--- Copyright (c) 2014 Jeremy Avigad. All rights reserved. --- Released under Apache 2.0 license as described in the file LICENSE. --- Author: Jeremy Avigad -- Theory nat2 -- =========== -- -- This is a continuation of the development of the natural numbers, with a general way of -- defining recursive functions, and definitions of div, mod, and gcd. -- TODO: replace the two uses of "not_or" by a constructive version import logic .sub struc.relation data.prod import logic.axioms.funext -- is this really needed? import tools.fake_simplifier using nat relation relation.iff_ops prod using fake_simplifier decidable namespace nat -- A general recursion principle -- ----------------------------- -- -- Data: -- -- dom, codom : Type -- default : codom -- measure : dom → ℕ -- rec_val : dom → (dom → codom) → codom -- -- and a proof -- -- rec_decreasing : ∀m, m ≥ measure x, rec_val x f = rec_val x (restrict f m) -- -- ... which says that the recursive call only depends on f at values with measure less than x, -- in the sense that changing other values to the default value doesn't change the result. -- -- The result is a function f = rec_measure, satisfying -- -- f x = rec_val x f definition restrict {dom codom : Type} (default : codom) (measure : dom → ℕ) (f : dom → codom) (m : ℕ) (x : dom) := if measure x < m then f x else default theorem restrict_lt_eq {dom codom : Type} (default : codom) (measure : dom → ℕ) (f : dom → codom) (m : ℕ) (x : dom) (H : measure x < m) : restrict default measure f m x = f x := if_pos H _ _ theorem restrict_not_lt_eq {dom codom : Type} (default : codom) (measure : dom → ℕ) (f : dom → codom) (m : ℕ) (x : dom) (H : ¬ measure x < m) : restrict default measure f m x = default := if_neg H _ _ definition rec_measure_aux {dom codom : Type} (default : codom) (measure : dom → ℕ) (rec_val : dom → (dom → codom) → codom) : ℕ → dom → codom := nat_rec (λx, default) (λm g x, if measure x < succ m then rec_val x g else default) definition rec_measure {dom codom : Type} (default : codom) (measure : dom → ℕ) (rec_val : dom → (dom → codom) → codom) (x : dom) : codom := rec_measure_aux default measure rec_val (succ (measure x)) x -- TODO: is funext really needed here? theorem rec_measure_aux_spec {dom codom : Type} (default : codom) (measure : dom → ℕ) (rec_val : dom → (dom → codom) → codom) (rec_decreasing : ∀g m x, m ≥ measure x → rec_val x g = rec_val x (restrict default measure g m)) (m : ℕ) : let f' := rec_measure_aux default measure rec_val in let f := rec_measure default measure rec_val in f' m = restrict default measure f m := -- TODO: note the use of (need for) inline here let f' := rec_measure_aux default measure rec_val in let f := rec_measure default measure rec_val in case_strong_induction_on m (have H1 : f' 0 = (λx, default), from rfl, have H2 : restrict default measure f 0 = (λx, default), from funext (take x, have H3 [fact]: ¬ measure x < 0, from not_lt_zero _, show restrict default measure f 0 x = default, from if_neg H3 _ _), show f' 0 = restrict default measure f 