awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.VectorMeasure.WithDensity (goal)
Goal:
α : Type u_1
E : Type u_2
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
f : α → E
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
hfi : Integrable f μ
⊢ VectorMeasure.trim (μ.withDensityᵥ f) hm ≪ᵥ (μ.trim hm).toENNRealVectorMeasure
have refine : ∀ (j : Set α) (hj₁ : MeasurableSet j) (hj₂ : (↑(μ.trim hm).toENNRealVectorMeasure : Set α → ℝ≥0∞) j = 0),
  (↑(VectorMeasure.trim (μ.withDensityᵥ f) hm) : Set α → E) j = 0
Tactic: refine VectorMeasure.AbsolutelyContinuous.mk fun j hj₁ hj₂ => ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.VectorMeasure.WithDensity (hypothesis)
Goal:
α : Type u_1
E : Type u_2
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
f : α → E
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
hfi : Integrable f μ
j : Set α
hj₁ : MeasurableSet j
hj₂ : (↑(μ.trim hm).toENNRealVectorMeasure : Set α → ℝ≥0∞) j = 0
⊢ (↑(VectorMeasure.trim (μ.withDensityᵥ f) hm) : Set α → E) j = 0
have hj₃ : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) j : ℝ≥0∞) (0 : ℝ≥0∞)
Tactic: rw [Measure.toENNRealVectorMeasure_apply_measurable hj₁, trim_measurableSet_eq hm hj₁] at hj₂
Cleared hypotheses: hj₂
-- Lean 4 proof step from Mathlib.MeasureTheory.VectorMeasure.WithDensity (goal)
Goal:
α : Type u_1
E : Type u_2
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
f : α → E
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
hfi : Integrable f μ
j : Set α
hj₁ : MeasurableSet j
hj₂ : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) j : ℝ≥0∞) (0 : ℝ≥0∞)
⊢ (↑(VectorMeasure.trim (μ.withDensityᵥ f) hm) : Set α → E) j = 0
have rw : Eq.{u_2 + 1} (α := E) (∫ (x : α) in j, f x ∂μ) 0
Tactic: rw [VectorMeasure.trim_measurableSet_eq hm hj₁, withDensityᵥ_apply hfi (hm _ hj₁)]
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.Basic (goal)
Goal:
K : Type u_2
inst : Add K
inst_1 : Mul K
inst_2 : Neg K
inst_3 : Zero K
inst_4 : One K
inst_5 : CompatibleRing K
ax : FieldAxiom
⊢ K ⊨ ax.toSentence ↔ toProp K ax
have addAssoc : K ⊨ addAssoc.toSentence ↔ toProp K addAssoc
Tactic: cases ax
Cleared hypotheses: ax
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.Basic (goal)
Goal:
K : Type u_2
inst : Add K
inst_1 : Mul K
inst_2 : Neg K
inst_3 : Zero K
inst_4 : One K
inst_5 : CompatibleRing K
ax : FieldAxiom
addAssoc : K ⊨ FieldAxiom.addAssoc.toSentence ↔ toProp K FieldAxiom.addAssoc
⊢ K ⊨ ax.toSentence ↔ toProp K ax
have zeroAdd : K ⊨ zeroAdd.toSentence ↔ toProp K zeroAdd
Tactic: cases ax
Cleared hypotheses: ax
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.Basic (goal)
Goal:
K : Type u_2
inst : Add K
inst_1 : Mul K
inst_2 : Neg K
inst_3 : Zero K
inst_4 : One K
inst_5 : CompatibleRing K
ax : FieldAxiom
addAssoc : K ⊨ FieldAxiom.addAssoc.toSentence ↔ toProp K FieldAxiom.addAssoc
zeroAdd : K ⊨ FieldAxiom.zeroAdd.toSentence ↔ toProp K FieldAxiom.zeroAdd
⊢ K ⊨ ax.toSentence ↔ toProp K ax
