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 |
|---|---|---|---|---|---|---|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
t : α → Set β
s₁ s₂ : Set α
⊢ ⋃ x ∈ s₁, t x ⊆ ⋃ x ∈ s₂ ∪ s₁ \ s₂, t x | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | apply biUnion_subset_biUnion_left | theorem biUnion_diff_biUnion_subset (s₁ s₂ : Set α) :
((⋃ x ∈ s₁, t x) \ ⋃ x ∈ s₂, t x) ⊆ ⋃ x ∈ s₁ \ s₂, t x := by
simp only [diff_subset_iff, ← biUnion_union]
| Mathlib.Data.Set.Lattice.2291_0.5mONj49h3SYSDwc | theorem biUnion_diff_biUnion_subset (s₁ s₂ : Set α) :
((⋃ x ∈ s₁, t x) \ ⋃ x ∈ s₂, t x) ⊆ ⋃ x ∈ s₁ \ s₂, t x | Mathlib_Data_Set_Lattice |
case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
t : α → Set β
s₁ s₂ : Set α
⊢ s₁ ⊆ s₂ ∪ s₁ \ s₂ | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | rw [union_diff_self] | theorem biUnion_diff_biUnion_subset (s₁ s₂ : Set α) :
((⋃ x ∈ s₁, t x) \ ⋃ x ∈ s₂, t x) ⊆ ⋃ x ∈ s₁ \ s₂, t x := by
simp only [diff_subset_iff, ← biUnion_union]
apply biUnion_subset_biUnion_left
| Mathlib.Data.Set.Lattice.2291_0.5mONj49h3SYSDwc | theorem biUnion_diff_biUnion_subset (s₁ s₂ : Set α) :
((⋃ x ∈ s₁, t x) \ ⋃ x ∈ s₂, t x) ⊆ ⋃ x ∈ s₁ \ s₂, t x | Mathlib_Data_Set_Lattice |
case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
t : α → Set β
s₁ s₂ : Set α
⊢ s₁ ⊆ s₂ ∪ s₁ | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | apply subset_union_right | theorem biUnion_diff_biUnion_subset (s₁ s₂ : Set α) :
((⋃ x ∈ s₁, t x) \ ⋃ x ∈ s₂, t x) ⊆ ⋃ x ∈ s₁ \ s₂, t x := by
simp only [diff_subset_iff, ← biUnion_union]
apply biUnion_subset_biUnion_left
rw [union_diff_self]
| Mathlib.Data.Set.Lattice.2291_0.5mONj49h3SYSDwc | theorem biUnion_diff_biUnion_subset (s₁ s₂ : Set α) :
((⋃ x ∈ s₁, t x) \ ⋃ x ∈ s₂, t x) ⊆ ⋃ x ∈ s₁ \ s₂, t x | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
t : α → Set β
b : β
hb : b ∈ ⋃ i, t i
⊢ ∃ a, b ∈ t a | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simpa using hb | theorem sigmaToiUnion_surjective : Surjective (sigmaToiUnion t)
| ⟨b, hb⟩ =>
have : ∃ a, b ∈ t a := by | Mathlib.Data.Set.Lattice.2305_0.5mONj49h3SYSDwc | theorem sigmaToiUnion_surjective : Surjective (sigmaToiUnion t)
| ⟨b, hb⟩ =>
have : ∃ a, b ∈ t a | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
t : α → Set β
h : ∀ (i j : α), i ≠ j → Disjoint (t i) (t j)
a₁ : α
b₁ : β
h₁ : b₁ ∈ t a₁
a₂ : α
b₂ : β
h₂ : b₂ ∈ t a₂
eq :
sigmaToiUnion t { fst := a₁, snd := { val ... | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | subst b_eq | theorem sigmaToiUnion_injective (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
Injective (sigmaToiUnion t)
| ⟨a₁, b₁, h₁⟩, ⟨a₂, b₂, h₂⟩, eq =>
have b_eq : b₁ = b₂ := congr_arg Subtype.val eq
have a_eq : a₁ = a₂ :=
by_contradiction fun ne =>
have : b₁ ∈ t a₁ ∩ t a₂ := ⟨h₁, b_eq.symm ▸ h₂⟩
... | Mathlib.Data.Set.Lattice.2312_0.5mONj49h3SYSDwc | theorem sigmaToiUnion_injective (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
Injective (sigmaToiUnion t)
| ⟨a₁, b₁, h₁⟩, ⟨a₂, b₂, h₂⟩, eq =>
have b_eq : b₁ = b₂ | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
t : α → Set β
h : ∀ (i j : α), i ≠ j → Disjoint (t i) (t j)
a₁ : α
b₁ : β
h₁ : b₁ ∈ t a₁
a₂ : α
a_eq : a₁ = a₂
h₂ : b₁ ∈ t a₂
eq :
sigmaToiUnion t { fst := a₁, snd :... | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | subst a_eq | theorem sigmaToiUnion_injective (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
Injective (sigmaToiUnion t)
| ⟨a₁, b₁, h₁⟩, ⟨a₂, b₂, h₂⟩, eq =>
have b_eq : b₁ = b₂ := congr_arg Subtype.val eq
have a_eq : a₁ = a₂ :=
by_contradiction fun ne =>
have : b₁ ∈ t a₁ ∩ t a₂ := ⟨h₁, b_eq.symm ▸ h₂⟩
... | Mathlib.Data.Set.Lattice.2312_0.5mONj49h3SYSDwc | theorem sigmaToiUnion_injective (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
Injective (sigmaToiUnion t)
| ⟨a₁, b₁, h₁⟩, ⟨a₂, b₂, h₂⟩, eq =>
have b_eq : b₁ = b₂ | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
t : α → Set β
h : ∀ (i j : α), i ≠ j → Disjoint (t i) (t j)
a₁ : α
b₁ : β
h₁ h₂ : b₁ ∈ t a₁
eq :
sigmaToiUnion t { fst := a₁, snd := { val := b₁, property := h₁ } } ... | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | rfl | theorem sigmaToiUnion_injective (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
Injective (sigmaToiUnion t)
| ⟨a₁, b₁, h₁⟩, ⟨a₂, b₂, h₂⟩, eq =>
have b_eq : b₁ = b₂ := congr_arg Subtype.val eq
have a_eq : a₁ = a₂ :=
by_contradiction fun ne =>
have : b₁ ∈ t a₁ ∩ t a₂ := ⟨h₁, b_eq.symm ▸ h₂⟩
... | Mathlib.Data.Set.Lattice.2312_0.5mONj49h3SYSDwc | theorem sigmaToiUnion_injective (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
Injective (sigmaToiUnion t)
| ⟨a₁, b₁, h₁⟩, ⟨a₂, b₂, h₂⟩, eq =>
have b_eq : b₁ = b₂ | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
inst✝ : CompleteLattice β
s : ι → Set α
f : α → β
⊢ ⨆ a ∈ ⋃ i, s i, f a = ⨆ i, ⨆ a ∈ s i, f a | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | rw [iSup_comm] | theorem iSup_iUnion (s : ι → Set α) (f : α → β) : ⨆ a ∈ ⋃ i, s i, f a = ⨆ (i) (a ∈ s i), f a := by
| Mathlib.Data.Set.Lattice.2374_0.5mONj49h3SYSDwc | theorem iSup_iUnion (s : ι → Set α) (f : α → β) : ⨆ a ∈ ⋃ i, s i, f a = ⨆ (i) (a ∈ s i), f a | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
