TSTP Solution File: NUM412+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : NUM412+1 : TPTP v8.1.2. Released v3.2.0.
% 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 : n014.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 11:56:09 EDT 2023
% Result : Theorem 0.20s 0.66s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : NUM412+1 : TPTP v8.1.2. Released v3.2.0.
% 0.00/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.35 % Computer : n014.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Fri Aug 25 13:24:18 EDT 2023
% 0.14/0.35 % CPUTime :
% 0.20/0.66 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.20/0.66
% 0.20/0.66 % SZS status Theorem
% 0.20/0.66
% 0.20/0.67 % SZS output start Proof
% 0.20/0.67 Take the following subset of the input axioms:
% 0.20/0.67 fof(d8_ordinal1, axiom, ![B, A2]: ((relation(B) & (function(B) & transfinite_sequence(B))) => (transfinite_sequence_of(B, A2) <=> subset(relation_rng(B), A2)))).
% 0.20/0.67 fof(dt_k2_ordinal1, axiom, ![B2, A2_2]: ((relation(A2_2) & (function(A2_2) & (transfinite_sequence(A2_2) & ordinal(B2)))) => transfinite_sequence_of(tseq_dom_restriction(A2_2, B2), relation_rng(A2_2)))).
% 0.20/0.68 fof(dt_m1_ordinal1, axiom, ![A, B2]: (transfinite_sequence_of(B2, A) => (relation(B2) & (function(B2) & transfinite_sequence(B2))))).
% 0.20/0.68 fof(t47_ordinal1, axiom, ![B2, A2_2]: (subset(A2_2, B2) => ![C]: (transfinite_sequence_of(C, A2_2) => transfinite_sequence_of(C, B2)))).
% 0.20/0.68 fof(t48_ordinal1, conjecture, ![A3, B2]: (transfinite_sequence_of(B2, A3) => ![C2]: (ordinal(C2) => transfinite_sequence_of(tseq_dom_restriction(B2, C2), A3)))).
% 0.20/0.68
% 0.20/0.68 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.68 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.68 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.68 fresh(y, y, x1...xn) = u
% 0.20/0.68 C => fresh(s, t, x1...xn) = v
% 0.20/0.69 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.69 variables of u and v.
% 0.20/0.69 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.69 input problem has no model of domain size 1).
% 0.20/0.69
% 0.20/0.70 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.70
% 0.20/0.70 Axiom 1 (t48_ordinal1): ordinal(c) = true2.
% 0.20/0.70 Axiom 2 (t48_ordinal1_1): transfinite_sequence_of(b, a) = true2.
% 0.20/0.70 Axiom 3 (dt_m1_ordinal1): fresh26(X, X, Y) = true2.
% 0.20/0.70 Axiom 4 (dt_m1_ordinal1_1): fresh25(X, X, Y) = true2.
% 0.20/0.70 Axiom 5 (dt_m1_ordinal1_2): fresh24(X, X, Y) = true2.
% 0.20/0.70 Axiom 6 (d8_ordinal1): fresh51(X, X, Y, Z) = true2.
% 0.20/0.70 Axiom 7 (d8_ordinal1): fresh49(X, X, Y, Z) = subset(relation_rng(Z), Y).
% 0.20/0.70 Axiom 8 (dt_k2_ordinal1): fresh47(X, X, Y, Z) = true2.
% 0.20/0.70 Axiom 9 (t47_ordinal1): fresh6(X, X, Y, Z) = true2.
% 0.20/0.70 Axiom 10 (dt_k2_ordinal1): fresh45(X, X, Y, Z) = transfinite_sequence_of(tseq_dom_restriction(Y, Z), relation_rng(Y)).
% 0.20/0.70 Axiom 11 (d8_ordinal1): fresh50(X, X, Y, Z) = fresh51(function(Z), true2, Y, Z).
% 0.20/0.70 Axiom 12 (d8_ordinal1): fresh48(X, X, Y, Z) = fresh49(relation(Z), true2, Y, Z).
% 0.20/0.70 Axiom 13 (dt_k2_ordinal1): fresh46(X, X, Y, Z) = fresh47(function(Y), true2, Y, Z).
