%------------------------------------------------------------------------------ % File : ePrincess---1.0 % Problem : SEU098+1 : TPTP v8.1.0. Released v3.2.0. % Transfm : none % Format : tptp:raw % Command : ePrincess-casc -timeout=%d %s % Computer : n015.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 : 600s % DateTime : Tue Jul 19 08:46:35 EDT 2022 % Result : Theorem 4.29s 1.74s % Output : Proof 7.13s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.13/0.14 % Problem : SEU098+1 : TPTP v8.1.0. Released v3.2.0. % 0.13/0.15 % Command : ePrincess-casc -timeout=%d %s % 0.15/0.37 % Computer : n015.cluster.edu % 0.15/0.37 % Model : x86_64 x86_64 % 0.15/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.15/0.37 % Memory : 8042.1875MB % 0.15/0.37 % OS : Linux 3.10.0-693.el7.x86_64 % 0.15/0.37 % CPULimit : 300 % 0.15/0.37 % WCLimit : 600 % 0.15/0.37 % DateTime : Sat Jun 18 23:44:14 EDT 2022 % 0.15/0.37 % CPUTime : % 0.51/0.63 ____ _ % 0.51/0.63 ___ / __ \_____(_)___ ________ __________ % 0.51/0.63 / _ \/ /_/ / ___/ / __ \/ ___/ _ \/ ___/ ___/ % 0.51/0.63 / __/ ____/ / / / / / / /__/ __(__ |__ ) % 0.51/0.63 \___/_/ /_/ /_/_/ /_/\___/\___/____/____/ % 0.51/0.63 % 0.51/0.63 A Theorem Prover for First-Order Logic % 0.51/0.63 (ePrincess v.1.0) % 0.51/0.63 % 0.51/0.63 (c) Philipp Rümmer, 2009-2015 % 0.51/0.63 (c) Peter Backeman, 2014-2015 % 0.51/0.63 (contributions by Angelo Brillout, Peter Baumgartner) % 0.51/0.63 Free software under GNU Lesser General Public License (LGPL). % 0.51/0.63 Bug reports to peter@backeman.se % 0.51/0.63 % 0.51/0.63 For more information, visit http://user.uu.se/~petba168/breu/ % 0.51/0.63 % 0.51/0.63 Loading /export/starexec/sandbox2/benchmark/theBenchmark.p ... % 0.73/0.68 Prover 0: Options: -triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMaximal -resolutionMethod=nonUnifying +ignoreQuantifiers -generateTriggers=all % 1.87/1.08 Prover 0: Preprocessing ... % 2.98/1.41 Prover 0: Warning: ignoring some quantifiers % 2.98/1.44 Prover 0: Constructing countermodel ... % 4.29/1.74 Prover 0: proved (1055ms) % 4.29/1.74 % 4.29/1.74 No countermodel exists, formula is valid % 4.29/1.74 % SZS status Theorem for theBenchmark % 4.29/1.74 % 4.29/1.74 Generating proof ... Warning: ignoring some quantifiers % 6.39/2.21 found it (size 40) % 6.39/2.21 % 6.39/2.21 % SZS output start Proof for theBenchmark % 6.39/2.21 Assumed formulas after preprocessing and simplification: % 6.39/2.21 | (0) ? [v0] : ? [v1] : ? [v2] : ? [v3] : ? [v4] : ? [v5] : ? [v6] : ? [v7] : ? [v8] : ? [v9] : ? [v10] : ? [v11] : ? [v12] : ? [v13] : ? [v14] : ? [v15] : ? [v16] : ? [v17] : ? [v18] : ? [v19] : ? [v20] : ? [v21] : ? [v22] : (relation_dom(v0) = v1 & ordinal_yielding(v11) & being_limit_ordinal(v17) & function_yielding(v20) & relation_non_empty(v2) & transfinite_sequence(v11) & transfinite_sequence(v3) & relation_empty_yielding(v5) & relation_empty_yielding(v4) & relation_empty_yielding(empty_set) & one_to_one(v12) & one_to_one(v7) & one_to_one(empty_set) & natural(v22) & natural(v8) & relation(v20) & relation(v19) & relation(v16) & relation(v13) & relation(v12) & relation(v11) & relation(v10) & relation(v7) & relation(v5) & relation(v4) & relation(v3) & relation(v2) & relation(v0) & relation(empty_set) & function(v20) & function(v19) & function(v13) & function(v12) & function(v11) & function(v7) & function(v4) & function(v3) & function(v2) & function(v0) & function(empty_set) & finite(v21) & empty(v16) & empty(v15) & empty(v13) & empty(v12) & empty(v8) & empty(empty_set) & epsilon_connected(v22) & epsilon_connected(v18) & epsilon_connected(v17) & epsilon_connected(v14) & epsilon_connected(v12) & epsilon_connected(v8) & epsilon_connected(v6) & epsilon_connected(empty_set) & epsilon_transitive(v22) & epsilon_transitive(v18) & epsilon_transitive(v17) & epsilon_transitive(v14) & epsilon_transitive(v12) & epsilon_transitive(v8) & epsilon_transitive(v6) & epsilon_transitive(empty_set) & element(v14, positive_rationals) & element(v8, positive_rationals) & ordinal(v22) & ordinal(v18) & ordinal(v17) & ordinal(v14) & ordinal(v12) & ordinal(v8) & ordinal(v6) & ordinal(empty_set) & ~ empty(v22) & ~ empty(v21) & ~ empty(v14) & ~ empty(v10) & ~ empty(v9) & ~ empty(v6) & ~ empty(positive_rationals) & ! [v23] : ! [v24] : ! [v25] : ! [v26] : ! [v27] : ! [v28] : (v28 = v24 | ~ (relation_rng(v23) = v25) | ~ (relation_dom(v23) = v24) | ~ (first_projection_as_func_of(v24, v25) = v27) | ~ (function_image(v26, v24, v27, v23) = v28) | ~ (cartesian_product2(v24, v25) = v26) | ~ relation(v23) | ~ function(v23)) & ! [v23] : ! [v24] : ! [v25] : ! [v26] : ! [v27] : ! [v28] : (v24 = v23 | ~ (function_image(v28, v27, v26, v25) = v24) | ~ (function_image(v28, v27, v26, v25) = v23)) & ! [v23] : ! [v24] : ! [v25] : ! [v26] : ! [v27] : ( ~ (function_image(v23, v24, v25, v26) = v27) | ~ relation_of2(v25, v23, v24) | ~ quasi_total(v25, v23, v24) | ~ function(v25) | relation_image(v25, v26) = v27) & ! [v23] : ! [v24] : ! [v25] : ! [v26] : ! [v27] : ( ~ (function_image(v23, v24, v25, v26) = v27) | ~ relation_of2(v25, v23, v24) | ~ quasi_total(v25, v23, v24) | ~ function(v25) | ? [v28] : (powerset(v24) = v28 & element(v27, v28))) & ! [v23] : ! [v24] : ! [v25] : ! [v26] : (v24 = v23 | ~ (relation_image(v26, v25) = v24) | ~ (relation_image(v26, v25) = v23)) & ! [v23] : ! [v24] : ! [v25] : ! [v26] : (v24 = v23 | ~ (first_projection_as_func_of(v26, v25) = v24) | ~ (first_projection_as_func_of(v26, v25) = v23)) & ! [v23] : ! [v24] : ! [v25] : ! [v26] : (v24 = v23 | ~ (first_projection(v26, v25) = v24) | ~ (first_projection(v26, v25) = v23)) & ! [v23] : ! [v24] : ! [v25] : ! [v26] : (v24 = v23 | ~ (cartesian_product2(v26, v25) = v24) | ~ (cartesian_product2(v26, v25) = v23)) & ! [v23] : ! [v24] : ! [v25] : ! [v26] : ( ~ (relation_rng(v23) = v25) | ~ (relation_dom(v23) = v24) | ~ (cartesian_product2(v24, v25) = v26) | ~ relation(v23) | subset(v23, v26)) & ! [v23] : ! [v24] : ! [v25] : ! [v26] : ( ~ (cartesian_product2(v23, v24) = v26) | ~ relation_of2_as_subset(v25, v23, v24) | ? [v27] : (powerset(v26) = v27 & element(v25, v27))) & ! [v23] : ! [v24] : ! [v25] : ! [v26] : ( ~ (powerset(v25) = v26) | ~ empty(v25) | ~ element(v24, v26) | ~ in(v23, v24)) & ! [v23] : ! [v24] : ! [v25] : ! [v26] : ( ~ (powerset(v25) = v26) | ~ element(v24, v26) | ~ in(v23, v24) | element(v23, v25)) & ? [v23] : ! [v24] : ! [v25] : ! [v26] : ( ~ (cartesian_product2(v24, v25) = v26) | relation(v23) | ? [v27] : (powerset(v26) = v27 & ~ element(v23, v27))) & ! [v23] : ! [v24] : ! [v25] : (v24 = v23 | ~ (relation_rng(v25) = v24) | ~ (relation_rng(v25) = v23)) & ! [v23] : ! [v24] : ! [v25] : (v24 = v23 | ~ (relation_dom(v25) = v24) | ~ (relation_dom(v25) = v23)) & ! [v23] : ! [v24] : ! [v25] : (v24 = v23 | ~ (powerset(v25) = v24) | ~ (powerset(v25) = v23)) & ! [v23] : ! [v24] : ! [v25] : ( ~ (relation_image(v24, v23) = v25) | ~ relation(v24) | ~ function(v24) | ~ finite(v23) | finite(v25)) & ! [v23] : ! [v24] : ! [v25] : ( ~ (relation_image(v23, v24) = v25) | ~ relation(v23) | ~ function(v23) | ~ finite(v24) | finite(v25)) & ! [v23] : ! [v24] : ! [v25] : ( ~ (first_projection_as_func_of(v23, v24) = v25) | first_projection(v23, v24) = v25) & ! [v23] : ! [v24] : ! [v25] : ( ~ (first_projection_as_func_of(v23, v24) = v25) | function(v25)) & ! [v23] : ! [v24] : ! [v25] : ( ~ (first_projection_as_func_of(v23, v24) = v25) | ? [v26] : (cartesian_product2(v23, v24) = v26 & relation_of2_as_subset(v25, v26, v23))) & ! [v23] : ! [v24] : ! [v25] : ( ~ (first_projection_as_func_of(v23, v24) = v25) | ? [v26] : (cartesian_product2(v23, v24) = v26 & quasi_total(v25, v26, v23))) & ! [v23] : ! [v24] : ! [v25] : ( ~ (first_projection(v23, v24) = v25) | first_projection_as_func_of(v23, v24) = v25) & ! [v23] : ! [v24] : ! [v25] : ( ~ (first_projection(v23, v24) = v25) | relation(v25)) & ! [v23] : ! [v24] : ! [v25] : ( ~ (first_projection(v23, v24) = v25) | function(v25)) & ! [v23] : ! [v24] : ! [v25] : ( ~ (cartesian_product2(v23, v24) = v25) | ~ finite(v24) | ~ finite(v23) | finite(v25)) & ! [v23] : ! [v24] : ! [v25] : ( ~ (cartesian_product2(v23, v24) = v25) | ~ empty(v25) | empty(v24) | empty(v23)) & ! [v23] : ! [v24] : ! [v25] : ( ~ (cartesian_product2(v23, v24) = v25) | ? [v26] : (first_projection_as_func_of(v23, v24) = v26 & relation_of2_as_subset(v26, v25, v23))) & ! [v23] : ! [v24] : ! [v25] : ( ~ (cartesian_product2(v23, v24) = v25) | ? [v26] : (first_projection_as_func_of(v23, v24) = v26 & quasi_total(v26, v25, v23))) & ! [v23] : ! [v24] : ! [v25] : ( ~ (powerset(v24) = v25) | ~ subset(v23, v24) | element(v23, v25)) & ! [v23] : ! [v24] : ! [v25] : ( ~ (powerset(v24) = v25) | ~ element(v23, v25) | subset(v23, v24)) & ! [v23] : ! [v24] : ! [v25] : ( ~ (powerset(v23) = v24) | ~ finite(v23) | ~ element(v25, v24) | finite(v25)) & ! [v23] : ! [v24] : ! [v25] : ( ~ relation_of2_as_subset(v25, v23, v24) | relation_of2(v25, v23, v24)) & ! [v23] : ! [v24] : ! [v25] : ( ~ relation_of2(v25, v23, v24) | relation_of2_as_subset(v25, v23, v24)) & ! [v23] : ! [v24] : (v24 = v23 | ~ empty(v24) | ~ empty(v23)) & ! [v23] : ! [v24] : ( ~ (relation_rng(v23) = v24) | ~ relation_non_empty(v23) | ~ relation(v23) | ~ function(v23) | with_non_empty_elements(v24)) & ! [v23] : ! [v24] : ( ~ (relation_rng(v23) = v24) | ~ relation(v23) | ~ function(v23) | finite(v24) | ? [v25] : (relation_dom(v23) = v25 & ~ finite(v25))) & ! [v23] : ! [v24] : ( ~ (relation_rng(v23) = v24) | ~ relation(v23) | ~ empty(v24) | empty(v23)) & ! [v23] : ! [v24] : ( ~ (relation_rng(v23) = v24) | ~ empty(v23) | relation(v24)) & ! [v23] : ! [v24] : ( ~ (relation_rng(v23) = v24) | ~ empty(v23) | empty(v24)) & ! [v23] : ! [v24] : ( ~ (relation_dom(v23) = v24) | ~ transfinite_sequence(v23) | ~ relation(v23) | ~ function(v23) | epsilon_connected(v24)) & ! [v23] : ! [v24] : ( ~ (relation_dom(v23) = v24) | ~ transfinite_sequence(v23) | ~ relation(v23) | ~ function(v23) | epsilon_transitive(v24)) & ! [v23] : ! [v24] : ( ~ (relation_dom(v23) = v24) | ~ transfinite_sequence(v23) | ~ relation(v23) | ~ function(v23) | ordinal(v24)) & ! [v23] : ! [v24] : ( ~ (relation_dom(v23) = v24) | ~ relation(v23) | ~ function(v23) | ~ finite(v24) | ? [v25] : (relation_rng(v23) = v25 & finite(v25))) & ! [v23] : ! [v24] : ( ~ (relation_dom(v23) = v24) | ~ relation(v23) | ~ function(v23) | ? [v25] : ? [v26] : ? [v27] : (relation_rng(v23) = v25 & first_projection_as_func_of(v24, v25) = v27 & function_image(v26, v24, v27, v23) = v24 & cartesian_product2(v24, v25) = v26)) & ! [v23] : ! [v24] : ( ~ (relation_dom(v23) = v24) | ~ relation(v23) | ~ empty(v24) | empty(v23)) & ! [v23] : ! [v24] : ( ~ (relation_dom(v23) = v24) | ~ empty(v23) | relation(v24)) & ! [v23] : ! [v24] : ( ~ (relation_dom(v23) = v24) | ~ empty(v23) | empty(v24)) & ! [v23] : ! [v24] : ( ~ (powerset(v23) = v24) | ~ empty(v24)) & ! [v23] : ! [v24] : ( ~ (powerset(v23) = v24) | empty(v23) | ? [v25] : (finite(v25) & element(v25, v24) & ~ empty(v25))) & ! [v23] : ! [v24] : ( ~ (powerset(v23) = v24) | empty(v23) | ? [v25] : (element(v25, v24) & ~ empty(v25))) & ! [v23] : ! [v24] : ( ~ (powerset(v23) = v24) | ? [v25] : (one_to_one(v25) & natural(v25) & relation(v25) & function(v25) & finite(v25) & empty(v25) & epsilon_connected(v25) & epsilon_transitive(v25) & element(v25, v24) & ordinal(v25))) & ! [v23] : ! [v24] : ( ~ (powerset(v23) = v24) | ? [v25] : (empty(v25) & element(v25, v24))) & ! [v23] : ! [v24] : ( ~ subset(v23, v24) | ~ finite(v24) | finite(v23)) & ! [v23] : ! [v24] : ( ~ empty(v24) | ~ in(v23, v24)) & ! [v23] : ! [v24] : ( ~ element(v24, v23) | ~ ordinal(v23) | epsilon_connected(v24)) & ! [v23] : ! [v24] : ( ~ element(v24, v23) | ~ ordinal(v23) | epsilon_transitive(v24)) & ! [v23] : ! [v24] : ( ~ element(v24, v23) | ~ ordinal(v23) | ordinal(v24)) & ! [v23] : ! [v24] : ( ~ element(v23, v24) | empty(v24) | in(v23, v24)) & ! [v23] : ! [v24] : ( ~ in(v24, v23) | ~ in(v23, v24)) & ! [v23] : ! [v24] : ( ~ in(v23, v24) | element(v23, v24)) & ! [v23] : (v23 = empty_set | ~ empty(v23)) & ! [v23] : ( ~ relation(v23) | ~ function(v23) | ~ empty(v23) | one_to_one(v23)) & ! [v23] : ( ~ empty(v23) | ~ ordinal(v23) | natural(v23)) & ! [v23] : ( ~ empty(v23) | ~ ordinal(v23) | epsilon_connected(v23)) & ! [v23] : ( ~ empty(v23) | ~ ordinal(v23) | epsilon_transitive(v23)) & ! [v23] : ( ~ empty(v23) | relation(v23)) & ! [v23] : ( ~ empty(v23) | function(v23)) & ! [v23] : ( ~ empty(v23) | finite(v23)) & ! [v23] : ( ~ empty(v23) | epsilon_connected(v23)) & ! [v23] : ( ~ empty(v23) | epsilon_transitive(v23)) & ! [v23] : ( ~ empty(v23) | ordinal(v23)) & ! [v23] : ( ~ epsilon_connected(v23) | ~ epsilon_transitive(v23) | ordinal(v23)) & ! [v23] : ( ~ element(v23, positive_rationals) | ~ ordinal(v23) | natural(v23)) & ! [v23] : ( ~ element(v23, positive_rationals) | ~ ordinal(v23) | epsilon_connected(v23)) & ! [v23] : ( ~ element(v23, positive_rationals) | ~ ordinal(v23) | epsilon_transitive(v23)) & ! [v23] : ( ~ ordinal(v23) | epsilon_connected(v23)) & ! [v23] : ( ~ ordinal(v23) | epsilon_transitive(v23)) & ? [v23] : ? [v24] : ? [v25] : relation_of2_as_subset(v25, v23, v24) & ? [v23] : ? [v24] : ? [v25] : relation_of2(v25, v23, v24) & ? [v23] : ? [v24] : element(v24, v23) & ? [v23] : subset(v23, v23) & ((finite(v1) & ~ finite(v0)) | (finite(v0) & ~ finite(v1)))) % 6.77/2.25 | Instantiating (0) with all_0_0_0, all_0_1_1, all_0_2_2, all_0_3_3, all_0_4_4, all_0_5_5, all_0_6_6, all_0_7_7, all_0_8_8, all_0_9_9, all_0_10_10, all_0_11_11, all_0_12_12, all_0_13_13, all_0_14_14, all_0_15_15, all_0_16_16, all_0_17_17, all_0_18_18, all_0_19_19, all_0_20_20, all_0_21_21, all_0_22_22 yields: % 6.77/2.25 | (1) relation_dom(all_0_22_22) = all_0_21_21 & ordinal_yielding(all_0_11_11) & being_limit_ordinal(all_0_5_5) & function_yielding(all_0_2_2) & relation_non_empty(all_0_20_20) & transfinite_sequence(all_0_11_11) & transfinite_sequence(all_0_19_19) & relation_empty_yielding(all_0_17_17) & relation_empty_yielding(all_0_18_18) & relation_empty_yielding(empty_set) & one_to_one(all_0_10_10) & one_to_one(all_0_15_15) & one_to_one(empty_set) & natural(all_0_0_0) & natural(all_0_14_14) & relation(all_0_2_2) & relation(all_0_3_3) & relation(all_0_6_6) & relation(all_0_9_9) & relation(all_0_10_10) & relation(all_0_11_11) & relation(all_0_12_12) & relation(all_0_15_15) & relation(all_0_17_17) & relation(all_0_18_18) & relation(all_0_19_19) & relation(all_0_20_20) & relation(all_0_22_22) & relation(empty_set) & function(all_0_2_2) & function(all_0_3_3) & function(all_0_9_9) & function(all_0_10_10) & function(all_0_11_11) & function(all_0_15_15) & function(all_0_18_18) & function(all_0_19_19) & function(all_0_20_20) & function(all_0_22_22) & function(empty_set) & finite(all_0_1_1) & empty(all_0_6_6) & empty(all_0_7_7) & empty(all_0_9_9) & empty(all_0_10_10) & empty(all_0_14_14) & empty(empty_set) & epsilon_connected(all_0_0_0) & epsilon_connected(all_0_4_4) & epsilon_connected(all_0_5_5) & epsilon_connected(all_0_8_8) & epsilon_connected(all_0_10_10) & epsilon_connected(all_0_14_14) & epsilon_connected(all_0_16_16) & epsilon_connected(empty_set) & epsilon_transitive(all_0_0_0) & epsilon_transitive(all_0_4_4) & epsilon_transitive(all_0_5_5) & epsilon_transitive(all_0_8_8) & epsilon_transitive(all_0_10_10) & epsilon_transitive(all_0_14_14) & epsilon_transitive(all_0_16_16) & epsilon_transitive(empty_set) & element(all_0_8_8, positive_rationals) & element(all_0_14_14, positive_rationals) & ordinal(all_0_0_0) & ordinal(all_0_4_4) & ordinal(all_0_5_5) & ordinal(all_0_8_8) & ordinal(all_0_10_10) & ordinal(all_0_14_14) & ordinal(all_0_16_16) & ordinal(empty_set) & ~ empty(all_0_0_0) & ~ empty(all_0_1_1) & ~ empty(all_0_8_8) & ~ empty(all_0_12_12) & ~ empty(all_0_13_13) & ~ empty(all_0_16_16) & ~ empty(positive_rationals) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = v1 | ~ (relation_rng(v0) = v2) | ~ (relation_dom(v0) = v1) | ~ (first_projection_as_func_of(v1, v2) = v4) | ~ (function_image(v3, v1, v4, v0) = v5) | ~ (cartesian_product2(v1, v2) = v3) | ~ relation(v0) | ~ function(v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v1 = v0 | ~ (function_image(v5, v4, v3, v2) = v1) | ~ (function_image(v5, v4, v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (function_image(v0, v1, v2, v3) = v4) | ~ relation_of2(v2, v0, v1) | ~ quasi_total(v2, v0, v1) | ~ function(v2) | relation_image(v2, v3) = v4) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (function_image(v0, v1, v2, v3) = v4) | ~ relation_of2(v2, v0, v1) | ~ quasi_total(v2, v0, v1) | ~ function(v2) | ? [v5] : (powerset(v1) = v5 & element(v4, v5))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (relation_image(v3, v2) = v1) | ~ (relation_image(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (first_projection_as_func_of(v3, v2) = v1) | ~ (first_projection_as_func_of(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (first_projection(v3, v2) = v1) | ~ (first_projection(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (cartesian_product2(v3, v2) = v1) | ~ (cartesian_product2(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (relation_rng(v0) = v2) | ~ (relation_dom(v0) = v1) | ~ (cartesian_product2(v1, v2) = v3) | ~ relation(v0) | subset(v0, v3)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (cartesian_product2(v0, v1) = v3) | ~ relation_of2_as_subset(v2, v0, v1) | ? [v4] : (powerset(v3) = v4 & element(v2, v4))) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (powerset(v2) = v3) | ~ empty(v2) | ~ element(v1, v3) | ~ in(v0, v1)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (powerset(v2) = v3) | ~ element(v1, v3) | ~ in(v0, v1) | element(v0, v2)) & ? [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (cartesian_product2(v1, v2) = v3) | relation(v0) | ? [v4] : (powerset(v3) = v4 & ~ element(v0, v4))) & ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (relation_rng(v2) = v1) | ~ (relation_rng(v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (relation_dom(v2) = v1) | ~ (relation_dom(v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (powerset(v2) = v1) | ~ (powerset(v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (relation_image(v1, v0) = v2) | ~ relation(v1) | ~ function(v1) | ~ finite(v0) | finite(v2)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (relation_image(v0, v1) = v2) | ~ relation(v0) | ~ function(v0) | ~ finite(v1) | finite(v2)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (first_projection_as_func_of(v0, v1) = v2) | first_projection(v0, v1) = v2) & ! [v0] : ! [v1] : ! [v2] : ( ~ (first_projection_as_func_of(v0, v1) = v2) | function(v2)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (first_projection_as_func_of(v0, v1) = v2) | ? [v3] : (cartesian_product2(v0, v1) = v3 & relation_of2_as_subset(v2, v3, v0))) & ! [v0] : ! [v1] : ! [v2] : ( ~ (first_projection_as_func_of(v0, v1) = v2) | ? [v3] : (cartesian_product2(v0, v1) = v3 & quasi_total(v2, v3, v0))) & ! [v0] : ! [v1] : ! [v2] : ( ~ (first_projection(v0, v1) = v2) | first_projection_as_func_of(v0, v1) = v2) & ! [v0] : ! [v1] : ! [v2] : ( ~ (first_projection(v0, v1) = v2) | relation(v2)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (first_projection(v0, v1) = v2) | function(v2)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (cartesian_product2(v0, v1) = v2) | ~ finite(v1) | ~ finite(v0) | finite(v2)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (cartesian_product2(v0, v1) = v2) | ~ empty(v2) | empty(v1) | empty(v0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (cartesian_product2(v0, v1) = v2) | ? [v3] : (first_projection_as_func_of(v0, v1) = v3 & relation_of2_as_subset(v3, v2, v0))) & ! [v0] : ! [v1] : ! [v2] : ( ~ (cartesian_product2(v0, v1) = v2) | ? [v3] : (first_projection_as_func_of(v0, v1) = v3 & quasi_total(v3, v2, v0))) & ! [v0] : ! [v1] : ! [v2] : ( ~ (powerset(v1) = v2) | ~ subset(v0, v1) | element(v0, v2)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (powerset(v1) = v2) | ~ element(v0, v2) | subset(v0, v1)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (powerset(v0) = v1) | ~ finite(v0) | ~ element(v2, v1) | finite(v2)) & ! [v0] : ! [v1] : ! [v2] : ( ~ relation_of2_as_subset(v2, v0, v1) | relation_of2(v2, v0, v1)) & ! [v0] : ! [v1] : ! [v2] : ( ~ relation_of2(v2, v0, v1) | relation_of2_as_subset(v2, v0, v1)) & ! [v0] : ! [v1] : (v1 = v0 | ~ empty(v1) | ~ empty(v0)) & ! [v0] : ! [v1] : ( ~ (relation_rng(v0) = v1) | ~ relation_non_empty(v0) | ~ relation(v0) | ~ function(v0) | with_non_empty_elements(v1)) & ! [v0] : ! [v1] : ( ~ (relation_rng(v0) = v1) | ~ relation(v0) | ~ function(v0) | finite(v1) | ? [v2] : (relation_dom(v0) = v2 & ~ finite(v2))) & ! [v0] : ! [v1] : ( ~ (relation_rng(v0) = v1) | ~ relation(v0) | ~ empty(v1) | empty(v0)) & ! [v0] : ! [v1] : ( ~ (relation_rng(v0) = v1) | ~ empty(v0) | relation(v1)) & ! [v0] : ! [v1] : ( ~ (relation_rng(v0) = v1) | ~ empty(v0) | empty(v1)) & ! [v0] : ! [v1] : ( ~ (relation_dom(v0) = v1) | ~ transfinite_sequence(v0) | ~ relation(v0) | ~ function(v0) | epsilon_connected(v1)) & ! [v0] : ! [v1] : ( ~ (relation_dom(v0) = v1) | ~ transfinite_sequence(v0) | ~ relation(v0) | ~ function(v0) | epsilon_transitive(v1)) & ! [v0] : ! [v1] : ( ~ (relation_dom(v0) = v1) | ~ transfinite_sequence(v0) | ~ relation(v0) | ~ function(v0) | ordinal(v1)) & ! [v0] : ! [v1] : ( ~ (relation_dom(v0) = v1) | ~ relation(v0) | ~ function(v0) | ~ finite(v1) | ? [v2] : (relation_rng(v0) = v2 & finite(v2))) & ! [v0] : ! [v1] : ( ~ (relation_dom(v0) = v1) | ~ relation(v0) | ~ function(v0) | ? [v2] : ? [v3] : ? [v4] : (relation_rng(v0) = v2 & first_projection_as_func_of(v1, v2) = v4 & function_image(v3, v1, v4, v0) = v1 & cartesian_product2(v1, v2) = v3)) & ! [v0] : ! [v1] : ( ~ (relation_dom(v0) = v1) | ~ relation(v0) | ~ empty(v1) | empty(v0)) & ! [v0] : ! [v1] : ( ~ (relation_dom(v0) = v1) | ~ empty(v0) | relation(v1)) & ! [v0] : ! [v1] : ( ~ (relation_dom(v0) = v1) | ~ empty(v0) | empty(v1)) & ! [v0] : ! [v1] : ( ~ (powerset(v0) = v1) | ~ empty(v1)) & ! [v0] : ! [v1] : ( ~ (powerset(v0) = v1) | empty(v0) | ? [v2] : (finite(v2) & element(v2, v1) & ~ empty(v2))) & ! [v0] : ! [v1] : ( ~ (powerset(v0) = v1) | empty(v0) | ? [v2] : (element(v2, v1) & ~ empty(v2))) & ! [v0] : ! [v1] : ( ~ (powerset(v0) = v1) | ? [v2] : (one_to_one(v2) & natural(v2) & relation(v2) & function(v2) & finite(v2) & empty(v2) & epsilon_connected(v2) & epsilon_transitive(v2) & element(v2, v1) & ordinal(v2))) & ! [v0] : ! [v1] : ( ~ (powerset(v0) = v1) | ? [v2] : (empty(v2) & element(v2, v1))) & ! [v0] : ! [v1] : ( ~ subset(v0, v1) | ~ finite(v1) | finite(v0)) & ! [v0] : ! [v1] : ( ~ empty(v1) | ~ in(v0, v1)) & ! [v0] : ! [v1] : ( ~ element(v1, v0) | ~ ordinal(v0) | epsilon_connected(v1)) & ! [v0] : ! [v1] : ( ~ element(v1, v0) | ~ ordinal(v0) | epsilon_transitive(v1)) & ! [v0] : ! [v1] : ( ~ element(v1, v0) | ~ ordinal(v0) | ordinal(v1)) & ! [v0] : ! [v1] : ( ~ element(v0, v1) | empty(v1) | in(v0, v1)) & ! [v0] : ! [v1] : ( ~ in(v1, v0) | ~ in(v0, v1)) & ! [v0] : ! [v1] : ( ~ in(v0, v1) | element(v0, v1)) & ! [v0] : (v0 = empty_set | ~ empty(v0)) & ! [v0] : ( ~ relation(v0) | ~ function(v0) | ~ empty(v0) | one_to_one(v0)) & ! [v0] : ( ~ empty(v0) | ~ ordinal(v0) | natural(v0)) & ! [v0] : ( ~ empty(v0) | ~ ordinal(v0) | epsilon_connected(v0)) & ! [v0] : ( ~ empty(v0) | ~ ordinal(v0) | epsilon_transitive(v0)) & ! [v0] : ( ~ empty(v0) | relation(v0)) & ! [v0] : ( ~ empty(v0) | function(v0)) & ! [v0] : ( ~ empty(v0) | finite(v0)) & ! [v0] : ( ~ empty(v0) | epsilon_connected(v0)) & ! [v0] : ( ~ empty(v0) | epsilon_transitive(v0)) & ! [v0] : ( ~ empty(v0) | ordinal(v0)) & ! [v0] : ( ~ epsilon_connected(v0) | ~ epsilon_transitive(v0) | ordinal(v0)) & ! [v0] : ( ~ element(v0, positive_rationals) | ~ ordinal(v0) | natural(v0)) & ! [v0] : ( ~ element(v0, positive_rationals) | ~ ordinal(v0) | epsilon_connected(v0)) & ! [v0] : ( ~ element(v0, positive_rationals) | ~ ordinal(v0) | epsilon_transitive(v0)) & ! [v0] : ( ~ ordinal(v0) | epsilon_connected(v0)) & ! [v0] : ( ~ ordinal(v0) | epsilon_transitive(v0)) & ? [v0] : ? [v1] : ? [v2] : relation_of2_as_subset(v2, v0, v1) & ? [v0] : ? [v1] : ? [v2] : relation_of2(v2, v0, v1) & ? [v0] : ? [v1] : element(v1, v0) & ? [v0] : subset(v0, v0) & ((finite(all_0_21_21) & ~ finite(all_0_22_22)) | (finite(all_0_22_22) & ~ finite(all_0_21_21))) % 6.77/2.27 | % 6.77/2.27 | Applying alpha-rule on (1) yields: % 6.77/2.27 | (2) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (function_image(v0, v1, v2, v3) = v4) | ~ relation_of2(v2, v0, v1) | ~ quasi_total(v2, v0, v1) | ~ function(v2) | relation_image(v2, v3) = v4) % 6.77/2.27 | (3) ! [v0] : ! [v1] : ( ~ (relation_rng(v0) = v1) | ~ relation(v0) | ~ empty(v1) | empty(v0)) % 6.77/2.27 | (4) finite(all_0_1_1) % 6.77/2.27 | (5) ! [v0] : ( ~ empty(v0) | relation(v0)) % 6.77/2.27 | (6) ! [v0] : ( ~ relation(v0) | ~ function(v0) | ~ empty(v0) | one_to_one(v0)) % 6.77/2.27 | (7) ! [v0] : ( ~ element(v0, positive_rationals) | ~ ordinal(v0) | epsilon_transitive(v0)) % 6.77/2.27 | (8) ! [v0] : ! [v1] : ! [v2] : ( ~ (cartesian_product2(v0, v1) = v2) | ? [v3] : (first_projection_as_func_of(v0, v1) = v3 & relation_of2_as_subset(v3, v2, v0))) % 6.77/2.27 | (9) function(all_0_3_3) % 6.77/2.27 | (10) epsilon_connected(empty_set) % 6.77/2.27 | (11) function(all_0_2_2) % 6.77/2.27 | (12) ! [v0] : ! [v1] : ( ~ (relation_dom(v0) = v1) | ~ relation(v0) | ~ empty(v1) | empty(v0)) % 6.77/2.27 | (13) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v1 = v0 | ~ (function_image(v5, v4, v3, v2) = v1) | ~ (function_image(v5, v4, v3, v2) = v0)) % 6.77/2.28 | (14) ! [v0] : ! [v1] : ! [v2] : ( ~ (relation_image(v0, v1) = v2) | ~ relation(v0) | ~ function(v0) | ~ finite(v1) | finite(v2)) % 6.77/2.28 | (15) ! [v0] : ! [v1] : ! [v2] : ( ~ relation_of2(v2, v0, v1) | relation_of2_as_subset(v2, v0, v1)) % 6.77/2.28 | (16) ! [v0] : ! [v1] : ! [v2] : ( ~ relation_of2_as_subset(v2, v0, v1) | relation_of2(v2, v0, v1)) % 6.77/2.28 | (17) ~ empty(positive_rationals) % 6.77/2.28 | (18) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (cartesian_product2(v3, v2) = v1) | ~ (cartesian_product2(v3, v2) = v0)) % 6.77/2.28 | (19) function(all_0_15_15) % 6.77/2.28 | (20) ! [v0] : ( ~ ordinal(v0) | epsilon_connected(v0)) % 6.77/2.28 | (21) transfinite_sequence(all_0_11_11) % 6.77/2.28 | (22) natural(all_0_0_0) % 6.77/2.28 | (23) ? [v0] : subset(v0, v0) % 6.77/2.28 | (24) ordinal(all_0_16_16) % 6.77/2.28 | (25) relation(all_0_19_19) % 6.77/2.28 | (26) ordinal(all_0_14_14) % 6.77/2.28 | (27) epsilon_connected(all_0_10_10) % 6.77/2.28 | (28) relation(all_0_18_18) % 6.77/2.28 | (29) epsilon_transitive(all_0_8_8) % 6.77/2.28 | (30) ! [v0] : ! [v1] : ( ~ (relation_rng(v0) = v1) | ~ relation(v0) | ~ function(v0) | finite(v1) | ? [v2] : (relation_dom(v0) = v2 & ~ finite(v2))) % 6.77/2.28 | (31) function(all_0_9_9) % 6.77/2.28 | (32) ! [v0] : ! [v1] : ! [v2] : ( ~ (first_projection_as_func_of(v0, v1) = v2) | ? [v3] : (cartesian_product2(v0, v1) = v3 & quasi_total(v2, v3, v0))) % 6.77/2.28 | (33) element(all_0_8_8, positive_rationals) % 6.77/2.28 | (34) (finite(all_0_21_21) & ~ finite(all_0_22_22)) | (finite(all_0_22_22) & ~ finite(all_0_21_21)) % 6.77/2.28 | (35) ! [v0] : ! [v1] : ! [v2] : ( ~ (first_projection_as_func_of(v0, v1) = v2) | ? [v3] : (cartesian_product2(v0, v1) = v3 & relation_of2_as_subset(v2, v3, v0))) % 6.77/2.28 | (36) function(all_0_11_11) % 6.77/2.28 | (37) ! [v0] : ! [v1] : ( ~ empty(v1) | ~ in(v0, v1)) % 6.77/2.28 | (38) ! [v0] : ! [v1] : ! [v2] : ( ~ (powerset(v1) = v2) | ~ element(v0, v2) | subset(v0, v1)) % 6.77/2.28 | (39) ! [v0] : ! [v1] : ! [v2] : ( ~ (powerset(v1) = v2) | ~ subset(v0, v1) | element(v0, v2)) % 6.77/2.28 | (40) epsilon_transitive(empty_set) % 6.77/2.28 | (41) ! [v0] : ! [v1] : ! [v2] : ( ~ (cartesian_product2(v0, v1) = v2) | ~ finite(v1) | ~ finite(v0) | finite(v2)) % 6.77/2.28 | (42) epsilon_transitive(all_0_10_10) % 6.77/2.28 | (43) relation_empty_yielding(all_0_17_17) % 6.77/2.28 | (44) epsilon_transitive(all_0_16_16) % 6.77/2.28 | (45) ! [v0] : ! [v1] : ( ~ element(v1, v0) | ~ ordinal(v0) | epsilon_connected(v1)) % 6.77/2.28 | (46) ! [v0] : ! [v1] : ( ~ (relation_dom(v0) = v1) | ~ empty(v0) | empty(v1)) % 6.77/2.28 | (47) ! [v0] : ( ~ empty(v0) | finite(v0)) % 6.77/2.28 | (48) empty(all_0_10_10) % 6.77/2.28 | (49) ~ empty(all_0_13_13) % 6.77/2.28 | (50) relation_empty_yielding(all_0_18_18) % 6.77/2.28 | (51) ! [v0] : ! [v1] : ( ~ (relation_dom(v0) = v1) | ~ transfinite_sequence(v0) | ~ relation(v0) | ~ function(v0) | epsilon_transitive(v1)) % 6.77/2.28 | (52) ~ empty(all_0_12_12) % 6.77/2.28 | (53) ! [v0] : ! [v1] : ! [v2] : ( ~ (relation_image(v1, v0) = v2) | ~ relation(v1) | ~ function(v1) | ~ finite(v0) | finite(v2)) % 6.77/2.28 | (54) epsilon_transitive(all_0_4_4) % 6.77/2.28 | (55) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (first_projection(v3, v2) = v1) | ~ (first_projection(v3, v2) = v0)) % 6.77/2.28 | (56) ! [v0] : ! [v1] : ( ~ (relation_dom(v0) = v1) | ~ relation(v0) | ~ function(v0) | ? [v2] : ? [v3] : ? [v4] : (relation_rng(v0) = v2 & first_projection_as_func_of(v1, v2) = v4 & function_image(v3, v1, v4, v0) = v1 & cartesian_product2(v1, v2) = v3)) % 