0, from trans H1 (symm H2)) (take m, assume IH: ∀n, n ≤ m → f' n = restrict default measure f n, funext (take x : dom, show f' (succ m) x = restrict default measure f (succ m) x, from by_cases -- (measure x < succ m) (assume H1 : measure x < succ m, have H2 [fact] : f' (succ m) x = rec_val x f, from calc f' (succ m) x = if measure x < succ m then rec_val x (f' m) else default : rfl ... = rec_val x (f' m) : if_pos H1 _ _ ... = rec_val x (restrict default measure f m) : {IH m (le_refl m)} ... = rec_val x f : symm (rec_decreasing _ _ _ (lt_succ_imp_le H1)), have H3 : restrict default measure f (succ m) x = rec_val x f, from let m' := measure x in calc restrict default measure f (succ m) x = f x : if_pos H1 _ _ ... = f' (succ m') x : refl _ ... = if measure x < succ m' then rec_val x (f' m') else default : rfl ... = rec_val x (f' m') : if_pos (self_lt_succ _) _ _ ... = rec_val x (restrict default measure f m') : {IH m' (lt_succ_imp_le H1)} ... = rec_val x f : symm (rec_decreasing _ _ _ (le_refl _)), show f' (succ m) x = restrict default measure f (succ m) x, from trans H2 (symm H3)) (assume H1 : ¬ measure x < succ m, have H2 : f' (succ m) x = default, from calc f' (succ m) x = if measure x < succ m then rec_val x (f' m) else default : rfl ... = default : if_neg H1 _ _, have H3 : restrict default measure f (succ m) x = default, from if_neg H1 _ _, show f' (succ m) x = restrict default measure f (succ m) x, from trans H2 (symm H3)))) theorem rec_measure_spec {dom codom : Type} {default : codom} {measure : dom → ℕ} (rec_val : dom → (dom → codom) → codom) (rec_decreasing : ∀g m x, m ≥ measure x → rec_val x g = rec_val x (restrict default measure g m)) (x : dom): let f := rec_measure default measure rec_val in f x = rec_val x f := let f' := rec_measure_aux default measure rec_val in let f := rec_measure default measure rec_val in let m := measure x in calc f x = f' (succ m) x : refl _ ... = if measure x < succ m then rec_val x (f' m) else default : rfl ... = rec_val x (f' m) : if_pos (self_lt_succ _) _ _ ... = rec_val x (restrict default measure f m) : {rec_measure_aux_spec _ _ _ rec_decreasing _} ... = rec_val x f : symm (rec_decreasing _ _ _ (le_refl _)) -- Div and mod -- ----------- -- ### the definition of div -- for fixed y, recursive call for x div y definition div_aux_rec (y : ℕ) (x : ℕ) (div_aux' : ℕ → ℕ) : ℕ := if (y = 0 ∨ x < y) then 0 else succ (div_aux' (x - y)) definition div_aux (y : ℕ) : ℕ → ℕ := rec_measure 0 (fun x, x) (div_aux_rec y) theorem div_aux_decreasing (y : ℕ) (g : ℕ → ℕ) (m : ℕ) (x : ℕ) (H : m ≥ x) : div_aux_rec y x g = div_aux_rec y x (restrict 0 (fun x, x) g m) := let lhs := div_aux_rec y x g in let rhs := div_aux_rec y x (restrict 0 (fun x, x) g m) in show lhs = rhs, from by_cases -- (y = 0 ∨ x < y) (assume H1 : y = 0 ∨ x < y, calc lhs = 0 : if_pos H1 _ _ ... = rhs : symm (if_pos H1 _ _)) (assume H1 : ¬ (y = 0 ∨ x < y), have H2 : y ≠ 0 ∧ ¬ x < y, from sorry, -- subst (not_or _ _) H1, have ypos : y > 0, from ne_zero_imp_pos (and_elim_left H2), have xgey : x ≥ y, from not_lt_imp_ge (and_elim_right H2), have H4 : x - y < x, from sub_lt (lt_le_trans ypos xgey) ypos, have H5 : x - y < m, from lt_le_trans H4 H, symm (calc rhs = succ (restrict 0 (fun x, x) g m (x - y)) : if_neg H1 _ _ ... = succ (g (x - y)) : {restrict_lt_eq _ _ _ _ _ H5} ... = lhs : symm (if_neg H1 _ _))) theorem div_aux_spec (y : ℕ) (x : ℕ) : div_aux y x = if (y = 0 ∨ x < y) then 0 else succ (div_aux y (x - y)) := rec_measure_spec (div_aux_rec y) (div_aux_decreasing y) x definition idivide (x : ℕ) (y : ℕ) : ℕ := div_aux y x infixl `div` := idivide -- copied from Isabelle theorem div_zero (x : ℕ) : x div 0 = 0 := trans (div_aux_spec _ _) (if_pos (or_intro_left _ (refl _)) _ _) -- add_rewrite div_zero theorem div_less {x y : ℕ} (H : x < y) : x div y = 0 := trans (div_aux_spec _ _) (if_pos (or_intro_right _ H) _ _) -- add_rewrite div_less theorem zero_div (y : ℕ) : 0 div y = 0 := case y (div_zero 0) (take y', div_less (succ_pos _)) -- add_rewrite zero_div theorem div_rec {x y : ℕ} (H1 : y > 0) (H2 : x ≥ y) : x div y = succ ((x - y) div y) := have H3 : ¬ (y = 0 ∨ x < y), from not_intro (assume H4 : y = 0 ∨ x < y, or_elim H4 (assume H5 : y = 0, not_elim (lt_irrefl _) (subst H5 H1)) (assume H5 : x < y, not_elim (lt_imp_not_ge H5) H2)), trans (div_aux_spec _ _) (if_neg H3 _ _) theorem div_add_self_right (x : ℕ) {z : ℕ} (H : z > 0) : (x + z) div z = succ (x div z) := have H1 : z ≤ x + z, by simp, let H2 := div_rec H H1 in by simp theorem div_add_mul_self_right (x y : ℕ) {z : ℕ} (H : z > 0) : (x + y * z) div z = x div z + y := induction_on y (by simp) (take y, assume IH : (x + y * z) div z = x div z + y, calc (x + succ y * z) div z = (x + y * z + z) div z : by simp ... = succ ((x + y * z) div z) : div_add_self_right _ H ... = x div z + succ y : by simp) -- ### The definition of mod -- for fixed y, recursive call for x mod y definition mod_aux_rec (y : ℕ) (x : ℕ) (mod_aux' : ℕ → ℕ) : ℕ := if (y = 0 ∨ x < y) then x else mod_aux' (x - y) definition mod_aux (y : ℕ) : ℕ → ℕ := rec_measure 0 (fun x, x) (mod_aux_rec y) theorem mod_aux_decreasing (y : ℕ) (g : ℕ → ℕ) (m : ℕ) (x : ℕ) (H : m ≥ x) : mod_aux_rec y x g = mod_aux_rec y x (restrict 0 (fun x, x) g m) := let lhs := mod_aux_rec y x g in let rhs := mod_aux_rec y x (restrict 0 (fun x, x) g m) in show lhs = rhs, from by_cases -- (y = 0 ∨ x < y) (assume H1 : y = 0 ∨ x < y, calc lhs = x : if_pos H1 _ _ ... = rhs : symm (if_pos H1 _ _)) (assume H1 : ¬ (y = 0 ∨ x < y), have H2 : y ≠ 0 ∧ ¬ x < y, from sorry, -- subst (not_or _ _) H1, have ypos : y > 0, from ne_zero_imp_pos (and_elim_left H2), have xgey : x ≥ y, from not_lt_imp_ge (and_elim_right H2), have H4 : x - y < x, from sub_lt (lt_le_trans ypos xgey) ypos, have H5 : x - y < m, from lt_le_trans H4 H, symm (calc rhs = restrict 0 (fun x, x) g m (x - y) : if_neg H1 _ _ ... = g (x - y) : restrict_lt_eq _ _ _ _ _ H5 ... = lhs : symm (if_neg H1 _ _))) theorem mod_aux_spec (y : ℕ) (x : ℕ) : mod_aux y x = if (y = 0 ∨ x < y) then x else mod_aux y (x - y) := rec_measure_spec (mod_aux_rec