednK //; apply: dvdn_gt0 m_gt0 _; rewrite def_m dvdn_mulr.
have{def_m}: kn * d <= m.
have q_gt0 : 0 < q by rewrite def_m muln_gt0 n_gt0 ?andbT in m_gt0.
by rewrite /kn; case b; rewrite def_d def_m leq_pmul2r // leq_pred.
have{def_d}: km * d <= n by rewrite -[n]mul1n def_d leq_pmul2r // leq_b1.
move: km {q}kn m_gt0 n_gt0 defSpec; rewrite {}/b {}/d {}/bz.
elim: qs m n => [|q qs IHq] n r kn kr n_gt0 r_gt0 /=.
set d := gcdn n r; rewrite mul0n addn0 => <- le_kn_r _ def_d; split=> //.
have d_gt0: 0 < d by rewrite gcdn_gt0 n_gt0.
have /ltn_pmul2l<-: 0 < kn by rewrite -(ltn_pmul2r n_gt0) def_d ltn_addl.
by rewrite def_d -addn1 leq_add // mulnCA leq_mul2l le_kn_r orbT.
rewrite !natTrecE; set m := _ + r; set km := _ + kn; pose d := gcdn m n.
have ->: gcdn n r = d by rewrite [d]gcdnC gcdnMDl.
have m_gt0: 0 < m by rewrite addn_gt0 r_gt0 orbT.
have d_gt0: 0 < d by rewrite gcdn_gt0 m_gt0.
move=> {}/IHq IHq le_kn_r le_kr_n def_d; apply: IHq => //; rewrite -/d.
by rewrite mulnDl leq_add // -mulnA leq_mul2l le_kr_n orbT.
apply: (@addIn d); rewrite mulnDr -addnA addnACA -def_d addnACA mulnA.
rewrite -!mulnDl -mulnDr -addnA [kr * _]mulnC; congr addn.
by rewrite addnC addn_negb muln1 mul2n addnn.
Qed.
Lemma Bezoutl m n : m > 0 -> {a | a < m & m %| gcdn m n + a * n}.
Proof.
move=> m_gt0; case: (egcdnP n m_gt0) => km kn def_d lt_kn_m.
exists kn; last by rewrite addnC -def_d dvdn_mull.
apply: leq_ltn_trans lt_kn_m.
by rewrite -{1}[kn]muln1 leq_mul2l gcdn_gt0 m_gt0 orbT.
Qed.
Lemma Bezoutr m n : n > 0 -> {a | a < n & n %| gcdn m n + a * m}.
Proof. by rewrite gcdnC; apply: Bezoutl. Qed.
(* Back to the gcd. *)
Lemma dvdn_gcd p m n : p %| gcdn m n = (p %| m) && (p %| n).
Proof.
apply/idP/andP=> [dv_pmn | [dv_pm dv_pn]].
by rewrite !(dvdn_trans dv_pmn) ?dvdn_gcdl ?dvdn_gcdr.
have [->|n_gt0] := posnP n; first by rewrite gcdn0.
case: (Bezoutr m n_gt0) => // km _ /(dvdn_trans dv_pn).
by rewrite dvdn_addl // dvdn_mull.
Qed.
Lemma gcdnAC : right_commutative gcdn.
Proof.
suffices dvd m n p: gcdn (gcdn m n) p %| gcdn (gcdn m p) n.
by move=> m n p; apply/eqP; rewrite eqn_dvd !dvd.
rewrite !dvdn_gcd dvdn_gcdr.
by rewrite !(dvdn_trans (dvdn_gcdl _ p)) ?dvdn_gcdl ?dvdn_gcdr.
Qed.
Lemma gcdnA : associative gcdn.
Proof. by move=> m n p; rewrite !(gcdnC m) gcdnAC. Qed.
Lemma gcdnCA : left_commutative gcdn.
Proof. by move=> m n p; rewrite !gcdnA (gcdnC m). Qed.
Lemma gcdnACA : interchange gcdn gcdn.
Proof. by move=> m n p q; rewrite -!gcdnA (gcdnCA n). Qed.
Lemma muln_gcdr : right_distributive muln gcdn.
Proof.
move=> p m n; have [-> //|p_gt0] := posnP p.
elim/ltn_ind: m n => m IHm n; rewrite gcdnE [RHS]gcdnE muln_eq0 (gtn_eqF p_gt0).
by case: posnP => // m_gt0; rewrite -muln_modr //=; apply/IHm/ltn_pmod.
Qed.
Lemma muln_gcdl : left_distributive muln gcdn.
Proof. by move=> m n p; rewrite -!(mulnC p) muln_gcdr. Qed.
Lemma gcdn_def d m n :
d %| m -> d %| n -> (forall d', d' %| m -> d' %| n -> d' %| d) ->
gcdn m n = d.
Proof.
move=> dv_dm dv_dn gdv_d; apply/eqP.
by rewrite eqn_dvd dvdn_gcd dv_dm dv_dn gdv_d ?dvdn_gcdl ?dvdn_gcdr.
Qed.
Lemma muln_divCA_gcd n m : n * (m %/ gcdn n m) = m * (n %/ gcdn n m).
Proof. by rewrite muln_divCA ?dvdn_gcdl ?dvdn_gcdr. Qed.
(* We derive the lcm directly. *)
Definition lcmn m n := m * n %/ gcdn m n.
Lemma lcmnC : commutative lcmn.
Proof. by move=> m n; rewrite /lcmn mulnC gcdnC. Qed.
Lemma lcm0n : left_zero 0 lcmn. Proof. by move=> n; apply: div0n. Qed.
Lemma lcmn0 : right_zero 0 lcmn. Proof. by move=> n; rewrite lcmnC lcm0n. Qed.
Lemma lcm1n : left_id 1 lcmn.
Proof. by move=> n; rewrite /lcmn gcd1n mul1n divn1. Qed.
Lemma lcmn1 : right_id 1 lcmn.
Proof. by move=> n; rewrite lcmnC lcm1n. Qed.
Lemma muln_lcm_gcd m n : lcmn m n * gcdn m n = m * n.
Proof. by apply/eqP; rewrite divnK ?dvdn_mull ?dvdn_gcdr. Qed.
Lemma lcmn_gt0 m n : (0 < lcmn m n) = (0 < m) && (0 < n).
Proof. by rewrite -muln_gt0 ltn_divRL ?dvdn_mull ?dvdn_gcdr. Qed.
Lemma muln_lcmr : right_distributive muln lcmn.
Proof.
case=> // m n p; rewrite /lcmn -muln_gcdr -!mulnA divnMl // mulnCA.
by rewrite muln_divA ?dvdn_mull ?dvdn_gcdr.
Qed.
Lemma muln_lcml : left_distributive muln lcmn.
Proof. by move=> m n p; rewrite -!(mulnC p) muln_lcmr. Qed.
Lemma lcmnA : associative lcmn.
Proof.
move=> m n p; rewrite [LHS]/lcmn [RHS]/lcmn mulnC.
rewrite !divn_mulAC ?dvdn_mull ?dvdn_gcdr // -!divnMA ?dvdn_mulr ?dvdn_gcdl //.
rewrite mulnC mulnA !muln_gcdr; congr (_ %/ _).
by rewrite ![_ * lcmn _ _]mulnC !muln_lcm_gcd !muln_gcdl -!(mulnC m) gcdnA.
Qed.
Lemma lcmnCA : left_commutative lcmn.
Proof. by move=> m n p; rewrite !lcmnA (lcmnC m). Qed.
Lemma lcmnAC : right_commutative lcmn.
Proof. by move=> m n p; rewrite -!lcmnA (lcmnC n). Qed.
Lemma lcmnACA : interchange lcmn lcmn.
Proof. by move=> m n p q; rewrite -!lcmnA (lcmnCA n). Qed.
Lemma dvdn_lcml d1 d2 : d1 %| lcmn d1 d2.
Proof. by rewrite /lcmn -muln_divA ?dvdn_gcdr ?dvdn_mulr. Qed.
Lemma dvdn_lcmr d1 d2 : d2 %| lcmn d1 d2.
Proof. by rewrite lcmnC dvdn_lcml. Qed.
Lemma dvdn_lcm d1 d2 m : lcmn d1 d2 %| m = (d1 %| m) && (d2 %| m).
Proof.
case: d1 d2 => [|d1] [|d2]; try by case: m => [|m]; rewrite ?lcmn0 ?andbF.
rewrite -(@dvdn_pmul2r (gcdn d1.+1 d2.+1)) ?gcdn_gt0 // muln_lcm_gcd.
by rewrite muln_gcdr dvdn_gcd {1}mulnC andbC !dvdn_pmul2r.
Qed.
Lemma lcmnMl m n : lcmn m (m * n) = m * n.
Proof. by case: m => // m; rewrite /lcmn gcdnMr mulKn. Qed.
Lemma lcmnMr m n : lcmn n (m * n) = m * n.
Proof. by rewrite mulnC lcmnMl. Qed.
Lemma lcmn_idPr {m n} : reflect (lcmn m n = n) (m %| n).
Proof.
by apply: (iffP idP) => [/dvdnP[q ->] | <-]; rewrite (lcmnMr, dvdn_lcml).
Qed.
Lemma lcmn_idPl {m n} : reflect (lcmn m n = m) (n %| m).
Proof. by rewrite lcmnC; apply: lcmn_idPr. Qed.
Lemma expn_max e m n : e ^ maxn m n = lcmn (e ^ m) (e ^ n).
Proof. by case: leqP => [|/ltnW] /(dvdn_exp2l e) /lcmn_idPl; rewrite lcmnC. Qed.
(* Coprime factors *)
Definition coprime m n := gcdn m n == 1.
Lemma coprime1n n : coprime 1 n.
Proof. by rewrite /coprime gcd1n. Qed.
Lemma coprimen1 n : coprime n 1.
Proof. by rewrite /coprime gcdn1. Qed.
Lemma coprime_sym m n : coprime m n = coprime n m.
Proof. by rewrite /coprime gcdnC. Qed.
Lemma coprime_modl m n : coprime (m %% n) n = coprime m n.
Proof. by rewrite /coprime gcdn_modl. Qed.
Lemma coprime_modr m n : coprime m (n %% m) = coprime m n.
Proof. by rewrite /coprime gcdn_modr. Qed.
Lemma coprime2n n : coprime 2 n = odd n.
Proof. by rewrite -coprime_modr modn2; case: (odd n). Qed.
Lemma coprimen2 n : coprime n 2 = odd n.
Proof. by rewrite coprime_sym coprime2n. Qed.
Lemma coprimeSn n : coprime n.+1 n.
Proof. by rewrite -coprime_modl (modnDr 1) coprime_modl coprime1n. Qed.
Lemma coprimenS n : coprime n n.+1.
Proof. by rewrite coprime_sym coprimeSn. Qed.
Lemma coprimePn n : n > 0 -> coprime n.-1 n.
Proof. by case: n => // n _; rewrite coprimenS. Qed.
Lemma coprimenP n : n > 0 -> coprime n n.-1.
Proof. by case: n => // n _; rewrite coprimeSn. Qed.
Lemma coprimeP n m :
n > 0 -> reflect (exists u, u.1 * n - u.2 * m = 1) (coprime n m).
Proof.
move=> n_gt0; apply: (iffP eqP) => [<-| [[kn km] /= kn_km_1]].
by have [kn km kg _] := egcdnP m n_gt0; exists (kn, km); rewrite kg addKn.
apply gcdn_def; rewrite ?dvd1n // => d dv_d_n dv_d_m.
by rewrite -kn_km_1 dvdn_subr ?dvdn_mull // ltnW // -subn_gt0 kn_km_1.
Qed.
Lemma modn_coprime k n : 0 < k -> (exists u, (k * u) %% n = 1) -> coprime k n.
Proof.
move=> k_gt0 [u Hu]; apply/coprimeP=> //.
by exists (u, k * u %/ n); rewrite /= mulnC {1}(divn_eq (k * u) n) addKn.
Qed.
Lemma Gauss_dvd m n p : coprime m n -> (m * n %| p) = (m %| p) && (n %| p).
Proof. by move=> co_mn; rewrite -muln_lcm_gcd (eqnP co_mn) muln1 dvdn_lcm. Qed.
Lemma Gauss_dvdr m n p : coprime m n -> (m %| n * p) = (m %| p).
Proof.
case: n => [|n] co_mn; first by case: m co_mn => [|[]] // _; rewrite !dvd1n.
by symmetry; rewrite mulnC -(@dvdn_pmul2r n.+1) ?Gauss_dvd // andbC dvdn_mull.
Qed.
Lemma Gauss_dvdl m n p : coprime m p -> (m %| n * p) = (m %| n).
Proof. by rewrite mulnC; apply: Gauss_dvdr. Qed.
Lemma dvdn_double_leq m n : m %| n -> odd m -> ~~ odd n -> 0 < n -> m.*2 <= n.
Proof.
move=> m_dv_n odd_m even_n n_gt0.
by rewrite -muln2 dvdn_leq // Gauss_dvd ?coprimen2 ?m_dv_n ?dvdn2.
Qed.
Lemma dvdn_double_ltn m n : m %| n.-1 -> odd m -> odd n -> 1 < n -> m.*2 < n.
Proof. by case: n => //; apply: dvdn_double_leq. Qed.
Lemma Gauss_gcdr p m n : coprime p m -> gcdn p (m * n) = gcdn p n.
Proof.
move=> co_pm; apply/eqP; rewrite eqn_dvd !dvdn_gcd !dvdn_gcdl /=.
rewrite andbC dvdn_mull ?dvdn_gcdr //= -(@Gauss_dvdr _ m) ?dvdn_gcdr //.
by rewrite /coprime gcdnAC (eqnP co_pm) gcd1n.
Qed.
Lemma Gauss_gcdl p m n : coprime p n -> gcdn p (m * n) = gcdn p m.
Proof. by move=> co_pn; rewrite mulnC Gauss_gcdr. Qed.
Lemma coprimeMr p m n : coprime p (m * n) = coprime p m && coprime p n.
Proof.
case co_pm: (coprime p m) => /=; first by rewrite /coprime Gauss_gcdr.
apply/eqP=> co_p_mn; case/eqnP: co_pm; apply gcdn_def => // d dv_dp dv_dm.
by rewrite -co_p_mn dvdn_gcd dv_dp dvdn_mulr.
Qed.
Lemma coprimeMl p m n : coprime (m * n) p = coprime m p && coprime n p.
Proof. by rewrite -!(coprime_sym p) coprimeMr. Qed.
Lemma coprime_pexpl k m n : 0 < k -> coprime (m ^ k) n = coprime m n.
Proof.
case: k => // k _; elim: k => [|k IHk]; first by rewrite expn1.
by rewrite expnS coprimeMl -IHk; case coprime.
Qed.
Lemma coprime_pexpr k m n : 0 < k -> coprime m (n ^ k) = coprime m n.
Proof. by move=> k_gt0; rewrite !(coprime_sym m) coprime_pexpl. Qed.
Lemma coprimeXl k m n : coprime m n -> coprime (m ^ k) n.
Proof. by case: k => [|k] co_pm; rewrite ?coprime1n // coprime_pexpl. Qed.
Lemma coprimeXr k m n : coprime m n -> coprime m (n ^ k).
Proof. by rewrite !(coprime_sym m); apply: coprimeXl. Qed.
Lemma coprime_dvdl m n p : m %| n -> coprime n p -> coprime m p.
Proof. by case/dvdnP=> d ->; rewrite coprimeMl => /andP[]. Qed.
Lemma coprime_dvdr m n p : m %| n -> coprime p n -> coprime p m.
Proof. by rewrite !(coprime_sym p); apply: coprime_dvdl. Qed.
Lemma coprime_egcdn n m : n > 0 -> coprime (egcdn n m).1 (egcdn n m).2.
Proof.
move=> n_gt0; case: (egcdnP m n_gt0) => kn km /= /eqP.
have [/dvdnP[u defn] /dvdnP[v defm]] := (dvdn_gcdl n m, dvdn_gcdr n m).
rewrite -[gcdn n m]mul1n {1}defm {1}defn !mulnA -mulnDl addnC.
rewrite eqn_pmul2r ?gcdn_gt0 ?n_gt0 //; case: kn => // kn /eqP def_knu _.
by apply/coprimeP=> //; exists (u, v); rewrite mulnC def_knu mulnC addnK.
Qed.
Lemma dvdn_pexp2r m n k : k > 0 -> (m ^ k %| n ^ k) = (m %| n).
Proof.
move=> k_gt0; apply/idP/idP=> [dv_mn_k|]; last exact: dvdn_exp2r.
have [->|n_gt0] := posnP n; first by rewrite dvdn0.
have [n' def_n] := dvdnP (dvdn_gcdr m n); set d := gcdn m n in def_n.
have [m' def_m] := dvdnP (dvdn_gcdl m n); rewrite -/d in def_m.
have d_gt0: d > 0 by rewrite gcdn_gt0 n_gt0 orbT.
rewrite def_m def_n !expnMn dvdn_pmul2r ?expn_gt0 ?d_gt0 // in dv_mn_k.
have: coprime (m' ^ k) (n' ^ k).
rewrite coprime_pexpl // coprime_pexpr // /coprime -(eqn_pmul2r d_gt0) mul1n.
by rewrite muln_gcdl -def_m -def_n.
rewrite /coprime -gcdn_modr (eqnP dv_mn_k) gcdn0 -(exp1n k).
by rewrite (inj_eq (expIn k_gt0)) def_m; move/eqP->; rewrite mul1n dvdn_gcdr.
Qed.
Section Chinese.
(***********************************************************************)
(* The chinese remainder theorem *)
(***********************************************************************)
Variables m1 m2 : nat.
Hypothesis co_m12 : coprime m1 m2.
Lemma chinese_remainder x y :
(x == y %[mod m1 * m2]) = (x == y %[mod m1]) && (x == y %[mod m2]).
Proof.
wlog le_yx : x y / y <= x; last by rewrite !eqn_mod_dvd // Gauss_dvd.
by have [?|/ltnW ?] := leqP y x; last rewrite !(eq_sym (x %% _)); apply.
Qed.
(***********************************************************************)
(* A function that solves the chinese remainder problem *)
(***********************************************************************)
Definition chinese r1 r2 :=
r1 * m2 * (egcdn m2 m1).1 + r2 * m1 * (egcdn m1 m2).1.
Lemma chinese_modl r1 r2 : chinese r1 r2 = r1 %[mod m1].
Proof.
rewrite /chinese; case: (posnP m2) co_m12 => [-> /eqnP | m2_gt0 _].
by rewrite gcdn0 => ->; rewrite !modn1.
case: egcdnP => // k2 k1 def_m1 _.
rewrite mulnAC -mulnA def_m1 gcdnC (eqnP co_m12) mulnDr mulnA muln1.
by rewrite addnAC (mulnAC _ m1) -mulnDl modnMDl.
Qed.
Lemma chinese_modr r1 r2 : chinese r1 r2 = r2 %[mod m2].
Proof.
rewrite /chinese; case: (posnP m1) co_m12 => [-> /eqnP | m1_gt0 _].
by rewrite gcd0n => ->; rewrite !modn1.
case: (egcdnP m2) => // k1 k2 def_m2 _.
rewrite addnC mulnAC -mulnA def_m2 (eqnP co_m12) mulnDr mulnA muln1.
by rewrite addnAC (mulnAC _ m2) -mulnDl modnMDl.
Qed.
Lemma chinese_mod x : x = chinese (x %% m1) (x %% m2) %[mod m1 * m2].
Proof.
apply/eqP; rewrite chinese_remainder //.
by rewrite chinese_modl chinese_modr !modn_mod !eqxx.
Qed.
End Chinese.
|
LYM.lean
|
/-
Copyright (c) 2022 Bhavik Mehta, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Alena Gusakov, Yaël Dillies
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Field.Rat
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Combinatorics.SetFamily.Shadow
import Mathlib.Data.NNRat.Order
import Mathlib.Data.Nat.Cast.Order.Ring
/-!
# Lubell-Yamamoto-Meshalkin inequality and Sperner's theorem
This file proves the local LYM and LYM inequalities as well as Sperner's theorem.
## Main declarations
* `Finset.local_lubell_yamamoto_meshalkin_inequality_div`: Local Lubell-Yamamoto-Meshalkin
inequality. The shadow of a set `𝒜` in a layer takes a greater proportion of its layer than `𝒜`
does.
* `Finset.lubell_yamamoto_meshalkin_inequality_sum_card_div_choose`: Lubell-Yamamoto-Meshalkin
inequality. The sum of densities of `𝒜` in each layer is at most `1` for any antichain `𝒜`.
* `IsAntichain.sperner`: Sperner's theorem. The size of any antichain in `Finset α` is at most the
size of the maximal layer of `Finset α`. It is a corollary of
`lubell_yamamoto_meshalkin_inequality_sum_card_div_choose`.
## TODO
Prove upward local LYM.
Provide equality cases. Local LYM gives that the equality case of LYM and Sperner is precisely when
`𝒜` is a middle layer.
`falling` could be useful more generally in grade orders.
## References
* http://b-mehta.github.io/maths-notes/iii/mich/combinatorics.pdf
* http://discretemath.imp.fu-berlin.de/DMII-2015-16/kruskal.pdf
## Tags
shadow, lym, slice, sperner, antichain
-/
open Finset Nat
open scoped FinsetFamily
variable {𝕜 α : Type*} [Semifield 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜]
namespace Finset
/-! ### Local LYM inequality -/
section LocalLYM
variable [DecidableEq α] [Fintype α] {𝒜 : Finset (Finset α)} {r : ℕ}
/-- The downward **local LYM inequality**, with cancelled denominators. `𝒜` takes up less of `α^(r)`
(the finsets of card `r`) than `∂𝒜` takes up of `α^(r - 1)`. -/
theorem local_lubell_yamamoto_meshalkin_inequality_mul (h𝒜 : (𝒜 : Set (Finset α)).Sized r) :
#𝒜 * r ≤ #(∂ 𝒜) * (Fintype.card α - r + 1) := by
let i : DecidableRel ((· ⊆ ·) : Finset α → Finset α → Prop) := fun _ _ => Classical.dec _
refine card_mul_le_card_mul' (· ⊆ ·) (fun s hs => ?_) (fun s hs => ?_)
· rw [← h𝒜 hs, ← card_image_of_injOn s.erase_injOn]
refine card_le_card ?_
simp_rw [image_subset_iff, mem_bipartiteBelow]
exact fun a ha => ⟨erase_mem_shadow hs ha, erase_subset _ _⟩
refine le_trans ?_ tsub_tsub_le_tsub_add
rw [← (Set.Sized.shadow h𝒜) hs, ← card_compl, ← card_image_of_injOn (insert_inj_on' _)]
refine card_le_card fun t ht => ?_
rw [mem_bipartiteAbove] at ht
have : ∅ ∉ 𝒜 := by
rw [← mem_coe, h𝒜.empty_mem_iff, coe_eq_singleton]
rintro rfl
rw [shadow_singleton_empty] at hs
exact notMem_empty s hs
have h := exists_eq_insert_iff.2 ⟨ht.2, by
rw [(sized_shadow_iff this).1 (Set.Sized.shadow h𝒜) ht.1, (Set.Sized.shadow h𝒜) hs]⟩
rcases h with ⟨a, ha, rfl⟩
exact mem_image_of_mem _ (mem_compl.2 ha)
@[inherit_doc local_lubell_yamamoto_meshalkin_inequality_mul]
alias card_mul_le_card_shadow_mul := local_lubell_yamamoto_meshalkin_inequality_mul
/-- The downward **local LYM inequality**. `𝒜` takes up less of `α^(r)` (the finsets of card `r`)
than `∂𝒜` takes up of `α^(r - 1)`. -/
theorem local_lubell_yamamoto_meshalkin_inequality_div (hr : r ≠ 0)
(h𝒜 : (𝒜 : Set (Finset α)).Sized r) : (#𝒜 : 𝕜) / (Fintype.card α).choose r
≤ #(∂ 𝒜) / (Fintype.card α).choose (r - 1) := by
obtain hr' | hr' := lt_or_ge (Fintype.card α) r
· rw [choose_eq_zero_of_lt hr', cast_zero, div_zero]
exact div_nonneg (cast_nonneg _) (cast_nonneg _)
replace h𝒜 := local_lubell_yamamoto_meshalkin_inequality_mul h𝒜
rw [div_le_div_iff₀] <;> norm_cast
· rcases r with - | r
· exact (hr rfl).elim
rw [tsub_add_eq_add_tsub hr', add_tsub_add_eq_tsub_right] at h𝒜
apply le_of_mul_le_mul_right _ (pos_iff_ne_zero.2 hr)
convert Nat.mul_le_mul_right ((Fintype.card α).choose r) h𝒜 using 1
· simpa [mul_assoc, Nat.choose_succ_right_eq] using Or.inl (mul_comm _ _)
· simp only [mul_assoc, choose_succ_right_eq, mul_eq_mul_left_iff]
exact Or.inl (mul_comm _ _)
· exact Nat.choose_pos hr'
· exact Nat.choose_pos (r.pred_le.trans hr')
@[inherit_doc local_lubell_yamamoto_meshalkin_inequality_div]
alias card_div_choose_le_card_shadow_div_choose := local_lubell_yamamoto_meshalkin_inequality_div
end LocalLYM
/-! ### LYM inequality -/
section LYM
section Falling
variable [DecidableEq α] (k : ℕ) (𝒜 : Finset (Finset α))
/-- `falling k 𝒜` is all the finsets of cardinality `k` which are a subset of something in `𝒜`. -/
def falling : Finset (Finset α) :=
𝒜.sup <| powersetCard k
variable {𝒜 k} {s : Finset α}
theorem mem_falling : s ∈ falling k 𝒜 ↔ (∃ t ∈ 𝒜, s ⊆ t) ∧ #s = k := by
simp_rw [falling, mem_sup, mem_powersetCard]
aesop
variable (𝒜 k)
theorem sized_falling : (falling k 𝒜 : Set (Finset α)).Sized k := fun _ hs => (mem_falling.1 hs).2
theorem slice_subset_falling : 𝒜 # k ⊆ falling k 𝒜 := fun s hs =>
mem_falling.2 <| (mem_slice.1 hs).imp_left fun h => ⟨s, h, Subset.refl _⟩
theorem falling_zero_subset : falling 0 𝒜 ⊆ {∅} :=
subset_singleton_iff'.2 fun _ ht => card_eq_zero.1 <| sized_falling _ _ ht
theorem slice_union_shadow_falling_succ : 𝒜 # k ∪ ∂ (falling (k + 1) 𝒜) = falling k 𝒜 := by
ext s
simp_rw [mem_union, mem_slice, mem_shadow_iff, mem_falling]
constructor
· rintro (h | ⟨s, ⟨⟨t, ht, hst⟩, hs⟩, a, ha, rfl⟩)
· exact ⟨⟨s, h.1, Subset.refl _⟩, h.2⟩
refine ⟨⟨t, ht, (erase_subset _ _).trans hst⟩, ?_⟩
rw [card_erase_of_mem ha, hs]
rfl
· rintro ⟨⟨t, ht, hst⟩, hs⟩
by_cases h : s ∈ 𝒜
· exact Or.inl ⟨h, hs⟩
obtain ⟨a, ha, hst⟩ := ssubset_iff.1 (ssubset_of_subset_of_ne hst (ht.ne_of_notMem h).symm)
refine Or.inr ⟨insert a s, ⟨⟨t, ht, hst⟩, ?_⟩, a, mem_insert_self _ _, erase_insert ha⟩
rw [card_insert_of_notMem ha, hs]
variable {𝒜 k}
/-- The shadow of `falling m 𝒜` is disjoint from the `n`-sized elements of `𝒜`, thanks to the
antichain property. -/
theorem IsAntichain.disjoint_slice_shadow_falling {m n : ℕ}
(h𝒜 : IsAntichain (· ⊆ ·) (𝒜 : Set (Finset α))) : Disjoint (𝒜 # m) (∂ (falling n 𝒜)) :=
disjoint_right.2 fun s h₁ h₂ => by
simp_rw [mem_shadow_iff, mem_falling] at h₁
obtain ⟨s, ⟨⟨t, ht, hst⟩, _⟩, a, ha, rfl⟩ := h₁
refine h𝒜 (slice_subset h₂) ht ?_ ((erase_subset _ _).trans hst)
rintro rfl
exact notMem_erase _ _ (hst ha)
/-- A bound on any top part of the sum in LYM in terms of the size of `falling k 𝒜`. -/
theorem le_card_falling_div_choose [Fintype α] (hk : k ≤ Fintype.card α)
(h𝒜 : IsAntichain (· ⊆ ·) (𝒜 : Set (Finset α))) :
(∑ r ∈ range (k + 1),
(#(𝒜 # (Fintype.card α - r)) : 𝕜) / (Fintype.card α).choose (Fintype.card α - r)) ≤
(falling (Fintype.card α - k) 𝒜).card / (Fintype.card α).choose (Fintype.card α - k) := by
induction k with
| zero =>
simp only [tsub_zero, cast_one, cast_le, sum_singleton, div_one, choose_self, range_one,
zero_add, range_one, sum_singleton, tsub_zero,
choose_self, cast_one, div_one, cast_le]
exact card_le_card (slice_subset_falling _ _)
| succ k ih =>
rw [sum_range_succ, ← slice_union_shadow_falling_succ,
card_union_of_disjoint (IsAntichain.disjoint_slice_shadow_falling h𝒜),
cast_add, _root_.add_div, add_comm]
rw [← tsub_tsub, tsub_add_cancel_of_le (le_tsub_of_add_le_left hk)]
exact add_le_add_left ((ih <| le_of_succ_le hk).trans <|
local_lubell_yamamoto_meshalkin_inequality_div
(tsub_pos_iff_lt.2 <| Nat.succ_le_iff.1 hk).ne' <| sized_falling _ _) _
end Falling
variable [Fintype α] {𝒜 : Finset (Finset α)}
/-- The **Lubell-Yamamoto-Meshalkin inequality**, also known as the **LYM inequality**.
If `𝒜` is an antichain, then the sum of the proportion of elements it takes from each layer is less
than `1`. -/
theorem lubell_yamamoto_meshalkin_inequality_sum_card_div_choose
(h𝒜 : IsAntichain (· ⊆ ·) (𝒜 : Set (Finset α))) :
∑ r ∈ range (Fintype.card α + 1), (#(𝒜 # r) / (Fintype.card α).choose r : 𝕜) ≤ 1 := by
classical
rw [← sum_flip]
refine (le_card_falling_div_choose le_rfl h𝒜).trans ?_
rw [div_le_iff₀] <;> norm_cast
· simpa only [Nat.sub_self, one_mul, Nat.choose_zero_right, falling] using
Set.Sized.card_le (sized_falling 0 𝒜)
· rw [tsub_self, choose_zero_right]
exact zero_lt_one
@[inherit_doc lubell_yamamoto_meshalkin_inequality_sum_card_div_choose]
alias sum_card_slice_div_choose_le_one := lubell_yamamoto_meshalkin_inequality_sum_card_div_choose
/-- The **Lubell-Yamamoto-Meshalkin inequality**, also known as the **LYM inequality**.
If `𝒜` is an antichain, then the sum of `(#α.choose #s)⁻¹` over `s ∈ 𝒜` is less than `1`. -/
theorem lubell_yamamoto_meshalkin_inequality_sum_inv_choose (h𝒜 : IsAntichain (· ⊆ ·) 𝒜.toSet) :
∑ s ∈ 𝒜, ((Fintype.card α).choose #s : 𝕜)⁻¹ ≤ 1 := by
calc
_ = ∑ r ∈ range (Fintype.card α + 1),
∑ s ∈ 𝒜 with #s = r, ((Fintype.card α).choose r : 𝕜)⁻¹ := by
rw [sum_fiberwise_of_maps_to']; simp [Nat.lt_succ_iff, card_le_univ]
_ = ∑ r ∈ range (Fintype.card α + 1), (#(𝒜 # r) / (Fintype.card α).choose r : 𝕜) := by
simp [slice, div_eq_mul_inv]
_ ≤ 1 := lubell_yamamoto_meshalkin_inequality_sum_card_div_choose h𝒜
/-! ### Sperner's theorem -/
/-- **Sperner's theorem**. The size of an antichain in `Finset α` is bounded by the size of the
maximal layer in `Finset α`. This precisely means that `Finset α` is a Sperner order. -/
theorem _root_.IsAntichain.sperner (h𝒜 : IsAntichain (· ⊆ ·) 𝒜.toSet) :
#𝒜 ≤ (Fintype.card α).choose (Fintype.card α / 2) := by
have : 0 < ((Fintype.card α).choose (Fintype.card α / 2) : ℚ≥0) :=
Nat.cast_pos.2 <| choose_pos (Nat.div_le_self _ _)
have h := calc
∑ s ∈ 𝒜, ((Fintype.card α).choose (Fintype.card α / 2) : ℚ≥0)⁻¹
_ ≤ ∑ s ∈ 𝒜, ((Fintype.card α).choose #s : ℚ≥0)⁻¹ := by
gcongr with s hs
· exact mod_cast choose_pos s.card_le_univ
· exact choose_le_middle _ _
_ ≤ 1 := lubell_yamamoto_meshalkin_inequality_sum_inv_choose h𝒜
simpa [mul_inv_le_iff₀' this] using h
end LYM
end Finset
|
Unit.lean
|
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.PUnit
import Mathlib.CategoryTheory.Limits.HasLimits
/-!
# `Discrete PUnit` has limits and colimits
Mostly for the sake of constructing trivial examples, we show all (co)cones into `Discrete PUnit`
are (co)limit (co)cones. We also show that such (co)cones exist, and that `Discrete PUnit` has all
(co)limits.
-/
universe v' v
open CategoryTheory
namespace CategoryTheory.Limits
variable {J : Type v} [Category.{v'} J] {F : J ⥤ Discrete PUnit}
/-- A trivial cone for a functor into `PUnit`. `punitConeIsLimit` shows it is a limit. -/
def punitCone : Cone F :=
⟨⟨⟨⟩⟩, (Functor.punitExt _ _).hom⟩
/-- A trivial cocone for a functor into `PUnit`. `punitCoconeIsLimit` shows it is a colimit. -/
def punitCocone : Cocone F :=
⟨⟨⟨⟩⟩, (Functor.punitExt _ _).hom⟩
/-- Any cone over a functor into `PUnit` is a limit cone.
-/
def punitConeIsLimit {c : Cone F} : IsLimit c where
lift := fun s => eqToHom (by simp [eq_iff_true_of_subsingleton])
/-- Any cocone over a functor into `PUnit` is a colimit cocone.
-/
def punitCoconeIsColimit {c : Cocone F} : IsColimit c where
desc := fun s => eqToHom (by simp [eq_iff_true_of_subsingleton])
instance : HasLimitsOfSize.{v', v} (Discrete PUnit) :=
⟨fun _ _ => ⟨fun _ => ⟨punitCone, punitConeIsLimit⟩⟩⟩
instance : HasColimitsOfSize.{v', v} (Discrete PUnit) :=
⟨fun _ _ => ⟨fun _ => ⟨punitCocone, punitCoconeIsColimit⟩⟩⟩
end CategoryTheory.Limits
|
Pointwise.lean
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Alex J. Best
-/
import Mathlib.MeasureTheory.Group.Arithmetic
/-!
# Pointwise set operations on `MeasurableSet`s
In this file we prove several versions of the following fact: if `s` is a measurable set, then so is
`a • s`. Note that the pointwise product of two measurable sets need not be measurable, so there is
no `MeasurableSet.mul` etc.
-/
open Pointwise
open Set
@[to_additive]
theorem MeasurableSet.const_smul {G α : Type*} [Group G] [MulAction G α] [MeasurableSpace G]
[MeasurableSpace α] [MeasurableSMul G α] {s : Set α} (hs : MeasurableSet s) (a : G) :
MeasurableSet (a • s) := by
rw [← preimage_smul_inv]
exact measurable_const_smul _ hs
theorem MeasurableSet.const_smul_of_ne_zero {G₀ α : Type*} [GroupWithZero G₀] [MulAction G₀ α]
[MeasurableSpace G₀] [MeasurableSpace α] [MeasurableSMul G₀ α] {s : Set α}
(hs : MeasurableSet s) {a : G₀} (ha : a ≠ 0) : MeasurableSet (a • s) := by
rw [← preimage_smul_inv₀ ha]
exact measurable_const_smul _ hs
theorem MeasurableSet.const_smul₀ {G₀ α : Type*} [GroupWithZero G₀] [Zero α]
[MulActionWithZero G₀ α] [MeasurableSpace G₀] [MeasurableSpace α] [MeasurableSMul G₀ α]
[MeasurableSingletonClass α] {s : Set α} (hs : MeasurableSet s) (a : G₀) :
MeasurableSet (a • s) := by
rcases eq_or_ne a 0 with (rfl | ha)
exacts [(subsingleton_zero_smul_set s).measurableSet, hs.const_smul_of_ne_zero ha]
|
Prorepresentability.lean
|
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.Algebra.Category.Grp.Limits
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.CategoryTheory.Galois.Decomposition
import Mathlib.CategoryTheory.Limits.IndYoneda
import Mathlib.CategoryTheory.Limits.Preserves.Ulift
/-!
# Pro-Representability of fiber functors
We show that any fiber functor is pro-representable, i.e. there exists a pro-object
`X : I ⥤ C` such that `F` is naturally isomorphic to the colimit of `X ⋙ coyoneda`.
From this we deduce the canonical isomorphism of `Aut F` with the limit over the automorphism
groups of all Galois objects.
## Main definitions
- `PointedGaloisObject`: the category of pointed Galois objects
- `PointedGaloisObject.cocone`: a cocone on `(PointedGaloisObject.incl F).op ≫ coyoneda` with
point `F ⋙ FintypeCat.incl`.
- `autGaloisSystem`: the system of automorphism groups indexed by the pointed Galois objects.
## Main results
- `PointedGaloisObject.isColimit`: the cocone `PointedGaloisObject.cocone` is a colimit cocone.
- `autMulEquivAutGalois`: `Aut F` is canonically isomorphic to the limit over the automorphism
groups of all Galois objects.
- `FiberFunctor.isPretransitive_of_isConnected`: The `Aut F` action on the fiber of a connected
object is transitive.
## Implementation details
The pro-representability statement and the isomorphism of `Aut F` with the limit over the
automorphism groups of all Galois objects naturally forces `F` to take values in `FintypeCat.{u₂}`
where `u₂` is the `Hom`-universe of `C`. Since this is used to show that `Aut F` acts
transitively on `F.obj X` for connected `X`, we a priori only obtain this result for
the mentioned specialized universe setup. To obtain the result for `F` taking values in an arbitrary
`FintypeCat.{w}`, we postcompose with an equivalence `FintypeCat.{w} ≌ FintypeCat.{u₂}` and apply
the specialized result.
In the following the section `Specialized` is reserved for the setup where `F` takes values in
`FintypeCat.{u₂}` and the section `General` contains results holding for `F` taking values in
an arbitrary `FintypeCat.{w}`.
## References
* [lenstraGSchemes]: H. W. Lenstra. Galois theory for schemes.
-/
universe u₁ u₂ w
namespace CategoryTheory
namespace PreGaloisCategory
open Limits Functor
variable {C : Type u₁} [Category.{u₂} C] [GaloisCategory C]
/-- A pointed Galois object is a Galois object with a fixed point of its fiber. -/
structure PointedGaloisObject (F : C ⥤ FintypeCat.{w}) : Type (max u₁ u₂ w) where
/-- The underlying object of `C`. -/
obj : C
/-- An element of the fiber of `obj`. -/
pt : F.obj obj
/-- `obj` is Galois. -/
isGalois : IsGalois obj := by infer_instance
namespace PointedGaloisObject
section General
variable (F : C ⥤ FintypeCat.{w})
attribute [instance] isGalois
instance (X : PointedGaloisObject F) : CoeDep (PointedGaloisObject F) X C where
coe := X.obj
variable {F} in
/-- The type of homomorphisms between two pointed Galois objects. This is a homomorphism
of the underlying objects of `C` that maps the distinguished points to each other. -/
@[ext]
structure Hom (A B : PointedGaloisObject F) where
/-- The underlying homomorphism of `C`. -/
val : A.obj ⟶ B.obj
/-- The distinguished point of `A` is mapped to the distinguished point of `B`. -/
comp : F.map val A.pt = B.pt := by simp
attribute [simp] Hom.comp
/-- The category of pointed Galois objects. -/
instance : Category.{u₂} (PointedGaloisObject F) where
Hom A B := Hom A B
id A := { val := 𝟙 (A : C) }
comp {A B C} f g := { val := f.val ≫ g.val }
instance {A B : PointedGaloisObject F} : Coe (Hom A B) (A.obj ⟶ B.obj) where
coe f := f.val
variable {F}
@[ext]
lemma hom_ext {A B : PointedGaloisObject F} {f g : A ⟶ B} (h : f.val = g.val) : f = g :=
Hom.ext h
@[simp]
lemma id_val (A : PointedGaloisObject F) : 𝟙 A = 𝟙 A.obj :=
rfl
@[simp, reassoc]
lemma comp_val {A B C : PointedGaloisObject F} (f : A ⟶ B) (g : B ⟶ C) :
(f ≫ g).val = f.val ≫ g.val :=
rfl
variable (F)
/-- The canonical functor from pointed Galois objects to `C`. -/
def incl : PointedGaloisObject F ⥤ C where
obj := fun A ↦ A
map := fun ⟨f, _⟩ ↦ f
@[simp]
lemma incl_obj (A : PointedGaloisObject F) : (incl F).obj A = A :=
rfl
@[simp]
lemma incl_map {A B : PointedGaloisObject F} (f : A ⟶ B) : (incl F).map f = f.val :=
rfl
end General
section Specialized
variable (F : C ⥤ FintypeCat.{u₂})
/-- `F ⋙ FintypeCat.incl` as a cocone over `(can F).op ⋙ coyoneda`.
This is a colimit cocone (see `PreGaloisCategory.isColimìt`) -/
def cocone : Cocone ((incl F).op ⋙ coyoneda) where
pt := F ⋙ FintypeCat.incl
ι := {
app := fun ⟨A, a, _⟩ ↦ { app := fun X (f : (A : C) ⟶ X) ↦ F.map f a }
naturality := fun ⟨A, a, _⟩ ⟨B, b, _⟩ ⟨f, (hf : F.map f b = a)⟩ ↦ by
ext Y (g : (A : C) ⟶ Y)
suffices h : F.map g (F.map f b) = F.map g a by simpa
rw [hf]
}
@[simp]
lemma cocone_app (A : PointedGaloisObject F) (B : C) (f : (A : C) ⟶ B) :
((cocone F).ι.app ⟨A⟩).app B f = F.map f A.pt :=
rfl
variable [FiberFunctor F]
/-- The category of pointed Galois objects is cofiltered. -/
instance : IsCofilteredOrEmpty (PointedGaloisObject F) where
cone_objs := fun ⟨A, a, _⟩ ⟨B, b, _⟩ ↦ by
obtain ⟨Z, f, z, hgal, hfz⟩ := exists_hom_from_galois_of_fiber F (A ⨯ B)
<| (fiberBinaryProductEquiv F A B).symm (a, b)
refine ⟨⟨Z, z, hgal⟩, ⟨f ≫ prod.fst, ?_⟩, ⟨f ≫ prod.snd, ?_⟩, trivial⟩
· simp only [F.map_comp, hfz, FintypeCat.comp_apply, fiberBinaryProductEquiv_symm_fst_apply]
· simp only [F.map_comp, hfz, FintypeCat.comp_apply, fiberBinaryProductEquiv_symm_snd_apply]
cone_maps := fun ⟨A, a, _⟩ ⟨B, b, _⟩ ⟨f, hf⟩ ⟨g, hg⟩ ↦ by
obtain ⟨Z, h, z, hgal, hhz⟩ := exists_hom_from_galois_of_fiber F A a
refine ⟨⟨Z, z, hgal⟩, ⟨h, hhz⟩, hom_ext ?_⟩
apply evaluation_injective_of_isConnected F Z B z
simp [hhz, hf, hg]
/-- `cocone F` is a colimit cocone, i.e. `F` is pro-represented by `incl F`. -/
noncomputable def isColimit : IsColimit (cocone F) := by
refine evaluationJointlyReflectsColimits _ (fun X ↦ ?_)
refine Types.FilteredColimit.isColimitOf _ _ ?_ ?_
· intro (x : F.obj X)
obtain ⟨Y, i, y, h1, _, _⟩ := fiber_in_connected_component F X x
obtain ⟨Z, f, z, hgal, hfz⟩ := exists_hom_from_galois_of_fiber F Y y
refine ⟨⟨Z, z, hgal⟩, f ≫ i, ?_⟩
simp only [mapCocone_ι_app, evaluation_obj_map, cocone_app, map_comp,
← h1, FintypeCat.comp_apply, hfz]
· intro ⟨A, a, _⟩ ⟨B, b, _⟩ (u : (A : C) ⟶ X) (v : (B : C) ⟶ X) (h : F.map u a = F.map v b)
obtain ⟨⟨Z, z, _⟩, ⟨f, hf⟩, ⟨g, hg⟩, _⟩ :=
IsFilteredOrEmpty.cocone_objs (C := (PointedGaloisObject F)ᵒᵖ)
⟨{ obj := A, pt := a}⟩ ⟨{obj := B, pt := b}⟩
refine ⟨⟨{ obj := Z, pt := z }⟩, ⟨f, hf⟩, ⟨g, hg⟩, ?_⟩
apply evaluation_injective_of_isConnected F Z X z
change F.map (f ≫ u) z = F.map (g ≫ v) z
rw [map_comp, FintypeCat.comp_apply, hf, map_comp, FintypeCat.comp_apply, hg, h]
instance : HasColimit ((incl F).op ⋙ coyoneda) where
exists_colimit := ⟨cocone F, isColimit F⟩
end Specialized
end PointedGaloisObject
open PointedGaloisObject
section Specialized
variable (F : C ⥤ FintypeCat.{u₂})
/-- The diagram sending each pointed Galois object to its automorphism group
as an object of `C`. -/
@[simps]
noncomputable def autGaloisSystem : PointedGaloisObject F ⥤ Grp.{u₂} where
obj := fun A ↦ Grp.of <| Aut (A : C)
map := fun {A B} f ↦ Grp.ofHom (autMapHom f)
/-- The limit of `autGaloisSystem`. -/
noncomputable def AutGalois : Type (max u₁ u₂) :=
(autGaloisSystem F ⋙ forget _).sections
noncomputable instance : Group (AutGalois F) :=
inferInstanceAs <| Group (autGaloisSystem F ⋙ forget _).sections
/-- The canonical projection from `AutGalois F` to the `C`-automorphism group of each
pointed Galois object. -/
noncomputable def AutGalois.π (A : PointedGaloisObject F) : AutGalois F →* Aut (A : C) :=
Grp.sectionsπMonoidHom (autGaloisSystem F) A
/- Not a `simp` lemma, because we usually don't want to expose the internals here. -/
lemma AutGalois.π_apply (A : PointedGaloisObject F) (x : AutGalois F) :
AutGalois.π F A x = x.val A :=
rfl
lemma autGaloisSystem_map_surjective ⦃A B : PointedGaloisObject F⦄ (f : A ⟶ B) :
Function.Surjective ((autGaloisSystem F).map f) := by
intro (φ : Aut B.obj)
obtain ⟨ψ, hψ⟩ := autMap_surjective_of_isGalois f.val φ
use ψ
simp only [autGaloisSystem_map]
exact hψ
/-- Equality of elements of `AutGalois F` can be checked on the projections on each pointed
Galois object. -/
lemma AutGalois.ext {f g : AutGalois F}
(h : ∀ (A : PointedGaloisObject F), AutGalois.π F A f = AutGalois.π F A g) : f = g := by
dsimp only [AutGalois]
ext A
exact h A
variable [FiberFunctor F]
/-- `autGalois.π` is surjective for every pointed Galois object. -/
theorem AutGalois.π_surjective (A : PointedGaloisObject F) :
Function.Surjective (AutGalois.π F A) := fun (σ : Aut A.obj) ↦ by
have (i : PointedGaloisObject F) : Finite ((autGaloisSystem F ⋙ forget _).obj i) :=
inferInstanceAs <| Finite (Aut (i.obj))
exact eval_section_surjective_of_surjective
(autGaloisSystem F ⋙ forget _) (autGaloisSystem_map_surjective F) A σ
section EndAutGaloisIsomorphism
/-!
### Isomorphism between `Aut F` and `AutGalois F`
In this section we establish the isomorphism between the automorphism group of `F` and
the limit over the automorphism groups of all Galois objects.
We first establish the isomorphism between `End F` and `AutGalois F`, from which we deduce that
`End F` is a group, hence `End F = Aut F`. The isomorphism is built in multiple steps:
- `endEquivSectionsFibers : End F ≅ (incl F ⋙ F').sections`: the endomorphisms of
`F` are isomorphic to the limit over `F.obj A` for all Galois objects `A`.
This is obtained as the composition (slightly simplified):
`End F ≅ (colimit ((incl F).op ⋙ coyoneda) ⟶ F) ≅ (incl F ⋙ F).sections`
Where the first isomorphism is induced from the pro-representability of `F` and the second one
from the pro-coyoneda lemma.
- `endEquivAutGalois : End F ≅ AutGalois F`: this is the composition of `endEquivSectionsFibers`
with:
`(incl F ⋙ F).sections ≅ (autGaloisSystem F ⋙ forget Grp).sections`
which is induced from the level-wise equivalence `Aut A ≃ F.obj A` for a Galois object `A`.
-/
-- Local notation for `F` considered as a functor to types instead of finite types.
local notation "F'" => F ⋙ FintypeCat.incl
/-- The endomorphisms of `F` are isomorphic to the limit over the fibers of `F` on all
Galois objects. -/
noncomputable def endEquivSectionsFibers : End F ≃ (incl F ⋙ F').sections :=
let i1 : End F ≃ End F' :=
(FullyFaithful.whiskeringRight (FullyFaithful.ofFullyFaithful FintypeCat.incl) C).homEquiv
let i2 : End F' ≅ (colimit ((incl F).op ⋙ coyoneda) ⟶ F') :=
(yoneda.obj (F ⋙ FintypeCat.incl)).mapIso (colimit.isoColimitCocone ⟨cocone F, isColimit F⟩).op
let i3 : (colimit ((incl F).op ⋙ coyoneda) ⟶ F') ≅ limit ((incl F ⋙ F') ⋙ uliftFunctor.{u₁}) :=
colimitCoyonedaHomIsoLimit' (incl F) F'
let i4 : limit (incl F ⋙ F' ⋙ uliftFunctor.{u₁}) ≃ ((incl F ⋙ F') ⋙ uliftFunctor.{u₁}).sections :=
Types.limitEquivSections (incl F ⋙ (F ⋙ FintypeCat.incl) ⋙ uliftFunctor.{u₁, u₂})
let i5 : ((incl F ⋙ F') ⋙ uliftFunctor.{u₁}).sections ≃ (incl F ⋙ F').sections :=
(Types.sectionsEquiv (incl F ⋙ F')).symm
i1.trans <| i2.toEquiv.trans <| i3.toEquiv.trans <| i4.trans i5
@[simp]
lemma endEquivSectionsFibers_π (f : End F) (A : PointedGaloisObject F) :
(endEquivSectionsFibers F f).val A = f.app A A.pt := by
dsimp [endEquivSectionsFibers, Types.sectionsEquiv]
erw [Types.limitEquivSections_apply]
simp only [colimitCoyonedaHomIsoLimit'_π_apply, incl_obj, comp_obj, FintypeCat.incl_obj, op_obj,
FunctorToTypes.comp]
change (((FullyFaithful.whiskeringRight (FullyFaithful.ofFullyFaithful
FintypeCat.incl) C).homEquiv) f).app A
(((colimit.ι _ _) ≫ (colimit.isoColimitCocone ⟨cocone F, isColimit F⟩).hom).app
A _) = f.app A A.pt
simp
rfl
/-- Functorial isomorphism `Aut A ≅ F.obj A` for Galois objects `A`. -/
noncomputable def autIsoFibers :
autGaloisSystem F ⋙ forget Grp ≅ incl F ⋙ F' :=
NatIso.ofComponents (fun A ↦ ((evaluationEquivOfIsGalois F A A.pt).toIso))
(fun {A B} f ↦ by
ext (φ : Aut A.obj)
dsimp
erw [evaluationEquivOfIsGalois_apply, evaluationEquivOfIsGalois_apply]
simp [-Hom.comp, ← f.comp])
lemma autIsoFibers_inv_app (A : PointedGaloisObject F) (b : F.obj A) :
(autIsoFibers F).inv.app A b = (evaluationEquivOfIsGalois F A A.pt).symm b :=
rfl
/-- The equivalence between endomorphisms of `F` and the limit over the automorphism groups
of all Galois objects. -/
noncomputable def endEquivAutGalois : End F ≃ AutGalois F :=
let e1 := endEquivSectionsFibers F
let e2 := ((Functor.sectionsFunctor _).mapIso (autIsoFibers F).symm).toEquiv
e1.trans e2
lemma endEquivAutGalois_π (f : End F) (A : PointedGaloisObject F) :
F.map (AutGalois.π F A (endEquivAutGalois F f)).hom A.pt = f.app A A.pt := by
dsimp [endEquivAutGalois, AutGalois.π_apply]
change F.map ((((sectionsFunctor _).map (autIsoFibers F).inv) _).val A).hom A.pt = _
dsimp [autIsoFibers]
simp only [endEquivSectionsFibers_π]
erw [evaluationEquivOfIsGalois_symm_fiber]
@[simp]
theorem endEquivAutGalois_mul (f g : End F) :
(endEquivAutGalois F) (g ≫ f) = (endEquivAutGalois F g) * (endEquivAutGalois F f) := by
refine AutGalois.ext F (fun A ↦ evaluation_aut_injective_of_isConnected F A A.pt ?_)
simp only [map_mul, endEquivAutGalois_π, Aut.Aut_mul_def, NatTrans.comp_app, Iso.trans_hom]
simp only [map_comp, FintypeCat.comp_apply, endEquivAutGalois_π]
change f.app A (g.app A A.pt) =
(f.app A ≫ F.map ((AutGalois.π F A) ((endEquivAutGalois F) g)).hom) A.pt
rw [← f.naturality, FintypeCat.comp_apply, endEquivAutGalois_π]
/-- The monoid isomorphism between endomorphisms of `F` and the (multiplicative opposite of the)
limit of automorphism groups of all Galois objects. -/
noncomputable def endMulEquivAutGalois : End F ≃* (AutGalois F)ᵐᵒᵖ :=
MulEquiv.mk (Equiv.trans (endEquivAutGalois F) MulOpposite.opEquiv) (by simp)
lemma endMulEquivAutGalois_pi (f : End F) (A : PointedGaloisObject F) :
F.map (AutGalois.π F A (endMulEquivAutGalois F f).unop).hom A.2 = f.app A A.pt :=
endEquivAutGalois_π F f A
/-- Any endomorphism of a fiber functor is a unit. -/
theorem FibreFunctor.end_isUnit (f : End F) : IsUnit f :=
(isUnit_map_iff (endMulEquivAutGalois F) _).mp
(Group.isUnit ((endMulEquivAutGalois F) f))
/-- Any endomorphism of a fiber functor is an isomorphism. -/
instance FibreFunctor.end_isIso (f : End F) : IsIso f := by
rw [← isUnit_iff_isIso]
exact FibreFunctor.end_isUnit F f
/-- The automorphism group of `F` is multiplicatively isomorphic to
(the multiplicative opposite of) the limit over the automorphism groups of
the Galois objects. -/
noncomputable def autMulEquivAutGalois : Aut F ≃* (AutGalois F)ᵐᵒᵖ where
toFun := MonoidHom.comp (endMulEquivAutGalois F) (Aut.toEnd F)
invFun t := asIso ((endMulEquivAutGalois F).symm t)
left_inv t := by
simp only [MonoidHom.coe_comp, MonoidHom.coe_coe, Function.comp_apply,
MulEquiv.symm_apply_apply]
exact Aut.ext rfl
right_inv t := by
simp only [MonoidHom.coe_comp, MonoidHom.coe_coe, Function.comp_apply, Aut.toEnd_apply]
exact (MulEquiv.eq_symm_apply (endMulEquivAutGalois F)).mp rfl
map_mul' := by simp [map_mul]
lemma autMulEquivAutGalois_π (f : Aut F) (A : C) [IsGalois A] (a : F.obj A) :
F.map (AutGalois.π F { obj := A, pt := a } (autMulEquivAutGalois F f).unop).hom a =
f.hom.app A a := by
dsimp [autMulEquivAutGalois, endMulEquivAutGalois]
rw [endEquivAutGalois_π]
rfl
@[simp]
lemma autMulEquivAutGalois_symm_app (x : AutGalois F) (A : C) [IsGalois A] (a : F.obj A) :
((autMulEquivAutGalois F).symm ⟨x⟩).hom.app A a =
F.map (AutGalois.π F ⟨A, a, inferInstance⟩ x).hom a := by
rw [← autMulEquivAutGalois_π, MulEquiv.apply_symm_apply]
rfl
end EndAutGaloisIsomorphism
/-- The `Aut F` action on the fiber of a Galois object is transitive. See
`pretransitive_of_isConnected` for the same result for connected objects. -/
theorem FiberFunctor.isPretransitive_of_isGalois (X : C) [IsGalois X] :
MulAction.IsPretransitive (Aut F) (F.obj X) := by
refine ⟨fun x y ↦ ?_⟩
obtain ⟨(φ : Aut X), h⟩ := MulAction.IsPretransitive.exists_smul_eq (M := Aut X) x y
obtain ⟨a, ha⟩ := AutGalois.π_surjective F ⟨X, x, inferInstance⟩ φ
use (autMulEquivAutGalois F).symm ⟨a⟩
simpa [mulAction_def, ha]
/-- The `Aut F` action on the fiber of a connected object is transitive. For a version
with less restrictive universe assumptions, see `FiberFunctor.isPretransitive_of_isConnected`. -/
private instance FiberFunctor.isPretransitive_of_isConnected' (X : C) [IsConnected X] :
MulAction.IsPretransitive (Aut F) (F.obj X) := by
obtain ⟨A, f, hgal⟩ := exists_hom_from_galois_of_connected F X
have hs : Function.Surjective (F.map f) := surjective_of_nonempty_fiber_of_isConnected F f
refine ⟨fun x y ↦ ?_⟩
obtain ⟨a, ha⟩ := hs x
obtain ⟨b, hb⟩ := hs y
have : MulAction.IsPretransitive (Aut F) (F.obj A) := isPretransitive_of_isGalois F A
obtain ⟨σ, (hσ : σ.hom.app A a = b)⟩ := MulAction.exists_smul_eq (Aut F) a b
use σ
rw [← ha, ← hb]
change (F.map f ≫ σ.hom.app X) a = F.map f b
rw [σ.hom.naturality, FintypeCat.comp_apply, hσ]
end Specialized
section General
variable (F : C ⥤ FintypeCat.{w}) [FiberFunctor F]
/-- The `Aut F` action on the fiber of a connected object is transitive. -/
instance FiberFunctor.isPretransitive_of_isConnected (X : C) [IsConnected X] :
MulAction.IsPretransitive (Aut F) (F.obj X) where
exists_smul_eq x y := by
let F' : C ⥤ FintypeCat.{u₂} := F ⋙ FintypeCat.uSwitch.{w, u₂}
letI : FiberFunctor F' := FiberFunctor.comp_right _
let e (Y : C) : F'.obj Y ≃ F.obj Y := (F.obj Y).uSwitchEquiv
set x' : F'.obj X := (e X).symm x with hx'
set y' : F'.obj X := (e X).symm y with hy'
obtain ⟨g', (hg' : g'.hom.app X x' = y')⟩ := MulAction.exists_smul_eq (Aut F') x' y'
let gapp (Y : C) : F.obj Y ≅ F.obj Y := FintypeCat.equivEquivIso <|
(e Y).symm.trans <| (FintypeCat.equivEquivIso.symm (g'.app Y)).trans (e Y)
let g : F ≅ F := NatIso.ofComponents gapp <| fun {X Y} f ↦ by
ext x
simp only [FintypeCat.comp_apply, FintypeCat.equivEquivIso_apply_hom,
Equiv.trans_apply, FintypeCat.equivEquivIso_symm_apply_apply, Iso.app_hom, gapp, e]
erw [FintypeCat.uSwitchEquiv_naturality (F.map f)]
rw [← Functor.comp_map, ← FunctorToFintypeCat.naturality]
simp only [comp_obj, Functor.comp_map, F']
rw [FintypeCat.uSwitchEquiv_symm_naturality (F.map f)]
refine ⟨g, show (gapp X).hom x = y from ?_⟩
simp only [FintypeCat.equivEquivIso_apply_hom, Equiv.trans_apply,
FintypeCat.equivEquivIso_symm_apply_apply, Iso.app_hom, gapp]
rw [← hx', hg', hy', Equiv.apply_symm_apply]
end General
end PreGaloisCategory
end CategoryTheory
|
Identities.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, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
/-!
# Theory of univariate polynomials
The main def is `Polynomial.binomExpansion`.
-/
noncomputable section
namespace Polynomial
universe u v w x y z
variable {R : Type u} {S : Type v} {T : Type w} {ι : Type x} {k : Type y} {A : Type z} {a b : R}
{m n : ℕ}
section Identities
/- @TODO: `powAddExpansion` and `powSubPowFactor` are not specific to polynomials.
These belong somewhere else. But not in group_power because they depend on tactic.ring_exp
Maybe use `Data.Nat.Choose` to prove it.
-/
/-- `(x + y)^n` can be expressed as `x^n + n*x^(n-1)*y + k * y^2` for some `k` in the ring.
-/
def powAddExpansion {R : Type*} [CommSemiring R] (x y : R) :
∀ n : ℕ, { k // (x + y) ^ n = x ^ n + n * x ^ (n - 1) * y + k * y ^ 2 }
| 0 => ⟨0, by simp⟩
| 1 => ⟨0, by simp⟩
| n + 2 => by
obtain ⟨z, hz⟩ := (powAddExpansion x y (n + 1))
exists x * z + (n + 1) * x ^ n + z * y
calc
(x + y) ^ (n + 2) = (x + y) * (x + y) ^ (n + 1) := by ring
_ = (x + y) * (x ^ (n + 1) + ↑(n + 1) * x ^ (n + 1 - 1) * y + z * y ^ 2) := by rw [hz]
_ = x ^ (n + 2) + ↑(n + 2) * x ^ (n + 1) * y + (x * z + (n + 1) * x ^ n + z * y) * y ^ 2 := by
push_cast
ring!
variable [CommRing R]
private def polyBinomAux1 (x y : R) (e : ℕ) (a : R) :
{ k : R // a * (x + y) ^ e = a * (x ^ e + e * x ^ (e - 1) * y + k * y ^ 2) } := by
exists (powAddExpansion x y e).val
congr
apply (powAddExpansion _ _ _).property
private theorem poly_binom_aux2 (f : R[X]) (x y : R) :
f.eval (x + y) =
f.sum fun e a => a * (x ^ e + e * x ^ (e - 1) * y + (polyBinomAux1 x y e a).val * y ^ 2) := by
unfold eval; rw [eval₂_eq_sum]; congr with (n z)
apply (polyBinomAux1 x y _ _).property
private theorem poly_binom_aux3 (f : R[X]) (x y : R) :
f.eval (x + y) =
((f.sum fun e a => a * x ^ e) + f.sum fun e a => a * e * x ^ (e - 1) * y) +
f.sum fun e a => a * (polyBinomAux1 x y e a).val * y ^ 2 := by
rw [poly_binom_aux2]
simp [left_distrib, sum_add, mul_assoc]
/-- A polynomial `f` evaluated at `x + y` can be expressed as
the evaluation of `f` at `x`, plus `y` times the (polynomial) derivative of `f` at `x`,
plus some element `k : R` times `y^2`.
-/
def binomExpansion (f : R[X]) (x y : R) :
{ k : R // f.eval (x + y) = f.eval x + f.derivative.eval x * y + k * y ^ 2 } := by
exists f.sum fun e a => a * (polyBinomAux1 x y e a).val
rw [poly_binom_aux3]
congr
· rw [← eval_eq_sum]
· rw [derivative_eval]
exact (Finset.sum_mul ..).symm
· exact (Finset.sum_mul ..).symm
/-- `x^n - y^n` can be expressed as `z * (x - y)` for some `z` in the ring.
-/
def powSubPowFactor (x y : R) : ∀ i : ℕ, { z : R // x ^ i - y ^ i = z * (x - y) }
| 0 => ⟨0, by simp⟩
| 1 => ⟨1, by simp⟩
| k + 2 => by
obtain ⟨z, hz⟩ := @powSubPowFactor x y (k + 1)
exists z * x + y ^ (k + 1)
linear_combination (norm := ring) x * hz
/-- For any polynomial `f`, `f.eval x - f.eval y` can be expressed as `z * (x - y)`
for some `z` in the ring.
-/
def evalSubFactor (f : R[X]) (x y : R) : { z : R // f.eval x - f.eval y = z * (x - y) } := by
refine ⟨f.sum fun i r => r * (powSubPowFactor x y i).val, ?_⟩
delta eval; rw [eval₂_eq_sum, eval₂_eq_sum]
simp only [sum, ← Finset.sum_sub_distrib, Finset.sum_mul]
dsimp
congr with i
rw [mul_assoc, ← (powSubPowFactor x y _).prop, mul_sub]
end Identities
end Polynomial
|
FactorsThrough.lean
|
/-
Copyright (c) 2025 Etienne Marion. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Etienne Marion
-/
import Mathlib.MeasureTheory.Constructions.Polish.StronglyMeasurable
import Mathlib.Probability.Process.Filtration
/-!
# Factorization of a map from measurability
Consider `f : X → Y` and `g : X → Z` and assume that `g` is measurable with respect to the pullback
along `f`. Then `g` factors through `f`, which means that (if `Z` is nonempty)
there exists `h : Y → Z` such that `g = h ∘ f`.
If `Z` is completely metrizable, the factorization map `h` can be taken to be measurable.
This is the content of the [Doob-Dynkin lemma](https://en.wikipedia.org/wiki/Doob–Dynkin_lemma):
see `exists_eq_measurable_comp`.
-/
namespace MeasureTheory
open Filter Filtration Set TopologicalSpace
open scoped Topology
variable {X Y Z : Type*} [mY : MeasurableSpace Y] {f : X → Y} {g : X → Z}
section FactorsThrough
/-- If a function `g` is measurable with respect to the pullback along some function `f`,
then to prove `g x = g y` it is enough to prove `f x = f y`. -/
theorem _root_.Measurable.factorsThrough [MeasurableSpace Z] [MeasurableSingletonClass Z]
(hg : Measurable[mY.comap f] g) : g.FactorsThrough f := by
refine fun x₁ x₂ h ↦ eq_of_mem_singleton ?_
obtain ⟨s, -, hs⟩ := hg (measurableSet_singleton (g x₂))
rw [← mem_preimage, ← hs, mem_preimage, h, ← mem_preimage, hs, mem_preimage, mem_singleton_iff]
/-- If a function `g` is strongly measurable with respect to the pullback along some function `f`,
then to prove `g x = g y` it is enough to prove `f x = f y`.
If `Z` is not empty there exists `h : Y → Z` such that `g = h ∘ f`.
If `Z` is also completely metrizable, the factorization map `h` can be taken to be measurable
(see `exists_eq_measurable_comp`). -/
theorem StronglyMeasurable.factorsThrough [TopologicalSpace Z]
[PseudoMetrizableSpace Z] [T1Space Z] (hg : StronglyMeasurable[mY.comap f] g) :
g.FactorsThrough f := by
borelize Z
exact hg.measurable.factorsThrough
/-- If a function `g` is strongly measurable with respect to the pullback along some function `f`,
then there exists some strongly measurable function `h : Y → Z` such that `g = h ∘ f`. -/
theorem StronglyMeasurable.exists_eq_measurable_comp [Nonempty Z] [TopologicalSpace Z]
[IsCompletelyMetrizableSpace Z] (hg : StronglyMeasurable[mY.comap f] g) :
∃ h : Y → Z, StronglyMeasurable h ∧ g = h ∘ f := by
let mX : MeasurableSpace X := mY.comap f
induction g, hg using StronglyMeasurable.induction' with
| const z => exact ⟨fun _ ↦ z, stronglyMeasurable_const, rfl⟩
| @pcw g₁ g₂ s hg₁ hg₂ hs h₁ h₂ =>
obtain ⟨t, ht, rfl⟩ := hs
obtain ⟨h₁, mh₁, rfl⟩ := h₁
obtain ⟨h₂, mh₂, rfl⟩ := h₂
classical
exact ⟨t.piecewise h₁ h₂, mh₁.piecewise ht mh₂, by rw [piecewise_comp]⟩
| @lim g i hg hi h₁ h₂ =>
choose h mh hh using h₁
refine ⟨fun y ↦ _root_.limUnder atTop (h · y), StronglyMeasurable.limUnder mh, ?_⟩
ext x
rw [Function.comp_apply, Tendsto.limUnder_eq]
simp_all
/-- If a function `g` is measurable with respect to the pullback along some function `f`,
then there exists some measurable function `h : Y → Z` such that `g = h ∘ f`. -/
theorem _root_.Measurable.exists_eq_measurable_comp [Nonempty Z] [MeasurableSpace Z]
[StandardBorelSpace Z] (hg : Measurable[mY.comap f] g) :
∃ h : Y → Z, Measurable h ∧ g = h ∘ f := by
letI := upgradeStandardBorel Z
obtain ⟨h, mh, hh⟩ := hg.stronglyMeasurable.exists_eq_measurable_comp
exact ⟨h, mh.measurable, hh⟩
end FactorsThrough
variable {ι : Type*} {X : ι → Type*} [∀ i, MeasurableSpace (X i)] {f : (Π i, X i) → Z}
section piLE
variable [Preorder ι] {i : ι}
/-- If a function is measurable with respect to the σ-algebra generated by the
first coordinates, then it only depends on those first coordinates. -/
theorem _root_.Measurable.dependsOn_of_piLE [MeasurableSpace Z] [MeasurableSingletonClass Z]
(hf : Measurable[piLE i] f) : DependsOn f (Iic i) :=
dependsOn_iff_factorsThrough.2 hf.factorsThrough
/-- If a function is strongly measurable with respect to the σ-algebra generated by the
first coordinates, then it only depends on those first coordinates. -/
theorem StronglyMeasurable.dependsOn_of_piLE [TopologicalSpace Z] [PseudoMetrizableSpace Z]
[T1Space Z] (hf : StronglyMeasurable[piLE i] f) : DependsOn f (Iic i) :=
dependsOn_iff_factorsThrough.2 hf.factorsThrough
end piLE
section piFinset
variable {s : Finset ι}
/-- If a function is measurable with respect to the σ-algebra generated by the
first coordinates, then it only depends on those first coordinates. -/
theorem _root_.Measurable.dependsOn_of_piFinset [MeasurableSpace Z] [MeasurableSingletonClass Z]
(hf : Measurable[piFinset s] f) : DependsOn f s :=
dependsOn_iff_factorsThrough.2 hf.factorsThrough
/-- If a function is strongly measurable with respect to the σ-algebra generated by the
first coordinates, then it only depends on those first coordinates. -/
theorem StronglyMeasurable.dependsOn_of_piFinset [TopologicalSpace Z] [PseudoMetrizableSpace Z]
[T1Space Z] (hf : StronglyMeasurable[piFinset s] f) : DependsOn f s :=
dependsOn_iff_factorsThrough.2 hf.factorsThrough
end piFinset
end MeasureTheory
|
interval_inference.v
|
(* mathcomp analysis (c) 2017 Inria and AIST. License: CeCILL-C. *)
From HB Require Import structures.
From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice.
From mathcomp Require Import order ssralg ssrnum ssrint interval.
(**md**************************************************************************)
(* # Numbers within an interval *)
(* *)
(* This file develops tools to make the manipulation of numbers within *)
(* a known interval easier, thanks to canonical structures. This adds types *)
(* like {itv R & `[a, b]}, a notation e%:itv that infers an enclosing *)
(* interval for expression e according to existing canonical instances and *)
(* %:num to cast back from type {itv R & i} to R. *)
(* For instance, for x : {i01 R}, we have (1 - x%:num)%:itv : {i01 R} *)
(* automatically inferred. *)
(* *)
(* ## types for values within known interval *)
(* *)
(* ``` *)
(* {itv R & i} == generic type of values in interval i : interval int *)
(* See interval.v for notations that can be used for i. *)
(* R must have a numDomainType structure. This type is shown *)
(* to be a porderType. *)
(* {i01 R} := {itv R & `[0, 1]} *)
(* Allows to solve automatically goals of the form x >= 0 *)
(* and x <= 1 when x is canonically a {i01 R}. *)
(* {i01 R} is canonically stable by common operations. *)
(* {posnum R} := {itv R & `]0, +oo[) *)
(* {nonneg R} := {itv R & `[0, +oo[) *)
(* ``` *)
(* *)
(* ## casts from/to values within known interval *)
(* *)
(* Explicit casts of x to some {itv R & i} according to existing canonical *)
(* instances: *)
(* ``` *)
(* x%:itv == cast to the most precisely known {itv R & i} *)
(* x%:i01 == cast to {i01 R}, or fail *)
(* x%:pos == cast to {posnum R}, or fail *)
(* x%:nng == cast to {nonneg R}, or fail *)
(* ``` *)
(* *)
(* Explicit casts of x from some {itv R & i} to R: *)
(* ``` *)
(* x%:num == cast from {itv R & i} *)
(* x%:posnum == cast from {posnum R} *)
(* x%:nngnum == cast from {nonneg R} *)
(* ``` *)
(* *)
(* ## sign proofs *)
(* *)
(* ``` *)
(* [itv of x] == proof that x is in the interval inferred by x%:itv *)
(* [gt0 of x] == proof that x > 0 *)
(* [lt0 of x] == proof that x < 0 *)
(* [ge0 of x] == proof that x >= 0 *)
(* [le0 of x] == proof that x <= 0 *)
(* [cmp0 of x] == proof that 0 >=< x *)
(* [neq0 of x] == proof that x != 0 *)
(* ``` *)
(* *)
(* ## constructors *)
(* *)
(* ``` *)
(* ItvNum xr lx xu == builds a {itv R & i} from proofs xr : x \in Num.real, *)
(* lx : map_itv_bound (Itv.num_sem R) l <= BLeft x *)
(* xu : BRight x <= map_itv_bound (Itv.num_sem R) u *)
(* where x : R with R : numDomainType *)
(* and l u : itv_bound int *)
(* ItvReal lx xu == builds a {itv R & i} from proofs *)
(* lx : map_itv_bound (Itv.num_sem R) l <= BLeft x *)
(* xu : BRight x <= map_itv_bound (Itv.num_sem R) u *)
(* where x : R with R : realDomainType *)
(* and l u : itv_bound int *)
(* Itv01 x0 x1 == builds a {i01 R} from proofs x0 : 0 <= x and x1 : x <= 1*)
(* where x : R with R : numDomainType *)
(* PosNum x0 == builds a {posnum R} from a proof x0 : x > 0 where x : R *)
(* NngNum x0 == builds a {posnum R} from a proof x0 : x >= 0 where x : R*)
(* ``` *)
(* *)
(* A number of canonical instances are provided for common operations, if *)
(* your favorite operator is missing, look below for examples on how to add *)
(* the appropriate Canonical. *)
(* Also note that all provided instances aren't necessarily optimal, *)
(* improvements welcome! *)
(* Canonical instances are also provided according to types, as a *)
(* fallback when no known operator appears in the expression. Look to top_typ *)
(* below for an example on how to add your favorite type. *)
(* *)
(******************************************************************************)
Reserved Notation "{ 'itv' R & i }"
(R at level 200, i at level 200, format "{ 'itv' R & i }").
Reserved Notation "{ 'i01' R }"
(R at level 200, format "{ 'i01' R }").
Reserved Notation "{ 'posnum' R }" (format "{ 'posnum' R }").
Reserved Notation "{ 'nonneg' R }" (format "{ 'nonneg' R }").
Reserved Notation "x %:itv" (format "x %:itv").
Reserved Notation "x %:i01" (format "x %:i01").
Reserved Notation "x %:pos" (format "x %:pos").
Reserved Notation "x %:nng" (format "x %:nng").
Reserved Notation "x %:inum" (format "x %:inum").
Reserved Notation "x %:num" (format "x %:num").
Reserved Notation "x %:posnum" (format "x %:posnum").
Reserved Notation "x %:nngnum" (format "x %:nngnum").
Reserved Notation "[ 'itv' 'of' x ]" (format "[ 'itv' 'of' x ]").
Reserved Notation "[ 'gt0' 'of' x ]" (format "[ 'gt0' 'of' x ]").
Reserved Notation "[ 'lt0' 'of' x ]" (format "[ 'lt0' 'of' x ]").
Reserved Notation "[ 'ge0' 'of' x ]" (format "[ 'ge0' 'of' x ]").
Reserved Notation "[ 'le0' 'of' x ]" (format "[ 'le0' 'of' x ]").
Reserved Notation "[ 'cmp0' 'of' x ]" (format "[ 'cmp0' 'of' x ]").
Reserved Notation "[ 'neq0' 'of' x ]" (format "[ 'neq0' 'of' x ]").
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Import Order.TTheory Order.Syntax.
Import GRing.Theory Num.Theory.
Local Open Scope ring_scope.
Local Open Scope order_scope.
Definition map_itv_bound S T (f : S -> T) (b : itv_bound S) : itv_bound T :=
match b with
| BSide b x => BSide b (f x)
| BInfty b => BInfty _ b
end.
Lemma map_itv_bound_comp S T U (f : T -> S) (g : U -> T) (b : itv_bound U) :
map_itv_bound (f \o g) b = map_itv_bound f (map_itv_bound g b).
Proof. by case: b. Qed.
Definition map_itv S T (f : S -> T) (i : interval S) : interval T :=
let 'Interval l u := i in Interval (map_itv_bound f l) (map_itv_bound f u).
Lemma map_itv_comp S T U (f : T -> S) (g : U -> T) (i : interval U) :
map_itv (f \o g) i = map_itv f (map_itv g i).
Proof. by case: i => l u /=; rewrite -!map_itv_bound_comp. Qed.
(* First, the interval arithmetic operations we will later use *)
Module IntItv.
Implicit Types (b : itv_bound int) (i j : interval int).
Definition opp_bound b :=
match b with
| BSide b x => BSide (~~ b) (intZmod.oppz x)
| BInfty b => BInfty _ (~~ b)
end.
Lemma opp_bound_ge0 b : (BLeft 0%R <= opp_bound b)%O = (b <= BRight 0%R)%O.
Proof. by case: b => [[] b | []//]; rewrite /= !bnd_simp oppr_ge0. Qed.
Lemma opp_bound_gt0 b : (BRight 0%R <= opp_bound b)%O = (b <= BLeft 0%R)%O.
Proof.
by case: b => [[] b | []//]; rewrite /= !bnd_simp ?oppr_ge0 ?oppr_gt0.
Qed.
Definition opp i :=
let: Interval l u := i in Interval (opp_bound u) (opp_bound l).
Arguments opp /.
Definition add_boundl b1 b2 :=
match b1, b2 with
| BSide b1 x1, BSide b2 x2 => BSide (b1 && b2) (intZmod.addz x1 x2)
| _, _ => BInfty _ true
end.
Definition add_boundr b1 b2 :=
match b1, b2 with
| BSide b1 x1, BSide b2 x2 => BSide (b1 || b2) (intZmod.addz x1 x2)
| _, _ => BInfty _ false
end.
Definition add i1 i2 :=
let: Interval l1 u1 := i1 in let: Interval l2 u2 := i2 in
Interval (add_boundl l1 l2) (add_boundr u1 u2).
Arguments add /.
Variant signb := EqZero | NonNeg | NonPos.
Definition sign_boundl b :=
let: b0 := BLeft 0%Z in
if b == b0 then EqZero else if (b <= b0)%O then NonPos else NonNeg.
Definition sign_boundr b :=
let: b0 := BRight 0%Z in
if b == b0 then EqZero else if (b <= b0)%O then NonPos else NonNeg.
Variant signi := Known of signb | Unknown | Empty.
Definition sign i : signi :=
let: Interval l u := i in
match sign_boundl l, sign_boundr u with
| EqZero, NonPos
| NonNeg, EqZero
| NonNeg, NonPos => Empty
| EqZero, EqZero => Known EqZero
| NonPos, EqZero
| NonPos, NonPos => Known NonPos
| EqZero, NonNeg
| NonNeg, NonNeg => Known NonNeg
| NonPos, NonNeg => Unknown
end.
Definition mul_boundl b1 b2 :=
match b1, b2 with
| BInfty _, _
| _, BInfty _
| BLeft 0%Z, _
| _, BLeft 0%Z => BLeft 0%Z
| BSide b1 x1, BSide b2 x2 => BSide (b1 && b2) (intRing.mulz x1 x2)
end.
Definition mul_boundr b1 b2 :=
match b1, b2 with
| BLeft 0%Z, _
| _, BLeft 0%Z => BLeft 0%Z
| BRight 0%Z, _
| _, BRight 0%Z => BRight 0%Z
| BSide b1 x1, BSide b2 x2 => BSide (b1 || b2) (intRing.mulz x1 x2)
| _, BInfty _
| BInfty _, _ => +oo%O
end.
Lemma mul_boundrC b1 b2 : mul_boundr b1 b2 = mul_boundr b2 b1.
Proof.
by move: b1 b2 => [[] [[|?]|?] | []] [[] [[|?]|?] | []] //=; rewrite mulnC.
Qed.
Lemma mul_boundr_gt0 b1 b2 :
(BRight 0%Z <= b1 -> BRight 0%Z <= b2 -> BRight 0%Z <= mul_boundr b1 b2)%O.
Proof.
case: b1 b2 => [b1b b1 | []] [b2b b2 | []]//=.
- by case: b1b b2b => -[]; case: b1 b2 => [[|b1] | b1] [[|b2] | b2].
- by case: b1b b1 => -[[] |].
- by case: b2b b2 => -[[] |].
Qed.
Definition mul i1 i2 :=
let: Interval l1 u1 := i1 in let: Interval l2 u2 := i2 in
let: opp := opp_bound in
let: mull := mul_boundl in let: mulr := mul_boundr in
match sign i1, sign i2 with
| Empty, _ | _, Empty => `[1, 0]
| Known EqZero, _ | _, Known EqZero => `[0, 0]
| Known NonNeg, Known NonNeg =>
Interval (mull l1 l2) (mulr u1 u2)
| Known NonPos, Known NonPos =>
Interval (mull (opp u1) (opp u2)) (mulr (opp l1) (opp l2))
| Known NonNeg, Known NonPos =>
Interval (opp (mulr u1 (opp l2))) (opp (mull l1 (opp u2)))
| Known NonPos, Known NonNeg =>
Interval (opp (mulr (opp l1) u2)) (opp (mull (opp u1) l2))
| Known NonNeg, Unknown =>
Interval (opp (mulr u1 (opp l2))) (mulr u1 u2)
| Known NonPos, Unknown =>
Interval (opp (mulr (opp l1) u2)) (mulr (opp l1) (opp l2))
| Unknown, Known NonNeg =>
Interval (opp (mulr (opp l1) u2)) (mulr u1 u2)
| Unknown, Known NonPos =>
Interval (opp (mulr u1 (opp l2))) (mulr (opp l1) (opp l2))
| Unknown, Unknown =>
Interval
(Order.min (opp (mulr (opp l1) u2)) (opp (mulr u1 (opp l2))))
(Order.max (mulr (opp l1) (opp l2)) (mulr u1 u2))
end.
Arguments mul /.
Definition min i j :=
let: Interval li ui := i in let: Interval lj uj := j in
Interval (Order.min li lj) (Order.min ui uj).
Arguments min /.
Definition max i j :=
let: Interval li ui := i in let: Interval lj uj := j in
Interval (Order.max li lj) (Order.max ui uj).
Arguments max /.
Definition keep_nonneg_bound b :=
match b with
| BSide _ (Posz _) => BLeft 0%Z
| BSide _ (Negz _) => -oo%O
| BInfty _ => -oo%O
end.
Arguments keep_nonneg_bound /.
Definition keep_pos_bound b :=
match b with
| BSide b 0%Z => BSide b 0%Z
| BSide _ (Posz (S _)) => BRight 0%Z
| BSide _ (Negz _) => -oo
| BInfty _ => -oo
end.
Arguments keep_pos_bound /.
Definition keep_nonpos_bound b :=
match b with
| BSide _ (Negz _) | BSide _ (Posz 0) => BRight 0%Z
| BSide _ (Posz (S _)) => +oo%O
| BInfty _ => +oo%O
end.
Arguments keep_nonpos_bound /.
Definition keep_neg_bound b :=
match b with
| BSide b 0%Z => BSide b 0%Z
| BSide _ (Negz _) => BLeft 0%Z
| BSide _ (Posz _) => +oo
| BInfty _ => +oo
end.
Arguments keep_neg_bound /.
Definition inv i :=
let: Interval l u := i in
Interval (keep_pos_bound l) (keep_neg_bound u).
Arguments inv /.
Definition exprn_le1_bound b1 b2 :=
if b2 isn't BSide _ 1%Z then +oo
else if (BLeft (-1)%Z <= b1)%O then BRight 1%Z else +oo.
Arguments exprn_le1_bound /.
Definition exprn i :=
let: Interval l u := i in
Interval (keep_pos_bound l) (exprn_le1_bound l u).
Arguments exprn /.
Definition exprz i1 i2 :=
let: Interval l2 _ := i2 in
if l2 is BSide _ (Posz _) then exprn i1 else
let: Interval l u := i1 in
Interval (keep_pos_bound l) +oo.
Arguments exprz /.
Definition keep_sign i :=
let: Interval l u := i in
Interval (keep_nonneg_bound l) (keep_nonpos_bound u).
(* used in ereal.v *)
Definition keep_nonpos i :=
let 'Interval l u := i in
Interval -oo%O (keep_nonpos_bound u).
Arguments keep_nonpos /.
(* used in ereal.v *)
Definition keep_nonneg i :=
let 'Interval l u := i in
Interval (keep_nonneg_bound l) +oo%O.
Arguments keep_nonneg /.
End IntItv.
Module Itv.
Variant t := Top | Real of interval int.
Definition sub (x y : t) :=
match x, y with
| _, Top => true
| Top, Real _ => false
| Real xi, Real yi => subitv xi yi
end.
Section Itv.
Context T (sem : interval int -> T -> bool).
Definition spec (i : t) (x : T) := if i is Real i then sem i x else true.
Record def (i : t) := Def {
r : T;
#[canonical=no]
P : spec i r
}.
End Itv.
Record typ i := Typ {
sort : Type;
#[canonical=no]
sort_sem : interval int -> sort -> bool;
#[canonical=no]
allP : forall x : sort, spec sort_sem i x
}.
Definition mk {T f} i x P : @def T f i := @Def T f i x P.
Definition from {T f i} {x : @def T f i} (phx : phantom T (r x)) := x.
Definition fromP {T f i} {x : @def T f i} (phx : phantom T (r x)) := P x.
Definition num_sem (R : numDomainType) (i : interval int) (x : R) : bool :=
(x \in Num.real) && (x \in map_itv intr i).
Definition nat_sem (i : interval int) (x : nat) : bool := Posz x \in i.
Definition posnum (R : numDomainType) of phant R :=
def (@num_sem R) (Real `]0, +oo[).
Definition nonneg (R : numDomainType) of phant R :=
def (@num_sem R) (Real `[0, +oo[).
(* a few lifting helper functions *)
Definition real1 (op1 : interval int -> interval int) (x : Itv.t) : Itv.t :=
match x with Itv.Top => Itv.Top | Itv.Real x => Itv.Real (op1 x) end.
Definition real2 (op2 : interval int -> interval int -> interval int)
(x y : Itv.t) : Itv.t :=
match x, y with
| Itv.Top, _ | _, Itv.Top => Itv.Top
| Itv.Real x, Itv.Real y => Itv.Real (op2 x y)
end.
Lemma spec_real1 T f (op1 : T -> T) (op1i : interval int -> interval int) :
forall (x : T), (forall xi, f xi x = true -> f (op1i xi) (op1 x) = true) ->
forall xi, spec f xi x -> spec f (real1 op1i xi) (op1 x).
Proof. by move=> x + [//| xi]; apply. Qed.
Lemma spec_real2 T f (op2 : T -> T -> T)
(op2i : interval int -> interval int -> interval int) (x y : T) :
(forall xi yi, f xi x = true -> f yi y = true ->
f (op2i xi yi) (op2 x y) = true) ->
forall xi yi, spec f xi x -> spec f yi y ->
spec f (real2 op2i xi yi) (op2 x y).
Proof. by move=> + [//| xi] [//| yi]; apply. Qed.
Module Exports.
Arguments r {T sem i}.
Notation "{ 'itv' R & i }" := (def (@num_sem R) (Itv.Real i%Z)) : type_scope.
Notation "{ 'i01' R }" := {itv R & `[0, 1]} : type_scope.
Notation "{ 'posnum' R }" := (@posnum _ (Phant R)) : ring_scope.
Notation "{ 'nonneg' R }" := (@nonneg _ (Phant R)) : ring_scope.
Notation "x %:itv" := (from (Phantom _ x)) : ring_scope.
Notation "[ 'itv' 'of' x ]" := (fromP (Phantom _ x)) : ring_scope.
Notation num := r.
Notation "x %:inum" := (r x) (only parsing) : ring_scope.
Notation "x %:num" := (r x) : ring_scope.
Notation "x %:posnum" := (@r _ _ (Real `]0%Z, +oo[) x) : ring_scope.
Notation "x %:nngnum" := (@r _ _ (Real `[0%Z, +oo[) x) : ring_scope.
End Exports.
End Itv.
Export Itv.Exports.
Local Notation num_spec := (Itv.spec (@Itv.num_sem _)).
Local Notation num_def R := (Itv.def (@Itv.num_sem R)).
Local Notation num_itv_bound R := (@map_itv_bound _ R intr).
Local Notation nat_spec := (Itv.spec Itv.nat_sem).
Local Notation nat_def := (Itv.def Itv.nat_sem).
Section POrder.
Context d (T : porderType d) (f : interval int -> T -> bool) (i : Itv.t).
Local Notation itv := (Itv.def f i).
HB.instance Definition _ := [isSub for @Itv.r T f i].
HB.instance Definition _ : Order.POrder d itv := [POrder of itv by <:].
End POrder.
Section Order.
Variables (R : numDomainType) (i : interval int).
Local Notation nR := (num_def R (Itv.Real i)).
Lemma itv_le_total_subproof : total (<=%O : rel nR).
Proof.
move=> x y; apply: real_comparable.
- by case: x => [x /=/andP[]].
- by case: y => [y /=/andP[]].
Qed.
HB.instance Definition _ := Order.POrder_isTotal.Build ring_display nR
itv_le_total_subproof.
End Order.
Module TypInstances.
Lemma top_typ_spec T f (x : T) : Itv.spec f Itv.Top x.
Proof. by []. Qed.
Canonical top_typ T f := Itv.Typ (@top_typ_spec T f).
Lemma real_domain_typ_spec (R : realDomainType) (x : R) :
num_spec (Itv.Real `]-oo, +oo[) x.
Proof. by rewrite /Itv.num_sem/= num_real. Qed.
Canonical real_domain_typ (R : realDomainType) :=
Itv.Typ (@real_domain_typ_spec R).
Lemma real_field_typ_spec (R : realFieldType) (x : R) :
num_spec (Itv.Real `]-oo, +oo[) x.
Proof. exact: real_domain_typ_spec. Qed.
Canonical real_field_typ (R : realFieldType) :=
Itv.Typ (@real_field_typ_spec R).
Lemma nat_typ_spec (x : nat) : nat_spec (Itv.Real `[0, +oo[) x.
Proof. by []. Qed.
Canonical nat_typ := Itv.Typ nat_typ_spec.
Lemma typ_inum_spec (i : Itv.t) (xt : Itv.typ i) (x : Itv.sort xt) :
Itv.spec (@Itv.sort_sem _ xt) i x.
Proof. by move: xt x => []. Qed.
(* This adds _ <- Itv.r ( typ_inum )
to canonical projections (c.f., Print Canonical Projections
Itv.r) meaning that if no other canonical instance (with a
registered head symbol) is found, a canonical instance of
Itv.typ, like the ones above, will be looked for. *)
Canonical typ_inum (i : Itv.t) (xt : Itv.typ i) (x : Itv.sort xt) :=
Itv.mk (typ_inum_spec x).
End TypInstances.
Export (canonicals) TypInstances.
Class unify {T} f (x y : T) := Unify : f x y = true.
#[export] Hint Mode unify + + + + : typeclass_instances.
Class unify' {T} f (x y : T) := Unify' : f x y = true.
#[export] Instance unify'P {T} f (x y : T) : unify' f x y -> unify f x y := id.
#[export]
Hint Extern 0 (unify' _ _ _) => vm_compute; reflexivity : typeclass_instances.
Notation unify_itv ix iy := (unify Itv.sub ix iy).
#[export] Instance top_wider_anything i : unify_itv i Itv.Top.
Proof. by case: i. Qed.
#[export] Instance real_wider_anyreal i :
unify_itv (Itv.Real i) (Itv.Real `]-oo, +oo[).
Proof. by case: i => [l u]; apply/andP; rewrite !bnd_simp. Qed.
Section NumDomainTheory.
Context {R : numDomainType} {i : Itv.t}.
Implicit Type x : num_def R i.
Lemma le_num_itv_bound (x y : itv_bound int) :
(num_itv_bound R x <= num_itv_bound R y)%O = (x <= y)%O.
Proof.
by case: x y => [[] x | x] [[] y | y]//=; rewrite !bnd_simp ?ler_int ?ltr_int.
Qed.
Lemma num_itv_bound_le_BLeft (x : itv_bound int) (y : int) :
(num_itv_bound R x <= BLeft (y%:~R : R))%O = (x <= BLeft y)%O.
Proof.
rewrite -[BLeft y%:~R]/(map_itv_bound intr (BLeft y)).
by rewrite le_num_itv_bound.
Qed.
Lemma BRight_le_num_itv_bound (x : int) (y : itv_bound int) :
(BRight (x%:~R : R) <= num_itv_bound R y)%O = (BRight x <= y)%O.
Proof.
rewrite -[BRight x%:~R]/(map_itv_bound intr (BRight x)).
by rewrite le_num_itv_bound.
Qed.
Lemma num_spec_sub (x y : Itv.t) : Itv.sub x y ->
forall z : R, num_spec x z -> num_spec y z.
Proof.
case: x y => [| x] [| y] //= x_sub_y z /andP[rz]; rewrite /Itv.num_sem rz/=.
move: x y x_sub_y => [lx ux] [ly uy] /andP[lel leu] /=.
move=> /andP[lxz zux]; apply/andP; split.
- by apply: le_trans lxz; rewrite le_num_itv_bound.
- by apply: le_trans zux _; rewrite le_num_itv_bound.
Qed.
Definition empty_itv := Itv.Real `[1, 0]%Z.
Lemma bottom x : ~ unify_itv i empty_itv.
Proof.
case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=.
by rewrite in_itv/= => /andP[] /le_trans /[apply]; rewrite ler10.
Qed.
Lemma gt0 x : unify_itv i (Itv.Real `]0%Z, +oo[) -> 0 < x%:num :> R.
Proof.
case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_].
by rewrite /= in_itv/= andbT.
Qed.
Lemma le0F x : unify_itv i (Itv.Real `]0%Z, +oo[) -> x%:num <= 0 :> R = false.
Proof.
case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=.
by rewrite in_itv/= andbT => /lt_geF.
Qed.
Lemma lt0 x : unify_itv i (Itv.Real `]-oo, 0%Z[) -> x%:num < 0 :> R.
Proof.
by case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=; rewrite in_itv.
Qed.
Lemma ge0F x : unify_itv i (Itv.Real `]-oo, 0%Z[) -> 0 <= x%:num :> R = false.
Proof.
case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=.
by rewrite in_itv/= => /lt_geF.
Qed.
Lemma ge0 x : unify_itv i (Itv.Real `[0%Z, +oo[) -> 0 <= x%:num :> R.
Proof.
case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=.
by rewrite in_itv/= andbT.
Qed.
Lemma lt0F x : unify_itv i (Itv.Real `[0%Z, +oo[) -> x%:num < 0 :> R = false.
Proof.
case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=.
by rewrite in_itv/= andbT => /le_gtF.
Qed.
Lemma le0 x : unify_itv i (Itv.Real `]-oo, 0%Z]) -> x%:num <= 0 :> R.
Proof.
by case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=; rewrite in_itv.
Qed.
Lemma gt0F x : unify_itv i (Itv.Real `]-oo, 0%Z]) -> 0 < x%:num :> R = false.
Proof.
case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=.
by rewrite in_itv/= => /le_gtF.
Qed.
Lemma cmp0 x : unify_itv i (Itv.Real `]-oo, +oo[) -> 0 >=< x%:num.
Proof. by case: i x => [//| i' [x /=/andP[]]]. Qed.
Lemma neq0 x :
unify (fun ix iy => ~~ Itv.sub ix iy) (Itv.Real `[0%Z, 0%Z]) i ->
x%:num != 0 :> R.
Proof.
case: i x => [//| [l u] [x /= Px]]; apply: contra => /eqP x0 /=.
move: Px; rewrite x0 => /and3P[_ /= l0 u0]; apply/andP; split.
- by case: l l0 => [[] l /= |//]; rewrite !bnd_simp ?lerz0 ?ltrz0.
- by case: u u0 => [[] u /= |//]; rewrite !bnd_simp ?ler0z ?ltr0z.
Qed.
Lemma eq0F x :
unify (fun ix iy => ~~ Itv.sub ix iy) (Itv.Real `[0%Z, 0%Z]) i ->
x%:num == 0 :> R = false.
Proof. by move=> u; apply/negbTE/neq0. Qed.
Lemma lt1 x : unify_itv i (Itv.Real `]-oo, 1%Z[) -> x%:num < 1 :> R.
Proof.
by case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=; rewrite in_itv.
Qed.
Lemma ge1F x : unify_itv i (Itv.Real `]-oo, 1%Z[) -> 1 <= x%:num :> R = false.
Proof.
case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=.
by rewrite in_itv/= => /lt_geF.
Qed.
Lemma le1 x : unify_itv i (Itv.Real `]-oo, 1%Z]) -> x%:num <= 1 :> R.
Proof.
by case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=; rewrite in_itv.
Qed.
Lemma gt1F x : unify_itv i (Itv.Real `]-oo, 1%Z]) -> 1 < x%:num :> R = false.
Proof.
case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=.
by rewrite in_itv/= => /le_gtF.
Qed.
Lemma widen_itv_subproof x i' : Itv.sub i i' -> num_spec i' x%:num.
Proof. by case: x => x /= /[swap] /num_spec_sub; apply. Qed.
Definition widen_itv x i' (uni : unify_itv i i') :=
Itv.mk (widen_itv_subproof x uni).
Lemma widen_itvE x (uni : unify_itv i i) : @widen_itv x i uni = x.
Proof. exact/val_inj. Qed.
Lemma posE x (uni : unify_itv i (Itv.Real `]0%Z, +oo[)) :
(widen_itv x%:num%:itv uni)%:num = x%:num.
Proof. by []. Qed.
Lemma nngE x (uni : unify_itv i (Itv.Real `[0%Z, +oo[)) :
(widen_itv x%:num%:itv uni)%:num = x%:num.
Proof. by []. Qed.
End NumDomainTheory.
Arguments bottom {R i} _ {_}.
Arguments gt0 {R i} _ {_}.
Arguments le0F {R i} _ {_}.
Arguments lt0 {R i} _ {_}.
Arguments ge0F {R i} _ {_}.
Arguments ge0 {R i} _ {_}.
Arguments lt0F {R i} _ {_}.
Arguments le0 {R i} _ {_}.
Arguments gt0F {R i} _ {_}.
Arguments cmp0 {R i} _ {_}.
Arguments neq0 {R i} _ {_}.
Arguments eq0F {R i} _ {_}.
Arguments lt1 {R i} _ {_}.
Arguments ge1F {R i} _ {_}.
Arguments le1 {R i} _ {_}.
Arguments gt1F {R i} _ {_}.
Arguments widen_itv {R i} _ {_ _}.
Arguments widen_itvE {R i} _ {_}.
Arguments posE {R i} _ {_}.
Arguments nngE {R i} _ {_}.
Notation "[ 'gt0' 'of' x ]" := (ltac:(refine (gt0 x%:itv))) (only parsing).
Notation "[ 'lt0' 'of' x ]" := (ltac:(refine (lt0 x%:itv))) (only parsing).
Notation "[ 'ge0' 'of' x ]" := (ltac:(refine (ge0 x%:itv))) (only parsing).
Notation "[ 'le0' 'of' x ]" := (ltac:(refine (le0 x%:itv))) (only parsing).
Notation "[ 'cmp0' 'of' x ]" := (ltac:(refine (cmp0 x%:itv))) (only parsing).
Notation "[ 'neq0' 'of' x ]" := (ltac:(refine (neq0 x%:itv))) (only parsing).
#[export] Hint Extern 0 (is_true (0%R < _)%R) => solve [apply: gt0] : core.
#[export] Hint Extern 0 (is_true (_ < 0%R)%R) => solve [apply: lt0] : core.
#[export] Hint Extern 0 (is_true (0%R <= _)%R) => solve [apply: ge0] : core.
#[export] Hint Extern 0 (is_true (_ <= 0%R)%R) => solve [apply: le0] : core.
#[export] Hint Extern 0 (is_true (_ \is Num.real)) => solve [apply: cmp0]
: core.
#[export] Hint Extern 0 (is_true (0%R >=< _)%R) => solve [apply: cmp0] : core.
#[export] Hint Extern 0 (is_true (_ != 0%R)) => solve [apply: neq0] : core.
#[export] Hint Extern 0 (is_true (_ < 1%R)%R) => solve [apply: lt1] : core.
#[export] Hint Extern 0 (is_true (_ <= 1%R)%R) => solve [apply: le1] : core.
Notation "x %:i01" := (widen_itv x%:itv : {i01 _}) (only parsing) : ring_scope.
Notation "x %:i01" := (@widen_itv _ _
(@Itv.from _ _ _ (Phantom _ x)) (Itv.Real `[0, 1]%Z) _)
(only printing) : ring_scope.
Notation "x %:pos" := (widen_itv x%:itv : {posnum _}) (only parsing)
: ring_scope.
Notation "x %:pos" := (@widen_itv _ _
(@Itv.from _ _ _ (Phantom _ x)) (Itv.Real `]0%Z, +oo[) _)
(only printing) : ring_scope.
Notation "x %:nng" := (widen_itv x%:itv : {nonneg _}) (only parsing)
: ring_scope.
Notation "x %:nng" := (@widen_itv _ _
(@Itv.from _ _ _ (Phantom _ x)) (Itv.Real `[0%Z, +oo[) _)
(only printing) : ring_scope.
Local Open Scope ring_scope.
Module Instances.
Import IntItv.
Section NumDomainInstances.
Context {R : numDomainType}.
Lemma num_spec_zero : num_spec (Itv.Real `[0, 0]) (0 : R).
Proof. by apply/andP; split; [exact: real0 | rewrite /= in_itv/= lexx]. Qed.
Canonical zero_inum := Itv.mk num_spec_zero.
Lemma num_spec_one : num_spec (Itv.Real `[1, 1]) (1 : R).
Proof. by apply/andP; split; [exact: real1 | rewrite /= in_itv/= lexx]. Qed.
Canonical one_inum := Itv.mk num_spec_one.
Lemma opp_boundr (x : R) b :
(BRight (- x)%R <= num_itv_bound R (opp_bound b))%O
= (num_itv_bound R b <= BLeft x)%O.
Proof.
by case: b => [[] b | []//]; rewrite /= !bnd_simp mulrNz ?lerN2 // ltrN2.
Qed.
Lemma opp_boundl (x : R) b :
(num_itv_bound R (opp_bound b) <= BLeft (- x)%R)%O
= (BRight x <= num_itv_bound R b)%O.
Proof.
by case: b => [[] b | []//]; rewrite /= !bnd_simp mulrNz ?lerN2 // ltrN2.
Qed.
Lemma num_spec_opp (i : Itv.t) (x : num_def R i) (r := Itv.real1 opp i) :
num_spec r (- x%:num).
Proof.
apply: Itv.spec_real1 (Itv.P x).
case: x => x /= _ [l u] /and3P[xr lx xu].
rewrite /Itv.num_sem/= realN xr/=; apply/andP.
by rewrite opp_boundl opp_boundr.
Qed.
Canonical opp_inum (i : Itv.t) (x : num_def R i) := Itv.mk (num_spec_opp x).
Lemma num_itv_add_boundl (x1 x2 : R) b1 b2 :
(num_itv_bound R b1 <= BLeft x1)%O -> (num_itv_bound R b2 <= BLeft x2)%O ->
(num_itv_bound R (add_boundl b1 b2) <= BLeft (x1 + x2)%R)%O.
Proof.
case: b1 b2 => [bb1 b1 |//] [bb2 b2 |//].
case: bb1; case: bb2; rewrite /= !bnd_simp mulrzDr.
- exact: lerD.
- exact: ler_ltD.
- exact: ltr_leD.
- exact: ltrD.
Qed.
Lemma num_itv_add_boundr (x1 x2 : R) b1 b2 :
(BRight x1 <= num_itv_bound R b1)%O -> (BRight x2 <= num_itv_bound R b2)%O ->
(BRight (x1 + x2)%R <= num_itv_bound R (add_boundr b1 b2))%O.
Proof.
case: b1 b2 => [bb1 b1 |//] [bb2 b2 |//].
case: bb1; case: bb2; rewrite /= !bnd_simp mulrzDr.
- exact: ltrD.
- exact: ltr_leD.
- exact: ler_ltD.
- exact: lerD.
Qed.
Lemma num_spec_add (xi yi : Itv.t) (x : num_def R xi) (y : num_def R yi)
(r := Itv.real2 add xi yi) :
num_spec r (x%:num + y%:num).
Proof.
apply: Itv.spec_real2 (Itv.P x) (Itv.P y).
case: x y => [x /= _] [y /= _] => {xi yi r} -[lx ux] [ly uy]/=.
move=> /andP[xr /=/andP[lxx xux]] /andP[yr /=/andP[lyy yuy]].
rewrite /Itv.num_sem realD//=; apply/andP.
by rewrite num_itv_add_boundl ?num_itv_add_boundr.
Qed.
Canonical add_inum (xi yi : Itv.t) (x : num_def R xi) (y : num_def R yi) :=
Itv.mk (num_spec_add x y).
Variant sign_spec (l u : itv_bound int) (x : R) : signi -> Set :=
| ISignEqZero : l = BLeft 0 -> u = BRight 0 -> x = 0 ->
sign_spec l u x (Known EqZero)
| ISignNonNeg : (BLeft 0%:Z <= l)%O -> (BRight 0%:Z < u)%O -> 0 <= x ->
sign_spec l u x (Known NonNeg)
| ISignNonPos : (l < BLeft 0%:Z)%O -> (u <= BRight 0%:Z)%O -> x <= 0 ->
sign_spec l u x (Known NonPos)
| ISignBoth : (l < BLeft 0%:Z)%O -> (BRight 0%:Z < u)%O -> x \in Num.real ->
sign_spec l u x Unknown.
Lemma signP (l u : itv_bound int) (x : R) :
(num_itv_bound R l <= BLeft x)%O -> (BRight x <= num_itv_bound R u)%O ->
x \in Num.real ->
sign_spec l u x (sign (Interval l u)).
Proof.
move=> + + xr; rewrite /sign/sign_boundl/sign_boundr.
have [lneg|lpos|->] := ltgtP l; have [uneg|upos|->] := ltgtP u => lx xu.
- apply: ISignNonPos => //; first exact: ltW.
have:= le_trans xu (eqbRL (le_num_itv_bound _ _) (ltW uneg)).
by rewrite bnd_simp.
- exact: ISignBoth.
- exact: ISignNonPos.
- have:= @ltxx _ _ (num_itv_bound R l).
rewrite (le_lt_trans lx) -?leBRight_ltBLeft ?(le_trans xu)//.
by rewrite le_num_itv_bound (le_trans (ltW uneg)).
- apply: ISignNonNeg => //; first exact: ltW.
have:= le_trans (eqbRL (le_num_itv_bound _ _) (ltW lpos)) lx.
by rewrite bnd_simp.
- have:= @ltxx _ _ (num_itv_bound R l).
rewrite (le_lt_trans lx) -?leBRight_ltBLeft ?(le_trans xu)//.
by rewrite le_num_itv_bound ?leBRight_ltBLeft.
- have:= @ltxx _ _ (num_itv_bound R (BLeft 0%Z)).
rewrite (le_lt_trans lx) -?leBRight_ltBLeft ?(le_trans xu)//.
by rewrite le_num_itv_bound -?ltBRight_leBLeft.
- exact: ISignNonNeg.
- apply: ISignEqZero => //.
by apply/le_anti/andP; move: lx xu; rewrite !bnd_simp.
Qed.
Lemma num_itv_mul_boundl b1 b2 (x1 x2 : R) :
(BLeft 0%:Z <= b1 -> BLeft 0%:Z <= b2 ->
num_itv_bound R b1 <= BLeft x1 ->
num_itv_bound R b2 <= BLeft x2 ->
num_itv_bound R (mul_boundl b1 b2) <= BLeft (x1 * x2))%O.
Proof.
move: b1 b2 => [[] b1 | []//] [[] b2 | []//] /=; rewrite 4!bnd_simp.
- set bl := match b1 with 0%Z => _ | _ => _ end.
have -> : bl = BLeft (b1 * b2).
rewrite {}/bl; move: b1 b2 => [[|p1]|p1] [[|p2]|p2]; congr BLeft.
by rewrite mulr0.
by rewrite bnd_simp intrM -2!(ler0z R); apply: ler_pM.
- case: b1 => [[|b1] | b1]; rewrite !bnd_simp// => b1p b2p sx1 sx2.
+ by rewrite mulr_ge0 ?(le_trans _ (ltW sx2)) ?ler0z.
+ rewrite intrM (@lt_le_trans _ _ (b1.+1%:~R * x2)) ?ltr_pM2l//.
by rewrite ler_pM2r// (le_lt_trans _ sx2) ?ler0z.
- case: b2 => [[|b2] | b2]; rewrite !bnd_simp// => b1p b2p sx1 sx2.
+ by rewrite mulr_ge0 ?(le_trans _ (ltW sx1)) ?ler0z.
+ rewrite intrM (@le_lt_trans _ _ (b1%:~R * x2)) ?ler_wpM2l ?ler0z//.
by rewrite ltr_pM2r ?(lt_le_trans _ sx2).
- by rewrite -2!(ler0z R) bnd_simp intrM; apply: ltr_pM.
Qed.
Lemma num_itv_mul_boundr b1 b2 (x1 x2 : R) :
(0 <= x1 -> 0 <= x2 ->
BRight x1 <= num_itv_bound R b1 ->
BRight x2 <= num_itv_bound R b2 ->
BRight (x1 * x2) <= num_itv_bound R (mul_boundr b1 b2))%O.
Proof.
case: b1 b2 => [b1b b1 | []] [b2b b2 | []] //= x1p x2p; last first.
- case: b2b b2 => -[[|//] | //] _ x20.
+ have:= @ltxx _ (itv_bound R) (BLeft 0%:~R).
by rewrite (lt_le_trans _ x20).
+ have -> : x2 = 0 by apply/le_anti/andP.
by rewrite mulr0.
- case: b1b b1 => -[[|//] |//] x10 _.
+ have:= @ltxx _ (itv_bound R) (BLeft 0%Z%:~R).
by rewrite (lt_le_trans _ x10).
+ by have -> : x1 = 0; [apply/le_anti/andP | rewrite mul0r].
case: b1b b2b => -[]; rewrite -[intRing.mulz]/GRing.mul.
- case: b1 => [[|b1] | b1]; rewrite !bnd_simp => x1b x2b.
+ by have:= @ltxx _ R 0; rewrite (le_lt_trans x1p x1b).
+ case: b2 x2b => [[| b2] | b2] => x2b; rewrite bnd_simp.
* by have:= @ltxx _ R 0; rewrite (le_lt_trans x2p x2b).
* by rewrite intrM ltr_pM.
* have:= @ltxx _ R 0; rewrite (le_lt_trans x2p)//.
by rewrite (lt_le_trans x2b) ?lerz0.
+ have:= @ltxx _ R 0; rewrite (le_lt_trans x1p)//.
by rewrite (lt_le_trans x1b) ?lerz0.
- case: b1 => [[|b1] | b1]; rewrite !bnd_simp => x1b x2b.
+ by have:= @ltxx _ R 0; rewrite (le_lt_trans x1p x1b).
+ case: b2 x2b => [[| b2] | b2] x2b; rewrite bnd_simp.
* exact: mulr_ge0_le0.
* by rewrite intrM (le_lt_trans (ler_wpM2l x1p x2b)) ?ltr_pM2r.
* have:= @ltxx _ _ x2.
by rewrite (le_lt_trans x2b) ?(lt_le_trans _ x2p) ?ltrz0.
+ have:= @ltxx _ _ x1.
by rewrite (lt_le_trans x1b) ?(le_trans _ x1p) ?lerz0.
- case: b1 => [[|b1] | b1]; rewrite !bnd_simp => x1b x2b.
+ case: b2 x2b => [[|b2] | b2] x2b; rewrite bnd_simp.
* by have:= @ltxx _ _ x2; rewrite (lt_le_trans x2b).
* by have -> : x1 = 0; [apply/le_anti/andP | rewrite mul0r].
* have:= @ltxx _ _ x2.
by rewrite (lt_le_trans x2b) ?(le_trans _ x2p) ?lerz0.
+ case: b2 x2b => [[|b2] | b2] x2b; rewrite bnd_simp.
* by have:= @ltxx _ _ x2; rewrite (lt_le_trans x2b).
* by rewrite intrM (le_lt_trans (ler_wpM2r x2p x1b)) ?ltr_pM2l.
* have:= @ltxx _ _ x2.
by rewrite (lt_le_trans x2b) ?(le_trans _ x2p) ?lerz0.
+ have:= @ltxx _ _ x1.
by rewrite (le_lt_trans x1b) ?(lt_le_trans _ x1p) ?ltrz0.
- case: b1 => [[|b1] | b1]; rewrite !bnd_simp => x1b x2b.
+ by have -> : x1 = 0; [apply/le_anti/andP | rewrite mul0r].
+ case: b2 x2b => [[| b2] | b2] x2b; rewrite bnd_simp.
* by have -> : x2 = 0; [apply/le_anti/andP | rewrite mulr0].
* by rewrite intrM ler_pM.
* have:= @ltxx _ _ x2.
by rewrite (le_lt_trans x2b) ?(lt_le_trans _ x2p) ?ltrz0.
+ have:= @ltxx _ _ x1.
by rewrite (le_lt_trans x1b) ?(lt_le_trans _ x1p) ?ltrz0.
Qed.
Lemma BRight_le_mul_boundr b1 b2 (x1 x2 : R) :
(0 <= x1 -> x2 \in Num.real -> BRight 0%Z <= b2 ->
BRight x1 <= num_itv_bound R b1 ->
BRight x2 <= num_itv_bound R b2 ->
BRight (x1 * x2) <= num_itv_bound R (mul_boundr b1 b2))%O.
Proof.
move=> x1ge0 x2r b2ge0 lex1b1 lex2b2.
have /orP[x2ge0 | x2le0] := x2r; first exact: num_itv_mul_boundr.
have lem0 : (BRight (x1 * x2) <= BRight 0%R)%O.
by rewrite bnd_simp mulr_ge0_le0 // ltW.
apply: le_trans lem0 _.
rewrite -(mulr0z 1) BRight_le_num_itv_bound.
apply: mul_boundr_gt0 => //.
by rewrite -(@BRight_le_num_itv_bound R) (le_trans _ lex1b1).
Qed.
Lemma comparable_num_itv_bound (x y : itv_bound int) :
(num_itv_bound R x >=< num_itv_bound R y)%O.
Proof.
by case: x y => [[] x | []] [[] y | []]//; apply/orP;
rewrite !bnd_simp ?ler_int ?ltr_int;
case: leP => xy; apply/orP => //; rewrite ltW ?orbT.
Qed.
Lemma num_itv_bound_min (x y : itv_bound int) :
num_itv_bound R (Order.min x y)
= Order.min (num_itv_bound R x) (num_itv_bound R y).
Proof.
have [lexy | ltyx] := leP x y; [by rewrite !minEle le_num_itv_bound lexy|].
rewrite minElt -if_neg -comparable_leNgt ?le_num_itv_bound ?ltW//.
exact: comparable_num_itv_bound.
Qed.
Lemma num_itv_bound_max (x y : itv_bound int) :
num_itv_bound R (Order.max x y)
= Order.max (num_itv_bound R x) (num_itv_bound R y).
Proof.
have [lexy | ltyx] := leP x y; [by rewrite !maxEle le_num_itv_bound lexy|].
rewrite maxElt -if_neg -comparable_leNgt ?le_num_itv_bound ?ltW//.
exact: comparable_num_itv_bound.
Qed.
Lemma num_spec_mul (xi yi : Itv.t) (x : num_def R xi) (y : num_def R yi)
(r := Itv.real2 mul xi yi) :
num_spec r (x%:num * y%:num).
Proof.
rewrite {}/r; case: xi yi x y => [//| [xl xu]] [//| [yl yu]].
case=> [x /=/and3P[xr /= xlx xxu]] [y /=/and3P[yr /= yly yyu]].
rewrite -/(sign (Interval xl xu)) -/(sign (Interval yl yu)).
have ns000 : @Itv.num_sem R `[0, 0] 0 by apply/and3P.
have xyr : x * y \in Num.real by exact: realM.
case: (signP xlx xxu xr) => xlb xub xs.
- by rewrite xs mul0r; case: (signP yly yyu yr).
- case: (signP yly yyu yr) => ylb yub ys.
+ by rewrite ys mulr0.
+ apply/and3P; split=> //=.
* exact: num_itv_mul_boundl xlx yly.
* exact: num_itv_mul_boundr xxu yyu.
+ apply/and3P; split=> //=; rewrite -[x * y]opprK -mulrN.
* by rewrite opp_boundl num_itv_mul_boundr ?oppr_ge0// opp_boundr.
* by rewrite opp_boundr num_itv_mul_boundl ?opp_boundl// opp_bound_ge0.
+ apply/and3P; split=> //=.
* rewrite -[x * y]opprK -mulrN opp_boundl.
by rewrite BRight_le_mul_boundr ?realN ?opp_boundr// opp_bound_gt0 ltW.
* by rewrite BRight_le_mul_boundr// ltW.
- case: (signP yly yyu yr) => ylb yub ys.
+ by rewrite ys mulr0.
+ apply/and3P; split=> //=; rewrite -[x * y]opprK -mulNr.
* rewrite opp_boundl.
by rewrite num_itv_mul_boundr ?oppr_ge0 ?opp_boundr.
* by rewrite opp_boundr num_itv_mul_boundl ?opp_boundl// opp_bound_ge0.
+ apply/and3P; split=> //=; rewrite -mulrNN.
* by rewrite num_itv_mul_boundl ?opp_bound_ge0 ?opp_boundl.
* by rewrite num_itv_mul_boundr ?oppr_ge0 ?opp_boundr.
+ apply/and3P; split=> //=; rewrite -[x * y]opprK.
* rewrite -mulNr opp_boundl BRight_le_mul_boundr ?oppr_ge0 ?opp_boundr//.
exact: ltW.
* rewrite opprK -mulrNN.
by rewrite BRight_le_mul_boundr ?opp_boundr
?oppr_ge0 ?realN ?opp_bound_gt0// ltW.
- case: (signP yly yyu yr) => ylb yub ys.
+ by rewrite ys mulr0.
+ apply/and3P; split=> //=; rewrite mulrC mul_boundrC.
* rewrite -[y * x]opprK -mulrN opp_boundl.
rewrite BRight_le_mul_boundr ?oppr_ge0 ?realN ?opp_boundr//.
by rewrite opp_bound_gt0 ltW.
* by rewrite BRight_le_mul_boundr// ltW.
+ apply/and3P; split=> //=; rewrite mulrC mul_boundrC.
* rewrite -[y * x]opprK -mulNr opp_boundl.
by rewrite BRight_le_mul_boundr ?oppr_ge0 ?opp_boundr// ltW.
* rewrite -mulrNN BRight_le_mul_boundr ?oppr_ge0 ?realN ?opp_boundr//.
by rewrite opp_bound_gt0 ltW.
apply/and3P; rewrite xyr/= num_itv_bound_min num_itv_bound_max.
rewrite (comparable_ge_min _ (comparable_num_itv_bound _ _)).
rewrite (comparable_le_max _ (comparable_num_itv_bound _ _)).
case: (comparable_leP xr) => [x0 | /ltW x0]; split=> //.
- apply/orP; right; rewrite -[x * y]opprK -mulrN opp_boundl.
by rewrite BRight_le_mul_boundr ?realN ?opp_boundr// opp_bound_gt0 ltW.
- by apply/orP; right; rewrite BRight_le_mul_boundr// ltW.
- apply/orP; left; rewrite -[x * y]opprK -mulNr opp_boundl.
by rewrite BRight_le_mul_boundr ?oppr_ge0 ?opp_boundr// ltW.
- apply/orP; left; rewrite -mulrNN.
rewrite BRight_le_mul_boundr ?oppr_ge0 ?realN ?opp_boundr//.
by rewrite opp_bound_gt0 ltW.
Qed.
Canonical mul_inum (xi yi : Itv.t) (x : num_def R xi) (y : num_def R yi) :=
Itv.mk (num_spec_mul x y).
Lemma num_spec_min (xi yi : Itv.t) (x : num_def R xi) (y : num_def R yi)
(r := Itv.real2 min xi yi) :
num_spec r (Order.min x%:num y%:num).
Proof.
apply: Itv.spec_real2 (Itv.P x) (Itv.P y).
case: x y => [x /= _] [y /= _] => {xi yi r} -[lx ux] [ly uy]/=.
move=> /andP[xr /=/andP[lxx xux]] /andP[yr /=/andP[lyy yuy]].
apply/and3P; split; rewrite ?min_real//= num_itv_bound_min real_BSide_min//.
- apply: (comparable_le_min2 (comparable_num_itv_bound _ _)) => //.
exact: real_comparable.
- apply: (comparable_le_min2 _ (comparable_num_itv_bound _ _)) => //.
exact: real_comparable.
Qed.
Lemma num_spec_max (xi yi : Itv.t) (x : num_def R xi) (y : num_def R yi)
(r := Itv.real2 max xi yi) :
num_spec r (Order.max x%:num y%:num).
Proof.
apply: Itv.spec_real2 (Itv.P x) (Itv.P y).
case: x y => [x /= _] [y /= _] => {xi yi r} -[lx ux] [ly uy]/=.
move=> /andP[xr /=/andP[lxx xux]] /andP[yr /=/andP[lyy yuy]].
apply/and3P; split; rewrite ?max_real//= num_itv_bound_max real_BSide_max//.
- apply: (comparable_le_max2 (comparable_num_itv_bound _ _)) => //.
exact: real_comparable.
- apply: (comparable_le_max2 _ (comparable_num_itv_bound _ _)) => //.
exact: real_comparable.
Qed.
(* We can't directly put an instance on Order.min for R : numDomainType
since we may want instances for other porderType
(typically \bar R or even nat). So we resort on this additional
canonical structure. *)
Record min_max_typ d := MinMaxTyp {
min_max_sort : porderType d;
#[canonical=no]
min_max_sem : interval int -> min_max_sort -> bool;
#[canonical=no]
min_max_minP : forall (xi yi : Itv.t) (x : Itv.def min_max_sem xi)
(y : Itv.def min_max_sem yi),
let: r := Itv.real2 min xi yi in
Itv.spec min_max_sem r (Order.min x%:num y%:num);
#[canonical=no]
min_max_maxP : forall (xi yi : Itv.t) (x : Itv.def min_max_sem xi)
(y : Itv.def min_max_sem yi),
let: r := Itv.real2 max xi yi in
Itv.spec min_max_sem r (Order.max x%:num y%:num);
}.
(* The default instances on porderType, for min... *)
Canonical min_typ_inum d (t : min_max_typ d) (xi yi : Itv.t)
(x : Itv.def (@min_max_sem d t) xi) (y : Itv.def (@min_max_sem d t) yi)
(r := Itv.real2 min xi yi) :=
Itv.mk (min_max_minP x y).
(* ...and for max *)
Canonical max_typ_inum d (t : min_max_typ d) (xi yi : Itv.t)
(x : Itv.def (@min_max_sem d t) xi) (y : Itv.def (@min_max_sem d t) yi)
(r := Itv.real2 min xi yi) :=
Itv.mk (min_max_maxP x y).
(* Instance of the above structure for numDomainType *)
Canonical num_min_max_typ := MinMaxTyp num_spec_min num_spec_max.
Lemma nat_num_spec (i : Itv.t) (n : nat) : nat_spec i n = num_spec i (n%:R : R).
Proof.
case: i => [//| [l u]]; rewrite /= /Itv.num_sem realn/=; congr (_ && _).
- by case: l => [[] l |//]; rewrite !bnd_simp ?pmulrn ?ler_int ?ltr_int.
- by case: u => [[] u |//]; rewrite !bnd_simp ?pmulrn ?ler_int ?ltr_int.
Qed.
Definition natmul_itv (i1 i2 : Itv.t) : Itv.t :=
match i1, i2 with
| Itv.Top, _ => Itv.Top
| _, Itv.Top => Itv.Real `]-oo, +oo[
| Itv.Real i1, Itv.Real i2 => Itv.Real (mul i1 i2)
end.
Arguments natmul_itv /.
Lemma num_spec_natmul (xi ni : Itv.t) (x : num_def R xi) (n : nat_def ni)
(r := natmul_itv xi ni) :
num_spec r (x%:num *+ n%:num).
Proof.
rewrite {}/r; case: xi x ni n => [//| xi] x [| ni] n.
by apply/and3P; case: n%:num => [|?]; rewrite ?mulr0n ?realrMn.
have Pn : num_spec (Itv.Real ni) (n%:num%:R : R).
by case: n => /= n; rewrite [Itv.nat_sem ni n](nat_num_spec (Itv.Real ni)).
rewrite -mulr_natr -[n%:num%:R]/((Itv.Def Pn)%:num).
by rewrite (@num_spec_mul (Itv.Real xi) (Itv.Real ni)).
Qed.
Canonical natmul_inum (xi ni : Itv.t) (x : num_def R xi) (n : nat_def ni) :=
Itv.mk (num_spec_natmul x n).
Lemma num_spec_int (i : Itv.t) (n : int) :
num_spec i n = num_spec i (n%:~R : R).
Proof.
case: i => [//| [l u]]; rewrite /= /Itv.num_sem num_real realz/=.
congr (andb _ _).
- by case: l => [[] l |//]; rewrite !bnd_simp intz ?ler_int ?ltr_int.
- by case: u => [[] u |//]; rewrite !bnd_simp intz ?ler_int ?ltr_int.
Qed.
Lemma num_spec_intmul (xi ii : Itv.t) (x : num_def R xi) (i : num_def int ii)
(r := natmul_itv xi ii) :
num_spec r (x%:num *~ i%:num).
Proof.
rewrite {}/r; case: xi x ii i => [//| xi] x [| ii] i.
by apply/and3P; case: i%:inum => [[|n] | n]; rewrite ?mulr0z ?realN ?realrMn.
have Pi : num_spec (Itv.Real ii) (i%:num%:~R : R).
by case: i => /= i; rewrite [Itv.num_sem ii i](num_spec_int (Itv.Real ii)).
rewrite -mulrzr -[i%:num%:~R]/((Itv.Def Pi)%:num).
by rewrite (@num_spec_mul (Itv.Real xi) (Itv.Real ii)).
Qed.
Canonical intmul_inum (xi ni : Itv.t) (x : num_def R xi) (n : num_def int ni) :=
Itv.mk (num_spec_intmul x n).
Lemma num_itv_bound_keep_pos (op : R -> R) (x : R) b :
{homo op : x / 0 <= x} -> {homo op : x / 0 < x} ->
(num_itv_bound R b <= BLeft x)%O ->
(num_itv_bound R (keep_pos_bound b) <= BLeft (op x))%O.
Proof.
case: b => [[] [] [| b] // | []//] hle hlt; rewrite !bnd_simp.
- exact: hle.
- by move=> blex; apply: le_lt_trans (hlt _ _) => //; apply: lt_le_trans blex.
- exact: hlt.
- by move=> bltx; apply: le_lt_trans (hlt _ _) => //; apply: le_lt_trans bltx.
Qed.
Lemma num_itv_bound_keep_neg (op : R -> R) (x : R) b :
{homo op : x / x <= 0} -> {homo op : x / x < 0} ->
(BRight x <= num_itv_bound R b)%O ->
(BRight (op x) <= num_itv_bound R (keep_neg_bound b))%O.
Proof.
case: b => [[] [[|//] | b] | []//] hge hgt; rewrite !bnd_simp.
- exact: hgt.
- by move=> xltb; apply: hgt; apply: lt_le_trans xltb _; rewrite lerz0.
- exact: hge.
- by move=> xleb; apply: hgt; apply: le_lt_trans xleb _; rewrite ltrz0.
Qed.
Lemma num_spec_inv (i : Itv.t) (x : num_def R i) (r := Itv.real1 inv i) :
num_spec r (x%:num^-1).
Proof.
apply: Itv.spec_real1 (Itv.P x).
case: x => x /= _ [l u] /and3P[xr /= lx xu].
rewrite /Itv.num_sem/= realV xr/=; apply/andP; split.
- apply: num_itv_bound_keep_pos lx.
+ by move=> ?; rewrite invr_ge0.
+ by move=> ?; rewrite invr_gt0.
- apply: num_itv_bound_keep_neg xu.
+ by move=> ?; rewrite invr_le0.
+ by move=> ?; rewrite invr_lt0.
Qed.
Canonical inv_inum (i : Itv.t) (x : num_def R i) := Itv.mk (num_spec_inv x).
Lemma num_itv_bound_exprn_le1 (x : R) n l u :
(num_itv_bound R l <= BLeft x)%O ->
(BRight x <= num_itv_bound R u)%O ->
(BRight (x ^+ n) <= num_itv_bound R (exprn_le1_bound l u))%O.
Proof.
case: u => [bu [[//|[|//]] |//] | []//].
rewrite /exprn_le1_bound; case: (leP _ l) => [lge1 /= |//] lx xu.
rewrite bnd_simp; case: n => [| n]; rewrite ?expr0//.
have xN1 : -1 <= x.
case: l lge1 lx => [[] l | []//]; rewrite !bnd_simp -(@ler_int R).
- exact: le_trans.
- by move=> + /ltW; apply: le_trans.
have x1 : x <= 1 by case: bu xu; rewrite bnd_simp// => /ltW.
have xr : x \is Num.real by exact: ler1_real.
case: (real_ge0P xr) => x0; first by rewrite expr_le1.
rewrite -[x]opprK exprNn; apply: le_trans (ler_piMl _ _) _.
- by rewrite exprn_ge0 ?oppr_ge0 1?ltW.
- suff: -1 <= (-1) ^+ n.+1 :> R /\ (-1) ^+ n.+1 <= 1 :> R => [[]//|].
elim: n => [|n [IHn1 IHn2]]; rewrite ?expr1// ![_ ^+ n.+2]exprS !mulN1r.
by rewrite lerNl opprK lerNl.
- by rewrite expr_le1 ?oppr_ge0 1?lerNl// ltW.
Qed.
Lemma num_spec_exprn (i : Itv.t) (x : num_def R i) n (r := Itv.real1 exprn i) :
num_spec r (x%:num ^+ n).
Proof.
apply: (@Itv.spec_real1 _ _ (fun x => x^+n) _ _ _ _ (Itv.P x)).
case: x => x /= _ [l u] /and3P[xr /= lx xu].
rewrite /Itv.num_sem realX//=; apply/andP; split.
- apply: (@num_itv_bound_keep_pos (fun x => x^+n)) lx.
+ exact: exprn_ge0.
+ exact: exprn_gt0.
- exact: num_itv_bound_exprn_le1 lx xu.
Qed.
Canonical exprn_inum (i : Itv.t) (x : num_def R i) n :=
Itv.mk (num_spec_exprn x n).
Lemma num_spec_exprz (xi ki : Itv.t) (x : num_def R xi) (k : num_def int ki)
(r := Itv.real2 exprz xi ki) :
num_spec r (x%:num ^ k%:num).
Proof.
rewrite {}/r; case: ki k => [|[lk uk]] k; first by case: xi x.
case: xi x => [//|xi x]; rewrite /Itv.real2.
have P : Itv.num_sem
(let 'Interval l _ := xi in Interval (keep_pos_bound l) +oo)
(x%:num ^ k%:num).
case: xi x => lx ux x; apply/and3P; split=> [||//].
have xr : x%:num \is Num.real by case: x => x /=/andP[].
by case: k%:num => n; rewrite ?realV realX.
apply: (@num_itv_bound_keep_pos (fun x => x ^ k%:num));
[exact: exprz_ge0 | exact: exprz_gt0 |].
by case: x => x /=/and3P[].
case: lk k P => [slk [lk | lk] | slk] k P; [|exact: P..].
case: k P => -[k | k] /= => [_ _|]; rewrite -/(exprn xi); last first.
by move=> /and3P[_ /=]; case: slk; rewrite bnd_simp -pmulrn natz.
exact: (@num_spec_exprn (Itv.Real xi)).
Qed.
Canonical exprz_inum (xi ki : Itv.t) (x : num_def R xi) (k : num_def int ki) :=
Itv.mk (num_spec_exprz x k).
Lemma num_spec_norm {V : normedZmodType R} (x : V) :
num_spec (Itv.Real `[0, +oo[) `|x|.
Proof. by apply/and3P; split; rewrite //= ?normr_real ?bnd_simp ?normr_ge0. Qed.
Canonical norm_inum {V : normedZmodType R} (x : V) := Itv.mk (num_spec_norm x).
End NumDomainInstances.
Section RcfInstances.
Context {R : rcfType}.
Definition sqrt_itv (i : Itv.t) : Itv.t :=
match i with
| Itv.Top => Itv.Real `[0%Z, +oo[
| Itv.Real (Interval l u) =>
match l with
| BSide b 0%Z => Itv.Real (Interval (BSide b 0%Z) +oo)
| BSide b (Posz (S _)) => Itv.Real `]0%Z, +oo[
| _ => Itv.Real `[0, +oo[
end
end.
Arguments sqrt_itv /.
Lemma num_spec_sqrt (i : Itv.t) (x : num_def R i) (r := sqrt_itv i) :
num_spec r (Num.sqrt x%:num).
Proof.
have: Itv.num_sem `[0%Z, +oo[ (Num.sqrt x%:num).
by apply/and3P; rewrite /= num_real !bnd_simp sqrtr_ge0.
rewrite {}/r; case: i x => [//| [[bl [l |//] |//] u]] [x /= +] _.
case: bl l => -[| l] /and3P[xr /= bx _]; apply/and3P; split=> //=;
move: bx; rewrite !bnd_simp ?sqrtr_ge0// sqrtr_gt0;
[exact: lt_le_trans | exact: le_lt_trans..].
Qed.
Canonical sqrt_inum (i : Itv.t) (x : num_def R i) := Itv.mk (num_spec_sqrt x).
End RcfInstances.
Section NumClosedFieldInstances.
Context {R : numClosedFieldType}.
Definition sqrtC_itv (i : Itv.t) : Itv.t :=
match i with
| Itv.Top => Itv.Top
| Itv.Real (Interval l u) =>
match l with
| BSide b (Posz _) => Itv.Real (Interval (BSide b 0%Z) +oo)
| _ => Itv.Top
end
end.
Arguments sqrtC_itv /.
Lemma num_spec_sqrtC (i : Itv.t) (x : num_def R i) (r := sqrtC_itv i) :
num_spec r (sqrtC x%:num).
Proof.
rewrite {}/r; case: i x => [//| [l u] [x /=/and3P[xr /= lx xu]]].
case: l lx => [bl [l |//] |[]//] lx; apply/and3P; split=> //=.
by apply: sqrtC_real; case: bl lx => /[!bnd_simp] [|/ltW]; apply: le_trans.
case: bl lx => /[!bnd_simp] lx.
- by rewrite sqrtC_ge0; apply: le_trans lx.
- by rewrite sqrtC_gt0; apply: le_lt_trans lx.
Qed.
Canonical sqrtC_inum (i : Itv.t) (x : num_def R i) := Itv.mk (num_spec_sqrtC x).
End NumClosedFieldInstances.
Section NatInstances.
Local Open Scope nat_scope.
Implicit Type (n : nat).
Lemma nat_spec_zero : nat_spec (Itv.Real `[0, 0]%Z) 0. Proof. by []. Qed.
Canonical zeron_inum := Itv.mk nat_spec_zero.
Lemma nat_spec_add (xi yi : Itv.t) (x : nat_def xi) (y : nat_def yi)
(r := Itv.real2 add xi yi) :
nat_spec r (x%:num + y%:num).
Proof.
have Px : num_spec xi (x%:num%:R : int).
by case: x => /= x; rewrite (@nat_num_spec int).
have Py : num_spec yi (y%:num%:R : int).
by case: y => /= y; rewrite (@nat_num_spec int).
rewrite (@nat_num_spec int) natrD.
rewrite -[x%:num%:R]/((Itv.Def Px)%:num) -[y%:num%:R]/((Itv.Def Py)%:num).
exact: num_spec_add.
Qed.
Canonical addn_inum (xi yi : Itv.t) (x : nat_def xi) (y : nat_def yi) :=
Itv.mk (nat_spec_add x y).
Lemma nat_spec_succ (i : Itv.t) (n : nat_def i)
(r := Itv.real2 add i (Itv.Real `[1, 1]%Z)) :
nat_spec r (S n%:num).
Proof.
pose i1 := Itv.Real `[1, 1]%Z; have P1 : nat_spec i1 1 by [].
by rewrite -addn1 -[1%N]/((Itv.Def P1)%:num); apply: nat_spec_add.
Qed.
Canonical succn_inum (i : Itv.t) (n : nat_def i) := Itv.mk (nat_spec_succ n).
Lemma nat_spec_double (i : Itv.t) (n : nat_def i) (r := Itv.real2 add i i) :
nat_spec r (n%:num.*2).
Proof. by rewrite -addnn nat_spec_add. Qed.
Canonical double_inum (i : Itv.t) (x : nat_def i) := Itv.mk (nat_spec_double x).
Lemma nat_spec_mul (xi yi : Itv.t) (x : nat_def xi) (y : nat_def yi)
(r := Itv.real2 mul xi yi) :
nat_spec r (x%:num * y%:num).
Proof.
have Px : num_spec xi (x%:num%:R : int).
by case: x => /= x; rewrite (@nat_num_spec int).
have Py : num_spec yi (y%:num%:R : int).
by case: y => /= y; rewrite (@nat_num_spec int).
rewrite (@nat_num_spec int) natrM.
rewrite -[x%:num%:R]/((Itv.Def Px)%:num) -[y%:num%:R]/((Itv.Def Py)%:num).
exact: num_spec_mul.
Qed.
Canonical muln_inum (xi yi : Itv.t) (x : nat_def xi) (y : nat_def yi) :=
Itv.mk (nat_spec_mul x y).
Lemma nat_spec_exp (i : Itv.t) (x : nat_def i) n (r := Itv.real1 exprn i) :
nat_spec r (x%:num ^ n).
Proof.
have Px : num_spec i (x%:num%:R : int).
by case: x => /= x; rewrite (@nat_num_spec int).
rewrite (@nat_num_spec int) natrX -[x%:num%:R]/((Itv.Def Px)%:num).
exact: num_spec_exprn.
Qed.
Canonical expn_inum (i : Itv.t) (x : nat_def i) n := Itv.mk (nat_spec_exp x n).
Lemma nat_spec_min (xi yi : Itv.t) (x : nat_def xi) (y : nat_def yi)
(r := Itv.real2 min xi yi) :
nat_spec r (minn x%:num y%:num).
Proof.
have Px : num_spec xi (x%:num%:R : int).
by case: x => /= x; rewrite (@nat_num_spec int).
have Py : num_spec yi (y%:num%:R : int).
by case: y => /= y; rewrite (@nat_num_spec int).
rewrite (@nat_num_spec int) -minEnat natr_min.
rewrite -[x%:num%:R]/((Itv.Def Px)%:num) -[y%:num%:R]/((Itv.Def Py)%:num).
exact: num_spec_min.
Qed.
Canonical minn_inum (xi yi : Itv.t) (x : nat_def xi) (y : nat_def yi) :=
Itv.mk (nat_spec_min x y).
Lemma nat_spec_max (xi yi : Itv.t) (x : nat_def xi) (y : nat_def yi)
(r := Itv.real2 max xi yi) :
nat_spec r (maxn x%:num y%:num).
Proof.
have Px : num_spec xi (x%:num%:R : int).
by case: x => /= x; rewrite (@nat_num_spec int).
have Py : num_spec yi (y%:num%:R : int).
by case: y => /= y; rewrite (@nat_num_spec int).
rewrite (@nat_num_spec int) -maxEnat natr_max.
rewrite -[x%:num%:R]/((Itv.Def Px)%:num) -[y%:num%:R]/((Itv.Def Py)%:num).
exact: num_spec_max.
Qed.
Canonical maxn_inum (xi yi : Itv.t) (x : nat_def xi) (y : nat_def yi) :=
Itv.mk (nat_spec_max x y).
Canonical nat_min_max_typ := MinMaxTyp nat_spec_min nat_spec_max.
Lemma nat_spec_factorial (n : nat) : nat_spec (Itv.Real `[1%Z, +oo[) n`!.
Proof. by apply/andP; rewrite bnd_simp lez_nat fact_gt0. Qed.
Canonical factorial_inum n := Itv.mk (nat_spec_factorial n).
End NatInstances.
Section IntInstances.
Lemma num_spec_Posz n : num_spec (Itv.Real `[0, +oo[) (Posz n).
Proof. by apply/and3P; rewrite /= num_real !bnd_simp. Qed.
Canonical Posz_inum n := Itv.mk (num_spec_Posz n).
Lemma num_spec_Negz n : num_spec (Itv.Real `]-oo, (-1)]) (Negz n).
Proof. by apply/and3P; rewrite /= num_real !bnd_simp. Qed.
Canonical Negz_inum n := Itv.mk (num_spec_Negz n).
End IntInstances.
End Instances.
Export (canonicals) Instances.
Section Morph.
Context {R : numDomainType} {i : Itv.t}.
Local Notation nR := (num_def R i).
Implicit Types x y : nR.
Local Notation num := (@num R (@Itv.num_sem R) i).
Lemma num_eq : {mono num : x y / x == y}. Proof. by []. Qed.
Lemma num_le : {mono num : x y / (x <= y)%O}. Proof. by []. Qed.
Lemma num_lt : {mono num : x y / (x < y)%O}. Proof. by []. Qed.
Lemma num_min : {morph num : x y / Order.min x y}.
Proof. by move=> x y; rewrite !minEle num_le -fun_if. Qed.
Lemma num_max : {morph num : x y / Order.max x y}.
Proof. by move=> x y; rewrite !maxEle num_le -fun_if. Qed.
End Morph.
Section MorphNum.
Context {R : numDomainType}.
Lemma num_abs_eq0 (a : R) : (`|a|%:nng == 0%:nng) = (a == 0).
Proof. by rewrite -normr_eq0. Qed.
End MorphNum.
Section MorphReal.
Context {R : numDomainType} {i : interval int}.
Local Notation nR := (num_def R (Itv.Real i)).
Implicit Type x y : nR.
Local Notation num := (@num R (@Itv.num_sem R) i).
Lemma num_le_max a x y :
a <= Num.max x%:num y%:num = (a <= x%:num) || (a <= y%:num).
Proof. by rewrite -comparable_le_max// real_comparable. Qed.
Lemma num_ge_max a x y :
Num.max x%:num y%:num <= a = (x%:num <= a) && (y%:num <= a).
Proof. by rewrite -comparable_ge_max// real_comparable. Qed.
Lemma num_le_min a x y :
a <= Num.min x%:num y%:num = (a <= x%:num) && (a <= y%:num).
Proof. by rewrite -comparable_le_min// real_comparable. Qed.
Lemma num_ge_min a x y :
Num.min x%:num y%:num <= a = (x%:num <= a) || (y%:num <= a).
Proof. by rewrite -comparable_ge_min// real_comparable. Qed.
Lemma num_lt_max a x y :
a < Num.max x%:num y%:num = (a < x%:num) || (a < y%:num).
Proof. by rewrite -comparable_lt_max// real_comparable. Qed.
Lemma num_gt_max a x y :
Num.max x%:num y%:num < a = (x%:num < a) && (y%:num < a).
Proof. by rewrite -comparable_gt_max// real_comparable. Qed.
Lemma num_lt_min a x y :
a < Num.min x%:num y%:num = (a < x%:num) && (a < y%:num).
Proof. by rewrite -comparable_lt_min// real_comparable. Qed.
Lemma num_gt_min a x y :
Num.min x%:num y%:num < a = (x%:num < a) || (y%:num < a).
Proof. by rewrite -comparable_gt_min// real_comparable. Qed.
End MorphReal.
Section MorphGe0.
Context {R : numDomainType}.
Local Notation nR := (num_def R (Itv.Real `[0%Z, +oo[)).
Implicit Type x y : nR.
Local Notation num := (@num R (@Itv.num_sem R) (Itv.Real `[0%Z, +oo[)).
Lemma num_abs_le a x : 0 <= a -> (`|a|%:nng <= x) = (a <= x%:num).
Proof. by move=> a0; rewrite -num_le//= ger0_norm. Qed.
Lemma num_abs_lt a x : 0 <= a -> (`|a|%:nng < x) = (a < x%:num).
Proof. by move=> a0; rewrite -num_lt/= ger0_norm. Qed.
End MorphGe0.
Section ItvNum.
Context (R : numDomainType).
Context (x : R) (l u : itv_bound int).
Context (x_real : x \in Num.real).
Context (l_le_x : (num_itv_bound R l <= BLeft x)%O).
Context (x_le_u : (BRight x <= num_itv_bound R u)%O).
Lemma itvnum_subdef : num_spec (Itv.Real (Interval l u)) x.
Proof. by apply/and3P. Qed.
Definition ItvNum : num_def R (Itv.Real (Interval l u)) := Itv.mk itvnum_subdef.
End ItvNum.
Section ItvReal.
Context (R : realDomainType).
Context (x : R) (l u : itv_bound int).
Context (l_le_x : (num_itv_bound R l <= BLeft x)%O).
Context (x_le_u : (BRight x <= num_itv_bound R u)%O).
Lemma itvreal_subdef : num_spec (Itv.Real (Interval l u)) x.
Proof. by apply/and3P; split; first exact: num_real. Qed.
Definition ItvReal : num_def R (Itv.Real (Interval l u)) :=
Itv.mk itvreal_subdef.
End ItvReal.
Section Itv01.
Context (R : numDomainType).
Context (x : R) (x_ge0 : 0 <= x) (x_le1 : x <= 1).
Lemma itv01_subdef : num_spec (Itv.Real `[0%Z, 1%Z]) x.
Proof. by apply/and3P; split; rewrite ?bnd_simp// ger0_real. Qed.
Definition Itv01 : num_def R (Itv.Real `[0%Z, 1%Z]) := Itv.mk itv01_subdef.
End Itv01.
Section Posnum.
Context (R : numDomainType) (x : R) (x_gt0 : 0 < x).
Lemma posnum_subdef : num_spec (Itv.Real `]0, +oo[) x.
Proof. by apply/and3P; rewrite /= gtr0_real. Qed.
Definition PosNum : {posnum R} := Itv.mk posnum_subdef.
End Posnum.
Section Nngnum.
Context (R : numDomainType) (x : R) (x_ge0 : 0 <= x).
Lemma nngnum_subdef : num_spec (Itv.Real `[0, +oo[) x.
Proof. by apply/and3P; rewrite /= ger0_real. Qed.
Definition NngNum : {nonneg R} := Itv.mk nngnum_subdef.
End Nngnum.
Variant posnum_spec (R : numDomainType) (x : R) :
R -> bool -> bool -> bool -> Type :=
| IsPosnum (p : {posnum R}) : posnum_spec x (p%:num) false true true.
Lemma posnumP (R : numDomainType) (x : R) : 0 < x ->
posnum_spec x x (x == 0) (0 <= x) (0 < x).
Proof.
move=> x_gt0; case: real_ltgt0P (x_gt0) => []; rewrite ?gtr0_real // => _ _.
by rewrite -[x]/(PosNum x_gt0)%:num; constructor.
Qed.
Variant nonneg_spec (R : numDomainType) (x : R) : R -> bool -> Type :=
| IsNonneg (p : {nonneg R}) : nonneg_spec x (p%:num) true.
Lemma nonnegP (R : numDomainType) (x : R) : 0 <= x -> nonneg_spec x x (0 <= x).
Proof. by move=> xge0; rewrite xge0 -[x]/(NngNum xge0)%:num; constructor. Qed.
Section Test1.
Variable R : numDomainType.
Variable x : {i01 R}.
Goal 0%:i01 = 1%:i01 :> {i01 R}.
Proof.
Abort.
Goal (- x%:num)%:itv = (- x%:num)%:itv :> {itv R & `[-1, 0]}.
Proof.
Abort.
Goal (1 - x%:num)%:i01 = x.
Proof.
Abort.
End Test1.
Section Test2.
Variable R : realDomainType.
Variable x y : {i01 R}.
Goal (x%:num * y%:num)%:i01 = x%:num%:i01.
Proof.
Abort.
End Test2.
Module Test3.
Section Test3.
Variable R : realDomainType.
Definition s_of_pq (p q : {i01 R}) : {i01 R} :=
(1 - ((1 - p%:num)%:i01%:num * (1 - q%:num)%:i01%:num))%:i01.
Lemma s_of_p0 (p : {i01 R}) : s_of_pq p 0%:i01 = p.
Proof. by apply/val_inj; rewrite /= subr0 mulr1 subKr. Qed.
End Test3.
End Test3.
|
Basic.lean
|
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ComplexShape
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Algebra.Group.Nat.Defs
import Mathlib.Tactic.ByContra
/-! # Embeddings of complex shapes
Given two complex shapes `c : ComplexShape ι` and `c' : ComplexShape ι'`,
an embedding from `c` to `c'` (`e : c.Embedding c'`) consists of the data
of an injective map `f : ι → ι'` such that for all `i₁ i₂ : ι`,
`c.Rel i₁ i₂` implies `c'.Rel (e.f i₁) (e.f i₂)`.
We define a type class `e.IsRelIff` to express that this implication is an equivalence.
Other type classes `e.IsTruncLE` and `e.IsTruncGE` are introduced in order to
formalize truncation functors.
This notion first appeared in the Liquid Tensor Experiment, and was developed there
mostly by Johan Commelin, Adam Topaz and Joël Riou. It shall be used in order to
relate the categories `CochainComplex C ℕ` and `ChainComplex C ℕ` to `CochainComplex C ℤ`.
It shall also be used in the construction of the canonical t-structure on the derived
category of an abelian category (TODO).
## Description of the API
- The extension functor `e.extendFunctor C : HomologicalComplex C c ⥤ HomologicalComplex C c'`
(extending by the zero object outside of the image of `e.f`) is defined in
the file `Embedding.Extend`;
- assuming `e.IsRelIff`, the restriction functor
`e.restrictionFunctor C : HomologicalComplex C c' ⥤ HomologicalComplex C c`
is defined in the file `Embedding.Restriction`;
- the stupid truncation functor
`e.stupidTruncFunctor C : HomologicalComplex C c' ⥤ HomologicalComplex C c'`
which is the composition of the two previous functors is defined in the file
`Embedding.StupidTrunc`.
- assuming `e.IsTruncGE`, we have truncation functors
`e.truncGE'Functor C : HomologicalComplex C c' ⥤ HomologicalComplex C c` and
`e.truncGEFunctor C : HomologicalComplex C c' ⥤ HomologicalComplex C c'`
(see the file `Embedding.TruncGE`), and a natural
transformation `e.πTruncGENatTrans : 𝟭 _ ⟶ e.truncGEFunctor C` which is a quasi-isomorphism
in degrees in the image of `e.f` (TODO);
- assuming `e.IsTruncLE`, we have truncation functors
`e.truncLE'Functor C : HomologicalComplex C c' ⥤ HomologicalComplex C c` and
`e.truncLEFunctor C : HomologicalComplex C c' ⥤ HomologicalComplex C c'`, and a natural
transformation `e.ιTruncLENatTrans : e.truncGEFunctor C ⟶ 𝟭 _` which is a quasi-isomorphism
in degrees in the image of `e.f` (TODO);
-/
assert_not_exists Nat.instAddMonoidWithOne Nat.instMulZeroClass
variable {ι ι' : Type*} (c : ComplexShape ι) (c' : ComplexShape ι')
namespace ComplexShape
/-- An embedding of a complex shape `c : ComplexShape ι` into a complex shape
`c' : ComplexShape ι'` consists of a injective map `f : ι → ι'` which satisfies
a compatibility with respect to the relations `c.Rel` and `c'.Rel`. -/
structure Embedding where
/-- the map between the underlying types of indices -/
f : ι → ι'
injective_f : Function.Injective f
rel {i₁ i₂ : ι} (h : c.Rel i₁ i₂) : c'.Rel (f i₁) (f i₂)
namespace Embedding
variable {c c'}
variable (e : Embedding c c')
/-- The opposite embedding in `Embedding c.symm c'.symm` of `e : Embedding c c'`. -/
@[simps]
def op : Embedding c.symm c'.symm where
f := e.f
injective_f := e.injective_f
rel h := e.rel h
/-- An embedding of complex shapes `e` satisfies `e.IsRelIff` if the implication
`e.rel` is an equivalence. -/
class IsRelIff : Prop where
rel' (i₁ i₂ : ι) (h : c'.Rel (e.f i₁) (e.f i₂)) : c.Rel i₁ i₂
lemma rel_iff [e.IsRelIff] (i₁ i₂ : ι) : c'.Rel (e.f i₁) (e.f i₂) ↔ c.Rel i₁ i₂ := by
constructor
· apply IsRelIff.rel'
· exact e.rel
instance [e.IsRelIff] : e.op.IsRelIff where
rel' i₁ i₂ h := (e.rel_iff i₂ i₁).1 h
section
variable (c c')
variable (f : ι → ι') (hf : Function.Injective f)
(iff : ∀ (i₁ i₂ : ι), c.Rel i₁ i₂ ↔ c'.Rel (f i₁) (f i₂))
/-- Constructor for embeddings between complex shapes when we have an equivalence
`∀ (i₁ i₂ : ι), c.Rel i₁ i₂ ↔ c'.Rel (f i₁) (f i₂)`. -/
@[simps]
def mk' : Embedding c c' where
f := f
injective_f := hf
rel h := (iff _ _).1 h
instance : (mk' c c' f hf iff).IsRelIff where
rel' _ _ h := (iff _ _).2 h
end
/-- The condition that the image of the map `e.f` of an embedding of
complex shapes `e : Embedding c c'` is stable by `c'.next`. -/
class IsTruncGE : Prop extends e.IsRelIff where
mem_next {j : ι} {k' : ι'} (h : c'.Rel (e.f j) k') :
∃ k, e.f k = k'
lemma mem_next [e.IsTruncGE] {j : ι} {k' : ι'} (h : c'.Rel (e.f j) k') : ∃ k, e.f k = k' :=
IsTruncGE.mem_next h
/-- The condition that the image of the map `e.f` of an embedding of
complex shapes `e : Embedding c c'` is stable by `c'.prev`. -/
class IsTruncLE : Prop extends e.IsRelIff where
mem_prev {i' : ι'} {j : ι} (h : c'.Rel i' (e.f j)) :
∃ i, e.f i = i'
lemma mem_prev [e.IsTruncLE] {i' : ι'} {j : ι} (h : c'.Rel i' (e.f j)) : ∃ i, e.f i = i' :=
IsTruncLE.mem_prev h
instance [e.IsTruncGE] : e.op.IsTruncLE where
mem_prev h := e.mem_next h
instance [e.IsTruncLE] : e.op.IsTruncGE where
mem_next h := e.mem_prev h
open Classical in
/-- The map `ι' → Option ι` which sends `e.f i` to `some i` and the other elements to `none`. -/
noncomputable def r (i' : ι') : Option ι :=
if h : ∃ (i : ι), e.f i = i'
then some h.choose
else none
lemma r_eq_some {i : ι} {i' : ι'} (hi : e.f i = i') :
e.r i' = some i := by
have h : ∃ (i : ι), e.f i = i' := ⟨i, hi⟩
have : h.choose = i := e.injective_f (h.choose_spec.trans (hi.symm))
dsimp [r]
rw [dif_pos ⟨i, hi⟩, this]
lemma r_eq_none (i' : ι') (hi : ∀ i, e.f i ≠ i') :
e.r i' = none :=
dif_neg (by
rintro ⟨i, hi'⟩
exact hi i hi')
@[simp] lemma r_f (i : ι) : e.r (e.f i) = some i := r_eq_some _ rfl
lemma f_eq_of_r_eq_some {i : ι} {i' : ι'} (hi : e.r i' = some i) :
e.f i = i' := by
by_cases h : ∃ (k : ι), e.f k = i'
· obtain ⟨k, rfl⟩ := h
rw [r_f] at hi
congr 1
simpa using hi.symm
· simp [e.r_eq_none i' (by simpa using h)] at hi
end Embedding
section
variable {A : Type*} [AddCommSemigroup A] [IsRightCancelAdd A] [One A]
/-- The embedding from `up' a` to itself via (· + b). -/
@[simps!]
def embeddingUp'Add (a b : A) : Embedding (up' a) (up' a) :=
Embedding.mk' _ _ (· + b)
(fun _ _ h => by simpa using h)
(by dsimp; simp_rw [add_right_comm _ b a, add_right_cancel_iff, implies_true])
instance (a b : A) : (embeddingUp'Add a b).IsRelIff := by dsimp [embeddingUp'Add]; infer_instance
instance (a b : A) : (embeddingUp'Add a b).IsTruncGE where
mem_next {j _} h := ⟨j + a, (add_right_comm _ _ _).trans h⟩
/-- The embedding from `down' a` to itself via (· + b). -/
@[simps!]
def embeddingDown'Add (a b : A) : Embedding (down' a) (down' a) :=
Embedding.mk' _ _ (· + b)
(fun _ _ h => by simpa using h)
(by dsimp; simp_rw [add_right_comm _ b a, add_right_cancel_iff, implies_true])
instance (a b : A) : (embeddingDown'Add a b).IsRelIff := by
dsimp [embeddingDown'Add]; infer_instance
instance (a b : A) : (embeddingDown'Add a b).IsTruncLE where
mem_prev {_ x} h := ⟨x + a, (add_right_comm _ _ _).trans h⟩
end
/-- The obvious embedding from `up ℕ` to `up ℤ`. -/
@[simps!]
def embeddingUpNat : Embedding (up ℕ) (up ℤ) :=
Embedding.mk' _ _ (fun n => n)
(fun _ _ h => by simpa using h)
(by dsimp; omega)
instance : embeddingUpNat.IsRelIff := by dsimp [embeddingUpNat]; infer_instance
instance : embeddingUpNat.IsTruncGE where
mem_next {j _} h := ⟨j + 1, h⟩
/-- The embedding from `down ℕ` to `up ℤ` with sends `n` to `-n`. -/
@[simps!]
def embeddingDownNat : Embedding (down ℕ) (up ℤ) :=
Embedding.mk' _ _ (fun n => -n)
(fun _ _ h => by simpa using h)
(by dsimp; omega)
instance : embeddingDownNat.IsRelIff := by dsimp [embeddingDownNat]; infer_instance
instance : embeddingDownNat.IsTruncLE where
mem_prev {i j} h := ⟨j + 1, by dsimp at h ⊢; omega⟩
variable (p : ℤ)
/-- The embedding from `up ℕ` to `up ℤ` which sends `n : ℕ` to `p + n`. -/
@[simps!]
def embeddingUpIntGE : Embedding (up ℕ) (up ℤ) :=
Embedding.mk' _ _ (fun n => p + n)
(fun _ _ h => by dsimp at h; omega)
(by dsimp; omega)
instance : (embeddingUpIntGE p).IsRelIff := by dsimp [embeddingUpIntGE]; infer_instance
instance : (embeddingUpIntGE p).IsTruncGE where
mem_next {j _} h := ⟨j + 1, by dsimp at h ⊢; omega⟩
/-- The embedding from `down ℕ` to `up ℤ` which sends `n : ℕ` to `p - n`. -/
@[simps!]
def embeddingUpIntLE : Embedding (down ℕ) (up ℤ) :=
Embedding.mk' _ _ (fun n => p - n)
(fun _ _ h => by dsimp at h; omega)
(by dsimp; omega)
instance : (embeddingUpIntLE p).IsRelIff := by dsimp [embeddingUpIntLE]; infer_instance
instance : (embeddingUpIntLE p).IsTruncLE where
mem_prev {_ k} h := ⟨k + 1, by dsimp at h ⊢; omega⟩
lemma notMem_range_embeddingUpIntLE_iff (n : ℤ) :
(∀ (i : ℕ), (embeddingUpIntLE p).f i ≠ n) ↔ p < n := by
constructor
· intro h
by_contra!
exact h (p - n).natAbs (by simp; omega)
· intros
dsimp
omega
@[deprecated (since := "2025-05-23")]
alias not_mem_range_embeddingUpIntLE_iff := notMem_range_embeddingUpIntLE_iff
lemma notMem_range_embeddingUpIntGE_iff (n : ℤ) :
(∀ (i : ℕ), (embeddingUpIntGE p).f i ≠ n) ↔ n < p := by
constructor
· intro h
by_contra!
exact h (n - p).natAbs (by simp; omega)
· intros
dsimp
omega
@[deprecated (since := "2025-05-23")]
alias not_mem_range_embeddingUpIntGE_iff := notMem_range_embeddingUpIntGE_iff
end ComplexShape
|
Basic.lean
|
/-
Copyright (c) 2025 Robin Carlier. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robin Carlier
-/
import Mathlib.AlgebraicTopology.SimplexCategory.Basic
import Mathlib.CategoryTheory.PathCategory.Basic
/-! # Presentation of the simplex category by generators and relations.
We introduce `SimplexCategoryGenRel` as the category presented by generating
morphisms `δ i : [n] ⟶ [n + 1]` and `σ i : [n + 1] ⟶ [n]` and subject to the
simplicial identities, and we provide induction principles for reasoning about
objects and morphisms in this category.
This category admits a canonical functor `toSimplexCategory` to the usual simplex category.
The fact that this functor is an equivalence will be recorded in a separate file.
-/
open CategoryTheory
/-- The objects of the free simplex quiver are the natural numbers. -/
def FreeSimplexQuiver := ℕ
/-- Making an object of `FreeSimplexQuiver` out of a natural number. -/
def FreeSimplexQuiver.mk (n : ℕ) : FreeSimplexQuiver := n
/-- Getting back the natural number from the objects. -/
def FreeSimplexQuiver.len (x : FreeSimplexQuiver) : ℕ := x
namespace FreeSimplexQuiver
/-- A morphism in `FreeSimplexQuiver` is either a face map (`δ`) or a degeneracy map (`σ`). -/
inductive Hom : FreeSimplexQuiver → FreeSimplexQuiver → Type
| δ {n : ℕ} (i : Fin (n + 2)) : Hom (.mk n) (.mk (n + 1))
| σ {n : ℕ} (i : Fin (n + 1)) : Hom (.mk (n + 1)) (.mk n)
instance quiv : Quiver FreeSimplexQuiver where
Hom := FreeSimplexQuiver.Hom
/-- `FreeSimplexQuiver.δ i` represents the `i`-th face map `.mk n ⟶ .mk (n + 1)`. -/
abbrev δ {n : ℕ} (i : Fin (n + 2)) : FreeSimplexQuiver.mk n ⟶ .mk (n + 1) :=
FreeSimplexQuiver.Hom.δ i
/-- `FreeSimplexQuiver.σ i` represents `i`-th degeneracy map `.mk (n + 1) ⟶ .mk n`. -/
abbrev σ {n : ℕ} (i : Fin (n + 1)) : FreeSimplexQuiver.mk (n + 1) ⟶ .mk n :=
FreeSimplexQuiver.Hom.σ i
/-- `FreeSimplexQuiver.homRel` is the relation on morphisms freely generated on the
five simplicial identities. -/
inductive homRel : HomRel (Paths FreeSimplexQuiver)
| δ_comp_δ {n : ℕ} {i j : Fin (n + 2)} (H : i ≤ j) : homRel
((Paths.of FreeSimplexQuiver).map (δ i) ≫ (Paths.of FreeSimplexQuiver).map (δ j.succ))
((Paths.of FreeSimplexQuiver).map (δ j) ≫ (Paths.of FreeSimplexQuiver).map (δ i.castSucc))
| δ_comp_σ_of_le {n : ℕ} {i : Fin (n + 2)} {j : Fin (n + 1)} (H : i ≤ j.castSucc) : homRel
((Paths.of FreeSimplexQuiver).map (δ i.castSucc) ≫ (Paths.of FreeSimplexQuiver).map (σ j.succ))
((Paths.of FreeSimplexQuiver).map (σ j) ≫ (Paths.of FreeSimplexQuiver).map (δ i))
| δ_comp_σ_self {n : ℕ} {i : Fin (n + 1)} : homRel
((Paths.of FreeSimplexQuiver).map (δ i.castSucc) ≫ (Paths.of FreeSimplexQuiver).map (σ i)) (𝟙 _)
| δ_comp_σ_succ {n : ℕ} {i : Fin (n + 1)} : homRel
((Paths.of FreeSimplexQuiver).map (δ i.succ) ≫ (Paths.of FreeSimplexQuiver).map (σ i)) (𝟙 _)
| δ_comp_σ_of_gt {n : ℕ} {i : Fin (n + 2)} {j : Fin (n + 1)} (H : j.castSucc < i) : homRel
((Paths.of FreeSimplexQuiver).map (δ i.succ) ≫ (Paths.of FreeSimplexQuiver).map (σ j.castSucc))
((Paths.of FreeSimplexQuiver).map (σ j) ≫ (Paths.of FreeSimplexQuiver).map (δ i))
| σ_comp_σ {n : ℕ} {i j : Fin (n + 1)} (H : i ≤ j) : homRel
((Paths.of FreeSimplexQuiver).map (σ i.castSucc) ≫ (Paths.of FreeSimplexQuiver).map (σ j))
((Paths.of FreeSimplexQuiver).map (σ j.succ) ≫ (Paths.of FreeSimplexQuiver).map (σ i))
end FreeSimplexQuiver
/-- SimplexCategory is the category presented by generators and relation by the simplicial
identities. -/
def SimplexCategoryGenRel := Quotient FreeSimplexQuiver.homRel
deriving Category
/-- `SimplexCategoryGenRel.mk` is the main constructor for objects of `SimplexCategoryGenRel`. -/
def SimplexCategoryGenRel.mk (n : ℕ) : SimplexCategoryGenRel where
as := (Paths.of FreeSimplexQuiver).obj n
namespace SimplexCategoryGenRel
/-- `SimplexCategoryGenRel.δ i` is the `i`-th face map `.mk n ⟶ .mk (n + 1)`. -/
abbrev δ {n : ℕ} (i : Fin (n + 2)) : mk n ⟶ mk (n + 1) :=
(Quotient.functor FreeSimplexQuiver.homRel).map <| (Paths.of FreeSimplexQuiver).map (.δ i)
/-- `SimplexCategoryGenRel.σ i` is the `i`-th degeneracy map `.mk (n + 1) ⟶ .mk n`. -/
abbrev σ {n : ℕ} (i : Fin (n + 1)) : mk (n + 1) ⟶ mk n :=
(Quotient.functor FreeSimplexQuiver.homRel).map <| (Paths.of FreeSimplexQuiver).map (.σ i)
/-- The length of an object of `SimplexCategoryGenRel`. -/
def len (x : SimplexCategoryGenRel) : ℕ := by rcases x with ⟨n⟩; exact n
@[simp]
lemma mk_len (n : ℕ) : len (mk n) = n := rfl
section InductionPrinciples
/-- A morphism is called a face if it is a `δ i` for some `i : Fin (n + 2)`. -/
inductive faces : MorphismProperty SimplexCategoryGenRel
| δ {n : ℕ} (i : Fin (n + 2)) : faces (δ i)
/-- A morphism is called a degeneracy if it is a `σ i` for some `i : Fin (n + 1)`. -/
inductive degeneracies : MorphismProperty SimplexCategoryGenRel
| σ {n : ℕ} (i : Fin (n + 1)) : degeneracies (σ i)
/-- A morphism is a generator if it is either a face or a degeneracy. -/
abbrev generators := faces ⊔ degeneracies
namespace generators
lemma δ {n : ℕ} (i : Fin (n + 2)) : generators (δ i) := le_sup_left (a := faces) _ (.δ i)
lemma σ {n : ℕ} (i : Fin (n + 1)) : generators (σ i) := le_sup_right (a := faces) _ (.σ i)
end generators
/-- A property is true for every morphism iff it holds for generators and is multiplicative. -/
lemma multiplicativeClosure_isGenerator_eq_top : generators.multiplicativeClosure = ⊤ := by
apply le_antisymm (by simp)
intro x y f _
apply CategoryTheory.Quotient.induction
apply Paths.induction
· exact generators.multiplicativeClosure.id_mem _
· rintro _ _ _ _ ⟨⟩ h
· exact generators.multiplicativeClosure.comp_mem _ _ h <| .of _ <| .δ _
· exact generators.multiplicativeClosure.comp_mem _ _ h <| .of _ <| .σ _
/-- An unrolled version of the induction principle obtained in the previous lemma. -/
@[elab_as_elim, cases_eliminator, induction_eliminator]
lemma hom_induction (P : MorphismProperty SimplexCategoryGenRel)
(id : ∀ {n : ℕ}, P (𝟙 (mk n)))
(comp_δ : ∀ {n m : ℕ} (u : mk n ⟶ mk m) (i : Fin (m + 2)), P u → P (u ≫ δ i))
(comp_σ : ∀ {n m : ℕ} (u : mk n ⟶ mk (m + 1)) (i : Fin (m + 1)), P u → P (u ≫ σ i))
{a b : SimplexCategoryGenRel} (f : a ⟶ b) :
P f :=
by
suffices generators.multiplicativeClosure ≤ P by
rw [multiplicativeClosure_isGenerator_eq_top, top_le_iff] at this
rw [this]
apply MorphismProperty.top_apply
intro _ _ f hf
induction hf with
| of f h =>
rcases h with ⟨⟨i⟩⟩ | ⟨⟨i⟩⟩
· simpa using (comp_δ (𝟙 _) i id)
· simpa using (comp_σ (𝟙 _) i id)
| id n => exact id
| comp_of f g hf hg hrec =>
rcases hg with ⟨⟨i⟩⟩ | ⟨⟨i⟩⟩
· simpa using (comp_δ f i hrec)
· simpa using (comp_σ f i hrec)
/-- An induction principle for reasonning about morphisms in SimplexCategoryGenRel, where we compose
with generators on the right. -/
lemma hom_induction' (P : MorphismProperty SimplexCategoryGenRel)
(id : ∀ {n : ℕ}, P (𝟙 (mk n)))
(δ_comp : ∀ {n m : ℕ} (u : mk (m + 1) ⟶ mk n)
(i : Fin (m + 2)), P u → P (δ i ≫ u))
(σ_comp : ∀ {n m : ℕ} (u : mk m ⟶ mk n)
(i : Fin (m + 1)), P u → P (σ i ≫ u)) {a b : SimplexCategoryGenRel} (f : a ⟶ b) :
P f := by
suffices generators.multiplicativeClosure' ≤ P by
rw [← MorphismProperty.multiplicativeClosure_eq_multiplicativeClosure',
multiplicativeClosure_isGenerator_eq_top, top_le_iff] at this
rw [this]
apply MorphismProperty.top_apply
intro _ _ f hf
induction hf with
| of f h =>
rcases h with ⟨⟨i⟩⟩ | ⟨⟨i⟩⟩
· simpa using (δ_comp (𝟙 _) i id)
· simpa using (σ_comp (𝟙 _) i id)
| id n => exact id
| of_comp f g hf hg hrec =>
rcases hf with ⟨⟨i⟩⟩ | ⟨⟨i⟩⟩
· simpa using (δ_comp g i hrec)
· simpa using (σ_comp g i hrec)
/-- An induction principle for reasonning about objects in `SimplexCategoryGenRel`. This should be
used instead of identifying an object with `mk` of its `len`. -/
@[elab_as_elim, cases_eliminator]
protected def rec {P : SimplexCategoryGenRel → Sort*}
(H : ∀ n : ℕ, P (.mk n)) :
∀ x : SimplexCategoryGenRel, P x := by
intro x
exact H x.len
/-- A basic `ext` lemma for objects of `SimplexCategoryGenRel`. -/
@[ext]
lemma ext {x y : SimplexCategoryGenRel} (h : x.len = y.len) : x = y := by
cases x
cases y
simp only [mk_len] at h
congr
end InductionPrinciples
section SimplicialIdentities
@[reassoc]
theorem δ_comp_δ {n} {i j : Fin (n + 2)} (H : i ≤ j) :
δ i ≫ δ j.succ = δ j ≫ δ i.castSucc := by
apply CategoryTheory.Quotient.sound
exact FreeSimplexQuiver.homRel.δ_comp_δ H
@[reassoc]
theorem δ_comp_σ_of_le {n} {i : Fin (n + 2)} {j : Fin (n + 1)} (H : i ≤ j.castSucc) :
δ i.castSucc ≫ σ j.succ = σ j ≫ δ i := by
apply CategoryTheory.Quotient.sound
exact FreeSimplexQuiver.homRel.δ_comp_σ_of_le H
@[reassoc]
theorem δ_comp_σ_self {n} {i : Fin (n + 1)} :
δ i.castSucc ≫ σ i = 𝟙 (mk n) := by
apply CategoryTheory.Quotient.sound
exact FreeSimplexQuiver.homRel.δ_comp_σ_self
@[reassoc]
theorem δ_comp_σ_succ {n} {i : Fin (n + 1)} : δ i.succ ≫ σ i = 𝟙 (mk n) := by
apply CategoryTheory.Quotient.sound
exact FreeSimplexQuiver.homRel.δ_comp_σ_succ
@[reassoc]
theorem δ_comp_σ_of_gt {n} {i : Fin (n + 2)} {j : Fin (n + 1)} (H : j.castSucc < i) :
δ i.succ ≫ σ j.castSucc = σ j ≫ δ i := by
apply CategoryTheory.Quotient.sound
exact FreeSimplexQuiver.homRel.δ_comp_σ_of_gt H
@[reassoc]
theorem σ_comp_σ {n} {i j : Fin (n + 1)} (H : i ≤ j) :
σ i.castSucc ≫ σ j = σ j.succ ≫ σ i := by
apply CategoryTheory.Quotient.sound
exact FreeSimplexQuiver.homRel.σ_comp_σ H
/-- A version of δ_comp_δ with indices in ℕ satisfying relevant inequalities. -/
lemma δ_comp_δ_nat {n} (i j : ℕ) (hi : i < n + 2) (hj : j < n + 2) (H : i ≤ j) :
δ ⟨i, hi⟩ ≫ δ ⟨j + 1, by omega⟩ = δ ⟨j, hj⟩ ≫ δ ⟨i, by omega⟩ :=
δ_comp_δ (n := n) (i := ⟨i, by omega⟩) (j := ⟨j, by omega⟩) (by simpa)
/-- A version of σ_comp_σ with indices in ℕ satisfying relevant inequalities. -/
lemma σ_comp_σ_nat {n} (i j : ℕ) (hi : i < n + 1) (hj : j < n + 1) (H : i ≤ j) :
σ ⟨i, by omega⟩ ≫ σ ⟨j, hj⟩ = σ ⟨j + 1, by omega⟩ ≫ σ ⟨i, hi⟩ :=
σ_comp_σ (n := n) (i := ⟨i, by omega⟩) (j := ⟨j, by omega⟩) (by simpa)
end SimplicialIdentities
/-- The canonical functor from `SimplexCategoryGenRel` to SimplexCategory, which exists as the
simplicial identities hold in `SimplexCategory`. -/
def toSimplexCategory : SimplexCategoryGenRel ⥤ SimplexCategory :=
CategoryTheory.Quotient.lift _
(Paths.lift
{ obj := .mk
map f := match f with
| FreeSimplexQuiver.Hom.δ i => SimplexCategory.δ i
| FreeSimplexQuiver.Hom.σ i => SimplexCategory.σ i })
(fun _ _ _ _ h ↦ match h with
| .δ_comp_δ H => SimplexCategory.δ_comp_δ H
| .δ_comp_σ_of_le H => SimplexCategory.δ_comp_σ_of_le H
| .δ_comp_σ_self => SimplexCategory.δ_comp_σ_self
| .δ_comp_σ_succ => SimplexCategory.δ_comp_σ_succ
| .δ_comp_σ_of_gt H => SimplexCategory.δ_comp_σ_of_gt H
| .σ_comp_σ H => SimplexCategory.σ_comp_σ H)
@[simp]
lemma toSimplexCategory_obj_mk (n : ℕ) : toSimplexCategory.obj (mk n) = .mk n := rfl
@[simp]
lemma toSimplexCategory_map_δ {n : ℕ} (i : Fin (n + 2)) :
toSimplexCategory.map (δ i) = SimplexCategory.δ i := rfl
@[simp]
lemma toSimplexCategory_map_σ {n : ℕ} (i : Fin (n + 1)) :
toSimplexCategory.map (σ i) = SimplexCategory.σ i := rfl
@[simp]
lemma toSimplexCategory_len {x : SimplexCategoryGenRel} : (toSimplexCategory.obj x).len = x.len :=
rfl
end SimplexCategoryGenRel
|
Imo2020Q2.lean
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Yury Kudryashov
-/
import Mathlib.Analysis.MeanInequalities
/-!
# IMO 2020 Q2
The real numbers `a`, `b`, `c`, `d` are such that `a ≥ b ≥ c ≥ d > 0` and `a + b + c + d = 1`.
Prove that `(a + 2b + 3c + 4d) a^a b^b c^c d^d < 1`.
A solution is to eliminate the powers using weighted AM-GM and replace
`1` by `(a+b+c+d)^3`, leaving a homogeneous inequality that can be
proved in many ways by expanding, rearranging and comparing individual
terms. The version here using factors such as `a+3b+3c+3d` is from
the official solutions.
-/
open Real
theorem imo2020_q2 (a b c d : ℝ) (hd0 : 0 < d) (hdc : d ≤ c) (hcb : c ≤ b) (hba : b ≤ a)
(h1 : a + b + c + d = 1) : (a + 2 * b + 3 * c + 4 * d) * a ^ a * b ^ b * c ^ c * d ^ d < 1 := by
have hp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d := by
refine geom_mean_le_arith_mean4_weighted ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ h1 <;> linarith
calc
(a + 2 * b + 3 * c + 4 * d) * a ^ a * b ^ b * c ^ c * d ^ d =
(a + 2 * b + 3 * c + 4 * d) * (a ^ a * b ^ b * c ^ c * d ^ d) := by ac_rfl
_ ≤ (a + 2 * b + 3 * c + 4 * d) * (a * a + b * b + c * c + d * d) := by gcongr; linarith
_ = (a + 2 * b + 3 * c + 4 * d) * a ^ 2 + (a + 2 * b + 3 * c + 4 * d) * b ^ 2
+ (a + 2 * b + 3 * c + 4 * d) * c ^ 2 + (a + 2 * b + 3 * c + 4 * d) * d ^ 2 := by ring
_ ≤ (a + 3 * b + 3 * c + 3 * d) * a ^ 2 + (3 * a + b + 3 * c + 3 * d) * b ^ 2
+ (3 * a + 3 * b + c + 3 * d) * c ^ 2 + (3 * a + 3 * b + 3 * c + d) * d ^ 2 := by
gcongr ?_ * _ + ?_ * _ + ?_ * _ + ?_ * _ <;> linarith
_ < (a + 3 * b + 3 * c + 3 * d) * a ^ 2 + (3 * a + b + 3 * c + 3 * d) * b ^ 2
+ (3 * a + 3 * b + c + 3 * d) * c ^ 2 + (3 * a + 3 * b + 3 * c + d) * d ^ 2
+ (6 * a * b * c + 6 * a * b * d + 6 * a * c * d + 6 * b * c * d) :=
(lt_add_of_pos_right _ (by apply_rules [add_pos, mul_pos, zero_lt_one] <;> linarith))
_ = (a + b + c + d) ^ 3 := by ring
_ = 1 := by simp [h1]
|
Lemmas.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, Kyle Miller
-/
import Mathlib.Data.Finset.Max
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Set.Lattice
import Mathlib.Data.Fintype.Powerset
import Mathlib.Logic.Embedding.Set
/-!
# Lemmas on finiteness of sets
This file should contain lemmas that prove some result under the *assumption* of `Set.Finite`.
If your proof has as *result* `Set.Finite`, then it should go to a more specific file.
## Tags
finite sets
-/
assert_not_exists OrderedRing MonoidWithZero
open Set Function
universe u v w x
variable {α : Type u} {β : Type v} {ι : Sort w} {γ : Type x}
namespace Set
/-! ### Properties -/
theorem Finite.fin_embedding {s : Set α} (h : s.Finite) :
∃ (n : ℕ) (f : Fin n ↪ α), range f = s :=
⟨_, (Fintype.equivFin (h.toFinset : Set α)).symm.asEmbedding, by
simp only [Finset.coe_sort_coe, Equiv.asEmbedding_range, Finite.coe_toFinset, setOf_mem_eq]⟩
theorem Finite.fin_param {s : Set α} (h : s.Finite) :
∃ (n : ℕ) (f : Fin n → α), Injective f ∧ range f = s :=
let ⟨n, f, hf⟩ := h.fin_embedding
⟨n, f, f.injective, hf⟩
/-- Induction up to a finite set `S`. -/
theorem Finite.induction_to {C : Set α → Prop} {S : Set α} (h : S.Finite)
(S0 : Set α) (hS0 : S0 ⊆ S) (H0 : C S0) (H1 : ∀ s ⊂ S, C s → ∃ a ∈ S \ s, C (insert a s)) :
C S := by
have : Finite S := Finite.to_subtype h
have : Finite {T : Set α // T ⊆ S} := Finite.of_equiv (Set S) (Equiv.Set.powerset S).symm
rw [← Subtype.coe_mk (p := (· ⊆ S)) _ le_rfl]
rw [← Subtype.coe_mk (p := (· ⊆ S)) _ hS0] at H0
refine Finite.to_wellFoundedGT.wf.induction_bot' (fun s hs hs' ↦ ?_) H0
obtain ⟨a, ⟨ha1, ha2⟩, ha'⟩ := H1 s (ssubset_of_ne_of_subset hs s.2) hs'
exact ⟨⟨insert a s.1, insert_subset ha1 s.2⟩, Set.ssubset_insert ha2, ha'⟩
/-- Induction up to `univ`. -/
theorem Finite.induction_to_univ [Finite α] {C : Set α → Prop} (S0 : Set α)
(H0 : C S0) (H1 : ∀ S ≠ univ, C S → ∃ a ∉ S, C (insert a S)) : C univ :=
finite_univ.induction_to S0 (subset_univ S0) H0 (by simpa [ssubset_univ_iff])
/-! ### Infinite sets -/
variable {s t : Set α}
/-! ### Order properties -/
theorem exists_min_image [LinearOrder β] (s : Set α) (f : α → β) (h1 : s.Finite) :
s.Nonempty → ∃ a ∈ s, ∀ b ∈ s, f a ≤ f b
| ⟨x, hx⟩ => by
simpa only [exists_prop, Finite.mem_toFinset] using
h1.toFinset.exists_min_image f ⟨x, h1.mem_toFinset.2 hx⟩
theorem exists_max_image [LinearOrder β] (s : Set α) (f : α → β) (h1 : s.Finite) :
s.Nonempty → ∃ a ∈ s, ∀ b ∈ s, f b ≤ f a
| ⟨x, hx⟩ => by
simpa only [exists_prop, Finite.mem_toFinset] using
h1.toFinset.exists_max_image f ⟨x, h1.mem_toFinset.2 hx⟩
theorem exists_lower_bound_image [Nonempty α] [LinearOrder β] (s : Set α) (f : α → β)
(h : s.Finite) : ∃ a : α, ∀ b ∈ s, f a ≤ f b := by
rcases s.eq_empty_or_nonempty with rfl | hs
· exact ‹Nonempty α›.elim fun a => ⟨a, fun _ => False.elim⟩
· rcases Set.exists_min_image s f h hs with ⟨x₀, _, hx₀⟩
exact ⟨x₀, fun x hx => hx₀ x hx⟩
theorem exists_upper_bound_image [Nonempty α] [LinearOrder β] (s : Set α) (f : α → β)
(h : s.Finite) : ∃ a : α, ∀ b ∈ s, f b ≤ f a :=
exists_lower_bound_image (β := βᵒᵈ) s f h
end Set
|
ZMod.lean
|
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Algebra.Group.EvenFunction
import Mathlib.Analysis.SpecialFunctions.Complex.CircleAddChar
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.NumberTheory.DirichletCharacter.GaussSum
/-!
# Fourier theory on `ZMod N`
Basic definitions and properties of the discrete Fourier transform for functions on `ZMod N`
(taking values in an arbitrary `ℂ`-vector space).
### Main definitions and results
* `ZMod.dft`: the Fourier transform, with respect to the standard additive character
`ZMod.stdAddChar` (mapping `j mod N` to `exp (2 * π * I * j / N)`). The notation `𝓕`, scoped in
namespace `ZMod`, is available for this.
* `ZMod.dft_dft`: the Fourier inversion formula.
* `DirichletCharacter.fourierTransform_eq_inv_mul_gaussSum`: the discrete Fourier transform of a
primitive Dirichlet character `χ` is a Gauss sum times `χ⁻¹`.
-/
open MeasureTheory Finset AddChar ZMod
namespace ZMod
variable {N : ℕ} [NeZero N] {E : Type*} [AddCommGroup E] [Module ℂ E]
section private_defs
/-
It doesn't _quite_ work to define the Fourier transform as a `LinearEquiv` in one go, because that
leads to annoying repetition between the proof fields. So we set up a private definition first,
prove a minimal set of lemmas about it, and then define the `LinearEquiv` using that.
**Do not add more lemmas about `auxDFT`**: it should be invisible to end-users.
-/
/--
The discrete Fourier transform on `ℤ / N ℤ` (with the counting measure). This definition is
private because it is superseded by the bundled `LinearEquiv` version.
-/
private noncomputable def auxDFT (Φ : ZMod N → E) (k : ZMod N) : E :=
∑ j : ZMod N, stdAddChar (-(j * k)) • Φ j
private lemma auxDFT_neg (Φ : ZMod N → E) : auxDFT (fun j ↦ Φ (-j)) = fun k ↦ auxDFT Φ (-k) := by
ext1 k; simpa only [auxDFT] using
Fintype.sum_equiv (Equiv.neg _) _ _ (fun j ↦ by rw [Equiv.neg_apply, neg_mul_neg])
/-- Fourier inversion formula, discrete case. -/
private lemma auxDFT_auxDFT (Φ : ZMod N → E) : auxDFT (auxDFT Φ) = fun j ↦ (N : ℂ) • Φ (-j) := by
ext1 j
simp only [auxDFT, mul_comm _ j, smul_sum, ← smul_assoc, smul_eq_mul, ← map_add_eq_mul, ←
neg_add, ← add_mul]
rw [sum_comm]
simp only [← sum_smul, ← neg_mul]
have h1 (t : ZMod N) : ∑ i, stdAddChar (t * i) = if t = 0 then ↑N else 0 := by
split_ifs with h
· simp only [h, zero_mul, map_zero_eq_one, sum_const, card_univ, card,
nsmul_eq_mul, mul_one]
· exact sum_eq_zero_of_ne_one (isPrimitive_stdAddChar N h)
have h2 (x j : ZMod N) : -(j + x) = 0 ↔ x = -j := by
rw [neg_add, add_comm, add_eq_zero_iff_neg_eq, neg_neg]
simp only [h1, h2, ite_smul, zero_smul, sum_ite_eq', mem_univ, ite_true]
private lemma auxDFT_smul (c : ℂ) (Φ : ZMod N → E) :
auxDFT (c • Φ) = c • auxDFT Φ := by
ext; simp only [Pi.smul_def, auxDFT, smul_sum, smul_comm c]
end private_defs
section defs
/--
The discrete Fourier transform on `ℤ / N ℤ` (with the counting measure), bundled as a linear
equivalence. Denoted as `𝓕` within the `ZMod` namespace.
-/
noncomputable def dft : (ZMod N → E) ≃ₗ[ℂ] (ZMod N → E) where
toFun := auxDFT
map_add' Φ₁ Φ₂ := by
ext; simp only [auxDFT, Pi.add_def, smul_add, sum_add_distrib]
map_smul' c Φ := by
ext; simp only [auxDFT, Pi.smul_apply, RingHom.id_apply, smul_sum, smul_comm c]
invFun Φ k := (N : ℂ)⁻¹ • auxDFT Φ (-k)
left_inv Φ := by
simp only [auxDFT_auxDFT, neg_neg, ← mul_smul, inv_mul_cancel₀ (NeZero.ne _), one_smul]
right_inv Φ := by
ext1 j
simp only [← Pi.smul_def, auxDFT_smul, auxDFT_neg, auxDFT_auxDFT, neg_neg, ← mul_smul,
inv_mul_cancel₀ (NeZero.ne _), one_smul]
@[inherit_doc] scoped notation "𝓕" => dft
/-- The inverse Fourier transform on `ZMod N`. -/
scoped notation "𝓕⁻" => LinearEquiv.symm dft
lemma dft_apply (Φ : ZMod N → E) (k : ZMod N) :
𝓕 Φ k = ∑ j : ZMod N, stdAddChar (-(j * k)) • Φ j :=
rfl
lemma dft_def (Φ : ZMod N → E) :
𝓕 Φ = fun k ↦ ∑ j : ZMod N, stdAddChar (-(j * k)) • Φ j :=
rfl
lemma invDFT_apply (Ψ : ZMod N → E) (k : ZMod N) :
𝓕⁻ Ψ k = (N : ℂ)⁻¹ • ∑ j : ZMod N, stdAddChar (j * k) • Ψ j := by
simp only [dft, LinearEquiv.coe_symm_mk, auxDFT, mul_neg, neg_neg]
lemma invDFT_def (Ψ : ZMod N → E) :
𝓕⁻ Ψ = fun k ↦ (N : ℂ)⁻¹ • ∑ j : ZMod N, stdAddChar (j * k) • Ψ j :=
funext <| invDFT_apply Ψ
lemma invDFT_apply' (Ψ : ZMod N → E) (k : ZMod N) : 𝓕⁻ Ψ k = (N : ℂ)⁻¹ • 𝓕 Ψ (-k) :=
rfl
lemma invDFT_def' (Ψ : ZMod N → E) : 𝓕⁻ Ψ = fun k ↦ (N : ℂ)⁻¹ • 𝓕 Ψ (-k) :=
rfl
lemma dft_apply_zero (Φ : ZMod N → E) : 𝓕 Φ 0 = ∑ j, Φ j := by
simp only [dft_apply, mul_zero, neg_zero, map_zero_eq_one, one_smul]
/--
The discrete Fourier transform agrees with the general one (assuming the target space is a complete
normed space).
-/
lemma dft_eq_fourier {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
(Φ : ZMod N → E) (k : ZMod N) :
𝓕 Φ k = Fourier.fourierIntegral toCircle Measure.count Φ k := by
simp only [dft_apply, stdAddChar_apply, Fourier.fourierIntegral_def, Circle.smul_def,
integral_countable' <| .of_finite .., count_real_singleton, one_smul, tsum_fintype]
end defs
section arith
/-!
## Compatibility with scalar multiplication
These lemmas are more general than `LinearEquiv.map_mul` etc, since they allow any scalars that
commute with the `ℂ`-action, rather than just `ℂ` itself.
-/
lemma dft_const_smul {R : Type*} [DistribSMul R E] [SMulCommClass R ℂ E] (r : R) (Φ : ZMod N → E) :
𝓕 (r • Φ) = r • 𝓕 Φ := by
simp only [Pi.smul_def, dft_def, smul_sum, smul_comm]
lemma dft_smul_const {R : Type*} [Ring R] [Module ℂ R] [Module R E] [IsScalarTower ℂ R E]
(Φ : ZMod N → R) (e : E) :
𝓕 (fun j ↦ Φ j • e) = fun k ↦ 𝓕 Φ k • e := by
simp only [dft_def, sum_smul, smul_assoc]
lemma dft_const_mul {R : Type*} [Ring R] [Algebra ℂ R] (r : R) (Φ : ZMod N → R) :
𝓕 (fun j ↦ r * Φ j) = fun k ↦ r * 𝓕 Φ k :=
dft_const_smul r Φ
lemma dft_mul_const {R : Type*} [Ring R] [Algebra ℂ R] (Φ : ZMod N → R) (r : R) :
𝓕 (fun j ↦ Φ j * r) = fun k ↦ 𝓕 Φ k * r :=
dft_smul_const Φ r
end arith
section inversion
lemma dft_comp_neg (Φ : ZMod N → E) : 𝓕 (fun j ↦ Φ (-j)) = fun k ↦ 𝓕 Φ (-k) :=
auxDFT_neg ..
/-- Fourier inversion formula, discrete case. -/
lemma dft_dft (Φ : ZMod N → E) : 𝓕 (𝓕 Φ) = fun j ↦ (N : ℂ) • Φ (-j) :=
auxDFT_auxDFT ..
end inversion
lemma dft_comp_unitMul (Φ : ZMod N → E) (u : (ZMod N)ˣ) (k : ZMod N) :
𝓕 (fun j ↦ Φ (u.val * j)) k = 𝓕 Φ (u⁻¹.val * k) := by
refine Fintype.sum_equiv u.mulLeft _ _ fun x ↦ ?_
simp only [mul_comm u.val, u.mulLeft_apply, ← mul_assoc, u.mul_inv_cancel_right]
section signs
/-- The discrete Fourier transform of `Φ` is even if and only if `Φ` itself is. -/
lemma dft_even_iff {Φ : ZMod N → ℂ} : (𝓕 Φ).Even ↔ Φ.Even := by
have h {f : ZMod N → ℂ} (hf : f.Even) : (𝓕 f).Even := by
simp only [Function.Even, ← congr_fun (dft_comp_neg f), funext hf, implies_true]
refine ⟨fun hΦ x ↦ ?_, h⟩
simpa only [neg_neg, smul_right_inj (NeZero.ne (N : ℂ)), dft_dft] using h hΦ (-x)
/-- The discrete Fourier transform of `Φ` is odd if and only if `Φ` itself is. -/
lemma dft_odd_iff {Φ : ZMod N → ℂ} : (𝓕 Φ).Odd ↔ Φ.Odd := by
have h {f : ZMod N → ℂ} (hf : f.Odd) : (𝓕 f).Odd := by
simp only [Function.Odd, ← congr_fun (dft_comp_neg f), funext hf, ← Pi.neg_apply, map_neg,
implies_true]
refine ⟨fun hΦ x ↦ ?_, h⟩
simpa only [neg_neg, dft_dft, ← smul_neg, smul_right_inj (NeZero.ne (N : ℂ))] using h hΦ (-x)
end signs
end ZMod
namespace DirichletCharacter
variable {N : ℕ} [NeZero N]
lemma fourierTransform_eq_gaussSum_mulShift (χ : DirichletCharacter ℂ N) (k : ZMod N) :
𝓕 χ k = gaussSum χ (stdAddChar.mulShift (-k)) := by
simp only [dft_apply, smul_eq_mul]
congr 1 with j
rw [mulShift_apply, mul_comm j, neg_mul, stdAddChar_apply, mul_comm (χ _)]
/-- For a primitive Dirichlet character `χ`, the Fourier transform of `χ` is a constant multiple
of `χ⁻¹` (and the constant is essentially the Gauss sum). -/
lemma IsPrimitive.fourierTransform_eq_inv_mul_gaussSum {χ : DirichletCharacter ℂ N}
(hχ : IsPrimitive χ) (k : ZMod N) :
𝓕 χ k = χ⁻¹ (-k) * gaussSum χ stdAddChar := by
rw [fourierTransform_eq_gaussSum_mulShift, gaussSum_mulShift_of_isPrimitive _ hχ]
end DirichletCharacter
|
Odd.lean
|
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Order.Group.Defs
import Mathlib.Algebra.Order.Monoid.OrderDual
import Mathlib.Order.Monotone.Union
/-!
# Monotonicity of odd functions
An odd function on a linear ordered additive commutative group `G` is monotone on the whole group
provided that it is monotone on `Set.Ici 0`, see `monotone_of_odd_of_monotoneOn_nonneg`. We also
prove versions of this lemma for `Antitone`, `StrictMono`, and `StrictAnti`.
-/
open Set
variable {G H : Type*} [AddCommGroup G] [LinearOrder G] [IsOrderedAddMonoid G]
[AddCommGroup H] [PartialOrder H] [IsOrderedAddMonoid H]
/-- An odd function on a linear ordered additive commutative group is strictly monotone on the whole
group provided that it is strictly monotone on `Set.Ici 0`. -/
theorem strictMono_of_odd_strictMonoOn_nonneg {f : G → H} (h₁ : ∀ x, f (-x) = -f x)
(h₂ : StrictMonoOn f (Ici 0)) : StrictMono f := by
refine StrictMonoOn.Iic_union_Ici (fun x hx y hy hxy => neg_lt_neg_iff.1 ?_) h₂
rw [← h₁, ← h₁]
exact h₂ (neg_nonneg.2 hy) (neg_nonneg.2 hx) (neg_lt_neg hxy)
/-- An odd function on a linear ordered additive commutative group is strictly antitone on the whole
group provided that it is strictly antitone on `Set.Ici 0`. -/
theorem strictAnti_of_odd_strictAntiOn_nonneg {f : G → H} (h₁ : ∀ x, f (-x) = -f x)
(h₂ : StrictAntiOn f (Ici 0)) : StrictAnti f :=
strictMono_of_odd_strictMonoOn_nonneg (H := Hᵒᵈ) h₁ h₂
/-- An odd function on a linear ordered additive commutative group is monotone on the whole group
provided that it is monotone on `Set.Ici 0`. -/
theorem monotone_of_odd_of_monotoneOn_nonneg {f : G → H} (h₁ : ∀ x, f (-x) = -f x)
(h₂ : MonotoneOn f (Ici 0)) : Monotone f := by
refine MonotoneOn.Iic_union_Ici (fun x hx y hy hxy => neg_le_neg_iff.1 ?_) h₂
rw [← h₁, ← h₁]
exact h₂ (neg_nonneg.2 hy) (neg_nonneg.2 hx) (neg_le_neg hxy)
/-- An odd function on a linear ordered additive commutative group is antitone on the whole group
provided that it is monotone on `Set.Ici 0`. -/
theorem antitone_of_odd_of_monotoneOn_nonneg {f : G → H} (h₁ : ∀ x, f (-x) = -f x)
(h₂ : AntitoneOn f (Ici 0)) : Antitone f :=
monotone_of_odd_of_monotoneOn_nonneg (H := Hᵒᵈ) h₁ h₂
|
jordanholder.v
|
(* (c) Copyright 2006-2016 Microsoft Corporation and Inria. *)
(* Distributed under the terms of CeCILL-B. *)
From HB Require Import structures.
From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.
From mathcomp Require Import choice fintype bigop finset fingroup morphism.
From mathcomp Require Import automorphism quotient action gseries.
(******************************************************************************)
(* This files establishes Jordan-Holder theorems for finite groups. These *)
(* theorems state the uniqueness up to permutation and isomorphism for the *)
(* series of quotient built from the successive elements of any composition *)
(* series of the same group. These quotients are also called factors of the *)
(* composition series. To avoid the heavy use of highly polymorphic lists *)
(* describing these quotient series, we introduce sections. *)
(* This library defines: *)
(* (G1 / G2)%sec == alias for the pair (G1, G2) of groups in the same *)
(* finGroupType, coerced to the actual quotient group*)
(* group G1 / G2. We call this pseudo-quotient a *)
(* section of G1 and G2. *)
(* section_isog s1 s2 == s1 and s2 respectively coerce to isomorphic *)
(* quotient groups. *)
(* section_repr s == canonical representative of the isomorphism class *)
(* of the section s. *)
(* mksrepr G1 G2 == canonical representative of the isomorphism class *)
(* of (G1 / G2)%sec. *)
(* mkfactors G s == if s is [:: s1, s2, ..., sn], constructs the list *)
(* [:: mksrepr G s1, mksrepr s1 s2, ..., mksrepr sn-1 sn] *)
(* comps G s == s is a composition series for G i.e. s is a *)
(* decreasing sequence of subgroups of G *)
(* in which two adjacent elements are maxnormal one *)
(* in the other and the last element of s is 1. *)
(* Given aT and rT two finGroupTypes, (D : {group rT}), (A : {group aT}) and *)
(* (to : groupAction A D) an external action. *)
(* maxainv to B C == C is a maximal proper normal subgroup of B *)
(* invariant by (the external action of A via) to. *)
(* asimple to B == the maximal proper normal subgroup of B invariant *)
(* by the external action to is trivial. *)
(* acomps to G s == s is a composition series for G invariant by to, *)
(* i.e. s is a decreasing sequence of subgroups of G *)
(* in which two adjacent elements are maximally *)
(* invariant by to one in the other and the *)
(* last element of s is 1. *)
(* We prove two versions of the result: *)
(* - JordanHolderUniqueness establishes the uniqueness up to permutation *)
(* and isomorphism of the lists of factors in composition series of a *)
(* given group. *)
(* - StrongJordanHolderUniqueness extends the result to composition series *)
(* invariant by an external group action. *)
(* See also "The Rooster and the Butterflies", proceedings of Calculemus 2013,*)
(* by Assia Mahboubi. *)
(******************************************************************************)
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Declare Scope section_scope.
Import GroupScope.
Inductive section (gT : finGroupType) := GSection of {group gT} * {group gT}.
Delimit Scope section_scope with sec.
Bind Scope section_scope with section.
Definition mkSec (gT : finGroupType) (G1 G2 : {group gT}) := GSection (G1, G2).
Infix "/" := mkSec : section_scope.
Coercion pair_of_section gT (s : section gT) := let: GSection u := s in u.
Coercion quotient_of_section gT (s : section gT) : GroupSet.sort _ := s.1 / s.2.
Coercion section_group gT (s : section gT) : {group (coset_of s.2)} :=
Eval hnf in [group of s].
Section Sections.
Variables (gT : finGroupType).
Implicit Types (G : {group gT}) (s : section gT).
HB.instance Definition _ := [isNew for (@pair_of_section gT)].
HB.instance Definition _ := [Finite of section gT by <:].
Canonical section_group.
(* Isomorphic sections *)
Definition section_isog := [rel x y : section gT | x \isog y].
(* A witness of the isomorphism class of a section *)
Definition section_repr s := odflt (1 / 1)%sec (pick (section_isog ^~ s)).
Definition mksrepr G1 G2 := section_repr (mkSec G1 G2).
Lemma section_reprP s : section_repr s \isog s.
Proof.
by rewrite /section_repr; case: pickP => //= /(_ s); rewrite isog_refl.
Qed.
Lemma section_repr_isog s1 s2 :
s1 \isog s2 -> section_repr s1 = section_repr s2.
Proof.
by move=> iso12; congr (odflt _ _); apply: eq_pick => s; apply: isog_transr.
Qed.
Definition mkfactors (G : {group gT}) (s : seq {group gT}) :=
map section_repr (pairmap (@mkSec _) G s).
End Sections.
Section CompositionSeries.
Variable gT : finGroupType.
Local Notation gTg := {group gT}.
Implicit Types (G : gTg) (s : seq gTg).
Local Notation compo := [rel x y : {set gT} | maxnormal y x x].
Definition comps G s := ((last G s) == 1%G) && compo.-series G s.
Lemma compsP G s :
reflect (last G s = 1%G /\ path [rel x y : gTg | maxnormal y x x] G s)
(comps G s).
Proof. by apply: (iffP andP) => [] [/eqP]. Qed.
Lemma trivg_comps G s : comps G s -> (G :==: 1) = (s == [::]).
Proof.
case/andP=> ls cs; apply/eqP/eqP=> [G1 | s1]; last first.
by rewrite s1 /= in ls; apply/eqP.
by case: s {ls} cs => //= H s /andP[/maxgroupp]; rewrite G1 /proper sub1G andbF.
Qed.
Lemma comps_cons G H s : comps G (H :: s) -> comps H s.
Proof. by case/andP => /= ls /andP[_]; rewrite /comps ls. Qed.
Lemma simple_compsP G s : comps G s -> reflect (s = [:: 1%G]) (simple G).
Proof.
move=> cs; apply: (iffP idP) => [|s1]; last first.
by rewrite s1 /comps eqxx /= andbT -simple_maxnormal in cs.
case: s cs => [/trivg_comps/eqP-> | H s]; first by case/simpleP; rewrite eqxx.
rewrite [comps _ _]andbCA /= => /andP[/maxgroupp maxH /trivg_comps/esym nil_s].
rewrite simple_maxnormal => /maxgroupP[_ simG].
have H1: H = 1%G by apply/val_inj/simG; rewrite // sub1G.
by move: nil_s; rewrite H1 eqxx => /eqP->.
Qed.
Lemma exists_comps (G : gTg) : exists s, comps G s.
Proof.
elim: {G} #|G| {1 3}G (leqnn #|G|) => [G | n IHn G cG].
by rewrite leqNgt cardG_gt0.
have [sG | nsG] := boolP (simple G).
by exists [:: 1%G]; rewrite /comps eqxx /= -simple_maxnormal andbT.
have [-> | ntG] := eqVneq G 1%G; first by exists [::]; rewrite /comps eqxx.
have [N maxN] := ex_maxnormal_ntrivg ntG.
have [|s /andP[ls cs]] := IHn N.
by rewrite -ltnS (leq_trans _ cG) // proper_card // (maxnormal_proper maxN).
by exists (N :: s); apply/and3P.
Qed.
(******************************************************************************)
(* The factors associated to two composition series of the same group are *)
(* the same up to isomorphism and permutation *)
(******************************************************************************)
Lemma JordanHolderUniqueness (G : gTg) (s1 s2 : seq gTg) :
comps G s1 -> comps G s2 -> perm_eq (mkfactors G s1) (mkfactors G s2).
Proof.
have [n] := ubnP #|G|; elim: n G => // n Hi G in s1 s2 * => /ltnSE-cG cs1 cs2.
have [G1 | ntG] := boolP (G :==: 1).
have -> : s1 = [::] by apply/eqP; rewrite -(trivg_comps cs1).
have -> : s2 = [::] by apply/eqP; rewrite -(trivg_comps cs2).
by rewrite /= perm_refl.
have [sG | nsG] := boolP (simple G).
by rewrite (simple_compsP cs1 sG) (simple_compsP cs2 sG) perm_refl.
case es1: s1 cs1 => [|N1 st1] cs1.
by move: (trivg_comps cs1); rewrite eqxx; move/negP:ntG.
case es2: s2 cs2 => [|N2 st2] cs2 {s1 es1}.
by move: (trivg_comps cs2); rewrite eqxx; move/negP:ntG.
case/andP: cs1 => /= lst1; case/andP=> maxN_1 pst1.
case/andP: cs2 => /= lst2; case/andP=> maxN_2 pst2.
have cN1 : #|N1| < n.
by rewrite (leq_trans _ cG) ?proper_card ?(maxnormal_proper maxN_1).
have cN2 : #|N2| < n.
by rewrite (leq_trans _ cG) ?proper_card ?(maxnormal_proper maxN_2).
case: (N1 =P N2) {s2 es2} => [eN12 |].
by rewrite eN12 /= perm_cons Hi // /comps ?lst2 //= -eN12 lst1.
move/eqP; rewrite -val_eqE /=; move/eqP=> neN12.
have nN1G : N1 <| G by apply: maxnormal_normal.
have nN2G : N2 <| G by apply: maxnormal_normal.
pose N := (N1 :&: N2)%G.
have nNG : N <| G.
by rewrite /normal subIset ?(normal_sub nN1G) //= normsI ?normal_norm.
have iso1 : (G / N1)%G \isog (N2 / N)%G.
rewrite isog_sym /= -(maxnormalM maxN_1 maxN_2) //.
rewrite (@normC _ N1 N2) ?(subset_trans (normal_sub nN1G)) ?normal_norm //.
by rewrite weak_second_isog ?(subset_trans (normal_sub nN2G)) ?normal_norm.
have iso2 : (G / N2)%G \isog (N1 / N)%G.
rewrite isog_sym /= -(maxnormalM maxN_1 maxN_2) // setIC.
by rewrite weak_second_isog ?(subset_trans (normal_sub nN1G)) ?normal_norm.
have [sN /andP[lsN csN]] := exists_comps N.
have i1 : perm_eq (mksrepr G N1 :: mkfactors N1 st1)
[:: mksrepr G N1, mksrepr N1 N & mkfactors N sN].
rewrite perm_cons -[mksrepr _ _ :: _]/(mkfactors N1 [:: N & sN]).
apply: Hi=> //; rewrite /comps ?lst1 //= lsN csN andbT /=.
rewrite -quotient_simple.
by rewrite -(isog_simple iso2) quotient_simple.
by rewrite (normalS (subsetIl N1 N2) (normal_sub nN1G)).
have i2 : perm_eq (mksrepr G N2 :: mkfactors N2 st2)
[:: mksrepr G N2, mksrepr N2 N & mkfactors N sN].
rewrite perm_cons -[mksrepr _ _ :: _]/(mkfactors N2 [:: N & sN]).
apply: Hi=> //; rewrite /comps ?lst2 //= lsN csN andbT /=.
rewrite -quotient_simple.
by rewrite -(isog_simple iso1) quotient_simple.
by rewrite (normalS (subsetIr N1 N2) (normal_sub nN2G)).
pose fG1 := [:: mksrepr G N1, mksrepr N1 N & mkfactors N sN].
pose fG2 := [:: mksrepr G N2, mksrepr N2 N & mkfactors N sN].
have i3 : perm_eq fG1 fG2.
rewrite (@perm_catCA _ [::_] [::_]) /mksrepr.
rewrite (@section_repr_isog _ (mkSec _ _) (mkSec _ _) iso1).
rewrite -(@section_repr_isog _ (mkSec _ _) (mkSec _ _) iso2).
exact: perm_refl.
apply: (perm_trans i1); apply: (perm_trans i3); rewrite perm_sym.
by apply: perm_trans i2; apply: perm_refl.
Qed.
End CompositionSeries.
(******************************************************************************)
(* Helper lemmas for group actions. *)
(******************************************************************************)
Section MoreGroupAction.
Variables (aT rT : finGroupType).
Variables (A : {group aT}) (D : {group rT}).
Variable to : groupAction A D.
Lemma gactsP (G : {set rT}) : reflect {acts A, on G | to} [acts A, on G | to].
Proof.
apply: (iffP idP) => [nGA x|nGA]; first exact: acts_act.
apply/subsetP=> a Aa /[!inE]; rewrite Aa.
by apply/subsetP=> x; rewrite inE nGA.
Qed.
Lemma gactsM (N1 N2 : {set rT}) :
N1 \subset D -> N2 \subset D ->
[acts A, on N1 | to] -> [acts A, on N2 | to] -> [acts A, on N1 * N2 | to].
Proof.
move=> sN1D sN2D aAN1 aAN2; apply/gactsP=> x Ax y.
apply/idP/idP; case/mulsgP=> y1 y2 N1y1 N2y2 e.
move: (actKin to Ax y); rewrite e; move<-.
rewrite gactM ?groupV ?(subsetP sN1D y1) ?(subsetP sN2D) //.
by apply: mem_mulg; rewrite ?(gactsP _ aAN1) ?(gactsP _ aAN2) // groupV.
rewrite e gactM // ?(subsetP sN1D y1) ?(subsetP sN2D) //.
by apply: mem_mulg; rewrite ?(gactsP _ aAN1) // ?(gactsP _ aAN2).
Qed.
Lemma gactsI (N1 N2 : {set rT}) :
[acts A, on N1 | to] -> [acts A, on N2 | to] -> [acts A, on N1 :&: N2 | to].
Proof.
move=> aAN1 aAN2.
apply/subsetP=> x Ax; rewrite !inE Ax /=; apply/subsetP=> y Ny /[1!inE].
case/setIP: Ny=> N1y N2y; rewrite inE ?astabs_act ?N1y ?N2y //.
- by move/subsetP: aAN2; move/(_ x Ax).
- by move/subsetP: aAN1; move/(_ x Ax).
Qed.
Lemma gastabsP (S : {set rT}) (a : aT) :
a \in A -> reflect (forall x, (to x a \in S) = (x \in S)) (a \in 'N(S | to)).
Proof.
move=> Aa; apply: (iffP idP) => [nSa x|nSa]; first exact: astabs_act.
by rewrite !inE Aa; apply/subsetP=> x; rewrite inE nSa.
Qed.
End MoreGroupAction.
(******************************************************************************)
(* Helper lemmas for quotient actions. *)
(******************************************************************************)
Section MoreQuotientAction.
Variables (aT rT : finGroupType).
Variables (A : {group aT})(D : {group rT}).
Variable to : groupAction A D.
Lemma qact_dom_doms (H : {group rT}) : H \subset D -> qact_dom to H \subset A.
Proof.
by move=> sHD; apply/subsetP=> x; rewrite qact_domE // inE; case/andP.
Qed.
Lemma acts_qact_doms (H : {group rT}) :
H \subset D -> [acts A, on H | to] -> qact_dom to H :=: A.
Proof.
move=> sHD aH; apply/eqP; rewrite eqEsubset; apply/andP.
split; first exact: qact_dom_doms.
apply/subsetP=> x Ax; rewrite qact_domE //; apply/gastabsP=> //.
by move/gactsP: aH; move/(_ x Ax).
Qed.
Lemma qacts_cosetpre (H : {group rT}) (K' : {group coset_of H}) :
H \subset D -> [acts A, on H | to] ->
[acts qact_dom to H, on K' | to / H] ->
[acts A, on coset H @*^-1 K' | to].
Proof.
move=> sHD aH aK'; apply/subsetP=> x Ax; move: (Ax) (subsetP aK').
rewrite -{1}(acts_qact_doms sHD aH) => qdx; move/(_ x qdx) => nx.
rewrite !inE Ax; apply/subsetP=> y; case/morphpreP=> Ny /= K'Hy /[1!inE].
apply/morphpreP; split; first by rewrite acts_qact_dom_norm.
by move/gastabsP: nx; move/(_ qdx (coset H y)); rewrite K'Hy qactE.
Qed.
Lemma qacts_coset (H K : {group rT}) :
H \subset D -> [acts A, on K | to] ->
[acts qact_dom to H, on (coset H) @* K | to / H].
Proof.
move=> sHD aK.
apply/subsetP=> x qdx; rewrite inE qdx inE; apply/subsetP=> y.
case/morphimP=> z Nz Kz /= e; rewrite e inE qactE // imset_f // inE.
move/gactsP: aK; move/(_ x (subsetP (qact_dom_doms sHD) _ qdx) z); rewrite Kz.
move->; move/acts_act: (acts_qact_dom to H); move/(_ x qdx z).
by rewrite Nz andbT.
Qed.
End MoreQuotientAction.
Section StableCompositionSeries.
Variables (aT rT : finGroupType).
Variables (D : {group rT})(A : {group aT}).
Variable to : groupAction A D.
Definition maxainv (B C : {set rT}) :=
[max C of H |
[&& (H <| B), ~~ (B \subset H) & [acts A, on H | to]]].
Section MaxAinvProps.
Variables K N : {group rT}.
Lemma maxainv_norm : maxainv K N -> N <| K.
Proof. by move/maxgroupp; case/andP. Qed.
Lemma maxainv_proper : maxainv K N -> N \proper K.
Proof.
by move/maxgroupp; case/andP; rewrite properE; move/normal_sub->; case/andP.
Qed.
Lemma maxainv_sub : maxainv K N -> N \subset K.
Proof. by move=> h; apply: proper_sub; apply: maxainv_proper. Qed.
Lemma maxainv_ainvar : maxainv K N -> A \subset 'N(N | to).
Proof. by move/maxgroupp; case/and3P. Qed.
Lemma maxainvS : maxainv K N -> N \subset K.
Proof. by move=> pNN; rewrite proper_sub // maxainv_proper. Qed.
Lemma maxainv_exists : K :!=: 1 -> {N : {group rT} | maxainv K N}.
Proof.
move=> nt; apply: ex_maxgroup. exists [1 rT]%G.
rewrite /= normal1 subG1 nt /=.
apply/subsetP=> a Da; rewrite !inE Da /= sub1set !inE.
by rewrite /= -actmE // morph1 eqxx.
Qed.
End MaxAinvProps.
Lemma maxainvM (G H K : {group rT}) :
H \subset D -> K \subset D -> maxainv G H -> maxainv G K ->
H :<>: K -> H * K = G.
Proof.
move: H K => N1 N2 sN1D sN2D pmN1 pmN2 neN12.
have cN12 : commute N1 N2.
apply: normC; apply: (subset_trans (maxainv_sub pmN1)).
by rewrite normal_norm ?maxainv_norm.
wlog nsN21 : G N1 N2 sN1D sN2D pmN1 pmN2 neN12 cN12/ ~~(N1 \subset N2).
move/eqP: (neN12); rewrite eqEsubset negb_and; case/orP=> ns; first by apply.
by rewrite cN12; apply=> //; apply: sym_not_eq.
have nP : N1 * N2 <| G by rewrite normalM ?maxainv_norm.
have sN2P : N2 \subset N1 * N2 by rewrite mulg_subr ?group1.
case/maxgroupP: (pmN1); case/andP=> nN1G pN1G mN1.
case/maxgroupP: (pmN2); case/andP=> nN2G pN2G mN2.
case/andP: pN1G=> nsGN1 ha1; case/andP: pN2G=> nsGN2 ha2.
case e : (G \subset N1 * N2).
by apply/eqP; rewrite eqEsubset e mulG_subG !normal_sub.
have: N1 <*> N2 = N2 by apply: mN2; rewrite /= ?comm_joingE // nP e /= gactsM.
by rewrite comm_joingE // => h; move: nsN21; rewrite -h mulg_subl.
Qed.
Definition asimple (K : {set rT}) := maxainv K 1.
Implicit Types (H K : {group rT}) (s : seq {group rT}).
Lemma asimpleP K :
reflect [/\ K :!=: 1
& forall H, H <| K -> [acts A, on H | to] -> H :=: 1 \/ H :=: K]
(asimple K).
Proof.
apply: (iffP idP).
case/maxgroupP; rewrite normal1 /=; case/andP=> nsK1 aK H1.
rewrite eqEsubset negb_and nsK1 /=; split => // H nHK ha.
case eHK : (H :==: K); first by right; apply/eqP.
left; apply: H1; rewrite ?sub1G // nHK; move/negbT: eHK.
by rewrite eqEsubset negb_and normal_sub //=; move->.
case=> ntK h; apply/maxgroupP; split.
move: ntK; rewrite eqEsubset sub1G andbT normal1; move->.
apply/subsetP=> a Da; rewrite !inE Da /= sub1set !inE.
by rewrite /= -actmE // morph1 eqxx.
move=> H /andP[nHK /andP[nsKH ha]] _.
case: (h _ nHK ha)=> // /eqP; rewrite eqEsubset.
by rewrite (negbTE nsKH) andbF.
Qed.
Definition acomps K s :=
((last K s) == 1%G) && path [rel x y : {group rT} | maxainv x y] K s.
Lemma acompsP K s :
reflect (last K s = 1%G /\ path [rel x y : {group rT} | maxainv x y] K s)
(acomps K s).
Proof. by apply: (iffP andP); case; move/eqP. Qed.
Lemma trivg_acomps K s : acomps K s -> (K :==: 1) = (s == [::]).
Proof.
case/andP=> ls cs; apply/eqP/eqP; last first.
by move=> se; rewrite se /= in ls; apply/eqP.
move=> G1; case: s ls cs => // H s _ /=; case/andP; case/maxgroupP.
by rewrite G1 sub1G andbF.
Qed.
Lemma acomps_cons K H s : acomps K (H :: s) -> acomps H s.
Proof. by case/andP => /= ls; case/andP=> _ p; rewrite /acomps ls. Qed.
Lemma asimple_acompsP K s : acomps K s -> reflect (s = [:: 1%G]) (asimple K).
Proof.
move=> cs; apply: (iffP idP); last first.
by move=> se; move: cs; rewrite se /=; case/andP=> /=; rewrite andbT.
case: s cs.
by rewrite /acomps /= andbT; move/eqP->; case/asimpleP; rewrite eqxx.
move=> H s cs sG; apply/eqP.
rewrite eqseq_cons -(trivg_acomps (acomps_cons cs)) andbC andbb.
case/acompsP: cs => /= ls; case/andP=> mH ps.
case/maxgroupP: sG; case/and3P => _ ntG _ ->; rewrite ?sub1G //.
rewrite (maxainv_norm mH); case/andP: (maxainv_proper mH)=> _ ->.
exact: (maxainv_ainvar mH).
Qed.
Lemma exists_acomps K : exists s, acomps K s.
Proof.
elim: {K} #|K| {1 3}K (leqnn #|K|) => [K | n Hi K cK].
by rewrite leqNgt cardG_gt0.
case/orP: (orbN (asimple K)) => [sK | nsK].
by exists [:: (1%G : {group rT})]; rewrite /acomps eqxx /= andbT.
case/orP: (orbN (K :==: 1))=> [tK | ntK].
by exists (Nil _); rewrite /acomps /= andbT.
case: (maxainv_exists ntK)=> N pmN.
have cN: #|N| <= n.
by rewrite -ltnS (leq_trans _ cK) // proper_card // (maxainv_proper pmN).
case: (Hi _ cN)=> s; case/andP=> lasts ps; exists [:: N & s]; rewrite /acomps.
by rewrite last_cons lasts /= pmN.
Qed.
End StableCompositionSeries.
Arguments maxainv {aT rT D%_G A%_G} to%_gact B%_g C%_g.
Arguments asimple {aT rT D%_G A%_G} to%_gact K%_g.
Section StrongJordanHolder.
Section AuxiliaryLemmas.
Variables aT rT : finGroupType.
Variables (A : {group aT}) (D : {group rT}) (to : groupAction A D).
Lemma maxainv_asimple_quo (G H : {group rT}) :
H \subset D -> maxainv to G H -> asimple (to / H) (G / H).
Proof.
move=> sHD /maxgroupP[/and3P[nHG pHG aH] Hmax].
apply/asimpleP; split; first by rewrite -subG1 quotient_sub1 ?normal_norm.
move=> K' nK'Q aK'.
have: (K' \proper (G / H)) || (G / H == K').
by rewrite properE eqEsubset andbC (normal_sub nK'Q) !andbT orbC orbN.
case/orP=> [ pHQ | eQH]; last by right; apply sym_eq; apply/eqP.
left; pose K := ((coset H) @*^-1 K')%G.
have eK'I : K' \subset (coset H) @* 'N(H).
by rewrite (subset_trans (normal_sub nK'Q)) ?morphimS ?normal_norm.
have eKK' : K' :=: K / H by rewrite /(K / H) morphpreK //=.
suff eKH : K :=: H by rewrite -trivg_quotient eKK' eKH.
have sHK : H \subset K by rewrite -ker_coset kerE morphpreS // sub1set group1.
apply: Hmax => //; apply/and3P; split; last exact: qacts_cosetpre.
by rewrite -(quotientGK nHG) cosetpre_normal.
by move: (proper_subn pHQ); rewrite sub_morphim_pre ?normal_norm.
Qed.
Lemma asimple_quo_maxainv (G H : {group rT}) :
H \subset D -> G \subset D -> [acts A, on G | to] -> [acts A, on H | to] ->
H <| G -> asimple (to / H) (G / H) ->
maxainv to G H.
Proof.
move=> sHD sGD aG aH nHG /asimpleP[ntQ maxQ]; apply/maxgroupP; split.
by rewrite nHG -quotient_sub1 ?normal_norm // subG1 ntQ.
move=> K /and3P[nKG nsGK aK] sHK.
pose K' := (K / H)%G.
have K'dQ : K' <| (G / H)%G by apply: morphim_normal.
have nKH : H <| K by rewrite (normalS _ _ nHG) // normal_sub.
have: K' :=: 1%G \/ K' :=: (G / H).
apply: (maxQ K' K'dQ) => /=.
apply/subsetP=> x Adx. rewrite inE Adx /= inE. apply/subsetP=> y.
rewrite quotientE; case/morphimP=> z Nz Kz ->; rewrite /= !inE qactE //.
have ntoyx : to z x \in 'N(H) by rewrite (acts_qact_dom_norm Adx).
apply/morphimP; exists (to z x) => //.
suff h: qact_dom to H \subset A.
by rewrite astabs_act // (subsetP aK) //; apply: (subsetP h).
by apply/subsetP=> t; rewrite qact_domE // inE; case/andP.
case=> [|/quotient_injG /[!inE]/(_ nKH nHG) c]; last by rewrite c subxx in nsGK.
rewrite /= -trivg_quotient => tK'; apply: (congr1 (@gval _)); move: tK'.
by apply: (@quotient_injG _ H); rewrite ?inE /= ?normal_refl.
Qed.
Lemma asimpleI (N1 N2 : {group rT}) :
N2 \subset 'N(N1) -> N1 \subset D ->
[acts A, on N1 | to] -> [acts A, on N2 | to] ->
asimple (to / N1) (N2 / N1) ->
asimple (to / (N2 :&: N1)) (N2 / (N2 :&: N1)).
Proof.
move=> nN21 sN1D aN1 aN2 /asimpleP[ntQ1 max1].
have [f1 [f1e f1ker f1pre f1im]] := restrmP (coset_morphism N1) nN21.
have hf2' : N2 \subset 'N(N2 :&: N1) by apply: normsI => //; rewrite normG.
have hf2'' : 'ker (coset (N2 :&: N1)) \subset 'ker f1.
by rewrite f1ker !ker_coset.
pose f2 := factm_morphism hf2'' hf2'.
apply/asimpleP; split.
rewrite /= setIC; apply/negP; move: (second_isog nN21); move/isog_eq1->.
by apply/negP.
move=> H nHQ2 aH; pose K := f2 @* H.
have nKQ1 : K <| N2 / N1.
rewrite (_ : N2 / N1 = f2 @* (N2 / (N2 :&: N1))) ?morphim_normal //.
by rewrite morphim_factm f1im.
have sqA : qact_dom to N1 \subset A.
by apply/subsetP=> t; rewrite qact_domE // inE; case/andP.
have nNN2 : (N2 :&: N1) <| N2.
by rewrite /normal subsetIl; apply: normsI => //; apply: normG.
have aKQ1 : [acts qact_dom to N1, on K | to / N1].
pose H':= coset (N2 :&: N1)@*^-1 H.
have eHH' : H :=: H' / (N2 :&: N1) by rewrite cosetpreK.
have -> : K :=: f1 @* H' by rewrite /K eHH' morphim_factm.
have sH'N2 : H' \subset N2.
rewrite /H' eHH' quotientGK ?normal_cosetpre //=.
by rewrite sub_cosetpre_quo ?normal_sub.
have -> : f1 @* H' = coset N1 @* H' by rewrite f1im //=.
apply: qacts_coset => //; apply: qacts_cosetpre => //; last exact: gactsI.
by apply: (subset_trans (subsetIr _ _)).
have injf2 : 'injm f2.
by rewrite ker_factm f1ker /= ker_coset /= subG1 /= -quotientE trivg_quotient.
have iHK : H \isog K.
apply/isogP; pose f3 := restrm_morphism (normal_sub nHQ2) f2.
by exists f3; rewrite 1?injm_restrm // morphim_restrm setIid.
case: (max1 _ nKQ1 aKQ1).
by move/eqP; rewrite -(isog_eq1 iHK); move/eqP->; left.
move=> he /=; right; apply/eqP; rewrite eqEcard normal_sub //=.
move: (second_isog nN21); rewrite setIC; move/card_isog->; rewrite -he.
by move/card_isog: iHK=> <-; rewrite leqnn.
Qed.
End AuxiliaryLemmas.
Variables (aT rT : finGroupType).
Variables (A : {group aT}) (D : {group rT}) (to : groupAction A D).
(******************************************************************************)
(* The factors associated to two A-stable composition series of the same *)
(* group are the same up to isomorphism and permutation *)
(******************************************************************************)
Lemma StrongJordanHolderUniqueness (G : {group rT}) (s1 s2 : seq {group rT}) :
G \subset D -> acomps to G s1 -> acomps to G s2 ->
perm_eq (mkfactors G s1) (mkfactors G s2).
