Add explanation of isMorePrecise for fixed-points, rename M0 into REAL and...

Add explanation of isMorePrecise for fixed-points, rename M0 into REAL and remove unused import from Expressions.v

coq/CertificateChecker.v

@@ -38,7 +38,7 @@ Theorem Certificate_checking_is_sound (e:expr Q) (absenv:analysisResult) P defVa
     CertificateChecker e absenv P defVars = true ->
     exists iv err vR vF m,
       FloverMap.find e absenv = Some (iv, err) /\
-      eval_expr E1 (toRMap defVars) (toREval (toRExp e)) vR M0 /\
+      eval_expr E1 (toRMap defVars) (toREval (toRExp e)) vR REAL /\
       eval_expr E2 defVars (toRExp e) vF m /\
       (forall vF m,
           eval_expr E2 defVars (toRExp e) vF m ->
@@ -93,7 +93,7 @@ Theorem Certificate_checking_cmds_is_sound (f:cmd Q) (absenv:analysisResult) P d
     CertificateCheckerCmd f absenv P defVars = true ->
     exists iv err vR vF m,
       FloverMap.find  (getRetExp f) absenv = Some (iv,err) /\
-    bstep (toREvalCmd (toRCmd f)) E1 (toRMap defVars) vR M0 /\
+    bstep (toREvalCmd (toRCmd f)) E1 (toRMap defVars) vR REAL /\
     bstep (toRCmd f) E2 defVars vF m /\
     (forall vF m,
         bstep (toRCmd f) E2 defVars vF m ->

coq/Commands.v

@@ -29,7 +29,7 @@ Fixpoint toRCmd (f:cmd Q) :=
 
 Fixpoint toREvalCmd (f:cmd R) :=
   match f with
-  |Let m x e g => Let M0 x (toREval e) (toREvalCmd g)
+  |Let m x e g => Let REAL x (toREval e) (toREvalCmd g)
   |Ret e => Ret (toREval e)
   end.
 

coq/ErrorBounds.v

@@ -8,7 +8,7 @@ Require Import Flover.Infra.Abbrevs Flover.Infra.RationalSimps Flover.Infra.Real
 Require Import Flover.Environments Flover.Infra.ExpressionAbbrevs.
 
 Lemma const_abs_err_bounded (n:R) (nR:R) (nF:R) (E1 E2:env) (m:mType) defVars:
-  eval_expr E1 (toRMap defVars) (Const M0 n) nR M0 ->
+  eval_expr E1 (toRMap defVars) (Const REAL n) nR REAL ->
   eval_expr E2 defVars (Const m n) nF m ->
   (Rabs (nR - nF) <= computeErrorR n m)%R.
 Proof.
@@ -31,11 +31,11 @@ Qed.
 
 Lemma add_abs_err_bounded (e1:expr Q) (e1R:R) (e1F:R) (e2:expr Q) (e2R:R) (e2F:R)
       (vR:R) (vF:R) (E1 E2:env) (err1 err2 :Q) (m m1 m2:mType) defVars:
-  eval_expr E1 (toRMap defVars) (toREval (toRExp e1)) e1R M0 ->
+  eval_expr E1 (toRMap defVars) (toREval (toRExp e1)) e1R REAL ->
   eval_expr E2 defVars (toRExp e1) e1F m1->
-  eval_expr E1 (toRMap defVars) (toREval (toRExp e2)) e2R M0 ->
+  eval_expr E1 (toRMap defVars) (toREval (toRExp e2)) e2R REAL ->
   eval_expr E2 defVars (toRExp e2) e2F m2 ->
-  eval_expr E1 (toRMap defVars) (toREval (Binop Plus (toRExp e1) (toRExp e2))) vR M0 ->
+  eval_expr E1 (toRMap defVars) (toREval (Binop Plus (toRExp e1) (toRExp e2))) vR REAL ->
   eval_expr (updEnv 2 e2F (updEnv 1 e1F emptyEnv))
            (updDefVars 2 m2 (updDefVars 1 m1 defVars))
            (Binop Plus (Var R 1) (Var R 2)) vF m ->
@@ -46,11 +46,11 @@ 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 (m0 = M0) by (eapply toRMap_eval_M0; eauto).
-  assert (m3 = M0) by (eapply toRMap_eval_M0; eauto).
+  assert (m0 = REAL) by (eapply toRMap_eval_REAL; eauto).
+  assert (m3 = REAL) by (eapply toRMap_eval_REAL; eauto).
   subst; simpl in H3; auto.
   rewrite delta_0_deterministic in plus_real; auto.
-  rewrite (delta_0_deterministic (evalBinop Plus v1 v2) (join M0 M0) delta); auto.
+  rewrite (delta_0_deterministic (evalBinop Plus v1 v2) (join REAL REAL) delta); auto.
   unfold evalBinop in *; simpl in *.
   clear delta H3.
   rewrite (meps_0_deterministic (toRExp e1) H5 e1_real);
@@ -104,11 +104,11 @@ Qed.
 **)
 Lemma subtract_abs_err_bounded (e1:expr Q) (e1R:R) (e1F:R) (e2:expr Q) (e2R:R)
       (e2F:R) (vR:R) (vF:R) (E1 E2:env) err1 err2 (m m1 m2:mType) defVars:
-  eval_expr E1 (toRMap defVars) (toREval (toRExp e1)) e1R M0 ->
+  eval_expr E1 (toRMap defVars) (toREval (toRExp e1)) e1R REAL ->
   eval_expr E2 defVars (toRExp e1) e1F m1 ->
-  eval_expr E1 (toRMap defVars) (toREval (toRExp e2)) e2R M0 ->
+  eval_expr E1 (toRMap defVars) (toREval (toRExp e2)) e2R REAL ->
   eval_expr E2 defVars (toRExp e2) e2F m2 ->
-  eval_expr E1 (toRMap defVars) (toREval (Binop Sub (toRExp e1) (toRExp e2))) vR M0 ->
+  eval_expr E1 (toRMap defVars) (toREval (Binop Sub (toRExp e1) (toRExp e2))) vR REAL ->
   eval_expr (updEnv 2 e2F (updEnv 1 e1F emptyEnv))
            (updDefVars 2 m2 (updDefVars 1 m1 defVars))
            (Binop Sub (Var R 1) (Var R 2)) vF m ->
@@ -119,8 +119,8 @@ 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 (m0 = M0) by (eapply toRMap_eval_M0; eauto).
-  assert (m3 = M0) by (eapply toRMap_eval_M0; eauto).
+  assert (m0 = REAL) by (eapply toRMap_eval_REAL; eauto).
+  assert (m3 = REAL) by (eapply toRMap_eval_REAL; eauto).
   subst; simpl in H3; auto.
   rewrite delta_0_deterministic in sub_real; auto.
   rewrite delta_0_deterministic; auto.
@@ -182,11 +182,11 @@ Qed.
 
