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Require Import List.
Require Import ZArith.
Require String. Open Scope string_scope.
Ltac Case s := let c := fresh "case" in set (c := s).
Set Implicit Arguments.
Unset Strict Implicit.
Inductive sv : Set :=
| I : Z -> sv
| S : list sv -> sv.
Section sv_induction.
Variables
(VP: sv -> Prop)
(LP: list sv -> Prop)
(VPint: forall n, VP (I n))
(VPset: forall vs, LP vs -> VP (S vs))
(lpcons: forall v vs, VP v -> LP vs -> LP (v::vs))
(lpnil: LP nil).
Fixpoint setl_value_indp (x:sv) {struct x}: VP x :=
match x as x return VP x with
| I n => VPint n
| S vs =>
VPset
((fix values_indp (vs:list sv) {struct vs}: (LP vs) :=
match vs as vs return LP vs with
| nil => lpnil
| v::vs => lpcons (setl_value_indp v) (values_indp vs)
end) vs)
end.
End sv_induction.
Inductive slt : sv -> sv -> Prop :=
| IC : forall z, slt (I z) (I z)
| IS : forall vs vs', slist_in vs vs' -> slt (S vs) (S vs')
with sin : sv -> list sv -> Prop :=
| Ihd : forall s s' sv', slt s s' -> sin s (s'::sv')
| Itl : forall s s' sv', sin s sv' -> sin s (s'::sv')
with slist_in : list sv -> list sv -> Prop :=
| Inil : forall sv',
slist_in nil sv'
| Icons : forall s sv sv',
sin s sv' ->
slist_in sv sv' ->
slist_in (s::sv) sv'.
Hint Constructors sin slt slist_in.
Require Import Program.
Program Fixpoint lt_dec (x y:sv) { struct x } : {slt x y}+{~slt x y} :=
match x with
| I x =>
match y with
| I y => if (Z_eq_dec x y) then in_left else in_right
| S ys => in_right
end
| S xs =>
match y with
| I y => in_right
| S ys =>
let fix list_in (xs ys:list sv) {struct xs} :
{slist_in xs ys} + {~slist_in xs ys} :=
match xs with
| nil => in_left
| x::xs =>
let fix elem_in (ys:list sv) : {sin x ys}+{~sin x ys} :=
match ys with
| nil => in_right
| y::ys => if lt_dec x y then in_left else if elem_in
ys then in_left else in_right
end
in
if elem_in ys then
if list_in xs ys then in_left else in_right
else in_right
end
in if list_in xs ys then in_left else in_right
end
end.
Next Obligation. intro H0. apply H; inversion H0; subst; trivial. Defined.
Next Obligation. intro H; inversion H. Defined.
Next Obligation. intro H; inversion H. Defined.
Next Obligation. intro H; inversion H; subst. Defined.
Next Obligation.
intro H1; contradict H. inversion H1; subst. assumption.
contradict H0; assumption. Defined.
Next Obligation.
intro H1; contradict H0. inversion H1; subst. assumption. Defined.
Next Obligation.
intro H1; contradict H. inversion H1; subst. assumption. Defined.
Next Obligation.
intro H0; contradict H. inversion H0; subst; auto. Defined.
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