inst✝ : CompleteLattice β
s : ι → Set α
f : α → β
⊢ ⨆ a ∈ ⋃ i, s i, f a = ⨆ j, ⨆ i, ⨆ (_ : j ∈ s i), f j | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp_rw [mem_iUnion, iSup_exists] | theorem iSup_iUnion (s : ι → Set α) (f : α → β) : ⨆ a ∈ ⋃ i, s i, f a = ⨆ (i) (a ∈ s i), f a := by
rw [iSup_comm]
| Mathlib.Data.Set.Lattice.2374_0.5mONj49h3SYSDwc | theorem iSup_iUnion (s : ι → Set α) (f : α → β) : ⨆ a ∈ ⋃ i, s i, f a = ⨆ (i) (a ∈ s i), f a | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
inst✝ : CompleteLattice β
s : Set (Set β)
⊢ sSup (⋃₀ s) = ⨆ t ∈ s, sSup t | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp only [sUnion_eq_biUnion, sSup_eq_iSup, iSup_iUnion] | theorem sSup_sUnion (s : Set (Set β)) : sSup (⋃₀ s) = ⨆ t ∈ s, sSup t := by
| Mathlib.Data.Set.Lattice.2383_0.5mONj49h3SYSDwc | theorem sSup_sUnion (s : Set (Set β)) : sSup (⋃₀ s) = ⨆ t ∈ s, sSup t | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
inst✝ : CompleteLattice β
S : Set (Set α)
f : α → β
⊢ ⨆ x ∈ ⋃₀ S, f x = ⨆ s ∈ S, ⨆ x ∈ s, f x | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | rw [sUnion_eq_iUnion, iSup_iUnion, ← iSup_subtype''] | lemma iSup_sUnion (S : Set (Set α)) (f : α → β) :
(⨆ x ∈ ⋃₀ S, f x) = ⨆ (s ∈ S) (x ∈ s), f x := by
| Mathlib.Data.Set.Lattice.2391_0.5mONj49h3SYSDwc | lemma iSup_sUnion (S : Set (Set α)) (f : α → β) :
(⨆ x ∈ ⋃₀ S, f x) = ⨆ (s ∈ S) (x ∈ s), f x | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
inst✝ : CompleteLattice β
S : Set (Set α)
f : α → β
⊢ ⨅ x ∈ ⋃₀ S, f x = ⨅ s ∈ S, ⨅ x ∈ s, f x | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | rw [sUnion_eq_iUnion, iInf_iUnion, ← iInf_subtype''] | lemma iInf_sUnion (S : Set (Set α)) (f : α → β) :
(⨅ x ∈ ⋃₀ S, f x) = ⨅ (s ∈ S) (x ∈ s), f x := by
| Mathlib.Data.Set.Lattice.2395_0.5mONj49h3SYSDwc | lemma iInf_sUnion (S : Set (Set α)) (f : α → β) :
(⨅ x ∈ ⋃₀ S, f x) = ⨅ (s ∈ S) (x ∈ s), f x | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
inst✝ : CompleteLattice β
S : Set (Set α)
p : α → Prop
⊢ (∀ x ∈ ⋃₀ S, p x) ↔ ∀ s ∈ S, ∀ x ∈ s, p x | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp_rw [← iInf_Prop_eq, iInf_sUnion] | lemma forall_sUnion {p : α → Prop} :
(∀ x ∈ ⋃₀ S, p x) ↔ ∀ s ∈ S, ∀ x ∈ s, p x := by
| Mathlib.Data.Set.Lattice.2399_0.5mONj49h3SYSDwc | lemma forall_sUnion {p : α → Prop} :
(∀ x ∈ ⋃₀ S, p x) ↔ ∀ s ∈ S, ∀ x ∈ s, p x | Mathlib_Data_Set_Lattice |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
inst✝ : CompleteLattice β
S : Set (Set α)
p : α → Prop
⊢ (∃ x ∈ ⋃₀ S, p x) ↔ ∃ s ∈ S, ∃ x ∈ s, p x | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#ali... | simp_rw [← exists_prop, ← iSup_Prop_eq, iSup_sUnion] | lemma exists_sUnion {p : α → Prop} :
(∃ x ∈ ⋃₀ S, p x) ↔ ∃ s ∈ S, ∃ x ∈ s, p x := by
| Mathlib.Data.Set.Lattice.2403_0.5mONj49h3SYSDwc | lemma exists_sUnion {p : α → Prop} :
(∃ x ∈ ⋃₀ S, p x) ↔ ∃ s ∈ S, ∃ x ∈ s, p x | Mathlib_Data_Set_Lattice |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasShift C ℤ
X : C
X✝ Y✝ : Triangle C
f : X✝ ⟶ Y✝
⊢ (Triangle.rotate X✝).mor₃ ≫ (shiftFunctor C 1).map f.hom₂ = (shiftFunctor C 1).map f.hom₁ ≫ (Triangle.rotate Y✝).mor₃ | /-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Triangulated.Basic
#align_import category_theory.triangulated.rotate from "leanpr... | dsimp | /-- Rotating triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def rotate : Triangle C ⥤ Triangle C
where
obj := Triangle.rotate
map f :=
{ hom₁ := f.hom₂
hom₂ := f.hom₃
hom₃ := f.hom₁⟦1⟧'
comm₃ := by
| Mathlib.CategoryTheory.Triangulated.Rotate.86_0.sGRpfSsY1fG2rGq | /-- Rotating triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def rotate : Triangle C ⥤ Triangle C
where
obj | Mathlib_CategoryTheory_Triangulated_Rotate |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasShift C ℤ
X : C
X✝ Y✝ : Triangle C
f : X✝ ⟶ Y✝
⊢ (-(shiftFunctor C 1).map X✝.mor₁) ≫ (shiftFunctor C 1).map f.hom₂ =
(shiftFunctor C 1).map f.hom₁ ≫ (-(shiftFunctor C 1).map Y✝.mor₁) | /-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Triangulated.Basic
#align_import category_theory.triangulated.rotate from "leanpr... | simp only [comp_neg, neg_comp, ← Functor.map_comp, f.comm₁] | /-- Rotating triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def rotate : Triangle C ⥤ Triangle C
where
obj := Triangle.rotate
map f :=
{ hom₁ := f.hom₂
hom₂ := f.hom₃
hom₃ := f.hom₁⟦1⟧'
comm₃ := by
dsimp
| Mathlib.CategoryTheory.Triangulated.Rotate.86_0.sGRpfSsY1fG2rGq | /-- Rotating triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def rotate : Triangle C ⥤ Triangle C
where
obj | Mathlib_CategoryTheory_Triangulated_Rotate |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasShift C ℤ
X : C
X✝ Y✝ : Triangle C
f : X✝ ⟶ Y✝
⊢ (Triangle.invRotate X✝).mor₁ ≫ f.hom₁ = (shiftFunctor C (-1)).map f.hom₃ ≫ (Triangle.invRotate Y✝).mor₁ | /-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Triangulated.Basic
#align_import category_theory.triangulated.rotate from "leanpr... | dsimp | /-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj := Triangle.invRotate
map f :=
{ hom₁ := f.hom₃⟦-1⟧'
hom₂ := f.hom₁
hom₃ := f.hom₂
comm₁ := by
