proposition: Lemma rswap_cmd_eq : ∀ (A₀ A₁ B : choiceType) (c₀ : command A₀) (c₁ : command A₁) (r : A₀ → A₁ → raw_code B), ⊢ ⦃ λ '(h₀, h₁), h₀ = h₁ ⦄ a₀ ← cmd c₀ ;; a₁ ← cmd c₁ ;; ret (a₀, a₁) ≈ a₁ ← cmd c₁ ;; a₀ ← cmd c₀ ;; ret (a₀, a₁) ⦃ eq ⦄ → ⊢ ⦃ λ '(h₀, h₁), h₀ = h₁ ⦄ a₀ ← cmd c₀ ;; a₁ ← cmd c₁ ;; r a₀ a₁ ≈ a₁ ← cmd c₁ ;; a₀ ← cmd c₀ ;; r a₀ a₁ ⦃ eq ⦄.
proof: Proof. intros A₀ A₁ B c₀ c₁ r h. eapply rswap_cmd. - intro. reflexivity. - intros a₀ a₁. eapply rsym_pre. 1: auto. apply rreflexivity_rule. - auto. 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)