def eval_lambda(f, d, is_formula=True):
"""
Evaluates lambda function f
Examples:
sage: assert InvMPP.eval_lambda('lambda x,y: max(x - 13,-3) >= y', {'x':11,'y':100}) == False
sage: assert InvMPP.eval_lambda('lambda x,y: x+y == 5', {'x':2,'y':3,'d':7})
sage: assert InvMPP.eval_lambda('lambda x,y: x+y == 6', {'x':2,'y':3,'d':7}) == False
sage: assert InvMPP.eval_lambda('lambda x,y: x+y', {'x':2,'y':3,'d':7}, is_formula=False) == 5
sage: assert InvMPP.eval_lambda('lambda x,y: x+y == 10 or x + y == 5', {'x':2,'y':3,'d':7})
sage: assert InvMPP.eval_lambda('lambda x,y: x+y == 1 or x + y == 2', {'x':2,'y':3,'d':7}) == False
"""
if __debug__:
assert is_str(f) and 'lambda' in f, f
assert is_dict(d), d
assert all(is_str(k) for k in d), d.keys()
f = sage_eval(f)
vs = f.func_code.co_varnames
assert set(vs) <= set(d.keys()), (vs, d.keys())
#if d has more keys than variables in f then remove those extra keys
d = dict([(k, d[k]) for k in vs])
rs = f(**d)
return bool(rs) if is_formula else rs
def slice_defs(prop, defs, assumes_state, assumes_trans):
"""
Return a new (potentially empty) def dictionary from the old one
consisting of only necessary variable definitions to prove property
"""
if __debug__:
assert is_dict(defs), defs
assert is_list(assumes_state), assumes_state
assert is_list(assumes_trans), assumes_trans
fs = [prop] + assumes_state + assumes_trans
fs = [f for f in fs if is_expr(f)]
vs = [get_vars(f) for f in fs]
vs = [cur(v_) if is_pre(v_) else v_ for v_ in vflatten(vs)]
vs = vset(vs, fhash)
vs_ = [Prog.get_influence_vs(v, defs, []) for v in vs]
vs = vset(vflatten(vs + vs_), fhash)
a_defs = OrderedDict()
for v in vs:
k = fhash(v)
if k in defs:
a_defs[k] = defs[k]
return a_defs
def eval_lambda(f, d, is_formula=True):
"""
Evaluates lambda function f
Examples:
sage: assert InvMPP.eval_lambda('lambda x,y: max(x - 13,-3) >= y', {'x':11,'y':100}) == False
sage: assert InvMPP.eval_lambda('lambda x,y: x+y == 5', {'x':2,'y':3,'d':7})
sage: assert InvMPP.eval_lambda('lambda x,y: x+y == 6', {'x':2,'y':3,'d':7}) == False
sage: assert InvMPP.eval_lambda('lambda x,y: x+y', {'x':2,'y':3,'d':7}, is_formula=False) == 5
sage: assert InvMPP.eval_lambda('lambda x,y: x+y == 10 or x + y == 5', {'x':2,'y':3,'d':7})
sage: assert InvMPP.eval_lambda('lambda x,y: x+y == 1 or x + y == 2', {'x':2,'y':3,'d':7}) == False
"""
if __debug__:
assert is_str(f) and 'lambda' in f, f
assert is_dict(d), d
assert all(is_str(k) for k in d), d.keys()
f = sage_eval(f)
vs = f.func_code.co_varnames
assert set(vs) <= set(d.keys()), (vs,d.keys())
#if d has more keys than variables in f then remove those extra keys
d=dict([(k,d[k]) for k in vs])
rs = f(**d)
return bool(rs) if is_formula else rs
def __init__(self, init_conds, defs, input_vars, assumes):
"""
This class models a program using
1. initial condition
2. transition (definitions of updated variables)
3. assumptions
Input variables:
- init_cond: list of initial conditions
e.g. [Block == Off,Reset == On,WaterPres == wp_init_val,
Overridden == False,SafetyInjection == On,Pressure == TooLow]
- defs: a dictionary consisting variables being updated by
the transition
- input_vars: variables that are INDEPDENT.
