Add a general hit recursion tactic.

The tactic resolves the induction principle for a HIT
using Canonical Structures, following the suggestion of @gallais

FinSets.v

@@ -1,5 +1,5 @@
-Require Import HoTT.
 Require Export HoTT.
+Require Import HitTactics.
 
 Module Export FinSet.
 
@@ -45,6 +45,10 @@ 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)
@@ -133,129 +137,66 @@ Axiom FinSets_beta_idem : forall
   idemP x.
 End FinSet.
 
-Definition isIn : forall 
+Section functions.
-  (A : Type)
+Parameter A : Type.
-  (eq : A -> A -> Bool),
+Parameter eq : A -> A -> Bool.
-  A -> FinSets A -> Bool.
+Definition isIn : A -> FinSets A -> Bool.
 Proof.
-intro A.
+intros a X.
-intro eq.
+hrecursion X.
-intro a.
+- exact false.
-refine (FinSets_rec A _ _ _ _ _ _ _ _ _).
+- intro a'. apply (eq a a').
-  Unshelve.
+- apply orb. 
-
+- intros x y z. compute. destruct x; reflexivity.
-  Focus 6.
+- intros x y. compute. destruct x, y; reflexivity.
-  apply false.
+- intros x. compute. reflexivity. 
-
+- intros x. compute. destruct x; reflexivity.
-  Focus 6.
+- intros a'. compute. destruct (eq a a'); reflexivity.
-  intro a'.
-  apply (eq a a').
-
-  Focus 6.
-  intro b.
-  intro b'.
-  apply (orb b b').
-
-  Focus 3.
-  intros.
-  compute.
-  reflexivity.
-
-  Focus 1.
-  intros.
-  compute.
-  destruct x.
-  reflexivity.
-  destruct y.
-  reflexivity.
-  reflexivity.
-
-  Focus 1.
-  intros.
-  compute.
-  destruct x.
-  destruct y.
-  reflexivity.
-  reflexivity.
-  destruct y.
-  reflexivity.
-  reflexivity.
-
-  Focus 1.
-  intros.
-  compute.
-  destruct x.
-  reflexivity.
-  reflexivity.
-
-  intros.
-  compute.
-  destruct (eq a x).
-  reflexivity.
-  reflexivity.
 Defined.
 
-Definition comprehension : forall
+
-  (A : Type)
+Definition comprehension : 
-  (eq : A -> A -> Bool),
   (A -> Bool) -> FinSets A -> FinSets A.
 Proof.
-intro A.
+intros P X.
-intro eq.
+hrecursion X.
-intro phi.
+- apply empty.
-refine (FinSets_rec A _ _ _ _ _ _ _ _ _).
+- intro a.
-  Unshelve.
+  refine (if (P a) then L A a else empty A).
-
+- intros X Y.
-  Focus 6.
+  apply (U A X Y).
-  apply empty.
+- intros. cbv. apply assoc.
-
+- intros. cbv. apply comm.
-  Focus 6.
+- intros. cbv. apply nl.
-  intro a.
+- intros. cbv. apply nr.
-  apply (if (phi a) then L A a else (empty A)).
+- intros. cbv. 
-
+  destruct (P x); simpl.
-  Focus 6.
+  + apply idem.
-  intro x.
+  + apply nl.
-  intro y.
-  apply (U A x y).
-
-  Focus 3.
-  intros.
-  compute.
-  apply nl.
-
-  Focus 1.
-  intros.
-  compute.
-  apply assoc.
-
-  Focus 1.
-  intros.
-  compute.
-  apply comm.
-
-  Focus 1.
-  intros.
-  compute.
-  apply nr.
-
-  intros.
-  compute.
-  destruct (phi x).
-  apply idem.
-  apply nl.
 Defined.
 
-Definition intersection : forall (A : Type) (eq : A -> A -> Bool), 
+Definition intersection : 
   FinSets A -> FinSets A -> FinSets A.
 Proof.
-intro A.
+intros X Y.
-intro eq.
+apply (comprehension (fun (a : A) => isIn a X) Y).
-intro x.
-intro y.
-apply (comprehension A eq (fun (a : A) => isIn A eq a x) y).
 Defined.
 
-Definition subset (A : Type) (eq : A -> A -> Bool) (x : FinSets A) (y : FinSets A) : Bool.
+Definition subset (x : FinSets A) (y : FinSets A) : Bool.
+Proof.
+hrecursion x.
+- apply true.
+- intro a. apply (isIn a y).
+- apply andb.
+- intros a b c. compute. destruct a; reflexivity.
+- intros a b. compute. destruct a, b; reflexivity.
+- intros x. compute. reflexivity.
+- intros x. compute. destruct x;reflexivity.
+- intros a. simpl. 
+  destruct (isIn a y); reflexivity.
+Defined.
+
+Definition subset' (x : FinSets A) (y : FinSets A) : Bool.
 Proof.
 refine (FinSets_rec A _ _ _ _ _ _ _ _ _ _).
   Unshelve.
@@ -268,7 +209,7 @@ refine (FinSets_rec A _ _ _ _ _ _ _ _ _ _).
 
   Focus 6.
   intro a.
-  apply (isIn A eq a y).
+  apply (isIn a y).
 
