awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Tutte (goal)
Goal:
V : Type u_1
G : SimpleGraph V
inst : Finite V
x a b c : V
M1 : (G ⊔ edge x b).Subgraph
M2 : (G ⊔ edge a c).Subgraph
hxa : G.Adj x a
hab : G.Adj a b
hnGxb : ¬G.Adj x b
hnGac : ¬G.Adj a c
hnxb : x ≠ b
hnxc : x ≠ c
hnac : a ≠ c
hnbc : b ≠ c
hM1 : M1.IsPerfectMatching
hM2 : M2.IsPerfectMatching
hM1xb : M1.Adj x b
hM2ac : M2.Adj a c
hM1nac : ¬M1.Adj a c
hsupG : G ⊔ edge x b ⊔ (G ⊔ edge a c) = G ⊔ edge a c ⊔ edge x b
cycles : SimpleGraph V := M1.spanningCoe ∆ M2.spanningCoe
hcalt : cycles.IsAlternating M2.spanningCoe
hcycles : (M1.spanningCoe ∆ M2.spanningCoe).IsCycles
hcac : cycles.Adj a c
hM1sub : M1.spanningCoe ≤ G ⊔ edge x b
hM2sub : M2.spanningCoe ≤ G ⊔ edge a c
induce_le :
  (cycles.connectedComponentMk c).toSimpleGraph.spanningCoe (s := (↑(cycles.connectedComponentMk c) : Set V)) ≤
    G ⊔ edge a c ⊔ edge x b
hxc : x ∈ (cycles.connectedComponentMk c).supp
hacc : a ∈ (cycles.connectedComponentMk c).supp
this : (G : SimpleGraph V) → G.LocallyFinite
hnM2 : ∀ (x' : V) (h : x' ≠ c), ¬M2.Adj x' a
p : (M1.spanningCoe ∆ M2.spanningCoe).Walk a a
hp : p.IsCycle ∧ p.toSubgraph.verts = (cycles.connectedComponentMk c).supp
p' : (M1.spanningCoe ∆ M2.spanningCoe).Walk a a
hp' : p'.IsCycle ∧ p'.snd = c ∧ p'.toSubgraph.verts = p.toSubgraph.verts
x' : V
hx' : Membership.mem (γ := Finset V) {x, b} x'
hx'p : x' ∈ p'.support
htw : ∀ (t : V), Membership.mem (γ := Finset V) {x, b} t → t ∈ (p'.takeUntil x' hx'p).support → t = x'
⊢ (p'.takeUntil x' hx'p).toSubgraph.Adj a (p'.takeUntil x' hx'p).snd
have h : ¬(p'.takeUntil x' hx'p).Nil
Tactic: apply Walk.toSubgraph_adj_snd
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Tutte (goal)
Goal:
V : Type u_1
G : SimpleGraph V
inst : Finite V
x a b c : V
M1 : (G ⊔ edge x b).Subgraph
M2 : (G ⊔ edge a c).Subgraph
hxa : G.Adj x a
hab : G.Adj a b
hnGxb : ¬G.Adj x b
hnGac : ¬G.Adj a c
hnxb : x ≠ b
hnxc : x ≠ c
hnac : a ≠ c
hnbc : b ≠ c
hM1 : M1.IsPerfectMatching
hM2 : M2.IsPerfectMatching
hM1xb : M1.Adj x b
hM2ac : M2.Adj a c
hM1nac : ¬M1.Adj a c
hsupG : G ⊔ edge x b ⊔ (G ⊔ edge a c) = G ⊔ edge a c ⊔ edge x b
cycles : SimpleGraph V := M1.spanningCoe ∆ M2.spanningCoe
hcalt : cycles.IsAlternating M2.spanningCoe
hcycles : (M1.spanningCoe ∆ M2.spanningCoe).IsCycles
hcac : cycles.Adj a c
hM1sub : M1.spanningCoe ≤ G ⊔ edge x b
hM2sub : M2.spanningCoe ≤ G ⊔ edge a c
induce_le :
  (cycles.connectedComponentMk c).toSimpleGraph.spanningCoe (s := (↑(cycles.connectedComponentMk c) : Set V)) ≤
    G ⊔ edge a c ⊔ edge x b
hxc : x ∈ (cycles.connectedComponentMk c).supp
hacc : a ∈ (cycles.connectedComponentMk c).supp
this : (G : SimpleGraph V) → G.LocallyFinite
hnM2 : ∀ (x' : V) (h : x' ≠ c), ¬M2.Adj x' a
p : (M1.spanningCoe ∆ M2.spanningCoe).Walk a a
hp : p.IsCycle ∧ p.toSubgraph.verts = (cycles.connectedComponentMk c).supp
p' : (M1.spanningCoe ∆ M2.spanningCoe).Walk a a
