TSTP Solution File: NUM024-1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : NUM024-1 : 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 : n023.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 11:55:02 EDT 2023
% Result : Unsatisfiable 4.14s 0.89s
% Output : Proof 4.34s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : NUM024-1 : TPTP v8.1.2. Bugfixed v4.0.0.
% 0.07/0.13 % 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 : n023.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 : Fri Aug 25 12:15:42 EDT 2023
% 0.13/0.34 % CPUTime :
% 4.14/0.89 Command-line arguments: --flip-ordering --lhs-weight 1 --depth-weight 60 --distributivity-heuristic
% 4.14/0.89
% 4.14/0.89 % SZS status Unsatisfiable
% 4.14/0.89
% 4.34/0.89 % SZS output start Proof
% 4.34/0.89 Take the following subset of the input axioms:
% 4.34/0.90 fof(adding_zero, axiom, ![A]: equalish(add(A, n0), A)).
% 4.34/0.90 fof(commutativity_of_plus, axiom, ![B, A3]: equalish(add(A3, B), add(B, A3))).
% 4.34/0.90 fof(impossible_a_is_less_than_itself, hypothesis, less(a, a)).
% 4.34/0.90 fof(less_lemma, axiom, ![A2, B2]: (~less(A2, B2) | equalish(add(successor(predecessor_of_1st_minus_2nd(B2, A2)), A2), B2))).
% 4.34/0.90 fof(plus_substitution, axiom, ![C, B2, A2_2]: (~equalish(add(A2_2, B2), add(C, B2)) | equalish(A2_2, C))).
% 4.34/0.90 fof(prove_a_contradiction, negated_conjecture, ![A3]: ~equalish(successor(A3), n0)).
% 4.34/0.90 fof(symmetry, axiom, ![X, Y]: (~equalish(X, Y) | equalish(Y, X))).
% 4.34/0.90 fof(transitivity, axiom, ![Z, X2, Y2]: (~equalish(X2, Y2) | (~equalish(Y2, Z) | equalish(X2, Z)))).
% 4.34/0.90
% 4.34/0.90 Now clausify the problem and encode Horn clauses using encoding 3 of
% 4.34/0.90 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 4.34/0.90 We repeatedly replace C & s=t => u=v by the two clauses:
% 4.34/0.90 fresh(y, y, x1...xn) = u
% 4.34/0.90 C => fresh(s, t, x1...xn) = v
% 4.34/0.90 where fresh is a fresh function symbol and x1..xn are the free
% 4.34/0.90 variables of u and v.
% 4.34/0.90 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 4.34/0.90 input problem has no model of domain size 1).
% 4.34/0.90
% 4.34/0.90 The encoding turns the above axioms into the following unit equations and goals:
% 4.34/0.90
% 4.34/0.90 Axiom 1 (impossible_a_is_less_than_itself): less(a, a) = true2.
% 4.34/0.90 Axiom 2 (adding_zero): equalish(add(X, n0), X) = true2.
% 4.34/0.90 Axiom 3 (plus_substitution): fresh10(X, X, Y, Z) = true2.
% 4.34/0.90 Axiom 4 (less_lemma): fresh9(X, X, Y, Z) = true2.
% 4.34/0.90 Axiom 5 (symmetry): fresh5(X, X, Y, Z) = true2.
% 4.34/0.90 Axiom 6 (transitivity): fresh3(X, X, Y, Z) = true2.
% 4.34/0.90 Axiom 7 (transitivity): fresh4(X, X, Y, Z, W) = equalish(Y, W).
% 4.34/0.90 Axiom 8 (commutativity_of_plus): equalish(add(X, Y), add(Y, X)) = true2.
% 4.34/0.90 Axiom 9 (symmetry): fresh5(equalish(X, Y), true2, X, Y) = equalish(Y, X).
% 4.34/0.90 Axiom 10 (less_lemma): fresh9(less(X, Y), true2, X, Y) = equalish(add(successor(predecessor_of_1st_minus_2nd(Y, X)), X), Y).
% 4.34/0.90 Axiom 11 (transitivity): fresh4(equalish(X, Y), true2, Z, X, Y) = fresh3(equalish(Z, X), true2, Z, Y).
% 4.34/0.90 Axiom 12 (plus_substitution): fresh10(equalish(add(X, Y), add(Z, Y)), true2, X, Z) = equalish(X, Z).
% 4.34/0.90
% 4.34/0.90 Goal 1 (prove_a_contradiction): equalish(successor(X), n0) = true2.
