state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
case h
V : Type u_1
inst✝¹ : Category.{u_2, u_1} V
inst✝ : Abelian V
X Y Z : Vᵒᵖ
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
⊢ imageToKernel g.unop f.unop (_ : g.unop ≫ f.unop = 0) ≫ Subobject.arrow (kernelSubobject f.unop) =
((imageSubobjectIso g.unop ≪≫ (Iso.unop (imageUnopOp g)).symm).hom ≫
(cokernel.desc f (facto... | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | simp only [Iso.trans_hom, Iso.symm_hom, Iso.trans_inv, kernelUnopUnop_inv, Category.assoc,
imageToKernel_arrow, kernelSubobject_arrow', kernel.lift_ι, cokernel.π_desc, Iso.unop_inv,
← unop_comp, factorThruImage_comp_imageUnopOp_inv, Quiver.Hom.unop_op, imageSubobject_arrow] | theorem imageToKernel_unop {X Y Z : Vᵒᵖ} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
imageToKernel g.unop f.unop (by rw [← unop_comp, w, unop_zero]) =
(imageSubobjectIso _ ≪≫ (imageUnopUnop _).symm).hom ≫
(cokernel.desc f (factorThruImage g)
(by rw [← cancel_mono (image.ι g), Category.asso... | Mathlib.Algebra.Homology.Opposite.53_0.Q6AwB66K8LWeNih | theorem imageToKernel_unop {X Y Z : Vᵒᵖ} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
imageToKernel g.unop f.unop (by rw [← unop_comp, w, unop_zero]) =
(imageSubobjectIso _ ≪≫ (imageUnopUnop _).symm).hom ≫
(cokernel.desc f (factorThruImage g)
(by rw [← cancel_mono (image.ι g), Category.asso... | Mathlib_Algebra_Homology_Opposite |
V : Type u_1
inst✝¹ : Category.{?u.23271, u_1} V
inst✝ : Abelian V
X Y Z : V
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
⊢ g.op ≫ f.op = 0 | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | rw [← op_comp, w, op_zero] | /-- Given `f, g` with `f ≫ g = 0`, the homology of `g.op, f.op` is the opposite of the homology of
`f, g`. -/
def homology'Op {X Y Z : V} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
homology' g.op f.op (by | Mathlib.Algebra.Homology.Opposite.66_0.Q6AwB66K8LWeNih | /-- Given `f, g` with `f ≫ g = 0`, the homology of `g.op, f.op` is the opposite of the homology of
`f, g`. -/
def homology'Op {X Y Z : V} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
homology' g.op f.op (by rw [← op_comp, w, op_zero]) ≅ Opposite.op (homology' f g w) | Mathlib_Algebra_Homology_Opposite |
V : Type u_1
inst✝¹ : Category.{?u.23271, u_1} V
inst✝ : Abelian V
X Y Z : V
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
⊢ cokernel.desc f g w = cokernel.desc f (factorThruImage g) (_ : f ≫ factorThruImage g = 0) ≫ image.ι g | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | ext | /-- Given `f, g` with `f ≫ g = 0`, the homology of `g.op, f.op` is the opposite of the homology of
`f, g`. -/
def homology'Op {X Y Z : V} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
homology' g.op f.op (by rw [← op_comp, w, op_zero]) ≅ Opposite.op (homology' f g w) :=
cokernelIsoOfEq (imageToKernel_op _ _ w) ≪≫ cok... | Mathlib.Algebra.Homology.Opposite.66_0.Q6AwB66K8LWeNih | /-- Given `f, g` with `f ≫ g = 0`, the homology of `g.op, f.op` is the opposite of the homology of
`f, g`. -/
def homology'Op {X Y Z : V} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
homology' g.op f.op (by rw [← op_comp, w, op_zero]) ≅ Opposite.op (homology' f g w) | Mathlib_Algebra_Homology_Opposite |
case h
V : Type u_1
inst✝¹ : Category.{?u.23271, u_1} V
inst✝ : Abelian V
X Y Z : V
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
⊢ coequalizer.π f 0 ≫ cokernel.desc f g w =
coequalizer.π f 0 ≫ cokernel.desc f (factorThruImage g) (_ : f ≫ factorThruImage g = 0) ≫ image.ι g | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | simp only [image.fac, cokernel.π_desc, cokernel.π_desc_assoc] | /-- Given `f, g` with `f ≫ g = 0`, the homology of `g.op, f.op` is the opposite of the homology of
`f, g`. -/
def homology'Op {X Y Z : V} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
homology' g.op f.op (by rw [← op_comp, w, op_zero]) ≅ Opposite.op (homology' f g w) :=
cokernelIsoOfEq (imageToKernel_op _ _ w) ≪≫ cok... | Mathlib.Algebra.Homology.Opposite.66_0.Q6AwB66K8LWeNih | /-- Given `f, g` with `f ≫ g = 0`, the homology of `g.op, f.op` is the opposite of the homology of
`f, g`. -/
def homology'Op {X Y Z : V} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
homology' g.op f.op (by rw [← op_comp, w, op_zero]) ≅ Opposite.op (homology' f g w) | Mathlib_Algebra_Homology_Opposite |
V : Type u_1
inst✝¹ : Category.{?u.44782, u_1} V
inst✝ : Abelian V
X Y Z : Vᵒᵖ
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
⊢ g.unop ≫ f.unop = 0 | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | rw [← unop_comp, w, unop_zero] | /-- Given morphisms `f, g` in `Vᵒᵖ` with `f ≫ g = 0`, the homology of `g.unop, f.unop` is the
opposite of the homology of `f, g`. -/
def homology'Unop {X Y Z : Vᵒᵖ} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
homology' g.unop f.unop (by | Mathlib.Algebra.Homology.Opposite.76_0.Q6AwB66K8LWeNih | /-- Given morphisms `f, g` in `Vᵒᵖ` with `f ≫ g = 0`, the homology of `g.unop, f.unop` is the
opposite of the homology of `f, g`. -/
def homology'Unop {X Y Z : Vᵒᵖ} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
homology' g.unop f.unop (by rw [← unop_comp, w, unop_zero]) ≅
Opposite.unop (homology' f g w) | Mathlib_Algebra_Homology_Opposite |
V : Type u_1
inst✝¹ : Category.{?u.44782, u_1} V
inst✝ : Abelian V
X Y Z : Vᵒᵖ
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
⊢ cokernel.desc f g w = cokernel.desc f (factorThruImage g) (_ : f ≫ factorThruImage g = 0) ≫ image.ι g | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | ext | /-- Given morphisms `f, g` in `Vᵒᵖ` with `f ≫ g = 0`, the homology of `g.unop, f.unop` is the
opposite of the homology of `f, g`. -/
def homology'Unop {X Y Z : Vᵒᵖ} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
homology' g.unop f.unop (by rw [← unop_comp, w, unop_zero]) ≅
Opposite.unop (homology' f g w) :=
coke... | Mathlib.Algebra.Homology.Opposite.76_0.Q6AwB66K8LWeNih | /-- Given morphisms `f, g` in `Vᵒᵖ` with `f ≫ g = 0`, the homology of `g.unop, f.unop` is the
opposite of the homology of `f, g`. -/
def homology'Unop {X Y Z : Vᵒᵖ} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
homology' g.unop f.unop (by rw [← unop_comp, w, unop_zero]) ≅
Opposite.unop (homology' f g w) | Mathlib_Algebra_Homology_Opposite |
case h
V : Type u_1
inst✝¹ : Category.{?u.44782, u_1} V
inst✝ : Abelian V
X Y Z : Vᵒᵖ
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
⊢ coequalizer.π f 0 ≫ cokernel.desc f g w =
coequalizer.π f 0 ≫ cokernel.desc f (factorThruImage g) (_ : f ≫ factorThruImage g = 0) ≫ image.ι g | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | simp only [image.fac, cokernel.π_desc, cokernel.π_desc_assoc] | /-- Given morphisms `f, g` in `Vᵒᵖ` with `f ≫ g = 0`, the homology of `g.unop, f.unop` is the
opposite of the homology of `f, g`. -/
def homology'Unop {X Y Z : Vᵒᵖ} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
homology' g.unop f.unop (by rw [← unop_comp, w, unop_zero]) ≅
Opposite.unop (homology' f g w) :=
coke... | Mathlib.Algebra.Homology.Opposite.76_0.Q6AwB66K8LWeNih | /-- Given morphisms `f, g` in `Vᵒᵖ` with `f ≫ g = 0`, the homology of `g.unop, f.unop` is the
opposite of the homology of `f, g`. -/
def homology'Unop {X Y Z : Vᵒᵖ} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
homology' g.unop f.unop (by rw [← unop_comp, w, unop_zero]) ≅
Opposite.unop (homology' f g w) | Mathlib_Algebra_Homology_Opposite |
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.66749, u_2} V
c : ComplexShape ι
inst✝ : Preadditive V
X : HomologicalComplex V c
i j : ι
hij : ¬ComplexShape.Rel (ComplexShape.symm c) i j
⊢ (fun i j => (d X j i).op) i j = 0 | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | simp only | /-- Sends a complex `X` with objects in `V` to the corresponding complex with objects in `Vᵒᵖ`. -/
@[simps]
protected def op (X : HomologicalComplex V c) : HomologicalComplex Vᵒᵖ c.symm where
X i := op (X.X i)
d i j := (X.d j i).op
shape i j hij := by | Mathlib.Algebra.Homology.Opposite.97_0.Q6AwB66K8LWeNih | /-- Sends a complex `X` with objects in `V` to the corresponding complex with objects in `Vᵒᵖ`. -/
@[simps]
protected def op (X : HomologicalComplex V c) : HomologicalComplex Vᵒᵖ c.symm where
X i | Mathlib_Algebra_Homology_Opposite |
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.66749, u_2} V
c : ComplexShape ι
inst✝ : Preadditive V
X : HomologicalComplex V c
i j : ι
hij : ¬ComplexShape.Rel (ComplexShape.symm c) i j
⊢ (d X j i).op = 0 | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | rw [X.shape j i hij, op_zero] | /-- Sends a complex `X` with objects in `V` to the corresponding complex with objects in `Vᵒᵖ`. -/
@[simps]
protected def op (X : HomologicalComplex V c) : HomologicalComplex Vᵒᵖ c.symm where
X i := op (X.X i)
d i j := (X.d j i).op
shape i j hij := by simp only; | Mathlib.Algebra.Homology.Opposite.97_0.Q6AwB66K8LWeNih | /-- Sends a complex `X` with objects in `V` to the corresponding complex with objects in `Vᵒᵖ`. -/
@[simps]
protected def op (X : HomologicalComplex V c) : HomologicalComplex Vᵒᵖ c.symm where
X i | Mathlib_Algebra_Homology_Opposite |
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.66749, u_2} V
c : ComplexShape ι
inst✝ : Preadditive V
X : HomologicalComplex V c
x✝⁴ x✝³ x✝² : ι
x✝¹ : ComplexShape.Rel (ComplexShape.symm c) x✝⁴ x✝³
x✝ : ComplexShape.Rel (ComplexShape.symm c) x✝³ x✝²
⊢ (fun i j => (d X j i).op) x✝⁴ x✝³ ≫ (fun i j => (d X j i).op) x✝³ x... | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | rw [← op_comp, X.d_comp_d, op_zero] | /-- Sends a complex `X` with objects in `V` to the corresponding complex with objects in `Vᵒᵖ`. -/
@[simps]
protected def op (X : HomologicalComplex V c) : HomologicalComplex Vᵒᵖ c.symm where
X i := op (X.X i)
d i j := (X.d j i).op
shape i j hij := by simp only; rw [X.shape j i hij, op_zero]
d_comp_d' _ _ _ _ _... | Mathlib.Algebra.Homology.Opposite.97_0.Q6AwB66K8LWeNih | /-- Sends a complex `X` with objects in `V` to the corresponding complex with objects in `Vᵒᵖ`. -/
@[simps]
protected def op (X : HomologicalComplex V c) : HomologicalComplex Vᵒᵖ c.symm where
X i | Mathlib_Algebra_Homology_Opposite |
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.68583, u_2} V
c : ComplexShape ι
inst✝ : Preadditive V
X : HomologicalComplex V (ComplexShape.symm c)
i j : ι
hij : ¬ComplexShape.Rel c i j
⊢ (fun i j => (d X j i).op) i j = 0 | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | simp only | /-- Sends a complex `X` with objects in `V` to the corresponding complex with objects in `Vᵒᵖ`. -/
@[simps]
protected def opSymm (X : HomologicalComplex V c.symm) : HomologicalComplex Vᵒᵖ c where
X i := op (X.X i)
d i j := (X.d j i).op
