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feat(library/unifier): do not fire type class resolution as last resort when type contains metavariables

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see discussion at #604
This commit is contained in:
Leonardo de Moura 2015-05-18 15:45:23 -07:00
parent 1fbc85f6df
commit 19361f0196
33 changed files with 83 additions and 79 deletions
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2
hott/algebra/category/functor.hlean
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@ -44,7 +44,7 @@ namespace functor
protected definition id [reducible] {C : Precategory} : functor C C :=
mk (λa, a) (λ a b f, f) (λ a, idp) (λ a b c f g, idp)
protected definition ID [reducible] (C : Precategory) : functor C C := id
protected definition ID [reducible] (C : Precategory) : functor C C := @functor.id C
definition functor_mk_eq' {F₁ F₂ : C → D} {H₁ : Π(a b : C), hom a b → hom (F₁ a) (F₁ b)}
{H₂ : Π(a b : C), hom a b → hom (F₂ a) (F₂ b)} (id₁ id₂ comp₁ comp₂)

2
hott/algebra/category/iso.hlean
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@ -231,7 +231,7 @@ namespace iso
(λ c g h H,
calc
g = g ∘ id : by rewrite id_right
... = g ∘ f ∘ section_of f : by rewrite -comp_section
... = g ∘ f ∘ section_of f : by rewrite -(comp_section f)
... = h ∘ f ∘ section_of f : by rewrite [assoc, H, -assoc]
... = h ∘ id : by rewrite comp_section
... = h : by rewrite id_right)

2
hott/algebra/category/nat_trans.hlean
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@ -36,7 +36,7 @@ namespace nat_trans
mk (λa, id) (λa b f, !id_right ⬝ !id_left⁻¹)
protected definition ID [reducible] {C D : Precategory} (F : functor C D) : nat_trans F F :=
id
(@id C D F)
definition nat_trans_mk_eq {η₁ η₂ : Π (a : C), hom (F a) (G a)}
(nat₁ : Π (a b : C) (f : hom a b), G f ∘ η₁ a = η₁ b ∘ F f)

17
hott/algebra/field.hlean
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@ -65,20 +65,21 @@ section division_ring
definition div_self (H : a ≠ 0) : a / a = 1 := mul_inv_cancel H
definition one_div_one : 1 / 1 = 1 := div_self (ne.symm zero_ne_one)
definition one_div_one : 1 / 1 = (1:A) :=
div_self (ne.symm zero_ne_one)
definition mul_div_assoc : (a * b) / c = a * (b / c) := !mul.assoc
definition one_div_ne_zero (H : a ≠ 0) : 1 / a ≠ 0 :=
assume H2 : 1 / a = 0,
have C1 : 0 = 1, from inverse (by rewrite [-(mul_one_div_cancel H), H2, mul_zero]),
have C1 : 0 = (1:A), from inverse (by rewrite [-(mul_one_div_cancel H), H2, mul_zero]),
absurd C1 zero_ne_one
-- definition 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
definition inv_one_is_one : 1⁻¹ = 1 :=
definition inv_one_is_one : 1⁻¹ = (1:A) :=
by rewrite [-mul_one, (inv_mul_cancel (ne.symm zero_ne_one))]
definition div_one : a / 1 = a :=
@ -100,8 +101,8 @@ section division_ring
-- make "left" and "right" versions?
definition eq_one_div_of_mul_eq_one (H : a * b = 1) : b = 1 / a :=
have H2 : a ≠ 0, from
(assume A : a = 0,
have B : 0 = 1, by rewrite [-(zero_mul b), -A, H],
(assume aeq0 : a = 0,
have B : 0 = (1:A), by rewrite [-(zero_mul b), -aeq0, H],
absurd B zero_ne_one),
show b = 1 / a, from inverse (calc
1 / a = (1 / a) * 1 : mul_one
@ -131,7 +132,7 @@ section division_ring
rewrite [mul.assoc, -(mul.assoc a), (mul_one_div_cancel Ha), one_mul, (mul_one_div_cancel Hb)],
eq_one_div_of_mul_eq_one H
definition one_div_neg_one_eq_neg_one : 1 / (-1) = -1 :=
definition one_div_neg_one_eq_neg_one : (1:A) / (-1) = -1 :=
have H : (-1) * (-1) = 1, by rewrite [-neg_eq_neg_one_mul, neg_neg],
inverse (eq_one_div_of_mul_eq_one H)
@ -140,7 +141,7 @@ section division_ring
(assume H2 : -1 = 0, absurd (inverse (calc
1 = -(-1) : neg_neg
... = -0 : H2
... = 0 : neg_zero)) zero_ne_one),
... = (0:A) : neg_zero)) zero_ne_one),
calc
1 / (- a) = 1 / ((-1) * a) : neg_eq_neg_one_mul
... = (1 / a) * (1 / (- 1)) : one_div_mul_one_div H H1
@ -339,7 +340,7 @@ section discrete_field
definition inv_zero : 0⁻¹ = (0 : A) := !discrete_field.inv_zero
definition one_div_zero : 1 / 0 = 0 :=
definition one_div_zero : 1 / 0 = (0:A) :=
calc
1 / 0 = 1 * 0⁻¹ : refl
... = 1 * 0 : discrete_field.inv_zero A