| Mathlib.CategoryTheory.Triangulated.Rotate.101_0.sGRpfSsY1fG2rGq | /-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj | Mathlib_CategoryTheory_Triangulated_Rotate |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasShift C ℤ
X : C
X✝ Y✝ : Triangle C
f : X✝ ⟶ Y✝
⊢ (-(shiftFunctor C (-1)).map X✝.mor₃ ≫ (shiftFunctorCompIsoId C 1 (-1) (_ : 1 + -1 = 0)).hom.app X✝.obj₁) ≫ f.hom₁ =
(shiftFunctor C (-1)).map f.hom₃ ≫
(-(shiftFunctor C (-1)).map Y✝.mor₃ ≫ ... | /-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Triangulated.Basic
#align_import category_theory.triangulated.rotate from "leanpr... | simp only [neg_comp, assoc, comp_neg, neg_inj, ← Functor.map_comp_assoc, ← f.comm₃] | /-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj := Triangle.invRotate
map f :=
{ hom₁ := f.hom₃⟦-1⟧'
hom₂ := f.hom₁
hom₃ := f.hom₂
comm₁ := by
dsimp
| Mathlib.CategoryTheory.Triangulated.Rotate.101_0.sGRpfSsY1fG2rGq | /-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj | Mathlib_CategoryTheory_Triangulated_Rotate |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasShift C ℤ
X : C
X✝ Y✝ : Triangle C
f : X✝ ⟶ Y✝
⊢ (shiftFunctor C (-1)).map X✝.mor₃ ≫ (shiftFunctorCompIsoId C 1 (-1) (_ : 1 + -1 = 0)).hom.app X✝.obj₁ ≫ f.hom₁ =
(shiftFunctor C (-1)).map (X✝.mor₃ ≫ (shiftFunctor C 1).map f.hom₁) ≫
(shift... | /-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Triangulated.Basic
#align_import category_theory.triangulated.rotate from "leanpr... | rw [Functor.map_comp, assoc] | /-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj := Triangle.invRotate
map f :=
{ hom₁ := f.hom₃⟦-1⟧'
hom₂ := f.hom₁
hom₃ := f.hom₂
comm₁ := by
dsimp
simp only [neg_comp, assoc,... | Mathlib.CategoryTheory.Triangulated.Rotate.101_0.sGRpfSsY1fG2rGq | /-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj | Mathlib_CategoryTheory_Triangulated_Rotate |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasShift C ℤ
X : C
X✝ Y✝ : Triangle C
f : X✝ ⟶ Y✝
⊢ (shiftFunctor C (-1)).map X✝.mor₃ ≫ (shiftFunctorCompIsoId C 1 (-1) (_ : 1 + -1 = 0)).hom.app X✝.obj₁ ≫ f.hom₁ =
(shiftFunctor C (-1)).map X✝.mor₃ ≫
(shiftFunctor C (-1)).map ((shiftFunctor... | /-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Triangulated.Basic
#align_import category_theory.triangulated.rotate from "leanpr... | erw [← NatTrans.naturality] | /-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj := Triangle.invRotate
map f :=
{ hom₁ := f.hom₃⟦-1⟧'
hom₂ := f.hom₁
hom₃ := f.hom₂
comm₁ := by
dsimp
simp only [neg_comp, assoc,... | Mathlib.CategoryTheory.Triangulated.Rotate.101_0.sGRpfSsY1fG2rGq | /-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj | Mathlib_CategoryTheory_Triangulated_Rotate |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasShift C ℤ
X : C
X✝ Y✝ : Triangle C
f : X✝ ⟶ Y✝
⊢ (shiftFunctor C (-1)).map X✝.mor₃ ≫
(shiftFunctor C 1 ⋙ shiftFunctor C (-1)).map f.1 ≫
(shiftFunctorCompIsoId C 1 (-1) (_ : 1 + -1 = 0)).hom.app Y✝.obj₁ =
(shiftFunctor C (-1)).map ... | /-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Triangulated.Basic
#align_import category_theory.triangulated.rotate from "leanpr... | rfl | /-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj := Triangle.invRotate
map f :=
{ hom₁ := f.hom₃⟦-1⟧'
hom₂ := f.hom₁
hom₃ := f.hom₂
comm₁ := by
dsimp
simp only [neg_comp, assoc,... | Mathlib.CategoryTheory.Triangulated.Rotate.101_0.sGRpfSsY1fG2rGq | /-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj | Mathlib_CategoryTheory_Triangulated_Rotate |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasShift C ℤ
X : C
X✝ Y✝ : Triangle C
f : X✝ ⟶ Y✝
⊢ (Triangle.invRotate X✝).mor₃ ≫ (shiftFunctor C 1).map ((shiftFunctor C (-1)).map f.hom₃) =
f.hom₂ ≫ (Triangle.invRotate Y✝).mor₃ | /-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Triangulated.Basic
#align_import category_theory.triangulated.rotate from "leanpr... | erw [← reassoc_of% f.comm₂, Category.assoc, ← NatTrans.naturality] | /-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj := Triangle.invRotate
map f :=
{ hom₁ := f.hom₃⟦-1⟧'
hom₂ := f.hom₁
hom₃ := f.hom₂
comm₁ := by
dsimp
simp only [neg_comp, assoc,... | Mathlib.CategoryTheory.Triangulated.Rotate.101_0.sGRpfSsY1fG2rGq | /-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj | Mathlib_CategoryTheory_Triangulated_Rotate |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasShift C ℤ
X : C
X✝ Y✝ : Triangle C
f : X✝ ⟶ Y✝
⊢ X✝.mor₂ ≫ (𝟭 C).map f.hom₃ ≫ (shiftEquiv C 1).counitIso.inv.app Y✝.obj₃ =
X✝.mor₂ ≫ f.hom₃ ≫ (shiftEquiv C 1).counitIso.inv.app Y✝.3 | /-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Triangulated.Basic
#align_import category_theory.triangulated.rotate from "leanpr... | rfl | /-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj := Triangle.invRotate
map f :=
{ hom₁ := f.hom₃⟦-1⟧'
hom₂ := f.hom₁
hom₃ := f.hom₂
comm₁ := by
dsimp
simp only [neg_comp, assoc,... | Mathlib.CategoryTheory.Triangulated.Rotate.101_0.sGRpfSsY1fG2rGq | /-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj | Mathlib_CategoryTheory_Triangulated_Rotate |
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
X : C
inst✝ : ∀ (n : ℤ), Functor.Additive (shiftFunctor C n)
⊢ IsEquivalence (rotate C) | /-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Triangulated.Basic
