Some more debugging of [Equations], unification still not perfect.

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@11396 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 217 insertions, 26 deletions

diff --git a/test-suite/success/Equations.v b/test-suite/success/Equations.v
index 1609401855..f452aeadb8 100644
--- a/test-suite/success/Equations.v
+++ b/test-suite/success/Equations.v
@@ -1,4 +1,3 @@
-Require Import Bvector.
 Require Import Program.
 
 Equations neg (b : bool) : bool :=
@@ -54,6 +53,19 @@ zip'' A f (cons a v) nil def := def.
 
 Eval compute in @zip''.
 
+Inductive fin : nat -> Set :=
+| fz : Π {n}, fin (S n)
+| fs : Π {n}, fin n -> fin (S n).
+
+Inductive finle : Π (n : nat) (x : fin n) (y : fin n), Prop :=
+| leqz : Π {n j}, finle (S n) fz j
+| leqs : Π {n i j}, finle n i j -> finle (S n) (fs i) (fs j).
+
+Implicit Arguments finle [[n]].
+
+
+Require Import Bvector.
+
 Implicit Arguments Vnil [[A]].
 Implicit Arguments Vcons [[A] [n]].
 
@@ -71,15 +83,169 @@ Eval compute in (vmap id (@Vnil nat)).
 Eval compute in (vmap id (@Vcons nat 2 _ Vnil)).
 Eval compute in @vmap.
 
-Inductive fin : nat -> Set :=
-| fz : Π {n}, fin (S n)
-| fs : Π {n}, fin n -> fin (S n).
+Record vlast_comp (A : Type) (n : nat) (v : vector A (S n)) : Type := { vlast_return : A }.
 
-Inductive finle : Π (n : nat) (x : fin n) (y : fin n), Prop :=
-| leqz : Π {n j}, finle (S n) fz j
-| leqs : Π {n i j}, finle n i j -> finle (S n) (fs i) (fs j).
+Class Comp (comp : Type) :=
+  return_type : comp -> Type ; call : Π c : comp, return_type c.
 
