def l_mp2df(ls, rs, is_max_plus, idx, is_eq):
if __debug__:
assert not is_empty(ls), ls
assert not is_empty(rs), rs
assert idx >= 0, idx
assert is_bool(is_max_plus), is_max_plus
ls_ = ls[idx:]
if __debug__:
assert ls_, ls_
first_elem = ls_[0]
first_elem_f = IeqMPP.r_mp2df(first_elem, rs, is_max_plus, 0, is_eq)
if len(ls_) == 1: #t <= max(x,y,z)
return first_elem_f
else:
op = ">=" if is_max_plus else "<="
rest_f = IeqMPP.l_mp2df(ls, rs, is_max_plus, idx + 1, is_eq)
others = ls[:idx] + ls[idx + 1:]
csts = ','.join('{} {} {}'.format(first_elem, op, t)
for t in others)
cond = ("And({})" if len(others) >= 2 else "{}").format(csts)
return "If({},{},{})".format(cond, first_elem_f, rest_f)
def r_mp2df(t, rs, is_max_plus, idx, is_eq):
"""
Convert max/min-plus format to disjunctive formula
Inputs:
t = single term
rs = list of terms
idx = operate over rs[idx:]
"""
if __debug__:
assert not is_empty(rs)
assert idx >= 0, idx
assert is_bool(is_max_plus), is_max_plus
rs_ = rs[idx:]
if __debug__:
assert not is_empty(rs_)
first_elem = rs_[0]
first_elem_f = ("{} {} {}".format(t, '==' if is_eq else '>=',
first_elem))
if len(rs_) == 1: # t <= x ==> t <= x
return first_elem_f
else: # t <= max(x,y,z) ==> If(x>=y and x >= z, t <= x, ...)
op = ">=" if is_max_plus else "<="
rest_f = IeqMPP.r_mp2df(t, rs, is_max_plus, idx + 1, is_eq)
others = rs[:idx] + rs[idx + 1:]
csts = ','.join('{} {} {}'.format(first_elem, op, t)
for t in others)
cond = ('And({})' if len(others) >= 2 else '{}').format(csts)
return "If({}, {}, {})".format(cond, first_elem_f, rest_f)
def get_invs(self, deg=2):
"""
Default method to obtain invariants
By default, DIG finds flat/nested relations over arr vars
and finds eqts/ieqs(deduction) over num vars.
"""
rs = []
if not is_empty(self.ss_arr):
logger.debug('*** Generate Array Invs over {} ***'.format(
self.ss_arr))
rs = rs + self.get_flat_array()
rs = rs + self.get_nested_array()
if not is_empty(self.ss_num):
logger.debug('*** Generate Polynomial Invs over {} ***'.format(
self.ss_num))
if __debug__:
assert deg >= 1, deg
terms = dig_miscs.gen_terms(deg, self.ss_num)
logger.debug('deg={}, |ss_num|={}, |terms|={}'.format(
deg, len(self.ss_num), len(terms)))
rs = rs + self.get_eqts(terms)
return rs
def l_mp2df(ls,rs,is_max_plus,idx,is_eq):
if __debug__:
assert not is_empty(ls), ls
assert not is_empty(rs), rs
assert idx >= 0, idx
assert is_bool(is_max_plus), is_max_plus
ls_ = ls[idx:]
if __debug__:
assert ls_, ls_
first_elem = ls_[0]
first_elem_f = IeqMPP.r_mp2df(first_elem,rs,is_max_plus,0,is_eq)
if len(ls_) == 1: #t <= max(x,y,z)
return first_elem_f
else:
op = ">=" if is_max_plus else "<="
rest_f = IeqMPP.l_mp2df(ls,rs,is_max_plus,idx+1,is_eq)
others = ls[:idx]+ls[idx+1:]
csts = ','.join('{} {} {}'.format(first_elem, op, t)
for t in others)
cond = ("And({})" if len(others) >= 2 else "{}").format(csts)
return "If({},{},{})".format(cond, first_elem_f, rest_f)
def r_mp2df(t,rs,is_max_plus,idx,is_eq):
"""
Convert max/min-plus format to disjunctive formula
Inputs:
t = single term
rs = list of terms
idx = operate over rs[idx:]
"""
if __debug__:
assert not is_empty(rs)
assert idx >= 0, idx
assert is_bool(is_max_plus), is_max_plus
rs_ = rs[idx:]
if __debug__:
assert not is_empty(rs_)
first_elem = rs_[0]
first_elem_f = ("{} {} {}"
.format(t, '==' if is_eq else '>=',first_elem))
if len(rs_) == 1: # t <= x ==> t <= x
return first_elem_f
else: # t <= max(x,y,z) ==> If(x>=y and x >= z, t <= x, ...)
