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Commit cad900bc authored by Heiko Becker's avatar Heiko Becker
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Merge branch 'errors_affine' into 'master' 

Rework type for delta map and fix issue #13

Closes #13

See merge request AVA/FloVer!18
parents 55c016cd f36a8d5d
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coq/CertificateChecker.v
View file @ cad900bc
......@@ -30,8 +30,8 @@ Definition CertificateChecker (e:expr Q) (absenv:analysisResult)
**)
Theorem Certificate_checking_is_sound (e:expr Q) (absenv:analysisResult) P defVars:
forall (E1 E2:env) DeltaMap,
(forall (e' : expr R) (m' : mType),
exists d : R, DeltaMap e' m' = Some d /\ (Rabs d <= mTypeToR m')%R) ->
(forall (v : R) (m' : mType),
exists d : R, DeltaMap v m' = Some d /\ (Rabs d <= mTypeToR m')%R) ->
(forall v, NatSet.In v (Expressions.usedVars e) ->
exists vR, E1 v = Some vR /\
(Q2R (fst (P v)) <= vR <= Q2R (snd (P v)))%R) ->
......@@ -81,10 +81,10 @@ Proof.
exists Gamma; intros approxE1E2.
assert (dVars_contained NatSet.empty (FloverMap.empty (affine_form Q))) as Hdvars
by (unfold dVars_contained; intros * Hset; clear - Hset; set_tac).
pose proof (RoundoffErrorValidator_sound e _ deltas_matched H approxE1E2 H1 eval_real R
pose proof (RoundoffErrorValidator_sound e H approxE1E2 H1 eval_real R
valid_e map_e Hdvars) as Hsound.
unfold validErrorBounds in Hsound.
eapply validErrorBounds_single in Hsound; eauto.
eapply validErrorBoundsRec_single in Hsound; eauto.
destruct Hsound as [[vF [mF eval_float]] err_bounded]; auto.
exists (elo, ehi), err_e, vR, vF, mF; repeat split; auto.
Qed.
......@@ -106,8 +106,8 @@ Definition CertificateCheckerCmd (f:cmd Q) (absenv:analysisResult) (P:precond)
Theorem Certificate_checking_cmds_is_sound (f:cmd Q) (absenv:analysisResult) P
defVars DeltaMap:
forall (E1 E2:env),
(forall (e' : expr R) (m' : mType),
exists d : R, DeltaMap e' m' = Some d /\ (Rabs d <= mTypeToR m')%R) ->
(forall (v : R) (m' : mType),
exists d : R, DeltaMap v m' = Some d /\ (Rabs d <= mTypeToR m')%R) ->
(forall v, NatSet.In v (freeVars f) ->
exists vR, E1 v = Some vR /\
(Q2R (fst (P v)) <= vR <= Q2R (snd (P v)))%R) ->
......@@ -158,7 +158,7 @@ Proof.
destruct valid_single as [iv [ err [vR [map_f [eval_real bounded_real_f]]]]].
destruct iv as [f_lo f_hi].
pose proof RoundoffErrorValidatorCmd_sound as Hsound.
eapply validErrorBoundsCmd_single in Hsound; eauto.
eapply validErrorBoundsCmdRec_single in Hsound; eauto.
2: unfold dVars_contained; intros; set_tac.
specialize Hsound as ((vF & mF & eval_float) & ?).
exists (f_lo, f_hi), err, vR, vF, mF; repeat split; try auto.
......
coq/Commands.v
View file @ cad900bc
......@@ -48,7 +48,7 @@ Inductive sstep : cmd R -> env -> R -> cmd R -> env -> Prop :=
Define big step semantics for the Flover language, terminating on a "returned"
result value
**)
Inductive bstep : cmd R -> env -> (expr R -> option mType) -> (expr R -> mType -> option R) ->
Inductive bstep : cmd R -> env -> (expr R -> option mType) -> (R -> mType -> option R) ->
R -> mType -> Prop :=
let_b m m' x e s E v res defVars DeltaMap:
eval_expr E defVars DeltaMap e v m ->
......
coq/ErrorAnalysis.v
View file @ cad900bc
......@@ -4,17 +4,17 @@ From Coq
From Flover
Require Import Commands ExpressionSemantics Environments RealRangeArith TypeValidator.