-Implicit Arguments finle [[n]].
+Instance vlast_Comp A n v : Comp (vlast_comp A n v) :=
+  return_type := λ c, A ;
+  call := λ c, vlast_return A n v c.
+
+Ltac ind v := 
+  intros until v ; generalize_eqs_vars v ; induction v ; simplify_dep_elim.
+
+
+Tactic Notation "rec" ident(v) "as" simple_intropattern(p) :=
+  (try intros until v) ; generalize_eqs_vars v ; induction v as p ; simplify_dep_elim ; 
+    simpl_IH_eqs.
+
+Tactic Notation "rec" hyp(v) :=
+  (try intros until v) ; generalize_eqs_vars v ; induction v ; simplify_dep_elim ; 
+    simpl_IH_eqs.
+
+Tactic Notation "case" "*" hyp(v) "as" simple_intropattern(p) :=
+  (try intros until v) ; generalize_eqs_vars v ; destruct v as p ; simplify_dep_elim.
+
+Tactic Notation "case" "*" hyp(v) :=
+  (try intros until v) ; generalize_eqs_vars v ; destruct v ; simplify_dep_elim.
+
+Tactic Notation "=>" constr(v) := 
+  constructor ; 
+    match goal with
+      | [ |- ?T ] => refine (v:T)
+    end.
+
+Ltac make_refls_term c app :=
+  match c with
+    | _ = _ -> ?c' => make_refls_term (app (@refl_equal _ _))
+    | @JMeq _ _ _ _ -> ?c' => make_refls_term (app (@JMeq_refl _ _))
+    | _ => constr:app
+  end.
+
+Ltac make_refls_IH c app :=
+  match c with
+    | Π x : _, ?c' => make_refls_IH (app _)
+    | _ => make_refls_term c app
+  end.
+
+Ltac simpl_IH H :=
+  let ty := type of H in
+    make_refls_IH ty H.
+
+Ltac move_top H :=
+  match reverse goal with [ H' : _ |- _ ] => move H after H' end.
+
+Ltac simplify_dep_elim_hyp H evhyp :=
+  let ev := eval compute in evhyp in
+    subst evhyp ; intros evhyp ; move_top evhyp ; simplify_dep_elim ; 
+      try clear H ; reverse ; intro evhyp ; eapply evhyp.
+
+(* Tactic Notation "strengthen" hyp(H) := *)
+(*   let strhyp := fresh "H'" in *)
+(*   let ty := type of H in *)
+(*     on_last_hyp ltac:(fun id => *)
+(*       reverse ; evar (strhyp:Type) ; intros until id). *)
+
+(*         assert(strhyp -> ty) ; [ simplify_dep_elim_hyp strhyp | intros ]). *)
+
+
+Equations Below_nat (P : nat -> Type) (n : nat) : Type :=
+Below_nat P 0 := unit ;
+Below_nat P (S n) := prod (P n) (Below_nat P n).
+
+Equations below_nat (P : nat -> Type) n (step : Π (n : nat), Below_nat P n -> P n) : Below_nat P n :=
+below_nat P 0 step := tt ;
+below_nat P (S n) step := let rest := below_nat P n step in
+  (step n rest, rest).
+
+Class BelowPack (A : Type) :=
+  Below : Type ; below : Below.
+
+Instance nat_BelowPack : BelowPack nat :=
+  Below := Π P n step, Below_nat P n ;
+  below := λ P n step, below_nat P n (step P).
+
+Definition rec_nat (P : nat -> Type) n (step : Π n, Below_nat P n -> P n) : P n :=
+  step n (below_nat P n step).
+
+Fixpoint Below_vector (P : Π A n, vector A n -> Type) A n (v : vector A n) : Type :=
+  match v with Vnil => unit | Vcons a n' v' => prod (P A n' v') (Below_vector P A n' v') end.
+
+(* Equations Below_vector (P : Π A n, vector A n -> Type) A n (v : vector A n) : Type := *)
+(* Below_vector P A ?(0) Vnil := unit ; *)
+(* Below_vector P A ?(S n) (Vcons a v) := prod (P A n v) (Below_vector P A n v). *)
+
+Equations below_vector (P : Π A n, vector A n -> Type) A n (v : vector A n)
+  (step : Π A n (v : vector A n), Below_vector P A n v -> P A n v) : Below_vector P A n v :=
+below_vector P A 0 Vnil step := tt ;
+below_vector P A (S n) (Vcons a v) step := 
+  let rest := below_vector P A n v step in
+    (step A n v rest, rest).
+
+Instance vector_BelowPack : BelowPack (Π A n, vector A n) :=
+  Below := Π P A n v step, Below_vector P A n v ;
+  below := λ P A n v step, below_vector P A n v (step P).
+
+Instance vector_noargs_BelowPack A n : BelowPack (vector A n) :=
+  Below := Π P v step, Below_vector P A n v ;
+  below := λ P v step, below_vector P A n v (step P).
+
+Definition rec_vector (P : Π A n, vector A n -> Type) A n v
+  (step : Π A n (v : vector A n), Below_vector P A n v -> P A n v) : P A n v :=
+  step A n v (below_vector P A n v step).
+
+Class Recursor (A : Type) (BP : BelowPack A) := 
+  rec_type : Π x : A, Type ; rec : Π x : A, rec_type x.
+
+Instance nat_Recursor : Recursor nat nat_BelowPack :=
+  rec_type := λ n, Π P step, P n ;
+  rec := λ n P step, rec_nat P n (step P).
+
+(* Instance vect_Recursor : Recursor (Π A n, vector A n) vector_BelowPack := *)
+(*   rec_type := Π (P : Π A n, vector A n -> Type) step A n v, P A n v; *)
+(*   rec := λ P step A n v, rec_vector P A n v step. *)
+
+Instance vect_Recursor_noargs A n : Recursor (vector A n) (vector_noargs_BelowPack A n) :=
+  rec_type := λ v, Π (P : Π A n, vector A n -> Type) step, P A n v;
+  rec := λ v P step, rec_vector P A n v step.
+
+Implicit Arguments Below_vector [P A n].
+
+Notation " x ~= y " := (@JMeq _ x _ y) (at level 70, no associativity).
+
+Ltac idify := match goal with [ |- ?T ] => change (id T) end.
+
+(*  induction v as [ | a v']; simplify_dep_elim. *)
+(*   on_last_hyp ltac:(fun id => *)
+(*     let strhyp := fresh "H'" in *)
+(*     let ty := type of IHv in *)
+(*       reverse ; evar (strhyp:Type) ; intros ; *)
+(*         let newhyp := fresh "IHv''" in *)
+(*           assert(ty = strhyp) ; [ idtac (* simplify_dep_elim_hyp IHv strhyp *) | idtac ]). *)
+(* (*         subst strhyp ; intros ; try (apply newhyp in IHv) ]). *) *)
+
+Ltac simpl_rec X :=
+  match type of X with
+    | ?f (?c ?x) =>  
+      let X1 := fresh in
+      let X2 := fresh in
+        hnf in X ; destruct X as [X1 X2] ; simplify_hyp X1
+    | _ => idtac
+  end.
+
+Ltac recur v :=
+  try intros until v ; generalize_eqs_vars v ; reverse ;
+  intros until v ; assert (recv:=rec v) ; simpl in recv ;
+    eapply recv; clear ; simplify_dep_elim.
+
+Definition vlast {A} {n} (v : vector A (S n)) : vlast_comp A n v.
+recur v. case* v0. case* v0. => a. simpl_rec X.
+=> (call H).
+Defined.
+  
+Record trans_comp {n:nat} {i j k : fin n} (p : finle i j) (q : finle j k) :=  {
+  return_comp : finle i k }.
 