have negAddCancel : K ⊨ negAddCancel.toSentence ↔ toProp K negAddCancel
Tactic: cases ax
Cleared hypotheses: ax
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.Basic (goal)
Goal:
K : Type u_2
inst : Add K
inst_1 : Mul K
inst_2 : Neg K
inst_3 : Zero K
inst_4 : One K
inst_5 : CompatibleRing K
ax : FieldAxiom
addAssoc : K ⊨ FieldAxiom.addAssoc.toSentence ↔ toProp K FieldAxiom.addAssoc
zeroAdd : K ⊨ FieldAxiom.zeroAdd.toSentence ↔ toProp K FieldAxiom.zeroAdd
negAddCancel : K ⊨ FieldAxiom.negAddCancel.toSentence ↔ toProp K FieldAxiom.negAddCancel
⊢ K ⊨ ax.toSentence ↔ toProp K ax
have mulAssoc : K ⊨ mulAssoc.toSentence ↔ toProp K mulAssoc
Tactic: cases ax
Cleared hypotheses: ax
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.Basic (goal)
Goal:
K : Type u_2
inst : Add K
inst_1 : Mul K
inst_2 : Neg K
inst_3 : Zero K
inst_4 : One K
inst_5 : CompatibleRing K
ax : FieldAxiom
addAssoc : K ⊨ FieldAxiom.addAssoc.toSentence ↔ toProp K FieldAxiom.addAssoc
zeroAdd : K ⊨ FieldAxiom.zeroAdd.toSentence ↔ toProp K FieldAxiom.zeroAdd
negAddCancel : K ⊨ FieldAxiom.negAddCancel.toSentence ↔ toProp K FieldAxiom.negAddCancel
mulAssoc : K ⊨ FieldAxiom.mulAssoc.toSentence ↔ toProp K FieldAxiom.mulAssoc
⊢ K ⊨ ax.toSentence ↔ toProp K ax
have mulComm : K ⊨ mulComm.toSentence ↔ toProp K mulComm
Tactic: cases ax
Cleared hypotheses: ax
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.Basic (goal)
Goal:
K : Type u_2
inst : Add K
inst_1 : Mul K
inst_2 : Neg K
inst_3 : Zero K
inst_4 : One K
inst_5 : CompatibleRing K
ax : FieldAxiom
addAssoc : K ⊨ FieldAxiom.addAssoc.toSentence ↔ toProp K FieldAxiom.addAssoc
zeroAdd : K ⊨ FieldAxiom.zeroAdd.toSentence ↔ toProp K FieldAxiom.zeroAdd
negAddCancel : K ⊨ FieldAxiom.negAddCancel.toSentence ↔ toProp K FieldAxiom.negAddCancel
mulAssoc : K ⊨ FieldAxiom.mulAssoc.toSentence ↔ toProp K FieldAxiom.mulAssoc
mulComm : K ⊨ FieldAxiom.mulComm.toSentence ↔ toProp K FieldAxiom.mulComm
⊢ K ⊨ ax.toSentence ↔ toProp K ax
have oneMul : K ⊨ oneMul.toSentence ↔ toProp K oneMul
Tactic: cases ax
Cleared hypotheses: ax
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.Basic (goal)
Goal:
K : Type u_2
inst : Add K
inst_1 : Mul K
inst_2 : Neg K
inst_3 : Zero K
inst_4 : One K
inst_5 : CompatibleRing K
ax : FieldAxiom
addAssoc : K ⊨ FieldAxiom.addAssoc.toSentence ↔ toProp K FieldAxiom.addAssoc
zeroAdd : K ⊨ FieldAxiom.zeroAdd.toSentence ↔ toProp K FieldAxiom.zeroAdd
negAddCancel : K ⊨ FieldAxiom.negAddCancel.toSentence ↔ toProp K FieldAxiom.negAddCancel
mulAssoc : K ⊨ FieldAxiom.mulAssoc.toSentence ↔ toProp K FieldAxiom.mulAssoc
mulComm : K ⊨ FieldAxiom.mulComm.toSentence ↔ toProp K FieldAxiom.mulComm
oneMul : K ⊨ FieldAxiom.oneMul.toSentence ↔ toProp K FieldAxiom.oneMul
⊢ K ⊨ ax.toSentence ↔ toProp K ax
have existsInv : K ⊨ existsInv.toSentence ↔ toProp K existsInv
Tactic: cases ax
Cleared hypotheses: ax
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.Basic (goal)
Goal:
K : Type u_2
