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https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.isIndContained_iff_exists_subgraph | [133, 1] | [142, 52] | exact e.isIndContained.trans hH'.isIndContained | case mpr.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
H' : H.Subgraph
e : G ≃g H'.coe
hH' : H'.IsInduced'
⊢ G ⊴ H | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_bot | [160, 1] | [177, 47] | rw [copyCount] | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝ : Fintype β
H : SimpleGraph β
⊢ ⊥.copyCount H = 1 | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝ : Fintype β
H : SimpleGraph β
⊢ (filter (fun H' => Nonempty (⊥ ≃g H'.coe)) univ).card = 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_bot | [160, 1] | [177, 47] | convert
card_singleton
({ verts := Set.univ
Adj := ⊥
adj_sub := False.elim
edge_vert := False.elim } : H.Subgraph) | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝ : Fintype β
H : SimpleGraph β
⊢ (filter (fun H' => Nonempty (⊥ ≃g H'.coe)) univ).card = 1 | case h.e'_2.h.e'_2
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝ : Fintype β
H : SimpleGraph β
⊢ filter (fun H' => Nonempty (⊥ ≃g H'.coe)) univ =
{{ verts := Set.univ, Adj := ⊥, adj_sub := ⋯, edge_vert := ⋯, symm := ⋯ }} |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_bot | [160, 1] | [177, 47] | simp only [eq_singleton_iff_unique_mem, mem_filter, mem_univ, Subgraph.coe_bot, true_and_iff,
Nonempty.forall] | case h.e'_2.h.e'_2
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝ : Fintype β
H : SimpleGraph β
⊢ filter (fun H' => Nonempty (⊥ ≃g H'.coe)) univ =
{{ verts := Set.univ, Adj := ⊥, adj_sub := ⋯, edge_vert := ⋯, symm := ⋯ }} | case h.e'_2.h.e'_2
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝ : Fintype β
H : SimpleGraph β
⊢ Nonempty (⊥ ≃g { verts := Set.univ, Adj := ⊥, adj_sub := ⋯, edge_vert := ⋯, symm := ⋯ }.coe) ∧
∀ (x : H.Subgraph), ⊥ ≃g x.coe → x = { verts := Set.univ, Adj := ⊥, adj_sub := ⋯, edge_vert := ⋯, symm := ⋯ } |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_bot | [160, 1] | [177, 47] | refine'
⟨⟨⟨(Equiv.Set.univ _).symm, by
simp only [Prop.bot_eq_false, Subgraph.coe_adj, Pi.bot_apply, bot_adj, iff_self_iff,
forall₂_true_iff]⟩⟩, fun H' e ↦
Subgraph.ext _ _ ((set_fintype_card_eq_univ_iff _).1 $ Fintype.card_congr e.toEquiv.symm) _⟩ | case h.e'_2.h.e'_2
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝ : Fintype β
H : SimpleGraph β
⊢ Nonempty (⊥ ≃g { verts := Set.univ, Adj := ⊥, adj_sub := ⋯, edge_vert := ⋯, symm := ⋯ }.coe) ∧
∀ (x : H.Subgraph), ⊥ ≃g x.coe → x = { verts := Set.univ, Adj := ⊥, adj_sub := ⋯, edge_vert := ⋯, symm := ⋯ } | case h.e'_2.h.e'_2
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝ : Fintype β
H : SimpleGraph β
H' : H.Subgraph
e : ⊥ ≃g H'.coe
⊢ H'.Adj = { verts := Set.univ, Adj := ⊥, adj_sub := ⋯, edge_vert := ⋯, symm := ⋯ }.Adj |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_bot | [160, 1] | [177, 47] | ext a b | case h.e'_2.h.e'_2
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝ : Fintype β
H : SimpleGraph β
H' : H.Subgraph
e : ⊥ ≃g H'.coe
⊢ H'.Adj = { verts := Set.univ, Adj := ⊥, adj_sub := ⋯, edge_vert := ⋯, symm := ⋯ }.Adj | case h.e'_2.h.e'_2.h.h.a
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝ : Fintype β
H : SimpleGraph β
H' : H.Subgraph
e : ⊥ ≃g H'.coe
a b : β
⊢ H'.Adj a b ↔ { verts := Set.univ, Adj := ⊥, adj_sub := ⋯, edge_vert := ⋯, symm := ⋯ }.Adj a b |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_bot | [160, 1] | [177, 47] | simp only [Prop.bot_eq_false, Pi.bot_apply, iff_false_iff] | case h.e'_2.h.e'_2.h.h.a
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝ : Fintype β
H : SimpleGraph β
H' : H.Subgraph
e : ⊥ ≃g H'.coe
a b : β
⊢ H'.Adj a b ↔ { verts := Set.univ, Adj := ⊥, adj_sub := ⋯, edge_vert := ⋯, symm := ⋯ }.Adj a b | case h.e'_2.h.e'_2.h.h.a
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝ : Fintype β
H : SimpleGraph β
H' : H.Subgraph
e : ⊥ ≃g H'.coe
a b : β
⊢ ¬H'.Adj a b |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_bot | [160, 1] | [177, 47] | exact fun hab ↦ e.symm.map_rel_iff.2 hab.coe | case h.e'_2.h.e'_2.h.h.a
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝ : Fintype β
H : SimpleGraph β
H' : H.Subgraph
e : ⊥ ≃g H'.coe
a b : β
⊢ ¬H'.Adj a b | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_bot | [160, 1] | [177, 47] | simp only [Prop.bot_eq_false, Subgraph.coe_adj, Pi.bot_apply, bot_adj, iff_self_iff,
forall₂_true_iff] | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝ : Fintype β
H : SimpleGraph β
⊢ ∀ {a b : β},
{ verts := Set.univ, Adj := ⊥, adj_sub := ⋯, edge_vert := ⋯, symm := ⋯ }.coe.Adj ((Equiv.Set.univ β).symm a)
((Equiv.Set.univ β).symm b) ↔
⊥.Adj a b | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_of_isEmpty | [179, 1] | [191, 49] | rw [copyCount] | α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
⊢ G.copyCount H = 1 | α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
⊢ (filter (fun H' => Nonempty (G ≃g H'.coe)) univ).card = 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_of_isEmpty | [179, 1] | [191, 49] | convert card_singleton (⊥ : H.Subgraph) | α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
⊢ (filter (fun H' => Nonempty (G ≃g H'.coe)) univ).card = 1 | case h.e'_2.h.e'_2
α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
⊢ filter (fun H' => Nonempty (G ≃g H'.coe)) univ = {⊥} |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_of_isEmpty | [179, 1] | [191, 49] | simp only [eq_singleton_iff_unique_mem, mem_filter, mem_univ, Subgraph.coe_bot, true_and_iff,
Nonempty.forall, Subsingleton.elim G ⊥] | case h.e'_2.h.e'_2
α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
