From abcc56a2a7aec6ec137075228923983ce577daed Mon Sep 17 00:00:00 2001 From: Rob Lewis Date: Fri, 27 Mar 2015 13:11:23 -0400 Subject: [PATCH] feat(library/algebra):refactor field and ordered_field more appropriately --- library/algebra/field.lean | 77 +++---- library/algebra/ordered_field.lean | 337 ++++++++++++++++++++--------- library/algebra/ordered_ring.lean | 15 ++ library/algebra/ring.lean | 19 ++ 4 files changed, 299 insertions(+), 149 deletions(-) diff --git a/library/algebra/field.lean b/library/algebra/field.lean index 8f13bf6673..d8ef619cdc 100644 --- a/library/algebra/field.lean +++ b/library/algebra/field.lean @@ -21,7 +21,7 @@ variable {A : Type} structure division_ring [class] (A : Type) extends ring A, has_inv A, zero_ne_one_class A := (mul_inv_cancel : ∀{a}, a ≠ zero → mul a (inv a) = one) (inv_mul_cancel : ∀{a}, a ≠ zero → mul (inv a) a = one) - (inv_zero : inv zero = zero) + --(inv_zero : inv zero = zero) section division_ring variables [s : division_ring A] {a b c : A} @@ -41,18 +41,20 @@ section division_ring theorem inv_eq_one_div : a⁻¹ = 1 / a := !one_mul⁻¹ - theorem inv_zero : 0⁻¹ = 0 := !division_ring.inv_zero +-- the following are only theorems if we assume inv_zero here +/- theorem inv_zero : 0⁻¹ = 0 := !division_ring.inv_zero theorem one_div_zero : 1 / 0 = 0 := calc 1 / 0 = 1 * 0⁻¹ : refl ... = 1 * 0 : division_ring.inv_zero A ... = 0 : mul_zero +-/ theorem div_eq_mul_one_div : a / b = a * (1 / b) := by rewrite [↑divide, one_mul] - theorem div_zero : a / 0 = 0 := by rewrite [div_eq_mul_one_div, one_div_zero, mul_zero] +-- theorem div_zero : a / 0 = 0 := by rewrite [div_eq_mul_one_div, one_div_zero, mul_zero] theorem mul_one_div_cancel (H : a ≠ 0) : a * (1 / a) = 1 := by rewrite [-inv_eq_one_div, (mul_inv_cancel H)] @@ -71,8 +73,8 @@ section division_ring have C1 : 0 = 1, from symm (by rewrite [-(mul_one_div_cancel H), H2, mul_zero]), absurd C1 zero_ne_one - theorem ne_zero_of_one_div_ne_zero (H : 1 / a ≠ 0) : a ≠ 0 := - assume Ha : a = 0, absurd (Ha⁻¹ ▸ one_div_zero) H +-- theorem ne_zero_of_one_div_ne_zero (H : 1 / a ≠ 0) : a ≠ 0 := +-- assume Ha : a = 0, absurd (Ha⁻¹ ▸ one_div_zero) H -- the analogue in group is called inv_one theorem inv_one_is_one : 1⁻¹ = 1 := @@ -89,24 +91,10 @@ section division_ring have C1 : a = 0, by rewrite [-mul_one, -(mul_one_div_cancel Hb), -mul.assoc, H, zero_mul], absurd C1 Ha - -- this belongs in ring? - theorem mul_ne_zero_imp_ne_zero (H : a * b ≠ 0) : a ≠ 0 ∧ b ≠ 0 := - have Ha : a ≠ 0, from - (assume Ha1 : a = 0, - have H1 : a * b = 0, by rewrite [Ha1, zero_mul], - absurd H1 H), - have Hb : b ≠ 0, from - (assume Hb1 : b = 0, - have H1 : a * b = 0, by rewrite [Hb1, mul_zero], - absurd H1 H), - and.intro Ha Hb - theorem mul_ne_zero_comm (H : a * b ≠ 0) : b * a ≠ 0 := have H2 : a ≠ 0 ∧ b ≠ 0, from mul_ne_zero_imp_ne_zero H, mul_ne_zero' (and.right H2) (and.left H2) --- theorem inv_zero_imp_zero (H : a⁻¹ = 0) : a = 0 := --- classical? -- make "left" and "right" versions? theorem eq_one_div_of_mul_eq_one (H : a * b = 1) : b = 1 / a := @@ -137,7 +125,6 @@ section division_ring ... = b * 1 : mul_one_div_cancel H2 ... = b : mul_one) - -- with 1 / 0 = 0, this should be provable without Ha and Hb. Needs decidable =? theorem one_div_mul_one_div (Ha : a ≠ 0) (Hb : b ≠ 0) : (1 / a) * (1 / b) = 1 / (b * a) := have H : (b * a) * ((1 / a) * (1 / b)) = 1, by rewrite [mul.assoc, -(mul.assoc a), (mul_one_div_cancel Ha), one_mul, (mul_one_div_cancel Hb)], @@ -147,15 +134,6 @@ section division_ring have H : (-1) * (-1) = 1, by rewrite [-neg_eq_neg_one_mul, neg_neg], symm (eq_one_div_of_mul_eq_one H) - -- this should be in ring - theorem mul_neg_one_eq_neg : a * (-1) = -a := - have H : a + a * -1 = 0, from calc - a + a * -1 = a * 1 + a * -1 : mul_one - ... = a * (1 + -1) : left_distrib - ... = a * 0 : add.right_inv - ... = 0 : mul_zero, - symm (neg_eq_of_add_eq_zero H) - theorem one_div_neg_eq_neg_one_div (H : a ≠ 0) : 1 / (- a) = - (1 / a) := have H1 : -1 ≠ 0, from (assume H2 : -1 = 0, absurd (symm (calc @@ -175,7 +153,6 @@ section division_ring ... = -(b * (1 / a)) : neg_mul_eq_mul_neg ... = - (b * a⁻¹) : inv_eq_one_div - -- can lose Ha theorem neg_div (Ha : a ≠ 0) : (-b) / a = - (b / a) := by rewrite [neg_eq_neg_one_mul, mul_div_assoc, -neg_eq_neg_one_mul] @@ -235,7 +212,7 @@ section division_ring have H : (a + b / c) * c = a * c + b, by rewrite [right_distrib, (div_mul_cancel Hc)], (iff.elim_right (eq_div_iff_mul_eq Hc)) H - theorem mul_mul_div (Hc : c ≠ 0) : a = a * c * (1 / c) := + theorem mul_mul_div (Hc : c ≠ 0) : a = a * c * (1 / c) := calc a = a * 1 : mul_one ... = a * (c * (1 / c)) : mul_one_div_cancel Hc @@ -333,13 +310,18 @@ section field end field structure discrete_field [class] (A : Type) extends field A := -(decidable_equality : ∀x y : A, decidable (x = y)) + (decidable_equality : ∀x y : A, decidable (x = y)) + (inv_zero : inv zero = zero) section discrete_field variable [s : discrete_field A] include s variables {a b c d : A} + -- many of the theorems in discrete_field are the same as theorems in field or division ring, + -- but with fewer hypotheses since 0⁻¹ = 0 and equality is decidable. + -- they are named with '. Is there a better convention? + -- name clash with order definition decidable_eq' [instance] (a b : A) : decidable (a = b) := @discrete_field.decidable_equality A s a b @@ -356,6 +338,19 @@ section discrete_field ⦃ integral_domain, s, eq_zero_or_eq_zero_of_mul_eq_zero := discrete_field.eq_zero_or_eq_zero_of_mul_eq_zero⦄ + theorem inv_zero : 0⁻¹ = 0 := !discrete_field.inv_zero + + theorem one_div_zero : 1 / 0 = 0 := + calc + 1 / 0 = 1 * 0⁻¹ : refl + ... = 1 * 0 : discrete_field.inv_zero A + ... = 0 : mul_zero + + theorem div_zero : a / 0 = 0 := by rewrite [div_eq_mul_one_div, one_div_zero, mul_zero] + + theorem ne_zero_of_one_div_ne_zero (H : 1 / a ≠ 0) : a ≠ 0 := + assume Ha : a = 0, absurd (Ha⁻¹ ▸ one_div_zero) H + theorem inv_zero_imp_zero (H : 1 / a = 0) : a = 0 := decidable.by_cases (assume Ha, Ha) @@ -374,17 +369,17 @@ section discrete_field theorem one_div_neg_eq_neg_one_div' : 