%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SEU342+1 : TPTP v9.0.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n010.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 : Sat Jun 21 05:27:39 AM UTC 2025
% Result : Theorem 0.19s 0.38s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : SEU342+1 : TPTP v9.0.0. Released v3.3.0.
% 0.11/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n010.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Fri Jun 20 06:51:12 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.19/0.38 % SZS status Theorem
% 0.19/0.38 % SZS output start Proof
% 0.19/0.38 tff(below_refl_type, type, (
% 0.19/0.38 below_refl: ( $i * $i * $i ) > $o)).
% 0.19/0.38 tff(tptp_fun_A_11_type, type, (
% 0.19/0.38 tptp_fun_A_11: $i)).
% 0.19/0.38 tff(ordered_pair_as_product_element_type, type, (
% 0.19/0.38 ordered_pair_as_product_element: ( $i * $i * $i * $i ) > $i)).
% 0.19/0.38 tff(the_carrier_type, type, (
% 0.19/0.38 the_carrier: $i > $i)).
% 0.19/0.38 tff(tptp_fun_C_13_type, type, (
% 0.19/0.38 tptp_fun_C_13: $i)).
% 0.19/0.38 tff(tptp_fun_B_12_type, type, (
% 0.19/0.38 tptp_fun_B_12: $i)).
% 0.19/0.38 tff(element_type, type, (
% 0.19/0.38 element: ( $i * $i ) > $o)).
% 0.19/0.38 tff(in_type, type, (
% 0.19/0.38 in: ( $i * $i ) > $o)).
% 0.19/0.38 tff(a_1_0_filter_1_type, type, (
% 0.19/0.38 a_1_0_filter_1: $i > $i)).
% 0.19/0.38 tff(relation_of_lattice_type, type, (
% 0.19/0.38 relation_of_lattice: $i > $i)).
% 0.19/0.38 tff(latt_str_type, type, (
% 0.19/0.38 latt_str: $i > $o)).
% 0.19/0.38 tff(lattice_type, type, (
% 0.19/0.38 lattice: $i > $o)).
% 0.19/0.38 tff(empty_carrier_type, type, (
% 0.19/0.38 empty_carrier: $i > $o)).
% 0.19/0.38 tff(tptp_fun_D_5_type, type, (
% 0.19/0.38 tptp_fun_D_5: ( $i * $i ) > $i)).
% 0.19/0.38 tff(tptp_fun_C_6_type, type, (
% 0.19/0.38 tptp_fun_C_6: ( $i * $i ) > $i)).
% 0.19/0.38 tff(ordered_pair_type, type, (
% 0.19/0.38 ordered_pair: ( $i * $i ) > $i)).
% 0.19/0.38 tff(empty_type, type, (
% 0.19/0.38 empty: $i > $o)).
% 0.19/0.38 tff(one_sorted_str_type, type, (
% 0.19/0.38 one_sorted_str: $i > $o)).
% 0.19/0.38 tff(join_semilatt_str_type, type, (
% 0.19/0.38 join_semilatt_str: $i > $o)).
% 0.19/0.38 tff(meet_semilatt_str_type, type, (
% 0.19/0.38 meet_semilatt_str: $i > $o)).
% 0.19/0.38 tff(1,plain,
% 0.19/0.38 ((((~empty_carrier(A!11)) & lattice(A!11) & latt_str(A!11)) & (element(B!12, the_carrier(A!11)) & (~((~element(C!13, the_carrier(A!11))) | (in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), relation_of_lattice(A!11)) <=> below_refl(A!11, B!12, C!13)))))) <=> ((~empty_carrier(A!11)) & lattice(A!11) & latt_str(A!11) & element(B!12, the_carrier(A!11)) & (~((~element(C!13, the_carrier(A!11))) | (in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), relation_of_lattice(A!11)) <=> below_refl(A!11, B!12, C!13)))))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(2,plain,
% 0.19/0.38 ((~![A: $i] : ((~((~empty_carrier(A)) & lattice(A) & latt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (in(ordered_pair_as_product_element(the_carrier(A), the_carrier(A), B, C), relation_of_lattice(A)) <=> below_refl(A, B, C)))))) <=> (~![A: $i] : ((~((~empty_carrier(A)) & lattice(A) & latt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (in(ordered_pair_as_product_element(the_carrier(A), the_carrier(A), B, C), relation_of_lattice(A)) <=> below_refl(A, B, C))))))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(3,plain,
% 0.19/0.38 ((~![A: $i] : ((((~empty_carrier(A)) & lattice(A)) & latt_str(A)) => ![B: $i] : (element(B, the_carrier(A)) => ![C: $i] : (element(C, the_carrier(A)) => (in(ordered_pair_as_product_element(the_carrier(A), the_carrier(A), B, C), relation_of_lattice(A)) <=> below_refl(A, B, C)))))) <=> (~![A: $i] : ((~((~empty_carrier(A)) & lattice(A) & latt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (in(ordered_pair_as_product_element(the_carrier(A), the_carrier(A), B, C), relation_of_lattice(A)) <=> below_refl(A, B, C))))))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(4,axiom,(~![A: $i] : ((((~empty_carrier(A)) & lattice(A)) & latt_str(A)) => ![B: $i] : (element(B, the_carrier(A)) => ![C: $i] : (element(C, the_carrier(A)) => (in(ordered_pair_as_product_element(the_carrier(A), the_carrier(A), B, C), relation_of_lattice(A)) <=> below_refl(A, B, C)))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t32_filter_1')).
% 0.19/0.38 tff(5,plain,
% 0.19/0.38 (~![A: $i] : ((~((~empty_carrier(A)) & lattice(A) & latt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (in(ordered_pair_as_product_element(the_carrier(A), the_carrier(A), B, C), relation_of_lattice(A)) <=> below_refl(A, B, C)))))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.19/0.38 tff(6,plain,
% 0.19/0.38 (~![A: $i] : ((~((~empty_carrier(A)) & lattice(A) & latt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (in(ordered_pair_as_product_element(the_carrier(A), the_carrier(A), B, C), relation_of_lattice(A)) <=> below_refl(A, B, C)))))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[5, 2])).
% 0.19/0.38 tff(7,plain,
% 0.19/0.38 (~![A: $i] : ((~((~empty_carrier(A)) & lattice(A) & latt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (in(ordered_pair_as_product_element(the_carrier(A), the_carrier(A), B, C), relation_of_lattice(A)) <=> below_refl(A, B, C)))))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.19/0.38 tff(8,plain,
% 0.19/0.38 (~![A: $i] : ((~((~empty_carrier(A)) & lattice(A) & latt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (in(ordered_pair_as_product_element(the_carrier(A), the_carrier(A), B, C), relation_of_lattice(A)) <=> below_refl(A, B, C)))))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[7, 2])).
% 0.19/0.38 tff(9,plain,
% 0.19/0.38 ((~empty_carrier(A!11)) & lattice(A!11) & latt_str(A!11) & element(B!12, the_carrier(A!11)) & (~((~element(C!13, the_carrier(A!11))) | (in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), relation_of_lattice(A!11)) <=> below_refl(A!11, B!12, C!13))))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[8, 1])).
% 0.19/0.38 tff(10,plain,
% 0.19/0.38 (latt_str(A!11)),
% 0.19/0.38 inference(and_elim,[status(thm)],[9])).
% 0.19/0.38 tff(11,plain,
% 0.19/0.38 (lattice(A!11)),
% 0.19/0.38 inference(and_elim,[status(thm)],[9])).
% 0.19/0.38 tff(12,plain,
% 0.19/0.38 (~empty_carrier(A!11)),
% 0.19/0.38 inference(and_elim,[status(thm)],[9])).
% 0.19/0.38 tff(13,plain,
% 0.19/0.38 (^[A: $i] : refl((empty_carrier(A) | (relation_of_lattice(A) = a_1_0_filter_1(A)) | (~latt_str(A)) | (~lattice(A))) <=> (empty_carrier(A) | (relation_of_lattice(A) = a_1_0_filter_1(A)) | (~latt_str(A)) | (~lattice(A))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(14,plain,
% 0.19/0.38 (![A: $i] : (empty_carrier(A) | (relation_of_lattice(A) = a_1_0_filter_1(A)) | (~latt_str(A)) | (~lattice(A))) <=> ![A: $i] : (empty_carrier(A) | (relation_of_lattice(A) = a_1_0_filter_1(A)) | (~latt_str(A)) | (~lattice(A)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[13])).
% 0.19/0.38 tff(15,plain,
% 0.19/0.38 (^[A: $i] : trans(monotonicity(trans(monotonicity(rewrite(((~empty_carrier(A)) & lattice(A) & latt_str(A)) <=> (~(empty_carrier(A) | (~latt_str(A)) | (~lattice(A))))), ((~((~empty_carrier(A)) & lattice(A) & latt_str(A))) <=> (~(~(empty_carrier(A) | (~latt_str(A)) | (~lattice(A))))))), rewrite((~(~(empty_carrier(A) | (~latt_str(A)) | (~lattice(A))))) <=> (empty_carrier(A) | (~latt_str(A)) | (~lattice(A)))), ((~((~empty_carrier(A)) & lattice(A) & latt_str(A))) <=> (empty_carrier(A) | (~latt_str(A)) | (~lattice(A))))), (((~((~empty_carrier(A)) & lattice(A) & latt_str(A))) | (relation_of_lattice(A) = a_1_0_filter_1(A))) <=> ((empty_carrier(A) | (~latt_str(A)) | (~lattice(A))) | (relation_of_lattice(A) = a_1_0_filter_1(A))))), rewrite(((empty_carrier(A) | (~latt_str(A)) | (~lattice(A))) | (relation_of_lattice(A) = a_1_0_filter_1(A))) <=> (empty_carrier(A) | (relation_of_lattice(A) = a_1_0_filter_1(A)) | (~latt_str(A)) | (~lattice(A)))), (((~((~empty_carrier(A)) & lattice(A) & latt_str(A))) | (relation_of_lattice(A) = a_1_0_filter_1(A))) <=> (empty_carrier(A) | (relation_of_lattice(A) = a_1_0_filter_1(A)) | (~latt_str(A)) | (~lattice(A)))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(16,plain,
% 0.19/0.38 (![A: $i] : ((~((~empty_carrier(A)) & lattice(A) & latt_str(A))) | (relation_of_lattice(A) = a_1_0_filter_1(A))) <=> ![A: $i] : (empty_carrier(A) | (relation_of_lattice(A) = a_1_0_filter_1(A)) | (~latt_str(A)) | (~lattice(A)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[15])).
% 0.19/0.38 tff(17,plain,
% 0.19/0.38 (![A: $i] : ((~((~empty_carrier(A)) & lattice(A) & latt_str(A))) | (relation_of_lattice(A) = a_1_0_filter_1(A))) <=> ![A: $i] : ((~((~empty_carrier(A)) & lattice(A) & latt_str(A))) | (relation_of_lattice(A) = a_1_0_filter_1(A)))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(18,plain,
% 0.19/0.38 (^[A: $i] : trans(monotonicity(rewrite((((~empty_carrier(A)) & lattice(A)) & latt_str(A)) <=> ((~empty_carrier(A)) & lattice(A) & latt_str(A))), (((((~empty_carrier(A)) & lattice(A)) & latt_str(A)) => (relation_of_lattice(A) = a_1_0_filter_1(A))) <=> (((~empty_carrier(A)) & lattice(A) & latt_str(A)) => (relation_of_lattice(A) = a_1_0_filter_1(A))))), rewrite((((~empty_carrier(A)) & lattice(A) & latt_str(A)) => (relation_of_lattice(A) = a_1_0_filter_1(A))) <=> ((~((~empty_carrier(A)) & lattice(A) & latt_str(A))) | (relation_of_lattice(A) = a_1_0_filter_1(A)))), (((((~empty_carrier(A)) & lattice(A)) & latt_str(A)) => (relation_of_lattice(A) = a_1_0_filter_1(A))) <=> ((~((~empty_carrier(A)) & lattice(A) & latt_str(A))) | (relation_of_lattice(A) = a_1_0_filter_1(A)))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(19,plain,
% 0.19/0.38 (![A: $i] : ((((~empty_carrier(A)) & lattice(A)) & latt_str(A)) => (relation_of_lattice(A) = a_1_0_filter_1(A))) <=> ![A: $i] : ((~((~empty_carrier(A)) & lattice(A) & latt_str(A))) | (relation_of_lattice(A) = a_1_0_filter_1(A)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[18])).