⊢ filter (fun H' => Nonempty (G ≃g H'.coe)) univ = {⊥} | case h.e'_2.h.e'_2
α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
⊢ Nonempty (⊥ ≃g ⊥) ∧ ∀ (x : H.Subgraph), ⊥ ≃g x.coe → x = ⊥ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_of_isEmpty | [179, 1] | [191, 49] | haveI : IsEmpty (⊥ : H.Subgraph).verts := by simp | case h.e'_2.h.e'_2
α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
⊢ Nonempty (⊥ ≃g ⊥) ∧ ∀ (x : H.Subgraph), ⊥ ≃g x.coe → x = ⊥ | case h.e'_2.h.e'_2
α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
this : IsEmpty ↑⊥.verts
⊢ Nonempty (⊥ ≃g ⊥) ∧ ∀ (x : H.Subgraph), ⊥ ≃g x.coe → x = ⊥ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_of_isEmpty | [179, 1] | [191, 49] | refine' ⟨⟨⟨⟨isEmptyElim, isEmptyElim, isEmptyElim, isEmptyElim⟩, fun {a} ↦ isEmptyElim a⟩⟩,
fun H' e ↦ Subgraph.ext _ _ _ $ funext₂ fun a b ↦ _⟩ | case h.e'_2.h.e'_2
α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
this : IsEmpty ↑⊥.verts
⊢ Nonempty (⊥ ≃g ⊥) ∧ ∀ (x : H.Subgraph), ⊥ ≃g x.coe → x = ⊥ | case h.e'_2.h.e'_2.refine'_1
α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
this : IsEmpty ↑⊥.verts
H' : H.Subgraph
e : ⊥ ≃g H'.coe
⊢ H'.verts = ⊥.verts
case h.e'_2.h.e'_2.refine'_2
α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
this : IsEmpty ↑⊥.verts
H' : H.Subgraph
e : ⊥ ≃g H'.coe
a b : β
⊢ H'.Adj a b = ⊥.Adj a b |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_of_isEmpty | [179, 1] | [191, 49] | simp | α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
⊢ IsEmpty ↑⊥.verts | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_of_isEmpty | [179, 1] | [191, 49] | simpa [Set.eq_empty_iff_forall_not_mem, filter_eq_empty_iff, ‹IsEmpty α›] using
e.toEquiv.symm.isEmpty_congr | case h.e'_2.h.e'_2.refine'_1
α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
this : IsEmpty ↑⊥.verts
H' : H.Subgraph
e : ⊥ ≃g H'.coe
⊢ H'.verts = ⊥.verts | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_of_isEmpty | [179, 1] | [191, 49] | simp only [Subgraph.not_bot_adj, eq_iff_iff, iff_false_iff] | case h.e'_2.h.e'_2.refine'_2
α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
this : IsEmpty ↑⊥.verts
H' : H.Subgraph
e : ⊥ ≃g H'.coe
a b : β
⊢ H'.Adj a b = ⊥.Adj a b | case h.e'_2.h.e'_2.refine'_2
α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
this : IsEmpty ↑⊥.verts
H' : H.Subgraph
e : ⊥ ≃g H'.coe
a b : β
⊢ ¬H'.Adj a b |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_of_isEmpty | [179, 1] | [191, 49] | exact fun hab ↦ e.symm.map_rel_iff.2 hab.coe | case h.e'_2.h.e'_2.refine'_2
α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
this : IsEmpty ↑⊥.verts
H' : H.Subgraph
e : ⊥ ≃g H'.coe
a b : β
⊢ ¬H'.Adj a b | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_eq_zero | [193, 1] | [194, 83] | simp [copyCount, isContained_iff_exists_subgraph, card_pos, filter_eq_empty_iff] | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝ : Fintype β
⊢ G.copyCount H = 0 ↔ ¬G ⊑ H | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_pos | [196, 1] | [197, 83] | simp [copyCount, isContained_iff_exists_subgraph, card_pos, filter_nonempty_iff] | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝ : Fintype β
⊢ 0 < G.copyCount H ↔ G ⊑ H | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.labelledCopyCount_of_isEmpty | [209, 1] | [212, 85] | convert Fintype.card_unique | α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝² : Fintype α
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
⊢ G.labelledCopyCount H = 1 | case convert_2
α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝² : Fintype α
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
⊢ Unique { f // Injective ⇑f } |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.labelledCopyCount_of_isEmpty | [209, 1] | [212, 85] | exact { default := ⟨default, isEmptyElim⟩, uniq := fun _ ↦ Subsingleton.elim _ _ } | case convert_2
α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝² : Fintype α
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
⊢ Unique { f // Injective ⇑f } | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.labelledCopyCount_eq_zero | [214, 1] | [215, 66] | simp [labelledCopyCount, IsContained, Fintype.card_eq_zero_iff] | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype α
inst✝ : Fintype β
⊢ G.labelledCopyCount H = 0 ↔ ¬G ⊑ H | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.labelledCopyCount_pos | [217, 1] | [218, 62] | simp [labelledCopyCount, IsContained, Fintype.card_pos_iff] | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype α
inst✝ : Fintype β
⊢ 0 < G.labelledCopyCount H ↔ G ⊑ H | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_le_labelledCopyCount | [220, 1] | [227, 8] | rw [copyCount, ← Fintype.card_coe] | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype α
inst✝ : Fintype β
⊢ G.copyCount H ≤ G.labelledCopyCount H | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype α
inst✝ : Fintype β
⊢ Fintype.card { x // x ∈ filter (fun H' => Nonempty (G ≃g H'.coe)) univ } ≤ G.labelledCopyCount H |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_le_labelledCopyCount | [220, 1] | [227, 8] | refine Fintype.card_le_of_injective (fun H' ↦
⟨H'.val.hom.comp (mem_filter.1 H'.2).2.some.toHom,
Subtype.coe_injective.comp (mem_filter.1 H'.2).2.some.injective⟩) ?_ | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype α
inst✝ : Fintype β
⊢ Fintype.card { x // x ∈ filter (fun H' => Nonempty (G ≃g H'.coe)) univ } ≤ G.labelledCopyCount H | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype α
inst✝ : Fintype β
⊢ Injective fun H' => ⟨(↑H').hom.comp ⋯.some.toHom, ⋯⟩ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_le_labelledCopyCount | [220, 1] | [227, 8] | sorry | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype α
