TSTP Solution File: SEU118+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SEU118+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 : n019.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 17:51:03 EDT 2023
% Result : Theorem 0.20s 0.47s
% 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 : SEU118+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
% 0.14/0.34 % Computer : n019.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Wed Aug 23 13:00:58 EDT 2023
% 0.14/0.34 % CPUTime :
% 0.20/0.47 Command-line arguments: --no-flatten-goal
% 0.20/0.47
% 0.20/0.47 % SZS status Theorem
% 0.20/0.47
% 0.20/0.49 % SZS output start Proof
% 0.20/0.49 Take the following subset of the input axioms:
% 0.20/0.49 fof(cc2_finset_1, axiom, ![A2]: (finite(A2) => ![B]: (element(B, powerset(A2)) => finite(B)))).
% 0.20/0.49 fof(d5_finsub_1, axiom, ![B2, A2_2]: (preboolean(B2) => (B2=finite_subsets(A2_2) <=> ![C]: (in(C, B2) <=> (subset(C, A2_2) & finite(C)))))).
% 0.20/0.49 fof(dt_k5_finsub_1, axiom, ![A]: preboolean(finite_subsets(A))).
% 0.20/0.49 fof(t1_subset, axiom, ![B2, A2_2]: (in(A2_2, B2) => element(A2_2, B2))).
% 0.20/0.49 fof(t34_finsub_1, conjecture, ![B2, A3]: (element(B2, powerset(A3)) => (finite(A3) => element(B2, finite_subsets(A3))))).
% 0.20/0.49 fof(t3_subset, axiom, ![B2, A2_2]: (element(A2_2, powerset(B2)) <=> subset(A2_2, B2))).
% 0.20/0.49
% 0.20/0.49 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.49 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.49 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.49 fresh(y, y, x1...xn) = u
% 0.20/0.49 C => fresh(s, t, x1...xn) = v
% 0.20/0.49 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.49 variables of u and v.
% 0.20/0.49 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.49 input problem has no model of domain size 1).
% 0.20/0.49
% 0.20/0.49 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.49
% 0.20/0.49 Axiom 1 (t34_finsub_1): finite(a) = true2.
% 0.20/0.49 Axiom 2 (dt_k5_finsub_1): preboolean(finite_subsets(X)) = true2.
% 0.20/0.49 Axiom 3 (cc2_finset_1): fresh24(X, X, Y) = true2.
% 0.20/0.49 Axiom 4 (t34_finsub_1_1): element(b, powerset(a)) = true2.
% 0.20/0.49 Axiom 5 (d5_finsub_1_1): fresh32(X, X, Y, Z) = true2.
% 0.20/0.49 Axiom 6 (cc2_finset_1): fresh25(X, X, Y, Z) = finite(Z).
% 0.20/0.49 Axiom 7 (d5_finsub_1_3): fresh18(X, X, Y, Z) = equiv(Y, Z).
% 0.20/0.49 Axiom 8 (d5_finsub_1_3): fresh17(X, X, Y, Z) = true2.
% 0.20/0.49 Axiom 9 (t1_subset): fresh9(X, X, Y, Z) = true2.
% 0.20/0.49 Axiom 10 (t3_subset): fresh8(X, X, Y, Z) = true2.
% 0.20/0.49 Axiom 11 (d5_finsub_1_1): fresh31(X, X, Y, Z, W) = fresh32(Z, finite_subsets(Y), Z, W).
% 0.20/0.49 Axiom 12 (d5_finsub_1_1): fresh19(X, X, Y, Z, W) = in(W, Z).
% 0.20/0.49 Axiom 13 (d5_finsub_1_3): fresh18(subset(X, Y), true2, Y, X) = fresh17(finite(X), true2, Y, X).
% 0.20/0.49 Axiom 14 (t1_subset): fresh9(in(X, Y), true2, X, Y) = element(X, Y).
% 0.20/0.49 Axiom 15 (d5_finsub_1_1): fresh31(equiv(X, Y), true2, X, Z, Y) = fresh19(preboolean(Z), true2, X, Z, Y).
% 0.20/0.49 Axiom 16 (cc2_finset_1): fresh25(element(X, powerset(Y)), true2, Y, X) = fresh24(finite(Y), true2, X).
% 0.20/0.49 Axiom 17 (t3_subset): fresh8(element(X, powerset(Y)), true2, X, Y) = subset(X, Y).
% 0.20/0.49
% 0.20/0.49 Goal 1 (t34_finsub_1_2): element(b, finite_subsets(a)) = true2.