hp' : p'.IsCycle ∧ p'.snd = c ∧ p'.toSubgraph.verts = p.toSubgraph.verts
x' : V
hx' : Membership.mem (γ := Finset V) {x, b} x'
hx'p : x' ∈ p'.support
htw : ∀ (t : V), Membership.mem (γ := Finset V) {x, b} t → t ∈ (p'.takeUntil x' hx'p).support → t = x'
⊢ ¬(p'.takeUntil x' hx'p).Nil
have h : ¬a = x'
Tactic: rw [Walk.nil_takeUntil]
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Tutte (hypothesis)
Goal:
V : Type u_1
G : SimpleGraph V
inst : Finite V
x a b c : V
M1 : (G ⊔ edge x b).Subgraph
M2 : (G ⊔ edge a c).Subgraph
hxa : G.Adj x a
hab : G.Adj a b
hnGxb : ¬G.Adj x b
hnGac : ¬G.Adj a c
hnxb : x ≠ b
hnxc : x ≠ c
hnac : a ≠ c
hnbc : b ≠ c
hM1 : M1.IsPerfectMatching
hM2 : M2.IsPerfectMatching
hM1xb : M1.Adj x b
hM2ac : M2.Adj a c
hM1nac : ¬M1.Adj a c
hsupG : G ⊔ edge x b ⊔ (G ⊔ edge a c) = G ⊔ edge a c ⊔ edge x b
cycles : SimpleGraph V := M1.spanningCoe ∆ M2.spanningCoe
hcalt : cycles.IsAlternating M2.spanningCoe
hcycles : (M1.spanningCoe ∆ M2.spanningCoe).IsCycles
hcac : cycles.Adj a c
hM1sub : M1.spanningCoe ≤ G ⊔ edge x b
hM2sub : M2.spanningCoe ≤ G ⊔ edge a c
induce_le :
  (cycles.connectedComponentMk c).toSimpleGraph.spanningCoe (s := (↑(cycles.connectedComponentMk c) : Set V)) ≤
    G ⊔ edge a c ⊔ edge x b
hxc : x ∈ (cycles.connectedComponentMk c).supp
hacc : a ∈ (cycles.connectedComponentMk c).supp
this : (G : SimpleGraph V) → G.LocallyFinite
hnM2 : ∀ (x' : V) (h : x' ≠ c), ¬M2.Adj x' a
p : (M1.spanningCoe ∆ M2.spanningCoe).Walk a a
hp : p.IsCycle ∧ p.toSubgraph.verts = (cycles.connectedComponentMk c).supp
p' : (M1.spanningCoe ∆ M2.spanningCoe).Walk a a
hp' : p'.IsCycle ∧ p'.snd = c ∧ p'.toSubgraph.verts = p.toSubgraph.verts
x' : V
hx' : Membership.mem (γ := Finset V) {x, b} x'
hx'p : x' ∈ p'.support
htw : ∀ (t : V), Membership.mem (γ := Finset V) {x, b} t → t ∈ (p'.takeUntil x' hx'p).support → t = x'
⊢ (p'.takeUntil x' hx'p).toSubgraph.Adj a c
have this₁ : (p'.takeUntil x' hx'p).toSubgraph.Adj a (p'.takeUntil x' hx'p).snd
Tactic: have : (p'.takeUntil x' hx'p).toSubgraph.Adj a (p'.takeUntil x' hx'p).snd := by
      apply Walk.toSubgraph_adj_snd
      rw [Walk.nil_takeUntil]
      grind
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Tutte (goal)
Goal:
V : Type u_1
G : SimpleGraph V
inst : Finite V
x a b c : V
M1 : (G ⊔ edge x b).Subgraph
M2 : (G ⊔ edge a c).Subgraph
hxa : G.Adj x a
hab : G.Adj a b
hnGxb : ¬G.Adj x b
hnGac : ¬G.Adj a c
hnxb : x ≠ b
hnxc : x ≠ c
hnac : a ≠ c
hnbc : b ≠ c
hM1 : M1.IsPerfectMatching
hM2 : M2.IsPerfectMatching
hM1xb : M1.Adj x b
hM2ac : M2.Adj a c
hM1nac : ¬M1.Adj a c
hsupG : G ⊔ edge x b ⊔ (G ⊔ edge a c) = G ⊔ edge a c ⊔ edge x b
cycles : SimpleGraph V := M1.spanningCoe ∆ M2.spanningCoe
hcalt : cycles.IsAlternating M2.spanningCoe
hcycles : (M1.spanningCoe ∆ M2.spanningCoe).IsCycles
hcac : cycles.Adj a c
hM1sub : M1.spanningCoe ≤ G ⊔ edge x b
hM2sub : M2.spanningCoe ≤ G ⊔ edge a c
induce_le :
