refactor(library/logic/cast): move heq declaration to a separate module

heq is be needed for some automatically generated constructions.
So, we want it available with the least number of dependencies.

library/logic/cast.lean

@@ -1,7 +1,7 @@
 -- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
 -- Released under Apache 2.0 license as described in the file LICENSE.
 -- Author: Leonardo de Moura
-import .eq .quantifiers
+import logic.eq logic.heq logic.quantifiers
 open eq.ops
 
 -- cast.lean
@@ -23,55 +23,17 @@ section
   rfl
 end
 
-inductive heq {A : Type} (a : A) : Π {B : Type}, B → Prop :=
-refl : heq a a
-infixl `==`:50 := heq
-
 namespace heq
   universe variable u
   variables {A B C : Type.{u}} {a a' : A} {b b' : B} {c : C}
-  theorem drec_on {C : Π {B : Type} (b : B), a == b → Type} (H₁ : a == b) (H₂ : C a (refl a)) : C b H₁ :=
-  rec (λ H₁ : a == a, show C a H₁, from H₂) H₁ H₁
-
-  theorem subst {P : ∀T : Type, T → Prop} (H₁ : a == b) (H₂ : P A a) : P B b :=
-  rec_on H₁ H₂
-
-  theorem symm (H : a == b) : b == a :=
-  subst H (refl a)
-
-  theorem type_eq (H : a == b) : A = B :=
-  subst H (eq.refl A)
-
-  theorem from_eq (H : a = a') : a == a' :=
-  eq.subst H (refl a)
-
-  theorem trans (H₁ : a == b) (H₂ : b == c) : a == c :=
-  subst H₂ H₁
-
-  theorem trans_left (H₁ : a == b) (H₂ : b = b') : a == b' :=
-  trans H₁ (from_eq H₂)
-
-  theorem trans_right (H₁ : a = a') (H₂ : a' == b) : a == b :=
-  trans (from_eq H₁) H₂
 
   theorem to_cast_eq (H : a == b) : cast (type_eq H) a = b :=
   drec_on H !cast_eq
 
-  theorem to_eq (H : a == a') : a = a' :=
-  calc
-    a = cast (eq.refl A) a : cast_eq
-  ... = a'                 : to_cast_eq H
-
   theorem elim {D : Type} (H₁ : a == b) (H₂ : ∀ (Hab : A = B), cast Hab a = b → D) : D :=
   H₂ (type_eq H₁) (to_cast_eq H₁)
-
 end heq
 
-calc_trans heq.trans
-calc_trans heq.trans_left
-calc_trans heq.trans_right
-calc_symm  heq.symm
-
 section
   universe variables u v
   variables {A A' B C : Type.{u}} {P P' : A → Type.{v}} {a a' : A} {b : B}
@@ -91,9 +53,6 @@ section
   theorem hproof_irrel {a b : Prop} (H : a = b) (H₁ : a) (H₂ : b) : H₁ == H₂ :=
   eq_rec_to_heq (proof_irrel (cast H H₁) H₂)
 
-  theorem heq.true_elim {a : Prop} (H : a == true) : a :=
-  eq_true_elim (heq.to_eq H)
-
   --TODO: generalize to eq.rec. This is a special case of rec_on_compose in eq.lean
   theorem cast_trans (Hab : A = B) (Hbc : B = C) (a : A) :
     cast Hbc (cast Hab a) = cast (Hab ⬝ Hbc) a :=

library/logic/heq.lean

@@ -0,0 +1,52 @@
+-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
+-- Released under Apache 2.0 license as described in the file LICENSE.
+-- Author: Leonardo de Moura
+import logic.eq
+
+inductive heq {A : Type} (a : A) : Π {B : Type}, B → Prop :=
+refl : heq a a
+infixl `==`:50 := heq
+
+namespace heq
+  universe variable u
+  variables {A B C : Type.{u}} {a a' : A} {b b' : B} {c : C}
+  theorem drec_on {C : Π {B : Type} (b : B), a == b → Type} (H₁ : a == b) (H₂ : C a (refl a)) : C b H₁ :=
+  rec (λ H₁ : a == a, show C a H₁, from H₂) H₁ H₁
+
+  theorem subst {P : ∀T : Type, T → Prop} (H₁ : a == b) (H₂ : P A a) : P B b :=
+  rec_on H₁ H₂
+
+  theorem symm (H : a == b) : b == a :=
+  subst H (refl a)
+
+  theorem type_eq (H : a == b) : A = B :=
+  subst H (eq.refl A)
+
+  theorem from_eq (H : a = a') : a == a' :=
+  eq.subst H (refl a)
+
+  theorem to_eq (H : a == a') : a = a' :=
+  have H₁ : ∀ (Ht : A = A), eq.rec_on Ht a = a, from
+    take Ht, eq.refl (eq.rec_on Ht a),
+  have H₂ : ∀ (Ht : A = A), eq.rec_on Ht a = a', from
+    heq.rec_on H H₁,
+  H₂ (type_eq H)
+
+  theorem trans (H₁ : a == b) (H₂ : b == c) : a == c :=
+  subst H₂ H₁
+
+  theorem trans_left (H₁ : a == b) (H₂ : b = b') : a == b' :=
+  trans H₁ (from_eq H₂)
+
+  theorem trans_right (H₁ : a = a') (H₂ : a' == b) : a == b :=
+  trans (from_eq H₁) H₂
+
+  theorem true_elim {a : Prop} (H : a == true) : a :=
+  eq_true_elim (heq.to_eq H)
+
+end heq
+
+calc_trans heq.trans
+calc_trans heq.trans_left
+calc_trans heq.trans_right
+calc_symm  heq.symm