Proof.
have [n] := ubnP #|G|; elim: n G => // n Hi G in s1 s2 * => cG hsD cs1 cs2.
case/orP: (orbN (G :==: 1)) => [tG | ntG].
have -> : s1 = [::] by apply/eqP; rewrite -(trivg_acomps cs1).
have -> : s2 = [::] by apply/eqP; rewrite -(trivg_acomps cs2).
by rewrite /= perm_refl.
case/orP: (orbN (asimple to G))=> [sG | nsG].
have -> : s1 = [:: 1%G ] by apply/(asimple_acompsP cs1).
have -> : s2 = [:: 1%G ] by apply/(asimple_acompsP cs2).
by rewrite /= perm_refl.
case es1: s1 cs1 => [|N1 st1] cs1.
by move: (trivg_comps cs1); rewrite eqxx; move/negP:ntG.
case es2: s2 cs2 => [|N2 st2] cs2 {s1 es1}.
by move: (trivg_comps cs2); rewrite eqxx; move/negP:ntG.
case/andP: cs1 => /= lst1; case/andP=> maxN_1 pst1.
case/andP: cs2 => /= lst2; case/andP=> maxN_2 pst2.
have sN1D : N1 \subset D.
by apply: subset_trans hsD; apply: maxainv_sub maxN_1.
have sN2D : N2 \subset D.
by apply: subset_trans hsD; apply: maxainv_sub maxN_2.
have cN1 : #|N1| < n.
by rewrite -ltnS (leq_trans _ cG) ?ltnS ?proper_card ?(maxainv_proper maxN_1).
have cN2 : #|N2| < n.
by rewrite -ltnS (leq_trans _ cG) ?ltnS ?proper_card ?(maxainv_proper maxN_2).
case: (N1 =P N2) {s2 es2} => [eN12 |].
by rewrite eN12 /= perm_cons Hi // /acomps ?lst2 //= -eN12 lst1.
move/eqP; rewrite -val_eqE /=; move/eqP=> neN12.
have nN1G : N1 <| G by apply: (maxainv_norm maxN_1).
have nN2G : N2 <| G by apply: (maxainv_norm maxN_2).
pose N := (N1 :&: N2)%G.
have nNG : N <| G.
by rewrite /normal subIset ?(normal_sub nN1G) //= normsI ?normal_norm.
have iso1 : (G / N1)%G \isog (N2 / N)%G.
rewrite isog_sym /= -(maxainvM _ _ maxN_1 maxN_2) //.
rewrite (@normC _ N1 N2) ?(subset_trans (normal_sub nN1G)) ?normal_norm //.
by rewrite weak_second_isog ?(subset_trans (normal_sub nN2G)) ?normal_norm.
have iso2 : (G / N2)%G \isog (N1 / N)%G.
rewrite isog_sym /= -(maxainvM _ _ maxN_1 maxN_2) // setIC.
by rewrite weak_second_isog ?(subset_trans (normal_sub nN1G)) ?normal_norm.
case: (exists_acomps to N)=> sN; case/andP=> lsN csN.
have aN1 : [acts A, on N1 | to].
by case/maxgroupP: maxN_1; case/and3P.
have aN2 : [acts A, on N2 | to].
by case/maxgroupP: maxN_2; case/and3P.
have nNN1 : N <| N1.
by apply: (normalS _ _ nNG); rewrite ?subsetIl ?normal_sub.
have nNN2 : N <| N2.
by apply: (normalS _ _ nNG); rewrite ?subsetIr ?normal_sub.
have aN : [ acts A, on N1 :&: N2 | to].
apply/subsetP=> x Ax; rewrite !inE Ax /=; apply/subsetP=> y Ny; rewrite inE.
case/setIP: Ny=> N1y N2y. rewrite inE ?astabs_act ?N1y ?N2y //.
by move/subsetP: aN2; move/(_ x Ax).
by move/subsetP: aN1; move/(_ x Ax).
have i1 : perm_eq (mksrepr G N1 :: mkfactors N1 st1)
[:: mksrepr G N1, mksrepr N1 N & mkfactors N sN].
rewrite perm_cons -[mksrepr _ _ :: _]/(mkfactors N1 [:: N & sN]).
apply: Hi=> //; rewrite /acomps ?lst1 //= lsN csN andbT /=.
apply: asimple_quo_maxainv=> //; first by apply: subIset; rewrite sN1D.
apply: asimpleI => //.
by apply: subset_trans (normal_norm nN2G); apply: normal_sub.
rewrite -quotientMidl (maxainvM _ _ maxN_2) //.
by apply: maxainv_asimple_quo.
by move=> e; apply: neN12.
have i2 : perm_eq (mksrepr G N2 :: mkfactors N2 st2)
[:: mksrepr G N2, mksrepr N2 N & mkfactors N sN].
rewrite perm_cons -[mksrepr _ _ :: _]/(mkfactors N2 [:: N & sN]).
apply: Hi=> //; rewrite /acomps ?lst2 //= lsN csN andbT /=.
apply: asimple_quo_maxainv=> //; first by apply: subIset; rewrite sN1D.
have e : N1 :&: N2 :=: N2 :&: N1 by rewrite setIC.
rewrite (group_inj (setIC N1 N2)); apply: asimpleI => //.
by apply: subset_trans (normal_norm nN1G); apply: normal_sub.
rewrite -quotientMidl (maxainvM _ _ maxN_1) //.
exact: maxainv_asimple_quo.
pose fG1 := [:: mksrepr G N1, mksrepr N1 N & mkfactors N sN].
pose fG2 := [:: mksrepr G N2, mksrepr N2 N & mkfactors N sN].
have i3 : perm_eq fG1 fG2.
rewrite (@perm_catCA _ [::_] [::_]) /mksrepr.
rewrite (@section_repr_isog _ (mkSec _ _) (mkSec _ _) iso1).
rewrite -(@section_repr_isog _ (mkSec _ _) (mkSec _ _) iso2).
exact: perm_refl.
apply: (perm_trans i1); apply: (perm_trans i3); rewrite perm_sym.
by apply: perm_trans i2; apply: perm_refl.
Qed.
End StrongJordanHolder.
|
IsTerminal.lean
|
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.PEmpty
import Mathlib.CategoryTheory.Limits.IsLimit
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Category.Preorder
/-!
# Initial and terminal objects in a category.
In this file we define the predicates `IsTerminal` and `IsInitial` as well as the class
`InitialMonoClass`.
The classes `HasTerminal` and `HasInitial` and the associated notations for terminal and initial
objects are defined in `Terminal.lean`.
## References
* [Stacks: Initial and final objects](https://stacks.math.columbia.edu/tag/002B)
-/
assert_not_exists CategoryTheory.Limits.HasLimit
noncomputable section
universe w w' v v₁ v₂ u u₁ u₂
open CategoryTheory
namespace CategoryTheory.Limits
variable {C : Type u₁} [Category.{v₁} C]
/-- Construct a cone for the empty diagram given an object. -/
@[simps]
def asEmptyCone (X : C) : Cone (Functor.empty.{0} C) :=
{ pt := X
π :=
{ app := by cat_disch } }
/-- Construct a cocone for the empty diagram given an object. -/
@[simps]
def asEmptyCocone (X : C) : Cocone (Functor.empty.{0} C) :=
{ pt := X
ι :=
{ app := by cat_disch } }
/-- `X` is terminal if the cone it induces on the empty diagram is limiting. -/
abbrev IsTerminal (X : C) :=
IsLimit (asEmptyCone X)
/-- `X` is initial if the cocone it induces on the empty diagram is colimiting. -/
abbrev IsInitial (X : C) :=
IsColimit (asEmptyCocone X)
/-- An object `Y` is terminal iff for every `X` there is a unique morphism `X ⟶ Y`. -/
def isTerminalEquivUnique (F : Discrete.{0} PEmpty.{1} ⥤ C) (Y : C) :
IsLimit (⟨Y, by cat_disch, by simp⟩ : Cone F) ≃ ∀ X : C, Unique (X ⟶ Y) where
toFun t X :=
{ default := t.lift ⟨X, ⟨by cat_disch, by simp⟩⟩
uniq := fun f =>
t.uniq ⟨X, ⟨by cat_disch, by simp⟩⟩ f (by simp) }
invFun u :=
{ lift := fun s => (u s.pt).default
uniq := fun s _ _ => (u s.pt).2 _ }
left_inv := by dsimp [Function.LeftInverse]; intro x; simp only [eq_iff_true_of_subsingleton]
right_inv := by
dsimp [Function.RightInverse,Function.LeftInverse]
subsingleton
/-- An object `Y` is terminal if for every `X` there is a unique morphism `X ⟶ Y`
(as an instance). -/
def IsTerminal.ofUnique (Y : C) [h : ∀ X : C, Unique (X ⟶ Y)] : IsTerminal Y where
lift s := (h s.pt).default
fac := fun _ ⟨j⟩ => j.elim
/-- An object `Y` is terminal if for every `X` there is a unique morphism `X ⟶ Y`
(as explicit arguments). -/
def IsTerminal.ofUniqueHom {Y : C} (h : ∀ X : C, X ⟶ Y) (uniq : ∀ (X : C) (m : X ⟶ Y), m = h X) :
IsTerminal Y :=
have : ∀ X : C, Unique (X ⟶ Y) := fun X ↦ ⟨⟨h X⟩, uniq X⟩
IsTerminal.ofUnique Y
/-- If `α` is a preorder with top, then `⊤` is a terminal object. -/
def isTerminalTop {α : Type*} [Preorder α] [OrderTop α] : IsTerminal (⊤ : α) :=
IsTerminal.ofUnique _
/-- Transport a term of type `IsTerminal` across an isomorphism. -/
def IsTerminal.ofIso {Y Z : C} (hY : IsTerminal Y) (i : Y ≅ Z) : IsTerminal Z :=
IsLimit.ofIsoLimit hY
{ hom := { hom := i.hom }
inv := { hom := i.inv } }
/-- If `X` and `Y` are isomorphic, then `X` is terminal iff `Y` is. -/
def IsTerminal.equivOfIso {X Y : C} (e : X ≅ Y) :
IsTerminal X ≃ IsTerminal Y where
toFun h := IsTerminal.ofIso h e
invFun h := IsTerminal.ofIso h e.symm
left_inv _ := Subsingleton.elim _ _
right_inv _ := Subsingleton.elim _ _
/-- An object `X` is initial iff for every `Y` there is a unique morphism `X ⟶ Y`. -/
def isInitialEquivUnique (F : Discrete.{0} PEmpty.{1} ⥤ C) (X : C) :
IsColimit (⟨X, ⟨by cat_disch, by simp⟩⟩ : Cocone F) ≃ ∀ Y : C, Unique (X ⟶ Y) where
toFun t X :=
{ default := t.desc ⟨X, ⟨by cat_disch, by simp⟩⟩
uniq := fun f => t.uniq ⟨X, ⟨by cat_disch, by simp⟩⟩ f (by simp) }
invFun u :=
{ desc := fun s => (u s.pt).default
uniq := fun s _ _ => (u s.pt).2 _ }
left_inv := by dsimp [Function.LeftInverse]; intro; simp only [eq_iff_true_of_subsingleton]
right_inv := by
#adaptation_note /-- 19-07-2025 grind stopped working -/
intro x; dsimp
/-- An object `X` is initial if for every `Y` there is a unique morphism `X ⟶ Y`
(as an instance). -/
def IsInitial.ofUnique (X : C) [h : ∀ Y : C, Unique (X ⟶ Y)] : IsInitial X where
desc s := (h s.pt).default
fac := fun _ ⟨j⟩ => j.elim
/-- An object `X` is initial if for every `Y` there is a unique morphism `X ⟶ Y`
(as explicit arguments). -/
def IsInitial.ofUniqueHom {X : C} (h : ∀ Y : C, X ⟶ Y) (uniq : ∀ (Y : C) (m : X ⟶ Y), m = h Y) :
IsInitial X :=
have : ∀ Y : C, Unique (X ⟶ Y) := fun Y ↦ ⟨⟨h Y⟩, uniq Y⟩
IsInitial.ofUnique X
/-- If `α` is a preorder with bot, then `⊥` is an initial object. -/
def isInitialBot {α : Type*} [Preorder α] [OrderBot α] : IsInitial (⊥ : α) :=
IsInitial.ofUnique _
/-- Transport a term of type `is_initial` across an isomorphism. -/
def IsInitial.ofIso {X Y : C} (hX : IsInitial X) (i : X ≅ Y) : IsInitial Y :=
IsColimit.ofIsoColimit hX
{ hom := { hom := i.hom }
inv := { hom := i.inv } }
/-- If `X` and `Y` are isomorphic, then `X` is initial iff `Y` is. -/
def IsInitial.equivOfIso {X Y : C} (e : X ≅ Y) :
IsInitial X ≃ IsInitial Y where
toFun h := IsInitial.ofIso h e
invFun h := IsInitial.ofIso h e.symm
left_inv _ := Subsingleton.elim _ _
right_inv _ := Subsingleton.elim _ _
/-- Give the morphism to a terminal object from any other. -/
def IsTerminal.from {X : C} (t : IsTerminal X) (Y : C) : Y ⟶ X :=
t.lift (asEmptyCone Y)
/-- Any two morphisms to a terminal object are equal. -/
theorem IsTerminal.hom_ext {X Y : C} (t : IsTerminal X) (f g : Y ⟶ X) : f = g :=
IsLimit.hom_ext t (by simp)
@[simp]
theorem IsTerminal.comp_from {Z : C} (t : IsTerminal Z) {X Y : C} (f : X ⟶ Y) :
f ≫ t.from Y = t.from X :=
t.hom_ext _ _
@[simp]
theorem IsTerminal.from_self {X : C} (t : IsTerminal X) : t.from X = 𝟙 X :=
t.hom_ext _ _
/-- Give the morphism from an initial object to any other. -/
def IsInitial.to {X : C} (t : IsInitial X) (Y : C) : X ⟶ Y :=
t.desc (asEmptyCocone Y)
/-- Any two morphisms from an initial object are equal. -/
theorem IsInitial.hom_ext {X Y : C} (t : IsInitial X) (f g : X ⟶ Y) : f = g :=
IsColimit.hom_ext t (by simp)
@[simp]
theorem IsInitial.to_comp {X : C} (t : IsInitial X) {Y Z : C} (f : Y ⟶ Z) : t.to Y ≫ f = t.to Z :=
t.hom_ext _ _
@[simp]
theorem IsInitial.to_self {X : C} (t : IsInitial X) : t.to X = 𝟙 X :=
t.hom_ext _ _
/-- Any morphism from a terminal object is split mono. -/
theorem IsTerminal.isSplitMono_from {X Y : C} (t : IsTerminal X) (f : X ⟶ Y) : IsSplitMono f :=
IsSplitMono.mk' ⟨t.from _, t.hom_ext _ _⟩
/-- Any morphism to an initial object is split epi. -/
theorem IsInitial.isSplitEpi_to {X Y : C} (t : IsInitial X) (f : Y ⟶ X) : IsSplitEpi f :=
IsSplitEpi.mk' ⟨t.to _, t.hom_ext _ _⟩
/-- Any morphism from a terminal object is mono. -/
theorem IsTerminal.mono_from {X Y : C} (t : IsTerminal X) (f : X ⟶ Y) : Mono f := by
haveI := t.isSplitMono_from f; infer_instance
/-- Any morphism to an initial object is epi. -/
theorem IsInitial.epi_to {X Y : C} (t : IsInitial X) (f : Y ⟶ X) : Epi f := by
haveI := t.isSplitEpi_to f; infer_instance
/-- If `T` and `T'` are terminal, they are isomorphic. -/
@[simps]
def IsTerminal.uniqueUpToIso {T T' : C} (hT : IsTerminal T) (hT' : IsTerminal T') : T ≅ T' where
hom := hT'.from _
inv := hT.from _
/-- If `I` and `I'` are initial, they are isomorphic. -/
@[simps]
def IsInitial.uniqueUpToIso {I I' : C} (hI : IsInitial I) (hI' : IsInitial I') : I ≅ I' where
hom := hI.to _
inv := hI'.to _
variable (C)
section Univ
variable (X : C) {F₁ : Discrete.{w} PEmpty ⥤ C} {F₂ : Discrete.{w'} PEmpty ⥤ C}
/-- Being terminal is independent of the empty diagram, its universe, and the cone over it,
as long as the cone points are isomorphic. -/
def isLimitChangeEmptyCone {c₁ : Cone F₁} (hl : IsLimit c₁) (c₂ : Cone F₂) (hi : c₁.pt ≅ c₂.pt) :
IsLimit c₂ where
lift c := hl.lift ⟨c.pt, by cat_disch, by simp⟩ ≫ hi.hom
uniq c f _ := by
dsimp
rw [← hl.uniq _ (f ≫ hi.inv) _]
· simp only [Category.assoc, Iso.inv_hom_id, Category.comp_id]
· simp
/-- Replacing an empty cone in `IsLimit` by another with the same cone point
is an equivalence. -/
def isLimitEmptyConeEquiv (c₁ : Cone F₁) (c₂ : Cone F₂) (h : c₁.pt ≅ c₂.pt) :
IsLimit c₁ ≃ IsLimit c₂ where
toFun hl := isLimitChangeEmptyCone C hl c₂ h
invFun hl := isLimitChangeEmptyCone C hl c₁ h.symm
left_inv := by dsimp [Function.LeftInverse]; intro; simp only [eq_iff_true_of_subsingleton]
right_inv := by
dsimp [Function.LeftInverse,Function.RightInverse]; intro
simp only [eq_iff_true_of_subsingleton]
/-- If `F` is an empty diagram, then a cone over `F` is limiting iff the cone point is terminal. -/
noncomputable
def isLimitEquivIsTerminalOfIsEmpty {J : Type*} [Category J] [IsEmpty J] {F : J ⥤ C} (c : Cone F) :
IsLimit c ≃ IsTerminal c.pt :=
(IsLimit.whiskerEquivalenceEquiv (equivalenceOfIsEmpty (Discrete PEmpty.{1}) _)).trans
(isLimitEmptyConeEquiv _ _ _ (.refl _))
/-- Being initial is independent of the empty diagram, its universe, and the cocone over it,
as long as the cocone points are isomorphic. -/
def isColimitChangeEmptyCocone {c₁ : Cocone F₁} (hl : IsColimit c₁) (c₂ : Cocone F₂)
(hi : c₁.pt ≅ c₂.pt) : IsColimit c₂ where
desc c := hi.inv ≫ hl.desc ⟨c.pt, by cat_disch, by simp⟩
uniq c f _ := by
dsimp
rw [← hl.uniq _ (hi.hom ≫ f) _]
· simp only [Iso.inv_hom_id_assoc]
· simp
/-- Replacing an empty cocone in `IsColimit` by another with the same cocone point
is an equivalence. -/
def isColimitEmptyCoconeEquiv (c₁ : Cocone F₁) (c₂ : Cocone F₂) (h : c₁.pt ≅ c₂.pt) :
IsColimit c₁ ≃ IsColimit c₂ where
toFun hl := isColimitChangeEmptyCocone C hl c₂ h
invFun hl := isColimitChangeEmptyCocone C hl c₁ h.symm
left_inv := by dsimp [Function.LeftInverse]; intro; simp only [eq_iff_true_of_subsingleton]
right_inv := by
dsimp [Function.LeftInverse,Function.RightInverse]; intro
simp only [eq_iff_true_of_subsingleton]
/-- If `F` is an empty diagram,
then a cocone over `F` is colimiting iff the cocone point is initial. -/
noncomputable
def isColimitEquivIsInitialOfIsEmpty {J : Type*} [Category J] [IsEmpty J]
{F : J ⥤ C} (c : Cocone F) : IsColimit c ≃ IsInitial c.pt :=
(IsColimit.whiskerEquivalenceEquiv (equivalenceOfIsEmpty (Discrete PEmpty.{1}) _)).trans
(isColimitEmptyCoconeEquiv _ _ _ (.refl _))
end Univ
section
variable {C}
/-- An initial object is terminal in the opposite category. -/
def terminalOpOfInitial {X : C} (t : IsInitial X) : IsTerminal (Opposite.op X) where
lift s := (t.to s.pt.unop).op
uniq _ _ _ := Quiver.Hom.unop_inj (t.hom_ext _ _)
/-- An initial object in the opposite category is terminal in the original category. -/
def terminalUnopOfInitial {X : Cᵒᵖ} (t : IsInitial X) : IsTerminal X.unop where
lift s := (t.to (Opposite.op s.pt)).unop
uniq _ _ _ := Quiver.Hom.op_inj (t.hom_ext _ _)
/-- A terminal object is initial in the opposite category. -/
def initialOpOfTerminal {X : C} (t : IsTerminal X) : IsInitial (Opposite.op X) where
desc s := (t.from s.pt.unop).op
uniq _ _ _ := Quiver.Hom.unop_inj (t.hom_ext _ _)
/-- A terminal object in the opposite category is initial in the original category. -/
def initialUnopOfTerminal {X : Cᵒᵖ} (t : IsTerminal X) : IsInitial X.unop where
desc s := (t.from (Opposite.op s.pt)).unop
uniq _ _ _ := Quiver.Hom.op_inj (t.hom_ext _ _)
/-- A category is an `InitialMonoClass` if the canonical morphism of an initial object is a
monomorphism. In practice, this is most useful when given an arbitrary morphism out of the chosen
initial object, see `initial.mono_from`.
Given a terminal object, this is equivalent to the assumption that the unique morphism from initial
to terminal is a monomorphism, which is the second of Freyd's axioms for an AT category.
TODO: This is a condition satisfied by categories with zero objects and morphisms.
-/
class InitialMonoClass (C : Type u₁) [Category.{v₁} C] : Prop where
/-- The map from the (any as stated) initial object to any other object is a
monomorphism -/
isInitial_mono_from : ∀ {I} (X : C) (hI : IsInitial I), Mono (hI.to X)
theorem IsInitial.mono_from [InitialMonoClass C] {I} {X : C} (hI : IsInitial I) (f : I ⟶ X) :
Mono f := by
rw [hI.hom_ext f (hI.to X)]
apply InitialMonoClass.isInitial_mono_from
/-- To show a category is an `InitialMonoClass` it suffices to give an initial object such that
every morphism out of it is a monomorphism. -/
theorem InitialMonoClass.of_isInitial {I : C} (hI : IsInitial I) (h : ∀ X, Mono (hI.to X)) :
InitialMonoClass C where
isInitial_mono_from {I'} X hI' := by
rw [hI'.hom_ext (hI'.to X) ((hI'.uniqueUpToIso hI).hom ≫ hI.to X)]
apply mono_comp
/-- To show a category is an `InitialMonoClass` it suffices to show the unique morphism from an
initial object to a terminal object is a monomorphism. -/
theorem InitialMonoClass.of_isTerminal {I T : C} (hI : IsInitial I) (hT : IsTerminal T)
(_ : Mono (hI.to T)) : InitialMonoClass C :=
InitialMonoClass.of_isInitial hI fun X => mono_of_mono_fac (hI.hom_ext (_ ≫ hT.from X) (hI.to T))
section Comparison
variable {D : Type u₂} [Category.{v₂} D] (G : C ⥤ D)
end Comparison
variable {J : Type u} [Category.{v} J]
/-- From a functor `F : J ⥤ C`, given an initial object of `J`, construct a cone for `J`.
In `limitOfDiagramInitial` we show it is a limit cone. -/
@[simps]
def coneOfDiagramInitial {X : J} (tX : IsInitial X) (F : J ⥤ C) : Cone F where
pt := F.obj X
π :=
{ app := fun j => F.map (tX.to j)
naturality := fun j j' k => by
dsimp
rw [← F.map_comp, Category.id_comp, tX.hom_ext (tX.to j ≫ k) (tX.to j')] }
/-- From a functor `F : J ⥤ C`, given an initial object of `J`, show the cone
`coneOfDiagramInitial` is a limit. -/
def limitOfDiagramInitial {X : J} (tX : IsInitial X) (F : J ⥤ C) :
IsLimit (coneOfDiagramInitial tX F) where
lift s := s.π.app X
uniq s m w := by
conv_lhs => dsimp
simp_rw [← w X, coneOfDiagramInitial_π_app, tX.hom_ext (tX.to X) (𝟙 _)]
simp
/-- From a functor `F : J ⥤ C`, given a terminal object of `J`, construct a cone for `J`,
provided that the morphisms in the diagram are isomorphisms.
In `limitOfDiagramTerminal` we show it is a limit cone. -/
@[simps]
def coneOfDiagramTerminal {X : J} (hX : IsTerminal X) (F : J ⥤ C)
[∀ (i j : J) (f : i ⟶ j), IsIso (F.map f)] : Cone F where
pt := F.obj X
π :=
{ app := fun _ => inv (F.map (hX.from _))
naturality := by
intro i j f
dsimp
simp only [IsIso.eq_inv_comp, IsIso.comp_inv_eq, Category.id_comp, ← F.map_comp,
hX.hom_ext (hX.from i) (f ≫ hX.from j)] }
/-- From a functor `F : J ⥤ C`, given a terminal object of `J` and that the morphisms in the
diagram are isomorphisms, show the cone `coneOfDiagramTerminal` is a limit. -/
def limitOfDiagramTerminal {X : J} (hX : IsTerminal X) (F : J ⥤ C)
[∀ (i j : J) (f : i ⟶ j), IsIso (F.map f)] : IsLimit (coneOfDiagramTerminal hX F) where
lift S := S.π.app _
/-- From a functor `F : J ⥤ C`, given a terminal object of `J`, construct a cocone for `J`.
In `colimitOfDiagramTerminal` we show it is a colimit cocone. -/
@[simps]
def coconeOfDiagramTerminal {X : J} (tX : IsTerminal X) (F : J ⥤ C) : Cocone F where
pt := F.obj X
ι :=
{ app := fun j => F.map (tX.from j)
naturality := fun j j' k => by
dsimp
rw [← F.map_comp, Category.comp_id, tX.hom_ext (k ≫ tX.from j') (tX.from j)] }
/-- From a functor `F : J ⥤ C`, given a terminal object of `J`, show the cocone
`coconeOfDiagramTerminal` is a colimit. -/
def colimitOfDiagramTerminal {X : J} (tX : IsTerminal X) (F : J ⥤ C) :
IsColimit (coconeOfDiagramTerminal tX F) where
desc s := s.ι.app X
uniq s m w := by
conv_rhs => dsimp -- Porting note: why do I need this much firepower?
rw [← w X, coconeOfDiagramTerminal_ι_app, tX.hom_ext (tX.from X) (𝟙 _)]
simp
lemma IsColimit.isIso_ι_app_of_isTerminal {F : J ⥤ C} {c : Cocone F} (hc : IsColimit c)
(X : J) (hX : IsTerminal X) :
IsIso (c.ι.app X) := by
change IsIso (coconePointUniqueUpToIso (colimitOfDiagramTerminal hX F) hc).hom
infer_instance
/-- From a functor `F : J ⥤ C`, given an initial object of `J`, construct a cocone for `J`,
provided that the morphisms in the diagram are isomorphisms.
In `colimitOfDiagramInitial` we show it is a colimit cocone. -/
@[simps]
def coconeOfDiagramInitial {X : J} (hX : IsInitial X) (F : J ⥤ C)
[∀ (i j : J) (f : i ⟶ j), IsIso (F.map f)] : Cocone F where
pt := F.obj X
ι :=
{ app := fun _ => inv (F.map (hX.to _))
naturality := by
intro i j f
dsimp
simp only [IsIso.eq_inv_comp, IsIso.comp_inv_eq, Category.comp_id, ← F.map_comp,
hX.hom_ext (hX.to i ≫ f) (hX.to j)] }
/-- From a functor `F : J ⥤ C`, given an initial object of `J` and that the morphisms in the
diagram are isomorphisms, show the cone `coconeOfDiagramInitial` is a colimit. -/
def colimitOfDiagramInitial {X : J} (hX : IsInitial X) (F : J ⥤ C)
[∀ (i j : J) (f : i ⟶ j), IsIso (F.map f)] : IsColimit (coconeOfDiagramInitial hX F) where
desc S := S.ι.app _
lemma IsLimit.isIso_π_app_of_isInitial {F : J ⥤ C} {c : Cone F} (hc : IsLimit c)
(X : J) (hX : IsInitial X) :
IsIso (c.π.app X) := by
change IsIso (conePointUniqueUpToIso hc (limitOfDiagramInitial hX F)).hom
infer_instance
/-- Any morphism between terminal objects is an isomorphism. -/
lemma isIso_of_isTerminal {X Y : C} (hX : IsTerminal X) (hY : IsTerminal Y) (f : X ⟶ Y) :
IsIso f := by
refine ⟨⟨IsTerminal.from hX Y, ?_⟩⟩
simp only [IsTerminal.comp_from, IsTerminal.from_self, true_and]
apply IsTerminal.hom_ext hY
/-- Any morphism between initial objects is an isomorphism. -/
lemma isIso_of_isInitial {X Y : C} (hX : IsInitial X) (hY : IsInitial Y) (f : X ⟶ Y) :
IsIso f := by
refine ⟨⟨IsInitial.to hY X, ?_⟩⟩
simp only [IsInitial.to_comp, IsInitial.to_self, and_true]
apply IsInitial.hom_ext hX
end
end CategoryTheory.Limits
|
Compatibility.lean
|
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Equivalence
/-! Tools for compatibilities between Dold-Kan equivalences
The purpose of this file is to introduce tools which will enable the
construction of the Dold-Kan equivalence `SimplicialObject C ≌ ChainComplex C ℕ`
for a pseudoabelian category `C` from the equivalence
`Karoubi (SimplicialObject C) ≌ Karoubi (ChainComplex C ℕ)` and the two
equivalences `simplicial_object C ≅ Karoubi (SimplicialObject C)` and
`ChainComplex C ℕ ≅ Karoubi (ChainComplex C ℕ)`.
It is certainly possible to get an equivalence `SimplicialObject C ≌ ChainComplex C ℕ`
using a compositions of the three equivalences above, but then neither the functor
nor the inverse would have good definitional properties. For example, it would be better
if the inverse functor of the equivalence was exactly the functor
`Γ₀ : SimplicialObject C ⥤ ChainComplex C ℕ` which was constructed in `FunctorGamma.lean`.
In this file, given four categories `A`, `A'`, `B`, `B'`, equivalences `eA : A ≅ A'`,
`eB : B ≅ B'`, `e' : A' ≅ B'`, functors `F : A ⥤ B'`, `G : B ⥤ A` equipped with certain
compatibilities, we construct successive equivalences:
- `equivalence₀` from `A` to `B'`, which is the composition of `eA` and `e'`.
- `equivalence₁` from `A` to `B'`, with the same inverse functor as `equivalence₀`,
but whose functor is `F`.
- `equivalence₂` from `A` to `B`, which is the composition of `equivalence₁` and the
inverse of `eB`:
- `equivalence` from `A` to `B`, which has the same functor `F ⋙ eB.inverse` as `equivalence₂`,
but whose inverse functor is `G`.
When extra assumptions are given, we shall also provide simplification lemmas for the
unit and counit isomorphisms of `equivalence`.
(See `Equivalence.lean` for the general strategy of proof of the Dold-Kan equivalence.)
-/
open CategoryTheory CategoryTheory.Category Functor
namespace AlgebraicTopology
namespace DoldKan
namespace Compatibility
variable {A A' B B' : Type*} [Category A] [Category A'] [Category B] [Category B'] (eA : A ≌ A')
(eB : B ≌ B') (e' : A' ≌ B') {F : A ⥤ B'} (hF : eA.functor ⋙ e'.functor ≅ F) {G : B ⥤ A}
(hG : eB.functor ⋙ e'.inverse ≅ G ⋙ eA.functor)
/-- A basic equivalence `A ≅ B'` obtained by composing `eA : A ≅ A'` and `e' : A' ≅ B'`. -/
@[simps! functor inverse unitIso_hom_app]
def equivalence₀ : A ≌ B' :=
eA.trans e'
variable {eA} {e'}
/-- An intermediate equivalence `A ≅ B'` whose functor is `F` and whose inverse is
`e'.inverse ⋙ eA.inverse`. -/
@[simps! functor]
def equivalence₁ : A ≌ B' := (equivalence₀ eA e').changeFunctor hF
theorem equivalence₁_inverse : (equivalence₁ hF).inverse = e'.inverse ⋙ eA.inverse :=
rfl
/-- The counit isomorphism of the equivalence `equivalence₁` between `A` and `B'`. -/
@[simps!]
def equivalence₁CounitIso : (e'.inverse ⋙ eA.inverse) ⋙ F ≅ 𝟭 B' :=
calc
(e'.inverse ⋙ eA.inverse) ⋙ F ≅ (e'.inverse ⋙ eA.inverse) ⋙ eA.functor ⋙ e'.functor :=
isoWhiskerLeft _ hF.symm
_ ≅ e'.inverse ⋙ (eA.inverse ⋙ eA.functor ⋙ e'.functor) := associator _ _ _
_ ≅ e'.inverse ⋙ (eA.inverse ⋙ eA.functor) ⋙ e'.functor :=
isoWhiskerLeft _ (associator _ _ _).symm
_ ≅ e'.inverse ⋙ 𝟭 _ ⋙ e'.functor := isoWhiskerLeft _ (isoWhiskerRight eA.counitIso _)
_ ≅ e'.inverse ⋙ e'.functor := isoWhiskerLeft _ (leftUnitor _)
_ ≅ 𝟭 B' := e'.counitIso
theorem equivalence₁CounitIso_eq : (equivalence₁ hF).counitIso = equivalence₁CounitIso hF := by
ext Y
simp [equivalence₁, equivalence₀]
/-- The unit isomorphism of the equivalence `equivalence₁` between `A` and `B'`. -/
@[simps!]
def equivalence₁UnitIso : 𝟭 A ≅ F ⋙ e'.inverse ⋙ eA.inverse :=
calc
𝟭 A ≅ eA.functor ⋙ eA.inverse := eA.unitIso
_ ≅ eA.functor ⋙ 𝟭 A' ⋙ eA.inverse := isoWhiskerLeft _ (leftUnitor _).symm
_ ≅ eA.functor ⋙ (e'.functor ⋙ e'.inverse) ⋙ eA.inverse :=
isoWhiskerLeft _ (isoWhiskerRight e'.unitIso _)
_ ≅ eA.functor ⋙ (e'.functor ⋙ e'.inverse ⋙ eA.inverse) :=
isoWhiskerLeft _ (associator _ _ _)
_ ≅ (eA.functor ⋙ e'.functor) ⋙ e'.inverse ⋙ eA.inverse := (associator _ _ _).symm
_ ≅ F ⋙ e'.inverse ⋙ eA.inverse := isoWhiskerRight hF _
theorem equivalence₁UnitIso_eq : (equivalence₁ hF).unitIso = equivalence₁UnitIso hF := by
ext X
simp [equivalence₁]
/-- An intermediate equivalence `A ≅ B` obtained as the composition of `equivalence₁` and
the inverse of `eB : B ≌ B'`. -/
@[simps! functor]
def equivalence₂ : A ≌ B :=
(equivalence₁ hF).trans eB.symm
theorem equivalence₂_inverse :
(equivalence₂ eB hF).inverse = eB.functor ⋙ e'.inverse ⋙ eA.inverse :=
rfl
/-- The counit isomorphism of the equivalence `equivalence₂` between `A` and `B`. -/
@[simps!]
def equivalence₂CounitIso : (eB.functor ⋙ e'.inverse ⋙ eA.inverse) ⋙ F ⋙ eB.inverse ≅ 𝟭 B :=
calc
(eB.functor ⋙ e'.inverse ⋙ eA.inverse) ⋙ F ⋙ eB.inverse
≅ eB.functor ⋙ (e'.inverse ⋙ eA.inverse) ⋙ F ⋙ eB.inverse := associator _ _ _
_ ≅ eB.functor ⋙ ((e'.inverse ⋙ eA.inverse) ⋙ F) ⋙ eB.inverse :=
isoWhiskerLeft _ (associator _ _ _).symm
_ ≅ eB.functor ⋙ 𝟭 _ ⋙ eB.inverse :=
isoWhiskerLeft _ (isoWhiskerRight (equivalence₁CounitIso hF) _)
_ ≅ eB.functor ⋙ eB.inverse := isoWhiskerLeft _ (leftUnitor _)
_ ≅ 𝟭 B := eB.unitIso.symm
theorem equivalence₂CounitIso_eq :
(equivalence₂ eB hF).counitIso = equivalence₂CounitIso eB hF := by
ext Y'
simp [equivalence₂, equivalence₁CounitIso_eq]
/-- The unit isomorphism of the equivalence `equivalence₂` between `A` and `B`. -/
@[simps!]
def equivalence₂UnitIso : 𝟭 A ≅ (F ⋙ eB.inverse) ⋙ eB.functor ⋙ e'.inverse ⋙ eA.inverse :=
calc
𝟭 A ≅ F ⋙ e'.inverse ⋙ eA.inverse := equivalence₁UnitIso hF
_ ≅ F ⋙ 𝟭 B' ⋙ e'.inverse ⋙ eA.inverse :=
isoWhiskerLeft _ (leftUnitor _).symm
_ ≅ F ⋙ (eB.inverse ⋙ eB.functor) ⋙ e'.inverse ⋙ eA.inverse :=
isoWhiskerLeft _ (isoWhiskerRight eB.counitIso.symm _)
_ ≅ (F ⋙ eB.inverse ⋙ eB.functor) ⋙ e'.inverse ⋙ eA.inverse :=
(associator _ _ _).symm
_ ≅ ((F ⋙ eB.inverse) ⋙ eB.functor) ⋙ e'.inverse ⋙ eA.inverse :=
isoWhiskerRight (associator _ _ _).symm _
_ ≅ (F ⋙ eB.inverse) ⋙ eB.functor ⋙ e'.inverse ⋙ eA.inverse :=
associator _ _ _
theorem equivalence₂UnitIso_eq : (equivalence₂ eB hF).unitIso = equivalence₂UnitIso eB hF := by
ext X
simp [equivalence₂, equivalence₁]
variable {eB}
/-- The equivalence `A ≅ B` whose functor is `F ⋙ eB.inverse` and
whose inverse is `G : B ≅ A`. -/
@[simps! inverse]
def equivalence : A ≌ B :=
((equivalence₂ eB hF).changeInverse
(calc eB.functor ⋙ e'.inverse ⋙ eA.inverse ≅
(eB.functor ⋙ e'.inverse) ⋙ eA.inverse := (associator _ _ _).symm
_ ≅ (G ⋙ eA.functor) ⋙ eA.inverse := isoWhiskerRight hG _
_ ≅ G ⋙ eA.functor ⋙ eA.inverse := associator _ _ _
_ ≅ G ⋙ 𝟭 A := isoWhiskerLeft _ eA.unitIso.symm
_ ≅ G := G.rightUnitor))
theorem equivalence_functor : (equivalence hF hG).functor = F ⋙ eB.inverse :=
rfl
/-- The isomorphism `eB.functor ⋙ e'.inverse ⋙ e'.functor ≅ eB.functor` deduced
from the counit isomorphism of `e'`. -/
@[simps! hom_app]
def τ₀ : eB.functor ⋙ e'.inverse ⋙ e'.functor ≅ eB.functor :=
calc
eB.functor ⋙ e'.inverse ⋙ e'.functor ≅ eB.functor ⋙ 𝟭 _ := isoWhiskerLeft _ e'.counitIso
_ ≅ eB.functor := Functor.rightUnitor _
/-- The isomorphism `eB.functor ⋙ e'.inverse ⋙ e'.functor ≅ eB.functor` deduced
from the isomorphisms `hF : eA.functor ⋙ e'.functor ≅ F`,
`hG : eB.functor ⋙ e'.inverse ≅ G ⋙ eA.functor` and the datum of
an isomorphism `η : G ⋙ F ≅ eB.functor`. -/
@[simps! hom_app]
def τ₁ (η : G ⋙ F ≅ eB.functor) : eB.functor ⋙ e'.inverse ⋙ e'.functor ≅ eB.functor :=
calc
eB.functor ⋙ e'.inverse ⋙ e'.functor ≅ (eB.functor ⋙ e'.inverse) ⋙ e'.functor :=
(associator _ _ _).symm
_ ≅ (G ⋙ eA.functor) ⋙ e'.functor := isoWhiskerRight hG _
_ ≅ G ⋙ eA.functor ⋙ e'.functor := associator _ _ _
_ ≅ G ⋙ F := isoWhiskerLeft _ hF
_ ≅ eB.functor := η
variable (η : G ⋙ F ≅ eB.functor)
/-- The counit isomorphism of `equivalence`. -/
@[simps!]
def equivalenceCounitIso : G ⋙ F ⋙ eB.inverse ≅ 𝟭 B :=
calc
G ⋙ F ⋙ eB.inverse ≅ (G ⋙ F) ⋙ eB.inverse := (associator _ _ _).symm
_ ≅ eB.functor ⋙ eB.inverse := isoWhiskerRight η _
_ ≅ 𝟭 B := eB.unitIso.symm
variable {η hF hG}
theorem equivalenceCounitIso_eq (hη : τ₀ = τ₁ hF hG η) :
(equivalence hF hG).counitIso = equivalenceCounitIso η := by
ext1; apply NatTrans.ext; ext Y
dsimp [equivalence]
simp only [comp_id, id_comp, Functor.map_comp, equivalence₂CounitIso_eq,
equivalence₂CounitIso_hom_app, assoc, equivalenceCounitIso_hom_app]
simp only [equivalence₂_inverse, comp_obj, ← τ₀_hom_app, hη, τ₁_hom_app, ←
eB.inverse.map_comp_assoc]
rw [hF.inv.naturality_assoc, hF.inv.naturality_assoc]
congr 2
simp only [← e'.functor.map_comp_assoc]
simp only [Functor.comp_map, Equivalence.fun_inv_map, comp_obj, id_obj, map_comp, assoc]
simp only [← e'.functor.map_comp_assoc]
simp only [Iso.inv_hom_id_app_assoc, Iso.inv_hom_id_app, comp_obj, comp_id,
Equivalence.functor_unit_comp, map_id, id_comp]
variable (hF)
/-- The isomorphism `eA.functor ≅ F ⋙ e'.inverse` deduced from the
unit isomorphism of `e'` and the isomorphism `hF : eA.functor ⋙ e'.functor ≅ F`. -/
@[simps!]
def υ : eA.functor ≅ F ⋙ e'.inverse :=
calc
eA.functor ≅ eA.functor ⋙ 𝟭 A' := (rightUnitor _).symm
_ ≅ eA.functor ⋙ e'.functor ⋙ e'.inverse := isoWhiskerLeft _ e'.unitIso
_ ≅ (eA.functor ⋙ e'.functor) ⋙ e'.inverse := (associator _ _ _).symm
_ ≅ F ⋙ e'.inverse := isoWhiskerRight hF _
variable (ε : eA.functor ≅ F ⋙ e'.inverse) (hG)
/-- The unit isomorphism of `equivalence`. -/
@[simps!]
def equivalenceUnitIso : 𝟭 A ≅ (F ⋙ eB.inverse) ⋙ G :=
calc
𝟭 A ≅ eA.functor ⋙ eA.inverse := eA.unitIso
_ ≅ (F ⋙ e'.inverse) ⋙ eA.inverse := isoWhiskerRight ε _
_ ≅ F ⋙ e'.inverse ⋙ eA.inverse := associator _ _ _
_ ≅ F ⋙ 𝟭 B' ⋙ e'.inverse ⋙ eA.inverse := isoWhiskerLeft _ (leftUnitor _).symm
_ ≅ F ⋙ (eB.inverse ⋙ eB.functor) ⋙ e'.inverse ⋙ eA.inverse :=
isoWhiskerLeft _ (isoWhiskerRight eB.counitIso.symm _)
_ ≅ (F ⋙ eB.inverse ⋙ eB.functor) ⋙ e'.inverse ⋙ eA.inverse := (associator _ _ _).symm
_ ≅ ((F ⋙ eB.inverse) ⋙ eB.functor) ⋙ e'.inverse ⋙ eA.inverse :=
isoWhiskerRight (associator _ _ _).symm _
_ ≅ (F ⋙ eB.inverse) ⋙ eB.functor ⋙ e'.inverse ⋙ eA.inverse := associator _ _ _
_ ≅ (F ⋙ eB.inverse) ⋙ (eB.functor ⋙ e'.inverse) ⋙ eA.inverse :=
isoWhiskerLeft _ (associator _ _ _).symm
_ ≅ (F ⋙ eB.inverse) ⋙ (G ⋙ eA.functor) ⋙ eA.inverse :=
isoWhiskerLeft _ (isoWhiskerRight hG _)
_ ≅ ((F ⋙ eB.inverse) ⋙ G ⋙ eA.functor) ⋙ eA.inverse := (associator _ _ _).symm
_ ≅ (((F ⋙ eB.inverse) ⋙ G) ⋙ eA.functor) ⋙ eA.inverse :=
isoWhiskerRight (associator _ _ _).symm _
_ ≅ ((F ⋙ eB.inverse) ⋙ G) ⋙ eA.functor ⋙ eA.inverse := associator _ _ _
_ ≅ ((F ⋙ eB.inverse) ⋙ G) ⋙ 𝟭 A := isoWhiskerLeft _ eA.unitIso.symm
_ ≅ (F ⋙ eB.inverse) ⋙ G := rightUnitor _
variable {ε hF hG}
theorem equivalenceUnitIso_eq (hε : υ hF = ε) :
(equivalence hF hG).unitIso = equivalenceUnitIso hG ε := by
ext1; apply NatTrans.ext; ext X
dsimp [equivalence]
simp only [assoc, comp_id, equivalenceUnitIso_hom_app, equivalence₂_inverse, Functor.comp_obj,
id_comp, equivalence₂UnitIso_eq eB hF, equivalence₂UnitIso_hom_app,
← eA.inverse.map_comp_assoc, assoc, ← hε, υ_hom_app]
end Compatibility
end DoldKan
end AlgebraicTopology
|
Pi.lean
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Tactic.Alias
import Mathlib.Data.Int.Notation
import Mathlib.Tactic.TypeStar
import Mathlib.Util.AssertExists
/-!
# Cast of integers to function types
This file provides a (pointwise) cast from `ℤ` to function types.
## Main declarations
* `Pi.instIntCast`: map `n : ℤ` to the constant function `n : ∀ i, π i`
-/
assert_not_exists OrderedCommMonoid RingHom
namespace Pi
variable {ι : Type*} {π : ι → Type*} [∀ i, IntCast (π i)]
instance instIntCast : IntCast (∀ i, π i) where intCast n _ := n
@[simp]
theorem intCast_apply (n : ℤ) (i : ι) : (n : ∀ i, π i) i = n :=
rfl
theorem intCast_def (n : ℤ) : (n : ∀ i, π i) = fun _ => ↑n :=
rfl
end Pi
@[simp]
theorem Sum.elim_intCast_intCast {α β γ : Type*} [IntCast γ] (n : ℤ) :
Sum.elim (n : α → γ) (n : β → γ) = n :=
Sum.elim_lam_const_lam_const (γ := γ) n
|
KrullTopology.lean
|
/-
Copyright (c) 2022 Sebastian Monnet. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sebastian Monnet
-/
import Mathlib.FieldTheory.Galois.Basic
import Mathlib.Topology.Algebra.FilterBasis
import Mathlib.Topology.Algebra.OpenSubgroup
/-!
# Krull topology
We define the Krull topology on `L ≃ₐ[K] L` for an arbitrary field extension `L/K`. In order to do
this, we first define a `GroupFilterBasis` on `L ≃ₐ[K] L`, whose sets are `E.fixingSubgroup` for
all intermediate fields `E` with `E/K` finite dimensional.
## Main Definitions
- `finiteExts K L`. Given a field extension `L/K`, this is the set of intermediate fields that are
finite-dimensional over `K`.
- `fixedByFinite K L`. Given a field extension `L/K`, `fixedByFinite K L` is the set of
subsets `Gal(L/E)` of `Gal(L/K)`, where `E/K` is finite
- `galBasis K L`. Given a field extension `L/K`, this is the filter basis on `L ≃ₐ[K] L` whose
sets are `Gal(L/E)` for intermediate fields `E` with `E/K` finite.
- `galGroupBasis K L`. This is the same as `galBasis K L`, but with the added structure
that it is a group filter basis on `L ≃ₐ[K] L`, rather than just a filter basis.
- `krullTopology K L`. Given a field extension `L/K`, this is the topology on `L ≃ₐ[K] L`, induced
by the group filter basis `galGroupBasis K L`.
## Main Results
- `krullTopology_t2 K L`. For an integral field extension `L/K`, the topology `krullTopology K L`
is Hausdorff.
- `krullTopology_totallyDisconnected K L`. For an integral field extension `L/K`, the topology
`krullTopology K L` is totally disconnected.
- `IntermediateField.finrank_eq_fixingSubgroup_index`: given a Galois extension `K/k` and an
intermediate field `L`, the `[L : k]` as a natural number is equal to the index of the
fixing subgroup of `L`.
## Notations
- In docstrings, we will write `Gal(L/E)` to denote the fixing subgroup of an intermediate field
`E`. That is, `Gal(L/E)` is the subgroup of `L ≃ₐ[K] L` consisting of automorphisms that fix
every element of `E`. In particular, we distinguish between `L ≃ₐ[E] L` and `Gal(L/E)`, since the
former is defined to be a subgroup of `L ≃ₐ[K] L`, while the latter is a group in its own right.
## Implementation Notes
- `krullTopology K L` is defined as an instance for type class inference.
-/
open scoped Pointwise
/-- Given a field extension `L/K`, `finiteExts K L` is the set of
intermediate field extensions `L/E/K` such that `E/K` is finite. -/
def finiteExts (K : Type*) [Field K] (L : Type*) [Field L] [Algebra K L] :
Set (IntermediateField K L) :=
{E | FiniteDimensional K E}
/-- Given a field extension `L/K`, `fixedByFinite K L` is the set of
subsets `Gal(L/E)` of `L ≃ₐ[K] L`, where `E/K` is finite. -/
def fixedByFinite (K L : Type*) [Field K] [Field L] [Algebra K L] : Set (Subgroup (L ≃ₐ[K] L)) :=
IntermediateField.fixingSubgroup '' finiteExts K L
@[deprecated (since := "2025-03-16")]
alias IntermediateField.finiteDimensional_bot := IntermediateField.instFiniteSubtypeMemBot
@[deprecated (since := "2025-03-12")]
alias IntermediateField.fixingSubgroup.bot := IntermediateField.fixingSubgroup_bot
/-- If `L/K` is a field extension, then we have `Gal(L/K) ∈ fixedByFinite K L`. -/
theorem top_fixedByFinite {K L : Type*} [Field K] [Field L] [Algebra K L] :
⊤ ∈ fixedByFinite K L :=
⟨⊥, IntermediateField.instFiniteSubtypeMemBot K, IntermediateField.fixingSubgroup_bot⟩
@[deprecated (since := "2025-03-16")]
alias finiteDimensional_sup := IntermediateField.finiteDimensional_sup
/-- Given a field extension `L/K`, `galBasis K L` is the filter basis on `L ≃ₐ[K] L` whose sets
are `Gal(L/E)` for intermediate fields `E` with `E/K` finite dimensional. -/
def galBasis (K L : Type*) [Field K] [Field L] [Algebra K L] : FilterBasis (L ≃ₐ[K] L) where
sets := (fun g => g.carrier) '' fixedByFinite K L
nonempty := ⟨⊤, ⊤, top_fixedByFinite, rfl⟩
inter_sets := by
rintro _ _ ⟨_, ⟨E1, h_E1, rfl⟩, rfl⟩ ⟨_, ⟨E2, h_E2, rfl⟩, rfl⟩
have : FiniteDimensional K E1 := h_E1
have : FiniteDimensional K E2 := h_E2
refine ⟨(E1 ⊔ E2).fixingSubgroup.carrier, ⟨_, ⟨_, E1.finiteDimensional_sup E2, rfl⟩, rfl⟩, ?_⟩
exact Set.subset_inter (E1.fixingSubgroup_le le_sup_left) (E2.fixingSubgroup_le le_sup_right)
/-- A subset of `L ≃ₐ[K] L` is a member of `galBasis K L` if and only if it is the underlying set
of `Gal(L/E)` for some finite subextension `E/K`. -/
theorem mem_galBasis_iff (K L : Type*) [Field K] [Field L] [Algebra K L] (U : Set (L ≃ₐ[K] L)) :
U ∈ galBasis K L ↔ U ∈ (fun g => g.carrier) '' fixedByFinite K L :=
Iff.rfl
/-- For a field extension `L/K`, `galGroupBasis K L` is the group filter basis on `L ≃ₐ[K] L`
whose sets are `Gal(L/E)` for finite subextensions `E/K`. -/
def galGroupBasis (K L : Type*) [Field K] [Field L] [Algebra K L] :
GroupFilterBasis (L ≃ₐ[K] L) where
toFilterBasis := galBasis K L
one' := fun ⟨H, _, h2⟩ => h2 ▸ H.one_mem
mul' {U} hU :=
⟨U, hU, by
rcases hU with ⟨H, _, rfl⟩
rintro x ⟨a, haH, b, hbH, rfl⟩
exact H.mul_mem haH hbH⟩
inv' {U} hU :=
⟨U, hU, by
rcases hU with ⟨H, _, rfl⟩
exact fun _ => H.inv_mem'⟩
conj' := by
rintro σ U ⟨H, ⟨E, hE, rfl⟩, rfl⟩
let F : IntermediateField K L := E.map σ.symm.toAlgHom
refine ⟨F.fixingSubgroup.carrier, ⟨⟨F.fixingSubgroup, ⟨F, ?_, rfl⟩, rfl⟩, fun g hg => ?_⟩⟩
· have : FiniteDimensional K E := hE
exact IntermediateField.finiteDimensional_map σ.symm.toAlgHom
change σ * g * σ⁻¹ ∈ E.fixingSubgroup
rw [IntermediateField.mem_fixingSubgroup_iff]
intro x hx
change σ (g (σ⁻¹ x)) = x
have h_in_F : σ⁻¹ x ∈ F := ⟨x, hx, by dsimp; rw [← AlgEquiv.invFun_eq_symm]; rfl⟩
have h_g_fix : g (σ⁻¹ x) = σ⁻¹ x := by
rw [Subgroup.mem_carrier, IntermediateField.mem_fixingSubgroup_iff F g] at hg
exact hg (σ⁻¹ x) h_in_F
rw [h_g_fix]
change σ (σ⁻¹ x) = x
exact AlgEquiv.apply_symm_apply σ x
/-- For a field extension `L/K`, `krullTopology K L` is the topological space structure on
`L ≃ₐ[K] L` induced by the group filter basis `galGroupBasis K L`. -/
instance krullTopology (K L : Type*) [Field K] [Field L] [Algebra K L] :
TopologicalSpace (L ≃ₐ[K] L) :=
GroupFilterBasis.topology (galGroupBasis K L)
/-- For a field extension `L/K`, the Krull topology on `L ≃ₐ[K] L` makes it a topological group. -/
@[stacks 0BMJ "We define Krull topology directly without proving the universal property"]
instance (K L : Type*) [Field K] [Field L] [Algebra K L] : IsTopologicalGroup (L ≃ₐ[K] L) :=
GroupFilterBasis.isTopologicalGroup (galGroupBasis K L)
open scoped Topology in
lemma krullTopology_mem_nhds_one_iff (K L : Type*) [Field K] [Field L] [Algebra K L]
(s : Set (L ≃ₐ[K] L)) : s ∈ 𝓝 1 ↔ ∃ E : IntermediateField K L,
FiniteDimensional K E ∧ (E.fixingSubgroup : Set (L ≃ₐ[K] L)) ⊆ s := by
rw [GroupFilterBasis.nhds_one_eq]
constructor
· rintro ⟨-, ⟨-, ⟨E, fin, rfl⟩, rfl⟩, hE⟩
exact ⟨E, fin, hE⟩
· rintro ⟨E, fin, hE⟩
exact ⟨E.fixingSubgroup, ⟨E.fixingSubgroup, ⟨E, fin, rfl⟩, rfl⟩, hE⟩
open scoped Topology in
lemma krullTopology_mem_nhds_one_iff_of_normal (K L : Type*) [Field K] [Field L] [Algebra K L]
[Normal K L] (s : Set (L ≃ₐ[K] L)) : s ∈ 𝓝 1 ↔ ∃ E : IntermediateField K L,
FiniteDimensional K E ∧ Normal K E ∧ (E.fixingSubgroup : Set (L ≃ₐ[K] L)) ⊆ s := by
rw [krullTopology_mem_nhds_one_iff]
refine ⟨fun ⟨E, _, hE⟩ ↦ ?_, fun ⟨E, hE⟩ ↦ ⟨E, hE.1, hE.2.2⟩⟩
use (IntermediateField.normalClosure K E L)
simp only [normalClosure.is_finiteDimensional K E L, normalClosure.normal K E L, true_and]
exact le_trans (E.fixingSubgroup_antitone E.le_normalClosure) hE
section KrullT2
open scoped Topology Filter
/-- Let `L/E/K` be a tower of fields with `E/K` finite. Then `Gal(L/E)` is an open subgroup of
`L ≃ₐ[K] L`. -/
theorem IntermediateField.fixingSubgroup_isOpen {K L : Type*} [Field K] [Field L] [Algebra K L]
(E : IntermediateField K L) [FiniteDimensional K E] :
IsOpen (E.fixingSubgroup : Set (L ≃ₐ[K] L)) := by
have h_basis : E.fixingSubgroup.carrier ∈ galGroupBasis K L :=
⟨E.fixingSubgroup, ⟨E, ‹_›, rfl⟩, rfl⟩
have h_nhds := GroupFilterBasis.mem_nhds_one (galGroupBasis K L) h_basis
exact Subgroup.isOpen_of_mem_nhds _ h_nhds
/-- Given a tower of fields `L/E/K`, with `E/K` finite, the subgroup `Gal(L/E) ≤ L ≃ₐ[K] L` is
closed. -/
theorem IntermediateField.fixingSubgroup_isClosed {K L : Type*} [Field K] [Field L] [Algebra K L]
(E : IntermediateField K L) [FiniteDimensional K E] :
IsClosed (E.fixingSubgroup : Set (L ≃ₐ[K] L)) :=
OpenSubgroup.isClosed ⟨E.fixingSubgroup, E.fixingSubgroup_isOpen⟩
/-- If `L/K` is an algebraic extension, then the Krull topology on `L ≃ₐ[K] L` is Hausdorff. -/
theorem krullTopology_t2 {K L : Type*} [Field K] [Field L] [Algebra K L]
[Algebra.IsIntegral K L] : T2Space (L ≃ₐ[K] L) :=
{ t2 := fun f g hfg => by
let φ := f⁻¹ * g
obtain ⟨x, hx⟩ := DFunLike.exists_ne hfg
have hφx : φ x ≠ x := by
apply ne_of_apply_ne f
change f (f.symm (g x)) ≠ f x
rw [AlgEquiv.apply_symm_apply f (g x), ne_comm]
exact hx
let E : IntermediateField K L := IntermediateField.adjoin K {x}
let h_findim : FiniteDimensional K E := IntermediateField.adjoin.finiteDimensional
(Algebra.IsIntegral.isIntegral x)
let H := E.fixingSubgroup
have h_basis : (H : Set (L ≃ₐ[K] L)) ∈ galGroupBasis K L := ⟨H, ⟨E, ⟨h_findim, rfl⟩⟩, rfl⟩
have h_nhds := GroupFilterBasis.mem_nhds_one (galGroupBasis K L) h_basis
rw [mem_nhds_iff] at h_nhds
rcases h_nhds with ⟨W, hWH, hW_open, hW_1⟩
refine ⟨f • W, g • W,
⟨hW_open.leftCoset f, hW_open.leftCoset g, ⟨1, hW_1, mul_one _⟩, ⟨1, hW_1, mul_one _⟩, ?_⟩⟩
rw [Set.disjoint_left]
rintro σ ⟨w1, hw1, h⟩ ⟨w2, hw2, rfl⟩
dsimp at h
rw [eq_inv_mul_iff_mul_eq.symm, ← mul_assoc, mul_inv_eq_iff_eq_mul.symm] at h
have h_in_H : w1 * w2⁻¹ ∈ H := H.mul_mem (hWH hw1) (H.inv_mem (hWH hw2))
rw [h] at h_in_H
change φ ∈ E.fixingSubgroup at h_in_H
rw [IntermediateField.mem_fixingSubgroup_iff] at h_in_H
specialize h_in_H x
have hxE : x ∈ E := by
apply IntermediateField.subset_adjoin
apply Set.mem_singleton
exact hφx (h_in_H hxE) }
end KrullT2
section TotallySeparated
instance {K L : Type*} [Field K] [Field L] [Algebra K L] [Algebra.IsIntegral K L] :
TotallySeparatedSpace (L ≃ₐ[K] L) := by
rw [totallySeparatedSpace_iff_exists_isClopen]
intro σ τ h_diff
have hστ : σ⁻¹ * τ ≠ 1 := by rwa [Ne, inv_mul_eq_one]
rcases DFunLike.exists_ne hστ with ⟨x, hx : (σ⁻¹ * τ) x ≠ x⟩
let E := IntermediateField.adjoin K ({x} : Set L)
haveI := IntermediateField.adjoin.finiteDimensional
(Algebra.IsIntegral.isIntegral (R := K) x)
refine ⟨σ • E.fixingSubgroup,
⟨E.fixingSubgroup_isClosed.leftCoset σ, E.fixingSubgroup_isOpen.leftCoset σ⟩,
⟨1, E.fixingSubgroup.one_mem', mul_one σ⟩, ?_⟩
simp only [Set.mem_compl_iff, mem_leftCoset_iff, SetLike.mem_coe,
IntermediateField.mem_fixingSubgroup_iff, not_forall]
exact ⟨x, IntermediateField.mem_adjoin_simple_self K x, hx⟩
/-- If `L/K` is an algebraic field extension, then the Krull topology on `L ≃ₐ[K] L` is
totally disconnected. -/
theorem krullTopology_isTotallySeparated {K L : Type*} [Field K] [Field L] [Algebra K L]
[Algebra.IsIntegral K L] : IsTotallySeparated (Set.univ : Set (L ≃ₐ[K] L)) :=
(totallySeparatedSpace_iff _).mp inferInstance
@[deprecated (since := "2025-04-03")]
alias krullTopology_totallyDisconnected := krullTopology_isTotallySeparated
end TotallySeparated
instance krullTopology_discreteTopology_of_finiteDimensional (K L : Type*) [Field K] [Field L]
[Algebra K L] [FiniteDimensional K L] : DiscreteTopology (L ≃ₐ[K] L) := by
rw [discreteTopology_iff_isOpen_singleton_one]
change IsOpen ((⊥ : Subgroup (L ≃ₐ[K] L)) : Set (L ≃ₐ[K] L))
rw [← IntermediateField.fixingSubgroup_top]
exact IntermediateField.fixingSubgroup_isOpen ⊤
namespace IntermediateField
variable {k E : Type*} (K : Type*) [Field k] [Field E] [Field K]
[Algebra k E] [Algebra k K] [Algebra E K] [IsScalarTower k E K] (L : IntermediateField k E)
/-- If `K / E / k` is a field extension tower with `E / k` normal,
`L` is an intermediate field of `E / k`, then the fixing subgroup of `L` viewed as an
intermediate field of `K / k` is equal to the preimage of the fixing subgroup of `L` viewed as an
intermediate field of `E / k` under the natural map `Aut(K / k) → Aut(E / k)`
(`AlgEquiv.restrictNormalHom`). -/
theorem map_fixingSubgroup [Normal k E] :
(L.map (IsScalarTower.toAlgHom k E K)).fixingSubgroup =
L.fixingSubgroup.comap (AlgEquiv.restrictNormalHom (F := k) (K₁ := K) E) := by
ext f
simp only [Subgroup.mem_comap, mem_fixingSubgroup_iff]
constructor
· rintro h x hx
change f.restrictNormal E x = x
apply_fun _ using (algebraMap E K).injective
rw [AlgEquiv.restrictNormal_commutes]
exact h _ ⟨x, hx, rfl⟩
· rintro h _ ⟨x, hx, rfl⟩
replace h := congr(algebraMap E K $(show f.restrictNormal E x = x from h x hx))
rwa [AlgEquiv.restrictNormal_commutes] at h
/-- If `K / E / k` is a field extension tower with `E / k` and `K / k` normal,
`L` is an intermediate field of `E / k`, then the index of the fixing subgroup of `L` viewed as an
intermediate field of `K / k` is equal to the index of the fixing subgroup of `L` viewed as an
intermediate field of `E / k`. -/
theorem map_fixingSubgroup_index [Normal k E] [Normal k K] :
(L.map (IsScalarTower.toAlgHom k E K)).fixingSubgroup.index = L.fixingSubgroup.index := by
rw [L.map_fixingSubgroup K, L.fixingSubgroup.index_comap_of_surjective
(AlgEquiv.restrictNormalHom_surjective _)]
variable {K} in
/-- If `K / k` is a Galois extension, `L` is an intermediate field of `K / k`, then `[L : k]`
as a natural number is equal to the index of the fixing subgroup of `L`. -/
theorem finrank_eq_fixingSubgroup_index (L : IntermediateField k K) [IsGalois k K] :
Module.finrank k L = L.fixingSubgroup.index := by
wlog hnfd : FiniteDimensional k L generalizing L
· rw [Module.finrank_of_infinite_dimensional hnfd]
by_contra! h
replace h : L.fixingSubgroup.FiniteIndex := ⟨h.symm⟩
obtain ⟨L', hfd, hL'⟩ :=
exists_lt_finrank_of_infinite_dimensional hnfd L.fixingSubgroup.index
let i := (liftAlgEquiv L').toLinearEquiv
replace hfd := i.finiteDimensional
rw [i.finrank_eq, this _ hfd] at hL'
exact (Subgroup.index_antitone <| fixingSubgroup_le <|
IntermediateField.lift_le L').not_gt hL'
let E := normalClosure k L K
have hle : L ≤ E := by simpa only [fieldRange_val] using L.val.fieldRange_le_normalClosure
let L' := restrict hle
have h := Module.finrank_mul_finrank k ↥L' ↥E
classical
rw [← IsGalois.card_fixingSubgroup_eq_finrank L', ← IsGalois.card_aut_eq_finrank k E] at h
rw [← L'.fixingSubgroup.index_mul_card, Nat.mul_left_inj Finite.card_pos.ne'] at h
rw [(restrict_algEquiv hle).toLinearEquiv.finrank_eq, h, ← L'.map_fixingSubgroup_index K]
congr 2
exact lift_restrict hle
end IntermediateField
|
action.v
|
(* (c) Copyright 2006-2016 Microsoft Corporation and Inria. *)
(* Distributed under the terms of CeCILL-B. *)
From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype.
From mathcomp Require Import ssrnat div seq prime fintype bigop finset.
From mathcomp Require Import fingroup morphism perm automorphism quotient.
(******************************************************************************)
(* Group action: orbits, stabilisers, transitivity. *)
(* is_action D to == the function to : T -> aT -> T defines an action *)
(* of D : {set aT} on T. *)
(* action D T == structure for a function defining an action of D. *)
(* act_dom to == the domain D of to : action D rT. *)
(* {action: aT &-> T} == structure for a total action. *)
(* := action [set: aT] T *)
(* TotalAction to1 toM == the constructor for total actions; to1 and toM *)
(* are the proofs of the action identities for 1 and *)
(* a * b, respectively. *)
(* is_groupAction R to == to is a group action on range R: for all a in D, *)
(* the permutation induced by to a is in Aut R. Thus *)
(* the action of D must be trivial outside R. *)
(* groupAction D R == the structure for group actions of D on R. This *)
(* is a telescope on action D rT. *)
(* gact_range to == the range R of to : groupAction D R. *)
(* GroupAction toAut == constructs a groupAction for action to from *)
(* toAut : actm to @* D \subset Aut R (actm to is *)
(* the morphism to {perm rT} associated to 'to'). *)
(* orbit to A x == the orbit of x under the action of A via to. *)
(* orbit_transversal to A S == a transversal of the partition orbit to A @: S *)
(* of S, provided A acts on S via to. *)
(* amove to A x y == the set of a in A whose action sends x to y. *)
(* 'C_A[x | to] == the stabiliser of x : rT in A :&: D. *)
(* 'C_A(S | to) == the pointwise stabiliser of S : {set rT} in D :&: A. *)
(* 'N_A(S | to) == the global stabiliser of S : {set rT} in D :&: A. *)
(* 'Fix_(S | to)[a] == the set of fixpoints of a in S. *)
(* 'Fix_(S | to)(A) == the set of fixpoints of A in S. *)
(* In the first three _A can be omitted and defaults to the domain D of to; *)
(* in the last two S can be omitted and defaults to [set: T], so 'Fix_to[a] *)
(* is the set of all fixpoints of a. *)
(* The domain restriction ensures that stabilisers have a canonical group *)
(* structure, but note that 'Fix sets are generally not groups. Indeed, we *)
(* provide alternative definitions when to is a group action on R: *)
(* 'C_(G | to)(A) == the centraliser in R :&: G of the group action of *)
(* D :&: A via to *)
(* 'C_(G | to)[a] == the centraliser in R :&: G of a \in D, via to. *)
(* These sets are groups when G is; G can be omitted: 'C(|to)(A) is the *)
(* centraliser in R of the action of D :&: A via to. *)
(* [acts A, on S | to] == A \subset D acts on the set S via to. *)
(* {acts A, on S | to} == A acts on the set S (Prop statement). *)
(* {acts A, on group G | to} == [acts A, on S | to] /\ G \subset R, i.e., *)
(* A \subset D acts on G \subset R, via *)
(* to : groupAction D R. *)
(* [transitive A, on S | to] == A acts transitively on S. *)
(* [faithful A, on S | to] == A acts faithfully on S. *)
(* acts_irreducibly to A G == A acts irreducibly via the groupAction to *)
(* on the nontrivial group G, i.e., A does *)
(* not act on any nontrivial subgroup of G. *)
(* Important caveat: the definitions of orbit, amove, 'Fix_(S | to)(A), *)
(* transitive and faithful assume that A is a subset of the domain D. As most *)
(* of the permutation actions we consider are total this is usually harmless. *)
(* (Note that the theory of partial actions is only partially developed.) *)
(* In all of the above, to is expected to be the actual action structure, *)
(* not merely the function. There is a special scope %act for actions, and *)
(* constructions and notations for many classical actions: *)
(* 'P == natural action of a permutation group via aperm. *)
(* 'J == internal group action (conjugation) via conjg (_ ^ _). *)
(* 'R == regular group action (right translation) via mulg (_ * _). *)
(* (However, to limit ambiguity, _ * _ is NOT a canonical action.) *)
(* to^* == the action induced by to on {set rT} via to^* (== setact to). *)
(* 'Js == the internal action on subsets via _ :^ _, equivalent to 'J^*. *)
(* 'Rs == the regular action on subsets via rcoset, equivalent to 'R^*. *)
(* 'JG == the conjugation action on {group rT} via (_ :^ _)%G. *)
(* to / H == the action induced by to on coset_of H via qact to H, and *)
(* restricted to (qact_dom to H) == 'N(rcosets H 'N(H) | to^* ). *)
(* 'Q == the action induced to cosets by conjugation; the domain is *)
(* qact_dom 'J H, which is provably equal to 'N(H). *)
(* to %% A == the action of coset_of A via modact to A, with domain D / A *)
(* and support restricted to 'C(D :&: A | to). *)
(* to \ sAD == the action of A via ract to sAD == to, if sAD : A \subset D. *)
(* [Aut G] == the permutation action restricted to Aut G, via autact G. *)
(* <[nRA]> == the action of A on R via actby nRA == to in A and on R, and *)
(* the trivial action elsewhere; here nRA : [acts A, on R | to] *)
(* or nRA : {acts A, on group R | to}. *)
(* to^? == the action induced by to on sT : @subType rT P, via subact to *)
(* with domain subact_dom P to == 'N([set x | P x] | to). *)
(* <<phi>> == the action of phi : D >-> {perm rT}, via mact phi. *)
(* to \o f == the composite action (with domain f @*^-1 D) of the action to *)
(* with f : {morphism G >-> aT}, via comp_act to f. Here f must *)
(* be the actual morphism object (e.g., coset_morphism H), not *)
(* the underlying function (e.g., coset H). *)
(* The explicit application of an action to is usually written (to%act x a), *)
(* but %act can be omitted if to is an abstract action or a set action to^*. *)
(* Note that this form will simplify and expose the acting function. *)
(* There is a %gact scope for group actions; the notations above are *)
(* recognised in %gact when they denote canonical group actions. *)
(* Actions can be used to define morphisms: *)
(* actperm to == the morphism D >-> {perm rT} induced by to. *)
(* actm to a == if a \in D the function on D induced by the action to, else *)
(* the identity function. If to is a group action with range R *)
(* then actm to a is canonically a morphism on R. *)
(* We also define here the restriction operation on permutations (the domain *)
(* of this operations is a stabiliser), and local automorphism groups: *)
(* restr_perm S p == if p acts on S, the permutation with support in S that *)
(* coincides with p on S; else the identity. Note that *)
(* restr_perm is a permutation group morphism that maps *)
(* Aut G to Aut S when S is a subgroup of G. *)
(* Aut_in A G == the local permutation group 'N_A(G | 'P) / 'C_A(G | 'P) *)
(* Usually A is an automorphism group, and then Aut_in A G *)
(* is isomorphic to a subgroup of Aut G, specifically *)
(* restr_perm @* A. *)
(* Finally, gproduct.v will provide a semi-direct group construction that *)
(* maps an external group action to an internal one; the theory of morphisms *)
(* between such products makes use of the following definition: *)
(* morph_act to to' f fA <=> the action of to' on the images of f and fA is *)
(* the image of the action of to, i.e., for all x and a we *)
(* have f (to x a) = to' (f x) (fA a). Note that there is *)
(* no mention of the domains of to and to'; if needed, this *)
(* predicate should be restricted via the {in ...} notation *)
(* and domain conditions should be added. *)
(******************************************************************************)
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Declare Scope action_scope.
Declare Scope groupAction_scope.
Import GroupScope.
Section ActionDef.
Variables (aT : finGroupType) (D : {set aT}) (rT : Type).
Implicit Types a b : aT.
Implicit Type x : rT.
Definition act_morph to x := forall a b, to x (a * b) = to (to x a) b.
Definition is_action to :=
left_injective to /\ forall x, {in D &, act_morph to x}.
Record action := Action {act :> rT -> aT -> rT; _ : is_action act}.
Definition clone_action to :=
let: Action _ toP := to return {type of Action for to} -> action in
fun k => k toP.
End ActionDef.
(* Need to close the Section here to avoid re-declaring all Argument Scopes *)
Delimit Scope action_scope with act.
Bind Scope action_scope with action.
Arguments act_morph {aT rT%_type} to x%_g.
Arguments is_action {aT} D%_g {rT} to.
Arguments act {aT D%_g rT%_type} to%_act x%_g a%_g : rename.
Arguments clone_action [aT D%_g rT%_type to%_act] _.
Notation "{ 'action' aT &-> T }" := (action [set: aT] T)
(format "{ 'action' aT &-> T }") : type_scope.
Notation "[ 'action' 'of' to ]" := (clone_action (@Action _ _ _ to))
(format "[ 'action' 'of' to ]") : form_scope.
Definition act_dom aT D rT of @action aT D rT := D.
Section TotalAction.
Variables (aT : finGroupType) (rT : Type) (to : rT -> aT -> rT).
Hypotheses (to1 : to^~ 1 =1 id) (toM : forall x, act_morph to x).
Lemma is_total_action : is_action setT to.
Proof.
split=> [a | x a b _ _] /=; last by rewrite toM.
by apply: can_inj (to^~ a^-1) _ => x; rewrite -toM ?mulgV.
Qed.
Definition TotalAction := Action is_total_action.
End TotalAction.
Section ActionDefs.
Variables (aT aT' : finGroupType) (D : {set aT}) (D' : {set aT'}).
Definition morph_act rT rT' (to : action D rT) (to' : action D' rT') f fA :=
forall x a, f (to x a) = to' (f x) (fA a).
Variable rT : finType. (* Most definitions require a finType structure on rT *)
Implicit Type to : action D rT.
Implicit Type A : {set aT}.
Implicit Type S : {set rT}.
Definition actm to a := if a \in D then to^~ a else id.
Definition setact to S a := [set to x a | x in S].
Definition orbit to A x := to x @: A.
Definition amove to A x y := [set a in A | to x a == y].
Definition afix to A := [set x | A \subset [set a | to x a == x]].
Definition astab S to := D :&: [set a | S \subset [set x | to x a == x]].
Definition astabs S to := D :&: [set a | S \subset to^~ a @^-1: S].
Definition acts_on A S to := {in A, forall a x, (to x a \in S) = (x \in S)}.
Definition atrans A S to := S \in orbit to A @: S.
Definition faithful A S to := A :&: astab S to \subset [1].
End ActionDefs.
Arguments setact {aT D%_g rT} to%_act S%_g a%_g.
Arguments orbit {aT D%_g rT} to%_act A%_g x%_g.
Arguments amove {aT D%_g rT} to%_act A%_g x%_g y%_g.
Arguments afix {aT D%_g rT} to%_act A%_g.
Arguments astab {aT D%_g rT} S%_g to%_act.
Arguments astabs {aT D%_g rT} S%_g to%_act.
Arguments acts_on {aT D%_g rT} A%_g S%_g to%_act.
Arguments atrans {aT D%_g rT} A%_g S%_g to%_act.
Arguments faithful {aT D%_g rT} A%_g S%_g to%_act.
Notation "to ^*" := (setact to) : function_scope.
Prenex Implicits orbit amove.
Notation "''Fix_' to ( A )" := (afix to A)
(to at level 2, format "''Fix_' to ( A )") : group_scope.
(* camlp4 grammar factoring *)
Notation "''Fix_' ( to ) ( A )" := 'Fix_to(A) (only parsing) : group_scope.
Notation "''Fix_' ( S | to ) ( A )" := (S :&: 'Fix_to(A))
(format "''Fix_' ( S | to ) ( A )") : group_scope.
Notation "''Fix_' to [ a ]" := ('Fix_to([set a]))
(to at level 2, format "''Fix_' to [ a ]") : group_scope.
Notation "''Fix_' ( S | to ) [ a ]" := (S :&: 'Fix_to[a])
(format "''Fix_' ( S | to ) [ a ]") : group_scope.
Notation "''C' ( S | to )" := (astab S to) : group_scope.
Notation "''C_' A ( S | to )" := (A :&: 'C(S | to)) : group_scope.
Notation "''C_' ( A ) ( S | to )" := 'C_A(S | to) (only parsing) : group_scope.
Notation "''C' [ x | to ]" := ('C([set x] | to)) : group_scope.
Notation "''C_' A [ x | to ]" := (A :&: 'C[x | to]) : group_scope.
Notation "''C_' ( A ) [ x | to ]" := 'C_A[x | to] (only parsing) : group_scope.
Notation "''N' ( S | to )" := (astabs S to)
(format "''N' ( S | to )") : group_scope.
Notation "''N_' A ( S | to )" := (A :&: 'N(S | to))
(A at level 2, format "''N_' A ( S | to )") : group_scope.
Notation "[ 'acts' A , 'on' S | to ]" := (A \subset pred_of_set 'N(S | to))
(format "[ 'acts' A , 'on' S | to ]") : form_scope.
Notation "{ 'acts' A , 'on' S | to }" := (acts_on A S to)
(format "{ 'acts' A , 'on' S | to }") : type_scope.
Notation "[ 'transitive' A , 'on' S | to ]" := (atrans A S to)
(format "[ 'transitive' A , 'on' S | to ]") : form_scope.
Notation "[ 'faithful' A , 'on' S | to ]" := (faithful A S to)
(format "[ 'faithful' A , 'on' S | to ]") : form_scope.
Section RawAction.
(* Lemmas that do not require the group structure on the action domain. *)
(* Some lemmas like actMin would be actually be valid for arbitrary rT, *)
(* e.g., for actions on a function type, but would be difficult to use *)
(* as a view due to the confusion between parameters and assumptions. *)
Variables (aT : finGroupType) (D : {set aT}) (rT : finType) (to : action D rT).
Implicit Types (a : aT) (x y : rT) (A B : {set aT}) (S T : {set rT}).
Lemma act_inj : left_injective to. Proof. by case: to => ? []. Qed.
Arguments act_inj : clear implicits.
Lemma actMin x : {in D &, act_morph to x}.
Proof. by case: to => ? []. Qed.
Lemma actmEfun a : a \in D -> actm to a = to^~ a.
Proof. by rewrite /actm => ->. Qed.
Lemma actmE a : a \in D -> actm to a =1 to^~ a.
Proof. by move=> Da; rewrite actmEfun. Qed.
Lemma setactE S a : to^* S a = [set to x a | x in S].
Proof. by []. Qed.
Lemma mem_setact S a x : x \in S -> to x a \in to^* S a.
Proof. exact: imset_f. Qed.
Lemma card_setact S a : #|to^* S a| = #|S|.
Proof. by apply: card_imset; apply: act_inj. Qed.
Lemma setact_is_action : is_action D to^*.
Proof.
split=> [a R S eqRS | a b Da Db S]; last first.
by rewrite /setact /= -imset_comp; apply: eq_imset => x; apply: actMin.
apply/setP=> x; apply/idP/idP=> /(mem_setact a).
by rewrite eqRS => /imsetP[y Sy /act_inj->].
by rewrite -eqRS => /imsetP[y Sy /act_inj->].
Qed.
Canonical set_action := Action setact_is_action.
Lemma orbitE A x : orbit to A x = to x @: A. Proof. by []. Qed.
Lemma orbitP A x y :
reflect (exists2 a, a \in A & to x a = y) (y \in orbit to A x).
Proof. by apply: (iffP imsetP) => [] [a]; exists a. Qed.
Lemma mem_orbit A x a : a \in A -> to x a \in orbit to A x.
Proof. exact: imset_f. Qed.
Lemma afixP A x : reflect (forall a, a \in A -> to x a = x) (x \in 'Fix_to(A)).
Proof.
rewrite inE; apply: (iffP subsetP) => [xfix a /xfix | xfix a Aa].
by rewrite inE => /eqP.
by rewrite inE xfix.
Qed.
Lemma afixS A B : A \subset B -> 'Fix_to(B) \subset 'Fix_to(A).
Proof. by move=> sAB; apply/subsetP=> u /[!inE]; apply: subset_trans. Qed.
Lemma afixU A B : 'Fix_to(A :|: B) = 'Fix_to(A) :&: 'Fix_to(B).
Proof. by apply/setP=> x; rewrite !inE subUset. Qed.
Lemma afix1P a x : reflect (to x a = x) (x \in 'Fix_to[a]).
Proof. by rewrite inE sub1set inE; apply: eqP. Qed.
Lemma astabIdom S : 'C_D(S | to) = 'C(S | to).
Proof. by rewrite setIA setIid. Qed.
Lemma astab_dom S : {subset 'C(S | to) <= D}.
Proof. by move=> a /setIP[]. Qed.
Lemma astab_act S a x : a \in 'C(S | to) -> x \in S -> to x a = x.
Proof.
rewrite 2!inE => /andP[_ cSa] Sx; apply/eqP.
by have /[1!inE] := subsetP cSa x Sx.
Qed.
Lemma astabS S1 S2 : S1 \subset S2 -> 'C(S2 | to) \subset 'C(S1 | to).
Proof.
by move=> sS12; apply/subsetP=> x /[!inE] /andP[->]; apply: subset_trans.
Qed.
Lemma astabsIdom S : 'N_D(S | to) = 'N(S | to).
Proof. by rewrite setIA setIid. Qed.
Lemma astabs_dom S : {subset 'N(S | to) <= D}.
Proof. by move=> a /setIdP[]. Qed.
Lemma astabs_act S a x : a \in 'N(S | to) -> (to x a \in S) = (x \in S).
Proof.
rewrite 2!inE subEproper properEcard => /andP[_].
rewrite (card_preimset _ (act_inj _)) ltnn andbF orbF => /eqP{2}->.
by rewrite inE.
Qed.
Lemma astab_sub S : 'C(S | to) \subset 'N(S | to).
Proof.
apply/subsetP=> a cSa; rewrite !inE (astab_dom cSa).
by apply/subsetP=> x Sx; rewrite inE (astab_act cSa).
Qed.
Lemma astabsC S : 'N(~: S | to) = 'N(S | to).
Proof.
apply/setP=> a; apply/idP/idP=> nSa; rewrite !inE (astabs_dom nSa).
by rewrite -setCS -preimsetC; apply/subsetP=> x; rewrite inE astabs_act.
by rewrite preimsetC setCS; apply/subsetP=> x; rewrite inE astabs_act.
Qed.
Lemma astabsI S T : 'N(S | to) :&: 'N(T | to) \subset 'N(S :&: T | to).
Proof.
apply/subsetP=> a; rewrite !inE -!andbA preimsetI => /and4P[-> nSa _ nTa] /=.
by rewrite setISS.
Qed.
Lemma astabs_setact S a : a \in 'N(S | to) -> to^* S a = S.
Proof.
move=> nSa; apply/eqP; rewrite eqEcard card_setact leqnn andbT.
by apply/subsetP=> _ /imsetP[x Sx ->]; rewrite astabs_act.
Qed.
Lemma astab1_set S : 'C[S | set_action] = 'N(S | to).
Proof.
apply/setP=> a; apply/idP/idP=> nSa.
case/setIdP: nSa => Da; rewrite !inE Da sub1set inE => /eqP defS.
by apply/subsetP=> x Sx; rewrite inE -defS mem_setact.
by rewrite !inE (astabs_dom nSa) sub1set inE /= astabs_setact.
Qed.
Lemma astabs_set1 x : 'N([set x] | to) = 'C[x | to].
Proof.
apply/eqP; rewrite eqEsubset astab_sub andbC setIS //.
by apply/subsetP=> a; rewrite ?(inE,sub1set).
Qed.
Lemma acts_dom A S : [acts A, on S | to] -> A \subset D.
Proof. by move=> nSA; rewrite (subset_trans nSA) ?subsetIl. Qed.
Lemma acts_act A S : [acts A, on S | to] -> {acts A, on S | to}.
Proof. by move=> nAS a Aa x; rewrite astabs_act ?(subsetP nAS). Qed.
Lemma astabCin A S :
A \subset D -> (A \subset 'C(S | to)) = (S \subset 'Fix_to(A)).
Proof.
move=> sAD; apply/subsetP/subsetP=> [sAC x xS | sSF a aA].
by apply/afixP=> a aA; apply: astab_act (sAC _ aA) xS.
rewrite !inE (subsetP sAD _ aA); apply/subsetP=> x xS.
by move/afixP/(_ _ aA): (sSF _ xS) => /[1!inE] ->.
Qed.
Section ActsSetop.
Variables (A : {set aT}) (S T : {set rT}).
Hypotheses (AactS : [acts A, on S | to]) (AactT : [acts A, on T | to]).
Lemma astabU : 'C(S :|: T | to) = 'C(S | to) :&: 'C(T | to).
Proof. by apply/setP=> a; rewrite !inE subUset; case: (a \in D). Qed.
Lemma astabsU : 'N(S | to) :&: 'N(T | to) \subset 'N(S :|: T | to).
Proof.
by rewrite -(astabsC S) -(astabsC T) -(astabsC (S :|: T)) setCU astabsI.
Qed.
Lemma astabsD : 'N(S | to) :&: 'N(T | to) \subset 'N(S :\: T| to).
Proof. by rewrite setDE -(astabsC T) astabsI. Qed.
Lemma actsI : [acts A, on S :&: T | to].
Proof. by apply: subset_trans (astabsI S T); rewrite subsetI AactS. Qed.
Lemma actsU : [acts A, on S :|: T | to].
Proof. by apply: subset_trans astabsU; rewrite subsetI AactS. Qed.
Lemma actsD : [acts A, on S :\: T | to].
Proof. by apply: subset_trans astabsD; rewrite subsetI AactS. Qed.
End ActsSetop.
Lemma acts_in_orbit A S x y :
[acts A, on S | to] -> y \in orbit to A x -> x \in S -> y \in S.
Proof.
by move=> nSA/imsetP[a Aa ->{y}] Sx; rewrite (astabs_act _ (subsetP nSA a Aa)).
Qed.
Lemma subset_faithful A B S :
B \subset A -> [faithful A, on S | to] -> [faithful B, on S | to].
Proof. by move=> sAB; apply: subset_trans; apply: setSI. Qed.
Section Reindex.
Variables (vT : Type) (idx : vT) (op : Monoid.com_law idx) (S : {set rT}).
Lemma reindex_astabs a F : a \in 'N(S | to) ->
\big[op/idx]_(i in S) F i = \big[op/idx]_(i in S) F (to i a).
Proof.
move=> nSa; rewrite (reindex_inj (act_inj a)); apply: eq_bigl => x.
exact: astabs_act.
Qed.
Lemma reindex_acts A a F : [acts A, on S | to] -> a \in A ->
\big[op/idx]_(i in S) F i = \big[op/idx]_(i in S) F (to i a).
Proof. by move=> nSA /(subsetP nSA); apply: reindex_astabs. Qed.
End Reindex.
End RawAction.
Arguments act_inj {aT D rT} to a [x1 x2] : rename.
Notation "to ^*" := (set_action to) : action_scope.
Arguments orbitP {aT D rT to A x y}.
Arguments afixP {aT D rT to A x}.
Arguments afix1P {aT D rT to a x}.
Arguments reindex_astabs [aT D rT] to [vT idx op S] a [F].
Arguments reindex_acts [aT D rT] to [vT idx op S A a F].
Section PartialAction.
(* Lemmas that require a (partial) group domain. *)
Variables (aT : finGroupType) (D : {group aT}) (rT : finType).
Variable to : action D rT.
Implicit Types a : aT.
Implicit Types x y : rT.
Implicit Types A B : {set aT}.
Implicit Types G H : {group aT}.
Implicit Types S : {set rT}.
Lemma act1 x : to x 1 = x.
Proof. by apply: (act_inj to 1); rewrite -actMin ?mulg1. Qed.
Lemma actKin : {in D, right_loop invg to}.
Proof. by move=> a Da /= x; rewrite -actMin ?groupV // mulgV act1. Qed.
Lemma actKVin : {in D, rev_right_loop invg to}.
Proof. by move=> a Da /= x; rewrite -{2}(invgK a) actKin ?groupV. Qed.
Lemma setactVin S a : a \in D -> to^* S a^-1 = to^~ a @^-1: S.
Proof.
by move=> Da; apply: can2_imset_pre; [apply: actKVin | apply: actKin].
Qed.
Lemma actXin x a i : a \in D -> to x (a ^+ i) = iter i (to^~ a) x.
Proof.
move=> Da; elim: i => /= [|i <-]; first by rewrite act1.
by rewrite expgSr actMin ?groupX.
Qed.
Lemma afix1 : 'Fix_to(1) = setT.
Proof. by apply/setP=> x; rewrite !inE sub1set inE act1 eqxx. Qed.
Lemma afixD1 G : 'Fix_to(G^#) = 'Fix_to(G).
Proof. by rewrite -{2}(setD1K (group1 G)) afixU afix1 setTI. Qed.
Lemma orbit_refl G x : x \in orbit to G x.
Proof. by rewrite -{1}[x]act1 mem_orbit. Qed.
Local Notation orbit_rel A := (fun x y => x \in orbit to A y).
Lemma contra_orbit G x y : x \notin orbit to G y -> x != y.
Proof. by apply: contraNneq => ->; apply: orbit_refl. Qed.
Lemma orbit_in_sym G : G \subset D -> symmetric (orbit_rel G).
Proof.
move=> sGD; apply: symmetric_from_pre => x y /imsetP[a Ga].
by move/(canLR (actKin (subsetP sGD a Ga))) <-; rewrite mem_orbit ?groupV.
Qed.
Lemma orbit_in_trans G : G \subset D -> transitive (orbit_rel G).
Proof.
move=> sGD _ _ z /imsetP[a Ga ->] /imsetP[b Gb ->].
by rewrite -actMin ?mem_orbit ?groupM // (subsetP sGD).
Qed.
Lemma orbit_in_eqP G x y :
G \subset D -> reflect (orbit to G x = orbit to G y) (x \in orbit to G y).
Proof.
move=> sGD; apply: (iffP idP) => [yGx|<-]; last exact: orbit_refl.
by apply/setP=> z; apply/idP/idP=> /orbit_in_trans-> //; rewrite orbit_in_sym.
Qed.
Lemma orbit_in_transl G x y z :
G \subset D -> y \in orbit to G x ->
(y \in orbit to G z) = (x \in orbit to G z).
Proof.
by move=> sGD Gxy; rewrite !(orbit_in_sym sGD _ z) (orbit_in_eqP y x sGD Gxy).
Qed.
Lemma orbit_act_in x a G :
G \subset D -> a \in G -> orbit to G (to x a) = orbit to G x.
Proof. by move=> sGD /mem_orbit/orbit_in_eqP->. Qed.
Lemma orbit_actr_in x a G y :
G \subset D -> a \in G -> (to y a \in orbit to G x) = (y \in orbit to G x).
Proof. by move=> sGD /mem_orbit/orbit_in_transl->. Qed.
Lemma orbit_inv_in A x y :
A \subset D -> (y \in orbit to A^-1 x) = (x \in orbit to A y).
Proof.
move/subsetP=> sAD; apply/imsetP/imsetP=> [] [a Aa ->].
by exists a^-1; rewrite -?mem_invg ?actKin // -groupV sAD -?mem_invg.
by exists a^-1; rewrite ?memV_invg ?actKin // sAD.
Qed.
Lemma orbit_lcoset_in A a x :
A \subset D -> a \in D ->
orbit to (a *: A) x = orbit to A (to x a).
Proof.
move/subsetP=> sAD Da; apply/setP=> y; apply/imsetP/imsetP=> [] [b Ab ->{y}].
by exists (a^-1 * b); rewrite -?actMin ?mulKVg // ?sAD -?mem_lcoset.
by exists (a * b); rewrite ?mem_mulg ?set11 ?actMin // sAD.
Qed.
Lemma orbit_rcoset_in A a x y :
A \subset D -> a \in D ->
(to y a \in orbit to (A :* a) x) = (y \in orbit to A x).
Proof.
move=> sAD Da; rewrite -orbit_inv_in ?mul_subG ?sub1set // invMg.
by rewrite invg_set1 orbit_lcoset_in ?inv_subG ?groupV ?actKin ?orbit_inv_in.
Qed.
Lemma orbit_conjsg_in A a x y :
A \subset D -> a \in D ->
(to y a \in orbit to (A :^ a) (to x a)) = (y \in orbit to A x).
Proof.
move=> sAD Da; rewrite conjsgE.
by rewrite orbit_lcoset_in ?groupV ?mul_subG ?sub1set ?actKin ?orbit_rcoset_in.
Qed.
Lemma orbit1P G x : reflect (orbit to G x = [set x]) (x \in 'Fix_to(G)).
Proof.
apply: (iffP afixP) => [xfix | xfix a Ga].
apply/eqP; rewrite eq_sym eqEsubset sub1set -{1}[x]act1 imset_f //=.
by apply/subsetP=> y; case/imsetP=> a Ga ->; rewrite inE xfix.
by apply/set1P; rewrite -xfix imset_f.
Qed.
Lemma card_orbit1 G x : #|orbit to G x| = 1%N -> orbit to G x = [set x].
Proof.
move=> orb1; apply/eqP; rewrite eq_sym eqEcard {}orb1 cards1.
by rewrite sub1set orbit_refl.
Qed.
Lemma orbit_partition G S :
[acts G, on S | to] -> partition (orbit to G @: S) S.
Proof.
move=> actsGS; have sGD := acts_dom actsGS.
have eqiG: {in S & &, equivalence_rel [rel x y | y \in orbit to G x]}.
by move=> x y z * /=; rewrite orbit_refl; split=> // /orbit_in_eqP->.
congr (partition _ _): (equivalence_partitionP eqiG).
apply: eq_in_imset => x Sx; apply/setP=> y.
by rewrite inE /= andb_idl // => /acts_in_orbit->.
Qed.
Definition orbit_transversal A S := transversal (orbit to A @: S) S.
Lemma orbit_transversalP G S (P := orbit to G @: S)
(X := orbit_transversal G S) :
[acts G, on S | to] ->
[/\ is_transversal X P S, X \subset S,
{in X &, forall x y, (y \in orbit to G x) = (x == y)}
& forall x, x \in S -> exists2 a, a \in G & to x a \in X].
Proof.
move/orbit_partition; rewrite -/P => partP.
have [/eqP defS tiP _] := and3P partP.
have trXP: is_transversal X P S := transversalP partP.
have sXS: X \subset S := transversal_sub trXP.
split=> // [x y Xx Xy /= | x Sx].
have Sx := subsetP sXS x Xx.
rewrite -(inj_in_eq (pblock_inj trXP)) // eq_pblock ?defS //.
by rewrite (def_pblock tiP (imset_f _ Sx)) ?orbit_refl.
have /imsetP[y Xy defxG]: orbit to G x \in pblock P @: X.
by rewrite (pblock_transversal trXP) ?imset_f.
suffices /orbitP[a Ga def_y]: y \in orbit to G x by exists a; rewrite ?def_y.
by rewrite defxG mem_pblock defS (subsetP sXS).
Qed.
Lemma group_set_astab S : group_set 'C(S | to).
Proof.
apply/group_setP; split=> [|a b cSa cSb].
by rewrite !inE group1; apply/subsetP=> x _; rewrite inE act1.
rewrite !inE groupM ?(@astab_dom _ _ _ to S) //; apply/subsetP=> x Sx.
by rewrite inE actMin ?(@astab_dom _ _ _ to S) ?(astab_act _ Sx).
Qed.
Canonical astab_group S := group (group_set_astab S).
Lemma afix_gen_in A : A \subset D -> 'Fix_to(<<A>>) = 'Fix_to(A).
Proof.
move=> sAD; apply/eqP; rewrite eqEsubset afixS ?sub_gen //=.
by rewrite -astabCin gen_subG ?astabCin.
Qed.
Lemma afix_cycle_in a : a \in D -> 'Fix_to(<[a]>) = 'Fix_to[a].
Proof. by move=> Da; rewrite afix_gen_in ?sub1set. Qed.
Lemma afixYin A B :
A \subset D -> B \subset D -> 'Fix_to(A <*> B) = 'Fix_to(A) :&: 'Fix_to(B).
Proof. by move=> sAD sBD; rewrite afix_gen_in ?afixU // subUset sAD. Qed.
Lemma afixMin G H :
G \subset D -> H \subset D -> 'Fix_to(G * H) = 'Fix_to(G) :&: 'Fix_to(H).
Proof.
by move=> sGD sHD; rewrite -afix_gen_in ?mul_subG // genM_join afixYin.
Qed.
Lemma sub_astab1_in A x :
A \subset D -> (A \subset 'C[x | to]) = (x \in 'Fix_to(A)).
Proof. by move=> sAD; rewrite astabCin ?sub1set. Qed.
Lemma group_set_astabs S : group_set 'N(S | to).
Proof.
apply/group_setP; split=> [|a b cSa cSb].
by rewrite !inE group1; apply/subsetP=> x Sx; rewrite inE act1.
rewrite !inE groupM ?(@astabs_dom _ _ _ to S) //; apply/subsetP=> x Sx.
by rewrite inE actMin ?(@astabs_dom _ _ _ to S) ?astabs_act.
Qed.
Canonical astabs_group S := group (group_set_astabs S).
Lemma astab_norm S : 'N(S | to) \subset 'N('C(S | to)).
Proof.
apply/subsetP=> a nSa; rewrite inE sub_conjg; apply/subsetP=> b cSb.
have [Da Db] := (astabs_dom nSa, astab_dom cSb).
rewrite mem_conjgV !inE groupJ //; apply/subsetP=> x Sx.
rewrite inE !actMin ?groupM ?groupV //.
by rewrite (astab_act cSb) ?actKVin ?astabs_act ?groupV.
Qed.
Lemma astab_normal S : 'C(S | to) <| 'N(S | to).
Proof. by rewrite /normal astab_sub astab_norm. Qed.
Lemma acts_sub_orbit G S x :
[acts G, on S | to] -> (orbit to G x \subset S) = (x \in S).
Proof.
move/acts_act=> GactS.
apply/subsetP/idP=> [| Sx y]; first by apply; apply: orbit_refl.
by case/orbitP=> a Ga <-{y}; rewrite GactS.
Qed.
Lemma acts_orbit G x : G \subset D -> [acts G, on orbit to G x | to].
Proof.
move/subsetP=> sGD; apply/subsetP=> a Ga; rewrite !inE sGD //.
apply/subsetP=> _ /imsetP[b Gb ->].
by rewrite inE -actMin ?sGD // imset_f ?groupM.
Qed.
Lemma acts_subnorm_fix A : [acts 'N_D(A), on 'Fix_to(D :&: A) | to].
Proof.
apply/subsetP=> a nAa; have [Da _] := setIP nAa; rewrite !inE Da.
apply/subsetP=> x Cx /[1!inE]; apply/afixP=> b DAb.
have [Db _]:= setIP DAb; rewrite -actMin // conjgCV actMin ?groupJ ?groupV //.
by rewrite /= (afixP Cx) // memJ_norm // groupV (subsetP (normsGI _ _) _ nAa).
Qed.
Lemma atrans_orbit G x : [transitive G, on orbit to G x | to].
Proof. by apply: imset_f; apply: orbit_refl. Qed.
Section OrbitStabilizer.
Variables (G : {group aT}) (x : rT).
Hypothesis sGD : G \subset D.
Let ssGD := subsetP sGD.
Lemma amove_act a : a \in G -> amove to G x (to x a) = 'C_G[x | to] :* a.
Proof.
move=> Ga; apply/setP=> b; have Da := ssGD Ga.
rewrite mem_rcoset !(inE, sub1set) !groupMr ?groupV //.
by case Gb: (b \in G); rewrite //= actMin ?groupV ?ssGD ?(canF_eq (actKVin Da)).
Qed.
Lemma amove_orbit : amove to G x @: orbit to G x = rcosets 'C_G[x | to] G.
Proof.
apply/setP => Ha; apply/imsetP/rcosetsP=> [[y] | [a Ga ->]].
by case/imsetP=> b Gb -> ->{Ha y}; exists b => //; rewrite amove_act.
by rewrite -amove_act //; exists (to x a); first apply: mem_orbit.
Qed.
Lemma amoveK :
{in orbit to G x, cancel (amove to G x) (fun Ca => to x (repr Ca))}.
Proof.
move=> _ /orbitP[a Ga <-]; rewrite amove_act //= -[G :&: _]/(gval _).
case: repr_rcosetP => b; rewrite !(inE, sub1set)=> /and3P[Gb _ xbx].
by rewrite actMin ?ssGD ?(eqP xbx).
Qed.
Lemma orbit_stabilizer :
orbit to G x = [set to x (repr Ca) | Ca in rcosets 'C_G[x | to] G].
Proof.
rewrite -amove_orbit -imset_comp /=; apply/setP=> z.
by apply/idP/imsetP=> [xGz | [y xGy ->]]; first exists z; rewrite /= ?amoveK.
Qed.
Lemma act_reprK :
{in rcosets 'C_G[x | to] G, cancel (to x \o repr) (amove to G x)}.
Proof.
move=> _ /rcosetsP[a Ga ->] /=; rewrite amove_act ?rcoset_repr //.
rewrite -[G :&: _]/(gval _); case: repr_rcosetP => b /setIP[Gb _].
exact: groupM.
Qed.
End OrbitStabilizer.
Lemma card_orbit_in G x : G \subset D -> #|orbit to G x| = #|G : 'C_G[x | to]|.
Proof.
move=> sGD; rewrite orbit_stabilizer 1?card_in_imset //.
exact: can_in_inj (act_reprK _).
Qed.
Lemma card_orbit_in_stab G x :
G \subset D -> (#|orbit to G x| * #|'C_G[x | to]|)%N = #|G|.
Proof. by move=> sGD; rewrite mulnC card_orbit_in ?Lagrange ?subsetIl. Qed.
Lemma acts_sum_card_orbit G S :
[acts G, on S | to] -> \sum_(T in orbit to G @: S) #|T| = #|S|.
Proof. by move/orbit_partition/card_partition. Qed.
Lemma astab_setact_in S a : a \in D -> 'C(to^* S a | to) = 'C(S | to) :^ a.
Proof.
move=> Da; apply/setP=> b; rewrite mem_conjg !inE -mem_conjg conjGid //.
apply: andb_id2l => Db; rewrite sub_imset_pre; apply: eq_subset_r => x.
by rewrite !inE !actMin ?groupM ?groupV // invgK (canF_eq (actKVin Da)).
Qed.
Lemma astab1_act_in x a : a \in D -> 'C[to x a | to] = 'C[x | to] :^ a.
Proof. by move=> Da; rewrite -astab_setact_in // /setact imset_set1. Qed.
Theorem Frobenius_Cauchy G S : [acts G, on S | to] ->
\sum_(a in G) #|'Fix_(S | to)[a]| = (#|orbit to G @: S| * #|G|)%N.
Proof.
move=> GactS; have sGD := acts_dom GactS.
transitivity (\sum_(a in G) \sum_(x in 'Fix_(S | to)[a]) 1%N).
by apply: eq_bigr => a _; rewrite -sum1_card.
rewrite (exchange_big_dep [in S]) /= => [|a x _]; last by case/setIP.
rewrite (set_partition_big _ (orbit_partition GactS)) -sum_nat_const /=.
apply: eq_bigr => _ /imsetP[x Sx ->].
rewrite -(card_orbit_in_stab x sGD) -sum_nat_const.
apply: eq_bigr => y; rewrite orbit_in_sym // => /imsetP[a Ga defx].
rewrite defx astab1_act_in ?(subsetP sGD) //.
rewrite -{2}(conjGid Ga) -conjIg cardJg -sum1_card setIA (setIidPl sGD).
by apply: eq_bigl => b; rewrite !(sub1set, inE) -(acts_act GactS Ga) -defx Sx.
Qed.
Lemma atrans_dvd_index_in G S :
G \subset D -> [transitive G, on S | to] -> #|S| %| #|G : 'C_G(S | to)|.
Proof.
move=> sGD /imsetP[x Sx {1}->]; rewrite card_orbit_in //.
by rewrite indexgS // setIS // astabS // sub1set.
Qed.
Lemma atrans_dvd_in G S :
G \subset D -> [transitive G, on S | to] -> #|S| %| #|G|.
Proof.
move=> sGD transG; apply: dvdn_trans (atrans_dvd_index_in sGD transG) _.
exact: dvdn_indexg.
Qed.
Lemma atransPin G S :
G \subset D -> [transitive G, on S | to] ->
forall x, x \in S -> orbit to G x = S.
Proof. by move=> sGD /imsetP[y _ ->] x; apply/orbit_in_eqP. Qed.
Lemma atransP2in G S :
G \subset D -> [transitive G, on S | to] ->
{in S &, forall x y, exists2 a, a \in G & y = to x a}.
Proof. by move=> sGD transG x y /(atransPin sGD transG) <- /imsetP. Qed.
Lemma atrans_acts_in G S :
G \subset D -> [transitive G, on S | to] -> [acts G, on S | to].
Proof.
move=> sGD transG; apply/subsetP=> a Ga; rewrite !inE (subsetP sGD) //.
by apply/subsetP=> x /(atransPin sGD transG) <-; rewrite inE imset_f.
Qed.
Lemma subgroup_transitivePin G H S x :
x \in S -> H \subset G -> G \subset D -> [transitive G, on S | to] ->
reflect ('C_G[x | to] * H = G) [transitive H, on S | to].
Proof.
move=> Sx sHG sGD trG; have sHD := subset_trans sHG sGD.
apply: (iffP idP) => [trH | defG].
rewrite group_modr //; apply/setIidPl/subsetP=> a Ga.
have Sxa: to x a \in S by rewrite (acts_act (atrans_acts_in sGD trG)).
have [b Hb xab]:= atransP2in sHD trH Sxa Sx.
have Da := subsetP sGD a Ga; have Db := subsetP sHD b Hb.
rewrite -(mulgK b a) mem_mulg ?groupV // !inE groupM //= sub1set inE.
by rewrite actMin -?xab.
apply/imsetP; exists x => //; apply/setP=> y; rewrite -(atransPin sGD trG Sx).
apply/imsetP/imsetP=> [] [a]; last by exists a; first apply: (subsetP sHG).
rewrite -defG => /imset2P[c b /setIP[_ cxc] Hb ->] ->.
exists b; rewrite ?actMin ?(astab_dom cxc) ?(subsetP sHD) //.
by rewrite (astab_act cxc) ?inE.
Qed.
End PartialAction.
Arguments orbit_transversal {aT D%_g rT} to%_act A%_g S%_g.
Arguments orbit_in_eqP {aT D rT to G x y}.
Arguments orbit1P {aT D rT to G x}.
Arguments contra_orbit [aT D rT] to G [x y].
Notation "''C' ( S | to )" := (astab_group to S) : Group_scope.
Notation "''C_' A ( S | to )" := (setI_group A 'C(S | to)) : Group_scope.
Notation "''C_' ( A ) ( S | to )" := (setI_group A 'C(S | to))
(only parsing) : Group_scope.
Notation "''C' [ x | to ]" := (astab_group to [set x%g]) : Group_scope.
Notation "''C_' A [ x | to ]" := (setI_group A 'C[x | to]) : Group_scope.
Notation "''C_' ( A ) [ x | to ]" := (setI_group A 'C[x | to])
(only parsing) : Group_scope.
Notation "''N' ( S | to )" := (astabs_group to S) : Group_scope.
Notation "''N_' A ( S | to )" := (setI_group A 'N(S | to)) : Group_scope.
Section TotalActions.
(* These lemmas are only established for total actions (domain = [set: rT]) *)
Variable (aT : finGroupType) (rT : finType).
Variable to : {action aT &-> rT}.
Implicit Types (a b : aT) (x y z : rT) (A B : {set aT}) (G H : {group aT}).
Implicit Type S : {set rT}.
Lemma actM x a b : to x (a * b) = to (to x a) b.
Proof. by rewrite actMin ?inE. Qed.
Lemma actK : right_loop invg to.
Proof. by move=> a; apply: actKin; rewrite inE. Qed.
Lemma actKV : rev_right_loop invg to.
Proof. by move=> a; apply: actKVin; rewrite inE. Qed.
Lemma actX x a n : to x (a ^+ n) = iter n (to^~ a) x.
Proof. by elim: n => [|n /= <-]; rewrite ?act1 // -actM expgSr. Qed.
Lemma actCJ a b x : to (to x a) b = to (to x b) (a ^ b).
Proof. by rewrite !actM actK. Qed.
Lemma actCJV a b x : to (to x a) b = to (to x (b ^ a^-1)) a.
Proof. by rewrite (actCJ _ a) conjgKV. Qed.
Lemma orbit_sym G x y : (x \in orbit to G y) = (y \in orbit to G x).
Proof. exact/orbit_in_sym/subsetT. Qed.
Lemma orbit_trans G x y z :
x \in orbit to G y -> y \in orbit to G z -> x \in orbit to G z.
Proof. exact/orbit_in_trans/subsetT. Qed.
Lemma orbit_eqP G x y :
reflect (orbit to G x = orbit to G y) (x \in orbit to G y).
Proof. exact/orbit_in_eqP/subsetT. Qed.
Lemma orbit_transl G x y z :
y \in orbit to G x -> (y \in orbit to G z) = (x \in orbit to G z).
Proof. exact/orbit_in_transl/subsetT. Qed.
Lemma orbit_act G a x: a \in G -> orbit to G (to x a) = orbit to G x.
Proof. exact/orbit_act_in/subsetT. Qed.
Lemma orbit_actr G a x y :
a \in G -> (to y a \in orbit to G x) = (y \in orbit to G x).
Proof. by move/mem_orbit/orbit_transl; apply. Qed.
Lemma orbit_eq_mem G x y :
(orbit to G x == orbit to G y) = (x \in orbit to G y).
Proof. exact: sameP eqP (orbit_eqP G x y). Qed.
Lemma orbit_inv A x y : (y \in orbit to A^-1 x) = (x \in orbit to A y).
Proof. by rewrite orbit_inv_in ?subsetT. Qed.
Lemma orbit_lcoset A a x : orbit to (a *: A) x = orbit to A (to x a).
Proof. by rewrite orbit_lcoset_in ?subsetT ?inE. Qed.
Lemma orbit_rcoset A a x y :
(to y a \in orbit to (A :* a) x) = (y \in orbit to A x).
Proof. by rewrite orbit_rcoset_in ?subsetT ?inE. Qed.
Lemma orbit_conjsg A a x y :
(to y a \in orbit to (A :^ a) (to x a)) = (y \in orbit to A x).
Proof. by rewrite orbit_conjsg_in ?subsetT ?inE. Qed.
Lemma astabP S a : reflect (forall x, x \in S -> to x a = x) (a \in 'C(S | to)).
Proof.
apply: (iffP idP) => [cSa x|cSa]; first exact: astab_act.
by rewrite !inE; apply/subsetP=> x Sx; rewrite inE cSa.
Qed.
Lemma astab1P x a : reflect (to x a = x) (a \in 'C[x | to]).
Proof. by rewrite !inE sub1set inE; apply: eqP. Qed.
Lemma sub_astab1 A x : (A \subset 'C[x | to]) = (x \in 'Fix_to(A)).
Proof. by rewrite sub_astab1_in ?subsetT. Qed.
Lemma astabC A S : (A \subset 'C(S | to)) = (S \subset 'Fix_to(A)).
Proof. by rewrite astabCin ?subsetT. Qed.
Lemma afix_cycle a : 'Fix_to(<[a]>) = 'Fix_to[a].
Proof. by rewrite afix_cycle_in ?inE. Qed.
Lemma afix_gen A : 'Fix_to(<<A>>) = 'Fix_to(A).
Proof. by rewrite afix_gen_in ?subsetT. Qed.
Lemma afixM G H : 'Fix_to(G * H) = 'Fix_to(G) :&: 'Fix_to(H).
Proof. by rewrite afixMin ?subsetT. Qed.
Lemma astabsP S a :
reflect (forall x, (to x a \in S) = (x \in S)) (a \in 'N(S | to)).
Proof.
apply: (iffP idP) => [nSa x|nSa]; first exact: astabs_act.
by rewrite !inE; apply/subsetP=> x; rewrite inE nSa.
Qed.
Lemma card_orbit G x : #|orbit to G x| = #|G : 'C_G[x | to]|.
Proof. by rewrite card_orbit_in ?subsetT. Qed.
Lemma dvdn_orbit G x : #|orbit to G x| %| #|G|.
Proof. by rewrite card_orbit dvdn_indexg. Qed.
Lemma card_orbit_stab G x : (#|orbit to G x| * #|'C_G[x | to]|)%N = #|G|.
Proof. by rewrite mulnC card_orbit Lagrange ?subsetIl. Qed.
Lemma actsP A S : reflect {acts A, on S | to} [acts A, on S | to].
Proof.
apply: (iffP idP) => [nSA x|nSA]; first exact: acts_act.
by apply/subsetP=> a Aa /[!inE]; apply/subsetP=> x; rewrite inE nSA.
Qed.
Arguments actsP {A S}.
Lemma setact_orbit A x b : to^* (orbit to A x) b = orbit to (A :^ b) (to x b).
Proof.
apply/setP=> y; apply/idP/idP=> /imsetP[_ /imsetP[a Aa ->] ->{y}].
by rewrite actCJ mem_orbit ?memJ_conjg.
by rewrite -actCJ mem_setact ?mem_orbit.
Qed.
Lemma astab_setact S a : 'C(to^* S a | to) = 'C(S | to) :^ a.
Proof.
apply/setP=> b; rewrite mem_conjg.
apply/astabP/astabP=> stab x => [Sx|].
by rewrite conjgE invgK !actM stab ?actK //; apply/imsetP; exists x.
by case/imsetP=> y Sy ->{x}; rewrite -actM conjgCV actM stab.
Qed.
Lemma astab1_act x a : 'C[to x a | to] = 'C[x | to] :^ a.
Proof. by rewrite -astab_setact /setact imset_set1. Qed.
Lemma atransP G S : [transitive G, on S | to] ->
forall x, x \in S -> orbit to G x = S.
Proof. by case/imsetP=> x _ -> y; apply/orbit_eqP. Qed.
Lemma atransP2 G S : [transitive G, on S | to] ->
{in S &, forall x y, exists2 a, a \in G & y = to x a}.
Proof. by move=> GtrS x y /(atransP GtrS) <- /imsetP. Qed.
Lemma atrans_acts G S : [transitive G, on S | to] -> [acts G, on S | to].
Proof.
move=> GtrS; apply/subsetP=> a Ga; rewrite !inE.
by apply/subsetP=> x /(atransP GtrS) <-; rewrite inE imset_f.
Qed.
Lemma atrans_supgroup G H S :
G \subset H -> [transitive G, on S | to] ->
[transitive H, on S | to] = [acts H, on S | to].
Proof.
move=> sGH trG; apply/idP/idP=> [|actH]; first exact: atrans_acts.
case/imsetP: trG => x Sx defS; apply/imsetP; exists x => //.
by apply/eqP; rewrite eqEsubset acts_sub_orbit ?Sx // defS imsetS.
Qed.
Lemma atrans_acts_card G S :
[transitive G, on S | to] =
[acts G, on S | to] && (#|orbit to G @: S| == 1%N).
Proof.
apply/idP/andP=> [GtrS | [nSG]].
split; first exact: atrans_acts.
rewrite ((_ @: S =P [set S]) _) ?cards1 // eqEsubset sub1set.
apply/andP; split=> //; apply/subsetP=> _ /imsetP[x Sx ->].
by rewrite inE (atransP GtrS).
rewrite eqn_leq andbC lt0n => /andP[/existsP[X /imsetP[x Sx X_Gx]]].
rewrite (cardD1 X) {X}X_Gx imset_f // ltnS leqn0 => /eqP GtrS.
apply/imsetP; exists x => //; apply/eqP.
rewrite eqEsubset acts_sub_orbit // Sx andbT.
apply/subsetP=> y Sy; have:= card0_eq GtrS (orbit to G y).
by rewrite !inE /= imset_f // andbT => /eqP <-; apply: orbit_refl.
Qed.
Lemma atrans_dvd G S : [transitive G, on S | to] -> #|S| %| #|G|.
Proof. by case/imsetP=> x _ ->; apply: dvdn_orbit. Qed.
(* This is Aschbacher (5.2) *)
Lemma acts_fix_norm A B : A \subset 'N(B) -> [acts A, on 'Fix_to(B) | to].
Proof.
move=> nAB; have:= acts_subnorm_fix to B; rewrite !setTI.
exact: subset_trans.
Qed.
Lemma faithfulP A S :
reflect (forall a, a \in A -> {in S, to^~ a =1 id} -> a = 1)
[faithful A, on S | to].
Proof.
apply: (iffP subsetP) => [Cto1 a Aa Ca | Cto1 a].
by apply/set1P; rewrite Cto1 // inE Aa; apply/astabP.
by case/setIP=> Aa /astabP Ca; apply/set1P; apply: Cto1.
Qed.
(* This is the first part of Aschbacher (5.7) *)
Lemma astab_trans_gcore G S u :
[transitive G, on S | to] -> u \in S -> 'C(S | to) = gcore 'C[u | to] G.
Proof.
move=> transG Su; apply/eqP; rewrite eqEsubset.
rewrite gcore_max ?astabS ?sub1set //=; last first.
exact: subset_trans (atrans_acts transG) (astab_norm _ _).
apply/subsetP=> x cSx; apply/astabP=> uy.
case/(atransP2 transG Su) => y Gy ->{uy}.
by apply/astab1P; rewrite astab1_act (bigcapP cSx).
Qed.
(* This is Aschbacher (5.20) *)
Theorem subgroup_transitiveP G H S x :
x \in S -> H \subset G -> [transitive G, on S | to] ->
reflect ('C_G[x | to] * H = G) [transitive H, on S | to].
Proof. by move=> Sx sHG; apply: subgroup_transitivePin (subsetT G). Qed.
(* This is Aschbacher (5.21) *)
Lemma trans_subnorm_fixP x G H S :
let C := 'C_G[x | to] in let T := 'Fix_(S | to)(H) in
[transitive G, on S | to] -> x \in S -> H \subset C ->
reflect ((H :^: G) ::&: C = H :^: C) [transitive 'N_G(H), on T | to].
Proof.
move=> C T trGS Sx sHC; have actGS := acts_act (atrans_acts trGS).
have:= sHC; rewrite subsetI sub_astab1 => /andP[sHG cHx].
have Tx: x \in T by rewrite inE Sx.
apply: (iffP idP) => [trN | trC].
apply/setP=> Ha; apply/setIdP/imsetP=> [[]|[a Ca ->{Ha}]]; last first.
by rewrite conj_subG //; case/setIP: Ca => Ga _; rewrite imset_f.
case/imsetP=> a Ga ->{Ha}; rewrite subsetI !sub_conjg => /andP[_ sHCa].
have Txa: to x a^-1 \in T.
by rewrite inE -sub_astab1 astab1_act actGS ?Sx ?groupV.
have [b] := atransP2 trN Tx Txa; case/setIP=> Gb nHb cxba.
exists (b * a); last by rewrite conjsgM (normP nHb).
by rewrite inE groupM //; apply/astab1P; rewrite actM -cxba actKV.
apply/imsetP; exists x => //; apply/setP=> y; apply/idP/idP=> [Ty|].
have [Sy cHy]:= setIP Ty; have [a Ga defy] := atransP2 trGS Sx Sy.
have: H :^ a^-1 \in H :^: C.
rewrite -trC inE subsetI imset_f 1?conj_subG ?groupV // sub_conjgV.
by rewrite -astab1_act -defy sub_astab1.
case/imsetP=> b /setIP[Gb /astab1P cxb] defHb.
rewrite defy -{1}cxb -actM mem_orbit // inE groupM //.
by apply/normP; rewrite conjsgM -defHb conjsgKV.
case/imsetP=> a /setIP[Ga nHa] ->{y}.
by rewrite inE actGS // Sx (acts_act (acts_fix_norm _) nHa).
Qed.
End TotalActions.
Arguments astabP {aT rT to S a}.
Arguments orbit_eqP {aT rT to G x y}.
Arguments astab1P {aT rT to x a}.
Arguments astabsP {aT rT to S a}.
Arguments atransP {aT rT to G S}.
Arguments actsP {aT rT to A S}.
Arguments faithfulP {aT rT to A S}.
Section Restrict.
Variables (aT : finGroupType) (D : {set aT}) (rT : Type).
Variables (to : action D rT) (A : {set aT}).
Definition ract of A \subset D := act to.
Variable sAD : A \subset D.
Lemma ract_is_action : is_action A (ract sAD).
Proof.
rewrite /ract; case: to => f [injf fM].
by split=> // x; apply: (sub_in2 (subsetP sAD)).
Qed.
Canonical raction := Action ract_is_action.
Lemma ractE : raction =1 to. Proof. by []. Qed.
(* Other properties of raction need rT : finType; we defer them *)
(* until after the definition of actperm. *)
End Restrict.
Notation "to \ sAD" := (raction to sAD) (at level 50) : action_scope.
Section ActBy.
Variables (aT : finGroupType) (D : {set aT}) (rT : finType).
Definition actby_cond (A : {set aT}) R (to : action D rT) : Prop :=
[acts A, on R | to].
Definition actby A R to of actby_cond A R to :=
fun x a => if (x \in R) && (a \in A) then to x a else x.
Variables (A : {group aT}) (R : {set rT}) (to : action D rT).
Hypothesis nRA : actby_cond A R to.
Lemma actby_is_action : is_action A (actby nRA).
Proof.
rewrite /actby; split=> [a x y | x a b Aa Ab /=]; last first.
rewrite Aa Ab groupM // !andbT actMin ?(subsetP (acts_dom nRA)) //.
by case Rx: (x \in R); rewrite ?(acts_act nRA) ?Rx.
case Aa: (a \in A); rewrite ?andbF ?andbT //.
case Rx: (x \in R); case Ry: (y \in R) => // eqxy; first exact: act_inj eqxy.
by rewrite -eqxy (acts_act nRA Aa) Rx in Ry.
by rewrite eqxy (acts_act nRA Aa) Ry in Rx.
Qed.
Canonical action_by := Action actby_is_action.
Local Notation "<[nRA]>" := action_by : action_scope.
Lemma actbyE x a : x \in R -> a \in A -> <[nRA]>%act x a = to x a.
Proof. by rewrite /= /actby => -> ->. Qed.
Lemma afix_actby B : 'Fix_<[nRA]>(B) = ~: R :|: 'Fix_to(A :&: B).
Proof.
apply/setP=> x; rewrite !inE /= /actby.
case: (x \in R); last by apply/subsetP=> a _ /[!inE].
apply/subsetP/subsetP=> [cBx a | cABx a Ba] /[!inE].
by case/andP=> Aa /cBx; rewrite inE Aa.
by case: ifP => //= Aa; have:= cABx a; rewrite !inE Aa => ->.
Qed.
Lemma astab_actby S : 'C(S | <[nRA]>) = 'C_A(R :&: S | to).
Proof.
apply/setP=> a; rewrite setIA (setIidPl (acts_dom nRA)) !inE.
case Aa: (a \in A) => //=; apply/subsetP/subsetP=> cRSa x => [|Sx].
by case/setIP=> Rx /cRSa; rewrite !inE actbyE.
by have:= cRSa x; rewrite !inE /= /actby Aa Sx; case: (x \in R) => //; apply.
Qed.
Lemma astabs_actby S : 'N(S | <[nRA]>) = 'N_A(R :&: S | to).
Proof.
apply/setP=> a; rewrite setIA (setIidPl (acts_dom nRA)) !inE.
case Aa: (a \in A) => //=; apply/subsetP/subsetP=> nRSa x => [|Sx].
by case/setIP=> Rx /nRSa; rewrite !inE actbyE ?(acts_act nRA) ?Rx.
have:= nRSa x; rewrite !inE /= /actby Aa Sx ?(acts_act nRA) //.
by case: (x \in R) => //; apply.
Qed.
Lemma acts_actby (B : {set aT}) S :
[acts B, on S | <[nRA]>] = (B \subset A) && [acts B, on R :&: S | to].
Proof. by rewrite astabs_actby subsetI. Qed.
End ActBy.
Notation "<[ nRA ] >" := (action_by nRA) : action_scope.
Section SubAction.
Variables (aT : finGroupType) (D : {group aT}).
Variables (rT : finType) (sP : pred rT) (sT : subFinType sP) (to : action D rT).
Implicit Type A : {set aT}.
Implicit Type u : sT.
Implicit Type S : {set sT}.
Definition subact_dom := 'N([set x | sP x] | to).
Canonical subact_dom_group := [group of subact_dom].
Implicit Type Na : {a | a \in subact_dom}.
Lemma sub_act_proof u Na : sP (to (val u) (val Na)).
Proof. by case: Na => a /= /(astabs_act (val u)); rewrite !inE valP. Qed.
Definition subact u a :=
if insub a is Some Na then Sub _ (sub_act_proof u Na) else u.
Lemma val_subact u a :
val (subact u a) = if a \in subact_dom then to (val u) a else val u.
Proof.
by rewrite /subact -if_neg; case: insubP => [Na|] -> //=; rewrite SubK => ->.
Qed.
Lemma subact_is_action : is_action subact_dom subact.
Proof.
split=> [a u v eq_uv | u a b Na Nb]; apply: val_inj.
move/(congr1 val): eq_uv; rewrite !val_subact.
by case: (a \in _); first move/act_inj.
have Da := astabs_dom Na; have Db := astabs_dom Nb.
by rewrite !val_subact Na Nb groupM ?actMin.
Qed.
Canonical subaction := Action subact_is_action.
Lemma astab_subact S : 'C(S | subaction) = subact_dom :&: 'C(val @: S | to).
Proof.
apply/setP=> a; rewrite inE in_setI; apply: andb_id2l => sDa.
have [Da _] := setIP sDa; rewrite !inE Da.
apply/subsetP/subsetP=> [cSa _ /imsetP[x Sx ->] | cSa x Sx] /[!inE].
by have:= cSa x Sx; rewrite inE -val_eqE val_subact sDa.
by have:= cSa _ (imset_f val Sx); rewrite inE -val_eqE val_subact sDa.
Qed.
Lemma astabs_subact S : 'N(S | subaction) = subact_dom :&: 'N(val @: S | to).
Proof.
apply/setP=> a; rewrite inE in_setI; apply: andb_id2l => sDa.
have [Da _] := setIP sDa; rewrite !inE Da.
apply/subsetP/subsetP=> [nSa _ /imsetP[x Sx ->] | nSa x Sx] /[!inE].
by have /[1!inE]/(imset_f val) := nSa x Sx; rewrite val_subact sDa.
have /[1!inE]/imsetP[y Sy def_y] := nSa _ (imset_f val Sx).
by rewrite ((_ a =P y) _) // -val_eqE val_subact sDa def_y.
Qed.
Lemma afix_subact A :
A \subset subact_dom -> 'Fix_subaction(A) = val @^-1: 'Fix_to(A).
Proof.
move/subsetP=> sAD; apply/setP=> u.
rewrite !inE !(sameP setIidPl eqP); congr (_ == A).
apply/setP=> a /[!inE]; apply: andb_id2l => Aa.
by rewrite -val_eqE val_subact sAD.
Qed.
End SubAction.
Notation "to ^?" := (subaction _ to) (format "to ^?") : action_scope.
Section QuotientAction.
Variables (aT : finGroupType) (D : {group aT}) (rT : finGroupType).
Variables (to : action D rT) (H : {group rT}).
Definition qact_dom := 'N(rcosets H 'N(H) | to^*).
Canonical qact_dom_group := [group of qact_dom].
Local Notation subdom := (subact_dom (coset_range H) to^*).
Fact qact_subdomE : subdom = qact_dom.
Proof. by congr 'N(_|_); apply/setP=> Hx; rewrite !inE genGid. Qed.
Lemma qact_proof : qact_dom \subset subdom.
Proof. by rewrite qact_subdomE. Qed.
Definition qact : coset_of H -> aT -> coset_of H := act (to^*^? \ qact_proof).
Canonical quotient_action := [action of qact].
Lemma acts_qact_dom : [acts qact_dom, on 'N(H) | to].
Proof.
apply/subsetP=> a nNa; rewrite !inE (astabs_dom nNa); apply/subsetP=> x Nx.
have: H :* x \in rcosets H 'N(H) by rewrite -rcosetE imset_f.
rewrite inE -(astabs_act _ nNa) => /rcosetsP[y Ny defHy].
have: to x a \in H :* y by rewrite -defHy (imset_f (to^~a)) ?rcoset_refl.
by apply: subsetP; rewrite mul_subG ?sub1set ?normG.
Qed.
Lemma qactEcond x a :
x \in 'N(H) ->
quotient_action (coset H x) a
= coset H (if a \in qact_dom then to x a else x).
Proof.
move=> Nx; apply: val_inj; rewrite val_subact //= qact_subdomE.
have: H :* x \in rcosets H 'N(H) by rewrite -rcosetE imset_f.
case nNa: (a \in _); rewrite // -(astabs_act _ nNa).
rewrite !val_coset ?(acts_act acts_qact_dom nNa) //=.
case/rcosetsP=> y Ny defHy; rewrite defHy; apply: rcoset_eqP.
by rewrite rcoset_sym -defHy (imset_f (_^~_)) ?rcoset_refl.
Qed.
Lemma qactE x a :
x \in 'N(H) -> a \in qact_dom ->
quotient_action (coset H x) a = coset H (to x a).
Proof. by move=> Nx nNa; rewrite qactEcond ?nNa. Qed.
Lemma acts_quotient (A : {set aT}) (B : {set rT}) :
A \subset 'N_qact_dom(B | to) -> [acts A, on B / H | quotient_action].
Proof.
move=> nBA; apply: subset_trans {A}nBA _; apply/subsetP=> a /setIP[dHa nBa].
rewrite inE dHa inE; apply/subsetP=> _ /morphimP[x nHx Bx ->].
rewrite inE /= qactE //.
by rewrite mem_morphim ?(acts_act acts_qact_dom) ?(astabs_act _ nBa).
Qed.
Lemma astabs_quotient (G : {group rT}) :
H <| G -> 'N(G / H | quotient_action) = 'N_qact_dom(G | to).
Proof.
move=> nsHG; have [_ nHG] := andP nsHG.
apply/eqP; rewrite eqEsubset acts_quotient // andbT.
apply/subsetP=> a nGa; have dHa := astabs_dom nGa; have [Da _]:= setIdP dHa.
rewrite inE dHa 2!inE Da; apply/subsetP=> x Gx; have nHx := subsetP nHG x Gx.
rewrite -(quotientGK nsHG) 2!inE (acts_act acts_qact_dom) ?nHx //= inE.
by rewrite -qactE // (astabs_act _ nGa) mem_morphim.
Qed.
End QuotientAction.
Notation "to / H" := (quotient_action to H) : action_scope.
Section ModAction.
Variables (aT : finGroupType) (D : {group aT}) (rT : finType).
Variable to : action D rT.
Implicit Types (G : {group aT}) (S : {set rT}).
Section GenericMod.
Variable H : {group aT}.
Local Notation dom := 'N_D(H).
Local Notation range := 'Fix_to(D :&: H).
Let acts_dom : {acts dom, on range | to} := acts_act (acts_subnorm_fix to H).
Definition modact x (Ha : coset_of H) :=
if x \in range then to x (repr (D :&: Ha)) else x.
Lemma modactEcond x a :
a \in dom -> modact x (coset H a) = (if x \in range then to x a else x).
Proof.
case/setIP=> Da Na; case: ifP => Cx; rewrite /modact Cx //.
rewrite val_coset // -group_modr ?sub1set //.
case: (repr _) / (repr_rcosetP (D :&: H) a) => a' Ha'.
by rewrite actMin ?(afixP Cx _ Ha') //; case/setIP: Ha'.
Qed.
Lemma modactE x a :
a \in D -> a \in 'N(H) -> x \in range -> modact x (coset H a) = to x a.
Proof. by move=> Da Na Rx; rewrite modactEcond ?Rx // inE Da. Qed.
Lemma modact_is_action : is_action (D / H) modact.
Proof.
split=> [Ha x y | x Ha Hb]; last first.
case/morphimP=> a Na Da ->{Ha}; case/morphimP=> b Nb Db ->{Hb}.
rewrite -morphM //= !modactEcond // ?groupM ?(introT setIP _) //.
by case: ifP => Cx; rewrite ?(acts_dom, Cx, actMin, introT setIP _).
case: (set_0Vmem (D :&: Ha)) => [Da0 | [a /setIP[Da NHa]]].
by rewrite /modact Da0 repr_set0 !act1 !if_same.
have Na := subsetP (coset_norm _) _ NHa.
have NDa: a \in 'N_D(H) by rewrite inE Da.
rewrite -(coset_mem NHa) !modactEcond //.
do 2![case: ifP]=> Cy Cx // eqxy; first exact: act_inj eqxy.
by rewrite -eqxy acts_dom ?Cx in Cy.
by rewrite eqxy acts_dom ?Cy in Cx.
Qed.
Canonical mod_action := Action modact_is_action.
Section Stabilizers.
Variable S : {set rT}.
Hypothesis cSH : H \subset 'C(S | to).
Let fixSH : S \subset 'Fix_to(D :&: H).
Proof. by rewrite -astabCin ?subsetIl // subIset ?cSH ?orbT. Qed.
Lemma astabs_mod : 'N(S | mod_action) = 'N(S | to) / H.
Proof.
apply/setP=> Ha; apply/idP/morphimP=> [nSa | [a nHa nSa ->]].
case/morphimP: (astabs_dom nSa) => a nHa Da defHa.
exists a => //; rewrite !inE Da; apply/subsetP=> x Sx; rewrite !inE.
by have:= Sx; rewrite -(astabs_act x nSa) defHa /= modactE ?(subsetP fixSH).
have Da := astabs_dom nSa; rewrite !inE mem_quotient //; apply/subsetP=> x Sx.
by rewrite !inE /= modactE ?(astabs_act x nSa) ?(subsetP fixSH).
Qed.
Lemma astab_mod : 'C(S | mod_action) = 'C(S | to) / H.
Proof.
apply/setP=> Ha; apply/idP/morphimP=> [cSa | [a nHa cSa ->]].
case/morphimP: (astab_dom cSa) => a nHa Da defHa.
exists a => //; rewrite !inE Da; apply/subsetP=> x Sx; rewrite !inE.
by rewrite -{2}[x](astab_act cSa) // defHa /= modactE ?(subsetP fixSH).
have Da := astab_dom cSa; rewrite !inE mem_quotient //; apply/subsetP=> x Sx.
by rewrite !inE /= modactE ?(astab_act cSa) ?(subsetP fixSH).
Qed.
End Stabilizers.
Lemma afix_mod G S :
H \subset 'C(S | to) -> G \subset 'N_D(H) ->
'Fix_(S | mod_action)(G / H) = 'Fix_(S | to)(G).
Proof.
move=> cSH /subsetIP[sGD nHG].
apply/eqP; rewrite eqEsubset !subsetI !subsetIl /= -!astabCin ?quotientS //.
have cfixH F: H \subset 'C(S :&: F | to).
by rewrite (subset_trans cSH) // astabS ?subsetIl.
rewrite andbC astab_mod ?quotientS //=; last by rewrite astabCin ?subsetIr.
by rewrite -(quotientSGK nHG) //= -astab_mod // astabCin ?quotientS ?subsetIr.
Qed.
End GenericMod.
Lemma modact_faithful G S :
[faithful G / 'C_G(S | to), on S | mod_action 'C_G(S | to)].
Proof.
rewrite /faithful astab_mod ?subsetIr //=.
by rewrite -quotientIG ?subsetIr ?trivg_quotient.
Qed.
End ModAction.
Notation "to %% H" := (mod_action to H) : action_scope.
Section ActPerm.
(* Morphism to permutations induced by an action. *)
Variables (aT : finGroupType) (D : {set aT}) (rT : finType).
Variable to : action D rT.
Definition actperm a := perm (act_inj to a).
Lemma actpermM : {in D &, {morph actperm : a b / a * b}}.
Proof. by move=> a b Da Db; apply/permP=> x; rewrite permM !permE actMin. Qed.
Canonical actperm_morphism := Morphism actpermM.
Lemma actpermE a x : actperm a x = to x a.
Proof. by rewrite permE. Qed.
Lemma actpermK x a : aperm x (actperm a) = to x a.
Proof. exact: actpermE. Qed.
Lemma ker_actperm : 'ker actperm = 'C(setT | to).
Proof.
congr (_ :&: _); apply/setP=> a /[!inE]/=.
apply/eqP/subsetP=> [a1 x _ | a1]; first by rewrite inE -actpermE a1 perm1.
by apply/permP=> x; apply/eqP; have:= a1 x; rewrite !inE actpermE perm1 => ->.
Qed.
End ActPerm.
Section RestrictActionTheory.
Variables (aT : finGroupType) (D : {set aT}) (rT : finType).
Variables (to : action D rT).
Lemma faithful_isom (A : {group aT}) S (nSA : actby_cond A S to) :
[faithful A, on S | to] -> isom A (actperm <[nSA]> @* A) (actperm <[nSA]>).
Proof.
by move=> ffulAS; apply/isomP; rewrite ker_actperm astab_actby setIT.
Qed.
Variables (A : {set aT}) (sAD : A \subset D).
Lemma ractpermE : actperm (to \ sAD) =1 actperm to.
Proof. by move=> a; apply/permP=> x; rewrite !permE. Qed.
Lemma afix_ract B : 'Fix_(to \ sAD)(B) = 'Fix_to(B). Proof. by []. Qed.
Lemma astab_ract S : 'C(S | to \ sAD) = 'C_A(S | to).
Proof. by rewrite setIA (setIidPl sAD). Qed.
Lemma astabs_ract S : 'N(S | to \ sAD) = 'N_A(S | to).
Proof. by rewrite setIA (setIidPl sAD). Qed.
Lemma acts_ract (B : {set aT}) S :
[acts B, on S | to \ sAD] = (B \subset A) && [acts B, on S | to].
Proof. by rewrite astabs_ract subsetI. Qed.
End RestrictActionTheory.
Section MorphAct.
(* Action induced by a morphism to permutations. *)
Variables (aT : finGroupType) (D : {group aT}) (rT : finType).
Variable phi : {morphism D >-> {perm rT}}.
Definition mact x a := phi a x.
Lemma mact_is_action : is_action D mact.
Proof.
split=> [a x y | x a b Da Db]; first exact: perm_inj.
by rewrite /mact morphM //= permM.
Qed.
Canonical morph_action := Action mact_is_action.
Lemma mactE x a : morph_action x a = phi a x. Proof. by []. Qed.
Lemma injm_faithful : 'injm phi -> [faithful D, on setT | morph_action].
Proof.
move/injmP=> phi_inj; apply/subsetP=> a /setIP[Da /astab_act a1].
apply/set1P/phi_inj => //; apply/permP=> x.
by rewrite morph1 perm1 -mactE a1 ?inE.
Qed.
Lemma perm_mact a : actperm morph_action a = phi a.
Proof. by apply/permP=> x; rewrite permE. Qed.
End MorphAct.
Notation "<< phi >>" := (morph_action phi) : action_scope.
Section CompAct.
Variables (gT aT : finGroupType) (rT : finType).
Variables (D : {set aT}) (to : action D rT).
Variables (B : {set gT}) (f : {morphism B >-> aT}).
Definition comp_act x e := to x (f e).
Lemma comp_is_action : is_action (f @*^-1 D) comp_act.
Proof.
split=> [e | x e1 e2]; first exact: act_inj.
move=> /morphpreP[Be1 Dfe1] /morphpreP[Be2 Dfe2].
by rewrite /comp_act morphM ?actMin.
Qed.
Canonical comp_action := Action comp_is_action.
Lemma comp_actE x e : comp_action x e = to x (f e). Proof. by []. Qed.
Lemma afix_comp (A : {set gT}) :
A \subset B -> 'Fix_comp_action(A) = 'Fix_to(f @* A).
Proof.
move=> sAB; apply/setP=> x; rewrite !inE /morphim (setIidPr sAB).
apply/subsetP/subsetP; first by move=> + _ /imsetP[a + ->] => /[apply]/[!inE].
by move=> + a Aa => /(_ (f a)); rewrite !inE imset_f// => ->.
Qed.
Lemma astab_comp S : 'C(S | comp_action) = f @*^-1 'C(S | to).
Proof. by apply/setP=> x; rewrite !inE -andbA. Qed.
Lemma astabs_comp S : 'N(S | comp_action) = f @*^-1 'N(S | to).
Proof. by apply/setP=> x; rewrite !inE -andbA. Qed.
End CompAct.
Notation "to \o f" := (comp_action to f) : action_scope.
Section PermAction.
(* Natural action of permutation groups. *)
Variable rT : finType.
Local Notation gT := {perm rT}.
Implicit Types a b c : gT.
Lemma aperm_is_action : is_action setT (@aperm rT).
Proof.
by apply: is_total_action => [x|x a b]; rewrite apermE (perm1, permM).
Qed.
Canonical perm_action := Action aperm_is_action.
Lemma porbitE a : porbit a = orbit perm_action <[a]>%g.
Proof. by rewrite unlock. Qed.
Lemma perm_act1P a : reflect (forall x, aperm x a = x) (a == 1).
Proof.
apply: (iffP eqP) => [-> x | a1]; first exact: act1.
by apply/permP=> x; rewrite -apermE a1 perm1.
Qed.
Lemma perm_faithful A : [faithful A, on setT | perm_action].
Proof.
apply/subsetP=> a /setIP[Da crTa].
by apply/set1P; apply/permP=> x; rewrite -apermE perm1 (astabP crTa) ?inE.
Qed.
Lemma actperm_id p : actperm perm_action p = p.
Proof. by apply/permP=> x; rewrite permE. Qed.
End PermAction.
Arguments perm_act1P {rT a}.
Notation "'P" := (perm_action _) : action_scope.
Section ActpermOrbits.
Variables (aT : finGroupType) (D : {group aT}) (rT : finType).
Variable to : action D rT.
Lemma orbit_morphim_actperm (A : {set aT}) :
A \subset D -> orbit 'P (actperm to @* A) =1 orbit to A.
Proof.
move=> sAD x; rewrite morphimEsub // /orbit -imset_comp.
by apply: eq_imset => a //=; rewrite actpermK.
Qed.
Lemma porbit_actperm (a : aT) :
a \in D -> porbit (actperm to a) =1 orbit to <[a]>.
Proof.
move=> Da x.
by rewrite porbitE -orbit_morphim_actperm ?cycle_subG ?morphim_cycle.
Qed.
End ActpermOrbits.
Section RestrictPerm.
Variables (T : finType) (S : {set T}).
Definition restr_perm := actperm (<[subxx 'N(S | 'P)]>).
Canonical restr_perm_morphism := [morphism of restr_perm].
Lemma restr_perm_on p : perm_on S (restr_perm p).
Proof.
apply/subsetP=> x; apply: contraR => notSx.
by rewrite permE /= /actby (negPf notSx).
Qed.
Lemma triv_restr_perm p : p \notin 'N(S | 'P) -> restr_perm p = 1.
Proof.
move=> not_nSp; apply/permP=> x.
by rewrite !permE /= /actby (negPf not_nSp) andbF.
Qed.
Lemma restr_permE : {in 'N(S | 'P) & S, forall p, restr_perm p =1 p}.
Proof. by move=> y x nSp Sx; rewrite /= actpermE actbyE. Qed.
Lemma ker_restr_perm : 'ker restr_perm = 'C(S | 'P).
Proof. by rewrite ker_actperm astab_actby setIT (setIidPr (astab_sub _ _)). Qed.
Lemma im_restr_perm p : restr_perm p @: S = S.
Proof. exact: im_perm_on (restr_perm_on p). Qed.
Lemma restr_perm_commute s : commute (restr_perm s) s.
Proof.
have [sC|/triv_restr_perm->] := boolP (s \in 'N(S | 'P)); last first.
exact: (commute_sym (commute1 _)).
apply/permP => x; have /= xsS := astabsP sC x; rewrite !permM.
have [xS|xNS] := boolP (x \in S); first by rewrite ?(restr_permE) ?xsS.
by rewrite !(out_perm (restr_perm_on _)) ?xsS.
Qed.
End RestrictPerm.
Section Symmetry.
Variables (T : finType) (S : {set T}).
Lemma SymE : Sym S = 'C(~: S | 'P).
Proof.
apply/setP => s; rewrite inE; apply/idP/astabP => [sS x|/= S_id].
by rewrite inE /= apermE => /out_perm->.
by apply/subsetP => x; move=> /(contra_neqN (S_id _)); rewrite inE negbK.
Qed.
End Symmetry.
Section AutIn.
Variable gT : finGroupType.
Definition Aut_in A (B : {set gT}) := 'N_A(B | 'P) / 'C_A(B | 'P).
Variables G H : {group gT}.
Hypothesis sHG: H \subset G.
Lemma Aut_restr_perm a : a \in Aut G -> restr_perm H a \in Aut H.
Proof.
move=> AutGa.
case nHa: (a \in 'N(H | 'P)); last by rewrite triv_restr_perm ?nHa ?group1.
rewrite inE restr_perm_on; apply/morphicP=> x y Hx Hy /=.
by rewrite !restr_permE ?groupM // -(autmE AutGa) morphM ?(subsetP sHG).
Qed.
Lemma restr_perm_Aut : restr_perm H @* Aut G \subset Aut H.
Proof.
by apply/subsetP=> a'; case/morphimP=> a _ AutGa ->{a'}; apply: Aut_restr_perm.
Qed.
Lemma Aut_in_isog : Aut_in (Aut G) H \isog restr_perm H @* Aut G.
Proof.
rewrite /Aut_in -ker_restr_perm kerE -morphpreIdom -morphimIdom -kerE /=.
by rewrite setIA (setIC _ (Aut G)) first_isog_loc ?subsetIr.
Qed.
Lemma Aut_sub_fullP :
reflect (forall h : {morphism H >-> gT}, 'injm h -> h @* H = H ->
exists g : {morphism G >-> gT},
[/\ 'injm g, g @* G = G & {in H, g =1 h}])
(Aut_in (Aut G) H \isog Aut H).
Proof.
rewrite (isog_transl _ Aut_in_isog) /=; set rG := _ @* _.
apply: (iffP idP) => [iso_rG h injh hH| AutHinG].
have: aut injh hH \in rG; last case/morphimP=> g nHg AutGg def_g.
suffices ->: rG = Aut H by apply: Aut_aut.
by apply/eqP; rewrite eqEcard restr_perm_Aut /= (card_isog iso_rG).
exists (autm_morphism AutGg); rewrite injm_autm im_autm; split=> // x Hx.
by rewrite -(autE injh hH Hx) def_g actpermE actbyE.
suffices ->: rG = Aut H by apply: isog_refl.
apply/eqP; rewrite eqEsubset restr_perm_Aut /=.
apply/subsetP=> h AutHh; have hH := im_autm AutHh.
have [g [injg gG eq_gh]] := AutHinG _ (injm_autm AutHh) hH.
have [Ng AutGg]: aut injg gG \in 'N(H | 'P) /\ aut injg gG \in Aut G.
rewrite Aut_aut !inE; split=> //; apply/subsetP=> x Hx.
by rewrite inE /= /aperm autE ?(subsetP sHG) // -hH eq_gh ?mem_morphim.
apply/morphimP; exists (aut injg gG) => //; apply: (eq_Aut AutHh) => [|x Hx].
by rewrite (subsetP restr_perm_Aut) // mem_morphim.
by rewrite restr_permE //= /aperm autE ?eq_gh ?(subsetP sHG).
Qed.
End AutIn.
Arguments Aut_in {gT} A%_g B%_g.
Section InjmAutIn.
Variables (gT rT : finGroupType) (D G H : {group gT}) (f : {morphism D >-> rT}).
Hypotheses (injf : 'injm f) (sGD : G \subset D) (sHG : H \subset G).
Let sHD := subset_trans sHG sGD.
Local Notation fGisom := (Aut_isom injf sGD).
Local Notation fHisom := (Aut_isom injf sHD).
Local Notation inH := (restr_perm H).
Local Notation infH := (restr_perm (f @* H)).
Lemma astabs_Aut_isom a :
a \in Aut G -> (fGisom a \in 'N(f @* H | 'P)) = (a \in 'N(H | 'P)).
Proof.
move=> AutGa; rewrite !inE sub_morphim_pre // subsetI sHD /= /aperm.
rewrite !(sameP setIidPl eqP) !eqEsubset !subsetIl; apply: eq_subset_r => x.
rewrite !inE; apply: andb_id2l => Hx; have Gx: x \in G := subsetP sHG x Hx.
have Dax: a x \in D by rewrite (subsetP sGD) // Aut_closed.
by rewrite Aut_isomE // -!sub1set -morphim_set1 // injmSK ?sub1set.
Qed.
Lemma isom_restr_perm a : a \in Aut G -> fHisom (inH a) = infH (fGisom a).
Proof.
move=> AutGa; case nHa: (a \in 'N(H | 'P)); last first.
by rewrite !triv_restr_perm ?astabs_Aut_isom ?nHa ?morph1.
apply: (eq_Aut (Aut_Aut_isom injf sHD _)) => [|fx Hfx /=].
by rewrite (Aut_restr_perm (morphimS f sHG)) ?Aut_Aut_isom.
have [x Dx Hx def_fx] := morphimP Hfx; have Gx := subsetP sHG x Hx.
rewrite {1}def_fx Aut_isomE ?(Aut_restr_perm sHG) //.
by rewrite !restr_permE ?astabs_Aut_isom // def_fx Aut_isomE.
Qed.
Lemma restr_perm_isom : isom (inH @* Aut G) (infH @* Aut (f @* G)) fHisom.
Proof.
apply: sub_isom; rewrite ?restr_perm_Aut ?injm_Aut_isom //=.
rewrite -(im_Aut_isom injf sGD) -!morphim_comp.
apply: eq_in_morphim; last exact: isom_restr_perm.
(* TODO: investigate why rewrite does not match in the same order *)
apply/setP=> a; rewrite in_setI [in RHS]in_setI; apply: andb_id2r => AutGa.
(* the middle rewrite was rewrite 2!in_setI *)
rewrite /= inE andbC inE (Aut_restr_perm sHG) //=.
by symmetry; rewrite inE AutGa inE astabs_Aut_isom.
Qed.
Lemma injm_Aut_sub : Aut_in (Aut (f @* G)) (f @* H) \isog Aut_in (Aut G) H.
Proof.
do 2!rewrite isog_sym (isog_transl _ (Aut_in_isog _ _)).
by rewrite isog_sym (isom_isog _ _ restr_perm_isom) // restr_perm_Aut.
Qed.
Lemma injm_Aut_full :
(Aut_in (Aut (f @* G)) (f @* H) \isog Aut (f @* H))
= (Aut_in (Aut G) H \isog Aut H).
Proof.
by rewrite (isog_transl _ injm_Aut_sub) (isog_transr _ (injm_Aut injf sHD)).
Qed.
End InjmAutIn.
Section GroupAction.
Variables (aT rT : finGroupType) (D : {set aT}) (R : {set rT}).
Local Notation actT := (action D rT).
Definition is_groupAction (to : actT) :=
{in D, forall a, actperm to a \in Aut R}.
Structure groupAction := GroupAction {gact :> actT; _ : is_groupAction gact}.
Definition clone_groupAction to :=
let: GroupAction _ toA := to return {type of GroupAction for to} -> _ in
fun k => k toA : groupAction.
End GroupAction.
Delimit Scope groupAction_scope with gact.
Bind Scope groupAction_scope with groupAction.
Arguments is_groupAction {aT rT D%_g} R%_g to%_act.
Arguments groupAction {aT rT} D%_g R%_g.
Arguments gact {aT rT D%_g R%_g} to%_gact : rename.
Notation "[ 'groupAction' 'of' to ]" :=
(clone_groupAction (@GroupAction _ _ _ _ to))
(format "[ 'groupAction' 'of' to ]") : form_scope.
Section GroupActionDefs.
Variables (aT rT : finGroupType) (D : {set aT}) (R : {set rT}).
Implicit Type A : {set aT}.
Implicit Type S : {set rT}.
Implicit Type to : groupAction D R.
Definition gact_range of groupAction D R := R.
Definition gacent to A := 'Fix_(R | to)(D :&: A).
Definition acts_on_group A S to := [acts A, on S | to] /\ S \subset R.
Coercion actby_cond_group A S to : acts_on_group A S to -> actby_cond A S to :=
@proj1 _ _.
Definition acts_irreducibly A S to :=
[min S of G | G :!=: 1 & [acts A, on G | to]].
End GroupActionDefs.
Arguments gacent {aT rT D%_g R%_g} to%_gact A%_g.
Arguments acts_on_group {aT rT D%_g R%_g} A%_g S%_g to%_gact.
Arguments acts_irreducibly {aT rT D%_g R%_g} A%_g S%_g to%_gact.
Notation "''C_' ( | to ) ( A )" := (gacent to A) : group_scope.
Notation "''C_' ( G | to ) ( A )" := (G :&: 'C_(|to)(A)) : group_scope.
Notation "''C_' ( | to ) [ a ]" := 'C_(|to)([set a]) : group_scope.
Notation "''C_' ( G | to ) [ a ]" := 'C_(G | to)([set a]) : group_scope.
Notation "{ 'acts' A , 'on' 'group' G | to }" := (acts_on_group A G to)
(format "{ 'acts' A , 'on' 'group' G | to }") : type_scope.
Section RawGroupAction.
Variables (aT rT : finGroupType) (D : {set aT}) (R : {set rT}).
Variable to : groupAction D R.
Lemma actperm_Aut : is_groupAction R to. Proof. by case: to. Qed.
Lemma im_actperm_Aut : actperm to @* D \subset Aut R.
Proof. by apply/subsetP=> _ /morphimP[a _ Da ->]; apply: actperm_Aut. Qed.
Lemma gact_out x a : a \in D -> x \notin R -> to x a = x.
Proof. by move=> Da Rx; rewrite -actpermE (out_Aut _ Rx) ?actperm_Aut. Qed.
Lemma gactM : {in D, forall a, {in R &, {morph to^~ a : x y / x * y}}}.
Proof.
move=> a Da /= x y; rewrite -!(actpermE to); apply: morphicP x y.
by rewrite Aut_morphic ?actperm_Aut.
Qed.
Lemma actmM a : {in R &, {morph actm to a : x y / x * y}}.
Proof. by rewrite /actm; case: ifP => //; apply: gactM. Qed.
Canonical act_morphism a := Morphism (actmM a).
Lemma morphim_actm :
{in D, forall a (S : {set rT}), S \subset R -> actm to a @* S = to^* S a}.
Proof. by move=> a Da /= S sSR; rewrite /morphim /= actmEfun ?(setIidPr _). Qed.
Variables (a : aT) (A B : {set aT}) (S : {set rT}).
Lemma gacentIdom : 'C_(|to)(D :&: A) = 'C_(|to)(A).
Proof. by rewrite /gacent setIA setIid. Qed.
Lemma gacentIim : 'C_(R | to)(A) = 'C_(|to)(A).
Proof. by rewrite setIA setIid. Qed.
Lemma gacentS : A \subset B -> 'C_(|to)(B) \subset 'C_(|to)(A).
Proof. by move=> sAB; rewrite !(setIS, afixS). Qed.
Lemma gacentU : 'C_(|to)(A :|: B) = 'C_(|to)(A) :&: 'C_(|to)(B).
Proof. by rewrite -setIIr -afixU -setIUr. Qed.
Hypotheses (Da : a \in D) (sAD : A \subset D) (sSR : S \subset R).
Lemma gacentE : 'C_(|to)(A) = 'Fix_(R | to)(A).
Proof. by rewrite -{2}(setIidPr sAD). Qed.
Lemma gacent1E : 'C_(|to)[a] = 'Fix_(R | to)[a].
Proof. by rewrite /gacent [D :&: _](setIidPr _) ?sub1set. Qed.
Lemma subgacentE : 'C_(S | to)(A) = 'Fix_(S | to)(A).
Proof. by rewrite gacentE setIA (setIidPl sSR). Qed.
Lemma subgacent1E : 'C_(S | to)[a] = 'Fix_(S | to)[a].
Proof. by rewrite gacent1E setIA (setIidPl sSR). Qed.
End RawGroupAction.
Section GroupActionTheory.
Variables aT rT : finGroupType.
Variables (D : {group aT}) (R : {group rT}) (to : groupAction D R).
Implicit Type A B : {set aT}.
Implicit Types G H : {group aT}.
Implicit Type S : {set rT}.
Implicit Types M N : {group rT}.
Lemma gact1 : {in D, forall a, to 1 a = 1}.
Proof. by move=> a Da; rewrite /= -actmE ?morph1. Qed.
Lemma gactV : {in D, forall a, {in R, {morph to^~ a : x / x^-1}}}.
Proof. by move=> a Da /= x Rx; move; rewrite -!actmE ?morphV. Qed.
Lemma gactX : {in D, forall a n, {in R, {morph to^~ a : x / x ^+ n}}}.
Proof. by move=> a Da /= n x Rx; rewrite -!actmE // morphX. Qed.
Lemma gactJ : {in D, forall a, {in R &, {morph to^~ a : x y / x ^ y}}}.
Proof. by move=> a Da /= x Rx y Ry; rewrite -!actmE // morphJ. Qed.
Lemma gactR : {in D, forall a, {in R &, {morph to^~ a : x y / [~ x, y]}}}.
Proof. by move=> a Da /= x Rx y Ry; rewrite -!actmE // morphR. Qed.
Lemma gact_stable : {acts D, on R | to}.
Proof.
apply: acts_act; apply/subsetP=> a Da; rewrite !inE Da.
apply/subsetP=> x; rewrite inE; apply: contraLR => R'xa.
by rewrite -(actKin to Da x) gact_out ?groupV.
Qed.
Lemma group_set_gacent A : group_set 'C_(|to)(A).
Proof.
apply/group_setP; split=> [|x y].
by rewrite !inE group1; apply/subsetP=> a /setIP[Da _]; rewrite inE gact1.
case/setIP=> Rx /afixP cAx /setIP[Ry /afixP cAy].
rewrite inE groupM //; apply/afixP=> a Aa.
by rewrite gactM ?cAx ?cAy //; case/setIP: Aa.
Qed.
Canonical gacent_group A := Group (group_set_gacent A).
Lemma gacent1 : 'C_(|to)(1) = R.
Proof. by rewrite /gacent (setIidPr (sub1G _)) afix1 setIT. Qed.
Lemma gacent_gen A : A \subset D -> 'C_(|to)(<<A>>) = 'C_(|to)(A).
Proof.
by move=> sAD; rewrite /gacent  ?gen_subG ?afix_gen_in.
Qed.
Lemma gacentD1 A : 'C_(|to)(A^#) = 'C_(|to)(A).
Proof.
rewrite -gacentIdom -gacent_gen ?subsetIl // setIDA genD1 ?group1 //.
by rewrite gacent_gen ?subsetIl // gacentIdom.
Qed.
Lemma gacent_cycle a : a \in D -> 'C_(|to)(<[a]>) = 'C_(|to)[a].
Proof. by move=> Da; rewrite gacent_gen ?sub1set. Qed.
Lemma gacentY A B :
A \subset D -> B \subset D -> 'C_(|to)(A <*> B) = 'C_(|to)(A) :&: 'C_(|to)(B).
Proof. by move=> sAD sBD; rewrite gacent_gen ?gacentU // subUset sAD. Qed.
Lemma gacentM G H :
G \subset D -> H \subset D -> 'C_(|to)(G * H) = 'C_(|to)(G) :&: 'C_(|to)(H).
Proof.
by move=> sGD sHB; rewrite -gacent_gen ?mul_subG // genM_join gacentY.
Qed.
Lemma astab1 : 'C(1 | to) = D.
Proof.
by apply/setP=> x; rewrite ?(inE, sub1set) andb_idr //; move/gact1=> ->.
Qed.
Lemma astab_range : 'C(R | to) = 'C(setT | to).
Proof.
apply/eqP; rewrite eqEsubset andbC astabS ?subsetT //=.
apply/subsetP=> a cRa; have Da := astab_dom cRa; rewrite !inE Da.
apply/subsetP=> x; rewrite -(setUCr R) !inE.
by case/orP=> ?; [rewrite (astab_act cRa) | rewrite gact_out].
Qed.
Lemma gacentC A S :
A \subset D -> S \subset R ->
(S \subset 'C_(|to)(A)) = (A \subset 'C(S | to)).
Proof. by move=> sAD sSR; rewrite subsetI sSR astabCin // (setIidPr sAD). Qed.
Lemma astab_gen S : S \subset R -> 'C(<<S>> | to) = 'C(S | to).
Proof.
move=> sSR; apply/setP=> a; case Da: (a \in D); last by rewrite !inE Da.
by rewrite -!sub1set -!gacentC ?sub1set ?gen_subG.
Qed.
Lemma astabM M N :
M \subset R -> N \subset R -> 'C(M * N | to) = 'C(M | to) :&: 'C(N | to).
Proof.
move=> sMR sNR; rewrite -astabU -astab_gen ?mul_subG // genM_join.
by rewrite astab_gen // subUset sMR.
Qed.
Lemma astabs1 : 'N(1 | to) = D.
Proof. by rewrite astabs_set1 astab1. Qed.
Lemma astabs_range : 'N(R | to) = D.
Proof.
apply/setIidPl; apply/subsetP=> a Da; rewrite inE.
by apply/subsetP=> x Rx; rewrite inE gact_stable.
Qed.
Lemma astabsD1 S : 'N(S^# | to) = 'N(S | to).
Proof.
case S1: (1 \in S); last first.
by rewrite (setDidPl _) // disjoint_sym disjoints_subset sub1set inE S1.
apply/eqP; rewrite eqEsubset andbC -{1}astabsIdom -{1}astabs1 setIC astabsD /=.
by rewrite -{2}(setD1K S1) -astabsIdom -{1}astabs1 astabsU.
Qed.
Lemma gacts_range A : A \subset D -> {acts A, on group R | to}.
Proof. by move=> sAD; split; rewrite ?astabs_range. Qed.
Lemma acts_subnorm_gacent A : A \subset D ->
[acts 'N_D(A), on 'C_(| to)(A) | to].
Proof.
move=> sAD; rewrite gacentE // actsI ?astabs_range ?subsetIl //.
by rewrite -{2}(setIidPr sAD) acts_subnorm_fix.
Qed.
Lemma acts_subnorm_subgacent A B S :
A \subset D -> [acts B, on S | to] -> [acts 'N_B(A), on 'C_(S | to)(A) | to].
Proof.
move=> sAD actsB; rewrite actsI //; first by rewrite subIset ?actsB.
by rewrite (subset_trans _ (acts_subnorm_gacent sAD)) ?setSI ?(acts_dom actsB).
Qed.
Lemma acts_gen A S :
S \subset R -> [acts A, on S | to] -> [acts A, on <<S>> | to].
Proof.
move=> sSR actsA; apply: {A}subset_trans actsA _.
apply/subsetP=> a nSa; have Da := astabs_dom nSa; rewrite !inE Da.
apply: subset_trans (_ : <<S>> \subset actm to a @*^-1 <<S>>) _.
rewrite gen_subG subsetI sSR; apply/subsetP=> x Sx.
by rewrite inE /= actmE ?mem_gen // astabs_act.
by apply/subsetP=> x /[!inE]; case/andP=> Rx; rewrite /= actmE.
Qed.
Lemma acts_joing A M N :
M \subset R -> N \subset R -> [acts A, on M | to] -> [acts A, on N | to] ->
[acts A, on M <*> N | to].
Proof. by move=> sMR sNR nMA nNA; rewrite acts_gen ?actsU // subUset sMR. Qed.
Lemma injm_actm a : 'injm (actm to a).
Proof.
apply/injmP=> x y Rx Ry; rewrite /= /actm; case: ifP => Da //.
exact: act_inj.
Qed.
Lemma im_actm a : actm to a @* R = R.
Proof.
apply/eqP; rewrite eqEcard (card_injm (injm_actm a)) // leqnn andbT.
apply/subsetP=> _ /morphimP[x Rx _ ->] /=.
by rewrite /actm; case: ifP => // Da; rewrite gact_stable.
Qed.
Lemma acts_char G M : G \subset D -> M \char R -> [acts G, on M | to].
Proof.
move=> sGD /charP[sMR charM].
apply/subsetP=> a Ga; have Da := subsetP sGD a Ga; rewrite !inE Da.
apply/subsetP=> x Mx; have Rx := subsetP sMR x Mx.
by rewrite inE -(charM _ (injm_actm a) (im_actm a)) -actmE // mem_morphim.
Qed.
Lemma gacts_char G M :
G \subset D -> M \char R -> {acts G, on group M | to}.
(* TODO: investigate why rewrite does not match in the same order *)
Proof. by move=> sGD charM; split; rewrite ?acts_char// char_sub. Qed.
(* was ending with rewrite (acts_char, char_sub)// *)
Section Restrict.
Variables (A : {group aT}) (sAD : A \subset D).
Lemma ract_is_groupAction : is_groupAction R (to \ sAD).
Proof. by move=> a Aa /=; rewrite ractpermE actperm_Aut ?(subsetP sAD). Qed.
Canonical ract_groupAction := GroupAction ract_is_groupAction.
Lemma gacent_ract B : 'C_(|ract_groupAction)(B) = 'C_(|to)(A :&: B).
Proof. by rewrite /gacent afix_ract setIA (setIidPr sAD). Qed.
End Restrict.
Section ActBy.
Variables (A : {group aT}) (G : {group rT}) (nGAg : {acts A, on group G | to}).
Lemma actby_is_groupAction : is_groupAction G <[nGAg]>.
Proof.
move=> a Aa; rewrite /= inE; apply/andP; split.
apply/subsetP=> x; apply: contraR => Gx.
by rewrite actpermE /= /actby (negbTE Gx).
apply/morphicP=> x y Gx Gy; rewrite !actpermE /= /actby Aa groupM ?Gx ?Gy //=.
by case nGAg; move/acts_dom; do 2!move/subsetP=> ?; rewrite gactM; auto.
Qed.
Canonical actby_groupAction := GroupAction actby_is_groupAction.
Lemma gacent_actby B :
'C_(|actby_groupAction)(B) = 'C_(G | to)(A :&: B).
Proof.
rewrite /gacent afix_actby !setIA setIid setIUr setICr set0U.
by have [nAG sGR] := nGAg; rewrite (setIidPr (acts_dom nAG)) (setIidPl sGR).
Qed.
End ActBy.
Section Quotient.
Variable H : {group rT}.
Lemma acts_qact_dom_norm : {acts qact_dom to H, on 'N(H) | to}.
Proof.
move=> a HDa /= x; rewrite {2}(('N(H) =P to^~ a @^-1: 'N(H)) _) ?inE {x}//.
rewrite eqEcard (card_preimset _ (act_inj _ _)) leqnn andbT.
apply/subsetP=> x Nx; rewrite inE; move/(astabs_act (H :* x)): HDa.
rewrite mem_rcosets mulSGid ?normG // Nx => /rcosetsP[y Ny defHy].
suffices: to x a \in H :* y by apply: subsetP; rewrite mul_subG ?sub1set ?normG.
by rewrite -defHy; apply: imset_f; apply: rcoset_refl.
Qed.
Lemma qact_is_groupAction : is_groupAction (R / H) (to / H).
Proof.
move=> a HDa /=; have Da := astabs_dom HDa.
rewrite inE; apply/andP; split.
apply/subsetP=> Hx /=; case: (cosetP Hx) => x Nx ->{Hx}.
apply: contraR => R'Hx; rewrite actpermE qactE // gact_out //.
by apply: contra R'Hx; apply: mem_morphim.
apply/morphicP=> Hx Hy; rewrite !actpermE.
case/morphimP=> x Nx Gx ->{Hx}; case/morphimP=> y Ny Gy ->{Hy}.
by rewrite -morphM ?qactE ?groupM ?gactM // morphM ?acts_qact_dom_norm.
Qed.
Canonical quotient_groupAction := GroupAction qact_is_groupAction.
Lemma qact_domE : H \subset R -> qact_dom to H = 'N(H | to).
Proof.
move=> sHR; apply/setP=> a; apply/idP/idP=> nHa; have Da := astabs_dom nHa.
rewrite !inE Da; apply/subsetP=> x Hx; rewrite inE -(rcoset1 H).
have /rcosetsP[y Ny defHy]: to^~ a @: H \in rcosets H 'N(H).
by rewrite (astabs_act _ nHa); apply/rcosetsP; exists 1; rewrite ?mulg1.
by rewrite (rcoset_eqP (_ : 1 \in H :* y)) -defHy -1?(gact1 Da) mem_setact.
rewrite !inE Da; apply/subsetP=> Hx /[1!inE] /rcosetsP[x Nx ->{Hx}].
apply/imsetP; exists (to x a).
case Rx: (x \in R); last by rewrite gact_out ?Rx.
rewrite inE; apply/subsetP=> _ /imsetP[y Hy ->].
rewrite -(actKVin to Da y) -gactJ // ?(subsetP sHR, astabs_act, groupV) //.
by rewrite memJ_norm // astabs_act ?groupV.
apply/eqP; rewrite rcosetE eqEcard.
rewrite (card_imset _ (act_inj _ _)) !card_rcoset leqnn andbT.
apply/subsetP=> _ /imsetP[y Hxy ->]; rewrite !mem_rcoset in Hxy *.
have Rxy := subsetP sHR _ Hxy; rewrite -(mulgKV x y).
case Rx: (x \in R); last by rewrite !gact_out ?mulgK // 1?groupMl ?Rx.
by rewrite -gactV // -gactM 1?groupMr ?groupV // mulgK astabs_act.
Qed.
End Quotient.
Section Mod.
Variable H : {group aT}.
Lemma modact_is_groupAction : is_groupAction 'C_(|to)(H) (to %% H).
Proof.
move=> Ha /morphimP[a Na Da ->]; have NDa: a \in 'N_D(H) by apply/setIP.
rewrite inE; apply/andP; split.
apply/subsetP=> x; rewrite 2!inE andbC actpermE /= modactEcond //.
by apply: contraR; case: ifP => // E Rx; rewrite gact_out.
apply/morphicP=> x y /setIP[Rx cHx] /setIP[Ry cHy].
rewrite /= !actpermE /= !modactE ?gactM //.
suffices: x * y \in 'C_(|to)(H) by case/setIP.
by rewrite groupM //; apply/setIP.
Qed.
Canonical mod_groupAction := GroupAction modact_is_groupAction.
Lemma modgactE x a :
H \subset 'C(R | to) -> a \in 'N_D(H) -> (to %% H)%act x (coset H a) = to x a.
Proof.
move=> cRH NDa /=; have [Da Na] := setIP NDa.
have [Rx | notRx] := boolP (x \in R).
by rewrite modactE //; apply/afixP=> b /setIP[_ /(subsetP cRH)/astab_act->].
rewrite gact_out //= /modact; case: ifP => // _; rewrite gact_out //.
suffices: a \in D :&: coset H a by case/mem_repr/setIP.
by rewrite inE Da val_coset // rcoset_refl.
Qed.
Lemma gacent_mod G M :
H \subset 'C(M | to) -> G \subset 'N(H) ->
'C_(M | mod_groupAction)(G / H) = 'C_(M | to)(G).
Proof.
move=> cMH nHG; rewrite -gacentIdom gacentE ?subsetIl // setICA.
have sHD: H \subset D by rewrite (subset_trans cMH) ?subsetIl.
rewrite -quotientGI // afix_mod ?setIS // setICA -gacentIim (setIC R) -setIA.
rewrite -gacentE ?subsetIl // gacentIdom setICA (setIidPr _) //.
by rewrite gacentC // ?(subset_trans cMH) ?astabS ?subsetIl // setICA subsetIl.
Qed.
Lemma acts_irr_mod G M :
H \subset 'C(M | to) -> G \subset 'N(H) -> acts_irreducibly G M to ->
acts_irreducibly (G / H) M mod_groupAction.
Proof.
move=> cMH nHG /mingroupP[/andP[ntM nMG] minM].
apply/mingroupP; rewrite ntM astabs_mod ?quotientS //; split=> // L modL ntL.
have cLH: H \subset 'C(L | to) by rewrite (subset_trans cMH) ?astabS //.
apply: minM => //; case/andP: modL => ->; rewrite astabs_mod ?quotientSGK //.
by rewrite (subset_trans cLH) ?astab_sub.
Qed.
End Mod.
Lemma modact_coset_astab x a :
a \in D -> (to %% 'C(R | to))%act x (coset _ a) = to x a.
Proof.
move=> Da; apply: modgactE => {x}//.
rewrite !inE Da; apply/subsetP=> _ /imsetP[c Cc ->].
have Dc := astab_dom Cc; rewrite !inE groupJ //.
apply/subsetP=> x Rx; rewrite inE conjgE !actMin ?groupM ?groupV //.
by rewrite (astab_act Cc) ?actKVin // gact_stable ?groupV.
Qed.
Lemma acts_irr_mod_astab G M :
acts_irreducibly G M to ->
acts_irreducibly (G / 'C_G(M | to)) M (mod_groupAction _).
Proof.
move=> irrG; have /andP[_ nMG] := mingroupp irrG.
apply: acts_irr_mod irrG; first exact: subsetIr.
by rewrite normsI ?normG // (subset_trans nMG) // astab_norm.
Qed.
Section CompAct.
Variables (gT : finGroupType) (G : {group gT}) (f : {morphism G >-> aT}).
Lemma comp_is_groupAction : is_groupAction R (comp_action to f).
Proof.
move=> a /morphpreP[Ba Dfa]; apply: etrans (actperm_Aut to Dfa).
by congr (_ \in Aut R); apply/permP=> x; rewrite !actpermE.
Qed.
Canonical comp_groupAction := GroupAction comp_is_groupAction.
Lemma gacent_comp U : 'C_(|comp_groupAction)(U) = 'C_(|to)(f @* U).
Proof.
rewrite /gacent afix_comp ?subIset ?subxx //.
by rewrite -(setIC U) (setIC D) morphim_setIpre.
Qed.
End CompAct.
End GroupActionTheory.
Notation "''C_' ( | to ) ( A )" := (gacent_group to A) : Group_scope.
Notation "''C_' ( G | to ) ( A )" :=
(setI_group G 'C_(|to)(A)) : Group_scope.
Notation "''C_' ( | to ) [ a ]" := (gacent_group to [set a%g]) : Group_scope.
Notation "''C_' ( G | to ) [ a ]" :=
(setI_group G 'C_(|to)[a]) : Group_scope.
Notation "to \ sAD" := (ract_groupAction to sAD) : groupAction_scope.
Notation "<[ nGA ] >" := (actby_groupAction nGA) : groupAction_scope.
Notation "to / H" := (quotient_groupAction to H) : groupAction_scope.
Notation "to %% H" := (mod_groupAction to H) : groupAction_scope.
Notation "to \o f" := (comp_groupAction to f) : groupAction_scope.
(* Operator group isomorphism. *)
Section MorphAction.
Variables (aT1 aT2 : finGroupType) (rT1 rT2 : finType).
Variables (D1 : {group aT1}) (D2 : {group aT2}).
Variables (to1 : action D1 rT1) (to2 : action D2 rT2).
Variables (A : {set aT1}) (R S : {set rT1}).
Variables (h : rT1 -> rT2) (f : {morphism D1 >-> aT2}).
Hypotheses (actsDR : {acts D1, on R | to1}) (injh : {in R &, injective h}).
Hypothesis defD2 : f @* D1 = D2.
Hypotheses (sSR : S \subset R) (sAD1 : A \subset D1).
Hypothesis hfJ : {in S & D1, morph_act to1 to2 h f}.
Lemma morph_astabs : f @* 'N(S | to1) = 'N(h @: S | to2).
Proof.
apply/setP=> fx; apply/morphimP/idP=> [[x D1x nSx ->] | nSx].
rewrite 2!inE -{1}defD2 mem_morphim //=; apply/subsetP=> _ /imsetP[u Su ->].
by rewrite inE -hfJ ?imset_f // (astabs_act _ nSx).
have [|x D1x _ def_fx] := morphimP (_ : fx \in f @* D1).
by rewrite defD2 (astabs_dom nSx).
exists x => //; rewrite !inE D1x; apply/subsetP=> u Su.
have /imsetP[u' Su' /injh def_u']: h (to1 u x) \in h @: S.
by rewrite hfJ // -def_fx (astabs_act _ nSx) imset_f.
by rewrite inE def_u' ?actsDR ?(subsetP sSR).
Qed.
Lemma morph_astab : f @* 'C(S | to1) = 'C(h @: S | to2).
Proof.
apply/setP=> fx; apply/morphimP/idP=> [[x D1x cSx ->] | cSx].
rewrite 2!inE -{1}defD2 mem_morphim //=; apply/subsetP=> _ /imsetP[u Su ->].
by rewrite inE -hfJ // (astab_act cSx).
have [|x D1x _ def_fx] := morphimP (_ : fx \in f @* D1).
by rewrite defD2 (astab_dom cSx).
exists x => //; rewrite !inE D1x; apply/subsetP=> u Su.
rewrite inE -(inj_in_eq injh) ?actsDR ?(subsetP sSR) ?hfJ //.
by rewrite -def_fx (astab_act cSx) ?imset_f.
Qed.
Lemma morph_afix : h @: 'Fix_(S | to1)(A) = 'Fix_(h @: S | to2)(f @* A).
Proof.
apply/setP=> hu; apply/imsetP/setIP=> [[u /setIP[Su cAu] ->]|].
split; first by rewrite imset_f.
by apply/afixP=> _ /morphimP[x D1x Ax ->]; rewrite -hfJ ?(afixP cAu).
case=> /imsetP[u Su ->] /afixP c_hu_fA; exists u; rewrite // inE Su.
apply/afixP=> x Ax; have Dx := subsetP sAD1 x Ax.
by apply: injh; rewrite ?actsDR ?(subsetP sSR) ?hfJ // c_hu_fA ?mem_morphim.
Qed.
End MorphAction.
Section MorphGroupAction.
Variables (aT1 aT2 rT1 rT2 : finGroupType).
Variables (D1 : {group aT1}) (D2 : {group aT2}).
Variables (R1 : {group rT1}) (R2 : {group rT2}).
Variables (to1 : groupAction D1 R1) (to2 : groupAction D2 R2).
Variables (h : {morphism R1 >-> rT2}) (f : {morphism D1 >-> aT2}).
Hypotheses (iso_h : isom R1 R2 h) (iso_f : isom D1 D2 f).
Hypothesis hfJ : {in R1 & D1, morph_act to1 to2 h f}.
Implicit Types (A : {set aT1}) (S : {set rT1}) (M : {group rT1}).
Lemma morph_gastabs S : S \subset R1 -> f @* 'N(S | to1) = 'N(h @* S | to2).
Proof.
have [[_ defD2] [injh _]] := (isomP iso_f, isomP iso_h).
move=> sSR1; rewrite (morphimEsub _ sSR1).
apply: (morph_astabs (gact_stable to1) (injmP injh)) => // u x.
by move/(subsetP sSR1); apply: hfJ.
Qed.
Lemma morph_gastab S : S \subset R1 -> f @* 'C(S | to1) = 'C(h @* S | to2).
Proof.
have [[_ defD2] [injh _]] := (isomP iso_f, isomP iso_h).
move=> sSR1; rewrite (morphimEsub _ sSR1).
apply: (morph_astab (gact_stable to1) (injmP injh)) => // u x.
by move/(subsetP sSR1); apply: hfJ.
Qed.
Lemma morph_gacent A : A \subset D1 -> h @* 'C_(|to1)(A) = 'C_(|to2)(f @* A).
Proof.
have [[_ defD2] [injh defR2]] := (isomP iso_f, isomP iso_h).
move=> sAD1; rewrite !gacentE //; last by rewrite -defD2 morphimS.
rewrite morphimEsub ?subsetIl // -{1}defR2 morphimEdom.
exact: (morph_afix (gact_stable to1) (injmP injh)).
Qed.
Lemma morph_gact_irr A M :
A \subset D1 -> M \subset R1 ->
acts_irreducibly (f @* A) (h @* M) to2 = acts_irreducibly A M to1.
Proof.
move=> sAD1 sMR1.
have [[injf defD2] [injh defR2]] := (isomP iso_f, isomP iso_h).
have h_eq1 := morphim_injm_eq1 injh.
apply/mingroupP/mingroupP=> [] [/andP[ntM actAM] minM].
split=> [|U]; first by rewrite -h_eq1 // ntM -(injmSK injf) ?morph_gastabs.
case/andP=> ntU acts_fAU sUM; have sUR1 := subset_trans sUM sMR1.
apply: (injm_morphim_inj injh) => //; apply: minM; last exact: morphimS.
by rewrite h_eq1 // ntU -morph_gastabs ?morphimS.
split=> [|U]; first by rewrite h_eq1 // ntM -morph_gastabs ?morphimS.
case/andP=> ntU acts_fAU sUhM.
have sUhR1 := subset_trans sUhM (morphimS h sMR1).
have sU'M: h @*^-1 U \subset M by rewrite sub_morphpre_injm.
rewrite /= -(minM _ _ sU'M) ?morphpreK // -h_eq1 ?subsetIl // -(injmSK injf) //.
by rewrite morph_gastabs ?(subset_trans sU'M) // morphpreK ?ntU.
Qed.
End MorphGroupAction.
(* Conjugation and right translation actions. *)
Section InternalActionDefs.
Variable gT : finGroupType.
Implicit Type A : {set gT}.
Implicit Type G : {group gT}.
(* This is not a Canonical action because it is seldom used, and it would *)
(* cause too many spurious matches (any group product would be viewed as an *)
(* action!). *)
Definition mulgr_action := TotalAction (@mulg1 gT) (@mulgA gT).
Canonical conjg_action := TotalAction (@conjg1 gT) (@conjgM gT).
Lemma conjg_is_groupAction : is_groupAction setT conjg_action.
Proof.
move=> a _; rewrite inE; apply/andP; split; first by apply/subsetP=> x /[1!inE].
by apply/morphicP=> x y _ _; rewrite !actpermE /= conjMg.
Qed.
Canonical conjg_groupAction := GroupAction conjg_is_groupAction.
Lemma rcoset_is_action : is_action setT (@rcoset gT).
Proof.
by apply: is_total_action => [A|A x y]; rewrite !rcosetE (mulg1, rcosetM).
Qed.
Canonical rcoset_action := Action rcoset_is_action.
Canonical conjsg_action := TotalAction (@conjsg1 gT) (@conjsgM gT).
Lemma conjG_is_action : is_action setT (@conjG_group gT).
Proof.
apply: is_total_action => [G | G x y]; apply: val_inj; rewrite /= ?act1 //.
exact: actM.
Qed.
Definition conjG_action := Action conjG_is_action.
End InternalActionDefs.
Notation "'R" := (@mulgr_action _) : action_scope.
Notation "'Rs" := (@rcoset_action _) : action_scope.
Notation "'J" := (@conjg_action _) : action_scope.
Notation "'J" := (@conjg_groupAction _) : groupAction_scope.
Notation "'Js" := (@conjsg_action _) : action_scope.
Notation "'JG" := (@conjG_action _) : action_scope.
Notation "'Q" := ('J / _)%act : action_scope.
Notation "'Q" := ('J / _)%gact : groupAction_scope.
Section InternalGroupAction.
Variable gT : finGroupType.
Implicit Types A B : {set gT}.
Implicit Types G H : {group gT}.
Implicit Type x : gT.
(* Various identities for actions on groups. *)
Lemma orbitR G x : orbit 'R G x = x *: G.
Proof. by rewrite -lcosetE. Qed.
Lemma astab1R x : 'C[x | 'R] = 1.
Proof.
apply/trivgP/subsetP=> y cxy.
by rewrite -(mulKg x y) [x * y](astab1P cxy) mulVg set11.
Qed.
Lemma astabR G : 'C(G | 'R) = 1.
Proof.
apply/trivgP/subsetP=> x cGx.
by rewrite -(mul1g x) [1 * x](astabP cGx) group1.
Qed.
Lemma astabsR G : 'N(G | 'R) = G.
Proof.
apply/setP=> x; rewrite !inE -setactVin ?inE //=.
by rewrite -groupV -{1 3}(mulg1 G) rcoset_sym -sub1set -mulGS -!rcosetE.
Qed.
Lemma atransR G : [transitive G, on G | 'R].
Proof. by rewrite /atrans -{1}(mul1g G) -orbitR imset_f. Qed.
Lemma faithfulR G : [faithful G, on G | 'R].
Proof. by rewrite /faithful astabR subsetIr. Qed.
Definition Cayley_repr G := actperm <[atrans_acts (atransR G)]>.
Theorem Cayley_isom G : isom G (Cayley_repr G @* G) (Cayley_repr G).
Proof. exact: faithful_isom (faithfulR G). Qed.
Theorem Cayley_isog G : G \isog Cayley_repr G @* G.
Proof. exact: isom_isog (Cayley_isom G). Qed.
Lemma orbitJ G x : orbit 'J G x = x ^: G. Proof. by []. Qed.
Lemma afixJ A : 'Fix_('J)(A) = 'C(A).
Proof.
apply/setP=> x; apply/afixP/centP=> cAx y Ay /=.
by rewrite /commute conjgC cAx.
by rewrite conjgE cAx ?mulKg.
Qed.
Lemma astabJ A : 'C(A |'J) = 'C(A).
Proof.
apply/setP=> x; apply/astabP/centP=> cAx y Ay /=.
by apply: esym; rewrite conjgC cAx.
by rewrite conjgE -cAx ?mulKg.
Qed.
Lemma astab1J x : 'C[x |'J] = 'C[x].
Proof. by rewrite astabJ cent_set1. Qed.
Lemma astabsJ A : 'N(A | 'J) = 'N(A).
Proof. by apply/setP=> x; rewrite -2!groupV !inE -conjg_preim -sub_conjg. Qed.
Lemma setactJ A x : 'J^*%act A x = A :^ x. Proof. by []. Qed.
Lemma gacentJ A : 'C_(|'J)(A) = 'C(A).
Proof. by rewrite gacentE ?setTI ?subsetT ?afixJ. Qed.
Lemma orbitRs G A : orbit 'Rs G A = rcosets A G. Proof. by []. Qed.
Lemma sub_afixRs_norms G x A : (G :* x \in 'Fix_('Rs)(A)) = (A \subset G :^ x).
Proof.
rewrite inE /=; apply: eq_subset_r => a.
rewrite inE rcosetE -(can2_eq (rcosetKV x) (rcosetK x)) -!rcosetM.
rewrite eqEcard card_rcoset leqnn andbT mulgA (conjgCV x) mulgK.
by rewrite -{2 3}(mulGid G) mulGS sub1set -mem_conjg.
Qed.
Lemma sub_afixRs_norm G x : (G :* x \in 'Fix_('Rs)(G)) = (x \in 'N(G)).
Proof. by rewrite sub_afixRs_norms -groupV inE sub_conjgV. Qed.
Lemma afixRs_rcosets A G : 'Fix_(rcosets G A | 'Rs)(G) = rcosets G 'N_A(G).
Proof.
apply/setP=> Gx; apply/setIP/rcosetsP=> [[/rcosetsP[x Ax ->]]|[x]].
by rewrite sub_afixRs_norm => Nx; exists x; rewrite // inE Ax.
by case/setIP=> Ax Nx ->; rewrite -{1}rcosetE imset_f // sub_afixRs_norm.
Qed.
Lemma astab1Rs G : 'C[G : {set gT} | 'Rs] = G.
Proof.
apply/setP=> x.
by apply/astab1P/idP=> /= [<- | Gx]; rewrite rcosetE ?rcoset_refl ?rcoset_id.
Qed.
Lemma actsRs_rcosets H G : [acts G, on rcosets H G | 'Rs].
Proof. by rewrite -orbitRs acts_orbit ?subsetT. Qed.
Lemma transRs_rcosets H G : [transitive G, on rcosets H G | 'Rs].
Proof. by rewrite -orbitRs atrans_orbit. Qed.
(* This is the second part of Aschbacher (5.7) *)
Lemma astabRs_rcosets H G : 'C(rcosets H G | 'Rs) = gcore H G.
Proof.
have transGH := transRs_rcosets H G.
by rewrite (astab_trans_gcore transGH (orbit_refl _ G _)) astab1Rs.
Qed.
Lemma orbitJs G A : orbit 'Js G A = A :^: G. Proof. by []. Qed.
Lemma astab1Js A : 'C[A | 'Js] = 'N(A).
Proof. by apply/setP=> x; apply/astab1P/normP. Qed.
Lemma card_conjugates A G : #|A :^: G| = #|G : 'N_G(A)|.
Proof. by rewrite card_orbit astab1Js. Qed.
Lemma afixJG G A : (G \in 'Fix_('JG)(A)) = (A \subset 'N(G)).
Proof. by apply/afixP/normsP=> nG x Ax; apply/eqP; move/eqP: (nG x Ax). Qed.
Lemma astab1JG G : 'C[G | 'JG] = 'N(G).
Proof.
by apply/setP=> x; apply/astab1P/normP=> [/congr_group | /group_inj].
Qed.
Lemma dom_qactJ H : qact_dom 'J H = 'N(H).
Proof. by rewrite qact_domE ?subsetT ?astabsJ. Qed.
Lemma qactJ H (Hy : coset_of H) x :
'Q%act Hy x = if x \in 'N(H) then Hy ^ coset H x else Hy.
Proof.
case: (cosetP Hy) => y Ny ->{Hy}.
by rewrite qactEcond // dom_qactJ; case Nx: (x \in 'N(H)); rewrite ?morphJ.
Qed.
Lemma actsQ A B H :
A \subset 'N(H) -> A \subset 'N(B) -> [acts A, on B / H | 'Q].
Proof.
by move=> nHA nBA; rewrite acts_quotient // subsetI dom_qactJ nHA astabsJ.
Qed.
Lemma astabsQ G H : H <| G -> 'N(G / H | 'Q) = 'N(H) :&: 'N(G).
Proof. by move=> nsHG; rewrite astabs_quotient // dom_qactJ astabsJ. Qed.
Lemma astabQ H Abar : 'C(Abar |'Q) = coset H @*^-1 'C(Abar).
Proof.
apply/setP=> x; rewrite inE /= dom_qactJ morphpreE in_setI /=.
apply: andb_id2l => Nx; rewrite !inE -sub1set centsC cent_set1.
apply: eq_subset_r => {Abar} Hy; rewrite inE qactJ Nx (sameP eqP conjg_fixP).
by rewrite (sameP cent1P eqP) (sameP commgP eqP).
Qed.
Lemma sub_astabQ A H Bbar :
(A \subset 'C(Bbar | 'Q)) = (A \subset 'N(H)) && (A / H \subset 'C(Bbar)).
Proof.
rewrite astabQ -morphpreIdom subsetI; apply: andb_id2l => nHA.
by rewrite -sub_quotient_pre.
Qed.
Lemma sub_astabQR A B H :
A \subset 'N(H) -> B \subset 'N(H) ->
(A \subset 'C(B / H | 'Q)) = ([~: A, B] \subset H).
Proof.
move=> nHA nHB; rewrite sub_astabQ nHA /= (sameP commG1P eqP).
by rewrite eqEsubset sub1G andbT -quotientR // quotient_sub1 // comm_subG.
Qed.
Lemma astabQR A H : A \subset 'N(H) ->
'C(A / H | 'Q) = [set x in 'N(H) | [~: [set x], A] \subset H].
Proof.
move=> nHA; apply/setP=> x; rewrite astabQ -morphpreIdom 2!inE -astabQ.
by case nHx: (x \in _); rewrite //= -sub1set sub_astabQR ?sub1set.
Qed.
Lemma quotient_astabQ H Abar : 'C(Abar | 'Q) / H = 'C(Abar).
Proof. by rewrite astabQ cosetpreK. Qed.
Lemma conj_astabQ A H x :
x \in 'N(H) -> 'C(A / H | 'Q) :^ x = 'C(A :^ x / H | 'Q).
Proof.
move=> nHx; apply/setP=> y; rewrite !astabQ mem_conjg !in_setI -mem_conjg.
rewrite -normJ (normP nHx) quotientJ //; apply/andb_id2l => nHy.
by rewrite !inE centJ morphJ ?groupV ?morphV // -mem_conjg.
Qed.
Section CardClass.
Variable G : {group gT}.
Lemma index_cent1 x : #|G : 'C_G[x]| = #|x ^: G|.
Proof. by rewrite -astab1J -card_orbit. Qed.
Lemma classes_partition : partition (classes G) G.
Proof. by apply: orbit_partition; apply/actsP=> x Gx y; apply: groupJr. Qed.
Lemma sum_card_class : \sum_(C in classes G) #|C| = #|G|.
Proof. by apply: acts_sum_card_orbit; apply/actsP=> x Gx y; apply: groupJr. Qed.
Lemma class_formula : \sum_(C in classes G) #|G : 'C_G[repr C]| = #|G|.
Proof.
rewrite -sum_card_class; apply: eq_bigr => _ /imsetP[x Gx ->].
have: x \in x ^: G by rewrite -{1}(conjg1 x) imset_f.
by case/mem_repr/imsetP=> y Gy ->; rewrite index_cent1 classGidl.
Qed.
Lemma abelian_classP : reflect {in G, forall x, x ^: G = [set x]} (abelian G).
Proof.
rewrite /abelian -astabJ astabC.
by apply: (iffP subsetP) => cGG x Gx; apply/orbit1P; apply: cGG.
Qed.
Lemma card_classes_abelian : abelian G = (#|classes G| == #|G|).
Proof.
have cGgt0 C: C \in classes G -> 1 <= #|C| ?= iff (#|C| == 1)%N.
by case/imsetP=> x _ ->; rewrite eq_sym -index_cent1.
rewrite -sum_card_class -sum1_card (leqif_sum cGgt0).
apply/abelian_classP/forall_inP=> [cGG _ /imsetP[x Gx ->]| cGG x Gx].
by rewrite cGG ?cards1.
apply/esym/eqP; rewrite eqEcard sub1set cards1 class_refl leq_eqVlt cGG //.
exact: imset_f.
Qed.
End CardClass.
End InternalGroupAction.
Lemma gacentQ (gT : finGroupType) (H : {group gT}) (A : {set gT}) :
'C_(|'Q)(A) = 'C(A / H).
Proof.
apply/setP=> Hx; case: (cosetP Hx) => x Nx ->{Hx}.
rewrite -sub_cent1 -astab1J astabC sub1set -(quotientInorm H A).
have defD: qact_dom 'J H = 'N(H) by rewrite qact_domE ?subsetT ?astabsJ.
rewrite !(inE, mem_quotient) //= defD setIC.
apply/subsetP/subsetP=> [cAx _ /morphimP[a Na Aa ->] | cAx a Aa].
by move/cAx: Aa; rewrite !inE qactE ?defD ?morphJ.
have [_ Na] := setIP Aa; move/implyP: (cAx (coset H a)); rewrite mem_morphim //.
by rewrite !inE qactE ?defD ?morphJ.
Qed.
Section AutAct.
Variable (gT : finGroupType) (G : {set gT}).
Definition autact := act ('P \ subsetT (Aut G)).
Canonical aut_action := [action of autact].
Lemma autactK a : actperm aut_action a = a.
Proof. by apply/permP=> x; rewrite permE. Qed.
Lemma autact_is_groupAction : is_groupAction G aut_action.
Proof. by move=> a Aa /=; rewrite autactK. Qed.
Canonical aut_groupAction := GroupAction autact_is_groupAction.
Section perm_prime_orbit.
Variable (T : finType) (c : {perm T}).
Hypothesis Tp : prime #|T|.
Hypothesis cc : #[c]%g = #|T|.
Let cp : prime #[c]%g. Proof. by rewrite cc. Qed.
Lemma perm_prime_atrans : [transitive <[c]>, on setT | 'P].
Proof.
apply/imsetP; suff /existsP[x] : [exists x, ~~ (#|orbit 'P <[c]> x| < #[c])].
move=> oxT; suff /eqP orbit_x : orbit 'P <[c]> x == setT by exists x.
by rewrite eqEcard subsetT cardsT -cc leqNgt.
apply/forallP => olT; have o1 x : #|orbit 'P <[c]> x| == 1%N.
by case/primeP: cp => _ /(_ _ (dvdn_orbit 'P _ x))/orP[]//; rewrite ltn_eqF.
suff c1 : c = 1%g by rewrite c1 ?order1 in (cp).
apply/permP => x; rewrite perm1; apply/set1P.
by rewrite -(card_orbit1 (eqP (o1 _))) (mem_orbit 'P) ?cycle_id.
Qed.
Lemma perm_prime_orbit x : orbit 'P <[c]> x = [set: T].
Proof. by apply: atransP => //; apply: perm_prime_atrans. Qed.
Lemma perm_prime_astab x : 'C_<[c]>[x | 'P]%g = 1%g.
Proof.
by apply/card1_trivg/eqP; rewrite -(@eqn_pmul2l #|orbit 'P <[c]> x|)
?card_orbit_stab ?perm_prime_orbit ?cardsT ?muln1 ?prime_gt0// -cc.
Qed.
End perm_prime_orbit.
End AutAct.
Arguments autact {gT} G%_g.
Arguments aut_action {gT} G%_g.
Arguments aut_groupAction {gT} G%_g.
Notation "[ 'Aut' G ]" := (aut_action G) : action_scope.
Notation "[ 'Aut' G ]" := (aut_groupAction G) : groupAction_scope.
|
Field.lean
|
/-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.FieldTheory.PurelyInseparable.Basic
import Mathlib.RingTheory.Artinian.Ring
import Mathlib.RingTheory.LocalProperties.Basic
import Mathlib.Algebra.Polynomial.Taylor
import Mathlib.RingTheory.Unramified.Finite
/-!
# Unramified algebras over fields
## Main results
Let `K` be a field, `A` be a `K`-algebra and `L` be a field extension of `K`.
- `Algebra.FormallyUnramified.bijective_of_isAlgClosed_of_isLocalRing`:
If `A` is `K`-unramified and `K` is alg-closed, then `K = A`.
- `Algebra.FormallyUnramified.isReduced_of_field`:
If `A` is `K`-unramified then `A` is reduced.
- `Algebra.FormallyUnramified.iff_isSeparable`:
`L` is unramified over `K` iff `L` is separable over `K`.
## References
- [B. Iversen, *Generic Local Structure of the Morphisms in Commutative Algebra*][iversen]
-/
open Algebra Module Polynomial
open scoped TensorProduct
universe u
variable (K A L : Type*) [Field K] [Field L] [CommRing A] [Algebra K A] [Algebra K L]
namespace Algebra.FormallyUnramified
theorem of_isSeparable [Algebra.IsSeparable K L] : FormallyUnramified K L := by
rw [iff_comp_injective]
intros B _ _ I hI f₁ f₂ e
ext x
have : f₁ x - f₂ x ∈ I := by
simpa [Ideal.Quotient.mk_eq_mk_iff_sub_mem] using AlgHom.congr_fun e x
have := Polynomial.eval_add_of_sq_eq_zero ((minpoly K x).map (algebraMap K B)) (f₂ x)
(f₁ x - f₂ x) (show (f₁ x - f₂ x) ^ 2 ∈ ⊥ from hI ▸ Ideal.pow_mem_pow this 2)
simp only [add_sub_cancel, eval_map_algebraMap, aeval_algHom_apply, minpoly.aeval, map_zero,
derivative_map, zero_add] at this
rwa [eq_comm, ((isUnit_iff_ne_zero.mpr
((Algebra.IsSeparable.isSeparable K x).aeval_derivative_ne_zero
(minpoly.aeval K x))).map f₂).mul_right_eq_zero, sub_eq_zero] at this
variable [FormallyUnramified K A] [EssFiniteType K A]
variable [FormallyUnramified K L] [EssFiniteType K L]
theorem bijective_of_isAlgClosed_of_isLocalRing
[IsAlgClosed K] [IsLocalRing A] :
Function.Bijective (algebraMap K A) := by
have := finite_of_free (R := K) (S := A)
have : IsArtinianRing A := isArtinian_of_tower K inferInstance
have hA : IsNilpotent (IsLocalRing.maximalIdeal A) := by
rw [← IsLocalRing.jacobson_eq_maximalIdeal ⊥]
· exact IsArtinianRing.isNilpotent_jacobson_bot
· exact bot_ne_top
let e : K ≃ₐ[K] A ⧸ IsLocalRing.maximalIdeal A := {
__ := Algebra.ofId K (A ⧸ IsLocalRing.maximalIdeal A)
__ := Equiv.ofBijective _ IsAlgClosed.algebraMap_bijective_of_isIntegral }
let e' : A ⊗[K] (A ⧸ IsLocalRing.maximalIdeal A) ≃ₐ[A] A :=
(Algebra.TensorProduct.congr AlgEquiv.refl e.symm).trans (Algebra.TensorProduct.rid K A A)
let f : A ⧸ IsLocalRing.maximalIdeal A →ₗ[A] A := e'.toLinearMap.comp (sec K A _)
have hf : (Algebra.ofId _ _).toLinearMap ∘ₗ f = LinearMap.id := by
dsimp [f]
rw [← LinearMap.comp_assoc, ← comp_sec K A]
congr 1
apply LinearMap.restrictScalars_injective K
apply _root_.TensorProduct.ext'
intros r s
obtain ⟨s, rfl⟩ := e.surjective s
suffices s • (Ideal.Quotient.mk (IsLocalRing.maximalIdeal A)) r = r • e s by
simpa [ofId, e']
simp [Algebra.smul_def, e, ofId, mul_comm]
have hf₁ : f 1 • (1 : A ⧸ IsLocalRing.maximalIdeal A) = 1 := by
rw [← algebraMap_eq_smul_one]
exact LinearMap.congr_fun hf 1
have hf₂ : 1 - f 1 ∈ IsLocalRing.maximalIdeal A := by
rw [← Ideal.Quotient.eq_zero_iff_mem, map_sub, map_one, ← Ideal.Quotient.algebraMap_eq,
algebraMap_eq_smul_one, hf₁, sub_self]
have hf₃ : IsIdempotentElem (1 - f 1) := by
apply IsIdempotentElem.one_sub
rw [IsIdempotentElem, ← smul_eq_mul, ← map_smul, hf₁]
have hf₄ : f 1 = 1 := by
obtain ⟨n, hn⟩ := hA
have : (1 - f 1) ^ n = 0 := by
rw [← Ideal.mem_bot, ← Ideal.zero_eq_bot, ← hn]
exact Ideal.pow_mem_pow hf₂ n
rw [eq_comm, ← sub_eq_zero, ← hf₃.pow_succ_eq n, pow_succ, this, zero_mul]
refine Equiv.bijective ⟨algebraMap K A, ⇑e.symm ∘ ⇑(algebraMap A _), fun x ↦ by simp, fun x ↦ ?_⟩
have : ⇑(algebraMap K A) = ⇑f ∘ ⇑e := by
ext k
conv_rhs => rw [← mul_one k, ← smul_eq_mul, Function.comp_apply, map_smul,
LinearMap.map_smul_of_tower, map_one, hf₄, ← algebraMap_eq_smul_one]
rw [this]
simp only [Function.comp_apply, AlgEquiv.apply_symm_apply, algebraMap_eq_smul_one,
map_smul, hf₄, smul_eq_mul, mul_one]
theorem isField_of_isAlgClosed_of_isLocalRing
[IsAlgClosed K] [IsLocalRing A] : IsField A := by
rw [IsLocalRing.isField_iff_maximalIdeal_eq, eq_bot_iff]
intro x hx
obtain ⟨x, rfl⟩ := (bijective_of_isAlgClosed_of_isLocalRing K A).surjective x
change _ = 0
rw [← (algebraMap K A).map_zero]
by_contra hx'
exact hx ((isUnit_iff_ne_zero.mpr
(fun e ↦ hx' ((algebraMap K A).congr_arg e))).map (algebraMap K A))
include K in
theorem isReduced_of_field :
IsReduced A := by
constructor
intro x hx
let f := (Algebra.TensorProduct.includeRight (R := K) (A := AlgebraicClosure K) (B := A))
have : Function.Injective f := by
have : ⇑f = (LinearMap.rTensor A (Algebra.ofId K (AlgebraicClosure K)).toLinearMap).comp
(Algebra.TensorProduct.lid K A).symm.toLinearMap := by
ext x; simp [f]
rw [this]
suffices Function.Injective
(LinearMap.rTensor A (Algebra.ofId K (AlgebraicClosure K)).toLinearMap) by
exact this.comp (Algebra.TensorProduct.lid K A).symm.injective
apply Module.Flat.rTensor_preserves_injective_linearMap
exact (algebraMap K _).injective
apply this
rw [map_zero]
apply eq_zero_of_localization
intro M hM
have hy := (hx.map f).map (algebraMap _ (Localization.AtPrime M))
generalize algebraMap _ (Localization.AtPrime M) (f x) = y at *
have := EssFiniteType.of_isLocalization (Localization.AtPrime M) M.primeCompl
have := of_isLocalization (Rₘ := Localization.AtPrime M) M.primeCompl
have := EssFiniteType.comp (AlgebraicClosure K) (AlgebraicClosure K ⊗[K] A)
(Localization.AtPrime M)
have := comp (AlgebraicClosure K) (AlgebraicClosure K ⊗[K] A)
(Localization.AtPrime M)
letI := (isField_of_isAlgClosed_of_isLocalRing (AlgebraicClosure K)
(A := Localization.AtPrime M)).toField
exact hy.eq_zero
theorem range_eq_top_of_isPurelyInseparable
[IsPurelyInseparable K L] : (algebraMap K L).range = ⊤ := by
classical
have : Nontrivial (L ⊗[K] L) := by
rw [← not_subsingleton_iff_nontrivial, ← rank_zero_iff (R := K), rank_tensorProduct',
mul_eq_zero, or_self, rank_zero_iff, not_subsingleton_iff_nontrivial]
infer_instance
rw [← top_le_iff]
intro x _
obtain ⟨n, hn⟩ := IsPurelyInseparable.pow_mem K (ringExpChar K) x
have : ExpChar (L ⊗[K] L) (ringExpChar K) := by
refine expChar_of_injective_ringHom (algebraMap K _).injective (ringExpChar K)
have : (1 ⊗ₜ x - x ⊗ₜ 1 : L ⊗[K] L) ^ (ringExpChar K) ^ n = 0 := by
rw [sub_pow_expChar_pow, TensorProduct.tmul_pow, one_pow, TensorProduct.tmul_pow, one_pow]
obtain ⟨r, hr⟩ := hn
rw [← hr, algebraMap_eq_smul_one, TensorProduct.smul_tmul, sub_self]
have H : (1 ⊗ₜ x : L ⊗[K] L) = x ⊗ₜ 1 := by
have inst : IsReduced (L ⊗[K] L) := isReduced_of_field L _
exact sub_eq_zero.mp (IsNilpotent.eq_zero ⟨_, this⟩)
by_cases h' : LinearIndependent K ![1, x]
· have h := h'.linearIndepOn_id
let S := h.extend (Set.subset_univ _)
let a : S := ⟨1, h.subset_extend _ (by simp)⟩
have ha : Basis.extend h a = 1 := by simp [a]
let b : S := ⟨x, h.subset_extend _ (by simp)⟩
have hb : Basis.extend h b = x := by simp [b]
by_cases e : a = b
· obtain rfl : 1 = x := congr_arg Subtype.val e
exact ⟨1, map_one _⟩
have := DFunLike.congr_fun
(DFunLike.congr_arg ((Basis.extend h).tensorProduct (Basis.extend h)).repr H) (a, b)
simp only [Basis.tensorProduct_repr_tmul_apply, ← ha, ← hb, Basis.repr_self, smul_eq_mul,
Finsupp.single_apply, e, Ne.symm e, ↓reduceIte, mul_one, mul_zero, one_ne_zero] at this
· rw [LinearIndependent.pair_iff] at h'
simp only [not_forall, not_and, exists_prop] at h'
obtain ⟨a, b, e, hab⟩ := h'
have : IsUnit b := by
rw [isUnit_iff_ne_zero]
rintro rfl
rw [zero_smul, ← algebraMap_eq_smul_one, add_zero,
(injective_iff_map_eq_zero' _).mp (algebraMap K L).injective] at e
cases hab e rfl
use (-this.unit⁻¹ * a)
rw [map_mul, ← Algebra.smul_def, algebraMap_eq_smul_one, eq_neg_iff_add_eq_zero.mpr e,
smul_neg, neg_smul, neg_neg, smul_smul, this.val_inv_mul, one_smul]
theorem isSeparable : Algebra.IsSeparable K L := by
have := finite_of_free (R := K) (S := L)
rw [← separableClosure.eq_top_iff]
have := of_comp K (separableClosure K L) L
have := EssFiniteType.of_comp K (separableClosure K L) L
ext
change _ ↔ _ ∈ (⊤ : Subring _)
rw [← range_eq_top_of_isPurelyInseparable (separableClosure K L) L]
simp
theorem iff_isSeparable (L : Type u) [Field L] [Algebra K L] [EssFiniteType K L] :
FormallyUnramified K L ↔ Algebra.IsSeparable K L :=
⟨fun _ ↦ isSeparable K L, fun _ ↦ of_isSeparable K L⟩
end Algebra.FormallyUnramified
|
FunctorCategory.lean
|
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
import Mathlib.CategoryTheory.Monoidal.Cartesian.Basic
import Mathlib.CategoryTheory.Monoidal.Types.Basic
/-!
# Functor categories have chosen finite products
If `C` is a category with chosen finite products, then so is `J ⥤ C`.
-/
namespace CategoryTheory
open Limits MonoidalCategory Category CartesianMonoidalCategory
universe v
variable {J C D E : Type*} [Category J] [Category C] [Category D] [Category E]
[CartesianMonoidalCategory C] [CartesianMonoidalCategory E]
namespace Functor
variable (J C) in
/-- The chosen terminal object in `J ⥤ C`. -/
abbrev chosenTerminal : J ⥤ C := (Functor.const J).obj (𝟙_ C)
variable (J C) in
/-- The chosen terminal object in `J ⥤ C` is terminal. -/
def chosenTerminalIsTerminal : IsTerminal (chosenTerminal J C) :=
evaluationJointlyReflectsLimits _
fun _ ↦ isLimitChangeEmptyCone _ isTerminalTensorUnit _ (.refl _)
section
variable (F₁ F₂ : J ⥤ C)
/-- The chosen binary product on `J ⥤ C`. -/
@[simps]
def chosenProd : J ⥤ C where
obj j := F₁.obj j ⊗ F₂.obj j
map φ := F₁.map φ ⊗ₘ F₂.map φ
namespace chosenProd
/-- The first projection `chosenProd F₁ F₂ ⟶ F₁`. -/
def fst : chosenProd F₁ F₂ ⟶ F₁ where
app _ := CartesianMonoidalCategory.fst _ _
/-- The second projection `chosenProd F₁ F₂ ⟶ F₂`. -/
def snd : chosenProd F₁ F₂ ⟶ F₂ where
app _ := CartesianMonoidalCategory.snd _ _
/-- `Functor.chosenProd F₁ F₂` is a binary product of `F₁` and `F₂`. -/
def isLimit : IsLimit (BinaryFan.mk (fst F₁ F₂) (snd F₁ F₂)) :=
evaluationJointlyReflectsLimits _ (fun j =>
(IsLimit.postcomposeHomEquiv (mapPairIso (by exact Iso.refl _) (by exact Iso.refl _)) _).1
(IsLimit.ofIsoLimit
(tensorProductIsBinaryProduct (X := F₁.obj j) (Y := F₂.obj j))
(Cones.ext (Iso.refl _) (by rintro ⟨_ | _⟩; all_goals cat_disch))))
end chosenProd
end
instance cartesianMonoidalCategory : CartesianMonoidalCategory (J ⥤ C) :=
.ofChosenFiniteProducts ⟨_, chosenTerminalIsTerminal J C⟩
fun F₁ F₂ ↦ ⟨_, chosenProd.isLimit F₁ F₂⟩
namespace Monoidal
open CartesianMonoidalCategory
@[simp]
lemma tensorObj_obj (F₁ F₂ : J ⥤ C) (j : J) : (F₁ ⊗ F₂).obj j = (F₁.obj j) ⊗ (F₂.obj j) := rfl
@[simp]
lemma tensorObj_map (F₁ F₂ : J ⥤ C) {j j' : J} (f : j ⟶ j') :
(F₁ ⊗ F₂).map f = (F₁.map f) ⊗ₘ (F₂.map f) := rfl
@[simp]
lemma fst_app (F₁ F₂ : J ⥤ C) (j : J) : (fst F₁ F₂).app j = fst (F₁.obj j) (F₂.obj j) := rfl
@[simp]
lemma snd_app (F₁ F₂ : J ⥤ C) (j : J) : (snd F₁ F₂).app j = snd (F₁.obj j) (F₂.obj j) := rfl
@[simp]
lemma leftUnitor_hom_app (F : J ⥤ C) (j : J) :
(λ_ F).hom.app j = (λ_ (F.obj j)).hom := (leftUnitor_hom _).symm
@[simp]
lemma leftUnitor_inv_app (F : J ⥤ C) (j : J) :
(λ_ F).inv.app j = (λ_ (F.obj j)).inv := by
rw [← cancel_mono ((λ_ (F.obj j)).hom), Iso.inv_hom_id, ← leftUnitor_hom_app,
Iso.inv_hom_id_app]
@[simp]
lemma rightUnitor_hom_app (F : J ⥤ C) (j : J) :
(ρ_ F).hom.app j = (ρ_ (F.obj j)).hom := (rightUnitor_hom _).symm
@[simp]
lemma rightUnitor_inv_app (F : J ⥤ C) (j : J) :
(ρ_ F).inv.app j = (ρ_ (F.obj j)).inv := by
rw [← cancel_mono ((ρ_ (F.obj j)).hom), Iso.inv_hom_id, ← rightUnitor_hom_app,
Iso.inv_hom_id_app]
@[reassoc (attr := simp)]
lemma tensorHom_app_fst {F₁ F₁' F₂ F₂' : J ⥤ C} (f : F₁ ⟶ F₁') (g : F₂ ⟶ F₂') (j : J) :
(f ⊗ₘ g).app j ≫ fst _ _ = fst _ _ ≫ f.app j := by
change (f ⊗ₘ g).app j ≫ (fst F₁' F₂').app j = _
rw [← NatTrans.comp_app, tensorHom_fst, NatTrans.comp_app]
rfl
@[reassoc (attr := simp)]
lemma tensorHom_app_snd {F₁ F₁' F₂ F₂' : J ⥤ C} (f : F₁ ⟶ F₁') (g : F₂ ⟶ F₂') (j : J) :
(f ⊗ₘ g).app j ≫ snd _ _ = snd _ _ ≫ g.app j := by
change (f ⊗ₘ g).app j ≫ (snd F₁' F₂').app j = _
rw [← NatTrans.comp_app, tensorHom_snd, NatTrans.comp_app]
rfl
@[reassoc (attr := simp)]
lemma whiskerLeft_app_fst (F₁ : J ⥤ C) {F₂ F₂' : J ⥤ C} (g : F₂ ⟶ F₂') (j : J) :
(F₁ ◁ g).app j ≫ fst _ _ = fst _ _ :=
(tensorHom_app_fst (𝟙 F₁) g j).trans (by simp)
@[reassoc (attr := simp)]
lemma whiskerLeft_app_snd (F₁ : J ⥤ C) {F₂ F₂' : J ⥤ C} (g : F₂ ⟶ F₂') (j : J) :
(F₁ ◁ g).app j ≫ snd _ _ = snd _ _ ≫ g.app j :=
(tensorHom_app_snd (𝟙 F₁) g j)
@[reassoc (attr := simp)]
lemma whiskerRight_app_fst {F₁ F₁' : J ⥤ C} (f : F₁ ⟶ F₁') (F₂ : J ⥤ C) (j : J) :
(f ▷ F₂).app j ≫ fst _ _ = fst _ _ ≫ f.app j :=
(tensorHom_app_fst f (𝟙 F₂) j)
@[reassoc (attr := simp)]
lemma whiskerRight_app_snd {F₁ F₁' : J ⥤ C} (f : F₁ ⟶ F₁') (F₂ : J ⥤ C) (j : J) :
(f ▷ F₂).app j ≫ snd _ _ = snd _ _ :=
(tensorHom_app_snd f (𝟙 F₂) j).trans (by simp)
@[simp]
lemma associator_hom_app (F₁ F₂ F₃ : J ⥤ C) (j : J) :
(α_ F₁ F₂ F₃).hom.app j = (α_ _ _ _).hom := by
apply hom_ext
· rw [← fst_app, ← NatTrans.comp_app, associator_hom_fst]
simp
· apply hom_ext
· rw [← snd_app, ← NatTrans.comp_app, ← fst_app, ← NatTrans.comp_app, Category.assoc,
associator_hom_snd_fst]
simp
· rw [← snd_app, ← NatTrans.comp_app, ← snd_app, ← NatTrans.comp_app, Category.assoc,
associator_hom_snd_snd]
simp
@[simp]
lemma associator_inv_app (F₁ F₂ F₃ : J ⥤ C) (j : J) :
(α_ F₁ F₂ F₃).inv.app j = (α_ _ _ _).inv := by
rw [← cancel_mono ((α_ _ _ _).hom), Iso.inv_hom_id, ← associator_hom_app, Iso.inv_hom_id_app]
instance {K : Type*} [Category K] [HasColimitsOfShape K C]
[∀ X : C, PreservesColimitsOfShape K (tensorLeft X)] {F : J ⥤ C} :
PreservesColimitsOfShape K (tensorLeft F) := by
apply preservesColimitsOfShape_of_evaluation
intro k
haveI : tensorLeft F ⋙ (evaluation J C).obj k ≅ (evaluation J C).obj k ⋙ tensorLeft (F.obj k) :=
NatIso.ofComponents (fun _ ↦ Iso.refl _)
exact preservesColimitsOfShape_of_natIso this.symm
/-- A finite-products-preserving functor distributes over the tensor product of functors. -/
@[simps!]
noncomputable def tensorObjComp (F G : D ⥤ C) (H : C ⥤ E) [PreservesFiniteProducts H] :
(F ⊗ G) ⋙ H ≅ (F ⋙ H) ⊗ (G ⋙ H) :=
NatIso.ofComponents (fun X ↦ prodComparisonIso H (F.obj X) (G.obj X)) fun {X Y} f ↦ by
dsimp; ext <;> simp [← Functor.map_comp]
/-- A tensor product of representable functors is representable. -/
@[simps]
protected def RepresentableBy.tensorObj {F : Cᵒᵖ ⥤ Type v} {G : Cᵒᵖ ⥤ Type v} {X Y : C}
(h₁ : F.RepresentableBy X) (h₂ : G.RepresentableBy Y) : (F ⊗ G).RepresentableBy (X ⊗ Y) where
homEquiv {I} := homEquivToProd.trans (h₁.homEquiv.prodCongr h₂.homEquiv)
homEquiv_comp {I W} f g := by
refine Prod.ext ?_ ?_
· change h₁.homEquiv ((f ≫ g) ≫ fst X Y) = F.map f.op (h₁.homEquiv (g ≫ fst X Y))
simp [h₁.homEquiv_comp]
· change h₂.homEquiv ((f ≫ g) ≫ snd X Y) = G.map f.op (h₂.homEquiv (g ≫ snd X Y))
simp [h₂.homEquiv_comp]
end Monoidal
end Functor
end CategoryTheory
|
Generators.lean
|
/-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.MvPolynomial.Tower
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.RingTheory.Extension.Basic
/-!
# Generators of algebras
## Main definition
- `Algebra.Generators`: A family of generators of a `R`-algebra `S` consists of
1. `vars`: The type of variables.
2. `val : vars → S`: The assignment of each variable to a value.
3. `σ`: A set-theoretic section of the induced `R`-algebra homomorphism `R[X] → S`, where we
write `R[X]` for `R[vars]`.
- `Algebra.Generators.Hom`: Given a commuting square
```
R --→ P = R[X] ---→ S
| |
↓ ↓
R' -→ P' = R'[X'] → S
```
A hom between `P` and `P'` is an assignment `X → P'` such that the arrows commute.
- `Algebra.Generators.Cotangent`: The cotangent space wrt `P = R[X] → S`, i.e. the
space `I/I²` with `I` being the kernel of the presentation.
## TODOs
Currently, Lean does not see through the `vars` field of terms of `Generators R S` obtained
from constructions, e.g. composition. This causes fragile and cumbersome proofs, because
`simp` and `rw` often don't work properly. `Generators R S` (and `Presentation R S`, etc.) should
be refactored in a way that makes these equalities reducibly def-eq, for example
by unbundling the `vars` field or making the field globally reducible in constructions using
unification hints.
-/
universe w u v
open TensorProduct MvPolynomial
variable (R : Type u) (S : Type v) (ι : Type w) [CommRing R] [CommRing S] [Algebra R S]
/-- A family of generators of a `R`-algebra `S` consists of
1. `vars`: The type of variables.
2. `val : vars → S`: The assignment of each variable to a value in `S`.
3. `σ`: A section of `R[X] → S`. -/
structure Algebra.Generators where
/-- The assignment of each variable to a value in `S`. -/
val : ι → S
/-- A section of `R[X] → S`. -/
σ' : S → MvPolynomial ι R
aeval_val_σ' : ∀ s, aeval val (σ' s) = s
/-- An `R[X]`-algebra instance on `S`. The default is the one induced by the map `R[X] → S`,
but this causes a diamond if there is an existing instance. -/
algebra : Algebra (MvPolynomial ι R) S := (aeval val).toAlgebra
algebraMap_eq :
algebraMap (MvPolynomial ι R) S = aeval (R := R) val := by rfl
namespace Algebra.Generators
variable {R S ι}
variable (P : Generators R S ι)
set_option linter.unusedVariables false in
/-- The polynomial ring wrt a family of generators. -/
@[nolint unusedArguments]
protected
abbrev Ring (P : Generators R S ι) : Type (max w u) := MvPolynomial ι R
instance : Algebra P.Ring S := P.algebra
/-- The designated section of wrt a family of generators. -/
def σ : S → P.Ring := P.σ'
/-- See Note [custom simps projection] -/
def Simps.σ : S → P.Ring := P.σ
initialize_simps_projections Algebra.Generators (σ' → σ)
@[simp]
lemma aeval_val_σ (s) : aeval P.val (P.σ s) = s := P.aeval_val_σ' s
noncomputable instance {R₀} [CommRing R₀] [Algebra R₀ R] [Algebra R₀ S] [IsScalarTower R₀ R S] :
IsScalarTower R₀ P.Ring S := IsScalarTower.of_algebraMap_eq' <|
P.algebraMap_eq ▸ ((aeval (R := R) P.val).comp_algebraMap_of_tower R₀).symm
@[simp]
lemma algebraMap_apply (x) : algebraMap P.Ring S x = aeval (R := R) P.val x := by
simp [algebraMap_eq]
@[simp]
lemma σ_smul (x y) : P.σ x • y = x * y := by
rw [Algebra.smul_def, algebraMap_apply, aeval_val_σ]
lemma σ_injective : P.σ.Injective := by
intro x y e
rw [← P.aeval_val_σ x, ← P.aeval_val_σ y, e]
lemma algebraMap_surjective : Function.Surjective (algebraMap P.Ring S) :=
(⟨_, P.algebraMap_apply _ ▸ P.aeval_val_σ ·⟩)
section Construction
/-- Construct `Generators` from an assignment `I → S` such that `R[X] → S` is surjective. -/
@[simps val]
noncomputable
def ofSurjective (val : ι → S) (h : Function.Surjective (aeval (R := R) val)) :
Generators R S ι where
val := val
σ' x := (h x).choose
aeval_val_σ' x := (h x).choose_spec
/-- If `algebraMap R S` is surjective, the empty type generates `S`. -/
noncomputable def ofSurjectiveAlgebraMap (h : Function.Surjective (algebraMap R S)) :
Generators R S PEmpty.{w + 1} :=
ofSurjective PEmpty.elim <| fun s ↦ by
use C (h s).choose
simp [(h s).choose_spec]
/-- The canonical generators for `R` as an `R`-algebra. -/
noncomputable def id : Generators R R PEmpty.{w + 1} := ofSurjectiveAlgebraMap <| by
rw [algebraMap_self]
exact RingHomSurjective.is_surjective
/-- Construct `Generators` from an assignment `I → S` such that `R[X] → S` is surjective. -/
noncomputable
def ofAlgHom {I : Type*} (f : MvPolynomial I R →ₐ[R] S) (h : Function.Surjective f) :
Generators R S I :=
ofSurjective (f ∘ X) (by rwa [show aeval (f ∘ X) = f by ext; simp])
/-- Construct `Generators` from a family of generators of `S`. -/
noncomputable
def ofSet {s : Set S} (hs : Algebra.adjoin R s = ⊤) : Generators R S s := by
refine ofSurjective (Subtype.val : s → S) ?_
rwa [← AlgHom.range_eq_top, ← Algebra.adjoin_range_eq_range_aeval,
Subtype.range_coe_subtype, Set.setOf_mem_eq]
variable (R S) in
/-- The `Generators` containing the whole algebra, which induces the canonical map `R[S] → S`. -/
@[simps]
noncomputable
def self : Generators R S S where
val := _root_.id
σ' := X
aeval_val_σ' := aeval_X _
/-- The extension `R[X₁,...,Xₙ] → S` given a family of generators. -/
@[simps]
noncomputable
def toExtension : Extension R S where
Ring := P.Ring
σ := P.σ
algebraMap_σ := by simp
section Localization
variable (r : R) [IsLocalization.Away r S]
/-- If `S` is the localization of `R` away from `r`, we obtain a canonical generator mapping
to the inverse of `r`. -/
@[simps val, simps -isSimp σ]
noncomputable
def localizationAway : Generators R S Unit where
val _ := IsLocalization.Away.invSelf r
σ' s :=
letI a : R := (IsLocalization.Away.sec r s).1
letI n : ℕ := (IsLocalization.Away.sec r s).2
C a * X () ^ n
aeval_val_σ' s := by
rw [map_mul, algHom_C, map_pow, aeval_X]
simp only [← IsLocalization.Away.sec_spec, map_pow, IsLocalization.Away.invSelf]
rw [← IsLocalization.mk'_pow, one_pow, ← IsLocalization.mk'_one (M := Submonoid.powers r) S r]
rw [← IsLocalization.mk'_pow, one_pow, mul_assoc, ← IsLocalization.mk'_mul]
rw [mul_one, one_mul, IsLocalization.mk'_pow]
simp
end Localization
variable {ι' : Type*} {T} [CommRing T] [Algebra R T] [Algebra S T] [IsScalarTower R S T]
/-- Given two families of generators `S[X] → T` and `R[Y] → S`,
we may construct the family of generators `R[X, Y] → T`. -/
@[simps val, simps -isSimp σ]
noncomputable
def comp (Q : Generators S T ι') (P : Generators R S ι) : Generators R T (ι' ⊕ ι) where
val := Sum.elim Q.val (algebraMap S T ∘ P.val)
σ' x := (Q.σ x).sum (fun n r ↦ rename Sum.inr (P.σ r) * monomial (n.mapDomain Sum.inl) 1)
aeval_val_σ' s := by
have (x : P.Ring) : aeval (algebraMap S T ∘ P.val) x = algebraMap S T (aeval P.val x) := by
rw [map_aeval, aeval_def, coe_eval₂Hom, ← IsScalarTower.algebraMap_eq, Function.comp_def]
conv_rhs => rw [← Q.aeval_val_σ s, ← (Q.σ s).sum_single]
simp only [map_finsuppSum, map_mul, aeval_rename, Sum.elim_comp_inr, this, aeval_val_σ,
aeval_monomial, map_one, Finsupp.prod_mapDomain_index_inj Sum.inl_injective, Sum.elim_inl,
one_mul, single_eq_monomial]
variable (S) in
/-- If `R → S → T` is a tower of algebras, a family of generators `R[X] → T`
gives a family of generators `S[X] → T`. -/
@[simps val]
noncomputable
def extendScalars (P : Generators R T ι) : Generators S T ι where
val := P.val
σ' x := map (algebraMap R S) (P.σ x)
aeval_val_σ' s := by simp [@aeval_def S, ← IsScalarTower.algebraMap_eq, ← @aeval_def R]
/-- If `P` is a family of generators of `S` over `R` and `T` is an `R`-algebra, we
obtain a natural family of generators of `T ⊗[R] S` over `T`. -/
@[simps! val]
noncomputable
def baseChange {T} [CommRing T] [Algebra R T] (P : Generators R S ι) :
Generators T (T ⊗[R] S) ι := by
apply Generators.ofSurjective (fun x ↦ 1 ⊗ₜ[R] P.val x)
intro x
induction x using TensorProduct.induction_on with
| zero => exact ⟨0, map_zero _⟩
| tmul a b =>
let X := P.σ b
use a • MvPolynomial.map (algebraMap R T) X
simp only [LinearMapClass.map_smul, X, aeval_map_algebraMap]
have : ∀ y : P.Ring,
aeval (fun x ↦ (1 ⊗ₜ[R] P.val x : T ⊗[R] S)) y = 1 ⊗ₜ aeval (fun x ↦ P.val x) y := by
intro y
induction y using MvPolynomial.induction_on with
| C a =>
rw [aeval_C, aeval_C, TensorProduct.algebraMap_apply, algebraMap_eq_smul_one, smul_tmul,
algebraMap_eq_smul_one]
| add p q hp hq => simp [map_add, tmul_add, hp, hq]
| mul_X p i hp => simp [hp]
rw [this, P.aeval_val_σ, smul_tmul', smul_eq_mul, mul_one]
| add x y ex ey =>
obtain ⟨a, ha⟩ := ex
obtain ⟨b, hb⟩ := ey
use (a + b)
rw [map_add, ha, hb]
/-- Given generators `P` and an equivalence `ι ≃ P.vars`, these
are the induced generators indexed by `ι`. -/
noncomputable def reindex (P : Generators R S ι') (e : ι ≃ ι') :
Generators R S ι where
val := P.val ∘ e
σ' := rename e.symm ∘ P.σ
aeval_val_σ' s := by
conv_rhs => rw [← P.aeval_val_σ s]
rw [← MvPolynomial.aeval_rename]
simp
lemma reindex_val (P : Generators R S ι') (e : ι ≃ ι') :
(P.reindex e).val = P.val ∘ e :=
rfl
section
variable {σ : Type*} {I : Ideal (MvPolynomial σ R)}
(s : MvPolynomial σ R ⧸ I → MvPolynomial σ R)
(hs : ∀ x, Ideal.Quotient.mk _ (s x) = x)
/--
The naive generators for a quotient `R[Xᵢ] ⧸ I`.
If the definitional equality of the section matters, it can be explicitly provided.
-/
@[simps val]
noncomputable
def naive (s : MvPolynomial σ R ⧸ I → MvPolynomial σ R :=
Function.surjInv Ideal.Quotient.mk_surjective)
(hs : ∀ x, Ideal.Quotient.mk _ (s x) = x := by apply Function.surjInv_eq) :
Generators R (MvPolynomial σ R ⧸ I) σ where
val i := Ideal.Quotient.mk _ (X i)
σ' := s
aeval_val_σ' x := by
conv_rhs => rw [← hs x, ← Ideal.Quotient.mkₐ_eq_mk R, aeval_unique (Ideal.Quotient.mkₐ _ I)]
simp [Function.comp_def]
algebra := inferInstance
algebraMap_eq := by ext x <;> simp [IsScalarTower.algebraMap_apply R (MvPolynomial σ R)]
@[simp] lemma naive_σ : (Generators.naive s hs).σ = s := rfl
end
end Construction
variable {R' S' ι' : Type*} [CommRing R'] [CommRing S'] [Algebra R' S'] (P' : Generators R' S' ι')
variable {R'' S'' ι'' : Type*} [CommRing R''] [CommRing S''] [Algebra R'' S'']
(P'' : Generators R'' S'' ι'')
section Hom
section
variable [Algebra R R'] [Algebra R' R''] [Algebra R' S'']
variable [Algebra S S'] [Algebra S' S''] [Algebra S S'']
/-- Given a commuting square
R --→ P = R[X] ---→ S
| |
↓ ↓
R' -→ P' = R'[X'] → S
A hom between `P` and `P'` is an assignment `I → P'` such that the arrows commute.
