%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SEU292+2 : TPTP v9.0.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n015.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:27:30 AM UTC 2025
% Result : Theorem 29.45s 29.62s
% Output : Proof 29.47s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : SEU292+2 : TPTP v9.0.0. Released v3.3.0.
% 0.07/0.13 % Command : run_E %s %d THM
% 0.13/0.34 % Computer : n015.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Fri Jun 20 06:38:55 EDT 2025
% 0.13/0.34 % CPUTime :
% 29.45/29.62 % SZS status Theorem
% 29.45/29.62 % SZS output start Proof
% 29.45/29.62 tff(in_type, type, (
% 29.45/29.62 in: ( $i * $i ) > $o)).
% 29.45/29.62 tff(relation_dom_type, type, (
% 29.45/29.62 relation_dom: $i > $i)).
% 29.45/29.62 tff(tptp_fun_D_160_type, type, (
% 29.45/29.62 tptp_fun_D_160: $i)).
% 29.45/29.62 tff(tptp_fun_C_161_type, type, (
% 29.45/29.62 tptp_fun_C_161: $i)).
% 29.45/29.62 tff(tptp_fun_A_163_type, type, (
% 29.45/29.62 tptp_fun_A_163: $i)).
% 29.45/29.62 tff(relation_dom_as_subset_type, type, (
% 29.45/29.62 relation_dom_as_subset: ( $i * $i * $i ) > $i)).
% 29.45/29.62 tff(tptp_fun_B_162_type, type, (
% 29.45/29.62 tptp_fun_B_162: $i)).
% 29.45/29.62 tff(quasi_total_type, type, (
% 29.45/29.62 quasi_total: ( $i * $i * $i ) > $o)).
% 29.45/29.62 tff(empty_set_type, type, (
% 29.45/29.62 empty_set: $i)).
% 29.45/29.62 tff(relation_of2_as_subset_type, type, (
% 29.45/29.62 relation_of2_as_subset: ( $i * $i * $i ) > $o)).
% 29.45/29.62 tff(function_type, type, (
% 29.45/29.62 function: $i > $o)).
% 29.45/29.62 tff(tptp_fun_E_164_type, type, (
% 29.45/29.62 tptp_fun_E_164: $i)).
% 29.45/29.62 tff(relation_type, type, (
% 29.45/29.62 relation: $i > $o)).
% 29.45/29.62 tff(apply_type, type, (
% 29.45/29.62 apply: ( $i * $i ) > $i)).
% 29.45/29.62 tff(relation_composition_type, type, (
% 29.45/29.62 relation_composition: ( $i * $i ) > $i)).
% 29.45/29.62 tff(relation_of2_type, type, (
% 29.45/29.62 relation_of2: ( $i * $i * $i ) > $o)).
% 29.45/29.62 tff(element_type, type, (
% 29.45/29.62 element: ( $i * $i ) > $o)).
% 29.45/29.62 tff(powerset_type, type, (
% 29.45/29.62 powerset: $i > $i)).
% 29.45/29.62 tff(cartesian_product2_type, type, (
% 29.45/29.62 cartesian_product2: ( $i * $i ) > $i)).
% 29.45/29.62 tff(1,plain,
% 29.45/29.62 (((function(D!160) & quasi_total(D!160, A!163, B!162) & relation_of2_as_subset(D!160, A!163, B!162)) & (~((B!162 = empty_set) | (~in(C!161, A!163)) | (apply(relation_composition(D!160, E!164), C!161) = apply(E!164, apply(D!160, C!161))) | (~(relation(E!164) & function(E!164)))))) <=> (function(D!160) & quasi_total(D!160, A!163, B!162) & relation_of2_as_subset(D!160, A!163, B!162) & (~((B!162 = empty_set) | (~in(C!161, A!163)) | (apply(relation_composition(D!160, E!164), C!161) = apply(E!164, apply(D!160, C!161))) | (~(relation(E!164) & function(E!164))))))),
% 29.45/29.62 inference(rewrite,[status(thm)],[])).
% 29.45/29.62 tff(2,plain,
% 29.45/29.62 ((~((B!162 = empty_set) | (apply(relation_composition(D!160, E!164), C!161) = apply(E!164, apply(D!160, C!161))) | (~(relation(E!164) & function(E!164))) | (~in(C!161, A!163)))) <=> (~((B!162 = empty_set) | (~in(C!161, A!163)) | (apply(relation_composition(D!160, E!164), C!161) = apply(E!164, apply(D!160, C!161))) | (~(relation(E!164) & function(E!164)))))),
% 29.45/29.62 inference(rewrite,[status(thm)],[])).
% 29.45/29.62 tff(3,plain,
% 29.45/29.62 (((function(D!160) & quasi_total(D!160, A!163, B!162) & relation_of2_as_subset(D!160, A!163, B!162)) & (~((B!162 = empty_set) | (apply(relation_composition(D!160, E!164), C!161) = apply(E!164, apply(D!160, C!161))) | (~(relation(E!164) & function(E!164))) | (~in(C!161, A!163))))) <=> ((function(D!160) & quasi_total(D!160, A!163, B!162) & relation_of2_as_subset(D!160, A!163, B!162)) & (~((B!162 = empty_set) | (~in(C!161, A!163)) | (apply(relation_composition(D!160, E!164), C!161) = apply(E!164, apply(D!160, C!161))) | (~(relation(E!164) & function(E!164))))))),
% 29.45/29.62 inference(monotonicity,[status(thm)],[2])).
% 29.45/29.62 tff(4,plain,
% 29.45/29.62 (((function(D!160) & quasi_total(D!160, A!163, B!162) & relation_of2_as_subset(D!160, A!163, B!162)) & (~((B!162 = empty_set) | (apply(relation_composition(D!160, E!164), C!161) = apply(E!164, apply(D!160, C!161))) | (~(relation(E!164) & function(E!164))) | (~in(C!161, A!163))))) <=> (function(D!160) & quasi_total(D!160, A!163, B!162) & relation_of2_as_subset(D!160, A!163, B!162) & (~((B!162 = empty_set) | (~in(C!161, A!163)) | (apply(relation_composition(D!160, E!164), C!161) = apply(E!164, apply(D!160, C!161))) | (~(relation(E!164) & function(E!164))))))),
% 29.45/29.62 inference(transitivity,[status(thm)],[3, 1])).
% 29.45/29.62 tff(5,plain,
% 29.45/29.62 ((~![A: $i, B: $i, C: $i, D: $i] : ((~(function(D) & quasi_total(D, A, B) & relation_of2_as_subset(D, A, B))) | ![E: $i] : ((B = empty_set) | (apply(relation_composition(D, E), C) = apply(E, apply(D, C))) | (~(relation(E) & function(E))) | (~in(C, A))))) <=> (~![A: $i, B: $i, C: $i, D: $i] : ((~(function(D) & quasi_total(D, A, B) & relation_of2_as_subset(D, A, B))) | ![E: $i] : ((B = empty_set) | (apply(relation_composition(D, E), C) = apply(E, apply(D, C))) | (~(relation(E) & function(E))) | (~in(C, A)))))),
% 29.45/29.62 inference(rewrite,[status(thm)],[])).
% 29.45/29.62 tff(6,plain,
% 29.45/29.62 ((~![A: $i, B: $i, C: $i, D: $i] : (((function(D) & quasi_total(D, A, B)) & relation_of2_as_subset(D, A, B)) => ![E: $i] : ((relation(E) & function(E)) => (in(C, A) => ((B = empty_set) | (apply(relation_composition(D, E), C) = apply(E, apply(D, C)))))))) <=> (~![A: $i, B: $i, C: $i, D: $i] : ((~(function(D) & quasi_total(D, A, B) & relation_of2_as_subset(D, A, B))) | ![E: $i] : ((B = empty_set) | (apply(relation_composition(D, E), C) = apply(E, apply(D, C))) | (~(relation(E) & function(E))) | (~in(C, A)))))),
% 29.45/29.62 inference(rewrite,[status(thm)],[])).
% 29.45/29.62 tff(7,axiom,(~![A: $i, B: $i, C: $i, D: $i] : (((function(D) & quasi_total(D, A, B)) & relation_of2_as_subset(D, A, B)) => ![E: $i] : ((relation(E) & function(E)) => (in(C, A) => ((B = empty_set) | (apply(relation_composition(D, E), C) = apply(E, apply(D, C)))))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t21_funct_2')).
% 29.45/29.62 tff(8,plain,
% 29.45/29.62 (~![A: $i, B: $i, C: $i, D: $i] : ((~(function(D) & quasi_total(D, A, B) & relation_of2_as_subset(D, A, B))) | ![E: $i] : ((B = empty_set) | (apply(relation_composition(D, E), C) = apply(E, apply(D, C))) | (~(relation(E) & function(E))) | (~in(C, A))))),
% 29.45/29.62 inference(modus_ponens,[status(thm)],[7, 6])).
% 29.45/29.62 tff(9,plain,
% 29.45/29.62 (~![A: $i, B: $i, C: $i, D: $i] : ((~(function(D) & quasi_total(D, A, B) & relation_of2_as_subset(D, A, B))) | ![E: $i] : ((B = empty_set) | (apply(relation_composition(D, E), C) = apply(E, apply(D, C))) | (~(relation(E) & function(E))) | (~in(C, A))))),
% 29.45/29.62 inference(modus_ponens,[status(thm)],[8, 5])).
% 29.45/29.62 tff(10,plain,
% 29.45/29.62 (~![A: $i, B: $i, C: $i, D: $i] : ((~(function(D) & quasi_total(D, A, B) & relation_of2_as_subset(D, A, B))) | ![E: $i] : ((B = empty_set) | (apply(relation_composition(D, E), C) = apply(E, apply(D, C))) | (~(relation(E) & function(E))) | (~in(C, A))))),
% 29.45/29.62 inference(modus_ponens,[status(thm)],[9, 5])).
% 29.45/29.62 tff(11,plain,
% 29.45/29.62 (~![A: $i, B: $i, C: $i, D: $i] : ((~(function(D) & quasi_total(D, A, B) & relation_of2_as_subset(D, A, B))) | ![E: $i] : ((B = empty_set) | (apply(relation_composition(D, E), C) = apply(E, apply(D, C))) | (~(relation(E) & function(E))) | (~in(C, A))))),
% 29.45/29.62 inference(modus_ponens,[status(thm)],[10, 5])).