% 0.19/0.38 tff(20,axiom,(![A: $i] : ((((~empty_carrier(A)) & lattice(A)) & latt_str(A)) => (relation_of_lattice(A) = a_1_0_filter_1(A)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','d8_filter_1')).
% 0.19/0.38 tff(21,plain,
% 0.19/0.38 (![A: $i] : ((~((~empty_carrier(A)) & lattice(A) & latt_str(A))) | (relation_of_lattice(A) = a_1_0_filter_1(A)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[20, 19])).
% 0.19/0.38 tff(22,plain,
% 0.19/0.38 (![A: $i] : ((~((~empty_carrier(A)) & lattice(A) & latt_str(A))) | (relation_of_lattice(A) = a_1_0_filter_1(A)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[21, 17])).
% 0.19/0.38 tff(23,plain,(
% 0.19/0.38 ![A: $i] : ((~((~empty_carrier(A)) & lattice(A) & latt_str(A))) | (relation_of_lattice(A) = a_1_0_filter_1(A)))),
% 0.19/0.38 inference(skolemize,[status(sab)],[22])).
% 0.19/0.38 tff(24,plain,
% 0.19/0.38 (![A: $i] : (empty_carrier(A) | (relation_of_lattice(A) = a_1_0_filter_1(A)) | (~latt_str(A)) | (~lattice(A)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[23, 16])).
% 0.19/0.38 tff(25,plain,
% 0.19/0.38 (![A: $i] : (empty_carrier(A) | (relation_of_lattice(A) = a_1_0_filter_1(A)) | (~latt_str(A)) | (~lattice(A)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[24, 14])).
% 0.19/0.38 tff(26,plain,
% 0.19/0.38 (((~![A: $i] : (empty_carrier(A) | (relation_of_lattice(A) = a_1_0_filter_1(A)) | (~latt_str(A)) | (~lattice(A)))) | (empty_carrier(A!11) | (~latt_str(A!11)) | (~lattice(A!11)) | (relation_of_lattice(A!11) = a_1_0_filter_1(A!11)))) <=> ((~![A: $i] : (empty_carrier(A) | (relation_of_lattice(A) = a_1_0_filter_1(A)) | (~latt_str(A)) | (~lattice(A)))) | empty_carrier(A!11) | (~latt_str(A!11)) | (~lattice(A!11)) | (relation_of_lattice(A!11) = a_1_0_filter_1(A!11)))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(27,plain,
% 0.19/0.38 ((empty_carrier(A!11) | (relation_of_lattice(A!11) = a_1_0_filter_1(A!11)) | (~latt_str(A!11)) | (~lattice(A!11))) <=> (empty_carrier(A!11) | (~latt_str(A!11)) | (~lattice(A!11)) | (relation_of_lattice(A!11) = a_1_0_filter_1(A!11)))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(28,plain,
% 0.19/0.38 (((~![A: $i] : (empty_carrier(A) | (relation_of_lattice(A) = a_1_0_filter_1(A)) | (~latt_str(A)) | (~lattice(A)))) | (empty_carrier(A!11) | (relation_of_lattice(A!11) = a_1_0_filter_1(A!11)) | (~latt_str(A!11)) | (~lattice(A!11)))) <=> ((~![A: $i] : (empty_carrier(A) | (relation_of_lattice(A) = a_1_0_filter_1(A)) | (~latt_str(A)) | (~lattice(A)))) | (empty_carrier(A!11) | (~latt_str(A!11)) | (~lattice(A!11)) | (relation_of_lattice(A!11) = a_1_0_filter_1(A!11))))),
% 0.19/0.38 inference(monotonicity,[status(thm)],[27])).
% 0.19/0.38 tff(29,plain,
% 0.19/0.38 (((~![A: $i] : (empty_carrier(A) | (relation_of_lattice(A) = a_1_0_filter_1(A)) | (~latt_str(A)) | (~lattice(A)))) | (empty_carrier(A!11) | (relation_of_lattice(A!11) = a_1_0_filter_1(A!11)) | (~latt_str(A!11)) | (~lattice(A!11)))) <=> ((~![A: $i] : (empty_carrier(A) | (relation_of_lattice(A) = a_1_0_filter_1(A)) | (~latt_str(A)) | (~lattice(A)))) | empty_carrier(A!11) | (~latt_str(A!11)) | (~lattice(A!11)) | (relation_of_lattice(A!11) = a_1_0_filter_1(A!11)))),
% 0.19/0.38 inference(transitivity,[status(thm)],[28, 26])).
% 0.19/0.38 tff(30,plain,
% 0.19/0.38 ((~![A: $i] : (empty_carrier(A) | (relation_of_lattice(A) = a_1_0_filter_1(A)) | (~latt_str(A)) | (~lattice(A)))) | (empty_carrier(A!11) | (relation_of_lattice(A!11) = a_1_0_filter_1(A!11)) | (~latt_str(A!11)) | (~lattice(A!11)))),
% 0.19/0.38 inference(quant_inst,[status(thm)],[])).
% 0.19/0.38 tff(31,plain,
% 0.19/0.38 ((~![A: $i] : (empty_carrier(A) | (relation_of_lattice(A) = a_1_0_filter_1(A)) | (~latt_str(A)) | (~lattice(A)))) | empty_carrier(A!11) | (~latt_str(A!11)) | (~lattice(A!11)) | (relation_of_lattice(A!11) = a_1_0_filter_1(A!11))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[30, 29])).
% 0.19/0.38 tff(32,plain,
% 0.19/0.38 (relation_of_lattice(A!11) = a_1_0_filter_1(A!11)),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[31, 25, 12, 11, 10])).
% 0.19/0.38 tff(33,plain,
% 0.19/0.38 (a_1_0_filter_1(A!11) = relation_of_lattice(A!11)),
% 0.19/0.38 inference(symmetry,[status(thm)],[32])).
% 0.19/0.38 tff(34,plain,
% 0.19/0.38 (in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11)) <=> in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), relation_of_lattice(A!11))),
% 0.19/0.38 inference(monotonicity,[status(thm)],[33])).
% 0.19/0.38 tff(35,plain,
% 0.19/0.38 (in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), relation_of_lattice(A!11)) <=> in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11))),
% 0.19/0.38 inference(symmetry,[status(thm)],[34])).
% 0.19/0.38 tff(36,plain,
% 0.19/0.38 ((~in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), relation_of_lattice(A!11))) <=> (~in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11)))),
% 0.19/0.38 inference(monotonicity,[status(thm)],[35])).
% 0.19/0.38 tff(37,assumption,(~below_refl(A!11, B!12, C!13)), introduced(assumption)).
% 0.19/0.38 tff(38,plain,
% 0.19/0.38 (~((~element(C!13, the_carrier(A!11))) | (in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), relation_of_lattice(A!11)) <=> below_refl(A!11, B!12, C!13)))),
% 0.19/0.38 inference(and_elim,[status(thm)],[9])).
% 0.19/0.38 tff(39,plain,
% 0.19/0.38 (~(in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), relation_of_lattice(A!11)) <=> below_refl(A!11, B!12, C!13))),
% 0.19/0.38 inference(or_elim,[status(thm)],[38])).
% 0.19/0.38 tff(40,plain,
% 0.19/0.38 ((in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), relation_of_lattice(A!11)) <=> below_refl(A!11, B!12, C!13)) | in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), relation_of_lattice(A!11)) | below_refl(A!11, B!12, C!13)),
% 0.19/0.38 inference(tautology,[status(thm)],[])).
% 0.19/0.38 tff(41,plain,
% 0.19/0.38 (in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), relation_of_lattice(A!11)) | below_refl(A!11, B!12, C!13)),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[40, 39])).
% 0.19/0.38 tff(42,plain,
% 0.19/0.38 (in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), relation_of_lattice(A!11))),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[41, 37])).
% 0.19/0.38 tff(43,plain,
% 0.19/0.38 (in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[42, 35])).
% 0.19/0.38 tff(44,plain,
% 0.19/0.38 (^[A: $i, B: $i] : refl((empty_carrier(B) | (~latt_str(B)) | (~lattice(B)) | (~((~((~in(A, a_1_0_filter_1(B))) | (~((~element(tptp_fun_C_6(B, A), the_carrier(B))) | (~element(tptp_fun_D_5(B, A), the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) | (~below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A))))))) | (~(in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : ((~element(D, the_carrier(B))) | (~element(C, the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D))) | (~below_refl(B, C, D)))))))) <=> (empty_carrier(B) | (~latt_str(B)) | (~lattice(B)) | (~((~((~in(A, a_1_0_filter_1(B))) | (~((~element(tptp_fun_C_6(B, A), the_carrier(B))) | (~element(tptp_fun_D_5(B, A), the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) | (~below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A))))))) | (~(in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : ((~element(D, the_carrier(B))) | (~element(C, the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D))) | (~below_refl(B, C, D)))))))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(45,plain,
% 0.19/0.39 (![A: $i, B: $i] : (empty_carrier(B) | (~latt_str(B)) | (~lattice(B)) | (~((~((~in(A, a_1_0_filter_1(B))) | (~((~element(tptp_fun_C_6(B, A), the_carrier(B))) | (~element(tptp_fun_D_5(B, A), the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) | (~below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A))))))) | (~(in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : ((~element(D, the_carrier(B))) | (~element(C, the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D))) | (~below_refl(B, C, D)))))))) <=> ![A: $i, B: $i] : (empty_carrier(B) | (~latt_str(B)) | (~lattice(B)) | (~((~((~in(A, a_1_0_filter_1(B))) | (~((~element(tptp_fun_C_6(B, A), the_carrier(B))) | (~element(tptp_fun_D_5(B, A), the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) | (~below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A))))))) | (~(in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : ((~element(D, the_carrier(B))) | (~element(C, the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D))) | (~below_refl(B, C, D))))))))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[44])).
% 0.19/0.39 tff(46,plain,
% 0.19/0.39 (^[A: $i, B: $i] : rewrite((empty_carrier(B) | (~latt_str(B)) | (~lattice(B)) | (~((~((~in(A, a_1_0_filter_1(B))) | (~((~element(tptp_fun_C_6(B, A), the_carrier(B))) | (~element(tptp_fun_D_5(B, A), the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) | (~below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A))))))) | (~(in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : ((~element(D, the_carrier(B))) | (~element(C, the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D))) | (~below_refl(B, C, D)))))))) <=> (empty_carrier(B) | (~latt_str(B)) | (~lattice(B)) | (~((~((~in(A, a_1_0_filter_1(B))) | (~((~element(tptp_fun_C_6(B, A), the_carrier(B))) | (~element(tptp_fun_D_5(B, A), the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) | (~below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A))))))) | (~(in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : ((~element(D, the_carrier(B))) | (~element(C, the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D))) | (~below_refl(B, C, D)))))))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(47,plain,
% 0.19/0.39 (![A: $i, B: $i] : (empty_carrier(B) | (~latt_str(B)) | (~lattice(B)) | (~((~((~in(A, a_1_0_filter_1(B))) | (~((~element(tptp_fun_C_6(B, A), the_carrier(B))) | (~element(tptp_fun_D_5(B, A), the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) | (~below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A))))))) | (~(in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : ((~element(D, the_carrier(B))) | (~element(C, the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D))) | (~below_refl(B, C, D)))))))) <=> ![A: $i, B: $i] : (empty_carrier(B) | (~latt_str(B)) | (~lattice(B)) | (~((~((~in(A, a_1_0_filter_1(B))) | (~((~element(tptp_fun_C_6(B, A), the_carrier(B))) | (~element(tptp_fun_D_5(B, A), the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) | (~below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A))))))) | (~(in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : ((~element(D, the_carrier(B))) | (~element(C, the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D))) | (~below_refl(B, C, D))))))))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[46])).