SCR programs don't have these, because input (monitored) vars
are dependent due to OIA
- assumes: list of assumptions
Two types of assumptions:
(1) state assumes: those for each *state*
e.g.
And(0 <= WaterPres,WaterPres < 2000): WaterPres is in range 0,2000
at any state
(2) trans assumes: those for each *transition*
e.g.
One Input Assumption asserts only 1 var can changed at a time
or
And(pre(WaterPres) - 10 <= WaterPres,
WaterPres <= pre(WaterPres) + 10)
"""
if __debug__:
assert is_list(init_conds) and \
all(is_state(c) for c in init_conds), init_conds
assert is_dict(defs) and \
all(is_expr(v) for v in defs.values()), defs
assert is_list(input_vars) and \
all(is_expr_var(v) for v in input_vars), input_vars
assert is_list(assumes) and \
all(is_expr(a) for a in assumes), assumes
self.defs = defs
self.init_conds = init_conds
self.input_vars = input_vars
self.assumes_state = []
self.assumes_trans = []
for a in assumes:
Prog.append_f(a, self.assumes_state, self.assumes_trans)
#Known invariants (lemmas). Use add_inv() to add an inv as lemma
self.invs_state = []
self.invs_trans = []
def __init__(self, p):
"""
Ex:
('lambda A, B, A0: (A[A0]) + (-7*B[2*A0]) + (-3*A0) == 0',
[{'A0': 2}, {'A0': 0}, {'A0': 1}])
"""
if __debug__:
assert (is_tuple(p) and len(p) == 2 and is_str(p[0])
and all(s in p[0] for s in ['lambda', '[', ']'])
and is_list(p[1]) and all(is_dict(v) for v in p[1])), p
super(InvFlatArray, self).__init__(p[0].strip())
self.idx_info = p[1]
def __init__(self, p):
"""
Ex:
('lambda A, B, A0: (A[A0]) + (-7*B[2*A0]) + (-3*A0) == 0',
[{'A0': 2}, {'A0': 0}, {'A0': 1}])
"""
if __debug__:
assert (is_tuple(p) and len(p)==2 and
is_str(p[0]) and
all(s in p[0] for s in ['lambda','[' ,']']) and
is_list(p[1]) and all(is_dict(v) for v in p[1])), p
super(InvFlatArray,self).__init__(p[0].strip())
self.idx_info = p[1]
def __init__(self, terms, tcs, xinfo):
assert is_dict(xinfo), xinfo
logger.info('*** {} ***'.format(self.__class__.__name__))
self.terms = terms
self.tcs = tcs
self.xinfo = xinfo
self.do_refine = True
self.tcs_extra = []
self._sols = []
logger.debug('|terms|={}, |tcs|={}'.format(len(self.terms),
len(self.tcs)))
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 eval_lambda(f, idx_info, tc):
"""
Evaluate array expression p, e.g. p: A[i,j,k]=B[2i+3j+k]
if idx_info is specified then only test p on the idxs from idx_info
Assumes:
the first array in lambda is the pivot
lambda A,B,i,j,k: ... => i,j,k belong to A
inv = 'lambda B,C,D,i,j: B[i][j]=C[D[2i+3]]'
returns true if inv is true on tc
Examples:
sage: var('a,b,c,i,j')
(a, b, c, i, j)
sage: InvArray.eval_lambda('lambda a,b,c,i,j: a[i][j]==2*b[i]+c[j]', None, {'a':[[4,-5],[20,11]],'b':[1,9],'c':[2,-7]})
True