#align_import category_theory.triangulated.rotate from "leanpr... | change IsEquivalence (triangleRotation C).functor | instance : IsEquivalence (rotate C) := by
| Mathlib.CategoryTheory.Triangulated.Rotate.157_0.sGRpfSsY1fG2rGq | instance : IsEquivalence (rotate C) | Mathlib_CategoryTheory_Triangulated_Rotate |
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
X : C
inst✝ : ∀ (n : ℤ), Functor.Additive (shiftFunctor C n)
⊢ IsEquivalence (triangleRotation C).functor | /-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Triangulated.Basic
#align_import category_theory.triangulated.rotate from "leanpr... | infer_instance | instance : IsEquivalence (rotate C) := by
change IsEquivalence (triangleRotation C).functor
| Mathlib.CategoryTheory.Triangulated.Rotate.157_0.sGRpfSsY1fG2rGq | instance : IsEquivalence (rotate C) | Mathlib_CategoryTheory_Triangulated_Rotate |
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
X : C
inst✝ : ∀ (n : ℤ), Functor.Additive (shiftFunctor C n)
⊢ IsEquivalence (invRotate C) | /-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Triangulated.Basic
#align_import category_theory.triangulated.rotate from "leanpr... | change IsEquivalence (triangleRotation C).inverse | instance : IsEquivalence (invRotate C) := by
| Mathlib.CategoryTheory.Triangulated.Rotate.161_0.sGRpfSsY1fG2rGq | instance : IsEquivalence (invRotate C) | Mathlib_CategoryTheory_Triangulated_Rotate |
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
X : C
inst✝ : ∀ (n : ℤ), Functor.Additive (shiftFunctor C n)
⊢ IsEquivalence (triangleRotation C).inverse | /-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Triangulated.Basic
#align_import category_theory.triangulated.rotate from "leanpr... | infer_instance | instance : IsEquivalence (invRotate C) := by
change IsEquivalence (triangleRotation C).inverse
| Mathlib.CategoryTheory.Triangulated.Rotate.161_0.sGRpfSsY1fG2rGq | instance : IsEquivalence (invRotate C) | Mathlib_CategoryTheory_Triangulated_Rotate |
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
⊢ DevosMulRel x y ↔
card (x.1 * x.2) < card (y.1 * y.2) ∨
card (x.1 * x.2) = card (y.1 * y.2) ∧ card y.1 + card y.2 < card x.1 + card x.2 ∨
card (x.1 * x.2) = card (y.1 * y.2) ∧ card x.1 + card x.2 = card y.1 ... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | simp [DevosMulRel, Prod.lex_iff, and_or_left] | @[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.77_0.yGTPJO6UphimMFs | @[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
⊢ Set.WellFoundedOn {x | Finset.Nonempty x.1 ∧ Finset.Nonempty x.2} DevosMulRel | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn | @[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
| Mathlib.Combinatorics.SetFamily.CauchyDavenport.98_0.yGTPJO6UphimMFs | @[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
n : ℕ
⊢ Set.WellFoundedOn ({x | Finset.Nonempty x.1 ∧ Finset.Nonempty x.2} ∩ (fun x => card (x.1 * x.2)) ⁻¹' {n})
((fun x x_1 => x > x_1) on fun x => card x.1 + card x.2) | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2 | @[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_l... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.98_0.yGTPJO6UphimMFs | @[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
⊢ min (minOrder α) ↑(card s + card t - 1) ≤ ↑(card (s * t)) | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | set x := (s, t) with hx | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x✝ y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
x : Finset α × Finset α := (s, t)
hx : x = (s, t)
⊢ min (minOrder α) ↑(card s + card t - 1) ≤ ↑(card (s * t)) | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | clear_value x | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x✝ y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
x : Finset α × Finset α
hx : x = (s, t)
⊢ min (minOrder α) ↑(card s + card t - 1) ≤ ↑(card (s * t)) | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | simp only [Prod.ext_iff] at hx | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x✝ y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
x : Finset α × Finset α
hx : x.1 = s ∧ x.2 = t
⊢ min (minOrder α) ↑(card s + card t - 1) ≤ ↑(card (s * t)) | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | obtain ⟨rfl, rfl⟩ := hx | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
case intro
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x✝ y x : Finset α × Finset α
hs : Finset.Nonempty x.1
ht : Finset.Nonempty x.2
⊢ min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ ↑(card (x.1 * x.2)) | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _ | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
case intro
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x✝ y x : Finset α × Finset α
hs : Finset.Nonempty x.1
ht : Finset.Nonempty x.2
⊢ ∀ y ∈ {x | Finset.Nonempty x.1 ∧ Finset.Nonempty x.2},
(∀ z ∈ {x | Finset.Nonempty x.1 ∧ Finset.Nonempty x.2},
DevosMulRel z y → (fun x => min (minOrder α) ↑(card x... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | clear! x | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
case intro
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