op = ">=" if is_max_plus else "<="
rest_f = IeqMPP.r_mp2df(t,rs,is_max_plus,idx+1,is_eq)
others = rs[:idx]+rs[idx+1:]
csts = ','.join('{} {} {}'.
format(first_elem, op, t) for t in others)
cond = ('And({})' if len(others) >= 2 else '{}').format(csts)
return "If({}, {}, {})".format(cond, first_elem_f, rest_f)
def get_invs(self, deg=2):
"""
Default method to obtain invariants
By default, DIG finds flat/nested relations over arr vars
and finds eqts/ieqs(deduction) over num vars.
"""
rs = []
if not is_empty(self.ss_arr):
logger.debug('*** Generate Array Invs over {} ***\n'
.format(self.ss_arr))
rs = rs + self.get_flat_array()
rs = rs + self.get_nested_array()
if not is_empty(self.ss_num):
logger.debug('*** Generate Polynomial Invs over {} ***\n'
.format(self.ss_num))
if __debug__:
assert deg >= 1, deg
terms = dig_miscs.gen_terms(deg, self.ss_num)
logger.debug('deg={}, |ss_num|={}, |terms|={}'
.format(deg,len(self.ss_num),len(terms)))
rs = rs + self.get_eqts(terms)
return rs
def rfilter(self, tcs, do_parallel=True):
if is_empty(self.ps) or is_empty(tcs):
logger.debug('rfilter skips (|ps|={}, |tcs|={})'.format(
len(self.ps), len(tcs)))
return None
logger.debug('rfilter(|ps|={}, |tcs|={})'.format(
len(self.ps), len(tcs)))
if not isinstance(self.ps[0], InvExp):
from dig_miscs import Miscs
tcs = Miscs.keys_to_str(tcs)
def wprocess(tasks, Q):
rs = [p for p in tasks if all(p.seval(tc) for tc in tcs)]
if Q is None: #no multiprocessing
return rs
else:
Q.put(rs)
tasks = self.ps
if do_parallel:
from vu_common import get_workloads
from multiprocessing import (Process, Queue, current_process,
cpu_count)
Q = Queue()
workloads = get_workloads(tasks,
max_nprocesses=cpu_count(),
chunksiz=2)
logger.debug("workloads 'refine' {}: {}".format(
len(workloads), map(len, workloads)))
workers = [
Process(target=wprocess, args=(wl, Q)) for wl in workloads
]
for w in workers:
w.start()
wrs = []
for _ in workers:
wrs.extend(Q.get())
else:
wrs = wprocess(tasks, Q=None)
self.ps = wrs
Refine.print_diff('rfilter', len(tasks), len(self.ps))
def rem_dup_arrs(ps, ainfo):
"""
Remove relations that involve elements from same arrays
Examples:
sage: var('x_0 x_1 y_0 y_1')
(x_0, x_1, y_0, y_1)
sage: ainfo = {x_0:{'name':'x','idxs':[0]},x_1:{'name':'x','idxs':[1]}, y_0:{'name':'y','idxs':[0]},y_1:{'name':'y','idxs':[1]}}
sage: FlatArray.rem_dup_arrs([x_0 + x_1 == 0, x_1 + y_1 == 0, x_0 + y_1 + y_0==0, x_0 + x_1-2==0], ainfo)
dig_arrays:Warn:Removed 3 array eqts
x_0 + x_1 == 0
x_0 + y_0 + y_1 == 0
x_0 + x_1 - 2 == 0
[x_1 + y_1 == 0]
"""
get_anames = lambda p: [ainfo[v]['name'] for v in get_vars(p)]
ps_rem, ps = vpartition(ps, lambda p: vall_uniq(get_anames(p)))
if not is_empty(ps_rem):
logger.warn('Removed {} array eqts\n{}'
.format(len(ps_rem), list_str(ps_rem,'\n')))
return ps
def wprocess(ae,Q):
r = ae.peelme(data=self.tcs[0])
if not is_empty(r):
logger.debug('Nesting {} has {} rel(s)\n{}'
.format(ae,len(r),'\n'.join(r)))
Q.put(r)
def imply(fs, gs):
"""
Checks if the set f of formulas implies the set f of formulas
sage: var('x y')
(x, y)
sage: assert imply([x-6==0],x*x-36==0)
