Theorem: Lean4Axiomatic.Natural.pow_inputs_for_output_zero (Lean4Axiomatic/Natural/Exponentiation.lean)

State before:
ℕ : Type
inst✝¹¹ : Core ℕ
inst✝¹⁰ : Induction ℕ
inst✝⁹ : Addition ℕ
inst✝⁸ : Sign ℕ
inst✝⁷ : Multiplication ℕ
α : Type
mul : α → α → α
inst✝⁶ : EqvOp α
inst✝⁵ : OfNat α 1
inst✝⁴ : Exponentiation ℕ mul
inst✝³ : AA.Substitutive₂ (fun x x_1 => x * x_1) AA.tc (fun x x_1 => x ≃ x_1) fun x x_1 => x ≃ x_1
inst✝² : OfNat α 0
inst✝¹ : AP (1 ≄ 0)
inst✝ : AA.ZeroProduct fun x x_1 => x * x_1
x : α
n : ℕ
⊢ x ^ n ≃ 0 → x ≃ 0 ∧ n ≄ 0
Tactic:
apply ind_on (motive := λ m => x^m ≃ 0 → x ≃ 0 ∧ m ≄ 0) n
State after:
case zero
ℕ : Type
inst✝¹¹ : Core ℕ
inst✝¹⁰ : Induction ℕ
inst✝⁹ : Addition ℕ
inst✝⁸ : Sign ℕ
inst✝⁷ : Multiplication ℕ
α : Type
mul : α → α → α
inst✝⁶ : EqvOp α
inst✝⁵ : OfNat α 1
inst✝⁴ : Exponentiation ℕ mul
inst✝³ : AA.Substitutive₂ (fun x x_1 => x * x_1) AA.tc (fun x x_1 => x ≃ x_1) fun x x_1 => x ≃ x_1
inst✝² : OfNat α 0
inst✝¹ : AP (1 ≄ 0)
inst✝ : AA.ZeroProduct fun x x_1 => x * x_1
x : α
n : ℕ
⊢ x ^ 0 ≃ 0 → x ≃ 0 ∧ 0 ≄ 0

case step
ℕ : Type
inst✝¹¹ : Core ℕ
inst✝¹⁰ : Induction ℕ
inst✝⁹ : Addition ℕ
inst✝⁸ : Sign ℕ
inst✝⁷ : Multiplication ℕ
α : Type
mul : α → α → α
inst✝⁶ : EqvOp α
inst✝⁵ : OfNat α 1
inst✝⁴ : Exponentiation ℕ mul
inst✝³ : AA.Substitutive₂ (fun x x_1 => x * x_1) AA.tc (fun x x_1 => x ≃ x_1) fun x x_1 => x ≃ x_1
inst✝² : OfNat α 0
inst✝¹ : AP (1 ≄ 0)
inst✝ : AA.ZeroProduct fun x x_1 => x * x_1
x : α
n : ℕ
⊢ ∀ (m : ℕ), (x ^ m ≃ 0 → x ≃ 0 ∧ m ≄ 0) → x ^ step m ≃ 0 → x ≃ 0 ∧ step m ≄ 0

State before:
ℕ : Type
inst✝¹¹ : Core ℕ
inst✝¹⁰ : Induction ℕ
inst✝⁹ : Addition ℕ
inst✝⁸ : Sign ℕ
inst✝⁷ : Multiplication ℕ
α : Type
mul : α → α → α
inst✝⁶ : EqvOp α
inst✝⁵ : OfNat α 1
inst✝⁴ : Exponentiation ℕ mul
inst✝³ : AA.Substitutive₂ (fun x x_1 => x * x_1) AA.tc (fun x x_1 => x ≃ x_1) fun x x_1 => x ≃ x_1
inst✝² : OfNat α 0
inst✝¹ : AP (1 ≄ 0)
inst✝ : AA.ZeroProduct fun x x_1 => x * x_1
x : α
n : ℕ
⊢ x ^ 0 ≃ 0 → x ≃ 0 ∧ 0 ≄ 0
Tactic:
case zero =>
  intro (_ : x^(0:ℕ) ≃ 0)
  show x ≃ 0 ∧ 0 ≄ 0
  have : (1:α) ≃ 0 := calc
    _ ≃ 1   := Rel.refl
    _ ≃ x^0 := Rel.symm pow_zero
    _ ≃ 0   := ‹x^(0:ℕ) ≃ 0›
  exact absurd ‹(1:α) ≃ 0› ‹AP ((1:α) ≄ 0)›.ev
State after:
no goals

