feat(library/standard): add basic heq theorems

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

library/standard/classical.lean

@@ -80,7 +80,7 @@ theorem eq_id {A : Type} (a : A) : (a = a) = true
 := eqt_intro (refl a)
 
 theorem heq_id {A : Type} (a : A) : (a == a) = true
-:= eqt_intro (heq_refl a)
+:= eqt_intro (hrefl a)
 
 theorem not_or (a b : Bool) : (¬ (a ∨ b)) = (¬ a ∧ ¬ b)
 := boolext

library/standard/logic.lean

@@ -247,8 +247,42 @@ theorem heq_to_eq {A : Type} {a b : A} (H : a == b) : a = b
     calc a = cast w a : symm (cast_eq w a)
       ...  = b        : Hw
 
-theorem heq_refl {A : Type} (a : A) : a == a
+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,
+   obtains (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