inst✝ : Fintype β
⊢ Injective fun H' => ⟨(↑H').hom.comp ⋯.some.toHom, ⋯⟩ | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.aux | [242, 1] | [246, 71] | obtain ⟨e, he⟩ := edgeSet_nonempty.2 hG | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : H.Subgraph
⊢ Nonempty (G ≃g H'.coe) → H'.edgeSet.Nonempty | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : H.Subgraph
e : Sym2 α
he : e ∈ G.edgeSet
⊢ Nonempty (G ≃g H'.coe) → H'.edgeSet.Nonempty |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.aux | [242, 1] | [246, 71] | rw [← Subgraph.image_coe_edgeSet_coe] | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : H.Subgraph
e : Sym2 α
he : e ∈ G.edgeSet
⊢ Nonempty (G ≃g H'.coe) → H'.edgeSet.Nonempty | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : H.Subgraph
e : Sym2 α
he : e ∈ G.edgeSet
⊢ Nonempty (G ≃g H'.coe) → (Sym2.map Subtype.val '' H'.coe.edgeSet).Nonempty |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.aux | [242, 1] | [246, 71] | exact fun ⟨f⟩ ↦ Set.Nonempty.image _ ⟨_, f.map_mem_edgeSet_iff.2 he⟩ | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : H.Subgraph
e : Sym2 α
he : e ∈ G.edgeSet
⊢ Nonempty (G ≃g H'.coe) → (Sym2.map Subtype.val '' H'.coe.edgeSet).Nonempty | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.kill_le | [255, 1] | [256, 91] | rw [kill] | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
⊢ G.kill H ≤ H | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
⊢ (if hG : G = ⊥ then H else H.deleteEdges (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})) ≤ H |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.kill_le | [255, 1] | [256, 91] | split_ifs | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
⊢ (if hG : G = ⊥ then H else H.deleteEdges (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})) ≤ H | case pos
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
h✝ : G = ⊥
⊢ H ≤ H
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
h✝ : ¬G = ⊥
⊢ H.deleteEdges (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some}) ≤ H |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.kill_le | [255, 1] | [256, 91] | exacts [le_rfl, deleteEdges_le _] | case pos
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
h✝ : G = ⊥
⊢ H ≤ H
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
h✝ : ¬G = ⊥
⊢ H.deleteEdges (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some}) ≤ H | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.bot_kill | [258, 1] | [259, 31] | rw [kill] | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
H : SimpleGraph β
⊢ ⊥.kill H = H | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
H : SimpleGraph β
⊢ (if hG : ⊥ = ⊥ then H else H.deleteEdges (⋃ H', ⋃ (hH' : Nonempty (⊥ ≃g H'.coe)), {⋯.some})) = H |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.bot_kill | [258, 1] | [259, 31] | exact dif_pos rfl | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
H : SimpleGraph β
⊢ (if hG : ⊥ = ⊥ then H else H.deleteEdges (⋃ H', ⋃ (hH' : Nonempty (⊥ ≃g H'.coe)), {⋯.some})) = H | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.kill_of_ne_bot | [261, 1] | [264, 30] | rw [kill] | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H : SimpleGraph β
⊢ G.kill H = H.deleteEdges (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some}) | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H : SimpleGraph β
⊢ (if hG : G = ⊥ then H else H.deleteEdges (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})) =
H.deleteEdges (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some}) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.kill_of_ne_bot | [261, 1] | [264, 30] | exact dif_neg hG | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H : SimpleGraph β
⊢ (if hG : G = ⊥ then H else H.deleteEdges (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})) =
H.deleteEdges (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some}) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.kill_eq_right | [266, 1] | [271, 87] | simp only [kill_of_ne_bot hG, Set.disjoint_left, isContained_iff_exists_subgraph,
@forall_swap _ H.Subgraph, Set.iUnion_singleton_eq_range, deleteEdges_eq_self, Set.mem_iUnion,
Set.mem_range, not_exists, not_nonempty_iff, Nonempty.forall] | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
⊢ G.kill H = H ↔ ¬G ⊑ H | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
⊢ (∀ (y : H.Subgraph), ∀ x ∈ H.edgeSet, ∀ (a : G ≃g y.coe), x ∉ {⋯.some}) ↔ ∀ (x : H.Subgraph), IsEmpty (G ≃g x.coe) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.kill_eq_right | [266, 1] | [271, 87] | exact forall_congr' fun H' ↦ ⟨fun h ↦ ⟨fun f ↦ h _
(Subgraph.edgeSet_subset _ $ (aux hG ⟨f⟩).choose_spec) f rfl⟩, fun h _ _ ↦ h.elim⟩ | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
⊢ (∀ (y : H.Subgraph), ∀ x ∈ H.edgeSet, ∀ (a : G ≃g y.coe), x ∉ {⋯.some}) ↔ ∀ (x : H.Subgraph), IsEmpty (G ≃g x.coe) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.kill_of_not_isContained | [273, 1] | [276, 35] | obtain rfl | hG := eq_or_ne G ⊥ | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hGH : ¬G ⊑ H
⊢ G.kill H = H | case inl
α : Type u_1
β : Type u_2
γ : Type u_3
G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hGH : ¬⊥ ⊑ H
⊢ ⊥.kill H = H
case inr
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hGH : ¬G ⊑ H
hG : G ≠ ⊥
⊢ G.kill H = H |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.kill_of_not_isContained | [273, 1] | [276, 35] | exact bot_kill _ | case inl
α : Type u_1
β : Type u_2
γ : Type u_3
G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hGH : ¬⊥ ⊑ H
