awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (hypothesis)
Goal:
V : Type u
G : SimpleGraph V
inst : Nonempty V
n : ℕ
hc : G.Colorable n
m : ℕ
hm : Membership.mem (γ := Set ℕ) {n : ℕ | G.Colorable n} m
h' : m < Nat.succ 0
i : ℕ
hi : i < m
⊢ False
have h₁ : i < 0
Tactic: have h₁ : i < 0 := lt_of_lt_of_le hi (Nat.le_of_lt_succ h')
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (hypothesis)
Goal:
V : Type u
G : SimpleGraph V
h : G.chromaticNumber ≠ ⊤
⊢ G.Colorable G.chromaticNumber.toNat
have h₁ : ∃ (n : ℕ), G.Colorable n
Tactic: rw [chromaticNumber_ne_top_iff_exists] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
G : SimpleGraph V
h : ∃ (n : ℕ), G.Colorable n
⊢ G.Colorable G.chromaticNumber.toNat
have obtain : ∀ (n : ℕ) (hn : G.Colorable n), G.Colorable G.chromaticNumber.toNat
Tactic: obtain ⟨n, hn⟩ := h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
G : SimpleGraph V
V' : Type u_4
G' : SimpleGraph V'
h : ∀ (n : ℕ), G'.Colorable n → G.Colorable n
⊢ LE.le (α := ℕ∞) (⨅ n ∈ setOf G.Colorable, (↑n : ℕ∞)) (⨅ n ∈ setOf G'.Colorable, (↑n : ℕ∞))
have simp : ∀ (i : ℕ) (i_1 : G'.Colorable i), LE.le (α := ℕ∞) (⨅ (n : ℕ), ⨅ (_ : G.Colorable n), (↑n : ℕ∞)) (↑i : ℕ∞)
Tactic: simp only [Set.mem_setOf_eq, le_iInf_iff]
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (hypothesis)
Goal:
V : Type u
G : SimpleGraph V
V' : Type u_4
G' : SimpleGraph V'
h : ∀ (n : ℕ), G'.Colorable n → G.Colorable n
m : ℕ
hc : G'.Colorable m
⊢ LE.le (α := ℕ∞) (⨅ (n : ℕ), ⨅ (_ : G.Colorable n), (↑n : ℕ∞)) (↑m : ℕ∞)
have this : G.Colorable m
Tactic: have := h _ hc
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (hypothesis)
Goal:
V : Type u
G : SimpleGraph V
V' : Type u_4
G' : SimpleGraph V'
h : ∀ (n : ℕ), G'.Colorable n → G.Colorable n
m : ℕ
hc : G'.Colorable m
this : G.Colorable m
⊢ LE.le (α := ℕ∞) (⨅ (n : ℕ), ⨅ (_ : G.Colorable n), (↑n : ℕ∞)) (↑m : ℕ∞)
have this₁ : G.chromaticNumber ≤ (↑m : ℕ∞)
Tactic: rw [← chromaticNumber_le_iff_colorable] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
G : SimpleGraph V
α : Type u_2
inst : Fintype α
⊢ (↑(card α) : ℕ∞) ≤ G.chromaticNumber ↔
    ∀ (C : G.Coloring α), Surjective.{u + 1, u_2 + 1} (α := V) (β := α) (⇑C : (a : V) → α)
have refine_1 : ∀ (h : (↑(card α) : ℕ∞) ≤ G.chromaticNumber) (C : G.Coloring α),
  Surjective.{u + 1, u_2 + 1} (α := V) (β := α) (⇑C : (a : V) → α)
Tactic: refine ⟨fun h C ↦ ?_, fun h ↦ ?_⟩
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
G : SimpleGraph V
α : Type u_2
inst : Fintype α
refine_1 :
  ∀ (h : (↑(card α) : ℕ∞) ≤ G.chromaticNumber) (C : G.Coloring α),
    Surjective.{u + 1, u_2 + 1} (α := V) (β := α) (⇑C : (a : V) → α)
⊢ (↑(card α) : ℕ∞) ≤ G.chromaticNumber ↔