 Equations trans {n} {i j k : fin n} (p : finle i j) (q : finle j k) : finle i k :=
 trans (S n) fz j k leqz q := leqz ;
@@ -160,14 +326,14 @@ Equations tabulate {A} {n} (f : fin n -> A) : vector A n :=
 tabulate A 0 f := Vnil ;
 tabulate A (S n) f := Vcons (f fz) (tabulate (f ∘ fs)).
 
-Equations vlast {A} {n} (v : vector A (S n)) : A :=
-vlast A 0 (Vcons a Vnil) := a ;
-vlast A (S n) (Vcons a v) := vlast v.
+Equations vlast' {A} {n} (v : vector A (S n)) : A :=
+vlast' A 0 (Vcons a Vnil) := a ;
+vlast' A (S n) (Vcons a v) := vlast' v.
 
-Lemma vlast_equation1 A (a : A) : vlast (Vcons a Vnil) = a.
+Lemma vlast_equation1 A (a : A) : vlast' (Vcons a Vnil) = a.
 Proof. intros. compute ; reflexivity. Qed.
 
-Lemma vlast_equation2 A n a v : @vlast A (S n) (Vcons a v) = vlast v.
+Lemma vlast_equation2 A n a v : @vlast' A (S n) (Vcons a v) = vlast' v.
 Proof. intros. compute ; reflexivity. Qed.
 
 Print Assumptions vlast.
@@ -188,10 +354,10 @@ Lemma JMeq_Vcons_inj A n m a (x : vector A n) (y : vector A m) : n = m -> JMeq x
 Proof. intros until y. simplify_dep_elim. reflexivity. Qed.
 
