TSTP Solution File: LAT253-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LAT253-1 : TPTP v8.1.2. Released v3.1.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n015.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 06:27:57 EDT 2023
% Result : Unsatisfiable 0.21s 0.76s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : LAT253-1 : TPTP v8.1.2. Released v3.1.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.34 % Computer : n015.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Thu Aug 24 04:28:05 EDT 2023
% 0.14/0.34 % CPUTime :
% 0.21/0.76 Command-line arguments: --no-flatten-goal
% 0.21/0.76
% 0.21/0.76 % SZS status Unsatisfiable
% 0.21/0.76
% 0.21/0.76 % SZS output start Proof
% 0.21/0.76 Take the following subset of the input axioms:
% 0.21/0.77 fof(absorption1, axiom, ![X, Y]: meet(X, join(X, Y))=X).
% 0.21/0.77 fof(absorption2, axiom, ![X2, Y2]: join(X2, meet(X2, Y2))=X2).
% 0.21/0.77 fof(associativity_of_join, axiom, ![Z, X2, Y2]: join(join(X2, Y2), Z)=join(X2, join(Y2, Z))).
% 0.21/0.77 fof(associativity_of_meet, axiom, ![X2, Y2, Z2]: meet(meet(X2, Y2), Z2)=meet(X2, meet(Y2, Z2))).
% 0.21/0.77 fof(commutativity_of_join, axiom, ![X2, Y2]: join(X2, Y2)=join(Y2, X2)).
% 0.21/0.77 fof(commutativity_of_meet, axiom, ![X2, Y2]: meet(X2, Y2)=meet(Y2, X2)).
% 0.21/0.77 fof(complement_join, axiom, ![X2]: join(X2, complement(X2))=one).
% 0.21/0.77 fof(complement_meet, axiom, ![X2]: meet(X2, complement(X2))=zero).
% 0.21/0.77 fof(equation_H76, axiom, ![U, X2, Y2, Z2]: meet(X2, join(Y2, meet(Z2, join(Y2, U))))=meet(X2, join(Y2, meet(Z2, join(U, meet(X2, Y2)))))).
% 0.21/0.77 fof(meet_join_complement, axiom, ![X2, Y2]: (meet(X2, Y2)!=zero | (join(X2, Y2)!=one | complement(X2)=Y2))).
% 0.21/0.77 fof(prove_distributivity, negated_conjecture, join(complement(b), complement(a))!=complement(a)).
% 0.21/0.77 fof(prove_distributivity_hypothesis, hypothesis, meet(b, a)=a).
% 0.21/0.77
% 0.21/0.77 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.77 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.77 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.77 fresh(y, y, x1...xn) = u
% 0.21/0.77 C => fresh(s, t, x1...xn) = v
% 0.21/0.77 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.77 variables of u and v.
% 0.21/0.77 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.77 input problem has no model of domain size 1).
% 0.21/0.77
% 0.21/0.77 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.77
% 0.21/0.77 Axiom 1 (commutativity_of_join): join(X, Y) = join(Y, X).
% 0.21/0.77 Axiom 2 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
% 0.21/0.77 Axiom 3 (prove_distributivity_hypothesis): meet(b, a) = a.
% 0.21/0.77 Axiom 4 (complement_join): join(X, complement(X)) = one.
% 0.21/0.77 Axiom 5 (complement_meet): meet(X, complement(X)) = zero.
% 0.21/0.77 Axiom 6 (meet_join_complement): fresh(X, X, Y, Z) = Z.
% 0.21/0.77 Axiom 7 (meet_join_complement): fresh2(X, X, Y, Z) = complement(Y).
% 0.21/0.77 Axiom 8 (absorption2): join(X, meet(X, Y)) = X.
% 0.21/0.77 Axiom 9 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
% 0.21/0.77 Axiom 10 (absorption1): meet(X, join(X, Y)) = X.
% 0.21/0.77 Axiom 11 (associativity_of_meet): meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 0.21/0.77 Axiom 12 (meet_join_complement): fresh2(join(X, Y), one, X, Y) = fresh(meet(X, Y), zero, X, Y).
% 0.21/0.77 Axiom 13 (equation_H76): meet(X, join(Y, meet(Z, join(Y, W)))) = meet(X, join(Y, meet(Z, join(W, meet(X, Y))))).
% 0.21/0.77
% 0.21/0.77 Lemma 14: meet(X, one) = X.
% 0.21/0.77 Proof:
% 0.21/0.77 meet(X, one)
% 0.21/0.77 = { by axiom 4 (complement_join) R->L }
% 0.21/0.77 meet(X, join(X, complement(X)))
% 0.21/0.77 = { by axiom 10 (absorption1) }
% 0.21/0.77 X
% 0.21/0.77
% 0.21/0.77 Lemma 15: join(X, zero) = X.
% 0.21/0.77 Proof:
% 0.21/0.77 join(X, zero)
% 0.21/0.77 = { by axiom 5 (complement_meet) R->L }
% 0.21/0.77 join(X, meet(X, complement(X)))
% 0.21/0.77 = { by axiom 8 (absorption2) }
% 0.21/0.77 X
% 0.21/0.77
% 0.21/0.77 Goal 1 (prove_distributivity): join(complement(b), complement(a)) = complement(a).
