0.00/0.03 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.00/0.04 % Command : twee %s --tstp --casc --quiet --conditional-encoding if --smaller --drop-non-horn 0.03/0.23 % Computer : n063.star.cs.uiowa.edu 0.03/0.23 % Model : x86_64 x86_64 0.03/0.23 % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz 0.03/0.23 % Memory : 32218.625MB 0.03/0.23 % OS : Linux 3.10.0-693.2.2.el7.x86_64 0.06/0.23 % CPULimit : 300 0.06/0.23 % DateTime : Sat Jul 14 05:37:09 CDT 2018 0.06/0.23 % CPUTime : 0.44/0.66 % SZS status Theorem 0.44/0.66 0.44/0.67 % SZS output start Proof 0.44/0.67 Take the following subset of the input axioms: 0.44/0.67 fof(abstractness_v1_orders_2, axiom, 0.44/0.67 ![A]: 0.44/0.67 ((rel_str_of(the_carrier(A), the_InternalRel(A))=A 0.44/0.67 <= strict_rel_str(A)) 0.44/0.67 <= rel_str(A))). 0.44/0.67 fof(d1_yellow_1, axiom, 0.44/0.67 ![A]: incl_POSet(A)=rel_str_of(A, inclusion_order(A))). 0.44/0.67 fof(dt_k1_yellow_1, axiom, 0.44/0.67 ![A]: 0.44/0.67 (reflexive(inclusion_order(A)) 0.44/0.67 & (antisymmetric(inclusion_order(A)) 0.44/0.67 & (relation_of2_as_subset(inclusion_order(A), A, A) 0.44/0.67 & (v1_partfun1(inclusion_order(A), A, A) 0.44/0.67 & transitive(inclusion_order(A))))))). 0.44/0.67 fof(dt_k2_yellow_1, axiom, 0.44/0.67 ![A]: (rel_str(incl_POSet(A)) & strict_rel_str(incl_POSet(A)))). 0.44/0.67 fof(fc5_yellow_1, axiom, 0.44/0.67 ![A]: 0.44/0.67 (strict_rel_str(incl_POSet(A)) 0.44/0.67 & (reflexive_relstr(incl_POSet(A)) 0.44/0.67 & (antisymmetric_relstr(incl_POSet(A)) 0.44/0.67 & transitive_relstr(incl_POSet(A)))))). 0.44/0.67 fof(free_g1_orders_2, axiom, 0.44/0.67 ![A, B]: 0.44/0.67 (![C, D]: ((A=C & D=B) <= rel_str_of(A, B)=rel_str_of(C, D)) 0.44/0.67 <= relation_of2(B, A, A))). 0.44/0.67 fof(redefinition_m2_relset_1, axiom, 0.44/0.67 ![A, B, C]: 0.44/0.67 (relation_of2_as_subset(C, A, B) <=> relation_of2(C, A, B))). 0.44/0.67 fof(t4_waybel_7, conjecture, 0.44/0.67 ![A]: powerset(A)=the_carrier(boole_POSet(A))). 0.44/0.67 fof(t4_yellow_1, axiom, 0.44/0.67 ![A]: incl_POSet(powerset(A))=boole_POSet(A)). 0.44/0.67 0.44/0.67 Now clausify the problem and encode Horn clauses using encoding 3 of 0.44/0.67 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 0.44/0.67 We repeatedly replace C & s=t => u=v by the two clauses: 0.44/0.67 $$fresh(y, y, x1...xn) = u 0.44/0.67 C => $$fresh(s, t, x1...xn) = v 0.44/0.67 where $$fresh is a fresh function symbol and x1..xn are the free 0.44/0.67 variables of u and v. 0.44/0.67 A predicate p(X) is encoded as p(X)=$$true (this is sound, because the 0.44/0.67 input problem has no model of domain size 1). 