refactor(library/data/set): remove [reducible] annotation from set operations

library/data/set/basic.lean

@@ -6,7 +6,7 @@ Author: Jeremy Avigad, Leonardo de Moura
 import logic.connectives logic.identities algebra.binary
 open eq.ops binary
 
-definition set [reducible] (X : Type) := X → Prop
+definition set (X : Type) := X → Prop
 
 namespace set
 
@@ -14,7 +14,7 @@ variable {X : Type}
 
 /- membership and subset -/
 
-definition mem [reducible] (x : X) (a : set X) := a x
+definition mem (x : X) (a : set X) := a x
 infix ∈ := mem
 notation a ∉ b := ¬ mem a b
 
@@ -24,7 +24,7 @@ funext (take x, propext (H x))
 definition subset (a b : set X) := ∀⦃x⦄, x ∈ a → x ∈ b
 infix ⊆ := subset
 
-definition superset [reducible] (s t : set X) : Prop := t ⊆ s
+definition superset (s t : set X) : Prop := t ⊆ s
 infix ⊇ := superset
 
 theorem subset.refl (a : set X) : a ⊆ a := take x, assume H, H
@@ -66,7 +66,7 @@ exists.intro x (and.intro xs Px)
 
 /- empty set -/
 
-definition empty [reducible] : set X := λx, false
+definition empty : set X := λx, false
 notation `∅` := empty
 
 theorem not_mem_empty (x : X) : ¬ (x ∈ ∅) :=
@@ -112,7 +112,7 @@ ext (take x, iff.intro (assume H', trivial) (assume H', H x))
 
 /- union -/
 
-definition union [reducible] (a b : set X) : set X := λx, x ∈ a ∨ x ∈ b
+definition union (a b : set X) : set X := λx, x ∈ a ∨ x ∈ b
 notation a ∪ b := union a b
 
 theorem mem_union_left {x : X} {a : set X} (b : set X) : x ∈ a → x ∈ a ∪ b :=
@@ -167,7 +167,7 @@ theorem union_subset {s t r : set X} (sr : s ⊆ r) (tr : t ⊆ r) : s ∪ t ⊆
 
 /- intersection -/
 
-definition inter [reducible] (a b : set X) : set X := λx, x ∈ a ∧ x ∈ b
+definition inter (a b : set X) : set X := λx, x ∈ a ∧ x ∈ b
 notation a ∩ b := inter a b
 
 theorem mem_inter_iff (x : X) (a b : set X) : x ∈ a ∩ b ↔ x ∈ a ∧ x ∈ b := !iff.refl
@@ -234,7 +234,7 @@ ext (take x, !or.right_distrib)
 /- set-builder notation -/
 
 -- {x : X | P}
-definition set_of [reducible] (P : X → Prop) : set X := P
+definition set_of (P : X → Prop) : set X := P
 notation `{` binder ` | ` r:(scoped:1 P, set_of P) `}` := r
 
 -- {x ∈ s | P}

library/theories/analysis/real_limit.lean

@@ -241,6 +241,8 @@ exists.intro x
   (have x ∈ X ∧ ¬ b ≤ x, by rewrite [-not_implies_iff_and_not]; apply Hx,
      and.intro (and.left this) (lt_of_not_ge (and.right this)))
 
+section
+local attribute mem [quasireducible]
 -- TODO: is there a better place to put this?
 proposition image_neg_eq (X : set ℝ) : (λ x, -x) '[X] = {x | -x ∈ X} :=
 set.ext (take x, iff.intro
@@ -290,7 +292,7 @@ have HX : X = {x | -x ∈ negX},
   from set.ext (take x, by rewrite [↑set_of, ↑mem, +neg_neg]),
 show inf {x | -x ∈ X} = - sup X,
   using HX Hb' nonempty_negX, by rewrite [HX at {2}, sup_neg nonempty_negX Hb', neg_neg]
-
+end
 end
 
 /- limits under pointwise operations -/

library/theories/group_theory/hom.lean

@@ -116,17 +116,19 @@ variable {H : set A}
 variable [is_subgH : is_subgroup H]
 include is_subgH
 
+section mem_reducible
+local attribute mem [reducible]
 theorem hom_map_subgroup : is_subgroup (f '[H]) :=
-        have Pone : 1 ∈ f '[H], from mem_image subg_has_one (hom_map_one f),
-        have Pclosed : mul_closed_on (f '[H]), from hom_map_mul_closed f H subg_mul_closed,
-        assert Pinv : ∀ b, b ∈ f '[H] → b⁻¹ ∈ f '[H], from
-          assume b, assume Pimg,
-          obtain a (Pa : a ∈ H ∧ f a = b), from Pimg,
-          assert Painv : a⁻¹ ∈ H, from subg_has_inv a (and.left Pa),
-          assert Pfainv : (f a)⁻¹ ∈ f '[H], from mem_image Painv (hom_map_inv f a),
-          and.right Pa ▸ Pfainv,
-        is_subgroup.mk Pone Pclosed Pinv
-
+  have Pone : 1 ∈ f '[H], from mem_image subg_has_one (hom_map_one f),
+  have Pclosed : mul_closed_on (f '[H]), from hom_map_mul_closed f H subg_mul_closed,
+  assert Pinv : ∀ b, b ∈ f '[H] → b⁻¹ ∈ f '[H], from
+  assume b, assume Pimg,
+  obtain a (Pa : a ∈ H ∧ f a = b), from Pimg,
+  assert Painv : a⁻¹ ∈ H, from subg_has_inv a (and.left Pa),
+  assert Pfainv : (f a)⁻¹ ∈ f '[H], from mem_image Painv (hom_map_inv f a),
+    and.right Pa ▸ Pfainv,
+  is_subgroup.mk Pone Pclosed Pinv
+end mem_reducible
 end
 
 section hom_theorem

library/theories/group_theory/subgroup.lean

@@ -205,7 +205,8 @@ include s
 variable {H : set A}
 variable [is_subg : is_subgroup H]
 include is_subg
-
+section set_reducible
+local attribute set [reducible]
 lemma subg_has_one : H (1 : A) := @is_subgroup.has_one A s H is_subg
 lemma subg_mul_closed : mul_closed_on H := @is_subgroup.mul_closed A s H is_subg
 lemma subg_has_inv : subgroup.has_inv H := @is_subgroup.has_inv A s H is_subg
@@ -240,6 +241,7 @@ lemma subg_in_coset_refl (a : A) : a ∈ a ∘> H ∧ a ∈ H <∘ a :=
       assert PHinvaa : H (a⁻¹*a), from (eq.symm (mul.left_inv a)) ▸ PH1,
       assert PHainva : H (a*a⁻¹), from (eq.symm (mul.right_inv a)) ▸ PH1,
       and.intro PHinvaa PHainva
+end set_reducible
 lemma subg_in_lcoset_same_lcoset (a b : A) : in_lcoset H a b → same_lcoset H a b :=
       assume Pa_in_b : H (b⁻¹*a),
       have Pbinva : b⁻¹*a ∘> H = H, from subgroup_lcoset_id (b⁻¹*a) Pa_in_b,

tests/lean/run/congr_tac2.lean

@@ -30,6 +30,7 @@ example (A : Type) (s₁ s₂ : set A) [fxs₁ : finite_set s₁] [fxs₂ : fini
 begin
   intros,
   congruence,
+  unfold set at *,
   assumption
 end