TSTP Solution File: LCL678+1.001 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : LCL678+1.001 : TPTP v8.1.2. Released 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 : n026.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 08:20:07 EDT 2023
% Result : Theorem 0.13s 0.39s
% Output : Proof 0.13s
% 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 : LCL678+1.001 : TPTP v8.1.2. Released 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
% 0.13/0.35 % Computer : n026.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Thu Aug 24 20:38:02 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.13/0.39 Command-line arguments: --ground-connectedness --complete-subsets
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% 0.13/0.39 % SZS status Theorem
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% 0.13/0.39 % SZS output start Proof
% 0.13/0.39 Take the following subset of the input axioms:
% 0.13/0.39 fof(main, conjecture, ~?[X]: ~($false | ~![Y]: (~r1(X, Y) | ($false | ~![X2]: (~r1(Y, X2) | (![Y3]: (~r1(X2, Y3) | p1(Y3)) | ~![Y2]: (~r1(X2, Y2) | p1(Y2)))))))).
% 0.13/0.39 fof(reflexivity, axiom, ![X2]: r1(X2, X2)).
% 0.13/0.39
% 0.13/0.39 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.13/0.39 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.13/0.39 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.13/0.39 fresh(y, y, x1...xn) = u
% 0.13/0.39 C => fresh(s, t, x1...xn) = v
% 0.13/0.39 where fresh is a fresh function symbol and x1..xn are the free
% 0.13/0.39 variables of u and v.
% 0.13/0.39 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.13/0.39 input problem has no model of domain size 1).
% 0.13/0.39
% 0.13/0.39 The encoding turns the above axioms into the following unit equations and goals:
% 0.13/0.39
% 0.13/0.39 Axiom 1 (reflexivity): r1(X, X) = true2.
% 0.13/0.39 Axiom 2 (main_1): fresh6(X, X, Y) = true2.
% 0.13/0.39 Axiom 3 (main_2): fresh3(X, X, Y) = true2.
% 0.13/0.39 Axiom 4 (main_2): fresh4(X, X, Y, Z) = p1(Z).
% 0.13/0.39 Axiom 5 (main_1): fresh6(r1(x2, X), true2, X) = r1(x(X), y(X)).
% 0.13/0.39 Axiom 6 (main_2): fresh4(r1(x(X), Y), true2, X, Y) = fresh3(r1(x2, X), true2, Y).
% 0.13/0.39
% 0.13/0.39 Goal 1 (main_3): tuple(r1(x2, X), p1(y(X))) = tuple(true2, true2).
% 0.13/0.39 The goal is true when:
% 0.13/0.39 X = x2
% 0.13/0.39
% 0.13/0.39 Proof:
% 0.13/0.39 tuple(r1(x2, x2), p1(y(x2)))
% 0.13/0.39 = { by axiom 4 (main_2) R->L }
% 0.13/0.39 tuple(r1(x2, x2), fresh4(true2, true2, x2, y(x2)))
% 0.13/0.39 = { by axiom 2 (main_1) R->L }
% 0.13/0.39 tuple(r1(x2, x2), fresh4(fresh6(true2, true2, x2), true2, x2, y(x2)))
% 0.13/0.39 = { by axiom 1 (reflexivity) R->L }
% 0.13/0.39 tuple(r1(x2, x2), fresh4(fresh6(r1(x2, x2), true2, x2), true2, x2, y(x2)))
% 0.13/0.39 = { by axiom 5 (main_1) }
% 0.13/0.39 tuple(r1(x2, x2), fresh4(r1(x(x2), y(x2)), true2, x2, y(x2)))
% 0.13/0.39 = { by axiom 6 (main_2) }
% 0.13/0.39 tuple(r1(x2, x2), fresh3(r1(x2, x2), true2, y(x2)))
% 0.13/0.39 = { by axiom 1 (reflexivity) }
% 0.13/0.39 tuple(r1(x2, x2), fresh3(true2, true2, y(x2)))
% 0.13/0.39 = { by axiom 3 (main_2) }
% 0.13/0.39 tuple(r1(x2, x2), true2)
% 0.13/0.39 = { by axiom 1 (reflexivity) }
% 0.13/0.39 tuple(true2, true2)
% 0.13/0.39 % SZS output end Proof
% 0.13/0.39
% 0.13/0.39 RESULT: Theorem (the conjecture is true).
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