proposition: Lemma rswap_cmd_bind_eq : ∀ {A₀ A₁ B : choiceType} c₀ c₁ (r : A₀ → A₁ → raw_code B), ⊢ ⦃ λ '(h₀, h₁), h₀ = h₁ ⦄ a₀ ← cmd c₀ ;; a₁ ← c₁ ;; ret (a₀, a₁) ≈ a₁ ← c₁ ;; a₀ ← cmd c₀ ;; ret (a₀, a₁) ⦃ eq ⦄ → ⊢ ⦃ λ '(h₀, h₁), h₀ = h₁ ⦄ a₀ ← cmd c₀ ;; a₁ ← c₁ ;; r a₀ a₁ ≈ a₁ ← c₁ ;; a₀ ← cmd c₀ ;; r a₀ a₁ ⦃ eq ⦄.
proof: Proof. intros A₀ A₁ B c₀ c₁ r h. eapply from_sem_jdg. simpl. setoid_rewrite repr_cmd_bind. setoid_rewrite repr_bind. simpl. setoid_rewrite repr_cmd_bind. eapply (swap_ruleR (λ a₀ a₁, repr (r a₀ a₁)) (repr_cmd c₀) (repr c₁)). - intros a₀ a₁. eapply to_sem_jdg. apply rsym_pre. 1: auto. apply rreflexivity_rule. - auto. - intro s. unshelve eapply coupling_eq. + exact: (λ '(h₀, h₁), h₀ = h₁). + eapply to_sem_jdg in h. setoid_rewrite repr_cmd_bind in h. simpl in h. setoid_rewrite repr_bind in h. simpl in h. setoid_rewrite repr_cmd_bind in h. simpl in h. auto. + reflexivity. Qed.
imports: Coq(Utf8) Relational(OrderEnrichedCategory OrderEnrichedRelativeMonadExamples) mathcomp(ssrnat ssreflect ssrfun ssrbool ssrnum eqtype choice reals distr seq all_algebra fintype realsum) extructures(ord fset fmap) Mon(SPropBase) Crypt(Prelude Axioms ChoiceAsOrd SubDistr Couplings RulesStateProb UniformStateProb UniformDistrLemmas StateTransfThetaDens StateTransformingLaxMorph choice_type pkg_core_definition pkg_notation pkg_tactics pkg_composition pkg_heap pkg_semantics pkg_lookup pkg_advantage pkg_invariants pkg_distr) Equations.Prop.DepElim Equations(Equations) Crypt(FreeProbProg)