sage: InvArray.eval_lambda('lambda c,a,b,xor,i: c[i] == xor(a[i],b[i])', None, {'a': [147, 156, 184, 76], 'b': [51, 26, 247, 189], 'c': [160, 334, 79, 281]})
False
sage: InvArray.eval_lambda('lambda c,a,b,xor,i1: c[i1] == xor(a[i1],b[i1])', None, {'a': [147, 156, 184, 76], 'b': [51, 26, 247, 189], 'c': [160, 134, 79, 241]})
True
sage: InvArray.eval_lambda('lambda rvu, t, rvu1, rvu0: (rvu[rvu0][rvu1]) + (-t[4*rvu0 + rvu1]) == 0', None, {'t': [28, 131, 11, 85, 133, 46, 179, 20, 227, 148, 225, 197, 38, 221, 221, 126], 'rvu': [[28, 131, 11, 85], [133, 46, 179, 20], [227, 148, 225, 197], [38, 221, 221, 126]]})
True
#The following illustrate the use of idxVals,
#i.e. p is only true under certain array rages
sage: InvArray.eval_lambda('lambda st, rvu, st0, st1: (-st[st0][st1]) + (rvu[4*st0 + st1]) == 0', None, tc = {'rvu': [28, 131, 11, 85, 193, 124, 103, 215, 66, 26, 68, 54, 176, 102, 15, 237], 'st': [[28, 131, 11, 85, 133, 46, 179, 20, 227, 148, 225, 197, 38, 221, 221, 126], [193, 124, 103, 215, 106, 229, 162, 168, 166, 78, 144, 234, 199, 254, 152, 250], [66, 26, 68, 54, 206, 16, 155, 248, 231, 198, 240, 43, 208, 205, 213, 26], [176, 102, 15, 237, 49, 141, 213, 97, 137, 155, 50, 243, 112, 51, 124, 107]]})
False
sage: InvArray.eval_lambda('lambda st, rvu, st0, st1: (-st[st0][st1]) + (rvu[4*st0 + st1]) == 0', idx_info = [{'st0': 0, 'st1': 0}, {'st0': 0, 'st1': 1}, {'st0': 2, 'st1': 2}, {'st0': 2, 'st1': 3}, {'st0': 3, 'st1': 0}, {'st0': 3, 'st1': 1}, {'st0': 3, 'st1': 2}, {'st0': 3, 'st1': 3}, {'st0': 0, 'st1': 2}, {'st0': 0, 'st1': 3}, {'st0': 1, 'st1': 0}, {'st0': 1, 'st1': 1}, {'st0': 1, 'st1': 2}, {'st0': 1, 'st1': 3}, {'st0': 2, 'st1': 0}, {'st0': 2, 'st1': 1}], tc = {'rvu': [28, 131, 11, 85, 193, 124, 103, 215, 66, 26, 68, 54, 176, 102, 15, 237], 'st': [[28, 131, 11, 85, 133, 46, 179, 20, 227, 148, 225, 197, 38, 221, 221, 126], [193, 124, 103, 215, 106, 229, 162, 168, 166, 78, 144, 234, 199, 254, 152, 250], [66, 26, 68, 54, 206, 16, 155, 248, 231, 198, 240, 43, 208, 205, 213, 26], [176, 102, 15, 237, 49, 141, 213, 97, 137, 155, 50, 243, 112, 51, 124, 107]]})
True
"""
"""
Note: sage_eval vs eval
sage_eval works on str of the format 'lambda x,y: 2*x+y'
whereas eval works on str of the format 2*x+y directly (no lambda)
Also, the result of sage_eval can apply on dicts whose keys are str
e.g. f(**{'x':2,'y':3})
whereas the result of eval applies on dict whose keys are variables
e.g. f(**{x:2,y:3})
"""
if __debug__:
assert is_str(f) and 'lambda' in f, f
assert (idx_info is None
or is_list(idx_info) and all(is_dict(v)
for v in idx_info)), indx_info
assert is_dict(tc), tc
assert all(is_str(k) for k in tc), tc.keys()
f = sage_eval(f)
vs = f.func_code.co_varnames
arrs = [v for v in vs if v in tc] #A,B
extfuns = [v for v in vs if v in ExtFun.efdict]