Also see `Algebra.Generators.Hom.equivAlgHom`.
-/
@[ext]
structure Hom where
/-- The assignment of each variable in `I` to a value in `P' = R'[X']`. -/
val : ι → P'.Ring
aeval_val : ∀ i, aeval P'.val (val i) = algebraMap S S' (P.val i)
attribute [simp] Hom.aeval_val
variable {P P'}
/-- A hom between two families of generators gives
an algebra homomorphism between the polynomial rings. -/
noncomputable
def Hom.toAlgHom (f : Hom P P') : P.Ring →ₐ[R] P'.Ring := MvPolynomial.aeval f.val
variable [Algebra R S'] [IsScalarTower R R' S'] [IsScalarTower R S S'] in
@[simp]
lemma Hom.algebraMap_toAlgHom (f : Hom P P') (x) : MvPolynomial.aeval P'.val (f.toAlgHom x) =
algebraMap S S' (MvPolynomial.aeval P.val x) := by
suffices ((MvPolynomial.aeval P'.val).restrictScalars R).comp f.toAlgHom =
(IsScalarTower.toAlgHom R S S').comp (MvPolynomial.aeval P.val) from
DFunLike.congr_fun this x
apply MvPolynomial.algHom_ext
intro i
simp [Hom.toAlgHom]
@[simp]
lemma Hom.toAlgHom_X (f : Hom P P') (i) : f.toAlgHom (.X i) = f.val i :=
MvPolynomial.aeval_X f.val i
lemma Hom.toAlgHom_C (f : Hom P P') (r) : f.toAlgHom (.C r) = .C (algebraMap _ _ r) :=
MvPolynomial.aeval_C f.val r
lemma Hom.toAlgHom_monomial (f : Generators.Hom P P') (v r) :
f.toAlgHom (monomial v r) = r • v.prod (f.val · ^ ·) := by
rw [toAlgHom, aeval_monomial, Algebra.smul_def]
variable [Algebra R S'] [IsScalarTower R R' S'] [IsScalarTower R S S'] in
/-- Giving a hom between two families of generators is equivalent to
giving an algebra homomorphism between the polynomial rings. -/
@[simps]
noncomputable
def Hom.equivAlgHom :
Hom P P' ≃ { f : P.Ring →ₐ[R] P'.Ring //
∀ x, aeval P'.val (f x) = algebraMap S S' (aeval P.val x) } where
toFun f := ⟨f.toAlgHom, f.algebraMap_toAlgHom⟩
invFun f := ⟨fun i ↦ f.1 (.X i), fun i ↦ by simp [f.2]⟩
left_inv f := by ext; simp
right_inv f := by ext; simp
variable (P P')
/-- The hom from `P` to `P'` given by the designated section of `P'`. -/
@[simps]
def defaultHom : Hom P P' := ⟨P'.σ ∘ algebraMap S S' ∘ P.val, fun x ↦ by simp⟩
instance : Inhabited (Hom P P') := ⟨defaultHom P P'⟩
/-- The identity hom. -/
@[simps]
protected noncomputable def Hom.id : Hom P P := ⟨X, by simp⟩
@[simp]
lemma Hom.toAlgHom_id : Hom.toAlgHom (.id P) = AlgHom.id _ _ := by ext1; simp
variable {P P' P''}
/-- The composition of two homs. -/
@[simps]
noncomputable def Hom.comp [IsScalarTower R' R'' S''] [IsScalarTower R' S' S'']
[IsScalarTower S S' S''] (f : Hom P' P'') (g : Hom P P') : Hom P P'' where
val x := aeval f.val (g.val x)
aeval_val x := by
rw [IsScalarTower.algebraMap_apply S S' S'', ← g.aeval_val]
induction g.val x using MvPolynomial.induction_on with
| C r => simp [← IsScalarTower.algebraMap_apply]
| add x y hx hy => simp only [map_add, hx, hy]
| mul_X p i hp => simp only [map_mul, hp, aeval_X, aeval_val]
@[simp]
lemma Hom.comp_id [Algebra R S'] [IsScalarTower R R' S'] [IsScalarTower R S S'] (f : Hom P P') :
f.comp (Hom.id P) = f := by ext; simp
end
@[simp]
lemma Hom.id_comp [Algebra S S'] (f : Hom P P') : (Hom.id P').comp f = f := by
ext; simp [Hom.id, aeval_X_left]
variable [Algebra R R'] [Algebra R' R''] [Algebra R' S'']
variable [Algebra S S'] [Algebra S' S''] [Algebra S S'']
@[simp]
lemma Hom.toAlgHom_comp_apply
[Algebra R R''] [IsScalarTower R R' R''] [IsScalarTower R' R'' S'']
[IsScalarTower R' S' S''] [IsScalarTower S S' S'']
(f : Hom P P') (g : Hom P' P'') (x) :
(g.comp f).toAlgHom x = g.toAlgHom (f.toAlgHom x) := by
induction x using MvPolynomial.induction_on with
| C r => simp only [← MvPolynomial.algebraMap_eq, AlgHom.map_algebraMap]
| add x y hx hy => simp only [map_add, hx, hy]
| mul_X p i hp => simp only [map_mul, hp, toAlgHom_X, comp_val]; rfl
variable {T : Type*} [CommRing T] [Algebra R T] [Algebra S T] [IsScalarTower R S T]
/-- Given families of generators `X ⊆ T` over `S` and `Y ⊆ S` over `R`,
there is a map of generators `R[Y] → R[X, Y]`. -/
@[simps]
noncomputable
def toComp (Q : Generators S T ι') (P : Generators R S ι) : Hom P (Q.comp P) where
val i := X (.inr i)
aeval_val i := by simp
lemma toComp_toAlgHom (Q : Generators S T ι') (P : Generators R S ι) :
(Q.toComp P).toAlgHom = rename Sum.inr := rfl
/-- Given families of generators `X ⊆ T` over `S` and `Y ⊆ S` over `R`,
there is a map of generators `R[X, Y] → S[X]`. -/
@[simps]
noncomputable
def ofComp (Q : Generators S T ι') (P : Generators R S ι) : Hom (Q.comp P) Q where
val i := i.elim X (C ∘ P.val)
aeval_val i := by cases i <;> simp
lemma ofComp_toAlgHom_monomial_sumElim (Q : Generators S T ι') (P : Generators R S ι) (v₁ v₂ a) :
(Q.ofComp P).toAlgHom (monomial (Finsupp.sumElim v₁ v₂) a) =
monomial v₁ (aeval P.val (monomial v₂ a)) := by
rw [Hom.toAlgHom_monomial, monomial_eq]
simp only [ofComp_val, aeval_monomial]
rw [Finsupp.prod_sumElim]
simp only [Function.comp_def, Sum.elim_inl, Sum.elim_inr, ← map_pow, ← map_finsuppProd,
C_mul, Algebra.smul_def, MvPolynomial.algebraMap_apply, mul_assoc]
nth_rw 2 [mul_comm]
lemma toComp_toAlgHom_monomial (Q : Generators S T ι') (P : Generators R S ι) (j a) :
(Q.toComp P).toAlgHom (monomial j a) =
monomial (Finsupp.sumElim 0 j) a := by
convert rename_monomial _ _ _
ext f (i₁ | i₂) <;>
simp [Finsupp.mapDomain_notin_range, Finsupp.mapDomain_apply Sum.inr_injective]
@[simp]
lemma toAlgHom_ofComp_rename (Q : Generators S T ι') (P : Generators R S ι) (p : P.Ring) :
(Q.ofComp P).toAlgHom ((rename Sum.inr) p) = C (algebraMap _ _ p) :=
have : (Q.ofComp P).toAlgHom.comp (rename Sum.inr) =
(IsScalarTower.toAlgHom R S Q.Ring).comp (IsScalarTower.toAlgHom R P.Ring S) := by ext; simp
DFunLike.congr_fun this p
lemma toAlgHom_ofComp_surjective (Q : Generators S T ι') (P : Generators R S ι) :
Function.Surjective (Q.ofComp P).toAlgHom := by
intro p
induction p using MvPolynomial.induction_on with
| C a =>
use MvPolynomial.rename Sum.inr (P.σ a)
simp only [Hom.toAlgHom, ofComp, Generators.comp, MvPolynomial.aeval_rename,
Sum.elim_comp_inr]
simp_rw [Function.comp_def, ← MvPolynomial.algebraMap_eq, ← IsScalarTower.toAlgHom_apply R,
← MvPolynomial.comp_aeval]
simp
| add p q hp hq =>
obtain ⟨p, rfl⟩ := hp
obtain ⟨q, rfl⟩ := hq
use p + q
simp
| mul_X p i hp =>
obtain ⟨(p : MvPolynomial (ι' ⊕ ι) R), rfl⟩ := hp
use p * MvPolynomial.X (R := R) (Sum.inl i)
simp [Algebra.Generators.ofComp, Algebra.Generators.Hom.toAlgHom]
/-- Given families of generators `X ⊆ T`, there is a map `R[X] → S[X]`. -/
@[simps]
noncomputable
def toExtendScalars (P : Generators R T ι) : Hom P (P.extendScalars S) where
val := X
aeval_val i := by simp
variable {P P'} in
/-- Reinterpret a hom between generators as a hom between extensions. -/
@[simps]
noncomputable
def Hom.toExtensionHom [Algebra R S'] [IsScalarTower R R' S'] [IsScalarTower R S S']
(f : P.Hom P') : P.toExtension.Hom P'.toExtension where
toRingHom := f.toAlgHom.toRingHom
toRingHom_algebraMap x := by simp
algebraMap_toRingHom x := by simp
@[simp]
lemma Hom.toExtensionHom_id : Hom.toExtensionHom (.id P) = .id _ := by ext; simp
@[simp]
lemma Hom.toExtensionHom_comp [Algebra R S'] [IsScalarTower R S S']
[Algebra R R''] [Algebra R S''] [IsScalarTower R R'' S'']
[IsScalarTower R S S''] [IsScalarTower R' R'' S''] [IsScalarTower R' S' S'']
[IsScalarTower S S' S''] [IsScalarTower R R' R''] [IsScalarTower R R' S']
(f : P'.Hom P'') (g : P.Hom P') :
toExtensionHom (f.comp g) = f.toExtensionHom.comp g.toExtensionHom := by ext; simp
lemma Hom.toExtensionHom_toAlgHom_apply [Algebra R S'] [IsScalarTower R R' S']
[IsScalarTower R S S'] (f : P.Hom P') (x) :
f.toExtensionHom.toAlgHom x = f.toAlgHom x := rfl
/-- The kernel of a presentation. -/
noncomputable abbrev ker : Ideal P.Ring := P.toExtension.ker
lemma ker_eq_ker_aeval_val : P.ker = RingHom.ker (aeval P.val) := by
simp only [ker, Extension.ker, toExtension_Ring, algebraMap_eq]
rfl
variable {P} in
lemma aeval_val_eq_zero {x} (hx : x ∈ P.ker) : aeval P.val x = 0 := by rwa [← algebraMap_apply]
lemma ker_naive {σ : Type*} {I : Ideal (MvPolynomial σ R)}
(s : MvPolynomial σ R ⧸ I → MvPolynomial σ R) (hs : ∀ x, Ideal.Quotient.mk _ (s x) = x) :
(Generators.naive s hs).ker = I :=
I.mk_ker
lemma map_toComp_ker (Q : Generators S T ι') (P : Generators R S ι) :
P.ker.map (Q.toComp P).toAlgHom = RingHom.ker (Q.ofComp P).toAlgHom := by
letI : DecidableEq (ι' →₀ ℕ) := Classical.decEq _
apply le_antisymm
· rw [Ideal.map_le_iff_le_comap]
rintro x (hx : algebraMap P.Ring S x = 0)
have : (Q.ofComp P).toAlgHom.comp (Q.toComp P).toAlgHom = IsScalarTower.toAlgHom R _ _ := by
ext1; simp
simp only [Ideal.mem_comap,
RingHom.mem_ker, ← AlgHom.comp_apply, this, IsScalarTower.toAlgHom_apply]
rw [IsScalarTower.algebraMap_apply P.Ring S, hx, map_zero]
· rintro x (h₂ : (Q.ofComp P).toAlgHom x = 0)
let e : (ι' ⊕ ι →₀ ℕ) ≃+ (ι' →₀ ℕ) × (ι →₀ ℕ) :=
Finsupp.sumFinsuppAddEquivProdFinsupp
suffices ∑ v ∈ (support x).map e, (monomial (e.symm v)) (coeff (e.symm v) x) ∈
Ideal.map (Q.toComp P).toAlgHom.toRingHom P.ker by
simpa only [AlgHom.toRingHom_eq_coe, Finset.sum_map, Equiv.coe_toEmbedding,
EquivLike.coe_coe, AddEquiv.symm_apply_apply, support_sum_monomial_coeff] using this
rw [← Finset.sum_fiberwise_of_maps_to (fun i ↦ Finset.mem_image_of_mem Prod.fst)]
refine sum_mem fun i hi ↦ ?_
convert_to monomial (e.symm (i, 0)) 1 * (Q.toComp P).toAlgHom.toRingHom
(∑ j ∈ (support x).map e.toEmbedding with j.1 = i, monomial j.2 (coeff (e.symm j) x)) ∈ _
· rw [map_sum, Finset.mul_sum]
refine Finset.sum_congr rfl fun j hj ↦ ?_
obtain rfl := (Finset.mem_filter.mp hj).2
obtain ⟨i, j⟩ := j
clear hj hi
have : (Q.toComp P).toAlgHom (monomial j (coeff (e.symm (i, j)) x)) =
monomial (e.symm (0, j)) (coeff (e.symm (i, j)) x) :=
toComp_toAlgHom_monomial ..
simp only [AlgHom.toRingHom_eq_coe, RingHom.coe_coe,
this]
rw [monomial_mul, ← map_add, Prod.mk_add_mk, add_zero, zero_add, one_mul]
· apply Ideal.mul_mem_left
refine Ideal.mem_map_of_mem _ ?_
simp only [ker_eq_ker_aeval_val, AddEquiv.toEquiv_eq_coe, RingHom.mem_ker, map_sum]
rw [← coeff_zero i, ← h₂]
clear h₂ hi
have (x : (Q.comp P).Ring) : (Function.support fun a ↦ if a.1 = i then aeval P.val
(monomial a.2 (coeff (e.symm a) x)) else 0) ⊆ ((support x).map e).toSet := by
rw [← Set.compl_subset_compl]
intro j
obtain ⟨j, rfl⟩ := e.surjective j
simp_all
rw [Finset.sum_filter, ← finsum_eq_sum_of_support_subset _ (this x)]
induction x using MvPolynomial.induction_on' with
| monomial v a =>
rw [finsum_eq_sum_of_support_subset _ (this _), ← Finset.sum_filter]
obtain ⟨v, rfl⟩ := e.symm.surjective v
-- Rewrite `e` in the right hand side only.
conv_rhs => simp only [e, Finsupp.sumFinsuppAddEquivProdFinsupp,
Finsupp.sumFinsuppEquivProdFinsupp, AddEquiv.symm_mk, AddEquiv.coe_mk,
Equiv.coe_fn_symm_mk, ofComp_toAlgHom_monomial_sumElim]
classical
simp only [coeff_monomial, ← e.injective.eq_iff,
map_zero, AddEquiv.apply_symm_apply, apply_ite]
rw [← apply_ite, Finset.sum_ite_eq]
simp only [Finset.mem_filter, Finset.mem_map_equiv, AddEquiv.coe_toEquiv_symm,
mem_support_iff, coeff_monomial, ↓reduceIte, ne_eq, ite_and, ite_not]
split
· simp only [*, map_zero, ite_self]
· congr
| add p q hp hq =>
simp only [coeff_add, map_add, ite_add_zero]
rw [finsum_add_distrib, hp, hq]
· refine (((support p).map e).finite_toSet.subset ?_)
convert this p
· refine (((support q).map e).finite_toSet.subset ?_)
convert this q
/--
Given `R[X] → S` and `S[Y] → T`, this is the lift of an element in `ker(S[Y] → T)`
to `ker(R[X][Y] → S[Y] → T)` constructed from `P.σ`.
-/
noncomputable
def kerCompPreimage (Q : Generators S T ι') (P : Generators R S ι) (x : Q.ker) :
(Q.comp P).ker := by
refine ⟨x.1.sum fun n r ↦ ?_, ?_⟩
· -- The use of `refine` is intentional to control the elaboration order
-- so that the term has type `(Q.comp P).Ring` and not `MvPolynomial (Q.vars ⊕ P.vars) R`
refine rename ?_ (P.σ r) * monomial ?_ 1
exacts [Sum.inr, n.mapDomain Sum.inl]
· simp only [ker_eq_ker_aeval_val, RingHom.mem_ker]
conv_rhs => rw [← aeval_val_eq_zero x.2, ← x.1.support_sum_monomial_coeff]
simp only [Finsupp.sum, map_sum, map_mul, aeval_rename, Function.comp_def, comp_val,
Sum.elim_inr, aeval_monomial, map_one, Finsupp.prod_mapDomain_index_inj Sum.inl_injective,
Sum.elim_inl, one_mul]
congr! with v i
simp_rw [← IsScalarTower.toAlgHom_apply R, ← comp_aeval, AlgHom.comp_apply, P.aeval_val_σ,
coeff]
lemma ofComp_kerCompPreimage (Q : Generators S T ι') (P : Generators R S ι) (x : Q.ker) :
(Q.ofComp P).toAlgHom (kerCompPreimage Q P x) = x := by
conv_rhs => rw [← x.1.support_sum_monomial_coeff]
rw [kerCompPreimage, map_finsuppSum, Finsupp.sum]
refine Finset.sum_congr rfl fun j _ ↦ ?_
simp only [map_mul, Hom.toAlgHom_monomial]
rw [one_smul, Finsupp.prod_mapDomain_index_inj Sum.inl_injective]
rw [rename, ← AlgHom.comp_apply, comp_aeval]
simp only [ofComp_val, Sum.elim_inr, Function.comp_apply,
Sum.elim_inl, monomial_eq, Hom.toAlgHom_X]
congr 1
rw [aeval_def, IsScalarTower.algebraMap_eq R S, ← MvPolynomial.algebraMap_eq,
← coe_eval₂Hom, ← map_aeval, P.aeval_val_σ]
simp [coeff]
lemma map_ofComp_ker (Q : Generators S T ι') (P : Generators R S ι) :
Ideal.map (Q.ofComp P).toAlgHom (Q.comp P).ker = Q.ker := by
ext x
rw [Ideal.mem_map_iff_of_surjective _ (toAlgHom_ofComp_surjective Q P)]
constructor
· rintro ⟨x, hx, rfl⟩
simp only [ker_eq_ker_aeval_val,
RingHom.mem_ker] at hx ⊢
rw [← hx, Hom.algebraMap_toAlgHom, algebraMap_self_apply]
· intro hx
exact ⟨_, (kerCompPreimage Q P ⟨x, hx⟩).2, ofComp_kerCompPreimage Q P ⟨x, hx⟩⟩
lemma ker_comp_eq_sup (Q : Generators S T ι') (P : Generators R S ι) :
(Q.comp P).ker =
Ideal.map (Q.toComp P).toAlgHom P.ker ⊔ Ideal.comap (Q.ofComp P).toAlgHom Q.ker := by
rw [← map_ofComp_ker Q P,
Ideal.comap_map_of_surjective _ (toAlgHom_ofComp_surjective Q P)]
rw [← sup_assoc, Algebra.Generators.map_toComp_ker, ← RingHom.ker_eq_comap_bot]
apply le_antisymm (le_trans le_sup_right le_sup_left)
simp only [le_sup_left, sup_of_le_left, sup_le_iff, le_refl, and_true]
intro x hx
simp only [RingHom.mem_ker] at hx
rw [Generators.ker_eq_ker_aeval_val, RingHom.mem_ker,
← algebraMap_self_apply (MvPolynomial.aeval _ x)]
rw [← Generators.Hom.algebraMap_toAlgHom (Q.ofComp P), hx, map_zero]
end Hom
end Algebra.Generators
|
T0Sierpinski.lean
|
/-
Copyright (c) 2022 Ivan Sadofschi Costa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ivan Sadofschi Costa
-/
import Mathlib.Topology.Order
import Mathlib.Topology.Sets.Opens
import Mathlib.Topology.ContinuousMap.Basic
/-!
# Any T0 space embeds in a product of copies of the Sierpinski space.
We consider `Prop` with the Sierpinski topology. If `X` is a topological space, there is a
continuous map `productOfMemOpens` from `X` to `Opens X → Prop` which is the product of the maps
`X → Prop` given by `x ↦ x ∈ u`.
The map `productOfMemOpens` is always inducing. Whenever `X` is T0, `productOfMemOpens` is
also injective and therefore an embedding.
-/
open Topology
noncomputable section
namespace TopologicalSpace
theorem eq_induced_by_maps_to_sierpinski (X : Type*) [t : TopologicalSpace X] :
t = ⨅ u : Opens X, sierpinskiSpace.induced (· ∈ u) := by
apply le_antisymm
· rw [le_iInf_iff]
exact fun u => Continuous.le_induced (isOpen_iff_continuous_mem.mp u.2)
· intro u h
rw [← generateFrom_iUnion_isOpen]
apply isOpen_generateFrom_of_mem
simp only [Set.mem_iUnion, Set.mem_setOf_eq, isOpen_induced_iff]
exact ⟨⟨u, h⟩, {True}, isOpen_singleton_true, by simp [Set.preimage]⟩
variable (X : Type*) [TopologicalSpace X]
/-- The continuous map from `X` to the product of copies of the Sierpinski space, (one copy for each
open subset `u` of `X`). The `u` coordinate of `productOfMemOpens x` is given by `x ∈ u`.
-/
def productOfMemOpens : C(X, Opens X → Prop) where
toFun x u := x ∈ u
continuous_toFun := continuous_pi_iff.2 fun u => continuous_Prop.2 u.isOpen
theorem productOfMemOpens_isInducing : IsInducing (productOfMemOpens X) := by
convert inducing_iInf_to_pi fun (u : Opens X) (x : X) => x ∈ u
apply eq_induced_by_maps_to_sierpinski
theorem productOfMemOpens_injective [T0Space X] : Function.Injective (productOfMemOpens X) := by
intro x1 x2 h
apply Inseparable.eq
rw [← IsInducing.inseparable_iff (productOfMemOpens_isInducing X), h]
theorem productOfMemOpens_isEmbedding [T0Space X] : IsEmbedding (productOfMemOpens X) :=
.mk (productOfMemOpens_isInducing X) (productOfMemOpens_injective X)
end TopologicalSpace
|
all.v
|
From mathcomp Require Export all_boot.
From mathcomp Require Export all_order.
From mathcomp Require Export all_fingroup.
From mathcomp Require Export all_algebra.
From mathcomp Require Export all_solvable.
From mathcomp Require Export all_field.
From mathcomp Require Export all_character.
|
BigOperators.lean
|
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Star.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Defs
/-! # Big-operators lemmas about `star` algebraic operations
These results are kept separate from `Algebra.Star.Basic` to avoid it needing to import `Finset`.
-/
variable {R : Type*}
@[simp]
theorem star_prod [CommMonoid R] [StarMul R] {α : Type*} (s : Finset α) (f : α → R) :
star (∏ x ∈ s, f x) = ∏ x ∈ s, star (f x) := map_prod (starMulAut : R ≃* R) _ _
@[simp]
theorem star_sum [AddCommMonoid R] [StarAddMonoid R] {α : Type*} (s : Finset α) (f : α → R) :
star (∑ x ∈ s, f x) = ∑ x ∈ s, star (f x) := map_sum (starAddEquiv : R ≃+ R) _ _
|
LocallyFinite.lean
|
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Order.Filter.SmallSets
import Mathlib.Topology.ContinuousOn
/-!
### Locally finite families of sets
We say that a family of sets in a topological space is *locally finite* if at every point `x : X`,
there is a neighborhood of `x` which meets only finitely many sets in the family.
In this file we give the definition and prove basic properties of locally finite families of sets.
-/
-- locally finite family [General Topology (Bourbaki, 1995)]
open Set Function Filter Topology
variable {ι ι' α X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] {f g : ι → Set X}
/-- A family of sets in `Set X` is locally finite if at every point `x : X`,
there is a neighborhood of `x` which meets only finitely many sets in the family. -/
def LocallyFinite (f : ι → Set X) :=
∀ x : X, ∃ t ∈ 𝓝 x, { i | (f i ∩ t).Nonempty }.Finite
theorem locallyFinite_of_finite [Finite ι] (f : ι → Set X) : LocallyFinite f := fun _ =>
⟨univ, univ_mem, toFinite _⟩
namespace LocallyFinite
theorem point_finite (hf : LocallyFinite f) (x : X) : { b | x ∈ f b }.Finite :=
let ⟨_t, hxt, ht⟩ := hf x
ht.subset fun _b hb => ⟨x, hb, mem_of_mem_nhds hxt⟩
protected theorem subset (hf : LocallyFinite f) (hg : ∀ i, g i ⊆ f i) : LocallyFinite g := fun a =>
let ⟨t, ht₁, ht₂⟩ := hf a
⟨t, ht₁, ht₂.subset fun i hi => hi.mono <| inter_subset_inter (hg i) Subset.rfl⟩
theorem comp_injOn {g : ι' → ι} (hf : LocallyFinite f) (hg : InjOn g { i | (f (g i)).Nonempty }) :
LocallyFinite (f ∘ g) := fun x => by
let ⟨t, htx, htf⟩ := hf x
refine ⟨t, htx, htf.preimage <| ?_⟩
exact hg.mono fun i (hi : Set.Nonempty _) => hi.left
theorem comp_injective {g : ι' → ι} (hf : LocallyFinite f) (hg : Injective g) :
LocallyFinite (f ∘ g) :=
hf.comp_injOn hg.injOn
theorem _root_.locallyFinite_iff_smallSets :
LocallyFinite f ↔ ∀ x, ∀ᶠ s in (𝓝 x).smallSets, { i | (f i ∩ s).Nonempty }.Finite :=
forall_congr' fun _ => Iff.symm <|
eventually_smallSets' fun _s _t hst ht =>
ht.subset fun _i hi => hi.mono <| inter_subset_inter_right _ hst
protected theorem eventually_smallSets (hf : LocallyFinite f) (x : X) :
∀ᶠ s in (𝓝 x).smallSets, { i | (f i ∩ s).Nonempty }.Finite :=
locallyFinite_iff_smallSets.mp hf x
theorem exists_mem_basis {ι' : Sort*} (hf : LocallyFinite f) {p : ι' → Prop} {s : ι' → Set X}
{x : X} (hb : (𝓝 x).HasBasis p s) : ∃ i, p i ∧ { j | (f j ∩ s i).Nonempty }.Finite :=
let ⟨i, hpi, hi⟩ := hb.smallSets.eventually_iff.mp (hf.eventually_smallSets x)
⟨i, hpi, hi Subset.rfl⟩
protected theorem nhdsWithin_iUnion (hf : LocallyFinite f) (a : X) :
𝓝[⋃ i, f i] a = ⨆ i, 𝓝[f i] a := by
rcases hf a with ⟨U, haU, hfin⟩
refine le_antisymm ?_ (Monotone.le_map_iSup fun _ _ ↦ nhdsWithin_mono _)
calc
𝓝[⋃ i, f i] a = 𝓝[⋃ i, f i ∩ U] a := by
rw [← iUnion_inter, ← nhdsWithin_inter_of_mem' (nhdsWithin_le_nhds haU)]
_ = 𝓝[⋃ i ∈ {j | (f j ∩ U).Nonempty}, (f i ∩ U)] a := by
simp only [mem_setOf_eq, iUnion_nonempty_self]
_ = ⨆ i ∈ {j | (f j ∩ U).Nonempty}, 𝓝[f i ∩ U] a := nhdsWithin_biUnion hfin _ _
_ ≤ ⨆ i, 𝓝[f i ∩ U] a := iSup₂_le_iSup _ _
_ ≤ ⨆ i, 𝓝[f i] a := iSup_mono fun i ↦ nhdsWithin_mono _ inter_subset_left
theorem continuousOn_iUnion' {g : X → Y} (hf : LocallyFinite f)
(hc : ∀ i x, x ∈ closure (f i) → ContinuousWithinAt g (f i) x) :
ContinuousOn g (⋃ i, f i) := by
rintro x -
rw [ContinuousWithinAt, hf.nhdsWithin_iUnion, tendsto_iSup]
intro i
by_cases hx : x ∈ closure (f i)
· exact hc i _ hx
· rw [mem_closure_iff_nhdsWithin_neBot, not_neBot] at hx
rw [hx]
exact tendsto_bot
theorem continuousOn_iUnion {g : X → Y} (hf : LocallyFinite f) (h_cl : ∀ i, IsClosed (f i))
(h_cont : ∀ i, ContinuousOn g (f i)) : ContinuousOn g (⋃ i, f i) :=
hf.continuousOn_iUnion' fun i x hx ↦ h_cont i x <| (h_cl i).closure_subset hx
protected theorem continuous' {g : X → Y} (hf : LocallyFinite f) (h_cov : ⋃ i, f i = univ)
(hc : ∀ i x, x ∈ closure (f i) → ContinuousWithinAt g (f i) x) :
Continuous g :=
continuousOn_univ.1 <| h_cov ▸ hf.continuousOn_iUnion' hc
protected theorem continuous {g : X → Y} (hf : LocallyFinite f) (h_cov : ⋃ i, f i = univ)
(h_cl : ∀ i, IsClosed (f i)) (h_cont : ∀ i, ContinuousOn g (f i)) :
Continuous g :=
continuousOn_univ.1 <| h_cov ▸ hf.continuousOn_iUnion h_cl h_cont
protected theorem closure (hf : LocallyFinite f) : LocallyFinite fun i => closure (f i) := by
intro x
rcases hf x with ⟨s, hsx, hsf⟩
refine ⟨interior s, interior_mem_nhds.2 hsx, hsf.subset fun i hi => ?_⟩
exact (hi.mono isOpen_interior.closure_inter).of_closure.mono
(inter_subset_inter_right _ interior_subset)
theorem closure_iUnion (h : LocallyFinite f) : closure (⋃ i, f i) = ⋃ i, closure (f i) := by
ext x
simp only [mem_closure_iff_nhdsWithin_neBot, h.nhdsWithin_iUnion, iSup_neBot, mem_iUnion]
theorem isClosed_iUnion (hf : LocallyFinite f) (hc : ∀ i, IsClosed (f i)) :
IsClosed (⋃ i, f i) := by
simp only [← closure_eq_iff_isClosed, hf.closure_iUnion, (hc _).closure_eq]
/-- If `f : β → Set α` is a locally finite family of closed sets, then for any `x : α`, the
intersection of the complements to `f i`, `x ∉ f i`, is a neighbourhood of `x`. -/
theorem iInter_compl_mem_nhds (hf : LocallyFinite f) (hc : ∀ i, IsClosed (f i)) (x : X) :
(⋂ (i) (_ : x ∉ f i), (f i)ᶜ) ∈ 𝓝 x := by
refine IsOpen.mem_nhds ?_ (mem_iInter₂.2 fun i => id)
suffices IsClosed (⋃ i : { i // x ∉ f i }, f i) by
rwa [← isOpen_compl_iff, compl_iUnion, iInter_subtype] at this
exact (hf.comp_injective Subtype.val_injective).isClosed_iUnion fun i => hc _
/-- Let `f : ℕ → Π a, β a` be a sequence of (dependent) functions on a topological space. Suppose
that the family of sets `s n = {x | f (n + 1) x ≠ f n x}` is locally finite. Then there exists a
function `F : Π a, β a` such that for any `x`, we have `f n x = F x` on the product of an infinite
interval `[N, +∞)` and a neighbourhood of `x`.
We formulate the conclusion in terms of the product of filter `Filter.atTop` and `𝓝 x`. -/
theorem exists_forall_eventually_eq_prod {π : X → Sort*} {f : ℕ → ∀ x : X, π x}
(hf : LocallyFinite fun n => { x | f (n + 1) x ≠ f n x }) :
∃ F : ∀ x : X, π x, ∀ x, ∀ᶠ p : ℕ × X in atTop ×ˢ 𝓝 x, f p.1 p.2 = F p.2 := by
choose U hUx hU using hf
choose N hN using fun x => (hU x).bddAbove
replace hN : ∀ (x), ∀ n > N x, ∀ y ∈ U x, f (n + 1) y = f n y :=
fun x n hn y hy => by_contra fun hne => hn.lt.not_ge <| hN x ⟨y, hne, hy⟩
replace hN : ∀ (x), ∀ n ≥ N x + 1, ∀ y ∈ U x, f n y = f (N x + 1) y :=
fun x n hn y hy => Nat.le_induction rfl (fun k hle => (hN x _ hle _ hy).trans) n hn
refine ⟨fun x => f (N x + 1) x, fun x => ?_⟩
filter_upwards [Filter.prod_mem_prod (eventually_gt_atTop (N x)) (hUx x)]
rintro ⟨n, y⟩ ⟨hn : N x < n, hy : y ∈ U x⟩
calc
f n y = f (N x + 1) y := hN _ _ hn _ hy
_ = f (max (N x + 1) (N y + 1)) y := (hN _ _ (le_max_left _ _) _ hy).symm
_ = f (N y + 1) y := hN _ _ (le_max_right _ _) _ (mem_of_mem_nhds <| hUx y)
/-- Let `f : ℕ → Π a, β a` be a sequence of (dependent) functions on a topological space. Suppose
that the family of sets `s n = {x | f (n + 1) x ≠ f n x}` is locally finite. Then there exists a
function `F : Π a, β a` such that for any `x`, for sufficiently large values of `n`, we have
`f n y = F y` in a neighbourhood of `x`. -/
theorem exists_forall_eventually_atTop_eventually_eq' {π : X → Sort*} {f : ℕ → ∀ x : X, π x}
(hf : LocallyFinite fun n => { x | f (n + 1) x ≠ f n x }) :
∃ F : ∀ x : X, π x, ∀ x, ∀ᶠ n : ℕ in atTop, ∀ᶠ y : X in 𝓝 x, f n y = F y :=
hf.exists_forall_eventually_eq_prod.imp fun _F hF x => (hF x).curry
/-- Let `f : ℕ → α → β` be a sequence of functions on a topological space. Suppose
that the family of sets `s n = {x | f (n + 1) x ≠ f n x}` is locally finite. Then there exists a
function `F : α → β` such that for any `x`, for sufficiently large values of `n`, we have
`f n =ᶠ[𝓝 x] F`. -/
theorem exists_forall_eventually_atTop_eventuallyEq {f : ℕ → X → α}
(hf : LocallyFinite fun n => { x | f (n + 1) x ≠ f n x }) :
∃ F : X → α, ∀ x, ∀ᶠ n : ℕ in atTop, f n =ᶠ[𝓝 x] F :=
hf.exists_forall_eventually_atTop_eventually_eq'
theorem preimage_continuous {g : Y → X} (hf : LocallyFinite f) (hg : Continuous g) :
LocallyFinite (g ⁻¹' f ·) := fun x =>
let ⟨s, hsx, hs⟩ := hf (g x)
⟨g ⁻¹' s, hg.continuousAt hsx, hs.subset fun _ ⟨y, hy⟩ => ⟨g y, hy⟩⟩
theorem prod_right (hf : LocallyFinite f) (g : ι → Set Y) : LocallyFinite (fun i ↦ f i ×ˢ g i) :=
(hf.preimage_continuous continuous_fst).subset fun _ ↦ prod_subset_preimage_fst _ _
theorem prod_left {g : ι → Set Y} (hg : LocallyFinite g) (f : ι → Set X) :
LocallyFinite (fun i ↦ f i ×ˢ g i) :=
(hg.preimage_continuous continuous_snd).subset fun _ ↦ prod_subset_preimage_snd _ _
end LocallyFinite
@[simp]
theorem Equiv.locallyFinite_comp_iff (e : ι' ≃ ι) : LocallyFinite (f ∘ e) ↔ LocallyFinite f :=
⟨fun h => by simpa only [comp_def, e.apply_symm_apply] using h.comp_injective e.symm.injective,
fun h => h.comp_injective e.injective⟩
theorem locallyFinite_sum {f : ι ⊕ ι' → Set X} :
LocallyFinite f ↔ LocallyFinite (f ∘ Sum.inl) ∧ LocallyFinite (f ∘ Sum.inr) := by
simp only [locallyFinite_iff_smallSets, ← forall_and, ← finite_preimage_inl_and_inr,
preimage_setOf_eq, (· ∘ ·), eventually_and]
theorem LocallyFinite.sumElim {g : ι' → Set X} (hf : LocallyFinite f) (hg : LocallyFinite g) :
LocallyFinite (Sum.elim f g) :=
locallyFinite_sum.mpr ⟨hf, hg⟩
@[deprecated (since := "2025-02-20")] alias LocallyFinite.sum_elim := LocallyFinite.sumElim
theorem locallyFinite_option {f : Option ι → Set X} :
LocallyFinite f ↔ LocallyFinite (f ∘ some) := by
rw [← (Equiv.optionEquivSumPUnit.{0, _} ι).symm.locallyFinite_comp_iff, locallyFinite_sum]
simp only [locallyFinite_of_finite, and_true]
rfl
theorem LocallyFinite.option_elim' (hf : LocallyFinite f) (s : Set X) :
LocallyFinite (Option.elim' s f) :=
locallyFinite_option.2 hf
theorem LocallyFinite.eventually_subset {s : ι → Set X}
(hs : LocallyFinite s) (hs' : ∀ i, IsClosed (s i)) (x : X) :
∀ᶠ y in 𝓝 x, {i | y ∈ s i} ⊆ {i | x ∈ s i} := by
filter_upwards [hs.iInter_compl_mem_nhds hs' x] with y hy i hi
simp only [mem_iInter, mem_compl_iff] at hy
exact not_imp_not.mp (hy i) hi
|
Card.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
-/
import Mathlib.LinearAlgebra.Quotient.Defs
import Mathlib.SetTheory.Cardinal.Finite
import Mathlib.GroupTheory.Coset.Basic
/-! Results about the cardinality of a quotient module. -/
namespace Submodule
open LinearMap QuotientAddGroup
variable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M]
theorem card_eq_card_quotient_mul_card (S : Submodule R M) :
Nat.card M = Nat.card S * Nat.card (M ⧸ S) := by
rw [mul_comm, ← Nat.card_prod]
exact Nat.card_congr AddSubgroup.addGroupEquivQuotientProdAddSubgroup
end Submodule
|
Atlas.lean
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions
import Mathlib.Geometry.Manifold.VectorBundle.Tangent
/-!
# Differentiability of models with corners and (extended) charts
In this file, we analyse the differentiability of charts, models with corners and extended charts.
We show that
* models with corners are differentiable
* charts are differentiable on their source
* `mdifferentiableOn_extChartAt`: `extChartAt` is differentiable on its source
Suppose a partial homeomorphism `e` is differentiable. This file shows
* `PartialHomeomorph.MDifferentiable.mfderiv`: its derivative is a continuous linear equivalence
* `PartialHomeomorph.MDifferentiable.mfderiv_bijective`: its derivative is bijective;
there are also spelling with trivial kernel and full range
In particular, (extended) charts have bijective differential.
## Tags
charts, differentiable, bijective
-/
noncomputable section
open scoped Manifold ContDiff
open Bundle Set Topology
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
{E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace H]
{I : ModelWithCorners 𝕜 E H} {M : Type*} [TopologicalSpace M] [ChartedSpace H M]
{E' : Type*} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type*} [TopologicalSpace H']
{I' : ModelWithCorners 𝕜 E' H'} {M' : Type*} [TopologicalSpace M'] [ChartedSpace H' M']
{E'' : Type*} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type*} [TopologicalSpace H'']
{I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type*} [TopologicalSpace M''] [ChartedSpace H'' M'']
section ModelWithCorners
namespace ModelWithCorners
/- In general, the model with corner `I` is implicit in most theorems in differential geometry, but
this section is about `I` as a map, not as a parameter. Therefore, we make it explicit. -/
variable (I)
/-! #### Model with corners -/
protected theorem hasMFDerivAt {x} : HasMFDerivAt I 𝓘(𝕜, E) I x (ContinuousLinearMap.id _ _) :=
⟨I.continuousAt, (hasFDerivWithinAt_id _ _).congr' I.rightInvOn (mem_range_self _)⟩
protected theorem hasMFDerivWithinAt {s x} :
HasMFDerivWithinAt I 𝓘(𝕜, E) I s x (ContinuousLinearMap.id _ _) :=
I.hasMFDerivAt.hasMFDerivWithinAt
protected theorem mdifferentiableWithinAt {s x} : MDifferentiableWithinAt I 𝓘(𝕜, E) I s x :=
I.hasMFDerivWithinAt.mdifferentiableWithinAt
protected theorem mdifferentiableAt {x} : MDifferentiableAt I 𝓘(𝕜, E) I x :=
I.hasMFDerivAt.mdifferentiableAt
protected theorem mdifferentiableOn {s} : MDifferentiableOn I 𝓘(𝕜, E) I s := fun _ _ =>
I.mdifferentiableWithinAt
protected theorem mdifferentiable : MDifferentiable I 𝓘(𝕜, E) I := fun _ => I.mdifferentiableAt
theorem hasMFDerivWithinAt_symm {x} (hx : x ∈ range I) :
HasMFDerivWithinAt 𝓘(𝕜, E) I I.symm (range I) x (ContinuousLinearMap.id _ _) :=
⟨I.continuousWithinAt_symm,
(hasFDerivWithinAt_id _ _).congr' (fun _y hy => I.rightInvOn hy.1) ⟨hx, mem_range_self _⟩⟩
theorem mdifferentiableOn_symm : MDifferentiableOn 𝓘(𝕜, E) I I.symm (range I) := fun _x hx =>
(I.hasMFDerivWithinAt_symm hx).mdifferentiableWithinAt
theorem mdifferentiableWithinAt_symm {z : E} (hz : z ∈ range I) :
MDifferentiableWithinAt 𝓘(𝕜, E) I I.symm (range I) z :=
I.mdifferentiableOn_symm z hz
end ModelWithCorners
end ModelWithCorners
section Charts
variable [IsManifold I 1 M] [IsManifold I' 1 M']
[IsManifold I'' 1 M''] {e : PartialHomeomorph M H}
theorem mdifferentiableAt_atlas (h : e ∈ atlas H M) {x : M} (hx : x ∈ e.source) :
MDifferentiableAt I I e x := by
rw [mdifferentiableAt_iff]
refine ⟨(e.continuousOn x hx).continuousAt (e.open_source.mem_nhds hx), ?_⟩
have mem :
I ((chartAt H x : M → H) x) ∈ I.symm ⁻¹' ((chartAt H x).symm ≫ₕ e).source ∩ range I := by
simp only [hx, mfld_simps]
have : (chartAt H x).symm.trans e ∈ contDiffGroupoid 1 I :=
HasGroupoid.compatible (chart_mem_atlas H x) h
have A :
ContDiffOn 𝕜 1 (I ∘ (chartAt H x).symm.trans e ∘ I.symm)
(I.symm ⁻¹' ((chartAt H x).symm.trans e).source ∩ range I) :=
this.1
have B := A.differentiableOn le_rfl (I ((chartAt H x : M → H) x)) mem
simp only [mfld_simps] at B
rw [inter_comm, differentiableWithinAt_inter] at B
· simpa only [mfld_simps]
· apply IsOpen.mem_nhds ((PartialHomeomorph.open_source _).preimage I.continuous_symm) mem.1
theorem mdifferentiableOn_atlas (h : e ∈ atlas H M) : MDifferentiableOn I I e e.source :=
fun _x hx => (mdifferentiableAt_atlas h hx).mdifferentiableWithinAt
theorem mdifferentiableAt_atlas_symm (h : e ∈ atlas H M) {x : H} (hx : x ∈ e.target) :
MDifferentiableAt I I e.symm x := by
rw [mdifferentiableAt_iff]
refine ⟨(e.continuousOn_symm x hx).continuousAt (e.open_target.mem_nhds hx), ?_⟩
have mem : I x ∈ I.symm ⁻¹' (e.symm ≫ₕ chartAt H (e.symm x)).source ∩ range I := by
simp only [hx, mfld_simps]
have : e.symm.trans (chartAt H (e.symm x)) ∈ contDiffGroupoid 1 I :=
HasGroupoid.compatible h (chart_mem_atlas H _)
have A :
ContDiffOn 𝕜 1 (I ∘ e.symm.trans (chartAt H (e.symm x)) ∘ I.symm)
(I.symm ⁻¹' (e.symm.trans (chartAt H (e.symm x))).source ∩ range I) :=
this.1
have B := A.differentiableOn le_rfl (I x) mem
simp only [mfld_simps] at B
rw [inter_comm, differentiableWithinAt_inter] at B
· simpa only [mfld_simps]
· apply IsOpen.mem_nhds ((PartialHomeomorph.open_source _).preimage I.continuous_symm) mem.1
theorem mdifferentiableOn_atlas_symm (h : e ∈ atlas H M) : MDifferentiableOn I I e.symm e.target :=
fun _x hx => (mdifferentiableAt_atlas_symm h hx).mdifferentiableWithinAt
theorem mdifferentiable_of_mem_atlas (h : e ∈ atlas H M) : e.MDifferentiable I I :=
⟨mdifferentiableOn_atlas h, mdifferentiableOn_atlas_symm h⟩
theorem mdifferentiable_chart (x : M) : (chartAt H x).MDifferentiable I I :=
mdifferentiable_of_mem_atlas (chart_mem_atlas _ _)
end Charts
/-! ### Differentiable partial homeomorphisms -/
namespace PartialHomeomorph.MDifferentiable
variable {e : PartialHomeomorph M M'} (he : e.MDifferentiable I I') {e' : PartialHomeomorph M' M''}
include he
nonrec theorem symm : e.symm.MDifferentiable I' I := he.symm
protected theorem mdifferentiableAt {x : M} (hx : x ∈ e.source) : MDifferentiableAt I I' e x :=
(he.1 x hx).mdifferentiableAt (e.open_source.mem_nhds hx)
theorem mdifferentiableAt_symm {x : M'} (hx : x ∈ e.target) : MDifferentiableAt I' I e.symm x :=
(he.2 x hx).mdifferentiableAt (e.open_target.mem_nhds hx)
theorem symm_comp_deriv {x : M} (hx : x ∈ e.source) :
(mfderiv I' I e.symm (e x)).comp (mfderiv I I' e x) =
ContinuousLinearMap.id 𝕜 (TangentSpace I x) := by
have : mfderiv I I (e.symm ∘ e) x = (mfderiv I' I e.symm (e x)).comp (mfderiv I I' e x) :=
mfderiv_comp x (he.mdifferentiableAt_symm (e.map_source hx)) (he.mdifferentiableAt hx)
rw [← this]
have : mfderiv I I (_root_.id : M → M) x = ContinuousLinearMap.id _ _ := mfderiv_id
rw [← this]
apply Filter.EventuallyEq.mfderiv_eq
have : e.source ∈ 𝓝 x := e.open_source.mem_nhds hx
exact Filter.mem_of_superset this (by mfld_set_tac)
theorem comp_symm_deriv {x : M'} (hx : x ∈ e.target) :
(mfderiv I I' e (e.symm x)).comp (mfderiv I' I e.symm x) =
ContinuousLinearMap.id 𝕜 (TangentSpace I' x) :=
he.symm.symm_comp_deriv hx
/-- The derivative of a differentiable partial homeomorphism, as a continuous linear equivalence
between the tangent spaces at `x` and `e x`. -/
protected def mfderiv (he : e.MDifferentiable I I') {x : M} (hx : x ∈ e.source) :
TangentSpace I x ≃L[𝕜] TangentSpace I' (e x) :=
{ mfderiv I I' e x with
invFun := mfderiv I' I e.symm (e x)
continuous_toFun := (mfderiv I I' e x).cont
continuous_invFun := (mfderiv I' I e.symm (e x)).cont
left_inv := fun y => by
have : (ContinuousLinearMap.id _ _ : TangentSpace I x →L[𝕜] TangentSpace I x) y = y := rfl
conv_rhs => rw [← this, ← he.symm_comp_deriv hx]
rfl
right_inv := fun y => by
have :
(ContinuousLinearMap.id 𝕜 _ : TangentSpace I' (e x) →L[𝕜] TangentSpace I' (e x)) y = y :=
rfl
conv_rhs => rw [← this, ← he.comp_symm_deriv (e.map_source hx)]
rw [e.left_inv hx]
rfl }
theorem mfderiv_bijective {x : M} (hx : x ∈ e.source) : Function.Bijective (mfderiv I I' e x) :=
(he.mfderiv hx).bijective
theorem mfderiv_injective {x : M} (hx : x ∈ e.source) : Function.Injective (mfderiv I I' e x) :=
(he.mfderiv hx).injective
theorem mfderiv_surjective {x : M} (hx : x ∈ e.source) : Function.Surjective (mfderiv I I' e x) :=
(he.mfderiv hx).surjective
theorem ker_mfderiv_eq_bot {x : M} (hx : x ∈ e.source) : LinearMap.ker (mfderiv I I' e x) = ⊥ :=
(he.mfderiv hx).toLinearEquiv.ker
theorem range_mfderiv_eq_top {x : M} (hx : x ∈ e.source) : LinearMap.range (mfderiv I I' e x) = ⊤ :=
(he.mfderiv hx).toLinearEquiv.range
theorem range_mfderiv_eq_univ {x : M} (hx : x ∈ e.source) : range (mfderiv I I' e x) = univ :=
(he.mfderiv_surjective hx).range_eq
theorem trans (he' : e'.MDifferentiable I' I'') : (e.trans e').MDifferentiable I I'' := by
constructor
· intro x hx
simp only [mfld_simps] at hx
exact
((he'.mdifferentiableAt hx.2).comp _ (he.mdifferentiableAt hx.1)).mdifferentiableWithinAt
· intro x hx
simp only [mfld_simps] at hx
exact
((he.symm.mdifferentiableAt hx.2).comp _
(he'.symm.mdifferentiableAt hx.1)).mdifferentiableWithinAt
end PartialHomeomorph.MDifferentiable
/-! ### Differentiability of `extChartAt` -/
section extChartAt
variable [IsManifold I 1 M] {s : Set M} {x y : M} {z : E}
theorem hasMFDerivAt_extChartAt (h : y ∈ (chartAt H x).source) :
HasMFDerivAt I 𝓘(𝕜, E) (extChartAt I x) y (mfderiv I I (chartAt H x) y :) :=
I.hasMFDerivAt.comp y ((mdifferentiable_chart x).mdifferentiableAt h).hasMFDerivAt
theorem hasMFDerivWithinAt_extChartAt (h : y ∈ (chartAt H x).source) :
HasMFDerivWithinAt I 𝓘(𝕜, E) (extChartAt I x) s y (mfderiv I I (chartAt H x) y :) :=
(hasMFDerivAt_extChartAt h).hasMFDerivWithinAt
theorem mdifferentiableAt_extChartAt (h : y ∈ (chartAt H x).source) :
MDifferentiableAt I 𝓘(𝕜, E) (extChartAt I x) y :=
(hasMFDerivAt_extChartAt h).mdifferentiableAt
theorem mdifferentiableOn_extChartAt :
MDifferentiableOn I 𝓘(𝕜, E) (extChartAt I x) (chartAt H x).source := fun _y hy =>
(hasMFDerivWithinAt_extChartAt hy).mdifferentiableWithinAt
theorem mdifferentiableWithinAt_extChartAt_symm (h : z ∈ (extChartAt I x).target) :
MDifferentiableWithinAt 𝓘(𝕜, E) I (extChartAt I x).symm (range I) z := by
have Z := I.mdifferentiableWithinAt_symm (extChartAt_target_subset_range x h)
apply MDifferentiableAt.comp_mdifferentiableWithinAt (I' := I) _ _ Z
apply mdifferentiableAt_atlas_symm (ChartedSpace.chart_mem_atlas x)
simp only [extChartAt, PartialHomeomorph.extend, PartialEquiv.trans_target,
ModelWithCorners.target_eq, ModelWithCorners.toPartialEquiv_coe_symm, mem_inter_iff, mem_range,
mem_preimage] at h
exact h.2
theorem mdifferentiableOn_extChartAt_symm :
MDifferentiableOn 𝓘(𝕜, E) I (extChartAt I x).symm (extChartAt I x).target := by
intro y hy
exact (mdifferentiableWithinAt_extChartAt_symm hy).mono (extChartAt_target_subset_range x)
/-- The composition of the derivative of `extChartAt` with the derivative of the inverse of
`extChartAt` gives the identity.
Version where the basepoint belongs to `(extChartAt I x).target`. -/
lemma mfderiv_extChartAt_comp_mfderivWithin_extChartAt_symm {x : M}
{y : E} (hy : y ∈ (extChartAt I x).target) :
(mfderiv I 𝓘(𝕜, E) (extChartAt I x) ((extChartAt I x).symm y)) ∘L
(mfderivWithin 𝓘(𝕜, E) I (extChartAt I x).symm (range I) y) = ContinuousLinearMap.id _ _ := by
have U : UniqueMDiffWithinAt 𝓘(𝕜, E) (range ↑I) y := by
apply I.uniqueMDiffOn
exact extChartAt_target_subset_range x hy
have h'y : (extChartAt I x).symm y ∈ (extChartAt I x).source := (extChartAt I x).map_target hy
have h''y : (extChartAt I x).symm y ∈ (chartAt H x).source := by
rwa [← extChartAt_source (I := I)]
rw [← mfderiv_comp_mfderivWithin]; rotate_left
· apply mdifferentiableAt_extChartAt h''y
· exact mdifferentiableWithinAt_extChartAt_symm hy
· exact U
rw [← mfderivWithin_id U]
apply Filter.EventuallyEq.mfderivWithin_eq
· filter_upwards [extChartAt_target_mem_nhdsWithin_of_mem hy] with z hz
simp only [Function.comp_def, PartialEquiv.right_inv (extChartAt I x) hz, id_eq]
· simp only [Function.comp_def, PartialEquiv.right_inv (extChartAt I x) hy, id_eq]
/-- The composition of the derivative of `extChartAt` with the derivative of the inverse of
`extChartAt` gives the identity.
Version where the basepoint belongs to `(extChartAt I x).source`. -/
lemma mfderiv_extChartAt_comp_mfderivWithin_extChartAt_symm' {x : M}
{y : M} (hy : y ∈ (extChartAt I x).source) :
(mfderiv I 𝓘(𝕜, E) (extChartAt I x) y) ∘L
(mfderivWithin 𝓘(𝕜, E) I (extChartAt I x).symm (range I) (extChartAt I x y))
= ContinuousLinearMap.id _ _ := by
have : y = (extChartAt I x).symm (extChartAt I x y) := ((extChartAt I x).left_inv hy).symm
convert mfderiv_extChartAt_comp_mfderivWithin_extChartAt_symm ((extChartAt I x).map_source hy)
/-- The composition of the derivative of the inverse of `extChartAt` with the derivative of
`extChartAt` gives the identity.
Version where the basepoint belongs to `(extChartAt I x).target`. -/
lemma mfderivWithin_extChartAt_symm_comp_mfderiv_extChartAt
{y : E} (hy : y ∈ (extChartAt I x).target) :
(mfderivWithin 𝓘(𝕜, E) I (extChartAt I x).symm (range I) y) ∘L
(mfderiv I 𝓘(𝕜, E) (extChartAt I x) ((extChartAt I x).symm y))
= ContinuousLinearMap.id _ _ := by
have h'y : (extChartAt I x).symm y ∈ (extChartAt I x).source := (extChartAt I x).map_target hy
have h''y : (extChartAt I x).symm y ∈ (chartAt H x).source := by
rwa [← extChartAt_source (I := I)]
have U' : UniqueMDiffWithinAt I (extChartAt I x).source ((extChartAt I x).symm y) :=
(isOpen_extChartAt_source x).uniqueMDiffWithinAt h'y
have : mfderiv I 𝓘(𝕜, E) (extChartAt I x) ((extChartAt I x).symm y)
= mfderivWithin I 𝓘(𝕜, E) (extChartAt I x) (extChartAt I x).source
((extChartAt I x).symm y) := by
rw [mfderivWithin_eq_mfderiv U']
exact mdifferentiableAt_extChartAt h''y
rw [this, ← mfderivWithin_comp_of_eq]; rotate_left
· exact mdifferentiableWithinAt_extChartAt_symm hy
· exact (mdifferentiableAt_extChartAt h''y).mdifferentiableWithinAt
· intro z hz
apply extChartAt_target_subset_range x
exact PartialEquiv.map_source (extChartAt I x) hz
· exact U'
· exact PartialEquiv.right_inv (extChartAt I x) hy
rw [← mfderivWithin_id U']
apply Filter.EventuallyEq.mfderivWithin_eq
· filter_upwards [extChartAt_source_mem_nhdsWithin' h'y] with z hz
simp only [Function.comp_def, PartialEquiv.left_inv (extChartAt I x) hz, id_eq]
· simp only [Function.comp_def, PartialEquiv.right_inv (extChartAt I x) hy, id_eq]
/-- The composition of the derivative of the inverse of `extChartAt` with the derivative of
`extChartAt` gives the identity.
Version where the basepoint belongs to `(extChartAt I x).source`. -/
lemma mfderivWithin_extChartAt_symm_comp_mfderiv_extChartAt'
{y : M} (hy : y ∈ (extChartAt I x).source) :
(mfderivWithin 𝓘(𝕜, E) I (extChartAt I x).symm (range I) (extChartAt I x y)) ∘L
(mfderiv I 𝓘(𝕜, E) (extChartAt I x) y)
= ContinuousLinearMap.id _ _ := by
have : y = (extChartAt I x).symm (extChartAt I x y) := ((extChartAt I x).left_inv hy).symm
convert mfderivWithin_extChartAt_symm_comp_mfderiv_extChartAt ((extChartAt I x).map_source hy)
lemma isInvertible_mfderivWithin_extChartAt_symm {y : E} (hy : y ∈ (extChartAt I x).target) :
(mfderivWithin 𝓘(𝕜, E) I (extChartAt I x).symm (range I) y).IsInvertible :=
ContinuousLinearMap.IsInvertible.of_inverse
(mfderivWithin_extChartAt_symm_comp_mfderiv_extChartAt hy)
(mfderiv_extChartAt_comp_mfderivWithin_extChartAt_symm hy)
lemma isInvertible_mfderiv_extChartAt {y : M} (hy : y ∈ (extChartAt I x).source) :
(mfderiv I 𝓘(𝕜, E) (extChartAt I x) y).IsInvertible := by
have h'y : extChartAt I x y ∈ (extChartAt I x).target := (extChartAt I x).map_source hy
have Z := ContinuousLinearMap.IsInvertible.of_inverse
(mfderiv_extChartAt_comp_mfderivWithin_extChartAt_symm h'y)
(mfderivWithin_extChartAt_symm_comp_mfderiv_extChartAt h'y)
have : (extChartAt I x).symm ((extChartAt I x) y) = y := (extChartAt I x).left_inv hy
rwa [this] at Z
/-- The trivialization of the tangent bundle at a point is the manifold derivative of the
extended chart.
Use with care as this abuses the defeq `TangentSpace 𝓘(𝕜, E) y = E` for `y : E`. -/
theorem TangentBundle.continuousLinearMapAt_trivializationAt
{x₀ x : M} (hx : x ∈ (chartAt H x₀).source) :
(trivializationAt E (TangentSpace I) x₀).continuousLinearMapAt 𝕜 x =
mfderiv I 𝓘(𝕜, E) (extChartAt I x₀) x := by
have : MDifferentiableAt I 𝓘(𝕜, E) (extChartAt I x₀) x := mdifferentiableAt_extChartAt hx
simp only [extChartAt, PartialHomeomorph.extend, PartialEquiv.coe_trans,
ModelWithCorners.toPartialEquiv_coe, PartialHomeomorph.toFun_eq_coe] at this
simp [hx, mfderiv, this]
/-- The inverse trivialization of the tangent bundle at a point is the manifold derivative of the
inverse of the extended chart.
Use with care as this abuses the defeq `TangentSpace 𝓘(𝕜, E) y = E` for `y : E`. -/
theorem TangentBundle.symmL_trivializationAt
{x₀ x : M} (hx : x ∈ (chartAt H x₀).source) :
(trivializationAt E (TangentSpace I) x₀).symmL 𝕜 x =
mfderivWithin 𝓘(𝕜, E) I (extChartAt I x₀).symm (range I) (extChartAt I x₀ x) := by
have : MDifferentiableWithinAt 𝓘(𝕜, E) I (extChartAt I x₀).symm (range I) (extChartAt I x₀ x) :=
mdifferentiableWithinAt_extChartAt_symm (by simp [hx])
simp? at this says
simp only [extChartAt, PartialHomeomorph.extend, PartialEquiv.coe_trans_symm,
PartialHomeomorph.coe_coe_symm, ModelWithCorners.toPartialEquiv_coe_symm,
PartialEquiv.coe_trans, ModelWithCorners.toPartialEquiv_coe, PartialHomeomorph.toFun_eq_coe,
Function.comp_apply] at this
simp [hx, mfderivWithin, this]
end extChartAt
|
archimedean.v
|
(* (c) Copyright 2006-2016 Microsoft Corporation and Inria. *)
(* Distributed under the terms of CeCILL-B. *)
From HB Require Import structures.
From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice.
From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint.
(******************************************************************************)
(* Archimedean structures *)
(* *)
(* NB: See CONTRIBUTING.md for an introduction to HB concepts and commands. *)
(* *)
(* This file defines some numeric structures extended with the Archimedean *)
(* axiom. To use this file, insert "Import Num.Theory." and optionally *)
(* "Import Num.Def." before your scripts as in the ssrnum library. *)
(* The modules provided by this library subsume those from ssrnum. *)
(* *)
(* This file defines the following structures: *)
(* *)
(* archiNumDomainType == numDomainType with the Archimedean axiom *)
(* The HB class is called ArchiNumDomain. *)
(* archiNumFieldType == numFieldType with the Archimedean axiom *)
(* The HB class is called ArchiNumField. *)
(* archiClosedFieldType == closedFieldType with the Archimedean axiom *)
(* The HB class is called ArchiClosedField. *)
(* archiRealDomainType == realDomainType with the Archimedean axiom *)
(* The HB class is called ArchiRealDomain. *)
(* archiRealFieldType == realFieldType with the Archimedean axiom *)
(* The HB class is called ArchiRealField. *)
(* archiRcfType == rcfType with the Archimedean axiom *)
(* The HB class is called ArchiRealClosedField. *)
(* *)
(* Over these structures, we have the following operations: *)
(* x \is a Num.int <=> x is an integer, i.e., x = m%:~R for some m : int *)
(* x \is a Num.nat <=> x is a natural number, i.e., x = m%:R for some m : nat*)
(* Num.floor x == the m : int such that m%:~R <= x < (m + 1)%:~R *)
(* when x \is a Num.real, otherwise 0%Z *)
(* Num.ceil x == the m : int such that (m - 1)%:~R < x <= m%:~R *)
(* when x \is a Num.real, otherwise 0%Z *)
(* Num.truncn x == the n : nat such that n%:R <= x < n.+1%:R *)
(* when 0 <= n, otherwise 0%N *)
(* Num.bound x == an upper bound for x, i.e., an n such that `|x| < n%:R *)
(******************************************************************************)
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Import Order.TTheory GRing.Theory Num.Theory.
Module Num.
Import Num.Def.
HB.mixin Record NumDomain_hasFloorCeilTruncn R of Num.NumDomain R := {
floor : R -> int;
ceil : R -> int;
truncn : R -> nat;
int_num_subdef : pred R;
nat_num_subdef : pred R;
floor_subproof :
forall x,
if x \is Rreal then (floor x)%:~R <= x < (floor x + 1)%:~R
else floor x == 0;
ceil_subproof : forall x, ceil x = - floor (- x);
truncn_subproof : forall x, truncn x = if floor x is Posz n then n else 0;
int_num_subproof : forall x, reflect (exists n, x = n%:~R) (int_num_subdef x);
nat_num_subproof : forall x, reflect (exists n, x = n%:R) (nat_num_subdef x);
}.
#[short(type="archiNumDomainType")]
HB.structure Definition ArchiNumDomain :=
{ R of NumDomain_hasFloorCeilTruncn R & Num.NumDomain R }.
Module ArchiNumDomainExports.
Bind Scope ring_scope with ArchiNumDomain.sort.
End ArchiNumDomainExports.
HB.export ArchiNumDomainExports.
#[short(type="archiNumFieldType")]
HB.structure Definition ArchiNumField :=
{ R of NumDomain_hasFloorCeilTruncn R & Num.NumField R }.
Module ArchiNumFieldExports.
Bind Scope ring_scope with ArchiNumField.sort.
End ArchiNumFieldExports.
HB.export ArchiNumFieldExports.
#[short(type="archiClosedFieldType")]
HB.structure Definition ArchiClosedField :=
{ R of NumDomain_hasFloorCeilTruncn R & Num.ClosedField R }.
Module ArchiClosedFieldExports.
Bind Scope ring_scope with ArchiClosedField.sort.
End ArchiClosedFieldExports.
HB.export ArchiClosedFieldExports.
#[short(type="archiRealDomainType")]
HB.structure Definition ArchiRealDomain :=
{ R of NumDomain_hasFloorCeilTruncn R & Num.RealDomain R }.
Module ArchiRealDomainExports.
Bind Scope ring_scope with ArchiRealDomain.sort.
End ArchiRealDomainExports.
HB.export ArchiRealDomainExports.
#[short(type="archiRealFieldType")]
HB.structure Definition ArchiRealField :=
{ R of NumDomain_hasFloorCeilTruncn R & Num.RealField R }.
Module ArchiRealFieldExports.
Bind Scope ring_scope with ArchiRealField.sort.
End ArchiRealFieldExports.
HB.export ArchiRealFieldExports.
#[short(type="archiRcfType")]
HB.structure Definition ArchiRealClosedField :=
{ R of NumDomain_hasFloorCeilTruncn R & Num.RealClosedField R }.
Module ArchiRealClosedFieldExports.
Bind Scope ring_scope with ArchiRealClosedField.sort.
End ArchiRealClosedFieldExports.
HB.export ArchiRealClosedFieldExports.
Section Def.
Context {R : archiNumDomainType}.
Definition nat_num : qualifier 1 R := [qualify a x : R | nat_num_subdef x].
Definition int_num : qualifier 1 R := [qualify a x : R | int_num_subdef x].
Definition bound (x : R) := (truncn `|x|).+1.
End Def.
Arguments floor {R} : rename, simpl never.
Arguments ceil {R} : rename, simpl never.
Arguments truncn {R} : rename, simpl never.
Arguments nat_num {R} : simpl never.
Arguments int_num {R} : simpl never.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to truncn.")]
Notation trunc := truncn.
Module Def.
Export ssrnum.Num.Def.
Notation truncn := truncn.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to truncn.")]
Notation trunc := truncn.
Notation floor := floor.
Notation ceil := ceil.
Notation nat_num := nat_num.
Notation int_num := int_num.
Notation archi_bound := bound.
End Def.
Module intArchimedean.
Section intArchimedean.
Implicit Types n : int.
Lemma floorP n : if n \is Rreal then n%:~R <= n < (n + 1)%:~R else n == 0.
Proof. by rewrite num_real !intz ltzD1 lexx. Qed.
Lemma intrP n : reflect (exists m, n = m%:~R) true.
Proof. by apply: ReflectT; exists n; rewrite intz. Qed.
Lemma natrP n : reflect (exists m, n = m%:R) (0 <= n).
Proof.
apply: (iffP idP); last by case=> m ->; rewrite ler0n.
by case: n => // n _; exists n; rewrite natz.
Qed.
End intArchimedean.
End intArchimedean.
#[export]
HB.instance Definition _ :=
@NumDomain_hasFloorCeilTruncn.Build int id id _ xpredT Rnneg_pred
intArchimedean.floorP (fun=> esym (opprK _)) (fun=> erefl)
intArchimedean.intrP intArchimedean.natrP.
Module Import Theory.
Export ssrnum.Num.Theory.
Section ArchiNumDomainTheory.
Variable R : archiNumDomainType.
Implicit Types x y z : R.
Local Notation truncn := (@truncn R).
Local Notation floor := (@floor R).
Local Notation ceil := (@ceil R).
Local Notation nat_num := (@Def.nat_num R).
Local Notation int_num := (@Def.int_num R).
Local Lemma floorP x :
if x \is Rreal then (floor x)%:~R <= x < (floor x + 1)%:~R else floor x == 0.
Proof. exact: floor_subproof. Qed.
Lemma floorNceil x : floor x = - ceil (- x).
Proof. by rewrite ceil_subproof !opprK. Qed.
Lemma ceilNfloor x : ceil x = - floor (- x).
Proof. exact: ceil_subproof. Qed.
Lemma truncEfloor x : truncn x = if floor x is Posz n then n else 0.
Proof. exact: truncn_subproof. Qed.
Lemma natrP x : reflect (exists n, x = n%:R) (x \is a nat_num).
Proof. exact: nat_num_subproof. Qed.
Lemma intrP x : reflect (exists m, x = m%:~R) (x \is a int_num).
Proof. exact: int_num_subproof. Qed.
(* int_num and nat_num *)
Lemma intr_int m : m%:~R \is a int_num. Proof. by apply/intrP; exists m. Qed.
Lemma natr_nat n : n%:R \is a nat_num. Proof. by apply/natrP; exists n. Qed.
#[local] Hint Resolve intr_int natr_nat : core.
Lemma rpred_int_num (S : subringClosed R) x : x \is a int_num -> x \in S.
Proof. by move=> /intrP[n ->]; rewrite rpred_int. Qed.
Lemma rpred_nat_num (S : semiringClosed R) x : x \is a nat_num -> x \in S.
Proof. by move=> /natrP[n ->]; apply: rpred_nat. Qed.
Lemma int_num0 : 0 \is a int_num. Proof. exact: (intr_int 0). Qed.
Lemma int_num1 : 1 \is a int_num. Proof. exact: (intr_int 1). Qed.
Lemma nat_num0 : 0 \is a nat_num. Proof. exact: (natr_nat 0). Qed.
Lemma nat_num1 : 1 \is a nat_num. Proof. exact: (natr_nat 1). Qed.
#[local] Hint Resolve int_num0 int_num1 nat_num0 nat_num1 : core.
Fact int_num_subring : subring_closed int_num.
Proof.
by split=> // _ _ /intrP[n ->] /intrP[m ->]; rewrite -(intrB, intrM).
Qed.
#[export]
HB.instance Definition _ := GRing.isSubringClosed.Build R int_num_subdef
int_num_subring.
Fact nat_num_semiring : semiring_closed nat_num.
Proof.
by do 2![split] => //= _ _ /natrP[n ->] /natrP[m ->]; rewrite -(natrD, natrM).
Qed.
#[export]
HB.instance Definition _ := GRing.isSemiringClosed.Build R nat_num_subdef
nat_num_semiring.
Lemma Rreal_nat : {subset nat_num <= Rreal}. Proof. exact: rpred_nat_num. Qed.
Lemma intr_nat : {subset nat_num <= int_num}.
Proof. by move=> _ /natrP[n ->]; rewrite pmulrn intr_int. Qed.
Lemma Rreal_int : {subset int_num <= Rreal}. Proof. exact: rpred_int_num. Qed.
Lemma intrE x : (x \is a int_num) = (x \is a nat_num) || (- x \is a nat_num).
Proof.
apply/idP/orP => [/intrP[[n|n] ->]|[]/intr_nat]; rewrite ?rpredN //.
by left; apply/natrP; exists n.
by rewrite NegzE intrN opprK; right; apply/natrP; exists n.+1.
Qed.
Lemma intr_normK x : x \is a int_num -> `|x| ^+ 2 = x ^+ 2.
Proof. by move/Rreal_int/real_normK. Qed.
Lemma natr_normK x : x \is a nat_num -> `|x| ^+ 2 = x ^+ 2.
Proof. by move/Rreal_nat/real_normK. Qed.
Lemma natr_norm_int x : x \is a int_num -> `|x| \is a nat_num.
Proof. by move=> /intrP[m ->]; rewrite -intr_norm rpred_nat_num ?natr_nat. Qed.
Lemma natr_ge0 x : x \is a nat_num -> 0 <= x.
Proof. by move=> /natrP[n ->]; apply: ler0n. Qed.
Lemma natr_gt0 x : x \is a nat_num -> (0 < x) = (x != 0).
Proof. by move/natr_ge0; case: comparableP. Qed.
Lemma natrEint x : (x \is a nat_num) = (x \is a int_num) && (0 <= x).
Proof.
apply/idP/andP=> [Nx | [Zx x_ge0]]; first by rewrite intr_nat ?natr_ge0.
by rewrite -(ger0_norm x_ge0) natr_norm_int.
Qed.
Lemma intrEge0 x : 0 <= x -> (x \is a int_num) = (x \is a nat_num).
Proof. by rewrite natrEint andbC => ->. Qed.
Lemma intrEsign x : x \is a int_num -> x = (-1) ^+ (x < 0)%R * `|x|.
Proof. by move/Rreal_int/realEsign. Qed.
Lemma norm_natr x : x \is a nat_num -> `|x| = x.
Proof. by move/natr_ge0/ger0_norm. Qed.
Lemma natr_exp_even x n : ~~ odd n -> x \is a int_num -> x ^+ n \is a nat_num.
Proof.
move=> n_oddF x_intr.
by rewrite natrEint rpredX //= real_exprn_even_ge0 // Rreal_int.
Qed.
Lemma norm_intr_ge1 x : x \is a int_num -> x != 0 -> 1 <= `|x|.
Proof.
rewrite -normr_eq0 => /natr_norm_int/natrP[n ->].
by rewrite pnatr_eq0 ler1n lt0n.
Qed.
Lemma sqr_intr_ge1 x : x \is a int_num -> x != 0 -> 1 <= x ^+ 2.
Proof.
by move=> Zx nz_x; rewrite -intr_normK // expr_ge1 ?normr_ge0 ?norm_intr_ge1.
Qed.
Lemma intr_ler_sqr x : x \is a int_num -> x <= x ^+ 2.
Proof.
move=> Zx; have [-> | nz_x] := eqVneq x 0; first by rewrite expr0n.
apply: le_trans (_ : `|x| <= _); first by rewrite real_ler_norm ?Rreal_int.
by rewrite -intr_normK // ler_eXnr // norm_intr_ge1.
Qed.
(* floor and int_num *)
Lemma real_floor_itv x : x \is Rreal -> (floor x)%:~R <= x < (floor x + 1)%:~R.
Proof. by case: ifP (floorP x). Qed.
Lemma real_floor_le x : x \is Rreal -> (floor x)%:~R <= x.
Proof. by case/real_floor_itv/andP. Qed.
Lemma real_floorD1_gt x : x \is Rreal -> x < (floor x + 1)%:~R.
Proof. by case/real_floor_itv/andP. Qed.
Lemma floor_def x m : m%:~R <= x < (m + 1)%:~R -> floor x = m.
Proof.
case/andP=> lemx ltxm1; apply/eqP; rewrite eq_le -!ltzD1.
move: (ger_real lemx); rewrite realz => /real_floor_itv/andP[lefx ltxf1].
by rewrite -!(ltr_int R) 2?(@le_lt_trans _ _ x).
Qed.
(* TODO: rename to real_floor_ge_int,
once the currently deprecated one has been removed *)
Lemma real_floor_ge_int_tmp x n : x \is Rreal -> (n <= floor x) = (n%:~R <= x).
Proof.
move=> /real_floor_itv /andP[lefx ltxf1]; apply/idP/idP => lenx.
by apply: le_trans lefx; rewrite ler_int.
by rewrite -ltzD1 -(ltr_int R); apply: le_lt_trans ltxf1.
Qed.
#[deprecated(since="mathcomp 2.4.0", note="Use real_floor_ge_int_tmp instead.")]
Lemma real_floor_ge_int x n : x \is Rreal -> (n%:~R <= x) = (n <= floor x).
Proof. by move=> ?; rewrite real_floor_ge_int_tmp. Qed.
Lemma real_floor_lt_int x n : x \is Rreal -> (floor x < n) = (x < n%:~R).
Proof.
by move=> ?; rewrite [RHS]real_ltNge ?realz -?real_floor_ge_int_tmp -?ltNge.
Qed.
Lemma real_floor_eq x n : x \is Rreal ->
(floor x == n) = (n%:~R <= x < (n + 1)%:~R).
Proof.
by move=> xr; apply/eqP/idP => [<-|]; [exact: real_floor_itv|exact: floor_def].
Qed.
Lemma le_floor : {homo floor : x y / x <= y}.
Proof.
move=> x y lexy; move: (floorP x) (floorP y); rewrite (ger_real lexy).
case: ifP => [_ /andP[lefx _] /andP[_] | _ /eqP-> /eqP-> //].
by move=> /(le_lt_trans lexy) /(le_lt_trans lefx); rewrite ltr_int ltzD1.
Qed.
Lemma intrKfloor : cancel intr floor.
Proof. by move=> m; apply: floor_def; rewrite lexx rmorphD ltrDl ltr01. Qed.
Lemma natr_int n : n%:R \is a int_num.
Proof. by rewrite intrE natr_nat. Qed.
#[local] Hint Resolve natr_int : core.
Lemma intrEfloor x : x \is a int_num = ((floor x)%:~R == x).
Proof.
by apply/intrP/eqP => [[n ->] | <-]; [rewrite intrKfloor | exists (floor x)].
Qed.
Lemma floorK : {in int_num, cancel floor intr}.
Proof. by move=> z; rewrite intrEfloor => /eqP. Qed.
Lemma floor0 : floor 0 = 0. Proof. exact: intrKfloor 0. Qed.
Lemma floor1 : floor 1 = 1. Proof. exact: intrKfloor 1. Qed.
#[local] Hint Resolve floor0 floor1 : core.
Lemma real_floorDzr : {in int_num & Rreal, {morph floor : x y / x + y}}.
Proof.
move=> _ y /intrP[m ->] Ry; apply: floor_def.
by rewrite -addrA 2!rmorphD /= intrKfloor lerD2l ltrD2l real_floor_itv.
Qed.
Lemma real_floorDrz : {in Rreal & int_num, {morph floor : x y / x + y}}.
Proof. by move=> x y xr yz; rewrite addrC real_floorDzr // addrC. Qed.
Lemma floorN : {in int_num, {morph floor : x / - x}}.
Proof. by move=> _ /intrP[m ->]; rewrite -rmorphN !intrKfloor. Qed.
Lemma floorM : {in int_num &, {morph floor : x y / x * y}}.
Proof.
by move=> _ _ /intrP[m1 ->] /intrP[m2 ->]; rewrite -rmorphM !intrKfloor.
Qed.
Lemma floorX n : {in int_num, {morph floor : x / x ^+ n}}.
Proof. by move=> _ /intrP[m ->]; rewrite -rmorphXn !intrKfloor. Qed.
Lemma real_floor_ge0 x : x \is Rreal -> (0 <= floor x) = (0 <= x).
Proof. by move=> ?; rewrite real_floor_ge_int_tmp. Qed.
Lemma floor_lt0 x : (floor x < 0) = (x < 0).
Proof.
case: ifP (floorP x) => [xr _ | xr /eqP <-]; first by rewrite real_floor_lt_int.
by rewrite ltxx; apply/esym/(contraFF _ xr)/ltr0_real.
Qed.
Lemma real_floor_le0 x : x \is Rreal -> (floor x <= 0) = (x < 1).
Proof. by move=> ?; rewrite -ltzD1 add0r real_floor_lt_int. Qed.
Lemma floor_gt0 x : (floor x > 0) = (x >= 1).
Proof.
case: ifP (floorP x) => [xr _ | xr /eqP->].
by rewrite gtz0_ge1 real_floor_ge_int_tmp.
by rewrite ltxx; apply/esym/(contraFF _ xr)/ger1_real.
Qed.
Lemma floor_neq0 x : (floor x != 0) = (x < 0) || (x >= 1).
Proof.
case: ifP (floorP x) => [xr _ | xr /eqP->]; rewrite ?eqxx/=.
by rewrite neq_lt floor_lt0 floor_gt0.
by apply/esym/(contraFF _ xr) => /orP[/ltr0_real|/ger1_real].
Qed.
Lemma floorpK : {in polyOver int_num, cancel (map_poly floor) (map_poly intr)}.
Proof.
move=> p /(all_nthP 0) Zp; apply/polyP=> i.
rewrite coef_map coef_map_id0 //= -[p]coefK coef_poly.
by case: ifP => [/Zp/floorK // | _]; rewrite floor0.
Qed.
Lemma floorpP (p : {poly R}) :
p \is a polyOver int_num -> {q | p = map_poly intr q}.
Proof. by exists (map_poly floor p); rewrite floorpK. Qed.
(* ceil and int_num *)
Lemma real_ceil_itv x : x \is Rreal -> (ceil x - 1)%:~R < x <= (ceil x)%:~R.
Proof.
rewrite ceilNfloor -opprD !intrN ltrNl lerNr andbC -realN.
exact: real_floor_itv.
Qed.
Lemma real_ceilB1_lt x : x \is Rreal -> (ceil x - 1)%:~R < x.
Proof. by case/real_ceil_itv/andP. Qed.
Lemma real_ceil_ge x : x \is Rreal -> x <= (ceil x)%:~R.
Proof. by case/real_ceil_itv/andP. Qed.
Lemma ceil_def x m : (m - 1)%:~R < x <= m%:~R -> ceil x = m.
Proof.
rewrite -ltrN2 -lerN2 andbC -!intrN opprD opprK ceilNfloor.
by move=> /floor_def ->; rewrite opprK.
Qed.
(* TODO: rename to real_ceil_le_int,
once the currently deprecated one has been removed *)
Lemma real_ceil_le_int_tmp x n : x \is Rreal -> (ceil x <= n) = (x <= n%:~R).
Proof.
rewrite ceilNfloor lerNl -realN => /real_floor_ge_int_tmp ->.
by rewrite intrN lerN2.
Qed.
#[deprecated(since="mathcomp 2.4.0", note="Use real_ceil_le_int_tmp instead.")]
Lemma real_ceil_le_int x n : x \is Rreal -> x <= n%:~R = (ceil x <= n).
Proof. by move=> ?; rewrite real_ceil_le_int_tmp. Qed.
Lemma real_ceil_gt_int x n : x \is Rreal -> (n < ceil x) = (n%:~R < x).
Proof.
by move=> ?; rewrite [RHS]real_ltNge ?realz -?real_ceil_le_int_tmp ?ltNge.
Qed.
Lemma real_ceil_eq x n : x \is Rreal ->
(ceil x == n) = ((n - 1)%:~R < x <= n%:~R).
Proof.
by move=> xr; apply/eqP/idP => [<-|]; [exact: real_ceil_itv|exact: ceil_def].
Qed.
(* TODO: rename to le_ceil,
once the currently deprecated one has been removed *)
Lemma le_ceil_tmp : {homo ceil : x y / x <= y}.
Proof. by move=> x y lexy; rewrite !ceilNfloor lerN2 le_floor ?lerN2. Qed.
Lemma intrKceil : cancel intr ceil.
Proof. by move=> m; rewrite ceilNfloor -intrN intrKfloor opprK. Qed.
Lemma intrEceil x : x \is a int_num = ((ceil x)%:~R == x).
Proof. by rewrite -rpredN intrEfloor -eqr_oppLR -intrN -ceilNfloor. Qed.
Lemma ceilK : {in int_num, cancel ceil intr}.
Proof. by move=> z; rewrite intrEceil => /eqP. Qed.
Lemma ceil0 : ceil 0 = 0. Proof. exact: intrKceil 0. Qed.
Lemma ceil1 : ceil 1 = 1. Proof. exact: intrKceil 1. Qed.
#[local] Hint Resolve ceil0 ceil1 : core.
Lemma real_ceilDzr : {in int_num & Rreal, {morph ceil : x y / x + y}}.
Proof.
move=> x y x_int y_real.
by rewrite ceilNfloor opprD real_floorDzr ?rpredN // opprD -!ceilNfloor.
Qed.
Lemma real_ceilDrz : {in Rreal & int_num, {morph ceil : x y / x + y}}.
Proof. by move=> x y xr yz; rewrite addrC real_ceilDzr // addrC. Qed.
Lemma ceilN : {in int_num, {morph ceil : x / - x}}.
Proof. by move=> ? ?; rewrite !ceilNfloor !opprK floorN. Qed.
Lemma ceilM : {in int_num &, {morph ceil : x y / x * y}}.
Proof.
by move=> _ _ /intrP[m1 ->] /intrP[m2 ->]; rewrite -rmorphM !intrKceil.
Qed.
Lemma ceilX n : {in int_num, {morph ceil : x / x ^+ n}}.
Proof. by move=> _ /intrP[m ->]; rewrite -rmorphXn !intrKceil. Qed.
Lemma real_ceil_ge0 x : x \is Rreal -> (0 <= ceil x) = (-1 < x).
Proof.
by move=> ?; rewrite ceilNfloor oppr_ge0 real_floor_le0 ?realN 1?ltrNl.
Qed.
Lemma ceil_lt0 x : (ceil x < 0) = (x <= -1).
Proof. by rewrite ceilNfloor oppr_lt0 floor_gt0 lerNr. Qed.
Lemma real_ceil_le0 x : x \is Rreal -> (ceil x <= 0) = (x <= 0).
Proof. by move=> ?; rewrite real_ceil_le_int_tmp. Qed.
Lemma ceil_gt0 x : (ceil x > 0) = (x > 0).
Proof. by rewrite ceilNfloor oppr_gt0 floor_lt0 oppr_lt0. Qed.
Lemma ceil_neq0 x : (ceil x != 0) = (x <= -1) || (x > 0).
Proof. by rewrite ceilNfloor oppr_eq0 floor_neq0 oppr_lt0 lerNr orbC. Qed.
Lemma real_ceil_floor x : x \is Rreal ->
ceil x = floor x + (x \isn't a int_num).
Proof.
case Ix: (x \is a int_num) => Rx /=.
by apply/eqP; rewrite addr0 ceilNfloor eqr_oppLR floorN.
apply/ceil_def; rewrite addrK; move: (real_floor_itv Rx).
by rewrite le_eqVlt -intrEfloor Ix /= => /andP[-> /ltW].
Qed.
(* Relating Cnat and oldCnat. *)
Lemma truncn_floor x : truncn x = if 0 <= x then `|floor x|%N else 0%N.
Proof.
move: (floorP x); rewrite truncEfloor realE.
have [/le_floor|_]/= := boolP (0 <= x); first by rewrite floor0; case: floor.
by case: ifP => [/le_floor|_ /eqP->//]; rewrite floor0; case: floor => [[]|].
Qed.
(* trunc and nat_num *)
Local Lemma truncnP x :
if 0 <= x then (truncn x)%:R <= x < (truncn x).+1%:R else truncn x == 0%N.
Proof.
rewrite truncn_floor.
case: (boolP (0 <= x)) => //= /[dup] /le_floor + /ger0_real/real_floor_itv.
by rewrite floor0; case: (floor x) => // n _; rewrite absz_nat addrC -intS.
Qed.
Lemma truncn_itv x : 0 <= x -> (truncn x)%:R <= x < (truncn x).+1%:R.
Proof. by move=> x_ge0; move: (truncnP x); rewrite x_ge0. Qed.
Lemma truncn_le x : (truncn x)%:R <= x = (0 <= x).
Proof. by case: ifP (truncnP x) => [+ /andP[] | + /eqP->//]. Qed.
Lemma real_truncnS_gt x : x \is Rreal -> x < (truncn x).+1%:R.
Proof. by move/real_ge0P => [/truncn_itv/andP[]|/lt_le_trans->]. Qed.
Lemma truncn_def x n : n%:R <= x < n.+1%:R -> truncn x = n.
Proof.
case/andP=> lemx ltxm1; apply/eqP; rewrite eqn_leq -ltnS -[(n <= _)%N]ltnS.
have/truncn_itv/andP[lefx ltxf1]: 0 <= x by apply: le_trans lemx; apply: ler0n.
by rewrite -!(ltr_nat R) 2?(@le_lt_trans _ _ x).
Qed.
Lemma truncn_ge_nat x n : 0 <= x -> (n <= truncn x)%N = (n%:R <= x).
Proof.
move=> /truncn_itv /andP[letx ltxt1]; apply/idP/idP => lenx.
by apply: le_trans letx; rewrite ler_nat.
by rewrite -ltnS -(ltr_nat R); apply: le_lt_trans ltxt1.
Qed.
Lemma truncn_gt_nat x n : (n < truncn x)%N = (n.+1%:R <= x).
Proof.
case: ifP (truncnP x) => [x0 _ | x0 /eqP->]; first by rewrite truncn_ge_nat.
by rewrite ltn0; apply/esym/(contraFF _ x0)/le_trans.
Qed.
Lemma truncn_lt_nat x n : 0 <= x -> (truncn x < n)%N = (x < n%:R).
Proof. by move=> ?; rewrite real_ltNge ?ger0_real// ltnNge truncn_ge_nat. Qed.
Lemma real_truncn_le_nat x n : x \is Rreal -> (truncn x <= n)%N = (x < n.+1%:R).
Proof. by move=> ?; rewrite real_ltNge// leqNgt truncn_gt_nat. Qed.
Lemma truncn_eq x n : 0 <= x -> (truncn x == n) = (n%:R <= x < n.+1%:R).
Proof.
by move=> xr; apply/eqP/idP => [<-|]; [exact: truncn_itv|exact: truncn_def].
Qed.
Lemma le_truncn : {homo truncn : x y / x <= y >-> (x <= y)%N}.
Proof.
move=> x y lexy; move: (truncnP x) (truncnP y).
case: ifP => [x0 /andP[letx _] | x0 /eqP->//].
case: ifP => [y0 /andP[_] | y0 /eqP->]; [|by rewrite (le_trans x0 lexy) in y0].
by move=> /(le_lt_trans lexy) /(le_lt_trans letx); rewrite ltr_nat ltnS.
Qed.
Lemma natrK : cancel (GRing.natmul 1) truncn.
Proof. by move=> m; apply: truncn_def; rewrite ler_nat ltr_nat ltnS leqnn. Qed.
Lemma natrEtruncn x : (x \is a nat_num) = ((truncn x)%:R == x).
Proof.
by apply/natrP/eqP => [[n ->]|<-]; [rewrite natrK | exists (truncn x)].
Qed.
Lemma archi_boundP x : 0 <= x -> x < (bound x)%:R.
Proof.
move=> x_ge0; case/truncn_itv/andP: (normr_ge0 x) => _.
exact/le_lt_trans/real_ler_norm/ger0_real.
Qed.
Lemma truncnK : {in nat_num, cancel truncn (GRing.natmul 1)}.
Proof. by move=> x; rewrite natrEtruncn => /eqP. Qed.
Lemma truncn0 : truncn 0 = 0%N. Proof. exact: natrK 0%N. Qed.
Lemma truncn1 : truncn 1 = 1%N. Proof. exact: natrK 1%N. Qed.
#[local] Hint Resolve truncn0 truncn1 : core.
Lemma truncnD :
{in nat_num & Rnneg, {morph truncn : x y / x + y >-> (x + y)%N}}.
Proof.
move=> _ y /natrP[n ->] y_ge0; apply: truncn_def.
by rewrite -addnS !natrD !natrK lerD2l ltrD2l truncn_itv.
Qed.
Lemma truncnM : {in nat_num &, {morph truncn : x y / x * y >-> (x * y)%N}}.
Proof. by move=> _ _ /natrP[n1 ->] /natrP[n2 ->]; rewrite -natrM !natrK. Qed.
Lemma truncnX n : {in nat_num, {morph truncn : x / x ^+ n >-> (x ^ n)%N}}.
Proof. by move=> _ /natrP[n1 ->]; rewrite -natrX !natrK. Qed.
Lemma truncn_gt0 x : (0 < truncn x)%N = (1 <= x).
Proof.
case: ifP (truncnP x) => [x0 | x0 /eqP<-]; first by rewrite truncn_ge_nat.
by rewrite ltnn; apply/esym/(contraFF _ x0)/le_trans.
Qed.
Lemma truncn0Pn x : reflect (truncn x = 0%N) (~~ (1 <= x)).
Proof. by rewrite -truncn_gt0 -eqn0Ngt; apply: eqP. Qed.
Lemma sum_truncnK I r (P : pred I) F : (forall i, P i -> F i \is a nat_num) ->
(\sum_(i <- r | P i) truncn (F i))%:R = \sum_(i <- r | P i) F i.
Proof. by rewrite natr_sum => natr; apply: eq_bigr => i /natr /truncnK. Qed.
Lemma prod_truncnK I r (P : pred I) F : (forall i, P i -> F i \is a nat_num) ->
(\prod_(i <- r | P i) truncn (F i))%:R = \prod_(i <- r | P i) F i.
Proof. by rewrite natr_prod => natr; apply: eq_bigr => i /natr /truncnK. Qed.
Lemma natr_sum_eq1 (I : finType) (P : pred I) (F : I -> R) :
(forall i, P i -> F i \is a nat_num) -> \sum_(i | P i) F i = 1 ->
{i : I | [/\ P i, F i = 1 & forall j, j != i -> P j -> F j = 0]}.
Proof.
move=> natF /eqP; rewrite -sum_truncnK// -[1]/1%:R eqr_nat => /sum_nat_eq1 exi.
have [i /and3P[Pi /eqP f1 /forallP a]] : {i : I | [&& P i, truncn (F i) == 1
& [forall j : I, ((j != i) ==> P j ==> (truncn (F j) == 0))]]}.
apply/sigW; have [i [Pi /eqP f1 a]] := exi; exists i; apply/and3P; split=> //.
by apply/forallP => j; apply/implyP => ji; apply/implyP => Pj; apply/eqP/a.
exists i; split=> [//||j ji Pj]; rewrite -[LHS]truncnK ?natF ?f1//; apply/eqP.
by rewrite -[0]/0%:R eqr_nat; apply: implyP Pj; apply: implyP ji; apply: a.
Qed.
Lemma natr_mul_eq1 x y :
x \is a nat_num -> y \is a nat_num -> (x * y == 1) = (x == 1) && (y == 1).
Proof. by do 2!move/truncnK <-; rewrite -natrM !pnatr_eq1 muln_eq1. Qed.
Lemma natr_prod_eq1 (I : finType) (P : pred I) (F : I -> R) :
(forall i, P i -> F i \is a nat_num) -> \prod_(i | P i) F i = 1 ->
forall i, P i -> F i = 1.
Proof.
move=> natF /eqP; rewrite -prod_truncnK// -[1]/1%:R eqr_nat prod_nat_seq_eq1.
move/allP => a i Pi; apply/eqP; rewrite -[F i]truncnK ?natF// eqr_nat.
by apply: implyP Pi; apply: a; apply: mem_index_enum.
Qed.
(* predCmod *)
Variables (U V : lmodType R) (f : {additive U -> V}).
Lemma raddfZ_nat a u : a \is a nat_num -> f (a *: u) = a *: f u.
Proof. by move=> /natrP[n ->]; apply: raddfZnat. Qed.
Lemma rpredZ_nat (S : addrClosed V) :
{in nat_num & S, forall z u, z *: u \in S}.
Proof. by move=> _ u /natrP[n ->]; apply: rpredZnat. Qed.
Lemma raddfZ_int a u : a \is a int_num -> f (a *: u) = a *: f u.
Proof. by move=> /intrP[m ->]; rewrite !scaler_int raddfMz. Qed.
Lemma rpredZ_int (S : zmodClosed V) :
{in int_num & S, forall z u, z *: u \in S}.
Proof. by move=> _ u /intrP[m ->] ?; rewrite scaler_int rpredMz. Qed.
(* autC *)
Implicit Type nu : {rmorphism R -> R}.
Lemma aut_natr nu : {in nat_num, nu =1 id}.
Proof. by move=> _ /natrP[n ->]; apply: rmorph_nat. Qed.
Lemma aut_intr nu : {in int_num, nu =1 id}.
Proof. by move=> _ /intrP[m ->]; apply: rmorph_int. Qed.
End ArchiNumDomainTheory.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to truncn_itv.")]
Notation trunc_itv := truncn_itv.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to truncn_def.")]
Notation trunc_def := truncn_def.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to truncnK.")]
Notation truncK := truncnK.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to truncn0.")]
Notation trunc0 := truncn0.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to truncn1.")]
Notation trunc1 := truncn1.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to truncnD.")]
Notation truncD := truncnD.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to truncnM.")]
Notation truncM := truncnM.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to truncnX.")]
Notation truncX := truncnX.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to truncn_gt0.")]
Notation trunc_gt0 := truncn_gt0.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to truncn0Pn.")]
Notation trunc0Pn := truncn0Pn.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to sum_truncnK.")]
Notation sum_truncK := sum_truncnK.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to prod_truncnK.")]
Notation prod_truncK := prod_truncnK.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to truncn_floor.")]
Notation trunc_floor := truncn_floor.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to real_floor_le.")]
Notation real_ge_floor := real_floor_le.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to real_floorD1_gt.")]
Notation real_lt_succ_floor := real_floorD1_gt.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to real_ceilB1_lt.")]
Notation real_gt_pred_ceil := real_floorD1_gt.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to real_ceil_ge.")]
Notation real_le_ceil := real_ceil_ge.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to le_floor.")]
Notation floor_le := le_floor.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to le_ceil.")]
Notation ceil_le := le_ceil_tmp.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to natrEtruncn.")]
Notation natrE := natrEtruncn.
Arguments natrK {R} _%_N.
Arguments intrKfloor {R}.
Arguments intrKceil {R}.
Arguments natrP {R x}.
Arguments intrP {R x}.
#[global] Hint Resolve truncn0 truncn1 : core.
#[global] Hint Resolve floor0 floor1 : core.
#[global] Hint Resolve ceil0 ceil1 : core.
#[global] Hint Extern 0 (is_true (_%:R \is a nat_num)) => apply: natr_nat : core.
#[global] Hint Extern 0 (is_true (_%:R \in nat_num_subdef)) => apply: natr_nat : core.
#[global] Hint Extern 0 (is_true (_%:~R \is a int_num)) => apply: intr_int : core.
#[global] Hint Extern 0 (is_true (_%:~R \in int_num_subdef)) => apply: intr_int : core.
#[global] Hint Extern 0 (is_true (_%:R \is a int_num)) => apply: natr_int : core.
#[global] Hint Extern 0 (is_true (_%:R \in int_num_subdef)) => apply: natr_int : core.
#[global] Hint Extern 0 (is_true (0 \is a nat_num)) => apply: nat_num0 : core.
#[global] Hint Extern 0 (is_true (0 \in nat_num_subdef)) => apply: nat_num0 : core.
#[global] Hint Extern 0 (is_true (1 \is a nat_num)) => apply: nat_num1 : core.
#[global] Hint Extern 0 (is_true (1 \in int_num_subdef)) => apply: nat_num1 : core.
#[global] Hint Extern 0 (is_true (0 \is a int_num)) => apply: int_num0 : core.
#[global] Hint Extern 0 (is_true (0 \in int_num_subdef)) => apply: int_num0 : core.
#[global] Hint Extern 0 (is_true (1 \is a int_num)) => apply: int_num1 : core.
#[global] Hint Extern 0 (is_true (1 \in int_num_subdef)) => apply: int_num1 : core.
Section ArchiRealDomainTheory.
Variables (R : archiRealDomainType).
Implicit Type x : R.
Lemma upper_nthrootP x i : (bound x <= i)%N -> x < 2%:R ^+ i.
Proof.
case/truncn_itv/andP: (normr_ge0 x) => _ /ltr_normlW xlt le_b_i.
by rewrite (lt_le_trans xlt) // -natrX ler_nat (ltn_trans le_b_i) // ltn_expl.
Qed.
Lemma truncnS_gt x : x < (truncn x).+1%:R.
Proof. exact: real_truncnS_gt. Qed.
Lemma truncn_le_nat x n : (truncn x <= n)%N = (x < n.+1%:R).
Proof. exact: real_truncn_le_nat. Qed.
Lemma floor_itv x : (floor x)%:~R <= x < (floor x + 1)%:~R.
Proof. exact: real_floor_itv. Qed.
(* TODO: rename to floor_le, once the deprecated one has been removed *)
Lemma floor_le_tmp x : (floor x)%:~R <= x. Proof. exact: real_floor_le. Qed.
Lemma floorD1_gt x : x < (floor x + 1)%:~R.
Proof. exact: real_floorD1_gt. Qed.
#[deprecated(since="mathcomp 2.4.0", note="Use floor_ge_int_tmp instead.")]
Lemma floor_ge_int x n : n%:~R <= x = (n <= floor x).
Proof. by rewrite real_floor_ge_int_tmp. Qed.
(* TODO: rename to floor_ge_int,
once the currently deprecated one has been removed *)
Lemma floor_ge_int_tmp x n : (n <= floor x) = (n%:~R <= x).
Proof. exact: real_floor_ge_int_tmp. Qed.
Lemma floor_lt_int x n : (floor x < n) = (x < n%:~R).
Proof. exact: real_floor_lt_int. Qed.
Lemma floor_eq x n : (floor x == n) = (n%:~R <= x < (n + 1)%:~R).
Proof. exact: real_floor_eq. Qed.
Lemma floorDzr : {in @int_num R, {morph floor : x y / x + y}}.
Proof. by move=> x xz y; apply/real_floorDzr/num_real. Qed.
Lemma floorDrz x y : y \is a int_num -> floor (x + y) = floor x + floor y.
Proof. by move=> yz; apply/real_floorDrz/yz/num_real. Qed.
Lemma floor_ge0 x : (0 <= floor x) = (0 <= x).
Proof. exact: real_floor_ge0. Qed.
Lemma floor_le0 x : (floor x <= 0) = (x < 1).
Proof. exact: real_floor_le0. Qed.
Lemma ceil_itv x : (ceil x - 1)%:~R < x <= (ceil x)%:~R.
Proof. exact: real_ceil_itv. Qed.
Lemma ceilB1_lt x : (ceil x - 1)%:~R < x.
Proof. exact: real_ceilB1_lt. Qed.
Lemma ceil_ge x : x <= (ceil x)%:~R. Proof. exact: real_ceil_ge. Qed.
#[deprecated(since="mathcomp 2.4.0", note="Use ceil_le_int_tmp instead.")]
Lemma ceil_le_int x n : x <= n%:~R = (ceil x <= n).
Proof. by rewrite real_ceil_le_int_tmp. Qed.
(* TODO: rename to ceil_le_int,
once the currently deprecated one has been removed *)
Lemma ceil_le_int_tmp x n : (ceil x <= n) = (x <= n%:~R).
Proof. exact: real_ceil_le_int_tmp. Qed.
Lemma ceil_gt_int x n : (n < ceil x) = (n%:~R < x).
Proof. exact: real_ceil_gt_int. Qed.
Lemma ceil_eq x n : (ceil x == n) = ((n - 1)%:~R < x <= n%:~R).
Proof. exact: real_ceil_eq. Qed.
Lemma ceilDzr : {in @int_num R, {morph ceil : x y / x + y}}.
Proof. by move=> x xz y; apply/real_ceilDzr/num_real. Qed.
Lemma ceilDrz x y : y \is a int_num -> ceil (x + y) = ceil x + ceil y.
Proof. by move=> yz; apply/real_ceilDrz/yz/num_real. Qed.
Lemma ceil_ge0 x : (0 <= ceil x) = (-1 < x).
Proof. exact: real_ceil_ge0. Qed.
Lemma ceil_le0 x : (ceil x <= 0) = (x <= 0).
Proof. exact: real_ceil_le0. Qed.
Lemma ceil_floor x : ceil x = floor x + (x \isn't a int_num).
Proof. exact: real_ceil_floor. Qed.
End ArchiRealDomainTheory.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to floor_le_tmp.")]
Notation ge_floor := floor_le_tmp.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to floorD1_gt.")]
Notation lt_succ_floor := floorD1_gt.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to ceilB1_lt.")]
Notation gt_pred_ceil := ceilB1_lt.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to ceil_ge.")]
Notation le_ceil := ceil_ge.
Section ArchiNumFieldTheory.
(* autLmodC *)
Variables (R : archiNumFieldType) (nu : {rmorphism R -> R}).
Lemma natr_aut x : (nu x \is a nat_num) = (x \is a nat_num).
Proof. by apply/idP/idP=> /[dup] ? /(aut_natr nu) => [/fmorph_inj <-| ->]. Qed.
Lemma intr_aut x : (nu x \is a int_num) = (x \is a int_num).
Proof. by rewrite !intrE -rmorphN !natr_aut. Qed.
End ArchiNumFieldTheory.
Section ArchiClosedFieldTheory.
Variable R : archiClosedFieldType.
Implicit Type x : R.
Lemma conj_natr x : x \is a nat_num -> x^* = x.
Proof. by move/Rreal_nat/CrealP. Qed.
Lemma conj_intr x : x \is a int_num -> x^* = x.
Proof. by move/Rreal_int/CrealP. Qed.
End ArchiClosedFieldTheory.
Section ZnatPred.
Lemma Znat_def (n : int) : (n \is a nat_num) = (0 <= n).
Proof. by []. Qed.
Lemma ZnatP (m : int) : reflect (exists n : nat, m = n) (m \is a nat_num).
Proof. by case: m => m; constructor; [exists m | case]. Qed.
End ZnatPred.
End Theory.
(* Factories *)
HB.factory Record NumDomain_hasTruncn R of Num.NumDomain R := {
trunc : R -> nat;
nat_num : pred R;
int_num : pred R;
truncP : forall x,
if 0 <= x then (trunc x)%:R <= x < (trunc x).+1%:R else trunc x == 0;
natrE : forall x, nat_num x = ((trunc x)%:R == x);
intrE : forall x, int_num x = nat_num x || nat_num (- x);
}.
#[deprecated(since="mathcomp 2.4.0",
note="Use NumDomain_hasTruncn instead.")]
Notation NumDomain_isArchimedean R := (NumDomain_hasTruncn R) (only parsing).
Module NumDomain_isArchimedean.
#[deprecated(since="mathcomp 2.4.0",
note="Use NumDomain_hasTruncn.Build instead.")]
Notation Build T U := (NumDomain_hasTruncn.Build T U) (only parsing).
End NumDomain_isArchimedean.
HB.builders Context R of NumDomain_hasTruncn R.
Fact trunc_itv x : 0 <= x -> (trunc x)%:R <= x < (trunc x).+1%:R.
Proof. by move=> x_ge0; move: (truncP x); rewrite x_ge0. Qed.
Definition floor (x : R) : int :=
if 0 <= x then Posz (trunc x)
else if x < 0 then - Posz (trunc (- x) + ~~ int_num x) else 0.
Fact floorP x :
if x \is Rreal then (floor x)%:~R <= x < (floor x + 1)%:~R else floor x == 0.
Proof.
rewrite /floor intrE !natrE negb_or realE.
case: (comparableP x 0) (@trunc_itv x) => //=;
try by rewrite -PoszD addn1 -pmulrn => _ ->.
move=> x_lt0 _; move: (truncP x); rewrite lt_geF // => /eqP ->.
rewrite gt_eqF //=; move: x_lt0.
rewrite [_ + 1]addrC -opprB !intrN lerNl ltrNr andbC -oppr_gt0.
move: {x}(- x) => x x_gt0; rewrite PoszD -addrA -PoszD.
have ->: Posz ((trunc x)%:R != x) - 1 = - Posz ((trunc x)%:R == x) by case: eqP.
have := trunc_itv (ltW x_gt0); rewrite le_eqVlt.
case: eqVneq => /=; last first.
by rewrite subr0 addn1 -!pmulrn => _ /andP[-> /ltW ->].
by rewrite intrB mulr1z addn0 -!pmulrn => -> _; rewrite gtrBl lexx andbT.
Qed.
Fact truncE x : trunc x = if floor x is Posz n then n else 0.
Proof.
rewrite /floor.
case: (comparableP x 0) (truncP x) => [+ /eqP ->| |_ /eqP ->|] //=.
by case: (_ + _)%N.
Qed.
Fact trunc_def x n : n%:R <= x < n.+1%:R -> trunc x = n.
Proof.
case/andP=> lemx ltxm1; apply/eqP; rewrite eqn_leq -ltnS -[(n <= _)%N]ltnS.
have/trunc_itv/andP[lefx ltxf1]: 0 <= x by apply: le_trans lemx; apply: ler0n.
by rewrite -!(ltr_nat R) 2?(@le_lt_trans _ _ x).
Qed.
Fact natrK : cancel (GRing.natmul 1) trunc.
Proof. by move=> m; apply: trunc_def; rewrite ler_nat ltr_nat ltnS leqnn. Qed.
Fact intrP x : reflect (exists n, x = n%:~R) (int_num x).
Proof.
rewrite intrE !natrE; apply: (iffP idP) => [|[n ->]]; last first.
by case: n => n; rewrite ?NegzE ?opprK natrK eqxx // orbT.
rewrite -eqr_oppLR !pmulrn -intrN.
by move=> /orP[] /eqP<-; [exists (trunc x) | exists (- Posz (trunc (- x)))].
Qed.
Fact natrP x : reflect (exists n, x = n%:R) (nat_num x).
Proof.
rewrite natrE.
by apply: (iffP eqP) => [<-|[n ->]]; [exists (trunc x) | rewrite natrK].
Qed.
HB.instance Definition _ :=
@NumDomain_hasFloorCeilTruncn.Build R floor _ trunc int_num nat_num
floorP (fun=> erefl) truncE intrP natrP.
HB.end.
HB.factory Record NumDomain_bounded_isArchimedean R of Num.NumDomain R := {
archi_bound_subproof : Num.archimedean_axiom R
}.
HB.builders Context R of NumDomain_bounded_isArchimedean R.
Implicit Type x : R.
Definition bound x := sval (sigW (archi_bound_subproof x)).
Lemma boundP x : 0 <= x -> x < (bound x)%:R.
Proof. by move/ger0_norm=> {1}<-; rewrite /bound; case: (sigW _). Qed.
Fact truncn_subproof x : {m | 0 <= x -> m%:R <= x < m.+1%:R }.
Proof.
have [Rx | _] := boolP (0 <= x); last by exists 0%N.
have/ex_minnP[n lt_x_n1 min_n]: exists n, x < n.+1%:R.
by exists (bound x); rewrite (lt_trans (boundP Rx)) ?ltr_nat.
exists n => _; rewrite {}lt_x_n1 andbT; case: n min_n => //= n min_n.
rewrite real_leNgt ?rpred_nat ?ger0_real //; apply/negP => /min_n.
by rewrite ltnn.
Qed.
Definition truncn x := if 0 <= x then sval (truncn_subproof x) else 0%N.
Lemma truncnP x :
if 0 <= x then (truncn x)%:R <= x < (truncn x).+1%:R else truncn x == 0%N.
Proof.
rewrite /truncn; case: truncn_subproof => // n hn.
by case: ifP => x_ge0; rewrite ?(ifT _ _ x_ge0) ?(ifF _ _ x_ge0) // hn.
Qed.
HB.instance Definition _ := NumDomain_hasTruncn.Build R
truncnP (fun => erefl) (fun => erefl).
HB.end.
Module Exports. HB.reexport. End Exports.
(* Not to pollute the local namespace, we define Num.nat and Num.int here. *)
Notation nat := nat_num.
Notation int := int_num.
#[deprecated(since="mathcomp 2.3.0", note="Use Num.ArchiRealDomain instead.")]
Notation ArchiDomain T := (ArchiRealDomain T).
Module ArchiDomain.
#[deprecated(since="mathcomp 2.3.0",
note="Use Num.ArchiRealDomain.type instead.")]
Notation type := ArchiRealDomain.type.
#[deprecated(since="mathcomp 2.3.0",
note="Use Num.ArchiRealDomain.copy instead.")]
Notation copy T C := (ArchiRealDomain.copy T C).
#[deprecated(since="mathcomp 2.3.0",
note="Use Num.ArchiRealDomain.on instead.")]
Notation on T := (ArchiRealDomain.on T).
End ArchiDomain.
#[deprecated(since="mathcomp 2.3.0", note="Use Num.ArchiRealField instead.")]
Notation ArchiField T := (ArchiRealField T).
Module ArchiField.
#[deprecated(since="mathcomp 2.3.0",
note="Use Num.ArchiRealField.type instead.")]
Notation type := ArchiRealField.type.
#[deprecated(since="mathcomp 2.3.0",
note="Use Num.ArchiRealField.copy instead.")]
Notation copy T C := (ArchiRealField.copy T C).
#[deprecated(since="mathcomp 2.3.0", note="Use Num.ArchiRealField.on instead.")]
Notation on T := (ArchiRealField.on T).
End ArchiField.
#[deprecated(since="mathcomp 2.3.0", note="Use real_floorDzr instead.")]
Notation floorD := real_floorDzr.
#[deprecated(since="mathcomp 2.3.0", note="Use real_ceilDzr instead.")]
Notation ceilD := real_ceilDzr.
#[deprecated(since="mathcomp 2.3.0", note="Use real_ceilDzr instead.")]
Notation real_ceilD := real_ceilDzr.
End Num.
Export Num.Exports.
#[deprecated(since="mathcomp 2.3.0", note="Use archiRealDomainType instead.")]
Notation archiDomainType := archiRealDomainType (only parsing).
#[deprecated(since="mathcomp 2.3.0", note="Use archiRealFieldType instead.")]
Notation archiFieldType := archiRealFieldType (only parsing).
|
ZMod.lean
|
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.ZMod
import Mathlib.Analysis.NormedSpace.Connected
import Mathlib.NumberTheory.LSeries.RiemannZeta
/-!
# L-series of functions on `ZMod N`
We show that if `N` is a positive integer and `Φ : ZMod N → ℂ`, then the L-series of `Φ` has
analytic continuation (away from a pole at `s = 1` if `∑ j, Φ j ≠ 0`) and satisfies a functional
equation. We also define completed L-functions (given by multiplying the naive L-function by a
Gamma-factor), and prove analytic continuation and functional equations for these too, assuming `Φ`
is either even or odd.
The most familiar case is when `Φ` is a Dirichlet character, but the results here are valid
for general functions; for the specific case of Dirichlet characters see
`Mathlib/NumberTheory/LSeries/DirichletContinuation.lean`.
## Main definitions
* `ZMod.LFunction Φ s`: the meromorphic continuation of the function `∑ n : ℕ, Φ n * n ^ (-s)`.
* `ZMod.completedLFunction Φ s`: the completed L-function, which for *almost* all `s` is equal to
`LFunction Φ s` multiplied by an Archimedean Gamma-factor.
Note that `ZMod.completedLFunction Φ s` is only mathematically well-defined if `Φ` is either even
or odd. Here we extend it to all functions `Φ` by linearity (but the functional equation only holds
if `Φ` is either even or odd).
## Main theorems
Results for non-completed L-functions:
* `ZMod.LFunction_eq_LSeries`: if `1 < re s` then the `LFunction` coincides with the naive
`LSeries`.
* `ZMod.differentiableAt_LFunction`: `ZMod.LFunction Φ` is differentiable at `s ∈ ℂ` if either
`s ≠ 1` or `∑ j, Φ j = 0`.
* `ZMod.LFunction_one_sub`: the functional equation relating `LFunction Φ (1 - s)` to
`LFunction (𝓕 Φ) s`, where `𝓕` is the Fourier transform.
Results for completed L-functions:
* `ZMod.LFunction_eq_completed_div_gammaFactor_even` and
`LFunction_eq_completed_div_gammaFactor_odd`: we have
`LFunction Φ s = completedLFunction Φ s / Gammaℝ s` for `Φ` even, and
`LFunction Φ s = completedLFunction Φ s / Gammaℝ (s + 1)` for `Φ` odd. (We formulate it this way
around so it is still valid at the poles of the Gamma factor.)
* `ZMod.differentiableAt_completedLFunction`: `ZMod.completedLFunction Φ` is differentiable at
`s ∈ ℂ`, unless `s = 1` and `∑ j, Φ j ≠ 0`, or `s = 0` and `Φ 0 ≠ 0`.
* `ZMod.completedLFunction_one_sub_even` and `ZMod.completedLFunction_one_sub_odd`:
the functional equation relating `completedLFunction Φ (1 - s)` to `completedLFunction (𝓕 Φ) s`.
-/
open HurwitzZeta Complex ZMod Finset Topology Filter Set
open scoped Real
namespace ZMod
variable {N : ℕ} [NeZero N]
/-- If `Φ` is a periodic function, then the L-series of `Φ` converges for `1 < re s`. -/
lemma LSeriesSummable_of_one_lt_re (Φ : ZMod N → ℂ) {s : ℂ} (hs : 1 < re s) :
LSeriesSummable (Φ ·) s := by
let c := max' _ <| univ_nonempty.image (norm ∘ Φ)
refine LSeriesSummable_of_bounded_of_one_lt_re (fun n _ ↦ le_max' _ _ ?_) (m := c) hs
exact mem_image_of_mem _ (mem_univ _)
/--
The unique meromorphic function `ℂ → ℂ` which agrees with `∑' n : ℕ, Φ n / n ^ s` wherever the
latter is convergent. This is constructed as a linear combination of Hurwitz zeta functions.
Note that this is not the same as `LSeries Φ`: they agree in the convergence range, but
`LSeries Φ s` is defined to be `0` if `re s ≤ 1`.
-/
noncomputable def LFunction (Φ : ZMod N → ℂ) (s : ℂ) : ℂ :=
N ^ (-s) * ∑ j : ZMod N, Φ j * hurwitzZeta (toAddCircle j) s
/-- The L-function of a function on `ZMod 1` is a scalar multiple of the Riemann zeta function. -/
lemma LFunction_modOne_eq (Φ : ZMod 1 → ℂ) (s : ℂ) :
LFunction Φ s = Φ 0 * riemannZeta s := by
simp only [LFunction, Nat.cast_one, one_cpow, ← singleton_eq_univ (0 : ZMod 1), sum_singleton,
map_zero, hurwitzZeta_zero, one_mul]
/-- For `1 < re s` the congruence L-function agrees with the sum of the Dirichlet series. -/
lemma LFunction_eq_LSeries (Φ : ZMod N → ℂ) {s : ℂ} (hs : 1 < re s) :
LFunction Φ s = LSeries (Φ ·) s := by
rw [LFunction, LSeries, mul_sum, Nat.sumByResidueClasses (LSeriesSummable_of_one_lt_re Φ hs) N]
congr 1 with j
have : (j.val / N : ℝ) ∈ Set.Icc 0 1 := mem_Icc.mpr ⟨by positivity,
(div_le_one (Nat.cast_pos.mpr <| NeZero.pos _)).mpr <| Nat.cast_le.mpr (val_lt j).le⟩
rw [toAddCircle_apply, ← (hasSum_hurwitzZeta_of_one_lt_re this hs).tsum_eq, ← mul_assoc,
← tsum_mul_left]
congr 1 with m
-- The following manipulation is slightly delicate because `(x * y) ^ s = x ^ s * y ^ s` is
-- false for general complex `x`, `y`, but it is true if `x` and `y` are non-negative reals, so
-- we have to carefully juggle coercions `ℕ → ℝ → ℂ`.
calc N ^ (-s) * Φ j * (1 / (m + (j.val / N : ℝ)) ^ s)
_ = Φ j * (N ^ (-s) * (1 / (m + (j.val / N : ℝ)) ^ s)) := by
rw [← mul_assoc, mul_comm _ (Φ _)]
_ = Φ j * (1 / (N : ℝ) ^ s * (1 / ((j.val + N * m) / N : ℝ) ^ s)) := by
simp only [cpow_neg, ← one_div, ofReal_div, ofReal_natCast, add_comm, add_div, ofReal_add,
ofReal_mul, mul_div_cancel_left₀ (m : ℂ) (Nat.cast_ne_zero.mpr (NeZero.ne N))]
_ = Φ j / ((N : ℝ) * ((j.val + N * m) / N : ℝ)) ^ s := by -- this is the delicate step!
rw [one_div_mul_one_div, mul_one_div, mul_cpow_ofReal_nonneg] <;> positivity
_ = Φ j / (N * (j.val + N * m) / N) ^ s := by
simp only [ofReal_natCast, ofReal_div, ofReal_add, ofReal_mul, mul_div_assoc]
_ = Φ j / (j.val + N * m) ^ s := by
rw [mul_div_cancel_left₀ _ (Nat.cast_ne_zero.mpr (NeZero.ne N))]
_ = Φ ↑(j.val + N * m) / (↑(j.val + N * m)) ^ s := by
simp only [Nat.cast_add, Nat.cast_mul, natCast_zmod_val, natCast_self, zero_mul, add_zero]
_ = LSeries.term (Φ ·) s (j.val + N * m) := by
rw [LSeries.term_of_ne_zero' (ne_zero_of_one_lt_re hs)]
lemma differentiableAt_LFunction (Φ : ZMod N → ℂ) (s : ℂ) (hs : s ≠ 1 ∨ ∑ j, Φ j = 0) :
DifferentiableAt ℂ (LFunction Φ) s := by
refine .mul (by fun_prop) ?_
rcases ne_or_eq s 1 with hs' | rfl
· exact .fun_sum fun j _ ↦ (differentiableAt_hurwitzZeta _ hs').const_mul _
· have := DifferentiableAt.fun_sum (u := univ) fun j _ ↦
(differentiableAt_hurwitzZeta_sub_one_div (toAddCircle j)).const_mul (Φ j)
simpa only [mul_sub, sum_sub_distrib, ← sum_mul, hs.neg_resolve_left rfl, zero_mul, sub_zero]
lemma differentiable_LFunction_of_sum_zero {Φ : ZMod N → ℂ} (hΦ : ∑ j, Φ j = 0) :
Differentiable ℂ (LFunction Φ) :=
fun s ↦ differentiableAt_LFunction Φ s (Or.inr hΦ)
/-- The L-function of `Φ` has a residue at `s = 1` equal to the average value of `Φ`. -/
lemma LFunction_residue_one (Φ : ZMod N → ℂ) :
Tendsto (fun s ↦ (s - 1) * LFunction Φ s) (𝓝[≠] 1) (𝓝 (∑ j, Φ j / N)) := by
simp only [LFunction, mul_sum]
refine tendsto_finset_sum _ fun j _ ↦ ?_
rw [(by ring : Φ j / N = Φ j * (1 / N * 1)), one_div, ← cpow_neg_one]
simp only [show ∀ a b c d : ℂ, a * (b * (c * d)) = c * (b * (a * d)) by intros; ring]
refine tendsto_const_nhds.mul (.mul ?_ <| hurwitzZeta_residue_one _)
exact ((continuous_neg.const_cpow (Or.inl <| NeZero.ne _)).tendsto _).mono_left
nhdsWithin_le_nhds
local notation "𝕖" => stdAddChar
/--
The `LFunction` of the function `x ↦ e (j * x)`, where `e : ZMod N → ℂ` is the standard additive
character, agrees with `expZeta (j / N)` on `1 < re s`. Private since it is a stepping-stone to
the more general result `LFunction_stdAddChar_eq_expZeta` below.
-/
private lemma LFunction_stdAddChar_eq_expZeta_of_one_lt_re (j : ZMod N) {s : ℂ} (hs : 1 < s.re) :
LFunction (fun k ↦ 𝕖 (j * k)) s = expZeta (ZMod.toAddCircle j) s := by
rw [toAddCircle_apply, ← (hasSum_expZeta_of_one_lt_re (j.val / N) hs).tsum_eq,
LFunction_eq_LSeries _ hs, LSeries]
congr 1 with n
rw [LSeries.term_of_ne_zero' (ne_zero_of_one_lt_re hs), ofReal_div, ofReal_natCast,
ofReal_natCast, mul_assoc, div_mul_eq_mul_div, stdAddChar_apply]
have := ZMod.toCircle_intCast (N := N) (j.val * n)
conv_rhs at this => rw [Int.cast_mul, Int.cast_natCast, Int.cast_natCast, mul_div_assoc]
rw [← this, Int.cast_mul, Int.cast_natCast, Int.cast_natCast, natCast_zmod_val]
/--
The `LFunction` of the function `x ↦ e (j * x)`, where `e : ZMod N → ℂ` is the standard additive
character, is `expZeta (j / N)`.
Note this is not at all obvious from the definitions, and we prove it by analytic continuation
from the convergence range.
-/
lemma LFunction_stdAddChar_eq_expZeta (j : ZMod N) (s : ℂ) (hjs : j ≠ 0 ∨ s ≠ 1) :
LFunction (fun k ↦ 𝕖 (j * k)) s = expZeta (ZMod.toAddCircle j) s := by
let U := if j = 0 then {z : ℂ | z ≠ 1} else univ -- region of analyticity of both functions
let V := {z : ℂ | 1 < re z} -- convergence region
have hUo : IsOpen U := by
by_cases h : j = 0
· simpa only [h, ↓reduceIte, U] using isOpen_compl_singleton
· simp only [h, ↓reduceIte, isOpen_univ, U]
let f := LFunction (fun k ↦ stdAddChar (j * k))
let g := expZeta (toAddCircle j)
have hU {u} : u ∈ U ↔ u ≠ 1 ∨ j ≠ 0 := by simp only [mem_ite_univ_right, U]; tauto
-- hypotheses for uniqueness of analytic continuation
have hf : AnalyticOnNhd ℂ f U := by
refine DifferentiableOn.analyticOnNhd (fun u hu ↦ ?_) hUo
refine (differentiableAt_LFunction _ _ ((hU.mp hu).imp_right fun h ↦ ?_)).differentiableWithinAt
simp only [mul_comm j, AddChar.sum_mulShift _ (isPrimitive_stdAddChar _), h,
↓reduceIte, CharP.cast_eq_zero]
have hg : AnalyticOnNhd ℂ g U := by
refine DifferentiableOn.analyticOnNhd (fun u hu ↦ ?_) hUo
refine (differentiableAt_expZeta _ _ ((hU.mp hu).imp_right fun h ↦ ?_)).differentiableWithinAt
rwa [ne_eq, toAddCircle_eq_zero]
have hUc : IsPreconnected U := by
by_cases h : j = 0
· simpa only [h, ↓reduceIte, U] using
(isConnected_compl_singleton_of_one_lt_rank (by simp) _).isPreconnected
· simpa only [h, ↓reduceIte, U] using isPreconnected_univ
have hV : V ∈ 𝓝 2 := (continuous_re.isOpen_preimage _ isOpen_Ioi).mem_nhds (by simp)
have hUmem : 2 ∈ U := by simp [U]
have hUmem' : s ∈ U := hU.mpr hjs.symm
-- apply uniqueness result
refine hf.eqOn_of_preconnected_of_eventuallyEq hg hUc hUmem ?_ hUmem'
-- now remains to prove equality on `1 < re s`
filter_upwards [hV] with z using LFunction_stdAddChar_eq_expZeta_of_one_lt_re _
/-- Explicit formula for the L-function of `𝓕 Φ`, where `𝓕` is the discrete Fourier transform. -/
lemma LFunction_dft (Φ : ZMod N → ℂ) {s : ℂ} (hs : Φ 0 = 0 ∨ s ≠ 1) :
LFunction (𝓕 Φ) s = ∑ j : ZMod N, Φ j * expZeta (toAddCircle (-j)) s := by
have (j : ZMod N) : Φ j * LFunction (fun k ↦ 𝕖 (-j * k)) s =
Φ j * expZeta (toAddCircle (-j)) s := by
by_cases h : -j ≠ 0 ∨ s ≠ 1
· rw [LFunction_stdAddChar_eq_expZeta _ _ h]
· simp only [neg_ne_zero, not_or, not_not] at h
rw [h.1, show Φ 0 = 0 by tauto, zero_mul, zero_mul]
simp only [LFunction, ← this, mul_sum]
rw [dft_def, sum_comm]
simp only [sum_mul, mul_sum, smul_eq_mul, stdAddChar_apply, ← mul_assoc]
congr 1 with j
congr 1 with k
rw [mul_assoc (Φ _), mul_comm (Φ _), neg_mul]
/-- Functional equation for `ZMod` L-functions, in terms of discrete Fourier transform. -/
theorem LFunction_one_sub (Φ : ZMod N → ℂ) {s : ℂ}
(hs : ∀ (n : ℕ), s ≠ -n) (hs' : Φ 0 = 0 ∨ s ≠ 1) :
LFunction Φ (1 - s) = N ^ (s - 1) * (2 * π) ^ (-s) * Gamma s *
(cexp (π * I * s / 2) * LFunction (𝓕 Φ) s
+ cexp (-π * I * s / 2) * LFunction (𝓕 fun x ↦ Φ (-x)) s) := by
rw [LFunction]
have (j : ZMod N) : Φ j * hurwitzZeta (toAddCircle j) (1 - s) = Φ j *
((2 * π) ^ (-s) * Gamma s * (cexp (-π * I * s / 2) *
expZeta (toAddCircle j) s + cexp (π * I * s / 2) * expZeta (-toAddCircle j) s)) := by
rcases eq_or_ne j 0 with rfl | hj
· rcases hs' with hΦ | hs'
· simp only [hΦ, zero_mul]
· rw [hurwitzZeta_one_sub _ hs (Or.inr hs')]
· rw [hurwitzZeta_one_sub _ hs (Or.inl <| toAddCircle_eq_zero.not.mpr hj)]
simp only [this, mul_assoc _ _ (Gamma s)]
-- get rid of Gamma terms and power of N
generalize (2 * π) ^ (-s) * Gamma s = C
simp_rw [← mul_assoc, mul_comm _ C, mul_assoc, ← mul_sum, ← mul_assoc, mul_comm _ C, mul_assoc,
neg_sub]
congr 2
-- now gather sum terms
rw [LFunction_dft _ hs', LFunction_dft _ (hs'.imp_left <| by simp only [neg_zero, imp_self])]
conv_rhs => enter [2, 2]; rw [← (Equiv.neg _).sum_comp _ _ (by simp), Equiv.neg_apply]
simp_rw [neg_neg, mul_sum, ← sum_add_distrib, ← mul_assoc, mul_comm _ (Φ _), mul_assoc,
← mul_add, map_neg, add_comm]
section signed
variable {Φ : ZMod N → ℂ}
lemma LFunction_def_even (hΦ : Φ.Even) (s : ℂ) :
LFunction Φ s = N ^ (-s) * ∑ j : ZMod N, Φ j * hurwitzZetaEven (toAddCircle j) s := by
simp only [LFunction, hurwitzZeta, mul_add (Φ _), sum_add_distrib]
congr 1
simp only [add_eq_left, ← neg_eq_self ℂ, ← sum_neg_distrib]
refine Fintype.sum_equiv (.neg _) _ _ fun i ↦ ?_
simp only [Equiv.neg_apply, hΦ i, map_neg, hurwitzZetaOdd_neg, mul_neg]
lemma LFunction_def_odd (hΦ : Φ.Odd) (s : ℂ) :
LFunction Φ s = N ^ (-s) * ∑ j : ZMod N, Φ j * hurwitzZetaOdd (toAddCircle j) s := by
simp only [LFunction, hurwitzZeta, mul_add (Φ _), sum_add_distrib]
congr 1
simp only [add_eq_right, ← neg_eq_self ℂ, ← sum_neg_distrib]
refine Fintype.sum_equiv (.neg _) _ _ fun i ↦ ?_
simp only [Equiv.neg_apply, hΦ i, map_neg, hurwitzZetaEven_neg, neg_mul]
/-- Explicit formula for `LFunction Φ 0` when `Φ` is even. -/
@[simp] lemma LFunction_apply_zero_of_even (hΦ : Φ.Even) :
LFunction Φ 0 = -Φ 0 / 2 := by
simp only [LFunction_def_even hΦ, neg_zero, cpow_zero, hurwitzZetaEven_apply_zero,
toAddCircle_eq_zero, mul_ite, mul_div, mul_neg_one, mul_zero, sum_ite_eq', Finset.mem_univ,
↓reduceIte, one_mul]
/-- The L-function of an even function vanishes at negative even integers. -/
@[simp] lemma LFunction_neg_two_mul_nat_add_one (hΦ : Φ.Even) (n : ℕ) :
LFunction Φ (-(2 * (n + 1))) = 0 := by
simp only [LFunction_def_even hΦ, hurwitzZetaEven_neg_two_mul_nat_add_one, mul_zero,
sum_const_zero, ← neg_mul]
/-- The L-function of an odd function vanishes at negative odd integers. -/
@[simp] lemma LFunction_neg_two_mul_nat_sub_one (hΦ : Φ.Odd) (n : ℕ) :
LFunction Φ (-(2 * n) - 1) = 0 := by
simp only [LFunction_def_odd hΦ, hurwitzZetaOdd_neg_two_mul_nat_sub_one, mul_zero, ← neg_mul,
sum_const_zero]
/--
The completed L-function of a function `Φ : ZMod N → ℂ`.
This is only mathematically meaningful if `Φ` is either even, or odd; here we extend this to all `Φ`
by linearity.
-/
noncomputable def completedLFunction (Φ : ZMod N → ℂ) (s : ℂ) : ℂ :=
N ^ (-s) * ∑ j, Φ j * completedHurwitzZetaEven (toAddCircle j) s
+ N ^ (-s) * ∑ j, Φ j * completedHurwitzZetaOdd (toAddCircle j) s
@[simp] lemma completedLFunction_zero (s : ℂ) : completedLFunction (0 : ZMod N → ℂ) s = 0 := by
simp only [completedLFunction, Pi.zero_apply, zero_mul, sum_const_zero, mul_zero, zero_add]
lemma completedLFunction_const_mul (a : ℂ) (Φ : ZMod N → ℂ) (s : ℂ) :
completedLFunction (fun j ↦ a * Φ j) s = a * completedLFunction Φ s := by
simp only [completedLFunction, mul_add, mul_sum]
congr with i <;> ring
lemma completedLFunction_def_even (hΦ : Φ.Even) (s : ℂ) :
completedLFunction Φ s = N ^ (-s) * ∑ j, Φ j * completedHurwitzZetaEven (toAddCircle j) s := by
suffices ∑ j, Φ j * completedHurwitzZetaOdd (toAddCircle j) s = 0 by
rw [completedLFunction, this, mul_zero, add_zero]
refine (hΦ.mul_odd fun j ↦ ?_).sum_eq_zero
rw [map_neg, completedHurwitzZetaOdd_neg]
lemma completedLFunction_def_odd (hΦ : Φ.Odd) (s : ℂ) :
completedLFunction Φ s = N ^ (-s) * ∑ j, Φ j * completedHurwitzZetaOdd (toAddCircle j) s := by
suffices ∑ j, Φ j * completedHurwitzZetaEven (toAddCircle j) s = 0 by
rw [completedLFunction, this, mul_zero, zero_add]
refine (hΦ.mul_even fun j ↦ ?_).sum_eq_zero
rw [map_neg, completedHurwitzZetaEven_neg]
/--
The completed L-function of a function `ZMod 1 → ℂ` is a scalar multiple of the completed Riemann
zeta function.
-/
lemma completedLFunction_modOne_eq (Φ : ZMod 1 → ℂ) (s : ℂ) :
completedLFunction Φ s = Φ 1 * completedRiemannZeta s := by
rw [completedLFunction_def_even (show Φ.Even from fun _ ↦ congr_arg Φ (Subsingleton.elim ..)),
Nat.cast_one, one_cpow, one_mul, ← singleton_eq_univ 0, sum_singleton, map_zero,
completedHurwitzZetaEven_zero, Subsingleton.elim 0 1]
/--
The completed L-function of a function `ZMod N → ℂ`, modified by adding multiples of `N ^ (-s) / s`
and `N ^ (-s) / (1 - s)` to make it entire.
-/
noncomputable def completedLFunction₀ (Φ : ZMod N → ℂ) (s : ℂ) : ℂ :=
N ^ (-s) * ∑ j : ZMod N, Φ j * completedHurwitzZetaEven₀ (toAddCircle j) s
+ N ^ (-s) * ∑ j : ZMod N, Φ j * completedHurwitzZetaOdd (toAddCircle j) s
/-- The function `completedLFunction₀ Φ` is differentiable. -/
lemma differentiable_completedLFunction₀ (Φ : ZMod N → ℂ) :
Differentiable ℂ (completedLFunction₀ Φ) := by
refine .add ?_ ?_ <;>
refine .mul (by fun_prop) (.fun_sum fun i _ ↦ .const_mul ?_ _)
exacts [differentiable_completedHurwitzZetaEven₀ _, differentiable_completedHurwitzZetaOdd _]
lemma completedLFunction_eq (Φ : ZMod N → ℂ) (s : ℂ) :
completedLFunction Φ s =
completedLFunction₀ Φ s - N ^ (-s) * Φ 0 / s - N ^ (-s) * (∑ j, Φ j) / (1 - s) := by
simp only [completedLFunction, completedHurwitzZetaEven_eq, toAddCircle_eq_zero, div_eq_mul_inv,
ite_mul, one_mul, zero_mul, mul_sub, mul_ite, mul_zero, sum_sub_distrib, Fintype.sum_ite_eq',
← sum_mul, completedLFunction₀, mul_assoc]
abel
/--
The completed L-function of a function `ZMod N → ℂ` is differentiable, with the following
exceptions: at `s = 1` if `∑ j, Φ j ≠ 0`; and at `s = 0` if `Φ 0 ≠ 0`.
-/
lemma differentiableAt_completedLFunction (Φ : ZMod N → ℂ) (s : ℂ) (hs₀ : s ≠ 0 ∨ Φ 0 = 0)
(hs₁ : s ≠ 1 ∨ ∑ j, Φ j = 0) : DifferentiableAt ℂ (completedLFunction Φ) s := by
simp only [funext (completedLFunction_eq Φ), mul_div_assoc]
-- We know `completedLFunction₀` is differentiable everywhere, so it suffices to show that the
-- correction terms from `completedLFunction_eq` are differentiable at `s`.
refine ((differentiable_completedLFunction₀ _ _).sub ?_).sub ?_
· -- term with `1 / s`
refine .mul (by fun_prop) (hs₀.elim ?_ ?_)
· exact fun h ↦ (differentiableAt_const _).div differentiableAt_id h
· exact fun h ↦ by simp only [h, funext zero_div, differentiableAt_const]
· -- term with `1 / (1 - s)`
refine .mul (by fun_prop) (hs₁.elim ?_ ?_)
· exact fun h ↦ .div (by fun_prop) (by fun_prop) (by rwa [sub_ne_zero, ne_comm])
· exact fun h ↦ by simp only [h, zero_div, differentiableAt_const]
/--
Special case of `differentiableAt_completedLFunction` asserting differentiability everywhere
under suitable hypotheses.
-/
lemma differentiable_completedLFunction (hΦ₂ : Φ 0 = 0) (hΦ₃ : ∑ j, Φ j = 0) :
Differentiable ℂ (completedLFunction Φ) :=
fun s ↦ differentiableAt_completedLFunction Φ s (.inr hΦ₂) (.inr hΦ₃)
/--
Relation between the completed L-function and the usual one (even case).
We state it this way around so it holds at the poles of the gamma factor as well
(except at `s = 0`, where it is genuinely false if `N > 1` and `Φ 0 ≠ 0`).
-/
lemma LFunction_eq_completed_div_gammaFactor_even (hΦ : Φ.Even) (s : ℂ) (hs : s ≠ 0 ∨ Φ 0 = 0) :
LFunction Φ s = completedLFunction Φ s / Gammaℝ s := by
simp only [completedLFunction_def_even hΦ, LFunction_def_even hΦ, mul_div_assoc, sum_div]
congr 2 with i
rcases ne_or_eq i 0 with hi | rfl
· rw [hurwitzZetaEven_def_of_ne_or_ne (.inl (hi ∘ toAddCircle_eq_zero.mp))]
· rcases hs with hs | hΦ'
· rw [hurwitzZetaEven_def_of_ne_or_ne (.inr hs)]
· simp only [hΦ', map_zero, zero_mul]
/--
Relation between the completed L-function and the usual one (odd case).
We state it this way around so it holds at the poles of the gamma factor as well.
-/
lemma LFunction_eq_completed_div_gammaFactor_odd (hΦ : Φ.Odd) (s : ℂ) :
LFunction Φ s = completedLFunction Φ s / Gammaℝ (s + 1) := by
simp only [LFunction_def_odd hΦ, completedLFunction_def_odd hΦ, hurwitzZetaOdd, mul_div_assoc,
sum_div]
/--
First form of functional equation for completed L-functions (even case).
Private because it is superseded by `completedLFunction_one_sub_even` below, which is valid for a
much wider range of `s`.
-/
private lemma completedLFunction_one_sub_of_one_lt_even (hΦ : Φ.Even) {s : ℂ} (hs : 1 < re s) :
completedLFunction Φ (1 - s) = N ^ (s - 1) * completedLFunction (𝓕 Φ) s := by
have hs₀ : s ≠ 0 := ne_zero_of_one_lt_re hs
have hs₁ : s ≠ 1 := (lt_irrefl _ <| one_re ▸ · ▸ hs)
-- strip down to the key equality:
suffices ∑ x, Φ x * completedCosZeta (toAddCircle x) s = completedLFunction (𝓕 Φ) s by
simp only [completedLFunction_def_even hΦ, neg_sub, completedHurwitzZetaEven_one_sub, this]
-- reduce to equality with un-completed L-functions:
suffices ∑ x, Φ x * cosZeta (toAddCircle x) s = LFunction (𝓕 Φ) s by
simpa only [cosZeta, Function.update_of_ne hs₀, ← mul_div_assoc, ← sum_div,
LFunction_eq_completed_div_gammaFactor_even (dft_even_iff.mpr hΦ) _ (.inl hs₀),
div_left_inj' (Gammaℝ_ne_zero_of_re_pos (zero_lt_one.trans hs))]
-- expand out `LFunction (𝓕 Φ)` and use parity:
simp only [cosZeta_eq, ← mul_div_assoc _ _ (2 : ℂ), mul_add, ← sum_div, sum_add_distrib,
LFunction_dft Φ (.inr hs₁), map_neg, div_eq_iff (two_ne_zero' ℂ), mul_two, add_left_inj]
exact Fintype.sum_equiv (.neg _) _ _ (by simp [hΦ _])
/--
First form of functional equation for completed L-functions (odd case).
Private because it is superseded by `completedLFunction_one_sub_odd` below, which is valid for a
much wider range of `s`.
-/
private lemma completedLFunction_one_sub_of_one_lt_odd (hΦ : Φ.Odd) {s : ℂ} (hs : 1 < re s) :
completedLFunction Φ (1 - s) = N ^ (s - 1) * I * completedLFunction (𝓕 Φ) s := by
-- strip down to the key equality:
suffices ∑ x, Φ x * completedSinZeta (toAddCircle x) s = I * completedLFunction (𝓕 Φ) s by
simp only [completedLFunction_def_odd hΦ, neg_sub, completedHurwitzZetaOdd_one_sub, this,
mul_assoc]
-- reduce to equality with un-completed L-functions:
suffices ∑ x, Φ x * sinZeta (toAddCircle x) s = I * LFunction (𝓕 Φ) s by
have hs' : 0 < re (s + 1) := by simp only [add_re, one_re]; linarith
simpa only [sinZeta, ← mul_div_assoc, ← sum_div, div_left_inj' (Gammaℝ_ne_zero_of_re_pos hs'),
LFunction_eq_completed_div_gammaFactor_odd (dft_odd_iff.mpr hΦ)]
-- now calculate:
calc ∑ x, Φ x * sinZeta (toAddCircle x) s
_ = (∑ x, Φ x * expZeta (toAddCircle x) s) / (2 * I)
- (∑ x, Φ x * expZeta (toAddCircle (-x)) s) / (2 * I) := by
simp only [sinZeta_eq, ← mul_div_assoc, mul_sub, sub_div, sum_sub_distrib, sum_div, map_neg]
_ = (∑ x, Φ (-x) * expZeta (toAddCircle (-x)) s) / (_) - (_) := by
congrm ?_ / _ - _
exact (Fintype.sum_equiv (.neg _) _ _ fun x ↦ by rfl).symm
_ = -I⁻¹ * LFunction (𝓕 Φ) s := by
simp only [hΦ _, neg_mul, sum_neg_distrib, LFunction_dft Φ (.inl hΦ.map_zero)]
ring
_ = I * LFunction (𝓕 Φ) s := by rw [inv_I, neg_neg]
/--
Functional equation for completed L-functions (even case), valid at all points of differentiability.
-/
theorem completedLFunction_one_sub_even (hΦ : Φ.Even) (s : ℂ)
(hs₀ : s ≠ 0 ∨ ∑ j, Φ j = 0) (hs₁ : s ≠ 1 ∨ Φ 0 = 0) :
completedLFunction Φ (1 - s) = N ^ (s - 1) * completedLFunction (𝓕 Φ) s := by
-- We prove this using `AnalyticOnNhd.eqOn_of_preconnected_of_eventuallyEq`, so we need to
-- gather up the ingredients for this big theorem.
-- First set up some notations:
let F (t) := completedLFunction Φ (1 - t)
let G (t) := ↑N ^ (t - 1) * completedLFunction (𝓕 Φ) t
-- Set on which F, G are analytic:
let U := {t : ℂ | (t ≠ 0 ∨ ∑ j, Φ j = 0) ∧ (t ≠ 1 ∨ Φ 0 = 0)}
-- Properties of U:
have hsU : s ∈ U := ⟨hs₀, hs₁⟩
have h2U : 2 ∈ U := ⟨.inl two_ne_zero, .inl (OfNat.ofNat_ne_one _)⟩
have hUo : IsOpen U := (isOpen_compl_singleton.union isOpen_const).inter
(isOpen_compl_singleton.union isOpen_const)
have hUp : IsPreconnected U := by
-- need to write `U` as the complement of an obviously countable set
let Uc : Set ℂ := (if ∑ j, Φ j = 0 then ∅ else {0}) ∪ (if Φ 0 = 0 then ∅ else {1})
have : Uc.Countable := by
apply Countable.union <;>
split_ifs <;>
simp only [countable_singleton, countable_empty]
convert (this.isConnected_compl_of_one_lt_rank ?_).isPreconnected using 1
· ext x
by_cases h : Φ 0 = 0 <;>
by_cases h' : ∑ j, Φ j = 0 <;>
simp [U, Uc, h, h', and_comm]
· simp only [rank_real_complex, Nat.one_lt_ofNat]
-- Analyticity on U:
have hF : AnalyticOnNhd ℂ F U := by
refine DifferentiableOn.analyticOnNhd
(fun t ht ↦ DifferentiableAt.differentiableWithinAt ?_) hUo
refine (differentiableAt_completedLFunction Φ _ ?_ ?_).comp t (differentiableAt_id.const_sub 1)
exacts [ht.2.imp_left (sub_ne_zero.mpr ∘ Ne.symm), ht.1.imp_left sub_eq_self.not.mpr]
have hG : AnalyticOnNhd ℂ G U := by
refine DifferentiableOn.analyticOnNhd
(fun t ht ↦ DifferentiableAt.differentiableWithinAt ?_) hUo
apply ((differentiableAt_id.sub_const 1).const_cpow (.inl (NeZero.ne _))).mul
apply differentiableAt_completedLFunction _ _ (ht.1.imp_right fun h ↦ dft_apply_zero Φ ▸ h)
exact ht.2.imp_right (fun h ↦ by simp only [← dft_apply_zero, dft_dft, neg_zero, h, smul_zero])
-- set where we know equality
have hV : {z | 1 < re z} ∈ 𝓝 2 := (continuous_re.isOpen_preimage _ isOpen_Ioi).mem_nhds (by simp)
have hFG : F =ᶠ[𝓝 2] G := eventually_of_mem hV <| fun t ht ↦ by
simpa only [F, G, pow_zero, mul_one] using completedLFunction_one_sub_of_one_lt_even hΦ ht
-- now apply the big hammer to finish
exact hF.eqOn_of_preconnected_of_eventuallyEq hG hUp h2U hFG hsU
/-- Functional equation for completed L-functions (odd case), valid for all `s`. -/
theorem completedLFunction_one_sub_odd (hΦ : Φ.Odd) (s : ℂ) :
completedLFunction Φ (1 - s) = N ^ (s - 1) * I * completedLFunction (𝓕 Φ) s := by
-- This is much easier than the even case since both functions are entire.
-- First set up some notations:
let F (t) := completedLFunction Φ (1 - t)
let G (t) := ↑N ^ (t - 1) * I * completedLFunction (𝓕 Φ) t
-- check F, G globally differentiable
have hF : Differentiable ℂ F := (differentiable_completedLFunction hΦ.map_zero
hΦ.sum_eq_zero).comp (differentiable_id.const_sub 1)
have hG : Differentiable ℂ G := by
apply (((differentiable_id.sub_const 1).const_cpow (.inl (NeZero.ne _))).mul_const _).mul
rw [← dft_odd_iff] at hΦ
exact differentiable_completedLFunction hΦ.map_zero hΦ.sum_eq_zero
-- set where we know equality
have : {z | 1 < re z} ∈ 𝓝 2 := (continuous_re.isOpen_preimage _ isOpen_Ioi).mem_nhds (by simp)
have hFG : F =ᶠ[𝓝 2] G := by filter_upwards [this] with t ht
using completedLFunction_one_sub_of_one_lt_odd hΦ ht
-- now apply the big hammer to finish
rw [← analyticOnNhd_univ_iff_differentiable] at hF hG
exact congr_fun (hF.eq_of_eventuallyEq hG hFG) s
end signed
end ZMod
|
MulAction.lean
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.AddTorsor.Defs
import Mathlib.GroupTheory.GroupAction.SubMulAction
import Mathlib.Topology.Algebra.Constructions
import Mathlib.Topology.Algebra.ConstMulAction
import Mathlib.Topology.Connected.Basic
/-!
# Continuous monoid action
In this file we define class `ContinuousSMul`. We say `ContinuousSMul M X` if `M` acts on `X` and
the map `(c, x) ↦ c • x` is continuous on `M × X`. We reuse this class for topological
(semi)modules, vector spaces and algebras.
## Main definitions
* `ContinuousSMul M X` : typeclass saying that the map `(c, x) ↦ c • x` is continuous
on `M × X`;
* `Units.continuousSMul`: scalar multiplication by `Mˣ` is continuous when scalar
multiplication by `M` is continuous. This allows `Homeomorph.smul` to be used with on monoids
with `G = Mˣ`.
## Main results
Besides homeomorphisms mentioned above, in this file we provide lemmas like `Continuous.smul`
or `Filter.Tendsto.smul` that provide dot-syntax access to `ContinuousSMul`.
-/
open Topology Pointwise
open Filter
/-- Class `ContinuousSMul M X` says that the scalar multiplication `(•) : M → X → X`
is continuous in both arguments. We use the same class for all kinds of multiplicative actions,
including (semi)modules and algebras. -/
class ContinuousSMul (M X : Type*) [SMul M X] [TopologicalSpace M] [TopologicalSpace X] :
Prop where
/-- The scalar multiplication `(•)` is continuous. -/
continuous_smul : Continuous fun p : M × X => p.1 • p.2
export ContinuousSMul (continuous_smul)
/-- Class `ContinuousVAdd M X` says that the additive action `(+ᵥ) : M → X → X`
is continuous in both arguments. We use the same class for all kinds of additive actions,
including (semi)modules and algebras. -/
class ContinuousVAdd (M X : Type*) [VAdd M X] [TopologicalSpace M] [TopologicalSpace X] :
Prop where
/-- The additive action `(+ᵥ)` is continuous. -/
continuous_vadd : Continuous fun p : M × X => p.1 +ᵥ p.2
export ContinuousVAdd (continuous_vadd)
attribute [to_additive] ContinuousSMul
attribute [continuity, fun_prop] continuous_smul continuous_vadd
section Main
variable {M X Y α : Type*} [TopologicalSpace M] [TopologicalSpace X] [TopologicalSpace Y]
section SMul
variable [SMul M X] [ContinuousSMul M X]
lemma IsScalarTower.continuousSMul {M : Type*} (N : Type*) {α : Type*} [Monoid N] [SMul M N]
[MulAction N α] [SMul M α] [IsScalarTower M N α] [TopologicalSpace M] [TopologicalSpace N]
[TopologicalSpace α] [ContinuousSMul M N] [ContinuousSMul N α] : ContinuousSMul M α :=
{ continuous_smul := by
suffices Continuous (fun p : M × α ↦ (p.1 • (1 : N)) • p.2) by simpa
fun_prop }
@[to_additive]
instance : ContinuousSMul (ULift M) X :=
⟨(continuous_smul (M := M)).comp₂ (continuous_uliftDown.comp continuous_fst) continuous_snd⟩
@[to_additive]
instance (priority := 100) ContinuousSMul.continuousConstSMul : ContinuousConstSMul M X where
continuous_const_smul _ := continuous_smul.comp (continuous_const.prodMk continuous_id)
theorem ContinuousSMul.induced {R : Type*} {α : Type*} {β : Type*} {F : Type*} [FunLike F α β]
[Semiring R] [AddCommMonoid α] [AddCommMonoid β] [Module R α] [Module R β]
[TopologicalSpace R] [LinearMapClass F R α β] [tβ : TopologicalSpace β] [ContinuousSMul R β]
(f : F) : @ContinuousSMul R α _ _ (tβ.induced f) := by
let tα := tβ.induced f
refine ⟨continuous_induced_rng.2 ?_⟩
simp only [Function.comp_def, map_smul]
fun_prop
@[to_additive]
theorem Filter.Tendsto.smul {f : α → M} {g : α → X} {l : Filter α} {c : M} {a : X}
(hf : Tendsto f l (𝓝 c)) (hg : Tendsto g l (𝓝 a)) :
Tendsto (fun x => f x • g x) l (𝓝 <| c • a) :=
(continuous_smul.tendsto _).comp (hf.prodMk_nhds hg)
@[to_additive]
theorem Filter.Tendsto.smul_const {f : α → M} {l : Filter α} {c : M} (hf : Tendsto f l (𝓝 c))
(a : X) : Tendsto (fun x => f x • a) l (𝓝 (c • a)) :=
hf.smul tendsto_const_nhds
variable {f : Y → M} {g : Y → X} {b : Y} {s : Set Y}
@[to_additive]
theorem ContinuousWithinAt.smul (hf : ContinuousWithinAt f s b) (hg : ContinuousWithinAt g s b) :
ContinuousWithinAt (fun x => f x • g x) s b :=
Filter.Tendsto.smul hf hg
@[to_additive (attr := fun_prop)]
theorem ContinuousAt.smul (hf : ContinuousAt f b) (hg : ContinuousAt g b) :
ContinuousAt (fun x => f x • g x) b :=
Filter.Tendsto.smul hf hg
@[to_additive (attr := fun_prop)]
theorem ContinuousOn.smul (hf : ContinuousOn f s) (hg : ContinuousOn g s) :
ContinuousOn (fun x => f x • g x) s := fun x hx => (hf x hx).smul (hg x hx)
@[to_additive (attr := continuity, fun_prop)]
theorem Continuous.smul (hf : Continuous f) (hg : Continuous g) : Continuous fun x => f x • g x :=
continuous_smul.comp (hf.prodMk hg)
/-- If a scalar action is central, then its right action is continuous when its left action is. -/
@[to_additive /-- If an additive action is central, then its right action is continuous when its
left action is. -/]
instance ContinuousSMul.op [SMul Mᵐᵒᵖ X] [IsCentralScalar M X] : ContinuousSMul Mᵐᵒᵖ X :=
⟨by
suffices Continuous fun p : M × X => MulOpposite.op p.fst • p.snd from
this.comp (MulOpposite.continuous_unop.prodMap continuous_id)
simpa only [op_smul_eq_smul] using (continuous_smul : Continuous fun p : M × X => _)⟩
@[to_additive]
instance MulOpposite.continuousSMul : ContinuousSMul M Xᵐᵒᵖ :=
⟨MulOpposite.continuous_op.comp <|
continuous_smul.comp <| continuous_id.prodMap MulOpposite.continuous_unop⟩
@[to_additive]
protected theorem Specializes.smul {a b : M} {x y : X} (h₁ : a ⤳ b) (h₂ : x ⤳ y) :
(a • x) ⤳ (b • y) :=
(h₁.prod h₂).map continuous_smul
@[to_additive]
protected theorem Inseparable.smul {a b : M} {x y : X} (h₁ : Inseparable a b)
(h₂ : Inseparable x y) : Inseparable (a • x) (b • y) :=
(h₁.prod h₂).map continuous_smul
@[to_additive]
lemma IsCompact.smul_set {k : Set M} {u : Set X} (hk : IsCompact k) (hu : IsCompact u) :
IsCompact (k • u) := by
rw [← Set.image_smul_prod]
exact IsCompact.image (hk.prod hu) continuous_smul
@[to_additive]
lemma smul_set_closure_subset (K : Set M) (L : Set X) :
closure K • closure L ⊆ closure (K • L) :=
Set.smul_subset_iff.2 fun _x hx _y hy ↦ map_mem_closure₂ continuous_smul hx hy fun _a ha _b hb ↦
Set.smul_mem_smul ha hb
/-- Suppose that `N` acts on `X` and `M` continuously acts on `Y`.
Suppose that `g : Y → X` is an action homomorphism in the following sense:
there exists a continuous function `f : N → M` such that `g (c • x) = f c • g x`.
Then the action of `N` on `X` is continuous as well.
In many cases, `f = id` so that `g` is an action homomorphism in the sense of `MulActionHom`.
However, this version also works for semilinear maps and `f = Units.val`. -/
@[to_additive
/-- Suppose that `N` additively acts on `X` and `M` continuously additively acts on `Y`.
Suppose that `g : Y → X` is an additive action homomorphism in the following sense:
there exists a continuous function `f : N → M` such that `g (c +ᵥ x) = f c +ᵥ g x`.
Then the action of `N` on `X` is continuous as well.
In many cases, `f = id` so that `g` is an action homomorphism in the sense of `AddActionHom`.
However, this version also works for `f = AddUnits.val`. -/]
lemma Topology.IsInducing.continuousSMul {N : Type*} [SMul N Y] [TopologicalSpace N] {f : N → M}
(hg : IsInducing g) (hf : Continuous f) (hsmul : ∀ {c x}, g (c • x) = f c • g x) :
ContinuousSMul N Y where
continuous_smul := by
simpa only [hg.continuous_iff, Function.comp_def, hsmul]
using (hf.comp continuous_fst).smul <| hg.continuous.comp continuous_snd
@[to_additive]
instance SMulMemClass.continuousSMul {S : Type*} [SetLike S X] [SMulMemClass S M X] (s : S) :
ContinuousSMul M s :=
IsInducing.subtypeVal.continuousSMul continuous_id rfl
end SMul
section Monoid
variable [Monoid M] [MulAction M X] [ContinuousSMul M X]
@[to_additive]
instance Units.continuousSMul : ContinuousSMul Mˣ X :=
IsInducing.id.continuousSMul Units.continuous_val rfl
/-- If an action is continuous, then composing this action with a continuous homomorphism gives
again a continuous action. -/
@[to_additive]
theorem MulAction.continuousSMul_compHom
{N : Type*} [TopologicalSpace N] [Monoid N] {f : N →* M} (hf : Continuous f) :
letI : MulAction N X := MulAction.compHom _ f
ContinuousSMul N X := by
let _ : MulAction N X := MulAction.compHom _ f
exact ⟨(hf.comp continuous_fst).smul continuous_snd⟩
@[to_additive]
instance Submonoid.continuousSMul {S : Submonoid M} : ContinuousSMul S X :=
IsInducing.id.continuousSMul continuous_subtype_val rfl
end Monoid
section Group
variable [Group M] [MulAction M X] [ContinuousSMul M X]
@[to_additive]
instance Subgroup.continuousSMul {S : Subgroup M} : ContinuousSMul S X :=
S.toSubmonoid.continuousSMul
variable (M)
/-- The stabilizer of a continuous group action on a discrete space is an open subgroup. -/
lemma stabilizer_isOpen [DiscreteTopology X] (x : X) : IsOpen (MulAction.stabilizer M x : Set M) :=
IsOpen.preimage (f := fun g ↦ g • x) (by fun_prop) (isOpen_discrete {x})
end Group
@[to_additive]
instance Prod.continuousSMul [SMul M X] [SMul M Y] [ContinuousSMul M X] [ContinuousSMul M Y] :
ContinuousSMul M (X × Y) :=
⟨(continuous_fst.smul (continuous_fst.comp continuous_snd)).prodMk
(continuous_fst.smul (continuous_snd.comp continuous_snd))⟩
@[to_additive]
instance {ι : Type*} {γ : ι → Type*} [∀ i, TopologicalSpace (γ i)] [∀ i, SMul M (γ i)]
[∀ i, ContinuousSMul M (γ i)] : ContinuousSMul M (∀ i, γ i) :=
⟨continuous_pi fun i =>
(continuous_fst.smul continuous_snd).comp <|
continuous_fst.prodMk ((continuous_apply i).comp continuous_snd)⟩
end Main
section LatticeOps
variable {ι : Sort*} {M X : Type*} [TopologicalSpace M] [SMul M X]
@[to_additive]
theorem continuousSMul_sInf {ts : Set (TopologicalSpace X)}
(h : ∀ t ∈ ts, @ContinuousSMul M X _ _ t) : @ContinuousSMul M X _ _ (sInf ts) :=
-- Porting note: {} doesn't work because `sInf ts` isn't found by TC search. `(_)` finds it by
-- unification instead.
@ContinuousSMul.mk M X _ _ (_) <| by
-- Porting note: needs `( :)`
rw [← (@sInf_singleton _ _ ‹TopologicalSpace M›:)]
exact
continuous_sInf_rng.2 fun t ht =>
continuous_sInf_dom₂ (Eq.refl _) ht
(@ContinuousSMul.continuous_smul _ _ _ _ t (h t ht))
@[to_additive]
theorem continuousSMul_iInf {ts' : ι → TopologicalSpace X}
(h : ∀ i, @ContinuousSMul M X _ _ (ts' i)) : @ContinuousSMul M X _ _ (⨅ i, ts' i) :=
continuousSMul_sInf <| Set.forall_mem_range.mpr h
@[to_additive]
theorem continuousSMul_inf {t₁ t₂ : TopologicalSpace X} [@ContinuousSMul M X _ _ t₁]
[@ContinuousSMul M X _ _ t₂] : @ContinuousSMul M X _ _ (t₁ ⊓ t₂) := by
rw [inf_eq_iInf]
refine continuousSMul_iInf fun b => ?_
cases b <;> assumption
end LatticeOps
section AddTorsor
variable (G : Type*) (P : Type*) [AddGroup G] [AddTorsor G P] [TopologicalSpace G]
variable [PreconnectedSpace G] [TopologicalSpace P] [ContinuousVAdd G P]
include G in
/-- An `AddTorsor` for a connected space is a connected space. This is not an instance because
it loops for a group as a torsor over itself. -/
protected theorem AddTorsor.connectedSpace : ConnectedSpace P :=
{ isPreconnected_univ := by
convert
isPreconnected_univ.image (Equiv.vaddConst (Classical.arbitrary P) : G → P)
(continuous_id.vadd continuous_const).continuousOn
rw [Set.image_univ, Equiv.range_eq_univ]
toNonempty := inferInstance }
end AddTorsor
|
delaborators.lean
|
import Mathlib.Util.Delaborators
import Mathlib.Data.Set.Lattice
section PiNotation
variable (P : Nat → Prop) (α : Nat → Type) (s : Set ℕ)
/-- info: ∀ x > 0, P x : Prop -/
#guard_msgs in
#check ∀ x, x > 0 → P x
/-- info: ∀ x > 0, P x : Prop -/
#guard_msgs in
#check ∀ x > 0, P x
/-- info: ∀ x ≥ 0, P x : Prop -/
#guard_msgs in
#check ∀ x, x ≥ 0 → P x
/-- info: ∀ x ≥ 0, P x : Prop -/
#guard_msgs in
#check ∀ x ≥ 0, P x
/-- info: ∀ x < 0, P x : Prop -/
#guard_msgs in
#check ∀ x < 0, P x
/-- info: ∀ x < 0, P x : Prop -/
#guard_msgs in
#check ∀ x, x < 0 → P x
/-- info: ∀ x ≤ 0, P x : Prop -/
#guard_msgs in
#check ∀ x ≤ 0, P x
/-- info: ∀ x ≤ 0, P x : Prop -/
#guard_msgs in
#check ∀ x, x ≤ 0 → P x
/-- info: ∀ x ∈ s, P x : Prop -/
#guard_msgs in
#check ∀ x ∈ s, P x
/-- info: ∀ x ∈ s, P x : Prop -/
#guard_msgs in
#check ∀ x, x ∈ s → P x
/-- info: ∀ x ∉ s, P x : Prop -/
#guard_msgs in
#check ∀ x ∉ s,P x
/-- info: ∀ x ∉ s, P x : Prop -/
#guard_msgs in
#check ∀ x, x ∉ s → P x
variable (Q : Set ℕ → Prop)
/-- info: ∀ t ⊆ s, Q t : Prop -/
#guard_msgs in
#check ∀ t ⊆ s, Q t
/-- info: ∀ t ⊆ s, Q t : Prop -/
#guard_msgs in
#check ∀ t, t ⊆ s → Q t
/-- info: ∀ t ⊂ s, Q t : Prop -/
#guard_msgs in
#check ∀ t ⊂ s, Q t
/-- info: ∀ t ⊂ s, Q t : Prop -/
#guard_msgs in
#check ∀ t, t ⊂ s → Q t
/-- info: ∀ t ⊇ s, Q t : Prop -/
#guard_msgs in
#check ∀ t ⊇ s, Q t
/-- info: ∀ t ⊇ s, Q t : Prop -/
#guard_msgs in
#check ∀ t, t ⊇ s → Q t
/-- info: ∀ t ⊃ s, Q t : Prop -/
#guard_msgs in
#check ∀ t ⊃ s, Q t
/-- info: ∀ t ⊃ s, Q t : Prop -/
#guard_msgs in
#check ∀ t, t ⊃ s → Q t
/-- info: (x : ℕ) → α x : Type -/
#guard_msgs in
#check (x : Nat) → α x
/-!
Disabling binder predicates
-/
/-- info: ∀ x > 0, P x : Prop -/
#guard_msgs in
#check ∀ x > 0, P x
/-- info: ∀ (x : ℕ), x > 0 → P x : Prop -/
#guard_msgs in
set_option pp.mathlib.binderPredicates false in
#check ∀ x > 0, P x
/-- info: ∃ x > 0, P x : Prop -/
#guard_msgs in
#check ∃ x > 0, P x
/-- info: ∃ x, x > 0 ∧ P x : Prop -/
#guard_msgs in
set_option pp.mathlib.binderPredicates false in
#check ∃ x > 0, P x
/-!
Opening the `PiNotation` namespace enables `Π` notation.
-/
open PiNotation
/-- info: Π (x : ℕ), α x : Type -/
#guard_msgs in
#check (x : Nat) → α x
/-- info: ∀ (x : ℕ), P x : Prop -/
#guard_msgs in
#check (x : Nat) → P x
/-!
Note that the implementation of the `Π` delaborator in `Mathlib/Util/Delaborators.lean`
does not (yet?) make use of binder predicates.
-/
/-- info: Π (x : ℕ), x > 0 → α x : Type -/
#guard_msgs in
#check Π x > 0, α x
/-- info: Π (x : ℕ), x > 0 → α x : Type -/
#guard_msgs in
set_option pp.mathlib.binderPredicates false in
#check Π x > 0, α x
end PiNotation
section UnionInter
variable (s : ℕ → Set ℕ) (u : Set ℕ) (p : ℕ → Prop)
/-- info: ⋃ n ∈ u, s n : Set ℕ -/
#guard_msgs in
#check ⋃ n ∈ u, s n
/-- info: ⋂ n ∈ u, s n : Set ℕ -/
#guard_msgs in
#check ⋂ n ∈ u, s n
end UnionInter
section CompleteLattice
/-- info: ⨆ i ∈ Set.univ, (i = i) : Prop -/
#guard_msgs in
#check ⨆ (i : Nat) (_ : i ∈ Set.univ), (i = i)
/-- info: ⨅ i ∈ Set.univ, (i = i) : Prop -/
#guard_msgs in
#check ⨅ (i : Nat) (_ : i ∈ Set.univ), (i = i)
end CompleteLattice
section existential
/-- info: ∃ i ≥ 3, i = i : Prop -/
#guard_msgs in
#check ∃ (i : Nat), i ≥ 3 ∧ i = i
/-- info: ∃ i > 3, i = i : Prop -/
#guard_msgs in
#check ∃ (i : Nat), i > 3 ∧ i = i
/-- info: ∃ i ≤ 3, i = i : Prop -/
#guard_msgs in
#check ∃ (i : Nat), i ≤ 3 ∧ i = i
/-- info: ∃ i < 3, i = i : Prop -/
#guard_msgs in
#check ∃ (i : Nat), i < 3 ∧ i = i
variable (s : Set ℕ) (P : ℕ → Prop) (Q : Set ℕ → Prop)
/-- info: ∃ x ∉ s, P x : Prop -/
#guard_msgs in
#check ∃ x ∉ s, P x
/-- info: ∃ x ∉ s, P x : Prop -/
#guard_msgs in
#check ∃ x, x ∉ s ∧ P x
variable (Q : Set ℕ → Prop)
/-- info: ∃ t ⊆ s, Q t : Prop -/
#guard_msgs in
#check ∃ t ⊆ s, Q t
/-- info: ∃ t ⊆ s, Q t : Prop -/
#guard_msgs in
#check ∃ t, t ⊆ s ∧ Q t
/-- info: ∃ t ⊂ s, Q t : Prop -/
#guard_msgs in
#check ∃ t ⊂ s, Q t
/-- info: ∃ t ⊂ s, Q t : Prop -/
#guard_msgs in
#check ∃ t, t ⊂ s ∧ Q t
/-- info: ∃ t ⊇ s, Q t : Prop -/
#guard_msgs in
#check ∃ t ⊇ s, Q t
/-- info: ∃ t ⊇ s, Q t : Prop -/
#guard_msgs in
#check ∃ t, t ⊇ s ∧ Q t
/-- info: ∃ t ⊃ s, Q t : Prop -/
#guard_msgs in
#check ∃ t ⊃ s, Q t
/-- info: ∃ t ⊃ s, Q t : Prop -/
#guard_msgs in
#check ∃ t, t ⊃ s ∧ Q t
/-- info: ∃ n k, n = k : Prop -/
#guard_msgs in
#check ∃ n k, n = k
/-- info: ∃ n k, n = k : Prop -/
#guard_msgs in
#check ∃ n, ∃ k, n = k
section merging
/-- info: ∃ (_ : True), True : Prop -/
#guard_msgs in #check ∃ (_ : True), True
/-- info: ∃ (_ : True), ∃ x, x = x : Prop -/
#guard_msgs in #check ∃ (_ : True) (x : Nat), x = x
/-- info: ∃ (_ : True), ∃ x y, x = y : Prop -/
#guard_msgs in #check ∃ (_ : True) (x y : Nat), x = y
/-- info: ∃ (_ : True) (_ : False), True : Prop -/
#guard_msgs in #check ∃ (_ : True) (_ : False), True
set_option pp.funBinderTypes true in
/-- info: ∃ (x : ℕ) (x : ℕ), True : Prop -/
#guard_msgs in #check ∃ (_ : Nat) (_ : Nat), True
end merging
end existential
section prod
variable (x : ℕ × ℕ)
/-- info: x.1 : ℕ -/
#guard_msgs in
#check x.1
variable (p : (ℕ → ℕ) × (ℕ → ℕ))
/-- info: p.1 22 : ℕ -/
#guard_msgs in
#check p.1 22
set_option pp.numericProj.prod false in
/-- info: x.fst : ℕ -/
#guard_msgs in
#check x.1
set_option pp.numericProj.prod false in
/-- info: x.snd : ℕ -/
#guard_msgs in
#check x.2
set_option pp.explicit true in
/-- info: @Prod.fst Nat Nat x : Nat -/
#guard_msgs in
#check x.1
set_option pp.explicit true in
/-- info: @Prod.snd Nat Nat x : Nat -/
#guard_msgs in
#check x.2
set_option pp.fieldNotation false in
/-- info: Prod.fst x : ℕ -/
#guard_msgs in
#check x.1
set_option pp.fieldNotation false in
/-- info: Prod.snd x : ℕ -/
#guard_msgs in
#check x.2
end prod
|
Mod_.lean
|
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Paul Lezeau, Robin Carlier
-/
import Mathlib.CategoryTheory.Monoidal.Mon_
import Mathlib.CategoryTheory.Monoidal.Action.Basic
/-!
# The category of module objects over a monoid object.
-/
universe v₁ v₂ u₁ u₂
open CategoryTheory MonoidalCategory Mon_Class
variable {C : Type u₁} [Category.{v₁} C] [MonoidalCategory.{v₁} C]
{D : Type u₂} [Category.{v₂} D] [MonoidalLeftAction C D]
section Mod_Class
open Mon_Class
variable (M : C) [Mon_Class M]
open scoped MonoidalLeftAction
/-- Given an action of a monoidal category `C` on a category `D`,
an action of a monoid object `M` in `C` on an object `X` in `D` is the data of a
map `smul : M ⊙ₗ X ⟶ X` that satisfies unitality and associativity with
multiplication.
See `MulAction` for the non-categorical version. -/
class Mod_Class (X : D) where
/-- The action map -/
smul : M ⊙ₗ X ⟶ X
/-- The identity acts trivially. -/
one_smul' (X) : η ⊵ₗ X ≫ smul = (λₗ X).hom := by cat_disch
/-- The action map is compatible with multiplication. -/
mul_smul' (X) : μ ⊵ₗ X ≫ smul = (αₗ M M X).hom ≫ M ⊴ₗ smul ≫ smul := by cat_disch
attribute [reassoc] Mod_Class.mul_smul' Mod_Class.one_smul'
@[inherit_doc] scoped[Mon_Class] notation "γ" => Mod_Class.smul
@[inherit_doc] scoped[Mon_Class] notation "γ["Y"]" => Mod_Class.smul (X := Y)
@[inherit_doc] scoped[Mon_Class] notation "γ["N","Y"]" =>
Mod_Class.smul (M := N) (X := Y)
variable {M}
namespace Mod_Class
@[reassoc (attr := simp)]
theorem one_smul (X : D) [Mod_Class M X] :
η ⊵ₗ X ≫ γ[M,X] = (λₗ[C] X).hom :=
Mod_Class.one_smul' X
@[reassoc (attr := simp)]
theorem mul_smul (X : D) [Mod_Class M X] :
μ ⊵ₗ X ≫ γ = (αₗ M M X).hom ≫ M ⊴ₗ γ ≫ γ := Mod_Class.mul_smul' X
theorem assoc_flip (X : D) [Mod_Class M X] : M ⊴ₗ γ ≫ γ =
(αₗ M M X).inv ≫ μ[M] ⊵ₗ X ≫ γ := by
simp
variable (M) in
/-- The action of a monoid object on itself. -/
-- See note [reducible non instances]
abbrev regular : Mod_Class M M where
smul := μ
attribute [local instance] regular in
@[simp] lemma smul_eq_mul (M : C) [Mon_Class M] : γ[M, M] = μ[M] := rfl
/-- If `C` acts monoidally on `D`, then every object of `D` is canonically a
module over the trivial monoid. -/
@[simps]
instance (X : D) : Mod_Class (𝟙_ C) X where
smul := (λₗ _).hom
@[ext]
theorem ext {X : C} (h₁ h₂ : Mod_Class M X) (H : h₁.smul = h₂.smul) :
h₁ = h₂ := by
cases h₁
cases h₂
subst H
rfl
end Mod_Class
end Mod_Class
open scoped Mod_Class MonoidalLeftAction
variable (A : C) [Mon_Class A]
/-- A morphism in `D` is a morphism of `A`-module objects if it commutes with
the action maps -/
class IsMod_Hom {M N : D} [Mod_Class A M] [Mod_Class A N] (f : M ⟶ N) where
smul_hom : γ[M] ≫ f = A ⊴ₗ f ≫ γ[N] := by cat_disch
attribute [reassoc (attr := simp)] IsMod_Hom.smul_hom
variable {M N O : D} [Mod_Class A M] [Mod_Class A N] [Mod_Class A O]
instance : IsMod_Hom A (𝟙 M) where
instance (f : M ⟶ N) (g : N ⟶ O) [IsMod_Hom A f] [IsMod_Hom A g] :
IsMod_Hom A (f ≫ g) where
instance (f : M ≅ N) [IsMod_Hom A f.hom] :
IsMod_Hom A f.inv where
smul_hom := by simp [Iso.comp_inv_eq]
variable (D) in
/-- A module object for a monoid object in a monoidal category acting on the
ambient category. -/
structure Mod_ (A : C) [Mon_Class A] where
/-- The underlying object in the ambient category -/
X : D
[mod : Mod_Class A X]
attribute [instance] Mod_.mod
namespace Mod_
variable {A : C} [Mon_Class A] (M : Mod_ D A)
theorem assoc_flip : A ⊴ₗ γ ≫ γ = (αₗ A A M.X).inv ≫ μ ⊵ₗ M.X ≫ γ := by simp
/-- A morphism of module objects. -/
@[ext]
structure Hom (M N : Mod_ D A) where
/-- The underlying morphism -/
hom : M.X ⟶ N.X
[isMod_Hom : IsMod_Hom A hom]
attribute [instance] Hom.isMod_Hom
/-- An alternative constructor for `Hom`,
taking a morphism without a [isMod_Hom] instance, as well as the relevant
equality to put such an instance. -/
@[simps!]
def Hom.mk' {M N : Mod_ D A} (f : M.X ⟶ N.X)
(smul_hom : γ[M.X] ≫ f = A ⊴ₗ f ≫ γ[N.X] := by cat_disch) : Hom M N :=
letI : IsMod_Hom A f := ⟨smul_hom⟩
⟨f⟩
/-- An alternative constructor for `Hom`,
taking a morphism without a [isMod_Hom] instance, between objects with
a `Mod_Class` instance (rather than bundled as `Mod_`),
as well as the relevant equality to put such an instance. -/
@[simps!]
def Hom.mk'' {M N : D} [Mod_Class A M] [Mod_Class A N] (f : M ⟶ N)
(smul_hom : γ[M] ≫ f = A ⊴ₗ f ≫ γ[N] := by cat_disch) :
Hom (.mk (A := A) M) (.mk (A := A) N) :=
letI : IsMod_Hom A f := ⟨smul_hom⟩
⟨f⟩
/-- The identity morphism on a module object. -/
@[simps]
def id (M : Mod_ D A) : Hom M M where hom := 𝟙 M.X
instance homInhabited (M : Mod_ D A) : Inhabited (Hom M M) :=
⟨id M⟩
/-- Composition of module object morphisms. -/
@[simps]
def comp {M N O : Mod_ D A} (f : Hom M N) (g : Hom N O) :
Hom M O where
hom := f.hom ≫ g.hom
instance : Category (Mod_ D A) where
Hom M N := Hom M N
id := id
comp f g := comp f g
@[ext]
lemma hom_ext {M N : Mod_ D A} (f₁ f₂ : M ⟶ N) (h : f₁.hom = f₂.hom) :
f₁ = f₂ :=
Hom.ext h
@[simp]
theorem id_hom' (M : Mod_ D A) : (𝟙 M : M ⟶ M).hom = 𝟙 M.X := by
rfl
@[simp]
theorem comp_hom' {M N K : Mod_ D A} (f : M ⟶ N) (g : N ⟶ K) :
(f ≫ g).hom = f.hom ≫ g.hom :=
rfl
variable (A)
/-- A monoid object as a module over itself. -/
@[simps]
def regular : Mod_ C A :=
letI : Mod_Class A A := .regular A
⟨A⟩
instance : Inhabited (Mod_ C A) :=
⟨regular A⟩
/-- The forgetful functor from module objects to the ambient category. -/
@[simps]
def forget : Mod_ D A ⥤ D where
obj A := A.X
map f := f.hom
section comap
variable {A B : C} [Mon_Class A] [Mon_Class B] (f : A ⟶ B) [IsMon_Hom f]
open MonoidalLeftAction in
/-- When `M` is a `B`-module in `D` and `f : A ⟶ B` is a morphism of internal
monoid objects, `M` inherits an `A`-module structure via
"restriction of scalars", i.e `γ[A, M] = f.hom ⊵ₗ M ≫ γ[B, M]`. -/
@[simps!]
def scalarRestriction (M : D) [Mod_Class B M] : Mod_Class A M where
smul := f ⊵ₗ M ≫ γ[B, M]
one_smul' := by
rw [← comp_actionHomLeft_assoc]
rw [IsMon_Hom.one_hom, Mod_Class.one_smul]
mul_smul' := by
-- oh, for homotopy.io in a widget!
slice_rhs 2 3 => rw [action_exchange]
simp only [actionHomLeft_action_assoc, Category.assoc, Iso.hom_inv_id_assoc,
actionHomRight_comp]
slice_rhs 4 6 => rw [Mod_Class.assoc_flip]
slice_rhs 2 4 => rw [← whiskerLeft_actionHomLeft]
slice_rhs 1 2 => rw [← comp_actionHomLeft]
rw [← comp_actionHomLeft, Category.assoc, ← comp_actionHomLeft_assoc,
IsMon_Hom.mul_hom, tensorHom_def, Category.assoc]
open MonoidalLeftAction in
/-- If `g : M ⟶ N` is a `B`-linear morphisms of `B`-modules, then it induces an
`A`-linear morphism when `M` and `N` have an `A`-module structure obtained
by restricting scalars along a monoid morphism `A ⟶ B`. -/
lemma scalarRestriction_hom
(M N : D) [Mod_Class B M] [Mod_Class B N] (g : M ⟶ N) [IsMod_Hom B g] :
letI := scalarRestriction f M
letI := scalarRestriction f N
IsMod_Hom A g :=
letI := scalarRestriction f M
letI := scalarRestriction f N
{ smul_hom := by
dsimp
slice_rhs 1 2 => rw [action_exchange]
slice_rhs 2 3 => rw [← IsMod_Hom.smul_hom]
rw [Category.assoc] }
/-- A morphism of monoid objects induces a "restriction" or "comap" functor
between the categories of module objects.
-/
@[simps]
def comap {A B : C} [Mon_Class A] [Mon_Class B] (f : A ⟶ B) [IsMon_Hom f] :
Mod_ D B ⥤ Mod_ D A where
obj M :=
letI := scalarRestriction f M.X
⟨M.X⟩
map {M N} g :=
letI := scalarRestriction_hom f M.X N.X g.hom
⟨g.hom⟩
-- Lots more could be said about `comap`, e.g. how it interacts with
-- identities, compositions, and equalities of monoid object morphisms.
end comap
end Mod_
|
tuple.v
|
(* (c) Copyright 2006-2016 Microsoft Corporation and Inria. *)
(* Distributed under the terms of CeCILL-B. *)
From HB Require Import structures.
From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat.
From mathcomp Require Import seq choice fintype path.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
(******************************************************************************)
(* This file defines tuples, i.e., sequences with a fixed (known) length, *)
(* and sequences with bounded length. *)
(* For tuples we define: *)
(* n.-tuple T == the type of n-tuples of elements of type T *)
(* [tuple of s] == the tuple whose underlying sequence (value) is s *)
(* The size of s must be known: specifically, Coq must *)
(* be able to infer a Canonical tuple projecting on s. *)
(* in_tuple s == the (size s).-tuple with value s *)
(* [tuple] == the empty tuple *)
(* [tuple x1; ..; xn] == the explicit n.-tuple <x1; ..; xn> *)
(* [tuple E | i < n] == the n.-tuple with general term E (i : 'I_n is bound *)
(* in E) *)
(* tcast Emn t == the m.-tuple t cast as an n.-tuple using Emn : m = n *)
(* As n.-tuple T coerces to seq t, all seq operations (size, nth, ...) can be *)
(* applied to t : n.-tuple T; we provide a few specialized instances when *)
(* avoids the need for a default value. *)
(* tsize t == the size of t (the n in n.-tuple T) *)
(* tnth t i == the i'th component of t, where i : 'I_n *)
(* [tnth t i] == the i'th component of t, where i : nat and i < n *)
(* is convertible to true *)
(* thead t == the first element of t, when n is m.+1 for some m *)
(* For bounded sequences we define: *)
(* n.-bseq T == the type of bounded sequences of elements of type T, *)
(* the length of a bounded sequence is smaller or *)
(* or equal to n *)
(* [bseq of s] == the bounded sequence whose underlying value is s *)
(* The size of s must be known. *)
(* in_bseq s == the (size s).-bseq with value s *)
(* [bseq] == the empty bseq *)
(* insub_bseq n s == the n.-bseq of value s if size s <= n, else [bseq] *)
(* [bseq x1; ..; xn] == the explicit n.-bseq <x1; ..; xn> *)
(* cast_bseq Emn t == the m.-bseq t cast as an n.-tuple using Emn : m = n *)
(* widen_bseq Lmn t == the m.-bseq t cast as an n.-tuple using Lmn : m <= n *)
(* Most seq constructors (cons, behead, cat, rcons, belast, take, drop, rot, *)
(* rotr, map, ...) can be used to build tuples and bounded sequences via *)
(* the [tuple of s] and [bseq of s] constructs respectively. *)
(* Tuples and bounded sequences are actually instances of subType of seq, *)
(* and inherit all combinatorial structures, including the finType structure. *)
(* Some useful lemmas and definitions: *)
(* tuple0 : [tuple] is the only 0.-tuple *)
(* bseq0 : [bseq] is the only 0.-bseq *)
(* tupleP : elimination view for n.+1.-tuple *)
(* ord_tuple n : the n.-tuple of all i : 'I_n *)
(******************************************************************************)
Section TupleDef.
Variables (n : nat) (T : Type).
Structure tuple_of : Type := Tuple {tval :> seq T; _ : size tval == n}.
HB.instance Definition _ := [isSub for tval].
Implicit Type t : tuple_of.
Definition tsize of tuple_of := n.
Lemma size_tuple t : size t = n.
Proof. exact: (eqP (valP t)). Qed.
Lemma tnth_default t : 'I_n -> T.
Proof. by rewrite -(size_tuple t); case: (tval t) => [|//] []. Qed.
Definition tnth t i := nth (tnth_default t i) t i.
Lemma tnth_nth x t i : tnth t i = nth x t i.
Proof. by apply: set_nth_default; rewrite size_tuple. Qed.
Lemma tnth_onth x t i : tnth t i = x <-> onth t i = Some x.
Proof.
rewrite (tnth_nth x) onthE (nth_map x) ?size_tuple//.
by split; [move->|case].
Qed.
Lemma map_tnth_enum t : map (tnth t) (enum 'I_n) = t.
Proof.
case def_t: {-}(val t) => [|x0 t'].
by rewrite [enum _]size0nil // -cardE card_ord -(size_tuple t) def_t.
apply: (@eq_from_nth _ x0) => [|i]; rewrite size_map.
by rewrite -cardE size_tuple card_ord.
move=> lt_i_e; have lt_i_n: i < n by rewrite -cardE card_ord in lt_i_e.
by rewrite (nth_map (Ordinal lt_i_n)) // (tnth_nth x0) nth_enum_ord.
Qed.
Lemma eq_from_tnth t1 t2 : tnth t1 =1 tnth t2 -> t1 = t2.
Proof.
by move/eq_map=> eq_t; apply: val_inj; rewrite /= -!map_tnth_enum eq_t.
Qed.
Definition tuple t mkT : tuple_of :=
mkT (let: Tuple _ tP := t return size t == n in tP).
Lemma tupleE t : tuple (fun sP => @Tuple t sP) = t.
Proof. by case: t. Qed.
End TupleDef.
Notation "n .-tuple" := (tuple_of n) (format "n .-tuple") : type_scope.
Notation "{ 'tuple' n 'of' T }" := (n.-tuple T : predArgType)
(only parsing) : type_scope.
Notation "[ 'tuple' 'of' s ]" := (tuple (fun sP => @Tuple _ _ s sP))
(format "[ 'tuple' 'of' s ]") : form_scope.
Notation "[ 'tnth' t i ]" := (tnth t (@Ordinal (tsize t) i (erefl true)))
(t, i at level 8, format "[ 'tnth' t i ]") : form_scope.
Canonical nil_tuple T := Tuple (isT : @size T [::] == 0).
Canonical cons_tuple n T x (t : n.-tuple T) :=
Tuple (valP t : size (x :: t) == n.+1).
Notation "[ 'tuple' x1 ; .. ; xn ]" := [tuple of x1 :: .. [:: xn] ..]
(format "[ 'tuple' '[' x1 ; '/' .. ; '/' xn ']' ]") : form_scope.
Notation "[ 'tuple' ]" := [tuple of [::]] (format "[ 'tuple' ]") : form_scope.
Section CastTuple.
Variable T : Type.
Definition in_tuple (s : seq T) := Tuple (eqxx (size s)).
Definition tcast m n (eq_mn : m = n) t :=
let: erefl in _ = n := eq_mn return n.-tuple T in t.
Lemma tcastE m n (eq_mn : m = n) t i :
tnth (tcast eq_mn t) i = tnth t (cast_ord (esym eq_mn) i).
Proof. by case: n / eq_mn in i *; rewrite cast_ord_id. Qed.
Lemma tcast_id n (eq_nn : n = n) t : tcast eq_nn t = t.
Proof. by rewrite (eq_axiomK eq_nn). Qed.
Lemma tcastK m n (eq_mn : m = n) : cancel (tcast eq_mn) (tcast (esym eq_mn)).
Proof. by case: n / eq_mn. Qed.
Lemma tcastKV m n (eq_mn : m = n) : cancel (tcast (esym eq_mn)) (tcast eq_mn).
Proof. by case: n / eq_mn. Qed.
Lemma tcast_trans m n p (eq_mn : m = n) (eq_np : n = p) t:
tcast (etrans eq_mn eq_np) t = tcast eq_np (tcast eq_mn t).
Proof. by case: n / eq_mn eq_np; case: p /. Qed.
Lemma tvalK n (t : n.-tuple T) : in_tuple t = tcast (esym (size_tuple t)) t.
Proof. by apply: val_inj => /=; case: _ / (esym _). Qed.
Lemma val_tcast m n (eq_mn : m = n) (t : m.-tuple T) :
tcast eq_mn t = t :> seq T.
Proof. by case: n / eq_mn. Qed.
Lemma in_tupleE s : in_tuple s = s :> seq T. Proof. by []. Qed.
End CastTuple.
Section SeqTuple.
Variables (n m : nat) (T U rT : Type).
Implicit Type t : n.-tuple T.
Lemma rcons_tupleP t x : size (rcons t x) == n.+1.
Proof. by rewrite size_rcons size_tuple. Qed.
Canonical rcons_tuple t x := Tuple (rcons_tupleP t x).
Lemma nseq_tupleP x : @size T (nseq n x) == n.
Proof. by rewrite size_nseq. Qed.
Canonical nseq_tuple x := Tuple (nseq_tupleP x).
Lemma iota_tupleP : size (iota m n) == n.
Proof. by rewrite size_iota. Qed.
Canonical iota_tuple := Tuple iota_tupleP.
Lemma behead_tupleP t : size (behead t) == n.-1.
Proof. by rewrite size_behead size_tuple. Qed.
Canonical behead_tuple t := Tuple (behead_tupleP t).
Lemma belast_tupleP x t : size (belast x t) == n.
Proof. by rewrite size_belast size_tuple. Qed.
Canonical belast_tuple x t := Tuple (belast_tupleP x t).
Lemma cat_tupleP t (u : m.-tuple T) : size (t ++ u) == n + m.
Proof. by rewrite size_cat !size_tuple. Qed.
Canonical cat_tuple t u := Tuple (cat_tupleP t u).
Lemma take_tupleP t : size (take m t) == minn m n.
Proof. by rewrite size_take size_tuple eqxx. Qed.
Canonical take_tuple t := Tuple (take_tupleP t).
Lemma drop_tupleP t : size (drop m t) == n - m.
Proof. by rewrite size_drop size_tuple. Qed.
Canonical drop_tuple t := Tuple (drop_tupleP t).
Lemma rev_tupleP t : size (rev t) == n.
Proof. by rewrite size_rev size_tuple. Qed.
Canonical rev_tuple t := Tuple (rev_tupleP t).
Lemma rot_tupleP t : size (rot m t) == n.
Proof. by rewrite size_rot size_tuple. Qed.
Canonical rot_tuple t := Tuple (rot_tupleP t).
Lemma rotr_tupleP t : size (rotr m t) == n.
Proof. by rewrite size_rotr size_tuple. Qed.
Canonical rotr_tuple t := Tuple (rotr_tupleP t).
Lemma map_tupleP f t : @size rT (map f t) == n.
Proof. by rewrite size_map size_tuple. Qed.
Canonical map_tuple f t := Tuple (map_tupleP f t).
Lemma scanl_tupleP f x t : @size rT (scanl f x t) == n.
Proof. by rewrite size_scanl size_tuple. Qed.
Canonical scanl_tuple f x t := Tuple (scanl_tupleP f x t).
Lemma pairmap_tupleP f x t : @size rT (pairmap f x t) == n.
Proof. by rewrite size_pairmap size_tuple. Qed.
Canonical pairmap_tuple f x t := Tuple (pairmap_tupleP f x t).
Lemma zip_tupleP t (u : n.-tuple U) : size (zip t u) == n.
Proof. by rewrite size1_zip !size_tuple. Qed.
Canonical zip_tuple t u := Tuple (zip_tupleP t u).
Lemma allpairs_tupleP f t (u : m.-tuple U) : @size rT (allpairs f t u) == n * m.
Proof. by rewrite size_allpairs !size_tuple. Qed.
Canonical allpairs_tuple f t u := Tuple (allpairs_tupleP f t u).
Lemma sort_tupleP r t : size (sort r t) == n.
Proof. by rewrite size_sort size_tuple. Qed.
Canonical sort_tuple r t := Tuple (sort_tupleP r t).
Definition thead (u : n.+1.-tuple T) := tnth u ord0.
Lemma tnth0 x t : tnth [tuple of x :: t] ord0 = x.
Proof. by []. Qed.
Lemma tnthS x t i : tnth [tuple of x :: t] (lift ord0 i) = tnth t i.
Proof. by rewrite (tnth_nth (tnth_default t i)). Qed.
Lemma theadE x t : thead [tuple of x :: t] = x.
Proof. by []. Qed.
Lemma tuple0 : all_equal_to ([tuple] : 0.-tuple T).
Proof. by move=> t; apply: val_inj; case: t => [[]]. Qed.
Variant tuple1_spec : n.+1.-tuple T -> Type :=
Tuple1spec x t : tuple1_spec [tuple of x :: t].
Lemma tupleP u : tuple1_spec u.
Proof.
case: u => [[|x s] //= sz_s]; pose t := @Tuple n _ s sz_s.
by rewrite (_ : Tuple _ = [tuple of x :: t]) //; apply: val_inj.
Qed.
Lemma tnth_map f t i : tnth [tuple of map f t] i = f (tnth t i) :> rT.
Proof. by apply: nth_map; rewrite size_tuple. Qed.
Lemma tnth_nseq x i : tnth [tuple of nseq n x] i = x.
Proof.
by rewrite !(tnth_nth (tnth_default (nseq_tuple x) i)) nth_nseq ltn_ord.
Qed.
End SeqTuple.
Lemma tnth_behead n T (t : n.+1.-tuple T) i :
tnth [tuple of behead t] i = tnth t (inord i.+1).
Proof. by case/tupleP: t => x t; rewrite !(tnth_nth x) inordK ?ltnS. Qed.
Lemma tuple_eta n T (t : n.+1.-tuple T) : t = [tuple of thead t :: behead t].
Proof. by case/tupleP: t => x t; apply: val_inj. Qed.
Section tnth_shift.
Context {T : Type} {n1 n2} (t1 : n1.-tuple T) (t2 : n2.-tuple T).
Lemma tnth_lshift i : tnth [tuple of t1 ++ t2] (lshift n2 i) = tnth t1 i.
Proof.
have x0 := tnth_default t1 i; rewrite !(tnth_nth x0).
by rewrite nth_cat size_tuple /= ltn_ord.
Qed.
Lemma tnth_rshift j : tnth [tuple of t1 ++ t2] (rshift n1 j) = tnth t2 j.
Proof.
have x0 := tnth_default t2 j; rewrite !(tnth_nth x0).
by rewrite nth_cat size_tuple ltnNge leq_addr /= addKn.
Qed.
End tnth_shift.
Section TupleQuantifiers.
Variables (n : nat) (T : Type).
Implicit Types (a : pred T) (t : n.-tuple T).
Lemma forallb_tnth a t : [forall i, a (tnth t i)] = all a t.
Proof.
apply: negb_inj; rewrite -has_predC -has_map negb_forall.
apply/existsP/(has_nthP true) => [[i a_t_i] | [i lt_i_n a_t_i]].
by exists i; rewrite ?size_tuple // -tnth_nth tnth_map.
rewrite size_tuple in lt_i_n; exists (Ordinal lt_i_n).
by rewrite -tnth_map (tnth_nth true).
Qed.
Lemma existsb_tnth a t : [exists i, a (tnth t i)] = has a t.
Proof. by apply: negb_inj; rewrite negb_exists -all_predC -forallb_tnth. Qed.
Lemma all_tnthP a t : reflect (forall i, a (tnth t i)) (all a t).
Proof. by rewrite -forallb_tnth; apply: forallP. Qed.
Lemma has_tnthP a t : reflect (exists i, a (tnth t i)) (has a t).
Proof. by rewrite -existsb_tnth; apply: existsP. Qed.
End TupleQuantifiers.
Arguments all_tnthP {n T a t}.
Arguments has_tnthP {n T a t}.
Section EqTuple.
Variables (n : nat) (T : eqType).
HB.instance Definition _ : hasDecEq (n.-tuple T) :=
[Equality of n.-tuple T by <:].
Canonical tuple_predType := PredType (pred_of_seq : n.-tuple T -> pred T).
Lemma eqEtuple (t1 t2 : n.-tuple T) :
(t1 == t2) = [forall i, tnth t1 i == tnth t2 i].
Proof. by apply/eqP/'forall_eqP => [->|/eq_from_tnth]. Qed.
Lemma memtE (t : n.-tuple T) : mem t = mem (tval t).
Proof. by []. Qed.
Lemma mem_tnth i (t : n.-tuple T) : tnth t i \in t.
Proof. by rewrite mem_nth ?size_tuple. Qed.
Lemma memt_nth x0 (t : n.-tuple T) i : i < n -> nth x0 t i \in t.
Proof. by move=> i_lt_n; rewrite mem_nth ?size_tuple. Qed.
Lemma tnthP (t : n.-tuple T) x : reflect (exists i, x = tnth t i) (x \in t).
Proof.
apply: (iffP idP) => [/(nthP x)[i ltin <-] | [i ->]]; last exact: mem_tnth.
by rewrite size_tuple in ltin; exists (Ordinal ltin); rewrite (tnth_nth x).
Qed.
Lemma seq_tnthP (s : seq T) x : x \in s -> {i | x = tnth (in_tuple s) i}.
Proof.
move=> s_x; pose i := index x s; have lt_i: i < size s by rewrite index_mem.
by exists (Ordinal lt_i); rewrite (tnth_nth x) nth_index.
Qed.
Lemma tuple_uniqP (t : n.-tuple T) : reflect (injective (tnth t)) (uniq t).
Proof.
case: {+}n => [|m] in t *; first by rewrite tuple0; constructor => -[].
pose x0 := tnth t ord0; apply/(equivP (uniqP x0)); split=> tinj i j.
by rewrite !(tnth_nth x0) => /tinj/val_inj; apply; rewrite size_tuple inE.
rewrite !size_tuple !inE => im jm; have := tinj (Ordinal im) (Ordinal jm).
by rewrite !(tnth_nth x0) => /[apply]-[].
Qed.
End EqTuple.
HB.instance Definition _ n (T : choiceType) :=
[Choice of n.-tuple T by <:].
HB.instance Definition _ n (T : countType) :=
[Countable of n.-tuple T by <:].
Module Type FinTupleSig.
Section FinTupleSig.
Variables (n : nat) (T : finType).
Parameter enum : seq (n.-tuple T).
Axiom enumP : Finite.axiom enum.
Axiom size_enum : size enum = #|T| ^ n.
End FinTupleSig.
End FinTupleSig.
Module FinTuple : FinTupleSig.
Section FinTuple.
Variables (n : nat) (T : finType).
Definition enum : seq (n.-tuple T) :=
let extend e := flatten (codom (fun x => map (cons x) e)) in
pmap insub (iter n extend [::[::]]).
Lemma enumP : Finite.axiom enum.
Proof.
case=> /= t t_n; rewrite -(count_map _ (pred1 t)) (pmap_filter (insubK _)).
rewrite count_filter -(@eq_count _ (pred1 t)) => [|s /=]; last first.
by rewrite isSome_insub; case: eqP=> // ->.
elim: n t t_n => [|m IHm] [|x t] //= {}/IHm; move: (iter m _ _) => em IHm.
transitivity (x \in T : nat); rewrite // -mem_enum codomE.
elim: (fintype.enum T) (enum_uniq T) => //= y e IHe /andP[/negPf ney].
rewrite count_cat count_map inE /preim /= [in LHS]/eq_op /= eq_sym => /IHe->.
by case: eqP => [->|_]; rewrite ?(ney, count_pred0, IHm).
Qed.
Lemma size_enum : size enum = #|T| ^ n.
Proof.
rewrite /= cardE size_pmap_sub; elim: n => //= m IHm.
rewrite expnS /codom /image_mem; elim: {2 3}(fintype.enum T) => //= x e IHe.
by rewrite count_cat {}IHe count_map IHm.
Qed.
End FinTuple.
End FinTuple.
Section UseFinTuple.
Variables (n : nat) (T : finType).
(* tuple_finMixin could, in principle, be made Canonical to allow for folding *)
(* Finite.enum of a finite tuple type (see comments around eqE in eqtype.v), *)
(* but in practice it will not work because the mixin_enum projector *)
(* has been buried under an opaque alias, to avoid some performance issues *)
(* during type inference. *)
HB.instance Definition _ := isFinite.Build (n.-tuple T) (@FinTuple.enumP n T).
Lemma card_tuple : #|{:n.-tuple T}| = #|T| ^ n.
Proof. by rewrite [#|_|]cardT enumT unlock FinTuple.size_enum. Qed.
Lemma enum_tupleP (A : {pred T}) : size (enum A) == #|A|.
Proof. by rewrite -cardE. Qed.
Canonical enum_tuple A := Tuple (enum_tupleP A).
Definition ord_tuple : n.-tuple 'I_n := Tuple (introT eqP (size_enum_ord n)).
Lemma val_ord_tuple : val ord_tuple = enum 'I_n. Proof. by []. Qed.
Lemma tuple_map_ord U (t : n.-tuple U) : t = [tuple of map (tnth t) ord_tuple].
Proof. by apply: val_inj => /=; rewrite map_tnth_enum. Qed.
Lemma tnth_ord_tuple i : tnth ord_tuple i = i.
Proof.
apply: val_inj; rewrite (tnth_nth i) -(nth_map _ 0) ?size_tuple //.
by rewrite /= enumT unlock val_ord_enum nth_iota.
Qed.
Section ImageTuple.
Variables (T' : Type) (f : T -> T') (A : {pred T}).
Canonical image_tuple : #|A|.-tuple T' := [tuple of image f A].
Canonical codom_tuple : #|T|.-tuple T' := [tuple of codom f].
End ImageTuple.
Section MkTuple.
Variables (T' : Type) (f : 'I_n -> T').
Definition mktuple := map_tuple f ord_tuple.
Lemma tnth_mktuple i : tnth mktuple i = f i.
Proof. by rewrite tnth_map tnth_ord_tuple. Qed.
Lemma nth_mktuple x0 (i : 'I_n) : nth x0 mktuple i = f i.
Proof. by rewrite -tnth_nth tnth_mktuple. Qed.
End MkTuple.
Lemma eq_mktuple T' (f1 f2 : 'I_n -> T') :
f1 =1 f2 -> mktuple f1 = mktuple f2.
Proof. by move=> eq_f; apply eq_from_tnth=> i; rewrite !tnth_map eq_f. Qed.
End UseFinTuple.
Notation "[ 'tuple' F | i < n ]" := (mktuple (fun i : 'I_n => F))
(i at level 0,
format "[ '[hv' 'tuple' F '/' | i < n ] ']'") : form_scope.
Arguments eq_mktuple {n T'} [f1] f2 eq_f12.
Section BseqDef.
Variables (n : nat) (T : Type).
Structure bseq_of : Type := Bseq {bseqval :> seq T; _ : size bseqval <= n}.
HB.instance Definition _ := [isSub for bseqval].
Implicit Type bs : bseq_of.
Lemma size_bseq bs : size bs <= n.
Proof. by case: bs. Qed.
Definition bseq bs mkB : bseq_of :=
mkB (let: Bseq _ bsP := bs return size bs <= n in bsP).
Lemma bseqE bs : bseq (fun sP => @Bseq bs sP) = bs.
Proof. by case: bs. Qed.
End BseqDef.
Canonical nil_bseq n T := Bseq (isT : @size T [::] <= n).
Canonical cons_bseq n T x (t : bseq_of n T) :=
Bseq (valP t : size (x :: t) <= n.+1).
Notation "n .-bseq" := (bseq_of n) (format "n .-bseq") : type_scope.
Notation "{ 'bseq' n 'of' T }" := (n.-bseq T : predArgType)
(only parsing) : type_scope.
Notation "[ 'bseq' 'of' s ]" := (bseq (fun sP => @Bseq _ _ s sP))
(format "[ 'bseq' 'of' s ]") : form_scope.
Notation "[ 'bseq' x1 ; .. ; xn ]" := [bseq of x1 :: .. [:: xn] ..]
(format "[ 'bseq' '[' x1 ; '/' .. ; '/' xn ']' ]") : form_scope.
Notation "[ 'bseq' ]" := [bseq of [::]] (format "[ 'bseq' ]") : form_scope.
Coercion bseq_of_tuple n T (t : n.-tuple T) : n.-bseq T :=
Bseq (eq_leq (size_tuple t)).
Definition insub_bseq n T (s : seq T) : n.-bseq T := insubd [bseq] s.
Lemma size_insub_bseq n T (s : seq T) : size (insub_bseq n s) <= size s.
Proof. by rewrite /insub_bseq /insubd; case: insubP => // ? ? ->. Qed.
Section CastBseq.
Variable T : Type.
Definition in_bseq (s : seq T) : (size s).-bseq T := Bseq (leqnn (size s)).
Definition cast_bseq m n (eq_mn : m = n) bs :=
let: erefl in _ = n := eq_mn return n.-bseq T in bs.
Definition widen_bseq m n (lemn : m <= n) (bs : m.-bseq T) : n.-bseq T :=
@Bseq n T bs (leq_trans (size_bseq bs) lemn).
Lemma cast_bseq_id n (eq_nn : n = n) bs : cast_bseq eq_nn bs = bs.
Proof. by rewrite (eq_axiomK eq_nn). Qed.
Lemma cast_bseqK m n (eq_mn : m = n) :
cancel (cast_bseq eq_mn) (cast_bseq (esym eq_mn)).
Proof. by case: n / eq_mn. Qed.
Lemma cast_bseqKV m n (eq_mn : m = n) :
cancel (cast_bseq (esym eq_mn)) (cast_bseq eq_mn).
Proof. by case: n / eq_mn. Qed.
Lemma cast_bseq_trans m n p (eq_mn : m = n) (eq_np : n = p) bs :
cast_bseq (etrans eq_mn eq_np) bs = cast_bseq eq_np (cast_bseq eq_mn bs).
Proof. by case: n / eq_mn eq_np; case: p /. Qed.
Lemma size_cast_bseq m n (eq_mn : m = n) (bs : m.-bseq T) :
size (cast_bseq eq_mn bs) = size bs.
Proof. by case: n / eq_mn. Qed.
Lemma widen_bseq_id n (lenn : n <= n) (bs : n.-bseq T) :
widen_bseq lenn bs = bs.
Proof. exact: val_inj. Qed.
Lemma cast_bseqEwiden m n (eq_mn : m = n) (bs : m.-bseq T) :
cast_bseq eq_mn bs = widen_bseq (eq_leq eq_mn) bs.
Proof. by case: n / eq_mn; rewrite widen_bseq_id. Qed.
Lemma widen_bseqK m n (lemn : m <= n) (lenm : n <= m) :
cancel (@widen_bseq m n lemn) (widen_bseq lenm).
Proof. by move=> t; apply: val_inj. Qed.
Lemma widen_bseq_trans m n p (lemn : m <= n) (lenp : n <= p) (bs : m.-bseq T) :
widen_bseq (leq_trans lemn lenp) bs = widen_bseq lenp (widen_bseq lemn bs).
Proof. exact/val_inj. Qed.
Lemma size_widen_bseq m n (lemn : m <= n) (bs : m.-bseq T) :
size (widen_bseq lemn bs) = size bs.
Proof. by []. Qed.
Lemma in_bseqE s : in_bseq s = s :> seq T. Proof. by []. Qed.
Lemma widen_bseq_in_bseq n (bs : n.-bseq T) :
widen_bseq (size_bseq bs) (in_bseq bs) = bs.
Proof. exact: val_inj. Qed.
End CastBseq.
Section SeqBseq.
Variables (n m : nat) (T U rT : Type).
Implicit Type s : n.-bseq T.
Lemma rcons_bseqP s x : size (rcons s x) <= n.+1.
Proof. by rewrite size_rcons ltnS size_bseq. Qed.
Canonical rcons_bseq s x := Bseq (rcons_bseqP s x).
Lemma behead_bseqP s : size (behead s) <= n.-1.
Proof. rewrite size_behead -!subn1; apply/leq_sub2r/size_bseq. Qed.
Canonical behead_bseq s := Bseq (behead_bseqP s).
Lemma belast_bseqP x s : size (belast x s) <= n.
Proof. by rewrite size_belast; apply/size_bseq. Qed.
Canonical belast_bseq x s := Bseq (belast_bseqP x s).
Lemma cat_bseqP s (s' : m.-bseq T) : size (s ++ s') <= n + m.
Proof. by rewrite size_cat; apply/leq_add/size_bseq/size_bseq. Qed.
Canonical cat_bseq s (s' : m.-bseq T) := Bseq (cat_bseqP s s').
Lemma take_bseqP s : size (take m s) <= n.
Proof.
by rewrite size_take_min (leq_trans _ (size_bseq s)) // geq_minr.
Qed.
Canonical take_bseq s := Bseq (take_bseqP s).
Lemma drop_bseqP s : size (drop m s) <= n - m.
Proof. by rewrite size_drop; apply/leq_sub2r/size_bseq. Qed.
Canonical drop_bseq s := Bseq (drop_bseqP s).
Lemma rev_bseqP s : size (rev s) <= n.
Proof. by rewrite size_rev size_bseq. Qed.
Canonical rev_bseq s := Bseq (rev_bseqP s).
Lemma rot_bseqP s : size (rot m s) <= n.
Proof. by rewrite size_rot size_bseq. Qed.
Canonical rot_bseq s := Bseq (rot_bseqP s).
Lemma rotr_bseqP s : size (rotr m s) <= n.
Proof. by rewrite size_rotr size_bseq. Qed.
Canonical rotr_bseq s := Bseq (rotr_bseqP s).
Lemma map_bseqP f s : @size rT (map f s) <= n.
Proof. by rewrite size_map size_bseq. Qed.
Canonical map_bseq f s := Bseq (map_bseqP f s).
Lemma scanl_bseqP f x s : @size rT (scanl f x s) <= n.
Proof. by rewrite size_scanl size_bseq. Qed.
Canonical scanl_bseq f x s := Bseq (scanl_bseqP f x s).
Lemma pairmap_bseqP f x s : @size rT (pairmap f x s) <= n.
Proof. by rewrite size_pairmap size_bseq. Qed.
Canonical pairmap_bseq f x s := Bseq (pairmap_bseqP f x s).
Lemma allpairs_bseqP f s (s' : m.-bseq U) : @size rT (allpairs f s s') <= n * m.
Proof. by rewrite size_allpairs; apply/leq_mul/size_bseq/size_bseq. Qed.
Canonical allpairs_bseq f s (s' : m.-bseq U) := Bseq (allpairs_bseqP f s s').
Lemma sort_bseqP r s : size (sort r s) <= n.
Proof. by rewrite size_sort size_bseq. Qed.
Canonical sort_bseq r s := Bseq (sort_bseqP r s).
Lemma bseq0 : all_equal_to ([bseq] : 0.-bseq T).
Proof. by move=> s; apply: val_inj; case: s => [[]]. Qed.
End SeqBseq.
HB.instance Definition bseq_hasDecEq n (T : eqType) :=
[Equality of n.-bseq T by <:].
Canonical bseq_predType n (T : eqType) :=
Eval hnf in PredType (fun t : n.-bseq T => mem_seq t).
Lemma membsE n (T : eqType) (bs : n.-bseq T) : mem bs = mem (bseqval bs).
Proof. by []. Qed.
HB.instance Definition bseq_hasChoice n (T : choiceType) :=
[Choice of n.-bseq T by <:].
HB.instance Definition bseq_isCountable n (T : countType) :=
[Countable of n.-bseq T by <:].
Definition bseq_tagged_tuple n T (s : n.-bseq T) : {k : 'I_n.+1 & k.-tuple T} :=
Tagged _ (in_tuple s : (Ordinal (size_bseq s : size s < n.+1)).-tuple _).
Arguments bseq_tagged_tuple {n T}.
Definition tagged_tuple_bseq n T (t : {k : 'I_n.+1 & k.-tuple T}) : n.-bseq T :=
widen_bseq (leq_ord (tag t)) (tagged t).
Arguments tagged_tuple_bseq {n T}.
Lemma bseq_tagged_tupleK {n T} :
cancel (@bseq_tagged_tuple n T) tagged_tuple_bseq.
Proof. by move=> bs; apply/val_inj. Qed.
Lemma tagged_tuple_bseqK {n T} :
cancel (@tagged_tuple_bseq n T) bseq_tagged_tuple.
Proof.
move=> [[k lt_kn] t]; apply: eq_existT_curried => [|k_eq]; apply/val_inj.
by rewrite /= size_tuple.
by refine (let: erefl := k_eq in _).
Qed.
Lemma bseq_tagged_tuple_bij {n T} : bijective (@bseq_tagged_tuple n T).
Proof. exact/Bijective/tagged_tuple_bseqK/bseq_tagged_tupleK. Qed.
Lemma tagged_tuple_bseq_bij {n T} : bijective (@tagged_tuple_bseq n T).
Proof. exact/Bijective/bseq_tagged_tupleK/tagged_tuple_bseqK. Qed.
#[global] Hint Resolve bseq_tagged_tuple_bij tagged_tuple_bseq_bij : core.
#[non_forgetful_inheritance]
HB.instance Definition _ n (T : finType) := isFinite.Build (n.-bseq T)
(pcan_enumP (can_pcan (@bseq_tagged_tupleK n T))).
|
ArtinianObject.lean
|
/-
Copyright (c) 2025 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou, Kim Morrison
-/
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.CategoryTheory.ObjectProperty.ContainsZero
import Mathlib.CategoryTheory.ObjectProperty.EpiMono
import Mathlib.CategoryTheory.Limits.Constructions.EventuallyConstant
import Mathlib.Order.OrderIsoNat
import Mathlib.CategoryTheory.Simple
/-!
# Artinian objects
We shall say that an object `X` in a category `C` is Artinian
(type class `IsArtinianObject X`) if the ordered type `Subobject X`
satisfies the descending chain condition. The corresponding property of
objects `isArtinianObject : ObjectProperty C` is always
closed under subobjects.
## Future works
* when `C` is an abelian category, relate `IsArtinianObject` in `C`
with `IsNoetherianObject` in `Cᵒᵖ`.
-/
universe v u
namespace CategoryTheory
open Limits ZeroObject
variable {C : Type u} [Category.{v} C]
/-- An object `X` in a category `C` is Artinian if `Subobject X`
satisfies the descending chain condition. This definition is a
term in `ObjectProperty C` which allows to study the stability
properties of Artinian objects. For statements regarding
specific objects, it is advisable to use the type class
`IsArtinianObject` instead. -/
@[stacks 0FCF]
def isArtinianObject : ObjectProperty C :=
fun X ↦ WellFoundedLT (Subobject X)
variable (X Y : C)
/-- An object `X` in a category `C` is Artinian if `Subobject X`
satisfies the descending chain condition. -/
@[stacks 0FCF]
abbrev IsArtinianObject : Prop := isArtinianObject.Is X
instance [IsArtinianObject X] : WellFoundedLT (Subobject X) :=
isArtinianObject.prop_of_is X
lemma isArtinianObject_iff_antitone_chain_condition :
IsArtinianObject X ↔ ∀ (f : ℕ →o (Subobject X)ᵒᵈ),
∃ (n : ℕ), ∀ (m : ℕ), n ≤ m → f n = f m := by
dsimp only [IsArtinianObject]
rw [ObjectProperty.is_iff, isArtinianObject,
← wellFoundedGT_dual_iff,
wellFoundedGT_iff_monotone_chain_condition]
variable {X} in
lemma antitone_chain_condition_of_isArtinianObject
[IsArtinianObject X] (f : ℕ →o (Subobject X)ᵒᵈ) :
∃ (n : ℕ), ∀ (m : ℕ), n ≤ m → f n = f m :=
(isArtinianObject_iff_antitone_chain_condition X).1 inferInstance f
lemma isArtinianObject_iff_not_strictAnti :
IsArtinianObject X ↔ ∀ (f : ℕ → Subobject X), ¬ StrictAnti f := by
refine ⟨fun _ ↦ not_strictAnti_of_wellFoundedLT, fun h ↦ ?_⟩
dsimp only [IsArtinianObject]
rw [ObjectProperty.is_iff, isArtinianObject, WellFoundedLT,
isWellFounded_iff, RelEmbedding.wellFounded_iff_isEmpty]
exact ⟨fun f ↦ h f.toFun (fun a b h ↦ f.map_rel_iff.2 h)⟩
variable {X} in
lemma not_strictAnti_of_isArtinianObject
[IsArtinianObject X] (f : ℕ → Subobject X) :
¬ StrictAnti f :=
(isArtinianObject_iff_not_strictAnti X).1 inferInstance f
lemma isArtinianObject_iff_isEventuallyConstant :
IsArtinianObject X ↔ ∀ (F : ℕ ⥤ (MonoOver X)ᵒᵖ),
IsFiltered.IsEventuallyConstant F := by
rw [isArtinianObject_iff_antitone_chain_condition]
refine ⟨fun h G ↦ ?_, fun h F ↦ ?_⟩
· obtain ⟨n, hn⟩ := h ⟨_, (G ⋙ (Subobject.equivMonoOver X).inverse.op ⋙
(orderDualEquivalence _).inverse).monotone⟩
refine ⟨n, fun m hm ↦ ?_⟩
rw [← isIso_unop_iff, MonoOver.isIso_iff_subobjectMk_eq]
exact (hn m (leOfHom hm)).symm
· obtain ⟨n, hn⟩ := h (F.monotone.functor ⋙ (orderDualEquivalence _).functor ⋙
Subobject.representative.op)
refine ⟨n, fun m hm ↦ Eq.symm ?_⟩
simpa [isIso_op_iff, isIso_iff_of_reflects_iso, PartialOrder.isIso_iff_eq]
using hn (homOfLE hm)
variable {X} in
lemma isEventuallyConstant_of_isArtinianObject [IsArtinianObject X]
(F : ℕ ⥤ (MonoOver X)ᵒᵖ) : IsFiltered.IsEventuallyConstant F :=
(isArtinianObject_iff_isEventuallyConstant X).1 inferInstance F
variable {X Y}
lemma isArtinianObject_of_isZero (hX : IsZero X) : IsArtinianObject X := by
rw [isArtinianObject_iff_antitone_chain_condition]
have := Subobject.subsingleton_of_isZero hX
intro f
exact ⟨0, fun m hm ↦ Subsingleton.elim _ _⟩
instance [HasZeroObject C] : (isArtinianObject (C := C)).ContainsZero where
exists_zero := ⟨0, isZero_zero _, by
rw [← isArtinianObject.is_iff]
exact isArtinianObject_of_isZero (isZero_zero C)⟩
lemma isArtinianObject_of_mono (i : X ⟶ Y) [Mono i] [IsArtinianObject Y] :
IsArtinianObject X := by
rw [isArtinianObject_iff_antitone_chain_condition]
intro f
obtain ⟨n, hn⟩ := antitone_chain_condition_of_isArtinianObject
⟨fun n ↦ (Subobject.map i).obj (f n),
fun _ _ h ↦ (Subobject.map i).monotone (f.2 h)⟩
exact ⟨n, fun m hm ↦ Subobject.map_obj_injective i (hn m hm)⟩
instance : (isArtinianObject (C := C)).IsClosedUnderSubobjects where
prop_of_mono f _ hY := by
rw [← isArtinianObject.is_iff] at hY ⊢
exact isArtinianObject_of_mono f
open Subobject
variable [HasZeroMorphisms C] [HasZeroObject C]
theorem exists_simple_subobject {X : C} [IsArtinianObject X] (h : ¬IsZero X) :
∃ Y : Subobject X, Simple (Y : C) := by
haveI : Nontrivial (Subobject X) := nontrivial_of_not_isZero h
obtain ⟨Y, s⟩ := (IsAtomic.eq_bot_or_exists_atom_le (⊤ : Subobject X)).resolve_left top_ne_bot
exact ⟨Y, (subobject_simple_iff_isAtom _).mpr s.1⟩
/-- Choose an arbitrary simple subobject of a non-zero Artinian object. -/
noncomputable def simpleSubobject {X : C} [IsArtinianObject X] (h : ¬IsZero X) : C :=
(exists_simple_subobject h).choose
/-- The monomorphism from the arbitrary simple subobject of a non-zero artinian object. -/
noncomputable def simpleSubobjectArrow {X : C} [IsArtinianObject X] (h : ¬IsZero X) :
simpleSubobject h ⟶ X :=
(exists_simple_subobject h).choose.arrow
instance mono_simpleSubobjectArrow {X : C} [IsArtinianObject X] (h : ¬IsZero X) :
Mono (simpleSubobjectArrow h) := by
dsimp only [simpleSubobjectArrow]
infer_instance
instance {X : C} [IsArtinianObject X] (h : ¬IsZero X) : Simple (simpleSubobject h) :=
(exists_simple_subobject h).choose_spec
end CategoryTheory
|
archimedean.v
|
(* (c) Copyright 2006-2016 Microsoft Corporation and Inria. *)
(* Distributed under the terms of CeCILL-B. *)
From HB Require Import structures.
From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice.
From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint.
(******************************************************************************)
(* Archimedean structures *)
(* *)
(* NB: See CONTRIBUTING.md for an introduction to HB concepts and commands. *)
(* *)
(* This file defines some numeric structures extended with the Archimedean *)
(* axiom. To use this file, insert "Import Num.Theory." and optionally *)
(* "Import Num.Def." before your scripts as in the ssrnum library. *)
(* The modules provided by this library subsume those from ssrnum. *)
(* *)
(* This file defines the following structures: *)
(* *)
(* archiNumDomainType == numDomainType with the Archimedean axiom *)
(* The HB class is called ArchiNumDomain. *)
(* archiNumFieldType == numFieldType with the Archimedean axiom *)
(* The HB class is called ArchiNumField. *)
(* archiClosedFieldType == closedFieldType with the Archimedean axiom *)
(* The HB class is called ArchiClosedField. *)
(* archiRealDomainType == realDomainType with the Archimedean axiom *)
(* The HB class is called ArchiRealDomain. *)
(* archiRealFieldType == realFieldType with the Archimedean axiom *)
(* The HB class is called ArchiRealField. *)
(* archiRcfType == rcfType with the Archimedean axiom *)
(* The HB class is called ArchiRealClosedField. *)
(* *)
(* Over these structures, we have the following operations: *)
(* x \is a Num.int <=> x is an integer, i.e., x = m%:~R for some m : int *)
(* x \is a Num.nat <=> x is a natural number, i.e., x = m%:R for some m : nat*)
(* Num.floor x == the m : int such that m%:~R <= x < (m + 1)%:~R *)
(* when x \is a Num.real, otherwise 0%Z *)
(* Num.ceil x == the m : int such that (m - 1)%:~R < x <= m%:~R *)
(* when x \is a Num.real, otherwise 0%Z *)
(* Num.truncn x == the n : nat such that n%:R <= x < n.+1%:R *)
(* when 0 <= n, otherwise 0%N *)
(* Num.bound x == an upper bound for x, i.e., an n such that `|x| < n%:R *)
(******************************************************************************)
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Import Order.TTheory GRing.Theory Num.Theory.
Module Num.
Import Num.Def.
HB.mixin Record NumDomain_hasFloorCeilTruncn R of Num.NumDomain R := {
floor : R -> int;
ceil : R -> int;
truncn : R -> nat;
int_num_subdef : pred R;
nat_num_subdef : pred R;
floor_subproof :
forall x,
if x \is Rreal then (floor x)%:~R <= x < (floor x + 1)%:~R
else floor x == 0;
ceil_subproof : forall x, ceil x = - floor (- x);
truncn_subproof : forall x, truncn x = if floor x is Posz n then n else 0;
int_num_subproof : forall x, reflect (exists n, x = n%:~R) (int_num_subdef x);
nat_num_subproof : forall x, reflect (exists n, x = n%:R) (nat_num_subdef x);
}.
#[short(type="archiNumDomainType")]
HB.structure Definition ArchiNumDomain :=
{ R of NumDomain_hasFloorCeilTruncn R & Num.NumDomain R }.
Module ArchiNumDomainExports.
Bind Scope ring_scope with ArchiNumDomain.sort.
End ArchiNumDomainExports.
HB.export ArchiNumDomainExports.
#[short(type="archiNumFieldType")]
HB.structure Definition ArchiNumField :=
{ R of NumDomain_hasFloorCeilTruncn R & Num.NumField R }.
Module ArchiNumFieldExports.
Bind Scope ring_scope with ArchiNumField.sort.
End ArchiNumFieldExports.
HB.export ArchiNumFieldExports.
#[short(type="archiClosedFieldType")]
HB.structure Definition ArchiClosedField :=
{ R of NumDomain_hasFloorCeilTruncn R & Num.ClosedField R }.
Module ArchiClosedFieldExports.
Bind Scope ring_scope with ArchiClosedField.sort.
End ArchiClosedFieldExports.
HB.export ArchiClosedFieldExports.
#[short(type="archiRealDomainType")]
HB.structure Definition ArchiRealDomain :=
{ R of NumDomain_hasFloorCeilTruncn R & Num.RealDomain R }.
Module ArchiRealDomainExports.
Bind Scope ring_scope with ArchiRealDomain.sort.
End ArchiRealDomainExports.
HB.export ArchiRealDomainExports.
#[short(type="archiRealFieldType")]
HB.structure Definition ArchiRealField :=
{ R of NumDomain_hasFloorCeilTruncn R & Num.RealField R }.
Module ArchiRealFieldExports.
Bind Scope ring_scope with ArchiRealField.sort.
End ArchiRealFieldExports.
HB.export ArchiRealFieldExports.
#[short(type="archiRcfType")]
HB.structure Definition ArchiRealClosedField :=
{ R of NumDomain_hasFloorCeilTruncn R & Num.RealClosedField R }.
Module ArchiRealClosedFieldExports.
Bind Scope ring_scope with ArchiRealClosedField.sort.
End ArchiRealClosedFieldExports.
HB.export ArchiRealClosedFieldExports.
Section Def.
Context {R : archiNumDomainType}.
Definition nat_num : qualifier 1 R := [qualify a x : R | nat_num_subdef x].
Definition int_num : qualifier 1 R := [qualify a x : R | int_num_subdef x].
Definition bound (x : R) := (truncn `|x|).+1.
End Def.
Arguments floor {R} : rename, simpl never.
Arguments ceil {R} : rename, simpl never.
Arguments truncn {R} : rename, simpl never.
Arguments nat_num {R} : simpl never.
Arguments int_num {R} : simpl never.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to truncn.")]
Notation trunc := truncn.
Module Def.
Export ssrnum.Num.Def.
Notation truncn := truncn.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to truncn.")]
Notation trunc := truncn.
Notation floor := floor.
Notation ceil := ceil.
Notation nat_num := nat_num.
Notation int_num := int_num.
Notation archi_bound := bound.
End Def.
Module intArchimedean.
Section intArchimedean.
Implicit Types n : int.
Lemma floorP n : if n \is Rreal then n%:~R <= n < (n + 1)%:~R else n == 0.
Proof. by rewrite num_real !intz ltzD1 lexx. Qed.
Lemma intrP n : reflect (exists m, n = m%:~R) true.
Proof. by apply: ReflectT; exists n; rewrite intz. Qed.
Lemma natrP n : reflect (exists m, n = m%:R) (0 <= n).
Proof.
apply: (iffP idP); last by case=> m ->; rewrite ler0n.
by case: n => // n _; exists n; rewrite natz.
Qed.
End intArchimedean.
End intArchimedean.
#[export]
HB.instance Definition _ :=
@NumDomain_hasFloorCeilTruncn.Build int id id _ xpredT Rnneg_pred
intArchimedean.floorP (fun=> esym (opprK _)) (fun=> erefl)
intArchimedean.intrP intArchimedean.natrP.
Module Import Theory.
Export ssrnum.Num.Theory.
Section ArchiNumDomainTheory.
Variable R : archiNumDomainType.
Implicit Types x y z : R.
Local Notation truncn := (@truncn R).
Local Notation floor := (@floor R).
Local Notation ceil := (@ceil R).
Local Notation nat_num := (@Def.nat_num R).
Local Notation int_num := (@Def.int_num R).
Local Lemma floorP x :
if x \is Rreal then (floor x)%:~R <= x < (floor x + 1)%:~R else floor x == 0.
Proof. exact: floor_subproof. Qed.
Lemma floorNceil x : floor x = - ceil (- x).
Proof. by rewrite ceil_subproof !opprK. Qed.
Lemma ceilNfloor x : ceil x = - floor (- x).
Proof. exact: ceil_subproof. Qed.
Lemma truncEfloor x : truncn x = if floor x is Posz n then n else 0.
Proof. exact: truncn_subproof. Qed.
Lemma natrP x : reflect (exists n, x = n%:R) (x \is a nat_num).
Proof. exact: nat_num_subproof. Qed.
Lemma intrP x : reflect (exists m, x = m%:~R) (x \is a int_num).
Proof. exact: int_num_subproof. Qed.
(* int_num and nat_num *)
Lemma intr_int m : m%:~R \is a int_num. Proof. by apply/intrP; exists m. Qed.
Lemma natr_nat n : n%:R \is a nat_num. Proof. by apply/natrP; exists n. Qed.
#[local] Hint Resolve intr_int natr_nat : core.
Lemma rpred_int_num (S : subringClosed R) x : x \is a int_num -> x \in S.
Proof. by move=> /intrP[n ->]; rewrite rpred_int. Qed.
Lemma rpred_nat_num (S : semiringClosed R) x : x \is a nat_num -> x \in S.
Proof. by move=> /natrP[n ->]; apply: rpred_nat. Qed.
Lemma int_num0 : 0 \is a int_num. Proof. exact: (intr_int 0). Qed.
Lemma int_num1 : 1 \is a int_num. Proof. exact: (intr_int 1). Qed.
Lemma nat_num0 : 0 \is a nat_num. Proof. exact: (natr_nat 0). Qed.
Lemma nat_num1 : 1 \is a nat_num. Proof. exact: (natr_nat 1). Qed.