% 0.21/0.77 Proof:
% 0.21/0.77 join(complement(b), complement(a))
% 0.21/0.77 = { by axiom 6 (meet_join_complement) R->L }
% 0.21/0.77 fresh(zero, zero, a, join(complement(b), complement(a)))
% 0.21/0.77 = { by axiom 5 (complement_meet) R->L }
% 0.21/0.77 fresh(meet(a, complement(a)), zero, a, join(complement(b), complement(a)))
% 0.21/0.77 = { by lemma 15 R->L }
% 0.21/0.77 fresh(meet(a, join(complement(a), zero)), zero, a, join(complement(b), complement(a)))
% 0.21/0.77 = { by axiom 8 (absorption2) R->L }
% 0.21/0.77 fresh(meet(a, join(complement(a), join(zero, meet(zero, a)))), zero, a, join(complement(b), complement(a)))
% 0.21/0.77 = { by axiom 1 (commutativity_of_join) R->L }
% 0.21/0.77 fresh(meet(a, join(complement(a), join(meet(zero, a), zero))), zero, a, join(complement(b), complement(a)))
% 0.21/0.77 = { by lemma 15 }
% 0.21/0.77 fresh(meet(a, join(complement(a), meet(zero, a))), zero, a, join(complement(b), complement(a)))
% 0.21/0.77 = { by axiom 5 (complement_meet) R->L }
% 0.21/0.77 fresh(meet(a, join(complement(a), meet(meet(b, complement(b)), a))), zero, a, join(complement(b), complement(a)))
% 0.21/0.77 = { by axiom 11 (associativity_of_meet) }
% 0.21/0.77 fresh(meet(a, join(complement(a), meet(b, meet(complement(b), a)))), zero, a, join(complement(b), complement(a)))
% 0.21/0.77 = { by axiom 2 (commutativity_of_meet) R->L }
% 0.21/0.77 fresh(meet(a, join(complement(a), meet(b, meet(a, complement(b))))), zero, a, join(complement(b), complement(a)))
% 0.21/0.77 = { by axiom 11 (associativity_of_meet) R->L }
% 0.21/0.77 fresh(meet(a, join(complement(a), meet(meet(b, a), complement(b)))), zero, a, join(complement(b), complement(a)))
% 0.21/0.77 = { by axiom 3 (prove_distributivity_hypothesis) }
% 0.21/0.77 fresh(meet(a, join(complement(a), meet(a, complement(b)))), zero, a, join(complement(b), complement(a)))
% 0.21/0.77 = { by axiom 2 (commutativity_of_meet) R->L }
% 0.21/0.77 fresh(meet(a, join(complement(a), meet(complement(b), a))), zero, a, join(complement(b), complement(a)))
% 0.21/0.77 = { by axiom 8 (absorption2) R->L }
% 0.21/0.77 fresh(meet(a, join(complement(a), meet(complement(b), join(a, meet(a, complement(a)))))), zero, a, join(complement(b), complement(a)))
% 0.21/0.77 = { by axiom 13 (equation_H76) R->L }
% 0.21/0.77 fresh(meet(a, join(complement(a), meet(complement(b), join(complement(a), a)))), zero, a, join(complement(b), complement(a)))
% 0.21/0.77 = { by axiom 1 (commutativity_of_join) }
% 0.21/0.77 fresh(meet(a, join(complement(a), meet(complement(b), join(a, complement(a))))), zero, a, join(complement(b), complement(a)))
% 0.21/0.77 = { by axiom 4 (complement_join) }
% 0.21/0.77 fresh(meet(a, join(complement(a), meet(complement(b), one))), zero, a, join(complement(b), complement(a)))
% 0.21/0.77 = { by lemma 14 }
% 0.21/0.77 fresh(meet(a, join(complement(a), complement(b))), zero, a, join(complement(b), complement(a)))
% 0.21/0.77 = { by axiom 1 (commutativity_of_join) }
% 0.21/0.77 fresh(meet(a, join(complement(b), complement(a))), zero, a, join(complement(b), complement(a)))
% 0.21/0.77 = { by axiom 12 (meet_join_complement) R->L }
% 0.21/0.77 fresh2(join(a, join(complement(b), complement(a))), one, a, join(complement(b), complement(a)))
% 0.21/0.77 = { by axiom 1 (commutativity_of_join) R->L }
% 0.21/0.77 fresh2(join(a, join(complement(a), complement(b))), one, a, join(complement(b), complement(a)))
% 0.21/0.77 = { by axiom 9 (associativity_of_join) R->L }
% 0.21/0.77 fresh2(join(join(a, complement(a)), complement(b)), one, a, join(complement(b), complement(a)))
% 0.21/0.77 = { by axiom 4 (complement_join) }
% 0.21/0.77 fresh2(join(one, complement(b)), one, a, join(complement(b), complement(a)))
% 0.21/0.77 = { by lemma 14 R->L }
% 0.21/0.77 fresh2(join(one, meet(complement(b), one)), one, a, join(complement(b), complement(a)))
% 0.21/0.77 = { by axiom 2 (commutativity_of_meet) }
% 0.21/0.77 fresh2(join(one, meet(one, complement(b))), one, a, join(complement(b), complement(a)))
% 0.21/0.77 = { by axiom 8 (absorption2) }
% 0.21/0.77 fresh2(one, one, a, join(complement(b), complement(a)))
% 0.21/0.77 = { by axiom 7 (meet_join_complement) }
% 0.21/0.77 complement(a)
% 0.21/0.77 % SZS output end Proof
% 0.21/0.77
% 0.21/0.77 RESULT: Unsatisfiable (the axioms are contradictory).
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