    ∀ (C : G.Coloring α), Surjective.{u + 1, u_2 + 1} (α := V) (β := α) (⇑C : (a : V) → α)
have refine_2 : ∀ (h : ∀ (C : G.Coloring α), Surjective.{u + 1, u_2 + 1} (α := V) (β := α) (⇑C : (a : V) → α)),
  (↑(card α) : ℕ∞) ≤ G.chromaticNumber
Tactic: refine ⟨fun h C ↦ ?_, fun h ↦ ?_⟩
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
G : SimpleGraph V
α : Type u_2
inst : Fintype α
h : (↑(card α) : ℕ∞) ≤ G.chromaticNumber
C : G.Coloring α
⊢ Surjective.{u + 1, u_2 + 1} (α := V) (β := α) (⇑C : (a : V) → α)
have refine_1 : ∀ (h : card α ≤ sInf {n' : ℕ | G.Colorable n'}) (C : G.Coloring α),
  Surjective.{u + 1, u_2 + 1} (α := V) (β := α) (⇑C : (a : V) → α)
Tactic: rw [C.colorable.chromaticNumber_eq_sInf, Nat.cast_le] at h
Cleared hypotheses: h, C
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
G : SimpleGraph V
α : Type u_2
inst : Fintype α
h : card α ≤ sInf {n' : ℕ | G.Colorable n'}
C : G.Coloring α
i : α
⊢ ∃ (a : V), (C : (a : V) → α) a = i
have refine_1 : ∀ (hi : ∀ (a : V), (C : (a : V) → α) a ≠ i), False
Tactic: by_contra! hi
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
G : SimpleGraph V
α : Type u_2
inst : Fintype α
h : ∀ (C : G.Coloring α), Surjective.{u + 1, u_2 + 1} (α := V) (β := α) (⇑C : (a : V) → α)
⊢ (↑(card α) : ℕ∞) ≤ G.chromaticNumber
have refine_2 : ∀ (i : ℕ) (i_1 : G.Colorable i), card α ≤ i
Tactic: simp only [chromaticNumber, Set.mem_setOf_eq, le_iInf_iff, Nat.cast_le]
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
G : SimpleGraph V
α : Type u_2
inst : Fintype α
h : ∀ (C : G.Coloring α), Surjective.{u + 1, u_2 + 1} (α := V) (β := α) (⇑C : (a : V) → α)
i : ℕ
C : G.Coloring (Fin i)
⊢ card α ≤ i
have refine_2 : ∀ (h : i < card α), ∃ (C : G.Coloring α), ¬Surjective.{u + 1, u_2 + 1} (α := V) (β := α) (⇑C : (a : V) → α)
Tactic: contrapose! h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
G : SimpleGraph V
α : Type u_2
inst : Fintype α
i : ℕ
C : G.Coloring (Fin i)
h : i < card α
⊢ ∃ (C : G.Coloring α), ¬Surjective.{u + 1, u_2 + 1} (α := V) (β := α) (⇑C : (a : V) → α)
have refine_2 : ∀
  (hC :
    Surjective.{u + 1, u_2 + 1} (α := V) (β := α)
      (⇑((G.recolorOfCardLE sorry : (a : G.Coloring (Fin i)) → G.Coloring α) C : G.Coloring α) : (a : V) → α)),
  False
Tactic: refine ⟨G.recolorOfCardLE (by simpa using h.le) C, fun hC ↦ ?_⟩
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
G : SimpleGraph V
α : Type u_2
inst : Fintype α
hG : G.Colorable (card α)
⊢ G.chromaticNumber = (↑(card α) : ℕ∞) ↔
    ∀ (C : G.Coloring α), Surjective.{u + 1, u_2 + 1} (α := V) (β := α) (⇑C : (a : V) → α)
have rw : (↑(card α) : ℕ∞) ≤ G.chromaticNumber ↔
  ∀ (C : G.Coloring α), Surjective.{u + 1, u_2 + 1} (α := V) (β := α) (⇑C : (a : V) → α)