% 0.19/0.39 tff(48,plain,
% 0.19/0.39 (![A: $i, B: $i] : (empty_carrier(B) | (~latt_str(B)) | (~lattice(B)) | (~((~((~in(A, a_1_0_filter_1(B))) | (~((~element(tptp_fun_C_6(B, A), the_carrier(B))) | (~element(tptp_fun_D_5(B, A), the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) | (~below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A))))))) | (~(in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : ((~element(D, the_carrier(B))) | (~element(C, the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D))) | (~below_refl(B, C, D)))))))) <=> ![A: $i, B: $i] : (empty_carrier(B) | (~latt_str(B)) | (~lattice(B)) | (~((~((~in(A, a_1_0_filter_1(B))) | (~((~element(tptp_fun_C_6(B, A), the_carrier(B))) | (~element(tptp_fun_D_5(B, A), the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) | (~below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A))))))) | (~(in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : ((~element(D, the_carrier(B))) | (~element(C, the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D))) | (~below_refl(B, C, D))))))))),
% 0.19/0.39 inference(transitivity,[status(thm)],[47, 45])).
% 0.19/0.39 tff(49,plain,
% 0.19/0.39 (^[A: $i, B: $i] : trans(monotonicity(trans(monotonicity(rewrite(((~empty_carrier(B)) & lattice(B) & latt_str(B)) <=> (~(empty_carrier(B) | (~latt_str(B)) | (~lattice(B))))), ((~((~empty_carrier(B)) & lattice(B) & latt_str(B))) <=> (~(~(empty_carrier(B) | (~latt_str(B)) | (~lattice(B))))))), rewrite((~(~(empty_carrier(B) | (~latt_str(B)) | (~lattice(B))))) <=> (empty_carrier(B) | (~latt_str(B)) | (~lattice(B)))), ((~((~empty_carrier(B)) & lattice(B) & latt_str(B))) <=> (empty_carrier(B) | (~latt_str(B)) | (~lattice(B))))), trans(monotonicity(rewrite(((~in(A, a_1_0_filter_1(B))) | (element(tptp_fun_C_6(B, A), the_carrier(B)) & element(tptp_fun_D_5(B, A), the_carrier(B)) & (A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A))) & below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) <=> ((~in(A, a_1_0_filter_1(B))) | (~((~element(tptp_fun_C_6(B, A), the_carrier(B))) | (~element(tptp_fun_D_5(B, A), the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) | (~below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A))))))), rewrite((in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : (~(element(C, the_carrier(B)) & element(D, the_carrier(B)) & (A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D)) & below_refl(B, C, D)))) <=> (in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : ((~element(D, the_carrier(B))) | (~element(C, the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D))) | (~below_refl(B, C, D))))), ((((~in(A, a_1_0_filter_1(B))) | (element(tptp_fun_C_6(B, A), the_carrier(B)) & element(tptp_fun_D_5(B, A), the_carrier(B)) & (A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A))) & below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) & (in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : (~(element(C, the_carrier(B)) & element(D, the_carrier(B)) & (A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D)) & below_refl(B, C, D))))) <=> (((~in(A, a_1_0_filter_1(B))) | (~((~element(tptp_fun_C_6(B, A), the_carrier(B))) | (~element(tptp_fun_D_5(B, A), the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) | (~below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))))) & (in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : ((~element(D, the_carrier(B))) | (~element(C, the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D))) | (~below_refl(B, C, D))))))), rewrite((((~in(A, a_1_0_filter_1(B))) | (~((~element(tptp_fun_C_6(B, A), the_carrier(B))) | (~element(tptp_fun_D_5(B, A), the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) | (~below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))))) & (in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : ((~element(D, the_carrier(B))) | (~element(C, the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D))) | (~below_refl(B, C, D))))) <=> (~((~((~in(A, a_1_0_filter_1(B))) | (~((~element(tptp_fun_C_6(B, A), the_carrier(B))) | (~element(tptp_fun_D_5(B, A), the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) | (~below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A))))))) | (~(in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : ((~element(D, the_carrier(B))) | (~element(C, the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D))) | (~below_refl(B, C, D)))))))), ((((~in(A, a_1_0_filter_1(B))) | (element(tptp_fun_C_6(B, A), the_carrier(B)) & element(tptp_fun_D_5(B, A), the_carrier(B)) & (A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A))) & below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) & (in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : (~(element(C, the_carrier(B)) & element(D, the_carrier(B)) & (A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D)) & below_refl(B, C, D))))) <=> (~((~((~in(A, a_1_0_filter_1(B))) | (~((~element(tptp_fun_C_6(B, A), the_carrier(B))) | (~element(tptp_fun_D_5(B, A), the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) | (~below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A))))))) | (~(in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : ((~element(D, the_carrier(B))) | (~element(C, the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D))) | (~below_refl(B, C, D))))))))), (((~((~empty_carrier(B)) & lattice(B) & latt_str(B))) | (((~in(A, a_1_0_filter_1(B))) | (element(tptp_fun_C_6(B, A), the_carrier(B)) & element(tptp_fun_D_5(B, A), the_carrier(B)) & (A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A))) & below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) & (in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : (~(element(C, the_carrier(B)) & element(D, the_carrier(B)) & (A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D)) & below_refl(B, C, D)))))) <=> ((empty_carrier(B) | (~latt_str(B)) | (~lattice(B))) | (~((~((~in(A, a_1_0_filter_1(B))) | (~((~element(tptp_fun_C_6(B, A), the_carrier(B))) | (~element(tptp_fun_D_5(B, A), the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) | (~below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A))))))) | (~(in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : ((~element(D, the_carrier(B))) | (~element(C, the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D))) | (~below_refl(B, C, D)))))))))), rewrite(((empty_carrier(B) | (~latt_str(B)) | (~lattice(B))) | (~((~((~in(A, a_1_0_filter_1(B))) | (~((~element(tptp_fun_C_6(B, A), the_carrier(B))) | (~element(tptp_fun_D_5(B, A), the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) | (~below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A))))))) | (~(in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : ((~element(D, the_carrier(B))) | (~element(C, the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D))) | (~below_refl(B, C, D)))))))) <=> (empty_carrier(B) | (~latt_str(B)) | (~lattice(B)) | (~((~((~in(A, a_1_0_filter_1(B))) | (~((~element(tptp_fun_C_6(B, A), the_carrier(B))) | (~element(tptp_fun_D_5(B, A), the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) | (~below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A))))))) | (~(in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : ((~element(D, the_carrier(B))) | (~element(C, the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D))) | (~below_refl(B, C, D))))))))), (((~((~empty_carrier(B)) & lattice(B) & latt_str(B))) | (((~in(A, a_1_0_filter_1(B))) | (element(tptp_fun_C_6(B, A), the_carrier(B)) & element(tptp_fun_D_5(B, A), the_carrier(B)) & (A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A))) & below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) & (in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : (~(element(C, the_carrier(B)) & element(D, the_carrier(B)) & (A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D)) & below_refl(B, C, D)))))) <=> (empty_carrier(B) | (~latt_str(B)) | (~lattice(B)) | (~((~((~in(A, a_1_0_filter_1(B))) | (~((~element(tptp_fun_C_6(B, A), the_carrier(B))) | (~element(tptp_fun_D_5(B, A), the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) | (~below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A))))))) | (~(in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : ((~element(D, the_carrier(B))) | (~element(C, the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D))) | (~below_refl(B, C, D))))))))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(50,plain,
% 0.19/0.39 (![A: $i, B: $i] : ((~((~empty_carrier(B)) & lattice(B) & latt_str(B))) | (((~in(A, a_1_0_filter_1(B))) | (element(tptp_fun_C_6(B, A), the_carrier(B)) & element(tptp_fun_D_5(B, A), the_carrier(B)) & (A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A))) & below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) & (in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : (~(element(C, the_carrier(B)) & element(D, the_carrier(B)) & (A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D)) & below_refl(B, C, D)))))) <=> ![A: $i, B: $i] : (empty_carrier(B) | (~latt_str(B)) | (~lattice(B)) | (~((~((~in(A, a_1_0_filter_1(B))) | (~((~element(tptp_fun_C_6(B, A), the_carrier(B))) | (~element(tptp_fun_D_5(B, A), the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) | (~below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A))))))) | (~(in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : ((~element(D, the_carrier(B))) | (~element(C, the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D))) | (~below_refl(B, C, D))))))))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[49])).
% 0.19/0.39 tff(51,plain,
% 0.19/0.39 (![A: $i, B: $i] : ((~((~empty_carrier(B)) & lattice(B) & latt_str(B))) | (in(A, a_1_0_filter_1(B)) <=> ?[C: $i, D: $i] : (element(C, the_carrier(B)) & element(D, the_carrier(B)) & (A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D)) & below_refl(B, C, D)))) <=> ![A: $i, B: $i] : ((~((~empty_carrier(B)) & lattice(B) & latt_str(B))) | (in(A, a_1_0_filter_1(B)) <=> ?[C: $i, D: $i] : (element(C, the_carrier(B)) & element(D, the_carrier(B)) & (A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D)) & below_refl(B, C, D))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(52,plain,
% 0.19/0.39 (^[A: $i, B: $i] : trans(monotonicity(rewrite((((~empty_carrier(B)) & lattice(B)) & latt_str(B)) <=> ((~empty_carrier(B)) & lattice(B) & latt_str(B))), rewrite((in(A, a_1_0_filter_1(B)) <=> ?[C: $i, D: $i] : (((element(C, the_carrier(B)) & element(D, the_carrier(B))) & (A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D))) & below_refl(B, C, D))) <=> (in(A, a_1_0_filter_1(B)) <=> ?[C: $i, D: $i] : (element(C, the_carrier(B)) & element(D, the_carrier(B)) & (A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D)) & below_refl(B, C, D)))), (((((~empty_carrier(B)) & lattice(B)) & latt_str(B)) => (in(A, a_1_0_filter_1(B)) <=> ?[C: $i, D: $i] : (((element(C, the_carrier(B)) & element(D, the_carrier(B))) & (A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D))) & below_refl(B, C, D)))) <=> (((~empty_carrier(B)) & lattice(B) & latt_str(B)) => (in(A, a_1_0_filter_1(B)) <=> ?[C: $i, D: $i] : (element(C, the_carrier(B)) & element(D, the_carrier(B)) & (A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D)) & below_refl(B, C, D)))))), rewrite((((~empty_carrier(B)) & lattice(B) & latt_str(B)) => (in(A, a_1_0_filter_1(B)) <=> ?[C: $i, D: $i] : (element(C, the_carrier(B)) & element(D, the_carrier(B)) & (A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D)) & below_refl(B, C, D)))) <=> ((~((~empty_carrier(B)) & lattice(B) & latt_str(B))) | (in(A, a_1_0_filter_1(B)) <=> ?[C: $i, D: $i] : (element(C, the_carrier(B)) & element(D, the_carrier(B)) & (A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D)) & below_refl(B, C, D))))), (((((~empty_carrier(B)) & lattice(B)) & latt_str(B)) => (in(A, a_1_0_filter_1(B)) <=> ?[C: $i, D: $i] : (((element(C, the_carrier(B)) & element(D, the_carrier(B))) & (A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D))) & below_refl(B, C, D)))) <=> ((~((~empty_carrier(B)) & lattice(B) & latt_str(B))) | (in(A, a_1_0_filter_1(B)) <=> ?[C: $i, D: $i] : (element(C, the_carrier(B)) & element(D, the_carrier(B)) & (A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D)) & below_refl(B, C, D))))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(53,plain,
% 0.19/0.39 (![A: $i, B: $i] : ((((~empty_carrier(B)) & lattice(B)) & latt_str(B)) => (in(A, a_1_0_filter_1(B)) <=> ?[C: $i, D: $i] : (((element(C, the_carrier(B)) & element(D, the_carrier(B))) & (A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D))) & below_refl(B, C, D)))) <=> ![A: $i, B: $i] : ((~((~empty_carrier(B)) & lattice(B) & latt_str(B))) | (in(A, a_1_0_filter_1(B)) <=> ?[C: $i, D: $i] : (element(C, the_carrier(B)) & element(D, the_carrier(B)) & (A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D)) & below_refl(B, C, D))))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[52])).