sage: assert imply([x-6==0,x+6==0],x*x-36==0)
sage: assert not imply([x*x-36==0],x-6==0)
sage: assert not imply([x-6==0],x-36==0)
sage: assert imply(x-7>=0,x>=6)
sage: assert not imply(x-7>=0,x>=8)
sage: assert not imply(x-6>=0,x-7>=0)
sage: assert not imply([x-7>=0,y+5>=0],x+y-3>=0)
sage: assert imply([x-7>=0,y+5>=0],x+y-2>=0)
sage: assert imply([x-2*y>=0,y-1>=0],x-2>=0)
sage: assert not imply([],x-2>=0)
sage: assert imply([x-7>=0,y+5>=0],x+y-2>=0)
sage: assert imply([x^2-9>=0,x>=0],x-3>=0)
sage: assert not imply([1/2*x**2 - 3/28*x + 1 >= 0],1/20*x**2 - 9/20*x + 1 >= 0)
sage: assert imply([1/20*x**2 - 9/20*x + 1 >= 0],1/2*x**2 - 3/28*x + 1 >= 0)
sage: assert imply([x-6==0],x*x-36==0)
sage: assert not imply([x+7>=0,y+5>=0],x*y+36>=0)
sage: assert not imply([x+7>=0,y+5>=0],x*y+35>=0)
sage: assert not imply([x+7>=0,y+5>=0],x*y-35>=0)
sage: assert not imply([x+7>=0],x-8>=0)
sage: assert imply([x+7>=0],x+8>=0)
sage: assert imply([x+7>=0],x+8.9>=0)
sage: assert imply([x>=7,y>=5],x*y>=35)
sage: assert not imply([x>=-7,y>=-5],x*y>=35)
sage: assert imply([x-7>=0,y+5>=0],[x+y-2>=0,x>=2-y])
"""
if is_empty(fs) or is_empty(gs):
return False #conservative approach
if not isinstance(fs, list): fs = [fs]
if not isinstance(gs, list): gs = [gs]
fs = [toZ3(f, is_real=True) for f in fs]
gs = [toZ3(g, is_real=True) for g in gs]
claim = z3.Implies(z3.And(fs), z3.And(gs))
is_proved, _ = vprove(claim)
return is_proved
def rfilter(self,tcs,do_parallel=True):
if is_empty(self.ps) or is_empty(tcs):
logger.debug('rfilter skips (|ps|={}, |tcs|={})'
.format(len(self.ps), len(tcs)))
return None
logger.debug('rfilter(|ps|={}, |tcs|={})'
.format(len(self.ps), len(tcs)))
if not isinstance(self.ps[0],InvExp):
from dig_miscs import Miscs
tcs = Miscs.keys_to_str(tcs)
def wprocess(tasks,Q):
rs = [p for p in tasks if all(p.seval(tc) for tc in tcs)]
if Q is None: #no multiprocessing
return rs
else:
Q.put(rs)
tasks = self.ps
if do_parallel:
from vu_common import get_workloads
from multiprocessing import (Process, Queue,
current_process, cpu_count)
Q=Queue()
workloads = get_workloads(tasks,
max_nprocesses=cpu_count(),
chunksiz=2)
logger.debug("workloads 'refine' {}: {}"
.format(len(workloads),map(len,workloads)))
workers = [Process(target=wprocess,args=(wl,Q))
for wl in workloads]
for w in workers: w.start()
wrs = []
for _ in workers: wrs.extend(Q.get())
else:
wrs = wprocess(tasks,Q=None)
self.ps = wrs
Refine.print_diff('rfilter', len(tasks), len(self.ps))
def solve(self): #FlatArray
#Create new variables and traces
logger.info('Compute new traces (treating array elems as new vars)')
ainfo = {} #{A_0_0=[0,0],B_1_2_3=[1,2,3]}
tsinfo = {} #{A: [A_0,A_1, ..], B:[B_0_0,B_0_1,..]}
print self.tcs
tcs = [FlatArray.compute_new_trace(tc, ainfo, tsinfo)
for tc in self.tcs]
ps = self.get_rels_elems1(tcs)
#ps = self.get_rels_elems2(tcs,tsinfo)
#Group arrays and in each group, find rels among array idxs
gs = FlatArray.group_arr_eqts(ps, ainfo)
logger.info('Partition {} eqts into {} groups'
.format(len(ps), len(gs)))
sols = []
for i,(gns,gps) in enumerate(gs.iteritems()):