shape i j hij := by | Mathlib.Algebra.Homology.Opposite.106_0.Q6AwB66K8LWeNih | /-- Sends a complex `X` with objects in `V` to the corresponding complex with objects in `Vᵒᵖ`. -/
@[simps]
protected def opSymm (X : HomologicalComplex V c.symm) : HomologicalComplex Vᵒᵖ c where
X i | Mathlib_Algebra_Homology_Opposite |
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.68583, u_2} V
c : ComplexShape ι
inst✝ : Preadditive V
X : HomologicalComplex V (ComplexShape.symm c)
i j : ι
hij : ¬ComplexShape.Rel c i j
⊢ (d X j i).op = 0 | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | rw [X.shape j i hij, op_zero] | /-- Sends a complex `X` with objects in `V` to the corresponding complex with objects in `Vᵒᵖ`. -/
@[simps]
protected def opSymm (X : HomologicalComplex V c.symm) : HomologicalComplex Vᵒᵖ c where
X i := op (X.X i)
d i j := (X.d j i).op
shape i j hij := by simp only; | Mathlib.Algebra.Homology.Opposite.106_0.Q6AwB66K8LWeNih | /-- Sends a complex `X` with objects in `V` to the corresponding complex with objects in `Vᵒᵖ`. -/
@[simps]
protected def opSymm (X : HomologicalComplex V c.symm) : HomologicalComplex Vᵒᵖ c where
X i | Mathlib_Algebra_Homology_Opposite |
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.68583, u_2} V
c : ComplexShape ι
inst✝ : Preadditive V
X : HomologicalComplex V (ComplexShape.symm c)
x✝⁴ x✝³ x✝² : ι
x✝¹ : ComplexShape.Rel c x✝⁴ x✝³
x✝ : ComplexShape.Rel c x✝³ x✝²
⊢ (fun i j => (d X j i).op) x✝⁴ x✝³ ≫ (fun i j => (d X j i).op) x✝³ x✝² = 0 | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | rw [← op_comp, X.d_comp_d, op_zero] | /-- Sends a complex `X` with objects in `V` to the corresponding complex with objects in `Vᵒᵖ`. -/
@[simps]
protected def opSymm (X : HomologicalComplex V c.symm) : HomologicalComplex Vᵒᵖ c where
X i := op (X.X i)
d i j := (X.d j i).op
shape i j hij := by simp only; rw [X.shape j i hij, op_zero]
d_comp_d' _ _ _... | Mathlib.Algebra.Homology.Opposite.106_0.Q6AwB66K8LWeNih | /-- Sends a complex `X` with objects in `V` to the corresponding complex with objects in `Vᵒᵖ`. -/
@[simps]
protected def opSymm (X : HomologicalComplex V c.symm) : HomologicalComplex Vᵒᵖ c where
X i | Mathlib_Algebra_Homology_Opposite |
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.70387, u_2} V
c : ComplexShape ι
inst✝ : Preadditive V
X : HomologicalComplex Vᵒᵖ c
i j : ι
hij : ¬ComplexShape.Rel (ComplexShape.symm c) i j
⊢ (fun i j => (d X j i).unop) i j = 0 | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | simp only | /-- Sends a complex `X` with objects in `Vᵒᵖ` to the corresponding complex with objects in `V`. -/
@[simps]
protected def unop (X : HomologicalComplex Vᵒᵖ c) : HomologicalComplex V c.symm where
X i := unop (X.X i)
d i j := (X.d j i).unop
shape i j hij := by | Mathlib.Algebra.Homology.Opposite.115_0.Q6AwB66K8LWeNih | /-- Sends a complex `X` with objects in `Vᵒᵖ` to the corresponding complex with objects in `V`. -/
@[simps]
protected def unop (X : HomologicalComplex Vᵒᵖ c) : HomologicalComplex V c.symm where
X i | Mathlib_Algebra_Homology_Opposite |
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.70387, u_2} V
c : ComplexShape ι
inst✝ : Preadditive V
X : HomologicalComplex Vᵒᵖ c
i j : ι
hij : ¬ComplexShape.Rel (ComplexShape.symm c) i j
⊢ (d X j i).unop = 0 | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | rw [X.shape j i hij, unop_zero] | /-- Sends a complex `X` with objects in `Vᵒᵖ` to the corresponding complex with objects in `V`. -/
@[simps]
protected def unop (X : HomologicalComplex Vᵒᵖ c) : HomologicalComplex V c.symm where
X i := unop (X.X i)
d i j := (X.d j i).unop
shape i j hij := by simp only; | Mathlib.Algebra.Homology.Opposite.115_0.Q6AwB66K8LWeNih | /-- Sends a complex `X` with objects in `Vᵒᵖ` to the corresponding complex with objects in `V`. -/
@[simps]
protected def unop (X : HomologicalComplex Vᵒᵖ c) : HomologicalComplex V c.symm where
X i | Mathlib_Algebra_Homology_Opposite |
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.70387, u_2} V
c : ComplexShape ι
inst✝ : Preadditive V
X : HomologicalComplex Vᵒᵖ c
x✝⁴ x✝³ x✝² : ι
x✝¹ : ComplexShape.Rel (ComplexShape.symm c) x✝⁴ x✝³
x✝ : ComplexShape.Rel (ComplexShape.symm c) x✝³ x✝²
⊢ (fun i j => (d X j i).unop) x✝⁴ x✝³ ≫ (fun i j => (d X j i).unop)... | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | rw [← unop_comp, X.d_comp_d, unop_zero] | /-- Sends a complex `X` with objects in `Vᵒᵖ` to the corresponding complex with objects in `V`. -/
@[simps]
protected def unop (X : HomologicalComplex Vᵒᵖ c) : HomologicalComplex V c.symm where
X i := unop (X.X i)
d i j := (X.d j i).unop
shape i j hij := by simp only; rw [X.shape j i hij, unop_zero]
d_comp_d' _... | Mathlib.Algebra.Homology.Opposite.115_0.Q6AwB66K8LWeNih | /-- Sends a complex `X` with objects in `Vᵒᵖ` to the corresponding complex with objects in `V`. -/
@[simps]
protected def unop (X : HomologicalComplex Vᵒᵖ c) : HomologicalComplex V c.symm where
X i | Mathlib_Algebra_Homology_Opposite |
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.72203, u_2} V
c : ComplexShape ι
inst✝ : Preadditive V
X : HomologicalComplex Vᵒᵖ (ComplexShape.symm c)
i j : ι
hij : ¬ComplexShape.Rel c i j
⊢ (fun i j => (d X j i).unop) i j = 0 | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | simp only | /-- Sends a complex `X` with objects in `Vᵒᵖ` to the corresponding complex with objects in `V`. -/
@[simps]
protected def unopSymm (X : HomologicalComplex Vᵒᵖ c.symm) : HomologicalComplex V c where
X i := unop (X.X i)
d i j := (X.d j i).unop
shape i j hij := by | Mathlib.Algebra.Homology.Opposite.124_0.Q6AwB66K8LWeNih | /-- Sends a complex `X` with objects in `Vᵒᵖ` to the corresponding complex with objects in `V`. -/
@[simps]
protected def unopSymm (X : HomologicalComplex Vᵒᵖ c.symm) : HomologicalComplex V c where
X i | Mathlib_Algebra_Homology_Opposite |
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.72203, u_2} V
c : ComplexShape ι
inst✝ : Preadditive V
X : HomologicalComplex Vᵒᵖ (ComplexShape.symm c)
i j : ι
hij : ¬ComplexShape.Rel c i j
⊢ (d X j i).unop = 0 | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | rw [X.shape j i hij, unop_zero] | /-- Sends a complex `X` with objects in `Vᵒᵖ` to the corresponding complex with objects in `V`. -/
@[simps]
protected def unopSymm (X : HomologicalComplex Vᵒᵖ c.symm) : HomologicalComplex V c where
X i := unop (X.X i)
d i j := (X.d j i).unop
shape i j hij := by simp only; | Mathlib.Algebra.Homology.Opposite.124_0.Q6AwB66K8LWeNih | /-- Sends a complex `X` with objects in `Vᵒᵖ` to the corresponding complex with objects in `V`. -/
@[simps]
protected def unopSymm (X : HomologicalComplex Vᵒᵖ c.symm) : HomologicalComplex V c where
X i | Mathlib_Algebra_Homology_Opposite |
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.72203, u_2} V
c : ComplexShape ι
inst✝ : Preadditive V
X : HomologicalComplex Vᵒᵖ (ComplexShape.symm c)
x✝⁴ x✝³ x✝² : ι
x✝¹ : ComplexShape.Rel c x✝⁴ x✝³
x✝ : ComplexShape.Rel c x✝³ x✝²
⊢ (fun i j => (d X j i).unop) x✝⁴ x✝³ ≫ (fun i j => (d X j i).unop) x✝³ x✝² = 0 | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | rw [← unop_comp, X.d_comp_d, unop_zero] | /-- Sends a complex `X` with objects in `Vᵒᵖ` to the corresponding complex with objects in `V`. -/
@[simps]
protected def unopSymm (X : HomologicalComplex Vᵒᵖ c.symm) : HomologicalComplex V c where
X i := unop (X.X i)
d i j := (X.d j i).unop
shape i j hij := by simp only; rw [X.shape j i hij, unop_zero]
d_comp_... | Mathlib.Algebra.Homology.Opposite.124_0.Q6AwB66K8LWeNih | /-- Sends a complex `X` with objects in `Vᵒᵖ` to the corresponding complex with objects in `V`. -/
@[simps]
protected def unopSymm (X : HomologicalComplex Vᵒᵖ c.symm) : HomologicalComplex V c where
X i | Mathlib_Algebra_Homology_Opposite |
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.74051, u_2} V
c : ComplexShape ι
inst✝ : Preadditive V
X✝ Y✝ : (HomologicalComplex V c)ᵒᵖ
f : X✝ ⟶ Y✝
i j : ι
x✝ : ComplexShape.Rel (ComplexShape.symm c) i j
⊢ (fun i => (Hom.f f.unop i).op) i ≫ d ((fun X => HomologicalComplex.op X.unop) Y✝) i j =
d ((fun X => Homolog... | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | simp only [op_d, ← op_comp, f.unop.comm] | /-- Auxiliary definition for `opEquivalence`. -/
@[simps]
def opFunctor : (HomologicalComplex V c)ᵒᵖ ⥤ HomologicalComplex Vᵒᵖ c.symm where
obj X := (unop X).op
map f :=
{ f := fun i => (f.unop.f i).op
comm' := fun i j _ => by | Mathlib.Algebra.Homology.Opposite.135_0.Q6AwB66K8LWeNih | /-- Auxiliary definition for `opEquivalence`. -/
@[simps]
def opFunctor : (HomologicalComplex V c)ᵒᵖ ⥤ HomologicalComplex Vᵒᵖ c.symm where
obj X | Mathlib_Algebra_Homology_Opposite |
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.82677, u_2} V
c : ComplexShape ι
inst✝ : Preadditive V
X✝ Y✝ : HomologicalComplex Vᵒᵖ (ComplexShape.symm c)
f : X✝ ⟶ Y✝
i j : ι
x✝ : ComplexShape.Rel c i j
⊢ (fun i => (Hom.f f i).unop) i ≫ d (HomologicalComplex.unopSymm X✝) i j =
d (HomologicalComplex.unopSymm Y✝) i ... | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | simp only [unopSymm_d, ← unop_comp, f.comm] | /-- Auxiliary definition for `opEquivalence`. -/
@[simps]
def opInverse : HomologicalComplex Vᵒᵖ c.symm ⥤ (HomologicalComplex V c)ᵒᵖ where
obj X := op X.unopSymm
map f := Quiver.Hom.op
{ f := fun i => (f.f i).unop
comm' := fun i j _ => by | Mathlib.Algebra.Homology.Opposite.144_0.Q6AwB66K8LWeNih | /-- Auxiliary definition for `opEquivalence`. -/
@[simps]
def opInverse : HomologicalComplex Vᵒᵖ c.symm ⥤ (HomologicalComplex V c)ᵒᵖ where
obj X | Mathlib_Algebra_Homology_Opposite |
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.90768, u_2} V
c : ComplexShape ι
inst✝ : Preadditive V
X : (HomologicalComplex V c)ᵒᵖ
i j : ι
x✝ : ComplexShape.Rel c i j
⊢ ((fun i => Iso.refl (HomologicalComplex.X (HomologicalComplex.unopSymm (HomologicalComplex.op X.unop)) i)) i).hom ≫
d X.unop i j =
d (Homo... | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | simp only [Iso.refl_hom, Category.id_comp, unopSymm_d, op_d, Quiver.Hom.unop_op,
Category.comp_id] | /-- Auxiliary definition for `opEquivalence`. -/
def opUnitIso : 𝟭 (HomologicalComplex V c)ᵒᵖ ≅ opFunctor V c ⋙ opInverse V c :=
NatIso.ofComponents
(fun X =>
(HomologicalComplex.Hom.isoOfComponents (fun i => Iso.refl _) fun i j _ => by
| Mathlib.Algebra.Homology.Opposite.153_0.Q6AwB66K8LWeNih | /-- Auxiliary definition for `opEquivalence`. -/
def opUnitIso : 𝟭 (HomologicalComplex V c)ᵒᵖ ≅ opFunctor V c ⋙ opInverse V c | Mathlib_Algebra_Homology_Opposite |
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.90768, u_2} V
c : ComplexShape ι
inst✝ : Preadditive V
⊢ ∀ {X Y : (HomologicalComplex V c)ᵒᵖ} (f : X ⟶ Y),
(𝟭 (HomologicalComplex V c)ᵒᵖ).map f ≫
((fun X =>
Iso.op
(Hom.isoOfComponents fun i =>
