-- 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 -- int.order -- ========= -- The order relation on the integers, and the sign function. import .basic using nat (hiding case) using decidable using fake_simplifier using int namespace int -- ## le definition le (a b : ℤ) : Prop := ∃n : ℕ, a + n = b infix `<=` := int.le infix `≤` := int.le theorem le_intro {a b : ℤ} {n : ℕ} (H : a + n = b) : a ≤ b := exists_intro n H theorem le_elim {a b : ℤ} (H : a ≤ b) : ∃n : ℕ, a + n = b := H -- ### partial order theorem le_refl (a : ℤ) : a ≤ a := le_intro (add_zero_right a) theorem le_of_nat (n m : ℕ) : (of_nat n ≤ of_nat m) ↔ (n ≤ m) := iff_intro (assume H : of_nat n ≤ of_nat m, obtain (k : ℕ) (Hk : of_nat n + of_nat k = of_nat m), from le_elim H, have H2 : n + k = m, from of_nat_inj (trans (symm (add_of_nat n k)) Hk), nat.le_intro H2) (assume H : n ≤ m, obtain (k : ℕ) (Hk : n + k = m), from nat.le_elim H, have H2 : of_nat n + of_nat k = of_nat m, from subst Hk (add_of_nat n k), le_intro H2) theorem le_trans {a b c : ℤ} (H1 : a ≤ b) (H2 : b ≤ c) : a ≤ c := obtain (n : ℕ) (Hn : a + n = b), from le_elim H1, obtain (m : ℕ) (Hm : b + m = c), from le_elim H2, have H3 : a + of_nat (n + m) = c, from calc a + of_nat (n + m) = a + (of_nat n + m) : {symm (add_of_nat n m)} ... = a + n + m : symm (add_assoc a n m) ... = b + m : {Hn} ... = c : Hm, le_intro H3 theorem le_antisym {a b : ℤ} (H1 : a ≤ b) (H2 : b ≤ a) : a = b := obtain (n : ℕ) (Hn : a + n = b), from le_elim H1, obtain (m : ℕ) (Hm : b + m = a), from le_elim H2, have H3 : a + of_nat (n + m) = a + 0, from calc a + of_nat (n + m) = a + (of_nat n + m) : {symm (add_of_nat n m)} ... = a + n + m : symm (add_assoc a n m) ... = b + m : {Hn} ... = a : Hm ... = a + 0 : symm (add_zero_right a), have H4 : of_nat (n + m) = of_nat 0, from add_cancel_left H3, have H5 : n + m = 0, from of_nat_inj H4, have H6 : n = 0, from nat.add_eq_zero_left H5, show a = b, from calc a = a + of_nat 0 : symm (add_zero_right a) ... = a + n : {symm H6} ... = b : Hn -- ### interaction with add theorem le_add_of_nat_right (a : ℤ) (n : ℕ) : a ≤ a + n := le_intro (refl (a + n)) theorem le_add_of_nat_left (a : ℤ) (n : ℕ) : a ≤ n + a := le_intro (add_comm a n) theorem add_le_left {a b : ℤ} (H : a ≤ b) (c : ℤ) : c + a ≤ c + b := obtain (n : ℕ) (Hn : a + n = b), from le_elim H, have H2 : c + a + n = c + b, from calc c + a + n = c + (a + n) : add_assoc c a n ... = c + b : {Hn}, le_intro H2 theorem add_le_right {a b : ℤ} (H : a ≤ b) (c : ℤ) : a + c ≤ b + c := subst (add_comm c b) (subst (add_comm c a) (add_le_left H c)) theorem add_le {a b c d : ℤ} (H1 : a ≤ b) (H2 : c ≤ d) : a + c ≤ b + d := le_trans (add_le_right H1 c) (add_le_left H2 b) theorem add_le_cancel_right {a b c : ℤ} (H : a + c ≤ b + c) : a ≤ b := have H1 : a + c + -c ≤ b + c + -c, from add_le_right H (-c), have H2 : a + c - c ≤ b + c - c, from subst (add_neg_right _ _) (subst (add_neg_right _ _) H1), subst (add_sub_inverse b c) (subst (add_sub_inverse a c) H2) theorem add_le_cancel_left {a b c : ℤ} (H : c + a ≤ c + b) : a ≤ b := add_le_cancel_right (subst (add_comm c b) (subst (add_comm c a) H)) theorem add_le_inv {a b c d : ℤ} (H1 : a + b ≤ c + d) (H2 : c ≤ a) : b ≤ d := obtain (n : ℕ) (Hn : c + n = a), from le_elim H2, have H3 : c + (n + b) ≤ c + d, from subst (add_assoc c n b) (subst (symm Hn) H1), have H4 : n + b ≤ d, from add_le_cancel_left H3, show b ≤ d, from le_trans (le_add_of_nat_left b n) H4 theorem le_add_of_nat_right_trans {a b : ℤ} (H : a ≤ b) (n : ℕ) : a ≤ b + n := le_trans H (le_add_of_nat_right b n) theorem le_imp_succ_le_or_eq {a b : ℤ} (H : a ≤ b) : a + 1 ≤ b ∨ a = b := obtain (n : ℕ) (Hn : a + n = b), from le_elim H, discriminate (assume H2 : n = 0, have H3 : a = b, from calc a = a + 0 : symm (add_zero_right a) ... = a + n : {symm H2} ... = b : Hn, or_intro_right _ H3) (take k : ℕ, assume H2 : n = succ k, have H3 : a + 1 + k = b, from calc a + 1 + k = a + succ k : by simp ... = a + n : by simp ... = b : Hn, or_intro_left _ (le_intro H3)) -- ### interaction with neg and sub theorem le_neg {a b : ℤ} (H : a ≤ b) : -b ≤ -a := obtain (n : ℕ) (Hn : a + n = b), from le_elim H, have H2 : b - n = a, from add_imp_sub_right Hn, have H3 : -b + n = -a, from calc -b + n = -b + -(-n) : {symm (neg_neg n)} ... = -(b + -n) : symm (neg_add_distr b (-n)) ... = -(b - n) : {add_neg_right _ _} ... = -a : {H2}, le_intro H3 theorem neg_le_zero {a : ℤ} (H : 0 ≤ a) : -a ≤ 0 := subst neg_zero (le_neg H) theorem zero_le_neg {a : ℤ} (H : a ≤ 0) : 0 ≤ -a := subst neg_zero (le_neg H) theorem le_neg_inv {a b : ℤ} (H : -a ≤ -b) : b ≤ a := subst (neg_neg b) (subst (neg_neg a) (le_neg H)) theorem le_sub_of_nat (a : ℤ) (n : ℕ) : a - n ≤ a := le_intro (sub_add_inverse a n) theorem sub_le_right {a b : ℤ} (H : a ≤ b) (c : ℤ) : a - c ≤ b - c := subst (add_neg_right _ _) (subst (add_neg_right _ _) (add_le_right H (-c))) theorem sub_le_left {a b : ℤ} (H : a ≤ b) (c : ℤ) : c - b ≤ c - a := subst (add_neg_right _ _) (subst (add_neg_right _ _) (add_le_left (le_neg H) c)) theorem sub_le {a b c d : ℤ} (H1 : a ≤ b) (H2 : d ≤ c) : a - c ≤ b - d := subst (add_neg_right _ _) (subst (add_neg_right _ _) (add_le H1 (le_neg H2))) theorem sub_le_right_inv {a b c : ℤ} (H : a - c ≤ b - c) : a ≤ b := add_le_cancel_right (subst (symm (add_neg_right _ _)) (subst (symm (add_neg_right _ _)) H)) theorem sub_le_left_inv {a b c : ℤ} (H : c - a ≤ c - b) : b ≤ a := le_neg_inv (add_le_cancel_left (subst (symm (add_neg_right _ _)) (subst (symm (add_neg_right _ _)) H))) theorem le_iff_sub_nonneg (a b : ℤ) : a ≤ b ↔ 0 ≤ b - a := iff_intro (assume H, subst (sub_self _) (sub_le_right H a)) (assume H, subst (sub_add_inverse _ _) (subst (add_zero_left _) (add_le_right H