if __debug__:
assert not is_empty(gps)
# Modify/reformat if necessary
gps = FlatArray.modify_arr_eqts(gps, ainfo)
#Find rels over array indices
logger.debug("{}. Find rels over idx from {} eqts (group {})"
.format(i,len(gps),gns))
gps = [FlatArray.parse_arr_eqt(p,ainfo) for p in gps]
gsols = FlatArray.find_rels(gps)
sols.extend(gsols)
if is_empty(sols):
logger.warn('No rels found over arr idxs, use orig results')
sols = flatten(ps)
self.sols = map(InvEqt, sols)
self.do_refine = False
else:
self.sols = map(InvFlatArray, sols)
self.print_sols()
def get_eqts(self, terms=None):
if terms is None:
terms = self.ss_num
from dig_polynomials import Eqt
solver = Eqt(terms, self.tcs_num, self.xinfo)
solver.go()
sols = solver.sols
if not (is_empty(sols) or is_empty(self.xinfo['Assume'])):
#deduce new info if external knowledge (assumes) is avail
from dig_polynomials import IeqDeduce
solver = IeqDeduce(xinfo=self.xinfo, eqts=[s.p for s in sols])
solver.solve()
solver.print_sols()
sols = sols + solver.sols
return sols
def get_eqts(self,terms=None):
if terms is None:
terms = self.ss_num
from dig_polynomials import Eqt
solver = Eqt(terms, self.tcs_num, self.xinfo)
solver.go()
sols = solver.sols
if not (is_empty(sols) or is_empty(self.xinfo['Assume'])):
#deduce new info if external knowledge (assumes) is avail
from dig_polynomials import IeqDeduce
solver = IeqDeduce(xinfo = self.xinfo,
eqts = [s.p for s in sols])
solver.solve()
solver.print_sols()
sols = sols + solver.sols
return sols
def rinfer(self):
"""
Return a (preferably minimal) subset of ps that
infers all properties in ps.
sage: logger.set_level(VLog.DEBUG)
sage: var('a s t y')
(a, s, t, y)
sage: rf = Refine(map(InvEqt,[a*a - s + t == 0, t*t - 4*s + 2*t + 1 == 0, \
a*s - 1/2*s*t + 1/2*s == 0, a*x - 1/2*x*t + 1/2*x == 0, \
a - 1/2*t + 1/2 == 0, a*t - 2*s + 3/2*t + 1/2 == 0]))
sage: rf.rinfer()
refine:Debug:rinfer(|ps|=6)
refine:Debug:rinfer (before 6, after 2, diff 4)
sage: assert \
set([p.p for p in rf.ps]) == set([a - 1/2*t + 1/2 == 0, t^2 - 4*s + 2*t + 1 == 0]) or \
set([p.p for p in rf.ps]) == set([a - 1/2*t + 1/2 == 0, a^2 - s + t == 0])
sage: rf = Refine(map(InvIeq,[x-7>=0, x + y -2>=0, y+5 >= 0, -7 + x >= 0])); rf.rinfer(); sorted(rf.ps,key=str)
refine:Debug:vset (before 4, after 3, diff 1)
refine:Debug:rinfer(|ps|=3)
refine:Debug:rinfer (before 3, after 2, diff 1)
[x - 7 >= 0, y + 5 >= 0]
sage: rf = Refine(map(InvEqt,[x+y==0])); rf.rinfer(); sorted(rf.ps,key=str)
refine:Debug:rinfer skips (|ps|=1)
[x + y == 0]
"""
if len(self.ps) <= 1:
logger.debug('rinfer skips (|ps|={})'.format(len(self.ps)))
return None
logger.debug('rinfer(|ps|={})'.format(len(self.ps)))
ps = [p.p for p in self.ps if isinstance(p, InvExp)]
if is_empty(ps):
logger.warn('cannot do inferrence on non polynomial rels')
return None
self.ps = [
InvEqt(p) if p.operator() == operator.eq else InvIeq(p)
for p in Refine.rinfer_helper(ps)
]
Refine.print_diff('rinfer', len(ps), len(self.ps))
def rinfer(self):
"""
Return a (preferably minimal) subset of ps that
infers all properties in ps.
