TSTP Solution File: SEU275+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SEU275+1 : TPTP v8.1.2. Released v3.3.0.
% 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 17:51:57 EDT 2023
% Result : Theorem 0.19s 0.40s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : SEU275+1 : TPTP v8.1.2. Released v3.3.0.
% 0.07/0.13 % 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 : Wed Aug 23 14:12:37 EDT 2023
% 0.12/0.34 % CPUTime :
% 0.19/0.40 Command-line arguments: --no-flatten-goal
% 0.19/0.40
% 0.19/0.40 % SZS status Theorem
% 0.19/0.40
% 0.19/0.41 % SZS output start Proof
% 0.19/0.41 Take the following subset of the input axioms:
% 0.19/0.41 fof(d4_wellord1, axiom, ![A2]: (relation(A2) => (well_ordering(A2) <=> (reflexive(A2) & (transitive(A2) & (antisymmetric(A2) & (connected(A2) & well_founded_relation(A2)))))))).
% 0.19/0.41 fof(dt_k1_wellord2, axiom, ![A]: relation(inclusion_relation(A))).
% 0.19/0.41 fof(t2_wellord2, axiom, ![A3]: reflexive(inclusion_relation(A3))).
% 0.19/0.41 fof(t3_wellord2, axiom, ![A3]: transitive(inclusion_relation(A3))).
% 0.19/0.41 fof(t4_wellord2, axiom, ![A2_2]: (ordinal(A2_2) => connected(inclusion_relation(A2_2)))).
% 0.19/0.41 fof(t5_wellord2, axiom, ![A3]: antisymmetric(inclusion_relation(A3))).
% 0.19/0.41 fof(t6_wellord2, axiom, ![A2_2]: (ordinal(A2_2) => well_founded_relation(inclusion_relation(A2_2)))).
% 0.19/0.41 fof(t7_wellord2, conjecture, ![A3]: (ordinal(A3) => well_ordering(inclusion_relation(A3)))).
% 0.19/0.41
% 0.19/0.41 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.41 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.41 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.41 fresh(y, y, x1...xn) = u
% 0.19/0.41 C => fresh(s, t, x1...xn) = v
% 0.19/0.41 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.41 variables of u and v.
% 0.19/0.41 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.41 input problem has no model of domain size 1).
% 0.19/0.41
% 0.19/0.41 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.41
% 0.19/0.41 Axiom 1 (t7_wellord2): ordinal(a) = true.
% 0.19/0.41 Axiom 2 (t2_wellord2): reflexive(inclusion_relation(X)) = true.
% 0.19/0.41 Axiom 3 (t3_wellord2): transitive(inclusion_relation(X)) = true.
% 0.19/0.41 Axiom 4 (t5_wellord2): antisymmetric(inclusion_relation(X)) = true.
% 0.19/0.41 Axiom 5 (dt_k1_wellord2): relation(inclusion_relation(X)) = true.
% 0.19/0.41 Axiom 6 (t6_wellord2): fresh(X, X, Y) = true.
% 0.19/0.41 Axiom 7 (d4_wellord1_5): fresh22(X, X, Y) = true.
% 0.19/0.41 Axiom 8 (d4_wellord1_5): fresh20(X, X, Y) = well_ordering(Y).
% 0.19/0.41 Axiom 9 (t4_wellord2): fresh2(X, X, Y) = true.
% 0.19/0.41 Axiom 10 (t6_wellord2): fresh(ordinal(X), true, X) = well_founded_relation(inclusion_relation(X)).
% 0.19/0.41 Axiom 11 (d4_wellord1_5): fresh21(X, X, Y) = fresh22(relation(Y), true, Y).
% 0.19/0.41 Axiom 12 (d4_wellord1_5): fresh18(X, X, Y) = fresh21(transitive(Y), true, Y).
% 0.19/0.41 Axiom 13 (d4_wellord1_5): fresh19(X, X, Y) = fresh20(reflexive(Y), true, Y).