Iso.refl (Homolog... | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | intro X Y f | /-- Auxiliary definition for `opEquivalence`. -/
def opUnitIso : 𝟭 (HomologicalComplex V c)ᵒᵖ ≅ opFunctor V c ⋙ opInverse V c :=
NatIso.ofComponents
(fun X =>
(HomologicalComplex.Hom.isoOfComponents (fun i => Iso.refl _) fun i j _ => by
simp only [Iso.refl_hom, Category.id_comp, unopSymm_d, op_... | Mathlib.Algebra.Homology.Opposite.153_0.Q6AwB66K8LWeNih | /-- Auxiliary definition for `opEquivalence`. -/
def opUnitIso : 𝟭 (HomologicalComplex V c)ᵒᵖ ≅ opFunctor V c ⋙ opInverse V c | Mathlib_Algebra_Homology_Opposite |
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.90768, u_2} V
c : ComplexShape ι
inst✝ : Preadditive V
X Y : (HomologicalComplex V c)ᵒᵖ
f : X ⟶ Y
⊢ (𝟭 (HomologicalComplex V c)ᵒᵖ).map f ≫
((fun X =>
Iso.op
(Hom.isoOfComponents fun i =>
Iso.refl (HomologicalComplex.X (Homo... | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | refine' Quiver.Hom.unop_inj _ | /-- Auxiliary definition for `opEquivalence`. -/
def opUnitIso : 𝟭 (HomologicalComplex V c)ᵒᵖ ≅ opFunctor V c ⋙ opInverse V c :=
NatIso.ofComponents
(fun X =>
(HomologicalComplex.Hom.isoOfComponents (fun i => Iso.refl _) fun i j _ => by
simp only [Iso.refl_hom, Category.id_comp, unopSymm_d, op_... | Mathlib.Algebra.Homology.Opposite.153_0.Q6AwB66K8LWeNih | /-- Auxiliary definition for `opEquivalence`. -/
def opUnitIso : 𝟭 (HomologicalComplex V c)ᵒᵖ ≅ opFunctor V c ⋙ opInverse V c | Mathlib_Algebra_Homology_Opposite |
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.90768, u_2} V
c : ComplexShape ι
inst✝ : Preadditive V
X Y : (HomologicalComplex V c)ᵒᵖ
f : X ⟶ Y
⊢ ((𝟭 (HomologicalComplex V c)ᵒᵖ).map f ≫
((fun X =>
Iso.op
(Hom.isoOfComponents fun i =>
Iso.refl (HomologicalComple... | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | ext x | /-- Auxiliary definition for `opEquivalence`. -/
def opUnitIso : 𝟭 (HomologicalComplex V c)ᵒᵖ ≅ opFunctor V c ⋙ opInverse V c :=
NatIso.ofComponents
(fun X =>
(HomologicalComplex.Hom.isoOfComponents (fun i => Iso.refl _) fun i j _ => by
simp only [Iso.refl_hom, Category.id_comp, unopSymm_d, op_... | Mathlib.Algebra.Homology.Opposite.153_0.Q6AwB66K8LWeNih | /-- Auxiliary definition for `opEquivalence`. -/
def opUnitIso : 𝟭 (HomologicalComplex V c)ᵒᵖ ≅ opFunctor V c ⋙ opInverse V c | Mathlib_Algebra_Homology_Opposite |
case h
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.90768, u_2} V
c : ComplexShape ι
inst✝ : Preadditive V
X Y : (HomologicalComplex V c)ᵒᵖ
f : X ⟶ Y
x : ι
⊢ Hom.f
((𝟭 (HomologicalComplex V c)ᵒᵖ).map f ≫
((fun X =>
Iso.op
(Hom.isoOfComponents fun i =>
... | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | simp only [Quiver.Hom.unop_op, Functor.id_map, Iso.op_hom, Functor.comp_map, unop_comp,
comp_f, Hom.isoOfComponents_hom_f] | /-- Auxiliary definition for `opEquivalence`. -/
def opUnitIso : 𝟭 (HomologicalComplex V c)ᵒᵖ ≅ opFunctor V c ⋙ opInverse V c :=
NatIso.ofComponents
(fun X =>
(HomologicalComplex.Hom.isoOfComponents (fun i => Iso.refl _) fun i j _ => by
simp only [Iso.refl_hom, Category.id_comp, unopSymm_d, op_... | Mathlib.Algebra.Homology.Opposite.153_0.Q6AwB66K8LWeNih | /-- Auxiliary definition for `opEquivalence`. -/
def opUnitIso : 𝟭 (HomologicalComplex V c)ᵒᵖ ≅ opFunctor V c ⋙ opInverse V c | Mathlib_Algebra_Homology_Opposite |
case h
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.90768, u_2} V
c : ComplexShape ι
inst✝ : Preadditive V
X Y : (HomologicalComplex V c)ᵒᵖ
f : X ⟶ Y
x : ι
⊢ (Iso.refl (HomologicalComplex.X (HomologicalComplex.unopSymm (HomologicalComplex.op Y.unop)) x)).hom ≫
Hom.f f.unop x =
Hom.f ((opInverse V c).map ((... | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | erw [Category.id_comp, Category.comp_id (f.unop.f x)] | /-- Auxiliary definition for `opEquivalence`. -/
def opUnitIso : 𝟭 (HomologicalComplex V c)ᵒᵖ ≅ opFunctor V c ⋙ opInverse V c :=
NatIso.ofComponents
(fun X =>
(HomologicalComplex.Hom.isoOfComponents (fun i => Iso.refl _) fun i j _ => by
simp only [Iso.refl_hom, Category.id_comp, unopSymm_d, op_... | Mathlib.Algebra.Homology.Opposite.153_0.Q6AwB66K8LWeNih | /-- Auxiliary definition for `opEquivalence`. -/
def opUnitIso : 𝟭 (HomologicalComplex V c)ᵒᵖ ≅ opFunctor V c ⋙ opInverse V c | Mathlib_Algebra_Homology_Opposite |
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.110202, u_2} V
c : ComplexShape ι
inst✝ : Preadditive V
X : (HomologicalComplex V c)ᵒᵖ
⊢ (opFunctor V c).map ((opUnitIso V c).hom.app X) ≫ (opCounitIso V c).hom.app ((opFunctor V c).obj X) =
𝟙 ((opFunctor V c).obj X) | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | ext | /-- Given a category of complexes with objects in `V`, there is a natural equivalence between its
opposite category and a category of complexes with objects in `Vᵒᵖ`. -/
@[simps]
def opEquivalence : (HomologicalComplex V c)ᵒᵖ ≌ HomologicalComplex Vᵒᵖ c.symm where
functor := opFunctor V c
inverse := opInverse V c
... | Mathlib.Algebra.Homology.Opposite.176_0.Q6AwB66K8LWeNih | /-- Given a category of complexes with objects in `V`, there is a natural equivalence between its
opposite category and a category of complexes with objects in `Vᵒᵖ`. -/
@[simps]
def opEquivalence : (HomologicalComplex V c)ᵒᵖ ≌ HomologicalComplex Vᵒᵖ c.symm where
functor | Mathlib_Algebra_Homology_Opposite |
case h
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.110202, u_2} V
c : ComplexShape ι
inst✝ : Preadditive V
X : (HomologicalComplex V c)ᵒᵖ
i✝ : ι
⊢ Hom.f ((opFunctor V c).map ((opUnitIso V c).hom.app X) ≫ (opCounitIso V c).hom.app ((opFunctor V c).obj X)) i✝ =
Hom.f (𝟙 ((opFunctor V c).obj X)) i✝ | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | simp only [opUnitIso, opCounitIso, NatIso.ofComponents_hom_app, Iso.op_hom, comp_f,
opFunctor_map_f, Quiver.Hom.unop_op, Hom.isoOfComponents_hom_f] | /-- Given a category of complexes with objects in `V`, there is a natural equivalence between its
opposite category and a category of complexes with objects in `Vᵒᵖ`. -/
@[simps]
def opEquivalence : (HomologicalComplex V c)ᵒᵖ ≌ HomologicalComplex Vᵒᵖ c.symm where
functor := opFunctor V c
inverse := opInverse V c
... | Mathlib.Algebra.Homology.Opposite.176_0.Q6AwB66K8LWeNih | /-- Given a category of complexes with objects in `V`, there is a natural equivalence between its
opposite category and a category of complexes with objects in `Vᵒᵖ`. -/
@[simps]
def opEquivalence : (HomologicalComplex V c)ᵒᵖ ≌ HomologicalComplex Vᵒᵖ c.symm where
functor | Mathlib_Algebra_Homology_Opposite |
case h
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.110202, u_2} V
c : ComplexShape ι
inst✝ : Preadditive V
X : (HomologicalComplex V c)ᵒᵖ
i✝ : ι
⊢ (Iso.refl (HomologicalComplex.X (HomologicalComplex.unopSymm (HomologicalComplex.op X.unop)) i✝)).hom.op ≫
(Iso.refl (HomologicalComplex.X ((opInverse V c ⋙ opFunc... | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | exact Category.comp_id _ | /-- Given a category of complexes with objects in `V`, there is a natural equivalence between its
opposite category and a category of complexes with objects in `Vᵒᵖ`. -/
@[simps]
def opEquivalence : (HomologicalComplex V c)ᵒᵖ ≌ HomologicalComplex Vᵒᵖ c.symm where
functor := opFunctor V c
inverse := opInverse V c
... | Mathlib.Algebra.Homology.Opposite.176_0.Q6AwB66K8LWeNih | /-- Given a category of complexes with objects in `V`, there is a natural equivalence between its
opposite category and a category of complexes with objects in `Vᵒᵖ`. -/
@[simps]
def opEquivalence : (HomologicalComplex V c)ᵒᵖ ≌ HomologicalComplex Vᵒᵖ c.symm where
functor | Mathlib_Algebra_Homology_Opposite |
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.112048, u_2} V
c : ComplexShape ι
inst✝ : Preadditive V
X✝ Y✝ : (HomologicalComplex Vᵒᵖ c)ᵒᵖ
f : X✝ ⟶ Y✝
i j : ι
x✝ : ComplexShape.Rel (ComplexShape.symm c) i j
⊢ (fun i => (Hom.f f.unop i).unop) i ≫ d ((fun X => HomologicalComplex.unop X.unop) Y✝) i j =
d ((fun X => ... | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | simp only [unop_d, ← unop_comp, f.unop.comm] | /-- Auxiliary definition for `unopEquivalence`. -/
@[simps]
def unopFunctor : (HomologicalComplex Vᵒᵖ c)ᵒᵖ ⥤ HomologicalComplex V c.symm where
obj X := (unop X).unop
map f :=
{ f := fun i => (f.unop.f i).unop
comm' := fun i j _ => by | Mathlib.Algebra.Homology.Opposite.191_0.Q6AwB66K8LWeNih | /-- Auxiliary definition for `unopEquivalence`. -/
@[simps]
def unopFunctor : (HomologicalComplex Vᵒᵖ c)ᵒᵖ ⥤ HomologicalComplex V c.symm where
obj X | Mathlib_Algebra_Homology_Opposite |
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.120694, u_2} V
c : ComplexShape ι
inst✝ : Preadditive V
X✝ Y✝ : HomologicalComplex V (ComplexShape.symm c)
f : X✝ ⟶ Y✝
i j : ι
x✝ : ComplexShape.Rel c i j
⊢ (fun i => (Hom.f f i).op) i ≫ d (HomologicalComplex.opSymm X✝) i j =
d (HomologicalComplex.opSymm Y✝) i j ≫ (fu... | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | simp only [opSymm_d, ← op_comp, f.comm] | /-- Auxiliary definition for `unopEquivalence`. -/
@[simps]
def unopInverse : HomologicalComplex V c.symm ⥤ (HomologicalComplex Vᵒᵖ c)ᵒᵖ where
obj X := op X.opSymm
map f := Quiver.Hom.op
{ f := fun i => (f.f i).op
comm' := fun i j _ => by | Mathlib.Algebra.Homology.Opposite.200_0.Q6AwB66K8LWeNih | /-- Auxiliary definition for `unopEquivalence`. -/
@[simps]
def unopInverse : HomologicalComplex V c.symm ⥤ (HomologicalComplex Vᵒᵖ c)ᵒᵖ where
obj X | Mathlib_Algebra_Homology_Opposite |
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.128766, u_2} V
c : ComplexShape ι
inst✝ : Preadditive V
X : (HomologicalComplex Vᵒᵖ c)ᵒᵖ
i j : ι
x✝ : ComplexShape.Rel c i j
⊢ ((fun i => Iso.refl (HomologicalComplex.X (HomologicalComplex.unopSymm (HomologicalComplex.op X.unop)) i)) i).hom ≫
d X.unop i j =
d (H... | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | simp only [Iso.refl_hom, Category.id_comp, unopSymm_d, op_d, Quiver.Hom.unop_op,
Category.comp_id] | /-- Auxiliary definition for `unopEquivalence`. -/
def unopUnitIso : 𝟭 (HomologicalComplex Vᵒᵖ c)ᵒᵖ ≅ unopFunctor V c ⋙ unopInverse V c :=
NatIso.ofComponents
(fun X =>
(HomologicalComplex.Hom.isoOfComponents (fun i => Iso.refl _) fun i j _ => by
| Mathlib.Algebra.Homology.Opposite.209_0.Q6AwB66K8LWeNih | /-- Auxiliary definition for `unopEquivalence`. -/
def unopUnitIso : 𝟭 (HomologicalComplex Vᵒᵖ c)ᵒᵖ ≅ unopFunctor V c ⋙ unopInverse V c | Mathlib_Algebra_Homology_Opposite |
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.128766, u_2} V
c : ComplexShape ι
inst✝ : Preadditive V
⊢ ∀ {X Y : (HomologicalComplex Vᵒᵖ c)ᵒᵖ} (f : X ⟶ Y),
(𝟭 (HomologicalComplex Vᵒᵖ c)ᵒᵖ).map f ≫
((fun X =>
Iso.op
(Hom.isoOfComponents fun i =>
