TSTP Solution File: SET776+4 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SET776+4 : TPTP v8.1.2. Released v2.2.0.
% 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 : n003.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 15:33:12 EDT 2023
% Result : Theorem 5.67s 1.11s
% Output : Proof 5.67s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SET776+4 : TPTP v8.1.2. Released v2.2.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.34 % Computer : n003.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Sat Aug 26 16:14:39 EDT 2023
% 0.12/0.34 % CPUTime :
% 5.67/1.11 Command-line arguments: --kbo-weight0 --lhs-weight 5 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10 --goal-heuristic
% 5.67/1.11
% 5.67/1.11 % SZS status Theorem
% 5.67/1.11
% 5.67/1.12 % SZS output start Proof
% 5.67/1.12 Take the following subset of the input axioms:
% 5.67/1.13 fof(pre_order, axiom, ![E, R]: (pre_order(R, E) <=> (![X]: (member(X, E) => apply(R, X, X)) & ![Y, Z, X3]: ((member(X3, E) & (member(Y, E) & member(Z, E))) => ((apply(R, X3, Y) & apply(R, Y, Z)) => apply(R, X3, Z)))))).
% 5.67/1.13 fof(thIII12, conjecture, ![P, E2, R2]: ((pre_order(P, E2) & ![B, A2]: ((member(A2, E2) & member(B, E2)) => (apply(R2, A2, B) <=> (apply(P, A2, B) & apply(P, B, A2))))) => ![X1, X2, Y1, Y2]: ((member(X1, E2) & (member(X2, E2) & (member(Y1, E2) & member(Y2, E2)))) => ((apply(R2, X1, Y1) & (apply(R2, X2, Y2) & apply(P, X1, X2))) => apply(P, Y1, Y2))))).
% 5.67/1.13
% 5.67/1.13 Now clausify the problem and encode Horn clauses using encoding 3 of
% 5.67/1.13 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 5.67/1.13 We repeatedly replace C & s=t => u=v by the two clauses:
% 5.67/1.13 fresh(y, y, x1...xn) = u
% 5.67/1.13 C => fresh(s, t, x1...xn) = v
% 5.67/1.13 where fresh is a fresh function symbol and x1..xn are the free
% 5.67/1.13 variables of u and v.
% 5.67/1.13 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 5.67/1.13 input problem has no model of domain size 1).
% 5.67/1.13
% 5.67/1.13 The encoding turns the above axioms into the following unit equations and goals:
% 5.67/1.13
% 5.67/1.13 Axiom 1 (thIII12_1): member(y1, e) = true2.
% 5.67/1.13 Axiom 2 (thIII12_3): member(y2, e) = true2.
% 5.67/1.13 Axiom 3 (thIII12): member(x2, e) = true2.
% 5.67/1.13 Axiom 4 (thIII12_2): member(x1, e) = true2.
% 5.67/1.13 Axiom 5 (thIII12_7): pre_order(p, e) = true2.
% 5.67/1.13 Axiom 6 (thIII12_4): apply(p, x1, x2) = true2.
% 5.67/1.13 Axiom 7 (thIII12_5): apply(r, x2, y2) = true2.
% 5.67/1.13 Axiom 8 (thIII12_6): apply(r, x1, y1) = true2.
% 5.67/1.13 Axiom 9 (thIII12_9): fresh55(X, X, Y, Z) = true2.
% 5.67/1.13 Axiom 10 (thIII12_10): fresh53(X, X, Y, Z) = true2.
% 5.67/1.13 Axiom 11 (thIII12_10): fresh5(X, X, Y, Z) = apply(p, Z, Y).
% 5.67/1.13 Axiom 12 (thIII12_9): fresh4(X, X, Y, Z) = apply(p, Y, Z).
% 5.67/1.13 Axiom 13 (pre_order_6): fresh65(X, X, Y, Z, W) = true2.
% 5.67/1.13 Axiom 14 (thIII12_9): fresh54(X, X, Y, Z) = fresh55(member(Y, e), true2, Y, Z).
% 5.67/1.13 Axiom 15 (thIII12_10): fresh52(X, X, Y, Z) = fresh53(member(Y, e), true2, Y, Z).
% 5.67/1.13 Axiom 16 (pre_order_6): fresh63(X, X, Y, Z, W, V) = apply(Y, W, V).
% 5.67/1.13 Axiom 17 (thIII12_9): fresh54(apply(r, X, Y), true2, X, Y) = fresh4(member(Y, e), true2, X, Y).