 Lemma mult_abs_err_bounded (e1:expr Q) (e1R:R) (e1F:R) (e2:expr Q) (e2R:R) (e2F:R)
       (vR:R) (vF:R) (E1 E2:env) (m m1 m2:mType) defVars:
-  eval_expr E1 (toRMap defVars) (toREval (toRExp e1)) e1R M0 ->
+  eval_expr E1 (toRMap defVars) (toREval (toRExp e1)) e1R REAL ->
   eval_expr E2 defVars (toRExp e1) e1F m1 ->
-  eval_expr E1 (toRMap defVars) (toREval (toRExp e2)) e2R M0 ->
+  eval_expr E1 (toRMap defVars) (toREval (toRExp e2)) e2R REAL ->
   eval_expr E2 defVars (toRExp e2) e2F m2 ->
-  eval_expr E1 (toRMap defVars) (toREval (Binop Mult (toRExp e1) (toRExp e2))) vR M0 ->
+  eval_expr E1 (toRMap defVars) (toREval (Binop Mult (toRExp e1) (toRExp e2))) vR REAL ->
   eval_expr (updEnv 2 e2F (updEnv 1 e1F emptyEnv))
            (updDefVars 2 m2 (updDefVars 1 m1 defVars))
            (Binop Mult (Var R 1) (Var R 2)) vF m ->
@@ -195,8 +195,8 @@ 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 (m0 = M0) by (eapply toRMap_eval_M0; eauto).
-  assert (m3 = M0) by (eapply toRMap_eval_M0; eauto).
+  assert (m0 = REAL) by (eapply toRMap_eval_REAL; eauto).
+  assert (m3 = REAL) by (eapply toRMap_eval_REAL; eauto).
   subst; simpl in H3; auto.
   rewrite delta_0_deterministic in mult_real; auto.
   rewrite delta_0_deterministic; auto.
@@ -231,11 +231,11 @@ Qed.
 
