315 lines
10 KiB
Text
315 lines
10 KiB
Text
/-
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Copyright (c) 2014 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Floris van Doorn, Leonardo de Moura, Jeremy Avigad
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Basic operations on the natural numbers.
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-/
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import ..num algebra.ring
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open binary eq.ops
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namespace nat
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/- a variant of add, defined by recursion on the first argument -/
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definition addl (x y : ℕ) : ℕ :=
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nat.rec y (λ n r, succ r) x
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infix ` ⊕ `:65 := addl
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theorem addl_succ_right (n m : ℕ) : n ⊕ succ m = succ (n ⊕ m) :=
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nat.induction_on n
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rfl
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(λ n₁ ih, calc
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succ n₁ ⊕ succ m = succ (n₁ ⊕ succ m) : rfl
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... = succ (succ (n₁ ⊕ m)) : ih
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... = succ (succ n₁ ⊕ m) : rfl)
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theorem add_eq_addl (x : ℕ) : ∀y, x + y = x ⊕ y :=
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nat.induction_on x
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(λ y, nat.induction_on y
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rfl
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(λ y₁ ih, calc
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0 + succ y₁ = succ (0 + y₁) : rfl
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... = succ (0 ⊕ y₁) : {ih}
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... = 0 ⊕ (succ y₁) : rfl))
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(λ x₁ ih₁ y, nat.induction_on y
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(calc
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succ x₁ + 0 = succ (x₁ + 0) : rfl
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... = succ (x₁ ⊕ 0) : {ih₁ 0}
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... = succ x₁ ⊕ 0 : rfl)
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(λ y₁ ih₂, calc
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succ x₁ + succ y₁ = succ (succ x₁ + y₁) : rfl
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... = succ (succ x₁ ⊕ y₁) : {ih₂}
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... = succ x₁ ⊕ succ y₁ : addl_succ_right))
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/- successor and predecessor -/
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theorem succ_ne_zero (n : ℕ) : succ n ≠ 0 :=
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by contradiction
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-- add_rewrite succ_ne_zero
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theorem pred_zero [simp] : pred 0 = 0 :=
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rfl
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theorem pred_succ [simp] (n : ℕ) : pred (succ n) = n :=
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rfl
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theorem eq_zero_or_eq_succ_pred (n : ℕ) : n = 0 ∨ n = succ (pred n) :=
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nat.induction_on n
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(or.inl rfl)
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(take m IH, or.inr
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(show succ m = succ (pred (succ m)), from congr_arg succ !pred_succ⁻¹))
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theorem exists_eq_succ_of_ne_zero {n : ℕ} (H : n ≠ 0) : ∃k : ℕ, n = succ k :=
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exists.intro _ (or_resolve_right !eq_zero_or_eq_succ_pred H)
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theorem succ.inj {n m : ℕ} (H : succ n = succ m) : n = m :=
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nat.no_confusion H imp.id
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abbreviation eq_of_succ_eq_succ := @succ.inj
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theorem succ_ne_self {n : ℕ} : succ n ≠ n :=
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nat.induction_on n
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(take H : 1 = 0,
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have ne : 1 ≠ 0, from !succ_ne_zero,
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absurd H ne)
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(take k IH H, IH (succ.inj H))
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theorem discriminate {B : Prop} {n : ℕ} (H1: n = 0 → B) (H2 : ∀m, n = succ m → B) : B :=
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have H : n = n → B, from nat.cases_on n H1 H2,
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H rfl
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theorem two_step_induction_on {P : ℕ → Prop} (a : ℕ) (H1 : P 0) (H2 : P 1)
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(H3 : ∀ (n : ℕ) (IH1 : P n) (IH2 : P (succ n)), P (succ (succ n))) : P a :=
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have stronger : P a ∧ P (succ a), from
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nat.induction_on a
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(and.intro H1 H2)
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(take k IH,
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have IH1 : P k, from and.elim_left IH,
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have IH2 : P (succ k), from and.elim_right IH,
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and.intro IH2 (H3 k IH1 IH2)),
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and.elim_left stronger
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theorem sub_induction {P : ℕ → ℕ → Prop} (n m : ℕ) (H1 : ∀m, P 0 m)
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(H2 : ∀n, P (succ n) 0) (H3 : ∀n m, P n m → P (succ n) (succ m)) : P n m :=
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have general : ∀m, P n m, from nat.induction_on n H1
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(take k : ℕ,
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assume IH : ∀m, P k m,
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take m : ℕ,
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nat.cases_on m (H2 k) (take l, (H3 k l (IH l)))),
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general m
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/- addition -/
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protected theorem add_zero [simp] (n : ℕ) : n + 0 = n :=
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rfl
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theorem add_succ [simp] (n m : ℕ) : n + succ m = succ (n + m) :=
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rfl
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protected theorem zero_add [simp] (n : ℕ) : 0 + n = n :=
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nat.induction_on n
