---------------------------------------------------------------------------------------------------- -- Copyright (c) 2014 Floris van Doorn. All rights reserved. -- Released under Apache 2.0 license as described in the file LICENSE. -- Author: Floris van Doorn ---------------------------------------------------------------------------------------------------- import logic struc.binary using tactic num binary eq_ops using decidable namespace nat inductive nat : Type := zero : nat, succ : nat → nat notation `ℕ`:max := nat abbreviation plus (x y : ℕ) : ℕ := nat_rec x (λ n r, succ r) y definition to_nat [coercion] [inline] (n : num) : ℕ := num_rec zero (λ n, pos_num_rec (succ zero) (λ n r, plus r (plus r (succ zero))) (λ n r, plus r r) n) n namespace helper_tactics definition apply_refl := apply @refl tactic_hint apply_refl end helper_tactics using helper_tactics theorem nat_rec_zero {P : ℕ → Type} (x : P 0) (f : ∀m, P m → P (succ m)) : nat_rec x f 0 = x theorem nat_rec_succ {P : ℕ → Type} (x : P 0) (f : ∀m, P m → P (succ m)) (n : ℕ) : nat_rec x f (succ n) = f n (nat_rec x f n) theorem induction_on {P : ℕ → Prop} (a : ℕ) (H1 : P 0) (H2 : ∀ (n : ℕ) (IH : P n), P (succ n)) : P a := nat_rec H1 H2 a definition rec_on {P : ℕ → Type} (n : ℕ) (H1 : P 0) (H2 : ∀m, P m → P (succ m)) : P n := nat_rec H1 H2 n -------------------------------------------------- succ pred theorem succ_ne_zero (n : ℕ) : succ n ≠ 0 := assume H : succ n = 0, have H2 : true = false, from let f := (nat_rec false (fun a b, true)) in calc true = f (succ n) : _ ... = f 0 : {H} ... = false : _, absurd H2 true_ne_false definition pred (n : ℕ) := nat_rec 0 (fun m x, m) n theorem pred_zero : pred 0 = 0 theorem pred_succ (n : ℕ) : pred (succ n) = n theorem zero_or_succ (n : ℕ) : n = 0 ∨ n = succ (pred n) := induction_on n (or_intro_left _ (refl 0)) (take m IH, or_intro_right _ (show succ m = succ (pred (succ m)), from congr_arg succ (pred_succ m⁻¹))) theorem zero_or_succ2 (n : ℕ) : n = 0 ∨ ∃k, n = succ k := or_imp_or (zero_or_succ n) (assume H, H) (assume H : n = succ (pred n), exists_intro (pred n) H) theorem case {P : ℕ → Prop} (n : ℕ) (H1: P 0) (H2 : ∀m, P (succ m)) : P n := induction_on n H1 (take m IH, H2 m) theorem discriminate {B : Prop} {n : ℕ} (H1: n = 0 → B) (H2 : ∀m, n = succ m → B) : B := or_elim (zero_or_succ n) (take H3 : n = 0, H1 H3) (take H3 : n = succ (pred n), H2 (pred n) H3) theorem succ_inj {n m : ℕ} (H : succ n = succ m) : n = m := calc n = pred (succ n) : pred_succ n⁻¹ ... = pred (succ m) : {H} ... = m : pred_succ m theorem succ_ne_self (n : ℕ) : succ n ≠ n := induction_on n (take H : 1 = 0, have ne : 1 ≠ 0, from succ_ne_zero 0, absurd H ne) (take k IH H, IH (succ_inj H)) theorem decidable_eq [instance] (n m : ℕ) : decidable (n = m) := have general : ∀n, decidable (n = m), from rec_on m (take n, rec_on n (inl (refl 0)) (λ m iH, inr (succ_ne_zero m))) (λ (m' : ℕ) (iH1 : ∀n, decidable (n = m')), take n, rec_on n (inr (ne_symm (succ_ne_zero m'))) (λ (n' : ℕ) (iH2 : decidable (n' = succ m')), have d1 : decidable (n' = m'), from iH1 n', decidable.rec_on d1 (assume Heq : n' = m', inl (congr_arg succ Heq)) (assume Hne : n' ≠ m', have H1 : succ n' ≠ succ m', from assume Heq, absurd (succ_inj Heq) Hne, inr H1))), general n theorem two_step_induction_on {P : ℕ → Prop} (a : ℕ) (H1 : P 0) (H2 : P 1) (H3 : ∀ (n : ℕ) (IH1 : P n) (IH2 : P (succ n)), P (succ (succ n))) : P a := have stronger : P a ∧ P (succ a), from induction_on a (and_intro H1 H2) (take k IH, have IH1 : P k, from and_elim_left IH, have IH2 : P (succ k), from and_elim_right IH, and_intro IH2 (H3 k IH1 IH2)), and_elim_left stronger theorem sub_induction {P : ℕ → ℕ → Prop} (n m : ℕ) (H1 : ∀m, P 0 m) (H2 : ∀n, P (succ n) 0) (H3 : ∀n m, P n m → P (succ n) (succ m)) : P n m := have general : ∀m, P n m, from induction_on n (take m : ℕ, H1 m) (take k : ℕ, assume IH : ∀m, P k m, take m : ℕ, discriminate (assume Hm : m = 0, Hm⁻¹ ▸ (H2 k)) (take l : ℕ, assume Hm : m = succ l, Hm⁻¹ ▸ (H3 k l (IH l)))), general m -------------------------------------------------- add definition add (x y : ℕ) : ℕ := plus x y infixl `+`:65 := add theorem add_zero_right (n : ℕ) : n + 0 = n theorem add_succ_right (n m : ℕ) : n + succ m = succ (n + m) ---------- comm, assoc theorem add_zero_left (n : ℕ) : 0 + n = n := induction_on n (add_zero_right 0) (take m IH, show 0 + succ m = succ m, from calc 0 + succ m = succ (0 + m) : add_succ_right _ _ ... = succ m : {IH}) theorem add_succ_left (n m : ℕ) : (succ n) + m = succ (n + m) := induction_on m (calc succ n + 0 = succ n : add_zero_right (succ n) ... = succ (n + 0) : {symm (add_zero_right n)}) (take k IH, calc succ n + succ k = succ (succ n + k) : add_succ_right _ _ ... = succ (succ (n + k)) : {IH} ... = succ (n + succ k) : {symm (add_succ_right _ _)}) theorem add_comm (n m : ℕ) : n + m = m + n := induction_on m (trans (add_zero_right _) (symm (add_zero_left _))) (take k IH, calc n + succ k = succ (n+k) : add_succ_right _ _ ... = succ (k + n) : {IH} ... = succ k + n : symm (add_succ_left _ _)) theorem add_move_succ (n m : ℕ) : succ n + m = n + succ m := calc succ n + m = succ (n + m) : add_succ_left n m ... = n +succ m : symm (add_succ_right n m) theorem add_comm_succ (n m : ℕ) : n + succ m = m + succ n := calc n + succ m = succ n + m : symm (add_move_succ n m) ... = m + succ n : add_comm (succ n) m theorem add_assoc (n m k : ℕ) : (n + m) + k = n + (m + k) := induction_on k (calc (n + m) + 