%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SEU009+1 : TPTP v9.0.0. Released v3.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n012.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 05:26:47 AM UTC 2025
% Result : Theorem 0.85s 1.05s
% Output : Proof 0.85s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.12 % Problem : SEU009+1 : TPTP v9.0.0. Released v3.2.0.
% 0.12/0.12 % Command : run_E %s %d THM
% 0.12/0.34 % Computer : n012.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Fri Jun 20 05:56:10 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.85/1.05 % SZS status Theorem
% 0.85/1.05 % SZS output start Proof
% 0.85/1.05 tff(in_type, type, (
% 0.85/1.05 in: ( $i * $i ) > $o)).
% 0.85/1.05 tff(relation_dom_type, type, (
% 0.85/1.05 relation_dom: $i > $i)).
% 0.85/1.05 tff(identity_relation_type, type, (
% 0.85/1.05 identity_relation: $i > $i)).
% 0.85/1.05 tff(tptp_fun_A_12_type, type, (
% 0.85/1.05 tptp_fun_A_12: $i)).
% 0.85/1.05 tff(apply_type, type, (
% 0.85/1.05 apply: ( $i * $i ) > $i)).
% 0.85/1.05 tff(tptp_fun_B_11_type, type, (
% 0.85/1.05 tptp_fun_B_11: $i)).
% 0.85/1.05 tff(tptp_fun_C_10_type, type, (
% 0.85/1.05 tptp_fun_C_10: $i)).
% 0.85/1.05 tff(function_type, type, (
% 0.85/1.05 function: $i > $o)).
% 0.85/1.05 tff(relation_type, type, (
% 0.85/1.05 relation: $i > $o)).
% 0.85/1.05 tff(tptp_fun_C_9_type, type, (
% 0.85/1.05 tptp_fun_C_9: ( $i * $i ) > $i)).
% 0.85/1.05 tff(relation_composition_type, type, (
% 0.85/1.05 relation_composition: ( $i * $i ) > $i)).
% 0.85/1.05 tff(1,plain,
% 0.85/1.05 (^[A: $i] : refl((~((~relation(identity_relation(A))) | (~function(identity_relation(A))))) <=> (~((~relation(identity_relation(A))) | (~function(identity_relation(A))))))),
% 0.85/1.05 inference(bind,[status(th)],[])).
% 0.85/1.05 tff(2,plain,
% 0.85/1.05 (![A: $i] : (~((~relation(identity_relation(A))) | (~function(identity_relation(A))))) <=> ![A: $i] : (~((~relation(identity_relation(A))) | (~function(identity_relation(A)))))),
% 0.85/1.05 inference(quant_intro,[status(thm)],[1])).
% 0.85/1.05 tff(3,plain,
% 0.85/1.05 (^[A: $i] : rewrite((relation(identity_relation(A)) & function(identity_relation(A))) <=> (~((~relation(identity_relation(A))) | (~function(identity_relation(A))))))),
% 0.85/1.05 inference(bind,[status(th)],[])).
% 0.85/1.05 tff(4,plain,
% 0.85/1.05 (![A: $i] : (relation(identity_relation(A)) & function(identity_relation(A))) <=> ![A: $i] : (~((~relation(identity_relation(A))) | (~function(identity_relation(A)))))),
% 0.85/1.05 inference(quant_intro,[status(thm)],[3])).
% 0.85/1.05 tff(5,plain,
% 0.85/1.05 (![A: $i] : (relation(identity_relation(A)) & function(identity_relation(A))) <=> ![A: $i] : (relation(identity_relation(A)) & function(identity_relation(A)))),
% 0.85/1.05 inference(rewrite,[status(thm)],[])).
% 0.85/1.05 tff(6,axiom,(![A: $i] : (relation(identity_relation(A)) & function(identity_relation(A)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','fc2_funct_1')).
% 0.85/1.05 tff(7,plain,
% 0.85/1.05 (![A: $i] : (relation(identity_relation(A)) & function(identity_relation(A)))),
% 0.85/1.05 inference(modus_ponens,[status(thm)],[6, 5])).
