Package both the induction and recursion principles in the same record

FinSets.v

@@ -45,10 +45,6 @@ Fixpoint FinSets_rec
            (FinSets_rec A P e l u assocP commP nlP nrP idemP z)
       end) assocP commP nlP nrP idemP.
 
-Definition FinSetsCL A : HitRec.class (FinSets A) _ := 
-   HitRec.Class (FinSets A) (fun x P e l u aP cP lP rP iP => FinSets_rec A P e l u aP cP lP rP iP x).
-Canonical Structure FinSetsTy A : HitRec.type := HitRec.Pack _ _ (FinSetsCL A).
-
 Axiom FinSets_beta_assoc : forall
   (A : Type)
   (P : Type)
@@ -135,6 +131,12 @@ Axiom FinSets_beta_idem : forall
   ap (FinSets_rec A P e l u assocP commP nlP nrP idemP) (idem A x)
   =
   idemP x.
+
+(* TODO: add an induction principle *)
+Definition FinSetsCL A : HitRec.class (FinSets A) _ _ := 
+   HitRec.Class (FinSets A) (fun x P e l u aP cP lP rP iP => FinSets_rec A P e l u aP cP lP rP iP x) (fun x P e l u aP cP lP rP iP => FinSets_rec A P e l u aP cP lP rP iP x).
+Canonical Structure FinSetsTy A : HitRec.type := HitRec.Pack _ _ _ (FinSetsCL A).
+
 End FinSet.
 
 Section functions.

HitTactics.v

@@ -1,45 +1,29 @@
 Module HitRec.
 Record class (H : Type) (* The HIT type itself *)
+             (indTy : H -> Type) (* The type of the induciton principle *)
              (recTy : H -> Type) (* The type of the recursion principle *)
-             := Class { recursion_term : forall (x : H), recTy x }.
+             := Class { 
-Structure type := Pack { obj : Type ; rec : obj -> Type ; class_of : class obj rec }.
+    H_recursor : forall (x : H), recTy x;
+    H_inductor : forall (x : H), indTy x  }.
+Structure type := Pack { obj : Type ; ind : obj -> Type ; rec : obj -> Type ; class_of : class obj ind rec }.
 Definition hrecursion (e : type) : forall x, rec e x :=
-  let 'Pack _ _ (Class r) := e
+  let 'Pack _ _ _ (Class r _) := e
   in r.
+Definition hinduction (e : type) : forall x, ind e x :=
+  let 'Pack _ _ _ (Class _ i) := e
+  in i.
 Arguments hrecursion {e} x : simpl never.
-Arguments Class H {recTy} recursion_term.
+Arguments hinduction {e} x : simpl never.
+Arguments Class H {indTy recTy} H_recursor H_inductor.
 End HitRec.
 