0 = n + m : add_zero_right _ ... = n + (m + 0) : {symm (add_zero_right m)}) (take l IH, calc (n + m) + succ l = succ ((n + m) + l) : add_succ_right _ _ ... = succ (n + (m + l)) : {IH} ... = n + succ (m + l) : symm (add_succ_right _ _) ... = n + (m + succ l) : {symm (add_succ_right _ _)}) theorem add_left_comm (n m k : ℕ) : n + (m + k) = m + (n + k) := left_comm add_comm add_assoc n m k theorem add_right_comm (n m k : ℕ) : n + m + k = n + k + m := right_comm add_comm add_assoc n m k ---------- inversion theorem add_cancel_left {n m k : ℕ} : n + m = n + k → m = k := induction_on n (take H : 0 + m = 0 + k, calc m = 0 + m : symm (add_zero_left m) ... = 0 + k : H ... = k : add_zero_left k) (take (n : ℕ) (IH : n + m = n + k → m = k) (H : succ n + m = succ n + k), have H2 : succ (n + m) = succ (n + k), from calc succ (n + m) = succ n + m : symm (add_succ_left n m) ... = succ n + k : H ... = succ (n + k) : add_succ_left n k, have H3 : n + m = n + k, from succ_inj H2, IH H3) --rename to and_cancel_right theorem add_cancel_right {n m k : ℕ} (H : n + m = k + m) : n = k := have H2 : m + n = m + k, from calc m + n = n + m : add_comm m n ... = k + m : H ... = m + k : add_comm k m, add_cancel_left H2 theorem add_eq_zero_left {n m : ℕ} : n + m = 0 → n = 0 := induction_on n (take (H : 0 + m = 0), refl 0) (take k IH, assume (H : succ k + m = 0), absurd_elim (succ k = 0) (show succ (k + m) = 0, from calc succ (k + m) = succ k + m : symm (add_succ_left k m) ... = 0 : H) (succ_ne_zero (k + m))) theorem add_eq_zero_right {n m : ℕ} (H : n + m = 0) : m = 0 := add_eq_zero_left (trans (add_comm m n) H) theorem add_eq_zero {n m : ℕ} (H : n + m = 0) : n = 0 ∧ m = 0 := and_intro (add_eq_zero_left H) (add_eq_zero_right H) -- add_eq_self below ---------- misc theorem add_one (n:ℕ) : n + 1 = succ n := calc n + 1 = succ (n + 0) : add_succ_right _ _ ... = succ n : {add_zero_right _} theorem add_one_left (n:ℕ) : 1 + n = succ n := calc 1 + n = succ (0 + n) : add_succ_left _ _ ... = succ n : {add_zero_left _} --the following theorem has a terrible name, but since the name is not a substring or superstring of another name, it is at least easy to globally replace it theorem induction_plus_one {P : ℕ → Prop} (a : ℕ) (H1 : P 0) (H2 : ∀ (n : ℕ) (IH : P n), P (n + 1)) : P a := nat_rec H1 (take n IH, (add_one n) ▸ (H2 n IH)) a -------------------------------------------------- mul definition mul (n m : ℕ) := nat_rec 0 (fun m x, x + n) m infixl `*`:75 := mul theorem mul_zero_right (n:ℕ) : n * 0 = 0 theorem mul_succ_right (n m:ℕ) : n * succ m = n * m + n set_option unifier.max_steps 100000 ---------- comm, distr, assoc, identity theorem mul_zero_left (n:ℕ) : 0 * n = 0 := induction_on n (mul_zero_right 0) (take m IH, calc 0 * succ m = 0 * m + 0 : mul_succ_right _ _ ... = 0 * m : add_zero_right _ ... = 0 : IH) theorem mul_succ_left (n m:ℕ) : (succ n) * m = (n * m) + m := induction_on m (calc succ n * 0 = 0 : mul_zero_right _ ... = n * 0 : symm (mul_zero_right _) ... = n * 0 + 0 : symm (add_zero_right _)) (take k IH, calc succ n * succ k = (succ n * k) + succ n : mul_succ_right _ _ ... = (n * k) + k + succ n : { IH } ... = (n * k) + (k + succ n) : add_assoc _ _ _ ... = (n * k) + (n + succ k) : {add_comm_succ _ _} ... = (n * k) + n + succ k : symm (add_assoc _ _ _) ... = (n * succ k) + succ k : {symm (mul_succ_right n k)}) theorem mul_comm (n m:ℕ) : n * m = m * n := induction_on m (trans (mul_zero_right _) (symm (mul_zero_left _))) (take k IH, calc n * succ k = n * k + n : mul_succ_right _ _ ... = k * n + n : {IH} ... = (succ k) * n : symm (mul_succ_left _ _)) theorem mul_add_distr_left (n m k : ℕ) : (n + m) * k = n * k + m * k := induction_on k (calc (n + m) * 0 = 0 : mul_zero_right _ ... = 0 + 0 : symm (add_zero_right _) ... = n * 0 + 0 : refl _ ... = n * 0 + m * 0 : refl _) (take l IH, calc (n + m) * succ l = (n + m) * l + (n + m) : mul_succ_right _ _ ... = n * l + m * l + (n + m) : {IH} ... = n * l + m * l + n + m : symm (add_assoc _ _ _) ... = n * l + n + m * l + m : {add_right_comm _ _ _} ... = n * l + n + (m * l + m) : add_assoc _ _ _ ... = n * succ l + (m * l + m) : {symm (mul_succ_right _ _)} ... = n * succ l + m * succ l : {symm (mul_succ_right _ _)}) theorem mul_add_distr_right (n m k : ℕ) : n * (m + k) = n * m + n * k := calc n * (m + k) = (m + k) * n : mul_comm _ _ ... = m * n + k * n : mul_add_distr_left _ _ _ ... = n * m + k * n : {mul_comm _ _} ... = n * m + n * k : {mul_comm _ _} theorem mul_assoc (n m k:ℕ) : (n * m) * k = n * (m * k) := induction_on k (calc (n * m) * 0 = 0 : mul_zero_right _ ... = n * 0 : symm (mul_zero_right _) ... = n * (m * 0) : {symm (mul_zero_right _)}) (take l IH, calc (n * m) * succ l = (n * m) * l + n * m : mul_succ_right _ _ ... = n * (m * l) + n * m : {IH} ... = n * (m * l + m) : symm (mul_add_distr_right _ _ _) ... = n * (m * succ l) : {symm (mul_succ_right _ _)}) theorem mul_comm_left (n m k : ℕ) : n * (m * k) = m * (n * k) := left_comm mul_comm mul_assoc n m k theorem mul_comm_right (n m k : ℕ) : n * m * k = n * k * m := right_comm mul_comm mul_assoc n m k theorem mul_one_right (n : ℕ) : n * 1 = n := calc n * 1 = n * 0 + n : mul_succ_right n 0 ... = 0 + n : {mul_zero_right n} ... = n : add_zero_left n theorem mul_one_left (n : ℕ) : 1 * n = n := calc 1 * n = n * 1 : mul_comm _ _ ... = n : mul_one_right n ---------- inversion theorem mul_eq_zero {n m : ℕ} (H : n * m = 0) : n = 0 ∨ m = 0 := discriminate (take Hn : n = 0, or_intro_left _ Hn) (take (k : ℕ), assume (Hk : n = succ k), discriminate (take (Hm : m = 0), or_intro_right _ Hm) (take (l : ℕ), assume (Hl : m = succ l), have Heq : succ (k * succ l + l) = n * m, from symm (calc n * m = n * succ l : { Hl } ... = succ k * succ l : { Hk } ... = k * succ l + succ l : mul_succ_left _ _ ... = succ (k * succ l + l) : add_succ_right _ _), absurd_elim _ (trans Heq H) (succ_ne_zero _))) -- see more under "positivity" below -------------------------------------------------- le definition le (n m:ℕ) : Prop := ∃k, n + k = m infix `<=`:50 := le infix `≤`:50 := le theorem le_intro {n m k : ℕ} (H : n + k = m) : n ≤ m := exists_intro k H theorem le_elim {n m : ℕ} (H : n ≤ m) : ∃ k, n + k = m := H ---------- partial order (totality is part of lt) theorem le_intro2 (n m : ℕ) : n ≤ n + m := le_intro (refl (n + m)) theorem le_refl (n : ℕ) : n ≤ n := le_intro (add_zero_right n) theorem zero_le (n : ℕ) : 0 ≤ n := le_intro (add_zero_left n) theorem le_zero {n : ℕ} (H : n ≤ 0) : n = 0 := obtain (k : ℕ) (Hk : n + k = 0), from le_elim H, add_eq_zero_left Hk theorem not_succ_zero_le (n : ℕ) : ¬ succ n ≤ 0 := assume H : succ n ≤ 0, have H2 : succ n = 0, from le_zero H, absurd H2 (succ_ne_zero n) theorem le_zero_inv {n : ℕ} (H : n ≤ 0) : n = 0 := obtain (k : ℕ) (Hk : n + k = 0), from le_elim H, add_eq_zero_left Hk theorem le_trans {n m k : ℕ} (H1 : n ≤ m) (H2 : m ≤ k) : n ≤ k := obtain (l1 : ℕ) (Hl1 : n + l1 = m), from le_elim H1, obtain (l2 : ℕ) (Hl2 : m + l2 = k), from le_elim H2, le_intro (calc n + (l1 + l2) = n + l1 + l2 : symm (add_assoc n l1 l2) ... = m + l2 : { Hl1 } ... = k : Hl2) theorem le_antisym {n m : ℕ} (H1 : n ≤ m) (H2 : m ≤ n) : n = m := obtain (k : ℕ) (Hk : n + k = m), from (le_elim H1), obtain (l : ℕ) (Hl : m + l = n), from (le_elim H2), have L1 : k + l = 0, from add_cancel_left (calc n + (k + l) = n + k + l : { symm (add_assoc n k l) } ... = m + l : { Hk } ... = n : Hl ... = n + 0 : symm (add_zero_right n)), have L2 : k = 0, from add_eq_zero_left L1, calc n = n + 0 : symm (add_zero_right n) ... = n + k : { symm L2 } ... = m : Hk ---------- interaction with add theorem add_le_left {n m : ℕ} (H : n ≤ m) (k : ℕ) : k + n ≤ k + m := obtain (l : ℕ) (Hl : n + l = m), from (le_elim H), le_intro (calc k + n + l = k + (n + l) : add_assoc k n l ... = k + m : { Hl }) theorem add_le_right {n m : ℕ} (H : n ≤ m) (k : ℕ) : n + k ≤ m + k := (add_comm k m) ▸ (add_comm k n) ▸ (add_le_left H k) theorem add_le {n m k l : ℕ} (H1 : n ≤ k) (H2 : m ≤ l) : n + m ≤ k + l := le_trans (add_le_right H1 m) (add_le_left H2 k) theorem add_le_left_inv {n m k : ℕ} (H : k + n ≤ k + m) : n ≤ m := obtain (l : ℕ) (Hl : k + n + l = k + m), from (le_elim H), le_intro (add_cancel_left (calc k + (n + l) = k + n + l : symm (add_assoc k n l) ... = k + m : Hl)) theorem add_le_right_inv {n m k : ℕ} (H : n + k ≤ m + k) : n ≤ m := add_le_left_inv (add_comm m k ▸ add_comm n k ▸ H) ---------- interaction with succ and pred theorem succ_le {n m : ℕ} (H : n ≤ m) : succ n ≤ succ m := add_one m ▸ add_one n ▸ add_le_right H 1 theorem succ_le_cancel {n m : ℕ} (H : succ n ≤ succ m) : n ≤ m := add_le_right_inv (add_one m⁻¹ ▸ add_one n⁻¹ ▸ H) theorem self_le_succ (n : ℕ) : n ≤ succ n := le_intro (add_one n) theorem le_imp_le_succ {n m : ℕ} (H : n ≤ m) : n ≤ succ m := le_trans H (self_le_succ m) theorem succ_le_left_or {n m : ℕ} (H : n ≤ m) : succ n ≤ m ∨ n = m := obtain (k : ℕ) (Hk : n + k = m), from (le_elim H), discriminate (assume H3 : k = 0, have Heq : n = m, from calc n = n + 0 : (add_zero_right n)⁻¹ ... = n + k : {H3⁻¹} ... = m : Hk, or_intro_right _ Heq) (take l:ℕ, assume H3 : k = succ l, have Hlt : succ n ≤ m, from (le_intro (calc succ n + l = n + succ l : add_move_succ n l ... = n + k : {H3⁻¹} ... = m : Hk)), or_intro_left _ Hlt) theorem succ_le_left {n m : ℕ} (H1 : n ≤ m) (H2 : n ≠ m) : succ n ≤ m := resolve_left (succ_le_left_or H1) H2 theorem succ_le_right_inv {n m : ℕ} (H : n ≤ succ m) : n ≤ m ∨ n = succ m := or_imp_or (succ_le_left_or H) (take H2 : succ n ≤ succ m, show n ≤ m, from succ_le_cancel H2) (take H2 : n = succ m, H2) theorem succ_le_left_inv {n m : ℕ} (H : succ n ≤ m) : n ≤ m ∧ n ≠ m := obtain (k : ℕ) (H2 : succ n + k = m), from (le_elim H), and_intro (have H3 : n + succ k = m, from calc n + succ k = succ n + k : symm (add_move_succ n k) ... = m : H2, show n ≤ m, from le_intro H3) (assume H3 : n = m, have H4 : succ n ≤ n, from subst (symm H3) H, have H5 : succ n = n, from le_antisym H4 (self_le_succ n), show false, from absurd H5 (succ_ne_self n)) theorem le_pred_self (n : ℕ) : pred n ≤ n := case n (subst (symm pred_zero) (le_refl 0)) (take k : ℕ, subst (symm (pred_succ k)) (self_le_succ k)) theorem pred_le {n m : ℕ} (H : n ≤ m) : pred n ≤ pred m := discriminate (take Hn : n = 0, have H2 : pred n = 0, from calc pred n = pred 0 : {Hn} ... = 0 : pred_zero, subst (symm H2) (zero_le (pred m))) (take k : ℕ, assume Hn : n = succ k, obtain (l : ℕ) (Hl : n + l = m), from le_elim H, have H2 : pred n + l = pred m, from calc pred n + l = pred (succ k) + l : {Hn} ... = k + l : {pred_succ k} ... = pred (succ (k + l)) : symm (pred_succ (k + l)) ... = pred (succ k + l) : {symm (add_succ_left k l)} ... = pred (n + l) : {symm Hn} ... = pred m : {Hl}, le_intro H2) theorem pred_le_left_inv {n m : ℕ} (H : pred n ≤ m) : n ≤ m ∨ n = succ m := discriminate (take Hn : n = 0, or_intro_left _ (subst (symm Hn) (zero_le m))) (take k : ℕ, assume Hn : n = succ k, have H2 : pred