Iso.refl (Ho... | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | intro X Y f | /-- Auxiliary definition for `unopEquivalence`. -/
def unopUnitIso : 𝟭 (HomologicalComplex Vᵒᵖ c)ᵒᵖ ≅ unopFunctor V c ⋙ unopInverse V c :=
NatIso.ofComponents
(fun X =>
(HomologicalComplex.Hom.isoOfComponents (fun i => Iso.refl _) fun i j _ => by
simp only [Iso.refl_hom, Category.id_comp, unopS... | Mathlib.Algebra.Homology.Opposite.209_0.Q6AwB66K8LWeNih | /-- Auxiliary definition for `unopEquivalence`. -/
def unopUnitIso : 𝟭 (HomologicalComplex Vᵒᵖ c)ᵒᵖ ≅ unopFunctor V c ⋙ unopInverse V c | Mathlib_Algebra_Homology_Opposite |
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.128766, u_2} V
c : ComplexShape ι
inst✝ : Preadditive V
X Y : (HomologicalComplex Vᵒᵖ c)ᵒᵖ
f : X ⟶ Y
⊢ (𝟭 (HomologicalComplex Vᵒᵖ c)ᵒᵖ).map f ≫
((fun X =>
Iso.op
(Hom.isoOfComponents fun i =>
Iso.refl (HomologicalComplex.X ... | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | refine' Quiver.Hom.unop_inj _ | /-- Auxiliary definition for `unopEquivalence`. -/
def unopUnitIso : 𝟭 (HomologicalComplex Vᵒᵖ c)ᵒᵖ ≅ unopFunctor V c ⋙ unopInverse V c :=
NatIso.ofComponents
(fun X =>
(HomologicalComplex.Hom.isoOfComponents (fun i => Iso.refl _) fun i j _ => by
simp only [Iso.refl_hom, Category.id_comp, unopS... | Mathlib.Algebra.Homology.Opposite.209_0.Q6AwB66K8LWeNih | /-- Auxiliary definition for `unopEquivalence`. -/
def unopUnitIso : 𝟭 (HomologicalComplex Vᵒᵖ c)ᵒᵖ ≅ unopFunctor V c ⋙ unopInverse V c | Mathlib_Algebra_Homology_Opposite |
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.128766, u_2} V
c : ComplexShape ι
inst✝ : Preadditive V
X Y : (HomologicalComplex Vᵒᵖ c)ᵒᵖ
f : X ⟶ Y
⊢ ((𝟭 (HomologicalComplex Vᵒᵖ c)ᵒᵖ).map f ≫
((fun X =>
Iso.op
(Hom.isoOfComponents fun i =>
Iso.refl (HomologicalC... | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | ext x | /-- Auxiliary definition for `unopEquivalence`. -/
def unopUnitIso : 𝟭 (HomologicalComplex Vᵒᵖ c)ᵒᵖ ≅ unopFunctor V c ⋙ unopInverse V c :=
NatIso.ofComponents
(fun X =>
(HomologicalComplex.Hom.isoOfComponents (fun i => Iso.refl _) fun i j _ => by
simp only [Iso.refl_hom, Category.id_comp, unopS... | Mathlib.Algebra.Homology.Opposite.209_0.Q6AwB66K8LWeNih | /-- Auxiliary definition for `unopEquivalence`. -/
def unopUnitIso : 𝟭 (HomologicalComplex Vᵒᵖ c)ᵒᵖ ≅ unopFunctor V c ⋙ unopInverse V c | Mathlib_Algebra_Homology_Opposite |
case h
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.128766, u_2} V
c : ComplexShape ι
inst✝ : Preadditive V
X Y : (HomologicalComplex Vᵒᵖ c)ᵒᵖ
f : X ⟶ Y
x : ι
⊢ Hom.f
((𝟭 (HomologicalComplex Vᵒᵖ c)ᵒᵖ).map f ≫
((fun X =>
Iso.op
(Hom.isoOfComponents fun i =>
... | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | simp only [Quiver.Hom.unop_op, Functor.id_map, Iso.op_hom, Functor.comp_map, unop_comp,
comp_f, Hom.isoOfComponents_hom_f] | /-- Auxiliary definition for `unopEquivalence`. -/
def unopUnitIso : 𝟭 (HomologicalComplex Vᵒᵖ c)ᵒᵖ ≅ unopFunctor V c ⋙ unopInverse V c :=
NatIso.ofComponents
(fun X =>
(HomologicalComplex.Hom.isoOfComponents (fun i => Iso.refl _) fun i j _ => by
simp only [Iso.refl_hom, Category.id_comp, unopS... | Mathlib.Algebra.Homology.Opposite.209_0.Q6AwB66K8LWeNih | /-- Auxiliary definition for `unopEquivalence`. -/
def unopUnitIso : 𝟭 (HomologicalComplex Vᵒᵖ c)ᵒᵖ ≅ unopFunctor V c ⋙ unopInverse V c | Mathlib_Algebra_Homology_Opposite |
case h
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.128766, u_2} V
c : ComplexShape ι
inst✝ : Preadditive V
X Y : (HomologicalComplex Vᵒᵖ c)ᵒᵖ
f : X ⟶ Y
x : ι
⊢ (Iso.refl (HomologicalComplex.X (HomologicalComplex.unopSymm (HomologicalComplex.op Y.unop)) x)).hom ≫
Hom.f f.unop x =
Hom.f ((unopInverse V c).m... | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | erw [Category.id_comp, Category.comp_id (f.unop.f x)] | /-- Auxiliary definition for `unopEquivalence`. -/
def unopUnitIso : 𝟭 (HomologicalComplex Vᵒᵖ c)ᵒᵖ ≅ unopFunctor V c ⋙ unopInverse V c :=
NatIso.ofComponents
(fun X =>
(HomologicalComplex.Hom.isoOfComponents (fun i => Iso.refl _) fun i j _ => by
simp only [Iso.refl_hom, Category.id_comp, unopS... | Mathlib.Algebra.Homology.Opposite.209_0.Q6AwB66K8LWeNih | /-- Auxiliary definition for `unopEquivalence`. -/
def unopUnitIso : 𝟭 (HomologicalComplex Vᵒᵖ c)ᵒᵖ ≅ unopFunctor V c ⋙ unopInverse V c | Mathlib_Algebra_Homology_Opposite |
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.148250, u_2} V
c : ComplexShape ι
inst✝ : Preadditive V
X : (HomologicalComplex Vᵒᵖ c)ᵒᵖ
⊢ (unopFunctor V c).map ((unopUnitIso V c).hom.app X) ≫ (unopCounitIso V c).hom.app ((unopFunctor V c).obj X) =
𝟙 ((unopFunctor V c).obj X) | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | ext | /-- Given a category of complexes with objects in `Vᵒᵖ`, there is a natural equivalence between its
opposite category and a category of complexes with objects in `V`. -/
@[simps]
def unopEquivalence : (HomologicalComplex Vᵒᵖ c)ᵒᵖ ≌ HomologicalComplex V c.symm where
functor := unopFunctor V c
inverse := unopInverse ... | Mathlib.Algebra.Homology.Opposite.232_0.Q6AwB66K8LWeNih | /-- Given a category of complexes with objects in `Vᵒᵖ`, there is a natural equivalence between its
opposite category and a category of complexes with objects in `V`. -/
@[simps]
def unopEquivalence : (HomologicalComplex Vᵒᵖ c)ᵒᵖ ≌ HomologicalComplex V c.symm where
functor | Mathlib_Algebra_Homology_Opposite |
case h
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.148250, u_2} V
c : ComplexShape ι
inst✝ : Preadditive V
X : (HomologicalComplex Vᵒᵖ c)ᵒᵖ
i✝ : ι
⊢ Hom.f ((unopFunctor V c).map ((unopUnitIso V c).hom.app X) ≫ (unopCounitIso V c).hom.app ((unopFunctor V c).obj X))
i✝ =
Hom.f (𝟙 ((unopFunctor V c).obj X))... | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | simp only [opUnitIso, opCounitIso, NatIso.ofComponents_hom_app, Iso.op_hom, comp_f,
opFunctor_map_f, Quiver.Hom.unop_op, Hom.isoOfComponents_hom_f] | /-- Given a category of complexes with objects in `Vᵒᵖ`, there is a natural equivalence between its
opposite category and a category of complexes with objects in `V`. -/
@[simps]
def unopEquivalence : (HomologicalComplex Vᵒᵖ c)ᵒᵖ ≌ HomologicalComplex V c.symm where
functor := unopFunctor V c
inverse := unopInverse ... | Mathlib.Algebra.Homology.Opposite.232_0.Q6AwB66K8LWeNih | /-- Given a category of complexes with objects in `Vᵒᵖ`, there is a natural equivalence between its
opposite category and a category of complexes with objects in `V`. -/
@[simps]
def unopEquivalence : (HomologicalComplex Vᵒᵖ c)ᵒᵖ ≌ HomologicalComplex V c.symm where
functor | Mathlib_Algebra_Homology_Opposite |
case h
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.148250, u_2} V
c : ComplexShape ι
inst✝ : Preadditive V
X : (HomologicalComplex Vᵒᵖ c)ᵒᵖ
i✝ : ι
⊢ Hom.f ((unopFunctor V c).map ((unopUnitIso V c).hom.app X)) i✝ ≫
Hom.f ((unopCounitIso V c).hom.app ((unopFunctor V c).obj X)) i✝ =
Hom.f (𝟙 ((unopFunctor V... | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | exact Category.comp_id _ | /-- Given a category of complexes with objects in `Vᵒᵖ`, there is a natural equivalence between its
opposite category and a category of complexes with objects in `V`. -/
@[simps]
def unopEquivalence : (HomologicalComplex Vᵒᵖ c)ᵒᵖ ≌ HomologicalComplex V c.symm where
functor := unopFunctor V c
inverse := unopInverse ... | Mathlib.Algebra.Homology.Opposite.232_0.Q6AwB66K8LWeNih | /-- Given a category of complexes with objects in `Vᵒᵖ`, there is a natural equivalence between its
opposite category and a category of complexes with objects in `V`. -/
@[simps]
def unopEquivalence : (HomologicalComplex Vᵒᵖ c)ᵒᵖ ≌ HomologicalComplex V c.symm where
functor | Mathlib_Algebra_Homology_Opposite |
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.157635, u_2} V
c : ComplexShape ι
inst✝ : Abelian V
C : HomologicalComplex V c
i : ι
⊢ (dFrom C i).op ≫ (dTo C i).op = 0 | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | rw [← op_comp, C.dTo_comp_dFrom i, op_zero] | /-- Auxiliary tautological definition for `homologyOp`. -/
def homology'OpDef : C.op.homology' i ≅
_root_.homology' (C.dFrom i).op (C.dTo i).op (by | Mathlib.Algebra.Homology.Opposite.279_0.Q6AwB66K8LWeNih | /-- Auxiliary tautological definition for `homologyOp`. -/
def homology'OpDef : C.op.homology' i ≅
_root_.homology' (C.dFrom i).op (C.dTo i).op (by rw [← op_comp, C.dTo_comp_dFrom i, op_zero]) | Mathlib_Algebra_Homology_Opposite |
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.162814, u_2} V
c : ComplexShape ι
inst✝ : Abelian V
C✝ : HomologicalComplex V c
i : ι
C : HomologicalComplex Vᵒᵖ c
⊢ (dFrom C i).unop ≫ (dTo C i).unop = 0 | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | rw [← unop_comp, C.dTo_comp_dFrom i, unop_zero] | /-- Auxiliary tautological definition for `homologyUnop`. -/
def homology'UnopDef (C : HomologicalComplex Vᵒᵖ c) :
C.unop.homology' i ≅
_root_.homology' (C.dFrom i).unop (C.dTo i).unop
(by | Mathlib.Algebra.Homology.Opposite.291_0.Q6AwB66K8LWeNih | /-- Auxiliary tautological definition for `homologyUnop`. -/
def homology'UnopDef (C : HomologicalComplex Vᵒᵖ c) :
C.unop.homology' i ≅
_root_.homology' (C.dFrom i).unop (C.dTo i).unop
(by rw [← unop_comp, C.dTo_comp_dFrom i, unop_zero]) | Mathlib_Algebra_Homology_Opposite |
α : Type u_1
inst✝² : Zero α
p : Prop
inst✝¹ : Decidable p
a b : α
inst✝ : LT α
ha : 0 < a
hb : 0 < b
⊢ 0 < if p then a else b | /-
Copyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Heather Macbeth, Yaël Dillies
-/
import Std.Lean.Parser
import Mathlib.Data.Int.Order.Basic
import Mathlib.Data.Int.CharZero
import Mathlib.Data.Nat.Fa... | by_cases p | private lemma ite_pos [LT α] (ha : 0 < a) (hb : 0 < b) : 0 < ite p a b := by
| Mathlib.Tactic.Positivity.Basic.33_0.hOM93nWOlZMew4l | private lemma ite_pos [LT α] (ha : 0 < a) (hb : 0 < b) : 0 < ite p a b | Mathlib_Tactic_Positivity_Basic |
case pos
α : Type u_1
inst✝² : Zero α
p : Prop
inst✝¹ : Decidable p
a b : α
inst✝ : LT α
ha : 0 < a
hb : 0 < b
h✝ : p
⊢ 0 < if p then a else b | /-
Copyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Heather Macbeth, Yaël Dillies
-/
import Std.Lean.Parser
import Mathlib.Data.Int.Order.Basic
import Mathlib.Data.Int.CharZero
import Mathlib.Data.Nat.Fa... | simp [*] | private lemma ite_pos [LT α] (ha : 0 < a) (hb : 0 < b) : 0 < ite p a b := by
by_cases p <;> | Mathlib.Tactic.Positivity.Basic.33_0.hOM93nWOlZMew4l | private lemma ite_pos [LT α] (ha : 0 < a) (hb : 0 < b) : 0 < ite p a b | Mathlib_Tactic_Positivity_Basic |
case neg
α : Type u_1
inst✝² : Zero α
p : Prop
inst✝¹ : Decidable p
a b : α
inst✝ : LT α
ha : 0 < a
hb : 0 < b
h✝ : ¬p
⊢ 0 < if p then a else b | /-
Copyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Heather Macbeth, Yaël Dillies
-/
import Std.Lean.Parser
import Mathlib.Data.Int.Order.Basic