% 0.85/1.05 tff(8,plain,(
% 0.85/1.05 ![A: $i] : (relation(identity_relation(A)) & function(identity_relation(A)))),
% 0.85/1.05 inference(skolemize,[status(sab)],[7])).
% 0.85/1.05 tff(9,plain,
% 0.85/1.05 (![A: $i] : (~((~relation(identity_relation(A))) | (~function(identity_relation(A)))))),
% 0.85/1.05 inference(modus_ponens,[status(thm)],[8, 4])).
% 0.85/1.05 tff(10,plain,
% 0.85/1.05 (![A: $i] : (~((~relation(identity_relation(A))) | (~function(identity_relation(A)))))),
% 0.85/1.05 inference(modus_ponens,[status(thm)],[9, 2])).
% 0.85/1.05 tff(11,plain,
% 0.85/1.05 ((~![A: $i] : (~((~relation(identity_relation(A))) | (~function(identity_relation(A)))))) | (~((~relation(identity_relation(A!12))) | (~function(identity_relation(A!12)))))),
% 0.85/1.05 inference(quant_inst,[status(thm)],[])).
% 0.85/1.05 tff(12,plain,
% 0.85/1.05 (~((~relation(identity_relation(A!12))) | (~function(identity_relation(A!12))))),
% 0.85/1.05 inference(unit_resolution,[status(thm)],[11, 10])).
% 0.85/1.05 tff(13,plain,
% 0.85/1.05 (((~relation(identity_relation(A!12))) | (~function(identity_relation(A!12)))) | function(identity_relation(A!12))),
% 0.85/1.05 inference(tautology,[status(thm)],[])).
% 0.85/1.05 tff(14,plain,
% 0.85/1.05 (function(identity_relation(A!12))),
% 0.85/1.05 inference(unit_resolution,[status(thm)],[13, 12])).
% 0.85/1.05 tff(15,plain,
% 0.85/1.05 (^[A: $i] : refl(relation(identity_relation(A)) <=> relation(identity_relation(A)))),
% 0.85/1.05 inference(bind,[status(th)],[])).
% 0.85/1.05 tff(16,plain,
% 0.85/1.05 (![A: $i] : relation(identity_relation(A)) <=> ![A: $i] : relation(identity_relation(A))),
% 0.85/1.05 inference(quant_intro,[status(thm)],[15])).
% 0.85/1.05 tff(17,plain,
% 0.85/1.05 (![A: $i] : relation(identity_relation(A)) <=> ![A: $i] : relation(identity_relation(A))),
% 0.85/1.05 inference(rewrite,[status(thm)],[])).
% 0.85/1.05 tff(18,axiom,(![A: $i] : relation(identity_relation(A))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','dt_k6_relat_1')).
% 0.85/1.05 tff(19,plain,
% 0.85/1.05 (![A: $i] : relation(identity_relation(A))),
% 0.85/1.05 inference(modus_ponens,[status(thm)],[18, 17])).
% 0.85/1.05 tff(20,plain,(
% 0.85/1.05 ![A: $i] : relation(identity_relation(A))),
% 0.85/1.05 inference(skolemize,[status(sab)],[19])).
% 0.85/1.05 tff(21,plain,
% 0.85/1.05 (![A: $i] : relation(identity_relation(A))),
% 0.85/1.05 inference(modus_ponens,[status(thm)],[20, 16])).
% 0.85/1.05 tff(22,plain,
% 0.85/1.05 ((~![A: $i] : relation(identity_relation(A))) | relation(identity_relation(A!12))),
% 0.85/1.05 inference(quant_inst,[status(thm)],[])).
% 0.85/1.05 tff(23,plain,
% 0.85/1.05 (relation(identity_relation(A!12))),
% 0.85/1.05 inference(unit_resolution,[status(thm)],[22, 21])).