State before:
ℕ : Type
inst✝¹¹ : Core ℕ
inst✝¹⁰ : Induction ℕ
inst✝⁹ : Addition ℕ
inst✝⁸ : Sign ℕ
inst✝⁷ : Multiplication ℕ
α : Type
mul : α → α → α
inst✝⁶ : EqvOp α
inst✝⁵ : OfNat α 1
inst✝⁴ : Exponentiation ℕ mul
inst✝³ : AA.Substitutive₂ (fun x x_1 => x * x_1) AA.tc (fun x x_1 => x ≃ x_1) fun x x_1 => x ≃ x_1
inst✝² : OfNat α 0
inst✝¹ : AP (1 ≄ 0)
inst✝ : AA.ZeroProduct fun x x_1 => x * x_1
x : α
n : ℕ
⊢ ∀ (m : ℕ), (x ^ m ≃ 0 → x ≃ 0 ∧ m ≄ 0) → x ^ step m ≃ 0 → x ≃ 0 ∧ step m ≄ 0
Tactic:
case step =>
  intro (n' : ℕ) (ih : x^n' ≃ 0 → x ≃ 0 ∧ n' ≄ 0) (_ : x^(step n') ≃ 0)
  show x ≃ 0 ∧ step n' ≄ 0
  have : x^n' * x ≃ 0 := AA.eqv_substL pow_step ‹x^(step n') ≃ 0›
  have : x^n' ≃ 0 ∨ x ≃ 0 := AA.zero_prod this
  have : x ≃ 0 :=
    match this with
    | Or.inl (_ : x^n' ≃ 0) => (ih ‹x^n' ≃ 0›).left
    | Or.inr (_ : x ≃ 0) => ‹x ≃ 0›
  have : step n' ≄ 0 := step_neqv_zero
  exact And.intro ‹x ≃ 0› ‹step n' ≄ 0›
State after:
no goals

State before:
ℕ : Type
inst✝¹¹ : Core ℕ
inst✝¹⁰ : Induction ℕ
inst✝⁹ : Addition ℕ
inst✝⁸ : Sign ℕ
inst✝⁷ : Multiplication ℕ
α : Type
mul : α → α → α
inst✝⁶ : EqvOp α
inst✝⁵ : OfNat α 1
inst✝⁴ : Exponentiation ℕ mul
inst✝³ : AA.Substitutive₂ (fun x x_1 => x * x_1) AA.tc (fun x x_1 => x ≃ x_1) fun x x_1 => x ≃ x_1
inst✝² : OfNat α 0
inst✝¹ : AP (1 ≄ 0)
inst✝ : AA.ZeroProduct fun x x_1 => x * x_1
x : α
n : ℕ
⊢ x ^ 0 ≃ 0 → x ≃ 0 ∧ 0 ≄ 0
Tactic:
intro (_ : x^(0:ℕ) ≃ 0)
State after:
ℕ : Type
inst✝¹¹ : Core ℕ
inst✝¹⁰ : Induction ℕ
inst✝⁹ : Addition ℕ
inst✝⁸ : Sign ℕ
inst✝⁷ : Multiplication ℕ
α : Type
mul : α → α → α
inst✝⁶ : EqvOp α
inst✝⁵ : OfNat α 1
inst✝⁴ : Exponentiation ℕ mul
inst✝³ : AA.Substitutive₂ (fun x x_1 => x * x_1) AA.tc (fun x x_1 => x ≃ x_1) fun x x_1 => x ≃ x_1
inst✝² : OfNat α 0
inst✝¹ : AP (1 ≄ 0)
inst✝ : AA.ZeroProduct fun x x_1 => x * x_1
x : α
n : ℕ
x✝ : x ^ 0 ≃ 0
⊢ x ≃ 0 ∧ 0 ≄ 0

