proposition: Lemma update_pre_spec : ∀ l pre s₀ s₁, update_pre l pre (s₀, s₁) → ∃ s₀' s₁', pre (s₀', s₁') ∧ (s₀, s₁) = update_heaps l s₀' s₁'.
proof: Proof. intros l pre s₀ s₁ h. induction l as [| [] l ih] in pre, s₀, s₁, h |- *. - simpl in h. simpl. eexists _, _. intuition eauto. - simpl in h. simpl. destruct h as [s [h ?]]. subst. eapply ih in h. destruct h as [? [? [? e]]]. eexists _, _. split. 1: eauto. rewrite -e. reflexivity. - simpl in h. simpl. destruct h as [s [h ?]]. subst. eapply ih in h. destruct h as [? [? [? e]]]. eexists _, _. split. 1: eauto. rewrite -e. 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) Equations.Prop.DepElim Equations(Equations) Crypt(FreeProbProg)