⊢ ∀ y ∈ {x | Finset.Nonempty x.1 ∧ Finset.Nonempty x.2},
(∀ z ∈ {x | Finset.Nonempty x.1 ∧ Finset.Nonempty x.2},
DevosMulRel z y → (fun x => min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ ↑(card (x.1 * x.2))) z) →
... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | rintro ⟨s, t⟩ ⟨hs, ht⟩ ih | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
case intro.mk.intro
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty (s, t).1
ht : Finset.Nonempty (s, t).2
ih :
∀ z ∈ {x | Finset.Nonempty x.1 ∧ Finset.Nonempty x.2},
DevosMulRel z (s, t) → (fun x => min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ ↑(... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at * | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
case intro.mk.intro
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | obtain hts | hst := lt_or_le t.card s.card | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
case intro.mk.intro.inl
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ car... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and]) | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a * b) + 1
hts : card... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | simpa only [← mul_inv_rev, add_comm, card_inv, true_and] | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
case intro.mk.intro.inr
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ car... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
case intro.mk.intro.inr.inl.intro
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
t : Finset α
ht : Finset.Nonempty t
a : α
hs : Finset.Nonempty {a}
ih :
∀ (a_1 b : Finset α),
Finset.Nonempty a_1 →
Finset.Nonempty b →
DevosMulRel (a_1, b) ({a}, t) → minOrder α ≤ ↑(card ... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | simp [add_comm] | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
case intro.mk.intro.inr.inr.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a *... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩ | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a * b) + 1
hst : card... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | simp | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
case intro.mk.intro.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | obtain hsg | hsg := eq_or_ne (op g • s) s | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
case intro.mk.intro.inr.inr.intro.intro.intro.intro.intro.intro.inl
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rw... | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a * b) + 1
hst : card... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_ | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
case refine_1
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a * b) ... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | rw [← hsg, coe_smul_finset, smul_comm] | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
case refine_1
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a * b) ... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | exact Set.smul_mem_smul_set hc | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
case refine_2
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a * b) ... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | simp only | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
case refine_2
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a * b) ... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg] | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
case intro.mk.intro.inr.inr.intro.intro.intro.intro.intro.intro.inl
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht) | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
case intro.mk.intro.inr.inr.intro.intro.intro.intro.intro.intro.inl
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | rw [← coe_smul_finset] | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
case intro.mk.intro.inr.inr.intro.intro.intro.intro.intro.intro.inl
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | simp [-coe_smul_finset] | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
case intro.mk.intro.inr.inr.intro.intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | replace hsg : (s ∩ op g • s).card < s.card := card_lt_card ⟨inter_subset_left _ _, fun h ↦
hsg <| eq_of_superset_of_card_ge (h.trans <| inter_subset_right _ _) (card_smul_finset _ _).le⟩ | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
case intro.mk.intro.inr.inr.intro.intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | replace aux1 := card_mono $ mulETransformLeft.fst_mul_snd_subset g (s, t) | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
case intro.mk.intro.inr.inr.intro.intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | replace aux2 := card_mono $ mulETransformRight.fst_mul_snd_subset g (s, t) | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
case intro.mk.intro.inr.inr.intro.intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | obtain hgt | hgt := disjoint_or_nonempty_inter t (g⁻¹ • t) | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
case intro.mk.intro.inr.inr.intro.intro.intro.intro.intro.intro.inr.inl