 Lemma div_abs_err_bounded (e1:expr Q) (e1R:R) (e1F:R) (e2:expr Q) (e2R:R) (e2F:R)
       (vR:R) (vF:R) (E1 E2:env) (m m1 m2:mType) defVars:
-  eval_expr E1 (toRMap defVars) (toREval (toRExp e1)) e1R M0 ->
+  eval_expr E1 (toRMap defVars) (toREval (toRExp e1)) e1R REAL ->
   eval_expr E2 defVars (toRExp e1) e1F m1 ->
-  eval_expr E1 (toRMap defVars) (toREval (toRExp e2)) e2R M0 ->
+  eval_expr E1 (toRMap defVars) (toREval (toRExp e2)) e2R REAL ->
   eval_expr E2 defVars (toRExp e2) e2F m2 ->
-  eval_expr E1 (toRMap defVars) (toREval (Binop Div (toRExp e1) (toRExp e2))) vR M0 ->
+  eval_expr E1 (toRMap defVars) (toREval (Binop Div (toRExp e1) (toRExp e2))) vR REAL ->
   eval_expr (updEnv 2 e2F (updEnv 1 e1F emptyEnv))
            (updDefVars 2 m2 (updDefVars 1 m1 defVars))
            (Binop Div (Var R 1) (Var R 2)) vF m ->
@@ -244,8 +244,8 @@ 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 (m0 = M0) by (eapply toRMap_eval_M0; eauto).
-  assert (m3 = M0) by (eapply toRMap_eval_M0; eauto).
+  assert (m0 = REAL) by (eapply toRMap_eval_REAL; eauto).
+  assert (m3 = REAL) by (eapply toRMap_eval_REAL; eauto).
   subst; simpl in H3; auto.
   rewrite delta_0_deterministic in div_real; auto.
   rewrite delta_0_deterministic; auto.
@@ -280,13 +280,13 @@ Qed.
 Lemma fma_abs_err_bounded (e1:expr Q) (e1R:R) (e1F:R) (e2:expr Q) (e2R:R) (e2F:R)
       (e3:expr Q) (e3R:R) (e3F:R)
       (vR:R) (vF:R) (E1 E2:env) (m m1 m2 m3:mType) defVars:
-  eval_expr E1 (toRMap defVars) (toREval (toRExp e1)) e1R M0 ->
+  eval_expr E1 (toRMap defVars) (toREval (toRExp e1)) e1R REAL ->
   eval_expr E2 defVars (toRExp e1) e1F m1->
-  eval_expr E1 (toRMap defVars) (toREval (toRExp e2)) e2R M0 ->
+  eval_expr E1 (toRMap defVars) (toREval (toRExp e2)) e2R REAL ->
   eval_expr E2 defVars (toRExp e2) e2F m2 ->
-  eval_expr E1 (toRMap defVars) (toREval (toRExp e3)) e3R M0 ->
+  eval_expr E1 (toRMap defVars) (toREval (toRExp e3)) e3R REAL ->
   eval_expr E2 defVars (toRExp e3) e3F m3->
-  eval_expr E1 (toRMap defVars) (toREval (Fma (toRExp e1) (toRExp e2) (toRExp e3))) vR M0 ->
+  eval_expr E1 (toRMap defVars) (toREval (Fma (toRExp e1) (toRExp e2) (toRExp e3))) vR REAL ->
   eval_expr (updEnv 3 e3F (updEnv 2 e2F (updEnv 1 e1F emptyEnv)))
            (updDefVars 3 m3 (updDefVars 2 m2 (updDefVars 1 m1 defVars)))
            (Fma (Var R 1) (Var R 2) (Var R 3)) vF m ->
@@ -294,9 +294,9 @@ Lemma fma_abs_err_bounded (e1:expr Q) (e1R:R) (e1F:R) (e2:expr Q) (e2R:R) (e2F:R
 Proof.
   intros e1_real e1_float e2_real e2_float e3_real e3_float fma_real fma_float.
   inversion fma_real; subst;
-  assert (m0 = M0) by (eapply toRMap_eval_M0; eauto).
-  assert (m4 = M0) by (eapply toRMap_eval_M0; eauto).
-  assert (m5 = M0) by (eapply toRMap_eval_M0; eauto).
+  assert (m0 = REAL) by (eapply toRMap_eval_REAL; eauto).
+  assert (m4 = REAL) by (eapply toRMap_eval_REAL; eauto).
+  assert (m5 = REAL) by (eapply toRMap_eval_REAL; eauto).
   subst; simpl in H3; auto.
   rewrite delta_0_deterministic in fma_real; auto.
   rewrite delta_0_deterministic; auto.
@@ -345,7 +345,7 @@ Proof.
 Qed.
 
 Lemma round_abs_err_bounded (e:expr R) (nR nF1 nF:R) (E1 E2: env) (err:R) (mEps m:mType) defVars:
-  eval_expr E1 (toRMap defVars) (toREval e) nR M0 ->
+  eval_expr E1 (toRMap defVars) (toREval e) nR REAL ->
   eval_expr E2 defVars e nF1 m ->
   eval_expr (updEnv 1 nF1 emptyEnv)
            (updDefVars 1 m defVars)

coq/ErrorValidation.v

@@ -151,7 +151,7 @@ Qed.
 
 Lemma validErrorboundCorrectVariable_eval E1 E2 A (v:nat) e nR nlo nhi P fVars
       dVars Gamma exprTypes:
-    eval_expr E1 (toRMap Gamma) (toREval (toRExp (Var Q v))) nR M0 ->
+    eval_expr E1 (toRMap Gamma) (toREval (toRExp (Var Q v))) nR REAL ->
     typeCheck (Var Q v) Gamma exprTypes = true ->
     approxEnv E1 Gamma A fVars dVars E2 ->
     validIntervalbounds (Var Q v) A P dVars = true ->
@@ -181,7 +181,7 @@ Qed.
 
 Lemma validErrorboundCorrectVariable:
   forall E1 E2 A (v:nat) e nR nF mF nlo nhi P fVars dVars Gamma exprTypes,
-    eval_expr E1 (toRMap Gamma) (toREval (toRExp (Var Q v))) nR M0 ->
+    eval_expr E1 (toRMap Gamma) (toREval (toRExp (Var Q v))) nR REAL ->
     eval_expr E2 Gamma (toRExp (Var Q v)) nF mF ->
     typeCheck (Var Q v) Gamma exprTypes = true ->
     approxEnv E1 Gamma A fVars dVars E2 ->
@@ -226,7 +226,7 @@ Proof.
 Qed.
 
 Lemma validErrorboundCorrectConstant_eval E1 E2 m n nR  Gamma:
-    eval_expr E1 (toRMap Gamma) (toREval (toRExp (Const m n))) nR M0 ->
+    eval_expr E1 (toRMap Gamma) (toREval (toRExp (Const m n))) nR REAL ->
     exists nF m',
       eval_expr E2 Gamma (toRExp (Const m n)) nF m'.
 Proof.
@@ -238,7 +238,7 @@ Proof.
 Qed.
 