4
hott/algebra/group.hlean
View file

@ -163,7 +163,7 @@ section group
theorem inv_eq_of_mul_eq_one {a b : A} (H : a * b = 1) : a⁻¹ = b :=
by rewrite [-mul_one a⁻¹, -H, inv_mul_cancel_left]
theorem inv_one : 1⁻¹ = 1 := inv_eq_of_mul_eq_one (one_mul 1)
theorem inv_one : 1⁻¹ = (1:A) := inv_eq_of_mul_eq_one (one_mul 1)
theorem inv_inv (a : A) : (a⁻¹)⁻¹ = a := inv_eq_of_mul_eq_one (mul.left_inv a)
@ -281,7 +281,7 @@ section add_group
theorem neg_eq_of_add_eq_zero {a b : A} (H : a + b = 0) : -a = b :=
by rewrite [-add_zero, -H, neg_add_cancel_left]
theorem neg_zero : -0 = 0 := neg_eq_of_add_eq_zero (zero_add 0)
theorem neg_zero : -0 = (0:A) := neg_eq_of_add_eq_zero (zero_add 0)
theorem neg_neg (a : A) : -(-a) = a := neg_eq_of_add_eq_zero (add.left_inv a)

2
hott/algebra/ordered_group.hlean
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@ -424,7 +424,7 @@ section
definition abs_of_neg (H : a < 0) : abs a = -a := if_neg (not_le_of_lt H)
definition abs_zero : abs 0 = 0 := abs_of_nonneg (le.refl _)
definition abs_zero : abs 0 = (0:A) := abs_of_nonneg (le.refl _)
definition abs_of_nonpos (H : a ≤ 0) : abs a = -a :=
decidable.by_cases

12
hott/algebra/ordered_ring.hlean
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@ -339,8 +339,8 @@ section
(assume H : a ≥ 0, mul_nonneg H H)
(assume H : a ≤ 0, mul_nonneg_of_nonpos_of_nonpos H H)
definition zero_le_one : 0 ≤ 1 := one_mul 1 ▸ mul_self_nonneg (1 : A)
definition zero_lt_one : 0 < 1 := lt_of_le_of_ne zero_le_one zero_ne_one
definition zero_le_one : 0 ≤ (1:A) := one_mul 1 ▸ mul_self_nonneg (1 : A)
definition zero_lt_one : 0 < (1:A) := lt_of_le_of_ne zero_le_one zero_ne_one
definition pos_and_pos_or_neg_and_neg_of_mul_pos {a b : A} (Hab : a * b > 0) :
(a > 0 × b > 0) ⊎ (a < 0 × b < 0) :=
@ -400,13 +400,13 @@ section
definition sign_of_neg (H : a < 0) : sign a = -1 := lt.cases_of_lt H
definition sign_zero : sign 0 = 0 := lt.cases_of_eq rfl
definition sign_zero : sign 0 = (0:A) := lt.cases_of_eq rfl
definition sign_of_pos (H : a > 0) : sign a = 1 := lt.cases_of_gt H
definition sign_one : sign 1 = 1 := sign_of_pos zero_lt_one
definition sign_one : sign 1 = (1:A) := sign_of_pos zero_lt_one
definition sign_neg_one : sign (-1) = -1 := sign_of_neg (neg_neg_of_pos zero_lt_one)
definition sign_neg_one : sign (-1) = -(1:A) := sign_of_neg (neg_neg_of_pos zero_lt_one)
definition sign_sign (a : A) : sign (sign a) = sign a :=
lt.by_cases
@ -454,7 +454,7 @@ section
have H2 : -1 = 1, from H⁻¹ ⬝ (sign_of_pos H1),
absurd ((eq_zero_of_neg_eq H2)⁻¹) zero_ne_one)
(assume H1 : 0 = a,
have H2 : 0 = -1,
have H2 : (0:A) = -1,
begin
rewrite [-H1 at H, sign_zero at H],
exact H

