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refactor(library/algebra/field.lean): rename has_decidable_eq and declare instance

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Jeremy Avigad 2015-04-18 13:48:27 -04:00 • committed by Leonardo de Moura
parent dfaeb475cc
commit 6359132b67
2 changed files with 44 additions and 46 deletions
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8
library/algebra/field.lean
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@ -310,9 +310,11 @@ section field
end field end field
structure discrete_field [class] (A : Type) extends field A := structure discrete_field [class] (A : Type) extends field A :=
(decidable_equality : ∀x y : A, decidable (x = y)) (has_decidable_eq : decidable_eq A)
(inv_zero : inv zero = zero) (inv_zero : inv zero = zero)
attribute discrete_field.has_decidable_eq [instance]
section discrete_field section discrete_field
variable [s : discrete_field A] variable [s : discrete_field A]
include s include s
@ -322,10 +324,6 @@ section discrete_field
-- but with fewer hypotheses since 0⁻¹ = 0 and equality is decidable. -- but with fewer hypotheses since 0⁻¹ = 0 and equality is decidable.
-- they are named with '. Is there a better convention? -- 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
theorem discrete_field.eq_zero_or_eq_zero_of_mul_eq_zero theorem discrete_field.eq_zero_or_eq_zero_of_mul_eq_zero
(x y : A) (H : x * y = 0) : x = 0 ∨ y = 0 := (x y : A) (H : x * y = 0) : x = 0 ∨ y = 0 :=
decidable.by_cases decidable.by_cases

82
library/algebra/ordered_field.lean
View file

@ -11,15 +11,15 @@ import algebra.ordered_ring algebra.field
open eq eq.ops open eq eq.ops
namespace algebra namespace algebra
structure linear_ordered_field [class] (A : Type) extends linear_ordered_ring A, field A structure linear_ordered_field [class] (A : Type) extends linear_ordered_ring A, field A
section linear_ordered_field section linear_ordered_field
variable {A : Type} variable {A : Type}
variables [s : linear_ordered_field A] {a b c d : A} variables [s : linear_ordered_field A] {a b c d : A}
include s include s
-- helpers for following -- helpers for following
theorem mul_zero_lt_mul_inv_of_pos (H : 0 < a) : a * 0 < a * (1 / a) := theorem mul_zero_lt_mul_inv_of_pos (H : 0 < a) : a * 0 < a * (1 / a) :=
calc calc
@ -27,17 +27,17 @@ section linear_ordered_field
... < 1 : zero_lt_one ... < 1 : zero_lt_one
... = a * a⁻¹ : mul_inv_cancel (ne.symm (ne_of_lt H)) ... = a * a⁻¹ : mul_inv_cancel (ne.symm (ne_of_lt H))
... = a * (1 / a) : inv_eq_one_div ... = a * (1 / a) : inv_eq_one_div
theorem mul_zero_lt_mul_inv_of_neg (H : a < 0) : a * 0 < a * (1 / a) := theorem mul_zero_lt_mul_inv_of_neg (H : a < 0) : a * 0 < a * (1 / a) :=
calc calc
a * 0 = 0 : mul_zero a * 0 = 0 : mul_zero
... < 1 : zero_lt_one ... < 1 : zero_lt_one
... = a * a⁻¹ : mul_inv_cancel (ne_of_lt H) ... = a * a⁻¹ : mul_inv_cancel (ne_of_lt H)
... = a * (1 / a) : inv_eq_one_div ... = a * (1 / a) : inv_eq_one_div
theorem div_pos_of_pos (H : 0 < a) : 0 < 1 / a := 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) lt_of_mul_lt_mul_left (mul_zero_lt_mul_inv_of_pos H) (le_of_lt H)
theorem div_neg_of_neg (H : a < 0) : 1 / a < 0 := 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) gt_of_mul_lt_mul_neg_left (mul_zero_lt_mul_inv_of_neg H) (le_of_lt H)
@ -48,7 +48,7 @@ section linear_ordered_field
theorem lt_mul_of_gt_one_right (Hb : b > 0) (H : a > 1) : b < b * a := theorem lt_mul_of_gt_one_right (Hb : b > 0) (H : a > 1) : b < b * a :=
mul_one _ ▸ (mul_lt_mul_of_pos_left H Hb) mul_one _ ▸ (mul_lt_mul_of_pos_left H Hb)
theorem one_le_div_iff_le (Hb : b > 0) : 1 ≤ a / b ↔ b ≤ a := theorem one_le_div_iff_le (Hb : b > 0) : 1 ≤ a / b ↔ b ≤ a :=
have Hb' : b ≠ 0, from ne.symm (ne_of_lt Hb), have Hb' : b ≠ 0, from ne.symm (ne_of_lt Hb),
iff.intro iff.intro
(assume H : 1 ≤ a / b, (assume H : 1 ≤ a / b,
@ -88,20 +88,20 @@ section linear_ordered_field
theorem one_lt_div_of_lt (Hb : b > 0) (H : b < a) : 1 < a / b := theorem one_lt_div_of_lt (Hb : b > 0) (H : b < a) : 1 < a / b :=
(iff.mp' (one_lt_div_iff_lt Hb)) H (iff.mp' (one_lt_div_iff_lt Hb)) H
theorem exists_lt : ∃ x, x < a := theorem exists_lt : ∃ x, x < a :=
have H : a - 1 < a, from add_lt_of_le_of_neg (le.refl _) zero_gt_neg_one, have H : a - 1 < a, from add_lt_of_le_of_neg (le.refl _) zero_gt_neg_one,
exists.intro _ H exists.intro _ H
theorem exists_gt : ∃ x, x > a := theorem exists_gt : ∃ x, x > a :=
have H : a + 1 > a, from lt_add_of_le_of_pos (le.refl _) zero_lt_one, have H : a + 1 > a, from lt_add_of_le_of_pos (le.refl _) zero_lt_one,
exists.intro _ H exists.intro _ H
-- the following theorems amount to four iffs, for <, ≤, ≥, >. -- the following theorems amount to four iffs, for <, ≤, ≥, >.
