feat(hit): start using induction tactic

hott/hit/circle.hlean

@@ -25,13 +25,13 @@ namespace circle
   definition rec2 {P : circle → Type} (Pb1 : P base1) (Pb2 : P base2)
     (Ps1 : seg1 ▸ Pb1 = Pb2) (Ps2 : seg2 ▸ Pb1 = Pb2) (x : circle) : P x :=
   begin
-    fapply (suspension.rec_on x),
+    induction x with b,
     { exact Pb1},
     { exact Pb2},
-    { esimp, intro b, fapply (suspension.rec_on b),
+    { esimp at *, induction b with y,
       { exact Ps1},
       { exact Ps2},
-      { intro x, cases x}},
+      { cases y}},
   end
 
   definition rec2_on [reducible] {P : circle → Type} (x : circle) (Pb1 : P base1) (Pb2 : P base2)
@@ -198,7 +198,7 @@ namespace circle
 
   protected definition decode {x : circle} : circle.code x → base = x :=
   begin
-    refine circle.rec_on x _ _,
+    induction x,
     { exact power loop},
     { apply eq_of_homotopy, intro a,
       refine !arrow.arrow_transport ⬝ !transport_eq_r ⬝ _,
@@ -208,7 +208,7 @@ namespace circle
   --remove this theorem after #484
   theorem encode_decode {x : circle} : Π(a : circle.code x), circle.encode (circle.decode a) = a :=
   begin
-    unfold circle.decode, refine circle.rec_on x _ _,
+    unfold circle.decode, induction x,
     { intro a, esimp [base,base1], --simplify after #587
       apply rec_nat_on a,
       { exact idp},

hott/hit/coeq.hlean

@@ -33,9 +33,9 @@ parameters {A B : Type.{u}} (f g : A → B)
   protected definition rec {P : coeq → Type} (P_i : Π(x : B), P (coeq_i x))
     (Pcp : Π(x : A), cp x ▸ P_i (f x) = P_i (g x)) (y : coeq) : P y :=
   begin
-    fapply (type_quotient.rec_on y),
+    induction y,
-    { intro a, apply P_i},
+    { apply P_i},
-    { intro a a' H, cases H, apply Pcp}
+    { cases H, apply Pcp}
   end
 
   protected definition rec_on [reducible] {P : coeq → Type} (y : coeq)

hott/hit/cylinder.hlean

@@ -37,11 +37,11 @@ parameters {A B : Type.{u}} (f : A → B)
     (Pbase : Π(b : B), P (base b)) (Ptop : Π(a : A), P (top a))
     (Pseg : Π(a : A), seg a ▸ Pbase (f a) = Ptop a) (x : cylinder) : P x :=
   begin
-    fapply (type_quotient.rec_on x),
+    induction x,
-    { intro a, cases a,
+    { cases a,
-       apply Pbase,
+        apply Pbase,
-       apply Ptop},
+        apply Ptop},
-    { intro a a' H, cases H, apply Pseg}
+    { cases H, apply Pseg}
   end
 
   protected definition rec_on [reducible] {P : cylinder → Type} (x : cylinder)

hott/hit/pushout.hlean

@@ -36,11 +36,11 @@ parameters {TL BL TR : Type} (f : TL → BL) (g : TL → TR)
     (Pinr : Π(x : TR), P (inr x)) (Pglue : Π(x : TL), glue x ▸ Pinl (f x) = Pinr (g x))
       (y : pushout) : P y :=
   begin
-    fapply (type_quotient.rec_on y),
+    induction y,
-    { intro a, cases a,
+    { cases a,
-       apply Pinl,
+        apply Pinl,
-       apply Pinr},
+        apply Pinr},
-    { intro a a' H, cases H, apply Pglue}
+    { cases H, apply Pglue}
   end
 
   protected definition rec_on [reducible] {P : pushout → Type} (y : pushout)
@@ -83,31 +83,6 @@ parameters {TL BL TR : Type} (f : TL → BL) (g : TL → TR)
   by rewrite [tr_eq_cast_ap_fn,↑elim_type,elim_glue];apply cast_ua_fn
 
 end
-
-
-
-  namespace test
-    open pushout equiv is_equiv unit bool
-    private definition unit_of_empty (u : empty) : unit := star
-
-    example : pushout unit_of_empty unit_of_empty ≃ bool :=
-    begin
-      fapply equiv.MK,
-      { intro x, fapply (pushout.rec_on _ _ x),
-          intro u, exact ff,
-          intro u, exact tt,
-          intro c, cases c},
-      { intro b, cases b,
-          exact (inl _ _ ⋆),
-          exact (inr _ _ ⋆)},
-      { intro b, cases b, esimp, esimp},
-      { intro x, fapply (pushout.rec_on _ _ x),
-          intro u, cases u, esimp,
-          intro u, cases u, esimp,
-          intro c, cases c},
-    end
-
-  end test
 end pushout
 
 attribute pushout.inl pushout.inr [constructor]

hott/hit/quotient.hlean

@@ -72,5 +72,5 @@ end
 end quotient
 
 attribute quotient.class_of [constructor]
-attribute quotient.elim quotient.rec [unfold-c 7] [recursor]
+attribute quotient.elim quotient.rec [unfold-c 7] [recursor 7]
 attribute quotient.rec_on quotient.elim_on [unfold-c 4]