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|
(* Check forward dependencies *)
Check [P:nat->Prop][Q][A:(P O)->Q][B:(n:nat)(P (S n))->Q][x]
<[_]Q>Cases x of
| (exist O H) => (A H)
| (exist (S n) H) => (B n H)
end.
(* Check dependencies in anonymous arguments (from FTA/listn.v) *)
Inductive listn [A:Set] : nat->Set :=
niln: (listn A O)
| consn: (a:A)(n:nat)(listn A n)->(listn A (S n)).
Section Folding.
Variables B, C : Set.
Variable g : B -> C -> C.
Variable c : C.
Fixpoint foldrn [n:nat; bs:(listn B n)] : C :=
Cases bs of niln => c
| (consn b _ tl) => (g b (foldrn ? tl))
end.
End Folding.
(* -------------------------------------------------------------------- *)
(* Example to test patterns matching on dependent families *)
(* This exemple extracted from the developement done by Nacira Chabane *)
(* (equipe Paris 6) *)
(* -------------------------------------------------------------------- *)
Require Prelude.
Require Logic_Type.
Section Orderings.
Variable U: Type.
Definition Relation := U -> U -> Prop.
Variable R: Relation.
Definition Reflexive : Prop := (x: U) (R x x).
Definition Transitive : Prop := (x,y,z: U) (R x y) -> (R y z) -> (R x z).
Definition Symmetric : Prop := (x,y: U) (R x y) -> (R y x).
Definition Antisymmetric : Prop :=
(x,y: U) (R x y) -> (R y x) -> x==y.
Definition contains : Relation -> Relation -> Prop :=
[R,R': Relation] (x,y: U) (R' x y) -> (R x y).
Definition same_relation : Relation -> Relation -> Prop :=
[R,R': Relation] (contains R R') /\ (contains R' R).
Inductive Equivalence : Prop :=
Build_Equivalence:
Reflexive -> Transitive -> Symmetric -> Equivalence.
Inductive PER : Prop :=
Build_PER: Symmetric -> Transitive -> PER.
End Orderings.
(***** Setoid *******)
Inductive Setoid : Type
:= Build_Setoid : (S:Type)(R:(Relation S))(Equivalence ? R) -> Setoid.
Definition elem := [A:Setoid] let (S,R,e)=A in S.
Grammar constr constr1 :=
elem [ "|" constr0($s) "|"] -> [ (elem $s) ].
Definition equal := [A:Setoid]
<[s:Setoid](Relation |s|)>let (S,R,e)=A in R.
Grammar constr constr1 :=
equal [ constr0($c) "=" "%" "S" constr0($c2) ] ->
[ (equal ? $c $c2) ].
Axiom prf_equiv : (A:Setoid)(Equivalence |A| (equal A)).
Axiom prf_refl : (A:Setoid)(Reflexive |A| (equal A)).
Axiom prf_sym : (A:Setoid)(Symmetric |A| (equal A)).
Axiom prf_trans : (A:Setoid)(Transitive |A| (equal A)).
Section Maps.
Variables A,B: Setoid.
Definition Map_law := [f:|A| -> |B|]
(x,y:|A|) x =%S y -> (f x) =%S (f y).
Inductive Map : Type :=
Build_Map : (f:|A| -> |B|)(p:(Map_law f))Map.
Definition explicit_ap := [m:Map] <|A| -> |B|>Match m with
[f:?][p:?]f end.
Axiom pres : (m:Map)(Map_law (explicit_ap m)).
Definition ext := [f,g:Map]
(x:|A|) (explicit_ap f x) =%S (explicit_ap g x).
Axiom Equiv_map_eq : (Equivalence Map ext).
Definition Map_setoid := (Build_Setoid Map ext Equiv_map_eq).
End Maps.
Notation ap := (explicit_ap ? ?).
Grammar constr constr8 :=
map_setoid [ constr7($c1) "=>" constr8($c2) ]
-> [ (Map_setoid $c1 $c2) ].
Definition ap2 := [A,B,C:Setoid][f:|(A=>(B=>C))|][a:|A|] (ap (ap f a)).
(***** posint ******)
Inductive posint : Type
:= Z : posint | Suc : posint -> posint.
Axiom f_equal : (A,B:Type)(f:A->B)(x,y:A) x==y -> (f x)==(f y).
Axiom eq_Suc : (n,m:posint) n==m -> (Suc n)==(Suc m).
