From Coq Require Import Reals.Reals. From Flover.Infra Require Import RealRationalProps RationalSimps Ltacs. From Flover.Infra Require Export ExpressionAbbrevs. (** Finally, define an error function that computes an errorneous value for a given type. For a fixed-point datatype, truncation is used and any floating-point type is perturbed. As we need not compute on this function we define it in Prop. **) Definition perturb (rVal:R) (m:mType) (delta:R) :R := match m with (* The Real-type has no error *) |REAL => rVal (* Fixed-point numbers have an absolute error *) |F w f => rVal + delta (* Floating-point numbers have a relative error *) | _ => rVal * (1 + delta) end. Hint Unfold perturb. (** Define expression evaluation relation parametric by an "error" epsilon. The result value exprresses float computations according to the IEEE standard, using a perturbation of the real valued computation by (1 + delta), where |delta| <= machine epsilon. **) Open Scope R_scope. Inductive eval_expr (E:env) (Gamma: expr R -> option mType) :(expr R) -> R -> mType -> Prop := | Var_load m x v: Gamma (Var R x) = Some m -> E x = Some v -> eval_expr E Gamma (Var R x) v m | Const_dist m n delta: Rabs delta <= mTypeToR m -> eval_expr E Gamma (Const m n) (perturb n m delta) m | Unop_neg m mN f1 v1: Gamma (Unop Neg f1) = Some mN -> isCompat m mN = true -> eval_expr E Gamma f1 v1 m -> eval_expr E Gamma (Unop Neg f1) (evalUnop Neg v1) mN | Unop_inv m mN f1 v1 delta: Gamma (Unop Inv f1) = Some mN -> isCompat m mN = true -> Rabs delta <= mTypeToR mN -> eval_expr E Gamma f1 v1 m -> (~ v1 = 0)%R -> eval_expr E Gamma (Unop Inv f1) (perturb (evalUnop Inv v1) mN delta) mN | Downcast_dist m m1 f1 v1 delta: Gamma (Downcast m f1) = Some m -> isMorePrecise m1 m = true -> Rabs delta <= mTypeToR m -> eval_expr E Gamma f1 v1 m1 -> eval_expr E Gamma (Downcast m f1) (perturb v1 m delta) m | Binop_dist m1 m2 op f1 f2 v1 v2 delta m: Gamma (Binop op f1 f2) = Some m -> isJoin m1 m2 m = true -> Rabs delta <= mTypeToR m -> eval_expr E Gamma f1 v1 m1 -> eval_expr E Gamma f2 v2 m2 -> ((op = Div) -> (~ v2 = 0)%R) -> eval_expr E Gamma (Binop op f1 f2) (perturb (evalBinop op v1 v2) m delta) m | Fma_dist m1 m2 m3 m f1 f2 f3 v1 v2 v3 delta: Gamma (Fma f1 f2 f3) = Some m -> isJoin3 m1 m2 m3 m = true -> Rabs delta <= mTypeToR m -> eval_expr E Gamma f1 v1 m1 -> eval_expr E Gamma f2 v2 m2 -> eval_expr E Gamma f3 v3 m3 -> eval_expr E Gamma (Fma f1 f2 f3) (perturb (evalFma v1 v2 v3) m delta) m. Close Scope R_scope. Hint Constructors eval_expr. (** *) (* Show some simpler (more general) rule lemmata *) (* **) Lemma Const_dist' m n delta v m' E Gamma: Rle (Rabs delta) (mTypeToR m') -> v = perturb n m delta -> m' = m -> eval_expr E Gamma (Const m n) v m'. Proof. intros; subst; auto. Qed. Hint Resolve Const_dist'. Lemma Unop_neg' m mN f1 v1 v m' E Gamma: eval_expr E Gamma f1 v1 m -> v = evalUnop Neg v1 -> Gamma (Unop Neg