Theorem: Lean4Axiomatic.Natural.pow_compatL_add (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 : AA.Associative fun x x_1 => x * x_1
inst✝¹ : AA.Commutative fun x x_1 => x * x_1
inst✝ : AA.Identity 1 fun x x_1 => x * x_1
x : α
n m : ℕ
⊢ x ^ (n + m) ≃ x ^ n * x ^ m
Tactic:
apply ind_on 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 : AA.Associative fun x x_1 => x * x_1
inst✝¹ : AA.Commutative fun x x_1 => x * x_1
inst✝ : AA.Identity 1 fun x x_1 => x * x_1
x : α
n m : ℕ
⊢ x ^ (0 + m) ≃ x ^ 0 * x ^ m

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 : AA.Associative fun x x_1 => x * x_1
inst✝¹ : AA.Commutative fun x x_1 => x * x_1
inst✝ : AA.Identity 1 fun x x_1 => x * x_1
x : α
n m : ℕ
⊢ ∀ (m_1 : ℕ), x ^ (m_1 + m) ≃ x ^ m_1 * x ^ m → x ^ (step m_1 + m) ≃ x ^ step m_1 * x ^ m

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 : AA.Associative fun x x_1 => x * x_1
inst✝¹ : AA.Commutative fun x x_1 => x * x_1
inst✝ : AA.Identity 1 fun x x_1 => x * x_1
x : α
n m : ℕ
⊢ x ^ (0 + m) ≃ x ^ 0 * x ^ m
Tactic:
case zero =>
  show x^(0 + m) ≃ x^(0:ℕ) * x^m
  calc
    _ ≃ x^(0 + m)     := Rel.refl
    _ ≃ x^m           := pow_substR AA.identL
    _ ≃ 1 * x^m       := Rel.symm AA.identL
    _ ≃ x^(0:ℕ) * x^m := AA.substL (Rel.symm pow_zero)
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 : AA.Associative fun x x_1 => x * x_1
inst✝¹ : AA.Commutative fun x x_1 => x * x_1
inst✝ : AA.Identity 1 fun x x_1 => x * x_1
x : α
n m : ℕ
⊢ ∀ (m_1 : ℕ), x ^ (m_1 + m) ≃ x ^ m_1 * x ^ m → x ^ (step m_1 + m) ≃ x ^ step m_1 * x ^ m
Tactic:
case step =>
  intro n' (ih : x^(n' + m) ≃ x^n' * x^m)
  show x^(step n' + m) ≃ x^(step n') * x^m
  calc
    _ ≃ x^(step n' + m)   := Rel.refl
    _ ≃ x^(step (n' + m)) := pow_substR (Rel.symm AA.scompatL)
    _ ≃ x^(n' + m) * x    := pow_step
    _ ≃ (x^n' * x^m) * x  := AA.substL ih
    _ ≃ x^n' * (x^m * x)  := AA.assoc
    _ ≃ x^n' * (x * x^m)  := AA.substR AA.comm
    _ ≃ (x^n' * x) * x^m  := Rel.symm AA.assoc
    _ ≃ x^(step n') * x^m := AA.substL (Rel.symm pow_step)
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 : AA.Associative fun x x_1 => x * x_1
inst✝¹ : AA.Commutative fun x x_1 => x * x_1
inst✝ : AA.Identity 1 fun x x_1 => x * x_1
x : α
n m : ℕ
⊢ x ^ (0 + m) ≃ x ^ 0 * x ^ m
Tactic:
calc
  _ ≃ x^(0 + m)     := Rel.refl
  _ ≃ x^m           := pow_substR AA.identL
  _ ≃ 1 * x^m       := Rel.symm AA.identL
  _ ≃ x^(0:ℕ) * x^m := AA.substL (Rel.symm pow_zero)
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 : AA.Associative fun x x_1 => x * x_1
inst✝¹ : AA.Commutative fun x x_1 => x * x_1
inst✝ : AA.Identity 1 fun x x_1 => x * x_1
x : α
n m : ℕ
⊢ ∀ (m_1 : ℕ), x ^ (m_1 + m) ≃ x ^ m_1 * x ^ m → x ^ (step m_1 + m) ≃ x ^ step m_1 * x ^ m
Tactic:
intro n' (ih : x^(n' + m) ≃ x^n' * x^m)
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 : AA.Associative fun x x_1 => x * x_1
inst✝¹ : AA.Commutative fun x x_1 => x * x_1
inst✝ : AA.Identity 1 fun x x_1 => x * x_1
x : α
n m n' : ℕ
ih : x ^ (n' + m) ≃ x ^ n' * x ^ m
⊢ x ^ (step n' + m) ≃ x ^ step n' * x ^ m

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 : AA.Associative fun x x_1 => x * x_1
inst✝¹ : AA.Commutative fun x x_1 => x * x_1
inst✝ : AA.Identity 1 fun x x_1 => x * x_1
x : α
n m n' : ℕ
ih : x ^ (n' + m) ≃ x ^ n' * x ^ m
⊢ x ^ (step n' + m) ≃ x ^ step n' * x ^ m
Tactic:
calc
  _ ≃ x^(step n' + m)   := Rel.refl
  _ ≃ x^(step (n' + m)) := pow_substR (Rel.symm AA.scompatL)
  _ ≃ x^(n' + m) * x    := pow_step
  _ ≃ (x^n' * x^m) * x  := AA.substL ih
  _ ≃ x^n' * (x^m * x)  := AA.assoc
  _ ≃ x^n' * (x * x^m)  := AA.substR AA.comm
  _ ≃ (x^n' * x) * x^m  := Rel.symm AA.assoc
  _ ≃ x^(step n') * x^m := AA.substL (Rel.symm pow_step)
State after:
no goals