% 0.85/1.05 tff(24,plain,
% 0.85/1.05 (^[A: $i, B: $i] : trans(monotonicity(rewrite((~((~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))) | (~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A)))))))) <=> (~((~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A)))))) | (~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C)))))))))), (((~relation(B)) | (~function(B)) | (~((~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))) | (~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A))))))))) <=> ((~relation(B)) | (~function(B)) | (~((~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A)))))) | (~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C)))))))))))), rewrite(((~relation(B)) | (~function(B)) | (~((~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A)))))) | (~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C)))))))))) <=> ((~relation(B)) | (~function(B)) | (~((~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A)))))) | (~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))))))), (((~relation(B)) | (~function(B)) | (~((~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))) | (~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A))))))))) <=> ((~relation(B)) | (~function(B)) | (~((~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A)))))) | (~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))))))))),
% 0.85/1.05 inference(bind,[status(th)],[])).
% 0.85/1.05 tff(25,plain,
% 0.85/1.05 (![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | (~((~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))) | (~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A))))))))) <=> ![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | (~((~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A)))))) | (~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))))))),
% 0.85/1.05 inference(quant_intro,[status(thm)],[24])).
% 0.85/1.05 tff(26,plain,
% 0.85/1.05 (^[A: $i, B: $i] : refl(((~relation(B)) | (~function(B)) | (~((~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))) | (~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A))))))))) <=> ((~relation(B)) | (~function(B)) | (~((~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))) | (~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A))))))))))),
% 0.85/1.05 inference(bind,[status(th)],[])).
% 0.85/1.05 tff(27,plain,
% 0.85/1.05 (![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | (~((~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))) | (~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A))))))))) <=> ![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | (~((~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))) | (~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A)))))))))),
% 0.85/1.05 inference(quant_intro,[status(thm)],[26])).
% 0.85/1.05 tff(28,plain,
% 0.85/1.05 (^[A: $i, B: $i] : rewrite(((~relation(B)) | (~function(B)) | (~((~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))) | (~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A))))))))) <=> ((~relation(B)) | (~function(B)) | (~((~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))) | (~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A))))))))))),
% 0.85/1.05 inference(bind,[status(th)],[])).
% 0.85/1.05 tff(29,plain,
% 0.85/1.05 (![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | (~((~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))) | (~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A))))))))) <=> ![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | (~((~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))) | (~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A)))))))))),
% 0.85/1.05 inference(quant_intro,[status(thm)],[28])).
% 0.85/1.05 tff(30,plain,
% 0.85/1.05 (![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | (~((~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))) | (~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A))))))))) <=> ![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | (~((~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))) | (~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A)))))))))),
% 0.85/1.05 inference(transitivity,[status(thm)],[29, 27])).
% 0.85/1.05 tff(31,plain,
% 0.85/1.05 (^[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))))), trans(monotonicity(rewrite(((~(B = identity_relation(A))) | ((relation_dom(B) = A) & ![C: $i] : ((~in(C, A)) | (apply(B, C) = C)))) <=> ((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))), rewrite(((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A))))) <=> ((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A)))))), ((((~(B = identity_relation(A))) | ((relation_dom(B) = A) & ![C: $i] : ((~in(C, A)) | (apply(B, C) = C)))) & ((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A)))))) <=> (((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C)))))) & ((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A)))))))), rewrite((((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C)))))) & ((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A)))))) <=> (~((~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))) | (~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A))))))))), ((((~(B = identity_relation(A))) | ((relation_dom(B) = A) & ![C: $i] : ((~in(C, A)) | (apply(B, C) = C)))) & ((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A)))))) <=> (~((~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))) | (~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A)))))))))), (((~(relation(B) & function(B))) | (((~(B = identity_relation(A))) | ((relation_dom(B) = A) & ![C: $i] : ((~in(C, A)) | (apply(B, C) = C)))) & ((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A))))))) <=> (((~relation(B)) | (~function(B))) | (~((~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))) | (~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A))))))))))), rewrite((((~relation(B)) | (~function(B))) | (~((~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))) | (~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A))))))))) <=> ((~relation(B)) | (~function(B)) | (~((~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))) | (~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A)))))))))), (((~(relation(B) & function(B))) | (((~(B = identity_relation(A))) | ((relation_dom(B) = A) & ![C: $i] : ((~in(C, A)) | (apply(B, C) = C)))) & ((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A))))))) <=> ((~relation(B)) | (~function(B)) | (~((~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))) | (~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A)))))))))))),
% 0.85/1.05 inference(bind,[status(th)],[])).