import Mathlib.Data.Int.CharZero
import Mathlib.Data.Nat.Fa... | simp [*] | private lemma ite_pos [LT α] (ha : 0 < a) (hb : 0 < b) : 0 < ite p a b := by
by_cases p <;> | Mathlib.Tactic.Positivity.Basic.33_0.hOM93nWOlZMew4l | private lemma ite_pos [LT α] (ha : 0 < a) (hb : 0 < b) : 0 < ite p a b | Mathlib_Tactic_Positivity_Basic |
α : Type u_1
inst✝² : Zero α
p : Prop
inst✝¹ : Decidable p
a b : α
inst✝ : LE α
ha : 0 ≤ a
hb : 0 ≤ b
⊢ 0 ≤ if p then a else b | /-
Copyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Heather Macbeth, Yaël Dillies
-/
import Std.Lean.Parser
import Mathlib.Data.Int.Order.Basic
import Mathlib.Data.Int.CharZero
import Mathlib.Data.Nat.Fa... | by_cases p | private lemma ite_nonneg [LE α] (ha : 0 ≤ a) (hb : 0 ≤ b) : 0 ≤ ite p a b := by
| Mathlib.Tactic.Positivity.Basic.36_0.hOM93nWOlZMew4l | private lemma ite_nonneg [LE α] (ha : 0 ≤ a) (hb : 0 ≤ b) : 0 ≤ ite p a b | Mathlib_Tactic_Positivity_Basic |
case pos
α : Type u_1
inst✝² : Zero α
p : Prop
inst✝¹ : Decidable p
a b : α
inst✝ : LE α
ha : 0 ≤ a
hb : 0 ≤ b
h✝ : p
⊢ 0 ≤ if p then a else b | /-
Copyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Heather Macbeth, Yaël Dillies
-/
import Std.Lean.Parser
import Mathlib.Data.Int.Order.Basic
import Mathlib.Data.Int.CharZero
import Mathlib.Data.Nat.Fa... | simp [*] | private lemma ite_nonneg [LE α] (ha : 0 ≤ a) (hb : 0 ≤ b) : 0 ≤ ite p a b := by
by_cases p <;> | Mathlib.Tactic.Positivity.Basic.36_0.hOM93nWOlZMew4l | private lemma ite_nonneg [LE α] (ha : 0 ≤ a) (hb : 0 ≤ b) : 0 ≤ ite p a b | Mathlib_Tactic_Positivity_Basic |
case neg
α : Type u_1
inst✝² : Zero α
p : Prop
inst✝¹ : Decidable p
a b : α
inst✝ : LE α
ha : 0 ≤ a
hb : 0 ≤ b
h✝ : ¬p
⊢ 0 ≤ if p then a else b | /-
Copyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Heather Macbeth, Yaël Dillies
-/
import Std.Lean.Parser
import Mathlib.Data.Int.Order.Basic
import Mathlib.Data.Int.CharZero
import Mathlib.Data.Nat.Fa... | simp [*] | private lemma ite_nonneg [LE α] (ha : 0 ≤ a) (hb : 0 ≤ b) : 0 ≤ ite p a b := by
by_cases p <;> | Mathlib.Tactic.Positivity.Basic.36_0.hOM93nWOlZMew4l | private lemma ite_nonneg [LE α] (ha : 0 ≤ a) (hb : 0 ≤ b) : 0 ≤ ite p a b | Mathlib_Tactic_Positivity_Basic |
α : Type u_1
inst✝¹ : Zero α
p : Prop
inst✝ : Decidable p
a b : α
ha : a ≠ 0
hb : b ≠ 0
⊢ (if p then a else b) ≠ 0 | /-
Copyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Heather Macbeth, Yaël Dillies
-/
import Std.Lean.Parser
import Mathlib.Data.Int.Order.Basic
import Mathlib.Data.Int.CharZero
import Mathlib.Data.Nat.Fa... | by_cases p | private lemma ite_ne_zero (ha : a ≠ 0) (hb : b ≠ 0) : ite p a b ≠ 0 := by | Mathlib.Tactic.Positivity.Basic.45_0.hOM93nWOlZMew4l | private lemma ite_ne_zero (ha : a ≠ 0) (hb : b ≠ 0) : ite p a b ≠ 0 | Mathlib_Tactic_Positivity_Basic |
case pos
α : Type u_1
inst✝¹ : Zero α
p : Prop
inst✝ : Decidable p
a b : α
ha : a ≠ 0
hb : b ≠ 0
h✝ : p
⊢ (if p then a else b) ≠ 0 | /-
Copyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Heather Macbeth, Yaël Dillies
-/
import Std.Lean.Parser
import Mathlib.Data.Int.Order.Basic
import Mathlib.Data.Int.CharZero
import Mathlib.Data.Nat.Fa... | simp [*] | private lemma ite_ne_zero (ha : a ≠ 0) (hb : b ≠ 0) : ite p a b ≠ 0 := by by_cases p <;> | Mathlib.Tactic.Positivity.Basic.45_0.hOM93nWOlZMew4l | private lemma ite_ne_zero (ha : a ≠ 0) (hb : b ≠ 0) : ite p a b ≠ 0 | Mathlib_Tactic_Positivity_Basic |
case neg
α : Type u_1
inst✝¹ : Zero α
p : Prop
inst✝ : Decidable p
a b : α
ha : a ≠ 0
hb : b ≠ 0
h✝ : ¬p
⊢ (if p then a else b) ≠ 0 | /-
Copyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Heather Macbeth, Yaël Dillies
-/
import Std.Lean.Parser
import Mathlib.Data.Int.Order.Basic
import Mathlib.Data.Int.CharZero
import Mathlib.Data.Nat.Fa... | simp [*] | private lemma ite_ne_zero (ha : a ≠ 0) (hb : b ≠ 0) : ite p a b ≠ 0 := by by_cases p <;> | Mathlib.Tactic.Positivity.Basic.45_0.hOM93nWOlZMew4l | private lemma ite_ne_zero (ha : a ≠ 0) (hb : b ≠ 0) : ite p a b ≠ 0 | Mathlib_Tactic_Positivity_Basic |
R : Type u_1
inst✝ : LinearOrder R
a b c : R
ha : a ≠ c
hb : b ≠ c
⊢ min a b ≠ c | /-
Copyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Heather Macbeth, Yaël Dillies
-/
import Std.Lean.Parser
import Mathlib.Data.Int.Order.Basic
import Mathlib.Data.Int.CharZero
import Mathlib.Data.Nat.Fa... | rw [min_def] | private lemma min_ne (ha : a ≠ c) (hb : b ≠ c) : min a b ≠ c := by
| Mathlib.Tactic.Positivity.Basic.95_0.hOM93nWOlZMew4l | private lemma min_ne (ha : a ≠ c) (hb : b ≠ c) : min a b ≠ c | Mathlib_Tactic_Positivity_Basic |
R : Type u_1
inst✝ : LinearOrder R
a b c : R
ha : a ≠ c
hb : b ≠ c
⊢ (if a ≤ b then a else b) ≠ c | /-
Copyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Heather Macbeth, Yaël Dillies
-/
import Std.Lean.Parser
import Mathlib.Data.Int.Order.Basic
import Mathlib.Data.Int.CharZero
import Mathlib.Data.Nat.Fa... | split_ifs | private lemma min_ne (ha : a ≠ c) (hb : b ≠ c) : min a b ≠ c := by
rw [min_def]; | Mathlib.Tactic.Positivity.Basic.95_0.hOM93nWOlZMew4l | private lemma min_ne (ha : a ≠ c) (hb : b ≠ c) : min a b ≠ c | Mathlib_Tactic_Positivity_Basic |
case pos
R : Type u_1
inst✝ : LinearOrder R
a b c : R
ha : a ≠ c
hb : b ≠ c
h✝ : a ≤ b
⊢ a ≠ c | /-
Copyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Heather Macbeth, Yaël Dillies
-/
import Std.Lean.Parser
import Mathlib.Data.Int.Order.Basic
import Mathlib.Data.Int.CharZero
import Mathlib.Data.Nat.Fa... | assumption | private lemma min_ne (ha : a ≠ c) (hb : b ≠ c) : min a b ≠ c := by
rw [min_def]; split_ifs <;> | Mathlib.Tactic.Positivity.Basic.95_0.hOM93nWOlZMew4l | private lemma min_ne (ha : a ≠ c) (hb : b ≠ c) : min a b ≠ c | Mathlib_Tactic_Positivity_Basic |
case neg
R : Type u_1
inst✝ : LinearOrder R
a b c : R
ha : a ≠ c
hb : b ≠ c
h✝ : ¬a ≤ b
⊢ b ≠ c | /-
Copyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Heather Macbeth, Yaël Dillies
-/
import Std.Lean.Parser
import Mathlib.Data.Int.Order.Basic
import Mathlib.Data.Int.CharZero
import Mathlib.Data.Nat.Fa... | assumption | private lemma min_ne (ha : a ≠ c) (hb : b ≠ c) : min a b ≠ c := by
rw [min_def]; split_ifs <;> | Mathlib.Tactic.Positivity.Basic.95_0.hOM93nWOlZMew4l | private lemma min_ne (ha : a ≠ c) (hb : b ≠ c) : min a b ≠ c | Mathlib_Tactic_Positivity_Basic |
R : Type u_1
inst✝ : LinearOrder R
a b c : R
ha : a ≠ c
hb : b ≠ c
⊢ max a b ≠ c | /-
Copyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Heather Macbeth, Yaël Dillies
-/
import Std.Lean.Parser
import Mathlib.Data.Int.Order.Basic
import Mathlib.Data.Int.CharZero
import Mathlib.Data.Nat.Fa... | rw [max_def] | private lemma max_ne (ha : a ≠ c) (hb : b ≠ c) : max a b ≠ c := by
| Mathlib.Tactic.Positivity.Basic.101_0.hOM93nWOlZMew4l | private lemma max_ne (ha : a ≠ c) (hb : b ≠ c) : max a b ≠ c | Mathlib_Tactic_Positivity_Basic |
R : Type u_1
inst✝ : LinearOrder R
a b c : R
ha : a ≠ c
hb : b ≠ c
⊢ (if a ≤ b then b else a) ≠ c | /-
Copyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Heather Macbeth, Yaël Dillies
-/
import Std.Lean.Parser
import Mathlib.Data.Int.Order.Basic
import Mathlib.Data.Int.CharZero
import Mathlib.Data.Nat.Fa... | split_ifs | private lemma max_ne (ha : a ≠ c) (hb : b ≠ c) : max a b ≠ c := by
rw [max_def]; | Mathlib.Tactic.Positivity.Basic.101_0.hOM93nWOlZMew4l | private lemma max_ne (ha : a ≠ c) (hb : b ≠ c) : max a b ≠ c | Mathlib_Tactic_Positivity_Basic |
case pos
R : Type u_1
inst✝ : LinearOrder R
a b c : R
ha : a ≠ c
hb : b ≠ c
h✝ : a ≤ b
⊢ b ≠ c | /-
Copyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Heather Macbeth, Yaël Dillies
-/
import Std.Lean.Parser
import Mathlib.Data.Int.Order.Basic
import Mathlib.Data.Int.CharZero
import Mathlib.Data.Nat.Fa... | assumption | private lemma max_ne (ha : a ≠ c) (hb : b ≠ c) : max a b ≠ c := by
rw [max_def]; split_ifs <;> | Mathlib.Tactic.Positivity.Basic.101_0.hOM93nWOlZMew4l | private lemma max_ne (ha : a ≠ c) (hb : b ≠ c) : max a b ≠ c | Mathlib_Tactic_Positivity_Basic |
case neg
R : Type u_1
inst✝ : LinearOrder R
a b c : R
ha : a ≠ c
hb : b ≠ c
h✝ : ¬a ≤ b
⊢ a ≠ c | /-
Copyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Heather Macbeth, Yaël Dillies
-/
import Std.Lean.Parser
import Mathlib.Data.Int.Order.Basic
import Mathlib.Data.Int.CharZero
import Mathlib.Data.Nat.Fa... | assumption | private lemma max_ne (ha : a ≠ c) (hb : b ≠ c) : max a b ≠ c := by
rw [max_def]; split_ifs <;> | Mathlib.Tactic.Positivity.Basic.101_0.hOM93nWOlZMew4l | private lemma max_ne (ha : a ≠ c) (hb : b ≠ c) : max a b ≠ c | Mathlib_Tactic_Positivity_Basic |
a : ℤ
ha : 0 < a
⊢ 0 < a / a | /-
Copyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Heather Macbeth, Yaël Dillies
-/
import Std.Lean.Parser
import Mathlib.Data.Int.Order.Basic
import Mathlib.Data.Int.CharZero
import Mathlib.Data.Nat.Fa... | rw [Int.ediv_self ha.ne'] | private lemma int_div_self_pos {a : ℤ} (ha : 0 < a) : 0 < a / a := by
| Mathlib.Tactic.Positivity.Basic.222_0.hOM93nWOlZMew4l | private lemma int_div_self_pos {a : ℤ} (ha : 0 < a) : 0 < a / a | Mathlib_Tactic_Positivity_Basic |
a : ℤ
ha : 0 < a
⊢ 0 < 1 | /-
Copyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Heather Macbeth, Yaël Dillies
-/
import Std.Lean.Parser
import Mathlib.Data.Int.Order.Basic
import Mathlib.Data.Int.CharZero
import Mathlib.Data.Nat.Fa... | exact zero_lt_one | private lemma int_div_self_pos {a : ℤ} (ha : 0 < a) : 0 < a / a := by
rw [Int.ediv_self ha.ne']; | Mathlib.Tactic.Positivity.Basic.222_0.hOM93nWOlZMew4l | private lemma int_div_self_pos {a : ℤ} (ha : 0 < a) : 0 < a / a | Mathlib_Tactic_Positivity_Basic |
u : Level
α : Q(Type u)
zα : Q(Zero «$α»)
pα : Q(PartialOrder «$α»)
e : Q(«$α»)
fn✝ : Expr
a : Q(«$α»)
b : Q(ℕ)
n : ℕ
x✝ : PUnit.{1}
m : Q(ℕ)
this✝ : «$b» =Q bit0 «$m»
_a : Q(LinearOrderedRing «$α»)
this : «$e» =Q «$a» ^ «$b»
zα_eq✝ : «$zα» =Q MonoidWithZero.toZero
pα_eq✝ : «$pα» =Q StrictOrderedRing.toPartialOrder
⊢ S... | /-
Copyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Heather Macbeth, Yaël Dillies
-/
import Std.Lean.Parser
import Mathlib.Data.Int.Order.Basic
import Mathlib.Data.Int.CharZero
import Mathlib.Data.Nat.Fa... | exact .nonnegative q(pow_bit0_nonneg $a $m) | set_option linter.deprecated false in
/-- The `positivity` extension which identifies expressions of the form `a ^ (b : ℕ)`,
such that `positivity` successfully recognises both `a` and `b`. -/
@[positivity (_ : α) ^ (_ : ℕ), Pow.pow _ (_ : ℕ)]
def evalPow : PositivityExt where eval {u α} zα pα e := do
let .app (.app ... | Mathlib.Tactic.Positivity.Basic.329_0.hOM93nWOlZMew4l | set_option linter.deprecated false in
/-- The `positivity` extension which identifies expressions of the form `a ^ (b : ℕ)`,
such that `positivity` successfully recognises both `a` and `b`. -/
