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AVA
FloVer
Commits
e257e52b
Commit
e257e52b
authored
Feb 07, 2017
by
Heiko Becker
Browse files
Some more type annotations and remove unused lemma
parent
53aba077
Changes
2
Hide whitespace changes
Inline
Side-by-side
coq/IntervalValidation.v
View file @
e257e52b
...
...
@@ -21,7 +21,7 @@ Fixpoint freeVars (V:Type) (f:exp V) : list nat:=
|
Binop
o
f1
f2
=>
(
freeVars
V
f1
)
++
(
freeVars
V
f2
)
end
.
Fixpoint
validIntervalbounds
(
e
:
exp
Q
)
(
absenv
:
analysisResult
)
(
P
:
precond
)
validVars
:=
Fixpoint
validIntervalbounds
(
e
:
exp
Q
)
(
absenv
:
analysisResult
)
(
P
:
precond
)
(
validVars
:
NatSet
.
t
)
:=
let
(
intv
,
_
)
:=
absenv
e
in
match
e
with
|
Var
_
v
=>
NatSet
.
mem
v
validVars
...
...
@@ -70,7 +70,7 @@ Fixpoint validIntervalbounds (e:exp Q) (absenv:analysisResult) (P:precond) valid
andb
rec
opres
end
.
Fixpoint
validIntervalboundsCmd
(
f
:
cmd
Q
)
(
absenv
:
analysisResult
)
(
P
:
precond
)
validVars
{
struct
f
}
:
bool
:=
Fixpoint
validIntervalboundsCmd
(
f
:
cmd
Q
)
(
absenv
:
analysisResult
)
(
P
:
precond
)
(
validVars
:
NatSet
.
t
)
:
bool
:=
match
f
with
|
Let
_
x
e
g
=>
validIntervalbounds
e
absenv
P
validVars
&&
...
...
@@ -156,7 +156,6 @@ Qed.
Theorem
validIntervalbounds_sound
(
f
:
exp
Q
)
(
absenv
:
analysisResult
)
(
P
:
precond
)
V
VarEnv
ParamEnv
:
forall
vR
,
(
*
precondValidForExec
P
cenv
->*
)
validIntervalbounds
f
absenv
P
V
=
true
->
(
forall
v
,
NatSet
.
mem
v
V
=
true
->
(
Q2R
(
fst
(
fst
(
absenv
(
Var
Q
v
))))
<=
VarEnv
v
<=
Q2R
(
snd
(
fst
(
absenv
(
Var
Q
v
)))))
%
R
)
->
...
...
@@ -458,72 +457,6 @@ Proof.
rewrite
<-
Q2R_max4
in
valid_div_hi
;
auto
.
}
Qed
.
Theorem
ssaVars_are_sound
(
f
:
cmd
Q
)
freeVars
outVars
(
absenv
:
analysisResult
)
(
v_lo
v_hi
err
:
R
)
VarEnv
ParamEnv
P
TEnv
:
ssaPrg
Q
f
(
freeVars
)
(
outVars
)
->
bstep
(
toRCmd
f
)
VarEnv
ParamEnv
P
0
%
R
(
Nop
R
)
TEnv
->
(
forall
v
,
NatSet
.
mem
v
freeVars
=
true
->
(
Q2R
(
fst
(
fst
(
absenv
(
Var
Q
v
))))
<=
VarEnv
v
<=
Q2R
(
snd
(
fst
(
absenv
(
Var
Q
v
)))))
%
R
)
->
validIntervalboundsCmd
f
absenv
P
(
freeVars
)
=
true
->
forall
v
:
nat
,
NatSet
.
mem
v
outVars
=
true
->
(
Q2R
(
fst
(
fst
(
absenv
(
Var
Q
v
))))
<=
TEnv
v
<=
Q2R
(
snd
(
fst
(
absenv
(
Var
Q
v
)))))
%
R
.
Proof
.
intros
ssa_f
.
revert
VarEnv
.
induction
ssa_f
;
intros
VarEnv
bstep_f
freeVars_sound
validBounds
v
in_outVars
;
unfold
validIntervalbounds
in
validBounds
;
andb_to_prop
validBounds
.