⊢ ⊥.kill H = H | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.kill_of_not_isContained | [273, 1] | [276, 35] | exact (kill_eq_right hG).2 hGH | case inr
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hGH : ¬G ⊑ H
hG : G ≠ ⊥
⊢ G.kill H = H | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | rw [kill_of_ne_bot hG, deleteEdges, isContained_iff_exists_subgraph] | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
⊢ ¬G ⊑ G.kill H | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
⊢ ¬∃ H', Nonempty (G ≃g H'.coe) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | rintro ⟨H', hGH'⟩ | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
⊢ ¬∃ H', Nonempty (G ≃g H'.coe) | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | have hH' : (H'.map $ Hom.ofLE (sdiff_le : H \ _ ≤ H)).edgeSet.Nonempty := by
rw [Subgraph.edgeSet_map]
exact (aux hG hGH').image _ | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
⊢ False | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | set e := hH'.some with he | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
⊢ False | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 β := hH'.some
he : e = hH'.some
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | have : e ∈ _ := hH'.some_mem | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 β := hH'.some
he : e = hH'.some
⊢ False | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 β := hH'.some
he : e = hH'.some
this : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').edgeSet
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | clear_value e | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 β := hH'.some
he : e = hH'.some
this : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').edgeSet
⊢ False | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 β
he : e = hH'.some
this : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').edgeSet
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | rw [← Subgraph.image_coe_edgeSet_coe] at this | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 β
he : e = hH'.some
this : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').edgeSet
⊢ False | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 β
he : e = hH'.some
this : e ∈ Sym2.map Subtype.val '' (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | subst he | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 β
he : e = hH'.some
this : e ∈ Sym2.map Subtype.val '' (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
⊢ False | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
this : hH'.some ∈ Sym2.map Subtype.val '' (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | obtain ⟨e, he₀, he₁⟩ := this | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
this : hH'.some ∈ Sym2.map Subtype.val '' (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
⊢ False | case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | let e' : Sym2 H'.verts := Sym2.map (Subgraph.isoMap (Hom.ofLE _) injective_id _).symm e | case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
⊢ False | case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
e' : Sym2 ↑H'.verts := Sym2.map (⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | have he' : e' ∈ H'.coe.edgeSet := (Iso.map_mem_edgeSet_iff _).2 he₀ | case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
e' : Sym2 ↑H'.verts := Sym2.map (⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e
⊢ False | case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
e' : Sym2 ↑H'.verts := Sym2.map (⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e
he' : e' ∈ H'.coe.edgeSet
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | rw [Subgraph.edgeSet_coe] at he' | case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
e' : Sym2 ↑H'.verts := Sym2.map (⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e
he' : e' ∈ H'.coe.edgeSet
⊢ False | case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
e' : Sym2 ↑H'.verts := Sym2.map (⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e
he' : e' ∈ Sym2.map Subtype.val ⁻¹' H'.edgeSet
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | have := Subgraph.edgeSet_subset _ he' | case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
e' : Sym2 ↑H'.verts := Sym2.map (⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e
he' : e' ∈ Sym2.map Subtype.val ⁻¹' H'.edgeSet
⊢ False | case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
e' : Sym2 ↑H'.verts := Sym2.map (⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e
he' : e' ∈ Sym2.map Subtype.val ⁻¹' H'.edgeSet
this : Sym2.map Subtype.val e' ∈ (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).edgeSet
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | simp only [edgeSet_sdiff, edgeSet_fromEdgeSet, edgeSet_sdiff_sdiff_isDiag, Set.mem_diff,
Set.mem_iUnion, not_exists] at this | case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
e' : Sym2 ↑H'.verts := Sym2.map (⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e
he' : e' ∈ Sym2.map Subtype.val ⁻¹' H'.edgeSet
this : Sym2.map Subtype.val e' ∈ (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).edgeSet
⊢ False | case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
e' : Sym2 ↑H'.verts := Sym2.map (⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e
he' : e' ∈ Sym2.map Subtype.val ⁻¹' H'.edgeSet
this :
Sym2.map Subtype.val e' ∈ H.edgeSet ∧
∀ (x : H.Subgraph) (x_1 : Nonempty (G ≃g x.coe)), Sym2.map Subtype.val e' ∉ {⋯.some}
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | refine' this.2 (H'.map $ Hom.ofLE sdiff_le)
⟨(Subgraph.isoMap (Hom.ofLE _) injective_id _).comp hGH'.some⟩ _ | case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