% 0.19/0.41 Axiom 14 (d4_wellord1_5): fresh17(X, X, Y) = fresh19(antisymmetric(Y), true, Y).
% 0.19/0.41 Axiom 15 (d4_wellord1_5): fresh17(well_founded_relation(X), true, X) = fresh18(connected(X), true, X).
% 0.19/0.41 Axiom 16 (t4_wellord2): fresh2(ordinal(X), true, X) = connected(inclusion_relation(X)).
% 0.19/0.41
% 0.19/0.41 Goal 1 (t7_wellord2_1): well_ordering(inclusion_relation(a)) = true.
% 0.19/0.41 Proof:
% 0.19/0.41 well_ordering(inclusion_relation(a))
% 0.19/0.41 = { by axiom 8 (d4_wellord1_5) R->L }
% 0.19/0.41 fresh20(true, true, inclusion_relation(a))
% 0.19/0.41 = { by axiom 2 (t2_wellord2) R->L }
% 0.19/0.41 fresh20(reflexive(inclusion_relation(a)), true, inclusion_relation(a))
% 0.19/0.41 = { by axiom 13 (d4_wellord1_5) R->L }
% 0.19/0.41 fresh19(true, true, inclusion_relation(a))
% 0.19/0.41 = { by axiom 4 (t5_wellord2) R->L }
% 0.19/0.41 fresh19(antisymmetric(inclusion_relation(a)), true, inclusion_relation(a))
% 0.19/0.41 = { by axiom 14 (d4_wellord1_5) R->L }
% 0.19/0.41 fresh17(true, true, inclusion_relation(a))
% 0.19/0.41 = { by axiom 6 (t6_wellord2) R->L }
% 0.19/0.41 fresh17(fresh(true, true, a), true, inclusion_relation(a))
% 0.19/0.41 = { by axiom 1 (t7_wellord2) R->L }
% 0.19/0.41 fresh17(fresh(ordinal(a), true, a), true, inclusion_relation(a))
% 0.19/0.41 = { by axiom 10 (t6_wellord2) }
% 0.19/0.41 fresh17(well_founded_relation(inclusion_relation(a)), true, inclusion_relation(a))
% 0.19/0.41 = { by axiom 15 (d4_wellord1_5) }
% 0.19/0.41 fresh18(connected(inclusion_relation(a)), true, inclusion_relation(a))
% 0.19/0.41 = { by axiom 16 (t4_wellord2) R->L }
% 0.19/0.41 fresh18(fresh2(ordinal(a), true, a), true, inclusion_relation(a))
% 0.19/0.41 = { by axiom 1 (t7_wellord2) }
% 0.19/0.41 fresh18(fresh2(true, true, a), true, inclusion_relation(a))
% 0.19/0.41 = { by axiom 9 (t4_wellord2) }
% 0.19/0.41 fresh18(true, true, inclusion_relation(a))
% 0.19/0.41 = { by axiom 12 (d4_wellord1_5) }
% 0.19/0.41 fresh21(transitive(inclusion_relation(a)), true, inclusion_relation(a))
% 0.19/0.41 = { by axiom 3 (t3_wellord2) }
% 0.19/0.41 fresh21(true, true, inclusion_relation(a))
% 0.19/0.41 = { by axiom 11 (d4_wellord1_5) }
% 0.19/0.41 fresh22(relation(inclusion_relation(a)), true, inclusion_relation(a))
% 0.19/0.41 = { by axiom 5 (dt_k1_wellord2) }
% 0.19/0.41 fresh22(true, true, inclusion_relation(a))
% 0.19/0.41 = { by axiom 7 (d4_wellord1_5) }
% 0.19/0.41 true
% 0.19/0.41 % SZS output end Proof
% 0.19/0.41
% 0.19/0.41 RESULT: Theorem (the conjecture is true).
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