   Focus 6.
   intro b.
@@ -308,15 +249,16 @@ refine (FinSets_rec A _ _ _ _ _ _ _ _ _ _).
   reflexivity.
 
   intros.
-  destruct (isIn A eq x0 y).
+  destruct (isIn x0 y).
   compute.
   reflexivity.
   compute.
   reflexivity.
 Defined.
+(* TODO: subset = subset' *)
 
-Definition equal_set (A : Type) (eq : A -> A -> Bool) (x : FinSets A) (y : FinSets A) : Bool
+Definition equal_set (x : FinSets A) (y : FinSets A) : Bool
-  := andb (subset A eq x y) (subset A eq y x).
+  := andb (subset x y) (subset y x).
 
 Fixpoint eq_nat n m : Bool :=
   match n, m with
@@ -324,4 +266,6 @@ Fixpoint eq_nat n m : Bool :=
     | O, S _ => false
     | S _, O => false
     | S n1, S m1 => eq_nat n1 m1
-  end.
+  end.
+
+End functions.

HitTactics.v

@@ -0,0 +1,72 @@
+Module HitRec.
+Record class (H : Type) (* The HIT type itself *)
+             (recTy : H -> Type) (* The type of the recursion principle *)
+             := Class { recursion_term : forall (x : H), recTy x }.
+Structure type := Pack { obj : Type ; rec : obj -> Type ; class_of : class obj rec }.
+Definition hrecursion (e : type) : forall x, rec e x :=
+  let 'Pack _ _ (Class r) := e
+  in r.
+Arguments hrecursion {e} x : simpl never.
+Arguments Class H {recTy} recursion_term.
+End HitRec.
+
+Ltac hrecursion_ target :=
+  generalize dependent target;
+  match goal with
+  | [ |- forall 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  =>
+        simple refine (HitRec.hrecursion target P _)
+      | forall Y _ _, Y =>
+        simple refine (HitRec.hrecursion target P _ _)
+      | forall Y _ _ _, Y  =>
+        simple refine (HitRec.hrecursion target P _ _ _)
+      | forall Y _ _ _ _, Y  =>
+        simple refine (HitRec.hrecursion target P _ _ _ _)
+      | forall Y _ _ _ _ _, Y =>
+        simple refine (HitRec.hrecursion target P _ _ _ _ _)
+      | forall Y _ _ _ _ _ _, Y =>
+        simple refine (HitRec.hrecursion target P _ _ _ _ _ _)
+      | forall Y _ _ _ _ _ _ _, Y =>
+        simple refine (HitRec.hrecursion target P _ _ _ _ _ _ _)
+      | forall Y _ _ _ _ _ _ _ _, Y =>
+        simple refine (HitRec.hrecursion target P _ _ _ _ _ _ _ _)
+      | forall Y _ _ _ _ _ _ _ _ _, Y =>
+        simple refine (HitRec.hrecursion target P _ _ _ _ _ _ _ _ _)
+      | forall Y _ _ _ _ _ _ _ _ _ _, Y =>
+        simple refine (HitRec.hrecursion target P _ _ _ _ _ _ _ _ _ _)
+      | _ => fail "Cannot handle the recursion principle (too many parameters?) :("
+     end
+    end
+  end.
+
+Ltac hrecursion x := hrecursion_ x; simpl; try (clear x).
+
+

Mod2.v

@@ -1,6 +1,6 @@
-Require Import HoTT.
 Require Export HoTT.
 
+Require Import HitTactics.
 Module Export modulo.
 
 Private Inductive Mod2 : Type0 :=
@@ -51,44 +51,38 @@ Axiom Mod2_rec_beta_mod : forall
   (mod' : a = s (s a))
   , ap (Mod2_rec P a s mod') mod = mod'.
 
+
+Definition Mod2CL : HitRec.class Mod2 _ := 
+   HitRec.Class Mod2 (fun x P a s p => Mod2_ind P a s p x).
+Canonical Structure Mod2ty : HitRec.type := HitRec.Pack Mod2 _ Mod2CL.
+
 End modulo.
 
-Definition negate : Mod2 -> Mod2.
-Proof.
-refine (Mod2_ind _ _ _ _).
- Unshelve.
- Focus 2.
- apply (succ Z).
-
- Focus 2.
- intros.
- apply (succ H).
-
- simpl.
- rewrite transport_const.
- rewrite <- mod.
- reflexivity.
-Defined.
 
 Theorem modulo2 : forall n : Mod2, n = succ(succ n).
 Proof.
-refine (Mod2_ind _ _ _ _).
+intro n.
- Unshelve.
+hrecursion n.
- Focus 2.
+- apply mod.
- apply mod.
+- intros n p.
+  apply (ap succ p).
+- simpl.
+  etransitivity.
+  eapply (@transport_paths_FlFr _ _ idmap (fun n => succ (succ n))).
+  hott_simpl.
+  apply ap_compose.
+Defined.
 
- Focus 2.
+Definition negate : Mod2 -> Mod2.
- intro n.
+Proof.
- intro p.
+intro x.
- apply (ap succ p).
+hrecursion x.
-
+- apply Z. 
- simpl.
+- intros. apply (succ H).
- rewrite @HoTT.Types.Paths.transport_paths_FlFr.
+- simpl.
- rewrite ap_idmap.
+  etransitivity.
- rewrite concat_Vp.
+  eapply transport_const.
- rewrite concat_1p.
+  eapply modulo2.
- rewrite ap_compose.
- reflexivity.
 Defined.
 
 Definition plus : Mod2 -> Mod2 -> Mod2.

_CoqProject

@@ -1,5 +1,6 @@
 -R . "" COQC = hoqc COQDEP = hoqdep
 
+HitTactics.v
 Mod2.v
 FinSets.v
 Expressions.v