 Lemma validErrorboundCorrectConstant E1 E2 A m n nR nF e nlo nhi dVars Gamma defVars:
-    eval_expr E1 (toRMap defVars) (toREval (toRExp (Const m n))) nR M0 ->
+    eval_expr E1 (toRMap defVars) (toREval (toRExp (Const m n))) nR REAL ->
     eval_expr E2 defVars (toRExp (Const m n)) nF m ->
     typeCheck (Const m n) defVars Gamma = true ->
     validErrorbound (Const m n) Gamma A dVars = true ->
@@ -265,9 +265,9 @@ Lemma validErrorboundCorrectAddition E1 E2 A
       (e1:expr Q) (e2:expr Q) (nR nR1 nR2 nF nF1 nF2 :R) (e err1 err2 :error)
       (alo ahi e1lo e1hi e2lo e2hi:Q) dVars m m1 m2 Gamma defVars:
   m = join m1 m2 ->
-  eval_expr E1 (toRMap defVars) (toREval (toRExp e1)) nR1 M0 ->
-  eval_expr E1 (toRMap defVars) (toREval (toRExp e2)) nR2 M0 ->
-  eval_expr E1 (toRMap defVars) (toREval (toRExp (Binop Plus e1 e2))) nR M0 ->
+  eval_expr E1 (toRMap defVars) (toREval (toRExp e1)) nR1 REAL ->
+  eval_expr E1 (toRMap defVars) (toREval (toRExp e2)) nR2 REAL ->
+  eval_expr E1 (toRMap defVars) (toREval (toRExp (Binop Plus e1 e2))) nR REAL ->
   eval_expr E2 defVars (toRExp e1) nF1 m1 ->
   eval_expr E2 defVars (toRExp e2) nF2 m2 ->
   eval_expr (updEnv 2 nF2 (updEnv 1 nF1 emptyEnv))
@@ -327,9 +327,9 @@ Lemma validErrorboundCorrectSubtraction E1 E2 A
       (e1:expr Q) (e2:expr Q) (nR nR1 nR2 nF nF1 nF2 :R) (e err1 err2 :error)
       (alo ahi e1lo e1hi e2lo e2hi:Q) dVars (m m1 m2:mType) Gamma defVars:
   m = join m1 m2 ->
-  eval_expr E1 (toRMap defVars) (toREval (toRExp e1)) nR1 M0 ->
-  eval_expr E1 (toRMap defVars) (toREval (toRExp e2)) nR2 M0 ->
-  eval_expr E1 (toRMap defVars) (toREval (toRExp (Binop Sub e1 e2))) nR M0 ->
+  eval_expr E1 (toRMap defVars) (toREval (toRExp e1)) nR1 REAL ->
+  eval_expr E1 (toRMap defVars) (toREval (toRExp e2)) nR2 REAL ->
+  eval_expr E1 (toRMap defVars) (toREval (toRExp (Binop Sub e1 e2))) nR REAL ->
   eval_expr E2 defVars (toRExp e1) nF1 m1->
   eval_expr E2 defVars (toRExp e2) nF2 m2 ->
   eval_expr (updEnv 2 nF2 (updEnv 1 nF1 emptyEnv))
@@ -859,9 +859,9 @@ Lemma validErrorboundCorrectMult E1 E2 A
       (e1:expr Q) (e2:expr Q) (nR nR1 nR2 nF nF1 nF2 :R) (e err1 err2 :error)
       (alo ahi e1lo e1hi e2lo e2hi:Q) dVars (m m1 m2:mType) Gamma defVars:
   m = join m1 m2 ->
-  eval_expr E1 (toRMap defVars) (toREval (toRExp e1)) nR1 M0 ->
-  eval_expr E1 (toRMap defVars) (toREval (toRExp e2)) nR2 M0 ->
-  eval_expr E1 (toRMap defVars) (toREval (toRExp (Binop Mult e1 e2))) nR M0 ->
+  eval_expr E1 (toRMap defVars) (toREval (toRExp e1)) nR1 REAL ->
+  eval_expr E1 (toRMap defVars) (toREval (toRExp e2)) nR2 REAL ->
+  eval_expr E1 (toRMap defVars) (toREval (toRExp (Binop Mult e1 e2))) nR REAL ->
   eval_expr E2 defVars (toRExp e1) nF1 m1 ->
   eval_expr E2 defVars (toRExp e2) nF2 m2 ->
   eval_expr (updEnv 2 nF2 (updEnv 1 nF1 emptyEnv))
@@ -930,9 +930,9 @@ Lemma validErrorboundCorrectDiv E1 E2 A
       (e1:expr Q) (e2:expr Q) (nR nR1 nR2 nF nF1 nF2 :R) (e err1 err2 :error)
       (alo ahi e1lo e1hi e2lo e2hi:Q) dVars (m m1 m2:mType) Gamma defVars:
   m = join m1 m2 ->
-  eval_expr E1 (toRMap defVars) (toREval (toRExp e1)) nR1 M0 ->
-  eval_expr E1 (toRMap defVars) (toREval (toRExp e2)) nR2 M0 ->
-  eval_expr E1 (toRMap defVars) (toREval (toRExp (Binop Div e1 e2))) nR M0 ->
+  eval_expr E1 (toRMap defVars) (toREval (toRExp e1)) nR1 REAL ->
+  eval_expr E1 (toRMap defVars) (toREval (toRExp e2)) nR2 REAL ->
+  eval_expr E1 (toRMap defVars) (toREval (toRExp (Binop Div e1 e2))) nR REAL ->
   eval_expr E2 defVars (toRExp e1) nF1 m1 ->
   eval_expr E2 defVars (toRExp e2) nF2 m2 ->
   eval_expr (updEnv 2 nF2 (updEnv 1 nF1 emptyEnv))
@@ -1845,10 +1845,10 @@ Lemma validErrorboundCorrectFma E1 E2 A
       (e1:expr Q) (e2:expr Q) (e3: expr Q) (nR nR1 nR2 nR3 nF nF1 nF2 nF3: R) (e err1 err2 err3 :error)
       (alo ahi e1lo e1hi e2lo e2hi e3lo e3hi:Q) dVars (m m1 m2 m3:mType) Gamma defVars:
   m = join3 m1 m2 m3 ->
-  eval_expr E1 (toRMap defVars) (toREval (toRExp e1)) nR1 M0 ->
-  eval_expr E1 (toRMap defVars) (toREval (toRExp e2)) nR2 M0 ->
-  eval_expr E1 (toRMap defVars) (toREval (toRExp e3)) nR3 M0 ->
-  eval_expr E1 (toRMap defVars) (toREval (toRExp (Fma e1 e2 e3))) nR M0 ->
+  eval_expr E1 (toRMap defVars) (toREval (toRExp e1)) nR1 REAL ->
+  eval_expr E1 (toRMap defVars) (toREval (toRExp e2)) nR2 REAL ->
+  eval_expr E1 (toRMap defVars) (toREval (toRExp e3)) nR3 REAL ->
+  eval_expr E1 (toRMap defVars) (toREval (toRExp (Fma e1 e2 e3))) nR REAL ->
   eval_expr E2 defVars (toRExp e1) nF1 m1 ->
   eval_expr E2 defVars (toRExp e2) nF2 m2 ->
   eval_expr E2 defVars (toRExp e3) nF3 m3 ->
@@ -1938,7 +1938,7 @@ Proof.
 Qed.
 
