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AVA
Dandelion
Commits
2c652a21
Commit
2c652a21
authored
3 years ago
by
Heiko Becker
Browse files
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Add support for +, unary and binary - to approximations
parent
204830d8
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Pipeline
#54714
failed
3 years ago
Changes
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2 changed files
checkerScript.sml
+165
-108
165 additions, 108 deletions
checkerScript.sml
transcLangScript.sml
+11
-3
11 additions, 3 deletions
transcLangScript.sml
with
176 additions
and
111 deletions
checkerScript.sml
+
165
−
108
View file @
2c652a21
...
...
@@ -77,7 +77,7 @@ End
Definition
approxPoly_def
:
approxPoly
transc
(
iv
:
real
#real
)
(
hs
:
hint
list
)
approxSteps
:(
poly
#real
)
option
=
case
transc
of
|
F
UN
tr
V
AR
=>
|
F
un
tr
V
ar
=>
if
tr
=
"exp"
then
case
getExpHint
hs
of
|
NONE
=>
...
...
@@ -93,6 +93,27 @@ Definition approxPoly_def:
else
if
tr
=
"sin"
then
SOME
(
sin_poly
approxSteps
,
sin_err
iv
approxSteps
)
else
NONE
|
Add
tr1
tr2
=>
(
case
approxPoly
tr1
iv
hs
approxSteps
of
|
NONE
=>
NONE
|
SOME
(
appr1
,
err1
)
=>
case
approxPoly
tr2
iv
hs
approxSteps
of
|
NONE
=>
NONE
|
SOME
(
appr2
,
err2
)
=>
SOME
(
appr1
+p
appr2
,
err1
+
err2
))
|
Sub
tr1
tr2
=>
(
case
approxPoly
tr1
iv
hs
approxSteps
of
|
NONE
=>
NONE
|
SOME
(
appr1
,
err1
)
=>
case
approxPoly
tr2
iv
hs
approxSteps
of
|
NONE
=>
NONE
|
SOME
(
appr2
,
err2
)
=>
SOME
(
appr1
-p
appr2
,
err1
+
err2
))
|
Neg
tr1
=>
(
case
approxPoly
tr1
iv
hs
approxSteps
of
|
NONE
=>
NONE
|
SOME
(
appr1
,
err1
)
=>
SOME
(
--p
appr1
,
err1
))
|
_
=>
NONE
End
...
...
@@ -362,135 +383,171 @@ Theorem approxPoly_soundness:
FST
iv
≤
x
∧
x
≤
SND
iv
⇒
abs
(
interp
transc
x
-
evalPoly
p
x
)
≤
err
Proof
simp
[
approxPoly_def
,
approxPolySideCond_def
,
CaseEq
"transc"
]
ho_match_mp_tac
approxPoly_ind
>>
rpt
strip_tac
>>
‘
(
tr
=
"exp"
∧
((
iv
=
(
0
,
1
*
inv
2
)
∧
getExpHint
hs
=
NONE
)
∨
∃
n
.
getExpHint
hs
=
SOME
n
∧
iv
=
(
0
,
&n
*
inv
2
)))
∨
tr
=
"cos"
∨
tr
=
"sin"
’
by
(
every_case_tac
>>
gs
[
getExpHint_SOME_MEM
])
(*
exp function, 0 to 1/2 *)
>-
(
gs
[
interp_def
,
getFun_def
]
>>
qspecl_then
[
‘
x
’
,
‘
approxSteps
’
]
strip_assume_tac
MCLAURIN_EXP_LE
>>
pop_assum
$
rewrite_tac
o
single
>>
rpt
VAR_EQ_TAC
>>
rewrite_tac
[
exp_sum_to_poly
]
>>
qmatch_goalsub_abbrev_tac
‘
abs
(
exp_taylor
+
taylor_rem
-
exp_taylor
)
≤
_
’
>>
‘
exp_taylor
+
taylor_rem
-
exp_taylor
=
taylor_rem
’
by
real_tac
>>
pop_assum
$
rewrite_tac
o
single
>>
unabbrev_all_tac
>>
‘
exp_err_small