Fixpoint validErrorBounds (e:expr Q) E1 E2 A Gamma DeltaMap :Prop :=
Fixpoint validErrorBoundsRec (e:expr Q) E1 E2 A Gamma DeltaMap :Prop :=
(match e with
| Unop u e => validErrorBounds e E1 E2 A Gamma DeltaMap
| Downcast m e => validErrorBounds e E1 E2 A Gamma DeltaMap
| Unop u e => validErrorBoundsRec e E1 E2 A Gamma DeltaMap
| Downcast m e => validErrorBoundsRec e E1 E2 A Gamma DeltaMap
| Binop b e1 e2 =>
validErrorBounds e1 E1 E2 A Gamma DeltaMap /\
validErrorBounds e2 E1 E2 A Gamma DeltaMap
validErrorBoundsRec e1 E1 E2 A Gamma DeltaMap /\
validErrorBoundsRec e2 E1 E2 A Gamma DeltaMap
| Fma e1 e2 e3 =>
validErrorBounds e1 E1 E2 A Gamma DeltaMap /\
validErrorBounds e2 E1 E2 A Gamma DeltaMap /\
validErrorBounds e3 E1 E2 A Gamma DeltaMap
validErrorBoundsRec e1 E1 E2 A Gamma DeltaMap /\
validErrorBoundsRec e2 E1 E2 A Gamma DeltaMap /\
validErrorBoundsRec e3 E1 E2 A Gamma DeltaMap
| _ => True
end) /\
forall v__R (iv: intv) (err: error),
......@@ -26,8 +26,14 @@ Fixpoint validErrorBounds (e:expr Q) E1 E2 A Gamma DeltaMap :Prop :=
eval_expr E2 (toRExpMap Gamma) DeltaMap (toRExp e) v__FP m__FP ->
(Rabs (v__R - v__FP) <= (Q2R err))%R).
Lemma validErrorBounds_single e E1 E2 A Gamma DeltaMap:
validErrorBounds e E1 E2 A Gamma DeltaMap ->
Definition validErrorBounds e E1 E2 A Gamma: Prop :=
forall DeltaMap,
(forall (v : R) (m' : mType),
exists d : R, DeltaMap v m' = Some d /\ (Rabs d <= mTypeToR m')%R) ->
validErrorBoundsRec e E1 E2 A Gamma DeltaMap.
Lemma validErrorBoundsRec_single e E1 E2 A Gamma DeltaMap:
validErrorBoundsRec e E1 E2 A Gamma DeltaMap ->
forall v__R iv err,
eval_expr E1 (toRTMap (toRExpMap Gamma)) DeltaMapR (toREval (toRExp e)) v__R REAL ->
FloverMap.find e A = Some (iv, err) ->
......@@ -39,13 +45,13 @@ Lemma validErrorBounds_single e E1 E2 A Gamma DeltaMap:
Proof.
intros validError_e;
intros; destruct e; cbn in *; split;
destruct validError_e as (? & ? & ?); eauto.
edestruct validError_e as (? & ? & ?); eauto.
Qed.
Fixpoint validErrorBoundsCmd (c: cmd Q) E1 E2 A Gamma DeltaMap: Prop :=
Fixpoint validErrorBoundsCmdRec (c: cmd Q) E1 E2 A Gamma DeltaMap: Prop :=
match c with
| Let m x e k =>
validErrorBounds e E1 E2 A Gamma DeltaMap /\
validErrorBoundsRec e E1 E2 A Gamma DeltaMap /\
(exists iv_e err_e iv_x err_x,
FloverMap.find e A = Some (iv_e, err_e) /\
FloverMap.find (Var Q x) A = Some (iv_x, err_x) /\
......@@ -53,8 +59,8 @@ Fixpoint validErrorBoundsCmd (c: cmd Q) E1 E2 A Gamma DeltaMap: Prop :=
(forall v__R v__FP,
eval_expr E1 (toRTMap (toRExpMap Gamma)) DeltaMapR (toREval (toRExp e)) v__R REAL ->
eval_expr E2 (toRExpMap Gamma) DeltaMap (toRExp e) v__FP m ->
validErrorBoundsCmd k (updEnv x v__R E1) (updEnv x v__FP E2) A Gamma DeltaMap)
| Ret e => validErrorBounds e E1 E2 A Gamma DeltaMap
validErrorBoundsCmdRec k (updEnv x v__R E1) (updEnv x v__FP E2) A Gamma DeltaMap)
| Ret e => validErrorBoundsRec e E1 E2 A Gamma DeltaMap
end /\
forall v__R (iv: intv) (err: error),
bstep (toREvalCmd (toRCmd c)) E1 (toRTMap (toRExpMap Gamma)) DeltaMapR v__R REAL ->
......@@ -65,8 +71,14 @@ Fixpoint validErrorBoundsCmd (c: cmd Q) E1 E2 A Gamma DeltaMap: Prop :=
bstep (toRCmd c) E2 (toRExpMap Gamma) DeltaMap v__FP m__FP ->
(Rabs (v__R - v__FP) <= (Q2R err))%R).