% 29.45/29.62 tff(12,plain,
% 29.45/29.62 (function(D!160) & quasi_total(D!160, A!163, B!162) & relation_of2_as_subset(D!160, A!163, B!162) & (~((B!162 = empty_set) | (~in(C!161, A!163)) | (apply(relation_composition(D!160, E!164), C!161) = apply(E!164, apply(D!160, C!161))) | (~(relation(E!164) & function(E!164)))))),
% 29.45/29.62 inference(modus_ponens,[status(thm)],[11, 4])).
% 29.45/29.62 tff(13,plain,
% 29.45/29.62 (relation_of2_as_subset(D!160, A!163, B!162)),
% 29.45/29.62 inference(and_elim,[status(thm)],[12])).
% 29.45/29.62 tff(14,plain,
% 29.45/29.62 (^[A: $i, B: $i, C: $i] : refl(((~relation_of2_as_subset(C, A, B)) | (~((~((~((~(B = empty_set)) | (A = empty_set))) | (quasi_total(C, A, B) <=> (A = relation_dom_as_subset(A, B, C))))) | (~((quasi_total(C, A, B) <=> (C = empty_set)) | (A = empty_set) | (~(B = empty_set))))))) <=> ((~relation_of2_as_subset(C, A, B)) | (~((~((~((~(B = empty_set)) | (A = empty_set))) | (quasi_total(C, A, B) <=> (A = relation_dom_as_subset(A, B, C))))) | (~((quasi_total(C, A, B) <=> (C = empty_set)) | (A = empty_set) | (~(B = empty_set))))))))),
% 29.45/29.62 inference(bind,[status(th)],[])).
% 29.45/29.62 tff(15,plain,
% 29.45/29.62 (![A: $i, B: $i, C: $i] : ((~relation_of2_as_subset(C, A, B)) | (~((~((~((~(B = empty_set)) | (A = empty_set))) | (quasi_total(C, A, B) <=> (A = relation_dom_as_subset(A, B, C))))) | (~((quasi_total(C, A, B) <=> (C = empty_set)) | (A = empty_set) | (~(B = empty_set))))))) <=> ![A: $i, B: $i, C: $i] : ((~relation_of2_as_subset(C, A, B)) | (~((~((~((~(B = empty_set)) | (A = empty_set))) | (quasi_total(C, A, B) <=> (A = relation_dom_as_subset(A, B, C))))) | (~((quasi_total(C, A, B) <=> (C = empty_set)) | (A = empty_set) | (~(B = empty_set)))))))),
% 29.45/29.62 inference(quant_intro,[status(thm)],[14])).
% 29.45/29.62 tff(16,plain,
% 29.45/29.62 (^[A: $i, B: $i, C: $i] : rewrite(((~relation_of2_as_subset(C, A, B)) | (((~((~(B = empty_set)) | (A = empty_set))) | (quasi_total(C, A, B) <=> (A = relation_dom_as_subset(A, B, C)))) & ((quasi_total(C, A, B) <=> (C = empty_set)) | (A = empty_set) | (~(B = empty_set))))) <=> ((~relation_of2_as_subset(C, A, B)) | (~((~((~((~(B = empty_set)) | (A = empty_set))) | (quasi_total(C, A, B) <=> (A = relation_dom_as_subset(A, B, C))))) | (~((quasi_total(C, A, B) <=> (C = empty_set)) | (A = empty_set) | (~(B = empty_set))))))))),
% 29.45/29.62 inference(bind,[status(th)],[])).
% 29.45/29.62 tff(17,plain,
% 29.45/29.62 (![A: $i, B: $i, C: $i] : ((~relation_of2_as_subset(C, A, B)) | (((~((~(B = empty_set)) | (A = empty_set))) | (quasi_total(C, A, B) <=> (A = relation_dom_as_subset(A, B, C)))) & ((quasi_total(C, A, B) <=> (C = empty_set)) | (A = empty_set) | (~(B = empty_set))))) <=> ![A: $i, B: $i, C: $i] : ((~relation_of2_as_subset(C, A, B)) | (~((~((~((~(B = empty_set)) | (A = empty_set))) | (quasi_total(C, A, B) <=> (A = relation_dom_as_subset(A, B, C))))) | (~((quasi_total(C, A, B) <=> (C = empty_set)) | (A = empty_set) | (~(B = empty_set)))))))),
% 29.45/29.62 inference(quant_intro,[status(thm)],[16])).
% 29.45/29.62 tff(18,plain,
% 29.45/29.62 (![A: $i, B: $i, C: $i] : ((~relation_of2_as_subset(C, A, B)) | (((~((~(B = empty_set)) | (A = empty_set))) | (quasi_total(C, A, B) <=> (A = relation_dom_as_subset(A, B, C)))) & ((quasi_total(C, A, B) <=> (C = empty_set)) | (A = empty_set) | (~(B = empty_set))))) <=> ![A: $i, B: $i, C: $i] : ((~relation_of2_as_subset(C, A, B)) | (((~((~(B = empty_set)) | (A = empty_set))) | (quasi_total(C, A, B) <=> (A = relation_dom_as_subset(A, B, C)))) & ((quasi_total(C, A, B) <=> (C = empty_set)) | (A = empty_set) | (~(B = empty_set)))))),
% 29.45/29.62 inference(rewrite,[status(thm)],[])).
% 29.45/29.62 tff(19,plain,
% 29.45/29.62 (^[A: $i, B: $i, C: $i] : trans(monotonicity(rewrite(((((B = empty_set) => (A = empty_set)) => (quasi_total(C, A, B) <=> (A = relation_dom_as_subset(A, B, C)))) & ((B = empty_set) => ((A = empty_set) | (quasi_total(C, A, B) <=> (C = empty_set))))) <=> (((~((~(B = empty_set)) | (A = empty_set))) | (quasi_total(C, A, B) <=> (A = relation_dom_as_subset(A, B, C)))) & ((quasi_total(C, A, B) <=> (C = empty_set)) | (A = empty_set) | (~(B = empty_set))))), ((relation_of2_as_subset(C, A, B) => ((((B = empty_set) => (A = empty_set)) => (quasi_total(C, A, B) <=> (A = relation_dom_as_subset(A, B, C)))) & ((B = empty_set) => ((A = empty_set) | (quasi_total(C, A, B) <=> (C = empty_set)))))) <=> (relation_of2_as_subset(C, A, B) => (((~((~(B = empty_set)) | (A = empty_set))) | (quasi_total(C, A, B) <=> (A = relation_dom_as_subset(A, B, C)))) & ((quasi_total(C, A, B) <=> (C = empty_set)) | (A = empty_set) | (~(B = empty_set))))))), rewrite((relation_of2_as_subset(C, A, B) => (((~((~(B = empty_set)) | (A = empty_set))) | (quasi_total(C, A, B) <=> (A = relation_dom_as_subset(A, B, C)))) & ((quasi_total(C, A, B) <=> (C = empty_set)) | (A = empty_set) | (~(B = empty_set))))) <=> ((~relation_of2_as_subset(C, A, B)) | (((~((~(B = empty_set)) | (A = empty_set))) | (quasi_total(C, A, B) <=> (A = relation_dom_as_subset(A, B, C)))) & ((quasi_total(C, A, B) <=> (C = empty_set)) | (A = empty_set) | (~(B = empty_set)))))), ((relation_of2_as_subset(C, A, B) => ((((B = empty_set) => (A = empty_set)) => (quasi_total(C, A, B) <=> (A = relation_dom_as_subset(A, B, C)))) & ((B = empty_set) => ((A = empty_set) | (quasi_total(C, A, B) <=> (C = empty_set)))))) <=> ((~relation_of2_as_subset(C, A, B)) | (((~((~(B = empty_set)) | (A = empty_set))) | (quasi_total(C, A, B) <=> (A = relation_dom_as_subset(A, B, C)))) & ((quasi_total(C, A, B) <=> (C = empty_set)) | (A = empty_set) | (~(B = empty_set)))))))),
% 29.45/29.62 inference(bind,[status(th)],[])).
% 29.45/29.62 tff(20,plain,
% 29.45/29.62 (![A: $i, B: $i, C: $i] : (relation_of2_as_subset(C, A, B) => ((((B = empty_set) => (A = empty_set)) => (quasi_total(C, A, B) <=> (A = relation_dom_as_subset(A, B, C)))) & ((B = empty_set) => ((A = empty_set) | (quasi_total(C, A, B) <=> (C = empty_set)))))) <=> ![A: $i, B: $i, C: $i] : ((~relation_of2_as_subset(C, A, B)) | (((~((~(B = empty_set)) | (A = empty_set))) | (quasi_total(C, A, B) <=> (A = relation_dom_as_subset(A, B, C)))) & ((quasi_total(C, A, B) <=> (C = empty_set)) | (A = empty_set) | (~(B = empty_set)))))),
% 29.45/29.62 inference(quant_intro,[status(thm)],[19])).
% 29.45/29.62 tff(21,axiom,(![A: $i, B: $i, C: $i] : (relation_of2_as_subset(C, A, B) => ((((B = empty_set) => (A = empty_set)) => (quasi_total(C, A, B) <=> (A = relation_dom_as_subset(A, B, C)))) & ((B = empty_set) => ((A = empty_set) | (quasi_total(C, A, B) <=> (C = empty_set))))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','d1_funct_2')).
% 29.45/29.62 tff(22,plain,
% 29.45/29.62 (![A: $i, B: $i, C: $i] : ((~relation_of2_as_subset(C, A, B)) | (((~((~(B = empty_set)) | (A = empty_set))) | (quasi_total(C, A, B) <=> (A = relation_dom_as_subset(A, B, C)))) & ((quasi_total(C, A, B) <=> (C = empty_set)) | (A = empty_set) | (~(B = empty_set)))))),
% 29.45/29.62 inference(modus_ponens,[status(thm)],[21, 20])).
