Add some small docstrings

coq/CertificateChecker.v

+(**
+Full Certtificate Checker.
+This function must be run on the daisy output.
+**)
 Require Import Coq.Reals.Reals Coq.QArith.Qreals.
 Require Import Daisy.Infra.RealSimps Daisy.Infra.RealRationalProps.
 Require Import Daisy.IntervalValidation Daisy.ErrorValidation Daisy.PreconditionValidation.
@@ -8,6 +12,9 @@ Require Export Daisy.Infra.ExpressionAbbrevs.
 Definition CertificateChecker (e:exp Q) (absenv:analysisResult) (P:precond) :=
   andb (andb (validIntervalbounds e absenv P) (approximatesPrecond e absenv P)) (validErrorbound e absenv).
 
+(**
+Soundness proof.
+**)
 Theorem Certificate_checking_is_sound (e:exp Q) (absenv:analysisResult) P:
   forall (cenv:env_ty) (vR:R) (vF:R),
     precondValidForExec P cenv ->

coq/ErrorBounds.v

-Require Import Coq.Reals.Reals Coq.micromega.Psatz Interval.Interval_tactic.
-Require Import Daisy.Infra.Abbrevs Daisy.Infra.RealSimps Daisy.Expressions.
 (**
-  TODO: Check wether we need Rabs (n * machineEpsilon) instead
+Proofs of general bounds on the error of arithmetic expressions.
+This shortens soundness proofs later.
 **)
+Require Import Coq.Reals.Reals Coq.micromega.Psatz Interval.Interval_tactic.
+Require Import Daisy.Infra.Abbrevs Daisy.Infra.RealSimps Daisy.Expressions.
+
 Lemma const_abs_err_bounded (n:R) (nR:R) (nF:R):
   forall cenv:nat -> R,
   eval_exp 0%R cenv (Const n) nR ->
@@ -19,10 +21,6 @@ Proof.
   apply Rmult_le_compat_l; [apply Rabs_pos | auto].
 Qed.
 
-(**
-  TODO: Maybe improve bound by adding interval constraint and proving that it is leq than maxAbs of bounds
-  (nlo <= cenv n <= nhi)%R ->   (Rabs (nR - nF) <= Rabs ((Rmax (Rabs nlo) (Rabs nhi)) * machineEpsilon))%R.
-**)
 Lemma param_abs_err_bounded (n:nat) (nR:R) (nF:R) (cenv:nat->R):
   eval_exp 0%R cenv (Param R n) nR ->
   eval_exp machineEpsilon cenv (Param R n) nF ->

coq/ErrorValidation.v

+(**
+Error bound checker and its soundness proof
+**)
 Require Import Coq.QArith.QArith Coq.QArith.Qminmax Coq.QArith.Qabs Coq.QArith.Qreals Coq.Lists.List.
-(*Coq.QArith.Qcanon.*)
 Require Import Coq.micromega.Psatz Coq.Reals.Reals Interval.Interval_tactic.
 Require Import Daisy.Infra.Abbrevs Daisy.Infra.RationalSimps Daisy.Infra.RealRationalProps Daisy.Infra.RealSimps.
 Require Import Daisy.Infra.ExpressionAbbrevs Daisy.IntervalArith Daisy.IntervalArithQ.
@@ -30,10 +32,8 @@ Fixpoint validErrorbound (e:exp Q) (env:analysisResult) :=
        end
    in andb (andb rec errPos) theVal
   end.
-(*
-  Functional Scheme validErrorbound_ind := Induction for validErrorbound Sort Prop.
- *)
 
+(** Since errors are intervals with 0 as center, we track them as single value since this eases proving upper bounds **)
 Lemma err_always_positive e (absenv:analysisResult) iv err:
   validErrorbound e absenv = true ->
   (iv,err) = absenv e ->

coq/Infra/RealRationalProps.v

+(**
+Some rewriting properties for translating between rationals and real numbers.
+These are used in the soundness proofs since we want to have the final theorem on the real valued evaluation.
+**)
 Require Import Coq.QArith.QArith Coq.QArith.Qminmax Coq.QArith.Qabs Coq.QArith.Qreals.
 Require Import Coq.Reals.Reals Coq.micromega.Psatz.
 Require Import Flocq.Core.Fcore_Raux.
@@ -167,17 +171,4 @@ Proof.
   unfold machineEpsilon.
   apply Qle_bool_iff.
   unfold Qle_bool; auto.
-Qed.
-
-(*
-Lemma errorIsPos:
-  forall e:error, (0 <= Q2R e)%R.
-Proof.
-  intros e.
-  unfold error in e.
-  pose proof (QposGeq0 e).
-  apply Qle_bool_iff in H.
-  apply Qle_Rle in H.
-  rewrite Q2R0_is_0 in H; auto.
-Qed.
-*)
\ No newline at end of file
+Qed.
\ No newline at end of file

coq/IntervalArith.v

@@ -179,18 +179,6 @@ Proof.
   apply interval_negation_valid; auto.
 Qed.
 