e' : Sym2 ↑H'.verts := Sym2.map (⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e
he' : e' ∈ Sym2.map Subtype.val ⁻¹' H'.edgeSet
this :
Sym2.map Subtype.val e' ∈ H.edgeSet ∧
∀ (x : H.Subgraph) (x_1 : Nonempty (G ≃g x.coe)), Sym2.map Subtype.val e' ∉ {⋯.some}
⊢ False | case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
e' : Sym2 ↑H'.verts := Sym2.map (⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e
he' : e' ∈ Sym2.map Subtype.val ⁻¹' H'.edgeSet
this :
Sym2.map Subtype.val e' ∈ H.edgeSet ∧
∀ (x : H.Subgraph) (x_1 : Nonempty (G ≃g x.coe)), Sym2.map Subtype.val e' ∉ {⋯.some}
⊢ Sym2.map Subtype.val e' ∈ {⋯.some} |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | rw [Sym2.map_map, Set.mem_singleton_iff, ← he₁] | case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
e' : Sym2 ↑H'.verts := Sym2.map (⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e
he' : e' ∈ Sym2.map Subtype.val ⁻¹' H'.edgeSet
this :
Sym2.map Subtype.val e' ∈ H.edgeSet ∧
∀ (x : H.Subgraph) (x_1 : Nonempty (G ≃g x.coe)), Sym2.map Subtype.val e' ∉ {⋯.some}
⊢ Sym2.map Subtype.val e' ∈ {⋯.some} | case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
e' : Sym2 ↑H'.verts := Sym2.map (⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e
he' : e' ∈ Sym2.map Subtype.val ⁻¹' H'.edgeSet
this :
Sym2.map Subtype.val e' ∈ H.edgeSet ∧
∀ (x : H.Subgraph) (x_1 : Nonempty (G ≃g x.coe)), Sym2.map Subtype.val e' ∉ {⋯.some}
⊢ Sym2.map (Subtype.val ∘ ⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e = Sym2.map Subtype.val e |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | congr 1 with x | case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
e' : Sym2 ↑H'.verts := Sym2.map (⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e
he' : e' ∈ Sym2.map Subtype.val ⁻¹' H'.edgeSet
this :
Sym2.map Subtype.val e' ∈ H.edgeSet ∧
∀ (x : H.Subgraph) (x_1 : Nonempty (G ≃g x.coe)), Sym2.map Subtype.val e' ∉ {⋯.some}
⊢ Sym2.map (Subtype.val ∘ ⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e = Sym2.map Subtype.val e | case intro.intro.intro.e_f.h
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
e' : Sym2 ↑H'.verts := Sym2.map (⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e
he' : e' ∈ Sym2.map Subtype.val ⁻¹' H'.edgeSet
this :
Sym2.map Subtype.val e' ∈ H.edgeSet ∧
∀ (x : H.Subgraph) (x_1 : Nonempty (G ≃g x.coe)), Sym2.map Subtype.val e' ∉ {⋯.some}
x : ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
⊢ (Subtype.val ∘ ⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) x = ↑x |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | refine' congr_arg (↑) (Equiv.Set.image_symm_apply _ _ injective_id _ _) | case intro.intro.intro.e_f.h
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
e' : Sym2 ↑H'.verts := Sym2.map (⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e
he' : e' ∈ Sym2.map Subtype.val ⁻¹' H'.edgeSet
this :
Sym2.map Subtype.val e' ∈ H.edgeSet ∧
∀ (x : H.Subgraph) (x_1 : Nonempty (G ≃g x.coe)), Sym2.map Subtype.val e' ∉ {⋯.some}
x : ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
⊢ (Subtype.val ∘ ⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) x = ↑x | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | rw [Subgraph.edgeSet_map] | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
⊢ (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
⊢ (Sym2.map ⇑(Hom.ofLE ⋯) '' H'.edgeSet).Nonempty |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | exact (aux hG hGH').image _ | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
⊢ (Sym2.map ⇑(Hom.ofLE ⋯) '' H'.edgeSet).Nonempty | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.le_card_edgeFinset_kill | [309, 1] | [329, 27] | obtain rfl | hG := eq_or_ne G ⊥ | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype ↑H.edgeSet
inst✝ : Fintype β
⊢ H.edgeFinset.card - G.copyCount H ≤ (G.kill H).edgeFinset.card | case inl
α : Type u_1
β : Type u_2
γ : Type u_3
G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype ↑H.edgeSet
inst✝ : Fintype β
⊢ H.edgeFinset.card - ⊥.copyCount H ≤ (⊥.kill H).edgeFinset.card
case inr
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype ↑H.edgeSet
inst✝ : Fintype β
hG : G ≠ ⊥
⊢ H.edgeFinset.card - G.copyCount H ≤ (G.kill H).edgeFinset.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.le_card_edgeFinset_kill | [309, 1] | [329, 27] | let f (H' : {H' : H.Subgraph // Nonempty (G ≃g H'.coe)}) := (aux hG H'.2).some | case inr
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype ↑H.edgeSet
inst✝ : Fintype β
hG : G ≠ ⊥
⊢ H.edgeFinset.card - G.copyCount H ≤ (G.kill H).edgeFinset.card | case inr
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype ↑H.edgeSet
inst✝ : Fintype β
hG : G ≠ ⊥
f : { H' // Nonempty (G ≃g H'.coe) } → Sym2 β := fun H' => ⋯.some
⊢ H.edgeFinset.card - G.copyCount H ≤ (G.kill H).edgeFinset.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.le_card_edgeFinset_kill | [309, 1] | [329, 27] | calc
_ = H.edgeFinset.card - card {H' : H.Subgraph // Nonempty (G ≃g H'.coe)} := ?_
_ ≤ H.edgeFinset.card - (univ.image f).card := Nat.sub_le_sub_left card_image_le _
_ = H.edgeFinset.card - (Set.range f).toFinset.card := by rw [Set.toFinset_range]
_ ≤ (H.edgeFinset \ (Set.range f).toFinset).card := le_card_sdiff ..
_ = (G.kill H).edgeFinset.card := ?_ | case inr
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype ↑H.edgeSet
inst✝ : Fintype β
hG : G ≠ ⊥
f : { H' // Nonempty (G ≃g H'.coe) } → Sym2 β := fun H' => ⋯.some
⊢ H.edgeFinset.card - G.copyCount H ≤ (G.kill H).edgeFinset.card | case inr.calc_1
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype ↑H.edgeSet
inst✝ : Fintype β
hG : G ≠ ⊥
f : { H' // Nonempty (G ≃g H'.coe) } → Sym2 β := fun H' => ⋯.some