#[local] Hint Resolve int_num0 int_num1 nat_num0 nat_num1 : core.
Fact int_num_subring : subring_closed int_num.
Proof.
by split=> // _ _ /intrP[n ->] /intrP[m ->]; rewrite -(intrB, intrM).
Qed.
#[export]
HB.instance Definition _ := GRing.isSubringClosed.Build R int_num_subdef
int_num_subring.
Fact nat_num_semiring : semiring_closed nat_num.
Proof.
by do 2![split] => //= _ _ /natrP[n ->] /natrP[m ->]; rewrite -(natrD, natrM).
Qed.
#[export]
HB.instance Definition _ := GRing.isSemiringClosed.Build R nat_num_subdef
nat_num_semiring.
Lemma Rreal_nat : {subset nat_num <= Rreal}. Proof. exact: rpred_nat_num. Qed.
Lemma intr_nat : {subset nat_num <= int_num}.
Proof. by move=> _ /natrP[n ->]; rewrite pmulrn intr_int. Qed.
Lemma Rreal_int : {subset int_num <= Rreal}. Proof. exact: rpred_int_num. Qed.
Lemma intrE x : (x \is a int_num) = (x \is a nat_num) || (- x \is a nat_num).
Proof.
apply/idP/orP => [/intrP[[n|n] ->]|[]/intr_nat]; rewrite ?rpredN //.
by left; apply/natrP; exists n.
by rewrite NegzE intrN opprK; right; apply/natrP; exists n.+1.
Qed.
Lemma intr_normK x : x \is a int_num -> `|x| ^+ 2 = x ^+ 2.
Proof. by move/Rreal_int/real_normK. Qed.
Lemma natr_normK x : x \is a nat_num -> `|x| ^+ 2 = x ^+ 2.
Proof. by move/Rreal_nat/real_normK. Qed.
Lemma natr_norm_int x : x \is a int_num -> `|x| \is a nat_num.
Proof. by move=> /intrP[m ->]; rewrite -intr_norm rpred_nat_num ?natr_nat. Qed.
Lemma natr_ge0 x : x \is a nat_num -> 0 <= x.
Proof. by move=> /natrP[n ->]; apply: ler0n. Qed.
Lemma natr_gt0 x : x \is a nat_num -> (0 < x) = (x != 0).
Proof. by move/natr_ge0; case: comparableP. Qed.
Lemma natrEint x : (x \is a nat_num) = (x \is a int_num) && (0 <= x).
Proof.
apply/idP/andP=> [Nx | [Zx x_ge0]]; first by rewrite intr_nat ?natr_ge0.
by rewrite -(ger0_norm x_ge0) natr_norm_int.
Qed.
Lemma intrEge0 x : 0 <= x -> (x \is a int_num) = (x \is a nat_num).
Proof. by rewrite natrEint andbC => ->. Qed.
Lemma intrEsign x : x \is a int_num -> x = (-1) ^+ (x < 0)%R * `|x|.
Proof. by move/Rreal_int/realEsign. Qed.
Lemma norm_natr x : x \is a nat_num -> `|x| = x.
Proof. by move/natr_ge0/ger0_norm. Qed.
Lemma natr_exp_even x n : ~~ odd n -> x \is a int_num -> x ^+ n \is a nat_num.
Proof.
move=> n_oddF x_intr.
by rewrite natrEint rpredX //= real_exprn_even_ge0 // Rreal_int.
Qed.
Lemma norm_intr_ge1 x : x \is a int_num -> x != 0 -> 1 <= `|x|.
Proof.
rewrite -normr_eq0 => /natr_norm_int/natrP[n ->].
by rewrite pnatr_eq0 ler1n lt0n.
Qed.
Lemma sqr_intr_ge1 x : x \is a int_num -> x != 0 -> 1 <= x ^+ 2.
Proof.
by move=> Zx nz_x; rewrite -intr_normK // expr_ge1 ?normr_ge0 ?norm_intr_ge1.
Qed.
Lemma intr_ler_sqr x : x \is a int_num -> x <= x ^+ 2.
Proof.
move=> Zx; have [-> | nz_x] := eqVneq x 0; first by rewrite expr0n.
apply: le_trans (_ : `|x| <= _); first by rewrite real_ler_norm ?Rreal_int.
by rewrite -intr_normK // ler_eXnr // norm_intr_ge1.
Qed.
(* floor and int_num *)
Lemma real_floor_itv x : x \is Rreal -> (floor x)%:~R <= x < (floor x + 1)%:~R.
Proof. by case: ifP (floorP x). Qed.
Lemma real_floor_le x : x \is Rreal -> (floor x)%:~R <= x.
Proof. by case/real_floor_itv/andP. Qed.
Lemma real_floorD1_gt x : x \is Rreal -> x < (floor x + 1)%:~R.
Proof. by case/real_floor_itv/andP. Qed.
Lemma floor_def x m : m%:~R <= x < (m + 1)%:~R -> floor x = m.
Proof.
case/andP=> lemx ltxm1; apply/eqP; rewrite eq_le -!ltzD1.
move: (ger_real lemx); rewrite realz => /real_floor_itv/andP[lefx ltxf1].
by rewrite -!(ltr_int R) 2?(@le_lt_trans _ _ x).
Qed.
(* TODO: rename to real_floor_ge_int,
once the currently deprecated one has been removed *)
Lemma real_floor_ge_int_tmp x n : x \is Rreal -> (n <= floor x) = (n%:~R <= x).
Proof.
move=> /real_floor_itv /andP[lefx ltxf1]; apply/idP/idP => lenx.
by apply: le_trans lefx; rewrite ler_int.
by rewrite -ltzD1 -(ltr_int R); apply: le_lt_trans ltxf1.
Qed.
#[deprecated(since="mathcomp 2.4.0", note="Use real_floor_ge_int_tmp instead.")]
Lemma real_floor_ge_int x n : x \is Rreal -> (n%:~R <= x) = (n <= floor x).
Proof. by move=> ?; rewrite real_floor_ge_int_tmp. Qed.
Lemma real_floor_lt_int x n : x \is Rreal -> (floor x < n) = (x < n%:~R).
Proof.
by move=> ?; rewrite [RHS]real_ltNge ?realz -?real_floor_ge_int_tmp -?ltNge.
Qed.
Lemma real_floor_eq x n : x \is Rreal ->
(floor x == n) = (n%:~R <= x < (n + 1)%:~R).
Proof.
by move=> xr; apply/eqP/idP => [<-|]; [exact: real_floor_itv|exact: floor_def].
Qed.
Lemma le_floor : {homo floor : x y / x <= y}.
Proof.
move=> x y lexy; move: (floorP x) (floorP y); rewrite (ger_real lexy).
case: ifP => [_ /andP[lefx _] /andP[_] | _ /eqP-> /eqP-> //].
by move=> /(le_lt_trans lexy) /(le_lt_trans lefx); rewrite ltr_int ltzD1.
Qed.
Lemma intrKfloor : cancel intr floor.
Proof. by move=> m; apply: floor_def; rewrite lexx rmorphD ltrDl ltr01. Qed.
Lemma natr_int n : n%:R \is a int_num.
Proof. by rewrite intrE natr_nat. Qed.
#[local] Hint Resolve natr_int : core.
Lemma intrEfloor x : x \is a int_num = ((floor x)%:~R == x).
Proof.
by apply/intrP/eqP => [[n ->] | <-]; [rewrite intrKfloor | exists (floor x)].
Qed.
Lemma floorK : {in int_num, cancel floor intr}.
Proof. by move=> z; rewrite intrEfloor => /eqP. Qed.
Lemma floor0 : floor 0 = 0. Proof. exact: intrKfloor 0. Qed.
Lemma floor1 : floor 1 = 1. Proof. exact: intrKfloor 1. Qed.
#[local] Hint Resolve floor0 floor1 : core.
Lemma real_floorDzr : {in int_num & Rreal, {morph floor : x y / x + y}}.
Proof.
move=> _ y /intrP[m ->] Ry; apply: floor_def.
by rewrite -addrA 2!rmorphD /= intrKfloor lerD2l ltrD2l real_floor_itv.
Qed.
Lemma real_floorDrz : {in Rreal & int_num, {morph floor : x y / x + y}}.
Proof. by move=> x y xr yz; rewrite addrC real_floorDzr // addrC. Qed.
Lemma floorN : {in int_num, {morph floor : x / - x}}.
Proof. by move=> _ /intrP[m ->]; rewrite -rmorphN !intrKfloor. Qed.
Lemma floorM : {in int_num &, {morph floor : x y / x * y}}.
Proof.
by move=> _ _ /intrP[m1 ->] /intrP[m2 ->]; rewrite -rmorphM !intrKfloor.
Qed.
Lemma floorX n : {in int_num, {morph floor : x / x ^+ n}}.
Proof. by move=> _ /intrP[m ->]; rewrite -rmorphXn !intrKfloor. Qed.
Lemma real_floor_ge0 x : x \is Rreal -> (0 <= floor x) = (0 <= x).
Proof. by move=> ?; rewrite real_floor_ge_int_tmp. Qed.
Lemma floor_lt0 x : (floor x < 0) = (x < 0).
Proof.
case: ifP (floorP x) => [xr _ | xr /eqP <-]; first by rewrite real_floor_lt_int.
by rewrite ltxx; apply/esym/(contraFF _ xr)/ltr0_real.
Qed.
Lemma real_floor_le0 x : x \is Rreal -> (floor x <= 0) = (x < 1).
Proof. by move=> ?; rewrite -ltzD1 add0r real_floor_lt_int. Qed.
Lemma floor_gt0 x : (floor x > 0) = (x >= 1).
Proof.
case: ifP (floorP x) => [xr _ | xr /eqP->].
by rewrite gtz0_ge1 real_floor_ge_int_tmp.
by rewrite ltxx; apply/esym/(contraFF _ xr)/ger1_real.
Qed.
Lemma floor_neq0 x : (floor x != 0) = (x < 0) || (x >= 1).
Proof.
case: ifP (floorP x) => [xr _ | xr /eqP->]; rewrite ?eqxx/=.
by rewrite neq_lt floor_lt0 floor_gt0.
by apply/esym/(contraFF _ xr) => /orP[/ltr0_real|/ger1_real].
Qed.
Lemma floorpK : {in polyOver int_num, cancel (map_poly floor) (map_poly intr)}.
Proof.
move=> p /(all_nthP 0) Zp; apply/polyP=> i.
rewrite coef_map coef_map_id0 //= -[p]coefK coef_poly.
by case: ifP => [/Zp/floorK // | _]; rewrite floor0.
Qed.
Lemma floorpP (p : {poly R}) :
p \is a polyOver int_num -> {q | p = map_poly intr q}.
Proof. by exists (map_poly floor p); rewrite floorpK. Qed.
(* ceil and int_num *)
Lemma real_ceil_itv x : x \is Rreal -> (ceil x - 1)%:~R < x <= (ceil x)%:~R.
Proof.
rewrite ceilNfloor -opprD !intrN ltrNl lerNr andbC -realN.
exact: real_floor_itv.
Qed.
Lemma real_ceilB1_lt x : x \is Rreal -> (ceil x - 1)%:~R < x.
Proof. by case/real_ceil_itv/andP. Qed.
Lemma real_ceil_ge x : x \is Rreal -> x <= (ceil x)%:~R.
Proof. by case/real_ceil_itv/andP. Qed.
Lemma ceil_def x m : (m - 1)%:~R < x <= m%:~R -> ceil x = m.
Proof.
rewrite -ltrN2 -lerN2 andbC -!intrN opprD opprK ceilNfloor.
by move=> /floor_def ->; rewrite opprK.
Qed.
(* TODO: rename to real_ceil_le_int,
once the currently deprecated one has been removed *)
Lemma real_ceil_le_int_tmp x n : x \is Rreal -> (ceil x <= n) = (x <= n%:~R).
Proof.
rewrite ceilNfloor lerNl -realN => /real_floor_ge_int_tmp ->.
by rewrite intrN lerN2.
Qed.
#[deprecated(since="mathcomp 2.4.0", note="Use real_ceil_le_int_tmp instead.")]
Lemma real_ceil_le_int x n : x \is Rreal -> x <= n%:~R = (ceil x <= n).
Proof. by move=> ?; rewrite real_ceil_le_int_tmp. Qed.
Lemma real_ceil_gt_int x n : x \is Rreal -> (n < ceil x) = (n%:~R < x).
Proof.
by move=> ?; rewrite [RHS]real_ltNge ?realz -?real_ceil_le_int_tmp ?ltNge.
Qed.
Lemma real_ceil_eq x n : x \is Rreal ->
(ceil x == n) = ((n - 1)%:~R < x <= n%:~R).
Proof.
by move=> xr; apply/eqP/idP => [<-|]; [exact: real_ceil_itv|exact: ceil_def].
Qed.
(* TODO: rename to le_ceil,
once the currently deprecated one has been removed *)
Lemma le_ceil_tmp : {homo ceil : x y / x <= y}.
Proof. by move=> x y lexy; rewrite !ceilNfloor lerN2 le_floor ?lerN2. Qed.
Lemma intrKceil : cancel intr ceil.
Proof. by move=> m; rewrite ceilNfloor -intrN intrKfloor opprK. Qed.
Lemma intrEceil x : x \is a int_num = ((ceil x)%:~R == x).
Proof. by rewrite -rpredN intrEfloor -eqr_oppLR -intrN -ceilNfloor. Qed.
Lemma ceilK : {in int_num, cancel ceil intr}.
Proof. by move=> z; rewrite intrEceil => /eqP. Qed.
Lemma ceil0 : ceil 0 = 0. Proof. exact: intrKceil 0. Qed.
Lemma ceil1 : ceil 1 = 1. Proof. exact: intrKceil 1. Qed.
#[local] Hint Resolve ceil0 ceil1 : core.
Lemma real_ceilDzr : {in int_num & Rreal, {morph ceil : x y / x + y}}.
Proof.
move=> x y x_int y_real.
by rewrite ceilNfloor opprD real_floorDzr ?rpredN // opprD -!ceilNfloor.
Qed.
Lemma real_ceilDrz : {in Rreal & int_num, {morph ceil : x y / x + y}}.
Proof. by move=> x y xr yz; rewrite addrC real_ceilDzr // addrC. Qed.
Lemma ceilN : {in int_num, {morph ceil : x / - x}}.
Proof. by move=> ? ?; rewrite !ceilNfloor !opprK floorN. Qed.
Lemma ceilM : {in int_num &, {morph ceil : x y / x * y}}.
Proof.
by move=> _ _ /intrP[m1 ->] /intrP[m2 ->]; rewrite -rmorphM !intrKceil.
Qed.
Lemma ceilX n : {in int_num, {morph ceil : x / x ^+ n}}.
Proof. by move=> _ /intrP[m ->]; rewrite -rmorphXn !intrKceil. Qed.
Lemma real_ceil_ge0 x : x \is Rreal -> (0 <= ceil x) = (-1 < x).
Proof.
by move=> ?; rewrite ceilNfloor oppr_ge0 real_floor_le0 ?realN 1?ltrNl.
Qed.
Lemma ceil_lt0 x : (ceil x < 0) = (x <= -1).
Proof. by rewrite ceilNfloor oppr_lt0 floor_gt0 lerNr. Qed.
Lemma real_ceil_le0 x : x \is Rreal -> (ceil x <= 0) = (x <= 0).
Proof. by move=> ?; rewrite real_ceil_le_int_tmp. Qed.
Lemma ceil_gt0 x : (ceil x > 0) = (x > 0).
Proof. by rewrite ceilNfloor oppr_gt0 floor_lt0 oppr_lt0. Qed.
Lemma ceil_neq0 x : (ceil x != 0) = (x <= -1) || (x > 0).
Proof. by rewrite ceilNfloor oppr_eq0 floor_neq0 oppr_lt0 lerNr orbC. Qed.
Lemma real_ceil_floor x : x \is Rreal ->
ceil x = floor x + (x \isn't a int_num).
Proof.
case Ix: (x \is a int_num) => Rx /=.
by apply/eqP; rewrite addr0 ceilNfloor eqr_oppLR floorN.
apply/ceil_def; rewrite addrK; move: (real_floor_itv Rx).
by rewrite le_eqVlt -intrEfloor Ix /= => /andP[-> /ltW].
Qed.
(* Relating Cnat and oldCnat. *)
Lemma truncn_floor x : truncn x = if 0 <= x then `|floor x|%N else 0%N.
Proof.
move: (floorP x); rewrite truncEfloor realE.
have [/le_floor|_]/= := boolP (0 <= x); first by rewrite floor0; case: floor.
by case: ifP => [/le_floor|_ /eqP->//]; rewrite floor0; case: floor => [[]|].
Qed.
(* trunc and nat_num *)
Local Lemma truncnP x :
if 0 <= x then (truncn x)%:R <= x < (truncn x).+1%:R else truncn x == 0%N.
Proof.
rewrite truncn_floor.
case: (boolP (0 <= x)) => //= /[dup] /le_floor + /ger0_real/real_floor_itv.
by rewrite floor0; case: (floor x) => // n _; rewrite absz_nat addrC -intS.
Qed.
Lemma truncn_itv x : 0 <= x -> (truncn x)%:R <= x < (truncn x).+1%:R.
Proof. by move=> x_ge0; move: (truncnP x); rewrite x_ge0. Qed.
Lemma truncn_le x : (truncn x)%:R <= x = (0 <= x).
Proof. by case: ifP (truncnP x) => [+ /andP[] | + /eqP->//]. Qed.
Lemma real_truncnS_gt x : x \is Rreal -> x < (truncn x).+1%:R.
Proof. by move/real_ge0P => [/truncn_itv/andP[]|/lt_le_trans->]. Qed.
Lemma truncn_def x n : n%:R <= x < n.+1%:R -> truncn x = n.
Proof.
case/andP=> lemx ltxm1; apply/eqP; rewrite eqn_leq -ltnS -[(n <= _)%N]ltnS.
have/truncn_itv/andP[lefx ltxf1]: 0 <= x by apply: le_trans lemx; apply: ler0n.
by rewrite -!(ltr_nat R) 2?(@le_lt_trans _ _ x).
Qed.
Lemma truncn_ge_nat x n : 0 <= x -> (n <= truncn x)%N = (n%:R <= x).
Proof.
move=> /truncn_itv /andP[letx ltxt1]; apply/idP/idP => lenx.
by apply: le_trans letx; rewrite ler_nat.
by rewrite -ltnS -(ltr_nat R); apply: le_lt_trans ltxt1.
Qed.
Lemma truncn_gt_nat x n : (n < truncn x)%N = (n.+1%:R <= x).
Proof.
case: ifP (truncnP x) => [x0 _ | x0 /eqP->]; first by rewrite truncn_ge_nat.
by rewrite ltn0; apply/esym/(contraFF _ x0)/le_trans.
Qed.
Lemma truncn_lt_nat x n : 0 <= x -> (truncn x < n)%N = (x < n%:R).
Proof. by move=> ?; rewrite real_ltNge ?ger0_real// ltnNge truncn_ge_nat. Qed.
Lemma real_truncn_le_nat x n : x \is Rreal -> (truncn x <= n)%N = (x < n.+1%:R).
Proof. by move=> ?; rewrite real_ltNge// leqNgt truncn_gt_nat. Qed.
Lemma truncn_eq x n : 0 <= x -> (truncn x == n) = (n%:R <= x < n.+1%:R).
Proof.
by move=> xr; apply/eqP/idP => [<-|]; [exact: truncn_itv|exact: truncn_def].
Qed.
Lemma le_truncn : {homo truncn : x y / x <= y >-> (x <= y)%N}.
Proof.
move=> x y lexy; move: (truncnP x) (truncnP y).
case: ifP => [x0 /andP[letx _] | x0 /eqP->//].
case: ifP => [y0 /andP[_] | y0 /eqP->]; [|by rewrite (le_trans x0 lexy) in y0].
by move=> /(le_lt_trans lexy) /(le_lt_trans letx); rewrite ltr_nat ltnS.
Qed.
Lemma natrK : cancel (GRing.natmul 1) truncn.
Proof. by move=> m; apply: truncn_def; rewrite ler_nat ltr_nat ltnS leqnn. Qed.
Lemma natrEtruncn x : (x \is a nat_num) = ((truncn x)%:R == x).
Proof.
by apply/natrP/eqP => [[n ->]|<-]; [rewrite natrK | exists (truncn x)].
Qed.
Lemma archi_boundP x : 0 <= x -> x < (bound x)%:R.
Proof.
move=> x_ge0; case/truncn_itv/andP: (normr_ge0 x) => _.
exact/le_lt_trans/real_ler_norm/ger0_real.
Qed.
Lemma truncnK : {in nat_num, cancel truncn (GRing.natmul 1)}.
Proof. by move=> x; rewrite natrEtruncn => /eqP. Qed.
Lemma truncn0 : truncn 0 = 0%N. Proof. exact: natrK 0%N. Qed.
Lemma truncn1 : truncn 1 = 1%N. Proof. exact: natrK 1%N. Qed.
#[local] Hint Resolve truncn0 truncn1 : core.
Lemma truncnD :
{in nat_num & Rnneg, {morph truncn : x y / x + y >-> (x + y)%N}}.
Proof.
move=> _ y /natrP[n ->] y_ge0; apply: truncn_def.
by rewrite -addnS !natrD !natrK lerD2l ltrD2l truncn_itv.
Qed.
Lemma truncnM : {in nat_num &, {morph truncn : x y / x * y >-> (x * y)%N}}.
Proof. by move=> _ _ /natrP[n1 ->] /natrP[n2 ->]; rewrite -natrM !natrK. Qed.
Lemma truncnX n : {in nat_num, {morph truncn : x / x ^+ n >-> (x ^ n)%N}}.
Proof. by move=> _ /natrP[n1 ->]; rewrite -natrX !natrK. Qed.
Lemma truncn_gt0 x : (0 < truncn x)%N = (1 <= x).
Proof.
case: ifP (truncnP x) => [x0 | x0 /eqP<-]; first by rewrite truncn_ge_nat.
by rewrite ltnn; apply/esym/(contraFF _ x0)/le_trans.
Qed.
Lemma truncn0Pn x : reflect (truncn x = 0%N) (~~ (1 <= x)).
Proof. by rewrite -truncn_gt0 -eqn0Ngt; apply: eqP. Qed.
Lemma sum_truncnK I r (P : pred I) F : (forall i, P i -> F i \is a nat_num) ->
(\sum_(i <- r | P i) truncn (F i))%:R = \sum_(i <- r | P i) F i.
Proof. by rewrite natr_sum => natr; apply: eq_bigr => i /natr /truncnK. Qed.
Lemma prod_truncnK I r (P : pred I) F : (forall i, P i -> F i \is a nat_num) ->
(\prod_(i <- r | P i) truncn (F i))%:R = \prod_(i <- r | P i) F i.
Proof. by rewrite natr_prod => natr; apply: eq_bigr => i /natr /truncnK. Qed.
Lemma natr_sum_eq1 (I : finType) (P : pred I) (F : I -> R) :
(forall i, P i -> F i \is a nat_num) -> \sum_(i | P i) F i = 1 ->
{i : I | [/\ P i, F i = 1 & forall j, j != i -> P j -> F j = 0]}.
Proof.
move=> natF /eqP; rewrite -sum_truncnK// -[1]/1%:R eqr_nat => /sum_nat_eq1 exi.
have [i /and3P[Pi /eqP f1 /forallP a]] : {i : I | [&& P i, truncn (F i) == 1
& [forall j : I, ((j != i) ==> P j ==> (truncn (F j) == 0))]]}.
apply/sigW; have [i [Pi /eqP f1 a]] := exi; exists i; apply/and3P; split=> //.
by apply/forallP => j; apply/implyP => ji; apply/implyP => Pj; apply/eqP/a.
exists i; split=> [//||j ji Pj]; rewrite -[LHS]truncnK ?natF ?f1//; apply/eqP.
by rewrite -[0]/0%:R eqr_nat; apply: implyP Pj; apply: implyP ji; apply: a.
Qed.
Lemma natr_mul_eq1 x y :
x \is a nat_num -> y \is a nat_num -> (x * y == 1) = (x == 1) && (y == 1).
Proof. by do 2!move/truncnK <-; rewrite -natrM !pnatr_eq1 muln_eq1. Qed.
Lemma natr_prod_eq1 (I : finType) (P : pred I) (F : I -> R) :
(forall i, P i -> F i \is a nat_num) -> \prod_(i | P i) F i = 1 ->
forall i, P i -> F i = 1.
Proof.
move=> natF /eqP; rewrite -prod_truncnK// -[1]/1%:R eqr_nat prod_nat_seq_eq1.
move/allP => a i Pi; apply/eqP; rewrite -[F i]truncnK ?natF// eqr_nat.
by apply: implyP Pi; apply: a; apply: mem_index_enum.
Qed.
(* predCmod *)
Variables (U V : lmodType R) (f : {additive U -> V}).
Lemma raddfZ_nat a u : a \is a nat_num -> f (a *: u) = a *: f u.
Proof. by move=> /natrP[n ->]; apply: raddfZnat. Qed.
Lemma rpredZ_nat (S : addrClosed V) :
{in nat_num & S, forall z u, z *: u \in S}.
Proof. by move=> _ u /natrP[n ->]; apply: rpredZnat. Qed.
Lemma raddfZ_int a u : a \is a int_num -> f (a *: u) = a *: f u.
Proof. by move=> /intrP[m ->]; rewrite !scaler_int raddfMz. Qed.
Lemma rpredZ_int (S : zmodClosed V) :
{in int_num & S, forall z u, z *: u \in S}.
Proof. by move=> _ u /intrP[m ->] ?; rewrite scaler_int rpredMz. Qed.
(* autC *)
Implicit Type nu : {rmorphism R -> R}.
Lemma aut_natr nu : {in nat_num, nu =1 id}.
Proof. by move=> _ /natrP[n ->]; apply: rmorph_nat. Qed.
Lemma aut_intr nu : {in int_num, nu =1 id}.
Proof. by move=> _ /intrP[m ->]; apply: rmorph_int. Qed.
End ArchiNumDomainTheory.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to truncn_itv.")]
Notation trunc_itv := truncn_itv.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to truncn_def.")]
Notation trunc_def := truncn_def.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to truncnK.")]
Notation truncK := truncnK.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to truncn0.")]
Notation trunc0 := truncn0.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to truncn1.")]
Notation trunc1 := truncn1.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to truncnD.")]
Notation truncD := truncnD.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to truncnM.")]
Notation truncM := truncnM.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to truncnX.")]
Notation truncX := truncnX.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to truncn_gt0.")]
Notation trunc_gt0 := truncn_gt0.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to truncn0Pn.")]
Notation trunc0Pn := truncn0Pn.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to sum_truncnK.")]
Notation sum_truncK := sum_truncnK.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to prod_truncnK.")]
Notation prod_truncK := prod_truncnK.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to truncn_floor.")]
Notation trunc_floor := truncn_floor.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to real_floor_le.")]
Notation real_ge_floor := real_floor_le.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to real_floorD1_gt.")]
Notation real_lt_succ_floor := real_floorD1_gt.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to real_ceilB1_lt.")]
Notation real_gt_pred_ceil := real_floorD1_gt.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to real_ceil_ge.")]
Notation real_le_ceil := real_ceil_ge.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to le_floor.")]
Notation floor_le := le_floor.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to le_ceil.")]
Notation ceil_le := le_ceil_tmp.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to natrEtruncn.")]
Notation natrE := natrEtruncn.
Arguments natrK {R} _%_N.
Arguments intrKfloor {R}.
Arguments intrKceil {R}.
Arguments natrP {R x}.
Arguments intrP {R x}.
#[global] Hint Resolve truncn0 truncn1 : core.
#[global] Hint Resolve floor0 floor1 : core.
#[global] Hint Resolve ceil0 ceil1 : core.
#[global] Hint Extern 0 (is_true (_%:R \is a nat_num)) => apply: natr_nat : core.
#[global] Hint Extern 0 (is_true (_%:R \in nat_num_subdef)) => apply: natr_nat : core.
#[global] Hint Extern 0 (is_true (_%:~R \is a int_num)) => apply: intr_int : core.
#[global] Hint Extern 0 (is_true (_%:~R \in int_num_subdef)) => apply: intr_int : core.
#[global] Hint Extern 0 (is_true (_%:R \is a int_num)) => apply: natr_int : core.
#[global] Hint Extern 0 (is_true (_%:R \in int_num_subdef)) => apply: natr_int : core.
#[global] Hint Extern 0 (is_true (0 \is a nat_num)) => apply: nat_num0 : core.
#[global] Hint Extern 0 (is_true (0 \in nat_num_subdef)) => apply: nat_num0 : core.
#[global] Hint Extern 0 (is_true (1 \is a nat_num)) => apply: nat_num1 : core.
#[global] Hint Extern 0 (is_true (1 \in int_num_subdef)) => apply: nat_num1 : core.
#[global] Hint Extern 0 (is_true (0 \is a int_num)) => apply: int_num0 : core.
#[global] Hint Extern 0 (is_true (0 \in int_num_subdef)) => apply: int_num0 : core.
#[global] Hint Extern 0 (is_true (1 \is a int_num)) => apply: int_num1 : core.
#[global] Hint Extern 0 (is_true (1 \in int_num_subdef)) => apply: int_num1 : core.
Section ArchiRealDomainTheory.
Variables (R : archiRealDomainType).
Implicit Type x : R.
Lemma upper_nthrootP x i : (bound x <= i)%N -> x < 2%:R ^+ i.
Proof.
case/truncn_itv/andP: (normr_ge0 x) => _ /ltr_normlW xlt le_b_i.
by rewrite (lt_le_trans xlt) // -natrX ler_nat (ltn_trans le_b_i) // ltn_expl.
Qed.
Lemma truncnS_gt x : x < (truncn x).+1%:R.
Proof. exact: real_truncnS_gt. Qed.
Lemma truncn_le_nat x n : (truncn x <= n)%N = (x < n.+1%:R).
Proof. exact: real_truncn_le_nat. Qed.
Lemma floor_itv x : (floor x)%:~R <= x < (floor x + 1)%:~R.
Proof. exact: real_floor_itv. Qed.
(* TODO: rename to floor_le, once the deprecated one has been removed *)
Lemma floor_le_tmp x : (floor x)%:~R <= x. Proof. exact: real_floor_le. Qed.
Lemma floorD1_gt x : x < (floor x + 1)%:~R.
Proof. exact: real_floorD1_gt. Qed.
#[deprecated(since="mathcomp 2.4.0", note="Use floor_ge_int_tmp instead.")]
Lemma floor_ge_int x n : n%:~R <= x = (n <= floor x).
Proof. by rewrite real_floor_ge_int_tmp. Qed.
(* TODO: rename to floor_ge_int,
once the currently deprecated one has been removed *)
Lemma floor_ge_int_tmp x n : (n <= floor x) = (n%:~R <= x).
Proof. exact: real_floor_ge_int_tmp. Qed.
Lemma floor_lt_int x n : (floor x < n) = (x < n%:~R).
Proof. exact: real_floor_lt_int. Qed.
Lemma floor_eq x n : (floor x == n) = (n%:~R <= x < (n + 1)%:~R).
Proof. exact: real_floor_eq. Qed.
Lemma floorDzr : {in @int_num R, {morph floor : x y / x + y}}.
Proof. by move=> x xz y; apply/real_floorDzr/num_real. Qed.
Lemma floorDrz x y : y \is a int_num -> floor (x + y) = floor x + floor y.
Proof. by move=> yz; apply/real_floorDrz/yz/num_real. Qed.
Lemma floor_ge0 x : (0 <= floor x) = (0 <= x).
Proof. exact: real_floor_ge0. Qed.
Lemma floor_le0 x : (floor x <= 0) = (x < 1).
Proof. exact: real_floor_le0. Qed.
Lemma ceil_itv x : (ceil x - 1)%:~R < x <= (ceil x)%:~R.
Proof. exact: real_ceil_itv. Qed.
Lemma ceilB1_lt x : (ceil x - 1)%:~R < x.
Proof. exact: real_ceilB1_lt. Qed.
Lemma ceil_ge x : x <= (ceil x)%:~R. Proof. exact: real_ceil_ge. Qed.
#[deprecated(since="mathcomp 2.4.0", note="Use ceil_le_int_tmp instead.")]
Lemma ceil_le_int x n : x <= n%:~R = (ceil x <= n).
Proof. by rewrite real_ceil_le_int_tmp. Qed.
(* TODO: rename to ceil_le_int,
once the currently deprecated one has been removed *)
Lemma ceil_le_int_tmp x n : (ceil x <= n) = (x <= n%:~R).
Proof. exact: real_ceil_le_int_tmp. Qed.
Lemma ceil_gt_int x n : (n < ceil x) = (n%:~R < x).
Proof. exact: real_ceil_gt_int. Qed.
Lemma ceil_eq x n : (ceil x == n) = ((n - 1)%:~R < x <= n%:~R).
Proof. exact: real_ceil_eq. Qed.
Lemma ceilDzr : {in @int_num R, {morph ceil : x y / x + y}}.
Proof. by move=> x xz y; apply/real_ceilDzr/num_real. Qed.
Lemma ceilDrz x y : y \is a int_num -> ceil (x + y) = ceil x + ceil y.
Proof. by move=> yz; apply/real_ceilDrz/yz/num_real. Qed.
Lemma ceil_ge0 x : (0 <= ceil x) = (-1 < x).
Proof. exact: real_ceil_ge0. Qed.
Lemma ceil_le0 x : (ceil x <= 0) = (x <= 0).
Proof. exact: real_ceil_le0. Qed.
Lemma ceil_floor x : ceil x = floor x + (x \isn't a int_num).
Proof. exact: real_ceil_floor. Qed.
End ArchiRealDomainTheory.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to floor_le_tmp.")]
Notation ge_floor := floor_le_tmp.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to floorD1_gt.")]
Notation lt_succ_floor := floorD1_gt.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to ceilB1_lt.")]
Notation gt_pred_ceil := ceilB1_lt.
#[deprecated(since="mathcomp 2.4.0", note="Renamed to ceil_ge.")]
Notation le_ceil := ceil_ge.
Section ArchiNumFieldTheory.
(* autLmodC *)
Variables (R : archiNumFieldType) (nu : {rmorphism R -> R}).
Lemma natr_aut x : (nu x \is a nat_num) = (x \is a nat_num).
Proof. by apply/idP/idP=> /[dup] ? /(aut_natr nu) => [/fmorph_inj <-| ->]. Qed.
Lemma intr_aut x : (nu x \is a int_num) = (x \is a int_num).
Proof. by rewrite !intrE -rmorphN !natr_aut. Qed.
End ArchiNumFieldTheory.
Section ArchiClosedFieldTheory.
Variable R : archiClosedFieldType.
Implicit Type x : R.
Lemma conj_natr x : x \is a nat_num -> x^* = x.
Proof. by move/Rreal_nat/CrealP. Qed.
Lemma conj_intr x : x \is a int_num -> x^* = x.
Proof. by move/Rreal_int/CrealP. Qed.
End ArchiClosedFieldTheory.
Section ZnatPred.
Lemma Znat_def (n : int) : (n \is a nat_num) = (0 <= n).
Proof. by []. Qed.
Lemma ZnatP (m : int) : reflect (exists n : nat, m = n) (m \is a nat_num).
Proof. by case: m => m; constructor; [exists m | case]. Qed.
End ZnatPred.
End Theory.
(* Factories *)
HB.factory Record NumDomain_hasTruncn R of Num.NumDomain R := {
trunc : R -> nat;
nat_num : pred R;
int_num : pred R;
truncP : forall x,
if 0 <= x then (trunc x)%:R <= x < (trunc x).+1%:R else trunc x == 0;
natrE : forall x, nat_num x = ((trunc x)%:R == x);
intrE : forall x, int_num x = nat_num x || nat_num (- x);
}.
#[deprecated(since="mathcomp 2.4.0",
note="Use NumDomain_hasTruncn instead.")]
Notation NumDomain_isArchimedean R := (NumDomain_hasTruncn R) (only parsing).
Module NumDomain_isArchimedean.
#[deprecated(since="mathcomp 2.4.0",
note="Use NumDomain_hasTruncn.Build instead.")]
Notation Build T U := (NumDomain_hasTruncn.Build T U) (only parsing).
End NumDomain_isArchimedean.
HB.builders Context R of NumDomain_hasTruncn R.
Fact trunc_itv x : 0 <= x -> (trunc x)%:R <= x < (trunc x).+1%:R.
Proof. by move=> x_ge0; move: (truncP x); rewrite x_ge0. Qed.
Definition floor (x : R) : int :=
if 0 <= x then Posz (trunc x)
else if x < 0 then - Posz (trunc (- x) + ~~ int_num x) else 0.
Fact floorP x :
if x \is Rreal then (floor x)%:~R <= x < (floor x + 1)%:~R else floor x == 0.
Proof.
rewrite /floor intrE !natrE negb_or realE.
case: (comparableP x 0) (@trunc_itv x) => //=;
try by rewrite -PoszD addn1 -pmulrn => _ ->.
move=> x_lt0 _; move: (truncP x); rewrite lt_geF // => /eqP ->.
rewrite gt_eqF //=; move: x_lt0.
rewrite [_ + 1]addrC -opprB !intrN lerNl ltrNr andbC -oppr_gt0.
move: {x}(- x) => x x_gt0; rewrite PoszD -addrA -PoszD.
have ->: Posz ((trunc x)%:R != x) - 1 = - Posz ((trunc x)%:R == x) by case: eqP.
have := trunc_itv (ltW x_gt0); rewrite le_eqVlt.
case: eqVneq => /=; last first.
by rewrite subr0 addn1 -!pmulrn => _ /andP[-> /ltW ->].
by rewrite intrB mulr1z addn0 -!pmulrn => -> _; rewrite gtrBl lexx andbT.
Qed.
Fact truncE x : trunc x = if floor x is Posz n then n else 0.
Proof.
rewrite /floor.
case: (comparableP x 0) (truncP x) => [+ /eqP ->| |_ /eqP ->|] //=.
by case: (_ + _)%N.
Qed.
Fact trunc_def x n : n%:R <= x < n.+1%:R -> trunc x = n.
Proof.
case/andP=> lemx ltxm1; apply/eqP; rewrite eqn_leq -ltnS -[(n <= _)%N]ltnS.
have/trunc_itv/andP[lefx ltxf1]: 0 <= x by apply: le_trans lemx; apply: ler0n.
by rewrite -!(ltr_nat R) 2?(@le_lt_trans _ _ x).
Qed.
Fact natrK : cancel (GRing.natmul 1) trunc.
Proof. by move=> m; apply: trunc_def; rewrite ler_nat ltr_nat ltnS leqnn. Qed.
Fact intrP x : reflect (exists n, x = n%:~R) (int_num x).
Proof.
rewrite intrE !natrE; apply: (iffP idP) => [|[n ->]]; last first.
by case: n => n; rewrite ?NegzE ?opprK natrK eqxx // orbT.
rewrite -eqr_oppLR !pmulrn -intrN.
by move=> /orP[] /eqP<-; [exists (trunc x) | exists (- Posz (trunc (- x)))].
Qed.
Fact natrP x : reflect (exists n, x = n%:R) (nat_num x).
Proof.
rewrite natrE.
by apply: (iffP eqP) => [<-|[n ->]]; [exists (trunc x) | rewrite natrK].
Qed.
HB.instance Definition _ :=
@NumDomain_hasFloorCeilTruncn.Build R floor _ trunc int_num nat_num
floorP (fun=> erefl) truncE intrP natrP.
HB.end.
HB.factory Record NumDomain_bounded_isArchimedean R of Num.NumDomain R := {
archi_bound_subproof : Num.archimedean_axiom R
}.
HB.builders Context R of NumDomain_bounded_isArchimedean R.
Implicit Type x : R.
Definition bound x := sval (sigW (archi_bound_subproof x)).
Lemma boundP x : 0 <= x -> x < (bound x)%:R.
Proof. by move/ger0_norm=> {1}<-; rewrite /bound; case: (sigW _). Qed.
Fact truncn_subproof x : {m | 0 <= x -> m%:R <= x < m.+1%:R }.
Proof.
have [Rx | _] := boolP (0 <= x); last by exists 0%N.
have/ex_minnP[n lt_x_n1 min_n]: exists n, x < n.+1%:R.
by exists (bound x); rewrite (lt_trans (boundP Rx)) ?ltr_nat.
exists n => _; rewrite {}lt_x_n1 andbT; case: n min_n => //= n min_n.
rewrite real_leNgt ?rpred_nat ?ger0_real //; apply/negP => /min_n.
by rewrite ltnn.
Qed.
Definition truncn x := if 0 <= x then sval (truncn_subproof x) else 0%N.
Lemma truncnP x :
if 0 <= x then (truncn x)%:R <= x < (truncn x).+1%:R else truncn x == 0%N.
Proof.
rewrite /truncn; case: truncn_subproof => // n hn.
by case: ifP => x_ge0; rewrite ?(ifT _ _ x_ge0) ?(ifF _ _ x_ge0) // hn.
Qed.
HB.instance Definition _ := NumDomain_hasTruncn.Build R
truncnP (fun => erefl) (fun => erefl).
HB.end.
Module Exports. HB.reexport. End Exports.
(* Not to pollute the local namespace, we define Num.nat and Num.int here. *)
Notation nat := nat_num.
Notation int := int_num.
#[deprecated(since="mathcomp 2.3.0", note="Use Num.ArchiRealDomain instead.")]
Notation ArchiDomain T := (ArchiRealDomain T).
Module ArchiDomain.
#[deprecated(since="mathcomp 2.3.0",
note="Use Num.ArchiRealDomain.type instead.")]
Notation type := ArchiRealDomain.type.
#[deprecated(since="mathcomp 2.3.0",
note="Use Num.ArchiRealDomain.copy instead.")]
Notation copy T C := (ArchiRealDomain.copy T C).
#[deprecated(since="mathcomp 2.3.0",
note="Use Num.ArchiRealDomain.on instead.")]
Notation on T := (ArchiRealDomain.on T).
End ArchiDomain.
#[deprecated(since="mathcomp 2.3.0", note="Use Num.ArchiRealField instead.")]
Notation ArchiField T := (ArchiRealField T).
Module ArchiField.
#[deprecated(since="mathcomp 2.3.0",
note="Use Num.ArchiRealField.type instead.")]
Notation type := ArchiRealField.type.
#[deprecated(since="mathcomp 2.3.0",
note="Use Num.ArchiRealField.copy instead.")]
Notation copy T C := (ArchiRealField.copy T C).
#[deprecated(since="mathcomp 2.3.0", note="Use Num.ArchiRealField.on instead.")]
Notation on T := (ArchiRealField.on T).
End ArchiField.
#[deprecated(since="mathcomp 2.3.0", note="Use real_floorDzr instead.")]
Notation floorD := real_floorDzr.
#[deprecated(since="mathcomp 2.3.0", note="Use real_ceilDzr instead.")]
Notation ceilD := real_ceilDzr.
#[deprecated(since="mathcomp 2.3.0", note="Use real_ceilDzr instead.")]
Notation real_ceilD := real_ceilDzr.
End Num.
Export Num.Exports.
#[deprecated(since="mathcomp 2.3.0", note="Use archiRealDomainType instead.")]
Notation archiDomainType := archiRealDomainType (only parsing).
#[deprecated(since="mathcomp 2.3.0", note="Use archiRealFieldType instead.")]
Notation archiFieldType := archiRealFieldType (only parsing).
|
List.lean
|
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Support
/-!
# Permutations from a list
A list `l : List α` can be interpreted as an `Equiv.Perm α` where each element in the list
is permuted to the next one, defined as `formPerm`. When we have that `Nodup l`,
we prove that `Equiv.Perm.support (formPerm l) = l.toFinset`, and that
`formPerm l` is rotationally invariant, in `formPerm_rotate`.
When there are duplicate elements in `l`, how and in what arrangement with respect to the other
elements they appear in the list determines the formed permutation.
This is because `List.formPerm` is implemented as a product of `Equiv.swap`s.
That means that presence of a sublist of two adjacent duplicates like `[..., x, x, ...]`
will produce the same permutation as if the adjacent duplicates were not present.
The `List.formPerm` definition is meant to primarily be used with `Nodup l`, so that
the resulting permutation is cyclic (if `l` has at least two elements).
The presence of duplicates in a particular placement can lead `List.formPerm` to produce a
nontrivial permutation that is noncyclic.
-/
namespace List
variable {α β : Type*}
section FormPerm
variable [DecidableEq α] (l : List α)
open Equiv Equiv.Perm
/-- A list `l : List α` can be interpreted as an `Equiv.Perm α` where each element in the list
is permuted to the next one, defined as `formPerm`. When we have that `Nodup l`,
we prove that `Equiv.Perm.support (formPerm l) = l.toFinset`, and that
`formPerm l` is rotationally invariant, in `formPerm_rotate`.
-/
def formPerm : Equiv.Perm α :=
(zipWith Equiv.swap l l.tail).prod
@[simp]
theorem formPerm_nil : formPerm ([] : List α) = 1 :=
rfl
@[simp]
theorem formPerm_singleton (x : α) : formPerm [x] = 1 :=
rfl
@[simp]
theorem formPerm_cons_cons (x y : α) (l : List α) :
formPerm (x :: y :: l) = swap x y * formPerm (y :: l) :=
rfl
theorem formPerm_pair (x y : α) : formPerm [x, y] = swap x y :=
rfl
theorem mem_or_mem_of_zipWith_swap_prod_ne : ∀ {l l' : List α} {x : α},
(zipWith swap l l').prod x ≠ x → x ∈ l ∨ x ∈ l'
| [], _, _ => by simp
| _, [], _ => by simp
| a::l, b::l', x => fun hx ↦
if h : (zipWith swap l l').prod x = x then
(eq_or_eq_of_swap_apply_ne_self (a := a) (b := b) (x := x) (by simpa [h] using hx)).imp
(by rintro rfl; exact .head _) (by rintro rfl; exact .head _)
else
(mem_or_mem_of_zipWith_swap_prod_ne h).imp (.tail _) (.tail _)
theorem zipWith_swap_prod_support' (l l' : List α) :
{ x | (zipWith swap l l').prod x ≠ x } ≤ l.toFinset ⊔ l'.toFinset := fun _ h ↦ by
simpa using mem_or_mem_of_zipWith_swap_prod_ne h
theorem zipWith_swap_prod_support [Fintype α] (l l' : List α) :
(zipWith swap l l').prod.support ≤ l.toFinset ⊔ l'.toFinset := by
intro x hx
have hx' : x ∈ { x | (zipWith swap l l').prod x ≠ x } := by simpa using hx
simpa using zipWith_swap_prod_support' _ _ hx'
theorem support_formPerm_le' : { x | formPerm l x ≠ x } ≤ l.toFinset := by
refine (zipWith_swap_prod_support' l l.tail).trans ?_
simpa [Finset.subset_iff] using tail_subset l
theorem support_formPerm_le [Fintype α] : support (formPerm l) ≤ l.toFinset := by
intro x hx
have hx' : x ∈ { x | formPerm l x ≠ x } := by simpa using hx
simpa using support_formPerm_le' _ hx'
variable {l} {x : α}
theorem mem_of_formPerm_apply_ne (h : l.formPerm x ≠ x) : x ∈ l := by
simpa [or_iff_left_of_imp mem_of_mem_tail] using mem_or_mem_of_zipWith_swap_prod_ne h
theorem formPerm_apply_of_notMem (h : x ∉ l) : formPerm l x = x :=
not_imp_comm.1 mem_of_formPerm_apply_ne h
@[deprecated (since := "2025-05-23")] alias formPerm_apply_of_not_mem := formPerm_apply_of_notMem
theorem formPerm_apply_mem_of_mem (h : x ∈ l) : formPerm l x ∈ l := by
rcases l with - | ⟨y, l⟩
· simp at h
induction l generalizing x y with
| nil => simpa using h
| cons z l IH =>
by_cases hx : x ∈ z :: l
· rw [formPerm_cons_cons, mul_apply, swap_apply_def]
split_ifs
· simp
· simp
· simp [*]
· replace h : x = y := Or.resolve_right (mem_cons.1 h) hx
simp [formPerm_apply_of_notMem hx, ← h]
theorem mem_of_formPerm_apply_mem (h : l.formPerm x ∈ l) : x ∈ l := by
contrapose h
rwa [formPerm_apply_of_notMem h]
@[simp]
theorem formPerm_mem_iff_mem : l.formPerm x ∈ l ↔ x ∈ l :=
⟨l.mem_of_formPerm_apply_mem, l.formPerm_apply_mem_of_mem⟩
@[simp]
theorem formPerm_cons_concat_apply_last (x y : α) (xs : List α) :
formPerm (x :: (xs ++ [y])) y = x := by
induction xs generalizing x y with
| nil => simp
| cons z xs IH => simp [IH]
@[simp]
theorem formPerm_apply_getLast (x : α) (xs : List α) :
formPerm (x :: xs) ((x :: xs).getLast (cons_ne_nil x xs)) = x := by
induction xs using List.reverseRecOn generalizing x <;> simp
@[simp]
theorem formPerm_apply_getElem_length (x : α) (xs : List α) :
formPerm (x :: xs) (x :: xs)[xs.length] = x := by
rw [getElem_cons_length rfl, formPerm_apply_getLast]
theorem formPerm_apply_head (x y : α) (xs : List α) (h : Nodup (x :: y :: xs)) :
formPerm (x :: y :: xs) x = y := by simp [formPerm_apply_of_notMem h.notMem]
theorem formPerm_apply_getElem_zero (l : List α) (h : Nodup l) (hl : 1 < l.length) :
formPerm l l[0] = l[1] := by
rcases l with (_ | ⟨x, _ | ⟨y, tl⟩⟩)
· simp at hl
· simp at hl
· rw [getElem_cons_zero, formPerm_apply_head _ _ _ h, getElem_cons_succ, getElem_cons_zero]
variable (l)
theorem formPerm_eq_head_iff_eq_getLast (x y : α) :
formPerm (y :: l) x = y ↔ x = getLast (y :: l) (cons_ne_nil _ _) :=
Iff.trans (by rw [formPerm_apply_getLast]) (formPerm (y :: l)).injective.eq_iff
theorem formPerm_apply_lt_getElem (xs : List α) (h : Nodup xs) (n : ℕ) (hn : n + 1 < xs.length) :
formPerm xs xs[n] = xs[n+1] := by
induction n generalizing xs with
| zero => simpa using formPerm_apply_getElem_zero _ h _
| succ n IH =>
rcases xs with (_ | ⟨x, _ | ⟨y, l⟩⟩)
· simp at hn
· rw [formPerm_singleton, getElem_singleton, getElem_singleton, one_apply]
· specialize IH (y :: l) h.of_cons _
· simpa [Nat.succ_lt_succ_iff] using hn
simp only [swap_apply_eq_iff, coe_mul, formPerm_cons_cons, Function.comp]
simp only [getElem_cons_succ] at *
rw [← IH, swap_apply_of_ne_of_ne] <;>
· intro hx
rw [← hx, IH] at h
simp [getElem_mem] at h
theorem formPerm_apply_getElem (xs : List α) (w : Nodup xs) (i : ℕ) (h : i < xs.length) :
formPerm xs xs[i] =
xs[(i + 1) % xs.length]'(Nat.mod_lt _ (i.zero_le.trans_lt h)) := by
rcases xs with - | ⟨x, xs⟩
· simp at h
· have : i ≤ xs.length := by
refine Nat.le_of_lt_succ ?_
simpa using h
rcases this.eq_or_lt with (rfl | hn')
· simp
· rw [formPerm_apply_lt_getElem (x :: xs) w _ (Nat.succ_lt_succ hn')]
congr
rw [Nat.mod_eq_of_lt]; simpa [Nat.succ_eq_add_one]
theorem support_formPerm_of_nodup' (l : List α) (h : Nodup l) (h' : ∀ x : α, l ≠ [x]) :
{ x | formPerm l x ≠ x } = l.toFinset := by
apply _root_.le_antisymm
· exact support_formPerm_le' l
· intro x hx
simp only [Finset.mem_coe, mem_toFinset] at hx
obtain ⟨n, hn, rfl⟩ := getElem_of_mem hx
rw [Set.mem_setOf_eq, formPerm_apply_getElem _ h]
intro H
rw [nodup_iff_injective_get, Function.Injective] at h
specialize h H
rcases (Nat.succ_le_of_lt hn).eq_or_lt with hn' | hn'
· simp only [← hn', Nat.mod_self] at h
refine not_exists.mpr h' ?_
rw [← length_eq_one_iff, ← hn', (Fin.mk.inj_iff.mp h).symm]
· simp [Nat.mod_eq_of_lt hn'] at h
theorem support_formPerm_of_nodup [Fintype α] (l : List α) (h : Nodup l) (h' : ∀ x : α, l ≠ [x]) :
support (formPerm l) = l.toFinset := by
rw [← Finset.coe_inj]
convert support_formPerm_of_nodup' _ h h'
simp [Set.ext_iff]
theorem formPerm_rotate_one (l : List α) (h : Nodup l) : formPerm (l.rotate 1) = formPerm l := by
have h' : Nodup (l.rotate 1) := by simpa using h
ext x
by_cases hx : x ∈ l.rotate 1
· obtain ⟨k, hk, rfl⟩ := getElem_of_mem hx
rw [formPerm_apply_getElem _ h', getElem_rotate l, getElem_rotate l, formPerm_apply_getElem _ h]
simp
· rw [formPerm_apply_of_notMem hx, formPerm_apply_of_notMem]
simpa using hx
theorem formPerm_rotate (l : List α) (h : Nodup l) (n : ℕ) :
formPerm (l.rotate n) = formPerm l := by
induction n with
| zero => simp
| succ n hn =>
rw [← rotate_rotate, formPerm_rotate_one, hn]
rwa [IsRotated.nodup_iff]
exact IsRotated.forall l n
theorem formPerm_eq_of_isRotated {l l' : List α} (hd : Nodup l) (h : l ~r l') :
formPerm l = formPerm l' := by
obtain ⟨n, rfl⟩ := h
exact (formPerm_rotate l hd n).symm
theorem formPerm_append_pair : ∀ (l : List α) (a b : α),
formPerm (l ++ [a, b]) = formPerm (l ++ [a]) * swap a b
| [], _, _ => rfl
| [_], _, _ => rfl
| x::y::l, a, b => by
simpa [mul_assoc] using formPerm_append_pair (y::l) a b
theorem formPerm_reverse : ∀ l : List α, formPerm l.reverse = (formPerm l)⁻¹
| [] => rfl
| [_] => rfl
| a::b::l => by
simp [formPerm_append_pair, swap_comm, ← formPerm_reverse (b::l)]
theorem formPerm_pow_apply_getElem (l : List α) (w : Nodup l) (n : ℕ) (i : ℕ) (h : i < l.length) :
(formPerm l ^ n) l[i] =
l[(i + n) % l.length]'(Nat.mod_lt _ (i.zero_le.trans_lt h)) := by
induction n with
| zero => simp [Nat.mod_eq_of_lt h]
| succ n hn =>
simp [pow_succ', mul_apply, hn, formPerm_apply_getElem _ w,
← Nat.add_assoc]
theorem formPerm_pow_apply_head (x : α) (l : List α) (h : Nodup (x :: l)) (n : ℕ) :
(formPerm (x :: l) ^ n) x =
(x :: l)[(n % (x :: l).length)]'(Nat.mod_lt _ (Nat.zero_lt_succ _)) := by
convert formPerm_pow_apply_getElem _ h n 0 (Nat.succ_pos _)
simp
theorem formPerm_ext_iff {x y x' y' : α} {l l' : List α} (hd : Nodup (x :: y :: l))
(hd' : Nodup (x' :: y' :: l')) :
formPerm (x :: y :: l) = formPerm (x' :: y' :: l') ↔ (x :: y :: l) ~r (x' :: y' :: l') := by
refine ⟨fun h => ?_, fun hr => formPerm_eq_of_isRotated hd hr⟩
rw [Equiv.Perm.ext_iff] at h
have hx : x' ∈ x :: y :: l := by
have : x' ∈ { z | formPerm (x :: y :: l) z ≠ z } := by
rw [Set.mem_setOf_eq, h x', formPerm_apply_head _ _ _ hd']
simp only [mem_cons, nodup_cons] at hd'
push_neg at hd'
exact hd'.left.left.symm
simpa using support_formPerm_le' _ this
obtain ⟨⟨n, hn⟩, hx'⟩ := get_of_mem hx
have hl : (x :: y :: l).length = (x' :: y' :: l').length := by
rw [← dedup_eq_self.mpr hd, ← dedup_eq_self.mpr hd', ← card_toFinset, ← card_toFinset]
refine congr_arg Finset.card ?_
rw [← Finset.coe_inj, ← support_formPerm_of_nodup' _ hd (by simp), ←
support_formPerm_of_nodup' _ hd' (by simp)]
simp only [h]
use n
apply List.ext_getElem
· rw [length_rotate, hl]
· intro k hk hk'
rw [getElem_rotate]
induction k with
| zero =>
refine Eq.trans ?_ hx'
congr
simpa using hn
| succ k IH =>
conv => congr <;> · arg 2; (rw [← Nat.mod_eq_of_lt hk'])
rw [← formPerm_apply_getElem _ hd' k (k.lt_succ_self.trans hk'),
← IH (k.lt_succ_self.trans hk), ← h, formPerm_apply_getElem _ hd]
congr 1
rw [hl, Nat.mod_eq_of_lt hk', add_right_comm]
apply Nat.add_mod
theorem formPerm_apply_mem_eq_self_iff (hl : Nodup l) (x : α) (hx : x ∈ l) :
formPerm l x = x ↔ length l ≤ 1 := by
obtain ⟨k, hk, rfl⟩ := getElem_of_mem hx
rw [formPerm_apply_getElem _ hl k hk, hl.getElem_inj_iff]
cases hn : l.length
· exact absurd k.zero_le (hk.trans_le hn.le).not_ge
· rw [hn] at hk
rcases (Nat.le_of_lt_succ hk).eq_or_lt with hk' | hk'
· simp [← hk', eq_comm]
· simpa [Nat.mod_eq_of_lt (Nat.succ_lt_succ hk'), Nat.succ_lt_succ_iff] using
(k.zero_le.trans_lt hk').ne.symm
theorem formPerm_apply_mem_ne_self_iff (hl : Nodup l) (x : α) (hx : x ∈ l) :
formPerm l x ≠ x ↔ 2 ≤ l.length := by
rw [Ne, formPerm_apply_mem_eq_self_iff _ hl x hx, not_le]
exact ⟨Nat.succ_le_of_lt, Nat.lt_of_succ_le⟩
theorem mem_of_formPerm_ne_self (l : List α) (x : α) (h : formPerm l x ≠ x) : x ∈ l := by
suffices x ∈ { y | formPerm l y ≠ y } by
rw [← mem_toFinset]
exact support_formPerm_le' _ this
simpa using h
theorem formPerm_eq_self_of_notMem (l : List α) (x : α) (h : x ∉ l) : formPerm l x = x :=
by_contra fun H => h <| mem_of_formPerm_ne_self _ _ H
@[deprecated (since := "2025-05-23")]
alias formPerm_eq_self_of_not_mem := formPerm_eq_self_of_notMem
theorem formPerm_eq_one_iff (hl : Nodup l) : formPerm l = 1 ↔ l.length ≤ 1 := by
rcases l with - | ⟨hd, tl⟩
· simp
· rw [← formPerm_apply_mem_eq_self_iff _ hl hd mem_cons_self]
constructor
· simp +contextual
· intro h
simp only [(hd :: tl).formPerm_apply_mem_eq_self_iff hl hd mem_cons_self,
add_le_iff_nonpos_left, length, nonpos_iff_eq_zero, length_eq_zero_iff] at h
simp [h]
theorem formPerm_eq_formPerm_iff {l l' : List α} (hl : l.Nodup) (hl' : l'.Nodup) :
l.formPerm = l'.formPerm ↔ l ~r l' ∨ l.length ≤ 1 ∧ l'.length ≤ 1 := by
rcases l with (_ | ⟨x, _ | ⟨y, l⟩⟩)
· suffices l'.length ≤ 1 ↔ l' = nil ∨ l'.length ≤ 1 by
simpa [eq_comm, formPerm_eq_one_iff, hl, hl', length_eq_zero_iff]
refine ⟨fun h => Or.inr h, ?_⟩
rintro (rfl | h)
· simp
· exact h
· suffices l'.length ≤ 1 ↔ [x] ~r l' ∨ l'.length ≤ 1 by
simpa [eq_comm, formPerm_eq_one_iff, hl, hl', length_eq_zero_iff, le_rfl]
refine ⟨fun h => Or.inr h, ?_⟩
rintro (h | h)
· simp [← h.perm.length_eq]
· exact h
· rcases l' with (_ | ⟨x', _ | ⟨y', l'⟩⟩)
· simp [formPerm_eq_one_iff _ hl, -formPerm_cons_cons]
· simp [formPerm_eq_one_iff _ hl, -formPerm_cons_cons]
· simp [-formPerm_cons_cons, formPerm_ext_iff hl hl']
theorem form_perm_zpow_apply_mem_imp_mem (l : List α) (x : α) (hx : x ∈ l) (n : ℤ) :
(formPerm l ^ n) x ∈ l := by
by_cases h : (l.formPerm ^ n) x = x
· simpa [h] using hx
· have h : x ∈ { x | (l.formPerm ^ n) x ≠ x } := h
rw [← set_support_apply_mem] at h
replace h := set_support_zpow_subset _ _ h
simpa using support_formPerm_le' _ h
theorem formPerm_pow_length_eq_one_of_nodup (hl : Nodup l) : formPerm l ^ length l = 1 := by
ext x
by_cases hx : x ∈ l
· obtain ⟨k, hk, rfl⟩ := getElem_of_mem hx
simp [formPerm_pow_apply_getElem _ hl, Nat.mod_eq_of_lt hk]
· have : x ∉ { x | (l.formPerm ^ l.length) x ≠ x } := by
intro H
refine hx ?_
replace H := set_support_zpow_subset l.formPerm l.length H
simpa using support_formPerm_le' _ H
simpa using this
end FormPerm
end List
|
linear_combination.lean
|
import Mathlib.Tactic.Abel
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Module
set_option autoImplicit true
private axiom test_sorry : ∀ {α}, α
-- We deliberately mock R here so that we don't have to import the deps
axiom Real : Type
notation "ℝ" => Real
@[instance] axiom Real.field : Field ℝ
@[instance] axiom Real.linearOrder : LinearOrder ℝ
@[instance] axiom Real.isStrictOrderedRing : IsStrictOrderedRing ℝ
/-! ### Simple Cases with ℤ and two or less equations -/
example (x y : ℤ) (h1 : 3 * x + 2 * y = 10) : 3 * x + 2 * y = 10 := by
linear_combination 1 * h1
example (x y : ℤ) (h1 : 3 * x + 2 * y = 10) : 3 * x + 2 * y = 10 := by
linear_combination h1
example (x y : ℤ) (h1 : x + 2 = -3) (_h2 : y = 10) : 2 * x + 4 = -6 := by
linear_combination 2 * h1
example (x y : ℤ) (h1 : x * y + 2 * x = 1) (h2 : x = y) : x * y = -2 * y + 1 := by
linear_combination 1 * h1 - 2 * h2
example (x y : ℤ) (h1 : x * y + 2 * x = 1) (h2 : x = y) : x * y = -2 * y + 1 := by
linear_combination -2 * h2 + h1
example (x y : ℤ) (h1 : x + 2 = -3) (h2 : y = 10) : 2 * x + 4 - y = -16 := by
linear_combination 2 * h1 + -1 * h2
example (x y : ℤ) (h1 : x + 2 = -3) (h2 : y = 10) : -y + 2 * x + 4 = -16 := by
linear_combination -h2 + 2 * h1
example (x y : ℤ) (h1 : 3 * x + 2 * y = 10) (h2 : 2 * x + 5 * y = 3) : 11 * y = -11 := by
linear_combination -2 * h1 + 3 * h2
example (x y : ℤ) (h1 : 3 * x + 2 * y = 10) (h2 : 2 * x + 5 * y = 3) : -11 * y = 11 := by
linear_combination 2 * h1 - 3 * h2
example (x y : ℤ) (h1 : 3 * x + 2 * y = 10) (h2 : 2 * x + 5 * y = 3) : -11 * y = 11 + 1 - 1 := by
linear_combination 2 * h1 + -3 * h2
example (x y : ℤ) (h1 : 10 = 3 * x + 2 * y) (h2 : 3 = 2 * x + 5 * y) : 11 + 1 - 1 = -11 * y := by
linear_combination 2 * h1 - 3 * h2
/-! ### More complicated cases with two equations -/
example (x y : ℤ) (h1 : x + 2 = -3) (h2 : y = 10) : -y + 2 * x + 4 = -16 := by
linear_combination 2 * h1 - h2
example (x y : ℚ) (h1 : 3 * x + 2 * y = 10) (h2 : 2 * x + 5 * y = 3) : -11 * y + 1 = 11 + 1 := by
linear_combination 2 * h1 - 3 * h2
example (a b : ℝ) (ha : 2 * a = 4) (hab : 2 * b = a - b) : b = 2 / 3 := by
linear_combination ha / 6 + hab / 3
/-! ### Cases with more than 2 equations -/
example (a b : ℝ) (ha : 2 * a = 4) (hab : 2 * b = a - b) (hignore : 3 = a + b) : b = 2 / 3 := by
linear_combination 1 / 6 * ha + 1 / 3 * hab + 0 * hignore
example (x y z : ℝ) (ha : x + 2 * y - z = 4) (hb : 2 * x + y + z = -2) (hc : x + 2 * y + z = 2) :
-3 * x - 3 * y - 4 * z = 2 := by linear_combination ha - hb - 2 * hc
example (x y z : ℝ) (ha : x + 2 * y - z = 4) (hb : 2 * x + y + z = -2) (hc : x + 2 * y + z = 2) :
6 * x = -10 := by
linear_combination 1 * ha + 4 * hb - 3 * hc
example (x y z : ℝ) (ha : x + 2 * y - z = 4) (hb : 2 * x + y + z = -2) (hc : x + 2 * y + z = 2) :
10 = 6 * -x := by
linear_combination ha + 4 * hb - 3 * hc
example (w x y z : ℝ) (h1 : x + 2.1 * y + 2 * z = 2) (h2 : x + 8 * z + 5 * w = -6.5)
(h3 : x + y + 5 * z + 5 * w = 3) : x + 2.2 * y + 2 * z - 5 * w = -8.5 := by
linear_combination 2 * h1 + 1 * h2 - 2 * h3
example (w x y z : ℝ) (h1 : x + 2.1 * y + 2 * z = 2) (h2 : x + 8 * z + 5 * w = -6.5)
(h3 : x + y + 5 * z + 5 * w = 3) : x + 2.2 * y + 2 * z - 5 * w = -8.5 := by
linear_combination 2 * h1 + h2 - 2 * h3
example (a b c d : ℚ) (h1 : a = 4) (h2 : 3 = b) (h3 : c * 3 = d) (h4 : -d = a) :
2 * a - 3 + 9 * c + 3 * d = 8 - b + 3 * d - 3 * a := by
linear_combination 2 * h1 - 1 * h2 + 3 * h3 - 3 * h4
example (a b c d : ℚ) (h1 : a = 4) (h2 : 3 = b) (h3 : c * 3 = d) (h4 : -d = a) :
6 - 3 * c + 3 * a + 3 * d = 2 * b - d + 12 - 3 * a := by
linear_combination 2 * h2 - h3 + 3 * h1 - 3 * h4
/-! ### Cases with non-hypothesis inputs -/
axiom qc : ℚ
axiom hqc : qc = 2 * qc
example (a b : ℚ) (h : ∀ p q : ℚ, p = q) : 3 * a + qc = 3 * b + 2 * qc := by
linear_combination 3 * h a b + hqc
axiom bad (q : ℚ) : q = 0
example (a b : ℚ) : a + b ^ 3 = 0 := by linear_combination bad a + b * bad (b * b)
/-! ### Cases with arbitrary coefficients -/
example (a b : ℤ) (h : a = b) : a * a = a * b := by linear_combination a * h
example (a b c : ℤ) (h : a = b) : a * c = b * c := by linear_combination c * h
example (a b c : ℤ) (h1 : a = b) (h2 : b = 1) : c * a + b = c * b + 1 := by
linear_combination c * h1 + h2
example (x y : ℚ) (h1 : x + y = 3) (h2 : 3 * x = 7) :
x * x * y + y * x * y + 6 * x = 3 * x * y + 14 := by
linear_combination x * y * h1 + 2 * h2
example {α} [h : CommRing α] {a b c d e f : α} (h1 : a * d = b * c) (h2 : c * f = e * d) :
c * (a * f - b * e) = 0 := by linear_combination e * h1 + a * h2
example (x y z w : ℚ) (hzw : z = w) : x * z + 2 * y * z = x * w + 2 * y * w := by
linear_combination (x + 2 * y) * hzw
example (x y : ℤ) (h : x = 0) : y ^ 2 * x = 0 := by linear_combination y ^ 2 * h
/-! ### Scalar multiplication -/
section
variable {K V : Type*}
section
variable [AddCommGroup V] [Field K] [CharZero K] [Module K V] {a b μ ν : K} {v w x y : V}
example (h : a ^ 2 + b ^ 2 = 1) : a • (a • x - b • y) + (b • a • y + b • b • x) = x := by
linear_combination (norm := module) h • x
example (h1 : a • x + b • y = 0) (h2 : a • μ • x + b • ν • y = 0) : (μ - ν) • a • x = 0 := by
linear_combination (norm := module) h2 - ν • h1
example (h₁ : x - y = -(v - w)) (h₂ : x + y = v + w) : x = w := by
linear_combination (norm := module) (2:K)⁻¹ • h₁ + (2:K)⁻¹ • h₂
example (h : a + b ≠ 0) (H : a • x = b • y) : x = (b / (a + b)) • (x + y) := by
linear_combination (norm := match_scalars) (a + b)⁻¹ • H
· field_simp
ring
· ring
end
example [CommSemiring K] [PartialOrder K] [IsOrderedRing K]
[AddCommMonoid V] [PartialOrder V] [IsOrderedCancelAddMonoid V] [Module K V] [OrderedSMul K V]
{x y r : V} (hx : x < r) (hy : y < r) {a b : K} (ha : 0 < a) (hb : 0 ≤ b) (hab : a + b = 1) :
a • x + b • y < r := by
linear_combination (norm := skip) a • hx + b • hy + hab • r
apply le_of_eq
module
example [CommSemiring K] [PartialOrder K] [IsOrderedRing K]
[AddCommMonoid V] [PartialOrder V] [IsOrderedCancelAddMonoid V] [Module K V] [OrderedSMul K V]
{x y z : V} (hyz : y ≤ z) {a b : K} (hb : 0 ≤ b) (hab : a + b = 1) (H : z ≤ a • x + b • y) :
a • z ≤ a • x := by
linear_combination (norm := skip) b • hyz + hab • z + H
apply le_of_eq
module
example [CommRing K] [PartialOrder K] [IsOrderedRing K]
[AddCommGroup V] [PartialOrder V] [IsOrderedAddMonoid V] [Module K V] [OrderedSMul K V]
{x y : V} (hx : 0 < x) (hxy : x < y) {a b c : K} (hc : 0 < c) (hac : c < a) (hab : a + b ≤ 1):
c • x + b • y < y := by
have := hx.trans hxy
linear_combination (norm := skip) hab • y + hac • y + c • hxy
apply le_of_eq
module
end
/-! ### Tests in semirings -/
example (a _b : ℕ) (h1 : a = 3) : a = 3 := by
linear_combination h1
example {a b : ℕ} (h1 : a = b + 4) (h2 : b = 2) : a = 6 := by
linear_combination h1 + h2
example {a : ℕ} (h : a = 3) : 3 = a := by linear_combination -h
example {a b : ℕ} (h1 : 3 * a = b + 5) (h2 : 2 * a = b + 3) : a = 2 := by
linear_combination h1 - h2
/- Note: currently negation/subtraction is handled differently in "constants" than in "proofs", so
in particular negation/subtraction does not "distribute". The following four tests record the
current behaviour, without taking a stance on whether this should be considered a feature or a bug.
-/
example {a : ℕ} (h : a = 3) : a ^ 2 + 3 = 4 * a := by
linear_combination a * h - h
/--
error: ring failed, ring expressions not equal
a b : ℕ
h : a = 3
⊢ 3 + a ^ 2 + (a - 1) * 3 = a * 4 + a * (a - 1)
-/
#guard_msgs in
example {a b : ℕ} (h : a = 3) : a ^ 2 + 3 = 4 * a := by
linear_combination (a - 1) * h
example {a b c : ℕ} (h1 : c = 1) (h2 : a - b = 4) : (a - b) * c = 4 := by
linear_combination (a - b) * h1 + h2
/--
error: ring failed, ring expressions not equal
a b c : ℕ
h1 : c = 1
h2 : a - b = 4
⊢ 4 + (a - b) * c + c * b + a = 4 + (a - b) + c * a + b
-/
#guard_msgs in
example {a b c : ℕ} (h1 : c = 1) (h2 : a - b = 4) : (a - b) * c = 4 := by
linear_combination a * h1 - b * h1 + h2
/-! ### Cases that explicitly use a config -/
example (x y : ℚ) (h1 : 3 * x + 2 * y = 10) (h2 : 2 * x + 5 * y = 3) : -11 * y + 1 = 11 + 1 := by
linear_combination (norm := ring) 2 * h1 - 3 * h2
example (x y : ℚ) (h1 : 3 * x + 2 * y = 10) (h2 : 2 * x + 5 * y = 3) : -11 * y + 1 = 11 + 1 := by
linear_combination (norm := ring1) 2 * h1 + -3 * h2
example (a b : ℝ) (ha : 2 * a = 4) (hab : 2 * b = a - b) : b = 2 / 3 := by
linear_combination (norm := ring_nf) 1 / 6 * ha + 1 / 3 * hab
example (x y : ℤ) (h1 : 3 * x + 2 * y = 10) : 3 * x + 2 * y = 10 := by
linear_combination (norm := skip) h1
ring1
/-! ### Cases that have linear_combination skip normalization -/
example (a b : ℝ) (ha : 2 * a = 4) (hab : 2 * b = a - b) : b = 2 / 3 := by
linear_combination (norm := skip) 1 / 6 * ha + 1 / 3 * hab
linarith
example (x y : ℤ) (h1 : x = -3) (_h2 : y = 10) : 2 * x = -6 := by
linear_combination (norm := skip) 2 * h1
ring1
/-! ### Cases without any arguments provided -/
-- the corner case is "just apply the normalization procedure".
example {x y z w : ℤ} (_h₁ : 3 * x = 4 + y) (_h₂ : x + 2 * y = 1) : z + w = w + z := by
linear_combination
example (x : ℤ) : x ^ 2 = x ^ 2 := by linear_combination
-- this interacts as expected with options
example {x y z w : ℤ} (_h₁ : 3 * x = 4 + y) (_h₂ : x + 2 * y = 1) : z + w = w + z := by
linear_combination (norm := skip)
guard_target = z + w + 0 - (w + z + 0) = 0
simp [add_comm]
example {x y z w : ℤ} (_h₁ : 3 * x = 4 + y) (_h₂ : x + 2 * y = 1) : z + w = w + z := by
linear_combination (norm := simp [add_comm])
/-! ### Cases where the goal is not closed -/
example (x y : ℚ) (h1 : x + y = 3) (h2 : 3 * x = 7) :
x * x * y + y * x * y + 6 * x = 3 * x * y + 14 := by
linear_combination (norm := ring_nf) x * y * h1 + h2
guard_target = -7 + x * 3 = 0
linear_combination h2
example (a b c d : ℚ) (h1 : a = 4) (h2 : 3 = b) (h3 : c * 3 = d) (h4 : -d = a) :
6 - 3 * c + 3 * a + 3 * d = 2 * b - d + 12 - 3 * a := by
linear_combination (norm := ring_nf) 2 * h2
linear_combination (norm := ring_nf) -h3
linear_combination (norm := ring_nf) 3 * h1
linear_combination (norm := ring_nf) -3 * h4
example (x y : ℤ) (h1 : x * y + 2 * x = 1) (h2 : x = y) : x * y = -2 * y + 1 := by
linear_combination (norm := ring_nf)
linear_combination h1 - 2 * h2
example {A : Type*} [AddCommGroup A] {x y z : A} (h1 : x + y = 10 • z) (h2 : x - y = 6 • z) :
2 • x = 2 • (8 • z) := by
linear_combination (norm := abel) h1 + h2
/-! ### Cases that should fail -/
/--
error: ring failed, ring expressions not equal
a : ℤ
ha : a = 1
⊢ -1 = 0
-/
#guard_msgs in
example (a : ℤ) (ha : a = 1) : a = 2 := by linear_combination ha
/--
error: ring failed, ring expressions not equal
a : ℚ
ha : a = 1
⊢ -1 = 0
-/
#guard_msgs in
example (a : ℚ) (ha : a = 1) : a = 2 := by linear_combination ha
-- This should fail because the second coefficient has a different type than
-- the equations it is being combined with. This was a design choice for the
-- sake of simplicity, but the tactic could potentially be modified to allow
-- this behavior.
/--
error: Application type mismatch: The argument
0
has type
ℝ
but is expected to have type
ℤ
in the application
Mathlib.Tactic.LinearCombination.mul_const_eq h2 0
-/
#guard_msgs in
example (x y : ℤ) (h1 : x * y + 2 * x = 1) (h2 : x = y) : x * y + 2 * x = 1 := by
linear_combination h1 + (0 : ℝ) * h2
set_option linter.unusedVariables false in
example (a b : ℤ) (x y : ℝ) (hab : a = b) (hxy : x = y) : 2 * x = 2 * y := by
fail_if_success linear_combination 2 * hab
linear_combination 2 * hxy
/--
warning: this constant has no effect on the linear combination; it can be dropped from the term
-/
#guard_msgs in
example (x y : ℤ) (h1 : 3 * x + 2 * y = 10) : 3 * x + 2 * y = 10 := by
linear_combination h1 + 3
/--
warning: this constant has no effect on the linear combination; it can be dropped from the term
-/
#guard_msgs in
example (x : ℤ) : x ^ 2 = x ^ 2 := by linear_combination x ^ 2
/-- error: 'linear_combination' supports only linear operations -/
#guard_msgs in
example {x y : ℤ} (h : x = y) : x ^ 2 = y ^ 2 := by linear_combination h * h
/-- error: 'linear_combination' supports only linear operations -/
#guard_msgs in
example {x y : ℤ} (h : x = y) : 3 / x = 3 / y := by linear_combination 3 / h
/-! ### Cases with exponent -/
example (x y z : ℚ) (h : x = y) (h2 : x * y = 0) : x + y*z = 0 := by
linear_combination (exp := 2) (-y * z ^ 2 + x) * h + (z ^ 2 + 2 * z + 1) * h2
example (x y z : ℚ) (h : x = y) (h2 : x * y = 0) : y*z = -x := by
linear_combination (norm := skip) (exp := 2) (-y * z ^ 2 + x) * h + (z ^ 2 + 2 * z + 1) * h2
ring
example (K : Type)
[Field K]
[CharZero K]
{x y z : K}
(h₂ : y ^ 3 + x * (3 * z ^ 2) = 0)
(h₁ : x ^ 3 + z * (3 * y ^ 2) = 0)
(h₀ : y * (3 * x ^ 2) + z ^ 3 = 0)
(h : x ^ 3 * y + y ^ 3 * z + z ^ 3 * x = 0) :
x = 0 := by
linear_combination (exp := 6) 2 * y * z ^ 2 * h₂ / 7 + (x ^ 3 - y ^ 2 * z / 7) * h₁ -
x * y * z * h₀ + y * z * h / 7
/-! ### Linear inequalities -/
example : (3:ℤ) ≤ 4 := by linear_combination
example (x : ℚ) (hx : x ≤ 3) : x - 1 ≤ 5 := by linear_combination hx
example (x : ℝ) (hx : x ≤ 3) : x - 1 ≤ 5 := by linear_combination hx
example (a b : ℚ) (h1 : a ≤ 1) (h2 : b ≤ 1) : a + b ≤ 2 := by linear_combination h1 + h2
example (a b : ℚ) (h1 : a ≤ 1) (h2 : b = 1) : a + b < 3 := by linear_combination h1 + h2
example (a b : ℚ) (h1 : a ≤ 1) (h2 : b ≥ 2) : a ≤ b := by linear_combination h1 + h2
example (a : ℚ) (ha : 0 ≤ a) : 0 ≤ 2 * a := by linear_combination 2 * ha
example {x y : ℚ} (h : x + 1 < y) : x < y := by linear_combination h
example {x y : ℚ} (h : x < y) : x < y := by linear_combination h
example (a b : ℚ) (h1 : a ≤ 1) (h2 : b = 1) : (a + b) / 2 ≤ 1 := by linear_combination (h1 + h2) / 2
example {x y : ℤ} (hx : x + 3 ≤ 2) (hy : y + 2 * x ≥ 3) : y > 3 := by linear_combination hy + 2 * hx
example {x y : ℕ} (hx : x + 3 ≤ 2) (hy : y + 2 * x ≥ 3) : y > 3 := by linear_combination hy + 2 * hx
example {x y : ℤ} (h : x + 1 ≤ y) : x < y := by linear_combination h
example {x y z : ℚ} (h1 : 4 * x + y + 3 * z ≤ 25) (h2 : -x + 2 * y + z = 3)
(h3 : 5 * x + 7 * z = 43) :
x ≤ 4 := by
linear_combination (14 * h1 - 7 * h2 - 5 * h3) / 38
example {a b c d e : ℚ}
(h1 : 3 * a + 4 * b - 2 * c + d = 15)
(h2 : a + 2 * b + c - 2 * d + 2 * e ≤ 3)
(h3 : 5 * a + 5 * b - c + d + 4 * e = 31)
(h4 : 8 * a + b - c - 2 * d + 2 * e = 8)
(h5 : 1 - 2 * b + 3 * c - 4 * d + 5 * e = -4) :
a ≤ 1 := by
linear_combination (-155 * h1 + 68 * h2 + 49 * h3 + 59 * h4 - 90 * h5) / 320
example {a b c d e : ℚ}
(h1 : 3 * a + 4 * b - 2 * c + d = 15)
(h2 : a + 2 * b + c - 2 * d + 2 * e ≤ 3)
(h3 : 5 * a + 5 * b - c + d + 4 * e = 31)
(h4 : 8 * a + b - c - 2 * d + 2 * e = 8)
(h5 : 1 - 2 * b + 3 * c - 4 * d + 5 * e > -4) :
a < 1 := by
linear_combination (-155 * h1 + 68 * h2 + 49 * h3 + 59 * h4 + 90 * h5) / 320
/--
error: comparison failed, LHS is larger
a b : ℚ
h1 : a ≤ 1
h2 : b ≥ 0
⊢ 1 ≤ 0
-/
#guard_msgs in
example (a b : ℚ) (h1 : a ≤ 1) (h2 : b ≥ 0) : a ≤ b := by linear_combination h1 + h2
/--
error: ring failed, ring expressions not equal up to an additive constant
a b : ℚ
h1 : a ≤ 1
h2 : b ≥ 0
⊢ 1 - b ≤ 0
-/
#guard_msgs in
example (a b : ℚ) (h1 : a ≤ 1) (h2 : b ≥ 0) : a ≤ b := by linear_combination h1
/-- error: coefficients of inequalities in 'linear_combination' must be nonnegative -/
#guard_msgs in
example (x y : ℤ) (h : x ≤ y) : -x ≤ -y := by linear_combination 4 - h
/-! ### Nonlinear inequalities -/
example {a b : ℝ} (ha : 0 ≤ a) (hb : b < 1) : a * b ≤ a := by linear_combination a * hb
example {a b : ℝ} (ha : 0 ≤ a) (hb : b < 1) : a * b ≤ a := by linear_combination hb * a
/-- error: could not establish the nonnegativity of a -/
#guard_msgs in
example {a b : ℝ} (hb : b < 1) : a * b ≤ a := by linear_combination a * hb
example {u v x y A B : ℝ} (_ : 0 ≤ u) (_ : 0 ≤ v) (h2 : A ≤ 1) (h3 : 1 ≤ B) (h4 : x ≤ B)
(h5 : y ≤ B) (h8 : u < A) (h9 : v < A) :
u * y + v * x + u * v < 3 * A * B := by
linear_combination v * h2 + v * h3 + v * h4 + u * h5 + (v + B) * h8 + 2 * B * h9
example {t : ℚ} (ht : t ≥ 10) : t ^ 2 - 3 * t - 17 ≥ 5 := by linear_combination (t + 7) * ht
example {n : ℤ} (hn : n ≥ 5) : n ^ 2 > 2 * n + 11 := by linear_combination (n + 3) * hn
example {a b : ℚ} : a * b ≤ (a ^ 2 + b ^ 2) / 2 := by linear_combination sq_nonneg (a - b) / 2
example {a b c : ℚ} (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) :
a * b * c ≤ (a ^ 3 + b ^ 3 + c ^ 3) / 3 := by
have h : (a - b) ^ 2 + (b - c) ^ 2 + (c - a) ^ 2 ≥ 0 := by positivity
linear_combination (a + b + c) * h / 6
example {a b c x : ℚ} (h : a * x ^ 2 + b * x + c = 0) : b ^ 2 ≥ 4 * a * c := by
linear_combination 4 * a * h + sq_nonneg (2 * a * x + b)
/-! ### Regression tests -/
def g (a : ℤ) : ℤ := a ^ 2
example (h : g a = g b) : a ^ 4 = b ^ 4 := by
dsimp [g] at h
linear_combination (a ^ 2 + b ^ 2) * h
example {r s a b : ℕ} (h₁ : (r : ℤ) = a + 1) (h₂ : (s : ℤ) = b + 1) :
r * s = (a + 1 : ℤ) * (b + 1) := by
linear_combination (↑b + 1) * h₁ + ↑r * h₂
-- Implementation at the time of the port (Nov 2022) was 110,000 heartbeats.
-- Eagerly elaborating leaf nodes brings this to 7,540 heartbeats.
set_option maxHeartbeats 10000 in
example (K : Type*) [Field K] [CharZero K] {x y z p q : K}
(h₀ : 3 * x ^ 2 + z ^ 2 * p = 0)
(h₁ : z * (2 * y) = 0)
(h₂ : -y ^ 2 + p * x * (2 * z) + q * (3 * z ^ 2) = 0) :
((27 * q ^ 2 + 4 * p ^ 3) * x) ^ 4 = 0 := by
linear_combination (norm := skip)
(256 / 3 * p ^ 12 * x ^ 2 + 128 * q * p ^ 11 * x * z + 2304 * q ^ 2 * p ^ 9 * x ^ 2 +
2592 * q ^ 3 * p ^ 8 * x * z -
64 * q * p ^ 10 * y ^ 2 +
23328 * q ^ 4 * p ^ 6 * x ^ 2 +
17496 * q ^ 5 * p ^ 5 * x * z -
1296 * q ^ 3 * p ^ 7 * y ^ 2 +
104976 * q ^ 6 * p ^ 3 * x ^ 2 +
39366 * q ^ 7 * p ^ 2 * x * z -
8748 * q ^ 5 * p ^ 4 * y ^ 2 +
177147 * q ^ 8 * x ^ 2 -
19683 * q ^ 7 * p * y ^ 2) *
h₀ +
(-(64 / 3 * p ^ 12 * x * y) + 32 * q * p ^ 11 * z * y - 432 * q ^ 2 * p ^ 9 * x * y +
648 * q ^ 3 * p ^ 8 * z * y -
2916 * q ^ 4 * p ^ 6 * x * y +
4374 * q ^ 5 * p ^ 5 * z * y -
6561 * q ^ 6 * p ^ 3 * x * y +
19683 / 2 * q ^ 7 * p ^ 2 * z * y) *
h₁ +
(-(128 / 3 * p ^ 12 * x * z) - 192 * q * p ^ 10 * x ^ 2 - 864 * q ^ 2 * p ^ 9 * x * z -
3888 * q ^ 3 * p ^ 7 * x ^ 2 -
5832 * q ^ 4 * p ^ 6 * x * z -
26244 * q ^ 5 * p ^ 4 * x ^ 2 -
13122 * q ^ 6 * p ^ 3 * x * z -
59049 * q ^ 7 * p * x ^ 2) *
h₂
exact test_sorry
/- When `linear_combination` is used to prove inequalities, its speed is very sensitive to how much
typeclass inference is demanded by the lemmas it orchestrates. This example took 2146 heartbeats
(and 73 ms on a good laptop) on an implementation with "minimal" typeclasses everywhere, e.g. lots of
`CovariantClass`/`ContravariantClass`, and takes 206 heartbeats (10 ms on a good laptop) on the
implementation at the time of joining Mathlib (November 2024). -/
set_option maxHeartbeats 1200 in
example {a b : ℝ} (h : a < b) : 0 < b - a := by
linear_combination (norm := skip) h
exact test_sorry
|
Basic.lean
|
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.LinearAlgebra.Dimension.DivisionRing
import Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
import Mathlib.RingTheory.Finiteness.Quotient
import Mathlib.RingTheory.Ideal.Norm.AbsNorm
/-!
# Ramification index and inertia degree
Given `P : Ideal S` lying over `p : Ideal R` for the ring extension `f : R →+* S`
(assuming `P` and `p` are prime or maximal where needed),
the **ramification index** `Ideal.ramificationIdx f p P` is the multiplicity of `P` in `map f p`,
and the **inertia degree** `Ideal.inertiaDeg f p P` is the degree of the field extension
`(S / P) : (R / p)`.
## Main results
The main theorem `Ideal.sum_ramification_inertia` states that for all coprime `P` lying over `p`,
`Σ P, ramification_idx f p P * inertia_deg f p P` equals the degree of the field extension
`Frac(S) : Frac(R)`.
## Implementation notes
Often the above theory is set up in the case where:
* `R` is the ring of integers of a number field `K`,
* `L` is a finite separable extension of `K`,
* `S` is the integral closure of `R` in `L`,
* `p` and `P` are maximal ideals,
* `P` is an ideal lying over `p`
We will try to relax the above hypotheses as much as possible.
## Notation
In this file, `e` stands for the ramification index and `f` for the inertia degree of `P` over `p`,
leaving `p` and `P` implicit.
-/
namespace Ideal
universe u v
variable {R : Type u} [CommRing R]
variable {S : Type v} [CommRing S] (f : R →+* S)
variable (p : Ideal R) (P : Ideal S)
open Module
open UniqueFactorizationMonoid
attribute [local instance] Ideal.Quotient.field
section DecEq
/-- The ramification index of `P` over `p` is the largest exponent `n` such that
`p` is contained in `P^n`.
In particular, if `p` is not contained in `P^n`, then the ramification index is 0.
If there is no largest such `n` (e.g. because `p = ⊥`), then `ramificationIdx` is
defined to be 0.
-/
noncomputable def ramificationIdx : ℕ := sSup {n | map f p ≤ P ^ n}
variable {f p P}
theorem ramificationIdx_eq_find [DecidablePred fun n ↦ ∀ (k : ℕ), map f p ≤ P ^ k → k ≤ n]
(h : ∃ n, ∀ k, map f p ≤ P ^ k → k ≤ n) :
ramificationIdx f p P = Nat.find h := by
convert Nat.sSup_def h
theorem ramificationIdx_eq_zero (h : ∀ n : ℕ, ∃ k, map f p ≤ P ^ k ∧ n < k) :
ramificationIdx f p P = 0 :=
dif_neg (by push_neg; exact h)
theorem ramificationIdx_spec {n : ℕ} (hle : map f p ≤ P ^ n) (hgt : ¬map f p ≤ P ^ (n + 1)) :
ramificationIdx f p P = n := by
classical
let Q : ℕ → Prop := fun m => ∀ k : ℕ, map f p ≤ P ^ k → k ≤ m
have : Q n := by
intro k hk
refine le_of_not_gt fun hnk => ?_
exact hgt (hk.trans (Ideal.pow_le_pow_right hnk))
rw [ramificationIdx_eq_find ⟨n, this⟩]
refine le_antisymm (Nat.find_min' _ this) (le_of_not_gt fun h : Nat.find _ < n => ?_)
obtain this' := Nat.find_spec ⟨n, this⟩
exact h.not_ge (this' _ hle)
theorem ramificationIdx_lt {n : ℕ} (hgt : ¬map f p ≤ P ^ n) : ramificationIdx f p P < n := by
classical
rcases n with - | n
· simp at hgt
· rw [Nat.lt_succ_iff]
have : ∀ k, map f p ≤ P ^ k → k ≤ n := by
refine fun k hk => le_of_not_gt fun hnk => ?_
exact hgt (hk.trans (Ideal.pow_le_pow_right hnk))
rw [ramificationIdx_eq_find ⟨n, this⟩]
exact Nat.find_min' ⟨n, this⟩ this
@[simp]
theorem ramificationIdx_bot : ramificationIdx f ⊥ P = 0 :=
dif_neg <| not_exists.mpr fun n hn => n.lt_succ_self.not_ge (hn _ (by simp))
@[simp]
theorem ramificationIdx_of_not_le (h : ¬map f p ≤ P) : ramificationIdx f p P = 0 :=
ramificationIdx_spec (by simp) (by simpa using h)
theorem ramificationIdx_ne_zero {e : ℕ} (he : e ≠ 0) (hle : map f p ≤ P ^ e)
(hnle : ¬map f p ≤ P ^ (e + 1)) : ramificationIdx f p P ≠ 0 := by
rwa [ramificationIdx_spec hle hnle]
theorem le_pow_of_le_ramificationIdx {n : ℕ} (hn : n ≤ ramificationIdx f p P) :
map f p ≤ P ^ n := by
contrapose! hn
exact ramificationIdx_lt hn
theorem le_pow_ramificationIdx : map f p ≤ P ^ ramificationIdx f p P :=
le_pow_of_le_ramificationIdx (le_refl _)
theorem le_comap_pow_ramificationIdx : p ≤ comap f (P ^ ramificationIdx f p P) :=
map_le_iff_le_comap.mp le_pow_ramificationIdx
theorem le_comap_of_ramificationIdx_ne_zero (h : ramificationIdx f p P ≠ 0) : p ≤ comap f P :=
Ideal.map_le_iff_le_comap.mp <| le_pow_ramificationIdx.trans <| Ideal.pow_le_self <| h
variable {S₁ : Type*} [CommRing S₁] [Algebra R S₁]
variable (p) in
lemma ramificationIdx_comap_eq [Algebra R S] (e : S ≃ₐ[R] S₁) (P : Ideal S₁) :
ramificationIdx (algebraMap R S) p (P.comap e) = ramificationIdx (algebraMap R S₁) p P := by
dsimp only [ramificationIdx]
congr 1
ext n
simp only [Set.mem_setOf_eq, Ideal.map_le_iff_le_comap]
rw [← comap_coe e, ← e.toRingEquiv_toRingHom, comap_coe, ← RingEquiv.symm_symm (e : S ≃+* S₁),
← map_comap_of_equiv, ← Ideal.map_pow, map_comap_of_equiv, ← comap_coe (RingEquiv.symm _),
comap_comap, RingEquiv.symm_symm, e.toRingEquiv_toRingHom, ← e.toAlgHom_toRingHom,
AlgHom.comp_algebraMap]
variable (p) in
lemma ramificationIdx_map_eq [Algebra R S] {E : Type*} [EquivLike E S S₁] [AlgEquivClass E R S S₁]
(P : Ideal S) (e : E) :
ramificationIdx (algebraMap R S₁) p (P.map e) = ramificationIdx (algebraMap R S) p P := by
rw [show P.map e = _ from P.map_comap_of_equiv (e : S ≃+* S₁)]
exact p.ramificationIdx_comap_eq (e : S ≃ₐ[R] S₁).symm P
lemma ramificationIdx_ne_one_iff (hp : map f p ≤ P) :
ramificationIdx f p P ≠ 1 ↔ p.map f ≤ P ^ 2 := by
classical
by_cases H : ∀ n : ℕ, ∃ k, p.map f ≤ P ^ k ∧ n < k
· obtain ⟨k, hk, h2k⟩ := H 2
simp [Ideal.ramificationIdx_eq_zero H, hk.trans (Ideal.pow_le_pow_right h2k.le)]
push_neg at H
rw [Ideal.ramificationIdx_eq_find H]
constructor
· intro he
have : 1 ≤ Nat.find H := Nat.find_spec H 1 (by simpa)
have := Nat.find_min H (m := 1) (by omega)
push_neg at this
obtain ⟨k, hk, h1k⟩ := this
exact hk.trans (Ideal.pow_le_pow_right (Nat.succ_le.mpr h1k))
· intro he
have := Nat.find_spec H 2 he
omega
open IsLocalRing in
/-- The converse is true when `S` is a Dedekind domain.
See `Ideal.ramificationIdx_eq_one_iff_of_isDedekindDomain`. -/
lemma ramificationIdx_eq_one_of_map_localization
[Algebra R S] {p : Ideal R} {P : Ideal S} [P.IsPrime] [IsNoetherianRing S]
(hpP : map (algebraMap R S) p ≤ P) (hp : P ≠ ⊥) (hp' : P.primeCompl ≤ nonZeroDivisors S)
(H : p.map (algebraMap R (Localization.AtPrime P)) = maximalIdeal (Localization.AtPrime P)) :
ramificationIdx (algebraMap R S) p P = 1 := by
rw [← not_ne_iff (b := 1), Ideal.ramificationIdx_ne_one_iff hpP]
intro h₂
replace h₂ := Ideal.map_mono (f := algebraMap S (Localization.AtPrime P)) h₂
rw [Ideal.map_pow, Localization.AtPrime.map_eq_maximalIdeal, Ideal.map_map,
← IsScalarTower.algebraMap_eq, H, pow_two] at h₂
have := Submodule.eq_bot_of_le_smul_of_le_jacobson_bot _ _ (IsNoetherian.noetherian _) h₂
(maximalIdeal_le_jacobson _)
rw [← Localization.AtPrime.map_eq_maximalIdeal, Ideal.map_eq_bot_iff_of_injective] at this
· exact hp this
· exact IsLocalization.injective _ hp'
namespace IsDedekindDomain
variable [IsDedekindDomain S]
theorem ramificationIdx_eq_normalizedFactors_count [DecidableEq (Ideal S)]
(hp0 : map f p ≠ ⊥) (hP : P.IsPrime)
(hP0 : P ≠ ⊥) : ramificationIdx f p P = (normalizedFactors (map f p)).count P := by
have hPirr := (Ideal.prime_of_isPrime hP0 hP).irreducible
refine ramificationIdx_spec (Ideal.le_of_dvd ?_) (mt Ideal.dvd_iff_le.mpr ?_) <;>
rw [dvd_iff_normalizedFactors_le_normalizedFactors (pow_ne_zero _ hP0) hp0,
normalizedFactors_pow, normalizedFactors_irreducible hPirr, normalize_eq,
Multiset.nsmul_singleton, ← Multiset.le_count_iff_replicate_le]
exact (Nat.lt_succ_self _).not_ge
theorem ramificationIdx_eq_factors_count [DecidableEq (Ideal S)]
(hp0 : map f p ≠ ⊥) (hP : P.IsPrime) (hP0 : P ≠ ⊥) :
ramificationIdx f p P = (factors (map f p)).count P := by
rw [IsDedekindDomain.ramificationIdx_eq_normalizedFactors_count hp0 hP hP0,
factors_eq_normalizedFactors]
theorem ramificationIdx_ne_zero (hp0 : map f p ≠ ⊥) (hP : P.IsPrime) (le : map f p ≤ P) :
ramificationIdx f p P ≠ 0 := by
classical
have hP0 : P ≠ ⊥ := by
rintro rfl
exact hp0 (le_bot_iff.mp le)
have hPirr := (Ideal.prime_of_isPrime hP0 hP).irreducible
rw [IsDedekindDomain.ramificationIdx_eq_normalizedFactors_count hp0 hP hP0]
obtain ⟨P', hP', P'_eq⟩ :=
exists_mem_normalizedFactors_of_dvd hp0 hPirr (Ideal.dvd_iff_le.mpr le)
rwa [Multiset.count_ne_zero, associated_iff_eq.mp P'_eq]
open IsLocalRing in
lemma ramificationIdx_eq_one_iff
[Algebra R S] {p : Ideal R} {P : Ideal S} [P.IsPrime]
(hp : P ≠ ⊥) (hpP : p.map (algebraMap R S) ≤ P) :
ramificationIdx (algebraMap R S) p P = 1 ↔
p.map (algebraMap R (Localization.AtPrime P)) = maximalIdeal (Localization.AtPrime P) := by
refine ⟨?_, ramificationIdx_eq_one_of_map_localization hpP hp
(primeCompl_le_nonZeroDivisors _)⟩
let Sₚ := Localization.AtPrime P
rw [← not_ne_iff (b := 1), ramificationIdx_ne_one_iff hpP, pow_two]
intro H₁
obtain ⟨a, ha⟩ : P ∣ p.map (algebraMap R S) := Ideal.dvd_iff_le.mpr hpP
have ha' : ¬ a ≤ P := fun h ↦ H₁ (ha.trans_le (Ideal.mul_mono_right h))
rw [IsScalarTower.algebraMap_eq _ S, ← Ideal.map_map, ha, Ideal.map_mul,
Localization.AtPrime.map_eq_maximalIdeal]
convert Ideal.mul_top _
rw [← not_ne_iff, IsLocalization.map_algebraMap_ne_top_iff_disjoint P.primeCompl]
simpa [primeCompl, Set.disjoint_compl_left_iff_subset]
end IsDedekindDomain
variable (f p P) [Algebra R S]
local notation "f" => algebraMap R S
open Classical in
/-- The inertia degree of `P : Ideal S` lying over `p : Ideal R` is the degree of the
extension `(S / P) : (R / p)`.
We do not assume `P` lies over `p` in the definition; we return `0` instead.
See `inertiaDeg_algebraMap` for the common case where `f = algebraMap R S`
and there is an algebra structure `R / p → S / P`.
-/
noncomputable def inertiaDeg : ℕ :=
if hPp : comap f P = p then
letI : Algebra (R ⧸ p) (S ⧸ P) := Quotient.algebraQuotientOfLEComap hPp.ge
finrank (R ⧸ p) (S ⧸ P)
else 0
-- Useful for the `nontriviality` tactic using `comap_eq_of_scalar_tower_quotient`.
@[simp]
theorem inertiaDeg_of_subsingleton [hp : p.IsMaximal] [hQ : Subsingleton (S ⧸ P)] :
inertiaDeg p P = 0 := by
have := Ideal.Quotient.subsingleton_iff.mp hQ
subst this
exact dif_neg fun h => hp.ne_top <| h.symm.trans comap_top
@[simp]
theorem inertiaDeg_algebraMap [P.LiesOver p] [p.IsMaximal] :
inertiaDeg p P = finrank (R ⧸ p) (S ⧸ P) := by
nontriviality S ⧸ P using inertiaDeg_of_subsingleton, finrank_zero_of_subsingleton
rw [inertiaDeg, dif_pos (over_def P p).symm]
theorem inertiaDeg_pos [p.IsMaximal] [Module.Finite R S]
[P.LiesOver p] : 0 < inertiaDeg p P :=
haveI : Nontrivial (S ⧸ P) := Quotient.nontrivial_of_liesOver_of_isPrime P p
finrank_pos.trans_eq (inertiaDeg_algebraMap p P).symm
lemma inertiaDeg_comap_eq (e : S ≃ₐ[R] S₁) (P : Ideal S₁) [p.IsMaximal] :
inertiaDeg p (P.comap e) = inertiaDeg p P := by
have he : (P.comap e).comap (algebraMap R S) = p ↔ P.comap (algebraMap R S₁) = p := by
rw [← comap_coe e, comap_comap, ← e.toAlgHom_toRingHom, AlgHom.comp_algebraMap]
by_cases h : P.LiesOver p
· rw [inertiaDeg_algebraMap, inertiaDeg_algebraMap]
exact (Quotient.algEquivOfEqComap p e rfl).toLinearEquiv.finrank_eq
· rw [inertiaDeg, dif_neg (fun eq => h ⟨(he.mp eq).symm⟩)]
rw [inertiaDeg, dif_neg (fun eq => h ⟨eq.symm⟩)]
lemma inertiaDeg_map_eq [p.IsMaximal] (P : Ideal S)
{E : Type*} [EquivLike E S S₁] [AlgEquivClass E R S S₁] (e : E) :
inertiaDeg p (P.map e) = inertiaDeg p P := by
rw [show P.map e = _ from map_comap_of_equiv (e : S ≃+* S₁)]
exact p.inertiaDeg_comap_eq (e : S ≃ₐ[R] S₁).symm P
end DecEq
section absNorm
/-- The absolute norm of an ideal `P` above a rational prime `p` is
`|p| ^ ((span {p}).inertiaDeg P)`. -/
lemma absNorm_eq_pow_inertiaDeg [IsDedekindDomain R] [Module.Free ℤ R] [Module.Finite ℤ R] {p : ℤ}
(P : Ideal R) [P.LiesOver (span {p})] (hp : Prime p) :
absNorm P = p.natAbs ^ ((span {p}).inertiaDeg P) := by
have : (span {p}).IsMaximal :=
(isPrime_of_prime (prime_span_singleton_iff.mpr hp)).isMaximal (by simp [hp.ne_zero])
have h : Module.Finite (ℤ ⧸ span {p}) (R ⧸ P) := module_finite_of_liesOver P (span {p})
let _ : Field (ℤ ⧸ span {p}) := Quotient.field (span {p})
rw [inertiaDeg_algebraMap, absNorm_apply, Submodule.cardQuot_apply,
Module.natCard_eq_pow_finrank (K := ℤ ⧸ span {p})]
simp [Nat.card_congr (Int.quotientSpanEquivZMod p).toEquiv]
end absNorm
section FinrankQuotientMap
open scoped nonZeroDivisors
variable [Algebra R S]
variable {K : Type*} [Field K] [Algebra R K]
variable {L : Type*} [Field L] [Algebra S L] [IsFractionRing S L]
variable {V V' V'' : Type*}
variable [AddCommGroup V] [Module R V] [Module K V] [IsScalarTower R K V]
variable [AddCommGroup V'] [Module R V'] [Module S V'] [IsScalarTower R S V']
variable [AddCommGroup V''] [Module R V'']
variable (K)
open scoped Matrix
variable {K} in
/-- If `b` mod `p` spans `S/p` as `R/p`-space, then `b` itself spans `Frac(S)` as `K`-space.
Here,
* `p` is an ideal of `R` such that `R / p` is nontrivial
* `K` is a field that has an embedding of `R` (in particular we can take `K = Frac(R)`)
* `L` is a field extension of `K`
* `S` is the integral closure of `R` in `L`
More precisely, we avoid quotients in this statement and instead require that `b ∪ pS` spans `S`.
-/
theorem FinrankQuotientMap.span_eq_top [IsDomain R] [IsDomain S] [Algebra K L] [Module.Finite R S]
[Algebra R L] [IsScalarTower R S L] [IsScalarTower R K L] [Algebra.IsAlgebraic R S]
[NoZeroSMulDivisors R K] (hp : p ≠ ⊤) (b : Set S)
(hb' : Submodule.span R b ⊔ (p.map (algebraMap R S)).restrictScalars R = ⊤) :
Submodule.span K (algebraMap S L '' b) = ⊤ := by
have hRL : Function.Injective (algebraMap R L) := by
rw [IsScalarTower.algebraMap_eq R K L]
exact (algebraMap K L).injective.comp (FaithfulSMul.algebraMap_injective R K)
-- Let `M` be the `R`-module spanned by the proposed basis elements.
let M : Submodule R S := Submodule.span R b
-- Then `S / M` is generated by some finite set of `n` vectors `a`.
obtain ⟨n, a, ha⟩ := @Module.Finite.exists_fin R (S ⧸ M) _ _ _ _
-- Because the image of `p` in `S / M` is `⊤`,
have smul_top_eq : p • (⊤ : Submodule R (S ⧸ M)) = ⊤ := by
calc
p • ⊤ = Submodule.map M.mkQ (p • ⊤) := by
rw [Submodule.map_smul'', Submodule.map_top, M.range_mkQ]
_ = ⊤ := by rw [Ideal.smul_top_eq_map, (Submodule.map_mkQ_eq_top M _).mpr hb']
-- we can write the elements of `a` as `p`-linear combinations of other elements of `a`.
have exists_sum : ∀ x : S ⧸ M, ∃ a' : Fin n → R, (∀ i, a' i ∈ p) ∧ ∑ i, a' i • a i = x := by
intro x
obtain ⟨a'', ha'', hx⟩ := (Submodule.mem_ideal_smul_span_iff_exists_sum p a x).1
(by { rw [ha, smul_top_eq]; exact Submodule.mem_top } :
x ∈ p • Submodule.span R (Set.range a))
· refine ⟨fun i => a'' i, fun i => ha'' _, ?_⟩
rw [← hx, Finsupp.sum_fintype]
exact fun _ => zero_smul _ _
choose A' hA'p hA' using fun i => exists_sum (a i)
-- This gives us a(n invertible) matrix `A` such that `det A ∈ (M = span R b)`,
let A : Matrix (Fin n) (Fin n) R := Matrix.of A' - 1
let B := A.adjugate
have A_smul : ∀ i, ∑ j, A i j • a j = 0 := by
intros
simp [A, Matrix.sub_apply, Matrix.of_apply, Matrix.one_apply, sub_smul,
Finset.sum_sub_distrib, hA', sub_self]
-- since `span S {det A} / M = 0`.
have d_smul : ∀ i, A.det • a i = 0 := by
intro i
calc
A.det • a i = ∑ j, (B * A) i j • a j := ?_
_ = ∑ k, B i k • ∑ j, A k j • a j := ?_
_ = 0 := Finset.sum_eq_zero fun k _ => ?_
· simp only [B, Matrix.adjugate_mul, Matrix.smul_apply, Matrix.one_apply, smul_eq_mul, ite_true,
mul_ite, mul_one, mul_zero, ite_smul, zero_smul, Finset.sum_ite_eq, Finset.mem_univ]
· simp only [Matrix.mul_apply, Finset.smul_sum, Finset.sum_smul, smul_smul]
rw [Finset.sum_comm]
· rw [A_smul, smul_zero]
-- In the rings of integers we have the desired inclusion.
have span_d : (Submodule.span S ({algebraMap R S A.det} : Set S)).restrictScalars R ≤ M := by
intro x hx
rw [Submodule.restrictScalars_mem] at hx
obtain ⟨x', rfl⟩ := Submodule.mem_span_singleton.mp hx
rw [smul_eq_mul, mul_comm, ← Algebra.smul_def] at hx ⊢
rw [← Submodule.Quotient.mk_eq_zero, Submodule.Quotient.mk_smul]
obtain ⟨a', _, quot_x_eq⟩ := exists_sum (Submodule.Quotient.mk x')
rw [← quot_x_eq, Finset.smul_sum]
conv =>
lhs; congr; next => skip
intro x; rw [smul_comm A.det, d_smul, smul_zero]
exact Finset.sum_const_zero
refine top_le_iff.mp
(calc
⊤ = (Ideal.span {algebraMap R L A.det}).restrictScalars K := ?_
_ ≤ Submodule.span K (algebraMap S L '' b) := ?_)
-- Because `det A ≠ 0`, we have `span L {det A} = ⊤`.
· rw [eq_comm, Submodule.restrictScalars_eq_top_iff, Ideal.span_singleton_eq_top]
refine IsUnit.mk0 _ ((map_ne_zero_iff (algebraMap R L) hRL).mpr ?_)
refine ne_zero_of_map (f := Ideal.Quotient.mk p) ?_
haveI := Ideal.Quotient.nontrivial hp
calc
Ideal.Quotient.mk p A.det = Matrix.det ((Ideal.Quotient.mk p).mapMatrix A) := by
rw [RingHom.map_det]
_ = Matrix.det ((Ideal.Quotient.mk p).mapMatrix (Matrix.of A' - 1)) := rfl
_ = Matrix.det fun i j =>
(Ideal.Quotient.mk p) (A' i j) - (1 : Matrix (Fin n) (Fin n) (R ⧸ p)) i j := ?_
_ = Matrix.det (-1 : Matrix (Fin n) (Fin n) (R ⧸ p)) := ?_
_ = (-1 : R ⧸ p) ^ n := by rw [Matrix.det_neg, Fintype.card_fin, Matrix.det_one, mul_one]
_ ≠ 0 := IsUnit.ne_zero (isUnit_one.neg.pow _)
· refine congr_arg Matrix.det (Matrix.ext fun i j => ?_)
rw [map_sub, RingHom.mapMatrix_apply, map_one]
rfl
· refine congr_arg Matrix.det (Matrix.ext fun i j => ?_)
rw [Ideal.Quotient.eq_zero_iff_mem.mpr (hA'p i j), zero_sub]
rfl
-- And we conclude `L = span L {det A} ≤ span K b`, so `span K b` spans everything.
· intro x hx
rw [Submodule.restrictScalars_mem, IsScalarTower.algebraMap_apply R S L] at hx
exact IsFractionRing.ideal_span_singleton_map_subset R hRL span_d hx
variable [hRK : IsFractionRing R K]
/-- Let `V` be a vector space over `K = Frac(R)`, `S / R` a ring extension
and `V'` a module over `S`. If `b`, in the intersection `V''` of `V` and `V'`,
is linear independent over `S` in `V'`, then it is linear independent over `R` in `V`.
The statement we prove is actually slightly more general:
* it suffices that the inclusion `algebraMap R S : R → S` is nontrivial
* the function `f' : V'' → V'` doesn't need to be injective
-/
theorem FinrankQuotientMap.linearIndependent_of_nontrivial [IsDedekindDomain R]
(hRS : RingHom.ker (algebraMap R S) ≠ ⊤) (f : V'' →ₗ[R] V) (hf : Function.Injective f)
(f' : V'' →ₗ[R] V') {ι : Type*} {b : ι → V''} (hb' : LinearIndependent S (f' ∘ b)) :
LinearIndependent K (f ∘ b) := by
contrapose! hb' with hb
-- Informally, if we have a nontrivial linear dependence with coefficients `g` in `K`,
-- then we can find a linear dependence with coefficients `I.Quotient.mk g'` in `R/I`,
-- where `I = ker (algebraMap R S)`.
-- We make use of the same principle but stay in `R` everywhere.
simp only [linearIndependent_iff', not_forall] at hb ⊢
obtain ⟨s, g, eq, j', hj's, hj'g⟩ := hb
use s
obtain ⟨a, hag, j, hjs, hgI⟩ := Ideal.exist_integer_multiples_notMem hRS s g hj's hj'g
choose g'' hg'' using hag
letI := Classical.propDecidable
let g' i := if h : i ∈ s then g'' i h else 0
have hg' : ∀ i ∈ s, algebraMap _ _ (g' i) = a * g i := by
intro i hi; exact (congr_arg _ (dif_pos hi)).trans (hg'' i hi)
-- Because `R/I` is nontrivial, we can lift `g` to a nontrivial linear dependence in `S`.
have hgI : algebraMap R S (g' j) ≠ 0 := by
simp only [FractionalIdeal.mem_coeIdeal, not_exists, not_and'] at hgI
exact hgI _ (hg' j hjs)
refine ⟨fun i => algebraMap R S (g' i), ?_, j, hjs, hgI⟩
have eq : f (∑ i ∈ s, g' i • b i) = 0 := by
rw [map_sum, ← smul_zero a, ← eq, Finset.smul_sum]
refine Finset.sum_congr rfl ?_
intro i hi
rw [LinearMap.map_smul, ← IsScalarTower.algebraMap_smul K, hg' i hi, ← smul_assoc,
smul_eq_mul, Function.comp_apply]
simp only [IsScalarTower.algebraMap_smul, ← map_smul, ← map_sum,
(f.map_eq_zero_iff hf).mp eq, LinearMap.map_zero, (· ∘ ·)]
variable (L)
/-- If `p` is a maximal ideal of `R`, and `S` is the integral closure of `R` in `L`,
then the dimension `[S/pS : R/p]` is equal to `[Frac(S) : Frac(R)]`. -/
theorem finrank_quotient_map [IsDomain S] [IsDedekindDomain R] [Algebra K L]
[Algebra R L] [IsScalarTower R K L] [IsScalarTower R S L]
[hp : p.IsMaximal] [Module.Finite R S] :
finrank (R ⧸ p) (S ⧸ map (algebraMap R S) p) = finrank K L := by
-- Choose an arbitrary basis `b` for `[S/pS : R/p]`.
-- We'll use the previous results to turn it into a basis on `[Frac(S) : Frac(R)]`.
let ι := Module.Free.ChooseBasisIndex (R ⧸ p) (S ⧸ map (algebraMap R S) p)
let b : Basis ι (R ⧸ p) (S ⧸ map (algebraMap R S) p) := Module.Free.chooseBasis _ _
-- Namely, choose a representative `b' i : S` for each `b i : S / pS`.
let b' : ι → S := fun i => (Ideal.Quotient.mk_surjective (b i)).choose
have b_eq_b' : ⇑b = (Submodule.mkQ (map (algebraMap R S) p)).restrictScalars R ∘ b' :=
funext fun i => (Ideal.Quotient.mk_surjective (b i)).choose_spec.symm
-- We claim `b'` is a basis for `Frac(S)` over `Frac(R)` because it is linear independent
-- and spans the whole of `Frac(S)`.
let b'' : ι → L := algebraMap S L ∘ b'
have b''_li : LinearIndependent K b'' := ?_
· have b''_sp : Submodule.span K (Set.range b'') = ⊤ := ?_
-- Since the two bases have the same index set, the spaces have the same dimension.
· let c : Basis ι K L := Basis.mk b''_li b''_sp.ge
rw [finrank_eq_card_basis b, finrank_eq_card_basis c]
-- It remains to show that the basis is indeed linear independent and spans the whole space.
· rw [Set.range_comp]
refine FinrankQuotientMap.span_eq_top p hp.ne_top _ (top_le_iff.mp ?_)
-- The nicest way to show `S ≤ span b' ⊔ pS` is by reducing both sides modulo pS.
-- However, this would imply distinguishing between `pS` as `S`-ideal,
-- and `pS` as `R`-submodule, since they have different (non-defeq) quotients.
-- Instead we'll lift `x mod pS ∈ span b` to `y ∈ span b'` for some `y - x ∈ pS`.
intro x _
have mem_span_b : ((Submodule.mkQ (map (algebraMap R S) p)) x : S ⧸ map (algebraMap R S) p) ∈
Submodule.span (R ⧸ p) (Set.range b) := b.mem_span _
rw [← @Submodule.restrictScalars_mem R,
Submodule.restrictScalars_span R (R ⧸ p) Ideal.Quotient.mk_surjective, b_eq_b',
Set.range_comp, ← Submodule.map_span] at mem_span_b
obtain ⟨y, y_mem, y_eq⟩ := Submodule.mem_map.mp mem_span_b
suffices y + -(y - x) ∈ _ by simpa
rw [LinearMap.restrictScalars_apply, Submodule.mkQ_apply, Submodule.mkQ_apply,
Submodule.Quotient.eq] at y_eq
exact add_mem (Submodule.mem_sup_left y_mem) (neg_mem <| Submodule.mem_sup_right y_eq)
· have := b.linearIndependent; rw [b_eq_b'] at this
convert FinrankQuotientMap.linearIndependent_of_nontrivial K _
((Algebra.linearMap S L).restrictScalars R) _ ((Submodule.mkQ _).restrictScalars R) this
· rw [Quotient.algebraMap_eq, Ideal.mk_ker]
exact hp.ne_top
· exact IsFractionRing.injective S L
end FinrankQuotientMap
section FactLeComap
variable [Algebra R S]
local notation "f" => algebraMap R S
local notation "e" => ramificationIdx f p P
/-- `R / p` has a canonical map to `S / (P ^ e)`, where `e` is the ramification index
of `P` over `p`. -/
noncomputable instance Quotient.algebraQuotientPowRamificationIdx : Algebra (R ⧸ p) (S ⧸ P ^ e) :=
Quotient.algebraQuotientOfLEComap (Ideal.map_le_iff_le_comap.mp le_pow_ramificationIdx)
@[simp]
theorem Quotient.algebraMap_quotient_pow_ramificationIdx (x : R) :
algebraMap (R ⧸ p) (S ⧸ P ^ e) (Ideal.Quotient.mk p x) = Ideal.Quotient.mk (P ^ e) (f x) := rfl
/-- If `P` lies over `p`, then `R / p` has a canonical map to `S / P`.
This can't be an instance since the map `f : R → S` is generally not inferable.
-/
def Quotient.algebraQuotientOfRamificationIdxNeZero [hfp : NeZero e] :
Algebra (R ⧸ p) (S ⧸ P) :=
Quotient.algebraQuotientOfLEComap (le_comap_of_ramificationIdx_ne_zero hfp.out)
attribute [local instance] Ideal.Quotient.algebraQuotientOfRamificationIdxNeZero
@[simp]
theorem Quotient.algebraMap_quotient_of_ramificationIdx_neZero
[NeZero e] (x : R) :
algebraMap (R ⧸ p) (S ⧸ P) (Ideal.Quotient.mk p x) = Ideal.Quotient.mk P (f x) := rfl
/-- The inclusion `(P^(i + 1) / P^e) ⊂ (P^i / P^e)`. -/
@[simps]
noncomputable def powQuotSuccInclusion (i : ℕ) :
Ideal.map (Ideal.Quotient.mk (P ^ e)) (P ^ (i + 1)) →ₗ[R ⧸ p]
Ideal.map (Ideal.Quotient.mk (P ^ e)) (P ^ i) where
toFun x := ⟨x, Ideal.map_mono (Ideal.pow_le_pow_right i.le_succ) x.2⟩
map_add' _ _ := rfl
map_smul' _ _ := rfl
theorem powQuotSuccInclusion_injective (i : ℕ) :
Function.Injective (powQuotSuccInclusion p P i) := by
rw [← LinearMap.ker_eq_bot, LinearMap.ker_eq_bot']
rintro ⟨x, hx⟩ hx0
rw [Subtype.ext_iff] at hx0 ⊢
rwa [powQuotSuccInclusion_apply_coe] at hx0
/-- `S ⧸ P` embeds into the quotient by `P^(i+1) ⧸ P^e` as a subspace of `P^i ⧸ P^e`.
See `quotientToQuotientRangePowQuotSucc` for this as a linear map,
and `quotientRangePowQuotSuccInclusionEquiv` for this as a linear equivalence.
-/
noncomputable def quotientToQuotientRangePowQuotSuccAux {i : ℕ} {a : S} (a_mem : a ∈ P ^ i) :
S ⧸ P →
(P ^ i).map (Ideal.Quotient.mk (P ^ e)) ⧸ LinearMap.range (powQuotSuccInclusion p P i) :=
Quotient.map' (fun x : S => ⟨_, Ideal.mem_map_of_mem _ (Ideal.mul_mem_right x _ a_mem)⟩)
fun x y h => by
rw [Submodule.quotientRel_def] at h ⊢
simp only [map_mul, LinearMap.mem_range]
refine ⟨⟨_, Ideal.mem_map_of_mem _ (Ideal.mul_mem_mul a_mem h)⟩, ?_⟩
ext
rw [powQuotSuccInclusion_apply_coe, Subtype.coe_mk, Submodule.coe_sub, Subtype.coe_mk,
Subtype.coe_mk, map_mul, map_sub, mul_sub]
theorem quotientToQuotientRangePowQuotSuccAux_mk {i : ℕ} {a : S} (a_mem : a ∈ P ^ i) (x : S) :
quotientToQuotientRangePowQuotSuccAux p P a_mem (Submodule.Quotient.mk x) =
Submodule.Quotient.mk ⟨_, Ideal.mem_map_of_mem _ (Ideal.mul_mem_right x _ a_mem)⟩ := by
apply Quotient.map'_mk''
section
variable [hfp : NeZero (ramificationIdx (algebraMap R S) p P)]
/-- `S ⧸ P` embeds into the quotient by `P^(i+1) ⧸ P^e` as a subspace of `P^i ⧸ P^e`. -/
noncomputable def quotientToQuotientRangePowQuotSucc
{i : ℕ} {a : S} (a_mem : a ∈ P ^ i) :
S ⧸ P →ₗ[R ⧸ p]
(P ^ i).map (Ideal.Quotient.mk (P ^ e)) ⧸ LinearMap.range (powQuotSuccInclusion p P i) where
toFun := quotientToQuotientRangePowQuotSuccAux p P a_mem
map_add' := by
intro x y; refine Quotient.inductionOn' x fun x => Quotient.inductionOn' y fun y => ?_
simp only [Submodule.Quotient.mk''_eq_mk, ← Submodule.Quotient.mk_add,
quotientToQuotientRangePowQuotSuccAux_mk, mul_add]
exact congr_arg Submodule.Quotient.mk rfl
map_smul' := by
intro x y; refine Quotient.inductionOn' x fun x => Quotient.inductionOn' y fun y => ?_
simp only [Submodule.Quotient.mk''_eq_mk, RingHom.id_apply,
quotientToQuotientRangePowQuotSuccAux_mk]
refine congr_arg Submodule.Quotient.mk ?_
ext
simp only [map_mul, Quotient.mk_eq_mk, Submodule.coe_smul_of_tower,
Algebra.smul_def, Quotient.algebraMap_quotient_pow_ramificationIdx]
ring
theorem quotientToQuotientRangePowQuotSucc_mk {i : ℕ} {a : S} (a_mem : a ∈ P ^ i) (x : S) :
quotientToQuotientRangePowQuotSucc p P a_mem (Submodule.Quotient.mk x) =
Submodule.Quotient.mk ⟨_, Ideal.mem_map_of_mem _ (Ideal.mul_mem_right x _ a_mem)⟩ :=
quotientToQuotientRangePowQuotSuccAux_mk p P a_mem x
theorem quotientToQuotientRangePowQuotSucc_injective [IsDedekindDomain S] [P.IsPrime]
{i : ℕ} (hi : i < e) {a : S} (a_mem : a ∈ P ^ i) (a_notMem : a ∉ P ^ (i + 1)) :
Function.Injective (quotientToQuotientRangePowQuotSucc p P a_mem) := fun x =>
Quotient.inductionOn' x fun x y =>
Quotient.inductionOn' y fun y h => by
have Pe_le_Pi1 : P ^ e ≤ P ^ (i + 1) := Ideal.pow_le_pow_right hi
simp only [Submodule.Quotient.mk''_eq_mk, quotientToQuotientRangePowQuotSucc_mk,
Submodule.Quotient.eq, LinearMap.mem_range, Subtype.ext_iff,
Submodule.coe_sub] at h ⊢
rcases h with ⟨⟨⟨z⟩, hz⟩, h⟩
rw [Submodule.Quotient.quot_mk_eq_mk, Ideal.Quotient.mk_eq_mk, Ideal.mem_quotient_iff_mem_sup,
sup_eq_left.mpr Pe_le_Pi1] at hz
rw [powQuotSuccInclusion_apply_coe, Subtype.coe_mk, Submodule.Quotient.quot_mk_eq_mk,
Ideal.Quotient.mk_eq_mk, ← map_sub, Ideal.Quotient.eq, ← mul_sub] at h
exact
(Ideal.IsPrime.mem_pow_mul _
((Submodule.sub_mem_iff_right _ hz).mp (Pe_le_Pi1 h))).resolve_left
a_notMem
theorem quotientToQuotientRangePowQuotSucc_surjective [IsDedekindDomain S]
(hP0 : P ≠ ⊥) [hP : P.IsPrime] {i : ℕ} (hi : i < e) {a : S} (a_mem : a ∈ P ^ i)
(a_notMem : a ∉ P ^ (i + 1)) :
Function.Surjective (quotientToQuotientRangePowQuotSucc p P a_mem) := by
rintro ⟨⟨⟨x⟩, hx⟩⟩
have Pe_le_Pi : P ^ e ≤ P ^ i := Ideal.pow_le_pow_right hi.le
rw [Submodule.Quotient.quot_mk_eq_mk, Ideal.Quotient.mk_eq_mk, Ideal.mem_quotient_iff_mem_sup,
sup_eq_left.mpr Pe_le_Pi] at hx
suffices hx' : x ∈ Ideal.span {a} ⊔ P ^ (i + 1) by
obtain ⟨y', hy', z, hz, rfl⟩ := Submodule.mem_sup.mp hx'
obtain ⟨y, rfl⟩ := Ideal.mem_span_singleton.mp hy'
refine ⟨Submodule.Quotient.mk y, ?_⟩
simp only [Submodule.Quotient.quot_mk_eq_mk, quotientToQuotientRangePowQuotSucc_mk,
Submodule.Quotient.eq, LinearMap.mem_range, Subtype.ext_iff,
Submodule.coe_sub]
refine ⟨⟨_, Ideal.mem_map_of_mem _ (Submodule.neg_mem _ hz)⟩, ?_⟩
rw [powQuotSuccInclusion_apply_coe, Subtype.coe_mk, Ideal.Quotient.mk_eq_mk, map_add,
sub_add_cancel_left, map_neg]
letI := Classical.decEq (Ideal S)
rw [sup_eq_prod_inf_factors _ (pow_ne_zero _ hP0), normalizedFactors_pow,
normalizedFactors_irreducible ((Ideal.prime_iff_isPrime hP0).mpr hP).irreducible, normalize_eq,
Multiset.nsmul_singleton, Multiset.inter_replicate, Multiset.prod_replicate]
· rw [← Submodule.span_singleton_le_iff_mem, Ideal.submodule_span_eq] at a_mem a_notMem
rwa [Ideal.count_normalizedFactors_eq a_mem a_notMem, min_eq_left i.le_succ]
· intro ha
rw [Ideal.span_singleton_eq_bot.mp ha] at a_notMem
have := (P ^ (i + 1)).zero_mem
contradiction
/-- Quotienting `P^i / P^e` by its subspace `P^(i+1) ⧸ P^e` is
`R ⧸ p`-linearly isomorphic to `S ⧸ P`. -/
noncomputable def quotientRangePowQuotSuccInclusionEquiv [IsDedekindDomain S]
[P.IsPrime] (hP : P ≠ ⊥) {i : ℕ} (hi : i < e) :
((P ^ i).map (Ideal.Quotient.mk (P ^ e)) ⧸ LinearMap.range (powQuotSuccInclusion p P i))
≃ₗ[R ⧸ p] S ⧸ P := by
choose a a_mem a_notMem using
SetLike.exists_of_lt
(Ideal.pow_right_strictAnti P hP (Ideal.IsPrime.ne_top inferInstance) (le_refl i.succ))
refine (LinearEquiv.ofBijective ?_ ⟨?_, ?_⟩).symm
· exact quotientToQuotientRangePowQuotSucc p P a_mem
· exact quotientToQuotientRangePowQuotSucc_injective p P hi a_mem a_notMem
· exact quotientToQuotientRangePowQuotSucc_surjective p P hP hi a_mem a_notMem
/-- Since the inclusion `(P^(i + 1) / P^e) ⊂ (P^i / P^e)` has a kernel isomorphic to `P / S`,
`[P^i / P^e : R / p] = [P^(i+1) / P^e : R / p] + [P / S : R / p]` -/
theorem rank_pow_quot_aux [IsDedekindDomain S] [p.IsMaximal] [P.IsPrime] (hP0 : P ≠ ⊥)
{i : ℕ} (hi : i < e) :
Module.rank (R ⧸ p) (Ideal.map (Ideal.Quotient.mk (P ^ e)) (P ^ i)) =
Module.rank (R ⧸ p) (S ⧸ P) +
Module.rank (R ⧸ p) (Ideal.map (Ideal.Quotient.mk (P ^ e)) (P ^ (i + 1))) := by
rw [← rank_range_of_injective _ (powQuotSuccInclusion_injective p P i),
(quotientRangePowQuotSuccInclusionEquiv p P hP0 hi).symm.rank_eq]
exact (Submodule.rank_quotient_add_rank (LinearMap.range (powQuotSuccInclusion p P i))).symm
theorem rank_pow_quot [IsDedekindDomain S] [p.IsMaximal] [P.IsPrime] (hP0 : P ≠ ⊥)
(i : ℕ) (hi : i ≤ e) :
Module.rank (R ⧸ p) (Ideal.map (Ideal.Quotient.mk (P ^ e)) (P ^ i)) =
(e - i) • Module.rank (R ⧸ p) (S ⧸ P) := by
let Q : ℕ → Prop :=
fun i => Module.rank (R ⧸ p) { x // x ∈ map (Quotient.mk (P ^ e)) (P ^ i) }
= (e - i) • Module.rank (R ⧸ p) (S ⧸ P)
refine Nat.decreasingInduction' (P := Q) (fun j lt_e _le_j ih => ?_) hi ?_
· dsimp only [Q]
rw [rank_pow_quot_aux p P _ lt_e, ih, ← succ_nsmul', Nat.sub_succ, ← Nat.succ_eq_add_one,
Nat.succ_pred_eq_of_pos (Nat.sub_pos_of_lt lt_e)]
assumption
· dsimp only [Q]
rw [Nat.sub_self, zero_nsmul, map_quotient_self]
exact rank_bot (R ⧸ p) (S ⧸ P ^ e)
end
/-- If `p` is a maximal ideal of `R`, `S` extends `R` and `P^e` lies over `p`,
then the dimension `[S/(P^e) : R/p]` is equal to `e * [S/P : R/p]`. -/
theorem rank_prime_pow_ramificationIdx [IsDedekindDomain S] [p.IsMaximal] [P.IsPrime]
(hP0 : P ≠ ⊥) (he : e ≠ 0) :
Module.rank (R ⧸ p) (S ⧸ P ^ e) =
e •
@Module.rank (R ⧸ p) (S ⧸ P) _ _
(@Algebra.toModule _ _ _ _ <|
@Quotient.algebraQuotientOfRamificationIdxNeZero _ _ _ _ _ _ _ ⟨he⟩) := by
letI : NeZero e := ⟨he⟩
have := rank_pow_quot p P hP0 0 (Nat.zero_le e)
rw [pow_zero, Nat.sub_zero, Ideal.one_eq_top, Ideal.map_top] at this
exact (rank_top (R ⧸ p) _).symm.trans this
/-- If `p` is a maximal ideal of `R`, `S` extends `R` and `P^e` lies over `p`,
then the dimension `[S/(P^e) : R/p]`, as a natural number, is equal to `e * [S/P : R/p]`. -/
theorem finrank_prime_pow_ramificationIdx [IsDedekindDomain S] (hP0 : P ≠ ⊥)
[p.IsMaximal] [P.IsPrime] (he : e ≠ 0) :
finrank (R ⧸ p) (S ⧸ P ^ e) =
e *
@finrank (R ⧸ p) (S ⧸ P) _ _
(@Algebra.toModule _ _ _ _ <|
@Quotient.algebraQuotientOfRamificationIdxNeZero _ _ _ _ _ _ _ ⟨he⟩) := by
letI : NeZero e := ⟨he⟩
letI : Algebra (R ⧸ p) (S ⧸ P) := Quotient.algebraQuotientOfRamificationIdxNeZero p P
have hdim := rank_prime_pow_ramificationIdx _ _ hP0 he
by_cases hP : FiniteDimensional (R ⧸ p) (S ⧸ P)
· haveI := (finiteDimensional_iff_of_rank_eq_nsmul he hdim).mpr hP
apply @Nat.cast_injective Cardinal
rw [finrank_eq_rank', Nat.cast_mul, finrank_eq_rank', hdim, nsmul_eq_mul]
have hPe := mt (finiteDimensional_iff_of_rank_eq_nsmul he hdim).mp hP
simp only [finrank_of_infinite_dimensional hP, finrank_of_infinite_dimensional hPe,
mul_zero]
end FactLeComap
section FactorsMap
/-! ## Properties of the factors of `p.map (algebraMap R S)` -/
variable [IsDedekindDomain S] [Algebra R S]
open scoped Classical in
theorem Factors.ne_bot (P : (factors (map (algebraMap R S) p)).toFinset) : (P : Ideal S) ≠ ⊥ :=
(prime_of_factor _ (Multiset.mem_toFinset.mp P.2)).ne_zero
open scoped Classical in
instance Factors.isPrime (P : (factors (map (algebraMap R S) p)).toFinset) :
IsPrime (P : Ideal S) :=
Ideal.isPrime_of_prime (prime_of_factor _ (Multiset.mem_toFinset.mp P.2))
open scoped Classical in
theorem Factors.ramificationIdx_ne_zero (P : (factors (map (algebraMap R S) p)).toFinset) :
ramificationIdx (algebraMap R S) p P ≠ 0 :=
IsDedekindDomain.ramificationIdx_ne_zero (ne_zero_of_mem_factors (Multiset.mem_toFinset.mp P.2))
(Factors.isPrime p P) (Ideal.le_of_dvd (dvd_of_mem_factors (Multiset.mem_toFinset.mp P.2)))
open scoped Classical in
instance Factors.fact_ramificationIdx_neZero (P : (factors (map (algebraMap R S) p)).toFinset) :
NeZero (ramificationIdx (algebraMap R S) p P) :=
⟨Factors.ramificationIdx_ne_zero p P⟩
attribute [local instance] Quotient.algebraQuotientOfRamificationIdxNeZero
open scoped Classical in
instance Factors.isScalarTower (P : (factors (map (algebraMap R S) p)).toFinset) :
IsScalarTower R (R ⧸ p) (S ⧸ (P : Ideal S)) :=
IsScalarTower.of_algebraMap_eq' rfl
open scoped Classical in
instance Factors.liesOver [p.IsMaximal] (P : (factors (map (algebraMap R S) p)).toFinset) :
P.1.LiesOver p :=
⟨(comap_eq_of_scalar_tower_quotient (algebraMap (R ⧸ p) (S ⧸ P.1)).injective).symm⟩
open scoped Classical in
theorem Factors.finrank_pow_ramificationIdx [p.IsMaximal]
(P : (factors (map (algebraMap R S) p)).toFinset) :
finrank (R ⧸ p) (S ⧸ (P : Ideal S) ^ ramificationIdx (algebraMap R S) p P) =
ramificationIdx (algebraMap R S) p P * inertiaDeg p (P : Ideal S) := by
rw [finrank_prime_pow_ramificationIdx, inertiaDeg_algebraMap]
exacts [Factors.ne_bot p P, NeZero.ne _]
open scoped Classical in
instance Factors.finiteDimensional_quotient_pow [Module.Finite R S] [p.IsMaximal]
(P : (factors (map (algebraMap R S) p)).toFinset) :
FiniteDimensional (R ⧸ p) (S ⧸ (P : Ideal S) ^ ramificationIdx (algebraMap R S) p P) := by
refine .of_finrank_pos ?_
rw [pos_iff_ne_zero, Factors.finrank_pow_ramificationIdx]
exact mul_ne_zero (Factors.ramificationIdx_ne_zero p P) (inertiaDeg_pos p P.1).ne'
universe w
open scoped Classical in
/-- **Chinese remainder theorem** for a ring of integers: if the prime ideal `p : Ideal R`
factors in `S` as `∏ i, P i ^ e i`, then `S ⧸ I` factors as `Π i, R ⧸ (P i ^ e i)`. -/
noncomputable def Factors.piQuotientEquiv (p : Ideal R) (hp : map (algebraMap R S) p ≠ ⊥) :
S ⧸ map (algebraMap R S) p ≃+*
∀ P : (factors (map (algebraMap R S) p)).toFinset,
S ⧸ (P : Ideal S) ^ ramificationIdx (algebraMap R S) p P :=
(IsDedekindDomain.quotientEquivPiFactors hp).trans <|
@RingEquiv.piCongrRight (factors (map (algebraMap R S) p)).toFinset
(fun P => S ⧸ (P : Ideal S) ^ (factors (map (algebraMap R S) p)).count (P : Ideal S))
(fun P => S ⧸ (P : Ideal S) ^ ramificationIdx (algebraMap R S) p P) _ _
fun P : (factors (map (algebraMap R S) p)).toFinset =>
Ideal.quotEquivOfEq <| by
rw [IsDedekindDomain.ramificationIdx_eq_factors_count hp (Factors.isPrime p P)
(Factors.ne_bot p P)]
@[simp]
theorem Factors.piQuotientEquiv_mk (p : Ideal R) (hp : map (algebraMap R S) p ≠ ⊥) (x : S) :
Factors.piQuotientEquiv p hp (Ideal.Quotient.mk _ x) = fun _ => Ideal.Quotient.mk _ x := rfl
@[simp]
theorem Factors.piQuotientEquiv_map (p : Ideal R) (hp : map (algebraMap R S) p ≠ ⊥) (x : R) :
Factors.piQuotientEquiv p hp (algebraMap _ _ x) = fun _ =>
Ideal.Quotient.mk _ (algebraMap _ _ x) := rfl
variable (S)
open scoped Classical in
/-- **Chinese remainder theorem** for a ring of integers: if the prime ideal `p : Ideal R`
factors in `S` as `∏ i, P i ^ e i`,
then `S ⧸ I` factors `R ⧸ I`-linearly as `Π i, R ⧸ (P i ^ e i)`. -/
noncomputable def Factors.piQuotientLinearEquiv (p : Ideal R) (hp : map (algebraMap R S) p ≠ ⊥) :
(S ⧸ map (algebraMap R S) p) ≃ₗ[R ⧸ p]
∀ P : (factors (map (algebraMap R S) p)).toFinset,
S ⧸ (P : Ideal S) ^ ramificationIdx (algebraMap R S) p P :=
{ Factors.piQuotientEquiv p hp with
map_smul' := by
rintro ⟨c⟩ ⟨x⟩; ext P
simp only [Submodule.Quotient.quot_mk_eq_mk, Quotient.mk_eq_mk, Algebra.smul_def,
Quotient.algebraMap_quotient_map_quotient, Quotient.mk_algebraMap,
RingHomCompTriple.comp_apply, Pi.mul_apply, Pi.algebraMap_apply]
congr }
open scoped Classical in
/-- The **fundamental identity** of ramification index `e` and inertia degree `f`:
for `P` ranging over the primes lying over `p`, `∑ P, e P * f P = [Frac(S) : Frac(R)]`;
here `S` is a finite `R`-module (and thus `Frac(S) : Frac(R)` is a finite extension) and `p`
is maximal. -/
theorem sum_ramification_inertia (K L : Type*) [Field K] [Field L] [IsDedekindDomain R]
[Algebra R K] [IsFractionRing R K] [Algebra S L] [IsFractionRing S L] [Algebra K L]
[Algebra R L] [IsScalarTower R S L] [IsScalarTower R K L] [Module.Finite R S]
[p.IsMaximal] (hp0 : p ≠ ⊥) :
(∑ P ∈ (factors (map (algebraMap R S) p)).toFinset,
ramificationIdx (algebraMap R S) p P * inertiaDeg p P) =
finrank K L := by
set e := ramificationIdx (algebraMap R S) p
set f := inertiaDeg p (S := S)
calc
(∑ P ∈ (factors (map (algebraMap R S) p)).toFinset, e P * f P) =
∑ P ∈ (factors (map (algebraMap R S) p)).toFinset.attach,
finrank (R ⧸ p) (S ⧸ (P : Ideal S) ^ e P) := ?_
_ = finrank (R ⧸ p)
(∀ P : (factors (map (algebraMap R S) p)).toFinset, S ⧸ (P : Ideal S) ^ e P) :=
(finrank_pi_fintype (R ⧸ p)).symm
_ = finrank (R ⧸ p) (S ⧸ map (algebraMap R S) p) := ?_
_ = finrank K L := ?_
· rw [← Finset.sum_attach]
refine Finset.sum_congr rfl fun P _ => ?_
rw [Factors.finrank_pow_ramificationIdx]
· refine LinearEquiv.finrank_eq (Factors.piQuotientLinearEquiv S p ?_).symm
rwa [Ne, Ideal.map_eq_bot_iff_le_ker, (RingHom.injective_iff_ker_eq_bot _).mp <|
algebraMap_injective_of_field_isFractionRing R S K L, le_bot_iff]
· exact finrank_quotient_map p K L
end FactorsMap
section tower
variable {R S T : Type*} [CommRing R] [CommRing S] [CommRing T]
theorem ramificationIdx_tower [IsDedekindDomain S] [IsDedekindDomain T] {f : R →+* S} {g : S →+* T}
{p : Ideal R} {P : Ideal S} {Q : Ideal T} [hpm : P.IsPrime] [hqm : Q.IsPrime]
(hg0 : map g P ≠ ⊥) (hfg : map (g.comp f) p ≠ ⊥) (hg : map g P ≤ Q) :
ramificationIdx (g.comp f) p Q = ramificationIdx f p P * ramificationIdx g P Q := by
classical
have hf0 : map f p ≠ ⊥ :=
ne_bot_of_map_ne_bot (Eq.mp (congrArg (fun I ↦ I ≠ ⊥) (map_map f g).symm) hfg)
have hp0 : P ≠ ⊥ := ne_bot_of_map_ne_bot hg0
have hq0 : Q ≠ ⊥ := ne_bot_of_le_ne_bot hg0 hg
letI : P.IsMaximal := Ring.DimensionLEOne.maximalOfPrime hp0 hpm
rw [IsDedekindDomain.ramificationIdx_eq_normalizedFactors_count hf0 hpm hp0,
IsDedekindDomain.ramificationIdx_eq_normalizedFactors_count hg0 hqm hq0,
IsDedekindDomain.ramificationIdx_eq_normalizedFactors_count hfg hqm hq0, ← map_map]
rcases eq_prime_pow_mul_coprime hf0 P with ⟨I, hcp, heq⟩
have hcp : ⊤ = map g P ⊔ map g I := by rw [← map_sup, hcp, map_top g]
have hntq : ¬ ⊤ ≤ Q := fun ht ↦ IsPrime.ne_top hqm (Iff.mpr (eq_top_iff_one Q) (ht trivial))
nth_rw 1 [heq, Ideal.map_mul, Ideal.map_pow, normalizedFactors_mul (pow_ne_zero _ hg0) <| by
by_contra h
simp only [h, Submodule.zero_eq_bot, bot_le, sup_of_le_left] at hcp
exact hntq (hcp.trans_le hg), Multiset.count_add, normalizedFactors_pow, Multiset.count_nsmul]
exact add_eq_left.mpr <| Decidable.byContradiction fun h ↦ hntq <| hcp.trans_le <|
sup_le hg <| le_of_dvd <| dvd_of_mem_normalizedFactors <| Multiset.count_ne_zero.mp h
variable [Algebra R S] [Algebra S T] [Algebra R T] [IsScalarTower R S T]
/-- Let `T / S / R` be a tower of algebras, `p, P, Q` be ideals in `R, S, T` respectively,
and `P` and `Q` are prime. If `P = Q ∩ S`, then `e (Q | p) = e (P | p) * e (Q | P)`. -/
theorem ramificationIdx_algebra_tower [IsDedekindDomain S] [IsDedekindDomain T]
{p : Ideal R} {P : Ideal S} {Q : Ideal T} [hpm : P.IsPrime] [hqm : Q.IsPrime]
(hg0 : map (algebraMap S T) P ≠ ⊥)
(hfg : map (algebraMap R T) p ≠ ⊥) (hg : map (algebraMap S T) P ≤ Q) :
ramificationIdx (algebraMap R T) p Q =
ramificationIdx (algebraMap R S) p P * ramificationIdx (algebraMap S T) P Q := by
rw [IsScalarTower.algebraMap_eq R S T] at hfg ⊢
exact ramificationIdx_tower hg0 hfg hg
/-- Let `T / S / R` be a tower of algebras, `p, P, I` be ideals in `R, S, T`, respectively,
and `p` and `P` are maximal. If `p = P ∩ S` and `P = I ∩ S`,
then `f (I | p) = f (P | p) * f (I | P)`. -/
theorem inertiaDeg_algebra_tower (p : Ideal R) (P : Ideal S) (I : Ideal T) [p.IsMaximal]
[P.IsMaximal] [P.LiesOver p] [I.LiesOver P] : inertiaDeg p I =
inertiaDeg p P * inertiaDeg P I := by
have h₁ := P.over_def p
have h₂ := I.over_def P
have h₃ := (LiesOver.trans I P p).over
simp only [inertiaDeg, dif_pos h₁.symm, dif_pos h₂.symm, dif_pos h₃.symm]
letI : Algebra (R ⧸ p) (S ⧸ P) := Ideal.Quotient.algebraQuotientOfLEComap h₁.le
letI : Algebra (S ⧸ P) (T ⧸ I) := Ideal.Quotient.algebraQuotientOfLEComap h₂.le
letI : Algebra (R ⧸ p) (T ⧸ I) := Ideal.Quotient.algebraQuotientOfLEComap h₃.le
letI : IsScalarTower (R ⧸ p) (S ⧸ P) (T ⧸ I) := IsScalarTower.of_algebraMap_eq <| by
rintro ⟨x⟩; exact congr_arg _ (IsScalarTower.algebraMap_apply R S T x)
exact (finrank_mul_finrank (R ⧸ p) (S ⧸ P) (T ⧸ I)).symm
end tower
end Ideal
|
ShiftSequence.lean
|
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.InducedShiftSequence
import Mathlib.CategoryTheory.Shift.Localization
import Mathlib.Algebra.Homology.HomotopyCategory.Shift
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Algebra.Homology.QuasiIso
/-! # Compatibilities of the homology functor with the shift
This files studies how homology of cochain complexes behaves with respect to
the shift: there is a natural isomorphism `(K⟦n⟧).homology a ≅ K.homology a`
when `n + a = a'`. This is summarized by instances
`(homologyFunctor C (ComplexShape.up ℤ) 0).ShiftSequence ℤ` in the `CochainComplex`
and `HomotopyCategory` namespaces.
-/
assert_not_exists TwoSidedIdeal
open CategoryTheory Category ComplexShape Limits
variable (C : Type*) [Category C] [Preadditive C]
namespace CochainComplex
open HomologicalComplex
attribute [local simp] XIsoOfEq_hom_naturality smul_smul
/-- The natural isomorphism `(K⟦n⟧).sc' i j k ≅ K.sc' i' j' k'` when `n + i = i'`,
`n + j = j'` and `n + k = k'`. -/
@[simps!]
def shiftShortComplexFunctor' (n i j k i' j' k' : ℤ)
(hi : n + i = i') (hj : n + j = j') (hk : n + k = k') :
(CategoryTheory.shiftFunctor (CochainComplex C ℤ) n) ⋙ shortComplexFunctor' C _ i j k ≅
shortComplexFunctor' C _ i' j' k' :=
NatIso.ofComponents (fun K => ShortComplex.isoMk
(n.negOnePow • ((shiftEval C n i i' hi).app K))
((shiftEval C n j j' hj).app K) (n.negOnePow • ((shiftEval C n k k' hk).app K))
(by simp) (by simp))
(fun f ↦ by ext <;> simp)
/-- The natural isomorphism `(K⟦n⟧).sc i ≅ K.sc i'` when `n + i = i'`. -/
@[simps!]
noncomputable def shiftShortComplexFunctorIso (n i i' : ℤ) (hi : n + i = i') :
shiftFunctor C n ⋙ shortComplexFunctor C _ i ≅ shortComplexFunctor C _ i' :=
shiftShortComplexFunctor' C n _ i _ _ i' _
(by simp only [prev]; omega) hi (by simp only [next]; omega)
variable {C}
lemma shiftShortComplexFunctorIso_zero_add_hom_app (a : ℤ) (K : CochainComplex C ℤ) :
(shiftShortComplexFunctorIso C 0 a a (zero_add a)).hom.app K =
(shortComplexFunctor C (ComplexShape.up ℤ) a).map
((shiftFunctorZero (CochainComplex C ℤ) ℤ).hom.app K) := by
ext <;> simp [one_smul, shiftFunctorZero_hom_app_f]
lemma shiftShortComplexFunctorIso_add'_hom_app
(n m mn : ℤ) (hmn : m + n = mn) (a a' a'' : ℤ) (ha' : n + a = a') (ha'' : m + a' = a'')
(K : CochainComplex C ℤ) :
(shiftShortComplexFunctorIso C mn a a'' (by rw [← ha'', ← ha', ← add_assoc, hmn])).hom.app K =
(shortComplexFunctor C (ComplexShape.up ℤ) a).map
((CategoryTheory.shiftFunctorAdd' (CochainComplex C ℤ) m n mn hmn).hom.app K) ≫
(shiftShortComplexFunctorIso C n a a' ha').hom.app (K⟦m⟧) ≫
(shiftShortComplexFunctorIso C m a' a'' ha'' ).hom.app K := by
ext <;> dsimp <;> simp only [← hmn, Int.negOnePow_add, shiftFunctorAdd'_hom_app_f',
XIsoOfEq_shift, Linear.comp_units_smul, Linear.units_smul_comp,
XIsoOfEq_hom_comp_XIsoOfEq_hom, smul_smul]
variable [CategoryWithHomology C]
namespace ShiftSequence
variable (C) in
/-- The natural isomorphism `(K⟦n⟧).homology a ≅ K.homology a'`when `n + a = a`. -/
noncomputable def shiftIso (n a a' : ℤ) (ha' : n + a = a') :
(CategoryTheory.shiftFunctor _ n) ⋙ homologyFunctor C (ComplexShape.up ℤ) a ≅
homologyFunctor C (ComplexShape.up ℤ) a' :=
Functor.isoWhiskerLeft _ (homologyFunctorIso C (ComplexShape.up ℤ) a) ≪≫
(Functor.associator _ _ _).symm ≪≫
Functor.isoWhiskerRight (shiftShortComplexFunctorIso C n a a' ha')
(ShortComplex.homologyFunctor C) ≪≫
(homologyFunctorIso C (ComplexShape.up ℤ) a').symm
lemma shiftIso_hom_app (n a a' : ℤ) (ha' : n + a = a') (K : CochainComplex C ℤ) :
(shiftIso C n a a' ha').hom.app K =
ShortComplex.homologyMap ((shiftShortComplexFunctorIso C n a a' ha').hom.app K) := by
dsimp [shiftIso]
rw [id_comp, id_comp]
-- This `erw` is required to bridge the gap between
-- `((shortComplexFunctor C (up ℤ) a').obj K).homology`
-- (the target of the first morphism)
-- and
-- `homology K a'`
-- (the source of the identity morphism).
erw [comp_id]
lemma shiftIso_inv_app (n a a' : ℤ) (ha' : n + a = a') (K : CochainComplex C ℤ) :
(shiftIso C n a a' ha').inv.app K =
ShortComplex.homologyMap ((shiftShortComplexFunctorIso C n a a' ha').inv.app K) := by
dsimp [shiftIso]
rw [id_comp, comp_id]
-- This `erw` is required as above in `shiftIso_hom_app`.
erw [comp_id]
end ShiftSequence
noncomputable instance :
(homologyFunctor C (ComplexShape.up ℤ) 0).ShiftSequence ℤ where
sequence n := homologyFunctor C (ComplexShape.up ℤ) n
isoZero := Iso.refl _
shiftIso n a a' ha' := ShiftSequence.shiftIso C n a a' ha'
shiftIso_zero a := by
ext K
dsimp [homologyMap]
simp only [ShiftSequence.shiftIso_hom_app, comp_id,
shiftShortComplexFunctorIso_zero_add_hom_app]
shiftIso_add n m a a' a'' ha' ha'' := by
ext K
dsimp [homologyMap]
simp only [ShiftSequence.shiftIso_hom_app, id_comp,
← ShortComplex.homologyMap_comp, shiftFunctorAdd'_eq_shiftFunctorAdd,
shiftShortComplexFunctorIso_add'_hom_app n m _ rfl a a' a'' ha' ha'' K]
lemma quasiIsoAt_shift_iff {K L : CochainComplex C ℤ} (φ : K ⟶ L) (n i j : ℤ) (h : n + i = j) :
QuasiIsoAt (φ⟦n⟧') i ↔ QuasiIsoAt φ j := by
simp only [quasiIsoAt_iff_isIso_homologyMap]
exact (NatIso.isIso_map_iff
((homologyFunctor C (ComplexShape.up ℤ) 0).shiftIso n i j h) φ)
lemma quasiIso_shift_iff {K L : CochainComplex C ℤ} (φ : K ⟶ L) (n : ℤ) :
QuasiIso (φ⟦n⟧') ↔ QuasiIso φ := by
simp only [quasiIso_iff, fun i ↦ quasiIsoAt_shift_iff φ n i _ rfl]
constructor
· intro h j
obtain ⟨i, rfl⟩ : ∃ i, j = n + i := ⟨j - n, by omega⟩
exact h i
· intro h i
exact h (n + i)
instance {K L : CochainComplex C ℤ} (φ : K ⟶ L) (n : ℤ) [QuasiIso φ] :
QuasiIso (φ⟦n⟧') := by
rw [quasiIso_shift_iff]
infer_instance
instance : (HomologicalComplex.quasiIso C (ComplexShape.up ℤ)).IsCompatibleWithShift ℤ where
condition n := by ext; apply quasiIso_shift_iff
variable (C) in
lemma homologyFunctor_shift (n : ℤ) :
(homologyFunctor C (ComplexShape.up ℤ) 0).shift n =
homologyFunctor C (ComplexShape.up ℤ) n := rfl
@[reassoc]
lemma liftCycles_shift_homologyπ
(K : CochainComplex C ℤ) {A : C} {n i : ℤ} (f : A ⟶ (K⟦n⟧).X i) (j : ℤ)
(hj : (up ℤ).next i = j) (hf : f ≫ (K⟦n⟧).d i j = 0) (i' : ℤ) (hi' : n + i = i') (j' : ℤ)
(hj' : (up ℤ).next i' = j') :
(K⟦n⟧).liftCycles f j hj hf ≫ (K⟦n⟧).homologyπ i =
K.liftCycles (f ≫ (K.shiftFunctorObjXIso n i i' (by omega)).hom) j' hj' (by
simp only [next] at hj hj'
obtain rfl : i' = i + n := by omega
obtain rfl : j' = j + n := by omega
dsimp at hf ⊢
simp only [Linear.comp_units_smul] at hf
apply (one_smul (M := ℤˣ) _).symm.trans _
rw [← Int.units_mul_self n.negOnePow, mul_smul, comp_id, hf, smul_zero]) ≫
K.homologyπ i' ≫
((HomologicalComplex.homologyFunctor C (up ℤ) 0).shiftIso n i i' hi').inv.app K := by
simp only [liftCycles, homologyπ,
shiftFunctorObjXIso, Functor.shiftIso, Functor.ShiftSequence.shiftIso,
ShiftSequence.shiftIso_inv_app, ShortComplex.homologyπ_naturality,
ShortComplex.liftCycles_comp_cyclesMap_assoc, shiftShortComplexFunctorIso_inv_app_τ₂,
assoc, Iso.hom_inv_id, comp_id]
rfl
end CochainComplex
namespace HomotopyCategory
variable [CategoryWithHomology C]
noncomputable instance :
(homologyFunctor C (ComplexShape.up ℤ) 0).ShiftSequence ℤ :=
Functor.ShiftSequence.induced (homologyFunctorFactors C (ComplexShape.up ℤ) 0) ℤ
(homologyFunctor C (ComplexShape.up ℤ))
(homologyFunctorFactors C (ComplexShape.up ℤ))
variable {C}
lemma homologyShiftIso_hom_app (n a a' : ℤ) (ha' : n + a = a') (K : CochainComplex C ℤ) :
((homologyFunctor C (ComplexShape.up ℤ) 0).shiftIso n a a' ha').hom.app
((quotient _ _).obj K) =
(homologyFunctor _ _ a).map (((quotient _ _).commShiftIso n).inv.app K) ≫
(homologyFunctorFactors _ _ a).hom.app (K⟦n⟧) ≫
((HomologicalComplex.homologyFunctor _ _ 0).shiftIso n a a' ha').hom.app K ≫
(homologyFunctorFactors _ _ a').inv.app K := by
apply Functor.ShiftSequence.induced_shiftIso_hom_app_obj
@[reassoc]
lemma homologyFunctor_shiftMap
{K L : CochainComplex C ℤ} {n : ℤ} (f : K ⟶ L⟦n⟧) (a a' : ℤ) (h : n + a = a') :
(homologyFunctor C (ComplexShape.up ℤ) 0).shiftMap
((quotient _ _).map f ≫ ((quotient _ _).commShiftIso n).hom.app _) a a' h =
(homologyFunctorFactors _ _ a).hom.app K ≫
(HomologicalComplex.homologyFunctor C (ComplexShape.up ℤ) 0).shiftMap f a a' h ≫
(homologyFunctorFactors _ _ a').inv.app L := by
apply Functor.ShiftSequence.induced_shiftMap
end HomotopyCategory
|
Simplex.lean
|
/-
Copyright (c) 2025 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.Normed.Affine.Simplex
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Simplices in Euclidean spaces.
This file defines properties of simplices in a Euclidean space.
## Main definitions
* `Affine.Simplex.AcuteAngled`
-/
namespace Affine
open EuclideanGeometry
open scoped Real
variable {V P : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P]
variable [NormedAddTorsor V P]
namespace Simplex
variable {m n : ℕ}
lemma Equilateral.angle_eq_pi_div_three {s : Simplex ℝ P n} (he : s.Equilateral)
{i₁ i₂ i₃ : Fin (n + 1)} (h₁₂ : i₁ ≠ i₂) (h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃) :
∠ (s.points i₁) (s.points i₂) (s.points i₃) = π / 3 := by
rcases he with ⟨r, hr⟩
rw [angle, InnerProductGeometry.angle,
real_inner_eq_norm_mul_self_add_norm_mul_self_sub_norm_sub_mul_self_div_two]
refine Real.arccos_eq_of_eq_cos (by linarith [Real.pi_nonneg]) (by linarith [Real.pi_nonneg]) ?_
simp only [vsub_sub_vsub_cancel_right, ← dist_eq_norm_vsub, hr _ _ h₁₂, hr _ _ h₁₃,
hr _ _ h₂₃.symm, Real.cos_pi_div_three]
have hr0 : r ≠ 0 := by
rintro rfl
replace hr := hr _ _ h₁₂
rw [dist_eq_zero] at hr
exact h₁₂ (s.independent.injective hr)
field_simp [mul_comm]
/-- The property of all angles of a simplex being acute. -/
def AcuteAngled (s : Simplex ℝ P n) : Prop :=
∀ i₁ i₂ i₃ : Fin (n + 1), i₁ ≠ i₂ → i₁ ≠ i₃ → i₂ ≠ i₃ →
∠ (s.points i₁) (s.points i₂) (s.points i₃) < π / 2
@[simp] lemma acuteAngled_reindex_iff {s : Simplex ℝ P m} (e : Fin (m + 1) ≃ Fin (n + 1)) :
(s.reindex e).AcuteAngled ↔ s.AcuteAngled := by
refine ⟨fun h {i₁ i₂ i₃} h₁₂ h₁₃ h₂₃ ↦ ?_, fun h {i₁ i₂ i₃} h₁₂ h₁₃ h₂₃ ↦ ?_⟩
· convert h (i₁ := e i₁) (i₂ := e i₂) (i₃ := e i₃) ?_ ?_ ?_ using 1 <;> simp [*]
· convert h (i₁ := e.symm i₁) (i₂ := e.symm i₂) (i₃ := e.symm i₃) ?_ ?_ ?_ using 1 <;> simp [*]
lemma Equilateral.acuteAngled {s : Simplex ℝ P n} (he : s.Equilateral) : s.AcuteAngled := by
intro i₁ i₂ i₃ h₁₂ h₁₃ h₂₃
rw [he.angle_eq_pi_div_three h₁₂ h₁₃ h₂₃]
linarith [Real.pi_pos]
end Simplex
namespace Triangle
lemma acuteAngled_iff_angle_lt {t : Triangle ℝ P} : t.AcuteAngled ↔
∠ (t.points 0) (t.points 1) (t.points 2) < π / 2 ∧
∠ (t.points 1) (t.points 2) (t.points 0) < π / 2 ∧
∠ (t.points 2) (t.points 0) (t.points 1) < π / 2 := by
refine ⟨fun h ↦ ⟨h _ _ _ (by decide) (by decide) (by decide),
h _ _ _ (by decide) (by decide) (by decide),
h _ _ _ (by decide) (by decide) (by decide)⟩,
fun ⟨h012, h120, h201⟩ ↦ ?_⟩
have h210 := angle_comm (t.points 0) _ _ ▸ h012
have h021 := angle_comm (t.points 1) _ _ ▸ h120
have h102 := angle_comm (t.points 2) _ _ ▸ h201
intro i₁ i₂ i₃ h₁₂ h₁₃ h₂₃
fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃ <;> simp [*] at *
end Triangle
end Affine
|
Instances.lean
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Group.Defs
import Mathlib.Algebra.Order.Monoid.OrderDual
import Mathlib.Tactic.Linter.DeprecatedModule
deprecated_module (since := "2025-04-16")
|
NonUnital.lean
|
/-
Copyright (c) 2024 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.Spectrum.Quasispectrum
import Mathlib.Topology.ContinuousMap.Compact
import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital
import Mathlib.Topology.UniformSpace.CompactConvergence
/-!
# The continuous functional calculus for non-unital algebras
This file defines a generic API for the *continuous functional calculus* in *non-unital* algebras
which is suitable in a wide range of settings. The design is intended to match as closely as
possible that for unital algebras in
`Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Unital.lean`. Changes to either file
should be mirrored in its counterpart whenever possible. The underlying reasons for the design
decisions in the unital case apply equally in the non-unital case. See the module documentation in
that file for more information.
A continuous functional calculus for an element `a : A` in a non-unital topological `R`-algebra is
a continuous extension of the polynomial functional calculus (i.e., `Polynomial.aeval`) for
polynomials with no constant term to continuous `R`-valued functions on `quasispectrum R a` which
vanish at zero. More precisely, it is a continuous star algebra homomorphism
`C(quasispectrum R a, R)₀ →⋆ₙₐ[R] A` that sends `(ContinuousMap.id R).restrict (quasispectrum R a)`
to `a`. In all cases of interest (e.g., when `quasispectrum R a` is compact and `R` is `ℝ≥0`, `ℝ`,
or `ℂ`), this is sufficient to uniquely determine the continuous functional calculus which is
encoded in the `ContinuousMapZero.UniqueHom` class.
## Main declarations
+ `NonUnitalContinuousFunctionalCalculus R A (p : A → Prop)`: a class stating that every `a : A`
satisfying `p a` has a non-unital star algebra homomorphism from the continuous `R`-valued
functions on the `R`-quasispectrum of `a` vanishing at zero into the algebra `A`. This map is a
closed embedding, and satisfies the **spectral mapping theorem**.
+ `cfcₙHom : p a → C(quasispectrum R a, R)₀ →⋆ₐ[R] A`: the underlying non-unital star algebra
homomorphism for an element satisfying property `p`.
+ `cfcₙ : (R → R) → A → A`: an unbundled version of `cfcₙHom` which takes the junk value `0` when
`cfcₙHom` is not defined.
## Main theorems
+ `cfcₙ_comp : cfcₙ (x ↦ g (f x)) a = cfcₙ g (cfcₙ f a)`
-/
local notation "σₙ" => quasispectrum
open Topology ContinuousMapZero
/-- A non-unital star `R`-algebra `A` has a *continuous functional calculus* for elements
satisfying the property `p : A → Prop` if
+ for every such element `a : A` there is a non-unital star algebra homomorphism
`cfcₙHom : C(quasispectrum R a, R)₀ →⋆ₙₐ[R] A` sending the (restriction of) the identity map
to `a`.
+ `cfcₙHom` is a closed embedding for which the quasispectrum of the image of function `f` is its
range.
+ `cfcₙHom` preserves the property `p`.
The property `p` is marked as an `outParam` so that the user need not specify it. In practice,
+ for `R := ℂ`, we choose `p := IsStarNormal`,
+ for `R := ℝ`, we choose `p := IsSelfAdjoint`,
+ for `R := ℝ≥0`, we choose `p := (0 ≤ ·)`.
Instead of directly providing the data we opt instead for a `Prop` class. In all relevant cases,
the continuous functional calculus is uniquely determined, and utilizing this approach
prevents diamonds or problems arising from multiple instances. -/
class NonUnitalContinuousFunctionalCalculus (R A : Type*) (p : outParam (A → Prop))
[CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [IsTopologicalSemiring R]
[ContinuousStar R] [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module R A]
[IsScalarTower R A A] [SMulCommClass R A A] : Prop where
predicate_zero : p 0
[compactSpace_quasispectrum : ∀ a : A, CompactSpace (σₙ R a)]
exists_cfc_of_predicate : ∀ a, p a → ∃ φ : C(σₙ R a, R)₀ →⋆ₙₐ[R] A,
IsClosedEmbedding φ ∧ φ ⟨(ContinuousMap.id R).restrict <| σₙ R a, rfl⟩ = a ∧
(∀ f, σₙ R (φ f) = Set.range f) ∧ ∀ f, p (φ f)
-- this instance should not be activated everywhere but it is useful when developing generic API
-- for the continuous functional calculus
scoped[NonUnitalContinuousFunctionalCalculus]
attribute [instance] NonUnitalContinuousFunctionalCalculus.compactSpace_quasispectrum
/-- A class guaranteeing that the non-unital continuous functional calculus is uniquely determined
by the properties that it is a continuous non-unital star algebra homomorphism mapping the
(restriction of) the identity to `a`. This is the necessary tool used to establish `cfcₙHom_comp`
and the more common variant `cfcₙ_comp`.
This class will have instances in each of the common cases `ℂ`, `ℝ` and `ℝ≥0` as a consequence of
the Stone-Weierstrass theorem. -/
class ContinuousMapZero.UniqueHom (R A : Type*) [CommSemiring R] [StarRing R]
[MetricSpace R] [IsTopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A]
[TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] : Prop where
eq_of_continuous_of_map_id (s : Set R) [CompactSpace s] [Zero s] (h0 : (0 : s) = (0 : R))
(φ ψ : C(s, R)₀ →⋆ₙₐ[R] A) (hφ : Continuous φ) (hψ : Continuous ψ)
(h : φ (⟨.restrict s <| .id R, h0⟩) = ψ (⟨.restrict s <| .id R, h0⟩)) :
φ = ψ
section Main
variable {R A : Type*} {p : A → Prop} [CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R]
variable [IsTopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A]
variable [TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]
variable [instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p]
include instCFCₙ in
lemma NonUnitalContinuousFunctionalCalculus.isCompact_quasispectrum (a : A) :
IsCompact (σₙ R a) :=
isCompact_iff_compactSpace.mpr inferInstance
lemma NonUnitalStarAlgHom.ext_continuousMap [UniqueHom R A]
(a : A) [CompactSpace (σₙ R a)] (φ ψ : C(σₙ R a, R)₀ →⋆ₙₐ[R] A)
(hφ : Continuous φ) (hψ : Continuous ψ)
(h : φ ⟨.restrict (σₙ R a) <| .id R, rfl⟩ = ψ ⟨.restrict (σₙ R a) <| .id R, rfl⟩) :
φ = ψ :=
UniqueHom.eq_of_continuous_of_map_id _ (by simp) φ ψ hφ hψ h
section cfcₙHom
variable {a : A} (ha : p a)
/-- The non-unital star algebra homomorphism underlying a instance of the continuous functional
calculus for non-unital algebras; a version for continuous functions on the quasispectrum.
In this case, the user must supply the fact that `a` satisfies the predicate `p`.
While `NonUnitalContinuousFunctionalCalculus` is stated in terms of these homomorphisms, in practice
the user should instead prefer `cfcₙ` over `cfcₙHom`.
-/
noncomputable def cfcₙHom : C(σₙ R a, R)₀ →⋆ₙₐ[R] A :=
(NonUnitalContinuousFunctionalCalculus.exists_cfc_of_predicate a ha).choose
lemma cfcₙHom_isClosedEmbedding :
IsClosedEmbedding <| (cfcₙHom ha : C(σₙ R a, R)₀ →⋆ₙₐ[R] A) :=
(NonUnitalContinuousFunctionalCalculus.exists_cfc_of_predicate a ha).choose_spec.1
@[fun_prop]
lemma cfcₙHom_continuous : Continuous (cfcₙHom ha : C(σₙ R a, R)₀ →⋆ₙₐ[R] A) :=
cfcₙHom_isClosedEmbedding ha |>.continuous
lemma cfcₙHom_id :
cfcₙHom ha ⟨(ContinuousMap.id R).restrict <| σₙ R a, rfl⟩ = a :=
(NonUnitalContinuousFunctionalCalculus.exists_cfc_of_predicate a ha).choose_spec.2.1
/-- The **spectral mapping theorem** for the non-unital continuous functional calculus. -/
lemma cfcₙHom_map_quasispectrum (f : C(σₙ R a, R)₀) :
σₙ R (cfcₙHom ha f) = Set.range f :=
(NonUnitalContinuousFunctionalCalculus.exists_cfc_of_predicate a ha).choose_spec.2.2.1 f
lemma cfcₙHom_predicate (f : C(σₙ R a, R)₀) :
p (cfcₙHom ha f) :=
(NonUnitalContinuousFunctionalCalculus.exists_cfc_of_predicate a ha).choose_spec.2.2.2 f
open scoped NonUnitalContinuousFunctionalCalculus in
lemma cfcₙHom_eq_of_continuous_of_map_id [UniqueHom R A]
(φ : C(σₙ R a, R)₀ →⋆ₙₐ[R] A) (hφ₁ : Continuous φ)
(hφ₂ : φ ⟨.restrict (σₙ R a) <| .id R, rfl⟩ = a) : cfcₙHom ha = φ :=
(cfcₙHom ha).ext_continuousMap a φ (cfcₙHom_isClosedEmbedding ha).continuous hφ₁ <| by
rw [cfcₙHom_id ha, hφ₂]
theorem cfcₙHom_comp [UniqueHom R A] (f : C(σₙ R a, R)₀)
(f' : C(σₙ R a, σₙ R (cfcₙHom ha f))₀)
(hff' : ∀ x, f x = f' x) (g : C(σₙ R (cfcₙHom ha f), R)₀) :
cfcₙHom ha (g.comp f') = cfcₙHom (cfcₙHom_predicate ha f) g := by
let ψ : C(σₙ R (cfcₙHom ha f), R)₀ →⋆ₙₐ[R] C(σₙ R a, R)₀ :=
{ toFun := (ContinuousMapZero.comp · f')
map_smul' := fun _ _ ↦ rfl
map_add' := fun _ _ ↦ rfl
map_mul' := fun _ _ ↦ rfl
map_zero' := rfl
map_star' := fun _ ↦ rfl }
let φ : C(σₙ R (cfcₙHom ha f), R)₀ →⋆ₙₐ[R] A := (cfcₙHom ha).comp ψ
suffices cfcₙHom (cfcₙHom_predicate ha f) = φ from DFunLike.congr_fun this.symm g
refine cfcₙHom_eq_of_continuous_of_map_id (cfcₙHom_predicate ha f) φ ?_ ?_
· refine (cfcₙHom_isClosedEmbedding ha).continuous.comp <| continuous_induced_rng.mpr ?_
exact f'.toContinuousMap.continuous_precomp.comp continuous_induced_dom
· simp only [φ, ψ, NonUnitalStarAlgHom.comp_apply, NonUnitalStarAlgHom.coe_mk',
NonUnitalAlgHom.coe_mk]
congr
ext x
simp [hff']
end cfcₙHom
section cfcₙL
/-- `cfcₙHom` bundled as a continuous linear map. -/
@[simps apply]
noncomputable def cfcₙL {a : A} (ha : p a) : C(σₙ R a, R)₀ →L[R] A :=
{ cfcₙHom ha with
toFun := cfcₙHom ha
map_smul' := map_smul _
cont := (cfcₙHom_isClosedEmbedding ha).continuous }
end cfcₙL
section CFCn
open Classical in
/-- This is the *continuous functional calculus* of an element `a : A` in a non-unital algebra
applied to bare functions. When either `a` does not satisfy the predicate `p` (i.e., `a` is not
`IsStarNormal`, `IsSelfAdjoint`, or `0 ≤ a` when `R` is `ℂ`, `ℝ`, or `ℝ≥0`, respectively), or when
`f : R → R` is not continuous on the quasispectrum of `a` or `f 0 ≠ 0`, then `cfcₙ f a` returns the
junk value `0`.
This is the primary declaration intended for widespread use of the continuous functional calculus
for non-unital algebras, and all the API applies to this declaration. For more information, see the
module documentation for `Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital`. -/
noncomputable irreducible_def cfcₙ (f : R → R) (a : A) : A :=
if h : p a ∧ ContinuousOn f (σₙ R a) ∧ f 0 = 0
then cfcₙHom h.1 ⟨⟨_, h.2.1.restrict⟩, h.2.2⟩
else 0
variable (f g : R → R) (a : A)
variable (hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac) (hf0 : f 0 = 0 := by cfc_zero_tac)
variable (hg : ContinuousOn g (σₙ R a) := by cfc_cont_tac) (hg0 : g 0 = 0 := by cfc_zero_tac)
variable (ha : p a := by cfc_tac)
lemma cfcₙ_apply : cfcₙ f a = cfcₙHom (a := a) ha ⟨⟨_, hf.restrict⟩, hf0⟩ := by
rw [cfcₙ_def, dif_pos ⟨ha, hf, hf0⟩]
lemma cfcₙ_apply_pi {ι : Type*} (f : ι → R → R) (a : A) (ha := by cfc_tac)
(hf : ∀ i, ContinuousOn (f i) (σₙ R a) := by cfc_cont_tac)
(hf0 : ∀ i, f i 0 = 0 := by cfc_zero_tac) :
(fun i => cfcₙ (f i) a) = (fun i => cfcₙHom (a := a) ha ⟨⟨_, (hf i).restrict⟩, hf0 i⟩) := by
ext i
simp only [cfcₙ_apply (f i) a (hf i) (hf0 i)]
lemma cfcₙ_apply_of_not_and_and {f : R → R} (a : A)
(ha : ¬ (p a ∧ ContinuousOn f (σₙ R a) ∧ f 0 = 0)) :
cfcₙ f a = 0 := by
rw [cfcₙ_def, dif_neg ha]
lemma cfcₙ_apply_of_not_predicate {f : R → R} (a : A) (ha : ¬ p a) :
cfcₙ f a = 0 := by
rw [cfcₙ_def, dif_neg (not_and_of_not_left _ ha)]
lemma cfcₙ_apply_of_not_continuousOn {f : R → R} (a : A) (hf : ¬ ContinuousOn f (σₙ R a)) :
cfcₙ f a = 0 := by
rw [cfcₙ_def, dif_neg (not_and_of_not_right _ (not_and_of_not_left _ hf))]
lemma cfcₙ_apply_of_not_map_zero {f : R → R} (a : A) (hf : ¬ f 0 = 0) :
cfcₙ f a = 0 := by
rw [cfcₙ_def, dif_neg (not_and_of_not_right _ (not_and_of_not_right _ hf))]
lemma cfcₙHom_eq_cfcₙ_extend {a : A} (g : R → R) (ha : p a) (f : C(σₙ R a, R)₀) :
cfcₙHom ha f = cfcₙ (Function.extend Subtype.val f g) a := by
have h : f = (σₙ R a).restrict (Function.extend Subtype.val f g) := by
ext; simp
have hg : ContinuousOn (Function.extend Subtype.val f g) (σₙ R a) :=
continuousOn_iff_continuous_restrict.mpr <| h ▸ map_continuous f
have hg0 : (Function.extend Subtype.val f g) 0 = 0 := by
rw [← quasispectrum.coe_zero (R := R) a, Subtype.val_injective.extend_apply]
exact map_zero f
rw [cfcₙ_apply ..]
congr!
lemma cfcₙ_eq_cfcₙL {a : A} {f : R → R} (ha : p a) (hf : ContinuousOn f (σₙ R a)) (hf0 : f 0 = 0) :
cfcₙ f a = cfcₙL ha ⟨⟨_, hf.restrict⟩, hf0⟩ := by
rw [cfcₙ_def, dif_pos ⟨ha, hf, hf0⟩, cfcₙL_apply]
/-- A version of `cfcₙ_apply` in terms of `ContinuousMapZero.mkD` -/
lemma cfcₙ_apply_mkD :
cfcₙ f a = cfcₙHom (a := a) ha (mkD ((quasispectrum R a).restrict f) 0) := by
by_cases f_cont : ContinuousOn f (quasispectrum R a)
· by_cases f_zero : f 0 = 0
· rw [cfcₙ_apply f a, mkD_of_continuousOn f_cont f_zero]
· rw [cfcₙ_apply_of_not_map_zero a f_zero, mkD_of_not_zero, map_zero]
exact f_zero
· rw [cfcₙ_apply_of_not_continuousOn a f_cont, mkD_of_not_continuousOn f_cont, map_zero]
/-- A version of `cfcₙ_eq_cfcₙL` in terms of `ContinuousMapZero.mkD` -/
lemma cfcₙ_eq_cfcₙL_mkD :
cfcₙ f a = cfcₙL (a := a) ha (mkD ((quasispectrum R a).restrict f) 0) :=
cfcₙ_apply_mkD _ _
lemma cfcₙ_cases (P : A → Prop) (a : A) (f : R → R) (h₀ : P 0)
(haf : ∀ (hf : ContinuousOn f (σₙ R a)) h0 ha, P (cfcₙHom ha ⟨⟨_, hf.restrict⟩, h0⟩)) :
P (cfcₙ f a) := by
by_cases h : ContinuousOn f (σₙ R a) ∧ f 0 = 0 ∧ p a
· rw [cfcₙ_apply f a h.1 h.2.1 h.2.2]
exact haf h.1 h.2.1 h.2.2
· simp only [not_and_or] at h
obtain (h | h | h) := h
· rwa [cfcₙ_apply_of_not_continuousOn _ h]
· rwa [cfcₙ_apply_of_not_map_zero _ h]
· rwa [cfcₙ_apply_of_not_predicate _ h]
lemma cfcₙ_commute_cfcₙ (f g : R → R) (a : A) : Commute (cfcₙ f a) (cfcₙ g a) := by
refine cfcₙ_cases (fun x ↦ Commute x (cfcₙ g a)) a f (by simp) fun hf hf0 ha ↦ ?_
refine cfcₙ_cases (fun x ↦ Commute _ x) a g (by simp) fun hg hg0 _ ↦ ?_
exact Commute.all _ _ |>.map _
variable (R) in
include ha in
lemma cfcₙ_id : cfcₙ (id : R → R) a = a :=
cfcₙ_apply (id : R → R) a ▸ cfcₙHom_id (p := p) ha
variable (R) in
include ha in
lemma cfcₙ_id' : cfcₙ (fun x : R ↦ x) a = a := cfcₙ_id R a
include ha hf hf0 in
/-- The **spectral mapping theorem** for the non-unital continuous functional calculus. -/
lemma cfcₙ_map_quasispectrum : σₙ R (cfcₙ f a) = f '' σₙ R a := by
simp [cfcₙ_apply f a, cfcₙHom_map_quasispectrum (p := p)]
variable (R) in
include R in
lemma cfcₙ_predicate_zero : p 0 :=
NonUnitalContinuousFunctionalCalculus.predicate_zero (R := R)
lemma cfcₙ_predicate (f : R → R) (a : A) : p (cfcₙ f a) :=
cfcₙ_cases p a f (cfcₙ_predicate_zero R) fun _ _ _ ↦ cfcₙHom_predicate ..
lemma cfcₙ_congr {f g : R → R} {a : A} (hfg : (σₙ R a).EqOn f g) :
cfcₙ f a = cfcₙ g a := by
by_cases h : p a ∧ ContinuousOn g (σₙ R a) ∧ g 0 = 0
· rw [cfcₙ_apply f a (h.2.1.congr hfg) (hfg (quasispectrum.zero_mem R a) ▸ h.2.2) h.1,
cfcₙ_apply g a h.2.1 h.2.2 h.1]
congr 3
exact Set.restrict_eq_iff.mpr hfg
· simp only [not_and_or] at h
obtain (ha | hg | h0) := h
· simp [cfcₙ_apply_of_not_predicate a ha]
· rw [cfcₙ_apply_of_not_continuousOn a hg, cfcₙ_apply_of_not_continuousOn]
exact fun hf ↦ hg (hf.congr hfg.symm)
· rw [cfcₙ_apply_of_not_map_zero a h0, cfcₙ_apply_of_not_map_zero]
exact fun hf ↦ h0 (hfg (quasispectrum.zero_mem R a) ▸ hf)
lemma eqOn_of_cfcₙ_eq_cfcₙ {f g : R → R} {a : A} (h : cfcₙ f a = cfcₙ g a) (ha : p a := by cfc_tac)
(hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac) (hf0 : f 0 = 0 := by cfc_zero_tac)
(hg : ContinuousOn g (σₙ R a) := by cfc_cont_tac) (hg0 : g 0 = 0 := by cfc_zero_tac) :
(σₙ R a).EqOn f g := by
rw [cfcₙ_apply f a, cfcₙ_apply g a] at h
have := (cfcₙHom_isClosedEmbedding (show p a from ha) (R := R)).injective h
intro x hx
congrm($(this) ⟨x, hx⟩)
lemma cfcₙ_eq_cfcₙ_iff_eqOn {f g : R → R} {a : A} (ha : p a := by cfc_tac)
(hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac) (hf0 : f 0 = 0 := by cfc_zero_tac)
(hg : ContinuousOn g (σₙ R a) := by cfc_cont_tac) (hg0 : g 0 = 0 := by cfc_zero_tac) :
cfcₙ f a = cfcₙ g a ↔ (σₙ R a).EqOn f g :=
⟨eqOn_of_cfcₙ_eq_cfcₙ, cfcₙ_congr⟩
variable (R)
@[simp]
lemma cfcₙ_zero : cfcₙ (0 : R → R) a = 0 := by
by_cases ha : p a
· exact cfcₙ_apply (0 : R → R) a ▸ map_zero (cfcₙHom ha)
· rw [cfcₙ_apply_of_not_predicate a ha]
@[simp]
lemma cfcₙ_const_zero : cfcₙ (fun _ : R ↦ 0) a = 0 := cfcₙ_zero R a
variable {R}
include hf hf0 hg hg0 in
lemma cfcₙ_mul : cfcₙ (fun x ↦ f x * g x) a = cfcₙ f a * cfcₙ g a := by
by_cases ha : p a
· rw [cfcₙ_apply f a, cfcₙ_apply g a, ← map_mul, cfcₙ_apply _ a]
congr
· simp [cfcₙ_apply_of_not_predicate a ha]
include hf hf0 hg hg0 in
lemma cfcₙ_add : cfcₙ (fun x ↦ f x + g x) a = cfcₙ f a + cfcₙ g a := by
by_cases ha : p a
· rw [cfcₙ_apply f a, cfcₙ_apply g a, cfcₙ_apply _ a]
simp_rw [← map_add]
congr
· simp [cfcₙ_apply_of_not_predicate a ha]
open Finset in
lemma cfcₙ_sum {ι : Type*} (f : ι → R → R) (a : A) (s : Finset ι)
(hf : ∀ i ∈ s, ContinuousOn (f i) (σₙ R a) := by cfc_cont_tac)
(hf0 : ∀ i ∈ s, f i 0 = 0 := by cfc_zero_tac) :
cfcₙ (∑ i ∈ s, f i) a = ∑ i ∈ s, cfcₙ (f i) a := by
by_cases ha : p a
· have hsum : s.sum f = fun z => ∑ i ∈ s, f i z := by ext; simp
have hf' : ContinuousOn (∑ i : s, f i) (σₙ R a) := by
rw [sum_coe_sort s, hsum]
exact continuousOn_finset_sum s fun i hi => hf i hi
rw [← sum_coe_sort s, ← sum_coe_sort s]
rw [cfcₙ_apply_pi _ a _ (fun ⟨i, hi⟩ => hf i hi), ← map_sum, cfcₙ_apply _ a hf']
congr 1
ext
simp
· simp [cfcₙ_apply_of_not_predicate a ha]
open Finset in
lemma cfcₙ_sum_univ {ι : Type*} [Fintype ι] (f : ι → R → R) (a : A)
(hf : ∀ i, ContinuousOn (f i) (σₙ R a) := by cfc_cont_tac)
(hf0 : ∀ i, f i 0 = 0 := by cfc_zero_tac) :
cfcₙ (∑ i, f i) a = ∑ i, cfcₙ (f i) a :=
cfcₙ_sum f a _ (fun i _ ↦ hf i) (fun i _ ↦ hf0 i)
lemma cfcₙ_smul {S : Type*} [SMulZeroClass S R] [ContinuousConstSMul S R]
[SMulZeroClass S A] [IsScalarTower S R A] [IsScalarTower S R (R → R)]
(s : S) (f : R → R) (a : A) (hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac)
(h0 : f 0 = 0 := by cfc_zero_tac) :
cfcₙ (fun x ↦ s • f x) a = s • cfcₙ f a := by
by_cases ha : p a
· rw [cfcₙ_apply f a, cfcₙ_apply _ a]
simp_rw [← Pi.smul_def, ← smul_one_smul R s _]
rw [← map_smul]
congr
· simp [cfcₙ_apply_of_not_predicate a ha]
lemma cfcₙ_const_mul (r : R) (f : R → R) (a : A) (hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac)
(h0 : f 0 = 0 := by cfc_zero_tac) :
cfcₙ (fun x ↦ r * f x) a = r • cfcₙ f a :=
cfcₙ_smul r f a
lemma cfcₙ_star : cfcₙ (fun x ↦ star (f x)) a = star (cfcₙ f a) := by
by_cases h : p a ∧ ContinuousOn f (σₙ R a) ∧ f 0 = 0
· obtain ⟨ha, hf, h0⟩ := h
rw [cfcₙ_apply f a, ← map_star, cfcₙ_apply _ a]
congr
· simp only [not_and_or] at h
obtain (ha | hf | h0) := h
· simp [cfcₙ_apply_of_not_predicate a ha]
· rw [cfcₙ_apply_of_not_continuousOn a hf, cfcₙ_apply_of_not_continuousOn, star_zero]
exact fun hf_star ↦ hf <| by simpa using hf_star.star
· rw [cfcₙ_apply_of_not_map_zero a h0, cfcₙ_apply_of_not_map_zero, star_zero]
exact fun hf0 ↦ h0 <| by simpa using congr(star $(hf0))
lemma cfcₙ_smul_id {S : Type*} [SMulZeroClass S R] [ContinuousConstSMul S R]
[SMulZeroClass S A] [IsScalarTower S R A] [IsScalarTower S R (R → R)]
(s : S) (a : A) (ha : p a := by cfc_tac) : cfcₙ (s • · : R → R) a = s • a := by
rw [cfcₙ_smul s _ a, cfcₙ_id' R a]
lemma cfcₙ_const_mul_id (r : R) (a : A) (ha : p a := by cfc_tac) : cfcₙ (r * ·) a = r • a :=
cfcₙ_smul_id r a
include ha in
lemma cfcₙ_star_id : cfcₙ (star · : R → R) a = star a := by
rw [cfcₙ_star _ a, cfcₙ_id' R a]
section Comp
variable [UniqueHom R A]
lemma cfcₙ_comp (g f : R → R) (a : A)
(hg : ContinuousOn g (f '' σₙ R a) := by cfc_cont_tac) (hg0 : g 0 = 0 := by cfc_zero_tac)
(hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac) (hf0 : f 0 = 0 := by cfc_zero_tac)
(ha : p a := by cfc_tac) :
cfcₙ (g ∘ f) a = cfcₙ g (cfcₙ f a) := by
have := hg.comp hf <| (σₙ R a).mapsTo_image f
have sp_eq :
σₙ R (cfcₙHom (show p a from ha) ⟨ContinuousMap.mk _ hf.restrict, hf0⟩) = f '' (σₙ R a) := by
rw [cfcₙHom_map_quasispectrum (by exact ha) _]
ext
simp
rw [cfcₙ_apply .., cfcₙ_apply f a,
cfcₙ_apply _ _ (by convert hg) (ha := cfcₙHom_predicate (show p a from ha) _),
← cfcₙHom_comp _ _]
swap
· exact ⟨.mk _ <| hf.restrict.codRestrict fun x ↦ by rw [sp_eq]; use x.1; simp, Subtype.ext hf0⟩
· congr
· exact fun _ ↦ rfl
lemma cfcₙ_comp' (g f : R → R) (a : A)
(hg : ContinuousOn g (f '' σₙ R a) := by cfc_cont_tac) (hg0 : g 0 = 0 := by cfc_zero_tac)
(hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac) (hf0 : f 0 = 0 := by cfc_zero_tac)
(ha : p a := by cfc_tac) :
cfcₙ (g <| f ·) a = cfcₙ g (cfcₙ f a) :=
cfcₙ_comp g f a
lemma cfcₙ_comp_smul {S : Type*} [SMulZeroClass S R] [ContinuousConstSMul S R]
[SMulZeroClass S A] [IsScalarTower S R A] [IsScalarTower S R (R → R)]
(s : S) (f : R → R) (a : A) (hf : ContinuousOn f ((s • ·) '' (σₙ R a)) := by cfc_cont_tac)
(hf0 : f 0 = 0 := by cfc_zero_tac) (ha : p a := by cfc_tac) :
cfcₙ (f <| s • ·) a = cfcₙ f (s • a) := by
rw [cfcₙ_comp' f (s • ·) a, cfcₙ_smul_id s a]
lemma cfcₙ_comp_const_mul (r : R) (f : R → R) (a : A)
(hf : ContinuousOn f ((r * ·) '' (σₙ R a)) := by cfc_cont_tac)
(hf0 : f 0 = 0 := by cfc_zero_tac) (ha : p a := by cfc_tac) :
cfcₙ (f <| r * ·) a = cfcₙ f (r • a) := by
rw [cfcₙ_comp' f (r * ·) a, cfcₙ_const_mul_id r a]
lemma cfcₙ_comp_star (hf : ContinuousOn f (star '' (σₙ R a)) := by cfc_cont_tac)
(hf0 : f 0 = 0 := by cfc_zero_tac) (ha : p a := by cfc_tac) :
cfcₙ (f <| star ·) a = cfcₙ f (star a) := by
rw [cfcₙ_comp' f star a, cfcₙ_star_id a]
end Comp
lemma CFC.eq_zero_of_quasispectrum_eq_zero (h_spec : σₙ R a ⊆ {0}) (ha : p a := by cfc_tac) :
a = 0 := by
simpa [cfcₙ_id R a] using cfcₙ_congr (a := a) (f := id) (g := fun _ : R ↦ 0) fun x ↦ by simp_all
include instCFCₙ in
lemma CFC.quasispectrum_zero_eq : σₙ R (0 : A) = {0} := by
refine Set.eq_singleton_iff_unique_mem.mpr ⟨quasispectrum.zero_mem R 0, fun x hx ↦ ?_⟩
rw [← cfcₙ_zero R (0 : A),
cfcₙ_map_quasispectrum _ _ (by cfc_cont_tac) (by cfc_zero_tac) (cfcₙ_predicate_zero R)] at hx
simp_all
@[simp] lemma cfcₙ_apply_zero {f : R → R} : cfcₙ f (0 : A) = 0 := by
by_cases hf0 : f 0 = 0
· nth_rw 2 [← cfcₙ_zero R 0]
apply cfcₙ_congr
simpa [CFC.quasispectrum_zero_eq]
· exact cfcₙ_apply_of_not_map_zero _ hf0
@[simp]
instance IsStarNormal.cfcₙ_map (f : R → R) (a : A) : IsStarNormal (cfcₙ f a) where
star_comm_self := by
refine cfcₙ_cases (fun x ↦ Commute (star x) x) _ _ (Commute.zero_right _) fun _ _ _ ↦ ?_
simp only [Commute, SemiconjBy]
rw [← cfcₙ_apply f a, ← cfcₙ_star, ← cfcₙ_mul .., ← cfcₙ_mul ..]
congr! 2
exact mul_comm _ _
-- The following two lemmas are just `cfcₙ_predicate`, but specific enough for the `@[simp]` tag.
@[simp]
protected lemma IsSelfAdjoint.cfcₙ
[NonUnitalContinuousFunctionalCalculus R A IsSelfAdjoint] {f : R → R} {a : A} :
IsSelfAdjoint (cfcₙ f a) :=
cfcₙ_predicate _ _
@[simp]
lemma cfcₙ_nonneg_of_predicate [LE A]
[NonUnitalContinuousFunctionalCalculus R A (0 ≤ ·)] {f : R → R} {a : A} :
0 ≤ cfcₙ f a :=
cfcₙ_predicate _ _
end CFCn
end Main
section Neg
variable {R A : Type*} {p : A → Prop} [CommRing R] [Nontrivial R] [StarRing R] [MetricSpace R]
variable [IsTopologicalRing R] [ContinuousStar R] [TopologicalSpace A] [NonUnitalRing A]
variable [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]
variable [NonUnitalContinuousFunctionalCalculus R A p]
variable (f g : R → R) (a : A)
variable (hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac) (hf0 : f 0 = 0 := by cfc_zero_tac)
variable (hg : ContinuousOn g (σₙ R a) := by cfc_cont_tac) (hg0 : g 0 = 0 := by cfc_zero_tac)
include hf hf0 hg hg0 in
lemma cfcₙ_sub : cfcₙ (fun x ↦ f x - g x) a = cfcₙ f a - cfcₙ g a := by
by_cases ha : p a
· rw [cfcₙ_apply f a, cfcₙ_apply g a, ← map_sub, cfcₙ_apply ..]
congr
· simp [cfcₙ_apply_of_not_predicate a ha]
lemma cfcₙ_neg : cfcₙ (fun x ↦ - (f x)) a = - (cfcₙ f a) := by
by_cases h : p a ∧ ContinuousOn f (σₙ R a) ∧ f 0 = 0
· obtain ⟨ha, hf, h0⟩ := h
rw [cfcₙ_apply f a, ← map_neg, cfcₙ_apply ..]
congr
· simp only [not_and_or] at h
obtain (ha | hf | h0) := h
· simp [cfcₙ_apply_of_not_predicate a ha]
· rw [cfcₙ_apply_of_not_continuousOn a hf, cfcₙ_apply_of_not_continuousOn, neg_zero]
exact fun hf_neg ↦ hf <| by simpa using hf_neg.neg
· rw [cfcₙ_apply_of_not_map_zero a h0, cfcₙ_apply_of_not_map_zero, neg_zero]
exact (h0 <| neg_eq_zero.mp ·)
lemma cfcₙ_neg_id (ha : p a := by cfc_tac) :
cfcₙ (- · : R → R) a = -a := by
rw [cfcₙ_neg .., cfcₙ_id' R a]
variable [UniqueHom R A]
lemma cfcₙ_comp_neg (hf : ContinuousOn f ((-·) '' (σₙ R a)) := by cfc_cont_tac)
(h0 : f 0 = 0 := by cfc_zero_tac) (ha : p a := by cfc_tac) :
cfcₙ (f <| - ·) a = cfcₙ f (-a) := by
rw [cfcₙ_comp' .., cfcₙ_neg_id _]
end Neg
section Order
section Semiring
variable {R A : Type*} {p : A → Prop} [CommSemiring R] [PartialOrder R] [Nontrivial R]
variable [StarRing R] [MetricSpace R] [IsTopologicalSemiring R] [ContinuousStar R]
variable [ContinuousSqrt R] [StarOrderedRing R] [NoZeroDivisors R]
variable [TopologicalSpace A] [NonUnitalRing A] [StarRing A] [PartialOrder A] [StarOrderedRing A]
variable [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]
variable [NonUnitalContinuousFunctionalCalculus R A p]
lemma cfcₙHom_mono {a : A} (ha : p a) {f g : C(σₙ R a, R)₀} (hfg : f ≤ g) :
cfcₙHom ha f ≤ cfcₙHom ha g :=
OrderHomClass.mono (cfcₙHom ha) hfg
lemma cfcₙHom_nonneg_iff [NonnegSpectrumClass R A] {a : A} (ha : p a) {f : C(σₙ R a, R)₀} :
0 ≤ cfcₙHom ha f ↔ 0 ≤ f := by
constructor
· exact fun hf x ↦
(cfcₙHom_map_quasispectrum ha (R := R) _ ▸ quasispectrum_nonneg_of_nonneg (cfcₙHom ha f) hf)
_ ⟨x, rfl⟩
· simpa using (cfcₙHom_mono ha (f := 0) (g := f) ·)
lemma cfcₙ_mono {f g : R → R} {a : A} (h : ∀ x ∈ σₙ R a, f x ≤ g x)
(hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac)
(hg : ContinuousOn g (σₙ R a) := by cfc_cont_tac)
(hf0 : f 0 = 0 := by cfc_zero_tac) (hg0 : g 0 = 0 := by cfc_zero_tac) :
cfcₙ f a ≤ cfcₙ g a := by
by_cases ha : p a
· rw [cfcₙ_apply f a, cfcₙ_apply g a]
exact cfcₙHom_mono ha fun x ↦ h x.1 x.2
· simp only [cfcₙ_apply_of_not_predicate _ ha, le_rfl]
lemma cfcₙ_nonneg_iff [NonnegSpectrumClass R A] (f : R → R) (a : A)
(hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac)
(h0 : f 0 = 0 := by cfc_zero_tac) (ha : p a := by cfc_tac) :
0 ≤ cfcₙ f a ↔ ∀ x ∈ σₙ R a, 0 ≤ f x := by
rw [cfcₙ_apply .., cfcₙHom_nonneg_iff, ContinuousMapZero.le_def]
simp only [ContinuousMapZero.coe_mk, ContinuousMap.coe_mk, Set.restrict_apply, Subtype.forall]
congr!
lemma StarOrderedRing.nonneg_iff_quasispectrum_nonneg [NonnegSpectrumClass R A] (a : A)
(ha : p a := by cfc_tac) : 0 ≤ a ↔ ∀ x ∈ quasispectrum R a, 0 ≤ x := by
have := cfcₙ_nonneg_iff (id : R → R) a (by fun_prop)
simpa [cfcₙ_id _ a ha] using this
lemma cfcₙ_nonneg {f : R → R} {a : A} (h : ∀ x ∈ σₙ R a, 0 ≤ f x) :
0 ≤ cfcₙ f a := by
by_cases hf : ContinuousOn f (σₙ R a) ∧ f 0 = 0
· obtain ⟨h₁, h₂⟩ := hf
simpa using cfcₙ_mono h
· simp only [not_and_or] at hf
obtain (hf | hf) := hf
· simp only [cfcₙ_apply_of_not_continuousOn _ hf, le_rfl]
· simp only [cfcₙ_apply_of_not_map_zero _ hf, le_rfl]
lemma cfcₙ_nonpos (f : R → R) (a : A) (h : ∀ x ∈ σₙ R a, f x ≤ 0) :
cfcₙ f a ≤ 0 := by
by_cases hf : ContinuousOn f (σₙ R a) ∧ f 0 = 0
· obtain ⟨h₁, h₂⟩ := hf
simpa using cfcₙ_mono h
· simp only [not_and_or] at hf
obtain (hf | hf) := hf
· simp only [cfcₙ_apply_of_not_continuousOn _ hf, le_rfl]
· simp only [cfcₙ_apply_of_not_map_zero _ hf, le_rfl]
end Semiring
section Ring
variable {R A : Type*} {p : A → Prop} [CommRing R] [PartialOrder R] [Nontrivial R]
variable [StarRing R] [MetricSpace R] [IsTopologicalRing R] [ContinuousStar R]
variable [ContinuousSqrt R] [StarOrderedRing R] [NoZeroDivisors R]
variable [TopologicalSpace A] [NonUnitalRing A] [StarRing A] [PartialOrder A] [StarOrderedRing A]
variable [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]
variable [NonUnitalContinuousFunctionalCalculus R A p] [NonnegSpectrumClass R A]
lemma cfcₙHom_le_iff {a : A} (ha : p a) {f g : C(σₙ R a, R)₀} :
cfcₙHom ha f ≤ cfcₙHom ha g ↔ f ≤ g := by
rw [← sub_nonneg, ← map_sub, cfcₙHom_nonneg_iff, sub_nonneg]
lemma cfcₙ_le_iff (f g : R → R) (a : A) (hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac)
(hg : ContinuousOn g (σₙ R a) := by cfc_cont_tac) (hf0 : f 0 = 0 := by cfc_zero_tac)
(hg0 : g 0 = 0 := by cfc_zero_tac) (ha : p a := by cfc_tac) :
cfcₙ f a ≤ cfcₙ g a ↔ ∀ x ∈ σₙ R a, f x ≤ g x := by
rw [cfcₙ_apply f a, cfcₙ_apply g a, cfcₙHom_le_iff (show p a from ha), ContinuousMapZero.le_def]
simp
lemma cfcₙ_nonpos_iff (f : R → R) (a : A) (hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac)
(h0 : f 0 = 0 := by cfc_zero_tac) (ha : p a := by cfc_tac) :
cfcₙ f a ≤ 0 ↔ ∀ x ∈ σₙ R a, f x ≤ 0 := by
simp_rw [← neg_nonneg, ← cfcₙ_neg]
exact cfcₙ_nonneg_iff (fun x ↦ -f x) a
end Ring
end Order
/-! ### `cfcₙHom` on a superset of the quasispectrum -/
section Superset
open ContinuousMapZero
variable {R A : Type*} {p : A → Prop} [CommSemiring R] [Nontrivial R] [StarRing R]
[MetricSpace R] [IsTopologicalSemiring R] [ContinuousStar R] [NonUnitalRing A] [StarRing A]
[TopologicalSpace A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]
[instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p]
/-- The composition of `cfcₙHom` with the natural embedding `C(s, R)₀ → C(quasispectrum R a, R)₀`
whenever `quasispectrum R a ⊆ s`.
This is sometimes necessary in order to consider the same continuous functions applied to multiple
distinct elements, with the added constraint that `cfcₙ` does not suffice. This can occur, for
example, if it is necessary to use uniqueness of this continuous functional calculus. A practical
example can be found in the proof of `CFC.posPart_negPart_unique`. -/
@[simps!]
noncomputable def cfcₙHomSuperset {a : A} (ha : p a) {s : Set R} (hs : σₙ R a ⊆ s) :
letI : Zero s := ⟨0, hs (quasispectrum.zero_mem R a)⟩
C(s, R)₀ →⋆ₙₐ[R] A :=
letI : Zero s := ⟨0, hs (quasispectrum.zero_mem R a)⟩
cfcₙHom ha (R := R) |>.comp <| ContinuousMapZero.nonUnitalStarAlgHom_precomp R <|
⟨⟨_, continuous_id.subtype_map hs⟩, rfl⟩
lemma cfcₙHomSuperset_continuous {a : A} (ha : p a) {s : Set R} (hs : σₙ R a ⊆ s) :
Continuous (cfcₙHomSuperset ha hs) :=
letI : Zero s := ⟨0, hs (quasispectrum.zero_mem R a)⟩
(cfcₙHom_continuous ha).comp <| ContinuousMapZero.continuous_comp_left _
lemma cfcₙHomSuperset_id {a : A} (ha : p a) {s : Set R} (hs : σₙ R a ⊆ s) :
letI : Zero s := ⟨0, hs (quasispectrum.zero_mem R a)⟩
cfcₙHomSuperset ha hs ⟨.restrict s <| .id R, rfl⟩ = a :=
cfcₙHom_id ha
/-- this version uses `ContinuousMapZero.id`. -/
lemma cfcₙHomSuperset_id' {a : A} (ha : p a) {s : Set R} (hs : σₙ R a ⊆ s) :
letI : Zero s := ⟨0, hs (quasispectrum.zero_mem R a)⟩
cfcₙHomSuperset ha hs (.id rfl) = a :=
cfcₙHom_id ha
end Superset
/-! ### Obtain a non-unital continuous functional calculus from a unital one -/
section UnitalToNonUnital
open ContinuousMapZero Set Uniformity ContinuousMap
variable {R A : Type*} {p : A → Prop} [Semifield R] [StarRing R] [MetricSpace R]
variable [IsTopologicalSemiring R] [ContinuousStar R] [Ring A] [StarRing A] [TopologicalSpace A]
variable [Algebra R A] [ContinuousFunctionalCalculus R A p]
variable (R) in
/-- The non-unital continuous functional calculus obtained by restricting a unital calculus
to functions that map zero to zero. This is an auxiliary definition and is not
intended for use outside this file. The equality between the non-unital and unital
calculi in this case is encoded in the lemma `cfcₙ_eq_cfc`. -/
noncomputable def cfcₙHom_of_cfcHom {a : A} (ha : p a) : C(σₙ R a, R)₀ →⋆ₙₐ[R] A :=
let e := ContinuousMapZero.toContinuousMapHom (X := σₙ R a) (R := R)
let f : C(spectrum R a, quasispectrum R a) :=
⟨_, continuous_inclusion <| spectrum_subset_quasispectrum R a⟩
let ψ := ContinuousMap.compStarAlgHom' R R f
(cfcHom ha (R := R) : C(spectrum R a, R) →⋆ₙₐ[R] A).comp <|
(ψ : C(σₙ R a, R) →⋆ₙₐ[R] C(spectrum R a, R)).comp e
lemma cfcₙHom_of_cfcHom_map_quasispectrum {a : A} (ha : p a) :
∀ f : C(σₙ R a, R)₀, σₙ R (cfcₙHom_of_cfcHom R ha f) = range f := by
intro f
simp only [cfcₙHom_of_cfcHom]
rw [quasispectrum_eq_spectrum_union_zero]
simp only [NonUnitalStarAlgHom.comp_apply, NonUnitalStarAlgHom.coe_coe]
rw [cfcHom_map_spectrum ha]
ext x
constructor
· rintro (⟨x, rfl⟩ | rfl)
· exact ⟨⟨x.1, spectrum_subset_quasispectrum R a x.2⟩, rfl⟩
· exact ⟨0, map_zero f⟩
· rintro ⟨x, rfl⟩
have hx := x.2
simp_rw [quasispectrum_eq_spectrum_union_zero R a] at hx
obtain (hx | hx) := hx
· exact Or.inl ⟨⟨x.1, hx⟩, rfl⟩
· apply Or.inr
simp only [Set.mem_singleton_iff] at hx ⊢
rw [show x = 0 from Subtype.val_injective hx, map_zero]
variable [CompleteSpace R]
-- gives access to the `ContinuousFunctionalCalculus.compactSpace_spectrum` instance
open scoped ContinuousFunctionalCalculus
lemma isClosedEmbedding_cfcₙHom_of_cfcHom {a : A} (ha : p a) :
IsClosedEmbedding (cfcₙHom_of_cfcHom R ha) := by
let f : C(spectrum R a, σₙ R a) :=
⟨_, continuous_inclusion <| spectrum_subset_quasispectrum R a⟩
refine (cfcHom_isClosedEmbedding ha).comp <|
(IsUniformInducing.isUniformEmbedding ⟨?_⟩).isClosedEmbedding
have := uniformSpace_eq_inf_precomp_of_cover (β := R) f (0 : C(Unit, σₙ R a))
(map_continuous f).isProperMap (map_continuous 0).isProperMap <| by
simp only [← Subtype.val_injective.image_injective.eq_iff, f, ContinuousMap.coe_mk,
ContinuousMap.coe_zero, range_zero, image_union, image_singleton,
quasispectrum.coe_zero, ← range_comp, val_comp_inclusion, image_univ, Subtype.range_coe,
quasispectrum_eq_spectrum_union_zero]
simp_rw [ContinuousMapZero.instUniformSpace, this, uniformity_comap,
@inf_uniformity _ (.comap _ _) (.comap _ _), uniformity_comap, Filter.comap_inf,
Filter.comap_comap]
refine .symm <| inf_eq_left.mpr <| le_top.trans <| eq_top_iff.mp ?_
have : ∀ U ∈ 𝓤 (C(Unit, R)), (0, 0) ∈ U := fun U hU ↦ refl_mem_uniformity hU
convert Filter.comap_const_of_mem this with ⟨u, v⟩ <;>
ext ⟨x, rfl⟩ <;> [exact map_zero u; exact map_zero v]
instance ContinuousFunctionalCalculus.toNonUnital :
NonUnitalContinuousFunctionalCalculus R A p where
predicate_zero := cfc_predicate_zero R
compactSpace_quasispectrum a := by
have h_cpct : CompactSpace (spectrum R a) := inferInstance
simp only [← isCompact_iff_compactSpace, quasispectrum_eq_spectrum_union_zero] at h_cpct ⊢
exact h_cpct |>.union isCompact_singleton
exists_cfc_of_predicate _ ha :=
⟨cfcₙHom_of_cfcHom R ha,
isClosedEmbedding_cfcₙHom_of_cfcHom ha,
cfcHom_id ha,
cfcₙHom_of_cfcHom_map_quasispectrum ha,
fun _ ↦ cfcHom_predicate ha _⟩
open scoped NonUnitalContinuousFunctionalCalculus in
lemma cfcₙHom_eq_cfcₙHom_of_cfcHom [ContinuousMapZero.UniqueHom R A] {a : A}
(ha : p a) : cfcₙHom (R := R) ha = cfcₙHom_of_cfcHom R ha := by
refine ContinuousMapZero.UniqueHom.eq_of_continuous_of_map_id
(σₙ R a) ?_ _ _ ?_ ?_ ?_
· simp
· exact (cfcₙHom_isClosedEmbedding (R := R) ha).continuous
· exact (isClosedEmbedding_cfcₙHom_of_cfcHom ha).continuous
· simpa only [cfcₙHom_id (R := R) ha] using (cfcHom_id ha).symm
/-- When `cfc` is applied to a function that maps zero to zero, it is equivalent to using
`cfcₙ`. -/
lemma cfcₙ_eq_cfc [ContinuousMapZero.UniqueHom R A] {f : R → R} {a : A}
(hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac) (hf0 : f 0 = 0 := by cfc_zero_tac) :
cfcₙ f a = cfc f a := by
by_cases ha : p a
· have hf' := hf.mono <| spectrum_subset_quasispectrum R a
rw [cfc_apply f a ha hf', cfcₙ_apply f a hf, cfcₙHom_eq_cfcₙHom_of_cfcHom, cfcₙHom_of_cfcHom]
dsimp only [NonUnitalStarAlgHom.comp_apply, toContinuousMapHom_apply,
NonUnitalStarAlgHom.coe_coe, compStarAlgHom'_apply]
congr
· simp [cfc_apply_of_not_predicate a ha, cfcₙ_apply_of_not_predicate (R := R) a ha]
end UnitalToNonUnital
|
binomial.v
|
(* (c) Copyright 2006-2016 Microsoft Corporation and Inria. *)
(* Distributed under the terms of CeCILL-B. *)
From HB Require Import structures.
From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.
From mathcomp Require Import div fintype tuple finfun bigop prime finset.
(******************************************************************************)
(* This files contains the definition of: *)
(* 'C(n, m) == the binomial coefficient n choose m. *)
(* n ^_ m == the falling (or lower) factorial of n with m terms, i.e., *)
(* the product n * (n - 1) * ... * (n - m + 1). *)
(* Note that n ^_ m = 0 if m > n, and 'C(n, m) = n ^_ m %/ m`!. *)
(* *)
(* In additions to the properties of these functions, we prove a few seminal *)
(* results such as bin2_sum, Wilson and expnDn; their proofs are good *)
(* examples of how to manipulate expressions with bigops. *)
(******************************************************************************)
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
(** More properties of the factorial **)
Lemma fact_prod n : n`! = \prod_(1 <= i < n.+1) i.
Proof.
elim: n => [|n IHn] //; first by rewrite big_nil.
by apply/esym; rewrite factS IHn // !big_add1 big_nat_recr //= mulnC.
Qed.
Lemma fact_split n m : m <= n -> n`! = m`! * \prod_(m.+1 <= k < n.+1) k.
Proof. by move=> leq_mn; rewrite !fact_prod -big_cat_nat. Qed.
Lemma logn_fact p n : prime p -> logn p n`! = \sum_(1 <= k < n.+1) n %/ p ^ k.
Proof.
move=> p_prime; transitivity (\sum_(1 <= i < n.+1) logn p i).
rewrite big_add1; elim: n => /= [|n IHn]; first by rewrite logn1 big_geq.
by rewrite big_nat_recr // -IHn /= factS mulnC lognM ?fact_gt0.
transitivity (\sum_(1 <= i < n.+1) \sum_(1 <= k < n.+1) (p ^ k %| i)).
apply: eq_big_nat => i /andP[i_gt0 le_i_n]; rewrite logn_count_dvd //.
rewrite -!big_mkcond (big_nat_widen _ _ n.+1) 1?ltnW //; apply: eq_bigl => k.
by apply: andb_idr => /dvdn_leq/(leq_trans (ltn_expl _ (prime_gt1 _)))->.
by rewrite exchange_big_nat; apply: eq_bigr => i _; rewrite divn_count_dvd.
Qed.
Theorem Wilson p : p > 1 -> prime p = (p %| ((p.-1)`!).+1).
Proof.
have dFact n: 0 < n -> (n.-1)`! = \prod_(0 <= i < n | i != 0) i.
move=> n_gt0; rewrite -big_filter fact_prod; symmetry; apply: congr_big => //.
rewrite /index_iota subn1 -[n]prednK //=; apply/all_filterP.
by rewrite all_predC has_pred1 mem_iota.
move=> lt1p; have p_gt0 := ltnW lt1p.
apply/idP/idP=> [pr_p | dv_pF]; last first.
apply/primeP; split=> // d dv_dp; have: d <= p by apply: dvdn_leq.
rewrite orbC leq_eqVlt => /orP[-> // | ltdp].
have:= dvdn_trans dv_dp dv_pF; rewrite dFact // big_mkord.
rewrite (bigD1 (Ordinal ltdp)) /=; last by rewrite -lt0n (dvdn_gt0 p_gt0).
by rewrite orbC -addn1 dvdn_addr ?dvdn_mulr // dvdn1 => ->.
pose Fp1 := Ordinal lt1p; pose Fp0 := Ordinal p_gt0.
have ltp1p: p.-1 < p by [rewrite prednK]; pose Fpn1 := Ordinal ltp1p.
case eqF1n1: (Fp1 == Fpn1); first by rewrite -{1}[p]prednK -1?((1 =P p.-1) _).
have toFpP m: m %% p < p by rewrite ltn_mod.
pose toFp := Ordinal (toFpP _); pose mFp (i j : 'I_p) := toFp (i * j).
have Fp_mod (i : 'I_p) : i %% p = i by apply: modn_small.
have mFpA: associative mFp.
by move=> i j k; apply: val_inj; rewrite /= modnMml modnMmr mulnA.
have mFpC: commutative mFp by move=> i j; apply: val_inj; rewrite /= mulnC.
have mFp1: left_id Fp1 mFp by move=> i; apply: val_inj; rewrite /= mul1n.
have mFp1r: right_id Fp1 mFp by move=> i; apply: val_inj; rewrite /= muln1.
pose mFpcM := Monoid.isComLaw.Build 'I_p Fp1 mFp mFpA mFpC mFp1.
pose mFpCL : Monoid.com_law _ := HB.pack mFp mFpcM.
pose mFpM := Monoid.Law.sort mFpCL.
pose vFp (i : 'I_p) := toFp (egcdn i p).1.
have vFpV i: i != Fp0 -> mFp (vFp i) i = Fp1.
rewrite -val_eqE /= -lt0n => i_gt0; apply: val_inj => /=.
rewrite modnMml; case: egcdnP => //= _ km -> _; rewrite {km}modnMDl.
suffices: coprime i p by move/eqnP->; rewrite modn_small.
rewrite coprime_sym prime_coprime //; apply/negP=> /(dvdn_leq i_gt0).
by rewrite leqNgt ltn_ord.
have vFp0 i: i != Fp0 -> vFp i != Fp0.
by move/vFpV; apply/contra_eq_neq => ->; rewrite -val_eqE /= mul0n mod0n.
have vFpK: {in predC1 Fp0, involutive vFp}.
move=> i n0i; rewrite /= -[vFp _]mFp1r -(vFpV _ n0i) mFpA.
by rewrite vFpV (vFp0, mFp1).
have le_pmFp (i : 'I_p) m: i <= p + m.
by apply: leq_trans (ltnW _) (leq_addr _ _).
have eqFp (i j : 'I_p): (i == j) = (p %| p + i - j).
by rewrite -eqn_mod_dvd ?(modnDl, Fp_mod).
have vFpId i: (vFp i == i :> nat) = xpred2 Fp1 Fpn1 i.
have [->{i} | ni0] := eqVneq i Fp0.
by rewrite -!val_eqE /= egcd0n modn_small //= -(subnKC lt1p).
rewrite 2![i == _]eqFp -Euclid_dvdM // -[_ - p.-1]subSS prednK //.
have lt0i: 0 < i by rewrite lt0n.
rewrite -addnS addKn -addnBA // mulnDl -{2}(addn1 i) -subn_sqr.
rewrite addnBA ?leq_sqr // mulnS -addnA -mulnn -mulnDl.
rewrite -(subnK (le_pmFp (vFp i) i)) mulnDl addnCA.
rewrite -[1 ^ 2]/(Fp1 : nat) -addnBA // dvdn_addl.
by rewrite Euclid_dvdM // -eqFp eq_sym orbC /dvdn Fp_mod eqn0Ngt lt0i.
by rewrite -eqn_mod_dvd // Fp_mod modnDl -(vFpV _ ni0).
suffices [mod_fact]: toFp (p.-1)`! = Fpn1.
by rewrite /dvdn -addn1 -modnDml mod_fact addn1 prednK // modnn.
rewrite dFact //; rewrite ((big_morph toFp) Fp1 mFpM) //; first last.
- by apply: val_inj; rewrite /= modn_small.
- by move=> i j; apply: val_inj; rewrite /= modnMm.
rewrite big_mkord (eq_bigr id) => [|i _]; last by apply: val_inj => /=.
pose ltv i := vFp i < i; rewrite (bigID ltv) -/mFpM [mFpM _ _]mFpC.
rewrite (bigD1 Fp1) -/mFpM; last by rewrite [ltv _]ltn_neqAle vFpId.
rewrite [mFpM _ _]mFp1 (bigD1 Fpn1) -?mFpA -/mFpM; last first.
rewrite -lt0n -ltnS prednK // lt1p.
by rewrite [ltv _]ltn_neqAle vFpId eqxx orbT eq_sym eqF1n1.
rewrite (reindex_onto vFp vFp) -/mFpM => [|i]; last by do 3!case/andP; auto.
rewrite (eq_bigl (xpredD1 ltv Fp0)) => [|i]; last first.
rewrite andbC -!andbA -2!negb_or -vFpId orbC -leq_eqVlt -ltnNge.
have [->|ni0] := eqVneq i; last by rewrite vFpK // eqxx vFp0.
by case: eqP => // ->; rewrite !andbF.
rewrite -{2}[mFp]/mFpM -[mFpM _ _]big_split -/mFpM.
by rewrite big1 ?mFp1r //= => i /andP [/vFpV].
Qed.
(** The falling factorial *)
Fixpoint ffact_rec n m := if m is m'.+1 then n * ffact_rec n.-1 m' else 1.
Definition falling_factorial := ffact_rec.
Arguments falling_factorial : simpl never.
Notation "n ^_ m" := (falling_factorial n m)
(at level 30, right associativity) : nat_scope.
Lemma ffactE : falling_factorial = ffact_rec. Proof. by []. Qed.
Lemma ffactn0 n : n ^_ 0 = 1. Proof. by []. Qed.
Lemma ffact0n m : 0 ^_ m = (m == 0). Proof. by case: m. Qed.
Lemma ffactnS n m : n ^_ m.+1 = n * n.-1 ^_ m. Proof. by []. Qed.
Lemma ffactSS n m : n.+1 ^_ m.+1 = n.+1 * n ^_ m. Proof. by []. Qed.
Lemma ffactn1 n : n ^_ 1 = n. Proof. exact: muln1. Qed.
Lemma ffactnSr n m : n ^_ m.+1 = n ^_ m * (n - m).
Proof.
elim: n m => [|n IHn] [|m] //=; first by rewrite ffactn1 mul1n.
by rewrite !ffactSS IHn mulnA.
Qed.
Lemma ffact_prod n m : n ^_ m = \prod_(i < m) (n - i).
Proof.
elim: m n => [n | m IH [|n] //]; first by rewrite ffactn0 big_ord0.
by rewrite big_ord_recr /= sub0n muln0.
by rewrite ffactSS IH big_ord_recl subn0.
Qed.
Lemma ffact_gt0 n m : (0 < n ^_ m) = (m <= n).
Proof. by elim: n m => [|n IHn] [|m] //=; rewrite ffactSS muln_gt0 IHn. Qed.
Lemma ffact_small n m : n < m -> n ^_ m = 0.
Proof. by rewrite ltnNge -ffact_gt0; case: posnP. Qed.
Lemma ffactnn n : n ^_ n = n`!.
Proof. by elim: n => [|n IHn] //; rewrite ffactnS IHn. Qed.
Lemma ffact_fact n m : m <= n -> n ^_ m * (n - m)`! = n`!.
Proof.
by elim: n m => [|n IHn] [|m] //= le_m_n; rewrite ?mul1n // -mulnA IHn.
Qed.
Lemma ffact_factd n m : m <= n -> n ^_ m = n`! %/ (n - m)`!.
Proof. by move/ffact_fact <-; rewrite mulnK ?fact_gt0. Qed.
(** Binomial coefficients *)
Fixpoint binomial n m :=
match n, m with
| n'.+1, m'.+1 => binomial n' m + binomial n' m'
| _, 0 => 1
| 0, _.+1 => 0
end.
Arguments binomial : simpl never.
Notation "''C' ( n , m )" := (binomial n m) : nat_scope.
Lemma binE n m : binomial n m =
match n, m with
| n'.+1, m'.+1 => binomial n' m + binomial n' m'
| _, 0 => 1
| 0, _.+1 => 0
end.
Proof. by case: n. Qed.
Lemma bin0 n : 'C(n, 0) = 1. Proof. by case: n. Qed.
Lemma bin0n m : 'C(0, m) = (m == 0). Proof. by case: m. Qed.
Lemma binS n m : 'C(n.+1, m.+1) = 'C(n, m.+1) + 'C(n, m). Proof. by []. Qed.
Lemma bin1 n : 'C(n, 1) = n.
Proof. by elim: n => //= n IHn; rewrite binS bin0 IHn addn1. Qed.
Lemma bin_gt0 n m : (0 < 'C(n, m)) = (m <= n).
Proof.
by elim: n m => [|n IHn] [|m] //; rewrite addn_gt0 !IHn orbC ltn_neqAle andKb.
Qed.
Lemma leq_bin2l n1 n2 m : n1 <= n2 -> 'C(n1, m) <= 'C(n2, m).
Proof.
by elim: n1 n2 m => [|n1 IHn] [|n2] [|n] le_n12 //; rewrite leq_add ?IHn.
Qed.
Lemma bin_small n m : n < m -> 'C(n, m) = 0.
Proof. by rewrite ltnNge -bin_gt0; case: posnP. Qed.
Lemma binn n : 'C(n, n) = 1.
Proof. by elim: n => [|n IHn] //; rewrite binS bin_small. Qed.
(* Multiply to move diagonally down and right in the Pascal triangle. *)
Lemma mul_bin_diag n m : n * 'C(n.-1, m) = m.+1 * 'C(n, m.+1).
Proof.
rewrite [RHS]mulnC; elim: n m => [|[|n] IHn] [|m] //=; first by rewrite bin1.
by rewrite mulSn [in _ * _]binS mulnDr addnCA !IHn -mulnS -mulnDl -binS.
Qed.
Lemma bin_fact n m : m <= n -> 'C(n, m) * (m`! * (n - m)`!) = n`!.
Proof.
elim: n m => [|n IHn] [|m] // le_m_n; first by rewrite bin0 !mul1n.
by rewrite !factS -!mulnA mulnCA mulnA -mul_bin_diag -mulnA IHn.
Qed.
(* In fact the only exception for bin_factd is n = 0 and m = 1 *)
Lemma bin_factd n m : 0 < n -> 'C(n, m) = n`! %/ (m`! * (n - m)`!).
Proof.
have [/bin_fact<-|*] := leqP m n; first by rewrite mulnK ?muln_gt0 ?fact_gt0.
by rewrite divnMA bin_small ?divn_small ?fact_gt0 ?ltn_fact.
Qed.
Lemma bin_ffact n m : 'C(n, m) * m`! = n ^_ m.
Proof.
have [lt_n_m | le_m_n] := ltnP n m; first by rewrite bin_small ?ffact_small.
by rewrite ffact_factd // -(bin_fact le_m_n) mulnA mulnK ?fact_gt0.
Qed.
Lemma bin_ffactd n m : 'C(n, m) = n ^_ m %/ m`!.
Proof. by rewrite -bin_ffact mulnK ?fact_gt0. Qed.
Lemma bin_sub n m : m <= n -> 'C(n, n - m) = 'C(n, m).
Proof.
by move=> le_m_n; rewrite !bin_ffactd !ffact_factd ?leq_subr // divnAC subKn.
Qed.
(* Multiply to move down in the Pascal triangle. *)
Lemma mul_bin_down n m : n * 'C(n.-1, m) = (n - m) * 'C(n, m).
Proof.
case: n => //= n; have [lt_n_m | le_m_n] := ltnP n m.
by rewrite (eqnP lt_n_m) mulnC bin_small.
by rewrite -!['C(_, m)]bin_sub ?leqW ?subSn ?mul_bin_diag.
Qed.
(* Multiply to move left in the Pascal triangle. *)
Lemma mul_bin_left n m : m.+1 * 'C(n, m.+1) = (n - m) * 'C(n, m).
Proof. by rewrite -mul_bin_diag mul_bin_down. Qed.
Lemma binSn n : 'C(n.+1, n) = n.+1.
Proof. by rewrite -bin_sub ?leqnSn // subSnn bin1. Qed.
Lemma bin2 n : 'C(n, 2) = (n * n.-1)./2.
Proof. by rewrite -[n.-1]bin1 mul_bin_diag -divn2 mulKn. Qed.
Lemma bin2odd n : odd n -> 'C(n, 2) = n * n.-1./2.
Proof. by case: n => // n oddn; rewrite bin2 -!divn2 muln_divA ?dvdn2. Qed.
Lemma prime_dvd_bin k p : prime p -> 0 < k < p -> p %| 'C(p, k).
Proof.
move=> p_pr /andP[k_gt0 lt_k_p].
suffices /Gauss_dvdr<-: coprime p (p - k) by rewrite -mul_bin_down dvdn_mulr.
by rewrite prime_coprime // dvdn_subr 1?ltnW // gtnNdvd.
Qed.
Lemma bin2_sum n : \sum_(0 <= i < n) i = 'C(n, 2).
Proof.
elim: n => [|n IHn]; first by rewrite big_geq.
by rewrite big_nat_recr // IHn binS bin1.
Qed.
#[deprecated(since="mathcomp 2.3.0", note="Use bin2_sum instead.")]
Notation triangular_sum := bin2_sum (only parsing).
(* textbook proof of `bin2_sum`. Should be moved out of the main
library, to a dedicated "showcase" library.
Lemma textbook_triangular_sum n : \sum_(0 <= i < n) i = 'C(n, 2).
Proof.
rewrite bin2; apply: canRL half_double _.
rewrite -addnn {1}big_nat_rev -big_split big_mkord /= ?add0n.
rewrite (eq_bigr (fun _ => n.-1)); first by rewrite sum_nat_const card_ord.
by case: n => [|n] [i le_i_n] //=; rewrite subSS subnK.
Qed. *)
Theorem expnDn a b n :
(a + b) ^ n = \sum_(i < n.+1) 'C(n, i) * (a ^ (n - i) * b ^ i).