theorem mul_le_of_le_div (Hc : 0 < c) (H : a ≤ b / c) : a * c ≤ b := theorem mul_le_of_le_div (Hc : 0 < c) (H : a ≤ b / c) : a * c ≤ b :=
div_mul_cancel (ne.symm (ne_of_lt Hc)) ▸ mul_le_mul_of_nonneg_right H (le_of_lt Hc) div_mul_cancel (ne.symm (ne_of_lt Hc)) ▸ mul_le_mul_of_nonneg_right H (le_of_lt Hc)
theorem le_div_of_mul_le (Hc : 0 < c) (H : a * c ≤ b) : a ≤ b / c := theorem le_div_of_mul_le (Hc : 0 < c) (H : a * c ≤ b) : a ≤ b / c :=
calc calc
a = a * c * (1 / c) : mul_mul_div (ne.symm (ne_of_lt Hc)) a = a * c * (1 / c) : mul_mul_div (ne.symm (ne_of_lt Hc))
@ -120,12 +120,12 @@ section linear_ordered_field
theorem mul_le_of_ge_div_neg (Hc : c < 0) (H : a ≥ b / c) : a * c ≤ b := theorem mul_le_of_ge_div_neg (Hc : c < 0) (H : a ≥ b / c) : a * c ≤ b :=
div_mul_cancel (ne_of_lt Hc) ▸ mul_le_mul_of_nonpos_right H (le_of_lt Hc) div_mul_cancel (ne_of_lt Hc) ▸ mul_le_mul_of_nonpos_right H (le_of_lt Hc)
theorem ge_div_of_mul_le_neg (Hc : c < 0) (H : a * c ≤ b) : a ≥ b / c := theorem ge_div_of_mul_le_neg (Hc : c < 0) (H : a * c ≤ b) : a ≥ b / c :=
calc calc
a = a * c * (1 / c) : mul_mul_div (ne_of_lt Hc) a = a * c * (1 / c) : mul_mul_div (ne_of_lt Hc)
... ≥ b * (1 / c) : mul_le_mul_of_nonpos_right H (le_of_lt (div_neg_of_neg Hc)) ... ≥ b * (1 / c) : mul_le_mul_of_nonpos_right H (le_of_lt (div_neg_of_neg Hc))
... = b / c : div_eq_mul_one_div ... = b / c : div_eq_mul_one_div
theorem mul_lt_of_gt_div_neg (Hc : c < 0) (H : a > b / c) : a * c < b := theorem mul_lt_of_gt_div_neg (Hc : c < 0) (H : a > b / c) : a * c < b :=
div_mul_cancel (ne_of_lt Hc) ▸ mul_lt_mul_of_neg_right H Hc div_mul_cancel (ne_of_lt Hc) ▸ mul_lt_mul_of_neg_right H Hc
@ -158,16 +158,16 @@ section linear_ordered_field
... = (a * d - c * b) / (c * d) : div_sub_div Hc Hd ... = (a * d - c * b) / (c * d) : div_sub_div Hc Hd
... = (a * d - b * c) / (c * d) : mul.comm ... = (a * d - b * c) / (c * d) : mul.comm
theorem div_lt_div_of_mul_sub_mul_div_neg (Hc : c ≠ 0) (Hd : d ≠ 0) 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 := (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 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 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 _, 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 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) 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 := (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 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 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 _, 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 begin rewrite [zero_add at H3, neg_add_cancel_right at H3], exact H3 end
@ -181,7 +181,7 @@ section linear_ordered_field
exact Hb exact Hb
end end
theorem nonneg_div_of_nonneg_of_pos (Ha : 0 ≤ a) (Hb : 0 < b) : 0 ≤ a / b := theorem nonneg_div_of_nonneg_of_pos (Ha : 0 ≤ a) (Hb : 0 < b) : 0 ≤ a / b :=
begin begin
rewrite div_eq_mul_one_div, rewrite div_eq_mul_one_div,
apply mul_nonneg, apply mul_nonneg,
@ -191,7 +191,7 @@ section linear_ordered_field
exact Hb exact Hb
end end
theorem neg_div_of_neg_of_pos (Ha : a < 0) (Hb : 0 < b) : a / b < 0:= theorem neg_div_of_neg_of_pos (Ha : a < 0) (Hb : 0 < b) : a / b < 0:=