a))) -- Less than, Greater than, Greater than or equal -- ---------------------------------------------- definition lt (a b : ℤ) := a + 1 ≤ b infix `<` := int.lt definition ge (a b : ℤ) := b ≤ a infix `>=` := int.ge infix `≥` := int.ge definition gt (a b : ℤ) := b < a infix `>` := int.gt theorem lt_def (a b : ℤ) : a < b ↔ a + 1 ≤ b := iff_refl (a < b) theorem gt_def (n m : ℕ) : n > m ↔ m < n := iff_refl (n > m) theorem ge_def (n m : ℕ) : n ≥ m ↔ m ≤ n := iff_refl (n ≥ m) -- add_rewrite gt_def ge_def --it might be possible to remove this in Lean 0.2 theorem lt_add_succ (a : ℤ) (n : ℕ) : a < a + succ n := le_intro (show a + 1 + n = a + succ n, by simp) theorem lt_intro {a b : ℤ} {n : ℕ} (H : a + succ n = b) : a < b := subst H (lt_add_succ a n) theorem lt_elim {a b : ℤ} (H : a < b) : ∃n : ℕ, a + succ n = b := obtain (n : ℕ) (Hn : a + 1 + n = b), from le_elim H, have H2 : a + succ n = b, from calc a + succ n = a + 1 + n : by simp ... = b : Hn, exists_intro n H2 -- -- ### basic facts theorem lt_irrefl (a : ℤ) : ¬ a < a := not_intro (assume H : a < a, obtain (n : ℕ) (Hn : a + succ n = a), from lt_elim H, have H2 : a + succ n = a + 0, from calc a + succ n = a : Hn ... = a + 0 : by simp, have H3 : succ n = 0, from add_cancel_left H2, have H4 : succ n = 0, from of_nat_inj H3, absurd H4 (succ_ne_zero n)) theorem lt_imp_ne {a b : ℤ} (H : a < b) : a ≠ b := not_intro (assume H2 : a = b, absurd (subst H2 H) (lt_irrefl b)) theorem lt_of_nat (n m : ℕ) : (of_nat n < of_nat m) ↔ (n < m) := calc (of_nat n + 1 ≤ of_nat m) ↔ (of_nat (succ n) ≤ of_nat m) : by simp ... ↔ (succ n ≤ m) : le_of_nat (succ n) m ... ↔ (n < m) : iff_symm (eq_to_iff (nat.lt_def n m)) theorem gt_of_nat (n m : ℕ) : (of_nat n > of_nat m) ↔ (n > m) := lt_of_nat m n -- ### interaction with le theorem lt_imp_le_succ {a b : ℤ} (H : a < b) : a + 1 ≤ b := H theorem le_succ_imp_lt {a b : ℤ} (H : a + 1 ≤ b) : a < b := H theorem self_lt_succ (a : ℤ) : a < a + 1 := le_refl (a + 1) theorem lt_imp_le {a b : ℤ} (H : a < b) : a ≤ b := obtain (n : ℕ) (Hn : a + succ n = b), from lt_elim H, le_intro Hn theorem le_imp_lt_or_eq {a b : ℤ} (H : a ≤ b) : a < b ∨ a = b := le_imp_succ_le_or_eq H theorem le_ne_imp_lt {a b : ℤ} (H1 : a ≤ b) (H2 : a ≠ b) : a < b := resolve_left (le_imp_lt_or_eq H1) H2 theorem le_imp_lt_succ {a b : ℤ} (H : a ≤ b) : a < b + 1 := add_le_right H 1 theorem lt_succ_imp_le {a b : ℤ} (H : a < b + 1) : a ≤ b := add_le_cancel_right H -- ### transitivity, antisymmmetry theorem lt_le_trans {a b c : ℤ} (H1 : a < b) (H2 : b ≤ c) : a < c := le_trans H1 H2 theorem le_lt_trans {a b c : ℤ} (H1 : a ≤ b) (H2 : b < c) : a < c := le_trans (add_le_right H1 1) H2 theorem lt_trans {a b c : ℤ} (H1 : a < b) (H2 : b < c) : a < c := lt_le_trans H1 (lt_imp_le H2) theorem le_imp_not_gt {a b : ℤ} (H : a ≤ b) : ¬ a > b := not_intro (assume H2 : a > b, absurd (le_lt_trans