Lemma validErrorBoundsCmd_single c E1 E2 A Gamma DeltaMap:
validErrorBoundsCmd c E1 E2 A Gamma DeltaMap ->
Definition validErrorBoundsCmd (c: cmd Q) E1 E2 A Gamma: Prop :=
forall DeltaMap,
(forall (v : R) (m' : mType),
exists d : R, DeltaMap v m' = Some d /\ (Rabs d <= mTypeToR m')%R) ->
validErrorBoundsCmdRec c E1 E2 A Gamma DeltaMap.
Lemma validErrorBoundsCmdRec_single c E1 E2 A Gamma DeltaMap:
validErrorBoundsCmdRec c E1 E2 A Gamma DeltaMap ->
forall v__R (iv: intv) (err: error),
bstep (toREvalCmd (toRCmd c)) E1 (toRTMap (toRExpMap Gamma)) DeltaMapR v__R REAL ->
FloverMap.find (getRetExp c) A = Some (iv, err) ->
......@@ -78,5 +90,5 @@ Lemma validErrorBoundsCmd_single c E1 E2 A Gamma DeltaMap:
Proof.
intros validError_e;
intros; destruct c; cbn in *; split;
destruct validError_e as (? & ? & ?); eauto.
edestruct validError_e as (? & ? & ?); eauto.
Qed.
coq/ErrorBounds.v
View file @ cad900bc
......@@ -43,8 +43,8 @@ Lemma add_abs_err_bounded (e1:expr Q) (e1R:R) (e1F:R) (e2:expr Q) (e2R:R) (e2F:R
eval_expr (updEnv 2 e2F (updEnv 1 e1F emptyEnv))
(updDefVars (Binop Plus (Var R 1) (Var R 2)) m
(updDefVars (Var R 2) m2 (updDefVars (Var R 1) m1 defVars)))
(fun x _ => if R_orderedExps.eq_dec x (Binop Plus (Var R 1) (Var R 2))
then DeltaMap (Binop Plus (toRExp e1) (toRExp e2)) m
(fun x _ => if Req_dec_sum x (evalBinop Plus e1F e2F)
then DeltaMap (evalBinop Plus e1F e2F) m
else None)
(Binop Plus (Var R 1) (Var R 2)) vF m ->
(Rabs (e1R - e1F) <= Q2R err1)%R ->
......@@ -116,8 +116,8 @@ Lemma subtract_abs_err_bounded (e1:expr Q) (e1R:R) (e1F:R) (e2:expr Q) (e2R:R)
eval_expr (updEnv 2 e2F (updEnv 1 e1F emptyEnv))
(updDefVars (Binop Sub (Var R 1) (Var R 2)) m
(updDefVars (Var R 2) m2 (updDefVars (Var R 1) m1 defVars)))
(fun x _ => if R_orderedExps.eq_dec x (Binop Sub (Var R 1) (Var R 2))
then DeltaMap (Binop Sub (toRExp e1) (toRExp e2)) m
(fun x _ => if Req_dec_sum x (evalBinop Sub e1F e2F)
then DeltaMap (evalBinop Sub e1F e2F) m
else None)
(Binop Sub (Var R 1) (Var R 2)) vF m ->
(Rabs (e1R - e1F) <= Q2R err1)%R ->
......@@ -194,8 +194,8 @@ Lemma mult_abs_err_bounded (e1:expr Q) (e1R:R) (e1F:R) (e2:expr Q) (e2R:R) (e2F:
eval_expr (updEnv 2 e2F (updEnv 1 e1F emptyEnv))
(updDefVars (Binop Mult (Var R 1) (Var R 2)) m
(updDefVars (Var R 2) m2 (updDefVars (Var R 1) m1 defVars)))
(fun x _ => if R_orderedExps.eq_dec x (Binop Mult (Var R 1) (Var R 2))
then DeltaMap (Binop Mult (toRExp e1) (toRExp e2)) m
(fun x _ => if Req_dec_sum x (evalBinop Mult e1F e2F)
then DeltaMap (evalBinop Mult e1F e2F) m
else None)
(Binop Mult (Var R 1) (Var R 2)) vF m ->
(Rabs (vR - vF) <= Rabs (e1R * e2R - e1F * e2F) + computeErrorR (e1F * e2F) m)%R.