⊢ H.edgeFinset.card - G.copyCount H = H.edgeFinset.card - Fintype.card { H' // Nonempty (G ≃g H'.coe) }
case inr.calc_2
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype ↑H.edgeSet
inst✝ : Fintype β
hG : G ≠ ⊥
f : { H' // Nonempty (G ≃g H'.coe) } → Sym2 β := fun H' => ⋯.some
⊢ (H.edgeFinset \ (Set.range f).toFinset).card = (G.kill H).edgeFinset.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.le_card_edgeFinset_kill | [309, 1] | [329, 27] | simp only [kill_of_ne_bot, hG, Ne, not_false_iff, Set.iUnion_singleton_eq_range,
Set.toFinset_card, Fintype.card_ofFinset, edgeSet_deleteEdges] | case inr.calc_2
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype ↑H.edgeSet
inst✝ : Fintype β
hG : G ≠ ⊥
f : { H' // Nonempty (G ≃g H'.coe) } → Sym2 β := fun H' => ⋯.some
⊢ (H.edgeFinset \ (Set.range f).toFinset).card = (G.kill H).edgeFinset.card | case inr.calc_2
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype ↑H.edgeSet
inst✝ : Fintype β
hG : G ≠ ⊥
f : { H' // Nonempty (G ≃g H'.coe) } → Sym2 β := fun H' => ⋯.some
⊢ (H.edgeFinset \ (Set.range f).toFinset).card =
Fintype.card ↑(H.edgeSet \ ⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some}) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.le_card_edgeFinset_kill | [309, 1] | [329, 27] | simp only [Finset.sdiff_eq_inter_compl, Set.diff_eq, ← Set.iUnion_singleton_eq_range, coe_sdiff,
Set.coe_toFinset, coe_filter, Set.sep_mem_eq, Set.iUnion_subtype, ← Fintype.card_coe,
← Finset.coe_sort_coe, coe_inter, coe_compl, Set.coe_toFinset, Set.compl_iUnion,
Fintype.card_ofFinset] | case inr.calc_2
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype ↑H.edgeSet
inst✝ : Fintype β
hG : G ≠ ⊥
f : { H' // Nonempty (G ≃g H'.coe) } → Sym2 β := fun H' => ⋯.some
⊢ (H.edgeFinset \ (Set.range f).toFinset).card =
Fintype.card ↑(H.edgeSet \ ⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some}) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.le_card_edgeFinset_kill | [309, 1] | [329, 27] | simp | case inl
α : Type u_1
β : Type u_2
γ : Type u_3
G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype ↑H.edgeSet
inst✝ : Fintype β
⊢ H.edgeFinset.card - ⊥.copyCount H ≤ (⊥.kill H).edgeFinset.card | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.le_card_edgeFinset_kill | [309, 1] | [329, 27] | rw [Set.toFinset_range] | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype ↑H.edgeSet
inst✝ : Fintype β
hG : G ≠ ⊥
f : { H' // Nonempty (G ≃g H'.coe) } → Sym2 β := fun H' => ⋯.some
⊢ H.edgeFinset.card - (image f univ).card = H.edgeFinset.card - (Set.range f).toFinset.card | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.le_card_edgeFinset_kill | [309, 1] | [329, 27] | simp only [Set.toFinset_card] | case inr.calc_1
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype ↑H.edgeSet
inst✝ : Fintype β
hG : G ≠ ⊥
f : { H' // Nonempty (G ≃g H'.coe) } → Sym2 β := fun H' => ⋯.some
⊢ H.edgeFinset.card - G.copyCount H = H.edgeFinset.card - Fintype.card { H' // Nonempty (G ≃g H'.coe) } | case inr.calc_1
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype ↑H.edgeSet
inst✝ : Fintype β
hG : G ≠ ⊥
f : { H' // Nonempty (G ≃g H'.coe) } → Sym2 β := fun H' => ⋯.some
⊢ Fintype.card ↑H.edgeSet - G.copyCount H = Fintype.card ↑H.edgeSet - Fintype.card { H' // Nonempty (G ≃g H'.coe) } |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.le_card_edgeFinset_kill | [309, 1] | [329, 27] | rw [← Set.toFinset_card, ← edgeFinset, copyCount, ← card_subtype, subtype_univ, card_univ] | case inr.calc_1
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype ↑H.edgeSet
inst✝ : Fintype β
hG : G ≠ ⊥
f : { H' // Nonempty (G ≃g H'.coe) } → Sym2 β := fun H' => ⋯.some
⊢ Fintype.card ↑H.edgeSet - G.copyCount H = Fintype.card ↑H.edgeSet - Fintype.card { H' // Nonempty (G ≃g H'.coe) } | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_ofFintype | [16, 1] | [33, 52] | have :
((univ : Finset β) : Set β).PairwiseDisjoint fun b ↦
univ.filter fun a : α ↦ g a = b := by
refine' fun b₁ _ b₂ _ hb ↦ disjoint_left.2 fun a ha₁ ha₂ ↦ _
simp only [mem_filter, mem_univ, true_and_iff] at ha₁ ha₂
exact hb (ha₁.symm.trans ha₂) | α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : α → ℝ≥0∞
h : ∑ a : α, f a = 1
g : α → β
⊢ ∑ a : β, (fun b => ∑ a ∈ Finset.filter (fun a => g a = b) univ, f a) a = 1 | α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : α → ℝ≥0∞
h : ∑ a : α, f a = 1
g : α → β
this : (↑univ).PairwiseDisjoint fun b => Finset.filter (fun a => g a = b) univ
⊢ ∑ a : β, (fun b => ∑ a ∈ Finset.filter (fun a => g a = b) univ, f a) a = 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_ofFintype | [16, 1] | [33, 52] | rw [← sum_disjiUnion _ _ this] | α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : α → ℝ≥0∞
h : ∑ a : α, f a = 1
g : α → β
this : (↑univ).PairwiseDisjoint fun b => Finset.filter (fun a => g a = b) univ
⊢ ∑ a : β, (fun b => ∑ a ∈ Finset.filter (fun a => g a = b) univ, f a) a = 1 | α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : α → ℝ≥0∞
h : ∑ a : α, f a = 1
g : α → β
this : (↑univ).PairwiseDisjoint fun b => Finset.filter (fun a => g a = b) univ
⊢ ∑ x ∈ univ.disjiUnion (fun b => Finset.filter (fun a => g a = b) univ) this, f x = 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_ofFintype | [16, 1] | [33, 52] | convert h | α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : α → ℝ≥0∞
h : ∑ a : α, f a = 1
g : α → β
this : (↑univ).PairwiseDisjoint fun b => Finset.filter (fun a => g a = b) univ
⊢ ∑ x ∈ univ.disjiUnion (fun b => Finset.filter (fun a => g a = b) univ) this, f x = 1 | case h.e'_2.h
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : α → ℝ≥0∞
h : ∑ a : α, f a = 1
g : α → β