2
hott/algebra/ring.hlean
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@ -230,7 +230,7 @@ section
... = 0 : mul_zero,
inverse (neg_eq_of_add_eq_zero H)
definition mul_ne_zero_imp_ne_zero {a b} (H : a * b ≠ 0) : a ≠ 0 × b ≠ 0 :=
definition mul_ne_zero_imp_ne_zero {a b : A} (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],

2
hott/hit/trunc.hlean
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@ -47,7 +47,7 @@ namespace trunc
equiv.mk tr _
definition is_trunc_of_is_equiv_tr [H : is_equiv (@tr n A)] : is_trunc n A :=
is_trunc_is_equiv_closed n tr⁻¹
is_trunc_is_equiv_closed n (@tr n _)⁻¹
definition untrunc_of_is_trunc [reducible] [H : is_trunc n A] : trunc n A → A :=
tr⁻¹

2
hott/init/trunc.hlean
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@ -233,7 +233,7 @@ namespace is_trunc
definition is_trunc_is_equiv_closed (n : trunc_index) (f : A → B) [H : is_equiv f]
[HA : is_trunc n A] : is_trunc n B :=
trunc_index.rec_on n
(λA (HA : is_contr A) B f (H : is_equiv f), !is_contr_is_equiv_closed)
(λA (HA : is_contr A) B f (H : is_equiv f), is_contr_is_equiv_closed f)
(λn IH A (HA : is_trunc n.+1 A) B f (H : is_equiv f), @is_trunc_succ_intro _ _ (λ x y : B,
IH (f⁻¹ x = f⁻¹ y) _ (x = y) (ap f⁻¹)⁻¹ !is_equiv_inv))
A HA B f H

2
hott/types/eq.hlean
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@ -134,7 +134,7 @@ namespace eq
definition equiv_ap (f : A → B) [H : is_equiv f] (a1 a2 : A)
: (a1 = a2) ≃ (f a1 = f a2) :=
equiv.mk _ _
equiv.mk (ap f) _
/- Path operations are equivalences -/

2
library/algebra/category/basic.lean
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@ -23,7 +23,7 @@ namespace category
variables {a b c d : ob}
include C
definition compose := comp
definition compose := @comp ob _
definition id [reducible] {a : ob} : hom a a := ID a

6
library/algebra/category/functor.lean
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@ -47,11 +47,11 @@ namespace functor
protected definition id [reducible] {C : Category} : functor C C :=
mk (λa, a) (λ a b f, f) (λ a, rfl) (λ a b c f g, rfl)
protected definition ID [reducible] (C : Category) : functor C C := id
protected definition ID [reducible] (C : Category) : functor C C := @id C
protected theorem id_left (F : functor C D) : id ∘f F = F :=
protected theorem id_left (F : functor C D) : (@id D) ∘f F = F :=
functor.rec (λ obF homF idF compF, dcongr_arg4 mk rfl rfl !proof_irrel !proof_irrel) F
protected theorem id_right (F : functor C D) : F ∘f id = F :=
protected theorem id_right (F : functor C D) : F ∘f (@id C) = F :=
functor.rec (λ obF homF idF compF, dcongr_arg4 mk rfl rfl !proof_irrel !proof_irrel) F
end functor