1 / (- a) = - (1 / a) := decidable.by_cases - (assume Ha : a = 0, by rewrite [Ha, neg_zero, div_zero, -neg_zero, -(@div_zero A s 1)]) + (assume Ha : a = 0, by rewrite [Ha, neg_zero, 2 div_zero, neg_zero]) (assume Ha : a ≠ 0, one_div_neg_eq_neg_one_div Ha) theorem neg_div' : (-b) / a = - (b / a) := decidable.by_cases - (assume Ha : a = 0, by rewrite [Ha, div_zero, -neg_zero, -(@div_zero A s b)]) + (assume Ha : a = 0, by rewrite [Ha, 2 div_zero, neg_zero]) (assume Ha : a ≠ 0, neg_div Ha) theorem neg_div_neg_eq_div' : (-a) / (-b) = a / b := decidable.by_cases - (assume Hb : b = 0, by rewrite [Hb, neg_zero, div_zero, -(@div_zero A s a)]) + (assume Hb : b = 0, by rewrite [Hb, neg_zero, 2 div_zero]) (assume Hb : b ≠ 0, neg_div_neg_eq_div Hb) theorem div_div' : 1 / (1 / a) = a := @@ -403,10 +398,10 @@ section discrete_field theorem mul_inv' : (b * a)⁻¹ = a⁻¹ * b⁻¹ := decidable.by_cases - (assume Ha : a = 0, by rewrite [Ha, mul_zero, inv_zero, -(zero_mul b⁻¹), -inv_zero]) + (assume Ha : a = 0, by rewrite [Ha, mul_zero, 2 inv_zero, zero_mul]) (assume Ha : a ≠ 0, decidable.by_cases - (assume Hb : b = 0, by rewrite [Hb, zero_mul, inv_zero, -(mul_zero a⁻¹), -inv_zero]) + (assume Hb : b = 0, by rewrite [Hb, zero_mul, 2 inv_zero, mul_zero]) (assume Hb : b ≠ 0, mul_inv Ha Hb)) -- the following are specifically for fields @@ -415,7 +410,7 @@ section discrete_field theorem div_mul_right' (Ha : a ≠ 0) : a / (a * b) = 1 / b := decidable.by_cases - (assume Hb : b = 0, by rewrite [Hb, mul_zero, div_zero, -(@div_zero A s 1)]) + (assume Hb : b = 0, by rewrite [Hb, mul_zero, 2 div_zero]) (assume Hb : b ≠ 0, div_mul_right Hb (mul_ne_zero Ha Hb)) theorem div_mul_left' (Hb : b ≠ 0) : b / (a * b) = 1 / a := @@ -431,7 +426,7 @@ section discrete_field theorem mul_div_mul_left' (Hc : c ≠ 0) : (c * a) / (c * b) = a / b := decidable.by_cases - (assume Hb : b = 0, by rewrite [Hb, mul_zero, div_zero, -(@div_zero A s a)]) + (assume Hb : b = 0, by rewrite [Hb, mul_zero, 2 div_zero]) (assume Hb : b ≠ 0, mul_div_mul_left Hb Hc) theorem mul_div_mul_right' (Hc : c ≠ 0) : (a * c) / (b * c) = a / b := @@ -445,10 +440,10 @@ section discrete_field theorem one_div_div' : 1 / (a / b) = b / a := decidable.by_cases - (assume Ha : a = 0, by rewrite [Ha, zero_div, div_zero, -(@div_zero A s b)]) + (assume Ha : a = 0, by rewrite [Ha, zero_div, 2 div_zero]) (assume Ha : a ≠ 0, decidable.by_cases - (assume Hb : b = 0, by rewrite [Hb, 2 div_zero, -(@zero_div A s a)]) + (assume Hb : b = 0, by rewrite [Hb, 2 div_zero, zero_div]) (assume Hb : b ≠ 0, one_div_div Ha Hb)) theorem div_div_eq_mul_div' : a / (b / c) = (a * c) / b := diff --git a/library/algebra/ordered_field.lean b/library/algebra/ordered_field.lean index 16cd427079..fc1aebbce3 100644 --- a/library/algebra/ordered_field.lean +++ b/library/algebra/ordered_field.lean @@ -17,20 +17,9 @@ structure linear_ordered_field [class] (A : Type) extends linear_ordered_ring A, section linear_ordered_field variable {A : Type} - variables [s : linear_ordered_field A] {a b c : A} + variables [s : linear_ordered_field A] {a b c d : A} include s - -- ordered ring theorem? - -- split H3 into its own lemma - theorem gt_of_mul_lt_mul_neg_left (H : c * a < c * b) (Hc : c ≤ 0) : a > b := - have nhc : -c ≥ 0, from neg_nonneg_of_nonpos Hc, - have H2 : -(c * b) < -(c * a), from iff.mp' (neg_lt_neg_iff_lt _ _) H, - have H3 : (-c) * b < (-c) * a, from calc - (-c) * b = - (c * b) : neg_mul_eq_neg_mul - ... < -(c * a) : H2 - ... = (-c) * a : neg_mul_eq_neg_mul, - lt_of_mul_lt_mul_left H3 nhc - -- helpers for following theorem mul_zero_lt_mul_inv_of_pos (H : 0 < a) : a * 0 < a * (1 / a) := calc @@ -48,24 +37,10 @@ section linear_ordered_field theorem div_pos_of_pos (H : 0 < a) : 0 < 1 / a := lt_of_mul_lt_mul_left (mul_zero_lt_mul_inv_of_pos H) (le_of_lt H) - - theorem pos_of_div_pos (H : 0 < 1 / a) : 0 < a := - have H1 : 0 < 1 / (1 / a), from div_pos_of_pos H, - have H2 : 1 / a ≠ 0, from - (assume H3 : 1 / a = 0, - have H4 : 1 / (1 / a) = 0, from H3⁻¹ ▸ div_zero, - absurd H4 (ne.symm (ne_of_lt H1))), - (div_div (ne_zero_of_one_div_ne_zero H2)) ▸ H1 - + theorem div_neg_of_neg (H : a < 0) : 1 / a < 0 := gt_of_mul_lt_mul_neg_left (mul_zero_lt_mul_inv_of_neg H) (le_of_lt H) - theorem neg_of_div_neg (H : 1 / a < 0) : a < 0 := - have H1 : 0 < - (1 / a), from neg_pos_of_neg H, - have Ha : a ≠ 0, from ne_zero_of_one_div_ne_zero (ne_of_lt H), - have H2 : 0 < 1 / (-a), from (one_div_neg_eq_neg_one_div Ha)⁻¹ ▸ H1, - have H3 : 0 < -a, from pos_of_div_pos H2, - neg_of_neg_pos H3 theorem le_mul_of_ge_one_right (Hb : b ≥ 0) (H : a ≥ 1) : b ≤ b * a := mul_one _ ▸ (mul_le_mul_of_nonneg_left H Hb) @@ -113,74 +88,6 @@ section linear_ordered_field theorem one_lt_div_of_lt (Hb : b > 0) (H : b < a) : 1 < a / b := (iff.mp' (one_lt_div_iff_lt Hb)) H --- why is mul_le_mul under ordered_ring namespace? - theorem le_of_div_le (H : 0 < a) (Hl : 1 / a ≤ 1 / b) : b ≤ a := - have Hb : 0 < b, from pos_of_div_pos (calc - 0 < 1 / a : div_pos_of_pos H - ... ≤ 1 / b : Hl), - have H' : 1 ≤ a / b, from (calc - 1 = a / a : div_self (ne.symm (ne_of_lt H)) - ... = a * (1 / a) : div_eq_mul_one_div - ... ≤ a * (1 / b) : ordered_ring.mul_le_mul_of_nonneg_left Hl (le_of_lt H) - ... = a / b : div_eq_mul_one_div - ), le_of_one_le_div Hb H' - - - theorem lt_of_div_lt (H : 0 < a) (Hl : 1 / a < 1 / b) : b < a := - have Hb : 0 < b, from pos_of_div_pos (calc - 0 < 1 / a : div_pos_of_pos H - ... < 1 / b : Hl), - have H : 1 < a / b, from (calc - 1 = a / a : div_self (ne.symm (ne_of_lt H)) - ... = a * (1 / a) : div_eq_mul_one_div - ... < a * (1 / b) : mul_lt_mul_of_pos_left Hl H - ... = a / b : div_eq_mul_one_div), - lt_of_one_lt_div Hb H - - theorem le_of_div_le_neg (H : b < 0) (Hl : 1 / a ≤ 1 / b) : b ≤ a := - have Ha : a ≠ 0, from ne_of_lt (neg_of_div_neg (calc - 1 / a ≤ 1 / b : Hl - ... < 0 : div_neg_of_neg H)), - have H' : -b > 0, from neg_pos_of_neg H, - have Hl' : - (1 / b) ≤ - (1 / a), from neg_le_neg Hl, - have Hl'' : 1 / - b ≤ 1 / - a, from calc - 1 / -b = - (1 / b) : one_div_neg_eq_neg_one_div (ne_of_lt H) - ... ≤ - (1 / a) : Hl' - ... = 1 / -a : one_div_neg_eq_neg_one_div Ha, - le_of_neg_le_neg (le_of_div_le H' Hl'') - - theorem lt_of_div_lt_neg (H : b < 0) (Hl : 1 / a < 1 / b) : b < a := - have H1 : b ≤ a, from le_of_div_le_neg H (le_of_lt Hl), - have Hn : b ≠ a, from - (assume Hn' : b = a, - have Hl' : 1 / a = 1 / b, from Hn' ▸ refl _, - absurd Hl' (ne_of_lt Hl)), - lt_of_le_of_ne H1 Hn - - theorem div_lt_div_of_lt (Ha : 0 < a) (H : a < b) : 1 / b < 1 / a := - lt_of_not_le - (assume H', - absurd H (not_lt_of_le (le_of_div_le Ha H'))) - - theorem div_le_div_of_le (Ha : 0 < a) (H : a ≤ b) : 1 / b ≤ 1 / a := - le_of_not_lt - (assume H', - absurd H (not_le_of_lt (lt_of_div_lt Ha H'))) - - theorem div_lt_div_of_lt_neg (Hb : b < 0) (H : a < b) : 1 / b < 1 / a := - lt_of_not_le - (assume H', - absurd H (not_lt_of_le (le_of_div_le_neg Hb H'))) - - theorem div_le_div_of_le_neg (Hb : b < 0) (H : a ≤ b) : 1 / b ≤ 1 / a := - le_of_not_lt - (assume H', - absurd H (not_le_of_lt (lt_of_div_lt_neg Hb H'))) - - -- belongs in ordered ring? - theorem zero_gt_neg_one : -1 < 0 := - neg_zero ▸ (neg_lt_neg zero_lt_one) - theorem exists_lt : ∃ x, x < a := have H : a - 1 < a, from add_lt_of_le_of_neg (le.refl _) zero_gt_neg_one, exists.intro _ H @@ -189,18 +96,6 @@ section linear_ordered_field have H : a + 1 > a, from lt_add_of_le_of_pos (le.refl _) zero_lt_one, exists.intro _ H - theorem one_lt_div (H1 : 0 < a) (H2 : a < 1) : 1 < 1 / a := - one_div_one ▸ div_lt_div_of_lt H1 H2 - - theorem one_le_div (H1 : 0 < a) (H2 : a ≤ 1) : 1 ≤ 1 / a := - one_div_one ▸ div_le_div_of_le H1 H2 - - theorem neg_one_lt_div_neg (H1 : a < 0) (H2 : -1 < a) : 1 / a < -1 := - one_div_neg_one_eq_neg_one ▸ div_lt_div_of_lt_neg H1 H2 - - theorem neg_one_le_div_neg (H1 : a < 0) (H2 : -1 ≤ a) : 1 / a ≤ -1 := - one_div_neg_one_eq_neg_one ▸ div_le_div_of_le_neg H1 H2 - -- the following theorems amount to four iffs, for <, ≤, ≥, >. @@ -240,10 +135,132 @@ section linear_ordered_field ... > b * (1 / c) : mul_lt_mul_of_neg_right H (div_neg_of_neg Hc) ... = b / c : div_eq_mul_one_div + -- following these in the isabelle file, there are 8 biconditionals for the above with - signs + -- skipping for now + + theorem mul_sub_mul_div_mul_neg (Hc : c ≠ 0) (Hd : d ≠ 0) (H : a / c < b / d) : + (a * d - b * c) / (c * d) < 0 := + have H1 : a / c - b / d < 0, from calc + a / c - b / d < b / d - b / d : sub_lt_sub_right H + ... = 0 : sub_self, + calc + 0 > a / c - b / d : H1 + ... = (a * d - c * b) / (c * d) : div_sub_div Hc Hd + ... = (a * d - b * c) / (c * d) : mul.comm + + theorem mul_sub_mul_div_mul_nonpos (Hc : c ≠ 0) (Hd : d ≠ 0) (H : a / c ≤ b / d) : + (a * d - b * c) / (c * d) ≤ 0 := + have H1 : a / c - b / d ≤ 0, from calc + a / c - b / d ≤ b / d - b / d : sub_le_sub_right H + ... = 0 : sub_self, + calc + 0 ≥ a / c - b / d : H1 + ... = (a * d - c * b) / (c * d) : div_sub_div Hc Hd + ... = (a * d - b * c) / (c * d) : mul.comm + + theorem div_lt_div_of_mul_sub_mul_div_neg (Hc : c ≠ 0) (Hd : d ≠ 0) + (H : (a * d - b * c) / (c * d) < 0) : a / c < b / d := + have H1 : (a * d - c * b) / (c * d) < 0, from !mul.comm ▸ H, + have H2 : a / c - b / d < 0, from (div_sub_div Hc Hd)⁻¹ ▸ H1, + have H3 [visible] : a / c - b / d + b / d < 0 + b / d, from add_lt_add_right H2 _, + begin rewrite [zero_add at H3, neg_add_cancel_right at H3], exact H3 end + + theorem div_le_div_of_mul_sub_mul_div_nonpos (Hc : c ≠ 0) (Hd : d ≠ 0) + (H : (a * d - b * c) / (c * d) ≤ 0) : a / c ≤ b / d := + have H1 : (a * d - c * b) / (c * d) ≤ 0, from !mul.comm ▸ H, + have H2 : a / c - b / d ≤ 0, from (div_sub_div Hc Hd)⁻¹ ▸ H1, + have H3 [visible] : a / c - b / d + b / d ≤ 0 + b / d, from add_le_add_right H2 _, + begin rewrite [zero_add at H3, neg_add_cancel_right at H3], exact H3 end + + theorem pos_div_of_pos_of_pos (Ha : 0 < a) (Hb : 0 < b) : 0 < a / b := + begin + rewrite div_eq_mul_one_div, + apply mul_pos, + exact Ha, + apply div_pos_of_pos, + exact Hb + end + + theorem nonneg_div_of_nonneg_of_pos (Ha : 0 ≤ a) (Hb : 0 < b) : 0 ≤ a / b := + begin + rewrite div_eq_mul_one_div, + apply mul_nonneg, + exact Ha, + apply le_of_lt, + apply div_pos_of_pos, + exact Hb + end + + theorem neg_div_of_neg_of_pos (Ha : a < 0) (Hb : 0 < b) : a / b < 0:= + begin + rewrite div_eq_mul_one_div, + apply mul_neg_of_neg_of_pos, + exact Ha, + apply div_pos_of_pos, + exact Hb + end + + theorem nonpos_div_of_nonpos_of_pos (Ha : a ≤ 0) (Hb : 0 < b) : a / b ≤ 0 := + begin + rewrite div_eq_mul_one_div, + apply mul_nonpos_of_nonpos_of_nonneg, + exact Ha, + apply le_of_lt, + apply div_pos_of_pos, + exact Hb + end + + theorem neg_div_of_pos_of_neg (Ha : 0 < a) (Hb : b < 0) : a / b < 0 := + begin + rewrite div_eq_mul_one_div, + apply mul_neg_of_pos_of_neg, + exact Ha, + apply div_neg_of_neg, + exact Hb + end + + theorem nonpos_div_of_nonneg_of_neg (Ha : 0 ≤ a) (Hb : b < 0) : a / b ≤ 0 := + begin + rewrite div_eq_mul_one_div, + apply mul_nonpos_of_nonneg_of_nonpos, + exact Ha, + apply le_of_lt, + apply div_neg_of_neg, + exact Hb + end + + theorem pos_div_of_neg_of_neg (Ha : a < 0) (Hb : b < 0) : 0 < a / b := + begin + rewrite div_eq_mul_one_div, + apply mul_pos_of_neg_of_neg, + exact Ha, + apply div_neg_of_neg, + exact Hb + end + + + theorem nonneg_div_of_nonpos_of_neg (Ha : a ≤ 0) (Hb : b < 0) : 0 ≤ a / b := + begin + rewrite div_eq_mul_one_div, + apply mul_nonneg_of_nonpos_of_nonpos, + exact Ha, + apply le_of_lt, + apply div_neg_of_neg, + exact Hb + end + + theorem div_lt_div_of_lt_of_pos (H : a < b) (Hc : 0 < c) : a / c < b / c := + div_eq_mul_one_div⁻¹ ▸ div_eq_mul_one_div⁻¹ ▸ mul_lt_mul_of_pos_right H (div_pos_of_pos Hc) + + theorem div_lt_div_of_lt_of_neg (H : b < a) (Hc : c < 0) : a / c < b / c := + div_eq_mul_one_div⁻¹ ▸ div_eq_mul_one_div⁻¹ ▸ mul_lt_mul_of_neg_right H (div_neg_of_neg Hc) + + end linear_ordered_field structure discrete_linear_ordered_field [class] (A : Type) extends linear_ordered_field A, - decidable_linear_ordered_comm_ring A + decidable_linear_ordered_comm_ring A := + (inv_zero : inv zero = zero) section discrete_linear_ordered_field @@ -266,7 +283,111 @@ section discrete_linear_ordered_field [s : discrete_linear_ordered_field A] : discrete_field A := ⦃ discrete_field, s, decidable_equality := dec_eq_of_dec_lt⦄ + theorem pos_of_div_pos (H : 0 < 1 / a) : 0 < a := + have H1 : 0 < 1 / (1 / a), from div_pos_of_pos H, + have H2 : 1 / a ≠ 0, from + (assume H3 : 1 / a = 0, + have H4 : 1 / (1 / a) = 0, from H3⁻¹ ▸ div_zero, + absurd H4 (ne.symm (ne_of_lt H1))), + (div_div (ne_zero_of_one_div_ne_zero H2)) ▸ H1 + theorem neg_of_div_neg (H : 1 / a < 0) : a < 0 := + have H1 : 0 < - (1 / a), from neg_pos_of_neg H, + have Ha : a ≠ 0, from ne_zero_of_one_div_ne_zero (ne_of_lt H), + have H2 : 0 < 1 / (-a), from (one_div_neg_eq_neg_one_div Ha)⁻¹ ▸ H1, + have H3 : 0 < -a, from pos_of_div_pos H2, + neg_of_neg_pos H3 + +-- why is mul_le_mul under ordered_ring namespace? + theorem le_of_div_le (H : 0 < a) (Hl : 1 / a ≤ 1 / b) : b ≤ a := + have Hb : 0 < b, from pos_of_div_pos (calc + 0 < 1 / a : div_pos_of_pos H + ... ≤ 1 / b : Hl), + have H' : 1 ≤ a / b, from (calc + 1 = a / a : div_self (ne.symm (ne_of_lt H)) + ... = a * (1 / a) : div_eq_mul_one_div + ... ≤ a * (1 / b) : ordered_ring.mul_le_mul_of_nonneg_left Hl (le_of_lt H) + ... = a / b : div_eq_mul_one_div + ), le_of_one_le_div Hb H' + + + theorem le_of_div_le_neg (H : b < 0) (Hl : 1 / a ≤ 1 / b) : b ≤ a := + have Ha : a ≠ 0, from ne_of_lt (neg_of_div_neg (calc + 1 / a ≤ 1 / b : Hl + ... < 0 : div_neg_of_neg H)), + have H' : -b > 0, from neg_pos_of_neg H, + have Hl' : - (1 / b) ≤ - (1 / a), from neg_le_neg Hl, + have Hl'' : 1 / - b ≤ 1 / - a, from calc + 1 / -b = - (1 / b) : one_div_neg_eq_neg_one_div (ne_of_lt H) + ... ≤ - (1 / a) : Hl' + ... = 1 / -a : one_div_neg_eq_neg_one_div Ha, + le_of_neg_le_neg (le_of_div_le H' Hl'') + + theorem lt_of_div_lt (H : 0 < a) (Hl : 1 / a < 1 / b) : b < a := + have Hb : 0 < b, from pos_of_div_pos (calc + 0 < 1 / a : div_pos_of_pos H + ... < 1 / b : Hl), + have H : 1 < a / b, from (calc + 1 = a / a : div_self (ne.symm (ne_of_lt H)) + ... = a * (1 / a) : div_eq_mul_one_div + ... < a * (1 / b) : mul_lt_mul_of_pos_left Hl H + ... = a / b : div_eq_mul_one_div), + lt_of_one_lt_div Hb H + + + theorem lt_of_div_lt_neg (H : b < 0) (Hl : 1 / a < 1 / b) : b < a := + have H1 : b ≤ a, from le_of_div_le_neg H (le_of_lt Hl), + have Hn : b ≠ a, from + (assume Hn' : b = a, + have Hl' : 1 / a = 1 / b, from Hn' ▸ refl _, + absurd Hl' (ne_of_lt Hl)), + lt_of_le_of_ne H1 Hn + + + theorem div_lt_div_of_lt (Ha : 0 < a) (H : a < b) : 1 / b < 1 / a := + lt_of_not_le + (assume H', + absurd H (not_lt_of_le (le_of_div_le Ha H'))) + + theorem div_le_div_of_le (Ha : 0 < a) (H : a ≤ b) : 1 / b ≤ 1 / a := + le_of_not_lt + (assume H', + absurd H (not_le_of_lt (lt_of_div_lt Ha H'))) + + theorem div_lt_div_of_lt_neg (Hb : b < 0) (H : a < b) : 1 / b < 1 / a := + lt_of_not_le + (assume H', + absurd H (not_lt_of_le (le_of_div_le_neg Hb H'))) + + theorem div_le_div_of_le_neg (Hb : b < 0) (H : a ≤ b) : 1 / b ≤ 1 / a := + le_of_not_lt + (assume H', + absurd H (not_le_of_lt (lt_of_div_lt_neg Hb H'))) + + + theorem one_lt_div (H1 : 0 < a) (H2 : a < 1) : 1 < 1 / a := + one_div_one ▸ div_lt_div_of_lt H1 H2 + + theorem one_le_div (H1 : 0 < a) (H2 : a ≤ 1) : 1 ≤ 1 / a := + one_div_one ▸ div_le_div_of_le H1 H2 + + theorem neg_one_lt_div_neg (H1 : a < 0) (H2 : -1 < a) : 1 / a < -1 := + one_div_neg_one_eq_neg_one ▸ div_lt_div_of_lt_neg H1 H2 + + theorem neg_one_le_div_neg (H1 : a < 0) (H2 : -1 ≤ a) : 1 / a ≤ -1 := + one_div_neg_one_eq_neg_one ▸ div_le_div_of_le_neg H1 H2 + + theorem div_lt_div_of_pos_of_lt_of_pos (Hb : 0 < b) (H : b < a) (Hc : 0 < c) : c / a < c / b := + begin + apply (iff.mp (sub_neg_iff_lt _ _)), + rewrite [div_eq_mul_one_div, {c / b}div_eq_mul_one_div], + rewrite -mul_sub_left_distrib, + apply mul_neg_of_pos_of_neg, + exact Hc, + apply (iff.mp' (sub_neg_iff_lt _ _)), + apply div_lt_div_of_lt, + exact Hb, exact H + end end discrete_linear_ordered_field end algebra diff --git a/library/algebra/ordered_ring.lean b/library/algebra/ordered_ring.lean index 5009690a03..db711c5ea5 100644 --- a/library/algebra/ordered_ring.lean +++ b/library/algebra/ordered_ring.lean @@ -100,6 +100,7 @@ section rewrite zero_mul at H, exact H end + end structure linear_ordered_semiring [class] (A : Type) @@ -258,6 +259,7 @@ section rewrite zero_mul at H, exact H end + end -- TODO: we can eliminate mul_pos_of_pos, but now it is not worth the effort to redeclare the @@ -366,6 +368,19 @@ section apply (absurd_a_lt_a Hab) end) (assume Hb : b < 0, or.inr (and.intro Ha Hb))) + + theorem gt_of_mul_lt_mul_neg_left {a b c} (H : c * a < c * b) (Hc : c ≤ 0) : a > b := + have nhc : -c ≥ 0, from neg_nonneg_of_nonpos Hc, + have H2 : -(c * b) < -(c * a), from iff.mp' (neg_lt_neg_iff_lt _ _) H, + have H3 : (-c) * b < (-c) * a, from calc + (-c) * b = - (c * b) : neg_mul_eq_neg_mul + ... < -(c * a) : H2 + ... = (-c) * a : neg_mul_eq_neg_mul, + lt_of_mul_lt_mul_left H3 nhc + + theorem zero_gt_neg_one : -1 < 0 := + neg_zero ▸ (neg_lt_neg zero_lt_one) + end /- TODO: Isabelle's library has all kinds of cancelation rules for the simplifier. diff --git a/library/algebra/ring.lean b/library/algebra/ring.lean index 5ae8ef411b..4c51bffafe 100644 --- a/library/algebra/ring.lean +++ b/library/algebra/ring.lean @@ -220,6 +220,25 @@ section ... ↔ a * e + c - b * e = d : iff.symm !sub_eq_iff_eq_add ... ↔ a * e - b * e + c = d : by rewrite sub_add_eq_add_sub ... ↔ (a - b) * e + c = d : by rewrite mul_sub_right_distrib + + theorem mul_neg_one_eq_neg : a * (-1) = -a := + have H : a + a * -1 = 0, from calc + a + a * -1 = a * 1 + a * -1 : mul_one + ... = a * (1 + -1) : left_distrib + ... = a * 0 : add.right_inv + ... = 0 : mul_zero, + symm (neg_eq_of_add_eq_zero H) + + theorem mul_ne_zero_imp_ne_zero {a b} (H : a * b ≠ 0) : a ≠ 0 ∧ b ≠ 0 := + have Ha : a ≠ 0, from + (assume Ha1 : a = 0, + have H1 : a * b = 0, by rewrite [Ha1, zero_mul], + absurd H1 H), + have Hb : b ≠ 0, from + (assume Hb1 : b = 0, + have H1 : a * b = 0, by rewrite [Hb1, mul_zero], + absurd H1 H), + and.intro Ha Hb end structure comm_ring [class] (A : Type) extends ring A, comm_semigroup A