% 5.67/1.13 Axiom 18 (thIII12_10): fresh52(apply(r, X, Y), true2, X, Y) = fresh5(member(Y, e), true2, X, Y).
% 5.67/1.13 Axiom 19 (pre_order_6): fresh64(X, X, Y, Z, W, V, U) = fresh65(member(W, Z), true2, Y, W, U).
% 5.67/1.13 Axiom 20 (pre_order_6): fresh62(X, X, Y, Z, W, V, U) = fresh63(member(V, Z), true2, Y, Z, W, U).
% 5.67/1.13 Axiom 21 (pre_order_6): fresh61(X, X, Y, Z, W, V, U) = fresh64(member(U, Z), true2, Y, Z, W, V, U).
% 5.67/1.13 Axiom 22 (pre_order_6): fresh60(X, X, Y, Z, W, V, U) = fresh62(apply(Y, W, V), true2, Y, Z, W, V, U).
% 5.67/1.13 Axiom 23 (pre_order_6): fresh60(pre_order(X, Y), true2, X, Y, Z, W, V) = fresh61(apply(X, W, V), true2, X, Y, Z, W, V).
% 5.67/1.13
% 5.67/1.13 Goal 1 (thIII12_11): apply(p, y1, y2) = true2.
% 5.67/1.13 Proof:
% 5.67/1.13 apply(p, y1, y2)
% 5.67/1.13 = { by axiom 16 (pre_order_6) R->L }
% 5.67/1.13 fresh63(true2, true2, p, e, y1, y2)
% 5.67/1.13 = { by axiom 3 (thIII12) R->L }
% 5.67/1.13 fresh63(member(x2, e), true2, p, e, y1, y2)
% 5.67/1.13 = { by axiom 20 (pre_order_6) R->L }
% 5.67/1.13 fresh62(true2, true2, p, e, y1, x2, y2)
% 5.67/1.13 = { by axiom 13 (pre_order_6) R->L }
% 5.67/1.13 fresh62(fresh65(true2, true2, p, y1, x2), true2, p, e, y1, x2, y2)
% 5.67/1.13 = { by axiom 1 (thIII12_1) R->L }
% 5.67/1.13 fresh62(fresh65(member(y1, e), true2, p, y1, x2), true2, p, e, y1, x2, y2)
% 5.67/1.13 = { by axiom 19 (pre_order_6) R->L }
% 5.67/1.13 fresh62(fresh64(true2, true2, p, e, y1, x1, x2), true2, p, e, y1, x2, y2)
% 5.67/1.13 = { by axiom 3 (thIII12) R->L }
% 5.67/1.13 fresh62(fresh64(member(x2, e), true2, p, e, y1, x1, x2), true2, p, e, y1, x2, y2)
% 5.67/1.13 = { by axiom 21 (pre_order_6) R->L }
% 5.67/1.13 fresh62(fresh61(true2, true2, p, e, y1, x1, x2), true2, p, e, y1, x2, y2)
% 5.67/1.13 = { by axiom 6 (thIII12_4) R->L }
% 5.67/1.13 fresh62(fresh61(apply(p, x1, x2), true2, p, e, y1, x1, x2), true2, p, e, y1, x2, y2)
% 5.67/1.13 = { by axiom 23 (pre_order_6) R->L }
% 5.67/1.13 fresh62(fresh60(pre_order(p, e), true2, p, e, y1, x1, x2), true2, p, e, y1, x2, y2)
% 5.67/1.13 = { by axiom 5 (thIII12_7) }
% 5.67/1.13 fresh62(fresh60(true2, true2, p, e, y1, x1, x2), true2, p, e, y1, x2, y2)
% 5.67/1.13 = { by axiom 22 (pre_order_6) }
% 5.67/1.13 fresh62(fresh62(apply(p, y1, x1), true2, p, e, y1, x1, x2), true2, p, e, y1, x2, y2)
% 5.67/1.13 = { by axiom 11 (thIII12_10) R->L }
% 5.67/1.13 fresh62(fresh62(fresh5(true2, true2, x1, y1), true2, p, e, y1, x1, x2), true2, p, e, y1, x2, y2)
% 5.67/1.13 = { by axiom 1 (thIII12_1) R->L }
% 5.67/1.13 fresh62(fresh62(fresh5(member(y1, e), true2, x1, y1), true2, p, e, y1, x1, x2), true2, p, e, y1, x2, y2)
% 5.67/1.13 = { by axiom 18 (thIII12_10) R->L }
% 5.67/1.13 fresh62(fresh62(fresh52(apply(r, x1, y1), true2, x1, y1), true2, p, e, y1, x1, x2), true2, p, e, y1, x2, y2)