7
library/algebra/category/morphism.lean
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@ -137,7 +137,8 @@ namespace morphism
theorem refl (a : ob) : a ≅ a := mk id
theorem symm ⦃a b : ob⦄ (H : a ≅ b) : b ≅ a := mk (inverse (iso H))
theorem trans ⦃a b c : ob⦄ (H1 : a ≅ b) (H2 : b ≅ c) : a ≅ c := mk (iso H2 ∘ iso H1)
theorem is_equivalence_eq [instance] (T : Type) : is_equivalence isomorphic :=
theorem is_equivalence_eq [instance] (T : Type) : is_equivalence (isomorphic : ob → ob → Type) :=
is_equivalence.mk refl symm trans
end isomorphic
@ -160,7 +161,7 @@ namespace morphism
is_mono.mk
(λ c g h H, calc
g = id ∘ g : by rewrite id_left
... = (retraction_of f ∘ f) ∘ g : by rewrite -retraction_compose
... = (retraction_of f ∘ f) ∘ g : by rewrite -(retraction_compose f)
... = (retraction_of f ∘ f) ∘ h : by rewrite [-assoc, H, -assoc]
... = id ∘ h : by rewrite retraction_compose
... = h : by rewrite id_left)
@ -169,7 +170,7 @@ namespace morphism
is_epi.mk
(λ c g h H, calc
g = g ∘ id : by rewrite id_right
... = g ∘ f ∘ section_of f : by rewrite -compose_section
... = g ∘ f ∘ section_of f : by rewrite -(compose_section f)
... = h ∘ f ∘ section_of f : by rewrite [assoc, H, -assoc]
... = h ∘ id : by rewrite compose_section
... = h : by rewrite id_right)

6
library/algebra/category/natural_transformation.lean
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@ -43,11 +43,11 @@ namespace natural_transformation
protected definition id {C D : Category} {F : functor C D} : natural_transformation F F :=
mk (λa, id) (λa b f, !id_right ⬝ symm !id_left)
protected definition ID {C D : Category} (F : functor C D) : natural_transformation F F := id
protected definition ID {C D : Category} (F : functor C D) : natural_transformation F F := natural_transformation.id
protected theorem id_left (η : F ⟹ G) : natural_transformation.compose id η = η :=
protected theorem id_left (η : F ⟹ G) : natural_transformation.compose natural_transformation.id η = η :=
natural_transformation.rec (λf H, dcongr_arg2 mk (funext (take x, !id_left)) !proof_irrel) η
protected theorem id_right (η : F ⟹ G) : natural_transformation.compose η id = η :=
protected theorem id_right (η : F ⟹ G) : natural_transformation.compose η natural_transformation.id = η :=
natural_transformation.rec (λf H, dcongr_arg2 mk (funext (take x, !id_right)) !proof_irrel) η
end natural_transformation

22
library/algebra/field.lean
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@ -64,21 +64,21 @@ section division_ring
theorem div_self (H : a ≠ 0) : a / a = 1 := mul_inv_cancel H
theorem one_div_one : 1 / 1 = 1 := div_self (ne.symm zero_ne_one)
theorem one_div_one : 1 / 1 = (1:A) := div_self (ne.symm zero_ne_one)
theorem mul_div_assoc : (a * b) / c = a * (b / c) := !mul.assoc
theorem one_div_ne_zero (H : a ≠ 0) : 1 / a ≠ 0 :=
assume H2 : 1 / a = 0,
have C1 : 0 = 1, from symm (by rewrite [-(mul_one_div_cancel H), H2, mul_zero]),
have C1 : 0 = (1:A), 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
-- the analogue in group is called inv_one
theorem inv_one_is_one : 1⁻¹ = 1 :=
by rewrite [-mul_one, (inv_mul_cancel (ne.symm zero_ne_one))]
theorem inv_one_is_one : 1⁻¹ = (1:A) :=
by rewrite [-mul_one, inv_mul_cancel (ne.symm (@zero_ne_one A _))]
theorem div_one : a / 1 = a :=
by rewrite [↑divide, inv_one_is_one, mul_one]
@ -99,8 +99,8 @@ section division_ring
-- make "left" and "right" versions?
theorem eq_one_div_of_mul_eq_one (H : a * b = 1) : b = 1 / a :=
have H2 : a ≠ 0, from
(assume A : a = 0,
have B : 0 = 1, by rewrite [-(zero_mul b), -A, H],
(assume aeq0 : a = 0,
have B : 0 = (1:A), by rewrite [-(zero_mul b), -aeq0, H],
absurd B zero_ne_one),
show b = 1 / a, from symm (calc
1 / a = (1 / a) * 1 : mul_one
@ -130,8 +130,8 @@ section division_ring
rewrite [mul.assoc, -(mul.assoc a), (mul_one_div_cancel Ha), one_mul, (mul_one_div_cancel Hb)],
eq_one_div_of_mul_eq_one H
theorem one_div_neg_one_eq_neg_one : 1 / (-1) = -1 :=
have H : (-1) * (-1) = 1, by rewrite [-neg_eq_neg_one_mul, neg_neg],
theorem one_div_neg_one_eq_neg_one : (1:A) / (-1) = -1 :=
have H : (-1) * (-1) = (1:A), by rewrite [-neg_eq_neg_one_mul, neg_neg],
symm (eq_one_div_of_mul_eq_one H)
theorem one_div_neg_eq_neg_one_div (H : a ≠ 0) : 1 / (- a) = - (1 / a) :=
@ -139,7 +139,7 @@ section division_ring
(assume H2 : -1 = 0, absurd (symm (calc
1 = -(-1) : neg_neg
... = -0 : H2
... = 0 : neg_zero)) zero_ne_one),
... = (0:A) : neg_zero)) zero_ne_one),
calc
1 / (- a) = 1 / ((-1) * a) : neg_eq_neg_one_mul
... = (1 / a) * (1 / (- 1)) : one_div_mul_one_div H H1
@ -336,9 +336,9 @@ 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 inv_zero : 0⁻¹ = (0:A) := !discrete_field.inv_zero
theorem one_div_zero : 1 / 0 = 0 :=
theorem one_div_zero : 1 / 0 = (0:A) :=
calc
1 / 0 = 1 * 0⁻¹ : refl
... = 1 * 0 : discrete_field.inv_zero A