......@@ -245,8 +245,8 @@ Lemma div_abs_err_bounded (e1:expr Q) (e1R:R) (e1F:R) (e2:expr Q) (e2R:R) (e2F:R
eval_expr (updEnv 2 e2F (updEnv 1 e1F emptyEnv))
(updDefVars (Binop Div (Var R 1) (Var R 2)) m
(updDefVars (Var R 2) m2 (updDefVars (Var R 1) m1 defVars)))
(fun x _ => if R_orderedExps.eq_dec x (Binop Div (Var R 1) (Var R 2))
then DeltaMap (Binop Div (toRExp e1) (toRExp e2)) m
(fun x _ => if Req_dec_sum x (evalBinop Div e1F e2F)
then DeltaMap (evalBinop Div e1F e2F) m
else None)
(Binop Div (Var R 1) (Var R 2)) vF m ->
(Rabs (vR - vF) <= Rabs (e1R / e2R - e1F / e2F) + computeErrorR (e1F / e2F) m)%R.
......@@ -301,8 +301,8 @@ Lemma fma_abs_err_bounded (e1:expr Q) (e1R:R) (e1F:R) (e2:expr Q) (e2R:R) (e2F:R
(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 defVars))))
(fun x _ => if R_orderedExps.eq_dec x (Fma (Var R 1) (Var R 2) (Var R 3))
then DeltaMap (Fma (toRExp e1) (toRExp e2) (toRExp e3)) m
(fun x _ => if Req_dec_sum x (evalFma e1F e2F e3F)
then DeltaMap (evalFma e1F e2F e3F) m
else None)
(Fma (Var R 1) (Var R 2) (Var R 3)) vF m ->
(Rabs (vR - vF) <= Rabs ((e1R - e1F) + (e2R * e3R - e2F * e3F)) + computeErrorR (e1F + e2F * e3F ) m)%R.
......@@ -366,8 +366,8 @@ Lemma round_abs_err_bounded (e:expr R) (nR nF1 nF:R) (E1 E2: env) (err:R)
eval_expr (updEnv 1 nF1 emptyEnv)
(updDefVars (Downcast mEps (Var R 1)) mEps
(updDefVars (Var R 1) m defVars))
(fun x _ => if R_orderedExps.eq_dec x (Downcast mEps (Var R 1))
then DeltaMap (Downcast mEps e) mEps
(fun x _ => if Req_dec_sum x nF1
then DeltaMap nF1 mEps
else None)
(toRExp (Downcast mEps (Var Q 1))) nF mEps->
(Rabs (nR - nF1) <= err)%R ->
......
coq/ErrorValidation.v
View file @ cad900bc
This diff is collapsed.
coq/ErrorValidationAA.v
View file @ cad900bc
......@@ -236,6 +236,11 @@ Fixpoint validErrorboundAACmd (f: cmd Q) (* analyzed cmd with let's *)
| Ret e => validErrorboundAA e typeMap A dVars currNoise errMap
end.
(* Notation for the universal case of the soundness statement, to help reason
about memoization cases. This allows us to show several monotonicity lemmas
that simplify the soundness proofs. This definition is just an extract from
the full soundness statement, for elaboration on the used assumptions and the
goal, find that statement below *)
Definition checked_error_expressions (e: expr Q) E1 E2 A Gamma DeltaMap
noise_map noise expr_map :=
forall v__R v__FP m__FP (iv__A: intv) (err__A: error),
......@@ -777,12 +782,17 @@ Proof.
* apply flover_map_in_add.
Qed.
(* The soundness statements starts off with assumption that the checking
function succeeds and then also maintains several invariants. We require
these invariants because of the checker function dependency on the current
noise index and its memoization of the previosly computed polynomials. We
explain these invariants in-line below. *)
Definition validErrorboundAA_sound_statement e E1 E2 A Gamma DeltaMap fVars dVars :=
forall (expr_map1 expr_map2 : FloverMap.t (affine_form Q))
(noise_map1 : nat -> option noise_type) (noise1 noise2 : nat) (v__R : R)
(iv__A : intv) (err__A : error),
(forall (e' : expr R) (m' : mType),
exists d : R, DeltaMap e' m' = Some d /\ (Rabs d <= mTypeToR m')%R) ->
(forall (v : R) (m' : mType),
exists d : R, DeltaMap v m' = Some d /\ (Rabs d <= mTypeToR m')%R) ->
approxEnv E1 (toRExpMap Gamma) A fVars dVars E2 ->
eval_expr E1 (toRTMap (toRExpMap Gamma)) DeltaMapR (toREval (toRExp e)) v__R REAL ->
validRanges e A E1 (toRTMap (toRExpMap Gamma)) ->
......@@ -790,39 +800,74 @@ Definition validErrorboundAA_sound_statement e E1 E2 A Gamma DeltaMap fVars dVar
NatSet.Subset (usedVars e -- dVars) fVars ->
validTypes e Gamma ->
FloverMap.find e A = Some (iv__A, err__A) ->
(* Starting noise index is greater than 0 and the noise mapping doesn't
map it and anything above to a noise value *)