  (cycles.connectedComponentMk c).toSimpleGraph.spanningCoe (s := (↑(cycles.connectedComponentMk c) : Set V)) ≤
    G ⊔ edge a c ⊔ edge x b
hxc : x ∈ (cycles.connectedComponentMk c).supp
hacc : a ∈ (cycles.connectedComponentMk c).supp
this_1 : (G : SimpleGraph V) → G.LocallyFinite
hnM2 : ∀ (x' : V) (h : x' ≠ c), ¬M2.Adj x' a
p : (M1.spanningCoe ∆ M2.spanningCoe).Walk a a
hp : p.IsCycle ∧ p.toSubgraph.verts = (cycles.connectedComponentMk c).supp
p' : (M1.spanningCoe ∆ M2.spanningCoe).Walk a a
hp' : p'.IsCycle ∧ p'.snd = c ∧ p'.toSubgraph.verts = p.toSubgraph.verts
x' : V
hx' : Membership.mem (γ := Finset V) {x, b} x'
hx'p : x' ∈ p'.support
htw : ∀ (t : V), Membership.mem (γ := Finset V) {x, b} t → t ∈ (p'.takeUntil x' hx'p).support → t = x'
this : (p'.takeUntil x' hx'p).toSubgraph.Adj a (p'.takeUntil x' hx'p).snd
⊢ (p'.takeUntil x' hx'p).toSubgraph.Adj a c
have refine_1 : x' ≠ a
Tactic: rwa [Walk.snd_takeUntil, hp'.2.1] at this
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Tutte (hypothesis)
Goal:
V : Type u_1
G : SimpleGraph V
inst : Finite V
x a b c : V
M1 : (G ⊔ edge x b).Subgraph
M2 : (G ⊔ edge a c).Subgraph
hxa : G.Adj x a
hab : G.Adj a b
hnGxb : ¬G.Adj x b
hnGac : ¬G.Adj a c
hnxb : x ≠ b
hnxc : x ≠ c
hnac : a ≠ c
hnbc : b ≠ c
hM1 : M1.IsPerfectMatching
hM2 : M2.IsPerfectMatching
hM1xb : M1.Adj x b
hM2ac : M2.Adj a c
hM1nac : ¬M1.Adj a c
hsupG : G ⊔ edge x b ⊔ (G ⊔ edge a c) = G ⊔ edge a c ⊔ edge x b
cycles : SimpleGraph V := M1.spanningCoe ∆ M2.spanningCoe
hcalt : cycles.IsAlternating M2.spanningCoe
hcycles : (M1.spanningCoe ∆ M2.spanningCoe).IsCycles
hcac : cycles.Adj a c
hM1sub : M1.spanningCoe ≤ G ⊔ edge x b
hM2sub : M2.spanningCoe ≤ G ⊔ edge a c
induce_le :
  (cycles.connectedComponentMk c).toSimpleGraph.spanningCoe (s := (↑(cycles.connectedComponentMk c) : Set V)) ≤
    G ⊔ edge a c ⊔ edge x b
hxc : x ∈ (cycles.connectedComponentMk c).supp
hacc : a ∈ (cycles.connectedComponentMk c).supp
this_1 : (G : SimpleGraph V) → G.LocallyFinite
hnM2 : ∀ (x' : V) (h : x' ≠ c), ¬M2.Adj x' a
p : (M1.spanningCoe ∆ M2.spanningCoe).Walk a a
hp : p.IsCycle ∧ p.toSubgraph.verts = (cycles.connectedComponentMk c).supp
p' : (M1.spanningCoe ∆ M2.spanningCoe).Walk a a
hp' : p'.IsCycle ∧ p'.snd = c ∧ p'.toSubgraph.verts = p.toSubgraph.verts
x' : V
hx' : Membership.mem (γ := Finset V) {x, b} x'
hx'p : x' ∈ p'.support
htw : ∀ (t : V), Membership.mem (γ := Finset V) {x, b} t → t ∈ (p'.takeUntil x' hx'p).support → t = x'
this : (p'.takeUntil x' hx'p).toSubgraph.Adj a (p'.takeUntil x' hx'p).snd
⊢ x' ≠ a
have hx'₁ : x' = x ∨ x' = b
Tactic: simp only [Finset.mem_insert, Finset.mem_singleton] at hx'
Cleared hypotheses: hx'
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Tutte (goal)
Goal:
V : Type u_1
G : SimpleGraph V
inst : Finite V
x a b c : V
M1 : (G ⊔ edge x b).Subgraph
M2 : (G ⊔ edge a c).Subgraph
hxa : G.Adj x a
hab : G.Adj a b
hnGxb : ¬G.Adj x b
hnGac : ¬G.Adj a c
hnxb : x ≠ b
hnxc : x ≠ c
hnac : a ≠ c