 Equations NoConfusion_fin (P : Prop) {n : nat} (x y : fin n) : Prop :=
-NoConfusion_fin P (S n) fz fz := P -> P ;
-NoConfusion_fin P (S n) fz (fs y) := P ;
-NoConfusion_fin P (S n) (fs x) fz := P ;
-NoConfusion_fin P (S n) (fs x) (fs y) := (x = y -> P) -> P.
+NoConfusion_fin P ?(S n) fz fz := P -> P ;
+NoConfusion_fin P ?(S n) fz (fs y) := P ;
+NoConfusion_fin P ?(S n) (fs x) fz := P ;
+NoConfusion_fin P ?(S n) (fs x) (fs y) := (x = y -> P) -> P.
 
 Equations noConfusion_fin P (n : nat) (x y : fin n) (H : x = y) : NoConfusion_fin P x y :=
 noConfusion_fin P (S n) fz fz refl := λ p, p ;
@@ -201,14 +367,39 @@ Equations NoConfusion_vect (P : Prop) {A n} (x y : vector A n) : Prop :=
 NoConfusion_vect P A 0 Vnil Vnil := P -> P ;
 NoConfusion_vect P A (S n) (Vcons a x) (Vcons b y) := (a = b -> x = y -> P) -> P.
 
-Equations noConfusion_vect (P : Prop) A n (x y : vector A n) (H : x = y) : NoConfusion_vect P x y :=
-noConfusion_vect P A 0 Vnil Vnil refl := λ p, p ;
-noConfusion_vect P A (S n) (Vcons a v) (Vcons a v) refl := λ p : a = a -> v = v -> P, p refl refl.
+(* Equations noConfusion_vect (P : Prop) A n (x y : vector A n) (H : x = y) : NoConfusion_vect P x y := *)
+(* noConfusion_vect P A 0 Vnil Vnil refl := λ p, p ; *)
+(* noConfusion_vect P A (S n) (Vcons a v) (Vcons a v) refl := λ p : a = a -> v = v -> P, p refl refl. *)
+
+(* Instance fin_noconf n : NoConfusionPackage (fin n) := *)
+(*   NoConfusion := λ P, Π x y, x = y -> NoConfusion_fin P x y ; *)
+(*   noConfusion := λ P x y, noConfusion_fin P n x y. *)
+
+(* Instance vect_noconf A n : NoConfusionPackage (vector A n) := *)
+(*   NoConfusion := λ P, Π x y, x = y -> NoConfusion_vect P x y ; *)
+(*   noConfusion := λ P x y, noConfusion_vect P A n x y. *)
+
+Equations fog {n} (f : fin n) : nat :=
+fog (S n) fz := 0 ; fog (S n) (fs f) := S (fog f).
+
+About vapp.
+
+Inductive Split {X : Set}{m n : nat} : vector X (m + n) -> Set :=
+  append : Π (xs : vector X m)(ys : vector X n), Split (vapp xs ys).
+
+Implicit Arguments Split [[X]].
+
+Equations split {X : Set}(m n : nat) (xs : vector X (m + n)) : Split m n xs :=
+split X 0    n xs := append Vnil xs ;
+split X (S m) n (Vcons x xs) := 
+  let 'append xs' ys' in Split _ _ vec := split m n xs return Split (S m) n (Vcons x vec) in
+    append (Vcons x xs') ys'.
+
+Eval compute in (split 0 1 (vapp Vnil (Vcons 2 Vnil))).
+Eval compute in (split _ _ (vapp (Vcons 3 Vnil) (Vcons 2 Vnil))).
+
+Extraction Inline split_obligation_1 split_obligation_2.
 
-Instance fin_noconf n : NoConfusionPackage (fin n) :=
-  NoConfusion := λ P, Π x y, x = y -> NoConfusion_fin P x y ;
-  noConfusion := λ P x y, noConfusion_fin P n x y.
+Recursive Extraction split.
 
-Instance vect_noconf A n : NoConfusionPackage (vector A n) :=
-  NoConfusion := λ P, Π x y, x = y -> NoConfusion_vect P x y ;
-  noConfusion := λ P x y, noConfusion_vect P A n x y.
+Eval compute in @split.