inst : Add K
inst_1 : Mul K
inst_2 : Neg K
inst_3 : Zero K
inst_4 : One K
inst_5 : CompatibleRing K
ax : FieldAxiom
addAssoc : K ⊨ FieldAxiom.addAssoc.toSentence ↔ toProp K FieldAxiom.addAssoc
zeroAdd : K ⊨ FieldAxiom.zeroAdd.toSentence ↔ toProp K FieldAxiom.zeroAdd
negAddCancel : K ⊨ FieldAxiom.negAddCancel.toSentence ↔ toProp K FieldAxiom.negAddCancel
mulAssoc : K ⊨ FieldAxiom.mulAssoc.toSentence ↔ toProp K FieldAxiom.mulAssoc
mulComm : K ⊨ FieldAxiom.mulComm.toSentence ↔ toProp K FieldAxiom.mulComm
oneMul : K ⊨ FieldAxiom.oneMul.toSentence ↔ toProp K FieldAxiom.oneMul
existsInv : K ⊨ FieldAxiom.existsInv.toSentence ↔ toProp K FieldAxiom.existsInv
⊢ K ⊨ ax.toSentence ↔ toProp K ax
have leftDistrib : K ⊨ leftDistrib.toSentence ↔ toProp K leftDistrib
Tactic: cases ax
Cleared hypotheses: ax
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.Basic (goal)
Goal:
K : Type u_2
inst : Add K
inst_1 : Mul K
inst_2 : Neg K
inst_3 : Zero K
inst_4 : One K
inst_5 : CompatibleRing K
ax : FieldAxiom
addAssoc : K ⊨ FieldAxiom.addAssoc.toSentence ↔ toProp K FieldAxiom.addAssoc
zeroAdd : K ⊨ FieldAxiom.zeroAdd.toSentence ↔ toProp K FieldAxiom.zeroAdd
negAddCancel : K ⊨ FieldAxiom.negAddCancel.toSentence ↔ toProp K FieldAxiom.negAddCancel
mulAssoc : K ⊨ FieldAxiom.mulAssoc.toSentence ↔ toProp K FieldAxiom.mulAssoc
mulComm : K ⊨ FieldAxiom.mulComm.toSentence ↔ toProp K FieldAxiom.mulComm
oneMul : K ⊨ FieldAxiom.oneMul.toSentence ↔ toProp K FieldAxiom.oneMul
existsInv : K ⊨ FieldAxiom.existsInv.toSentence ↔ toProp K FieldAxiom.existsInv
leftDistrib : K ⊨ FieldAxiom.leftDistrib.toSentence ↔ toProp K FieldAxiom.leftDistrib
⊢ K ⊨ ax.toSentence ↔ toProp K ax
have existsPairNE : K ⊨ existsPairNE.toSentence ↔ toProp K existsPairNE
Tactic: cases ax
Cleared hypotheses: ax
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : CompatibleRing K
⊢ ∀ φ ∈ Theory.field, K ⊨ φ
have simp : ∀ (φ : ring.Sentence) (x : FieldAxiom), x.toSentence = φ → K ⊨ φ
Tactic: simp only [Theory.field, Set.mem_range, exists_imp]
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : CompatibleRing K
a : FieldAxiom
⊢ K ⊨ a.toSentence
have rw : toProp K a
Tactic: rw [a.realize_toSentence_iff_toProp (K := K)]
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : CompatibleRing K
a : FieldAxiom
⊢ toProp K a
have existsPairNE : toProp K existsPairNE
Tactic: cases a with
      | existsPairNE => exact exists_pair_ne K
      | existsInv => exact fun x hx0 => ⟨x⁻¹, mul_inv_cancel₀ hx0⟩
      | addAssoc => exact add_assoc
      | zeroAdd => exact zero_add
      | negAddCancel => exact neg_add_cancel
      | mulAssoc => exact mul_assoc
      | mulComm => exact mul_comm
      | oneMul => exact one_mul
      | leftDistrib => exact mul_add
Cleared hypotheses: a
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : CompatibleRing K
a : FieldAxiom
existsPairNE : toProp K FieldAxiom.existsPairNE
⊢ toProp K a
have existsInv : toProp K existsInv
Tactic: cases a with