% 0.20/0.49 Proof:
% 0.20/0.49 element(b, finite_subsets(a))
% 0.20/0.49 = { by axiom 14 (t1_subset) R->L }
% 0.20/0.49 fresh9(in(b, finite_subsets(a)), true2, b, finite_subsets(a))
% 0.20/0.49 = { by axiom 12 (d5_finsub_1_1) R->L }
% 0.20/0.49 fresh9(fresh19(true2, true2, a, finite_subsets(a), b), true2, b, finite_subsets(a))
% 0.20/0.49 = { by axiom 2 (dt_k5_finsub_1) R->L }
% 0.20/0.49 fresh9(fresh19(preboolean(finite_subsets(a)), true2, a, finite_subsets(a), b), true2, b, finite_subsets(a))
% 0.20/0.49 = { by axiom 15 (d5_finsub_1_1) R->L }
% 0.20/0.49 fresh9(fresh31(equiv(a, b), true2, a, finite_subsets(a), b), true2, b, finite_subsets(a))
% 0.20/0.49 = { by axiom 7 (d5_finsub_1_3) R->L }
% 0.20/0.49 fresh9(fresh31(fresh18(true2, true2, a, b), true2, a, finite_subsets(a), b), true2, b, finite_subsets(a))
% 0.20/0.49 = { by axiom 10 (t3_subset) R->L }
% 0.20/0.49 fresh9(fresh31(fresh18(fresh8(true2, true2, b, a), true2, a, b), true2, a, finite_subsets(a), b), true2, b, finite_subsets(a))
% 0.20/0.49 = { by axiom 4 (t34_finsub_1_1) R->L }
% 0.20/0.49 fresh9(fresh31(fresh18(fresh8(element(b, powerset(a)), true2, b, a), true2, a, b), true2, a, finite_subsets(a), b), true2, b, finite_subsets(a))
% 0.20/0.49 = { by axiom 17 (t3_subset) }
% 0.20/0.49 fresh9(fresh31(fresh18(subset(b, a), true2, a, b), true2, a, finite_subsets(a), b), true2, b, finite_subsets(a))
% 0.20/0.49 = { by axiom 13 (d5_finsub_1_3) }
% 0.20/0.49 fresh9(fresh31(fresh17(finite(b), true2, a, b), true2, a, finite_subsets(a), b), true2, b, finite_subsets(a))
% 0.20/0.49 = { by axiom 6 (cc2_finset_1) R->L }
% 0.20/0.49 fresh9(fresh31(fresh17(fresh25(true2, true2, a, b), true2, a, b), true2, a, finite_subsets(a), b), true2, b, finite_subsets(a))
% 0.20/0.49 = { by axiom 4 (t34_finsub_1_1) R->L }
% 0.20/0.49 fresh9(fresh31(fresh17(fresh25(element(b, powerset(a)), true2, a, b), true2, a, b), true2, a, finite_subsets(a), b), true2, b, finite_subsets(a))
% 0.20/0.49 = { by axiom 16 (cc2_finset_1) }
% 0.20/0.49 fresh9(fresh31(fresh17(fresh24(finite(a), true2, b), true2, a, b), true2, a, finite_subsets(a), b), true2, b, finite_subsets(a))
% 0.20/0.49 = { by axiom 1 (t34_finsub_1) }
% 0.20/0.49 fresh9(fresh31(fresh17(fresh24(true2, true2, b), true2, a, b), true2, a, finite_subsets(a), b), true2, b, finite_subsets(a))
% 0.20/0.49 = { by axiom 3 (cc2_finset_1) }
% 0.20/0.49 fresh9(fresh31(fresh17(true2, true2, a, b), true2, a, finite_subsets(a), b), true2, b, finite_subsets(a))
% 0.20/0.49 = { by axiom 8 (d5_finsub_1_3) }
% 0.20/0.49 fresh9(fresh31(true2, true2, a, finite_subsets(a), b), true2, b, finite_subsets(a))
% 0.20/0.49 = { by axiom 11 (d5_finsub_1_1) }
% 0.20/0.50 fresh9(fresh32(finite_subsets(a), finite_subsets(a), finite_subsets(a), b), true2, b, finite_subsets(a))
% 0.20/0.50 = { by axiom 5 (d5_finsub_1_1) }
% 0.20/0.50 fresh9(true2, true2, b, finite_subsets(a))
% 0.20/0.50 = { by axiom 9 (t1_subset) }
% 0.20/0.50 true2
% 0.20/0.50 % SZS output end Proof
% 0.20/0.50
% 0.20/0.50 RESULT: Theorem (the conjecture is true).
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