4
library/algebra/group.lean
View file

@ -174,7 +174,7 @@ section group
theorem inv_eq_of_mul_eq_one {a b : A} (H : a * b = 1) : a⁻¹ = b :=
by rewrite [-mul_one a⁻¹, -H, inv_mul_cancel_left]
theorem inv_one : 1⁻¹ = 1 := inv_eq_of_mul_eq_one (one_mul 1)
theorem inv_one : 1⁻¹ = (1 : A) := inv_eq_of_mul_eq_one (one_mul 1)
theorem inv_inv (a : A) : (a⁻¹)⁻¹ = a := inv_eq_of_mul_eq_one (mul.left_inv a)
@ -292,7 +292,7 @@ section add_group
theorem neg_eq_of_add_eq_zero {a b : A} (H : a + b = 0) : -a = b :=
by rewrite [-add_zero, -H, neg_add_cancel_left]
theorem neg_zero : -0 = 0 := neg_eq_of_add_eq_zero (zero_add 0)
theorem neg_zero : -0 = (0 : A) := neg_eq_of_add_eq_zero (zero_add 0)
theorem neg_neg (a : A) : -(-a) = a := neg_eq_of_add_eq_zero (add.left_inv a)

8
library/algebra/group_bigops.lean
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@ -62,7 +62,7 @@ section monoid
by rewrite [union_eq_append d, Prodl_append]
end deceqA
theorem Prodl_one : ∀(l : list A), Prodl l (λ x, 1) = 1
theorem Prodl_one : ∀(l : list A), Prodl l (λ x, 1) = (1:B)
| [] := rfl
| (a::l) := by rewrite [Prodl_cons, Prodl_one, mul_one]
end monoid
@ -139,7 +139,7 @@ section comm_monoid
by rewrite [Prod_insert_of_not_mem f H1, Prod_insert_of_not_mem g H1, IH H3, H4])
end deceqA
theorem Prod_one (s : finset A) : Prod s (λ x, 1) = 1 :=
theorem Prod_one (s : finset A) : Prod s (λ x, 1) = (1:B) :=
quot.induction_on s (take u, !Prodl_one)
end comm_monoid
@ -169,7 +169,7 @@ section add_monoid
Suml (union l₁ l₂) f = Suml l₁ f + Suml l₂ f := Prodl_union f d
end deceqA
theorem Suml_zero (l : list A) : Suml l (λ x, 0) = 0 := Prodl_one l
theorem Suml_zero (l : list A) : Suml l (λ x, 0) = (0:B) := Prodl_one l
end add_monoid
section add_comm_monoid
@ -209,7 +209,7 @@ section add_comm_monoid
Sum s f = Sum s g := Prod_ext H
end deceqA
theorem Sum_zero (s : finset A) : Sum s (λ x, 0) = 0 := Prod_one s
theorem Sum_zero (s : finset A) : Sum s (λ x, 0) = (0:B) := Prod_one s
end add_comm_monoid
end algebra