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minO... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | rw [← card_smul_finset g⁻¹ t] | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
case intro.mk.intro.inr.inr.intro.intro.intro.intro.intro.intro.inr.inl
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minO... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | refine' Or.inr ((add_le_add_right hst _).trans _) | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
case intro.mk.intro.inr.inr.intro.intro.intro.intro.intro.intro.inr.inl
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minO... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | rw [← card_union_eq hgt] | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
case intro.mk.intro.inr.inr.intro.intro.intro.intro.intro.intro.inr.inl
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minO... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | exact (card_le_card_mul_left _ hgs).trans (le_add_of_le_left aux1) | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
case intro.mk.intro.inr.inr.intro.intro.intro.intro.intro.intro.inr.inr
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minO... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | obtain hstg | hstg := le_or_lt_of_add_le_add (MulETransform.card g (s, t)).ge | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
case intro.mk.intro.inr.inr.intro.intro.intro.intro.intro.intro.inr.inr.inl
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → ... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | exact (ih _ _ hgs (hgt.mono inter_subset_union) <| devosMulRel_of_le_of_le aux1 hstg hsg).imp
(WithTop.coe_le_coe.2 aux1).trans' fun h ↦ hstg.trans <| h.trans <| add_le_add_right aux1 _ | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
case intro.mk.intro.inr.inr.intro.intro.intro.intro.intro.intro.inr.inr.inr
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → ... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | exact (ih _ _ (hgs.mono inter_subset_union) hgt <| devosMulRel_of_le aux2 hstg).imp
(WithTop.coe_le_coe.2 aux2).trans' fun h ↦
hstg.le.trans <| h.trans <| add_le_add_right aux2 _ | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
h : IsTorsionFree α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
⊢ card s + card t - 1 ≤ card (s * t) | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | simpa only [h.minOrder, min_eq_right, le_top, Nat.cast_le] using Finset.min_le_card_mul hs ht | /-- The **Cauchy-Davenport Theorem** for torsion-free groups. The size of `s * t` is lower-bounded
by `|s| + |t| - 1`. -/
@[to_additive "The **Cauchy-Davenport theorem** for torsion-free groups. The size of `s + t` is
lower-bounded by `|s| + |t| - 1`."]
lemma Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul (h : ... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.182_0.yGTPJO6UphimMFs | /-- The **Cauchy-Davenport Theorem** for torsion-free groups. The size of `s * t` is lower-bounded
by `|s| + |t| - 1`. -/
@[to_additive "The **Cauchy-Davenport theorem** for torsion-free groups. The size of `s + t` is
lower-bounded by `|s| + |t| - 1`."]
lemma Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul (h : ... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
α : Type u_1
p : ℕ
hp : Nat.Prime p
s t : Finset (ZMod p)
hs : Finset.Nonempty s
ht : Finset.Nonempty t
⊢ min p (Finset.card s + Finset.card t - 1) ≤ Finset.card (s + t) | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | simpa only [ZMod.minOrder_of_prime hp, min_le_iff, Nat.cast_le] using Finset.min_le_card_add hs ht | /-- The **Cauchy-Davenport Theorem**. If `s`, `t` are nonempty sets in $$ℤ/pℤ$$, then the size of
`s + t` is lower-bounded by `|s| + |t| - 1`, unless this quantity is greater than `p`. -/
lemma ZMod.min_le_card_add {p : ℕ} (hp : p.Prime) {s t : Finset (ZMod p)} (hs : s.Nonempty)
(ht : t.Nonempty) : min p (s.card + ... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.194_0.yGTPJO6UphimMFs | /-- The **Cauchy-Davenport Theorem**. If `s`, `t` are nonempty sets in $$ℤ/pℤ$$, then the size of
`s + t` is lower-bounded by `|s| + |t| - 1`, unless this quantity is greater than `p`. -/
lemma ZMod.min_le_card_add {p : ℕ} (hp : p.Prime) {s t : Finset (ZMod p)} (hs : s.Nonempty)
(ht : t.Nonempty) : min p (s.card + ... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
α : Type u_1
inst✝⁴ : LinearOrder α
inst✝³ : Semigroup α
inst✝² : IsCancelMul α
inst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
⊢ card s + card t - 1 ≤ card (s... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | suffices s * {t.min' ht} ∩ ({s.max' hs} * t) = {s.max' hs * t.min' ht} by
rw [← card_singleton_mul t (s.max' hs), ← card_mul_singleton s (t.min' ht),
← card_union_add_card_inter, ← card_singleton _, ← this, Nat.add_sub_cancel]
exact card_mono (union_subset (mul_subset_mul_left <| singleton_subset_iff.2 <|... | /-- The **Cauchy-Davenport Theorem** for linearly ordered cancellative semigroups. The size of
`s * t` is lower-bounded by `|s| + |t| - 1`. -/
@[to_additive
"The **Cauchy-Davenport theorem** for linearly ordered additive cancellative semigroups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1`."]