sage: logger.set_level(VLog.DEBUG)
sage: var('a s t y')
(a, s, t, y)
sage: rf = Refine(map(InvEqt,[a*a - s + t == 0, t*t - 4*s + 2*t + 1 == 0, \
a*s - 1/2*s*t + 1/2*s == 0, a*x - 1/2*x*t + 1/2*x == 0, \
a - 1/2*t + 1/2 == 0, a*t - 2*s + 3/2*t + 1/2 == 0]))
sage: rf.rinfer()
refine:Debug:rinfer(|ps|=6)
refine:Debug:rinfer (before 6, after 2, diff 4)
sage: assert \
set([p.p for p in rf.ps]) == set([a - 1/2*t + 1/2 == 0, t^2 - 4*s + 2*t + 1 == 0]) or \
set([p.p for p in rf.ps]) == set([a - 1/2*t + 1/2 == 0, a^2 - s + t == 0])
sage: rf = Refine(map(InvIeq,[x-7>=0, x + y -2>=0, y+5 >= 0, -7 + x >= 0])); rf.rinfer(); sorted(rf.ps,key=str)
refine:Debug:vset (before 4, after 3, diff 1)
refine:Debug:rinfer(|ps|=3)
refine:Debug:rinfer (before 3, after 2, diff 1)
[x - 7 >= 0, y + 5 >= 0]
sage: rf = Refine(map(InvEqt,[x+y==0])); rf.rinfer(); sorted(rf.ps,key=str)
refine:Debug:rinfer skips (|ps|=1)
[x + y == 0]
"""
if len(self.ps) <= 1:
logger.debug('rinfer skips (|ps|={})'.format(len(self.ps)))
return None
logger.debug('rinfer(|ps|={})'.format(len(self.ps)))
ps = [p.p for p in self.ps if isinstance(p,InvExp)]
if is_empty(ps):
logger.warn('cannot do inferrence on non polynomial rels')
return None
self.ps = [InvEqt(p) if p.operator() == operator.eq else InvIeq(p)
for p in Refine.rinfer_helper(ps)]
Refine.print_diff('rinfer', len(ps), len(self.ps))
def _get_clp(self, typ, terms=None, subset_siz=None):
if terms is None:
terms = self.ss_num
if is_empty(terms):
return []
import dig_polynomials as dp
if 'gen' in typ:
solver = dp.IeqCLPGen(terms, self.tcs_num, self.xinfo)
else:
solver = dp.IeqCLPFixed(terms, self.tcs_num, self.xinfo)
solver.subset_siz = subset_siz
solver.go()
return solver.sols
def toZ3(p, is_real):
"""
Convert a Sage expression to a Z3 expression
Initially implements with a dictionary containing variables
e.g. {x:Real('x')} but the code is longer and more complicated.
This implemention does not require a dictionary pass in.
Todo: cache this function
sage: toZ3(x*x*x, False)
x*x*x
"""
assert is_sage_expr(p), p
def retval(p):
if p.is_symbol():
_f = z3.Real if is_real else z3.Int
else:
_f = z3.RealVal if is_real else z3.IntVal
return _f(str(p))
try:
oprs = p.operands()
except Exception:
return retval(p)
if is_empty(oprs):
return retval(p)
else:
op = p.operator()
#z3 has problem w/ t^c , so use t*t*t..
if op == operator.pow:
assert len(oprs) == 2, oprs
t, c = oprs
t = toZ3(t, is_real)
vs = [t] * c
z3exp = reduce(operator.mul, vs)
else:
oprs = [toZ3(o, is_real) for o in oprs]
z3exp = reduce(op, oprs)
assert z3.is_expr(z3exp), z3exp
return z3exp
def print_summary(filename,expect,sols_filename,sols_run):
if is_empty(sols_filename):
return
print("\nSUMMARY ({})".format(filename))
print("Expect ({}): {}".format(len(expect), ', '.join(map(str,expect))))
time_total = 0.0
for (nr,rs,deg,etime) in sols_filename:
print '* Run {}, time {}, deg {}'.format(nr, etime, deg)
Inv.print_invs(rs)
time_total = time_total + etime
print('\nObtain {} unique results over {} run'
.format(len(sols_run), len(sols_filename)))
Inv.print_invs(sols_run,nice_format=False)
time_str = '{0:.1f}'.format(time_total/len(sols_filename))
print('TIME AVG {}, #invs {}, ({})\n'
.format(time_str,len(sols_run), filename))