hnbc : b ≠ c
hM1 : M1.IsPerfectMatching
hM2 : M2.IsPerfectMatching
hM1xb : M1.Adj x b
hM2ac : M2.Adj a c
hM1nac : ¬M1.Adj a c
hsupG : G ⊔ edge x b ⊔ (G ⊔ edge a c) = G ⊔ edge a c ⊔ edge x b
cycles : SimpleGraph V := M1.spanningCoe ∆ M2.spanningCoe
hcalt : cycles.IsAlternating M2.spanningCoe
hcycles : (M1.spanningCoe ∆ M2.spanningCoe).IsCycles
hcac : cycles.Adj a c
hM1sub : M1.spanningCoe ≤ G ⊔ edge x b
hM2sub : M2.spanningCoe ≤ G ⊔ edge a c
induce_le :
  (cycles.connectedComponentMk c).toSimpleGraph.spanningCoe (s := (↑(cycles.connectedComponentMk c) : Set V)) ≤
    G ⊔ edge a c ⊔ edge x b
hxc : x ∈ (cycles.connectedComponentMk c).supp
hacc : a ∈ (cycles.connectedComponentMk c).supp
this_1 : (G : SimpleGraph V) → G.LocallyFinite
hnM2 : ∀ (x' : V) (h : x' ≠ c), ¬M2.Adj x' a
p : (M1.spanningCoe ∆ M2.spanningCoe).Walk a a
hp : p.IsCycle ∧ p.toSubgraph.verts = (cycles.connectedComponentMk c).supp
p' : (M1.spanningCoe ∆ M2.spanningCoe).Walk a a
hp' : p'.IsCycle ∧ p'.snd = c ∧ p'.toSubgraph.verts = p.toSubgraph.verts
x' : V
hx'p : x' ∈ p'.support
htw : ∀ (t : V), Membership.mem (γ := Finset V) {x, b} t → t ∈ (p'.takeUntil x' hx'p).support → t = x'
this : (p'.takeUntil x' hx'p).toSubgraph.Adj a (p'.takeUntil x' hx'p).snd
hx' : x' = x ∨ x' = b
⊢ x' ≠ a
have refine_1 : ∀ {M1 : (G ⊔ edge x' b).Subgraph} (hxa : G.Adj x' a) (hnGxb : ¬G.Adj x' b) (hnxb : x' ≠ b) (hnxc : x' ≠ c)
  (hM1 : M1.IsPerfectMatching) (hM1xb : M1.Adj x' b) (hM1nac : ¬M1.Adj a c)
  (hsupG : G ⊔ edge x' b ⊔ (G ⊔ edge a c) = G ⊔ edge a c ⊔ edge x' b),
  let cycles := M1.spanningCoe ∆ M2.spanningCoe;
  ∀ (hcalt : cycles.IsAlternating M2.spanningCoe) (hcycles : (M1.spanningCoe ∆ M2.spanningCoe).IsCycles)
    (hcac : cycles.Adj a c) (hM1sub : M1.spanningCoe ≤ G ⊔ edge x' b)
    (induce_le :
      (cycles.connectedComponentMk c).toSimpleGraph.spanningCoe (s := (↑(cycles.connectedComponentMk c) : Set V)) ≤
        G ⊔ edge a c ⊔ edge x' b)
    (hxc : x' ∈ (cycles.connectedComponentMk c).supp) (hacc : a ∈ (cycles.connectedComponentMk c).supp)
    (p : (M1.spanningCoe ∆ M2.spanningCoe).Walk a a)
    (hp : p.IsCycle ∧ p.toSubgraph.verts = (cycles.connectedComponentMk c).supp)
    (p' : (M1.spanningCoe ∆ M2.spanningCoe).Walk a a)
    (hp' : p'.IsCycle ∧ p'.snd = c ∧ p'.toSubgraph.verts = p.toSubgraph.verts) (hx'p : x' ∈ p'.support)
    (htw : ∀ (t : V), Membership.mem (γ := Finset V) {x', b} t → t ∈ (p'.takeUntil x' hx'p).support → t = x')
    (this : (p'.takeUntil x' hx'p).toSubgraph.Adj a (p'.takeUntil x' hx'p).snd), x' ≠ a
Tactic: obtain rfl | rfl := hx'
Cleared hypotheses: x, M1, hxa, hnGxb, hnxb, hnxc, hM1, hM1xb, hM1nac, hsupG, cycles, hcalt, hcycles, hcac, hM1sub, induce_le, hxc, hacc, p, hp, p', hp', hx'p, htw, this, hx'
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Tutte (goal)
Goal:
V : Type u_1
G : SimpleGraph V
inst : Finite V
x a b c : V
M1 : (G ⊔ edge x b).Subgraph
M2 : (G ⊔ edge a c).Subgraph