lemma Finset.ca... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.202_0.yGTPJO6UphimMFs | /-- The **Cauchy-Davenport Theorem** for linearly ordered cancellative semigroups. The size of
`s * t` is lower-bounded by `|s| + |t| - 1`. -/
@[to_additive
"The **Cauchy-Davenport theorem** for linearly ordered additive cancellative semigroups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1`."]
lemma Finset.ca... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
α : Type u_1
inst✝⁴ : LinearOrder α
inst✝³ : Semigroup α
inst✝² : IsCancelMul α
inst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
this : s * {min' t ht} ∩ ({max'... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | rw [← card_singleton_mul t (s.max' hs), ← card_mul_singleton s (t.min' ht),
← card_union_add_card_inter, ← card_singleton _, ← this, Nat.add_sub_cancel] | /-- The **Cauchy-Davenport Theorem** for linearly ordered cancellative semigroups. The size of
`s * t` is lower-bounded by `|s| + |t| - 1`. -/
@[to_additive
"The **Cauchy-Davenport theorem** for linearly ordered additive cancellative semigroups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1`."]
lemma Finset.ca... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.202_0.yGTPJO6UphimMFs | /-- The **Cauchy-Davenport Theorem** for linearly ordered cancellative semigroups. The size of
`s * t` is lower-bounded by `|s| + |t| - 1`. -/
@[to_additive
"The **Cauchy-Davenport theorem** for linearly ordered additive cancellative semigroups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1`."]
lemma Finset.ca... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
α : Type u_1
inst✝⁴ : LinearOrder α
inst✝³ : Semigroup α
inst✝² : IsCancelMul α
inst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
this : s * {min' t ht} ∩ ({max'... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | exact card_mono (union_subset (mul_subset_mul_left <| singleton_subset_iff.2 <| min'_mem _ _) <|
mul_subset_mul_right <| singleton_subset_iff.2 <| max'_mem _ _) | /-- The **Cauchy-Davenport Theorem** for linearly ordered cancellative semigroups. The size of
`s * t` is lower-bounded by `|s| + |t| - 1`. -/
@[to_additive
"The **Cauchy-Davenport theorem** for linearly ordered additive cancellative semigroups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1`."]
lemma Finset.ca... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.202_0.yGTPJO6UphimMFs | /-- The **Cauchy-Davenport Theorem** for linearly ordered cancellative semigroups. The size of
`s * t` is lower-bounded by `|s| + |t| - 1`. -/
@[to_additive
"The **Cauchy-Davenport theorem** for linearly ordered additive cancellative semigroups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1`."]
lemma Finset.ca... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
α : Type u_1
inst✝⁴ : LinearOrder α
inst✝³ : Semigroup α
inst✝² : IsCancelMul α
inst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
⊢ s * {min' t ht} ∩ ({max' s hs... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | refine' eq_singleton_iff_unique_mem.2 ⟨mem_inter.2 ⟨mul_mem_mul (max'_mem _ _) <|
mem_singleton_self _, mul_mem_mul (mem_singleton_self _) <| min'_mem _ _⟩, _⟩ | /-- The **Cauchy-Davenport Theorem** for linearly ordered cancellative semigroups. The size of
`s * t` is lower-bounded by `|s| + |t| - 1`. -/
@[to_additive
"The **Cauchy-Davenport theorem** for linearly ordered additive cancellative semigroups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1`."]
lemma Finset.ca... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.202_0.yGTPJO6UphimMFs | /-- The **Cauchy-Davenport Theorem** for linearly ordered cancellative semigroups. The size of
`s * t` is lower-bounded by `|s| + |t| - 1`. -/
@[to_additive
"The **Cauchy-Davenport theorem** for linearly ordered additive cancellative semigroups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1`."]
lemma Finset.ca... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
α : Type u_1
inst✝⁴ : LinearOrder α
inst✝³ : Semigroup α
inst✝² : IsCancelMul α
inst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
⊢ ∀ x ∈ s * {min' t ht} ∩ ({max... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | simp only [mem_inter, and_imp, mem_mul, mem_singleton, exists_and_left, exists_eq_left,
forall_exists_index, and_imp, forall_apply_eq_imp_iff₂, mul_left_inj] | /-- The **Cauchy-Davenport Theorem** for linearly ordered cancellative semigroups. The size of
`s * t` is lower-bounded by `|s| + |t| - 1`. -/
@[to_additive
"The **Cauchy-Davenport theorem** for linearly ordered additive cancellative semigroups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1`."]
lemma Finset.ca... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.202_0.yGTPJO6UphimMFs | /-- The **Cauchy-Davenport Theorem** for linearly ordered cancellative semigroups. The size of
`s * t` is lower-bounded by `|s| + |t| - 1`. -/
@[to_additive
"The **Cauchy-Davenport theorem** for linearly ordered additive cancellative semigroups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1`."]