State before:
ℕ : Type
inst✝¹¹ : Core ℕ
inst✝¹⁰ : Induction ℕ
inst✝⁹ : Addition ℕ
inst✝⁸ : Sign ℕ
inst✝⁷ : Multiplication ℕ
α : Type
mul : α → α → α
inst✝⁶ : EqvOp α
inst✝⁵ : OfNat α 1
inst✝⁴ : Exponentiation ℕ mul
inst✝³ : AA.Substitutive₂ (fun x x_1 => x * x_1) AA.tc (fun x x_1 => x ≃ x_1) fun x x_1 => x ≃ x_1
inst✝² : OfNat α 0
inst✝¹ : AP (1 ≄ 0)
inst✝ : AA.ZeroProduct fun x x_1 => x * x_1
x : α
n : ℕ
x✝ : x ^ 0 ≃ 0
⊢ x ≃ 0 ∧ 0 ≄ 0
Tactic:
have : (1:α) ≃ 0 := calc
  _ ≃ 1   := Rel.refl
  _ ≃ x^0 := Rel.symm pow_zero
  _ ≃ 0   := ‹x^(0:ℕ) ≃ 0›
State after:
ℕ : Type
inst✝¹¹ : Core ℕ
inst✝¹⁰ : Induction ℕ
inst✝⁹ : Addition ℕ
inst✝⁸ : Sign ℕ
inst✝⁷ : Multiplication ℕ
α : Type
mul : α → α → α
inst✝⁶ : EqvOp α
inst✝⁵ : OfNat α 1
inst✝⁴ : Exponentiation ℕ mul
inst✝³ : AA.Substitutive₂ (fun x x_1 => x * x_1) AA.tc (fun x x_1 => x ≃ x_1) fun x x_1 => x ≃ x_1
inst✝² : OfNat α 0
inst✝¹ : AP (1 ≄ 0)
inst✝ : AA.ZeroProduct fun x x_1 => x * x_1
x : α
n : ℕ
x✝ : x ^ 0 ≃ 0
this : 1 ≃ 0
⊢ x ≃ 0 ∧ 0 ≄ 0

State before:
ℕ : Type
inst✝¹¹ : Core ℕ
inst✝¹⁰ : Induction ℕ
inst✝⁹ : Addition ℕ
inst✝⁸ : Sign ℕ
inst✝⁷ : Multiplication ℕ
α : Type
mul : α → α → α
inst✝⁶ : EqvOp α
inst✝⁵ : OfNat α 1
inst✝⁴ : Exponentiation ℕ mul
inst✝³ : AA.Substitutive₂ (fun x x_1 => x * x_1) AA.tc (fun x x_1 => x ≃ x_1) fun x x_1 => x ≃ x_1
inst✝² : OfNat α 0
inst✝¹ : AP (1 ≄ 0)
inst✝ : AA.ZeroProduct fun x x_1 => x * x_1
x : α
n : ℕ
x✝ : x ^ 0 ≃ 0
this : 1 ≃ 0
⊢ x ≃ 0 ∧ 0 ≄ 0
Tactic:
exact absurd ‹(1:α) ≃ 0› ‹AP ((1:α) ≄ 0)›.ev
State after:
no goals