hxa : G.Adj x a
hab : G.Adj a b
hnGxb : ¬G.Adj x b
hnGac : ¬G.Adj a c
hnxb : x ≠ b
hnxc : x ≠ c
hnac : a ≠ c
hnbc : b ≠ c
hM1 : M1.IsPerfectMatching
hM2 : M2.IsPerfectMatching
hM1xb : M1.Adj x b
hM2ac : M2.Adj a c
hM1nac : ¬M1.Adj a c
hsupG : G ⊔ edge x b ⊔ (G ⊔ edge a c) = G ⊔ edge a c ⊔ edge x b
cycles : SimpleGraph V := M1.spanningCoe ∆ M2.spanningCoe
hcalt : cycles.IsAlternating M2.spanningCoe
hcycles : (M1.spanningCoe ∆ M2.spanningCoe).IsCycles
hcac : cycles.Adj a c
hM1sub : M1.spanningCoe ≤ G ⊔ edge x b
hM2sub : M2.spanningCoe ≤ G ⊔ edge a c
induce_le :
  (cycles.connectedComponentMk c).toSimpleGraph.spanningCoe (s := (↑(cycles.connectedComponentMk c) : Set V)) ≤
    G ⊔ edge a c ⊔ edge x b
hxc : x ∈ (cycles.connectedComponentMk c).supp
hacc : a ∈ (cycles.connectedComponentMk c).supp
this : (G : SimpleGraph V) → G.LocallyFinite
hnM2 : ∀ (x' : V) (h : x' ≠ c), ¬M2.Adj x' a
⊢ ∃ (G' : SimpleGraph V), G'.IsAlternating M2.spanningCoe ∧ G'.IsCycles ∧ ¬G'.Adj x b ∧ G'.Adj a c ∧ G' ≤ G ⊔ edge a c
have neg : ∀ (x' : V) (hx' : Membership.mem (γ := Finset V) {x, b} x') (p : cycles.Walk a x') (hp : p.IsPath)
  (hpac : p.toSubgraph.Adj a c) (hnpxb : ¬p.toSubgraph.Adj x b),
  ∃ (G' : SimpleGraph V), G'.IsAlternating M2.spanningCoe ∧ G'.IsCycles ∧ ¬G'.Adj x b ∧ G'.Adj a c ∧ G' ≤ G ⊔ edge a c
Tactic: obtain ⟨x', hx', p, hp, hpac, hnpxb⟩ :
      ∃ x' ∈ ({x, b} : Finset V), ∃ (p : cycles.Walk a x'), p.IsPath ∧
        p.toSubgraph.Adj a c ∧ ¬p.toSubgraph.Adj x b := by
    obtain ⟨p, hp⟩ := hcycles.exists_cycle_toSubgraph_verts_eq_connectedComponentSupp hacc
      ⟨_, hcac⟩
    obtain ⟨p', hp'⟩ := hp.1.exists_isCycle_snd_verts_eq (by
      rwa [hp.1.adj_toSubgraph_iff_of_isCycles hcycles (hp.2 ▸ hacc)])
    obtain ⟨x', hx', hx'p, htw⟩ := Walk.exists_mem_support_forall_mem_support_imp_eq {x, b} <| by
      use x
      simp only [Finset.mem_filter, Finset.mem_insert, Finset.mem_singleton, true_or, true_and]
      rwa [← @Walk.mem_verts_toSubgraph, hp'.2.2, hp.2]
    refine ⟨x', hx', p'.takeUntil x' hx'p, hp'.1.isPath_takeUntil hx'p, ?_, fun h ↦ ?_⟩; swap
    · simp [htw _ (by simp) (Walk.mem_support_of_adj_toSubgraph h.symm),
        htw _ (by simp) (Walk.mem_support_of_adj_toSubgraph h)] at hnxb
    have : (p'.takeUntil x' hx'p).toSubgraph.Adj a (p'.takeUntil x' hx'p).snd := by
      apply Walk.toSubgraph_adj_snd
      rw [Walk.nil_takeUntil]
      grind
    rwa [Walk.snd_takeUntil, hp'.2.1] at this
    simp only [Finset.mem_insert, Finset.mem_singleton] at hx'
    obtain rfl | rfl := hx'
    exacts [hxa.ne, hab.ne.symm]
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Tutte (goal)
Goal:
V : Type u_1
G : SimpleGraph V
inst : Finite V
x a b c : V
M1 : (G ⊔ edge x b).Subgraph
M2 : (G ⊔ edge a c).Subgraph
hxa : G.Adj x a
hab : G.Adj a b
hnGxb : ¬G.Adj x b
hnGac : ¬G.Adj a c
hnxb : x ≠ b
hnxc : x ≠ c
hnac : a ≠ c
hnbc : b ≠ c
hM1 : M1.IsPerfectMatching
hM2 : M2.IsPerfectMatching
