TSTP Solution File: ITP012+1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : ITP012+1 : TPTP v8.1.2. Bugfixed v7.5.0.
% 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 : n010.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 04:16:00 EDT 2023
% Result : Theorem 30.85s 4.31s
% Output : Proof 30.85s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP012+1 : TPTP v8.1.2. Bugfixed v7.5.0.
% 0.12/0.13 % 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 : n010.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 : Sun Aug 27 10:34:34 EDT 2023
% 0.13/0.35 % CPUTime :
% 30.85/4.31 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 30.85/4.31
% 30.85/4.31 % SZS status Theorem
% 30.85/4.31
% 30.85/4.31 % SZS output start Proof
% 30.85/4.31 Take the following subset of the input axioms:
% 30.85/4.31 fof(reserved_2Eho_2Etruth, axiom, p(s(tyop_2Emin_2Ebool, c_2Ebool_2ET_2E0))).
% 30.85/4.31 fof(thm_2Ebool_2EEQ__CLAUSES, axiom, ![V0t_2E0]: ((s(tyop_2Emin_2Ebool, c_2Ebool_2ET_2E0)=s(tyop_2Emin_2Ebool, V0t_2E0) <=> p(s(tyop_2Emin_2Ebool, V0t_2E0))) & ((s(tyop_2Emin_2Ebool, V0t_2E0)=s(tyop_2Emin_2Ebool, c_2Ebool_2ET_2E0) <=> p(s(tyop_2Emin_2Ebool, V0t_2E0))) & ((s(tyop_2Emin_2Ebool, c_2Ebool_2EF_2E0)=s(tyop_2Emin_2Ebool, V0t_2E0) <=> ~p(s(tyop_2Emin_2Ebool, V0t_2E0))) & (s(tyop_2Emin_2Ebool, V0t_2E0)=s(tyop_2Emin_2Ebool, c_2Ebool_2EF_2E0) <=> ~p(s(tyop_2Emin_2Ebool, V0t_2E0))))))).
% 30.85/4.31 fof(thm_2Einteger_2EINT__DIVIDES__NEG, axiom, ![V0p_2E0, V1q_2E0]: (s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, V0p_2E0), s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, V1q_2E0)))))=s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, V0p_2E0), s(tyop_2Einteger_2Eint, V1q_2E0))) & s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, V0p_2E0))), s(tyop_2Einteger_2Eint, V1q_2E0)))=s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, V0p_2E0), s(tyop_2Einteger_2Eint, V1q_2E0))))).
% 30.85/4.31 fof(thm_2Einteger_2EINT__DIVIDES__RADD, axiom, ![V2r_2E0, V0p_2E0_2, V1q_2E0_2]: (p(s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, V0p_2E0_2), s(tyop_2Einteger_2Eint, V1q_2E0_2)))) => s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, V0p_2E0_2), s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__add_2E2(s(tyop_2Einteger_2Eint, V2r_2E0), s(tyop_2Einteger_2Eint, V1q_2E0_2)))))=s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, V0p_2E0_2), s(tyop_2Einteger_2Eint, V2r_2E0))))).
% 30.85/4.31 fof(thm_2Einteger_2EINT__DIVIDES__RSUB, conjecture, ![V0p_2E0_2, V1q_2E0_2, V2r_2E0_2]: (p(s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, V0p_2E0_2), s(tyop_2Einteger_2Eint, V1q_2E0_2)))) => s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, V0p_2E0_2), s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__sub_2E2(s(tyop_2Einteger_2Eint, V2r_2E0_2), s(tyop_2Einteger_2Eint, V1q_2E0_2)))))=s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, V0p_2E0_2), s(tyop_2Einteger_2Eint, V2r_2E0_2))))).
% 30.85/4.31 fof(thm_2Einteger_2Eint__sub, axiom, ![V0x_2E0, V1y_2E0]: s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__sub_2E2(s(tyop_2Einteger_2Eint, V0x_2E0), s(tyop_2Einteger_2Eint, V1y_2E0)))=s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__add_2E2(s(tyop_2Einteger_2Eint, V0x_2E0), s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, V1y_2E0)))))).
% 30.85/4.31
% 30.85/4.31 Now clausify the problem and encode Horn clauses using encoding 3 of
% 30.85/4.31 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 30.85/4.31 We repeatedly replace C & s=t => u=v by the two clauses:
% 30.85/4.31 fresh(y, y, x1...xn) = u
% 30.85/4.31 C => fresh(s, t, x1...xn) = v
% 30.85/4.31 where fresh is a fresh function symbol and x1..xn are the free
% 30.85/4.31 variables of u and v.
