TSTP Solution File: SET020-7 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SET020-7 : 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 : 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 15:30:39 EDT 2023
% Result : Unsatisfiable 0.20s 0.72s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SET020-7 : TPTP v8.1.2. Bugfixed v2.1.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n001.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 : Sat Aug 26 10:25:30 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.20/0.72 Command-line arguments: --flip-ordering --lhs-weight 1 --depth-weight 60 --distributivity-heuristic
% 0.20/0.72
% 0.20/0.72 % SZS status Unsatisfiable
% 0.20/0.72
% 0.20/0.72 % SZS output start Proof
% 0.20/0.72 Take the following subset of the input axioms:
% 0.20/0.72 fof(cartesian_product4, axiom, ![X, Y, Z]: (~member(Z, cross_product(X, Y)) | ordered_pair(first(Z), second(Z))=Z)).
% 0.20/0.72 fof(corollary_1_to_cartesian_product, axiom, ![U, V, X2, Y2]: (~member(ordered_pair(U, V), cross_product(X2, Y2)) | member(U, universal_class))).
% 0.20/0.72 fof(ordered_pair_determines_components1, axiom, ![W, X2, Y2, Z2]: (ordered_pair(W, X2)!=ordered_pair(Y2, Z2) | (~member(W, universal_class) | W=Y2))).
% 0.20/0.72 fof(prove_unique_1st_and_2nd_in_pair_of_sets1_1, negated_conjecture, member(ordered_pair(u, v), cross_product(universal_class, universal_class))).
% 0.20/0.72 fof(prove_unique_1st_and_2nd_in_pair_of_sets1_2, negated_conjecture, first(ordered_pair(u, v))!=u).
% 0.20/0.72
% 0.20/0.72 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.72 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.72 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.72 fresh(y, y, x1...xn) = u
% 0.20/0.72 C => fresh(s, t, x1...xn) = v
% 0.20/0.72 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.72 variables of u and v.
% 0.20/0.72 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.72 input problem has no model of domain size 1).
% 0.20/0.72
% 0.20/0.72 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.72
% 0.20/0.72 Axiom 1 (corollary_1_to_cartesian_product): fresh86(X, X, Y) = true2.
% 0.20/0.72 Axiom 2 (cartesian_product4): fresh19(X, X, Y) = Y.
% 0.20/0.72 Axiom 3 (ordered_pair_determines_components1): fresh4(X, X, Y, Z) = Z.
% 0.20/0.72 Axiom 4 (prove_unique_1st_and_2nd_in_pair_of_sets1_1): member(ordered_pair(u, v), cross_product(universal_class, universal_class)) = true2.
% 0.20/0.72 Axiom 5 (ordered_pair_determines_components1): fresh5(X, X, Y, Z, W, V) = Y.
% 0.20/0.72 Axiom 6 (cartesian_product4): fresh19(member(X, cross_product(Y, Z)), true2, X) = ordered_pair(first(X), second(X)).
% 0.20/0.72 Axiom 7 (ordered_pair_determines_components1): fresh5(member(X, universal_class), true2, X, Y, Z, W) = fresh4(ordered_pair(X, Y), ordered_pair(Z, W), X, Z).
% 0.20/0.72 Axiom 8 (corollary_1_to_cartesian_product): fresh86(member(ordered_pair(X, Y), cross_product(Z, W)), true2, X) = member(X, universal_class).
% 0.20/0.72
% 0.20/0.72 Goal 1 (prove_unique_1st_and_2nd_in_pair_of_sets1_2): first(ordered_pair(u, v)) = u.
% 0.20/0.72 Proof:
% 0.20/0.72 first(ordered_pair(u, v))
% 0.20/0.72 = { by axiom 3 (ordered_pair_determines_components1) R->L }
% 0.20/0.72 fresh4(ordered_pair(u, v), ordered_pair(u, v), u, first(ordered_pair(u, v)))
% 0.20/0.72 = { by axiom 2 (cartesian_product4) R->L }
% 0.20/0.72 fresh4(ordered_pair(u, v), fresh19(true2, true2, ordered_pair(u, v)), u, first(ordered_pair(u, v)))
% 0.20/0.72 = { by axiom 4 (prove_unique_1st_and_2nd_in_pair_of_sets1_1) R->L }
% 0.20/0.72 fresh4(ordered_pair(u, v), fresh19(member(ordered_pair(u, v), cross_product(universal_class, universal_class)), true2, ordered_pair(u, v)), u, first(ordered_pair(u, v)))
% 0.20/0.72 = { by axiom 6 (cartesian_product4) }
% 0.20/0.72 fresh4(ordered_pair(u, v), ordered_pair(first(ordered_pair(u, v)), second(ordered_pair(u, v))), u, first(ordered_pair(u, v)))
% 0.20/0.72 = { by axiom 7 (ordered_pair_determines_components1) R->L }
% 0.20/0.72 fresh5(member(u, universal_class), true2, u, v, first(ordered_pair(u, v)), second(ordered_pair(u, v)))
% 0.20/0.72 = { by axiom 8 (corollary_1_to_cartesian_product) R->L }
% 0.20/0.72 fresh5(fresh86(member(ordered_pair(u, v), cross_product(universal_class, universal_class)), true2, u), true2, u, v, first(ordered_pair(u, v)), second(ordered_pair(u, v)))
% 0.20/0.72 = { by axiom 4 (prove_unique_1st_and_2nd_in_pair_of_sets1_1) }
% 0.20/0.72 fresh5(fresh86(true2, true2, u), true2, u, v, first(ordered_pair(u, v)), second(ordered_pair(u, v)))
% 0.20/0.72 = { by axiom 1 (corollary_1_to_cartesian_product) }
% 0.20/0.72 fresh5(true2, true2, u, v, first(ordered_pair(u, v)), second(ordered_pair(u, v)))
% 0.20/0.72 = { by axiom 5 (ordered_pair_determines_components1) }
% 0.20/0.72 u
% 0.20/0.72 % SZS output end Proof
% 0.20/0.72
% 0.20/0.72 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------