TSTP Solution File: SET118-6 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : SET118-6 : TPTP v8.1.2. Bugfixed v2.1.0.
% 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 : n009.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 15:31:22 EDT 2023
% Result : Unsatisfiable 0.22s 0.54s
% Output : Proof 0.22s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.13/0.13 % Problem : SET118-6 : TPTP v8.1.2. Bugfixed v2.1.0.
% 0.13/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.15/0.36 % Computer : n009.cluster.edu
% 0.15/0.36 % Model : x86_64 x86_64
% 0.15/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.36 % Memory : 8042.1875MB
% 0.15/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.36 % CPULimit : 300
% 0.15/0.36 % WCLimit : 300
% 0.15/0.36 % DateTime : Sat Aug 26 12:39:35 EDT 2023
% 0.15/0.36 % CPUTime :
% 0.22/0.54 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.22/0.54
% 0.22/0.54 % SZS status Unsatisfiable
% 0.22/0.54
% 0.22/0.54 % SZS output start Proof
% 0.22/0.54 Take the following subset of the input axioms:
% 0.22/0.54 fof(cartesian_product4, axiom, ![X, Y, Z]: (~member(Z, cross_product(X, Y)) | ordered_pair(first(Z), second(Z))=Z)).
% 0.22/0.54 fof(ordered_pair, axiom, ![X2, Y2]: unordered_pair(singleton(X2), unordered_pair(X2, singleton(Y2)))=ordered_pair(X2, Y2)).
% 0.22/0.54 fof(prove_corollary_2_to_ordered_pairs_are_sets_1, negated_conjecture, member(x, cross_product(universal_class, universal_class))).
% 0.22/0.54 fof(prove_corollary_2_to_ordered_pairs_are_sets_2, negated_conjecture, ~member(x, universal_class)).
% 0.22/0.54 fof(unordered_pairs_in_universal, axiom, ![X2, Y2]: member(unordered_pair(X2, Y2), universal_class)).
% 0.22/0.54
% 0.22/0.54 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.22/0.54 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.22/0.54 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.22/0.54 fresh(y, y, x1...xn) = u
% 0.22/0.54 C => fresh(s, t, x1...xn) = v
% 0.22/0.54 where fresh is a fresh function symbol and x1..xn are the free
% 0.22/0.54 variables of u and v.
% 0.22/0.54 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.22/0.54 input problem has no model of domain size 1).
% 0.22/0.54
% 0.22/0.54 The encoding turns the above axioms into the following unit equations and goals:
% 0.22/0.54
% 0.22/0.54 Axiom 1 (cartesian_product4): fresh2(X, X, Y) = Y.
% 0.22/0.54 Axiom 2 (prove_corollary_2_to_ordered_pairs_are_sets_1): member(x, cross_product(universal_class, universal_class)) = true2.
% 0.22/0.54 Axiom 3 (unordered_pairs_in_universal): member(unordered_pair(X, Y), universal_class) = true2.
% 0.22/0.54 Axiom 4 (ordered_pair): unordered_pair(singleton(X), unordered_pair(X, singleton(Y))) = ordered_pair(X, Y).
% 0.22/0.54 Axiom 5 (cartesian_product4): fresh2(member(X, cross_product(Y, Z)), true2, X) = ordered_pair(first(X), second(X)).
% 0.22/0.54
% 0.22/0.54 Goal 1 (prove_corollary_2_to_ordered_pairs_are_sets_2): member(x, universal_class) = true2.
% 0.22/0.54 Proof:
% 0.22/0.54 member(x, universal_class)
% 0.22/0.54 = { by axiom 1 (cartesian_product4) R->L }
% 0.22/0.54 member(fresh2(true2, true2, x), universal_class)
% 0.22/0.54 = { by axiom 2 (prove_corollary_2_to_ordered_pairs_are_sets_1) R->L }
% 0.22/0.54 member(fresh2(member(x, cross_product(universal_class, universal_class)), true2, x), universal_class)
% 0.22/0.54 = { by axiom 5 (cartesian_product4) }
% 0.22/0.54 member(ordered_pair(first(x), second(x)), universal_class)
% 0.22/0.54 = { by axiom 4 (ordered_pair) R->L }
% 0.22/0.54 member(unordered_pair(singleton(first(x)), unordered_pair(first(x), singleton(second(x)))), universal_class)
% 0.22/0.54 = { by axiom 3 (unordered_pairs_in_universal) }
% 0.22/0.54 true2
% 0.22/0.54 % SZS output end Proof
% 0.22/0.54
% 0.22/0.54 RESULT: Unsatisfiable (the axioms are contradictory).
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