% 29.45/29.62 tff(23,plain,
% 29.45/29.62 (![A: $i, B: $i, C: $i] : ((~relation_of2_as_subset(C, A, B)) | (((~((~(B = empty_set)) | (A = empty_set))) | (quasi_total(C, A, B) <=> (A = relation_dom_as_subset(A, B, C)))) & ((quasi_total(C, A, B) <=> (C = empty_set)) | (A = empty_set) | (~(B = empty_set)))))),
% 29.45/29.62 inference(modus_ponens,[status(thm)],[22, 18])).
% 29.45/29.62 tff(24,plain,(
% 29.45/29.62 ![A: $i, B: $i, C: $i] : ((~relation_of2_as_subset(C, A, B)) | (((~((~(B = empty_set)) | (A = empty_set))) | (quasi_total(C, A, B) <=> (A = relation_dom_as_subset(A, B, C)))) & ((quasi_total(C, A, B) <=> (C = empty_set)) | (A = empty_set) | (~(B = empty_set)))))),
% 29.45/29.62 inference(skolemize,[status(sab)],[23])).
% 29.45/29.62 tff(25,plain,
% 29.45/29.62 (![A: $i, B: $i, C: $i] : ((~relation_of2_as_subset(C, A, B)) | (~((~((~((~(B = empty_set)) | (A = empty_set))) | (quasi_total(C, A, B) <=> (A = relation_dom_as_subset(A, B, C))))) | (~((quasi_total(C, A, B) <=> (C = empty_set)) | (A = empty_set) | (~(B = empty_set)))))))),
% 29.45/29.62 inference(modus_ponens,[status(thm)],[24, 17])).
% 29.45/29.62 tff(26,plain,
% 29.45/29.62 (![A: $i, B: $i, C: $i] : ((~relation_of2_as_subset(C, A, B)) | (~((~((~((~(B = empty_set)) | (A = empty_set))) | (quasi_total(C, A, B) <=> (A = relation_dom_as_subset(A, B, C))))) | (~((quasi_total(C, A, B) <=> (C = empty_set)) | (A = empty_set) | (~(B = empty_set)))))))),
% 29.45/29.62 inference(modus_ponens,[status(thm)],[25, 15])).
% 29.45/29.62 tff(27,plain,
% 29.45/29.62 (((~![A: $i, B: $i, C: $i] : ((~relation_of2_as_subset(C, A, B)) | (~((~((~((~(B = empty_set)) | (A = empty_set))) | (quasi_total(C, A, B) <=> (A = relation_dom_as_subset(A, B, C))))) | (~((quasi_total(C, A, B) <=> (C = empty_set)) | (A = empty_set) | (~(B = empty_set)))))))) | ((~relation_of2_as_subset(D!160, A!163, B!162)) | (~((~((~((~(B!162 = empty_set)) | (A!163 = empty_set))) | (quasi_total(D!160, A!163, B!162) <=> (A!163 = relation_dom_as_subset(A!163, B!162, D!160))))) | (~((~(B!162 = empty_set)) | (A!163 = empty_set) | (quasi_total(D!160, A!163, B!162) <=> (D!160 = empty_set)))))))) <=> ((~![A: $i, B: $i, C: $i] : ((~relation_of2_as_subset(C, A, B)) | (~((~((~((~(B = empty_set)) | (A = empty_set))) | (quasi_total(C, A, B) <=> (A = relation_dom_as_subset(A, B, C))))) | (~((quasi_total(C, A, B) <=> (C = empty_set)) | (A = empty_set) | (~(B = empty_set)))))))) | (~relation_of2_as_subset(D!160, A!163, B!162)) | (~((~((~((~(B!162 = empty_set)) | (A!163 = empty_set))) | (quasi_total(D!160, A!163, B!162) <=> (A!163 = relation_dom_as_subset(A!163, B!162, D!160))))) | (~((~(B!162 = empty_set)) | (A!163 = empty_set) | (quasi_total(D!160, A!163, B!162) <=> (D!160 = empty_set)))))))),
% 29.45/29.62 inference(rewrite,[status(thm)],[])).
% 29.45/29.62 tff(28,plain,
% 29.45/29.62 (((~relation_of2_as_subset(D!160, A!163, B!162)) | (~((~((~((~(B!162 = empty_set)) | (A!163 = empty_set))) | (quasi_total(D!160, A!163, B!162) <=> (A!163 = relation_dom_as_subset(A!163, B!162, D!160))))) | (~((quasi_total(D!160, A!163, B!162) <=> (D!160 = empty_set)) | (A!163 = empty_set) | (~(B!162 = empty_set))))))) <=> ((~relation_of2_as_subset(D!160, A!163, B!162)) | (~((~((~((~(B!162 = empty_set)) | (A!163 = empty_set))) | (quasi_total(D!160, A!163, B!162) <=> (A!163 = relation_dom_as_subset(A!163, B!162, D!160))))) | (~((~(B!162 = empty_set)) | (A!163 = empty_set) | (quasi_total(D!160, A!163, B!162) <=> (D!160 = empty_set)))))))),
% 29.45/29.62 inference(rewrite,[status(thm)],[])).
% 29.45/29.62 tff(29,plain,
% 29.45/29.62 (((~![A: $i, B: $i, C: $i] : ((~relation_of2_as_subset(C, A, B)) | (~((~((~((~(B = empty_set)) | (A = empty_set))) | (quasi_total(C, A, B) <=> (A = relation_dom_as_subset(A, B, C))))) | (~((quasi_total(C, A, B) <=> (C = empty_set)) | (A = empty_set) | (~(B = empty_set)))))))) | ((~relation_of2_as_subset(D!160, A!163, B!162)) | (~((~((~((~(B!162 = empty_set)) | (A!163 = empty_set))) | (quasi_total(D!160, A!163, B!162) <=> (A!163 = relation_dom_as_subset(A!163, B!162, D!160))))) | (~((quasi_total(D!160, A!163, B!162) <=> (D!160 = empty_set)) | (A!163 = empty_set) | (~(B!162 = empty_set)))))))) <=> ((~![A: $i, B: $i, C: $i] : ((~relation_of2_as_subset(C, A, B)) | (~((~((~((~(B = empty_set)) | (A = empty_set))) | (quasi_total(C, A, B) <=> (A = relation_dom_as_subset(A, B, C))))) | (~((quasi_total(C, A, B) <=> (C = empty_set)) | (A = empty_set) | (~(B = empty_set)))))))) | ((~relation_of2_as_subset(D!160, A!163, B!162)) | (~((~((~((~(B!162 = empty_set)) | (A!163 = empty_set))) | (quasi_total(D!160, A!163, B!162) <=> (A!163 = relation_dom_as_subset(A!163, B!162, D!160))))) | (~((~(B!162 = empty_set)) | (A!163 = empty_set) | (quasi_total(D!160, A!163, B!162) <=> (D!160 = empty_set))))))))),
% 29.45/29.63 inference(monotonicity,[status(thm)],[28])).
% 29.45/29.63 tff(30,plain,
% 29.45/29.63 (((~![A: $i, B: $i, C: $i] : ((~relation_of2_as_subset(C, A, B)) | (~((~((~((~(B = empty_set)) | (A = empty_set))) | (quasi_total(C, A, B) <=> (A = relation_dom_as_subset(A, B, C))))) | (~((quasi_total(C, A, B) <=> (C = empty_set)) | (A = empty_set) | (~(B = empty_set)))))))) | ((~relation_of2_as_subset(D!160, A!163, B!162)) | (~((~((~((~(B!162 = empty_set)) | (A!163 = empty_set))) | (quasi_total(D!160, A!163, B!162) <=> (A!163 = relation_dom_as_subset(A!163, B!162, D!160))))) | (~((quasi_total(D!160, A!163, B!162) <=> (D!160 = empty_set)) | (A!163 = empty_set) | (~(B!162 = empty_set)))))))) <=> ((~![A: $i, B: $i, C: $i] : ((~relation_of2_as_subset(C, A, B)) | (~((~((~((~(B = empty_set)) | (A = empty_set))) | (quasi_total(C, A, B) <=> (A = relation_dom_as_subset(A, B, C))))) | (~((quasi_total(C, A, B) <=> (C = empty_set)) | (A = empty_set) | (~(B = empty_set)))))))) | (~relation_of2_as_subset(D!160, A!163, B!162)) | (~((~((~((~(B!162 = empty_set)) | (A!163 = empty_set))) | (quasi_total(D!160, A!163, B!162) <=> (A!163 = relation_dom_as_subset(A!163, B!162, D!160))))) | (~((~(B!162 = empty_set)) | (A!163 = empty_set) | (quasi_total(D!160, A!163, B!162) <=> (D!160 = empty_set)))))))),
% 29.45/29.63 inference(transitivity,[status(thm)],[29, 27])).
% 29.45/29.63 tff(31,plain,
% 29.45/29.63 ((~![A: $i, B: $i, C: $i] : ((~relation_of2_as_subset(C, A, B)) | (~((~((~((~(B = empty_set)) | (A = empty_set))) | (quasi_total(C, A, B) <=> (A = relation_dom_as_subset(A, B, C))))) | (~((quasi_total(C, A, B) <=> (C = empty_set)) | (A = empty_set) | (~(B = empty_set)))))))) | ((~relation_of2_as_subset(D!160, A!163, B!162)) | (~((~((~((~(B!162 = empty_set)) | (A!163 = empty_set))) | (quasi_total(D!160, A!163, B!162) <=> (A!163 = relation_dom_as_subset(A!163, B!162, D!160))))) | (~((quasi_total(D!160, A!163, B!162) <=> (D!160 = empty_set)) | (A!163 = empty_set) | (~(B!162 = empty_set)))))))),
% 29.45/29.63 inference(quant_inst,[status(thm)],[])).