Tactic: rw [← hG.chromaticNumber_le.ge_iff_eq, card_le_chromaticNumber_iff_forall_surjective]
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
G : SimpleGraph V
α : Type u_2
inst : Fintype α
hG : G.Colorable (card α)
rw :
  (↑(card α) : ℕ∞) ≤ G.chromaticNumber ↔
    ∀ (C : G.Coloring α), Surjective.{u + 1, u_2 + 1} (α := V) (β := α) (⇑C : (a : V) → α)
⊢ G.chromaticNumber = (↑(card α) : ℕ∞) ↔
    ∀ (C : G.Coloring α), Surjective.{u + 1, u_2 + 1} (α := V) (β := α) (⇑C : (a : V) → α)
have rw₁ : (∀ (C : G.Coloring α), Surjective.{u + 1, u_2 + 1} (α := V) (β := α) (⇑C : (a : V) → α)) ↔
  ∀ (C : G.Coloring α), Surjective.{u + 1, u_2 + 1} (α := V) (β := α) (⇑C : (a : V) → α)
Tactic: rw [← hG.chromaticNumber_le.ge_iff_eq, card_le_chromaticNumber_iff_forall_surjective]
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
G : SimpleGraph V
n : ℕ
hG : G.Colorable n
⊢ G.chromaticNumber = (↑n : ℕ∞) ↔ ∀ (C : G.Coloring (Fin n)), Surjective (α := V) (β := Fin n) (⇑C : (a : V) → Fin n)
have rw : (↑n : ℕ∞) ≤ G.chromaticNumber ↔ ∀ (C : G.Coloring (Fin n)), Surjective (α := V) (β := Fin n) (⇑C : (a : V) → Fin n)
Tactic: rw [← hG.chromaticNumber_le.ge_iff_eq, le_chromaticNumber_iff_forall_surjective]
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
G : SimpleGraph V
n : ℕ
hG : G.Colorable n
rw : (↑n : ℕ∞) ≤ G.chromaticNumber ↔ ∀ (C : G.Coloring (Fin n)), Surjective (α := V) (β := Fin n) (⇑C : (a : V) → Fin n)
⊢ G.chromaticNumber = (↑n : ℕ∞) ↔ ∀ (C : G.Coloring (Fin n)), Surjective (α := V) (β := Fin n) (⇑C : (a : V) → Fin n)
have rw₁ : (∀ (C : G.Coloring (Fin n)), Surjective (α := V) (β := Fin n) (⇑C : (a : V) → Fin n)) ↔
  ∀ (C : G.Coloring (Fin n)), Surjective (α := V) (β := Fin n) (⇑C : (a : V) → Fin n)
Tactic: rw [← hG.chromaticNumber_le.ge_iff_eq, le_chromaticNumber_iff_forall_surjective]
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
inst : Fintype V
⊢ chromaticNumber.{u} (V := V) ⊤ = (↑(card V) : ℕ∞)
have rw : ∀ (C : Coloring (V := V) ⊤ V), Surjective (α := V) (β := V) (⇑C : (a : V) → V)
Tactic: rw [chromaticNumber_eq_card_iff_forall_surjective (selfColoring _).colorable]
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
inst : Fintype V
C : Coloring (V := V) ⊤ V
⊢ Surjective (α := V) (β := V) (⇑C : (a : V) → V)
have rw : Injective (α := V) (β := V) (⇑C : (a : V) → V)
Tactic: rw [← Finite.injective_iff_surjective]
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u_4
inst : Infinite V
⊢ chromaticNumber.{u_4} (V := V) ⊤ = ⊤
have by_contra : ∀ (hc : ¬chromaticNumber.{u_4} (V := V) ⊤ = ⊤), False
Tactic: by_contra hc
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (hypothesis)
Goal:
V : Type u_4
inst : Infinite V
hc : ¬chromaticNumber.{u_4} (V := V) ⊤ = ⊤
⊢ False