 Ltac hrecursion_ target :=
   generalize dependent target;
   match goal with
-  | [ |- forall target, ?P] => intro target;
+  | [ |- _ -> ?P ] => intro target;
     match type of (HitRec.hrecursion target) with
     | ?Q => 
       match (eval simpl in Q) with
-      | forall Y , Y target =>
-        simple refine (HitRec.hrecursion target (fun target => P) _)
-      | forall Y _, Y target  =>
-        simple refine (HitRec.hrecursion target (fun target => P) _)
-      | forall Y _ _, Y target  =>
-        simple refine (HitRec.hrecursion target (fun target => P) _ _)
-      | forall Y _ _ _, Y target  =>
-        let hrec:=(eval hnf in (HitRec.hrecursion target)) in
-        simple refine (hrec (fun target => P) _ _ _)
-      | forall Y _ _ _ _, Y target  =>
-        simple refine (HitRec.hrecursion target (fun target => P) _ _ _ _)
-      | forall Y _ _ _ _ _, Y target  =>
-        simple refine (HitRec.hrecursion target (fun target => P) _ _ _ _ _)
-      | forall Y _ _ _ _ _ _, Y target  =>
-        simple refine (HitRec.hrecursion target (fun target => P) _ _ _ _ _ _)
-      | forall Y _ _ _ _ _ _ _, Y target  =>
-        simple refine (HitRec.hrecursion target (fun target => P) _ _ _ _ _ _ _)
-      | forall Y _ _ _ _ _ _ _, Y target  =>
-        simple refine (HitRec.hrecursion target (fun target => P) _ _ _ _ _ _ _)
-      | forall Y _ _ _ _ _ _ _ _, Y target  =>
-        simple refine (HitRec.hrecursion target (fun target => P) _ _ _ _ _ _ _ _)
-      | forall Y _ _ _ _ _ _ _ _ _, Y target  =>
-        simple refine (HitRec.hrecursion target (fun target => P) _ _ _ _ _ _ _ _ _)
       | forall Y, Y =>
         simple refine (HitRec.hrecursion target P)
       | forall Y _, Y  =>
@@ -63,10 +47,40 @@ Ltac hrecursion_ target :=
       | forall Y _ _ _ _ _ _ _ _ _ _, Y =>
         simple refine (HitRec.hrecursion target P _ _ _ _ _ _ _ _ _ _)
       | _ => fail "Cannot handle the recursion principle (too many parameters?) :("
+      end
+    end
+  | [ |- forall target, ?P] => intro target;
+    match type of (HitRec.hinduction target) with
+    | ?Q => 
+      match (eval simpl in Q) with
+      | forall Y , Y target =>
+        simple refine (HitRec.hinduction target (fun target => P) _)
+      | forall Y _, Y target  =>
+        simple refine (HitRec.hinduction target (fun target => P) _)
+      | forall Y _ _, Y target  =>
+        simple refine (HitRec.hinduction target (fun target => P) _ _)
+      | forall Y _ _ _, Y target  =>
+        let hrec:=(eval hnf in (HitRec.hinduction target)) in
+        simple refine (hrec (fun target => P) _ _ _)
+      | forall Y _ _ _ _, Y target  =>
+        simple refine (HitRec.hinduction target (fun target => P) _ _ _ _)
+      | forall Y _ _ _ _ _, Y target  =>
+        simple refine (HitRec.hinduction target (fun target => P) _ _ _ _ _)
+      | forall Y _ _ _ _ _ _, Y target  =>
+        simple refine (HitRec.hinduction target (fun target => P) _ _ _ _ _ _)
+      | forall Y _ _ _ _ _ _ _, Y target  =>
+        simple refine (HitRec.hinduction target (fun target => P) _ _ _ _ _ _ _)
+      | forall Y _ _ _ _ _ _ _, Y target  =>
+        simple refine (HitRec.hinduction target (fun target => P) _ _ _ _ _ _ _)
+      | forall Y _ _ _ _ _ _ _ _, Y target  =>
+        simple refine (HitRec.hinduction target (fun target => P) _ _ _ _ _ _ _ _)
+      | forall Y _ _ _ _ _ _ _ _ _, Y target  =>
+        simple refine (HitRec.hinduction target (fun target => P) _ _ _ _ _ _ _ _ _)
+      | _ => fail "Cannot handle the induction principle (too many parameters?) :("
      end
     end
+  (*| _ => fail "I am sorry, but something went wrong!"*)
   end.
 
 Ltac hrecursion x := hrecursion_ x; simpl; try (clear x).
-
+Ltac hinduction x := hrecursion x.
-

Mod2.v

@@ -52,9 +52,9 @@ Axiom Mod2_rec_beta_mod : forall
   , ap (Mod2_rec P a s mod') mod = mod'.
 
 
-Definition Mod2CL : HitRec.class Mod2 _ := 
+Definition Mod2CL : HitRec.class Mod2 _ _ := 
-   HitRec.Class Mod2 (fun x P a s p => Mod2_ind P a s p x).
+   HitRec.Class Mod2 (fun x P a s p => Mod2_rec P a s p x) (fun x P a s p => Mod2_ind P a s p x).
-Canonical Structure Mod2ty : HitRec.type := HitRec.Pack Mod2 _ Mod2CL.
+Canonical Structure Mod2ty : HitRec.type := HitRec.Pack Mod2 _ _ Mod2CL.
 
 End modulo.
 
@@ -79,132 +79,72 @@ intro x.
 hrecursion x.
 - apply Z. 
 - intros. apply (succ H).
-- simpl.
+- simpl. apply mod.
-  etransitivity.
-  eapply transport_const.
-  eapply modulo2.
 Defined.
 