this : (↑univ).PairwiseDisjoint fun b => Finset.filter (fun a => g a = b) univ
⊢ univ.disjiUnion (fun b => Finset.filter (fun a => g a = b) univ) this = univ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_ofFintype | [16, 1] | [33, 52] | exact eq_univ_of_forall fun a ↦
mem_disjiUnion.2 ⟨_, mem_univ _, mem_filter.2 ⟨mem_univ _, rfl⟩⟩ | case h.e'_2.h
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : α → ℝ≥0∞
h : ∑ a : α, f a = 1
g : α → β
this : (↑univ).PairwiseDisjoint fun b => Finset.filter (fun a => g a = b) univ
⊢ univ.disjiUnion (fun b => Finset.filter (fun a => g a = b) univ) this = univ | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_ofFintype | [16, 1] | [33, 52] | refine' fun b₁ _ b₂ _ hb ↦ disjoint_left.2 fun a ha₁ ha₂ ↦ _ | α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : α → ℝ≥0∞
h : ∑ a : α, f a = 1
g : α → β
⊢ (↑univ).PairwiseDisjoint fun b => Finset.filter (fun a => g a = b) univ | α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : α → ℝ≥0∞
h : ∑ a : α, f a = 1
g : α → β
b₁ : β
x✝¹ : b₁ ∈ ↑univ
b₂ : β
x✝ : b₂ ∈ ↑univ
hb : b₁ ≠ b₂
a : α
ha₁ : a ∈ (fun b => Finset.filter (fun a => g a = b) univ) b₁
ha₂ : a ∈ (fun b => Finset.filter (fun a => g a = b) univ) b₂
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_ofFintype | [16, 1] | [33, 52] | simp only [mem_filter, mem_univ, true_and_iff] at ha₁ ha₂ | α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : α → ℝ≥0∞
h : ∑ a : α, f a = 1
g : α → β
b₁ : β
x✝¹ : b₁ ∈ ↑univ
b₂ : β
x✝ : b₂ ∈ ↑univ
hb : b₁ ≠ b₂
a : α
ha₁ : a ∈ (fun b => Finset.filter (fun a => g a = b) univ) b₁
ha₂ : a ∈ (fun b => Finset.filter (fun a => g a = b) univ) b₂
⊢ False | α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : α → ℝ≥0∞
h : ∑ a : α, f a = 1
g : α → β
b₁ : β
x✝¹ : b₁ ∈ ↑univ
b₂ : β
x✝ : b₂ ∈ ↑univ
hb : b₁ ≠ b₂
a : α
ha₁ : g a = b₁
ha₂ : g a = b₂
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_ofFintype | [16, 1] | [33, 52] | exact hb (ha₁.symm.trans ha₂) | α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : α → ℝ≥0∞
h : ∑ a : α, f a = 1
g : α → β
b₁ : β
x✝¹ : b₁ ∈ ↑univ
b₂ : β
x✝ : b₂ ∈ ↑univ
hb : b₁ ≠ b₂
a : α
ha₁ : g a = b₁
ha₂ : g a = b₂
⊢ False | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_ofFintype | [16, 1] | [33, 52] | ext b : 1 | α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : α → ℝ≥0∞
h : ∑ a : α, f a = 1
g : α → β
⊢ map g (ofFintype f h) = ofFintype (fun b => ∑ a ∈ Finset.filter (fun a => g a = b) univ, f a) ⋯ | case h
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : α → ℝ≥0∞
h : ∑ a : α, f a = 1
g : α → β
b : β
⊢ (map g (ofFintype f h)) b = (ofFintype (fun b => ∑ a ∈ Finset.filter (fun a => g a = b) univ, f a) ⋯) b |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_ofFintype | [16, 1] | [33, 52] | simp only [sum_filter, eq_comm, map_apply, ofFintype_apply] | case h
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : α → ℝ≥0∞
h : ∑ a : α, f a = 1
g : α → β
b : β
⊢ (map g (ofFintype f h)) b = (ofFintype (fun b => ∑ a ∈ Finset.filter (fun a => g a = b) univ, f a) ⋯) b | case h
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : α → ℝ≥0∞
h : ∑ a : α, f a = 1
g : α → β
b : β
⊢ (∑' (a : α), if b = g a then f a else 0) = ∑ a : α, if b = g a then f a else 0 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_ofFintype | [16, 1] | [33, 52] | exact tsum_eq_sum fun _ h ↦ (h $ mem_univ _).elim | case h
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : α → ℝ≥0∞
h : ∑ a : α, f a = 1
g : α → β
b : β
⊢ (∑' (a : α), if b = g a then f a else 0) = ∑ a : α, if b = g a then f a else 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_toMeasure | [41, 1] | [43, 82] | ext s hs : 1 | α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p : PMF α
hf : Measurable f
⊢ Measure.map f p.toMeasure = (map f p).toMeasure | case h
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p : PMF α
hf : Measurable f
s : Set β
hs : MeasurableSet s
⊢ (Measure.map f p.toMeasure) s = (map f p).toMeasure s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_toMeasure | [41, 1] | [43, 82] | rw [PMF.toMeasure_map_apply _ _ _ hf hs, Measure.map_apply hf hs] | case h
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p : PMF α
hf : Measurable f
s : Set β
hs : MeasurableSet s
⊢ (Measure.map f p.toMeasure) s = (map f p).toMeasure s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.support_bernoulli' | [56, 1] | [57, 80] | by_cases b <;> simp [*, hp.lt_iff_ne, tsub_eq_zero_iff_le] | α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p : ℝ≥0
hp : p ≤ 1
b✝ b : Prop
⊢ b ∈ (bernoulli' p hp).support ↔ b ∈ {b | if b then p ≠ 0 else p ≠ 1} | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.mem_support_bernoulli'_iff | [59, 1] | [60, 7] | simp | α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p : ℝ≥0
hp : p ≤ 1
b : Prop
⊢ b ∈ (bernoulli' p hp).support ↔ if b then p ≠ 0 else p ≠ 1 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_not_bernoulli' | [62, 1] | [72, 91] | have : ∀ p : Prop, (Finset.univ.filter fun q : Prop ↦ ¬ q ↔ p) = {¬ p} := by
rintro p
ext q
by_cases p <;> by_cases q <;> simp [*] | α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p : ℝ≥0
hp : p ≤ 1
b : Prop
⊢ map Not (bernoulli' p hp) = bernoulli' (1 - p) ⋯ | α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p : ℝ≥0
hp : p ≤ 1
b : Prop
this : ∀ (p : Prop), Finset.filter (fun q => ¬q ↔ p) Finset.univ = {¬p}
⊢ map Not (bernoulli' p hp) = bernoulli' (1 - p) ⋯ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_not_bernoulli' | [62, 1] | [72, 91] | refine' (map_ofFintype _ _ _).trans _ | α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p : ℝ≥0
hp : p ≤ 1
b : Prop
this : ∀ (p : Prop), Finset.filter (fun q => ¬q ↔ p) Finset.univ = {¬p}