% 29.45/29.63 tff(32,plain,
% 29.45/29.63 ((~![A: $i, B: $i, C: $i] : ((~relation_of2_as_subset(C, A, B)) | (~((~((~((~(B = empty_set)) | (A = empty_set))) | (quasi_total(C, A, B) <=> (A = relation_dom_as_subset(A, B, C))))) | (~((quasi_total(C, A, B) <=> (C = empty_set)) | (A = empty_set) | (~(B = empty_set)))))))) | (~relation_of2_as_subset(D!160, A!163, B!162)) | (~((~((~((~(B!162 = empty_set)) | (A!163 = empty_set))) | (quasi_total(D!160, A!163, B!162) <=> (A!163 = relation_dom_as_subset(A!163, B!162, D!160))))) | (~((~(B!162 = empty_set)) | (A!163 = empty_set) | (quasi_total(D!160, A!163, B!162) <=> (D!160 = empty_set))))))),
% 29.45/29.63 inference(modus_ponens,[status(thm)],[31, 30])).
% 29.45/29.63 tff(33,plain,
% 29.45/29.63 (~((~((~((~(B!162 = empty_set)) | (A!163 = empty_set))) | (quasi_total(D!160, A!163, B!162) <=> (A!163 = relation_dom_as_subset(A!163, B!162, D!160))))) | (~((~(B!162 = empty_set)) | (A!163 = empty_set) | (quasi_total(D!160, A!163, B!162) <=> (D!160 = empty_set)))))),
% 29.45/29.63 inference(unit_resolution,[status(thm)],[32, 26, 13])).
% 29.45/29.63 tff(34,plain,
% 29.45/29.63 (((~((~((~(B!162 = empty_set)) | (A!163 = empty_set))) | (quasi_total(D!160, A!163, B!162) <=> (A!163 = relation_dom_as_subset(A!163, B!162, D!160))))) | (~((~(B!162 = empty_set)) | (A!163 = empty_set) | (quasi_total(D!160, A!163, B!162) <=> (D!160 = empty_set))))) | ((~((~(B!162 = empty_set)) | (A!163 = empty_set))) | (quasi_total(D!160, A!163, B!162) <=> (A!163 = relation_dom_as_subset(A!163, B!162, D!160))))),
% 29.45/29.63 inference(tautology,[status(thm)],[])).
% 29.45/29.63 tff(35,plain,
% 29.45/29.63 ((~((~(B!162 = empty_set)) | (A!163 = empty_set))) | (quasi_total(D!160, A!163, B!162) <=> (A!163 = relation_dom_as_subset(A!163, B!162, D!160)))),
% 29.45/29.63 inference(unit_resolution,[status(thm)],[34, 33])).
% 29.45/29.63 tff(36,plain,
% 29.45/29.63 (~((B!162 = empty_set) | (~in(C!161, A!163)) | (apply(relation_composition(D!160, E!164), C!161) = apply(E!164, apply(D!160, C!161))) | (~(relation(E!164) & function(E!164))))),
% 29.45/29.63 inference(and_elim,[status(thm)],[12])).
% 29.45/29.63 tff(37,plain,
% 29.45/29.63 (~(B!162 = empty_set)),
% 29.45/29.63 inference(or_elim,[status(thm)],[36])).
% 29.45/29.63 tff(38,plain,
% 29.45/29.63 (((~(B!162 = empty_set)) | (A!163 = empty_set)) | (B!162 = empty_set)),
% 29.45/29.63 inference(tautology,[status(thm)],[])).
% 29.45/29.63 tff(39,plain,
% 29.45/29.63 ((~(B!162 = empty_set)) | (A!163 = empty_set)),
% 29.45/29.63 inference(unit_resolution,[status(thm)],[38, 37])).
% 29.45/29.63 tff(40,plain,
% 29.45/29.63 ((~((~((~(B!162 = empty_set)) | (A!163 = empty_set))) | (quasi_total(D!160, A!163, B!162) <=> (A!163 = relation_dom_as_subset(A!163, B!162, D!160))))) | (~((~(B!162 = empty_set)) | (A!163 = empty_set))) | (quasi_total(D!160, A!163, B!162) <=> (A!163 = relation_dom_as_subset(A!163, B!162, D!160)))),
% 29.45/29.63 inference(tautology,[status(thm)],[])).
% 29.45/29.63 tff(41,plain,
% 29.45/29.63 ((~((~((~(B!162 = empty_set)) | (A!163 = empty_set))) | (quasi_total(D!160, A!163, B!162) <=> (A!163 = relation_dom_as_subset(A!163, B!162, D!160))))) | (quasi_total(D!160, A!163, B!162) <=> (A!163 = relation_dom_as_subset(A!163, B!162, D!160)))),
% 29.45/29.63 inference(unit_resolution,[status(thm)],[40, 39])).
% 29.45/29.63 tff(42,plain,
% 29.45/29.63 (quasi_total(D!160, A!163, B!162) <=> (A!163 = relation_dom_as_subset(A!163, B!162, D!160))),
% 29.45/29.63 inference(unit_resolution,[status(thm)],[41, 35])).
% 29.45/29.63 tff(43,plain,
% 29.45/29.63 (quasi_total(D!160, A!163, B!162)),
% 29.45/29.63 inference(and_elim,[status(thm)],[12])).
% 29.45/29.63 tff(44,plain,
% 29.45/29.63 ((~(quasi_total(D!160, A!163, B!162) <=> (A!163 = relation_dom_as_subset(A!163, B!162, D!160)))) | (~quasi_total(D!160, A!163, B!162)) | (A!163 = relation_dom_as_subset(A!163, B!162, D!160))),
% 29.45/29.63 inference(tautology,[status(thm)],[])).
% 29.45/29.63 tff(45,plain,
% 29.45/29.63 ((~(quasi_total(D!160, A!163, B!162) <=> (A!163 = relation_dom_as_subset(A!163, B!162, D!160)))) | (A!163 = relation_dom_as_subset(A!163, B!162, D!160))),
% 29.45/29.63 inference(unit_resolution,[status(thm)],[44, 43])).
% 29.45/29.63 tff(46,plain,
% 29.45/29.63 (A!163 = relation_dom_as_subset(A!163, B!162, D!160)),
% 29.45/29.63 inference(unit_resolution,[status(thm)],[45, 42])).
% 29.45/29.63 tff(47,plain,
% 29.45/29.63 (relation_dom_as_subset(A!163, B!162, D!160) = A!163),
% 29.45/29.63 inference(symmetry,[status(thm)],[46])).
% 29.45/29.63 tff(48,plain,
% 29.45/29.63 (^[A: $i, B: $i, C: $i] : refl((relation_of2_as_subset(C, A, B) <=> relation_of2(C, A, B)) <=> (relation_of2_as_subset(C, A, B) <=> relation_of2(C, A, B)))),
% 29.45/29.63 inference(bind,[status(th)],[])).
% 29.45/29.63 tff(49,plain,
% 29.45/29.63 (![A: $i, B: $i, C: $i] : (relation_of2_as_subset(C, A, B) <=> relation_of2(C, A, B)) <=> ![A: $i, B: $i, C: $i] : (relation_of2_as_subset(C, A, B) <=> relation_of2(C, A, B))),
% 29.45/29.63 inference(quant_intro,[status(thm)],[48])).
% 29.45/29.63 tff(50,plain,
% 29.45/29.63 (![A: $i, B: $i, C: $i] : (relation_of2_as_subset(C, A, B) <=> relation_of2(C, A, B)) <=> ![A: $i, B: $i, C: $i] : (relation_of2_as_subset(C, A, B) <=> relation_of2(C, A, B))),
% 29.45/29.63 inference(rewrite,[status(thm)],[])).
% 29.45/29.63 tff(51,axiom,(![A: $i, B: $i, C: $i] : (relation_of2_as_subset(C, A, B) <=> relation_of2(C, A, B))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','redefinition_m2_relset_1')).
% 29.45/29.63 tff(52,plain,
% 29.45/29.63 (![A: $i, B: $i, C: $i] : (relation_of2_as_subset(C, A, B) <=> relation_of2(C, A, B))),
% 29.45/29.63 inference(modus_ponens,[status(thm)],[51, 50])).
% 29.45/29.63 tff(53,plain,(
% 29.45/29.63 ![A: $i, B: $i, C: $i] : (relation_of2_as_subset(C, A, B) <=> relation_of2(C, A, B))),
% 29.45/29.63 inference(skolemize,[status(sab)],[52])).
% 29.45/29.63 tff(54,plain,
% 29.45/29.63 (![A: $i, B: $i, C: $i] : (relation_of2_as_subset(C, A, B) <=> relation_of2(C, A, B))),
% 29.45/29.63 inference(modus_ponens,[status(thm)],[53, 49])).
% 29.45/29.63 tff(55,plain,
% 29.45/29.63 ((~![A: $i, B: $i, C: $i] : (relation_of2_as_subset(C, A, B) <=> relation_of2(C, A, B))) | (relation_of2_as_subset(D!160, A!163, B!162) <=> relation_of2(D!160, A!163, B!162))),
% 29.45/29.63 inference(quant_inst,[status(thm)],[])).
% 29.45/29.63 tff(56,plain,
% 29.45/29.63 (relation_of2_as_subset(D!160, A!163, B!162) <=> relation_of2(D!160, A!163, B!162)),
% 29.45/29.63 inference(unit_resolution,[status(thm)],[55, 54])).
% 29.45/29.63 tff(57,plain,
% 29.45/29.63 ((~(relation_of2_as_subset(D!160, A!163, B!162) <=> relation_of2(D!160, A!163, B!162))) | (~relation_of2_as_subset(D!160, A!163, B!162)) | relation_of2(D!160, A!163, B!162)),
% 29.45/29.63 inference(tautology,[status(thm)],[])).
% 29.45/29.63 tff(58,plain,
% 29.45/29.63 ((~(relation_of2_as_subset(D!160, A!163, B!162) <=> relation_of2(D!160, A!163, B!162))) | relation_of2(D!160, A!163, B!162)),
% 29.45/29.63 inference(unit_resolution,[status(thm)],[57, 13])).
% 29.45/29.63 tff(59,plain,
% 29.45/29.63 (relation_of2(D!160, A!163, B!162)),
% 29.45/29.63 inference(unit_resolution,[status(thm)],[58, 56])).