@[positivity (_ : α) ^ (_ : ℕ), Pow.pow _ (_ : ℕ)]
def evalPow : PositivityExt where eval {u α} zα pα e | Mathlib_Tactic_Positivity_Basic |
u : Level
α : Q(Type u)
zα : Q(Zero «$α»)
pα : Q(PartialOrder «$α»)
e : Q(«$α»)
fn✝ : Expr
a : Q(«$α»)
b : Q(ℕ)
result : Strictness zα pα e
ra : Strictness zα pα a
pa : Q(0 ≤ «$a»)
_oα : Q(OrderedSemiring «$α»)
this : «$e» =Q «$a» ^ «$b»
zα_eq✝ : «$zα» =Q MonoidWithZero.toZero
pα_eq✝ : «$pα» =Q OrderedSemiring.toPartia... | /-
Copyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Heather Macbeth, Yaël Dillies
-/
import Std.Lean.Parser
import Mathlib.Data.Int.Order.Basic
import Mathlib.Data.Int.CharZero
import Mathlib.Data.Nat.Fa... | exact .nonnegative (q(pow_nonneg $pa $b)) | set_option linter.deprecated false in
/-- The `positivity` extension which identifies expressions of the form `a ^ (b : ℕ)`,
such that `positivity` successfully recognises both `a` and `b`. -/
@[positivity (_ : α) ^ (_ : ℕ), Pow.pow _ (_ : ℕ)]
def evalPow : PositivityExt where eval {u α} zα pα e := do
let .app (.app ... | Mathlib.Tactic.Positivity.Basic.329_0.hOM93nWOlZMew4l | set_option linter.deprecated false in
/-- The `positivity` extension which identifies expressions of the form `a ^ (b : ℕ)`,
such that `positivity` successfully recognises both `a` and `b`. -/
@[positivity (_ : α) ^ (_ : ℕ), Pow.pow _ (_ : ℕ)]
def evalPow : PositivityExt where eval {u α} zα pα e | Mathlib_Tactic_Positivity_Basic |
u : Level
α : Q(Type u)
zα : Q(Zero «$α»)
pα : Q(PartialOrder «$α»)
e : Q(«$α»)
fn✝ : Expr
a : Q(«$α»)
b : Q(ℕ)
result : Strictness zα pα e
ra : Strictness zα pα a
ofNonneg : Q(0 ≤ «$a») → Q(OrderedSemiring «$α») → MetaM (Strictness zα pα e) :=
fun pa _oα =>
let_fun zα_eq := (_ : «$zα» =Q MonoidWithZero.toZero);
... | /-
Copyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Heather Macbeth, Yaël Dillies
-/
import Std.Lean.Parser
import Mathlib.Data.Int.Order.Basic
import Mathlib.Data.Int.CharZero
import Mathlib.Data.Nat.Fa... | exact q(pow_ne_zero $b $pa) | set_option linter.deprecated false in
/-- The `positivity` extension which identifies expressions of the form `a ^ (b : ℕ)`,
such that `positivity` successfully recognises both `a` and `b`. -/
@[positivity (_ : α) ^ (_ : ℕ), Pow.pow _ (_ : ℕ)]
def evalPow : PositivityExt where eval {u α} zα pα e := do
let .app (.app ... | Mathlib.Tactic.Positivity.Basic.329_0.hOM93nWOlZMew4l | set_option linter.deprecated false in
/-- The `positivity` extension which identifies expressions of the form `a ^ (b : ℕ)`,
such that `positivity` successfully recognises both `a` and `b`. -/
@[positivity (_ : α) ^ (_ : ℕ), Pow.pow _ (_ : ℕ)]
def evalPow : PositivityExt where eval {u α} zα pα e | Mathlib_Tactic_Positivity_Basic |
u : Level
α : Q(Type u)
zα : Q(Zero «$α»)
pα : Q(PartialOrder «$α»)
e : Q(«$α»)
fn✝ : Expr
a : Q(«$α»)
b : Q(ℕ)
result : Strictness zα pα e
ra : Strictness zα pα a
ofNonneg : Q(0 ≤ «$a») → Q(OrderedSemiring «$α») → MetaM (Strictness zα pα e) :=
fun pa _oα =>
let_fun zα_eq := (_ : «$zα» =Q MonoidWithZero.toZero);
... | /-
Copyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Heather Macbeth, Yaël Dillies
-/
import Std.Lean.Parser
import Mathlib.Data.Int.Order.Basic
import Mathlib.Data.Int.CharZero
import Mathlib.Data.Nat.Fa... | exact .positive (q(pow_pos $pa $b)) | set_option linter.deprecated false in
/-- The `positivity` extension which identifies expressions of the form `a ^ (b : ℕ)`,
such that `positivity` successfully recognises both `a` and `b`. -/
@[positivity (_ : α) ^ (_ : ℕ), Pow.pow _ (_ : ℕ)]
def evalPow : PositivityExt where eval {u α} zα pα e := do
let .app (.app ... | Mathlib.Tactic.Positivity.Basic.329_0.hOM93nWOlZMew4l | set_option linter.deprecated false in
/-- The `positivity` extension which identifies expressions of the form `a ^ (b : ℕ)`,
such that `positivity` successfully recognises both `a` and `b`. -/
@[positivity (_ : α) ^ (_ : ℕ), Pow.pow _ (_ : ℕ)]
def evalPow : PositivityExt where eval {u α} zα pα e | Mathlib_Tactic_Positivity_Basic |
u : Level
α : Q(Type u)
zα : Q(Zero «$α»)
pα : Q(PartialOrder «$α»)
e : Q(«$α»)
fn✝ : Expr
a : Q(«$α»)
b : Q(ℤ)
_a : Q(LinearOrderedField «$α»)
zα_eq✝ : «$zα» =Q CommMonoidWithZero.toZero
pα_eq✝ : «$pα» =Q StrictOrderedRing.toPartialOrder
us✝ : List Level
arg✝¹ : Expr
n : ℕ
arg✝ : Expr
x✝ : PUnit.{1}
m : Q(ℕ)
this✝ : «... | /-
Copyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Heather Macbeth, Yaël Dillies
-/
import Std.Lean.Parser
import Mathlib.Data.Int.Order.Basic
import Mathlib.Data.Int.CharZero
import Mathlib.Data.Nat.Fa... | exact .nonnegative q(zpow_bit0_nonneg $a $m) | /-- The `positivity` extension which identifies expressions of the form `a ^ (b : ℤ)`,
such that `positivity` successfully recognises both `a` and `b`. -/
@[positivity (_ : α) ^ (_ : ℤ), Pow.pow _ (_ : ℤ)]
def evalZpow : PositivityExt where eval {u α} zα pα e := do
let .app (.app _ (a : Q($α))) (b : Q(ℤ)) ← withReduc... | Mathlib.Tactic.Positivity.Basic.371_0.hOM93nWOlZMew4l | /-- The `positivity` extension which identifies expressions of the form `a ^ (b : ℤ)`,
such that `positivity` successfully recognises both `a` and `b`. -/
@[positivity (_ : α) ^ (_ : ℤ), Pow.pow _ (_ : ℤ)]
def evalZpow : PositivityExt where eval {u α} zα pα e | Mathlib_Tactic_Positivity_Basic |
u : Level
α : Q(Type u)
zα : Q(Zero «$α»)
pα : Q(PartialOrder «$α»)
e : Q(«$α»)
fn✝ : Expr
a : Q(«$α»)
b : Q(ℤ)
_a : Q(LinearOrderedField «$α»)
zα_eq✝ : «$zα» =Q CommMonoidWithZero.toZero
pα_eq✝ : «$pα» =Q StrictOrderedRing.toPartialOrder
us✝ : List Level
arg✝¹ arg✝ b'✝ b' : Expr
n : ℕ
x✝ : PUnit.{1}
m : Q(ℕ)
this✝ : «... | /-
Copyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Heather Macbeth, Yaël Dillies
-/
import Std.Lean.Parser
import Mathlib.Data.Int.Order.Basic
import Mathlib.Data.Int.CharZero
import Mathlib.Data.Nat.Fa... | exact .nonnegative q(zpow_bit0_nonneg $a (-$m)) | /-- The `positivity` extension which identifies expressions of the form `a ^ (b : ℤ)`,
such that `positivity` successfully recognises both `a` and `b`. -/
@[positivity (_ : α) ^ (_ : ℤ), Pow.pow _ (_ : ℤ)]
def evalZpow : PositivityExt where eval {u α} zα pα e := do
let .app (.app _ (a : Q($α))) (b : Q(ℤ)) ← withReduc... | Mathlib.Tactic.Positivity.Basic.371_0.hOM93nWOlZMew4l | /-- The `positivity` extension which identifies expressions of the form `a ^ (b : ℤ)`,
such that `positivity` successfully recognises both `a` and `b`. -/
@[positivity (_ : α) ^ (_ : ℤ), Pow.pow _ (_ : ℤ)]
def evalZpow : PositivityExt where eval {u α} zα pα e | Mathlib_Tactic_Positivity_Basic |
u : Level
α : Q(Type u)
zα : Q(Zero «$α»)
pα : Q(PartialOrder «$α»)
e : Q(«$α»)
fn✝ : Expr
a : Q(«$α»)
b : Q(ℤ)
result : Strictness zα pα e
ra : Strictness zα pα a
pa : Q(0 ≤ «$a»)
_oα : Q(LinearOrderedSemifield «$α»)
this : «$e» =Q «$a» ^ «$b»
zα_eq✝ : «$zα» =Q CommMonoidWithZero.toZero
pα_eq✝ : «$pα» =Q StrictOrdered... | /-
Copyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Heather Macbeth, Yaël Dillies
-/
import Std.Lean.Parser
import Mathlib.Data.Int.Order.Basic
import Mathlib.Data.Int.CharZero
import Mathlib.Data.Nat.Fa... | exact .nonnegative (q(zpow_nonneg $pa $b)) | /-- The `positivity` extension which identifies expressions of the form `a ^ (b : ℤ)`,
such that `positivity` successfully recognises both `a` and `b`. -/
@[positivity (_ : α) ^ (_ : ℤ), Pow.pow _ (_ : ℤ)]
def evalZpow : PositivityExt where eval {u α} zα pα e := do
let .app (.app _ (a : Q($α))) (b : Q(ℤ)) ← withReduc... | Mathlib.Tactic.Positivity.Basic.371_0.hOM93nWOlZMew4l | /-- The `positivity` extension which identifies expressions of the form `a ^ (b : ℤ)`,
such that `positivity` successfully recognises both `a` and `b`. -/
@[positivity (_ : α) ^ (_ : ℤ), Pow.pow _ (_ : ℤ)]
def evalZpow : PositivityExt where eval {u α} zα pα e | Mathlib_Tactic_Positivity_Basic |
u : Level
α : Q(Type u)
zα : Q(Zero «$α»)
pα : Q(PartialOrder «$α»)
e : Q(«$α»)
fn✝ : Expr
a : Q(«$α»)
b : Q(ℤ)
result : Strictness zα pα e
ra : Strictness zα pα a
ofNonneg : Q(0 ≤ «$a») → Q(LinearOrderedSemifield «$α») → MetaM (Strictness zα pα e) :=
fun pa _oα =>
let_fun zα_eq := (_ : «$zα» =Q CommMonoidWithZer... | /-
Copyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Heather Macbeth, Yaël Dillies
-/
import Std.Lean.Parser
import Mathlib.Data.Int.Order.Basic
import Mathlib.Data.Int.CharZero
import Mathlib.Data.Nat.Fa... | exact q(zpow_ne_zero $b $pa) | /-- The `positivity` extension which identifies expressions of the form `a ^ (b : ℤ)`,
such that `positivity` successfully recognises both `a` and `b`. -/
@[positivity (_ : α) ^ (_ : ℤ), Pow.pow _ (_ : ℤ)]
def evalZpow : PositivityExt where eval {u α} zα pα e := do
let .app (.app _ (a : Q($α))) (b : Q(ℤ)) ← withReduc... | Mathlib.Tactic.Positivity.Basic.371_0.hOM93nWOlZMew4l | /-- The `positivity` extension which identifies expressions of the form `a ^ (b : ℤ)`,
such that `positivity` successfully recognises both `a` and `b`. -/
@[positivity (_ : α) ^ (_ : ℤ), Pow.pow _ (_ : ℤ)]
def evalZpow : PositivityExt where eval {u α} zα pα e | Mathlib_Tactic_Positivity_Basic |
u : Level
α : Q(Type u)
zα : Q(Zero «$α»)
pα : Q(PartialOrder «$α»)
e : Q(«$α»)
fn✝ : Expr
a : Q(«$α»)
b : Q(ℤ)
result : Strictness zα pα e
ra : Strictness zα pα a
ofNonneg : Q(0 ≤ «$a») → Q(LinearOrderedSemifield «$α») → MetaM (Strictness zα pα e) :=
fun pa _oα =>
let_fun zα_eq := (_ : «$zα» =Q CommMonoidWithZer... | /-
Copyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Heather Macbeth, Yaël Dillies
-/
import Std.Lean.Parser
import Mathlib.Data.Int.Order.Basic
import Mathlib.Data.Int.CharZero
import Mathlib.Data.Nat.Fa... | exact .positive (q(zpow_pos_of_pos $pa $b)) | /-- The `positivity` extension which identifies expressions of the form `a ^ (b : ℤ)`,
such that `positivity` successfully recognises both `a` and `b`. -/
@[positivity (_ : α) ^ (_ : ℤ), Pow.pow _ (_ : ℤ)]
def evalZpow : PositivityExt where eval {u α} zα pα e := do
let .app (.app _ (a : Q($α))) (b : Q(ℤ)) ← withReduc... | Mathlib.Tactic.Positivity.Basic.371_0.hOM93nWOlZMew4l | /-- The `positivity` extension which identifies expressions of the form `a ^ (b : ℤ)`,
such that `positivity` successfully recognises both `a` and `b`. -/
@[positivity (_ : α) ^ (_ : ℤ), Pow.pow _ (_ : ℤ)]
def evalZpow : PositivityExt where eval {u α} zα pα e | Mathlib_Tactic_Positivity_Basic |
ι : Type u_1
R : Type u_2
M : Type u_3
M₁ : Type u_4
M₂ : Type u_5
M₃ : Type u_6
M₄ : Type u_7
inst✝¹⁰ : CommSemiring R
inst✝⁹ : AddCommMonoid M
inst✝⁸ : AddCommMonoid M₁
inst✝⁷ : AddCommMonoid M₂
inst✝⁶ : AddCommMonoid M₃
inst✝⁵ : AddCommMonoid M₄
inst✝⁴ : Module R M
inst✝³ : Module R M₁
inst✝² : Module R M₂
inst✝¹ : ... | /-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.QuadraticForm.Basic
/-!
