proposition: Theorem pre_weaken_rule {A1 A2 : ord_choiceType} {S1 S2 : choiceType} {d1 : FrStP S1 A1} {d2 : FrStP S2 A2} : ∀ (pre pre' : S1 * S2 → Prop) post, (⊨ ⦃ pre ⦄ d1 ≈ d2 ⦃ post ⦄) → (∀ st1 st2, pre' (st1, st2) → pre (st1, st2) ) → (⊨ ⦃ pre' ⦄ d1 ≈ d2 ⦃ post ⦄).
proof: Proof. move => w w' post Hjudg Hleq. move => [st1 st2]. move => π [H1 H2]. simpl in π. apply: Hjudg. rewrite /fromPrePost /=. split. - by apply: Hleq. - assumption. Qed.
imports: Coq(RelationClasses Morphisms Utf8) Mon(SPropMonadicStructures SpecificationMonads MonadExamples SPropBase FiniteProbabilities) Relational(OrderEnrichedCategory OrderEnrichedRelativeMonadExamples Commutativity GenericRulesSimple) mathcomp(all_ssreflect all_algebra reals distr realsum finmap.set finmap.finmap xfinmap) Crypt(Axioms ChoiceAsOrd SubDistr Couplings Theta_dens Theta_exCP LaxComp FreeProbProg RelativeMonadMorph_prod StateTransformingLaxMorph choice_type)