have hc₁ : ∃ (n : ℕ), Colorable.{u_4} (V := V) ⊤ n
Tactic: rw [← Ne, chromaticNumber_ne_top_iff_exists] at hc
Cleared hypotheses: hc
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u_4
inst : Infinite V
hc : ∃ (n : ℕ), Colorable.{u_4} (V := V) ⊤ n
⊢ False
have obtain : ∀ (n : ℕ) (hn : Coloring (V := V) ⊤ (Fin n)), False
Tactic: obtain ⟨n, ⟨hn⟩⟩ := hc
Cleared hypotheses: hc
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
⊢ chromaticNumber.{u} (V := V) ⊤ = ENat.card V
have inl : Finite V → chromaticNumber.{u} (V := V) ⊤ = ENat.card V
Tactic: cases finite_or_infinite V
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
inl : Finite V → chromaticNumber.{u} (V := V) ⊤ = ENat.card V
⊢ chromaticNumber.{u} (V := V) ⊤ = ENat.card V
have inr : Infinite V → chromaticNumber.{u} (V := V) ⊤ = ENat.card V
Tactic: cases finite_or_infinite V
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
h : Finite V
⊢ chromaticNumber.{u} (V := V) ⊤ = ENat.card V
have inl : ∀ (this : Fintype V), chromaticNumber.{u} (V := V) ⊤ = ENat.card V
Tactic: have := Fintype.ofFinite ‹_›
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
G : SimpleGraph V
inst : Fintype V
h : G.chromaticNumber = (↑(card V) : ℕ∞)
⊢ G = ⊤
have by_contra! : ∀ (hh : G ≠ ⊤), False
Tactic: by_contra! hh
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
G : SimpleGraph V
inst : Fintype V
h : G.chromaticNumber = (↑(card V) : ℕ∞)
hh : G ≠ ⊤
⊢ G.chromaticNumber ≤ HSub.hSub (α := ℕ∞) (↑(card V) : ℕ∞) 1
have obtain : ∀ (a b : V) (hne : a ≠ b), ¬G.Adj a b → G.chromaticNumber ≤ HSub.hSub (α := ℕ∞) (↑(card V) : ℕ∞) 1
Tactic: obtain ⟨a, b, hne, _⟩ := ne_top_iff_exists_not_adj.mp hh
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
G : SimpleGraph V
inst : Fintype V
h : G.chromaticNumber = (↑(card V) : ℕ∞)
hh : G ≠ ⊤
a b : V
hne : a ≠ b
right : ¬G.Adj a b
⊢ G.chromaticNumber ≤ HSub.hSub (α := ℕ∞) (↑(card V) : ℕ∞) 1
have apply : G.Colorable (card V - 1)
Tactic: apply chromaticNumber_le_iff_colorable.mpr
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
G : SimpleGraph V
inst : Fintype V
h : G.chromaticNumber = (↑(card V) : ℕ∞)
hh : G ≠ ⊤
a b : V
hne : a ≠ b
right : ¬G.Adj a b
⊢ G.Colorable (card V - 1)
have suffices : G.Coloring ↥(Finset.erase Finset.univ b)
Tactic: suffices G.Coloring (Finset.univ.erase b) by simpa using Coloring.colorable this
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
G : SimpleGraph V
inst : Fintype V
h : G.chromaticNumber = (↑(card V) : ℕ∞)
hh : G ≠ ⊤
a b : V
hne : a ≠ b
right : ¬G.Adj a b
⊢ G.Coloring ↥(Finset.erase Finset.univ b)
have apply : ∀ {v w : V},
  G.Adj v w →
    Ne.{u + 1} (α := ↥(Finset.erase Finset.univ b))
      (if h' : v ≠ b then Subtype.mk (p := fun (x : V) ↦ x ∈ Finset.erase Finset.univ b) v sorry
