Proofs done until validErrorbound_sound included

coq/ErrorBounds.v

@@ -58,29 +58,28 @@ Lemma add_abs_err_bounded (e1:exp Q) (e1R:R) (e1F:R) (e2:exp Q) (e2R:R) (e2F:R)
 Proof.
   intros e1_real e1_float e2_real e2_float plus_real plus_float bound_e1 bound_e2.
   (* Prove that e1R and e2R are the correct values and that vR is e1R + e2R *)
-  inversion plus_real; subst;
-    assert (m3 = M0) by (apply (ifM0isJoin_l M0 m3 m4); auto);
-    assert (m4 = M0) by (apply (ifM0isJoin_r M0 m3 m4); auto); subst;
-      simpl (meps M0) in H3; rewrite Q2R0_is_0 in H3; auto.
+  inversion plus_real; subst.
+  destruct m0; destruct m3; inversion H2;
+      simpl in H4; rewrite Q2R0_is_0 in H4; auto.
   rewrite delta_0_deterministic in plus_real; auto.
   rewrite (delta_0_deterministic (evalBinop Plus v1 v2) delta); auto.
   unfold evalBinop in *; simpl in *.
-  clear delta H3.
-  rewrite (meps_0_deterministic (toRExp e1) H6 e1_real);
-    rewrite (meps_0_deterministic (toRExp e2) H7 e2_real).
-  rewrite (meps_0_deterministic (toRExp e1) H6 e1_real) in plus_real.
-  rewrite (meps_0_deterministic (toRExp e2) H7 e2_real) in plus_real.
-  clear H6 H7 v1 v2.
+  clear delta H4.
+  rewrite (meps_0_deterministic (toRExp e1) H5 e1_real);
+    rewrite (meps_0_deterministic (toRExp e2) H6 e2_real).
+  rewrite (meps_0_deterministic (toRExp e1) H5 e1_real) in plus_real.
+  rewrite (meps_0_deterministic (toRExp e2) H6 e2_real) in plus_real.
+  clear H5 H6 v1 v2.
   (* Now unfold the float valued evaluation to get the deltas we need for the inequality *)
   inversion plus_float; subst.
   unfold perturb; simpl.
-  inversion H7; subst; inversion H8; subst.
+  inversion H6; subst; inversion H7; subst.
   unfold updEnv; simpl.
-  unfold updEnv in H6,H9; simpl in *.
-  symmetry in H6,H9.
-  inversion H6; inversion H9; subst.
+  unfold updEnv in H5,H8; simpl in *.
+  symmetry in H5,H8.
+  inversion H5; inversion H8; subst.
   (* We have now obtained all necessary values from the evaluations --> remove them for readability *)
-  clear plus_float H7 H8 plus_real e1_real e1_float e2_real e2_float H9 H6.
+  clear plus_float H7 H8 plus_real e1_real e1_float e2_real e2_float H5 H8.
   repeat rewrite Rmult_plus_distr_l.
   rewrite Rmult_1_r.
   rewrite Rsub_eq_Ropp_Rplus.
@@ -103,7 +102,7 @@ Proof.
     eapply Rle_trans.
     eapply Rmult_le_compat_l.
     apply Rabs_pos.
-    apply H4.
+    apply H3.
     apply Req_le; auto.
 Qed.
 