Proof.
elim: n => [|n IHn]; rewrite big_ord_recl muln1 ?big_ord0 //.
rewrite expnS {}IHn /= mulnDl !big_distrr /= big_ord_recl muln1 subn0.
rewrite !big_ord_recr /= !binn !subnn bin0 !subn0 !mul1n -!expnS -addnA.
congr (_ + _); rewrite addnA -big_split /=; congr (_ + _).
apply: eq_bigr => i _; rewrite mulnCA (mulnA a) -expnS subnSK //=.
by rewrite (mulnC b) -2!mulnA -expnSr -mulnDl.
Qed.
#[deprecated(since="mathcomp 2.3.0", note="Use expnDn instead.")]
Definition Pascal := expnDn.
Lemma Vandermonde k l i :
\sum_(j < i.+1) 'C(k, j) * 'C(l, i - j) = 'C(k + l , i).
Proof.
pose f k i := \sum_(j < i.+1) 'C(k, j) * 'C(l, i - j).
suffices{k i} fxx k i: f k.+1 i.+1 = f k i.+1 + f k i.
elim: k i => [i | k IHk [|i]]; last by rewrite -/(f _ _) fxx /f !IHk -binS.
by rewrite big_ord_recl big1_eq addn0 mul1n subn0.
by rewrite big_ord_recl big_ord0 addn0 !bin0 muln1.
rewrite {}/f big_ord_recl (big_ord_recl (i.+1)) !bin0 !mul1n.
rewrite -addnA -big_split /=; congr (_ + _).
by apply: eq_bigr => j _; rewrite -mulnDl.
Qed.
Lemma subn_exp m n k :
m ^ k - n ^ k = (m - n) * (\sum_(i < k) m ^ (k.-1 -i) * n ^ i).
Proof.
case: k => [|k]; first by rewrite big_ord0 muln0.
rewrite mulnBl !big_distrr big_ord_recl big_ord_recr /= subn0 muln1.
rewrite subnn mul1n -!expnS subnDA; congr (_ - _); apply: canRL (addnK _) _.
congr (_ + _); apply: eq_bigr => i _.
by rewrite (mulnCA n) -expnS mulnA -expnS subnSK /=.
Qed.
Lemma predn_exp m k : (m ^ k).-1 = m.-1 * (\sum_(i < k) m ^ i).
Proof.
rewrite -!subn1 -[in LHS](exp1n k) subn_exp; congr (_ * _).
symmetry; rewrite (reindex_inj rev_ord_inj); apply: eq_bigr => i _ /=.
by rewrite -subn1 -subnDA exp1n muln1.
Qed.
Lemma dvdn_pred_predX n e : (n.-1 %| (n ^ e).-1)%N.
Proof. by rewrite predn_exp dvdn_mulr. Qed.
Lemma modn_summ I r (P : pred I) F d :
\sum_(i <- r | P i) F i %% d = \sum_(i <- r | P i) F i %[mod d].
Proof.
by apply/eqP; elim/big_rec2: _ => // i m n _; rewrite modnDml eqn_modDl.
Qed.
Lemma prime_modn_expSn p n : prime p -> n.+1 ^ p = (n ^ p).+1 %[mod p].
Proof.
case: p => // p pP.
rewrite -[(_ ^ _).+1]addn0 (expnDn 1) big_ord_recr big_ord_recl /=.
rewrite subnn binn exp1n !mul1n addnAC -modnDmr; congr ((_ + _) %% _).
apply/eqP/dvdn_sum => -[i ?] _; exact/dvdn_mulr/prime_dvd_bin.
Qed.
Lemma fermat_little a p : prime p -> a ^ p = a %[mod p].
Proof.
move=> pP.
elim: a => [|a IH]; first by rewrite exp0n // prime_gt0.
by rewrite prime_modn_expSn // -addn1 -modnDml IH modnDml addn1.
Qed.
(* Combinatorial characterizations. *)
Section Combinations.
Implicit Types T D : finType.
Lemma card_uniq_tuples T n (A : pred T) :
#|[set t : n.-tuple T | all A t & uniq t]| = #|A| ^_ n.
Proof.
elim: n A => [|n IHn] A.
by rewrite (@eq_card1 _ [tuple]) // => t; rewrite [t]tuple0 inE.
rewrite -sum1dep_card (partition_big (@thead _ _) A) /= => [|t]; last first.
by case/tupleP: t => x t; do 2!case/andP.
rewrite ffactnS -sum_nat_const; apply: eq_bigr => x Ax.
rewrite (cardD1 x) [x \in A]Ax /= -(IHn [predD1 A & x]) -sum1dep_card.
rewrite (reindex (fun t : n.-tuple T => [tuple of x :: t])) /=; last first.
pose ttail (t : n.+1.-tuple T) := [tuple of behead t].
exists ttail => [t _ | t /andP[_ /eqP <-]]; first exact: val_inj.
by rewrite -tuple_eta.
apply: eq_bigl=> t; rewrite Ax theadE eqxx andbT /= andbA; congr (_ && _).
by rewrite all_predI all_predC has_pred1 andbC.
Qed.
Lemma card_inj_ffuns_on D T (R : pred T) :
#|[set f : {ffun D -> T} in ffun_on R | injectiveb f]| = #|R| ^_ #|D|.
Proof.
rewrite -card_uniq_tuples.
have bijFF: {on (_ : pred _), bijective (@Finfun D T)}.
by exists fgraph => x _; [apply: FinfunK | apply: fgraphK].
rewrite -(on_card_preimset (bijFF _)); apply: eq_card => /= t.
rewrite !inE -(big_andE predT) -big_image /= big_all.
by rewrite -[t in RHS]FinfunK -codom_ffun.
Qed.
Lemma card_inj_ffuns D T :
#|[set f : {ffun D -> T} | injectiveb f]| = #|T| ^_ #|D|.
Proof.
rewrite -card_inj_ffuns_on; apply: eq_card => f.
by rewrite 2!inE; case: ffun_onP.
Qed.
Lemma cards_draws T (B : {set T}) k :
#|[set A : {set T} | A \subset B & #|A| == k]| = 'C(#|B|, k).
Proof.
have [ltTk | lekT] := ltnP #|B| k.
rewrite bin_small // eq_card0 // => A.
rewrite inE eqn_leq [k <= _]leqNgt.
have [AsubB /=|//] := boolP (A \subset B).
by rewrite (leq_ltn_trans (subset_leq_card AsubB)) ?andbF.
apply/eqP; rewrite -(eqn_pmul2r (fact_gt0 k)) bin_ffact // eq_sym.
rewrite -sum_nat_cond_const -{1 3}(card_ord k).
rewrite -card_inj_ffuns_on -sum1dep_card.
pose imIk (f : {ffun 'I_k -> T}) := f @: 'I_k.
rewrite (partition_big imIk (fun A => (A \subset B) && (#|A| == k))) /=
=> [|f]; last first.
move=> /andP [/ffun_onP f_ffun /injectiveP inj_f].
rewrite card_imset ?card_ord // eqxx andbT.
by apply/subsetP => x /imsetP [i _ ->]; rewrite f_ffun.
apply/eqP; apply: eq_bigr => A /andP [AsubB /eqP cardAk].
have [f0 inj_f0 im_f0]: exists2 f, injective f & f @: 'I_k = A.
rewrite -cardAk; exists enum_val; first exact: enum_val_inj.
apply/setP=> a; apply/imsetP/idP=> [[i _ ->] | Aa]; first exact: enum_valP.
by exists (enum_rank_in Aa a); rewrite ?enum_rankK_in.
rewrite (reindex (fun p : {ffun _} => [ffun i => f0 (p i)])) /=; last first.
pose ff0' f i := odflt i [pick j | f i == f0 j].
exists (fun f => [ffun i => ff0' f i]) => [p _ | f].
apply/ffunP=> i; rewrite ffunE /ff0'; case: pickP => [j | /(_ (p i))].
by rewrite ffunE (inj_eq inj_f0) => /eqP.
by rewrite ffunE eqxx.
rewrite -im_f0 => /andP[/andP[/ffun_onP f_ffun /injectiveP injf] /eqP im_f].
apply/ffunP=> i; rewrite !ffunE /ff0'; case: pickP => [y /eqP //|].
have /imsetP[j _ eq_f0j_fi]: f i \in f0 @: 'I_k by rewrite -im_f imset_f.
by move/(_ j)/eqP.
rewrite -ffactnn -card_inj_ffuns -sum1dep_card; apply: eq_bigl => p.
rewrite -andbA.
apply/and3P/injectiveP=> [[_ /injectiveP inj_f0p _] i j eq_pij | inj_p].
by apply: inj_f0p; rewrite !ffunE eq_pij.
set f := finfun _.
have injf: injective f by move=> i j /[!ffunE] /inj_f0; apply: inj_p.
have imIkf : imIk f == A.
rewrite eqEcard card_imset // cardAk card_ord leqnn andbT -im_f0.
by apply/subsetP=> x /imsetP[i _ ->]; rewrite ffunE imset_f.
split; [|exact/injectiveP|exact: imIkf].
by apply/ffun_onP => x; apply: (subsetP AsubB); rewrite -(eqP imIkf) imset_f.
Qed.
Lemma card_draws T k : #|[set A : {set T} | #|A| == k]| = 'C(#|T|, k).
Proof.
by rewrite -cardsT -cards_draws; apply: eq_card => A; rewrite !inE subsetT.
Qed.
Lemma card_ltn_sorted_tuples m n :
#|[set t : m.-tuple 'I_n | sorted ltn (map val t)]| = 'C(n, m).
Proof.
have [-> | n_gt0] := posnP n; last pose i0 := Ordinal n_gt0.
case: m => [|m]; last by apply: eq_card0; case/tupleP=> [[]].
by apply: (@eq_card1 _ [tuple]) => t; rewrite [t]tuple0 inE.
rewrite -[n in RHS]card_ord -card_draws.
pose f_t (t : m.-tuple 'I_n) := [set i in t].
pose f_A (A : {set 'I_n}) := [tuple of mkseq (nth i0 (enum A)) m].
have val_fA (A : {set 'I_n}) : #|A| = m -> val (f_A A) = enum A.
by move=> Am; rewrite -[enum _](mkseq_nth i0) -cardE Am.
have inc_A (A : {set 'I_n}) : sorted ltn (map val (enum A)).
rewrite -[enum _](eq_filter (mem_enum _)).
rewrite -(eq_filter (mem_map val_inj _)) -filter_map.
by rewrite (sorted_filter ltn_trans) // unlock val_ord_enum iota_ltn_sorted.
rewrite -!sum1dep_card (reindex_onto f_t f_A) /= => [|A]; last first.
by move/eqP=> cardAm; apply/setP=> x; rewrite inE -(mem_enum A) -val_fA.
apply: eq_bigl => t.
apply/idP/idP => [inc_t|/andP [/eqP t_m /eqP <-]]; last by rewrite val_fA.
have ft_m: #|f_t t| = m.
rewrite cardsE (card_uniqP _) ?size_tuple // -(map_inj_uniq val_inj).
exact: (sorted_uniq ltn_trans ltnn).
rewrite ft_m eqxx -val_eqE val_fA // -(inj_eq (inj_map val_inj)) /=.
apply/eqP/(irr_sorted_eq ltn_trans ltnn) => // y.
by apply/mapP/mapP=> [] [x t_x ->]; exists x; rewrite // mem_enum inE in t_x *.
Qed.
Lemma card_sorted_tuples m n :
#|[set t : m.-tuple 'I_n.+1 | sorted leq (map val t)]| = 'C(m + n, m).
Proof.
set In1 := 'I_n.+1; pose x0 : In1 := ord0.
have add_mnP (i : 'I_m) (x : In1) : i + x < m + n.
by rewrite -ltnS -addSn -!addnS leq_add.
pose add_mn t i := Ordinal (add_mnP i (tnth t i)).
pose add_mn_nat (t : m.-tuple In1) i := i + nth x0 t i.
have add_mnC t: val \o add_mn t =1 add_mn_nat t \o val.
by move=> i; rewrite /= (tnth_nth x0).
pose f_add t := [tuple of map (add_mn t) (ord_tuple m)].
rewrite -card_ltn_sorted_tuples -!sum1dep_card (reindex f_add) /=.
apply: eq_bigl => t; rewrite -map_comp (eq_map (add_mnC t)) map_comp.
rewrite enumT unlock val_ord_enum -[in LHS](drop0 t).
have [m0 | m_gt0] := posnP m.
by rewrite {2}m0 /= drop_oversize // size_tuple m0.
have def_m := subnK m_gt0; rewrite -{2}def_m addn1 /= {1}/add_mn_nat.
move: 0 (m - 1) def_m => i k; rewrite -[in RHS](size_tuple t) => def_m.
rewrite (drop_nth x0) /=; last by rewrite -def_m leq_addl.
elim: k i (nth x0 t i) def_m => [|k IHk] i x /=.
by rewrite add0n => ->; rewrite drop_size.
rewrite addSnnS => def_m; rewrite -addSn leq_add2l -IHk //.
by rewrite (drop_nth x0) // -def_m leq_addl.
pose sub_mn (t : m.-tuple 'I_(m + n)) i : In1 := inord (tnth t i - i).
exists (fun t => [tuple of map (sub_mn t) (ord_tuple m)]) => [t _ | t].
apply: eq_from_tnth => i; apply: val_inj.
by rewrite /sub_mn !(tnth_ord_tuple, tnth_map) addKn inord_val.
rewrite inE /= => inc_t; apply: eq_from_tnth => i; apply: val_inj.
rewrite tnth_map tnth_ord_tuple /= tnth_map tnth_ord_tuple.
suffices [le_i_ti le_ti_ni]: i <= tnth t i /\ tnth t i <= i + n.
by rewrite /sub_mn inordK ?subnKC // ltnS leq_subLR.
pose y0 := tnth t i; rewrite (tnth_nth y0) -(nth_map _ (val i)) ?size_tuple //.
case def_e: (map _ _) => [|x e] /=; first by rewrite nth_nil ?leq_addr.
set nth_i := nth (i : nat); rewrite def_e in inc_t; split.
have: i < size (x :: e) by rewrite -def_e size_map size_tuple ltn_ord.
elim: (val i) => //= j IHj lt_j_e.
by apply: leq_trans (pathP (val i) inc_t _ lt_j_e); rewrite ltnS IHj 1?ltnW.
move: (_ - _) (subnK (valP i)) => k /=.
elim: k (val i) => /= [|k IHk] j; rewrite -ltnS -addSn ?add0n => def_m.
by rewrite def_m -def_e /nth_i (nth_map y0) ?ltn_ord // size_tuple -def_m.
rewrite (leq_trans _ (IHk _ _)) -1?addSnnS //; apply: (pathP _ inc_t).
rewrite -ltnS (leq_trans (leq_addl k _)) // -addSnnS def_m.
by rewrite -(size_tuple t) -(size_map val) def_e.
Qed.
Lemma card_partial_ord_partitions m n :
#|[set t : m.-tuple 'I_n.+1 | \sum_(i <- t) i <= n]| = 'C(m + n, m).
Proof.
symmetry; set In1 := 'I_n.+1; pose x0 : In1 := ord0.
pose add_mn (i j : In1) : In1 := inord (i + j).
pose f_add (t : m.-tuple In1) := [tuple of scanl add_mn x0 t].
rewrite -card_sorted_tuples -!sum1dep_card (reindex f_add) /=.
apply: eq_bigl => t; rewrite -[\sum_(i <- t) i]add0n.
transitivity (path leq x0 (map val (f_add t))) => /=; first by case: map.
rewrite -{1 2}[0]/(val x0); elim: {t}(val t) (x0) => /= [|x t IHt] s.
by rewrite big_nil addn0 -ltnS ltn_ord.
rewrite big_cons addnA IHt /= val_insubd ltnS.
have [_ | ltn_n_sx] := leqP (s + x) n; first by rewrite leq_addr.
rewrite -(leq_add2r x) leqNgt (leq_trans (valP x)) //=.
by rewrite leqNgt (leq_trans ltn_n_sx) ?leq_addr.
pose sub_mn (i j : In1) := Ordinal (leq_ltn_trans (leq_subr i j) (valP j)).
exists (fun t : m.-tuple In1 => [tuple of pairmap sub_mn x0 t]) => /= t inc_t.
apply: val_inj => /=; have{inc_t}: path leq x0 (map val (f_add t)).
by move: inc_t; rewrite inE /=; case: map.
rewrite [map _ _]/=; elim: {t}(val t) (x0) => //= x t IHt s.
case/andP=> le_s_sx /IHt->; congr (_ :: _); apply: val_inj => /=.
move: le_s_sx; rewrite val_insubd.
case le_sx_n: (_ < n.+1); first by rewrite addKn.
by case: (val s) le_sx_n; rewrite ?ltn_ord.
apply: val_inj => /=; have{inc_t}: path leq x0 (map val t).
by move: inc_t; rewrite inE /=; case: map.
elim: {t}(val t) (x0) => //= x t IHt s /andP[le_s_sx inc_t].
suffices ->: add_mn s (sub_mn s x) = x by rewrite IHt.
by apply: val_inj; rewrite /add_mn /= subnKC ?inord_val.
Qed.
Lemma card_ord_partitions m n :
#|[set t : m.+1.-tuple 'I_n.+1 | \sum_(i <- t) i == n]| = 'C(m + n, m).
Proof.
symmetry; set In1 := 'I_n.+1; pose x0 : In1 := ord0.
pose f_add (t : m.-tuple In1) := [tuple of sub_ord (\sum_(x <- t) x) :: t].
rewrite -card_partial_ord_partitions -!sum1dep_card (reindex f_add) /=.
by apply: eq_bigl => t; rewrite big_cons /= addnC (sameP maxn_idPr eqP) maxnE.
exists (fun t : m.+1.-tuple In1 => [tuple of behead t]) => [t _|].
exact: val_inj.
case/tupleP=> x t /[!(inE, big_cons)] /eqP def_n.
by apply: val_inj; congr (_ :: _); apply: val_inj; rewrite /= -{1}def_n addnK.
Qed.
End Combinations.
|
Ideal.lean
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.RingTheory.Finiteness.Finsupp
import Mathlib.RingTheory.Ideal.Maps
/-!
# Finitely generated ideals
Lemmas about finiteness of ideal operations.
-/
open Function (Surjective)
open Finsupp
namespace Ideal
variable {R : Type*} {M : Type*} [Semiring R] [AddCommMonoid M] [Module R M]
/-- The image of a finitely generated ideal is finitely generated.
This is the `Ideal` version of `Submodule.FG.map`. -/
theorem FG.map {R S : Type*} [Semiring R] [Semiring S] {I : Ideal R} (h : I.FG) (f : R →+* S) :
(I.map f).FG := by
classical
obtain ⟨s, hs⟩ := h
refine ⟨s.image f, ?_⟩
rw [Finset.coe_image, ← Ideal.map_span, hs]
theorem fg_ker_comp {R S A : Type*} [CommRing R] [CommRing S] [CommRing A] (f : R →+* S)
(g : S →+* A) (hf : (RingHom.ker f).FG) (hg : (RingHom.ker g).FG)
(hsur : Function.Surjective f) :
(RingHom.ker (g.comp f)).FG := by
letI : Algebra R S := RingHom.toAlgebra f
letI : Algebra R A := RingHom.toAlgebra (g.comp f)
letI : Algebra S A := RingHom.toAlgebra g
letI : IsScalarTower R S A := IsScalarTower.of_algebraMap_eq fun _ => rfl
let f₁ := Algebra.linearMap R S
let g₁ := (IsScalarTower.toAlgHom R S A).toLinearMap
exact Submodule.fg_ker_comp f₁ g₁ hf (Submodule.fg_restrictScalars (RingHom.ker g) hg hsur) hsur
theorem exists_radical_pow_le_of_fg {R : Type*} [CommSemiring R] (I : Ideal R) (h : I.radical.FG) :
∃ n : ℕ, I.radical ^ n ≤ I := by
have := le_refl I.radical; revert this
refine Submodule.fg_induction _ _ (fun J => J ≤ I.radical → ∃ n : ℕ, J ^ n ≤ I) ?_ ?_ _ h
· intro x hx
obtain ⟨n, hn⟩ := hx (subset_span (Set.mem_singleton x))
exact ⟨n, by rwa [← Ideal.span, span_singleton_pow, span_le, Set.singleton_subset_iff]⟩
· intro J K hJ hK hJK
obtain ⟨n, hn⟩ := hJ fun x hx => hJK <| Ideal.mem_sup_left hx
obtain ⟨m, hm⟩ := hK fun x hx => hJK <| Ideal.mem_sup_right hx
use n + m
rw [← Ideal.add_eq_sup, add_pow, Ideal.sum_eq_sup, Finset.sup_le_iff]
refine fun i _ => Ideal.mul_le_right.trans ?_
obtain h | h := le_or_gt n i
· apply Ideal.mul_le_right.trans ((Ideal.pow_le_pow_right h).trans hn)
· apply Ideal.mul_le_left.trans
refine (Ideal.pow_le_pow_right ?_).trans hm
rw [add_comm, Nat.add_sub_assoc h.le]
apply Nat.le_add_right
theorem exists_pow_le_of_le_radical_of_fg_radical {R : Type*} [CommSemiring R] {I J : Ideal R}
(hIJ : I ≤ J.radical) (hJ : J.radical.FG) :
∃ k : ℕ, I ^ k ≤ J := by
obtain ⟨k, hk⟩ := J.exists_radical_pow_le_of_fg hJ
use k
calc
I ^ k ≤ J.radical ^ k := Ideal.pow_right_mono hIJ _
_ ≤ J := hk
lemma exists_pow_le_of_le_radical_of_fg {R : Type*} [CommSemiring R] {I J : Ideal R}
(h' : I ≤ J.radical) (h : I.FG) :
∃ n : ℕ, I ^ n ≤ J := by
revert h'
apply Submodule.fg_induction _ _ _ _ _ I h
· intro x hJ
simp only [Ideal.submodule_span_eq, Ideal.span_le,
Set.singleton_subset_iff, SetLike.mem_coe] at hJ
obtain ⟨n, hn⟩ := hJ
refine ⟨n, by simpa [Ideal.span_singleton_pow, Ideal.span_le]⟩
· intros I₁ I₂ h₁ h₂ hJ
obtain ⟨n₁, hn₁⟩ := h₁ (le_sup_left.trans hJ)
obtain ⟨n₂, hn₂⟩ := h₂ (le_sup_right.trans hJ)
use n₁ + n₂
exact Ideal.sup_pow_add_le_pow_sup_pow.trans (sup_le hn₁ hn₂)
end Ideal
|
SymmDiff.lean
|
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra.Basic
import Mathlib.Logic.Equiv.Basic
/-!
# Symmetric difference and bi-implication
This file defines the symmetric difference and bi-implication operators in (co-)Heyting algebras.
## Examples
Some examples are
* The symmetric difference of two sets is the set of elements that are in either but not both.
* The symmetric difference on propositions is `Xor'`.
* The symmetric difference on `Bool` is `Bool.xor`.
* The equivalence of propositions. Two propositions are equivalent if they imply each other.
* The symmetric difference translates to addition when considering a Boolean algebra as a Boolean
ring.
## Main declarations
* `symmDiff`: The symmetric difference operator, defined as `(a \ b) ⊔ (b \ a)`
* `bihimp`: The bi-implication operator, defined as `(b ⇨ a) ⊓ (a ⇨ b)`
In generalized Boolean algebras, the symmetric difference operator is:
* `symmDiff_comm`: commutative, and
* `symmDiff_assoc`: associative.
## Notations
* `a ∆ b`: `symmDiff a b`
* `a ⇔ b`: `bihimp a b`
## References
The proof of associativity follows the note "Associativity of the Symmetric Difference of Sets: A
Proof from the Book" by John McCuan:
* <https://people.math.gatech.edu/~mccuan/courses/4317/symmetricdifference.pdf>
## Tags
boolean ring, generalized boolean algebra, boolean algebra, symmetric difference, bi-implication,
Heyting
-/
assert_not_exists RelIso
open Function OrderDual
variable {ι α β : Type*} {π : ι → Type*}
/-- The symmetric difference operator on a type with `⊔` and `\` is `(A \ B) ⊔ (B \ A)`. -/
def symmDiff [Max α] [SDiff α] (a b : α) : α :=
a \ b ⊔ b \ a
/-- The Heyting bi-implication is `(b ⇨ a) ⊓ (a ⇨ b)`. This generalizes equivalence of
propositions. -/
def bihimp [Min α] [HImp α] (a b : α) : α :=
(b ⇨ a) ⊓ (a ⇨ b)
/-- Notation for symmDiff -/
scoped[symmDiff] infixl:100 " ∆ " => symmDiff
/-- Notation for bihimp -/
scoped[symmDiff] infixl:100 " ⇔ " => bihimp
open scoped symmDiff
theorem symmDiff_def [Max α] [SDiff α] (a b : α) : a ∆ b = a \ b ⊔ b \ a :=
rfl
theorem bihimp_def [Min α] [HImp α] (a b : α) : a ⇔ b = (b ⇨ a) ⊓ (a ⇨ b) :=
rfl
theorem symmDiff_eq_Xor' (p q : Prop) : p ∆ q = Xor' p q :=
rfl
@[simp]
theorem bihimp_iff_iff {p q : Prop} : p ⇔ q ↔ (p ↔ q) :=
iff_iff_implies_and_implies.symm.trans Iff.comm
@[simp]
theorem Bool.symmDiff_eq_xor : ∀ p q : Bool, p ∆ q = xor p q := by decide
section GeneralizedCoheytingAlgebra
variable [GeneralizedCoheytingAlgebra α] (a b c : α)
@[simp]
theorem toDual_symmDiff : toDual (a ∆ b) = toDual a ⇔ toDual b :=
rfl
@[simp]
theorem ofDual_bihimp (a b : αᵒᵈ) : ofDual (a ⇔ b) = ofDual a ∆ ofDual b :=
rfl
theorem symmDiff_comm : a ∆ b = b ∆ a := by simp only [symmDiff, sup_comm]
instance symmDiff_isCommutative : Std.Commutative (α := α) (· ∆ ·) :=
⟨symmDiff_comm⟩
@[simp]
theorem symmDiff_self : a ∆ a = ⊥ := by rw [symmDiff, sup_idem, sdiff_self]
@[simp]
theorem symmDiff_bot : a ∆ ⊥ = a := by rw [symmDiff, sdiff_bot, bot_sdiff, sup_bot_eq]
@[simp]
theorem bot_symmDiff : ⊥ ∆ a = a := by rw [symmDiff_comm, symmDiff_bot]
@[simp]
theorem symmDiff_eq_bot {a b : α} : a ∆ b = ⊥ ↔ a = b := by
simp_rw [symmDiff, sup_eq_bot_iff, sdiff_eq_bot_iff, le_antisymm_iff]
theorem symmDiff_of_le {a b : α} (h : a ≤ b) : a ∆ b = b \ a := by
rw [symmDiff, sdiff_eq_bot_iff.2 h, bot_sup_eq]
theorem symmDiff_of_ge {a b : α} (h : b ≤ a) : a ∆ b = a \ b := by
rw [symmDiff, sdiff_eq_bot_iff.2 h, sup_bot_eq]
theorem symmDiff_le {a b c : α} (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ∆ b ≤ c :=
sup_le (sdiff_le_iff.2 ha) <| sdiff_le_iff.2 hb
theorem symmDiff_le_iff {a b c : α} : a ∆ b ≤ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c := by
simp_rw [symmDiff, sup_le_iff, sdiff_le_iff]
@[simp]
theorem symmDiff_le_sup {a b : α} : a ∆ b ≤ a ⊔ b :=
sup_le_sup sdiff_le sdiff_le
theorem symmDiff_eq_sup_sdiff_inf : a ∆ b = (a ⊔ b) \ (a ⊓ b) := by simp [sup_sdiff, symmDiff]
theorem Disjoint.symmDiff_eq_sup {a b : α} (h : Disjoint a b) : a ∆ b = a ⊔ b := by
rw [symmDiff, h.sdiff_eq_left, h.sdiff_eq_right]
theorem symmDiff_sdiff : a ∆ b \ c = a \ (b ⊔ c) ⊔ b \ (a ⊔ c) := by
rw [symmDiff, sup_sdiff_distrib, sdiff_sdiff_left, sdiff_sdiff_left]
@[simp]
theorem symmDiff_sdiff_inf : a ∆ b \ (a ⊓ b) = a ∆ b := by
rw [symmDiff_sdiff]
simp [symmDiff]
@[simp]
theorem symmDiff_sdiff_eq_sup : a ∆ (b \ a) = a ⊔ b := by
rw [symmDiff, sdiff_idem]
exact
le_antisymm (sup_le_sup sdiff_le sdiff_le)
(sup_le le_sdiff_sup <| le_sdiff_sup.trans <| sup_le le_sup_right le_sdiff_sup)
@[simp]
theorem sdiff_symmDiff_eq_sup : (a \ b) ∆ b = a ⊔ b := by
rw [symmDiff_comm, symmDiff_sdiff_eq_sup, sup_comm]
@[simp]
theorem symmDiff_sup_inf : a ∆ b ⊔ a ⊓ b = a ⊔ b := by
refine le_antisymm (sup_le symmDiff_le_sup inf_le_sup) ?_
rw [sup_inf_left, symmDiff]
refine sup_le (le_inf le_sup_right ?_) (le_inf ?_ le_sup_right)
· rw [sup_right_comm]
exact le_sup_of_le_left le_sdiff_sup
· rw [sup_assoc]
exact le_sup_of_le_right le_sdiff_sup
@[simp]
theorem inf_sup_symmDiff : a ⊓ b ⊔ a ∆ b = a ⊔ b := by rw [sup_comm, symmDiff_sup_inf]
@[simp]
theorem symmDiff_symmDiff_inf : a ∆ b ∆ (a ⊓ b) = a ⊔ b := by
rw [← symmDiff_sdiff_inf a, sdiff_symmDiff_eq_sup, symmDiff_sup_inf]
@[simp]
theorem inf_symmDiff_symmDiff : (a ⊓ b) ∆ (a ∆ b) = a ⊔ b := by
rw [symmDiff_comm, symmDiff_symmDiff_inf]
theorem symmDiff_triangle : a ∆ c ≤ a ∆ b ⊔ b ∆ c := by
refine (sup_le_sup (sdiff_triangle a b c) <| sdiff_triangle _ b _).trans_eq ?_
rw [sup_comm (c \ b), sup_sup_sup_comm, symmDiff, symmDiff]
theorem le_symmDiff_sup_right (a b : α) : a ≤ (a ∆ b) ⊔ b := by
convert symmDiff_triangle a b ⊥ <;> rw [symmDiff_bot]
theorem le_symmDiff_sup_left (a b : α) : b ≤ (a ∆ b) ⊔ a :=
symmDiff_comm a b ▸ le_symmDiff_sup_right ..
end GeneralizedCoheytingAlgebra
section GeneralizedHeytingAlgebra
variable [GeneralizedHeytingAlgebra α] (a b c : α)
@[simp]
theorem toDual_bihimp : toDual (a ⇔ b) = toDual a ∆ toDual b :=
rfl
@[simp]
theorem ofDual_symmDiff (a b : αᵒᵈ) : ofDual (a ∆ b) = ofDual a ⇔ ofDual b :=
rfl
theorem bihimp_comm : a ⇔ b = b ⇔ a := by simp only [(· ⇔ ·), inf_comm]
instance bihimp_isCommutative : Std.Commutative (α := α) (· ⇔ ·) :=
⟨bihimp_comm⟩
@[simp]
theorem bihimp_self : a ⇔ a = ⊤ := by rw [bihimp, inf_idem, himp_self]
@[simp]
theorem bihimp_top : a ⇔ ⊤ = a := by rw [bihimp, himp_top, top_himp, inf_top_eq]
@[simp]
theorem top_bihimp : ⊤ ⇔ a = a := by rw [bihimp_comm, bihimp_top]
@[simp]
theorem bihimp_eq_top {a b : α} : a ⇔ b = ⊤ ↔ a = b :=
@symmDiff_eq_bot αᵒᵈ _ _ _
theorem bihimp_of_le {a b : α} (h : a ≤ b) : a ⇔ b = b ⇨ a := by
rw [bihimp, himp_eq_top_iff.2 h, inf_top_eq]
theorem bihimp_of_ge {a b : α} (h : b ≤ a) : a ⇔ b = a ⇨ b := by
rw [bihimp, himp_eq_top_iff.2 h, top_inf_eq]
theorem le_bihimp {a b c : α} (hb : a ⊓ b ≤ c) (hc : a ⊓ c ≤ b) : a ≤ b ⇔ c :=
le_inf (le_himp_iff.2 hc) <| le_himp_iff.2 hb
theorem le_bihimp_iff {a b c : α} : a ≤ b ⇔ c ↔ a ⊓ b ≤ c ∧ a ⊓ c ≤ b := by
simp_rw [bihimp, le_inf_iff, le_himp_iff, and_comm]
@[simp]
theorem inf_le_bihimp {a b : α} : a ⊓ b ≤ a ⇔ b :=
inf_le_inf le_himp le_himp
theorem bihimp_eq_sup_himp_inf : a ⇔ b = a ⊔ b ⇨ a ⊓ b := by simp [himp_inf_distrib, bihimp]
@[deprecated (since := "2025-06-05")] alias bihimp_eq_inf_himp_inf := bihimp_eq_sup_himp_inf
theorem Codisjoint.bihimp_eq_inf {a b : α} (h : Codisjoint a b) : a ⇔ b = a ⊓ b := by
rw [bihimp, h.himp_eq_left, h.himp_eq_right]
theorem himp_bihimp : a ⇨ b ⇔ c = (a ⊓ c ⇨ b) ⊓ (a ⊓ b ⇨ c) := by
rw [bihimp, himp_inf_distrib, himp_himp, himp_himp]
@[simp]
theorem sup_himp_bihimp : a ⊔ b ⇨ a ⇔ b = a ⇔ b := by
rw [himp_bihimp]
simp [bihimp]
@[simp]
theorem bihimp_himp_eq_inf : a ⇔ (a ⇨ b) = a ⊓ b :=
@symmDiff_sdiff_eq_sup αᵒᵈ _ _ _
@[simp]
theorem himp_bihimp_eq_inf : (b ⇨ a) ⇔ b = a ⊓ b :=
@sdiff_symmDiff_eq_sup αᵒᵈ _ _ _
@[simp]
theorem bihimp_inf_sup : a ⇔ b ⊓ (a ⊔ b) = a ⊓ b :=
@symmDiff_sup_inf αᵒᵈ _ _ _
@[simp]
theorem sup_inf_bihimp : (a ⊔ b) ⊓ a ⇔ b = a ⊓ b :=
@inf_sup_symmDiff αᵒᵈ _ _ _
@[simp]
theorem bihimp_bihimp_sup : a ⇔ b ⇔ (a ⊔ b) = a ⊓ b :=
@symmDiff_symmDiff_inf αᵒᵈ _ _ _
@[simp]
theorem sup_bihimp_bihimp : (a ⊔ b) ⇔ (a ⇔ b) = a ⊓ b :=
@inf_symmDiff_symmDiff αᵒᵈ _ _ _
theorem bihimp_triangle : a ⇔ b ⊓ b ⇔ c ≤ a ⇔ c :=
@symmDiff_triangle αᵒᵈ _ _ _ _
end GeneralizedHeytingAlgebra
section CoheytingAlgebra
variable [CoheytingAlgebra α] (a : α)
@[simp]
theorem symmDiff_top' : a ∆ ⊤ = ￢a := by simp [symmDiff]
@[simp]
theorem top_symmDiff' : ⊤ ∆ a = ￢a := by simp [symmDiff]
@[simp]
theorem hnot_symmDiff_self : (￢a) ∆ a = ⊤ := by
rw [eq_top_iff, symmDiff, hnot_sdiff, sup_sdiff_self]
exact Codisjoint.top_le codisjoint_hnot_left
@[simp]
theorem symmDiff_hnot_self : a ∆ (￢a) = ⊤ := by rw [symmDiff_comm, hnot_symmDiff_self]
theorem IsCompl.symmDiff_eq_top {a b : α} (h : IsCompl a b) : a ∆ b = ⊤ := by
rw [h.eq_hnot, hnot_symmDiff_self]
end CoheytingAlgebra
section HeytingAlgebra
variable [HeytingAlgebra α] (a : α)
@[simp]
theorem bihimp_bot : a ⇔ ⊥ = aᶜ := by simp [bihimp]
@[simp]
theorem bot_bihimp : ⊥ ⇔ a = aᶜ := by simp [bihimp]
@[simp]
theorem compl_bihimp_self : aᶜ ⇔ a = ⊥ :=
@hnot_symmDiff_self αᵒᵈ _ _
@[simp]
theorem bihimp_hnot_self : a ⇔ aᶜ = ⊥ :=
@symmDiff_hnot_self αᵒᵈ _ _
theorem IsCompl.bihimp_eq_bot {a b : α} (h : IsCompl a b) : a ⇔ b = ⊥ := by
rw [h.eq_compl, compl_bihimp_self]
end HeytingAlgebra
section GeneralizedBooleanAlgebra
variable [GeneralizedBooleanAlgebra α] (a b c d : α)
@[simp]
theorem sup_sdiff_symmDiff : (a ⊔ b) \ a ∆ b = a ⊓ b :=
sdiff_eq_symm inf_le_sup (by rw [symmDiff_eq_sup_sdiff_inf])
theorem disjoint_symmDiff_inf : Disjoint (a ∆ b) (a ⊓ b) := by
rw [symmDiff_eq_sup_sdiff_inf]
exact disjoint_sdiff_self_left
theorem inf_symmDiff_distrib_left : a ⊓ b ∆ c = (a ⊓ b) ∆ (a ⊓ c) := by
rw [symmDiff_eq_sup_sdiff_inf, inf_sdiff_distrib_left, inf_sup_left, inf_inf_distrib_left,
symmDiff_eq_sup_sdiff_inf]
theorem inf_symmDiff_distrib_right : a ∆ b ⊓ c = (a ⊓ c) ∆ (b ⊓ c) := by
simp_rw [inf_comm _ c, inf_symmDiff_distrib_left]
theorem sdiff_symmDiff : c \ a ∆ b = c ⊓ a ⊓ b ⊔ c \ a ⊓ c \ b := by
simp only [(· ∆ ·), sdiff_sdiff_sup_sdiff']
theorem sdiff_symmDiff' : c \ a ∆ b = c ⊓ a ⊓ b ⊔ c \ (a ⊔ b) := by
rw [sdiff_symmDiff, sdiff_sup]
@[simp]
theorem symmDiff_sdiff_left : a ∆ b \ a = b \ a := by
rw [symmDiff_def, sup_sdiff, sdiff_idem, sdiff_sdiff_self, bot_sup_eq]
@[simp]
theorem symmDiff_sdiff_right : a ∆ b \ b = a \ b := by rw [symmDiff_comm, symmDiff_sdiff_left]
@[simp]
theorem sdiff_symmDiff_left : a \ a ∆ b = a ⊓ b := by simp [sdiff_symmDiff]
@[simp]
theorem sdiff_symmDiff_right : b \ a ∆ b = a ⊓ b := by
rw [symmDiff_comm, inf_comm, sdiff_symmDiff_left]
theorem symmDiff_eq_sup : a ∆ b = a ⊔ b ↔ Disjoint a b := by
refine ⟨fun h => ?_, Disjoint.symmDiff_eq_sup⟩
rw [symmDiff_eq_sup_sdiff_inf, sdiff_eq_self_iff_disjoint] at h
exact h.of_disjoint_inf_of_le le_sup_left
@[simp]
theorem le_symmDiff_iff_left : a ≤ a ∆ b ↔ Disjoint a b := by
refine ⟨fun h => ?_, fun h => h.symmDiff_eq_sup.symm ▸ le_sup_left⟩
rw [symmDiff_eq_sup_sdiff_inf] at h
exact disjoint_iff_inf_le.mpr (le_sdiff_right.1 <| inf_le_of_left_le h).le
@[simp]
theorem le_symmDiff_iff_right : b ≤ a ∆ b ↔ Disjoint a b := by
rw [symmDiff_comm, le_symmDiff_iff_left, disjoint_comm]
theorem symmDiff_symmDiff_left :
a ∆ b ∆ c = a \ (b ⊔ c) ⊔ b \ (a ⊔ c) ⊔ c \ (a ⊔ b) ⊔ a ⊓ b ⊓ c :=
calc
a ∆ b ∆ c = a ∆ b \ c ⊔ c \ a ∆ b := symmDiff_def _ _
_ = a \ (b ⊔ c) ⊔ b \ (a ⊔ c) ⊔ (c \ (a ⊔ b) ⊔ c ⊓ a ⊓ b) := by
{ rw [sdiff_symmDiff', sup_comm (c ⊓ a ⊓ b), symmDiff_sdiff] }
_ = a \ (b ⊔ c) ⊔ b \ (a ⊔ c) ⊔ c \ (a ⊔ b) ⊔ a ⊓ b ⊓ c := by ac_rfl
theorem symmDiff_symmDiff_right :
a ∆ (b ∆ c) = a \ (b ⊔ c) ⊔ b \ (a ⊔ c) ⊔ c \ (a ⊔ b) ⊔ a ⊓ b ⊓ c :=
calc
a ∆ (b ∆ c) = a \ b ∆ c ⊔ b ∆ c \ a := symmDiff_def _ _
_ = a \ (b ⊔ c) ⊔ a ⊓ b ⊓ c ⊔ (b \ (c ⊔ a) ⊔ c \ (b ⊔ a)) := by
{ rw [sdiff_symmDiff', sup_comm (a ⊓ b ⊓ c), symmDiff_sdiff] }
_ = a \ (b ⊔ c) ⊔ b \ (a ⊔ c) ⊔ c \ (a ⊔ b) ⊔ a ⊓ b ⊓ c := by ac_rfl
theorem symmDiff_assoc : a ∆ b ∆ c = a ∆ (b ∆ c) := by
rw [symmDiff_symmDiff_left, symmDiff_symmDiff_right]
instance symmDiff_isAssociative : Std.Associative (α := α) (· ∆ ·) :=
⟨symmDiff_assoc⟩
theorem symmDiff_left_comm : a ∆ (b ∆ c) = b ∆ (a ∆ c) := by
simp_rw [← symmDiff_assoc, symmDiff_comm]
theorem symmDiff_right_comm : a ∆ b ∆ c = a ∆ c ∆ b := by simp_rw [symmDiff_assoc, symmDiff_comm]
theorem symmDiff_symmDiff_symmDiff_comm : a ∆ b ∆ (c ∆ d) = a ∆ c ∆ (b ∆ d) := by
simp_rw [symmDiff_assoc, symmDiff_left_comm]
@[simp]
theorem symmDiff_symmDiff_cancel_left : a ∆ (a ∆ b) = b := by simp [← symmDiff_assoc]
@[simp]
theorem symmDiff_symmDiff_cancel_right : b ∆ a ∆ a = b := by simp [symmDiff_assoc]
@[simp]
theorem symmDiff_symmDiff_self' : a ∆ b ∆ a = b := by
rw [symmDiff_comm, symmDiff_symmDiff_cancel_left]
theorem symmDiff_left_involutive (a : α) : Involutive (· ∆ a) :=
symmDiff_symmDiff_cancel_right _
theorem symmDiff_right_involutive (a : α) : Involutive (a ∆ ·) :=
symmDiff_symmDiff_cancel_left _
theorem symmDiff_left_injective (a : α) : Injective (· ∆ a) :=
Function.Involutive.injective (symmDiff_left_involutive a)
theorem symmDiff_right_injective (a : α) : Injective (a ∆ ·) :=
Function.Involutive.injective (symmDiff_right_involutive _)
theorem symmDiff_left_surjective (a : α) : Surjective (· ∆ a) :=
Function.Involutive.surjective (symmDiff_left_involutive _)
theorem symmDiff_right_surjective (a : α) : Surjective (a ∆ ·) :=
Function.Involutive.surjective (symmDiff_right_involutive _)
variable {a b c}
@[simp]
theorem symmDiff_left_inj : a ∆ b = c ∆ b ↔ a = c :=
(symmDiff_left_injective _).eq_iff
@[simp]
theorem symmDiff_right_inj : a ∆ b = a ∆ c ↔ b = c :=
(symmDiff_right_injective _).eq_iff
@[simp]
theorem symmDiff_eq_left : a ∆ b = a ↔ b = ⊥ :=
calc
a ∆ b = a ↔ a ∆ b = a ∆ ⊥ := by rw [symmDiff_bot]
_ ↔ b = ⊥ := by rw [symmDiff_right_inj]
@[simp]
theorem symmDiff_eq_right : a ∆ b = b ↔ a = ⊥ := by rw [symmDiff_comm, symmDiff_eq_left]
protected theorem Disjoint.symmDiff_left (ha : Disjoint a c) (hb : Disjoint b c) :
Disjoint (a ∆ b) c := by
rw [symmDiff_eq_sup_sdiff_inf]
exact (ha.sup_left hb).disjoint_sdiff_left
protected theorem Disjoint.symmDiff_right (ha : Disjoint a b) (hb : Disjoint a c) :
Disjoint a (b ∆ c) :=
(ha.symm.symmDiff_left hb.symm).symm
theorem symmDiff_eq_iff_sdiff_eq (ha : a ≤ c) : a ∆ b = c ↔ c \ a = b := by
rw [← symmDiff_of_le ha]
exact ((symmDiff_right_involutive a).toPerm _).apply_eq_iff_eq_symm_apply.trans eq_comm
end GeneralizedBooleanAlgebra
section BooleanAlgebra
variable [BooleanAlgebra α] (a b c d : α)
/-! `CogeneralizedBooleanAlgebra` isn't actually a typeclass, but the lemmas in here are dual to
the `GeneralizedBooleanAlgebra` ones -/
section CogeneralizedBooleanAlgebra
@[simp]
theorem inf_himp_bihimp : a ⇔ b ⇨ a ⊓ b = a ⊔ b :=
@sup_sdiff_symmDiff αᵒᵈ _ _ _
theorem codisjoint_bihimp_sup : Codisjoint (a ⇔ b) (a ⊔ b) :=
@disjoint_symmDiff_inf αᵒᵈ _ _ _
@[simp]
theorem himp_bihimp_left : a ⇨ a ⇔ b = a ⇨ b :=
@symmDiff_sdiff_left αᵒᵈ _ _ _
@[simp]
theorem himp_bihimp_right : b ⇨ a ⇔ b = b ⇨ a :=
@symmDiff_sdiff_right αᵒᵈ _ _ _
@[simp]
theorem bihimp_himp_left : a ⇔ b ⇨ a = a ⊔ b :=
@sdiff_symmDiff_left αᵒᵈ _ _ _
@[simp]
theorem bihimp_himp_right : a ⇔ b ⇨ b = a ⊔ b :=
@sdiff_symmDiff_right αᵒᵈ _ _ _
@[simp]
theorem bihimp_eq_inf : a ⇔ b = a ⊓ b ↔ Codisjoint a b :=
@symmDiff_eq_sup αᵒᵈ _ _ _
@[simp]
theorem bihimp_le_iff_left : a ⇔ b ≤ a ↔ Codisjoint a b :=
@le_symmDiff_iff_left αᵒᵈ _ _ _
@[simp]
theorem bihimp_le_iff_right : a ⇔ b ≤ b ↔ Codisjoint a b :=
@le_symmDiff_iff_right αᵒᵈ _ _ _
theorem bihimp_assoc : a ⇔ b ⇔ c = a ⇔ (b ⇔ c) :=
@symmDiff_assoc αᵒᵈ _ _ _ _
instance bihimp_isAssociative : Std.Associative (α := α) (· ⇔ ·) :=
⟨bihimp_assoc⟩
theorem bihimp_left_comm : a ⇔ (b ⇔ c) = b ⇔ (a ⇔ c) := by simp_rw [← bihimp_assoc, bihimp_comm]
theorem bihimp_right_comm : a ⇔ b ⇔ c = a ⇔ c ⇔ b := by simp_rw [bihimp_assoc, bihimp_comm]
theorem bihimp_bihimp_bihimp_comm : a ⇔ b ⇔ (c ⇔ d) = a ⇔ c ⇔ (b ⇔ d) := by
simp_rw [bihimp_assoc, bihimp_left_comm]
@[simp]
theorem bihimp_bihimp_cancel_left : a ⇔ (a ⇔ b) = b := by simp [← bihimp_assoc]
@[simp]
theorem bihimp_bihimp_cancel_right : b ⇔ a ⇔ a = b := by simp [bihimp_assoc]
@[simp]
theorem bihimp_bihimp_self : a ⇔ b ⇔ a = b := by rw [bihimp_comm, bihimp_bihimp_cancel_left]
theorem bihimp_left_involutive (a : α) : Involutive (· ⇔ a) :=
bihimp_bihimp_cancel_right _
theorem bihimp_right_involutive (a : α) : Involutive (a ⇔ ·) :=
bihimp_bihimp_cancel_left _
theorem bihimp_left_injective (a : α) : Injective (· ⇔ a) :=
@symmDiff_left_injective αᵒᵈ _ _
theorem bihimp_right_injective (a : α) : Injective (a ⇔ ·) :=
@symmDiff_right_injective αᵒᵈ _ _
theorem bihimp_left_surjective (a : α) : Surjective (· ⇔ a) :=
@symmDiff_left_surjective αᵒᵈ _ _
theorem bihimp_right_surjective (a : α) : Surjective (a ⇔ ·) :=
@symmDiff_right_surjective αᵒᵈ _ _
variable {a b c}
@[simp]
theorem bihimp_left_inj : a ⇔ b = c ⇔ b ↔ a = c :=
(bihimp_left_injective _).eq_iff
@[simp]
theorem bihimp_right_inj : a ⇔ b = a ⇔ c ↔ b = c :=
(bihimp_right_injective _).eq_iff
@[simp]
theorem bihimp_eq_left : a ⇔ b = a ↔ b = ⊤ :=
@symmDiff_eq_left αᵒᵈ _ _ _
@[simp]
theorem bihimp_eq_right : a ⇔ b = b ↔ a = ⊤ :=
@symmDiff_eq_right αᵒᵈ _ _ _
protected theorem Codisjoint.bihimp_left (ha : Codisjoint a c) (hb : Codisjoint b c) :
Codisjoint (a ⇔ b) c :=
(ha.inf_left hb).mono_left inf_le_bihimp
protected theorem Codisjoint.bihimp_right (ha : Codisjoint a b) (hb : Codisjoint a c) :
Codisjoint a (b ⇔ c) :=
(ha.inf_right hb).mono_right inf_le_bihimp
end CogeneralizedBooleanAlgebra
theorem symmDiff_eq : a ∆ b = a ⊓ bᶜ ⊔ b ⊓ aᶜ := by simp only [(· ∆ ·), sdiff_eq]
theorem bihimp_eq : a ⇔ b = (a ⊔ bᶜ) ⊓ (b ⊔ aᶜ) := by simp only [(· ⇔ ·), himp_eq]
theorem symmDiff_eq' : a ∆ b = (a ⊔ b) ⊓ (aᶜ ⊔ bᶜ) := by
rw [symmDiff_eq_sup_sdiff_inf, sdiff_eq, compl_inf]
theorem bihimp_eq' : a ⇔ b = a ⊓ b ⊔ aᶜ ⊓ bᶜ :=
@symmDiff_eq' αᵒᵈ _ _ _
theorem symmDiff_top : a ∆ ⊤ = aᶜ :=
symmDiff_top' _
theorem top_symmDiff : ⊤ ∆ a = aᶜ :=
top_symmDiff' _
@[simp]
theorem compl_symmDiff : (a ∆ b)ᶜ = a ⇔ b := by
simp_rw [symmDiff, compl_sup_distrib, compl_sdiff, bihimp, inf_comm]
@[simp]
theorem compl_bihimp : (a ⇔ b)ᶜ = a ∆ b :=
@compl_symmDiff αᵒᵈ _ _ _
@[simp]
theorem compl_symmDiff_compl : aᶜ ∆ bᶜ = a ∆ b :=
(sup_comm _ _).trans <| by simp_rw [compl_sdiff_compl, sdiff_eq, symmDiff_eq]
@[simp]
theorem compl_bihimp_compl : aᶜ ⇔ bᶜ = a ⇔ b :=
@compl_symmDiff_compl αᵒᵈ _ _ _
@[simp]
theorem symmDiff_eq_top : a ∆ b = ⊤ ↔ IsCompl a b := by
rw [symmDiff_eq', ← compl_inf, inf_eq_top_iff, compl_eq_top, isCompl_iff, disjoint_iff,
codisjoint_iff, and_comm]
@[simp]
theorem bihimp_eq_bot : a ⇔ b = ⊥ ↔ IsCompl a b := by
rw [bihimp_eq', ← compl_sup, sup_eq_bot_iff, compl_eq_bot, isCompl_iff, disjoint_iff,
codisjoint_iff]
@[simp]
theorem compl_symmDiff_self : aᶜ ∆ a = ⊤ :=
hnot_symmDiff_self _
@[simp]
theorem symmDiff_compl_self : a ∆ aᶜ = ⊤ :=
symmDiff_hnot_self _
theorem symmDiff_symmDiff_right' :
a ∆ (b ∆ c) = a ⊓ b ⊓ c ⊔ a ⊓ bᶜ ⊓ cᶜ ⊔ aᶜ ⊓ b ⊓ cᶜ ⊔ aᶜ ⊓ bᶜ ⊓ c :=
calc
a ∆ (b ∆ c) = a ⊓ (b ⊓ c ⊔ bᶜ ⊓ cᶜ) ⊔ (b ⊓ cᶜ ⊔ c ⊓ bᶜ) ⊓ aᶜ := by
{ rw [symmDiff_eq, compl_symmDiff, bihimp_eq', symmDiff_eq] }
_ = a ⊓ b ⊓ c ⊔ a ⊓ bᶜ ⊓ cᶜ ⊔ b ⊓ cᶜ ⊓ aᶜ ⊔ c ⊓ bᶜ ⊓ aᶜ := by
{ rw [inf_sup_left, inf_sup_right, ← sup_assoc, ← inf_assoc, ← inf_assoc] }
_ = a ⊓ b ⊓ c ⊔ a ⊓ bᶜ ⊓ cᶜ ⊔ aᶜ ⊓ b ⊓ cᶜ ⊔ aᶜ ⊓ bᶜ ⊓ c := (by
congr 1
· congr 1
rw [inf_comm, inf_assoc]
· apply inf_left_right_swap)
variable {a b c}
theorem Disjoint.le_symmDiff_sup_symmDiff_left (h : Disjoint a b) : c ≤ a ∆ c ⊔ b ∆ c := by
trans c \ (a ⊓ b)
· rw [h.eq_bot, sdiff_bot]
· rw [sdiff_inf]
exact sup_le_sup le_sup_right le_sup_right
theorem Disjoint.le_symmDiff_sup_symmDiff_right (h : Disjoint b c) : a ≤ a ∆ b ⊔ a ∆ c := by
simp_rw [symmDiff_comm a]
exact h.le_symmDiff_sup_symmDiff_left
theorem Codisjoint.bihimp_inf_bihimp_le_left (h : Codisjoint a b) : a ⇔ c ⊓ b ⇔ c ≤ c :=
h.dual.le_symmDiff_sup_symmDiff_left
theorem Codisjoint.bihimp_inf_bihimp_le_right (h : Codisjoint b c) : a ⇔ b ⊓ a ⇔ c ≤ a :=
h.dual.le_symmDiff_sup_symmDiff_right
end BooleanAlgebra
/-! ### Prod -/
section Prod
@[simp]
theorem symmDiff_fst [GeneralizedCoheytingAlgebra α] [GeneralizedCoheytingAlgebra β]
(a b : α × β) : (a ∆ b).1 = a.1 ∆ b.1 :=
rfl
@[simp]
theorem symmDiff_snd [GeneralizedCoheytingAlgebra α] [GeneralizedCoheytingAlgebra β]
(a b : α × β) : (a ∆ b).2 = a.2 ∆ b.2 :=
rfl
@[simp]
theorem bihimp_fst [GeneralizedHeytingAlgebra α] [GeneralizedHeytingAlgebra β] (a b : α × β) :
(a ⇔ b).1 = a.1 ⇔ b.1 :=
rfl
@[simp]
theorem bihimp_snd [GeneralizedHeytingAlgebra α] [GeneralizedHeytingAlgebra β] (a b : α × β) :
(a ⇔ b).2 = a.2 ⇔ b.2 :=
rfl
end Prod
/-! ### Pi -/
namespace Pi
theorem symmDiff_def [∀ i, GeneralizedCoheytingAlgebra (π i)] (a b : ∀ i, π i) :
a ∆ b = fun i => a i ∆ b i :=
rfl
theorem bihimp_def [∀ i, GeneralizedHeytingAlgebra (π i)] (a b : ∀ i, π i) :
a ⇔ b = fun i => a i ⇔ b i :=
rfl
@[simp]
theorem symmDiff_apply [∀ i, GeneralizedCoheytingAlgebra (π i)] (a b : ∀ i, π i) (i : ι) :
(a ∆ b) i = a i ∆ b i :=
rfl
@[simp]
theorem bihimp_apply [∀ i, GeneralizedHeytingAlgebra (π i)] (a b : ∀ i, π i) (i : ι) :
(a ⇔ b) i = a i ⇔ b i :=
rfl
end Pi
|
character.v
|
(* (c) Copyright 2006-2016 Microsoft Corporation and Inria. *)
(* Distributed under the terms of CeCILL-B. *)
From HB Require Import structures.
From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq.
From mathcomp Require Import path div fintype tuple finfun bigop prime order.
From mathcomp Require Import ssralg poly finset gproduct fingroup morphism.
From mathcomp Require Import perm automorphism quotient finalg action zmodp.
From mathcomp Require Import commutator cyclic center pgroup nilpotent sylow.
From mathcomp Require Import abelian matrix mxalgebra mxpoly mxrepresentation.
From mathcomp Require Import vector ssrnum algC classfun archimedean.
(******************************************************************************)
(* This file contains the basic notions of character theory, based on Isaacs. *)
(* irr G == tuple of the elements of 'CF(G) that are irreducible *)
(* characters of G. *)
(* Nirr G == number of irreducible characters of G. *)
(* Iirr G == index type for the irreducible characters of G. *)
(* := 'I_(Nirr G). *)
(* 'chi_i == the i-th element of irr G, for i : Iirr G. *)
(* 'chi[G]_i Note that 'chi_0 = 1, the principal character of G. *)
(* 'Chi_i == an irreducible representation that affords 'chi_i. *)
(* socle_of_Iirr i == the Wedderburn component of the regular representation *)
(* of G, corresponding to 'Chi_i. *)
(* Iirr_of_socle == the inverse of socle_of_Iirr (which is one-to-one). *)
(* phi.[A]%CF == the image of A \in group_ring G under phi : 'CF(G). *)
(* cfRepr rG == the character afforded by the representation rG of G. *)
(* cfReg G == the regular character, afforded by the regular *)
(* representation of G. *)
(* detRepr rG == the linear character afforded by the determinant of rG. *)
(* cfDet phi == the linear character afforded by the determinant of a *)
(* representation affording phi. *)
(* 'o(phi) == the "determinential order" of phi (the multiplicative *)
(* order of cfDet phi. *)
(* phi \is a character <=> phi : 'CF(G) is a character of G or 0. *)
(* i \in irr_constt phi <=> 'chi_i is an irreducible constituent of phi: phi *)
(* has a non-zero coordinate on 'chi_i over the basis irr G. *)
(* xi \is a linear_char xi <=> xi : 'CF(G) is a linear character of G. *)
(* 'Z(chi)%CF == the center of chi when chi is a character of G, i.e., *)
(* rcenter rG where rG is a representation that affords phi. *)
(* If phi is not a character then 'Z(chi)%CF = cfker phi. *)
(* aut_Iirr u i == the index of cfAut u 'chi_i in irr G. *)
(* conjC_Iirr i == the index of 'chi_i^*%CF in irr G. *)
(* morph_Iirr i == the index of cfMorph 'chi[f @* G]_i in irr G. *)
(* isom_Iirr isoG i == the index of cfIsom isoG 'chi[G]_i in irr R. *)
(* mod_Iirr i == the index of ('chi[G / H]_i %% H)%CF in irr G. *)
(* quo_Iirr i == the index of ('chi[G]_i / H)%CF in irr (G / H). *)
(* Ind_Iirr G i == the index of 'Ind[G, H] 'chi_i, provided it is an *)
(* irreducible character (such as when if H is the inertia *)
(* group of 'chi_i). *)
(* Res_Iirr H i == the index of 'Res[H, G] 'chi_i, provided it is an *)
(* irreducible character (such as when 'chi_i is linear). *)
(* sdprod_Iirr defG i == the index of cfSdprod defG 'chi_i in irr G, given *)
(* defG : K ><| H = G. *)
(* And, for KxK : K \x H = G. *)
(* dprodl_Iirr KxH i == the index of cfDprodl KxH 'chi[K]_i in irr G. *)
(* dprodr_Iirr KxH j == the index of cfDprodr KxH 'chi[H]_j in irr G. *)
(* dprod_Iirr KxH (i, j) == the index of cfDprod KxH 'chi[K]_i 'chi[H]_j. *)
(* inv_dprod_Iirr KxH == the inverse of dprod_Iirr KxH. *)
(* The following are used to define and exploit the character table: *)
(* character_table G == the character table of G, whose i-th row lists the *)
(* values taken by 'chi_i on the conjugacy classes *)
(* of G; this is a square Nirr G x NirrG matrix. *)
(* irr_class i == the conjugacy class of G with index i : Iirr G. *)
(* class_Iirr xG == the index of xG \in classes G, in Iirr G. *)
(******************************************************************************)
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Import Order.TTheory GroupScope GRing.Theory Num.Theory.
Local Open Scope ring_scope.
Section AlgC.
Variable (gT : finGroupType).
Lemma groupC : group_closure_field algC gT.
Proof. exact: group_closure_closed_field. Qed.
End AlgC.
Section Tensor.
Variable (F : fieldType).
Fixpoint trow (n1 : nat) :
forall (A : 'rV[F]_n1) m2 n2 (B : 'M[F]_(m2,n2)), 'M[F]_(m2,n1 * n2) :=
if n1 is n'1.+1
then
fun (A : 'M[F]_(1,(1 + n'1))) m2 n2 (B : 'M[F]_(m2,n2)) =>
(row_mx (lsubmx A 0 0 *: B) (trow (rsubmx A) B))
else (fun _ _ _ _ => 0).
Lemma trow0 n1 m2 n2 B : @trow n1 0 m2 n2 B = 0.
Proof.
elim: n1=> //= n1 IH.
rewrite !mxE scale0r linear0.
rewrite IH //; apply/matrixP=> i j; rewrite !mxE.
by case: split=> *; rewrite mxE.
Qed.
Definition trowb n1 m2 n2 B A := @trow n1 A m2 n2 B.
Lemma trowbE n1 m2 n2 A B : trowb B A = @trow n1 A m2 n2 B.
Proof. by []. Qed.
Lemma trowb_is_linear n1 m2 n2 (B : 'M_(m2,n2)) : linear (@trowb n1 m2 n2 B).
Proof.
elim: n1=> [|n1 IH] //= k A1 A2 /=; first by rewrite scaler0 add0r.
rewrite !linearD /= !linearZ /= IH 2!mxE.
by rewrite scalerDl -scalerA -add_row_mx -scale_row_mx.
Qed.
HB.instance Definition _ n1 m2 n2 B :=
GRing.isSemilinear.Build _ _ _ _ (trowb B)
(GRing.semilinear_linear (@trowb_is_linear n1 m2 n2 B)).
Lemma trow_is_linear n1 m2 n2 (A : 'rV_n1) : linear (@trow n1 A m2 n2).
Proof.
elim: n1 A => [|n1 IH] //= A k A1 A2 /=; first by rewrite scaler0 add0r.
rewrite linearP /=; apply/matrixP=> i j; rewrite !mxE.
by case: split=> a; rewrite ?IH !mxE.
Qed.
HB.instance Definition _ n1 m2 n2 A :=
GRing.isSemilinear.Build _ _ _ _ (@trow n1 A m2 n2)
(GRing.semilinear_linear (@trow_is_linear n1 m2 n2 A)).
Fixpoint tprod (m1 : nat) :
forall n1 (A : 'M[F]_(m1,n1)) m2 n2 (B : 'M[F]_(m2,n2)),
'M[F]_(m1 * m2,n1 * n2) :=
if m1 is m'1.+1
return forall n1 (A : 'M[F]_(m1,n1)) m2 n2 (B : 'M[F]_(m2,n2)),
'M[F]_(m1 * m2,n1 * n2)
then
fun n1 (A : 'M[F]_(1 + m'1,n1)) m2 n2 B =>
(col_mx (trow (usubmx A) B) (tprod (dsubmx A) B))
else (fun _ _ _ _ _ => 0).
Lemma dsumx_mul m1 m2 n p A B :
dsubmx ((A *m B) : 'M[F]_(m1 + m2, n)) = dsubmx (A : 'M_(m1 + m2, p)) *m B.
Proof.
apply/matrixP=> i j /[!mxE]; apply: eq_bigr=> k _.
by rewrite !mxE.
Qed.
Lemma usumx_mul m1 m2 n p A B :
usubmx ((A *m B) : 'M[F]_(m1 + m2, n)) = usubmx (A : 'M_(m1 + m2, p)) *m B.
Proof.
by apply/matrixP=> i j /[!mxE]; apply: eq_bigr=> k _ /[!mxE].
Qed.
Let trow_mul (m1 m2 n2 p2 : nat)
(A : 'rV_m1) (B1: 'M[F]_(m2,n2)) (B2 :'M[F]_(n2,p2)) :
trow A (B1 *m B2) = B1 *m trow A B2.
Proof.
elim: m1 A => [|m1 IH] A /=; first by rewrite mulmx0.
by rewrite IH mul_mx_row -scalemxAr.
Qed.
Lemma tprodE m1 n1 p1 (A1 :'M[F]_(m1,n1)) (A2 :'M[F]_(n1,p1))
m2 n2 p2 (B1 :'M[F]_(m2,n2)) (B2 :'M[F]_(n2,p2)) :
tprod (A1 *m A2) (B1 *m B2) = (tprod A1 B1) *m (tprod A2 B2).
Proof.
elim: m1 n1 p1 A1 A2 m2 n2 p2 B1 B2 => /= [|m1 IH].
by move=> *; rewrite mul0mx.
move=> n1 p1 A1 A2 m2 n2 p2 B1 B2.
rewrite mul_col_mx -IH.
congr col_mx; last by rewrite dsumx_mul.
rewrite usumx_mul.
elim: n1 {A1}(usubmx (A1: 'M_(1 + m1, n1))) p1 A2=> //= [u p1 A2|].
by rewrite [A2](flatmx0) !mulmx0 -trowbE linear0.
move=> n1 IH1 A p1 A2 //=.
set Al := lsubmx _; set Ar := rsubmx _.
set Su := usubmx _; set Sd := dsubmx _.
rewrite mul_row_col -IH1.
rewrite -{1}(@hsubmxK F 1 1 n1 A).
rewrite -{1}(@vsubmxK F 1 n1 p1 A2).
rewrite (@mul_row_col F 1 1 n1 p1).
rewrite -trowbE linearD /= trowbE -/Al.
congr (_ + _).
rewrite {1}[Al]mx11_scalar mul_scalar_mx.
by rewrite -trowbE linearZ /= trowbE -/Su trow_mul scalemxAl.
Qed.
Let tprod_tr m1 n1 (A :'M[F]_(m1, 1 + n1)) m2 n2 (B :'M[F]_(m2, n2)) :
tprod A B = row_mx (trow (lsubmx A)^T B^T)^T (tprod (rsubmx A) B).
Proof.
elim: m1 n1 A m2 n2 B=> [|m1 IH] n1 A m2 n2 B //=.
by rewrite trmx0 row_mx0.
rewrite !IH.
pose A1 := A : 'M_(1 + m1, 1 + n1).
have F1: dsubmx (rsubmx A1) = rsubmx (dsubmx A1).
by apply/matrixP=> i j; rewrite !mxE.
have F2: rsubmx (usubmx A1) = usubmx (rsubmx A1).
by apply/matrixP=> i j; rewrite !mxE.
have F3: lsubmx (dsubmx A1) = dsubmx (lsubmx A1).
by apply/matrixP=> i j; rewrite !mxE.
rewrite tr_row_mx -block_mxEv -block_mxEh !(F1,F2,F3); congr block_mx.
- by rewrite !mxE linearZ /= trmxK.
by rewrite -trmx_dsub.
Qed.
Lemma tprod1 m n : tprod (1%:M : 'M[F]_(m,m)) (1%:M : 'M[F]_(n,n)) = 1%:M.
Proof.
elim: m n => [|m IH] n //=; first by rewrite [1%:M]flatmx0.
rewrite tprod_tr.
set u := rsubmx _; have->: u = 0.
apply/matrixP=> i j; rewrite !mxE.
by case: i; case: j=> /= j Hj; case.
set v := lsubmx (dsubmx _); have->: v = 0.
apply/matrixP=> i j; rewrite !mxE.
by case: i; case: j; case.
set w := rsubmx _; have->: w = 1%:M.
apply/matrixP=> i j; rewrite !mxE.
by case: i; case: j; case.
rewrite IH -!trowbE !linear0.
rewrite -block_mxEv.
set z := (lsubmx _) 0 0; have->: z = 1.
by rewrite /z !mxE eqxx.
by rewrite scale1r scalar_mx_block.
Qed.
Lemma mxtrace_prod m n (A :'M[F]_(m)) (B :'M[F]_(n)) :
\tr (tprod A B) = \tr A * \tr B.
Proof.
elim: m n A B => [|m IH] n A B //=.
by rewrite [A]flatmx0 mxtrace0 mul0r.
rewrite tprod_tr -block_mxEv mxtrace_block IH.
rewrite linearZ/= -mulrDl -trace_mx11; congr (_ * _).
pose A1 := A : 'M_(1 + m).
rewrite -[A in RHS](@submxK _ 1 m 1 m A1).
by rewrite (@mxtrace_block _ _ _ (ulsubmx A1)).
Qed.
End Tensor.
(* Representation sigma type and standard representations. *)
Section StandardRepresentation.
Variables (R : fieldType) (gT : finGroupType) (G : {group gT}).
Local Notation reprG := (mx_representation R G).
Record representation :=
Representation {rdegree; mx_repr_of_repr :> reprG rdegree}.
Lemma mx_repr0 : mx_repr G (fun _ : gT => 1%:M : 'M[R]_0).
Proof. by split=> // g h Hg Hx; rewrite mulmx1. Qed.
Definition grepr0 := Representation (MxRepresentation mx_repr0).
Lemma add_mx_repr (rG1 rG2 : representation) :
mx_repr G (fun g => block_mx (rG1 g) 0 0 (rG2 g)).
Proof.
split=> [|x y Hx Hy]; first by rewrite !repr_mx1 -scalar_mx_block.
by rewrite mulmx_block !(mulmx0, mul0mx, addr0, add0r, repr_mxM).
Qed.
Definition dadd_grepr rG1 rG2 :=
Representation (MxRepresentation (add_mx_repr rG1 rG2)).
Section DsumRepr.
Variables (n : nat) (rG : reprG n).
Lemma mx_rsim_dadd (U V W : 'M_n) (rU rV : representation)
(modU : mxmodule rG U) (modV : mxmodule rG V) (modW : mxmodule rG W) :
(U + V :=: W)%MS -> mxdirect (U + V) ->
mx_rsim (submod_repr modU) rU -> mx_rsim (submod_repr modV) rV ->
mx_rsim (submod_repr modW) (dadd_grepr rU rV).
Proof.
case: rU; case: rV=> nV rV nU rU defW dxUV /=.
have tiUV := mxdirect_addsP dxUV.
move=> [fU def_nU]; rewrite -{nU}def_nU in rU fU * => inv_fU hom_fU.
move=> [fV def_nV]; rewrite -{nV}def_nV in rV fV * => inv_fV hom_fV.
pose pU := in_submod U (proj_mx U V) *m fU.
pose pV := in_submod V (proj_mx V U) *m fV.
exists (val_submod 1%:M *m row_mx pU pV) => [||g Gg].
- by rewrite -defW (mxdirectP dxUV).
- apply/row_freeP.
pose pU' := invmx fU *m val_submod 1%:M.
pose pV' := invmx fV *m val_submod 1%:M.
exists (in_submod _ (col_mx pU' pV')).
rewrite in_submodE mulmxA -in_submodE -mulmxA mul_row_col mulmxDr.
rewrite -[pU *m _]mulmxA -[pV *m _]mulmxA !mulKVmx -?row_free_unit //.
rewrite addrC (in_submodE V) 2![val_submod 1%:M *m _]mulmxA -in_submodE.
rewrite addrC (in_submodE U) 2![val_submod 1%:M *m _ in X in X + _]mulmxA.
rewrite -in_submodE -!val_submodE !in_submodK ?proj_mx_sub //.
by rewrite add_proj_mx ?val_submodK // val_submod1 defW.
rewrite mulmxA -val_submodE -[submod_repr _ g]mul1mx val_submodJ //.
rewrite -(mulmxA _ (rG g)) mul_mx_row -[in RHS]mulmxA mul_row_block.
rewrite !mulmx0 addr0 add0r !mul_mx_row.
set W' := val_submod 1%:M; congr (row_mx _ _).
rewrite 3!mulmxA in_submodE mulmxA.
have hom_pU: (W' <= dom_hom_mx rG (proj_mx U V))%MS.
by rewrite val_submod1 -defW proj_mx_hom.
rewrite (hom_mxP hom_pU) // -in_submodE (in_submodJ modU) ?proj_mx_sub //.
rewrite -(mulmxA _ _ fU) hom_fU // in_submodE -2!(mulmxA W') -in_submodE.
by rewrite -mulmxA (mulmxA _ fU).
rewrite 3!mulmxA in_submodE mulmxA.
have hom_pV: (W' <= dom_hom_mx rG (proj_mx V U))%MS.
by rewrite val_submod1 -defW addsmxC proj_mx_hom // capmxC.
rewrite (hom_mxP hom_pV) // -in_submodE (in_submodJ modV) ?proj_mx_sub //.
rewrite -(mulmxA _ _ fV) hom_fV // in_submodE -2!(mulmxA W') -in_submodE.
by rewrite -mulmxA (mulmxA _ fV).
Qed.
Lemma mx_rsim_dsum (I : finType) (P : pred I) U rU (W : 'M_n)
(modU : forall i, mxmodule rG (U i)) (modW : mxmodule rG W) :
let S := (\sum_(i | P i) U i)%MS in (S :=: W)%MS -> mxdirect S ->
(forall i, mx_rsim (submod_repr (modU i)) (rU i : representation)) ->
mx_rsim (submod_repr modW) (\big[dadd_grepr/grepr0]_(i | P i) rU i).
Proof.
move=> /= defW dxW rsimU.
rewrite mxdirectE /= -!(big_filter _ P) in dxW defW *.
elim: {P}(filter P _) => [|i e IHe] in W modW dxW defW *.
rewrite !big_nil /= in defW *.
by exists 0 => [||? _]; rewrite ?mul0mx ?mulmx0 // /row_free -defW !mxrank0.
rewrite !big_cons /= in dxW defW *.
rewrite 2!(big_nth i) !big_mkord /= in IHe dxW defW.
set Wi := (\sum_i _)%MS in defW dxW IHe.
rewrite -mxdirectE mxdirect_addsE !mxdirectE eqxx /= -/Wi in dxW.
have modWi: mxmodule rG Wi by apply: sumsmx_module.
case/andP: dxW; move/(IHe Wi modWi) {IHe}; move/(_ (eqmx_refl _))=> rsimWi.
by move/eqP; move/mxdirect_addsP=> dxUiWi; apply: mx_rsim_dadd (rsimU i) rsimWi.
Qed.
Definition muln_grepr rW k := \big[dadd_grepr/grepr0]_(i < k) rW.
Lemma mx_rsim_socle (sG : socleType rG) (W : sG) (rW : representation) :
let modW : mxmodule rG W := component_mx_module rG (socle_base W) in
mx_rsim (socle_repr W) rW ->
mx_rsim (submod_repr modW) (muln_grepr rW (socle_mult W)).
Proof.
set M := socle_base W => modW rsimM.
have simM: mxsimple rG M := socle_simple W.
have rankM_gt0: (\rank M > 0)%N by rewrite lt0n mxrank_eq0; case: simM.
have [I /= U_I simU]: mxsemisimple rG W by apply: component_mx_semisimple.
pose U (i : 'I_#|I|) := U_I (enum_val i).
have reindexI := reindex _ (onW_bij I (enum_val_bij I)).
rewrite mxdirectE /= !reindexI -mxdirectE /= => defW dxW.
have isoU: forall i, mx_iso rG M (U i).
move=> i; have sUiW: (U i <= W)%MS by rewrite -defW (sumsmx_sup i).
exact: component_mx_iso (simU _) sUiW.
have ->: socle_mult W = #|I|.
rewrite -(mulnK #|I| rankM_gt0); congr (_ %/ _)%N.
rewrite -defW (mxdirectP dxW) /= -sum_nat_const reindexI /=.
by apply: eq_bigr => i _; rewrite -(mxrank_iso (isoU i)).
have modU: mxmodule rG (U _) := mxsimple_module (simU _).
suff: mx_rsim (submod_repr (modU _)) rW by apply: mx_rsim_dsum defW dxW.
by move=> i; apply: mx_rsim_trans (mx_rsim_sym _) rsimM; apply/mx_rsim_iso.
Qed.
End DsumRepr.
Section ProdRepr.
Variables (n1 n2 : nat) (rG1 : reprG n1) (rG2 : reprG n2).
Lemma prod_mx_repr : mx_repr G (fun g => tprod (rG1 g) (rG2 g)).
Proof.
split=>[|i j InG JnG]; first by rewrite !repr_mx1 tprod1.
by rewrite !repr_mxM // tprodE.
Qed.
Definition prod_repr := MxRepresentation prod_mx_repr.
End ProdRepr.
Lemma prod_repr_lin n2 (rG1 : reprG 1) (rG2 : reprG n2) :
{in G, forall x, let cast_n2 := esym (mul1n n2) in
prod_repr rG1 rG2 x = castmx (cast_n2, cast_n2) (rG1 x 0 0 *: rG2 x)}.
Proof.
move=> x Gx /=; set cast_n2 := esym _; rewrite /prod_repr /= !mxE !lshift0.
apply/matrixP=> i j; rewrite castmxE /=.
do 2![rewrite mxE; case: splitP => [? ? | []//]].
by congr ((_ *: rG2 x) _ _); apply: val_inj.
Qed.
End StandardRepresentation.
Arguments grepr0 {R gT G}.
Prenex Implicits dadd_grepr.
Section Char.
Variables (gT : finGroupType) (G : {group gT}).
Fact cfRepr_subproof n (rG : mx_representation algC G n) :
is_class_fun <<G>> [ffun x => \tr (rG x) *+ (x \in G)].
Proof.
rewrite genGid; apply: intro_class_fun => [x y Gx Gy | _ /negbTE-> //].
by rewrite groupJr // !repr_mxM ?groupM ?groupV // mxtrace_mulC repr_mxK.
Qed.
Definition cfRepr n rG := Cfun 0 (@cfRepr_subproof n rG).
Lemma cfRepr1 n rG : @cfRepr n rG 1%g = n%:R.
Proof. by rewrite cfunE group1 repr_mx1 mxtrace1. Qed.
Lemma cfRepr_sim n1 n2 rG1 rG2 :
mx_rsim rG1 rG2 -> @cfRepr n1 rG1 = @cfRepr n2 rG2.
Proof.
case/mx_rsim_def=> f12 [f21] fK def_rG1; apply/cfun_inP=> x Gx.
by rewrite !cfunE def_rG1 // mxtrace_mulC mulmxA fK mul1mx.
Qed.
Lemma cfRepr0 : cfRepr grepr0 = 0.
Proof. by apply/cfun_inP=> x Gx; rewrite !cfunE Gx mxtrace1. Qed.
Lemma cfRepr_dadd rG1 rG2 :
cfRepr (dadd_grepr rG1 rG2) = cfRepr rG1 + cfRepr rG2.
Proof. by apply/cfun_inP=> x Gx; rewrite !cfunE Gx mxtrace_block. Qed.
Lemma cfRepr_dsum I r (P : pred I) rG :
cfRepr (\big[dadd_grepr/grepr0]_(i <- r | P i) rG i)
= \sum_(i <- r | P i) cfRepr (rG i).
Proof. exact: (big_morph _ cfRepr_dadd cfRepr0). Qed.
Lemma cfRepr_muln rG k : cfRepr (muln_grepr rG k) = cfRepr rG *+ k.
Proof. by rewrite cfRepr_dsum /= sumr_const card_ord. Qed.
Section StandardRepr.
Variables (n : nat) (rG : mx_representation algC G n).
Let sG := DecSocleType rG.
Let iG : irrType algC G := DecSocleType _.
Definition standard_irr (W : sG) := irr_comp iG (socle_repr W).
Definition standard_socle i := pick [pred W | standard_irr W == i].
Local Notation soc := standard_socle.
Definition standard_irr_coef i := oapp (fun W => socle_mult W) 0 (soc i).
Definition standard_grepr :=
\big[dadd_grepr/grepr0]_i
muln_grepr (Representation (socle_repr i)) (standard_irr_coef i).
Lemma mx_rsim_standard : mx_rsim rG standard_grepr.
Proof.
pose W i := oapp val 0 (soc i); pose S := (\sum_i W i)%MS.
have C'G: [pchar algC]^'.-group G := algC'G_pchar G.
have [defS dxS]: (S :=: 1%:M)%MS /\ mxdirect S.
rewrite /S mxdirectE /= !(bigID soc xpredT) /=.
rewrite addsmxC big1 => [|i]; last by rewrite /W; case (soc i).
rewrite adds0mx_id addnC (@big1 nat) ?add0n => [|i]; last first.
by rewrite /W; case: (soc i); rewrite ?mxrank0.
have <-: Socle sG = 1%:M := reducible_Socle1 sG (mx_Maschke_pchar rG C'G).
have [W0 _ | noW] := pickP sG; last first.
suff no_i: (soc : pred iG) =1 xpred0 by rewrite /Socle !big_pred0 ?mxrank0.
by move=> i; rewrite /soc; case: pickP => // W0; have:= noW W0.
have irrK Wi: soc (standard_irr Wi) = Some Wi.
rewrite /soc; case: pickP => [W' | /(_ Wi)] /= /eqP // eqWi.
apply/eqP/socle_rsimP.
apply: mx_rsim_trans
(rsim_irr_comp_pchar iG C'G (socle_irr _)) (mx_rsim_sym _).
by rewrite [irr_comp _ _]eqWi; apply: rsim_irr_comp_pchar (socle_irr _).
have bij_irr: {on [pred i | soc i], bijective standard_irr}.
exists (odflt W0 \o soc) => [Wi _ | i]; first by rewrite /= irrK.
by rewrite inE /soc /=; case: pickP => //= Wi; move/eqP.
rewrite !(reindex standard_irr) {bij_irr}//=.
have all_soc Wi: soc (standard_irr Wi) by rewrite irrK.
rewrite (eq_bigr val) => [|Wi _]; last by rewrite /W irrK.
rewrite !(eq_bigl _ _ all_soc); split=> //.
rewrite (eq_bigr (mxrank \o val)) => [|Wi _]; last by rewrite /W irrK.
by rewrite -mxdirectE /= Socle_direct.
pose modW i : mxmodule rG (W i) :=
if soc i is Some Wi as oWi return mxmodule rG (oapp val 0 oWi) then
component_mx_module rG (socle_base Wi)
else mxmodule0 rG n.
apply: mx_rsim_trans (mx_rsim_sym (rsim_submod1 (mxmodule1 rG) _)) _ => //.
apply: mx_rsim_dsum (modW) _ defS dxS _ => i.
rewrite /W /standard_irr_coef /modW /soc; case: pickP => [Wi|_] /=; last first.
rewrite /muln_grepr big_ord0.
by exists 0 => [||x _]; rewrite /row_free ?mxrank0 ?mulmx0 ?mul0mx.
move/eqP=> <-; apply: mx_rsim_socle.
exact: rsim_irr_comp_pchar (socle_irr Wi).
Qed.
End StandardRepr.
Definition cfReg (B : {set gT}) : 'CF(B) := #|B|%:R *: '1_[1].
Lemma cfRegE x : @cfReg G x = #|G|%:R *+ (x == 1%g).
Proof. by rewrite cfunE cfuniE ?normal1 // inE mulr_natr. Qed.
(* This is Isaacs, Lemma (2.10). *)
Lemma cfReprReg : cfRepr (regular_repr algC G) = cfReg G.
Proof.
apply/cfun_inP=> x Gx; rewrite cfRegE.
have [-> | ntx] := eqVneq x 1%g; first by rewrite cfRepr1.
rewrite cfunE Gx [\tr _]big1 // => i _; rewrite 2!mxE /=.
rewrite -(inj_eq enum_val_inj) gring_indexK ?groupM ?enum_valP //.
by rewrite eq_mulVg1 mulKg (negbTE ntx).
Qed.
Definition xcfun (chi : 'CF(G)) A :=
(gring_row A *m (\col_(i < #|G|) chi (enum_val i))) 0 0.
Lemma xcfun_is_zmod_morphism phi : zmod_morphism (xcfun phi).
Proof. by move=> A B; rewrite /xcfun [gring_row _]linearB mulmxBl !mxE. Qed.
#[warning="-deprecated-since-mathcomp-2.5.0", deprecated(since="mathcomp 2.5.0",
note="use `xcfun_is_zmod_morphism` instead")]
Definition xcfun_is_additive := xcfun_is_zmod_morphism.
HB.instance Definition _ phi :=
GRing.isZmodMorphism.Build 'M_(gcard G) _ (xcfun phi) (xcfun_is_zmod_morphism phi).
Lemma xcfunZr a phi A : xcfun phi (a *: A) = a * xcfun phi A.
Proof. by rewrite /xcfun linearZ -scalemxAl mxE. Qed.
(* In order to add a second canonical structure on xcfun *)
Definition xcfun_r A phi := xcfun phi A.
Arguments xcfun_r A phi /.
Lemma xcfun_rE A chi : xcfun_r A chi = xcfun chi A. Proof. by []. Qed.
Fact xcfun_r_is_zmod_morphism A : zmod_morphism (xcfun_r A).
Proof.
move=> phi psi; rewrite /= /xcfun !mxE -sumrB; apply: eq_bigr => i _.
by rewrite !mxE !cfunE mulrBr.
Qed.
#[warning="-deprecated-since-mathcomp-2.5.0", deprecated(since="mathcomp 2.5.0",
note="use `xcfun_r_is_zmod_morphism` instead")]
Definition xcfun_r_is_additive := xcfun_r_is_zmod_morphism.
HB.instance Definition _ A := GRing.isZmodMorphism.Build _ _ (xcfun_r A)
(xcfun_r_is_zmod_morphism A).
Lemma xcfunZl a phi A : xcfun (a *: phi) A = a * xcfun phi A.
Proof.
rewrite /xcfun !mxE big_distrr; apply: eq_bigr => i _ /=.
by rewrite !mxE cfunE mulrCA.
Qed.
Lemma xcfun_repr n rG A : xcfun (@cfRepr n rG) A = \tr (gring_op rG A).
Proof.
rewrite gring_opE [gring_row A]row_sum_delta !linear_sum /xcfun !mxE.
apply: eq_bigr => i _; rewrite !mxE /= !linearZ cfunE enum_valP /=.
by congr (_ * \tr _); rewrite {A}/gring_mx /= -rowE rowK mxvecK.
Qed.
End Char.
Arguments xcfun_r {_ _} A phi /.
Notation "phi .[ A ]" := (xcfun phi A) : cfun_scope.
Definition pred_Nirr gT B := #|@classes gT B|.-1.
Arguments pred_Nirr {gT} B%_g.
Notation Nirr G := (pred_Nirr G).+1.
Notation Iirr G := 'I_(Nirr G).
Section IrrClassDef.
Variables (gT : finGroupType) (G : {group gT}).
Let sG := DecSocleType (regular_repr algC G).
Lemma NirrE : Nirr G = #|classes G|.
Proof. by rewrite /pred_Nirr (cardD1 [1]) classes1. Qed.
Fact Iirr_cast : Nirr G = #|sG|.
Proof. by rewrite NirrE ?card_irr_pchar ?algC'G_pchar //; apply: groupC. Qed.
Let offset := cast_ord (esym Iirr_cast) (enum_rank [1 sG]%irr).
Definition socle_of_Iirr (i : Iirr G) : sG :=
enum_val (cast_ord Iirr_cast (i + offset)).
Definition irr_of_socle (Wi : sG) : Iirr G :=
cast_ord (esym Iirr_cast) (enum_rank Wi) - offset.
Local Notation W := socle_of_Iirr.
Lemma socle_Iirr0 : W 0 = [1 sG]%irr.
Proof. by rewrite /W add0r cast_ordKV enum_rankK. Qed.
Lemma socle_of_IirrK : cancel W irr_of_socle.
Proof. by move=> i; rewrite /irr_of_socle enum_valK cast_ordK addrK. Qed.
Lemma irr_of_socleK : cancel irr_of_socle W.
Proof. by move=> Wi; rewrite /W subrK cast_ordKV enum_rankK. Qed.
Hint Resolve socle_of_IirrK irr_of_socleK : core.
Lemma irr_of_socle_bij (A : {pred (Iirr G)}) : {on A, bijective irr_of_socle}.
Proof. by apply: onW_bij; exists W. Qed.
Lemma socle_of_Iirr_bij (A : {pred sG}) : {on A, bijective W}.
Proof. by apply: onW_bij; exists irr_of_socle. Qed.
End IrrClassDef.
Prenex Implicits socle_of_IirrK irr_of_socleK.
Arguments socle_of_Iirr {gT G%_G} i%_R.
Notation "''Chi_' i" := (irr_repr (socle_of_Iirr i))
(at level 8, i at level 2, format "''Chi_' i").
HB.lock Definition irr gT B : (Nirr B).-tuple 'CF(B) :=
let irr_of i := 'Res[B, <<B>>] (@cfRepr gT _ _ 'Chi_(inord i)) in
[tuple of mkseq irr_of (Nirr B)].
Arguments irr {gT} B%_g.
Notation "''chi_' i" := (tnth (irr _) i%R)
(at level 8, i at level 2, format "''chi_' i") : ring_scope.
Notation "''chi[' G ]_ i" := (tnth (irr G) i%R)
(at level 8, i at level 2, only parsing) : ring_scope.
Section IrrClass.
Variable (gT : finGroupType) (G : {group gT}).
Implicit Types (i : Iirr G) (B : {set gT}).
Open Scope group_ring_scope.
Lemma congr_irr i1 i2 : i1 = i2 -> 'chi_i1 = 'chi_i2. Proof. by move->. Qed.
Lemma Iirr1_neq0 : G :!=: 1%g -> inord 1 != 0 :> Iirr G.
Proof. by rewrite -classes_gt1 -NirrE -val_eqE /= => /inordK->. Qed.
Lemma has_nonprincipal_irr : G :!=: 1%g -> {i : Iirr G | i != 0}.
Proof. by move/Iirr1_neq0; exists (inord 1). Qed.
Lemma irrRepr i : cfRepr 'Chi_i = 'chi_i.
Proof.
rewrite irr.unlock (tnth_nth 0) nth_mkseq // -[<<G>>]/(gval _) genGidG.
by rewrite cfRes_id inord_val.
Qed.
Lemma irr0 : 'chi[G]_0 = 1.
Proof.
apply/cfun_inP=> x Gx; rewrite -irrRepr cfun1E cfunE Gx.
by rewrite socle_Iirr0 irr1_repr // mxtrace1 degree_irr1.
Qed.
Lemma cfun1_irr : 1 \in irr G.
Proof. by rewrite -irr0 mem_tnth. Qed.
Lemma mem_irr i : 'chi_i \in irr G.
Proof. exact: mem_tnth. Qed.
Lemma irrP xi : reflect (exists i, xi = 'chi_i) (xi \in irr G).
Proof.
apply: (iffP idP) => [/(nthP 0)[i] | [i ->]]; last exact: mem_irr.
rewrite size_tuple => lt_i_G <-.
by exists (Ordinal lt_i_G); rewrite (tnth_nth 0).
Qed.
Let sG := DecSocleType (regular_repr algC G).
Let C'G := algC'G_pchar G.
Let closG := @groupC _ G.
Local Notation W i := (@socle_of_Iirr _ G i).
Local Notation "''n_' i" := 'n_(W i).
Local Notation "''R_' i" := 'R_(W i).
Local Notation "''e_' i" := 'e_(W i).
Lemma irr1_degree i : 'chi_i 1%g = ('n_i)%:R.
Proof. by rewrite -irrRepr cfRepr1. Qed.
Lemma Cnat_irr1 i : 'chi_i 1%g \in Num.nat.
Proof. by rewrite irr1_degree rpred_nat. Qed.
Lemma irr1_gt0 i : 0 < 'chi_i 1%g.
Proof. by rewrite irr1_degree ltr0n irr_degree_gt0. Qed.
Lemma irr1_neq0 i : 'chi_i 1%g != 0.
Proof. by rewrite eq_le lt_geF ?irr1_gt0. Qed.
Lemma irr_neq0 i : 'chi_i != 0.
Proof. by apply: contraNneq (irr1_neq0 i) => ->; rewrite cfunE. Qed.
Local Remark cfIirr_key : unit. Proof. by []. Qed.
Definition cfIirr : forall B, 'CF(B) -> Iirr B :=
locked_with cfIirr_key (fun B chi => inord (index chi (irr B))).
Lemma cfIirrE chi : chi \in irr G -> 'chi_(cfIirr chi) = chi.
Proof.
move=> chi_irr; rewrite (tnth_nth 0) [cfIirr]unlock inordK ?nth_index //.
by rewrite -index_mem size_tuple in chi_irr.
Qed.
Lemma cfIirrPE J (f : J -> 'CF(G)) (P : pred J) :
(forall j, P j -> f j \in irr G) ->
forall j, P j -> 'chi_(cfIirr (f j)) = f j.
Proof. by move=> irr_f j /irr_f; apply: cfIirrE. Qed.
(* This is Isaacs, Corollary (2.7). *)
Corollary irr_sum_square : \sum_i ('chi[G]_i 1%g) ^+ 2 = #|G|%:R.
Proof.
rewrite -(sum_irr_degree_pchar sG) // natr_sum.
rewrite (reindex _ (socle_of_Iirr_bij _)) /=.
by apply: eq_bigr => i _; rewrite irr1_degree natrX.
Qed.
(* This is Isaacs, Lemma (2.11). *)
Lemma cfReg_sum : cfReg G = \sum_i 'chi_i 1%g *: 'chi_i.
Proof.
apply/cfun_inP=> x Gx.
rewrite -cfReprReg cfunE Gx (mxtrace_regular_pchar sG) //=.
rewrite sum_cfunE (reindex _ (socle_of_Iirr_bij _)); apply: eq_bigr => i _.
by rewrite -irrRepr cfRepr1 !cfunE Gx mulr_natl.
Qed.
Let aG := regular_repr algC G.
Let R_G := group_ring algC G.
Lemma xcfun_annihilate i j A : i != j -> (A \in 'R_j)%MS -> ('chi_i).[A]%CF = 0.
Proof.
move=> neq_ij RjA; rewrite -irrRepr xcfun_repr.
rewrite (irr_repr'_op0_pchar _ _ RjA) ?raddf0 //.
by rewrite eq_sym (can_eq socle_of_IirrK).
Qed.
Lemma xcfunG phi x : x \in G -> phi.[aG x]%CF = phi x.
Proof.
by move=> Gx; rewrite /xcfun /gring_row rowK -rowE !mxE !(gring_indexK, mul1g).
Qed.
Lemma xcfun_mul_id i A :
(A \in R_G)%MS -> ('chi_i).['e_i *m A]%CF = ('chi_i).[A]%CF.
Proof.
move=> RG_A; rewrite -irrRepr !xcfun_repr gring_opM //.
by rewrite op_Wedderburn_id_pchar ?mul1mx.
Qed.
Lemma xcfun_id i j : ('chi_i).['e_j]%CF = 'chi_i 1%g *+ (i == j).
Proof.
have [<-{j} | /xcfun_annihilate->//] := eqVneq; last exact: Wedderburn_id_mem.
by rewrite -xcfunG // repr_mx1 -(xcfun_mul_id _ (envelop_mx1 _)) mulmx1.
Qed.
Lemma irr_free : free (irr G).
Proof.
apply/freeP=> s s0 i; apply: (mulIf (irr1_neq0 i)).
rewrite mul0r -(raddf0 (xcfun_r 'e_i)) -{}s0 raddf_sum /=.
rewrite (bigD1 i)//= -tnth_nth xcfunZl xcfun_id eqxx big1 ?addr0 // => j ne_ji.
by rewrite -tnth_nth xcfunZl xcfun_id (negbTE ne_ji) mulr0.
Qed.
Lemma irr_inj : injective (tnth (irr G)).
Proof. by apply/injectiveP/free_uniq; rewrite map_tnth_enum irr_free. Qed.
Lemma irrK : cancel (tnth (irr G)) (@cfIirr G).
Proof. by move=> i; apply: irr_inj; rewrite cfIirrE ?mem_irr. Qed.
Lemma irr_eq1 i : ('chi_i == 1) = (i == 0).
Proof. by rewrite -irr0 (inj_eq irr_inj). Qed.
Lemma cforder_irr_eq1 i : (#['chi_i]%CF == 1) = (i == 0).
Proof. by rewrite -dvdn1 dvdn_cforder irr_eq1. Qed.
Lemma irr_basis : basis_of 'CF(G)%VS (irr G).
Proof.
rewrite /basis_of irr_free andbT -dimv_leqif_eq ?subvf //.
by rewrite dim_cfun (eqnP irr_free) size_tuple NirrE.
Qed.
Lemma eq_sum_nth_irr a : \sum_i a i *: 'chi[G]_i = \sum_i a i *: (irr G)`_i.
Proof. by apply: eq_bigr => i; rewrite -tnth_nth. Qed.
(* This is Isaacs, Theorem (2.8). *)
Theorem cfun_irr_sum phi : {a | phi = \sum_i a i *: 'chi[G]_i}.
Proof.
rewrite (coord_basis irr_basis (memvf phi)) -eq_sum_nth_irr.
by exists ((coord (irr G))^~ phi).
Qed.
Lemma cfRepr_standard n (rG : mx_representation algC G n) :
cfRepr (standard_grepr rG)
= \sum_i (standard_irr_coef rG (W i))%:R *: 'chi_i.
Proof.
rewrite cfRepr_dsum (reindex _ (socle_of_Iirr_bij _)).
by apply: eq_bigr => i _; rewrite scaler_nat cfRepr_muln irrRepr.
Qed.
Lemma cfRepr_inj n1 n2 rG1 rG2 :
@cfRepr _ G n1 rG1 = @cfRepr _ G n2 rG2 -> mx_rsim rG1 rG2.
Proof.
move=> eq_repr12; pose c i : algC := (standard_irr_coef _ (W i))%:R.
have [rsim1 rsim2] := (mx_rsim_standard rG1, mx_rsim_standard rG2).
apply: mx_rsim_trans (rsim1) (mx_rsim_sym _).
suffices ->: standard_grepr rG1 = standard_grepr rG2 by [].
apply: eq_bigr => Wi _; congr (muln_grepr _ _); apply/eqP; rewrite -eqC_nat.
rewrite -[Wi]irr_of_socleK -!/(c _ _ _) -!(coord_sum_free (c _ _) _ irr_free).
rewrite -!eq_sum_nth_irr -!cfRepr_standard.
by rewrite -(cfRepr_sim rsim1) -(cfRepr_sim rsim2) eq_repr12.
Qed.
Lemma cfRepr_rsimP n1 n2 rG1 rG2 :
reflect (mx_rsim rG1 rG2) (@cfRepr _ G n1 rG1 == @cfRepr _ G n2 rG2).
Proof. by apply: (iffP eqP) => [/cfRepr_inj | /cfRepr_sim]. Qed.
Lemma irr_reprP xi :
reflect (exists2 rG : representation _ G, mx_irreducible rG & xi = cfRepr rG)
(xi \in irr G).
Proof.
apply: (iffP (irrP xi)) => [[i ->] | [[n rG] irr_rG ->]].
by exists (Representation 'Chi_i); [apply: socle_irr | rewrite irrRepr].
exists (irr_of_socle (irr_comp sG rG)); rewrite -irrRepr irr_of_socleK /=.
exact/cfRepr_sim/rsim_irr_comp_pchar.
Qed.
(* This is Isaacs, Theorem (2.12). *)
Lemma Wedderburn_id_expansion i :
'e_i = #|G|%:R^-1 *: (\sum_(x in G) 'chi_i 1%g * 'chi_i x^-1%g *: aG x).
Proof.
have Rei: ('e_i \in 'R_i)%MS by apply: Wedderburn_id_mem.
have /envelop_mxP[a def_e]: ('e_i \in R_G)%MS; last rewrite -/aG in def_e.
by move: Rei; rewrite genmxE mem_sub_gring => /andP[].
apply: canRL (scalerK (neq0CG _)) _; rewrite def_e linear_sum /=.
apply: eq_bigr => x Gx; have Gx' := groupVr Gx; rewrite scalerA; congr (_ *: _).
transitivity (cfReg G).['e_i *m aG x^-1%g]%CF.
rewrite def_e mulmx_suml raddf_sum (bigD1 x) //= -scalemxAl xcfunZr.
rewrite -repr_mxM // mulgV xcfunG // cfRegE eqxx mulrC big1 ?addr0 //.
move=> y /andP[Gy /negbTE neq_xy]; rewrite -scalemxAl xcfunZr -repr_mxM //.
by rewrite xcfunG ?groupM // cfRegE -eq_mulgV1 neq_xy mulr0.
rewrite cfReg_sum -xcfun_rE raddf_sum /= (bigD1 i) //= xcfunZl.
rewrite xcfun_mul_id ?envelop_mx_id ?xcfunG ?groupV ?big1 ?addr0 // => j ne_ji.
rewrite xcfunZl (xcfun_annihilate ne_ji) ?mulr0 //.
have /andP[_ /(submx_trans _)-> //] := Wedderburn_ideal (W i).
by rewrite mem_mulsmx // envelop_mx_id ?groupV.
Qed.
End IrrClass.
Arguments cfReg {gT} B%_g.
Prenex Implicits cfIirr irrK.
Arguments irrP {gT G xi}.
Arguments irr_reprP {gT G xi}.
Arguments irr_inj {gT G} [x1 x2].
Section IsChar.
Variable gT : finGroupType.
Definition character_pred {G : {set gT}} :=
fun phi : 'CF(G) => [forall i, coord (irr G) i phi \in Num.nat].
Arguments character_pred _ _ /.
Definition character {G : {set gT}} := [qualify a phi | @character_pred G phi].
Variable G : {group gT}.
Implicit Types (phi chi xi : 'CF(G)) (i : Iirr G).
Lemma irr_char i : 'chi_i \is a character.
Proof. by apply/forallP=> j; rewrite (tnth_nth 0) coord_free ?irr_free. Qed.
Lemma cfun1_char : (1 : 'CF(G)) \is a character.
Proof. by rewrite -irr0 irr_char. Qed.
Lemma cfun0_char : (0 : 'CF(G)) \is a character.
Proof. by apply/forallP=> i; rewrite linear0 rpred0. Qed.
Fact add_char : addr_closed (@character G).
Proof.
split=> [|chi xi /forallP-Nchi /forallP-Nxi]; first exact: cfun0_char.
by apply/forallP=> i; rewrite linearD rpredD /=.
Qed.
HB.instance Definition _ := GRing.isAddClosed.Build (classfun G) character_pred
add_char.
Lemma char_sum_irrP {phi} :
reflect (exists n, phi = \sum_i (n i)%:R *: 'chi_i) (phi \is a character).
Proof.
apply: (iffP idP)=> [/forallP-Nphi | [n ->]]; last first.
by apply: rpred_sum => i _; rewrite scaler_nat rpredMn // irr_char.
do [have [a ->] := cfun_irr_sum phi] in Nphi *; exists (Num.truncn \o a).
apply: eq_bigr => i _; congr (_ *: _); have:= eqP (Nphi i).
by rewrite eq_sum_nth_irr coord_sum_free ?irr_free.
Qed.
Lemma char_sum_irr chi :
chi \is a character -> {r | chi = \sum_(i <- r) 'chi_i}.
Proof.
move=> Nchi; apply: sig_eqW; case/char_sum_irrP: Nchi => n {chi}->.
elim/big_rec: _ => [|i _ _ [r ->]]; first by exists nil; rewrite big_nil.
exists (ncons (n i) i r); rewrite scaler_nat.
by elim: {n}(n i) => [|n IHn]; rewrite ?add0r //= big_cons mulrS -addrA IHn.
Qed.
Lemma Cnat_char1 chi : chi \is a character -> chi 1%g \in Num.nat.
Proof.
case/char_sum_irr=> r ->{chi}.
by elim/big_rec: _ => [|i chi _ Nchi1]; rewrite cfunE ?rpredD // Cnat_irr1.
Qed.
Lemma char1_ge0 chi : chi \is a character -> 0 <= chi 1%g.
Proof. by move/Cnat_char1/natr_ge0. Qed.
Lemma char1_eq0 chi : chi \is a character -> (chi 1%g == 0) = (chi == 0).
Proof.
case/char_sum_irr=> r ->; apply/idP/idP=> [|/eqP->]; last by rewrite cfunE.
case: r => [|i r]; rewrite ?big_nil // sum_cfunE big_cons.
rewrite paddr_eq0 ?sumr_ge0 => // [||j _]; rewrite 1?ltW ?irr1_gt0 //.
by rewrite (negbTE (irr1_neq0 i)).
Qed.
Lemma char1_gt0 chi : chi \is a character -> (0 < chi 1%g) = (chi != 0).
Proof. by move=> Nchi; rewrite -char1_eq0 // natr_gt0 ?Cnat_char1. Qed.
Lemma char_reprP phi :
reflect (exists rG : representation algC G, phi = cfRepr rG)
(phi \is a character).
Proof.
apply: (iffP char_sum_irrP) => [[n ->] | [[n rG] ->]]; last first.
exists (fun i => standard_irr_coef rG (socle_of_Iirr i)).
by rewrite -cfRepr_standard (cfRepr_sim (mx_rsim_standard rG)).
exists (\big[dadd_grepr/grepr0]_i muln_grepr (Representation 'Chi_i) (n i)).
rewrite cfRepr_dsum; apply: eq_bigr => i _.
by rewrite cfRepr_muln irrRepr scaler_nat.
Qed.
Local Notation reprG := (mx_representation algC G).
Lemma cfRepr_char n (rG : reprG n) : cfRepr rG \is a character.
Proof. by apply/char_reprP; exists (Representation rG). Qed.
Lemma cfReg_char : cfReg G \is a character.
Proof. by rewrite -cfReprReg cfRepr_char. Qed.
Lemma cfRepr_prod n1 n2 (rG1 : reprG n1) (rG2 : reprG n2) :
cfRepr rG1 * cfRepr rG2 = cfRepr (prod_repr rG1 rG2).
Proof. by apply/cfun_inP=> x Gx; rewrite !cfunE /= Gx mxtrace_prod. Qed.
Lemma mul_char : mulr_closed (@character G).
Proof.
split=> [|_ _ /char_reprP[rG1 ->] /char_reprP[rG2 ->]]; first exact: cfun1_char.
apply/char_reprP; exists (Representation (prod_repr rG1 rG2)).
by rewrite cfRepr_prod.
Qed.
HB.instance Definition _ := GRing.isMulClosed.Build (classfun G) character_pred
mul_char.
End IsChar.
Prenex Implicits character.
Arguments character_pred _ _ _ /.
Arguments char_reprP {gT G phi}.
Section AutChar.
Variables (gT : finGroupType) (G : {group gT}).
Implicit Type u : {rmorphism algC -> algC}.
Implicit Type chi : 'CF(G).
Lemma cfRepr_map u n (rG : mx_representation algC G n) :
cfRepr (map_repr u rG) = cfAut u (cfRepr rG).
Proof. by apply/cfun_inP=> x Gx; rewrite !cfunE Gx map_reprE trace_map_mx. Qed.
Lemma cfAut_char u chi : (cfAut u chi \is a character) = (chi \is a character).
Proof.
without loss /char_reprP[rG ->]: u chi / chi \is a character.
by move=> IHu; apply/idP/idP=> ?; first rewrite -(cfAutK u chi); rewrite IHu.
rewrite cfRepr_char; apply/char_reprP.
by exists (Representation (map_repr u rG)); rewrite cfRepr_map.
Qed.
Lemma cfConjC_char chi : (chi^*%CF \is a character) = (chi \is a character).
Proof. exact: cfAut_char. Qed.
Lemma cfAut_char1 u (chi : 'CF(G)) :
chi \is a character -> cfAut u chi 1%g = chi 1%g.
Proof. by move/Cnat_char1=> Nchi1; rewrite cfunE /= aut_natr. Qed.
Lemma cfAut_irr1 u i : (cfAut u 'chi[G]_i) 1%g = 'chi_i 1%g.
Proof. exact: cfAut_char1 (irr_char i). Qed.
Lemma cfConjC_char1 (chi : 'CF(G)) :
chi \is a character -> chi^*%CF 1%g = chi 1%g.
Proof. exact: cfAut_char1. Qed.
Lemma cfConjC_irr1 u i : ('chi[G]_i)^*%CF 1%g = 'chi_i 1%g.
Proof. exact: cfAut_irr1. Qed.
End AutChar.
Section Linear.
Variables (gT : finGroupType) (G : {group gT}).
Definition linear_char_pred {B : {set gT}} :=
fun phi : 'CF(B) => (phi \is a character) && (phi 1%g == 1).
Arguments linear_char_pred _ _ /.
Definition linear_char {B : {set gT}} :=
[qualify a phi | @linear_char_pred B phi].
Section OneChar.
Variable xi : 'CF(G).
Hypothesis CFxi : xi \is a linear_char.
Lemma lin_char1: xi 1%g = 1.
Proof. by case/andP: CFxi => _ /eqP. Qed.
Lemma lin_charW : xi \is a character.
Proof. by case/andP: CFxi. Qed.
Lemma cfun1_lin_char : (1 : 'CF(G)) \is a linear_char.
Proof. by rewrite qualifE/= cfun1_char /= cfun11. Qed.
Lemma lin_charM : {in G &, {morph xi : x y / (x * y)%g >-> x * y}}.
Proof.
move=> x y Gx Gy; case/andP: CFxi => /char_reprP[[n rG] -> /=].
rewrite cfRepr1 pnatr_eq1 => /eqP n1; rewrite {n}n1 in rG *.
rewrite !cfunE Gx Gy groupM //= !mulr1n repr_mxM //.
by rewrite [rG x]mx11_scalar [rG y]mx11_scalar -scalar_mxM !mxtrace_scalar.
Qed.
Lemma lin_char_prod I r (P : pred I) (x : I -> gT) :
(forall i, P i -> x i \in G) ->
xi (\prod_(i <- r | P i) x i)%g = \prod_(i <- r | P i) xi (x i).
Proof.
move=> Gx; elim/(big_load (fun y => y \in G)): _.
elim/big_rec2: _ => [|i a y Pi [Gy <-]]; first by rewrite lin_char1.
by rewrite groupM ?lin_charM ?Gx.
Qed.
Let xiMV x : x \in G -> xi x * xi (x^-1)%g = 1.
Proof. by move=> Gx; rewrite -lin_charM ?groupV // mulgV lin_char1. Qed.
Lemma lin_char_neq0 x : x \in G -> xi x != 0.
Proof.
by move/xiMV/(congr1 (predC1 0)); rewrite /= oner_eq0 mulf_eq0 => /norP[].
Qed.
Lemma lin_charV x : x \in G -> xi x^-1%g = (xi x)^-1.
Proof. by move=> Gx; rewrite -[_^-1]mulr1 -(xiMV Gx) mulKf ?lin_char_neq0. Qed.
Lemma lin_charX x n : x \in G -> xi (x ^+ n)%g = xi x ^+ n.
Proof.
move=> Gx; elim: n => [|n IHn]; first exact: lin_char1.
by rewrite expgS exprS lin_charM ?groupX ?IHn.
Qed.
Lemma lin_char_unity_root x : x \in G -> xi x ^+ #[x] = 1.
Proof. by move=> Gx; rewrite -lin_charX // expg_order lin_char1. Qed.
Lemma normC_lin_char x : x \in G -> `|xi x| = 1.
Proof.
move=> Gx; apply/eqP; rewrite -(@pexpr_eq1 _ _ #[x]) //.
by rewrite -normrX // lin_char_unity_root ?normr1.
Qed.
Lemma lin_charV_conj x : x \in G -> xi x^-1%g = (xi x)^*.
Proof.
move=> Gx; rewrite lin_charV // invC_norm mulrC normC_lin_char //.
by rewrite expr1n divr1.
Qed.
Lemma lin_char_irr : xi \in irr G.
Proof.
case/andP: CFxi => /char_reprP[rG ->]; rewrite cfRepr1 pnatr_eq1 => /eqP n1.
by apply/irr_reprP; exists rG => //; apply/mx_abs_irrW/linear_mx_abs_irr.
Qed.
Lemma mul_conjC_lin_char : xi * xi^*%CF = 1.
Proof.
apply/cfun_inP=> x Gx.
by rewrite !cfunE cfun1E Gx -normCK normC_lin_char ?expr1n.
Qed.
Lemma lin_char_unitr : xi \in GRing.unit.
Proof. by apply/unitrPr; exists xi^*%CF; apply: mul_conjC_lin_char. Qed.
Lemma invr_lin_char : xi^-1 = xi^*%CF.
Proof. by rewrite -[_^-1]mulr1 -mul_conjC_lin_char mulKr ?lin_char_unitr. Qed.
Lemma fful_lin_char_inj : cfaithful xi -> {in G &, injective xi}.
Proof.
move=> fful_phi x y Gx Gy xi_xy; apply/eqP; rewrite eq_mulgV1 -in_set1.
rewrite (subsetP fful_phi) // inE groupM ?groupV //=; apply/forallP=> z.
have [Gz | G'z] := boolP (z \in G); last by rewrite !cfun0 ?groupMl ?groupV.
by rewrite -mulgA lin_charM ?xi_xy -?lin_charM ?groupM ?groupV // mulKVg.
Qed.
End OneChar.
Lemma cfAut_lin_char u (xi : 'CF(G)) :
(cfAut u xi \is a linear_char) = (xi \is a linear_char).
Proof. by rewrite qualifE/= cfAut_char; apply/andb_id2l=> /cfAut_char1->. Qed.
Lemma cfConjC_lin_char (xi : 'CF(G)) :
(xi^*%CF \is a linear_char) = (xi \is a linear_char).
Proof. exact: cfAut_lin_char. Qed.
Lemma card_Iirr_abelian : abelian G -> #|Iirr G| = #|G|.
Proof. by rewrite card_ord NirrE card_classes_abelian => /eqP. Qed.
Lemma card_Iirr_cyclic : cyclic G -> #|Iirr G| = #|G|.
Proof. by move/cyclic_abelian/card_Iirr_abelian. Qed.
Lemma char_abelianP :
reflect (forall i : Iirr G, 'chi_i \is a linear_char) (abelian G).
Proof.
apply: (iffP idP) => [cGG i | CF_G].
rewrite qualifE/= irr_char /= irr1_degree.
by rewrite irr_degree_abelian //; last apply: groupC.
rewrite card_classes_abelian -NirrE -eqC_nat -irr_sum_square //.
rewrite -{1}[Nirr G]card_ord -sumr_const; apply/eqP/eq_bigr=> i _.
by rewrite lin_char1 ?expr1n ?CF_G.
Qed.
Lemma irr_repr_lin_char (i : Iirr G) x :
x \in G -> 'chi_i \is a linear_char ->
irr_repr (socle_of_Iirr i) x = ('chi_i x)%:M.
Proof.
move=> Gx CFi; rewrite -irrRepr cfunE Gx.
move: (_ x); rewrite -[irr_degree _](@natrK algC) -irr1_degree lin_char1 //.
by rewrite (natrK 1) => A; rewrite trace_mx11 -mx11_scalar.
Qed.
Fact linear_char_divr : divr_closed (@linear_char G).
Proof.
split=> [|chi xi Lchi Lxi]; first exact: cfun1_lin_char.
rewrite invr_lin_char // qualifE/= cfunE.
by rewrite rpredM ?lin_char1 ?mulr1 ?lin_charW //= cfConjC_lin_char.
Qed.
HB.instance Definition _ :=
GRing.isDivClosed.Build (classfun G) linear_char_pred linear_char_divr.
Lemma irr_cyclic_lin i : cyclic G -> 'chi[G]_i \is a linear_char.
Proof. by move/cyclic_abelian/char_abelianP. Qed.
Lemma irr_prime_lin i : prime #|G| -> 'chi[G]_i \is a linear_char.
Proof. by move/prime_cyclic/irr_cyclic_lin. Qed.
End Linear.
Prenex Implicits linear_char.
Arguments linear_char_pred _ _ _ /.
Section OrthogonalityRelations.
Variables aT gT : finGroupType.
(* This is Isaacs, Lemma (2.15) *)
Lemma repr_rsim_diag (G : {group gT}) f (rG : mx_representation algC G f) x :
x \in G -> let chi := cfRepr rG in
exists e,
[/\ (*a*) exists2 B, B \in unitmx & rG x = invmx B *m diag_mx e *m B,
(*b*) (forall i, e 0 i ^+ #[x] = 1) /\ (forall i, `|e 0 i| = 1),
(*c*) chi x = \sum_i e 0 i /\ `|chi x| <= chi 1%g
& (*d*) chi x^-1%g = (chi x)^*].
Proof.
move=> Gx; without loss cGG: G rG Gx / abelian G.
have sXG: <[x]> \subset G by rewrite cycle_subG.
move/(_ _ (subg_repr rG sXG) (cycle_id x) (cycle_abelian x)).
by rewrite /= !cfunE !groupV Gx (cycle_id x) !group1.
have [I U W simU W1 dxW]: mxsemisimple rG 1%:M.
rewrite -(reducible_Socle1 (DecSocleType rG)
(mx_Maschke_pchar _ (algC'G_pchar G))).
exact: Socle_semisimple.
have linU i: \rank (U i) = 1.
by apply: mxsimple_abelian_linear cGG (simU i); apply: groupC.
have castI: f = #|I|.
by rewrite -(mxrank1 algC f) -W1 (eqnP dxW) /= -sum1_card; apply/eq_bigr.
pose B := \matrix_j nz_row (U (enum_val (cast_ord castI j))).
have rowU i: (nz_row (U i) :=: U i)%MS.
apply/eqmxP; rewrite -(geq_leqif (mxrank_leqif_eq (nz_row_sub _))) linU.
by rewrite lt0n mxrank_eq0 (nz_row_mxsimple (simU i)).
have unitB: B \in unitmx.
rewrite -row_full_unit -sub1mx -W1; apply/sumsmx_subP=> i _.
pose j := cast_ord (esym castI) (enum_rank i).
by rewrite (submx_trans _ (row_sub j B)) // rowK cast_ordKV enum_rankK rowU.
pose e := \row_j row j (B *m rG x *m invmx B) 0 j.
have rGx: rG x = invmx B *m diag_mx e *m B.
rewrite -mulmxA; apply: canRL (mulKmx unitB) _.
apply/row_matrixP=> j; rewrite 2!row_mul; set u := row j B.
have /sub_rVP[a def_ux]: (u *m rG x <= u)%MS.
rewrite /u rowK rowU (eqmxMr _ (rowU _)).
exact: (mxmoduleP (mxsimple_module (simU _))).
rewrite def_ux [u]rowE scalemxAl; congr (_ *m _).
apply/rowP=> k; rewrite 5!mxE !row_mul def_ux [u]rowE scalemxAl mulmxK //.
by rewrite !mxE !eqxx !mulr_natr eq_sym.
have exp_e j: e 0 j ^+ #[x] = 1.
suffices: (diag_mx e j j) ^+ #[x] = (B *m rG (x ^+ #[x])%g *m invmx B) j j.
by rewrite expg_order repr_mx1 mulmx1 mulmxV // [e]lock !mxE eqxx.
elim: #[x] => [|n IHn]; first by rewrite repr_mx1 mulmx1 mulmxV // !mxE eqxx.
rewrite expgS repr_mxM ?groupX // {1}rGx -!mulmxA mulKVmx //.
by rewrite mul_diag_mx mulmxA [M in _ = M]mxE -IHn exprS {1}mxE eqxx.
have norm1_e j: `|e 0 j| = 1.
by apply/eqP; rewrite -(@pexpr_eq1 _ _ #[x]) // -normrX exp_e normr1.
exists e; split=> //; first by exists B.
rewrite cfRepr1 !cfunE Gx rGx mxtrace_mulC mulKVmx // mxtrace_diag.
split=> //=; apply: (le_trans (ler_norm_sum _ _ _)).
by rewrite (eq_bigr _ (in1W norm1_e)) sumr_const card_ord lexx.
rewrite !cfunE groupV !mulrb Gx rGx mxtrace_mulC mulKVmx //.
rewrite -trace_map_mx map_diag_mx; set d' := diag_mx _.
rewrite -[d'](mulKVmx unitB) mxtrace_mulC -[_ *m _](repr_mxK rG Gx) rGx.
rewrite -!mulmxA mulKVmx // (mulmxA d').
suffices->: d' *m diag_mx e = 1%:M by rewrite mul1mx mulKmx.
rewrite mulmx_diag -diag_const_mx; congr diag_mx; apply/rowP=> j.
by rewrite [e]lock !mxE mulrC -normCK -lock norm1_e expr1n.
Qed.
Variables (A : {group aT}) (G : {group gT}).
(* This is Isaacs, Lemma (2.15) (d). *)
Lemma char_inv (chi : 'CF(G)) x : chi \is a character -> chi x^-1%g = (chi x)^*.
Proof.
case Gx: (x \in G); last by rewrite !cfun0 ?rmorph0 ?groupV ?Gx.
by case/char_reprP=> rG ->; have [e [_ _ _]] := repr_rsim_diag rG Gx.
Qed.
Lemma irr_inv i x : 'chi[G]_i x^-1%g = ('chi_i x)^*.
Proof. exact/char_inv/irr_char. Qed.
(* This is Isaacs, Theorem (2.13). *)
Theorem generalized_orthogonality_relation y (i j : Iirr G) :
#|G|%:R^-1 * (\sum_(x in G) 'chi_i (x * y)%g * 'chi_j x^-1%g)
= (i == j)%:R * ('chi_i y / 'chi_i 1%g).
Proof.
pose W := @socle_of_Iirr _ G; pose e k := Wedderburn_id (W k).
pose aG := regular_repr algC G.
have [Gy | notGy] := boolP (y \in G); last first.
rewrite cfun0 // mul0r big1 ?mulr0 // => x Gx.
by rewrite cfun0 ?groupMl ?mul0r.
transitivity (('chi_i).[e j *m aG y]%CF / 'chi_j 1%g).
rewrite [e j]Wedderburn_id_expansion -scalemxAl xcfunZr -mulrA; congr (_ * _).
rewrite mulmx_suml raddf_sum big_distrl; apply: eq_bigr => x Gx /=.
rewrite -scalemxAl xcfunZr -repr_mxM // xcfunG ?groupM // mulrAC mulrC.
by congr (_ * _); rewrite mulrC mulKf ?irr1_neq0.
rewrite mulr_natl mulrb; have [<-{j} | neq_ij] := eqVneq.
by congr (_ / _); rewrite xcfun_mul_id ?envelop_mx_id ?xcfunG.
rewrite (xcfun_annihilate neq_ij) ?mul0r //.
case/andP: (Wedderburn_ideal (W j)) => _; apply: submx_trans.
by rewrite mem_mulsmx ?Wedderburn_id_mem ?envelop_mx_id.
Qed.
(* This is Isaacs, Corollary (2.14). *)
Corollary first_orthogonality_relation (i j : Iirr G) :
#|G|%:R^-1 * (\sum_(x in G) 'chi_i x * 'chi_j x^-1%g) = (i == j)%:R.
Proof.
have:= generalized_orthogonality_relation 1 i j.
rewrite mulrA mulfK ?irr1_neq0 // => <-; congr (_ * _).
by apply: eq_bigr => x; rewrite mulg1.
Qed.
(* The character table. *)
Definition irr_class i := enum_val (cast_ord (NirrE G) i).
Definition class_Iirr xG :=
cast_ord (esym (NirrE G)) (enum_rank_in (classes1 G) xG).
Local Notation c := irr_class.
Local Notation g i := (repr (c i)).
Local Notation iC := class_Iirr.
Definition character_table := \matrix_(i, j) 'chi[G]_i (g j).
Local Notation X := character_table.
Lemma irr_classP i : c i \in classes G.
Proof. exact: enum_valP. Qed.
Lemma repr_irr_classK i : g i ^: G = c i.
Proof. by case/repr_classesP: (irr_classP i). Qed.
Lemma irr_classK : cancel c iC.
Proof. by move=> i; rewrite /iC enum_valK_in cast_ordK. Qed.
Lemma class_IirrK : {in classes G, cancel iC c}.
Proof. by move=> xG GxG; rewrite /c cast_ordKV enum_rankK_in. Qed.
Lemma reindex_irr_class R idx (op : @Monoid.com_law R idx) F :
\big[op/idx]_(xG in classes G) F xG = \big[op/idx]_i F (c i).
Proof.
rewrite (reindex c); first by apply: eq_bigl => i; apply: enum_valP.
by exists iC; [apply: in1W; apply: irr_classK | apply: class_IirrK].
Qed.
(* The explicit value of the inverse is needed for the proof of the second *)
(* orthogonality relation. *)
Let X' := \matrix_(i, j) (#|'C_G[g i]|%:R^-1 * ('chi[G]_j (g i))^*).
Let XX'_1: X *m X' = 1%:M.
Proof.
apply/matrixP=> i j; rewrite !mxE -first_orthogonality_relation mulr_sumr.
rewrite sum_by_classes => [|u v Gu Gv]; last by rewrite -conjVg !cfunJ.
rewrite reindex_irr_class /=; apply/esym/eq_bigr=> k _.
rewrite !mxE irr_inv // -/(g k) -divg_index -indexgI /=.
rewrite (pchar0_natf_div Cpchar) ?dvdn_indexg // index_cent1 invfM invrK.
by rewrite repr_irr_classK mulrCA mulrA mulrCA.
Qed.
Lemma character_table_unit : X \in unitmx.
Proof. by case/mulmx1_unit: XX'_1. Qed.
Let uX := character_table_unit.
(* This is Isaacs, Theorem (2.18). *)
Theorem second_orthogonality_relation x y :
y \in G ->
\sum_i 'chi[G]_i x * ('chi_i y)^* = #|'C_G[x]|%:R *+ (x \in y ^: G).
Proof.
move=> Gy; pose i_x := iC (x ^: G); pose i_y := iC (y ^: G).
have [Gx | notGx] := boolP (x \in G); last first.
rewrite (contraNF (subsetP _ x) notGx) ?class_subG ?big1 // => i _.
by rewrite cfun0 ?mul0r.
transitivity ((#|'C_G[repr (y ^: G)]|%:R *: (X' *m X)) i_y i_x).
rewrite scalemxAl !mxE; apply: eq_bigr => k _; rewrite !mxE mulrC -!mulrA.
by rewrite !class_IirrK ?mem_classes // !cfun_repr mulVKf ?neq0CG.
rewrite mulmx1C // !mxE -!divg_index; do 2!rewrite -indexgI index_cent1.
rewrite (class_eqP (mem_repr y _)) ?class_refl // mulr_natr.
rewrite (can_in_eq class_IirrK) ?mem_classes //.
have [-> | not_yGx] := eqVneq; first by rewrite class_refl.
by rewrite [x \in _](contraNF _ not_yGx) // => /class_eqP->.
Qed.
Lemma eq_irr_mem_classP x y :
y \in G -> reflect (forall i, 'chi[G]_i x = 'chi_i y) (x \in y ^: G).
Proof.
move=> Gy; apply: (iffP idP) => [/imsetP[z Gz ->] i | xGy]; first exact: cfunJ.
have Gx: x \in G.
congr is_true: Gy; apply/eqP; rewrite -(can_eq oddb) -eqC_nat -!cfun1E.
by rewrite -irr0 xGy.
congr is_true: (class_refl G x); apply/eqP; rewrite -(can_eq oddb).
rewrite -(eqn_pmul2l (cardG_gt0 'C_G[x])) -eqC_nat !mulrnA; apply/eqP.
by rewrite -!second_orthogonality_relation //; apply/eq_bigr=> i _; rewrite xGy.
Qed.
(* This is Isaacs, Theorem (6.32) (due to Brauer). *)
Lemma card_afix_irr_classes (ito : action A (Iirr G)) (cto : action A _) a :
a \in A -> [acts A, on classes G | cto] ->
(forall i x y, x \in G -> y \in cto (x ^: G) a ->
'chi_i x = 'chi_(ito i a) y) ->
#|'Fix_ito[a]| = #|'Fix_(classes G | cto)[a]|.
Proof.
move=> Aa actsAG stabAchi; apply/eqP; rewrite -eqC_nat; apply/eqP.
have [[cP cK] iCK] := (irr_classP, irr_classK, class_IirrK).
pose icto b i := iC (cto (c i) b).
have Gca i: cto (c i) a \in classes G by rewrite (acts_act actsAG).
have inj_qa: injective (icto a).
by apply: can_inj (icto a^-1%g) _ => i; rewrite /icto iCK ?actKin ?cK.
pose Pa : 'M[algC]_(Nirr G) := perm_mx (actperm ito a).
pose qa := perm inj_qa; pose Qa : 'M[algC]_(Nirr G) := perm_mx qa^-1^-1%g.
transitivity (\tr Pa).
rewrite -sumr_const big_mkcond; apply: eq_bigr => i _.
by rewrite !mxE permE inE sub1set inE; case: ifP.
symmetry; transitivity (\tr Qa).
rewrite cardsE -sumr_const -big_filter_cond big_mkcond big_filter /=.
rewrite reindex_irr_class; apply: eq_bigr => i _; rewrite !mxE invgK permE.
by rewrite inE sub1set inE -(can_eq cK) iCK //; case: ifP.
rewrite -[Pa](mulmxK uX) -[Qa](mulKmx uX) mxtrace_mulC; congr (\tr(_ *m _)).
rewrite -row_permE -col_permE; apply/matrixP=> i j; rewrite !mxE.
rewrite -{2}[j](permKV qa); move: {j}(_ j) => j; rewrite !permE iCK //.
apply: stabAchi; first by case/repr_classesP: (cP j).
by rewrite repr_irr_classK (mem_repr_classes (Gca _)).
Qed.
End OrthogonalityRelations.
Prenex Implicits irr_class class_Iirr irr_classK.
Arguments class_IirrK {gT G%_G} [xG%_g] GxG : rename.
Arguments character_table {gT} G%_g.
Section InnerProduct.
Variable (gT : finGroupType) (G : {group gT}).
Lemma cfdot_irr i j : '['chi_i, 'chi_j]_G = (i == j)%:R.
Proof.
rewrite -first_orthogonality_relation; congr (_ * _).
by apply: eq_bigr => x Gx; rewrite irr_inv.
Qed.
Lemma cfnorm_irr i : '['chi[G]_i] = 1.
Proof. by rewrite cfdot_irr eqxx. Qed.
Lemma irr_orthonormal : orthonormal (irr G).
Proof.
apply/orthonormalP; split; first exact: free_uniq (irr_free G).
move=> _ _ /irrP[i ->] /irrP[j ->].
by rewrite cfdot_irr (inj_eq irr_inj).
Qed.
Lemma coord_cfdot phi i : coord (irr G) i phi = '[phi, 'chi_i].
Proof.
rewrite {2}(coord_basis (irr_basis G) (memvf phi)).
rewrite cfdot_suml (bigD1 i) // cfdotZl /= -tnth_nth cfdot_irr eqxx mulr1.
rewrite big1 ?addr0 // => j neq_ji; rewrite cfdotZl /= -tnth_nth cfdot_irr.
by rewrite (negbTE neq_ji) mulr0.
Qed.
Lemma cfun_sum_cfdot phi : phi = \sum_i '[phi, 'chi_i]_G *: 'chi_i.
Proof.
rewrite {1}(coord_basis (irr_basis G) (memvf phi)).
by apply: eq_bigr => i _; rewrite coord_cfdot -tnth_nth.
Qed.
Lemma cfdot_sum_irr phi psi :
'[phi, psi]_G = \sum_i '[phi, 'chi_i] * '[psi, 'chi_i]^*.
Proof.
rewrite {1}[phi]cfun_sum_cfdot cfdot_suml; apply: eq_bigr => i _.
by rewrite cfdotZl -cfdotC.
Qed.
Lemma Cnat_cfdot_char_irr i phi :
phi \is a character -> '[phi, 'chi_i]_G \in Num.nat.
Proof. by move/forallP/(_ i); rewrite coord_cfdot. Qed.
Lemma cfdot_char_r phi chi :
chi \is a character -> '[phi, chi]_G = \sum_i '[phi, 'chi_i] * '[chi, 'chi_i].
Proof.
move=> Nchi; rewrite cfdot_sum_irr; apply: eq_bigr => i _; congr (_ * _).
by rewrite conj_natr ?Cnat_cfdot_char_irr.
Qed.
Lemma Cnat_cfdot_char chi xi :
chi \is a character -> xi \is a character -> '[chi, xi]_G \in Num.nat.
Proof.
move=> Nchi Nxi; rewrite cfdot_char_r ?rpred_sum // => i _.
by rewrite rpredM ?Cnat_cfdot_char_irr.
Qed.
Lemma cfdotC_char chi xi :
chi \is a character-> xi \is a character -> '[chi, xi]_G = '[xi, chi].
Proof. by move=> Nchi Nxi; rewrite cfdotC conj_natr ?Cnat_cfdot_char. Qed.
Lemma irrEchar chi : (chi \in irr G) = (chi \is a character) && ('[chi] == 1).
Proof.
apply/irrP/andP=> [[i ->] | [Nchi]]; first by rewrite irr_char cfnorm_irr.
rewrite cfdot_sum_irr => /eqP/natr_sum_eq1[i _| i [_ ci1 cj0]].
by rewrite rpredM // ?conj_natr ?Cnat_cfdot_char_irr.
exists i; rewrite [chi]cfun_sum_cfdot (bigD1 i) //=.
rewrite -(normr_idP (natr_ge0 (Cnat_cfdot_char_irr i Nchi))).
rewrite normC_def {}ci1 sqrtC1 scale1r big1 ?addr0 // => j neq_ji.
by rewrite (('[_] =P 0) _) ?scale0r // -normr_eq0 normC_def cj0 ?sqrtC0.
Qed.
Lemma irrWchar chi : chi \in irr G -> chi \is a character.
Proof. by rewrite irrEchar => /andP[]. Qed.
Lemma irrWnorm chi : chi \in irr G -> '[chi] = 1.
Proof. by rewrite irrEchar => /andP[_ /eqP]. Qed.
Lemma mul_lin_irr xi chi :
xi \is a linear_char -> chi \in irr G -> xi * chi \in irr G.
Proof.
move=> Lxi; rewrite !irrEchar => /andP[Nphi /eqP <-].
rewrite rpredM // ?lin_charW //=; apply/eqP; congr (_ * _).
apply: eq_bigr=> x Gx; rewrite !cfunE rmorphM/= mulrACA -(lin_charV_conj Lxi)//.
by rewrite -lin_charM ?groupV // mulgV lin_char1 ?mul1r.
Qed.
Lemma eq_scaled_irr a b i j :
(a *: 'chi[G]_i == b *: 'chi_j) = (a == b) && ((a == 0) || (i == j)).
Proof.
apply/eqP/andP=> [|[/eqP-> /pred2P[]-> //]]; last by rewrite !scale0r.
move/(congr1 (cfdotr 'chi__)) => /= eq_ai_bj.
move: {eq_ai_bj}(eq_ai_bj i) (esym (eq_ai_bj j)); rewrite !cfdotZl !cfdot_irr.
by rewrite !mulr_natr !mulrb !eqxx eq_sym orbC; case: ifP => _ -> //= ->.
Qed.
Lemma eq_signed_irr (s t : bool) i j :
((-1) ^+ s *: 'chi[G]_i == (-1) ^+ t *: 'chi_j) = (s == t) && (i == j).
Proof. by rewrite eq_scaled_irr signr_eq0 (inj_eq signr_inj). Qed.
Lemma eq_scale_irr a (i j : Iirr G) :
(a *: 'chi_i == a *: 'chi_j) = (a == 0) || (i == j).
Proof. by rewrite eq_scaled_irr eqxx. Qed.
Lemma eq_addZ_irr a b (i j r t : Iirr G) :
(a *: 'chi_i + b *: 'chi_j == a *: 'chi_r + b *: 'chi_t)
= [|| [&& (a == 0) || (i == r) & (b == 0) || (j == t)],
[&& i == t, j == r & a == b] | [&& i == j, r == t & a == - b]].
Proof.
rewrite -!eq_scale_irr; apply/eqP/idP; last first.
case/orP; first by case/andP=> /eqP-> /eqP->.
case/orP=> /and3P[/eqP-> /eqP-> /eqP->]; first by rewrite addrC.
by rewrite !scaleNr !addNr.
have [-> /addrI/eqP-> // | /=] := eqVneq.
rewrite eq_scale_irr => /norP[/negP nz_a /negPf neq_ir].
move/(congr1 (cfdotr 'chi__))/esym/eqP => /= eq_cfdot.
move: {eq_cfdot}(eq_cfdot i) (eq_cfdot r); rewrite eq_sym !cfdotDl !cfdotZl.
rewrite !cfdot_irr !mulr_natr !mulrb !eqxx -!(eq_sym i) neq_ir !add0r.
have [<- _ | _] := i =P t; first by rewrite neq_ir addr0; case: ifP => // _ ->.
rewrite 2!fun_if if_arg addr0 addr_eq0; case: eqP => //= <- ->.
by rewrite neq_ir 2!fun_if if_arg eq_sym addr0; case: ifP.
Qed.
Lemma eq_subZnat_irr (a b : nat) (i j r t : Iirr G) :
(a%:R *: 'chi_i - b%:R *: 'chi_j == a%:R *: 'chi_r - b%:R *: 'chi_t)
= [|| a == 0 | i == r] && [|| b == 0 | j == t]
|| [&& i == j, r == t & a == b].
Proof.
rewrite -!scaleNr eq_addZ_irr oppr_eq0 opprK -addr_eq0 -natrD eqr_nat.
by rewrite !pnatr_eq0 addn_eq0; case: a b => [|a] [|b]; rewrite ?andbF.
Qed.
End InnerProduct.
Section IrrConstt.
Variable (gT : finGroupType) (G H : {group gT}).
Lemma char1_ge_norm (chi : 'CF(G)) x :
chi \is a character -> `|chi x| <= chi 1%g.
Proof.
case/char_reprP=> rG ->; case Gx: (x \in G); last first.
by rewrite cfunE cfRepr1 Gx normr0 ler0n.
by have [e [_ _ []]] := repr_rsim_diag rG Gx.
Qed.
Lemma max_cfRepr_norm_scalar n (rG : mx_representation algC G n) x :
x \in G -> `|cfRepr rG x| = cfRepr rG 1%g ->
exists2 c, `|c| = 1 & rG x = c%:M.
Proof.
move=> Gx; have [e [[B uB def_x] [_ e1] [-> _] _]] := repr_rsim_diag rG Gx.
rewrite cfRepr1 -[n in n%:R]card_ord -sumr_const -(eq_bigr _ (in1W e1)).
case/normC_sum_eq1=> [i _ | c /eqP norm_c_1 def_e]; first by rewrite e1.
have{} def_e: e = const_mx c by apply/rowP=> i; rewrite mxE def_e ?andbT.
by exists c => //; rewrite def_x def_e diag_const_mx scalar_mxC mulmxKV.
Qed.
Lemma max_cfRepr_mx1 n (rG : mx_representation algC G n) x :
x \in G -> cfRepr rG x = cfRepr rG 1%g -> rG x = 1%:M.
Proof.
move=> Gx kerGx; have [|c _ def_x] := @max_cfRepr_norm_scalar n rG x Gx.
by rewrite kerGx cfRepr1 normr_nat.
move/eqP: kerGx; rewrite cfRepr1 cfunE Gx {rG}def_x mxtrace_scalar.
case: n => [_|n]; first by rewrite ![_%:M]flatmx0.
rewrite mulrb -subr_eq0 -mulrnBl -mulr_natl mulf_eq0 pnatr_eq0 /=.
by rewrite subr_eq0 => /eqP->.
Qed.
Definition irr_constt (B : {set gT}) phi := [pred i | '[phi, 'chi_i]_B != 0].
Lemma irr_consttE i phi : (i \in irr_constt phi) = ('[phi, 'chi_i]_G != 0).
Proof. by []. Qed.
Lemma constt_charP (i : Iirr G) chi :
chi \is a character ->
reflect (exists2 chi', chi' \is a character & chi = 'chi_i + chi')
(i \in irr_constt chi).
Proof.
move=> Nchi; apply: (iffP idP) => [i_in_chi| [chi' Nchi' ->]]; last first.
rewrite inE /= cfdotDl cfdot_irr eqxx -(eqP (Cnat_cfdot_char_irr i Nchi')).
by rewrite -natrD pnatr_eq0.
exists (chi - 'chi_i); last by rewrite addrC subrK.
apply/forallP=> j; rewrite coord_cfdot cfdotBl cfdot_irr.
have [<- | _] := eqP; last by rewrite subr0 Cnat_cfdot_char_irr.
move: i_in_chi; rewrite inE; case/natrP: (Cnat_cfdot_char_irr i Nchi) => n ->.
by rewrite pnatr_eq0 -lt0n => /natrB <-; apply: rpred_nat.
Qed.
Lemma cfun_sum_constt (phi : 'CF(G)) :
phi = \sum_(i in irr_constt phi) '[phi, 'chi_i] *: 'chi_i.
Proof.
rewrite {1}[phi]cfun_sum_cfdot (bigID [pred i | '[phi, 'chi_i] == 0]) /=.
by rewrite big1 ?add0r // => i /eqP->; rewrite scale0r.
Qed.
Lemma neq0_has_constt (phi : 'CF(G)) :
phi != 0 -> exists i, i \in irr_constt phi.
Proof.
move=> nz_phi; apply/existsP; apply: contra nz_phi => /pred0P phi0.
by rewrite [phi]cfun_sum_constt big_pred0.
Qed.
Lemma constt_irr i : irr_constt 'chi[G]_i =i pred1 i.
Proof.
by move=> j; rewrite !inE cfdot_irr pnatr_eq0 (eq_sym j); case: (i == j).
Qed.
Lemma char1_ge_constt (i : Iirr G) chi :
chi \is a character -> i \in irr_constt chi -> 'chi_i 1%g <= chi 1%g.
Proof.
move=> {chi} _ /constt_charP[// | chi Nchi ->].
by rewrite cfunE addrC -subr_ge0 addrK char1_ge0.
Qed.
Lemma constt_ortho_char (phi psi : 'CF(G)) i j :
phi \is a character -> psi \is a character ->
i \in irr_constt phi -> j \in irr_constt psi ->
'[phi, psi] = 0 -> '['chi_i, 'chi_j] = 0.
Proof.
move=> _ _ /constt_charP[//|phi1 Nphi1 ->] /constt_charP[//|psi1 Npsi1 ->].
rewrite cfdot_irr; case: eqP => // -> /eqP/idPn[].
rewrite cfdotDl !cfdotDr cfnorm_irr -addrA gt_eqF ?ltr_wpDr ?ltr01 //.
by rewrite natr_ge0 ?rpredD ?Cnat_cfdot_char ?irr_char.
Qed.
End IrrConstt.
Arguments irr_constt {gT B%_g} phi%_CF.
Section Kernel.
Variable (gT : finGroupType) (G : {group gT}).
Implicit Types (phi chi xi : 'CF(G)) (H : {group gT}).
Lemma cfker_repr n (rG : mx_representation algC G n) :
cfker (cfRepr rG) = rker rG.
Proof.
apply/esym/setP=> x; rewrite inE mul1mx /=.
case Gx: (x \in G); last by rewrite inE Gx.
apply/eqP/idP=> Kx; last by rewrite max_cfRepr_mx1 // cfker1.
rewrite inE Gx; apply/forallP=> y; rewrite !cfunE !mulrb groupMl //.
by case: ifP => // Gy; rewrite repr_mxM // Kx mul1mx.
Qed.
Lemma cfkerEchar chi :
chi \is a character -> cfker chi = [set x in G | chi x == chi 1%g].
Proof.
move=> Nchi; apply/setP=> x; apply/idP/setIdP=> [Kx | [Gx /eqP chi_x]].
by rewrite (subsetP (cfker_sub chi)) // cfker1.
case/char_reprP: Nchi => rG -> in chi_x *; rewrite inE Gx; apply/forallP=> y.
rewrite !cfunE groupMl // !mulrb; case: ifP => // Gy.
by rewrite repr_mxM // max_cfRepr_mx1 ?mul1mx.
Qed.
Lemma cfker_nzcharE chi :
chi \is a character -> chi != 0 -> cfker chi = [set x | chi x == chi 1%g].
Proof.
move=> Nchi nzchi; apply/setP=> x; rewrite cfkerEchar // !inE andb_idl //.
by apply: contraLR => /cfun0-> //; rewrite eq_sym char1_eq0.
Qed.
Lemma cfkerEirr i : cfker 'chi[G]_i = [set x | 'chi_i x == 'chi_i 1%g].
Proof. by rewrite cfker_nzcharE ?irr_char ?irr_neq0. Qed.
Lemma cfker_irr0 : cfker 'chi[G]_0 = G.
Proof. by rewrite irr0 cfker_cfun1. Qed.
Lemma cfaithful_reg : cfaithful (cfReg G).
Proof.
apply/subsetP=> x; rewrite cfkerEchar ?cfReg_char // !inE !cfRegE eqxx.
by case/andP=> _; apply: contraLR => /negbTE->; rewrite eq_sym neq0CG.
Qed.
Lemma cfkerE chi :
chi \is a character ->
cfker chi = G :&: \bigcap_(i in irr_constt chi) cfker 'chi_i.
Proof.
move=> Nchi; rewrite cfkerEchar //; apply/setP=> x; rewrite !inE.
apply: andb_id2l => Gx; rewrite {1 2}[chi]cfun_sum_constt !sum_cfunE.
apply/eqP/bigcapP=> [Kx i Ci | Kx]; last first.
by apply: eq_bigr => i /Kx Kx_i; rewrite !cfunE cfker1.
rewrite cfkerEirr inE /= -(inj_eq (mulfI Ci)).
have:= (normC_sum_upper _ Kx) i; rewrite !cfunE => -> // {Ci}i _.
have chi_i_ge0: 0 <= '[chi, 'chi_i].
by rewrite natr_ge0 ?Cnat_cfdot_char_irr.
by rewrite !cfunE normrM (normr_idP _) ?ler_wpM2l ?char1_ge_norm ?irr_char.
Qed.
Lemma TI_cfker_irr : \bigcap_i cfker 'chi[G]_i = [1].
Proof.
apply/trivgP; apply: subset_trans cfaithful_reg; rewrite cfkerE ?cfReg_char //.
rewrite subsetI (bigcap_min 0) //=; last by rewrite cfker_irr0.
by apply/bigcapsP=> i _; rewrite bigcap_inf.
Qed.
Lemma cfker_constt i chi :
chi \is a character -> i \in irr_constt chi ->
cfker chi \subset cfker 'chi[G]_i.
Proof. by move=> Nchi Ci; rewrite cfkerE ?subIset ?(bigcap_min i) ?orbT. Qed.
Section KerLin.
Variable xi : 'CF(G).
Hypothesis lin_xi : xi \is a linear_char.
Let Nxi: xi \is a character. Proof. by have [] := andP lin_xi. Qed.
Lemma lin_char_der1 : G^`(1)%g \subset cfker xi.
Proof.
rewrite gen_subG /=; apply/subsetP=> _ /imset2P[x y Gx Gy ->].
rewrite cfkerEchar // inE groupR //= !lin_charM ?lin_charV ?in_group //.
by rewrite mulrCA mulKf ?mulVf ?lin_char_neq0 // lin_char1.
Qed.
Lemma cforder_lin_char : #[xi]%CF = exponent (G / cfker xi)%g.
Proof.
apply/eqP; rewrite eqn_dvd; apply/andP; split.
apply/dvdn_cforderP=> x Gx; rewrite -lin_charX // -cfQuoEker ?groupX //.
rewrite morphX ?(subsetP (cfker_norm xi)) //= expg_exponent ?mem_quotient //.
by rewrite cfQuo1 ?cfker_normal ?lin_char1.
have abGbar: abelian (G / cfker xi) := sub_der1_abelian lin_char_der1.
have [_ /morphimP[x Nx Gx ->] ->] := exponent_witness (abelian_nil abGbar).
rewrite order_dvdn -morphX //= coset_id cfkerEchar // !inE groupX //=.
by rewrite lin_charX ?lin_char1 // (dvdn_cforderP _ _ _).
Qed.
Lemma cforder_lin_char_dvdG : #[xi]%CF %| #|G|.
Proof.
by rewrite cforder_lin_char (dvdn_trans (exponent_dvdn _)) ?dvdn_morphim.
Qed.
Lemma cforder_lin_char_gt0 : (0 < #[xi]%CF)%N.
Proof. by rewrite cforder_lin_char exponent_gt0. Qed.
End KerLin.
End Kernel.
Section Restrict.
Variable (gT : finGroupType) (G H : {group gT}).
Lemma cfRepr_sub n (rG : mx_representation algC G n) (sHG : H \subset G) :
cfRepr (subg_repr rG sHG) = 'Res[H] (cfRepr rG).
Proof.
by apply/cfun_inP => x Hx; rewrite cfResE // !cfunE Hx (subsetP sHG).
Qed.
Lemma cfRes_char chi : chi \is a character -> 'Res[H, G] chi \is a character.
Proof.
have [sHG | not_sHG] := boolP (H \subset G).
by case/char_reprP=> rG ->; rewrite -(cfRepr_sub rG sHG) cfRepr_char.
by move/Cnat_char1=> Nchi1; rewrite cfResEout // rpredZ_nat ?rpred1.
Qed.
Lemma cfRes_eq0 phi : phi \is a character -> ('Res[H, G] phi == 0) = (phi == 0).
Proof. by move=> Nchi; rewrite -!char1_eq0 ?cfRes_char // cfRes1. Qed.
Lemma cfRes_lin_char chi :
chi \is a linear_char -> 'Res[H, G] chi \is a linear_char.
Proof. by case/andP=> Nchi; rewrite qualifE/= cfRes_char ?cfRes1. Qed.
Lemma Res_irr_neq0 i : 'Res[H, G] 'chi_i != 0.
Proof. by rewrite cfRes_eq0 ?irr_neq0 ?irr_char. Qed.
Lemma cfRes_lin_lin (chi : 'CF(G)) :
chi \is a character -> 'Res[H] chi \is a linear_char -> chi \is a linear_char.
Proof. by rewrite !qualifE/= !qualifE/= cfRes1 => -> /andP[]. Qed.
Lemma cfRes_irr_irr chi :
chi \is a character -> 'Res[H] chi \in irr H -> chi \in irr G.
Proof.
have [sHG /char_reprP[rG ->] | not_sHG Nchi] := boolP (H \subset G).
rewrite -(cfRepr_sub _ sHG) => /irr_reprP[rH irrH def_rH]; apply/irr_reprP.
suffices /subg_mx_irr: mx_irreducible (subg_repr rG sHG) by exists rG.
by apply: mx_rsim_irr irrH; apply/cfRepr_rsimP/eqP.
rewrite cfResEout // => /irrP[j Dchi_j]; apply/lin_char_irr/cfRes_lin_lin=> //.
suffices j0: j = 0 by rewrite cfResEout // Dchi_j j0 irr0 rpred1.
apply: contraNeq (irr1_neq0 j) => nz_j.
have:= xcfun_id j 0; rewrite -Dchi_j cfunE xcfunZl -irr0 xcfun_id eqxx => ->.
by rewrite (negPf nz_j).
Qed.
Definition Res_Iirr (A B : {set gT}) i := cfIirr ('Res[B, A] 'chi_i).
Lemma Res_Iirr0 : Res_Iirr H (0 : Iirr G) = 0.
Proof. by rewrite /Res_Iirr irr0 rmorph1 -irr0 irrK. Qed.
Lemma lin_Res_IirrE i : 'chi[G]_i 1%g = 1 -> 'chi_(Res_Iirr H i) = 'Res 'chi_i.
Proof.
move=> chi1; rewrite cfIirrE ?lin_char_irr ?cfRes_lin_char //.
by rewrite qualifE/= irr_char /= chi1.
Qed.
End Restrict.
Arguments Res_Iirr {gT A%_g} B%_g i%_R.
Section MoreConstt.
Variables (gT : finGroupType) (G H : {group gT}).
Lemma constt_Ind_Res i j :
i \in irr_constt ('Ind[G] 'chi_j) = (j \in irr_constt ('Res[H] 'chi_i)).
Proof. by rewrite !irr_consttE cfdotC conjC_eq0 -cfdot_Res_l. Qed.
Lemma cfdot_Res_ge_constt i j psi :
psi \is a character -> j \in irr_constt psi ->
'['Res[H, G] 'chi_j, 'chi_i] <= '['Res[H] psi, 'chi_i].
Proof.
move=> {psi} _ /constt_charP[// | psi Npsi ->].
rewrite linearD cfdotDl addrC -subr_ge0 addrK natr_ge0 //=.
by rewrite Cnat_cfdot_char_irr // cfRes_char.
Qed.
Lemma constt_Res_trans j psi :
psi \is a character -> j \in irr_constt psi ->
{subset irr_constt ('Res[H, G] 'chi_j) <= irr_constt ('Res[H] psi)}.
Proof.
move=> Npsi Cj i; apply: contraNneq; rewrite eq_le => {1}<-.
rewrite cfdot_Res_ge_constt ?natr_ge0 ?Cnat_cfdot_char_irr //.
by rewrite cfRes_char ?irr_char.
Qed.
End MoreConstt.
Section Morphim.
Variables (aT rT : finGroupType) (G D : {group aT}) (f : {morphism D >-> rT}).
Implicit Type chi : 'CF(f @* G).
Lemma cfRepr_morphim n (rfG : mx_representation algC (f @* G) n) sGD :
cfRepr (morphim_repr rfG sGD) = cfMorph (cfRepr rfG).
Proof.
apply/cfun_inP=> x Gx; have Dx: x \in D := subsetP sGD x Gx.
by rewrite cfMorphE // !cfunE ?mem_morphim ?Gx.
Qed.
Lemma cfMorph_char chi : chi \is a character -> cfMorph chi \is a character.
Proof.
have [sGD /char_reprP[rfG ->] | outGD Nchi] := boolP (G \subset D); last first.
by rewrite cfMorphEout // rpredZ_nat ?rpred1 ?Cnat_char1.
apply/char_reprP; exists (Representation (morphim_repr rfG sGD)).
by rewrite cfRepr_morphim.
Qed.
Lemma cfMorph_lin_char chi :
chi \is a linear_char -> cfMorph chi \is a linear_char.
Proof. by case/andP=> Nchi; rewrite qualifE/= cfMorph1 cfMorph_char. Qed.
Lemma cfMorph_charE chi :
G \subset D -> (cfMorph chi \is a character) = (chi \is a character).
Proof.
move=> sGD; apply/idP/idP=> [/char_reprP[[n rG] /=Dfchi] | /cfMorph_char//].
pose H := 'ker_G f; have kerH: H \subset rker rG.
by rewrite -cfker_repr -Dfchi cfker_morph // setIS // ker_sub_pre.
have nHG: G \subset 'N(H) by rewrite normsI // (subset_trans sGD) ?ker_norm.
have [h injh im_h] := first_isom_loc f sGD; rewrite -/H in h injh im_h.
have DfG: invm injh @*^-1 (G / H) == (f @* G)%g by rewrite morphpre_invm im_h.
pose rfG := eqg_repr (morphpre_repr _ (quo_repr kerH nHG)) DfG.
apply/char_reprP; exists (Representation rfG).
apply/cfun_inP=> _ /morphimP[x Dx Gx ->]; rewrite -cfMorphE // Dfchi !cfunE Gx.
pose xH := coset H x; have GxH: xH \in (G / H)%g by apply: mem_quotient.
suffices Dfx: f x = h xH by rewrite mem_morphim //= Dfx invmE ?quo_repr_coset.
by apply/set1_inj; rewrite -?morphim_set1 ?im_h ?(subsetP nHG) ?sub1set.
Qed.
Lemma cfMorph_lin_charE chi :
G \subset D -> (cfMorph chi \is a linear_char) = (chi \is a linear_char).
Proof. by rewrite qualifE/= cfMorph1 => /cfMorph_charE->. Qed.
Lemma cfMorph_irr chi :
G \subset D -> (cfMorph chi \in irr G) = (chi \in irr (f @* G)).
Proof. by move=> sGD; rewrite !irrEchar cfMorph_charE // cfMorph_iso. Qed.
Definition morph_Iirr i := cfIirr (cfMorph 'chi[f @* G]_i).
Lemma morph_Iirr0 : morph_Iirr 0 = 0.
Proof. by rewrite /morph_Iirr irr0 rmorph1 -irr0 irrK. Qed.
Hypothesis sGD : G \subset D.
Lemma morph_IirrE i : 'chi_(morph_Iirr i) = cfMorph 'chi_i.
Proof. by rewrite cfIirrE ?cfMorph_irr ?mem_irr. Qed.
Lemma morph_Iirr_inj : injective morph_Iirr.
Proof.
by move=> i j eq_ij; apply/irr_inj/cfMorph_inj; rewrite // -!morph_IirrE eq_ij.
Qed.
Lemma morph_Iirr_eq0 i : (morph_Iirr i == 0) = (i == 0).
Proof. by rewrite -!irr_eq1 morph_IirrE cfMorph_eq1. Qed.
End Morphim.
Section Isom.
Variables (aT rT : finGroupType) (G : {group aT}) (f : {morphism G >-> rT}).
Variables (R : {group rT}) (isoGR : isom G R f).
Implicit Type chi : 'CF(G).
Lemma cfIsom_char chi :
(cfIsom isoGR chi \is a character) = (chi \is a character).
Proof.
rewrite [cfIsom _]locked_withE cfMorph_charE //.
by rewrite (isom_im (isom_sym _)) cfRes_id.
Qed.
Lemma cfIsom_lin_char chi :
(cfIsom isoGR chi \is a linear_char) = (chi \is a linear_char).
Proof. by rewrite qualifE/= cfIsom_char cfIsom1. Qed.
Lemma cfIsom_irr chi : (cfIsom isoGR chi \in irr R) = (chi \in irr G).
Proof. by rewrite !irrEchar cfIsom_char cfIsom_iso. Qed.
Definition isom_Iirr i := cfIirr (cfIsom isoGR 'chi_i).
Lemma isom_IirrE i : 'chi_(isom_Iirr i) = cfIsom isoGR 'chi_i.
Proof. by rewrite cfIirrE ?cfIsom_irr ?mem_irr. Qed.
Lemma isom_Iirr_inj : injective isom_Iirr.
Proof.
by move=> i j eqij; apply/irr_inj/(cfIsom_inj isoGR); rewrite -!isom_IirrE eqij.
Qed.
Lemma isom_Iirr_eq0 i : (isom_Iirr i == 0) = (i == 0).
Proof. by rewrite -!irr_eq1 isom_IirrE cfIsom_eq1. Qed.
Lemma isom_Iirr0 : isom_Iirr 0 = 0.
Proof. by apply/eqP; rewrite isom_Iirr_eq0. Qed.
End Isom.
Arguments isom_Iirr_inj {aT rT G f R} isoGR [i1 i2] : rename.
Section IsomInv.
Variables (aT rT : finGroupType) (G : {group aT}) (f : {morphism G >-> rT}).
Variables (R : {group rT}) (isoGR : isom G R f).
Lemma isom_IirrK : cancel (isom_Iirr isoGR) (isom_Iirr (isom_sym isoGR)).
Proof. by move=> i; apply: irr_inj; rewrite !isom_IirrE cfIsomK. Qed.
Lemma isom_IirrKV : cancel (isom_Iirr (isom_sym isoGR)) (isom_Iirr isoGR).
Proof. by move=> i; apply: irr_inj; rewrite !isom_IirrE cfIsomKV. Qed.
End IsomInv.
Section Sdprod.
Variables (gT : finGroupType) (K H G : {group gT}).
Hypothesis defG : K ><| H = G.
Let nKG: G \subset 'N(K). Proof. by have [/andP[]] := sdprod_context defG. Qed.
Lemma cfSdprod_char chi :
(cfSdprod defG chi \is a character) = (chi \is a character).
Proof. by rewrite unlock cfMorph_charE // cfIsom_char. Qed.
Lemma cfSdprod_lin_char chi :
(cfSdprod defG chi \is a linear_char) = (chi \is a linear_char).
Proof. by rewrite qualifE/= cfSdprod_char cfSdprod1. Qed.
Lemma cfSdprod_irr chi : (cfSdprod defG chi \in irr G) = (chi \in irr H).
Proof. by rewrite !irrEchar cfSdprod_char cfSdprod_iso. Qed.
Definition sdprod_Iirr j := cfIirr (cfSdprod defG 'chi_j).
Lemma sdprod_IirrE j : 'chi_(sdprod_Iirr j) = cfSdprod defG 'chi_j.
Proof. by rewrite cfIirrE ?cfSdprod_irr ?mem_irr. Qed.
Lemma sdprod_IirrK : cancel sdprod_Iirr (Res_Iirr H).
Proof. by move=> j; rewrite /Res_Iirr sdprod_IirrE cfSdprodK irrK. Qed.
Lemma sdprod_Iirr_inj : injective sdprod_Iirr.
Proof. exact: can_inj sdprod_IirrK. Qed.
Lemma sdprod_Iirr_eq0 i : (sdprod_Iirr i == 0) = (i == 0).
Proof. by rewrite -!irr_eq1 sdprod_IirrE cfSdprod_eq1. Qed.
Lemma sdprod_Iirr0 : sdprod_Iirr 0 = 0.
Proof. by apply/eqP; rewrite sdprod_Iirr_eq0. Qed.
Lemma Res_sdprod_irr phi :
K \subset cfker phi -> phi \in irr G -> 'Res phi \in irr H.
Proof.
move=> kerK /irrP[i Dphi]; rewrite irrEchar -(cfSdprod_iso defG).
by rewrite cfRes_sdprodK // Dphi cfnorm_irr cfRes_char ?irr_char /=.
Qed.
Lemma sdprod_Res_IirrE i :
K \subset cfker 'chi[G]_i -> 'chi_(Res_Iirr H i) = 'Res 'chi_i.
Proof. by move=> kerK; rewrite cfIirrE ?Res_sdprod_irr ?mem_irr. Qed.
Lemma sdprod_Res_IirrK i :
K \subset cfker 'chi_i -> sdprod_Iirr (Res_Iirr H i) = i.
Proof.
by move=> kerK; rewrite /sdprod_Iirr sdprod_Res_IirrE ?cfRes_sdprodK ?irrK.
Qed.
End Sdprod.
Arguments sdprod_Iirr_inj {gT K H G} defG [i1 i2] : rename.
Section DProd.
Variables (gT : finGroupType) (G K H : {group gT}).
Hypothesis KxH : K \x H = G.
Lemma cfDprodKl_abelian j : abelian H -> cancel ((cfDprod KxH)^~ 'chi_j) 'Res.
Proof. by move=> cHH; apply: cfDprodKl; apply/lin_char1/char_abelianP. Qed.
Lemma cfDprodKr_abelian i : abelian K -> cancel (cfDprod KxH 'chi_i) 'Res.
Proof. by move=> cKK; apply: cfDprodKr; apply/lin_char1/char_abelianP. Qed.
Lemma cfDprodl_char phi :
(cfDprodl KxH phi \is a character) = (phi \is a character).
Proof. exact: cfSdprod_char. Qed.
Lemma cfDprodr_char psi :
(cfDprodr KxH psi \is a character) = (psi \is a character).
Proof. exact: cfSdprod_char. Qed.
Lemma cfDprod_char phi psi :
phi \is a character -> psi \is a character ->
cfDprod KxH phi psi \is a character.
Proof. by move=> Nphi Npsi; rewrite rpredM ?cfDprodl_char ?cfDprodr_char. Qed.
Lemma cfDprod_eq1 phi psi :
phi \is a character -> psi \is a character ->
(cfDprod KxH phi psi == 1) = (phi == 1) && (psi == 1).
Proof.
move=> /Cnat_char1 Nphi /Cnat_char1 Npsi.
apply/eqP/andP=> [phi_psi_1 | [/eqP-> /eqP->]]; last by rewrite cfDprod_cfun1.
have /andP[/eqP phi1 /eqP psi1]: (phi 1%g == 1) && (psi 1%g == 1).
by rewrite -natr_mul_eq1 // -(cfDprod1 KxH) phi_psi_1 cfun11.
rewrite -[phi](cfDprodKl KxH psi1) -{2}[psi](cfDprodKr KxH phi1) phi_psi_1.
by rewrite !rmorph1.
Qed.
Lemma cfDprodl_lin_char phi :
(cfDprodl KxH phi \is a linear_char) = (phi \is a linear_char).
Proof. exact: cfSdprod_lin_char. Qed.
Lemma cfDprodr_lin_char psi :
(cfDprodr KxH psi \is a linear_char) = (psi \is a linear_char).
Proof. exact: cfSdprod_lin_char. Qed.
Lemma cfDprod_lin_char phi psi :
phi \is a linear_char -> psi \is a linear_char ->
cfDprod KxH phi psi \is a linear_char.
Proof. by move=> Nphi Npsi; rewrite rpredM ?cfSdprod_lin_char. Qed.
Lemma cfDprodl_irr chi : (cfDprodl KxH chi \in irr G) = (chi \in irr K).
Proof. exact: cfSdprod_irr. Qed.
Lemma cfDprodr_irr chi : (cfDprodr KxH chi \in irr G) = (chi \in irr H).
Proof. exact: cfSdprod_irr. Qed.
Definition dprodl_Iirr i := cfIirr (cfDprodl KxH 'chi_i).
Lemma dprodl_IirrE i : 'chi_(dprodl_Iirr i) = cfDprodl KxH 'chi_i.
Proof. exact: sdprod_IirrE. Qed.
Lemma dprodl_IirrK : cancel dprodl_Iirr (Res_Iirr K).
Proof. exact: sdprod_IirrK. Qed.
Lemma dprodl_Iirr_eq0 i : (dprodl_Iirr i == 0) = (i == 0).
Proof. exact: sdprod_Iirr_eq0. Qed.
Lemma dprodl_Iirr0 : dprodl_Iirr 0 = 0.
Proof. exact: sdprod_Iirr0. Qed.
Definition dprodr_Iirr j := cfIirr (cfDprodr KxH 'chi_j).
Lemma dprodr_IirrE j : 'chi_(dprodr_Iirr j) = cfDprodr KxH 'chi_j.
Proof. exact: sdprod_IirrE. Qed.
Lemma dprodr_IirrK : cancel dprodr_Iirr (Res_Iirr H).
Proof. exact: sdprod_IirrK. Qed.
Lemma dprodr_Iirr_eq0 j : (dprodr_Iirr j == 0) = (j == 0).
Proof. exact: sdprod_Iirr_eq0. Qed.
Lemma dprodr_Iirr0 : dprodr_Iirr 0 = 0.
Proof. exact: sdprod_Iirr0. Qed.
Lemma cfDprod_irr i j : cfDprod KxH 'chi_i 'chi_j \in irr G.
Proof.
rewrite irrEchar cfDprod_char ?irr_char //=.
by rewrite cfdot_dprod !cfdot_irr !eqxx mul1r.
Qed.
Definition dprod_Iirr ij := cfIirr (cfDprod KxH 'chi_ij.1 'chi_ij.2).
Lemma dprod_IirrE i j : 'chi_(dprod_Iirr (i, j)) = cfDprod KxH 'chi_i 'chi_j.
Proof. by rewrite cfIirrE ?cfDprod_irr. Qed.
Lemma dprod_IirrEl i : 'chi_(dprod_Iirr (i, 0)) = cfDprodl KxH 'chi_i.
Proof. by rewrite dprod_IirrE /cfDprod irr0 rmorph1 mulr1. Qed.
Lemma dprod_IirrEr j : 'chi_(dprod_Iirr (0, j)) = cfDprodr KxH 'chi_j.
Proof. by rewrite dprod_IirrE /cfDprod irr0 rmorph1 mul1r. Qed.
Lemma dprod_Iirr_inj : injective dprod_Iirr.
Proof.
move=> [i1 j1] [i2 j2] /eqP; rewrite -[_ == _]oddb -(@natrK algC (_ == _)).
rewrite -cfdot_irr !dprod_IirrE cfdot_dprod !cfdot_irr -natrM mulnb.
by rewrite natrK oddb -xpair_eqE => /eqP.
Qed.
Lemma dprod_Iirr0 : dprod_Iirr (0, 0) = 0.
Proof. by apply/irr_inj; rewrite dprod_IirrE !irr0 cfDprod_cfun1. Qed.
Lemma dprod_Iirr0l j : dprod_Iirr (0, j) = dprodr_Iirr j.
Proof.
by apply/irr_inj; rewrite dprod_IirrE irr0 dprodr_IirrE cfDprod_cfun1l.
Qed.
Lemma dprod_Iirr0r i : dprod_Iirr (i, 0) = dprodl_Iirr i.
Proof.
by apply/irr_inj; rewrite dprod_IirrE irr0 dprodl_IirrE cfDprod_cfun1r.
Qed.
Lemma dprod_Iirr_eq0 i j : (dprod_Iirr (i, j) == 0) = (i == 0) && (j == 0).
Proof. by rewrite -xpair_eqE -(inj_eq dprod_Iirr_inj) dprod_Iirr0. Qed.
Lemma cfdot_dprod_irr i1 i2 j1 j2 :
'['chi_(dprod_Iirr (i1, j1)), 'chi_(dprod_Iirr (i2, j2))]
= ((i1 == i2) && (j1 == j2))%:R.
Proof. by rewrite cfdot_irr (inj_eq dprod_Iirr_inj). Qed.
Lemma dprod_Iirr_onto k : k \in codom dprod_Iirr.
Proof.
set D := codom _; have Df: dprod_Iirr _ \in D := codom_f dprod_Iirr _.
have: 'chi_k 1%g ^+ 2 != 0 by rewrite mulf_neq0 ?irr1_neq0.
apply: contraR => notDk; move/eqP: (irr_sum_square G).
rewrite (bigID [in D]) (reindex _ (bij_on_codom dprod_Iirr_inj (0, 0))) /=.
have ->: #|G|%:R = \sum_i \sum_j 'chi_(dprod_Iirr (i, j)) 1%g ^+ 2.
rewrite -(dprod_card KxH) natrM.
do 2![rewrite -irr_sum_square (mulr_suml, mulr_sumr); apply: eq_bigr => ? _].
by rewrite dprod_IirrE -exprMn -{3}(mulg1 1%g) cfDprodE.
rewrite (eq_bigl _ _ Df) pair_bigA addrC -subr_eq0 addrK.
by move/eqP/psumr_eq0P=> -> //= i _; rewrite irr1_degree -natrX ler0n.
Qed.
Definition inv_dprod_Iirr i := iinv (dprod_Iirr_onto i).
Lemma dprod_IirrK : cancel dprod_Iirr inv_dprod_Iirr.
Proof. by move=> p; apply: (iinv_f dprod_Iirr_inj). Qed.
Lemma inv_dprod_IirrK : cancel inv_dprod_Iirr dprod_Iirr.
Proof. by move=> i; apply: f_iinv. Qed.
Lemma inv_dprod_Iirr0 : inv_dprod_Iirr 0 = (0, 0).
Proof. by apply/(canLR dprod_IirrK); rewrite dprod_Iirr0. Qed.
End DProd.
Arguments dprod_Iirr_inj {gT G K H} KxH [i1 i2] : rename.
Lemma dprod_IirrC (gT : finGroupType) (G K H : {group gT})
(KxH : K \x H = G) (HxK : H \x K = G) i j :
dprod_Iirr KxH (i, j) = dprod_Iirr HxK (j, i).
Proof. by apply: irr_inj; rewrite !dprod_IirrE; apply: cfDprodC. Qed.
Section BigDprod.
Variables (gT : finGroupType) (I : finType) (P : pred I).
Variables (A : I -> {group gT}) (G : {group gT}).
Hypothesis defG : \big[dprod/1%g]_(i | P i) A i = G.
Let sAG i : P i -> A i \subset G.
Proof. by move=> Pi; rewrite -(bigdprodWY defG) (bigD1 i) ?joing_subl. Qed.
Lemma cfBigdprodi_char i (phi : 'CF(A i)) :
phi \is a character -> cfBigdprodi defG phi \is a character.
Proof. by move=> Nphi; rewrite cfDprodl_char cfRes_char. Qed.
Lemma cfBigdprodi_charE i (phi : 'CF(A i)) :
P i -> (cfBigdprodi defG phi \is a character) = (phi \is a character).
Proof. by move=> Pi; rewrite cfDprodl_char Pi cfRes_id. Qed.
Lemma cfBigdprod_char phi :
(forall i, P i -> phi i \is a character) ->
cfBigdprod defG phi \is a character.
Proof.
by move=> Nphi; apply: rpred_prod => i /Nphi; apply: cfBigdprodi_char.
Qed.
Lemma cfBigdprodi_lin_char i (phi : 'CF(A i)) :
phi \is a linear_char -> cfBigdprodi defG phi \is a linear_char.
Proof. by move=> Lphi; rewrite cfDprodl_lin_char ?cfRes_lin_char. Qed.
Lemma cfBigdprodi_lin_charE i (phi : 'CF(A i)) :
P i -> (cfBigdprodi defG phi \is a linear_char) = (phi \is a linear_char).
Proof. by move=> Pi; rewrite qualifE/= cfBigdprodi_charE // cfBigdprodi1. Qed.
Lemma cfBigdprod_lin_char phi :
(forall i, P i -> phi i \is a linear_char) ->
cfBigdprod defG phi \is a linear_char.
Proof.
by move=> Lphi; apply/rpred_prod=> i /Lphi; apply: cfBigdprodi_lin_char.
Qed.
Lemma cfBigdprodi_irr i chi :
P i -> (cfBigdprodi defG chi \in irr G) = (chi \in irr (A i)).
Proof. by move=> Pi; rewrite !irrEchar cfBigdprodi_charE ?cfBigdprodi_iso. Qed.
Lemma cfBigdprod_irr chi :
(forall i, P i -> chi i \in irr (A i)) -> cfBigdprod defG chi \in irr G.
Proof.
move=> Nchi; rewrite irrEchar cfBigdprod_char => [|i /Nchi/irrWchar] //=.
by rewrite cfdot_bigdprod big1 // => i /Nchi/irrWnorm.
Qed.
Lemma cfBigdprod_eq1 phi :
(forall i, P i -> phi i \is a character) ->
(cfBigdprod defG phi == 1) = [forall (i | P i), phi i == 1].
Proof.
move=> Nphi; set Phi := cfBigdprod defG phi.
apply/eqP/eqfun_inP=> [Phi1 i Pi | phi1]; last first.
by apply: big1 => i /phi1->; rewrite rmorph1.
have Phi1_1: Phi 1%g = 1 by rewrite Phi1 cfun1E group1.
have nz_Phi1: Phi 1%g != 0 by rewrite Phi1_1 oner_eq0.
have [_ <-] := cfBigdprodK nz_Phi1 Pi.
rewrite Phi1_1 divr1 -/Phi Phi1 rmorph1.
rewrite prod_cfunE // in Phi1_1; have := natr_prod_eq1 _ Phi1_1 Pi.
rewrite -(cfRes1 (A i)) cfBigdprodiK // => ->; first by rewrite scale1r.
by move=> {i Pi} j /Nphi Nphi_j; rewrite Cnat_char1 ?cfBigdprodi_char.
Qed.
Lemma cfBigdprod_Res_lin chi :
chi \is a linear_char -> cfBigdprod defG (fun i => 'Res[A i] chi) = chi.
Proof.
move=> Lchi; apply/cfun_inP=> _ /(mem_bigdprod defG)[x [Ax -> _]].
rewrite (lin_char_prod Lchi) ?cfBigdprodE // => [|i Pi]; last first.
by rewrite (subsetP (sAG Pi)) ?Ax.
by apply/eq_bigr=> i Pi; rewrite cfResE ?sAG ?Ax.
Qed.
Lemma cfBigdprodKlin phi :
(forall i, P i -> phi i \is a linear_char) ->
forall i, P i -> 'Res (cfBigdprod defG phi) = phi i.
Proof.
move=> Lphi i Pi; have Lpsi := cfBigdprod_lin_char Lphi.
have [_ <-] := cfBigdprodK (lin_char_neq0 Lpsi (group1 G)) Pi.
by rewrite !lin_char1 ?Lphi // divr1 scale1r.
Qed.
Lemma cfBigdprodKabelian Iphi (phi := fun i => 'chi_(Iphi i)) :
abelian G -> forall i, P i -> 'Res (cfBigdprod defG phi) = 'chi_(Iphi i).
Proof.
move=> /(abelianS _) cGG.
by apply: cfBigdprodKlin => i /sAG/cGG/char_abelianP->.
Qed.
End BigDprod.
Section Aut.
Variables (gT : finGroupType) (G : {group gT}).
Implicit Type u : {rmorphism algC -> algC}.
Lemma conjC_charAut u (chi : 'CF(G)) x :
chi \is a character -> (u (chi x))^* = u (chi x)^*.
Proof.
have [Gx | /cfun0->] := boolP (x \in G); last by rewrite !rmorph0.
case/char_reprP=> rG ->; have [e [_ [en1 _] [-> _] _]] := repr_rsim_diag rG Gx.
by rewrite !rmorph_sum; apply: eq_bigr => i _; apply: aut_unity_rootC (en1 i).
Qed.
Lemma conjC_irrAut u i x : (u ('chi[G]_i x))^* = u ('chi_i x)^*.
Proof. exact: conjC_charAut (irr_char i). Qed.
Lemma cfdot_aut_char u (phi chi : 'CF(G)) :
chi \is a character -> '[cfAut u phi, cfAut u chi] = u '[phi, chi].
Proof. by move/conjC_charAut=> Nchi; apply: cfdot_cfAut => _ /mapP[x _ ->]. Qed.
Lemma cfdot_aut_irr u phi i :
'[cfAut u phi, cfAut u 'chi[G]_i] = u '[phi, 'chi_i].
Proof. exact: cfdot_aut_char (irr_char i). Qed.
Lemma cfAut_irr u chi : (cfAut u chi \in irr G) = (chi \in irr G).
Proof.
rewrite !irrEchar cfAut_char; apply/andb_id2l=> /cfdot_aut_char->.
exact: fmorph_eq1.
Qed.
Lemma cfConjC_irr i : (('chi_i)^*)%CF \in irr G.
Proof. by rewrite cfAut_irr mem_irr. Qed.
Lemma irr_aut_closed u : cfAut_closed u (irr G).
Proof. by move=> chi; rewrite /= cfAut_irr. Qed.
Definition aut_Iirr u i := cfIirr (cfAut u 'chi[G]_i).
Lemma aut_IirrE u i : 'chi_(aut_Iirr u i) = cfAut u 'chi_i.
Proof. by rewrite cfIirrE ?cfAut_irr ?mem_irr. Qed.
Definition conjC_Iirr := aut_Iirr conjC.
Lemma conjC_IirrE i : 'chi[G]_(conjC_Iirr i) = ('chi_i)^*%CF.
Proof. exact: aut_IirrE. Qed.
Lemma conjC_IirrK : involutive conjC_Iirr.
Proof. by move=> i; apply: irr_inj; rewrite !conjC_IirrE cfConjCK. Qed.
Lemma aut_Iirr0 u : aut_Iirr u 0 = 0 :> Iirr G.
Proof. by apply/irr_inj; rewrite aut_IirrE irr0 cfAut_cfun1. Qed.
Lemma conjC_Iirr0 : conjC_Iirr 0 = 0 :> Iirr G.
Proof. exact: aut_Iirr0. Qed.
Lemma aut_Iirr_eq0 u i : (aut_Iirr u i == 0) = (i == 0).
Proof. by rewrite -!irr_eq1 aut_IirrE cfAut_eq1. Qed.
Lemma conjC_Iirr_eq0 i : (conjC_Iirr i == 0 :> Iirr G) = (i == 0).
Proof. exact: aut_Iirr_eq0. Qed.
Lemma aut_Iirr_inj u : injective (aut_Iirr u).
Proof.
by move=> i j eq_ij; apply/irr_inj/(cfAut_inj u); rewrite -!aut_IirrE eq_ij.
Qed.
End Aut.
Arguments aut_Iirr_inj {gT G} u [i1 i2] : rename.
Arguments conjC_IirrK {gT G} i : rename.
Section Coset.
Variable (gT : finGroupType).
Implicit Types G H : {group gT}.
Lemma cfQuo_char G H (chi : 'CF(G)) :
chi \is a character -> (chi / H)%CF \is a character.
Proof.
move=> Nchi; without loss kerH: / H \subset cfker chi.
move/contraNF=> IHchi; apply/wlog_neg=> N'chiH.
suffices ->: (chi / H)%CF = (chi 1%g)%:A.
by rewrite rpredZ_nat ?Cnat_char1 ?rpred1.
by apply/cfunP=> x; rewrite cfunE cfun1E mulr_natr cfunElock IHchi.
without loss nsHG: G chi Nchi kerH / H <| G.
move=> IHchi; have nsHN := normalSG (subset_trans kerH (cfker_sub chi)).
rewrite cfQuoInorm//; apply/cfRes_char/IHchi => //; first exact: cfRes_char.
by apply: sub_cfker_Res => //; apply: normal_sub.
have [rG Dchi] := char_reprP Nchi; rewrite Dchi cfker_repr in kerH.
apply/char_reprP; exists (Representation (quo_repr kerH (normal_norm nsHG))).
apply/cfun_inP=> _ /morphimP[x nHx Gx ->]; rewrite Dchi cfQuoE ?cfker_repr //=.
by rewrite !cfunE Gx quo_repr_coset ?mem_quotient.
Qed.
Lemma cfQuo_lin_char G H (chi : 'CF(G)) :
chi \is a linear_char -> (chi / H)%CF \is a linear_char.
Proof. by case/andP=> Nchi; rewrite qualifE/= cfQuo_char ?cfQuo1. Qed.
Lemma cfMod_char G H (chi : 'CF(G / H)) :
chi \is a character -> (chi %% H)%CF \is a character.
Proof. exact: cfMorph_char. Qed.
Lemma cfMod_lin_char G H (chi : 'CF(G / H)) :
chi \is a linear_char -> (chi %% H)%CF \is a linear_char.
Proof. exact: cfMorph_lin_char. Qed.
Lemma cfMod_charE G H (chi : 'CF(G / H)) :
H <| G -> (chi %% H \is a character)%CF = (chi \is a character).
Proof. by case/andP=> _; apply: cfMorph_charE. Qed.
Lemma cfMod_lin_charE G H (chi : 'CF(G / H)) :
H <| G -> (chi %% H \is a linear_char)%CF = (chi \is a linear_char).
Proof. by case/andP=> _; apply: cfMorph_lin_charE. Qed.
Lemma cfQuo_charE G H (chi : 'CF(G)) :
H <| G -> H \subset cfker chi ->
(chi / H \is a character)%CF = (chi \is a character).
Proof. by move=> nsHG kerH; rewrite -cfMod_charE ?cfQuoK. Qed.
Lemma cfQuo_lin_charE G H (chi : 'CF(G)) :
H <| G -> H \subset cfker chi ->
(chi / H \is a linear_char)%CF = (chi \is a linear_char).
Proof. by move=> nsHG kerH; rewrite -cfMod_lin_charE ?cfQuoK. Qed.
Lemma cfMod_irr G H chi :
H <| G -> (chi %% H \in irr G)%CF = (chi \in irr (G / H)).
Proof. by case/andP=> _; apply: cfMorph_irr. Qed.
Definition mod_Iirr G H i := cfIirr ('chi[G / H]_i %% H)%CF.
Lemma mod_Iirr0 G H : mod_Iirr (0 : Iirr (G / H)) = 0.
Proof. exact: morph_Iirr0. Qed.
Lemma mod_IirrE G H i : H <| G -> 'chi_(mod_Iirr i) = ('chi[G / H]_i %% H)%CF.
Proof. by move=> nsHG; rewrite cfIirrE ?cfMod_irr ?mem_irr. Qed.
Lemma mod_Iirr_eq0 G H i :
H <| G -> (mod_Iirr i == 0) = (i == 0 :> Iirr (G / H)).
Proof. by case/andP=> _ /morph_Iirr_eq0->. Qed.
Lemma cfQuo_irr G H chi :
H <| G -> H \subset cfker chi ->
((chi / H)%CF \in irr (G / H)) = (chi \in irr G).
Proof. by move=> nsHG kerH; rewrite -cfMod_irr ?cfQuoK. Qed.
Definition quo_Iirr G H i := cfIirr ('chi[G]_i / H)%CF.
Lemma quo_Iirr0 G H : quo_Iirr H (0 : Iirr G) = 0.
Proof. by rewrite /quo_Iirr irr0 cfQuo_cfun1 -irr0 irrK. Qed.
Lemma quo_IirrE G H i :
H <| G -> H \subset cfker 'chi[G]_i -> 'chi_(quo_Iirr H i) = ('chi_i / H)%CF.
Proof. by move=> nsHG kerH; rewrite cfIirrE ?cfQuo_irr ?mem_irr. Qed.
Lemma quo_Iirr_eq0 G H i :
H <| G -> H \subset cfker 'chi[G]_i -> (quo_Iirr H i == 0) = (i == 0).
Proof. by move=> nsHG kerH; rewrite -!irr_eq1 quo_IirrE ?cfQuo_eq1. Qed.
Lemma mod_IirrK G H : H <| G -> cancel (@mod_Iirr G H) (@quo_Iirr G H).
Proof.
move=> nsHG i; apply: irr_inj.
by rewrite quo_IirrE ?mod_IirrE ?cfker_mod // cfModK.
Qed.
Lemma quo_IirrK G H i :
H <| G -> H \subset cfker 'chi[G]_i -> mod_Iirr (quo_Iirr H i) = i.
Proof.
by move=> nsHG kerH; apply: irr_inj; rewrite mod_IirrE ?quo_IirrE ?cfQuoK.
Qed.
Lemma quo_IirrKeq G H :
H <| G ->
forall i, (mod_Iirr (quo_Iirr H i) == i) = (H \subset cfker 'chi[G]_i).
Proof.
move=> nsHG i; apply/eqP/idP=> [<- | ]; last exact: quo_IirrK.
by rewrite mod_IirrE ?cfker_mod.
Qed.
Lemma mod_Iirr_bij H G :
H <| G -> {on [pred i | H \subset cfker 'chi_i], bijective (@mod_Iirr G H)}.
Proof.
by exists (quo_Iirr H) => [i _ | i]; [apply: mod_IirrK | apply: quo_IirrK].
Qed.
Lemma sum_norm_irr_quo H G x :
x \in G -> H <| G ->
\sum_i `|'chi[G / H]_i (coset H x)| ^+ 2
= \sum_(i | H \subset cfker 'chi_i) `|'chi[G]_i x| ^+ 2.
Proof.
move=> Gx nsHG; rewrite (reindex _ (mod_Iirr_bij nsHG)) /=.
by apply/esym/eq_big=> [i | i _]; rewrite mod_IirrE ?cfker_mod ?cfModE.
Qed.
Lemma cap_cfker_normal G H :
H <| G -> \bigcap_(i | H \subset cfker 'chi[G]_i) (cfker 'chi_i) = H.
Proof.
move=> nsHG; have [sHG nHG] := andP nsHG; set lhs := \bigcap_(i | _) _.
have nHlhs: lhs \subset 'N(H) by rewrite (bigcap_min 0) ?cfker_irr0.
apply/esym/eqP; rewrite eqEsubset (introT bigcapsP) //= -quotient_sub1 //.
rewrite -(TI_cfker_irr (G / H)); apply/bigcapsP=> i _.
rewrite sub_quotient_pre // (bigcap_min (mod_Iirr i)) ?mod_IirrE ?cfker_mod //.
by rewrite cfker_morph ?subsetIr.
Qed.
Lemma cfker_reg_quo G H : H <| G -> cfker (cfReg (G / H)%g %% H) = H.
Proof.
move=> nsHG; have [sHG nHG] := andP nsHG.
apply/setP=> x; rewrite cfkerEchar ?cfMod_char ?cfReg_char //.
rewrite -[in RHS in _ = RHS](setIidPr sHG) !inE; apply: andb_id2l => Gx.
rewrite !cfModE // !cfRegE // morph1 eqxx.
rewrite (sameP eqP (kerP _ (subsetP nHG x Gx))) ker_coset.
by rewrite -!mulrnA eqr_nat eqn_pmul2l ?cardG_gt0 // (can_eq oddb) eqb_id.
Qed.
End Coset.
Section DerivedGroup.
Variable gT : finGroupType.
Implicit Types G H : {group gT}.
Lemma lin_irr_der1 G i :
('chi_i \is a linear_char) = (G^`(1)%g \subset cfker 'chi[G]_i).
Proof.
apply/idP/idP=> [|sG'K]; first exact: lin_char_der1.
have nsG'G: G^`(1) <| G := der_normal 1 G.
rewrite qualifE/= irr_char -[i](quo_IirrK nsG'G) // mod_IirrE //=.
by rewrite cfModE // morph1 lin_char1 //; apply/char_abelianP/der_abelian.
Qed.
Lemma subGcfker G i : (G \subset cfker 'chi[G]_i) = (i == 0).
Proof.
rewrite -irr_eq1; apply/idP/eqP=> [chiG1 | ->]; last by rewrite cfker_cfun1.
apply/cfun_inP=> x Gx; rewrite cfun1E Gx cfker1 ?(subsetP chiG1) ?lin_char1 //.
by rewrite lin_irr_der1 (subset_trans (der_sub 1 G)).
Qed.
Lemma irr_prime_injP G i :
prime #|G| -> reflect {in G &, injective 'chi[G]_i} (i != 0).
Proof.
move=> pr_G; apply: (iffP idP) => [nz_i | inj_chi].
apply: fful_lin_char_inj (irr_prime_lin i pr_G) _.
by rewrite cfaithfulE -(setIidPr (cfker_sub _)) prime_TIg // subGcfker.
have /trivgPn[x Gx ntx]: G :!=: 1%g by rewrite -cardG_gt1 prime_gt1.
apply: contraNneq ntx => i0; apply/eqP/inj_chi=> //.
by rewrite i0 irr0 !cfun1E Gx group1.
Qed.
(* This is Isaacs (2.23)(a). *)
Lemma cap_cfker_lin_irr G :
\bigcap_(i | 'chi[G]_i \is a linear_char) (cfker 'chi_i) = G^`(1)%g.
Proof.
rewrite -(cap_cfker_normal (der_normal 1 G)).
by apply: eq_bigl => i; rewrite lin_irr_der1.
Qed.
(* This is Isaacs (2.23)(b) *)
Lemma card_lin_irr G :
#|[pred i | 'chi[G]_i \is a linear_char]| = #|G : G^`(1)%g|.
Proof.
have nsG'G := der_normal 1 G; rewrite (eq_card (@lin_irr_der1 G)).
rewrite -(on_card_preimset (mod_Iirr_bij nsG'G)).
rewrite -card_quotient ?normal_norm //.
move: (der_abelian 0 G); rewrite card_classes_abelian; move/eqP<-.
rewrite -NirrE -[RHS]card_ord.
by apply: eq_card => i; rewrite !inE mod_IirrE ?cfker_mod.
(* Alternative: use the equivalent result in modular representation theory
transitivity #|@socle_of_Iirr _ G @^-1: linear_irr _|; last first.
rewrite (on_card_preimset (socle_of_Iirr_bij _)).
by rewrite card_linear_irr ?algC'G; last apply: groupC.
by apply: eq_card => i; rewrite !inE /lin_char irr_char irr1_degree -eqC_nat.
*)
Qed.
(* A non-trivial solvable group has a nonprincipal linear character. *)
Lemma solvable_has_lin_char G :
G :!=: 1%g -> solvable G ->
exists2 i, 'chi[G]_i \is a linear_char & 'chi_i != 1.
Proof.
move=> ntG solG.
suff /subsetPn[i]: ~~ ([pred i | 'chi[G]_i \is a linear_char] \subset pred1 0).
by rewrite !inE -(inj_eq irr_inj) irr0; exists i.
rewrite (contra (@subset_leq_card _ _ _)) // -ltnNge card1 card_lin_irr.
by rewrite indexg_gt1 proper_subn // (sol_der1_proper solG).
Qed.
(* A combinatorial group isommorphic to the linear characters. *)
Lemma lin_char_group G :
{linG : finGroupType & {cF : linG -> 'CF(G) |
[/\ injective cF, #|linG| = #|G : G^`(1)|,
forall u, cF u \is a linear_char
& forall phi, phi \is a linear_char -> exists u, phi = cF u]
& [/\ cF 1%g = 1%R,
{morph cF : u v / (u * v)%g >-> (u * v)%R},
forall k, {morph cF : u / (u^+ k)%g >-> u ^+ k},
{morph cF: u / u^-1%g >-> u^-1%CF}
& {mono cF: u / #[u]%g >-> #[u]%CF} ]}}.
Proof.
pose linT := {i : Iirr G | 'chi_i \is a linear_char}.
pose cF (u : linT) := 'chi_(sval u).
have cFlin u: cF u \is a linear_char := svalP u.
have cFinj: injective cF := inj_comp irr_inj val_inj.
have inT xi : xi \is a linear_char -> {u | cF u = xi}.
move=> lin_xi; have /irrP/sig_eqW[i Dxi] := lin_char_irr lin_xi.
by apply: (exist _ (Sub i _)) => //; rewrite -Dxi.
have [one cFone] := inT 1 (rpred1 _).
pose inv u := sval (inT _ (rpredVr (cFlin u))).
pose mul u v := sval (inT _ (rpredM (cFlin u) (cFlin v))).
have cFmul u v: cF (mul u v) = cF u * cF v := svalP (inT _ _).
have cFinv u: cF (inv u) = (cF u)^-1 := svalP (inT _ _).
have mulA: associative mul by move=> u v w; apply: cFinj; rewrite !cFmul mulrA.
have mul1: left_id one mul by move=> u; apply: cFinj; rewrite cFmul cFone mul1r.
have mulV: left_inverse one inv mul.
by move=> u; apply: cFinj; rewrite cFmul cFinv cFone mulVr ?lin_char_unitr.
pose imA := isMulGroup.Build linT mulA mul1 mulV.
pose linG : finGroupType := HB.pack linT imA.
have cFexp k: {morph cF : u / ((u : linG) ^+ k)%g >-> u ^+ k}.
by move=> u; elim: k => // k IHk; rewrite expgS exprS cFmul IHk.
do [exists linG, cF; split=> //] => [|xi /inT[u <-]|u]; first 2 [by exists u].
have inj_cFI: injective (cfIirr \o cF).
apply: can_inj (insubd one) _ => u; apply: val_inj.
by rewrite insubdK /= ?irrK //; apply: cFlin.
rewrite -(card_image inj_cFI) -card_lin_irr.
apply/eq_card=> i /[1!inE]; apply/codomP/idP=> [[u ->] | /inT[u Du]].
by rewrite /= irrK; apply: cFlin.
by exists u; apply: irr_inj; rewrite /= irrK.
apply/eqP; rewrite eqn_dvd; apply/andP; split.
by rewrite dvdn_cforder; rewrite -cFexp expg_order cFone.
by rewrite order_dvdn -(inj_eq cFinj) cFone cFexp exp_cforder.
Qed.
Lemma cfExp_prime_transitive G (i j : Iirr G) :
prime #|G| -> i != 0 -> j != 0 ->
exists2 k, coprime k #['chi_i]%CF & 'chi_j = 'chi_i ^+ k.
Proof.
set p := #|G| => pr_p nz_i nz_j; have cycG := prime_cyclic pr_p.
have [L [h [injh oL Lh h_ontoL]] [h1 hM hX _ o_h]] := lin_char_group G.
rewrite (derG1P (cyclic_abelian cycG)) indexg1 -/p in oL.
have /fin_all_exists[h' h'K] := h_ontoL _ (irr_cyclic_lin _ cycG).
have o_h' k: k != 0 -> #[h' k] = p.
rewrite -cforder_irr_eq1 h'K -o_h => nt_h'k.
by apply/prime_nt_dvdP=> //; rewrite cforder_lin_char_dvdG.
have{oL} genL k: k != 0 -> generator [set: L] (h' k).
move=> /o_h' o_h'k; rewrite /generator eq_sym eqEcard subsetT /=.
by rewrite cardsT oL -o_h'k.
have [/(_ =P <[_]>)-> gen_j] := (genL i nz_i, genL j nz_j).
have /cycleP[k Dj] := cycle_generator gen_j.
by rewrite !h'K Dj o_h hX generator_coprime coprime_sym in gen_j *; exists k.
Qed.
(* This is Isaacs (2.24). *)
Lemma card_subcent1_coset G H x :
x \in G -> H <| G -> (#|'C_(G / H)[coset H x]| <= #|'C_G[x]|)%N.
Proof.
move=> Gx nsHG; rewrite -leC_nat.
move: (second_orthogonality_relation x Gx); rewrite mulrb class_refl => <-.
have GHx: coset H x \in (G / H)%g by apply: mem_quotient.
move: (second_orthogonality_relation (coset H x) GHx).
rewrite mulrb class_refl => <-.
rewrite -2!(eq_bigr _ (fun _ _ => normCK _)) sum_norm_irr_quo // -subr_ge0.
rewrite (bigID (fun i => H \subset cfker 'chi[G]_i)) //= [X in X + _]addrC addrK.
by apply: sumr_ge0 => i _; rewrite normCK mul_conjC_ge0.
Qed.
End DerivedGroup.
Arguments irr_prime_injP {gT G i}.
(* Determinant characters and determinential order. *)
Section DetRepr.
Variables (gT : finGroupType) (G : {group gT}).
Variables (n : nat) (rG : mx_representation algC G n).
Definition det_repr_mx x : 'M_1 := (\det (rG x))%:M.
Fact det_is_repr : mx_repr G det_repr_mx.
Proof.
split=> [|g h Gg Gh]; first by rewrite /det_repr_mx repr_mx1 det1.
by rewrite /det_repr_mx repr_mxM // det_mulmx !mulmxE scalar_mxM.
Qed.
Canonical det_repr := MxRepresentation det_is_repr.
Definition detRepr := cfRepr det_repr.
Lemma detRepr_lin_char : detRepr \is a linear_char.
Proof.
by rewrite qualifE/= cfRepr_char cfunE group1 repr_mx1 mxtrace1 mulr1n /=.
Qed.
End DetRepr.
HB.lock
Definition cfDet (gT : finGroupType) (G : {group gT}) phi :=
\prod_i detRepr 'Chi_i ^+ Num.truncn '[phi, 'chi[G]_i].
Canonical cfDet_unlockable := Unlockable cfDet.unlock.
Section DetOrder.
Variables (gT : finGroupType) (G : {group gT}).
Local Notation cfDet := (@cfDet gT G).
Lemma cfDet_lin_char phi : cfDet phi \is a linear_char.
Proof.
rewrite unlock; apply: rpred_prod => i _; apply: rpredX.
exact: detRepr_lin_char.
Qed.
Lemma cfDetD :
{in character &, {morph cfDet : phi psi / phi + psi >-> phi * psi}}.
Proof.
move=> phi psi Nphi Npsi; rewrite unlock /= -big_split; apply: eq_bigr => i _ /=.
by rewrite -exprD cfdotDl truncnD ?nnegrE ?natr_ge0 // Cnat_cfdot_char_irr.
Qed.
Lemma cfDet0 : cfDet 0 = 1.
Proof. by rewrite unlock big1 // => i _; rewrite cfdot0l truncn0. Qed.
Lemma cfDetMn k :
{in character, {morph cfDet : phi / phi *+ k >-> phi ^+ k}}.
Proof.
move=> phi Nphi; elim: k => [|k IHk]; rewrite ?cfDet0 // mulrS exprS -{}IHk.
by rewrite cfDetD ?rpredMn.
Qed.
Lemma cfDetRepr n rG : cfDet (cfRepr rG) = @detRepr _ _ n rG.
Proof.
transitivity (\prod_W detRepr (socle_repr W) ^+ standard_irr_coef rG W).
rewrite (reindex _ (socle_of_Iirr_bij _)) unlock /=.
apply: eq_bigr => i _; congr (_ ^+ _).
rewrite (cfRepr_sim (mx_rsim_standard rG)) cfRepr_standard.
rewrite cfdot_suml (bigD1 i) ?big1 //= => [|j i'j]; last first.
by rewrite cfdotZl cfdot_irr (negPf i'j) mulr0.
by rewrite cfdotZl cfnorm_irr mulr1 addr0 natrK.
apply/cfun_inP=> x Gx; rewrite prod_cfunE //.
transitivity (detRepr (standard_grepr rG) x); last first.
rewrite !cfunE Gx !trace_mx11 !mxE eqxx !mulrb.
case: (standard_grepr rG) (mx_rsim_standard rG) => /= n1 rG1 [B Dn1].
rewrite -{n1}Dn1 in rG1 B *; rewrite row_free_unit => uB rG_B.
by rewrite -[rG x](mulmxK uB) rG_B // !det_mulmx mulrC -!det_mulmx mulKmx.
rewrite /standard_grepr; elim/big_rec2: _ => [|W y _ _ ->].
by rewrite cfunE trace_mx11 mxE Gx det1.
rewrite !cfunE Gx /= !{1}trace_mx11 !{1}mxE det_ublock; congr (_ * _).
rewrite exp_cfunE //; elim: (standard_irr_coef rG W) => /= [|k IHk].
by rewrite /muln_grepr big_ord0 det1.
rewrite exprS /muln_grepr big_ord_recl det_ublock -IHk; congr (_ * _).
by rewrite cfunE trace_mx11 mxE Gx.
Qed.
Lemma cfDet_id xi : xi \is a linear_char -> cfDet xi = xi.
Proof.
move=> lin_xi; have /irrP[i Dxi] := lin_char_irr lin_xi.
apply/cfun_inP=> x Gx; rewrite Dxi -irrRepr cfDetRepr !cfunE trace_mx11 mxE.
move: lin_xi (_ x) => /andP[_]; rewrite Dxi irr1_degree pnatr_eq1 => /eqP-> X.
by rewrite {1}[X]mx11_scalar det_scalar1 trace_mx11.
Qed.
Definition cfDet_order phi := #[cfDet phi]%CF.
Definition cfDet_order_lin xi :
xi \is a linear_char -> cfDet_order xi = #[xi]%CF.
Proof. by rewrite /cfDet_order => /cfDet_id->. Qed.
Definition cfDet_order_dvdG phi : cfDet_order phi %| #|G|.
Proof. by rewrite cforder_lin_char_dvdG ?cfDet_lin_char. Qed.
End DetOrder.
Notation "''o' ( phi )" := (cfDet_order phi)
(format "''o' ( phi )") : cfun_scope.
Section CfDetOps.
Implicit Types gT aT rT : finGroupType.
Lemma cfDetRes gT (G H : {group gT}) phi :
phi \is a character -> cfDet ('Res[H, G] phi) = 'Res (cfDet phi).
Proof.
move=> Nphi; have [sGH | not_sHG] := boolP (H \subset G); last first.
have /natrP[n Dphi1] := Cnat_char1 Nphi.
rewrite !cfResEout // Dphi1 lin_char1 ?cfDet_lin_char // scale1r.
by rewrite scaler_nat cfDetMn ?cfDet_id ?rpred1 // expr1n.
have [rG ->] := char_reprP Nphi; rewrite !(=^~ cfRepr_sub, cfDetRepr) //.
apply: cfRepr_sim; exists 1%:M; rewrite ?row_free_unit ?unitmx1 // => x Hx.
by rewrite mulmx1 mul1mx.
Qed.
Lemma cfDetMorph aT rT (D G : {group aT}) (f : {morphism D >-> rT})
(phi : 'CF(f @* G)) :
phi \is a character -> cfDet (cfMorph phi) = cfMorph (cfDet phi).
Proof.
move=> Nphi; have [sGD | not_sGD] := boolP (G \subset D); last first.
have /natrP[n Dphi1] := Cnat_char1 Nphi.
rewrite !cfMorphEout // Dphi1 lin_char1 ?cfDet_lin_char // scale1r.
by rewrite scaler_nat cfDetMn ?cfDet_id ?rpred1 // expr1n.
have [rG ->] := char_reprP Nphi; rewrite !(=^~ cfRepr_morphim, cfDetRepr) //.
apply: cfRepr_sim; exists 1%:M; rewrite ?row_free_unit ?unitmx1 // => x Hx.
by rewrite mulmx1 mul1mx.
Qed.
Lemma cfDetIsom aT rT (G : {group aT}) (R : {group rT})
(f : {morphism G >-> rT}) (isoGR : isom G R f) phi :
cfDet (cfIsom isoGR phi) = cfIsom isoGR (cfDet phi).
Proof.
rewrite unlock rmorph_prod (reindex (isom_Iirr isoGR)); last first.
by exists (isom_Iirr (isom_sym isoGR)) => i; rewrite ?isom_IirrK ?isom_IirrKV.
apply: eq_bigr=> i; rewrite -!cfDetRepr !irrRepr isom_IirrE rmorphXn cfIsom_iso.
by rewrite /= ![in cfIsom _]unlock cfDetMorph ?cfRes_char ?cfDetRes ?irr_char.
Qed.
Lemma cfDet_mul_lin gT (G : {group gT}) (lambda phi : 'CF(G)) :
lambda \is a linear_char -> phi \is a character ->
cfDet (lambda * phi) = lambda ^+ Num.truncn (phi 1%g) * cfDet phi.
Proof.
case/andP=> /char_reprP[[n1 rG1] ->] /= n1_1 /char_reprP[[n2 rG2] ->] /=.
do [rewrite !cfRepr1 pnatr_eq1 natrK; move/eqP] in n1_1 *.
rewrite {n1}n1_1 in rG1 *; rewrite cfRepr_prod cfDetRepr.
apply/cfun_inP=> x Gx; rewrite !cfunE cfDetRepr cfunE Gx !mulrb !trace_mx11.
rewrite !mxE prod_repr_lin ?mulrb //=; case: _ / (esym _); rewrite detZ.
congr (_ * _); case: {rG2}n2 => [|n2]; first by rewrite cfun1E Gx.
by rewrite expS_cfunE //= cfunE Gx trace_mx11.
Qed.
End CfDetOps.
Definition cfcenter (gT : finGroupType) (G : {set gT}) (phi : 'CF(G)) :=
if phi \is a character then [set g in G | `|phi g| == phi 1%g] else cfker phi.
Notation "''Z' ( phi )" := (cfcenter phi) : cfun_scope.
Section Center.
Variable (gT : finGroupType) (G : {group gT}).
Implicit Types (phi chi : 'CF(G)) (H : {group gT}).
(* This is Isaacs (2.27)(a). *)
Lemma cfcenter_repr n (rG : mx_representation algC G n) :
'Z(cfRepr rG)%CF = rcenter rG.
Proof.
rewrite /cfcenter /rcenter cfRepr_char /=.
apply/setP=> x /[!inE]; apply/andb_id2l=> Gx.
apply/eqP/is_scalar_mxP=> [|[c rG_c]].
by case/max_cfRepr_norm_scalar=> // c; exists c.
rewrite -(sqrCK (char1_ge0 (cfRepr_char rG))) normC_def; congr (sqrtC _).
rewrite expr2 -{2}(mulgV x) -char_inv ?cfRepr_char ?cfunE ?groupM ?groupV //.
rewrite Gx group1 repr_mx1 repr_mxM ?repr_mxV ?groupV // !mulrb rG_c.
by rewrite invmx_scalar -scalar_mxM !mxtrace_scalar mulrnAr mulrnAl mulr_natl.
Qed.
(* This is part of Isaacs (2.27)(b). *)
Fact cfcenter_group_set phi : group_set ('Z(phi))%CF.
Proof.
have [[rG ->] | /negbTE notNphi] := altP (@char_reprP _ G phi).
by rewrite cfcenter_repr groupP.
by rewrite /cfcenter notNphi groupP.
Qed.
Canonical cfcenter_group f := Group (cfcenter_group_set f).
Lemma char_cfcenterE chi x :
chi \is a character -> x \in G ->
(x \in ('Z(chi))%CF) = (`|chi x| == chi 1%g).
Proof. by move=> Nchi Gx; rewrite /cfcenter Nchi inE Gx. Qed.
Lemma irr_cfcenterE i x :
x \in G -> (x \in 'Z('chi[G]_i)%CF) = (`|'chi_i x| == 'chi_i 1%g).
Proof. by move/char_cfcenterE->; rewrite ?irr_char. Qed.
(* This is also Isaacs (2.27)(b). *)
Lemma cfcenter_sub phi : ('Z(phi))%CF \subset G.
Proof. by rewrite /cfcenter /cfker !setIdE -fun_if subsetIl. Qed.
Lemma cfker_center_normal phi : cfker phi <| 'Z(phi)%CF.
Proof.
apply: normalS (cfcenter_sub phi) (cfker_normal phi).
rewrite /= /cfcenter; case: ifP => // Hphi; rewrite cfkerEchar //.
apply/subsetP=> x /[!inE] /andP[-> /eqP->] /=.
by rewrite ger0_norm ?char1_ge0.
Qed.
Lemma cfcenter_normal phi : 'Z(phi)%CF <| G.
Proof.
have [[rG ->] | /negbTE notNphi] := altP (@char_reprP _ _ phi).
by rewrite cfcenter_repr rcenter_normal.
by rewrite /cfcenter notNphi cfker_normal.
Qed.
(* This is Isaacs (2.27)(c). *)
Lemma cfcenter_Res chi :
exists2 chi1, chi1 \is a linear_char & 'Res['Z(chi)%CF] chi = chi 1%g *: chi1.
Proof.
have [[rG ->] | /negbTE notNphi] := altP (@char_reprP _ _ chi); last first.
exists 1; first exact: cfun1_lin_char.
rewrite /cfcenter notNphi; apply/cfun_inP=> x Kx.
by rewrite cfunE cfun1E Kx mulr1 cfResE ?cfker_sub // cfker1.
rewrite cfcenter_repr -(cfRepr_sub _ (normal_sub (rcenter_normal _))).
case: rG => [[|n] rG] /=; rewrite cfRepr1.
exists 1; first exact: cfun1_lin_char.
by apply/cfun_inP=> x Zx; rewrite scale0r !cfunE flatmx0 raddf0 Zx.
pose rZmx x := ((rG x 0 0)%:M : 'M_(1,1)).
have rZmxP: mx_repr [group of rcenter rG] rZmx.
split=> [|x y]; first by rewrite /rZmx repr_mx1 mxE eqxx.
move=> /setIdP[Gx /is_scalar_mxP[a rGx]] /setIdP[Gy /is_scalar_mxP[b rGy]].
by rewrite /rZmx repr_mxM // rGx rGy -!scalar_mxM !mxE.
exists (cfRepr (MxRepresentation rZmxP)).
by rewrite qualifE/= cfRepr_char cfRepr1 eqxx.
apply/cfun_inP=> x Zx; rewrite !cfunE Zx /= /rZmx mulr_natl.
by case/setIdP: Zx => Gx /is_scalar_mxP[a ->]; rewrite mxE !mxtrace_scalar.
Qed.
(* This is Isaacs (2.27)(d). *)
Lemma cfcenter_cyclic chi : cyclic ('Z(chi)%CF / cfker chi)%g.
Proof.
case Nchi: (chi \is a character); last first.
by rewrite /cfcenter Nchi trivg_quotient cyclic1.
have [-> | nz_chi] := eqVneq chi 0.
rewrite quotientS1 ?cyclic1 //= /cfcenter cfkerEchar ?cfun0_char //.
by apply/subsetP=> x /setIdP[Gx _]; rewrite inE Gx /= !cfunE.
have [xi Lxi def_chi] := cfcenter_Res chi.
set Z := ('Z(_))%CF in xi Lxi def_chi *.
have sZG: Z \subset G by apply: cfcenter_sub.
have ->: cfker chi = cfker xi.
rewrite -(setIidPr (normal_sub (cfker_center_normal _))) -/Z.
rewrite !cfkerEchar // ?lin_charW //= -/Z.
apply/setP=> x /[!inE]; apply: andb_id2l => Zx.
rewrite (subsetP sZG) //= -!(cfResE chi sZG) ?group1 // def_chi !cfunE.
by rewrite (inj_eq (mulfI _)) ?char1_eq0.
have: abelian (Z / cfker xi) by rewrite sub_der1_abelian ?lin_char_der1.
have /irr_reprP[rG irrG ->] := lin_char_irr Lxi; rewrite cfker_repr.
apply: mx_faithful_irr_abelian_cyclic (kquo_mx_faithful rG) _.
exact/quo_mx_irr.
Qed.
(* This is Isaacs (2.27)(e). *)
Lemma cfcenter_subset_center chi :
('Z(chi)%CF / cfker chi)%g \subset 'Z(G / cfker chi)%g.
Proof.
case Nchi: (chi \is a character); last first.
by rewrite /cfcenter Nchi trivg_quotient sub1G.
rewrite subsetI quotientS ?cfcenter_sub // quotient_cents2r //=.
case/char_reprP: Nchi => rG ->{chi}; rewrite cfker_repr cfcenter_repr gen_subG.
apply/subsetP=> _ /imset2P[x y /setIdP[Gx /is_scalar_mxP[c rGx]] Gy ->].
rewrite inE groupR //= !repr_mxM ?groupM ?groupV // rGx -(scalar_mxC c) -rGx.
by rewrite !mulmxA !repr_mxKV.
Qed.
(* This is Isaacs (2.27)(f). *)
Lemma cfcenter_eq_center (i : Iirr G) :
('Z('chi_i)%CF / cfker 'chi_i)%g = 'Z(G / cfker 'chi_i)%g.
Proof.
apply/eqP; rewrite eqEsubset; rewrite cfcenter_subset_center ?irr_char //.
apply/subsetP=> _ /setIP[/morphimP[x /= _ Gx ->] cGx]; rewrite mem_quotient //=.
rewrite -irrRepr cfker_repr cfcenter_repr inE Gx in cGx *.
apply: mx_abs_irr_cent_scalar 'Chi_i _ _ _; first exact/groupC/socle_irr.
have nKG: G \subset 'N(rker 'Chi_i) by apply: rker_norm.
(* GG -- locking here is critical to prevent Coq kernel divergence. *)
apply/centgmxP=> y Gy; rewrite [eq]lock -2?(quo_repr_coset (subxx _) nKG) //.
move: (quo_repr _ _) => rG; rewrite -2?repr_mxM ?mem_quotient // -lock.
by rewrite (centP cGx) // mem_quotient.
Qed.
(* This is Isaacs (2.28). *)
Lemma cap_cfcenter_irr : \bigcap_i 'Z('chi[G]_i)%CF = 'Z(G).
Proof.
apply/esym/eqP; rewrite eqEsubset (introT bigcapsP) /= => [|i _]; last first.
rewrite -(quotientSGK _ (normal_sub (cfker_center_normal _))).
by rewrite cfcenter_eq_center morphim_center.
by rewrite subIset // normal_norm // cfker_normal.
set Z := \bigcap_i _.
have sZG: Z \subset G by rewrite (bigcap_min 0) ?cfcenter_sub.
rewrite subsetI sZG (sameP commG1P trivgP) -(TI_cfker_irr G).
apply/bigcapsP=> i _; have nKiG := normal_norm (cfker_normal 'chi_i).
rewrite -quotient_cents2 ?(subset_trans sZG) //.
rewrite (subset_trans (quotientS _ (bigcap_inf i _))) //.
by rewrite cfcenter_eq_center subsetIr.
Qed.
(* This is Isaacs (2.29). *)
Lemma cfnorm_Res_leif H phi :
H \subset G ->
'['Res[H] phi] <= #|G : H|%:R * '[phi] ?= iff (phi \in 'CF(G, H)).
Proof.
move=> sHG; rewrite cfun_onE mulrCA natf_indexg // -mulrA mulKf ?neq0CG //.
rewrite (big_setID H) (setIidPr sHG) /= addrC.
rewrite (mono_leif (ler_pM2l _)) ?invr_gt0 ?gt0CG // -leifBLR -sumrB.
rewrite big1 => [|x Hx]; last by rewrite !cfResE ?subrr.
have ->: (support phi \subset H) = (G :\: H \subset [set x | phi x == 0]).
rewrite subDset setUC -subDset; apply: eq_subset => x.
by rewrite !inE (andb_idr (contraR _)) // => /cfun0->.
rewrite (sameP subsetP forall_inP); apply: leif_0_sum => x _.
by rewrite !inE /<?=%R mul_conjC_ge0 eq_sym mul_conjC_eq0.
Qed.
(* This is Isaacs (2.30). *)
Lemma irr1_bound (i : Iirr G) :
('chi_i 1%g) ^+ 2 <= #|G : 'Z('chi_i)%CF|%:R
?= iff ('chi_i \in 'CF(G, 'Z('chi_i)%CF)).
Proof.
congr (_ <= _ ?= iff _): (cfnorm_Res_leif 'chi_i (cfcenter_sub 'chi_i)).
have [xi Lxi ->] := cfcenter_Res 'chi_i.
have /irrP[j ->] := lin_char_irr Lxi; rewrite cfdotZl cfdotZr cfdot_irr eqxx.
by rewrite mulr1 irr1_degree conjC_nat.
by rewrite cfdot_irr eqxx mulr1.
Qed.
(* This is Isaacs (2.31). *)
Lemma irr1_abelian_bound (i : Iirr G) :
abelian (G / 'Z('chi_i)%CF) -> ('chi_i 1%g) ^+ 2 = #|G : 'Z('chi_i)%CF|%:R.
Proof.
move=> AbGc; apply/eqP; rewrite irr1_bound cfun_onE; apply/subsetP=> x nz_chi_x.
have Gx: x \in G by apply: contraR nz_chi_x => /cfun0->.
have nKx := subsetP (normal_norm (cfker_normal 'chi_i)) _ Gx.
rewrite -(quotientGK (cfker_center_normal _)) inE nKx inE /=.
rewrite cfcenter_eq_center inE mem_quotient //=.
apply/centP=> _ /morphimP[y nKy Gy ->]; apply/commgP; rewrite -morphR //=.
set z := [~ x, y]; rewrite coset_id //.
have: z \in 'Z('chi_i)%CF.
apply: subsetP (mem_commg Gx Gy).
by rewrite der1_min // normal_norm ?cfcenter_normal.
rewrite -irrRepr cfker_repr cfcenter_repr !inE in nz_chi_x *.
case/andP=> Gz /is_scalar_mxP[c Chi_z]; rewrite Gz Chi_z mul1mx /=.
apply/eqP; congr _%:M; apply: (mulIf nz_chi_x); rewrite mul1r.
rewrite -{2}(cfunJ _ x Gy) conjg_mulR -/z !cfunE Gx groupM // !{1}mulrb.
by rewrite repr_mxM // Chi_z mul_mx_scalar mxtraceZ.
Qed.
(* This is Isaacs (2.32)(a). *)
Lemma irr_faithful_center i : cfaithful 'chi[G]_i -> cyclic 'Z(G).
Proof.
rewrite (isog_cyclic (isog_center (quotient1_isog G))) /=.
by move/trivgP <-; rewrite -cfcenter_eq_center cfcenter_cyclic.
Qed.
Lemma cfcenter_fful_irr i : cfaithful 'chi[G]_i -> 'Z('chi_i)%CF = 'Z(G).
Proof.
move/trivgP=> Ki1; have:= cfcenter_eq_center i; rewrite {}Ki1.
have inj1: 'injm (@coset gT 1%g) by rewrite ker_coset.
by rewrite -injm_center; first apply: injm_morphim_inj; rewrite ?norms1.
Qed.
(* This is Isaacs (2.32)(b). *)
Lemma pgroup_cyclic_faithful (p : nat) :
p.-group G -> cyclic 'Z(G) -> exists i, cfaithful 'chi[G]_i.
Proof.
pose Z := 'Ohm_1('Z(G)) => pG cycZG; have nilG := pgroup_nil pG.
have [-> | ntG] := eqsVneq G [1]; first by exists 0; apply: cfker_sub.
have{pG} [[p_pr _ _] pZ] := (pgroup_pdiv pG ntG, pgroupS (center_sub G) pG).
have ntZ: 'Z(G) != [1] by rewrite center_nil_eq1.
have{pZ} oZ: #|Z| = p by apply: Ohm1_cyclic_pgroup_prime.
apply/existsP; apply: contraR ntZ => /existsPn-not_ffulG.
rewrite -Ohm1_eq1 -subG1 /= -/Z -(TI_cfker_irr G); apply/bigcapsP=> i _.
rewrite prime_meetG ?oZ // setIC meet_Ohm1 // meet_center_nil ?cfker_normal //.
by rewrite -subG1 not_ffulG.
Qed.
End Center.
Section Induced.
Variables (gT : finGroupType) (G H : {group gT}).
Implicit Types (phi : 'CF(G)) (chi : 'CF(H)).
Lemma cfInd_char chi : chi \is a character -> 'Ind[G] chi \is a character.
Proof.
move=> Nchi; apply/forallP=> i; rewrite coord_cfdot -Frobenius_reciprocity //.
by rewrite Cnat_cfdot_char ?cfRes_char ?irr_char.
Qed.
Lemma cfInd_eq0 chi :
H \subset G -> chi \is a character -> ('Ind[G] chi == 0) = (chi == 0).
Proof.
move=> sHG Nchi; rewrite -!(char1_eq0) ?cfInd_char // cfInd1 //.
by rewrite (mulrI_eq0 _ (mulfI _)) ?neq0CiG.
Qed.
Lemma Ind_irr_neq0 i : H \subset G -> 'Ind[G, H] 'chi_i != 0.
Proof. by move/cfInd_eq0->; rewrite ?irr_neq0 ?irr_char. Qed.
Definition Ind_Iirr (A B : {set gT}) i := cfIirr ('Ind[B, A] 'chi_i).
Lemma constt_cfRes_irr i : {j | j \in irr_constt ('Res[H, G] 'chi_i)}.
Proof. apply/sigW/neq0_has_constt/Res_irr_neq0. Qed.
Lemma constt_cfInd_irr i :
H \subset G -> {j | j \in irr_constt ('Ind[G, H] 'chi_i)}.
Proof. by move=> sHG; apply/sigW/neq0_has_constt/Ind_irr_neq0. Qed.
Lemma cfker_Res phi :
H \subset G -> phi \is a character -> cfker ('Res[H] phi) = H :&: cfker phi.
Proof.
move=> sHG Nphi; apply/setP=> x; rewrite !cfkerEchar ?cfRes_char // !inE.
by apply/andb_id2l=> Hx; rewrite (subsetP sHG) ?cfResE.
Qed.
(* This is Isaacs Lemma (5.11). *)
Lemma cfker_Ind chi :
H \subset G -> chi \is a character -> chi != 0 ->
cfker ('Ind[G, H] chi) = gcore (cfker chi) G.
Proof.
move=> sHG Nchi nzchi; rewrite !cfker_nzcharE ?cfInd_char ?cfInd_eq0 //.
apply/setP=> x; rewrite inE cfIndE // (can2_eq (mulVKf _) (mulKf _)) ?neq0CG //.
rewrite cfInd1 // mulrA -natrM Lagrange // mulr_natl -sumr_const.
apply/eqP/bigcapP=> [/normC_sum_upper ker_chiG_x y Gy | ker_chiG_x].
by rewrite mem_conjg inE ker_chiG_x ?groupV // => z _; apply: char1_ge_norm.
by apply: eq_bigr => y /groupVr/ker_chiG_x; rewrite mem_conjgV inE => /eqP.
Qed.
Lemma cfker_Ind_irr i :
H \subset G -> cfker ('Ind[G, H] 'chi_i) = gcore (cfker 'chi_i) G.
Proof. by move/cfker_Ind->; rewrite ?irr_neq0 ?irr_char. Qed.
End Induced.
Arguments Ind_Iirr {gT A%_g} B%_g i%_R.
|
FreimanHom.lean
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.Group.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Submonoid.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Multiset
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.ZMod.Defs
/-!
# Freiman homomorphisms
In this file, we define Freiman homomorphisms and isomorphism.
An `n`-Freiman homomorphism from `A` to `B` is a function `f : α → β` such that `f '' A ⊆ B` and
`f x₁ * ... * f xₙ = f y₁ * ... * f yₙ` for all `x₁, ..., xₙ, y₁, ..., yₙ ∈ A` such that
`x₁ * ... * xₙ = y₁ * ... * yₙ`. In particular, any `MulHom` is a Freiman homomorphism.
Note a `0`- or `1`-Freiman homomorphism is simply a map, thus a `2`-Freiman homomorphism is the
first interesting case (and the most common). As `n` increases further, the property of being
an `n`-Freiman homomorphism between abelian groups becomes increasingly stronger.
An `n`-Freiman isomorphism from `A` to `B` is a function `f : α → β` bijective between `A` and `B`
such that `f x₁ * ... * f xₙ = f y₁ * ... * f yₙ ↔ x₁ * ... * xₙ = y₁ * ... * yₙ` for all
`x₁, ..., xₙ, y₁, ..., yₙ ∈ A`. In particular, any `MulEquiv` is a Freiman isomorphism.
They are of interest in additive combinatorics.
## Main declarations
* `IsMulFreimanHom`: Predicate for a function to be a multiplicative Freiman homomorphism.
* `IsAddFreimanHom`: Predicate for a function to be an additive Freiman homomorphism.
* `IsMulFreimanIso`: Predicate for a function to be a multiplicative Freiman isomorphism.
* `IsAddFreimanIso`: Predicate for a function to be an additive Freiman isomorphism.
## Main results
* `isMulFreimanHom_two`: Characterisation of `2`-Freiman homomorphisms.
* `IsMulFreimanHom.mono`: If `m ≤ n` and `f` is an `n`-Freiman homomorphism, then it is also an
`m`-Freiman homomorphism.
## Implementation notes
In the context of combinatorics, we are interested in Freiman homomorphisms over sets which are not
necessarily closed under addition/multiplication. This means we must parametrize them with a set in
an `AddMonoid`/`Monoid` instead of the `AddMonoid`/`Monoid` itself.
## References
[Yufei Zhao, *18.225: Graph Theory and Additive Combinatorics*](https://yufeizhao.com/gtac/)
## TODO
* `MonoidHomClass.isMulFreimanHom` could be relaxed to `MulHom.toFreimanHom` by proving
`(s.map f).prod = (t.map f).prod` directly by induction instead of going through `f s.prod`.
* Affine maps are Freiman homs.
