%------------------------------------------------------------------------------ % File : ePrincess---1.0 % Problem : SEU220+1 : TPTP v8.1.0. Released v3.3.0. % Transfm : none % Format : tptp:raw % Command : ePrincess-casc -timeout=%d %s % Computer : n023.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 08:47:47 EDT 2022 % Result : Theorem 4.11s 1.56s % Output : Proof 7.25s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.00/0.12 % Problem : SEU220+1 : TPTP v8.1.0. Released v3.3.0. % 0.12/0.12 % Command : ePrincess-casc -timeout=%d %s % 0.12/0.33 % Computer : n023.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 : 600 % 0.12/0.33 % DateTime : Mon Jun 20 01:56:48 EDT 2022 % 0.12/0.33 % CPUTime : % 0.18/0.57 ____ _ % 0.18/0.57 ___ / __ \_____(_)___ ________ __________ % 0.18/0.57 / _ \/ /_/ / ___/ / __ \/ ___/ _ \/ ___/ ___/ % 0.18/0.57 / __/ ____/ / / / / / / /__/ __(__ |__ ) % 0.18/0.57 \___/_/ /_/ /_/_/ /_/\___/\___/____/____/ % 0.18/0.57 % 0.18/0.57 A Theorem Prover for First-Order Logic % 0.18/0.58 (ePrincess v.1.0) % 0.18/0.58 % 0.18/0.58 (c) Philipp Rümmer, 2009-2015 % 0.18/0.58 (c) Peter Backeman, 2014-2015 % 0.18/0.58 (contributions by Angelo Brillout, Peter Baumgartner) % 0.18/0.58 Free software under GNU Lesser General Public License (LGPL). % 0.18/0.58 Bug reports to peter@backeman.se % 0.18/0.58 % 0.18/0.58 For more information, visit http://user.uu.se/~petba168/breu/ % 0.18/0.58 % 0.18/0.58 Loading /export/starexec/sandbox/benchmark/theBenchmark.p ... % 0.63/0.62 Prover 0: Options: -triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMaximal -resolutionMethod=nonUnifying +ignoreQuantifiers -generateTriggers=all % 1.67/0.93 Prover 0: Preprocessing ... % 2.75/1.24 Prover 0: Warning: ignoring some quantifiers % 2.75/1.27 Prover 0: Constructing countermodel ... % 4.11/1.56 Prover 0: proved (932ms) % 4.11/1.56 % 4.11/1.56 No countermodel exists, formula is valid % 4.11/1.56 % SZS status Theorem for theBenchmark % 4.11/1.56 % 4.11/1.56 Generating proof ... Warning: ignoring some quantifiers % 6.67/2.09 found it (size 58) % 6.67/2.09 % 6.67/2.09 % SZS output start Proof for theBenchmark % 6.67/2.09 Assumed formulas after preprocessing and simplification: % 6.67/2.09 | (0) ? [v0] : ? [v1] : ? [v2] : ? [v3] : ? [v4] : ? [v5] : ? [v6] : ? [v7] : ? [v8] : ? [v9] : ? [v10] : ? [v11] : ? [v12] : ? [v13] : ? [v14] : ? [v15] : (apply(v6, v0) = v7 & apply(v3, v0) = v4 & apply(v1, v4) = v5 & relation_rng(v1) = v2 & relation_composition(v3, v1) = 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] : (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 | ~ (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] : ( ~ (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_rng(v18) = v19 & relation_dom(v18) = v17 & relation_dom(v16) = v19 & function_inverse(v16) = v18)) & ! [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(v19) = v17 & relation_rng(v16) = v18 & relation_dom(v19) = v18 & function_inverse(v16) = v19)) & ! [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] : ( ~ (function_inverse(v16) = v17) | ~ one_to_one(v16) | ~ relation(v16) | ~ function(v16) | ? [v18] : ? [v19] : (relation_rng(v17) = v19 & relation_rng(v16) = v18 & relation_dom(v17) = v18 & relation_dom(v16) = v19)) & ! [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) & ( ~ (v7 = v0) | ~ (v5 = v0))) % 6.75/2.14 | 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: % 6.75/2.14 | (1) apply(all_0_9_9, all_0_15_15) = all_0_8_8 & apply(all_0_12_12, all_0_15_15) = all_0_11_11 & apply(all_0_14_14, all_0_11_11) = all_0_10_10 & relation_rng(all_0_14_14) = all_0_13_13 & relation_composition(all_0_12_12, all_0_14_14) = 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] : (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 | ~ (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] : ( ~ (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_rng(v2) = v3 & relation_dom(v2) = v1 & relation_dom(v0) = v3 & function_inverse(v0) = v2)) & ! [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(v3) = v1 & relation_rng(v0) = v2 & relation_dom(v3) = v2 & function_inverse(v0) = v3)) & ! [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] : ( ~ (function_inverse(v0) = v1) | ~ one_to_one(v0) | ~ relation(v0) | ~ function(v0) | ? [v2] : ? [v3] : (relation_rng(v1) = v3 & relation_rng(v0) = v2 & relation_dom(v1) = v2 & relation_dom(v0) = v3)) & ! [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) & ( ~ (all_0_8_8 = all_0_15_15) | ~ (all_0_10_10 = all_0_15_15)) % 6.75/2.16 | % 6.75/2.16 | Applying alpha-rule on (1) yields: % 6.75/2.16 | (2) relation(empty_set) % 6.75/2.16 | (3) ! [v0] : ! [v1] : ( ~ (relation_dom(v0) = v1) | ~ relation(v0) | ~ empty(v1) | empty(v0)) % 6.75/2.16 | (4) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (relation_rng(v2) = v1) | ~ (relation_rng(v2) = v0)) % 6.75/2.16 | (5) ! [v0] : ! [v1] : ( ~ in(v0, v1) | element(v0, v1)) % 6.75/2.16 | (6) ! [v0] : ( ~ relation(v0) | ~ function(v0) | ~ empty(v0) | one_to_one(v0)) % 6.75/2.16 | (7) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (relation_composition(v3, v2) = v1) | ~ (relation_composition(v3, v2) = v0)) % 6.75/2.16 | (8) function(all_0_0_0) % 6.75/2.16 | (9) ! [v0] : ! [v1] : ( ~ (relation_rng(v0) = v1) | ~ relation(v0) | ~ empty(v1) | empty(v0)) % 6.75/2.16 | (10) function(all_0_3_3) % 6.75/2.16 | (11) relation(all_0_14_14) % 6.75/2.16 | (12) ! [v0] : ! [v1] : ( ~ (function_inverse(v0) = v1) | ~ relation(v0) | ~ function(v0) | relation(v1)) % 6.75/2.16 | (13) ~ empty(all_0_5_5) % 6.75/2.16 | (14) ~ empty(all_0_4_4) % 6.75/2.16 | (15) ! [v0] : ! [v1] : (v1 = v0 | ~ empty(v1) | ~ empty(v0)) % 6.75/2.17 | (16) ! [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)))) % 6.75/2.17 | (17) ? [v0] : ? [v1] : element(v1, v0) % 6.75/2.17 | (18) ! [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)))) % 6.75/2.17 | (19) empty(all_0_2_2) % 6.75/2.17 | (20) apply(all_0_12_12, all_0_15_15) = all_0_11_11 % 6.75/2.17 | (21) relation_composition(all_0_12_12, all_0_14_14) = all_0_9_9 % 6.75/2.17 | (22) ! [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)))))))) % 6.75/2.17 | (23) ! [v0] : ! [v1] : ( ~ (function_inverse(v0) = v1) | ~ relation(v0) | ~ function(v0) | function(v1)) % 6.75/2.17 | (24) relation_rng(all_0_14_14) = all_0_13_13 % 6.75/2.17 | (25) ! [v0] : ! [v1] : ( ~ element(v0, v1) | empty(v1) | in(v0, v1)) % 6.75/2.17 | (26) relation_empty_yielding(empty_set) % 6.75/2.18 | (27) ! [v0] : ! [v1] : ( ~ (relation_rng(v0) = v1) | ~ empty(v0) | relation(v1)) % 6.75/2.18 | (28) ! [v0] : ( ~ empty(v0) | function(v0)) % 6.75/2.18 | (29) one_to_one(all_0_6_6) % 6.75/2.18 | (30) ! [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)))))))) % 6.75/2.18 | (31) ! [v0] : ! [v1] : ! [v2] : ( ~ (relation_composition(v0, v1) = v2) | ~ relation(v1) | ~ empty(v0) | empty(v2)) % 6.75/2.18 | (32) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (function_inverse(v2) = v1) | ~ (function_inverse(v2) = v0)) % 6.75/2.18 | (33) ! [v0] : ! [v1] : ( ~ (relation_dom(v0) = v1) | ~ one_to_one(v0) | ~ relation(v0) | ~ function(v0) | ? [v2] : ? [v3] : (relation_rng(v3) = v1 & relation_rng(v0) = v2 & relation_dom(v3) = v2 & function_inverse(v0) = v3)) % 6.75/2.18 | (34) ! [v0] : ! [v1] : ( ~ (function_inverse(v0) = v1) | ~ one_to_one(v0) | ~ relation(v0) | ~ function(v0) | ? [v2] : ? [v3] : (relation_rng(v1) = v3 & relation_rng(v0) = v2 & relation_dom(v1) = v2 & relation_dom(v0) = v3)) % 6.75/2.18 | (35) apply(all_0_14_14, all_0_11_11) = all_0_10_10 % 6.75/2.18 | (36) ! [v0] : ! [v1] : ! [v2] : ( ~ (relation_composition(v0, v1) = v2) | ~ relation(v1) | ~ empty(v0) | relation(v2)) % 6.75/2.18 | (37) relation(all_0_7_7) % 6.75/2.18 | (38) relation(all_0_4_4) % 6.75/2.18 | (39) ! [v0] : ! [v1] : ( ~ empty(v1) | ~ in(v0, v1)) % 6.75/2.18 | (40) relation(all_0_0_0) % 6.75/2.18 | (41) in(all_0_15_15, all_0_13_13) % 6.75/2.18 | (42) ! [v0] : ! [v1] : ( ~ (relation_dom(v0) = v1) | ~ empty(v0) | relation(v1)) % 6.75/2.18 | (43) ! [v0] : ! [v1] : ( ~ in(v1, v0) | ~ in(v0, v1)) % 6.75/2.18 | (44) one_to_one(all_0_14_14) % 6.75/2.18 | (45) ! [v0] : ! [v1] : ! [v2] : ( ~ (relation_composition(v0, v1) = v2) | ~ relation(v1) | ~ relation(v0) | ~ function(v1) | ~ function(v0) | relation(v2)) % 6.75/2.18 | (46) relation(all_0_6_6) % 6.75/2.18 | (47) relation(all_0_3_3) % 6.75/2.18 | (48) ! [v0] : ! [v1] : ( ~ (relation_rng(v0) = v1) | ~ one_to_one(v0) | ~ relation(v0) | ~ function(v0) | ? [v2] : ? [v3] : (relation_rng(v2) = v3 & relation_dom(v2) = v1 & relation_dom(v0) = v3 & function_inverse(v0) = v2)) % 6.75/2.18 | (49) function(all_0_14_14) % 6.75/2.18 | (50) ! [v0] : ! [v1] : ( ~ (relation_rng(v0) = v1) | ~ empty(v0) | empty(v1)) % 6.75/2.18 | (51) ~ (all_0_8_8 = all_0_15_15) | ~ (all_0_10_10 = all_0_15_15) % 6.75/2.18 | (52) ! [v0] : ! [v1] : ( ~ (relation_dom(v0) = v1) | ~ empty(v0) | empty(v1)) % 6.75/2.18 | (53) empty(all_0_1_1) % 6.75/2.18 | (54) ! [v0] : ! [v1] : ! [v2] : ( ~ (relation_composition(v1, v0) = v2) | ~ relation(v1) | ~ empty(v0) | relation(v2)) % 6.75/2.19 | (55) ! [v0] : ! [v1] : ! [v2] : ( ~ (relation_composition(v1, v0) = v2) | ~ relation(v1) | ~ empty(v0) | empty(v2)) % 6.75/2.19 | (56) ! [v0] : ( ~ empty(v0) | relation(v0)) % 6.75/2.19 | (57) function(all_0_6_6) % 6.75/2.19 | (58) ! [v0] : ! [v1] : ! [v2] : ( ~ (relation_composition(v0, v1) = v2) | ~ relation(v1) | ~ relation(v0) | relation(v2)) % 6.75/2.19 | (59) relation_empty_yielding(all_0_7_7) % 6.75/2.19 | (60) ! [v0] : ! [v1] : ! [v2] : ( ~ (relation_composition(v0, v1) = v2) | ~ relation(v1) | ~ relation(v0) | ~ function(v1) | ~ function(v0) | function(v2)) % 6.75/2.19 | (61) empty(all_0_3_3) % 6.75/2.19 | (62) apply(all_0_9_9, all_0_15_15) = all_0_8_8 % 6.75/2.19 | (63) empty(empty_set) % 6.75/2.19 | (64) ! [v0] : (v0 = empty_set | ~ empty(v0)) % 6.75/2.19 | (65) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (relation_dom(v2) = v1) | ~ (relation_dom(v2) = v0)) % 6.75/2.19 | (66) function_inverse(all_0_14_14) = all_0_12_12 % 6.75/2.19 | (67) relation(all_0_1_1) % 6.75/2.19 | (68) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (apply(v3, v2) = v1) | ~ (apply(v3, v2) = v0)) % 6.75/2.19 | % 6.75/2.19 | Instantiating formula (18) with all_0_10_10, all_0_14_14, all_0_11_11 and discharging atoms apply(all_0_14_14, all_0_11_11) = all_0_10_10, relation(all_0_14_14), function(all_0_14_14), yields: % 6.75/2.19 | (69) ? [v0] : (relation_dom(all_0_14_14) = v0 & ! [v1] : ! [v2] : ! [v3] : ( ~ (apply(v2, all_0_11_11) = v3) | ~ (relation_composition(all_0_14_14, 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_14_14, v1) = v3))) % 6.75/2.19 | % 6.75/2.19 | Instantiating formula (34) 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: % 6.75/2.19 | (70) ? [v0] : ? [v1] : (relation_rng(all_0_12_12) = v1 & relation_rng(all_0_14_14) = v0 & relation_dom(all_0_12_12) = v0 & relation_dom(all_0_14_14) = v1) % 6.75/2.19 | % 6.75/2.19 | Instantiating formula (22) 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: % 6.75/2.19 | (71) ? [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))))))) % 6.75/2.19 | % 6.75/2.19 | Instantiating formula (12) 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: % 6.75/2.19 | (72) relation(all_0_12_12) % 6.75/2.20 | % 6.75/2.20 | Instantiating formula (23) 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: % 6.75/2.20 | (73) function(all_0_12_12) % 6.75/2.20 | % 6.75/2.20 | Instantiating (70) with all_15_0_18, all_15_1_19 yields: % 6.75/2.20 | (74) relation_rng(all_0_12_12) = all_15_0_18 & relation_rng(all_0_14_14) = all_15_1_19 & relation_dom(all_0_12_12) = all_15_1_19 & relation_dom(all_0_14_14) = all_15_0_18 % 6.75/2.20 | % 6.75/2.20 | Applying alpha-rule on (74) yields: % 6.75/2.20 | (75) relation_rng(all_0_12_12) = all_15_0_18 % 6.75/2.20 | (76) relation_rng(all_0_14_14) = all_15_1_19 % 6.75/2.20 | (77) relation_dom(all_0_12_12) = all_15_1_19 % 6.75/2.20 | (78) relation_dom(all_0_14_14) = all_15_0_18 % 6.75/2.20 | % 6.75/2.20 | Instantiating (71) with