approxSteps
=
inv
(
&FACT
approxSteps
*
2
pow
(
approxSteps
-
1
))
’
by
EVAL_TAC
>>
qspecl_then
[
‘
approxSteps
’
,
‘
x
’
,
‘
t
’
]
mp_tac
exp_remainder_bounded_small
>>
impl_tac
>>
gs
[]
>>
real_tac
)
(*
exp function, 0 to 1 *)
>-
(
gs
[
interp_def
,
getFun_def
]
>>
‘
1
≠
inv
2
’
by
(
once_rewrite_tac
[
GSYM
REAL_INV1
]
>>
CCONTR_TAC
>>
pop_assum
$
mp_tac
o
SIMP_RULE
std_ss
[]
>>
rewrite_tac
[
REAL_INV_INJ
]
>>
real_tac
)
>>
rpt
VAR_EQ_TAC
>>
rewrite_tac
[
GSYM
poly_compat
,
eval_simps
]
>>
pop_assum
$
rewrite_tac
o
single
>>
rewrite_tac
[
exp_sum_to_poly
]
>>
qspecl_then
[
‘
x
’
,
‘
approxSteps
’
]
strip_assume_tac
MCLAURIN_EXP_LE
>>
pop_assum
$
rewrite_tac
o
single
>>
qmatch_goalsub_abbrev_tac
‘
abs
(
exp_taylor
+
taylor_rem
-
exp_taylor
)
≤
_
’
>>
‘
exp_taylor
+
taylor_rem
-
exp_taylor
=
taylor_rem
’
by
real_tac
>>
pop_assum
$
rewrite_tac
o
single
>>
unabbrev_all_tac
>>
‘
exp_err_big
n
approxSteps
=
2
pow
n
*
&n
pow
approxSteps
*
inv
(
&FACT
approxSteps
*
2
pow
approxSteps
)
’
by
(
rewrite_tac
[]
>>
EVAL_TAC
)
>>
pop_assum
$
rewrite_tac
o
single
>>
qspecl_then
[
‘
approxSteps
’
,
‘
n
’
,
‘
x
’
,
‘
t
’
]
mp_tac
exp_remainder_bounded_big
>>
impl_tac
>-
(
rpt
conj_tac
>>
gs
[]
>>
real_tac
)
>>
rewrite_tac
[])
(*
cos function *)
>>
qpat_x_assum
‘
approxPoly
_
_
_
_
=
SOME
_
’
mp_tac
>>
Cases_on
‘
transc
’
>>
simp
[
Once
approxPoly_def
,
CaseEq
"transc"
,
CaseEq
"option"
,
CaseEq
"prod"
]
>>
rpt
strip_tac
>>
gs
[
approxPolySideCond_def
]
>-
(
gs
[
interp_def
,
getFun_def
]
>>
rpt
VAR_EQ_TAC
>>
qspecl_then
[
‘
x
’
,
‘
approxSteps
’
]
strip_assume_tac
MCLAURIN_COS_LE
‘
(
s
=
"exp"
∧
((
iv
=
(
0
,
1
*
inv
2
)
∧
getExpHint
hs
=
NONE
)
∨
∃
n
.
getExpHint
hs
=
SOME
n
∧
iv
=
(
0
,
&n
*
inv
2
)))
∨
s
=
"cos"
∨
s
=
"sin"
’
by
(
every_case_tac
>>
gs
[
getExpHint_SOME_MEM
])
(*
exp function, 0 to 1/2 *)
>-
(
gs
[
interp_def
,
getFun_def
]
>>
qspecl_then
[
‘
x
’
,
‘
approxSteps
’
]
strip_assume_tac
MCLAURIN_EXP_LE
>>
pop_assum
$
rewrite_tac
o
single
>>
rpt
VAR_EQ_TAC
>>
rewrite_tac
[
exp_sum_to_poly
]
>>
qmatch_goalsub_abbrev_tac
‘
abs
(
exp_taylor
+
taylor_rem
-
exp_taylor
)
≤
_
’
>>
‘
exp_taylor
+
taylor_rem
-
exp_taylor
=
taylor_rem
’
by
real_tac
>>
pop_assum
$
rewrite_tac
o
single
>>
unabbrev_all_tac
>>
‘
exp_err_small
approxSteps
=
inv
(
&FACT
approxSteps
*
2
pow
(
approxSteps
-
1
))
’
by
EVAL_TAC
>>
rename1
‘
exp
t
’
>>
qspecl_then
[
‘
approxSteps
’
,
‘
x
’
,
‘
t
’
]
mp_tac
exp_remainder_bounded_small
>>
impl_tac
>>
gs
[]
>>
real_tac
)
(*
exp function, 0 to 1 *)
>-
(
gs
[
interp_def
,
getFun_def
]
>>
‘
1
≠
inv
2
’
by
(
once_rewrite_tac
[
GSYM