idxStr = [v for v in vs if v not in arrs + extfuns] #i,j,k
d_tc = dict([(v, tc[v]) for v in arrs])
d_extfun = dict([(v, ExtFun(v).get_fun()) for v in extfuns])
d_ = merge_dict([d_tc, d_extfun])
if idx_info is None: #obtain idxsVals from the pivot array
pivotContents = tc[arrs[0]]
idxVals = [idx for idx, _ in Miscs.travel(pivotContents)]
idx_info = [dict(zip(idxStr, idxV)) for idxV in idxVals]
ds = [merge_dict([d_, idx_info_]) for idx_info_ in idx_info]
try:
return all(f(**d) for d in ds)
except IndexError:
return False
except TypeError:
return False
except NameError as msg:
logger.warn(msg)
return False
def eval_lambda(f, idx_info, tc):
"""
Evaluate array expression p, e.g. p: A[i,j,k]=B[2i+3j+k]
if idx_info is specified then only test p on the idxs from idx_info
Assumes:
the first array in lambda is the pivot
lambda A,B,i,j,k: ... => i,j,k belong to A
inv = 'lambda B,C,D,i,j: B[i][j]=C[D[2i+3]]'
returns true if inv is true on tc
Examples:
sage: var('a,b,c,i,j')
(a, b, c, i, j)
sage: InvArray.eval_lambda('lambda a,b,c,i,j: a[i][j]==2*b[i]+c[j]', None, {'a':[[4,-5],[20,11]],'b':[1,9],'c':[2,-7]})
True
sage: InvArray.eval_lambda('lambda c,a,b,xor,i: c[i] == xor(a[i],b[i])', None, {'a': [147, 156, 184, 76], 'b': [51, 26, 247, 189], 'c': [160, 334, 79, 281]})
False
sage: InvArray.eval_lambda('lambda c,a,b,xor,i1: c[i1] == xor(a[i1],b[i1])', None, {'a': [147, 156, 184, 76], 'b': [51, 26, 247, 189], 'c': [160, 134, 79, 241]})
True
sage: InvArray.eval_lambda('lambda rvu, t, rvu1, rvu0: (rvu[rvu0][rvu1]) + (-t[4*rvu0 + rvu1]) == 0', None, {'t': [28, 131, 11, 85, 133, 46, 179, 20, 227, 148, 225, 197, 38, 221, 221, 126], 'rvu': [[28, 131, 11, 85], [133, 46, 179, 20], [227, 148, 225, 197], [38, 221, 221, 126]]})
True
#The following illustrate the use of idxVals,
#i.e. p is only true under certain array rages
sage: InvArray.eval_lambda('lambda st, rvu, st0, st1: (-st[st0][st1]) + (rvu[4*st0 + st1]) == 0', None, tc = {'rvu': [28, 131, 11, 85, 193, 124, 103, 215, 66, 26, 68, 54, 176, 102, 15, 237], 'st': [[28, 131, 11, 85, 133, 46, 179, 20, 227, 148, 225, 197, 38, 221, 221, 126], [193, 124, 103, 215, 106, 229, 162, 168, 166, 78, 144, 234, 199, 254, 152, 250], [66, 26, 68, 54, 206, 16, 155, 248, 231, 198, 240, 43, 208, 205, 213, 26], [176, 102, 15, 237, 49, 141, 213, 97, 137, 155, 50, 243, 112, 51, 124, 107]]})
False
sage: InvArray.eval_lambda('lambda st, rvu, st0, st1: (-st[st0][st1]) + (rvu[4*st0 + st1]) == 0', idx_info = [{'st0': 0, 'st1': 0}, {'st0': 0, 'st1': 1}, {'st0': 2, 'st1': 2}, {'st0': 2, 'st1': 3}, {'st0': 3, 'st1': 0}, {'st0': 3, 'st1': 1}, {'st0': 3, 'st1': 2}, {'st0': 3, 'st1': 3}, {'st0': 0, 'st1': 2}, {'st0': 0, 'st1': 3}, {'st0': 1, 'st1': 0}, {'st0': 1, 'st1': 1}, {'st0': 1, 'st1': 2}, {'st0': 1, 'st1': 3}, {'st0': 2, 'st1': 