(* The predecessor function *)
Definition pred : posint->posint
:= [n:posint](<posint>Case n of (* Z *) Z
(* Suc u *) [u:posint]u end).
Axiom pred_Sucn : (m:posint) m==(pred (Suc m)).
Axiom eq_add_Suc : (n,m:posint) (Suc n)==(Suc m) -> n==m.
Axiom not_eq_Suc : (n,m:posint) ~(n==m) -> ~((Suc n)==(Suc m)).
Definition IsSuc : posint->Prop
:= [n:posint](<Prop>Case n of (* Z *) False
(* Suc p *) [p:posint]True end).
Definition IsZero :posint->Prop :=
[n:posint]<Prop>Match n with
True
[p:posint][H:Prop]False end.
Axiom Z_Suc : (n:posint) ~(Z==(Suc n)).
Axiom Suc_Z: (n:posint) ~(Suc n)==Z.
Axiom n_Sucn : (n:posint) ~(n==(Suc n)).
Axiom Sucn_n : (n:posint) ~(Suc n)==n.
Axiom eqT_symt : (a,b:posint) ~(a==b)->~(b==a).
(******* Dsetoid *****)
Definition Decidable :=[A:Type][R:(Relation A)]
(x,y:A)(R x y) \/ ~(R x y).
Record DSetoid : Type :=
{Set_of : Setoid;
prf_decid : (Decidable |Set_of| (equal Set_of))}.
(* example de Dsetoide d'entiers *)
Axiom eqT_equiv : (Equivalence posint (eqT posint)).
Axiom Eq_posint_deci : (Decidable posint (eqT posint)).
(* Dsetoide des posint*)
Definition Set_of_posint := (Build_Setoid posint (eqT posint) eqT_equiv).
Definition Dposint := (Build_DSetoid Set_of_posint Eq_posint_deci).
(**************************************)
(* Definition des signatures *)
(* une signature est un ensemble d'operateurs muni
de l'arite de chaque operateur *)
Section Sig.
Record Signature :Type :=
{Sigma : DSetoid;
Arity : (Map (Set_of Sigma) (Set_of Dposint))}.
Variable S:Signature.
Variable Var : DSetoid.
Mutual Inductive TERM : Type :=
var : |(Set_of Var)| -> TERM
| oper : (op: |(Set_of (Sigma S))| ) (LTERM (ap (Arity S) op)) -> TERM
with
LTERM : posint -> Type :=
nil : (LTERM Z)
| cons : TERM -> (n:posint)(LTERM n) -> (LTERM (Suc n)).
(* -------------------------------------------------------------------- *)
(* Examples *)
(* -------------------------------------------------------------------- *)
Parameter t1,t2: TERM.
Type
Cases t1 t2 of
| (var v1) (var v2) => True
| (oper op1 l1) (oper op2 l2) => False
| _ _ => False
end.
Parameter n2:posint.
Parameter l1, l2:(LTERM n2).
Type
Cases l1 l2 of
nil nil => True
| (cons v m y) nil => False
| _ _ => False
end.
Type Cases l1 l2 of
nil nil => True
| (cons u n x) (cons v m y) =>False
| _ _ => False
end.
Definition equalT [t1:TERM]:TERM->Prop :=
[t2:TERM]
Cases t1 t2 of
(var v1) (var v2) => True
| (oper op1 l1) (oper op2 l2) => False
| _ _ => False
end.
Definition EqListT [n1:posint;l1:(LTERM n1)]: (n2:posint)(LTERM n2)->Prop :=
[n2:posint][l2:(LTERM n2)]
Cases l1 l2 of
nil nil => True
| (cons t1 n1' l1') (cons t2 n2' l2') => False
| _ _ => False
end.
Reset equalT.