REAL_INV1
]
>>
CCONTR_TAC
>>
pop_assum
$
mp_tac
o
SIMP_RULE
std_ss
[]
>>
rewrite_tac
[
REAL_INV_INJ
]
>>
real_tac
)
>>
rpt
VAR_EQ_TAC
>>
rewrite_tac
[
GSYM
poly_compat
,
eval_simps
]
>>
pop_assum
$
rewrite_tac
o
single
>>
rewrite_tac
[
exp_sum_to_poly
]
>>
qspecl_then
[
‘
x
’
,
‘
approxSteps
’
]
strip_assume_tac
MCLAURIN_EXP_LE
>>
pop_assum
$
rewrite_tac
o
single
>>
qmatch_goalsub_abbrev_tac
‘
abs
(
exp_taylor
+
taylor_rem
-
exp_taylor
)
≤
_
’
>>
‘
exp_taylor
+
taylor_rem
-
exp_taylor
=
taylor_rem
’
by
real_tac
>>
pop_assum
$
rewrite_tac
o
single
>>
unabbrev_all_tac
>>
‘
exp_err_big
n
approxSteps
=
2
pow
n
*
&n
pow
approxSteps
*
inv
(
&FACT
approxSteps
*
2
pow
approxSteps
)
’
by
(
rewrite_tac
[]
>>
EVAL_TAC
)
>>
pop_assum
$
rewrite_tac
o
single
>>
qspecl_then
[
‘
approxSteps
’
,
‘
n
’
,
‘
x
’
,
‘
t
’
]
mp_tac
exp_remainder_bounded_big
>>
impl_tac
>-
(
rpt
conj_tac
>>
gs
[]
>>
real_tac
)
>>
rewrite_tac
[])
(*
cos function *)
>-
(
gs
[
interp_def
,
getFun_def
]
>>
rpt
VAR_EQ_TAC
>>
qspecl_then
[
‘
x
’
,
‘
approxSteps
’
]
strip_assume_tac
MCLAURIN_COS_LE
>>
gs
[]
>>
pop_assum
$
rewrite_tac
o
single
>>
gs
[
cos_sum_to_poly
]
>>
qmatch_goalsub_abbrev_tac
‘
abs
(
cos_taylor
+
taylor_rem
-
cos_taylor
)
≤
_
’
>>
‘
cos_taylor
+
taylor_rem
-
cos_taylor
=
taylor_rem
’
by
real_tac
>>
pop_assum
$
rewrite_tac
o
single
>>
unabbrev_all_tac
>>
‘
(
x
pow
approxSteps
)
*
cos
t
*
inv
(
&FACT
approxSteps
)
=
(
cos
t
*
((
x
pow
approxSteps
)
*
inv
(
&FACT
approxSteps
)))
’
by
real_tac
>>
‘
-
(
x
pow
approxSteps
)
*
cos
t
*
inv
(
&FACT
approxSteps
)
=
-
(
cos
t
*
((
x
pow
approxSteps
)
*
inv
(
&FACT
approxSteps
)))
’
by
real_tac
>>
rewrite_tac
[]
>>
ntac
2
$
pop_assum
$
rewrite_tac
o
single
>>
rewrite_tac
[
GSYM
REAL_MUL_ASSOC
]
>>
qmatch_goalsub_abbrev_tac
‘
abs
(
cos
_
*
err_cos_concr
)
’
>>
irule
REAL_LE_TRANS
>>
qexists_tac
‘
1
*
abs
err_cos_concr
’
>>
conj_tac
>-
(
rewrite_tac
[
ABS_MUL
]
>>
irule
REAL_LE_RMUL_IMP
>>
unabbrev_all_tac
>>
gs
[
COS_BOUND
,
ABS_POS
])
>>
rewrite_tac
[
REAL_MUL_LID
]
>>
‘
abs
err_cos_concr
=
err_cos_concr
’
by
(
unabbrev_all_tac
>>
rewrite_tac
[
ABS_REFL
]
>>
irule
REAL_LE_MUL
>>
conj_tac
>-
(
irule
REAL_LE_INV
>>
gs
[
REAL_POS
])
>>
irule
REAL_LE_MUL
>>
conj_tac
>>
gs
[
REAL_POW_GE0
])
>>
pop_assum
$
rewrite_tac
o
single
>>
unabbrev_all_tac
>>
rewrite_tac
[
cos_err_def
]
>>
imp_res_tac
EVEN_ODD_EXISTS
>>
gs
[
POW_MINUS1
]
>>
irule
REAL_LE_LMUL_IMP
>>
gs
[
GSYM
POW_ABS
]
>>
irule
REAL_LE_TRANS
>>
qexists_tac
‘
abs
(
x
pow
(
2
*
m
))
’
>>
gs
[
ABS_LE
,
GSYM
POW_ABS
]
>>
irule
POW_LE
>>
gs
[
ABS_POS
]
>>
irule
RealSimpsTheory
.
