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Notation "( x & y )" := (@existS _ _ x y) : core_scope.
Unset Printing All.
Require Import Coq.Arith.Compare_dec.
Program Fixpoint euclid (a : nat) (b : { b : nat | b <> O }) {wf a lt} :
{ q : nat & { r : nat | a = b * q + r /\ r < b } } :=
if le_lt_dec b a then let (q', r) := euclid (a - b) b in
(S q' & r)
else (O & a).
Require Import Omega.
Obligations.
Obligation 1.
intros.
simpl in * ; auto with arith.
omega.
Defined.
Obligation 2 of euclid.
intros.
assert(x0 * S q' = x0 * q' + x0) by auto with arith ; omega.
Defined.
Obligation 4 of euclid.
exact Wf_nat.lt_wf.
Defined.
Obligation 3 of euclid.
intros.
omega.
Qed.
Eval cbv delta [euclid_obligation_1 euclid_obligation_2] in (euclid 0).
Extraction Inline Fix_sub Fix_F_sub.
Extract Inductive sigT => "pair" [ "" ].
Extract Inductive sumbool => "bool" [ "true" "false" ].
Extraction le_lt_dec.
Extraction euclid.
Program Definition testsig (a : nat) : { x : nat & { y : nat | x < y } } :=
(a & S a).
Solve Obligations using auto with zarith.
Extraction testsig.
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