% 5.67/1.13 = { by axiom 8 (thIII12_6) }
% 5.67/1.13 fresh62(fresh62(fresh52(true2, true2, x1, y1), true2, p, e, y1, x1, x2), true2, p, e, y1, x2, y2)
% 5.67/1.13 = { by axiom 15 (thIII12_10) }
% 5.67/1.13 fresh62(fresh62(fresh53(member(x1, e), true2, x1, y1), true2, p, e, y1, x1, x2), true2, p, e, y1, x2, y2)
% 5.67/1.13 = { by axiom 4 (thIII12_2) }
% 5.67/1.13 fresh62(fresh62(fresh53(true2, true2, x1, y1), true2, p, e, y1, x1, x2), true2, p, e, y1, x2, y2)
% 5.67/1.13 = { by axiom 10 (thIII12_10) }
% 5.67/1.13 fresh62(fresh62(true2, true2, p, e, y1, x1, x2), true2, p, e, y1, x2, y2)
% 5.67/1.13 = { by axiom 20 (pre_order_6) }
% 5.67/1.13 fresh62(fresh63(member(x1, e), true2, p, e, y1, x2), true2, p, e, y1, x2, y2)
% 5.67/1.13 = { by axiom 4 (thIII12_2) }
% 5.67/1.13 fresh62(fresh63(true2, true2, p, e, y1, x2), true2, p, e, y1, x2, y2)
% 5.67/1.13 = { by axiom 16 (pre_order_6) }
% 5.67/1.13 fresh62(apply(p, y1, x2), true2, p, e, y1, x2, y2)
% 5.67/1.13 = { by axiom 22 (pre_order_6) R->L }
% 5.67/1.13 fresh60(true2, true2, p, e, y1, x2, y2)
% 5.67/1.13 = { by axiom 5 (thIII12_7) R->L }
% 5.67/1.13 fresh60(pre_order(p, e), true2, p, e, y1, x2, y2)
% 5.67/1.13 = { by axiom 23 (pre_order_6) }
% 5.67/1.13 fresh61(apply(p, x2, y2), true2, p, e, y1, x2, y2)
% 5.67/1.13 = { by axiom 12 (thIII12_9) R->L }
% 5.67/1.13 fresh61(fresh4(true2, true2, x2, y2), true2, p, e, y1, x2, y2)
% 5.67/1.13 = { by axiom 2 (thIII12_3) R->L }
% 5.67/1.13 fresh61(fresh4(member(y2, e), true2, x2, y2), true2, p, e, y1, x2, y2)
% 5.67/1.13 = { by axiom 17 (thIII12_9) R->L }
% 5.67/1.13 fresh61(fresh54(apply(r, x2, y2), true2, x2, y2), true2, p, e, y1, x2, y2)
% 5.67/1.13 = { by axiom 7 (thIII12_5) }
% 5.67/1.13 fresh61(fresh54(true2, true2, x2, y2), true2, p, e, y1, x2, y2)
% 5.67/1.13 = { by axiom 14 (thIII12_9) }
% 5.67/1.13 fresh61(fresh55(member(x2, e), true2, x2, y2), true2, p, e, y1, x2, y2)
% 5.67/1.13 = { by axiom 3 (thIII12) }
% 5.67/1.13 fresh61(fresh55(true2, true2, x2, y2), true2, p, e, y1, x2, y2)
% 5.67/1.13 = { by axiom 9 (thIII12_9) }
% 5.67/1.13 fresh61(true2, true2, p, e, y1, x2, y2)
% 5.67/1.13 = { by axiom 21 (pre_order_6) }
% 5.67/1.13 fresh64(member(y2, e), true2, p, e, y1, x2, y2)
% 5.67/1.13 = { by axiom 2 (thIII12_3) }
% 5.67/1.13 fresh64(true2, true2, p, e, y1, x2, y2)
% 5.67/1.13 = { by axiom 19 (pre_order_6) }
% 5.67/1.13 fresh65(member(y1, e), true2, p, y1, y2)
% 5.67/1.13 = { by axiom 1 (thIII12_1) }
% 5.67/1.13 fresh65(true2, true2, p, y1, y2)
% 5.67/1.13 = { by axiom 13 (pre_order_6) }
% 5.67/1.13 true2
% 5.67/1.13 % SZS output end Proof
% 5.67/1.13
% 5.67/1.13 RESULT: Theorem (the conjecture is true).
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