y) (mod_aux_decreasing y) x definition modulo (x : ℕ) (y : ℕ) : ℕ := mod_aux y x infixl `mod` := modulo theorem mod_zero (x : ℕ) : x mod 0 = x := trans (mod_aux_spec _ _) (if_pos (or_intro_left _ (refl _)) _ _) -- add_rewrite mod_zero theorem mod_lt_eq {x y : ℕ} (H : x < y) : x mod y = x := trans (mod_aux_spec _ _) (if_pos (or_intro_right _ H) _ _) -- add_rewrite mod_lt_eq theorem zero_mod (y : ℕ) : 0 mod y = 0 := case y (mod_zero 0) (take y', mod_lt_eq (succ_pos _)) -- add_rewrite zero_mod theorem mod_rec {x y : ℕ} (H1 : y > 0) (H2 : x ≥ y) : x mod y = (x - y) mod y := have H3 : ¬ (y = 0 ∨ x < y), from not_intro (assume H4 : y = 0 ∨ x < y, or_elim H4 (assume H5 : y = 0, not_elim (lt_irrefl _) (subst H5 H1)) (assume H5 : x < y, not_elim (lt_imp_not_ge H5) H2)), trans (mod_aux_spec _ _) (if_neg H3 _ _) -- need more of these, add as rewrite rules theorem mod_add_self_right (x : ℕ) {z : ℕ} (H : z > 0) : (x + z) mod z = x mod z := have H1 : z ≤ x + z, by simp, let H2 := mod_rec H H1 in by simp theorem mod_add_mul_self_right (x y : ℕ) {z : ℕ} (H : z > 0) : (x + y * z) mod z = x mod z := induction_on y (by simp) (take y, assume IH : (x + y * z) mod z = x mod z, calc (x + succ y * z) mod z = (x + y * z + z) mod z : by simp ... = (x + y * z) mod z : mod_add_self_right _ H ... = x mod z : IH) theorem mod_mul_self_right (m n : ℕ) : (m * n) mod n = 0 := case_zero_pos n (by simp) (take n, assume npos : n > 0, subst (by simp) (mod_add_mul_self_right 0 m npos)) -- add_rewrite mod_mul_self_right theorem mod_mul_self_left (m n : ℕ) : (m * n) mod m = 0 := subst (mul_comm _ _) (mod_mul_self_right _ _) -- add_rewrite mod_mul_self_left -- ### properties of div and mod together theorem mod_lt (x : ℕ) {y : ℕ} (H : y > 0) : x mod y < y := case_strong_induction_on x (show 0 mod y < y, from subst (symm (zero_mod _)) H) (take x, assume IH : ∀x', x' ≤ x → x' mod y < y, show succ x mod y < y, from by_cases -- (succ x < y) (assume H1 : succ x < y, have H2 : succ x mod y = succ x, from mod_lt_eq H1, show succ x mod y < y, from subst (symm H2) H1) (assume H1 : ¬ succ x < y, have H2 : y ≤ succ x, from not_lt_imp_ge H1, have H3 : succ x mod y = (succ x - y) mod y, from mod_rec H H2, have H4 : succ x - y < succ x, from sub_lt (succ_pos _) H, have H5 : succ x - y ≤ x, from lt_succ_imp_le H4, show succ x mod y < y, from subst (symm H3) (IH _ H5))) theorem div_mod_eq (x y : ℕ) : x = x div y * y + x mod y := case_zero_pos y (show x = x div 0 * 0 + x mod 0, from symm (calc x div 0 * 0 + x mod 0 = 0 + x mod 0 : {mul_zero_right _} ... = x mod 0 : add_zero_left _ ... = x : mod_zero _)) (take y, assume H : y > 0, show x = x div y * y + x mod y, from case_strong_induction_on x (show 0 = (0 div y) * y + 0 mod y, by simp) (take x, assume IH : ∀x', x' ≤ x → x' = x' div y * y + x' mod y, show succ x = succ x div y * y + succ x mod y, from by_cases -- (succ x < y) (assume H1 : succ x < y, have H2 : succ x div y = 0, from div_less H1, have H3 : succ x mod y = succ x, from mod_lt_eq H1, by simp) (assume H1 : ¬ succ x < y, have H2 : y ≤ succ x, from not_lt_imp_ge H1, have H3 : succ x div y = succ ((succ x - y) div y), from div_rec H H2, have H4 : succ x mod y = (succ x - y) mod y, from mod_rec H H2, have H5 : succ x - y < succ x, from sub_lt (succ_pos _) H, have H6 : succ x - y ≤ x, from lt_succ_imp_le H5, symm (calc succ x div y * y + succ x mod y = succ ((succ x - y) div y) * y + succ x mod y : {H3} ... = ((succ x - y) div y) * y + y + succ x mod y : {mul_succ_left _ _} ... = ((succ x - y) div y) * y + y + (succ x - y) mod y : {H4} ... = ((succ x - y) div y) * y + (succ x - y) mod y + y : add_right_comm _ _ _ ... = succ x - y + y : {symm (IH _ H6)} ... = succ x : add_sub_ge_left H2)))) theorem mod_le (x y : ℕ) : x mod y ≤ x := subst (symm (div_mod_eq _ _)) (le_add_left (x mod y) _) --- a good example where simplifying using the context causes problems theorem remainder_unique {y : ℕ} (H : y > 0) {q1 r1 q2 r2 : ℕ} (H1 : r1 < y) (H2 : r2 < y) (H3 : q1 * y + r1 = q2 * y + r2) : r1 = r2 := calc r1 = r1 mod y : by simp ... = (r1 + q1 * y) mod y : symm (mod_add_mul_self_right _ _ H) ... = (q1 * y + r1) mod y : {add_comm _ _} ... = (r2 + q2 * y) mod y : by simp ... = r2 mod y : mod_add_mul_self_right _ _ H ... = r2 : by simp theorem quotient_unique {y : ℕ} (H : y > 0) {q1 r1 q2 r2 : ℕ} (H1 : r1 < y) (H2 : r2 < y) (H3 : q1 * y + r1 = q2 * y + r2) : q1 = q2 := have H4 : q1 * y + r2 = q2 * y + r2, from subst (remainder_unique H H1 H2 H3) H3, have H5 : q1 * y = q2 * y, from add_cancel_right H4, have H6 : y > 0, from le_lt_trans (zero_le _) H1, show q1 = q2, from mul_cancel_right H6 H5 theorem div_mul_mul {z : ℕ} (x y : ℕ) (zpos : z > 0) : (z * x) div (z * y) = x div y := by_cases -- (y = 0) (assume H : y = 0, by simp) (assume H : y ≠ 0, have ypos : y > 0, from ne_zero_imp_pos H, have zypos : z * y > 0, from mul_pos zpos ypos, have H1 : (z * x) mod (z * y) < z * y, from mod_lt _ zypos, have H2 : z * (x mod y) < z * y, from mul_lt_left zpos (mod_lt _ ypos), quotient_unique zypos H1 H2 (calc ((z * x) div (z * y)) * (z * y) + (z * x) mod (z * y) = z * x : symm (div_mod_eq _ _) ... = z * (x div y * y + x mod y) : {div_mod_eq _ _} ... = z * (x div y * y) + z * (x mod y) : mul_distr_left _ _ _ ... = (x div y) * (z * y) + z * (x mod y) : {mul_left_comm _ _ _})) --- something wrong with the term order --- ... = (x div y) * (z * y) + z * (x mod y) : by simp)) theorem mod_mul_mul {z : ℕ} (x y : ℕ) (zpos : z > 0) : (z * x) mod (z * y) = z * (x mod y) := by_cases -- (y = 0) (assume H : y = 0, by simp) (assume H : y ≠ 0, have ypos : y > 0, from ne_zero_imp_pos H, have zypos : z * y > 0, from mul_pos zpos ypos, have H1 : (z * x) mod (z * y) < z * y, from mod_lt _ zypos, have H2 : z * (x mod y) < z * y, from mul_lt_left zpos (mod_lt _ ypos), remainder_unique zypos H1 H2 (calc ((z * x) div (z * y)) * (z * y) + (z * x) mod (z * y) = z * x : symm (div_mod_eq _ _) ... = z * (x div y * y + x mod y) : {div_mod_eq _ _} ... = z * (x div y * y) + z * (x mod y) : mul_distr_left _ _ _ ... = (x div y) * (z * y) + z * (x mod y) : {mul_left_comm _ _ _})) theorem mod_one (x : ℕ) : x mod 1 = 0 := have H1 : x mod 1 < 1, from mod_lt _ (succ_pos 0), le_zero (lt_succ_imp_le H1) -- add_rewrite mod_one theorem mod_self (n : ℕ) : n mod n = 0 := case n (by simp) (take n, have H : (succ n * 1) mod (succ n * 1) = succ n * (1 mod 1), from mod_mul_mul 1 1 (succ_pos n), subst (by simp) H) -- add_rewrite mod_self theorem div_one (n : ℕ) : n div 1 = n := have H : n div 1 * 1 + n mod 1 = n, from symm (div_mod_eq n 1), subst (by simp) H -- add_rewrite div_one theorem pos_div_self {n : ℕ} (H : n > 0) : n div n = 1 := have H1 : (n * 1) div (n * 1) = 1 div 1, from div_mul_mul 1 1 H, subst (by simp) H1 -- add_rewrite pos_div_self -- Divides -- ------- definition dvd (x y : ℕ) : Prop := y mod x = 0 infix `|` := dvd theorem dvd_iff_mod_eq_zero (x y : ℕ) : x | y ↔ y mod x = 0 := refl _ theorem dvd_imp_div_mul_eq {x y : ℕ} (H : y | x) : x div y * y = x := symm (calc x = x div y * y + x mod y : div_mod_eq _ _ ... = x div y * y + 0 : {mp (dvd_iff_mod_eq_zero _ _) H} ... = x div y * y : add_zero_right _) -- add_rewrite dvd_imp_div_mul_eq theorem mul_eq_imp_dvd {z x y : ℕ} (H : z * y = x) : y | x := have H1 : z * y = x mod y + x div y * y, from trans (trans H (div_mod_eq x y)) (add_comm _ _), have H2 : (z - x div y) * y = x mod y, from calc (z - x div y) * y = z * y - x div y * y : mul_sub_distr_right _ _ _ ... = x mod y + x div y * y - x div y * y : {H1} ... = x mod y : sub_add_left _ _, show x mod y = 0, from by_cases -- (y = 0) (assume yz : y = 0, have xz : x = 0, from calc x = z * y : symm H ... = z * 0 : {yz} ... = 0 : mul_zero_right _, calc x mod y = x mod 0 : {yz} ... = x : mod_zero _ ... = 0 : xz) (assume ynz : y ≠ 0, have ypos : y > 0, from ne_zero_imp_pos ynz, have H3 : (z - x div y) * y < y, from subst (symm H2) (mod_lt x ypos), have H4 : (z - x div y) * y < 1 * y, from subst (symm (mul_one_left y)) H3, have H5 : z - x div y < 1, from mul_lt_cancel_right H4, have H6 : z - x div y = 0, from le_zero (lt_succ_imp_le H5), calc x mod y = (z - x div y) * y : symm H2 ... = 0 * y : {H6} ... = 0 : mul_zero_left _) theorem dvd_iff_exists_mul (x y : ℕ) : x | y ↔ ∃z, z * x = y := iff_intro (assume H : x | y, show ∃z, z * x = y, from exists_intro _ (dvd_imp_div_mul_eq H)) (assume H : ∃z, z * x = y, obtain (z : ℕ) (zx_eq : z * x = y), from H, show x | y, from mul_eq_imp_dvd zx_eq) theorem dvd_zero (n : ℕ) : n | 0 := sorry -- (by simp) (dvd_iff_mod_eq_zero n 0) -- add_rewrite dvd_zero theorem zero_dvd_iff {n : ℕ} : (0 | n) = (n = 0) := sorry -- (by simp) (dvd_iff_mod_eq_zero 0 n) -- add_rewrite zero_dvd_iff theorem one_dvd (n : ℕ) : 1 | n := sorry -- (by simp) (dvd_iff_mod_eq_zero 1 n) -- add_rewrite one_dvd theorem dvd_self (n : ℕ) : n | n := sorry -- (by simp) (dvd_iff_mod_eq_zero n n) -- add_rewrite dvd_self theorem dvd_mul_self_left (m n : ℕ) : m | (m * n) := sorry -- (by simp) (dvd_iff_mod_eq_zero m (m * n)) -- add_rewrite dvd_mul_self_left theorem dvd_mul_self_right (m n : ℕ) : m | (n * m) := sorry -- (by simp) (dvd_iff_mod_eq_zero m (n * m)) -- add_rewrite