-(*
-  FIXME: This needs to be beautified
-*)
-Definition interval_mult_err (I1:interval) (e1:err) (I2:interval) (e2:err) :=
-  addInterval
-    (addInterval
-       (addInterval
-          (absIntvUpd Rmult I1 I2)
-          (absIntvUpd Rmult I1 (e1,e1)))
-       (absIntvUpd Rmult I2 (e2,e2)))
-    (Rmult e1 e2, Rmult e1 e2).
-
 Definition multInterval (iv1:interval) (iv2:interval) :=
   absIntvUpd Rmult iv1 iv2.
 
@@ -275,19 +263,7 @@ Qed.
 Definition divideInterval (I1:interval) (I2:interval) :=
   multInterval I1 (mkInterval (/ (IVhi I2)) (/ (IVlo I2))).
 
-(*
-Corollary interval_divisision_valid (I1:interval) (I2:interval) :
-  validIntervalDiv I1 I2 (divideInterval I1 I2).
-Proof.
-  unfold validIntervalDiv.
-  intros a b contained_a contained_b; unfold divideInterval.
-  apply interval_multiplication_valid; auto.
-  unfold contained in *; simpl.
-  split. *)
-
-Definition intv_div_err (I1:interval) (e1:err) (I2:interval) (e2:err) :=
-  interval_mult_err I1 e1 ( (/ IVlo (I2))%R, (/ IVhi (I2))%R) e2.
-
+(** Define the maxAbs function on R and prove some minor properties of it. **)
 Definition RmaxAbsFun (iv:interval) :=
   Rmax (Rabs (fst iv)) (Rabs (snd iv)).
 

coq/IntervalArithQ.v

@@ -193,18 +193,6 @@ Proof.
   apply intv_negation_valid; auto.
 Qed.
 
-(*
-  FIXME: This needs to be beautified
-*)
-Definition intv_mult_err (I1:intv) (e1:error) (I2:intv) (e2:error) :=
-  addIntv
-    (addIntv
-       (addIntv
-          (absIvUpd Qmult I1 I2)
-          (absIvUpd Qmult I1 (e1,e1)))
-       (absIvUpd Qmult I2 (e2,e2)))
-    (Qmult e1 e2, Qmult e1 e2).
-
 Definition multIntv (iv1:intv) (iv2:intv) :=
   absIvUpd Qmult iv1 iv2.
 
@@ -298,17 +286,4 @@ Proof.
 Qed.
 
 Definition divideIntv (I1:intv) (I2:intv) :=
-  multIntv I1 (mkIntv (/ (ivhi I2)) (/ (ivlo I2))).
-
-(*
-Corollary intv_divisision_valid (I1:intv) (I2:intv) :
-  validIntvDiv I1 I2 (divideIntv I1 I2).
-Proof.
-  unfold validIntvDiv.
-  intros a b contained_a contained_b; unfold divideIntv.
-  apply intv_multiplication_valid; auto.
-  unfold contained in *; simpl.
-  split. *)
-
-Definition iv_div_err (I1:intv) (e1:error) (I2:intv) (e2:error) :=
-  intv_mult_err I1 e1 ( (/ ivlo (I2))%Q, (/ ivhi (I2))%Q) e2.
\ No newline at end of file
+  multIntv I1 (mkIntv (/ (ivhi I2)) (/ (ivlo I2))).
\ No newline at end of file

coq/IntervalValidation.v

+(**
+Interval arithmetic checker and its soundness proof
+**)
 Require Import Coq.QArith.QArith Coq.QArith.Qreals QArith.Qminmax Coq.Lists.List.
 Require Import Daisy.Infra.Abbrevs Daisy.Infra.RationalSimps Daisy.Infra.RealRationalProps.
 Require Import Daisy.Infra.ExpressionAbbrevs Daisy.IntervalArithQ Daisy.IntervalArith Daisy.Infra.RealSimps Daisy.PreconditionValidation.
@@ -6,8 +9,6 @@ Fixpoint validIntervalbounds (e:exp Q) (absenv:analysisResult) (P:precond):=
   let (intv, _) := absenv e in
     match e with
     | Var v => false
-                (* let bounds_v := P v in
-      andb (Qeq_bool (fst intv) (fst (bounds_v))) (Qeq_bool (snd bounds_v) (snd (intv))) *)
     | Param v =>
       isSupersetIntv (P v) intv
     | Const n =>

coq/PreconditionValidation.v

+(**
+Precondition agreement checker and its soundness proof
+**)
 Require Import Coq.Reals.Reals Coq.Lists.List Coq.QArith.QArith.
 Require Import Daisy.Infra.Abbrevs Daisy.Expressions Daisy.Infra.RationalSimps Daisy.Infra.ExpressionAbbrevs Daisy.IntervalArithQ.