 Lemma validErrorboundCorrectRounding E1 E2 A (e: expr Q) (nR nF nF1: R) (err err':error) (elo ehi alo ahi: Q) dVars (m: mType) (machineEpsilon:mType) Gamma defVars:
-  eval_expr E1 (toRMap defVars) (toREval (toRExp e)) nR M0 ->
+  eval_expr E1 (toRMap defVars) (toREval (toRExp e)) nR REAL ->
   eval_expr E2 defVars (toRExp e) nF1 m ->
   eval_expr (updEnv 1 nF1 emptyEnv)
            (updDefVars 1 m defVars)
@@ -1981,7 +1981,7 @@ Theorem validErrorbound_sound (e:expr Q):
     typeCheck e defVars Gamma = true ->
     approxEnv E1 defVars A fVars dVars E2 ->
     NatSet.Subset (NatSet.diff (Expressions.usedVars e) dVars) fVars ->
-    eval_expr E1 (toRMap defVars) (toREval (toRExp e)) nR M0 ->
+    eval_expr E1 (toRMap defVars) (toREval (toRExp e)) nR REAL ->
     validErrorbound e Gamma A dVars = true ->
     validIntervalbounds e A P dVars = true ->
     dVars_range_valid dVars E1 A ->
@@ -2029,7 +2029,7 @@ Proof.
   - cbn in *. rewrite A_eq in *.
     Flover_compute; try congruence; type_conv; subst; simpl in *.
     inversion eval_real; subst.
-    assert (m0 = M0 /\ m3 = M0) as [? ?] by (split; eapply toRMap_eval_M0; eauto); subst.
+    assert (m0 = REAL /\ m3 = REAL) as [? ?] by (split; eapply toRMap_eval_REAL; eauto); subst.
     destruct i as [ivlo1 ivhi1]; destruct i1 as [ivlo2 ivhi2];
       rename e into err1; rename e3 into err2.
     destruct (IHe1 E1 E2 fVars dVars A v1 err1 P ivlo1 ivhi1 Gamma defVars)
@@ -2120,7 +2120,7 @@ Proof.
   - cbn in *. rewrite A_eq in *.
     Flover_compute; try congruence; type_conv; subst; simpl in *.
     inversion eval_real; subst.
-    assert (m0 = M0 /\ m4 = M0 /\ m5 = M0) as [? [? ?]] by (split; try split; eapply toRMap_eval_M0; eauto); subst.
+    assert (m0 = REAL /\ m4 = REAL /\ m5 = REAL) as [? [? ?]] by (split; try split; eapply toRMap_eval_REAL; eauto); subst.
     destruct i as [ivlo1 ivhi1]; destruct i2 as [ivlo2 ivhi2]; destruct i1 as [ivlo3 ivhi3];
       rename e into err1; rename e5 into err2; rename e4 into err3.
     destruct (IHe1 E1 E2 fVars dVars A v1 err1 P ivlo1 ivhi1 Gamma defVars)
@@ -2175,7 +2175,7 @@ Proof.
         auto. }
   - cbn in *; Flover_compute; try congruence; type_conv; subst.
     inversion eval_real; subst.
-    apply M0_least_precision in H1.
+    apply REAL_least_precision in H1.
     subst.
     destruct i0 as [ivlo_e ivhi_e]; rename e1 into err_e.
     destruct (IHe E1 E2 fVars dVars A v1 err_e P ivlo_e ivhi_e Gamma defVars)
@@ -2212,7 +2212,7 @@ Theorem validErrorboundCmd_gives_eval (f:cmd Q) :
     approxEnv E1 defVars A fVars dVars E2 ->
     ssa f (NatSet.union fVars dVars) outVars ->
     NatSet.Subset (NatSet.diff (Commands.freeVars f) dVars) fVars ->
-    bstep (toREvalCmd (toRCmd f)) E1 (toRMap defVars) vR M0 ->
+    bstep (toREvalCmd (toRCmd f)) E1 (toRMap defVars) vR REAL ->
     validErrorboundCmd f Gamma A dVars = true ->
     validIntervalboundsCmd f A P dVars = true ->
     dVars_range_valid dVars E1 A ->
@@ -2273,7 +2273,7 @@ Proof.
         simpl.
         rewrite NatSet.diff_spec, NatSet.remove_spec, NatSet.union_spec.
         split; try auto. }
-      { eapply swap_Gamma_bstep with (Gamma1 := updDefVars n M0 (toRMap defVars));
+      { eapply swap_Gamma_bstep with (Gamma1 := updDefVars n REAL (toRMap defVars));
           eauto using Rmap_updVars_comm. }
       { intros v1 v1_mem.
         unfold updEnv.
@@ -2324,7 +2324,7 @@ Theorem validErrorboundCmd_sound (f:cmd Q):
     approxEnv E1 defVars A fVars dVars E2 ->
     ssa f (NatSet.union fVars dVars) outVars ->
     NatSet.Subset (NatSet.diff (Commands.freeVars f) dVars) fVars ->
-    bstep (toREvalCmd (toRCmd f)) E1 (toRMap defVars) vR M0 ->
+    bstep (toREvalCmd (toRCmd f)) E1 (toRMap defVars) vR REAL ->
     bstep (toRCmd f) E2 defVars vF mF ->
     validErrorboundCmd f Gamma A dVars = true ->
     validIntervalboundsCmd f A P dVars = true ->
@@ -2385,7 +2385,7 @@ Proof.
       simpl.
       rewrite NatSet.diff_spec, NatSet.remove_spec, NatSet.union_spec.
       split; try auto.
-    + eapply swap_Gamma_bstep with (Gamma1 := updDefVars n M0 (toRMap defVars));
+    + eapply swap_Gamma_bstep with (Gamma1 := updDefVars n REAL (toRMap defVars));
         eauto using Rmap_updVars_comm.
     + intros v1 v1_mem.
       unfold updEnv.