% 0.20/0.70 Axiom 14 (dt_k2_ordinal1): fresh44(X, X, Y, Z) = fresh45(ordinal(Z), true2, Y, Z).
% 0.20/0.70 Axiom 15 (dt_k2_ordinal1): fresh44(transfinite_sequence(X), true2, X, Y) = fresh46(relation(X), true2, X, Y).
% 0.20/0.70 Axiom 16 (dt_m1_ordinal1): fresh26(transfinite_sequence_of(X, Y), true2, X) = function(X).
% 0.20/0.70 Axiom 17 (dt_m1_ordinal1_1): fresh25(transfinite_sequence_of(X, Y), true2, X) = relation(X).
% 0.20/0.70 Axiom 18 (dt_m1_ordinal1_2): fresh24(transfinite_sequence_of(X, Y), true2, X) = transfinite_sequence(X).
% 0.20/0.70 Axiom 19 (t47_ordinal1): fresh7(X, X, Y, Z, W) = transfinite_sequence_of(W, Z).
% 0.20/0.70 Axiom 20 (d8_ordinal1): fresh48(transfinite_sequence_of(X, Y), true2, Y, X) = fresh50(transfinite_sequence(X), true2, Y, X).
% 0.20/0.70 Axiom 21 (t47_ordinal1): fresh7(subset(X, Y), true2, X, Y, Z) = fresh6(transfinite_sequence_of(Z, X), true2, Y, Z).
% 0.20/0.70
% 0.20/0.70 Lemma 22: function(b) = true2.
% 0.20/0.70 Proof:
% 0.20/0.70 function(b)
% 0.20/0.70 = { by axiom 16 (dt_m1_ordinal1) R->L }
% 0.20/0.70 fresh26(transfinite_sequence_of(b, a), true2, b)
% 0.20/0.70 = { by axiom 2 (t48_ordinal1_1) }
% 0.20/0.70 fresh26(true2, true2, b)
% 0.20/0.70 = { by axiom 3 (dt_m1_ordinal1) }
% 0.20/0.70 true2
% 0.20/0.70
% 0.20/0.70 Lemma 23: relation(b) = true2.
% 0.20/0.70 Proof:
% 0.20/0.70 relation(b)
% 0.20/0.70 = { by axiom 17 (dt_m1_ordinal1_1) R->L }
% 0.20/0.70 fresh25(transfinite_sequence_of(b, a), true2, b)
% 0.20/0.70 = { by axiom 2 (t48_ordinal1_1) }
% 0.20/0.70 fresh25(true2, true2, b)
% 0.20/0.70 = { by axiom 4 (dt_m1_ordinal1_1) }
% 0.20/0.70 true2
% 0.20/0.70
% 0.20/0.70 Lemma 24: transfinite_sequence(b) = true2.
% 0.20/0.70 Proof:
% 0.20/0.70 transfinite_sequence(b)
% 0.20/0.70 = { by axiom 18 (dt_m1_ordinal1_2) R->L }
% 0.20/0.70 fresh24(transfinite_sequence_of(b, a), true2, b)
% 0.20/0.70 = { by axiom 2 (t48_ordinal1_1) }
% 0.20/0.70 fresh24(true2, true2, b)
% 0.20/0.70 = { by axiom 5 (dt_m1_ordinal1_2) }
% 0.20/0.70 true2
% 0.20/0.70
% 0.20/0.71 Goal 1 (t48_ordinal1_2): transfinite_sequence_of(tseq_dom_restriction(b, c), a) = true2.