@@ -125,28 +124,27 @@ Proof.
   intros e1_real e1_float e2_real e2_float sub_real sub_float bound_e1 bound_e2.
   (* Prove that e1R and e2R are the correct values and that vR is e1R + e2R *)
   inversion sub_real; subst;
-    assert (m3 = M0) by (apply (ifM0isJoin_l M0 m3 m4); auto);
-    assert (m4 = M0) by (apply (ifM0isJoin_r M0 m3 m4); auto); subst;
-      simpl (meps M0) in H3; rewrite Q2R0_is_0 in H3; auto.
+  destruct m0; destruct m3; inversion H2;
+      simpl in H4; rewrite Q2R0_is_0 in H4; auto.
   rewrite delta_0_deterministic in sub_real; auto.
   rewrite delta_0_deterministic; auto.
   unfold evalBinop in *; simpl in *.
-  clear delta H3.
-  rewrite (meps_0_deterministic (toRExp e1) H6 e1_real);
-    rewrite (meps_0_deterministic (toRExp e2) H7 e2_real).
-  rewrite (meps_0_deterministic (toRExp e1) H6 e1_real) in sub_real.
-  rewrite (meps_0_deterministic (toRExp e2) H7 e2_real) in sub_real.
-  clear H6 H7 v1 v2.
+  clear delta H4.
+  rewrite (meps_0_deterministic (toRExp e1) H5 e1_real);
+    rewrite (meps_0_deterministic (toRExp e2) H6 e2_real).
+  rewrite (meps_0_deterministic (toRExp e1) H5 e1_real) in sub_real.
+  rewrite (meps_0_deterministic (toRExp e2) H6 e2_real) in sub_real.
+  clear H5 H6 v1 v2.
   (* Now unfold the float valued evaluation to get the deltas we need for the inequality *)
   inversion sub_float; subst.
   unfold perturb; simpl.
-  inversion H7; subst; inversion H8; subst.
+  inversion H6; subst; inversion H7; subst.
   unfold updEnv; simpl.
-  symmetry in H6, H9.
-  unfold updEnv in H6, H9; simpl in H6, H9.
-  inversion H6; inversion H9; subst.
+  symmetry in H5, H8.
+  unfold updEnv in H5, H8; simpl in H5, H8.
+  inversion H5; inversion H8; subst.
   (* We have now obtained all necessary values from the evaluations --> remove them for readability *)
-  clear sub_float H7 H8 sub_real e1_real e1_float e2_real e2_float H6 H9.
+  clear sub_float H7 H8 sub_real e1_real e1_float e2_real e2_float H5 H8.
   repeat rewrite Rmult_plus_distr_l.
   rewrite Rmult_1_r.
   repeat rewrite Rsub_eq_Ropp_Rplus.
@@ -182,27 +180,26 @@ Proof.
   intros e1_real e1_float e2_real e2_float mult_real mult_float.
   (* Prove that e1R and e2R are the correct values and that vR is e1R * e2R *)
   inversion mult_real; subst;
-    assert (m3 = M0) by (apply (ifM0isJoin_l M0 m3 m4); auto);
-    assert (m4 = M0) by (apply (ifM0isJoin_r M0 m3 m4); auto); subst;
-      simpl (meps M0) in H3; rewrite Q2R0_is_0 in H3; auto.
+    destruct m0; destruct m3; inversion H2;
+      simpl in H4; rewrite Q2R0_is_0 in H4; auto.
   rewrite delta_0_deterministic in mult_real; auto.
   rewrite delta_0_deterministic; auto.
   unfold evalBinop in *; simpl in *.
-  clear delta H3.
-  rewrite (meps_0_deterministic (toRExp e1) H6 e1_real);
-    rewrite (meps_0_deterministic (toRExp e2) H7 e2_real).
-  rewrite (meps_0_deterministic (toRExp e1) H6 e1_real) in mult_real.
-  rewrite (meps_0_deterministic (toRExp e2) H7 e2_real) in mult_real.
-  clear H6 H7 v1 v2.
+  clear delta H4.
+  rewrite (meps_0_deterministic (toRExp e1) H5 e1_real);
+    rewrite (meps_0_deterministic (toRExp e2) H6 e2_real).
+  rewrite (meps_0_deterministic (toRExp e1) H5 e1_real) in mult_real.
+  rewrite (meps_0_deterministic (toRExp e2) H6 e2_real) in mult_real.
+  clear H5 H6 v1 v2.
   (* Now unfold the float valued evaluation to get the deltas we need for the inequality *)
   inversion mult_float; subst.
   unfold perturb; simpl.
-  inversion H7; subst; inversion H8; subst.
-  symmetry in H6, H9;
+  inversion H6; subst; inversion H7; subst.
+  symmetry in H5, H8;
     unfold updEnv in *; simpl in *.
-  inversion H6; inversion H9; subst.
+  inversion H5; inversion H8; subst.
   (* We have now obtained all necessary values from the evaluations --> remove them for readability *)
-  clear mult_float H7 H8 mult_real e1_real e1_float e2_real e2_float H6 H9.
+  clear mult_float H7 H8 mult_real e1_real e1_float e2_real e2_float H5 H8.
   repeat rewrite Rmult_plus_distr_l.
   rewrite Rmult_1_r.
   rewrite Rsub_eq_Ropp_Rplus.
@@ -232,27 +229,26 @@ Proof.
   intros e1_real e1_float e2_real e2_float div_real div_float.
   (* Prove that e1R and e2R are the correct values and that vR is e1R * e2R *)
   inversion div_real; subst;
-    assert (m3 = M0) by (apply (ifM0isJoin_l M0 m3 m4); auto);
-    assert (m4 = M0) by (apply (ifM0isJoin_r M0 m3 m4); auto); subst;
-      simpl (meps M0) in H3; rewrite Q2R0_is_0 in H3; auto.
+  destruct m0; destruct m3; inversion H2;
+    simpl in H4; rewrite Q2R0_is_0 in H4; auto.
   rewrite delta_0_deterministic in div_real; auto.
   rewrite delta_0_deterministic; auto.
   unfold evalBinop in *; simpl in *.
-  clear delta H3.
-  rewrite (meps_0_deterministic (toRExp e1) H6 e1_real);
-    rewrite (meps_0_deterministic (toRExp e2) H7 e2_real).
-  rewrite (meps_0_deterministic (toRExp e1) H6 e1_real) in div_real.
-  rewrite (meps_0_deterministic (toRExp e2) H7 e2_real) in div_real.
-  clear H6 H7 v1 v2.
+  clear delta H4.
+  rewrite (meps_0_deterministic (toRExp e1) H5 e1_real);
+    rewrite (meps_0_deterministic (toRExp e2) H6 e2_real).
+  rewrite (meps_0_deterministic (toRExp e1) H5 e1_real) in div_real.
+  rewrite (meps_0_deterministic (toRExp e2) H6 e2_real) in div_real.
+  clear H5 H6 v1 v2.
   (* Now unfold the float valued evaluation to get the deltas we need for the inequality *)
   inversion div_float; subst.
   unfold perturb; simpl.
-  inversion H7; subst; inversion H8; subst.
-  symmetry in H6, H9;
+  inversion H6; subst; inversion H7; subst.
+  symmetry in H5, H8;
     unfold updEnv in *; simpl in *.
-  inversion H6; inversion H9; subst.
+  inversion H5; inversion H8; subst.
   (* We have now obtained all necessary values from the evaluations --> remove them for readability *)
-  clear div_float H7 H8 div_real e1_real e1_float e2_real e2_float H6 H9.
+  clear div_float H7 H8 div_real e1_real e1_float e2_real e2_float H5 H8.
   repeat rewrite Rmult_plus_distr_l.
   rewrite Rmult_1_r.
   rewrite Rsub_eq_Ropp_Rplus.
@@ -447,10 +443,10 @@ Proof.
   rewrite Q2R0_is_0; auto.
 Qed.
 