% 0.19/0.39 tff(54,axiom,(![A: $i, B: $i] : ((((~empty_carrier(B)) & lattice(B)) & latt_str(B)) => (in(A, a_1_0_filter_1(B)) <=> ?[C: $i, D: $i] : (((element(C, the_carrier(B)) & element(D, the_carrier(B))) & (A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D))) & below_refl(B, C, D))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','fraenkel_a_1_0_filter_1')).
% 0.19/0.39 tff(55,plain,
% 0.19/0.39 (![A: $i, B: $i] : ((~((~empty_carrier(B)) & lattice(B) & latt_str(B))) | (in(A, a_1_0_filter_1(B)) <=> ?[C: $i, D: $i] : (element(C, the_carrier(B)) & element(D, the_carrier(B)) & (A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D)) & below_refl(B, C, D))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[54, 53])).
% 0.19/0.39 tff(56,plain,
% 0.19/0.39 (![A: $i, B: $i] : ((~((~empty_carrier(B)) & lattice(B) & latt_str(B))) | (in(A, a_1_0_filter_1(B)) <=> ?[C: $i, D: $i] : (element(C, the_carrier(B)) & element(D, the_carrier(B)) & (A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D)) & below_refl(B, C, D))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[55, 51])).
% 0.19/0.39 tff(57,plain,(
% 0.19/0.39 ![A: $i, B: $i] : ((~((~empty_carrier(B)) & lattice(B) & latt_str(B))) | (((~in(A, a_1_0_filter_1(B))) | (element(tptp_fun_C_6(B, A), the_carrier(B)) & element(tptp_fun_D_5(B, A), the_carrier(B)) & (A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A))) & below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) & (in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : (~(element(C, the_carrier(B)) & element(D, the_carrier(B)) & (A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D)) & below_refl(B, C, D))))))),
% 0.19/0.39 inference(skolemize,[status(sab)],[56])).
% 0.19/0.39 tff(58,plain,
% 0.19/0.39 (![A: $i, B: $i] : (empty_carrier(B) | (~latt_str(B)) | (~lattice(B)) | (~((~((~in(A, a_1_0_filter_1(B))) | (~((~element(tptp_fun_C_6(B, A), the_carrier(B))) | (~element(tptp_fun_D_5(B, A), the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) | (~below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A))))))) | (~(in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : ((~element(D, the_carrier(B))) | (~element(C, the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D))) | (~below_refl(B, C, D))))))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[57, 50])).
% 0.19/0.39 tff(59,plain,
% 0.19/0.39 (![A: $i, B: $i] : (empty_carrier(B) | (~latt_str(B)) | (~lattice(B)) | (~((~((~in(A, a_1_0_filter_1(B))) | (~((~element(tptp_fun_C_6(B, A), the_carrier(B))) | (~element(tptp_fun_D_5(B, A), the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) | (~below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A))))))) | (~(in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : ((~element(D, the_carrier(B))) | (~element(C, the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D))) | (~below_refl(B, C, D))))))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[58, 48])).
% 0.19/0.39 tff(60,plain,
% 0.19/0.39 (((~![A: $i, B: $i] : (empty_carrier(B) | (~latt_str(B)) | (~lattice(B)) | (~((~((~in(A, a_1_0_filter_1(B))) | (~((~element(tptp_fun_C_6(B, A), the_carrier(B))) | (~element(tptp_fun_D_5(B, A), the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) | (~below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A))))))) | (~(in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : ((~element(D, the_carrier(B))) | (~element(C, the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D))) | (~below_refl(B, C, D))))))))) | (empty_carrier(A!11) | (~latt_str(A!11)) | (~lattice(A!11)) | (~((~(in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11)) | ![C: $i, D: $i] : ((~element(D, the_carrier(A!11))) | (~element(C, the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), C, D))) | (~below_refl(A!11, C, D))))) | (~((~in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11))) | (~((~below_refl(A!11, tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))))) | (~element(tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~element(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))))))))))) <=> ((~![A: $i, B: $i] : (empty_carrier(B) | (~latt_str(B)) | (~lattice(B)) | (~((~((~in(A, a_1_0_filter_1(B))) | (~((~element(tptp_fun_C_6(B, A), the_carrier(B))) | (~element(tptp_fun_D_5(B, A), the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) | (~below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A))))))) | (~(in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : ((~element(D, the_carrier(B))) | (~element(C, the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D))) | (~below_refl(B, C, D))))))))) | empty_carrier(A!11) | (~latt_str(A!11)) | (~lattice(A!11)) | (~((~(in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11)) | ![C: $i, D: $i] : ((~element(D, the_carrier(A!11))) | (~element(C, the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), C, D))) | (~below_refl(A!11, C, D))))) | (~((~in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11))) | (~((~below_refl(A!11, tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))))) | (~element(tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~element(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))))))))))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(61,plain,
% 0.19/0.40 ((empty_carrier(A!11) | (~latt_str(A!11)) | (~lattice(A!11)) | (~((~((~in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11))) | (~((~element(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~element(tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))))) | (~below_refl(A!11, tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))))))) | (~(in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11)) | ![C: $i, D: $i] : ((~element(D, the_carrier(A!11))) | (~element(C, the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), C, D))) | (~below_refl(A!11, C, D)))))))) <=> (empty_carrier(A!11) | (~latt_str(A!11)) | (~lattice(A!11)) | (~((~(in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11)) | ![C: $i, D: $i] : ((~element(D, the_carrier(A!11))) | (~element(C, the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), C, D))) | (~below_refl(A!11, C, D))))) | (~((~in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11))) | (~((~below_refl(A!11, tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))))) | (~element(tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~element(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))))))))))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(62,plain,
% 0.19/0.40 (((~![A: $i, B: $i] : (empty_carrier(B) | (~latt_str(B)) | (~lattice(B)) | (~((~((~in(A, a_1_0_filter_1(B))) | (~((~element(tptp_fun_C_6(B, A), the_carrier(B))) | (~element(tptp_fun_D_5(B, A), the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) | (~below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A))))))) | (~(in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : ((~element(D, the_carrier(B))) | (~element(C, the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D))) | (~below_refl(B, C, D))))))))) | (empty_carrier(A!11) | (~latt_str(A!11)) | (~lattice(A!11)) | (~((~((~in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11))) | (~((~element(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~element(tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))))) | (~below_refl(A!11, tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))))))) | (~(in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11)) | ![C: $i, D: $i] : ((~element(D, the_carrier(A!11))) | (~element(C, the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), C, D))) | (~below_refl(A!11, C, D))))))))) <=> ((~![A: $i, B: $i] : (empty_carrier(B) | (~latt_str(B)) | (~lattice(B)) | (~((~((~in(A, a_1_0_filter_1(B))) | (~((~element(tptp_fun_C_6(B, A), the_carrier(B))) | (~element(tptp_fun_D_5(B, A), the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) | (~below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A))))))) | (~(in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : ((~element(D, the_carrier(B))) | (~element(C, the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D))) | (~below_refl(B, C, D))))))))) | (empty_carrier(A!11) | (~latt_str(A!11)) | (~lattice(A!11)) | (~((~(in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11)) | ![C: $i, D: $i] : ((~element(D, the_carrier(A!11))) | (~element(C, the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), C, D))) | (~below_refl(A!11, C, D))))) | (~((~in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11))) | (~((~below_refl(A!11, tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))))) | (~element(tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~element(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11)))))))))))),
% 0.19/0.40 inference(monotonicity,[status(thm)],[61])).
% 0.19/0.40 tff(63,plain,
% 0.19/0.40 (((~![A: $i, B: $i] : (empty_carrier(B) | (~latt_str(B)) | (~lattice(B)) | (~((~((~in(A, a_1_0_filter_1(B))) | (~((~element(tptp_fun_C_6(B, A), the_carrier(B))) | (~element(tptp_fun_D_5(B, A), the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) | (~below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A))))))) | (~(in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : ((~element(D, the_carrier(B))) | (~element(C, the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D))) | (~below_refl(B, C, D))))))))) | (empty_carrier(A!11) | (~latt_str(A!11)) | (~lattice(A!11)) | (~((~((~in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11))) | (~((~element(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~element(tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))))) | (~below_refl(A!11, tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))))))) | (~(in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11)) | ![C: $i, D: $i] : ((~element(D, the_carrier(A!11))) | (~element(C, the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), C, D))) | (~below_refl(A!11, C, D))))))))) <=> ((~![A: $i, B: $i] : (empty_carrier(B) | (~latt_str(B)) | (~lattice(B)) | (~((~((~in(A, a_1_0_filter_1(B))) | (~((~element(tptp_fun_C_6(B, A), the_carrier(B))) | (~element(tptp_fun_D_5(B, A), the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) | (~below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A))))))) | (~(in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : ((~element(D, the_carrier(B))) | (~element(C, the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D))) | (~below_refl(B, C, D))))))))) | empty_carrier(A!11) | (~latt_str(A!11)) | (~lattice(A!11)) | (~((~(in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11)) | ![C: $i, D: $i] : ((~element(D, the_carrier(A!11))) | (~element(C, the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), C, D))) | (~below_refl(A!11, C, D))))) | (~((~in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11))) | (~((~below_refl(A!11, tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))))) | (~element(tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~element(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))))))))))),
% 0.19/0.41 inference(transitivity,[status(thm)],[62, 60])).
% 0.19/0.41 tff(64,plain,
% 0.19/0.41 ((~![A: $i, B: $i] : (empty_carrier(B) | (~latt_str(B)) | (~lattice(B)) | (~((~((~in(A, a_1_0_filter_1(B))) | (~((~element(tptp_fun_C_6(B, A), the_carrier(B))) | (~element(tptp_fun_D_5(B, A), the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) | (~below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A))))))) | (~(in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : ((~element(D, the_carrier(B))) | (~element(C, the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D))) | (~below_refl(B, C, D))))))))) | (empty_carrier(A!11) | (~latt_str(A!11)) | (~lattice(A!11)) | (~((~((~in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11))) | (~((~element(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~element(tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))))) | (~below_refl(A!11, tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))))))) | (~(in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11)) | ![C: $i, D: $i] : ((~element(D, the_carrier(A!11))) | (~element(C, the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), C, D))) | (~below_refl(A!11, C, D))))))))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(65,plain,
% 0.19/0.41 ((~![A: $i, B: $i] : (empty_carrier(B) | (~latt_str(B)) | (~lattice(B)) | (~((~((~in(A, a_1_0_filter_1(B))) | (~((~element(tptp_fun_C_6(B, A), the_carrier(B))) | (~element(tptp_fun_D_5(B, A), the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), tptp_fun_C_6(B, A), tptp_fun_D_5(B, A)))) | (~below_refl(B, tptp_fun_C_6(B, A), tptp_fun_D_5(B, A))))))) | (~(in(A, a_1_0_filter_1(B)) | ![C: $i, D: $i] : ((~element(D, the_carrier(B))) | (~element(C, the_carrier(B))) | (~(A = ordered_pair_as_product_element(the_carrier(B), the_carrier(B), C, D))) | (~below_refl(B, C, D))))))))) | empty_carrier(A!11) | (~latt_str(A!11)) | (~lattice(A!11)) | (~((~(in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11)) | ![C: $i, D: $i] : ((~element(D, the_carrier(A!11))) | (~element(C, the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), C, D))) | (~below_refl(A!11, C, D))))) | (~((~in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11))) | (~((~below_refl(A!11, tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))))) | (~element(tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~element(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11)))))))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[64, 63])).
