TSTP Solution File: SEU325+2 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SEU325+2 : 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 : n004.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 17:52:13 EDT 2023
% Result : Theorem 227.42s 29.61s
% Output : Proof 228.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU325+2 : TPTP v8.1.2. Released v3.3.0.
% 0.12/0.13 % 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 : n004.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 17:53:08 EDT 2023
% 0.12/0.34 % CPUTime :
% 227.42/29.61 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 227.42/29.61
% 227.42/29.61 % SZS status Theorem
% 227.42/29.61
% 228.10/29.61 % SZS output start Proof
% 228.10/29.61 Take the following subset of the input axioms:
% 228.10/29.62 fof(d1_struct_0, axiom, ![A2]: (one_sorted_str(A2) => (empty_carrier(A2) <=> empty(the_carrier(A2))))).
% 228.10/29.62 fof(d3_pre_topc, axiom, ![A2_2]: (one_sorted_str(A2_2) => cast_as_carrier_subset(A2_2)=the_carrier(A2_2))).
% 228.10/29.62 fof(d8_pre_topc, axiom, ![A2_2]: (one_sorted_str(A2_2) => ![B]: (element(B, powerset(powerset(the_carrier(A2_2)))) => (is_a_cover_of_carrier(A2_2, B) <=> cast_as_carrier_subset(A2_2)=union_of_subsets(the_carrier(A2_2), B))))).
% 228.10/29.62 fof(dt_k5_setfam_1, axiom, ![B2, A2_2]: (element(B2, powerset(powerset(A2_2))) => element(union_of_subsets(A2_2, B2), powerset(A2_2)))).
% 228.10/29.62 fof(fc12_relat_1, axiom, empty(empty_set) & (relation(empty_set) & relation_empty_yielding(empty_set))).
% 228.10/29.62 fof(rc2_finset_1, axiom, ![A]: ?[B2]: (element(B2, powerset(A)) & (empty(B2) & (relation(B2) & (function(B2) & (one_to_one(B2) & (epsilon_transitive(B2) & (epsilon_connected(B2) & (ordinal(B2) & (natural(B2) & finite(B2))))))))))).
% 228.10/29.62 fof(redefinition_k5_setfam_1, axiom, ![B2, A2_2]: (element(B2, powerset(powerset(A2_2))) => union_of_subsets(A2_2, B2)=union(B2))).
% 228.10/29.62 fof(t2_xboole_1, lemma, ![A3]: subset(empty_set, A3)).
% 228.10/29.62 fof(t3_subset, axiom, ![B2, A2_2]: (element(A2_2, powerset(B2)) <=> subset(A2_2, B2))).
% 228.10/29.62 fof(t3_xboole_1, lemma, ![A2_2]: (subset(A2_2, empty_set) => A2_2=empty_set)).
% 228.10/29.62 fof(t5_tops_2, conjecture, ![A3]: ((~empty_carrier(A3) & one_sorted_str(A3)) => ![B2]: (element(B2, powerset(powerset(the_carrier(A3)))) => ~(is_a_cover_of_carrier(A3, B2) & B2=empty_set)))).
% 228.10/29.62 fof(t6_boole, axiom, ![A2_2]: (empty(A2_2) => A2_2=empty_set)).
% 228.10/29.62
% 228.10/29.62 Now clausify the problem and encode Horn clauses using encoding 3 of
% 228.10/29.62 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 228.10/29.62 We repeatedly replace C & s=t => u=v by the two clauses:
% 228.10/29.62 fresh(y, y, x1...xn) = u
% 228.10/29.62 C => fresh(s, t, x1...xn) = v
% 228.10/29.62 where fresh is a fresh function symbol and x1..xn are the free
% 228.10/29.62 variables of u and v.
% 228.10/29.62 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 228.10/29.62 input problem has no model of domain size 1).
% 228.10/29.62
% 228.10/29.62 The encoding turns the above axioms into the following unit equations and goals:
% 228.10/29.62
% 228.10/29.62 Axiom 1 (t5_tops_2): b2 = empty_set.
% 228.10/29.62 Axiom 2 (fc12_relat_1): empty(empty_set) = true2.
