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+Require Import Program.
+
+Equations neg (b : bool) : bool :=
+neg true := false ;
+neg false := true.
+ 
+Eval compute in neg.
+
+Require Import Coq.Lists.List.
+
+Equations head A (default : A) (l : list A) : A :=
+head A default nil := default ;
+head A default (cons a v) := a.
+
+Eval compute in head.
+
+Equations tail {A} (l : list A) : (list A) :=
+tail A nil := nil ;
+tail A (cons a v) := v.
+
+Eval compute in @tail.
+
+Eval compute in (tail (cons 1 nil)).
+
+Reserved Notation " x ++ y " (at level 60, right associativity).
+
+Equations app' {A} (l l' : list A) : (list A) :=
+app' A nil l := l ;
+app' A (cons a v) l := cons a (app' v l).
+
+Equations app (l l' : list nat) : list nat :=
+  [] ++ l       := l ;
+  (a :: v) ++ l := a :: (v ++ l) 
+
+where " x ++ y " := (app x y).
+
+Eval compute in @app'.
+
+Equations zip' {A} (f : A -> A -> A) (l l' : list A) : (list A) :=
+zip' A f nil nil := nil ;
+zip' A f (cons a v) (cons b w) := cons (f a b) (zip' f v w) ;
+zip' A f nil (cons b w) := nil ;
+zip' A f (cons a v) nil := nil.
+
+
+Eval compute in @zip'.
+
+Equations zip'' {A} (f : A -> A -> A) (l l' : list A) (def : list A) : (list A) :=
+zip'' A f nil nil def := nil ;
+zip'' A f (cons a v) (cons b w) def := cons (f a b) (zip'' f v w def) ;
+zip'' A f nil (cons b w) def := def ;
+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).
+
+Scheme finle_ind_dep := Induction for finle Sort Prop.
+
+Instance finle_ind_pack n x y : DependentEliminationPackage (finle n x y) :=
+  { elim_type := _ ; elim := finle_ind_dep }.
+
+Implicit Arguments finle [[n]].
+
+Require Import Bvector.
+
+Implicit Arguments Vnil [[A]].
+Implicit Arguments Vcons [[A] [n]].
+
+Equations vhead {A n} (v : vector A (S n)) : A := 
+vhead A n (Vcons a v) := a.
+
+Equations vmap {A B} (f : A -> B) {n} (v : vector A n) : (vector B n) :=
+vmap A B f 0 Vnil := Vnil ;
+vmap A B f (S n) (Vcons a v) := Vcons (f a) (vmap f v).
+
+Eval compute in (vmap id (@Vnil nat)).
+Eval compute in (vmap id (@Vcons nat 2 _ Vnil)).
+Eval compute in @vmap.
+
+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)
+  (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).
+
+(** Won't pass the guardness check which diverges anyway. *)
+
+(* 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 ; *)
+(* trans ?(S n) ?(fs i) ?(fs j) ?(fs k) (leqs p) (leqs q) := leqs (trans p q). *)
+
+(* Lemma trans_eq1 n (j k : fin (S n)) (q : finle j k) : trans leqz q = leqz. *)
+(* Proof. intros. simplify_equations ; reflexivity. Qed. *)
+
+(* Lemma trans_eq2 n i j k p q : @trans (S n) (fs i) (fs j) (fs k) (leqs p) (leqs q) = leqs (trans p q). *)
+(* Proof. intros. simplify_equations ; reflexivity. Qed. *)
+
+Section Image.
+  Context {S T : Type}.
+  Variable f : S -> T.
+  
+  Inductive Imf : T -> Type := imf (s : S) : Imf (f s).
+
+  Equations inv (t : T) (im : Imf t) : S :=
+  inv (f s) (imf s) := s.
+
+End Image.
+
+Section Univ.
+
+  Inductive univ : Set :=
+  | ubool | unat | uarrow (from:univ) (to:univ).
+
+  Equations interp (u : univ) : Type :=
+  interp ubool := bool ; interp unat := nat ; 
+  interp (uarrow from to) := interp from -> interp to.
