refactor(library/standard): move cast and heq to separate file

Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>

library/standard/cast.lean

@@ -0,0 +1,77 @@
+-- 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
+
+definition cast {A B : Type} (H : A = B) (a : A) : B
+:= eq_rec a H
+
+theorem cast_refl {A : Type} (a : A) : cast (refl A) a = a
+:= refl (cast (refl A) a)
+
+theorem cast_proof_irrel {A B : Type} (H1 H2 : A = B) (a : A) : cast H1 a = cast H2 a
+:= refl (cast H1 a)
+
+theorem cast_eq {A : Type} (H : A = A) (a : A) : cast H a = a
+:= calc cast H a = cast (refl A) a : cast_proof_irrel H (refl A) a
+            ...  = a               : cast_refl a
+
+definition heq {A B : Type} (a : A) (b : B) := ∃ H, cast H a = b
+
+infixl `==`:50 := heq
+
+theorem heq_type_eq {A B : Type} {a : A} {b : B} (H : a == b) : A = B
+:= obtain w Hw, from H, w
+
+theorem eq_to_heq {A : Type} {a b : A} (H : a = b) : a == b
+:= exists_intro (refl A) (trans (cast_refl a) H)
+
+theorem heq_to_eq {A : Type} {a b : A} (H : a == b) : a = b
+:= obtain (w : A = A) (Hw : cast w a = b), from H,
+    calc a = cast w a : symm (cast_eq w a)
+      ...  = b        : Hw
+
+theorem hrefl {A : Type} (a : A) : a == a
+:= eq_to_heq (refl a)
+
+theorem heqt_elim {a : Bool} (H : a == true) : a
+:= eqt_elim (heq_to_eq H)
+
+opaque_hint (hiding cast)
+
+theorem hsubst {A B : Type} {a : A} {b : B} {P : ∀ (T : Type), T → Bool} (H1 : a == b) (H2 : P A a) : P B b
+:= have Haux1 : ∀ H : A = A, P A (cast H a), from
+     assume H : A = A, subst (symm (cast_eq H a)) H2,
+   obtain (Heq : A = B) (Hw : cast Heq a = b), from H1,
+   have Haux2 : P B (cast Heq a), from subst Heq Haux1 Heq,
+     subst Hw Haux2
+
+theorem hsymm {A B : Type} {a : A} {b : B} (H : a == b) : b == a
+:= hsubst H (hrefl a)
+
+theorem htrans {A B C : Type} {a : A} {b : B} {c : C} (H1 : a == b) (H2 : b == c) : a == c
+:= hsubst H2 H1
+
+theorem htrans_left {A B : Type} {a : A} {b c : B} (H1 : a == b) (H2 : b = c) : a == c
+:= htrans H1 (eq_to_heq H2)
+
+theorem htrans_right {A C : Type} {a b : A} {c : C} (H1 : a = b) (H2 : b == c) : a == c
+:= htrans (eq_to_heq H1) H2
+
+calc_trans htrans
+calc_trans htrans_left
+calc_trans htrans_right
+
+theorem cast_heq {A B : Type} (H : A = B) (a : A) : cast H a == a
+:= have H1 : ∀ (H : A = A) (a : A), cast H a == a, from
+     λ H a, eq_to_heq (cast_eq H a),
+   subst H H1 H a
+
+theorem cast_eq_to_heq {A B : Type} {a : A} {b : B} {H : A = B} (H1 : cast H a = b) : a == b
+:= calc a  == cast H a : hsymm (cast_heq H a)
+       ... =  b        : H1
+
+theorem cast_trans {A B C : Type} (Hab : A = B) (Hbc : B = C) (a : A) : cast Hbc (cast Hab a) = cast (trans Hab Hbc) a
+:= heq_to_eq (calc cast Hbc (cast Hab a)   == cast Hab a             : cast_heq Hbc (cast Hab a)
+                                      ...  == a                      : cast_heq Hab a
+                                      ...  == cast (trans Hab Hbc) a : hsymm (cast_heq (trans Hab Hbc) a))