      else Subtype.mk (p := fun (x : V) ↦ x ∈ Finset.erase Finset.univ b) a sorry)
      (if h' : w ≠ b then Subtype.mk (p := fun (x : V) ↦ x ∈ Finset.erase Finset.univ b) w sorry
      else Subtype.mk (p := fun (x : V) ↦ x ∈ Finset.erase Finset.univ b) a sorry)
Tactic: apply Coloring.mk (fun x ↦ if h' : x ≠ b then ⟨x, by simp [h']⟩ else ⟨a, by simp [hne]⟩)
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (hypothesis)
Goal:
V : Type u
G : SimpleGraph V
inst : Fintype V
h : G.chromaticNumber = (↑(card V) : ℕ∞)
hh : G ≠ ⊤
⊢ False
have this : G.chromaticNumber ≤ HSub.hSub (α := ℕ∞) (↑(card V) : ℕ∞) 1
Tactic: have : G.chromaticNumber ≤ Fintype.card V - 1 := by
    obtain ⟨a, b, hne, _⟩ := ne_top_iff_exists_not_adj.mp hh
    apply chromaticNumber_le_iff_colorable.mpr
    suffices G.Coloring (Finset.univ.erase b) by simpa using Coloring.colorable this
    apply Coloring.mk (fun x ↦ if h' : x ≠ b then ⟨x, by simp [h']⟩ else ⟨a, by simp [hne]⟩)
    grind [Adj.ne', adj_symm]
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (hypothesis)
Goal:
V : Type u
G : SimpleGraph V
inst : Fintype V
h : G.chromaticNumber = (↑(card V) : ℕ∞)
hh : G ≠ ⊤
this : G.chromaticNumber ≤ HSub.hSub (α := ℕ∞) (↑(card V) : ℕ∞) 1
⊢ False
have this₁ : card V ≤ card V - 1
Tactic: rw [h, ← ENat.coe_one, ← ENat.coe_sub, ENat.coe_le_coe] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (hypothesis)
Goal:
V : Type u
G : SimpleGraph V
inst : Fintype V
h : G.chromaticNumber = (↑(card V) : ℕ∞)
hh : G ≠ ⊤
this : card V ≤ card V - 1
⊢ False
have this₁ : 1 < card V
Tactic: have := Fintype.one_lt_card_iff_nontrivial.mpr <| SimpleGraph.nontrivial_iff.mp ⟨_, _, hh⟩
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
G : SimpleGraph V
inst : Fintype V
h : G.chromaticNumber = (↑(card V) : ℕ∞)
⊢ G = ⊤
have by_contra! : ¬G = ⊤ → False
Tactic: classical
  by_contra! hh
  have : G.chromaticNumber ≤ Fintype.card V - 1 := by
    obtain ⟨a, b, hne, _⟩ := ne_top_iff_exists_not_adj.mp hh
    apply chromaticNumber_le_iff_colorable.mpr
    suffices G.Coloring (Finset.univ.erase b) by simpa using Coloring.colorable this
    apply Coloring.mk (fun x ↦ if h' : x ≠ b then ⟨x, by simp [h']⟩ else ⟨a, by simp [hne]⟩)
    grind [Adj.ne', adj_symm]
  rw [h, ← ENat.coe_one, ← ENat.coe_sub, ENat.coe_le_coe] at this
  have := Fintype.one_lt_card_iff_nontrivial.mpr <| SimpleGraph.nontrivial_iff.mp ⟨_, _, hh⟩
  grind
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
G : SimpleGraph V
inst : Fintype V
h : G.chromaticNumber = (↑(card V) : ℕ∞)
by_contra! by_contra!₁ by_contra!₂ by_contra!₃ : ¬G = ⊤ → False
⊢ G = ⊤
have by_contra!₄ : G ≠ ⊤ → False
Tactic: classical
  by_contra! hh