(0 < noise1)%nat ->
(forall n0 : nat, (n0 >= noise1)%nat -> noise_map1 n0 = None) ->
(* If a var is a dVar, we know it's in expr_map and invoke the corresponding
preconditions (described below) *)
dVars_contained dVars expr_map1 ->
(* Precondition:
Memoization for the existential case of the goal: if we checked the expression
already, we know that there is an execution and certificate results for it *)
(forall e' : FloverMap.key,
FloverMap.In e' expr_map1 ->
(* Assumption needed for Cmd_sound proof *)
NatSet.Subset (usedVars e') (NatSet.union fVars dVars) /\
(exists iv__A' err__A', FloverMap.find e' A = Some (iv__A', err__A')) /\
exists (v__FP' : R) (m__FP' : mType),
eval_expr E2 (toRExpMap Gamma) DeltaMap (toRExp e') v__FP' m__FP') ->
(* Precondition:
Memoization for the universal case of the goal -- note that
`checked_error_expressions` track only that case: if we checked
the expression already, then, if we provide the execution in
finite precision and the certificate results, we will know that
the execution's error is bounded *)
(forall e' : FloverMap.key,
FloverMap.In e' expr_map1 ->
checked_error_expressions e' E1 E2 A Gamma DeltaMap noise_map1 noise1 expr_map1) ->
(* THE EXESTENTIAL CASE OF THE GOAL *)
((exists (v__FP : R) (m__FP : mType),
eval_expr E2 (toRExpMap Gamma) DeltaMap (toRExp e) v__FP m__FP) /\
(* Invariant of the existential case:
our given expr_map1 is updated only with the expressions for which
the existential part holds *)
(forall e' : FloverMap.key,
~ FloverMap.In e' expr_map1 -> FloverMap.In e' expr_map2 ->
NatSet.Subset (usedVars e') (NatSet.union fVars dVars) /\
(exists iv__A' err__A', FloverMap.find e' A = Some (iv__A', err__A')) /\
exists (v__FP' : R) (m__FP' : mType),
eval_expr E2 (toRExpMap Gamma) DeltaMap (toRExp e') v__FP' m__FP')) /\
(* UNIVERSAL CASE OF THE GOAL *)
(forall (v__FP : R) (m__FP : mType),
eval_expr E2 (toRExpMap Gamma) DeltaMap (toRExp e) v__FP m__FP ->
(* For the given evaluation we can find a polynomial and an
error bound with a monotonic extension of noise_map1 to
noise_map2 *)
exists (af : affine_form Q) (err__af : R) (noise_map2 : noise_mapping),
(* Invariant: the checker function only adds new polynomials
to the expr_map, never removes something from it *)
contained_flover_map expr_map1 expr_map2 /\
(* Noise index invariants: the checker function returns new
noise that is not used in already produced polynomials *)
(noise1 <= noise2)%nat /\
fresh noise2 (afQ2R af) /\
(* Invariant: noise_map is incrementally updated *)
contained_map noise_map1 noise_map2 /\
(forall n0 : nat, (n0 >= noise2)%nat -> noise_map2 n0 = None) /\
FloverMap.find (elt:=affine_form Q) e expr_map2 = Some af /\
(* To show that the error is bounded by the polynomial, we say that
the polynomial's range is bounded by err__af, which has a value
less or equal to the certificate error, and that the polynomial
captures any difference between real and finite-precision
evaluations *)
(0 <= err__af)%R /\
toInterval (afQ2R af) = ((- err__af)%R, err__af) /\
(err__af <= Q2R err__A)%R /\
af_evals (afQ2R af) (v__R - v__FP) noise_map2 /\
(* Invariant of the universal case:
our given expr_map1 is updated only with the expressions for which
the universal part (the conclusion directly above) holds *)
(forall e' : FloverMap.key,
~ FloverMap.In e' expr_map1 -> FloverMap.In e' expr_map2 ->
checked_error_expressions e' E1 E2 A Gamma DeltaMap noise_map2 noise2 expr_map2)).
......@@ -1008,7 +1053,7 @@ Proof.
eval_expr E2 (toRExpMap Gamma) DeltaMap
(toRExp (Expressions.Const m v)) v__FP m__FP).
{
specialize (Hdeltamap (toRExp (Expressions.Const m v)) m)
specialize (Hdeltamap (Q2R v) m)
as (delta' & Hdelta & Hdelta').
exists (perturb (Q2R v) m delta'), m.
eapply Const_dist' with (delta := delta'); auto.
