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(* Check the synthesis of predicate from a cast in case of matching of
the first component (here [list bool]) of a dependent type (here [sigS])
(Simplification of an example from file parsing2.v of the Coq'Art
exercises) *)
Require Import List.
Variable parse_rel : list bool -> list bool -> nat -> Prop.
Variables (l0 : list bool)
(rec :
forall l' : list bool,
length l' <= S (length l0) ->
{l'' : list bool &
{t : nat | parse_rel l' l'' t /\ length l'' <= length l'}} +
{(forall (l'' : list bool) (t : nat), ~ parse_rel l' l'' t)}).
Axiom HHH : forall A : Prop, A.
Check
(match rec l0 (HHH _) with
| inleft (existS _ (false :: l1) _) => inright _ (HHH _)
| inleft (existS _ (true :: l1) (exist _ t1 (conj Hp Hl))) =>
inright _ (HHH _)
| inleft (existS _ _ _) => inright _ (HHH _)
| inright Hnp => inright _ (HHH _)
end
:{l'' : list bool &
{t : nat | parse_rel (true :: l0) l'' t /\ length l'' <= S (length l0)}} +
{(forall (l'' : list bool) (t : nat), ~ parse_rel (true :: l0) l'' t)}).
(* The same but with relative links to l0 and rec *)
Check
(fun (l0 : list bool)
(rec : forall l' : list bool,
length l' <= S (length l0) ->
{l'' : list bool &
{t : nat | parse_rel l' l'' t /\ length l'' <= length l'}} +
{(forall (l'' : list bool) (t : nat), ~ parse_rel l' l'' t)}) =>
match rec l0 (HHH _) with
| inleft (existS _ (false :: l1) _) => inright _ (HHH _)
| inleft (existS _ (true :: l1) (exist _ t1 (conj Hp Hl))) =>
inright _ (HHH _)
| inleft (existS _ _ _) => inright _ (HHH _)
| inright Hnp => inright _ (HHH _)
end
:{l'' : list bool &
{t : nat | parse_rel (true :: l0) l'' t /\ length l'' <= S (length l0)}} +
{(forall (l'' : list bool) (t : nat), ~ parse_rel (true :: l0) l'' t)}).
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