TSTP Solution File: GRP001-5 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : GRP001-5 : TPTP v8.1.2. Released v1.0.0.
% 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 : n027.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 01:16:28 EDT 2023
% Result : Unsatisfiable 0.19s 0.41s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : GRP001-5 : TPTP v8.1.2. Released v1.0.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n027.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Mon Aug 28 21:40:52 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.19/0.41 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
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% 0.19/0.41 % SZS status Unsatisfiable
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% 0.19/0.42 % SZS output start Proof
% 0.19/0.42 Take the following subset of the input axioms:
% 0.19/0.42 fof(a_times_b_is_c, hypothesis, product(a, b, c)).
% 0.19/0.42 fof(associativity1, axiom, ![X, Y, U, Z, V, W]: (~product(X, Y, U) | (~product(Y, Z, V) | (~product(U, Z, W) | product(X, V, W))))).
% 0.19/0.42 fof(associativity2, axiom, ![X2, Y2, U2, Z2, V2, W2]: (~product(X2, Y2, U2) | (~product(Y2, Z2, V2) | (~product(X2, V2, W2) | product(U2, Z2, W2))))).
% 0.19/0.42 fof(left_identity, axiom, ![X2]: product(identity, X2, X2)).
% 0.19/0.42 fof(prove_b_times_a_is_c, negated_conjecture, ~product(b, a, c)).
% 0.19/0.42 fof(right_identity, axiom, ![X2]: product(X2, identity, X2)).
% 0.19/0.42 fof(square_element, hypothesis, ![X2]: product(X2, X2, identity)).
% 0.19/0.42
% 0.19/0.42 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.42 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.42 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.42 fresh(y, y, x1...xn) = u
% 0.19/0.42 C => fresh(s, t, x1...xn) = v
% 0.19/0.42 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.42 variables of u and v.
% 0.19/0.42 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.42 input problem has no model of domain size 1).
% 0.19/0.42
% 0.19/0.42 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.42
% 0.19/0.42 Axiom 1 (square_element): product(X, X, identity) = true.
% 0.19/0.42 Axiom 2 (right_identity): product(X, identity, X) = true.
% 0.19/0.42 Axiom 3 (a_times_b_is_c): product(a, b, c) = true.
% 0.19/0.42 Axiom 4 (left_identity): product(identity, X, X) = true.
% 0.19/0.42 Axiom 5 (associativity1): fresh6(X, X, Y, Z, W) = true.
% 0.19/0.42 Axiom 6 (associativity2): fresh4(X, X, Y, Z, W) = true.
% 0.19/0.42 Axiom 7 (associativity2): fresh(X, X, Y, Z, W, V, U) = product(W, V, U).
% 0.19/0.42 Axiom 8 (associativity1): fresh2(X, X, Y, Z, W, V, U) = product(Y, V, U).
% 0.19/0.42 Axiom 9 (associativity1): fresh5(X, X, Y, Z, W, V, U, T) = fresh6(product(Y, Z, W), true, Y, U, T).
% 0.19/0.42 Axiom 10 (associativity2): fresh3(X, X, Y, Z, W, V, U, T) = fresh4(product(Y, Z, W), true, W, V, T).
% 0.19/0.42 Axiom 11 (associativity1): fresh5(product(X, Y, Z), true, W, V, X, Y, U, Z) = fresh2(product(V, Y, U), true, W, V, X, U, Z).
% 0.19/0.42 Axiom 12 (associativity2): fresh3(product(X, Y, Z), true, W, X, V, Y, Z, U) = fresh(product(W, Z, U), true, W, X, V, Y, U).
% 0.19/0.42
% 0.19/0.42 Lemma 13: fresh4(product(X, Y, Z), true, Z, Y, X) = product(Z, Y, X).
% 0.19/0.42 Proof:
% 0.19/0.42 fresh4(product(X, Y, Z), true, Z, Y, X)
% 0.19/0.42 = { by axiom 10 (associativity2) R->L }
% 0.19/0.42 fresh3(true, true, X, Y, Z, Y, identity, X)
% 0.19/0.42 = { by axiom 1 (square_element) R->L }
% 0.19/0.42 fresh3(product(Y, Y, identity), true, X, Y, Z, Y, identity, X)
% 0.19/0.42 = { by axiom 12 (associativity2) }
% 0.19/0.42 fresh(product(X, identity, X), true, X, Y, Z, Y, X)
% 0.19/0.42 = { by axiom 2 (right_identity) }
% 0.19/0.42 fresh(true, true, X, Y, Z, Y, X)
% 0.19/0.42 = { by axiom 7 (associativity2) }
% 0.19/0.42 product(Z, Y, X)
% 0.19/0.42
% 0.19/0.42 Goal 1 (prove_b_times_a_is_c): product(b, a, c) = true.
% 0.19/0.42 Proof:
% 0.19/0.42 product(b, a, c)
% 0.19/0.42 = { by lemma 13 R->L }
% 0.19/0.42 fresh4(product(c, a, b), true, b, a, c)
% 0.19/0.42 = { by axiom 8 (associativity1) R->L }
% 0.19/0.42 fresh4(fresh2(true, true, c, c, identity, a, b), true, b, a, c)
% 0.19/0.42 = { by axiom 6 (associativity2) R->L }
% 0.19/0.42 fresh4(fresh2(fresh4(true, true, c, b, a), true, c, c, identity, a, b), true, b, a, c)
% 0.19/0.42 = { by axiom 3 (a_times_b_is_c) R->L }
% 0.19/0.42 fresh4(fresh2(fresh4(product(a, b, c), true, c, b, a), true, c, c, identity, a, b), true, b, a, c)
% 0.19/0.42 = { by lemma 13 }
% 0.19/0.42 fresh4(fresh2(product(c, b, a), true, c, c, identity, a, b), true, b, a, c)
% 0.19/0.42 = { by axiom 11 (associativity1) R->L }
% 0.19/0.42 fresh4(fresh5(product(identity, b, b), true, c, c, identity, b, a, b), true, b, a, c)
% 0.19/0.42 = { by axiom 4 (left_identity) }
% 0.19/0.42 fresh4(fresh5(true, true, c, c, identity, b, a, b), true, b, a, c)
% 0.19/0.42 = { by axiom 9 (associativity1) }
% 0.19/0.42 fresh4(fresh6(product(c, c, identity), true, c, a, b), true, b, a, c)
% 0.19/0.42 = { by axiom 1 (square_element) }
% 0.19/0.42 fresh4(fresh6(true, true, c, a, b), true, b, a, c)
% 0.19/0.42 = { by axiom 5 (associativity1) }
% 0.19/0.42 fresh4(true, true, b, a, c)
% 0.19/0.42 = { by axiom 6 (associativity2) }
% 0.19/0.42 true
% 0.19/0.42 % SZS output end Proof
% 0.19/0.42
% 0.19/0.42 RESULT: Unsatisfiable (the axioms are contradictory).
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