State before:
ℕ : Type
inst✝¹¹ : Core ℕ
inst✝¹⁰ : Induction ℕ
inst✝⁹ : Addition ℕ
inst✝⁸ : Sign ℕ
inst✝⁷ : Multiplication ℕ
α : Type
mul : α → α → α
inst✝⁶ : EqvOp α
inst✝⁵ : OfNat α 1
inst✝⁴ : Exponentiation ℕ mul
inst✝³ : AA.Substitutive₂ (fun x x_1 => x * x_1) AA.tc (fun x x_1 => x ≃ x_1) fun x x_1 => x ≃ x_1
inst✝² : OfNat α 0
inst✝¹ : AP (1 ≄ 0)
inst✝ : AA.ZeroProduct fun x x_1 => x * x_1
x : α
n : ℕ
⊢ ∀ (m : ℕ), (x ^ m ≃ 0 → x ≃ 0 ∧ m ≄ 0) → x ^ step m ≃ 0 → x ≃ 0 ∧ step m ≄ 0
Tactic:
intro (n' : ℕ) (ih : x^n' ≃ 0 → x ≃ 0 ∧ n' ≄ 0) (_ : x^(step n') ≃ 0)
State after:
ℕ : Type
inst✝¹¹ : Core ℕ
inst✝¹⁰ : Induction ℕ
inst✝⁹ : Addition ℕ
inst✝⁸ : Sign ℕ
inst✝⁷ : Multiplication ℕ
α : Type
mul : α → α → α
inst✝⁶ : EqvOp α
inst✝⁵ : OfNat α 1
inst✝⁴ : Exponentiation ℕ mul
inst✝³ : AA.Substitutive₂ (fun x x_1 => x * x_1) AA.tc (fun x x_1 => x ≃ x_1) fun x x_1 => x ≃ x_1
inst✝² : OfNat α 0
inst✝¹ : AP (1 ≄ 0)
inst✝ : AA.ZeroProduct fun x x_1 => x * x_1
x : α
n n' : ℕ
ih : x ^ n' ≃ 0 → x ≃ 0 ∧ n' ≄ 0
x✝ : x ^ step n' ≃ 0
⊢ x ≃ 0 ∧ step n' ≄ 0

State before:
ℕ : Type
inst✝¹¹ : Core ℕ
inst✝¹⁰ : Induction ℕ
inst✝⁹ : Addition ℕ
inst✝⁸ : Sign ℕ
inst✝⁷ : Multiplication ℕ
α : Type
mul : α → α → α
inst✝⁶ : EqvOp α
inst✝⁵ : OfNat α 1
inst✝⁴ : Exponentiation ℕ mul
inst✝³ : AA.Substitutive₂ (fun x x_1 => x * x_1) AA.tc (fun x x_1 => x ≃ x_1) fun x x_1 => x ≃ x_1
inst✝² : OfNat α 0
inst✝¹ : AP (1 ≄ 0)
inst✝ : AA.ZeroProduct fun x x_1 => x * x_1
x : α
n n' : ℕ
ih : x ^ n' ≃ 0 → x ≃ 0 ∧ n' ≄ 0
x✝ : x ^ step n' ≃ 0
⊢ x ≃ 0 ∧ step n' ≄ 0
Tactic:
have : x^n' * x ≃ 0 := AA.eqv_substL pow_step ‹x^(step n') ≃ 0›
State after:
ℕ : Type
inst✝¹¹ : Core ℕ
inst✝¹⁰ : Induction ℕ
inst✝⁹ : Addition ℕ
inst✝⁸ : Sign ℕ
inst✝⁷ : Multiplication ℕ
α : Type
mul : α → α → α
inst✝⁶ : EqvOp α
inst✝⁵ : OfNat α 1
inst✝⁴ : Exponentiation ℕ mul
inst✝³ : AA.Substitutive₂ (fun x x_1 => x * x_1) AA.tc (fun x x_1 => x ≃ x_1) fun x x_1 => x ≃ x_1
inst✝² : OfNat α 0
inst✝¹ : AP (1 ≄ 0)
inst✝ : AA.ZeroProduct fun x x_1 => x * x_1
x : α
n n' : ℕ
ih : x ^ n' ≃ 0 → x ≃ 0 ∧ n' ≄ 0
x✝ : x ^ step n' ≃ 0
this : x ^ n' * x ≃ 0
⊢ x ≃ 0 ∧ step n' ≄ 0