% 0.19/0.41 tff(66,plain,
% 0.19/0.41 (~((~(in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11)) | ![C: $i, D: $i] : ((~element(D, the_carrier(A!11))) | (~element(C, the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), C, D))) | (~below_refl(A!11, C, D))))) | (~((~in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11))) | (~((~below_refl(A!11, tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))))) | (~element(tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~element(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))))))))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[65, 59, 12, 11, 10])).
% 0.19/0.41 tff(67,plain,
% 0.19/0.41 (((~(in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11)) | ![C: $i, D: $i] : ((~element(D, the_carrier(A!11))) | (~element(C, the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), C, D))) | (~below_refl(A!11, C, D))))) | (~((~in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11))) | (~((~below_refl(A!11, tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))))) | (~element(tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~element(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11)))))))) | ((~in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11))) | (~((~below_refl(A!11, tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))))) | (~element(tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~element(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))))))),
% 0.19/0.41 inference(tautology,[status(thm)],[])).
% 0.19/0.41 tff(68,plain,
% 0.19/0.41 ((~in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11))) | (~((~below_refl(A!11, tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))))) | (~element(tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~element(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11)))))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[67, 66])).
% 0.19/0.41 tff(69,plain,
% 0.19/0.41 ((~((~in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11))) | (~((~below_refl(A!11, tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))))) | (~element(tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~element(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))))))) | (~in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11))) | (~((~below_refl(A!11, tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))))) | (~element(tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~element(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11)))))),
% 0.19/0.41 inference(tautology,[status(thm)],[])).
% 0.19/0.41 tff(70,plain,
% 0.19/0.41 ((~in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11))) | (~((~below_refl(A!11, tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))))) | (~element(tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~element(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11)))))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[69, 68])).
% 0.19/0.41 tff(71,plain,
% 0.19/0.41 (~((~below_refl(A!11, tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))))) | (~element(tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~element(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[70, 43])).
% 0.19/0.41 tff(72,plain,
% 0.19/0.41 (((~below_refl(A!11, tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))))) | (~element(tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~element(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11)))) | element(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))),
% 0.19/0.41 inference(tautology,[status(thm)],[])).
% 0.19/0.41 tff(73,plain,
% 0.19/0.41 (element(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[72, 71])).
% 0.19/0.41 tff(74,plain,
% 0.19/0.41 (((~below_refl(A!11, tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))))) | (~element(tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~element(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11)))) | element(tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))),
% 0.19/0.41 inference(tautology,[status(thm)],[])).
% 0.19/0.41 tff(75,plain,
% 0.19/0.41 (element(tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[74, 71])).
% 0.19/0.41 tff(76,plain,
% 0.19/0.41 (^[A: $i] : refl(((~latt_str(A)) | (~((~meet_semilatt_str(A)) | (~join_semilatt_str(A))))) <=> ((~latt_str(A)) | (~((~meet_semilatt_str(A)) | (~join_semilatt_str(A))))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(77,plain,
% 0.19/0.42 (![A: $i] : ((~latt_str(A)) | (~((~meet_semilatt_str(A)) | (~join_semilatt_str(A))))) <=> ![A: $i] : ((~latt_str(A)) | (~((~meet_semilatt_str(A)) | (~join_semilatt_str(A)))))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[76])).
% 0.19/0.42 tff(78,plain,
% 0.19/0.42 (^[A: $i] : rewrite(((~latt_str(A)) | (meet_semilatt_str(A) & join_semilatt_str(A))) <=> ((~latt_str(A)) | (~((~meet_semilatt_str(A)) | (~join_semilatt_str(A))))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(79,plain,
% 0.19/0.42 (![A: $i] : ((~latt_str(A)) | (meet_semilatt_str(A) & join_semilatt_str(A))) <=> ![A: $i] : ((~latt_str(A)) | (~((~meet_semilatt_str(A)) | (~join_semilatt_str(A)))))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[78])).
% 0.19/0.42 tff(80,plain,
% 0.19/0.42 (![A: $i] : ((~latt_str(A)) | (meet_semilatt_str(A) & join_semilatt_str(A))) <=> ![A: $i] : ((~latt_str(A)) | (meet_semilatt_str(A) & join_semilatt_str(A)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(81,plain,
% 0.19/0.42 (^[A: $i] : rewrite((latt_str(A) => (meet_semilatt_str(A) & join_semilatt_str(A))) <=> ((~latt_str(A)) | (meet_semilatt_str(A) & join_semilatt_str(A))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(82,plain,
% 0.19/0.42 (![A: $i] : (latt_str(A) => (meet_semilatt_str(A) & join_semilatt_str(A))) <=> ![A: $i] : ((~latt_str(A)) | (meet_semilatt_str(A) & join_semilatt_str(A)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[81])).
% 0.19/0.42 tff(83,axiom,(![A: $i] : (latt_str(A) => (meet_semilatt_str(A) & join_semilatt_str(A)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','dt_l3_lattices')).
% 0.19/0.42 tff(84,plain,
% 0.19/0.42 (![A: $i] : ((~latt_str(A)) | (meet_semilatt_str(A) & join_semilatt_str(A)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[83, 82])).
% 0.19/0.42 tff(85,plain,
% 0.19/0.42 (![A: $i] : ((~latt_str(A)) | (meet_semilatt_str(A) & join_semilatt_str(A)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[84, 80])).
% 0.19/0.42 tff(86,plain,(
% 0.19/0.42 ![A: $i] : ((~latt_str(A)) | (meet_semilatt_str(A) & join_semilatt_str(A)))),
% 0.19/0.42 inference(skolemize,[status(sab)],[85])).
% 0.19/0.42 tff(87,plain,
% 0.19/0.42 (![A: $i] : ((~latt_str(A)) | (~((~meet_semilatt_str(A)) | (~join_semilatt_str(A)))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[86, 79])).
% 0.19/0.42 tff(88,plain,
% 0.19/0.42 (![A: $i] : ((~latt_str(A)) | (~((~meet_semilatt_str(A)) | (~join_semilatt_str(A)))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[87, 77])).
% 0.19/0.42 tff(89,plain,
% 0.19/0.42 (((~![A: $i] : ((~latt_str(A)) | (~((~meet_semilatt_str(A)) | (~join_semilatt_str(A)))))) | ((~latt_str(A!11)) | (~((~meet_semilatt_str(A!11)) | (~join_semilatt_str(A!11)))))) <=> ((~![A: $i] : ((~latt_str(A)) | (~((~meet_semilatt_str(A)) | (~join_semilatt_str(A)))))) | (~latt_str(A!11)) | (~((~meet_semilatt_str(A!11)) | (~join_semilatt_str(A!11)))))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(90,plain,
% 0.19/0.42 ((~![A: $i] : ((~latt_str(A)) | (~((~meet_semilatt_str(A)) | (~join_semilatt_str(A)))))) | ((~latt_str(A!11)) | (~((~meet_semilatt_str(A!11)) | (~join_semilatt_str(A!11)))))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(91,plain,
% 0.19/0.42 ((~![A: $i] : ((~latt_str(A)) | (~((~meet_semilatt_str(A)) | (~join_semilatt_str(A)))))) | (~latt_str(A!11)) | (~((~meet_semilatt_str(A!11)) | (~join_semilatt_str(A!11))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[90, 89])).
% 0.19/0.42 tff(92,plain,
% 0.19/0.42 (~((~meet_semilatt_str(A!11)) | (~join_semilatt_str(A!11)))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[91, 88, 10])).
% 0.19/0.42 tff(93,plain,
% 0.19/0.42 (((~meet_semilatt_str(A!11)) | (~join_semilatt_str(A!11))) | join_semilatt_str(A!11)),
% 0.19/0.42 inference(tautology,[status(thm)],[])).
% 0.19/0.42 tff(94,plain,
% 0.19/0.42 (join_semilatt_str(A!11)),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[93, 92])).
% 0.19/0.42 tff(95,plain,
% 0.19/0.42 (^[A: $i] : refl((one_sorted_str(A) | (~join_semilatt_str(A))) <=> (one_sorted_str(A) | (~join_semilatt_str(A))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(96,plain,
% 0.19/0.42 (![A: $i] : (one_sorted_str(A) | (~join_semilatt_str(A))) <=> ![A: $i] : (one_sorted_str(A) | (~join_semilatt_str(A)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[95])).
% 0.19/0.42 tff(97,plain,
% 0.19/0.42 (![A: $i] : (one_sorted_str(A) | (~join_semilatt_str(A))) <=> ![A: $i] : (one_sorted_str(A) | (~join_semilatt_str(A)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(98,plain,
% 0.19/0.42 (^[A: $i] : rewrite((join_semilatt_str(A) => one_sorted_str(A)) <=> (one_sorted_str(A) | (~join_semilatt_str(A))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(99,plain,
% 0.19/0.42 (![A: $i] : (join_semilatt_str(A) => one_sorted_str(A)) <=> ![A: $i] : (one_sorted_str(A) | (~join_semilatt_str(A)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[98])).
% 0.19/0.42 tff(100,axiom,(![A: $i] : (join_semilatt_str(A) => one_sorted_str(A))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','dt_l2_lattices')).
% 0.19/0.42 tff(101,plain,
% 0.19/0.42 (![A: $i] : (one_sorted_str(A) | (~join_semilatt_str(A)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[100, 99])).
% 0.19/0.42 tff(102,plain,
% 0.19/0.42 (![A: $i] : (one_sorted_str(A) | (~join_semilatt_str(A)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[101, 97])).
% 0.19/0.42 tff(103,plain,(
% 0.19/0.42 ![A: $i] : (one_sorted_str(A) | (~join_semilatt_str(A)))),
% 0.19/0.42 inference(skolemize,[status(sab)],[102])).
% 0.19/0.42 tff(104,plain,
% 0.19/0.42 (![A: $i] : (one_sorted_str(A) | (~join_semilatt_str(A)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[103, 96])).
% 0.19/0.42 tff(105,plain,
% 0.19/0.42 (((~![A: $i] : (one_sorted_str(A) | (~join_semilatt_str(A)))) | (one_sorted_str(A!11) | (~join_semilatt_str(A!11)))) <=> ((~![A: $i] : (one_sorted_str(A) | (~join_semilatt_str(A)))) | one_sorted_str(A!11) | (~join_semilatt_str(A!11)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(106,plain,
% 0.19/0.42 ((~![A: $i] : (one_sorted_str(A) | (~join_semilatt_str(A)))) | (one_sorted_str(A!11) | (~join_semilatt_str(A!11)))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(107,plain,
% 0.19/0.42 ((~![A: $i] : (one_sorted_str(A) | (~join_semilatt_str(A)))) | one_sorted_str(A!11) | (~join_semilatt_str(A!11))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[106, 105])).
% 0.19/0.42 tff(108,plain,
% 0.19/0.42 (one_sorted_str(A!11)),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[107, 104, 94])).
% 0.19/0.42 tff(109,plain,
% 0.19/0.42 (^[A: $i] : refl((empty_carrier(A) | (~empty(the_carrier(A))) | (~one_sorted_str(A))) <=> (empty_carrier(A) | (~empty(the_carrier(A))) | (~one_sorted_str(A))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(110,plain,
% 0.19/0.42 (![A: $i] : (empty_carrier(A) | (~empty(the_carrier(A))) | (~one_sorted_str(A))) <=> ![A: $i] : (empty_carrier(A) | (~empty(the_carrier(A))) | (~one_sorted_str(A)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[109])).
