351 lines
12 KiB
Text
351 lines
12 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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Module: data.nat.basic
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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 logic.connectives data.num algebra.binary 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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zero + succ y₁ = succ (zero + y₁) : rfl
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... = succ (zero ⊕ y₁) : {ih}
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... = zero ⊕ (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 + zero = succ (x₁ + zero) : rfl
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... = succ (x₁ ⊕ zero) : {ih₁ zero}
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... = succ x₁ ⊕ zero : 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 : pred 0 = 0 :=
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rfl
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theorem pred_succ (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 (λe, e)
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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
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(take m : ℕ, H1 m)
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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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theorem add_zero (n : ℕ) : n + 0 = n :=
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rfl
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theorem add_succ (n m : ℕ) : n + succ m = succ (n + m) :=
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rfl
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theorem zero_add (n : ℕ) : 0 + n = n :=
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nat.induction_on n
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!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 (n m : ℕ) : (succ n) + m = succ (n + m) :=
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nat.induction_on m
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(!add_zero ▸ !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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theorem add.comm (n m : ℕ) : n + m = m + n :=
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nat.induction_on m
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(!add_zero ⬝ !zero_add⁻¹)
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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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theorem add.assoc (n m k : ℕ) : (n + m) + k = n + (m + k) :=
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nat.induction_on k
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(!add_zero ▸ !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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theorem add.left_comm (n m k : ℕ) : n + (m + k) = m + (n + k) :=
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left_comm add.comm add.assoc n m k
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theorem add.right_comm (n m k : ℕ) : n + m + k = n + k + m :=
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right_comm add.comm add.assoc n m k
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theorem add.cancel_left {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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!zero_add⁻¹ ⬝ H ⬝ !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 H2 : 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 H3 : n + m = n + k, from succ_inj H2,
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IH H3)
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theorem add.cancel_right {n m k : ℕ} (H : n + m = k + m) : n = k :=
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have H2 : m + n = m + k, from !add.comm ⬝ H ⬝ !add.comm,
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add.cancel_left 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 (!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 (n : ℕ) : n + 1 = succ n :=
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!add_zero ▸ !add_succ
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theorem one_add (n : ℕ) : 1 + n = succ n :=
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!zero_add ▸ !succ_add
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/- multiplication -/
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theorem mul_zero (n : ℕ) : n * 0 = 0 :=
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rfl
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theorem mul_succ (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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theorem zero_mul (n : ℕ) : 0 * n = 0 :=
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nat.induction_on n
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!mul_zero
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(take m IH, !mul_succ ⬝ !add_zero ⬝ IH)
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theorem succ_mul (n m : ℕ) : (succ n) * m = (n * m) + m :=
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nat.induction_on m
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(!mul_zero ⬝ !mul_zero⁻¹ ⬝ !add_zero⁻¹)
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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) : add.assoc
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... = n * k + (succ n + k) : 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 : add.assoc
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... = n * succ k + succ k : mul_succ)
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theorem mul.comm (n m : ℕ) : n * m = m * n :=
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nat.induction_on m
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(!mul_zero ⬝ !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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theorem mul.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 : mul_zero
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... = 0 + 0 : add_zero
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... = n * 0 + 0 : mul_zero
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... = n * 0 + m * 0 : 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 : add.assoc
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... = n * l + n + m * l + m : add.right_comm
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... = n * l + n + (m * l + m) : 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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theorem mul.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 : mul.comm