% 4.34/0.90 The goal is true when:
% 4.34/0.90 X = predecessor_of_1st_minus_2nd(a, a)
% 4.34/0.90
% 4.34/0.90 Proof:
% 4.34/0.90 equalish(successor(predecessor_of_1st_minus_2nd(a, a)), n0)
% 4.34/0.90 = { by axiom 12 (plus_substitution) R->L }
% 4.34/0.90 fresh10(equalish(add(successor(predecessor_of_1st_minus_2nd(a, a)), a), add(n0, a)), true2, successor(predecessor_of_1st_minus_2nd(a, a)), n0)
% 4.34/0.90 = { by axiom 7 (transitivity) R->L }
% 4.34/0.90 fresh10(fresh4(true2, true2, add(successor(predecessor_of_1st_minus_2nd(a, a)), a), a, add(n0, a)), true2, successor(predecessor_of_1st_minus_2nd(a, a)), n0)
% 4.34/0.90 = { by axiom 6 (transitivity) R->L }
% 4.34/0.90 fresh10(fresh4(fresh3(true2, true2, a, add(n0, a)), true2, add(successor(predecessor_of_1st_minus_2nd(a, a)), a), a, add(n0, a)), true2, successor(predecessor_of_1st_minus_2nd(a, a)), n0)
% 4.34/0.90 = { by axiom 5 (symmetry) R->L }
% 4.34/0.90 fresh10(fresh4(fresh3(fresh5(true2, true2, add(a, n0), a), true2, a, add(n0, a)), true2, add(successor(predecessor_of_1st_minus_2nd(a, a)), a), a, add(n0, a)), true2, successor(predecessor_of_1st_minus_2nd(a, a)), n0)
% 4.34/0.90 = { by axiom 2 (adding_zero) R->L }
% 4.34/0.90 fresh10(fresh4(fresh3(fresh5(equalish(add(a, n0), a), true2, add(a, n0), a), true2, a, add(n0, a)), true2, add(successor(predecessor_of_1st_minus_2nd(a, a)), a), a, add(n0, a)), true2, successor(predecessor_of_1st_minus_2nd(a, a)), n0)
% 4.34/0.90 = { by axiom 9 (symmetry) }
% 4.34/0.90 fresh10(fresh4(fresh3(equalish(a, add(a, n0)), true2, a, add(n0, a)), true2, add(successor(predecessor_of_1st_minus_2nd(a, a)), a), a, add(n0, a)), true2, successor(predecessor_of_1st_minus_2nd(a, a)), n0)
% 4.34/0.90 = { by axiom 11 (transitivity) R->L }
% 4.34/0.90 fresh10(fresh4(fresh4(equalish(add(a, n0), add(n0, a)), true2, a, add(a, n0), add(n0, a)), true2, add(successor(predecessor_of_1st_minus_2nd(a, a)), a), a, add(n0, a)), true2, successor(predecessor_of_1st_minus_2nd(a, a)), n0)
% 4.34/0.90 = { by axiom 8 (commutativity_of_plus) }
% 4.34/0.90 fresh10(fresh4(fresh4(true2, true2, a, add(a, n0), add(n0, a)), true2, add(successor(predecessor_of_1st_minus_2nd(a, a)), a), a, add(n0, a)), true2, successor(predecessor_of_1st_minus_2nd(a, a)), n0)
% 4.34/0.90 = { by axiom 7 (transitivity) }
% 4.34/0.90 fresh10(fresh4(equalish(a, add(n0, a)), true2, add(successor(predecessor_of_1st_minus_2nd(a, a)), a), a, add(n0, a)), true2, successor(predecessor_of_1st_minus_2nd(a, a)), n0)
% 4.34/0.90 = { by axiom 11 (transitivity) }
% 4.34/0.90 fresh10(fresh3(equalish(add(successor(predecessor_of_1st_minus_2nd(a, a)), a), a), true2, add(successor(predecessor_of_1st_minus_2nd(a, a)), a), add(n0, a)), true2, successor(predecessor_of_1st_minus_2nd(a, a)), n0)
% 4.34/0.90 = { by axiom 10 (less_lemma) R->L }
% 4.34/0.90 fresh10(fresh3(fresh9(less(a, a), true2, a, a), true2, add(successor(predecessor_of_1st_minus_2nd(a, a)), a), add(n0, a)), true2, successor(predecessor_of_1st_minus_2nd(a, a)), n0)
% 4.34/0.90 = { by axiom 1 (impossible_a_is_less_than_itself) }
% 4.34/0.90 fresh10(fresh3(fresh9(true2, true2, a, a), true2, add(successor(predecessor_of_1st_minus_2nd(a, a)), a), add(n0, a)), true2, successor(predecessor_of_1st_minus_2nd(a, a)), n0)
% 4.34/0.90 = { by axiom 4 (less_lemma) }
% 4.34/0.90 fresh10(fresh3(true2, true2, add(successor(predecessor_of_1st_minus_2nd(a, a)), a), add(n0, a)), true2, successor(predecessor_of_1st_minus_2nd(a, a)), n0)
% 4.34/0.90 = { by axiom 6 (transitivity) }
% 4.34/0.90 fresh10(true2, true2, successor(predecessor_of_1st_minus_2nd(a, a)), n0)
% 4.34/0.90 = { by axiom 3 (plus_substitution) }
% 4.34/0.90 true2
% 4.34/0.90 % SZS output end Proof
% 4.34/0.90
% 4.34/0.90 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------