% 29.45/29.63 tff(60,plain,
% 29.45/29.63 (^[A: $i, B: $i, C: $i] : refl(((~relation_of2(C, A, B)) | (relation_dom_as_subset(A, B, C) = relation_dom(C))) <=> ((~relation_of2(C, A, B)) | (relation_dom_as_subset(A, B, C) = relation_dom(C))))),
% 29.45/29.63 inference(bind,[status(th)],[])).
% 29.45/29.63 tff(61,plain,
% 29.45/29.63 (![A: $i, B: $i, C: $i] : ((~relation_of2(C, A, B)) | (relation_dom_as_subset(A, B, C) = relation_dom(C))) <=> ![A: $i, B: $i, C: $i] : ((~relation_of2(C, A, B)) | (relation_dom_as_subset(A, B, C) = relation_dom(C)))),
% 29.45/29.63 inference(quant_intro,[status(thm)],[60])).
% 29.45/29.63 tff(62,plain,
% 29.45/29.63 (![A: $i, B: $i, C: $i] : ((~relation_of2(C, A, B)) | (relation_dom_as_subset(A, B, C) = relation_dom(C))) <=> ![A: $i, B: $i, C: $i] : ((~relation_of2(C, A, B)) | (relation_dom_as_subset(A, B, C) = relation_dom(C)))),
% 29.45/29.63 inference(rewrite,[status(thm)],[])).
% 29.45/29.63 tff(63,plain,
% 29.45/29.63 (^[A: $i, B: $i, C: $i] : rewrite((relation_of2(C, A, B) => (relation_dom_as_subset(A, B, C) = relation_dom(C))) <=> ((~relation_of2(C, A, B)) | (relation_dom_as_subset(A, B, C) = relation_dom(C))))),
% 29.45/29.63 inference(bind,[status(th)],[])).
% 29.45/29.63 tff(64,plain,
% 29.45/29.63 (![A: $i, B: $i, C: $i] : (relation_of2(C, A, B) => (relation_dom_as_subset(A, B, C) = relation_dom(C))) <=> ![A: $i, B: $i, C: $i] : ((~relation_of2(C, A, B)) | (relation_dom_as_subset(A, B, C) = relation_dom(C)))),
% 29.45/29.63 inference(quant_intro,[status(thm)],[63])).
% 29.45/29.63 tff(65,axiom,(![A: $i, B: $i, C: $i] : (relation_of2(C, A, B) => (relation_dom_as_subset(A, B, C) = relation_dom(C)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','redefinition_k4_relset_1')).
% 29.45/29.63 tff(66,plain,
% 29.45/29.63 (![A: $i, B: $i, C: $i] : ((~relation_of2(C, A, B)) | (relation_dom_as_subset(A, B, C) = relation_dom(C)))),
% 29.45/29.63 inference(modus_ponens,[status(thm)],[65, 64])).
% 29.45/29.63 tff(67,plain,
% 29.45/29.63 (![A: $i, B: $i, C: $i] : ((~relation_of2(C, A, B)) | (relation_dom_as_subset(A, B, C) = relation_dom(C)))),
% 29.45/29.63 inference(modus_ponens,[status(thm)],[66, 62])).
% 29.45/29.63 tff(68,plain,(
% 29.45/29.63 ![A: $i, B: $i, C: $i] : ((~relation_of2(C, A, B)) | (relation_dom_as_subset(A, B, C) = relation_dom(C)))),
% 29.45/29.63 inference(skolemize,[status(sab)],[67])).
% 29.45/29.63 tff(69,plain,
% 29.45/29.63 (![A: $i, B: $i, C: $i] : ((~relation_of2(C, A, B)) | (relation_dom_as_subset(A, B, C) = relation_dom(C)))),
% 29.45/29.63 inference(modus_ponens,[status(thm)],[68, 61])).
% 29.45/29.63 tff(70,plain,
% 29.45/29.63 (((~![A: $i, B: $i, C: $i] : ((~relation_of2(C, A, B)) | (relation_dom_as_subset(A, B, C) = relation_dom(C)))) | ((~relation_of2(D!160, A!163, B!162)) | (relation_dom_as_subset(A!163, B!162, D!160) = relation_dom(D!160)))) <=> ((~![A: $i, B: $i, C: $i] : ((~relation_of2(C, A, B)) | (relation_dom_as_subset(A, B, C) = relation_dom(C)))) | (~relation_of2(D!160, A!163, B!162)) | (relation_dom_as_subset(A!163, B!162, D!160) = relation_dom(D!160)))),
% 29.45/29.63 inference(rewrite,[status(thm)],[])).
% 29.45/29.63 tff(71,plain,
% 29.45/29.63 ((~![A: $i, B: $i, C: $i] : ((~relation_of2(C, A, B)) | (relation_dom_as_subset(A, B, C) = relation_dom(C)))) | ((~relation_of2(D!160, A!163, B!162)) | (relation_dom_as_subset(A!163, B!162, D!160) = relation_dom(D!160)))),
% 29.47/29.63 inference(quant_inst,[status(thm)],[])).
% 29.47/29.63 tff(72,plain,
% 29.47/29.63 ((~![A: $i, B: $i, C: $i] : ((~relation_of2(C, A, B)) | (relation_dom_as_subset(A, B, C) = relation_dom(C)))) | (~relation_of2(D!160, A!163, B!162)) | (relation_dom_as_subset(A!163, B!162, D!160) = relation_dom(D!160))),
% 29.47/29.63 inference(modus_ponens,[status(thm)],[71, 70])).
% 29.47/29.63 tff(73,plain,
% 29.47/29.63 (relation_dom_as_subset(A!163, B!162, D!160) = relation_dom(D!160)),
% 29.47/29.63 inference(unit_resolution,[status(thm)],[72, 69, 59])).
% 29.47/29.63 tff(74,plain,
% 29.47/29.63 (relation_dom(D!160) = relation_dom_as_subset(A!163, B!162, D!160)),
% 29.47/29.63 inference(symmetry,[status(thm)],[73])).
% 29.47/29.63 tff(75,plain,
% 29.47/29.63 (relation_dom(D!160) = A!163),
% 29.47/29.63 inference(transitivity,[status(thm)],[74, 47])).
% 29.47/29.63 tff(76,plain,
% 29.47/29.63 (in(C!161, relation_dom(D!160)) <=> in(C!161, A!163)),
% 29.47/29.63 inference(monotonicity,[status(thm)],[75])).
% 29.47/29.63 tff(77,plain,
% 29.47/29.63 (in(C!161, A!163) <=> in(C!161, relation_dom(D!160))),
% 29.47/29.63 inference(symmetry,[status(thm)],[76])).
% 29.47/29.63 tff(78,plain,
% 29.47/29.63 (in(C!161, A!163)),
% 29.47/29.63 inference(or_elim,[status(thm)],[36])).
% 29.47/29.63 tff(79,plain,
% 29.47/29.63 (in(C!161, relation_dom(D!160))),
% 29.47/29.63 inference(modus_ponens,[status(thm)],[78, 77])).
% 29.47/29.63 tff(80,plain,
% 29.47/29.63 (^[A: $i, B: $i, C: $i] : refl(((~relation_of2_as_subset(C, A, B)) | element(C, powerset(cartesian_product2(A, B)))) <=> ((~relation_of2_as_subset(C, A, B)) | element(C, powerset(cartesian_product2(A, B)))))),
% 29.47/29.63 inference(bind,[status(th)],[])).
% 29.47/29.63 tff(81,plain,
% 29.47/29.63 (![A: $i, B: $i, C: $i] : ((~relation_of2_as_subset(C, A, B)) | element(C, powerset(cartesian_product2(A, B)))) <=> ![A: $i, B: $i, C: $i] : ((~relation_of2_as_subset(C, A, B)) | element(C, powerset(cartesian_product2(A, B))))),
% 29.47/29.63 inference(quant_intro,[status(thm)],[80])).
% 29.47/29.63 tff(82,plain,
% 29.47/29.63 (![A: $i, B: $i, C: $i] : ((~relation_of2_as_subset(C, A, B)) | element(C, powerset(cartesian_product2(A, B)))) <=> ![A: $i, B: $i, C: $i] : ((~relation_of2_as_subset(C, A, B)) | element(C, powerset(cartesian_product2(A, B))))),
% 29.47/29.63 inference(rewrite,[status(thm)],[])).
% 29.47/29.63 tff(83,plain,
% 29.47/29.63 (^[A: $i, B: $i, C: $i] : rewrite((relation_of2_as_subset(C, A, B) => element(C, powerset(cartesian_product2(A, B)))) <=> ((~relation_of2_as_subset(C, A, B)) | element(C, powerset(cartesian_product2(A, B)))))),
% 29.47/29.63 inference(bind,[status(th)],[])).
% 29.47/29.63 tff(84,plain,
% 29.47/29.63 (![A: $i, B: $i, C: $i] : (relation_of2_as_subset(C, A, B) => element(C, powerset(cartesian_product2(A, B)))) <=> ![A: $i, B: $i, C: $i] : ((~relation_of2_as_subset(C, A, B)) | element(C, powerset(cartesian_product2(A, B))))),
% 29.47/29.63 inference(quant_intro,[status(thm)],[83])).
% 29.47/29.63 tff(85,axiom,(![A: $i, B: $i, C: $i] : (relation_of2_as_subset(C, A, B) => element(C, powerset(cartesian_product2(A, B))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','dt_m2_relset_1')).
% 29.47/29.63 tff(86,plain,
% 29.47/29.63 (![A: $i, B: $i, C: $i] : ((~relation_of2_as_subset(C, A, B)) | element(C, powerset(cartesian_product2(A, B))))),
% 29.47/29.63 inference(modus_ponens,[status(thm)],[85, 84])).
% 29.47/29.63 tff(87,plain,
% 29.47/29.63 (![A: $i, B: $i, C: $i] : ((~relation_of2_as_subset(C, A, B)) | element(C, powerset(cartesian_product2(A, B))))),
% 29.47/29.63 inference(modus_ponens,[status(thm)],[86, 82])).
% 29.47/29.63 tff(88,plain,(
% 29.47/29.63 ![A: $i, B: $i, C: $i] : ((~relation_of2_as_subset(C, A, B)) | element(C, powerset(cartesian_product2(A, B))))),
% 29.47/29.63 inference(skolemize,[status(sab)],[87])).