f1) = Some mN -> isCompat m mN = true -> m' = mN -> eval_expr E Gamma (Unop Neg f1) v m'. Proof. intros; subst; eauto. Qed. Hint Resolve Unop_neg'. Lemma Unop_inv' m mN f1 v1 delta v m' E Gamma: Rle (Rabs delta) (mTypeToR m') -> eval_expr E Gamma f1 v1 m -> (~ v1 = 0)%R -> v = perturb (evalUnop Inv v1) mN delta -> Gamma (Unop Inv f1) = Some mN -> isCompat m mN = true -> m' = mN -> eval_expr E Gamma (Unop Inv f1) v m'. Proof. intros; subst; eauto. Qed. Hint Resolve Unop_inv'. Lemma Downcast_dist' m m1 f1 v1 delta v m' E Gamma: isMorePrecise m1 m = true -> Rle (Rabs delta) (mTypeToR m') -> eval_expr E Gamma f1 v1 m1 -> v = (perturb v1 m delta) -> Gamma (Downcast m f1) = Some m -> m' = m -> eval_expr E Gamma (Downcast m f1) v m'. Proof. intros; subst; eauto. Qed. Hint Resolve Downcast_dist'. Lemma Binop_dist' m1 m2 op f1 f2 v1 v2 delta v m m' E Gamma: Rle (Rabs delta) (mTypeToR m') -> eval_expr E Gamma f1 v1 m1 -> eval_expr E Gamma f2 v2 m2 -> ((op = Div) -> (~ v2 = 0)%R) -> v = perturb (evalBinop op v1 v2) m' delta -> Gamma (Binop op f1 f2) = Some m -> isJoin m1 m2 m = true -> m = m' -> eval_expr E Gamma (Binop op f1 f2) v m'. Proof. intros; subst; eauto. Qed. Hint Resolve Binop_dist'. Lemma Fma_dist' m1 m2 m3 f1 f2 f3 v1 v2 v3 delta v m' E Gamma m: Rle (Rabs delta) (mTypeToR m') -> eval_expr E Gamma f1 v1 m1 -> eval_expr E Gamma f2 v2 m2 -> eval_expr E Gamma f3 v3 m3 -> v = perturb (evalFma v1 v2 v3) m' delta -> Gamma (Fma f1 f2 f3) = Some m -> isJoin3 m1 m2 m3 m = true -> m = m' -> eval_expr E Gamma (Fma f1 f2 f3) v m'. Proof. intros; subst; eauto. Qed. Hint Resolve Fma_dist'. Lemma Gamma_det e E1 E2 Gamma v1 v2 m1 m2: eval_expr E1 Gamma e v1 m1 -> eval_expr E2 Gamma e v2 m2 -> m1 = m2. Proof. induction e; intros * eval_e1 eval_e2; inversion eval_e1; subst; inversion eval_e2; subst; try auto; match goal with | [H1: Gamma ?e = Some ?m1, H2: Gamma ?e = Some ?m2 |- _ ] => rewrite H1 in H2; inversion H2; subst end; auto. Qed. Lemma toRTMap_eval_REAL f: forall v E Gamma m, eval_expr E (toRTMap Gamma) (toREval f) v m -> m = REAL. Proof. induction f; intros * eval_f; inversion eval_f; subst. repeat match goal with | H: context[toRTMap _ _] |- _ => unfold toRTMap in H | H: context[match ?Gamma ?v with | _ => _ end ] |- _ => destruct (Gamma v) eqn:? | H: Some ?m1 = Some ?m2 |- _ => inversion H; try auto | H: None = Some ?m |- _ => inversion H end; try auto. - auto. - rewrite (IHf _ _ _ _ H5) in H2. unfold isCompat in H2. destruct m; type_conv; subst; try congruence; auto. - rewrite (IHf _ _ _ _ H4) in H2. unfold isCompat in H2. destruct m; type_conv; subst; try congruence; auto. - rewrite (IHf1 _ _ _ _ H5) in H3. rewrite (IHf2 _ _ _ _ H8) in H3. unfold isJoin in H3; simpl in H3. destruct m; try congruence; auto. - rewrite (IHf1 _ _ _ _ H5) in H3. rewrite (IHf2 _ _ _ _ H8) in H3. rewrite (IHf3 _ _ _ _ H9) in H3. unfold isJoin3 in H3; simpl in H3. destruct m; try