......@@ -1351,7 +1396,7 @@ Proof.
assert (exists (v__FP : R) (m__FP : mType),
eval_expr E2 (toRExpMap Gamma) DeltaMap (toRExp (Binop Plus e1 e2)) v__FP m__FP) as H.
{
specialize (Hdeltamap (toRExp (Binop Plus e1 e2)) m__e)
specialize (Hdeltamap (evalBinop Plus nF1 nF2) m__e)
as (delta' & Hdelta & Hdelta').
eexists; exists m__e.
eapply Binop_dist' with (delta := delta'); eauto.
......@@ -1615,7 +1660,7 @@ Proof.
assert (exists (v__FP : R) (m__FP : mType),
eval_expr E2 (toRExpMap Gamma) DeltaMap (toRExp (Binop Sub e1 e2)) v__FP m__FP).
{
specialize (Hdeltamap (toRExp (Binop Sub e1 e2)) m__e)
specialize (Hdeltamap (evalBinop Sub nF1 nF2) m__e)
as (delta' & Hdelta & Hdelta').
eexists; exists m__e.
eapply Binop_dist' with (delta := delta'); eauto.
......@@ -1867,7 +1912,7 @@ Proof.
assert (exists (v__FP : R) (m__FP : mType),
eval_expr E2 (toRExpMap Gamma) DeltaMap (toRExp (Binop Mult e1 e2)) v__FP m__FP).
{
specialize (Hdeltamap (toRExp (Binop Mult e1 e2)) m__e)
specialize (Hdeltamap (evalBinop Mult nF1 nF2) m__e)
as (delta' & Hdelta & Hdelta').
eexists; exists m__e.
eapply Binop_dist' with (delta := delta'); eauto.
......@@ -2180,7 +2225,7 @@ Proof.
assert (exists (v__FP : R) (m__FP : mType),
eval_expr E2 (toRExpMap Gamma) DeltaMap (toRExp (Binop Div e1 e2)) v__FP m__FP).
{
specialize (Hdeltamap (toRExp (Binop Div e1 e2)) m__e)
specialize (Hdeltamap (evalBinop Div nF1 nF2) m__e)
as (delta' & Hdelta & Hdelta').
eexists; exists m__e.
eapply Binop_dist' with (delta := delta'); eauto.
......@@ -2578,7 +2623,7 @@ Proof.
assert (exists (v__FP : R) (m__FP : mType),
eval_expr E2 (toRExpMap Gamma) DeltaMap (toRExp (Fma e1 e2 e3)) v__FP m__FP).
{
specialize (Hdeltamap (toRExp (Fma e1 e2 e3)) m__e)
specialize (Hdeltamap (evalFma nF1 nF2 nF3) m__e)
as (delta' & Hdelta & Hdelta').
eexists; exists m__e.
eapply Fma_dist' with (delta:= delta'); eauto.
......@@ -2879,7 +2924,7 @@ Proof.
assert (exists (v__FP : R) (m__FP : mType),
eval_expr E2 (toRExpMap Gamma) DeltaMap (toRExp (Downcast m__e e)) v__FP m__FP).
{
specialize (Hdeltamap (toRExp (Downcast m__e e)) m__e)
specialize (Hdeltamap nF m__e)
as (delta' & Hdelta & Hdelta').
exists (perturb nF m__e delta'), m__e.
eapply Downcast_dist' with (delta := delta'); eauto.
......@@ -3020,7 +3065,7 @@ Corollary validErrorbound_sound_validErrorBounds e E1 E2 A Gamma DeltaMap expr_m
eval_expr E2 (toRExpMap Gamma) DeltaMap (toRExp e') v__FP' m__FP') /\
checked_error_expressions e' E1 E2 A Gamma DeltaMap noise_map noise expr_map) ->
contained_subexpr e expr_map ->
validErrorBounds e E1 E2 A Gamma DeltaMap.
validErrorBoundsRec e E1 E2 A Gamma DeltaMap.
Proof.
induction e; intros Hchecked Hsubexpr.
- split; [trivial|].
......@@ -3058,6 +3103,7 @@ Proof.
- cbn.
specialize (IHe Hchecked).
specialize Hsubexpr as (Hsubexpr & Hin).
unfold validErrorBounds in IHe.
split; auto.
intros * Heval__R Hcert.
specialize Hchecked as (Hex & Hall); eauto.
......@@ -3077,6 +3123,7 @@ Proof.
specialize (IHe1 Hchecked).
specialize (IHe2 Hchecked).
specialize Hsubexpr as ((Hsubexpr1 & Hsubexpr2) & Hin).
unfold validErrorBounds in IHe1, IHe2.
split; auto.
intros * Heval__R Hcert.
specialize Hchecked as (Hex & Hall); eauto.