% 30.85/4.31 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 30.85/4.31 input problem has no model of domain size 1).
% 30.85/4.31
% 30.85/4.31 The encoding turns the above axioms into the following unit equations and goals:
% 30.85/4.31
% 30.85/4.31 Axiom 1 (thm_2Ebool_2EEQ__CLAUSES_6): fresh59(X, X, Y) = s(tyop_2Emin_2Ebool, Y).
% 30.85/4.31 Axiom 2 (reserved_2Eho_2Etruth): p(s(tyop_2Emin_2Ebool, c_2Ebool_2ET_2E0)) = true2.
% 30.85/4.31 Axiom 3 (thm_2Ebool_2EEQ__CLAUSES_6): fresh59(p(s(tyop_2Emin_2Ebool, X)), true2, X) = s(tyop_2Emin_2Ebool, c_2Ebool_2ET_2E0).
% 30.85/4.31 Axiom 4 (thm_2Einteger_2EINT__DIVIDES__RADD): fresh41(X, X, Y, Z, W) = s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, Y), s(tyop_2Einteger_2Eint, W))).
% 30.85/4.31 Axiom 5 (thm_2Einteger_2EINT__DIVIDES__RSUB): p(s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, v0p_2e0), s(tyop_2Einteger_2Eint, v1q_2e0)))) = true2.
% 30.85/4.31 Axiom 6 (thm_2Einteger_2Eint__sub): s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__sub_2E2(s(tyop_2Einteger_2Eint, X), s(tyop_2Einteger_2Eint, Y))) = s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__add_2E2(s(tyop_2Einteger_2Eint, X), s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, Y))))).
% 30.85/4.31 Axiom 7 (thm_2Einteger_2EINT__DIVIDES__NEG): s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, X), s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, Y))))) = s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, X), s(tyop_2Einteger_2Eint, Y))).
% 30.85/4.32 Axiom 8 (thm_2Einteger_2EINT__DIVIDES__NEG_1): s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, X))), s(tyop_2Einteger_2Eint, Y))) = s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, X), s(tyop_2Einteger_2Eint, Y))).
% 30.85/4.32 Axiom 9 (thm_2Einteger_2EINT__DIVIDES__RADD): fresh41(p(s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, X), s(tyop_2Einteger_2Eint, Y)))), true2, X, Y, Z) = s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, X), s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__add_2E2(s(tyop_2Einteger_2Eint, Z), s(tyop_2Einteger_2Eint, Y))))).
% 30.85/4.32
% 30.85/4.32 Lemma 10: fresh41(X, X, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, Y)), Z, W) = fresh41(V, V, Y, U, W).
% 30.85/4.32 Proof:
% 30.85/4.32 fresh41(X, X, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, Y)), Z, W)
% 30.85/4.32 = { by axiom 4 (thm_2Einteger_2EINT__DIVIDES__RADD) }
% 30.85/4.32 s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, Y))), s(tyop_2Einteger_2Eint, W)))
% 30.85/4.32 = { by axiom 8 (thm_2Einteger_2EINT__DIVIDES__NEG_1) }
% 30.85/4.32 s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, Y), s(tyop_2Einteger_2Eint, W)))
% 30.85/4.32 = { by axiom 4 (thm_2Einteger_2EINT__DIVIDES__RADD) R->L }
% 30.85/4.32 fresh41(V, V, Y, U, W)
% 30.85/4.32
% 30.85/4.32 Goal 1 (thm_2Einteger_2EINT__DIVIDES__RSUB_1): s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, v0p_2e0), s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__sub_2E2(s(tyop_2Einteger_2Eint, v2r_2e0), s(tyop_2Einteger_2Eint, v1q_2e0))))) = s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, v0p_2e0), s(tyop_2Einteger_2Eint, v2r_2e0))).