  have : G.chromaticNumber ≤ Fintype.card V - 1 := by
    obtain ⟨a, b, hne, _⟩ := ne_top_iff_exists_not_adj.mp hh
    apply chromaticNumber_le_iff_colorable.mpr
    suffices G.Coloring (Finset.univ.erase b) by simpa using Coloring.colorable this
    apply Coloring.mk (fun x ↦ if h' : x ≠ b then ⟨x, by simp [h']⟩ else ⟨a, by simp [hne]⟩)
    grind [Adj.ne', adj_symm]
  rw [h, ← ENat.coe_one, ← ENat.coe_sub, ENat.coe_le_coe] at this
  have := Fintype.one_lt_card_iff_nontrivial.mpr <| SimpleGraph.nontrivial_iff.mp ⟨_, _, hh⟩
  grind
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
G : SimpleGraph V
inst : Fintype V
h : G.chromaticNumber = (↑(card V) : ℕ∞)
by_contra! by_contra!₁ by_contra!₂ by_contra!₃ : ¬G = ⊤ → False
by_contra!₄ by_contra!₅ by_contra!₆ by_contra!₇ by_contra!₈ by_contra!₉ by_contra!₁₀ : ∀ (hh : G ≠ ⊤), False
⊢ G = ⊤
have by_contra!₁₁ : ∀ (hh : G ≠ ⊤) (this : G.chromaticNumber ≤ HSub.hSub (α := ℕ∞) (↑(card V) : ℕ∞) 1), False
Tactic: classical
  by_contra! hh
  have : G.chromaticNumber ≤ Fintype.card V - 1 := by
    obtain ⟨a, b, hne, _⟩ := ne_top_iff_exists_not_adj.mp hh
    apply chromaticNumber_le_iff_colorable.mpr
    suffices G.Coloring (Finset.univ.erase b) by simpa using Coloring.colorable this
    apply Coloring.mk (fun x ↦ if h' : x ≠ b then ⟨x, by simp [h']⟩ else ⟨a, by simp [hne]⟩)
    grind [Adj.ne', adj_symm]
  rw [h, ← ENat.coe_one, ← ENat.coe_sub, ENat.coe_le_coe] at this
  have := Fintype.one_lt_card_iff_nontrivial.mpr <| SimpleGraph.nontrivial_iff.mp ⟨_, _, hh⟩
  grind
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
G : SimpleGraph V
inst : Fintype V
h : G.chromaticNumber = (↑(card V) : ℕ∞)
by_contra! by_contra!₁ by_contra!₂ by_contra!₃ : ¬G = ⊤ → False
by_contra!₄ by_contra!₅ by_contra!₆ by_contra!₇ by_contra!₈ by_contra!₉ by_contra!₁₀ : ∀ (hh : G ≠ ⊤), False
by_contra!₁₁ : ∀ (hh : G ≠ ⊤) (this : G.chromaticNumber ≤ HSub.hSub (α := ℕ∞) (↑(card V) : ℕ∞) 1), False
⊢ G = ⊤
have by_contra!₁₂ : ∀ (hh : G ≠ ⊤) (this : card V ≤ card V - 1), False
Tactic: classical
  by_contra! hh
  have : G.chromaticNumber ≤ Fintype.card V - 1 := by
    obtain ⟨a, b, hne, _⟩ := ne_top_iff_exists_not_adj.mp hh
    apply chromaticNumber_le_iff_colorable.mpr
    suffices G.Coloring (Finset.univ.erase b) by simpa using Coloring.colorable this
    apply Coloring.mk (fun x ↦ if h' : x ≠ b then ⟨x, by simp [h']⟩ else ⟨a, by simp [hne]⟩)
    grind [Adj.ne', adj_symm]
  rw [h, ← ENat.coe_one, ← ENat.coe_sub, ENat.coe_le_coe] at this
  have := Fintype.one_lt_card_iff_nontrivial.mpr <| SimpleGraph.nontrivial_iff.mp ⟨_, _, hh⟩
  grind
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
G : SimpleGraph V
inst : Fintype V
h : G.chromaticNumber = (↑(card V) : ℕ∞)