-Lemma round_abs_err_bounded (e:exp R) (nR nF1 nF:R) (E: env) (err:R) (machineEpsilon m:mType):
-  eval_exp E (toREval e) nR M0 ->
-  eval_exp E e nF1 m ->
-  eval_exp (updEnv 1 m nF1 E) (toRExp (Downcast machineEpsilon (Var Q m 1))) nF machineEpsilon->
+Lemma round_abs_err_bounded (e:exp R) (nR nF1 nF:R) (E1 E2: env) (err:R) (machineEpsilon m:mType):
+  eval_exp E1 (toREval e) nR M0 ->
+  eval_exp E2 e nF1 m ->
+  eval_exp (updEnv 1 m nF1 emptyEnv) (toRExp (Downcast machineEpsilon (Var Q m 1))) nF machineEpsilon->
   (Rabs (nR - nF1) <= err)%R ->
   (Rabs (nR - nF) <= err + (Rabs nF1) * Q2R (meps machineEpsilon))%R.
 Proof.

coq/Expressions.v

@@ -235,12 +235,12 @@ Inductive eval_exp (E:env) :(exp R) -> R -> mType -> Prop :=
     Rle (Rabs delta) (Q2R (meps m)) ->
     eval_exp E f1 v1 m ->
     eval_exp E (Unop Inv f1) (perturb (evalUnop Inv v1) delta) m
-| Binop_dist m m1 m2 op f1 f2 v1 v2 delta:
-    isJoinOf m m1 m2 = true ->
-    Rle (Rabs delta) (Q2R (meps m)) ->
+| Binop_dist m1 m2 op f1 f2 v1 v2 delta:
+    (*isJoinOf m m1 m2 = true ->*)
+    Rle (Rabs delta) (Q2R (meps (computeJoin m1 m2))) ->
     eval_exp E f1 v1 m1 ->
     eval_exp E f2 v2 m2 ->
-    eval_exp E (Binop op f1 f2) (perturb (evalBinop op v1 v2) delta) m
+    eval_exp E (Binop op f1 f2) (perturb (evalBinop op v1 v2) delta)  (computeJoin m1 m2)
 | Downcast_dist m m1 f1 v1 delta:
     (* Downcast expression f1 (evaluating to machine type m1), to a machine type m, less precise than m1.*)
     isMorePrecise m1 m = true ->
@@ -296,18 +296,27 @@ Proof.
   - inversion eval_v1; inversion eval_v2; subst; auto;
       try repeat (repeat rewrite delta_0_deterministic; simpl in *; rewrite Q2R0_is_0 in *; subst; auto); simpl.
     assert (M0 = M0) as M00 by auto.
-    pose proof (ifM0isJoin_l M0 m0 m2 M00 H2); auto.
-    pose proof (ifM0isJoin_r M0 m0 m2 M00 H2); auto.
-    pose proof (ifM0isJoin_l M0 m4 m5 M00 H11); auto.
-    pose proof (ifM0isJoin_r M0 m4 m5 M00 H11); auto.
-    subst.
+    destruct m0; destruct m2; inversion H4.
+    destruct m3; destruct m4; inversion H10.
+    simpl in *.
     rewrite (IHf1 v0 v4 M0); auto.
     rewrite (IHf2 v5 v3 M0); auto.
+    rewrite Q2R0_is_0 in H2,H12.
+    rewrite delta_0_deterministic; auto.
+    rewrite delta_0_deterministic; auto.
   - simpl toREval in eval_v1.
     simpl toREval in eval_v2.
     apply (IHf v1 v2 m1); auto.
 Qed.
 