% 228.10/29.62 Axiom 3 (t5_tops_2_2): one_sorted_str(a) = true2.
% 228.10/29.62 Axiom 4 (t2_xboole_1): subset(empty_set, X) = true2.
% 228.10/29.62 Axiom 5 (rc2_finset_1_2): empty(b28(X)) = true2.
% 228.10/29.62 Axiom 6 (t5_tops_2_3): is_a_cover_of_carrier(a, b2) = true2.
% 228.10/29.62 Axiom 7 (d1_struct_0): fresh873(X, X, Y) = empty_carrier(Y).
% 228.10/29.62 Axiom 8 (d1_struct_0): fresh872(X, X, Y) = true2.
% 228.10/29.62 Axiom 9 (d3_pre_topc): fresh820(X, X, Y) = the_carrier(Y).
% 228.10/29.62 Axiom 10 (t3_xboole_1): fresh166(X, X, Y) = empty_set.
% 228.10/29.62 Axiom 11 (t6_boole): fresh122(X, X, Y) = empty_set.
% 228.10/29.62 Axiom 12 (rc2_finset_1): element(b28(X), powerset(X)) = true2.
% 228.10/29.62 Axiom 13 (d8_pre_topc_1): fresh1598(X, X, Y, Z) = union_of_subsets(the_carrier(Y), Z).
% 228.10/29.62 Axiom 14 (d3_pre_topc): fresh820(one_sorted_str(X), true2, X) = cast_as_carrier_subset(X).
% 228.10/29.62 Axiom 15 (d8_pre_topc_1): fresh718(X, X, Y, Z) = cast_as_carrier_subset(Y).
% 228.10/29.62 Axiom 16 (dt_k5_setfam_1): fresh689(X, X, Y, Z) = true2.
% 228.10/29.62 Axiom 17 (redefinition_k5_setfam_1): fresh494(X, X, Y, Z) = union(Z).
% 228.10/29.62 Axiom 18 (t3_subset): fresh168(X, X, Y, Z) = true2.
% 228.10/29.62 Axiom 19 (t3_subset_1): fresh167(X, X, Y, Z) = true2.
% 228.10/29.62 Axiom 20 (t6_boole): fresh122(empty(X), true2, X) = X.
% 228.10/29.62 Axiom 21 (t5_tops_2_1): element(b2, powerset(powerset(the_carrier(a)))) = true2.
% 228.10/29.62 Axiom 22 (d8_pre_topc_1): fresh1597(X, X, Y, Z) = fresh1598(one_sorted_str(Y), true2, Y, Z).
% 228.10/29.62 Axiom 23 (d1_struct_0): fresh873(one_sorted_str(X), true2, X) = fresh872(empty(the_carrier(X)), true2, X).
% 228.10/29.62 Axiom 24 (t3_xboole_1): fresh166(subset(X, empty_set), true2, X) = X.
% 228.10/29.62 Axiom 25 (t3_subset_1): fresh167(subset(X, Y), true2, X, Y) = element(X, powerset(Y)).
% 228.10/29.62 Axiom 26 (t3_subset): fresh168(element(X, powerset(Y)), true2, X, Y) = subset(X, Y).
% 228.10/29.62 Axiom 27 (dt_k5_setfam_1): fresh689(element(X, powerset(powerset(Y))), true2, Y, X) = element(union_of_subsets(Y, X), powerset(Y)).
% 228.10/29.62 Axiom 28 (redefinition_k5_setfam_1): fresh494(element(X, powerset(powerset(Y))), true2, Y, X) = union_of_subsets(Y, X).
% 228.10/29.62 Axiom 29 (d8_pre_topc_1): fresh1597(is_a_cover_of_carrier(X, Y), true2, X, Y) = fresh718(element(Y, powerset(powerset(the_carrier(X)))), true2, X, Y).
% 228.10/29.62
% 228.10/29.62 Lemma 30: element(empty_set, powerset(X)) = true2.