      | existsPairNE => exact exists_pair_ne K
      | existsInv => exact fun x hx0 => ⟨x⁻¹, mul_inv_cancel₀ hx0⟩
      | addAssoc => exact add_assoc
      | zeroAdd => exact zero_add
      | negAddCancel => exact neg_add_cancel
      | mulAssoc => exact mul_assoc
      | mulComm => exact mul_comm
      | oneMul => exact one_mul
      | leftDistrib => exact mul_add
Cleared hypotheses: a
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : CompatibleRing K
a : FieldAxiom
existsPairNE : toProp K FieldAxiom.existsPairNE
existsInv : toProp K FieldAxiom.existsInv
⊢ toProp K a
have addAssoc : toProp K addAssoc
Tactic: cases a with
      | existsPairNE => exact exists_pair_ne K
      | existsInv => exact fun x hx0 => ⟨x⁻¹, mul_inv_cancel₀ hx0⟩
      | addAssoc => exact add_assoc
      | zeroAdd => exact zero_add
      | negAddCancel => exact neg_add_cancel
      | mulAssoc => exact mul_assoc
      | mulComm => exact mul_comm
      | oneMul => exact one_mul
      | leftDistrib => exact mul_add
Cleared hypotheses: a
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : CompatibleRing K
a : FieldAxiom
existsPairNE : toProp K FieldAxiom.existsPairNE
existsInv : toProp K FieldAxiom.existsInv
addAssoc : toProp K FieldAxiom.addAssoc
⊢ toProp K a
have zeroAdd : toProp K zeroAdd
Tactic: cases a with
      | existsPairNE => exact exists_pair_ne K
      | existsInv => exact fun x hx0 => ⟨x⁻¹, mul_inv_cancel₀ hx0⟩
      | addAssoc => exact add_assoc
      | zeroAdd => exact zero_add
      | negAddCancel => exact neg_add_cancel
      | mulAssoc => exact mul_assoc
      | mulComm => exact mul_comm
      | oneMul => exact one_mul
      | leftDistrib => exact mul_add
Cleared hypotheses: a
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : CompatibleRing K
a : FieldAxiom
existsPairNE : toProp K FieldAxiom.existsPairNE
existsInv : toProp K FieldAxiom.existsInv
addAssoc : toProp K FieldAxiom.addAssoc
zeroAdd : toProp K FieldAxiom.zeroAdd
⊢ toProp K a
have negAddCancel : toProp K negAddCancel
Tactic: cases a with
      | existsPairNE => exact exists_pair_ne K
      | existsInv => exact fun x hx0 => ⟨x⁻¹, mul_inv_cancel₀ hx0⟩
      | addAssoc => exact add_assoc
      | zeroAdd => exact zero_add
      | negAddCancel => exact neg_add_cancel
      | mulAssoc => exact mul_assoc
      | mulComm => exact mul_comm
      | oneMul => exact one_mul
      | leftDistrib => exact mul_add
Cleared hypotheses: a
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : CompatibleRing K
a : FieldAxiom
existsPairNE : toProp K FieldAxiom.existsPairNE
existsInv : toProp K FieldAxiom.existsInv
addAssoc : toProp K FieldAxiom.addAssoc
zeroAdd : toProp K FieldAxiom.zeroAdd
negAddCancel : toProp K FieldAxiom.negAddCancel
⊢ toProp K a
have mulAssoc : toProp K mulAssoc
Tactic: cases a with
      | existsPairNE => exact exists_pair_ne K
      | existsInv => exact fun x hx0 => ⟨x⁻¹, mul_inv_cancel₀ hx0⟩
      | addAssoc => exact add_assoc
