TSTP Solution File: NUM020-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : NUM020-1 : TPTP v8.1.2. Bugfixed v4.0.0.
% 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 : n001.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 11:55:01 EDT 2023
% Result : Unsatisfiable 0.21s 0.43s
% Output : Proof 0.21s
% 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 : NUM020-1 : TPTP v8.1.2. Bugfixed v4.0.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.34 % Computer : n001.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 : Fri Aug 25 17:52:59 EDT 2023
% 0.14/0.34 % CPUTime :
% 0.21/0.43 Command-line arguments: --flip-ordering --lhs-weight 1 --depth-weight 60 --distributivity-heuristic
% 0.21/0.43
% 0.21/0.43 % SZS status Unsatisfiable
% 0.21/0.43
% 0.21/0.43 % SZS output start Proof
% 0.21/0.43 Take the following subset of the input axioms:
% 0.21/0.43 fof(adding_zero, axiom, ![A]: equalish(add(A, n0), A)).
% 0.21/0.43 fof(addition, axiom, ![B, A3]: equalish(add(A3, successor(B)), successor(add(A3, B)))).
% 0.21/0.43 fof(deny_addition_lemma, negated_conjecture, ~equalish(add(a, successor(n0)), successor(a))).
% 0.21/0.43 fof(successor_substitution, axiom, ![A2, B2]: (~equalish(A2, B2) | equalish(successor(A2), successor(B2)))).
% 0.21/0.43 fof(transitivity, axiom, ![X, Y, Z]: (~equalish(X, Y) | (~equalish(Y, Z) | equalish(X, Z)))).
% 0.21/0.43
% 0.21/0.43 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.43 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.43 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.43 fresh(y, y, x1...xn) = u
% 0.21/0.43 C => fresh(s, t, x1...xn) = v
% 0.21/0.43 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.43 variables of u and v.
% 0.21/0.43 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.43 input problem has no model of domain size 1).
% 0.21/0.44
% 0.21/0.44 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.44
% 0.21/0.44 Axiom 1 (adding_zero): equalish(add(X, n0), X) = true2.
% 0.21/0.44 Axiom 2 (transitivity): fresh(X, X, Y, Z) = true2.
% 0.21/0.44 Axiom 3 (successor_substitution): fresh5(X, X, Y, Z) = true2.
% 0.21/0.44 Axiom 4 (transitivity): fresh2(X, X, Y, Z, W) = equalish(Y, W).
% 0.21/0.44 Axiom 5 (successor_substitution): fresh5(equalish(X, Y), true2, X, Y) = equalish(successor(X), successor(Y)).
% 0.21/0.44 Axiom 6 (transitivity): fresh2(equalish(X, Y), true2, Z, X, Y) = fresh(equalish(Z, X), true2, Z, Y).
% 0.21/0.44 Axiom 7 (addition): equalish(add(X, successor(Y)), successor(add(X, Y))) = true2.
% 0.21/0.44
% 0.21/0.44 Goal 1 (deny_addition_lemma): equalish(add(a, successor(n0)), successor(a)) = true2.
% 0.21/0.44 Proof:
% 0.21/0.44 equalish(add(a, successor(n0)), successor(a))
% 0.21/0.44 = { by axiom 4 (transitivity) R->L }
% 0.21/0.44 fresh2(true2, true2, add(a, successor(n0)), successor(add(a, n0)), successor(a))
% 0.21/0.44 = { by axiom 3 (successor_substitution) R->L }
% 0.21/0.44 fresh2(fresh5(true2, true2, add(a, n0), a), true2, add(a, successor(n0)), successor(add(a, n0)), successor(a))
% 0.21/0.44 = { by axiom 1 (adding_zero) R->L }
% 0.21/0.44 fresh2(fresh5(equalish(add(a, n0), a), true2, add(a, n0), a), true2, add(a, successor(n0)), successor(add(a, n0)), successor(a))
% 0.21/0.44 = { by axiom 5 (successor_substitution) }
% 0.21/0.44 fresh2(equalish(successor(add(a, n0)), successor(a)), true2, add(a, successor(n0)), successor(add(a, n0)), successor(a))
% 0.21/0.44 = { by axiom 6 (transitivity) }
% 0.21/0.44 fresh(equalish(add(a, successor(n0)), successor(add(a, n0))), true2, add(a, successor(n0)), successor(a))
% 0.21/0.44 = { by axiom 7 (addition) }
% 0.21/0.44 fresh(true2, true2, add(a, successor(n0)), successor(a))
% 0.21/0.44 = { by axiom 2 (transitivity) }
% 0.21/0.44 true2
% 0.21/0.44 % SZS output end Proof
% 0.21/0.44
% 0.21/0.44 RESULT: Unsatisfiable (the axioms are contradictory).
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