% 30.85/4.32 Proof:
% 30.85/4.32 s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, v0p_2e0), s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__sub_2E2(s(tyop_2Einteger_2Eint, v2r_2e0), s(tyop_2Einteger_2Eint, v1q_2e0)))))
% 30.85/4.32 = { by axiom 6 (thm_2Einteger_2Eint__sub) }
% 30.85/4.32 s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, v0p_2e0), s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__add_2E2(s(tyop_2Einteger_2Eint, v2r_2e0), s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, v1q_2e0)))))))
% 30.85/4.32 = { by axiom 4 (thm_2Einteger_2EINT__DIVIDES__RADD) R->L }
% 30.85/4.32 fresh41(X, X, v0p_2e0, Y, c_2Einteger_2Eint__add_2E2(s(tyop_2Einteger_2Eint, v2r_2e0), s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, v1q_2e0)))))
% 30.85/4.32 = { by lemma 10 R->L }
% 30.85/4.32 fresh41(Z, Z, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, v0p_2e0)), W, c_2Einteger_2Eint__add_2E2(s(tyop_2Einteger_2Eint, v2r_2e0), s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, v1q_2e0)))))
% 30.85/4.32 = { by axiom 4 (thm_2Einteger_2EINT__DIVIDES__RADD) }
% 30.85/4.32 s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, v0p_2e0))), s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__add_2E2(s(tyop_2Einteger_2Eint, v2r_2e0), s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, v1q_2e0)))))))
% 30.85/4.32 = { by axiom 9 (thm_2Einteger_2EINT__DIVIDES__RADD) R->L }
% 30.85/4.32 fresh41(p(s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, v0p_2e0))), s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, v1q_2e0)))))), true2, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, v0p_2e0)), c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, v1q_2e0)), v2r_2e0)
% 30.85/4.32 = { by axiom 4 (thm_2Einteger_2EINT__DIVIDES__RADD) R->L }
% 30.85/4.32 fresh41(p(fresh41(V, V, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, v0p_2e0)), U, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, v1q_2e0)))), true2, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, v0p_2e0)), c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, v1q_2e0)), v2r_2e0)
% 30.85/4.32 = { by lemma 10 }
% 30.85/4.32 fresh41(p(fresh41(T, T, v0p_2e0, S, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, v1q_2e0)))), true2, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, v0p_2e0)), c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, v1q_2e0)), v2r_2e0)
% 30.85/4.32 = { by axiom 4 (thm_2Einteger_2EINT__DIVIDES__RADD) }
% 30.85/4.32 fresh41(p(s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, v0p_2e0), s(tyop_2Einteger_2Eint, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, v1q_2e0)))))), true2, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, v0p_2e0)), c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, v1q_2e0)), v2r_2e0)
% 30.85/4.32 = { by axiom 7 (thm_2Einteger_2EINT__DIVIDES__NEG) }
% 30.85/4.32 fresh41(p(s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, v0p_2e0), s(tyop_2Einteger_2Eint, v1q_2e0)))), true2, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, v0p_2e0)), c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, v1q_2e0)), v2r_2e0)
% 30.85/4.32 = { by axiom 1 (thm_2Ebool_2EEQ__CLAUSES_6) R->L }
% 30.85/4.32 fresh41(p(fresh59(true2, true2, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, v0p_2e0), s(tyop_2Einteger_2Eint, v1q_2e0)))), true2, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, v0p_2e0)), c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, v1q_2e0)), v2r_2e0)
% 30.85/4.32 = { by axiom 5 (thm_2Einteger_2EINT__DIVIDES__RSUB) R->L }
% 30.85/4.32 fresh41(p(fresh59(p(s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, v0p_2e0), s(tyop_2Einteger_2Eint, v1q_2e0)))), true2, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, v0p_2e0), s(tyop_2Einteger_2Eint, v1q_2e0)))), true2, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, v0p_2e0)), c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, v1q_2e0)), v2r_2e0)
% 30.85/4.32 = { by axiom 3 (thm_2Ebool_2EEQ__CLAUSES_6) }
% 30.85/4.32 fresh41(p(s(tyop_2Emin_2Ebool, c_2Ebool_2ET_2E0)), true2, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, v0p_2e0)), c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, v1q_2e0)), v2r_2e0)
% 30.85/4.32 = { by axiom 2 (reserved_2Eho_2Etruth) }
% 30.85/4.32 fresh41(true2, true2, c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, v0p_2e0)), c_2Einteger_2Eint__neg_2E1(s(tyop_2Einteger_2Eint, v1q_2e0)), v2r_2e0)
% 30.85/4.32 = { by lemma 10 }
% 30.85/4.32 fresh41(X2, X2, v0p_2e0, Y2, v2r_2e0)
% 30.85/4.32 = { by axiom 4 (thm_2Einteger_2EINT__DIVIDES__RADD) }
% 30.85/4.32 s(tyop_2Emin_2Ebool, c_2Einteger_2Eint__divides_2E2(s(tyop_2Einteger_2Eint, v0p_2e0), s(tyop_2Einteger_2Eint, v2r_2e0)))
% 30.85/4.32 % SZS output end Proof
% 30.85/4.32
% 30.85/4.32 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------