% 29.47/29.63 tff(89,plain,
% 29.47/29.63 (![A: $i, B: $i, C: $i] : ((~relation_of2_as_subset(C, A, B)) | element(C, powerset(cartesian_product2(A, B))))),
% 29.47/29.63 inference(modus_ponens,[status(thm)],[88, 81])).
% 29.47/29.63 tff(90,plain,
% 29.47/29.63 (((~![A: $i, B: $i, C: $i] : ((~relation_of2_as_subset(C, A, B)) | element(C, powerset(cartesian_product2(A, B))))) | ((~relation_of2_as_subset(D!160, A!163, B!162)) | element(D!160, powerset(cartesian_product2(A!163, B!162))))) <=> ((~![A: $i, B: $i, C: $i] : ((~relation_of2_as_subset(C, A, B)) | element(C, powerset(cartesian_product2(A, B))))) | (~relation_of2_as_subset(D!160, A!163, B!162)) | element(D!160, powerset(cartesian_product2(A!163, B!162))))),
% 29.47/29.63 inference(rewrite,[status(thm)],[])).
% 29.47/29.63 tff(91,plain,
% 29.47/29.63 ((~![A: $i, B: $i, C: $i] : ((~relation_of2_as_subset(C, A, B)) | element(C, powerset(cartesian_product2(A, B))))) | ((~relation_of2_as_subset(D!160, A!163, B!162)) | element(D!160, powerset(cartesian_product2(A!163, B!162))))),
% 29.47/29.63 inference(quant_inst,[status(thm)],[])).
% 29.47/29.63 tff(92,plain,
% 29.47/29.63 ((~![A: $i, B: $i, C: $i] : ((~relation_of2_as_subset(C, A, B)) | element(C, powerset(cartesian_product2(A, B))))) | (~relation_of2_as_subset(D!160, A!163, B!162)) | element(D!160, powerset(cartesian_product2(A!163, B!162)))),
% 29.47/29.63 inference(modus_ponens,[status(thm)],[91, 90])).
% 29.47/29.63 tff(93,plain,
% 29.47/29.63 (element(D!160, powerset(cartesian_product2(A!163, B!162)))),
% 29.47/29.63 inference(unit_resolution,[status(thm)],[92, 89, 13])).
% 29.47/29.63 tff(94,plain,
% 29.47/29.63 (^[A: $i, B: $i, C: $i] : refl((relation(C) | (~element(C, powerset(cartesian_product2(A, B))))) <=> (relation(C) | (~element(C, powerset(cartesian_product2(A, B))))))),
% 29.47/29.63 inference(bind,[status(th)],[])).
% 29.47/29.63 tff(95,plain,
% 29.47/29.63 (![A: $i, B: $i, C: $i] : (relation(C) | (~element(C, powerset(cartesian_product2(A, B))))) <=> ![A: $i, B: $i, C: $i] : (relation(C) | (~element(C, powerset(cartesian_product2(A, B)))))),
% 29.47/29.63 inference(quant_intro,[status(thm)],[94])).
% 29.47/29.63 tff(96,plain,
% 29.47/29.63 (![A: $i, B: $i, C: $i] : (relation(C) | (~element(C, powerset(cartesian_product2(A, B))))) <=> ![A: $i, B: $i, C: $i] : (relation(C) | (~element(C, powerset(cartesian_product2(A, B)))))),
% 29.47/29.63 inference(rewrite,[status(thm)],[])).
% 29.47/29.63 tff(97,plain,
% 29.47/29.63 (^[A: $i, B: $i, C: $i] : rewrite((element(C, powerset(cartesian_product2(A, B))) => relation(C)) <=> (relation(C) | (~element(C, powerset(cartesian_product2(A, B))))))),
% 29.47/29.63 inference(bind,[status(th)],[])).
% 29.47/29.63 tff(98,plain,
% 29.47/29.63 (![A: $i, B: $i, C: $i] : (element(C, powerset(cartesian_product2(A, B))) => relation(C)) <=> ![A: $i, B: $i, C: $i] : (relation(C) | (~element(C, powerset(cartesian_product2(A, B)))))),
% 29.47/29.63 inference(quant_intro,[status(thm)],[97])).
% 29.47/29.63 tff(99,axiom,(![A: $i, B: $i, C: $i] : (element(C, powerset(cartesian_product2(A, B))) => relation(C))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','cc1_relset_1')).
% 29.47/29.63 tff(100,plain,
% 29.47/29.63 (![A: $i, B: $i, C: $i] : (relation(C) | (~element(C, powerset(cartesian_product2(A, B)))))),
% 29.47/29.63 inference(modus_ponens,[status(thm)],[99, 98])).
% 29.47/29.63 tff(101,plain,
% 29.47/29.63 (![A: $i, B: $i, C: $i] : (relation(C) | (~element(C, powerset(cartesian_product2(A, B)))))),
% 29.47/29.63 inference(modus_ponens,[status(thm)],[100, 96])).
% 29.47/29.63 tff(102,plain,(
% 29.47/29.63 ![A: $i, B: $i, C: $i] : (relation(C) | (~element(C, powerset(cartesian_product2(A, B)))))),
% 29.47/29.63 inference(skolemize,[status(sab)],[101])).
% 29.47/29.63 tff(103,plain,
% 29.47/29.63 (![A: $i, B: $i, C: $i] : (relation(C) | (~element(C, powerset(cartesian_product2(A, B)))))),
% 29.47/29.63 inference(modus_ponens,[status(thm)],[102, 95])).
% 29.47/29.63 tff(104,plain,
% 29.47/29.63 (((~![A: $i, B: $i, C: $i] : (relation(C) | (~element(C, powerset(cartesian_product2(A, B)))))) | (relation(D!160) | (~element(D!160, powerset(cartesian_product2(A!163, B!162)))))) <=> ((~![A: $i, B: $i, C: $i] : (relation(C) | (~element(C, powerset(cartesian_product2(A, B)))))) | relation(D!160) | (~element(D!160, powerset(cartesian_product2(A!163, B!162)))))),
% 29.47/29.63 inference(rewrite,[status(thm)],[])).
% 29.47/29.63 tff(105,plain,
% 29.47/29.63 ((~![A: $i, B: $i, C: $i] : (relation(C) | (~element(C, powerset(cartesian_product2(A, B)))))) | (relation(D!160) | (~element(D!160, powerset(cartesian_product2(A!163, B!162)))))),
% 29.47/29.63 inference(quant_inst,[status(thm)],[])).
% 29.47/29.63 tff(106,plain,
% 29.47/29.63 ((~![A: $i, B: $i, C: $i] : (relation(C) | (~element(C, powerset(cartesian_product2(A, B)))))) | relation(D!160) | (~element(D!160, powerset(cartesian_product2(A!163, B!162))))),
% 29.47/29.63 inference(modus_ponens,[status(thm)],[105, 104])).
% 29.47/29.63 tff(107,plain,
% 29.47/29.63 (relation(D!160)),
% 29.47/29.63 inference(unit_resolution,[status(thm)],[106, 103, 93])).
% 29.47/29.63 tff(108,plain,
% 29.47/29.63 (^[A: $i, B: $i] : refl(((~relation(B)) | (~function(B)) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~relation(C)) | (~in(A, relation_dom(B))) | (~function(C)))) <=> ((~relation(B)) | (~function(B)) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~relation(C)) | (~in(A, relation_dom(B))) | (~function(C)))))),
% 29.47/29.63 inference(bind,[status(th)],[])).
% 29.47/29.63 tff(109,plain,
% 29.47/29.63 (![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~relation(C)) | (~in(A, relation_dom(B))) | (~function(C)))) <=> ![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~relation(C)) | (~in(A, relation_dom(B))) | (~function(C))))),
% 29.47/29.63 inference(quant_intro,[status(thm)],[108])).
% 29.47/29.63 tff(110,plain,
% 29.47/29.63 (^[A: $i, B: $i] : rewrite(((~relation(B)) | (~function(B)) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~relation(C)) | (~in(A, relation_dom(B))) | (~function(C)))) <=> ((~relation(B)) | (~function(B)) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~relation(C)) | (~in(A, relation_dom(B))) | (~function(C)))))),
% 29.47/29.63 inference(bind,[status(th)],[])).
% 29.47/29.63 tff(111,plain,
% 29.47/29.63 (![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~relation(C)) | (~in(A, relation_dom(B))) | (~function(C)))) <=> ![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~relation(C)) | (~in(A, relation_dom(B))) | (~function(C))))),
% 29.47/29.63 inference(quant_intro,[status(thm)],[110])).
% 29.47/29.63 tff(112,plain,
% 29.47/29.63 (![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~relation(C)) | (~in(A, relation_dom(B))) | (~function(C)))) <=> ![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~relation(C)) | (~in(A, relation_dom(B))) | (~function(C))))),
% 29.47/29.63 inference(transitivity,[status(thm)],[111, 109])).