hM1xb : M1.Adj x b
hM2ac : M2.Adj a c
hM1nac : ¬M1.Adj a c
hsupG : G ⊔ edge x b ⊔ (G ⊔ edge a c) = G ⊔ edge a c ⊔ edge x b
cycles : SimpleGraph V := M1.spanningCoe ∆ M2.spanningCoe
hcalt : cycles.IsAlternating M2.spanningCoe
hcycles : (M1.spanningCoe ∆ M2.spanningCoe).IsCycles
hcac : cycles.Adj a c
hM1sub : M1.spanningCoe ≤ G ⊔ edge x b
hM2sub : M2.spanningCoe ≤ G ⊔ edge a c
induce_le :
  (cycles.connectedComponentMk c).toSimpleGraph.spanningCoe (s := (↑(cycles.connectedComponentMk c) : Set V)) ≤
    G ⊔ edge a c ⊔ edge x b
hxc : x ∈ (cycles.connectedComponentMk c).supp
hacc : a ∈ (cycles.connectedComponentMk c).supp
this : (G : SimpleGraph V) → G.LocallyFinite
hnM2 : ∀ (x' : V) (h : x' ≠ c), ¬M2.Adj x' a
x' : V
hx' : Membership.mem (γ := Finset V) {x, b} x'
p : cycles.Walk a x'
hp : p.IsPath
hpac : p.toSubgraph.Adj a c
hnpxb : ¬p.toSubgraph.Adj x b
⊢ p.toSubgraph.spanningCoe ≤ G ⊔ edge a c
have rw : p.toSubgraph.spanningCoe ≤ G ⊔ edge a c ⊔ edge x b
Tactic: rw [← sdiff_edge _ (by simpa : ¬p.toSubgraph.spanningCoe.Adj x b), sdiff_le_iff']
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Tutte (goal)
Goal:
V : Type u_1
G : SimpleGraph V
inst : Finite V
x a b c : V
M1 : (G ⊔ edge x b).Subgraph
M2 : (G ⊔ edge a c).Subgraph
hxa : G.Adj x a
hab : G.Adj a b
hnGxb : ¬G.Adj x b
hnGac : ¬G.Adj a c
hnxb : x ≠ b
hnxc : x ≠ c
hnac : a ≠ c
hnbc : b ≠ c
hM1 : M1.IsPerfectMatching
hM2 : M2.IsPerfectMatching
hM1xb : M1.Adj x b
hM2ac : M2.Adj a c
hM1nac : ¬M1.Adj a c
hsupG : G ⊔ edge x b ⊔ (G ⊔ edge a c) = G ⊔ edge a c ⊔ edge x b
cycles : SimpleGraph V := M1.spanningCoe ∆ M2.spanningCoe
hcalt : cycles.IsAlternating M2.spanningCoe
hcycles : (M1.spanningCoe ∆ M2.spanningCoe).IsCycles
hcac : cycles.Adj a c
hM1sub : M1.spanningCoe ≤ G ⊔ edge x b
hM2sub : M2.spanningCoe ≤ G ⊔ edge a c
induce_le :
  (cycles.connectedComponentMk c).toSimpleGraph.spanningCoe (s := (↑(cycles.connectedComponentMk c) : Set V)) ≤
    G ⊔ edge a c ⊔ edge x b
hxc : x ∈ (cycles.connectedComponentMk c).supp
hacc : a ∈ (cycles.connectedComponentMk c).supp
this : (G : SimpleGraph V) → G.LocallyFinite
hnM2 : ∀ (x' : V) (h : x' ≠ c), ¬M2.Adj x' a
x' : V
hx' : Membership.mem (γ := Finset V) {x, b} x'
p : cycles.Walk a x'
hp : p.IsPath
hpac : p.toSubgraph.Adj a c
hnpxb : ¬p.toSubgraph.Adj x b
v w : V
hvw : p.toSubgraph.spanningCoe.Adj v w
⊢ (G ⊔ edge a c ⊔ edge x b).Adj v w
have a₁ : (M1.spanningCoe ⊔ M2.spanningCoe).Adj v w
Tactic: apply hsupG ▸ sup_le_sup hM1sub hM2sub
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Tutte (hypothesis)
Goal:
V : Type u_1
G : SimpleGraph V
inst : Finite V
x a b c : V
M1 : (G ⊔ edge x b).Subgraph
M2 : (G ⊔ edge a c).Subgraph
hxa : G.Adj x a
hab : G.Adj a b
hnGxb : ¬G.Adj x b
hnGac : ¬G.Adj a c
hnxb : x ≠ b
hnxc : x ≠ c
hnac : a ≠ c
hnbc : b ≠ c
hM1 : M1.IsPerfectMatching
hM2 : M2.IsPerfectMatching
hM1xb : M1.Adj x b
hM2ac : M2.Adj a c
hM1nac : ¬M1.Adj a c