⊢ map Not (bernoulli' p hp) = bernoulli' (1 - p) ⋯ | α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p : ℝ≥0
hp : p ≤ 1
b : Prop
this : ∀ (p : Prop), Finset.filter (fun q => ¬q ↔ p) Finset.univ = {¬p}
⊢ ofFintype (fun b => ∑ a ∈ Finset.filter (fun a => (¬a) = b) Finset.univ, if a then ↑p else 1 - ↑p) ⋯ =
bernoulli' (1 - p) ⋯ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_not_bernoulli' | [62, 1] | [72, 91] | simp only [this, bernoulli', Finset.mem_filter, Finset.mem_univ, true_and_iff,
Finset.mem_disjiUnion, tsub_le_self, eq_iff_iff, Finset.sum_singleton, WithTop.coe_sub,
ENNReal.coe_one] | α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p : ℝ≥0
hp : p ≤ 1
b : Prop
this : ∀ (p : Prop), Finset.filter (fun q => ¬q ↔ p) Finset.univ = {¬p}
⊢ ofFintype (fun b => ∑ a ∈ Finset.filter (fun a => (¬a) = b) Finset.univ, if a then ↑p else 1 - ↑p) ⋯ =
bernoulli' (1 - p) ⋯ | α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p : ℝ≥0
hp : p ≤ 1
b : Prop
this : ∀ (p : Prop), Finset.filter (fun q => ¬q ↔ p) Finset.univ = {¬p}
⊢ ofFintype (fun b => if ¬b then ↑p else 1 - ↑p) ⋯ = ofFintype (fun b => if b then ↑(1 - p) else 1 - ↑(1 - p)) ⋯ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_not_bernoulli' | [62, 1] | [72, 91] | congr 1 with q | α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p : ℝ≥0
hp : p ≤ 1
b : Prop
this : ∀ (p : Prop), Finset.filter (fun q => ¬q ↔ p) Finset.univ = {¬p}
⊢ ofFintype (fun b => if ¬b then ↑p else 1 - ↑p) ⋯ = ofFintype (fun b => if b then ↑(1 - p) else 1 - ↑(1 - p)) ⋯ | case e_f.h
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p : ℝ≥0
hp : p ≤ 1
b : Prop
this : ∀ (p : Prop), Finset.filter (fun q => ¬q ↔ p) Finset.univ = {¬p}
q : Prop
⊢ (if ¬q then ↑p else 1 - ↑p) = if q then ↑(1 - p) else 1 - ↑(1 - p) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_not_bernoulli' | [62, 1] | [72, 91] | split_ifs <;> simp [ENNReal.sub_sub_cancel WithTop.one_ne_top (ENNReal.coe_le_coe.2 hp)] | case e_f.h
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p : ℝ≥0
hp : p ≤ 1
b : Prop
this : ∀ (p : Prop), Finset.filter (fun q => ¬q ↔ p) Finset.univ = {¬p}
q : Prop
⊢ (if ¬q then ↑p else 1 - ↑p) = if q then ↑(1 - p) else 1 - ↑(1 - p) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_not_bernoulli' | [62, 1] | [72, 91] | rintro p | α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p : ℝ≥0
hp : p ≤ 1
b : Prop
⊢ ∀ (p : Prop), Finset.filter (fun q => ¬q ↔ p) Finset.univ = {¬p} | α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p✝ : ℝ≥0
hp : p✝ ≤ 1
b p : Prop
⊢ Finset.filter (fun q => ¬q ↔ p) Finset.univ = {¬p} |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_not_bernoulli' | [62, 1] | [72, 91] | ext q | α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p✝ : ℝ≥0
hp : p✝ ≤ 1
b p : Prop
⊢ Finset.filter (fun q => ¬q ↔ p) Finset.univ = {¬p} | case a
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p✝ : ℝ≥0
hp : p✝ ≤ 1
b p q : Prop
⊢ q ∈ Finset.filter (fun q => ¬q ↔ p) Finset.univ ↔ q ∈ {¬p} |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_not_bernoulli' | [62, 1] | [72, 91] | by_cases p <;> by_cases q <;> simp [*] | case a
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p✝ : ℝ≥0
hp : p✝ ≤ 1
b p q : Prop
⊢ q ∈ Finset.filter (fun q => ¬q ↔ p) Finset.univ ↔ q ∈ {¬p} | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Algebra/Order/Ring/Canonical.lean | mul_tsub_one | [7, 1] | [7, 84] | rw [mul_tsub, mul_one] | α : Type u_1
inst✝⁴ : CanonicallyOrderedCommSemiring α
inst✝³ : Sub α
inst✝² : OrderedSub α
inst✝¹ : IsTotal α fun x x_1 => x ≤ x_1
inst✝ : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
a b : α
⊢ a * (b - 1) = a * b - a | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Algebra/Order/Ring/Canonical.lean | tsub_one_mul | [8, 1] | [8, 84] | rw [tsub_mul, one_mul] | α : Type u_1
inst✝⁴ : CanonicallyOrderedCommSemiring α
inst✝³ : Sub α
inst✝² : OrderedSub α
inst✝¹ : IsTotal α fun x x_1 => x ≤ x_1
inst✝ : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
a b : α
⊢ (a - 1) * b = a * b - b | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mem_mulStab | [37, 1] | [41, 97] | rw [mulStab, mem_filter, mem_div, and_iff_right_of_imp] | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
hs : s.Nonempty
⊢ a ∈ s.mulStab ↔ a • s = s | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
hs : s.Nonempty
⊢ a • s = s → ∃ b ∈ s, ∃ c ∈ s, b / c = a |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mem_mulStab | [37, 1] | [41, 97] | obtain ⟨b, hb⟩ := hs | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a : α
hs : s.Nonempty
⊢ a • s = s → ∃ b ∈ s, ∃ c ∈ s, b / c = a | case intro
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a b : α
hb : b ∈ s
⊢ a • s = s → ∃ b ∈ s, ∃ c ∈ s, b / c = a |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mem_mulStab | [37, 1] | [41, 97] | exact fun h ↦ ⟨_, by rw [← h]; exact smul_mem_smul_finset hb, _, hb, mul_div_cancel_right _ _⟩ | case intro
ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a b : α
hb : b ∈ s
⊢ a • s = s → ∃ b ∈ s, ∃ c ∈ s, b / c = a | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mem_mulStab | [37, 1] | [41, 97] | rw [← h] | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a b : α
hb : b ∈ s
h : a • s = s
⊢ a * b ∈ s | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a b : α
hb : b ∈ s
h : a • s = s
⊢ a * b ∈ a • s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/MulStab.lean | Finset.mem_mulStab | [37, 1] | [41, 97] | exact smul_mem_smul_finset hb | ι : Type u_1
α : Type u_2
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
a b : α
hb : b ∈ s
h : a • s = s
⊢ a * b ∈ a • s | no goals |
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