dvd_mul_self_left theorem dvd_trans {m n k : ℕ} (H1 : m | n) (H2 : n | k) : m | k := have H3 : n = n div m * m, by simp, have H4 : k = k div n * (n div m) * m, from calc k = k div n * n : by simp ... = k div n * (n div m * m) : {H3} ... = k div n * (n div m) * m : symm (mul_assoc _ _ _), mp (symm (dvd_iff_exists_mul _ _)) (exists_intro (k div n * (n div m)) (symm H4)) theorem dvd_add {m n1 n2 : ℕ} (H1 : m | n1) (H2 : m | n2) : m | (n1 + n2) := have H : (n1 div m + n2 div m) * m = n1 + n2, by simp, mp (symm (dvd_iff_exists_mul _ _)) (exists_intro _ H) theorem dvd_add_cancel_left {m n1 n2 : ℕ} : m | (n1 + n2) → m | n1 → m | n2 := case_zero_pos m (assume H1 : 0 | n1 + n2, assume H2 : 0 | n1, have H3 : n1 + n2 = 0, from subst zero_dvd_iff H1, have H4 : n1 = 0, from subst zero_dvd_iff H2, have H5 : n2 = 0, from mp (by simp) (subst H4 H3), show 0 | n2, by simp) (take m, assume mpos : m > 0, assume H1 : m | (n1 + n2), assume H2 : m | n1, have H3 : n1 + n2 = n1 + n2 div m * m, from calc n1 + n2 = (n1 + n2) div m * m : by simp ... = (n1 div m * m + n2) div m * m : by simp ... = (n2 + n1 div m * m) div m * m : {add_comm _ _} ... = (n2 div m + n1 div m) * m : {div_add_mul_self_right _ _ mpos} ... = n2 div m * m + n1 div m * m : mul_distr_right _ _ _ ... = n1 div m * m + n2 div m * m : add_comm _ _ ... = n1 + n2 div m * m : by simp, have H4 : n2 = n2 div m * m, from add_cancel_left H3, mp (symm (dvd_iff_exists_mul _ _)) (exists_intro _ (symm H4))) theorem dvd_add_cancel_right {m n1 n2 : ℕ} (H : m | (n1 + n2)) : m | n2 → m | n1 := dvd_add_cancel_left (subst (add_comm _ _) H) theorem dvd_sub {m n1 n2 : ℕ} (H1 : m | n1) (H2 : m | n2) : m | (n1 - n2) := by_cases (assume H3 : n1 ≥ n2, have H4 : n1 = n1 - n2 + n2, from symm (add_sub_ge_left H3), show m | n1 - n2, from dvd_add_cancel_right (subst H4 H1) H2) (assume H3 : ¬ (n1 ≥ n2), have H4 : n1 - n2 = 0, from le_imp_sub_eq_zero (lt_imp_le (not_le_imp_gt H3)), show m | n1 - n2, from subst (symm H4) (dvd_zero _)) -- Gcd and lcm -- ----------- -- ### definition of gcd definition gcd_aux_measure (p : ℕ × ℕ) : ℕ := pr2 p definition gcd_aux_rec (p : ℕ × ℕ) (gcd_aux' : ℕ × ℕ → ℕ) : ℕ := let x := pr1 p, y := pr2 p in if y = 0 then x else gcd_aux' (pair y (x mod y)) definition gcd_aux : ℕ × ℕ → ℕ := rec_measure 0 gcd_aux_measure gcd_aux_rec theorem gcd_aux_decreasing (g : ℕ × ℕ → ℕ) (m : ℕ) (p : ℕ × ℕ) (H : m ≥ gcd_aux_measure p) : gcd_aux_rec p g = gcd_aux_rec p (restrict 0 gcd_aux_measure g m) := let x := pr1 p, y := pr2 p in let p' := pair y (x mod y) in let lhs := gcd_aux_rec p g in let rhs := gcd_aux_rec p (restrict 0 gcd_aux_measure g m) in show lhs = rhs, from by_cases -- (y = 0) (assume H1 : y = 0, calc lhs = x : if_pos H1 _ _ ... = rhs : symm (if_pos H1 _ _)) (assume H1 : y ≠ 0, have ypos : y > 0, from ne_zero_imp_pos H1, have H2 : gcd_aux_measure p' = x mod y, from pr2_pair _ _, have H3 : gcd_aux_measure p' < gcd_aux_measure p, from subst (symm H2) (mod_lt _ ypos), have H4: gcd_aux_measure p' < m, from lt_le_trans H3 H, symm (calc rhs = restrict 