# Isometric linear maps
## Main definitions
* `QuadraticForm.Isometry`: `LinearMap`s which map between two different... | cases f | instance instLinearMapClass : LinearMapClass (Q₁ →qᵢ Q₂) R M₁ M₂ where
coe f := f.toLinearMap
coe_injective' f g h := by | Mathlib.LinearAlgebra.QuadraticForm.Isometry.43_0.HsLQNA3L13iskC0 | instance instLinearMapClass : LinearMapClass (Q₁ →qᵢ Q₂) R M₁ M₂ where
coe f | Mathlib_LinearAlgebra_QuadraticForm_Isometry |
case mk
ι : Type u_1
R : Type u_2
M : Type u_3
M₁ : Type u_4
M₂ : Type u_5
M₃ : Type u_6
M₄ : Type u_7
inst✝¹⁰ : CommSemiring R
inst✝⁹ : AddCommMonoid M
inst✝⁸ : AddCommMonoid M₁
inst✝⁷ : AddCommMonoid M₂
inst✝⁶ : AddCommMonoid M₃
inst✝⁵ : AddCommMonoid M₄
inst✝⁴ : Module R M
inst✝³ : Module R M₁
inst✝² : Module R M₂
i... | /-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.QuadraticForm.Basic
/-!
# Isometric linear maps
## Main definitions
* `QuadraticForm.Isometry`: `LinearMap`s which map between two different... | cases g | instance instLinearMapClass : LinearMapClass (Q₁ →qᵢ Q₂) R M₁ M₂ where
coe f := f.toLinearMap
coe_injective' f g h := by cases f; | Mathlib.LinearAlgebra.QuadraticForm.Isometry.43_0.HsLQNA3L13iskC0 | instance instLinearMapClass : LinearMapClass (Q₁ →qᵢ Q₂) R M₁ M₂ where
coe f | Mathlib_LinearAlgebra_QuadraticForm_Isometry |
case mk.mk
ι : Type u_1
R : Type u_2
M : Type u_3
M₁ : Type u_4
M₂ : Type u_5
M₃ : Type u_6
M₄ : Type u_7
inst✝¹⁰ : CommSemiring R
inst✝⁹ : AddCommMonoid M
inst✝⁸ : AddCommMonoid M₁
inst✝⁷ : AddCommMonoid M₂
inst✝⁶ : AddCommMonoid M₃
inst✝⁵ : AddCommMonoid M₄
inst✝⁴ : Module R M
inst✝³ : Module R M₁
inst✝² : Module R M... | /-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.QuadraticForm.Basic
/-!
# Isometric linear maps
## Main definitions
* `QuadraticForm.Isometry`: `LinearMap`s which map between two different... | congr | instance instLinearMapClass : LinearMapClass (Q₁ →qᵢ Q₂) R M₁ M₂ where
coe f := f.toLinearMap
coe_injective' f g h := by cases f; cases g; | Mathlib.LinearAlgebra.QuadraticForm.Isometry.43_0.HsLQNA3L13iskC0 | instance instLinearMapClass : LinearMapClass (Q₁ →qᵢ Q₂) R M₁ M₂ where
coe f | Mathlib_LinearAlgebra_QuadraticForm_Isometry |
case mk.mk.e_toLinearMap
ι : Type u_1
R : Type u_2
M : Type u_3
M₁ : Type u_4
M₂ : Type u_5
M₃ : Type u_6
M₄ : Type u_7
inst✝¹⁰ : CommSemiring R
inst✝⁹ : AddCommMonoid M
inst✝⁸ : AddCommMonoid M₁
inst✝⁷ : AddCommMonoid M₂
inst✝⁶ : AddCommMonoid M₃
inst✝⁵ : AddCommMonoid M₄
inst✝⁴ : Module R M
inst✝³ : Module R M₁
inst✝... | /-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.QuadraticForm.Basic
/-!
# Isometric linear maps
## Main definitions
* `QuadraticForm.Isometry`: `LinearMap`s which map between two different... | exact FunLike.coe_injective h | instance instLinearMapClass : LinearMapClass (Q₁ →qᵢ Q₂) R M₁ M₂ where
coe f := f.toLinearMap
coe_injective' f g h := by cases f; cases g; congr; | Mathlib.LinearAlgebra.QuadraticForm.Isometry.43_0.HsLQNA3L13iskC0 | instance instLinearMapClass : LinearMapClass (Q₁ →qᵢ Q₂) R M₁ M₂ where
coe f | Mathlib_LinearAlgebra_QuadraticForm_Isometry |
ι : Type u_1
R : Type u_2
M : Type u_3
M₁ : Type u_4
M₂ : Type u_5
M₃ : Type u_6
M₄ : Type u_7
inst✝¹⁰ : CommSemiring R
inst✝⁹ : AddCommMonoid M
inst✝⁸ : AddCommMonoid M₁
inst✝⁷ : AddCommMonoid M₂
inst✝⁶ : AddCommMonoid M₃
inst✝⁵ : AddCommMonoid M₄
inst✝⁴ : Module R M
inst✝³ : Module R M₁
inst✝² : Module R M₂
inst✝¹ : ... | /-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.QuadraticForm.Basic
/-!
# Isometric linear maps
## Main definitions
* `QuadraticForm.Isometry`: `LinearMap`s which map between two different... | rw [← f.map_app, ← g.map_app] | /-- The composition of two isometries between quadratic forms. -/
@[simps]
def comp (g : Q₂ →qᵢ Q₃) (f : Q₁ →qᵢ Q₂) : Q₁ →qᵢ Q₃ where
toFun x := g (f x)
map_app' x := by | Mathlib.LinearAlgebra.QuadraticForm.Isometry.76_0.HsLQNA3L13iskC0 | /-- The composition of two isometries between quadratic forms. -/
@[simps]
def comp (g : Q₂ →qᵢ Q₃) (f : Q₁ →qᵢ Q₂) : Q₁ →qᵢ Q₃ where
toFun x | Mathlib_LinearAlgebra_QuadraticForm_Isometry |
α : Type u_1
ι : Sort u_2
κ : ι → Sort u_3
A B : Set (Finset α)
s : Finset α
r : ℕ
⊢ Sized r ∅ | /-
Copyright (c) 2021 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.BigOperators.Basic
import Mathlib.Data.Nat.Interval
import Mathlib.Order.Antichain
#align_import data.f... | simp [Sized] | @[simp] lemma sized_empty : (∅ : Set (Finset α)).Sized r := by | Mathlib.Data.Finset.Slice.53_0.8sUozAlxGvTTEvp | @[simp] lemma sized_empty : (∅ : Set (Finset α)).Sized r | Mathlib_Data_Finset_Slice |
α : Type u_1
ι : Sort u_2
κ : ι → Sort u_3
A B : Set (Finset α)
s : Finset α
r : ℕ
⊢ Sized r {s} ↔ card s = r | /-
Copyright (c) 2021 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.BigOperators.Basic
import Mathlib.Data.Nat.Interval
import Mathlib.Order.Antichain
#align_import data.f... | simp [Sized] | @[simp] lemma sized_singleton : ({s} : Set (Finset α)).Sized r ↔ s.card = r := by | Mathlib.Data.Finset.Slice.54_0.8sUozAlxGvTTEvp | @[simp] lemma sized_singleton : ({s} : Set (Finset α)).Sized r ↔ s.card = r | Mathlib_Data_Finset_Slice |
α : Type u_1
ι : Sort u_2
κ : ι → Sort u_3
A B : Set (Finset α)
s : Finset α
r : ℕ
f : ι → Set (Finset α)
⊢ Sized r (⋃ i, f i) ↔ ∀ (i : ι), Sized r (f i) | /-
Copyright (c) 2021 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.BigOperators.Basic
import Mathlib.Data.Nat.Interval
import Mathlib.Order.Antichain
#align_import data.f... | simp_rw [Set.Sized, Set.mem_iUnion, forall_exists_index] | @[simp]
theorem sized_iUnion {f : ι → Set (Finset α)} : (⋃ i, f i).Sized r ↔ ∀ i, (f i).Sized r := by
| Mathlib.Data.Finset.Slice.65_0.8sUozAlxGvTTEvp | @[simp]
theorem sized_iUnion {f : ι → Set (Finset α)} : (⋃ i, f i).Sized r ↔ ∀ i, (f i).Sized r | Mathlib_Data_Finset_Slice |
α : Type u_1
ι : Sort u_2
κ : ι → Sort u_3
A B : Set (Finset α)
s : Finset α
r : ℕ
f : ι → Set (Finset α)
⊢ (∀ ⦃x : Finset α⦄ (x_1 : ι), x ∈ f x_1 → card x = r) ↔ ∀ (i : ι) ⦃x : Finset α⦄, x ∈ f i → card x = r | /-
Copyright (c) 2021 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.BigOperators.Basic
import Mathlib.Data.Nat.Interval
import Mathlib.Order.Antichain
#align_import data.f... | exact forall_swap | @[simp]
theorem sized_iUnion {f : ι → Set (Finset α)} : (⋃ i, f i).Sized r ↔ ∀ i, (f i).Sized r := by
simp_rw [Set.Sized, Set.mem_iUnion, forall_exists_index]
| Mathlib.Data.Finset.Slice.65_0.8sUozAlxGvTTEvp | @[simp]
theorem sized_iUnion {f : ι → Set (Finset α)} : (⋃ i, f i).Sized r ↔ ∀ i, (f i).Sized r | Mathlib_Data_Finset_Slice |
α : Type u_1
ι : Sort u_2
κ : ι → Sort u_3
A B : Set (Finset α)
s : Finset α
r : ℕ
f : (i : ι) → κ i → Set (Finset α)
⊢ Sized r (⋃ i, ⋃ j, f i j) ↔ ∀ (i : ι) (j : κ i), Sized r (f i j) | /-
Copyright (c) 2021 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.BigOperators.Basic
import Mathlib.Data.Nat.Interval
import Mathlib.Order.Antichain
#align_import data.f... | simp only [Set.sized_iUnion] | theorem sized_iUnion₂ {f : ∀ i, κ i → Set (Finset α)} :
(⋃ (i) (j), f i j).Sized r ↔ ∀ i j, (f i j).Sized r :=
by | Mathlib.Data.Finset.Slice.72_0.8sUozAlxGvTTEvp | theorem sized_iUnion₂ {f : ∀ i, κ i → Set (Finset α)} :
(⋃ (i) (j), f i j).Sized r ↔ ∀ i j, (f i j).Sized r | Mathlib_Data_Finset_Slice |
α : Type u_1
ι : Sort u_2
κ : ι → Sort u_3
inst✝ : Fintype α
𝒜 : Finset (Finset α)
s : Finset α
r : ℕ
A : Finset α
⊢ A ∈ 𝒜 → A ∈ powersetCard r univ ↔ A ∈ ↑𝒜 → card A = r | /-
Copyright (c) 2021 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.BigOperators.Basic
import Mathlib.Data.Nat.Interval
import Mathlib.Order.Antichain
#align_import data.f... | rw [mem_powersetCard_univ, mem_coe] | theorem subset_powersetCard_univ_iff : 𝒜 ⊆ powersetCard r univ ↔ (𝒜 : Set (Finset α)).Sized r :=
forall_congr' fun A => by | Mathlib.Data.Finset.Slice.109_0.8sUozAlxGvTTEvp | theorem subset_powersetCard_univ_iff : 𝒜 ⊆ powersetCard r univ ↔ (𝒜 : Set (Finset α)).Sized r | Mathlib_Data_Finset_Slice |
α : Type u_1
ι : Sort u_2
κ : ι → Sort u_3
inst✝ : Fintype α
𝒜 : Finset (Finset α)
s : Finset α
r : ℕ
h𝒜 : Set.Sized r ↑𝒜
⊢ card 𝒜 ≤ Nat.choose (Fintype.card α) r | /-
Copyright (c) 2021 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.BigOperators.Basic
import Mathlib.Data.Nat.Interval
import Mathlib.Order.Antichain
#align_import data.f... | rw [Fintype.card, ← card_powersetCard] | theorem _root_.Set.Sized.card_le (h𝒜 : (𝒜 : Set (Finset α)).Sized r) :
card 𝒜 ≤ (Fintype.card α).choose r := by
| Mathlib.Data.Finset.Slice.116_0.8sUozAlxGvTTEvp | theorem _root_.Set.Sized.card_le (h𝒜 : (𝒜 : Set (Finset α)).Sized r) :
card 𝒜 ≤ (Fintype.card α).choose r | Mathlib_Data_Finset_Slice |
α : Type u_1
ι : Sort u_2
κ : ι → Sort u_3
inst✝ : Fintype α
𝒜 : Finset (Finset α)
s : Finset α
r : ℕ
h𝒜 : Set.Sized r ↑𝒜
⊢ card 𝒜 ≤ card (powersetCard r univ) | /-
Copyright (c) 2021 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.BigOperators.Basic
import Mathlib.Data.Nat.Interval
import Mathlib.Order.Antichain
#align_import data.f... | exact card_le_of_subset (subset_powersetCard_univ_iff.mpr h𝒜) | theorem _root_.Set.Sized.card_le (h𝒜 : (𝒜 : Set (Finset α)).Sized r) :
card 𝒜 ≤ (Fintype.card α).choose r := by
rw [Fintype.card, ← card_powersetCard]
| Mathlib.Data.Finset.Slice.116_0.8sUozAlxGvTTEvp | theorem _root_.Set.Sized.card_le (h𝒜 : (𝒜 : Set (Finset α)).Sized r) :
card 𝒜 ≤ (Fintype.card α).choose r | Mathlib_Data_Finset_Slice |
α : Type u_1
ι : Sort u_2
κ : ι → Sort u_3
𝒜 : Finset (Finset α)
A A₁ A₂ : Finset α
r r₁ r₂ : ℕ
inst✝ : Fintype α
⊢ ∑ r in Iic (Fintype.card α), card (𝒜 # r) = card 𝒜 | /-
Copyright (c) 2021 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.BigOperators.Basic
import Mathlib.Data.Nat.Interval
import Mathlib.Order.Antichain
#align_import data.f... | letI := Classical.decEq α | @[simp]
theorem sum_card_slice : (∑ r in Iic (Fintype.card α), (𝒜 # r).card) = 𝒜.card := by
| Mathlib.Data.Finset.Slice.176_0.8sUozAlxGvTTEvp | @[simp]
theorem sum_card_slice : (∑ r in Iic (Fintype.card α), (𝒜 # r).card) = 𝒜.card | Mathlib_Data_Finset_Slice |
α : Type u_1
ι : Sort u_2
κ : ι → Sort u_3
𝒜 : Finset (Finset α)
A A₁ A₂ : Finset α
r r₁ r₂ : ℕ
inst✝ : Fintype α
this : DecidableEq α := Classical.decEq α
⊢ ∑ r in Iic (Fintype.card α), card (𝒜 # r) = card 𝒜 | /-
Copyright (c) 2021 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.BigOperators.Basic