congruence; auto. - auto. Qed. (** If |delta| <= 0 then perturb v delta is exactly v. **) Lemma delta_0_deterministic (v:R) m (delta:R): (Rabs delta <= 0)%R -> perturb v m delta = v. Proof. intros abs_0; apply Rabs_0_impl_eq in abs_0; subst. unfold perturb. destruct m; lra. Qed. (** Evaluation with 0 as machine epsilon is deterministic **) Lemma meps_0_deterministic (f:expr R) (E:env) Gamma: forall v1 v2, eval_expr E (toRTMap Gamma) (toREval f) v1 REAL -> eval_expr E (toRTMap Gamma) (toREval f) v2 REAL -> v1 = v2. Proof. induction f; intros v1 v2 ev1 ev2. - inversion ev1; inversion ev2; subst. rewrite H1 in H6. inversion H6; auto. - inversion ev1; inversion ev2; subst. simpl in *; subst; auto. - inversion ev1; inversion ev2; subst; try congruence. + rewrite (IHf v0 v3); [ auto | |]; destruct m, m0; cbn in *; congruence. + cbn in *. Flover_compute; rewrite (IHf v0 v3); [auto | | ]; destruct m, m0; cbn in *; congruence. - inversion ev1; inversion ev2; subst. assert (m0 = REAL) by (eapply toRTMap_eval_REAL; eauto). assert (m3 = REAL) by (eapply toRTMap_eval_REAL; eauto). assert (m1 = REAL) by (eapply toRTMap_eval_REAL; eauto). assert (m2 = REAL) by (eapply toRTMap_eval_REAL; eauto). subst. rewrite (IHf1 v0 v4); try auto. rewrite (IHf2 v3 v5); try auto. - inversion ev1; inversion ev2; subst. assert (m0 = REAL) by (eapply toRTMap_eval_REAL; eauto). assert (m1 = REAL) by (eapply toRTMap_eval_REAL; eauto). assert (m2 = REAL) by (eapply toRTMap_eval_REAL; eauto). assert (m3 = REAL) by (eapply toRTMap_eval_REAL; eauto). assert (m4 = REAL) by (eapply toRTMap_eval_REAL; eauto). assert (m5 = REAL) by (eapply toRTMap_eval_REAL; eauto). subst. rewrite (IHf1 v0 v5); try auto. rewrite (IHf2 v3 v6); try auto. rewrite (IHf3 v4 v7); try auto. - inversion ev1; inversion ev2; subst. apply REAL_least_precision in H2; apply REAL_least_precision in H9; subst. rewrite (IHf v0 v3); try auto. Qed. (** Helping lemmas. Needed in soundness proof. For each evaluation of using an arbitrary epsilon, we can replace it by evaluating the subexprressions and then binding the result values to different variables in the Environment. **) Lemma binary_unfolding b f1 f2 E v1 v2 m1 m2 m Gamma delta: (b = Div -> ~(v2 = 0 )%R) -> (Rabs delta <= mTypeToR m)%R -> eval_expr E Gamma f1 v1 m1 -> eval_expr E Gamma f2 v2 m2 -> eval_expr E Gamma (Binop b f1 f2) (perturb (evalBinop b v1 v2) m delta) m -> eval_expr (updEnv 2 v2 (updEnv 1 v1 emptyEnv)) (updDefVars (Binop b (Var R 1) (Var R 2)) m (updDefVars (Var R 2) m2 (updDefVars (Var R 1) m1 Gamma))) (Binop b (Var R 1) (Var R 2)) (perturb (evalBinop b v1 v2) m delta) m. Proof. intros no_div_zero err_v eval_f1 eval_f2 eval_float. inversion eval_float; subst. rewrite H2 in *. repeat (match goal with | [H1: eval_expr ?E ?Gamma ?f ?v1 ?m1, H2: eval_expr ?E ?Gamma ?f ?v2 ?m2 |- _] => assert (m1 = m2) by (eapply Gamma_det; eauto); revert H1 H2 