begin begin
rewrite div_eq_mul_one_div, rewrite div_eq_mul_one_div,
apply mul_neg_of_neg_of_pos, apply mul_neg_of_neg_of_pos,
@ -200,7 +200,7 @@ section linear_ordered_field
exact Hb exact Hb
end end
theorem nonpos_div_of_nonpos_of_pos (Ha : a ≤ 0) (Hb : 0 < b) : a / b ≤ 0 := theorem nonpos_div_of_nonpos_of_pos (Ha : a ≤ 0) (Hb : 0 < b) : a / b ≤ 0 :=
begin begin
rewrite div_eq_mul_one_div, rewrite div_eq_mul_one_div,
apply mul_nonpos_of_nonpos_of_nonneg, apply mul_nonpos_of_nonpos_of_nonneg,
@ -210,7 +210,7 @@ section linear_ordered_field
exact Hb exact Hb
end end
theorem neg_div_of_pos_of_neg (Ha : 0 < a) (Hb : b < 0) : a / b < 0 := theorem neg_div_of_pos_of_neg (Ha : 0 < a) (Hb : b < 0) : a / b < 0 :=
begin begin
rewrite div_eq_mul_one_div, rewrite div_eq_mul_one_div,
apply mul_neg_of_pos_of_neg, apply mul_neg_of_pos_of_neg,
@ -219,7 +219,7 @@ section linear_ordered_field
exact Hb exact Hb
end end
theorem nonpos_div_of_nonneg_of_neg (Ha : 0 ≤ a) (Hb : b < 0) : a / b ≤ 0 := theorem nonpos_div_of_nonneg_of_neg (Ha : 0 ≤ a) (Hb : b < 0) : a / b ≤ 0 :=
begin begin
rewrite div_eq_mul_one_div, rewrite div_eq_mul_one_div,
apply mul_nonpos_of_nonneg_of_nonpos, apply mul_nonpos_of_nonneg_of_nonpos,
@ -229,7 +229,7 @@ section linear_ordered_field
exact Hb exact Hb
end end
theorem pos_div_of_neg_of_neg (Ha : a < 0) (Hb : b < 0) : 0 < a / b := theorem pos_div_of_neg_of_neg (Ha : a < 0) (Hb : b < 0) : 0 < a / b :=
begin begin
rewrite div_eq_mul_one_div, rewrite div_eq_mul_one_div,
apply mul_pos_of_neg_of_neg, apply mul_pos_of_neg_of_neg,
@ -237,9 +237,9 @@ section linear_ordered_field
apply div_neg_of_neg, apply div_neg_of_neg,
exact Hb exact Hb
end end
theorem nonneg_div_of_nonpos_of_neg (Ha : a ≤ 0) (Hb : b < 0) : 0 ≤ a / b :=
theorem nonneg_div_of_nonpos_of_neg (Ha : a ≤ 0) (Hb : b < 0) : 0 ≤ a / b :=
begin begin
rewrite div_eq_mul_one_div, rewrite div_eq_mul_one_div,
apply mul_nonneg_of_nonpos_of_nonpos, apply mul_nonneg_of_nonpos_of_nonpos,
@ -249,13 +249,13 @@ section linear_ordered_field
exact Hb exact Hb
end end
theorem div_lt_div_of_lt_of_pos (H : a < b) (Hc : 0 < c) : a / c < b / c := 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) 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 := 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) 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 end linear_ordered_field
structure discrete_linear_ordered_field [class] (A : Type) extends linear_ordered_field A, structure discrete_linear_ordered_field [class] (A : Type) extends linear_ordered_field A,
@ -263,36 +263,36 @@ structure discrete_linear_ordered_field [class] (A : Type) extends linear_ordere
(inv_zero : inv zero = zero) (inv_zero : inv zero = zero)
section discrete_linear_ordered_field section discrete_linear_ordered_field
variable {A : Type} variable {A : Type}
variables [s : discrete_linear_ordered_field A] {a b c : A} variables [s : discrete_linear_ordered_field A] {a b c : A}
include s include s
theorem dec_eq_of_dec_lt : ∀ x y : A, decidable (x = y) := theorem dec_eq_of_dec_lt : ∀ x y : A, decidable (x = y) :=
take x y, take x y,
decidable.by_cases decidable.by_cases
(assume H : x < y, decidable.inr (ne_of_lt H)) (assume H : x < y, decidable.inr (ne_of_lt H))