n = k, from calc pred n = pred (succ k) : {Hn} ... = k : pred_succ k, have H3 : k ≤ m, from subst H2 H, have H4 : succ k ≤ m ∨ k = m, from succ_le_left_or H3, show n ≤ m ∨ n = succ m, from or_imp_or H4 (take H5 : succ k ≤ m, show n ≤ m, from subst (symm Hn) H5) (take H5 : k = m, show n = succ m, from subst H5 Hn)) -- ### interaction with successor and predecessor theorem le_imp_succ_le_or_eq {n m : ℕ} (H : n ≤ m) : succ n ≤ m ∨ n = m := obtain (k : ℕ) (Hk : n + k = m), from (le_elim H), discriminate (assume H3 : k = 0, have Heq : n = m, from calc n = n + 0 : symm (add_zero_right n) ... = n + k : {symm H3} ... = m : Hk, or_intro_right _ Heq) (take l : nat, assume H3 : k = succ l, have Hlt : succ n ≤ m, from (le_intro (calc succ n + l = n + succ l : add_move_succ n l ... = n + k : {symm H3} ... = m : Hk)), or_intro_left _ Hlt) theorem le_ne_imp_succ_le {n m : ℕ} (H1 : n ≤ m) (H2 : n ≠ m) : succ n ≤ m := resolve_left (le_imp_succ_le_or_eq H1) H2 theorem le_succ_imp_le_or_eq {n m : ℕ} (H : n ≤ succ m) : n ≤ m ∨ n = succ m := or_imp_or_left (le_imp_succ_le_or_eq H) (take H2 : succ n ≤ succ m, show n ≤ m, from succ_le_cancel H2) theorem succ_le_imp_le_and_ne {n m : ℕ} (H : succ n ≤ m) : n ≤ m ∧ n ≠ m := and_intro (le_trans (self_le_succ n) H) (assume H2 : n = m, have H3 : succ n ≤ n, from subst (symm H2) H, have H4 : succ n = n, from le_antisym H3 (self_le_succ n), show false, from absurd H4 (succ_ne_self n)) theorem pred_le_self (n : ℕ) : pred n ≤ n := case n (subst (symm pred_zero) (le_refl 0)) (take k : nat, subst (symm (pred_succ k)) (self_le_succ k)) theorem pred_le_imp_le_or_eq {n m : ℕ} (H : pred n ≤ m) : n ≤ m ∨ n = succ m := discriminate (take Hn : n = 0, or_intro_left _ (subst (symm Hn) (zero_le m))) (take k : nat, assume Hn : n = succ k, have H2 : pred n = k, from calc pred n = pred (succ k) : {Hn} ... = k : pred_succ k, have H3 : k ≤ m, from subst H2 H, have H4 : succ k ≤ m ∨ k = m, from le_imp_succ_le_or_eq H3, show n ≤ m ∨ n = succ m, from or_imp_or H4 (take H5 : succ k ≤ m, show n ≤ m, from subst (symm Hn) H5) (take H5 : k = m, show n = succ m, from subst H5 Hn)) ---------- interaction with mul theorem mul_le_left {n m : ℕ} (H : n ≤ m) (k : ℕ) : k * n ≤ k * m := obtain (l : ℕ) (Hl : n + l = m), from (le_elim H), induction_on k (have H2 : 0 * n = 0 * m, from calc 0 * n = 0 : mul_zero_left n ... = 0 * m : symm (mul_zero_left m), show 0 * n ≤ 0 * m, from subst H2 (le_refl (0 * n))) (take (l : ℕ), assume IH : l * n ≤ l * m, have H2 : l * n + n ≤ l * m + m, from add_le IH H, have H3 : succ l * n ≤ l * m + m, from subst (symm (mul_succ_left l n)) H2, show succ l * n ≤ succ l * m, from subst (symm (mul_succ_left l m)) H3) theorem mul_le_right {n m : ℕ} (H : n ≤ m) (k : ℕ) : n * k ≤ m * k := mul_comm k m ▸ mul_comm k n ▸ (mul_le_left H k) theorem mul_le {n m k l : ℕ} (H1 : n ≤ k) (H2 : m ≤ l) : n * m ≤ k * l := le_trans (mul_le_right H1 m) (mul_le_left H2 k) -- mul_le_[left|right]_inv below -------------------------------------------------- lt definition lt (n m : ℕ) := succ n ≤ m infix `<`:50 := lt theorem lt_intro {n m k : ℕ} (H : succ n + k = m) : n < m := le_intro H theorem lt_elim {n m : ℕ} (H : n < m) : ∃ k, succ n + k = m := le_elim H theorem lt_intro2 (n m : ℕ) : n < n + succ m := lt_intro (add_move_succ n m) -------------------------------------------------- ge, gt definition ge (n m : ℕ) := m ≤ n infix `>=`:50 := ge infix `≥`:50 := ge definition gt (n m : ℕ) := m < n infix `>`:50 := gt ---------- basic facts theorem lt_ne {n m : ℕ} (H : n < m) : n ≠ m := and_elim_right (succ_le_left_inv H) theorem lt_irrefl (n : ℕ) : ¬ n < n := assume H : n < n, absurd (refl n) (lt_ne H) theorem lt_zero (n : ℕ) : 0 < succ n := succ_le (zero_le n) theorem lt_zero_inv (n : ℕ) : ¬ n < 0 := assume H : n < 0, have H2 : succ n = 0, from le_zero_inv H, absurd H2 (succ_ne_zero n) theorem lt_positive {n m : ℕ} (H : n < m) : ∃k, m = succ k := discriminate (take (Hm : m = 0), absurd_elim _ (subst Hm H) (lt_zero_inv n)) (take (l : ℕ) (Hm : m = succ l), exists_intro l Hm) ---------- interaction with le theorem lt_imp_le_succ {n m : ℕ} (H : n < m) : succ n ≤ m := H theorem le_succ_imp_lt {n m : ℕ} (H : succ n ≤ m) : n < m := H theorem self_lt_succ (n : ℕ) : n < succ n := le_refl (succ n) theorem lt_imp_le {n m : ℕ} (H : n < m) : n ≤ m := and_elim_left (succ_le_imp_le_and_ne H) theorem le_imp_lt_or_eq {n m : ℕ} (H : n ≤ m) : n < m ∨ n = m := le_imp_succ_le_or_eq H theorem le_ne_imp_lt {n m : ℕ} (H1 : n ≤ m) (H2 : n ≠ m) : n < m := le_ne_imp_succ_le H1 H2 theorem le_imp_lt_succ {n m : ℕ} (H : n ≤ m) : n < succ m := succ_le H theorem lt_succ_imp_le {n m : ℕ} (H : n < succ m) : n ≤ m := succ_le_cancel H ---------- trans, antisym theorem lt_le_trans {n m k : ℕ} (H1 : n < m) (H2 : m ≤ k) : n < k := le_trans H1 H2 theorem le_lt_trans {n m k : ℕ} (H1 : n ≤ m) (H2 : m < k) : n < k := le_trans (succ_le H1) H2 theorem lt_trans {n m k : ℕ} (H1 : n < m) (H2 : m < k) : n < k := lt_le_trans H1 (lt_imp_le H2) theorem le_imp_not_gt {n m : ℕ} (H : n ≤ m) : ¬ n > m := assume H2 : m < n, absurd (le_lt_trans H H2) (lt_irrefl n) theorem lt_imp_not_ge {n m : ℕ} (H : n < m) : ¬ n ≥ m := assume H2 : m ≤ n, absurd (lt_le_trans H H2) (lt_irrefl n) theorem lt_antisym {n m : ℕ} (H : n < m) : ¬ m < n := le_imp_not_gt (lt_imp_le H) ---------- interaction with add theorem add_lt_left {n m : ℕ} (H : n < m) (k : ℕ) : k + n < k + m := add_succ_right k n ▸ add_le_left H k theorem