+
+  Equations foo (u : univ) (el : interp u) : interp u :=
+  foo ubool true := false ;
+  foo ubool false := true ;
+  foo unat t := t ;
+  foo (uarrow from to) f := id ∘ f.
+
+  Eval lazy beta delta [ foo foo_obligation_1 foo_obligation_2 ] iota zeta in foo.
+
+End Univ.
+
+Eval compute in (foo ubool false).
+Eval compute in (foo (uarrow ubool ubool) negb).
+Eval compute in (foo (uarrow ubool ubool) id).
+
+Inductive foobar : Set := bar | baz.
+
+Equations bla (f : foobar) : bool :=
+bla bar := true ;
+bla baz := false.
+
+Eval simpl in bla.
+Print refl_equal.
+
+Notation "'refl'" := (@refl_equal _ _).
+
+Equations K {A} (x : A) (P : x = x -> Type) (p : P (refl_equal x)) (p : x = x) : P p :=
+K A x P p refl := p.
+
+Equations eq_sym {A} (x y : A) (H : x = y) : y = x :=
+eq_sym A x x refl := refl.
+
+Equations eq_trans {A} (x y z : A) (p : x = y) (q : y = z) : x = z :=
+eq_trans A x x x refl refl := refl.
+
+Lemma eq_trans_eq A x : @eq_trans A x x x refl refl = refl.
+Proof. reflexivity. Qed.
+
+Equations nth {A} {n} (v : vector A n) (f : fin n) : A :=
+nth A (S n) (Vcons a v) fz := a ;
+nth A (S n) (Vcons a v) (fs f) := nth v f.
+
+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 (n:=S n) v) := vlast v.
+
+Print Assumptions vlast.
+
+Equations vlast' {A} {n} (v : vector A (S n)) : A :=
+vlast' A ?(0) (Vcons a Vnil) := a ;
+vlast' A ?(S n) (Vcons a (n:=S n) v) := vlast' v.
+
+Lemma vlast_equation1 A (a : A) : vlast' (Vcons a Vnil) = a.
+Proof. intros. simplify_equations. reflexivity. Qed.
+
+Lemma vlast_equation2 A n a v : @vlast' A (S n) (Vcons a v) = vlast' v.
+Proof. intros. simplify_equations ; reflexivity. Qed.
+
+Print Assumptions vlast'.
+Print Assumptions nth. 
+Print Assumptions tabulate.
+
+Extraction vlast.
+Extraction vlast'.
+
+Equations vliat {A} {n} (v : vector A (S n)) : vector A n :=
+vliat A 0 (Vcons a Vnil) := Vnil ;
+vliat A (S n) (Vcons a v) := Vcons a (vliat v).
+
+Eval compute in (vliat (Vcons 2 (Vcons 5 (Vcons 4 Vnil)))).
+
+Equations vapp' {A} {n m} (v : vector A n) (w : vector A m) : vector A (n + m) :=
+vapp' A ?(0) m Vnil w := w ;
+vapp' A ?(S n) m (Vcons a v) w := Vcons a (vapp' v w).
+
+Eval compute in @vapp'.
+
+Fixpoint vapp {A n m} (v : vector A n) (w : vector A m) : vector A (n + m) :=
+  match v with
+    | Vnil => w
+    | Vcons a n' v' => Vcons a (vapp v' w)
+  end.
+
+Lemma JMeq_Vcons_inj A n m a (x : vector A n) (y : vector A m) : n = m -> JMeq x y -> JMeq (Vcons a x) (Vcons a y).
+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.
+
+Eval compute in NoConfusion_fin.
+Eval compute in NoConfusion_fin_comp.
+
+Print Assumptions NoConfusion_fin.
+
+Eval compute in (fun P n => NoConfusion_fin P (n:=S n) fz fz).
+
+(* 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 ; *)
+(* noConfusion_fin P (S n) (fs x) (fs x) refl := λ p : x = x -> P, p refl. *)
+
+Equations_nocomp 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.
+
+(* 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).
+
+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_nocomp 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.
+Recursive Extraction split.
+
+Eval compute in @split.