by_contra! by_contra!₁ by_contra!₂ by_contra!₃ : ¬G = ⊤ → False
by_contra!₄ by_contra!₅ by_contra!₆ by_contra!₇ by_contra!₈ by_contra!₉ by_contra!₁₀ : ∀ (hh : G ≠ ⊤), False
by_contra!₁₁ : ∀ (hh : G ≠ ⊤) (this : G.chromaticNumber ≤ HSub.hSub (α := ℕ∞) (↑(card V) : ℕ∞) 1), False
by_contra!₁₂ : ∀ (hh : G ≠ ⊤) (this : card V ≤ card V - 1), False
⊢ G = ⊤
have by_contra!₁₃ : ∀ (hh : G ≠ ⊤) (this : card V ≤ card V - 1) (this : 1 < card V), False
Tactic: classical
  by_contra! hh
  have : G.chromaticNumber ≤ Fintype.card V - 1 := by
    obtain ⟨a, b, hne, _⟩ := ne_top_iff_exists_not_adj.mp hh
    apply chromaticNumber_le_iff_colorable.mpr
    suffices G.Coloring (Finset.univ.erase b) by simpa using Coloring.colorable this
    apply Coloring.mk (fun x ↦ if h' : x ≠ b then ⟨x, by simp [h']⟩ else ⟨a, by simp [hne]⟩)
    grind [Adj.ne', adj_symm]
  rw [h, ← ENat.coe_one, ← ENat.coe_sub, ENat.coe_le_coe] at this
  have := Fintype.one_lt_card_iff_nontrivial.mpr <| SimpleGraph.nontrivial_iff.mp ⟨_, _, hh⟩
  grind
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
G : SimpleGraph V
inst : Fintype V
⊢ G.chromaticNumber ≤ (↑(card V) : ℕ∞)
have rw : G.chromaticNumber ≤ chromaticNumber.{u} (V := V) ⊤
Tactic: rw [← chromaticNumber_top]
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
G : SimpleGraph V
u v : V
hadj : G.Adj u v
⊢ 2 ≤ G.chromaticNumber
have refine : ∀ (h : G.chromaticNumber < 2), False
Tactic: refine le_of_not_gt fun h ↦ ?_
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
G : SimpleGraph V
u v : V
hadj : G.Adj u v
h : G.chromaticNumber < 2
⊢ False
have obtain : ∀ (c : G.Coloring (Fin 1)), False
Tactic: obtain ⟨c⟩ := chromaticNumber_le_iff_colorable.mp (Order.le_of_lt_add_one h)
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
G : SimpleGraph V
⊢ G.chromaticNumber = 1 ↔ G = ⊥ ∧ Nonempty V
have rw : G.chromaticNumber ≤ 1 ∧ ¬G.chromaticNumber < 1 ↔ G = ⊥ ∧ Nonempty V
Tactic: rw [eq_iff_le_not_lt, Order.lt_one_iff_nonpos, ← not_isEmpty_iff, ← Nat.cast_one, ← Nat.cast_zero,
    chromaticNumber_le_iff_colorable, chromaticNumber_le_iff_colorable, colorable_one_iff,
    colorable_zero_iff]
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
G : SimpleGraph V
rw : G.chromaticNumber ≤ 1 ∧ ¬G.chromaticNumber < 1 ↔ G = ⊥ ∧ Nonempty V
⊢ G.chromaticNumber = 1 ↔ G = ⊥ ∧ Nonempty V
have rw₁ : G.chromaticNumber ≤ 1 ∧ ¬G.chromaticNumber ≤ 0 ↔ G = ⊥ ∧ Nonempty V
Tactic: rw [eq_iff_le_not_lt, Order.lt_one_iff_nonpos, ← not_isEmpty_iff, ← Nat.cast_one, ← Nat.cast_zero,
    chromaticNumber_le_iff_colorable, chromaticNumber_le_iff_colorable, colorable_one_iff,
    colorable_zero_iff]
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
G : SimpleGraph V