add_lt_right {n m : ℕ} (H : n < m) (k : ℕ) : n + k < m + k := add_comm k m ▸ add_comm k n ▸ add_lt_left H k theorem add_le_lt {n m k l : ℕ} (H1 : n ≤ k) (H2 : m < l) : n + m < k + l := le_lt_trans (add_le_right H1 m) (add_lt_left H2 k) theorem add_lt_le {n m k l : ℕ} (H1 : n < k) (H2 : m ≤ l) : n + m < k + l := lt_le_trans (add_lt_right H1 m) (add_le_left H2 k) theorem add_lt {n m k l : ℕ} (H1 : n < k) (H2 : m < l) : n + m < k + l := add_lt_le H1 (lt_imp_le H2) theorem add_lt_left_inv {n m k : ℕ} (H : k + n < k + m) : n < m := add_le_left_inv (add_succ_right k n⁻¹ ▸ H) theorem add_lt_right_inv {n m k : ℕ} (H : n + k < m + k) : n < m := add_lt_left_inv (add_comm m k ▸ add_comm n k ▸ H) ---------- interaction with succ (see also the interaction with le) theorem succ_lt {n m : ℕ} (H : n < m) : succ n < succ m := add_one m ▸ add_one n ▸ add_lt_right H 1 theorem succ_lt_inv {n m : ℕ} (H : succ n < succ m) : n < m := add_lt_right_inv (add_one m⁻¹ ▸ add_one n⁻¹ ▸ H) theorem lt_self_succ (n : ℕ) : n < succ n := le_refl (succ n) theorem succ_lt_right {n m : ℕ} (H : n < m) : n < succ m := lt_trans H (lt_self_succ m) ---------- totality of lt and le theorem le_or_lt (n m : ℕ) : n ≤ m ∨ m < n := induction_on n (or_intro_left _ (zero_le m)) (take (k : ℕ), assume IH : k ≤ m ∨ m < k, or_elim IH (assume H : k ≤ m, obtain (l : ℕ) (Hl : k + l = m), from le_elim H, discriminate (assume H2 : l = 0, have H3 : m = k, from calc m = k + l : symm Hl ... = k + 0 : {H2} ... = k : add_zero_right k, have H4 : m < succ k, from subst H3 (lt_self_succ m), or_intro_right _ H4) (take l2 : ℕ, assume H2 : l = succ l2, have H3 : succ k + l2 = m, from calc succ k + l2 = k + succ l2 : add_move_succ k l2 ... = k + l : {symm H2} ... = m : Hl, or_intro_left _ (le_intro H3))) (assume H : m < k, or_intro_right _ (succ_lt_right H))) theorem trichotomy_alt (n m : ℕ) : (n < m ∨ n = m) ∨ m < n := or_imp_or (le_or_lt n m) (assume H : n ≤ m, le_imp_lt_or_eq H) (assume H : m < n, H) theorem trichotomy (n m : ℕ) : n < m ∨ n = m ∨ m < n := iff_elim_left (or_assoc _ _ _) (trichotomy_alt n m) theorem le_total (n m : ℕ) : n ≤ m ∨ m ≤ n := or_imp_or (le_or_lt n m) (assume H : n ≤ m, H) (assume H : m < n, lt_imp_le H) -- interaction with mul under "positivity" theorem strong_induction_on {P : ℕ → Prop} (n : ℕ) (IH : ∀n, (∀m, m < n → P m) → P n) : P n := have stronger : ∀k, k ≤ n → P k, from induction_on n (take (k : ℕ), assume H : k ≤ 0, have H2 : k = 0, from le_zero_inv H, have H3 : ∀m, m < k → P m, from (take m : ℕ, assume H4 : m < k, have H5 : m < 0, from subst H2 H4, absurd_elim _ H5 (lt_zero_inv m)), show P k, from IH k H3) (take l : ℕ, assume IHl : ∀k, k ≤ l → P k, take k : ℕ, assume H : k ≤ succ l, or_elim (succ_le_right_inv H) (assume H2 : k ≤ l, show P k, from IHl k H2) (assume H2 : k = succ l, have H3 : ∀m, m < k → P m, from (take m : ℕ, assume H4 : m < k, have H5 : m ≤ l, from lt_succ_imp_le (subst H2 H4), show P m, from IHl m H5), show P k, from IH k H3)), stronger n (le_refl n) theorem case_strong_induction_on {P : ℕ → Prop} (a : ℕ) (H0 : P 0) (Hind : ∀(n : ℕ), (∀m, m ≤ n → P m) → P (succ n)) : P a := strong_induction_on a (take n, case n (assume H : (∀m, m < 0 → P m), H0) (take n, assume H : (∀m, m < succ n → P m), Hind n (take m, assume H1 : m ≤ n, H m (le_imp_lt_succ H1)))) theorem add_eq_self {n m : ℕ} (H : n + m = n) : m = 0 := discriminate (take Hm : m = 0, Hm) (take k : ℕ, assume Hm : m = succ k, have H2 : succ n + k = n, from calc succ n + k = n + succ k : add_move_succ n k ... = n + m : {symm Hm} ... = n : H, have H3 : n < n, from lt_intro H2, have H4 : n ≠ n, from lt_ne H3, absurd_elim _ (refl n) H4) -------------------------------------------------- positivity -- we use " _ > 0" as canonical way of denoting that a number is positive ---------- basic theorem zero_or_positive (n : ℕ) : n = 0 ∨ n > 0 := or_imp_or (or_swap (le_imp_lt_or_eq (zero_le n))) (take H : 0 = n, symm H) (take H : n > 0, H) theorem succ_positive {n m : ℕ} (H : n = succ m) : n > 0 := subst (symm H) (lt_zero m) theorem ne_zero_positive {n : ℕ} (H : n ≠ 0) : n > 0 := or_elim (zero_or_positive n) (take H2 : n = 0, absurd_elim _ H2 H) (take H2 : n > 0, H2) theorem pos_imp_eq_succ {n : ℕ} (H : n > 0) : ∃l, n = succ l := discriminate (take H2, absurd_elim _ (subst H2 H) (lt_irrefl 0)) (take l Hl, exists_intro l Hl) theorem add_positive_right (n : ℕ) {k : ℕ} (H : k > 0) : n + k > n := obtain (l : ℕ) (Hl : k = succ l), from pos_imp_eq_succ H, subst (symm Hl) (lt_intro2 n l) theorem add_positive_left (n : ℕ) {k : ℕ} (H : k > 0) : k + n > n := subst (add_comm n k) (add_positive_right n H) -- Positivity -- --------- -- -- Writing "t > 0" is the preferred way to assert that a natural number is positive. -- ### basic -- See also succ_pos. theorem succ_pos (n : ℕ) : 0 < succ n := succ_le (zero_le n) theorem case_zero_pos {P : ℕ → Prop} (y : ℕ) (H0 : P 0) (H1 : ∀y, y > 0 → P y) : P y := case y H0 (take y', H1 _ (succ_pos _)) theorem zero_or_pos (n : ℕ) : n = 0 ∨ n > 0 := or_imp_or_left (or_swap (le_imp_lt_or_eq (zero_le n))) (take H : 0 = n, symm H) theorem succ_imp_pos {n m : ℕ} (H : n = succ m) : n > 0 := subst (symm H) (succ_pos m) theorem ne_zero_pos {n : ℕ} (H : n ≠ 0) : n > 0 := or_elim (zero_or_pos n) (take H2 : n = 0, absurd_elim _ H2 H) (take H2 : n > 0, H2) theorem add_pos_right (n : ℕ) {k : ℕ} (H : k > 0) : n + k > n := subst (add_zero_right n) (add_lt_left H n) theorem add_pos_left (n : ℕ) {k : ℕ} (H : k > 0) : k + n > n := subst (add_comm n k) (add_pos_right n H) ---------- mul theorem mul_positive {n m : ℕ} (Hn : n > 0) (Hm : m > 0) : n * m > 0 := obtain (k : ℕ) (Hk : n = succ k), from pos_imp_eq_succ Hn, obtain (l : ℕ) (Hl : m = succ l), from pos_imp_eq_succ Hm, succ_positive (calc n * m = succ k * m : {Hk} ... = succ k * succ l : {Hl} ... = succ k * l + succ k : mul_succ_right (succ k) l ... = succ (succ k * l + k) : add_succ_right _ _) theorem mul_positive_inv_left {n m : ℕ} (H : n * m > 0) : n > 0 := discriminate (assume H2 : n = 0, have H3 : n * m = 0, from calc n * m = 0 * m : {H2} ... = 0 : mul_zero_left m, have H4 : 0 > 0, from subst H3 H, absurd_elim _ H4 (lt_irrefl 0)) (take l : ℕ, assume Hl : n = succ l, subst (symm Hl) (lt_zero l)) theorem mul_positive_inv_right {n m : ℕ} (H : n * m > 0) : m > 0 := mul_positive_inv_left (subst (mul_comm n m) H) theorem mul_left_inj {n m k : ℕ} (Hn : n > 0) (H : n * m = n * k) : m = k := have general : ∀m, n * m = n * k → m = k, from induction_on k (take m:ℕ, assume H : n * m = n * 0, have H2 : n * m = 0, from calc n * m = n * 0 : H ... = 0 : mul_zero_right n, have H3 : n = 0 ∨ m = 0, from mul_eq_zero H2, resolve_right H3 (ne_symm (lt_ne Hn))) (take (l : ℕ), assume (IH : ∀ m, n * m = n * l → m = l), take (m : ℕ), assume (H : n * m = n * succ l), have H2 : n * succ l > 0, from mul_positive Hn (lt_zero l), have H3 : m > 0, from mul_positive_inv_right (subst (symm H) H2), obtain (l2:ℕ) (Hm : m = succ l2), from pos_imp_eq_succ H3, have H4 : n * l2 + n = n * l + n, from calc n * l2 + n = n * succ l2 : symm (mul_succ_right n l2) ... = n * m : {symm Hm} ... = n * succ l : H ... = n * l + n : mul_succ_right n l, have H5 : n * l2 = n * l, from add_cancel_right H4, calc m = succ l2 : Hm ... = succ l : {IH l2 H5}), general m H theorem mul_right_inj {n m k : ℕ} (Hm : m > 0) (H : n * m = k * m) : n = k := mul_left_inj Hm (subst (mul_comm k m) (subst (mul_comm n m) H)) -- mul_eq_one below ---------- interaction of mul with le and lt theorem mul_lt_left {n m k : ℕ} (Hk : k > 0) (H : n < m) : k * n < k * m := have H2 : k * n < k * n + k, from add_positive_right (k * n) Hk, have H3 : k * n + k ≤ k * m, from subst (mul_succ_right k n) (mul_le_left H k), lt_le_trans H2 H3 theorem mul_lt_right {n m k : ℕ} (Hk : k > 0) (H : n < m) : n * k < m * k := subst (mul_comm k m) (subst (mul_comm k n) (mul_lt_left Hk H)) theorem mul_le_lt {n m k l : ℕ} (Hk : k > 0) (H1 : n ≤ k) (H2 : m < l) : n * m < k * l := le_lt_trans (mul_le_right H1 m) (mul_lt_left Hk H2) theorem mul_lt_le {n m k l : ℕ} (Hl : l > 0) (H1 : n < k) (H2 : m ≤ l) : n * m < k * l := le_lt_trans (mul_le_left H2 n) (mul_lt_right Hl H1) theorem mul_lt {n m k l : ℕ} (H1 : n < k) (H2 : m < l) : n * m < k * l := have H3 : n * m ≤ k * m, from mul_le_right (lt_imp_le H1) m, have H4 : k * m < k * l, from mul_lt_left (le_lt_trans (zero_le n) H1) H2, le_lt_trans H3 H4 theorem mul_lt_left_inv {n m k : ℕ} (H : k * n < k * m) : n < m := have general : ∀ m, k * n < k * m → n < m, from induction_on n (take m : ℕ, assume H2 : k * 0 < k * m, have H3 : 0 < k * m, from mul_zero_right k ▸ H2, show 0 < m, from mul_positive_inv_right H3) (take l : ℕ, assume IH : ∀ m, k * l < k * m → l < m, take m : ℕ, assume H2 : k * succ l < k * m, have H3 : 0 < k * m, from le_lt_trans (zero_le _) H2, have H4 : 0 < m, from mul_positive_inv_right H3, obtain (l2 : ℕ) (Hl2 : m = succ l2), from pos_imp_eq_succ H4, have H5 : k * l + k < k * m, from mul_succ_right k l ▸ H2, have H6 : k * l + k < k * succ l2, from Hl2 ▸ H5, have H7 : k * l + k < k * l2 + k, from mul_succ_right k l2 ▸ H6, have H8 : k * l < k * l2, from add_lt_right_inv H7, have H9 : l < l2, from IH l2 H8, have H10 : succ l < succ l2, from succ_lt H9, show succ l < m, from Hl2⁻¹ ▸ H10), general m H theorem mul_lt_right_inv {n m k : ℕ} (H : n * k < m * k) : n < m := mul_lt_left_inv (mul_comm m k ▸ mul_comm n k ▸ H) theorem mul_le_left_inv {n m k : ℕ} (H : succ k * n ≤ succ k * m) : n ≤ m := have H2 : succ k * n < succ k * m + succ k, from le_lt_trans H (lt_intro2 _ _), have H3 : succ k * n < succ k * succ m, from subst (symm (mul_succ_right (succ k) m)) H2, have H4 : n < succ m, from mul_lt_left_inv H3, show n ≤ m, from lt_succ_imp_le H4 theorem mul_le_right_inv {n m k : ℕ} (H : n * succ m ≤ k * succ m) : n ≤ k := mul_le_left_inv (subst (mul_comm k (succ m)) (subst (mul_comm n (succ m)) H)) theorem mul_eq_one_left {n m : ℕ} (H : n * m = 1) : n = 1 := have H2 : n * m > 0, from subst (symm H) (lt_zero 0), have H3 : n > 0, from mul_positive_inv_left H2, have H4 : m > 0, from mul_positive_inv_right H2, or_elim (le_or_lt n 1) (assume H5 : n ≤ 1, show n = 1, from le_antisym H5 H3) (assume H5 : n > 1, have H6 : n * m ≥ 2 * 1, from mul_le H5 H4, have H7 : 1 ≥ 2, from subst (mul_one_right 2) (subst H H6), absurd_elim _ (self_lt_succ 1) (le_imp_not_gt H7)) theorem mul_eq_one_right {n m : ℕ} (H : n * m = 1) : m = 1 := mul_eq_one_left (subst (mul_comm n m) H) theorem mul_eq_one {n m : ℕ} (H : n * m = 1) : n = 1 ∧ m = 1 := and_intro (mul_eq_one_left H) (mul_eq_one_right H) -------------------------------------------------- sub definition sub (n m : ℕ) : ℕ := nat_rec n (fun m x, pred x) m infixl `-`:65 := sub theorem sub_zero_right (n : ℕ) : n - 0 = n theorem sub_succ_right (n m : ℕ) : n - succ m = pred (n - m) theorem sub_zero_left (n : ℕ) : 0 - n = 0 := induction_on n (sub_zero_right 0) (take k : ℕ, assume IH : 0 - k = 0, calc 0 - succ k = pred (0 - k) : sub_succ_right 0 k ... = pred 0 : {IH} ... = 0 : pred_zero) theorem sub_succ_succ (n m : ℕ) : succ n - succ m = n - m := induction_on m (calc succ n - 1 = pred (succ n - 0) : sub_succ_right (succ n) 0 ... = pred (succ n) : {sub_zero_right (succ n)} ... = n : pred_succ n ... = n - 0 : symm (sub_zero_right n)) (take k : ℕ, assume IH : succ n - succ k = n - k, calc succ n - succ (succ k) = pred (succ n - succ k) : sub_succ_right (succ n) (succ k) ... = pred (n - k) : {IH} ... = n - succ k : symm (sub_succ_right n k)) theorem sub_one (n : ℕ) : n - 1 = pred n := calc n - 1 = pred (n - 0) : sub_succ_right n 0 ... = pred n : {sub_zero_right n} theorem sub_self (n : ℕ) : n - n = 0 := induction_on n (sub_zero_right 0) (take k IH, trans (sub_succ_succ k k) IH) theorem sub_add_add_right (n m k : ℕ) : (n + k) - (m + k) = n - m := induction_on k (calc (n + 0) - (m + 0) = n - (m + 0) : {add_zero_right _} ... = n - m : {add_zero_right _}) (take l : ℕ, assume IH : (n + l) - (m + l) = n - m, calc (n + succ l) - (m + succ l) = succ (n + l) - (m + succ l) : {add_succ_right _ _} ... = succ (n + l) - succ (m + l) : {add_succ_right _ _} ... = (n + l) - (m + l) : sub_succ_succ _ _ ... = n - m : IH) theorem sub_add_add_left (n m k : ℕ) : (k + n) - (k + m) = n - m := subst (add_comm m k) (subst (add_comm n k) (sub_add_add_right n m k)) theorem sub_add_left (n m : ℕ) : n + m - m = n := induction_on m (subst (symm (add_zero_right n)) (sub_zero_right n)) (take k : ℕ, assume IH : n + k - k = n, calc n + succ k - succ k = succ (n + k) - succ k : {add_succ_right n k} ... = n + k - k : sub_succ_succ _ _ ... = n : IH) theorem sub_sub (n m k : ℕ) : n - m - k = n - (m + k) := induction_on k (calc n - m - 0 = n - m : sub_zero_right _ ... = n - (m + 0) : {symm (add_zero_right m)}) (take l : ℕ, assume IH : n - m - l = n - (m + l), calc n - m - succ l = pred (n - m - l) : sub_succ_right (n - m) l ... = pred (n - (m + l)) : {IH} ... = n - succ (m + l) : symm (sub_succ_right n (m + l)) ... = n - (m + succ l) : {symm (add_succ_right m l)}) theorem succ_sub_sub (n m k : ℕ) : succ n - m - succ k = n - m - k := calc succ n - m - succ k = succ n - (m + succ k) : sub_sub _ _ _ ... = succ n - succ (m + k) : {add_succ_right m k} ... = n - (m + k) : sub_succ_succ _ _ ... = n - m - k : symm (sub_sub n m k) theorem sub_add_right_eq_zero (n m : ℕ) : n - (n + m) = 0 := calc n - (n + m) = n - n - m : symm (sub_sub n n m) ... = 0 - m : {sub_self n} ... = 0 : sub_zero_left m theorem sub_comm (m n k : ℕ) : m - n - k = m - k - n := calc m - n - k = m - (n + k) : sub_sub m n k ... = m - (k + n) : {add_comm n k} ... = m - k - n : symm (sub_sub m k n) theorem succ_sub_one (n : ℕ) : succ n - 1 = n := sub_succ_succ n 0 ⬝ sub_zero_right n ---------- mul theorem mul_pred_left (n m : ℕ) : pred n * m = n * m - m := induction_on n (calc pred 0 * m = 0 * m : {pred_zero} ... = 0 : mul_zero_left _ ... = 0 - m : symm (sub_zero_left m) ... = 0 * m - m : {symm (mul_zero_left m)}) (take k : ℕ, assume IH : pred k * m = k * m - m, calc pred (succ k) * m = k * m : {pred_succ k} ... = k * m + m - m : symm (sub_add_left _ _) ... = succ k * m - m : {symm (mul_succ_left k m)}) theorem mul_pred_right (n m : ℕ) : n * pred m = n * m - n := calc n * pred m = pred m * n : mul_comm _ _ ... = m * n - n : mul_pred_left m n ... = n * m - n : {mul_comm m n} theorem mul_sub_distr_left (n m k : ℕ) : (n - m) * k = n * k - m * k := induction_on m (calc (n - 0) * k = n * k : {sub_zero_right n} ... = n * k - 0 : symm (sub_zero_right _) ... = n * k - 0 * k : {symm (mul_zero_left _)}) (take l : ℕ, assume IH : (n - l) * k = n * k - l * k, calc (n - succ l) * k = pred (n - l) * k : {sub_succ_right n l} ... = (n - l) * k - k : mul_pred_left _ _ ... = n * k - l * k - k : {IH} ... = n * k - (l * k + k) : sub_sub _ _ _ ... = n * k - (succ l * k) : {symm (mul_succ_left l k)}) theorem mul_sub_distr_right (n m k : ℕ) : n * (m - k) = n * m - n * k := calc n * (m - k) = (m - k) * n : mul_comm _ _ ... = m * n - k * n : mul_sub_distr_left _ _ _ ... = n * m - k * n : {mul_comm _ _} ... = n * m - n * k : {mul_comm _ _} -------------------------------------------------- max, min, iteration, maybe: sub, div theorem succ_sub {m n : ℕ} : m ≥ n → succ m - n = succ (m - n) := sub_induction n m (take k, assume H : 0 ≤ k, calc succ k - 0 = succ k : sub_zero_right (succ k) ... = succ (k - 0) : {symm (sub_zero_right k)}) (take k, assume H : succ k ≤ 0, absurd_elim _ H (not_succ_zero_le k)) (take k l, assume IH : k ≤ l → succ l - k = succ (l - k), take H : succ k ≤ succ l, calc succ (succ l) - succ k = succ l - k : sub_succ_succ (succ l) k ... = succ (l - k) : IH (succ_le_cancel H) ... = succ (succ l - succ k) : {symm (sub_succ_succ l k)}) theorem le_imp_sub_eq_zero {n m : ℕ} (H : n ≤ m) : n - m = 0 := obtain (k : ℕ) (Hk : n + k = m), from le_elim H, subst Hk (sub_add_right_eq_zero n k) theorem add_sub_le {n m : ℕ} : n ≤ m → n + (m - n) = m := sub_induction n m (take k, assume H : 0 ≤ k, calc 0 + (k - 0) = k - 0 : add_zero_left (k - 0) ... = k : sub_zero_right k) (take k, assume H : succ k ≤ 0, absurd_elim _ H (not_succ_zero_le k)) (take k l, assume IH : k ≤ l → k + (l - k) = l, take H : succ k ≤ succ l, calc succ k + (succ l - succ k) = succ k + (l - k) : {sub_succ_succ l k} ... = succ (k + (l - k)) : add_succ_left k (l - k) ... = succ l : {IH (succ_le_cancel H)}) theorem add_sub_ge_left {n m : ℕ} : n ≥ m → n - m + m = n := subst (add_comm m (n - m)) add_sub_le theorem add_sub_ge {n m : ℕ} (H : n ≥ m) : n + (m - n) = n := calc n + (m - n) = n + 0 : {le_imp_sub_eq_zero H} ... = n : add_zero_right n theorem add_sub_le_left {n m : ℕ} : n ≤ m → n - m + m = m := subst (add_comm m (n - m)) add_sub_ge theorem le_add_sub_left (n m : ℕ) : n ≤ n + (m - n) := or_elim (le_total n m) (assume H : n ≤ m, subst (symm (add_sub_le H)) H) (assume H : m ≤ n, subst (symm (add_sub_ge H)) (le_refl n)) theorem le_add_sub_right (n m : ℕ) : m ≤ n + (m - n) := or_elim (le_total n m) (assume H : n ≤ m, subst (symm (add_sub_le H)) (le_refl m)) (assume H : m ≤ n, subst (symm (add_sub_ge H)) H) theorem sub_split {P : ℕ → Prop} {n m : ℕ} (H1 : n ≤ m → P 0) (H2 : ∀k, m + k = n -> P k) : P (n - m) := or_elim (le_total n m) (assume H3 : n ≤ m, subst (symm (le_imp_sub_eq_zero H3)) (H1 H3)) (assume H3 : m ≤ n, H2 (n - m) (add_sub_le H3)) theorem sub_le_self (n m : ℕ) : n - m ≤ n := sub_split (assume H : n ≤ m, zero_le n) (take k : ℕ, assume H : m + k = n, le_intro (subst (add_comm m k) H)) theorem le_elim_sub (n m : ℕ) (H : n ≤ m) : ∃k, m - k = n := obtain (k : ℕ) (Hk : n + k = m), from le_elim H, exists_intro k (calc m - k = n + k - k : {symm Hk} ... = n : sub_add_left n k) theorem add_sub_assoc {m k : ℕ} (H : k ≤ m) (n : ℕ) : n + m - k = n + (m - k) := have l1 : k ≤ m → n + m - k = n + (m - k), from sub_induction k m (take m : ℕ, assume H : 0 ≤ m, calc n + m - 0 = n + m : sub_zero_right (n + m) ... = n + (m - 0) : {symm (sub_zero_right m)}) (take k : ℕ, assume H : succ k ≤ 0, absurd_elim _ H (not_succ_zero_le k)) (take k m, assume IH : k ≤ m → n + m - k = n + (m - k), take H : succ k ≤ succ m, calc n + succ m - succ k = succ (n + m) - succ k : {add_succ_right n m} ... = n + m - k : sub_succ_succ (n + m) k ... = n + (m - k) : IH (succ_le_cancel H) ... = n + (succ m - succ k) : {symm (sub_succ_succ m k)}), l1 H theorem sub_eq_zero_imp_le {n m : ℕ} : n - m = 0 → n ≤ m := sub_split (assume H1 : n ≤ m, assume H2 : 0 = 0, H1) (take k : ℕ, assume H1 : m + k = n, assume H2 : k = 0, have H3 : n = m, from subst (add_zero_right m) (subst H2 (symm H1)), subst H3 (le_refl n)) theorem sub_sub_split {P : ℕ → ℕ → Prop} {n m : ℕ} (H1 : ∀k, n = m + k -> P k 0) (H2 : ∀k, m = n + k → P 0 k) : P (n - m) (m - n) := or_elim (le_total n m) (assume H3 : n ≤ m, le_imp_sub_eq_zero H3⁻¹ ▸ (H2 (m - n) (add_sub_le H3⁻¹))) (assume H3 : m ≤ n, le_imp_sub_eq_zero H3⁻¹ ▸ (H1 (n - m) (add_sub_le H3⁻¹))) theorem sub_intro {n m k : ℕ} (H : n + m = k) : k - n = m := have H2 : k - n + n = m + n, from calc k - n + n = k : add_sub_ge_left (le_intro H) ... = n + m : symm H ... = m + n : add_comm n m, add_cancel_right H2 theorem sub_lt {x y : ℕ} (xpos : x > 0) (ypos : y > 0) : x - y < x := obtain (x' : ℕ) (xeq : x = succ x'), from pos_imp_eq_succ xpos, obtain (y' : ℕ) (yeq : y = succ y'), from pos_imp_eq_succ ypos, have xsuby_eq : x - y = x' - y', from calc x - y = succ x' - y : {xeq} ... = succ x' - succ y' : {yeq} ... = x' - y' : sub_succ_succ _ _, have H1 : x' - y' ≤ x', from sub_le_self _ _, have H2 : x' < succ x', from self_lt_succ _, show x - y < x, from xeq⁻¹ ▸ xsuby_eq⁻¹ ▸ le_lt_trans H1 H2 -- Max, min, iteration, and absolute difference -- -------------------------------------------- definition max (n m : ℕ) : ℕ := n + (m - n) definition min (n m : ℕ) : ℕ := m - (m - n) theorem max_le {n m : ℕ} (H : n ≤ m) : n + (m - n) = m := add_sub_le H theorem max_ge {n m : ℕ} (H : n ≥ m) : n + (m - n) = n := add_sub_ge H theorem left_le_max (n m : ℕ) : n ≤ n + (m - n) := le_add_sub_left n m theorem right_le_max (n m : ℕ) : m ≤ max n m := le_add_sub_right n m -- ### absolute difference -- This section is still incomplete definition dist (n m : ℕ) := (n - m) + (m - n) theorem dist_comm (n m : ℕ) : dist n m = dist m n := add_comm (n - m) (m - n) theorem dist_eq_zero {n m : ℕ} (H : dist n m = 0) : n = m := have H2 : n - m = 0, from add_eq_zero_left H, have H3 : n ≤ m, from sub_eq_zero_imp_le H2, have H4 : m - n = 0, from add_eq_zero_right H, have H5 : m ≤ n, from sub_eq_zero_imp_le H4, le_antisym H3 H5 theorem dist_le {n m : ℕ} (H : n ≤ m) : dist n m = m - n := calc dist n m = (n - m) + (m - n) : refl _ ... = 0 + (m - n) : {le_imp_sub_eq_zero H} ... = m - n : add_zero_left (m - n) theorem dist_ge {n m : ℕ} (H : n ≥ m) : dist n m = n - m := subst (dist_comm m n) (dist_le H) theorem dist_zero_right (n : ℕ) : dist n 0 = n := trans (dist_ge (zero_le n)) (sub_zero_right n) theorem dist_zero_left (n : ℕ) : dist 0 n = n := trans (dist_le (zero_le n)) (sub_zero_right n) theorem dist_intro {n m k : ℕ} (H : n + m = k) : dist k n = m := calc dist k n = k - n : dist_ge (le_intro H) ... = m : sub_intro H theorem dist_add_right (n k m : ℕ) : dist (n + k) (m + k) = dist n m := calc dist (n + k) (m + k) = ((n+k) - (m+k)) + ((m+k)-(n+k)) : refl _ ... = (n - m) + ((m + k) - (n + k)) : {sub_add_add_right _ _ _} ... = (n - m) + (m - n) : {sub_add_add_right _ _ _} theorem dist_add_left (k n m : ℕ) : dist (k + n) (k + m) = dist n m := subst (add_comm m k) (subst (add_comm n k) (dist_add_right n k m)) theorem dist_ge_add_right {n m : ℕ} (H : n ≥ m) : dist n m + m = n := calc dist n m + m = n - m + m : {dist_ge H} ... = n : add_sub_ge_left H theorem dist_eq_intro {n m k l : ℕ} (H : n + m = k + l) : dist n k = dist l m := calc dist n k = dist (n + m) (k + m) : symm (dist_add_right n m k) ... = dist (k + l) (k + m) : {H} ... = dist l m : dist_add_left k l m end nat