0}, {'st0': 2, 'st1': 1}], tc = {'rvu': [28, 131, 11, 85, 193, 124, 103, 215, 66, 26, 68, 54, 176, 102, 15, 237], 'st': [[28, 131, 11, 85, 133, 46, 179, 20, 227, 148, 225, 197, 38, 221, 221, 126], [193, 124, 103, 215, 106, 229, 162, 168, 166, 78, 144, 234, 199, 254, 152, 250], [66, 26, 68, 54, 206, 16, 155, 248, 231, 198, 240, 43, 208, 205, 213, 26], [176, 102, 15, 237, 49, 141, 213, 97, 137, 155, 50, 243, 112, 51, 124, 107]]})
True
"""
"""
Note: sage_eval vs eval
sage_eval works on str of the format 'lambda x,y: 2*x+y'
whereas eval works on str of the format 2*x+y directly (no lambda)
Also, the result of sage_eval can apply on dicts whose keys are str
e.g. f(**{'x':2,'y':3})
whereas the result of eval applies on dict whose keys are variables
e.g. f(**{x:2,y:3})
"""
if __debug__:
assert is_str(f) and 'lambda' in f, f
assert (idx_info is None or
is_list(idx_info) and all(is_dict(v) for v in idx_info)), indx_info
assert is_dict(tc), tc
assert all(is_str(k) for k in tc), tc.keys()
f = sage_eval(f)
vs = f.func_code.co_varnames
arrs = [v for v in vs if v in tc] #A,B
extfuns = [v for v in vs if v in ExtFun.efdict]
idxStr = [v for v in vs if v not in arrs+extfuns] #i,j,k
d_tc = dict([(v,tc[v]) for v in arrs])
d_extfun= dict([(v,ExtFun(v).get_fun()) for v in extfuns])
d_ = merge_dict([d_tc,d_extfun])
if idx_info is None: #obtain idxsVals from the pivot array
pivotContents = tc[arrs[0]]
idxVals = [idx for idx,_ in Miscs.travel(pivotContents)]
idx_info = [dict(zip(idxStr,idxV)) for idxV in idxVals]
ds = [merge_dict([d_, idx_info_]) for idx_info_ in idx_info]
try:
return all(f(**d) for d in ds)
except IndexError:
return False
except TypeError:
return False
except NameError as msg:
logger.warn(msg)
return False
def compute_new_trace(d, ainfo, tsinfo):
"""
Convert array elements into new variables and generate traces from these.
Examples:
sage: var('A B')
(A, B)
sage: ainfo = {}
sage: tc = FlatArray.compute_new_trace({A: [0,1,2], B: [[0, 1], [2, 3]]}, ainfo, {})
sage: sorted(tc.items(),key= lambda (x,_) : str(x))
[(A_0, 0), (A_1, 1), (A_2, 2), (B_0_0, 0), (B_0_1, 1), (B_1_0, 2), (B_1_1, 3)]
sage: sorted(ainfo.items(), key=lambda (x,_): str(x))
[(A_0, {'idx_': [(A0, 0)], 'name': 'A'}),
(A_1, {'idx_': [(A0, 1)], 'name': 'A'}),
(A_2, {'idx_': [(A0, 2)], 'name': 'A'}),
(B_0_0, {'idx_': [(B0, 0), (B1, 0)], 'name': 'B'}),
(B_0_1, {'idx_': [(B0, 0), (B1, 1)], 'name': 'B'}),
(B_1_0, {'idx_': [(B0, 1), (B1, 0)], 'name': 'B'}),
(B_1_1, {'idx_': [(B0, 1), (B1, 1)], 'name': 'B'})]
sage: ainfo = {}
sage: tc = FlatArray.compute_new_trace({'A':[['a','b'],['c','d'],['e','f',['z','w']]], \
'B':[1,2,[7,8]],'C':[100]}, ainfo, {})
sage: sorted(tc.items(),key= lambda (x,_) : str(x))
[(A_0_0, 'a'), (A_0_1, 'b'), (A_1_0, 'c'), (A_1_1, 'd'),
