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Fix completeness issue in SVSP
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parent
4179345956
commit
bb2024d254
1 changed files with 207 additions and 165 deletions
372
vsp.py
372
vsp.py
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@ -3,7 +3,7 @@ Check to see if the model has the variable
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sharing property.
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"""
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from itertools import product, chain, combinations
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from typing import List, Optional, Set, Tuple
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from typing import List, Generator, Optional, Set, Tuple
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from common import set_to_str
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from model import (
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Model, model_closure, ModelFunction, ModelValue
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@ -150,14 +150,15 @@ L: {set_to_str(self.L)}
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def powerset_minus_empty(s):
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return chain.from_iterable(combinations(s, r) for r in range(1, len(s) + 1))
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def find_k1_k2(model, impfunction: ModelFunction,
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negation_defined: bool) -> List[Tuple[Set[ModelValue], Set[ModelValue]]]:
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def find_k1_k2(model: Model, impfunction: ModelFunction,
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negation_defined: bool) -> Generator[Tuple[Set[ModelValue], Set[ModelValue]], None, None]:
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"""
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Returns a list of possible subalgebra pairs (K1, K2)
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for SVSP. This is less efficient than the VSP version
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due to interaction with the L and U sets in SVSP.
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"""
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assert model.ordering is not None, "Expected ordering table in model"
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result = []
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top = model.ordering.top()
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bottom = model.ordering.bottom()
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@ -165,39 +166,47 @@ def find_k1_k2(model, impfunction: ModelFunction,
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# x -> y does not take a designiated value
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I: List[Tuple[ModelValue, ModelValue]] = []
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for (x, y) in product(model.designated_values, model.designated_values):
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for (x, y) in product(model.carrier_set, model.carrier_set):
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if impfunction(x, y) not in model.designated_values:
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I.append((x, y))
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# Find the subalgebras which falsify implication
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for xys in I:
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Is = powerset_minus_empty(I)
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xi = xys[0]
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# Find the subalgebras which falsify implication
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for xys in Is:
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xs = {xy[0] for xy in xys}
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# Discard ({⊥} ∪ A', B) subalgebras
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if bottom is not None and xi == bottom:
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if bottom is not None and bottom in xs:
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continue
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# Discard ({⊤} ∪ A', B) subalgebras when negation is defined
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if top is not None and negation_defined and xi == top:
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if top is not None and negation_defined and top in xs:
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continue
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yi = xys[1]
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ys = {xy[1] for xy in xys}
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# Discard (A, {⊤} ∪ B') subalgebras
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if top is not None and yi == top:
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if top is not None and top in ys:
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continue
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# Discard (A, {⊥} ∪ B') subalgebras when negation is defined
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if bottom is not None and negation_defined and yi == bottom:
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if bottom is not None and negation_defined and bottom in ys:
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continue
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# Discard ({a} ∪ A', {b} ∪ B') subalgebras when a <= b
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if model.ordering.is_lt(xi, yi):
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continue
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order_dependent = False
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for (xi, yi) in product(xs, ys):
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# Discard ({a} ∪ A', {b} ∪ B') subalgebras when a <= b
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if model.ordering.is_lt(xi, yi):
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order_dependent = True
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break
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# Discard ({a} ∪ A', {b} ∪ B') subalgebras when b <= a and negation is defined
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if negation_defined and model.ordering.is_lt(yi, xi):
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order_dependent = True
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break
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# Discard ({a} ∪ A', {b} ∪ B') subalgebras when b <= a and negation is defined
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if negation_defined and model.ordering.is_lt(yi, xi):
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if order_dependent:
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continue
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# Compute the left closure of the set containing xi under all the operations
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@ -236,32 +245,66 @@ def find_k1_k2(model, impfunction: ModelFunction,
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break
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if falsified:
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result.append((carrier_set_left, carrier_set_right))
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yield (carrier_set_left, carrier_set_right)
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return result
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def find_candidate_u_l(
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model: Model, impfn: ModelFunction, negfn: Optional[ModelFunction],
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K1: Set[ModelValue], K2: Set[ModelValue]) -> Generator[Tuple[Set[ModelValue], Set[ModelValue]], None, None]:
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def find_candidate_u_l(model: Model, impfn: ModelFunction, negfn: Optional[ModelFunction]) -> List[Tuple[Set[ModelValue], Set[ModelValue]]]:
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result: List[Tuple[Set[ModelValue], Set[ModelValue]]] = []
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# Compute I the set of tuples (x, y) where
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# x -> y does not take a designiated value
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I: List[Tuple[ModelValue, ModelValue]] = []
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if negfn is None:
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# NOTE: K2 ∩ U = ∅ if ∀x(x → x) ∈ T
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# NOTE: K1 ∩ L = ∅ if ∀x(x → x) ∈ T
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for (x, y) in product(model.carrier_set - K2, model.carrier_set - K1):
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if impfn(x, y) not in model.designated_values:
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I.append((x, y))
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else:
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# NOTE: K1, K2, L, and U are pairwise distinct
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CmK1uK2 = model.carrier_set - (K1 | K2)
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for (x, y) in product(CmK1uK2, CmK1uK2):
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if impfn(x, y) not in model.designated_values:
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I.append((x, y))
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Is = powerset_minus_empty(I)
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F = model.carrier_set - model.designated_values
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Us = powerset_minus_empty(model.carrier_set)
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Ls = powerset_minus_empty(model.carrier_set)
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for (U, L) in product(Us, Ls):
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has_double_negation_eq = False
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if negfn is not None:
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has_double_negation_eq = True
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for x in model.carrier_set:
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if negfn(negfn(x)) != x:
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has_double_negation_eq = False
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break
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for ULs in Is:
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unsat = False
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U = set(U)
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L = set(L)
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U = {UL[0] for UL in ULs}
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L = {UL[1] for UL in ULs}
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# U and L are distinct
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if U.intersection(L):
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continue
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if has_double_negation_eq:
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# NOTE: U is the negation image of L, that is, U = {¬x | x ∈ L}, if ∀x(x = ¬¬x).