2
library/algebra/group_power.lean
View file

@ -37,7 +37,7 @@ theorem pow_succ' (a : A) : ∀n, a^(succ n) = a * a^n
| 0 := by rewrite [pow_succ, *pow_zero, one_mul, mul_one]
| (succ n) := by rewrite [pow_succ, pow_succ' at {1}, pow_succ, mul.assoc]
theorem one_pow : ∀ n : ℕ, 1^n = 1
theorem one_pow : ∀ n : ℕ, 1^n = (1:A)
| 0 := rfl
| (succ n) := by rewrite [pow_succ, mul_one, one_pow]

2
library/algebra/ordered_group.lean
View file

@ -424,7 +424,7 @@ section
theorem abs_of_neg (H : a < 0) : abs a = -a := if_neg (not_le_of_lt H)
theorem abs_zero : abs 0 = 0 := abs_of_nonneg (le.refl _)
theorem abs_zero : abs 0 = (0:A) := abs_of_nonneg (le.refl _)
theorem abs_of_nonpos (H : a ≤ 0) : abs a = -a :=
decidable.by_cases

16
library/algebra/ordered_ring.lean
View file

@ -337,8 +337,8 @@ section
(assume H : a ≥ 0, mul_nonneg H H)
(assume H : a ≤ 0, mul_nonneg_of_nonpos_of_nonpos H H)
theorem zero_le_one : 0 ≤ 1 := one_mul 1 ▸ mul_self_nonneg 1
theorem zero_lt_one : 0 < 1 := lt_of_le_of_ne zero_le_one zero_ne_one
theorem zero_le_one : 0 ≤ (1:A) := one_mul 1 ▸ mul_self_nonneg 1
theorem zero_lt_one : 0 < (1:A) := lt_of_le_of_ne zero_le_one zero_ne_one
theorem pos_and_pos_or_neg_and_neg_of_mul_pos {a b : A} (Hab : a * b > 0) :
(a > 0 ∧ b > 0) ∨ (a < 0 ∧ b < 0) :=
@ -369,7 +369,7 @@ section
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 :=
theorem gt_of_mul_lt_mul_neg_left {a b c : A} (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
@ -378,7 +378,7 @@ section
... = (-c) * a : neg_mul_eq_neg_mul,
lt_of_mul_lt_mul_left H3 nhc
theorem zero_gt_neg_one : -1 < 0 :=
theorem zero_gt_neg_one : -1 < (0:A) :=
neg_zero ▸ (neg_lt_neg zero_lt_one)
end
@ -398,13 +398,13 @@ section
theorem sign_of_neg (H : a < 0) : sign a = -1 := lt.cases_of_lt H
theorem sign_zero : sign 0 = 0 := lt.cases_of_eq rfl
theorem sign_zero : sign 0 = (0:A) := lt.cases_of_eq rfl
theorem sign_of_pos (H : a > 0) : sign a = 1 := lt.cases_of_gt H
theorem sign_one : sign 1 = 1 := sign_of_pos zero_lt_one
theorem sign_one : sign 1 = (1:A) := sign_of_pos zero_lt_one
theorem sign_neg_one : sign (-1) = -1 := sign_of_neg (neg_neg_of_pos zero_lt_one)
theorem sign_neg_one : sign (-1) = -(1:A) := sign_of_neg (neg_neg_of_pos zero_lt_one)
theorem sign_sign (a : A) : sign (sign a) = sign a :=
lt.by_cases
@ -452,7 +452,7 @@ section
have H2 : -1 = 1, from H⁻¹ ⬝ (sign_of_pos H1),
absurd ((eq_zero_of_neg_eq H2)⁻¹) zero_ne_one)
(assume H1 : 0 = a,
have H2 : 0 = -1,
have H2 : (0:A) = -1,
begin
rewrite [-H1 at H, sign_zero at H],
exact H

4
library/algebra/ring.lean
View file

@ -39,7 +39,7 @@ theorem mul_zero [s : mul_zero_class A] (a : A) : a * 0 = 0 := !mul_zero_class.m
structure zero_ne_one_class [class] (A : Type) extends has_zero A, has_one A :=
(zero_ne_one : zero ≠ one)
theorem zero_ne_one [s: zero_ne_one_class A] : 0 ≠ 1 := @zero_ne_one_class.zero_ne_one A s
theorem zero_ne_one [s: zero_ne_one_class A] : 0 ≠ (1:A) := @zero_ne_one_class.zero_ne_one A s
/- semiring -/
@ -229,7 +229,7 @@ section
... = 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 :=
theorem mul_ne_zero_imp_ne_zero {a b : A} (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],