% 29.47/29.63 tff(113,plain,
% 29.47/29.63 (^[A: $i, B: $i] : trans(monotonicity(trans(monotonicity(rewrite((relation(B) & function(B)) <=> (~((~relation(B)) | (~function(B))))), ((~(relation(B) & function(B))) <=> (~(~((~relation(B)) | (~function(B))))))), rewrite((~(~((~relation(B)) | (~function(B))))) <=> ((~relation(B)) | (~function(B)))), ((~(relation(B) & function(B))) <=> ((~relation(B)) | (~function(B))))), quant_intro(proof_bind(^[C: $i] : trans(monotonicity(trans(monotonicity(rewrite((relation(C) & function(C)) <=> (~((~relation(C)) | (~function(C))))), ((~(relation(C) & function(C))) <=> (~(~((~relation(C)) | (~function(C))))))), rewrite((~(~((~relation(C)) | (~function(C))))) <=> ((~relation(C)) | (~function(C)))), ((~(relation(C) & function(C))) <=> ((~relation(C)) | (~function(C))))), (((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~(relation(C) & function(C))) | (~in(A, relation_dom(B)))) <=> ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | ((~relation(C)) | (~function(C))) | (~in(A, relation_dom(B)))))), rewrite(((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | ((~relation(C)) | (~function(C))) | (~in(A, relation_dom(B)))) <=> ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~relation(C)) | (~in(A, relation_dom(B))) | (~function(C)))), (((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~(relation(C) & function(C))) | (~in(A, relation_dom(B)))) <=> ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~relation(C)) | (~in(A, relation_dom(B))) | (~function(C)))))), (![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~(relation(C) & function(C))) | (~in(A, relation_dom(B)))) <=> ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~relation(C)) | (~in(A, relation_dom(B))) | (~function(C))))), (((~(relation(B) & function(B))) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~(relation(C) & function(C))) | (~in(A, relation_dom(B))))) <=> (((~relation(B)) | (~function(B))) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~relation(C)) | (~in(A, relation_dom(B))) | (~function(C)))))), rewrite((((~relation(B)) | (~function(B))) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~relation(C)) | (~in(A, relation_dom(B))) | (~function(C)))) <=> ((~relation(B)) | (~function(B)) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~relation(C)) | (~in(A, relation_dom(B))) | (~function(C))))), (((~(relation(B) & function(B))) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~(relation(C) & function(C))) | (~in(A, relation_dom(B))))) <=> ((~relation(B)) | (~function(B)) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~relation(C)) | (~in(A, relation_dom(B))) | (~function(C))))))),
% 29.47/29.63 inference(bind,[status(th)],[])).
% 29.47/29.63 tff(114,plain,
% 29.47/29.63 (![A: $i, B: $i] : ((~(relation(B) & function(B))) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~(relation(C) & function(C))) | (~in(A, relation_dom(B))))) <=> ![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~relation(C)) | (~in(A, relation_dom(B))) | (~function(C))))),
% 29.47/29.63 inference(quant_intro,[status(thm)],[113])).
% 29.47/29.63 tff(115,plain,
% 29.47/29.63 (![A: $i, B: $i] : ((~(relation(B) & function(B))) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~(relation(C) & function(C))) | (~in(A, relation_dom(B))))) <=> ![A: $i, B: $i] : ((~(relation(B) & function(B))) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~(relation(C) & function(C))) | (~in(A, relation_dom(B)))))),
% 29.47/29.63 inference(rewrite,[status(thm)],[])).
% 29.47/29.63 tff(116,plain,
% 29.47/29.63 (^[A: $i, B: $i] : trans(monotonicity(quant_intro(proof_bind(^[C: $i] : trans(monotonicity(rewrite((in(A, relation_dom(B)) => (apply(relation_composition(B, C), A) = apply(C, apply(B, A)))) <=> ((~in(A, relation_dom(B))) | (apply(relation_composition(B, C), A) = apply(C, apply(B, A))))), (((relation(C) & function(C)) => (in(A, relation_dom(B)) => (apply(relation_composition(B, C), A) = apply(C, apply(B, A))))) <=> ((relation(C) & function(C)) => ((~in(A, relation_dom(B))) | (apply(relation_composition(B, C), A) = apply(C, apply(B, A))))))), rewrite(((relation(C) & function(C)) => ((~in(A, relation_dom(B))) | (apply(relation_composition(B, C), A) = apply(C, apply(B, A))))) <=> ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~(relation(C) & function(C))) | (~in(A, relation_dom(B))))), (((relation(C) & function(C)) => (in(A, relation_dom(B)) => (apply(relation_composition(B, C), A) = apply(C, apply(B, A))))) <=> ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~(relation(C) & function(C))) | (~in(A, relation_dom(B))))))), (![C: $i] : ((relation(C) & function(C)) => (in(A, relation_dom(B)) => (apply(relation_composition(B, C), A) = apply(C, apply(B, A))))) <=> ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~(relation(C) & function(C))) | (~in(A, relation_dom(B)))))), (((relation(B) & function(B)) => ![C: $i] : ((relation(C) & function(C)) => (in(A, relation_dom(B)) => (apply(relation_composition(B, C), A) = apply(C, apply(B, A)))))) <=> ((relation(B) & function(B)) => ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~(relation(C) & function(C))) | (~in(A, relation_dom(B))))))), rewrite(((relation(B) & function(B)) => ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~(relation(C) & function(C))) | (~in(A, relation_dom(B))))) <=> ((~(relation(B) & function(B))) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~(relation(C) & function(C))) | (~in(A, relation_dom(B)))))), (((relation(B) & function(B)) => ![C: $i] : ((relation(C) & function(C)) => (in(A, relation_dom(B)) => (apply(relation_composition(B, C), A) = apply(C, apply(B, A)))))) <=> ((~(relation(B) & function(B))) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~(relation(C) & function(C))) | (~in(A, relation_dom(B)))))))),
% 29.47/29.63 inference(bind,[status(th)],[])).
% 29.47/29.63 tff(117,plain,
% 29.47/29.63 (![A: $i, B: $i] : ((relation(B) & function(B)) => ![C: $i] : ((relation(C) & function(C)) => (in(A, relation_dom(B)) => (apply(relation_composition(B, C), A) = apply(C, apply(B, A)))))) <=> ![A: $i, B: $i] : ((~(relation(B) & function(B))) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~(relation(C) & function(C))) | (~in(A, relation_dom(B)))))),
% 29.47/29.63 inference(quant_intro,[status(thm)],[116])).
% 29.47/29.63 tff(118,axiom,(![A: $i, B: $i] : ((relation(B) & function(B)) => ![C: $i] : ((relation(C) & function(C)) => (in(A, relation_dom(B)) => (apply(relation_composition(B, C), A) = apply(C, apply(B, A))))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t23_funct_1')).
% 29.47/29.63 tff(119,plain,
% 29.47/29.63 (![A: $i, B: $i] : ((~(relation(B) & function(B))) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~(relation(C) & function(C))) | (~in(A, relation_dom(B)))))),
% 29.47/29.63 inference(modus_ponens,[status(thm)],[118, 117])).
% 29.47/29.63 tff(120,plain,
% 29.47/29.63 (![A: $i, B: $i] : ((~(relation(B) & function(B))) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~(relation(C) & function(C))) | (~in(A, relation_dom(B)))))),
% 29.47/29.63 inference(modus_ponens,[status(thm)],[119, 115])).
% 29.47/29.63 tff(121,plain,(
% 29.47/29.63 ![A: $i, B: $i] : ((~(relation(B) & function(B))) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~(relation(C) & function(C))) | (~in(A, relation_dom(B)))))),
% 29.47/29.63 inference(skolemize,[status(sab)],[120])).
% 29.47/29.63 tff(122,plain,
% 29.47/29.63 (![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~relation(C)) | (~in(A, relation_dom(B))) | (~function(C))))),
% 29.47/29.63 inference(modus_ponens,[status(thm)],[121, 114])).
% 29.47/29.63 tff(123,plain,
% 29.47/29.63 (![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~relation(C)) | (~in(A, relation_dom(B))) | (~function(C))))),
% 29.47/29.63 inference(modus_ponens,[status(thm)],[122, 112])).
% 29.47/29.63 tff(124,plain,
% 29.47/29.63 (function(D!160)),
% 29.47/29.63 inference(and_elim,[status(thm)],[12])).
% 29.47/29.63 tff(125,plain,
% 29.47/29.63 (((~![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~relation(C)) | (~in(A, relation_dom(B))) | (~function(C))))) | ((~relation(D!160)) | (~function(D!160)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (apply(relation_composition(D!160, C), C!161) = apply(C, apply(D!160, C!161))) | (~in(C!161, relation_dom(D!160)))))) <=> ((~![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~relation(C)) | (~in(A, relation_dom(B))) | (~function(C))))) | (~relation(D!160)) | (~function(D!160)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (apply(relation_composition(D!160, C), C!161) = apply(C, apply(D!160, C!161))) | (~in(C!161, relation_dom(D!160)))))),
% 29.47/29.63 inference(rewrite,[status(thm)],[])).
% 29.47/29.63 tff(126,plain,
% 29.47/29.63 (((~relation(D!160)) | (~function(D!160)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (apply(relation_composition(D!160, C), C!161) = apply(C, apply(D!160, C!161))) | (~in(C!161, relation_dom(D!160))))) <=> ((~relation(D!160)) | (~function(D!160)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (apply(relation_composition(D!160, C), C!161) = apply(C, apply(D!160, C!161))) | (~in(C!161, relation_dom(D!160)))))),
% 29.47/29.63 inference(rewrite,[status(thm)],[])).
% 29.47/29.63 tff(127,plain,
% 29.47/29.63 (^[C: $i] : rewrite(((apply(relation_composition(D!160, C), C!161) = apply(C, apply(D!160, C!161))) | (~relation(C)) | (~in(C!161, relation_dom(D!160))) | (~function(C))) <=> ((~relation(C)) | (~function(C)) | (apply(relation_composition(D!160, C), C!161) = apply(C, apply(D!160, C!161))) | (~in(C!161, relation_dom(D!160)))))),
% 29.47/29.63 inference(bind,[status(th)],[])).
% 29.47/29.63 tff(128,plain,
% 29.47/29.63 (![C: $i] : ((apply(relation_composition(D!160, C), C!161) = apply(C, apply(D!160, C!161))) | (~relation(C)) | (~in(C!161, relation_dom(D!160))) | (~function(C))) <=> ![C: $i] : ((~relation(C)) | (~function(C)) | (apply(relation_composition(D!160, C), C!161) = apply(C, apply(D!160, C!161))) | (~in(C!161, relation_dom(D!160))))),
% 29.47/29.63 inference(quant_intro,[status(thm)],[127])).
% 29.47/29.63 tff(129,plain,
% 29.47/29.63 (((~relation(D!160)) | (~function(D!160)) | ![C: $i] : ((apply(relation_composition(D!160, C), C!161) = apply(C, apply(D!160, C!161))) | (~relation(C)) | (~in(C!161, relation_dom(D!160))) | (~function(C)))) <=> ((~relation(D!160)) | (~function(D!160)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (apply(relation_composition(D!160, C), C!161) = apply(C, apply(D!160, C!161))) | (~in(C!161, relation_dom(D!160)))))),
% 29.47/29.63 inference(monotonicity,[status(thm)],[128])).