(A_2_0, 'e'), (A_2_1, 'f'), (A_2_2_0, 'z'), (A_2_2_1, 'w'),
(B_0, 1), (B_1, 2), (B_2_0, 7), (B_2_1, 8), (C_0, 100)]
sage: sorted(ainfo.items(), key=lambda (x,_): str(x))
[(A_0_0, {'idx_': [(A0, 0), (A1, 0)], 'name': 'A'}),
(A_0_1, {'idx_': [(A0, 0), (A1, 1)], 'name': 'A'}),
(A_1_0, {'idx_': [(A0, 1), (A1, 0)], 'name': 'A'}),
(A_1_1, {'idx_': [(A0, 1), (A1, 1)], 'name': 'A'}),
(A_2_0, {'idx_': [(A0, 2), (A1, 0)], 'name': 'A'}),
(A_2_1, {'idx_': [(A0, 2), (A1, 1)], 'name': 'A'}),
(A_2_2_0, {'idx_': [(A0, 2), (A1, 2), (A2, 0)], 'name': 'A'}),
(A_2_2_1, {'idx_': [(A0, 2), (A1, 2), (A2, 1)], 'name': 'A'}),
(B_0, {'idx_': [(B0, 0)], 'name': 'B'}),
(B_1, {'idx_': [(B0, 1)], 'name': 'B'}),
(B_2_0, {'idx_': [(B0, 2), (B1, 0)], 'name': 'B'}),
(B_2_1, {'idx_': [(B0, 2), (B1, 1)], 'name': 'B'}),
(C_0, {'idx_': [(C0, 0)], 'name': 'C'})]
"""
def compute_traces(aname, acontents, ainfo, tsinfo):
vi = Miscs.travel(acontents)
vals = Miscs.getVals(vi)
idxs = Miscs.getIdxs(vi)
aname = str(aname)
newvars = [var(aname + '_' + list_str(idx, '_')) for idx in idxs]
if aname not in tsinfo:
tsinfo[aname] = newvars
else:
assert tsinfo[aname] == newvars
dVals = dict(zip(newvars,vals)) #{A_0_0_1:'w'}
for nv,idx in zip(newvars,idxs):
if nv not in ainfo:
idx_ = zip([var('{}{}'.format(aname,li))
for li in srange(len(idx))],idx)
ainfo[nv]={'name':aname, 'idx_':idx_}
return dVals
if __debug__:
assert is_dict(d), d
assert is_dict(ainfo), d
tcs = [compute_traces(k,v,ainfo,tsinfo) for k,v in d.iteritems()]
tcs = merge_dict(tcs)
return tcs
def get_influence_vs(v, defs, rs):
"""
Return a list of variables that influences v
(i.e. the definition of v depends on these variables)
>>> from z3 import Bools, BoolVal
>>> from scr_miscs import mk_OIA
>>> s,t = Bools('s t')
>>> x,y,z = Bools('x y z')
>>> vs = [x,y,z]
>>> o = mk_OIA(vs)
>>> vv = [o,o,And(o,s)]
>>> vs2 = [t]
>>> vv2 = [BoolVal(True)]
>>> vs_k = map(fhash,vs + vs2)
>>> vs_v =vv + vv2
>>> defs = OrderedDict(zip(vs_k,vs_v))
>>> print Prog.get_influence_vs(x,defs,rs=[])
[s, y, z]
#>>> print Prog.get_influence_vs(x,defs,assumes=[x==s],rs=[])
#[s, y, z]
#>>> print Prog.get_influence_vs(x,defs,assumes=[y==t],rs=[])
#[s, y, z, t]
"""
if __debug__:
assert is_expr_var(v), v
assert is_dict(defs), defs
assert is_list(rs), rs
if is_pre(v):
v = cur(v)
#return if already in the result set
if expr_member(v, rs):
return rs
try:
vs = get_vars(defs[fhash(v)])
#print vs
except KeyError:
return rs
rs = rs + [v]
#convert v_pre to v
vs = [cur(v_) if is_pre(v_) else v_ for v_ in vs]
vs = vset(vs, fhash)
for v_ in vs:
rs_ = Prog.get_influence_vs(v_, defs, rs)
rs = rs + rs_
rs = rs + vs
rs = vset(rs, fhash)
#remove myself
v_idx = map(fhash, rs).index(fhash(v))
rs = rs[:v_idx] + rs[v_idx + 1:]
return sorted(rs, key=str)