      | zeroAdd => exact zero_add
      | negAddCancel => exact neg_add_cancel
      | mulAssoc => exact mul_assoc
      | mulComm => exact mul_comm
      | oneMul => exact one_mul
      | leftDistrib => exact mul_add
Cleared hypotheses: a
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : CompatibleRing K
a : FieldAxiom
existsPairNE : toProp K FieldAxiom.existsPairNE
existsInv : toProp K FieldAxiom.existsInv
addAssoc : toProp K FieldAxiom.addAssoc
zeroAdd : toProp K FieldAxiom.zeroAdd
negAddCancel : toProp K FieldAxiom.negAddCancel
mulAssoc : toProp K FieldAxiom.mulAssoc
⊢ toProp K a
have mulComm : toProp K mulComm
Tactic: cases a with
      | existsPairNE => exact exists_pair_ne K
      | existsInv => exact fun x hx0 => ⟨x⁻¹, mul_inv_cancel₀ hx0⟩
      | addAssoc => exact add_assoc
      | zeroAdd => exact zero_add
      | negAddCancel => exact neg_add_cancel
      | mulAssoc => exact mul_assoc
      | mulComm => exact mul_comm
      | oneMul => exact one_mul
      | leftDistrib => exact mul_add
Cleared hypotheses: a
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : CompatibleRing K
a : FieldAxiom
existsPairNE : toProp K FieldAxiom.existsPairNE
existsInv : toProp K FieldAxiom.existsInv
addAssoc : toProp K FieldAxiom.addAssoc
zeroAdd : toProp K FieldAxiom.zeroAdd
negAddCancel : toProp K FieldAxiom.negAddCancel
mulAssoc : toProp K FieldAxiom.mulAssoc
mulComm : toProp K FieldAxiom.mulComm
⊢ toProp K a
have oneMul : toProp K oneMul
Tactic: cases a with
      | existsPairNE => exact exists_pair_ne K
      | existsInv => exact fun x hx0 => ⟨x⁻¹, mul_inv_cancel₀ hx0⟩
      | addAssoc => exact add_assoc
      | zeroAdd => exact zero_add
      | negAddCancel => exact neg_add_cancel
      | mulAssoc => exact mul_assoc
      | mulComm => exact mul_comm
      | oneMul => exact one_mul
      | leftDistrib => exact mul_add
Cleared hypotheses: a
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : CompatibleRing K
a : FieldAxiom
existsPairNE : toProp K FieldAxiom.existsPairNE
existsInv : toProp K FieldAxiom.existsInv
addAssoc : toProp K FieldAxiom.addAssoc
zeroAdd : toProp K FieldAxiom.zeroAdd
negAddCancel : toProp K FieldAxiom.negAddCancel
mulAssoc : toProp K FieldAxiom.mulAssoc
mulComm : toProp K FieldAxiom.mulComm
oneMul : toProp K FieldAxiom.oneMul
⊢ toProp K a
have leftDistrib : toProp K leftDistrib
Tactic: cases a with
      | existsPairNE => exact exists_pair_ne K
      | existsInv => exact fun x hx0 => ⟨x⁻¹, mul_inv_cancel₀ hx0⟩
      | addAssoc => exact add_assoc
      | zeroAdd => exact zero_add
      | negAddCancel => exact neg_add_cancel
      | mulAssoc => exact mul_assoc
      | mulComm => exact mul_comm
      | oneMul => exact one_mul
      | leftDistrib => exact mul_add
Cleared hypotheses: a
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.CharP (goal)
Goal:
p : ℕ
K : Type u_1
inst : Field K
inst_1 : CompatibleRing K
inst_2 : CharP K p
⊢ K ⊨ Theory.fieldOfChar p
have refine : Theory.Model (L := ring) K