hsupG : G ⊔ edge x b ⊔ (G ⊔ edge a c) = G ⊔ edge a c ⊔ edge x b
cycles : SimpleGraph V := M1.spanningCoe ∆ M2.spanningCoe
hcalt : cycles.IsAlternating M2.spanningCoe
hcycles : (M1.spanningCoe ∆ M2.spanningCoe).IsCycles
hcac : cycles.Adj a c
hM1sub : M1.spanningCoe ≤ G ⊔ edge x b
hM2sub : M2.spanningCoe ≤ G ⊔ edge a c
induce_le :
  (cycles.connectedComponentMk c).toSimpleGraph.spanningCoe (s := (↑(cycles.connectedComponentMk c) : Set V)) ≤
    G ⊔ edge a c ⊔ edge x b
hxc : x ∈ (cycles.connectedComponentMk c).supp
hacc : a ∈ (cycles.connectedComponentMk c).supp
this : (G : SimpleGraph V) → G.LocallyFinite
hnM2 : ∀ (x' : V) (h : x' ≠ c), ¬M2.Adj x' a
x' : V
hx' : Membership.mem (γ := Finset V) {x, b} x'
p : cycles.Walk a x'
hp : p.IsPath
hpac : p.toSubgraph.Adj a c
hnpxb : ¬p.toSubgraph.Adj x b
v w : V
hvw : p.toSubgraph.spanningCoe.Adj v w
⊢ (M1.spanningCoe ⊔ M2.spanningCoe).Adj v w
have this₁ : cycles.Adj v w
Tactic: have := p.toSubgraph.spanningCoe_le hvw
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Tutte (hypothesis)
Goal:
V : Type u_1
G : SimpleGraph V
inst : Finite V
x a b c : V
M1 : (G ⊔ edge x b).Subgraph
M2 : (G ⊔ edge a c).Subgraph
hxa : G.Adj x a
hab : G.Adj a b
hnGxb : ¬G.Adj x b
hnGac : ¬G.Adj a c
hnxb : x ≠ b
hnxc : x ≠ c
hnac : a ≠ c
hnbc : b ≠ c
hM1 : M1.IsPerfectMatching
hM2 : M2.IsPerfectMatching
hM1xb : M1.Adj x b
hM2ac : M2.Adj a c
hM1nac : ¬M1.Adj a c
hsupG : G ⊔ edge x b ⊔ (G ⊔ edge a c) = G ⊔ edge a c ⊔ edge x b
cycles : SimpleGraph V := M1.spanningCoe ∆ M2.spanningCoe
hcalt : cycles.IsAlternating M2.spanningCoe
hcycles : (M1.spanningCoe ∆ M2.spanningCoe).IsCycles
hcac : cycles.Adj a c
hM1sub : M1.spanningCoe ≤ G ⊔ edge x b
hM2sub : M2.spanningCoe ≤ G ⊔ edge a c
induce_le :
  (cycles.connectedComponentMk c).toSimpleGraph.spanningCoe (s := (↑(cycles.connectedComponentMk c) : Set V)) ≤
    G ⊔ edge a c ⊔ edge x b
hxc : x ∈ (cycles.connectedComponentMk c).supp
hacc : a ∈ (cycles.connectedComponentMk c).supp
this : (G : SimpleGraph V) → G.LocallyFinite
hnM2 : ∀ (x' : V) (h : x' ≠ c), ¬M2.Adj x' a
x' : V
hx' : Membership.mem (γ := Finset V) {x, b} x'
p : cycles.Walk a x'
hp : p.IsPath
hpac : p.toSubgraph.Adj a c
hnpxb : ¬p.toSubgraph.Adj x b
⊢ ∃ (G' : SimpleGraph V), G'.IsAlternating M2.spanningCoe ∧ G'.IsCycles ∧ ¬G'.Adj x b ∧ G'.Adj a c ∧ G' ≤ G ⊔ edge a c
have hle : p.toSubgraph.spanningCoe ≤ G ⊔ edge a c
Tactic: have hle : p.toSubgraph.spanningCoe ≤ G ⊔ edge a c := by
    rw [← sdiff_edge _ (by simpa : ¬p.toSubgraph.spanningCoe.Adj x b), sdiff_le_iff']
    intro v w hvw
    apply hsupG ▸ sup_le_sup hM1sub hM2sub
    have := p.toSubgraph.spanningCoe_le hvw
    simp only [cycles, symmDiff_def] at this
    aesop
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Tutte (goal)
Goal:
V : Type u_1
G : SimpleGraph V
inst : Finite V
x a b c : V
M1 : (G ⊔ edge x b).Subgraph
M2 : (G ⊔ edge a c).Subgraph
hxa : G.Adj x a
hab : G.Adj a b
hnGxb : ¬G.Adj x b
