%------------------------------------------------------------------------------ % File : ePrincess---1.0 % Problem : SEU023+1 : TPTP v8.1.0. Released v3.2.0. % Transfm : none % Format : tptp:raw % Command : ePrincess-casc -timeout=%d %s % Computer : n011.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 08:46:15 EDT 2022 % Result : Theorem 3.98s 1.56s % Output : Proof 6.66s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.03/0.12 % Problem : SEU023+1 : TPTP v8.1.0. Released v3.2.0. % 0.03/0.13 % Command : ePrincess-casc -timeout=%d %s % 0.12/0.34 % Computer : n011.cluster.edu % 0.12/0.34 % Model : x86_64 x86_64 % 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.12/0.34 % Memory : 8042.1875MB % 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64 % 0.12/0.34 % CPULimit : 300 % 0.12/0.34 % WCLimit : 600 % 0.12/0.34 % DateTime : Sat Jun 18 22:56:38 EDT 2022 % 0.12/0.34 % CPUTime : % 0.56/0.59 ____ _ % 0.56/0.59 ___ / __ \_____(_)___ ________ __________ % 0.56/0.59 / _ \/ /_/ / ___/ / __ \/ ___/ _ \/ ___/ ___/ % 0.56/0.59 / __/ ____/ / / / / / / /__/ __(__ |__ ) % 0.56/0.59 \___/_/ /_/ /_/_/ /_/\___/\___/____/____/ % 0.56/0.59 % 0.56/0.59 A Theorem Prover for First-Order Logic % 0.56/0.59 (ePrincess v.1.0) % 0.56/0.59 % 0.56/0.59 (c) Philipp Rümmer, 2009-2015 % 0.56/0.59 (c) Peter Backeman, 2014-2015 % 0.56/0.59 (contributions by Angelo Brillout, Peter Baumgartner) % 0.56/0.59 Free software under GNU Lesser General Public License (LGPL). % 0.56/0.59 Bug reports to peter@backeman.se % 0.56/0.59 % 0.56/0.59 For more information, visit http://user.uu.se/~petba168/breu/ % 0.56/0.59 % 0.56/0.59 Loading /export/starexec/sandbox2/benchmark/theBenchmark.p ... % 0.77/0.64 Prover 0: Options: -triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMaximal -resolutionMethod=nonUnifying +ignoreQuantifiers -generateTriggers=all % 1.75/0.96 Prover 0: Preprocessing ... % 2.69/1.28 Prover 0: Warning: ignoring some quantifiers % 2.69/1.31 Prover 0: Constructing countermodel ... % 3.98/1.56 Prover 0: proved (917ms) % 3.98/1.56 % 3.98/1.56 No countermodel exists, formula is valid % 3.98/1.56 % SZS status Theorem for theBenchmark % 3.98/1.56 % 3.98/1.56 Generating proof ... Warning: ignoring some quantifiers % 6.07/2.02 found it (size 43) % 6.07/2.02 % 6.07/2.02 % SZS output start Proof for theBenchmark % 6.07/2.02 Assumed formulas after preprocessing and simplification: % 6.07/2.02 | (0) ? [v0] : ? [v1] : ? [v2] : ? [v3] : ? [v4] : ? [v5] : ? [v6] : ? [v7] : ? [v8] : ? [v9] : ? [v10] : ? [v11] : ? [v12] : ? [v13] : ? [v14] : ? [v15] : (apply(v6, v0) = v7 & apply(v3, v4) = v5 & apply(v1, v0) = v4 & relation_dom(v1) = v2 & relation_composition(v1, v3) = v6 & function_inverse(v1) = v3 & relation_empty_yielding(v8) & relation_empty_yielding(empty_set) & one_to_one(v9) & one_to_one(v1) & relation(v15) & relation(v14) & relation(v12) & relation(v11) & relation(v9) & relation(v8) & relation(v1) & relation(empty_set) & function(v15) & function(v12) & function(v9) & function(v1) & empty(v14) & empty(v13) & empty(v12) & empty(empty_set) & in(v0, v2) & ~ empty(v11) & ~ empty(v10) & ! [v16] : ! [v17] : ! [v18] : ! [v19] : (v17 = v16 | ~ (apply(v19, v18) = v17) | ~ (apply(v19, v18) = v16)) & ! [v16] : ! [v17] : ! [v18] : ! [v19] : (v17 = v16 | ~ (relation_composition(v19, v18) = v17) | ~ (relation_composition(v19, v18) = v16)) & ! [v16] : ! [v17] : ! [v18] : ! [v19] : ( ~ (powerset(v18) = v19) | ~ element(v17, v19) | ~ empty(v18) | ~ in(v16, v17)) & ! [v16] : ! [v17] : ! [v18] : ! [v19] : ( ~ (powerset(v18) = v19) | ~ element(v17, v19) | ~ in(v16, v17) | element(v16, v18)) & ! [v16] : ! [v17] : ! [v18] : (v17 = v16 | ~ (relation_rng(v18) = v17) | ~ (relation_rng(v18) = v16)) & ! [v16] : ! [v17] : ! [v18] : (v17 = v16 | ~ (relation_dom(v18) = v17) | ~ (relation_dom(v18) = v16)) & ! [v16] : ! [v17] : ! [v18] : (v17 = v16 | ~ (powerset(v18) = v17) | ~ (powerset(v18) = v16)) & ! [v16] : ! [v17] : ! [v18] : (v17 = v16 | ~ (function_inverse(v18) = v17) | ~ (function_inverse(v18) = v16)) & ! [v16] : ! [v17] : ! [v18] : ( ~ (apply(v17, v16) = v18) | ~ relation(v17) | ~ function(v17) | ? [v19] : (relation_dom(v17) = v19 & ! [v20] : ! [v21] : ! [v22] : ( ~ (apply(v21, v16) = v22) | ~ (relation_composition(v17, v20) = v21) | ~ relation(v20) | ~ function(v20) | ~ in(v16, v19) | apply(v20, v18) = v22) & ! [v20] : ! [v21] : ( ~ (apply(v20, v18) = v21) | ~ relation(v20) | ~ function(v20) | ~ in(v16, v19) | ? [v22] : (apply(v22, v16) = v21 & relation_composition(v17, v20) = v22)))) & ! [v16] : ! [v17] : ! [v18] : ( ~ (powerset(v17) = v18) | ~ subset(v16, v17) | element(v16, v18)) & ! [v16] : ! [v17] : ! [v18] : ( ~ (powerset(v17) = v18) | ~ element(v16, v18) | subset(v16, v17)) & ! [v16] : ! [v17] : ! [v18] : ( ~ (relation_composition(v17, v16) = v18) | ~ relation(v17) | ~ empty(v16) | relation(v18)) & ! [v16] : ! [v17] : ! [v18] : ( ~ (relation_composition(v17, v16) = v18) | ~ relation(v17) | ~ empty(v16) | empty(v18)) & ! [v16] : ! [v17] : ! [v18] : ( ~ (relation_composition(v16, v17) = v18) | ~ relation(v17) | ~ relation(v16) | ~ function(v17) | ~ function(v16) | relation(v18)) & ! [v16] : ! [v17] : ! [v18] : ( ~ (relation_composition(v16, v17) = v18) | ~ relation(v17) | ~ relation(v16) | ~ function(v17) | ~ function(v16) | function(v18)) & ! [v16] : ! [v17] : ! [v18] : ( ~ (relation_composition(v16, v17) = v18) | ~ relation(v17) | ~ relation(v16) | relation(v18)) & ! [v16] : ! [v17] : ! [v18] : ( ~ (relation_composition(v16, v17) = v18) | ~ relation(v17) | ~ empty(v16) | relation(v18)) & ! [v16] : ! [v17] : ! [v18] : ( ~ (relation_composition(v16, v17) = v18) | ~ relation(v17) | ~ empty(v16) | empty(v18)) & ! [v16] : ! [v17] : (v17 = v16 | ~ empty(v17) | ~ empty(v16)) & ! [v16] : ! [v17] : ( ~ (relation_rng(v16) = v17) | ~ one_to_one(v16) | ~ relation(v16) | ~ function(v16) | ? [v18] : ? [v19] : (relation_dom(v16) = v19 & function_inverse(v16) = v18 & ! [v20] : ! [v21] : ! [v22] : ! [v23] : (v23 = v22 | ~ (apply(v18, v21) = v23) | ~ (apply(v16, v22) = v21) | ~ (relation_dom(v18) = v20) | ~ relation(v18) | ~ function(v18) | ~ in(v22, v19)) & ! [v20] : ! [v21] : ! [v22] : ! [v23] : (v23 = v21 | ~ (apply(v18, v21) = v22) | ~ (apply(v16, v22) = v23) | ~ (relation_dom(v18) = v20) | ~ relation(v18) | ~ function(v18) | ~ in(v21, v17)) & ! [v20] : ! [v21] : ! [v22] : ! [v23] : ( ~ (apply(v18, v21) = v23) | ~ (apply(v16, v22) = v21) | ~ (relation_dom(v18) = v20) | ~ relation(v18) | ~ function(v18) | ~ in(v22, v19) | in(v21, v17)) & ! [v20] : ! [v21] : ! [v22] : ! [v23] : ( ~ (apply(v18, v21) = v22) | ~ (apply(v16, v22) = v23) | ~ (relation_dom(v18) = v20) | ~ relation(v18) | ~ function(v18) | ~ in(v21, v17) | in(v22, v19)) & ! [v20] : (v20 = v18 | ~ (relation_dom(v20) = v17) | ~ relation(v20) | ~ function(v20) | ? [v21] : ? [v22] : ? [v23] : ? [v24] : (apply(v20, v21) = v23 & apply(v16, v22) = v24 & ((v24 = v21 & in(v22, v19) & ( ~ (v23 = v22) | ~ in(v21, v17))) | (v23 = v22 & in(v21, v17) & ( ~ (v24 = v21) | ~ in(v22, v19)))))) & ! [v20] : (v20 = v17 | ~ (relation_dom(v18) = v20) | ~ relation(v18) | ~ function(v18)))) & ! [v16] : ! [v17] : ( ~ (relation_rng(v16) = v17) | ~ relation(v16) | ~ empty(v17) | empty(v16)) & ! [v16] : ! [v17] : ( ~ (relation_rng(v16) = v17) | ~ empty(v16) | relation(v17)) & ! [v16] : ! [v17] : ( ~ (relation_rng(v16) = v17) | ~ empty(v16) | empty(v17)) & ! [v16] : ! [v17] : ( ~ (relation_dom(v16) = v17) | ~ one_to_one(v16) | ~ relation(v16) | ~ function(v16) | ? [v18] : ? [v19] : (relation_rng(v16) = v19 & function_inverse(v16) = v18 & ! [v20] : ! [v21] : ! [v22] : ! [v23] : (v23 = v22 | ~ (apply(v18, v21) = v23) | ~ (apply(v16, v22) = v21) | ~ (relation_dom(v18) = v20) | ~ relation(v18) | ~ function(v18) | ~ in(v22, v17)) & ! [v20] : ! [v21] : ! [v22] : ! [v23] : (v23 = v21 | ~ (apply(v18, v21) = v22) | ~ (apply(v16, v22) = v23) | ~ (relation_dom(v18) = v20) | ~ relation(v18) | ~ function(v18) | ~ in(v21, v19)) & ! [v20] : ! [v21] : ! [v22] : ! [v23] : ( ~ (apply(v18, v21) = v23) | ~ (apply(v16, v22) = v21) | ~ (relation_dom(v18) = v20) | ~ relation(v18) | ~ function(v18) | ~ in(v22, v17) | in(v21, v19)) & ! [v20] : ! [v21] : ! [v22] : ! [v23] : ( ~ (apply(v18, v21) = v22) | ~ (apply(v16, v22) = v23) | ~ (relation_dom(v18) = v20) | ~ relation(v18) | ~ function(v18) | ~ in(v21, v19) | in(v22, v17)) & ! [v20] : (v20 = v19 | ~ (relation_dom(v18) = v20) | ~ relation(v18) | ~ function(v18)) & ! [v20] : (v20 = v18 | ~ (relation_dom(v20) = v19) | ~ relation(v20) | ~ function(v20) | ? [v21] : ? [v22] : ? [v23] : ? [v24] : (apply(v20, v21) = v23 & apply(v16, v22) = v24 & ((v24 = v21 & in(v22, v17) & ( ~ (v23 = v22) | ~ in(v21, v19))) | (v23 = v22 & in(v21, v19) & ( ~ (v24 = v21) | ~ in(v22, v17)))))))) & ! [v16] : ! [v17] : ( ~ (relation_dom(v16) = v17) | ~ relation(v16) | ~ empty(v17) | empty(v16)) & ! [v16] : ! [v17] : ( ~ (relation_dom(v16) = v17) | ~ empty(v16) | relation(v17)) & ! [v16] : ! [v17] : ( ~ (relation_dom(v16) = v17) | ~ empty(v16) | empty(v17)) & ! [v16] : ! [v17] : ( ~ (powerset(v16) = v17) | ~ empty(v17)) & ! [v16] : ! [v17] : ( ~ (powerset(v16) = v17) | empty(v16) | ? [v18] : (element(v18, v17) & ~ empty(v18))) & ! [v16] : ! [v17] : ( ~ (powerset(v16) = v17) | ? [v18] : (element(v18, v17) & empty(v18))) & ! [v16] : ! [v17] : ( ~ (function_inverse(v16) = v17) | ~ one_to_one(v16) | ~ relation(v16) | ~ function(v16) | ? [v18] : ? [v19] : (relation_rng(v16) = v18 & relation_dom(v16) = v19 & ! [v20] : ! [v21] : ! [v22] : ! [v23] : (v23 = v22 | ~ (apply(v17, v21) = v23) | ~ (apply(v16, v22) = v21) | ~ (relation_dom(v17) = v20) | ~ relation(v17) | ~ function(v17) | ~ in(v22, v19)) & ! [v20] : ! [v21] : ! [v22] : ! [v23] : (v23 = v21 | ~ (apply(v17, v21) = v22) | ~ (apply(v16, v22) = v23) | ~ (relation_dom(v17) = v20) | ~ relation(v17) | ~ function(v17) | ~ in(v21, v18)) & ! [v20] : ! [v21] : ! [v22] : ! [v23] : ( ~ (apply(v17, v21) = v23) | ~ (apply(v16, v22) = v21) | ~ (relation_dom(v17) = v20) | ~ relation(v17) | ~ function(v17) | ~ in(v22, v19) | in(v21, v18)) & ! [v20] : ! [v21] : ! [v22] : ! [v23] : ( ~ (apply(v17, v21) = v22) | ~ (apply(v16, v22) = v23) | ~ (relation_dom(v17) = v20) | ~ relation(v17) | ~ function(v17) | ~ in(v21, v18) | in(v22, v19)) & ! [v20] : (v20 = v18 | ~ (relation_dom(v17) = v20) | ~ relation(v17) | ~ function(v17)) & ! [v20] : (v20 = v17 | ~ (relation_dom(v20) = v18) | ~ relation(v20) | ~ function(v20) | ? [v21] : ? [v22] : ? [v23] : ? [v24] : (apply(v20, v21) = v23 & apply(v16, v22) = v24 & ((v24 = v21 & in(v22, v19) & ( ~ (v23 = v22) | ~ in(v21, v18))) | (v23 = v22 & in(v21, v18) & ( ~ (v24 = v21) | ~ in(v22, v19)))))))) & ! [v16] : ! [v17] : ( ~ (function_inverse(v16) = v17) | ~ relation(v16) | ~ function(v16) | relation(v17)) & ! [v16] : ! [v17] : ( ~ (function_inverse(v16) = v17) | ~ relation(v16) | ~ function(v16) | function(v17)) & ! [v16] : ! [v17] : ( ~ element(v16, v17) | empty(v17) | in(v16, v17)) & ! [v16] : ! [v17] : ( ~ empty(v17) | ~ in(v16, v17)) & ! [v16] : ! [v17] : ( ~ in(v17, v16) | ~ in(v16, v17)) & ! [v16] : ! [v17] : ( ~ in(v16, v17) | element(v16, v17)) & ! [v16] : (v16 = empty_set | ~ empty(v16)) & ! [v16] : ( ~ relation(v16) | ~ function(v16) | ~ empty(v16) | one_to_one(v16)) & ! [v16] : ( ~ empty(v16) | relation(v16)) & ! [v16] : ( ~ empty(v16) | function(v16)) & ? [v16] : ? [v17] : element(v17, v16) & ? [v16] : subset(v16, v16) & ( ~ (v7 = v0) | ~ (v5 = v0))) % 6.07/2.07 | 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 yields: % 6.07/2.07 | (1) apply(all_0_9_9, all_0_15_15) = all_0_8_8 & apply(all_0_12_12, all_0_11_11) = all_0_10_10 & apply(all_0_14_14, all_0_15_15) = all_0_11_11 & relation_dom(all_0_14_14) = all_0_13_13 & relation_composition(all_0_14_14, all_0_12_12) = all_0_9_9 & function_inverse(all_0_14_14) = all_0_12_12 & relation_empty_yielding(all_0_7_7) & relation_empty_yielding(empty_set) & one_to_one(all_0_6_6) & one_to_one(all_0_14_14) & relation(all_0_0_0) & relation(all_0_1_1) & relation(all_0_3_3) & relation(all_0_4_4) & relation(all_0_6_6) & relation(all_0_7_7) & relation(all_0_14_14) & relation(empty_set) & function(all_0_0_0) & function(all_0_3_3) & function(all_0_6_6) & function(all_0_14_14) & empty(all_0_1_1) & empty(all_0_2_2) & empty(all_0_3_3) & empty(empty_set) & in(all_0_15_15, all_0_13_13) & ~ empty(all_0_4_4) & ~ empty(all_0_5_5) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (apply(v3, v2) = v1) | ~ (apply(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (relation_composition(v3, v2) = v1) | ~ (relation_composition(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (powerset(v2) = v3) | ~ element(v1, v3) | ~ empty(v2) | ~ in(v0, v1)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (powerset(v2) = v3) | ~ element(v1, v3) | ~ in(v0, v1) | element(v0, v2)) & ! [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] : (v1 = v0 | ~ (function_inverse(v2) = v1) | ~ (function_inverse(v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (apply(v1, v0) = v2) | ~ relation(v1) | ~ function(v1) | ? [v3] : (relation_dom(v1) = v3 & ! [v4] : ! [v5] : ! [v6] : ( ~ (apply(v5, v0) = v6) | ~ (relation_composition(v1, v4) = v5) | ~ relation(v4) | ~ function(v4) | ~ in(v0, v3) | apply(v4, v2) = v6) & ! [v4] : ! [v5] : ( ~ (apply(v4, v2) = v5) | ~ relation(v4) | ~ function(v4) | ~ in(v0, v3) | ? [v6] : (apply(v6, v0) = v5 & relation_composition(v1, v4) = v6)))) & ! [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] : ( ~ (relation_composition(v1, v0) = v2) | ~ relation(v1) | ~ empty(v0) | relation(v2)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (relation_composition(v1, v0) = v2) | ~ relation(v1) | ~ empty(v0) | empty(v2)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (relation_composition(v0, v1) = v2) | ~ relation(v1) | ~ relation(v0) | ~ function(v1) | ~ function(v0) | relation(v2)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (relation_composition(v0, v1) = v2) | ~ relation(v1) | ~ relation(v0) | ~ function(v1) | ~ function(v0) | function(v2)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (relation_composition(v0, v1) = v2) | ~ relation(v1) | ~ relation(v0) | relation(v2)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (relation_composition(v0, v1) = v2) | ~ relation(v1) | ~ empty(v0) | relation(v2)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (relation_composition(v0, v1) = v2) | ~ relation(v1) | ~ empty(v0) | empty(v2)) & ! [v0] : ! [v1] : (v1 = v0 | ~ empty(v1) | ~ empty(v0)) & ! [v0] : ! [v1] : ( ~ (relation_rng(v0) = v1) | ~ one_to_one(v0) | ~ relation(v0) | ~ function(v0) | ? [v2] : ? [v3] : (relation_dom(v0) = v3 & function_inverse(v0) = v2 & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = v6 | ~ (apply(v2, v5) = v7) | ~ (apply(v0, v6) = v5) | ~ (relation_dom(v2) = v4) | ~ relation(v2) | ~ function(v2) | ~ in(v6, v3)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = v5 | ~ (apply(v2, v5) = v6) | ~ (apply(v0, v6) = v7) | ~ (relation_dom(v2) = v4) | ~ relation(v2) | ~ function(v2) | ~ in(v5, v1)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (apply(v2, v5) = v7) | ~ (apply(v0, v6) = v5) | ~ (relation_dom(v2) = v4) | ~ relation(v2) | ~ function(v2) | ~ in(v6, v3) | in(v5, v1)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (apply(v2, v5) = v6) | ~ (apply(v0, v6) = v7) | ~ (relation_dom(v2) = v4) | ~ relation(v2) | ~ function(v2) | ~ in(v5, v1) | in(v6, v3)) & ! [v4] : (v4 = v2 | ~ (relation_dom(v4) = v1) | ~ relation(v4) | ~ function(v4) | ? [v5] : ? [v6] : ? [v7] : ? [v8] : (apply(v4, v5) = v7 & apply(v0, v6) = v8 & ((v8 = v5 & in(v6, v3) & ( ~ (v7 = v6) | ~ in(v5, v1))) | (v7 = v6 & in(v5, v1) & ( ~ (v8 = v5) | ~ in(v6, v3)))))) & ! [v4] : (v4 = v1 | ~ (relation_dom(v2) = v4) | ~ relation(v2) | ~ function(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) | ~ one_to_one(v0) | ~ relation(v0) | ~ function(v0) | ? [v2] : ? [v3] : (relation_rng(v0) = v3 & function_inverse(v0) = v2 & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = v6 | ~ (apply(v2, v5) = v7) | ~ (apply(v0, v6) = v5) | ~ (relation_dom(v2) = v4) | ~ relation(v2) | ~ function(v2) | ~ in(v6, v1)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = v5 | ~ (apply(v2, v5) = v6) | ~ (apply(v0, v6) = v7) | ~ (relation_dom(v2) = v4) | ~ relation(v2) | ~ function(v2) | ~ in(v5, v3)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (apply(v2, v5) = v7) | ~ (apply(v0, v6) = v5) | ~ (relation_dom(v2) = v4) | ~ relation(v2) | ~ function(v2) | ~ in(v6, v1) | in(v5, v3)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (apply(v2, v5) = v6) | ~ (apply(v0, v6) = v7) | ~ (relation_dom(v2) = v4) | ~ relation(v2) | ~ function(v2) | ~ in(v5, v3) | in(v6, v1)) & ! [v4] : (v4 = v3 | ~ (relation_dom(v2) = v4) | ~ relation(v2) | ~ function(v2)) & ! [v4] : (v4 = v2 | ~ (relation_dom(v4) = v3) | ~ relation(v4) | ~ function(v4) | ? [v5] : ? [v6] : ? [v7] : ? [v8] : (apply(v4, v5) = v7 & apply(v0, v6) = v8 & ((v8 = v5 & in(v6, v1) & ( ~ (v7 = v6) | ~ in(v5, v3))) | (v7 = v6 & in(v5, v3) & ( ~ (v8 = v5) | ~ in(v6, v1)))))))) & ! [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] : (element(v2, v1) & ~ empty(v2))) & ! [v0] : ! [v1] : ( ~ (powerset(v0) = v1) | ? [v2] : (element(v2, v1) & empty(v2))) & ! [v0] : ! [v1] : ( ~ (function_inverse(v0) = v1) | ~ one_to_one(v0) | ~ relation(v0) | ~ function(v0) | ? [v2] : ? [v3] : (relation_rng(v0) = v2 & relation_dom(v0) = v3 & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = v6 | ~ (apply(v1, v5) = v7) | ~ (apply(v0, v6) = v5) | ~ (relation_dom(v1) = v4) | ~ relation(v1) | ~ function(v1) | ~ in(v6, v3)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = v5 | ~ (apply(v1, v5) = v6) | ~ (apply(v0, v6) = v7) | ~ (relation_dom(v1) = v4) | ~ relation(v1) | ~ function(v1) | ~ in(v5, v2)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (apply(v1, v5) = v7) | ~ (apply(v0, v6) = v5) | ~ (relation_dom(v1) = v4) | ~ relation(v1) | ~ function(v1) | ~ in(v6, v3) | in(v5, v2)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (apply(v1, v5) = v6) | ~ (apply(v0, v6) = v7) | ~ (relation_dom(v1) = v4) | ~ relation(v1) | ~ function(v1) | ~ in(v5, v2) | in(v6, v3)) & ! [v4] : (v4 = v2 | ~ (relation_dom(v1) = v4) | ~ relation(v1) | ~ function(v1)) & ! [v4] : (v4 = v1 | ~ (relation_dom(v4) = v2) | ~ relation(v4) | ~ function(v4) | ? [v5] : ? [v6] : ? [v7] : ? [v8] : (apply(v4, v5) = v7 & apply(v0, v6) = v8 & ((v8 = v5 & in(v6, v3) & ( ~ (v7 = v6) | ~ in(v5, v2))) | (v7 = v6 & in(v5, v2) & ( ~ (v8 = v5) | ~ in(v6, v3)))))))) & ! [v0] : ! [v1] : ( ~ (function_inverse(v0) = v1) | ~ relation(v0) | ~ function(v0) | relation(v1)) & ! [v0] : ! [v1] : ( ~ (function_inverse(v0) = v1) | ~ relation(v0) | ~ function(v0) | function(v1)) & ! [v0] : ! [v1] : ( ~ element(v0, v1) | empty(v1) | in(v0, v1)) & ! [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) | relation(v0)) & ! [v0] : ( ~ empty(v0) | function(v0)) & ? [v0] : ? [v1] : element(v1, v0) & ? [v0] : subset(v0, v0) & ( ~ (all_0_8_8 = all_0_15_15) | ~ (all_0_10_10 = all_0_15_15)) % 6.48/2.09 | % 6.48/2.09 | Applying alpha-rule on (1) yields: % 6.48/2.09 | (2) ! [v0] : ! [v1] : ! [v2] : ( ~ (powerset(v1) = v2) | ~ element(v0, v2) | subset(v0, v1)) % 6.48/2.09 | (3) ! [v0] : ! [v1] : ! [v2] : ( ~ (powerset(v1) = v2) | ~ subset(v0, v1) | element(v0, v2)) % 6.48/2.09 | (4) ! [v0] : ! [v1] : ( ~ (relation_rng(v0) = v1) | ~ relation(v0) | ~ empty(v1) | empty(v0)) % 6.48/2.09 | (5) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (relation_rng(v2) = v1) | ~ (relation_rng(v2) = v0)) % 6.48/2.09 | (6) ! [v0] : ! [v1] : ( ~ (powerset(v0) = v1) | empty(v0) | ? [v2] : (element(v2, v1) & ~ empty(v2))) % 6.48/2.09 | (7) function(all_0_6_6) % 6.48/2.09 | (8) relation(all_0_14_14) % 6.48/2.09 | (9) ! [v0] : ! [v1] : ( ~ (relation_dom(v0) = v1) | ~ empty(v0) | empty(v1)) % 6.48/2.09 | (10) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (apply(v3, v2) = v1) | ~ (apply(v3, v2) = v0)) % 6.48/2.10 | (11) ! [v0] : ( ~ relation(v0) | ~ function(v0) | ~ empty(v0) | one_to_one(v0)) % 6.48/2.10 | (12) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (relation_composition(v3, v2) = v1) | ~ (relation_composition(v3, v2) = v0)) % 6.48/2.10 | (13) ! [v0] : ! [v1] : ( ~ (function_inverse(v0) = v1) | ~ one_to_one(v0) | ~ relation(v0) | ~ function(v0) | ? [v2] : ? [v3] : (relation_rng(v0) = v2 & relation_dom(v0) = v3 & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = v6 | ~ (apply(v1, v5) = v7) | ~ (apply(v0, v6) = v5) | ~ (relation_dom(v1) = v4) | ~ relation(v1) | ~ function(v1) | ~ in(v6, v3)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = v5 | ~ (apply(v1, v5) = v6) | ~ (apply(v0, v6) = v7) | ~ (relation_dom(v1) = v4) | ~ relation(v1) | ~ function(v1) | ~ in(v5, v2)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (apply(v1, v5) = v7) | ~ (apply(v0, v6) = v5) | ~ (relation_dom(v1) = v4) | ~ relation(v1) | ~ function(v1) | ~ in(v6, v3) | in(v5, v2)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (apply(v1, v5) = v6) | ~ (apply(v0, v6) = v7) | ~ (relation_dom(v1) = v4) | ~ relation(v1) | ~ function(v1) | ~ in(v5, v2) | in(v6, v3)) & ! [v4] : (v4 = v2 | ~ (relation_dom(v1) = v4) | ~ relation(v1) | ~ function(v1)) & ! [v4] : (v4 = v1 | ~ (relation_dom(v4) = v2) | ~ relation(v4) | ~ function(v4) | ? [v5] : ? [v6] : ? [v7] : ? [v8] : (apply(v4, v5) = v7 & apply(v0, v6) = v8 & ((v8 = v5 & in(v6, v3) & ( ~ (v7 = v6) | ~ in(v5, v2))) | (v7 = v6 & in(v5, v2) & ( ~ (v8 = v5) | ~ in(v6, v3)))))))) % 6.48/2.10 | (14) relation(empty_set) % 6.48/2.10 | (15) function(all_0_3_3) % 6.48/2.10 | (16) relation(all_0_0_0) % 6.48/2.10 | (17) ! [v0] : ! [v1] : ! [v2] : ( ~ (relation_composition(v0, v1) = v2) | ~ relation(v1) | ~ empty(v0) | empty(v2)) % 6.48/2.10 | (18) ! [v0] : ! [v1] : (v1 = v0 | ~ empty(v1) | ~ empty(v0)) % 6.48/2.10 | (19) relation(all_0_1_1) % 6.48/2.10 | (20) ? [v0] : subset(v0, v0) % 6.48/2.10 | (21) ! [v0] : ! [v1] : ! [v2] : ( ~ (relation_composition(v1, v0) = v2) | ~ relation(v1) | ~ empty(v0) | empty(v2)) % 6.48/2.10 | (22) ! [v0] : ( ~ empty(v0) | function(v0)) % 6.58/2.10 | (23) ! [v0] : (v0 = empty_set | ~ empty(v0)) % 6.58/2.10 | (24) ! [v0] : ! [v1] : ( ~ (relation_rng(v0) = v1) | ~ empty(v0) | relation(v1)) % 6.58/2.10 | (25) relation(all_0_4_4) % 6.58/2.10 | (26) ! [v0] : ! [v1] : ( ~ (relation_dom(v0) = v1) | ~ relation(v0) | ~ empty(v1) | empty(v0)) % 6.58/2.10 | (27) ! [v0] : ! [v1] : ! [v2] : ( ~ (relation_composition(v0, v1) = v2) | ~ relation(v1) | ~ empty(v0) | relation(v2)) % 6.58/2.10 | (28) ! [v0] : ! [v1] : ( ~ (relation_dom(v0) = v1) | ~ empty(v0) | relation(v1)) % 6.58/2.10 | (29) ! [v0] : ! [v1] : ! [v2] : ( ~ (relation_composition(v1, v0) = v2) | ~ relation(v1) | ~ empty(v0) | relation(v2)) % 6.58/2.10 | (30) ! [v0] : ! [v1] : ( ~ (function_inverse(v0) = v1) | ~ relation(v0) | ~ function(v0) | function(v1)) % 6.58/2.10 | (31) relation(all_0_7_7) % 6.58/2.10 | (32) ! [v0] : ! [v1] : ( ~ element(v0, v1) | empty(v1) | in(v0, v1)) % 6.58/2.10 | (33) ! [v0] : ! [v1] : ( ~ empty(v1) | ~ in(v0, v1)) % 6.58/2.10 | (34) relation_composition(all_0_14_14, all_0_12_12) = all_0_9_9 % 6.58/2.10 | (35) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (function_inverse(v2) = v1) | ~ (function_inverse(v2) = v0)) % 6.58/2.10 | (36) ! [v0] : ! [v1] : ! [v2] : ( ~ (relation_composition(v0, v1) = v2) | ~ relation(v1) | ~ relation(v0) | ~ function(v1) | ~ function(v0) | relation(v2)) % 6.58/2.10 | (37) ~ (all_0_8_8 = all_0_15_15) | ~ (all_0_10_10 = all_0_15_15) % 6.58/2.10 | (38) ! [v0] : ! [v1] : ( ~ in(v1, v0) | ~ in(v0, v1)) % 6.58/2.10 | (39) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (powerset(v2) = v1) | ~ (powerset(v2) = v0)) % 6.58/2.10 | (40) ! [v0] : ! [v1] : ( ~ (relation_dom(v0) = v1) | ~ one_to_one(v0) | ~ relation(v0) | ~ function(v0) | ? [v2] : ? [v3] : (relation_rng(v0) = v3 & function_inverse(v0) = v2 & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = v6 | ~ (apply(v2, v5) = v7) | ~ (apply(v0, v6) = v5) | ~ (relation_dom(v2) = v4) | ~ relation(v2) | ~ function(v2) | ~ in(v6, v1)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = v5 | ~ (apply(v2, v5) = v6) | ~ (apply(v0, v6) = v7) | ~ (relation_dom(v2) = v4) | ~ relation(v2) | ~ function(v2) | ~ in(v5, v3)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (apply(v2, v5) = v7) | ~ (apply(v0, v6) = v5) | ~ (relation_dom(v2) = v4) | ~ relation(v2) | ~ function(v2) | ~ in(v6, v1) | in(v5, v3)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (apply(v2, v5) = v6) | ~ (apply(v0, v6) = v7) | ~ (relation_dom(v2) = v4) | ~ relation(v2) | ~ function(v2) | ~ in(v5, v3) | in(v6, v1)) & ! [v4] : (v4 = v3 | ~ (relation_dom(v2) = v4) | ~ relation(v2) | ~ function(v2)) & ! [v4] : (v4 = v2 | ~ (relation_dom(v4) = v3) | ~ relation(v4) | ~ function(v4) | ? [v5] : ? [v6] : ? [v7] : ? [v8] : (apply(v4, v5) = v7 & apply(v0, v6) = v8 & ((v8 = v5 & in(v6, v1) & ( ~ (v7 = v6) | ~ in(v5, v3))) | (v7 = v6 & in(v5, v3) & ( ~ (v8 = v5) | ~ in(v6, v1)))))))) % 6.58/2.11 | (41) empty(all_0_2_2) % 6.58/2.11 | (42) empty(empty_set) % 6.58/2.11 | (43) ! [v0] : ! [v1] : ( ~ (relation_rng(v0) = v1) | ~ empty(v0) | empty(v1)) % 6.58/2.11 | (44) ! [v0] : ! [v1] : ! [v2] : ( ~ (apply(v1, v0) = v2) | ~ relation(v1) | ~ function(v1) | ? [v3] : (relation_dom(v1) = v3 & ! [v4] : ! [v5] : ! [v6] : ( ~ (apply(v5, v0) = v6) | ~ (relation_composition(v1, v4) = v5) | ~ relation(v4) | ~ function(v4) | ~ in(v0, v3) | apply(v4, v2) = v6) & ! [v4] : ! [v5] : ( ~ (apply(v4, v2) = v5) | ~ relation(v4) | ~ function(v4) | ~ in(v0, v3) | ? [v6] : (apply(v6, v0) = v5 & relation_composition(v1, v4) = v6)))) % 6.58/2.11 | (45) ! [v0] : ! [v1] : ! [v2] : ( ~ (relation_composition(v0, v1) = v2) | ~ relation(v1) | ~ relation(v0) | relation(v2)) % 6.58/2.11 | (46) ! [v0] : ! [v1] : ( ~ (relation_rng(v0) = v1) | ~ one_to_one(v0) | ~ relation(v0) | ~ function(v0) | ? [v2] : ? [v3] : (relation_dom(v0) = v3 & function_inverse(v0) = v2 & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = v6 | ~ (apply(v2, v5) = v7) | ~ (apply(v0, v6) = v5) | ~ (relation_dom(v2) = v4) | ~ relation(v2) | ~ function(v2) | ~ in(v6, v3)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : (v7 = v5 | ~ (apply(v2, v5) = v6) | ~ (apply(v0, v6) = v7) | ~ (relation_dom(v2) = v4) | ~ relation(v2) | ~ function(v2) | ~ in(v5, v1)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (apply(v2, v5) = v7) | ~ (apply(v0, v6) = v5) | ~ (relation_dom(v2) = v4) | ~ relation(v2) | ~ function(v2) | ~ in(v6, v3) | in(v5, v1)) & ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (apply(v2, v5) = v6) | ~ (apply(v0, v6) = v7) | ~ (relation_dom(v2) = v4) | ~ relation(v2) | ~ function(v2) | ~ in(v5, v1) | in(v6, v3)) & ! [v4] : (v4 = v2 | ~ (relation_dom(v4) = v1) | ~ relation(v4) | ~ function(v4) | ? [v5] : ? [v6] : ? [v7] : ? [v8] : (apply(v4, v5) = v7 & apply(v0, v6) = v8 & ((v8 = v5 & in(v6, v3) & ( ~ (v7 = v6) | ~ in(v5, v1))) | (v7 = v6 & in(v5, v1) & ( ~ (v8 = v5) | ~ in(v6, v3)))))) & ! [v4] : (v4 = v1 | ~ (relation_dom(v2) = v4) | ~ relation(v2) | ~ function(v2)))) % 6.58/2.11 | (47) function(all_0_14_14) % 6.58/2.11 | (48) apply(all_0_14_14, all_0_15_15) = all_0_11_11 % 6.58/2.11 | (49) function(all_0_0_0) % 6.58/2.11 | (50) empty(all_0_1_1) % 6.58/2.11 | (51) relation_empty_yielding(empty_set) % 6.58/2.11 | (52) relation(all_0_6_6) % 6.58/2.11 | (53) in(all_0_15_15, all_0_13_13) % 6.58/2.11 | (54) ~ empty(all_0_5_5) % 6.58/2.11 | (55) ~ empty(all_0_4_4) % 6.58/2.11 | (56) apply(all_0_12_12, all_0_11_11) = all_0_10_10 % 6.58/2.11 | (57) ! [v0] : ! [v1] : ( ~ (powerset(v0) = v1) | ~ empty(v1)) % 6.58/2.11 | (58) ! [v0] : ! [v1] : ( ~ in(v0, v1) | element(v0, v1)) % 6.58/2.11 | (59) one_to_one(all_0_14_14) % 6.58/2.11 | (60) ! [v0] : ( ~ empty(v0) | relation(v0)) % 6.58/2.11 | (61) relation_dom(all_0_14_14) = all_0_13_13 % 6.58/2.12 | (62) apply(all_0_9_9, all_0_15_15) = all_0_8_8 % 6.58/2.12 | (63) ? [v0] : ? [v1] : element(v1, v0) % 6.58/2.12 | (64) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (relation_dom(v2) = v1) | ~ (relation_dom(v2) = v0)) % 6.58/2.12 | (65) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (powerset(v2) = v3) | ~ element(v1, v3) | ~ empty(v2) | ~ in(v0, v1)) % 6.58/2.12 | (66) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (powerset(v2) = v3) | ~ element(v1, v3) | ~ in(v0, v1) | element(v0, v2)) % 6.58/2.12 | (67) relation_empty_yielding(all_0_7_7) % 6.58/2.12 | (68) ! [v0] : ! [v1] : ( ~ (function_inverse(v0) = v1) | ~ relation(v0) | ~ function(v0) | relation(v1)) % 6.58/2.12 | (69) ! [v0] : ! [v1] : ( ~ (powerset(v0) = v1) | ? [v2] : (element(v2, v1) & empty(v2))) % 6.58/2.12 | (70) one_to_one(all_0_6_6) % 6.58/2.12 | (71) empty(all_0_3_3) % 6.58/2.12 | (72) relation(all_0_3_3) % 6.66/2.12 | (73) ! [v0] : ! [v1] : ! [v2] : ( ~ (relation_composition(v0, v1) = v2) | ~ relation(v1) | ~ relation(v0) | ~ function(v1) | ~ function(v0) | function(v2)) % 6.66/2.12 | (74) function_inverse(all_0_14_14) = all_0_12_12 % 6.66/2.12 | % 6.66/2.12 | Instantiating formula (44) with all_0_11_11, all_0_14_14, all_0_15_15 and discharging atoms apply(all_0_14_14, all_0_15_15) = all_0_11_11, relation(all_0_14_14), function(all_0_14_14), yields: % 6.66/2.12 | (75) ? [v0] : (relation_dom(all_0_14_14) = v0 & ! [v1] : ! [v2] : ! [v3] : ( ~ (apply(v2, all_0_15_15) = v3) | ~ (relation_composition(all_0_14_14, v1) = v2) | ~ relation(v1) | ~ function(v1) | ~ in(all_0_15_15, v0) | apply(v1, all_0_11_11) = v3) & ! [v1] : ! [v2] : ( ~ (apply(v1, all_0_11_11) = v2) | ~ relation(v1) | ~ function(v1) | ~ in(all_0_15_15, v0) | ? [v3] : (apply(v3, all_0_15_15) = v2 & relation_composition(all_0_14_14, v1) = v3))) % 6.66/2.12 | % 6.66/2.12 | Instantiating formula (13) with all_0_12_12, all_0_14_14 and discharging atoms function_inverse(all_0_14_14) = all_0_12_12, one_to_one(all_0_14_14), relation(all_0_14_14), function(all_0_14_14), yields: % 6.66/2.12 | (76) ? [v0] : ? [v1] : (relation_rng(all_0_14_14) = v0 & relation_dom(all_0_14_14) = v1 & ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = v4 | ~ (apply(all_0_12_12, v3) = v5) | ~ (apply(all_0_14_14, v4) = v3) | ~ (relation_dom(all_0_12_12) = v2) | ~ relation(all_0_12_12) | ~ function(all_0_12_12) | ~ in(v4, v1)) & ! [v2] : ! [v3] : ! [v4] : ! [v5] : (v5 = v3 | ~ (apply(all_0_12_12, v3) = v4) | ~ (apply(all_0_14_14, v4) = v5) | ~ (relation_dom(all_0_12_12) = v2) | ~ relation(all_0_12_12) | ~ function(all_0_12_12) | ~ in(v3, v0)) & ! [v2] : ! [v3] : ! [v4] : ! [v5] : ( ~ (apply(all_0_12_12, v3) = v5) | ~ (apply(all_0_14_14, v4) = v3) | ~ (relation_dom(all_0_12_12) = v2) | ~ relation(all_0_12_12) | ~ function(all_0_12_12) | ~ in(v4, v1) | in(v3, v0)) & ! [v2] : ! [v3] : ! [v4] : ! [v5] : ( ~ (apply(all_0_12_12, v3) = v4) | ~ (apply(all_0_14_14, v4) = v5) | ~ (relation_dom(all_0_12_12) = v2) | ~ relation(all_0_12_12) | ~ function(all_0_12_12) | ~ in(v3, v0) | in(v4, v1)) & ! [v2] : (v2 = v0 | ~ (relation_dom(all_0_12_12) = v2) | ~ relation(all_0_12_12) | ~ function(all_0_12_12)) & ! [v2] : (v2 = all_0_12_12 | ~ (relation_dom(v2) = v0) | ~ relation(v2) | ~ function(v2) | ? [v3] : ? [v4] : ? [v5] : ? [v6] : (apply(v2, v3) = v5 & apply(all_0_14_14, v4) = v6 & ((v6 = v3 & in(v4, v1) & ( ~ (v5 = v4) | ~ in(v3, v0))) | (v5 = v4 & in(v3, v0) & ( ~ (v6 = v3) | ~ in(v4, v1))))))) % 6.66/2.13 | % 6.66/2.13 | Instantiating formula (68) with all_0_12_12, all_0_14_14 and discharging atoms function_inverse(all_0_14_14) = all_0_12_12, relation(all_0_14_14), function(all_0_14_14), yields: % 6.66/2.13 | (77) relation(all_0_12_12) % 6.66/2.13 | % 6.66/2.13 | Instantiating formula (30) with all_0_12_12, all_0_14_14 and discharging atoms function_inverse(all_0_14_14) = all_0_12_12, relation(all_0_14_14), function(all_0_14_14), yields: % 6.66/2.13 | (78) function(all_0_12_12) % 6.66/2.13 | % 6.66/2.13 | Instantiating (75) with all_20_0_21 yields: % 6.66/2.13 | (79) relation_dom(all_0_14_14) = all_20_0_21 & ! [v0] : ! [v1] : ! [v2] : ( ~ (apply(v1, all_0_15_15) = v2) | ~ (relation_composition(all_0_14_14, v0) = v1) | ~ relation(v0) | ~ function(v0) | ~ in(all_0_15_15, all_20_0_21) | apply(v0, all_0_11_11) = v2) & ! [v0] : ! [v1] : ( ~ (apply(v0, all_0_11_11) = v1) | ~ relation(v0) | ~ function(v0) | ~ in(all_0_15_15, all_20_0_21) | ? [v2] : (apply(v2, all_0_15_15) = v1 & relation_composition(all_0_14_14, v0) = v2)) % 6.66/2.13 | % 6.66/2.13 | Applying alpha-rule on (79) yields: % 6.66/2.13 | (80) relation_dom(all_0_14_14) = all_20_0_21 % 6.66/2.13 | (81) ! [v0] : ! [v1] : ! [v2] : ( ~ (apply(v1, all_0_15_15) = v2) | ~ (relation_composition(all_0_14_14, v0) = v1) | ~ relation(v0) | ~ function(v0) | ~ in(all_0_15_15, all_20_0_21) | apply(v0, all_0_11_11) = v2) % 6.66/2.13 | (82) ! [v0] : ! [v1] : ( ~ (apply(v0, all_0_11_11) = v1) | ~ relation(v0) | ~ function(v0) | ~ in(all_0_15_15, all_20_0_21) | ? [v2] : (apply(v2, all_0_15_15) = v1 & relation_composition(all_0_14_14, v0) = v2)) % 6.66/2.13 | % 6.66/2.13 | Instantiating (76) with all_23_0_22, all_23_1_23 yields: % 6.66/2.13 | (83) relation_rng(all_0_14_14) = all_23_1_23 & relation_dom(all_0_14_14) = all_23_0_22 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = v2 | ~ (apply(all_0_12_12, v1) = v3) | ~ (apply(all_0_14_14, v2) = v1) | ~ (relation_dom(all_0_12_12) = v0) | ~ relation(all_0_12_12) | ~ function(all_0_12_12) | ~ in(v2, all_23_0_22)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = v1 | ~ (apply(all_0_12_12, v1) = v2) | ~ (apply(all_0_14_14, v2) = v3) | ~ (relation_dom(all_0_12_12) = v0) | ~ relation(all_0_12_12) | ~ function(all_0_12_12) | ~ in(v1, all_23_1_23)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (apply(all_0_12_12, v1) = v3) | ~ (apply(all_0_14_14, v2) = v1) | ~ (relation_dom(all_0_12_12) = v0) | ~ relation(all_0_12_12) | ~ function(all_0_12_12) | ~ in(v2, all_23_0_22) | in(v1, all_23_1_23)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (apply(all_0_12_12, v1) = v2) | ~ (apply(all_0_14_14, v2) = v3) | ~ (relation_dom(all_0_12_12) = v0) | ~ relation(all_0_12_12) | ~ function(all_0_12_12) | ~ in(v1, all_23_1_23) | in(v2, all_23_0_22)) & ! [v0] : (v0 = all_23_1_23 | ~ (relation_dom(all_0_12_12) = v0) | ~ relation(all_0_12_12) | ~ function(all_0_12_12)) & ! [v0] : (v0 = all_0_12_12 | ~ (relation_dom(v0) = all_23_1_23) | ~ relation(v0) | ~ function(v0) | ? [v1] : ? [v2] : ? [v3] : ? [v4] : (apply(v0, v1) = v3 & apply(all_0_14_14, v2) = v4 & ((v4 = v1 & in(v2, all_23_0_22) & ( ~ (v3 = v2) | ~ in(v1, all_23_1_23))) | (v3 = v2 & in(v1, all_23_1_23) & ( ~ (v4 = v1) | ~ in(v2, all_23_0_22)))))) % 6.66/2.13 | % 6.66/2.13 | Applying alpha-rule on (83) yields: % 6.66/2.13 | (84) relation_rng(all_0_14_14) = all_23_1_23 % 6.66/2.13 | (85) relation_dom(all_0_14_14) = all_23_0_22 % 6.66/2.13 | (86) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = v1 | ~ (apply(all_0_12_12, v1) = v2) | ~ (apply(all_0_14_14, v2) = v3) | ~ (relation_dom(all_0_12_12) = v0) | ~ relation(all_0_12_12) | ~ function(all_0_12_12) | ~ in(v1, all_23_1_23)) % 6.66/2.13 | (87) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (apply(all_0_12_12, v1) = v2) | ~ (apply(all_0_14_14, v2) = v3) | ~ (relation_dom(all_0_12_12) = v0) | ~ relation(all_0_12_12) | ~ function(all_0_12_12) | ~ in(v1, all_23_1_23) | in(v2, all_23_0_22)) % 6.66/2.13 | (88) ! [v0] : (v0 = all_23_1_23 | ~ (relation_dom(all_0_12_12) = v0) | ~ relation(all_0_12_12) | ~ function(all_0_12_12)) % 6.66/2.13 | (89) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (apply(all_0_12_12, v1) = v3) | ~ (apply(all_0_14_14, v2) = v1) | ~ (relation_dom(all_0_12_12) = v0) | ~ relation(all_0_12_12) | ~ function(all_0_12_12) | ~ in(v2, all_23_0_22) | in(v1, all_23_1_23)) % 6.66/2.13 | (90) ! [v0] : (v0 = all_0_12_12 | ~ (relation_dom(v0) = all_23_1_23) | ~ relation(v0) | ~ function(v0) | ? [v1] : ? [v2] : ? [v3] : ? [v4] : (apply(v0, v1) = v3 & apply(all_0_14_14, v2) = v4 & ((v4 = v1 & in(v2, all_23_0_22) & ( ~ (v3 = v2) | ~ in(v1, all_23_1_23))) | (v3 = v2 & in(v1, all_23_1_23) & ( ~ (v4 = v1) | ~ in(v2, all_23_0_22)))))) % 6.66/2.13 | (91) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v3 = v2 | ~ (apply(all_0_12_12, v1) = v3) | ~ (apply(all_0_14_14, v2) = v1) | ~ (relation_dom(all_0_12_12) = v0) | ~ relation(all_0_12_12) | ~ function(all_0_12_12) | ~ in(v2, all_23_0_22)) % 6.66/2.13 | % 6.66/2.13 | Instantiating formula (64) with all_0_14_14, all_23_0_22, all_0_13_13 and discharging atoms relation_dom(all_0_14_14) = all_23_0_22, relation_dom(all_0_14_14) = all_0_13_13, yields: % 6.66/2.13 | (92) all_23_0_22 = all_0_13_13 % 6.66/2.13 | % 6.66/2.13 | Instantiating formula (64) with all_0_14_14, all_20_0_21, all_23_0_22 and discharging atoms relation_dom(all_0_14_14) = all_23_0_22, relation_dom(all_0_14_14) = all_20_0_21, yields: % 6.66/2.13 | (93) all_23_0_22 = all_20_0_21 % 6.66/2.13 | % 6.66/2.13 | Combining equations (92,93) yields a new equation: % 6.66/2.13 | (94) all_20_0_21 = all_0_13_13 % 6.66/2.13 | % 6.66/2.13 | Combining equations (94,93) yields a new equation: % 6.66/2.13 | (92) all_23_0_22 = all_0_13_13 % 6.66/2.13 | % 6.66/2.13 | Instantiating formula (81) with all_0_8_8, all_0_9_9, all_0_12_12 and discharging atoms apply(all_0_9_9, all_0_15_15) = all_0_8_8, relation_composition(all_0_14_14, all_0_12_12) = all_0_9_9, relation(all_0_12_12), function(all_0_12_12), yields: % 6.66/2.13 | (96) ~ in(all_0_15_15, all_20_0_21) | apply(all_0_12_12, all_0_11_11) = all_0_8_8 % 6.66/2.14 | % 6.66/2.14 | Instantiating formula (44) with all_0_10_10, all_0_12_12, all_0_11_11 and discharging atoms apply(all_0_12_12, all_0_11_11) = all_0_10_10, relation(all_0_12_12), function(all_0_12_12), yields: % 6.66/2.14 | (97) ? [v0] : (relation_dom(all_0_12_12) = v0 & ! [v1] : ! [v2] : ! [v3] : ( ~ (apply(v2, all_0_11_11) = v3) | ~ (relation_composition(all_0_12_12, v1) = v2) | ~ relation(v1) | ~ function(v1) | ~ in(all_0_11_11, v0) | apply(v1, all_0_10_10) = v3) & ! [v1] : ! [v2] : ( ~ (apply(v1, all_0_10_10) = v2) | ~ relation(v1) | ~ function(v1) | ~ in(all_0_11_11, v0) | ? [v3] : (apply(v3, all_0_11_11) = v2 & relation_composition(all_0_12_12, v1) = v3))) % 6.66/2.14 | % 6.66/2.14 | Instantiating formula (82) with all_0_10_10, all_0_12_12 and discharging atoms apply(all_0_12_12, all_0_11_11) = all_0_10_10, relation(all_0_12_12), function(all_0_12_12), yields: % 6.66/2.14 | (98) ~ in(all_0_15_15, all_20_0_21) | ? [v0] : (apply(v0, all_0_15_15) = all_0_10_10 & relation_composition(all_0_14_14, all_0_12_12) = v0) % 6.66/2.14 | % 6.66/2.14 | Instantiating (97) with all_36_0_24 yields: % 6.66/2.14 | (99) relation_dom(all_0_12_12) = all_36_0_24 & ! [v0] : ! [v1] : ! [v2] : ( ~ (apply(v1, all_0_11_11) = v2) | ~ (relation_composition(all_0_12_12, v0) = v1) | ~ relation(v0) | ~ function(v0) | ~ in(all_0_11_11, all_36_0_24) | apply(v0, all_0_10_10) = v2) & ! [v0] : ! [v1] : ( ~ (apply(v0, all_0_10_10) = v1) | ~ relation(v0) | ~ function(v0) | ~ in(all_0_11_11, all_36_0_24) | ? [v2] : (apply(v2, all_0_11_11) = v1 & relation_composition(all_0_12_12, v0) = v2)) % 6.66/2.14 | % 6.66/2.14 | Applying alpha-rule on (99) yields: % 6.66/2.14 | (100) relation_dom(all_0_12_12) = all_36_0_24 % 6.66/2.14 | (101) ! [v0] : ! [v1] : ! [v2] : ( ~ (apply(v1, all_0_11_11) = v2) | ~ (relation_composition(all_0_12_12, v0) = v1) | ~ relation(v0) | ~ function(v0) | ~ in(all_0_11_11, all_36_0_24) | apply(v0, all_0_10_10) = v2) % 6.66/2.14 | (102) ! [v0] : ! [v1] : ( ~ (apply(v0, all_0_10_10) = v1) | ~ relation(v0) | ~ function(v0) | ~ in(all_0_11_11, all_36_0_24) | ? [v2] : (apply(v2, all_0_11_11) = v1 & relation_composition(all_0_12_12, v0) = v2)) % 6.66/2.14 | % 6.66/2.14 +-Applying beta-rule and splitting (98), into two cases. % 6.66/2.14 |-Branch one: % 6.66/2.14 | (103) ~ in(all_0_15_15, all_20_0_21) % 6.66/2.14 | % 6.66/2.14 | From (94) and (103) follows: % 6.66/2.14 | (104) ~ in(all_0_15_15, all_0_13_13) % 6.66/2.14 | % 6.66/2.14 | Using (53) and (104) yields: % 6.66/2.14 | (105) $false % 6.66/2.14 | % 6.66/2.14 |-The branch is then unsatisfiable % 6.66/2.14 |-Branch two: % 6.66/2.14 | (106) in(all_0_15_15, all_20_0_21) % 6.66/2.14 | (107) ? [v0] : (apply(v0, all_0_15_15) = all_0_10_10 & relation_composition(all_0_14_14, all_0_12_12) = v0) % 6.66/2.14 | % 6.66/2.14 | From (94) and (106) follows: % 6.66/2.14 | (53) in(all_0_15_15, all_0_13_13) % 6.66/2.14 | % 6.66/2.14 +-Applying beta-rule and splitting (96), into two cases. % 6.66/2.14 |-Branch one: % 6.66/2.14 | (103) ~ in(all_0_15_15, all_20_0_21) % 6.66/2.14 | % 6.66/2.14 | From (94) and (103) follows: % 6.66/2.14 | (104) ~ in(all_0_15_15, all_0_13_13) % 6.66/2.14 | % 6.66/2.14 | Using (53) and (104) yields: % 6.66/2.14 | (105) $false % 6.66/2.14 | % 6.66/2.14 |-The branch is then unsatisfiable % 6.66/2.14 |-Branch two: % 6.66/2.14 | (106) in(all_0_15_15, all_20_0_21) % 6.66/2.14 | (113) apply(all_0_12_12, all_0_11_11) = all_0_8_8 % 6.66/2.14 | % 6.66/2.14 | From (94) and (106) follows: % 6.66/2.14 | (53) in(all_0_15_15, all_0_13_13) % 6.66/2.14 | % 6.66/2.14 | Instantiating formula (10) with all_0_12_12, all_0_11_11, all_0_8_8, all_0_10_10 and discharging atoms apply(all_0_12_12, all_0_11_11) = all_0_8_8, apply(all_0_12_12, all_0_11_11) = all_0_10_10, yields: % 6.66/2.14 | (115) all_0_8_8 = all_0_10_10 % 6.66/2.14 | % 6.66/2.14 | Instantiating formula (91) with all_0_8_8, all_0_15_15, all_0_11_11, all_36_0_24 and discharging atoms apply(all_0_12_12, all_0_11_11) = all_0_8_8, apply(all_0_14_14, all_0_15_15) = all_0_11_11, relation_dom(all_0_12_12) = all_36_0_24, relation(all_0_12_12), function(all_0_12_12), yields: % 6.66/2.14 | (116) all_0_8_8 = all_0_15_15 | ~ in(all_0_15_15, all_23_0_22) % 6.66/2.14 | % 6.66/2.14 +-Applying beta-rule and splitting (37), into two cases. % 6.66/2.14 |-Branch one: % 6.66/2.14 | (117) ~ (all_0_8_8 = all_0_15_15) % 6.66/2.14 | % 6.66/2.14 | Equations (115) can reduce 117 to: % 6.66/2.14 | (118) ~ (all_0_10_10 = all_0_15_15) % 6.66/2.14 | % 6.66/2.14 +-Applying beta-rule and splitting (116), into two cases. % 6.66/2.14 |-Branch one: % 6.66/2.14 | (119) ~ in(all_0_15_15, all_23_0_22) % 6.66/2.14 | % 6.66/2.14 | From (92) and (119) follows: % 6.66/2.14 | (104) ~ in(all_0_15_15, all_0_13_13) % 6.66/2.14 | % 6.66/2.14 | Using (53) and (104) yields: % 6.66/2.14 | (105) $false % 6.66/2.14 | % 6.66/2.14 |-The branch is then unsatisfiable % 6.66/2.14 |-Branch two: % 6.66/2.14 | (122) in(all_0_15_15, all_23_0_22) % 6.66/2.14 | (123) all_0_8_8 = all_0_15_15 % 6.66/2.14 | % 6.66/2.14 | Combining equations (123,115) yields a new equation: % 6.66/2.14 | (124) all_0_10_10 = all_0_15_15 % 6.66/2.14 | % 6.66/2.14 | Equations (124) can reduce 118 to: % 6.66/2.14 | (125) $false % 6.66/2.14 | % 6.66/2.14 |-The branch is then unsatisfiable % 6.66/2.14 |-Branch two: % 6.66/2.14 | (123) all_0_8_8 = all_0_15_15 % 6.66/2.14 | (118) ~ (all_0_10_10 = all_0_15_15) % 6.66/2.14 | % 6.66/2.14 | Combining equations (123,115) yields a new equation: % 6.66/2.14 | (124) all_0_10_10 = all_0_15_15 % 6.66/2.14 | % 6.66/2.14 | Equations (124) can reduce 118 to: % 6.66/2.14 | (125) $false % 6.66/2.14 | % 6.66/2.14 |-The branch is then unsatisfiable % 6.66/2.14 % SZS output end Proof for theBenchmark % 6.66/2.14 % 6.66/2.14 1539ms %------------------------------------------------------------------------------