all_20_0_22, all_20_1_23 yields: % 6.75/2.20 | (79) relation_rng(all_0_14_14) = all_20_1_23 & relation_dom(all_0_14_14) = all_20_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_20_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_20_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_20_0_22) | in(v1, all_20_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_20_1_23) | in(v2, all_20_0_22)) & ! [v0] : (v0 = all_20_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_20_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_20_0_22) & ( ~ (v3 = v2) | ~ in(v1, all_20_1_23))) | (v3 = v2 & in(v1, all_20_1_23) & ( ~ (v4 = v1) | ~ in(v2, all_20_0_22)))))) % 6.75/2.20 | % 6.75/2.20 | Applying alpha-rule on (79) yields: % 6.75/2.20 | (80) relation_dom(all_0_14_14) = all_20_0_22 % 6.75/2.20 | (81) ! [v0] : (v0 = all_20_1_23 | ~ (relation_dom(all_0_12_12) = v0) | ~ relation(all_0_12_12) | ~ function(all_0_12_12)) % 6.75/2.20 | (82) ! [v0] : (v0 = all_0_12_12 | ~ (relation_dom(v0) = all_20_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_20_0_22) & ( ~ (v3 = v2) | ~ in(v1, all_20_1_23))) | (v3 = v2 & in(v1, all_20_1_23) & ( ~ (v4 = v1) | ~ in(v2, all_20_0_22)))))) % 6.75/2.20 | (83) ! [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_20_1_23)) % 6.75/2.20 | (84) ! [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_20_0_22) | in(v1, all_20_1_23)) % 6.75/2.20 | (85) ! [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_20_1_23) | in(v2, all_20_0_22)) % 6.75/2.20 | (86) ! [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_20_0_22)) % 6.75/2.20 | (87) relation_rng(all_0_14_14) = all_20_1_23 % 6.75/2.20 | % 6.75/2.20 | Instantiating (69) with all_25_0_26 yields: % 6.75/2.20 | (88) relation_dom(all_0_14_14) = all_25_0_26 & ! [v0] : ! [v1] : ! [v2] : ( ~ (apply(v1, all_0_11_11) = v2) | ~ (relation_composition(all_0_14_14, v0) = v1) | ~ relation(v0) | ~ function(v0) | ~ in(all_0_11_11, all_25_0_26) | 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_25_0_26) | ? [v2] : (apply(v2, all_0_11_11) = v1 & relation_composition(all_0_14_14, v0) = v2)) % 6.75/2.20 | % 6.75/2.20 | Applying alpha-rule on (88) yields: % 6.75/2.20 | (89) relation_dom(all_0_14_14) = all_25_0_26 % 6.75/2.20 | (90) ! [v0] : ! [v1] : ! [v2] : ( ~ (apply(v1, all_0_11_11) = v2) | ~ (relation_composition(all_0_14_14, v0) = v1) | ~ relation(v0) | ~ function(v0) | ~ in(all_0_11_11, all_25_0_26) | apply(v0, all_0_10_10) = v2) % 6.75/2.21 | (91) ! [v0] : ! [v1] : ( ~ (apply(v0, all_0_10_10) = v1) | ~ relation(v0) | ~ function(v0) | ~ in(all_0_11_11, all_25_0_26) | ? [v2] : (apply(v2, all_0_11_11) = v1 & relation_composition(all_0_14_14, v0) = v2)) % 6.75/2.21 | % 6.75/2.21 | Instantiating formula (4) with all_0_14_14, all_20_1_23, all_0_13_13 and discharging atoms relation_rng(all_0_14_14) = all_20_1_23, relation_rng(all_0_14_14) = all_0_13_13, yields: % 6.75/2.21 | (92) all_20_1_23 = all_0_13_13 % 6.75/2.21 | % 6.75/2.21 | Instantiating formula (81) with all_15_1_19 and discharging atoms