From Undecidability.L Require Import L Tactics.LTactics LBool.
Class eqbClass X (eqb : X -> X -> bool): Type :=
_eqb_spec : forall (x y:X), reflect (x=y) (eqb x y).
Hint Mode eqbClass + -: typeclass_instances.
Definition eqb X eqb `{H:eqbClass (X:=X) eqb} := eqb.
Arguments eqb {_ _ _}: simpl never.
Lemma eqb_spec {X} {f : X -> X -> bool} {_:eqbClass f}:
forall (x y:X), reflect (x=y) (eqb x y).
Proof.
intros. eapply _eqb_spec.
Qed.
Instance eqbBool_inst : eqbClass Bool.eqb.
Proof.
intros ? ?. eapply iff_reflect. rewrite eqb_true_iff. reflexivity.
Qed.
Lemma dec_reflect_remove {P Y} (d:dec P) b (H:reflect P b) (y y' : Y):
(if d then y else y') = (if b then y else y').
Proof.
destruct H,d;easy.
Qed.
Lemma eqDec_remove {X Y eqb} {H:eqbClass (X:=X) eqb} x x' (d:dec (x=x')) (a b : Y):
(if d then a else b) = (if eqb x x' then a else b).
Proof.
apply dec_reflect_remove. eapply eqb_spec.
Qed.
Class eqbCompT X {R:registered X} eqb {H:eqbClass (X:=X) eqb} :=
{ c__eqbComp :nat;
eqbTime x y:= min x y* c__eqbComp;
comp_eqb : computableTime' eqb (fun x _ =>(5,fun y _ => (eqbTime (size (enc x)) (size (enc y)),tt)))
}.
Arguments eqbCompT _ {_ _ _}.
Arguments c__eqbComp _ {_ _ _ _}.
Hint Mode eqbCompT + - - -: typeclass_instances.
Existing Instance comp_eqb.
Instance eqbComp_bool : eqbCompT bool.
Proof.
evar (c:nat). exists c. unfold Bool.eqb.
unfold enc;cbn.
extract.
solverec.
[c]:exact 3.
all:unfold c;try lia.
Qed.
Lemma eqbTime_le_l X {R : registered X} (eqb : X -> X -> bool) {H : eqbClass eqb}
{H' : eqbCompT X} x n':
eqbTime (X:=X) x n' <= x * c__eqbComp X.
Proof.
unfold eqbTime. rewrite Nat.le_min_l. easy.
Qed.
Lemma eqbTime_le_r X (R : registered X) (eqb : X -> X -> bool) (H : eqbClass eqb)
(eqbCompT : eqbCompT X) x n':
eqbTime (X:=X) n' x <= x * c__eqbComp X.
Proof.
unfold eqbTime. rewrite Nat.le_min_r. easy.
Qed.
Class eqbClass X (eqb : X -> X -> bool): Type :=
_eqb_spec : forall (x y:X), reflect (x=y) (eqb x y).
Hint Mode eqbClass + -: typeclass_instances.
Definition eqb X eqb `{H:eqbClass (X:=X) eqb} := eqb.
Arguments eqb {_ _ _}: simpl never.
Lemma eqb_spec {X} {f : X -> X -> bool} {_:eqbClass f}:
forall (x y:X), reflect (x=y) (eqb x y).
Proof.
intros. eapply _eqb_spec.
Qed.
Instance eqbBool_inst : eqbClass Bool.eqb.
Proof.
intros ? ?. eapply iff_reflect. rewrite eqb_true_iff. reflexivity.
Qed.
Lemma dec_reflect_remove {P Y} (d:dec P) b (H:reflect P b) (y y' : Y):
(if d then y else y') = (if b then y else y').
Proof.
destruct H,d;easy.
Qed.
Lemma eqDec_remove {X Y eqb} {H:eqbClass (X:=X) eqb} x x' (d:dec (x=x')) (a b : Y):
(if d then a else b) = (if eqb x x' then a else b).
Proof.
apply dec_reflect_remove. eapply eqb_spec.
Qed.
Class eqbCompT X {R:registered X} eqb {H:eqbClass (X:=X) eqb} :=
{ c__eqbComp :nat;
eqbTime x y:= min x y* c__eqbComp;
comp_eqb : computableTime' eqb (fun x _ =>(5,fun y _ => (eqbTime (size (enc x)) (size (enc y)),tt)))
}.
Arguments eqbCompT _ {_ _ _}.
Arguments c__eqbComp _ {_ _ _ _}.
Hint Mode eqbCompT + - - -: typeclass_instances.
Existing Instance comp_eqb.
Instance eqbComp_bool : eqbCompT bool.
Proof.
evar (c:nat). exists c. unfold Bool.eqb.
unfold enc;cbn.
extract.
solverec.
[c]:exact 3.
all:unfold c;try lia.
Qed.
Lemma eqbTime_le_l X {R : registered X} (eqb : X -> X -> bool) {H : eqbClass eqb}
{H' : eqbCompT X} x n':
eqbTime (X:=X) x n' <= x * c__eqbComp X.
Proof.
unfold eqbTime. rewrite Nat.le_min_l. easy.
Qed.
Lemma eqbTime_le_r X (R : registered X) (eqb : X -> X -> bool) (H : eqbClass eqb)
(eqbCompT : eqbCompT X) x n':
eqbTime (X:=X) n' x <= x * c__eqbComp X.
Proof.
unfold eqbTime. rewrite Nat.le_min_r. easy.
Qed.