def print_summary(filename, expect, sols_filename, sols_run):
if is_empty(sols_filename):
return
print("\nSUMMARY ({})".format(filename))
print("Expect ({}): {}".format(len(expect), ', '.join(map(str, expect))))
time_total = 0.0
for (nr, rs, deg, etime) in sols_filename:
print '* Run {}, time {}, deg {}'.format(nr, etime, deg)
Inv.print_invs(rs)
time_total = time_total + etime
print('\nObtain {} unique results over {} run'.format(
len(sols_run), len(sols_filename)))
Inv.print_invs(sols_run, nice_format=False)
time_str = '{0:.1f}'.format(time_total / len(sols_filename))
print('TIME AVG {}, #invs {}, ({})\n'.format(time_str, len(sols_run),
filename))
def preprocess(self, xinfo):
"""
Preprocess input data
1) transforms external functions to special arrays
1) change arr repr to value->index repr to speed up arr idx lookup
2) generate nodes
"""
if __debug__:
assert is_dict(xinfo), xinfo
evs = ExtFun.gen_extvars(xinfo=xinfo)
#arrays only
evs = [OrderedDict([(k,v) for k,v in ev.iteritems() if is_list(v)])
for ev in evs]
evs = Miscs.keys_to_str(evs)
if not is_empty(evs): #add to traces
self.tcs = [merge_dict(evs + [tc]) for tc in self.tcs]
self.tcs_extra = [merge_dict(evs + [tc]) for tc in self.tcs_extra]
mytcs = []
for tc in self.tcs:
#arrs reprent ext funs (already in new format)
efs = ExtFun.gen_extfuns(tc=tc,xinfo=xinfo)
#convert normal arr format to new format
arrs = [(k, Miscs.getListIdxs(v)) for k,v in tc.items()]
arrs = OrderedDict(arrs)
d = merge_dict(efs + [arrs])
mytcs.append(d)
self.tcs = mytcs
self.trees = [Tree({'root':k,
'children': [None] * len(c.values()[0][0]),
'commute': ExtFun(k).is_commute()})
for k,c in self.tcs[0].items()]
self.xinfo = xinfo
def _get_mpp(self, typ, terms=None, subset_siz=None):
if terms is None:
terms = self.ss_num
if is_empty(terms):
return []
import dig_polynomials as dp
solver = dp.IeqMPPGen if 'gen' in typ else dp.IeqMPPFixed
solver = solver(terms, self.tcs_num, self.xinfo)
if 'max_then_min' in typ:
mpp_opt = dp.IeqMPP.opt_max_then_min
elif 'min' in typ:
mpp_opt = dp.IeqMPP.opt_min_plus
else:
mpp_opt = dp.IeqMPP.opt_max_plus
solver.mpp_opt = mpp_opt
solver.subset_siz = subset_siz
solver.go()
return solver.sols
def to_z3exp(p, is_real):
"""
Convert a Sage expression to a Z3 expression
Initially implements with a dictionary containing variables
e.g. {x:Real('x')} but the code is longer and more complicated.
This implemention does not require a dictionary pass in.
Todo: cache this function
"""
assert is_sage_expr(p), p
def retval(p):
if p.is_symbol():
z3exp = (z3.Real if is_real else z3.Int)(str(p))
else:
z3exp = (z3.RealVal if is_real else z3.IntVal)(str(p))
if __debug__:
assert z3.is_expr(z3exp), z3exp
return z3exp
try:
oprs = p.operands()
except Exception:
return retval(p)
if is_empty(oprs):
return retval(p)
else:
oprs = [SMT_Z3.to_z3exp(o, is_real) for o in oprs]
z3exp = SMT_Z3._reduce(p.operator(),oprs)
assert z3.is_expr(z3exp), z3exp
return z3exp
def get_gid(ls):
k = [i for i, l in enumerate(ls) if not is_sage_inf(l) and l != 0]
return None if is_empty(k) else tuple(k)
def rfilter_old(self, tcs):
"""
Returns the subset of ps that satisfies all testcases tcs
Examples:
sage: logger.set_level(VLog.DEBUG)