H H2) (lt_irrefl a)) theorem lt_imp_not_ge {a b : ℤ} (H : a < b) : ¬ a ≥ b := not_intro (assume H2 : a ≥ b, absurd (lt_le_trans H H2) (lt_irrefl a)) theorem lt_antisym {a b : ℤ} (H : a < b) : ¬ b < a := le_imp_not_gt (lt_imp_le H) -- ### interaction with addition theorem add_lt_left {a b : ℤ} (H : a < b) (c : ℤ) : c + a < c + b := subst (symm (add_assoc c a 1)) (add_le_left H c) theorem add_lt_right {a b : ℤ} (H : a < b) (c : ℤ) : a + c < b + c := subst (add_comm c b) (subst (add_comm c a) (add_lt_left H c)) theorem add_le_lt {a b c d : ℤ} (H1 : a ≤ c) (H2 : b < d) : a + b < c + d := le_lt_trans (add_le_right H1 b) (add_lt_left H2 c) theorem add_lt_le {a b c d : ℤ} (H1 : a < c) (H2 : b ≤ d) : a + b < c + d := lt_le_trans (add_lt_right H1 b) (add_le_left H2 c) theorem add_lt {a b c d : ℤ} (H1 : a < c) (H2 : b < d) : a + b < c + d := add_lt_le H1 (lt_imp_le H2) theorem add_lt_cancel_left {a b c : ℤ} (H : c + a < c + b) : a < b := add_le_cancel_left (subst (add_assoc c a 1) (show c + a + 1 ≤ c + b, from H)) theorem add_lt_cancel_right {a b c : ℤ} (H : a + c < b + c) : a < b := add_lt_cancel_left (subst (add_comm b c) (subst (add_comm a c) H)) -- ### interaction with neg and sub theorem lt_neg {a b : ℤ} (H : a < b) : -b < -a := have H2 : -(a + 1) + 1 = -a, by simp, have H3 : -b ≤ -(a + 1), from le_neg H, have H4 : -b + 1 ≤ -(a + 1) + 1, from add_le_right H3 1, subst H2 H4 theorem neg_lt_zero {a : ℤ} (H : 0 < a) : -a < 0 := subst neg_zero (lt_neg H) theorem zero_lt_neg {a : ℤ} (H : a < 0) : 0 < -a := subst neg_zero (lt_neg H) theorem lt_neg_inv {a b : ℤ} (H : -a < -b) : b < a := subst (neg_neg b) (subst (neg_neg a) (lt_neg H)) theorem lt_sub_of_nat_succ (a : ℤ) (n : ℕ) : a - succ n < a := lt_intro (sub_add_inverse a (succ n)) theorem sub_lt_right {a b : ℤ} (H : a < b) (c : ℤ) : a - c < b - c := subst (add_neg_right _ _) (subst (add_neg_right _ _) (add_lt_right H (-c))) theorem sub_lt_left {a b : ℤ} (H : a < b) (c : ℤ) : c - b < c - a := subst (add_neg_right _ _) (subst (add_neg_right _ _) (add_lt_left (lt_neg H) c)) theorem sub_lt {a b c d : ℤ} (H1 : a < b) (H2 : d < c) : a - c < b - d := subst (add_neg_right _ _) (subst (add_neg_right _ _) (add_lt H1 (lt_neg H2))) theorem sub_lt_right_inv {a b c : ℤ} (H : a - c < b - c) : a < b := add_lt_cancel_right (subst (symm (add_neg_right _ _)) (subst (symm (add_neg_right _ _)) H)) theorem sub_lt_left_inv {a b c : ℤ} (H : c - a < c - b) : b < a := lt_neg_inv (add_lt_cancel_left (subst (symm (add_neg_right _ _)) (subst (symm (add_neg_right _ _)) H))) -- ### totality of lt and le -- add_rewrite succ_pos zero_le --move some of these to nat.lean -- add_rewrite le_of_nat lt_of_nat gt_of_nat --remove gt_of_nat in Lean 0.2 -- add_rewrite le_neg lt_neg neg_le_zero zero_le_neg zero_lt_neg neg_lt_zero theorem neg_le_pos (n m : ℕ) : -n ≤ m := have H1 : of_nat 0 ≤ of_nat