323 lines
10 KiB
Text
323 lines
10 KiB
Text
--- 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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--- Author: Floris van Doorn, Leonardo de Moura
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-- Basic operations on the natural numbers.
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import logic.connectives data.num algebra.binary
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open eq.ops binary
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namespace nat
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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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-- -------------------------
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theorem succ_ne_zero (n : ℕ) : succ n ≠ 0 :=
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assume H, no_confusion H
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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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irreducible pred
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theorem zero_or_succ_pred (n : ℕ) : n = 0 ∨ n = succ (pred n) :=
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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 zero_or_exists_succ (n : ℕ) : n = 0 ∨ ∃k, n = succ k :=
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or.imp_or (zero_or_succ_pred n) (assume H, H)
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(assume H : n = succ (pred n), exists_intro (pred n) H)
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theorem case {P : ℕ → Prop} (n : ℕ) (H1: P 0) (H2 : ∀m, P (succ m)) : P n :=
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induction_on n H1 (take m IH, H2 m)
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theorem succ.inj {n m : ℕ} (H : succ n = succ m) : n = m :=
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no_confusion H (λe, e)
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theorem succ.ne_self {n : ℕ} : succ n ≠ n :=
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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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or.elim (zero_or_succ_pred n)
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(take H3 : n = 0, H1 H3)
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(take H3 : n = succ (pred n), H2 (pred n) H3)
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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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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 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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discriminate
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(assume Hm : m = 0, Hm⁻¹ ▸ (H2 k))
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(take l : ℕ, assume Hm : m = succ l, Hm⁻¹ ▸ (H3 k l (IH l)))),
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general m
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-- Addition
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-- --------
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theorem add.zero_right (n : ℕ) : n + 0 = n :=
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rfl
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theorem add.succ_right (n m : ℕ) : n + succ m = succ (n + m) :=
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rfl
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irreducible add
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theorem add.zero_left (n : ℕ) : 0 + n = n :=
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induction_on n
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!add.zero_right
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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_right
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... = succ m : IH)
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theorem add.succ_left (n m : ℕ) : (succ n) + m = succ (n + m) :=
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induction_on m
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(!add.zero_right ▸ !add.zero_right)
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(take k IH, calc
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succ n + succ k = succ (succ n + k) : add.succ_right
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... = succ (succ (n + k)) : IH
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... = succ (n + succ k) : add.succ_right)
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theorem add.comm (n m : ℕ) : n + m = m + n :=
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induction_on m
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(!add.zero_right ⬝ !add.zero_left⁻¹)
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(take k IH, calc
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n + succ k = succ (n+k) : add.succ_right
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... = succ (k + n) : IH
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... = succ k + n : add.succ_left)
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theorem add.move_succ (n m : ℕ) : succ n + m = n + succ m :=
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!add.succ_left ⬝ !add.succ_right⁻¹
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theorem add.comm_succ (n m : ℕ) : n + succ m = m + succ n :=
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!add.move_succ⁻¹ ⬝ !add.comm
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theorem add.assoc (n m k : ℕ) : (n + m) + k = n + (m + k) :=
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induction_on k
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(!add.zero_right ▸ !add.zero_right)
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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_right
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... = succ (n + (m + l)) : IH
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... = n + succ (m + l) : add.succ_right
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... = n + (m + succ l) : add.succ_right)
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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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-- ### cancelation
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theorem add.cancel_left {n m k : ℕ} : n + m = n + k → m = k :=
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induction_on n
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(take H : 0 + m = 0 + k,
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!add.zero_left⁻¹ ⬝ H ⬝ !add.zero_left)
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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 : add.succ_left
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... = succ n + k : H
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... = succ (n + k) : add.succ_left,
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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 add.eq_zero_left {n m : ℕ} : n + m = 0 → n = 0 :=
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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 : add.succ_left
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... = 0 : H)
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!succ_ne_zero)
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theorem add.eq_zero_right {n m : ℕ} (H : n + m = 0) : m = 0 :=
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add.eq_zero_left (!add.comm ⬝ H)
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theorem add.eq_zero {n m : ℕ} (H : n + m = 0) : n = 0 ∧ m = 0 :=
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and.intro (add.eq_zero_left H) (add.eq_zero_right H)
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-- ### misc
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theorem add.one (n : ℕ) : n + 1 = succ n :=
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!add.zero_right ▸ !add.succ_right
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theorem add.one_left (n : ℕ) : 1 + n = succ n :=
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!add.zero_left ▸ !add.succ_left
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-- TODO: rename? remove?