% 0.19/0.42 tff(111,plain,
% 0.19/0.42 (^[A: $i] : trans(monotonicity(trans(monotonicity(rewrite(((~empty_carrier(A)) & one_sorted_str(A)) <=> (~(empty_carrier(A) | (~one_sorted_str(A))))), ((~((~empty_carrier(A)) & one_sorted_str(A))) <=> (~(~(empty_carrier(A) | (~one_sorted_str(A))))))), rewrite((~(~(empty_carrier(A) | (~one_sorted_str(A))))) <=> (empty_carrier(A) | (~one_sorted_str(A)))), ((~((~empty_carrier(A)) & one_sorted_str(A))) <=> (empty_carrier(A) | (~one_sorted_str(A))))), (((~empty(the_carrier(A))) | (~((~empty_carrier(A)) & one_sorted_str(A)))) <=> ((~empty(the_carrier(A))) | (empty_carrier(A) | (~one_sorted_str(A)))))), rewrite(((~empty(the_carrier(A))) | (empty_carrier(A) | (~one_sorted_str(A)))) <=> (empty_carrier(A) | (~empty(the_carrier(A))) | (~one_sorted_str(A)))), (((~empty(the_carrier(A))) | (~((~empty_carrier(A)) & one_sorted_str(A)))) <=> (empty_carrier(A) | (~empty(the_carrier(A))) | (~one_sorted_str(A)))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(112,plain,
% 0.19/0.42 (![A: $i] : ((~empty(the_carrier(A))) | (~((~empty_carrier(A)) & one_sorted_str(A)))) <=> ![A: $i] : (empty_carrier(A) | (~empty(the_carrier(A))) | (~one_sorted_str(A)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[111])).
% 0.19/0.42 tff(113,plain,
% 0.19/0.42 (![A: $i] : ((~empty(the_carrier(A))) | (~((~empty_carrier(A)) & one_sorted_str(A)))) <=> ![A: $i] : ((~empty(the_carrier(A))) | (~((~empty_carrier(A)) & one_sorted_str(A))))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(114,plain,
% 0.19/0.42 (^[A: $i] : rewrite((((~empty_carrier(A)) & one_sorted_str(A)) => (~empty(the_carrier(A)))) <=> ((~empty(the_carrier(A))) | (~((~empty_carrier(A)) & one_sorted_str(A)))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(115,plain,
% 0.19/0.42 (![A: $i] : (((~empty_carrier(A)) & one_sorted_str(A)) => (~empty(the_carrier(A)))) <=> ![A: $i] : ((~empty(the_carrier(A))) | (~((~empty_carrier(A)) & one_sorted_str(A))))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[114])).
% 0.19/0.42 tff(116,axiom,(![A: $i] : (((~empty_carrier(A)) & one_sorted_str(A)) => (~empty(the_carrier(A))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','fc1_struct_0')).
% 0.19/0.42 tff(117,plain,
% 0.19/0.42 (![A: $i] : ((~empty(the_carrier(A))) | (~((~empty_carrier(A)) & one_sorted_str(A))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[116, 115])).
% 0.19/0.42 tff(118,plain,
% 0.19/0.42 (![A: $i] : ((~empty(the_carrier(A))) | (~((~empty_carrier(A)) & one_sorted_str(A))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[117, 113])).
% 0.19/0.42 tff(119,plain,(
% 0.19/0.42 ![A: $i] : ((~empty(the_carrier(A))) | (~((~empty_carrier(A)) & one_sorted_str(A))))),
% 0.19/0.42 inference(skolemize,[status(sab)],[118])).
% 0.19/0.42 tff(120,plain,
% 0.19/0.42 (![A: $i] : (empty_carrier(A) | (~empty(the_carrier(A))) | (~one_sorted_str(A)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[119, 112])).
% 0.19/0.42 tff(121,plain,
% 0.19/0.42 (![A: $i] : (empty_carrier(A) | (~empty(the_carrier(A))) | (~one_sorted_str(A)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[120, 110])).
% 0.19/0.42 tff(122,plain,
% 0.19/0.42 (((~![A: $i] : (empty_carrier(A) | (~empty(the_carrier(A))) | (~one_sorted_str(A)))) | (empty_carrier(A!11) | (~empty(the_carrier(A!11))) | (~one_sorted_str(A!11)))) <=> ((~![A: $i] : (empty_carrier(A) | (~empty(the_carrier(A))) | (~one_sorted_str(A)))) | empty_carrier(A!11) | (~empty(the_carrier(A!11))) | (~one_sorted_str(A!11)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(123,plain,
% 0.19/0.42 ((~![A: $i] : (empty_carrier(A) | (~empty(the_carrier(A))) | (~one_sorted_str(A)))) | (empty_carrier(A!11) | (~empty(the_carrier(A!11))) | (~one_sorted_str(A!11)))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(124,plain,
% 0.19/0.42 ((~![A: $i] : (empty_carrier(A) | (~empty(the_carrier(A))) | (~one_sorted_str(A)))) | empty_carrier(A!11) | (~empty(the_carrier(A!11))) | (~one_sorted_str(A!11))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[123, 122])).
% 0.19/0.42 tff(125,plain,
% 0.19/0.42 ((~empty(the_carrier(A!11))) | (~one_sorted_str(A!11))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[124, 121, 12])).
% 0.19/0.42 tff(126,plain,
% 0.19/0.42 (~empty(the_carrier(A!11))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[125, 108])).
% 0.19/0.42 tff(127,plain,
% 0.19/0.42 (^[A: $i, B: $i, C: $i, D: $i] : refl((empty(B) | empty(A) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D)) | (~element(C, A)) | (~element(D, B))) <=> (empty(B) | empty(A) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D)) | (~element(C, A)) | (~element(D, B))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(128,plain,
% 0.19/0.42 (![A: $i, B: $i, C: $i, D: $i] : (empty(B) | empty(A) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D)) | (~element(C, A)) | (~element(D, B))) <=> ![A: $i, B: $i, C: $i, D: $i] : (empty(B) | empty(A) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D)) | (~element(C, A)) | (~element(D, B)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[127])).
% 0.19/0.42 tff(129,plain,
% 0.19/0.42 (^[A: $i, B: $i, C: $i, D: $i] : trans(monotonicity(trans(monotonicity(rewrite(((~empty(A)) & (~empty(B)) & element(C, A) & element(D, B)) <=> (~(empty(B) | empty(A) | (~element(C, A)) | (~element(D, B))))), ((~((~empty(A)) & (~empty(B)) & element(C, A) & element(D, B))) <=> (~(~(empty(B) | empty(A) | (~element(C, A)) | (~element(D, B))))))), rewrite((~(~(empty(B) | empty(A) | (~element(C, A)) | (~element(D, B))))) <=> (empty(B) | empty(A) | (~element(C, A)) | (~element(D, B)))), ((~((~empty(A)) & (~empty(B)) & element(C, A) & element(D, B))) <=> (empty(B) | empty(A) | (~element(C, A)) | (~element(D, B))))), (((~((~empty(A)) & (~empty(B)) & element(C, A) & element(D, B))) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D))) <=> ((empty(B) | empty(A) | (~element(C, A)) | (~element(D, B))) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D))))), rewrite(((empty(B) | empty(A) | (~element(C, A)) | (~element(D, B))) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D))) <=> (empty(B) | empty(A) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D)) | (~element(C, A)) | (~element(D, B)))), (((~((~empty(A)) & (~empty(B)) & element(C, A) & element(D, B))) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D))) <=> (empty(B) | empty(A) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D)) | (~element(C, A)) | (~element(D, B)))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(130,plain,
% 0.19/0.42 (![A: $i, B: $i, C: $i, D: $i] : ((~((~empty(A)) & (~empty(B)) & element(C, A) & element(D, B))) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D))) <=> ![A: $i, B: $i, C: $i, D: $i] : (empty(B) | empty(A) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D)) | (~element(C, A)) | (~element(D, B)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[129])).
% 0.19/0.42 tff(131,plain,
% 0.19/0.42 (![A: $i, B: $i, C: $i, D: $i] : ((~((~empty(A)) & (~empty(B)) & element(C, A) & element(D, B))) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D))) <=> ![A: $i, B: $i, C: $i, D: $i] : ((~((~empty(A)) & (~empty(B)) & element(C, A) & element(D, B))) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(132,plain,
% 0.19/0.42 (^[A: $i, B: $i, C: $i, D: $i] : trans(monotonicity(trans(monotonicity(rewrite((((~empty(A)) & (~empty(B))) & element(C, A)) <=> ((~empty(A)) & (~empty(B)) & element(C, A))), (((((~empty(A)) & (~empty(B))) & element(C, A)) & element(D, B)) <=> (((~empty(A)) & (~empty(B)) & element(C, A)) & element(D, B)))), rewrite((((~empty(A)) & (~empty(B)) & element(C, A)) & element(D, B)) <=> ((~empty(A)) & (~empty(B)) & element(C, A) & element(D, B))), (((((~empty(A)) & (~empty(B))) & element(C, A)) & element(D, B)) <=> ((~empty(A)) & (~empty(B)) & element(C, A) & element(D, B)))), ((((((~empty(A)) & (~empty(B))) & element(C, A)) & element(D, B)) => (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D))) <=> (((~empty(A)) & (~empty(B)) & element(C, A) & element(D, B)) => (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D))))), rewrite((((~empty(A)) & (~empty(B)) & element(C, A) & element(D, B)) => (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D))) <=> ((~((~empty(A)) & (~empty(B)) & element(C, A) & element(D, B))) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D)))), ((((((~empty(A)) & (~empty(B))) & element(C, A)) & element(D, B)) => (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D))) <=> ((~((~empty(A)) & (~empty(B)) & element(C, A) & element(D, B))) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D)))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(133,plain,
% 0.19/0.42 (![A: $i, B: $i, C: $i, D: $i] : (((((~empty(A)) & (~empty(B))) & element(C, A)) & element(D, B)) => (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D))) <=> ![A: $i, B: $i, C: $i, D: $i] : ((~((~empty(A)) & (~empty(B)) & element(C, A) & element(D, B))) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[132])).
% 0.19/0.42 tff(134,axiom,(![A: $i, B: $i, C: $i, D: $i] : (((((~empty(A)) & (~empty(B))) & element(C, A)) & element(D, B)) => (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','redefinition_k1_domain_1')).
% 0.19/0.42 tff(135,plain,
% 0.19/0.42 (![A: $i, B: $i, C: $i, D: $i] : ((~((~empty(A)) & (~empty(B)) & element(C, A) & element(D, B))) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[134, 133])).
% 0.19/0.42 tff(136,plain,
% 0.19/0.42 (![A: $i, B: $i, C: $i, D: $i] : ((~((~empty(A)) & (~empty(B)) & element(C, A) & element(D, B))) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[135, 131])).
% 0.19/0.42 tff(137,plain,(
% 0.19/0.42 ![A: $i, B: $i, C: $i, D: $i] : ((~((~empty(A)) & (~empty(B)) & element(C, A) & element(D, B))) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D)))),
% 0.19/0.42 inference(skolemize,[status(sab)],[136])).
% 0.19/0.42 tff(138,plain,
% 0.19/0.42 (![A: $i, B: $i, C: $i, D: $i] : (empty(B) | empty(A) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D)) | (~element(C, A)) | (~element(D, B)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[137, 130])).
% 0.19/0.42 tff(139,plain,
% 0.19/0.42 (![A: $i, B: $i, C: $i, D: $i] : (empty(B) | empty(A) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D)) | (~element(C, A)) | (~element(D, B)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[138, 128])).
