From ab9c51bd4bc47b89fc680bd4575f1c352d8fea40 Mon Sep 17 00:00:00 2001 From: Leonardo de Moura Date: Sat, 22 Nov 2014 17:19:24 -0800 Subject: [PATCH] refactor(library/data/nat/div): simplify 'gcd' definition --- library/data/nat/div.lean | 230 ++++++++++++++++---------------------- 1 file changed, 95 insertions(+), 135 deletions(-) diff --git a/library/data/nat/div.lean b/library/data/nat/div.lean index ddc48afba..612fc718b 100644 --- a/library/data/nat/div.lean +++ b/library/data/nat/div.lean @@ -92,7 +92,6 @@ calc (x + z) mod z ... = (x + z - z) mod z : if_pos (and.intro H (le_add_left z x)) ... = x mod z : sub_add_left - theorem mod_add_mul_self_right {x y z : ℕ} (H : z > 0) : (x + y * z) mod z = x mod z := induction_on y (calc (x + zero * z) mod z = (x + zero) mod z : mul.zero_left @@ -313,7 +312,7 @@ zero_mod n -- add_rewrite dvd_zero -theorem zero_dvd_iff {n : ℕ} : (0 | n) = (n = 0) := +theorem zero_dvd_eq (n : ℕ) : (0 | n) = (n = 0) := mod_zero n ▸ eq.refl (0 | n) -- add_rewrite zero_dvd_iff @@ -339,39 +338,45 @@ theorem dvd_mul_self_right (m n : ℕ) : 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 H3 : n = n div m * m, from (dvd_imp_div_mul_eq H1)⁻¹, 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 : !mul.assoc⁻¹, + k = k div n * n : dvd_imp_div_mul_eq H2 + ... = k div n * (n div m * m) : H3 + ... = k div n * (n div m) * m : mul.assoc, mp (!dvd_iff_exists_mul⁻¹) (exists_intro (k div n * (n div m)) (H4⁻¹)) 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, +have H : (n1 div m + n2 div m) * m = n1 + n2, from calc + (n1 div m + n2 div m) * m = n1 div m * m + n2 div m * m : mul.distr_right + ... = n1 + n2 div m * m : dvd_imp_div_mul_eq H1 + ... = n1 + n2 : dvd_imp_div_mul_eq H2, mp (!dvd_iff_exists_mul⁻¹) (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 zero_dvd_iff ▸ H1, - have H4 : n1 = 0, from zero_dvd_iff ▸ H2, - have H5 : n2 = 0, from mp (by simp) (H4 ▸ H3), - show 0 | n2, by simp) + have H3 : n1 + n2 = 0, from (zero_dvd_eq (n1 + n2)) ▸ H1, + have H4 : n1 = 0, from (zero_dvd_eq n1) ▸ H2, + have H5 : n2 = 0, from calc + n2 = 0 + n2 : add.zero_left + ... = n1 + n2 : H4 + ... = 0 : H3, + show 0 | n2, from H5 ▸ dvd_self n2) (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, + n1 + n2 = (n1 + n2) div m * m : dvd_imp_div_mul_eq H1 + ... = (n1 div m * m + n2) div m * m : dvd_imp_div_mul_eq H2 + ... = (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 : dvd_imp_div_mul_eq H2, have H4 : n2 = n2 div m * m, from add.cancel_left H3, mp (!dvd_iff_exists_mul⁻¹) (exists_intro _ (H4⁻¹))) @@ -390,146 +395,100 @@ by_cases -- Gcd and lcm -- ----------- -definition pair_nat.lt := lex lt lt -definition pair_nat.lt.wf [instance] : well_founded pair_nat.lt := -prod.lex.wf lt.wf lt.wf +private definition pair_nat.lt : nat × nat → nat × nat → Prop := measure pr₂ +private definition pair_nat.lt.wf : well_founded pair_nat.lt := +intro_k (measure.wf pr₂) 20 -- Remark: we use intro_k to be able to execute gcd efficiently in the kernel +instance pair_nat.lt.wf -- Remark: instance will not be saved in .olean infixl `≺`:50 := pair_nat.lt --- Lemma for justifying recursive call -private definition gcd.lt1 (x₁ y₁ : nat) : (x₁ - y₁, succ y₁) ≺ (succ x₁, succ y₁) := -!lex.left (le_imp_lt_succ (sub_le_self x₁ y₁)) +private definition gcd.lt.dec (x y₁ : nat) : (succ y₁, x mod succ y₁) ≺ (x, succ y₁) := +mod_lt (succ_pos y₁) --- Lemma for justifying recursive call -private definition gcd.lt2 (x₁ y₁ : nat) : (succ x₁, y₁ - x₁) ≺ (succ x₁, succ y₁) := -!lex.right (le_imp_lt_succ (sub_le_self y₁ x₁)) - -private definition gcd.F (p₁ : nat × nat) : (Π p₂ : nat × nat, p₂ ≺ p₁ → nat) → nat := -prod.cases_on p₁ (λ (x y : nat), -nat.cases_on x - (λ f, y) -- x = 0 - (λ x₁, nat.cases_on y - (λ f, succ x₁) -- y = 0 - (λ y₁ (f : (Π p₂ : nat × nat, p₂ ≺ (succ x₁, succ y₁) → nat)), - if y₁ ≤ x₁ then f (x₁ - y₁, succ y₁) !gcd.lt1 - else f (succ x₁, y₁ - x₁) !gcd.lt2))) +definition gcd.F (p₁ : nat × nat) : (Π p₂ : nat × nat, p₂ ≺ p₁ → nat) → nat := +prod.cases_on p₁ (λx y, cases_on y + (λ f, x) + (λ y₁ (f : Πp₂, p₂ ≺ (x, succ y₁) → nat), f (succ y₁, x mod succ y₁) !gcd.lt.dec)) definition gcd (x y : nat) := fix gcd.F (pair x y) -example : gcd 15 6 = 3 := -rfl - -theorem gcd_zero_left (y : nat) : gcd 0 y = y := -well_founded.fix_eq gcd.F (0, y) - -theorem gcd_succ_zero (x : nat) : gcd (succ x) 0 = succ x := -well_founded.fix_eq gcd.F (succ x, 0) - theorem gcd_zero (x : nat) : gcd x 0 = x := -cases_on x - (gcd_zero_left 0) - (λ x₁, !gcd_succ_zero) +well_founded.fix_eq gcd.F (x, 0) -theorem gcd_succ_succ (x y : nat) : gcd (succ x) (succ y) = if y ≤ x then gcd (x-y) (succ y) else gcd (succ x) (y-x) := -well_founded.fix_eq gcd.F (succ x, succ y) +theorem gcd_succ (x y : nat) : gcd x (succ y) = gcd (succ y) (x mod succ y) := +well_founded.fix_eq gcd.F (x, succ y) theorem gcd_one (n : ℕ) : gcd n 1 = 1 := -induction_on n - !gcd_zero_left - (λ n₁ ih, calc gcd (succ n₁) 1 - = if 0 ≤ n₁ then gcd (n₁ - 0) 1 else gcd (succ n₁) (0 - n₁) : gcd_succ_succ - ... = gcd (n₁ - 0) 1 : if_pos (zero_le n₁) - ... = gcd n₁ 1 : rfl - ... = 1 : ih) +calc gcd n 1 = gcd 1 (n mod 1) : gcd_succ n zero + ... = gcd 1 0 : mod_one + ... = 1 : gcd_zero + +theorem gcd_def (x y : ℕ) : gcd x y = if y = 0 then x else gcd y (x mod y) := +cases_on y + (calc gcd x 0 = x : gcd_zero x + ... = if 0 = 0 then x else gcd zero (x mod zero) : (if_pos rfl)⁻¹) + (λy₁, calc + gcd x (succ y₁) = gcd (succ y₁) (x mod succ y₁) : gcd_succ x y₁ + ... = if succ y₁ = 0 then x else gcd (succ y₁) (x mod succ y₁) : (if_neg (succ_ne_zero y₁))⁻¹) + +theorem gcd_pos (m : ℕ) {n : ℕ} (H : n > 0) : gcd m n = gcd n (m mod n) := +gcd_def m n ⬝ if_neg (pos_imp_ne_zero H) theorem gcd_self (n : ℕ) : gcd n n = n := -induction_on n - !gcd_zero_left - (λ n₁ ih, calc gcd (succ n₁) (succ n₁) - = if n₁ ≤ n₁ then gcd (n₁-n₁) (succ n₁) else gcd (succ n₁) (n₁-n₁) : gcd_succ_succ - ... = gcd (n₁-n₁) (succ n₁) : if_pos (le.refl n₁) - ... = gcd 0 (succ n₁) : sub_self - ... = succ n₁ : gcd_zero_left) +cases_on n + rfl + (λn₁, calc + gcd (succ n₁) (succ n₁) = gcd (succ n₁) (succ n₁ mod succ n₁) : gcd_succ (succ n₁) n₁ + ... = gcd (succ n₁) 0 : mod_self (succ n₁) + ... = succ n₁ : gcd_zero) -theorem gcd_left {m n : ℕ} (H : m < n) : gcd (succ m) (succ n) = gcd (succ m) (n - m) := -gcd_succ_succ m n ⬝ if_neg (lt_imp_not_ge H) +theorem gcd_zero_left (n : nat) : gcd 0 n = n := +cases_on n + rfl + (λ n₁, calc + gcd 0 (succ n₁) = gcd (succ n₁) (0 mod succ n₁) : gcd_succ + ... = gcd (succ n₁) 0 : zero_mod + ... = (succ n₁) : gcd_zero) -theorem gcd_right {m n : ℕ} (H : n < m) : gcd (succ m) (succ n) = gcd (m - n) (succ n) := -gcd_succ_succ m n ⬝ if_pos (le.of_lt H) - -private definition gcd_dvd_prop (m n : ℕ) : Prop := -(gcd m n | m) ∧ (gcd m n | n) - -private lemma gcd_arith_eq {m n : ℕ} (h : m > n) : m - n + succ n = succ m := -calc m - n + succ n = succ (m - n + n) : rfl - ... = succ m : @add_sub_ge_left m n (le.of_lt h) - -private lemma gcd_dvd.F (p₁ : nat × nat) : (∀p₂, p₂ ≺ p₁ → gcd_dvd_prop (pr₁ p₂) (pr₂ p₂)) → gcd_dvd_prop (pr₁ p₁) (pr₂ p₁) := -prod.cases_on p₁ (λ m n, cases_on m - (λ ih, and.intro !dvd_zero (!gcd_zero_left⁻¹ ▸ !dvd_self)) - (λ m₁, cases_on n - (λ ih, and.intro (!gcd_zero⁻¹ ▸ !dvd_self) !dvd_zero) - (λ n₁ (ih_core : ∀p₂, p₂ ≺ (succ m₁, succ n₁) → gcd_dvd_prop (pr₁ p₂) (pr₂ p₂)), - have ih : ∀{m₂ n₂}, (m₂, n₂) ≺ (succ m₁, succ n₁) → gcd m₂ n₂ | m₂ ∧ gcd m₂ n₂ | n₂, from - λ m₂ n₂ hlt, ih_core (m₂, n₂) hlt, - show (gcd (succ m₁) (succ n₁) | (succ m₁)) ∧ (gcd (succ m₁) (succ n₁) | (succ n₁)), from - or.elim (trichotomy n₁ m₁) - (λ hlt : n₁ < m₁, - have aux₁ : gcd (succ m₁) (succ n₁) = gcd (m₁ - n₁) (succ n₁), from gcd_right hlt, - have aux₂ : gcd (m₁ - n₁) (succ n₁) | (m₁ - n₁), from and.elim_left (ih !gcd.lt1), - have aux₃ : gcd (m₁ - n₁) (succ n₁) | succ n₁, from and.elim_right (ih !gcd.lt1), - have aux₄ : gcd (m₁ - n₁) (succ n₁) | succ m₁, from gcd_arith_eq hlt ▸ dvd_add aux₂ aux₃, - and.intro (aux₁⁻¹ ▸ aux₄) (aux₁⁻¹ ▸ aux₃)) - (λ h, or.elim h - (λ heq : n₁ = m₁, - have aux : gcd (succ n₁) (succ n₁) | (succ n₁), from gcd_self (succ n₁) ▸ !dvd_self, - eq.rec_on heq (and.intro aux aux)) - (λ hgt : n₁ > m₁, - have aux₁ : gcd (succ m₁) (succ n₁) = gcd (succ m₁) (n₁ - m₁), from gcd_left hgt, - have aux₂ : gcd (succ m₁) (n₁ - m₁) | succ m₁, from and.elim_left (ih !gcd.lt2), - have aux₃ : gcd (succ m₁) (n₁ - m₁) | (n₁ - m₁), from and.elim_right (ih !gcd.lt2), - have aux₄ : gcd (succ m₁) (n₁ - m₁) | succ n₁, from gcd_arith_eq hgt ▸ dvd_add aux₃ aux₂, - and.intro (aux₁⁻¹ ▸ aux₂) (aux₁⁻¹ ▸ aux₄)))))) +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 := +let Q : nat × nat → Prop := λ p : nat × nat, P (pr₁ p) (pr₂ p) in +have aux : Q (m, n), from + well_founded.induction (m, n) (λp, prod.cases_on p + (λm n, cases_on n + (λ ih, show P (pr₁ (m, 0)) (pr₂ (m, 0)), from H0 m) + (λ n₁ (ih : ∀p₂, p₂ ≺ (m, succ n₁) → P (pr₁ p₂) (pr₂ p₂)), + have hlt₁ : 0 < succ n₁, from succ_pos n₁, + have hlt₂ : (succ n₁, m mod succ n₁) ≺ (m, succ n₁), from gcd.lt.dec _ _, + have hp : P (succ n₁) (m mod succ n₁), from ih _ hlt₂, + show P m (succ n₁), from + H1 m (succ n₁) hlt₁ hp))), +aux theorem gcd_dvd (m n : ℕ) : (gcd m n | m) ∧ (gcd m n | n) := -well_founded.induction (m, n) gcd_dvd.F +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 div_mod_eq⁻¹ ▸ H, + have gcd_eq : gcd n (m mod n) = gcd m n, from (gcd_pos _ npos)⁻¹, + show (gcd m n | m) ∧ (gcd m n | n), from 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 -theorem gcd_greatest {m n k : ℕ} : k | m → k | n → k | (gcd m n) := -sorry - -end nat - -/- -theorem gcd_pos (m : ℕ) {n : ℕ} (H : n > 0) : gcd m n = gcd n (m mod n) := -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, (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) := -!gcd_def ⬝ if_neg !succ_ne_zero - 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, assume (h₁ : k | m) (h₂ : k | 0), + show k | gcd m 0, from !gcd_zero⁻¹ ▸ h₁) (take m n, assume npos : n > 0, assume IH : k | n → k | (m mod n) → k | gcd n (m mod n), @@ -539,4 +498,5 @@ gcd_induct m n 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 (gcd_pos _ npos)⁻¹, show k | gcd m n, from gcd_eq ▸ IH H2 H4) --/ + +end nat