m, by simp, have H2 : -n ≤ 0, by simp, le_trans H2 H1 theorem le_or_gt (a b : ℤ) : a ≤ b ∨ a > b := int_by_cases a (take n : ℕ, int_by_cases_succ b (take m : ℕ, show of_nat n ≤ m ∨ of_nat n > m, by simp) -- from (by simp) ◂ (le_or_gt n m)) (take m : ℕ, show n ≤ -succ m ∨ n > -succ m, from have H0 : -succ m < -m, from lt_neg (subst (symm (of_nat_succ m)) (self_lt_succ m)), have H : -succ m < n, from lt_le_trans H0 (neg_le_pos m n), or_intro_right _ H)) (take n : ℕ, int_by_cases_succ b (take m : ℕ, show -n ≤ m ∨ -n > m, from or_intro_left _ (neg_le_pos n m)) (take m : ℕ, show -n ≤ -succ m ∨ -n > -succ m, from or_imp_or (le_or_gt (succ m) n) (assume H : succ m ≤ n, le_neg (iff_elim_left (iff_symm (le_of_nat (succ m) n)) H)) (assume H : succ m > n, lt_neg (iff_elim_left (iff_symm (lt_of_nat n (succ m))) H)))) theorem trichotomy_alt (a b : ℤ) : (a < b ∨ a = b) ∨ a > b := or_imp_or_left (le_or_gt a b) (assume H : a ≤ b, le_imp_lt_or_eq H) theorem trichotomy (a b : ℤ) : a < b ∨ a = b ∨ a > b := iff_elim_left (or_assoc _ _ _) (trichotomy_alt a b) theorem le_total (a b : ℤ) : a ≤ b ∨ b ≤ a := or_imp_or_right (le_or_gt a b) (assume H : b < a, lt_imp_le H) theorem not_lt_imp_le {a b : ℤ} (H : ¬ a < b) : b ≤ a := resolve_left (le_or_gt b a) H theorem not_le_imp_lt {a b : ℤ} (H : ¬ a ≤ b) : b < a := resolve_right (le_or_gt a b) H -- (non)positivity and (non)negativity -- ------------------------------------- -- ### basic -- see also "int_by_cases" and similar theorems theorem pos_imp_exists_nat {a : ℤ} (H : a ≥ 0) : ∃n : ℕ, a = n := obtain (n : ℕ) (Hn : of_nat 0 + n = a), from le_elim H, exists_intro n (trans (symm Hn) (add_zero_left n)) theorem neg_imp_exists_nat {a : ℤ} (H : a ≤ 0) : ∃n : ℕ, a = -n := have H2 : -a ≥ 0, from zero_le_neg H, obtain (n : ℕ) (Hn : -a = n), from pos_imp_exists_nat H2, have H3 : a = -n, from symm (neg_move Hn), exists_intro n H3 theorem to_nat_nonneg_eq {a : ℤ} (H : a ≥ 0) : (to_nat a) = a := obtain (n : ℕ) (Hn : a = n), from pos_imp_exists_nat H, subst (symm Hn) (congr_arg of_nat (to_nat_of_nat n)) theorem of_nat_nonneg (n : ℕ) : of_nat n ≥ 0 := iff_mp (iff_symm (le_of_nat _ _)) (zero_le _) theorem le_decidable [instance] {a b : ℤ} : decidable (a ≤ b) := have aux : ∀x, decidable (0 ≤ x), from take x, have H : 0 ≤ x ↔ of_nat (to_nat x) = x, from iff_intro (assume H1, to_nat_nonneg_eq H1) (assume H1, subst H1 (of_nat_nonneg (to_nat x))), decidable_iff_equiv _ (iff_symm H), decidable_iff_equiv (aux _) (iff_symm (le_iff_sub_nonneg a b)) theorem ge_decidable [instance] {a b : ℤ} : decidable (a ≥ b) theorem lt_decidable [instance] {a b : ℤ} : decidable (a < b) theorem gt_decidable [instance] {a b : ℤ} : decidable (a > b) --to_nat_neg is already taken... rename? theorem to_nat_negative {a : ℤ} (H : a ≤ 0) : (to_nat a) = -a := obtain (n : ℕ) (Hn : a = -n), from neg_imp_exists_nat