  (if p = 0 then (fun (q : ℕ) ↦ ∼(eqZero q)) '' {q : ℕ | Nat.Prime q} else if Nat.Prime p then {eqZero p} else {⊥})
Tactic: refine Language.Theory.model_union_iff.2 ⟨inferInstance, ?_⟩
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.CharP (goal)
Goal:
p : ℕ
K : Type u_1
inst : Field K
inst_1 : CompatibleRing K
inst_2 : CharP K p
hp : p = 0
⊢ Theory.Model (L := ring) K
    (if p = 0 then (fun (q : ℕ) ↦ ∼(eqZero q)) '' {q : ℕ | Nat.Prime q} else if Nat.Prime p then {eqZero p} else {⊥})
have inr : ∀ [CharP K 0],
  Theory.Model (L := ring) K
    (if Eq (α := ℕ) 0 0 then (fun (q : ℕ) ↦ ∼(eqZero q)) '' {q : ℕ | Nat.Prime q}
    else if Nat.Prime 0 then {eqZero 0} else {⊥})
Tactic: subst hp
Cleared hypotheses: p, inst_2, hp
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.CharP (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : CompatibleRing K
inst_2 : CharP K 0
⊢ Theory.Model (L := ring) K
    (if Eq (α := ℕ) 0 0 then (fun (q : ℕ) ↦ ∼(eqZero q)) '' {q : ℕ | Nat.Prime q}
    else if Nat.Prime 0 then {eqZero 0} else {⊥})
have inr : CharZero K
Tactic: simp only [ite_true, Theory.model_iff, Set.mem_image, Set.mem_setOf_eq,
      Sentence.Realize, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂,
      Formula.realize_not, realize_eqZero, ← CharZero.charZero_iff_forall_prime_ne_zero]
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.CharP (hypothesis)
Goal:
p : ℕ
K : Type u_1
inst : Field K
inst_1 : CompatibleRing K
inst_2 : CharP K p
⊢ Theory.Model (L := ring) K
    (if p = 0 then (fun (q : ℕ) ↦ ∼(eqZero q)) '' {q : ℕ | Nat.Prime q} else if Nat.Prime p then {eqZero p} else {⊥})
have x : Nat.Prime p ∨ p = 0
Tactic: cases CharP.char_is_prime_or_zero K p with
  | inl hp =>
    simp [hp.ne_zero, hp, Sentence.Realize]
  | inr hp =>
    subst hp
    simp only [ite_true, Theory.model_iff, Set.mem_image, Set.mem_setOf_eq,
      Sentence.Realize, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂,
      Formula.realize_not, realize_eqZero, ← CharZero.charZero_iff_forall_prime_ne_zero]
    exact CharP.charP_to_charZero K
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.CharP (goal)
Goal:
p : ℕ
K : Type u_1
inst : Field K
inst_1 : CompatibleRing K
⊢ K ⊨ Theory.fieldOfChar p ↔ CharP K p
have simp : Theory.Model (L := ring) K
    (if p = 0 then (fun (q : ℕ) ↦ ∼(eqZero q)) '' {q : ℕ | Nat.Prime q} else if Nat.Prime p then {eqZero p} else {⊥}) ↔
  CharP K p
Tactic: simp only [Theory.fieldOfChar, Theory.model_union_iff,
    (show (Theory.field.Model K) by infer_instance), true_and]
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.CharP (goal)
Goal:
p : ℕ
K : Type u_1
inst : Field K
inst_1 : CompatibleRing K
⊢ Theory.Model (L := ring) K
      (if p = 0 then (fun (q : ℕ) ↦ ∼(eqZero q)) '' {q : ℕ | Nat.Prime q}
      else if Nat.Prime p then {eqZero p} else {⊥}) ↔
    CharP K p
have pos : ∀ (hp0 : p = 0), Theory.Model (L := ring) K ((fun (q : ℕ) ↦ ∼(eqZero q)) '' {q : ℕ | Nat.Prime q}) ↔ CharP K p