relation_dom(all_0_12_12) = all_15_1_19, relation(all_0_12_12), function(all_0_12_12), yields: % 6.75/2.21 | (93) all_20_1_23 = all_15_1_19 % 6.75/2.21 | % 6.75/2.21 | Combining equations (93,92) yields a new equation: % 6.75/2.21 | (94) all_15_1_19 = all_0_13_13 % 6.75/2.21 | % 6.75/2.21 | Simplifying 94 yields: % 6.75/2.21 | (95) all_15_1_19 = all_0_13_13 % 6.75/2.21 | % 6.75/2.21 | From (95) and (77) follows: % 6.75/2.21 | (96) relation_dom(all_0_12_12) = all_0_13_13 % 6.75/2.21 | % 6.75/2.21 | Instantiating formula (83) with all_0_10_10, all_0_11_11, all_0_15_15, all_0_13_13 and discharging atoms apply(all_0_12_12, all_0_15_15) = all_0_11_11, apply(all_0_14_14, all_0_11_11) = all_0_10_10, relation_dom(all_0_12_12) = all_0_13_13, relation(all_0_12_12), function(all_0_12_12), yields: % 6.75/2.21 | (97) all_0_10_10 = all_0_15_15 | ~ in(all_0_15_15, all_20_1_23) % 6.75/2.21 | % 6.75/2.21 | Instantiating formula (18) with all_0_11_11, all_0_12_12, all_0_15_15 and discharging atoms apply(all_0_12_12, all_0_15_15) = all_0_11_11, relation(all_0_12_12), function(all_0_12_12), yields: % 6.75/2.21 | (98) ? [v0] : (relation_dom(all_0_12_12) = v0 & ! [v1] : ! [v2] : ! [v3] : ( ~ (apply(v2, all_0_15_15) = v3) | ~ (relation_composition(all_0_12_12, 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_12_12, v1) = v3))) % 6.75/2.21 | % 6.75/2.21 | Instantiating formula (45) with all_0_9_9, all_0_14_14, all_0_12_12 and discharging atoms relation_composition(all_0_12_12, all_0_14_14) = all_0_9_9, relation(all_0_12_12), relation(all_0_14_14), function(all_0_12_12), function(all_0_14_14), yields: % 6.75/2.21 | (99) relation(all_0_9_9) % 6.75/2.21 | % 6.75/2.21 | Instantiating formula (60) with all_0_9_9, all_0_14_14, all_0_12_12 and discharging atoms relation_composition(all_0_12_12, all_0_14_14) = all_0_9_9, relation(all_0_12_12), relation(all_0_14_14), function(all_0_12_12), function(all_0_14_14), yields: % 6.75/2.21 | (100) function(all_0_9_9) % 6.75/2.21 | % 6.75/2.21 | Instantiating formula (85) with all_0_10_10, all_0_11_11, all_0_15_15, all_0_13_13 and discharging atoms apply(all_0_12_12, all_0_15_15) = all_0_11_11, apply(all_0_14_14, all_0_11_11) = all_0_10_10, relation_dom(all_0_12_12) = all_0_13_13, relation(all_0_12_12), function(all_0_12_12), yields: % 6.75/2.21 | (101) ~ in(all_0_15_15, all_20_1_23) | in(all_0_11_11, all_20_0_22) % 6.75/2.21 | % 6.75/2.21 | Instantiating (98) with all_41_0_29 yields: % 6.75/2.21 | (102) relation_dom(all_0_12_12) = all_41_0_29 & ! [v0] : ! [v1] : ! [v2] : ( ~ (apply(v1, all_0_15_15) = v2) | ~ (relation_composition(all_0_12_12, v0) = v1) | ~ relation(v0) | ~ function(v0) | ~ in(all_0_15_15, all_41_0_29) | 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_41_0_29) | ? [v2] : (apply(v2, all_0_15_15) = v1 & relation_composition(all_0_12_12, v0) = v2)) % 6.75/2.22 | % 6.75/2.22 | Applying alpha-rule on (102) yields: % 6.75/2.22 | (103) relation_dom(all_0_12_12) = all_41_0_29 % 6.75/2.22 | (104) ! [v0] : ! [v1] : ! [v2] : ( ~ (apply(v1, all_0_15_15) = v2) | ~ (relation_composition(all_0_12_12, v0) = v1) | ~ relation(v0) | ~ function(v0) | ~ in(all_0_15_15, all_41_0_29) | apply(v0, all_0_11_11) = v2) % 6.75/2.22 | (105) ! [v0] : ! [v1] : ( ~ (apply(v0, all_0_11_11) = v1) | ~ relation(v0) | ~ function(v0) | ~ in(all_0_15_15, all_41_0_29) | ? [v2] : (apply(v2, all_0_15_15) = v1 & relation_composition(all_0_12_12, v0) = v2)) % 6.75/2.22 | % 6.75/2.22 +-Applying beta-rule and splitting (101), into two cases. % 6.75/2.22 |-Branch one: % 6.75/2.22 | (106) ~ in(all_0_15_15, all_20_1_23) % 6.75/2.22 | % 6.75/2.22 | From (92) and (106) follows: % 6.75/2.22 | (107) ~ in(all_0_15_15, all_0_13_13) % 6.75/2.22 | % 7.25/2.22 | Using (41) and (107) yields: % 7.25/2.22 | (108) $false % 7.25/2.22 | % 7.25/2.22 |-The branch is then unsatisfiable % 7.25/2.22 |-Branch two: % 7.25/2.22 | (109) in(all_0_15_15, all_20_1_23) % 7.25/2.22 | (110) in(all_0_11_11, all_20_0_22) % 7.25/2.22 | % 7.25/2.22 | From (92) and (109) follows: % 7.25/2.22 | (41) in(all_0_15_15, all_0_13_13) % 7.25/2.22 | % 7.25/2.22 +-Applying beta-rule and splitting (97), into two cases. % 7.25/2.22 |-Branch one: % 7.25/2.22 | (106) ~ in(all_0_15_15, all_20_1_23) % 7.25/2.22 | % 7.25/2.22 | From (92) and (106) follows: % 7.25/2.22 | (107) ~ in(all_0_15_15, all_0_13_13) % 7.25/2.22 | % 7.25/2.22 | Using (41) and (107) yields: % 7.25/2.22 | (108) $false % 7.25/2.22 | % 7.25/2.22 |-The branch is then unsatisfiable % 7.25/2.22 |-Branch two: % 7.25/2.22 | (109) in(all_0_15_15, all_20_1_23) % 7.25/2.22 | (116) all_0_10_10 = all_0_15_15 % 7.25/2.22 | % 7.25/2.22 | From (116) and (35) follows: % 7.25/2.22 | (117) apply(all_0_14_14, all_0_11_11) = all_0_15_15 % 7.25/2.22 | % 7.25/2.22 | From (92) and (109) follows: % 7.25/2.22 | (41) in(all_0_15_15, all_0_13_13) % 7.25/2.22 | % 7.25/2.22 +-Applying beta-rule and splitting (51), into two cases. % 7.25/2.22 |-Branch one: % 7.25/2.22 | (119) ~ (all_0_8_8 = all_0_15_15) % 7.25/2.22 | % 7.25/2.22 | Instantiating formula (65) with all_0_12_12, all_41_0_29, all_0_13_13 and discharging atoms relation_dom(all_0_12_12) = all_41_0_29, relation_dom(all_0_12_12) = all_0_13_13, yields: % 7.25/2.22 | (120) all_41_0_29 = all_0_13_13 % 7.25/2.22 | % 7.25/2.22 | Instantiating formula (105) with all_0_15_15, all_0_14_14 and discharging atoms apply(all_0_14_14, all_0_11_11) = all_0_15_15, relation(all_0_14_14), function(all_0_14_14), yields: % 7.25/2.22 | (121) ~ in(all_0_15_15, all_41_0_29) | ? [v0] : (apply(v0, all_0_15_15) = all_0_15_15 & relation_composition(all_0_12_12, all_0_14_14) = v0) % 7.25/2.22 | % 7.25/2.22 | Instantiating formula (91) with all_0_8_8, all_0_9_9 and discharging atoms relation(all_0_9_9), function(all_0_9_9), yields: % 7.25/2.22 | (122) ~ (apply(all_0_9_9, all_0_10_10) = all_0_8_8) | ~ in(all_0_11_11, all_25_0_26) | ? [v0] : (apply(v0, all_0_11_11) = all_0_8_8 & relation_composition(all_0_14_14, all_0_9_9) = v0) % 7.25/2.22 | % 7.25/2.22 +-Applying beta-rule and splitting (121), into two cases. % 7.25/2.22 |-Branch one: % 7.25/2.22 | (123) ~ in(all_0_15_15, all_41_0_29) % 7.25/2.22 | % 7.25/2.22 | From (120) and (123) follows: % 7.25/2.22 | (107) ~ in(all_0_15_15, all_0_13_13) % 7.25/2.22 | % 7.25/2.22 | Using (41) and (107) yields: % 7.25/2.22 | (108) $false % 7.25/2.22 | % 7.25/2.22 |-The branch is then unsatisfiable % 7.25/2.22 |-Branch two: % 7.25/2.22 | (126) in(all_0_15_15, all_41_0_29) % 7.25/2.22 | (127) ? [v0] : (apply(v0, all_0_15_15) = all_0_15_15 & relation_composition(all_0_12_12, all_0_14_14) = v0) % 7.25/2.22 | % 7.25/2.22 | Instantiating (127) with all_81_0_31 yields: % 7.25/2.22 | (128) apply(all_81_0_31, all_0_15_15) = all_0_15_15 & relation_composition(all_0_12_12, all_0_14_14) = all_81_0_31 % 7.25/2.22 | % 7.25/2.22 | Applying alpha-rule on (128) yields: % 7.25/2.22 | (129) apply(all_81_0_31, all_0_15_15) = all_0_15_15 % 7.25/2.22 | (130) relation_composition(all_0_12_12, all_0_14_14) = all_81_0_31 % 7.25/2.22 | % 7.25/2.22 +-Applying beta-rule and splitting (122), into two cases. % 7.25/2.22 |-Branch one: % 7.25/2.23 | (131) ~ (apply(all_0_9_9, all_0_10_10) = all_0_8_8) % 7.25/2.23 | % 7.25/2.23 | From (116) and (131) follows: % 7.25/2.23 | (132) ~ (apply(all_0_9_9, all_0_15_15) = all_0_8_8) % 7.25/2.23 | % 7.25/2.23 | Using (62) and (132) yields: % 7.25/2.23 | (108) $false % 7.25/2.23 | % 7.25/2.23 |-The branch is then unsatisfiable % 7.25/2.23 |-Branch two: % 7.25/2.23 | (134) apply(all_0_9_9, all_0_10_10) = all_0_8_8 % 7.25/2.23 | (135) ~ in(all_0_11_11, all_25_0_26) | ? [v0] : (apply(v0, all_0_11_11) = all_0_8_8 & relation_composition(all_0_14_14, all_0_9_9) = v0) % 7.25/2.23 | % 7.25/2.23 | From (116) and (134) follows: % 7.25/2.23 | (62) apply(all_0_9_9, all_0_15_15) = all_0_8_8 % 7.25/2.23 | % 7.25/2.23 | Instantiating formula (7) with all_0_12_12, all_0_14_14, all_81_0_31, all_0_9_9 and discharging atoms relation_composition(all_0_12_12, all_0_14_14) = all_81_0_31, relation_composition(all_0_12_12, all_0_14_14) = all_0_9_9, yields: % 7.25/2.23 | (137) all_81_0_31 = all_0_9_9 % 7.25/2.23 | % 7.25/2.23 | From (137) and (129) follows: % 7.25/2.23 | (138) apply(all_0_9_9, all_0_15_15) = all_0_15_15 % 7.25/2.23 | % 7.25/2.23 | Instantiating formula (68) with all_0_9_9, all_0_15_15, all_0_15_15, all_0_8_8 and discharging atoms apply(all_0_9_9, all_0_15_15) = all_0_8_8, apply(all_0_9_9, all_0_15_15) = all_0_15_15, yields: % 7.25/2.23 | (139) all_0_8_8 = all_0_15_15 % 7.25/2.23 | % 7.25/2.23 | Equations (139) can reduce 119 to: % 7.25/2.23 | (140) $false % 7.25/2.23 | % 7.25/2.23 |-The branch is then unsatisfiable % 7.25/2.23 |-Branch two: % 7.25/2.23 | (139) all_0_8_8 = all_0_15_15 % 7.25/2.23 | (142) ~ (all_0_10_10 = all_0_15_15) % 7.25/2.23 | % 7.25/2.23 | Equations (116) can reduce 142 to: % 7.25/2.23 | (140) $false % 7.25/2.23 | % 7.25/2.23 |-The branch is then unsatisfiable % 7.25/2.23 % SZS output end Proof for theBenchmark % 7.25/2.23 % 7.25/2.23 1644ms %------------------------------------------------------------------------------