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sequence.v
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Load basics.
Definition SequenceRule (triple : D->D->D->Prop) :=
forall p q s q' r : D,
triple p q s ->
triple s q' r ->
triple p (q;q') r.
Hint Unfold SequenceRule.
Theorem ASH : AssociativityLaw -> SequenceRule HoareTriple.
Proof.
intro Assoc.
autounfold.
intuition.
rewrite <- Assoc.
eauto.
Qed.
Print ASH.
Theorem ASP: AssociativityLaw -> SequenceRule PlotkinReduction.
Proof.
intro Assoc.
autounfold.
intuition.
rewrite <- Assoc.
apply Transitivity with (y := s;q').
auto.
Qed.
Theorem ASM: AssociativityLaw -> SequenceRule MilnerTransition.
Proof.
intro Assoc.
autounfold.
intuition.
rewrite Assoc.
eauto.
Qed.
Theorem AST: AssociativityLaw -> SequenceRule TestGeneration.
Proof.
intro Assoc.
autounfold.
intuition.
rewrite Assoc.
eauto.
Qed.
Theorem SHSPA : SequenceRule HoareTriple /\ SequenceRule PlotkinReduction ->
AssociativityLaw.
Proof.
unfold SequenceRule, HoareTriple, PlotkinReduction, AssociativityLaw.
intros.
elim H.
intros.
intuition.
clear H2.
clear H3.
apply AntiSymmetry.
apply H1 with (p; q).
apply Reflexivity.
apply Reflexivity.
apply H0 with (p; q).
apply Reflexivity.
apply Reflexivity.
Qed.
Theorem SMSTA : SequenceRule MilnerTransition /\ SequenceRule TestGeneration ->
AssociativityLaw.
Proof.
unfold SequenceRule, TestGeneration, MilnerTransition, AssociativityLaw.
intros.
elim H.
intros.
intuition.
clear H2.
clear H3.
apply AntiSymmetry.
apply H0 with (q; r).
apply Reflexivity.
apply Reflexivity.
apply H1 with (q; r).
apply Reflexivity.
apply Reflexivity.
Qed.