lemma Finset.ca... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
α : Type u_1
inst✝⁴ : LinearOrder α
inst✝³ : Semigroup α
inst✝² : IsCancelMul α
inst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
⊢ ∀ a ∈ s, ∀ x ∈ t, max' s hs *... | /-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a ... | exact fun a' ha' b' hb' h ↦ (le_max' _ _ ha').eq_of_not_lt fun ha ↦
((mul_lt_mul_right' ha _).trans_eq' h).not_le <| mul_le_mul_left' (min'_le _ _ hb') _ | /-- The **Cauchy-Davenport Theorem** for linearly ordered cancellative semigroups. The size of
`s * t` is lower-bounded by `|s| + |t| - 1`. -/
@[to_additive
"The **Cauchy-Davenport theorem** for linearly ordered additive cancellative semigroups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1`."]
lemma Finset.ca... | Mathlib.Combinatorics.SetFamily.CauchyDavenport.202_0.yGTPJO6UphimMFs | /-- The **Cauchy-Davenport Theorem** for linearly ordered cancellative semigroups. The size of
`s * t` is lower-bounded by `|s| + |t| - 1`. -/
@[to_additive
"The **Cauchy-Davenport theorem** for linearly ordered additive cancellative semigroups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1`."]
lemma Finset.ca... | Mathlib_Combinatorics_SetFamily_CauchyDavenport |
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f g : α → F'
s : Set α
hm_not : ¬m... | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mat... | rw [condexp, dif_neg hm_not] | theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by | Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.106_0.yd50cWAuCo6hlry | theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 | Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic |
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f g : α → F'
s : Set α
hm : m ≤ m0... | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mat... | rw [condexp, dif_pos hm, dif_neg] | theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by | Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.109_0.yd50cWAuCo6hlry | theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 | Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic |
case hnc
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f g : α → F'
s : Set α
hm... | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mat... | push_neg | theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; | Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.109_0.yd50cWAuCo6hlry | theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 | Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic |
case hnc
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f g : α → F'
s : Set α
hm... | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mat... | exact fun h => absurd h hμm_not | theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; | Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.109_0.yd50cWAuCo6hlry | theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 | Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic |
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f g : α → F'
s : Set α
hm : m ≤ m0... | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mat... | rw [condexp, dif_pos hm] | theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
| Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.113_0.yd50cWAuCo6hlry | theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 | Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic |
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f g : α → F'
s : Set α
hm : m ≤ m0... | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mat... | simp only [hμm, Ne.def, true_and_iff] | theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
| Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.113_0.yd50cWAuCo6hlry | theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 | Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic |
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f g : α → F'
s : Set α
hm : m ≤ m0... | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mat... | by_cases hf : Integrable f μ | theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
| Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.113_0.yd50cWAuCo6hlry | theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 | Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic |
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f g : α → F'
s : Set α
hm... | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mat... | rw [dif_pos hf, if_pos hf] | theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by... | Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.113_0.yd50cWAuCo6hlry | theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 | Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic |
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f g : α → F'
s : Set α
hm... | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mat... | rw [dif_neg hf, if_neg hf] | theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by... | Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.113_0.yd50cWAuCo6hlry | theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 | Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic |
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
hm : m ≤ m... | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mat... | rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf] | theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
| Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.126_0.yd50cWAuCo6hlry | theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f | Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic |
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
hm : m ≤ m... | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mat... | rw [condexp_of_sigmaFinite hm] | theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
| Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.136_0.yd50cWAuCo6hlry | theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f | Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic |
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
hm : m ≤ m... | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mat... | by_cases hfi : Integrable f μ | theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
| Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.136_0.yd50cWAuCo6hlry | theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f | Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic |
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
h... | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mat... | rw [if_pos hfi] | theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· | Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.136_0.yd50cWAuCo6hlry | theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f | Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic |
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
h... | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mat... | by_cases hfm : StronglyMeasurable[m] f | theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
| Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.136_0.yd50cWAuCo6hlry | theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f | Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic |
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
h... | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mat... | rw [if_pos hfm] | theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· | Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.136_0.yd50cWAuCo6hlry | theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f | Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic |
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
h... | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mat... | exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm | theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
| Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.136_0.yd50cWAuCo6hlry | theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f | Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic |
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
h... | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mat... | rw [if_neg hfm] | theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyM... | Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.136_0.yd50cWAuCo6hlry | theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f | Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic |
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
h... | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mat... | exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm | theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyM... | Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.136_0.yd50cWAuCo6hlry | theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f | Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic |
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
h... | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mat... | rw [if_neg hfi, condexpL1_undef hfi] | theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyM... | Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.136_0.yd50cWAuCo6hlry | theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f | Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic |
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
h... | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mat... | exact (coeFn_zero _ _ _).symm | theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyM... | Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.136_0.yd50cWAuCo6hlry | theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f | Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic |
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace 𝕜 F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f g : α → F'
s : Set α
hm : m ≤ m... | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mat... | refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _) | theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
| Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.152_0.yd50cWAuCo6hlry | theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) | Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic |
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace 𝕜 F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f g : α → F'
s : Set α
hm : m ≤ m... | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mat... | rw [condexpL1_eq hf] | theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
| Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.152_0.yd50cWAuCo6hlry | theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) | Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic |
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f g : α → F'
s : Set α
hf : ¬Integ... | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mat... | by_cases hm : m ≤ m0 | theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
| Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.159_0.yd50cWAuCo6hlry | theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 | Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic |
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f g : α → F'
s : Set α
hf... | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mat... | swap | theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
| Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.159_0.yd50cWAuCo6hlry | theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 | Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic |
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f g : α → F'
s : Set α
hf... | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mat... | rw [condexp_of_not_le hm] | theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · | Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.159_0.yd50cWAuCo6hlry | theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 | Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic |
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f g : α → F'
s : Set α
hf... | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mat... | by_cases hμm : SigmaFinite (μ.trim hm) | theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
| Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.159_0.yd50cWAuCo6hlry | theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 | Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic |
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