Tactic: split_ifs with hp0 hp
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.CharP (goal)
Goal:
p : ℕ
K : Type u_1
inst : Field K
inst_1 : CompatibleRing K
pos : ∀ (hp0 : p = 0), Theory.Model (L := ring) K ((fun (q : ℕ) ↦ ∼(eqZero q)) '' {q : ℕ | Nat.Prime q}) ↔ CharP K p
⊢ Theory.Model (L := ring) K
      (if p = 0 then (fun (q : ℕ) ↦ ∼(eqZero q)) '' {q : ℕ | Nat.Prime q}
      else if Nat.Prime p then {eqZero p} else {⊥}) ↔
    CharP K p
have pos₁ : ∀ (hp0 : ¬p = 0) (hp : Nat.Prime p), Theory.Model (L := ring) K {eqZero p} ↔ CharP K p
Tactic: split_ifs with hp0 hp
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.CharP (goal)
Goal:
p : ℕ
K : Type u_1
inst : Field K
inst_1 : CompatibleRing K
pos : ∀ (hp0 : p = 0), Theory.Model (L := ring) K ((fun (q : ℕ) ↦ ∼(eqZero q)) '' {q : ℕ | Nat.Prime q}) ↔ CharP K p
pos₁ : ∀ (hp0 : ¬p = 0) (hp : Nat.Prime p), Theory.Model (L := ring) K {eqZero p} ↔ CharP K p
⊢ Theory.Model (L := ring) K
      (if p = 0 then (fun (q : ℕ) ↦ ∼(eqZero q)) '' {q : ℕ | Nat.Prime q}
      else if Nat.Prime p then {eqZero p} else {⊥}) ↔
    CharP K p
have neg : ∀ (hp0 : ¬p = 0) (hp : ¬Nat.Prime p), Theory.Model (L := ring) K {⊥} ↔ CharP K p
Tactic: split_ifs with hp0 hp
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.CharP (goal)
Goal:
p : ℕ
K : Type u_1
inst : Field K
inst_1 : CompatibleRing K
hp0 : p = 0
⊢ Theory.Model (L := ring) K ((fun (q : ℕ) ↦ ∼(eqZero q)) '' {q : ℕ | Nat.Prime q}) ↔ CharP K p
have pos : Theory.Model (L := ring) K ((fun (q : ℕ) ↦ ∼(eqZero q)) '' {q : ℕ | Nat.Prime q}) ↔ CharP K 0
Tactic: subst hp0
Cleared hypotheses: p, hp0
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.CharP (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : CompatibleRing K
⊢ Theory.Model (L := ring) K ((fun (q : ℕ) ↦ ∼(eqZero q)) '' {q : ℕ | Nat.Prime q}) ↔ CharP K 0
have pos : CharZero K ↔ CharP K 0
Tactic: simp only [Theory.model_iff, Set.mem_image, Set.mem_setOf_eq, Sentence.Realize,
      forall_exists_index, and_imp, forall_apply_eq_imp_iff₂, Formula.realize_not,
      realize_eqZero, ← CharZero.charZero_iff_forall_prime_ne_zero]
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.CharP (goal)
Goal:
p : ℕ
K : Type u_1
inst : Field K
inst_1 : CompatibleRing K
hp0 : ¬p = 0
hp : ¬Nat.Prime p
⊢ Theory.Model (L := ring) K {⊥} ↔ CharP K p
have neg : ¬CharP K p
Tactic: simp only [Theory.model_iff, Set.mem_singleton_iff, Sentence.Realize,
      forall_eq, Formula.realize_bot, false_iff]
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.IsAlgClosed (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : CompatibleRing K
n : ℕ
⊢ K ⊨ genericMonicPolyHasRoot n ↔ ∀ (p : { p : K[X] // p.Monic ∧ p.natDegree = n }), ∃ (x : K), eval x (↑p : K[X]) = 0
have rw : K ⊨ genericMonicPolyHasRoot n ↔
  ∀ (b : Fin n → K),
    ∃ (x : K),
      eval x
          (↑(((Equiv.trans (monicEquivDegreeLT n) (LinearEquiv.toEquiv (degreeLTEquiv K n))).symm :