coq/Expressions.v

 (**
   Formalization of the base exprression language for the flover framework
  **)
-From Coq
-     Require Import Structures.Orders.
-
 From Flover
      Require Import Infra.RealRationalProps Infra.Ltacs.
 
@@ -127,16 +124,16 @@ Fixpoint toRExp (e:expr Q) :=
 Fixpoint toREval (e:expr R) :=
   match e with
   | Var _ v => Var R v
-  | Const _ n => Const M0 n
+  | Const _ n => Const REAL n
   | Unop o e1 => Unop o (toREval e1)
   | Binop o e1 e2 => Binop o (toREval e1) (toREval e2)
   | Fma e1 e2 e3 => Fma (toREval e1) (toREval e2) (toREval e3)
-  | Downcast _ e1 =>   Downcast M0 (toREval e1)
+  | Downcast _ e1 =>   Downcast REAL (toREval e1)
   end.
 
 Definition toRMap (d:nat -> option mType) (n:nat) :=
   match d n with
-  | Some m => Some M0
+  | Some m => Some REAL
   | None => None
   end.
 
@@ -151,7 +148,7 @@ Arguments toRMap _ _/.
 Definition perturb (rVal:R) (m:mType) (delta:R) :R :=
   match m with
   (* The Real-type has no error *)
-  |M0 => rVal
+  |REAL => rVal
   (* Fixed-point numbers have an absolute error *)
   |F w f => rVal + delta
   (* Floating-point numbers have a relative error *)
@@ -299,8 +296,8 @@ Fixpoint usedVars (V:Type) (e:expr V) :NatSet.t :=
   | _ => NatSet.empty
   end.
 
-Lemma toRMap_eval_M0 f v E Gamma m:
-  eval_expr E (toRMap Gamma) (toREval f) v m -> m = M0.
+Lemma toRMap_eval_REAL f v E Gamma m:
+  eval_expr E (toRMap Gamma) (toREval f) v m -> m = REAL.
 Proof.
   revert v E Gamma m.
   induction f; intros * eval_f; inversion eval_f; subst;
@@ -313,16 +310,16 @@ Proof.
     end; try auto.
   - eapply IHf; eauto.
   - eapply IHf; eauto.
-  - assert (m1 = M0)
+  - assert (m1 = REAL)
       by (eapply IHf1; eauto).
-    assert (m2 = M0)
+    assert (m2 = REAL)
       by (eapply IHf2; eauto);
       subst; auto.
-  - assert (m1 = M0)
+  - assert (m1 = REAL)
       by (eapply IHf1; eauto).
-    assert (m2 = M0)
+    assert (m2 = REAL)
       by (eapply IHf2; eauto).
-    assert (m3 = M0)
+    assert (m3 = REAL)
       by (eapply IHf3; eauto);
       subst; auto.
 Qed.
@@ -343,8 +340,8 @@ Evaluation with 0 as machine epsilon is deterministic
 **)
 Lemma meps_0_deterministic (f:expr R) (E:env) Gamma:
   forall v1 v2,
-  eval_expr E (toRMap Gamma) (toREval f) v1 M0 ->
-  eval_expr E (toRMap Gamma) (toREval f) v2 M0 ->
+  eval_expr E (toRMap Gamma) (toREval f) v1 REAL ->
+  eval_expr E (toRMap Gamma) (toREval f) v2 REAL ->
   v1 = v2.
 Proof.
   induction f;
@@ -358,27 +355,27 @@ Proof.
     + rewrite (IHf v0 v3); eauto.
     + cbn in H1, H8. subst. rewrite (IHf v0 v3); eauto.
   - inversion ev1; inversion ev2; subst.
-    assert (m0 = M0) by (eapply toRMap_eval_M0; eauto).
-    assert (m3 = M0) by (eapply toRMap_eval_M0; eauto).
-    assert (m1 = M0) by (eapply toRMap_eval_M0; eauto).
-    assert (m2 = M0) by (eapply toRMap_eval_M0; eauto).
+    assert (m0 = REAL) by (eapply toRMap_eval_REAL; eauto).
+    assert (m3 = REAL) by (eapply toRMap_eval_REAL; eauto).
+    assert (m1 = REAL) by (eapply toRMap_eval_REAL; eauto).
+    assert (m2 = REAL) by (eapply toRMap_eval_REAL; eauto).
     subst.
     rewrite (IHf1 v0 v4); try auto.
     rewrite (IHf2 v3 v5); try auto.
   - inversion ev1; inversion ev2; subst.
-    assert (m0 = M0) by (eapply toRMap_eval_M0; eauto).
-    assert (m1 = M0) by (eapply toRMap_eval_M0; eauto).
-    assert (m2 = M0) by (eapply toRMap_eval_M0; eauto).
-    assert (m3 = M0) by (eapply toRMap_eval_M0; eauto).
-    assert (m4 = M0) by (eapply toRMap_eval_M0; eauto).
-    assert (m5 = M0) by (eapply toRMap_eval_M0; eauto).
+    assert (m0 = REAL) by (eapply toRMap_eval_REAL; eauto).
+    assert (m1 = REAL) by (eapply toRMap_eval_REAL; eauto).
+    assert (m2 = REAL) by (eapply toRMap_eval_REAL; eauto).
+    assert (m3 = REAL) by (eapply toRMap_eval_REAL; eauto).
+    assert (m4 = REAL) by (eapply toRMap_eval_REAL; eauto).
+    assert (m5 = REAL) by (eapply toRMap_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 M0_least_precision in H1;
-      apply M0_least_precision in H7; subst.
+    apply REAL_least_precision in H1;
+      apply REAL_least_precision in H7; subst.
     rewrite (IHf v0 v3); try auto.
 Qed.
 