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U2 = {negfn(x) for x in L}
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if U != U2:
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continue
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yield (U, L)
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LFi = F.intersection(L)
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# Required property: ∀x ∈ U, y ∈ L(x → y ∈ L ∩ F)
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for (x, y) in product(U, L):
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# Required property: ∀x ∈ U, y ∈ L(x → y ∈ L ∩ F)
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if impfn(x, y) not in LFi:
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unsat = True
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break
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if unsat:
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continue
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# Required Property: ∀x ∈ L, y ∈ U(x → y ∈ U)
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for (x, y) in product(L, U):
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if impfn(x, y) not in U:
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# Required Property: ∀x ∈ L, y ∈ U(x → y ∈ U)
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if impfn(y, x) not in U:
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unsat = True
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break
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@ -288,9 +331,8 @@ def find_candidate_u_l(model: Model, impfn: ModelFunction, negfn: Optional[Model
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continue
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# Passed all required properties
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result.append((U, L))
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yield (U, L)
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return result
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def has_svsp(model: Model, impfn: ModelFunction,
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conjfn: Optional[ModelFunction],
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@ -309,157 +351,157 @@ def has_svsp(model: Model, impfn: ModelFunction,
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starops = [conjfn, disjfn]
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K1K2s = find_k1_k2(model, impfn, negfn is not None)
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ULs = find_candidate_u_l(model, impfn, negfn)
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candidates = ((k1, k2, u, l) for (k1, k2), (u, l) in product(K1K2s, ULs))
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for K1, K2, U, L in candidates:
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unsat = False
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K1Uu = K1 | U
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K1Lu = K1 | L
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K1LuFi = K1Lu.intersection(F) # (K1 ∪ L) ∩ F
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K2Uu = K2 | U
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K2Lu = K2 | L
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K2LuFi = K2Lu.intersection(F) # (K2 ∪ L) ∩ F
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for K1, K2 in K1K2s:
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ULs = find_candidate_u_l(model, impfn, negfn, K1, K2)
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for U, L in ULs:
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unsat = False
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K1Uu = K1 | U
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K1Lu = K1 | L
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K1LuFi = K1Lu.intersection(F) # (K1 ∪ L) ∩ F
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K2Uu = K2 | U
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K2Lu = K2 | L
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K2LuFi = K2Lu.intersection(F) # (K2 ∪ L) ∩ F
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# (6)
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for x, y in product(K1, U):
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# b) x → y ∈ K1 ∪ U
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if impfn(x, y) not in K1Uu:
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unsat = True
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break
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# (6)
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for x, y in product(K1, U):
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# b) x → y ∈ K1 ∪ U
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if impfn(x, y) not in K1Uu:
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unsat = True
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break
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# c) y → x ∈ K1 ∪ L
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if impfn(y, x) not in K1Lu:
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unsat = True
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break
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# c) y → x ∈ K1 ∪ L
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if impfn(y, x) not in K1Lu:
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unsat = True
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break
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# a) x ∗ y, y ∗ x, y ∗ z ∈ K1 ∪ U
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for z in U:
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for op in starops:
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if op is not None:
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if op(x, y) not in K1Uu:
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unsat = True
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break
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if op(y, x) not in K1Uu:
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unsat = True
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break
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if op(y, z) not in K1Uu:
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unsat = True
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break
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if unsat:
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break
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if unsat:
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# Verification for these set of matrices failed
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break
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if unsat:
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# Move onto the next candidates K1, K2, U, L
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continue
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# (7)
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for x, y in product(K1, L):
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# b) x → y ∈ (K1 ∪ L) ∩ F
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if impfn(x, y) not in K1LuFi:
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unsat = True
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break
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# c) y → x ∈ K1 ∪ U
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if impfn(y, x) not in K1Uu:
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unsat = True
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break
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# a) x ∗ y, y ∗ x, y ∗ z ∈ K1 ∪ L
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for z in L:
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for op in starops:
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if op is not None:
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if op(x, y) not in K1Lu:
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unsat = True
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break
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if op(y, x) not in K1Lu:
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unsat = True
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break
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if op(y, z) not in K1Lu:
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unsat = True
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break
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if unsat:
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break
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# a) x ∗ y, y ∗ x, y ∗ z ∈ K1 ∪ U
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for z in U:
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for op in starops:
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if op is not None:
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if op(x, y) not in K1Uu:
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unsat = True
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break
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if op(y, x) not in K1Uu:
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unsat = True
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break
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if op(y, z) not in K1Uu:
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unsat = True
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break
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if unsat:
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break
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if unsat:
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# Verification for these set of matrices failed
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break
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continue