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theorem induction_plus_one {P : nat → Prop} (a : ℕ) (H1 : P 0)
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(H2 : ∀ (n : ℕ) (IH : P n), P (n + 1)) : P a :=
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nat.rec H1 (take n IH, !add.one ▸ (H2 n IH)) a
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-- Multiplication
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-- --------------
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theorem mul.zero_right (n : ℕ) : n * 0 = 0 :=
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rfl
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theorem mul.succ_right (n m : ℕ) : n * succ m = n * m + n :=
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rfl
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irreducible mul
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-- ### commutativity, distributivity, associativity, identity
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theorem mul.zero_left (n : ℕ) : 0 * n = 0 :=
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induction_on n
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!mul.zero_right
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(take m IH, !mul.succ_right ⬝ !add.zero_right ⬝ IH)
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theorem mul.succ_left (n m : ℕ) : (succ n) * m = (n * m) + m :=
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induction_on m
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(!mul.zero_right ⬝ !mul.zero_right⁻¹ ⬝ !add.zero_right⁻¹)
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(take k IH, calc
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succ n * succ k = (succ n * k) + succ n : mul.succ_right
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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) + (n + succ k) : add.comm_succ
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... = (n * k) + n + succ k : add.assoc
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... = (n * succ k) + succ k : mul.succ_right)
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theorem mul.comm (n m : ℕ) : n * m = m * n :=
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induction_on m
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(!mul.zero_right ⬝ !mul.zero_left⁻¹)
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(take k IH, calc
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n * succ k = n * k + n : mul.succ_right
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... = k * n + n : IH
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... = (succ k) * n : mul.succ_left)
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theorem mul.distr_right (n m k : ℕ) : (n + m) * k = n * k + m * k :=
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induction_on k
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(calc
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(n + m) * 0 = 0 : mul.zero_right
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... = 0 + 0 : add.zero_right
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... = n * 0 + 0 : mul.zero_right
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... = n * 0 + m * 0 : mul.zero_right)
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(take l IH, calc
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(n + m) * succ l = (n + m) * l + (n + m) : mul.succ_right
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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_right
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... = n * succ l + m * succ l : mul.succ_right)
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theorem mul.distr_left (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.distr_right
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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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induction_on k
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(calc
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(n * m) * 0 = 0 : mul.zero_right
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... = n * 0 : mul.zero_right
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... = n * (m * 0) : mul.zero_right)
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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_right
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... = n * (m * l) + n * m : IH
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... = n * (m * l + m) : mul.distr_left
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... = n * (m * succ l) : mul.succ_right)
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theorem mul.left_comm (n m k : ℕ) : n * (m * k) = m * (n * k) :=
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left_comm mul.comm mul.assoc n m k
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theorem mul.right_comm (n m k : ℕ) : n * m * k = n * k * m :=
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right_comm mul.comm mul.assoc n m k
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theorem mul.one_right (n : ℕ) : n * 1 = n :=
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calc
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n * 1 = n * 0 + n : mul.succ_right
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... = 0 + n : mul.zero_right
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... = n : add.zero_left
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theorem mul.one_left (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_right
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theorem mul.eq_zero {n m : ℕ} (H : n * m = 0) : n = 0 ∨ m = 0 :=
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discriminate
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(take Hn : n = 0, or.inl Hn)
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(take (k : ℕ),
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assume (Hk : n = succ k),
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discriminate
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(take (Hm : m = 0), or.inr Hm)
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(take (l : ℕ),
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assume (Hl : m = succ l),
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have Heq : succ (k * succ l + l) = n * m, from
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(calc
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n * m = n * succ l : Hl
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... = succ k * succ l : Hk
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... = k * succ l + succ l : mul.succ_left
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... = succ (k * succ l + l) : add.succ_right)⁻¹,
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absurd (Heq ⬝ H) !succ_ne_zero))
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end nat
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