import Mathlib.Data.Nat.Interval
import Mathlib.Order.Antichain
#align_import data.f... | rw [← card_biUnion, biUnion_slice] | @[simp]
theorem sum_card_slice : (∑ r in Iic (Fintype.card α), (𝒜 # r).card) = 𝒜.card := by
letI := Classical.decEq α
| Mathlib.Data.Finset.Slice.176_0.8sUozAlxGvTTEvp | @[simp]
theorem sum_card_slice : (∑ r in Iic (Fintype.card α), (𝒜 # r).card) = 𝒜.card | Mathlib_Data_Finset_Slice |
α : Type u_1
ι : Sort u_2
κ : ι → Sort u_3
𝒜 : Finset (Finset α)
A A₁ A₂ : Finset α
r r₁ r₂ : ℕ
inst✝ : Fintype α
this : DecidableEq α := Classical.decEq α
⊢ ∀ x ∈ Iic (Fintype.card α), ∀ y ∈ Iic (Fintype.card α), x ≠ y → Disjoint (𝒜 # x) (𝒜 # y) | /-
Copyright (c) 2021 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.BigOperators.Basic
import Mathlib.Data.Nat.Interval
import Mathlib.Order.Antichain
#align_import data.f... | exact Finset.pairwiseDisjoint_slice.subset (Set.subset_univ _) | @[simp]
theorem sum_card_slice : (∑ r in Iic (Fintype.card α), (𝒜 # r).card) = 𝒜.card := by
letI := Classical.decEq α
rw [← card_biUnion, biUnion_slice]
| Mathlib.Data.Finset.Slice.176_0.8sUozAlxGvTTEvp | @[simp]
theorem sum_card_slice : (∑ r in Iic (Fintype.card α), (𝒜 # r).card) = 𝒜.card | Mathlib_Data_Finset_Slice |
V : Type u
G : SimpleGraph V
P : Partition G
v : V
⊢ partOfVertex P v ∈ P.parts | /-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
#align_import combinatorics.simple_graph.partition from "leanprover-community/mathlib"@"2303b3e299f1c7... | obtain ⟨h, -⟩ := (P.isPartition.2 v).choose_spec.1 | theorem partOfVertex_mem (v : V) : P.partOfVertex v ∈ P.parts := by
| Mathlib.Combinatorics.SimpleGraph.Partition.88_0.83yZsHNZsmSPOSw | theorem partOfVertex_mem (v : V) : P.partOfVertex v ∈ P.parts | Mathlib_Combinatorics_SimpleGraph_Partition |
case intro
V : Type u
G : SimpleGraph V
P : Partition G
v : V
h : Exists.choose (_ : ∃! b x, v ∈ b) ∈ P.parts
⊢ partOfVertex P v ∈ P.parts | /-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
#align_import combinatorics.simple_graph.partition from "leanprover-community/mathlib"@"2303b3e299f1c7... | exact h | theorem partOfVertex_mem (v : V) : P.partOfVertex v ∈ P.parts := by
obtain ⟨h, -⟩ := (P.isPartition.2 v).choose_spec.1
| Mathlib.Combinatorics.SimpleGraph.Partition.88_0.83yZsHNZsmSPOSw | theorem partOfVertex_mem (v : V) : P.partOfVertex v ∈ P.parts | Mathlib_Combinatorics_SimpleGraph_Partition |
V : Type u
G : SimpleGraph V
P : Partition G
v : V
⊢ v ∈ partOfVertex P v | /-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
#align_import combinatorics.simple_graph.partition from "leanprover-community/mathlib"@"2303b3e299f1c7... | obtain ⟨⟨h1, h2⟩, _h3⟩ := (P.isPartition.2 v).choose_spec | theorem mem_partOfVertex (v : V) : v ∈ P.partOfVertex v := by
| Mathlib.Combinatorics.SimpleGraph.Partition.93_0.83yZsHNZsmSPOSw | theorem mem_partOfVertex (v : V) : v ∈ P.partOfVertex v | Mathlib_Combinatorics_SimpleGraph_Partition |
case intro.intro
V : Type u
G : SimpleGraph V
P : Partition G
v : V
_h3 : ∀ (y : Set V), (fun b => ∃! x, v ∈ b) y → y = Exists.choose (_ : ∃! b x, v ∈ b)
h1 : Exists.choose (_ : ∃! b x, v ∈ b) ∈ P.parts
h2 :
(fun x => v ∈ Exists.choose (_ : ∃! b x, v ∈ b)) h1 ∧
∀ (y : Exists.choose (_ : ∃! b x, v ∈ b) ∈ P.parts),... | /-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
#align_import combinatorics.simple_graph.partition from "leanprover-community/mathlib"@"2303b3e299f1c7... | exact h2.1 | theorem mem_partOfVertex (v : V) : v ∈ P.partOfVertex v := by
obtain ⟨⟨h1, h2⟩, _h3⟩ := (P.isPartition.2 v).choose_spec
| Mathlib.Combinatorics.SimpleGraph.Partition.93_0.83yZsHNZsmSPOSw | theorem mem_partOfVertex (v : V) : v ∈ P.partOfVertex v | Mathlib_Combinatorics_SimpleGraph_Partition |
V : Type u
G : SimpleGraph V
P : Partition G
v w : V
h : Adj G v w
⊢ partOfVertex P v ≠ partOfVertex P w | /-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
#align_import combinatorics.simple_graph.partition from "leanprover-community/mathlib"@"2303b3e299f1c7... | intro hn | theorem partOfVertex_ne_of_adj {v w : V} (h : G.Adj v w) : P.partOfVertex v ≠ P.partOfVertex w := by
| Mathlib.Combinatorics.SimpleGraph.Partition.98_0.83yZsHNZsmSPOSw | theorem partOfVertex_ne_of_adj {v w : V} (h : G.Adj v w) : P.partOfVertex v ≠ P.partOfVertex w | Mathlib_Combinatorics_SimpleGraph_Partition |
V : Type u
G : SimpleGraph V
P : Partition G
v w : V
h : Adj G v w
hn : partOfVertex P v = partOfVertex P w
⊢ False | /-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
#align_import combinatorics.simple_graph.partition from "leanprover-community/mathlib"@"2303b3e299f1c7... | have hw := P.mem_partOfVertex w | theorem partOfVertex_ne_of_adj {v w : V} (h : G.Adj v w) : P.partOfVertex v ≠ P.partOfVertex w := by
intro hn
| Mathlib.Combinatorics.SimpleGraph.Partition.98_0.83yZsHNZsmSPOSw | theorem partOfVertex_ne_of_adj {v w : V} (h : G.Adj v w) : P.partOfVertex v ≠ P.partOfVertex w | Mathlib_Combinatorics_SimpleGraph_Partition |
V : Type u
G : SimpleGraph V
P : Partition G
v w : V
h : Adj G v w
hn : partOfVertex P v = partOfVertex P w
hw : w ∈ partOfVertex P w
⊢ False | /-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
#align_import combinatorics.simple_graph.partition from "leanprover-community/mathlib"@"2303b3e299f1c7... | rw [← hn] at hw | theorem partOfVertex_ne_of_adj {v w : V} (h : G.Adj v w) : P.partOfVertex v ≠ P.partOfVertex w := by
intro hn
have hw := P.mem_partOfVertex w
| Mathlib.Combinatorics.SimpleGraph.Partition.98_0.83yZsHNZsmSPOSw | theorem partOfVertex_ne_of_adj {v w : V} (h : G.Adj v w) : P.partOfVertex v ≠ P.partOfVertex w | Mathlib_Combinatorics_SimpleGraph_Partition |
V : Type u
G : SimpleGraph V
P : Partition G
v w : V
h : Adj G v w
hn : partOfVertex P v = partOfVertex P w
hw : w ∈ partOfVertex P v
⊢ False | /-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
#align_import combinatorics.simple_graph.partition from "leanprover-community/mathlib"@"2303b3e299f1c7... | exact P.independent _ (P.partOfVertex_mem v) (P.mem_partOfVertex v) hw (G.ne_of_adj h) h | theorem partOfVertex_ne_of_adj {v w : V} (h : G.Adj v w) : P.partOfVertex v ≠ P.partOfVertex w := by
intro hn
have hw := P.mem_partOfVertex w
rw [← hn] at hw
| Mathlib.Combinatorics.SimpleGraph.Partition.98_0.83yZsHNZsmSPOSw | theorem partOfVertex_ne_of_adj {v w : V} (h : G.Adj v w) : P.partOfVertex v ≠ P.partOfVertex w | Mathlib_Combinatorics_SimpleGraph_Partition |
V : Type u
G : SimpleGraph V
P : Partition G
v✝ w✝ : V
hvw : Adj G v✝ w✝
⊢ (fun v => { val := partOfVertex P v, property := (_ : partOfVertex P v ∈ P.parts) }) v✝ ≠
(fun v => { val := partOfVertex P v, property := (_ : partOfVertex P v ∈ P.parts) }) w✝ | /-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
#align_import combinatorics.simple_graph.partition from "leanprover-community/mathlib"@"2303b3e299f1c7... | rw [Ne.def, Subtype.mk_eq_mk] | /-- Create a coloring using the parts themselves as the colors.
Each vertex is colored by the part it's contained in. -/
def toColoring : G.Coloring P.parts :=
Coloring.mk (fun v ↦ ⟨P.partOfVertex v, P.partOfVertex_mem v⟩) fun hvw ↦ by
| Mathlib.Combinatorics.SimpleGraph.Partition.105_0.83yZsHNZsmSPOSw | /-- Create a coloring using the parts themselves as the colors.
Each vertex is colored by the part it's contained in. -/
def toColoring : G.Coloring P.parts | Mathlib_Combinatorics_SimpleGraph_Partition |
V : Type u
G : SimpleGraph V
P : Partition G
v✝ w✝ : V
hvw : Adj G v✝ w✝
⊢ ¬partOfVertex P v✝ = partOfVertex P w✝ | /-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
#align_import combinatorics.simple_graph.partition from "leanprover-community/mathlib"@"2303b3e299f1c7... | exact P.partOfVertex_ne_of_adj hvw | /-- Create a coloring using the parts themselves as the colors.
Each vertex is colored by the part it's contained in. -/
def toColoring : G.Coloring P.parts :=
Coloring.mk (fun v ↦ ⟨P.partOfVertex v, P.partOfVertex_mem v⟩) fun hvw ↦ by
rw [Ne.def, Subtype.mk_eq_mk]
| Mathlib.Combinatorics.SimpleGraph.Partition.105_0.83yZsHNZsmSPOSw | /-- Create a coloring using the parts themselves as the colors.
Each vertex is colored by the part it's contained in. -/
def toColoring : G.Coloring P.parts | Mathlib_Combinatorics_SimpleGraph_Partition |
V : Type u
G : SimpleGraph V
α : Type v
C : Coloring G α
⊢ ∀ s ∈ colorClasses C, IsAntichain G.Adj s | /-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
#align_import combinatorics.simple_graph.partition from "leanprover-community/mathlib"@"2303b3e299f1c7... | rintro s ⟨c, rfl⟩ | /-- Creates a partition from a coloring. -/
@[simps]
def Coloring.toPartition {α : Type v} (C : G.Coloring α) : G.Partition
where
parts := C.colorClasses
isPartition := C.colorClasses_isPartition
independent := by
| Mathlib.Combinatorics.SimpleGraph.Partition.126_0.83yZsHNZsmSPOSw | /-- Creates a partition from a coloring. -/
@[simps]
def Coloring.toPartition {α : Type v} (C : G.Coloring α) : G.Partition
where
parts | Mathlib_Combinatorics_SimpleGraph_Partition |
case intro
V : Type u
G : SimpleGraph V
α : Type v
C : Coloring G α
c : V
⊢ IsAntichain G.Adj {x | Setoid.Rel (Setoid.ker ⇑C) x c} | /-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
#align_import combinatorics.simple_graph.partition from "leanprover-community/mathlib"@"2303b3e299f1c7... | apply C.color_classes_independent | /-- Creates a partition from a coloring. -/
@[simps]
def Coloring.toPartition {α : Type v} (C : G.Coloring α) : G.Partition
where
parts := C.colorClasses
isPartition := C.colorClasses_isPartition
independent := by
rintro s ⟨c, rfl⟩
| Mathlib.Combinatorics.SimpleGraph.Partition.126_0.83yZsHNZsmSPOSw | /-- Creates a partition from a coloring. -/
@[simps]
def Coloring.toPartition {α : Type v} (C : G.Coloring α) : G.Partition
where
parts | Mathlib_Combinatorics_SimpleGraph_Partition |
V : Type u
G : SimpleGraph V
n : ℕ
⊢ Partitionable G n ↔ Colorable G n | /-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
#align_import combinatorics.simple_graph.partition from "leanprover-community/mathlib"@"2303b3e299f1c7... | constructor | theorem partitionable_iff_colorable {n : ℕ} : G.Partitionable n ↔ G.Colorable n := by
| Mathlib.Combinatorics.SimpleGraph.Partition.141_0.83yZsHNZsmSPOSw | theorem partitionable_iff_colorable {n : ℕ} : G.Partitionable n ↔ G.Colorable n | Mathlib_Combinatorics_SimpleGraph_Partition |
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