TSTP Solution File: SEU307+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SEU307+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 : n005.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 17:52:08 EDT 2023
% Result : Theorem 0.20s 0.61s
% 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 : SEU307+1 : TPTP v8.1.2. Released v3.3.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.35 % Computer : n005.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Wed Aug 23 16:05:23 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.20/0.61 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.20/0.61
% 0.20/0.61 % SZS status Theorem
% 0.20/0.61
% 0.20/0.62 % SZS output start Proof
% 0.20/0.62 Take the following subset of the input axioms:
% 0.20/0.62 fof(d3_pre_topc, axiom, ![A2]: (one_sorted_str(A2) => cast_as_carrier_subset(A2)=the_carrier(A2))).
% 0.20/0.62 fof(redefinition_k5_subset_1, axiom, ![B, C, A2_2]: ((element(B, powerset(A2_2)) & element(C, powerset(A2_2))) => subset_intersection2(A2_2, B, C)=set_intersection2(B, C))).
% 0.20/0.62 fof(reflexivity_r1_tarski, axiom, ![A, B2]: subset(A, A)).
% 0.20/0.62 fof(t15_pre_topc, conjecture, ![A3]: (one_sorted_str(A3) => ![B2]: (element(B2, powerset(the_carrier(A3))) => subset_intersection2(the_carrier(A3), B2, cast_as_carrier_subset(A3))=B2))).
% 0.20/0.62 fof(t28_xboole_1, axiom, ![B2, A2_2]: (subset(A2_2, B2) => set_intersection2(A2_2, B2)=A2_2)).
% 0.20/0.62 fof(t3_subset, axiom, ![B2, A2_2]: (element(A2_2, powerset(B2)) <=> subset(A2_2, B2))).
% 0.20/0.62
% 0.20/0.62 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.62 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.62 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.62 fresh(y, y, x1...xn) = u
% 0.20/0.62 C => fresh(s, t, x1...xn) = v
% 0.20/0.62 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.62 variables of u and v.
% 0.20/0.62 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.62 input problem has no model of domain size 1).
% 0.20/0.62
% 0.20/0.62 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.62
% 0.20/0.62 Axiom 1 (t15_pre_topc_1): one_sorted_str(a) = true2.
% 0.20/0.62 Axiom 2 (reflexivity_r1_tarski): subset(X, X) = true2.
% 0.20/0.62 Axiom 3 (d3_pre_topc): fresh48(X, X, Y) = the_carrier(Y).
% 0.20/0.62 Axiom 4 (t28_xboole_1): fresh(X, X, Y, Z) = Y.
% 0.20/0.62 Axiom 5 (d3_pre_topc): fresh48(one_sorted_str(X), true2, X) = cast_as_carrier_subset(X).
% 0.20/0.62 Axiom 6 (t3_subset): fresh9(X, X, Y, Z) = true2.
% 0.20/0.62 Axiom 7 (t3_subset_1): fresh8(X, X, Y, Z) = true2.
% 0.20/0.62 Axiom 8 (t15_pre_topc): element(b, powerset(the_carrier(a))) = true2.
% 0.20/0.62 Axiom 9 (redefinition_k5_subset_1): fresh12(X, X, Y, Z, W) = set_intersection2(Z, W).
% 0.20/0.62 Axiom 10 (redefinition_k5_subset_1): fresh11(X, X, Y, Z, W) = subset_intersection2(Y, Z, W).
% 0.20/0.62 Axiom 11 (t28_xboole_1): fresh(subset(X, Y), true2, X, Y) = set_intersection2(X, Y).
% 0.20/0.62 Axiom 12 (t3_subset_1): fresh8(subset(X, Y), true2, X, Y) = element(X, powerset(Y)).
% 0.20/0.62 Axiom 13 (t3_subset): fresh9(element(X, powerset(Y)), true2, X, Y) = subset(X, Y).
% 0.20/0.62 Axiom 14 (redefinition_k5_subset_1): fresh11(element(X, powerset(Y)), true2, Y, Z, X) = fresh12(element(Z, powerset(Y)), true2, Y, Z, X).