maxAbs
>>
gs
[])
(*
sin *)
>>
gs
[
interp_def
,
getFun_def
]
>>
rpt
VAR_EQ_TAC
>>
qspecl_then
[
‘
x
’
,
‘
approxSteps
’
]
strip_assume_tac
MCLAURIN_SIN_LE
>>
gs
[]
>>
pop_assum
$
rewrite_tac
o
single
>>
gs
[
cos
_sum_to_poly
]
>>
qmatch_goalsub_abbrev_tac
‘
abs
(
cos
_taylor
+
taylor_rem
-
cos
_taylor
)
≤
_
’
>>
‘
cos
_taylor
+
taylor_rem
-
cos
_taylor
=
taylor_rem
’
by
real_tac
>>
gs
[
sin
_sum_to_poly
]
>>
qmatch_goalsub_abbrev_tac
‘
abs
(
sin
_taylor
+
taylor_rem
-
sin
_taylor
)
≤
_
’
>>
‘
sin
_taylor
+
taylor_rem
-
sin
_taylor
=
taylor_rem
’
by
real_tac
>>
pop_assum
$
rewrite_tac
o
single
>>
unabbrev_all_tac
>>
‘
(
x
pow
approxSteps
)
*
cos
t
*
inv
(
&FACT
approxSteps
)
=
(
cos
t
*
((
x
pow
approxSteps
)
*
inv
(
&FACT
approxSteps
)))
’
>>
‘
inv
(
&FACT
approxSteps
)
*
sin
t
*
x
pow
approxSteps
*
-
1
pow
(
approxSteps
DIV
2
)
=
(
sin
t
*
((
x
pow
approxSteps
)
*
inv
(
&FACT
approxSteps
)
*
-
1
pow
(
approxSteps
DIV
2
)
))
’
by
real_tac
>>
‘
-
(
x
pow
approxSteps
)
*
cos
t
*
inv
(
&FACT
approxSteps
)
=
-
(
cos
t
*
((
x
pow
approxSteps
)
*
inv
(
&FACT
approxSteps
)))
’
>>
‘
-
(
x
pow
approxSteps
)
*
inv
(
&FACT
approxSteps
)
*
sin
t
=
-
(
sin
t
*
((
x
pow
approxSteps
)
*
inv
(
&FACT
approxSteps
)))
’
by
real_tac
>>
rewrite_tac
[]
>>
ntac
2
$
pop_assum
$
rewrite_tac
o
single
>>
rewrite_tac
[
GSYM
REAL_MUL_ASSOC
]
>>
qmatch_goalsub_abbrev_tac
‘
abs
(
cos
_
*
err_cos_concr
)
’
>>
rewrite_tac
[
GSYM
REAL_MUL_ASSOC
]
>>
qmatch_goalsub_abbrev_tac
‘
_
*
err_sin_concr
’
>>
rewrite_tac
[
ABS_NEG
,
Once
ABS_MUL
]
>>
irule
REAL_LE_TRANS
>>
qexists_tac
‘
1
*
abs
err_
cos
_concr
’
>>
conj_tac
>-
(
rewrite_tac
[
ABS_MUL
]
>>
irule
REAL_LE_RMUL_IMP
>>
unabbrev_all_tac
>>
gs
[
COS
_BOUND
,
ABS_POS
])
>>
rewrite_tac
[
REAL_MUL_LID
]
>>
‘
abs
err_
cos
_concr
=
err_
cos
_concr
’
>>
qexists_tac
‘
1
*
abs
err_
sin
_concr
’
>>
conj_tac
>-
(
irule
REAL_LE_RMUL_IMP
>>
unabbrev_all_tac
>>
gs
[
SIN
_BOUND
,
ABS_POS
])
>>
rewrite_tac
[
REAL_MUL_LID
,
sin_err_def
,
ABS_MUL
]
>>
‘
abs
err_
sin
_concr
=
err_
sin
_concr
’
by
(
unabbrev_all_tac
>>
rewrite_tac
[
ABS_REFL
]
>>
irule
REAL_LE_MUL
>>
conj_tac
>-
(
irule
REAL_LE_INV
>>
gs
[
REAL_POS