(* ------------------------------------------------------------------*)
(* Initial exemple (without patterns) *)
(*-------------------------------------------------------------------*)
Fixpoint equalT [t1:TERM]:TERM->Prop :=
<TERM->Prop>Case t1 of
(*var*) [v1:|(Set_of Var)|][t2:TERM]
<Prop>Case t2 of
(*var*)[v2:|(Set_of Var)|] (v1 =%S v2)
(*oper*)[op2:|(Set_of (Sigma S))|][_:(LTERM (ap (Arity S) op2))]False
end
(*oper*)[op1:|(Set_of (Sigma S))|]
[l1:(LTERM (ap (Arity S) op1))][t2:TERM]
<Prop>Case t2 of
(*var*)[v2:|(Set_of Var)|]False
(*oper*)[op2:|(Set_of (Sigma S))|]
[l2:(LTERM (ap (Arity S) op2))]
((op1=%S op2)/\ (EqListT (ap (Arity S) op1) l1 (ap (Arity S) op2) l2))
end
end
with EqListT [n1:posint;l1:(LTERM n1)]: (n2:posint)(LTERM n2)->Prop :=
<[_:posint](n2:posint)(LTERM n2)->Prop>Case l1 of
(*nil*) [n2:posint][l2:(LTERM n2)]
<[_:posint]Prop>Case l2 of
(*nil*)True
(*cons*)[t2:TERM][n2':posint][l2':(LTERM n2')]False
end
(*cons*)[t1:TERM][n1':posint][l1':(LTERM n1')]
[n2:posint][l2:(LTERM n2)]
<[_:posint]Prop>Case l2 of
(*nil*) False
(*cons*)[t2:TERM][n2':posint][l2':(LTERM n2')]
((equalT t1 t2) /\ (EqListT n1' l1' n2' l2'))
end
end.
(* ---------------------------------------------------------------- *)
(* Version with simple patterns *)
(* ---------------------------------------------------------------- *)
Reset equalT.
Fixpoint equalT [t1:TERM]:TERM->Prop :=
Cases t1 of
(var v1) => [t2:TERM]
Cases t2 of
(var v2) => (v1 =%S v2)
| (oper op2 _) =>False
end
| (oper op1 l1) => [t2:TERM]
Cases t2 of
(var _) => False
| (oper op2 l2) => (op1=%S op2)
/\ (EqListT (ap (Arity S) op1) l1 (ap (Arity S) op2) l2)
end
end
with EqListT [n1:posint;l1:(LTERM n1)]: (n2:posint)(LTERM n2)->Prop :=
<[_:posint](n2:posint)(LTERM n2)->Prop>Cases l1 of
nil => [n2:posint][l2:(LTERM n2)]
Cases l2 of
nil => True
| _ => False
end
| (cons t1 n1' l1') => [n2:posint][l2:(LTERM n2)]
Cases l2 of
nil =>False
| (cons t2 n2' l2') => (equalT t1 t2)
/\ (EqListT n1' l1' n2' l2')
end
end.
Reset equalT.
Fixpoint equalT [t1:TERM]:TERM->Prop :=
Cases t1 of
(var v1) => [t2:TERM]
Cases t2 of
(var v2) => (v1 =%S v2)
| (oper op2 _) =>False
end
| (oper op1 l1) => [t2:TERM]
Cases t2 of
(var _) => False
| (oper op2 l2) => (op1=%S op2)
/\ (EqListT (ap (Arity S) op1) l1 (ap (Arity S) op2) l2)
end
end
with EqListT [n1:posint;l1:(LTERM n1)]: (n2:posint)(LTERM n2)->Prop :=
[n2:posint][l2:(LTERM n2)]
Cases l1 of
nil =>
Cases l2 of
nil => True
| _ => False
end
| (cons t1 n1' l1') => Cases l2 of
nil =>False
| (cons t2 n2' l2') => (equalT t1 t2)
/\ (EqListT n1' l1' n2' l2')
end
end.
(* ---------------------------------------------------------------- *)
(* Version with multiple patterns *)
(* ---------------------------------------------------------------- *)
Reset equalT.
Fixpoint equalT [t1:TERM]:TERM->Prop :=
[t2:TERM]
Cases t1 t2 of
(var v1) (var v2) => (v1 =%S v2)
| (oper op1 l1) (oper op2 l2) =>
(op1=%S op2) /\ (EqListT (ap (Arity S) op1) l1 (ap (Arity S) op2) l2)
| _ _ => False
end
with EqListT [n1:posint;l1:(LTERM n1)]: (n2:posint)(LTERM n2)->Prop :=
[n2:posint][l2:(LTERM n2)]
Cases l1 l2 of
nil nil => True
| (cons t1 n1' l1') (cons t2 n2' l2') => (equalT t1 t2)
/\ (EqListT n1' l1' n2' l2')
| _ _ => False
end.
(* ------------------------------------------------------------------ *)
End Sig.
(* Exemple soumis par Bruno *)
Definition bProp [b:bool] : Prop :=
if b then True else False.
Definition f0 [F:False;ty:bool]: (bProp ty) :=
<[_:bool][ty:bool](bProp ty)>Cases ty ty of
true true => I
| _ false => F
| _ true => I
end.
|