% 0.20/0.71 Proof:
% 0.20/0.71 transfinite_sequence_of(tseq_dom_restriction(b, c), a)
% 0.20/0.71 = { by axiom 19 (t47_ordinal1) R->L }
% 0.20/0.71 fresh7(true2, true2, relation_rng(b), a, tseq_dom_restriction(b, c))
% 0.20/0.71 = { by axiom 6 (d8_ordinal1) R->L }
% 0.20/0.71 fresh7(fresh51(true2, true2, a, b), true2, relation_rng(b), a, tseq_dom_restriction(b, c))
% 0.20/0.71 = { by lemma 22 R->L }
% 0.20/0.71 fresh7(fresh51(function(b), true2, a, b), true2, relation_rng(b), a, tseq_dom_restriction(b, c))
% 0.20/0.71 = { by axiom 11 (d8_ordinal1) R->L }
% 0.20/0.71 fresh7(fresh50(true2, true2, a, b), true2, relation_rng(b), a, tseq_dom_restriction(b, c))
% 0.20/0.71 = { by lemma 24 R->L }
% 0.20/0.71 fresh7(fresh50(transfinite_sequence(b), true2, a, b), true2, relation_rng(b), a, tseq_dom_restriction(b, c))
% 0.20/0.71 = { by axiom 20 (d8_ordinal1) R->L }
% 0.20/0.71 fresh7(fresh48(transfinite_sequence_of(b, a), true2, a, b), true2, relation_rng(b), a, tseq_dom_restriction(b, c))
% 0.20/0.71 = { by axiom 2 (t48_ordinal1_1) }
% 0.20/0.71 fresh7(fresh48(true2, true2, a, b), true2, relation_rng(b), a, tseq_dom_restriction(b, c))
% 0.20/0.71 = { by axiom 12 (d8_ordinal1) }
% 0.20/0.71 fresh7(fresh49(relation(b), true2, a, b), true2, relation_rng(b), a, tseq_dom_restriction(b, c))
% 0.20/0.71 = { by lemma 23 }
% 0.20/0.71 fresh7(fresh49(true2, true2, a, b), true2, relation_rng(b), a, tseq_dom_restriction(b, c))
% 0.20/0.71 = { by axiom 7 (d8_ordinal1) }
% 0.20/0.71 fresh7(subset(relation_rng(b), a), true2, relation_rng(b), a, tseq_dom_restriction(b, c))
% 0.20/0.71 = { by axiom 21 (t47_ordinal1) }
% 0.20/0.71 fresh6(transfinite_sequence_of(tseq_dom_restriction(b, c), relation_rng(b)), true2, a, tseq_dom_restriction(b, c))
% 0.20/0.71 = { by axiom 10 (dt_k2_ordinal1) R->L }
% 0.20/0.71 fresh6(fresh45(true2, true2, b, c), true2, a, tseq_dom_restriction(b, c))
% 0.20/0.71 = { by axiom 1 (t48_ordinal1) R->L }
% 0.20/0.71 fresh6(fresh45(ordinal(c), true2, b, c), true2, a, tseq_dom_restriction(b, c))
% 0.20/0.71 = { by axiom 14 (dt_k2_ordinal1) R->L }
% 0.20/0.71 fresh6(fresh44(true2, true2, b, c), true2, a, tseq_dom_restriction(b, c))
% 0.20/0.71 = { by lemma 24 R->L }
% 0.20/0.71 fresh6(fresh44(transfinite_sequence(b), true2, b, c), true2, a, tseq_dom_restriction(b, c))
% 0.20/0.71 = { by axiom 15 (dt_k2_ordinal1) }
% 0.20/0.71 fresh6(fresh46(relation(b), true2, b, c), true2, a, tseq_dom_restriction(b, c))
% 0.20/0.71 = { by lemma 23 }
% 0.20/0.71 fresh6(fresh46(true2, true2, b, c), true2, a, tseq_dom_restriction(b, c))
% 0.20/0.71 = { by axiom 13 (dt_k2_ordinal1) }
% 0.20/0.71 fresh6(fresh47(function(b), true2, b, c), true2, a, tseq_dom_restriction(b, c))
% 0.20/0.71 = { by lemma 22 }
% 0.20/0.71 fresh6(fresh47(true2, true2, b, c), true2, a, tseq_dom_restriction(b, c))
% 0.20/0.71 = { by axiom 8 (dt_k2_ordinal1) }
% 0.20/0.71 fresh6(true2, true2, a, tseq_dom_restriction(b, c))
% 0.20/0.71 = { by axiom 9 (t47_ordinal1) }
% 0.20/0.71 true2
% 0.20/0.71 % SZS output end Proof
% 0.20/0.71
% 0.20/0.71 RESULT: Theorem (the conjecture is true).
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