State before:
ℕ : Type
inst✝¹¹ : Core ℕ
inst✝¹⁰ : Induction ℕ
inst✝⁹ : Addition ℕ
inst✝⁸ : Sign ℕ
inst✝⁷ : Multiplication ℕ
α : Type
mul : α → α → α
inst✝⁶ : EqvOp α
inst✝⁵ : OfNat α 1
inst✝⁴ : Exponentiation ℕ mul
inst✝³ : AA.Substitutive₂ (fun x x_1 => x * x_1) AA.tc (fun x x_1 => x ≃ x_1) fun x x_1 => x ≃ x_1
inst✝² : OfNat α 0
inst✝¹ : AP (1 ≄ 0)
inst✝ : AA.ZeroProduct fun x x_1 => x * x_1
x : α
n n' : ℕ
ih : x ^ n' ≃ 0 → x ≃ 0 ∧ n' ≄ 0
x✝ : x ^ step n' ≃ 0
this : x ^ n' * x ≃ 0
⊢ x ≃ 0 ∧ step n' ≄ 0
Tactic:
have : x^n' ≃ 0 ∨ x ≃ 0 := AA.zero_prod this
State after:
ℕ : Type
inst✝¹¹ : Core ℕ
inst✝¹⁰ : Induction ℕ
inst✝⁹ : Addition ℕ
inst✝⁸ : Sign ℕ
inst✝⁷ : Multiplication ℕ
α : Type
mul : α → α → α
inst✝⁶ : EqvOp α
inst✝⁵ : OfNat α 1
inst✝⁴ : Exponentiation ℕ mul
inst✝³ : AA.Substitutive₂ (fun x x_1 => x * x_1) AA.tc (fun x x_1 => x ≃ x_1) fun x x_1 => x ≃ x_1
inst✝² : OfNat α 0
inst✝¹ : AP (1 ≄ 0)
inst✝ : AA.ZeroProduct fun x x_1 => x * x_1
x : α
n n' : ℕ
ih : x ^ n' ≃ 0 → x ≃ 0 ∧ n' ≄ 0
x✝ : x ^ step n' ≃ 0
this✝ : x ^ n' * x ≃ 0
this : x ^ n' ≃ 0 ∨ x ≃ 0
⊢ x ≃ 0 ∧ step n' ≄ 0

State before:
ℕ : Type
inst✝¹¹ : Core ℕ
inst✝¹⁰ : Induction ℕ
inst✝⁹ : Addition ℕ
inst✝⁸ : Sign ℕ
inst✝⁷ : Multiplication ℕ
α : Type
mul : α → α → α
inst✝⁶ : EqvOp α
inst✝⁵ : OfNat α 1
inst✝⁴ : Exponentiation ℕ mul
inst✝³ : AA.Substitutive₂ (fun x x_1 => x * x_1) AA.tc (fun x x_1 => x ≃ x_1) fun x x_1 => x ≃ x_1
inst✝² : OfNat α 0
inst✝¹ : AP (1 ≄ 0)
inst✝ : AA.ZeroProduct fun x x_1 => x * x_1
x : α
n n' : ℕ
ih : x ^ n' ≃ 0 → x ≃ 0 ∧ n' ≄ 0
x✝ : x ^ step n' ≃ 0
this✝ : x ^ n' * x ≃ 0
this : x ^ n' ≃ 0 ∨ x ≃ 0
⊢ x ≃ 0 ∧ step n' ≄ 0
Tactic:
have : x ≃ 0 :=
  match this with
  | Or.inl (_ : x^n' ≃ 0) => (ih ‹x^n' ≃ 0›).left
  | Or.inr (_ : x ≃ 0) => ‹x ≃ 0›
State after:
ℕ : Type
inst✝¹¹ : Core ℕ
inst✝¹⁰ : Induction ℕ
inst✝⁹ : Addition ℕ
inst✝⁸ : Sign ℕ
inst✝⁷ : Multiplication ℕ
α : Type
mul : α → α → α
inst✝⁶ : EqvOp α
inst✝⁵ : OfNat α 1
inst✝⁴ : Exponentiation ℕ mul
inst✝³ : AA.Substitutive₂ (fun x x_1 => x * x_1) AA.tc (fun x x_1 => x ≃ x_1) fun x x_1 => x ≃ x_1
inst✝² : OfNat α 0
inst✝¹ : AP (1 ≄ 0)
inst✝ : AA.ZeroProduct fun x x_1 => x * x_1
x : α
n n' : ℕ
ih : x ^ n' ≃ 0 → x ≃ 0 ∧ n' ≄ 0
x✝ : x ^ step n' ≃ 0
this✝¹ : x ^ n' * x ≃ 0
this✝ : x ^ n' ≃ 0 ∨ x ≃ 0
this : x ≃ 0
⊢ x ≃ 0 ∧ step n' ≄ 0

