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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