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!nat.add_zero
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(take m IH, show 0 + succ m = succ m, from
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calc
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0 + succ m = succ (0 + m) : add_succ
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... = succ m : IH)
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theorem succ_add [simp] (n m : ℕ) : (succ n) + m = succ (n + m) :=
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nat.induction_on m
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(!nat.add_zero ▸ !nat.add_zero)
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(take k IH, calc
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succ n + succ k = succ (succ n + k) : add_succ
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... = succ (succ (n + k)) : IH
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... = succ (n + succ k) : add_succ)
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protected theorem add_comm [simp] (n m : ℕ) : n + m = m + n :=
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nat.induction_on m
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(by rewrite [nat.add_zero, nat.zero_add])
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(take k IH, calc
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n + succ k = succ (n+k) : add_succ
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... = succ (k + n) : IH
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... = succ k + n : succ_add)
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theorem succ_add_eq_succ_add (n m : ℕ) : succ n + m = n + succ m :=
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!succ_add ⬝ !add_succ⁻¹
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protected theorem add_assoc [simp] (n m k : ℕ) : (n + m) + k = n + (m + k) :=
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nat.induction_on k
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(by rewrite +nat.add_zero)
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(take l IH,
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calc
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(n + m) + succ l = succ ((n + m) + l) : add_succ
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... = succ (n + (m + l)) : IH
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... = n + succ (m + l) : add_succ
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... = n + (m + succ l) : add_succ)
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protected theorem add_left_comm : Π (n m k : ℕ), n + (m + k) = m + (n + k) :=
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left_comm nat.add_comm nat.add_assoc
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protected theorem add_right_comm : Π (n m k : ℕ), n + m + k = n + k + m :=
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right_comm nat.add_comm nat.add_assoc
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protected theorem add_left_cancel {n m k : ℕ} : n + m = n + k → m = k :=
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nat.induction_on n
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(take H : 0 + m = 0 + k,
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!nat.zero_add⁻¹ ⬝ H ⬝ !nat.zero_add)
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(take (n : ℕ) (IH : n + m = n + k → m = k) (H : succ n + m = succ n + k),
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have succ (n + m) = succ (n + k),
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from calc
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succ (n + m) = succ n + m : succ_add
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... = succ n + k : H
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... = succ (n + k) : succ_add,
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have n + m = n + k, from succ.inj this,
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IH this)
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protected theorem add_right_cancel {n m k : ℕ} (H : n + m = k + m) : n = k :=
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have H2 : m + n = m + k, from !nat.add_comm ⬝ H ⬝ !nat.add_comm,
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nat.add_left_cancel H2
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theorem eq_zero_of_add_eq_zero_right {n m : ℕ} : n + m = 0 → n = 0 :=
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nat.induction_on n
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(take (H : 0 + m = 0), rfl)
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(take k IH,
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assume H : succ k + m = 0,
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absurd
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(show succ (k + m) = 0, from calc
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succ (k + m) = succ k + m : succ_add
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... = 0 : H)
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!succ_ne_zero)
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theorem eq_zero_of_add_eq_zero_left {n m : ℕ} (H : n + m = 0) : m = 0 :=
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eq_zero_of_add_eq_zero_right (!nat.add_comm ⬝ H)
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theorem eq_zero_and_eq_zero_of_add_eq_zero {n m : ℕ} (H : n + m = 0) : n = 0 ∧ m = 0 :=
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and.intro (eq_zero_of_add_eq_zero_right H) (eq_zero_of_add_eq_zero_left H)
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theorem add_one [simp] (n : ℕ) : n + 1 = succ n := rfl
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theorem one_add (n : ℕ) : 1 + n = succ n :=
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!nat.zero_add ▸ !succ_add
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/- multiplication -/
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protected theorem mul_zero [simp] (n : ℕ) : n * 0 = 0 :=
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rfl
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theorem mul_succ [simp] (n m : ℕ) : n * succ m = n * m + n :=
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rfl
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-- commutativity, distributivity, associativity, identity
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protected theorem zero_mul [simp] (n : ℕ) : 0 * n = 0 :=
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nat.induction_on n
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!nat.mul_zero
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(take m IH, !mul_succ ⬝ !nat.add_zero ⬝ IH)
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theorem succ_mul [simp] (n m : ℕ) : (succ n) * m = (n * m) + m :=
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nat.induction_on m
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(by rewrite nat.mul_zero)
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(take k IH, calc
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succ n * succ k = succ n * k + succ n : mul_succ
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... = n * k + k + succ n : IH
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... = n * k + (k + succ n) : nat.add_assoc
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... = n * k + (succ n + k) : nat.add_comm