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if unsat:
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# Move onto the next candidates K1, K2, U, L
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continue
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# (8)
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for x, y in product(K2, U):
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# b) x → y ∈ K2 ∪ U
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if impfn(x, y) not in K2Uu:
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unsat = True
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break
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# (7)
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for x, y in product(K1, L):
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# b) x → y ∈ (K1 ∪ L) ∩ F
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if impfn(x, y) not in K1LuFi:
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unsat = True
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break
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# c) y → x ∈ (K2 ∪ L) ∩ F
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if impfn(y, x) not in K2LuFi:
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unsat = True
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break
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# c) y → x ∈ K1 ∪ U
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if impfn(y, x) not in K1Uu:
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unsat = True
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break
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# a) x ∗ y, y ∗ x, y ∗ z ∈ K2 ∪ U
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for z in U:
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for op in starops:
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if op is not None:
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if op(x, y) not in K2Uu:
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unsat = True
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break
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if op(y, x) not in K2Uu:
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unsat = True
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break
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if op(y, z) not in K2Uu:
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unsat = True
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break
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if unsat:
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break
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# a) x ∗ y, y ∗ x, y ∗ z ∈ K1 ∪ L
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for z in L:
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for op in starops:
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if op is not None:
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if op(x, y) not in K1Lu:
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unsat = True
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break
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if op(y, x) not in K1Lu:
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unsat = True
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break
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if op(y, z) not in K1Lu:
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unsat = True
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break
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if unsat:
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break
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if unsat:
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break
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continue
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if unsat:
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continue
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# (9)
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for x, y in product(K2, L):
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# b) x → y ∈ K2 ∪ L
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if impfn(x, y) not in K2Lu:
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unsat = True
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break
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# (8)
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for x, y in product(K2, U):
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# b) x → y ∈ K2 ∪ U
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if impfn(x, y) not in K2Uu:
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unsat = True
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break
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# c) y → x ∈ K2 ∪ U
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if impfn(y, x) not in K2Uu:
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unsat = True
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break
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# c) y → x ∈ (K2 ∪ L) ∩ F
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if impfn(y, x) not in K2LuFi:
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unsat = True
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break
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# a) x ∗ y, y ∗ x, y ∗ z ∈ K2 ∪ L
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for z in L:
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for op in starops:
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if op is not None:
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if op(x, y) not in K2Lu:
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unsat = True
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break
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if op(y, x) not in K2Lu:
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unsat = True
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break
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if op(y, z) not in K2Lu:
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unsat = True
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break
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if unsat:
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break
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# a) x ∗ y, y ∗ x, y ∗ z ∈ K2 ∪ U
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for z in U:
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for op in starops:
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if op is not None:
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if op(x, y) not in K2Uu:
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unsat = True
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break
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if op(y, x) not in K2Uu:
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unsat = True
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break
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if op(y, z) not in K2Uu:
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unsat = True
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break
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if unsat:
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break
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if unsat:
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break
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if unsat:
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continue
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# (9)
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for x, y in product(K2, L):
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# b) x → y ∈ K2 ∪ L
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if impfn(x, y) not in K2Lu:
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unsat = True
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break
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# c) y → x ∈ K2 ∪ U
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if impfn(y, x) not in K2Uu:
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unsat = True
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break
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# a) x ∗ y, y ∗ x, y ∗ z ∈ K2 ∪ L
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for z in L:
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for op in starops:
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if op is not None:
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if op(x, y) not in K2Lu:
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unsat = True
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break
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if op(y, x) not in K2Lu:
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unsat = True
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break
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if op(y, z) not in K2Lu:
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unsat = True
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break
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if unsat:
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break
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if unsat:
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break
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if not unsat:
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return SVSP_Result(True, model.name, K1, K2, U, L)
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if not unsat:
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return SVSP_Result(True, model.name, K1, K2, U, L)
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return SVSP_Result(False, model.name)
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