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ProgramDerivation: ADT refinement reflexivity and transitivity

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Adam Chlipala 2018-05-05 14:11:37 -04:00
parent 4171f5c286
commit 2f5635938c
1 changed files with 54 additions and 5 deletions
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59
ProgramDerivation.v
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@ -6,6 +6,12 @@
Require Import FrapWithoutSets.
Require Import Program Setoids.Setoid Classes.RelationClasses Classes.Morphisms Morphisms_Prop.
Require Import Eqdep.
Ltac inv_pair :=
match goal with
| [ H : existT _ _ _ = existT _ _ _ |- _ ] => apply inj_pair2 in H; invert H
end.
(** * The computation monad *)
@ -210,6 +216,8 @@ Inductive RefineMethods {state1 state2} (R : state1 -> state2 -> Prop)
-> RefineMethods R (MethodsCons {| MethodName := name; MethodBody := f1 |} ms1)
(MethodsCons {| MethodName := name; MethodBody := f2 |} ms2).
Hint Constructors RefineMethods.
Record adt_refine {names} (adt1 adt2 : adt names) := {
ArRel : AdtState adt1 -> AdtState adt2 -> Prop;
ArConstructors : forall s2,
@ -272,6 +280,51 @@ Proof.
subst; eauto.
Qed.
Hint Immediate RefineMethods_refl.
Theorem refine_refl : forall names (adt1 : adt names),
adt_refine adt1 adt1.
Proof.
simplify.
choose_relation (@eq (AdtState adt1)); eauto.
Qed.
Lemma RefineMethods_trans : forall state1 state2 state3 names
R1 R2
(ms1 : methods state1 names)
(ms2 : methods state2 names)
(ms3 : methods state3 names),
RefineMethods R1 ms1 ms2
-> RefineMethods R2 ms2 ms3
-> RefineMethods (fun s1 s3 => exists s2, R1 s1 s2 /\ R2 s2 s3) ms1 ms3.
Proof.
induct 1; invert 1; repeat inv_pair; eauto.
econstructor; eauto.
first_order.
eapply H5 in H2; eauto.
first_order.
eapply H in H2; eauto.
first_order.
Qed.
Hint Resolve RefineMethods_trans.
Theorem refine_trans : forall names (adt1 adt2 adt3 : adt names),
adt_refine adt1 adt2
-> adt_refine adt2 adt3
-> adt_refine adt1 adt3.
Proof.
simplify.
invert X.
invert X0.
choose_relation (fun s1 s3 => exists s2, ArRel0 s1 s2 /\ ArRel1 s2 s3); eauto.
apply ArConstructors1 in H.
first_order.
Qed.
Theorem refine_constructor : forall names state constr1 constr2 (ms : methods _ names),
refine constr1 constr2
-> adt_refine {| AdtState := state;
@ -282,9 +335,5 @@ Theorem refine_constructor : forall names state constr1 constr2 (ms : methods _
AdtMethods := ms |}.
Proof.
simplify.
match goal with
| [ |- adt_refine ?a ?b ] => apply (Build_adt_refine names a b (@eq _))
end; simplify.
morphisms.
apply RefineMethods_refl.
choose_relation (@eq state); eauto.
Qed.