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... = n * k + (n + succ k) : succ_add_eq_succ_add
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... = n * k + n + succ k : nat.add_assoc
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... = n * succ k + succ k : mul_succ)
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protected theorem mul_comm [simp] (n m : ℕ) : n * m = m * n :=
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nat.induction_on m
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(!nat.mul_zero ⬝ !nat.zero_mul⁻¹)
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(take k IH, calc
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n * succ k = n * k + n : mul_succ
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... = k * n + n : IH
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... = (succ k) * n : succ_mul)
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protected theorem right_distrib (n m k : ℕ) : (n + m) * k = n * k + m * k :=
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nat.induction_on k
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(calc
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(n + m) * 0 = 0 : nat.mul_zero
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... = 0 + 0 : nat.add_zero
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... = n * 0 + 0 : nat.mul_zero
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... = n * 0 + m * 0 : nat.mul_zero)
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(take l IH, calc
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(n + m) * succ l = (n + m) * l + (n + m) : mul_succ
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... = n * l + m * l + (n + m) : IH
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... = n * l + m * l + n + m : nat.add_assoc
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... = n * l + n + m * l + m : nat.add_right_comm
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... = n * l + n + (m * l + m) : nat.add_assoc
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... = n * succ l + (m * l + m) : mul_succ
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... = n * succ l + m * succ l : mul_succ)
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protected theorem left_distrib (n m k : ℕ) : n * (m + k) = n * m + n * k :=
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calc
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n * (m + k) = (m + k) * n : nat.mul_comm
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... = m * n + k * n : nat.right_distrib
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... = n * m + k * n : nat.mul_comm
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... = n * m + n * k : nat.mul_comm
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protected theorem mul_assoc [simp] (n m k : ℕ) : (n * m) * k = n * (m * k) :=
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nat.induction_on k
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(calc
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(n * m) * 0 = n * (m * 0) : nat.mul_zero)
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(take l IH,
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calc
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(n * m) * succ l = (n * m) * l + n * m : mul_succ
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... = n * (m * l) + n * m : IH
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... = n * (m * l + m) : nat.left_distrib
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... = n * (m * succ l) : mul_succ)
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protected theorem mul_one [simp] (n : ℕ) : n * 1 = n :=
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calc
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n * 1 = n * 0 + n : mul_succ
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... = 0 + n : nat.mul_zero
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... = n : nat.zero_add
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protected theorem one_mul [simp] (n : ℕ) : 1 * n = n :=
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calc
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1 * n = n * 1 : nat.mul_comm
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... = n : nat.mul_one
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theorem eq_zero_or_eq_zero_of_mul_eq_zero {n m : ℕ} : n * m = 0 → n = 0 ∨ m = 0 :=
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nat.cases_on n
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(assume H, or.inl rfl)
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(take n',
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nat.cases_on m
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(assume H, or.inr rfl)
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(take m',
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assume H : succ n' * succ m' = 0,
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absurd
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(calc
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0 = succ n' * succ m' : H
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... = succ n' * m' + succ n' : mul_succ
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... = succ (succ n' * m' + n') : add_succ)⁻¹
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!succ_ne_zero))
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protected definition comm_semiring [reducible] [trans_instance] : comm_semiring nat :=
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⦃comm_semiring,
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add := nat.add,
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add_assoc := nat.add_assoc,
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zero := nat.zero,
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zero_add := nat.zero_add,
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add_zero := nat.add_zero,
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add_comm := nat.add_comm,
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mul := nat.mul,
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mul_assoc := nat.mul_assoc,
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one := nat.succ nat.zero,
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one_mul := nat.one_mul,
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mul_one := nat.mul_one,
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left_distrib := nat.left_distrib,
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right_distrib := nat.right_distrib,
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zero_mul := nat.zero_mul,
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mul_zero := nat.mul_zero,
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mul_comm := nat.mul_comm⦄
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end nat
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section
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open nat
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definition iterate {A : Type} (op : A → A) : ℕ → A → A
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| 0 := λ a, a
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| (succ k) := λ a, op (iterate k a)
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notation f`^[`n`]` := iterate f n
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end
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