hnGac : ¬G.Adj a c
hnxb : x ≠ b
hnxc : x ≠ c
hnac : a ≠ c
hnbc : b ≠ c
hM1 : M1.IsPerfectMatching
hM2 : M2.IsPerfectMatching
hM1xb : M1.Adj x b
hM2ac : M2.Adj a c
hM1nac : ¬M1.Adj a c
hsupG : G ⊔ edge x b ⊔ (G ⊔ edge a c) = G ⊔ edge a c ⊔ edge x b
cycles : SimpleGraph V := M1.spanningCoe ∆ M2.spanningCoe
hcalt : cycles.IsAlternating M2.spanningCoe
hcycles : (M1.spanningCoe ∆ M2.spanningCoe).IsCycles
hcac : cycles.Adj a c
hM1sub : M1.spanningCoe ≤ G ⊔ edge x b
hM2sub : M2.spanningCoe ≤ G ⊔ edge a c
induce_le :
  (cycles.connectedComponentMk c).toSimpleGraph.spanningCoe (s := (↑(cycles.connectedComponentMk c) : Set V)) ≤
    G ⊔ edge a c ⊔ edge x b
hxc : x ∈ (cycles.connectedComponentMk c).supp
hacc : a ∈ (cycles.connectedComponentMk c).supp
this : (G : SimpleGraph V) → G.LocallyFinite
hnM2 : ∀ (x' : V) (h : x' ≠ c), ¬M2.Adj x' a
x'_1 : V
hx'_1 : Membership.mem (γ := Finset V) {x, b} x'_1
p : cycles.Walk a x'_1
hp : p.IsPath
hpac : p.toSubgraph.Adj a c
hnpxb : ¬p.toSubgraph.Adj x b
hle : p.toSubgraph.spanningCoe ≤ G ⊔ edge a c
x' : V
hx' : Membership.mem (γ := Set V) {x, b} x'
c' : V
hc : c' ≠ a
hadj : p.toSubgraph.Adj c' x'
⊢ M2.Adj c' x'
have refine : ∀ (hl : (M1.spanningCoe \ M2.spanningCoe).Adj c' x'), p.toSubgraph.Adj x b
Tactic: refine (hadj.adj_sub.resolve_left fun hl ↦ hnpxb ?_).1
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Tutte (goal)
Goal:
V : Type u_1
G : SimpleGraph V
inst : Finite V
x a b c : V
M1 : (G ⊔ edge x b).Subgraph
M2 : (G ⊔ edge a c).Subgraph
hxa : G.Adj x a
hab : G.Adj a b
hnGxb : ¬G.Adj x b
hnGac : ¬G.Adj a c
hnxb : x ≠ b
hnxc : x ≠ c
hnac : a ≠ c
hnbc : b ≠ c
hM1 : M1.IsPerfectMatching
hM2 : M2.IsPerfectMatching
hM1xb : M1.Adj x b
hM2ac : M2.Adj a c
hM1nac : ¬M1.Adj a c
hsupG : G ⊔ edge x b ⊔ (G ⊔ edge a c) = G ⊔ edge a c ⊔ edge x b
cycles : SimpleGraph V := M1.spanningCoe ∆ M2.spanningCoe
hcalt : cycles.IsAlternating M2.spanningCoe
hcycles : (M1.spanningCoe ∆ M2.spanningCoe).IsCycles
hcac : cycles.Adj a c
hM1sub : M1.spanningCoe ≤ G ⊔ edge x b
hM2sub : M2.spanningCoe ≤ G ⊔ edge a c
induce_le :
  (cycles.connectedComponentMk c).toSimpleGraph.spanningCoe (s := (↑(cycles.connectedComponentMk c) : Set V)) ≤
    G ⊔ edge a c ⊔ edge x b
hxc : x ∈ (cycles.connectedComponentMk c).supp
hacc : a ∈ (cycles.connectedComponentMk c).supp
this : (G : SimpleGraph V) → G.LocallyFinite
hnM2 : ∀ (x' : V) (h : x' ≠ c), ¬M2.Adj x' a
x'_1 : V
hx'_1 : Membership.mem (γ := Finset V) {x, b} x'_1
p : cycles.Walk a x'_1
hp : p.IsPath
hpac : p.toSubgraph.Adj a c
hnpxb : ¬p.toSubgraph.Adj x b
hle : p.toSubgraph.spanningCoe ≤ G ⊔ edge a c
x' : V
hx' : Membership.mem (γ := Set V) {x, b} x'
c' : V
hc : c' ≠ a
hadj : p.toSubgraph.Adj c' x'
hl : (M1.spanningCoe \ M2.spanningCoe).Adj c' x'
⊢ p.toSubgraph.Adj x b
have obtain : ∀ (w : V) (hw : ∀ (y : V), (fun (w : V) ↦ M1.Adj x' w) y → y = w), p.toSubgraph.Adj x b
Tactic: obtain ⟨w, -, hw⟩ := hM1.1 (hM1.2 x')