% 228.10/29.62 Proof:
% 228.10/29.62 element(empty_set, powerset(X))
% 228.10/29.62 = { by axiom 25 (t3_subset_1) R->L }
% 228.10/29.62 fresh167(subset(empty_set, X), true2, empty_set, X)
% 228.10/29.62 = { by axiom 4 (t2_xboole_1) }
% 228.10/29.62 fresh167(true2, true2, empty_set, X)
% 228.10/29.62 = { by axiom 19 (t3_subset_1) }
% 228.10/29.62 true2
% 228.10/29.62
% 228.10/29.62 Goal 1 (t5_tops_2_4): empty_carrier(a) = true2.
% 228.10/29.62 Proof:
% 228.10/29.62 empty_carrier(a)
% 228.10/29.62 = { by axiom 7 (d1_struct_0) R->L }
% 228.10/29.62 fresh873(true2, true2, a)
% 228.10/29.62 = { by axiom 3 (t5_tops_2_2) R->L }
% 228.10/29.62 fresh873(one_sorted_str(a), true2, a)
% 228.10/29.62 = { by axiom 23 (d1_struct_0) }
% 228.10/29.62 fresh872(empty(the_carrier(a)), true2, a)
% 228.10/29.62 = { by axiom 9 (d3_pre_topc) R->L }
% 228.10/29.62 fresh872(empty(fresh820(true2, true2, a)), true2, a)
% 228.10/29.62 = { by axiom 3 (t5_tops_2_2) R->L }
% 228.10/29.62 fresh872(empty(fresh820(one_sorted_str(a), true2, a)), true2, a)
% 228.10/29.62 = { by axiom 14 (d3_pre_topc) }
% 228.10/29.62 fresh872(empty(cast_as_carrier_subset(a)), true2, a)
% 228.10/29.62 = { by axiom 15 (d8_pre_topc_1) R->L }
% 228.10/29.62 fresh872(empty(fresh718(true2, true2, a, empty_set)), true2, a)
% 228.10/29.62 = { by axiom 21 (t5_tops_2_1) R->L }
% 228.10/29.62 fresh872(empty(fresh718(element(b2, powerset(powerset(the_carrier(a)))), true2, a, empty_set)), true2, a)
% 228.10/29.62 = { by axiom 1 (t5_tops_2) }
% 228.10/29.62 fresh872(empty(fresh718(element(empty_set, powerset(powerset(the_carrier(a)))), true2, a, empty_set)), true2, a)
% 228.10/29.62 = { by axiom 29 (d8_pre_topc_1) R->L }
% 228.10/29.62 fresh872(empty(fresh1597(is_a_cover_of_carrier(a, empty_set), true2, a, empty_set)), true2, a)
% 228.10/29.62 = { by axiom 1 (t5_tops_2) R->L }
% 228.10/29.62 fresh872(empty(fresh1597(is_a_cover_of_carrier(a, b2), true2, a, empty_set)), true2, a)
% 228.10/29.62 = { by axiom 6 (t5_tops_2_3) }
% 228.10/29.62 fresh872(empty(fresh1597(true2, true2, a, empty_set)), true2, a)
% 228.10/29.62 = { by axiom 22 (d8_pre_topc_1) }
% 228.10/29.62 fresh872(empty(fresh1598(one_sorted_str(a), true2, a, empty_set)), true2, a)
% 228.10/29.62 = { by axiom 3 (t5_tops_2_2) }
% 228.10/29.62 fresh872(empty(fresh1598(true2, true2, a, empty_set)), true2, a)
% 228.10/29.62 = { by axiom 13 (d8_pre_topc_1) }
% 228.10/29.62 fresh872(empty(union_of_subsets(the_carrier(a), empty_set)), true2, a)
% 228.10/29.62 = { by axiom 28 (redefinition_k5_setfam_1) R->L }
% 228.10/29.62 fresh872(empty(fresh494(element(empty_set, powerset(powerset(the_carrier(a)))), true2, the_carrier(a), empty_set)), true2, a)
% 228.10/29.62 = { by lemma 30 }
% 228.10/29.62 fresh872(empty(fresh494(true2, true2, the_carrier(a), empty_set)), true2, a)
% 228.10/29.62 = { by axiom 17 (redefinition_k5_setfam_1) }
% 228.20/29.62 fresh872(empty(union(empty_set)), true2, a)
% 228.20/29.62 = { by axiom 24 (t3_xboole_1) R->L }
% 228.20/29.62 fresh872(empty(fresh166(subset(union(empty_set), empty_set), true2, union(empty_set))), true2, a)