                  (a : Fin n → K) → { p : K[X] // p.Monic ∧ p.natDegree = n })
                b) :
            K[X]) =
        0
Tactic: rw [Equiv.forall_congr_left ((monicEquivDegreeLT n).trans (degreeLTEquiv K n).toEquiv)]
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.IsAlgClosed (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : CompatibleRing K
p : ℕ
inst_2 : CharP K p
inst_3 : IsAlgClosed K
⊢ K ⊨ Theory.ACF p
have refine : K ⊨ genericMonicPolyHasRoot '' {n : ℕ | 0 < n}
Tactic: refine Theory.model_union_iff.2 ⟨inferInstance, ?_⟩
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.IsAlgClosed (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : CompatibleRing K
p : ℕ
inst_2 : CharP K p
inst_3 : IsAlgClosed K
⊢ K ⊨ genericMonicPolyHasRoot '' {n : ℕ | 0 < n}
have simp : ∀ (φ : Language.ring.Sentence) (x : ℕ),
  Membership.mem (γ := Set ℕ) {n : ℕ | 0 < n} x → genericMonicPolyHasRoot x = φ → K ⊨ φ
Tactic: simp only [Theory.model_iff, Set.mem_image,
    forall_exists_index, and_imp]
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.IsAlgClosed (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : CompatibleRing K
p : ℕ
inst_2 : CharP K p
inst_3 : IsAlgClosed K
n : ℕ
hn0 : Membership.mem (γ := Set ℕ) {n : ℕ | 0 < n} n
⊢ K ⊨ genericMonicPolyHasRoot n
have simp : ∀ (p : { p : K[X] // p.Monic ∧ p.natDegree = n }), ∃ (x : K), eval x (↑p : K[X]) = 0
Tactic: simp only [realize_genericMonicPolyHasRoot]
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.IsAlgClosed (hypothesis)
Goal:
K_1 : Type u_1
p : ℕ
K : Type u_2
inst : Language.ring.Structure K
h : K ⊨ Theory.ACF p
⊢ Field K
have this : K ⊨ Theory.field
Tactic: have := modelField_of_modelACF p K
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.IsAlgClosed (goal)
Goal:
p : ℕ
K : Type u_2
inst : Field K
inst_1 : CompatibleRing K
h : K ⊨ Theory.ACF p
⊢ IsAlgClosed K
have refine : ∀ (p : K[X]), p.Monic → Irreducible p → ∃ (x : K), eval x p = 0
Tactic: refine IsAlgClosed.of_exists_root _ ?_
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.IsAlgClosed (hypothesis)
Goal:
p_1 : ℕ
K : Type u_2
inst : Field K
inst_1 : CompatibleRing K
h : K ⊨ Theory.ACF p_1
p : K[X]
hpm : p.Monic
hpi : Irreducible p
⊢ ∃ (x : K), eval x p = 0
have h₁ : K ⊨ genericMonicPolyHasRoot '' {n : ℕ | 0 < n}
Tactic: have h : K ⊨ genericMonicPolyHasRoot '' {n | 0 < n} :=
    Theory.Model.mono h (by simp [Theory.ACF])
-- Lean 4 proof step from Mathlib.ModelTheory.Algebra.Field.IsAlgClosed (hypothesis)
Goal:
p_1 : ℕ
K : Type u_2
inst : Field K
inst_1 : CompatibleRing K
h_1 : K ⊨ Theory.ACF p_1
p : K[X]
hpm : p.Monic
hpi : Irreducible p
h : K ⊨ genericMonicPolyHasRoot '' {n : ℕ | 0 < n}
⊢ ∃ (x : K), eval x p = 0
have h₁ : ∀ (φ : Language.ring.Sentence) (x : ℕ),
  Membership.mem (γ := Set ℕ) {n : ℕ | 0 < n} x → genericMonicPolyHasRoot x = φ → K ⊨ φ
Tactic: simp only [Theory.model_iff, Set.mem_image,
    forall_exists_index, and_imp] at h
Cleared hypotheses: h