-/
assert_not_exists Field Ideal TwoSidedIdeal
open Multiset Set
open scoped Pointwise
variable {F α β γ : Type*}
section CommMonoid
variable [CommMonoid α] [CommMonoid β] [CommMonoid γ] {A A₁ A₂ : Set α}
{B B₁ B₂ : Set β} {C : Set γ} {f f₁ f₂ : α → β} {g : β → γ} {n : ℕ}
/-- An additive `n`-Freiman homomorphism from a set `A` to a set `B` is a map which preserves sums
of `n` elements. -/
structure IsAddFreimanHom [AddCommMonoid α] [AddCommMonoid β] (n : ℕ) (A : Set α) (B : Set β)
(f : α → β) : Prop where
mapsTo : MapsTo f A B
/-- An additive `n`-Freiman homomorphism preserves sums of `n` elements. -/
map_sum_eq_map_sum ⦃s t : Multiset α⦄ (hsA : ∀ ⦃x⦄, x ∈ s → x ∈ A) (htA : ∀ ⦃x⦄, x ∈ t → x ∈ A)
(hs : Multiset.card s = n) (ht : Multiset.card t = n) (h : s.sum = t.sum) :
(s.map f).sum = (t.map f).sum
/-- An `n`-Freiman homomorphism from a set `A` to a set `B` is a map which preserves products of `n`
elements. -/
@[to_additive]
structure IsMulFreimanHom (n : ℕ) (A : Set α) (B : Set β) (f : α → β) : Prop where
mapsTo : MapsTo f A B
/-- An `n`-Freiman homomorphism preserves products of `n` elements. -/
map_prod_eq_map_prod ⦃s t : Multiset α⦄ (hsA : ∀ ⦃x⦄, x ∈ s → x ∈ A) (htA : ∀ ⦃x⦄, x ∈ t → x ∈ A)
(hs : Multiset.card s = n) (ht : Multiset.card t = n) (h : s.prod = t.prod) :
(s.map f).prod = (t.map f).prod
/-- An additive `n`-Freiman homomorphism from a set `A` to a set `B` is a bijective map which
preserves sums of `n` elements. -/
structure IsAddFreimanIso [AddCommMonoid α] [AddCommMonoid β] (n : ℕ) (A : Set α) (B : Set β)
(f : α → β) : Prop where
bijOn : BijOn f A B
/-- An additive `n`-Freiman homomorphism preserves sums of `n` elements. -/
map_sum_eq_map_sum ⦃s t : Multiset α⦄ (hsA : ∀ ⦃x⦄, x ∈ s → x ∈ A) (htA : ∀ ⦃x⦄, x ∈ t → x ∈ A)
(hs : Multiset.card s = n) (ht : Multiset.card t = n) :
(s.map f).sum = (t.map f).sum ↔ s.sum = t.sum
/-- An `n`-Freiman homomorphism from a set `A` to a set `B` is a map which preserves products of `n`
elements. -/
@[to_additive]
structure IsMulFreimanIso (n : ℕ) (A : Set α) (B : Set β) (f : α → β) : Prop where
bijOn : BijOn f A B
/-- An `n`-Freiman homomorphism preserves products of `n` elements. -/
map_prod_eq_map_prod ⦃s t : Multiset α⦄ (hsA : ∀ ⦃x⦄, x ∈ s → x ∈ A) (htA : ∀ ⦃x⦄, x ∈ t → x ∈ A)
(hs : Multiset.card s = n) (ht : Multiset.card t = n) :
(s.map f).prod = (t.map f).prod ↔ s.prod = t.prod
@[to_additive]
lemma IsMulFreimanIso.isMulFreimanHom (hf : IsMulFreimanIso n A B f) : IsMulFreimanHom n A B f where
mapsTo := hf.bijOn.mapsTo
map_prod_eq_map_prod _s _t hsA htA hs ht := (hf.map_prod_eq_map_prod hsA htA hs ht).2
lemma IsMulFreimanHom.congr (hf₁ : IsMulFreimanHom n A B f₁) (h : EqOn f₁ f₂ A) :
IsMulFreimanHom n A B f₂ where
mapsTo := hf₁.mapsTo.congr h
map_prod_eq_map_prod s t hsA htA hs ht h' := by
rw [map_congr rfl fun x hx => (h (hsA hx)).symm, map_congr rfl fun x hx => (h (htA hx)).symm,
hf₁.map_prod_eq_map_prod hsA htA hs ht h']
lemma IsMulFreimanIso.congr (hf₁ : IsMulFreimanIso n A B f₁) (h : EqOn f₁ f₂ A) :
IsMulFreimanIso n A B f₂ where
bijOn := hf₁.bijOn.congr h
map_prod_eq_map_prod s t hsA htA hs ht := by
rw [map_congr rfl fun x hx => h.symm (hsA hx), map_congr rfl fun x hx => h.symm (htA hx),
hf₁.map_prod_eq_map_prod hsA htA hs ht]
@[to_additive]
lemma IsMulFreimanHom.mul_eq_mul (hf : IsMulFreimanHom 2 A B f) {a b c d : α}
(ha : a ∈ A) (hb : b ∈ A) (hc : c ∈ A) (hd : d ∈ A) (h : a * b = c * d) :
f a * f b = f c * f d := by
simp_rw [← prod_pair] at h ⊢
refine hf.map_prod_eq_map_prod ?_ ?_ (card_pair _ _) (card_pair _ _) h <;> simp [ha, hb, hc, hd]
@[to_additive]
lemma IsMulFreimanIso.mul_eq_mul (hf : IsMulFreimanIso 2 A B f) {a b c d : α}
(ha : a ∈ A) (hb : b ∈ A) (hc : c ∈ A) (hd : d ∈ A) :
f a * f b = f c * f d ↔ a * b = c * d := by
simp_rw [← prod_pair]
refine hf.map_prod_eq_map_prod ?_ ?_ (card_pair _ _) (card_pair _ _) <;> simp [ha, hb, hc, hd]
/-- Characterisation of `2`-Freiman homomorphisms. -/
@[to_additive /-- Characterisation of `2`-Freiman homomorphisms. -/]
lemma isMulFreimanHom_two :
IsMulFreimanHom 2 A B f ↔ MapsTo f A B ∧ ∀ a ∈ A, ∀ b ∈ A, ∀ c ∈ A, ∀ d ∈ A,
a * b = c * d → f a * f b = f c * f d where
mp hf := ⟨hf.mapsTo, fun _ ha _ hb _ hc _ hd ↦ hf.mul_eq_mul ha hb hc hd⟩
mpr hf := ⟨hf.1, by aesop (add simp card_eq_two)⟩
/-- Characterisation of `2`-Freiman homs. -/
@[to_additive /-- Characterisation of `2`-Freiman isomorphisms. -/]
lemma isMulFreimanIso_two :
IsMulFreimanIso 2 A B f ↔ BijOn f A B ∧ ∀ a ∈ A, ∀ b ∈ A, ∀ c ∈ A, ∀ d ∈ A,
f a * f b = f c * f d ↔ a * b = c * d where
mp hf := ⟨hf.bijOn, fun _ ha _ hb _ hc _ hd => hf.mul_eq_mul ha hb hc hd⟩
mpr hf := ⟨hf.1, by aesop (add simp card_eq_two)⟩
@[to_additive] lemma isMulFreimanHom_id (hA : A₁ ⊆ A₂) : IsMulFreimanHom n A₁ A₂ id where
mapsTo := hA
map_prod_eq_map_prod s t _ _ _ _ h := by simpa using h
@[to_additive] lemma isMulFreimanIso_id : IsMulFreimanIso n A A id where
bijOn := bijOn_id _
map_prod_eq_map_prod s t _ _ _ _ := by simp
@[to_additive] lemma IsMulFreimanHom.comp (hg : IsMulFreimanHom n B C g)
(hf : IsMulFreimanHom n A B f) : IsMulFreimanHom n A C (g ∘ f) where
mapsTo := hg.mapsTo.comp hf.mapsTo
map_prod_eq_map_prod s t hsA htA hs ht h := by
rw [← map_map, ← map_map]
refine hg.map_prod_eq_map_prod ?_ ?_ (by rwa [card_map]) (by rwa [card_map])
(hf.map_prod_eq_map_prod hsA htA hs ht h)
· simpa using fun a h ↦ hf.mapsTo (hsA h)
· simpa using fun a h ↦ hf.mapsTo (htA h)
@[to_additive] lemma IsMulFreimanIso.comp (hg : IsMulFreimanIso n B C g)
(hf : IsMulFreimanIso n A B f) : IsMulFreimanIso n A C (g ∘ f) where
bijOn := hg.bijOn.comp hf.bijOn
map_prod_eq_map_prod s t hsA htA hs ht := by
rw [← map_map, ← map_map]
rw [hg.map_prod_eq_map_prod _ _ (by rwa [card_map]) (by rwa [card_map]),
hf.map_prod_eq_map_prod hsA htA hs ht]
· simpa using fun a h ↦ hf.bijOn.mapsTo (hsA h)
· simpa using fun a h ↦ hf.bijOn.mapsTo (htA h)
@[to_additive] lemma IsMulFreimanHom.subset (hA : A₁ ⊆ A₂) (hf : IsMulFreimanHom n A₂ B₂ f)
(hf' : MapsTo f A₁ B₁) : IsMulFreimanHom n A₁ B₁ f where
mapsTo := hf'
__ := hf.comp (isMulFreimanHom_id hA)
@[to_additive] lemma IsMulFreimanHom.superset (hB : B₁ ⊆ B₂) (hf : IsMulFreimanHom n A B₁ f) :
IsMulFreimanHom n A B₂ f := (isMulFreimanHom_id hB).comp hf
@[to_additive] lemma IsMulFreimanIso.subset (hA : A₁ ⊆ A₂) (hf : IsMulFreimanIso n A₂ B₂ f)
(hf' : BijOn f A₁ B₁) : IsMulFreimanIso n A₁ B₁ f where
bijOn := hf'
map_prod_eq_map_prod s t hsA htA hs ht := by
refine hf.map_prod_eq_map_prod (fun a ha ↦ hA (hsA ha)) (fun a ha ↦ hA (htA ha)) hs ht
@[to_additive]
lemma isMulFreimanHom_const {b : β} (hb : b ∈ B) : IsMulFreimanHom n A B fun _ ↦ b where
mapsTo _ _ := hb
map_prod_eq_map_prod s t _ _ hs ht _ := by simp only [map_const', hs, prod_replicate, ht]
@[to_additive (attr := simp)]
lemma isMulFreimanHom_zero_iff : IsMulFreimanHom 0 A B f ↔ MapsTo f A B :=
⟨fun h => h.mapsTo, fun h => ⟨h, by aesop⟩⟩
@[to_additive (attr := simp)]
lemma isMulFreimanIso_zero_iff : IsMulFreimanIso 0 A B f ↔ BijOn f A B :=
⟨fun h => h.bijOn, fun h => ⟨h, by aesop⟩⟩
@[to_additive (attr := simp) isAddFreimanHom_one_iff]
lemma isMulFreimanHom_one_iff : IsMulFreimanHom 1 A B f ↔ MapsTo f A B :=
⟨fun h => h.mapsTo, fun h => ⟨h, by aesop (add simp card_eq_one)⟩⟩
@[to_additive (attr := simp) isAddFreimanIso_one_iff]
lemma isMulFreimanIso_one_iff : IsMulFreimanIso 1 A B f ↔ BijOn f A B :=
⟨fun h => h.bijOn, fun h => ⟨h, by aesop (add simp [card_eq_one, BijOn])⟩⟩
@[to_additive (attr := simp)]
lemma isMulFreimanHom_empty : IsMulFreimanHom n (∅ : Set α) B f where
mapsTo := mapsTo_empty f B
map_prod_eq_map_prod s t := by aesop (add simp eq_zero_of_forall_notMem)
@[to_additive (attr := simp)]
lemma isMulFreimanIso_empty : IsMulFreimanIso n (∅ : Set α) (∅ : Set β) f where
bijOn := bijOn_empty _
map_prod_eq_map_prod s t hs ht := by
simp [eq_zero_of_forall_notMem hs, eq_zero_of_forall_notMem ht]
@[to_additive] lemma IsMulFreimanHom.mul (h₁ : IsMulFreimanHom n A B₁ f₁)
(h₂ : IsMulFreimanHom n A B₂ f₂) : IsMulFreimanHom n A (B₁ * B₂) (f₁ * f₂) where
mapsTo := h₁.mapsTo.mul h₂.mapsTo
map_prod_eq_map_prod s t hsA htA hs ht h := by
rw [Pi.mul_def, prod_map_mul, prod_map_mul, h₁.map_prod_eq_map_prod hsA htA hs ht h,
h₂.map_prod_eq_map_prod hsA htA hs ht h]
@[to_additive] lemma MonoidHomClass.isMulFreimanHom [FunLike F α β] [MonoidHomClass F α β] (f : F)
(hfAB : MapsTo f A B) : IsMulFreimanHom n A B f where
mapsTo := hfAB
map_prod_eq_map_prod s t _ _ _ _ h := by rw [← map_multiset_prod, h, map_multiset_prod]
@[to_additive] lemma MulEquivClass.isMulFreimanIso [EquivLike F α β] [MulEquivClass F α β] (f : F)
(hfAB : BijOn f A B) : IsMulFreimanIso n A B f where
bijOn := hfAB
map_prod_eq_map_prod s t _ _ _ _ := by
rw [← map_multiset_prod, ← map_multiset_prod, EquivLike.apply_eq_iff_eq]
@[to_additive]
lemma IsMulFreimanHom.subtypeVal {S : Type*} [SetLike S α] [SubmonoidClass S α] {s : S} :
IsMulFreimanHom n (univ : Set s) univ Subtype.val :=
MonoidHomClass.isMulFreimanHom (SubmonoidClass.subtype s) (mapsTo_univ ..)
end CommMonoid
section CancelCommMonoid
variable [CommMonoid α] [CancelCommMonoid β] {A : Set α} {B : Set β} {f : α → β} {m n : ℕ}
@[to_additive]
lemma IsMulFreimanHom.mono (hmn : m ≤ n) (hf : IsMulFreimanHom n A B f) :
IsMulFreimanHom m A B f where
mapsTo := hf.mapsTo
map_prod_eq_map_prod s t hsA htA hs ht h := by
obtain rfl | hm := m.eq_zero_or_pos
· rw [card_eq_zero] at hs ht
rw [hs, ht]
simp only [← hs, card_pos_iff_exists_mem] at hm
obtain ⟨a, ha⟩ := hm
suffices ((s + replicate (n - m) a).map f).prod = ((t + replicate (n - m) a).map f).prod by
simp_rw [Multiset.map_add, prod_add] at this
exact mul_right_cancel this
replace ha := hsA ha
refine hf.map_prod_eq_map_prod (fun a ha ↦ ?_) (fun a ha ↦ ?_) ?_ ?_ ?_
· rw [Multiset.mem_add] at ha
obtain ha | ha := ha
· exact hsA ha
· rwa [eq_of_mem_replicate ha]
· rw [Multiset.mem_add] at ha
obtain ha | ha := ha
· exact htA ha
· rwa [eq_of_mem_replicate ha]
· rw [card_add, card_replicate, hs, Nat.add_sub_cancel' hmn]
· rw [card_add, card_replicate, ht, Nat.add_sub_cancel' hmn]
· rw [prod_add, prod_add, h]
end CancelCommMonoid
section CancelCommMonoid
variable [CancelCommMonoid α] [CancelCommMonoid β] {A : Set α} {B : Set β} {f : α → β} {m n : ℕ}
@[to_additive]
lemma IsMulFreimanIso.mono {hmn : m ≤ n} (hf : IsMulFreimanIso n A B f) :
IsMulFreimanIso m A B f where
bijOn := hf.bijOn
map_prod_eq_map_prod s t hsA htA hs ht := by
obtain rfl | hm := m.eq_zero_or_pos
· rw [card_eq_zero] at hs ht
simp [hs, ht]
simp only [← hs, card_pos_iff_exists_mem] at hm
obtain ⟨a, ha⟩ := hm
suffices
((s + replicate (n - m) a).map f).prod = ((t + replicate (n - m) a).map f).prod ↔
(s + replicate (n - m) a).prod = (t + replicate (n - m) a).prod by
simpa only [Multiset.map_add, prod_add, mul_right_cancel_iff] using this
replace ha := hsA ha
refine hf.map_prod_eq_map_prod (fun a ha ↦ ?_) (fun a ha ↦ ?_) ?_ ?_
· rw [Multiset.mem_add] at ha
obtain ha | ha := ha
· exact hsA ha
· rwa [eq_of_mem_replicate ha]
· rw [Multiset.mem_add] at ha
obtain ha | ha := ha
· exact htA ha
· rwa [eq_of_mem_replicate ha]
· rw [card_add, card_replicate, hs, Nat.add_sub_cancel' hmn]
· rw [card_add, card_replicate, ht, Nat.add_sub_cancel' hmn]
end CancelCommMonoid
section DivisionCommMonoid
variable [CommMonoid α] [DivisionCommMonoid β] {A : Set α} {B : Set β} {f : α → β} {n : ℕ}
@[to_additive]
lemma IsMulFreimanHom.inv (hf : IsMulFreimanHom n A B f) : IsMulFreimanHom n A B⁻¹ f⁻¹ where
mapsTo := hf.mapsTo.inv
map_prod_eq_map_prod s t hsA htA hs ht h := by
rw [Pi.inv_def, prod_map_inv, prod_map_inv, hf.map_prod_eq_map_prod hsA htA hs ht h]
@[to_additive] lemma IsMulFreimanHom.div {β : Type*} [DivisionCommMonoid β] {B₁ B₂ : Set β}
{f₁ f₂ : α → β} (h₁ : IsMulFreimanHom n A B₁ f₁) (h₂ : IsMulFreimanHom n A B₂ f₂) :
IsMulFreimanHom n A (B₁ / B₂) (f₁ / f₂) where
mapsTo := h₁.mapsTo.div h₂.mapsTo
map_prod_eq_map_prod s t hsA htA hs ht h := by
rw [Pi.div_def, prod_map_div, prod_map_div, h₁.map_prod_eq_map_prod hsA htA hs ht h,
h₂.map_prod_eq_map_prod hsA htA hs ht h]
end DivisionCommMonoid
section Prod
variable {α₁ α₂ β₁ β₂ : Type*} [CommMonoid α₁] [CommMonoid α₂] [CommMonoid β₁] [CommMonoid β₂]
{A₁ : Set α₁} {A₂ : Set α₂} {B₁ : Set β₁} {B₂ : Set β₂} {f₁ : α₁ → β₁} {f₂ : α₂ → β₂} {n : ℕ}
@[to_additive prodMap]
lemma IsMulFreimanHom.prodMap (h₁ : IsMulFreimanHom n A₁ B₁ f₁) (h₂ : IsMulFreimanHom n A₂ B₂ f₂) :
IsMulFreimanHom n (A₁ ×ˢ A₂) (B₁ ×ˢ B₂) (Prod.map f₁ f₂) where
mapsTo := h₁.mapsTo.prodMap h₂.mapsTo
map_prod_eq_map_prod s t hsA htA hs ht h := by
simp only [mem_prod, forall_and, Prod.forall] at hsA htA
simp only [Prod.ext_iff, fst_prod, snd_prod, map_map, Function.comp_apply, Prod.map_fst,
Prod.map_snd] at h ⊢
rw [← Function.comp_def, ← map_map, ← map_map, ← Function.comp_def f₂, ← map_map, ← map_map]
exact ⟨h₁.map_prod_eq_map_prod (by simpa using hsA.1) (by simpa using htA.1) (by simpa)
(by simpa) h.1, h₂.map_prod_eq_map_prod (by simpa [@forall_swap α₁] using hsA.2)
(by simpa [@forall_swap α₁] using htA.2) (by simpa) (by simpa) h.2⟩
@[deprecated (since := "2025-03-11")]
alias IsAddFreimanHom.sum := IsAddFreimanHom.prodMap
@[to_additive existing, deprecated (since := "2025-03-11")]
alias IsMulFreimanHom.prod := IsMulFreimanHom.prodMap
@[to_additive prodMap]
lemma IsMulFreimanIso.prodMap (h₁ : IsMulFreimanIso n A₁ B₁ f₁) (h₂ : IsMulFreimanIso n A₂ B₂ f₂) :
IsMulFreimanIso n (A₁ ×ˢ A₂) (B₁ ×ˢ B₂) (Prod.map f₁ f₂) where
bijOn := h₁.bijOn.prodMap h₂.bijOn
map_prod_eq_map_prod s t hsA htA hs ht := by
simp only [mem_prod, forall_and, Prod.forall] at hsA htA
simp only [Prod.ext_iff, fst_prod, map_map, Function.comp_apply, Prod.map_fst, snd_prod,
Prod.map_snd]
rw [← Function.comp_def, ← map_map, ← map_map, ← Function.comp_def f₂, ← map_map, ← map_map,
h₁.map_prod_eq_map_prod (by simpa using hsA.1) (by simpa using htA.1) (by simpa) (by simpa),
h₂.map_prod_eq_map_prod (by simpa [@forall_swap α₁] using hsA.2)
(by simpa [@forall_swap α₁] using htA.2) (by simpa) (by simpa)]
@[deprecated (since := "2025-03-11")]
alias IsAddFreimanIso.sum := IsAddFreimanIso.prodMap
@[to_additive existing, deprecated (since := "2025-03-11")]
alias IsMulFreimanIso.prod := IsMulFreimanIso.prodMap
end Prod
namespace Fin
variable {k m n : ℕ}
open Fin.CommRing
private lemma aux (hm : m ≠ 0) (hkmn : m * k ≤ n) : k < (n + 1) :=
Nat.lt_succ_iff.2 <| le_trans (Nat.le_mul_of_pos_left _ hm.bot_lt) hkmn
/-- **No wrap-around principle**.
The first `k + 1` elements of `Fin (n + 1)` are `m`-Freiman isomorphic to the first `k + 1` elements
of `ℕ` assuming there is no wrap-around. -/
lemma isAddFreimanIso_Iic (hm : m ≠ 0) (hkmn : m * k ≤ n) :
IsAddFreimanIso m (Iic (k : Fin (n + 1))) (Iic k) val where
bijOn.left := by simp [MapsTo, Fin.le_iff_val_le_val, Nat.mod_eq_of_lt, aux hm hkmn]
bijOn.right.left := val_injective.injOn
bijOn.right.right x (hx : x ≤ _) :=
⟨x, by simpa [le_iff_val_le_val, -val_fin_le, Nat.mod_eq_of_lt, aux hm hkmn, hx.trans_lt]⟩
map_sum_eq_map_sum s t hsA htA hs ht := by
have (u : Multiset (Fin (n + 1))) : Nat.castRingHom _ (u.map val).sum = u.sum := by simp
rw [← this, ← this]
have {u : Multiset (Fin (n + 1))} (huk : ∀ x ∈ u, x ≤ k) (hu : card u = m) :
(u.map val).sum < (n + 1) := Nat.lt_succ_iff.2 <| hkmn.trans' <| by
rw [← hu, ← card_map]
refine sum_le_card_nsmul (u.map val) k ?_
simpa [le_iff_val_le_val, -val_fin_le, Nat.mod_eq_of_lt, aux hm hkmn] using huk
exact ⟨congr_arg _, CharP.natCast_injOn_Iio _ (n + 1) (this hsA hs) (this htA ht)⟩
/-- **No wrap-around principle**.
The first `k` elements of `Fin (n + 1)` are `m`-Freiman isomorphic to the first `k` elements of `ℕ`
assuming there is no wrap-around. -/
lemma isAddFreimanIso_Iio (hm : m ≠ 0) (hkmn : m * k ≤ n) :
IsAddFreimanIso m (Iio (k : Fin (n + 1))) (Iio k) val := by
obtain _ | k := k
· simp [← bot_eq_zero]
have hkmn' : m * k ≤ n := (Nat.mul_le_mul_left _ k.le_succ).trans hkmn
convert isAddFreimanIso_Iic hm hkmn' using 1 <;> ext x
· simp [lt_iff_val_lt_val, le_iff_val_le_val, -val_fin_le, -val_fin_lt, Nat.mod_eq_of_lt,
aux hm hkmn']
simp_rw [← Nat.cast_add_one]
rw [Fin.val_cast_of_lt (aux hm hkmn), Nat.lt_succ_iff]
· simp [Nat.lt_succ_iff]
end Fin
|
CantorSet.lean
|
/-
Copyright (c) 2024 Jana Göken. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Artur Szafarczyk, Suraj Krishna M S, Jean-Baptiste Stiegler, Isabelle Dubois,
Tomáš Jakl, Lorenzo Zanichelli, Alina Yan, Emilie Uthaiwat, Jana Göken,
Filippo A. E. Nuccio
-/
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Algebra.Ring.Real
/-!
# Ternary Cantor Set
This file defines the Cantor ternary set and proves a few properties.
## Main Definitions
* `preCantorSet n`: The order `n` pre-Cantor set, defined inductively as the union of the images
under the functions `(· / 3)` and `((2 + ·) / 3)`, with `preCantorSet 0 := Set.Icc 0 1`, i.e.
`preCantorSet 0` is the unit interval [0,1].
* `cantorSet`: The ternary Cantor set, defined as the intersection of all pre-Cantor sets.
-/
/-- The order `n` pre-Cantor set, defined starting from `[0, 1]` and successively removing the
middle third of each interval. Formally, the order `n + 1` pre-Cantor set is the
union of the images under the functions `(· / 3)` and `((2 + ·) / 3)` of `preCantorSet n`.
-/
def preCantorSet : ℕ → Set ℝ
| 0 => Set.Icc 0 1
| n + 1 => (· / 3) '' preCantorSet n ∪ (fun x ↦ (2 + x) / 3) '' preCantorSet n
@[simp] lemma preCantorSet_zero : preCantorSet 0 = Set.Icc 0 1 := rfl
@[simp] lemma preCantorSet_succ (n : ℕ) :
preCantorSet (n + 1) = (· / 3) '' preCantorSet n ∪ (fun x ↦ (2 + x) / 3) '' preCantorSet n :=
rfl
/-- The Cantor set is the subset of the unit interval obtained as the intersection of all
pre-Cantor sets. This means that the Cantor set is obtained by iteratively removing the
open middle third of each subinterval, starting from the unit interval `[0, 1]`.
-/
def cantorSet : Set ℝ := ⋂ n, preCantorSet n
/-!
## Simple Properties
-/
lemma quarters_mem_preCantorSet (n : ℕ) : 1 / 4 ∈ preCantorSet n ∧ 3 / 4 ∈ preCantorSet n := by
induction n with
| zero =>
simp only [preCantorSet_zero]
refine ⟨⟨ ?_, ?_⟩, ?_, ?_⟩ <;> norm_num
| succ n ih =>
apply And.intro
· -- goal: 1 / 4 ∈ preCantorSet (n + 1)
-- follows by the inductive hyphothesis, since 3 / 4 ∈ preCantorSet n
exact Or.inl ⟨3 / 4, ih.2, by norm_num⟩
· -- goal: 3 / 4 ∈ preCantorSet (n + 1)
-- follows by the inductive hyphothesis, since 1 / 4 ∈ preCantorSet n
exact Or.inr ⟨1 / 4, ih.1, by norm_num⟩
lemma quarter_mem_preCantorSet (n : ℕ) : 1 / 4 ∈ preCantorSet n := (quarters_mem_preCantorSet n).1
theorem quarter_mem_cantorSet : 1 / 4 ∈ cantorSet :=
Set.mem_iInter.mpr quarter_mem_preCantorSet
lemma zero_mem_preCantorSet (n : ℕ) : 0 ∈ preCantorSet n := by
induction n with
| zero =>
simp [preCantorSet]
| succ n ih =>
exact Or.inl ⟨0, ih, by simp only [zero_div]⟩
theorem zero_mem_cantorSet : 0 ∈ cantorSet := by simp [cantorSet, zero_mem_preCantorSet]
theorem preCantorSet_antitone : Antitone preCantorSet := by
apply antitone_nat_of_succ_le
intro m
simp only [Set.le_eq_subset, preCantorSet_succ, Set.union_subset_iff]
induction m with
| zero =>
simp only [preCantorSet_zero]
constructor <;> intro x <;>
simp only [Set.mem_image, Set.mem_Icc, forall_exists_index, and_imp] <;>
intro y _ _ _ <;> constructor <;> linarith
| succ m ih => grind [preCantorSet_succ, Set.image_union]
lemma preCantorSet_subset_unitInterval {n : ℕ} : preCantorSet n ⊆ Set.Icc 0 1 := by
rw [← preCantorSet_zero]
exact preCantorSet_antitone (by simp)
/-- The ternary Cantor set is a subset of [0,1]. -/
lemma cantorSet_subset_unitInterval : cantorSet ⊆ Set.Icc 0 1 :=
Set.iInter_subset _ 0
/-- The ternary Cantor set satisfies the equation `C = C / 3 ∪ (2 / 3 + C / 3)`. -/
theorem cantorSet_eq_union_halves :
cantorSet = (· / 3) '' cantorSet ∪ (fun x ↦ (2 + x) / 3) '' cantorSet := by
simp only [cantorSet]
rw [Set.image_iInter, Set.image_iInter]
rotate_left
· exact (mulRight_bijective₀ 3⁻¹ (by norm_num)).comp (AddGroup.addLeft_bijective 2)
· exact mulRight_bijective₀ 3⁻¹ (by norm_num)
simp_rw [← Function.comp_def,
← Set.iInter_union_of_antitone
(Set.monotone_image.comp_antitone preCantorSet_antitone)
(Set.monotone_image.comp_antitone preCantorSet_antitone),
Function.comp_def, ← preCantorSet_succ]
exact (preCantorSet_antitone.iInter_nat_add _).symm
/-- The preCantor sets are closed. -/
lemma isClosed_preCantorSet (n : ℕ) : IsClosed (preCantorSet n) := by
let f := Homeomorph.mulLeft₀ (1 / 3 : ℝ) (by simp)
let g := (Homeomorph.addLeft (2 : ℝ)).trans f
induction n with
| zero => exact isClosed_Icc
| succ n ih =>
refine IsClosed.union ?_ ?_
· simpa [f, div_eq_inv_mul] using f.isClosedEmbedding.isClosed_iff_image_isClosed.mp ih
· simpa [g, f, div_eq_inv_mul] using g.isClosedEmbedding.isClosed_iff_image_isClosed.mp ih
/-- The ternary Cantor set is closed. -/
lemma isClosed_cantorSet : IsClosed cantorSet :=
isClosed_iInter isClosed_preCantorSet
/-- The ternary Cantor set is compact. -/
lemma isCompact_cantorSet : IsCompact cantorSet :=
isCompact_Icc.of_isClosed_subset isClosed_cantorSet cantorSet_subset_unitInterval
|
NatSqrt.lean
|
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kyle Miller
-/
import Mathlib.Tactic.NormNum
/-! # `norm_num` extension for `Nat.sqrt`
This module defines a `norm_num` extension for `Nat.sqrt`.
-/
namespace Tactic
namespace NormNum
open Qq Lean Elab.Tactic Mathlib.Meta.NormNum
lemma nat_sqrt_helper {x y r : ℕ} (hr : y * y + r = x) (hle : Nat.ble r (2 * y)) :
Nat.sqrt x = y := by
rw [← hr, ← pow_two]
rw [two_mul] at hle
exact Nat.sqrt_add_eq' _ (Nat.le_of_ble_eq_true hle)
theorem isNat_sqrt : {x nx z : ℕ} → IsNat x nx → Nat.sqrt nx = z → IsNat (Nat.sqrt x) z
| _, _, _, ⟨rfl⟩, rfl => ⟨rfl⟩
/-- Given the natural number literal `ex`, returns its square root as a natural number literal
and an equality proof. Panics if `ex` isn't a natural number literal. -/
def proveNatSqrt (ex : Q(ℕ)) : (ey : Q(ℕ)) × Q(Nat.sqrt $ex = $ey) :=
match ex.natLit! with
| 0 => show (ey : Q(ℕ)) × Q(Nat.sqrt 0 = $ey) from ⟨mkRawNatLit 0, q(Nat.sqrt_zero)⟩
| 1 => show (ey : Q(ℕ)) × Q(Nat.sqrt 1 = $ey) from ⟨mkRawNatLit 1, q(Nat.sqrt_one)⟩
| x =>
let y := Nat.sqrt x
have ey : Q(ℕ) := mkRawNatLit y
have er : Q(ℕ) := mkRawNatLit (x - y * y)
have hr : Q($ey * $ey + $er = $ex) := (q(Eq.refl $ex) : Expr)
have hle : Q(Nat.ble $er (2 * $ey)) := (q(Eq.refl true) : Expr)
⟨ey, q(nat_sqrt_helper $hr $hle)⟩
/-- Evaluates the `Nat.sqrt` function. -/
@[norm_num Nat.sqrt _]
def evalNatSqrt : NormNumExt where eval {_ _} e := do
let .app _ (x : Q(ℕ)) ← Meta.whnfR e | failure
let sℕ : Q(AddMonoidWithOne ℕ) := q(instAddMonoidWithOneNat)
let ⟨ex, p⟩ ← deriveNat x sℕ
let ⟨ey, pf⟩ := proveNatSqrt ex
let pf' : Q(IsNat (Nat.sqrt $x) $ey) := q(isNat_sqrt $p $pf)
return .isNat sℕ ey pf'
end NormNum
end Tactic
|
HullKernel.lean
|
/-
Copyright (c) 2024 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Data.Set.Subset
import Mathlib.Order.Irreducible
import Mathlib.Topology.Order.LowerUpperTopology
import Mathlib.Topology.Sets.Closeds
/-!
# Hull-Kernel Topology
Let `α` be a `CompleteLattice` and let `T` be a subset of `α`. The pair of maps
`S → sInf (Subtype.val '' S)` and `a → T ↓∩ Ici a` are often referred to as the `kernel` and the
`hull` respectively. They form an antitone Galois connection between the subsets of `T` and `α`.
When `α` can be generated from `T` by taking infs, this becomes a Galois insertion and the relative
topology (`Topology.lower`) on `T` takes on a particularly simple form: the relative-open sets are
exactly the sets of the form `(hull T a)ᶜ` for some `a` in `α`. The topological closure coincides
with the closure arising from the Galois insertion. For this reason the relative lower topology on
`T` is often referred to as the "hull-kernel topology". The names "Jacobson topology" and "structure
topology" also occur in the literature.
## Main statements
- `PrimitiveSpectrum.isTopologicalBasis_relativeLower` - the sets `(hull a)ᶜ` form a basis for the
relative lower topology on `T`.
- `PrimitiveSpectrum.isOpen_iff` - for a complete lattice, the sets `(hull a)ᶜ` are the relative
topology.
- `PrimitiveSpectrum.gc` - the `kernel` and the `hull` form a Galois connection
- `PrimitiveSpectrum.gi` - when `T` generates `α`, the Galois connection becomes an insertion.
## Implementation notes
The antitone Galois connection from `Set T` to `α` is implemented as a monotone Galois connection
between `Set T` to `αᵒᵈ`.
## Motivation
The motivating example for the study of a set `T` of prime elements which generate `α` is the
primitive spectrum of the lattice of M-ideals of a Banach space.
## References
* [Gierz et al, *A Compendium of Continuous Lattices*][GierzEtAl1980]
* [Henriksen et al, *Joincompact spaces, continuous lattices and C*-algebras*][henriksen_et_al1997]
## Tags
lower topology, hull-kernel topology, Jacobson topology, structure topology, primitive spectrum
-/
variable {α}
open TopologicalSpace
open Topology
open Set
open Set.Notation
section SemilatticeInf
variable [SemilatticeInf α]
namespace PrimitiveSpectrum
/-- For `a` of type `α` the set of element of `T` which dominate `a` is the `hull` of `a` in `T`. -/
abbrev hull (T : Set α) (a : α) := T ↓∩ Ici a
variable {T : Set α}
/- The set of relative-closed sets of the form `hull T a` for some `a` in `α` is closed under
pairwise union. -/
lemma hull_inf (hT : ∀ p ∈ T, InfPrime p) (a b : α) :
hull T (a ⊓ b) = hull T a ∪ hull T b := by
ext p
constructor <;> intro h
· exact (hT p p.2).2 h
· rcases h with (h1 | h3)
· exact inf_le_of_left_le h1
· exact inf_le_of_right_le h3
variable [DecidableEq α] [OrderTop α]
/- Every relative-closed set of the form `T ↓∩ (↑(upperClosure F))` for `F` finite is a
relative-closed set of the form `hull T a` where `a = ⨅ F`. -/
open Finset in
lemma hull_finsetInf (hT : ∀ p ∈ T, InfPrime p) (F : Finset α) :
hull T (inf F id) = T ↓∩ upperClosure F.toSet := by
rw [coe_upperClosure]
induction' F using Finset.induction_on with a F' _ I4
· simp only [coe_empty, mem_empty_iff_false, iUnion_of_empty, iUnion_empty, Set.preimage_empty,
inf_empty]
by_contra hf
rw [← Set.not_nonempty_iff_eq_empty, not_not] at hf
obtain ⟨x, hx⟩ := hf
exact (hT x (Subtype.coe_prop x)).1 (isMax_iff_eq_top.mpr (eq_top_iff.mpr hx))
· simp only [coe_insert, mem_insert_iff, mem_coe, iUnion_iUnion_eq_or_left, Set.preimage_union,
preimage_iUnion, inf_insert, id_eq, hull_inf hT, I4]
/- Every relative-open set of the form `T ↓∩ (↑(upperClosure F))ᶜ` for `F` finite is a relative-open
set of the form `(hull T a)ᶜ` where `a = ⨅ F`. -/
open Finset in
lemma preimage_upperClosure_compl_finset (hT : ∀ p ∈ T, InfPrime p) (F : Finset α) :
T ↓∩ (upperClosure F.toSet)ᶜ = (hull T (inf F id))ᶜ := by
rw [Set.preimage_compl, (hull_finsetInf hT)]
variable [TopologicalSpace α] [IsLower α]
/-
The relative-open sets of the form `(hull T a)ᶜ` for `a` in `α` form a basis for the relative
Lower topology.
-/
lemma isTopologicalBasis_relativeLower (hT : ∀ p ∈ T, InfPrime p) :
IsTopologicalBasis { S : Set T | ∃ (a : α), (hull T a)ᶜ = S } := by
convert isTopologicalBasis_subtype Topology.IsLower.isTopologicalBasis T
ext R
simp only [preimage_compl, mem_setOf_eq, IsLower.lowerBasis, mem_image, exists_exists_and_eq_and]
constructor <;> intro ha
· obtain ⟨a, ha'⟩ := ha
use {a}
rw [← (Function.Injective.preimage_image Subtype.val_injective R), ← ha']
simp only [finite_singleton, upperClosure_singleton, UpperSet.coe_Ici, image_val_compl,
Subtype.image_preimage_coe, diff_self_inter, preimage_diff, Subtype.coe_preimage_self,
true_and]
exact compl_eq_univ_diff (Subtype.val ⁻¹' Ici a)
· obtain ⟨F, hF⟩ := ha
lift F to Finset α using hF.1
use Finset.inf F id
ext
simp [hull_finsetInf hT, ← hF.2]
end PrimitiveSpectrum
end SemilatticeInf
namespace PrimitiveSpectrum
variable [CompleteLattice α] {T : Set α}
universe v
lemma hull_iSup {ι : Sort v} (s : ι → α) : hull T (iSup s) = ⋂ i, hull T (s i) := by aesop
lemma hull_sSup (S : Set α) : hull T (sSup S) = ⋂₀ { hull T a | a ∈ S } := by aesop
/- When `α` is complete, a set is Lower topology relative-open if and only if it is of the form
`(hull T a)ᶜ` for some `a` in `α`.-/
lemma isOpen_iff [TopologicalSpace α] [IsLower α] [DecidableEq α] (hT : ∀ p ∈ T, InfPrime p)
(S : Set T) : IsOpen S ↔ ∃ (a : α), S = (hull T a)ᶜ := by
constructor <;> intro h
· let R := {a : α | (hull T a)ᶜ ⊆ S}
use sSup R
rw [IsTopologicalBasis.open_eq_sUnion' (isTopologicalBasis_relativeLower hT) h]
aesop
· obtain ⟨a, ha⟩ := h
exact ⟨(Ici a)ᶜ, isClosed_Ici.isOpen_compl, ha.symm⟩
/- When `α` is complete, a set is closed in the relative lower topology if and only if it is of the
form `hull T a` for some `a` in `α`.-/
lemma isClosed_iff [TopologicalSpace α] [IsLower α] [DecidableEq α] (hT : ∀ p ∈ T, InfPrime p)
{S : Set T} : IsClosed S ↔ ∃ (a : α), S = hull T a := by
simp only [← isOpen_compl_iff, isOpen_iff hT, compl_inj_iff]
/-- For a subset `S` of `T`, `kernel S` is the infimum of `S` (considered as a set of `α`) -/
abbrev kernel (S : Set T) := sInf (Subtype.val '' S)
/- The pair of maps `kernel` and `hull` form an antitone Galois connection between the
subsets of `T` and `α`. -/
open OrderDual in
theorem gc : GaloisConnection (α := Set T) (β := αᵒᵈ)
(fun S => toDual (kernel S)) (fun a => hull T (ofDual a)) := fun S a => by
simp only [toDual_sInf, sSup_le_iff, mem_preimage, mem_image, Subtype.exists, exists_and_right,
exists_eq_right, ← ofDual_le_ofDual, forall_exists_index, OrderDual.forall, ofDual_toDual]
exact ⟨fun h b hbS => h _ (Subtype.coe_prop b) hbS, fun h b _ hbS => h hbS⟩
lemma gc_closureOperator (S : Set T) : gc.closureOperator S = hull T (kernel S) := by
simp only [toDual_sInf, GaloisConnection.closureOperator_apply, ofDual_sSup]
rw [← preimage_comp, ← OrderDual.toDual_symm_eq, Equiv.symm_comp_self, preimage_id_eq, id_eq]
variable (T)
/-- `T` order generates `α` if, for every `a` in `α`, there exists a subset of `T` such that `a` is
the `kernel` of `S`. -/
def OrderGenerates := ∀ (a : α), ∃ (S : Set T), a = kernel S
variable {T}
/--
When `T` is order generating, the `kernel` and the `hull` form a Galois insertion
-/
def gi (hG : OrderGenerates T) : GaloisInsertion (α := Set T) (β := αᵒᵈ)
(OrderDual.toDual ∘ kernel)
(hull T ∘ OrderDual.ofDual) :=
gc.toGaloisInsertion fun a ↦ (by
rw [OrderDual.le_toDual]
obtain ⟨S, hS⟩ := hG a
exact le_of_le_of_eq (sInf_le_sInf (image_val_mono (fun c hcS => mem_preimage.mpr (mem_Ici.mpr
(by rw [hS]; exact CompleteSemilatticeInf.sInf_le _ _ (mem_image_of_mem Subtype.val hcS))))))
hS.symm)
lemma kernel_hull (hG : OrderGenerates T) (a : α) : kernel (hull T a) = a := by
conv_rhs => rw [← OrderDual.ofDual_toDual a, ← (gi hG).l_u_eq a]
rfl
lemma hull_kernel_of_isClosed [TopologicalSpace α] [IsLower α] [DecidableEq α]
(hT : ∀ p ∈ T, InfPrime p) (hG : OrderGenerates T) {C : Set T} (h : IsClosed C) :
hull T (kernel C) = C := by
obtain ⟨a, ha⟩ := (isClosed_iff hT).mp h
rw [ha, kernel_hull hG]
lemma closedsGC_closureOperator [TopologicalSpace α] [IsLower α] [DecidableEq α]
(hT : ∀ p ∈ T, InfPrime p) (hG : OrderGenerates T) (S : Set T) :
(TopologicalSpace.Closeds.gc (α := T)).closureOperator S = hull T (kernel S) := by
simp only [GaloisConnection.closureOperator_apply, Closeds.coe_closure, closure, le_antisymm_iff]
constructor
· exact fun ⦃a⦄ a ↦ a (hull T (kernel S)) ⟨(isClosed_iff hT).mpr ⟨kernel S, rfl⟩,
image_subset_iff.mp (fun _ hbS => CompleteSemilatticeInf.sInf_le _ _ hbS)⟩
· simp_rw [le_eq_subset, subset_sInter_iff]
intro R hR
rw [← (hull_kernel_of_isClosed hT hG hR.1), ← gc_closureOperator]
exact ClosureOperator.monotone _ hR.2
end PrimitiveSpectrum
|
Bounded.lean
|
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.Order.Compact
import Mathlib.Topology.MetricSpace.ProperSpace
import Mathlib.Topology.MetricSpace.Cauchy
import Mathlib.Topology.MetricSpace.Defs
import Mathlib.Topology.EMetricSpace.Diam
/-!
## Boundedness in (pseudo)-metric spaces
This file contains one definition, and various results on boundedness in pseudo-metric spaces.
* `Metric.diam s` : The `iSup` of the distances of members of `s`.
Defined in terms of `EMetric.diam`, for better handling of the case when it should be infinite.
* `isBounded_iff_subset_closedBall`: a non-empty set is bounded if and only if
it is included in some closed ball
* describing the cobounded filter, relating to the cocompact filter
* `IsCompact.isBounded`: compact sets are bounded
* `TotallyBounded.isBounded`: totally bounded sets are bounded
* `isCompact_iff_isClosed_bounded`, the **Heine–Borel theorem**:
in a proper space, a set is compact if and only if it is closed and bounded.
* `cobounded_eq_cocompact`: in a proper space, cobounded and compact sets are the same
diameter of a subset, and its relation to boundedness
## Tags
metric, pseudo_metric, bounded, diameter, Heine-Borel theorem
-/
assert_not_exists Module.Basis
open Set Filter Bornology
open scoped ENNReal Uniformity Topology Pointwise
universe u v w
variable {α : Type u} {β : Type v} {X ι : Type*}
namespace Metric
section Bounded
variable {x : α} {s t : Set α} {r : ℝ}
variable [PseudoMetricSpace α]
/-- Closed balls are bounded -/
theorem isBounded_closedBall : IsBounded (closedBall x r) :=
isBounded_iff.2 ⟨r + r, fun y hy z hz =>
calc dist y z ≤ dist y x + dist z x := dist_triangle_right _ _ _
_ ≤ r + r := add_le_add hy hz⟩
/-- Open balls are bounded -/
theorem isBounded_ball : IsBounded (ball x r) :=
isBounded_closedBall.subset ball_subset_closedBall
/-- Spheres are bounded -/
theorem isBounded_sphere : IsBounded (sphere x r) :=
isBounded_closedBall.subset sphere_subset_closedBall
/-- Given a point, a bounded subset is included in some ball around this point -/
theorem isBounded_iff_subset_closedBall (c : α) : IsBounded s ↔ ∃ r, s ⊆ closedBall c r :=
⟨fun h ↦ (isBounded_iff.1 (h.insert c)).imp fun _r hr _x hx ↦ hr (.inr hx) (mem_insert _ _),
fun ⟨_r, hr⟩ ↦ isBounded_closedBall.subset hr⟩
theorem _root_.Bornology.IsBounded.subset_closedBall (h : IsBounded s) (c : α) :
∃ r, s ⊆ closedBall c r :=
(isBounded_iff_subset_closedBall c).1 h
theorem _root_.Bornology.IsBounded.subset_ball_lt (h : IsBounded s) (a : ℝ) (c : α) :
∃ r, a < r ∧ s ⊆ ball c r :=
let ⟨r, hr⟩ := h.subset_closedBall c
⟨max r a + 1, (le_max_right _ _).trans_lt (lt_add_one _), hr.trans <| closedBall_subset_ball <|
(le_max_left _ _).trans_lt (lt_add_one _)⟩
theorem _root_.Bornology.IsBounded.subset_ball (h : IsBounded s) (c : α) : ∃ r, s ⊆ ball c r :=
(h.subset_ball_lt 0 c).imp fun _ ↦ And.right
theorem isBounded_iff_subset_ball (c : α) : IsBounded s ↔ ∃ r, s ⊆ ball c r :=
⟨(IsBounded.subset_ball · c), fun ⟨_r, hr⟩ ↦ isBounded_ball.subset hr⟩
theorem _root_.Bornology.IsBounded.subset_closedBall_lt (h : IsBounded s) (a : ℝ) (c : α) :
∃ r, a < r ∧ s ⊆ closedBall c r :=
let ⟨r, har, hr⟩ := h.subset_ball_lt a c
⟨r, har, hr.trans ball_subset_closedBall⟩
theorem isBounded_closure_of_isBounded (h : IsBounded s) : IsBounded (closure s) :=
let ⟨C, h⟩ := isBounded_iff.1 h
isBounded_iff.2 ⟨C, fun _a ha _b hb => isClosed_Iic.closure_subset <|
map_mem_closure₂ continuous_dist ha hb h⟩
protected theorem _root_.Bornology.IsBounded.closure (h : IsBounded s) : IsBounded (closure s) :=
isBounded_closure_of_isBounded h
@[simp]
theorem isBounded_closure_iff : IsBounded (closure s) ↔ IsBounded s :=
⟨fun h => h.subset subset_closure, fun h => h.closure⟩
theorem hasBasis_cobounded_compl_closedBall (c : α) :
(cobounded α).HasBasis (fun _ ↦ True) (fun r ↦ (closedBall c r)ᶜ) :=
⟨compl_surjective.forall.2 fun _ ↦ (isBounded_iff_subset_closedBall c).trans <| by simp⟩
theorem hasAntitoneBasis_cobounded_compl_closedBall (c : α) :
(cobounded α).HasAntitoneBasis (fun r ↦ (closedBall c r)ᶜ) :=
⟨Metric.hasBasis_cobounded_compl_closedBall _, fun _ _ hr _ ↦ by simpa using hr.trans_lt⟩
theorem hasBasis_cobounded_compl_ball (c : α) :
(cobounded α).HasBasis (fun _ ↦ True) (fun r ↦ (ball c r)ᶜ) :=
⟨compl_surjective.forall.2 fun _ ↦ (isBounded_iff_subset_ball c).trans <| by simp⟩
theorem hasAntitoneBasis_cobounded_compl_ball (c : α) :
(cobounded α).HasAntitoneBasis (fun r ↦ (ball c r)ᶜ) :=
⟨Metric.hasBasis_cobounded_compl_ball _, fun _ _ hr _ ↦ by simpa using hr.trans⟩
@[simp]
theorem comap_dist_right_atTop (c : α) : comap (dist · c) atTop = cobounded α :=
(atTop_basis.comap _).eq_of_same_basis <| by
simpa only [compl_def, mem_ball, not_lt] using hasBasis_cobounded_compl_ball c
@[simp]
theorem comap_dist_left_atTop (c : α) : comap (dist c) atTop = cobounded α := by
simpa only [dist_comm _ c] using comap_dist_right_atTop c
@[simp]
theorem tendsto_dist_right_atTop_iff (c : α) {f : β → α} {l : Filter β} :
Tendsto (fun x ↦ dist (f x) c) l atTop ↔ Tendsto f l (cobounded α) := by
rw [← comap_dist_right_atTop c, tendsto_comap_iff, Function.comp_def]
@[simp]
theorem tendsto_dist_left_atTop_iff (c : α) {f : β → α} {l : Filter β} :
Tendsto (fun x ↦ dist c (f x)) l atTop ↔ Tendsto f l (cobounded α) := by
simp only [dist_comm c, tendsto_dist_right_atTop_iff]
theorem tendsto_dist_right_cobounded_atTop (c : α) : Tendsto (dist · c) (cobounded α) atTop :=
tendsto_iff_comap.2 (comap_dist_right_atTop c).ge
theorem tendsto_dist_left_cobounded_atTop (c : α) : Tendsto (dist c) (cobounded α) atTop :=
tendsto_iff_comap.2 (comap_dist_left_atTop c).ge
/-- A totally bounded set is bounded -/
theorem _root_.TotallyBounded.isBounded {s : Set α} (h : TotallyBounded s) : IsBounded s :=
-- We cover the totally bounded set by finitely many balls of radius 1,
-- and then argue that a finite union of bounded sets is bounded
let ⟨_t, fint, subs⟩ := (totallyBounded_iff.mp h) 1 zero_lt_one
((isBounded_biUnion fint).2 fun _ _ => isBounded_ball).subset subs
/-- A compact set is bounded -/
theorem _root_.IsCompact.isBounded {s : Set α} (h : IsCompact s) : IsBounded s :=
-- A compact set is totally bounded, thus bounded
h.totallyBounded.isBounded
theorem cobounded_le_cocompact : cobounded α ≤ cocompact α :=
hasBasis_cocompact.ge_iff.2 fun _s hs ↦ hs.isBounded
theorem isCobounded_iff_closedBall_compl_subset {s : Set α} (c : α) :
IsCobounded s ↔ ∃ (r : ℝ), (Metric.closedBall c r)ᶜ ⊆ s := by
rw [← isBounded_compl_iff, isBounded_iff_subset_closedBall c]
apply exists_congr
intro r
rw [compl_subset_comm]
theorem _root_.Bornology.IsCobounded.closedBall_compl_subset {s : Set α} (hs : IsCobounded s)
(c : α) : ∃ (r : ℝ), (Metric.closedBall c r)ᶜ ⊆ s :=
(isCobounded_iff_closedBall_compl_subset c).mp hs
theorem closedBall_compl_subset_of_mem_cocompact {s : Set α} (hs : s ∈ cocompact α) (c : α) :
∃ (r : ℝ), (Metric.closedBall c r)ᶜ ⊆ s :=
IsCobounded.closedBall_compl_subset (cobounded_le_cocompact hs) c
theorem mem_cocompact_of_closedBall_compl_subset [ProperSpace α] (c : α)
(h : ∃ r, (closedBall c r)ᶜ ⊆ s) : s ∈ cocompact α := by
rcases h with ⟨r, h⟩
rw [Filter.mem_cocompact]
exact ⟨closedBall c r, isCompact_closedBall c r, h⟩
theorem mem_cocompact_iff_closedBall_compl_subset [ProperSpace α] (c : α) :
s ∈ cocompact α ↔ ∃ r, (closedBall c r)ᶜ ⊆ s :=
⟨(closedBall_compl_subset_of_mem_cocompact · _), mem_cocompact_of_closedBall_compl_subset _⟩
/-- Characterization of the boundedness of the range of a function -/
theorem isBounded_range_iff {f : β → α} : IsBounded (range f) ↔ ∃ C, ∀ x y, dist (f x) (f y) ≤ C :=
isBounded_iff.trans <| by simp only [forall_mem_range]
theorem isBounded_image_iff {f : β → α} {s : Set β} :
IsBounded (f '' s) ↔ ∃ C, ∀ x ∈ s, ∀ y ∈ s, dist (f x) (f y) ≤ C :=
isBounded_iff.trans <| by simp only [forall_mem_image]
theorem isBounded_range_of_tendsto_cofinite_uniformity {f : β → α}
(hf : Tendsto (Prod.map f f) (.cofinite ×ˢ .cofinite) (𝓤 α)) : IsBounded (range f) := by
rcases (hasBasis_cofinite.prod_self.tendsto_iff uniformity_basis_dist).1 hf 1 zero_lt_one with
⟨s, hsf, hs1⟩
rw [← image_union_image_compl_eq_range]
refine (hsf.image f).isBounded.union (isBounded_image_iff.2 ⟨1, fun x hx y hy ↦ ?_⟩)
exact le_of_lt (hs1 (x, y) ⟨hx, hy⟩)
theorem isBounded_range_of_cauchy_map_cofinite {f : β → α} (hf : Cauchy (map f cofinite)) :
IsBounded (range f) :=
isBounded_range_of_tendsto_cofinite_uniformity <| (cauchy_map_iff.1 hf).2
theorem _root_.CauchySeq.isBounded_range {f : ℕ → α} (hf : CauchySeq f) : IsBounded (range f) :=
isBounded_range_of_cauchy_map_cofinite <| by rwa [Nat.cofinite_eq_atTop]
theorem isBounded_range_of_tendsto_cofinite {f : β → α} {a : α} (hf : Tendsto f cofinite (𝓝 a)) :
IsBounded (range f) :=
isBounded_range_of_tendsto_cofinite_uniformity <|
(hf.prodMap hf).mono_right <| nhds_prod_eq.symm.trans_le (nhds_le_uniformity a)
/-- In a compact space, all sets are bounded -/
theorem isBounded_of_compactSpace [CompactSpace α] : IsBounded s :=
isCompact_univ.isBounded.subset (subset_univ _)
theorem isBounded_range_of_tendsto (u : ℕ → α) {x : α} (hu : Tendsto u atTop (𝓝 x)) :
IsBounded (range u) :=
hu.cauchySeq.isBounded_range
theorem disjoint_nhds_cobounded (x : α) : Disjoint (𝓝 x) (cobounded α) :=
disjoint_of_disjoint_of_mem disjoint_compl_right (ball_mem_nhds _ one_pos) isBounded_ball
theorem disjoint_cobounded_nhds (x : α) : Disjoint (cobounded α) (𝓝 x) :=
(disjoint_nhds_cobounded x).symm
theorem disjoint_nhdsSet_cobounded {s : Set α} (hs : IsCompact s) : Disjoint (𝓝ˢ s) (cobounded α) :=
hs.disjoint_nhdsSet_left.2 fun _ _ ↦ disjoint_nhds_cobounded _
theorem disjoint_cobounded_nhdsSet {s : Set α} (hs : IsCompact s) : Disjoint (cobounded α) (𝓝ˢ s) :=
(disjoint_nhdsSet_cobounded hs).symm
theorem exists_isBounded_image_of_tendsto {α β : Type*} [PseudoMetricSpace β]
{l : Filter α} {f : α → β} {x : β} (hf : Tendsto f l (𝓝 x)) :
∃ s ∈ l, IsBounded (f '' s) :=
(l.basis_sets.map f).disjoint_iff_left.mp <| (disjoint_nhds_cobounded x).mono_left hf
/-- If a function is continuous within a set `s` at every point of a compact set `k`, then it is
bounded on some open neighborhood of `k` in `s`. -/
theorem exists_isOpen_isBounded_image_inter_of_isCompact_of_forall_continuousWithinAt
[TopologicalSpace β] {k s : Set β} {f : β → α} (hk : IsCompact k)
(hf : ∀ x ∈ k, ContinuousWithinAt f s x) :
∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' (t ∩ s)) := by
have : Disjoint (𝓝ˢ k ⊓ 𝓟 s) (comap f (cobounded α)) := by
rw [disjoint_assoc, inf_comm, hk.disjoint_nhdsSet_left]
exact fun x hx ↦ disjoint_left_comm.2 <|
tendsto_comap.disjoint (disjoint_cobounded_nhds _) (hf x hx)
rcases ((((hasBasis_nhdsSet _).inf_principal _)).disjoint_iff ((basis_sets _).comap _)).1 this
with ⟨U, ⟨hUo, hkU⟩, t, ht, hd⟩
refine ⟨U, hkU, hUo, (isBounded_compl_iff.2 ht).subset ?_⟩
rwa [image_subset_iff, preimage_compl, subset_compl_iff_disjoint_right]
/-- If a function is continuous at every point of a compact set `k`, then it is bounded on
some open neighborhood of `k`. -/
theorem exists_isOpen_isBounded_image_of_isCompact_of_forall_continuousAt [TopologicalSpace β]
{k : Set β} {f : β → α} (hk : IsCompact k) (hf : ∀ x ∈ k, ContinuousAt f x) :
∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' t) := by
simp_rw [← continuousWithinAt_univ] at hf
simpa only [inter_univ] using
exists_isOpen_isBounded_image_inter_of_isCompact_of_forall_continuousWithinAt hk hf
/-- If a function is continuous on a set `s` containing a compact set `k`, then it is bounded on
some open neighborhood of `k` in `s`. -/
theorem exists_isOpen_isBounded_image_inter_of_isCompact_of_continuousOn [TopologicalSpace β]
{k s : Set β} {f : β → α} (hk : IsCompact k) (hks : k ⊆ s) (hf : ContinuousOn f s) :
∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' (t ∩ s)) :=
exists_isOpen_isBounded_image_inter_of_isCompact_of_forall_continuousWithinAt hk fun x hx =>
hf x (hks hx)
/-- If a function is continuous on a neighborhood of a compact set `k`, then it is bounded on
some open neighborhood of `k`. -/
theorem exists_isOpen_isBounded_image_of_isCompact_of_continuousOn [TopologicalSpace β]
{k s : Set β} {f : β → α} (hk : IsCompact k) (hs : IsOpen s) (hks : k ⊆ s)
(hf : ContinuousOn f s) : ∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' t) :=
exists_isOpen_isBounded_image_of_isCompact_of_forall_continuousAt hk fun _x hx =>
hf.continuousAt (hs.mem_nhds (hks hx))
/-- The **Heine–Borel theorem**: In a proper space, a closed bounded set is compact. -/
theorem isCompact_of_isClosed_isBounded [ProperSpace α] (hc : IsClosed s) (hb : IsBounded s) :
IsCompact s := by
rcases eq_empty_or_nonempty s with (rfl | ⟨x, -⟩)
· exact isCompact_empty
· rcases hb.subset_closedBall x with ⟨r, hr⟩
exact (isCompact_closedBall x r).of_isClosed_subset hc hr
/-- The **Heine–Borel theorem**: In a proper space, the closure of a bounded set is compact. -/
theorem _root_.Bornology.IsBounded.isCompact_closure [ProperSpace α] (h : IsBounded s) :
IsCompact (closure s) :=
isCompact_of_isClosed_isBounded isClosed_closure h.closure
-- TODO: assume `[MetricSpace α]` instead of `[PseudoMetricSpace α] [T2Space α]`
/-- The **Heine–Borel theorem**:
In a proper Hausdorff space, a set is compact if and only if it is closed and bounded. -/
theorem isCompact_iff_isClosed_bounded [T2Space α] [ProperSpace α] :
IsCompact s ↔ IsClosed s ∧ IsBounded s :=
⟨fun h => ⟨h.isClosed, h.isBounded⟩, fun h => isCompact_of_isClosed_isBounded h.1 h.2⟩
theorem compactSpace_iff_isBounded_univ [ProperSpace α] :
CompactSpace α ↔ IsBounded (univ : Set α) :=
⟨@isBounded_of_compactSpace α _ _, fun hb => ⟨isCompact_of_isClosed_isBounded isClosed_univ hb⟩⟩
section CompactIccSpace
variable [Preorder α] [CompactIccSpace α]
theorem _root_.totallyBounded_Icc (a b : α) : TotallyBounded (Icc a b) :=
isCompact_Icc.totallyBounded
theorem _root_.totallyBounded_Ico (a b : α) : TotallyBounded (Ico a b) :=
(totallyBounded_Icc a b).subset Ico_subset_Icc_self
theorem _root_.totallyBounded_Ioc (a b : α) : TotallyBounded (Ioc a b) :=
(totallyBounded_Icc a b).subset Ioc_subset_Icc_self
theorem _root_.totallyBounded_Ioo (a b : α) : TotallyBounded (Ioo a b) :=
(totallyBounded_Icc a b).subset Ioo_subset_Icc_self
theorem isBounded_Icc (a b : α) : IsBounded (Icc a b) :=
(totallyBounded_Icc a b).isBounded
theorem isBounded_Ico (a b : α) : IsBounded (Ico a b) :=
(totallyBounded_Ico a b).isBounded
theorem isBounded_Ioc (a b : α) : IsBounded (Ioc a b) :=
(totallyBounded_Ioc a b).isBounded
theorem isBounded_Ioo (a b : α) : IsBounded (Ioo a b) :=
(totallyBounded_Ioo a b).isBounded
/-- In a pseudo metric space with a conditionally complete linear order such that the order and the
metric structure give the same topology, any order-bounded set is metric-bounded. -/
theorem isBounded_of_bddAbove_of_bddBelow {s : Set α} (h₁ : BddAbove s) (h₂ : BddBelow s) :
IsBounded s :=
let ⟨u, hu⟩ := h₁
let ⟨l, hl⟩ := h₂
(isBounded_Icc l u).subset (fun _x hx => mem_Icc.mpr ⟨hl hx, hu hx⟩)
end CompactIccSpace
end Bounded
section Diam
variable {s : Set α} {x y z : α}
section PseudoMetricSpace
variable [PseudoMetricSpace α]
/-- The diameter of a set in a metric space. To get controllable behavior even when the diameter
should be infinite, we express it in terms of the `EMetric.diam` -/
noncomputable def diam (s : Set α) : ℝ :=
ENNReal.toReal (EMetric.diam s)
/-- The diameter of a set is always nonnegative -/
theorem diam_nonneg : 0 ≤ diam s :=
ENNReal.toReal_nonneg
theorem diam_subsingleton (hs : s.Subsingleton) : diam s = 0 := by
simp only [diam, EMetric.diam_subsingleton hs, ENNReal.toReal_zero]
/-- The empty set has zero diameter -/
@[simp]
theorem diam_empty : diam (∅ : Set α) = 0 :=
diam_subsingleton subsingleton_empty
/-- A singleton has zero diameter -/
@[simp]
theorem diam_singleton : diam ({x} : Set α) = 0 :=
diam_subsingleton subsingleton_singleton
@[to_additive (attr := simp)]
theorem diam_one [One α] : diam (1 : Set α) = 0 :=
diam_singleton
-- Does not work as a simp-lemma, since {x, y} reduces to (insert y {x})
theorem diam_pair : diam ({x, y} : Set α) = dist x y := by
simp only [diam, EMetric.diam_pair, dist_edist]
-- Does not work as a simp-lemma, since {x, y, z} reduces to (insert z (insert y {x}))
theorem diam_triple :
Metric.diam ({x, y, z} : Set α) = max (max (dist x y) (dist x z)) (dist y z) := by
simp only [Metric.diam, EMetric.diam_triple, dist_edist]
rw [ENNReal.toReal_max, ENNReal.toReal_max] <;> apply_rules [ne_of_lt, edist_lt_top, max_lt]
/-- If the distance between any two points in a set is bounded by some constant `C`,
then `ENNReal.ofReal C` bounds the emetric diameter of this set. -/
theorem ediam_le_of_forall_dist_le {C : ℝ} (h : ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ C) :
EMetric.diam s ≤ ENNReal.ofReal C :=
EMetric.diam_le fun x hx y hy => (edist_dist x y).symm ▸ ENNReal.ofReal_le_ofReal (h x hx y hy)
/-- If the distance between any two points in a set is bounded by some non-negative constant,
this constant bounds the diameter. -/
theorem diam_le_of_forall_dist_le {C : ℝ} (h₀ : 0 ≤ C) (h : ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ C) :
diam s ≤ C :=
ENNReal.toReal_le_of_le_ofReal h₀ (ediam_le_of_forall_dist_le h)
/-- If the distance between any two points in a nonempty set is bounded by some constant,
this constant bounds the diameter. -/
theorem diam_le_of_forall_dist_le_of_nonempty (hs : s.Nonempty) {C : ℝ}
(h : ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ C) : diam s ≤ C :=
have h₀ : 0 ≤ C :=
let ⟨x, hx⟩ := hs
le_trans dist_nonneg (h x hx x hx)
diam_le_of_forall_dist_le h₀ h
/-- The distance between two points in a set is controlled by the diameter of the set. -/
theorem dist_le_diam_of_mem' (h : EMetric.diam s ≠ ⊤) (hx : x ∈ s) (hy : y ∈ s) :
dist x y ≤ diam s := by
rw [diam, dist_edist]
exact ENNReal.toReal_mono h <| EMetric.edist_le_diam_of_mem hx hy
/-- Characterize the boundedness of a set in terms of the finiteness of its emetric.diameter. -/
theorem isBounded_iff_ediam_ne_top : IsBounded s ↔ EMetric.diam s ≠ ⊤ :=
isBounded_iff.trans <| Iff.intro
(fun ⟨_C, hC⟩ => ne_top_of_le_ne_top ENNReal.ofReal_ne_top <| ediam_le_of_forall_dist_le hC)
fun h => ⟨diam s, fun _x hx _y hy => dist_le_diam_of_mem' h hx hy⟩
alias ⟨_root_.Bornology.IsBounded.ediam_ne_top, _⟩ := isBounded_iff_ediam_ne_top
theorem ediam_eq_top_iff_unbounded : EMetric.diam s = ⊤ ↔ ¬IsBounded s :=
isBounded_iff_ediam_ne_top.not_left.symm
theorem ediam_univ_eq_top_iff_noncompact [ProperSpace α] :
EMetric.diam (univ : Set α) = ∞ ↔ NoncompactSpace α := by
rw [← not_compactSpace_iff, compactSpace_iff_isBounded_univ, isBounded_iff_ediam_ne_top,
Classical.not_not]
@[simp]
theorem ediam_univ_of_noncompact [ProperSpace α] [NoncompactSpace α] :
EMetric.diam (univ : Set α) = ∞ :=
ediam_univ_eq_top_iff_noncompact.mpr ‹_›
@[simp]
theorem diam_univ_of_noncompact [ProperSpace α] [NoncompactSpace α] : diam (univ : Set α) = 0 := by
simp [diam]
/-- The distance between two points in a set is controlled by the diameter of the set. -/
theorem dist_le_diam_of_mem (h : IsBounded s) (hx : x ∈ s) (hy : y ∈ s) : dist x y ≤ diam s :=
dist_le_diam_of_mem' h.ediam_ne_top hx hy
theorem ediam_of_unbounded (h : ¬IsBounded s) : EMetric.diam s = ∞ := ediam_eq_top_iff_unbounded.2 h
/-- An unbounded set has zero diameter. If you would prefer to get the value ∞, use `EMetric.diam`.
This lemma makes it possible to avoid side conditions in some situations -/
theorem diam_eq_zero_of_unbounded (h : ¬IsBounded s) : diam s = 0 := by
rw [diam, ediam_of_unbounded h, ENNReal.toReal_top]
/-- If `s ⊆ t`, then the diameter of `s` is bounded by that of `t`, provided `t` is bounded. -/
theorem diam_mono {s t : Set α} (h : s ⊆ t) (ht : IsBounded t) : diam s ≤ diam t :=
ENNReal.toReal_mono ht.ediam_ne_top <| EMetric.diam_mono h
/-- The diameter of a union is controlled by the sum of the diameters, and the distance between
any two points in each of the sets. This lemma is true without any side condition, since it is
obviously true if `s ∪ t` is unbounded. -/
theorem diam_union {t : Set α} (xs : x ∈ s) (yt : y ∈ t) :
diam (s ∪ t) ≤ diam s + dist x y + diam t := by
simp only [diam, dist_edist]
refine (ENNReal.toReal_le_add' (EMetric.diam_union xs yt) ?_ ?_).trans
(add_le_add_right ENNReal.toReal_add_le _)
· simp only [ENNReal.add_eq_top, edist_ne_top, or_false]
exact fun h ↦ top_unique <| h ▸ EMetric.diam_mono subset_union_left
· exact fun h ↦ top_unique <| h ▸ EMetric.diam_mono subset_union_right
/-- If two sets intersect, the diameter of the union is bounded by the sum of the diameters. -/
theorem diam_union' {t : Set α} (h : (s ∩ t).Nonempty) : diam (s ∪ t) ≤ diam s + diam t := by
rcases h with ⟨x, ⟨xs, xt⟩⟩
simpa using diam_union xs xt
theorem diam_le_of_subset_closedBall {r : ℝ} (hr : 0 ≤ r) (h : s ⊆ closedBall x r) :
diam s ≤ 2 * r :=
diam_le_of_forall_dist_le (mul_nonneg zero_le_two hr) fun a ha b hb =>
calc
dist a b ≤ dist a x + dist b x := dist_triangle_right _ _ _
_ ≤ r + r := add_le_add (h ha) (h hb)
_ = 2 * r := by simp [mul_two, mul_comm]
/-- The diameter of a closed ball of radius `r` is at most `2 r`. -/
theorem diam_closedBall {r : ℝ} (h : 0 ≤ r) : diam (closedBall x r) ≤ 2 * r :=
diam_le_of_subset_closedBall h Subset.rfl
/-- The diameter of a ball of radius `r` is at most `2 r`. -/
theorem diam_ball {r : ℝ} (h : 0 ≤ r) : diam (ball x r) ≤ 2 * r :=
diam_le_of_subset_closedBall h ball_subset_closedBall
/-- If a family of complete sets with diameter tending to `0` is such that each finite intersection
is nonempty, then the total intersection is also nonempty. -/
theorem _root_.IsComplete.nonempty_iInter_of_nonempty_biInter {s : ℕ → Set α}
(h0 : IsComplete (s 0)) (hs : ∀ n, IsClosed (s n)) (h's : ∀ n, IsBounded (s n))
(h : ∀ N, (⋂ n ≤ N, s n).Nonempty) (h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)) :
(⋂ n, s n).Nonempty := by
let u N := (h N).some
have I : ∀ n N, n ≤ N → u N ∈ s n := by
intro n N hn
apply mem_of_subset_of_mem _ (h N).choose_spec
intro x hx
simp only [mem_iInter] at hx
exact hx n hn
have : CauchySeq u := by
apply cauchySeq_of_le_tendsto_0 _ _ h'
intro m n N hm hn
exact dist_le_diam_of_mem (h's N) (I _ _ hm) (I _ _ hn)
obtain ⟨x, -, xlim⟩ : ∃ x ∈ s 0, Tendsto (fun n : ℕ => u n) atTop (𝓝 x) :=
cauchySeq_tendsto_of_isComplete h0 (fun n => I 0 n (zero_le _)) this
refine ⟨x, mem_iInter.2 fun n => ?_⟩
apply (hs n).mem_of_tendsto xlim
filter_upwards [Ici_mem_atTop n] with p hp
exact I n p hp
/-- In a complete space, if a family of closed sets with diameter tending to `0` is such that each
finite intersection is nonempty, then the total intersection is also nonempty. -/
theorem nonempty_iInter_of_nonempty_biInter [CompleteSpace α] {s : ℕ → Set α}
(hs : ∀ n, IsClosed (s n)) (h's : ∀ n, IsBounded (s n)) (h : ∀ N, (⋂ n ≤ N, s n).Nonempty)
(h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)) : (⋂ n, s n).Nonempty :=
(hs 0).isComplete.nonempty_iInter_of_nonempty_biInter hs h's h h'
end PseudoMetricSpace
section MetricSpace
theorem diam_pos [MetricSpace α] (hs1 : s.Nontrivial) (hs2 : IsBounded s) : 0 < diam s := by
rcases hs1 with ⟨x, hx, y, hy, hxy⟩
exact (dist_pos.mpr hxy).trans_le <| Metric.dist_le_diam_of_mem hs2 hx hy
end MetricSpace
end Diam
end Metric
namespace Mathlib.Meta.Positivity
open Lean Meta Qq Function
/-- Extension for the `positivity` tactic: the diameter of a set is always nonnegative. -/
@[positivity Metric.diam _]
def evalDiam : PositivityExt where eval {u α} _zα _pα e := do
match u, α, e with
| 0, ~q(ℝ), ~q(@Metric.diam _ $inst $s) =>
assertInstancesCommute
pure (.nonnegative q(Metric.diam_nonneg))
| _, _, _ => throwError "not ‖ · ‖"
end Mathlib.Meta.Positivity
open Metric
variable [PseudoMetricSpace α]
theorem Metric.cobounded_eq_cocompact [ProperSpace α] : cobounded α = cocompact α := by
nontriviality α; inhabit α
exact cobounded_le_cocompact.antisymm <| (hasBasis_cobounded_compl_closedBall default).ge_iff.2
fun _ _ ↦ (isCompact_closedBall _ _).compl_mem_cocompact
theorem tendsto_dist_right_cocompact_atTop [ProperSpace α] (x : α) :
Tendsto (dist · x) (cocompact α) atTop :=
(tendsto_dist_right_cobounded_atTop x).mono_left cobounded_eq_cocompact.ge
theorem tendsto_dist_left_cocompact_atTop [ProperSpace α] (x : α) :
Tendsto (dist x) (cocompact α) atTop :=
(tendsto_dist_left_cobounded_atTop x).mono_left cobounded_eq_cocompact.ge
theorem comap_dist_left_atTop_eq_cocompact [ProperSpace α] (x : α) :
comap (dist x) atTop = cocompact α := by simp [cobounded_eq_cocompact]
theorem tendsto_cocompact_of_tendsto_dist_comp_atTop {f : β → α} {l : Filter β} (x : α)
(h : Tendsto (fun y => dist (f y) x) l atTop) : Tendsto f l (cocompact α) :=
((tendsto_dist_right_atTop_iff _).1 h).mono_right cobounded_le_cocompact
theorem Metric.finite_isBounded_inter_isClosed [ProperSpace α] {K s : Set α} [DiscreteTopology s]
(hK : IsBounded K) (hs : IsClosed s) : Set.Finite (K ∩ s) := by
refine Set.Finite.subset (IsCompact.finite ?_ ?_) (Set.inter_subset_inter_left s subset_closure)
· exact hK.isCompact_closure.inter_right hs
· exact DiscreteTopology.of_subset inferInstance Set.inter_subset_right
|
Hyperoperation.lean
|
/-
Copyright (c) 2023 Mark Andrew Gerads. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mark Andrew Gerads, Junyan Xu, Eric Wieser
-/
import Mathlib.Tactic.Ring
/-!
# Hyperoperation sequence
This file defines the Hyperoperation sequence.
`hyperoperation 0 m k = k + 1`
`hyperoperation 1 m k = m + k`
`hyperoperation 2 m k = m * k`
`hyperoperation 3 m k = m ^ k`
`hyperoperation (n + 3) m 0 = 1`
`hyperoperation (n + 1) m (k + 1) = hyperoperation n m (hyperoperation (n + 1) m k)`
## References
* <https://en.wikipedia.org/wiki/Hyperoperation>
## Tags
hyperoperation
-/
/-- Implementation of the hyperoperation sequence
where `hyperoperation n m k` is the `n`th hyperoperation between `m` and `k`.
-/
def hyperoperation : ℕ → ℕ → ℕ → ℕ
| 0, _, k => k + 1
| 1, m, 0 => m
| 2, _, 0 => 0
| _ + 3, _, 0 => 1
| n + 1, m, k + 1 => hyperoperation n m (hyperoperation (n + 1) m k)
attribute [local grind] hyperoperation
-- Basic hyperoperation lemmas
@[simp, grind =]
theorem hyperoperation_zero (m k : ℕ) : hyperoperation 0 m k = k + 1 := by
grind
@[grind =]
theorem hyperoperation_ge_three_eq_one (n m : ℕ) : hyperoperation (n + 3) m 0 = 1 := by
grind
@[grind =]
theorem hyperoperation_recursion (n m k : ℕ) :
hyperoperation (n + 1) m (k + 1) = hyperoperation n m (hyperoperation (n + 1) m k) := by
grind
-- Interesting hyperoperation lemmas
@[simp, grind =]
theorem hyperoperation_one (m k : ℕ) : hyperoperation 1 m k = m + k := by
induction k with grind
@[simp, grind =]
theorem hyperoperation_two (m k : ℕ) : hyperoperation 2 m k = m * k := by
induction k with grind
@[simp, grind =]
theorem hyperoperation_three (m k : ℕ) : hyperoperation 3 m k = m ^ k := by
induction k with grind
@[grind =]
theorem hyperoperation_ge_two_eq_self (n m : ℕ) : hyperoperation (n + 2) m 1 = m := by
induction n with grind
@[grind =]
theorem hyperoperation_two_two_eq_four (n : ℕ) : hyperoperation (n + 1) 2 2 = 4 := by
induction n with grind
@[grind =]
theorem hyperoperation_ge_three_one (n k : ℕ) : hyperoperation (n + 3) 1 k = 1 := by
induction n generalizing k with grind [cases Nat]
@[grind =]
theorem hyperoperation_ge_four_zero (n k : ℕ) :
hyperoperation (n + 4) 0 k = if Even k then 1 else 0 := by
induction k with grind
|
Archive.lean
|
import Archive.Arithcc
import Archive.Examples.Eisenstein
import Archive.Examples.IfNormalization.Result
import Archive.Examples.IfNormalization.Statement
import Archive.Examples.IfNormalization.WithoutAesop
import Archive.Examples.MersennePrimes
import Archive.Examples.PropEncodable
import Archive.Hairer
import Archive.Imo.Imo1959Q1
import Archive.Imo.Imo1959Q2
import Archive.Imo.Imo1960Q1
import Archive.Imo.Imo1960Q2
import Archive.Imo.Imo1961Q3
import Archive.Imo.Imo1962Q1
import Archive.Imo.Imo1962Q4
import Archive.Imo.Imo1963Q5
import Archive.Imo.Imo1964Q1
import Archive.Imo.Imo1969Q1
import Archive.Imo.Imo1972Q5
import Archive.Imo.Imo1975Q1
import Archive.Imo.Imo1977Q6
import Archive.Imo.Imo1981Q3
import Archive.Imo.Imo1982Q1
import Archive.Imo.Imo1982Q3
import Archive.Imo.Imo1985Q2
import Archive.Imo.Imo1986Q5
import Archive.Imo.Imo1987Q1
import Archive.Imo.Imo1988Q6
import Archive.Imo.Imo1994Q1
import Archive.Imo.Imo1997Q3
import Archive.Imo.Imo1998Q2
import Archive.Imo.Imo2001Q2
import Archive.Imo.Imo2001Q3
import Archive.Imo.Imo2001Q4
import Archive.Imo.Imo2001Q5
import Archive.Imo.Imo2001Q6
import Archive.Imo.Imo2005Q3
import Archive.Imo.Imo2005Q4
import Archive.Imo.Imo2006Q3
import Archive.Imo.Imo2006Q5
import Archive.Imo.Imo2008Q2
import Archive.Imo.Imo2008Q3
import Archive.Imo.Imo2008Q4
import Archive.Imo.Imo2011Q3
import Archive.Imo.Imo2011Q5
import Archive.Imo.Imo2013Q1
import Archive.Imo.Imo2013Q5
import Archive.Imo.Imo2015Q6
import Archive.Imo.Imo2019Q1
import Archive.Imo.Imo2019Q2
import Archive.Imo.Imo2019Q4
import Archive.Imo.Imo2020Q2
import Archive.Imo.Imo2021Q1
import Archive.Imo.Imo2024Q1
import Archive.Imo.Imo2024Q2
import Archive.Imo.Imo2024Q3
import Archive.Imo.Imo2024Q5
import Archive.Imo.Imo2024Q6
import Archive.MiuLanguage.Basic
import Archive.MiuLanguage.DecisionNec
import Archive.MiuLanguage.DecisionSuf
import Archive.OxfordInvariants.Summer2021.Week3P1
import Archive.Sensitivity
import Archive.Wiedijk100Theorems.AbelRuffini
import Archive.Wiedijk100Theorems.AreaOfACircle
import Archive.Wiedijk100Theorems.AscendingDescendingSequences
import Archive.Wiedijk100Theorems.BallotProblem
import Archive.Wiedijk100Theorems.BirthdayProblem
import Archive.Wiedijk100Theorems.BuffonsNeedle
import Archive.Wiedijk100Theorems.CubingACube
import Archive.Wiedijk100Theorems.FriendshipGraphs
import Archive.Wiedijk100Theorems.HeronsFormula
import Archive.Wiedijk100Theorems.InverseTriangleSum
import Archive.Wiedijk100Theorems.Konigsberg
import Archive.Wiedijk100Theorems.Partition
import Archive.Wiedijk100Theorems.PerfectNumbers
import Archive.Wiedijk100Theorems.SolutionOfCubicQuartic
import Archive.Wiedijk100Theorems.SumOfPrimeReciprocalsDiverges
import Archive.ZagierTwoSquares
|
FailIfNoProgress.lean
|
/-
Copyright (c) 2023 Thomas Murrills. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Murrills
-/
import Mathlib.Init
import Lean.Elab.Tactic.Basic
import Lean.Meta.Tactic.Util
/-!
# Fail if no progress
This implements the `fail_if_no_progress` tactic, which fails if no actual progress is made by the
following tactic sequence.
"Actual progress" means that either the number of goals has changed, that the
number or presence of expressions in the context has changed, or that the type of some expression
in the context or the type of the goal is no longer definitionally equal to what it used to be at
reducible transparency.
This means that, for example, `1 - 1` changing to `0` does not count as actual progress, since
```lean
example : (1 - 1 = 0) := by with_reducible rfl
```
This tactic is useful in situations where we want to stop iterating some tactics if they're not
having any effect, e.g. `repeat (fail_if_no_progress simp <;> ring_nf)`.
-/
namespace Mathlib.Tactic
open Lean Meta Elab Tactic
/-- `fail_if_no_progress tacs` evaluates `tacs`, and fails if no progress is made on the main goal
or the local context at reducible transparency. -/
syntax (name := failIfNoProgress) "fail_if_no_progress " tacticSeq : tactic
/-- `lctxIsDefEq l₁ l₂` compares two lists of `Option LocalDecl`s (as returned from e.g.
`(← (← getMainGoal).getDecl).lctx.decls.toList`). It returns `true` if they have the same
local declarations in the same order (up to defeq, without setting mvars), and `false` otherwise.
Assumption: this function is run with one of the local contexts as the current `MetaM` local
context, and one of the two lists consists of the `LocalDecl`s of that context. -/
def lctxIsDefEq : (l₁ l₂ : List (Option LocalDecl)) → MetaM Bool
| none :: l₁, l₂ => lctxIsDefEq l₁ l₂
| l₁, none :: l₂ => lctxIsDefEq l₁ l₂
| some d₁ :: l₁, some d₂ :: l₂ => do
unless d₁.isLet == d₂.isLet do
return false
unless d₁.fvarId == d₂.fvarId do
-- Without compatible fvarids, `isDefEq` checks for later entries will not make sense
return false
unless (← withNewMCtxDepth <| isDefEq d₁.type d₂.type) do
return false
if d₁.isLet then
unless (← withNewMCtxDepth <| isDefEq d₁.value d₂.value) do
return false
lctxIsDefEq l₁ l₂
| [], [] => return true
| _, _ => return false
/-- Run `tacs : TacticM Unit` on `goal`, and fail if no progress is made. -/
def runAndFailIfNoProgress (goal : MVarId) (tacs : TacticM Unit) : TacticM (List MVarId) := do
let l ← run goal tacs
try
let [newGoal] := l | failure
goal.withContext do
-- Check that the local contexts are compatible
let ctxDecls := (← goal.getDecl).lctx.decls.toList
let newCtxDecls := (← newGoal.getDecl).lctx.decls.toList
guard <|← withNewMCtxDepth <| withReducible <| lctxIsDefEq ctxDecls newCtxDecls
-- They are compatible, so now we can check that the goals are equivalent
guard <|← withNewMCtxDepth <| withReducible <| isDefEq (← newGoal.getType) (← goal.getType)
catch _ =>
return l
throwError "no progress made on\n{goal}"
elab_rules : tactic
| `(tactic| fail_if_no_progress $tacs) => do
let goal ← getMainGoal
let l ← runAndFailIfNoProgress goal (evalTactic tacs)
replaceMainGoal l
end Mathlib.Tactic
|
Nat.lean
|
/-
Copyright (c) 2014 Floris van Doorn (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Algebra.Group.Nat.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Tactic.Spread
/-!
# The natural numbers form a `CancelCommMonoidWithZero`
This file contains the `CancelCommMonoidWithZero` instance on the natural numbers.
See note [foundational algebra order theory].
-/
assert_not_exists Ring
namespace Nat
instance instMulZeroClass : MulZeroClass ℕ where
zero_mul := Nat.zero_mul
mul_zero := Nat.mul_zero
instance instSemigroupWithZero : SemigroupWithZero ℕ where
__ := instSemigroup
__ := instMulZeroClass
instance instMonoidWithZero : MonoidWithZero ℕ where
__ := instMonoid
__ := instMulZeroClass
__ := instSemigroupWithZero
instance instCommMonoidWithZero : CommMonoidWithZero ℕ where
__ := instCommMonoid
__ := instMonoidWithZero
instance instIsLeftCancelMulZero : IsLeftCancelMulZero ℕ where
mul_left_cancel_of_ne_zero h _ _ := Nat.eq_of_mul_eq_mul_left (Nat.pos_of_ne_zero h)
instance instCancelCommMonoidWithZero : CancelCommMonoidWithZero ℕ where
__ := instCommMonoidWithZero
__ := instIsLeftCancelMulZero
instance instMulDivCancelClass : MulDivCancelClass ℕ where
mul_div_cancel _ _b hb := Nat.mul_div_cancel _ (Nat.pos_iff_ne_zero.2 hb)
instance instMulZeroOneClass : MulZeroOneClass ℕ where
__ := instMulZeroClass
__ := instMulOneClass
end Nat
|
SingleTriangle.lean
|
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.DerivedCategory.ShortExact
/-!
# The distinguished triangle of a short exact sequence in an abelian category
Given a short exact short complex `S` in an abelian category, we construct
the associated distinguished triangle in the derived category:
`(singleFunctor C 0).obj S.X₁ ⟶ (singleFunctor C 0).obj S.X₂ ⟶ (singleFunctor C 0).obj S.X₃ ⟶ ...`
## TODO
* when the canonical t-structure on the derived category is formalized, refactor
this definition to make it a particular case of the triangle induced by a short
exact sequence in the heart of a t-structure
-/
assert_not_exists TwoSidedIdeal
universe w v u
namespace CategoryTheory
variable {C : Type u} [Category.{v} C] [Abelian C] [HasDerivedCategory.{w} C]
open Category DerivedCategory Pretriangulated
namespace ShortComplex
variable {S : ShortComplex C} (hS : S.ShortExact)
namespace ShortExact
/-- The connecting homomorphism
`(singleFunctor C 0).obj S.X₃ ⟶ ((singleFunctor C 0).obj S.X₁)⟦(1 : ℤ)⟧` in the derived
category of `C` when `S` is a short exact short complex in `C`. -/
noncomputable def singleδ : (singleFunctor C 0).obj S.X₃ ⟶
((singleFunctor C 0).obj S.X₁)⟦(1 : ℤ)⟧ :=
(((SingleFunctors.evaluation _ _ 0).mapIso (singleFunctorsPostcompQIso C)).hom.app S.X₃) ≫
triangleOfSESδ (hS.map_of_exact (HomologicalComplex.single C (ComplexShape.up ℤ) 0)) ≫
(((SingleFunctors.evaluation _ _ 0).mapIso
(singleFunctorsPostcompQIso C)).inv.app S.X₁)⟦(1 : ℤ)⟧'
/-- The (distinguished) triangle in the derived category of `C` given by a
short exact short complex in `C`. -/
@[simps!]
noncomputable def singleTriangle : Triangle (DerivedCategory C) :=
Triangle.mk ((singleFunctor C 0).map S.f)
((singleFunctor C 0).map S.g) hS.singleδ
/-- Given a short exact complex `S` in `C` that is short exact (`hS`), this is the
canonical isomorphism between the triangle `hS.singleTriangle` in the derived category
and the triangle attached to the corresponding short exact sequence of cochain complexes
after the application of the single functor. -/
@[simps!]
noncomputable def singleTriangleIso :
hS.singleTriangle ≅
triangleOfSES (hS.map_of_exact (HomologicalComplex.single C (ComplexShape.up ℤ) 0)) := by
let e := (SingleFunctors.evaluation _ _ 0).mapIso (singleFunctorsPostcompQIso C)
refine Triangle.isoMk _ _ (e.app S.X₁) (e.app S.X₂) (e.app S.X₃) ?_ ?_ ?_
· cat_disch
· cat_disch
· dsimp [singleδ, e]
rw [Category.assoc, Category.assoc, ← Functor.map_comp, SingleFunctors.inv_hom_id_hom_app]
erw [Functor.map_id]
rw [comp_id]
/-- The distinguished triangle in the derived category of `C` given by a
short exact short complex in `C`. -/
lemma singleTriangle_distinguished :
hS.singleTriangle ∈ distTriang (DerivedCategory C) :=
isomorphic_distinguished _ (triangleOfSES_distinguished (hS.map_of_exact
(HomologicalComplex.single C (ComplexShape.up ℤ) 0))) _ (singleTriangleIso hS)
end ShortExact
end ShortComplex
end CategoryTheory
|
PosPart.lean
|
/-
Copyright (c) 2021 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin, Yaël Dillies
-/
import Mathlib.Algebra.Order.Group.Unbundled.Abs
import Mathlib.Algebra.Notation
/-!
# Positive & negative parts
Mathematical structures possessing an absolute value often also possess a unique decomposition of
elements into "positive" and "negative" parts which are in some sense "disjoint" (e.g. the Jordan
decomposition of a measure).
This file provides instances of `PosPart` and `NegPart`, the positive and negative parts of an
element in a lattice ordered group.
## Main statements
* `posPart_sub_negPart`: Every element `a` can be decomposed into `a⁺ - a⁻`, the difference of its
positive and negative parts.
* `posPart_inf_negPart_eq_zero`: The positive and negative parts are coprime.
## References
* [Birkhoff, Lattice-ordered Groups][birkhoff1942]
* [Bourbaki, Algebra II][bourbaki1981]
* [Fuchs, Partially Ordered Algebraic Systems][fuchs1963]
* [Zaanen, Lectures on "Riesz Spaces"][zaanen1966]
* [Banasiak, Banach Lattices in Applications][banasiak]
## Tags
positive part, negative part
-/
open Function
variable {α : Type*}
section Lattice
variable [Lattice α]
section Group
variable [Group α] {a b : α}
/-- The *positive part* of an element `a` in a lattice ordered group is `a ⊔ 1`, denoted `a⁺ᵐ`. -/
@[to_additive
/-- The *positive part* of an element `a` in a lattice ordered group is `a ⊔ 0`, denoted `a⁺`. -/]
instance instOneLePart : OneLePart α where
oneLePart a := a ⊔ 1
/-- The *negative part* of an element `a` in a lattice ordered group is `a⁻¹ ⊔ 1`, denoted `a⁻ᵐ `.
-/
@[to_additive
/-- The *negative part* of an element `a` in a lattice ordered group is `(-a) ⊔ 0`, denoted `a⁻`.
-/]
instance instLeOnePart : LeOnePart α where
leOnePart a := a⁻¹ ⊔ 1
@[to_additive] lemma leOnePart_def (a : α) : a⁻ᵐ = a⁻¹ ⊔ 1 := rfl
@[to_additive] lemma oneLePart_def (a : α) : a⁺ᵐ = a ⊔ 1 := rfl
@[to_additive] lemma oneLePart_mono : Monotone (·⁺ᵐ : α → α) :=
fun _a _b hab ↦ sup_le_sup_right hab _
@[to_additive (attr := simp high)] lemma oneLePart_one : (1 : α)⁺ᵐ = 1 := sup_idem _
@[to_additive (attr := simp)] lemma leOnePart_one : (1 : α)⁻ᵐ = 1 := by simp [leOnePart]
@[to_additive posPart_nonneg] lemma one_le_oneLePart (a : α) : 1 ≤ a⁺ᵐ := le_sup_right
@[to_additive negPart_nonneg] lemma one_le_leOnePart (a : α) : 1 ≤ a⁻ᵐ := le_sup_right
-- TODO: `to_additive` guesses `nonposPart`
@[to_additive le_posPart] lemma le_oneLePart (a : α) : a ≤ a⁺ᵐ := le_sup_left
@[to_additive] lemma inv_le_leOnePart (a : α) : a⁻¹ ≤ a⁻ᵐ := le_sup_left
@[to_additive (attr := simp)] lemma oneLePart_eq_self : a⁺ᵐ = a ↔ 1 ≤ a := sup_eq_left
@[to_additive (attr := simp)] lemma oneLePart_eq_one : a⁺ᵐ = 1 ↔ a ≤ 1 := sup_eq_right
@[to_additive (attr := simp)] alias ⟨_, oneLePart_of_one_le⟩ := oneLePart_eq_self
@[to_additive (attr := simp)] alias ⟨_, oneLePart_of_le_one⟩ := oneLePart_eq_one
/-- See also `leOnePart_eq_inv`. -/
@[to_additive /-- See also `negPart_eq_neg`. -/]
lemma leOnePart_eq_inv' : a⁻ᵐ = a⁻¹ ↔ 1 ≤ a⁻¹ := sup_eq_left
/-- See also `leOnePart_eq_one`. -/
@[to_additive /-- See also `negPart_eq_zero`. -/]
lemma leOnePart_eq_one' : a⁻ᵐ = 1 ↔ a⁻¹ ≤ 1 := sup_eq_right
@[to_additive] lemma oneLePart_le_one : a⁺ᵐ ≤ 1 ↔ a ≤ 1 := by simp [oneLePart]
/-- See also `leOnePart_le_one`. -/
@[to_additive /-- See also `negPart_nonpos`. -/]
lemma leOnePart_le_one' : a⁻ᵐ ≤ 1 ↔ a⁻¹ ≤ 1 := by simp [leOnePart]
@[to_additive] lemma leOnePart_le_one : a⁻ᵐ ≤ 1 ↔ a⁻¹ ≤ 1 := by simp [leOnePart]
@[to_additive (attr := simp) posPart_pos] lemma one_lt_oneLePart (ha : 1 < a) : 1 < a⁺ᵐ := by
rwa [oneLePart_eq_self.2 ha.le]
@[to_additive (attr := simp)] lemma oneLePart_inv (a : α) : a⁻¹⁺ᵐ = a⁻ᵐ := rfl
@[to_additive (attr := simp)] lemma leOnePart_inv (a : α) : a⁻¹⁻ᵐ = a⁺ᵐ := by
simp [oneLePart, leOnePart]
section MulLeftMono
variable [MulLeftMono α]
@[to_additive (attr := simp)] lemma leOnePart_eq_inv : a⁻ᵐ = a⁻¹ ↔ a ≤ 1 := by simp [leOnePart]
@[to_additive (attr := simp)]
lemma leOnePart_eq_one : a⁻ᵐ = 1 ↔ 1 ≤ a := by simp [leOnePart_eq_one']
@[to_additive (attr := simp)] alias ⟨_, leOnePart_of_le_one⟩ := leOnePart_eq_inv
@[to_additive (attr := simp)] alias ⟨_, leOnePart_of_one_le⟩ := leOnePart_eq_one
@[to_additive (attr := simp) negPart_pos] lemma one_lt_ltOnePart (ha : a < 1) : 1 < a⁻ᵐ := by
rwa [leOnePart_eq_inv.2 ha.le, one_lt_inv']
-- Bourbaki A.VI.12 Prop 9 a)
@[to_additive (attr := simp)] lemma oneLePart_div_leOnePart (a : α) : a⁺ᵐ / a⁻ᵐ = a := by
rw [div_eq_mul_inv, mul_inv_eq_iff_eq_mul, leOnePart_def, mul_sup, mul_one, mul_inv_cancel,
sup_comm, oneLePart_def]
@[to_additive (attr := simp)] lemma leOnePart_div_oneLePart (a : α) : a⁻ᵐ / a⁺ᵐ = a⁻¹ := by
rw [← inv_div, oneLePart_div_leOnePart]
@[to_additive]
lemma oneLePart_leOnePart_injective : Injective fun a : α ↦ (a⁺ᵐ, a⁻ᵐ) := by
simp only [Injective, Prod.mk.injEq, and_imp]
rintro a b hpos hneg
rw [← oneLePart_div_leOnePart a, ← oneLePart_div_leOnePart b, hpos, hneg]
@[to_additive]
lemma oneLePart_leOnePart_inj : a⁺ᵐ = b⁺ᵐ ∧ a⁻ᵐ = b⁻ᵐ ↔ a = b :=
Prod.mk_inj.symm.trans oneLePart_leOnePart_injective.eq_iff
section MulRightMono
variable [MulRightMono α]
@[to_additive] lemma leOnePart_anti : Antitone (leOnePart : α → α) :=
fun _a _b hab ↦ sup_le_sup_right (inv_le_inv_iff.2 hab) _
@[to_additive]
lemma leOnePart_eq_inv_inf_one (a : α) : a⁻ᵐ = (a ⊓ 1)⁻¹ := by
rw [leOnePart_def, ← inv_inj, inv_sup, inv_inv, inv_inv, inv_one]
-- Bourbaki A.VI.12 Prop 9 d)
@[to_additive] lemma oneLePart_mul_leOnePart (a : α) : a⁺ᵐ * a⁻ᵐ = |a|ₘ := by
rw [oneLePart_def, sup_mul, one_mul, leOnePart_def, mul_sup, mul_one, mul_inv_cancel, sup_assoc,
← sup_assoc a, sup_eq_right.2 le_sup_right]
exact sup_eq_left.2 <| one_le_mabs a
@[to_additive] lemma leOnePart_mul_oneLePart (a : α) : a⁻ᵐ * a⁺ᵐ = |a|ₘ := by
rw [oneLePart_def, mul_sup, mul_one, leOnePart_def, sup_mul, one_mul, inv_mul_cancel, sup_assoc,
← @sup_assoc _ _ a, sup_eq_right.2 le_sup_right]
exact sup_eq_left.2 <| one_le_mabs a
-- Bourbaki A.VI.12 Prop 9 a)
-- a⁺ᵐ ⊓ a⁻ᵐ = 0 (`a⁺` and `a⁻` are co-prime, and, since they are positive, disjoint)
@[to_additive] lemma oneLePart_inf_leOnePart_eq_one (a : α) : a⁺ᵐ ⊓ a⁻ᵐ = 1 := by
rw [← mul_left_inj a⁻ᵐ⁻¹, inf_mul, one_mul, mul_inv_cancel, ← div_eq_mul_inv,
oneLePart_div_leOnePart, leOnePart_eq_inv_inf_one, inv_inv]
end MulRightMono
end MulLeftMono
end Group
section CommGroup
variable [CommGroup α] [MulLeftMono α]
-- Bourbaki A.VI.12 (with a and b swapped)
@[to_additive] lemma sup_eq_mul_oneLePart_div (a b : α) : a ⊔ b = b * (a / b)⁺ᵐ := by
simp [oneLePart, mul_sup]
-- Bourbaki A.VI.12 (with a and b swapped)
@[to_additive] lemma inf_eq_div_oneLePart_div (a b : α) : a ⊓ b = a / (a / b)⁺ᵐ := by
simp [oneLePart, div_sup, inf_comm]
-- Bourbaki A.VI.12 Prop 9 c)
@[to_additive] lemma le_iff_oneLePart_leOnePart (a b : α) : a ≤ b ↔ a⁺ᵐ ≤ b⁺ᵐ ∧ b⁻ᵐ ≤ a⁻ᵐ := by
refine ⟨fun h ↦ ⟨oneLePart_mono h, leOnePart_anti h⟩, fun h ↦ ?_⟩
rw [← oneLePart_div_leOnePart a, ← oneLePart_div_leOnePart b]
exact div_le_div'' h.1 h.2
@[to_additive abs_add_eq_two_nsmul_posPart]
lemma mabs_mul_eq_oneLePart_sq (a : α) : |a|ₘ * a = a⁺ᵐ ^ 2 := by
rw [sq, ← mul_mul_div_cancel a⁺ᵐ, oneLePart_mul_leOnePart, oneLePart_div_leOnePart]
@[to_additive add_abs_eq_two_nsmul_posPart]
lemma mul_mabs_eq_oneLePart_sq (a : α) : a * |a|ₘ = a⁺ᵐ ^ 2 := by
rw [mul_comm, mabs_mul_eq_oneLePart_sq]
@[to_additive abs_sub_eq_two_nsmul_negPart]
lemma mabs_div_eq_leOnePart_sq (a : α) : |a|ₘ / a = a⁻ᵐ ^ 2 := by
rw [sq, ← mul_div_div_cancel, oneLePart_mul_leOnePart, oneLePart_div_leOnePart]
@[to_additive sub_abs_eq_neg_two_nsmul_negPart]
lemma div_mabs_eq_inv_leOnePart_sq (a : α) : a / |a|ₘ = (a⁻ᵐ ^ 2)⁻¹ := by
rw [← mabs_div_eq_leOnePart_sq, inv_div]
end CommGroup
end Lattice
section LinearOrder
variable [LinearOrder α] [Group α] {a b : α}
@[to_additive] lemma oneLePart_eq_ite : a⁺ᵐ = if 1 ≤ a then a else 1 := by
rw [oneLePart_def, ← maxDefault, ← sup_eq_maxDefault]; simp_rw [sup_comm]
@[to_additive (attr := simp) posPart_pos_iff] lemma one_lt_oneLePart_iff : 1 < a⁺ᵐ ↔ 1 < a :=
lt_iff_lt_of_le_iff_le <| (one_le_oneLePart _).ge_iff_eq'.trans oneLePart_eq_one
@[to_additive posPart_eq_of_posPart_pos]
lemma oneLePart_of_one_lt_oneLePart (ha : 1 < a⁺ᵐ) : a⁺ᵐ = a := by
rw [oneLePart_def, right_lt_sup, not_le] at ha; exact oneLePart_eq_self.2 ha.le
@[to_additive (attr := simp)] lemma oneLePart_lt : a⁺ᵐ < b ↔ a < b ∧ 1 < b := sup_lt_iff
section covariantmul
variable [MulLeftMono α]
@[to_additive] lemma leOnePart_eq_ite : a⁻ᵐ = if a ≤ 1 then a⁻¹ else 1 := by
simp_rw [← one_le_inv']; rw [leOnePart_def, ← maxDefault, ← sup_eq_maxDefault]; simp_rw [sup_comm]
@[to_additive (attr := simp) negPart_pos_iff] lemma one_lt_ltOnePart_iff : 1 < a⁻ᵐ ↔ a < 1 :=
lt_iff_lt_of_le_iff_le <| (one_le_leOnePart _).ge_iff_eq'.trans leOnePart_eq_one
variable [MulRightMono α]
@[to_additive (attr := simp)] lemma leOnePart_lt : a⁻ᵐ < b ↔ b⁻¹ < a ∧ 1 < b :=
sup_lt_iff.trans <| by rw [inv_lt']
end covariantmul
end LinearOrder
namespace Pi
variable {ι : Type*} {α : ι → Type*} [∀ i, Lattice (α i)] [∀ i, Group (α i)]
@[to_additive (attr := simp)] lemma oneLePart_apply (f : ∀ i, α i) (i : ι) : f⁺ᵐ i = (f i)⁺ᵐ := rfl
@[to_additive (attr := simp)] lemma leOnePart_apply (f : ∀ i, α i) (i : ι) : f⁻ᵐ i = (f i)⁻ᵐ := rfl
@[to_additive] lemma oneLePart_def (f : ∀ i, α i) : f⁺ᵐ = fun i ↦ (f i)⁺ᵐ := rfl
@[to_additive] lemma leOnePart_def (f : ∀ i, α i) : f⁻ᵐ = fun i ↦ (f i)⁻ᵐ := rfl
end Pi
|
Order.lean
|
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Data.NNRat.Defs
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Algebra.Order.Nonneg.Ring
/-!
# Bundled ordered algebra structures on `ℚ≥0`
-/
instance : IsStrictOrderedRing ℚ≥0 := Nonneg.isStrictOrderedRing
-- TODO: `deriving instance OrderedSub for NNRat` doesn't work yet, so we add the instance manually
instance NNRat.instOrderedSub : OrderedSub ℚ≥0 := Nonneg.orderedSub
instance NNRat.instCanonicallyOrderedAdd : CanonicallyOrderedAdd ℚ≥0 := Nonneg.canonicallyOrderedAdd
|
Basic.lean
|
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Geißer, Michael Stoll
-/
import Mathlib.Data.Real.Irrational
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.Basic
/-!
# Diophantine Approximation
The first part of this file gives proofs of various versions of
**Dirichlet's approximation theorem** and its important consequence that when $\xi$ is an
irrational real number, then there are infinitely many rationals $x/y$ (in lowest terms)
such that
$$\left|\xi - \frac{x}{y}\right| < \frac{1}{y^2} \,.$$
The proof is based on the pigeonhole principle.
The second part of the file gives a proof of **Legendre's Theorem** on rational approximation,
which states that if $\xi$ is a real number and $x/y$ is a rational number such that
$$\left|\xi - \frac{x}{y}\right| < \frac{1}{2y^2} \,,$$
then $x/y$ must be a convergent of the continued fraction expansion of $\xi$.
## Main statements
The main results are three variants of Dirichlet's approximation theorem:
* `Real.exists_int_int_abs_mul_sub_le`, which states that for all real `ξ` and natural `0 < n`,
there are integers `j` and `k` with `0 < k ≤ n` and `|k*ξ - j| ≤ 1/(n+1)`,
* `Real.exists_nat_abs_mul_sub_round_le`, which replaces `j` by `round(k*ξ)` and uses
a natural number `k`,
* `Real.exists_rat_abs_sub_le_and_den_le`, which says that there is a rational number `q`
satisfying `|ξ - q| ≤ 1/((n+1)*q.den)` and `q.den ≤ n`,
and
* `Real.infinite_rat_abs_sub_lt_one_div_den_sq_of_irrational`, which states that
for irrational `ξ`, the set `{q : ℚ | |ξ - q| < 1/q.den^2}` is infinite.
We also show a converse,
* `Rat.finite_rat_abs_sub_lt_one_div_den_sq`, which states that the set above is finite
when `ξ` is a rational number.
Both statements are combined to give an equivalence,
`Real.infinite_rat_abs_sub_lt_one_div_den_sq_iff_irrational`.
There are two versions of Legendre's Theorem. One, `Real.exists_rat_eq_convergent`, uses
`Real.convergent`, a simple recursive definition of the convergents that is also defined
in this file, whereas the other, `Real.exists_convs_eq_rat` defined in the file
`Mathlib/NumberTheory/DiophantineApproximation/ContinuedFraction.lean`, uses
`GenContFract.convs` of `GenContFract.of ξ`.
## Implementation notes
We use the namespace `Real` for the results on real numbers and `Rat` for the results
on rational numbers. We introduce a secondary namespace `real.contfrac_legendre`
to separate off a definition and some technical auxiliary lemmas used in the proof
of Legendre's Theorem. For remarks on the proof of Legendre's Theorem, see below.
## References
<https://en.wikipedia.org/wiki/Dirichlet%27s_approximation_theorem>
<https://de.wikipedia.org/wiki/Kettenbruch> (The German Wikipedia page on continued
fractions is much more extensive than the English one.)
## Tags
Diophantine approximation, Dirichlet's approximation theorem, continued fraction
-/
namespace Real
section Dirichlet
/-!
### Dirichlet's approximation theorem
We show that for any real number `ξ` and positive natural `n`, there is a fraction `q`
such that `q.den ≤ n` and `|ξ - q| ≤ 1/((n+1)*q.den)`.
-/
open Finset Int
/-- *Dirichlet's approximation theorem:*
For any real number `ξ` and positive natural `n`, there are integers `j` and `k`,
with `0 < k ≤ n` and `|k*ξ - j| ≤ 1/(n+1)`.
See also `Real.exists_nat_abs_mul_sub_round_le`. -/
theorem exists_int_int_abs_mul_sub_le (ξ : ℝ) {n : ℕ} (n_pos : 0 < n) :
∃ j k : ℤ, 0 < k ∧ k ≤ n ∧ |↑k * ξ - j| ≤ 1 / (n + 1) := by
let f : ℤ → ℤ := fun m => ⌊fract (ξ * m) * (n + 1)⌋
have hn : 0 < (n : ℝ) + 1 := mod_cast Nat.succ_pos _
have hfu := fun m : ℤ => mul_lt_of_lt_one_left hn <| fract_lt_one (ξ * ↑m)
conv in |_| ≤ _ => rw [mul_comm, le_div_iff₀ hn, ← abs_of_pos hn, ← abs_mul]
let D := Icc (0 : ℤ) n
by_cases H : ∃ m ∈ D, f m = n
· obtain ⟨m, hm, hf⟩ := H
have hf' : ((n : ℤ) : ℝ) ≤ fract (ξ * m) * (n + 1) := hf ▸ floor_le (fract (ξ * m) * (n + 1))
have hm₀ : 0 < m := by
have hf₀ : f 0 = 0 := by
simp only [f, cast_zero, mul_zero, fract_zero, zero_mul, floor_zero]
refine Ne.lt_of_le (fun h => n_pos.ne ?_) (mem_Icc.mp hm).1
exact mod_cast hf₀.symm.trans (h.symm ▸ hf : f 0 = n)
refine ⟨⌊ξ * m⌋ + 1, m, hm₀, (mem_Icc.mp hm).2, ?_⟩
rw [cast_add, ← sub_sub, sub_mul, cast_one, one_mul, abs_le]
refine
⟨le_sub_iff_add_le.mpr ?_, sub_le_iff_le_add.mpr <| le_of_lt <| (hfu m).trans <| lt_one_add _⟩
simpa only [neg_add_cancel_comm_assoc] using hf'
· simp_rw [not_exists, not_and] at H
have hD : #(Ico (0 : ℤ) n) < #D := by rw [card_Icc, card_Ico]; exact lt_add_one n
have hfu' : ∀ m, f m ≤ n := fun m => lt_add_one_iff.mp (floor_lt.mpr (mod_cast hfu m))
have hwd : ∀ m : ℤ, m ∈ D → f m ∈ Ico (0 : ℤ) n := fun x hx =>
mem_Ico.mpr
⟨floor_nonneg.mpr (mul_nonneg (fract_nonneg (ξ * x)) hn.le), Ne.lt_of_le (H x hx) (hfu' x)⟩
obtain ⟨x, hx, y, hy, x_lt_y, hxy⟩ : ∃ x ∈ D, ∃ y ∈ D, x < y ∧ f x = f y := by
obtain ⟨x, hx, y, hy, x_ne_y, hxy⟩ := exists_ne_map_eq_of_card_lt_of_maps_to hD hwd
rcases lt_trichotomy x y with (h | h | h)
exacts [⟨x, hx, y, hy, h, hxy⟩, False.elim (x_ne_y h), ⟨y, hy, x, hx, h, hxy.symm⟩]
refine
⟨⌊ξ * y⌋ - ⌊ξ * x⌋, y - x, sub_pos_of_lt x_lt_y,
sub_le_iff_le_add.mpr <| le_add_of_le_of_nonneg (mem_Icc.mp hy).2 (mem_Icc.mp hx).1, ?_⟩
convert_to |fract (ξ * y) * (n + 1) - fract (ξ * x) * (n + 1)| ≤ 1
· congr; push_cast; simp only [fract]; ring
exact (abs_sub_lt_one_of_floor_eq_floor hxy.symm).le
/-- *Dirichlet's approximation theorem:*
For any real number `ξ` and positive natural `n`, there is a natural number `k`,
with `0 < k ≤ n` such that `|k*ξ - round(k*ξ)| ≤ 1/(n+1)`.
-/
theorem exists_nat_abs_mul_sub_round_le (ξ : ℝ) {n : ℕ} (n_pos : 0 < n) :
∃ k : ℕ, 0 < k ∧ k ≤ n ∧ |↑k * ξ - round (↑k * ξ)| ≤ 1 / (n + 1) := by
obtain ⟨j, k, hk₀, hk₁, h⟩ := exists_int_int_abs_mul_sub_le ξ n_pos
have hk := toNat_of_nonneg hk₀.le
rw [← hk] at hk₀ hk₁ h
exact ⟨k.toNat, natCast_pos.mp hk₀, Nat.cast_le.mp hk₁, (round_le (↑k.toNat * ξ) j).trans h⟩
/-- *Dirichlet's approximation theorem:*
For any real number `ξ` and positive natural `n`, there is a fraction `q`
such that `q.den ≤ n` and `|ξ - q| ≤ 1/((n+1)*q.den)`.
See also `AddCircle.exists_norm_nsmul_le`. -/
theorem exists_rat_abs_sub_le_and_den_le (ξ : ℝ) {n : ℕ} (n_pos : 0 < n) :
∃ q : ℚ, |ξ - q| ≤ 1 / ((n + 1) * q.den) ∧ q.den ≤ n := by
obtain ⟨j, k, hk₀, hk₁, h⟩ := exists_int_int_abs_mul_sub_le ξ n_pos
have hk₀' : (0 : ℝ) < k := Int.cast_pos.mpr hk₀
have hden : ((j / k : ℚ).den : ℤ) ≤ k := by
convert le_of_dvd hk₀ (Rat.den_dvd j k)
exact Rat.intCast_div_eq_divInt _ _
refine ⟨j / k, ?_, Nat.cast_le.mp (hden.trans hk₁)⟩
rw [← div_div, le_div_iff₀ (Nat.cast_pos.mpr <| Rat.pos _ : (0 : ℝ) < _)]
refine (mul_le_mul_of_nonneg_left (Int.cast_le.mpr hden : _ ≤ (k : ℝ)) (abs_nonneg _)).trans ?_
rwa [← abs_of_pos hk₀', Rat.cast_div, Rat.cast_intCast, Rat.cast_intCast, ← abs_mul, sub_mul,
div_mul_cancel₀ _ hk₀'.ne', mul_comm]
end Dirichlet
section RatApprox
/-!
### Infinitely many good approximations to irrational numbers
We show that an irrational real number `ξ` has infinitely many "good rational approximations",
i.e., fractions `x/y` in lowest terms such that `|ξ - x/y| < 1/y^2`.
-/
open Set
/-- Given any rational approximation `q` to the irrational real number `ξ`, there is
a good rational approximation `q'` such that `|ξ - q'| < |ξ - q|`. -/
theorem exists_rat_abs_sub_lt_and_lt_of_irrational {ξ : ℝ} (hξ : Irrational ξ) (q : ℚ) :
∃ q' : ℚ, |ξ - q'| < 1 / (q'.den : ℝ) ^ 2 ∧ |ξ - q'| < |ξ - q| := by
have h := abs_pos.mpr (sub_ne_zero.mpr <| Irrational.ne_rat hξ q)
obtain ⟨m, hm⟩ := exists_nat_gt (1 / |ξ - q|)
have m_pos : (0 : ℝ) < m := (one_div_pos.mpr h).trans hm
obtain ⟨q', hbd, hden⟩ := exists_rat_abs_sub_le_and_den_le ξ (Nat.cast_pos.mp m_pos)
have den_pos : (0 : ℝ) < q'.den := Nat.cast_pos.mpr q'.pos
have md_pos := mul_pos (add_pos m_pos zero_lt_one) den_pos
refine
⟨q', lt_of_le_of_lt hbd ?_,
lt_of_le_of_lt hbd <|
(one_div_lt md_pos h).mpr <|
hm.trans <|
lt_of_lt_of_le (lt_add_one _) <|
(le_mul_iff_one_le_right <| add_pos m_pos zero_lt_one).mpr <|
mod_cast (q'.pos : 1 ≤ q'.den)⟩
rw [sq, one_div_lt_one_div md_pos (mul_pos den_pos den_pos), mul_lt_mul_right den_pos]
exact lt_add_of_le_of_pos (Nat.cast_le.mpr hden) zero_lt_one
/-- If `ξ` is an irrational real number, then there are infinitely many good
rational approximations to `ξ`. -/
theorem infinite_rat_abs_sub_lt_one_div_den_sq_of_irrational {ξ : ℝ} (hξ : Irrational ξ) :
{q : ℚ | |ξ - q| < 1 / (q.den : ℝ) ^ 2}.Infinite := by
refine Or.resolve_left (Set.finite_or_infinite _) fun h => ?_
obtain ⟨q, _, hq⟩ :=
exists_min_image {q : ℚ | |ξ - q| < 1 / (q.den : ℝ) ^ 2} (fun q => |ξ - q|) h
⟨⌊ξ⌋, by simp [abs_of_nonneg, Int.fract_lt_one]⟩
obtain ⟨q', hmem, hbetter⟩ := exists_rat_abs_sub_lt_and_lt_of_irrational hξ q
exact lt_irrefl _ (lt_of_le_of_lt (hq q' hmem) hbetter)
end RatApprox
end Real
namespace Rat
/-!
### Finitely many good approximations to rational numbers
We now show that a rational number `ξ` has only finitely many good rational
approximations.
-/
open Set
/-- If `ξ` is rational, then the good rational approximations to `ξ` have bounded
numerator and denominator. -/
theorem den_le_and_le_num_le_of_sub_lt_one_div_den_sq {ξ q : ℚ}
(h : |ξ - q| < 1 / (q.den : ℚ) ^ 2) :
q.den ≤ ξ.den ∧ ⌈ξ * q.den⌉ - 1 ≤ q.num ∧ q.num ≤ ⌊ξ * q.den⌋ + 1 := by
have hq₀ : (0 : ℚ) < q.den := Nat.cast_pos.mpr q.pos
replace h : |ξ * q.den - q.num| < 1 / q.den := by
rw [← mul_lt_mul_right hq₀] at h
conv_lhs at h => rw [← abs_of_pos hq₀, ← abs_mul, sub_mul, mul_den_eq_num]
rwa [sq, div_mul, mul_div_cancel_left₀ _ hq₀.ne'] at h
constructor
· rcases eq_or_ne ξ q with (rfl | H)
· exact le_rfl
· have hξ₀ : (0 : ℚ) < ξ.den := Nat.cast_pos.mpr ξ.pos
rw [← Rat.num_div_den ξ, div_mul_eq_mul_div, div_sub' hξ₀.ne', abs_div, abs_of_pos hξ₀,
div_lt_iff₀ hξ₀, div_mul_comm, mul_one] at h
refine Nat.cast_le.mp ((one_lt_div hq₀).mp <| lt_of_le_of_lt ?_ h).le
norm_cast
rw [mul_comm _ q.num]
exact Int.one_le_abs (sub_ne_zero_of_ne <| mt Rat.eq_iff_mul_eq_mul.mpr H)
· obtain ⟨h₁, h₂⟩ :=
abs_sub_lt_iff.mp
(h.trans_le <|
(one_div_le zero_lt_one hq₀).mp <| (@one_div_one ℚ _).symm ▸ Nat.cast_le.mpr q.pos)
rw [sub_lt_iff_lt_add, add_comm] at h₁ h₂
rw [← sub_lt_iff_lt_add] at h₂
norm_cast at h₁ h₂
exact
⟨sub_le_iff_le_add.mpr (Int.ceil_le.mpr h₁.le), sub_le_iff_le_add.mp (Int.le_floor.mpr h₂.le)⟩
/-- A rational number has only finitely many good rational approximations. -/
theorem finite_rat_abs_sub_lt_one_div_den_sq (ξ : ℚ) :
{q : ℚ | |ξ - q| < 1 / (q.den : ℚ) ^ 2}.Finite := by
let f : ℚ → ℤ × ℕ := fun q => (q.num, q.den)
set s := {q : ℚ | |ξ - q| < 1 / (q.den : ℚ) ^ 2}
have hinj : Function.Injective f := by
intro a b hab
simp only [f, Prod.mk_inj] at hab
rw [← Rat.num_div_den a, ← Rat.num_div_den b, hab.1, hab.2]
have H : f '' s ⊆ ⋃ (y : ℕ) (_ : y ∈ Ioc 0 ξ.den), Icc (⌈ξ * y⌉ - 1) (⌊ξ * y⌋ + 1) ×ˢ {y} := by
intro xy hxy
simp only [mem_image] at hxy
obtain ⟨q, hq₁, hq₂⟩ := hxy
obtain ⟨hd, hn⟩ := den_le_and_le_num_le_of_sub_lt_one_div_den_sq hq₁
simp_rw [mem_iUnion]
refine ⟨q.den, Set.mem_Ioc.mpr ⟨q.pos, hd⟩, ?_⟩
simp only [prod_singleton, mem_image, mem_Icc]
exact ⟨q.num, hn, hq₂⟩
refine (Finite.subset ?_ H).of_finite_image hinj.injOn
exact Finite.biUnion (finite_Ioc _ _) fun x _ => Finite.prod (finite_Icc _ _) (finite_singleton _)
end Rat
/-- The set of good rational approximations to a real number `ξ` is infinite if and only if
`ξ` is irrational. -/
theorem Real.infinite_rat_abs_sub_lt_one_div_den_sq_iff_irrational (ξ : ℝ) :
{q : ℚ | |ξ - q| < 1 / (q.den : ℝ) ^ 2}.Infinite ↔ Irrational ξ := by
refine
⟨fun h => (irrational_iff_ne_rational ξ).mpr fun a b _ H => Set.not_infinite.mpr ?_ h,
Real.infinite_rat_abs_sub_lt_one_div_den_sq_of_irrational⟩
convert Rat.finite_rat_abs_sub_lt_one_div_den_sq ((a : ℚ) / b) with q
rw [H, (by (push_cast; rfl) : (1 : ℝ) / (q.den : ℝ) ^ 2 = (1 / (q.den : ℚ) ^ 2 : ℚ))]
norm_cast
/-!
### Legendre's Theorem on Rational Approximation
We prove **Legendre's Theorem** on rational approximation: If $\xi$ is a real number and
$x/y$ is a rational number such that $|\xi - x/y| < 1/(2y^2)$,
then $x/y$ is a convergent of the continued fraction expansion of $\xi$.
The proof is by induction. However, the induction proof does not work with the
statement as given, since the assumption is too weak to imply the corresponding
statement for the application of the induction hypothesis. This can be remedied
by making the statement slightly stronger. Namely, we assume that $|\xi - x/y| < 1/(y(2y-1))$
when $y \ge 2$ and $-\frac{1}{2} < \xi - x < 1$ when $y = 1$.
-/
section Convergent
namespace Real
open Int
/-!
### Convergents: definition and API lemmas
-/
/-- We give a direct recursive definition of the convergents of the continued fraction
expansion of a real number `ξ`. The main reason for that is that we want to have the
convergents as rational numbers; the versions `(GenContFract.of ξ).convs` and
`(GenContFract.of ξ).convs'` always give something of the same type as `ξ`.
We can then also use dot notation `ξ.convergent n`.
Another minor reason is that this demonstrates that the proof
of Legendre's theorem does not need anything beyond this definition.
We provide a proof that this definition agrees with the other one;
see `Real.convs_eq_convergent`.
(Note that we use the fact that `1/0 = 0` here to make it work for rational `ξ`.) -/
noncomputable def convergent : ℝ → ℕ → ℚ
| ξ, 0 => ⌊ξ⌋
| ξ, n + 1 => ⌊ξ⌋ + (convergent (fract ξ)⁻¹ n)⁻¹
/-- The zeroth convergent of `ξ` is `⌊ξ⌋`. -/
@[simp]
theorem convergent_zero (ξ : ℝ) : ξ.convergent 0 = ⌊ξ⌋ :=
rfl
/-- The `(n+1)`th convergent of `ξ` is the `n`th convergent of `1/(fract ξ)`. -/
@[simp]
theorem convergent_succ (ξ : ℝ) (n : ℕ) :
ξ.convergent (n + 1) = ⌊ξ⌋ + ((fract ξ)⁻¹.convergent n)⁻¹ :=
rfl
/-- All convergents of `0` are zero. -/
@[simp]
theorem convergent_of_zero (n : ℕ) : convergent 0 n = 0 := by
induction n with
| zero => simp only [convergent_zero, floor_zero, cast_zero]
| succ n ih =>
simp only [ih, convergent_succ, floor_zero, cast_zero, fract_zero, add_zero, inv_zero]
/-- If `ξ` is an integer, all its convergents equal `ξ`. -/
@[simp]
theorem convergent_of_int {ξ : ℤ} (n : ℕ) : convergent ξ n = ξ := by
cases n
· simp only [convergent_zero, floor_intCast]
· simp only [convergent_succ, floor_intCast, fract_intCast, convergent_of_zero, add_zero,
inv_zero]
end Real
end Convergent
/-!
### The key technical condition for the induction proof
-/
namespace Real
open Int
-- this is not `private`, as it is used in the public `exists_rat_eq_convergent'` below.
/-- Define the technical condition to be used as assumption in the inductive proof. -/
def ContfracLegendre.Ass (ξ : ℝ) (u v : ℤ) : Prop :=
IsCoprime u v ∧ (v = 1 → (-(1 / 2) : ℝ) < ξ - u) ∧
|ξ - u / v| < ((v : ℝ) * (2 * v - 1))⁻¹
-- ### Auxiliary lemmas
-- This saves a few lines below, as it is frequently needed.
private theorem aux₀ {v : ℤ} (hv : 0 < v) : (0 : ℝ) < v ∧ (0 : ℝ) < 2 * v - 1 :=
⟨cast_pos.mpr hv, by norm_cast; omega⟩
-- In the following, we assume that `ass ξ u v` holds and `v ≥ 2`.
variable {ξ : ℝ} {u v : ℤ}
section
variable (hv : 2 ≤ v) (h : ContfracLegendre.Ass ξ u v)
include hv h
-- The fractional part of `ξ` is positive.
private theorem aux₁ : 0 < fract ξ := by
have hv₀ : (0 : ℝ) < v := cast_pos.mpr (zero_lt_two.trans_le hv)
obtain ⟨hv₁, hv₂⟩ := aux₀ (zero_lt_two.trans_le hv)
obtain ⟨hcop, _, h⟩ := h
refine fract_pos.mpr fun hf => ?_
rw [hf] at h
have H : (2 * v - 1 : ℝ) < 1 := by
refine (mul_lt_iff_lt_one_right hv₀).1 ((inv_lt_inv₀ hv₀ (mul_pos hv₁ hv₂)).1 (h.trans_le' ?_))
have h' : (⌊ξ⌋ : ℝ) - u / v = (⌊ξ⌋ * v - u) / v := by field_simp
rw [h', abs_div, abs_of_pos hv₀, ← one_div, div_le_div_iff_of_pos_right hv₀]
norm_cast
rw [← zero_add (1 : ℤ), add_one_le_iff, abs_pos, sub_ne_zero]
rintro rfl
cases isUnit_iff.mp (isCoprime_self.mp (IsCoprime.mul_left_iff.mp hcop).2) <;> omega
norm_cast at H
linarith only [hv, H]
-- An auxiliary lemma for the inductive step.
private theorem aux₂ : 0 < u - ⌊ξ⌋ * v ∧ u - ⌊ξ⌋ * v < v := by
obtain ⟨hcop, _, h⟩ := h
obtain ⟨hv₀, hv₀'⟩ := aux₀ (zero_lt_two.trans_le hv)
have hv₁ : 0 < 2 * v - 1 := by linarith only [hv]
rw [← one_div, lt_div_iff₀ (mul_pos hv₀ hv₀'), ← abs_of_pos (mul_pos hv₀ hv₀'), ← abs_mul,
sub_mul, ← mul_assoc, ← mul_assoc, div_mul_cancel₀ _ hv₀.ne', abs_sub_comm, abs_lt,
lt_sub_iff_add_lt, sub_lt_iff_lt_add, mul_assoc] at h
have hu₀ : 0 ≤ u - ⌊ξ⌋ * v := by
refine (mul_nonneg_iff_of_pos_right hv₁).mp ?_
rw [← sub_one_lt_iff, zero_sub]
replace h := h.1
rw [← lt_sub_iff_add_lt, ← mul_assoc, ← sub_mul] at h
exact mod_cast
h.trans_le
((mul_le_mul_right <| hv₀').mpr <|
(sub_le_sub_iff_left (u : ℝ)).mpr ((mul_le_mul_right hv₀).mpr (floor_le ξ)))
have hu₁ : u - ⌊ξ⌋ * v ≤ v := by
refine _root_.le_of_mul_le_mul_right (le_of_lt_add_one ?_) hv₁
replace h := h.2
rw [← sub_lt_iff_lt_add, ← mul_assoc, ← sub_mul, ← add_lt_add_iff_right (v * (2 * v - 1) : ℝ),
add_comm (1 : ℝ)] at h
have :=
(mul_lt_mul_right <| hv₀').mpr
((sub_lt_sub_iff_left (u : ℝ)).mpr <|
(mul_lt_mul_right hv₀).mpr <| sub_right_lt_of_lt_add <| lt_floor_add_one ξ)
rw [sub_mul ξ, one_mul, ← sub_add, add_mul] at this
exact mod_cast this.trans h
have huv_cop : IsCoprime (u - ⌊ξ⌋ * v) v := by
rwa [sub_eq_add_neg, ← neg_mul, IsCoprime.add_mul_right_left_iff]
refine ⟨lt_of_le_of_ne' hu₀ fun hf => ?_, lt_of_le_of_ne hu₁ fun hf => ?_⟩ <;>
· rw [hf] at huv_cop
simp only [isCoprime_zero_left, isCoprime_self, isUnit_iff] at huv_cop
rcases huv_cop with huv_cop | huv_cop <;> linarith only [hv, huv_cop]
-- The key step: the relevant inequality persists in the inductive step.
private theorem aux₃ :
|(fract ξ)⁻¹ - v / (u - ⌊ξ⌋ * v)| < (((u : ℝ) - ⌊ξ⌋ * v) * (2 * (u - ⌊ξ⌋ * v) - 1))⁻¹ := by
obtain ⟨hu₀, huv⟩ := aux₂ hv h
have hξ₀ := aux₁ hv h
set u' := u - ⌊ξ⌋ * v with hu'
have hu'ℝ : (u' : ℝ) = u - ⌊ξ⌋ * v := mod_cast hu'
rw [← hu'ℝ]
replace hu'ℝ := (eq_sub_iff_add_eq.mp hu'ℝ).symm
obtain ⟨Hu, Hu'⟩ := aux₀ hu₀
obtain ⟨Hv, Hv'⟩ := aux₀ (zero_lt_two.trans_le hv)
have H₁ := div_pos (div_pos Hv Hu) hξ₀
replace h := h.2.2
have h' : |fract ξ - u' / v| < ((v : ℝ) * (2 * v - 1))⁻¹ := by
rwa [hu'ℝ, add_div, mul_div_cancel_right₀ _ Hv.ne', ← sub_sub, sub_right_comm] at h
have H : (2 * u' - 1 : ℝ) ≤ (2 * v - 1) * fract ξ := by
replace h := (abs_lt.mp h).1
have : (2 * (v : ℝ) - 1) * (-((v : ℝ) * (2 * v - 1))⁻¹ + u' / v) = 2 * u' - (1 + u') / v := by
field_simp; ring
rw [hu'ℝ, add_div, mul_div_cancel_right₀ _ Hv.ne', ← sub_sub, sub_right_comm, self_sub_floor,
lt_sub_iff_add_lt, ← mul_lt_mul_left Hv', this] at h
refine LE.le.trans ?_ h.le
rw [sub_le_sub_iff_left, div_le_one Hv, add_comm]
exact mod_cast huv
have help₁ {a b c : ℝ} : a ≠ 0 → b ≠ 0 → c ≠ 0 → |a⁻¹ - b / c| = |(a - c / b) * (b / c / a)| := by
intros; rw [abs_sub_comm]; congr 1; field_simp; ring
have help₂ :
∀ {a b c d : ℝ}, a ≠ 0 → b ≠ 0 → c ≠ 0 → d ≠ 0 → (b * c)⁻¹ * (b / d / a) = (d * c * a)⁻¹ := by
intros; field_simp; ring
calc
|(fract ξ)⁻¹ - v / u'| = |(fract ξ - u' / v) * (v / u' / fract ξ)| :=
help₁ hξ₀.ne' Hv.ne' Hu.ne'
_ = |fract ξ - u' / v| * (v / u' / fract ξ) := by rw [abs_mul, abs_of_pos H₁]
_ < ((v : ℝ) * (2 * v - 1))⁻¹ * (v / u' / fract ξ) := (mul_lt_mul_right H₁).mpr h'
_ = (u' * (2 * v - 1) * fract ξ)⁻¹ := help₂ hξ₀.ne' Hv.ne' Hv'.ne' Hu.ne'
_ ≤ ((u' : ℝ) * (2 * u' - 1))⁻¹ := by
rwa [inv_le_inv₀ (mul_pos (mul_pos Hu Hv') hξ₀) <| mul_pos Hu Hu', mul_assoc,
mul_le_mul_left Hu]
-- The conditions `ass ξ u v` persist in the inductive step.
private theorem invariant : ContfracLegendre.Ass (fract ξ)⁻¹ v (u - ⌊ξ⌋ * v) := by
refine ⟨?_, fun huv => ?_, mod_cast aux₃ hv h⟩
· rw [sub_eq_add_neg, ← neg_mul, isCoprime_comm, IsCoprime.add_mul_right_left_iff]
exact h.1
· obtain hv₀' := (aux₀ (zero_lt_two.trans_le hv)).2
have Hv : (v * (2 * v - 1) : ℝ)⁻¹ + (v : ℝ)⁻¹ = 2 / (2 * v - 1) := by
field_simp; ring
have Huv : (u / v : ℝ) = ⌊ξ⌋ + (v : ℝ)⁻¹ := by
rw [sub_eq_iff_eq_add'.mp huv]; field_simp
have h' := (abs_sub_lt_iff.mp h.2.2).1
rw [Huv, ← sub_sub, sub_lt_iff_lt_add, self_sub_floor, Hv] at h'
rwa [lt_sub_iff_add_lt', (by ring : (v : ℝ) + -(1 / 2) = (2 * v - 1) / 2),
lt_inv_comm₀ (div_pos hv₀' zero_lt_two) (aux₁ hv h), inv_div]
end
/-!
### The main result
-/
/-- The technical version of *Legendre's Theorem*. -/
theorem exists_rat_eq_convergent' {v : ℕ} (h : ContfracLegendre.Ass ξ u v) :
∃ n, (u / v : ℚ) = ξ.convergent n := by
induction v using Nat.strong_induction_on generalizing ξ u with | h v ih => ?_
rcases lt_trichotomy v 1 with (ht | rfl | ht)
· replace h := h.2.2
simp only [Nat.lt_one_iff.mp ht, Nat.cast_zero, div_zero, tsub_zero, zero_mul,
cast_zero, inv_zero] at h
exact False.elim (lt_irrefl _ <| (abs_nonneg ξ).trans_lt h)
· rw [Nat.cast_one, div_one]
obtain ⟨_, h₁, h₂⟩ := h
rcases le_or_gt (u : ℝ) ξ with ht | ht
· use 0
rw [convergent_zero, Rat.coe_int_inj, eq_comm, floor_eq_iff]
convert And.intro ht (sub_lt_iff_lt_add'.mp (abs_lt.mp h₂).2) <;> norm_num
· replace h₁ := lt_sub_iff_add_lt'.mp (h₁ rfl)
have hξ₁ : ⌊ξ⌋ = u - 1 := by
rw [floor_eq_iff, cast_sub, cast_one, sub_add_cancel]
exact ⟨(((sub_lt_sub_iff_left _).mpr one_half_lt_one).trans h₁).le, ht⟩
rcases eq_or_ne ξ ⌊ξ⌋ with Hξ | Hξ
· rw [Hξ, hξ₁, cast_sub, cast_one, ← sub_eq_add_neg, sub_lt_sub_iff_left] at h₁
exact False.elim (lt_irrefl _ <| h₁.trans one_half_lt_one)
· have hξ₂ : ⌊(fract ξ)⁻¹⌋ = 1 := by
rw [floor_eq_iff, cast_one, le_inv_comm₀ zero_lt_one (fract_pos.mpr Hξ), inv_one,
one_add_one_eq_two, inv_lt_comm₀ (fract_pos.mpr Hξ) zero_lt_two]
refine ⟨(fract_lt_one ξ).le, ?_⟩
rw [fract, hξ₁, cast_sub, cast_one, lt_sub_iff_add_lt', sub_add]
convert h₁ using 1
rw [sub_eq_add_neg]
norm_num
use 1
simp [convergent, hξ₁, hξ₂, cast_sub, cast_one]
· obtain ⟨huv₀, huv₁⟩ := aux₂ (Nat.cast_le.mpr ht) h
have Hv : (v : ℚ) ≠ 0 := (Nat.cast_pos.mpr (zero_lt_one.trans ht)).ne'
have huv₁' : (u - ⌊ξ⌋ * v).toNat < v := by zify; rwa [toNat_of_nonneg huv₀.le]
have inv : ContfracLegendre.Ass (fract ξ)⁻¹ v (u - ⌊ξ⌋ * ↑v).toNat :=
(toNat_of_nonneg huv₀.le).symm ▸ invariant (Nat.cast_le.mpr ht) h
obtain ⟨n, hn⟩ := ih (u - ⌊ξ⌋ * v).toNat huv₁' inv
use n + 1
rw [convergent_succ, ← hn,
(mod_cast toNat_of_nonneg huv₀.le : ((u - ⌊ξ⌋ * v).toNat : ℚ) = u - ⌊ξ⌋ * v),
cast_natCast, inv_div, sub_div, mul_div_cancel_right₀ _ Hv, add_sub_cancel]
/-- The main result, *Legendre's Theorem* on rational approximation:
if `ξ` is a real number and `q` is a rational number such that `|ξ - q| < 1/(2*q.den^2)`,
then `q` is a convergent of the continued fraction expansion of `ξ`.
This version uses `Real.convergent`. -/
theorem exists_rat_eq_convergent {q : ℚ} (h : |ξ - q| < 1 / (2 * (q.den : ℝ) ^ 2)) :
∃ n, q = ξ.convergent n := by
refine q.num_div_den ▸ exists_rat_eq_convergent' ⟨?_, fun hd => ?_, ?_⟩
· exact isCoprime_iff_nat_coprime.mpr (natAbs_natCast q.den ▸ q.reduced)
· rw [← q.den_eq_one_iff.mp (Nat.cast_eq_one.mp hd)] at h
simpa only [Rat.den_intCast, Nat.cast_one, one_pow, mul_one] using (abs_lt.mp h).1
· obtain ⟨hq₀, hq₁⟩ := aux₀ (Nat.cast_pos.mpr q.pos)
replace hq₁ := mul_pos hq₀ hq₁
have hq₂ : (0 : ℝ) < 2 * (q.den * q.den) := mul_pos zero_lt_two (mul_pos hq₀ hq₀)
rw [cast_natCast] at *
rw [(by norm_cast : (q.num / q.den : ℝ) = (q.num / q.den : ℚ)), Rat.num_div_den]
exact h.trans (by rw [← one_div, sq, one_div_lt_one_div hq₂ hq₁, ← sub_pos]; ring_nf; exact hq₀)
end Real
|
Preserves.lean
|
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.CategoryTheory.EffectiveEpi.Comp
import Mathlib.Data.Fintype.EquivFin
/-!
# Functors preserving effective epimorphisms
This file concerns functors which preserve and/or reflect effective epimorphisms and effective
epimorphic families.
## TODO
- Find nice sufficient conditions in terms of preserving/reflecting (co)limits, to preserve/reflect
effective epis, similar to `CategoryTheory.preserves_epi_of_preservesColimit`.
-/
universe u
namespace CategoryTheory
open Limits
variable {C : Type*} [Category C]
noncomputable section Equivalence
variable {D : Type*} [Category D] (e : C ≌ D) {B : C}
variable {α : Type*} (X : α → C) (π : (a : α) → (X a ⟶ B))
theorem effectiveEpiFamilyStructOfEquivalence_aux {W : D} (ε : (a : α) → e.functor.obj (X a) ⟶ W)
(h : ∀ {Z : D} (a₁ a₂ : α) (g₁ : Z ⟶ e.functor.obj (X a₁)) (g₂ : Z ⟶ e.functor.obj (X a₂)),
g₁ ≫ e.functor.map (π a₁) = g₂ ≫ e.functor.map (π a₂) → g₁ ≫ ε a₁ = g₂ ≫ ε a₂)
{Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂) (hg : g₁ ≫ π a₁ = g₂ ≫ π a₂) :
g₁ ≫ (fun a ↦ e.unit.app (X a) ≫ e.inverse.map (ε a)) a₁ =
g₂ ≫ (fun a ↦ e.unit.app (X a) ≫ e.inverse.map (ε a)) a₂ := by
have := h a₁ a₂ (e.functor.map g₁) (e.functor.map g₂)
simp only [← Functor.map_comp, hg] at this
simpa using congrArg e.inverse.map (this (by trivial))
variable [EffectiveEpiFamily X π]
/-- Equivalences preserve effective epimorphic families -/
def effectiveEpiFamilyStructOfEquivalence : EffectiveEpiFamilyStruct (fun a ↦ e.functor.obj (X a))
(fun a ↦ e.functor.map (π a)) where
desc ε h := (e.toAdjunction.homEquiv _ _).symm
(EffectiveEpiFamily.desc X π (fun a ↦ e.unit.app _ ≫ e.inverse.map (ε a))
(effectiveEpiFamilyStructOfEquivalence_aux e X π ε h))
fac ε h a := by
simp only [Functor.comp_obj, Adjunction.homEquiv_counit,
Equivalence.toAdjunction_counit]
have := congrArg ((fun f ↦ f ≫ e.counit.app _) ∘ e.functor.map)
(EffectiveEpiFamily.fac X π (fun a ↦ e.unit.app _ ≫ e.inverse.map (ε a))
(effectiveEpiFamilyStructOfEquivalence_aux e X π ε h) a)
simp only [Functor.id_obj, Functor.comp_obj, Function.comp_apply, Functor.map_comp,
Category.assoc, Equivalence.fun_inv_map, Iso.inv_hom_id_app, Category.comp_id] at this
simp [this]
uniq ε h m hm := by
simp only [Functor.comp_obj, Adjunction.homEquiv_counit,
Equivalence.toAdjunction_counit]
have := EffectiveEpiFamily.uniq X π (fun a ↦ e.unit.app _ ≫ e.inverse.map (ε a))
(effectiveEpiFamilyStructOfEquivalence_aux e X π ε h)
specialize this (e.unit.app _ ≫ e.inverse.map m) fun a ↦ ?_
· rw [← congrArg e.inverse.map (hm a)]
simp
· simp [← this]
instance (F : C ⥤ D) [F.IsEquivalence] :
EffectiveEpiFamily (fun a ↦ F.obj (X a)) (fun a ↦ F.map (π a)) :=
⟨⟨effectiveEpiFamilyStructOfEquivalence F.asEquivalence _ _⟩⟩
example {X B : C} (π : X ⟶ B) (F : C ⥤ D) [F.IsEquivalence] [EffectiveEpi π] :
EffectiveEpi <| F.map π := inferInstance
end Equivalence
namespace Functor
variable {D : Type*} [Category D]
section Preserves
/--
A class describing the property of preserving effective epimorphisms.
-/
class PreservesEffectiveEpis (F : C ⥤ D) : Prop where
/--
A functor preserves effective epimorphisms if it maps effective
epimorphisms to effective epimorphisms.
-/
preserves : ∀ {X Y : C} (f : X ⟶ Y) [EffectiveEpi f], EffectiveEpi (F.map f)
instance map_effectiveEpi (F : C ⥤ D) [F.PreservesEffectiveEpis] {X Y : C} (f : X ⟶ Y)
[EffectiveEpi f] : EffectiveEpi (F.map f) :=
PreservesEffectiveEpis.preserves f
/--
A class describing the property of preserving effective epimorphic families.
-/
class PreservesEffectiveEpiFamilies (F : C ⥤ D) : Prop where
/--
A functor preserves effective epimorphic families if it maps effective epimorphic families to
effective epimorphic families.
-/
preserves : ∀ {α : Type u} {B : C} (X : α → C) (π : (a : α) → (X a ⟶ B)) [EffectiveEpiFamily X π],
EffectiveEpiFamily (fun a ↦ F.obj (X a)) (fun a ↦ F.map (π a))
instance map_effectiveEpiFamily (F : C ⥤ D) [PreservesEffectiveEpiFamilies.{u} F]
{α : Type u} {B : C} (X : α → C) (π : (a : α) → (X a ⟶ B)) [EffectiveEpiFamily X π] :
EffectiveEpiFamily (fun a ↦ F.obj (X a)) (fun a ↦ F.map (π a)) :=
PreservesEffectiveEpiFamilies.preserves X π
/--
A class describing the property of preserving finite effective epimorphic families.
-/
class PreservesFiniteEffectiveEpiFamilies (F : C ⥤ D) : Prop where
/--
A functor preserves finite effective epimorphic families if it maps finite effective epimorphic
families to effective epimorphic families.
-/
preserves : ∀ {α : Type} [Finite α] {B : C} (X : α → C) (π : (a : α) → (X a ⟶ B))
[EffectiveEpiFamily X π],
EffectiveEpiFamily (fun a ↦ F.obj (X a)) (fun a ↦ F.map (π a))
instance map_finite_effectiveEpiFamily (F : C ⥤ D) [F.PreservesFiniteEffectiveEpiFamilies]
{α : Type} [Finite α] {B : C} (X : α → C) (π : (a : α) → (X a ⟶ B)) [EffectiveEpiFamily X π] :
EffectiveEpiFamily (fun a ↦ F.obj (X a)) (fun a ↦ F.map (π a)) :=
PreservesFiniteEffectiveEpiFamilies.preserves X π
instance (F : C ⥤ D) [PreservesEffectiveEpiFamilies.{0} F] :
PreservesFiniteEffectiveEpiFamilies F where
preserves _ _ := inferInstance
instance (F : C ⥤ D) [PreservesFiniteEffectiveEpiFamilies F] : PreservesEffectiveEpis F where
preserves _ := inferInstance
instance (F : C ⥤ D) [IsEquivalence F] : F.PreservesEffectiveEpiFamilies where
preserves _ _ := inferInstance
end Preserves
section Reflects
/--
A class describing the property of reflecting effective epimorphisms.
-/
class ReflectsEffectiveEpis (F : C ⥤ D) : Prop where
/--
A functor reflects effective epimorphisms if morphisms that are mapped to epimorphisms are
themselves effective epimorphisms.
-/
reflects : ∀ {X Y : C} (f : X ⟶ Y), EffectiveEpi (F.map f) → EffectiveEpi f
lemma effectiveEpi_of_map (F : C ⥤ D) [F.ReflectsEffectiveEpis] {X Y : C} (f : X ⟶ Y)
(h : EffectiveEpi (F.map f)) : EffectiveEpi f :=
ReflectsEffectiveEpis.reflects f h
/--
A class describing the property of reflecting effective epimorphic families.
-/
class ReflectsEffectiveEpiFamilies (F : C ⥤ D) : Prop where
/--
A functor reflects effective epimorphic families if families that are mapped to effective
epimorphic families are themselves effective epimorphic families.
-/
reflects : ∀ {α : Type u} {B : C} (X : α → C) (π : (a : α) → (X a ⟶ B)),
EffectiveEpiFamily (fun a ↦ F.obj (X a)) (fun a ↦ F.map (π a)) →
EffectiveEpiFamily X π
lemma effectiveEpiFamily_of_map (F : C ⥤ D) [ReflectsEffectiveEpiFamilies.{u} F]
{α : Type u} {B : C} (X : α → C) (π : (a : α) → (X a ⟶ B))
(h : EffectiveEpiFamily (fun a ↦ F.obj (X a)) (fun a ↦ F.map (π a))) :
EffectiveEpiFamily X π :=
ReflectsEffectiveEpiFamilies.reflects X π h
/--
A class describing the property of reflecting finite effective epimorphic families.
-/
class ReflectsFiniteEffectiveEpiFamilies (F : C ⥤ D) : Prop where
/--
A functor reflects finite effective epimorphic families if finite families that are
mapped to effective epimorphic families are themselves effective epimorphic families.
-/
reflects : ∀ {α : Type} [Finite α] {B : C} (X : α → C) (π : (a : α) → (X a ⟶ B)),
EffectiveEpiFamily (fun a ↦ F.obj (X a)) (fun a ↦ F.map (π a)) →
EffectiveEpiFamily X π
lemma finite_effectiveEpiFamily_of_map (F : C ⥤ D) [ReflectsFiniteEffectiveEpiFamilies F]
{α : Type} [Finite α] {B : C} (X : α → C) (π : (a : α) → (X a ⟶ B))
(h : EffectiveEpiFamily (fun a ↦ F.obj (X a)) (fun a ↦ F.map (π a))) :
EffectiveEpiFamily X π :=
ReflectsFiniteEffectiveEpiFamilies.reflects X π h
instance (F : C ⥤ D) [ReflectsEffectiveEpiFamilies.{0} F] :
ReflectsFiniteEffectiveEpiFamilies F where
reflects _ _ h := by
have := F.effectiveEpiFamily_of_map _ _ h
infer_instance
instance (F : C ⥤ D) [ReflectsFiniteEffectiveEpiFamilies F] : ReflectsEffectiveEpis F where
reflects _ h := by
rw [effectiveEpi_iff_effectiveEpiFamily] at h
have := F.finite_effectiveEpiFamily_of_map _ _ h
infer_instance
instance (F : C ⥤ D) [IsEquivalence F] : F.ReflectsEffectiveEpiFamilies where
reflects {α B} X π _ := by
let i : (a : α) → X a ⟶ (inv F).obj (F.obj (X a)) := fun a ↦ (asEquivalence F).unit.app _
have : EffectiveEpiFamily X (fun a ↦ (i a) ≫ (inv F).map (F.map (π a))) := inferInstance
simp only [inv_fun_map, Iso.hom_inv_id_app_assoc, i] at this
have : EffectiveEpiFamily X (fun a ↦ (π a ≫ (asEquivalence F).unit.app B) ≫
(asEquivalence F).unitInv.app _) := inferInstance
simpa
end Reflects
end Functor
end CategoryTheory
|
FinitePresentation.lean
|
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.RingTheory.Localization.Finiteness
import Mathlib.RingTheory.MvPolynomial.Localization
import Mathlib.RingTheory.RingHom.FiniteType
import Mathlib.RingTheory.Localization.Away.AdjoinRoot
/-!
# The meta properties of finitely-presented ring homomorphisms.
The main result is `RingHom.finitePresentation_isLocal`.
-/
open scoped Pointwise TensorProduct
namespace RingHom
attribute [local instance] MvPolynomial.algebraMvPolynomial
/-- Being finitely-presented is preserved by localizations. -/
theorem finitePresentation_localizationPreserves : LocalizationPreserves @FinitePresentation := by
introv R hf
letI := f.toAlgebra
letI := ((algebraMap S S').comp f).toAlgebra
let f' : R' →+* S' := IsLocalization.map S' f M.le_comap_map
letI := f'.toAlgebra
haveI : IsScalarTower R R' S' :=
IsScalarTower.of_algebraMap_eq' (IsLocalization.map_comp M.le_comap_map).symm
obtain ⟨n, g, hgsurj, hgker⟩ := hf
let MX : Submonoid (MvPolynomial (Fin n) R) :=
Algebra.algebraMapSubmonoid (MvPolynomial (Fin n) R) M
haveI : IsLocalization MX (MvPolynomial (Fin n) R') :=
inferInstanceAs <| IsLocalization (M.map MvPolynomial.C) (MvPolynomial (Fin n) R')
haveI : IsScalarTower R S S' := IsScalarTower.of_algebraMap_eq' rfl
haveI : IsLocalization (Algebra.algebraMapSubmonoid S M) S' :=
inferInstanceAs <| IsLocalization (M.map f) S'
let g' : MvPolynomial (Fin n) R' →ₐ[R'] S' := IsLocalization.mapₐ M R' _ S' g
let k : RingHom.ker g →ₗ[MvPolynomial (Fin n) R] RingHom.ker g' :=
AlgHom.toKerIsLocalization M R' _ S' g
have : IsLocalizedModule MX k := AlgHom.toKerIsLocalization_isLocalizedModule M _ _ _ g
have : Module.Finite (MvPolynomial (Fin n) R) (ker g) := Module.Finite.iff_fg.mpr hgker
exact ⟨n, g', IsLocalization.mapₐ_surjective_of_surjective M R' _ S' g hgsurj,
Module.Finite.iff_fg.mp (Module.Finite.of_isLocalizedModule MX k)⟩
/-- Being finitely-presented is stable under composition. -/
theorem finitePresentation_stableUnderComposition : StableUnderComposition @FinitePresentation := by
introv R hf hg
exact hg.comp hf
/-- If `R` is a ring, then `Rᵣ` is `R`-finitely-presented for any `r : R`. -/
theorem finitePresentation_holdsForLocalizationAway :
HoldsForLocalizationAway @FinitePresentation := by
introv R _
suffices Algebra.FinitePresentation R S by
rw [RingHom.FinitePresentation]
convert this; ext
rw [Algebra.smul_def]; rfl
exact IsLocalization.Away.finitePresentation r
/--
If `S` is an `R`-algebra with a surjection from a finitely-presented `R`-algebra `A`, such that
localized at a spanning set `{ r }` of elements of `A`, `Sᵣ` is finitely-presented, then
`S` is finitely presented.
This is almost `finitePresentation_ofLocalizationSpanTarget`. The difference is,
that here the set `t` generates the unit ideal of `A`, while in the general version,
it only generates a quotient of `A`.
-/
lemma finitePresentation_ofLocalizationSpanTarget_aux
{R S A : Type*} [CommRing R] [CommRing S] [CommRing A] [Algebra R S] [Algebra R A]
[Algebra.FinitePresentation R A] (f : A →ₐ[R] S) (hf : Function.Surjective f)
(t : Finset A) (ht : Ideal.span (t : Set A) = ⊤)
(H : ∀ g : t, Algebra.FinitePresentation R (Localization.Away (f g))) :
Algebra.FinitePresentation R S := by
apply Algebra.FinitePresentation.of_surjective hf
apply ker_fg_of_localizationSpan t ht
intro g
let f' : Localization.Away g.val →ₐ[R] Localization.Away (f g) :=
Localization.awayMapₐ f g.val
have (g : t) : Algebra.FinitePresentation R (Localization.Away g.val) :=
haveI : Algebra.FinitePresentation A (Localization.Away g.val) :=
IsLocalization.Away.finitePresentation g.val
Algebra.FinitePresentation.trans R A (Localization.Away g.val)
apply Algebra.FinitePresentation.ker_fG_of_surjective f'
exact IsLocalization.Away.mapₐ_surjective_of_surjective _ hf
/-- Finite-presentation can be checked on a standard covering of the target. -/
theorem finitePresentation_ofLocalizationSpanTarget :
OfLocalizationSpanTarget @FinitePresentation := by
rw [ofLocalizationSpanTarget_iff_finite]
introv R hs H
classical
letI := f.toAlgebra
replace H : ∀ r : s, Algebra.FinitePresentation R (Localization.Away (r : S)) := by
intro r; simp_rw [RingHom.FinitePresentation] at H
convert H r; ext; simp_rw [Algebra.smul_def]; rfl
/-
We already know that `S` is of finite type over `R`, so we have a surjection
`MvPolynomial (Fin n) R →ₐ[R] S`. To reason about the kernel, we want to check it on the stalks
of preimages of `s`. But the preimages do not necessarily span `MvPolynomial (Fin n) R`, so
we quotient out by an ideal and apply `finitePresentation_ofLocalizationSpanTarget_aux`.
-/
have hfintype : Algebra.FiniteType R S := by
apply finiteType_ofLocalizationSpanTarget f s hs
intro r
convert_to Algebra.FiniteType R (Localization.Away r.val)
· rw [RingHom.FiniteType]
constructor <;> intro h <;> convert h <;> ext <;> simp_rw [Algebra.smul_def] <;> rfl
· infer_instance
rw [RingHom.FinitePresentation]
obtain ⟨n, f, hf⟩ := Algebra.FiniteType.iff_quotient_mvPolynomial''.mp hfintype
obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_linearCombination S (s : Set S) 1).mp
(show (1 : S) ∈ Ideal.span (s : Set S) by rw [hs]; trivial)
choose g' hg' using (fun g : s ↦ hf g)
choose h' hh' using (fun g : s ↦ hf (l g))
let I : Ideal (MvPolynomial (Fin n) R) := Ideal.span { ∑ g : s, g' g * h' g - 1 }
let A := MvPolynomial (Fin n) R ⧸ I
have hfI : ∀ a ∈ I, f a = 0 := by
intro p hp
simp only [Finset.univ_eq_attach, I, Ideal.mem_span_singleton] at hp
obtain ⟨q, rfl⟩ := hp
simp only [map_mul, map_sub, map_sum, map_one, hg', hh']
rw [Finsupp.linearCombination_apply_of_mem_supported (α := (s : Set S)) S (s := s.attach)] at hl
· rw [← hl]
simp only [Finset.coe_sort_coe, smul_eq_mul, mul_comm, sub_self, zero_mul]
· rintro a -
simp
let f' : A →ₐ[R] S := Ideal.Quotient.liftₐ I f hfI
have hf' : Function.Surjective f' :=
Ideal.Quotient.lift_surjective_of_surjective I hfI hf
let t : Finset A := Finset.image (fun g ↦ g' g) Finset.univ
have ht : Ideal.span (t : Set A) = ⊤ := by
rw [Ideal.eq_top_iff_one]
have : ∑ g : { x // x ∈ s }, g' g * h' g = (1 : A) := by
apply eq_of_sub_eq_zero
rw [← map_one (Ideal.Quotient.mk I), ← map_sub, Ideal.Quotient.eq_zero_iff_mem]
apply Ideal.subset_span
simp
simp_rw [← this, Finset.univ_eq_attach, map_sum, map_mul]
refine Ideal.sum_mem _ (fun g _ ↦ Ideal.mul_mem_right _ _ <| Ideal.subset_span ?_)
simp [t]
have : Algebra.FinitePresentation R A := by
apply Algebra.FinitePresentation.quotient
simp only [Finset.univ_eq_attach, I]
exact ⟨{∑ g ∈ s.attach, g' g * h' g - 1}, by simp⟩
have Ht (g : t) : Algebra.FinitePresentation R (Localization.Away (f' g)) := by
have : ∃ (a : S) (hb : a ∈ s), (Ideal.Quotient.mk I) (g' ⟨a, hb⟩) = g.val := by
obtain ⟨g, hg⟩ := g
convert hg
simp [A, t]
obtain ⟨r, hr, hrr⟩ := this
simp only [f']
rw [← hrr, Ideal.Quotient.liftₐ_apply, Ideal.Quotient.lift_mk]
simp_rw [coe_coe]
rw [hg']
apply H
exact finitePresentation_ofLocalizationSpanTarget_aux f' hf' t ht Ht
/-- Being finitely-presented is a local property of rings. -/
theorem finitePresentation_isLocal : PropertyIsLocal @FinitePresentation :=
⟨finitePresentation_localizationPreserves.away,
finitePresentation_ofLocalizationSpanTarget,
finitePresentation_ofLocalizationSpanTarget.ofLocalizationSpan
(finitePresentation_stableUnderComposition.stableUnderCompositionWithLocalizationAway
finitePresentation_holdsForLocalizationAway).left,
(finitePresentation_stableUnderComposition.stableUnderCompositionWithLocalizationAway
finitePresentation_holdsForLocalizationAway).right⟩
/-- Being finitely-presented respects isomorphisms. -/
theorem finitePresentation_respectsIso : RingHom.RespectsIso @RingHom.FinitePresentation :=
RingHom.finitePresentation_isLocal.respectsIso
/-- Being finitely-presented is stable under base change. -/
theorem finitePresentation_isStableUnderBaseChange :
IsStableUnderBaseChange @FinitePresentation := by
apply IsStableUnderBaseChange.mk
· exact finitePresentation_respectsIso
· introv h
replace h : Algebra.FinitePresentation R T := by
rw [RingHom.FinitePresentation] at h; convert h; ext; simp_rw [Algebra.smul_def]; rfl
suffices Algebra.FinitePresentation S (S ⊗[R] T) by
rw [RingHom.FinitePresentation]; convert this; ext; simp_rw [Algebra.smul_def]; rfl
infer_instance
end RingHom
|
ssreflect.v
|
(* (c) Copyright 2006-2016 Microsoft Corporation and Inria. *)
(* Distributed under the terms of CeCILL-B. *)
From Corelib Require Export ssreflect.
Global Set SsrOldRewriteGoalsOrder.
Global Set Asymmetric Patterns.
Global Set Bullet Behavior "None".
#[deprecated(since="mathcomp 2.3.0", note="Use `Arguments def : simpl never` instead (should work fine since Coq 8.18).")]
Notation nosimpl t := (nosimpl t).
(**
Additions to be ported to Corelib ssreflect:
hide t == t, but hide t displays as <hidden>
hideT t == t, but both hideT t and its inferred type display as <hidden>
New ltac views:
=> /#[#hide#]# := insert hide in the type of the top assumption (and
its body if it is a 'let'), so it displays as <hidden>.
=> /#[#let#]# := if the type (or body) of the top assumption is a
'let', make that 'let' the top assumption, e.g.,
turn (let n := 1 in m < half n) -> G
into let n := 1 in m < half n -> G
=> /#[#fix#]# := names a 'fix' expression f in the goal G(f), by
replacing the goal with let fx := hide f in G(fx),
where fx is a fresh variable, and hide f displays
as <hidden>.
=> /#[#cofix#]# := similarly, names a 'cofix' expression in the goal.
**)
Definition hide {T} t : T := t.
Notation hideT := (@hide (hide _)) (only parsing).
Notation "<hidden >" := (hide _)
(at level 0, format "<hidden >", only printing) : ssr_scope.
Notation "'[' 'hide' ']'" := (ltac:(
move; lazymatch goal with
| |- forall x : ?A, ?G => change (forall x : hide A, G)
| |- let x : ?A := ?a in ?G => change (let x : hide A := @hide A a in G)
| _ => fail "[hide] : no top assumption"
end)) (at level 0, only parsing) : ssripat_scope.
Notation "'[' 'let' ']'" := (ltac:(
move; lazymatch goal with
| |- forall (H : let x : ?A := ?a in ?T), ?G =>
change (let x : A := a in forall H : T, G)
| |- let H : (let x : ?A := ?a in ?T) := ?t in ?G =>
change (let x : A := a in let H : T := t in G)
| |- let H : ?T := (let x : ?A := ?a in ?t) in ?G =>
change (let x : A := a in let H : T := t in G)
| _ => fail "[let]: top assumption type or body is not a 'let'"
end)) (at level 0, only parsing) : ssripat_scope.
Notation "'[' 'fix' ']'" := (ltac:(
match goal with
| |- context [?t] =>
is_fix t; let f := fresh "fix" in set f := t; move: @f => /[hide]
| _ => fail 1 "[fix]: no visible 'fix' in goal"
end)) (at level 0, only parsing) : ssripat_scope.
Notation "'[' 'cofix' ']'" := (ltac:(
match goal with
| |- context [?t] =>
is_cofix t; let z := fresh "cofix" in set z := t; move: @z => /[hide]
| _ => fail 1 "[cofix]: no visible 'cofix' in goal"
end)) (at level 0, only parsing) : ssripat_scope.
|
StructurePolynomial.lean
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.FieldTheory.Finite.Polynomial
import Mathlib.NumberTheory.Basic
import Mathlib.RingTheory.WittVector.WittPolynomial
/-!
# Witt structure polynomials
In this file we prove the main theorem that makes the whole theory of Witt vectors work.
Briefly, consider a polynomial `Φ : MvPolynomial idx ℤ` over the integers,
with polynomials variables indexed by an arbitrary type `idx`.
Then there exists a unique family of polynomials `φ : ℕ → MvPolynomial (idx × ℕ) Φ`
such that for all `n : ℕ` we have (`wittStructureInt_existsUnique`)
```
bind₁ φ (wittPolynomial p ℤ n) = bind₁ (fun i ↦ (rename (prod.mk i) (wittPolynomial p ℤ n))) Φ
```
In other words: evaluating the `n`-th Witt polynomial on the family `φ`
is the same as evaluating `Φ` on the (appropriately renamed) `n`-th Witt polynomials.
N.b.: As far as we know, these polynomials do not have a name in the literature,
so we have decided to call them the “Witt structure polynomials”. See `wittStructureInt`.
## Special cases
With the main result of this file in place, we apply it to certain special polynomials.
For example, by taking `Φ = X tt + X ff` resp. `Φ = X tt * X ff`
we obtain families of polynomials `witt_add` resp. `witt_mul`
(with type `ℕ → MvPolynomial (Bool × ℕ) ℤ`) that will be used in later files to define the
addition and multiplication on the ring of Witt vectors.
## Outline of the proof
The proof of `wittStructureInt_existsUnique` is rather technical, and takes up most of this file.
We start by proving the analogous version for polynomials with rational coefficients,
instead of integer coefficients.
In this case, the solution is rather easy,
since the Witt polynomials form a faithful change of coordinates
in the polynomial ring `MvPolynomial ℕ ℚ`.
We therefore obtain a family of polynomials `wittStructureRat Φ`
for every `Φ : MvPolynomial idx ℚ`.
If `Φ` has integer coefficients, then the polynomials `wittStructureRat Φ n` do so as well.
Proving this claim is the essential core of this file, and culminates in
`map_wittStructureInt`, which proves that upon mapping the coefficients
of `wittStructureInt Φ n` from the integers to the rationals,
one obtains `wittStructureRat Φ n`.
Ultimately, the proof of `map_wittStructureInt` relies on
```
dvd_sub_pow_of_dvd_sub {R : Type*} [CommRing R] {p : ℕ} {a b : R} :
(p : R) ∣ a - b → ∀ (k : ℕ), (p : R) ^ (k + 1) ∣ a ^ p ^ k - b ^ p ^ k
```
## Main results
* `wittStructureRat Φ`: the family of polynomials `ℕ → MvPolynomial (idx × ℕ) ℚ`
associated with `Φ : MvPolynomial idx ℚ` and satisfying the property explained above.
* `wittStructureRat_prop`: the proof that `wittStructureRat` indeed satisfies the property.
* `wittStructureInt Φ`: the family of polynomials `ℕ → MvPolynomial (idx × ℕ) ℤ`
associated with `Φ : MvPolynomial idx ℤ` and satisfying the property explained above.
* `map_wittStructureInt`: the proof that the integral polynomials `with_structure_int Φ`
are equal to `wittStructureRat Φ` when mapped to polynomials with rational coefficients.
* `wittStructureInt_prop`: the proof that `wittStructureInt` indeed satisfies the property.
* Five families of polynomials that will be used to define the ring structure
on the ring of Witt vectors:
- `WittVector.wittZero`
- `WittVector.wittOne`
- `WittVector.wittAdd`
- `WittVector.wittMul`
- `WittVector.wittNeg`
(We also define `WittVector.wittSub`, and later we will prove that it describes subtraction,
which is defined as `fun a b ↦ a + -b`. See `WittVector.sub_coeff` for this proof.)
## References
* [Hazewinkel, *Witt Vectors*][Haze09]
* [Commelin and Lewis, *Formalizing the Ring of Witt Vectors*][CL21]
-/
open MvPolynomial Set
open Finset (range)
open Finsupp (single)
-- This lemma reduces a bundled morphism to a "mere" function,
-- and consequently the simplifier cannot use a lot of powerful simp-lemmas.
-- We disable this locally, and probably it should be disabled globally in mathlib.
attribute [-simp] coe_eval₂Hom
variable {p : ℕ} {R : Type*} {idx : Type*} [CommRing R]
open scoped Witt
section PPrime
variable (p)
variable [hp : Fact p.Prime]
-- Notation with ring of coefficients explicit
set_option quotPrecheck false in
@[inherit_doc]
scoped[Witt] notation "W_" => wittPolynomial p
-- Notation with ring of coefficients implicit
set_option quotPrecheck false in
@[inherit_doc]
scoped[Witt] notation "W" => wittPolynomial p _
/-- `wittStructureRat Φ` is a family of polynomials `ℕ → MvPolynomial (idx × ℕ) ℚ`
that are uniquely characterised by the property that
```
bind₁ (wittStructureRat p Φ) (wittPolynomial p ℚ n) =
bind₁ (fun i ↦ (rename (prod.mk i) (wittPolynomial p ℚ n))) Φ
```
In other words: evaluating the `n`-th Witt polynomial on the family `wittStructureRat Φ`
is the same as evaluating `Φ` on the (appropriately renamed) `n`-th Witt polynomials.
See `wittStructureRat_prop` for this property,
and `wittStructureRat_existsUnique` for the fact that `wittStructureRat`
gives the unique family of polynomials with this property.
These polynomials turn out to have integral coefficients,
but it requires some effort to show this.
See `wittStructureInt` for the version with integral coefficients,
and `map_wittStructureInt` for the fact that it is equal to `wittStructureRat`
when mapped to polynomials over the rationals. -/
noncomputable def wittStructureRat (Φ : MvPolynomial idx ℚ) (n : ℕ) : MvPolynomial (idx × ℕ) ℚ :=
bind₁ (fun k => bind₁ (fun i => rename (Prod.mk i) (W_ ℚ k)) Φ) (xInTermsOfW p ℚ n)
theorem wittStructureRat_prop (Φ : MvPolynomial idx ℚ) (n : ℕ) :
bind₁ (wittStructureRat p Φ) (W_ ℚ n) = bind₁ (fun i => rename (Prod.mk i) (W_ ℚ n)) Φ :=
calc
bind₁ (wittStructureRat p Φ) (W_ ℚ n) =
bind₁ (fun k => bind₁ (fun i => (rename (Prod.mk i)) (W_ ℚ k)) Φ)
(bind₁ (xInTermsOfW p ℚ) (W_ ℚ n)) := by
rw [bind₁_bind₁]; exact eval₂Hom_congr (RingHom.ext_rat _ _) rfl rfl
_ = bind₁ (fun i => rename (Prod.mk i) (W_ ℚ n)) Φ := by
rw [bind₁_xInTermsOfW_wittPolynomial p _ n, bind₁_X_right]
theorem wittStructureRat_existsUnique (Φ : MvPolynomial idx ℚ) :
∃! φ : ℕ → MvPolynomial (idx × ℕ) ℚ,
∀ n : ℕ, bind₁ φ (W_ ℚ n) = bind₁ (fun i => rename (Prod.mk i) (W_ ℚ n)) Φ := by
refine ⟨wittStructureRat p Φ, ?_, ?_⟩
· intro n; apply wittStructureRat_prop
· intro φ H
funext n
rw [show φ n = bind₁ φ (bind₁ (W_ ℚ) (xInTermsOfW p ℚ n)) by
rw [bind₁_wittPolynomial_xInTermsOfW p, bind₁_X_right]]
rw [bind₁_bind₁]
exact eval₂Hom_congr (RingHom.ext_rat _ _) (funext H) rfl
theorem wittStructureRat_rec_aux (Φ : MvPolynomial idx ℚ) (n : ℕ) :
wittStructureRat p Φ n * C ((p : ℚ) ^ n) =
bind₁ (fun b => rename (fun i => (b, i)) (W_ ℚ n)) Φ -
∑ i ∈ range n, C ((p : ℚ) ^ i) * wittStructureRat p Φ i ^ p ^ (n - i) := by
have := xInTermsOfW_aux p ℚ n
replace := congr_arg (bind₁ fun k : ℕ => bind₁ (fun i => rename (Prod.mk i) (W_ ℚ k)) Φ) this
rw [map_mul, bind₁_C_right] at this
rw [wittStructureRat, this]; clear this
conv_lhs => simp only [map_sub, bind₁_X_right]
rw [sub_right_inj]
simp only [map_sum, map_mul, bind₁_C_right, map_pow]
rfl
/-- Write `wittStructureRat p φ n` in terms of `wittStructureRat p φ i` for `i < n`. -/
theorem wittStructureRat_rec (Φ : MvPolynomial idx ℚ) (n : ℕ) :
wittStructureRat p Φ n =
C (1 / (p : ℚ) ^ n) *
(bind₁ (fun b => rename (fun i => (b, i)) (W_ ℚ n)) Φ -
∑ i ∈ range n, C ((p : ℚ) ^ i) * wittStructureRat p Φ i ^ p ^ (n - i)) := by
calc
wittStructureRat p Φ n = C (1 / (p : ℚ) ^ n) * (wittStructureRat p Φ n * C ((p : ℚ) ^ n)) := ?_
_ = _ := by rw [wittStructureRat_rec_aux]
rw [mul_left_comm, ← C_mul, div_mul_cancel₀, C_1, mul_one]
exact pow_ne_zero _ (Nat.cast_ne_zero.2 hp.1.ne_zero)
/-- `wittStructureInt Φ` is a family of polynomials `ℕ → MvPolynomial (idx × ℕ) ℤ`
that are uniquely characterised by the property that
```
bind₁ (wittStructureInt p Φ) (wittPolynomial p ℤ n) =
bind₁ (fun i ↦ (rename (prod.mk i) (wittPolynomial p ℤ n))) Φ
```
In other words: evaluating the `n`-th Witt polynomial on the family `wittStructureInt Φ`
is the same as evaluating `Φ` on the (appropriately renamed) `n`-th Witt polynomials.
See `wittStructureInt_prop` for this property,
and `wittStructureInt_existsUnique` for the fact that `wittStructureInt`
gives the unique family of polynomials with this property. -/
noncomputable def wittStructureInt (Φ : MvPolynomial idx ℤ) (n : ℕ) : MvPolynomial (idx × ℕ) ℤ :=
Finsupp.mapRange Rat.num (Rat.num_intCast 0) (wittStructureRat p (map (Int.castRingHom ℚ) Φ) n)
variable {p}
theorem bind₁_rename_expand_wittPolynomial (Φ : MvPolynomial idx ℤ) (n : ℕ)
(IH :
∀ m : ℕ,
m < n + 1 →
map (Int.castRingHom ℚ) (wittStructureInt p Φ m) =
wittStructureRat p (map (Int.castRingHom ℚ) Φ) m) :
bind₁ (fun b => rename (fun i => (b, i)) (expand p (W_ ℤ n))) Φ =
bind₁ (fun i => expand p (wittStructureInt p Φ i)) (W_ ℤ n) := by
apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective
simp only [map_bind₁, map_rename, map_expand, rename_expand, map_wittPolynomial]
have key := (wittStructureRat_prop p (map (Int.castRingHom ℚ) Φ) n).symm
apply_fun expand p at key
simp only [expand_bind₁] at key
rw [key]; clear key
apply eval₂Hom_congr' rfl _ rfl
rintro i hi -
rw [wittPolynomial_vars, Finset.mem_range] at hi
simp only [IH i hi]
theorem C_p_pow_dvd_bind₁_rename_wittPolynomial_sub_sum (Φ : MvPolynomial idx ℤ) (n : ℕ)
(IH :
∀ m : ℕ,
m < n →
map (Int.castRingHom ℚ) (wittStructureInt p Φ m) =
wittStructureRat p (map (Int.castRingHom ℚ) Φ) m) :
(C ((p ^ n :) : ℤ) : MvPolynomial (idx × ℕ) ℤ) ∣
bind₁ (fun b : idx => rename (fun i => (b, i)) (wittPolynomial p ℤ n)) Φ -
∑ i ∈ range n, C ((p : ℤ) ^ i) * wittStructureInt p Φ i ^ p ^ (n - i) := by
rcases n with - | n
· simp only [isUnit_one, pow_zero, C_1, IsUnit.dvd,
Nat.cast_one]
-- prepare a useful equation for rewriting
have key := bind₁_rename_expand_wittPolynomial Φ n IH
apply_fun map (Int.castRingHom (ZMod (p ^ (n + 1)))) at key
conv_lhs at key => simp only [map_bind₁, map_rename, map_expand, map_wittPolynomial]
-- clean up and massage
rw [C_dvd_iff_zmod, RingHom.map_sub, sub_eq_zero, map_bind₁]
simp only [map_rename, map_wittPolynomial, wittPolynomial_zmod_self]
rw [key]; clear key IH
rw [bind₁, aeval_wittPolynomial, map_sum, map_sum, Finset.sum_congr rfl]
intro k hk
rw [Finset.mem_range, Nat.lt_succ_iff] at hk
rw [← sub_eq_zero, ← RingHom.map_sub, ← C_dvd_iff_zmod, C_eq_coe_nat, ← Nat.cast_pow,
← Nat.cast_pow, C_eq_coe_nat, ← mul_sub]
have : p ^ (n + 1) = p ^ k * p ^ (n - k + 1) := by
rw [← pow_add, ← add_assoc]; congr 2; rw [add_comm, ← tsub_eq_iff_eq_add_of_le hk]
rw [this]
rw [Nat.cast_mul, Nat.cast_pow, Nat.cast_pow]
apply mul_dvd_mul_left ((p : MvPolynomial (idx × ℕ) ℤ) ^ k)
rw [show p ^ (n + 1 - k) = p * p ^ (n - k) by rw [← pow_succ', ← tsub_add_eq_add_tsub hk]]
rw [pow_mul]
-- the machine!
apply dvd_sub_pow_of_dvd_sub
rw [← C_eq_coe_nat, C_dvd_iff_zmod, RingHom.map_sub, sub_eq_zero, map_expand, RingHom.map_pow,
MvPolynomial.expand_zmod]
variable (p)
@[simp]
theorem map_wittStructureInt (Φ : MvPolynomial idx ℤ) (n : ℕ) :
map (Int.castRingHom ℚ) (wittStructureInt p Φ n) =
wittStructureRat p (map (Int.castRingHom ℚ) Φ) n := by
induction n using Nat.strong_induction_on with | h n IH => ?_
rw [wittStructureInt, map_mapRange_eq_iff, Int.coe_castRingHom]
intro c
rw [wittStructureRat_rec, coeff_C_mul, mul_comm, mul_div_assoc', mul_one]
have sum_induction_steps :
map (Int.castRingHom ℚ)
(∑ i ∈ range n, C ((p : ℤ) ^ i) * wittStructureInt p Φ i ^ p ^ (n - i)) =
∑ i ∈ range n,
C ((p : ℚ) ^ i) * wittStructureRat p (map (Int.castRingHom ℚ) Φ) i ^ p ^ (n - i) := by
rw [map_sum]
apply Finset.sum_congr rfl
intro i hi
rw [Finset.mem_range] at hi
simp only [IH i hi, RingHom.map_mul, RingHom.map_pow, map_C]
rfl
simp only [← sum_induction_steps, ← map_wittPolynomial p (Int.castRingHom ℚ), ← map_rename, ←
map_bind₁, ← RingHom.map_sub, coeff_map]
rw [show (p : ℚ) ^ n = ((↑(p ^ n) : ℤ) : ℚ) by norm_cast]
rw [← Rat.den_eq_one_iff, eq_intCast, Rat.den_div_intCast_eq_one_iff]
swap; · exact mod_cast pow_ne_zero n hp.1.ne_zero
revert c; rw [← C_dvd_iff_dvd_coeff]
exact C_p_pow_dvd_bind₁_rename_wittPolynomial_sub_sum Φ n IH
theorem wittStructureInt_prop (Φ : MvPolynomial idx ℤ) (n) :
bind₁ (wittStructureInt p Φ) (wittPolynomial p ℤ n) =
bind₁ (fun i => rename (Prod.mk i) (W_ ℤ n)) Φ := by
apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective
have := wittStructureRat_prop p (map (Int.castRingHom ℚ) Φ) n
simpa only [map_bind₁, ← eval₂Hom_map_hom, eval₂Hom_C_left, map_rename, map_wittPolynomial,
AlgHom.coe_toRingHom, map_wittStructureInt]
theorem eq_wittStructureInt (Φ : MvPolynomial idx ℤ) (φ : ℕ → MvPolynomial (idx × ℕ) ℤ)
(h : ∀ n, bind₁ φ (wittPolynomial p ℤ n) = bind₁ (fun i => rename (Prod.mk i) (W_ ℤ n)) Φ) :
φ = wittStructureInt p Φ := by
funext k
apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective
rw [map_wittStructureInt]
-- Porting note: was `refine' congr_fun _ k`
revert k
refine congr_fun ?_
apply ExistsUnique.unique (wittStructureRat_existsUnique p (map (Int.castRingHom ℚ) Φ))
· intro n
specialize h n
apply_fun map (Int.castRingHom ℚ) at h
simpa only [map_bind₁, ← eval₂Hom_map_hom, eval₂Hom_C_left, map_rename, map_wittPolynomial,
AlgHom.coe_toRingHom] using h
· intro n; apply wittStructureRat_prop
theorem wittStructureInt_existsUnique (Φ : MvPolynomial idx ℤ) :
∃! φ : ℕ → MvPolynomial (idx × ℕ) ℤ,
∀ n : ℕ,
bind₁ φ (wittPolynomial p ℤ n) = bind₁ (fun i : idx => rename (Prod.mk i) (W_ ℤ n)) Φ :=
⟨wittStructureInt p Φ, wittStructureInt_prop _ _, eq_wittStructureInt _ _⟩
theorem witt_structure_prop (Φ : MvPolynomial idx ℤ) (n) :
aeval (fun i => map (Int.castRingHom R) (wittStructureInt p Φ i)) (wittPolynomial p ℤ n) =
aeval (fun i => rename (Prod.mk i) (W n)) Φ := by
convert congr_arg (map (Int.castRingHom R)) (wittStructureInt_prop p Φ n) using 1 <;>
rw [hom_bind₁] <;>
apply eval₂Hom_congr (RingHom.ext_int _ _) _ rfl
· rfl
· simp only [map_rename, map_wittPolynomial]
theorem wittStructureInt_rename {σ : Type*} (Φ : MvPolynomial idx ℤ) (f : idx → σ) (n : ℕ) :
wittStructureInt p (rename f Φ) n = rename (Prod.map f id) (wittStructureInt p Φ n) := by
apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective
simp only [map_rename, map_wittStructureInt, wittStructureRat, rename_bind₁, rename_rename,
bind₁_rename]
rfl
@[simp]
theorem constantCoeff_wittStructureRat_zero (Φ : MvPolynomial idx ℚ) :
constantCoeff (wittStructureRat p Φ 0) = constantCoeff Φ := by
simp only [wittStructureRat, bind₁, map_aeval, xInTermsOfW_zero, constantCoeff_rename,
constantCoeff_wittPolynomial, aeval_X, constantCoeff_comp_algebraMap, eval₂Hom_zero'_apply,
RingHom.id_apply]
theorem constantCoeff_wittStructureRat (Φ : MvPolynomial idx ℚ) (h : constantCoeff Φ = 0) (n : ℕ) :
constantCoeff (wittStructureRat p Φ n) = 0 := by
simp only [wittStructureRat, eval₂Hom_zero'_apply, h, bind₁, map_aeval, constantCoeff_rename,
constantCoeff_wittPolynomial, constantCoeff_comp_algebraMap, RingHom.id_apply,
constantCoeff_xInTermsOfW]
@[simp]
theorem constantCoeff_wittStructureInt_zero (Φ : MvPolynomial idx ℤ) :
constantCoeff (wittStructureInt p Φ 0) = constantCoeff Φ := by
have inj : Function.Injective (Int.castRingHom ℚ) := by intro m n; exact Int.cast_inj.mp
apply inj
rw [← constantCoeff_map, map_wittStructureInt, constantCoeff_wittStructureRat_zero,
constantCoeff_map]
theorem constantCoeff_wittStructureInt (Φ : MvPolynomial idx ℤ) (h : constantCoeff Φ = 0) (n : ℕ) :
constantCoeff (wittStructureInt p Φ n) = 0 := by
have inj : Function.Injective (Int.castRingHom ℚ) := by intro m n; exact Int.cast_inj.mp
apply inj
rw [← constantCoeff_map, map_wittStructureInt, constantCoeff_wittStructureRat, RingHom.map_zero]
rw [constantCoeff_map, h, RingHom.map_zero]
variable (R)
-- we could relax the fintype on `idx`, but then we need to cast from finset to set.
-- for our applications `idx` is always finite.
theorem wittStructureRat_vars [Fintype idx] (Φ : MvPolynomial idx ℚ) (n : ℕ) :
(wittStructureRat p Φ n).vars ⊆ Finset.univ ×ˢ Finset.range (n + 1) := by
rw [wittStructureRat]
intro x hx
simp only [Finset.mem_product, true_and, Finset.mem_univ, Finset.mem_range]
obtain ⟨k, hk, hx'⟩ := mem_vars_bind₁ _ _ hx
obtain ⟨i, -, hx''⟩ := mem_vars_bind₁ _ _ hx'
obtain ⟨j, hj, rfl⟩ := mem_vars_rename _ _ hx''
rw [wittPolynomial_vars, Finset.mem_range] at hj
replace hk := xInTermsOfW_vars_subset p _ hk
grind
-- we could relax the fintype on `idx`, but then we need to cast from finset to set.
-- for our applications `idx` is always finite.
theorem wittStructureInt_vars [Fintype idx] (Φ : MvPolynomial idx ℤ) (n : ℕ) :
(wittStructureInt p Φ n).vars ⊆ Finset.univ ×ˢ Finset.range (n + 1) := by
have : Function.Injective (Int.castRingHom ℚ) := Int.cast_injective
rw [← vars_map_of_injective _ this, map_wittStructureInt]
apply wittStructureRat_vars
end PPrime
|
Iso.lean
|
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullback
/-!
# The pullback of an isomorphism
This file provides some basic results about the pullback (and pushout) of an isomorphism.
-/
noncomputable section
open CategoryTheory
universe w v₁ v₂ v u u₂
namespace CategoryTheory.Limits
variable {C : Type u} [Category.{v} C] {X Y Z : C}
section PullbackLeftIso
open WalkingCospan
variable (f : X ⟶ Z) (g : Y ⟶ Z) [IsIso f]
/-- If `f : X ⟶ Z` is iso, then `X ×[Z] Y ≅ Y`. This is the explicit limit cone. -/
def pullbackConeOfLeftIso : PullbackCone f g :=
PullbackCone.mk (g ≫ inv f) (𝟙 _) <| by simp
@[simp]
theorem pullbackConeOfLeftIso_x : (pullbackConeOfLeftIso f g).pt = Y := rfl
@[simp]
theorem pullbackConeOfLeftIso_fst : (pullbackConeOfLeftIso f g).fst = g ≫ inv f := rfl
@[simp]
theorem pullbackConeOfLeftIso_snd : (pullbackConeOfLeftIso f g).snd = 𝟙 _ := rfl
theorem pullbackConeOfLeftIso_π_app_none : (pullbackConeOfLeftIso f g).π.app none = g := by simp
@[simp]
theorem pullbackConeOfLeftIso_π_app_left : (pullbackConeOfLeftIso f g).π.app left = g ≫ inv f :=
rfl
@[simp]
theorem pullbackConeOfLeftIso_π_app_right : (pullbackConeOfLeftIso f g).π.app right = 𝟙 _ := rfl
/-- Verify that the constructed limit cone is indeed a limit. -/
def pullbackConeOfLeftIsoIsLimit : IsLimit (pullbackConeOfLeftIso f g) :=
PullbackCone.isLimitAux' _ fun s => ⟨s.snd, by simp [← s.condition_assoc]⟩
theorem hasPullback_of_left_iso : HasPullback f g :=
⟨⟨⟨_, pullbackConeOfLeftIsoIsLimit f g⟩⟩⟩
attribute [local instance] hasPullback_of_left_iso
instance pullback_snd_iso_of_left_iso : IsIso (pullback.snd f g) := by
refine ⟨⟨pullback.lift (g ≫ inv f) (𝟙 _) (by simp), ?_, by simp⟩⟩
ext
· simp [← pullback.condition_assoc]
· simp
@[reassoc (attr := simp)]
lemma pullback_inv_snd_fst_of_left_isIso :
inv (pullback.snd f g) ≫ pullback.fst f g = g ≫ inv f := by
rw [IsIso.inv_comp_eq, ← pullback.condition_assoc, IsIso.hom_inv_id, Category.comp_id]
end PullbackLeftIso
section PullbackRightIso
open WalkingCospan
variable (f : X ⟶ Z) (g : Y ⟶ Z) [IsIso g]
/-- If `g : Y ⟶ Z` is iso, then `X ×[Z] Y ≅ X`. This is the explicit limit cone. -/
def pullbackConeOfRightIso : PullbackCone f g :=
PullbackCone.mk (𝟙 _) (f ≫ inv g) <| by simp
@[simp]
theorem pullbackConeOfRightIso_x : (pullbackConeOfRightIso f g).pt = X := rfl
@[simp]
theorem pullbackConeOfRightIso_fst : (pullbackConeOfRightIso f g).fst = 𝟙 _ := rfl
@[simp]
theorem pullbackConeOfRightIso_snd : (pullbackConeOfRightIso f g).snd = f ≫ inv g := rfl
theorem pullbackConeOfRightIso_π_app_none : (pullbackConeOfRightIso f g).π.app none = f := by simp
@[simp]
theorem pullbackConeOfRightIso_π_app_left : (pullbackConeOfRightIso f g).π.app left = 𝟙 _ :=
rfl
@[simp]
theorem pullbackConeOfRightIso_π_app_right : (pullbackConeOfRightIso f g).π.app right = f ≫ inv g :=
rfl
/-- Verify that the constructed limit cone is indeed a limit. -/
def pullbackConeOfRightIsoIsLimit : IsLimit (pullbackConeOfRightIso f g) :=
PullbackCone.isLimitAux' _ fun s => ⟨s.fst, by simp [s.condition_assoc]⟩
theorem hasPullback_of_right_iso : HasPullback f g :=
⟨⟨⟨_, pullbackConeOfRightIsoIsLimit f g⟩⟩⟩
attribute [local instance] hasPullback_of_right_iso
instance pullback_fst_iso_of_right_iso : IsIso (pullback.fst f g) := by
refine ⟨⟨pullback.lift (𝟙 _) (f ≫ inv g) (by simp), ?_, by simp⟩⟩
ext
· simp
· simp [pullback.condition_assoc]
@[reassoc (attr := simp)]
lemma pullback_inv_fst_snd_of_right_isIso :
inv (pullback.fst f g) ≫ pullback.snd f g = f ≫ inv g := by
rw [IsIso.inv_comp_eq, pullback.condition_assoc, IsIso.hom_inv_id, Category.comp_id]
end PullbackRightIso
section PushoutLeftIso
open WalkingSpan
variable (f : X ⟶ Y) (g : X ⟶ Z) [IsIso f]
/-- If `f : X ⟶ Y` is iso, then `Y ⨿[X] Z ≅ Z`. This is the explicit colimit cocone. -/
def pushoutCoconeOfLeftIso : PushoutCocone f g :=
PushoutCocone.mk (inv f ≫ g) (𝟙 _) <| by simp
@[simp]
theorem pushoutCoconeOfLeftIso_x : (pushoutCoconeOfLeftIso f g).pt = Z := rfl
@[simp]
theorem pushoutCoconeOfLeftIso_inl : (pushoutCoconeOfLeftIso f g).inl = inv f ≫ g := rfl
@[simp]
theorem pushoutCoconeOfLeftIso_inr : (pushoutCoconeOfLeftIso f g).inr = 𝟙 _ := rfl
theorem pushoutCoconeOfLeftIso_ι_app_none : (pushoutCoconeOfLeftIso f g).ι.app none = g := by
simp
@[simp]
theorem pushoutCoconeOfLeftIso_ι_app_left : (pushoutCoconeOfLeftIso f g).ι.app left = inv f ≫ g :=
rfl
@[simp]
theorem pushoutCoconeOfLeftIso_ι_app_right : (pushoutCoconeOfLeftIso f g).ι.app right = 𝟙 _ := rfl
/-- Verify that the constructed cocone is indeed a colimit. -/
def pushoutCoconeOfLeftIsoIsLimit : IsColimit (pushoutCoconeOfLeftIso f g) :=
PushoutCocone.isColimitAux' _ fun s => ⟨s.inr, by simp [← s.condition]⟩
theorem hasPushout_of_left_iso : HasPushout f g :=
⟨⟨⟨_, pushoutCoconeOfLeftIsoIsLimit f g⟩⟩⟩
attribute [local instance] hasPushout_of_left_iso
instance pushout_inr_iso_of_left_iso : IsIso (pushout.inr f g) := by
refine ⟨⟨pushout.desc (inv f ≫ g) (𝟙 _) (by simp), by simp, ?_⟩⟩
ext
· simp [← pushout.condition]
· simp
@[reassoc (attr := simp)]
lemma pushout_inl_inv_inr_of_right_isIso :
pushout.inl f g ≫ inv (pushout.inr f g) = inv f ≫ g := by
rw [IsIso.eq_inv_comp, pushout.condition_assoc, IsIso.hom_inv_id, Category.comp_id]
end PushoutLeftIso
section PushoutRightIso
open WalkingSpan
variable (f : X ⟶ Y) (g : X ⟶ Z) [IsIso g]
/-- If `f : X ⟶ Z` is iso, then `Y ⨿[X] Z ≅ Y`. This is the explicit colimit cocone. -/
def pushoutCoconeOfRightIso : PushoutCocone f g :=
PushoutCocone.mk (𝟙 _) (inv g ≫ f) <| by simp
@[simp]
theorem pushoutCoconeOfRightIso_x : (pushoutCoconeOfRightIso f g).pt = Y := rfl
@[simp]
theorem pushoutCoconeOfRightIso_inl : (pushoutCoconeOfRightIso f g).inl = 𝟙 _ := rfl
@[simp]
theorem pushoutCoconeOfRightIso_inr : (pushoutCoconeOfRightIso f g).inr = inv g ≫ f := rfl
theorem pushoutCoconeOfRightIso_ι_app_none : (pushoutCoconeOfRightIso f g).ι.app none = f := by
simp
@[simp]
theorem pushoutCoconeOfRightIso_ι_app_left : (pushoutCoconeOfRightIso f g).ι.app left = 𝟙 _ := rfl
@[simp]
theorem pushoutCoconeOfRightIso_ι_app_right :
(pushoutCoconeOfRightIso f g).ι.app right = inv g ≫ f := rfl
/-- Verify that the constructed cocone is indeed a colimit. -/
def pushoutCoconeOfRightIsoIsLimit : IsColimit (pushoutCoconeOfRightIso f g) :=
PushoutCocone.isColimitAux' _ fun s => ⟨s.inl, by simp [← s.condition]⟩
theorem hasPushout_of_right_iso : HasPushout f g :=
⟨⟨⟨_, pushoutCoconeOfRightIsoIsLimit f g⟩⟩⟩
attribute [local instance] hasPushout_of_right_iso
instance pushout_inl_iso_of_right_iso : IsIso (pushout.inl _ _ : _ ⟶ pushout f g) := by
refine ⟨⟨pushout.desc (𝟙 _) (inv g ≫ f) (by simp), by simp, ?_⟩⟩
ext
· simp
· simp [pushout.condition]
@[reassoc (attr := simp)]
lemma pushout_inr_inv_inl_of_right_isIso :
pushout.inr f g ≫ inv (pushout.inl f g) = inv g ≫ f := by
rw [IsIso.eq_inv_comp, ← pushout.condition_assoc, IsIso.hom_inv_id, Category.comp_id]
end PushoutRightIso
end CategoryTheory.Limits
|
center.v
|
(* (c) Copyright 2006-2016 Microsoft Corporation and Inria. *)
(* Distributed under the terms of CeCILL-B. *)
From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div.
From mathcomp Require Import fintype bigop finset fingroup morphism perm.
From mathcomp Require Import automorphism quotient action gproduct gfunctor.
From mathcomp Require Import cyclic.
(******************************************************************************)
(* Definition of the center of a group and of external central products: *)
(* 'Z(G) == the center of the group G, i.e., 'C_G(G). *)
(* cprod_by isoZ == the finGroupType for the central product of H and K *)
(* with centers identified by the isomorphism gz on 'Z(H); *)
(* here isoZ : isom 'Z(H) 'Z(K) gz. Note that the actual *)
(* central product is [set: cprod_by isoZ]. *)
(* cpairg1 isoZ == the isomorphism from H to cprod_by isoZ, isoZ as above. *)
(* cpair1g isoZ == the isomorphism from K to cprod_by isoZ, isoZ as above. *)
(* xcprod H K == the finGroupType for the external central product of H *)
(* and K with identified centers, provided the dynamically *)
(* tested condition 'Z(H) \isog 'Z(K) holds. *)
(* ncprod H n == the finGroupType for the central product of n copies of *)
(* H with their centers identified; [set: ncprod H 0] is *)
(* isomorphic to 'Z(H). *)
(* xcprodm cf eqf == the morphism induced on cprod_by isoZ, where as above *)
(* isoZ : isom 'Z(H) 'Z(K) gz, by fH : {morphism H >-> rT} *)
(* and fK : {morphism K >-> rT}, given both *)
(* cf : fH @* H \subset 'C(fK @* K) and *)
(* eqf : {in 'Z(H), fH =1 fK \o gz}. *)
(* Following Aschbacher, we only provide external central products with *)
(* identified centers, as these are well defined provided the local center *)
(* isomorphism group of one of the subgroups is full. Nevertheless the *)
(* entire construction could be carried out under the weaker assumption that *)
(* gz is an isomorphism between subgroups of 'Z(H) and 'Z(K), and even the *)
(* uniqueness theorem holds under the weaker assumption that gz map 'Z(H) to *)
(* a characteristic subgroup of 'Z(K) not isomorphic to any other subgroup of *)
(* 'Z(K), a condition that holds for example when K is cyclic, as in the *)
(* structure theorem for p-groups of symplectic type. *)
(******************************************************************************)
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Import GroupScope.
Section Defs.
Variable gT : finGroupType.
Definition center (A : {set gT}) := 'C_A(A).
Canonical center_group (G : {group gT}) : {group gT} :=
Eval hnf in [group of center G].
End Defs.
Arguments center {gT} A%_g.
Notation "''Z' ( A )" := (center A) : group_scope.
Notation "''Z' ( H )" := (center_group H) : Group_scope.
Lemma morphim_center : GFunctor.pcontinuous (@center).
Proof. by move=> gT rT G D f; apply: morphim_subcent. Qed.
Canonical center_igFun := [igFun by fun _ _ => subsetIl _ _ & morphim_center].
Canonical center_gFun := [gFun by morphim_center].
Canonical center_pgFun := [pgFun by morphim_center].
Section Center.
Variables gT : finGroupType.
Implicit Type rT : finGroupType.
Implicit Types (x y : gT) (A B : {set gT}) (G H K D : {group gT}).
Lemma subcentP A B x : reflect (x \in A /\ centralises x B) (x \in 'C_A(B)).
Proof.
rewrite inE. case: (x \in A); last by right; case.
by apply: (iffP centP) => [|[]].
Qed.
Lemma subcent_sub A B : 'C_A(B) \subset 'N_A(B).
Proof. by rewrite setIS ?cent_sub. Qed.
Lemma subcent_norm G B : 'N_G(B) \subset 'N('C_G(B)).
Proof. by rewrite normsI ?subIset ?normG // orbC cent_norm. Qed.
Lemma subcent_normal G B : 'C_G(B) <| 'N_G(B).
Proof. by rewrite /normal subcent_sub subcent_norm. Qed.
Lemma subcent_char G H K : H \char G -> K \char G -> 'C_H(K) \char G.
Proof.
case/charP=> sHG chHG /charP[sKG chKG]; apply/charP.
split=> [|f injf Gf]; first by rewrite subIset ?sHG.
by rewrite injm_subcent ?chHG ?chKG.
Qed.
Lemma centerP A x : reflect (x \in A /\ centralises x A) (x \in 'Z(A)).
Proof. exact: subcentP. Qed.
Lemma center_sub A : 'Z(A) \subset A.
Proof. exact: subsetIl. Qed.
Lemma center1 : 'Z(1) = 1 :> {set gT}.
Proof. exact: gF1. Qed.
Lemma centerC A : {in A, centralised 'Z(A)}.
Proof. by apply/centsP; rewrite centsC subsetIr. Qed.
Lemma center_normal G : 'Z(G) <| G.
Proof. exact: gFnormal. Qed.
Lemma sub_center_normal H G : H \subset 'Z(G) -> H <| G.
Proof. by rewrite subsetI centsC /normal => /andP[-> /cents_norm]. Qed.
Lemma center_abelian G : abelian 'Z(G).
Proof. by rewrite /abelian subIset // centsC subIset // subxx orbT. Qed.
Lemma center_char G : 'Z(G) \char G.
Proof. exact: gFchar. Qed.
Lemma center_idP A : reflect ('Z(A) = A) (abelian A).
Proof. exact: setIidPl. Qed.
Lemma center_class_formula G :
#|G| = #|'Z(G)| + \sum_(xG in [set x ^: G | x in G :\: 'C(G)]) #|xG|.
Proof.
by rewrite acts_sum_card_orbit ?cardsID // astabsJ normsD ?norms_cent ?normG.
Qed.
Lemma subcent1P A x y : reflect (y \in A /\ commute x y) (y \in 'C_A[x]).
Proof.
rewrite inE; case: (y \in A); last by right; case.
by apply: (iffP cent1P) => [|[]].
Qed.
Lemma subcent1_id x G : x \in G -> x \in 'C_G[x].
Proof. by move=> Gx; rewrite inE Gx; apply/cent1P. Qed.
Lemma subcent1_sub x G : 'C_G[x] \subset G.
Proof. exact: subsetIl. Qed.
Lemma subcent1C x y G : x \in G -> y \in 'C_G[x] -> x \in 'C_G[y].
Proof. by move=> Gx /subcent1P[_ cxy]; apply/subcent1P. Qed.
Lemma subcent1_cycle_sub x G : x \in G -> <[x]> \subset 'C_G[x].
Proof. by move=> Gx; rewrite cycle_subG ?subcent1_id. Qed.
Lemma subcent1_cycle_norm x G : 'C_G[x] \subset 'N(<[x]>).
Proof. by rewrite cents_norm // cent_gen cent_set1 subsetIr. Qed.
Lemma subcent1_cycle_normal x G : x \in G -> <[x]> <| 'C_G[x].
Proof.
by move=> Gx; rewrite /normal subcent1_cycle_norm subcent1_cycle_sub.
Qed.
(* Gorenstein. 1.3.4 *)
Lemma cyclic_center_factor_abelian G : cyclic (G / 'Z(G)) -> abelian G.
Proof.
case/cyclicP=> a Ga; case: (cosetP a) => /= z Nz def_a.
have G_Zz: G :=: 'Z(G) * <[z]>.
rewrite -quotientK ?cycle_subG ?quotient_cycle //=.
by rewrite -def_a -Ga quotientGK // center_normal.
rewrite G_Zz abelianM cycle_abelian center_abelian centsC /= G_Zz.
by rewrite subIset ?centS ?orbT ?mulG_subr.
Qed.
Lemma cyclic_factor_abelian H G :
H \subset 'Z(G) -> cyclic (G / H) -> abelian G.
Proof.
move=> sHZ cycGH; apply: cyclic_center_factor_abelian.
have /andP[_ nHG]: H <| G := sub_center_normal sHZ.
have [f <-]:= homgP (homg_quotientS nHG (gFnorm _ G) sHZ).
exact: morphim_cyclic cycGH.
Qed.
Section Injm.
Variables (rT : finGroupType) (D : {group gT}) (f : {morphism D >-> rT}).
Hypothesis injf : 'injm f.
Lemma injm_center G : G \subset D -> f @* 'Z(G) = 'Z(f @* G).
Proof. exact: injm_subcent. Qed.
End Injm.
End Center.
Arguments center_idP {gT A}.
Lemma isog_center (aT rT : finGroupType) (G : {group aT}) (H : {group rT}) :
G \isog H -> 'Z(G) \isog 'Z(H).
Proof. exact: gFisog. Qed.
Section Product.
Variable gT : finGroupType.
Implicit Types (A B C : {set gT}) (G H K : {group gT}).
Lemma center_prod H K : K \subset 'C(H) -> 'Z(H) * 'Z(K) = 'Z(H * K).
Proof.
move=> cHK; apply/setP=> z; rewrite {3}/center centM !inE.
have cKH: H \subset 'C(K) by rewrite centsC.
apply/imset2P/and3P=> [[x y /setIP[Hx cHx] /setIP[Ky cKy] ->{z}]| []].
by rewrite imset2_f ?groupM // ?(subsetP cHK) ?(subsetP cKH).
case/imset2P=> x y Hx Ky ->{z}.
rewrite groupMr => [cHx|]; last exact: subsetP Ky.
rewrite groupMl => [cKy|]; last exact: subsetP Hx.
by exists x y; rewrite ?inE ?Hx ?Ky.
Qed.
Lemma center_cprod A B G : A \* B = G -> 'Z(A) \* 'Z(B) = 'Z(G).
Proof.
case/cprodP => [[H K -> ->] <- cHK].
rewrite cprodE ?center_prod //= subIset ?(subset_trans cHK) //.
by rewrite centS ?center_sub.
Qed.
Lemma center_bigcprod I r P (F : I -> {set gT}) G :
\big[cprod/1]_(i <- r | P i) F i = G ->
\big[cprod/1]_(i <- r | P i) 'Z(F i) = 'Z(G).
Proof.
elim/big_ind2: _ G => [_ <-|A B C D IHA IHB G dG|_ _ G ->]; rewrite ?center1 //.
case/cprodP: dG IHA IHB (dG) => [[H K -> ->] _ _] IHH IHK dG.
by rewrite (IHH H) // (IHK K) // (center_cprod dG).
Qed.
Lemma cprod_center_id G : G \* 'Z(G) = G.
Proof. by rewrite cprodE ?subsetIr // mulGSid ?center_sub. Qed.
Lemma center_dprod A B G : A \x B = G -> 'Z(A) \x 'Z(B) = 'Z(G).
Proof.
case/dprodP=> [[H1 H2 -> ->] defG cH12 trH12].
move: defG; rewrite -cprodE // => /center_cprod/cprodP[_ /= <- cZ12].
by apply: dprodE; rewrite //= setIAC setIA -setIA trH12 (setIidPl _) ?sub1G.
Qed.
Lemma center_bigdprod I r P (F: I -> {set gT}) G :
\big[dprod/1]_(i <- r | P i) F i = G ->
\big[dprod/1]_(i <- r | P i) 'Z(F i) = 'Z(G).
Proof.
elim/big_ind2: _ G => [_ <-|A B C D IHA IHB G dG|_ _ G ->]; rewrite ?center1 //.
case/dprodP: dG IHA IHB (dG) => [[H K -> ->] _ _ _] IHH IHK dG.
by rewrite (IHH H) // (IHK K) // (center_dprod dG).
Qed.
Lemma Aut_cprod_full G H K :
H \* K = G -> 'Z(H) = 'Z(K) ->
Aut_in (Aut H) 'Z(H) \isog Aut 'Z(H) ->
Aut_in (Aut K) 'Z(K) \isog Aut 'Z(K) ->
Aut_in (Aut G) 'Z(G) \isog Aut 'Z(G).
Proof.
move=> defG eqZHK; have [_ defHK cHK] := cprodP defG.
have defZ: 'Z(G) = 'Z(H) by rewrite -defHK -center_prod // eqZHK mulGid.
have ziHK: H :&: K = 'Z(K).
by apply/eqP; rewrite eqEsubset subsetI -{1 2}eqZHK !center_sub setIS.
have AutZP := Aut_sub_fullP (@center_sub gT _).
move/AutZP=> AutZHfull /AutZP AutZKfull; apply/AutZP=> g injg gZ.
have [gH [def_gH ker_gH _ im_gH]] := domP g defZ.
have [gK [def_gK ker_gK _ im_gK]] := domP g (etrans defZ eqZHK).
have [injgH injgK]: 'injm gH /\ 'injm gK by rewrite ker_gH ker_gK.
have [gHH gKK]: gH @* 'Z(H) = 'Z(H) /\ gK @* 'Z(K) = 'Z(K).
by rewrite im_gH im_gK -eqZHK -defZ.
have [|fH [injfH im_fH fHZ]] := AutZHfull gH injgH.
by rewrite im_gH /= -defZ.
have [|fK [injfK im_fK fKZ]] := AutZKfull gK injgK.
by rewrite im_gK /= -eqZHK -defZ.
have cfHK: fK @* K \subset 'C(fH @* H) by rewrite im_fH im_fK.
have eq_fHK: {in H :&: K, fH =1 fK}.
by move=> z; rewrite ziHK => Zz; rewrite fHZ ?fKZ /= ?eqZHK // def_gH def_gK.
exists (cprodm_morphism defG cfHK eq_fHK).
rewrite injm_cprodm injfH injfK im_cprodm im_fH im_fK defHK.
rewrite -morphimIdom ziHK -eqZHK injm_center // im_fH eqxx.
split=> //= z; rewrite {1}defZ => Zz; have [Hz _] := setIP Zz.
by rewrite cprodmEl // fHZ // def_gH.
Qed.
End Product.
Section CprodBy.
Variables gTH gTK : finGroupType.
Variables (H : {group gTH}) (K : {group gTK}) (gz : {morphism 'Z(H) >-> gTK}).
Definition ker_cprod_by of isom 'Z(H) 'Z(K) gz :=
[set xy | let: (x, y) := xy in (x \in 'Z(H)) && (y == (gz x)^-1)].
Hypothesis isoZ : isom 'Z(H) 'Z(K) gz.
Let kerHK := ker_cprod_by isoZ.
Let injgz : 'injm gz. Proof. by case/isomP: isoZ. Qed.
Let gzZ : gz @* 'Z(H) = 'Z(K). Proof. by case/isomP: isoZ. Qed.
Let gzZchar : gz @* 'Z(H) \char 'Z(K). Proof. by rewrite gzZ. Qed.
Let sgzZZ : gz @* 'Z(H) \subset 'Z(K) := char_sub gzZchar.
Let sZH := center_sub H.
Let sZK := center_sub K.
Let sgzZG : gz @* 'Z(H) \subset K := subset_trans sgzZZ sZK.
Lemma ker_cprod_by_is_group : group_set kerHK.
Proof.
apply/group_setP; rewrite inE /= group1 morph1 invg1 /=.
split=> // [[x1 y1] [x2 y2]].
rewrite inE /= => /andP[Zx1 /eqP->]; have [_ cGx1] := setIP Zx1.
rewrite inE /= => /andP[Zx2 /eqP->]; have [Gx2 _] := setIP Zx2.
by rewrite inE /= groupM //= -invMg (centP cGx1) // morphM.
Qed.
Canonical ker_cprod_by_group := Group ker_cprod_by_is_group.
Lemma ker_cprod_by_central : kerHK \subset 'Z(setX H K).
Proof.
rewrite -(center_dprod (setX_dprod H K)) -morphim_pairg1 -morphim_pair1g.
rewrite -!injm_center ?subsetT ?injm_pair1g ?injm_pairg1 //=.
rewrite morphim_pairg1 morphim_pair1g setX_dprod.
apply/subsetP=> [[x y]] /[1!inE] /andP[Zx /eqP->].
by rewrite inE /= Zx groupV (subsetP sgzZZ) ?mem_morphim.
Qed.
Fact cprod_by_key : unit. Proof. by []. Qed.
Definition cprod_by_def : finGroupType := subg_of (setX H K / kerHK).
Definition cprod_by := locked_with cprod_by_key cprod_by_def.
Local Notation C := [set: FinGroup.sort cprod_by].
(*FIXME : Check if we need arg_sort instead of sort*)
Definition in_cprod : gTH * gTK -> cprod_by :=
let: tt as k := cprod_by_key return _ -> locked_with k cprod_by_def in
subg _ \o coset kerHK.
Lemma in_cprodM : {in setX H K &, {morph in_cprod : u v / u * v}}.
Proof.
rewrite /in_cprod /cprod_by; case: cprod_by_key => /= u v Gu Gv.
have nkerHKG := normal_norm (sub_center_normal ker_cprod_by_central).
by rewrite -!morphM ?mem_quotient // (subsetP nkerHKG).
Qed.
Canonical in_cprod_morphism := Morphism in_cprodM.
Lemma ker_in_cprod : 'ker in_cprod = kerHK.
Proof.
transitivity ('ker (subg [group of setX H K / kerHK] \o coset kerHK)).
rewrite /ker /morphpre /= /in_cprod /cprod_by; case: cprod_by_key => /=.
by rewrite ['N(_) :&: _]quotientGK ?sub_center_normal ?ker_cprod_by_central.
by rewrite ker_comp ker_subg -kerE ker_coset.
Qed.
Lemma cpairg1_dom : H \subset 'dom (in_cprod \o @pairg1 gTH gTK).
Proof. by rewrite -sub_morphim_pre ?subsetT // morphim_pairg1 setXS ?sub1G. Qed.
Lemma cpair1g_dom : K \subset 'dom (in_cprod \o @pair1g gTH gTK).
Proof. by rewrite -sub_morphim_pre ?subsetT // morphim_pair1g setXS ?sub1G. Qed.
Definition cpairg1 := tag (restrmP _ cpairg1_dom).
Definition cpair1g := tag (restrmP _ cpair1g_dom).
Local Notation CH := (mfun cpairg1 @* gval H).
Local Notation CK := (mfun cpair1g @* gval K).
Lemma injm_cpairg1 : 'injm cpairg1.
Proof.
rewrite /cpairg1; case: restrmP => _ [_ -> _ _].
rewrite ker_comp ker_in_cprod; apply/subsetP=> x; rewrite !inE /=.
by case/and3P=> _ Zx; rewrite eq_sym (inv_eq invgK) invg1 morph_injm_eq1.
Qed.
Let injH := injm_cpairg1.
Lemma injm_cpair1g : 'injm cpair1g.
Proof.
rewrite /cpair1g; case: restrmP => _ [_ -> _ _].
rewrite ker_comp ker_in_cprod; apply/subsetP=> y; rewrite !inE /= morph1 invg1.
by case/and3P.
Qed.
Let injK := injm_cpair1g.
Lemma im_cpair_cent : CK \subset 'C(CH).
Proof.
rewrite /cpairg1 /cpair1g; do 2!case: restrmP => _ [_ _ _ -> //].
rewrite !morphim_comp morphim_cents // morphim_pair1g morphim_pairg1.
by case/dprodP: (setX_dprod H K).
Qed.
Hint Resolve im_cpair_cent : core.
Lemma im_cpair : CH * CK = C.
Proof.
rewrite /cpairg1 /cpair1g; do 2!case: restrmP => _ [_ _ _ -> //].
rewrite !morphim_comp -morphimMl morphim_pairg1 ?setXS ?sub1G //.
rewrite morphim_pair1g setX_prod morphimEdom /= /in_cprod /cprod_by.
by case: cprod_by_key; rewrite /= imset_comp imset_coset -morphimEdom im_subg.
Qed.
Lemma im_cpair_cprod : CH \* CK = C. Proof. by rewrite cprodE ?im_cpair. Qed.
Lemma eq_cpairZ : {in 'Z(H), cpairg1 =1 cpair1g \o gz}.
Proof.
rewrite /cpairg1 /cpair1g => z1 Zz1; set z2 := gz z1.
have Zz2: z2 \in 'Z(K) by rewrite (subsetP sgzZZ) ?mem_morphim.
have [[Gz1 _] [/= Gz2 _]]:= (setIP Zz1, setIP Zz2).
do 2![case: restrmP => f /= [df _ _ _]; rewrite {f}df].
apply/rcoset_kerP; rewrite ?inE ?group1 ?andbT //.
by rewrite ker_in_cprod mem_rcoset inE /= invg1 mulg1 mul1g Zz1 /=.
Qed.
Lemma setI_im_cpair : CH :&: CK = 'Z(CH).
Proof.
apply/eqP; rewrite eqEsubset setIS //=.
rewrite subsetI center_sub -injm_center //.
rewrite (eq_in_morphim _ eq_cpairZ); first by rewrite morphim_comp morphimS.
by rewrite !(setIidPr _) // -sub_morphim_pre.
Qed.
Lemma cpair1g_center : cpair1g @* 'Z(K) = 'Z(C).
Proof.
case/cprodP: (center_cprod im_cpair_cprod) => _ <- _.
by rewrite injm_center // -setI_im_cpair mulSGid //= setIC setIS 1?centsC.
Qed.
(* Uses gzZ. *)
Lemma cpair_center_id : 'Z(CH) = 'Z(CK).
Proof.
rewrite -!injm_center // -gzZ -morphim_comp; apply: eq_in_morphim eq_cpairZ.
by rewrite !(setIidPr _) // -sub_morphim_pre.
Qed.
(* Uses gzZ. *)
Lemma cpairg1_center : cpairg1 @* 'Z(H) = 'Z(C).
Proof. by rewrite -cpair1g_center !injm_center // cpair_center_id. Qed.
Section ExtCprodm.
Variable rT : finGroupType.
Variables (fH : {morphism H >-> rT}) (fK : {morphism K >-> rT}).
Hypothesis cfHK : fK @* K \subset 'C(fH @* H).
Hypothesis eq_fHK : {in 'Z(H), fH =1 fK \o gz}.
Let gH := ifactm fH injm_cpairg1.
Let gK := ifactm fK injm_cpair1g.
Lemma xcprodm_cent : gK @* CK \subset 'C(gH @* CH).
Proof. by rewrite !im_ifactm. Qed.
Lemma xcprodmI : {in CH :&: CK, gH =1 gK}.
Proof.
rewrite setI_im_cpair -injm_center // => fHx; case/morphimP=> x Gx Zx ->{fHx}.
by rewrite {2}eq_cpairZ //= ?ifactmE ?eq_fHK //= (subsetP sgzZG) ?mem_morphim.
Qed.
Definition xcprodm := cprodm im_cpair_cprod xcprodm_cent xcprodmI.
Canonical xcprod_morphism := [morphism of xcprodm].
Lemma xcprodmEl : {in H, forall x, xcprodm (cpairg1 x) = fH x}.
Proof. by move=> x Hx; rewrite /xcprodm cprodmEl ?mem_morphim ?ifactmE. Qed.
Lemma xcprodmEr : {in K, forall y, xcprodm (cpair1g y) = fK y}.
Proof. by move=> y Ky; rewrite /xcprodm cprodmEr ?mem_morphim ?ifactmE. Qed.
Lemma xcprodmE :
{in H & K, forall x y, xcprodm (cpairg1 x * cpair1g y) = fH x * fK y}.
Proof.
by move=> x y Hx Ky; rewrite /xcprodm cprodmE ?mem_morphim ?ifactmE.
Qed.
Lemma im_xcprodm : xcprodm @* C = fH @* H * fK @* K.
Proof. by rewrite -im_cpair morphim_cprodm // !im_ifactm. Qed.
Lemma im_xcprodml A : xcprodm @* (cpairg1 @* A) = fH @* A.
Proof.
rewrite -!(morphimIdom _ A) morphim_cprodml ?morphimS ?subsetIl //.
by rewrite morphim_ifactm ?subsetIl.
Qed.
Lemma im_xcprodmr A : xcprodm @* (cpair1g @* A) = fK @* A.
Proof.
rewrite -!(morphimIdom _ A) morphim_cprodmr ?morphimS ?subsetIl //.
by rewrite morphim_ifactm ?subsetIl.
Qed.
Lemma injm_xcprodm : 'injm xcprodm = 'injm fH && 'injm fK.
Proof.
rewrite injm_cprodm !ker_ifactm !subG1 !morphim_injm_eq1 ?subsetIl // -!subG1.
apply: andb_id2l => /= injfH; apply: andb_idr => _.
rewrite !im_ifactm // -(morphimIdom gH) setI_im_cpair -injm_center //.
rewrite morphim_ifactm // eqEsubset subsetI morphimS //=.
rewrite {1}injm_center // setIS //=.
rewrite (eq_in_morphim _ eq_fHK); first by rewrite morphim_comp morphimS.
by rewrite !(setIidPr _) // -sub_morphim_pre.
Qed.
End ExtCprodm.
(* Uses gzZchar. *)
Lemma Aut_cprod_by_full :
Aut_in (Aut H) 'Z(H) \isog Aut 'Z(H) ->
Aut_in (Aut K) 'Z(K) \isog Aut 'Z(K) ->
Aut_in (Aut C) 'Z(C) \isog Aut 'Z(C).
Proof.
move=> AutZinH AutZinK.
have Cfull:= Aut_cprod_full im_cpair_cprod cpair_center_id.
by rewrite Cfull // -injm_center // injm_Aut_full ?center_sub.
Qed.
Section Isomorphism.
Let gzZ_lone (Y : {group gTK}) :
Y \subset 'Z(K) -> gz @* 'Z(H) \isog Y -> gz @* 'Z(H) = Y.
Proof.
move=> sYZ isoY; apply/eqP.
by rewrite eq_sym eqEcard (card_isog isoY) gzZ sYZ /=.
Qed.
Variables (rT : finGroupType) (GH GK G : {group rT}).
Hypotheses (defG : GH \* GK = G) (ziGHK : GH :&: GK = 'Z(GH)).
Hypothesis AutZHfull : Aut_in (Aut H) 'Z(H) \isog Aut 'Z(H).
Hypotheses (isoGH : GH \isog H) (isoGK : GK \isog K).
(* Uses gzZ_lone *)
Lemma cprod_by_uniq :
exists f : {morphism G >-> cprod_by},
[/\ isom G C f, f @* GH = CH & f @* GK = CK].
Proof.
have [_ defGHK cGKH] := cprodP defG.
have AutZinH := Aut_sub_fullP sZH AutZHfull.
have [fH injfH defGH]:= isogP (isog_symr isoGH).
have [fK injfK defGK]:= isogP (isog_symr isoGK).
have sfHZfK: fH @* 'Z(H) \subset fK @* K.
by rewrite injm_center //= defGH defGK -ziGHK subsetIr.
have gzZ_id: gz @* 'Z(H) = invm injfK @* (fH @* 'Z(H)).
apply: gzZ_lone => /=.
rewrite injm_center // defGH -ziGHK sub_morphim_pre /= ?defGK ?subsetIr //.
by rewrite setIC morphpre_invm injm_center // defGK setIS 1?centsC.
rewrite -morphim_comp.
apply: isog_trans (sub_isog _ _); first by rewrite isog_sym sub_isog.
by rewrite -sub_morphim_pre.
by rewrite !injm_comp ?injm_invm.
have: 'dom (invm injfH \o fK \o gz) = 'Z(H).
rewrite /dom /= -(morphpreIdom gz); apply/setIidPl.
by rewrite -2?sub_morphim_pre // gzZ_id morphim_invmE morphpreK ?morphimS.
case/domP=> gzH [def_gzH ker_gzH _ im_gzH].
have{ker_gzH} injgzH: 'injm gzH by rewrite ker_gzH !injm_comp ?injm_invm.
have{AutZinH} [|gH [injgH gH_H def_gH]] := AutZinH _ injgzH.
by rewrite im_gzH !morphim_comp /= gzZ_id !morphim_invmE morphpreK ?injmK.
have: 'dom (fH \o gH) = H by rewrite /dom /= -{3}gH_H injmK.
case/domP=> gfH [def_gfH ker_gfH _ im_gfH].
have{im_gfH} gfH_H: gfH @* H = GH by rewrite im_gfH morphim_comp gH_H.
have cgfHfK: fK @* K \subset 'C(gfH @* H) by rewrite gfH_H defGK.
have eq_gfHK: {in 'Z(H), gfH =1 fK \o gz}.
move=> z Zz; rewrite def_gfH /= def_gH //= def_gzH /= invmK //.
have {Zz}: gz z \in gz @* 'Z(H) by rewrite mem_morphim.
rewrite gzZ_id morphim_invmE; case/morphpreP=> _.
exact: (subsetP (morphimS _ _)).
pose f := xcprodm cgfHfK eq_gfHK.
have injf: 'injm f by rewrite injm_xcprodm ker_gfH injm_comp.
have fCH: f @* CH = GH by rewrite im_xcprodml gfH_H.
have fCK: f @* CK = GK by rewrite im_xcprodmr defGK.
have fC: f @* C = G by rewrite im_xcprodm gfH_H defGK defGHK.
have [f' [_ ker_f' _ im_f']] := domP (invm_morphism injf) fC.
exists f'; rewrite -fCH -fCK !{1}im_f' !{1}morphim_invm ?subsetT //.
by split=> //; apply/isomP; rewrite ker_f' injm_invm im_f' -fC im_invm.
Qed.
Lemma isog_cprod_by : G \isog C.
Proof. by have [f [isoG _ _]] := cprod_by_uniq; apply: isom_isog isoG. Qed.
End Isomorphism.
End CprodBy.
Section ExtCprod.
Import finfun.
Variables gTH gTK : finGroupType.
Variables (H : {group gTH}) (K : {group gTK}).
Let gt_ b := if b then gTK else gTH.
Local Notation isob := ('Z(H) \isog 'Z(K)) (only parsing).
Let G_ b := if b as b' return {group gt_ b'} then K else H.
Lemma xcprod_subproof :
{gz : {morphism 'Z(H) >-> gt_ isob} | isom 'Z(H) 'Z(G_ isob) gz}.
Proof.
case: (pickP [pred f : {ffun _} | misom 'Z(H) 'Z(K) f]) => [f isoZ | no_f].
rewrite (misom_isog isoZ); case/andP: isoZ => fM isoZ.
by exists [morphism of morphm fM].
move/pred0P: no_f => not_isoZ; rewrite [isob](congr1 negb not_isoZ).
by exists (idm_morphism _); apply/isomP; rewrite injm_idm im_idm.
Qed.
Definition xcprod := cprod_by (svalP xcprod_subproof).
Inductive xcprod_spec : finGroupType -> Prop :=
XcprodSpec gz isoZ : xcprod_spec (@cprod_by gTH gTK H K gz isoZ).
Lemma xcprodP : 'Z(H) \isog 'Z(K) -> xcprod_spec xcprod.
Proof. by rewrite /xcprod => isoZ; move: xcprod_subproof; rewrite isoZ. Qed.
Lemma isog_xcprod (rT : finGroupType) (GH GK G : {group rT}) :
Aut_in (Aut H) 'Z(H) \isog Aut 'Z(H) ->
GH \isog H -> GK \isog K -> GH \* GK = G -> 'Z(GH) = 'Z(GK) ->
G \isog [set: xcprod].
Proof.
move=> AutZinH isoGH isoGK defG eqZGHK; have [_ _ cGHK] := cprodP defG.
have [|gz isoZ] := xcprodP.
have [[fH injfH <-] [fK injfK <-]] := (isogP isoGH, isogP isoGK).
rewrite -!injm_center -?(isog_transl _ (sub_isog _ _)) ?center_sub //=.
by rewrite eqZGHK sub_isog ?center_sub.
rewrite (isog_cprod_by _ defG) //.
by apply/eqP; rewrite eqEsubset setIS // subsetI {2}eqZGHK !center_sub.
Qed.
End ExtCprod.
Section IterCprod.
Variables (gT : finGroupType) (G : {group gT}).
Fixpoint ncprod_def n : finGroupType :=
if n is n'.+1 then xcprod G [set: ncprod_def n']
else subg_of 'Z(G).
Fact ncprod_key : unit. Proof. by []. Qed.
Definition ncprod := locked_with ncprod_key ncprod_def.
Local Notation G_ n := [set: gsort (ncprod n)].
Lemma ncprod0 : G_ 0 \isog 'Z(G).
Proof. by rewrite [ncprod]unlock isog_sym isog_subg. Qed.
Lemma center_ncprod0 : 'Z(G_ 0) = G_ 0.
Proof. by apply: center_idP; rewrite (isog_abelian ncprod0) center_abelian. Qed.
Lemma center_ncprod n : 'Z(G_ n) \isog 'Z(G).
Proof.
elim: n => [|n]; first by rewrite center_ncprod0 ncprod0.
rewrite [ncprod]unlock=> /isog_symr/xcprodP[gz isoZ] /=.
by rewrite -cpairg1_center isog_sym sub_isog ?center_sub ?injm_cpairg1.
Qed.
Lemma ncprodS n : xcprod_spec G [set: ncprod n] (ncprod n.+1).
Proof.
by have:= xcprodP (isog_symr (center_ncprod n)); rewrite [ncprod]unlock.
Qed.
Lemma ncprod1 : G_ 1 \isog G.
Proof.
case: ncprodS => gz isoZ; rewrite isog_sym /= -im_cpair.
rewrite mulGSid /=; first by rewrite sub_isog ?injm_cpairg1.
rewrite -{3}center_ncprod0 injm_center ?injm_cpair1g //.
by rewrite -cpair_center_id center_sub.
Qed.
Lemma Aut_ncprod_full n :
Aut_in (Aut G) 'Z(G) \isog Aut 'Z(G) ->
Aut_in (Aut (G_ n)) 'Z(G_ n) \isog Aut 'Z(G_ n).
Proof.
move=> AutZinG; elim: n => [|n IHn].
by rewrite center_ncprod0; apply/Aut_sub_fullP=> // g injg gG0; exists g.
by case: ncprodS => gz isoZ; apply: Aut_cprod_by_full.
Qed.
End IterCprod.
|
Complete.lean
|
/-
Copyright (c) 2024 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.CategoryTheory.Adjunction.Lifting.Right
import Mathlib.CategoryTheory.Closed.FunctorCategory.Groupoid
import Mathlib.CategoryTheory.Groupoid.Discrete
import Mathlib.CategoryTheory.Limits.Preserves.FunctorCategory
import Mathlib.CategoryTheory.Monad.Comonadicity
/-!
# Functors into a complete monoidal closed category form a monoidal closed category.
TODO (in progress by Joël Riou): make a more explicit construction of the internal hom in functor
categories.
-/
universe v₁ v₂ u₁ u₂
open CategoryTheory MonoidalCategory MonoidalClosed Limits
noncomputable section
namespace CategoryTheory.Functor
section
variable (I : Type u₂) [Category.{v₂} I]
private abbrev incl : Discrete I ⥤ I := Discrete.functor id
variable (C : Type u₁) [Category.{v₁} C] [MonoidalCategory C] [MonoidalClosed C]
variable [∀ (F : Discrete I ⥤ C), (Discrete.functor id).HasRightKanExtension F]
-- is also implied by: `[HasLimitsOfSize.{u₂, max u₂ v₂} C]`
instance : ReflectsIsomorphisms <| (whiskeringLeft _ _ C).obj (incl I) where
reflects f h := by
simp only [NatTrans.isIso_iff_isIso_app] at *
intro X
exact h ⟨X⟩
variable [HasLimitsOfShape WalkingParallelPair C]
instance : Comonad.PreservesLimitOfIsCoreflexivePair ((whiskeringLeft _ _ C).obj (incl I)) :=
⟨inferInstance⟩
instance : ComonadicLeftAdjoint ((whiskeringLeft _ _ C).obj (incl I)) :=
Comonad.comonadicOfHasPreservesCoreflexiveEqualizersOfReflectsIsomorphisms
((incl I).ranAdjunction C)
instance (F : I ⥤ C) : IsLeftAdjoint (tensorLeft (incl I ⋙ F)) :=
(ihom.adjunction (incl I ⋙ F)).isLeftAdjoint
/-- Auxiliary definition for `functorCategoryMonoidalClosed` -/
def functorCategoryClosed (F : I ⥤ C) : Closed F :=
have := (ihom.adjunction (incl I ⋙ F)).isLeftAdjoint
have := isLeftAdjoint_square_lift_comonadic (tensorLeft F) ((whiskeringLeft _ _ C).obj (incl I))
((whiskeringLeft _ _ C).obj (incl I)) (tensorLeft (incl I ⋙ F)) (Iso.refl _)
{ rightAdj := (tensorLeft F).rightAdjoint
adj := Adjunction.ofIsLeftAdjoint (tensorLeft F) }
/--
Assuming the existence of certain limits, functors into a monoidal closed category form a
monoidal closed category.
Note: this is defined completely abstractly, and does not have any good definitional properties.
See the TODO in the module docstring.
-/
def functorCategoryMonoidalClosed : MonoidalClosed (I ⥤ C) where
closed F := functorCategoryClosed I C F
end
end CategoryTheory.Functor
|
hall.v
|
(* (c) Copyright 2006-2016 Microsoft Corporation and Inria. *)
(* Distributed under the terms of CeCILL-B. *)
From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div choice.
From mathcomp Require Import fintype finset prime fingroup morphism.
From mathcomp Require Import automorphism quotient action gproduct gfunctor.
From mathcomp Require Import commutator center pgroup finmodule nilpotent.
From mathcomp Require Import sylow abelian maximal.
(*****************************************************************************)
(* In this files we prove the Schur-Zassenhaus splitting and transitivity *)
(* theorems (under solvability assumptions), then derive P. Hall's *)
(* generalization of Sylow's theorem to solvable groups and its corollaries, *)
(* in particular the theory of coprime action. We develop both the theory of *)
(* coprime action of a solvable group on Sylow subgroups (as in Aschbacher *)
(* 18.7), and that of coprime action on Hall subgroups of a solvable group *)
(* as per B & G, Proposition 1.5; however we only support external group *)
(* action (as opposed to internal action by conjugation) for the latter case *)
(* because it is much harder to apply in practice. *)
(*****************************************************************************)
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Import GroupScope.
Section Hall.
Implicit Type gT : finGroupType.
Theorem SchurZassenhaus_split gT (G H : {group gT}) :
Hall G H -> H <| G -> [splits G, over H].
