Theorem: Lean4Axiomatic.Natural.pow_distribR_mul (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 y : α
n : ℕ
⊢ (x * y) ^ n ≃ x ^ n * y ^ n
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 y : α
n : ℕ
⊢ (x * y) ^ 0 ≃ x ^ 0 * y ^ 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 : 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 y : α
n : ℕ
⊢ ∀ (m : ℕ), (x * y) ^ m ≃ x ^ m * y ^ m → (x * y) ^ step m ≃ x ^ step m * y ^ step 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 y : α
n : ℕ
⊢ (x * y) ^ 0 ≃ x ^ 0 * y ^ 0
Tactic:
case zero =>
  show (x * y)^0 ≃ x^0 * y^0
  calc
    _ ≃ (x * y)^0 := Rel.refl
    _ ≃ 1         := pow_zero
    _ ≃ 1 * 1     := Rel.symm AA.identR
    _ ≃ x^0 * 1   := AA.substL (Rel.symm pow_zero)
    _ ≃ x^0 * y^0 := AA.substR (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 y : α
n : ℕ
⊢ ∀ (m : ℕ), (x * y) ^ m ≃ x ^ m * y ^ m → (x * y) ^ step m ≃ x ^ step m * y ^ step m
Tactic:
case step =>
  intro n' (ih : (x * y)^n' ≃ x^n' * y^n')
  show (x * y)^(step n') ≃ x^(step n') * y^(step n')
  calc
    _ ≃ (x * y)^(step n')         := Rel.refl
    _ ≃ (x * y)^n' * (x * y)      := pow_step
    _ ≃ (x^n' * y^n') * (x * y)   := AA.substL ih
    _ ≃ (x^n' * x) * (y^n' * y)   := AA.expr_xxfxxff_lr_swap_rl
    _ ≃ x^(step n') * (y^n' * y)  := AA.substL (Rel.symm pow_step)
    _ ≃ x^(step n') * y^(step n') := AA.substR (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 y : α
n : ℕ
⊢ (x * y) ^ 0 ≃ x ^ 0 * y ^ 0
Tactic:
calc
  _ ≃ (x * y)^0 := Rel.refl
  _ ≃ 1         := pow_zero
  _ ≃ 1 * 1     := Rel.symm AA.identR
  _ ≃ x^0 * 1   := AA.substL (Rel.symm pow_zero)
  _ ≃ x^0 * y^0 := AA.substR (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 y : α
n : ℕ
⊢ ∀ (m : ℕ), (x * y) ^ m ≃ x ^ m * y ^ m → (x * y) ^ step m ≃ x ^ step m * y ^ step m
Tactic:
intro n' (ih : (x * y)^n' ≃ x^n' * y^n')
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 y : α
n n' : ℕ
ih : (x * y) ^ n' ≃ x ^ n' * y ^ n'
⊢ (x * y) ^ step n' ≃ x ^ step n' * y ^ step n'

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 y : α
n n' : ℕ
ih : (x * y) ^ n' ≃ x ^ n' * y ^ n'
⊢ (x * y) ^ step n' ≃ x ^ step n' * y ^ step n'
Tactic:
calc
  _ ≃ (x * y)^(step n')         := Rel.refl
  _ ≃ (x * y)^n' * (x * y)      := pow_step
  _ ≃ (x^n' * y^n') * (x * y)   := AA.substL ih
  _ ≃ (x^n' * x) * (y^n' * y)   := AA.expr_xxfxxff_lr_swap_rl
  _ ≃ x^(step n') * (y^n' * y)  := AA.substL (Rel.symm pow_step)
  _ ≃ x^(step n') * y^(step n') := AA.substR (Rel.symm pow_step)
State after:
no goals