0 gcd_aux_measure g m p' : if_neg H1 _ _ ... = g p' : restrict_lt_eq _ _ _ _ _ H4 ... = lhs : symm (if_neg H1 _ _))) theorem gcd_aux_spec (p : ℕ × ℕ) : gcd_aux p = let x := pr1 p, y := pr2 p in if y = 0 then x else gcd_aux (pair y (x mod y)) := rec_measure_spec gcd_aux_rec gcd_aux_decreasing p definition gcd (x y : ℕ) : ℕ := gcd_aux (pair x y) theorem gcd_def (x y : ℕ) : gcd x y = if y = 0 then x else gcd y (x mod y) := let x' := pr1 (pair x y), y' := pr2 (pair x y) in calc gcd x y = if y' = 0 then x' else gcd_aux (pair y' (x' mod y')) : gcd_aux_spec (pair x y) ... = if y = 0 then x else gcd y (x mod y) : refl _ theorem gcd_zero (x : ℕ) : gcd x 0 = x := trans (gcd_def x 0) (if_pos (refl _) _ _) -- add_rewrite gcd_zero theorem gcd_pos (m : ℕ) {n : ℕ} (H : n > 0) : gcd m n = gcd n (m mod n) := trans (gcd_def m n) (if_neg (pos_imp_ne_zero H) _ _) theorem gcd_zero_left (x : ℕ) : gcd 0 x = x := case x (by simp) (take x, trans (gcd_def _ _) (by simp)) -- add_rewrite gcd_zero_left theorem gcd_induct {P : ℕ → ℕ → Prop} (m n : ℕ) (H0 : ∀m, P m 0) (H1 : ∀m n, 0 < n → P n (m mod n) → P m n) : P m n := have aux : ∀m, P m n, from case_strong_induction_on n H0 (take n, assume IH : ∀k, k ≤ n → ∀m, P m k, take m, have H2 : m mod succ n ≤ n, from lt_succ_imp_le (mod_lt _ (succ_pos _)), have H3 : P (succ n) (m mod succ n), from IH _ H2 _, show P m (succ n), from H1 _ _ (succ_pos _) H3), aux m theorem gcd_succ (m n : ℕ) : gcd m (succ n) = gcd (succ n) (m mod succ n) := trans (gcd_def _ _) (if_neg (succ_ne_zero n) _ _) theorem gcd_one (n : ℕ) : gcd n 1 = 1 := sorry -- (by simp) (gcd_succ n 0) theorem gcd_self (n : ℕ) : gcd n n = n := sorry -- case n (by simp) (take n, (by simp) (gcd_succ (succ n) n)) theorem gcd_dvd (m n : ℕ) : (gcd m n | m) ∧ (gcd m n | n) := gcd_induct m n (take m, show (gcd m 0 | m) ∧ (gcd m 0 | 0), by simp) (take m n, assume npos : 0 < n, assume IH : (gcd n (m mod n) | n) ∧ (gcd n (m mod n) | (m mod n)), have H : gcd n (m mod n) | (m div n * n + m mod n), from dvd_add (dvd_trans (and_elim_left IH) (dvd_mul_self_right _ _)) (and_elim_right IH), have H1 : gcd n (m mod n) | m, from subst (symm (div_mod_eq m n)) H, have gcd_eq : gcd n (m mod n) = gcd m n, from symm (gcd_pos _ npos), show (gcd m n | m) ∧ (gcd m n | n), from subst gcd_eq (and_intro H1 (and_elim_left IH))) theorem gcd_dvd_left (m n : ℕ) : (gcd m n | m) := and_elim_left (gcd_dvd _ _) theorem gcd_dvd_right (m n : ℕ) : (gcd m n | n) := and_elim_right (gcd_dvd _ _) -- add_rewrite gcd_dvd_left gcd_dvd_right theorem gcd_greatest {m n k : ℕ} : k | m → k | n → k | (gcd m n) := gcd_induct m n (take m, assume H : k | m, sorry) -- by simp) (take m n, assume npos : n > 0, assume IH : k | n → k | (m mod n) → k | gcd n (m mod n), assume H1 : k | m, assume H2 : k | n, have H3 : k | m div n * n + m mod n, from subst (div_mod_eq m n) H1, have H4 : k | m mod n, from dvd_add_cancel_left H3 (dvd_trans H2 (by simp)), have gcd_eq : gcd n (m mod n) = gcd m n, from symm (gcd_pos _ npos), show k | gcd m n, from subst gcd_eq (IH H2 H4)) end nat