coq/FPRangeValidator.v

@@ -196,7 +196,7 @@ Proof.
               - rewrite NatSet.add_spec, NatSet.union_spec in in_set;
                   rewrite NatSet.union_spec, NatSet.add_spec.
                 destruct in_set as [P1 | [ P2 | P3]]; auto. }
-            { eapply (swap_Gamma_bstep (Gamma1 := updDefVars n M0 (toRMap Gamma))); eauto.
+            { eapply (swap_Gamma_bstep (Gamma1 := updDefVars n REAL (toRMap Gamma))); eauto.
               eauto using Rmap_updVars_comm. }
             { set_tac; split.
               - split; try auto.

coq/IEEE_connection.v

@@ -911,7 +911,7 @@ Lemma bstep_gives_IEEE (f:cmd fl64) :
     validIntervalboundsCmd (B2Qcmd f) A P dVars = true ->
     validErrorboundCmd (B2Qcmd f) tMap A dVars = true ->
     FPRangeValidatorCmd (B2Qcmd f) A tMap dVars = true ->
-    bstep (toREvalCmd (toRCmd (B2Qcmd f))) E1 (toRMap Gamma) vR M0 ->
+    bstep (toREvalCmd (toRCmd (B2Qcmd f))) E1 (toRMap Gamma) vR REAL ->
     bstep (toRCmd (B2Qcmd f)) (toREnv E2) Gamma vF M64 ->
     NatSet.Subset (NatSet.diff (freeVars (B2Qcmd f)) dVars) fVars ->
     is64BitBstep (B2Qcmd f) ->
@@ -1017,7 +1017,7 @@ Proof.
         split; try auto. split; try auto.
         hnf; intros; subst. apply H2. rewrite NatSet.add_spec. auto.
         rewrite NatSet.add_spec in H2. hnf; intros; apply H2; auto.
-      - eapply (swap_Gamma_bstep (Gamma1:= updDefVars n M0 (toRMap Gamma)));
+      - eapply (swap_Gamma_bstep (Gamma1:= updDefVars n REAL (toRMap Gamma)));
           eauto.
         intros; unfold updDefVars, toRMap.
         destruct (n0 =? n); auto.
@@ -1077,7 +1077,7 @@ Proof.
       * rewrite NatSet.add_spec in H1;
           rewrite NatSet.union_spec, NatSet.add_spec in *;
           destruct H1; auto. destruct H1; auto.
-    + eapply (swap_Gamma_bstep (Gamma1:= updDefVars n M0 (toRMap Gamma)));
+    + eapply (swap_Gamma_bstep (Gamma1:= updDefVars n REAL (toRMap Gamma)));
         eauto.
       intros; unfold updDefVars, toRMap.
       destruct (n0 =? n); auto.
@@ -1180,7 +1180,7 @@ Theorem IEEE_connection_expr e A P E1 E2 defVars:
   CertificateChecker (B2Qexpr e) A P defVars = true ->
   exists iv err vR vF, (* m, currently = M64 *)
     FloverMap.find (B2Qexpr e) A = Some (iv, err) /\
-    eval_expr E1 (toRMap defVars) (toREval (toRExp (B2Qexpr e))) vR M0 /\
+    eval_expr E1 (toRMap defVars) (toREval (toRExp (B2Qexpr e))) vR REAL /\
     eval_expr_float e E2 = Some vF /\
     eval_expr (toREnv E2) defVars (toRExp (B2Qexpr e)) (Q2R (B2Q vF)) M64 /\
     (Rabs (vR - Q2R (B2Q vF )) <= Q2R err)%R.
@@ -1226,7 +1226,7 @@ Theorem IEEE_connection_cmd (f:cmd fl64) (A:analysisResult) P
     CertificateCheckerCmd (B2Qcmd f) A P defVars = true ->
     exists iv err vR vF m,
       FloverMap.find (getRetExp (B2Qcmd f)) A = Some (iv,err) /\
-    bstep (toREvalCmd (toRCmd (B2Qcmd f))) E1 (toRMap defVars) vR M0 /\
+    bstep (toREvalCmd (toRCmd (B2Qcmd f))) E1 (toRMap defVars) vR REAL /\
     bstep_float f E2 = Some vF /\
     bstep (toRCmd (B2Qcmd f)) (toREnv E2) defVars (Q2R (B2Q vF)) m /\
     (forall vF m,

coq/Infra/MachineType.v

@@ -18,7 +18,7 @@ From Flover
    like Flocq, where f:positive specifies the fraction size and w: the width of
    the base field.
  **)
-Inductive mType: Type := M0 | M16 | M32 | M64
+Inductive mType: Type := REAL | M16 | M32 | M64
                          | F (w:positive) (f:positive). (*| M128 | M256*)
 
