TSTP Solution File: NUM410+1 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : NUM410+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 : n024.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 0.50s
% 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
% 0.00/0.12 % Problem : NUM410+1 : TPTP v8.1.2. Released v3.2.0.
% 0.13/0.13 % 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 : n024.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 : Fri Aug 25 09:30:54 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.20/0.50 Command-line arguments: --no-flatten-goal
% 0.20/0.50
% 0.20/0.50 % SZS status Theorem
% 0.20/0.50
% 0.20/0.51 % SZS output start Proof
% 0.20/0.51 Take the following subset of the input axioms:
% 0.20/0.51 fof(d7_ordinal1, axiom, ![A2]: ((relation(A2) & function(A2)) => (transfinite_sequence(A2) <=> ordinal(relation_dom(A2))))).
% 0.20/0.51 fof(d8_ordinal1, axiom, ![B, A2_2]: ((relation(B) & (function(B) & transfinite_sequence(B))) => (transfinite_sequence_of(B, A2_2) <=> subset(relation_rng(B), A2_2)))).
% 0.20/0.51 fof(reflexivity_r1_tarski, axiom, ![A, B2]: subset(A, A)).
% 0.20/0.51 fof(t46_ordinal1, conjecture, ![A3]: ((relation(A3) & function(A3)) => (ordinal(relation_dom(A3)) => transfinite_sequence_of(A3, relation_rng(A3))))).
% 0.20/0.51
% 0.20/0.51 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.51 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.51 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.51 fresh(y, y, x1...xn) = u
% 0.20/0.51 C => fresh(s, t, x1...xn) = v
% 0.20/0.51 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.51 variables of u and v.
% 0.20/0.51 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.51 input problem has no model of domain size 1).
% 0.20/0.51
% 0.20/0.51 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.51
% 0.20/0.51 Axiom 1 (t46_ordinal1): function(a) = true2.
% 0.20/0.51 Axiom 2 (t46_ordinal1_2): relation(a) = true2.
% 0.20/0.51 Axiom 3 (reflexivity_r1_tarski): subset(X, X) = true2.
% 0.20/0.51 Axiom 4 (t46_ordinal1_1): ordinal(relation_dom(a)) = true2.
% 0.20/0.51 Axiom 5 (d7_ordinal1): fresh44(X, X, Y) = true2.
% 0.20/0.51 Axiom 6 (d7_ordinal1): fresh22(X, X, Y) = transfinite_sequence(Y).
% 0.20/0.51 Axiom 7 (d7_ordinal1): fresh43(X, X, Y) = fresh44(function(Y), true2, Y).
% 0.20/0.51 Axiom 8 (d8_ordinal1_1): fresh42(X, X, Y, Z) = true2.
% 0.20/0.51 Axiom 9 (d8_ordinal1_1): fresh40(X, X, Y, Z) = transfinite_sequence_of(Z, Y).
% 0.20/0.51 Axiom 10 (d8_ordinal1_1): fresh41(X, X, Y, Z) = fresh42(function(Z), true2, Y, Z).
% 0.20/0.51 Axiom 11 (d8_ordinal1_1): fresh39(X, X, Y, Z) = fresh40(relation(Z), true2, Y, Z).
% 0.20/0.51 Axiom 12 (d7_ordinal1): fresh43(relation(X), true2, X) = fresh22(ordinal(relation_dom(X)), true2, X).
% 0.20/0.51 Axiom 13 (d8_ordinal1_1): fresh39(subset(relation_rng(X), Y), true2, Y, X) = fresh41(transfinite_sequence(X), true2, Y, X).
% 0.20/0.51
% 0.20/0.51 Goal 1 (t46_ordinal1_3): transfinite_sequence_of(a, relation_rng(a)) = true2.
% 0.20/0.51 Proof:
% 0.20/0.51 transfinite_sequence_of(a, relation_rng(a))
% 0.20/0.51 = { by axiom 9 (d8_ordinal1_1) R->L }
% 0.20/0.51 fresh40(true2, true2, relation_rng(a), a)
% 0.20/0.51 = { by axiom 2 (t46_ordinal1_2) R->L }
% 0.20/0.51 fresh40(relation(a), true2, relation_rng(a), a)
% 0.20/0.51 = { by axiom 11 (d8_ordinal1_1) R->L }
% 0.20/0.51 fresh39(true2, true2, relation_rng(a), a)
% 0.20/0.51 = { by axiom 3 (reflexivity_r1_tarski) R->L }
% 0.20/0.51 fresh39(subset(relation_rng(a), relation_rng(a)), true2, relation_rng(a), a)
% 0.20/0.51 = { by axiom 13 (d8_ordinal1_1) }
% 0.20/0.51 fresh41(transfinite_sequence(a), true2, relation_rng(a), a)
% 0.20/0.51 = { by axiom 6 (d7_ordinal1) R->L }
% 0.20/0.51 fresh41(fresh22(true2, true2, a), true2, relation_rng(a), a)
% 0.20/0.51 = { by axiom 4 (t46_ordinal1_1) R->L }
% 0.20/0.51 fresh41(fresh22(ordinal(relation_dom(a)), true2, a), true2, relation_rng(a), a)
% 0.20/0.51 = { by axiom 12 (d7_ordinal1) R->L }
% 0.20/0.51 fresh41(fresh43(relation(a), true2, a), true2, relation_rng(a), a)
% 0.20/0.51 = { by axiom 2 (t46_ordinal1_2) }
% 0.20/0.51 fresh41(fresh43(true2, true2, a), true2, relation_rng(a), a)
% 0.20/0.51 = { by axiom 7 (d7_ordinal1) }
% 0.20/0.51 fresh41(fresh44(function(a), true2, a), true2, relation_rng(a), a)
% 0.20/0.51 = { by axiom 1 (t46_ordinal1) }
% 0.20/0.51 fresh41(fresh44(true2, true2, a), true2, relation_rng(a), a)
% 0.20/0.51 = { by axiom 5 (d7_ordinal1) }
% 0.20/0.51 fresh41(true2, true2, relation_rng(a), a)
% 0.20/0.51 = { by axiom 10 (d8_ordinal1_1) }
% 0.20/0.51 fresh42(function(a), true2, relation_rng(a), a)
% 0.20/0.51 = { by axiom 1 (t46_ordinal1) }
% 0.20/0.51 fresh42(true2, true2, relation_rng(a), a)
% 0.20/0.51 = { by axiom 8 (d8_ordinal1_1) }
% 0.20/0.51 true2
% 0.20/0.51 % SZS output end Proof
% 0.20/0.51
% 0.20/0.51 RESULT: Theorem (the conjecture is true).
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