......@@ -3097,6 +3144,7 @@ Proof.
specialize (IHe2 Hchecked).
specialize (IHe3 Hchecked).
specialize Hsubexpr as ((Hsubexpr1 & Hsubexpr2 & Hsubexpr3) & Hin).
unfold validErrorBounds in IHe1, IHe2, IHe3.
split; auto.
intros * Heval__R Hcert.
specialize Hchecked as (Hex & Hall); eauto.
......@@ -3115,6 +3163,7 @@ Proof.
- cbn.
specialize (IHe Hchecked).
specialize Hsubexpr as (Hsubexpr & Hin).
unfold validErrorBounds in IHe.
split; auto.
intros * Heval__R Hcert.
specialize Hchecked as (Hex & Hall); eauto.
......@@ -3285,8 +3334,8 @@ Theorem validErrorboundAACmd_sound c:
forall E1 E2 A Gamma DeltaMap fVars dVars outVars
(expr_map1 expr_map2 : FloverMap.t (affine_form Q))
(noise_map1 : nat -> option noise_type) (noise1 noise2 : nat) (v__R : R),
(forall (e' : expr R) (m' : mType),
exists d : R, DeltaMap e' m' = Some d /\ (Rabs d <= mTypeToR m')%R) ->
(forall (v : R) (m' : mType),
exists d : R, DeltaMap v m' = Some d /\ (Rabs d <= mTypeToR m')%R) ->
approxEnv E1 (toRExpMap Gamma) A fVars dVars E2 ->
bstep (toREvalCmd (toRCmd c)) E1 (toRTMap (toRExpMap Gamma)) DeltaMapR v__R REAL ->
validRangesCmd c A E1 (toRTMap (toRExpMap Gamma)) ->
......@@ -3331,7 +3380,7 @@ Theorem validErrorboundAACmd_sound c:
toInterval (afQ2R af) = ((- err__af)%R, err__af) /\
(err__af <= Q2R err__A)%R /\
af_evals (afQ2R af) (v__R - v__FP) noise_map2 /\
validErrorBoundsCmd c E1 E2 A Gamma DeltaMap /\
validErrorBoundsCmdRec c E1 E2 A Gamma DeltaMap /\
(forall e' : FloverMap.key,
~ FloverMap.In (elt:=affine_form Q) e' expr_map1 ->
FloverMap.In (elt:=affine_form Q) e' expr_map2 ->
......@@ -3737,7 +3786,7 @@ Proof.
replace m__FP0 with m__FP in * by eauto using bstep_Gamma_det.
assert (v__FP0 = v__FP) by eauto using bstep_det.
assert (v__R0 = v__R) by eauto using bstep_det.
cbn in Hcert.
cbn in Hcert'.
assert (err = err__A) by congruence.
subst.
eapply Rle_trans; eauto.
......
coq/ExpressionSemantics.v
View file @ cad900bc
......@@ -34,16 +34,16 @@ Open Scope R_scope.
Inductive eval_expr (E:env)
(Gamma: expr R -> option mType)
(DeltaMap: expr R -> mType -> option R)
(DeltaMap: R -> mType -> option R)
:(expr R) -> R -> mType -> Prop :=
| Var_load m x v:
Gamma (Var R x) = Some m ->
E x = Some v ->
eval_expr E Gamma DeltaMap (Var R x) v m
| Const_dist m n delta:
DeltaMap (Const m n) m = Some delta ->
| Const_dist m c delta:
DeltaMap c m = Some delta ->
Rabs delta <= mTypeToR m ->
eval_expr E Gamma DeltaMap (Const m n) (perturb n m delta) m
eval_expr E Gamma DeltaMap (Const m c) (perturb c m delta) m
| Unop_neg m mN f1 v1:
Gamma (Unop Neg f1) = Some mN ->
isCompat m mN = true ->
......@@ -51,7 +51,7 @@ Inductive eval_expr (E:env)
eval_expr E Gamma DeltaMap (Unop Neg f1) (evalUnop Neg v1) mN
| Unop_inv m mN f1 v1 delta:
Gamma (Unop Inv f1) = Some mN ->
DeltaMap (Unop Inv f1) mN = Some delta ->
DeltaMap (evalUnop Inv v1) mN = Some delta ->
isCompat m mN = true ->
Rabs delta <= mTypeToR mN ->
eval_expr E Gamma DeltaMap f1 v1 m ->
......@@ -59,14 +59,14 @@ Inductive eval_expr (E:env)
eval_expr E Gamma DeltaMap (Unop Inv f1) (perturb (evalUnop Inv v1) mN delta) mN
| Downcast_dist m m1 f1 v1 delta:
Gamma (Downcast m f1) = Some m ->
DeltaMap (Downcast m f1) m = Some delta ->
DeltaMap v1 m = Some delta ->
isMorePrecise m1 m = true ->
Rabs delta <= mTypeToR m ->
eval_expr E Gamma DeltaMap f1 v1 m1 ->
eval_expr E Gamma DeltaMap (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 ->
DeltaMap (Binop op f1 f2) m = Some delta ->
DeltaMap (evalBinop op v1 v2) m = Some delta ->
isJoin m1 m2 m = true ->
Rabs delta <= mTypeToR m ->
eval_expr E Gamma DeltaMap f1 v1 m1 ->
......@@ -75,7 +75,7 @@ Inductive eval_expr (E:env)
eval_expr E Gamma DeltaMap (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 ->
DeltaMap (Fma f1 f2 f3) m = Some delta ->
DeltaMap (evalFma v1 v2 v3) m = Some delta ->
isJoin3 m1 m2 m3 m = true ->
Rabs delta <= mTypeToR m ->
eval_expr E Gamma DeltaMap f1 v1 m1 ->
......@@ -83,7 +83,7 @@ Inductive eval_expr (E:env)
eval_expr E Gamma DeltaMap f3 v3 m3 ->
eval_expr E Gamma DeltaMap (Fma f1 f2 f3) (perturb (evalFma v1 v2 v3) m delta) m.