rw : G.chromaticNumber ≤ 1 ∧ ¬G.chromaticNumber < 1 ↔ G = ⊥ ∧ Nonempty V
rw₁ : G.chromaticNumber ≤ 1 ∧ ¬G.chromaticNumber ≤ 0 ↔ G = ⊥ ∧ Nonempty V
⊢ G.chromaticNumber = 1 ↔ G = ⊥ ∧ Nonempty V
have rw₂ : G.chromaticNumber ≤ 1 ∧ ¬G.chromaticNumber ≤ 0 ↔ G = ⊥ ∧ ¬IsEmpty V
Tactic: rw [eq_iff_le_not_lt, Order.lt_one_iff_nonpos, ← not_isEmpty_iff, ← Nat.cast_one, ← Nat.cast_zero,
    chromaticNumber_le_iff_colorable, chromaticNumber_le_iff_colorable, colorable_one_iff,
    colorable_zero_iff]
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
G : SimpleGraph V
rw : G.chromaticNumber ≤ 1 ∧ ¬G.chromaticNumber < 1 ↔ G = ⊥ ∧ Nonempty V
rw₁ : G.chromaticNumber ≤ 1 ∧ ¬G.chromaticNumber ≤ 0 ↔ G = ⊥ ∧ Nonempty V
rw₂ : G.chromaticNumber ≤ 1 ∧ ¬G.chromaticNumber ≤ 0 ↔ G = ⊥ ∧ ¬IsEmpty V
⊢ G.chromaticNumber = 1 ↔ G = ⊥ ∧ Nonempty V
have rw₃ : G.chromaticNumber ≤ (↑1 : ℕ∞) ∧ ¬G.chromaticNumber ≤ 0 ↔ G = ⊥ ∧ ¬IsEmpty V
Tactic: rw [eq_iff_le_not_lt, Order.lt_one_iff_nonpos, ← not_isEmpty_iff, ← Nat.cast_one, ← Nat.cast_zero,
    chromaticNumber_le_iff_colorable, chromaticNumber_le_iff_colorable, colorable_one_iff,
    colorable_zero_iff]
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
G : SimpleGraph V
rw : G.chromaticNumber ≤ 1 ∧ ¬G.chromaticNumber < 1 ↔ G = ⊥ ∧ Nonempty V
rw₁ : G.chromaticNumber ≤ 1 ∧ ¬G.chromaticNumber ≤ 0 ↔ G = ⊥ ∧ Nonempty V
rw₂ : G.chromaticNumber ≤ 1 ∧ ¬G.chromaticNumber ≤ 0 ↔ G = ⊥ ∧ ¬IsEmpty V
rw₃ : G.chromaticNumber ≤ (↑1 : ℕ∞) ∧ ¬G.chromaticNumber ≤ 0 ↔ G = ⊥ ∧ ¬IsEmpty V
⊢ G.chromaticNumber = 1 ↔ G = ⊥ ∧ Nonempty V
have rw₄ : G.chromaticNumber ≤ (↑1 : ℕ∞) ∧ ¬G.chromaticNumber ≤ (↑0 : ℕ∞) ↔ G = ⊥ ∧ ¬IsEmpty V
Tactic: rw [eq_iff_le_not_lt, Order.lt_one_iff_nonpos, ← not_isEmpty_iff, ← Nat.cast_one, ← Nat.cast_zero,
    chromaticNumber_le_iff_colorable, chromaticNumber_le_iff_colorable, colorable_one_iff,
    colorable_zero_iff]
-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring (goal)
Goal:
V : Type u
G : SimpleGraph V
rw : G.chromaticNumber ≤ 1 ∧ ¬G.chromaticNumber < 1 ↔ G = ⊥ ∧ Nonempty V
rw₁ : G.chromaticNumber ≤ 1 ∧ ¬G.chromaticNumber ≤ 0 ↔ G = ⊥ ∧ Nonempty V
rw₂ : G.chromaticNumber ≤ 1 ∧ ¬G.chromaticNumber ≤ 0 ↔ G = ⊥ ∧ ¬IsEmpty V
rw₃ : G.chromaticNumber ≤ (↑1 : ℕ∞) ∧ ¬G.chromaticNumber ≤ 0 ↔ G = ⊥ ∧ ¬IsEmpty V
rw₄ : G.chromaticNumber ≤ (↑1 : ℕ∞) ∧ ¬G.chromaticNumber ≤ (↑0 : ℕ∞) ↔ G = ⊥ ∧ ¬IsEmpty V
⊢ G.chromaticNumber = 1 ↔ G = ⊥ ∧ Nonempty V
have rw₅ : G.Colorable 1 ∧ ¬G.chromaticNumber ≤ (↑0 : ℕ∞) ↔ G = ⊥ ∧ ¬IsEmpty V
Tactic: rw [eq_iff_le_not_lt, Order.lt_one_iff_nonpos, ← not_isEmpty_iff, ← Nat.cast_one, ← Nat.cast_zero,
    chromaticNumber_le_iff_colorable, chromaticNumber_le_iff_colorable, colorable_one_iff,
    colorable_zero_iff]