library/standard/classical.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 logic
+import logic cast
 
 axiom boolcomplete (a : Bool) : a = true ∨ a = false
 

library/standard/logic.lean

@@ -218,71 +218,3 @@ theorem inhabited_Bool [instance] : inhabited Bool
 
 theorem inhabited_fun [instance] (A : Type) {B : Type} (H : inhabited B) : inhabited (A → B)
 := inhabited_elim H (take (b : B), inhabited_intro (λ a : A, b))
-
-definition cast {A B : Type} (H : A = B) (a : A) : B
-:= eq_rec a H
-
-theorem cast_refl {A : Type} (a : A) : cast (refl A) a = a
-:= refl (cast (refl A) a)
-
-theorem cast_proof_irrel {A B : Type} (H1 H2 : A = B) (a : A) : cast H1 a = cast H2 a
-:= refl (cast H1 a)
-
-theorem cast_eq {A : Type} (H : A = A) (a : A) : cast H a = a
-:= calc cast H a = cast (refl A) a : cast_proof_irrel H (refl A) a
-            ...  = a               : cast_refl a
-
-definition heq {A B : Type} (a : A) (b : B) := ∃ H, cast H a = b
-
-infixl `==`:50 := heq
-
-theorem heq_type_eq {A B : Type} {a : A} {b : B} (H : a == b) : A = B
-:= obtain w Hw, from H, w
-
-theorem eq_to_heq {A : Type} {a b : A} (H : a = b) : a == b
-:= exists_intro (refl A) (trans (cast_refl a) H)
-
-theorem heq_to_eq {A : Type} {a b : A} (H : a == b) : a = b
-:= obtain (w : A = A) (Hw : cast w a = b), from H,
-    calc a = cast w a : symm (cast_eq w a)
-      ...  = b        : Hw
-
-theorem hrefl {A : Type} (a : A) : a == a
-:= eq_to_heq (refl a)
-
-theorem heqt_elim {a : Bool} (H : a == true) : a
-:= eqt_elim (heq_to_eq H)
-
-opaque_hint (hiding cast)
-
-theorem hsubst {A B : Type} {a : A} {b : B} {P : ∀ (T : Type), T → Bool} (H1 : a == b) (H2 : P A a) : P B b
-:= have Haux1 : ∀ H : A = A, P A (cast H a), from
-     assume H : A = A, subst (symm (cast_eq H a)) H2,
-   obtain (Heq : A = B) (Hw : cast Heq a = b), from H1,
-   have Haux2 : P B (cast Heq a), from subst Heq Haux1 Heq,
-     subst Hw Haux2
-
-theorem hsymm {A B : Type} {a : A} {b : B} (H : a == b) : b == a
-:= hsubst H (hrefl a)
-
-theorem htrans {A B C : Type} {a : A} {b : B} {c : C} (H1 : a == b) (H2 : b == c) : a == c
-:= hsubst H2 H1
-
-theorem htrans_left {A B : Type} {a : A} {b c : B} (H1 : a == b) (H2 : b = c) : a == c
-:= htrans H1 (eq_to_heq H2)
-
-theorem htrans_right {A C : Type} {a b : A} {c : C} (H1 : a = b) (H2 : b == c) : a == c
-:= htrans (eq_to_heq H1) H2
-
-calc_trans htrans
-calc_trans htrans_left
-calc_trans htrans_right
-
-theorem cast_heq {A B : Type} (H : A = B) (a : A) : cast H a == a
-:= have H1 : ∀ (H : A = A) (a : A), cast H a == a, from
-     λ H a, eq_to_heq (cast_eq H a),
-   subst H H1 H a
-
-theorem cast_eq_to_heq {A B : Type} {a : A} {b : B} {H : A = B} (H1 : cast H a = b) : a == b
-:= calc a  == cast H a : hsymm (cast_heq H a)
-       ... =  b        : H1

library/standard/standard.lean

@@ -1,2 +1 @@
-import logic tactic num string pair
+import logic tactic num string pair cast
-