sage: var('y z')
(y, z)
sage: rf = Refine(map(InvIeq,[x^2 >= 0 , x-y >= 7]));rf.rfilter([{x:1,y:0}]); sorted(rf.ps,key=str)
refine:Debug:rfilter(|ps|=2, |tcs|=1)
refine:Debug:rfilter (before 2, after 1, diff 1)
[x^2 >= 0]
sage: rf = Refine(map(InvIeq,[2*x -y >= 0])); rf.rfilter([{y: 14, x: 7}, {y: 13, x: 7}, {y: 6, x: 4}, {y: 1, x: 1}, {y: 2, x: 1}, {y: 5, x: 100}]); sorted(rf.ps,key=str)
refine:Debug:rfilter(|ps|=1, |tcs|=6)
[2*x - y >= 0]
sage: rf = Refine(map(InvIeq,[2*x -y >= 0])); rf.rfilter([{y: 14, x: 7}, {y: 13, x: 7}, {y: 6, x: 4}, {y: 1, x: 1}, {y: 2, x: 1}, {y: 25, x: 9}, {y:25 , x*y: 15, x: 9}]); sorted(rf.ps,key=str)
refine:Debug:rfilter(|ps|=1, |tcs|=7)
refine:Debug:rfilter (before 1, after 0, diff 1)
[]
** This is by design
sage: rf = Refine(map(InvIeq,[x^3 >= 0 , x-y >= 7])); rf.rfilter([{z:1}])
refine:Debug:rfilter(|ps|=2, |tcs|=1)
refine:Debug:rfilter (before 2, after 0, diff 2)
sage: assert(rf.ps == [])
sage: rf = Refine(map(InvMPP,[('lambda x: x>=10',''), ('lambda x,y: max(x,y)>12',''), ('lambda x: x>=10','')])); rf.rfilter([{x:20,y:0},{x:9,y:13}]); sorted(rf.ps,key=str)
refine:Debug:vset (before 3, after 2, diff 1)
refine:Debug:rfilter(|ps|=2, |tcs|=2)
refine:Debug:rfilter (before 2, after 1, diff 1)
['lambda x,y: max(x,y)>12']
sage: rf = Refine(map(InvMPP,[('lambda x: x>=10', ''), ('lambda x,y: max(x,y)>12','')])); rf.rfilter([{x:20,y:0}]); sorted(rf.ps,key=str)
refine:Debug:rfilter(|ps|=2, |tcs|=1)
['lambda x: x>=10', 'lambda x,y: max(x,y)>12']
sage: rf = Refine(map(InvMPP,[('lambda x: x>=10',''), ('lambda x,y: max(x,y)>12','')])); rf.rfilter([]); sorted(rf.ps,key=str)
refine:Debug:rfilter skips (|ps|=2, |tcs|=0)
['lambda x: x>=10', 'lambda x,y: max(x,y)>12']
"""
if is_empty(self.ps) or is_empty(tcs):
logger.debug('rfilter skips (|ps|={}, |tcs|={})'
.format(len(self.ps), len(tcs)))
return None
logger.debug('rfilter(|ps|={}, |tcs|={})'
.format(len(self.ps), len(tcs)))
if not isinstance(self.ps[0],InvExp):
from dig_miscs import Miscs
tcs = Miscs.keys_to_str(tcs)
old_len = len(self.ps)
self.ps = [p for p in self.ps if all(p.seval(tc) for tc in tcs)]
Refine.print_diff('rfilter', old_len, len(self.ps))
def rfilter_old(self, tcs):
"""
Returns the subset of ps that satisfies all testcases tcs
Examples:
sage: logger.set_level(VLog.DEBUG)
sage: var('y z')
(y, z)
sage: rf = Refine(map(InvIeq,[x^2 >= 0 , x-y >= 7]));rf.rfilter([{x:1,y:0}]); sorted(rf.ps,key=str)
refine:Debug:rfilter(|ps|=2, |tcs|=1)
refine:Debug:rfilter (before 2, after 1, diff 1)
[x^2 >= 0]
sage: rf = Refine(map(InvIeq,[2*x -y >= 0])); rf.rfilter([{y: 14, x: 7}, {y: 13, x: 7}, {y: 6, x: 4}, {y: 1, x: 1}, {y: 2, x: 1}, {y: 5, x: 100}]); sorted(rf.ps,key=str)
refine:Debug:rfilter(|ps|=1, |tcs|=6)
[2*x - y >= 0]
sage: rf = Refine(map(InvIeq,[2*x -y >= 0])); rf.rfilter([{y: 14, x: 7}, {y: 13, x: 7}, {y: 6, x: 4}, {y: 1, x: 1}, {y: 2, x: 1}, {y: 25, x: 9}, {y:25 , x*y: 15, x: 9}]); sorted(rf.ps,key=str)
refine:Debug:rfilter(|ps|=1, |tcs|=7)
refine:Debug:rfilter (before 1, after 0, diff 1)
[]
** This is by design
sage: rf = Refine(map(InvIeq,[x^3 >= 0 , x-y >= 7])); rf.rfilter([{z:1}])
refine:Debug:rfilter(|ps|=2, |tcs|=1)
refine:Debug:rfilter (before 2, after 0, diff 2)
sage: assert(rf.ps == [])