% 29.47/29.63 tff(130,plain,
% 29.47/29.63 (((~relation(D!160)) | (~function(D!160)) | ![C: $i] : ((apply(relation_composition(D!160, C), C!161) = apply(C, apply(D!160, C!161))) | (~relation(C)) | (~in(C!161, relation_dom(D!160))) | (~function(C)))) <=> ((~relation(D!160)) | (~function(D!160)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (apply(relation_composition(D!160, C), C!161) = apply(C, apply(D!160, C!161))) | (~in(C!161, relation_dom(D!160)))))),
% 29.47/29.63 inference(transitivity,[status(thm)],[129, 126])).
% 29.47/29.63 tff(131,plain,
% 29.47/29.63 (((~![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~relation(C)) | (~in(A, relation_dom(B))) | (~function(C))))) | ((~relation(D!160)) | (~function(D!160)) | ![C: $i] : ((apply(relation_composition(D!160, C), C!161) = apply(C, apply(D!160, C!161))) | (~relation(C)) | (~in(C!161, relation_dom(D!160))) | (~function(C))))) <=> ((~![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~relation(C)) | (~in(A, relation_dom(B))) | (~function(C))))) | ((~relation(D!160)) | (~function(D!160)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (apply(relation_composition(D!160, C), C!161) = apply(C, apply(D!160, C!161))) | (~in(C!161, relation_dom(D!160))))))),
% 29.47/29.63 inference(monotonicity,[status(thm)],[130])).
% 29.47/29.63 tff(132,plain,
% 29.47/29.63 (((~![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~relation(C)) | (~in(A, relation_dom(B))) | (~function(C))))) | ((~relation(D!160)) | (~function(D!160)) | ![C: $i] : ((apply(relation_composition(D!160, C), C!161) = apply(C, apply(D!160, C!161))) | (~relation(C)) | (~in(C!161, relation_dom(D!160))) | (~function(C))))) <=> ((~![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~relation(C)) | (~in(A, relation_dom(B))) | (~function(C))))) | (~relation(D!160)) | (~function(D!160)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (apply(relation_composition(D!160, C), C!161) = apply(C, apply(D!160, C!161))) | (~in(C!161, relation_dom(D!160)))))),
% 29.47/29.63 inference(transitivity,[status(thm)],[131, 125])).
% 29.47/29.63 tff(133,plain,
% 29.47/29.63 ((~![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~relation(C)) | (~in(A, relation_dom(B))) | (~function(C))))) | ((~relation(D!160)) | (~function(D!160)) | ![C: $i] : ((apply(relation_composition(D!160, C), C!161) = apply(C, apply(D!160, C!161))) | (~relation(C)) | (~in(C!161, relation_dom(D!160))) | (~function(C))))),
% 29.47/29.63 inference(quant_inst,[status(thm)],[])).
% 29.47/29.63 tff(134,plain,
% 29.47/29.63 ((~![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | ![C: $i] : ((apply(relation_composition(B, C), A) = apply(C, apply(B, A))) | (~relation(C)) | (~in(A, relation_dom(B))) | (~function(C))))) | (~relation(D!160)) | (~function(D!160)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (apply(relation_composition(D!160, C), C!161) = apply(C, apply(D!160, C!161))) | (~in(C!161, relation_dom(D!160))))),
% 29.47/29.63 inference(modus_ponens,[status(thm)],[133, 132])).
% 29.47/29.64 tff(135,plain,
% 29.47/29.64 ((~relation(D!160)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (apply(relation_composition(D!160, C), C!161) = apply(C, apply(D!160, C!161))) | (~in(C!161, relation_dom(D!160))))),
% 29.47/29.64 inference(unit_resolution,[status(thm)],[134, 124, 123])).
% 29.47/29.64 tff(136,plain,
% 29.47/29.64 (![C: $i] : ((~relation(C)) | (~function(C)) | (apply(relation_composition(D!160, C), C!161) = apply(C, apply(D!160, C!161))) | (~in(C!161, relation_dom(D!160))))),
% 29.47/29.64 inference(unit_resolution,[status(thm)],[135, 107])).
% 29.47/29.64 tff(137,plain,
% 29.47/29.64 (relation(E!164) & function(E!164)),
% 29.47/29.64 inference(or_elim,[status(thm)],[36])).
% 29.47/29.64 tff(138,plain,
% 29.47/29.64 (function(E!164)),
% 29.47/29.64 inference(and_elim,[status(thm)],[137])).
% 29.47/29.64 tff(139,plain,
% 29.47/29.64 (relation(E!164)),
% 29.47/29.64 inference(and_elim,[status(thm)],[137])).
% 29.47/29.64 tff(140,plain,
% 29.47/29.64 (~(apply(relation_composition(D!160, E!164), C!161) = apply(E!164, apply(D!160, C!161)))),
% 29.47/29.64 inference(or_elim,[status(thm)],[36])).
% 29.47/29.64 tff(141,plain,
% 29.47/29.64 (((~![C: $i] : ((~relation(C)) | (~function(C)) | (apply(relation_composition(D!160, C), C!161) = apply(C, apply(D!160, C!161))) | (~in(C!161, relation_dom(D!160))))) | ((apply(relation_composition(D!160, E!164), C!161) = apply(E!164, apply(D!160, C!161))) | (~relation(E!164)) | (~function(E!164)) | (~in(C!161, relation_dom(D!160))))) <=> ((~![C: $i] : ((~relation(C)) | (~function(C)) | (apply(relation_composition(D!160, C), C!161) = apply(C, apply(D!160, C!161))) | (~in(C!161, relation_dom(D!160))))) | (apply(relation_composition(D!160, E!164), C!161) = apply(E!164, apply(D!160, C!161))) | (~relation(E!164)) | (~function(E!164)) | (~in(C!161, relation_dom(D!160))))),
% 29.47/29.64 inference(rewrite,[status(thm)],[])).
% 29.47/29.64 tff(142,plain,
% 29.47/29.64 (((~relation(E!164)) | (~function(E!164)) | (apply(relation_composition(D!160, E!164), C!161) = apply(E!164, apply(D!160, C!161))) | (~in(C!161, relation_dom(D!160)))) <=> ((apply(relation_composition(D!160, E!164), C!161) = apply(E!164, apply(D!160, C!161))) | (~relation(E!164)) | (~function(E!164)) | (~in(C!161, relation_dom(D!160))))),
% 29.47/29.64 inference(rewrite,[status(thm)],[])).
% 29.47/29.64 tff(143,plain,
% 29.47/29.64 (((~![C: $i] : ((~relation(C)) | (~function(C)) | (apply(relation_composition(D!160, C), C!161) = apply(C, apply(D!160, C!161))) | (~in(C!161, relation_dom(D!160))))) | ((~relation(E!164)) | (~function(E!164)) | (apply(relation_composition(D!160, E!164), C!161) = apply(E!164, apply(D!160, C!161))) | (~in(C!161, relation_dom(D!160))))) <=> ((~![C: $i] : ((~relation(C)) | (~function(C)) | (apply(relation_composition(D!160, C), C!161) = apply(C, apply(D!160, C!161))) | (~in(C!161, relation_dom(D!160))))) | ((apply(relation_composition(D!160, E!164), C!161) = apply(E!164, apply(D!160, C!161))) | (~relation(E!164)) | (~function(E!164)) | (~in(C!161, relation_dom(D!160)))))),
% 29.47/29.64 inference(monotonicity,[status(thm)],[142])).
% 29.47/29.64 tff(144,plain,
% 29.47/29.64 (((~![C: $i] : ((~relation(C)) | (~function(C)) | (apply(relation_composition(D!160, C), C!161) = apply(C, apply(D!160, C!161))) | (~in(C!161, relation_dom(D!160))))) | ((~relation(E!164)) | (~function(E!164)) | (apply(relation_composition(D!160, E!164), C!161) = apply(E!164, apply(D!160, C!161))) | (~in(C!161, relation_dom(D!160))))) <=> ((~![C: $i] : ((~relation(C)) | (~function(C)) | (apply(relation_composition(D!160, C), C!161) = apply(C, apply(D!160, C!161))) | (~in(C!161, relation_dom(D!160))))) | (apply(relation_composition(D!160, E!164), C!161) = apply(E!164, apply(D!160, C!161))) | (~relation(E!164)) | (~function(E!164)) | (~in(C!161, relation_dom(D!160))))),
% 29.47/29.64 inference(transitivity,[status(thm)],[143, 141])).
% 29.47/29.64 tff(145,plain,
% 29.47/29.64 ((~![C: $i] : ((~relation(C)) | (~function(C)) | (apply(relation_composition(D!160, C), C!161) = apply(C, apply(D!160, C!161))) | (~in(C!161, relation_dom(D!160))))) | ((~relation(E!164)) | (~function(E!164)) | (apply(relation_composition(D!160, E!164), C!161) = apply(E!164, apply(D!160, C!161))) | (~in(C!161, relation_dom(D!160))))),
% 29.47/29.64 inference(quant_inst,[status(thm)],[])).
% 29.47/29.64 tff(146,plain,
% 29.47/29.64 ((~![C: $i] : ((~relation(C)) | (~function(C)) | (apply(relation_composition(D!160, C), C!161) = apply(C, apply(D!160, C!161))) | (~in(C!161, relation_dom(D!160))))) | (apply(relation_composition(D!160, E!164), C!161) = apply(E!164, apply(D!160, C!161))) | (~relation(E!164)) | (~function(E!164)) | (~in(C!161, relation_dom(D!160)))),
% 29.47/29.67 inference(modus_ponens,[status(thm)],[145, 144])).
% 29.47/29.67 tff(147,plain,
% 29.47/29.67 (~in(C!161, relation_dom(D!160))),
% 29.47/29.67 inference(unit_resolution,[status(thm)],[146, 140, 139, 138, 136])).
% 29.47/29.67 tff(148,plain,
% 29.47/29.67 ($false),
% 29.47/29.67 inference(unit_resolution,[status(thm)],[147, 79])).
% 29.47/29.67 % SZS output end Proof
% 29.57/29.75 % E exiting
%------------------------------------------------------------------------------