proposition: Theorem eq_impl_comp_spec : forall (A B : Set){eqda : EqDec A}{eqdb : EqDec B}(c1 : Comp A)(c2 : Comp B) x y, well_formed_comp c1 -> well_formed_comp c2 -> evalDist c1 x == evalDist c2 y -> comp_spec (fun a b => a = x <-> b = y) c1 c2.
proof: Proof. intuition. destruct (eq_Rat_dec (evalDist c1 x) 0). exists (a <-$ c1; b <-$ c2; ret (a, b)). intuition. unfold marginal_l. dist_inline_first. rewrite <- evalDist_right_ident. dist_skip. dist_inline_first. dist_irr_r. dist_compute. unfold marginal_r. dist_inline_first. rewrite <- evalDist_right_ident. dist_irr_r. dist_inline_first. dist_skip. dist_compute. repeat simp_in_support. simpl in *. exfalso. eapply getSupport_not_In_evalDist. eapply e. trivial. repeat simp_in_support. simpl in *. exfalso. eapply getSupport_not_In_evalDist. rewrite <- H1. trivial. trivial. destruct (eq_Rat_dec (evalDist c1 x) 1). exists (a <-$ c1; b <-$ c2; ret (a, b)). split. intuition. unfold marginal_l. dist_inline_first. rewrite <- evalDist_right_ident. dist_skip. dist_inline_first. dist_irr_r. dist_compute. split. intuition. unfold marginal_r. dist_inline_first. rewrite <- evalDist_right_ident. dist_irr_r. dist_inline_first. dist_skip. dist_compute. intros. repeat simp_in_support; simpl in *. erewrite (evalDist_1) in H5; intuition. 2:{ rewrite <- H1. trivial. } simpl in *; intuition. erewrite (evalDist_1) in H3; eauto. simpl in *; intuition. exists (a <-$ c1; b <-$ if (eqb a x) then (ret y) else (Repeat c2 (fun b => negb (eqb b y))); ret (a, b)). split. intuition. unfold marginal_l. dist_inline_first. rewrite <- evalDist_right_ident. dist_skip. dist_inline_first. dist_irr_r. case_eq (eqb x0 x); intuition; wftac. remember (getSupport c2) as s. destruct s. specialize (getSupport_length_nz H0); intuition. rewrite <- Heqs in H4. simpl in *. lia. destruct (EqDec_dec _ b y); subst. destruct s. exfalso. eapply n0. rewrite H1. rewrite <- (evalDist_lossless H0). rewrite <- Heqs. rewrite sumList_cons. unfold sumList; simpl. eapply ratAdd_0_r. eapply (@well_formed_Repeat _ _ _ _ b). trivial. eapply filter_In; intuition. rewrite <- Heqs. simpl. intuition. case_eq (eqb b y); intuition. rewrite eqb_leibniz in H4. subst. specialize (getSupport_NoDup c2); intuition. rewrite <- Heqs in H4. inversion H4; clear H4; subst. simpl in *. intuition. eapply (@well_formed_Repeat _ _ _ _ b). trivial. eapply filter_In; intuition. rewrite <- Heqs; simpl; intuition. case_eq (eqb b y); intuition. rewrite eqb_leibniz in H4; subst. intuition. dist_compute. split. intuition. unfold marginal_r. dist_inline_first. rewrite evalDist_seq_step. rewrite (sumList_filter_partition (eqb x)). symmetry. eapply eqRat_trans. eapply ratAdd_eqRat_compat. eapply sumList_body_eq. intros. eapply ratMult_eqRat_compat. eapply eqRat_refl. apply filter_In in H2. intuition. rewrite eqb_leibniz in H4. subst. rewrite eqb_refl. dist_inline_l. simpl snd. eapply eqRat_refl. eapply sumList_body_eq; intuition. apply filter_In in H2. intuition. case_eq (eqb x a); intuition. rewrite eqb_leibniz in H2. subst. rewrite eqb_refl in H4. simpl in *. discriminate. rewrite evalDist_assoc at 1. case_eq (eqb a x); intuition. rewrite eqb_leibniz in H5. subst. rewrite eqb_refl in H2. discriminate. eapply ratMult_eqRat_compat. eapply eqRat_refl. eapply evalDist_seq_eq. intuition. eapply eqRat_refl. intuition. dist_simp. unfold snd. eapply eqRat_refl. intuition. rewrite (@sumList_exactly_one _ x). case_eq (eqb y r2); intuition. rewrite eqb_leibniz in H2. subst. assert (Pr[ret true] == 1). dist_compute. rewrite H2. rewrite ratMult_1_r. clear H2. rewrite H1. symmetry. rewrite ratAdd_0_r at 1. eapply ratAdd_eqRat_compat; intuition. symmetry. eapply sumList_0. intuition. apply filter_In in H2. intuition. assert (Pr [x0 <-$ Repeat c2 (fun b : B => negb (eqb b r2)); ret eqb x0 r2 ] == Pr [x0 <-$ Repeat c2 (fun b : B => negb (eqb b r2)); ret eqb r2 x0 ] ). dist_skip; dist_compute. rewrite H2. clear H2. rewrite <- evalDist_event_equiv. eapply ratMult_0. right. eapply evalDist_Repeat_0; intuition. subst. rewrite eqb_refl in H5. simpl in *; discriminate. assert (Pr [ret false] == 0). dist_compute. rewrite H3. clear H3. rewrite ratMult_0_r. rewrite <- ratAdd_0_l. rewrite sumList_factor_constant_r. rewrite sumList_filter_complement. rewrite H1. rewrite evalDist_seq_step. destruct (in_dec (EqDec_dec _) r2 (getSupport c2)). rewrite (@sumList_exactly_one _ r2). assert (Pr [ret eqb r2 r2 ] == 1). dist_compute. rewrite H3. clear H3. rewrite ratMult_1_r. simpl. unfold indicator. case_eq (eqb r2 y); intuition. rewrite eqb_leibniz in H3; subst. rewrite eqb_refl in H2. discriminate. simpl. rewrite ratMult_1_l. symmetry. rewrite <- ratMult_1_l at 1. rewrite <- ratMult_assoc. eapply ratMult_eqRat_compat; intuition. symmetry. eapply eqRat_trans. eapply ratMult_eqRat_compat. eapply eqRat_refl. eapply ratInverse_eqRat_compat. intuition. assert ( (filter (fun b : B => negb (eqb b y)) (getSupport c2)) = (filter (fun b : B => negb (eqb y b)) (getSupport c2))). eapply filter_ext; intuition. rewrite eqb_symm. trivial. rewrite H5 in H4. clear H5. eapply n0. eapply leRat_impl_eqRat. eapply evalDist_le_1. eapply ratSubtract_0_inv. rewrite sumList_filter_complement in H4. rewrite H1. intuition. trivial. assert ((filter (fun b : B => negb (eqb b y)) (getSupport c2)) = (filter (fun b : B => negb (eqb y b)) (getSupport c2))). eapply filter_ext. intuition. rewrite eqb_symm. intuition. rewrite H4. clear H4. eapply sumList_filter_complement. trivial. rewrite ratMult_comm. eapply ratInverse_prod_1. intuition. eapply n0. eapply leRat_impl_eqRat. eapply evalDist_le_1. eapply ratSubtract_0_inv. rewrite H1. intuition. eapply getSupport_NoDup. simpl. eapply filter_In; intuition. rewrite eqb_symm. rewrite H2. trivial. intuition. dist_compute. eapply eqRat_trans. eapply ratMult_eqRat_compat. eapply eqRat_refl. eapply sumList_0. intuition. simpl in H3. apply filter_In in H3. intuition. case_eq (eqb a r2); intuition. rewrite eqb_leibniz in H3; subst. intuition. dist_compute. rewrite ratMult_0_r. symmetry. eapply getSupport_not_In_evalDist. trivial. trivial. eapply filter_NoDup. eapply getSupport_NoDup. eapply filter_In; intuition. eapply getSupport_In_evalDist; intuition. eapply eqb_refl. intuition. apply filter_In in H2. intuition. rewrite eqb_leibniz in H5. intuition. intros. repeat simp_in_support; simpl in *; intuition. rewrite eqb_refl in Heqx2. discriminate. symmetry in Heqx2. rewrite eqb_leibniz in Heqx2. trivial. simpl in *. eapply filter_In in H5. intuition. rewrite eqb_refl in H4. simpl in *. discriminate. Unshelve. intuition. intuition. intuition. intuition. intuition. unfold eq_dec; intuition. eapply (EqDec_dec _). unfold eq_dec; intuition. eapply (EqDec_dec _). Qed.
imports: FCF.Crypto FCF.Bernoulli FCF.NotationV1