6.77/2.28 | (57) empty(all_0_7_7) % 6.77/2.28 | (58) ? [v0] : ? [v1] : ? [v2] : relation_of2_as_subset(v2, v0, v1) % 6.77/2.28 | (59) ! [v0] : ! [v1] : ! [v2] : ( ~ (first_projection(v0, v1) = v2) | relation(v2)) % 6.77/2.28 | (60) epsilon_connected(all_0_16_16) % 6.77/2.28 | (61) one_to_one(all_0_15_15) % 6.77/2.28 | (62) ordinal(all_0_4_4) % 6.77/2.28 | (63) ! [v0] : ! [v1] : ! [v2] : ( ~ (cartesian_product2(v0, v1) = v2) | ~ empty(v2) | empty(v1) | empty(v0)) % 6.77/2.28 | (64) empty(all_0_6_6) % 6.77/2.28 | (65) function(all_0_22_22) % 6.77/2.28 | (66) ! [v0] : ! [v1] : ( ~ (relation_rng(v0) = v1) | ~ empty(v0) | relation(v1)) % 6.77/2.28 | (67) ? [v0] : ? [v1] : ? [v2] : relation_of2(v2, v0, v1) % 6.77/2.28 | (68) ! [v0] : ! [v1] : ( ~ (relation_dom(v0) = v1) | ~ empty(v0) | relation(v1)) % 6.77/2.28 | (69) ! [v0] : ( ~ empty(v0) | ~ ordinal(v0) | epsilon_connected(v0)) % 6.77/2.28 | (70) ordinal(all_0_0_0) % 6.77/2.28 | (71) ! [v0] : ! [v1] : ( ~ (powerset(v0) = v1) | ~ empty(v1)) % 6.77/2.28 | (72) ! [v0] : ( ~ empty(v0) | ordinal(v0)) % 6.77/2.28 | (73) relation(all_0_22_22) % 6.77/2.29 | (74) ! [v0] : ( ~ empty(v0) | epsilon_transitive(v0)) % 6.77/2.29 | (75) relation_empty_yielding(empty_set) % 6.77/2.29 | (76) ! [v0] : ! [v1] : ( ~ (powerset(v0) = v1) | ? [v2] : (empty(v2) & element(v2, v1))) % 6.77/2.29 | (77) ~ empty(all_0_0_0) % 6.77/2.29 | (78) ! [v0] : ! [v1] : ( ~ (relation_dom(v0) = v1) | ~ transfinite_sequence(v0) | ~ relation(v0) | ~ function(v0) | ordinal(v1)) % 6.77/2.29 | (79) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (relation_image(v3, v2) = v1) | ~ (relation_image(v3, v2) = v0)) % 6.77/2.29 | (80) ! [v0] : ! [v1] : ! [v2] : ( ~ (powerset(v0) = v1) | ~ finite(v0) | ~ element(v2, v1) | finite(v2)) % 6.77/2.29 | (81) function(all_0_19_19) % 6.77/2.29 | (82) ~ empty(all_0_8_8) % 6.77/2.29 | (83) ordinal_yielding(all_0_11_11) % 6.77/2.29 | (84) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (powerset(v2) = v1) | ~ (powerset(v2) = v0)) % 6.77/2.29 | (85) ! [v0] : (v0 = empty_set | ~ empty(v0)) % 6.77/2.29 | (86) ordinal(all_0_5_5) % 6.77/2.29 | (87) ! [v0] : ( ~ element(v0, positive_rationals) | ~ ordinal(v0) | natural(v0)) % 6.77/2.29 | (88) ! [v0] : ! [v1] : ( ~ in(v1, v0) | ~ in(v0, v1)) % 6.77/2.29 | (89) ordinal(all_0_8_8) % 6.77/2.29 | (90) function(all_0_10_10) % 6.77/2.29 | (91) ~ empty(all_0_1_1) % 6.77/2.29 | (92) epsilon_transitive(all_0_5_5) % 6.77/2.29 | (93) relation_dom(all_0_22_22) = all_0_21_21 % 6.77/2.29 | (94) relation(all_0_12_12) % 6.77/2.29 | (95) ! [v0] : ( ~ epsilon_connected(v0) | ~ epsilon_transitive(v0) | ordinal(v0)) % 6.77/2.29 | (96) ! [v0] : ! [v1] : ( ~ element(v1, v0) | ~ ordinal(v0) | epsilon_transitive(v1)) % 6.77/2.29 | (97) relation(all_0_6_6) % 6.77/2.29 | (98) ! [v0] : ! [v1] : ( ~ (relation_rng(v0) = v1) | ~ empty(v0) | empty(v1)) % 6.77/2.29 | (99) function(empty_set) % 6.77/2.29 | (100) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = v1 | ~ (relation_rng(v0) = v2) | ~ (relation_dom(v0) = v1) | ~ (first_projection_as_func_of(v1, v2) = v4) | ~ (function_image(v3, v1, v4, v0) = v5) | ~ (cartesian_product2(v1, v2) = v3) | ~ relation(v0) | ~ function(v0)) % 6.77/2.29 | (101) relation(all_0_10_10) % 6.77/2.29 | (102) transfinite_sequence(all_0_19_19) % 6.77/2.29 | (103) ! [v0] : ! [v1] : ! [v2] : ( ~ (cartesian_product2(v0, v1) = v2) | ? [v3] : (first_projection_as_func_of(v0, v1) = v3 & quasi_total(v3, v2, v0))) % 6.77/2.29 | (104) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (relation_dom(v2) = v1) | ~ (relation_dom(v2) = v0)) % 6.77/2.29 | (105) empty(empty_set) % 6.77/2.29 | (106) ! [v0] : ( ~ empty(v0) | epsilon_connected(v0)) % 6.77/2.29 | (107) ! [v0] : ! [v1] : ! [v2] : ( ~ (first_projection_as_func_of(v0, v1) = v2) | function(v2)) % 6.77/2.29 | (108) empty(all_0_14_14) % 6.77/2.29 | (109) relation(all_0_20_20) % 6.77/2.29 | (110) ! [v0] : ( ~ empty(v0) | function(v0)) % 6.77/2.29 | (111) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (powerset(v2) = v3) | ~ empty(v2) | ~ element(v1, v3) | ~ in(v0, v1)) % 6.77/2.29 | (112) epsilon_connected(all_0_8_8) % 6.77/2.29 | (113) ! [v0] : ! [v1] : ( ~ subset(v0, v1) | ~ finite(v1) | finite(v0)) % 6.77/2.29 | (114) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (powerset(v2) = v3) | ~ element(v1, v3) | ~ in(v0, v1) | element(v0, v2)) % 6.77/2.29 | (115) empty(all_0_9_9) % 6.77/2.29 | (116) ! [v0] : ! [v1] : ( ~ (powerset(v0) = v1) | ? [v2] : (one_to_one(v2) & natural(v2) & relation(v2) & function(v2) & finite(v2) & empty(v2) & epsilon_connected(v2) & epsilon_transitive(v2) & element(v2, v1) & ordinal(v2))) % 6.77/2.29 | (117) natural(all_0_14_14) % 6.77/2.29 | (118) ! [v0] : ! [v1] : (v1 = v0 | ~ empty(v1) | ~ empty(v0)) % 6.77/2.29 | (119) ! [v0] : ( ~ empty(v0) | ~ ordinal(v0) | natural(v0)) % 6.77/2.29 | (120) function_yielding(all_0_2_2) % 6.77/2.29 | (121) ! [v0] : ! [v1] : ( ~ element(v1, v0) | ~ ordinal(v0) | ordinal(v1)) % 7.13/2.29 | (122) ! [v0] : ! [v1] : ( ~ (relation_dom(v0) = v1) | ~ transfinite_sequence(v0) | ~ relation(v0) | ~ function(v0) | epsilon_connected(v1)) % 7.13/2.29 | (123) being_limit_ordinal(all_0_5_5) % 7.13/2.29 | (124) ! [v0] : ! [v1] : ( ~ (powerset(v0) = v1) | empty(v0) | ? [v2] : (element(v2, v1) & ~ empty(v2))) % 7.13/2.30 | (125) relation(all_0_17_17) % 7.13/2.30 | (126) ! [v0] : ! [v1] : ( ~ (relation_dom(v0) = v1) | ~ relation(v0) | ~ function(v0) | ~ finite(v1) | ? [v2] : (relation_rng(v0) = v2 & finite(v2))) % 7.13/2.30 | (127) relation(all_0_11_11) % 7.13/2.30 | (128) epsilon_connected(all_0_14_14) % 7.13/2.30 | (129) element(all_0_14_14, positive_rationals) % 7.13/2.30 | (130) ! [v0] : ! [v1] : ( ~ (relation_rng(v0) = v1) | ~ relation_non_empty(v0) | ~ relation(v0) | ~ function(v0) | with_non_empty_elements(v1)) % 7.13/2.30 | (131) ! [v0] : ( ~ empty(v0) | ~ ordinal(v0) | epsilon_transitive(v0)) % 7.13/2.30 | (132) ? [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (cartesian_product2(v1, v2) = v3) | relation(v0) | ? [v4] : (powerset(v3) = v4 & ~ element(v0, v4))) % 7.13/2.30 | (133) ! [v0] : ! [v1] : ( ~ in(v0, v1) | element(v0, v1)) % 7.13/2.30 | (134) ordinal(all_0_10_10) % 7.13/2.30 | (135) epsilon_connected(all_0_0_0) % 7.13/2.30 | (136) epsilon_transitive(all_0_14_14) % 7.13/2.30 | (137) function(all_0_18_18) % 7.13/2.30 | (138) relation(all_0_9_9) % 7.13/2.30 | (139) one_to_one(all_0_10_10) % 7.13/2.30 | (140) ! [v0] : ! [v1] : ! [v2] : ( ~ (first_projection(v0, v1) = v2) | first_projection_as_func_of(v0, v1) = v2) % 7.13/2.30 | (141) ! [v0] : ! [v1] : ! [v2] : ( ~ (first_projection_as_func_of(v0, v1) = v2) | first_projection(v0, v1) = v2) % 7.13/2.30 | (142) ! [v0] : ! [v1] : ( ~ (powerset(v0) = v1) | empty(v0) | ? [v2] : (finite(v2) & element(v2, v1) & ~ empty(v2))) % 7.13/2.30 | (143) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (relation_rng(v0) = v2) | ~ (relation_dom(v0) = v1) | ~ (cartesian_product2(v1, v2) = v3) | ~ relation(v0) | subset(v0, v3)) % 7.13/2.30 | (144) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (function_image(v0, v1, v2, v3) = v4) | ~ relation_of2(v2, v0, v1) | ~ quasi_total(v2, v0, v1) | ~ function(v2) | ? [v5] : (powerset(v1) = v5 & element(v4, v5))) % 7.13/2.30 | (145) ! [v0] : ! [v1] : ! [v2] : ( ~ (first_projection(v0, v1) = v2) | function(v2)) % 7.13/2.30 | (146) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (cartesian_product2(v0, v1) = v3) | ~ relation_of2_as_subset(v2, v0, v1) | ? [v4] : (powerset(v3) = v4 & element(v2, v4))) % 7.13/2.30 | (147) ordinal(empty_set) % 7.13/2.30 | (148) epsilon_connected(all_0_5_5) % 7.13/2.30 | (149) one_to_one(empty_set) % 7.13/2.30 | (150) relation(all_0_15_15) % 7.13/2.30 | (151) relation(all_0_2_2) % 7.13/2.30 | (152) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (first_projection_as_func_of(v3, v2) = v1) | ~ (first_projection_as_func_of(v3, v2) = v0)) % 7.13/2.30 | (153) relation_non_empty(all_0_20_20) % 7.13/2.30 | (154) epsilon_transitive(all_0_0_0) % 7.13/2.30 | (155) ! [v0] : ( ~ element(v0, positive_rationals) | ~ ordinal(v0) | epsilon_connected(v0)) % 7.13/2.30 | (156) ? [v0] : ? [v1] : element(v1, v0) % 7.13/2.30 | (157) function(all_0_20_20) % 7.13/2.30 | (158) ~ empty(all_0_16_16) % 7.13/2.30 | (159) epsilon_connected(all_0_4_4) % 7.13/2.30 | (160) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (relation_rng(v2) = v1) | ~ (relation_rng(v2) = v0)) % 7.13/2.30 | (161) ! [v0] : ( ~ ordinal(v0) | epsilon_transitive(v0)) % 7.13/2.30 | (162) relation(all_0_3_3) % 7.13/2.30 | (163) relation(empty_set) % 7.13/2.30 | (164) ! [v0] : ! [v1] : ( ~ element(v0, v1) | empty(v1) | in(v0, v1)) % 7.13/2.30 | % 7.13/2.30 | Instantiating formula (56) with all_0_21_21, all_0_22_22 and discharging atoms relation_dom(all_0_22_22) = all_0_21_21, relation(all_0_22_22), function(all_0_22_22), yields: % 7.13/2.30 | (165) ? [v0] : ? [v1] : ? [v2] : (relation_rng(all_0_22_22) = v0 & first_projection_as_func_of(all_0_21_21, v0) = v2 & function_image(v1, all_0_21_21, v2, all_0_22_22) = all_0_21_21 & cartesian_product2(all_0_21_21, v0) = v1) % 7.13/2.30 | % 7.13/2.30 | Instantiating (165) with all_23_0_33, all_23_1_34, all_23_2_35 yields: % 7.13/2.30 | (166) relation_rng(all_0_22_22) = all_23_2_35 & first_projection_as_func_of(all_0_21_21, all_23_2_35) = all_23_0_33 & function_image(all_23_1_34, all_0_21_21, all_23_0_33, all_0_22_22) = all_0_21_21 & cartesian_product2(all_0_21_21, all_23_2_35) = all_23_1_34 % 7.13/2.31 | % 7.13/2.31 | Applying alpha-rule on (166) yields: % 7.13/2.31 | (167) relation_rng(all_0_22_22) = all_23_2_35 % 7.13/2.31 | (168) first_projection_as_func_of(all_0_21_21, all_23_2_35) = all_23_0_33 % 7.13/2.31 | (169) function_image(all_23_1_34, all_0_21_21, all_23_0_33, all_0_22_22) = all_0_21_21 % 7.13/2.31 | (170) cartesian_product2(all_0_21_21, all_23_2_35) = all_23_1_34 % 7.13/2.31 | % 7.13/2.31 | Instantiating formula (30) with all_23_2_35, all_0_22_22 and discharging atoms relation_rng(all_0_22_22) = all_23_2_35, relation(all_0_22_22), function(all_0_22_22), yields: % 7.13/2.31 | (171) finite(all_23_2_35) | ? [v0] : (relation_dom(all_0_22_22) = v0 & ~ finite(v0)) % 7.13/2.31 | % 7.13/2.31 | Instantiating formula (141) with all_23_0_33, all_23_2_35, all_0_21_21 and discharging atoms first_projection_as_func_of(all_0_21_21, all_23_2_35) = all_23_0_33, yields: % 7.13/2.31 | (172) first_projection(all_0_21_21, all_23_2_35) = all_23_0_33 % 7.13/2.31 | % 7.13/2.31 | Instantiating formula (107) with all_23_0_33, all_23_2_35, all_0_21_21 and discharging atoms first_projection_as_func_of(all_0_21_21, all_23_2_35) = all_23_0_33, yields: % 7.13/2.31 | (173) function(all_23_0_33) % 7.13/2.31 | % 7.13/2.31 | Instantiating formula (35) with all_23_0_33, all_23_2_35, all_0_21_21 and discharging atoms first_projection_as_func_of(all_0_21_21, all_23_2_35) = all_23_0_33, yields: % 7.13/2.31 | (174) ? [v0] : (cartesian_product2(all_0_21_21, all_23_2_35) = v0 & relation_of2_as_subset(all_23_0_33, v0, all_0_21_21)) % 7.13/2.31 | % 7.13/2.31 | Instantiating formula (32) with all_23_0_33, all_23_2_35, all_0_21_21 and discharging atoms first_projection_as_func_of(all_0_21_21, all_23_2_35) = all_23_0_33, yields: % 7.13/2.31 | (175) ? [v0] : (cartesian_product2(all_0_21_21, all_23_2_35) = v0 & quasi_total(all_23_0_33, v0, all_0_21_21)) % 7.13/2.31 | % 7.13/2.31 | Instantiating formula (143) with all_23_1_34, all_23_2_35, all_0_21_21, all_0_22_22 and discharging atoms relation_rng(all_0_22_22) = all_23_2_35, relation_dom(all_0_22_22) = all_0_21_21, cartesian_product2(all_0_21_21, all_23_2_35) = all_23_1_34, relation(all_0_22_22), yields: % 7.13/2.31 | (176) subset(all_0_22_22, all_23_1_34) % 