% 0.19/0.42 tff(140,plain,
% 0.19/0.42 (((~![A: $i, B: $i, C: $i, D: $i] : (empty(B) | empty(A) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D)) | (~element(C, A)) | (~element(D, B)))) | (empty(the_carrier(A!11)) | (~element(tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~element(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))) = ordered_pair(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))))) <=> ((~![A: $i, B: $i, C: $i, D: $i] : (empty(B) | empty(A) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D)) | (~element(C, A)) | (~element(D, B)))) | empty(the_carrier(A!11)) | (~element(tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~element(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))) = ordered_pair(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(141,plain,
% 0.19/0.42 ((empty(the_carrier(A!11)) | empty(the_carrier(A!11)) | (ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))) = ordered_pair(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))) | (~element(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~element(tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11)))) <=> (empty(the_carrier(A!11)) | (~element(tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~element(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))) = ordered_pair(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(142,plain,
% 0.19/0.43 (((~![A: $i, B: $i, C: $i, D: $i] : (empty(B) | empty(A) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D)) | (~element(C, A)) | (~element(D, B)))) | (empty(the_carrier(A!11)) | empty(the_carrier(A!11)) | (ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))) = ordered_pair(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))) | (~element(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~element(tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))))) <=> ((~![A: $i, B: $i, C: $i, D: $i] : (empty(B) | empty(A) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D)) | (~element(C, A)) | (~element(D, B)))) | (empty(the_carrier(A!11)) | (~element(tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~element(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))) = ordered_pair(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))))))),
% 0.19/0.43 inference(monotonicity,[status(thm)],[141])).
% 0.19/0.43 tff(143,plain,
% 0.19/0.43 (((~![A: $i, B: $i, C: $i, D: $i] : (empty(B) | empty(A) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D)) | (~element(C, A)) | (~element(D, B)))) | (empty(the_carrier(A!11)) | empty(the_carrier(A!11)) | (ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))) = ordered_pair(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))) | (~element(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~element(tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))))) <=> ((~![A: $i, B: $i, C: $i, D: $i] : (empty(B) | empty(A) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D)) | (~element(C, A)) | (~element(D, B)))) | empty(the_carrier(A!11)) | (~element(tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~element(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))) = ordered_pair(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))))),
% 0.19/0.43 inference(transitivity,[status(thm)],[142, 140])).
% 0.19/0.43 tff(144,plain,
% 0.19/0.43 ((~![A: $i, B: $i, C: $i, D: $i] : (empty(B) | empty(A) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D)) | (~element(C, A)) | (~element(D, B)))) | (empty(the_carrier(A!11)) | empty(the_carrier(A!11)) | (ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))) = ordered_pair(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))) | (~element(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~element(tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(145,plain,
% 0.19/0.43 ((~![A: $i, B: $i, C: $i, D: $i] : (empty(B) | empty(A) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D)) | (~element(C, A)) | (~element(D, B)))) | empty(the_carrier(A!11)) | (~element(tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~element(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))) = ordered_pair(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[144, 143])).
% 0.19/0.43 tff(146,plain,
% 0.19/0.43 (ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))) = ordered_pair(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[145, 139, 126, 75, 73])).
% 0.19/0.43 tff(147,plain,
% 0.19/0.43 (((~below_refl(A!11, tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))))) | (~element(tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~element(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11)))) | (ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))))),
% 0.19/0.43 inference(tautology,[status(thm)],[])).
% 0.19/0.43 tff(148,plain,
% 0.19/0.43 (ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[147, 71])).
% 0.19/0.43 tff(149,plain,
% 0.19/0.43 (element(C!13, the_carrier(A!11))),
% 0.19/0.43 inference(or_elim,[status(thm)],[38])).
% 0.19/0.43 tff(150,plain,
% 0.19/0.43 (element(B!12, the_carrier(A!11))),
% 0.19/0.43 inference(and_elim,[status(thm)],[9])).
% 0.19/0.43 tff(151,plain,
% 0.19/0.43 (((~![A: $i, B: $i, C: $i, D: $i] : (empty(B) | empty(A) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D)) | (~element(C, A)) | (~element(D, B)))) | ((~element(B!12, the_carrier(A!11))) | (~element(C!13, the_carrier(A!11))) | empty(the_carrier(A!11)) | (ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair(B!12, C!13)))) <=> ((~![A: $i, B: $i, C: $i, D: $i] : (empty(B) | empty(A) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D)) | (~element(C, A)) | (~element(D, B)))) | (~element(B!12, the_carrier(A!11))) | (~element(C!13, the_carrier(A!11))) | empty(the_carrier(A!11)) | (ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair(B!12, C!13)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(152,plain,
% 0.19/0.43 ((empty(the_carrier(A!11)) | empty(the_carrier(A!11)) | (ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair(B!12, C!13)) | (~element(B!12, the_carrier(A!11))) | (~element(C!13, the_carrier(A!11)))) <=> ((~element(B!12, the_carrier(A!11))) | (~element(C!13, the_carrier(A!11))) | empty(the_carrier(A!11)) | (ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair(B!12, C!13)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(153,plain,
% 0.19/0.43 (((~![A: $i, B: $i, C: $i, D: $i] : (empty(B) | empty(A) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D)) | (~element(C, A)) | (~element(D, B)))) | (empty(the_carrier(A!11)) | empty(the_carrier(A!11)) | (ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair(B!12, C!13)) | (~element(B!12, the_carrier(A!11))) | (~element(C!13, the_carrier(A!11))))) <=> ((~![A: $i, B: $i, C: $i, D: $i] : (empty(B) | empty(A) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D)) | (~element(C, A)) | (~element(D, B)))) | ((~element(B!12, the_carrier(A!11))) | (~element(C!13, the_carrier(A!11))) | empty(the_carrier(A!11)) | (ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair(B!12, C!13))))),
% 0.19/0.43 inference(monotonicity,[status(thm)],[152])).
% 0.19/0.43 tff(154,plain,
% 0.19/0.43 (((~![A: $i, B: $i, C: $i, D: $i] : (empty(B) | empty(A) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D)) | (~element(C, A)) | (~element(D, B)))) | (empty(the_carrier(A!11)) | empty(the_carrier(A!11)) | (ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair(B!12, C!13)) | (~element(B!12, the_carrier(A!11))) | (~element(C!13, the_carrier(A!11))))) <=> ((~![A: $i, B: $i, C: $i, D: $i] : (empty(B) | empty(A) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D)) | (~element(C, A)) | (~element(D, B)))) | (~element(B!12, the_carrier(A!11))) | (~element(C!13, the_carrier(A!11))) | empty(the_carrier(A!11)) | (ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair(B!12, C!13)))),
% 0.19/0.43 inference(transitivity,[status(thm)],[153, 151])).
% 0.19/0.43 tff(155,plain,
% 0.19/0.43 ((~![A: $i, B: $i, C: $i, D: $i] : (empty(B) | empty(A) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D)) | (~element(C, A)) | (~element(D, B)))) | (empty(the_carrier(A!11)) | empty(the_carrier(A!11)) | (ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair(B!12, C!13)) | (~element(B!12, the_carrier(A!11))) | (~element(C!13, the_carrier(A!11))))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(156,plain,
% 0.19/0.43 ((~![A: $i, B: $i, C: $i, D: $i] : (empty(B) | empty(A) | (ordered_pair_as_product_element(A, B, C, D) = ordered_pair(C, D)) | (~element(C, A)) | (~element(D, B)))) | (~element(B!12, the_carrier(A!11))) | (~element(C!13, the_carrier(A!11))) | empty(the_carrier(A!11)) | (ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair(B!12, C!13))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[155, 154])).
% 0.19/0.44 tff(157,plain,
% 0.19/0.44 (empty(the_carrier(A!11)) | (ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair(B!12, C!13))),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[156, 139, 150, 149])).
% 0.19/0.44 tff(158,plain,
% 0.19/0.44 (ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair(B!12, C!13)),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[157, 126])).
% 0.19/0.44 tff(159,plain,
% 0.19/0.44 (ordered_pair(B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)),
% 0.19/0.44 inference(symmetry,[status(thm)],[158])).
% 0.19/0.44 tff(160,plain,
% 0.19/0.44 (ordered_pair(B!12, C!13) = ordered_pair(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))),
% 0.19/0.44 inference(transitivity,[status(thm)],[159, 148, 146])).
% 0.19/0.44 tff(161,plain,
% 0.19/0.44 (^[A: $i, B: $i, C: $i, D: $i] : refl(((~(ordered_pair(A, B) = ordered_pair(C, D))) | (~((~(A = C)) | (~(B = D))))) <=> ((~(ordered_pair(A, B) = ordered_pair(C, D))) | (~((~(A = C)) | (~(B = D))))))),
% 0.19/0.44 inference(bind,[status(th)],[])).
% 0.19/0.44 tff(162,plain,
% 0.19/0.44 (![A: $i, B: $i, C: $i, D: $i] : ((~(ordered_pair(A, B) = ordered_pair(C, D))) | (~((~(A = C)) | (~(B = D))))) <=> ![A: $i, B: $i, C: $i, D: $i] : ((~(ordered_pair(A, B) = ordered_pair(C, D))) | (~((~(A = C)) | (~(B = D)))))),
% 0.19/0.44 inference(quant_intro,[status(thm)],[161])).
% 0.19/0.44 tff(163,plain,
% 0.19/0.44 (^[A: $i, B: $i, C: $i, D: $i] : rewrite(((~(ordered_pair(A, B) = ordered_pair(C, D))) | ((A = C) & (B = D))) <=> ((~(ordered_pair(A, B) = ordered_pair(C, D))) | (~((~(A = C)) | (~(B = D))))))),
% 0.19/0.44 inference(bind,[status(th)],[])).
% 0.19/0.44 tff(164,plain,
% 0.19/0.44 (![A: $i, B: $i, C: $i, D: $i] : ((~(ordered_pair(A, B) = ordered_pair(C, D))) | ((A = C) & (B = D))) <=> ![A: $i, B: $i, C: $i, D: $i] : ((~(ordered_pair(A, B) = ordered_pair(C, D))) | (~((~(A = C)) | (~(B = D)))))),
% 0.19/0.44 inference(quant_intro,[status(thm)],[163])).
% 0.19/0.44 tff(165,plain,
% 0.19/0.44 (![A: $i, B: $i, C: $i, D: $i] : ((~(ordered_pair(A, B) = ordered_pair(C, D))) | ((A = C) & (B = D))) <=> ![A: $i, B: $i, C: $i, D: $i] : ((~(ordered_pair(A, B) = ordered_pair(C, D))) | ((A = C) & (B = D)))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(166,plain,
% 0.19/0.44 (^[A: $i, B: $i, C: $i, D: $i] : rewrite(((ordered_pair(A, B) = ordered_pair(C, D)) => ((A = C) & (B = D))) <=> ((~(ordered_pair(A, B) = ordered_pair(C, D))) | ((A = C) & (B = D))))),
% 0.19/0.44 inference(bind,[status(th)],[])).
% 0.19/0.44 tff(167,plain,
% 0.19/0.44 (![A: $i, B: $i, C: $i, D: $i] : ((ordered_pair(A, B) = ordered_pair(C, D)) => ((A = C) & (B = D))) <=> ![A: $i, B: $i, C: $i, D: $i] : ((~(ordered_pair(A, B) = ordered_pair(C, D))) | ((A = C) & (B = D)))),
% 0.19/0.44 inference(quant_intro,[status(thm)],[166])).
% 0.19/0.44 tff(168,axiom,(![A: $i, B: $i, C: $i, D: $i] : ((ordered_pair(A, B) = ordered_pair(C, D)) => ((A = C) & (B = D)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t33_zfmisc_1')).
% 0.19/0.44 tff(169,plain,
% 0.19/0.44 (![A: $i, B: $i, C: $i, D: $i] : ((~(ordered_pair(A, B) = ordered_pair(C, D))) | ((A = C) & (B = D)))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[168, 167])).
% 0.19/0.44 tff(170,plain,
% 0.19/0.44 (![A: $i, B: $i, C: $i, D: $i] : ((~(ordered_pair(A, B) = ordered_pair(C, D))) | ((A = C) & (B = D)))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[169, 165])).
% 0.19/0.44 tff(171,plain,(
% 0.19/0.44 ![A: $i, B: $i, C: $i, D: $i] : ((~(ordered_pair(A, B) = ordered_pair(C, D))) | ((A = C) & (B = D)))),
% 0.19/0.44 inference(skolemize,[status(sab)],[170])).
% 0.19/0.44 tff(172,plain,
% 0.19/0.44 (![A: $i, B: $i, C: $i, D: $i] : ((~(ordered_pair(A, B) = ordered_pair(C, D))) | (~((~(A = C)) | (~(B = D)))))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[171, 164])).