4
library/data/finset/bigops.lean
View file

@ -54,7 +54,7 @@ section deceqA
Unionl (list.insert a l) f = f a ∪ Unionl l f := algebra.Prodl_insert_of_not_mem f H
theorem Unionl_union {l₁ l₂ : list A} (f : A → finset B) (d : list.disjoint l₁ l₂) :
Unionl (list.union l₁ l₂) f = Unionl l₁ f ∪ Unionl l₂ f := algebra.Prodl_union f d
theorem Unionl_empty (l : list A) : Unionl l (λ x, ∅) = ∅ := algebra.Prodl_one l
theorem Unionl_empty (l : list A) : Unionl l (λ x, ∅) = (∅ : finset B) := algebra.Prodl_one l
end deceqA
theorem Union_empty (f : A → finset B) : Union ∅ f = ∅ := algebra.Prod_empty f
@ -70,7 +70,7 @@ section deceqA
Union (s₁ ∪ s₂) f = Union s₁ f ∪ Union s₂ f := algebra.Prod_union f disj
theorem Union_ext {s : finset A} {f g : A → finset B} (H : ∀x, x ∈ s → f x = g x) :
Union s f = Union s g := algebra.Prod_ext H
theorem Union_empty' (s : finset A) : Union s (λ x, ∅) = ∅ := algebra.Prod_one s
theorem Union_empty' (s : finset A) : Union s (λ x, ∅) = (∅ : finset B) := algebra.Prod_one s
-- this will eventually be an instance of something more general
theorem inter_Union (s : finset B) (t : finset A) (f : A → finset B) :

4
library/data/finset/card.lean
View file

@ -120,8 +120,8 @@ finset.induction_on s
H1 !mem_insert (!mem_insert_of_mem xs') anex,
assert H8 : f a ∩ (⋃ (x : A) ∈ s', f x) = ∅, from
calc
f a ∩ (⋃ (x : A) ∈ s', f x) = (⋃ (x : A) ∈ s', f a ∩ f x) : inter_Union
... = (⋃ (x : A) ∈ s', ∅) : Union_ext H7
f a ∩ (⋃ (x : A) ∈ s', f x) = (⋃ (x : A) ∈ s', f a ∩ f x) : inter_Union
... = (⋃ (x : A) ∈ s', ∅) : Union_ext H7
... = ∅ : Union_empty',
by rewrite [Union_insert_of_not_mem _ H, Sum_insert_of_not_mem _ H,
card_union_of_disjoint H8, H6])

8
library/data/finset/to_set.lean
View file

@ -15,18 +15,18 @@ variable [deceq : decidable_eq A]
include deceq
definition to_set (s : finset A) : set A := λx, x ∈ s
abbreviation ts := to_set -- until coercion is working
abbreviation ts := @to_set A _ -- until coercion is working
variables (s t : finset A) (x : A)
/- operations -/
theorem mem_to_set_empty : (x ∈ ts ∅) = (x ∈ ∅) := rfl
theorem to_set_empty : ts ∅ = ∅ := rfl
theorem to_set_empty : ts ∅ = (∅ : set A) := rfl
theorem mem_to_set_univ [h : fintype A] : (x ∈ ts univ) = (x ∈ set.univ) :=
propext (iff.intro (assume H, trivial) (assume H, !mem_univ))
theorem to_set_univ [h : fintype A] : ts univ = set.univ := funext (λ x, !mem_to_set_univ)
theorem to_set_univ [h : fintype A] : ts univ = (set.univ : set A) := funext (λ x, !mem_to_set_univ)
theorem mem_to_set_union : (x ∈ ts (s ∪ t)) = (x ∈ ts s ∪ ts t) := !finset.mem_union_eq
theorem to_set_union : ts (s ∪ t) = ts s ∪ ts t := funext (λ x, !mem_to_set_union)
@ -51,7 +51,7 @@ theorem to_set_image {B : Type} [h : decidable_eq B] (f : A → B) (s : finset A
theorem mem_eq_mem_to_set : (x ∈ s) = (x ∈ ts s) := rfl
definition decidable_mem_to_set [instance] (x s) : decidable (x ∈ ts s) :=
definition decidable_mem_to_set [instance] (x : A) (s : finset A) : decidable (x ∈ ts s) :=
decidable_of_decidable_of_eq _ !mem_eq_mem_to_set
theorem eq_eq_to_set_eq : (s = t) = (ts s = ts t) :=