+(* Lemma rnd_0_deterministic f E m v: *)
+(*   eval_exp E (toREval (Downcast m f)) v M0 <-> *)
+(*   eval_exp E (toREval f) v M0. *)
+(* Proof. *)
+(*   split; intros. *)
+(*   - simpl in H. auto. *)
+(*   - simpl; auto. *)
+(* Qed. *)
 
   
 (**
@@ -334,10 +343,11 @@ variables in the Environment.
 Lemma binary_unfolding b f1 f2 m E vF:
   eval_exp E (Binop b f1 f2) vF m ->
   exists vF1 vF2 m1 m2,
-  eval_exp E f1 vF1 m1 /\
-  eval_exp E f2 vF2 m2 /\
-  eval_exp  (updEnv 2 m2 vF2 (updEnv 1 m1 vF1 emptyEnv))
-           (Binop b (Var R m1 1) (Var R m2 2)) vF m.
+    m = computeJoin m1 m2 /\
+    eval_exp E f1 vF1 m1 /\
+    eval_exp E f2 vF2 m2 /\
+    eval_exp  (updEnv 2 m2 vF2 (updEnv 1 m1 vF1 emptyEnv))
+              (Binop b (Var R m1 1) (Var R m2 2)) vF m.
 Proof.
   intros eval_float.
   inversion eval_float; subst.

coq/Infra/MachineType.v

@@ -272,6 +272,15 @@ Proof.
   - apply EquivEqBoolEq in H; auto.
 Qed.
 
+Lemma ifM0isJoin (m1:mType) (m2:mType):
+  isJoinOf M0 m1 m2 = true -> m1 = M0 /\ m2 = M0.
+Proof.
+  assert (M0 = M0) by auto.
+  intros; split.
+  - apply (ifM0isJoin_l M0 m1 m2); auto.
+  - apply (ifM0isJoin_r M0 m1 m2); auto.  
+Qed.
+
 Lemma computeJoinIsJoin (m1:mType) (m2:mType) :
   isJoinOf (computeJoin m1 m2) m1 m2 = true.
 Proof.

coq/IntervalValidation.v

@@ -476,7 +476,7 @@ Proof.
       rewrite NatSet.diff_spec in in_diff_e1.
       destruct in_diff_e1 as [ in_usedVars not_dVar].
       split; try auto.
-      assert (m1 = M0) by (apply (ifM0isJoin_l M0 m1 m2); auto); subst; auto.
+      destruct m1; destruct m2; inversion H2; subst; auto.
     + assert (Q2R (fst (fst (iv2, err2))) <= v2 <= Q2R (snd (fst (iv2, err2))))%R as valid_bounds_e2.
       * apply IHf2; try auto.
         intros v in_diff_e2.
@@ -484,7 +484,7 @@ Proof.
         simpl. rewrite NatSet.diff_spec, NatSet.union_spec.
         rewrite NatSet.diff_spec in in_diff_e2.
         destruct in_diff_e2; split; auto.
-        assert (m2 = M0) by (apply (ifM0isJoin_r M0 m1 m2); auto); subst; auto.
+        destruct m1; destruct m2; inversion H2; auto.
       * destruct b; simpl in *.
         { pose proof (interval_addition_valid (iv1 :=(Q2R (fst iv1),Q2R (snd iv1))) (iv2 :=(Q2R (fst iv2), Q2R (snd iv2)))) as valid_add.
           unfold validIntervalAdd in valid_add.
@@ -614,8 +614,10 @@ Proof.
                     rewrite <- Q2R_inv in valid_div_hi; [ | auto].
                     repeat rewrite <- Q2R_mult in valid_div_hi.
                     rewrite <- Q2R_max4 in valid_div_hi; auto. } }
-    + simpl in H3; rewrite Q2R0_is_0 in H3; auto.
-    + simpl in H3; rewrite Q2R0_is_0 in H3; auto.
+    + destruct m1; destruct m2; inversion H2.
+      simpl in H4; rewrite Q2R0_is_0 in H4; auto.
+    + destruct m1; destruct m2; inversion H2.
+      simpl in H4; rewrite Q2R0_is_0 in H4; auto.
   - unfold validIntervalbounds in valid_bounds.
     (*simpl erasure in valid_bounds.*)
     simpl in *; destruct (absenv (Downcast m f)); destruct (absenv f); simpl in *.