Proof.
have [n] := ubnP #|G|; elim: n => // n IHn in gT G H * => /ltnSE-Gn hallH nsHG.
have [sHG nHG] := andP nsHG.
have [-> | [p pr_p pH]] := trivgVpdiv H.
by apply/splitsP; exists G; rewrite inE -subG1 subsetIl mul1g eqxx.
have [P sylP] := Sylow_exists p H.
case nPG: (P <| G); last first.
pose N := ('N_G(P))%G; have sNG: N \subset G by rewrite subsetIl.
have eqHN_G: H * N = G by apply: Frattini_arg sylP.
pose H' := (H :&: N)%G.
have nsH'N: H' <| N.
by rewrite /normal subsetIr normsI ?normG ?(subset_trans sNG).
have eq_iH: #|G : H| = #|N| %/ #|H'|.
rewrite -divgS // -(divnMl (cardG_gt0 H')) mulnC -eqHN_G.
by rewrite -mul_cardG (mulnC #|H'|) divnMl // cardG_gt0.
have hallH': Hall N H'.
rewrite /Hall -divgS subsetIr //= -eq_iH.
by case/andP: hallH => _; apply: coprimeSg; apply: subsetIl.
have: [splits N, over H'].
apply: IHn hallH' nsH'N; apply: {n}leq_trans Gn.
rewrite proper_card // properEneq sNG andbT; apply/eqP=> eqNG.
by rewrite -eqNG normal_subnorm (subset_trans (pHall_sub sylP)) in nPG.
case/splitsP=> K /complP[tiKN eqH'K].
have sKN: K \subset N by rewrite -(mul1g K) -eqH'K mulSg ?sub1set.
apply/splitsP; exists K; rewrite inE -subG1; apply/andP; split.
by rewrite /= -(setIidPr sKN) setIA tiKN.
by rewrite eqEsubset -eqHN_G mulgS // -eqH'K mulGS mulSg ?subsetIl.
pose Z := 'Z(P); pose Gbar := G / Z; pose Hbar := H / Z.
have sZP: Z \subset P by apply: center_sub.
have sZH: Z \subset H by apply: subset_trans (pHall_sub sylP).
have sZG: Z \subset G by apply: subset_trans sHG.
have nZG: Z <| G by apply: gFnormal_trans nPG.
have nZH: Z <| H by apply: normalS nZG.
have nHGbar: Hbar <| Gbar by apply: morphim_normal.
have hallHbar: Hall Gbar Hbar by apply: morphim_Hall (normal_norm _) _.
have: [splits Gbar, over Hbar].
apply: IHn => //; apply: {n}leq_trans Gn; rewrite ltn_quotient //.
apply/eqP=> /(trivg_center_pgroup (pHall_pgroup sylP))/eqP.
rewrite trivg_card1 (card_Hall sylP) p_part -(expn0 p).
by rewrite eqn_exp2l ?prime_gt1 // lognE pH pr_p cardG_gt0.
case/splitsP=> Kbar /complP[tiHKbar eqHKbar].
have: Kbar \subset Gbar by rewrite -eqHKbar mulG_subr.
case/inv_quotientS=> //= ZK quoZK sZZK sZKG.
have nZZK: Z <| ZK by apply: normalS nZG.
have cardZK: #|ZK| = (#|Z| * #|G : H|)%N.
rewrite -(Lagrange sZZK); congr (_ * _)%N.
rewrite -card_quotient -?quoZK; last by case/andP: nZZK.
rewrite -(divgS sHG) -(Lagrange sZG) -(Lagrange sZH) divnMl //.
rewrite -!card_quotient ?normal_norm //= -/Gbar -/Hbar.
by rewrite -eqHKbar (TI_cardMg tiHKbar) mulKn.
have: [splits ZK, over Z].
rewrite (Gaschutz_split nZZK _ sZZK) ?center_abelian //; last first.
rewrite -divgS // cardZK mulKn ?cardG_gt0 //.
by case/andP: hallH => _; apply: coprimeSg.
by apply/splitsP; exists 1%G; rewrite inE -subG1 subsetIr mulg1 eqxx.
case/splitsP=> K /complP[tiZK eqZK].
have sKZK: K \subset ZK by rewrite -(mul1g K) -eqZK mulSg ?sub1G.
have tiHK: H :&: K = 1.
apply/trivgP; rewrite /= -(setIidPr sKZK) setIA -tiZK setSI //.
rewrite -quotient_sub1; last by rewrite subIset 1?normal_norm.
by rewrite /= quotientGI //= -quoZK tiHKbar.
apply/splitsP; exists K; rewrite inE tiHK ?eqEcard subxx leqnn /=.
rewrite mul_subG ?(subset_trans sKZK) //= TI_cardMg //.
rewrite -(@mulKn #|K| #|Z|) ?cardG_gt0 // -TI_cardMg // eqZK.
by rewrite cardZK mulKn ?cardG_gt0 // Lagrange.
Qed.
Theorem SchurZassenhaus_trans_sol gT (H K K1 : {group gT}) :
solvable H -> K \subset 'N(H) -> K1 \subset H * K ->
coprime #|H| #|K| -> #|K1| = #|K| ->
exists2 x, x \in H & K1 :=: K :^ x.
Proof.
have [n] := ubnP #|H|.
elim: n => // n IHn in gT H K K1 * => /ltnSE-leHn solH nHK.
have [-> | ] := eqsVneq H 1.
rewrite mul1g => sK1K _ eqK1K; exists 1; first exact: set11.
by apply/eqP; rewrite conjsg1 eqEcard sK1K eqK1K /=.
pose G := (H <*> K)%G.
have defG: G :=: H * K by rewrite -normC // -norm_joinEl // joingC.
have sHG: H \subset G by apply: joing_subl.
have sKG: K \subset G by apply: joing_subr.
have nsHG: H <| G by rewrite /(H <| G) sHG join_subG normG.
case/(solvable_norm_abelem solH nsHG)=> M [sMH nsMG ntM] /and3P[_ abelM _].
have [sMG nMG] := andP nsMG; rewrite -defG => sK1G coHK oK1K.
have nMsG (L : {set gT}): L \subset G -> L \subset 'N(M).
by move/subset_trans->.
have [coKM coHMK]: coprime #|M| #|K| /\ coprime #|H / M| #|K|.
by apply/andP; rewrite -coprimeMl card_quotient ?nMsG ?Lagrange.
have oKM (K' : {group gT}): K' \subset G -> #|K'| = #|K| -> #|K' / M| = #|K|.
move=> sK'G oK'.
rewrite -quotientMidr -?norm_joinEl ?card_quotient ?nMsG //; last first.
by rewrite gen_subG subUset sK'G.
rewrite -divgS /=; last by rewrite -gen_subG genS ?subsetUr.
by rewrite norm_joinEl ?nMsG // coprime_cardMg ?mulnK // oK' coprime_sym.
have [xb]: exists2 xb, xb \in H / M & K1 / M = (K / M) :^ xb.
apply: IHn; try by rewrite (quotient_sol, morphim_norms, oKM K) ?(oKM K1).
by apply: leq_trans leHn; rewrite ltn_quotient.
by rewrite -morphimMl ?nMsG // -defG morphimS.
case/morphimP=> x nMx Hx ->{xb} eqK1Kx; pose K2 := (K :^ x)%G.
have{eqK1Kx} eqK12: K1 / M = K2 / M by rewrite quotientJ.
suff [y My ->]: exists2 y, y \in M & K1 :=: K2 :^ y.
by exists (x * y); [rewrite groupMl // (subsetP sMH) | rewrite conjsgM].
have nMK1: K1 \subset 'N(M) by apply: nMsG.
have defMK: M * K1 = M <*> K1 by rewrite -normC // -norm_joinEl // joingC.
have sMKM: M \subset M <*> K1 by rewrite joing_subl.
have nMKM: M <| M <*> K1 by rewrite normalYl.
have trMK1: M :&: K1 = 1 by rewrite coprime_TIg ?oK1K.
have trMK2: M :&: K2 = 1 by rewrite coprime_TIg ?cardJg ?oK1K.
apply: (Gaschutz_transitive nMKM _ sMKM) => //=; last 2 first.
- by rewrite inE trMK1 defMK !eqxx.
- by rewrite -!(setIC M) trMK1.
- by rewrite -divgS //= -defMK coprime_cardMg oK1K // mulKn.
rewrite inE trMK2 eqxx eq_sym eqEcard /= -defMK andbC.
by rewrite !coprime_cardMg ?cardJg ?oK1K ?leqnn //= mulGS -quotientSK -?eqK12.
Qed.
Lemma SchurZassenhaus_trans_actsol gT (G A B : {group gT}) :
solvable A -> A \subset 'N(G) -> B \subset A <*> G ->
coprime #|G| #|A| -> #|A| = #|B| ->
exists2 x, x \in G & B :=: A :^ x.
Proof.
set AG := A <*> G; have [n] := ubnP #|AG|.
elim: n => // n IHn in gT A B G AG * => /ltnSE-leAn solA nGA sB_AG coGA oAB.
have [A1 | ntA] := eqsVneq A 1.
by exists 1; rewrite // conjsg1 A1 (@card1_trivg _ B) // -oAB A1 cards1.
have [M [sMA nsMA ntM]] := solvable_norm_abelem solA (normal_refl A) ntA.
case/is_abelemP=> q q_pr /abelem_pgroup qM; have nMA := normal_norm nsMA.
have defAG: AG = A * G := norm_joinEl nGA.
have sA_AG: A \subset AG := joing_subl _ _.
have sG_AG: G \subset AG := joing_subr _ _.
have sM_AG := subset_trans sMA sA_AG.
have oAG: #|AG| = (#|A| * #|G|)%N by rewrite defAG coprime_cardMg 1?coprime_sym.
have q'G: #|G|`_q = 1%N.
rewrite part_p'nat ?p'natE -?prime_coprime // coprime_sym.
have [_ _ [k oM]] := pgroup_pdiv qM ntM.
by rewrite -(@coprime_pexpr k.+1) // -oM (coprimegS sMA).
have coBG: coprime #|B| #|G| by rewrite -oAB coprime_sym.
have defBG: B * G = AG.
by apply/eqP; rewrite eqEcard mul_subG ?sG_AG //= oAG oAB coprime_cardMg.
case nMG: (G \subset 'N(M)).
have nsM_AG: M <| AG by rewrite /normal sM_AG join_subG nMA.
have nMB: B \subset 'N(M) := subset_trans sB_AG (normal_norm nsM_AG).
have sMB: M \subset B.
have [Q sylQ]:= Sylow_exists q B; have sQB := pHall_sub sylQ.
apply: subset_trans (normal_sub_max_pgroup (Hall_max _) qM nsM_AG) (sQB).
rewrite pHallE (subset_trans sQB) //= oAG partnM // q'G muln1 oAB.
by rewrite (card_Hall sylQ).
have defAGq: AG / M = (A / M) <*> (G / M).
by rewrite quotient_gen ?quotientU ?subUset ?nMA.
have: B / M \subset (A / M) <*> (G / M) by rewrite -defAGq quotientS.
case/IHn; rewrite ?morphim_sol ?quotient_norms ?coprime_morph //.
- by rewrite -defAGq (leq_trans _ leAn) ?ltn_quotient.
- by rewrite !card_quotient // -!divgS // oAB.
move=> Mx; case/morphimP=> x Nx Gx ->{Mx} //; rewrite -quotientJ //= => defBq.
exists x => //; apply: quotient_inj defBq; first by rewrite /normal sMB.
by rewrite -(normsP nMG x Gx) /normal normJ !conjSg.
pose K := M <*> G; pose R := K :&: B; pose N := 'N_G(M).
have defK: K = M * G by rewrite -norm_joinEl ?(subset_trans sMA).
have oK: #|K| = (#|M| * #|G|)%N.
by rewrite defK coprime_cardMg // coprime_sym (coprimegS sMA).
have sylM: q.-Sylow(K) M.
by rewrite pHallE joing_subl /= oK partnM // q'G muln1 part_pnat_id.
have sylR: q.-Sylow(K) R.
rewrite pHallE subsetIl /= -(card_Hall sylM) -(@eqn_pmul2r #|G|) // -oK.
rewrite -coprime_cardMg ?(coprimeSg _ coBG) ?subsetIr //=.
by rewrite group_modr ?joing_subr ?(setIidPl _) // defBG join_subG sM_AG.
have [mx] := Sylow_trans sylM sylR.
rewrite /= -/K defK; case/imset2P=> m x Mm Gx ->{mx}.
rewrite conjsgM (conjGid Mm) {m Mm} => defR.
have sNG: N \subset G := subsetIl _ _.
have pNG: N \proper G by rewrite /proper sNG subsetI subxx nMG.
have nNA: A \subset 'N(N) by rewrite normsI ?norms_norm.
have: B :^ x^-1 \subset A <*> N.
rewrite norm_joinEl ?group_modl // -defAG subsetI !sub_conjgV -normJ -defR.
rewrite conjGid ?(subsetP sG_AG) // normsI ?normsG // (subset_trans sB_AG) //.
by rewrite join_subG normsM // -defK normsG ?joing_subr.
do [case/IHn; rewrite ?cardJg ?(coprimeSg _ coGA) //= -/N] => [|y Ny defB].
rewrite joingC norm_joinEr // coprime_cardMg ?(coprimeSg sNG) //.
by rewrite (leq_trans _ leAn) // oAG mulnC ltn_pmul2l // proper_card.
exists (y * x); first by rewrite groupM // (subsetP sNG).
by rewrite conjsgM -defB conjsgKV.
Qed.
Lemma Hall_exists_subJ pi gT (G : {group gT}) :
solvable G -> exists2 H : {group gT}, pi.-Hall(G) H
& forall K : {group gT}, K \subset G -> pi.-group K ->
exists2 x, x \in G & K \subset H :^ x.
Proof.
have [n] := ubnP #|G|; elim: n gT G => // n IHn gT G /ltnSE-leGn solG.
have [-> | ntG] := eqsVneq G 1.
exists 1%G => [|_ /trivGP-> _]; last by exists 1; rewrite ?set11 ?sub1G.
by rewrite pHallE sub1G cards1 part_p'nat.
case: (solvable_norm_abelem solG (normal_refl _)) => // M [sMG nsMG ntM].
case/is_abelemP=> p pr_p /and3P[pM cMM _].
pose Gb := (G / M)%G; case: (IHn _ Gb) => [||Hb]; try exact: quotient_sol.
by rewrite (leq_trans (ltn_quotient _ _)).
case/and3P=> [sHbGb piHb pi'Hb'] transHb.
case: (inv_quotientS nsMG sHbGb) => H def_H sMH sHG.
have nMG := normal_norm nsMG; have nMH := subset_trans sHG nMG.
have{transHb} transH (K : {group gT}):
K \subset G -> pi.-group K -> exists2 x, x \in G & K \subset H :^ x.
- move=> sKG piK; have nMK := subset_trans sKG nMG.
case: (transHb (K / M)%G) => [||xb Gxb sKHxb]; first exact: morphimS.
exact: morphim_pgroup.
case/morphimP: Gxb => x Nx Gx /= def_x; exists x => //.
apply/subsetP=> y Ky.
have: y \in coset M y by rewrite val_coset (subsetP nMK, rcoset_refl).
have: coset M y \in (H :^ x) / M.
rewrite /quotient morphimJ //=.
by rewrite def_x def_H in sKHxb; apply/(subsetP sKHxb)/mem_quotient.
case/morphimP=> z Nz Hxz ->.
rewrite val_coset //; case/rcosetP=> t Mt ->; rewrite groupMl //.
by rewrite mem_conjg (subsetP sMH) // -mem_conjg (normP Nx).
have{pi'Hb'} pi'H': pi^'.-nat #|G : H|.
move: pi'Hb'; rewrite -!divgS // def_H !card_quotient //.
by rewrite -(divnMl (cardG_gt0 M)) !Lagrange.
have [pi_p | pi'p] := boolP (p \in pi).
exists H => //; apply/and3P; split=> //; rewrite /pgroup.
by rewrite -(Lagrange sMH) -card_quotient // pnatM -def_H (pi_pnat pM).
have [ltHG | leGH {n IHn leGn transH}] := ltnP #|H| #|G|.
case: (IHn _ H (leq_trans ltHG leGn)) => [|H1]; first exact: solvableS solG.
case/and3P=> sH1H piH1 pi'H1' transH1.
have sH1G: H1 \subset G by apply: subset_trans sHG.
exists H1 => [|K sKG piK].
apply/and3P; split => //.
rewrite -divgS // -(Lagrange sHG) -(Lagrange sH1H) -mulnA.
by rewrite mulKn // pnatM pi'H1'.
case: (transH K sKG piK) => x Gx def_K.
case: (transH1 (K :^ x^-1)%G) => [||y Hy def_K1].
- by rewrite sub_conjgV.
- by rewrite /pgroup cardJg.
exists (y * x); first by rewrite groupMr // (subsetP sHG).
by rewrite -(conjsgKV x K) conjsgM conjSg.
have{leGH Gb sHbGb sHG sMH pi'H'} eqHG: H = G.
by apply/eqP; rewrite -val_eqE eqEcard sHG.
have{H Hb def_H eqHG piHb nMH} hallM: pi^'.-Hall(G) M.
rewrite /pHall /pgroup sMG pnatNK -card_quotient //=.
by rewrite -eqHG -def_H (pi_pnat pM).
case/splitsP: (SchurZassenhaus_split (pHall_Hall hallM) nsMG) => H.
case/complP=> trMH defG.
have sHG: H \subset G by rewrite -defG mulG_subr.
exists H => [|K sKG piK].
apply: etrans hallM; rewrite /pHall sMG sHG /= -!divgS // -defG andbC.
by rewrite (TI_cardMg trMH) mulKn ?mulnK // pnatNK.
pose G1 := (K <*> M)%G; pose K1 := (H :&: G1)%G.
have nMK: K \subset 'N(M) by apply: subset_trans sKG nMG.
have defG1: M * K = G1 by rewrite -normC -?norm_joinEl.
have sK1G1: K1 \subset M * K by rewrite defG1 subsetIr.
have coMK: coprime #|M| #|K|.
by rewrite coprime_sym (pnat_coprime piK) //; apply: (pHall_pgroup hallM).
case: (SchurZassenhaus_trans_sol _ nMK sK1G1 coMK) => [||x Mx defK1].
- exact: solvableS solG.
- apply/eqP; rewrite -(eqn_pmul2l (cardG_gt0 M)) -TI_cardMg //; last first.
by apply/trivgP; rewrite -trMH /= setIA subsetIl.
rewrite -coprime_cardMg // defG1; apply/eqP; congr #|(_ : {set _})|.
rewrite group_modl; last by rewrite -defG1 mulG_subl.
by apply/setIidPr; rewrite defG gen_subG subUset sKG.
exists x^-1; first by rewrite groupV (subsetP sMG).
by rewrite -(_ : K1 :^ x^-1 = K) ?(conjSg, subsetIl) // defK1 conjsgK.
Qed.
End Hall.
Section HallCorollaries.
Variable gT : finGroupType.
Corollary Hall_exists pi (G : {group gT}) :
solvable G -> exists H : {group gT}, pi.-Hall(G) H.
Proof. by case/(Hall_exists_subJ pi) => H; exists H. Qed.
Corollary Hall_trans pi (G H1 H2 : {group gT}) :
solvable G -> pi.-Hall(G) H1 -> pi.-Hall(G) H2 ->
exists2 x, x \in G & H1 :=: H2 :^ x.
Proof.
move=> solG; have [H hallH transH] := Hall_exists_subJ pi solG.
have conjH (K : {group gT}):
pi.-Hall(G) K -> exists2 x, x \in G & K = (H :^ x)%G.
- move=> hallK; have [sKG piK _] := and3P hallK.
case: (transH K sKG piK) => x Gx sKH; exists x => //.
apply/eqP; rewrite -val_eqE eqEcard sKH cardJg.
by rewrite (card_Hall hallH) (card_Hall hallK) /=.
case/conjH=> x1 Gx1 ->{H1}; case/conjH=> x2 Gx2 ->{H2}.
exists (x2^-1 * x1); first by rewrite groupMl ?groupV.
by apply: val_inj; rewrite /= conjsgM conjsgK.
Qed.
Corollary Hall_superset pi (G K : {group gT}) :
solvable G -> K \subset G -> pi.-group K ->
exists2 H : {group gT}, pi.-Hall(G) H & K \subset H.
Proof.
move=> solG sKG; have [H hallH transH] := Hall_exists_subJ pi solG.
by case/transH=> // x Gx sKHx; exists (H :^ x)%G; rewrite ?pHallJ.
Qed.
Corollary Hall_subJ pi (G H K : {group gT}) :
solvable G -> pi.-Hall(G) H -> K \subset G -> pi.-group K ->
exists2 x, x \in G & K \subset H :^ x.
Proof.
move=> solG HallH sKG piK; have [M HallM sKM]:= Hall_superset solG sKG piK.
have [x Gx defM] := Hall_trans solG HallM HallH.
by exists x; rewrite // -defM.
Qed.
Corollary Hall_Jsub pi (G H K : {group gT}) :
solvable G -> pi.-Hall(G) H -> K \subset G -> pi.-group K ->
exists2 x, x \in G & K :^ x \subset H.
Proof.
move=> solG HallH sKG piK; have [x Gx sKHx] := Hall_subJ solG HallH sKG piK.
by exists x^-1; rewrite ?groupV // sub_conjgV.
Qed.
Lemma Hall_Frattini_arg pi (G K H : {group gT}) :
solvable K -> K <| G -> pi.-Hall(K) H -> K * 'N_G(H) = G.
Proof.
move=> solK /andP[sKG nKG] hallH.
have sHG: H \subset G by apply: subset_trans sKG; case/andP: hallH.
rewrite setIC group_modl //; apply/setIidPr/subsetP=> x Gx.
pose H1 := (H :^ x^-1)%G.
have hallH1: pi.-Hall(K) H1 by rewrite pHallJnorm // groupV (subsetP nKG).
case: (Hall_trans solK hallH hallH1) => y Ky defH.
rewrite -(mulKVg y x) mem_mulg //; apply/normP.
by rewrite conjsgM {1}defH conjsgK conjsgKV.
Qed.
End HallCorollaries.
Section InternalAction.
Variables (pi : nat_pred) (gT : finGroupType).
Implicit Types G H K A X : {group gT}.
(* Part of Aschbacher (18.7.4). *)
Lemma coprime_norm_cent A G :
A \subset 'N(G) -> coprime #|G| #|A| -> 'N_G(A) = 'C_G(A).
Proof.
move=> nGA coGA; apply/eqP; rewrite eqEsubset andbC setIS ?cent_sub //=.
rewrite subsetI subsetIl /= (sameP commG1P trivgP) -(coprime_TIg coGA).
rewrite subsetI commg_subr subsetIr andbT.
move: nGA; rewrite -commg_subl; apply: subset_trans.
by rewrite commSg ?subsetIl.
Qed.
(* This is B & G, Proposition 1.5(a) *)
Proposition coprime_Hall_exists A G :
A \subset 'N(G) -> coprime #|G| #|A| -> solvable G ->
exists2 H : {group gT}, pi.-Hall(G) H & A \subset 'N(H).
Proof.
move=> nGA coGA solG; have [H hallH] := Hall_exists pi solG.
have sG_AG: G \subset A <*> G by rewrite joing_subr.
have nG_AG: A <*> G \subset 'N(G) by rewrite join_subG nGA normG.
pose N := 'N_(A <*> G)(H)%G.
have nGN: N \subset 'N(G) by rewrite subIset ?nG_AG.
have nGN_N: G :&: N <| N by rewrite /(_ <| N) subsetIr normsI ?normG.
have NG_AG: G * N = A <*> G.
by apply: Hall_Frattini_arg hallH => //; apply/andP.
have iGN_A: #|N| %/ #|G :&: N| = #|A|.
rewrite setIC divgI -card_quotient // -quotientMidl NG_AG.
rewrite card_quotient -?divgS //= norm_joinEl //.
by rewrite coprime_cardMg 1?coprime_sym // mulnK.
have hallGN: Hall N (G :&: N).
by rewrite /Hall -divgS subsetIr //= iGN_A (coprimeSg _ coGA) ?subsetIl.
case/splitsP: {hallGN nGN_N}(SchurZassenhaus_split hallGN nGN_N) => B.
case/complP=> trBGN defN.
have{trBGN iGN_A} oBA: #|B| = #|A|.
by rewrite -iGN_A -{1}defN (TI_cardMg trBGN) mulKn.
have sBN: B \subset N by rewrite -defN mulG_subr.
case: (SchurZassenhaus_trans_sol solG nGA _ coGA oBA) => [|x Gx defB].
by rewrite -(normC nGA) -norm_joinEl // -NG_AG -(mul1g B) mulgSS ?sub1G.
exists (H :^ x^-1)%G; first by rewrite pHallJ ?groupV.
apply/subsetP=> y Ay; have: y ^ x \in B by rewrite defB memJ_conjg.
move/(subsetP sBN)=> /setIP[_ /normP nHyx].
by apply/normP; rewrite -conjsgM conjgCV invgK conjsgM nHyx.
Qed.
(* This is B & G, Proposition 1.5(c) *)
Proposition coprime_Hall_trans A G H1 H2 :
A \subset 'N(G) -> coprime #|G| #|A| -> solvable G ->
pi.-Hall(G) H1 -> A \subset 'N(H1) ->
pi.-Hall(G) H2 -> A \subset 'N(H2) ->
exists2 x, x \in 'C_G(A) & H1 :=: H2 :^ x.
Proof.
move: H1 => H nGA coGA solG hallH nHA hallH2.
have{H2 hallH2} [x Gx -> nH1xA] := Hall_trans solG hallH2 hallH.
have sG_AG: G \subset A <*> G by rewrite -{1}genGid genS ?subsetUr.
have nG_AG: A <*> G \subset 'N(G) by rewrite gen_subG subUset nGA normG.
pose N := 'N_(A <*> G)(H)%G.
have nGN: N \subset 'N(G) by rewrite subIset ?nG_AG.
have nGN_N: G :&: N <| N.
apply/normalP; rewrite subsetIr; split=> // y Ny.
by rewrite conjIg (normP _) // (subsetP nGN, conjGid).
have NG_AG : G * N = A <*> G.
by apply: Hall_Frattini_arg hallH => //; apply/andP.
have iGN_A: #|N : G :&: N| = #|A|.
rewrite -card_quotient //; last by case/andP: nGN_N.
rewrite (card_isog (second_isog nGN)) /= -quotientMidr (normC nGN) NG_AG.
rewrite card_quotient // -divgS //= joingC norm_joinEr //.
by rewrite coprime_cardMg // mulnC mulnK.
have solGN: solvable (G :&: N) by apply: solvableS solG; apply: subsetIl.
have oAxA: #|A :^ x^-1| = #|A| by apply: cardJg.
have sAN: A \subset N by rewrite subsetI -{1}genGid genS // subsetUl.
have nGNA: A \subset 'N(G :&: N).
by apply/normsP=> y ?; rewrite conjIg (normsP nGA) ?(conjGid, subsetP sAN).
have coGNA: coprime #|G :&: N| #|A| := coprimeSg (subsetIl _ _) coGA.
case: (SchurZassenhaus_trans_sol solGN nGNA _ coGNA oAxA) => [|y GNy defAx].
have ->: (G :&: N) * A = N.
apply/eqP; rewrite eqEcard -{2}(mulGid N) mulgSS ?subsetIr //=.
by rewrite coprime_cardMg // -iGN_A Lagrange ?subsetIr.
rewrite sub_conjgV conjIg -normJ subsetI conjGid ?joing_subl //.
by rewrite mem_gen // inE Gx orbT.
case/setIP: GNy => Gy; case/setIP=> _; move/normP=> nHy.
exists (y * x)^-1.
rewrite -coprime_norm_cent // groupV inE groupM //=; apply/normP.
by rewrite conjsgM -defAx conjsgKV.
by apply: val_inj; rewrite /= -{2}nHy -(conjsgM _ y) conjsgK.
Qed.
(* A complement to the above: 'C(A) acts on 'Nby(A) *)
Lemma norm_conj_cent A G x : x \in 'C(A) ->
(A \subset 'N(G :^ x)) = (A \subset 'N(G)).
Proof. by move=> cAx; rewrite norm_conj_norm ?(subsetP (cent_sub A)). Qed.
(* Strongest version of the centraliser lemma -- not found in textbooks! *)
(* Obviously, the solvability condition could be removed once we have the *)
(* Odd Order Theorem. *)
Lemma strongest_coprime_quotient_cent A G H :
let R := H :&: [~: G, A] in
A \subset 'N(H) -> R \subset G -> coprime #|R| #|A| ->
solvable R || solvable A ->
'C_G(A) / H = 'C_(G / H)(A / H).
Proof.
move=> R nHA sRG coRA solRA.
have nRA: A \subset 'N(R) by rewrite normsI ?commg_normr.
apply/eqP; rewrite eqEsubset subsetI morphimS ?subsetIl //=.
rewrite (subset_trans _ (morphim_cent _ _)) ?morphimS ?subsetIr //=.
apply/subsetP=> _ /setIP[/morphimP[x Nx Gx ->] cAHx].
have{cAHx} cAxR y: y \in A -> [~ x, y] \in R.
move=> Ay; have Ny: y \in 'N(H) by apply: subsetP Ay.
rewrite inE mem_commg // andbT coset_idr ?groupR // morphR //=.
by apply/eqP; apply/commgP; apply: (centP cAHx); rewrite mem_quotient.
have AxRA: A :^ x \subset R * A.
apply/subsetP=> _ /imsetP[y Ay ->].
rewrite -normC // -(mulKVg y (y ^ x)) -commgEl mem_mulg //.
by rewrite -groupV invg_comm cAxR.
have [y Ry def_Ax]: exists2 y, y \in R & A :^ x = A :^ y.
have oAx: #|A :^ x| = #|A| by rewrite cardJg.
have [solR | solA] := orP solRA; first exact: SchurZassenhaus_trans_sol.
by apply: SchurZassenhaus_trans_actsol; rewrite // joingC norm_joinEr.
rewrite -imset_coset; apply/imsetP; exists (x * y^-1); last first.
by rewrite conjgCV mkerl // ker_coset memJ_norm groupV; case/setIP: Ry.
rewrite /= inE groupMl // ?(groupV, subsetP sRG) //=.
apply/centP=> z Az; apply/commgP/eqP/set1P.
rewrite -[[set 1]](coprime_TIg coRA) inE {1}commgEl commgEr /= -/R.
rewrite invMg -mulgA invgK (@groupMl _ R) // conjMg mulgA -commgEl.
rewrite groupMl ?cAxR // memJ_norm ?(groupV, subsetP nRA) // Ry /=.
by rewrite groupMr // conjVg groupV conjgM -mem_conjg -def_Ax memJ_conjg.
Qed.
(* A weaker but more practical version, still stronger than the usual form *)
(* (viz. Aschbacher 18.7.4), similar to the one needed in Aschbacher's *)
(* proof of Thompson factorization. Note that the coprime and solvability *)
(* assumptions could be further weakened to H :&: G (and hence become *)
(* trivial if H and G are TI). However, the assumption that A act on G is *)
(* needed in this case. *)
Lemma coprime_norm_quotient_cent A G H :
A \subset 'N(G) -> A \subset 'N(H) -> coprime #|H| #|A| -> solvable H ->
'C_G(A) / H = 'C_(G / H)(A / H).
Proof.
move=> nGA nHA coHA solH; have sRH := subsetIl H [~: G, A].
rewrite strongest_coprime_quotient_cent ?(coprimeSg sRH) 1?(solvableS sRH) //.
by rewrite subIset // commg_subl nGA orbT.
Qed.
(* A useful consequence (similar to Ex. 6.1 in Aschbacher) of the stronger *)
(* theorem. *)
Lemma coprime_cent_mulG A G H :
A \subset 'N(G) -> A \subset 'N(H) -> G \subset 'N(H) ->
coprime #|H| #|A| -> solvable H ->
'C_(H * G)(A) = 'C_H(A) * 'C_G(A).
Proof.
move=> nHA nGA nHG coHA solH; rewrite -norm_joinEr //.
have nsHG: H <| H <*> G by rewrite /normal joing_subl join_subG normG.
rewrite -{2}(setIidPr (normal_sub nsHG)) setIAC.
rewrite group_modr ?setSI ?joing_subr //=; symmetry; apply/setIidPl.
rewrite -quotientSK ?subIset 1?normal_norm //.
by rewrite !coprime_norm_quotient_cent ?normsY //= norm_joinEr ?quotientMidl.
Qed.
(* Another special case of the strong coprime quotient lemma; not found in *)
(* textbooks, but nevertheless used implicitly throughout B & G, sometimes *)
(* justified by switching to external action. *)
Lemma quotient_TI_subcent K G H :
G \subset 'N(K) -> G \subset 'N(H) -> K :&: H = 1 ->
'C_K(G) / H = 'C_(K / H)(G / H).
Proof.
move=> nGK nGH tiKH.
have tiHR: H :&: [~: K, G] = 1.
by apply/trivgP; rewrite /= setIC -tiKH setSI ?commg_subl.
apply: strongest_coprime_quotient_cent; rewrite ?tiHR ?sub1G ?solvable1 //.
by rewrite cards1 coprime1n.
Qed.
(* This is B & G, Proposition 1.5(d): the more traditional form of the lemma *)
(* above, with the assumption H <| G weakened to H \subset G. The stronger *)
(* coprime and solvability assumptions are easier to satisfy in practice. *)
Proposition coprime_quotient_cent A G H :
H \subset G -> A \subset 'N(H) -> coprime #|G| #|A| -> solvable G ->
'C_G(A) / H = 'C_(G / H)(A / H).
Proof.
move=> sHG nHA coGA solG.
have sRG: H :&: [~: G, A] \subset G by rewrite subIset ?sHG.
by rewrite strongest_coprime_quotient_cent ?(coprimeSg sRG) 1?(solvableS sRG).
Qed.
(* This is B & G, Proposition 1.5(e). *)
Proposition coprime_comm_pcore A G K :
A \subset 'N(G) -> coprime #|G| #|A| -> solvable G ->
pi^'.-Hall(G) K -> K \subset 'C_G(A) ->
[~: G, A] \subset 'O_pi(G).
Proof.
move=> nGA coGA solG hallK cKA.
case: (coprime_Hall_exists nGA) => // H hallH nHA.
have sHG: H \subset G by case/andP: hallH.
have sKG: K \subset G by case/andP: hallK.
have coKH: coprime #|K| #|H|.
case/and3P: hallH=> _ piH _; case/and3P: hallK => _ pi'K _.
by rewrite coprime_sym (pnat_coprime piH pi'K).
have defG: G :=: K * H.
apply/eqP; rewrite eq_sym eqEcard coprime_cardMg //.
rewrite -{1}(mulGid G) mulgSS //= (card_Hall hallH) (card_Hall hallK).
by rewrite mulnC partnC.
have sGA_H: [~: G, A] \subset H.
rewrite gen_subG defG.
apply/subsetP=> _ /imset2P[_ a /imset2P[x y Kx Hy ->] Aa ->].
rewrite commMgJ (([~ x, a] =P 1) _) ?(conj1g, mul1g).
by rewrite groupMl ?groupV // memJ_norm ?(subsetP nHA).
by rewrite subsetI sKG in cKA; apply/commgP/(centsP cKA).
apply: pcore_max; last first.
by rewrite /(_ <| G) /= commg_norml commGC commg_subr nGA.
by case/and3P: hallH => _ piH _; apply: pgroupS piH.
Qed.
End InternalAction.
(* This is B & G, Proposition 1.5(b). *)
Proposition coprime_Hall_subset pi (gT : finGroupType) (A G X : {group gT}) :
A \subset 'N(G) -> coprime #|G| #|A| -> solvable G ->
X \subset G -> pi.-group X -> A \subset 'N(X) ->
exists H : {group gT}, [/\ pi.-Hall(G) H, A \subset 'N(H) & X \subset H].
Proof.
have [n] := ubnP #|G|.
elim: n => // n IHn in gT A G X * => /ltnSE-leGn nGA coGA solG sXG piX nXA.
have [G1 | ntG] := eqsVneq G 1.
case: (coprime_Hall_exists pi nGA) => // H hallH nHA.
by exists H; split; rewrite // (subset_trans sXG) // G1 sub1G.
have sG_AG: G \subset A <*> G by rewrite joing_subr.
have sA_AG: A \subset A <*> G by rewrite joing_subl.
have nG_AG: A <*> G \subset 'N(G) by rewrite join_subG nGA normG.
have nsG_AG: G <| A <*> G by apply/andP.
case: (solvable_norm_abelem solG nsG_AG) => // M [sMG nsMAG ntM].
have{nsMAG} [nMA nMG]: A \subset 'N(M) /\ G \subset 'N(M).
by apply/andP; rewrite -join_subG normal_norm.
have nMX: X \subset 'N(M) by apply: subset_trans nMG.
case/is_abelemP=> p pr_p; case/and3P=> pM cMM _.
have: #|G / M| < n by rewrite (leq_trans (ltn_quotient _ _)).
move/(IHn _ (A / M)%G _ (X / M)%G); rewrite !(quotient_norms, quotientS) //.
rewrite !(coprime_morph, quotient_sol, morphim_pgroup) //.
case=> //= Hq []; case/and3P=> sHGq piHq pi'Hq' nHAq sXHq.
case/inv_quotientS: (sHGq) => [|HM defHM sMHM sHMG]; first exact/andP.
have nMHM := subset_trans sHMG nMG.
have{sXHq} sXHM: X \subset HM by rewrite -(quotientSGK nMX) -?defHM.
have{pi'Hq' sHGq} pi'HM': pi^'.-nat #|G : HM|.
move: pi'Hq'; rewrite -!divgS // defHM !card_quotient //.
by rewrite -(divnMl (cardG_gt0 M)) !Lagrange.
have{nHAq} nHMA: A \subset 'N(HM).
by rewrite -(quotientSGK nMA) ?normsG ?quotient_normG -?defHM //; apply/andP.
case/orP: (orbN (p \in pi)) => pi_p.
exists HM; split=> //; apply/and3P; split; rewrite /pgroup //.
by rewrite -(Lagrange sMHM) pnatM -card_quotient // -defHM (pi_pnat pM).
case: (ltnP #|HM| #|G|) => [ltHG | leGHM {n IHn leGn}].
case: (IHn _ A HM X (leq_trans ltHG leGn)) => // [||H [hallH nHA sXH]].
- exact: coprimeSg coGA.
- exact: solvableS solG.
case/and3P: hallH => sHHM piH pi'H'.
have sHG: H \subset G by apply: subset_trans sHMG.
exists H; split=> //; apply/and3P; split=> //.
rewrite -divgS // -(Lagrange sHMG) -(Lagrange sHHM) -mulnA mulKn //.
by rewrite pnatM pi'H'.
have{leGHM nHMA sHMG sMHM sXHM pi'HM'} eqHMG: HM = G.
by apply/eqP; rewrite -val_eqE eqEcard sHMG.
have pi'M: pi^'.-group M by rewrite /pgroup (pi_pnat pM).
have{HM Hq nMHM defHM eqHMG piHq} hallM: pi^'.-Hall(G) M.
apply/and3P; split; rewrite // /pgroup pnatNK.
by rewrite -card_quotient // -eqHMG -defHM.
case: (coprime_Hall_exists pi nGA) => // H hallH nHA.
pose XM := (X <*> M)%G; pose Y := (H :&: XM)%G.
case/and3P: (hallH) => sHG piH _.
have sXXM: X \subset XM by rewrite joing_subl.
have co_pi_M (B : {group gT}): pi.-group B -> coprime #|B| #|M|.
by move=> piB; rewrite (pnat_coprime piB).
have hallX: pi.-Hall(XM) X.
rewrite /pHall piX sXXM -divgS //= norm_joinEl //.
by rewrite coprime_cardMg ?co_pi_M // mulKn.
have sXMG: XM \subset G by rewrite join_subG sXG.
have hallY: pi.-Hall(XM) Y.
have sYXM: Y \subset XM by rewrite subsetIr.
have piY: pi.-group Y by apply: pgroupS piH; apply: subsetIl.
rewrite /pHall sYXM piY -divgS // -(_ : Y * M = XM).
by rewrite coprime_cardMg ?co_pi_M // mulKn //.
rewrite /= setIC group_modr ?joing_subr //=; apply/setIidPl.
rewrite ((H * M =P G) _) // eqEcard mul_subG //= coprime_cardMg ?co_pi_M //.
by rewrite (card_Hall hallM) (card_Hall hallH) partnC.
have nXMA: A \subset 'N(XM) by rewrite normsY.
have:= coprime_Hall_trans nXMA _ _ hallX nXA hallY.
rewrite !(coprimeSg sXMG, solvableS sXMG, normsI) //.
case=> // x /setIP[XMx cAx] ->.
exists (H :^ x)%G; split; first by rewrite pHallJ ?(subsetP sXMG).
by rewrite norm_conj_cent.
by rewrite conjSg subsetIl.
Qed.
Section ExternalAction.
Variables (pi : nat_pred) (aT gT : finGroupType).
Variables (A : {group aT}) (G : {group gT}) (to : groupAction A G).
Section FullExtension.
Local Notation inA := (sdpair2 to).
Local Notation inG := (sdpair1 to).
Local Notation A' := (inA @* gval A).
Local Notation G' := (inG @* gval G).
Let injG : 'injm inG := injm_sdpair1 _.
Let injA : 'injm inA := injm_sdpair2 _.
Hypotheses (coGA : coprime #|G| #|A|) (solG : solvable G).
Lemma external_action_im_coprime : coprime #|G'| #|A'|.
Proof. by rewrite !card_injm. Qed.
Let coGA' := external_action_im_coprime.
Let solG' : solvable G' := morphim_sol _ solG.
Let nGA' := im_sdpair_norm to.
Lemma ext_coprime_Hall_exists :
exists2 H : {group gT}, pi.-Hall(G) H & [acts A, on H | to].
Proof.
have [H' hallH' nHA'] := coprime_Hall_exists pi nGA' coGA' solG'.
have sHG' := pHall_sub hallH'.
exists (inG @*^-1 H')%G => /=.
by rewrite -(morphim_invmE injG) -{1}(im_invm injG) morphim_pHall.
by rewrite actsEsd ?morphpreK // subsetIl.
Qed.
Lemma ext_coprime_Hall_trans (H1 H2 : {group gT}) :
pi.-Hall(G) H1 -> [acts A, on H1 | to] ->
pi.-Hall(G) H2 -> [acts A, on H2 | to] ->
exists2 x, x \in 'C_(G | to)(A) & H1 :=: H2 :^ x.
Proof.
move=> hallH1 nH1A hallH2 nH2A.
have sH1G := pHall_sub hallH1; have sH2G := pHall_sub hallH2.
rewrite !actsEsd // in nH1A nH2A.
have hallH1': pi.-Hall(G') (inG @* H1) by rewrite morphim_pHall.
have hallH2': pi.-Hall(G') (inG @* H2) by rewrite morphim_pHall.
have [x'] := coprime_Hall_trans nGA' coGA' solG' hallH1' nH1A hallH2' nH2A.
case/setIP=> /= Gx' cAx' /eqP defH1; pose x := invm injG x'.
have Gx: x \in G by rewrite -(im_invm injG) mem_morphim.
have def_x': x' = inG x by rewrite invmK.
exists x; first by rewrite inE Gx gacentEsd mem_morphpre /= -?def_x'.
apply/eqP; move: defH1; rewrite def_x' /= -morphimJ //=.
by rewrite !eqEsubset !injmSK // conj_subG.
Qed.
Lemma ext_norm_conj_cent (H : {group gT}) x :
H \subset G -> x \in 'C_(G | to)(A) ->
[acts A, on H :^ x | to] = [acts A, on H | to].
Proof.
move=> sHG /setIP[Gx].
rewrite gacentEsd !actsEsd ?conj_subG ?morphimJ // 2!inE Gx /=.
exact: norm_conj_cent.
Qed.
Lemma ext_coprime_Hall_subset (X : {group gT}) :
X \subset G -> pi.-group X -> [acts A, on X | to] ->
exists H : {group gT},
[/\ pi.-Hall(G) H, [acts A, on H | to] & X \subset H].
Proof.
move=> sXG piX; rewrite actsEsd // => nXA'.
case: (coprime_Hall_subset nGA' coGA' solG' _ (morphim_pgroup _ piX) nXA').
exact: morphimS.
move=> H' /= [piH' nHA' sXH']; have sHG' := pHall_sub piH'.
exists (inG @*^-1 H')%G; rewrite actsEsd ?subsetIl ?morphpreK // nHA'.
rewrite -sub_morphim_pre //= sXH'; split=> //.
by rewrite -(morphim_invmE injG) -{1}(im_invm injG) morphim_pHall.
Qed.
End FullExtension.
(* We only prove a weaker form of the coprime group action centraliser *)
(* lemma, because it is more convenient in practice to make G the range *)
(* of the action, whence G both contains H and is stable under A. *)
(* However we do restrict the coprime/solvable assumptions to H, and *)
(* we do not require that G normalize H. *)
Lemma ext_coprime_quotient_cent (H : {group gT}) :
H \subset G -> [acts A, on H | to] -> coprime #|H| #|A| -> solvable H ->
'C_(|to)(A) / H = 'C_(|to / H)(A).
Proof.
move=> sHG nHA coHA solH; pose N := 'N_G(H).
have nsHN: H <| N by rewrite normal_subnorm.
have [sHN nHn] := andP nsHN.
have sNG: N \subset G by apply: subsetIl.
have nNA: {acts A, on group N | to}.
split; rewrite // actsEsd // injm_subnorm ?injm_sdpair1 //=.
by rewrite normsI ?norms_norm ?im_sdpair_norm -?actsEsd.
rewrite -!(gacentIdom _ A) -quotientInorm -gacentIim setIAC.
rewrite -(gacent_actby nNA) gacentEsd -morphpreIim /= -/N.
have:= (injm_sdpair1 <[nNA]>, injm_sdpair2 <[nNA]>).
set inG := sdpair1 _; set inA := sdpair2 _ => [[injG injA]].
set G' := inG @* N; set A' := inA @* A; pose H' := inG @* H.
have defN: 'N(H | to) = A by apply/eqP; rewrite eqEsubset subsetIl.
have def_Dq: qact_dom to H = A by rewrite qact_domE.
have sAq: A \subset qact_dom to H by rewrite def_Dq.
rewrite {2}def_Dq -(gacent_ract _ sAq); set to_q := (_ \ _)%gact.
have:= And3 (sdprod_sdpair to_q) (injm_sdpair1 to_q) (injm_sdpair2 to_q).
rewrite gacentEsd; set inAq := sdpair2 _; set inGq := sdpair1 _ => /=.
set Gq := inGq @* _; set Aq := inAq @* _ => [[q_d iGq iAq]].
have nH': 'N(H') = setT.
apply/eqP; rewrite -subTset -im_sdpair mulG_subG morphim_norms //=.
by rewrite -actsEsd // acts_actby subxx /= (setIidPr sHN).
have: 'dom (coset H' \o inA \o invm iAq) = Aq.
by rewrite ['dom _]morphpre_invm /= nH' morphpreT.
case/domP=> /= qA [def_qA ker_qA _ im_qA].
have{coHA} coHA': coprime #|H'| #|A'| by rewrite !card_injm.
have{ker_qA} injAq: 'injm qA.
rewrite {}ker_qA !ker_comp ker_coset morphpre_invm -morphpreIim /= setIC.
by rewrite coprime_TIg // -kerE (trivgP injA) morphim1.
have{im_qA} im_Aq : qA @* Aq = A' / H'.
by rewrite {}im_qA !morphim_comp im_invm.
have: 'dom (quotm (sdpair1_morphism <[nNA]>) nsHN \o invm iGq) = Gq.
by rewrite ['dom _]morphpre_invm /= quotientInorm.
case/domP=> /= qG [def_qG ker_qG _ im_qG].
have{ker_qG} injGq: 'injm qG.
rewrite {}ker_qG ker_comp ker_quotm morphpre_invm (trivgP injG).
by rewrite quotient1 morphim1.
have im_Gq: qG @* Gq = G' / H'.
rewrite {}im_qG morphim_comp im_invm morphim_quotm //= -/inG -/H'.
by rewrite -morphimIdom setIAC setIid.
have{def_qA def_qG} q_J : {in Gq & Aq, morph_act 'J 'J qG qA}.
move=> x' a'; case/morphimP=> Hx; case/morphimP=> x nHx Gx -> GHx ->{Hx x'}.
case/morphimP=> a _ Aa ->{a'} /=; rewrite -/inAq -/inGq.
rewrite !{}def_qG {}def_qA /= !invmE // -sdpair_act //= -/inG -/inA.
have Nx: x \in N by rewrite inE Gx.
have Nxa: to x a \in N by case: (nNA); move/acts_act->.
have [Gxa nHxa] := setIP Nxa.
rewrite invmE qactE ?quotmE ?mem_morphim ?def_Dq //=.
by rewrite -morphJ /= ?nH' ?inE // -sdpair_act //= actbyE.
pose q := sdprodm q_d q_J.
have{injAq injGq} injq: 'injm q.
rewrite injm_sdprodm injAq injGq /= {}im_Aq {}im_Gq -/Aq .
by rewrite -quotientGI ?im_sdpair_TI ?morphimS //= quotient1.
rewrite -[inGq @*^-1 _]morphpreIim -/Gq.
have sC'G: inG @*^-1 'C_G'(A') \subset G by rewrite !subIset ?subxx.
rewrite -[_ / _](injmK iGq) ?quotientS //= -/inGq; congr (_ @*^-1 _).
apply: (injm_morphim_inj injq); rewrite 1?injm_subcent ?subsetT //= -/q.
rewrite 2?morphim_sdprodml ?morphimS //= im_Gq.
rewrite morphim_sdprodmr ?morphimS //= im_Aq.
rewrite {}im_qG morphim_comp morphim_invm ?morphimS //.
rewrite morphim_quotm morphpreK ?subsetIl //= -/H'.
rewrite coprime_norm_quotient_cent ?im_sdpair_norm ?nH' ?subsetT //=.
exact: morphim_sol.
Qed.
End ExternalAction.
Section SylowSolvableAct.
Variables (gT : finGroupType) (p : nat).
Implicit Types A B G X : {group gT}.
Lemma sol_coprime_Sylow_exists A G :
solvable A -> A \subset 'N(G) -> coprime #|G| #|A| ->
exists2 P : {group gT}, p.-Sylow(G) P & A \subset 'N(P).
Proof.
move=> solA nGA coGA; pose AG := A <*> G.
have nsG_AG: G <| AG by rewrite /normal joing_subr join_subG nGA normG.
have [sG_AG nG_AG]:= andP nsG_AG.
have [P sylP] := Sylow_exists p G; pose N := 'N_AG(P); pose NG := G :&: N.
have nGN: N \subset 'N(G) by rewrite subIset ?nG_AG.
have sNG_G: NG \subset G := subsetIl G N.
have nsNG_N: NG <| N by rewrite /normal subsetIr normsI ?normG.
have defAG: G * N = AG := Frattini_arg nsG_AG sylP.
have oA : #|A| = #|N| %/ #|NG|.
rewrite /NG setIC divgI -card_quotient // -quotientMidl defAG.
rewrite card_quotient -?divgS //= norm_joinEl //.
by rewrite coprime_cardMg 1?coprime_sym // mulnK.
have: [splits N, over NG].
rewrite SchurZassenhaus_split // /Hall -divgS subsetIr //.
by rewrite -oA (coprimeSg sNG_G).
case/splitsP=> B; case/complP=> tNG_B defN.
have [nPB]: B \subset 'N(P) /\ B \subset AG.
by apply/andP; rewrite andbC -subsetI -/N -defN mulG_subr.
case/SchurZassenhaus_trans_actsol => // [|x Gx defB].
by rewrite oA -defN TI_cardMg // mulKn.
exists (P :^ x^-1)%G; first by rewrite pHallJ ?groupV.
by rewrite normJ -sub_conjg -defB.
Qed.
Lemma sol_coprime_Sylow_trans A G :
solvable A -> A \subset 'N(G) -> coprime #|G| #|A| ->
[transitive 'C_G(A), on [set P in 'Syl_p(G) | A \subset 'N(P)] | 'JG].
Proof.
move=> solA nGA coGA; pose AG := A <*> G; set FpA := finset _.
have nG_AG: AG \subset 'N(G) by rewrite join_subG nGA normG.
have [P sylP nPA] := sol_coprime_Sylow_exists solA nGA coGA.
pose N := 'N_AG(P); have sAN: A \subset N by rewrite subsetI joing_subl.
have trNPA: A :^: AG ::&: N = A :^: N.
pose NG := 'N_G(P); have sNG_G : NG \subset G := subsetIl _ _.
have nNGA: A \subset 'N(NG) by rewrite normsI ?norms_norm.
apply/setP=> Ax; apply/setIdP/imsetP=> [[]|[x Nx ->{Ax}]]; last first.
by rewrite conj_subG //; case/setIP: Nx => AGx; rewrite imset_f.
have ->: N = A <*> NG by rewrite /N /AG !norm_joinEl // -group_modl.
have coNG_A := coprimeSg sNG_G coGA; case/imsetP=> x AGx ->{Ax}.
case/SchurZassenhaus_trans_actsol; rewrite ?cardJg // => y Ny /= ->.
by exists y; rewrite // mem_gen 1?inE ?Ny ?orbT.
have{trNPA}: [transitive 'N_AG(A), on FpA | 'JG].
have ->: FpA = 'Fix_('Syl_p(G) | 'JG)(A).
by apply/setP=> Q; rewrite 4!inE afixJG.
have SylP : P \in 'Syl_p(G) by rewrite inE.
apply/(trans_subnorm_fixP _ SylP); rewrite ?astab1JG //.
rewrite (atrans_supgroup _ (Syl_trans _ _)) ?joing_subr //= -/AG.
by apply/actsP=> x /= AGx Q /=; rewrite !inE -{1}(normsP nG_AG x) ?pHallJ2.
rewrite {1}/AG norm_joinEl // -group_modl ?normG ?coprime_norm_cent //=.
rewrite -cent_joinEr ?subsetIr // => trC_FpA.
have FpA_P: P \in FpA by rewrite !inE sylP.
apply/(subgroup_transitiveP FpA_P _ trC_FpA); rewrite ?joing_subr //=.
rewrite astab1JG cent_joinEr ?subsetIr // -group_modl // -mulgA.
by congr (_ * _); rewrite mulSGid ?subsetIl.
Qed.
Lemma sol_coprime_Sylow_subset A G X :
A \subset 'N(G) -> coprime #|G| #|A| -> solvable A ->
X \subset G -> p.-group X -> A \subset 'N(X) ->
exists P : {group gT}, [/\ p.-Sylow(G) P, A \subset 'N(P) & X \subset P].
Proof.
move=> nGA coGA solA sXG pX nXA.
pose nAp (Q : {group gT}) := [&& p.-group Q, Q \subset G & A \subset 'N(Q)].
have: nAp X by apply/and3P.
case/maxgroup_exists=> R; case/maxgroupP; case/and3P=> pR sRG nRA maxR sXR.
have [P sylP sRP]:= Sylow_superset sRG pR.
suffices defP: P :=: R by exists P; rewrite sylP defP.
case/and3P: sylP => sPG pP _; apply: (nilpotent_sub_norm (pgroup_nil pP)) => //.
pose N := 'N_G(R); have{sPG} sPN_N: 'N_P(R) \subset N by apply: setSI.
apply: norm_sub_max_pgroup (pgroupS (subsetIl _ _) pP) sPN_N (subsetIr _ _).
have nNA: A \subset 'N(N) by rewrite normsI ?norms_norm.
have coNA: coprime #|N| #|A| by apply: coprimeSg coGA; rewrite subsetIl.
have{solA coNA} [Q sylQ nQA] := sol_coprime_Sylow_exists solA nNA coNA.
suffices defQ: Q :=: R by rewrite max_pgroup_Sylow -{2}defQ.
apply: maxR; first by apply/and3P; case/and3P: sylQ; rewrite subsetI; case/andP.
by apply: normal_sub_max_pgroup (Hall_max sylQ) pR _; rewrite normal_subnorm.
Qed.
End SylowSolvableAct.
|
Degrees.lean
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
/-!
# Degrees of polynomials
This file establishes many results about the degree of a multivariate polynomial.
The *degree set* of a polynomial $P \in R[X]$ is a `Multiset` containing, for each $x$ in the
variable set, $n$ copies of $x$, where $n$ is the maximum number of copies of $x$ appearing in a
monomial of $P$.
## Main declarations
* `MvPolynomial.degrees p` : the multiset of variables representing the union of the multisets
corresponding to each non-zero monomial in `p`.
For example if `7 ≠ 0` in `R` and `p = x²y+7y³` then `degrees p = {x, x, y, y, y}`
* `MvPolynomial.degreeOf n p : ℕ` : the total degree of `p` with respect to the variable `n`.
For example if `p = x⁴y+yz` then `degreeOf y p = 1`.
* `MvPolynomial.totalDegree p : ℕ` :
the max of the sizes of the multisets `s` whose monomials `X^s` occur in `p`.
For example if `p = x⁴y+yz` then `totalDegree p = 5`.
## Notation
As in other polynomial files, we typically use the notation:
+ `σ τ : Type*` (indexing the variables)
+ `R : Type*` `[CommSemiring R]` (the coefficients)
+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.
This will give rise to a monomial in `MvPolynomial σ R` which mathematicians might call `X^s`
+ `r : R`
+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians
+ `p : MvPolynomial σ R`
-/
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
variable {σ τ : Type*} {r : R} {e : ℕ} {n m : σ} {s : σ →₀ ℕ}
section CommSemiring
variable [CommSemiring R] {p q : MvPolynomial σ R}
section Degrees
/-! ### `degrees` -/
/-- The maximal degrees of each variable in a multi-variable polynomial, expressed as a multiset.
(For example, `degrees (x^2 * y + y^3)` would be `{x, x, y, y, y}`.)
-/
def degrees (p : MvPolynomial σ R) : Multiset σ :=
letI := Classical.decEq σ
p.support.sup fun s : σ →₀ ℕ => toMultiset s
theorem degrees_def [DecidableEq σ] (p : MvPolynomial σ R) :
p.degrees = p.support.sup fun s : σ →₀ ℕ => Finsupp.toMultiset s := by rw [degrees]; convert rfl
theorem degrees_monomial (s : σ →₀ ℕ) (a : R) : degrees (monomial s a) ≤ toMultiset s := by
classical
refine (supDegree_single s a).trans_le ?_
split_ifs
exacts [bot_le, le_rfl]
theorem degrees_monomial_eq (s : σ →₀ ℕ) (a : R) (ha : a ≠ 0) :
degrees (monomial s a) = toMultiset s := by
classical
exact (supDegree_single s a).trans (if_neg ha)
theorem degrees_C (a : R) : degrees (C a : MvPolynomial σ R) = 0 :=
Multiset.le_zero.1 <| degrees_monomial _ _
theorem degrees_X' (n : σ) : degrees (X n : MvPolynomial σ R) ≤ {n} :=
le_trans (degrees_monomial _ _) <| le_of_eq <| toMultiset_single _ _
@[simp]
theorem degrees_X [Nontrivial R] (n : σ) : degrees (X n : MvPolynomial σ R) = {n} :=
(degrees_monomial_eq _ (1 : R) one_ne_zero).trans (toMultiset_single _ _)
@[simp]
theorem degrees_zero : degrees (0 : MvPolynomial σ R) = 0 := by
rw [← C_0]
exact degrees_C 0
@[simp]
theorem degrees_one : degrees (1 : MvPolynomial σ R) = 0 :=
degrees_C 1
theorem degrees_add_le [DecidableEq σ] {p q : MvPolynomial σ R} :
(p + q).degrees ≤ p.degrees ⊔ q.degrees := by
simp_rw [degrees_def]; exact supDegree_add_le
theorem degrees_sum_le {ι : Type*} [DecidableEq σ] (s : Finset ι) (f : ι → MvPolynomial σ R) :
(∑ i ∈ s, f i).degrees ≤ s.sup fun i => (f i).degrees := by
simp_rw [degrees_def]; exact supDegree_sum_le
theorem degrees_mul_le {p q : MvPolynomial σ R} : (p * q).degrees ≤ p.degrees + q.degrees := by
classical
simp_rw [degrees_def]
exact supDegree_mul_le (map_add _)
theorem degrees_prod_le {ι : Type*} {s : Finset ι} {f : ι → MvPolynomial σ R} :
(∏ i ∈ s, f i).degrees ≤ ∑ i ∈ s, (f i).degrees := by
classical exact supDegree_prod_le (map_zero _) (map_add _)
theorem degrees_pow_le {p : MvPolynomial σ R} {n : ℕ} : (p ^ n).degrees ≤ n • p.degrees := by
simpa using degrees_prod_le (s := .range n) (f := fun _ ↦ p)
theorem mem_degrees {p : MvPolynomial σ R} {i : σ} :
i ∈ p.degrees ↔ ∃ d, p.coeff d ≠ 0 ∧ i ∈ d.support := by
classical
simp only [degrees_def, Multiset.mem_sup, ← mem_support_iff, Finsupp.mem_toMultiset]
theorem le_degrees_add_left (h : Disjoint p.degrees q.degrees) : p.degrees ≤ (p + q).degrees := by
classical
apply Finset.sup_le
intro d hd
rw [Multiset.disjoint_iff_ne] at h
obtain rfl | h0 := eq_or_ne d 0
· rw [toMultiset_zero]; apply Multiset.zero_le
· refine Finset.le_sup_of_le (b := d) ?_ le_rfl
rw [mem_support_iff, coeff_add]
suffices q.coeff d = 0 by rwa [this, add_zero, coeff, ← Finsupp.mem_support_iff]
rw [Ne, ← Finsupp.support_eq_empty, ← Ne, ← Finset.nonempty_iff_ne_empty] at h0
obtain ⟨j, hj⟩ := h0
contrapose! h
rw [mem_support_iff] at hd
refine ⟨j, ?_, j, ?_, rfl⟩
all_goals rw [mem_degrees]; refine ⟨d, ?_, hj⟩; assumption
lemma le_degrees_add_right (h : Disjoint p.degrees q.degrees) : q.degrees ≤ (p + q).degrees := by
simpa [add_comm] using le_degrees_add_left h.symm
theorem degrees_add_of_disjoint [DecidableEq σ] (h : Disjoint p.degrees q.degrees) :
(p + q).degrees = p.degrees ∪ q.degrees :=
degrees_add_le.antisymm <| Multiset.union_le (le_degrees_add_left h) (le_degrees_add_right h)
lemma degrees_map_le [CommSemiring S] {f : R →+* S} : (map f p).degrees ≤ p.degrees := by
classical exact Finset.sup_mono <| support_map_subset ..
theorem degrees_rename (f : σ → τ) (φ : MvPolynomial σ R) :
(rename f φ).degrees ⊆ φ.degrees.map f := by
classical
intro i
rw [mem_degrees, Multiset.mem_map]
rintro ⟨d, hd, hi⟩
obtain ⟨x, rfl, hx⟩ := coeff_rename_ne_zero _ _ _ hd
simp only [Finsupp.mapDomain, Finsupp.mem_support_iff] at hi
rw [sum_apply, Finsupp.sum] at hi
contrapose! hi
rw [Finset.sum_eq_zero]
intro j hj
simp only [mem_degrees] at hi
specialize hi j ⟨x, hx, hj⟩
rw [Finsupp.single_apply, if_neg hi]
theorem degrees_map_of_injective [CommSemiring S] (p : MvPolynomial σ R) {f : R →+* S}
(hf : Injective f) : (map f p).degrees = p.degrees := by
simp only [degrees, MvPolynomial.support_map_of_injective _ hf]
theorem degrees_rename_of_injective {p : MvPolynomial σ R} {f : σ → τ} (h : Function.Injective f) :
degrees (rename f p) = (degrees p).map f := by
classical
simp only [degrees, Multiset.map_finset_sup p.support Finsupp.toMultiset f h,
support_rename_of_injective h, Finset.sup_image]
refine Finset.sup_congr rfl fun x _ => ?_
exact (Finsupp.toMultiset_map _ _).symm
end Degrees
section DegreeOf
/-! ### `degreeOf` -/
/-- `degreeOf n p` gives the highest power of X_n that appears in `p` -/
def degreeOf (n : σ) (p : MvPolynomial σ R) : ℕ :=
letI := Classical.decEq σ
p.degrees.count n
theorem degreeOf_def [DecidableEq σ] (n : σ) (p : MvPolynomial σ R) :
p.degreeOf n = p.degrees.count n := by rw [degreeOf]; convert rfl
theorem degreeOf_eq_sup (n : σ) (f : MvPolynomial σ R) :
degreeOf n f = f.support.sup fun m => m n := by
classical
rw [degreeOf_def, degrees, Multiset.count_finset_sup]
congr
ext
simp only [count_toMultiset]
theorem degreeOf_lt_iff {n : σ} {f : MvPolynomial σ R} {d : ℕ} (h : 0 < d) :
degreeOf n f < d ↔ ∀ m : σ →₀ ℕ, m ∈ f.support → m n < d := by
rwa [degreeOf_eq_sup, Finset.sup_lt_iff]
lemma degreeOf_le_iff {n : σ} {f : MvPolynomial σ R} {d : ℕ} :
degreeOf n f ≤ d ↔ ∀ m ∈ support f, m n ≤ d := by
rw [degreeOf_eq_sup, Finset.sup_le_iff]
@[simp]
theorem degreeOf_zero (n : σ) : degreeOf n (0 : MvPolynomial σ R) = 0 := by
classical simp only [degreeOf_def, degrees_zero, Multiset.count_zero]
@[simp]
theorem degreeOf_C (a : R) (x : σ) : degreeOf x (C a : MvPolynomial σ R) = 0 := by
classical simp [degreeOf_def, degrees_C]
theorem degreeOf_X [DecidableEq σ] (i j : σ) [Nontrivial R] :
degreeOf i (X j : MvPolynomial σ R) = if i = j then 1 else 0 := by
classical
by_cases c : i = j
· simp only [c, if_true, degreeOf_def, degrees_X, Multiset.count_singleton]
simp [c, degreeOf_def, degrees_X]
theorem degreeOf_add_le (n : σ) (f g : MvPolynomial σ R) :
degreeOf n (f + g) ≤ max (degreeOf n f) (degreeOf n g) := by
simp_rw [degreeOf_eq_sup]; exact supDegree_add_le
theorem monomial_le_degreeOf (i : σ) {f : MvPolynomial σ R} {m : σ →₀ ℕ} (h_m : m ∈ f.support) :
m i ≤ degreeOf i f := by
rw [degreeOf_eq_sup i]
apply Finset.le_sup h_m
lemma degreeOf_monomial_eq (s : σ →₀ ℕ) (i : σ) {a : R} (ha : a ≠ 0) :
(monomial s a).degreeOf i = s i := by
classical rw [degreeOf_def, degrees_monomial_eq _ _ ha, Finsupp.count_toMultiset]
-- TODO we can prove equality with `NoZeroDivisors R`
theorem degreeOf_mul_le (i : σ) (f g : MvPolynomial σ R) :
degreeOf i (f * g) ≤ degreeOf i f + degreeOf i g := by
classical
simp only [degreeOf]
convert Multiset.count_le_of_le i degrees_mul_le
rw [Multiset.count_add]
theorem degreeOf_sum_le {ι : Type*} (i : σ) (s : Finset ι) (f : ι → MvPolynomial σ R) :
degreeOf i (∑ j ∈ s, f j) ≤ s.sup fun j => degreeOf i (f j) := by
simp_rw [degreeOf_eq_sup]
exact supDegree_sum_le
-- TODO we can prove equality with `NoZeroDivisors R`
theorem degreeOf_prod_le {ι : Type*} (i : σ) (s : Finset ι) (f : ι → MvPolynomial σ R) :
degreeOf i (∏ j ∈ s, f j) ≤ ∑ j ∈ s, (f j).degreeOf i := by
simp_rw [degreeOf_eq_sup]
exact supDegree_prod_le (by simp only [coe_zero, Pi.zero_apply]) (by simp)
-- TODO we can prove equality with `NoZeroDivisors R`
theorem degreeOf_pow_le (i : σ) (p : MvPolynomial σ R) (n : ℕ) :
degreeOf i (p ^ n) ≤ n * degreeOf i p := by
simpa using degreeOf_prod_le i (Finset.range n) (fun _ => p)
theorem degreeOf_mul_X_of_ne {i j : σ} (f : MvPolynomial σ R) (h : i ≠ j) :
degreeOf i (f * X j) = degreeOf i f := by
classical
simp only [degreeOf_eq_sup i, support_mul_X, Finset.sup_map]
congr
ext
simp only [Finsupp.single, add_eq_left, addRightEmbedding_apply, coe_mk,
Pi.add_apply, comp_apply, Finsupp.coe_add, Pi.single_eq_of_ne h]
theorem degreeOf_mul_X_self (j : σ) (f : MvPolynomial σ R) :
degreeOf j (f * X j) ≤ degreeOf j f + 1 := by
classical
simp only [degreeOf]
apply (Multiset.count_le_of_le j degrees_mul_le).trans
simp only [Multiset.count_add, add_le_add_iff_left]
convert Multiset.count_le_of_le j <| degrees_X' j
rw [Multiset.count_singleton_self]
theorem degreeOf_mul_X_eq_degreeOf_add_one_iff (j : σ) (f : MvPolynomial σ R) :
degreeOf j (f * X j) = degreeOf j f + 1 ↔ f ≠ 0 := by
refine ⟨fun h => by by_contra ha; simp [ha] at h, fun h => ?_⟩
apply Nat.le_antisymm (degreeOf_mul_X_self j f)
have : (f.support.sup fun m ↦ m j) + 1 = (f.support.sup fun m ↦ (m j + 1)) :=
Finset.comp_sup_eq_sup_comp_of_nonempty @Nat.succ_le_succ (support_nonempty.mpr h)
simp only [degreeOf_eq_sup, support_mul_X, this]
apply Finset.sup_le
intro x hx
simp only [Finset.sup_map, bot_eq_zero', add_pos_iff, zero_lt_one, or_true, Finset.le_sup_iff]
use x
simpa using mem_support_iff.mp hx
theorem degreeOf_C_mul_le (p : MvPolynomial σ R) (i : σ) (c : R) :
(C c * p).degreeOf i ≤ p.degreeOf i := by
unfold degreeOf
convert Multiset.count_le_of_le i degrees_mul_le
simp only [degrees_C, zero_add]
theorem degreeOf_mul_C_le (p : MvPolynomial σ R) (i : σ) (c : R) :
(p * C c).degreeOf i ≤ p.degreeOf i := by
unfold degreeOf
convert Multiset.count_le_of_le i degrees_mul_le
simp only [degrees_C, add_zero]
theorem degreeOf_rename_of_injective {p : MvPolynomial σ R} {f : σ → τ} (h : Function.Injective f)
(i : σ) : degreeOf (f i) (rename f p) = degreeOf i p := by
classical
simp only [degreeOf, degrees_rename_of_injective h, Multiset.count_map_eq_count' f p.degrees h]
end DegreeOf
section TotalDegree
/-! ### `totalDegree` -/
/-- `totalDegree p` gives the maximum |s| over the monomials X^s in `p` -/
def totalDegree (p : MvPolynomial σ R) : ℕ :=
p.support.sup fun s => s.sum fun _ e => e
theorem totalDegree_eq (p : MvPolynomial σ R) :
p.totalDegree = p.support.sup fun m => Multiset.card (toMultiset m) := by
rw [totalDegree]
congr; funext m
exact (Finsupp.card_toMultiset _).symm
theorem le_totalDegree {p : MvPolynomial σ R} {s : σ →₀ ℕ} (h : s ∈ p.support) :
(s.sum fun _ e => e) ≤ totalDegree p :=
Finset.le_sup (α := ℕ) (f := fun s => sum s fun _ e => e) h
theorem totalDegree_le_degrees_card (p : MvPolynomial σ R) :
p.totalDegree ≤ Multiset.card p.degrees := by
classical
rw [totalDegree_eq]
exact Finset.sup_le fun s hs => Multiset.card_le_card <| Finset.le_sup hs
theorem totalDegree_le_of_support_subset (h : p.support ⊆ q.support) :
totalDegree p ≤ totalDegree q :=
Finset.sup_mono h
@[simp]
theorem totalDegree_C (a : R) : (C a : MvPolynomial σ R).totalDegree = 0 :=
(supDegree_single 0 a).trans <| by rw [sum_zero_index, bot_eq_zero', ite_self]
@[simp]
theorem totalDegree_zero : (0 : MvPolynomial σ R).totalDegree = 0 := by
rw [← C_0]; exact totalDegree_C (0 : R)
@[simp]
theorem totalDegree_one : (1 : MvPolynomial σ R).totalDegree = 0 :=
totalDegree_C (1 : R)
@[simp]
theorem totalDegree_X {R} [CommSemiring R] [Nontrivial R] (s : σ) :
(X s : MvPolynomial σ R).totalDegree = 1 := by
rw [totalDegree, support_X]
simp only [Finset.sup, Finsupp.sum_single_index, Finset.fold_singleton, sup_bot_eq]
theorem totalDegree_add (a b : MvPolynomial σ R) :
(a + b).totalDegree ≤ max a.totalDegree b.totalDegree :=
sup_support_add_le _ _ _
theorem totalDegree_add_eq_left_of_totalDegree_lt {p q : MvPolynomial σ R}
(h : q.totalDegree < p.totalDegree) : (p + q).totalDegree = p.totalDegree := by
classical
apply le_antisymm
· rw [← max_eq_left_of_lt h]
exact totalDegree_add p q
by_cases hp : p = 0
· simp [hp]
obtain ⟨b, hb₁, hb₂⟩ :=
p.support.exists_mem_eq_sup (Finsupp.support_nonempty_iff.mpr hp) fun m : σ →₀ ℕ =>
Multiset.card (toMultiset m)
have hb : b ∉ q.support := by
contrapose! h
rw [totalDegree_eq p, hb₂, totalDegree_eq]
apply Finset.le_sup h
have hbb : b ∈ (p + q).support := by
apply support_sdiff_support_subset_support_add
rw [Finset.mem_sdiff]
exact ⟨hb₁, hb⟩
rw [totalDegree_eq, hb₂, totalDegree_eq]
exact Finset.le_sup (f := fun m => Multiset.card (Finsupp.toMultiset m)) hbb
theorem totalDegree_add_eq_right_of_totalDegree_lt {p q : MvPolynomial σ R}
(h : q.totalDegree < p.totalDegree) : (q + p).totalDegree = p.totalDegree := by
rw [add_comm, totalDegree_add_eq_left_of_totalDegree_lt h]
theorem totalDegree_mul (a b : MvPolynomial σ R) :
(a * b).totalDegree ≤ a.totalDegree + b.totalDegree :=
sup_support_mul_le (by exact (Finsupp.sum_add_index' (fun _ => rfl) (fun _ _ _ => rfl)).le) _ _
theorem totalDegree_smul_le [CommSemiring S] [DistribMulAction R S] (a : R) (f : MvPolynomial σ S) :
(a • f).totalDegree ≤ f.totalDegree :=
Finset.sup_mono support_smul
theorem totalDegree_pow (a : MvPolynomial σ R) (n : ℕ) :
(a ^ n).totalDegree ≤ n * a.totalDegree := by
rw [Finset.pow_eq_prod_const, ← Nat.nsmul_eq_mul, Finset.nsmul_eq_sum_const]
refine supDegree_prod_le rfl (fun _ _ => ?_)
exact Finsupp.sum_add_index' (fun _ => rfl) (fun _ _ _ => rfl)
@[simp]
theorem totalDegree_monomial (s : σ →₀ ℕ) {c : R} (hc : c ≠ 0) :
(monomial s c : MvPolynomial σ R).totalDegree = s.sum fun _ e => e := by
classical simp [totalDegree, support_monomial, if_neg hc]
theorem totalDegree_monomial_le (s : σ →₀ ℕ) (c : R) :
(monomial s c).totalDegree ≤ s.sum fun _ ↦ id := by
if hc : c = 0 then
simp only [hc, map_zero, totalDegree_zero, zero_le]
else
rw [totalDegree_monomial _ hc, Function.id_def]
@[simp]
theorem totalDegree_X_pow [Nontrivial R] (s : σ) (n : ℕ) :
(X s ^ n : MvPolynomial σ R).totalDegree = n := by simp [X_pow_eq_monomial, one_ne_zero]
theorem totalDegree_list_prod :
∀ s : List (MvPolynomial σ R), s.prod.totalDegree ≤ (s.map MvPolynomial.totalDegree).sum
| [] => by rw [List.prod_nil, totalDegree_one, List.map_nil, List.sum_nil]
| p::ps => by
rw [List.prod_cons, List.map, List.sum_cons]
exact le_trans (totalDegree_mul _ _) (add_le_add_left (totalDegree_list_prod ps) _)
theorem totalDegree_multiset_prod (s : Multiset (MvPolynomial σ R)) :
s.prod.totalDegree ≤ (s.map MvPolynomial.totalDegree).sum := by
refine Quotient.inductionOn s fun l => ?_
rw [Multiset.quot_mk_to_coe, Multiset.prod_coe, Multiset.map_coe, Multiset.sum_coe]
exact totalDegree_list_prod l
theorem totalDegree_finset_prod {ι : Type*} (s : Finset ι) (f : ι → MvPolynomial σ R) :
(s.prod f).totalDegree ≤ ∑ i ∈ s, (f i).totalDegree := by
refine le_trans (totalDegree_multiset_prod _) ?_
simp only [Multiset.map_map, comp_apply, Finset.sum_map_val, le_refl]
theorem totalDegree_finset_sum {ι : Type*} (s : Finset ι) (f : ι → MvPolynomial σ R) :
(s.sum f).totalDegree ≤ Finset.sup s fun i => (f i).totalDegree := by
induction' s using Finset.cons_induction with a s has hind
· exact zero_le _
· rw [Finset.sum_cons, Finset.sup_cons]
exact (MvPolynomial.totalDegree_add _ _).trans (max_le_max le_rfl hind)
lemma totalDegree_finsetSum_le {ι : Type*} {s : Finset ι} {f : ι → MvPolynomial σ R} {d : ℕ}
(hf : ∀ i ∈ s, (f i).totalDegree ≤ d) : (s.sum f).totalDegree ≤ d :=
(totalDegree_finset_sum ..).trans <| Finset.sup_le hf
lemma degreeOf_le_totalDegree (f : MvPolynomial σ R) (i : σ) : f.degreeOf i ≤ f.totalDegree :=
degreeOf_le_iff.mpr fun d hd ↦ (eq_or_ne (d i) 0).elim (by omega) fun h ↦
(Finset.single_le_sum (by omega) <| Finsupp.mem_support_iff.mpr h).trans
(le_totalDegree hd)
theorem exists_degree_lt [Fintype σ] (f : MvPolynomial σ R) (n : ℕ)
(h : f.totalDegree < n * Fintype.card σ) {d : σ →₀ ℕ} (hd : d ∈ f.support) : ∃ i, d i < n := by
contrapose! h
calc
n * Fintype.card σ = ∑ _s : σ, n := by
rw [Finset.sum_const, Nat.nsmul_eq_mul, mul_comm, Finset.card_univ]
_ ≤ ∑ s, d s := Finset.sum_le_sum fun s _ => h s
_ ≤ d.sum fun _ e => e := by
rw [Finsupp.sum_fintype]
intros
rfl
_ ≤ f.totalDegree := le_totalDegree hd
theorem coeff_eq_zero_of_totalDegree_lt {f : MvPolynomial σ R} {d : σ →₀ ℕ}
(h : f.totalDegree < ∑ i ∈ d.support, d i) : coeff d f = 0 := by
classical
rw [totalDegree, Finset.sup_lt_iff] at h
· specialize h d
rw [mem_support_iff] at h
refine not_not.mp (mt h ?_)
exact lt_irrefl _
· exact lt_of_le_of_lt (Nat.zero_le _) h
theorem totalDegree_eq_zero_iff_eq_C {p : MvPolynomial σ R} :
p.totalDegree = 0 ↔ p = C (p.coeff 0) := by
constructor <;> intro h
· ext m; classical rw [coeff_C]; split_ifs with hm; · rw [← hm]
apply coeff_eq_zero_of_totalDegree_lt; rw [h]
exact Finset.sum_pos (fun i hi ↦ Nat.pos_of_ne_zero <| Finsupp.mem_support_iff.mp hi)
(Finsupp.support_nonempty_iff.mpr <| Ne.symm hm)
· rw [h, totalDegree_C]
theorem totalDegree_rename_le (f : σ → τ) (p : MvPolynomial σ R) :
(rename f p).totalDegree ≤ p.totalDegree :=
Finset.sup_le fun b => by
classical
intro h
rw [rename_eq] at h
have h' := Finsupp.mapDomain_support h
rw [Finset.mem_image] at h'
rcases h' with ⟨s, hs, rfl⟩
exact (sum_mapDomain_index (fun _ => rfl) (fun _ _ _ => rfl)).trans_le (le_totalDegree hs)
lemma totalDegree_renameEquiv (f : σ ≃ τ) (p : MvPolynomial σ R) :
(renameEquiv R f p).totalDegree = p.totalDegree :=
(totalDegree_rename_le f p).antisymm (le_trans (by simp) (totalDegree_rename_le f.symm _))
end TotalDegree
section degreesLE
variable {s t : Multiset σ}
variable (R σ s) in
/-- The submodule of multivariate polynomials of degrees bounded by a monomial `s`. -/
def degreesLE : Submodule R (MvPolynomial σ R) where
carrier := {p | p.degrees ≤ s}
add_mem' {a b} ha hb := by classical exact degrees_add_le.trans (sup_le ha hb)
zero_mem' := by simp
smul_mem' c {x} hx := by
dsimp
rw [Algebra.smul_def]
refine degrees_mul_le.trans ?_
simpa [degrees_C] using hx
@[simp] lemma mem_degreesLE : p ∈ degreesLE R σ s ↔ p.degrees ≤ s := Iff.rfl
variable (s t) in
lemma degreesLE_add : degreesLE R σ (s + t) = degreesLE R σ s * degreesLE R σ t := by
classical
rw [le_antisymm_iff, Submodule.mul_le]
refine ⟨fun x hx ↦ x.as_sum ▸ sum_mem fun i hi ↦ ?_,
fun x hx y hy ↦ degrees_mul_le.trans (add_le_add hx hy)⟩
replace hi : i.toMultiset ≤ s + t := (Finset.le_sup hi).trans hx
let a := (i.toMultiset - t).toFinsupp
let b := (i.toMultiset ⊓ t).toFinsupp
have : a + b = i := Multiset.toFinsupp.symm.injective (by simp [a, b, Multiset.sub_add_inter])
have ha : a.toMultiset ≤ s := by simpa [a, add_comm (a := t)] using hi
have hb : b.toMultiset ≤ t := by simp [b, Multiset.inter_le_right]
rw [show monomial i (x.coeff i) = monomial a (x.coeff i) * monomial b 1 by simp [this]]
exact Submodule.mul_mem_mul ((degrees_monomial _ _).trans ha) ((degrees_monomial _ _).trans hb)
@[simp] lemma degreesLE_zero : degreesLE R σ 0 = 1 := by
refine le_antisymm (fun x hx ↦ ?_) (by simp)
simp only [mem_degreesLE, nonpos_iff_eq_zero] at hx
have := (totalDegree_eq_zero_iff_eq_C (p := x)).mp
(Nat.eq_zero_of_le_zero (x.totalDegree_le_degrees_card.trans (by simp [hx])))
exact ⟨x.coeff 0, by simp [Algebra.smul_def, ← this]⟩
variable (s) in
lemma degreesLE_nsmul : ∀ n, degreesLE R σ (n • s) = degreesLE R σ s ^ n
| 0 => by simp
| k + 1 => by simp only [pow_succ, degreesLE_nsmul, degreesLE_add, add_smul, one_smul]
end degreesLE
end CommSemiring
end MvPolynomial
|
all_fingroup.v
|
From mathcomp Require Export action.
From mathcomp Require Export automorphism.
From mathcomp Require Export fingroup.
From mathcomp Require Export gproduct.
From mathcomp Require Export morphism.
From mathcomp Require Export perm.
From mathcomp Require Export presentation.
From mathcomp Require Export quotient.
|
Basic.lean
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.Ring.GeomSum
import Mathlib.RingTheory.Ideal.Quotient.Defs
import Mathlib.RingTheory.Ideal.Span
/-!
# Basic results in number theory
This file should contain basic results in number theory. So far, it only contains the essential
lemma in the construction of the ring of Witt vectors.
## Main statement
`dvd_sub_pow_of_dvd_sub` proves that for elements `a` and `b` in a commutative ring `R` and for
all natural numbers `p` and `k` if `p` divides `a-b` in `R`, then `p ^ (k + 1)` divides
`a ^ (p ^ k) - b ^ (p ^ k)`.
-/
section
open Ideal Ideal.Quotient
theorem dvd_sub_pow_of_dvd_sub {R : Type*} [CommRing R] {p : ℕ} {a b : R} (h : (p : R) ∣ a - b)
(k : ℕ) : (p ^ (k + 1) : R) ∣ a ^ p ^ k - b ^ p ^ k := by
induction k with
| zero => rwa [pow_one, pow_zero, pow_one, pow_one]
| succ k ih =>
rw [pow_succ p k, pow_mul, pow_mul, ← geom_sum₂_mul, pow_succ']
refine mul_dvd_mul ?_ ih
let f : R →+* R ⧸ span {(p : R)} := mk (span {(p : R)})
have hf : ∀ r : R, (p : R) ∣ r ↔ f r = 0 := fun r ↦ by rw [eq_zero_iff_mem, mem_span_singleton]
rw [hf, map_sub, sub_eq_zero] at h
rw [hf, RingHom.map_geom_sum₂, map_pow, map_pow, h, geom_sum₂_self, mul_eq_zero_of_left]
rw [← map_natCast f, eq_zero_iff_mem, mem_span_singleton]
end
|
Prod.lean
|
/-
Copyright (c) 2019 Jan-David Salchow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo
-/
import Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear
/-!
# Operator norm: Cartesian products
Interaction of operator norm with Cartesian products.
-/
variable {𝕜 E F G : Type*} [NontriviallyNormedField 𝕜]
open Set Real Metric ContinuousLinearMap
section SemiNormed
variable [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G]
variable [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] [NormedSpace 𝕜 G]
namespace ContinuousLinearMap
section FirstSecond
variable (𝕜 E F)
/-- The operator norm of the first projection `E × F → E` is at most 1. (It is 0 if `E` is zero, so
the inequality cannot be improved without further assumptions.) -/
lemma norm_fst_le : ‖fst 𝕜 E F‖ ≤ 1 :=
opNorm_le_bound _ zero_le_one (fun ⟨e, f⟩ ↦ by simpa only [one_mul] using le_max_left ‖e‖ ‖f‖)
/-- The operator norm of the second projection `E × F → F` is at most 1. (It is 0 if `F` is zero, so
the inequality cannot be improved without further assumptions.) -/
lemma norm_snd_le : ‖snd 𝕜 E F‖ ≤ 1 :=
opNorm_le_bound _ zero_le_one (fun ⟨e, f⟩ ↦ by simpa only [one_mul] using le_max_right ‖e‖ ‖f‖)
end FirstSecond
section OpNorm
@[simp]
theorem opNorm_prod (f : E →L[𝕜] F) (g : E →L[𝕜] G) : ‖f.prod g‖ = ‖(f, g)‖ :=
le_antisymm
(opNorm_le_bound _ (norm_nonneg _) fun x => by
simpa only [prod_apply, Prod.norm_def, max_mul_of_nonneg, norm_nonneg] using
max_le_max (le_opNorm f x) (le_opNorm g x)) <|
max_le
(opNorm_le_bound _ (norm_nonneg _) fun x =>
(le_max_left _ _).trans ((f.prod g).le_opNorm x))
(opNorm_le_bound _ (norm_nonneg _) fun x =>
(le_max_right _ _).trans ((f.prod g).le_opNorm x))
@[simp]
theorem opNNNorm_prod (f : E →L[𝕜] F) (g : E →L[𝕜] G) : ‖f.prod g‖₊ = ‖(f, g)‖₊ :=
Subtype.ext <| opNorm_prod f g
/-- `ContinuousLinearMap.prod` as a `LinearIsometryEquiv`. -/
noncomputable def prodₗᵢ (R : Type*)
[Semiring R] [Module R F] [Module R G] [ContinuousConstSMul R F]
[ContinuousConstSMul R G] [SMulCommClass 𝕜 R F] [SMulCommClass 𝕜 R G] :
(E →L[𝕜] F) × (E →L[𝕜] G) ≃ₗᵢ[R] E →L[𝕜] F × G :=
⟨prodₗ R, fun ⟨f, g⟩ => opNorm_prod f g⟩
end OpNorm
section Prod
variable (𝕜)
variable (M₁ M₂ M₃ M₄ : Type*)
[SeminormedAddCommGroup M₁] [NormedSpace 𝕜 M₁]
[SeminormedAddCommGroup M₂] [NormedSpace 𝕜 M₂]
[SeminormedAddCommGroup M₃] [NormedSpace 𝕜 M₃]
[SeminormedAddCommGroup M₄] [NormedSpace 𝕜 M₄]
/-- `ContinuousLinearMap.prodMap` as a continuous linear map. -/
noncomputable def prodMapL : (M₁ →L[𝕜] M₂) × (M₃ →L[𝕜] M₄) →L[𝕜] M₁ × M₃ →L[𝕜] M₂ × M₄ :=
ContinuousLinearMap.copy
(have Φ₁ : (M₁ →L[𝕜] M₂) →L[𝕜] M₁ →L[𝕜] M₂ × M₄ :=
ContinuousLinearMap.compL 𝕜 M₁ M₂ (M₂ × M₄) (ContinuousLinearMap.inl 𝕜 M₂ M₄)
have Φ₂ : (M₃ →L[𝕜] M₄) →L[𝕜] M₃ →L[𝕜] M₂ × M₄ :=
ContinuousLinearMap.compL 𝕜 M₃ M₄ (M₂ × M₄) (ContinuousLinearMap.inr 𝕜 M₂ M₄)
have Φ₁' :=
(ContinuousLinearMap.compL 𝕜 (M₁ × M₃) M₁ (M₂ × M₄)).flip (ContinuousLinearMap.fst 𝕜 M₁ M₃)
have Φ₂' :=
(ContinuousLinearMap.compL 𝕜 (M₁ × M₃) M₃ (M₂ × M₄)).flip (ContinuousLinearMap.snd 𝕜 M₁ M₃)
have Ψ₁ : (M₁ →L[𝕜] M₂) × (M₃ →L[𝕜] M₄) →L[𝕜] M₁ →L[𝕜] M₂ :=
ContinuousLinearMap.fst 𝕜 (M₁ →L[𝕜] M₂) (M₃ →L[𝕜] M₄)
have Ψ₂ : (M₁ →L[𝕜] M₂) × (M₃ →L[𝕜] M₄) →L[𝕜] M₃ →L[𝕜] M₄ :=
ContinuousLinearMap.snd 𝕜 (M₁ →L[𝕜] M₂) (M₃ →L[𝕜] M₄)
Φ₁' ∘L Φ₁ ∘L Ψ₁ + Φ₂' ∘L Φ₂ ∘L Ψ₂)
(fun p : (M₁ →L[𝕜] M₂) × (M₃ →L[𝕜] M₄) => p.1.prodMap p.2) (by
apply funext
rintro ⟨φ, ψ⟩
refine ContinuousLinearMap.ext fun ⟨x₁, x₂⟩ => ?_
simp)
variable {M₁ M₂ M₃ M₄}
@[simp]
theorem prodMapL_apply (p : (M₁ →L[𝕜] M₂) × (M₃ →L[𝕜] M₄)) :
ContinuousLinearMap.prodMapL 𝕜 M₁ M₂ M₃ M₄ p = p.1.prodMap p.2 :=
rfl
variable {X : Type*} [TopologicalSpace X]
theorem _root_.Continuous.prod_mapL {f : X → M₁ →L[𝕜] M₂} {g : X → M₃ →L[𝕜] M₄} (hf : Continuous f)
(hg : Continuous g) : Continuous fun x => (f x).prodMap (g x) :=
(prodMapL 𝕜 M₁ M₂ M₃ M₄).continuous.comp (hf.prodMk hg)
theorem _root_.Continuous.prod_map_equivL {f : X → M₁ ≃L[𝕜] M₂} {g : X → M₃ ≃L[𝕜] M₄}
(hf : Continuous fun x => (f x : M₁ →L[𝕜] M₂)) (hg : Continuous fun x => (g x : M₃ →L[𝕜] M₄)) :
Continuous fun x => ((f x).prodCongr (g x) : M₁ × M₃ →L[𝕜] M₂ × M₄) :=
(prodMapL 𝕜 M₁ M₂ M₃ M₄).continuous.comp (hf.prodMk hg)
theorem _root_.ContinuousOn.prod_mapL {f : X → M₁ →L[𝕜] M₂} {g : X → M₃ →L[𝕜] M₄} {s : Set X}
(hf : ContinuousOn f s) (hg : ContinuousOn g s) :
ContinuousOn (fun x => (f x).prodMap (g x)) s :=
((prodMapL 𝕜 M₁ M₂ M₃ M₄).continuous.comp_continuousOn (hf.prodMk hg) :)
theorem _root_.ContinuousOn.prod_map_equivL {f : X → M₁ ≃L[𝕜] M₂} {g : X → M₃ ≃L[𝕜] M₄} {s : Set X}
(hf : ContinuousOn (fun x => (f x : M₁ →L[𝕜] M₂)) s)
(hg : ContinuousOn (fun x => (g x : M₃ →L[𝕜] M₄)) s) :
ContinuousOn (fun x => ((f x).prodCongr (g x) : M₁ × M₃ →L[𝕜] M₂ × M₄)) s :=
hf.prod_mapL _ hg
end Prod
end ContinuousLinearMap
end SemiNormed
section Normed
namespace ContinuousLinearMap
section FirstSecond
variable (𝕜 E F)
/-- The operator norm of the first projection `E × F → E` is exactly 1 if `E` is nontrivial. -/
@[simp] lemma norm_fst [NormedAddCommGroup E] [NormedSpace 𝕜 E]
[SeminormedAddCommGroup F] [NormedSpace 𝕜 F] [Nontrivial E] :
‖fst 𝕜 E F‖ = 1 := by
refine le_antisymm (norm_fst_le ..) ?_
let ⟨e, he⟩ := exists_ne (0 : E)
have : ‖e‖ ≤ _ * max ‖e‖ ‖(0 : F)‖ := (fst 𝕜 E F).le_opNorm (e, 0)
rw [norm_zero, max_eq_left (norm_nonneg e)] at this
rwa [← mul_le_mul_iff_of_pos_right (norm_pos_iff.mpr he), one_mul]
/-- The operator norm of the second projection `E × F → F` is exactly 1 if `F` is nontrivial. -/
@[simp] lemma norm_snd [SeminormedAddCommGroup E] [NormedSpace 𝕜 E]
[NormedAddCommGroup F] [NormedSpace 𝕜 F] [Nontrivial F] :
‖snd 𝕜 E F‖ = 1 := by
refine le_antisymm (norm_snd_le ..) ?_
let ⟨f, hf⟩ := exists_ne (0 : F)
have : ‖f‖ ≤ _ * max ‖(0 : E)‖ ‖f‖ := (snd 𝕜 E F).le_opNorm (0, f)
rw [norm_zero, max_eq_right (norm_nonneg f)] at this
rwa [← mul_le_mul_iff_of_pos_right (norm_pos_iff.mpr hf), one_mul]
end FirstSecond
end ContinuousLinearMap
end Normed
|
ring.lean
|
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Ring
private axiom test_sorry : ∀ {α}, α
set_option autoImplicit true
-- We deliberately mock R here so that we don't have to import the deps
axiom Real : Type
notation "ℝ" => Real
@[instance] axiom Real.field : Field ℝ
@[instance] axiom Real.linearOrder : LinearOrder ℝ
@[instance] axiom Real.isStrictOrderedRing : IsStrictOrderedRing ℝ
example {a b c : ℝ} {f : ℝ → ℝ} (h : f (a * c * b) * f (c + b + a) = 1) :
f (a + b + c) * f (b * a * c) = 1 := by
ring_nf at *
exact h
example (x y : ℕ) : x + y = y + x := by ring
example (x y : ℕ) : x + y + y = 2 * y + x := by ring
example (x y : ℕ) : x + id y = y + id x := by ring!
example (x y : ℕ+) : x + y = y + x := by ring
example {α} [CommRing α] (x y : α) : x + y + y - x = 2 * y := by ring
example {α} [CommSemiring α] (x y : α) : (x + y)^2 = x^2 + 2 • x * y + y^2 := by ring
example (x y : ℕ) : (x + y) ^ 3 = x ^ 3 + y ^ 3 + 3 * (x * y ^ 2 + x ^ 2 * y) := by ring
example (x y : ℕ+) : (x + y) ^ 3 = x ^ 3 + y ^ 3 + 3 * (x * y ^ 2 + x ^ 2 * y) := by ring
example (x y : ℝ) : (x + y) ^ 3 = x ^ 3 + y ^ 3 + 3 * (x * y ^ 2 + x ^ 2 * y) := by ring
example {α} [CommSemiring α] (x : α) : (x + 1) ^ 6 = (1 + x) ^ 6 := by ring
example (n : ℕ) : (n / 2) + (n / 2) = 2 * (n / 2) := by ring
example {α} [Field α] [CharZero α] (a : α) : a / 2 = a / 2 := by ring
example {α} [Field α] [LinearOrder α] [IsStrictOrderedRing α] (a b c : α) :
a * (-c / b) * (-c / b) + -c + c = a * (c / b * (c / b)) := by ring
example {α} [Field α] [LinearOrder α] [IsStrictOrderedRing α] (a b c : α) :
b ^ 2 - 4 * c * a = -(4 * c * a) + b ^ 2 := by ring
example {α} [CommSemiring α] (x : α) : x ^ (2 + 2) = x^4 := by ring1
example {α} [CommSemiring α] (x : α) : x ^ (2 + 2) = x^4 := by ring
example {α} [CommRing α] (x : α) : x ^ 2 = x * x := by ring
-- example {α} [CommRing α] (x : α) : x ^ (2 : ℤ) = x * x := by ring
example {α} [Field α] [LinearOrder α] [IsStrictOrderedRing α] (a b c : α) :
b ^ 2 - 4 * c * a = -(4 * c * a) + b ^ 2 := by ring
example {α} [Field α] [LinearOrder α] [IsStrictOrderedRing α] (a b c : α) :
b ^ 2 - 4 * a * c = 4 * a * 0 + b * b - 4 * a * c := by ring
example {α} [CommSemiring α] (x y z : α) (n : ℕ) :
(x + y) * (z * (y * y) + (x * x ^ n + (1 + ↑n) * x ^ n * y)) =
x * (x * x ^ n) + ((2 + ↑n) * (x * x ^ n) * y +
(x * z + (z * y + (1 + ↑n) * x ^ n)) * (y * y)) := by ring
example {α} [CommRing α] (a b c d e : α) :
(-(a * b) + c + d) * e = (c + (d + -a * b)) * e := by ring
example (a n s : ℕ) : a * (n - s) = (n - s) * a := by ring
section Rat
variable [Field α]
example (x : ℚ) : x / 2 + x / 2 = x := by ring
example (x : α) : x / 2 = x / 2 := by ring1
example : (1 + 1)⁻¹ = (2⁻¹ : α) := by ring1
example (x : α) : x⁻¹ ^ 2 = (x ^ 2)⁻¹ := by ring1
example (x : α) : x⁻¹ ^ 2 = (x ^ 2)⁻¹ := by ring1
example (x y : α) : x * y⁻¹ = y⁻¹ * x := by ring1
example (x y : α) : (x^2 * y)⁻¹ = (y * x^2)⁻¹ := by ring1
example (x y : α) : (x^2)⁻¹ / y = (y * x^2)⁻¹ := by ring1
example (x y : α) : 3 / (x - x + y)⁻¹ = 3 * (x + y⁻¹ - x)⁻¹ := by ring1
variable [CharZero α]
example (x : α) : x / 2 = x / 2 := by ring
example (x : α) : (x : α) = x * (5/3)*(3/5) := by ring1
end Rat
example (A : ℕ) : (2 * A) ^ 2 = (2 * A) ^ 2 := by ring
example (x y z : ℚ) (hx : x ≠ 0) (hy : y ≠ 0) :
x / (y / z) + y ⁻¹ + 1 / (y * -x) = -1/ (x * y) + (x * z + 1) / y := by
field_simp
ring
example (a b c d x y : ℚ) (hx : x ≠ 0) (hy : y ≠ 0) :
a + b / x - c / x^2 + d / x^3 = a + x⁻¹ * (y * b / y + (d / x - c) / x) := by
field_simp
ring
example : (876544 : ℤ) * -1 + (1000000 - 123456) = 0 := by ring
example (x : ℝ) (hx : x ≠ 0) :
2 * x ^ 3 * 2 / (24 * x) = x ^ 2 / 6 := by
field_simp
ring
-- As reported at
-- https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/ring_nf.20failing.20to.20fully.20normalize
example (x : ℤ) (h : x - x + x = 0) : x = 0 := by
ring_nf at h
exact h
-- this proof style is not recommended practice
example (A B : ℕ) (H : B * A = 2) : A * B = 2 := by ring_nf at H ⊢; exact H
example (f : ℕ → ℕ) :
2 + f (2 * f 3 * f 3) + f 3 = 1 + f (f 3 ^ 2 + f 3 * f 3) + 1 + f (2 + 1) := by ring_nf
example (n : ℕ) (m : ℤ) : 2^(n + 1) * m = 2 * 2^n * m := by ring
example (a b : ℤ) (n : ℕ) : (a + b)^(n + 2) = (a^2 + b^2 + a * b + b * a) * (a + b)^n := by ring
example (x y : ℕ) : x + id y = y + id x := by ring!
-- Example with ring discharging the goal
example : 22 + 7 * 4 + 3 * 8 = 0 + 7 * 4 + 46 := by
conv => ring
trivial -- FIXME: not needed in lean 3
-- Example with ring failing to discharge, to normalizing the goal
/-- info: Try this: ring_nf -/
#guard_msgs in
example : (22 + 7 * 4 + 3 * 8 = 0 + 7 * 4 + 47) = (74 = 75) := by
conv => ring
trivial
-- Example with ring discharging the goal
example (x : ℕ) : 22 + 7 * x + 3 * 8 = 0 + 7 * x + 46 := by
conv => ring
trivial
-- Example with ring failing to discharge, to normalizing the goal
/-- info: Try this: ring_nf -/
#guard_msgs in
example (x : ℕ) : (22 + 7 * x + 3 * 8 = 0 + 7 * x + 46 + 1)
= (7 * x + 46 = 7 * x + 47) := by
conv => ring
trivial
-- check that mdata is consumed
noncomputable def f : Nat → Nat := test_sorry
example (a : Nat) : 1 * f a * 1 = f (a + 0) := by
have ha : a + 0 = a := by ring
rw [ha] -- goal has mdata
ring1
-- check that mdata is consumed by ring_nf
example (a b : ℤ) : a+b=0 ↔ b+a=0 := by
have : 3 = 3 := rfl
ring_nf -- reduced to `True` with mdata
-- Powers in the exponent get evaluated correctly
example (X : ℤ) : (X^5 + 1) * (X^2^3 + X) = X^13 + X^8 + X^6 + X := by ring
-- simulate the type of MvPolynomial
def R : Type u → Type v → Sort (max (u+1) (v+1)) := test_sorry
noncomputable instance : CommRing (R a b) := test_sorry
example (p : R PUnit.{u + 1} PUnit.{v + 1}) : p + 0 = p := by
ring
example (p q : R PUnit.{u + 1} PUnit.{v + 1}) : p + q = q + p := by
ring
example (p : R PUnit.{u + 1} PUnit.{v + 1}) : p + 0 = p := by
ring_nf
example (p q : R PUnit.{u + 1} PUnit.{v + 1}) : p + q = q + p := by
ring_nf
-- https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/ring_nf.20returns.20ugly.20literals/near/400988184
example {n : ℝ} :
(n + 1 / 2) ^ 2 * (n + 1 + 1 / 3) = 1 / 3 + n * (19 / 12) + n ^ 2 * (7 / 3) + n ^ 3 := by
-- `conv_lhs` prevents `ring_nf` picking a bad normalization for both sides.
conv_lhs => ring_nf
-- We can't use `guard_target =ₛ` here, as while it does detect stray `OfNat`s, it also complains
-- about differing instance paths.
/--
trace: n : ℝ
_hn : 0 ≤ n
⊢ 1 / 3 + n * (19 / 12) + n ^ 2 * (7 / 3) + n ^ 3 ≤ 1 / 3 + n * (5 / 3) + n ^ 2 * (7 / 3) + n ^ 3
-/
#guard_msgs (trace) in
example {n : ℝ} (_hn : 0 ≤ n) : (n + 1 / 2) ^ 2 * (n + 1 + 1 / 3) ≤ (n + 1 / 3) * (n + 1) ^ 2 := by
ring_nf
trace_state
exact test_sorry
section
abbrev myId (a : ℤ) : ℤ := a
/-
Test that when `ring_nf` normalizes multiple expressions which contain a particular atom, it uses a
form for that atom which is consistent between expressions.
We can't use `guard_hyp h :ₛ` here, as while it does tell apart `x` and `myId x`, it also complains
about differing instance paths.
-/
/--
trace: x : ℤ
R : ℤ → ℤ → Prop
h : R (myId x * 2) (myId x * 2)
⊢ True
-/
#guard_msgs (trace) in
example (x : ℤ) (R : ℤ → ℤ → Prop) : True := by
have h : R (myId x + x) (x + myId x) := test_sorry
ring_nf at h
trace_state
trivial
end
-- Test that `ring_nf` doesn't unfold local let expressions, and `ring_nf!` does
set_option linter.unusedTactic false in
example (x : ℝ) (f : ℝ → ℝ) : True := by
let y := x
/-
Two of these fail, and two of these succeed in rewriting the instance, so it's not a good idea
to use `fail_if_success` since the instances could change without warning.
-/
have : x = y := by
ring_nf -failIfUnchanged
ring_nf!
have : x - y = 0 := by
ring_nf -failIfUnchanged
ring_nf!
have : f x = f y := by
ring_nf -failIfUnchanged
ring_nf!
have : f x - f y = 0 := by
ring_nf -failIfUnchanged
ring_nf!
trivial
-- Test that `ring_nf` doesn't get confused about bound variables
example : (fun x : ℝ => x * x^2) = (fun y => y^2 * y) := by
ring_nf
-- Test that `ring` works for division without subtraction
example {R : Type} [Semifield R] [CharZero R] {x : R} : x / 2 + x / 2 = x := by ring
|
Solvable.lean
|
/-
Copyright (c) 2021 Jordan Brown, Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jordan Brown, Thomas Browning, Patrick Lutz
-/
import Mathlib.Data.Fin.VecNotation
import Mathlib.GroupTheory.Abelianization.Defs
import Mathlib.GroupTheory.Perm.ViaEmbedding
import Mathlib.GroupTheory.Subgroup.Simple
import Mathlib.SetTheory.Cardinal.Order
/-!
# Solvable Groups
In this file we introduce the notion of a solvable group. We define a solvable group as one whose
derived series is eventually trivial. This requires defining the commutator of two subgroups and
the derived series of a group.
## Main definitions
* `derivedSeries G n` : the `n`th term in the derived series of `G`, defined by iterating
`general_commutator` starting with the top subgroup
* `IsSolvable G` : the group `G` is solvable
-/
open Subgroup
variable {G G' : Type*} [Group G] [Group G'] {f : G →* G'}
section derivedSeries
variable (G)
/-- The derived series of the group `G`, obtained by starting from the subgroup `⊤` and repeatedly
taking the commutator of the previous subgroup with itself for `n` times. -/
def derivedSeries : ℕ → Subgroup G
| 0 => ⊤
| n + 1 => ⁅derivedSeries n, derivedSeries n⁆
@[simp]
theorem derivedSeries_zero : derivedSeries G 0 = ⊤ :=
rfl
@[simp]
theorem derivedSeries_succ (n : ℕ) :
derivedSeries G (n + 1) = ⁅derivedSeries G n, derivedSeries G n⁆ :=
rfl
theorem derivedSeries_normal (n : ℕ) : (derivedSeries G n).Normal := by
induction n with
| zero => exact (⊤ : Subgroup G).normal_of_characteristic
| succ n ih => exact Subgroup.commutator_normal (derivedSeries G n) (derivedSeries G n)
@[simp 1100]
theorem derivedSeries_one : derivedSeries G 1 = commutator G :=
rfl
theorem derivedSeries_antitone : Antitone (derivedSeries G) :=
antitone_nat_of_succ_le fun n => (derivedSeries G n).commutator_le_self
instance derivedSeries_characteristic (n : ℕ) : (derivedSeries G n).Characteristic := by
induction n with
| zero => exact Subgroup.topCharacteristic
| succ n _ => exact Subgroup.commutator_characteristic _ _
end derivedSeries
section CommutatorMap
section DerivedSeriesMap
variable (f) in
theorem map_derivedSeries_le_derivedSeries (n : ℕ) :
(derivedSeries G n).map f ≤ derivedSeries G' n := by
induction n with
| zero => exact le_top
| succ n ih => simp only [derivedSeries_succ, map_commutator, commutator_mono, ih]
theorem derivedSeries_le_map_derivedSeries (hf : Function.Surjective f) (n : ℕ) :
derivedSeries G' n ≤ (derivedSeries G n).map f := by
induction n with
| zero => exact (map_top_of_surjective f hf).ge
| succ n ih => exact commutator_le_map_commutator ih ih
theorem map_derivedSeries_eq (hf : Function.Surjective f) (n : ℕ) :
(derivedSeries G n).map f = derivedSeries G' n :=
le_antisymm (map_derivedSeries_le_derivedSeries f n) (derivedSeries_le_map_derivedSeries hf n)
end DerivedSeriesMap
end CommutatorMap
section Solvable
variable (G)
/-- A group `G` is solvable if its derived series is eventually trivial. We use this definition
because it's the most convenient one to work with. -/
@[mk_iff isSolvable_def]
class IsSolvable : Prop where
/-- A group `G` is solvable if its derived series is eventually trivial. -/
solvable : ∃ n : ℕ, derivedSeries G n = ⊥
instance (priority := 100) CommGroup.isSolvable {G : Type*} [CommGroup G] : IsSolvable G :=
⟨⟨1, le_bot_iff.mp (Abelianization.commutator_subset_ker (MonoidHom.id G))⟩⟩
theorem isSolvable_of_comm {G : Type*} [hG : Group G] (h : ∀ a b : G, a * b = b * a) :
IsSolvable G := by
letI hG' : CommGroup G := { hG with mul_comm := h }
cases hG
exact CommGroup.isSolvable
theorem isSolvable_of_top_eq_bot (h : (⊤ : Subgroup G) = ⊥) : IsSolvable G :=
⟨⟨0, h⟩⟩
instance (priority := 100) isSolvable_of_subsingleton [Subsingleton G] : IsSolvable G :=
isSolvable_of_top_eq_bot G (by simp [eq_iff_true_of_subsingleton])
variable {G}
theorem solvable_of_ker_le_range {G' G'' : Type*} [Group G'] [Group G''] (f : G' →* G)
(g : G →* G'') (hfg : g.ker ≤ f.range) [hG' : IsSolvable G'] [hG'' : IsSolvable G''] :
IsSolvable G := by
obtain ⟨n, hn⟩ := id hG''
obtain ⟨m, hm⟩ := id hG'
refine ⟨⟨n + m, le_bot_iff.mp (Subgroup.map_bot f ▸ hm ▸ ?_)⟩⟩
clear hm
induction m with
| zero =>
exact f.range_eq_map ▸ ((derivedSeries G n).map_eq_bot_iff.mp
(le_bot_iff.mp ((map_derivedSeries_le_derivedSeries g n).trans hn.le))).trans hfg
| succ m hm => exact commutator_le_map_commutator hm hm
theorem solvable_of_solvable_injective (hf : Function.Injective f) [IsSolvable G'] :
IsSolvable G :=
solvable_of_ker_le_range (1 : G' →* G) f ((f.ker_eq_bot_iff.mpr hf).symm ▸ bot_le)
instance subgroup_solvable_of_solvable (H : Subgroup G) [IsSolvable G] : IsSolvable H :=
solvable_of_solvable_injective H.subtype_injective
theorem solvable_of_surjective (hf : Function.Surjective f) [IsSolvable G] : IsSolvable G' :=
solvable_of_ker_le_range f (1 : G' →* G) (f.range_eq_top_of_surjective hf ▸ le_top)
instance solvable_quotient_of_solvable (H : Subgroup G) [H.Normal] [IsSolvable G] :
IsSolvable (G ⧸ H) :=
solvable_of_surjective (QuotientGroup.mk'_surjective H)
instance solvable_prod {G' : Type*} [Group G'] [IsSolvable G] [IsSolvable G'] :
IsSolvable (G × G') :=
solvable_of_ker_le_range (MonoidHom.inl G G') (MonoidHom.snd G G') fun x hx =>
⟨x.1, Prod.ext rfl hx.symm⟩
variable (G) in
theorem IsSolvable.commutator_lt_top_of_nontrivial [hG : IsSolvable G] [Nontrivial G] :
commutator G < ⊤ := by
rw [lt_top_iff_ne_top]
obtain ⟨n, hn⟩ := hG
contrapose! hn
refine ne_of_eq_of_ne ?_ top_ne_bot
induction n with
| zero => exact derivedSeries_zero G
| succ n h => rwa [derivedSeries_succ, h]
theorem IsSolvable.commutator_lt_of_ne_bot [IsSolvable G] {H : Subgroup G} (hH : H ≠ ⊥) :
⁅H, H⁆ < H := by
rw [← nontrivial_iff_ne_bot] at hH
rw [← H.range_subtype, MonoidHom.range_eq_map, ← map_commutator, map_subtype_lt_map_subtype]
exact commutator_lt_top_of_nontrivial H
theorem isSolvable_iff_commutator_lt [WellFoundedLT (Subgroup G)] :
IsSolvable G ↔ ∀ H : Subgroup G, H ≠ ⊥ → ⁅H, H⁆ < H := by
refine ⟨fun _ _ ↦ IsSolvable.commutator_lt_of_ne_bot, fun h ↦ ?_⟩
suffices h : IsSolvable (⊤ : Subgroup G) from
solvable_of_surjective (MonoidHom.range_eq_top.mp (range_subtype ⊤))
refine WellFoundedLT.induction (C := fun (H : Subgroup G) ↦ IsSolvable H) ⊤ fun H hH ↦ ?_
rcases eq_or_ne H ⊥ with rfl | h'
· infer_instance
· obtain ⟨n, hn⟩ := hH ⁅H, H⁆ (h H h')
use n + 1
rw [← (map_injective (subtype_injective _)).eq_iff, Subgroup.map_bot] at hn ⊢
rw [← hn]
clear hn
induction n with
| zero =>
rw [derivedSeries_succ, derivedSeries_zero, derivedSeries_zero, map_commutator,
← MonoidHom.range_eq_map, ← MonoidHom.range_eq_map, range_subtype, range_subtype]
| succ n ih => rw [derivedSeries_succ, map_commutator, ih, derivedSeries_succ, map_commutator]
end Solvable
section IsSimpleGroup
variable [IsSimpleGroup G]
theorem IsSimpleGroup.derivedSeries_succ {n : ℕ} : derivedSeries G n.succ = commutator G := by
induction n with
| zero => exact derivedSeries_one G
| succ n ih =>
rw [_root_.derivedSeries_succ, ih, _root_.commutator]
rcases (commutator_normal (⊤ : Subgroup G) (⊤ : Subgroup G)).eq_bot_or_eq_top with h | h
· rw [h, commutator_bot_left]
· rwa [h]
theorem IsSimpleGroup.comm_iff_isSolvable : (∀ a b : G, a * b = b * a) ↔ IsSolvable G :=
⟨isSolvable_of_comm, fun ⟨⟨n, hn⟩⟩ => by
cases n
· intro a b
refine (mem_bot.1 ?_).trans (mem_bot.1 ?_).symm <;>
· rw [← hn]
exact mem_top _
· rw [IsSimpleGroup.derivedSeries_succ] at hn
intro a b
rw [← mul_inv_eq_one, mul_inv_rev, ← mul_assoc, ← mem_bot, ← hn, commutator_eq_closure]
exact subset_closure ⟨a, b, rfl⟩⟩
end IsSimpleGroup
section PermNotSolvable
theorem not_solvable_of_mem_derivedSeries {g : G} (h1 : g ≠ 1)
(h2 : ∀ n : ℕ, g ∈ derivedSeries G n) : ¬IsSolvable G :=
mt (isSolvable_def _).mp
(not_exists_of_forall_not fun n h =>
h1 (Subgroup.mem_bot.mp ((congr_arg (g ∈ ·) h).mp (h2 n))))
theorem Equiv.Perm.fin_5_not_solvable : ¬IsSolvable (Equiv.Perm (Fin 5)) := by
let x : Equiv.Perm (Fin 5) := ⟨![1, 2, 0, 3, 4], ![2, 0, 1, 3, 4], by decide, by decide⟩
let y : Equiv.Perm (Fin 5) := ⟨![3, 4, 2, 0, 1], ![3, 4, 2, 0, 1], by decide, by decide⟩
let z : Equiv.Perm (Fin 5) := ⟨![0, 3, 2, 1, 4], ![0, 3, 2, 1, 4], by decide, by decide⟩
have key : x = z * ⁅x, y * x * y⁻¹⁆ * z⁻¹ := by unfold x y z; decide
refine not_solvable_of_mem_derivedSeries (show x ≠ 1 by decide) fun n => ?_
induction n with
| zero => exact mem_top x
| succ n ih =>
rw [key, (derivedSeries_normal _ _).mem_comm_iff, inv_mul_cancel_left]
exact commutator_mem_commutator ih ((derivedSeries_normal _ _).conj_mem _ ih _)
theorem Equiv.Perm.not_solvable (X : Type*) (hX : 5 ≤ Cardinal.mk X) :
¬IsSolvable (Equiv.Perm X) := by
intro h
have key : Nonempty (Fin 5 ↪ X) := by
rwa [← Cardinal.lift_mk_le, Cardinal.mk_fin, Cardinal.lift_natCast, Cardinal.lift_id]
exact
Equiv.Perm.fin_5_not_solvable
(solvable_of_solvable_injective (Equiv.Perm.viaEmbeddingHom_injective (Nonempty.some key)))
end PermNotSolvable
|
EulerMascheroni.lean
|
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Data.Complex.ExponentialBounds
import Mathlib.NumberTheory.Harmonic.Defs
import Mathlib.Analysis.Normed.Order.Lattice
import Mathlib.Analysis.SpecialFunctions.Pow.Real
/-!
# The Euler-Mascheroni constant `γ`
We define the constant `γ`, and give upper and lower bounds for it.
## Main definitions and results
* `Real.eulerMascheroniConstant`: the constant `γ`
* `Real.tendsto_harmonic_sub_log`: the sequence `n ↦ harmonic n - log n` tends to `γ` as `n → ∞`
* `one_half_lt_eulerMascheroniConstant` and `eulerMascheroniConstant_lt_two_thirds`: upper and
lower bounds.
## Outline of proofs
We show that
* the sequence `eulerMascheroniSeq` given by `n ↦ harmonic n - log (n + 1)` is strictly increasing;
* the sequence `eulerMascheroniSeq'` given by `n ↦ harmonic n - log n`, modified with a junk value
for `n = 0`, is strictly decreasing;
* the difference `eulerMascheroniSeq' n - eulerMascheroniSeq n` is non-negative and tends to 0.
It follows that both sequences tend to a common limit `γ`, and we have the inequality
`eulerMascheroniSeq n < γ < eulerMascheroniSeq' n` for all `n`. Taking `n = 6` gives the bounds
`1 / 2 < γ < 2 / 3`.
-/
open Filter Topology
namespace Real
section LowerSequence
/-- The sequence with `n`-th term `harmonic n - log (n + 1)`. -/
noncomputable def eulerMascheroniSeq (n : ℕ) : ℝ := harmonic n - log (n + 1)
lemma eulerMascheroniSeq_zero : eulerMascheroniSeq 0 = 0 := by
simp [eulerMascheroniSeq, harmonic_zero]
lemma strictMono_eulerMascheroniSeq : StrictMono eulerMascheroniSeq := by
refine strictMono_nat_of_lt_succ (fun n ↦ ?_)
rw [eulerMascheroniSeq, eulerMascheroniSeq, ← sub_pos, sub_sub_sub_comm,
harmonic_succ, add_comm, Rat.cast_add, add_sub_cancel_right,
← log_div (by positivity) (by positivity), add_div, Nat.cast_add_one,
Nat.cast_add_one, div_self (by positivity), sub_pos, one_div, Rat.cast_inv, Rat.cast_add,
Rat.cast_one, Rat.cast_natCast]
refine (log_lt_sub_one_of_pos ?_ (ne_of_gt <| lt_add_of_pos_right _ ?_)).trans_le (le_of_eq ?_)
· positivity
· positivity
· simp only [add_sub_cancel_left]
lemma one_half_lt_eulerMascheroniSeq_six : 1 / 2 < eulerMascheroniSeq 6 := by
have : eulerMascheroniSeq 6 = 49 / 20 - log 7 := by
rw [eulerMascheroniSeq]
norm_num
rw [this, lt_sub_iff_add_lt, ← lt_sub_iff_add_lt', log_lt_iff_lt_exp (by positivity)]
refine lt_of_lt_of_le ?_ (Real.sum_le_exp_of_nonneg (by norm_num) 7)
simp_rw [Finset.sum_range_succ, Nat.factorial_succ]
norm_num
end LowerSequence
section UpperSequence
/-- The sequence with `n`-th term `harmonic n - log n`. We use a junk value for `n = 0`, in order
to have the sequence be strictly decreasing. -/
noncomputable def eulerMascheroniSeq' (n : ℕ) : ℝ :=
if n = 0 then 2 else ↑(harmonic n) - log n
lemma eulerMascheroniSeq'_one : eulerMascheroniSeq' 1 = 1 := by
simp [eulerMascheroniSeq']
lemma strictAnti_eulerMascheroniSeq' : StrictAnti eulerMascheroniSeq' := by
refine strictAnti_nat_of_succ_lt (fun n ↦ ?_)
rcases Nat.eq_zero_or_pos n with rfl | hn
· simp [eulerMascheroniSeq']
simp_rw [eulerMascheroniSeq', eq_false_intro hn.ne', reduceCtorEq, if_false]
rw [← sub_pos, sub_sub_sub_comm,
harmonic_succ, Rat.cast_add, ← sub_sub, sub_self, zero_sub, sub_eq_add_neg, neg_sub,
← sub_eq_neg_add, sub_pos, ← log_div (by positivity) (by positivity), ← neg_lt_neg_iff,
← log_inv]
refine (log_lt_sub_one_of_pos ?_ ?_).trans_le (le_of_eq ?_)
· positivity
· field_simp
· field_simp
lemma eulerMascheroniSeq'_six_lt_two_thirds : eulerMascheroniSeq' 6 < 2 / 3 := by
have h1 : eulerMascheroniSeq' 6 = 49 / 20 - log 6 := by
rw [eulerMascheroniSeq']
norm_num
rw [h1, sub_lt_iff_lt_add, ← sub_lt_iff_lt_add', lt_log_iff_exp_lt (by positivity)]
norm_num
have := rpow_lt_rpow (exp_pos _).le exp_one_lt_d9 (by simp : (0 : ℝ) < 107 / 60)
rw [exp_one_rpow] at this
refine lt_trans this ?_
rw [← rpow_lt_rpow_iff (z := 60), ← rpow_mul, div_mul_cancel₀, ← Nat.cast_ofNat,
← Nat.cast_ofNat, rpow_natCast, Nat.cast_ofNat, ← Nat.cast_ofNat (n := 60), rpow_natCast]
· norm_num
all_goals positivity
lemma eulerMascheroniSeq_lt_eulerMascheroniSeq' (m n : ℕ) :
eulerMascheroniSeq m < eulerMascheroniSeq' n := by
have (r : ℕ) : eulerMascheroniSeq r < eulerMascheroniSeq' r := by
rcases eq_zero_or_pos r with rfl | hr
· simp [eulerMascheroniSeq, eulerMascheroniSeq']
simp only [eulerMascheroniSeq, eulerMascheroniSeq', hr.ne', if_false]
gcongr
linarith
apply (strictMono_eulerMascheroniSeq.monotone (le_max_left m n)).trans_lt
exact (this _).trans_le (strictAnti_eulerMascheroniSeq'.antitone (le_max_right m n))
end UpperSequence
/-- The Euler-Mascheroni constant `γ`. -/
noncomputable def eulerMascheroniConstant : ℝ := limUnder atTop eulerMascheroniSeq
lemma tendsto_eulerMascheroniSeq :
Tendsto eulerMascheroniSeq atTop (𝓝 eulerMascheroniConstant) := by
have := tendsto_atTop_ciSup strictMono_eulerMascheroniSeq.monotone ?_
· rwa [eulerMascheroniConstant, this.limUnder_eq]
· exact ⟨_, fun _ ⟨_, hn⟩ ↦ hn ▸ (eulerMascheroniSeq_lt_eulerMascheroniSeq' _ 1).le⟩
lemma tendsto_harmonic_sub_log_add_one :
Tendsto (fun n : ℕ ↦ harmonic n - log (n + 1)) atTop (𝓝 eulerMascheroniConstant) :=
tendsto_eulerMascheroniSeq
lemma tendsto_eulerMascheroniSeq' :
Tendsto eulerMascheroniSeq' atTop (𝓝 eulerMascheroniConstant) := by
suffices Tendsto (fun n ↦ eulerMascheroniSeq' n - eulerMascheroniSeq n) atTop (𝓝 0) by
simpa using this.add tendsto_eulerMascheroniSeq
suffices Tendsto (fun x : ℝ ↦ log (x + 1) - log x) atTop (𝓝 0) by
apply (this.comp tendsto_natCast_atTop_atTop).congr'
filter_upwards [eventually_ne_atTop 0] with n hn
simp [eulerMascheroniSeq, eulerMascheroniSeq', eq_false_intro hn]
exact tendsto_log_comp_add_sub_log 1
lemma tendsto_harmonic_sub_log :
Tendsto (fun n : ℕ ↦ harmonic n - log n) atTop (𝓝 eulerMascheroniConstant) := by
apply tendsto_eulerMascheroniSeq'.congr'
filter_upwards [eventually_ne_atTop 0] with n hn
simp_rw [eulerMascheroniSeq', hn, if_false]
lemma eulerMascheroniSeq_lt_eulerMascheroniConstant (n : ℕ) :
eulerMascheroniSeq n < eulerMascheroniConstant := by
refine (strictMono_eulerMascheroniSeq (Nat.lt_succ_self n)).trans_le ?_
apply strictMono_eulerMascheroniSeq.monotone.ge_of_tendsto tendsto_eulerMascheroniSeq
lemma eulerMascheroniConstant_lt_eulerMascheroniSeq' (n : ℕ) :
eulerMascheroniConstant < eulerMascheroniSeq' n := by
refine lt_of_le_of_lt ?_ (strictAnti_eulerMascheroniSeq' (Nat.lt_succ_self n))
apply strictAnti_eulerMascheroniSeq'.antitone.le_of_tendsto tendsto_eulerMascheroniSeq'
/-- Lower bound for `γ`. (The true value is about 0.57.) -/
lemma one_half_lt_eulerMascheroniConstant : 1 / 2 < eulerMascheroniConstant :=
one_half_lt_eulerMascheroniSeq_six.trans (eulerMascheroniSeq_lt_eulerMascheroniConstant _)
/-- Upper bound for `γ`. (The true value is about 0.57.) -/
lemma eulerMascheroniConstant_lt_two_thirds : eulerMascheroniConstant < 2 / 3 :=
(eulerMascheroniConstant_lt_eulerMascheroniSeq' _).trans eulerMascheroniSeq'_six_lt_two_thirds
end Real
|
SetLike.lean
|
/-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Init
import Aesop
/-!
# SetLike Rule Set
This module defines the `SetLike` and `SetLike!` Aesop rule sets.
Aesop rule sets only become visible once the file in which they're declared is imported,
so we must put this declaration into its own file.
-/
declare_aesop_rule_sets [SetLike] (default := true)
declare_aesop_rule_sets [SetLike!] (default := false)
library_note "SetLike Aesop ruleset"/--
The Aesop tactic (`aesop`) can automatically prove obvious facts about membership in structures
such as subgroups and subrings. Certain lemmas regarding membership in algebraic substructures
are given the `aesop` attribute according to the following principles:
- Rules are in the `SetLike` ruleset: (rule_sets := [SetLike]).
- Apply-style rules with trivial hypotheses are registered both as `simp` rules and as
`safe` Aesop rules. The latter is needed in case there are metavariables in the goal.
For instance, Aesop can use the rule `one_mem` to prove
`(M : Type*) [Monoid M] (s : Submonoid M) ⊢ ∃ m : M, m ∈ s`.
- Apply-style rules with nontrivial hypotheses are marked `unsafe`. This is because applying them
might not be provability-preserving in the context of more complex membership rules.
For instance, `mul_mem` is marked `unsafe`.
- Unsafe rules are given a probability no higher than 90%. This is the same probability
Aesop gives to safe rules when they generate metavariables. If the priority is too high, loops
generated in the presence of metavariables will time out Aesop.
- Rules that cause loops (even in the absence of metavariables) are given a low priority of 5%.
These rules are placed in the `SetLike!` ruleset instead of the `SetLike` ruleset so that
they are not invoked by default. An example is `SetLike.mem_of_subset`.
- Simplifying the left hand side of a membership goal is prioritised over simplifying the
right hand side. By default, rules simplifying the LHS (e.g., `mul_mem`) are given
probability 90% and rules simplifying the RHS are given probability 80%
(e.g., `Subgroup.mem_closure_of_mem`).
- These default probabilities are for rules with simple hypotheses that fail quickly when
not satisfied, such as `mul_mem`. Rules with more complicated hypotheses, or rules that are
less likely to progress the proof state towards a solution, are given a lower priority.
- To optimise performance and avoid timeouts, Aesop should not be invoking low-priority rules
unless it can make no other progress. If common usage patterns cause Aesop to invoke such rules,
additional lemmas are added at a higher priority to cover that pattern.
For example, `Subgroup.mem_closure_of_mem` covers a common use case of `SetLike.mem_of_subset`.
Some examples of membership-related goals which Aesop with this ruleset is designed to close
can be found in the file MathlibTest/set_like.lean.
-/
|
Int.lean
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Order.Group.Unbundled.Abs
import Mathlib.Algebra.Group.Int.Defs
import Mathlib.Data.Int.Basic
/-!
# Facts about `ℤ` as an (unbundled) ordered group
See note [foundational algebra order theory].
## Recursors
* `Int.rec`: Sign disjunction. Something is true/defined on `ℤ` if it's true/defined for nonnegative
and for negative values. (Defined in core Lean 3)
* `Int.inductionOn`: Simple growing induction on positive numbers, plus simple decreasing induction
on negative numbers. Note that this recursor is currently only `Prop`-valued.
* `Int.inductionOn'`: Simple growing induction for numbers greater than `b`, plus simple decreasing
induction on numbers less than `b`.
-/
-- We should need only a minimal development of sets in order to get here.
assert_not_exists Set.Subsingleton Ring
open Function Nat
namespace Int
theorem natCast_strictMono : StrictMono (· : ℕ → ℤ) := fun _ _ ↦ Int.ofNat_lt.2
/-! ### Miscellaneous lemmas -/
theorem abs_eq_natAbs : ∀ a : ℤ, |a| = natAbs a
| (n : ℕ) => abs_of_nonneg <| ofNat_zero_le _
| -[_+1] => abs_of_nonpos <| le_of_lt <| negSucc_lt_zero _
@[norm_cast] lemma natCast_natAbs (n : ℤ) : (n.natAbs : ℤ) = |n| := n.abs_eq_natAbs.symm
theorem natAbs_abs (a : ℤ) : natAbs |a| = natAbs a := by rw [abs_eq_natAbs]; rfl
theorem sign_mul_abs (a : ℤ) : sign a * |a| = a := by
rw [abs_eq_natAbs, sign_mul_natAbs a]
theorem sign_mul_self_eq_abs (a : ℤ) : sign a * a = |a| := by
rw [abs_eq_natAbs, sign_mul_self_eq_natAbs]
lemma natAbs_le_self_sq (a : ℤ) : (Int.natAbs a : ℤ) ≤ a ^ 2 := by
rw [← Int.natAbs_sq a, sq]
norm_cast
apply Nat.le_mul_self
alias natAbs_le_self_pow_two := natAbs_le_self_sq
lemma le_self_sq (b : ℤ) : b ≤ b ^ 2 := le_trans le_natAbs (natAbs_le_self_sq _)
alias le_self_pow_two := le_self_sq
@[norm_cast] lemma abs_natCast (n : ℕ) : |(n : ℤ)| = n := abs_of_nonneg (natCast_nonneg n)
theorem natAbs_sub_pos_iff {i j : ℤ} : 0 < natAbs (i - j) ↔ i ≠ j := by
rw [natAbs_pos, ne_eq, sub_eq_zero]
theorem natAbs_sub_ne_zero_iff {i j : ℤ} : natAbs (i - j) ≠ 0 ↔ i ≠ j :=
Nat.ne_zero_iff_zero_lt.trans natAbs_sub_pos_iff
@[simp]
theorem abs_lt_one_iff {a : ℤ} : |a| < 1 ↔ a = 0 := by
rw [← zero_add 1, lt_add_one_iff, abs_nonpos_iff]
theorem abs_le_one_iff {a : ℤ} : |a| ≤ 1 ↔ a = 0 ∨ a = 1 ∨ a = -1 := by
rw [le_iff_lt_or_eq, abs_lt_one_iff]
match a with
| (n : ℕ) => simp [abs_eq_natAbs]
| -[n+1] =>
simp only [negSucc_ne_zero, abs_eq_natAbs, natAbs_negSucc, succ_eq_add_one,
Int.natCast_add, cast_ofNat_Int, add_eq_right, natCast_eq_zero, false_or, reduceNeg]
rw [negSucc_eq]
omega
theorem one_le_abs {z : ℤ} (h₀ : z ≠ 0) : 1 ≤ |z| :=
add_one_le_iff.mpr (abs_pos.mpr h₀)
lemma eq_zero_of_abs_lt_dvd {m x : ℤ} (h1 : m ∣ x) (h2 : |x| < m) : x = 0 := by
by_contra h
have := Int.natAbs_le_of_dvd_ne_zero h1 h
rw [Int.abs_eq_natAbs] at h2
omega
lemma abs_sub_lt_of_lt_lt {m a b : ℕ} (ha : a < m) (hb : b < m) : |(b : ℤ) - a| < m := by
rw [abs_lt]; omega
/-! #### `/` -/
theorem ediv_eq_zero_of_lt_abs {a b : ℤ} (H1 : 0 ≤ a) (H2 : a < |b|) : a / b = 0 :=
match b, |b|, abs_eq_natAbs b, H2 with
| (n : ℕ), _, rfl, H2 => ediv_eq_zero_of_lt H1 H2
| -[n+1], _, rfl, H2 => neg_injective <| by rw [← Int.ediv_neg]; exact ediv_eq_zero_of_lt H1 H2
/-! #### mod -/
@[simp]
theorem emod_abs (a b : ℤ) : a % |b| = a % b :=
abs_by_cases (fun i => a % i = a % b) rfl (emod_neg _ _)
theorem emod_lt_abs (a : ℤ) {b : ℤ} (H : b ≠ 0) : a % b < |b| := by
rw [← emod_abs]; exact emod_lt_of_pos _ (abs_pos.2 H)
/-! ### properties of `/` and `%` -/
theorem abs_ediv_le_abs : ∀ a b : ℤ, |a / b| ≤ |a| :=
suffices ∀ (a : ℤ) (n : ℕ), |a / n| ≤ |a| from fun a b =>
match b, Int.eq_nat_or_neg b with
| _, ⟨n, Or.inl rfl⟩ => this _ _
| _, ⟨n, Or.inr rfl⟩ => by rw [Int.ediv_neg, abs_neg]; apply this
fun a n => by
rw [abs_eq_natAbs, abs_eq_natAbs];
exact ofNat_le_ofNat_of_le
(match a, n with
| (m : ℕ), n => Nat.div_le_self _ _
| -[m+1], 0 => Nat.zero_le _
| -[m+1], n + 1 => Nat.succ_le_succ (Nat.div_le_self _ _))
theorem abs_sign_of_nonzero {z : ℤ} (hz : z ≠ 0) : |z.sign| = 1 := by
rw [abs_eq_natAbs, natAbs_sign_of_ne_zero hz, Int.ofNat_one]
protected theorem sign_eq_ediv_abs' (a : ℤ) : sign a = a / |a| :=
if az : a = 0 then by simp [az]
else (Int.ediv_eq_of_eq_mul_left (mt abs_eq_zero.1 az) (sign_mul_abs _).symm).symm
@[deprecated (since := "2025-03-10")] alias sign_eq_ediv_abs := Int.sign_eq_ediv_abs'
protected theorem sign_eq_abs_ediv (a : ℤ) : sign a = |a| / a :=
if az : a = 0 then by simp [az]
else (Int.ediv_eq_of_eq_mul_left az (sign_mul_self_eq_abs _).symm).symm
end Int
section Group
variable {G : Type*} [Group G]
@[to_additive (attr := simp) abs_zsmul_eq_zero]
lemma zpow_abs_eq_one (a : G) (n : ℤ) : a ^ |n| = 1 ↔ a ^ n = 1 := by
rw [← Int.natCast_natAbs, zpow_natCast, pow_natAbs_eq_one]
end Group
|
FunctorCategory.lean
|
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Finite
import Mathlib.CategoryTheory.Limits.Yoneda
import Mathlib.CategoryTheory.Limits.Presheaf
/-!
# Preservation of (co)limits in the functor category
* Show that if `X ⨯ -` preserves colimits in `D` for any `X : D`, then the product functor `F ⨯ -`
for `F : C ⥤ D` preserves colimits.
The idea of the proof is simply that products and colimits in the functor category are computed
pointwise, so pointwise preservation implies general preservation.
* Show that `F ⋙ -` preserves limits if the target category has limits.
* Show that `F : C ⥤ D` preserves limits of a certain shape
if `Lan F.op : Cᵒᵖ ⥤ Type*` preserves such limits.
# References
https://ncatlab.org/nlab/show/commutativity+of+limits+and+colimits#preservation_by_functor_categories_and_localizations
-/
universe w w' v v₁ v₂ v₃ u u₁ u₂ u₃
noncomputable section
namespace CategoryTheory
open Category Limits Functor
section
variable {C : Type u} [Category.{v₁} C]
variable {D : Type u₂} [Category.{u} D]
variable {E : Type u} [Category.{v₂} E]
/-- If `X × -` preserves colimits in `D` for any `X : D`, then the product functor `F ⨯ -` for
`F : C ⥤ D` also preserves colimits.
Note this is (mathematically) a special case of the statement that
"if limits commute with colimits in `D`, then they do as well in `C ⥤ D`"
but the story in Lean is a bit more complex, and this statement isn't directly a special case.
That is, even with a formalised proof of the general statement, there would still need to be some
work to convert to this version: namely, the natural isomorphism
`(evaluation C D).obj k ⋙ prod.functor.obj (F.obj k) ≅
prod.functor.obj F ⋙ (evaluation C D).obj k`
-/
lemma FunctorCategory.prod_preservesColimits [HasBinaryProducts D] [HasColimits D]
[∀ X : D, PreservesColimits (prod.functor.obj X)] (F : C ⥤ D) :
PreservesColimits (prod.functor.obj F) where
preservesColimitsOfShape {J : Type u} [Category.{u, u} J] :=
{
preservesColimit := fun {K : J ⥤ C ⥤ D} => ({
preserves := fun {c : Cocone K} (t : IsColimit c) => ⟨by
apply evaluationJointlyReflectsColimits _ fun {k} => ?_
change IsColimit ((prod.functor.obj F ⋙ (evaluation _ _).obj k).mapCocone c)
let this :=
isColimitOfPreserves ((evaluation C D).obj k ⋙ prod.functor.obj (F.obj k)) t
apply IsColimit.mapCoconeEquiv _ this
apply (NatIso.ofComponents _ _).symm
· intro G
apply asIso (prodComparison ((evaluation C D).obj k) F G)
· intro G G'
apply prodComparison_natural ((evaluation C D).obj k) (𝟙 F)⟩ } ) }
end
section
variable {C : Type u₁} [Category.{v₁} C]
variable {D : Type u₂} [Category.{v₂} D]
variable {E : Type u₃} [Category.{v₃} E]
instance whiskeringLeft_preservesLimitsOfShape (J : Type u) [Category.{v} J]
[HasLimitsOfShape J D] (F : C ⥤ E) :
PreservesLimitsOfShape J ((whiskeringLeft C E D).obj F) :=
⟨fun {K} =>
⟨fun c {hc} => ⟨by
apply evaluationJointlyReflectsLimits
intro Y
change IsLimit (((evaluation E D).obj (F.obj Y)).mapCone c)
exact isLimitOfPreserves _ hc⟩⟩⟩
instance whiskeringLeft_preservesColimitsOfShape (J : Type u) [Category.{v} J]
[HasColimitsOfShape J D] (F : C ⥤ E) :
PreservesColimitsOfShape J ((whiskeringLeft C E D).obj F) :=
⟨fun {K} =>
⟨fun c {hc} => ⟨by
apply evaluationJointlyReflectsColimits
intro Y
change IsColimit (((evaluation E D).obj (F.obj Y)).mapCocone c)
exact isColimitOfPreserves _ hc⟩⟩⟩
instance whiskeringLeft_preservesLimits [HasLimitsOfSize.{w, w'} D] (F : C ⥤ E) :
PreservesLimitsOfSize.{w, w'} ((whiskeringLeft C E D).obj F) :=
⟨fun {J} _ => whiskeringLeft_preservesLimitsOfShape J F⟩
instance whiskeringLeft_preservesColimit [HasColimitsOfSize.{w, w'} D] (F : C ⥤ E) :
PreservesColimitsOfSize.{w, w'} ((whiskeringLeft C E D).obj F) :=
⟨fun {J} _ => whiskeringLeft_preservesColimitsOfShape J F⟩
instance whiskeringRight_preservesLimitsOfShape {C : Type*} [Category C] {D : Type*}
[Category D] {E : Type*} [Category E] {J : Type*} [Category J]
[HasLimitsOfShape J D] (F : D ⥤ E) [PreservesLimitsOfShape J F] :
PreservesLimitsOfShape J ((whiskeringRight C D E).obj F) :=
⟨fun {K} =>
⟨fun c {hc} => ⟨by
apply evaluationJointlyReflectsLimits _ (fun k => ?_)
change IsLimit (((evaluation _ _).obj k ⋙ F).mapCone c)
exact isLimitOfPreserves _ hc⟩⟩⟩
/-- Whiskering right and then taking a limit is the same as taking the limit and applying the
functor. -/
def limitCompWhiskeringRightIsoLimitComp {C : Type*} [Category C] {D : Type*}
[Category D] {E : Type*} [Category E] {J : Type*} [Category J]
[HasLimitsOfShape J D] (F : D ⥤ E) [PreservesLimitsOfShape J F] (G : J ⥤ C ⥤ D) :
limit (G ⋙ (whiskeringRight _ _ _).obj F) ≅ limit G ⋙ F :=
(preservesLimitIso _ _).symm
@[reassoc (attr := simp)]
theorem limitCompWhiskeringRightIsoLimitComp_inv_π {C : Type*} [Category C] {D : Type*}
[Category D] {E : Type*} [Category E] {J : Type*} [Category J]
[HasLimitsOfShape J D] (F : D ⥤ E) [PreservesLimitsOfShape J F] (G : J ⥤ C ⥤ D) (j : J) :
(limitCompWhiskeringRightIsoLimitComp F G).inv ≫
limit.π (G ⋙ (whiskeringRight _ _ _).obj F) j = whiskerRight (limit.π G j) F := by
simp [limitCompWhiskeringRightIsoLimitComp]
@[reassoc (attr := simp)]
theorem limitCompWhiskeringRightIsoLimitComp_hom_whiskerRight_π {C : Type*} [Category C] {D : Type*}
[Category D] {E : Type*} [Category E] {J : Type*} [Category J]
[HasLimitsOfShape J D] (F : D ⥤ E) [PreservesLimitsOfShape J F] (G : J ⥤ C ⥤ D) (j : J) :
(limitCompWhiskeringRightIsoLimitComp F G).hom ≫ whiskerRight (limit.π G j) F =
limit.π (G ⋙ (whiskeringRight _ _ _).obj F) j := by
simp [← Iso.eq_inv_comp]
instance whiskeringRight_preservesColimitsOfShape {C : Type*} [Category C] {D : Type*}
[Category D] {E : Type*} [Category E] {J : Type*} [Category J]
[HasColimitsOfShape J D] (F : D ⥤ E) [PreservesColimitsOfShape J F] :
PreservesColimitsOfShape J ((whiskeringRight C D E).obj F) :=
⟨fun {K} =>
⟨fun c {hc} => ⟨by
apply evaluationJointlyReflectsColimits _ (fun k => ?_)
change IsColimit (((evaluation _ _).obj k ⋙ F).mapCocone c)
exact isColimitOfPreserves _ hc⟩⟩⟩
/-- Whiskering right and then taking a colimit is the same as taking the colimit and applying the
functor. -/
def colimitCompWhiskeringRightIsoColimitComp {C : Type*} [Category C] {D : Type*}
[Category D] {E : Type*} [Category E] {J : Type*} [Category J]
[HasColimitsOfShape J D] (F : D ⥤ E) [PreservesColimitsOfShape J F] (G : J ⥤ C ⥤ D) :
colimit (G ⋙ (whiskeringRight _ _ _).obj F) ≅ colimit G ⋙ F :=
(preservesColimitIso _ _).symm
@[reassoc (attr := simp)]
theorem ι_colimitCompWhiskeringRightIsoColimitComp_hom {C : Type*} [Category C] {D : Type*}
[Category D] {E : Type*} [Category E] {J : Type*} [Category J]
[HasColimitsOfShape J D] (F : D ⥤ E) [PreservesColimitsOfShape J F] (G : J ⥤ C ⥤ D) (j : J) :
colimit.ι (G ⋙ (whiskeringRight _ _ _).obj F) j ≫
(colimitCompWhiskeringRightIsoColimitComp F G).hom = whiskerRight (colimit.ι G j) F := by
simp [colimitCompWhiskeringRightIsoColimitComp]
@[reassoc (attr := simp)]
theorem whiskerRight_ι_colimitCompWhiskeringRightIsoColimitComp_inv {C : Type*} [Category C]
{D : Type*} [Category D] {E : Type*} [Category E] {J : Type*} [Category J]
[HasColimitsOfShape J D] (F : D ⥤ E) [PreservesColimitsOfShape J F] (G : J ⥤ C ⥤ D) (j : J) :
whiskerRight (colimit.ι G j) F ≫ (colimitCompWhiskeringRightIsoColimitComp F G).inv =
colimit.ι (G ⋙ (whiskeringRight _ _ _).obj F) j := by
simp [Iso.comp_inv_eq]
instance whiskeringRightPreservesLimits {C : Type*} [Category C] {D : Type*} [Category D]
{E : Type*} [Category E] (F : D ⥤ E) [HasLimitsOfSize.{w, w'} D]
[PreservesLimitsOfSize.{w, w'} F] :
PreservesLimitsOfSize.{w, w'} ((whiskeringRight C D E).obj F) :=
⟨inferInstance⟩
instance whiskeringRightPreservesColimits {C : Type*} [Category C] {D : Type*} [Category D]
{E : Type*} [Category E] (F : D ⥤ E) [HasColimitsOfSize.{w, w'} D]
[PreservesColimitsOfSize.{w, w'} F] :
PreservesColimitsOfSize.{w, w'} ((whiskeringRight C D E).obj F) :=
⟨inferInstance⟩
/-- If `Lan F.op : (Cᵒᵖ ⥤ Type*) ⥤ (Dᵒᵖ ⥤ Type*)` preserves limits of shape `J`, so will `F`. -/
lemma preservesLimit_of_lan_preservesLimit {C D : Type u} [SmallCategory C]
[SmallCategory D] (F : C ⥤ D) (J : Type u) [SmallCategory J]
[PreservesLimitsOfShape J (F.op.lan : _ ⥤ Dᵒᵖ ⥤ Type u)] : PreservesLimitsOfShape J F := by
apply @preservesLimitsOfShape_of_reflects_of_preserves _ _ _ _ _ _ _ _ F yoneda ?_
exact preservesLimitsOfShape_of_natIso (Presheaf.compYonedaIsoYonedaCompLan F).symm
/-- `F : C ⥤ D ⥤ E` preserves finite limits if it does for each `d : D`. -/
lemma preservesFiniteLimits_of_evaluation {D : Type*} [Category D] {E : Type*} [Category E]
(F : C ⥤ D ⥤ E) (h : ∀ d : D, PreservesFiniteLimits (F ⋙ (evaluation D E).obj d)) :
PreservesFiniteLimits F :=
⟨fun J _ _ => preservesLimitsOfShape_of_evaluation F J fun k => (h k).preservesFiniteLimits _⟩
/-- `F : C ⥤ D ⥤ E` preserves finite limits if it does for each `d : D`. -/
lemma preservesFiniteColimits_of_evaluation {D : Type*} [Category D] {E : Type*} [Category E]
(F : C ⥤ D ⥤ E) (h : ∀ d : D, PreservesFiniteColimits (F ⋙ (evaluation D E).obj d)) :
PreservesFiniteColimits F :=
⟨fun J _ _ => preservesColimitsOfShape_of_evaluation F J fun k => (h k).preservesFiniteColimits _⟩
end
section
variable {C : Type u} [Category.{v} C]
variable {J : Type u₁} [Category.{v₁} J]
variable {K : Type u₂} [Category.{v₂} K]
variable {D : Type u₃} [Category.{v₃} D]
section
variable [HasLimitsOfShape J C] [HasColimitsOfShape K C]
variable [PreservesLimitsOfShape J (colim : (K ⥤ C) ⥤ _)]
noncomputable instance : PreservesLimitsOfShape J (colim : (K ⥤ D ⥤ C) ⥤ _) :=
preservesLimitsOfShape_of_evaluation _ _ (fun d =>
let i : (colim : (K ⥤ D ⥤ C) ⥤ _) ⋙ (evaluation D C).obj d ≅
colimit ((whiskeringRight K (D ⥤ C) C).obj ((evaluation D C).obj d)).flip :=
NatIso.ofComponents (fun X => (colimitObjIsoColimitCompEvaluation _ _) ≪≫
(by exact HasColimit.isoOfNatIso (Iso.refl _)) ≪≫
(colimitObjIsoColimitCompEvaluation _ _).symm)
(fun {F G} η => colimit_obj_ext (fun j => by simp [← NatTrans.comp_app_assoc]))
preservesLimitsOfShape_of_natIso (i ≪≫ colimitFlipIsoCompColim _).symm)
end
section
variable [HasColimitsOfShape J C] [HasLimitsOfShape K C]
variable [PreservesColimitsOfShape J (lim : (K ⥤ C) ⥤ _)]
noncomputable instance : PreservesColimitsOfShape J (lim : (K ⥤ D ⥤ C) ⥤ _) :=
preservesColimitsOfShape_of_evaluation _ _ (fun d =>
let i : (lim : (K ⥤ D ⥤ C) ⥤ _) ⋙ (evaluation D C).obj d ≅
limit ((whiskeringRight K (D ⥤ C) C).obj ((evaluation D C).obj d)).flip :=
NatIso.ofComponents (fun X => (limitObjIsoLimitCompEvaluation _ _) ≪≫
(by exact HasLimit.isoOfNatIso (Iso.refl _)) ≪≫
(limitObjIsoLimitCompEvaluation _ _).symm)
(fun {F G} η => limit_obj_ext (fun j => by simp [← NatTrans.comp_app]))
preservesColimitsOfShape_of_natIso (i ≪≫ limitFlipIsoCompLim _).symm)
end
end
end CategoryTheory
|
OmegaCompletePartialOrder.lean
|
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Order.OmegaCompletePartialOrder
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
import Mathlib.CategoryTheory.ConcreteCategory.Basic
/-!
# Category of types with an omega complete partial order
In this file, we bundle the class `OmegaCompletePartialOrder` into a
concrete category and prove that continuous functions also form
an `OmegaCompletePartialOrder`.
## Main definitions
* `ωCPO`
* an instance of `Category` and `ConcreteCategory`
-/
open CategoryTheory
universe u v
/-- The category of types with an omega complete partial order. -/
structure ωCPO : Type (u + 1) where
/-- The underlying type. -/
carrier : Type u
[str : OmegaCompletePartialOrder carrier]
attribute [instance] ωCPO.str
namespace ωCPO
open OmegaCompletePartialOrder
instance : CoeSort ωCPO Type* :=
⟨carrier⟩
/-- Construct a bundled ωCPO from the underlying type and typeclass. -/
abbrev of (α : Type*) [OmegaCompletePartialOrder α] : ωCPO where
carrier := α
theorem coe_of (α : Type*) [OmegaCompletePartialOrder α] : ↥(of α) = α :=
rfl
instance : LargeCategory.{u} ωCPO where
Hom X Y := ContinuousHom X Y
id X := ContinuousHom.id
comp f g := g.comp f
instance : ConcreteCategory ωCPO (ContinuousHom · ·) where
hom f := f
ofHom f := f
instance : Inhabited ωCPO :=
⟨of PUnit⟩
section
open CategoryTheory.Limits
namespace HasProducts
/-- The pi-type gives a cone for a product. -/
def product {J : Type v} (f : J → ωCPO.{v}) : Fan f :=
Fan.mk (of (∀ j, f j)) fun j => .mk (Pi.evalOrderHom j) fun _ => rfl
/-- The pi-type is a limit cone for the product. -/
def isProduct (J : Type v) (f : J → ωCPO) : IsLimit (product f) where
lift s :=
⟨⟨fun t j => (s.π.app ⟨j⟩) t, fun _ _ h j => (s.π.app ⟨j⟩).monotone h⟩,
fun x => funext fun j => (s.π.app ⟨j⟩).continuous x⟩
uniq s m w := by
ext t; funext j -- Porting note (https://github.com/leanprover-community/mathlib4/issues/11041): Originally `ext t j`
change m t j = (s.π.app ⟨j⟩) t
rw [← w ⟨j⟩]
rfl
fac _ _ := rfl
instance (J : Type v) (f : J → ωCPO.{v}) : HasProduct f :=
HasLimit.mk ⟨_, isProduct _ f⟩
end HasProducts
instance omegaCompletePartialOrderEqualizer {α β : Type*} [OmegaCompletePartialOrder α]
[OmegaCompletePartialOrder β] (f g : α →𝒄 β) :
OmegaCompletePartialOrder { a : α // f a = g a } :=
OmegaCompletePartialOrder.subtype _ fun c hc => by
rw [f.continuous, g.continuous]
congr 1
apply OrderHom.ext; funext x -- Porting note (https://github.com/leanprover-community/mathlib4/issues/11041): Originally `ext`
apply hc _ ⟨_, rfl⟩
namespace HasEqualizers
/-- The equalizer inclusion function as a `ContinuousHom`. -/
def equalizerι {α β : Type*} [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β]
(f g : α →𝒄 β) : { a : α // f a = g a } →𝒄 α :=
.mk (OrderHom.Subtype.val _) fun _ => rfl
/-- A construction of the equalizer fork. -/
def equalizer {X Y : ωCPO.{v}} (f g : X ⟶ Y) : Fork f g :=
Fork.ofι (P := ωCPO.of { a // f a = g a }) (equalizerι f g)
(ContinuousHom.ext _ _ fun x => x.2)
/-- The equalizer fork is a limit. -/
def isEqualizer {X Y : ωCPO.{v}} (f g : X ⟶ Y) : IsLimit (equalizer f g) :=
Fork.IsLimit.mk' _ fun s =>
⟨{ toFun := fun x => ⟨s.ι x, by apply ContinuousHom.congr_fun s.condition⟩
monotone' := fun _ _ h => s.ι.monotone h
map_ωSup' := fun x => Subtype.ext (s.ι.continuous x)
}, by ext; rfl, fun hm => by
ext x : 2; apply Subtype.ext ?_ -- Porting note (https://github.com/leanprover-community/mathlib4/issues/11041): Originally `ext`
apply ContinuousHom.congr_fun hm⟩
end HasEqualizers
instance : HasProducts.{v} ωCPO.{v} :=
fun _ => { has_limit := fun _ => hasLimit_of_iso Discrete.natIsoFunctor.symm }
instance {X Y : ωCPO.{v}} (f g : X ⟶ Y) : HasLimit (parallelPair f g) :=
HasLimit.mk ⟨_, HasEqualizers.isEqualizer f g⟩
instance : HasEqualizers ωCPO.{v} :=
hasEqualizers_of_hasLimit_parallelPair _
instance : HasLimits ωCPO.{v} :=
has_limits_of_hasEqualizers_and_products
end
end ωCPO
|
qpoly.v
|
From HB Require Import structures.
From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice.
From mathcomp Require Import fintype tuple bigop binomial finset finfun ssralg.
From mathcomp Require Import countalg finalg poly polydiv perm fingroup matrix.
From mathcomp Require Import mxalgebra mxpoly vector countalg.
(******************************************************************************)
(* This file defines the algebras R[X]/<p> and their theory. *)
(* It mimics the zmod file for polynomials *)
(* First, it defines polynomials of bounded size (equivalent of 'I_n), *)
(* gives it a structure of choice, finite and countable ring, ..., and *)
(* lmodule, when possible. *)
(* Internally, the construction uses poly_rV and rVpoly, but they should not *)
(* be exposed. *)
(* We provide two bases: the 'X^i and the lagrange polynomials. *)
(* {poly_n R} == the type of polynomial of size at most n *)
(* irreducibleb p == boolean decision procedure for irreducibility *)
(* of a bounded size polynomial over a finite idomain *)
(* Considering {poly_n F} over a field F, it is a vectType and *)
(* 'nX^i == 'X^i as an element of {poly_n R} *)
(* polynX == [tuple 'X^0, ..., 'X^(n - 1)], basis of {poly_n R} *)
(* x.-lagrange == lagrange basis of {poly_n R} wrt x : nat -> F *)
(* x.-lagrange_ i == the ith lagrange polynomial wrt the sampling points x *)
(* Second, it defines polynomials quotiented by a poly (equivalent of 'Z_p), *)
(* as bounded polynomial. As we are aiming to build a ring structure we need *)
(* the polynomial to be monic and of size greater than one. If it is not the *)
(* case we quotient by 'X *)
(* mk_monic p == the actual polynomial on which we quotient *)
(* if p is monic and of size > 1 it is p otherwise 'X *)
(* {poly %/ p} == defined as {poly_(size (mk_poly p)).-1 R} on which *)
(* there is a ring structure *)
(* in_qpoly q == turn the polynomial q into an element of {poly %/ p} by *)
(* taking a modulo *)
(* 'qX == in_qpoly 'X *)
(* The last part that defines the field structure when the quotient is an *)
(* irreducible polynomial is defined in field/qfpoly *)
(******************************************************************************)
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Import GRing.Theory.
Import Pdiv.CommonRing.
Import Pdiv.RingMonic.
Import Pdiv.Field.
Import FinRing.Theory.
Local Open Scope ring_scope.
Reserved Notation "'{poly_' n R }" (n at level 2, format "'{poly_' n R }").
Reserved Notation "''nX^' i" (at level 1, format "''nX^' i").
Reserved Notation "x .-lagrange" (format "x .-lagrange").
Reserved Notation "x .-lagrange_" (format "x .-lagrange_").
Reserved Notation "'qX".
Reserved Notation "{ 'poly' '%/' p }"
(p at level 2, format "{ 'poly' '%/' p }").
Section poly_of_size_zmod.
Context {R : nzRingType}.
Implicit Types (n : nat).
Section poly_of_size.
Variable (n : nat).
Definition poly_of_size_pred := fun p : {poly R} => size p <= n.
Arguments poly_of_size_pred _ /.
Definition poly_of_size := [qualify a p | poly_of_size_pred p].
Lemma npoly_submod_closed : submod_closed poly_of_size.
Proof.
split=> [|x p q sp sq]; rewrite qualifE/= ?size_polyC ?eqxx//.
rewrite (leq_trans (size_polyD _ _)) // geq_max.
by rewrite (leq_trans (size_scale_leq _ _)).
Qed.
HB.instance Definition _ :=
GRing.isSubmodClosed.Build R {poly R} poly_of_size_pred npoly_submod_closed.
End poly_of_size.
Arguments poly_of_size_pred _ _ /.
Section npoly.
Variable (n : nat).
Record npoly : predArgType := NPoly {
polyn :> {poly R};
_ : polyn \is a poly_of_size n
}.
HB.instance Definition _ := [isSub for @polyn].
Lemma npoly_is_a_poly_of_size (p : npoly) : val p \is a poly_of_size n.
Proof. by case: p. Qed.
Hint Resolve npoly_is_a_poly_of_size : core.
Lemma size_npoly (p : npoly) : size p <= n.
Proof. exact: npoly_is_a_poly_of_size. Qed.
Hint Resolve size_npoly : core.
HB.instance Definition _ := [Choice of npoly by <:].
HB.instance Definition _ := [SubChoice_isSubLmodule of npoly by <:].
Definition npoly_rV : npoly -> 'rV[R]_n := poly_rV \o val.
Definition rVnpoly : 'rV[R]_n -> npoly :=
insubd (0 : npoly) \o rVpoly.
Arguments rVnpoly /.
Arguments npoly_rV /.
Lemma npoly_rV_K : cancel npoly_rV rVnpoly.
Proof.
move=> p /=; apply/val_inj.
by rewrite val_insubd [_ \is a _]size_poly ?poly_rV_K.
Qed.
Lemma rVnpolyK : cancel rVnpoly npoly_rV.
Proof. by move=> p /=; rewrite val_insubd [_ \is a _]size_poly rVpolyK. Qed.
Hint Resolve npoly_rV_K rVnpolyK : core.
Lemma npoly_vect_axiom : Vector.axiom n npoly.
Proof. by exists npoly_rV; [exact:linearPZ | exists rVnpoly]. Qed.
HB.instance Definition _ := Lmodule_hasFinDim.Build R npoly npoly_vect_axiom.
End npoly.
End poly_of_size_zmod.
Arguments npoly {R}%_type n%_N.
Notation "'{poly_' n R }" := (@npoly R n) : type_scope.
#[global]
Hint Resolve size_npoly npoly_is_a_poly_of_size : core.
Arguments poly_of_size_pred _ _ _ /.
Arguments npoly : clear implicits.
HB.instance Definition _ (R : countNzRingType) n :=
[Countable of {poly_n R} by <:].
HB.instance Definition _ (R : finNzRingType) n : isFinite {poly_n R} :=
CanIsFinite (@npoly_rV_K R n).
Section npoly_theory.
Context (R : nzRingType) {n : nat}.
Lemma polyn_is_linear : linear (@polyn _ _ : {poly_n R} -> _).
Proof. by []. Qed.
HB.instance Definition _ :=
GRing.isSemilinear.Build R {poly_n R} {poly R} _ (polyn (n:=n))
(GRing.semilinear_linear polyn_is_linear).
Canonical mk_npoly (E : nat -> R) : {poly_n R} :=
@NPoly R _ (\poly_(i < n) E i) (size_poly _ _).
Fact size_npoly0 : size (0 : {poly R}) <= n.
Proof. by rewrite size_poly0. Qed.
Definition npoly0 := NPoly (size_npoly0).
Fact npolyp_key : unit. Proof. exact: tt. Qed.
Definition npolyp : {poly R} -> {poly_n R} :=
locked_with npolyp_key (mk_npoly \o (nth 0)).
Definition npoly_of_seq := npolyp \o Poly.
Lemma npolyP (p q : {poly_n R}) : nth 0 p =1 nth 0 q <-> p = q.
Proof. by split => [/polyP/val_inj|->]. Qed.
Lemma coef_npolyp (p : {poly R}) i : (npolyp p)`_i = if i < n then p`_i else 0.
Proof. by rewrite /npolyp unlock /= coef_poly. Qed.
Lemma big_coef_npoly (p : {poly_n R}) i : n <= i -> p`_i = 0.
Proof.
by move=> i_big; rewrite nth_default // (leq_trans _ i_big) ?size_npoly.
Qed.
Lemma npolypK (p : {poly R}) : size p <= n -> npolyp p = p :> {poly R}.
Proof.
move=> spn; apply/polyP=> i; rewrite coef_npolyp.
by have [i_big|i_small] // := ltnP; rewrite nth_default ?(leq_trans spn).
Qed.
Lemma coefn_sum (I : Type) (r : seq I) (P : pred I)
(F : I -> {poly_n R}) (k : nat) :
(\sum_(i <- r | P i) F i)`_k = \sum_(i <- r | P i) (F i)`_k.
Proof. by rewrite !raddf_sum //= coef_sum. Qed.
End npoly_theory.
Arguments mk_npoly {R} n E.
Arguments npolyp {R} n p.
Section fin_npoly.
Variable R : finNzRingType.
Variable n : nat.
Implicit Types p q : {poly_n R}.
Definition npoly_enum : seq {poly_n R} :=
if n isn't n.+1 then [:: npoly0 _] else
pmap insub [seq \poly_(i < n.+1) c (inord i) | c : (R ^ n.+1)%type].
Lemma npoly_enum_uniq : uniq npoly_enum.
Proof.
rewrite /npoly_enum; case: n=> [|k] //.
rewrite pmap_sub_uniq // map_inj_uniq => [|f g eqfg]; rewrite ?enum_uniq //.
apply/ffunP => /= i; have /(congr1 (fun p : {poly _} => p`_i)) := eqfg.
by rewrite !coef_poly ltn_ord inord_val.
Qed.
Lemma mem_npoly_enum p : p \in npoly_enum.
Proof.
rewrite /npoly_enum; case: n => [|k] // in p *.
case: p => [p sp] /=.
by rewrite in_cons -val_eqE /= -size_poly_leq0 [size _ <= _]sp.
rewrite mem_pmap_sub; apply/mapP.
eexists [ffun i : 'I__ => p`_i]; first by rewrite mem_enum.
apply/polyP => i; rewrite coef_poly.
have [i_small|i_big] := ltnP; first by rewrite ffunE /= inordK.
by rewrite nth_default // 1?(leq_trans _ i_big) // size_npoly.
Qed.
Lemma card_npoly : #|{poly_n R}| = (#|R| ^ n)%N.
Proof.
rewrite -(card_imset _ (can_inj (@npoly_rV_K _ _))) eq_cardT.
by rewrite -cardT /= card_mx mul1n.
by move=> v; apply/imsetP; exists (rVnpoly v); rewrite ?rVnpolyK //.
Qed.
End fin_npoly.
Section Irreducible.
Variable R : finIdomainType.
Variable p : {poly R}.
Definition irreducibleb :=
((1 < size p) &&
[forall q : {poly_((size p).-1) R},
(Pdiv.Ring.rdvdp q p)%R ==> (size q <= 1)])%N.
Lemma irreducibleP : reflect (irreducible_poly p) irreducibleb.
Proof.
rewrite /irreducibleb /irreducible_poly.
apply: (iffP idP) => [/andP[sp /'forall_implyP /= Fp]|[sp Fpoly]].
have sp_gt0 : size p > 0 by case: size sp.
have p_neq0 : p != 0 by rewrite -size_poly_eq0; case: size sp.
split => // q sq_neq1 dvd_qp; rewrite -dvdp_size_eqp // eqn_leq dvdp_leq //=.
apply: contraNT sq_neq1; rewrite -ltnNge => sq_lt_sp.
have q_small: (size q <= (size p).-1)%N by rewrite -ltnS prednK.
rewrite Pdiv.Idomain.dvdpE in dvd_qp.
have /= := Fp (NPoly q_small) dvd_qp.
rewrite leq_eqVlt ltnS => /orP[//|]; rewrite size_poly_leq0 => /eqP q_eq0.
by rewrite -Pdiv.Idomain.dvdpE q_eq0 dvd0p (negPf p_neq0) in dvd_qp.
have sp_gt0 : size p > 0 by case: size sp.
rewrite sp /=; apply/'forall_implyP => /= q.
rewrite -Pdiv.Idomain.dvdpE=> dvd_qp.
have [/eqP->//|/Fpoly/(_ dvd_qp)/eqp_size sq_eq_sp] := boolP (size q == 1%N).
by have := size_npoly q; rewrite sq_eq_sp -ltnS prednK ?ltnn.
Qed.
End Irreducible.
Section Vspace.
Variable (K : fieldType) (n : nat).
Lemma dim_polyn : \dim (fullv : {vspace {poly_n K}}) = n.
Proof. by rewrite [LHS]mxrank_gen mxrank1. Qed.
Definition npolyX : n.-tuple {poly_n K} := [tuple npolyp n 'X^i | i < n].
Notation "''nX^' i" := (tnth npolyX i).
Lemma npolyXE (i : 'I_n) : 'nX^i = 'X^i :> {poly _}.
Proof. by rewrite tnth_map tnth_ord_tuple npolypK // size_polyXn. Qed.
Lemma nth_npolyX (i : 'I_n) : npolyX`_i = 'nX^i.
Proof. by rewrite -tnth_nth. Qed.
Lemma npolyX_free : free npolyX.
Proof.
apply/freeP=> u /= sum_uX_eq0 i; have /npolyP /(_ i) := sum_uX_eq0.
rewrite (@big_morph _ _ _ 0%R +%R) // coef_sum coef0.
rewrite (bigD1 i) ?big1 /= ?addr0 ?coefZ ?(nth_map 0%N) ?size_iota //.
by rewrite nth_npolyX npolyXE coefXn eqxx mulr1.
move=> j; rewrite -val_eqE /= => neq_ji.
by rewrite nth_npolyX npolyXE coefZ coefXn eq_sym (negPf neq_ji) mulr0.
Qed.
Lemma npolyX_full : basis_of fullv npolyX.
Proof.
by rewrite basisEfree npolyX_free subvf size_map size_enum_ord dim_polyn /=.
Qed.
Lemma npolyX_coords (p : {poly_n K}) i : coord npolyX i p = p`_i.
Proof.
rewrite [p in RHS](coord_basis npolyX_full) ?memvf // coefn_sum.
rewrite (bigD1 i) //= coefZ nth_npolyX npolyXE coefXn eqxx mulr1 big1 ?addr0//.
move=> j; rewrite -val_eqE => /= neq_ji.
by rewrite coefZ nth_npolyX npolyXE coefXn eq_sym (negPf neq_ji) mulr0.
Qed.
Lemma npolyX_gen (p : {poly K}) : (size p <= n)%N ->
p = \sum_(i < n) p`_i *: 'nX^i.
Proof.
move=> sp; rewrite -[p](@npolypK _ n) //.
rewrite [npolyp _ _ in LHS](coord_basis npolyX_full) ?memvf //.
rewrite (@big_morph _ _ _ 0%R +%R) // !raddf_sum.
by apply: eq_bigr=> i _; rewrite npolyX_coords //= nth_npolyX npolyXE.
Qed.
Section lagrange.
Variables (x : nat -> K).
Notation lagrange_def := (fun i :'I_n =>
let k := i in let p := \prod_(j < n | j != k) ('X - (x j)%:P)
in (p.[x k]^-1)%:P * p).
Fact lagrange_key : unit. Proof. exact: tt. Qed.
Definition lagrange := locked_with lagrange_key
[tuple npolyp n (lagrange_def i) | i < n].
Notation lagrange_ := (tnth lagrange).
Hypothesis n_gt0 : (0 < n)%N.
Hypothesis x_inj : injective x.
Let lagrange_def_sample (i j : 'I_n) : (lagrange_def i).[x j] = (i == j)%:R.
Proof.
clear n_gt0; rewrite hornerM hornerC; set p := (\prod_(_ < _ | _) _).
have [<-|neq_ij] /= := altP eqP.
rewrite mulVf // horner_prod; apply/prodf_neq0 => k neq_ki.
by rewrite hornerXsubC subr_eq0 inj_eq // eq_sym.
rewrite [X in _ * X]horner_prod (bigD1 j) 1?eq_sym //=.
by rewrite hornerXsubC subrr mul0r mulr0.
Qed.
Let size_lagrange_def i : size (lagrange_def i) = n.
Proof.
rewrite size_Cmul; last first.
suff : (lagrange_def i).[x i] != 0.
by rewrite hornerE mulf_eq0 => /norP [].
by rewrite lagrange_def_sample ?eqxx ?oner_eq0.
rewrite size_prod /=; last first.
by move=> j neq_ji; rewrite polyXsubC_eq0.
rewrite (eq_bigr (fun=> (2 * 1)%N)); last first.
by move=> j neq_ji; rewrite size_XsubC.
rewrite -big_distrr /= sum1_card cardC1 card_ord /=.
by case: (n) {i} n_gt0 => ?; rewrite mul2n -addnn -addSn addnK.
Qed.
Lemma lagrangeE i : lagrange_ i = lagrange_def i :> {poly _}.
Proof.
rewrite [lagrange]unlock tnth_map.
by rewrite [val _]npolypK tnth_ord_tuple // size_lagrange_def.
Qed.
Lemma nth_lagrange (i : 'I_n) : lagrange`_i = lagrange_ i.
Proof. by rewrite -tnth_nth. Qed.
Lemma size_lagrange_ i : size (lagrange_ i) = n.
Proof. by rewrite lagrangeE size_lagrange_def. Qed.
Lemma size_lagrange : size lagrange = n.
Proof. by rewrite size_tuple. Qed.
Lemma lagrange_sample (i j : 'I_n) : (lagrange_ i).[x j] = (i == j)%:R.
Proof. by rewrite lagrangeE lagrange_def_sample. Qed.
Lemma lagrange_free : free lagrange.
Proof.
apply/freeP=> lambda eq_l i.
have /(congr1 (fun p : {poly__ _} => p.[x i])) := eq_l.
rewrite (@big_morph _ _ _ 0%R +%R) // horner_sum horner0.
rewrite (bigD1 i) // big1 => [|j /= /negPf ji] /=;
by rewrite ?hornerE nth_lagrange lagrange_sample ?eqxx ?ji ?mulr1 ?mulr0.
Qed.
Lemma lagrange_full : basis_of fullv lagrange.
Proof.
by rewrite basisEfree lagrange_free subvf size_lagrange dim_polyn /=.
Qed.
Lemma lagrange_coords (p : {poly_n K}) i : coord lagrange i p = p.[x i].
Proof.
rewrite [p in RHS](coord_basis lagrange_full) ?memvf //.
rewrite (@big_morph _ _ _ 0%R +%R) // horner_sum.
rewrite (bigD1 i) // big1 => [|j /= /negPf ji] /=;
by rewrite ?hornerE nth_lagrange lagrange_sample ?eqxx ?ji ?mulr1 ?mulr0.
Qed.
Lemma lagrange_gen (p : {poly K}) : (size p <= n)%N ->
p = \sum_(i < n) p.[x i]%:P * lagrange_ i.
Proof.
move=> sp; rewrite -[p](@npolypK _ n) //.
rewrite [npolyp _ _ in LHS](coord_basis lagrange_full) ?memvf //.
rewrite (@big_morph _ _ _ 0%R +%R) //; apply: eq_bigr=> i _.
by rewrite lagrange_coords mul_polyC nth_lagrange.
Qed.
End lagrange.
End Vspace.
Notation "''nX^' i" := (tnth (npolyX _) i) : ring_scope.
Notation "x .-lagrange" := (lagrange x) : ring_scope.
Notation "x .-lagrange_" := (tnth x.-lagrange) : ring_scope.
Section Qpoly.
Variable R : nzRingType.
Variable h : {poly R}.
Definition mk_monic :=
if (1 < size h)%N && (h \is monic) then h else 'X.
Definition qpoly := {poly_(size mk_monic).-1 R}.
End Qpoly.
Notation "{ 'poly' '%/' p }" := (qpoly p) : type_scope.
Section QpolyProp.
Variable R : nzRingType.
Variable h : {poly R}.
Lemma monic_mk_monic : (mk_monic h) \is monic.
Proof.
rewrite /mk_monic; case: leqP=> [_|/=]; first by apply: monicX.
by case E : (h \is monic) => [->//|] => _; apply: monicX.
Qed.
Lemma size_mk_monic_gt1 : (1 < size (mk_monic h))%N.
Proof.
by rewrite !fun_if size_polyX; case: leqP => //=; rewrite if_same.
Qed.
Lemma size_mk_monic_gt0 : (0 < size (mk_monic h))%N.
Proof. by rewrite (leq_trans _ size_mk_monic_gt1). Qed.
Lemma mk_monic_neq0 : mk_monic h != 0.
Proof. by rewrite -size_poly_gt0 size_mk_monic_gt0. Qed.
Lemma size_mk_monic (p : {poly %/ h}) : size p < size (mk_monic h).
Proof.
have: (p : {poly R}) \is a poly_of_size (size (mk_monic h)).-1 by case: p.
by rewrite qualifE/= -ltnS prednK // size_mk_monic_gt0.
Qed.
(* standard inject *)
Lemma poly_of_size_mod p :
rmodp p (mk_monic h) \is a poly_of_size (size (mk_monic h)).-1.
Proof.
rewrite qualifE/= -ltnS prednK ?size_mk_monic_gt0 //.
by apply: ltn_rmodpN0; rewrite mk_monic_neq0.
Qed.
Definition in_qpoly p : {poly %/ h} := NPoly (poly_of_size_mod p).
Lemma in_qpoly_small (p : {poly R}) :
size p < size (mk_monic h) -> in_qpoly p = p :> {poly R}.
Proof. exact: rmodp_small. Qed.
Lemma in_qpoly0 : in_qpoly 0 = 0.
Proof. by apply/val_eqP; rewrite /= rmod0p. Qed.
Lemma in_qpolyD p q : in_qpoly (p + q) = in_qpoly p + in_qpoly q.
Proof. by apply/val_eqP=> /=; rewrite rmodpD ?monic_mk_monic. Qed.
Lemma in_qpolyZ a p : in_qpoly (a *: p) = a *: in_qpoly p.
Proof. apply/val_eqP=> /=; rewrite rmodpZ ?monic_mk_monic //. Qed.
Fact in_qpoly_is_linear : linear in_qpoly.
Proof. by move=> k p q; rewrite in_qpolyD in_qpolyZ. Qed.
HB.instance Definition _ :=
GRing.isSemilinear.Build R {poly R} {poly_(size (mk_monic h)).-1 R} _ in_qpoly
(GRing.semilinear_linear in_qpoly_is_linear).
Lemma qpolyC_proof k :
(k%:P : {poly R}) \is a poly_of_size (size (mk_monic h)).-1.
Proof.
rewrite qualifE/= -ltnS size_polyC prednK ?size_mk_monic_gt0 //.
by rewrite (leq_ltn_trans _ size_mk_monic_gt1) //; case: eqP.
Qed.
Definition qpolyC k : {poly %/ h} := NPoly (qpolyC_proof k).
Lemma qpolyCE k : qpolyC k = k%:P :> {poly R}.
Proof. by []. Qed.
Lemma qpolyC0 : qpolyC 0 = 0.
Proof. by apply/val_eqP/eqP. Qed.
Definition qpoly1 := qpolyC 1.
Definition qpoly_mul (q1 q2 : {poly %/ h}) : {poly %/ h} :=
in_qpoly ((q1 : {poly R}) * q2).
Lemma qpoly_mul1z : left_id qpoly1 qpoly_mul.
Proof.
by move=> x; apply: val_inj; rewrite /= mul1r rmodp_small // size_mk_monic.
Qed.
Lemma qpoly_mulz1 : right_id qpoly1 qpoly_mul.
Proof.
by move=> x; apply: val_inj; rewrite /= mulr1 rmodp_small // size_mk_monic.
Qed.
Lemma qpoly_nontrivial : qpoly1 != 0.
Proof. by apply/eqP/val_eqP; rewrite /= oner_eq0. Qed.
Definition qpolyX := in_qpoly 'X.
Notation "'qX" := qpolyX.
Lemma qpolyXE : 2 < size h -> h \is monic -> 'qX = 'X :> {poly R}.
Proof.
move=> sh_gt2 h_mo.
by rewrite in_qpoly_small // size_polyX /mk_monic ifT // (ltn_trans _ sh_gt2).
Qed.
End QpolyProp.
Notation "'qX" := (qpolyX _) : ring_scope.
Lemma mk_monic_X (R : nzRingType) : mk_monic 'X = 'X :> {poly R}.
Proof. by rewrite /mk_monic size_polyX monicX. Qed.
Lemma mk_monic_Xn (R : nzRingType) n : mk_monic 'X^n = 'X^(n.-1.+1) :> {poly R}.
Proof. by case: n => [|n]; rewrite /mk_monic size_polyXn monicXn /= ?expr1. Qed.
Lemma card_qpoly (R : finNzRingType) (h : {poly R}):
#|{poly %/ h}| = #|R| ^ (size (mk_monic h)).-1.
Proof. by rewrite card_npoly. Qed.
Lemma card_monic_qpoly (R : finNzRingType) (h : {poly R}):
1 < size h -> h \is monic -> #|{poly %/ h}| = #|R| ^ (size h).-1.
Proof. by move=> sh_gt1 hM; rewrite card_qpoly /mk_monic sh_gt1 hM. Qed.
Section QRing.
Variable A : comNzRingType.
Variable h : {poly A}.
(* Ring operations *)
Lemma qpoly_mulC : commutative (@qpoly_mul A h).
Proof. by move=> p q; apply: val_inj; rewrite /= mulrC. Qed.
Lemma qpoly_mulA : associative (@qpoly_mul A h).
Proof.
have rPM := monic_mk_monic h; move=> p q r; apply: val_inj.
by rewrite /= rmodp_mulml // rmodp_mulmr // mulrA.
Qed.
Lemma qpoly_mul_addr : right_distributive (@qpoly_mul A h) +%R.
Proof.
have rPM := monic_mk_monic h; move=> p q r; apply: val_inj.
by rewrite /= !(mulrDr, rmodp_mulmr, rmodpD).
Qed.
Lemma qpoly_mul_addl : left_distributive (@qpoly_mul A h) +%R.
Proof. by move=> p q r; rewrite -!(qpoly_mulC r) qpoly_mul_addr. Qed.
HB.instance Definition _ := GRing.Zmodule_isComNzRing.Build {poly__ A} qpoly_mulA
qpoly_mulC (@qpoly_mul1z _ h) qpoly_mul_addl (@qpoly_nontrivial _ h).
HB.instance Definition _ := GRing.ComNzRing.on {poly %/ h}.
Lemma in_qpoly1 : in_qpoly h 1 = 1.
Proof.
apply/val_eqP/eqP/in_qpoly_small.
by rewrite size_polyC oner_eq0 /= size_mk_monic_gt1.
Qed.
Lemma in_qpolyM q1 q2 : in_qpoly h (q1 * q2) = in_qpoly h q1 * in_qpoly h q2.
Proof.
apply/val_eqP => /=.
by rewrite rmodp_mulml ?rmodp_mulmr // monic_mk_monic.
Qed.
Fact in_qpoly_monoid_morphism : monoid_morphism (in_qpoly h).
Proof. by split; [ apply: in_qpoly1 | apply: in_qpolyM]. Qed.
#[warning="-deprecated-since-mathcomp-2.5.0", deprecated(since="mathcomp 2.5.0",
note="use `in_qpoly_is_monoid_morphism` instead")]
Definition in_qpoly_is_multiplicative :=
(fun g => (g.2,g.1)) in_qpoly_monoid_morphism.
HB.instance Definition _ :=
GRing.isMonoidMorphism.Build {poly A} {poly %/ h} (in_qpoly h)
in_qpoly_monoid_morphism.
Lemma poly_of_qpoly_sum I (r : seq I) (P1 : pred I) (F : I -> {poly %/ h}) :
((\sum_(i <- r | P1 i) F i) =
\sum_(p <- r | P1 p) ((F p) : {poly A}) :> {poly A})%R.
Proof. by elim/big_rec2: _ => // i p q IH <-. Qed.
Lemma poly_of_qpolyD (p q : {poly %/ h}) :
p + q= (p : {poly A}) + q :> {poly A}.
Proof. by []. Qed.
Lemma qpolyC_natr p : (p%:R : {poly %/ h}) = p%:R :> {poly A}.
Proof. by elim: p => //= p IH; rewrite !mulrS poly_of_qpolyD IH. Qed.
Lemma pchar_qpoly : [pchar {poly %/ h}] =i [pchar A].
Proof.
move=> p; rewrite !inE; congr (_ && _).
apply/eqP/eqP=> [/(congr1 val) /=|pE]; last first.
by apply: val_inj => //=; rewrite qpolyC_natr /= -polyC_natr pE.
rewrite !qpolyC_natr -!polyC_natr => /(congr1 val) /=.
by rewrite polyseqC polyseq0; case: eqP.
Qed.
Lemma poly_of_qpolyM (p q : {poly %/ h}) :
p * q = rmodp ((p : {poly A}) * q) (mk_monic h) :> {poly A}.
Proof. by []. Qed.
Lemma poly_of_qpolyX (p : {poly %/ h}) n :
p ^+ n = rmodp ((p : {poly A}) ^+ n) (mk_monic h) :> {poly A}.
Proof.
have HhQ := monic_mk_monic h.
elim: n => //= [|n IH].
rewrite rmodp_small // size_polyC ?(leq_ltn_trans _ (size_mk_monic_gt1 _)) //.
by case: eqP.
by rewrite exprS /= IH // rmodp_mulmr // -exprS.
Qed.
Lemma qpolyCN (a : A) : qpolyC h (- a) = -(qpolyC h a).
Proof. apply: val_inj; rewrite /= raddfN //= raddfN. Qed.
Lemma qpolyCD : {morph (qpolyC h) : a b / a + b >-> a + b}%R.
Proof. by move=> a b; apply/val_eqP/eqP=> /=; rewrite -!raddfD. Qed.
Lemma qpolyCM : {morph (qpolyC h) : a b / a * b >-> a * b}%R.
Proof.
move=> a b; apply/val_eqP/eqP=> /=; rewrite -polyCM rmodp_small //=.
have := qpolyC_proof h (a * b).
by rewrite qualifE/= -ltnS prednK // size_mk_monic_gt0.
Qed.
Lemma qpolyC_is_zmod_morphism : zmod_morphism (qpolyC h).
Proof. by move=> x y; rewrite qpolyCD qpolyCN. Qed.
#[warning="-deprecated-since-mathcomp-2.5.0", deprecated(since="mathcomp 2.5.0",
note="use `qpolyC_is_zmod_morphism` instead")]
Definition qpolyC_is_additive := qpolyC_is_zmod_morphism.
Lemma qpolyC_is_monoid_morphism : monoid_morphism (qpolyC h).
Proof. by split=> // x y; rewrite qpolyCM. Qed.
#[warning="-deprecated-since-mathcomp-2.5.0", deprecated(since="mathcomp 2.5.0",
note="use `qpolyC_is_monoid_morphism` instead")]
Definition qpolyC_is_multiplicative :=
(fun g => (g.2,g.1)) qpolyC_is_monoid_morphism.
HB.instance Definition _ := GRing.isZmodMorphism.Build A {poly %/ h} (qpolyC h)
qpolyC_is_zmod_morphism.
HB.instance Definition _ :=
GRing.isMonoidMorphism.Build A {poly %/ h} (qpolyC h)
qpolyC_is_monoid_morphism.
Definition qpoly_scale k (p : {poly %/ h}) : {poly %/ h} := (k *: p)%R.
Fact qpoly_scaleA a b p :
qpoly_scale a (qpoly_scale b p) = qpoly_scale (a * b) p.
Proof. by apply/val_eqP; rewrite /= scalerA. Qed.
Fact qpoly_scale1l : left_id 1%R qpoly_scale.
Proof. by move=> p; apply/val_eqP; rewrite /= scale1r. Qed.
Fact qpoly_scaleDr a : {morph qpoly_scale a : p q / (p + q)%R}.
Proof. by move=> p q; apply/val_eqP; rewrite /= scalerDr. Qed.
Fact qpoly_scaleDl p : {morph qpoly_scale^~ p : a b / a + b}%R.
Proof. by move=> a b; apply/val_eqP; rewrite /= scalerDl. Qed.
Fact qpoly_scaleAl a p q : qpoly_scale a (p * q) = (qpoly_scale a p * q).
Proof. by apply/val_eqP; rewrite /= -scalerAl rmodpZ // monic_mk_monic. Qed.
Fact qpoly_scaleAr a p q : qpoly_scale a (p * q) = p * (qpoly_scale a q).
Proof. by apply/val_eqP; rewrite /= -scalerAr rmodpZ // monic_mk_monic. Qed.
HB.instance Definition _ := GRing.Lmodule_isLalgebra.Build A {poly__ A}
qpoly_scaleAl.
HB.instance Definition _ := GRing.Lalgebra.on {poly %/ h}.
HB.instance Definition _ := GRing.Lalgebra_isAlgebra.Build A {poly__ A}
qpoly_scaleAr.
HB.instance Definition _ := GRing.Algebra.on {poly %/ h}.
Lemma poly_of_qpolyZ (p : {poly %/ h}) a :
a *: p = a *: (p : {poly A}) :> {poly A}.
Proof. by []. Qed.
End QRing.
#[deprecated(since="mathcomp 2.4.0", note="Use pchar_qpoly instead.")]
Notation char_qpoly := (pchar_qpoly) (only parsing).
Section Field.
Variable R : fieldType.
Variable h : {poly R}.
Local Notation hQ := (mk_monic h).
Definition qpoly_inv (p : {poly %/ h}) :=
if coprimep hQ p then let v : {poly %/ h} := in_qpoly h (egcdp hQ p).2 in
((lead_coef (v * p)) ^-1 *: v) else p.
(* Ugly *)
Lemma qpoly_mulVz (p : {poly %/ h}) : coprimep hQ p -> (qpoly_inv p * p = 1)%R.
Proof.
have hQM := monic_mk_monic h.
move=> hCp; apply: val_inj; rewrite /qpoly_inv /in_qpoly hCp /=.
have p_neq0 : p != 0%R.
apply/eqP=> pZ; move: hCp; rewrite pZ.
rewrite coprimep0 -size_poly_eq1.
by case: size (size_mk_monic_gt1 h) => [|[]].
have F : (egcdp hQ p).1 * hQ + (egcdp hQ p).2 * p %= 1.
apply: eqp_trans _ (_ : gcdp hQ p %= _).
rewrite eqp_sym.
by case: (egcdpP (mk_monic_neq0 h) p_neq0).
by rewrite -size_poly_eq1.
rewrite rmodp_mulml // -scalerAl rmodpZ // rmodp_mulml //.
rewrite -[rmodp]/Pdiv.Ring.rmodp -!Pdiv.IdomainMonic.modpE //.
have := eqp_modpl hQ F.
rewrite modpD // modp_mull add0r // .
rewrite [(1 %% _)%R]modp_small => // [egcdE|]; last first.
by rewrite size_polyC oner_eq0 size_mk_monic_gt1.
rewrite {2}(eqpfP egcdE) lead_coefC divr1 alg_polyC scale_polyC mulVf //.
rewrite lead_coef_eq0.
apply/eqP => egcdZ.
by move: egcdE; rewrite -size_poly_eq1 egcdZ size_polyC eq_sym eqxx.
Qed.
Lemma qpoly_mulzV (p : {poly %/ h}) :
coprimep hQ p -> (p * (qpoly_inv p) = 1)%R.
Proof. by move=> hCp; rewrite /= mulrC qpoly_mulVz. Qed.
Lemma qpoly_intro_unit (p q : {poly %/ h}) : (q * p = 1)%R -> coprimep hQ p.
Proof.
have hQM := monic_mk_monic h.
case; rewrite -[rmodp]/Pdiv.Ring.rmodp -!Pdiv.IdomainMonic.modpE // => qp1.
have:= coprimep1 hQ.
rewrite -coprimep_modr -[1%R]qp1 !coprimep_modr coprimepMr; by case/andP.
Qed.
Lemma qpoly_inv_out (p : {poly %/ h}) : ~~ coprimep hQ p -> qpoly_inv p = p.
Proof. by rewrite /qpoly_inv => /negPf->. Qed.
HB.instance Definition _ := GRing.ComNzRing_hasMulInverse.Build {poly__ _}
qpoly_mulVz qpoly_intro_unit qpoly_inv_out.
HB.instance Definition _ := GRing.ComUnitAlgebra.on {poly %/ h}.
Lemma irreducible_poly_coprime (A : idomainType) (p q : {poly A}) :
irreducible_poly p -> coprimep p q = ~~(p %| q)%R.
Proof.
case => H1 H2; apply/coprimepP/negP.
move=> sPq H.
by have := sPq p (dvdpp _) H; rewrite -size_poly_eq1; case: size H1 => [|[]].
move=> pNDq d dDp dPq.
rewrite -size_poly_eq1; case: eqP => // /eqP /(H2 _) => /(_ dDp) dEp.
by case: pNDq; rewrite -(eqp_dvdl _ dEp).
Qed.
End Field.
|
PartialHomeomorph.lean
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Logic.Equiv.PartialEquiv
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.Sets.Opens
/-!
# Partial homeomorphisms
This file defines homeomorphisms between open subsets of topological spaces. An element `e` of
`PartialHomeomorph X Y` is an extension of `PartialEquiv X Y`, i.e., it is a pair of functions
`e.toFun` and `e.invFun`, inverse of each other on the sets `e.source` and `e.target`.
Additionally, we require that these sets are open, and that the functions are continuous on them.
Equivalently, they are homeomorphisms there.
As in equivs, we register a coercion to functions, and we use `e x` and `e.symm x` throughout
instead of `e.toFun x` and `e.invFun x`.
## Main definitions
* `Homeomorph.toPartialHomeomorph`: associating a partial homeomorphism to a homeomorphism, with
`source = target = Set.univ`;
* `PartialHomeomorph.symm`: the inverse of a partial homeomorphism
* `PartialHomeomorph.trans`: the composition of two partial homeomorphisms
* `PartialHomeomorph.refl`: the identity partial homeomorphism
* `PartialHomeomorph.const`: a partial homeomorphism which is a constant map,
whose source and target are necessarily singleton sets
* `PartialHomeomorph.ofSet`: the identity on a set `s`
* `PartialHomeomorph.restr s`: restrict a partial homeomorphism `e` to `e.source ∩ interior s`
* `PartialHomeomorph.EqOnSource`: equivalence relation describing the "right" notion of equality
for partial homeomorphisms
* `PartialHomeomorph.prod`: the product of two partial homeomorphisms,
as a partial homeomorphism on the product space
* `PartialHomeomorph.pi`: the product of a finite family of partial homeomorphisms
* `PartialHomeomorph.disjointUnion`: combine two partial homeomorphisms with disjoint sources
and disjoint targets
* `PartialHomeomorph.lift_openEmbedding`: extend a partial homeomorphism `X → Y`
under an open embedding `X → X'`, to a partial homeomorphism `X' → Z`.
(This is used to define the disjoint union of charted spaces.)
## Implementation notes
Most statements are copied from their `PartialEquiv` versions, although some care is required
especially when restricting to subsets, as these should be open subsets.
For design notes, see `PartialEquiv.lean`.
### Local coding conventions
If a lemma deals with the intersection of a set with either source or target of a `PartialEquiv`,
then it should use `e.source ∩ s` or `e.target ∩ t`, not `s ∩ e.source` or `t ∩ e.target`.
-/
open Function Set Filter Topology
variable {X X' : Type*} {Y Y' : Type*} {Z Z' : Type*}
[TopologicalSpace X] [TopologicalSpace X'] [TopologicalSpace Y] [TopologicalSpace Y']
[TopologicalSpace Z] [TopologicalSpace Z']
/-- Partial homeomorphisms, defined on open subsets of the space -/
structure PartialHomeomorph (X : Type*) (Y : Type*) [TopologicalSpace X]
[TopologicalSpace Y] extends PartialEquiv X Y where
open_source : IsOpen source
open_target : IsOpen target
continuousOn_toFun : ContinuousOn toFun source
continuousOn_invFun : ContinuousOn invFun target
namespace PartialHomeomorph
variable (e : PartialHomeomorph X Y)
/-! Basic properties; inverse (symm instance) -/
section Basic
/-- Coercion of a partial homeomorphisms to a function. We don't use `e.toFun` because it is
actually `e.toPartialEquiv.toFun`, so `simp` will apply lemmas about `toPartialEquiv`.
While we may want to switch to this behavior later, doing it mid-port will break a lot of proofs. -/
@[coe] def toFun' : X → Y := e.toFun
/-- Coercion of a `PartialHomeomorph` to function.
Note that a `PartialHomeomorph` is not `DFunLike`. -/
instance : CoeFun (PartialHomeomorph X Y) fun _ => X → Y :=
⟨fun e => e.toFun'⟩
/-- The inverse of a partial homeomorphism -/
@[symm]
protected def symm : PartialHomeomorph Y X where
toPartialEquiv := e.toPartialEquiv.symm
open_source := e.open_target
open_target := e.open_source
continuousOn_toFun := e.continuousOn_invFun
continuousOn_invFun := e.continuousOn_toFun
/-- See Note [custom simps projection]. We need to specify this projection explicitly in this case,
because it is a composition of multiple projections. -/
def Simps.apply (e : PartialHomeomorph X Y) : X → Y := e
/-- See Note [custom simps projection] -/
def Simps.symm_apply (e : PartialHomeomorph X Y) : Y → X := e.symm
initialize_simps_projections PartialHomeomorph (toFun → apply, invFun → symm_apply)
protected theorem continuousOn : ContinuousOn e e.source :=
e.continuousOn_toFun
theorem continuousOn_symm : ContinuousOn e.symm e.target :=
e.continuousOn_invFun
@[simp, mfld_simps]
theorem mk_coe (e : PartialEquiv X Y) (a b c d) : (PartialHomeomorph.mk e a b c d : X → Y) = e :=
rfl
@[simp, mfld_simps]
theorem mk_coe_symm (e : PartialEquiv X Y) (a b c d) :
((PartialHomeomorph.mk e a b c d).symm : Y → X) = e.symm :=
rfl
theorem toPartialEquiv_injective :
Injective (toPartialEquiv : PartialHomeomorph X Y → PartialEquiv X Y)
| ⟨_, _, _, _, _⟩, ⟨_, _, _, _, _⟩, rfl => rfl
/- Register a few simp lemmas to make sure that `simp` puts the application of a local
homeomorphism in its normal form, i.e., in terms of its coercion to a function. -/
@[simp, mfld_simps]
theorem toFun_eq_coe (e : PartialHomeomorph X Y) : e.toFun = e :=
rfl
@[simp, mfld_simps]
theorem invFun_eq_coe (e : PartialHomeomorph X Y) : e.invFun = e.symm :=
rfl
@[simp, mfld_simps]
theorem coe_coe : (e.toPartialEquiv : X → Y) = e :=
rfl
@[simp, mfld_simps]
theorem coe_coe_symm : (e.toPartialEquiv.symm : Y → X) = e.symm :=
rfl
@[simp, mfld_simps]
theorem map_source {x : X} (h : x ∈ e.source) : e x ∈ e.target :=
e.map_source' h
/-- Variant of `map_source`, stated for images of subsets of `source`. -/
lemma map_source'' : e '' e.source ⊆ e.target :=
fun _ ⟨_, hx, hex⟩ ↦ mem_of_eq_of_mem (id hex.symm) (e.map_source' hx)
@[simp, mfld_simps]
theorem map_target {x : Y} (h : x ∈ e.target) : e.symm x ∈ e.source :=
e.map_target' h
@[simp, mfld_simps]
theorem left_inv {x : X} (h : x ∈ e.source) : e.symm (e x) = x :=
e.left_inv' h
@[simp, mfld_simps]
theorem right_inv {x : Y} (h : x ∈ e.target) : e (e.symm x) = x :=
e.right_inv' h
theorem eq_symm_apply {x : X} {y : Y} (hx : x ∈ e.source) (hy : y ∈ e.target) :
x = e.symm y ↔ e x = y :=
e.toPartialEquiv.eq_symm_apply hx hy
protected theorem mapsTo : MapsTo e e.source e.target := fun _ => e.map_source
protected theorem symm_mapsTo : MapsTo e.symm e.target e.source :=
e.symm.mapsTo
protected theorem leftInvOn : LeftInvOn e.symm e e.source := fun _ => e.left_inv
protected theorem rightInvOn : RightInvOn e.symm e e.target := fun _ => e.right_inv
protected theorem invOn : InvOn e.symm e e.source e.target :=
⟨e.leftInvOn, e.rightInvOn⟩
protected theorem injOn : InjOn e e.source :=
e.leftInvOn.injOn
protected theorem bijOn : BijOn e e.source e.target :=
e.invOn.bijOn e.mapsTo e.symm_mapsTo
protected theorem surjOn : SurjOn e e.source e.target :=
e.bijOn.surjOn
end Basic
/-- Interpret a `Homeomorph` as a `PartialHomeomorph` by restricting it
to an open set `s` in the domain and to `t` in the codomain. -/
@[simps! -fullyApplied apply symm_apply toPartialEquiv,
simps! -isSimp source target]
def _root_.Homeomorph.toPartialHomeomorphOfImageEq (e : X ≃ₜ Y) (s : Set X) (hs : IsOpen s)
(t : Set Y) (h : e '' s = t) : PartialHomeomorph X Y where
toPartialEquiv := e.toPartialEquivOfImageEq s t h
open_source := hs
open_target := by simpa [← h]
continuousOn_toFun := e.continuous.continuousOn
continuousOn_invFun := e.symm.continuous.continuousOn
/-- A homeomorphism induces a partial homeomorphism on the whole space -/
@[simps! (config := mfld_cfg)]
def _root_.Homeomorph.toPartialHomeomorph (e : X ≃ₜ Y) : PartialHomeomorph X Y :=
e.toPartialHomeomorphOfImageEq univ isOpen_univ univ <| by rw [image_univ, e.surjective.range_eq]
/-- Replace `toPartialEquiv` field to provide better definitional equalities. -/
def replaceEquiv (e : PartialHomeomorph X Y) (e' : PartialEquiv X Y) (h : e.toPartialEquiv = e') :
PartialHomeomorph X Y where
toPartialEquiv := e'
open_source := h ▸ e.open_source
open_target := h ▸ e.open_target
continuousOn_toFun := h ▸ e.continuousOn_toFun
continuousOn_invFun := h ▸ e.continuousOn_invFun
theorem replaceEquiv_eq_self (e' : PartialEquiv X Y)
(h : e.toPartialEquiv = e') : e.replaceEquiv e' h = e := by
cases e
subst e'
rfl
theorem source_preimage_target : e.source ⊆ e ⁻¹' e.target :=
e.mapsTo
theorem eventually_left_inverse {x} (hx : x ∈ e.source) :
∀ᶠ y in 𝓝 x, e.symm (e y) = y :=
(e.open_source.eventually_mem hx).mono e.left_inv'
theorem eventually_left_inverse' {x} (hx : x ∈ e.target) :
∀ᶠ y in 𝓝 (e.symm x), e.symm (e y) = y :=
e.eventually_left_inverse (e.map_target hx)
theorem eventually_right_inverse {x} (hx : x ∈ e.target) :
∀ᶠ y in 𝓝 x, e (e.symm y) = y :=
(e.open_target.eventually_mem hx).mono e.right_inv'
theorem eventually_right_inverse' {x} (hx : x ∈ e.source) :
∀ᶠ y in 𝓝 (e x), e (e.symm y) = y :=
e.eventually_right_inverse (e.map_source hx)
theorem eventually_ne_nhdsWithin {x} (hx : x ∈ e.source) :
∀ᶠ x' in 𝓝[≠] x, e x' ≠ e x :=
eventually_nhdsWithin_iff.2 <|
(e.eventually_left_inverse hx).mono fun x' hx' =>
mt fun h => by rw [mem_singleton_iff, ← e.left_inv hx, ← h, hx']
theorem nhdsWithin_source_inter {x} (hx : x ∈ e.source) (s : Set X) : 𝓝[e.source ∩ s] x = 𝓝[s] x :=
nhdsWithin_inter_of_mem (mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds e.open_source hx)
theorem nhdsWithin_target_inter {x} (hx : x ∈ e.target) (s : Set Y) : 𝓝[e.target ∩ s] x = 𝓝[s] x :=
e.symm.nhdsWithin_source_inter hx s
theorem image_eq_target_inter_inv_preimage {s : Set X} (h : s ⊆ e.source) :
e '' s = e.target ∩ e.symm ⁻¹' s :=
e.toPartialEquiv.image_eq_target_inter_inv_preimage h
theorem image_source_inter_eq' (s : Set X) : e '' (e.source ∩ s) = e.target ∩ e.symm ⁻¹' s :=
e.toPartialEquiv.image_source_inter_eq' s
theorem image_source_inter_eq (s : Set X) :
e '' (e.source ∩ s) = e.target ∩ e.symm ⁻¹' (e.source ∩ s) :=
e.toPartialEquiv.image_source_inter_eq s
theorem symm_image_eq_source_inter_preimage {s : Set Y} (h : s ⊆ e.target) :
e.symm '' s = e.source ∩ e ⁻¹' s :=
e.symm.image_eq_target_inter_inv_preimage h
theorem symm_image_target_inter_eq (s : Set Y) :
e.symm '' (e.target ∩ s) = e.source ∩ e ⁻¹' (e.target ∩ s) :=
e.symm.image_source_inter_eq _
theorem source_inter_preimage_inv_preimage (s : Set X) :
e.source ∩ e ⁻¹' (e.symm ⁻¹' s) = e.source ∩ s :=
e.toPartialEquiv.source_inter_preimage_inv_preimage s
theorem target_inter_inv_preimage_preimage (s : Set Y) :
e.target ∩ e.symm ⁻¹' (e ⁻¹' s) = e.target ∩ s :=
e.symm.source_inter_preimage_inv_preimage _
theorem source_inter_preimage_target_inter (s : Set Y) :
e.source ∩ e ⁻¹' (e.target ∩ s) = e.source ∩ e ⁻¹' s :=
e.toPartialEquiv.source_inter_preimage_target_inter s
theorem image_source_eq_target : e '' e.source = e.target :=
e.toPartialEquiv.image_source_eq_target
theorem symm_image_target_eq_source : e.symm '' e.target = e.source :=
e.symm.image_source_eq_target
/-- Two partial homeomorphisms are equal when they have equal `toFun`, `invFun` and `source`.
It is not sufficient to have equal `toFun` and `source`, as this only determines `invFun` on
the target. This would only be true for a weaker notion of equality, arguably the right one,
called `EqOnSource`. -/
@[ext]
protected theorem ext (e' : PartialHomeomorph X Y) (h : ∀ x, e x = e' x)
(hinv : ∀ x, e.symm x = e'.symm x) (hs : e.source = e'.source) : e = e' :=
toPartialEquiv_injective (PartialEquiv.ext h hinv hs)
@[simp, mfld_simps]
theorem symm_toPartialEquiv : e.symm.toPartialEquiv = e.toPartialEquiv.symm :=
rfl
-- The following lemmas are already simp via `PartialEquiv`
theorem symm_source : e.symm.source = e.target :=
rfl
theorem symm_target : e.symm.target = e.source :=
rfl
@[simp, mfld_simps] theorem symm_symm : e.symm.symm = e := rfl
theorem symm_bijective : Function.Bijective
(PartialHomeomorph.symm : PartialHomeomorph X Y → PartialHomeomorph Y X) :=
Function.bijective_iff_has_inverse.mpr ⟨_, symm_symm, symm_symm⟩
/-- A partial homeomorphism is continuous at any point of its source -/
protected theorem continuousAt {x : X} (h : x ∈ e.source) : ContinuousAt e x :=
(e.continuousOn x h).continuousAt (e.open_source.mem_nhds h)
/-- A partial homeomorphism inverse is continuous at any point of its target -/
theorem continuousAt_symm {x : Y} (h : x ∈ e.target) : ContinuousAt e.symm x :=
e.symm.continuousAt h
theorem tendsto_symm {x} (hx : x ∈ e.source) : Tendsto e.symm (𝓝 (e x)) (𝓝 x) := by
simpa only [ContinuousAt, e.left_inv hx] using e.continuousAt_symm (e.map_source hx)
theorem map_nhds_eq {x} (hx : x ∈ e.source) : map e (𝓝 x) = 𝓝 (e x) :=
le_antisymm (e.continuousAt hx) <|
le_map_of_right_inverse (e.eventually_right_inverse' hx) (e.tendsto_symm hx)
theorem symm_map_nhds_eq {x} (hx : x ∈ e.source) : map e.symm (𝓝 (e x)) = 𝓝 x :=
(e.symm.map_nhds_eq <| e.map_source hx).trans <| by rw [e.left_inv hx]
theorem image_mem_nhds {x} (hx : x ∈ e.source) {s : Set X} (hs : s ∈ 𝓝 x) : e '' s ∈ 𝓝 (e x) :=
e.map_nhds_eq hx ▸ Filter.image_mem_map hs
theorem map_nhdsWithin_eq {x} (hx : x ∈ e.source) (s : Set X) :
map e (𝓝[s] x) = 𝓝[e '' (e.source ∩ s)] e x :=
calc
map e (𝓝[s] x) = map e (𝓝[e.source ∩ s] x) :=
congr_arg (map e) (e.nhdsWithin_source_inter hx _).symm
_ = 𝓝[e '' (e.source ∩ s)] e x :=
(e.leftInvOn.mono inter_subset_left).map_nhdsWithin_eq (e.left_inv hx)
(e.continuousAt_symm (e.map_source hx)).continuousWithinAt
(e.continuousAt hx).continuousWithinAt
theorem map_nhdsWithin_preimage_eq {x} (hx : x ∈ e.source) (s : Set Y) :
map e (𝓝[e ⁻¹' s] x) = 𝓝[s] e x := by
rw [e.map_nhdsWithin_eq hx, e.image_source_inter_eq', e.target_inter_inv_preimage_preimage,
e.nhdsWithin_target_inter (e.map_source hx)]
theorem eventually_nhds {x : X} (p : Y → Prop) (hx : x ∈ e.source) :
(∀ᶠ y in 𝓝 (e x), p y) ↔ ∀ᶠ x in 𝓝 x, p (e x) :=
Iff.trans (by rw [e.map_nhds_eq hx]) eventually_map
theorem eventually_nhds' {x : X} (p : X → Prop) (hx : x ∈ e.source) :
(∀ᶠ y in 𝓝 (e x), p (e.symm y)) ↔ ∀ᶠ x in 𝓝 x, p x := by
rw [e.eventually_nhds _ hx]
refine eventually_congr ((e.eventually_left_inverse hx).mono fun y hy => ?_)
rw [hy]
theorem eventually_nhdsWithin {x : X} (p : Y → Prop) {s : Set X}
(hx : x ∈ e.source) : (∀ᶠ y in 𝓝[e.symm ⁻¹' s] e x, p y) ↔ ∀ᶠ x in 𝓝[s] x, p (e x) := by
refine Iff.trans ?_ eventually_map
rw [e.map_nhdsWithin_eq hx, e.image_source_inter_eq', e.nhdsWithin_target_inter (e.mapsTo hx)]
theorem eventually_nhdsWithin' {x : X} (p : X → Prop) {s : Set X}
(hx : x ∈ e.source) : (∀ᶠ y in 𝓝[e.symm ⁻¹' s] e x, p (e.symm y)) ↔ ∀ᶠ x in 𝓝[s] x, p x := by
rw [e.eventually_nhdsWithin _ hx]
refine eventually_congr <|
(eventually_nhdsWithin_of_eventually_nhds <| e.eventually_left_inverse hx).mono fun y hy => ?_
rw [hy]
/-- This lemma is useful in the manifold library in the case that `e` is a chart. It states that
locally around `e x` the set `e.symm ⁻¹' s` is the same as the set intersected with the target
of `e` and some other neighborhood of `f x` (which will be the source of a chart on `Z`). -/
theorem preimage_eventuallyEq_target_inter_preimage_inter {e : PartialHomeomorph X Y} {s : Set X}
{t : Set Z} {x : X} {f : X → Z} (hf : ContinuousWithinAt f s x) (hxe : x ∈ e.source)
(ht : t ∈ 𝓝 (f x)) :
e.symm ⁻¹' s =ᶠ[𝓝 (e x)] (e.target ∩ e.symm ⁻¹' (s ∩ f ⁻¹' t) : Set Y) := by
rw [eventuallyEq_set, e.eventually_nhds _ hxe]
filter_upwards [e.open_source.mem_nhds hxe,
mem_nhdsWithin_iff_eventually.mp (hf.preimage_mem_nhdsWithin ht)]
intro y hy hyu
simp_rw [mem_inter_iff, mem_preimage, mem_inter_iff, e.mapsTo hy, true_and, iff_self_and,
e.left_inv hy, iff_true_intro hyu]
theorem isOpen_inter_preimage {s : Set Y} (hs : IsOpen s) : IsOpen (e.source ∩ e ⁻¹' s) :=
e.continuousOn.isOpen_inter_preimage e.open_source hs
theorem isOpen_inter_preimage_symm {s : Set X} (hs : IsOpen s) : IsOpen (e.target ∩ e.symm ⁻¹' s) :=
e.symm.continuousOn.isOpen_inter_preimage e.open_target hs
/-- A partial homeomorphism is an open map on its source:
the image of an open subset of the source is open. -/
lemma isOpen_image_of_subset_source {s : Set X} (hs : IsOpen s) (hse : s ⊆ e.source) :
IsOpen (e '' s) := by
rw [(image_eq_target_inter_inv_preimage (e := e) hse)]
exact e.continuousOn_invFun.isOpen_inter_preimage e.open_target hs
/-- The image of the restriction of an open set to the source is open. -/
theorem isOpen_image_source_inter {s : Set X} (hs : IsOpen s) :
IsOpen (e '' (e.source ∩ s)) :=
e.isOpen_image_of_subset_source (e.open_source.inter hs) inter_subset_left
/-- The inverse of a partial homeomorphism `e` is an open map on `e.target`. -/
lemma isOpen_image_symm_of_subset_target {t : Set Y} (ht : IsOpen t) (hte : t ⊆ e.target) :
IsOpen (e.symm '' t) :=
isOpen_image_of_subset_source e.symm ht (e.symm_source ▸ hte)
lemma isOpen_symm_image_iff_of_subset_target {t : Set Y} (hs : t ⊆ e.target) :
IsOpen (e.symm '' t) ↔ IsOpen t := by
refine ⟨fun h ↦ ?_, fun h ↦ e.symm.isOpen_image_of_subset_source h hs⟩
have hs' : e.symm '' t ⊆ e.source := by
rw [e.symm_image_eq_source_inter_preimage hs]
apply Set.inter_subset_left
rw [← e.image_symm_image_of_subset_target hs]
exact e.isOpen_image_of_subset_source h hs'
theorem isOpen_image_iff_of_subset_source {s : Set X} (hs : s ⊆ e.source) :
IsOpen (e '' s) ↔ IsOpen s := by
rw [← e.symm.isOpen_symm_image_iff_of_subset_target hs, e.symm_symm]
section IsImage
/-!
### `PartialHomeomorph.IsImage` relation
We say that `t : Set Y` is an image of `s : Set X` under a partial homeomorphism `e` if any of the
following equivalent conditions hold:
* `e '' (e.source ∩ s) = e.target ∩ t`;
* `e.source ∩ e ⁻¹ t = e.source ∩ s`;
* `∀ x ∈ e.source, e x ∈ t ↔ x ∈ s` (this one is used in the definition).
This definition is a restatement of `PartialEquiv.IsImage` for partial homeomorphisms.
In this section we transfer API about `PartialEquiv.IsImage` to partial homeomorphisms and
add a few `PartialHomeomorph`-specific lemmas like `PartialHomeomorph.IsImage.closure`.
-/
/-- We say that `t : Set Y` is an image of `s : Set X` under a partial homeomorphism `e`
if any of the following equivalent conditions hold:
* `e '' (e.source ∩ s) = e.target ∩ t`;
* `e.source ∩ e ⁻¹ t = e.source ∩ s`;
* `∀ x ∈ e.source, e x ∈ t ↔ x ∈ s` (this one is used in the definition).
-/
def IsImage (s : Set X) (t : Set Y) : Prop :=
∀ ⦃x⦄, x ∈ e.source → (e x ∈ t ↔ x ∈ s)
namespace IsImage
variable {e} {s : Set X} {t : Set Y} {x : X} {y : Y}
theorem toPartialEquiv (h : e.IsImage s t) : e.toPartialEquiv.IsImage s t :=
h
theorem apply_mem_iff (h : e.IsImage s t) (hx : x ∈ e.source) : e x ∈ t ↔ x ∈ s :=
h hx
protected theorem symm (h : e.IsImage s t) : e.symm.IsImage t s :=
h.toPartialEquiv.symm
theorem symm_apply_mem_iff (h : e.IsImage s t) (hy : y ∈ e.target) : e.symm y ∈ s ↔ y ∈ t :=
h.symm hy
@[simp]
theorem symm_iff : e.symm.IsImage t s ↔ e.IsImage s t :=
⟨fun h => h.symm, fun h => h.symm⟩
protected theorem mapsTo (h : e.IsImage s t) : MapsTo e (e.source ∩ s) (e.target ∩ t) :=
h.toPartialEquiv.mapsTo
theorem symm_mapsTo (h : e.IsImage s t) : MapsTo e.symm (e.target ∩ t) (e.source ∩ s) :=
h.symm.mapsTo
theorem image_eq (h : e.IsImage s t) : e '' (e.source ∩ s) = e.target ∩ t :=
h.toPartialEquiv.image_eq
theorem symm_image_eq (h : e.IsImage s t) : e.symm '' (e.target ∩ t) = e.source ∩ s :=
h.symm.image_eq
theorem iff_preimage_eq : e.IsImage s t ↔ e.source ∩ e ⁻¹' t = e.source ∩ s :=
PartialEquiv.IsImage.iff_preimage_eq
alias ⟨preimage_eq, of_preimage_eq⟩ := iff_preimage_eq
theorem iff_symm_preimage_eq : e.IsImage s t ↔ e.target ∩ e.symm ⁻¹' s = e.target ∩ t :=
symm_iff.symm.trans iff_preimage_eq
alias ⟨symm_preimage_eq, of_symm_preimage_eq⟩ := iff_symm_preimage_eq
theorem iff_symm_preimage_eq' :
e.IsImage s t ↔ e.target ∩ e.symm ⁻¹' (e.source ∩ s) = e.target ∩ t := by
rw [iff_symm_preimage_eq, ← image_source_inter_eq, ← image_source_inter_eq']
alias ⟨symm_preimage_eq', of_symm_preimage_eq'⟩ := iff_symm_preimage_eq'
theorem iff_preimage_eq' : e.IsImage s t ↔ e.source ∩ e ⁻¹' (e.target ∩ t) = e.source ∩ s :=
symm_iff.symm.trans iff_symm_preimage_eq'
alias ⟨preimage_eq', of_preimage_eq'⟩ := iff_preimage_eq'
theorem of_image_eq (h : e '' (e.source ∩ s) = e.target ∩ t) : e.IsImage s t :=
PartialEquiv.IsImage.of_image_eq h
theorem of_symm_image_eq (h : e.symm '' (e.target ∩ t) = e.source ∩ s) : e.IsImage s t :=
PartialEquiv.IsImage.of_symm_image_eq h
protected theorem compl (h : e.IsImage s t) : e.IsImage sᶜ tᶜ := fun _ hx => (h hx).not
protected theorem inter {s' t'} (h : e.IsImage s t) (h' : e.IsImage s' t') :
e.IsImage (s ∩ s') (t ∩ t') := fun _ hx => (h hx).and (h' hx)
protected theorem union {s' t'} (h : e.IsImage s t) (h' : e.IsImage s' t') :
e.IsImage (s ∪ s') (t ∪ t') := fun _ hx => (h hx).or (h' hx)
protected theorem diff {s' t'} (h : e.IsImage s t) (h' : e.IsImage s' t') :
e.IsImage (s \ s') (t \ t') :=
h.inter h'.compl
theorem leftInvOn_piecewise {e' : PartialHomeomorph X Y} [∀ i, Decidable (i ∈ s)]
[∀ i, Decidable (i ∈ t)] (h : e.IsImage s t) (h' : e'.IsImage s t) :
LeftInvOn (t.piecewise e.symm e'.symm) (s.piecewise e e') (s.ite e.source e'.source) :=
h.toPartialEquiv.leftInvOn_piecewise h'
theorem inter_eq_of_inter_eq_of_eqOn {e' : PartialHomeomorph X Y} (h : e.IsImage s t)
(h' : e'.IsImage s t) (hs : e.source ∩ s = e'.source ∩ s) (Heq : EqOn e e' (e.source ∩ s)) :
e.target ∩ t = e'.target ∩ t :=
h.toPartialEquiv.inter_eq_of_inter_eq_of_eqOn h' hs Heq
theorem symm_eqOn_of_inter_eq_of_eqOn {e' : PartialHomeomorph X Y} (h : e.IsImage s t)
(hs : e.source ∩ s = e'.source ∩ s) (Heq : EqOn e e' (e.source ∩ s)) :
EqOn e.symm e'.symm (e.target ∩ t) :=
h.toPartialEquiv.symm_eq_on_of_inter_eq_of_eqOn hs Heq
theorem map_nhdsWithin_eq (h : e.IsImage s t) (hx : x ∈ e.source) : map e (𝓝[s] x) = 𝓝[t] e x := by
rw [e.map_nhdsWithin_eq hx, h.image_eq, e.nhdsWithin_target_inter (e.map_source hx)]
protected theorem closure (h : e.IsImage s t) : e.IsImage (closure s) (closure t) := fun x hx => by
simp only [mem_closure_iff_nhdsWithin_neBot, ← h.map_nhdsWithin_eq hx, map_neBot_iff]
protected theorem interior (h : e.IsImage s t) : e.IsImage (interior s) (interior t) := by
simpa only [closure_compl, compl_compl] using h.compl.closure.compl
protected theorem frontier (h : e.IsImage s t) : e.IsImage (frontier s) (frontier t) :=
h.closure.diff h.interior
theorem isOpen_iff (h : e.IsImage s t) : IsOpen (e.source ∩ s) ↔ IsOpen (e.target ∩ t) :=
⟨fun hs => h.symm_preimage_eq' ▸ e.symm.isOpen_inter_preimage hs, fun hs =>
h.preimage_eq' ▸ e.isOpen_inter_preimage hs⟩
/-- Restrict a `PartialHomeomorph` to a pair of corresponding open sets. -/
@[simps toPartialEquiv]
def restr (h : e.IsImage s t) (hs : IsOpen (e.source ∩ s)) : PartialHomeomorph X Y where
toPartialEquiv := h.toPartialEquiv.restr
open_source := hs
open_target := h.isOpen_iff.1 hs
continuousOn_toFun := e.continuousOn.mono inter_subset_left
continuousOn_invFun := e.symm.continuousOn.mono inter_subset_left
end IsImage
theorem isImage_source_target : e.IsImage e.source e.target :=
e.toPartialEquiv.isImage_source_target
theorem isImage_source_target_of_disjoint (e' : PartialHomeomorph X Y)
(hs : Disjoint e.source e'.source) (ht : Disjoint e.target e'.target) :
e.IsImage e'.source e'.target :=
e.toPartialEquiv.isImage_source_target_of_disjoint e'.toPartialEquiv hs ht
/-- Preimage of interior or interior of preimage coincide for partial homeomorphisms,
when restricted to the source. -/
theorem preimage_interior (s : Set Y) :
e.source ∩ e ⁻¹' interior s = e.source ∩ interior (e ⁻¹' s) :=
(IsImage.of_preimage_eq rfl).interior.preimage_eq
theorem preimage_closure (s : Set Y) : e.source ∩ e ⁻¹' closure s = e.source ∩ closure (e ⁻¹' s) :=
(IsImage.of_preimage_eq rfl).closure.preimage_eq
theorem preimage_frontier (s : Set Y) :
e.source ∩ e ⁻¹' frontier s = e.source ∩ frontier (e ⁻¹' s) :=
(IsImage.of_preimage_eq rfl).frontier.preimage_eq
end IsImage
/-- A `PartialEquiv` with continuous open forward map and open source is a `PartialHomeomorph`. -/
def ofContinuousOpenRestrict (e : PartialEquiv X Y) (hc : ContinuousOn e e.source)
(ho : IsOpenMap (e.source.restrict e)) (hs : IsOpen e.source) : PartialHomeomorph X Y where
toPartialEquiv := e
open_source := hs
open_target := by simpa only [range_restrict, e.image_source_eq_target] using ho.isOpen_range
continuousOn_toFun := hc
continuousOn_invFun := e.image_source_eq_target ▸ ho.continuousOn_image_of_leftInvOn e.leftInvOn
/-- A `PartialEquiv` with continuous open forward map and open source is a `PartialHomeomorph`. -/
def ofContinuousOpen (e : PartialEquiv X Y) (hc : ContinuousOn e e.source) (ho : IsOpenMap e)
(hs : IsOpen e.source) : PartialHomeomorph X Y :=
ofContinuousOpenRestrict e hc (ho.restrict hs) hs
/-- Restricting a partial homeomorphism `e` to `e.source ∩ s` when `s` is open.
This is sometimes hard to use because of the openness assumption, but it has the advantage that
when it can be used then its `PartialEquiv` is defeq to `PartialEquiv.restr`. -/
protected def restrOpen (s : Set X) (hs : IsOpen s) : PartialHomeomorph X Y :=
(@IsImage.of_symm_preimage_eq X Y _ _ e s (e.symm ⁻¹' s) rfl).restr
(IsOpen.inter e.open_source hs)
@[simp, mfld_simps]
theorem restrOpen_toPartialEquiv (s : Set X) (hs : IsOpen s) :
(e.restrOpen s hs).toPartialEquiv = e.toPartialEquiv.restr s :=
rfl
-- Already simp via `PartialEquiv`
theorem restrOpen_source (s : Set X) (hs : IsOpen s) : (e.restrOpen s hs).source = e.source ∩ s :=
rfl
/-- Restricting a partial homeomorphism `e` to `e.source ∩ interior s`. We use the interior to make
sure that the restriction is well defined whatever the set s, since partial homeomorphisms are by
definition defined on open sets. In applications where `s` is open, this coincides with the
restriction of partial equivalences -/
@[simps! (config := mfld_cfg) apply symm_apply, simps! -isSimp source target]
protected def restr (s : Set X) : PartialHomeomorph X Y :=
e.restrOpen (interior s) isOpen_interior
@[simp, mfld_simps]
theorem restr_toPartialEquiv (s : Set X) :
(e.restr s).toPartialEquiv = e.toPartialEquiv.restr (interior s) :=
rfl
theorem restr_source' (s : Set X) (hs : IsOpen s) : (e.restr s).source = e.source ∩ s := by
rw [e.restr_source, hs.interior_eq]
theorem restr_toPartialEquiv' (s : Set X) (hs : IsOpen s) :
(e.restr s).toPartialEquiv = e.toPartialEquiv.restr s := by
rw [e.restr_toPartialEquiv, hs.interior_eq]
theorem restr_eq_of_source_subset {e : PartialHomeomorph X Y} {s : Set X} (h : e.source ⊆ s) :
e.restr s = e :=
toPartialEquiv_injective <| PartialEquiv.restr_eq_of_source_subset <|
interior_maximal h e.open_source
@[simp, mfld_simps]
theorem restr_univ {e : PartialHomeomorph X Y} : e.restr univ = e :=
restr_eq_of_source_subset (subset_univ _)
theorem restr_source_inter (s : Set X) : e.restr (e.source ∩ s) = e.restr s := by
refine PartialHomeomorph.ext _ _ (fun x => rfl) (fun x => rfl) ?_
simp [e.open_source.interior_eq, ← inter_assoc]
/-- The identity on the whole space as a partial homeomorphism. -/
@[simps! (config := mfld_cfg) apply, simps! -isSimp source target]
protected def refl (X : Type*) [TopologicalSpace X] : PartialHomeomorph X X :=
(Homeomorph.refl X).toPartialHomeomorph
@[simp, mfld_simps]
theorem refl_partialEquiv : (PartialHomeomorph.refl X).toPartialEquiv = PartialEquiv.refl X :=
rfl
@[simp, mfld_simps]
theorem refl_symm : (PartialHomeomorph.refl X).symm = PartialHomeomorph.refl X :=
rfl
/-! const: `PartialEquiv.const` as a partial homeomorphism -/
section const
variable {a : X} {b : Y}
/--
This is `PartialEquiv.single` as a partial homeomorphism: a constant map,
whose source and target are necessarily singleton sets.
-/
def const (ha : IsOpen {a}) (hb : IsOpen {b}) : PartialHomeomorph X Y where
toPartialEquiv := PartialEquiv.single a b
open_source := ha
open_target := hb
continuousOn_toFun := by simp
continuousOn_invFun := by simp
@[simp, mfld_simps]
lemma const_apply (ha : IsOpen {a}) (hb : IsOpen {b}) (x : X) : (const ha hb) x = b := rfl
@[simp, mfld_simps]
lemma const_source (ha : IsOpen {a}) (hb : IsOpen {b}) : (const ha hb).source = {a} := rfl
@[simp, mfld_simps]
lemma const_target (ha : IsOpen {a}) (hb : IsOpen {b}) : (const ha hb).target = {b} := rfl
end const
/-! ofSet: the identity on a set `s` -/
section ofSet
variable {s : Set X} (hs : IsOpen s)
/-- The identity partial equivalence on a set `s` -/
@[simps! (config := mfld_cfg) apply, simps! -isSimp source target]
def ofSet (s : Set X) (hs : IsOpen s) : PartialHomeomorph X X where
toPartialEquiv := PartialEquiv.ofSet s
open_source := hs
open_target := hs
continuousOn_toFun := continuous_id.continuousOn
continuousOn_invFun := continuous_id.continuousOn
@[simp, mfld_simps]
theorem ofSet_toPartialEquiv : (ofSet s hs).toPartialEquiv = PartialEquiv.ofSet s :=
rfl
@[simp, mfld_simps]
theorem ofSet_symm : (ofSet s hs).symm = ofSet s hs :=
rfl
@[simp, mfld_simps]
theorem ofSet_univ_eq_refl : ofSet univ isOpen_univ = PartialHomeomorph.refl X := by ext <;> simp
end ofSet
/-! `trans`: composition of two partial homeomorphisms -/
section trans
variable (e' : PartialHomeomorph Y Z)
/-- Composition of two partial homeomorphisms when the target of the first and the source of
the second coincide. -/
@[simps! apply symm_apply toPartialEquiv, simps! -isSimp source target]
protected def trans' (h : e.target = e'.source) : PartialHomeomorph X Z where
toPartialEquiv := PartialEquiv.trans' e.toPartialEquiv e'.toPartialEquiv h
open_source := e.open_source
open_target := e'.open_target
continuousOn_toFun := e'.continuousOn.comp e.continuousOn <| h ▸ e.mapsTo
continuousOn_invFun := e.continuousOn_symm.comp e'.continuousOn_symm <| h.symm ▸ e'.symm_mapsTo
/-- Composing two partial homeomorphisms, by restricting to the maximal domain where their
composition is well defined.