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... = m * n + k * n : mul.right_distrib
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... = n * m + k * n : mul.comm
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... = n * m + n * k : mul.comm
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theorem mul.assoc (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) : 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) : mul.left_distrib
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... = n * (m * succ l) : mul_succ)
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theorem mul_one (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 : mul_zero
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... = n : zero_add
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theorem one_mul (n : ℕ) : 1 * n = n :=
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calc
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1 * n = n * 1 : mul.comm
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... = n : 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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section
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open [classes] algebra
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protected definition comm_semiring [instance] [reducible] : algebra.comm_semiring nat :=
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⦃algebra.comm_semiring,
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add := add,
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add_assoc := add.assoc,
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zero := zero,
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zero_add := zero_add,
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add_zero := add_zero,
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add_comm := add.comm,
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mul := mul,
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mul_assoc := mul.assoc,
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one := succ zero,
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one_mul := one_mul,
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mul_one := mul_one,
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left_distrib := mul.left_distrib,
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right_distrib := mul.right_distrib,
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zero_mul := zero_mul,
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mul_zero := mul_zero,
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mul_comm := mul.comm⦄
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end
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section port_algebra
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open [classes] algebra
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theorem mul.left_comm : ∀a b c : ℕ, a * (b * c) = b * (a * c) := algebra.mul.left_comm
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theorem mul.right_comm : ∀a b c : ℕ, (a * b) * c = (a * c) * b := algebra.mul.right_comm
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definition dvd (a b : ℕ) : Prop := algebra.dvd a b
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notation a ∣ b := dvd a b
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theorem dvd.intro : ∀{a b c : ℕ} (H : a * c = b), a ∣ b := @algebra.dvd.intro _ _
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theorem dvd.intro_left : ∀{a b c : ℕ} (H : c * a = b), a ∣ b := @algebra.dvd.intro_left _ _
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theorem exists_eq_mul_right_of_dvd : ∀{a b : ℕ} (H : a ∣ b), ∃c, b = a * c :=
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@algebra.exists_eq_mul_right_of_dvd _ _
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theorem dvd.elim : ∀{P : Prop} {a b : ℕ} (H₁ : a ∣ b) (H₂ : ∀c, b = a * c → P), P :=
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@algebra.dvd.elim _ _
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theorem exists_eq_mul_left_of_dvd : ∀{a b : ℕ} (H : a ∣ b), ∃c, b = c * a :=
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@algebra.exists_eq_mul_left_of_dvd _ _
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theorem dvd.elim_left : ∀{P : Prop} {a b : ℕ} (H₁ : a ∣ b) (H₂ : ∀c, b = c * a → P), P :=
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@algebra.dvd.elim_left _ _
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theorem dvd.refl : ∀a : ℕ, a ∣ a := algebra.dvd.refl
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theorem dvd.trans : ∀{a b c : ℕ}, a ∣ b → b ∣ c → a ∣ c := @algebra.dvd.trans _ _
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theorem eq_zero_of_zero_dvd : ∀{a : ℕ}, 0 ∣ a → a = 0 := @algebra.eq_zero_of_zero_dvd _ _
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theorem dvd_zero : ∀a : ℕ, a ∣ 0 := algebra.dvd_zero
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theorem one_dvd : ∀a : ℕ, 1 ∣ a := algebra.one_dvd
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theorem dvd_mul_right : ∀a b : ℕ, a ∣ a * b := algebra.dvd_mul_right
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theorem dvd_mul_left : ∀a b : ℕ, a ∣ b * a := algebra.dvd_mul_left
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theorem dvd_mul_of_dvd_left : ∀{a b : ℕ} (H : a ∣ b) (c : ℕ), a ∣ b * c :=
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@algebra.dvd_mul_of_dvd_left _ _
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theorem dvd_mul_of_dvd_right : ∀{a b : ℕ} (H : a ∣ b) (c : ℕ), a ∣ c * b :=
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@algebra.dvd_mul_of_dvd_right _ _
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theorem mul_dvd_mul : ∀{a b c d : ℕ}, a ∣ b → c ∣ d → a * c ∣ b * d :=
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@algebra.mul_dvd_mul _ _
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theorem dvd_of_mul_right_dvd : ∀{a b c : ℕ}, a * b ∣ c → a ∣ c :=
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@algebra.dvd_of_mul_right_dvd _ _
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theorem dvd_of_mul_left_dvd : ∀{a b c : ℕ}, a * b ∣ c → b ∣ c :=
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@algebra.dvd_of_mul_left_dvd _ _
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theorem dvd_add : ∀{a b c : ℕ}, a ∣ b → a ∣ c → a ∣ b + c := @algebra.dvd_add _ _
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end port_algebra
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end nat
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