])
>>
irule
REAL_LE_MUL
>>
conj_tac
>>
gs
[
REAL_POW_GE0
])
>>
gs
[
REAL_POW_GE0
]
>>
irule
REAL_LE_MUL
>>
gs
[
REAL_POS
,
REAL_POW_GE0
])
>>
pop_assum
$
rewrite_tac
o
single
>>
unabbrev_all_tac
>>
rewrite_tac
[
cos
_err_def
]
>>
rewrite_tac
[
sin
_err_def
]
>>
imp_res_tac
EVEN_ODD_EXISTS
>>
gs
[
POW_MINUS1
]
>>
irule
REAL_LE_LMUL_IMP
>>
gs
[
GSYM
POW_ABS
]
>>
irule
REAL_LE_TRANS
>>
qexists_tac
‘
abs
(
x
pow
(
2
*
m
'
))
’
>>
gs
[
ABS_LE
,
GSYM
POW_ABS
]
>>
qexists_tac
‘
abs
(
x
pow
(
2
*
m
))
’
>>
gs
[
ABS_LE
,
GSYM
POW_ABS
]
>>
irule
POW_LE
>>
gs
[
ABS_POS
]
>>
irule
RealSimpsTheory
.
maxAbs
>>
gs
[])
(*
sin *)
>>
gs
[
interp_def
,
getFun_def
]
>>
rpt
VAR_EQ_TAC
>>
qspecl_then
[
‘
x
’
,
‘
approxSteps
’
]
strip_assume_tac
MCLAURIN_SIN_LE
>>
gs
[]
>>
pop_assum
$
rewrite_tac
o
single
>>
gs
[
sin_sum_to_poly
]
>>
qmatch_goalsub_abbrev_tac
‘
abs
(
sin_taylor
+
taylor_rem
-
sin_taylor
)
≤
_
’
>>
‘
sin_taylor
+
taylor_rem
-
sin_taylor
=
taylor_rem
’
by
real_tac
>>
pop_assum
$
rewrite_tac
o
single
>>
unabbrev_all_tac
>>
‘
inv
(
&FACT
approxSteps
)
*
sin
t
*
x
pow
approxSteps
*
-
1
pow
(
approxSteps
DIV
2
)
=
(
sin
t
*
((
x
pow
approxSteps
)
*
inv
(
&FACT
approxSteps
)
*
-
1
pow
(
approxSteps
DIV
2
)))
’
by
real_tac
>>
‘
-
(
x
pow
approxSteps
)
*
inv
(
&FACT
approxSteps
)
*
sin
t
=
-
(
sin
t
*
((
x
pow
approxSteps
)
*
inv
(
&FACT
approxSteps
)))
’
by
real_tac
>>
rewrite_tac
[]
>>
ntac
2
$
pop_assum
$
rewrite_tac
o
single
>>
rewrite_tac
[
GSYM
REAL_MUL_ASSOC
]
>>
qmatch_goalsub_abbrev_tac
‘
_
*
err_sin_concr
’
>>
rewrite_tac
[
ABS_NEG
,
Once
ABS_MUL
]
>>
irule
REAL_LE_TRANS
>>
qexists_tac
‘
1
*
abs
err_sin_concr
’
>>
conj_tac
>-
(
irule
REAL_LE_RMUL_IMP
>>
unabbrev_all_tac
>>
gs
[
SIN_BOUND
,
ABS_POS
])
>>
rewrite_tac
[
REAL_MUL_LID
,
sin_err_def
,
ABS_MUL
]
>>
‘
abs
err_sin_concr
=
err_sin_concr
’
by
(
unabbrev_all_tac
>>
rewrite_tac
[
ABS_REFL
]
>>
irule
REAL_LE_MUL
>>
conj_tac
>>
gs
[
REAL_POW_GE0
]
>>
irule
REAL_LE_MUL
>>
gs
[
REAL_POS
,
REAL_POW_GE0
])
>-
(
rpt
VAR_EQ_TAC
>>
gs
[
interp_def
,
eval_simps
]
>>
qmatch_goalsub_abbrev_tac
‘
abs
(
f_x
+
g_x
-
(
f_poly_x
+
g_poly_x
))
≤
_
’
>>
‘
f_x
+
g_x
-
(
f_poly_x
+