 Definition plus : Mod2 -> Mod2 -> Mod2.
 Proof.
-intro n.
+intros n m.
-refine (Mod2_ind _ _ _ _).
+hrecursion m.
-  Unshelve.
+- exact n.
-
+- apply succ.
-  Focus 2.
+- apply modulo2.
-  apply n.
-
-  Focus 2.
-  intro m.
-  intro k.
-  apply (succ k).
-
-  simpl.
-  rewrite transport_const.
-  apply modulo2.
 Defined.
 
 Definition Bool_to_Mod2 : Bool -> Mod2.
 Proof.
 intro b.
 destruct b.
- apply (succ Z).
++ apply (succ Z).
-
++ apply Z.
- apply Z.
 Defined.
 
 Definition Mod2_to_Bool : Mod2 -> Bool.
 Proof.
-refine (Mod2_ind _ _ _ _).
+intro x.
- Unshelve.
+hrecursion x.
- Focus 2.
+- exact false.
- apply false.
+- exact negb.
-
+- simpl. reflexivity.
- Focus 2.
- intro n.
- apply negb.
-
- Focus 1.
- simpl.
- apply transport_const.
 Defined.
 
 Theorem eq1 : forall n : Bool, Mod2_to_Bool (Bool_to_Mod2 n) = n.
 Proof.
 intro b.
-destruct b.
+destruct b; compute; reflexivity.
- Focus 1.
- compute.
- reflexivity.
-
- compute.
- reflexivity.
 Qed.
 
 Theorem Bool_to_Mod2_negb : forall x : Bool, 
   succ (Bool_to_Mod2 x) = Bool_to_Mod2 (negb x).
 Proof.
 intros.
-destruct x.
+destruct x; compute.
- compute.
++ apply mod^.
- apply mod^.
++ apply reflexivity.
-
- compute.
- apply reflexivity.
 Defined.
 
 Theorem eq2 : forall n : Mod2, Bool_to_Mod2 (Mod2_to_Bool n) = n.
 Proof.
-refine (Mod2_ind _ _ _ _).
+intro n.
-  Unshelve.
+hinduction n.
-  Focus 2.
+- reflexivity.
-  compute.
+- intros n IHn.
-  reflexivity.
+  symmetry. etransitivity. apply (ap succ IHn^). 
-
+  etransitivity. apply Bool_to_Mod2_negb.
-  Focus 2.
+  hott_simpl.
-  intro n.
+- rewrite @HoTT.Types.Paths.transport_paths_FlFr.
-  intro IHn.
+  hott_simpl.
-  symmetry.
+  rewrite ap_compose. 
-  transitivity (succ (Bool_to_Mod2 (Mod2_to_Bool n))).
+  enough (ap Mod2_to_Bool mod = idpath).
-
+  + rewrite X. hott_simpl.
-    Focus 1.
+  + unfold Mod2_to_Bool. unfold HitRec.hrecursion. simpl. 
-    symmetry.
+    apply (Mod2_rec_beta_mod Bool false negb 1).
-    apply (ap succ IHn).
-
-    transitivity (Bool_to_Mod2 (negb (Mod2_to_Bool n))).
-    apply Bool_to_Mod2_negb.
-    enough (negb (Mod2_to_Bool n) = Mod2_to_Bool (succ n)).
-    apply (ap Bool_to_Mod2 X).
-
-    compute.
-    reflexivity.
-    simpl.
-    rewrite concat_p1.
-    rewrite concat_1p.
-    rewrite @HoTT.Types.Paths.transport_paths_FlFr.
-    rewrite concat_p1.
-    rewrite ap_idmap.
-    rewrite ap_compose.
-
-    enough (ap Mod2_to_Bool mod = reflexivity false).
-    rewrite X.
-    simpl.
-    rewrite concat_1p.
-    rewrite inv_V.
-    reflexivity.
-
-    enough (IsHSet Bool).
-    apply axiomK_hset.
-    apply X.
-    apply hset_bool.
 Defined.
 
 Theorem adj : 
   forall x : Mod2, eq1 (Mod2_to_Bool x) = ap Mod2_to_Bool (eq2 x).
 Proof.
 intro x.
-enough (IsHSet Bool).
-apply set_path2.
 apply hset_bool.
 Defined.