sage: rf = Refine(map(InvMPP,[('lambda x: x>=10',''), ('lambda x,y: max(x,y)>12',''), ('lambda x: x>=10','')])); rf.rfilter([{x:20,y:0},{x:9,y:13}]); sorted(rf.ps,key=str)
refine:Debug:vset (before 3, after 2, diff 1)
refine:Debug:rfilter(|ps|=2, |tcs|=2)
refine:Debug:rfilter (before 2, after 1, diff 1)
['lambda x,y: max(x,y)>12']
sage: rf = Refine(map(InvMPP,[('lambda x: x>=10', ''), ('lambda x,y: max(x,y)>12','')])); rf.rfilter([{x:20,y:0}]); sorted(rf.ps,key=str)
refine:Debug:rfilter(|ps|=2, |tcs|=1)
['lambda x: x>=10', 'lambda x,y: max(x,y)>12']
sage: rf = Refine(map(InvMPP,[('lambda x: x>=10',''), ('lambda x,y: max(x,y)>12','')])); rf.rfilter([]); sorted(rf.ps,key=str)
refine:Debug:rfilter skips (|ps|=2, |tcs|=0)
['lambda x: x>=10', 'lambda x,y: max(x,y)>12']
"""
if is_empty(self.ps) or is_empty(tcs):
logger.debug('rfilter skips (|ps|={}, |tcs|={})'
.format(len(self.ps), len(tcs)))
return
logger.debug('rfilter(|ps|={}, |tcs|={})'
.format(len(self.ps), len(tcs)))
if not isinstance(self.ps[0],InvExp):
from dig_miscs import Miscs
tcs = Miscs.keys_to_str(tcs)
old_len = len(self.ps)
self.ps = [p for p in self.ps if all(p.seval(tc) for tc in tcs)]
Refine.print_diff('rfilter', old_len, len(self.ps))
def imply(fs, gs):
"""
Checks if the set f of formulas implies the set f of formulas
sage: var('x y')
(x, y)
sage: SMT_Z3.imply([x-6==0],x*x-36==0)
True
sage: SMT_Z3.imply([x-6==0,x+6==0],x*x-36==0)
True
sage: SMT_Z3.imply([x*x-36==0],x-6==0)
False
sage: SMT_Z3.imply([x-6==0],x-36==0)
False
sage: SMT_Z3.imply(x-7>=0,x>=6)
True
sage: SMT_Z3.imply(x-7>=0,x>=8)
False
sage: SMT_Z3.imply(x-6>=0,x-7>=0)
False
sage: SMT_Z3.imply([x-7>=0,y+5>=0],x+y-3>=0)
False
sage: SMT_Z3.imply([x-7>=0,y+5>=0],x+y-2>=0)
True
sage: SMT_Z3.imply([x-2*y>=0,y-1>=0],x-2>=0)
True
sage: SMT_Z3.imply([],x-2>=0)
False
sage: SMT_Z3.imply([x-7>=0,y+5>=0],x+y-2>=0)
True
sage: SMT_Z3.imply([x^2-9>=0,x>=0],x-3>=0)
True
sage: SMT_Z3.imply([1/2*x**2 - 3/28*x + 1 >= 0],1/20*x**2 - 9/20*x + 1 >= 0)
False
sage: SMT_Z3.imply([1/20*x**2 - 9/20*x + 1 >= 0],1/2*x**2 - 3/28*x + 1 >= 0)
True
sage: SMT_Z3.imply([x-6==0],x*x-36==0)
True
sage: SMT_Z3.imply([x+7>=0,y+5>=0],x*y+36>=0)
False
sage: SMT_Z3.imply([x+7>=0,y+5>=0],x*y+35>=0)
False
sage: SMT_Z3.imply([x+7>=0,y+5>=0],x*y-35>=0)
False
sage: SMT_Z3.imply([x+7>=0],x-8>=0)
False
sage: SMT_Z3.imply([x+7>=0],x+8>=0)
True
sage: SMT_Z3.imply([x+7>=0],x+8.9>=0)
True
sage: SMT_Z3.imply([x>=7,y>=5],x*y>=35)
True
sage: SMT_Z3.imply([x>=-7,y>=-5],x*y>=35)
False
sage: SMT_Z3.imply([x-7>=0,y+5>=0],[x+y-2>=0,x>=2-y])
True
"""
if __debug__:
assert fs is not None, fs
assert gs is not None, gs
if is_empty(fs) or is_empty(gs):
return False #conservative approach
if not is_list(fs): fs = [fs]
if not is_list(gs): gs = [gs]
fs = [SMT_Z3.to_z3exp(f, is_real=True) for f in fs]
gs = [SMT_Z3.to_z3exp(g, is_real=True) for g in gs]
claim = z3.Implies(z3.And(fs), z3.And(gs))
is_proved, _ = vprove(claim)
return is_proved
def get_gid(ls):
k = [i for i,l in enumerate(ls) if not SR(l).is_infinity() and l != 0]
return None if is_empty(k) else tuple(k)
def get_gid(ls):
k = [
i for i, l in enumerate(ls)
if not SR(l).is_infinity() and l != 0
]
return None if is_empty(k) else tuple(k)