-
(
*
First
rename
auto
-
generated
hyp
names
*
)
rename
L
into
eq_lo
;
rename
R1
into
eq_hi
;
rename
L0
into
validBounds_e
.
inversion
bstep_f
;
subst
.
eapply
IHssa_f
;
eauto
.
+
intros
v1
mem_Vx
.
rewrite
NatSet
.
mem_spec
,
NatSet
.
add_spec
in
mem_Vx
.
unfold
updEnv
.
case_eq
(
v1
=?
x
);
intros
v1_eq_dec
.
*
assert
(
Q2R
(
fst
(
fst
(
absenv
e
)))
<=
v0
<=
Q2R
(
snd
(
fst
(
absenv
e
))))
%
R
as
validIV_e
by
(
eapply
validIntervalbounds_sound
;
eauto
).
rewrite
Nat
.
eqb_eq
in
v1_eq_dec
.
rewrite
v1_eq_dec
.
apply
Qeq_bool_iff
in
eq_lo
.
apply
Qeq_eqR
in
eq_lo
.
apply
Qeq_bool_iff
in
eq_hi
.
apply
Qeq_eqR
in
eq_hi
.
rewrite
<-
eq_lo
,
<-
eq_hi
.
auto
.
*
destruct
mem_Vx
.
{
subst
.
rewrite
Nat
.
eqb_neq
in
v1_eq_dec
.
hnf
in
v1_eq_dec
.
exfalso
.
apply
v1_eq_dec
.
reflexivity
.
}
{
apply
freeVars_sound
.
rewrite
NatSet
.
mem_spec
;
auto
.
}
-
rename
H
into
eq_V_Vterm
.
rewrite
NatSet
.
equal_spec
in
eq_V_Vterm
.
rewrite
NatSet
.
mem_spec
in
in_outVars
.
hnf
in
eq_V_Vterm
.
rewrite
<-
eq_V_Vterm
in
in_outVars
.
rewrite
<-
NatSet
.
mem_spec
in
in_outVars
.
inversion
bstep_f
;
subst
.
unfold
updEnv
.
case_eq
(
v
=?
0
);
intros
v_eq
.
+
assert
(
Q2R
(
fst
(
fst
(
absenv
e
)))
<=
v0
<=
Q2R
(
snd
(
fst
(
absenv
e
))))
%
R
by
(
eapply
validIntervalbounds_sound
;
eauto
).
rename
L0
into
eq_lo
;
rename
R0
into
eq_hi
.
apply
Qeq_bool_iff
in
eq_lo
;
apply
Qeq_eqR
in
eq_lo
.
apply
Qeq_bool_iff
in
eq_hi
;
apply
Qeq_eqR
in
eq_hi
.
rewrite
Nat
.
eqb_eq
in
v_eq
.
subst
.
rewrite
<-
eq_lo
,
<-
eq_hi
.
assumption
.
+
apply
freeVars_sound
;
auto
.
Qed
.
Theorem
validIntervalboundsCmd_sound
(
f
:
cmd
Q
)
(
absenv
:
analysisResult
)
:
forall
VarEnv
ParamEnv
envR
inVars
outVars
elo
ehi
err
P
,
ssaPrg
Q
f
inVars
outVars
->
...
...
coq/ssaPrgs.v
View file @
e257e52b
...
...
@@ -2,7 +2,7 @@ Require Import Coq.MSets.MSets Coq.Arith.PeanoNat.
Require
Export
Daisy
.
Commands
.
(
**
Module
for
an
ordered
type
with
leibniz
,
based
on
code
from
coq
-
club
code
Module
for
an
ordered
type
with
leibniz
,
based
on
code
from
coq
-
club
mailing
list
http:
//coq-club.inria.narkive.com/zptqoou2/how-to-use-msets
**
)
Module
OWL
.
...
...
@@ -40,6 +40,7 @@ Fixpoint validVars (V:Type) (f:exp V) Vs :bool :=
|
Binop
o
f1
f2
=>
validVars
V
f1
Vs
&&
validVars
V
f2
Vs
end
.
(
*
TODO
:
This
still
allows
overwriting
of
return
value
*
)
Inductive
ssaPrg
(
V
:
Type
)
:
(
cmd
V
)
->
(
NatSet
.
t
)
->
(
NatSet
.
t
)
->
Prop
:=
ssaLet
x
e
s
inVars
Vterm
:
validVars
V
e
inVars
=
true
->
...
...
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