end); intros; subst. eapply Binop_dist' with (v1:=v1) (v2:=v2) (delta:=delta); try eauto. - eapply Var_load; eauto. - eapply Var_load; eauto. - unfold updDefVars. unfold R_orderedExps.compare; rewrite R_orderedExps.exprCompare_refl; auto. Qed. Lemma fma_unfolding f1 f2 f3 E v1 v2 v3 m1 m2 m3 m Gamma delta: (Rabs delta <= mTypeToR m)%R -> eval_expr E Gamma f1 v1 m1 -> eval_expr E Gamma f2 v2 m2 -> eval_expr E Gamma f3 v3 m3 -> eval_expr E Gamma (Fma f1 f2 f3) (perturb (evalFma v1 v2 v3) m delta) m -> eval_expr (updEnv 3 v3 (updEnv 2 v2 (updEnv 1 v1 emptyEnv))) (updDefVars (Fma (Var R 1) (Var R 2) (Var R 3) ) m (updDefVars (Var R 3) m3 (updDefVars (Var R 2) m2 (updDefVars (Var R 1) m1 Gamma)))) (Fma (Var R 1) (Var R 2) (Var R 3)) (perturb (evalFma v1 v2 v3) m delta) m. Proof. intros err_v eval_f1 eval_f2 eval_f3 eval_float. inversion eval_float; subst. repeat (match goal with | [H1: eval_expr ?E ?Gamma ?f ?v1 ?m1, H2: eval_expr ?E ?Gamma ?f ?v2 ?m2 |- _] => assert (m1 = m2) by (eapply Gamma_det; eauto); revert H1 H2 end). intros; subst. rewrite H2. eapply Fma_dist' with (v1:=v1) (v2:=v2) (v3:=v3) (delta:=delta); try eauto. - eapply Var_load; eauto. - eapply Var_load; eauto. - eapply Var_load; eauto. - cbn; auto. Qed. Lemma eval_eq_env e: forall E1 E2 Gamma v m, (forall x, E1 x = E2 x) -> eval_expr E1 Gamma e v m -> eval_expr E2 Gamma e v m. Proof. induction e; intros; (match_pat (eval_expr _ _ _ _ _) (fun H => inversion H; subst; simpl in H)); try eauto. eapply Var_load; auto. rewrite <- (H n); auto. Qed. Lemma eval_expr_ignore_bind e: forall x v m Gamma E, eval_expr E Gamma e v m -> ~ NatSet.In x (usedVars e) -> forall v_new, eval_expr (updEnv x v_new E) Gamma e v m. Proof. induction e; intros * eval_e no_usedVar *; cbn in *; inversion eval_e; subst; try eauto. - assert (n <> x). { hnf. intros. subst. apply no_usedVar; set_tac. } rewrite <- Nat.eqb_neq in H. eapply Var_load. + unfold updDefVars. cbn. apply beq_nat_false in H. destruct (n ?= x)%nat eqn:?; try auto. + unfold updEnv. rewrite H; auto. - eapply Binop_dist'; eauto; [ eapply IHe1 | eapply IHe2]; eauto; hnf; intros; eapply no_usedVar; set_tac. - eapply Fma_dist'; eauto; [eapply IHe1 | eapply IHe2 | eapply IHe3]; eauto; hnf; intros; eapply no_usedVar; set_tac. Qed. Lemma swap_Gamma_eval_expr e E vR m Gamma1 Gamma2: (forall n, Gamma1 n = Gamma2 n) -> eval_expr E Gamma1 e vR m -> eval_expr E Gamma2 e vR m. Proof. revert E vR Gamma1 Gamma2 m; induction e; intros * Gamma_eq eval_e; inversion eval_e; subst; simpl in *; [ eapply Var_load | eapply Const_dist' | eapply Unop_neg' | eapply Unop_inv' | eapply Binop_dist' | eapply Fma_dist' | eapply Downcast_dist' ]; try eauto; rewrite <- Gamma_eq; auto. Qed. Lemma Rmap_updVars_comm Gamma n m: forall x, updDefVars n REAL (toRMap Gamma) x = toRMap (updDefVars n m Gamma) x. Proof. unfold updDefVars, toRMap; simpl. intros x; destruct (R_orderedExps.compare x n); auto. Qed.