 (**
@@ -26,7 +26,7 @@ Inductive mType: Type := M0 | M16 | M32 | M64
 **)
 Definition mTypeToR (m:mType) :R :=
   match m with
-  | M0 => 0%R
+  | REAL => 0%R
   | M16 => 1 / 2^11
   | M32 => 1/ 2^24
   | M64 => 1/ 2^53
@@ -40,7 +40,7 @@ Definition mTypeToR (m:mType) :R :=
 
 Definition mTypeToQ (m:mType) :Q :=
   match m with
-  | M0 => 0
+  | REAL => 0
   | M16 => (Qpower (2#1) (Zneg 11))
   | M32 => (Qpower (2#1) (Zneg 24))
   | M64 => (Qpower (2#1) (Zneg 53))
@@ -54,14 +54,14 @@ Definition mTypeToQ (m:mType) :Q :=
 
 Definition computeErrorR v m :R :=
   match m with
-  |M0 => 0
+  |REAL => 0
   |F w f => mTypeToR m
   |_ => Rabs v * mTypeToR m
   end.
 
 Definition computeErrorQ v m :Q :=
   match m with
-  |M0 => 0
+  |REAL => 0
   |F w f => mTypeToQ m
   |_ => Qabs v * mTypeToQ m
   end.
@@ -159,7 +159,7 @@ Qed.
 
 Definition mTypeEq (m1:mType) (m2:mType) :=
   match m1, m2 with
-  | M0, M0 => true
+  | REAL, REAL => true
   | M16, M16 => true
   | M32, M32 => true
   | M64, M64 => true
@@ -218,14 +218,17 @@ Qed.
 
 (**
   Check if machine precision m1 is more precise than machine precision m2.
-  M0 is real-valued evaluation, hence the most precise.
+  REAL is real-valued evaluation, hence the most precise.
   All floating-point types are compared by
     mTypeToQ m1 <= mTypeToQ m2
-  For the moment, fixed-point and floating-point formats are incomparable
+  All fixed-point types are compared by comparing only the word size
+  We consider a 32 bit fixed-point number to be more precise than a 16 bit
+  fixed-point number, no matter what the fractional bits may be.
+  For the moment, fixed-point and floating-point formats are incomparable.
  **)
 Definition isMorePrecise (m1:mType) (m2:mType) :=
   match m1, m2 with
-  |M0, _ => true
+  |REAL, _ => true
   | F w1 f1, F w2 f2 => (w1 <=? w2)%positive (*&& (f1 <=? f2)%positive *)
   | F w f, _ => false
   | _ , F w f => false
@@ -236,14 +239,14 @@ Definition isMorePrecise (m1:mType) (m2:mType) :=
    for fixed-points *)
 Definition morePrecise (m1:mType) (m2:mType) :=
   match m1, m2 with
-  | M0, _ => true
+  | REAL, _ => true
   | F w1 f1, F w2 f2 => (w1 <=? w2)%positive (*&& (f1 <=? f2)%positive*)
   | _ , F w f => false
   | F w f, _ => false
   | M16, M16 => true
   | M32, M32 => true
   | M32, M16 => true
-  | M64, M0 => false
+  | M64, REAL => false
   | M64, _ => true
   | _, _ => false
   end.
@@ -291,16 +294,16 @@ Proof.
   apply Pos.leb_le; apply Pos.le_refl.
 Qed.
 
-Lemma M0_least_precision (m:mType) :
-  isMorePrecise m M0 = true -> m = M0.
+Lemma REAL_least_precision (m:mType) :
+  isMorePrecise m REAL = true -> m = REAL.
 Proof.
   intro m_morePrecise.
   unfold isMorePrecise in *.
   destruct m; cbv in m_morePrecise; congruence.
 Qed.
 
-Lemma M0_lower_bound (m:mType) :
-  isMorePrecise M0 m = true.
+Lemma REAL_lower_bound (m:mType) :
+  isMorePrecise REAL m = true.
 Proof.
   destruct m; unfold isMorePrecise; cbv; auto.
 Qed.
@@ -316,8 +319,8 @@ Definition join (m1:mType) (m2:mType) :=
 Definition join3 (m1:mType) (m2:mType) (m3:mType) :=
   join m1 (join m2 m3).
 
-(* Lemma M0_join_is_M0 m1 m2: *)
-(*   join m1 m2 = M0 -> m1 = M0 /\ m2 = M0. *)
+(* Lemma REAL_join_is_REAL m1 m2: *)
+(*   join m1 m2 = REAL -> m1 = REAL /\ m2 = REAL. *)
 (* Proof. *)
 (*   unfold join. *)
 (*   intros. *)
@@ -326,7 +329,7 @@ Definition join3 (m1:mType) (m2:mType) (m3:mType) :=