H, calc (to_nat a) = (to_nat ( -n)) : {Hn} ... = (to_nat n) : {to_nat_neg n} ... = n : {to_nat_of_nat n} ... = -a : symm (neg_move (symm Hn)) theorem to_nat_cases (a : ℤ) : a = (to_nat a) ∨ a = - (to_nat a) := or_imp_or (le_total 0 a) (assume H : a ≥ 0, symm (to_nat_nonneg_eq H)) (assume H : a ≤ 0, symm (neg_move (symm (to_nat_negative H)))) -- ### interaction of mul with le and lt theorem mul_le_left_nonneg {a b c : ℤ} (Ha : a ≥ 0) (H : b ≤ c) : a * b ≤ a * c := obtain (n : ℕ) (Hn : b + n = c), from le_elim H, have H2 : a * b + of_nat ((to_nat a) * n) = a * c, from calc a * b + of_nat ((to_nat a) * n) = a * b + (to_nat a) * of_nat n : by simp ... = a * b + a * n : {to_nat_nonneg_eq Ha} ... = a * (b + n) : by simp ... = a * c : by simp, le_intro H2 theorem mul_le_right_nonneg {a b c : ℤ} (Hb : b ≥ 0) (H : a ≤ c) : a * b ≤ c * b := subst (mul_comm b c) (subst (mul_comm b a) (mul_le_left_nonneg Hb H)) theorem mul_le_left_nonpos {a b c : ℤ} (Ha : a ≤ 0) (H : b ≤ c) : a * c ≤ a * b := have H2 : -a * b ≤ -a * c, from mul_le_left_nonneg (zero_le_neg Ha) H, have H3 : -(a * b) ≤ -(a * c), from subst (mul_neg_left a c) (subst (mul_neg_left a b) H2), le_neg_inv H3 theorem mul_le_right_nonpos {a b c : ℤ} (Hb : b ≤ 0) (H : c ≤ a) : a * b ≤ c * b := subst (mul_comm b c) (subst (mul_comm b a) (mul_le_left_nonpos Hb H)) ---this theorem can be made more general by replacing either Ha with 0 ≤ a or Hb with 0 ≤ d... theorem mul_le_nonneg {a b c d : ℤ} (Ha : a ≥ 0) (Hb : b ≥ 0) (Hc : a ≤ c) (Hd : b ≤ d) : a * b ≤ c * d := le_trans (mul_le_right_nonneg Hb Hc) (mul_le_left_nonneg (le_trans Ha Hc) Hd) theorem mul_le_nonpos {a b c d : ℤ} (Ha : a ≤ 0) (Hb : b ≤ 0) (Hc : c ≤ a) (Hd : d ≤ b) : a * b ≤ c * d := le_trans (mul_le_right_nonpos Hb Hc) (mul_le_left_nonpos (le_trans Hc Ha) Hd) theorem mul_lt_left_pos {a b c : ℤ} (Ha : a > 0) (H : b < c) : a * b < a * c := have H2 : a * b < a * b + a, from subst (add_zero_right (a * b)) (add_lt_left Ha (a * b)), have H3 : a * b + a ≤ a * c, from subst (by simp) (mul_le_left_nonneg (lt_imp_le Ha) H), lt_le_trans H2 H3 theorem mul_lt_right_pos {a b c : ℤ} (Hb : b > 0) (H : a < c) : a * b < c * b := subst (mul_comm b c) (subst (mul_comm b a) (mul_lt_left_pos Hb H)) theorem mul_lt_left_neg {a b c : ℤ} (Ha : a < 0) (H : b < c) : a * c < a * b := have H2 : -a * b < -a * c, from mul_lt_left_pos (zero_lt_neg Ha) H, have H3 : -(a * b) < -(a * c), from subst (mul_neg_left a c) (subst (mul_neg_left a b) H2), lt_neg_inv H3 theorem mul_lt_right_neg {a b c : ℤ} (Hb : b < 0) (H : c < a) : a * b < c * b := subst (mul_comm b c) (subst (mul_comm b a) (mul_lt_left_neg Hb H)) theorem mul_le_lt_pos {a b c d : ℤ} (Ha : a > 0) (Hb : b ≥ 0) (Hc : a ≤ c) (Hd : b < d) : a * b < c * d := le_lt_trans (mul_le_right_nonneg Hb Hc) (mul_lt_left_pos (lt_le_trans Ha Hc) Hd) theorem mul_lt_le_pos {a b c d : ℤ} (Ha : a ≥ 0) (Hb : b > 0) (Hc : a < c) (Hd : b ≤ d) : a * b < c * d := lt_le_trans (mul_lt_right_pos Hb Hc) (mul_le_left_nonneg (le_trans Ha (lt_imp_le Hc)) Hd) theorem mul_lt_pos {a b c d : ℤ} (Ha : a > 0) (Hb : b > 0) (Hc : a < c) (Hd : b < d) : a * b < c * d := mul_lt_le_pos (lt_imp_le Ha) Hb Hc (lt_imp_le Hd) theorem mul_lt_neg {a b c d : ℤ} (Ha : a < 0) (Hb : b < 0) (Hc : c < a) (Hd : d < b) : a * b < c * d := lt_trans (mul_lt_right_neg Hb Hc) (mul_lt_left_neg (lt_trans Hc Ha) Hd) -- theorem mul_le_lt_neg and mul_lt_le_neg? theorem mul_lt_cancel_left_nonneg {a b c : ℤ} (Hc : c ≥ 0) (H : c * a < c * b) : a < b := or_elim (le_or_gt b a) (assume H2 : b ≤ a, have H3 : c * b ≤ c * a, from mul_le_left_nonneg Hc H2, absurd_elim _ H3 (lt_imp_not_ge H)) (assume H2 : a < b, H2) theorem mul_lt_cancel_right_nonneg {a b c : ℤ} (Hc : c ≥ 0) (H : a * c < b * c) : a < b := mul_lt_cancel_left_nonneg Hc (subst (mul_comm b c) (subst (mul_comm a c) H)) theorem mul_lt_cancel_left_nonpos {a b c : ℤ} (Hc : c ≤ 0) (H : c * b < c * a) : a < b := have H2 : -(c * a) < -(c * b), from lt_neg H, have H3 : -c * a < -c * b, from subst (symm (mul_neg_left c b)) (subst (symm (mul_neg_left c a)) H2), have H4 : -c ≥ 0, from zero_le_neg Hc, mul_lt_cancel_left_nonneg H4 H3 theorem mul_lt_cancel_right_nonpos {a b c : ℤ} (Hc : c ≤ 0) (H : b * c < a * c) : a < b := mul_lt_cancel_left_nonpos Hc (subst (mul_comm b c) (subst (mul_comm a c) H)) theorem mul_le_cancel_left_pos {a b c : ℤ} (Hc : c > 0) (H : c * a ≤ c * b) : a ≤ b := or_elim (le_or_gt a b) (assume H2 : a ≤ b, H2) (assume H2 : a > b, have H3 : c * a > c * b, from mul_lt_left_pos Hc H2, absurd_elim _ H3 (le_imp_not_gt H)) theorem mul_le_cancel_right_pos {a b c : ℤ} (Hc : c > 0) (H : a * c ≤ b * c) : a ≤ b := mul_le_cancel_left_pos Hc (subst (mul_comm b c) (subst (mul_comm a c) H)) theorem mul_le_cancel_left_neg {a b c : ℤ} (Hc : c < 0) (H : c * b ≤ c * a) : a ≤ b := have H2 : -(c * a) ≤ -(c * b), from le_neg H, have H3 : -c * a ≤ -c * b, from subst (symm (mul_neg_left c b)) (subst (symm (mul_neg_left c a)) H2), have H4 : -c > 0, from zero_lt_neg Hc, mul_le_cancel_left_pos H4 H3 theorem mul_le_cancel_right_neg {a b c : ℤ} (Hc : c < 0) (H : b * c ≤ a * c) : a ≤ b := mul_le_cancel_left_neg Hc (subst (mul_comm b c) (subst (mul_comm a c) H)) theorem mul_eq_one_left {a b : ℤ} (H : a * b = 1) : a = 1 ∨ a = - 1 := have H2 : (to_nat a) * (to_nat b) = 1, from calc (to_nat a) * (to_nat b) = (to_nat (a * b)) : symm (mul_to_nat a b) ... = (to_nat 1) : {H} ... = 1 : to_nat_of_nat 1, have H3 : (to_nat a) = 1, from mul_eq_one_left H2, or_imp_or (to_nat_cases a) (assume H4 : a = (to_nat a), subst H3 H4) (assume H4 : a = - (to_nat a), subst H3 H4) theorem mul_eq_one_right {a b : ℤ} (H : a * b = 1) : b = 1 ∨ b = - 1 := mul_eq_one_left (subst (mul_comm a b) H) -- sign function -- ------------- definition sign (a : ℤ) : ℤ := if a > 0 then 1 else (if a < 0 then - 1 else 0) -- TODO: for kernel theorem or_elim3 {a b c d : Prop} (H : a ∨ b ∨ c) (Ha : a → d) (Hb : b → d) (Hc : c → d) : d := or_elim H Ha (assume H2,or_elim H2 Hb Hc) theorem sign_pos {a : ℤ} (H : a > 0) : sign a = 1 := if_pos H _ _ theorem sign_negative {a : ℤ} (H : a < 0) : sign a = - 1 := trans (if_neg (lt_antisym H) _ _) (if_pos H _ _) theorem sign_zero : sign 0 = 0 := trans (if_neg (lt_irrefl 0) _ _) (if_neg (lt_irrefl 0) _ _) -- add_rewrite sign_negative sign_pos to_nat_negative to_nat_nonneg_eq sign_zero mul_to_nat theorem mul_sign_to_nat (a : ℤ) : sign a * (to_nat a) = a := have temp1 : ∀a : ℤ, a < 0 → a ≤ 0, from take a, lt_imp_le, have temp2 : ∀a : ℤ, a > 0 → a ≥ 0, from take a, lt_imp_le, or_elim3 (trichotomy a 0) (assume H : a < 0, by simp) (assume H : a = 0, by simp) (assume H : a > 0, by simp) -- TODO: show decidable for equality (and avoid classical library) theorem sign_mul (a b : ℤ) : sign (a * b) = sign a * sign b := or_elim (em (a = 0)) (assume Ha : a = 0, by simp) (assume Ha : a ≠ 0, or_elim (em (b = 0)) (assume Hb : b = 0, by simp) (assume Hb : b ≠ 0, have H : sign (a * b) * (to_nat (a * b)) = sign a * sign b * (to_nat (a * b)), from calc sign (a * b) * (to_nat (a * b)) = a * b : mul_sign_to_nat (a * b) ... = sign a * (to_nat a) * b : {symm (mul_sign_to_nat a)} ... = sign a * (to_nat a) * (sign b * (to_nat b)) : {symm (mul_sign_to_nat b)} ... = sign a * sign b * (to_nat (a * b)) : by simp, have H2 : (to_nat (a * b)) ≠ 0, from take H2', mul_ne_zero Ha Hb (to_nat_eq_zero H2'), have H3 : (to_nat (a * b)) ≠ of_nat 0, from contrapos of_nat_inj H2, mul_cancel_right H3 H)) --set_option pp::coercion true theorem sign_idempotent (a : ℤ) : sign (sign a) = sign a := have temp : of_nat 1 > 0, from iff_elim_left (iff_symm (lt_of_nat 0 1)) (succ_pos 0), --this should be done with simp or_elim3 (trichotomy a 0) sorry sorry sorry -- (by simp) -- (by simp) -- (by simp) theorem sign_succ (n : ℕ) : sign (succ n) = 1 := sign_pos (iff_elim_left (iff_symm (lt_of_nat 0 (succ n))) (succ_pos n)) --this should be done with simp theorem sign_neg (a : ℤ) : sign (-a) = - sign a := have temp1 : a > 0 → -a < 0, from neg_lt_zero, have temp2 : a < 0 → -a > 0, from zero_lt_neg, or_elim3 (trichotomy a 0) sorry sorry sorry -- (by simp) -- (by simp) -- (by simp) -- add_rewrite sign_neg theorem to_nat_sign_ne_zero {a : ℤ} (H : a ≠ 0) : (to_nat (sign a)) = 1 := or_elim3 (trichotomy a 0) sorry -- (by simp) (assume H2 : a = 0, absurd_elim _ H2 H) sorry -- (by simp) theorem sign_to_nat (a : ℤ) : sign (to_nat a) = to_nat (sign a) := have temp1 : ∀a : ℤ, a < 0 → a ≤ 0, from take a, lt_imp_le, have temp2 : ∀a : ℤ, a > 0 → a ≥ 0, from take a, lt_imp_le, or_elim3 (trichotomy a 0) sorry sorry sorry -- (by simp) -- (by simp) -- (by simp) end int