(assume H : ¬ x < y, (assume H : ¬ x < y,
decidable.by_cases decidable.by_cases
(assume H' : y < x, decidable.inr (ne.symm (ne_of_lt H'))) (assume H' : y < x, decidable.inr (ne.symm (ne_of_lt H')))
(assume H' : ¬ y < x, (assume H' : ¬ y < x,
decidable.inl (le.antisymm (le_of_not_lt H') (le_of_not_lt H)))) decidable.inl (le.antisymm (le_of_not_lt H') (le_of_not_lt H))))
definition discrete_linear_ordered_field.to_discrete_field [instance] [reducible] [coercion] definition discrete_linear_ordered_field.to_discrete_field [instance] [reducible] [coercion]
[s : discrete_linear_ordered_field A] : discrete_field A := [s : discrete_linear_ordered_field A] : discrete_field A :=
⦃ discrete_field, s, decidable_equality := dec_eq_of_dec_lt⦄ ⦃ discrete_field, s, has_decidable_eq := dec_eq_of_dec_lt⦄
theorem pos_of_div_pos (H : 0 < 1 / a) : 0 < a := theorem pos_of_div_pos (H : 0 < 1 / a) : 0 < a :=
have H1 : 0 < 1 / (1 / a), from div_pos_of_pos H, have H1 : 0 < 1 / (1 / a), from div_pos_of_pos H,
have H2 : 1 / a ≠ 0, from have H2 : 1 / a ≠ 0, from
(assume H3 : 1 / a = 0, (assume H3 : 1 / a = 0,
have H4 : 1 / (1 / a) = 0, from H3⁻¹ ▸ div_zero, have H4 : 1 / (1 / a) = 0, from H3⁻¹ ▸ div_zero,
absurd H4 (ne.symm (ne_of_lt H1))), absurd H4 (ne.symm (ne_of_lt H1))),
(div_div (ne_zero_of_one_div_ne_zero H2)) ▸ H1 (div_div (ne_zero_of_one_div_ne_zero H2)) ▸ H1
theorem neg_of_div_neg (H : 1 / a < 0) : a < 0 := theorem neg_of_div_neg (H : 1 / a < 0) : a < 0 :=
have H1 : 0 < - (1 / a), from neg_pos_of_neg H, 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 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 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, have H3 : 0 < -a, from pos_of_div_pos H2,
@ -308,8 +308,8 @@ section discrete_linear_ordered_field
... = a * (1 / a) : div_eq_mul_one_div ... = 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 * (1 / b) : ordered_ring.mul_le_mul_of_nonneg_left Hl (le_of_lt H)
... = a / b : div_eq_mul_one_div ... = a / b : div_eq_mul_one_div
), le_of_one_le_div Hb H' ), le_of_one_le_div Hb H'
theorem le_of_div_le_neg (H : b < 0) (Hl : 1 / a ≤ 1 / b) : b ≤ a := 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 have Ha : a ≠ 0, from ne_of_lt (neg_of_div_neg (calc
@ -339,17 +339,17 @@ section discrete_linear_ordered_field
have H1 : b ≤ a, from le_of_div_le_neg H (le_of_lt Hl), have H1 : b ≤ a, from le_of_div_le_neg H (le_of_lt Hl),
have Hn : b ≠ a, from have Hn : b ≠ a, from
(assume Hn' : b = a, (assume Hn' : b = a,
have Hl' : 1 / a = 1 / b, from Hn' ▸ refl _, have Hl' : 1 / a = 1 / b, from Hn' ▸ refl _,
absurd Hl' (ne_of_lt Hl)), absurd Hl' (ne_of_lt Hl)),
lt_of_le_of_ne H1 Hn lt_of_le_of_ne H1 Hn
theorem div_lt_div_of_lt (Ha : 0 < a) (H : a < b) : 1 / b < 1 / a := theorem div_lt_div_of_lt (Ha : 0 < a) (H : a < b) : 1 / b < 1 / a :=
lt_of_not_le lt_of_not_le
(assume H', (assume H',
absurd H (not_lt_of_le (le_of_div_le Ha 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 := theorem div_le_div_of_le (Ha : 0 < a) (H : a ≤ b) : 1 / b ≤ 1 / a :=
le_of_not_lt le_of_not_lt
(assume H', (assume H',
absurd H (not_le_of_lt (lt_of_div_lt Ha H'))) absurd H (not_le_of_lt (lt_of_div_lt Ha H')))