% 0.19/0.44 tff(173,plain,
% 0.19/0.44 (![A: $i, B: $i, C: $i, D: $i] : ((~(ordered_pair(A, B) = ordered_pair(C, D))) | (~((~(A = C)) | (~(B = D)))))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[172, 162])).
% 0.19/0.44 tff(174,plain,
% 0.19/0.44 (((~![A: $i, B: $i, C: $i, D: $i] : ((~(ordered_pair(A, B) = ordered_pair(C, D))) | (~((~(A = C)) | (~(B = D)))))) | ((~(ordered_pair(B!12, C!13) = ordered_pair(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))))) | (~((~(B!12 = tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))) | (~(C!13 = tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))))))) <=> ((~![A: $i, B: $i, C: $i, D: $i] : ((~(ordered_pair(A, B) = ordered_pair(C, D))) | (~((~(A = C)) | (~(B = D)))))) | (~(ordered_pair(B!12, C!13) = ordered_pair(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))))) | (~((~(B!12 = tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))) | (~(C!13 = tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))))))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(175,plain,
% 0.19/0.44 ((~![A: $i, B: $i, C: $i, D: $i] : ((~(ordered_pair(A, B) = ordered_pair(C, D))) | (~((~(A = C)) | (~(B = D)))))) | ((~(ordered_pair(B!12, C!13) = ordered_pair(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))))) | (~((~(B!12 = tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))) | (~(C!13 = tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))))))),
% 0.19/0.44 inference(quant_inst,[status(thm)],[])).
% 0.19/0.44 tff(176,plain,
% 0.19/0.44 ((~![A: $i, B: $i, C: $i, D: $i] : ((~(ordered_pair(A, B) = ordered_pair(C, D))) | (~((~(A = C)) | (~(B = D)))))) | (~(ordered_pair(B!12, C!13) = ordered_pair(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))))) | (~((~(B!12 = tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))) | (~(C!13 = tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))))))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[175, 174])).
% 0.19/0.44 tff(177,plain,
% 0.19/0.44 (~((~(B!12 = tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))) | (~(C!13 = tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))))),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[176, 173, 160])).
% 0.19/0.44 tff(178,plain,
% 0.19/0.44 (((~(B!12 = tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))) | (~(C!13 = tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))))) | (C!13 = tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))),
% 0.19/0.44 inference(tautology,[status(thm)],[])).
% 0.19/0.44 tff(179,plain,
% 0.19/0.44 (C!13 = tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[178, 177])).
% 0.19/0.44 tff(180,plain,
% 0.19/0.44 (((~(B!12 = tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))) | (~(C!13 = tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))))) | (B!12 = tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))),
% 0.19/0.44 inference(tautology,[status(thm)],[])).
% 0.19/0.44 tff(181,plain,
% 0.19/0.44 (B!12 = tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[180, 177])).
% 0.19/0.44 tff(182,plain,
% 0.19/0.44 (below_refl(A!11, B!12, C!13) <=> below_refl(A!11, tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))),
% 0.19/0.44 inference(monotonicity,[status(thm)],[181, 179])).
% 0.19/0.44 tff(183,plain,
% 0.19/0.44 (below_refl(A!11, tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))) <=> below_refl(A!11, B!12, C!13)),
% 0.19/0.44 inference(symmetry,[status(thm)],[182])).
% 0.19/0.44 tff(184,plain,
% 0.19/0.44 (((~below_refl(A!11, tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))))) | (~element(tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~element(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11)))) | below_refl(A!11, tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))),
% 0.19/0.44 inference(tautology,[status(thm)],[])).
% 0.19/0.44 tff(185,plain,
% 0.19/0.44 (below_refl(A!11, tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[184, 71])).
% 0.19/0.44 tff(186,plain,
% 0.19/0.44 (below_refl(A!11, B!12, C!13)),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[185, 183])).
% 0.19/0.44 tff(187,plain,
% 0.19/0.44 ($false),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[37, 186])).
% 0.19/0.44 tff(188,plain,(below_refl(A!11, B!12, C!13)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.44 tff(189,plain,
% 0.19/0.44 ((in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), relation_of_lattice(A!11)) <=> below_refl(A!11, B!12, C!13)) | (~in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), relation_of_lattice(A!11))) | (~below_refl(A!11, B!12, C!13))),
% 0.19/0.44 inference(tautology,[status(thm)],[])).
% 0.19/0.44 tff(190,plain,
% 0.19/0.44 ((~in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), relation_of_lattice(A!11))) | (~below_refl(A!11, B!12, C!13))),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[189, 39])).
% 0.19/0.44 tff(191,plain,
% 0.19/0.44 (~in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), relation_of_lattice(A!11))),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[190, 188])).
% 0.19/0.44 tff(192,plain,
% 0.19/0.44 (~in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[191, 36])).
% 0.19/0.44 tff(193,plain,
% 0.19/0.44 (((~(in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11)) | ![C: $i, D: $i] : ((~element(D, the_carrier(A!11))) | (~element(C, the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), C, D))) | (~below_refl(A!11, C, D))))) | (~((~in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11))) | (~((~below_refl(A!11, tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))))) | (~element(tptp_fun_D_5(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11))) | (~element(tptp_fun_C_6(A!11, ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)), the_carrier(A!11)))))))) | (in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11)) | ![C: $i, D: $i] : ((~element(D, the_carrier(A!11))) | (~element(C, the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), C, D))) | (~below_refl(A!11, C, D))))),
% 0.19/0.44 inference(tautology,[status(thm)],[])).
% 0.19/0.44 tff(194,plain,
% 0.19/0.44 (in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11)) | ![C: $i, D: $i] : ((~element(D, the_carrier(A!11))) | (~element(C, the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), C, D))) | (~below_refl(A!11, C, D)))),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[193, 66])).
% 0.19/0.44 tff(195,plain,
% 0.19/0.44 ((~(in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11)) | ![C: $i, D: $i] : ((~element(D, the_carrier(A!11))) | (~element(C, the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), C, D))) | (~below_refl(A!11, C, D))))) | in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11)) | ![C: $i, D: $i] : ((~element(D, the_carrier(A!11))) | (~element(C, the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), C, D))) | (~below_refl(A!11, C, D)))),
% 0.19/0.44 inference(tautology,[status(thm)],[])).
% 0.19/0.44 tff(196,plain,
% 0.19/0.44 (in(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13), a_1_0_filter_1(A!11)) | ![C: $i, D: $i] : ((~element(D, the_carrier(A!11))) | (~element(C, the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), C, D))) | (~below_refl(A!11, C, D)))),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[195, 194])).
% 0.19/0.44 tff(197,plain,
% 0.19/0.44 (![C: $i, D: $i] : ((~element(D, the_carrier(A!11))) | (~element(C, the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), C, D))) | (~below_refl(A!11, C, D)))),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[196, 192])).
% 0.19/0.44 tff(198,plain,
% 0.19/0.44 (((~![C: $i, D: $i] : ((~element(D, the_carrier(A!11))) | (~element(C, the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), C, D))) | (~below_refl(A!11, C, D)))) | ((~element(B!12, the_carrier(A!11))) | (~element(C!13, the_carrier(A!11))) | (~below_refl(A!11, B!12, C!13)))) <=> ((~![C: $i, D: $i] : ((~element(D, the_carrier(A!11))) | (~element(C, the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), C, D))) | (~below_refl(A!11, C, D)))) | (~element(B!12, the_carrier(A!11))) | (~element(C!13, the_carrier(A!11))) | (~below_refl(A!11, B!12, C!13)))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(199,plain,
% 0.19/0.45 (((~element(C!13, the_carrier(A!11))) | (~element(B!12, the_carrier(A!11))) | $false | (~below_refl(A!11, B!12, C!13))) <=> ((~element(B!12, the_carrier(A!11))) | (~element(C!13, the_carrier(A!11))) | (~below_refl(A!11, B!12, C!13)))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(200,plain,
% 0.19/0.45 ((~$true) <=> $false),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(201,plain,
% 0.19/0.45 ((ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13)) <=> $true),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(202,plain,
% 0.19/0.45 ((~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))) <=> (~$true)),
% 0.19/0.45 inference(monotonicity,[status(thm)],[201])).
% 0.19/0.45 tff(203,plain,
% 0.19/0.45 ((~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))) <=> $false),
% 0.19/0.45 inference(transitivity,[status(thm)],[202, 200])).
% 0.19/0.45 tff(204,plain,
% 0.19/0.45 (((~element(C!13, the_carrier(A!11))) | (~element(B!12, the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))) | (~below_refl(A!11, B!12, C!13))) <=> ((~element(C!13, the_carrier(A!11))) | (~element(B!12, the_carrier(A!11))) | $false | (~below_refl(A!11, B!12, C!13)))),
% 0.19/0.45 inference(monotonicity,[status(thm)],[203])).
% 0.19/0.45 tff(205,plain,
% 0.19/0.45 (((~element(C!13, the_carrier(A!11))) | (~element(B!12, the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))) | (~below_refl(A!11, B!12, C!13))) <=> ((~element(B!12, the_carrier(A!11))) | (~element(C!13, the_carrier(A!11))) | (~below_refl(A!11, B!12, C!13)))),
% 0.19/0.45 inference(transitivity,[status(thm)],[204, 199])).
% 0.19/0.45 tff(206,plain,
% 0.19/0.45 (((~![C: $i, D: $i] : ((~element(D, the_carrier(A!11))) | (~element(C, the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), C, D))) | (~below_refl(A!11, C, D)))) | ((~element(C!13, the_carrier(A!11))) | (~element(B!12, the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))) | (~below_refl(A!11, B!12, C!13)))) <=> ((~![C: $i, D: $i] : ((~element(D, the_carrier(A!11))) | (~element(C, the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), C, D))) | (~below_refl(A!11, C, D)))) | ((~element(B!12, the_carrier(A!11))) | (~element(C!13, the_carrier(A!11))) | (~below_refl(A!11, B!12, C!13))))),
% 0.19/0.45 inference(monotonicity,[status(thm)],[205])).
% 0.19/0.45 tff(207,plain,
% 0.19/0.45 (((~![C: $i, D: $i] : ((~element(D, the_carrier(A!11))) | (~element(C, the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), C, D))) | (~below_refl(A!11, C, D)))) | ((~element(C!13, the_carrier(A!11))) | (~element(B!12, the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))) | (~below_refl(A!11, B!12, C!13)))) <=> ((~![C: $i, D: $i] : ((~element(D, the_carrier(A!11))) | (~element(C, the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), C, D))) | (~below_refl(A!11, C, D)))) | (~element(B!12, the_carrier(A!11))) | (~element(C!13, the_carrier(A!11))) | (~below_refl(A!11, B!12, C!13)))),
% 0.19/0.45 inference(transitivity,[status(thm)],[206, 198])).
% 0.19/0.45 tff(208,plain,
% 0.19/0.45 ((~![C: $i, D: $i] : ((~element(D, the_carrier(A!11))) | (~element(C, the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), C, D))) | (~below_refl(A!11, C, D)))) | ((~element(C!13, the_carrier(A!11))) | (~element(B!12, the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13))) | (~below_refl(A!11, B!12, C!13)))),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(209,plain,
% 0.19/0.45 ((~![C: $i, D: $i] : ((~element(D, the_carrier(A!11))) | (~element(C, the_carrier(A!11))) | (~(ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), B!12, C!13) = ordered_pair_as_product_element(the_carrier(A!11), the_carrier(A!11), C, D))) | (~below_refl(A!11, C, D)))) | (~element(B!12, the_carrier(A!11))) | (~element(C!13, the_carrier(A!11))) | (~below_refl(A!11, B!12, C!13))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[208, 207])).
% 0.19/0.45 tff(210,plain,
% 0.19/0.45 ($false),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[209, 150, 149, 188, 197])).
% 0.19/0.45 % SZS output end Proof
% 0.19/0.45 % E exiting
%------------------------------------------------------------------------------