coq/ssaPrgs.v

@@ -78,9 +78,10 @@ Proof.
           apply NatSet.union_spec; auto. }
       * destruct eval_e1_def as [vR1 eval_e1_def];
           destruct eval_e2_def as [vR2 eval_e2_def].
-        exists (perturb (evalBinop b vR1 vR2) 0); econstructor; eauto.
-        auto.
-        simpl. rewrite Q2R0_is_0. rewrite Rabs_R0; lra.
+        exists (perturb (evalBinop b vR1 vR2) 0).
+        replace M0 with (computeJoin M0 M0) by auto.
+        constructor; auto.
+        simpl meps; rewrite Q2R0_is_0. rewrite Rabs_R0; lra.
   - assert (exists vR, eval_exp (toREvalEnv E) (toREval (toRExp e))  vR M0) as eval_r_def by (eapply IHe; eauto).
     destruct eval_r_def as [vr eval_r_def].
     exists vr.
@@ -220,13 +221,10 @@ Proof.
     + erewrite <- IHe; eauto.
     + erewrite <- IHe; eauto.
   - split; intros eval_Binop; inversion eval_Binop; subst; econstructor; eauto;
-      assert (M0 = M0) as M00 by auto;
-      pose proof (ifM0isJoin_l M0 m1 m2 M00 H2);
-      pose proof (ifM0isJoin_r M0 m1 m2 M00 H2);
-      subst;
+    destruct m1; destruct m2; inversion H2;
       try (erewrite IHe1; eauto);
       try (erewrite IHe2; eauto); auto.
-  - split; intros eval_Downcast; inversion eval_Downcast; subst; try auto; erewrite IHe; eauto.
+  - split; intros eval_Downcast; simpl; simpl in eval_Downcast; erewrite IHe; eauto.
 Qed.
 
 Lemma shadowing_free_rewriting_cmd f E1 E2 vR:
@@ -285,9 +283,7 @@ Proof.
   - inversion eval_e; subst; econstructor; eauto.
   - simpl in valid_vars.
     inversion eval_e; subst; econstructor; eauto;
-      assert (M0 = M0) as M00 by auto;
-      pose proof (ifM0isJoin_l M0 m1 m2 M00 H2);
-      pose proof (ifM0isJoin_r M0 m1 m2 M00 H2);
+    destruct m1; destruct m2; inversion H2;
       subst.
     + eapply IHe1; eauto.
       hnf; intros a in_e1.
@@ -347,15 +343,11 @@ Proof.
         rewrite IHe; auto.
   - intros v_res; split; [intros eval_upd | intros eval_subst].
     + inversion eval_upd; econstructor; auto;
-        assert (M0 = M0) as M00 by auto; pose proof (ifM0isJoin_l M0 m1 m2 M00 H2);
-          pose proof (ifM0isJoin_r M0 m1 m2 M00 H2); subst.
-      * apply H2.
+      destruct m1; destruct m2; inversion H2.
       * rewrite <- IHe1; auto.
       * rewrite <- IHe2; auto.
     + inversion eval_subst; econstructor; auto;
-        assert (M0 = M0) as M00 by auto; pose proof (ifM0isJoin_l M0 m1 m2 M00 H2);
-          pose proof (ifM0isJoin_r M0 m1 m2 M00 H2); subst.
-      * apply H2.
+        destruct m1; destruct m2; inversion H2.
       * rewrite IHe1; auto.
       * rewrite IHe2; auto.
   - split; [intros eval_upd | intros eval_subst].