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(* The "?" of cons and eq should be inferred *)
Variable list : Set -> Set.
Variable cons : forall T : Set, T -> list T -> list T.
Check (forall n : list nat, exists l : _, (exists x : _, n = cons _ x l)).
(* Examples provided by Eduardo Gimenez *)
Definition c A (Q : (nat * A -> Prop) -> Prop) P :=
Q (fun p : nat * A => let (i, v) := p in P i v).
(* What does this test ? *)
Require Import List.
Definition list_forall_bool (A : Set) (p : A -> bool)
(l : list A) : bool :=
fold_right (fun a r => if p a then r else false) true l.
(* Checks that solvable ? in the lambda prefix of the definition are harmless*)
Parameter A1 A2 F B C : Set.
Parameter f : F -> A1 -> B.
Definition f1 frm0 a1 : B := f frm0 a1.
(* Checks that solvable ? in the type part of the definition are harmless *)
Definition f2 frm0 a1 : B := f frm0 a1.
(* Checks that sorts that are evars are handled correctly (bug 705) *)
Require Import List.
Fixpoint build (nl : list nat) :
match nl with
| nil => True
| _ => False
end -> unit :=
match nl return (match nl with
| nil => True
| _ => False
end -> unit) with
| nil => fun _ => tt
| n :: rest =>
match n with
| O => fun _ => tt
| S m => fun a => build rest (False_ind _ a)
end
end.
(* Checks that disjoint contexts are correctly set by restrict_hyp *)
(* Bug de 1999 corrigé en déc 2004 *)
Check
(let p :=
fun (m : nat) f (n : nat) =>
match f m n with
| exist a b => exist _ a b
end in
p
:forall x : nat,
(forall y n : nat, {q : nat | y = q * n}) ->
forall n : nat, {q : nat | x = q * n}).
(* Check instantiation of nested evars (bug #1089) *)
Check (fun f:(forall (v:Type->Type), v (v nat) -> nat) => f _ (Some (Some O))).
(* This used to fail with anomaly "evar was not declared" in V8.0pl3 *)
Theorem contradiction : forall p, ~ p -> p -> False.
Proof. trivial. Qed.
Hint Resolve contradiction.
Goal False.
eauto.
Abort.
(* This used to fail in V8.1beta because first-order unification was
used before using type information *)
Check (exist _ O (refl_equal 0) : {n:nat|n=0}).
Check (exist _ O I : {n:nat|True}).
(* An example (initially from Marseille/Fairisle) that involves an evar with
different solutions (Input, Output or bool) that may or may not be
considered distinct depending on which kind of conversion is used *)
Section A.
Definition STATE := (nat * bool)%type.
Let Input := bool.
Let Output := bool.
Parameter Out : STATE -> Output.
Check fun (s : STATE) (reg : Input) => reg = Out s.
End A.
(* The return predicate found should be: "in _=U return U" *)
(* (feature already available in V8.0) *)
Definition g (T1 T2:Type) (x:T1) (e:T1=T2) : T2 :=
match e with
| refl_equal => x
end.
(* An example extracted from FMapAVL which (may) test restriction on
evars problems of the form ?n[args1]=?n[args2] with distinct args1
and args2 *)
Set Implicit Arguments.
Parameter t:Set->Set.
Parameter map:forall elt elt' : Set, (elt -> elt') -> t elt -> t elt'.
Parameter avl: forall elt : Set, t elt -> Prop.
Parameter bst: forall elt : Set, t elt -> Prop.
Parameter map_avl: forall (elt elt' : Set) (f : elt -> elt') (m : t elt),
avl m -> avl (map f m).
Parameter map_bst: forall (elt elt' : Set) (f : elt -> elt') (m : t elt),
bst m -> bst (map f m).
Record bbst (elt:Set) : Set :=
Bbst {this :> t elt; is_bst : bst this; is_avl: avl this}.
Definition t' := bbst.
Section B.
Variables elt elt': Set.
Definition map' f (m:t' elt) : t' elt' :=
Bbst (map_bst f m.(is_bst)) (map_avl f m.(is_avl)).
End B.
Unset Implicit Arguments.
(* An example from Lexicographic_Exponentiation that tests the
contraction of reducible fixpoints in type inference *)
Require Import List.
Check (fun (A:Set) (a b x:A) (l:list A)
(H : l ++ cons x nil = cons b (cons a nil)) =>
app_inj_tail l (cons b nil) _ _ H).
(* An example from NMake (simplified), that uses restriction in solve_refl *)
Parameter h:(nat->nat)->(nat->nat).
Fixpoint G p cont {struct p} :=
h (fun n => match p with O => cont | S p => G p cont end n).
(* An example from Bordeaux/Cantor that applies evar restriction
below a binder *)
Require Import Relations.
Parameter lex : forall (A B : Set), (forall (a1 a2:A), {a1=a2}+{a1<>a2})
-> relation A -> relation B -> A * B -> A * B -> Prop.
Check
forall (A B : Set) eq_A_dec o1 o2,
antisymmetric A o1 -> transitive A o1 -> transitive B o2 ->
transitive _ (lex _ _ eq_A_dec o1 o2).
(* Another example from Julien Forest that tests unification below binders *)
Require Import List.
Set Implicit Arguments.
Parameter
merge : forall (A B : Set) (eqA : forall (a1 a2 : A), {a1=a2}+{a1<>a2})
(eqB : forall (b1 b2 : B), {b1=b2}+{b1<>b2})
(partial_res l : list (A*B)), option (list (A*B)).
Axiom merge_correct :
forall (A B : Set) eqA eqB (l1 l2 : list (A*B)),
(forall a2 b2 c2, In (a2,b2) l2 -> In (a2,c2) l2 -> b2 = c2) ->
match merge eqA eqB l1 l2 with _ => True end.
Unset Implicit Arguments.
(* An example from Bordeaux/Additions that tests restriction below binders *)
Section Additions_while.
Variable A : Set.
Variables P Q : A -> Prop.
Variable le : A -> A -> Prop.
Hypothesis Q_dec : forall s : A, P s -> {Q s} + {~ Q s}.
Hypothesis le_step : forall s : A, ~ Q s -> P s -> {s' | P s' /\ le s' s}.
Hypothesis le_wf : well_founded le.
Lemma loopexec : forall s : A, P s -> {s' : A | P s' /\ Q s'}.
refine
(well_founded_induction_type le_wf (fun s => _ -> {s' : A | _ /\ _})
(fun s hr i =>
match Q_dec s i with
| left _ => _
| right _ =>
match le_step s _ _ with
| exist s' h' =>
match hr s' _ _ with
| exist s'' _ => exist _ s'' _
end
end
end)).
Abort.
End Additions_while.
(* Two examples from G. Melquiond (bugs #1878 and #1884) *)
Parameter F1 G1 : nat -> Prop.
Goal forall x : nat, F1 x -> G1 x.
refine (fun x H => proj2 (_ x H)).
Abort.
Goal forall x : nat, F1 x -> G1 x.
refine (fun x H => proj2 (_ x H) _).
Abort.
(* Remark: the following example does not succeed any longer in 8.2 because,
the algorithm is more general and does exclude a solution that it should
exclude for typing reason. Handling of types and backtracking is still to
be done
Section S.
Variables A B : nat -> Prop.
Goal forall x : nat, A x -> B x.
refine (fun x H => proj2 (_ x H) _).
Abort.
End S.
*)
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