% 0.20/0.62
% 0.20/0.62 Lemma 15: the_carrier(a) = cast_as_carrier_subset(a).
% 0.20/0.62 Proof:
% 0.20/0.62 the_carrier(a)
% 0.20/0.62 = { by axiom 3 (d3_pre_topc) R->L }
% 0.20/0.62 fresh48(true2, true2, a)
% 0.20/0.62 = { by axiom 1 (t15_pre_topc_1) R->L }
% 0.20/0.62 fresh48(one_sorted_str(a), true2, a)
% 0.20/0.62 = { by axiom 5 (d3_pre_topc) }
% 0.20/0.62 cast_as_carrier_subset(a)
% 0.20/0.62
% 0.20/0.62 Goal 1 (t15_pre_topc_2): subset_intersection2(the_carrier(a), b, cast_as_carrier_subset(a)) = b.
% 0.20/0.62 Proof:
% 0.20/0.62 subset_intersection2(the_carrier(a), b, cast_as_carrier_subset(a))
% 0.20/0.62 = { by lemma 15 R->L }
% 0.20/0.62 subset_intersection2(the_carrier(a), b, the_carrier(a))
% 0.20/0.62 = { by axiom 10 (redefinition_k5_subset_1) R->L }
% 0.20/0.62 fresh11(true2, true2, the_carrier(a), b, the_carrier(a))
% 0.20/0.62 = { by axiom 7 (t3_subset_1) R->L }
% 0.20/0.62 fresh11(fresh8(true2, true2, the_carrier(a), the_carrier(a)), true2, the_carrier(a), b, the_carrier(a))
% 0.20/0.62 = { by axiom 2 (reflexivity_r1_tarski) R->L }
% 0.20/0.62 fresh11(fresh8(subset(the_carrier(a), the_carrier(a)), true2, the_carrier(a), the_carrier(a)), true2, the_carrier(a), b, the_carrier(a))
% 0.20/0.62 = { by axiom 12 (t3_subset_1) }
% 0.20/0.62 fresh11(element(the_carrier(a), powerset(the_carrier(a))), true2, the_carrier(a), b, the_carrier(a))
% 0.20/0.62 = { by axiom 14 (redefinition_k5_subset_1) }
% 0.20/0.62 fresh12(element(b, powerset(the_carrier(a))), true2, the_carrier(a), b, the_carrier(a))
% 0.20/0.62 = { by axiom 8 (t15_pre_topc) }
% 0.20/0.62 fresh12(true2, true2, the_carrier(a), b, the_carrier(a))
% 0.20/0.62 = { by axiom 9 (redefinition_k5_subset_1) }
% 0.20/0.62 set_intersection2(b, the_carrier(a))
% 0.20/0.62 = { by lemma 15 }
% 0.20/0.62 set_intersection2(b, cast_as_carrier_subset(a))
% 0.20/0.62 = { by axiom 11 (t28_xboole_1) R->L }
% 0.20/0.62 fresh(subset(b, cast_as_carrier_subset(a)), true2, b, cast_as_carrier_subset(a))
% 0.20/0.62 = { by lemma 15 R->L }
% 0.20/0.62 fresh(subset(b, the_carrier(a)), true2, b, cast_as_carrier_subset(a))
% 0.20/0.62 = { by axiom 13 (t3_subset) R->L }
% 0.20/0.62 fresh(fresh9(element(b, powerset(the_carrier(a))), true2, b, the_carrier(a)), true2, b, cast_as_carrier_subset(a))
% 0.20/0.62 = { by axiom 8 (t15_pre_topc) }
% 0.20/0.62 fresh(fresh9(true2, true2, b, the_carrier(a)), true2, b, cast_as_carrier_subset(a))
% 0.20/0.62 = { by axiom 6 (t3_subset) }
% 0.20/0.62 fresh(true2, true2, b, cast_as_carrier_subset(a))
% 0.20/0.62 = { by axiom 4 (t28_xboole_1) }
% 0.20/0.62 b
% 0.20/0.62 % SZS output end Proof
% 0.20/0.62
% 0.20/0.62 RESULT: Theorem (the conjecture is true).
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