7.13/2.31 | % 7.13/2.31 | Instantiating (175) with all_31_0_36 yields: % 7.13/2.31 | (177) cartesian_product2(all_0_21_21, all_23_2_35) = all_31_0_36 & quasi_total(all_23_0_33, all_31_0_36, all_0_21_21) % 7.13/2.31 | % 7.13/2.31 | Applying alpha-rule on (177) yields: % 7.13/2.31 | (178) cartesian_product2(all_0_21_21, all_23_2_35) = all_31_0_36 % 7.13/2.31 | (179) quasi_total(all_23_0_33, all_31_0_36, all_0_21_21) % 7.13/2.31 | % 7.13/2.31 | Instantiating (174) with all_37_0_39 yields: % 7.13/2.31 | (180) cartesian_product2(all_0_21_21, all_23_2_35) = all_37_0_39 & relation_of2_as_subset(all_23_0_33, all_37_0_39, all_0_21_21) % 7.13/2.31 | % 7.13/2.31 | Applying alpha-rule on (180) yields: % 7.13/2.31 | (181) cartesian_product2(all_0_21_21, all_23_2_35) = all_37_0_39 % 7.13/2.31 | (182) relation_of2_as_subset(all_23_0_33, all_37_0_39, all_0_21_21) % 7.13/2.31 | % 7.13/2.31 | Instantiating formula (18) with all_0_21_21, all_23_2_35, all_37_0_39, all_23_1_34 and discharging atoms cartesian_product2(all_0_21_21, all_23_2_35) = all_37_0_39, cartesian_product2(all_0_21_21, all_23_2_35) = all_23_1_34, yields: % 7.13/2.31 | (183) all_37_0_39 = all_23_1_34 % 7.13/2.31 | % 7.13/2.31 | Instantiating formula (18) with all_0_21_21, all_23_2_35, all_31_0_36, all_37_0_39 and discharging atoms cartesian_product2(all_0_21_21, all_23_2_35) = all_37_0_39, cartesian_product2(all_0_21_21, all_23_2_35) = all_31_0_36, yields: % 7.13/2.31 | (184) all_37_0_39 = all_31_0_36 % 7.13/2.31 | % 7.13/2.31 | Combining equations (183,184) yields a new equation: % 7.13/2.31 | (185) all_31_0_36 = all_23_1_34 % 7.13/2.31 | % 7.13/2.31 | Combining equations (185,184) yields a new equation: % 7.13/2.31 | (183) all_37_0_39 = all_23_1_34 % 7.13/2.31 | % 7.13/2.31 | From (185) and (178) follows: % 7.13/2.31 | (170) cartesian_product2(all_0_21_21, all_23_2_35) = all_23_1_34 % 7.13/2.31 | % 7.13/2.31 | From (183) and (182) follows: % 7.13/2.31 | (188) relation_of2_as_subset(all_23_0_33, all_23_1_34, all_0_21_21) % 7.13/2.31 | % 7.13/2.31 | From (185) and (179) follows: % 7.13/2.31 | (189) quasi_total(all_23_0_33, all_23_1_34, all_0_21_21) % 7.13/2.31 | % 7.13/2.31 | Instantiating formula (59) with all_23_0_33, all_23_2_35, all_0_21_21 and discharging atoms first_projection(all_0_21_21, all_23_2_35) = all_23_0_33, yields: % 7.13/2.31 | (190) relation(all_23_0_33) % 7.13/2.31 | % 7.13/2.31 | Instantiating formula (16) with all_23_0_33, all_0_21_21, all_23_1_34 and discharging atoms relation_of2_as_subset(all_23_0_33, all_23_1_34, all_0_21_21), yields: % 7.13/2.31 | (191) relation_of2(all_23_0_33, all_23_1_34, all_0_21_21) % 7.13/2.31 | % 7.13/2.31 | Instantiating formula (2) with all_0_21_21, all_0_22_22, all_23_0_33, all_0_21_21, all_23_1_34 and discharging atoms function_image(all_23_1_34, all_0_21_21, all_23_0_33, all_0_22_22) = all_0_21_21, relation_of2(all_23_0_33, all_23_1_34, all_0_21_21), quasi_total(all_23_0_33, all_23_1_34, all_0_21_21), function(all_23_0_33), yields: % 7.13/2.31 | (192) relation_image(all_23_0_33, all_0_22_22) = all_0_21_21 % 7.13/2.31 | % 7.13/2.31 +-Applying beta-rule and splitting (34), into two cases. % 7.13/2.31 |-Branch one: % 7.13/2.31 | (193) finite(all_0_21_21) & ~ finite(all_0_22_22) % 7.13/2.31 | % 7.13/2.32 | Applying alpha-rule on (193) yields: % 7.13/2.32 | (194) finite(all_0_21_21) % 7.13/2.32 | (195) ~ finite(all_0_22_22) % 7.13/2.32 | % 7.13/2.32 +-Applying beta-rule and splitting (171), into two cases. % 7.13/2.32 |-Branch one: % 7.13/2.32 | (196) finite(all_23_2_35) % 7.13/2.32 | % 7.13/2.32 | Instantiating formula (41) with all_23_1_34, all_23_2_35, all_0_21_21 and discharging atoms cartesian_product2(all_0_21_21, all_23_2_35) = all_23_1_34, finite(all_23_2_35), finite(all_0_21_21), yields: % 7.13/2.32 | (197) finite(all_23_1_34) % 7.13/2.32 | % 7.13/2.32 | Instantiating formula (113) with all_23_1_34, all_0_22_22 and discharging atoms subset(all_0_22_22, all_23_1_34), finite(all_23_1_34), ~ finite(all_0_22_22), yields: % 7.13/2.32 | (198) $false % 7.13/2.32 | % 7.13/2.32 |-The branch is then unsatisfiable % 7.13/2.32 |-Branch two: % 7.13/2.32 | (199) ~ finite(all_23_2_35) % 7.13/2.32 | (200) ? [v0] : (relation_dom(all_0_22_22) = v0 & ~ finite(v0)) % 7.13/2.32 | % 7.13/2.32 | Instantiating (200) with all_88_0_43 yields: % 7.13/2.32 | (201) relation_dom(all_0_22_22) = all_88_0_43 & ~ finite(all_88_0_43) % 7.13/2.32 | % 7.13/2.32 | Applying alpha-rule on (201) yields: % 7.13/2.32 | (202) relation_dom(all_0_22_22) = all_88_0_43 % 7.13/2.32 | (203) ~ finite(all_88_0_43) % 7.13/2.32 | % 7.13/2.32 | Instantiating formula (104) with all_0_22_22, all_88_0_43, all_0_21_21 and discharging atoms relation_dom(all_0_22_22) = all_88_0_43, relation_dom(all_0_22_22) = all_0_21_21, yields: % 7.13/2.32 | (204) all_88_0_43 = all_0_21_21 % 7.13/2.32 | % 7.13/2.32 | From (204) and (203) follows: % 7.13/2.32 | (205) ~ finite(all_0_21_21) % 7.13/2.32 | % 7.13/2.32 | Using (194) and (205) yields: % 7.13/2.32 | (198) $false % 7.13/2.32 | % 7.13/2.32 |-The branch is then unsatisfiable % 7.13/2.32 |-Branch two: % 7.13/2.32 | (207) finite(all_0_22_22) & ~ finite(all_0_21_21) % 7.13/2.32 | % 7.13/2.32 | Applying alpha-rule on (207) yields: % 7.13/2.32 | (208) finite(all_0_22_22) % 7.13/2.32 | (205) ~ finite(all_0_21_21) % 7.13/2.32 | % 7.13/2.32 | Instantiating formula (53) with all_0_21_21, all_23_0_33, all_0_22_22 and discharging atoms relation_image(all_23_0_33, all_0_22_22) = all_0_21_21, relation(all_23_0_33), function(all_23_0_33), finite(all_0_22_22), ~ finite(all_0_21_21), yields: % 7.13/2.32 | (198) $false % 7.13/2.32 | % 7.13/2.32 |-The branch is then unsatisfiable % 7.13/2.32 % SZS output end Proof for theBenchmark % 7.13/2.32 % 7.13/2.32 1678ms %------------------------------------------------------------------------------