Within the `Manifold` namespace, there is the notation `e ≫ₕ f` for this. -/
@[trans]
protected def trans : PartialHomeomorph X Z :=
PartialHomeomorph.trans' (e.symm.restrOpen e'.source e'.open_source).symm
(e'.restrOpen e.target e.open_target) (by simp [inter_comm])
@[simp, mfld_simps]
theorem trans_toPartialEquiv :
(e.trans e').toPartialEquiv = e.toPartialEquiv.trans e'.toPartialEquiv :=
rfl
@[simp, mfld_simps]
theorem coe_trans : (e.trans e' : X → Z) = e' ∘ e :=
rfl
@[simp, mfld_simps]
theorem coe_trans_symm : ((e.trans e').symm : Z → X) = e.symm ∘ e'.symm :=
rfl
theorem trans_apply {x : X} : (e.trans e') x = e' (e x) :=
rfl
theorem trans_symm_eq_symm_trans_symm : (e.trans e').symm = e'.symm.trans e.symm := rfl
/- This could be considered as a simp lemma, but there are many situations where it makes something
simple into something more complicated. -/
theorem trans_source : (e.trans e').source = e.source ∩ e ⁻¹' e'.source :=
PartialEquiv.trans_source e.toPartialEquiv e'.toPartialEquiv
theorem trans_source' : (e.trans e').source = e.source ∩ e ⁻¹' (e.target ∩ e'.source) :=
PartialEquiv.trans_source' e.toPartialEquiv e'.toPartialEquiv
theorem trans_source'' : (e.trans e').source = e.symm '' (e.target ∩ e'.source) :=
PartialEquiv.trans_source'' e.toPartialEquiv e'.toPartialEquiv
theorem image_trans_source : e '' (e.trans e').source = e.target ∩ e'.source :=
PartialEquiv.image_trans_source e.toPartialEquiv e'.toPartialEquiv
theorem trans_target : (e.trans e').target = e'.target ∩ e'.symm ⁻¹' e.target :=
rfl
theorem trans_target' : (e.trans e').target = e'.target ∩ e'.symm ⁻¹' (e'.source ∩ e.target) :=
trans_source' e'.symm e.symm
theorem trans_target'' : (e.trans e').target = e' '' (e'.source ∩ e.target) :=
trans_source'' e'.symm e.symm
theorem inv_image_trans_target : e'.symm '' (e.trans e').target = e'.source ∩ e.target :=
image_trans_source e'.symm e.symm
theorem trans_assoc (e'' : PartialHomeomorph Z Z') :
(e.trans e').trans e'' = e.trans (e'.trans e'') :=
toPartialEquiv_injective <| e.1.trans_assoc _ _
@[simp, mfld_simps]
theorem trans_refl : e.trans (PartialHomeomorph.refl Y) = e :=
toPartialEquiv_injective e.1.trans_refl
@[simp, mfld_simps]
theorem refl_trans : (PartialHomeomorph.refl X).trans e = e :=
toPartialEquiv_injective e.1.refl_trans
theorem trans_ofSet {s : Set Y} (hs : IsOpen s) : e.trans (ofSet s hs) = e.restr (e ⁻¹' s) :=
PartialHomeomorph.ext _ _ (fun _ => rfl) (fun _ => rfl) <| by
rw [trans_source, restr_source, ofSet_source, ← preimage_interior, hs.interior_eq]
theorem trans_of_set' {s : Set Y} (hs : IsOpen s) :
e.trans (ofSet s hs) = e.restr (e.source ∩ e ⁻¹' s) := by rw [trans_ofSet, restr_source_inter]
theorem ofSet_trans {s : Set X} (hs : IsOpen s) : (ofSet s hs).trans e = e.restr s :=
PartialHomeomorph.ext _ _ (fun _ => rfl) (fun _ => rfl) <| by simp [hs.interior_eq, inter_comm]
theorem ofSet_trans' {s : Set X} (hs : IsOpen s) :
(ofSet s hs).trans e = e.restr (e.source ∩ s) := by
rw [ofSet_trans, restr_source_inter]
@[simp, mfld_simps]
theorem ofSet_trans_ofSet {s : Set X} (hs : IsOpen s) {s' : Set X} (hs' : IsOpen s') :
(ofSet s hs).trans (ofSet s' hs') = ofSet (s ∩ s') (IsOpen.inter hs hs') := by
rw [(ofSet s hs).trans_ofSet hs']
ext <;> simp [hs'.interior_eq]
theorem restr_trans (s : Set X) : (e.restr s).trans e' = (e.trans e').restr s :=
toPartialEquiv_injective <|
PartialEquiv.restr_trans e.toPartialEquiv e'.toPartialEquiv (interior s)
end trans
/-! `EqOnSource`: equivalence on their source -/
section EqOnSource
/-- `EqOnSource e e'` means that `e` and `e'` have the same source, and coincide there. They
should really be considered the same partial equivalence. -/
def EqOnSource (e e' : PartialHomeomorph X Y) : Prop :=
e.source = e'.source ∧ EqOn e e' e.source
theorem eqOnSource_iff (e e' : PartialHomeomorph X Y) :
EqOnSource e e' ↔ PartialEquiv.EqOnSource e.toPartialEquiv e'.toPartialEquiv :=
Iff.rfl
/-- `EqOnSource` is an equivalence relation. -/
instance eqOnSourceSetoid : Setoid (PartialHomeomorph X Y) :=
{ PartialEquiv.eqOnSourceSetoid.comap toPartialEquiv with r := EqOnSource }
theorem eqOnSource_refl : e ≈ e := Setoid.refl _
/-- If two partial homeomorphisms are equivalent, so are their inverses. -/
theorem EqOnSource.symm' {e e' : PartialHomeomorph X Y} (h : e ≈ e') : e.symm ≈ e'.symm :=
PartialEquiv.EqOnSource.symm' h
/-- Two equivalent partial homeomorphisms have the same source. -/
theorem EqOnSource.source_eq {e e' : PartialHomeomorph X Y} (h : e ≈ e') : e.source = e'.source :=
h.1
/-- Two equivalent partial homeomorphisms have the same target. -/
theorem EqOnSource.target_eq {e e' : PartialHomeomorph X Y} (h : e ≈ e') : e.target = e'.target :=
h.symm'.1
/-- Two equivalent partial homeomorphisms have coinciding `toFun` on the source -/
theorem EqOnSource.eqOn {e e' : PartialHomeomorph X Y} (h : e ≈ e') : EqOn e e' e.source :=
h.2
/-- Two equivalent partial homeomorphisms have coinciding `invFun` on the target -/
theorem EqOnSource.symm_eqOn_target {e e' : PartialHomeomorph X Y} (h : e ≈ e') :
EqOn e.symm e'.symm e.target :=
h.symm'.2
/-- Composition of partial homeomorphisms respects equivalence. -/
theorem EqOnSource.trans' {e e' : PartialHomeomorph X Y} {f f' : PartialHomeomorph Y Z}
(he : e ≈ e') (hf : f ≈ f') : e.trans f ≈ e'.trans f' :=
PartialEquiv.EqOnSource.trans' he hf
/-- Restriction of partial homeomorphisms respects equivalence -/
theorem EqOnSource.restr {e e' : PartialHomeomorph X Y} (he : e ≈ e') (s : Set X) :
e.restr s ≈ e'.restr s :=
PartialEquiv.EqOnSource.restr he _
/-- Two equivalent partial homeomorphisms are equal when the source and target are `univ`. -/
theorem Set.EqOn.restr_eqOn_source {e e' : PartialHomeomorph X Y}
(h : EqOn e e' (e.source ∩ e'.source)) : e.restr e'.source ≈ e'.restr e.source := by
constructor
· rw [e'.restr_source' _ e.open_source]
rw [e.restr_source' _ e'.open_source]
exact Set.inter_comm _ _
· rw [e.restr_source' _ e'.open_source]
refine (EqOn.trans ?_ h).trans ?_ <;> simp only [mfld_simps, eqOn_refl]
/-- Composition of a partial homeomorphism and its inverse is equivalent to the restriction of the
identity to the source -/
theorem self_trans_symm : e.trans e.symm ≈ PartialHomeomorph.ofSet e.source e.open_source :=
PartialEquiv.self_trans_symm _
theorem symm_trans_self : e.symm.trans e ≈ PartialHomeomorph.ofSet e.target e.open_target :=
e.symm.self_trans_symm
theorem eq_of_eqOnSource_univ {e e' : PartialHomeomorph X Y} (h : e ≈ e') (s : e.source = univ)
(t : e.target = univ) : e = e' :=
toPartialEquiv_injective <| PartialEquiv.eq_of_eqOnSource_univ _ _ h s t
end EqOnSource
/-! product of two partial homeomorphisms -/
section Prod
/-- The product of two partial homeomorphisms, as a partial homeomorphism on the product space. -/
@[simps! (config := mfld_cfg) toPartialEquiv apply,
simps! -isSimp source target symm_apply]
def prod (eX : PartialHomeomorph X X') (eY : PartialHomeomorph Y Y') :
PartialHomeomorph (X × Y) (X' × Y') where
open_source := eX.open_source.prod eY.open_source
open_target := eX.open_target.prod eY.open_target
continuousOn_toFun := eX.continuousOn.prodMap eY.continuousOn
continuousOn_invFun := eX.continuousOn_symm.prodMap eY.continuousOn_symm
toPartialEquiv := eX.toPartialEquiv.prod eY.toPartialEquiv
@[simp, mfld_simps]
theorem prod_symm (eX : PartialHomeomorph X X') (eY : PartialHomeomorph Y Y') :
(eX.prod eY).symm = eX.symm.prod eY.symm :=
rfl
@[simp]
theorem refl_prod_refl :
(PartialHomeomorph.refl X).prod (PartialHomeomorph.refl Y) = PartialHomeomorph.refl (X × Y) :=
PartialHomeomorph.ext _ _ (fun _ => rfl) (fun _ => rfl) univ_prod_univ
@[simp, mfld_simps]
theorem prod_trans (e : PartialHomeomorph X Y) (f : PartialHomeomorph Y Z)
(e' : PartialHomeomorph X' Y') (f' : PartialHomeomorph Y' Z') :
(e.prod e').trans (f.prod f') = (e.trans f).prod (e'.trans f') :=
toPartialEquiv_injective <| e.1.prod_trans ..
theorem prod_eq_prod_of_nonempty {eX eX' : PartialHomeomorph X X'} {eY eY' : PartialHomeomorph Y Y'}
(h : (eX.prod eY).source.Nonempty) : eX.prod eY = eX'.prod eY' ↔ eX = eX' ∧ eY = eY' := by
obtain ⟨⟨x, y⟩, -⟩ := id h
haveI : Nonempty X := ⟨x⟩
haveI : Nonempty X' := ⟨eX x⟩
haveI : Nonempty Y := ⟨y⟩
haveI : Nonempty Y' := ⟨eY y⟩
simp_rw [PartialHomeomorph.ext_iff, prod_apply, prod_symm_apply, prod_source, Prod.ext_iff,
Set.prod_eq_prod_iff_of_nonempty h, forall_and, Prod.forall, forall_const,
and_assoc, and_left_comm]
theorem prod_eq_prod_of_nonempty'
{eX eX' : PartialHomeomorph X X'} {eY eY' : PartialHomeomorph Y Y'}
(h : (eX'.prod eY').source.Nonempty) : eX.prod eY = eX'.prod eY' ↔ eX = eX' ∧ eY = eY' := by
rw [eq_comm, prod_eq_prod_of_nonempty h, eq_comm, @eq_comm _ eY']
end Prod
/-! finite product of partial homeomorphisms -/
section Pi
variable {ι : Type*} [Finite ι] {X Y : ι → Type*} [∀ i, TopologicalSpace (X i)]
[∀ i, TopologicalSpace (Y i)] (ei : ∀ i, PartialHomeomorph (X i) (Y i))
/-- The product of a finite family of `PartialHomeomorph`s. -/
@[simps! toPartialEquiv apply symm_apply source target]
def pi : PartialHomeomorph (∀ i, X i) (∀ i, Y i) where
toPartialEquiv := PartialEquiv.pi fun i => (ei i).toPartialEquiv
open_source := isOpen_set_pi finite_univ fun i _ => (ei i).open_source
open_target := isOpen_set_pi finite_univ fun i _ => (ei i).open_target
continuousOn_toFun := continuousOn_pi.2 fun i =>
(ei i).continuousOn.comp (continuous_apply _).continuousOn fun _f hf => hf i trivial
continuousOn_invFun := continuousOn_pi.2 fun i =>
(ei i).continuousOn_symm.comp (continuous_apply _).continuousOn fun _f hf => hf i trivial
end Pi
/-! combining two partial homeomorphisms using `Set.piecewise` -/
section Piecewise
/-- Combine two `PartialHomeomorph`s using `Set.piecewise`. The source of the new
`PartialHomeomorph` is `s.ite e.source e'.source = e.source ∩ s ∪ e'.source \ s`, and similarly for
target. The function sends `e.source ∩ s` to `e.target ∩ t` using `e` and
`e'.source \ s` to `e'.target \ t` using `e'`, and similarly for the inverse function.
To ensure the maps `toFun` and `invFun` are inverse of each other on the new `source` and `target`,
the definition assumes that the sets `s` and `t` are related both by `e.is_image` and `e'.is_image`.
To ensure that the new maps are continuous on `source`/`target`, it also assumes that `e.source` and
`e'.source` meet `frontier s` on the same set and `e x = e' x` on this intersection. -/
@[simps! -fullyApplied toPartialEquiv apply]
def piecewise (e e' : PartialHomeomorph X Y) (s : Set X) (t : Set Y) [∀ x, Decidable (x ∈ s)]
[∀ y, Decidable (y ∈ t)] (H : e.IsImage s t) (H' : e'.IsImage s t)
(Hs : e.source ∩ frontier s = e'.source ∩ frontier s)
(Heq : EqOn e e' (e.source ∩ frontier s)) : PartialHomeomorph X Y where
toPartialEquiv := e.toPartialEquiv.piecewise e'.toPartialEquiv s t H H'
open_source := e.open_source.ite e'.open_source Hs
open_target :=
e.open_target.ite e'.open_target <| H.frontier.inter_eq_of_inter_eq_of_eqOn H'.frontier Hs Heq
continuousOn_toFun := continuousOn_piecewise_ite e.continuousOn e'.continuousOn Hs Heq
continuousOn_invFun :=
continuousOn_piecewise_ite e.continuousOn_symm e'.continuousOn_symm
(H.frontier.inter_eq_of_inter_eq_of_eqOn H'.frontier Hs Heq)
(H.frontier.symm_eqOn_of_inter_eq_of_eqOn Hs Heq)
@[simp]
theorem symm_piecewise (e e' : PartialHomeomorph X Y) {s : Set X} {t : Set Y}
[∀ x, Decidable (x ∈ s)] [∀ y, Decidable (y ∈ t)] (H : e.IsImage s t) (H' : e'.IsImage s t)
(Hs : e.source ∩ frontier s = e'.source ∩ frontier s)
(Heq : EqOn e e' (e.source ∩ frontier s)) :
(e.piecewise e' s t H H' Hs Heq).symm =
e.symm.piecewise e'.symm t s H.symm H'.symm
(H.frontier.inter_eq_of_inter_eq_of_eqOn H'.frontier Hs Heq)
(H.frontier.symm_eqOn_of_inter_eq_of_eqOn Hs Heq) :=
rfl
/-- Combine two `PartialHomeomorph`s with disjoint sources and disjoint targets. We reuse
`PartialHomeomorph.piecewise` then override `toPartialEquiv` to `PartialEquiv.disjointUnion`.
This way we have better definitional equalities for `source` and `target`. -/
def disjointUnion (e e' : PartialHomeomorph X Y) [∀ x, Decidable (x ∈ e.source)]
[∀ y, Decidable (y ∈ e.target)] (Hs : Disjoint e.source e'.source)
(Ht : Disjoint e.target e'.target) : PartialHomeomorph X Y :=
(e.piecewise e' e.source e.target e.isImage_source_target
(e'.isImage_source_target_of_disjoint e Hs.symm Ht.symm)
(by rw [e.open_source.inter_frontier_eq, (Hs.symm.frontier_right e'.open_source).inter_eq])
(by
rw [e.open_source.inter_frontier_eq]
exact eqOn_empty _ _)).replaceEquiv
(e.toPartialEquiv.disjointUnion e'.toPartialEquiv Hs Ht)
(PartialEquiv.disjointUnion_eq_piecewise _ _ _ _).symm
end Piecewise
section Continuity
/-- Continuity within a set at a point can be read under right composition with a local
homeomorphism, if the point is in its target -/
theorem continuousWithinAt_iff_continuousWithinAt_comp_right {f : Y → Z} {s : Set Y} {x : Y}
(h : x ∈ e.target) :
ContinuousWithinAt f s x ↔ ContinuousWithinAt (f ∘ e) (e ⁻¹' s) (e.symm x) := by
simp_rw [ContinuousWithinAt, ← @tendsto_map'_iff _ _ _ _ e,
e.map_nhdsWithin_preimage_eq (e.map_target h), (· ∘ ·), e.right_inv h]
/-- Continuity at a point can be read under right composition with a partial homeomorphism, if the
point is in its target -/
theorem continuousAt_iff_continuousAt_comp_right {f : Y → Z} {x : Y} (h : x ∈ e.target) :
ContinuousAt f x ↔ ContinuousAt (f ∘ e) (e.symm x) := by
rw [← continuousWithinAt_univ, e.continuousWithinAt_iff_continuousWithinAt_comp_right h,
preimage_univ, continuousWithinAt_univ]
/-- A function is continuous on a set if and only if its composition with a partial homeomorphism
on the right is continuous on the corresponding set. -/
theorem continuousOn_iff_continuousOn_comp_right {f : Y → Z} {s : Set Y} (h : s ⊆ e.target) :
ContinuousOn f s ↔ ContinuousOn (f ∘ e) (e.source ∩ e ⁻¹' s) := by
simp only [← e.symm_image_eq_source_inter_preimage h, ContinuousOn, forall_mem_image]
refine forall₂_congr fun x hx => ?_
rw [e.continuousWithinAt_iff_continuousWithinAt_comp_right (h hx),
e.symm_image_eq_source_inter_preimage h, inter_comm, continuousWithinAt_inter]
exact IsOpen.mem_nhds e.open_source (e.map_target (h hx))
/-- Continuity within a set at a point can be read under left composition with a local
homeomorphism if a neighborhood of the initial point is sent to the source of the local
homeomorphism -/
theorem continuousWithinAt_iff_continuousWithinAt_comp_left {f : Z → X} {s : Set Z} {x : Z}
(hx : f x ∈ e.source) (h : f ⁻¹' e.source ∈ 𝓝[s] x) :
ContinuousWithinAt f s x ↔ ContinuousWithinAt (e ∘ f) s x := by
refine ⟨(e.continuousAt hx).comp_continuousWithinAt, fun fe_cont => ?_⟩
rw [← continuousWithinAt_inter' h] at fe_cont ⊢
have : ContinuousWithinAt (e.symm ∘ e ∘ f) (s ∩ f ⁻¹' e.source) x :=
haveI : ContinuousWithinAt e.symm univ (e (f x)) :=
(e.continuousAt_symm (e.map_source hx)).continuousWithinAt
ContinuousWithinAt.comp this fe_cont (subset_univ _)
exact this.congr (fun y hy => by simp [e.left_inv hy.2]) (by simp [e.left_inv hx])
/-- Continuity at a point can be read under left composition with a partial homeomorphism if a
neighborhood of the initial point is sent to the source of the partial homeomorphism -/
theorem continuousAt_iff_continuousAt_comp_left {f : Z → X} {x : Z} (h : f ⁻¹' e.source ∈ 𝓝 x) :
ContinuousAt f x ↔ ContinuousAt (e ∘ f) x := by
have hx : f x ∈ e.source := (mem_of_mem_nhds h :)
have h' : f ⁻¹' e.source ∈ 𝓝[univ] x := by rwa [nhdsWithin_univ]
rw [← continuousWithinAt_univ, ← continuousWithinAt_univ,
e.continuousWithinAt_iff_continuousWithinAt_comp_left hx h']
/-- A function is continuous on a set if and only if its composition with a partial homeomorphism
on the left is continuous on the corresponding set. -/
theorem continuousOn_iff_continuousOn_comp_left {f : Z → X} {s : Set Z} (h : s ⊆ f ⁻¹' e.source) :
ContinuousOn f s ↔ ContinuousOn (e ∘ f) s :=
forall₂_congr fun _x hx =>
e.continuousWithinAt_iff_continuousWithinAt_comp_left (h hx)
(mem_of_superset self_mem_nhdsWithin h)
/-- A function is continuous if and only if its composition with a partial homeomorphism
on the left is continuous and its image is contained in the source. -/
theorem continuous_iff_continuous_comp_left {f : Z → X} (h : f ⁻¹' e.source = univ) :
Continuous f ↔ Continuous (e ∘ f) := by
simp only [← continuousOn_univ]
exact e.continuousOn_iff_continuousOn_comp_left (Eq.symm h).subset
end Continuity
/-- The homeomorphism obtained by restricting a `PartialHomeomorph` to a subset of the source. -/
@[simps]
def homeomorphOfImageSubsetSource {s : Set X} {t : Set Y} (hs : s ⊆ e.source) (ht : e '' s = t) :
s ≃ₜ t :=
have h₁ : MapsTo e s t := mapsTo'.2 ht.subset
have h₂ : t ⊆ e.target := ht ▸ e.image_source_eq_target ▸ image_mono hs
have h₃ : MapsTo e.symm t s := ht ▸ forall_mem_image.2 fun _x hx =>
(e.left_inv (hs hx)).symm ▸ hx
{ toFun := MapsTo.restrict e s t h₁
invFun := MapsTo.restrict e.symm t s h₃
left_inv := fun a => Subtype.ext (e.left_inv (hs a.2))
right_inv := fun b => Subtype.eq <| e.right_inv (h₂ b.2)
continuous_toFun := (e.continuousOn.mono hs).restrict_mapsTo h₁
continuous_invFun := (e.continuousOn_symm.mono h₂).restrict_mapsTo h₃ }
/-- A partial homeomorphism defines a homeomorphism between its source and target. -/
@[simps!]
def toHomeomorphSourceTarget : e.source ≃ₜ e.target :=
e.homeomorphOfImageSubsetSource subset_rfl e.image_source_eq_target
theorem secondCountableTopology_source [SecondCountableTopology Y] :
SecondCountableTopology e.source :=
e.toHomeomorphSourceTarget.secondCountableTopology
theorem nhds_eq_comap_inf_principal {x} (hx : x ∈ e.source) :
𝓝 x = comap e (𝓝 (e x)) ⊓ 𝓟 e.source := by
lift x to e.source using hx
rw [← e.open_source.nhdsWithin_eq x.2, ← map_nhds_subtype_val, ← map_comap_setCoe_val,
e.toHomeomorphSourceTarget.nhds_eq_comap, nhds_subtype_eq_comap]
simp only [Function.comp_def, toHomeomorphSourceTarget_apply_coe, comap_comap]
/-- If a partial homeomorphism has source and target equal to univ, then it induces a homeomorphism
between the whole spaces, expressed in this definition. -/
@[simps (config := mfld_cfg) apply symm_apply]
-- TODO: add a `PartialEquiv` version
def toHomeomorphOfSourceEqUnivTargetEqUniv (h : e.source = (univ : Set X)) (h' : e.target = univ) :
X ≃ₜ Y where
toFun := e
invFun := e.symm
left_inv x :=
e.left_inv <| by
rw [h]
exact mem_univ _
right_inv x :=
e.right_inv <| by
rw [h']
exact mem_univ _
continuous_toFun := by
simpa only [continuousOn_univ, h] using e.continuousOn
continuous_invFun := by
simpa only [continuousOn_univ, h'] using e.continuousOn_symm
theorem isOpenEmbedding_restrict : IsOpenEmbedding (e.source.restrict e) := by
refine .of_continuous_injective_isOpenMap (e.continuousOn.comp_continuous
continuous_subtype_val Subtype.prop) e.injOn.injective fun V hV ↦ ?_
rw [Set.restrict_eq, Set.image_comp]
exact e.isOpen_image_of_subset_source (e.open_source.isOpenMap_subtype_val V hV)
fun _ ⟨x, _, h⟩ ↦ h ▸ x.2
/-- A partial homeomorphism whose source is all of `X` defines an open embedding of `X` into `Y`.
The converse is also true; see `IsOpenEmbedding.toPartialHomeomorph`. -/
theorem to_isOpenEmbedding (h : e.source = Set.univ) : IsOpenEmbedding e :=
e.isOpenEmbedding_restrict.comp
((Homeomorph.setCongr h).trans <| Homeomorph.Set.univ X).symm.isOpenEmbedding
end PartialHomeomorph
namespace Homeomorph
variable (e : X ≃ₜ Y) (e' : Y ≃ₜ Z)
/- Register as simp lemmas that the fields of a partial homeomorphism built from a homeomorphism
correspond to the fields of the original homeomorphism. -/
@[simp, mfld_simps]
theorem refl_toPartialHomeomorph :
(Homeomorph.refl X).toPartialHomeomorph = PartialHomeomorph.refl X :=
rfl
@[simp, mfld_simps]
theorem symm_toPartialHomeomorph : e.symm.toPartialHomeomorph = e.toPartialHomeomorph.symm :=
rfl
@[simp, mfld_simps]
theorem trans_toPartialHomeomorph :
(e.trans e').toPartialHomeomorph = e.toPartialHomeomorph.trans e'.toPartialHomeomorph :=
PartialHomeomorph.toPartialEquiv_injective <| Equiv.trans_toPartialEquiv _ _
/-- Precompose a partial homeomorphism with a homeomorphism.
We modify the source and target to have better definitional behavior. -/
@[simps! -fullyApplied]
def transPartialHomeomorph (e : X ≃ₜ Y) (f' : PartialHomeomorph Y Z) : PartialHomeomorph X Z where
toPartialEquiv := e.toEquiv.transPartialEquiv f'.toPartialEquiv
open_source := f'.open_source.preimage e.continuous
open_target := f'.open_target
continuousOn_toFun := f'.continuousOn.comp e.continuous.continuousOn fun _ => id
continuousOn_invFun := e.symm.continuous.comp_continuousOn f'.symm.continuousOn
theorem transPartialHomeomorph_eq_trans (e : X ≃ₜ Y) (f' : PartialHomeomorph Y Z) :
e.transPartialHomeomorph f' = e.toPartialHomeomorph.trans f' :=
PartialHomeomorph.toPartialEquiv_injective <| Equiv.transPartialEquiv_eq_trans _ _
@[simp, mfld_simps]
theorem transPartialHomeomorph_trans (e : X ≃ₜ Y) (f : PartialHomeomorph Y Z)
(f' : PartialHomeomorph Z Z') :
(e.transPartialHomeomorph f).trans f' = e.transPartialHomeomorph (f.trans f') := by
simp only [transPartialHomeomorph_eq_trans, PartialHomeomorph.trans_assoc]
@[simp, mfld_simps]
theorem trans_transPartialHomeomorph (e : X ≃ₜ Y) (e' : Y ≃ₜ Z) (f'' : PartialHomeomorph Z Z') :
(e.trans e').transPartialHomeomorph f'' =
e.transPartialHomeomorph (e'.transPartialHomeomorph f'') := by
simp only [transPartialHomeomorph_eq_trans, PartialHomeomorph.trans_assoc,
trans_toPartialHomeomorph]
end Homeomorph
namespace Topology.IsOpenEmbedding
variable (f : X → Y) (h : IsOpenEmbedding f)
/-- An open embedding of `X` into `Y`, with `X` nonempty, defines a partial homeomorphism
whose source is all of `X`. The converse is also true; see
`PartialHomeomorph.to_isOpenEmbedding`. -/
@[simps! (config := mfld_cfg) apply source target]
noncomputable def toPartialHomeomorph [Nonempty X] : PartialHomeomorph X Y :=
PartialHomeomorph.ofContinuousOpen (h.isEmbedding.injective.injOn.toPartialEquiv f univ)
h.continuous.continuousOn h.isOpenMap isOpen_univ
variable [Nonempty X]
lemma toPartialHomeomorph_left_inv {x : X} : (h.toPartialHomeomorph f).symm (f x) = x := by
rw [← congr_fun (h.toPartialHomeomorph_apply f), PartialHomeomorph.left_inv]
exact Set.mem_univ _
lemma toPartialHomeomorph_right_inv {x : Y} (hx : x ∈ Set.range f) :
f ((h.toPartialHomeomorph f).symm x) = x := by
rw [← congr_fun (h.toPartialHomeomorph_apply f), PartialHomeomorph.right_inv]
rwa [toPartialHomeomorph_target]
end Topology.IsOpenEmbedding
/-! inclusion of an open set in a topological space -/
namespace TopologicalSpace.Opens
/- `Nonempty s` is not a type class argument because `s`, being a subset, rarely comes with a type
class instance. Then we'd have to manually provide the instance every time we use the following
lemmas, tediously using `haveI := ...` or `@foobar _ _ _ ...`. -/
variable (s : Opens X) (hs : Nonempty s)
/-- The inclusion of an open subset `s` of a space `X` into `X` is a partial homeomorphism from the
subtype `s` to `X`. -/
noncomputable def partialHomeomorphSubtypeCoe : PartialHomeomorph s X :=
IsOpenEmbedding.toPartialHomeomorph _ s.2.isOpenEmbedding_subtypeVal
@[simp, mfld_simps]
theorem partialHomeomorphSubtypeCoe_coe : (s.partialHomeomorphSubtypeCoe hs : s → X) = (↑) :=
rfl
@[simp, mfld_simps]
theorem partialHomeomorphSubtypeCoe_source : (s.partialHomeomorphSubtypeCoe hs).source = Set.univ :=
rfl
@[simp, mfld_simps]
theorem partialHomeomorphSubtypeCoe_target : (s.partialHomeomorphSubtypeCoe hs).target = s := by
simp only [partialHomeomorphSubtypeCoe, Subtype.range_coe_subtype, mfld_simps]
rfl
end TopologicalSpace.Opens
namespace PartialHomeomorph
/- post-compose with a partial homeomorphism -/
section transHomeomorph
/-- Postcompose a partial homeomorphism with a homeomorphism.
We modify the source and target to have better definitional behavior. -/
@[simps! -fullyApplied]
def transHomeomorph (e : PartialHomeomorph X Y) (f' : Y ≃ₜ Z) : PartialHomeomorph X Z where
toPartialEquiv := e.toPartialEquiv.transEquiv f'.toEquiv
open_source := e.open_source
open_target := e.open_target.preimage f'.symm.continuous
continuousOn_toFun := f'.continuous.comp_continuousOn e.continuousOn
continuousOn_invFun := e.symm.continuousOn.comp f'.symm.continuous.continuousOn fun _ => id
theorem transHomeomorph_eq_trans (e : PartialHomeomorph X Y) (f' : Y ≃ₜ Z) :
e.transHomeomorph f' = e.trans f'.toPartialHomeomorph :=
toPartialEquiv_injective <| PartialEquiv.transEquiv_eq_trans _ _
@[simp, mfld_simps]
theorem transHomeomorph_transHomeomorph (e : PartialHomeomorph X Y) (f' : Y ≃ₜ Z) (f'' : Z ≃ₜ Z') :
(e.transHomeomorph f').transHomeomorph f'' = e.transHomeomorph (f'.trans f'') := by
simp only [transHomeomorph_eq_trans, trans_assoc, Homeomorph.trans_toPartialHomeomorph]
@[simp, mfld_simps]
theorem trans_transHomeomorph (e : PartialHomeomorph X Y) (e' : PartialHomeomorph Y Z)
(f'' : Z ≃ₜ Z') :
(e.trans e').transHomeomorph f'' = e.trans (e'.transHomeomorph f'') := by
simp only [transHomeomorph_eq_trans, trans_assoc]
end transHomeomorph
/-! `subtypeRestr`: restriction to a subtype -/
section subtypeRestr
open TopologicalSpace
variable (e : PartialHomeomorph X Y)
variable {s : Opens X} (hs : Nonempty s)
/-- The restriction of a partial homeomorphism `e` to an open subset `s` of the domain type
produces a partial homeomorphism whose domain is the subtype `s`. -/
noncomputable def subtypeRestr : PartialHomeomorph s Y :=
(s.partialHomeomorphSubtypeCoe hs).trans e
theorem subtypeRestr_def : e.subtypeRestr hs = (s.partialHomeomorphSubtypeCoe hs).trans e :=
rfl
@[simp, mfld_simps]
theorem subtypeRestr_coe :
((e.subtypeRestr hs : PartialHomeomorph s Y) : s → Y) = Set.restrict ↑s (e : X → Y) :=
rfl
@[simp, mfld_simps]
theorem subtypeRestr_source : (e.subtypeRestr hs).source = (↑) ⁻¹' e.source := by
simp only [subtypeRestr_def, mfld_simps]
theorem map_subtype_source {x : s} (hxe : (x : X) ∈ e.source) :
e x ∈ (e.subtypeRestr hs).target := by
refine ⟨e.map_source hxe, ?_⟩
rw [s.partialHomeomorphSubtypeCoe_target, mem_preimage, e.leftInvOn hxe]
exact x.prop
/-- This lemma characterizes the transition functions of an open subset in terms of the transition
functions of the original space. -/
theorem subtypeRestr_symm_trans_subtypeRestr (f f' : PartialHomeomorph X Y) :
(f.subtypeRestr hs).symm.trans (f'.subtypeRestr hs) ≈
(f.symm.trans f').restr (f.target ∩ f.symm ⁻¹' s) := by
simp only [subtypeRestr_def, trans_symm_eq_symm_trans_symm]
have openness₁ : IsOpen (f.target ∩ f.symm ⁻¹' s) := f.isOpen_inter_preimage_symm s.2
rw [← ofSet_trans _ openness₁, ← trans_assoc, ← trans_assoc]
refine EqOnSource.trans' ?_ (eqOnSource_refl _)
-- f' has been eliminated !!!
have set_identity : f.symm.source ∩ (f.target ∩ f.symm ⁻¹' s) = f.symm.source ∩ f.symm ⁻¹' s := by
mfld_set_tac
have openness₂ : IsOpen (s : Set X) := s.2
rw [ofSet_trans', set_identity, ← trans_of_set' _ openness₂, trans_assoc]
refine EqOnSource.trans' (eqOnSource_refl _) ?_
-- f has been eliminated !!!
refine Setoid.trans (symm_trans_self (s.partialHomeomorphSubtypeCoe hs)) ?_
simp only [mfld_simps, Setoid.refl]
theorem subtypeRestr_symm_eqOn {U : Opens X} (hU : Nonempty U) :
EqOn e.symm (Subtype.val ∘ (e.subtypeRestr hU).symm) (e.subtypeRestr hU).target := by
intro y hy
rw [eq_comm, eq_symm_apply _ _ hy.1]
· change restrict _ e _ = _
rw [← subtypeRestr_coe, (e.subtypeRestr hU).right_inv hy]
· have := map_target _ hy; rwa [subtypeRestr_source] at this
theorem subtypeRestr_symm_eqOn_of_le {U V : Opens X} (hU : Nonempty U) (hV : Nonempty V)
(hUV : U ≤ V) : EqOn (e.subtypeRestr hV).symm (Set.inclusion hUV ∘ (e.subtypeRestr hU).symm)
(e.subtypeRestr hU).target := by
set i := Set.inclusion hUV
intro y hy
dsimp [PartialHomeomorph.subtypeRestr_def] at hy ⊢
have hyV : e.symm y ∈ (V.partialHomeomorphSubtypeCoe hV).target := by
rw [Opens.partialHomeomorphSubtypeCoe_target] at hy ⊢
exact hUV hy.2
refine (V.partialHomeomorphSubtypeCoe hV).injOn ?_ trivial ?_
· rw [← PartialHomeomorph.symm_target]
apply PartialHomeomorph.map_source
rw [PartialHomeomorph.symm_source]
exact hyV
· rw [(V.partialHomeomorphSubtypeCoe hV).right_inv hyV]
change _ = U.partialHomeomorphSubtypeCoe hU _
rw [(U.partialHomeomorphSubtypeCoe hU).right_inv hy.2]
end subtypeRestr
variable {X X' Z : Type*} [TopologicalSpace X] [TopologicalSpace X'] [TopologicalSpace Z]
[Nonempty Z] {f : X → X'}
/-- Extend a partial homeomorphism `e : X → Z` to `X' → Z`, using an open embedding `ι : X → X'`.
On `ι(X)`, the extension is specified by `e`; its value elsewhere is arbitrary (and uninteresting).
-/
noncomputable def lift_openEmbedding (e : PartialHomeomorph X Z) (hf : IsOpenEmbedding f) :
PartialHomeomorph X' Z where
toFun := extend f e (fun _ ↦ (Classical.arbitrary Z))
invFun := f ∘ e.invFun
source := f '' e.source
target := e.target
map_source' := by
rintro x ⟨x₀, hx₀, hxx₀⟩
rw [← hxx₀, hf.injective.extend_apply e]
exact e.map_source' hx₀
map_target' z hz := mem_image_of_mem f (e.map_target' hz)
left_inv' := by
intro x ⟨x₀, hx₀, hxx₀⟩
rw [← hxx₀, hf.injective.extend_apply e, comp_apply]
congr
exact e.left_inv' hx₀
right_inv' z hz := by simpa only [comp_apply, hf.injective.extend_apply e] using e.right_inv' hz
open_source := hf.isOpenMap _ e.open_source
open_target := e.open_target
continuousOn_toFun := by
by_cases Nonempty X; swap
· intro x hx; simp_all
set F := (extend f e (fun _ ↦ (Classical.arbitrary Z))) with F_eq
have heq : EqOn F (e ∘ (hf.toPartialHomeomorph).symm) (f '' e.source) := by
intro x ⟨x₀, hx₀, hxx₀⟩
rw [← hxx₀, F_eq, hf.injective.extend_apply e, comp_apply, hf.toPartialHomeomorph_left_inv]
have : ContinuousOn (e ∘ (hf.toPartialHomeomorph).symm) (f '' e.source) := by
apply e.continuousOn_toFun.comp; swap
· intro x' ⟨x, hx, hx'x⟩
rw [← hx'x, hf.toPartialHomeomorph_left_inv]; exact hx
have : ContinuousOn (hf.toPartialHomeomorph).symm (f '' univ) :=
(hf.toPartialHomeomorph).continuousOn_invFun
exact this.mono <| image_mono <| subset_univ _
exact ContinuousOn.congr this heq
continuousOn_invFun := hf.continuous.comp_continuousOn e.continuousOn_invFun
@[simp, mfld_simps]
lemma lift_openEmbedding_toFun (e : PartialHomeomorph X Z) (hf : IsOpenEmbedding f) :
(e.lift_openEmbedding hf) = extend f e (fun _ ↦ (Classical.arbitrary Z)) := rfl
lemma lift_openEmbedding_apply (e : PartialHomeomorph X Z) (hf : IsOpenEmbedding f) {x : X} :
(lift_openEmbedding e hf) (f x) = e x := by
simp_rw [e.lift_openEmbedding_toFun]
apply hf.injective.extend_apply
@[simp, mfld_simps]
lemma lift_openEmbedding_source (e : PartialHomeomorph X Z) (hf : IsOpenEmbedding f) :
(e.lift_openEmbedding hf).source = f '' e.source := rfl
@[simp, mfld_simps]
lemma lift_openEmbedding_target (e : PartialHomeomorph X Z) (hf : IsOpenEmbedding f) :
(e.lift_openEmbedding hf).target = e.target := rfl
@[simp, mfld_simps]
lemma lift_openEmbedding_symm (e : PartialHomeomorph X Z) (hf : IsOpenEmbedding f) :
(e.lift_openEmbedding hf).symm = f ∘ e.symm := rfl
@[simp, mfld_simps]
lemma lift_openEmbedding_symm_source (e : PartialHomeomorph X Z) (hf : IsOpenEmbedding f) :
(e.lift_openEmbedding hf).symm.source = e.target := rfl
@[simp, mfld_simps]
lemma lift_openEmbedding_symm_target (e : PartialHomeomorph X Z) (hf : IsOpenEmbedding f) :
(e.lift_openEmbedding hf).symm.target = f '' e.source := by
rw [PartialHomeomorph.symm_target, e.lift_openEmbedding_source]
lemma lift_openEmbedding_trans_apply
(e e' : PartialHomeomorph X Z) (hf : IsOpenEmbedding f) (z : Z) :
(e.lift_openEmbedding hf).symm.trans (e'.lift_openEmbedding hf) z = (e.symm.trans e') z := by
simp [hf.injective.extend_apply e']
@[simp, mfld_simps]
lemma lift_openEmbedding_trans (e e' : PartialHomeomorph X Z) (hf : IsOpenEmbedding f) :
(e.lift_openEmbedding hf).symm.trans (e'.lift_openEmbedding hf) = e.symm.trans e' := by
ext z
· exact e.lift_openEmbedding_trans_apply e' hf z
· simp [hf.injective.extend_apply e]
· simp_rw [PartialHomeomorph.trans_source, e.lift_openEmbedding_symm_source, e.symm_source,
e.lift_openEmbedding_symm, e'.lift_openEmbedding_source]
refine ⟨fun ⟨hx, ⟨y, hy, hxy⟩⟩ ↦ ⟨hx, ?_⟩, fun ⟨hx, hx'⟩ ↦ ⟨hx, mem_image_of_mem f hx'⟩⟩
rw [mem_preimage]; rw [comp_apply] at hxy
exact (hf.injective hxy) ▸ hy
end PartialHomeomorph
|
IdealQuotient.lean
|
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Quotient
/-! # Ideals in free modules over PIDs
## Main results
- `Ideal.quotientEquivPiSpan`: `S ⧸ I`, if `S` is finite free as a module over a PID `R`,
can be written as a product of quotients of `R` by principal ideals.
-/
open Module
open scoped DirectSum
namespace Ideal
variable {ι R S : Type*} [CommRing R] [CommRing S] [Algebra R S]
variable [IsDomain R] [IsPrincipalIdealRing R] [IsDomain S] [Finite ι]
/-- We can write the quotient of an ideal over a PID as a product of quotients by principal ideals.
-/
noncomputable def quotientEquivPiSpan (I : Ideal S) (b : Basis ι R S) (hI : I ≠ ⊥) :
(S ⧸ I) ≃ₗ[R] ∀ i, R ⧸ span ({I.smithCoeffs b hI i} : Set R) :=
Submodule.quotientEquivPiSpan (I.restrictScalars R) b <| finrank_eq_finrank b I hI
/-- Ideal quotients over a free finite extension of `ℤ` are isomorphic to a direct product of
`ZMod`. -/
noncomputable def quotientEquivPiZMod (I : Ideal S) (b : Basis ι ℤ S) (hI : I ≠ ⊥) :
S ⧸ I ≃+ ∀ i, ZMod (I.smithCoeffs b hI i).natAbs :=
Submodule.quotientEquivPiZMod (I.restrictScalars ℤ) b <| finrank_eq_finrank b I hI
/--
A nonzero ideal over a free finite extension of `ℤ` has a finite quotient.
It can't be an instance because of the side condition `I ≠ ⊥`.
-/
theorem finiteQuotientOfFreeOfNeBot [Module.Free ℤ S] [Module.Finite ℤ S]
(I : Ideal S) (hI : I ≠ ⊥) : Finite (S ⧸ I) :=
let b := Module.Free.chooseBasis ℤ S
Submodule.finiteQuotientOfFreeOfRankEq (I.restrictScalars ℤ) <| finrank_eq_finrank b I hI
@[deprecated (since := "2025-03-15")] alias fintypeQuotientOfFreeOfNeBot :=
finiteQuotientOfFreeOfNeBot
variable (F : Type*) [CommRing F] [Algebra F R] [Algebra F S] [IsScalarTower F R S]
(b : Basis ι R S) {I : Ideal S} (hI : I ≠ ⊥)
/-- Decompose `S⧸I` as a direct sum of cyclic `R`-modules
(quotients by the ideals generated by Smith coefficients of `I`). -/
noncomputable def quotientEquivDirectSum :
(S ⧸ I) ≃ₗ[F] ⨁ i, R ⧸ span ({I.smithCoeffs b hI i} : Set R) :=
Submodule.quotientEquivDirectSum F b (N := (I.restrictScalars R)) <| finrank_eq_finrank b I hI
theorem finrank_quotient_eq_sum {ι} [Fintype ι] (b : Basis ι R S) [Nontrivial F]
[∀ i, Module.Free F (R ⧸ span ({I.smithCoeffs b hI i} : Set R))]
[∀ i, Module.Finite F (R ⧸ span ({I.smithCoeffs b hI i} : Set R))] :
Module.finrank F (S ⧸ I) =
∑ i, Module.finrank F (R ⧸ span ({I.smithCoeffs b hI i} : Set R)) := by
-- slow, and dot notation doesn't work
rw [LinearEquiv.finrank_eq <| quotientEquivDirectSum F b hI, Module.finrank_directSum]
end Ideal
|
IsKan.lean
|
/-
Copyright (c) 2023 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno
-/
import Mathlib.CategoryTheory.Bicategory.Extension
/-!
# Kan extensions and Kan lifts in bicategories
The left Kan extension of a 1-morphism `g : a ⟶ c` along a 1-morphism `f : a ⟶ b` is the initial
object in the category of left extensions `LeftExtension f g`. The universal property can be
accessed by the following definition and lemmas:
* `LeftExtension.IsKan.desc`: the family of 2-morphisms out of the left Kan extension.
* `LeftExtension.IsKan.fac`: the unit of any left extension factors through the left Kan extension.
* `LeftExtension.IsKan.hom_ext`: two 2-morphisms out of the left Kan extension are equal if their
compositions with each unit are equal.
We also define left Kan lifts, right Kan extensions, and right Kan lifts.
## Implementation Notes
We use the Is-Has design pattern, which is used for the implementation of limits and colimits in
the category theory library. This means that `IsKan t` is a structure containing the data of
2-morphisms which ensure that `t` is a Kan extension, while `HasKan f g` defined in
`CategoryTheory.Bicategory.Kan.HasKan` is a `Prop`-valued typeclass asserting that a Kan extension
of `g` along `f` exists.
We define `LeftExtension.IsKan t` for an extension `t : LeftExtension f g` (which is an
abbreviation of `t : StructuredArrow g (precomp _ f)`) to be an abbreviation for
`StructuredArrow.IsUniversal t`. This means that we can use the definitions and lemmas living
in the namespace `StructuredArrow.IsUniversal`.
## References
https://ncatlab.org/nlab/show/Kan+extension
-/
namespace CategoryTheory
namespace Bicategory
universe w v u
variable {B : Type u} [Bicategory.{w, v} B] {a b c : B}
namespace LeftExtension
variable {f : a ⟶ b} {g : a ⟶ c}
/-- A left Kan extension of `g` along `f` is an initial object in `LeftExtension f g`. -/
abbrev IsKan (t : LeftExtension f g) := t.IsUniversal
/-- An absolute left Kan extension is a Kan extension that commutes with any 1-morphism. -/
abbrev IsAbsKan (t : LeftExtension f g) :=
∀ {x : B} (h : c ⟶ x), IsKan (t.whisker h)
namespace IsKan
variable {s t : LeftExtension f g}
/-- To show that a left extension `t` is a Kan extension, we need to show that for every left
extension `s` there is a unique morphism `t ⟶ s`. -/
abbrev mk (desc : ∀ s, t ⟶ s) (w : ∀ s τ, τ = desc s) :
IsKan t :=
.ofUniqueHom desc w
/-- The family of 2-morphisms out of a left Kan extension. -/
abbrev desc (H : IsKan t) (s : LeftExtension f g) : t.extension ⟶ s.extension :=
StructuredArrow.IsUniversal.desc H s
@[reassoc (attr := simp)]
theorem fac (H : IsKan t) (s : LeftExtension f g) :
t.unit ≫ f ◁ H.desc s = s.unit :=
StructuredArrow.IsUniversal.fac H s
/-- Two 2-morphisms out of a left Kan extension are equal if their compositions with
each triangle 2-morphism are equal. -/
theorem hom_ext (H : IsKan t) {k : b ⟶ c} {τ τ' : t.extension ⟶ k}
(w : t.unit ≫ f ◁ τ = t.unit ≫ f ◁ τ') : τ = τ' :=
StructuredArrow.IsUniversal.hom_ext H w
/-- Kan extensions on `g` along `f` are unique up to isomorphism. -/
def uniqueUpToIso (P : IsKan s) (Q : IsKan t) : s ≅ t :=
Limits.IsInitial.uniqueUpToIso P Q
@[simp]
theorem uniqueUpToIso_hom_right (P : IsKan s) (Q : IsKan t) :
(uniqueUpToIso P Q).hom.right = P.desc t := rfl
@[simp]
theorem uniqueUpToIso_inv_right (P : IsKan s) (Q : IsKan t) :
(uniqueUpToIso P Q).inv.right = Q.desc s := rfl
/-- Transport evidence that a left extension is a Kan extension across an isomorphism
of extensions. -/
def ofIsoKan (P : IsKan s) (i : s ≅ t) : IsKan t :=
Limits.IsInitial.ofIso P i
/-- If `t : LeftExtension f (g ≫ 𝟙 c)` is a Kan extension, then `t.ofCompId : LeftExtension f g`
is also a Kan extension. -/
def ofCompId (t : LeftExtension f (g ≫ 𝟙 c)) (P : IsKan t) : IsKan t.ofCompId :=
.mk (fun s ↦ t.whiskerIdCancel <| P.to (s.whisker (𝟙 c))) <| by
intro s τ
ext
apply P.hom_ext
simp [← LeftExtension.w τ]
/-- If `s ≅ t` and `IsKan (s.whisker h)`, then `IsKan (t.whisker h)`. -/
def whiskerOfCommute (s t : LeftExtension f g) (i : s ≅ t) {x : B} (h : c ⟶ x)
(P : IsKan (s.whisker h)) :
IsKan (t.whisker h) :=
P.ofIsoKan <| whiskerIso i h
end IsKan
namespace IsAbsKan
variable {s t : LeftExtension f g}
/-- The family of 2-morphisms out of an absolute left Kan extension. -/
abbrev desc (H : IsAbsKan t) {x : B} {h : c ⟶ x} (s : LeftExtension f (g ≫ h)) :
t.extension ≫ h ⟶ s.extension :=
(H h).desc s
/-- An absolute left Kan extension is a left Kan extension. -/
def isKan (H : IsAbsKan t) : IsKan t :=
((H (𝟙 c)).ofCompId _).ofIsoKan <| whiskerOfCompIdIsoSelf t
/-- Transport evidence that a left extension is a Kan extension across an isomorphism
of extensions. -/
def ofIsoAbsKan (P : IsAbsKan s) (i : s ≅ t) : IsAbsKan t :=
fun h ↦ (P h).ofIsoKan (whiskerIso i h)
end IsAbsKan
end LeftExtension
namespace LeftLift
variable {f : b ⟶ a} {g : c ⟶ a}
/-- A left Kan lift of `g` along `f` is an initial object in `LeftLift f g`. -/
abbrev IsKan (t : LeftLift f g) := t.IsUniversal
/-- An absolute left Kan lift is a Kan lift such that every 1-morphism commutes with it. -/
abbrev IsAbsKan (t : LeftLift f g) :=
∀ {x : B} (h : x ⟶ c), IsKan (t.whisker h)
namespace IsKan
variable {s t : LeftLift f g}
/-- To show that a left lift `t` is a Kan lift, we need to show that for every left lift `s`
there is a unique morphism `t ⟶ s`. -/
abbrev mk (desc : ∀ s, t ⟶ s) (w : ∀ s τ, τ = desc s) :
IsKan t :=
.ofUniqueHom desc w
/-- The family of 2-morphisms out of a left Kan lift. -/
abbrev desc (H : IsKan t) (s : LeftLift f g) : t.lift ⟶ s.lift :=
StructuredArrow.IsUniversal.desc H s
@[reassoc (attr := simp)]
theorem fac (H : IsKan t) (s : LeftLift f g) :
t.unit ≫ H.desc s ▷ f = s.unit :=
StructuredArrow.IsUniversal.fac H s
/-- Two 2-morphisms out of a left Kan lift are equal if their compositions with
each triangle 2-morphism are equal. -/
theorem hom_ext (H : IsKan t) {k : c ⟶ b} {τ τ' : t.lift ⟶ k}
(w : t.unit ≫ τ ▷ f = t.unit ≫ τ' ▷ f) : τ = τ' :=
StructuredArrow.IsUniversal.hom_ext H w
/-- Kan lifts on `g` along `f` are unique up to isomorphism. -/
def uniqueUpToIso (P : IsKan s) (Q : IsKan t) : s ≅ t :=
Limits.IsInitial.uniqueUpToIso P Q
@[simp]
theorem uniqueUpToIso_hom_right (P : IsKan s) (Q : IsKan t) :
(uniqueUpToIso P Q).hom.right = P.desc t := rfl
@[simp]
theorem uniqueUpToIso_inv_right (P : IsKan s) (Q : IsKan t) :
(uniqueUpToIso P Q).inv.right = Q.desc s := rfl
/-- Transport evidence that a left lift is a Kan lift across an isomorphism of lifts. -/
def ofIsoKan (P : IsKan s) (i : s ≅ t) : IsKan t :=
Limits.IsInitial.ofIso P i
/-- If `t : LeftLift f (𝟙 c ≫ g)` is a Kan lift, then `t.ofIdComp : LeftLift f g` is also
a Kan lift. -/
def ofIdComp (t : LeftLift f (𝟙 c ≫ g)) (P : IsKan t) : IsKan t.ofIdComp :=
.mk (fun s ↦ t.whiskerIdCancel <| P.to (s.whisker (𝟙 c))) <| by
intro s τ
ext
apply P.hom_ext
simp [← LeftLift.w τ]
/-- If `s ≅ t` and `IsKan (s.whisker h)`, then `IsKan (t.whisker h)`. -/
def whiskerOfCommute (s t : LeftLift f g) (i : s ≅ t) {x : B} (h : x ⟶ c)
(P : IsKan (s.whisker h)) :
IsKan (t.whisker h) :=
P.ofIsoKan <| whiskerIso i h
end IsKan
namespace IsAbsKan
variable {s t : LeftLift f g}
/-- The family of 2-morphisms out of an absolute left Kan lift. -/
abbrev desc (H : IsAbsKan t) {x : B} {h : x ⟶ c} (s : LeftLift f (h ≫ g)) :
h ≫ t.lift ⟶ s.lift :=
(H h).desc s
/-- An absolute left Kan lift is a left Kan lift. -/
def isKan (H : IsAbsKan t) : IsKan t :=
((H (𝟙 c)).ofIdComp _).ofIsoKan <| whiskerOfIdCompIsoSelf t
/-- Transport evidence that a left lift is a Kan lift across an isomorphism of lifts. -/
def ofIsoAbsKan (P : IsAbsKan s) (i : s ≅ t) : IsAbsKan t :=
fun h ↦ (P h).ofIsoKan (whiskerIso i h)
end IsAbsKan
end LeftLift
namespace RightExtension
variable {f : a ⟶ b} {g : a ⟶ c}
/-- A right Kan extension of `g` along `f` is a terminal object in `RightExtension f g`. -/
abbrev IsKan (t : RightExtension f g) := t.IsUniversal
end RightExtension
namespace RightLift
variable {f : b ⟶ a} {g : c ⟶ a}
/-- A right Kan lift of `g` along `f` is a terminal object in `RightLift f g`. -/
abbrev IsKan (t : RightLift f g) := t.IsUniversal
end RightLift
end Bicategory
end CategoryTheory
|
Localization.lean
|
/-
Copyright (c) 2025 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Center.Basic
import Mathlib.CategoryTheory.Localization.Predicate
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
/-!
# Localization of the center of a category
Given a localization functor `L : C ⥤ D` with respect to `W : MorphismProperty C`,
we define a localization map `CatCenter C → CatCenter D` for the centers
of these categories. In case `L` is an additive functor between preadditive
categories, we promote this to a ring morphism `CatCenter C →+* CatCenter D`.
-/
universe w v₁ v₂ u₁ u₂
namespace CategoryTheory
variable {C : Type u₁} [Category.{v₁} C] {D : Type u₂} [Category.{v₂} D]
(r s : CatCenter C) (L : C ⥤ D) (W : MorphismProperty C) [L.IsLocalization W]
namespace CatCenter
/-- Given `r : CatCenter C` and `L : C ⥤ D` a localization functor with respect
to `W : MorphismProperty D`, this is the induced element in `CatCenter D`
obtained by localization. -/
noncomputable def localization : CatCenter D :=
Localization.liftNatTrans L W L L (𝟭 D) (𝟭 D) (Functor.whiskerRight r L)
@[simp]
lemma localization_app (X : C) :
(r.localization L W).app (L.obj X) = L.map (r.app X) := by
dsimp [localization]
simp only [Localization.liftNatTrans_app, Functor.id_obj, Functor.whiskerRight_app,
NatTrans.naturality, Functor.comp_map, Functor.id_map, Iso.hom_inv_id_app_assoc]
include W
lemma ext_of_localization (r s : CatCenter D)
(h : ∀ (X : C), r.app (L.obj X) = s.app (L.obj X)) : r = s :=
Localization.natTrans_ext L W h
lemma localization_one :
(1 : CatCenter C).localization L W = 1 :=
ext_of_localization L W _ _ (fun X => by simp)
lemma localization_mul :
(r * s).localization L W = r.localization L W * s.localization L W :=
ext_of_localization L W _ _ (fun X => by simp)
section Preadditive
variable [Preadditive C] [Preadditive D] [L.Additive]
lemma localization_zero :
(0 : CatCenter C).localization L W = 0 :=
ext_of_localization L W _ _ (fun X => by simp)
lemma localization_add :
(r + s).localization L W = r.localization L W + s.localization L W :=
ext_of_localization L W _ _ (fun X => by
rw [localization_app, NatTrans.app_add, NatTrans.app_add, L.map_add,
localization_app, localization_app])
/-- The morphism of rings `CatCenter C →+* CatCenter D` when `L : C ⥤ D`
is an additive localization functor between preadditive categories. -/
noncomputable def localizationRingHom : CatCenter C →+* CatCenter D where
toFun r := r.localization L W
map_zero' := localization_zero L W
map_one' := localization_one L W
map_add' _ _ := localization_add _ _ _ _
map_mul' _ _ := localization_mul _ _ _ _
end Preadditive
end CatCenter
end CategoryTheory
|
EqLocus.lean
|
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Module.Submodule.Ker
/-!
# The submodule of elements `x : M` such that `f x = g x`
## Main declarations
* `LinearMap.eqLocus`: the submodule of elements `x : M` such that `f x = g x`
## Tags
linear algebra, vector space, module
-/
variable {R : Type*} {R₂ : Type*}
variable {M : Type*} {M₂ : Type*}
/-! ### Properties of linear maps -/
namespace LinearMap
section AddCommMonoid
variable [Semiring R] [Semiring R₂]
variable [AddCommMonoid M] [AddCommMonoid M₂]
variable [Module R M] [Module R₂ M₂]
open Submodule
variable {τ₁₂ : R →+* R₂}
section
variable {F : Type*} [FunLike F M M₂] [SemilinearMapClass F τ₁₂ M M₂]
/-- A linear map version of `AddMonoidHom.eqLocusM` -/
def eqLocus (f g : F) : Submodule R M :=
{ (f : M →+ M₂).eqLocusM g with
carrier := { x | f x = g x }
smul_mem' := fun {r} {x} (hx : _ = _) => show _ = _ by
-- Note: https://github.com/leanprover-community/mathlib4/pull/8386 changed `map_smulₛₗ` into `map_smulₛₗ _`
simpa only [map_smulₛₗ _] using congr_arg (τ₁₂ r • ·) hx }
@[simp]
theorem mem_eqLocus {x : M} {f g : F} : x ∈ eqLocus f g ↔ f x = g x :=
Iff.rfl
theorem eqLocus_toAddSubmonoid (f g : F) :
(eqLocus f g).toAddSubmonoid = (f : M →+ M₂).eqLocusM g :=
rfl
@[simp]
theorem eqLocus_eq_top {f g : F} : eqLocus f g = ⊤ ↔ f = g := by
simp [SetLike.ext_iff, DFunLike.ext_iff]
@[simp]
theorem eqLocus_same (f : F) : eqLocus f f = ⊤ := eqLocus_eq_top.2 rfl
theorem le_eqLocus {f g : F} {S : Submodule R M} : S ≤ eqLocus f g ↔ Set.EqOn f g S := Iff.rfl
include τ₁₂ in
theorem eqOn_sup {f g : F} {S T : Submodule R M} (hS : Set.EqOn f g S) (hT : Set.EqOn f g T) :
Set.EqOn f g ↑(S ⊔ T) := by
rw [← le_eqLocus] at hS hT ⊢
exact sup_le hS hT
include τ₁₂ in
theorem ext_on_codisjoint {f g : F} {S T : Submodule R M} (hST : Codisjoint S T)
(hS : Set.EqOn f g S) (hT : Set.EqOn f g T) : f = g :=
DFunLike.ext _ _ fun _ ↦ eqOn_sup hS hT <| hST.eq_top.symm ▸ trivial
end
end AddCommMonoid
section Ring
variable [Ring R] [Ring R₂]
variable [AddCommGroup M] [AddCommGroup M₂]
variable [Module R M] [Module R₂ M₂]
variable {τ₁₂ : R →+* R₂}
open Submodule
theorem eqLocus_eq_ker_sub (f g : M →ₛₗ[τ₁₂] M₂) : eqLocus f g = ker (f - g) :=
SetLike.ext fun _ => sub_eq_zero.symm
end Ring
end LinearMap
|
PartialSups.lean
|
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Data.Set.Finite.Lattice
import Mathlib.Order.ConditionallyCompleteLattice.Indexed
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Order.SuccPred.Basic
/-!
# The monotone sequence of partial supremums of a sequence
For `ι` a preorder in which all bounded-above intervals are finite (such as `ℕ`), and `α` a
`⊔`-semilattice, we define `partialSups : (ι → α) → ι →o α` by the formula
`partialSups f i = (Finset.Iic i).sup' ⋯ f`, where the `⋯` denotes a proof that `Finset.Iic i` is
nonempty. This is a way of spelling `⊔ k ≤ i, f k` which does not require a `α` to have a bottom
element, and makes sense in conditionally-complete lattices (where indexed suprema over sets are
badly-behaved).
Under stronger hypotheses on `α` and `ι`, we show that this coincides with other candidate
definitions, see e.g. `partialSups_eq_biSup`, `partialSups_eq_sup_range`,
and `partialSups_eq_sup'_range`.
We show this construction gives a Galois insertion between functions `ι → α` and monotone functions
`ι →o α`, see `partialSups.gi`.
## Notes
One might dispute whether this sequence should start at `f 0` or `⊥`. We choose the former because:
* Starting at `⊥` requires... having a bottom element.
* `fun f i ↦ (Finset.Iio i).sup f` is already effectively the sequence starting at `⊥`.
* If we started at `⊥` we wouldn't have the Galois insertion. See `partialSups.gi`.
-/
open Finset
variable {α β ι : Type*}
section SemilatticeSup
variable [SemilatticeSup α] [SemilatticeSup β]
section Preorder
variable [Preorder ι] [LocallyFiniteOrderBot ι]
/-- The monotone sequence whose value at `i` is the supremum of the `f j` where `j ≤ i`. -/
def partialSups (f : ι → α) : ι →o α where
toFun i := (Iic i).sup' nonempty_Iic f
monotone' _ _ hmn := sup'_mono f (Iic_subset_Iic.mpr hmn) nonempty_Iic
lemma partialSups_apply (f : ι → α) (i : ι) :
partialSups f i = (Iic i).sup' nonempty_Iic f :=
rfl
lemma partialSups_iff_forall {f : ι → α} (p : α → Prop)
(hp : ∀ {a b}, p (a ⊔ b) ↔ p a ∧ p b) {i : ι} :
p (partialSups f i) ↔ ∀ j ≤ i, p (f j) := by
classical
rw [partialSups_apply, comp_sup'_eq_sup'_comp (γ := Propᵒᵈ) _ p, sup'_eq_sup]
· change (Iic i).inf (p ∘ f) ↔ _
simp [Finset.inf_eq_iInf]
· intro x y
rw [hp]
rfl
@[simp]
lemma partialSups_le_iff {f : ι → α} {i : ι} {a : α} :
partialSups f i ≤ a ↔ ∀ j ≤ i, f j ≤ a :=
partialSups_iff_forall (· ≤ a) sup_le_iff
theorem le_partialSups_of_le (f : ι → α) {i j : ι} (h : i ≤ j) :
f i ≤ partialSups f j :=
partialSups_le_iff.1 le_rfl i h
theorem le_partialSups (f : ι → α) :
f ≤ partialSups f :=
fun _ => le_partialSups_of_le f le_rfl
theorem partialSups_le (f : ι → α) (i : ι) (a : α) (w : ∀ j ≤ i, f j ≤ a) :
partialSups f i ≤ a :=
partialSups_le_iff.2 w
@[simp]
lemma upperBounds_range_partialSups (f : ι → α) :
upperBounds (Set.range (partialSups f)) = upperBounds (Set.range f) := by
ext a
simp only [mem_upperBounds, Set.forall_mem_range, partialSups_le_iff]
exact ⟨fun h _ ↦ h _ _ le_rfl, fun h _ _ _ ↦ h _⟩
@[simp]
theorem bddAbove_range_partialSups {f : ι → α} :
BddAbove (Set.range (partialSups f)) ↔ BddAbove (Set.range f) :=
.of_eq <| congr_arg Set.Nonempty <| upperBounds_range_partialSups f
theorem Monotone.partialSups_eq {f : ι → α} (hf : Monotone f) :
partialSups f = f :=
funext fun i ↦ le_antisymm (partialSups_le _ _ _ (@hf · i)) (le_partialSups _ _)
theorem partialSups_mono :
Monotone (partialSups : (ι → α) → ι →o α) :=
fun _ _ h _ ↦ partialSups_le_iff.2 fun j hj ↦ (h j).trans (le_partialSups_of_le _ hj)
lemma partialSups_monotone (f : ι → α) :
Monotone (partialSups f) :=
fun i _ hnm ↦ partialSups_le f i _ (fun _ hm'n ↦ le_partialSups_of_le _ (hm'n.trans hnm))
/-- `partialSups` forms a Galois insertion with the coercion from monotone functions to functions.
-/
def partialSups.gi :
GaloisInsertion (partialSups : (ι → α) → ι →o α) (↑) where
choice f h :=
⟨f, by convert (partialSups f).monotone using 1; exact (le_partialSups f).antisymm h⟩
gc f g := by
refine ⟨(le_partialSups f).trans, fun h ↦ ?_⟩
convert partialSups_mono h
exact OrderHom.ext _ _ g.monotone.partialSups_eq.symm
le_l_u f := le_partialSups f
choice_eq f h := OrderHom.ext _ _ ((le_partialSups f).antisymm h)
protected lemma Pi.partialSups_apply {τ : Type*} {π : τ → Type*} [∀ t, SemilatticeSup (π t)]
(f : ι → (t : τ) → π t) (i : ι) (t : τ) :
partialSups f i t = partialSups (f · t) i := by
simp only [partialSups_apply, Finset.sup'_apply]
lemma comp_partialSups {F : Type*} [FunLike F α β] [SupHomClass F α β] (f : ι → α) (g : F) :
partialSups (g ∘ f) = g ∘ partialSups f := by
funext _; simp [partialSups]
end Preorder
@[simp]
theorem partialSups_succ [LinearOrder ι] [LocallyFiniteOrderBot ι] [SuccOrder ι]
(f : ι → α) (i : ι) :
partialSups f (Order.succ i) = partialSups f i ⊔ f (Order.succ i) := by
suffices Iic (Order.succ i) = Iic i ∪ {Order.succ i} by simp only [partialSups_apply, this,
sup'_union nonempty_Iic ⟨_, mem_singleton_self _⟩ f, sup'_singleton]
ext
simp only [mem_Iic, mem_union, mem_singleton]
constructor
· exact fun h ↦ (Order.le_succ_iff_eq_or_le.mp h).symm
· exact fun h ↦ h.elim (le_trans · <| Order.le_succ _) le_of_eq
@[simp]
theorem partialSups_bot [PartialOrder ι] [LocallyFiniteOrder ι] [OrderBot ι]
(f : ι → α) : partialSups f ⊥ = f ⊥ := by
simp only [partialSups_apply]
-- should we add a lemma `Finset.Iic_bot`?
suffices Iic (⊥ : ι) = {⊥} by simp only [this, sup'_singleton]
simp only [← coe_eq_singleton, coe_Iic, Set.Iic_bot]
/-!
### Functions out of `ℕ`
-/
@[simp]
theorem partialSups_zero (f : ℕ → α) : partialSups f 0 = f 0 :=
partialSups_bot f
theorem partialSups_eq_sup'_range (f : ℕ → α) (n : ℕ) :
partialSups f n = (Finset.range (n + 1)).sup' nonempty_range_succ f :=
eq_of_forall_ge_iff fun _ ↦ by simp [Nat.lt_succ_iff]
theorem partialSups_eq_sup_range [OrderBot α] (f : ℕ → α) (n : ℕ) :
partialSups f n = (Finset.range (n + 1)).sup f :=
eq_of_forall_ge_iff fun _ ↦ by simp [Nat.lt_succ_iff]
end SemilatticeSup
section DistribLattice
/-!
### Functions valued in a distributive lattice
These lemmas require the target to be a distributive lattice, so they are not useful (or true) in
situations such as submodules.
-/
variable [Preorder ι] [LocallyFiniteOrderBot ι] [DistribLattice α] [OrderBot α]
@[simp]
lemma disjoint_partialSups_left {f : ι → α} {i : ι} {x : α} :
Disjoint (partialSups f i) x ↔ ∀ j ≤ i, Disjoint (f j) x :=
partialSups_iff_forall (Disjoint · x) disjoint_sup_left
@[simp]
lemma disjoint_partialSups_right {f : ι → α} {i : ι} {x : α} :
Disjoint x (partialSups f i) ↔ ∀ j ≤ i, Disjoint x (f j) :=
partialSups_iff_forall (Disjoint x) disjoint_sup_right
open scoped Function in -- required for scoped `on` notation
/- Note this lemma requires a distributive lattice, so is not useful (or true) in situations such as
submodules. -/
theorem partialSups_disjoint_of_disjoint (f : ι → α) (h : Pairwise (Disjoint on f))
{i j : ι} (hij : i < j) :
Disjoint (partialSups f i) (f j) :=
disjoint_partialSups_left.2 fun _ hk ↦ h (hk.trans_lt hij).ne
end DistribLattice
section ConditionallyCompleteLattice
/-!
### Lemmas about the supremum over the whole domain
These lemmas require some completeness assumptions on the target space.
-/
variable [Preorder ι] [LocallyFiniteOrderBot ι]
theorem partialSups_eq_ciSup_Iic [ConditionallyCompleteLattice α] (f : ι → α) (i : ι) :
partialSups f i = ⨆ i : Set.Iic i, f i := by
simp only [partialSups_apply]
apply le_antisymm
· exact sup'_le _ _ fun j hj ↦ le_ciSup_of_le (Set.finite_range _).bddAbove
⟨j, by simpa only [Set.mem_Iic, mem_Iic] using hj⟩ le_rfl
· exact ciSup_le fun ⟨j, hj⟩ ↦ le_sup' f (by simpa only [mem_Iic, Set.mem_Iic] using hj)
@[simp]
theorem ciSup_partialSups_eq [ConditionallyCompleteLattice α]
{f : ι → α} (h : BddAbove (Set.range f)) :
⨆ i, partialSups f i = ⨆ i, f i := by
by_cases hι : Nonempty ι
· refine (ciSup_le fun i ↦ ?_).antisymm (ciSup_mono ?_ <| le_partialSups f)
· simpa only [partialSups_eq_ciSup_Iic] using ciSup_le fun i ↦ le_ciSup h _
· rwa [bddAbove_range_partialSups]
· exact congr_arg _ (funext (not_nonempty_iff.mp hι).elim)
/-- Version of `ciSup_partialSups_eq` without boundedness assumptions, but requiring a
`ConditionallyCompleteLinearOrder` rather than just a `ConditionallyCompleteLattice`. -/
@[simp]
theorem ciSup_partialSups_eq' [ConditionallyCompleteLinearOrder α] (f : ι → α) :
⨆ i, partialSups f i = ⨆ i, f i := by
by_cases h : BddAbove (Set.range f)
· exact ciSup_partialSups_eq h
· rw [iSup, iSup, ConditionallyCompleteLinearOrder.csSup_of_not_bddAbove _ h,
ConditionallyCompleteLinearOrder.csSup_of_not_bddAbove _
(bddAbove_range_partialSups.not.mpr h)]
end ConditionallyCompleteLattice
section CompleteLattice
variable [Preorder ι] [LocallyFiniteOrderBot ι] [CompleteLattice α]
/-- Version of `ciSup_partialSups_eq` without boundedness assumptions, but requiring a
`CompleteLattice` rather than just a `ConditionallyCompleteLattice`. -/
theorem iSup_partialSups_eq (f : ι → α) :
⨆ i, partialSups f i = ⨆ i, f i :=
ciSup_partialSups_eq <| OrderTop.bddAbove _
theorem partialSups_eq_biSup (f : ι → α) (i : ι) :
partialSups f i = ⨆ j ≤ i, f j := by
simpa only [iSup_subtype] using partialSups_eq_ciSup_Iic f i
theorem iSup_le_iSup_of_partialSups_le_partialSups {f g : ι → α}
(h : partialSups f ≤ partialSups g) : ⨆ i, f i ≤ ⨆ i, g i := by
rw [← iSup_partialSups_eq f, ← iSup_partialSups_eq g]
exact iSup_mono h
theorem iSup_eq_iSup_of_partialSups_eq_partialSups {f g : ι → α}
(h : partialSups f = partialSups g) : ⨆ i, f i = ⨆ i, g i := by
simp_rw [← iSup_partialSups_eq f, ← iSup_partialSups_eq g, h]
end CompleteLattice
section Set
/-!
### Functions into `Set α`
-/
lemma partialSups_eq_sUnion_image [DecidableEq (Set α)] (s : ℕ → Set α) (n : ℕ) :
partialSups s n = ⋃₀ ↑((Finset.range (n + 1)).image s) := by
ext; simp [partialSups_eq_biSup, Nat.lt_succ_iff]
lemma partialSups_eq_biUnion_range (s : ℕ → Set α) (n : ℕ) :
partialSups s n = ⋃ i ∈ Finset.range (n + 1), s i := by
ext; simp [partialSups_eq_biSup, Nat.lt_succ]
end Set
|
Tannaka.lean
|
/-
Copyright (c) 2025 Yacine Benmeuraiem. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yacine Benmeuraiem
-/
import Mathlib.RepresentationTheory.FDRep
/-!
# Tannaka duality for finite groups
In this file we prove Tannaka duality for finite groups.
The theorem can be formulated as follows: for any integral domain `k`, a finite group `G` can be
recovered from `FDRep k G`, the monoidal category of finite dimensional `k`-linear representations
of `G`, and the monoidal forgetful functor `forget : FDRep k G ⥤ FGModuleCat k`.
The main result is the isomorphism `equiv : G ≃* Aut (forget k G)`.
## Reference
<https://math.leidenuniv.nl/scripties/1bachCommelin.pdf>
-/
noncomputable section
open CategoryTheory MonoidalCategory ModuleCat Finset Pi
universe u
namespace TannakaDuality
namespace FiniteGroup
variable {k G : Type u} [CommRing k] [Group G]
section definitions
instance : (forget₂ (FDRep k G) (FGModuleCat k)).Monoidal := by
change (Action.forget _ _).Monoidal; infer_instance
variable (k G) in
/-- The monoidal forgetful functor from `FDRep k G` to `FGModuleCat k`. -/
def forget := LaxMonoidalFunctor.of (forget₂ (FDRep k G) (FGModuleCat k))
@[simp] lemma forget_obj (X : FDRep k G) : (forget k G).obj X = X.V := rfl
@[simp] lemma forget_map (X Y : FDRep k G) (f : X ⟶ Y) : (forget k G).map f = f.hom := rfl
/-- Definition of `equivHom g : Aut (forget k G)` by its components. -/
@[simps]
def equivApp (g : G) (X : FDRep k G) : X.V ≅ X.V where
hom := ofHom (X.ρ g)
inv := ofHom (X.ρ g⁻¹)
hom_inv_id := by
ext x
simp
inv_hom_id := by
ext x
simp
variable (k G) in
/-- The group homomorphism `G →* Aut (forget k G)` shown to be an isomorphism. -/
@[simps]
def equivHom : G →* Aut (forget k G) where
toFun g :=
LaxMonoidalFunctor.isoOfComponents (equivApp g) (fun f ↦ (f.comm g).symm) rfl (by intros; rfl)
map_one' := by ext; simp; rfl
map_mul' _ _ := by ext; simp; rfl
/-- The representation on `G → k` induced by multiplication on the right in `G`. -/
def rightRegular : Representation k G (G → k) where
toFun s :=
{ toFun f t := f (t * s)
map_add' _ _ := rfl
map_smul' _ _ := rfl }
map_one' := by
ext
simp
map_mul' _ _ := by
ext
simp [mul_assoc]
@[simp]
lemma rightRegular_apply (s t : G) (f : G → k) : rightRegular s f t = f (t * s) := rfl
/-- The representation on `G → k` induced by multiplication on the left in `G`. -/
def leftRegular : Representation k G (G → k) where
toFun s :=
{ toFun f t := f (s⁻¹ * t)
map_add' _ _ := rfl
map_smul' _ _ := rfl }
map_one' := by
ext
simp
map_mul' _ _ := by
ext
simp [mul_assoc]
@[simp]
lemma leftRegular_apply (s t : G) (f : G → k) : leftRegular s f t = f (s⁻¹ * t) := rfl
/-- The right regular representation `rightRegular` on `G → k` as a `FDRep k G`. -/
@[simp]
def rightFDRep [Finite G] : FDRep k G := FDRep.of rightRegular
end definitions
variable [Finite G]
lemma equivHom_injective [Nontrivial k] : Function.Injective (equivHom k G) := by
intro s t h
classical
apply_fun (fun x ↦ (x.hom.hom.app rightFDRep).hom (single t 1) 1) at h
simp_all [single_apply]
@[deprecated (since := "2025-04-27")]
alias equivHom_inj := equivHom_injective
/-- The `FDRep k G` morphism induced by multiplication on `G → k`. -/
def mulRepHom : rightFDRep (k := k) (G := G) ⊗ rightFDRep ⟶ rightFDRep where
hom := ofHom (LinearMap.mul' k (G → k))
comm := by
intro
ext u
refine TensorProduct.induction_on u rfl (fun _ _ ↦ rfl) (fun _ _ hx hy ↦ ?_)
simp only [map_add, hx, hy]
/-- The `rightFDRep` component of `η : Aut (forget k G)` preserves multiplication -/
lemma map_mul_toRightFDRepComp (η : Aut (forget k G)) (f g : G → k) :
let α : (G → k) →ₗ[k] (G → k) := (η.hom.hom.app rightFDRep).hom
α (f * g) = (α f) * (α g) := by
have nat := η.hom.hom.naturality mulRepHom
have tensor (X Y) : η.hom.hom.app (X ⊗ Y) = (η.hom.hom.app X ⊗ₘ η.hom.hom.app Y) :=
η.hom.isMonoidal.tensor X Y
rw [tensor] at nat
apply_fun (Hom.hom · (f ⊗ₜ[k] g)) at nat
exact nat
/-- The `rightFDRep` component of `η : Aut (forget k G)` gives rise to
an algebra morphism `(G → k) →ₐ[k] (G → k)`. -/
def algHomOfRightFDRepComp (η : Aut (forget k G)) : (G → k) →ₐ[k] (G → k) := by
let α : (G → k) →ₗ[k] (G → k) := (η.hom.hom.app rightFDRep).hom
let α_inv : (G → k) →ₗ[k] (G → k) := (η.inv.hom.app rightFDRep).hom
refine AlgHom.ofLinearMap α ?_ (map_mul_toRightFDRepComp η)
suffices α (α_inv 1) = (1 : G → k) by
have h := this
rwa [← one_mul (α_inv 1), map_mul_toRightFDRepComp, h, mul_one] at this
have := η.inv_hom_id
apply_fun (fun x ↦ (x.hom.app rightFDRep).hom (1 : G → k)) at this
exact this
/-- For `v : X` and `G` a finite group, the `G`-equivariant linear map from the right
regular representation `rightFDRep` to `X` sending `single 1 1` to `v`. -/
@[simps]
def sumSMulInv [Fintype G] {X : FDRep k G} (v : X) : (G → k) →ₗ[k] X where
toFun f := ∑ s : G, (f s) • (X.ρ s⁻¹ v)
map_add' _ _ := by simp [add_smul, sum_add_distrib]
map_smul' _ _ := by simp [smul_sum, smul_smul]
omit [Finite G] in
@[simp]
lemma sumSMulInv_single_id [Fintype G] [DecidableEq G] {X : FDRep k G} (v : X) :
∑ s : G, (single 1 1 : G → k) s • (X.ρ s⁻¹) v = v := by
rw [Fintype.sum_eq_single 1]
· simp
· simp_all
/-- For `v : X` and `G` a finite group, the representation morphism from the right
regular representation `rightFDRep` to `X` sending `single 1 1` to `v`. -/
@[simps]
def ofRightFDRep [Fintype G] (X : FDRep k G) (v : X) : rightFDRep ⟶ X where
hom := ofHom (sumSMulInv v)
comm t := by
ext f
let φ_term (X : FDRep k G) (f : G → k) v s := (f s) • (X.ρ s⁻¹ v)
have := sum_map univ (mulRightEmbedding t⁻¹) (φ_term X (rightRegular t f) v)
simpa [φ_term] using this
lemma toRightFDRepComp_injective {η₁ η₂ : Aut (forget k G)}
(h : η₁.hom.hom.app rightFDRep = η₂.hom.hom.app rightFDRep) : η₁ = η₂ := by
have := Fintype.ofFinite G
classical
ext X v
have h1 := η₁.hom.hom.naturality (ofRightFDRep X v)
have h2 := η₂.hom.hom.naturality (ofRightFDRep X v)
rw [h, ← h2] at h1
simpa using congr(($h1).hom (single 1 1))
/-- `leftRegular` as a morphism `rightFDRep k G ⟶ rightFDRep k G` in `FDRep k G`. -/
def leftRegularFDRepHom (s : G) : End (rightFDRep : FDRep k G) where
hom := ofHom (leftRegular s)
comm _ := by
ext f
funext _
apply congrArg f
exact mul_assoc ..
lemma toRightFDRepComp_in_rightRegular [IsDomain k] (η : Aut (forget k G)) :
∃ (s : G), (η.hom.hom.app rightFDRep).hom = rightRegular s := by
classical
obtain ⟨s, hs⟩ := ((evalAlgHom _ _ 1).comp (algHomOfRightFDRepComp η)).eq_piEvalAlgHom
refine ⟨s, (basisFun k G).ext fun u ↦ ?_⟩
simp only [rightFDRep, forget_obj]
ext t
have nat := η.hom.hom.naturality (leftRegularFDRepHom t⁻¹)
calc
_ = leftRegular t⁻¹ ((η.hom.hom.app rightFDRep).hom (single u 1)) 1 := by simp
_ = (η.hom.hom.app rightFDRep).hom (leftRegular t⁻¹ (single u 1)) 1 :=
congrFun congr(($nat.symm).hom (single u 1)) 1
_ = evalAlgHom _ _ s (leftRegular t⁻¹ (single u 1)) :=
congr($hs (leftRegular t⁻¹ (single u 1)))
_ = _ := by by_cases u = t * s <;> simp_all [single_apply]
lemma equivHom_surjective [IsDomain k] : Function.Surjective (equivHom k G) := by
intro η
obtain ⟨s, h⟩ := toRightFDRepComp_in_rightRegular η
exact ⟨s, toRightFDRepComp_injective (hom_ext h.symm)⟩
variable (k G) in
/-- Tannaka duality for finite groups:
A finite group `G` is isomorphic to `Aut (forget k G)`, where `k` is any integral domain,
and `forget k G` is the monoidal forgetful functor `FDRep k G ⥤ FGModuleCat k G`. -/
def equiv [IsDomain k] : G ≃* Aut (forget k G) :=
MulEquiv.ofBijective (equivHom k G) ⟨equivHom_injective, equivHom_surjective⟩
end FiniteGroup
end TannakaDuality
end
|
Subquiver.lean
|
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.Order.Notation
import Mathlib.Combinatorics.Quiver.Basic
/-!
## Wide subquivers
A wide subquiver `H` of a quiver `H` consists of a subset of the edge set `a ⟶ b` for
every pair of vertices `a b : V`. We include 'wide' in the name to emphasize that these
subquivers by definition contain all vertices.
-/
universe v u
/--
A wide subquiver `H` of `G` picks out a set `H a b` of arrows from `a` to `b`
for every pair of vertices `a b`.
NB: this does not work for `Prop`-valued quivers. It requires `G : Quiver.{v+1} V`. -/
def WideSubquiver (V) [Quiver.{v + 1} V] :=
∀ a b : V, Set (a ⟶ b)
/-- A type synonym for `V`, when thought of as a quiver having only the arrows from
some `WideSubquiver`. -/
@[nolint unusedArguments]
def WideSubquiver.toType (V) [Quiver V] (_ : WideSubquiver V) : Type u :=
V
instance wideSubquiverHasCoeToSort {V} [Quiver V] :
CoeSort (WideSubquiver V) (Type u) where coe H := WideSubquiver.toType V H
/-- A wide subquiver viewed as a quiver on its own. -/
instance WideSubquiver.quiver {V} [Quiver V] (H : WideSubquiver V) : Quiver H :=
⟨fun a b ↦ { f // f ∈ H a b }⟩
namespace Quiver
instance {V} [Quiver V] : Bot (WideSubquiver V) :=
⟨fun _ _ ↦ ∅⟩
instance {V} [Quiver V] : Top (WideSubquiver V) :=
⟨fun _ _ ↦ Set.univ⟩
noncomputable instance {V} [Quiver V] : Inhabited (WideSubquiver V) :=
⟨⊤⟩
-- TODO Unify with `CategoryTheory.Arrow`? (The fields have been named to match.)
/-- `Total V` is the type of _all_ arrows of `V`. -/
-- Porting note: no hasNonemptyInstance linter yet https://github.com/leanprover-community/mathlib4/issues/5171
@[ext]
structure Total (V : Type u) [Quiver.{v} V] : Sort max (u + 1) v where
/-- the source vertex of an arrow -/
left : V
/-- the target vertex of an arrow -/
right : V
/-- an arrow -/
hom : left ⟶ right
/-- A wide subquiver of `G` can equivalently be viewed as a total set of arrows. -/
def wideSubquiverEquivSetTotal {V} [Quiver V] :
WideSubquiver V ≃
Set (Total V) where
toFun H := { e | e.hom ∈ H e.left e.right }
invFun S a b := { e | Total.mk a b e ∈ S }
/-- An `L`-labelling of a quiver assigns to every arrow an element of `L`. -/
def Labelling (V : Type u) [Quiver V] (L : Sort*) :=
∀ ⦃a b : V⦄, (a ⟶ b) → L
instance {V : Type u} [Quiver V] (L) [Inhabited L] : Inhabited (Labelling V L) :=
⟨fun _ _ _ ↦ default⟩
end Quiver
|
SmallComplete.lean
|
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.SetTheory.Cardinal.Basic
/-!
# Any small complete category is a preorder
We show that any small category which has all (small) limits is a preorder: In particular, we show
that if a small category `C` in universe `u` has products of size `u`, then for any `X Y : C`
there is at most one morphism `X ⟶ Y`.
Note that in Lean, a preorder category is strictly one where the morphisms are in `Prop`, so
we instead show that the homsets are subsingleton.
## References
* https://ncatlab.org/nlab/show/complete+small+category#in_classical_logic
## Tags
small complete, preorder, Freyd
-/
namespace CategoryTheory
open Category Limits
open Cardinal
universe u
variable {C : Type u} [SmallCategory C] [HasProducts.{u} C]
/-- A small category with products is a thin category.
in Lean, a preorder category is one where the morphisms are in Prop, which is weaker than the usual
notion of a preorder/thin category which says that each homset is subsingleton; we show the latter
rather than providing a `Preorder C` instance.
-/
instance (priority := 100) : Quiver.IsThin C := fun X Y =>
⟨fun r s => by
classical
by_contra r_ne_s
have z : (2 : Cardinal) ≤ #(X ⟶ Y) := by
rw [Cardinal.two_le_iff]
exact ⟨_, _, r_ne_s⟩
let md := Σ Z W : C, Z ⟶ W
let α := #md
apply not_le_of_gt (Cardinal.cantor α)
let yp : C := ∏ᶜ fun _ : md => Y
apply _root_.trans _ _
· exact #(X ⟶ yp)
· apply le_trans (Cardinal.power_le_power_right z)
rw [Cardinal.power_def]
apply le_of_eq
rw [Cardinal.eq]
refine ⟨⟨Pi.lift, fun f k => f ≫ Pi.π _ k, ?_, ?_⟩⟩
· intro f
ext k
simp [yp]
· intro f
ext ⟨j⟩
simp [yp]
· apply Cardinal.mk_le_of_injective _
· intro f
exact ⟨_, _, f⟩
· rintro f g k
cases k
rfl⟩
end CategoryTheory
|
WeakBilin.lean
|
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä, Moritz Doll
-/
import Mathlib.Topology.Algebra.Module.LinearMap
import Mathlib.LinearAlgebra.BilinearMap
/-!
# Weak dual topology
This file defines the weak topology given two vector spaces `E` and `F` over a commutative semiring
`𝕜` and a bilinear form `B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜`. The weak topology on `E` is the coarsest topology
such that for all `y : F` every map `fun x => B x y` is continuous.
## Main definitions
The main definition is the type `WeakBilin B`.
* Given `B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜`, the type `WeakBilin B` is a type synonym for `E`.
* The instance `WeakBilin.instTopologicalSpace` is the weak topology induced by the bilinear form
`B`.
## Main results
We establish that `WeakBilin B` has the following structure:
* `WeakBilin.instContinuousAdd`: The addition in `WeakBilin B` is continuous.
* `WeakBilin.instContinuousSMul`: The scalar multiplication in `WeakBilin B` is continuous.
We prove the following results characterizing the weak topology:
* `eval_continuous`: For any `y : F`, the evaluation mapping `fun x => B x y` is continuous.
* `continuous_of_continuous_eval`: For a mapping to `WeakBilin B` to be continuous,
it suffices that its compositions with pairing with `B` at all points `y : F` is continuous.
* `tendsto_iff_forall_eval_tendsto`: Convergence in `WeakBilin B` can be characterized
in terms of convergence of the evaluations at all points `y : F`.
## Notations
No new notation is introduced.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
weak-star, weak dual, duality
-/
noncomputable section
open Filter
open Topology
variable {α 𝕜 𝕝 E F : Type*}
section WeakTopology
/-- The space `E` equipped with the weak topology induced by the bilinear form `B`. -/
@[nolint unusedArguments]
def WeakBilin [CommSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F]
(_ : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) := E
deriving AddCommMonoid, Module 𝕜
namespace WeakBilin
instance instAddCommGroup [CommSemiring 𝕜] [a : AddCommGroup E] [Module 𝕜 E] [AddCommMonoid F]
[Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) : AddCommGroup (WeakBilin B) := a
instance (priority := 100) instModule' [CommSemiring 𝕜] [CommSemiring 𝕝] [AddCommMonoid E]
[Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] [m : Module 𝕝 E] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) :
Module 𝕝 (WeakBilin B) := m
instance instIsScalarTower [CommSemiring 𝕜] [CommSemiring 𝕝] [AddCommMonoid E] [Module 𝕜 E]
[AddCommMonoid F] [Module 𝕜 F] [SMul 𝕝 𝕜] [Module 𝕝 E] [s : IsScalarTower 𝕝 𝕜 E]
(B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) : IsScalarTower 𝕝 𝕜 (WeakBilin B) := s
section Semiring
variable [TopologicalSpace 𝕜] [CommSemiring 𝕜]
variable [AddCommMonoid E] [Module 𝕜 E]
variable [AddCommMonoid F] [Module 𝕜 F]
variable (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜)
instance instTopologicalSpace : TopologicalSpace (WeakBilin B) :=
TopologicalSpace.induced (fun x y => B x y) Pi.topologicalSpace
/-- The coercion `(fun x y => B x y) : E → (F → 𝕜)` is continuous. -/
theorem coeFn_continuous : Continuous fun (x : WeakBilin B) y => B x y :=
continuous_induced_dom
@[fun_prop]
theorem eval_continuous (y : F) : Continuous fun x : WeakBilin B => B x y :=
(continuous_pi_iff.mp (coeFn_continuous B)) y
theorem continuous_of_continuous_eval [TopologicalSpace α] {g : α → WeakBilin B}
(h : ∀ y, Continuous fun a => B (g a) y) : Continuous g :=
continuous_induced_rng.2 (continuous_pi_iff.mpr h)
/-- The coercion `(fun x y => B x y) : E → (F → 𝕜)` is an embedding. -/
theorem isEmbedding {B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜} (hB : Function.Injective B) :
IsEmbedding fun (x : WeakBilin B) y => B x y :=
Function.Injective.isEmbedding_induced <| LinearMap.coe_injective.comp hB
theorem tendsto_iff_forall_eval_tendsto {l : Filter α} {f : α → WeakBilin B} {x : WeakBilin B}
(hB : Function.Injective B) :
Tendsto f l (𝓝 x) ↔ ∀ y, Tendsto (fun i => B (f i) y) l (𝓝 (B x y)) := by
rw [← tendsto_pi_nhds, (isEmbedding hB).tendsto_nhds_iff]
rfl
/-- Addition in `WeakBilin B` is continuous. -/
instance instContinuousAdd [ContinuousAdd 𝕜] : ContinuousAdd (WeakBilin B) := by
refine ⟨continuous_induced_rng.2 ?_⟩
refine
cast (congr_arg _ ?_)
(((coeFn_continuous B).comp continuous_fst).add ((coeFn_continuous B).comp continuous_snd))
ext
simp only [Function.comp_apply, Pi.add_apply, map_add, LinearMap.add_apply]
/-- Scalar multiplication by `𝕜` on `WeakBilin B` is continuous. -/
instance instContinuousSMul [ContinuousSMul 𝕜 𝕜] : ContinuousSMul 𝕜 (WeakBilin B) := by
refine ⟨continuous_induced_rng.2 ?_⟩
refine cast (congr_arg _ ?_) (continuous_fst.smul ((coeFn_continuous B).comp continuous_snd))
ext
simp only [Function.comp_apply, Pi.smul_apply, LinearMap.map_smulₛₗ, RingHom.id_apply,
LinearMap.smul_apply]
/--
Map `F` into the topological dual of `E` with the weak topology induced by `F`
-/
def eval [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] :
F →ₗ[𝕜] WeakBilin B →L[𝕜] 𝕜 where
toFun f := ⟨B.flip f, by fun_prop⟩
map_add' _ _ := by ext; simp
map_smul' _ _ := by ext; simp
end Semiring
section Ring
variable [TopologicalSpace 𝕜] [CommRing 𝕜]
variable [AddCommGroup E] [Module 𝕜 E]
variable [AddCommGroup F] [Module 𝕜 F]
variable (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜)
/-- `WeakBilin B` is a `IsTopologicalAddGroup`, meaning that addition and negation are
continuous. -/
instance instIsTopologicalAddGroup [ContinuousAdd 𝕜] : IsTopologicalAddGroup (WeakBilin B) where
toContinuousAdd := by infer_instance
continuous_neg := by
refine continuous_induced_rng.2 (continuous_pi_iff.mpr fun y => ?_)
refine cast (congr_arg _ ?_) (eval_continuous B (-y))
ext x
simp only [map_neg, Function.comp_apply, LinearMap.neg_apply]
end Ring
end WeakBilin
end WeakTopology
|
Coequalizer.lean
|
/-
Copyright (c) 2025 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.Logic.Function.Defs
/-!
# Coequalizer of a pair of functions
The coequalizer of two functions `f g : α → β` is the pair (`μ`, `p : β → μ`) that
satisfies the following universal property: Every function `u : β → γ`
with `u ∘ f = u ∘ g` factors uniquely via `p`.
In this file we define the coequalizer and provide the basic API.
-/
universe v
namespace Function
/-- The relation generating the equivalence relation used for defining `Function.coequalizer`. -/
inductive Coequalizer.Rel {α β : Type*} (f g : α → β) : β → β → Prop where
| intro (x : α) : Rel f g (f x) (g x)
/-- The coequalizer of two functions `f g : α → β` is the pair (`μ`, `p : β → μ`) that
satisfies the following universal property: Every function `u : β → γ`
with `u ∘ f = u ∘ g` factors uniquely via `p`. -/
def Coequalizer {α : Type*} {β : Type v} (f g : α → β) : Type v :=
Quot (Function.Coequalizer.Rel f g)
namespace Coequalizer
variable {α β : Type*} (f g : α → β)
/-- The canonical projection to the coequalizer. -/
def mk (x : β) : Coequalizer f g :=
Quot.mk _ x
lemma condition (x : α) : mk f g (f x) = mk f g (g x) :=
Quot.sound (.intro x)
lemma mk_surjective : Function.Surjective (mk f g) :=
Quot.exists_rep
/-- Any map `u : β → γ` with `u ∘ f = u ∘ g` factors via `Function.Coequalizer.mk`. -/
def desc {γ : Type*} (u : β → γ) (hu : u ∘ f = u ∘ g) : Coequalizer f g → γ :=
Quot.lift u (fun _ _ (.intro e) ↦ congrFun hu e)
@[simp] lemma desc_mk {γ : Type*} (u : β → γ) (hu : u ∘ f = u ∘ g) (x : β) :
desc f g u hu (mk f g x) = u x :=
rfl
end Function.Coequalizer
|
Injective.lean
|
/-
Copyright (c) 2024 Andrew Yang, Qi Ge, Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Qi Ge, Christian Merten
-/
import Mathlib.RingTheory.RingHomProperties
/-! # Meta properties of injective ring homomorphisms -/
lemma _root_.RingHom.injective_stableUnderComposition :
RingHom.StableUnderComposition (fun f ↦ Function.Injective f) := by
intro R S T _ _ _ f g hf hg
simp only [RingHom.coe_comp]
exact Function.Injective.comp hg hf
lemma _root_.RingHom.injective_respectsIso :
RingHom.RespectsIso (fun f ↦ Function.Injective f) := by
apply RingHom.injective_stableUnderComposition.respectsIso
intro R S _ _ e
exact e.bijective.injective
|
Basic.lean
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Data.ENat.Basic
import Mathlib.RingTheory.UniqueFactorizationDomain.Defs
/-!
# Basic results un unique factorization monoids
## Main results
* `prime_factors_unique`: the prime factors of an element in a cancellative
commutative monoid with zero (e.g. an integral domain) are unique up to associates
* `UniqueFactorizationMonoid.factors_unique`: the irreducible factors of an element
in a unique factorization monoid (e.g. a UFD) are unique up to associates
* `UniqueFactorizationMonoid.iff_exists_prime_factors`: unique factorization exists iff each nonzero
elements factors into a product of primes
* `UniqueFactorizationMonoid.dvd_of_dvd_mul_left_of_no_prime_factors`: Euclid's lemma:
if `a ∣ b * c` and `a` and `c` have no common prime factors, `a ∣ b`.
* `UniqueFactorizationMonoid.dvd_of_dvd_mul_right_of_no_prime_factors`: Euclid's lemma:
if `a ∣ b * c` and `a` and `b` have no common prime factors, `a ∣ c`.
* `UniqueFactorizationMonoid.exists_reduced_factors`: in a UFM, we can divide out a common factor
to get relatively prime elements.
-/
assert_not_exists Field
variable {α : Type*}
local infixl:50 " ~ᵤ " => Associated
namespace WfDvdMonoid
variable [CommMonoidWithZero α]
open Associates Nat
theorem of_wfDvdMonoid_associates (_ : WfDvdMonoid (Associates α)) : WfDvdMonoid α :=
⟨(mk_surjective.wellFounded_iff mk_dvdNotUnit_mk_iff.symm).2 wellFounded_dvdNotUnit⟩
variable [WfDvdMonoid α]
instance wfDvdMonoid_associates : WfDvdMonoid (Associates α) :=
⟨(mk_surjective.wellFounded_iff mk_dvdNotUnit_mk_iff.symm).1 wellFounded_dvdNotUnit⟩
theorem wellFoundedLT_associates : WellFoundedLT (Associates α) :=
⟨Subrelation.wf dvdNotUnit_of_lt wellFounded_dvdNotUnit⟩
end WfDvdMonoid
theorem WfDvdMonoid.of_wellFoundedLT_associates [CancelCommMonoidWithZero α]
(h : WellFoundedLT (Associates α)) : WfDvdMonoid α :=
WfDvdMonoid.of_wfDvdMonoid_associates
⟨by
convert h.wf
ext
exact Associates.dvdNotUnit_iff_lt⟩
theorem WfDvdMonoid.iff_wellFounded_associates [CancelCommMonoidWithZero α] :
WfDvdMonoid α ↔ WellFoundedLT (Associates α) :=
⟨by apply WfDvdMonoid.wellFoundedLT_associates, WfDvdMonoid.of_wellFoundedLT_associates⟩
instance Associates.ufm [CancelCommMonoidWithZero α] [UniqueFactorizationMonoid α] :
UniqueFactorizationMonoid (Associates α) :=
{ (WfDvdMonoid.wfDvdMonoid_associates : WfDvdMonoid (Associates α)) with
irreducible_iff_prime := by
rw [← Associates.irreducible_iff_prime_iff]
apply UniqueFactorizationMonoid.irreducible_iff_prime }
theorem prime_factors_unique [CancelCommMonoidWithZero α] :
∀ {f g : Multiset α},
(∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g := by
classical
intro f
induction' f using Multiset.induction_on with p f ih
· intros g _ hg h
exact Multiset.rel_zero_left.2 <|
Multiset.eq_zero_of_forall_notMem fun x hx =>
have : IsUnit g.prod := by simpa [associated_one_iff_isUnit] using h.symm
(hg x hx).not_unit <|
isUnit_iff_dvd_one.2 <| (Multiset.dvd_prod hx).trans (isUnit_iff_dvd_one.1 this)
· intros g hf hg hfg
let ⟨b, hbg, hb⟩ :=
(exists_associated_mem_of_dvd_prod (hf p (by simp)) fun q hq => hg _ hq) <|
hfg.dvd_iff_dvd_right.1 (show p ∣ (p ::ₘ f).prod by simp)
haveI := Classical.decEq α
rw [← Multiset.cons_erase hbg]
exact
Multiset.Rel.cons hb
(ih (fun q hq => hf _ (by simp [hq]))
(fun {q} (hq : q ∈ g.erase b) => hg q (Multiset.mem_of_mem_erase hq))
(Associated.of_mul_left
(by rwa [← Multiset.prod_cons, ← Multiset.prod_cons, Multiset.cons_erase hbg]) hb
(hf p (by simp)).ne_zero))
namespace UniqueFactorizationMonoid
variable [CancelCommMonoidWithZero α] [UniqueFactorizationMonoid α]
theorem factors_unique {f g : Multiset α} (hf : ∀ x ∈ f, Irreducible x)
(hg : ∀ x ∈ g, Irreducible x) (h : f.prod ~ᵤ g.prod) : Multiset.Rel Associated f g :=
prime_factors_unique (fun x hx => UniqueFactorizationMonoid.irreducible_iff_prime.mp (hf x hx))
(fun x hx => UniqueFactorizationMonoid.irreducible_iff_prime.mp (hg x hx)) h
end UniqueFactorizationMonoid
/-- If an irreducible has a prime factorization,
then it is an associate of one of its prime factors. -/
theorem prime_factors_irreducible [CommMonoidWithZero α] {a : α} {f : Multiset α}
(ha : Irreducible a) (pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a) : ∃ p, a ~ᵤ p ∧ f = {p} := by
haveI := Classical.decEq α
refine @Multiset.induction_on _
(fun g => (g.prod ~ᵤ a) → (∀ b ∈ g, Prime b) → ∃ p, a ~ᵤ p ∧ g = {p}) f ?_ ?_ pfa.2 pfa.1
· intro h; exact (ha.not_isUnit (associated_one_iff_isUnit.1 (Associated.symm h))).elim
· rintro p s _ ⟨u, hu⟩ hs
use p
have hs0 : s = 0 := by
by_contra hs0
obtain ⟨q, hq⟩ := Multiset.exists_mem_of_ne_zero hs0
apply (hs q (by simp [hq])).2.1
refine (ha.isUnit_or_isUnit (?_ : _ = p * ↑u * (s.erase q).prod * _)).resolve_left ?_
· rw [mul_right_comm _ _ q, mul_assoc, ← Multiset.prod_cons, Multiset.cons_erase hq, ← hu,
mul_comm, mul_comm p _, mul_assoc]
simp
apply mt isUnit_of_mul_isUnit_left (mt isUnit_of_mul_isUnit_left _)
apply (hs p (Multiset.mem_cons_self _ _)).2.1
simp only [mul_one, Multiset.prod_cons, Multiset.prod_zero, hs0] at *
exact ⟨Associated.symm ⟨u, hu⟩, rfl⟩
theorem irreducible_iff_prime_of_existsUnique_irreducible_factors [CancelCommMonoidWithZero α]
(eif : ∀ a : α, a ≠ 0 → ∃ f : Multiset α, (∀ b ∈ f, Irreducible b) ∧ f.prod ~ᵤ a)
(uif :
∀ f g : Multiset α,
(∀ x ∈ f, Irreducible x) →
(∀ x ∈ g, Irreducible x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g)
(p : α) : Irreducible p ↔ Prime p :=
letI := Classical.decEq α
⟨ fun hpi =>
⟨hpi.ne_zero, hpi.1, fun a b ⟨x, hx⟩ =>
if hab0 : a * b = 0 then
(eq_zero_or_eq_zero_of_mul_eq_zero hab0).elim (fun ha0 => by simp [ha0]) fun hb0 => by
simp [hb0]
else by
have hx0 : x ≠ 0 := fun hx0 => by simp_all
have ha0 : a ≠ 0 := left_ne_zero_of_mul hab0
have hb0 : b ≠ 0 := right_ne_zero_of_mul hab0
obtain ⟨fx, hfx⟩ := eif x hx0
obtain ⟨fa, hfa⟩ := eif a ha0
obtain ⟨fb, hfb⟩ := eif b hb0
have h : Multiset.Rel Associated (p ::ₘ fx) (fa + fb) := by
apply uif
· exact fun i hi => (Multiset.mem_cons.1 hi).elim (fun hip => hip.symm ▸ hpi) (hfx.1 _)
· exact fun i hi => (Multiset.mem_add.1 hi).elim (hfa.1 _) (hfb.1 _)
calc
Multiset.prod (p ::ₘ fx) ~ᵤ a * b := by
rw [hx, Multiset.prod_cons]; exact hfx.2.mul_left _
_ ~ᵤ fa.prod * fb.prod := hfa.2.symm.mul_mul hfb.2.symm
_ = _ := by rw [Multiset.prod_add]
exact
let ⟨q, hqf, hq⟩ := Multiset.exists_mem_of_rel_of_mem h (Multiset.mem_cons_self p _)
(Multiset.mem_add.1 hqf).elim
(fun hqa =>
Or.inl <| hq.dvd_iff_dvd_left.2 <| hfa.2.dvd_iff_dvd_right.1 (Multiset.dvd_prod hqa))
fun hqb =>
Or.inr <| hq.dvd_iff_dvd_left.2 <| hfb.2.dvd_iff_dvd_right.1 (Multiset.dvd_prod hqb)⟩,
Prime.irreducible⟩
namespace UniqueFactorizationMonoid
variable [CancelCommMonoidWithZero α]
variable [UniqueFactorizationMonoid α]
@[simp]
theorem factors_one : factors (1 : α) = 0 := by
nontriviality α using factors
rw [← Multiset.rel_zero_right]
refine factors_unique irreducible_of_factor (fun x hx => (Multiset.notMem_zero x hx).elim) ?_
rw [Multiset.prod_zero]
exact factors_prod one_ne_zero
theorem exists_mem_factors_of_dvd {a p : α} (ha0 : a ≠ 0) (hp : Irreducible p) :
p ∣ a → ∃ q ∈ factors a, p ~ᵤ q := fun ⟨b, hb⟩ =>
have hb0 : b ≠ 0 := fun hb0 => by simp_all
have : Multiset.Rel Associated (p ::ₘ factors b) (factors a) :=
factors_unique
(fun _ hx => (Multiset.mem_cons.1 hx).elim (fun h => h.symm ▸ hp) (irreducible_of_factor _))
irreducible_of_factor
(Associated.symm <|
calc
Multiset.prod (factors a) ~ᵤ a := factors_prod ha0
_ = p * b := hb
_ ~ᵤ Multiset.prod (p ::ₘ factors b) := by
rw [Multiset.prod_cons]; exact (factors_prod hb0).symm.mul_left _
)
Multiset.exists_mem_of_rel_of_mem this (by simp)
theorem exists_mem_factors {x : α} (hx : x ≠ 0) (h : ¬IsUnit x) : ∃ p, p ∈ factors x := by
obtain ⟨p', hp', hp'x⟩ := WfDvdMonoid.exists_irreducible_factor h hx
obtain ⟨p, hp, _⟩ := exists_mem_factors_of_dvd hx hp' hp'x
exact ⟨p, hp⟩
open Classical in
theorem factors_mul {x y : α} (hx : x ≠ 0) (hy : y ≠ 0) :
Multiset.Rel Associated (factors (x * y)) (factors x + factors y) := by
refine
factors_unique irreducible_of_factor
(fun a ha =>
(Multiset.mem_add.mp ha).by_cases (irreducible_of_factor _) (irreducible_of_factor _))
((factors_prod (mul_ne_zero hx hy)).trans ?_)
rw [Multiset.prod_add]
exact (Associated.mul_mul (factors_prod hx) (factors_prod hy)).symm
theorem factors_pow {x : α} (n : ℕ) :
Multiset.Rel Associated (factors (x ^ n)) (n • factors x) := by
match n with
| 0 => rw [zero_nsmul, pow_zero, factors_one, Multiset.rel_zero_right]
| n + 1 =>
by_cases h0 : x = 0
· simp [h0, zero_pow n.succ_ne_zero, nsmul_zero]
· rw [pow_succ', succ_nsmul']
refine Multiset.Rel.trans _ (factors_mul h0 (pow_ne_zero n h0)) ?_
refine Multiset.Rel.add ?_ <| factors_pow n
exact Multiset.rel_refl_of_refl_on fun y _ => Associated.refl _
@[simp]
theorem factors_pos (x : α) (hx : x ≠ 0) : 0 < factors x ↔ ¬IsUnit x := by
constructor
· intro h hx
obtain ⟨p, hp⟩ := Multiset.exists_mem_of_ne_zero h.ne'
exact (prime_of_factor _ hp).not_unit (isUnit_of_dvd_unit (dvd_of_mem_factors hp) hx)
· intro h
obtain ⟨p, hp⟩ := exists_mem_factors hx h
exact
bot_lt_iff_ne_bot.mpr
(mt Multiset.eq_zero_iff_forall_notMem.mp (not_forall.mpr ⟨p, not_not.mpr hp⟩))
open Multiset in
theorem factors_pow_count_prod [DecidableEq α] {x : α} (hx : x ≠ 0) :
(∏ p ∈ (factors x).toFinset, p ^ (factors x).count p) ~ᵤ x :=
calc
_ = prod (∑ a ∈ toFinset (factors x), count a (factors x) • {a}) := by
simp only [prod_sum, prod_nsmul, prod_singleton]
_ = prod (factors x) := by rw [toFinset_sum_count_nsmul_eq (factors x)]
_ ~ᵤ x := factors_prod hx
theorem factors_rel_of_associated {a b : α} (h : Associated a b) :
Multiset.Rel Associated (factors a) (factors b) := by
rcases iff_iff_and_or_not_and_not.mp h.eq_zero_iff with (⟨rfl, rfl⟩ | ⟨ha, hb⟩)
· simp
· refine factors_unique irreducible_of_factor irreducible_of_factor ?_
exact ((factors_prod ha).trans h).trans (factors_prod hb).symm
end UniqueFactorizationMonoid
namespace Associates
attribute [local instance] Associated.setoid
open Multiset UniqueFactorizationMonoid
variable [CancelCommMonoidWithZero α] [UniqueFactorizationMonoid α]
theorem unique' {p q : Multiset (Associates α)} :
(∀ a ∈ p, Irreducible a) → (∀ a ∈ q, Irreducible a) → p.prod = q.prod → p = q := by
apply Multiset.induction_on_multiset_quot p
apply Multiset.induction_on_multiset_quot q
intro s t hs ht eq
refine Multiset.map_mk_eq_map_mk_of_rel (UniqueFactorizationMonoid.factors_unique ?_ ?_ ?_)
· exact fun a ha => irreducible_mk.1 <| hs _ <| Multiset.mem_map_of_mem _ ha
· exact fun a ha => irreducible_mk.1 <| ht _ <| Multiset.mem_map_of_mem _ ha
have eq' : (Quot.mk Setoid.r : α → Associates α) = Associates.mk := funext quot_mk_eq_mk
rwa [eq', prod_mk, prod_mk, mk_eq_mk_iff_associated] at eq
theorem prod_le_prod_iff_le [Nontrivial α] {p q : Multiset (Associates α)}
(hp : ∀ a ∈ p, Irreducible a) (hq : ∀ a ∈ q, Irreducible a) : p.prod ≤ q.prod ↔ p ≤ q := by
refine ⟨?_, prod_le_prod⟩
rintro ⟨c, eqc⟩
refine Multiset.le_iff_exists_add.2 ⟨factors c, unique' hq (fun x hx ↦ ?_) ?_⟩
· obtain h | h := Multiset.mem_add.1 hx
· exact hp x h
· exact irreducible_of_factor _ h
· rw [eqc, Multiset.prod_add]
congr
refine associated_iff_eq.mp (factors_prod fun hc => ?_).symm
refine not_irreducible_zero (hq _ ?_)
rw [← prod_eq_zero_iff, eqc, hc, mul_zero]
end Associates
section ExistsPrimeFactors
variable [CancelCommMonoidWithZero α]
variable (pf : ∀ a : α, a ≠ 0 → ∃ f : Multiset α, (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a)
include pf
theorem WfDvdMonoid.of_exists_prime_factors : WfDvdMonoid α :=
⟨by
classical
refine RelHomClass.wellFounded
(RelHom.mk ?_ ?_ : (DvdNotUnit : α → α → Prop) →r ((· < ·) : ℕ∞ → ℕ∞ → Prop)) wellFounded_lt
· intro a
by_cases h : a = 0
· exact ⊤
exact ↑(Multiset.card (Classical.choose (pf a h)))
rintro a b ⟨ane0, ⟨c, hc, b_eq⟩⟩
rw [dif_neg ane0]
by_cases h : b = 0
· simp [h, lt_top_iff_ne_top]
· rw [dif_neg h, Nat.cast_lt]
have cne0 : c ≠ 0 := by
refine mt (fun con => ?_) h
rw [b_eq, con, mul_zero]
calc
Multiset.card (Classical.choose (pf a ane0)) <
_ + Multiset.card (Classical.choose (pf c cne0)) :=
lt_add_of_pos_right _
(Multiset.card_pos.mpr fun con => hc (associated_one_iff_isUnit.mp ?_))
_ = Multiset.card (Classical.choose (pf a ane0) + Classical.choose (pf c cne0)) :=
(Multiset.card_add _ _).symm
_ = Multiset.card (Classical.choose (pf b h)) :=
Multiset.card_eq_card_of_rel
(prime_factors_unique ?_ (Classical.choose_spec (pf _ h)).1 ?_)
· convert (Classical.choose_spec (pf c cne0)).2.symm
rw [con, Multiset.prod_zero]
· intro x hadd
rw [Multiset.mem_add] at hadd
rcases hadd with h | h <;> apply (Classical.choose_spec (pf _ _)).1 _ h <;> assumption
· rw [Multiset.prod_add]
trans a * c
· apply Associated.mul_mul <;> apply (Classical.choose_spec (pf _ _)).2 <;> assumption
· rw [← b_eq]
apply (Classical.choose_spec (pf _ _)).2.symm; assumption⟩
theorem irreducible_iff_prime_of_exists_prime_factors {p : α} : Irreducible p ↔ Prime p := by
by_cases hp0 : p = 0
· simp [hp0]
refine ⟨fun h => ?_, Prime.irreducible⟩
obtain ⟨f, hf⟩ := pf p hp0
obtain ⟨q, hq, rfl⟩ := prime_factors_irreducible h hf
rw [hq.prime_iff]
exact hf.1 q (Multiset.mem_singleton_self _)
theorem UniqueFactorizationMonoid.of_exists_prime_factors : UniqueFactorizationMonoid α :=
{ WfDvdMonoid.of_exists_prime_factors pf with
irreducible_iff_prime := irreducible_iff_prime_of_exists_prime_factors pf }
end ExistsPrimeFactors
theorem UniqueFactorizationMonoid.iff_exists_prime_factors [CancelCommMonoidWithZero α] :
UniqueFactorizationMonoid α ↔
∀ a : α, a ≠ 0 → ∃ f : Multiset α, (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a :=
⟨fun h => @UniqueFactorizationMonoid.exists_prime_factors _ _ h,
UniqueFactorizationMonoid.of_exists_prime_factors⟩
section
variable {β : Type*} [CancelCommMonoidWithZero α] [CancelCommMonoidWithZero β]
theorem MulEquiv.uniqueFactorizationMonoid (e : α ≃* β) (hα : UniqueFactorizationMonoid α) :
UniqueFactorizationMonoid β := by
rw [UniqueFactorizationMonoid.iff_exists_prime_factors] at hα ⊢
intro a ha
obtain ⟨w, hp, u, h⟩ :=
hα (e.symm a) fun h =>
ha <| by
convert← map_zero e
simp [← h]
exact
⟨w.map e, fun b hb =>
let ⟨c, hc, he⟩ := Multiset.mem_map.1 hb
he ▸ (prime_iff e).2 (hp c hc),
Units.map e.toMonoidHom u,
by
rw [Multiset.prod_hom, toMonoidHom_eq_coe, Units.coe_map, MonoidHom.coe_coe, ← map_mul e, h,
apply_symm_apply]⟩
theorem MulEquiv.uniqueFactorizationMonoid_iff (e : α ≃* β) :
UniqueFactorizationMonoid α ↔ UniqueFactorizationMonoid β :=
⟨e.uniqueFactorizationMonoid, e.symm.uniqueFactorizationMonoid⟩
end
namespace UniqueFactorizationMonoid
theorem of_existsUnique_irreducible_factors [CancelCommMonoidWithZero α]
(eif : ∀ a : α, a ≠ 0 → ∃ f : Multiset α, (∀ b ∈ f, Irreducible b) ∧ f.prod ~ᵤ a)
(uif :
∀ f g : Multiset α,
(∀ x ∈ f, Irreducible x) →
(∀ x ∈ g, Irreducible x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g) :
UniqueFactorizationMonoid α :=
UniqueFactorizationMonoid.of_exists_prime_factors
(by
convert eif using 7
simp_rw [irreducible_iff_prime_of_existsUnique_irreducible_factors eif uif])
variable {R : Type*} [CancelCommMonoidWithZero R] [UniqueFactorizationMonoid R]
theorem isRelPrime_iff_no_prime_factors {a b : R} (ha : a ≠ 0) :
IsRelPrime a b ↔ ∀ ⦃d⦄, d ∣ a → d ∣ b → ¬Prime d :=
⟨fun h _ ha hb ↦ (·.not_unit <| h ha hb), fun h ↦ WfDvdMonoid.isRelPrime_of_no_irreducible_factors
(ha ·.1) fun _ irr ha hb ↦ h ha hb (UniqueFactorizationMonoid.irreducible_iff_prime.mp irr)⟩
/-- Euclid's lemma: if `a ∣ b * c` and `a` and `c` have no common prime factors, `a ∣ b`.
Compare `IsCoprime.dvd_of_dvd_mul_left`. -/
theorem dvd_of_dvd_mul_left_of_no_prime_factors {a b c : R} (ha : a ≠ 0)
(h : ∀ ⦃d⦄, d ∣ a → d ∣ c → ¬Prime d) : a ∣ b * c → a ∣ b :=
((isRelPrime_iff_no_prime_factors ha).mpr h).dvd_of_dvd_mul_right
/-- Euclid's lemma: if `a ∣ b * c` and `a` and `b` have no common prime factors, `a ∣ c`.
Compare `IsCoprime.dvd_of_dvd_mul_right`. -/
theorem dvd_of_dvd_mul_right_of_no_prime_factors {a b c : R} (ha : a ≠ 0)
(no_factors : ∀ {d}, d ∣ a → d ∣ b → ¬Prime d) : a ∣ b * c → a ∣ c := by
simpa [mul_comm b c] using dvd_of_dvd_mul_left_of_no_prime_factors ha @no_factors
/-- If `a ≠ 0, b` are elements of a unique factorization domain, then dividing
out their common factor `c'` gives `a'` and `b'` with no factors in common. -/
theorem exists_reduced_factors :
∀ a ≠ (0 : R), ∀ b,
∃ a' b' c', IsRelPrime a' b' ∧ c' * a' = a ∧ c' * b' = b := by
intro a
refine induction_on_prime a ?_ ?_ ?_
· intros
contradiction
· intro a a_unit _ b
use a, b, 1
constructor
· intro p p_dvd_a _
exact isUnit_of_dvd_unit p_dvd_a a_unit
· simp
· intro a p a_ne_zero p_prime ih_a pa_ne_zero b
by_cases h : p ∣ b
· rcases h with ⟨b, rfl⟩
obtain ⟨a', b', c', no_factor, ha', hb'⟩ := ih_a a_ne_zero b
refine ⟨a', b', p * c', @no_factor, ?_, ?_⟩
· rw [mul_assoc, ha']
· rw [mul_assoc, hb']
· obtain ⟨a', b', c', coprime, rfl, rfl⟩ := ih_a a_ne_zero b
refine ⟨p * a', b', c', ?_, mul_left_comm _ _ _, rfl⟩
intro q q_dvd_pa' q_dvd_b'
rcases p_prime.left_dvd_or_dvd_right_of_dvd_mul q_dvd_pa' with p_dvd_q | q_dvd_a'
· have : p ∣ c' * b' := dvd_mul_of_dvd_right (p_dvd_q.trans q_dvd_b') _
contradiction
exact coprime q_dvd_a' q_dvd_b'
theorem exists_reduced_factors' (a b : R) (hb : b ≠ 0) :
∃ a' b' c', IsRelPrime a' b' ∧ c' * a' = a ∧ c' * b' = b :=
let ⟨b', a', c', no_factor, hb, ha⟩ := exists_reduced_factors b hb a
⟨a', b', c', fun _ hpb hpa => no_factor hpa hpb, ha, hb⟩
end UniqueFactorizationMonoid
|
Integer.lean
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.Algebra.Group.Pointwise.Set.Scalar
import Mathlib.Algebra.Ring.Subsemiring.Basic
import Mathlib.RingTheory.Localization.Defs
/-!
# Integer elements of a localization
## Main definitions
* `IsLocalization.IsInteger` is a predicate stating that `x : S` is in the image of `R`
## Implementation notes
See `Mathlib/RingTheory/Localization/Basic.lean` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variable {R : Type*} [CommSemiring R] {M : Submonoid R} {S : Type*} [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
open Function
namespace IsLocalization
section
variable (R)
-- TODO: define a subalgebra of `IsInteger`s
/-- Given `a : S`, `S` a localization of `R`, `IsInteger R a` iff `a` is in the image of
the localization map from `R` to `S`. -/
def IsInteger (a : S) : Prop :=
a ∈ (algebraMap R S).rangeS
end
theorem isInteger_zero : IsInteger R (0 : S) :=
Subsemiring.zero_mem _
theorem isInteger_one : IsInteger R (1 : S) :=
Subsemiring.one_mem _
theorem isInteger_add {a b : S} (ha : IsInteger R a) (hb : IsInteger R b) : IsInteger R (a + b) :=
Subsemiring.add_mem _ ha hb
theorem isInteger_mul {a b : S} (ha : IsInteger R a) (hb : IsInteger R b) : IsInteger R (a * b) :=
Subsemiring.mul_mem _ ha hb
theorem isInteger_smul {a : R} {b : S} (hb : IsInteger R b) : IsInteger R (a • b) := by
rcases hb with ⟨b', hb⟩
use a * b'
rw [← hb, (algebraMap R S).map_mul, Algebra.smul_def]
variable (M)
variable [IsLocalization M S]
/-- Each element `a : S` has an `M`-multiple which is an integer.
This version multiplies `a` on the right, matching the argument order in `LocalizationMap.surj`.
-/
theorem exists_integer_multiple' (a : S) : ∃ b : M, IsInteger R (a * algebraMap R S b) :=
let ⟨⟨Num, denom⟩, h⟩ := IsLocalization.surj _ a
⟨denom, Set.mem_range.mpr ⟨Num, h.symm⟩⟩
/-- Each element `a : S` has an `M`-multiple which is an integer.
This version multiplies `a` on the left, matching the argument order in the `SMul` instance.
-/
theorem exists_integer_multiple (a : S) : ∃ b : M, IsInteger R ((b : R) • a) := by
simp_rw [Algebra.smul_def, mul_comm _ a]
apply exists_integer_multiple'
/-- We can clear the denominators of a `Finset`-indexed family of fractions. -/
theorem exist_integer_multiples {ι : Type*} (s : Finset ι) (f : ι → S) :
∃ b : M, ∀ i ∈ s, IsLocalization.IsInteger R ((b : R) • f i) := by
haveI := Classical.propDecidable
refine ⟨∏ i ∈ s, (sec M (f i)).2, fun i hi => ⟨?_, ?_⟩⟩
· exact (∏ j ∈ s.erase i, (sec M (f j)).2) * (sec M (f i)).1
rw [RingHom.map_mul, sec_spec', ← mul_assoc, ← (algebraMap R S).map_mul, ← Algebra.smul_def]
congr 2
refine _root_.trans ?_ (map_prod (Submonoid.subtype M) _ _).symm
rw [mul_comm,Submonoid.coe_finset_prod,
-- Porting note: explicitly supplied `f`
← Finset.prod_insert (f := fun i => ((sec M (f i)).snd : R)) (s.notMem_erase i),
Finset.insert_erase hi]
rfl
/-- We can clear the denominators of a finite indexed family of fractions. -/
theorem exist_integer_multiples_of_finite {ι : Type*} [Finite ι] (f : ι → S) :
∃ b : M, ∀ i, IsLocalization.IsInteger R ((b : R) • f i) := by
cases nonempty_fintype ι
obtain ⟨b, hb⟩ := exist_integer_multiples M Finset.univ f
exact ⟨b, fun i => hb i (Finset.mem_univ _)⟩
/-- We can clear the denominators of a finite set of fractions. -/
theorem exist_integer_multiples_of_finset (s : Finset S) :
∃ b : M, ∀ a ∈ s, IsInteger R ((b : R) • a) :=
exist_integer_multiples M s id
/-- A choice of a common multiple of the denominators of a `Finset`-indexed family of fractions. -/
noncomputable def commonDenom {ι : Type*} (s : Finset ι) (f : ι → S) : M :=
(exist_integer_multiples M s f).choose
/-- The numerator of a fraction after clearing the denominators
of a `Finset`-indexed family of fractions. -/
noncomputable def integerMultiple {ι : Type*} (s : Finset ι) (f : ι → S) (i : s) : R :=
((exist_integer_multiples M s f).choose_spec i i.prop).choose
@[simp]
theorem map_integerMultiple {ι : Type*} (s : Finset ι) (f : ι → S) (i : s) :
algebraMap R S (integerMultiple M s f i) = commonDenom M s f • f i :=
((exist_integer_multiples M s f).choose_spec _ i.prop).choose_spec
/-- A choice of a common multiple of the denominators of a finite set of fractions. -/
noncomputable def commonDenomOfFinset (s : Finset S) : M :=
commonDenom M s id
/-- The finset of numerators after clearing the denominators of a finite set of fractions. -/
noncomputable def finsetIntegerMultiple [DecidableEq R] (s : Finset S) : Finset R :=
s.attach.image fun t => integerMultiple M s id t
open Pointwise
theorem finsetIntegerMultiple_image [DecidableEq R] (s : Finset S) :
algebraMap R S '' finsetIntegerMultiple M s = commonDenomOfFinset M s • (s : Set S) := by
delta finsetIntegerMultiple commonDenom
rw [Finset.coe_image]
ext
constructor
· rintro ⟨_, ⟨x, -, rfl⟩, rfl⟩
rw [map_integerMultiple]
exact Set.mem_image_of_mem _ x.prop
· rintro ⟨x, hx, rfl⟩
exact ⟨_, ⟨⟨x, hx⟩, s.mem_attach _, rfl⟩, map_integerMultiple M s id _⟩
end IsLocalization
|
extraspecial.v
|
(* (c) Copyright 2006-2016 Microsoft Corporation and Inria. *)
(* Distributed under the terms of CeCILL-B. *)
From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div.
From mathcomp Require Import choice fintype bigop finset prime binomial.
From mathcomp Require Import fingroup morphism perm automorphism presentation.
From mathcomp Require Import quotient action commutator gproduct gfunctor.
From mathcomp Require Import ssralg finalg zmodp cyclic pgroup center gseries.
From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal.
From mathcomp Require Import extremal.
(******************************************************************************)
(* This file contains the fine structure thorems for extraspecial p-groups. *)
(* Together with the material in the maximal and extremal libraries, it *)
(* completes the coverage of Aschbacher, section 23. *)
(* We define canonical representatives for the group classes that cover the *)
(* extremal p-groups (non-abelian p-groups with a cyclic maximal subgroup): *)
(* 'Mod_m == the modular group of order m, for m = p ^ n, p prime and n >= 3. *)
(* 'D_m == the dihedral group of order m, for m = 2n >= 4. *)
(* 'Q_m == the generalized quaternion group of order m, for q = 2 ^ n >= 8. *)
(* 'SD_m == the semi-dihedral group of order m, for m = 2 ^ n >= 16. *)
(* In each case the notation is defined in the %type, %g and %G scopes, where *)
(* it denotes a finGroupType, a full gset and the full group for that type. *)
(* However each notation is only meaningful under the given conditions, in *)
(* We construct and study the following extraspecial groups: *)
(* p^{1+2} == if p is prime, an extraspecial group of order p^3 that has *)
(* exponent p or 4, and p-rank 2: thus p^{1+2} is isomorphic to *)
(* 'D_8 if p - 2, and NOT isomorphic to 'Mod_(p^3) if p is odd. *)
(* p^{1+2*n} == the central product of n copies of p^{1+2}, thus of order *)
(* p^(1+2*n) if p is a prime, and, when n > 0, a representative *)
(* of the (unique) isomorphism class of extraspecial groups of *)
(* order p^(1+2*n), of exponent p or 4, and p-rank n+1. *)
(* 'D^n == an alternative (and preferred) notation for 2^{1+2*n}, which *)
(* is isomorphic to the central product of n copies od 'D_8. *)
(* 'D^n*Q == the central product of 'D^n with 'Q_8, thus isomorphic to *)
(* all extraspecial groups of order 2 ^ (2 * n + 3) that are *)
(* not isomorphic to 'D^n.+1 (or, equivalently, have 2-rank n). *)
(* As in extremal.v, these notations are simultaneously defined in the %type, *)
(* %g and %G scopes -- depending on the syntactic context, they denote either *)
(* a finGroupType, the set, or the group of all its elements. *)
(******************************************************************************)
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Notation "n %:R" := (n %:R%R).
Import GroupScope GRing.Theory.
Reserved Notation "p ^{1+2}" (format "p ^{1+2}").
Reserved Notation "p ^{1+2* n }" (n at level 2, format "p ^{1+2* n }").
Reserved Notation "''D^' n" (at level 0, n at level 2, format "''D^' n").
Reserved Notation "''D^' n * 'Q'" (format "''D^' n * 'Q'").
Module Pextraspecial.
Section Construction.
Variable p : nat.
Definition act ij (k : 'Z_p) := let: (i, j) := ij in (i + k * j, j)%R.
Lemma actP : is_action [set: 'Z_p] act.
Proof.
apply: is_total_action=> [] [i j] => [|k1 k2] /=; first by rewrite mul0r addr0.
by rewrite mulrDl addrA.
Qed.
Canonical action := Action actP.
Lemma gactP : is_groupAction [set: 'Z_p * 'Z_p] action.
Proof.
move=> k _ /[1!inE]; apply/andP; split; first by apply/subsetP=> ij _ /[1!inE].
apply/morphicP=> /= [[i1 j1] [i2 j2] _ _].
by rewrite !permE /= mulrDr -addrA (addrCA i2) (addrA i1).
Qed.
Definition groupAction := GroupAction gactP.
Fact gtype_key : unit. Proof. by []. Qed.
Definition gtype := locked_with gtype_key (sdprod_groupType groupAction).
Definition ngtype := ncprod [set: gtype].
End Construction.
Definition ngtypeQ n := xcprod [set: ngtype 2 n] 'Q_8.
End Pextraspecial.
Notation "p ^{1+2}" := (Pextraspecial.gtype p) : type_scope.
Notation "p ^{1+2}" := [set: gsort p^{1+2}] : group_scope.
Notation "p ^{1+2}" := [set: gsort p^{1+2}]%G : Group_scope.
Notation "p ^{1+2* n }" := (Pextraspecial.ngtype p n) : type_scope.
Notation "p ^{1+2* n }" := [set: gsort p^{1+2*n}] : group_scope.
Notation "p ^{1+2* n }" := [set: gsort p^{1+2*n}]%G : Group_scope.
Notation "''D^' n" := (Pextraspecial.ngtype 2 n) : type_scope.
Notation "''D^' n" := [set: gsort 'D^n] : group_scope.
Notation "''D^' n" := [set: gsort 'D^n]%G : Group_scope.
Notation "''D^' n * 'Q'" := (Pextraspecial.ngtypeQ n) : type_scope.
Notation "''D^' n * 'Q'" := [set: gsort 'D^n*Q] : group_scope.
Notation "''D^' n * 'Q'" := [set: gsort 'D^n*Q]%G : Group_scope.
Section ExponentPextraspecialTheory.
Variable p : nat.
Hypothesis p_pr : prime p.
Let p_gt1 := prime_gt1 p_pr.
Let p_gt0 := ltnW p_gt1.
Local Notation gtype := Pextraspecial.gtype.
Local Notation actp := (Pextraspecial.groupAction p).
Lemma card_pX1p2 : #|p^{1+2}| = (p ^ 3)%N.
Proof.
rewrite [@gtype _]unlock -(sdprod_card (sdprod_sdpair _)).
rewrite !card_injm ?injm_sdpair1 ?injm_sdpair2 // !cardsT card_prod card_ord.
by rewrite -mulnA Zp_cast.
Qed.
Lemma Grp_pX1p2 :
p^{1+2} \isog Grp (x : y : x ^+ p, y ^+ p, [~ x, y, x], [~ x, y, y]).
Proof.
rewrite [@gtype _]unlock; apply: intro_isoGrp => [|rT H].
apply/existsP; pose x := sdpair1 actp (0, 1)%R; pose y := sdpair2 actp 1%R.
exists (x, y); rewrite /= !xpair_eqE; set z := [~ x, y]; set G := _ <*> _.
have def_z: z = sdpair1 actp (1, 0)%R.
rewrite [z]commgEl -sdpair_act ?inE //=.
rewrite -morphV -?morphM ?inE //=; congr (sdpair1 _ (_, _)) => /=.
by rewrite mulr1 mulKg.
by rewrite mulVg.
have def_xi i: x ^+ i = sdpair1 actp (0, i%:R)%R. (* FIXME: had to explicitly give actp (instead of _) *)
rewrite -morphX ?inE //; congr (sdpair1 _ _).
by apply/eqP; rewrite /eq_op /= !morphX ?inE ?expg1n //=.
have def_yi i: y ^+ i = sdpair2 actp i%:R. (* FIXME: had to explicitly give actp (instead of _) *)
by rewrite -morphX ?inE //.
have def_zi i: z ^+ i = sdpair1 actp (i%:R, 0)%R. (* FIXME: had to explicitly give actp (instead of _) *)
rewrite def_z -morphX ?inE //; congr (sdpair1 _ _).
by apply/eqP; rewrite /eq_op /= !morphX ?inE ?expg1n ?andbT //=.
rewrite def_xi def_yi pchar_Zp ?morph1 //.
rewrite def_z -morphR ?inE // !commgEl -sdpair_act ?inE //= mulr0 addr0.
rewrite mulVg -[_ * _]/(_ , _) /= !invg1 mulg1 !mul1g mulVg morph1 !andbT.
have Gx: x \in G by rewrite -cycle_subG joing_subl.
have Gy: y \in G by rewrite -cycle_subG joing_subr.
rewrite eqEsubset subsetT -im_sdpair mulG_subG /= -/G; apply/andP; split.
apply/subsetP=> u /morphimP[[i j] _ _ def_u].
suffices ->: u = z ^+ i * x ^+ j.
rewrite groupMl; apply/groupX; first exact: Gx.
by apply/groupR; first exact: Gx.
rewrite def_zi def_xi !natr_Zp -morphM ?inE // def_u.
by congr (sdpair1 _ (_, _)); rewrite ?mulg1 ?mul1g.
apply/subsetP=> v /morphimP[k _ _ def_v].
suffices ->: v = y ^+ k by rewrite groupX.
by rewrite def_yi natr_Zp.
case/existsP=> [[x y] /=]; set z := [~ x, y].
case/eqP=> defH xp yp /eqP/commgP czx /eqP/commgP czy.
have zp: z ^+ p = 1 by rewrite -commXg // xp comm1g.
pose f1 (ij : 'Z_p * 'Z_p) := let: (i, j) := ij in z ^+ i * x ^+ j.
have f1M: {in setT &, {morph f1 : u v / u * v}}.
case=> /= [i1 j1] [i2 j2] _ _ /=; rewrite {3 6}Zp_cast // !expg_mod //.
rewrite !expgD !mulgA; congr (_ * _); rewrite -!mulgA; congr (_ * _).
by apply: commuteX2.
pose f2 (k : 'Z_p) := y ^+ k.
have f2M: {in setT &, {morph f2 : u v / u * v}}.
by move=> k1 k2 _ _; rewrite /f2 /= {3}Zp_cast // expg_mod // expgD.
have actf: {in setT & setT, morph_act actp 'J (Morphism f1M) (Morphism f2M)}.
case=> /= i j k _ _; rewrite modnDmr {4}Zp_cast // expg_mod // expgD.
rewrite /f2 conjMg {1}/conjg (commuteX2 i k czy) mulKg -mulgA.
congr (_ * _); rewrite (commuteX2 _ _ czx) mulnC expgM.
by rewrite -commXg // -commgX ?mulKVg // commXg // /commute commuteX.
apply/homgP; exists (xsdprod_morphism actf).
apply/eqP; rewrite eqEsubset -{2}defH -genM_join gen_subG /= im_xsdprodm.
have Hx: x \in H by rewrite -cycle_subG -defH joing_subl.
have Hy: y \in H by rewrite -cycle_subG -defH joing_subr.
rewrite mulG_subG -andbA; apply/and3P; split.
- apply/subsetP=> _ /morphimP[[i j] _ _ -> /=].
by rewrite groupMl groupX ?groupR.
- by apply/subsetP=> _ /morphimP[k _ _ ->]; rewrite groupX.
rewrite mulgSS ?cycle_subG //= morphimEdom; apply/imsetP.
by exists (0, 1)%R; rewrite ?inE //= mul1g.
by exists 1%R; rewrite ?inE.
Qed.
Lemma pX1p2_pgroup : p.-group p^{1+2}.
Proof. by rewrite /pgroup card_pX1p2 pnatX pnat_id. Qed.
(* This is part of the existence half of Aschbacher ex. (8.7)(1) *)
Lemma pX1p2_extraspecial : extraspecial p^{1+2}.
Proof.
apply: (p3group_extraspecial pX1p2_pgroup); last first.
by rewrite card_pX1p2 pfactorK.
case/existsP: (isoGrp_hom Grp_pX1p2) card_pX1p2 => [[x y]] /=.
case/eqP=> <- xp yp _ _ oXY.
apply: contraL (dvdn_cardMg <[x]> <[y]>) => cXY_XY.
rewrite -cent_joinEl ?(sub_abelian_cent2 cXY_XY) ?joing_subl ?joing_subr //.
rewrite oXY -!orderE pfactor_dvdn ?muln_gt0 ?order_gt0 // -leqNgt.
rewrite -(pfactorK 2 p_pr) dvdn_leq_log ?expn_gt0 ?p_gt0 //.
by rewrite dvdn_mul ?order_dvdn ?xp ?yp.
Qed.
(* This is part of the existence half of Aschbacher ex. (8.7)(1) *)
Lemma exponent_pX1p2 : odd p -> exponent p^{1+2} %| p.
Proof.
move=> p_odd; have pG := pX1p2_pgroup.
have ->: p^{1+2} = 'Ohm_1(p^{1+2}).
apply/eqP; rewrite eqEsubset Ohm_sub andbT (OhmE 1 pG).
case/existsP: (isoGrp_hom Grp_pX1p2) => [[x y]] /=.
case/eqP=> <- xp yp _ _; rewrite joing_idl joing_idr genS //.
by rewrite subsetI subset_gen subUset !sub1set !inE xp yp!eqxx.
rewrite exponent_Ohm1_class2 ?card_pX1p2 ?oddX // nil_class2.
by have [[_ ->] _ ] := pX1p2_extraspecial.
Qed.
(* This is the uniqueness half of Aschbacher ex. (8.7)(1) *)
Lemma isog_pX1p2 (gT : finGroupType) (G : {group gT}) :
extraspecial G -> exponent G %| p -> #|G| = (p ^ 3)%N -> G \isog p^{1+2}.
Proof.
move=> esG expGp oG; apply/(isoGrpP _ Grp_pX1p2).
rewrite card_pX1p2; split=> //.
have pG: p.-group G by rewrite /pgroup oG pnatX pnat_id.
have oZ := card_center_extraspecial pG esG.
have [x Gx notZx]: exists2 x, x \in G & x \notin 'Z(G).
apply/subsetPn; rewrite proper_subn // properEcard center_sub oZ oG.
by rewrite (ltn_exp2l 1 3).
have ox: #[x] = p.
by apply: nt_prime_order; rewrite ?(exponentP expGp) ?(group1_contra notZx).
have [y Gy not_cxy]: exists2 y, y \in G & y \notin 'C[x].
by apply/subsetPn; rewrite sub_cent1; rewrite inE Gx in notZx.
apply/existsP; exists (x, y) => /=; set z := [~ x, y].
have [[defPhiG defG'] _] := esG.
have Zz: z \in 'Z(G) by rewrite -defG' mem_commg.
have [Gz cGz] := setIP Zz; rewrite !xpair_eqE !(exponentP expGp) //.
have [_ nZG] := andP (center_normal G).
rewrite /commg /conjg !(centP cGz) // !mulKg mulVg !eqxx !andbT.
have sXY_G: <[x]> <*> <[y]> \subset G by rewrite join_subG !cycle_subG Gx.
have defZ: <[z]> = 'Z(G).
apply/eqP; rewrite eqEcard cycle_subG Zz oZ /= -orderE.
rewrite (nt_prime_order p_pr) ?(exponentP expGp) //.
by rewrite (sameP commgP cent1P) cent1C.
have sZ_XY: 'Z(G) \subset <[x]> <*> <[y]>.
by rewrite -defZ cycle_subG groupR // mem_gen // inE cycle_id ?orbT.
rewrite eqEcard sXY_G /= oG -(Lagrange sZ_XY) oZ leq_pmul2l //.
rewrite -card_quotient ?(subset_trans sXY_G) //.
rewrite quotientY ?quotient_cycle ?cycle_subG ?(subsetP nZG) //.
have abelGz: p.-abelem (G / 'Z(G)) by rewrite -defPhiG Phi_quotient_abelem.
have [cGzGz expGz] := abelemP p_pr abelGz.
rewrite cent_joinEr ?(sub_abelian_cent2 cGzGz) ?cycle_subG ?mem_quotient //.
have oZx: #|<[coset 'Z(G) x]>| = p.
rewrite -orderE (nt_prime_order p_pr) ?expGz ?mem_quotient //.
by apply: contra notZx; move/eqP=> Zx; rewrite coset_idr ?(subsetP nZG).
rewrite TI_cardMg ?oZx -?orderE ?(nt_prime_order p_pr) ?expGz ?mem_quotient //.
apply: contra not_cxy; move/eqP=> Zy.
rewrite -cent_cycle (subsetP _ y (coset_idr _ Zy)) ?(subsetP nZG) //.
by rewrite subIset ?centS ?orbT ?cycle_subG.
rewrite prime_TIg ?oZx // cycle_subG; apply: contra not_cxy.
case/cycleP=> i; rewrite -morphX ?(subsetP nZG) // => /rcoset_kercosetP.
rewrite groupX ?(subsetP nZG) // cent1C => /(_ isT isT); apply: subsetP.
rewrite mul_subG ?sub1set ?groupX ?cent1id //= -cent_cycle subIset // orbC.
by rewrite centS ?cycle_subG.
Qed.
End ExponentPextraspecialTheory.
Section GeneralExponentPextraspecialTheory.
Variable p : nat.
Lemma pX1p2id : p^{1+2*1} \isog p^{1+2}.
Proof. exact: ncprod1. Qed.
Lemma pX1p2S n : xcprod_spec p^{1+2} p^{1+2*n} p^{1+2*n.+1}%type.
Proof. exact: ncprodS. Qed.
Lemma card_pX1p2n n : prime p -> #|p^{1+2*n}| = (p ^ n.*2.+1)%N.
Proof.
move=> p_pr; have pG := pX1p2_pgroup p_pr.
have oG := card_pX1p2 p_pr; have esG := pX1p2_extraspecial p_pr.
have oZ := card_center_extraspecial pG esG.
elim: n => [|n IHn]; first by rewrite (card_isog (ncprod0 _)) oZ.
case: pX1p2S => gz isoZ; rewrite -im_cpair cardMg_divn setI_im_cpair.
rewrite -injm_center ?{1}card_injm ?injm_cpairg1 ?injm_cpair1g ?center_sub //.
by rewrite oG oZ IHn -expnD mulKn ?prime_gt0.
Qed.
Lemma pX1p2n_pgroup n : prime p -> p.-group p^{1+2*n}.
Proof. by move=> p_pr; rewrite /pgroup card_pX1p2n // pnatX pnat_id. Qed.
(* This is part of the existence half of Aschbacher (23.13) *)
Lemma exponent_pX1p2n n : prime p -> odd p -> exponent p^{1+2*n} = p.
Proof.
move=> p_pr odd_p; apply: prime_nt_dvdP => //.
rewrite -dvdn1 -trivg_exponent -cardG_gt1 card_pX1p2n //.
by rewrite (ltn_exp2l 0) // prime_gt1.
elim: n => [|n IHn].
by rewrite (dvdn_trans (exponent_dvdn _)) ?card_pX1p2n.
case: pX1p2S => gz isoZ; rewrite -im_cpair /=.
apply/exponentP=> xy; case/imset2P=> x y C1x C2y ->{xy}.
rewrite expgMn; last by red; rewrite -(centsP (im_cpair_cent isoZ)).
rewrite (exponentP _ y C2y) ?exponent_injm ?injm_cpair1g // mulg1.
by rewrite (exponentP _ x C1x) ?exponent_injm ?injm_cpairg1 // exponent_pX1p2.
Qed.
(* This is part of the existence half of Aschbacher (23.13) and (23.14) *)
Lemma pX1p2n_extraspecial n : prime p -> n > 0 -> extraspecial p^{1+2*n}.
Proof.
move=> p_pr; elim: n => [//|n IHn _].
have esG := pX1p2_extraspecial p_pr.
have [n0 | n_gt0] := posnP n.
by apply: isog_extraspecial esG; rewrite isog_sym n0 pX1p2id.
case: pX1p2S (pX1p2n_pgroup n.+1 p_pr) => gz isoZ pGn.
apply: (cprod_extraspecial pGn (im_cpair_cprod isoZ) (setI_im_cpair isoZ)).
by apply: injm_extraspecial esG; rewrite ?injm_cpairg1.
by apply: injm_extraspecial (IHn n_gt0); rewrite ?injm_cpair1g.
Qed.
(* This is Aschbacher (23.12) *)
Lemma Ohm1_extraspecial_odd (gT : finGroupType) (G : {group gT}) :
p.-group G -> extraspecial G -> odd #|G| ->
let Y := 'Ohm_1(G) in
[/\ exponent Y = p, #|G : Y| %| p
& Y != G ->
exists E : {group gT},
[/\ #|G : Y| = p, #|E| = p \/ extraspecial E,
exists2 X : {group gT}, #|X| = p & X \x E = Y
& exists M : {group gT},
[/\ M \isog 'Mod_(p ^ 3), M \* E = G & M :&: E = 'Z(M)]]].
Proof.
move=> pG esG oddG Y; have [spG _] := esG.
have [defPhiG defG'] := spG; set Z := 'Z(G) in defPhiG defG'.
have{spG} expG: exponent G %| p ^ 2 by apply: exponent_special.
have p_pr := extraspecial_prime pG esG.
have p_gt1 := prime_gt1 p_pr; have p_gt0 := ltnW p_gt1.
have oZ: #|Z| = p := card_center_extraspecial pG esG.
have nsZG: Z <| G := center_normal G; have [sZG nZG] := andP nsZG.
have nsYG: Y <| G := Ohm_normal 1 G; have [sYG nYG] := andP nsYG.
have ntZ: Z != 1 by rewrite -cardG_gt1 oZ.
have sZY: Z \subset Y.
by apply: contraR ntZ => ?; rewrite -(setIidPl sZG) TI_Ohm1 ?prime_TIg ?oZ.
have ntY: Y != 1 by apply: subG1_contra ntZ.
have p_odd: odd p by rewrite -oZ (oddSg sZG).
have expY: exponent Y %| p by rewrite exponent_Ohm1_class2 // nil_class2 defG'.
rewrite (prime_nt_dvdP p_pr _ expY) -?dvdn1 -?trivg_exponent //.
have [-> | neYG] := eqVneq Y G; first by rewrite indexgg dvd1n; split.
have sG1Z: 'Mho^1(G) \subset Z by rewrite -defPhiG (Phi_joing pG) joing_subr.
have Z_Gp: {in G, forall x, x ^+ p \in Z}.
by move=> x Gx; rewrite /= (subsetP sG1Z) ?(Mho_p_elt 1) ?(mem_p_elt pG).
have{expG} oY': {in G :\: Y, forall u, #[u] = (p ^ 2)%N}.
move=> u /setDP[Gu notYu]; apply/eqP.
have [k ou] := p_natP (mem_p_elt pG Gu).
rewrite eqn_dvd order_dvdn (exponentP expG) // eqxx ou dvdn_Pexp2l // ltnNge.
apply: contra notYu => k_le_1; rewrite [Y](OhmE _ pG) mem_gen // !inE Gu /=.
by rewrite -order_dvdn ou dvdn_exp2l.
have isoMod3 (M : {group gT}):
M \subset G -> ~~ abelian M -> ~~ (M \subset Y) -> #|M| = (p ^ 3)%N ->
M \isog 'Mod_(p ^ 3).
- move=> sMG not_cMM /subsetPn[u Mu notYu oM].
have pM := pgroupS sMG pG; have sUM: <[u]> \subset M by rewrite cycle_subG.
have Y'u: u \in G :\: Y by rewrite inE notYu (subsetP sMG).
have iUM: #|M : <[u]>| = p by rewrite -divgS // oM expnS -(oY' u) ?mulnK.
have cM := maximal_cycle_extremal pM not_cMM (cycle_cyclic u) sUM iUM.
rewrite (sameP eqP (prime_oddPn p_pr)) p_odd orbF in cM.
rewrite /extremal_class oM pdiv_pfactor // pfactorK //= in cM.
by do 3!case: ifP => // _ in cM.
have iYG: #|G : Y| = p.
have [V maxV sYV]: {V : {group gT} | maximal V G & Y \subset V}.
by apply: maxgroup_exists; rewrite properEneq neYG.
have [sVG [u Gu notVu]] := properP (maxgroupp maxV).
without loss [v Vv notYv]: / exists2 v, v \in V & v \notin Y.
have [->| ] := eqVneq Y V; first by rewrite (p_maximal_index pG).
by rewrite eqEsubset sYV => not_sVY; apply; apply/subsetPn.
pose U := <[u]> <*> <[v]>; have Gv := subsetP sVG v Vv.
have sUG: U \subset G by rewrite join_subG !cycle_subG Gu.
have Uu: u \in U by rewrite -cycle_subG joing_subl.
have Uv: v \in U by rewrite -cycle_subG joing_subr.
have not_sUY: ~~ (U \subset Y) by apply/subsetPn; exists v.
have sU1U: 'Ohm_1(U) \subset U := Ohm_sub 1 _.
have sU1Y: 'Ohm_1(U) \subset Y := OhmS 1 sUG.
suffices defUV: U :&: V = 'Ohm_1(U).
by rewrite (subsetP sU1Y) // -defUV inE Uv in notYv.
suffices iU1U: #|U : 'Ohm_1(U)| = p.
have: maximal 'Ohm_1(U) U by rewrite p_index_maximal ?Ohm_sub ?iU1U.
case/maxgroupP=> _; apply; rewrite /= -/U.
by apply/properP; split; last exists u; rewrite ?subsetIl ?inE ?Uu.
by rewrite subsetI Ohm_sub (subset_trans sU1Y).
apply/prime_nt_dvdP=> //.
by apply: contra not_sUY; rewrite /U; move/eqP; move/(index1g sU1U)=> <-.
have ov: #[v] = (p ^ 2)%N by rewrite oY' // inE notYv.
have sZv: Z \subset <[v]>.
suffices defZ: <[v ^+ p]> == Z by rewrite -(eqP defZ) cycleX.
by rewrite eqEcard cycle_subG Z_Gp //= oZ -orderE (orderXexp 1 ov).
have nvG: G \subset 'N(<[v]>) by rewrite sub_der1_norm ?cycle_subG // defG'.
have [cUU | not_cUU] := orP (orbN (abelian U)).
rewrite -divgS ?Ohm_sub // -(mul_card_Ohm_Mho_abelian 1 cUU) /= -/U.
by rewrite mulKn ?cardG_gt0 //= -oZ cardSg ?(subset_trans (MhoS 1 sUG)).
have oU: #|U| = (p ^ 3)%N.
have nvu := subsetP nvG u Gu; have nvU := subset_trans sUG nvG.
rewrite -(Lagrange (joing_subr _ _)) -orderE ov mulnC; congr (_ * _)%N.
rewrite -card_quotient //= quotientYidr ?cycle_subG //=.
rewrite quotient_cycle // -orderE; apply: nt_prime_order => //.
by rewrite -morphX //= coset_id // (subsetP sZv) // Z_Gp.
have svV: <[v]> \subset V by rewrite cycle_subG.
by apply: contra notVu; move/eqP=> v_u; rewrite (subsetP svV) // coset_idr.
have isoU := isoMod3 _ sUG not_cUU not_sUY oU; rewrite /= -/U in isoU.
have [//|[x y] genU modU] := generators_modular_group p_pr _ isoU.
case/modular_group_structure: genU => // _ _ _ _.
case: eqP (p_odd) => [[-> //] | _ _]; case/(_ 1%N)=> // _ oU1.
by rewrite -divgS // oU oU1 mulnK // muln_gt0 p_gt0.
have iC1U (U : {group gT}) x:
U \subset G -> x \in G :\: 'C(U) -> #|U : 'C_U[x]| = p.
- move=> sUG /setDP[Gx not_cUx]; apply/prime_nt_dvdP=> //.
apply: contra not_cUx; rewrite -sub_cent1 => /eqP sUCx.
by rewrite -(index1g _ sUCx) ?subsetIl ?subsetIr.
rewrite -(@dvdn_pmul2l (#|U| * #|'C_G[x]|)) ?muln_gt0 ?cardG_gt0 //.
have maxCx: maximal 'C_G[x] G.
rewrite cent1_extraspecial_maximal //; apply: contra not_cUx.
by rewrite inE Gx; apply: subsetP (centS sUG) _.
rewrite {1}mul_cardG setIA (setIidPl sUG) -(p_maximal_index pG maxCx) -!mulnA.
rewrite !Lagrange ?subsetIl // mulnC dvdn_pmul2l //.
have [sCxG nCxG] := andP (p_maximal_normal pG maxCx).
by rewrite -norm_joinEl ?cardSg ?join_subG ?(subset_trans sUG).
have oCG (U : {group gT}):
Z \subset U -> U \subset G -> #|'C_G(U)| = (p * #|G : U|)%N.
- have [m] := ubnP #|U|; elim: m U => // m IHm U leUm sZU sUG.
have [<- | neZU] := eqVneq Z U.
by rewrite -oZ Lagrange // (setIidPl _) // centsC subsetIr.
have{neZU} [x Gx not_cUx]: exists2 x, x \in G & x \notin 'C(U).
by apply/subsetPn; rewrite eqEsubset sZU subsetI sUG centsC in neZU.
pose W := 'C_U[x]; have iWU: #|U : W| = p by rewrite iC1U // inE not_cUx.
have maxW: maximal W U by rewrite p_index_maximal ?subsetIl ?iWU.
have ltWU: W \proper U by apply: maxgroupp maxW.
have [sWU [u Uu notWu]] := properP ltWU.
have defU: W * <[u]> = U.
have nsWU: W <| U := p_maximal_normal (pgroupS sUG pG) maxW.
by rewrite (mulg_normal_maximal nsWU) ?cycle_subG.
have sWG := subset_trans sWU sUG.
have sZW: Z \subset W.
by rewrite subsetI sZU -cent_set1 subIset ?centS ?orbT ?sub1set.
have iCW_CU: #|'C_G(W) : 'C_G(U)| = p.
rewrite -defU centM cent_cycle setIA /= -/W.
rewrite iC1U ?subsetIl ?setIS ?centS // inE andbC (subsetP sUG) //=.
rewrite -sub_cent1; apply/subsetPn; exists x.
by rewrite inE Gx -sub_cent1 subsetIr.
by rewrite -defU centM cent_cycle inE -sub_cent1 subsetIr in not_cUx.
apply/eqP; rewrite -(eqn_pmul2r p_gt0) -{1}iCW_CU Lagrange ?setIS ?centS //.
rewrite IHm ?(leq_trans (proper_card ltWU)) //= -/W.
by rewrite -(Lagrange_index sUG sWU) iWU mulnA.
have oCY: #|'C_G(Y)| = (p ^ 2)%N by rewrite oCG // iYG.
have [x cYx notZx]: exists2 x, x \in 'C_G(Y) & x \notin Z.
apply/subsetPn; rewrite proper_subn // properEcard setIS ?centS //=.
by rewrite oZ oCY (ltn_exp2l 1 2).
have{cYx} [Gx cYx] := setIP cYx; have nZx := subsetP nZG x Gx.
have defCx: 'C_G[x] = Y.
apply/eqP; rewrite eq_sym eqEcard subsetI sYG sub_cent1 cYx /=.
rewrite -(leq_pmul2r p_gt0) -{2}iYG -(iC1U G x) ?Lagrange ?subsetIl //.
by rewrite !inE Gx ?andbT in notZx *.
have Yx: x \in Y by rewrite -defCx inE Gx cent1id.
have ox: #[x] = p.
by apply: nt_prime_order; rewrite ?(exponentP expY) // (group1_contra notZx).
have defCy: 'C_G(Y) = Z * <[x]>.
apply/eqP; rewrite eq_sym eqEcard mulG_subG setIS ?centS //=.
rewrite cycle_subG inE Gx cYx oCY TI_cardMg ?oZ -?orderE ?ox //=.
by rewrite setIC prime_TIg -?orderE ?ox ?cycle_subG.
have abelYt: p.-abelem (Y / Z).
by rewrite (abelemS (quotientS _ sYG)) //= -/Z -defPhiG Phi_quotient_abelem.
have Yxt: coset Z x \in Y / Z by rewrite mem_quotient.
have{Yxt} [Et [sEtYt oEt defYt]] := p_abelem_split1 abelYt Yxt.
have nsZY: Z <| Y := normalS sZY sYG nsZG.
have [E defEt sZE sEY] := inv_quotientS nsZY sEtYt.
have{defYt} [_ defYt _ tiXEt] := dprodP defYt.
have defY: <[x]> \x E = Y.
have nZX: <[x]> \subset 'N(Z) by rewrite cycle_subG.
have TIxE: <[x]> :&: E = 1.
rewrite prime_TIg -?orderE ?ox // -(quotientSGK _ sZE) ?quotient_cycle //.
rewrite (sameP setIidPl eqP) eq_sym -defEt tiXEt -quotient_cycle //.
by rewrite -subG1 quotient_sub1 // cycle_subG.
rewrite dprodE //; last 1 first.
by rewrite cent_cycle (subset_trans sEY) //= -/Y -defCx subsetIr.
rewrite -[Y](quotientGK nsZY) -defYt cosetpreM -quotient_cycle //.
rewrite quotientK // -(normC nZX) defEt quotientGK ?(normalS _ sEY) //.
by rewrite -mulgA (mulSGid sZE).
have sEG := subset_trans sEY sYG; have nZE := subset_trans sEG nZG.
have defZE: 'Z(E) = Z.
apply/eqP; rewrite eqEsubset andbC subsetI sZE subIset ?centS ?orbT //.
rewrite -quotient_sub1 ?subIset ?nZE //= -tiXEt defEt subsetI andbC.
rewrite quotientS ?center_sub //= -quotient_cycle //.
rewrite -(quotientMidl _ <[x]>) /= -defCy quotientS // /Y.
by case/dprodP: defY => _ <- _ _; rewrite centM setIA cent_cycle defCx setSI.
have pE := pgroupS sEG pG.
rewrite iYG; split=> // _; exists E.
split=> //; first 2 [by exists [group of <[x]>]].
have:= sZE; rewrite subEproper; case/predU1P=> [<- | ltZE]; [by left | right].
split; rewrite /special defZE ?oZ // (Phi_joing pE).
have defE': E^`(1) = Z.
have sE'Z: E^`(1) \subset Z by rewrite -defG' dergS.
apply/eqP; rewrite eqEcard sE'Z -(prime_nt_dvdP _ _ (cardSg sE'Z)) ?oZ //=.
rewrite -trivg_card1 (sameP eqP commG1P).
by rewrite /proper sZE /= -/Z -defZE subsetI subxx in ltZE.
split=> //; rewrite -defE'; apply/joing_idPl.
by rewrite /= defE' -defPhiG (Phi_joing pG) joingC sub_gen ?subsetU ?MhoS.
have iEG: #|G : E| = (p ^ 2)%N.
apply/eqP; rewrite -(@eqn_pmul2l #|E|) // Lagrange // -(Lagrange sYG) iYG.
by rewrite -(dprod_card defY) -mulnA mulnCA -orderE ox.
pose M := 'C_G(E); exists [group of M] => /=.
have sMG: M \subset G := subsetIl _ _; have pM: p.-group M := pgroupS sMG pG.
have sZM: Z \subset M by rewrite setIS ?centS.
have oM: #|M| = (p ^ 3)%N by rewrite oCG ?iEG.
have defME: M * E = G.
apply/eqP; rewrite eqEcard mulG_subG sMG sEG /= -(leq_pmul2r p_gt0).
rewrite -{2}oZ -defZE /('Z(E)) -{2}(setIidPr sEG) setIAC -mul_cardG /= -/M.
by rewrite -(Lagrange sEG) mulnAC -mulnA mulnC iEG oM.
have defZM: 'Z(M) = Z.
apply/eqP; rewrite eqEsubset andbC subsetI sZM subIset ?centS ?orbT //=.
by rewrite /Z /('Z(G)) -{2}defME centM setIA setIAC.
rewrite cprodE 1?centsC ?subsetIr //.
rewrite defME setIAC (setIidPr sEG) defZM isoMod3 //.
rewrite abelianE (sameP setIidPl eqP) eqEcard subsetIl /= -/('Z(M)) -/M.
by rewrite defZM oZ oM (leq_exp2l 3 1).
by apply: contra neYG => sMY; rewrite eqEsubset sYG -defME mulG_subG sMY.
Qed.
(* This is the uniqueness half of Aschbacher (23.13); the proof incorporates *)
(* in part the proof that symplectic spaces are hyperbolic (19.16). *)
Lemma isog_pX1p2n n (gT : finGroupType) (G : {group gT}) :
prime p -> extraspecial G -> #|G| = (p ^ n.*2.+1)%N -> exponent G %| p ->
G \isog p^{1+2*n}.
Proof.
move=> p_pr esG oG expG; have p_gt1 := prime_gt1 p_pr.
have not_le_p3_p: ~~ (p ^ 3 <= p) by rewrite (leq_exp2l 3 1).
have pG: p.-group G by rewrite /pgroup oG pnatX pnat_id.
have oZ := card_center_extraspecial pG esG.
have{pG esG} [Es p3Es defG] := extraspecial_structure pG esG.
set Z := 'Z(G) in oZ defG p3Es.
elim: Es {+}G => [|E Es IHs] S in n oG expG p3Es defG *.
rewrite big_nil cprod1g in defG; rewrite -defG.
have ->: n = 0.
apply: double_inj; apply/eqP.
by rewrite -eqSS -(eqn_exp2l _ _ p_gt1) -oG -defG oZ.
by rewrite isog_cyclic_card prime_cyclic ?oZ ?card_pX1p2n //=.
rewrite big_cons -cprodA in defG; rewrite /= -andbA in p3Es.
have [[_ T _ defT] defET cTE] := cprodP defG; rewrite defT in defET cTE defG.
move: p3Es => /and3P[/eqP oE /eqP defZE /IHs{}IHs].
have not_cEE: ~~ abelian E.
by apply: contra not_le_p3_p => cEE; rewrite -oE -oZ -defZE (center_idP _).
have sES: E \subset S by rewrite -defET mulG_subl.
have sTS: T \subset S by rewrite -defET mulG_subr.
have expE: exponent E %| p by apply: dvdn_trans (exponentS sES) expG.
have expT: exponent T %| p by apply: dvdn_trans (exponentS sTS) expG.
have{expE not_cEE} isoE: E \isog p^{1+2}.
apply: isog_pX1p2 => //.
by apply: card_p3group_extraspecial p_pr oE _; rewrite defZE.
have sZT: 'Z(E) \subset T.
by case/cprodP: defT => [[U _ -> _] <- _]; rewrite defZE mulG_subr.
case def_n: n => [|n'].
case/negP: not_le_p3_p; rewrite def_n in oG; rewrite -oE -[p]oG.
exact: subset_leq_card.
have zI_ET: E :&: T = 'Z(E).
by apply/eqP; rewrite eqEsubset subsetI sZT subsetIl setIS // centsC.
have{n def_n oG} oT: #|T| = (p ^ n'.*2.+1)%N.
apply/eqP; rewrite -(eqn_pmul2l (cardG_gt0 E)) mul_cardG zI_ET defET.
by rewrite defZE oE oG oZ -expnSr -expnD def_n.
have{IHs oT expT defT Es} isoT: T \isog p^{1+2*n'} by rewrite IHs.
case: pX1p2S => gz isoZ; rewrite (isog_cprod_by _ defG) //.
exact: Aut_extraspecial_full (pX1p2_pgroup p_pr) (pX1p2_extraspecial p_pr).
Qed.
End GeneralExponentPextraspecialTheory.
Lemma isog_2X1p2 : 2^{1+2} \isog 'D_8.
Proof.
have pr2: prime 2 by []; have oG := card_pX1p2 pr2; rewrite -[8]oG.
case/existsP: (isoGrp_hom (Grp_pX1p2 pr2)) => [[x y]] /=.
rewrite -/2^{1+2}; case/eqP=> defG x2 y2 _ _.
have not_oG_2: ~~ (#|2^{1+2}| %| 2) by rewrite oG.
have ox: #[x] = 2.
apply: nt_prime_order => //; apply: contra not_oG_2 => x1.
by rewrite -defG (eqP x1) cycle1 joing1G order_dvdn y2.
have oy: #[y] = 2.
apply: nt_prime_order => //; apply: contra not_oG_2 => y1.
by rewrite -defG (eqP y1) cycle1 joingG1 order_dvdn x2.
rewrite -defG joing_idl joing_idr involutions_gen_dihedral //.
apply: contra not_oG_2 => eq_xy; rewrite -defG (eqP eq_xy) (joing_idPl _) //.
by rewrite -orderE oy.
Qed.
Lemma Q8_extraspecial : extraspecial 'Q_8.
Proof.
have gt32: 3 > 2 by []; have isoQ: 'Q_8 \isog 'Q_(2 ^ 3) by apply: isog_refl.
have [[x y] genQ _] := generators_quaternion gt32 isoQ.
have [_ [defQ' defPhiQ _ _]] := quaternion_structure gt32 genQ isoQ.
case=> defZ oZ _ _ _ _ _; split; last by rewrite oZ.
by split; rewrite ?defPhiQ defZ.
Qed.
Lemma DnQ_P n : xcprod_spec 'D^n 'Q_8 ('D^n*Q)%type.
Proof.
have pQ: 2.-group 'Q_(2 ^ 3) by rewrite /pgroup card_quaternion.
have{pQ} oZQ := card_center_extraspecial pQ Q8_extraspecial.
suffices oZDn: #|'Z('D^n)| = 2.
by apply: xcprodP; rewrite isog_cyclic_card ?prime_cyclic ?oZQ ?oZDn.
have [-> | n_gt0] := posnP n; first by rewrite center_ncprod0 card_pX1p2n.
have pr2: prime 2 by []; have pDn := pX1p2n_pgroup n pr2.
exact: card_center_extraspecial (pX1p2n_extraspecial pr2 n_gt0).
Qed.
Lemma card_DnQ n : #|'D^n*Q| = (2 ^ n.+1.*2.+1)%N.
Proof.
have oQ: #|'Q_(2 ^ 3)| = 8 by rewrite card_quaternion.
have pQ: 2.-group 'Q_8 by rewrite /pgroup oQ.
case: DnQ_P => gz isoZ.
rewrite -im_cpair cardMg_divn setI_im_cpair cpair_center_id.
rewrite -injm_center//; last exact: injm_cpair1g.
rewrite (card_injm (injm_cpairg1 _))//= (card_injm (injm_cpair1g _))//.
rewrite (card_injm (injm_cpair1g _))//; last exact: center_sub.
rewrite oQ card_pX1p2n // (card_center_extraspecial pQ Q8_extraspecial).
by rewrite -muln_divA // mulnC -(expnD 2 2).
Qed.
Lemma DnQ_pgroup n : 2.-group 'D^n*Q.
Proof. by rewrite /pgroup card_DnQ pnatX. Qed.
(* Final part of the existence half of Aschbacher (23.14). *)
Lemma DnQ_extraspecial n : extraspecial 'D^n*Q.
Proof.
case: DnQ_P (DnQ_pgroup n) => gz isoZ pDnQ.
have [injDn injQ] := (injm_cpairg1 isoZ, injm_cpair1g isoZ).
have [n0 | n_gt0] := posnP n.
rewrite -im_cpair mulSGid; first exact: injm_extraspecial Q8_extraspecial.
apply/setIidPl; rewrite setI_im_cpair -injm_center //=.
by congr (_ @* _); rewrite n0 center_ncprod0.
apply: (cprod_extraspecial pDnQ (im_cpair_cprod isoZ) (setI_im_cpair _)).
exact: injm_extraspecial (pX1p2n_extraspecial _ _).
exact: injm_extraspecial Q8_extraspecial.
Qed.
(* A special case of the uniqueness half of Achsbacher (23.14). *)
Lemma card_isog8_extraspecial (gT : finGroupType) (G : {group gT}) :
#|G| = 8 -> extraspecial G -> (G \isog 'D_8) || (G \isog 'Q_8).
Proof.
move=> oG esG; have pG: 2.-group G by rewrite /pgroup oG.
apply/norP=> [[notG_D8 notG_Q8]].
have not_extG: extremal_class G = NotExtremal.
by rewrite /extremal_class oG andFb (negPf notG_D8) (negPf notG_Q8).
have [x Gx ox] := exponent_witness (pgroup_nil pG).
pose X := <[x]>; have cycX: cyclic X := cycle_cyclic x.
have sXG: X \subset G by rewrite cycle_subG.
have iXG: #|G : X| = 2.
by rewrite -divgS // oG -orderE -ox exponent_2extraspecial.
have not_cGG := extraspecial_nonabelian esG.
have:= maximal_cycle_extremal pG not_cGG cycX sXG iXG.
by rewrite /extremal2 not_extG.
Qed.
(* The uniqueness half of Achsbacher (23.14). The proof incorporates in part *)
(* the proof that symplectic spces are hyperbolic (Aschbacher (19.16)), and *)
(* the determination of quadratic spaces over 'F_2 (21.2); however we use *)
(* the second part of exercise (8.4) to avoid resorting to Witt's lemma and *)
(* Galois theory as in (20.9) and (21.1). *)
Lemma isog_2extraspecial (gT : finGroupType) (G : {group gT}) n :
#|G| = (2 ^ n.*2.+1)%N -> extraspecial G -> G \isog 'D^n \/ G \isog 'D^n.-1*Q.
Proof.
elim: n G => [|n IHn] G oG esG.
case/negP: (extraspecial_nonabelian esG).
by rewrite cyclic_abelian ?prime_cyclic ?oG.
have pG: 2.-group G by rewrite /pgroup oG pnatX.
have oZ:= card_center_extraspecial pG esG.
have: 'Z(G) \subset 'Ohm_1(G).
apply/subsetP=> z Zz; rewrite (OhmE _ pG) mem_gen //.
by rewrite !inE -order_dvdn -oZ order_dvdG ?(subsetP (center_sub G)).
rewrite subEproper; case/predU1P=> [defG1 | ltZG1].
have [n' n'_gt2 isoG]: exists2 n', n' > 2 & G \isog 'Q_(2 ^ n').
apply/quaternion_classP; apply/eqP.
have not_cycG: ~~ cyclic G.
by apply: contra (extraspecial_nonabelian esG); apply: cyclic_abelian.
move: oZ; rewrite defG1; move/prime_Ohm1P; rewrite (negPf not_cycG) /=.
by apply=> //; apply: contra not_cycG; move/eqP->; apply: cyclic1.
have [n0 n'3]: n = 0 /\ n' = 3.
have [[x y] genG _] := generators_quaternion n'_gt2 isoG.
have n'3: n' = 3.
have [_ [_ _ oG' _] _ _ _] := quaternion_structure n'_gt2 genG isoG.
apply/eqP; rewrite -(subnKC (ltnW n'_gt2)) subn2 !eqSS -(@eqn_exp2l 2) //.
by rewrite -oG' -oZ; case: esG => [[_ ->]].
by move/eqP: oG; have [-> _ _ _] := genG; rewrite n'3 eqn_exp2l //; case n.
right; rewrite (isog_trans isoG) // n'3 n0 /=.
case: DnQ_P => z isoZ; rewrite -im_cpair mulSGid ?sub_isog ?injm_cpair1g //.
apply/setIidPl; rewrite setI_im_cpair -injm_center ?injm_cpairg1 //.
by rewrite center_ncprod0.
case/andP: ltZG1 => _; rewrite (OhmE _ pG) gen_subG.
case/subsetPn=> x; case/LdivP=> Gx x2 notZx.
have ox: #[x] = 2 by apply: nt_prime_order (group1_contra notZx).
have Z'x: x \in G :\: 'Z(G) by rewrite inE notZx.
have [E [R [[oE oR] [defG ziER]]]] := split1_extraspecial pG esG Z'x.
case=> defZE defZR [esE Ex] esR.
have isoE: E \isog 2^{1+2}.
apply: isog_trans (isog_symr isog_2X1p2).
case/orP: (card_isog8_extraspecial oE esE) => // isoE; case/negP: notZx.
have gt32: 3 > 2 by [].
have [[y z] genE _] := generators_quaternion gt32 isoE.
have [_ _ [defZx _ eq_y2 _ _] _ _] := quaternion_structure gt32 genE isoE.
by rewrite (eq_y2 x) // -cycle_subG -defZx defZE.
rewrite oG doubleS 2!expnS divnMl ?mulKn // in oR.
case: ifP esR => [_ defR | _ esR].
have ->: n = 0 by move/eqP: oR; rewrite defR oZ (eqn_exp2l 1) //; case n.
left; apply: isog_trans (isog_symr (ncprod1 _)).
by rewrite -defG defR -defZE cprod_center_id.
have AutZin2_1p2: Aut_in (Aut 2^{1+2}) 'Z(2^{1+2}) \isog Aut 'Z(2^{1+2}).
exact: Aut_extraspecial_full (pX1p2_pgroup _) (pX1p2_extraspecial _).
have [isoR | isoR] := IHn R oR esR.
by left; case: pX1p2S => gz isoZ; rewrite (isog_cprod_by _ defG).
have n_gt0: n > 0.
have pR: 2.-group R by rewrite /pgroup oR pnatX.
have:= min_card_extraspecial pR esR.
by rewrite oR leq_exp2l // ltnS (leq_double 1).
case: DnQ_P isoR => gR isoZR /=; rewrite isog_sym; case/isogP=> fR injfR im_fR.
have [injDn injQ] := (injm_cpairg1 isoZR, injm_cpair1g isoZR).
pose Dn1 := cpairg1 isoZR @* 'D^n.-1; pose Q := cpair1g isoZR @* 'Q_8.
have defR: fR @* Dn1 \* fR @* Q = R.
rewrite cprodE ?morphim_cents ?im_cpair_cent //.
by rewrite -morphimMl ?subsetT ?im_cpair.
rewrite -defR cprodA in defG.
have [[Dn _ defDn _] _ _] := cprodP defG; rewrite defDn in defG.
have isoDn: Dn \isog 'D^n.
rewrite -(prednK n_gt0); case: pX1p2S => gz isoZ.
rewrite (isog_cprod_by _ defDn) //; last 1 first.
by rewrite isog_sym (isog_trans _ (sub_isog _ _)) ?subsetT // sub_isog.
rewrite /= -morphimIim im_fR setIA ziER; apply/setIidPl.
rewrite defZE -defZR -{1}im_fR -injm_center // morphimS //.
by rewrite -cpairg1_center morphimS // center_sub.
right; case: DnQ_P => gz isoZ; rewrite (isog_cprod_by _ defG) //; first 1 last.
- exact: Aut_extraspecial_full (pX1p2n_pgroup _ _) (pX1p2n_extraspecial _ _).
- by rewrite isog_sym (isog_trans _ (sub_isog _ _)) ?subsetT // sub_isog.
rewrite /= -morphimIim; case/cprodP: defDn => _ defDn cDn1E.
rewrite setICA setIA -defDn -group_modr ?morphimS ?subsetT //.
rewrite /= im_fR (setIC R) ziER -center_prod // defZE -defZR.
rewrite mulSGid /=; last first.
by rewrite -{1}im_fR -injm_center // -cpairg1_center !morphimS ?center_sub.
rewrite -injm_center ?subsetT // -injmI // setI_im_cpair.
by rewrite -injm_center // cpairg1_center injm_center // im_fR mulGid.
Qed.
(* The first concluding remark of Aschbacher (23.14). *)
Lemma rank_Dn n : 'r_2('D^n) = n.+1.
Proof.
elim: n => [|n IHn]; first by rewrite p_rank_abelem ?prime_abelem ?card_pX1p2n.
have oDDn: #|'D^n.+1| = (2 ^ n.+1.*2.+1)%N by apply: card_pX1p2n.
have esDDn: extraspecial 'D^n.+1 by apply: pX1p2n_extraspecial.
do [case: pX1p2S => gz isoZ; set DDn := [set: _]] in oDDn esDDn *.
have pDDn: 2.-group DDn by rewrite /pgroup oDDn pnatX.
apply/eqP; rewrite eqn_leq; apply/andP; split.
have [E EprE]:= p_rank_witness 2 [group of DDn].
have [sEDDn abelE <-] := pnElemP EprE; have [pE cEE _]:= and3P abelE.
rewrite -(@leq_exp2l 2) // -p_part part_pnat_id // -leq_sqr -expnM -mulnn.
rewrite muln2 doubleS expnS -oDDn -(@leq_pmul2r #|'C_DDn(E)|) ?cardG_gt0 //.
rewrite {1}(card_subcent_extraspecial pDDn) // mulnCA -mulnA Lagrange //=.
rewrite mulnAC mulnA leq_pmul2r ?cardG_gt0 // setTI.
have ->: (2 * #|'C(E)| = #|'Z(DDn)| * #|'C(E)|)%N.
by rewrite (card_center_extraspecial pDDn).
by rewrite leq_mul ?subset_leq_card ?subsetIl.
have [inj1 injn] := (injm_cpairg1 isoZ, injm_cpair1g isoZ).
pose D := cpairg1 isoZ @* 2^{1+2}; pose Dn := cpair1g isoZ @* 'D^n.
have [E EprE] := p_rank_witness 2 [group of Dn].
rewrite injm_p_rank //= IHn in EprE; have [sEDn abelE dimE]:= pnElemP EprE.
have [x [Dx ox] notDnx]: exists x, [/\ x \in D, #[x] = 2 & x \notin Dn].
have isoD: D \isog 'D_(2 ^ 3).
by rewrite isog_sym -(isog_transl _ isog_2X1p2) sub_isog.
have [//| [x y] genD [oy _]] := generators_2dihedral _ isoD.
have [_ _ _ X'y] := genD; case/setDP: X'y; rewrite /= -/D => Dy notXy.
exists y; split=> //; apply: contra notXy => Dny.
case/dihedral2_structure: genD => // _ _ _ _ [defZD _ _ _ _].
by rewrite (subsetP (cycleX x 2)) // -defZD -setI_im_cpair inE Dy.
have def_xE: <[x]> \x E = <[x]> <*> E.
rewrite dprodEY ?prime_TIg -?orderE ?ox //.
by rewrite (centSS sEDn _ (im_cpair_cent _)) ?cycle_subG.
by rewrite cycle_subG (contra (subsetP sEDn x)).
apply/p_rank_geP; exists (<[x]> <*> E)%G.
rewrite 2!inE subsetT (dprod_abelem _ def_xE) abelE -(dprod_card def_xE).
by rewrite prime_abelem -?orderE ?ox //= lognM ?cardG_gt0 ?dimE.
Qed.
(* The second concluding remark of Aschbacher (23.14). *)
Lemma rank_DnQ n : 'r_2('D^n*Q) = n.+1.
Proof.
have pDnQ: 2.-group 'D^n*Q := DnQ_pgroup n.
have esDnQ: extraspecial 'D^n*Q := DnQ_extraspecial n.
do [case: DnQ_P => gz isoZ; set DnQ := setT] in pDnQ esDnQ *.
suffices [E]: exists2 E, E \in 'E*_2(DnQ) & logn 2 #|E| = n.+1.
by rewrite (pmaxElem_extraspecial pDnQ esDnQ); case/pnElemP=> _ _ <-.
have oZ: #|'Z(DnQ)| = 2 by apply: card_center_extraspecial.
pose Dn := cpairg1 isoZ @* 'D^n; pose Q := cpair1g isoZ @* 'Q_8.
have [injDn injQ] := (injm_cpairg1 isoZ, injm_cpair1g isoZ).
have [E EprE]:= p_rank_witness 2 [group of Dn].
have [sEDn abelE dimE] := pnElemP EprE; have [pE cEE eE]:= and3P abelE.
rewrite injm_p_rank // rank_Dn in dimE; exists E => //.
have sZE: 'Z(DnQ) \subset E.
have maxE := subsetP (p_rankElem_max _ _) E EprE.
have abelZ: 2.-abelem 'Z(DnQ) by rewrite prime_abelem ?oZ.
rewrite -(Ohm1_id abelZ) (OhmE _ (abelem_pgroup abelZ)) gen_subG.
rewrite -(pmaxElem_LdivP _ maxE) // setSI //=.
by rewrite -cpairg1_center injm_center // setIS ?centS.
have scE: 'C_Dn(E) = E.
apply/eqP; rewrite eq_sym eqEcard subsetI sEDn -abelianE cEE /=.
have [n0 | n_gt0] := posnP n.
rewrite subset_leq_card // subIset // (subset_trans _ sZE) //.
by rewrite -cpairg1_center morphimS // n0 center_ncprod0.
have pDn: 2.-group Dn by rewrite morphim_pgroup ?pX1p2n_pgroup.
have esDn: extraspecial Dn.
exact: injm_extraspecial (pX1p2n_extraspecial _ _).
rewrite dvdn_leq ?cardG_gt0 // (card_subcent_extraspecial pDn) //=.
rewrite -injm_center // cpairg1_center (setIidPl sZE) oZ.
rewrite -(dvdn_pmul2l (cardG_gt0 E)) mulnn mulnCA Lagrange //.
rewrite card_injm ?card_pX1p2n // -expnS pfactor_dvdn ?expn_gt0 ?cardG_gt0 //.
by rewrite lognX dimE mul2n.
apply/pmaxElemP; split=> [|F E2F sEF]; first by rewrite inE subsetT abelE.
have{E2F} [_ abelF] := pElemP E2F; have [pF cFF eF] := and3P abelF.
apply/eqP; rewrite eqEsubset sEF andbT; apply/subsetP=> x Fx.
have DnQx: x \in Dn * Q by rewrite im_cpair inE.
have{DnQx} [y z Dn_y Qz def_x]:= imset2P DnQx.
have{Dn_y} Ey: y \in E.
have cEz: z \in 'C(E).
by rewrite (subsetP (centS sEDn)) // (subsetP (im_cpair_cent _)).
rewrite -scE inE Dn_y -(groupMr _ cEz) -def_x (subsetP (centS sEF)) //.
by rewrite (subsetP cFF).
rewrite def_x groupMl // (subsetP sZE) // -cpair1g_center injm_center //= -/Q.
have: z \in 'Ohm_1(Q).
rewrite (OhmE 1 (pgroupS (subsetT Q) pDnQ)) mem_gen // !inE Qz /=.
rewrite -[z](mulKg y) -def_x (exponentP eF) ?groupM //.
by rewrite groupV (subsetP sEF).
have isoQ: Q \isog 'Q_(2 ^ 3) by rewrite isog_sym sub_isog.
have [//|[u v] genQ _] := generators_quaternion _ isoQ.
by case/quaternion_structure: genQ => // _ _ [-> _ _ [-> _] _] _ _.
Qed.
(* The final concluding remark of Aschbacher (23.14). *)
Lemma not_isog_Dn_DnQ n : ~~ ('D^n \isog 'D^n.-1*Q).
Proof.
case: n => [|n] /=; first by rewrite isogEcard card_pX1p2n // card_DnQ andbF.
apply: contraL (leqnn n.+1) => isoDn1DnQ.
by rewrite -ltnNge -rank_Dn (isog_p_rank isoDn1DnQ) rank_DnQ.
Qed.
|
SynthesizeUsing.lean
|
/-
Copyright (c) 2022 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Init
import Lean.Elab.Tactic.Basic
import Qq
/-!
# `SynthesizeUsing`
This is a slight simplification of the `solve_aux` tactic in Lean3.
-/
open Lean Elab Tactic Meta Qq
/--
`synthesizeUsing type tac` synthesizes an element of type `type` using tactic `tac`.
The tactic `tac` is allowed to leave goals open, and these remain as metavariables in the
returned expression.
-/
-- In Lean3 this was called `solve_aux`,
-- and took a `TacticM α` and captured the produced value in `α`.
-- As this was barely used, we've simplified here.
def synthesizeUsing {u : Level} (type : Q(Sort u)) (tac : TacticM Unit) :
MetaM (List MVarId × Q($type)) := do
let m ← mkFreshExprMVar type
let goals ← (Term.withoutErrToSorry <| run m.mvarId! tac).run'
return (goals, ← instantiateMVars m)
/--
`synthesizeUsing type tac` synthesizes an element of type `type` using tactic `tac`.
The tactic must solve for all goals, in contrast to `synthesizeUsing`.
-/
def synthesizeUsing' {u : Level} (type : Q(Sort u)) (tac : TacticM Unit) : MetaM Q($type) := do
let (goals, e) ← synthesizeUsing type tac
-- Note: doesn't use `tac *> Tactic.done` since that just adds a message
-- rather than raising an error.
unless goals.isEmpty do
throwError m!"synthesizeUsing': unsolved goals\n{goalsToMessageData goals}"
return e
/--
`synthesizeUsing type tacticSyntax` synthesizes an element of type `type` by evaluating the
given tactic syntax.
Example:
```lean
let (gs, e) ← synthesizeUsingTactic ty (← `(tactic| congr!))
```
The tactic `tac` is allowed to leave goals open, and these remain as metavariables in the
returned expression.
-/
def synthesizeUsingTactic {u : Level} (type : Q(Sort u)) (tac : Syntax) :
MetaM (List MVarId × Q($type)) := do
synthesizeUsing type (do evalTactic tac)
/--
`synthesizeUsing' type tacticSyntax` synthesizes an element of type `type` by evaluating the
given tactic syntax.
Example:
```lean
let e ← synthesizeUsingTactic' ty (← `(tactic| norm_num))
```
The tactic must solve for all goals, in contrast to `synthesizeUsingTactic`.
If you need to insert expressions into a tactic proof, then you might use `synthesizeUsing'`
directly, since the `TacticM` monad has access to the `TermElabM` monad. For example, here
is a term elaborator that wraps the `simp at ...` tactic:
```
def simpTerm (e : Expr) : MetaM Expr := do
let mvar ← Meta.mkFreshTypeMVar
let e' ← synthesizeUsing' mvar
(do evalTactic (← `(tactic| have h := $(← Term.exprToSyntax e); simp at h; exact h)))
-- Note: `simp` does not always insert type hints, so to ensure that we get a term
-- with the simplified type (as opposed to one that is merely defeq), we should add
-- a type hint ourselves.
Meta.mkExpectedTypeHint e' mvar
elab "simpTerm% " t:term : term => do simpTerm (← Term.elabTerm t none)
```
-/
def synthesizeUsingTactic' {u : Level} (type : Q(Sort u)) (tac : Syntax) : MetaM Q($type) := do
synthesizeUsing' type (do evalTactic tac)
|
jordanholder.v
|
(* (c) Copyright 2006-2016 Microsoft Corporation and Inria. *)
(* Distributed under the terms of CeCILL-B. *)
From HB Require Import structures.
From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path.
From mathcomp Require Import choice fintype bigop finset fingroup morphism.
From mathcomp Require Import automorphism quotient action gseries.
(******************************************************************************)
(* This files establishes Jordan-Holder theorems for finite groups. These *)
(* theorems state the uniqueness up to permutation and isomorphism for the *)
(* series of quotient built from the successive elements of any composition *)
(* series of the same group. These quotients are also called factors of the *)
(* composition series. To avoid the heavy use of highly polymorphic lists *)
(* describing these quotient series, we introduce sections. *)
(* This library defines: *)
(* (G1 / G2)%sec == alias for the pair (G1, G2) of groups in the same *)
(* finGroupType, coerced to the actual quotient group*)
(* group G1 / G2. We call this pseudo-quotient a *)
(* section of G1 and G2. *)
(* section_isog s1 s2 == s1 and s2 respectively coerce to isomorphic *)
(* quotient groups. *)
(* section_repr s == canonical representative of the isomorphism class *)
(* of the section s. *)
(* mksrepr G1 G2 == canonical representative of the isomorphism class *)
(* of (G1 / G2)%sec. *)
(* mkfactors G s == if s is [:: s1, s2, ..., sn], constructs the list *)
(* [:: mksrepr G s1, mksrepr s1 s2, ..., mksrepr sn-1 sn] *)
(* comps G s == s is a composition series for G i.e. s is a *)
(* decreasing sequence of subgroups of G *)
(* in which two adjacent elements are maxnormal one *)
(* in the other and the last element of s is 1. *)
(* Given aT and rT two finGroupTypes, (D : {group rT}), (A : {group aT}) and *)
(* (to : groupAction A D) an external action. *)
(* maxainv to B C == C is a maximal proper normal subgroup of B *)
(* invariant by (the external action of A via) to. *)
(* asimple to B == the maximal proper normal subgroup of B invariant *)
(* by the external action to is trivial. *)
(* acomps to G s == s is a composition series for G invariant by to, *)
(* i.e. s is a decreasing sequence of subgroups of G *)
(* in which two adjacent elements are maximally *)
(* invariant by to one in the other and the *)
(* last element of s is 1. *)
(* We prove two versions of the result: *)
(* - JordanHolderUniqueness establishes the uniqueness up to permutation *)
(* and isomorphism of the lists of factors in composition series of a *)
(* given group. *)
(* - StrongJordanHolderUniqueness extends the result to composition series *)
(* invariant by an external group action. *)
(* See also "The Rooster and the Butterflies", proceedings of Calculemus 2013,*)
(* by Assia Mahboubi. *)
(******************************************************************************)
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Declare Scope section_scope.
Import GroupScope.
Inductive section (gT : finGroupType) := GSection of {group gT} * {group gT}.
Delimit Scope section_scope with sec.
Bind Scope section_scope with section.
Definition mkSec (gT : finGroupType) (G1 G2 : {group gT}) := GSection (G1, G2).
Infix "/" := mkSec : section_scope.
Coercion pair_of_section gT (s : section gT) := let: GSection u := s in u.
Coercion quotient_of_section gT (s : section gT) : GroupSet.sort _ := s.1 / s.2.
Coercion section_group gT (s : section gT) : {group (coset_of s.2)} :=
Eval hnf in [group of s].
Section Sections.
Variables (gT : finGroupType).
Implicit Types (G : {group gT}) (s : section gT).
HB.instance Definition _ := [isNew for (@pair_of_section gT)].
HB.instance Definition _ := [Finite of section gT by <:].
Canonical section_group.
(* Isomorphic sections *)
Definition section_isog := [rel x y : section gT | x \isog y].
(* A witness of the isomorphism class of a section *)
Definition section_repr s := odflt (1 / 1)%sec (pick (section_isog ^~ s)).
Definition mksrepr G1 G2 := section_repr (mkSec G1 G2).
Lemma section_reprP s : section_repr s \isog s.
Proof.
by rewrite /section_repr; case: pickP => //= /(_ s); rewrite isog_refl.
Qed.
Lemma section_repr_isog s1 s2 :
s1 \isog s2 -> section_repr s1 = section_repr s2.
Proof.
by move=> iso12; congr (odflt _ _); apply: eq_pick => s; apply: isog_transr.
Qed.
Definition mkfactors (G : {group gT}) (s : seq {group gT}) :=
map section_repr (pairmap (@mkSec _) G s).
End Sections.
Section CompositionSeries.
Variable gT : finGroupType.
Local Notation gTg := {group gT}.
Implicit Types (G : gTg) (s : seq gTg).
Local Notation compo := [rel x y : {set gT} | maxnormal y x x].
Definition comps G s := ((last G s) == 1%G) && compo.-series G s.
Lemma compsP G s :
reflect (last G s = 1%G /\ path [rel x y : gTg | maxnormal y x x] G s)
(comps G s).
Proof. by apply: (iffP andP) => [] [/eqP]. Qed.
Lemma trivg_comps G s : comps G s -> (G :==: 1) = (s == [::]).
Proof.
case/andP=> ls cs; apply/eqP/eqP=> [G1 | s1]; last first.
by rewrite s1 /= in ls; apply/eqP.
by case: s {ls} cs => //= H s /andP[/maxgroupp]; rewrite G1 /proper sub1G andbF.
Qed.
Lemma comps_cons G H s : comps G (H :: s) -> comps H s.
Proof. by case/andP => /= ls /andP[_]; rewrite /comps ls. Qed.
Lemma simple_compsP G s : comps G s -> reflect (s = [:: 1%G]) (simple G).
Proof.
move=> cs; apply: (iffP idP) => [|s1]; last first.
by rewrite s1 /comps eqxx /= andbT -simple_maxnormal in cs.
case: s cs => [/trivg_comps/eqP-> | H s]; first by case/simpleP; rewrite eqxx.
rewrite [comps _ _]andbCA /= => /andP[/maxgroupp maxH /trivg_comps/esym nil_s].
rewrite simple_maxnormal => /maxgroupP[_ simG].
have H1: H = 1%G by apply/val_inj/simG; rewrite // sub1G.
by move: nil_s; rewrite H1 eqxx => /eqP->.
Qed.
Lemma exists_comps (G : gTg) : exists s, comps G s.
Proof.
elim: {G} #|G| {1 3}G (leqnn #|G|) => [G | n IHn G cG].
by rewrite leqNgt cardG_gt0.
have [sG | nsG] := boolP (simple G).
by exists [:: 1%G]; rewrite /comps eqxx /= -simple_maxnormal andbT.
have [-> | ntG] := eqVneq G 1%G; first by exists [::]; rewrite /comps eqxx.
have [N maxN] := ex_maxnormal_ntrivg ntG.
have [|s /andP[ls cs]] := IHn N.
by rewrite -ltnS (leq_trans _ cG) // proper_card // (maxnormal_proper maxN).
by exists (N :: s); apply/and3P.
Qed.
(******************************************************************************)
(* The factors associated to two composition series of the same group are *)
(* the same up to isomorphism and permutation *)
(******************************************************************************)
Lemma JordanHolderUniqueness (G : gTg) (s1 s2 : seq gTg) :
comps G s1 -> comps G s2 -> perm_eq (mkfactors G s1) (mkfactors G s2).
Proof.
have [n] := ubnP #|G|; elim: n G => // n Hi G in s1 s2 * => /ltnSE-cG cs1 cs2.
have [G1 | ntG] := boolP (G :==: 1).
have -> : s1 = [::] by apply/eqP; rewrite -(trivg_comps cs1).
have -> : s2 = [::] by apply/eqP; rewrite -(trivg_comps cs2).
by rewrite /= perm_refl.
have [sG | nsG] := boolP (simple G).
by rewrite (simple_compsP cs1 sG) (simple_compsP cs2 sG) perm_refl.
case es1: s1 cs1 => [|N1 st1] cs1.
by move: (trivg_comps cs1); rewrite eqxx; move/negP:ntG.
case es2: s2 cs2 => [|N2 st2] cs2 {s1 es1}.
by move: (trivg_comps cs2); rewrite eqxx; move/negP:ntG.
case/andP: cs1 => /= lst1; case/andP=> maxN_1 pst1.
case/andP: cs2 => /= lst2; case/andP=> maxN_2 pst2.
have cN1 : #|N1| < n.
by rewrite (leq_trans _ cG) ?proper_card ?(maxnormal_proper maxN_1).
have cN2 : #|N2| < n.
by rewrite (leq_trans _ cG) ?proper_card ?(maxnormal_proper maxN_2).
case: (N1 =P N2) {s2 es2} => [eN12 |].
by rewrite eN12 /= perm_cons Hi // /comps ?lst2 //= -eN12 lst1.
move/eqP; rewrite -val_eqE /=; move/eqP=> neN12.
have nN1G : N1 <| G by apply: maxnormal_normal.
have nN2G : N2 <| G by apply: maxnormal_normal.
pose N := (N1 :&: N2)%G.
have nNG : N <| G.
by rewrite /normal subIset ?(normal_sub nN1G) //= normsI ?normal_norm.
have iso1 : (G / N1)%G \isog (N2 / N)%G.
rewrite isog_sym /= -(maxnormalM maxN_1 maxN_2) //.
rewrite (@normC _ N1 N2) ?(subset_trans (normal_sub nN1G)) ?normal_norm //.
by rewrite weak_second_isog ?(subset_trans (normal_sub nN2G)) ?normal_norm.
have iso2 : (G / N2)%G \isog (N1 / N)%G.
rewrite isog_sym /= -(maxnormalM maxN_1 maxN_2) // setIC.
by rewrite weak_second_isog ?(subset_trans (normal_sub nN1G)) ?normal_norm.
have [sN /andP[lsN csN]] := exists_comps N.
have i1 : perm_eq (mksrepr G N1 :: mkfactors N1 st1)
[:: mksrepr G N1, mksrepr N1 N & mkfactors N sN].
rewrite perm_cons -[mksrepr _ _ :: _]/(mkfactors N1 [:: N & sN]).
apply: Hi=> //; rewrite /comps ?lst1 //= lsN csN andbT /=.
rewrite -quotient_simple.
by rewrite -(isog_simple iso2) quotient_simple.
by rewrite (normalS (subsetIl N1 N2) (normal_sub nN1G)).
have i2 : perm_eq (mksrepr G N2 :: mkfactors N2 st2)
[:: mksrepr G N2, mksrepr N2 N & mkfactors N sN].
rewrite perm_cons -[mksrepr _ _ :: _]/(mkfactors N2 [:: N & sN]).
apply: Hi=> //; rewrite /comps ?lst2 //= lsN csN andbT /=.
rewrite -quotient_simple.
by rewrite -(isog_simple iso1) quotient_simple.
by rewrite (normalS (subsetIr N1 N2) (normal_sub nN2G)).
pose fG1 := [:: mksrepr G N1, mksrepr N1 N & mkfactors N sN].
pose fG2 := [:: mksrepr G N2, mksrepr N2 N & mkfactors N sN].
have i3 : perm_eq fG1 fG2.
rewrite (@perm_catCA _ [::_] [::_]) /mksrepr.
rewrite (@section_repr_isog _ (mkSec _ _) (mkSec _ _) iso1).
rewrite -(@section_repr_isog _ (mkSec _ _) (mkSec _ _) iso2).
exact: perm_refl.
apply: (perm_trans i1); apply: (perm_trans i3); rewrite perm_sym.
by apply: perm_trans i2; apply: perm_refl.
Qed.
End CompositionSeries.
(******************************************************************************)
(* Helper lemmas for group actions. *)
(******************************************************************************)
Section MoreGroupAction.
Variables (aT rT : finGroupType).
Variables (A : {group aT}) (D : {group rT}).
Variable to : groupAction A D.
Lemma gactsP (G : {set rT}) : reflect {acts A, on G | to} [acts A, on G | to].
Proof.
apply: (iffP idP) => [nGA x|nGA]; first exact: acts_act.
apply/subsetP=> a Aa /[!inE]; rewrite Aa.
by apply/subsetP=> x; rewrite inE nGA.
Qed.
Lemma gactsM (N1 N2 : {set rT}) :
N1 \subset D -> N2 \subset D ->
[acts A, on N1 | to] -> [acts A, on N2 | to] -> [acts A, on N1 * N2 | to].
Proof.
move=> sN1D sN2D aAN1 aAN2; apply/gactsP=> x Ax y.
apply/idP/idP; case/mulsgP=> y1 y2 N1y1 N2y2 e.
move: (actKin to Ax y); rewrite e; move<-.
rewrite gactM ?groupV ?(subsetP sN1D y1) ?(subsetP sN2D) //.
by apply: mem_mulg; rewrite ?(gactsP _ aAN1) ?(gactsP _ aAN2) // groupV.
rewrite e gactM // ?(subsetP sN1D y1) ?(subsetP sN2D) //.
by apply: mem_mulg; rewrite ?(gactsP _ aAN1) // ?(gactsP _ aAN2).
Qed.
Lemma gactsI (N1 N2 : {set rT}) :
[acts A, on N1 | to] -> [acts A, on N2 | to] -> [acts A, on N1 :&: N2 | to].
Proof.
move=> aAN1 aAN2.
apply/subsetP=> x Ax; rewrite !inE Ax /=; apply/subsetP=> y Ny /[1!inE].
case/setIP: Ny=> N1y N2y; rewrite inE ?astabs_act ?N1y ?N2y //.
- by move/subsetP: aAN2; move/(_ x Ax).
- by move/subsetP: aAN1; move/(_ x Ax).
Qed.
Lemma gastabsP (S : {set rT}) (a : aT) :
a \in A -> reflect (forall x, (to x a \in S) = (x \in S)) (a \in 'N(S | to)).
Proof.
move=> Aa; apply: (iffP idP) => [nSa x|nSa]; first exact: astabs_act.
by rewrite !inE Aa; apply/subsetP=> x; rewrite inE nSa.
Qed.
End MoreGroupAction.
(******************************************************************************)
(* Helper lemmas for quotient actions. *)
(******************************************************************************)
Section MoreQuotientAction.
Variables (aT rT : finGroupType).
Variables (A : {group aT})(D : {group rT}).
Variable to : groupAction A D.
Lemma qact_dom_doms (H : {group rT}) : H \subset D -> qact_dom to H \subset A.
Proof.
by move=> sHD; apply/subsetP=> x; rewrite qact_domE // inE; case/andP.
Qed.
Lemma acts_qact_doms (H : {group rT}) :
H \subset D -> [acts A, on H | to] -> qact_dom to H :=: A.
Proof.
move=> sHD aH; apply/eqP; rewrite eqEsubset; apply/andP.
split; first exact: qact_dom_doms.
apply/subsetP=> x Ax; rewrite qact_domE //; apply/gastabsP=> //.
by move/gactsP: aH; move/(_ x Ax).
Qed.
Lemma qacts_cosetpre (H : {group rT}) (K' : {group coset_of H}) :
H \subset D -> [acts A, on H | to] ->
[acts qact_dom to H, on K' | to / H] ->
[acts A, on coset H @*^-1 K' | to].
Proof.
move=> sHD aH aK'; apply/subsetP=> x Ax; move: (Ax) (subsetP aK').
rewrite -{1}(acts_qact_doms sHD aH) => qdx; move/(_ x qdx) => nx.
rewrite !inE Ax; apply/subsetP=> y; case/morphpreP=> Ny /= K'Hy /[1!inE].
apply/morphpreP; split; first by rewrite acts_qact_dom_norm.
by move/gastabsP: nx; move/(_ qdx (coset H y)); rewrite K'Hy qactE.
Qed.
Lemma qacts_coset (H K : {group rT}) :
H \subset D -> [acts A, on K | to] ->
[acts qact_dom to H, on (coset H) @* K | to / H].
Proof.
move=> sHD aK.
apply/subsetP=> x qdx; rewrite inE qdx inE; apply/subsetP=> y.
case/morphimP=> z Nz Kz /= e; rewrite e inE qactE // imset_f // inE.
move/gactsP: aK; move/(_ x (subsetP (qact_dom_doms sHD) _ qdx) z); rewrite Kz.
move->; move/acts_act: (acts_qact_dom to H); move/(_ x qdx z).
by rewrite Nz andbT.
Qed.
End MoreQuotientAction.
Section StableCompositionSeries.
Variables (aT rT : finGroupType).
Variables (D : {group rT})(A : {group aT}).
Variable to : groupAction A D.
Definition maxainv (B C : {set rT}) :=
[max C of H |
[&& (H <| B), ~~ (B \subset H) & [acts A, on H | to]]].
Section MaxAinvProps.
Variables K N : {group rT}.
Lemma maxainv_norm : maxainv K N -> N <| K.
Proof. by move/maxgroupp; case/andP. Qed.
Lemma maxainv_proper : maxainv K N -> N \proper K.
Proof.
by move/maxgroupp; case/andP; rewrite properE; move/normal_sub->; case/andP.
Qed.
Lemma maxainv_sub : maxainv K N -> N \subset K.
Proof. by move=> h; apply: proper_sub; apply: maxainv_proper. Qed.
Lemma maxainv_ainvar : maxainv K N -> A \subset 'N(N | to).
Proof. by move/maxgroupp; case/and3P. Qed.
Lemma maxainvS : maxainv K N -> N \subset K.
Proof. by move=> pNN; rewrite proper_sub // maxainv_proper. Qed.
Lemma maxainv_exists : K :!=: 1 -> {N : {group rT} | maxainv K N}.
Proof.
move=> nt; apply: ex_maxgroup. exists [1 rT]%G.
rewrite /= normal1 subG1 nt /=.
apply/subsetP=> a Da; rewrite !inE Da /= sub1set !inE.
by rewrite /= -actmE // morph1 eqxx.
Qed.
End MaxAinvProps.
Lemma maxainvM (G H K : {group rT}) :
H \subset D -> K \subset D -> maxainv G H -> maxainv G K ->
H :<>: K -> H * K = G.
Proof.
move: H K => N1 N2 sN1D sN2D pmN1 pmN2 neN12.
have cN12 : commute N1 N2.
apply: normC; apply: (subset_trans (maxainv_sub pmN1)).
by rewrite normal_norm ?maxainv_norm.
wlog nsN21 : G N1 N2 sN1D sN2D pmN1 pmN2 neN12 cN12/ ~~(N1 \subset N2).
move/eqP: (neN12); rewrite eqEsubset negb_and; case/orP=> ns; first by apply.
by rewrite cN12; apply=> //; apply: sym_not_eq.
have nP : N1 * N2 <| G by rewrite normalM ?maxainv_norm.
have sN2P : N2 \subset N1 * N2 by rewrite mulg_subr ?group1.
case/maxgroupP: (pmN1); case/andP=> nN1G pN1G mN1.
case/maxgroupP: (pmN2); case/andP=> nN2G pN2G mN2.
case/andP: pN1G=> nsGN1 ha1; case/andP: pN2G=> nsGN2 ha2.
case e : (G \subset N1 * N2).
by apply/eqP; rewrite eqEsubset e mulG_subG !normal_sub.
have: N1 <*> N2 = N2 by apply: mN2; rewrite /= ?comm_joingE // nP e /= gactsM.
by rewrite comm_joingE // => h; move: nsN21; rewrite -h mulg_subl.
Qed.
Definition asimple (K : {set rT}) := maxainv K 1.
Implicit Types (H K : {group rT}) (s : seq {group rT}).
Lemma asimpleP K :
reflect [/\ K :!=: 1
& forall H, H <| K -> [acts A, on H | to] -> H :=: 1 \/ H :=: K]
(asimple K).
Proof.
apply: (iffP idP).
case/maxgroupP; rewrite normal1 /=; case/andP=> nsK1 aK H1.
rewrite eqEsubset negb_and nsK1 /=; split => // H nHK ha.
case eHK : (H :==: K); first by right; apply/eqP.
left; apply: H1; rewrite ?sub1G // nHK; move/negbT: eHK.
by rewrite eqEsubset negb_and normal_sub //=; move->.
case=> ntK h; apply/maxgroupP; split.
move: ntK; rewrite eqEsubset sub1G andbT normal1; move->.
apply/subsetP=> a Da; rewrite !inE Da /= sub1set !inE.
by rewrite /= -actmE // morph1 eqxx.
move=> H /andP[nHK /andP[nsKH ha]] _.
case: (h _ nHK ha)=> // /eqP; rewrite eqEsubset.
by rewrite (negbTE nsKH) andbF.
Qed.
Definition acomps K s :=
((last K s) == 1%G) && path [rel x y : {group rT} | maxainv x y] K s.
Lemma acompsP K s :
reflect (last K s = 1%G /\ path [rel x y : {group rT} | maxainv x y] K s)
(acomps K s).
Proof. by apply: (iffP andP); case; move/eqP. Qed.
Lemma trivg_acomps K s : acomps K s -> (K :==: 1) = (s == [::]).
Proof.
case/andP=> ls cs; apply/eqP/eqP; last first.
by move=> se; rewrite se /= in ls; apply/eqP.
move=> G1; case: s ls cs => // H s _ /=; case/andP; case/maxgroupP.
by rewrite G1 sub1G andbF.
Qed.
Lemma acomps_cons K H s : acomps K (H :: s) -> acomps H s.
Proof. by case/andP => /= ls; case/andP=> _ p; rewrite /acomps ls. Qed.
Lemma asimple_acompsP K s : acomps K s -> reflect (s = [:: 1%G]) (asimple K).
Proof.
move=> cs; apply: (iffP idP); last first.
by move=> se; move: cs; rewrite se /=; case/andP=> /=; rewrite andbT.
case: s cs.
by rewrite /acomps /= andbT; move/eqP->; case/asimpleP; rewrite eqxx.
move=> H s cs sG; apply/eqP.
rewrite eqseq_cons -(trivg_acomps (acomps_cons cs)) andbC andbb.
case/acompsP: cs => /= ls; case/andP=> mH ps.
case/maxgroupP: sG; case/and3P => _ ntG _ ->; rewrite ?sub1G //.
rewrite (maxainv_norm mH); case/andP: (maxainv_proper mH)=> _ ->.
exact: (maxainv_ainvar mH).
Qed.
Lemma exists_acomps K : exists s, acomps K s.
Proof.
elim: {K} #|K| {1 3}K (leqnn #|K|) => [K | n Hi K cK].
by rewrite leqNgt cardG_gt0.
case/orP: (orbN (asimple K)) => [sK | nsK].
by exists [:: (1%G : {group rT})]; rewrite /acomps eqxx /= andbT.
case/orP: (orbN (K :==: 1))=> [tK | ntK].
by exists (Nil _); rewrite /acomps /= andbT.
case: (maxainv_exists ntK)=> N pmN.
have cN: #|N| <= n.
by rewrite -ltnS (leq_trans _ cK) // proper_card // (maxainv_proper pmN).
case: (Hi _ cN)=> s; case/andP=> lasts ps; exists [:: N & s]; rewrite /acomps.
by rewrite last_cons lasts /= pmN.
Qed.
End StableCompositionSeries.
Arguments maxainv {aT rT D%_G A%_G} to%_gact B%_g C%_g.
Arguments asimple {aT rT D%_G A%_G} to%_gact K%_g.
Section StrongJordanHolder.
Section AuxiliaryLemmas.
Variables aT rT : finGroupType.
Variables (A : {group aT}) (D : {group rT}) (to : groupAction A D).
Lemma maxainv_asimple_quo (G H : {group rT}) :
H \subset D -> maxainv to G H -> asimple (to / H) (G / H).
Proof.
move=> sHD /maxgroupP[/and3P[nHG pHG aH] Hmax].
apply/asimpleP; split; first by rewrite -subG1 quotient_sub1 ?normal_norm.
move=> K' nK'Q aK'.
have: (K' \proper (G / H)) || (G / H == K').
by rewrite properE eqEsubset andbC (normal_sub nK'Q) !andbT orbC orbN.
case/orP=> [ pHQ | eQH]; last by right; apply sym_eq; apply/eqP.
left; pose K := ((coset H) @*^-1 K')%G.
have eK'I : K' \subset (coset H) @* 'N(H).
by rewrite (subset_trans (normal_sub nK'Q)) ?morphimS ?normal_norm.
have eKK' : K' :=: K / H by rewrite /(K / H) morphpreK //=.
suff eKH : K :=: H by rewrite -trivg_quotient eKK' eKH.
have sHK : H \subset K by rewrite -ker_coset kerE morphpreS // sub1set group1.
apply: Hmax => //; apply/and3P; split; last exact: qacts_cosetpre.
by rewrite -(quotientGK nHG) cosetpre_normal.
by move: (proper_subn pHQ); rewrite sub_morphim_pre ?normal_norm.
Qed.
Lemma asimple_quo_maxainv (G H : {group rT}) :
H \subset D -> G \subset D -> [acts A, on G | to] -> [acts A, on H | to] ->
H <| G -> asimple (to / H) (G / H) ->
maxainv to G H.
Proof.
move=> sHD sGD aG aH nHG /asimpleP[ntQ maxQ]; apply/maxgroupP; split.
by rewrite nHG -quotient_sub1 ?normal_norm // subG1 ntQ.
move=> K /and3P[nKG nsGK aK] sHK.
pose K' := (K / H)%G.
have K'dQ : K' <| (G / H)%G by apply: morphim_normal.
have nKH : H <| K by rewrite (normalS _ _ nHG) // normal_sub.
have: K' :=: 1%G \/ K' :=: (G / H).
apply: (maxQ K' K'dQ) => /=.
apply/subsetP=> x Adx. rewrite inE Adx /= inE. apply/subsetP=> y.
rewrite quotientE; case/morphimP=> z Nz Kz ->; rewrite /= !inE qactE //.
have ntoyx : to z x \in 'N(H) by rewrite (acts_qact_dom_norm Adx).
apply/morphimP; exists (to z x) => //.
suff h: qact_dom to H \subset A.
by rewrite astabs_act // (subsetP aK) //; apply: (subsetP h).
by apply/subsetP=> t; rewrite qact_domE // inE; case/andP.
case=> [|/quotient_injG /[!inE]/(_ nKH nHG) c]; last by rewrite c subxx in nsGK.
rewrite /= -trivg_quotient => tK'; apply: (congr1 (@gval _)); move: tK'.
by apply: (@quotient_injG _ H); rewrite ?inE /= ?normal_refl.
Qed.
Lemma asimpleI (N1 N2 : {group rT}) :
N2 \subset 'N(N1) -> N1 \subset D ->
[acts A, on N1 | to] -> [acts A, on N2 | to] ->
asimple (to / N1) (N2 / N1) ->
asimple (to / (N2 :&: N1)) (N2 / (N2 :&: N1)).
Proof.
move=> nN21 sN1D aN1 aN2 /asimpleP[ntQ1 max1].
have [f1 [f1e f1ker f1pre f1im]] := restrmP (coset_morphism N1) nN21.
have hf2' : N2 \subset 'N(N2 :&: N1) by apply: normsI => //; rewrite normG.
have hf2'' : 'ker (coset (N2 :&: N1)) \subset 'ker f1.
by rewrite f1ker !ker_coset.
pose f2 := factm_morphism hf2'' hf2'.
apply/asimpleP; split.
rewrite /= setIC; apply/negP; move: (second_isog nN21); move/isog_eq1->.
by apply/negP.
move=> H nHQ2 aH; pose K := f2 @* H.
have nKQ1 : K <| N2 / N1.
rewrite (_ : N2 / N1 = f2 @* (N2 / (N2 :&: N1))) ?morphim_normal //.
by rewrite morphim_factm f1im.
have sqA : qact_dom to N1 \subset A.
by apply/subsetP=> t; rewrite qact_domE // inE; case/andP.
have nNN2 : (N2 :&: N1) <| N2.
by rewrite /normal subsetIl; apply: normsI => //; apply: normG.
have aKQ1 : [acts qact_dom to N1, on K | to / N1].
pose H':= coset (N2 :&: N1)@*^-1 H.
have eHH' : H :=: H' / (N2 :&: N1) by rewrite cosetpreK.
have -> : K :=: f1 @* H' by rewrite /K eHH' morphim_factm.
have sH'N2 : H' \subset N2.
rewrite /H' eHH' quotientGK ?normal_cosetpre //=.
by rewrite sub_cosetpre_quo ?normal_sub.
have -> : f1 @* H' = coset N1 @* H' by rewrite f1im //=.
apply: qacts_coset => //; apply: qacts_cosetpre => //; last exact: gactsI.
by apply: (subset_trans (subsetIr _ _)).
have injf2 : 'injm f2.
by rewrite ker_factm f1ker /= ker_coset /= subG1 /= -quotientE trivg_quotient.
have iHK : H \isog K.
apply/isogP; pose f3 := restrm_morphism (normal_sub nHQ2) f2.
by exists f3; rewrite 1?injm_restrm // morphim_restrm setIid.
case: (max1 _ nKQ1 aKQ1).
by move/eqP; rewrite -(isog_eq1 iHK); move/eqP->; left.
move=> he /=; right; apply/eqP; rewrite eqEcard normal_sub //=.
move: (second_isog nN21); rewrite setIC; move/card_isog->; rewrite -he.
by move/card_isog: iHK=> <-; rewrite leqnn.
Qed.
End AuxiliaryLemmas.
Variables (aT rT : finGroupType).
Variables (A : {group aT}) (D : {group rT}) (to : groupAction A D).
(******************************************************************************)
(* The factors associated to two A-stable composition series of the same *)
(* group are the same up to isomorphism and permutation *)
(******************************************************************************)
Lemma StrongJordanHolderUniqueness (G : {group rT}) (s1 s2 : seq {group rT}) :
G \subset D -> acomps to G s1 -> acomps to G s2 ->
perm_eq (mkfactors G s1) (mkfactors G s2).
Proof.
have [n] := ubnP #|G|; elim: n G => // n Hi G in s1 s2 * => cG hsD cs1 cs2.
case/orP: (orbN (G :==: 1)) => [tG | ntG].
have -> : s1 = [::] by apply/eqP; rewrite -(trivg_acomps cs1).
have -> : s2 = [::] by apply/eqP; rewrite -(trivg_acomps cs2).
by rewrite /= perm_refl.
case/orP: (orbN (asimple to G))=> [sG | nsG].
have -> : s1 = [:: 1%G ] by apply/(asimple_acompsP cs1).
have -> : s2 = [:: 1%G ] by apply/(asimple_acompsP cs2).
by rewrite /= perm_refl.
case es1: s1 cs1 => [|N1 st1] cs1.
by move: (trivg_comps cs1); rewrite eqxx; move/negP:ntG.
case es2: s2 cs2 => [|N2 st2] cs2 {s1 es1}.
by move: (trivg_comps cs2); rewrite eqxx; move/negP:ntG.
case/andP: cs1 => /= lst1; case/andP=> maxN_1 pst1.
case/andP: cs2 => /= lst2; case/andP=> maxN_2 pst2.
have sN1D : N1 \subset D.
by apply: subset_trans hsD; apply: maxainv_sub maxN_1.
have sN2D : N2 \subset D.
by apply: subset_trans hsD; apply: maxainv_sub maxN_2.
have cN1 : #|N1| < n.
by rewrite -ltnS (leq_trans _ cG) ?ltnS ?proper_card ?(maxainv_proper maxN_1).
have cN2 : #|N2| < n.
by rewrite -ltnS (leq_trans _ cG) ?ltnS ?proper_card ?(maxainv_proper maxN_2).
case: (N1 =P N2) {s2 es2} => [eN12 |].
by rewrite eN12 /= perm_cons Hi // /acomps ?lst2 //= -eN12 lst1.
move/eqP; rewrite -val_eqE /=; move/eqP=> neN12.
have nN1G : N1 <| G by apply: (maxainv_norm maxN_1).
have nN2G : N2 <| G by apply: (maxainv_norm maxN_2).
pose N := (N1 :&: N2)%G.
have nNG : N <| G.
by rewrite /normal subIset ?(normal_sub nN1G) //= normsI ?normal_norm.
have iso1 : (G / N1)%G \isog (N2 / N)%G.
rewrite isog_sym /= -(maxainvM _ _ maxN_1 maxN_2) //.
rewrite (@normC _ N1 N2) ?(subset_trans (normal_sub nN1G)) ?normal_norm //.
by rewrite weak_second_isog ?(subset_trans (normal_sub nN2G)) ?normal_norm.
have iso2 : (G / N2)%G \isog (N1 / N)%G.
rewrite isog_sym /= -(maxainvM _ _ maxN_1 maxN_2) // setIC.
by rewrite weak_second_isog ?(subset_trans (normal_sub nN1G)) ?normal_norm.
case: (exists_acomps to N)=> sN; case/andP=> lsN csN.
have aN1 : [acts A, on N1 | to].
by case/maxgroupP: maxN_1; case/and3P.
have aN2 : [acts A, on N2 | to].
by case/maxgroupP: maxN_2; case/and3P.
have nNN1 : N <| N1.
by apply: (normalS _ _ nNG); rewrite ?subsetIl ?normal_sub.
have nNN2 : N <| N2.
by apply: (normalS _ _ nNG); rewrite ?subsetIr ?normal_sub.
have aN : [ acts A, on N1 :&: N2 | to].
apply/subsetP=> x Ax; rewrite !inE Ax /=; apply/subsetP=> y Ny; rewrite inE.
case/setIP: Ny=> N1y N2y. rewrite inE ?astabs_act ?N1y ?N2y //.
by move/subsetP: aN2; move/(_ x Ax).
by move/subsetP: aN1; move/(_ x Ax).
have i1 : perm_eq (mksrepr G N1 :: mkfactors N1 st1)
[:: mksrepr G N1, mksrepr N1 N & mkfactors N sN].
rewrite perm_cons -[mksrepr _ _ :: _]/(mkfactors N1 [:: N & sN]).
apply: Hi=> //; rewrite /acomps ?lst1 //= lsN csN andbT /=.
apply: asimple_quo_maxainv=> //; first by apply: subIset; rewrite sN1D.
apply: asimpleI => //.
by apply: subset_trans (normal_norm nN2G); apply: normal_sub.
rewrite -quotientMidl (maxainvM _ _ maxN_2) //.
by apply: maxainv_asimple_quo.
by move=> e; apply: neN12.
have i2 : perm_eq (mksrepr G N2 :: mkfactors N2 st2)
[:: mksrepr G N2, mksrepr N2 N & mkfactors N sN].
rewrite perm_cons -[mksrepr _ _ :: _]/(mkfactors N2 [:: N & sN]).
apply: Hi=> //; rewrite /acomps ?lst2 //= lsN csN andbT /=.
apply: asimple_quo_maxainv=> //; first by apply: subIset; rewrite sN1D.
have e : N1 :&: N2 :=: N2 :&: N1 by rewrite setIC.
rewrite (group_inj (setIC N1 N2)); apply: asimpleI => //.
by apply: subset_trans (normal_norm nN1G); apply: normal_sub.
rewrite -quotientMidl (maxainvM _ _ maxN_1) //.
exact: maxainv_asimple_quo.
pose fG1 := [:: mksrepr G N1, mksrepr N1 N & mkfactors N sN].
pose fG2 := [:: mksrepr G N2, mksrepr N2 N & mkfactors N sN].
have i3 : perm_eq fG1 fG2.
rewrite (@perm_catCA _ [::_] [::_]) /mksrepr.
rewrite (@section_repr_isog _ (mkSec _ _) (mkSec _ _) iso1).
rewrite -(@section_repr_isog _ (mkSec _ _) (mkSec _ _) iso2).
exact: perm_refl.
apply: (perm_trans i1); apply: (perm_trans i3); rewrite perm_sym.
by apply: perm_trans i2; apply: perm_refl.
Qed.
End StrongJordanHolder.
|
TwoP.lean
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.CategoryTheory.Category.Bipointed
import Mathlib.Data.TwoPointing
/-!
# The category of two-pointed types
This defines `TwoP`, the category of two-pointed types.
## References
* [nLab, *coalgebra of the real interval*]
(https://ncatlab.org/nlab/show/coalgebra+of+the+real+interval)
-/
open CategoryTheory Option
universe u
variable {α β : Type*}
/-- The category of two-pointed types. -/
structure TwoP : Type (u + 1) where
/-- The underlying type of a two-pointed type. -/
protected X : Type u
/-- The two points of a bipointed type, bundled together as a pair of distinct elements. -/
toTwoPointing : TwoPointing X
namespace TwoP
instance : CoeSort TwoP Type* :=
⟨TwoP.X⟩
/-- Turns a two-pointing into a two-pointed type. -/
abbrev of {X : Type*} (toTwoPointing : TwoPointing X) : TwoP :=
⟨X, toTwoPointing⟩
theorem coe_of {X : Type*} (toTwoPointing : TwoPointing X) : ↥(of toTwoPointing) = X :=
rfl
alias _root_.TwoPointing.TwoP := of
instance : Inhabited TwoP :=
⟨of TwoPointing.bool⟩
/-- Turns a two-pointed type into a bipointed type, by forgetting that the pointed elements are
distinct. -/
noncomputable def toBipointed (X : TwoP) : Bipointed :=
X.toTwoPointing.toProd.Bipointed
@[simp]
theorem coe_toBipointed (X : TwoP) : ↥X.toBipointed = ↥X :=
rfl
noncomputable instance largeCategory : LargeCategory TwoP :=
InducedCategory.category toBipointed
noncomputable instance concreteCategory : ConcreteCategory TwoP
(fun X Y => Bipointed.HomSubtype X.toBipointed Y.toBipointed) :=
InducedCategory.concreteCategory toBipointed
noncomputable instance hasForgetToBipointed : HasForget₂ TwoP Bipointed :=
InducedCategory.hasForget₂ toBipointed
/-- Swaps the pointed elements of a two-pointed type. `TwoPointing.swap` as a functor. -/
@[simps]
noncomputable def swap : TwoP ⥤ TwoP where
obj X := ⟨X, X.toTwoPointing.swap⟩
map f := ⟨f.toFun, f.map_snd, f.map_fst⟩
/-- The equivalence between `TwoP` and itself induced by `Prod.swap` both ways. -/
@[simps!]
noncomputable def swapEquiv : TwoP ≌ TwoP where
functor := swap
inverse := swap
unitIso := Iso.refl _
counitIso := Iso.refl _
@[simp]
theorem swapEquiv_symm : swapEquiv.symm = swapEquiv :=
rfl
end TwoP
@[simp]
theorem TwoP_swap_comp_forget_to_Bipointed :
TwoP.swap ⋙ forget₂ TwoP Bipointed = forget₂ TwoP Bipointed ⋙ Bipointed.swap :=
rfl
/-- The functor from `Pointed` to `TwoP` which adds a second point. -/
@[simps]
noncomputable def pointedToTwoPFst : Pointed.{u} ⥤ TwoP where
obj X := ⟨Option X, ⟨X.point, none⟩, some_ne_none _⟩
map f := ⟨Option.map f.toFun, congr_arg _ f.map_point, rfl⟩
map_id _ := Bipointed.Hom.ext Option.map_id
map_comp f g := Bipointed.Hom.ext (Option.map_comp_map f.1 g.1).symm
/-- The functor from `Pointed` to `TwoP` which adds a first point. -/
@[simps]
noncomputable def pointedToTwoPSnd : Pointed.{u} ⥤ TwoP where
obj X := ⟨Option X, ⟨none, X.point⟩, (some_ne_none _).symm⟩
map f := ⟨Option.map f.toFun, rfl, congr_arg _ f.map_point⟩
map_id _ := Bipointed.Hom.ext Option.map_id
map_comp f g := Bipointed.Hom.ext (Option.map_comp_map f.1 g.1).symm
@[simp]
theorem pointedToTwoPFst_comp_swap : pointedToTwoPFst ⋙ TwoP.swap = pointedToTwoPSnd :=
rfl
@[simp]
theorem pointedToTwoPSnd_comp_swap : pointedToTwoPSnd ⋙ TwoP.swap = pointedToTwoPFst :=
rfl
@[simp]
theorem pointedToTwoPFst_comp_forget_to_bipointed :
pointedToTwoPFst ⋙ forget₂ TwoP Bipointed = pointedToBipointedFst :=
rfl
@[simp]
theorem pointedToTwoPSnd_comp_forget_to_bipointed :
pointedToTwoPSnd ⋙ forget₂ TwoP Bipointed = pointedToBipointedSnd :=
rfl
/-- Adding a second point is left adjoint to forgetting the second point. -/
noncomputable def pointedToTwoPFstForgetCompBipointedToPointedFstAdjunction :
pointedToTwoPFst ⊣ forget₂ TwoP Bipointed ⋙ bipointedToPointedFst :=
Adjunction.mkOfHomEquiv
{ homEquiv := fun X Y =>
{ toFun := fun f => ⟨f.toFun ∘ Option.some, f.map_fst⟩
invFun := fun f => ⟨fun o => o.elim Y.toTwoPointing.toProd.2 f.toFun, f.map_point, rfl⟩
left_inv := fun f => by
apply Bipointed.Hom.ext
funext x
cases x
· exact f.map_snd.symm
· rfl }
homEquiv_naturality_left_symm := fun f g => by
apply Bipointed.Hom.ext
funext x
cases x <;> rfl }
/-- Adding a first point is left adjoint to forgetting the first point. -/
noncomputable def pointedToTwoPSndForgetCompBipointedToPointedSndAdjunction :
pointedToTwoPSnd ⊣ forget₂ TwoP Bipointed ⋙ bipointedToPointedSnd :=
Adjunction.mkOfHomEquiv
{ homEquiv := fun X Y =>
{ toFun := fun f => ⟨f.toFun ∘ Option.some, f.map_snd⟩
invFun := fun f => ⟨fun o => o.elim Y.toTwoPointing.toProd.1 f.toFun, rfl, f.map_point⟩
left_inv := fun f => by
apply Bipointed.Hom.ext
funext x
cases x
· exact f.map_fst.symm
· rfl }
homEquiv_naturality_left_symm := fun f g => by
apply Bipointed.Hom.ext
funext x
cases x <;> rfl }
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