2
library/data/list/set.lean
View file

@ -325,7 +325,7 @@ definition erase_dup [H : decidable_eq A] : list A → list A
| [] := []
| (x :: xs) := if x ∈ xs then erase_dup xs else x :: erase_dup xs
theorem erase_dup_nil [H : decidable_eq A] : erase_dup [] = []
theorem erase_dup_nil [H : decidable_eq A] : erase_dup [] = ([] : list A)
theorem erase_dup_cons_of_mem [H : decidable_eq A] {a : A} {l : list A} : a ∈ l → erase_dup (a::l) = erase_dup l :=
assume ainl, calc

2
src/library/unifier.cpp
View file

@ -2198,7 +2198,7 @@ struct unifier_fn {
constraint c = p->first;
unsigned cidx = p->second;
if (cidx >= get_group_first_index(cnstr_group::ClassInstance) &&
!m_config.m_discard && is_choice_cnstr(c) && cnstr_on_demand(c)) {
is_choice_cnstr(c) && cnstr_on_demand(c)) {
// we postpone class-instance constraints whose type still contains metavariables
m_cnstrs.erase_min();
postpone(c);

2
tests/lean/const.lean
View file

@ -11,7 +11,7 @@ set_option pp.implicit true
section
variable A : Type
variable S : inhabited A
variable B : bla
variable B : @bla A _
check B
check @foo A _
end

2
tests/lean/run/alg_rw.lean
View file

@ -6,7 +6,7 @@ section
variable [s : comm_monoid A]
include s
theorem one_mul_one : 1 * 1 = 1 :=
theorem one_mul_one : 1 * 1 = (1:A) :=
mul_one 1
end

2
tests/lean/run/dec_trivial_loop.lean Normal file
View file

@ -0,0 +1,2 @@
check @dec_trivial
print dec_trivial

2
tests/lean/run/finset.lean
View file

@ -186,7 +186,7 @@ namespace finset
theorem eq_clean {A : Type} [H : decidable_eq A] : ∀ s : finset A, clean s = s :=
take s, quot.induction_on s (λ l, eq.symm (quot.sound (eqv_norep l)))
theorem eq_of_clean_eq_clean {A : Type} [H : decidable_eq A] : ∀ s₁ s₂, clean s₁ = clean s₂ → s₁ = s₂ :=
theorem eq_of_clean_eq_clean {A : Type} [H : decidable_eq A] : ∀ s₁ s₂ : finset A, clean s₁ = clean s₂ → s₁ = s₂ :=
take s₁ s₂, by rewrite *eq_clean; intro H; apply H
example : to_finset [(1:nat), 1, 2, 3] = to_finset [1, 2, 2, 2, 3, 3] :=

4
tests/lean/run/num_bug2.lean
View file

@ -3,8 +3,8 @@ open algebra
variable A : Type
variable s : ring A
variable H0 : 0 = 0 -- since algebra defines notation '0' it should have precedence over num
variable H1 : 1 = 1 -- since algebra defines notation '1' it should have precedence over num
variable H0 : 0 = (0:A) -- since algebra defines notation '0' it should have precedence over num
variable H1 : 1 = (1:A) -- since algebra defines notation '1' it should have precedence over num
example : has_zero.zero A = has_zero.zero A :=
H0

2
tests/lean/run/rewriter2.lean
View file

@ -13,7 +13,7 @@ check @mul_zero
axiom Ax {A : Type} [s₁ : has_mul A] [s₂ : has_zero A] (a : A) : f (a * 0) (a * 0) = 0
theorem test2 {A : Type} [s : comm_ring A] (a b c : A) : f 0 0 = 0 :=
theorem test2 {A : Type} [s : comm_ring A] (a b c : A) : f 0 0 = (0:A) :=
begin
rewrite [
-(mul_zero a) at {1, 2}, -- - means apply symmetry, rewrite 0 ==> a * 0 at 1st and 2nd occurrences