State before:
ℕ : Type
inst✝¹¹ : Core ℕ
inst✝¹⁰ : Induction ℕ
inst✝⁹ : Addition ℕ
inst✝⁸ : Sign ℕ
inst✝⁷ : Multiplication ℕ
α : Type
mul : α → α → α
inst✝⁶ : EqvOp α
inst✝⁵ : OfNat α 1
inst✝⁴ : Exponentiation ℕ mul
inst✝³ : AA.Substitutive₂ (fun x x_1 => x * x_1) AA.tc (fun x x_1 => x ≃ x_1) fun x x_1 => x ≃ x_1
inst✝² : OfNat α 0
inst✝¹ : AP (1 ≄ 0)
inst✝ : AA.ZeroProduct fun x x_1 => x * x_1
x : α
n n' : ℕ
ih : x ^ n' ≃ 0 → x ≃ 0 ∧ n' ≄ 0
x✝ : x ^ step n' ≃ 0
this✝¹ : x ^ n' * x ≃ 0
this✝ : x ^ n' ≃ 0 ∨ x ≃ 0
this : x ≃ 0
⊢ x ≃ 0 ∧ step n' ≄ 0
Tactic:
have : step n' ≄ 0 := step_neqv_zero
State after:
ℕ : Type
inst✝¹¹ : Core ℕ
inst✝¹⁰ : Induction ℕ
inst✝⁹ : Addition ℕ
inst✝⁸ : Sign ℕ
inst✝⁷ : Multiplication ℕ
α : Type
mul : α → α → α
inst✝⁶ : EqvOp α
inst✝⁵ : OfNat α 1
inst✝⁴ : Exponentiation ℕ mul
inst✝³ : AA.Substitutive₂ (fun x x_1 => x * x_1) AA.tc (fun x x_1 => x ≃ x_1) fun x x_1 => x ≃ x_1
inst✝² : OfNat α 0
inst✝¹ : AP (1 ≄ 0)
inst✝ : AA.ZeroProduct fun x x_1 => x * x_1
x : α
n n' : ℕ
ih : x ^ n' ≃ 0 → x ≃ 0 ∧ n' ≄ 0
x✝ : x ^ step n' ≃ 0
this✝² : x ^ n' * x ≃ 0
this✝¹ : x ^ n' ≃ 0 ∨ x ≃ 0
this✝ : x ≃ 0
this : step n' ≄ 0
⊢ x ≃ 0 ∧ step n' ≄ 0

State before:
ℕ : Type
inst✝¹¹ : Core ℕ
inst✝¹⁰ : Induction ℕ
inst✝⁹ : Addition ℕ
inst✝⁸ : Sign ℕ
inst✝⁷ : Multiplication ℕ
α : Type
mul : α → α → α
inst✝⁶ : EqvOp α
inst✝⁵ : OfNat α 1
inst✝⁴ : Exponentiation ℕ mul
inst✝³ : AA.Substitutive₂ (fun x x_1 => x * x_1) AA.tc (fun x x_1 => x ≃ x_1) fun x x_1 => x ≃ x_1
inst✝² : OfNat α 0
inst✝¹ : AP (1 ≄ 0)
inst✝ : AA.ZeroProduct fun x x_1 => x * x_1
x : α
n n' : ℕ
ih : x ^ n' ≃ 0 → x ≃ 0 ∧ n' ≄ 0
x✝ : x ^ step n' ≃ 0
this✝² : x ^ n' * x ≃ 0
this✝¹ : x ^ n' ≃ 0 ∨ x ≃ 0
this✝ : x ≃ 0
this : step n' ≄ 0
⊢ x ≃ 0 ∧ step n' ≄ 0
Tactic:
exact And.intro ‹x ≃ 0› ‹step n' ≄ 0›
State after:
no goals