Definition DeltaMapR: expr R -> mType -> option R := (fun x m => Some 0).
Definition DeltaMapR: R -> mType -> option R := (fun x m => Some 0).
Close Scope R_scope.
......@@ -92,12 +92,12 @@ Hint Constructors eval_expr.
(** *)
(* Show some simpler (more general) rule lemmata *)
(* **)
Lemma Const_dist' DeltaMap m n delta v m' E Gamma:
Lemma Const_dist' DeltaMap m c delta v m' E Gamma:
Rle (Rabs delta) (mTypeToR m') ->
DeltaMap (Const m n) m = Some delta ->
v = perturb n m delta ->
DeltaMap c m = Some delta ->
v = perturb c m delta ->
m' = m ->
eval_expr E Gamma DeltaMap (Const m n) v m'.
eval_expr E Gamma DeltaMap (Const m c) v m'.
Proof.
intros; subst; auto.
Qed.
......@@ -120,7 +120,7 @@ Hint Resolve Unop_neg'.
Lemma Unop_inv' DeltaMap m mN f1 v1 delta v m' E Gamma:
Rle (Rabs delta) (mTypeToR m') ->
eval_expr E Gamma DeltaMap f1 v1 m ->
DeltaMap (Unop Inv f1) m' = Some delta ->
DeltaMap (evalUnop Inv v1) m' = Some delta ->
(~ v1 = 0)%R ->
v = perturb (evalUnop Inv v1) mN delta ->
Gamma (Unop Inv f1) = Some mN ->
......@@ -137,7 +137,7 @@ Lemma Downcast_dist' DeltaMap m m1 f1 v1 delta v m' E Gamma:
isMorePrecise m1 m = true ->
Rle (Rabs delta) (mTypeToR m') ->
eval_expr E Gamma DeltaMap f1 v1 m1 ->
DeltaMap (Downcast m f1) m' = Some delta ->
DeltaMap v1 m' = Some delta ->
v = (perturb v1 m delta) ->
Gamma (Downcast m f1) = Some m ->
m' = m ->
......@@ -152,7 +152,7 @@ Lemma Binop_dist' DeltaMap m1 m2 op f1 f2 v1 v2 delta v m m' E Gamma:
Rle (Rabs delta) (mTypeToR m') ->
eval_expr E Gamma DeltaMap f1 v1 m1 ->
eval_expr E Gamma DeltaMap f2 v2 m2 ->
DeltaMap (Binop op f1 f2) m' = Some delta ->
DeltaMap (evalBinop op v1 v2) m' = Some delta ->
((op = Div) -> (~ v2 = 0)%R) ->
v = perturb (evalBinop op v1 v2) m' delta ->
Gamma (Binop op f1 f2) = Some m ->
......@@ -170,7 +170,7 @@ Lemma Fma_dist' DeltaMap m1 m2 m3 f1 f2 f3 v1 v2 v3 delta v m' E Gamma m:
eval_expr E Gamma DeltaMap f1 v1 m1 ->
eval_expr E Gamma DeltaMap f2 v2 m2 ->
eval_expr E Gamma DeltaMap f3 v3 m3 ->
DeltaMap (Fma f1 f2 f3) m' = Some delta ->
DeltaMap (evalFma v1 v2 v3) m' = Some delta ->
v = perturb (evalFma v1 v2 v3) m' delta ->