g_poly_x
)
=
(
f_x
-
f_poly_x
)
+
(
g_x
-
g_poly_x
)
’
by
real_tac
>>
pop_assum
$
rewrite_tac
o
single
>>
irule
REAL_LE_TRANS
>>
qexists_tac
‘
abs
(
f_x
-
f_poly_x
)
+
abs
(
g_x
-
g_poly_x
)
’
>>
gs
[
REAL_ABS_TRIANGLE
]
>>
irule
REAL_LE_ADD2
>>
unabbrev_all_tac
>>
gs
[])
>-
(
rpt
VAR_EQ_TAC
>>
gs
[
interp_def
,
eval_simps
]
>>
qmatch_goalsub_abbrev_tac
‘
abs
(
f_x
-
g_x
-
(
f_poly_x
-
g_poly_x
))
≤
_
’
>>
‘
f_x
-
g_x
-
(
f_poly_x
-
g_poly_x
)
=
(
f_x
-
f_poly_x
)
-
(
g_x
-
g_poly_x
)
’
by
real_tac
>>
pop_assum
$
rewrite_tac
o
single
>>
irule
REAL_LE_TRANS
>>
qexists_tac
‘
abs
(
f_x
-
f_poly_x
)
+
abs
(
-
(
g_x
-
g_poly_x
))
’
>>
conj_tac
>-
(
rewrite_tac
[
real_sub
]
>>
irule
REAL_ABS_TRIANGLE
)
>>
gs
[
real_sub
]
>>
irule
REAL_LE_ADD2
>>
unabbrev_all_tac
>>
gs
[])
>>
rpt
VAR_EQ_TAC
>>
gs
[
interp_def
,
eval_simps
,
real_sub
]
>>
qmatch_goalsub_abbrev_tac
‘
abs
(
-
f_x
+
f_poly_x
)
’
>>
‘
-
f_x
+
f_poly_x
=
f_poly_x
-
f_x
’
by
real_tac
>>
pop_assum
$
rewrite_tac
o
single
>>
unabbrev_all_tac
>>
rewrite_tac
[
sin_err_def
]
>>
imp_res_tac
EVEN_ODD_EXISTS
>>
gs
[
POW_MINUS1
]
>>
irule
REAL_LE_LMUL_IMP
>>
gs
[
GSYM
POW_ABS
]
>>
irule
REAL_LE_TRANS
>>
qexists_tac
‘
abs
(
x
pow
(
2
*
m'
))
’
>>
gs
[
ABS_LE
,
GSYM
POW_ABS
]
>>
irule
POW_LE
>>
gs
[
ABS_POS
]
>>
irule
RealSimpsTheory
.
maxAbs
>>
gs
[]
>>
once_rewrite_tac
[
ABS_SUB
]
>>
unabbrev_all_tac
>>
gs
[
real_sub
]
QED
(*
...
...
This diff is collapsed.
Click to expand it.
transcLangScript.sml
+
11
−
3
View file @
2c652a21
...
...
@@ -9,7 +9,12 @@ open preamble;
val
_
=
new_theory
"transcLang"
;
Datatype
:
transc
=
FUN
string
transc
|
VAR
transc
=
Fun
string
transc
|
Add
transc
transc
|
Sub
transc
transc
|
Neg
transc
|
Var
End
Definition
getFun_def
:
...
...
@@ -24,8 +29,11 @@ Definition getFun_def:
End
Definition
interp_def
:
interp
VAR
x
=
x
∧
interp
(
FUN
s
trans
)
x
=
interp
Var
x
=
x
∧
interp
(
Neg
t
)
x
=
-
(
interp
t
x
)
∧
interp
(
Add
t1
t2
)
x
=
(
interp
t1
x
)
+
(
interp
t2
x
)
∧
interp
(
Sub
t1
t2
)
x
=
(
interp
t1
x
)
-
(
interp
t2
x
)
∧
interp
(
Fun
s
trans
)
x
=
(
getFun
s
)
(
interp
trans
x
)
End
...
...
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