% 228.20/29.62 = { by axiom 26 (t3_subset) R->L }
% 228.20/29.62 fresh872(empty(fresh166(fresh168(element(union(empty_set), powerset(empty_set)), true2, union(empty_set), empty_set), true2, union(empty_set))), true2, a)
% 228.20/29.62 = { by axiom 17 (redefinition_k5_setfam_1) R->L }
% 228.20/29.62 fresh872(empty(fresh166(fresh168(element(fresh494(true2, true2, empty_set, empty_set), powerset(empty_set)), true2, union(empty_set), empty_set), true2, union(empty_set))), true2, a)
% 228.20/29.62 = { by lemma 30 R->L }
% 228.20/29.62 fresh872(empty(fresh166(fresh168(element(fresh494(element(empty_set, powerset(powerset(empty_set))), true2, empty_set, empty_set), powerset(empty_set)), true2, union(empty_set), empty_set), true2, union(empty_set))), true2, a)
% 228.20/29.62 = { by axiom 28 (redefinition_k5_setfam_1) }
% 228.20/29.62 fresh872(empty(fresh166(fresh168(element(union_of_subsets(empty_set, empty_set), powerset(empty_set)), true2, union(empty_set), empty_set), true2, union(empty_set))), true2, a)
% 228.20/29.62 = { by axiom 11 (t6_boole) R->L }
% 228.20/29.62 fresh872(empty(fresh166(fresh168(element(union_of_subsets(empty_set, fresh122(true2, true2, b28(powerset(empty_set)))), powerset(empty_set)), true2, union(empty_set), empty_set), true2, union(empty_set))), true2, a)
% 228.20/29.62 = { by axiom 5 (rc2_finset_1_2) R->L }
% 228.20/29.62 fresh872(empty(fresh166(fresh168(element(union_of_subsets(empty_set, fresh122(empty(b28(powerset(empty_set))), true2, b28(powerset(empty_set)))), powerset(empty_set)), true2, union(empty_set), empty_set), true2, union(empty_set))), true2, a)
% 228.20/29.62 = { by axiom 20 (t6_boole) }
% 228.20/29.62 fresh872(empty(fresh166(fresh168(element(union_of_subsets(empty_set, b28(powerset(empty_set))), powerset(empty_set)), true2, union(empty_set), empty_set), true2, union(empty_set))), true2, a)
% 228.20/29.62 = { by axiom 27 (dt_k5_setfam_1) R->L }
% 228.20/29.62 fresh872(empty(fresh166(fresh168(fresh689(element(b28(powerset(empty_set)), powerset(powerset(empty_set))), true2, empty_set, b28(powerset(empty_set))), true2, union(empty_set), empty_set), true2, union(empty_set))), true2, a)
% 228.20/29.62 = { by axiom 12 (rc2_finset_1) }
% 228.20/29.62 fresh872(empty(fresh166(fresh168(fresh689(true2, true2, empty_set, b28(powerset(empty_set))), true2, union(empty_set), empty_set), true2, union(empty_set))), true2, a)
% 228.20/29.62 = { by axiom 16 (dt_k5_setfam_1) }
% 228.20/29.62 fresh872(empty(fresh166(fresh168(true2, true2, union(empty_set), empty_set), true2, union(empty_set))), true2, a)
% 228.20/29.62 = { by axiom 18 (t3_subset) }
% 228.20/29.62 fresh872(empty(fresh166(true2, true2, union(empty_set))), true2, a)
% 228.20/29.62 = { by axiom 10 (t3_xboole_1) }
% 228.20/29.62 fresh872(empty(empty_set), true2, a)
% 228.20/29.62 = { by axiom 2 (fc12_relat_1) }
% 228.20/29.62 fresh872(true2, true2, a)
% 228.20/29.62 = { by axiom 8 (d1_struct_0) }
% 228.20/29.62 true2
% 228.20/29.62 % SZS output end Proof
% 228.20/29.62
% 228.20/29.62 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------