0.44/0.67 0.44/0.67 The encoding turns the above axioms into the following unit equations and goals: 0.44/0.67 0.44/0.67 Axiom 1 (abstractness_v1_orders_2): $$fresh(X, X, Y) = Y. 0.44/0.67 Axiom 2 (abstractness_v1_orders_2): $$fresh46(X, X, Y) = rel_str_of(the_carrier(Y), the_InternalRel(Y)). 0.44/0.67 Axiom 36 (free_g1_orders_2): $$fresh3(X, X, Y, Z, W, V) = Y. 0.44/0.67 Axiom 37 (free_g1_orders_2): $$fresh2(X, X, Y, Z) = Z. 0.44/0.67 Axiom 45 (redefinition_m2_relset_1): $$fresh15(X, X, Y, Z, W) = $$true2. 0.44/0.67 Axiom 65 (fc5_yellow_1): strict_rel_str(incl_POSet(X)) = $$true2. 0.44/0.67 Axiom 101 (t4_yellow_1): incl_POSet(powerset(X)) = boole_POSet(X). 0.44/0.67 Axiom 126 (redefinition_m2_relset_1): $$fresh15(relation_of2_as_subset(X, Y, Z), $$true2, Y, Z, X) = relation_of2(X, Y, Z). 0.44/0.67 Axiom 127 (d1_yellow_1): incl_POSet(X) = rel_str_of(X, inclusion_order(X)). 0.44/0.67 Axiom 129 (free_g1_orders_2): $$fresh3(relation_of2(X, Y, Y), $$true2, Y, X, Z, W) = $$fresh2(rel_str_of(Y, X), rel_str_of(Z, W), Y, Z). 0.44/0.67 Axiom 174 (dt_k2_yellow_1_1): rel_str(incl_POSet(X)) = $$true2. 0.44/0.67 Axiom 188 (dt_k1_yellow_1): relation_of2_as_subset(inclusion_order(X), X, X) = $$true2. 0.44/0.67 Axiom 223 (abstractness_v1_orders_2): $$fresh46(rel_str(X), $$true2, X) = $$fresh(strict_rel_str(X), $$true2, X). 0.44/0.67 0.44/0.67 Goal 1 (t4_waybel_7): powerset(sK17_t4_waybel_7_A) = the_carrier(boole_POSet(sK17_t4_waybel_7_A)). 0.44/0.67 Proof: 0.44/0.67 powerset(sK17_t4_waybel_7_A) 0.44/0.67 = { by axiom 36 (free_g1_orders_2) } 0.44/0.67 $$fresh3($$true2, $$true2, powerset(sK17_t4_waybel_7_A), inclusion_order(powerset(sK17_t4_waybel_7_A)), the_carrier(incl_POSet(powerset(sK17_t4_waybel_7_A))), the_InternalRel(incl_POSet(powerset(sK17_t4_waybel_7_A)))) 0.44/0.67 = { by axiom 45 (redefinition_m2_relset_1) } 0.44/0.67 $$fresh3($$fresh15($$true2, $$true2, powerset(sK17_t4_waybel_7_A), powerset(sK17_t4_waybel_7_A), inclusion_order(powerset(sK17_t4_waybel_7_A))), $$true2, powerset(sK17_t4_waybel_7_A), inclusion_order(powerset(sK17_t4_waybel_7_A)), the_carrier(incl_POSet(powerset(sK17_t4_waybel_7_A))), the_InternalRel(incl_POSet(powerset(sK17_t4_waybel_7_A)))) 0.44/0.67 = { by axiom 188 (dt_k1_yellow_1) } 0.44/0.67 $$fresh3($$fresh15(relation_of2_as_subset(inclusion_order(powerset(sK17_t4_waybel_7_A)), powerset(sK17_t4_waybel_7_A), powerset(sK17_t4_waybel_7_A)), $$true2, powerset(sK17_t4_waybel_7_A), powerset(sK17_t4_waybel_7_A), inclusion_order(powerset(sK17_t4_waybel_7_A))), $$true2, powerset(sK17_t4_waybel_7_A), inclusion_order(powerset(sK17_t4_waybel_7_A)), the_carrier(incl_POSet(powerset(sK17_t4_waybel_7_A))), the_InternalRel(incl_POSet(powerset(sK17_t4_waybel_7_A)))) 0.44/0.67 = { by axiom 126 (redefinition_m2_relset_1) } 0.44/0.67 $$fresh3(relation_of2(inclusion_order(powerset(sK17_t4_waybel_7_A)), powerset(sK17_t4_waybel_7_A), powerset(sK17_t4_waybel_7_A)), $$true2, powerset(sK17_t4_waybel_7_A), inclusion_order(powerset(sK17_t4_waybel_7_A)), the_carrier(incl_POSet(powerset(sK17_t4_waybel_7_A))), the_InternalRel(incl_POSet(powerset(sK17_t4_waybel_7_A)))) 0.44/0.67 = { by axiom 129 (free_g1_orders_2) } 0.44/0.67 $$fresh2(rel_str_of(powerset(sK17_t4_waybel_7_A), inclusion_order(powerset(sK17_t4_waybel_7_A))), rel_str_of(the_carrier(incl_POSet(powerset(sK17_t4_waybel_7_A))), the_InternalRel(incl_POSet(powerset(sK17_t4_waybel_7_A)))), powerset(sK17_t4_waybel_7_A), the_carrier(incl_POSet(powerset(sK17_t4_waybel_7_A)))) 0.44/0.67 = { by axiom 127 (d1_yellow_1) } 0.44/0.67 $$fresh2(incl_POSet(powerset(sK17_t4_waybel_7_A)), rel_str_of(the_carrier(incl_POSet(powerset(sK17_t4_waybel_7_A))), the_InternalRel(incl_POSet(powerset(sK17_t4_waybel_7_A)))), powerset(sK17_t4_waybel_7_A), the_carrier(incl_POSet(powerset(sK17_t4_waybel_7_A)))) 0.44/0.67 = { by axiom 2 (abstractness_v1_orders_2) } 0.44/0.67 $$fresh2(incl_POSet(powerset(sK17_t4_waybel_7_A)), $$fresh46($$true2, $$true2, incl_POSet(powerset(sK17_t4_waybel_7_A))), powerset(sK17_t4_waybel_7_A), the_carrier(incl_POSet(powerset(sK17_t4_waybel_7_A)))) 0.44/0.67 = { by axiom 174 (dt_k2_yellow_1_1) } 0.44/0.67 $$fresh2(incl_POSet(powerset(sK17_t4_waybel_7_A)), $$fresh46(rel_str(incl_POSet(powerset(sK17_t4_waybel_7_A))), $$true2, incl_POSet(powerset(sK17_t4_waybel_7_A))), powerset(sK17_t4_waybel_7_A), the_carrier(incl_POSet(powerset(sK17_t4_waybel_7_A)))) 0.44/0.67 = { by axiom 223 (abstractness_v1_orders_2) } 0.44/0.67 $$fresh2(incl_POSet(powerset(sK17_t4_waybel_7_A)), $$fresh(strict_rel_str(incl_POSet(powerset(sK17_t4_waybel_7_A))), $$true2, incl_POSet(powerset(sK17_t4_waybel_7_A))), powerset(sK17_t4_waybel_7_A), the_carrier(incl_POSet(powerset(sK17_t4_waybel_7_A)))) 0.44/0.67 = { by axiom 65 (fc5_yellow_1) } 0.44/0.67 $$fresh2(incl_POSet(powerset(sK17_t4_waybel_7_A)), $$fresh($$true2, $$true2, incl_POSet(powerset(sK17_t4_waybel_7_A))), powerset(sK17_t4_waybel_7_A), the_carrier(incl_POSet(powerset(sK17_t4_waybel_7_A)))) 0.44/0.67 = { by axiom 1 (abstractness_v1_orders_2) } 0.44/0.67 $$fresh2(incl_POSet(powerset(sK17_t4_waybel_7_A)), incl_POSet(powerset(sK17_t4_waybel_7_A)), powerset(sK17_t4_waybel_7_A), the_carrier(incl_POSet(powerset(sK17_t4_waybel_7_A)))) 0.44/0.67 = { by axiom 37 (free_g1_orders_2) } 0.44/0.67 the_carrier(incl_POSet(powerset(sK17_t4_waybel_7_A))) 0.44/0.67 = { by axiom 101 (t4_yellow_1) } 0.44/0.67 the_carrier(boole_POSet(sK17_t4_waybel_7_A)) 0.44/0.67 % SZS output end Proof 0.44/0.67 0.44/0.67 RESULT: Theorem (the conjecture is true). 0.51/0.67 EOF