% 0.85/1.05 tff(32,plain,
% 0.85/1.05 (![A: $i, B: $i] : ((~(relation(B) & function(B))) | (((~(B = identity_relation(A))) | ((relation_dom(B) = A) & ![C: $i] : ((~in(C, A)) | (apply(B, C) = C)))) & ((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A))))))) <=> ![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | (~((~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))) | (~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A)))))))))),
% 0.85/1.05 inference(quant_intro,[status(thm)],[31])).
% 0.85/1.05 tff(33,plain,
% 0.85/1.05 (^[A: $i, B: $i] : rewrite(((~(relation(B) & function(B))) | (((~(B = identity_relation(A))) | ((relation_dom(B) = A) & ![C: $i] : ((~in(C, A)) | (apply(B, C) = C)))) & ((B = identity_relation(A)) | ((~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A)))))))) <=> ((~(relation(B) & function(B))) | (((~(B = identity_relation(A))) | ((relation_dom(B) = A) & ![C: $i] : ((~in(C, A)) | (apply(B, C) = C)))) & ((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A))))))))),
% 0.85/1.05 inference(bind,[status(th)],[])).
% 0.85/1.05 tff(34,plain,
% 0.85/1.05 (![A: $i, B: $i] : ((~(relation(B) & function(B))) | (((~(B = identity_relation(A))) | ((relation_dom(B) = A) & ![C: $i] : ((~in(C, A)) | (apply(B, C) = C)))) & ((B = identity_relation(A)) | ((~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A)))))))) <=> ![A: $i, B: $i] : ((~(relation(B) & function(B))) | (((~(B = identity_relation(A))) | ((relation_dom(B) = A) & ![C: $i] : ((~in(C, A)) | (apply(B, C) = C)))) & ((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A)))))))),
% 0.85/1.05 inference(quant_intro,[status(thm)],[33])).
% 0.85/1.05 tff(35,plain,
% 0.85/1.05 (![A: $i, B: $i] : ((~(relation(B) & function(B))) | ((B = identity_relation(A)) <=> ((relation_dom(B) = A) & ![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))) <=> ![A: $i, B: $i] : ((~(relation(B) & function(B))) | ((B = identity_relation(A)) <=> ((relation_dom(B) = A) & ![C: $i] : ((~in(C, A)) | (apply(B, C) = C)))))),
% 0.85/1.05 inference(rewrite,[status(thm)],[])).
% 0.85/1.05 tff(36,plain,
% 0.85/1.05 (^[A: $i, B: $i] : trans(monotonicity(rewrite(((B = identity_relation(A)) <=> ((relation_dom(B) = A) & ![C: $i] : (in(C, A) => (apply(B, C) = C)))) <=> ((B = identity_relation(A)) <=> ((relation_dom(B) = A) & ![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))), (((relation(B) & function(B)) => ((B = identity_relation(A)) <=> ((relation_dom(B) = A) & ![C: $i] : (in(C, A) => (apply(B, C) = C))))) <=> ((relation(B) & function(B)) => ((B = identity_relation(A)) <=> ((relation_dom(B) = A) & ![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))), rewrite(((relation(B) & function(B)) => ((B = identity_relation(A)) <=> ((relation_dom(B) = A) & ![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))) <=> ((~(relation(B) & function(B))) | ((B = identity_relation(A)) <=> ((relation_dom(B) = A) & ![C: $i] : ((~in(C, A)) | (apply(B, C) = C)))))), (((relation(B) & function(B)) => ((B = identity_relation(A)) <=> ((relation_dom(B) = A) & ![C: $i] : (in(C, A) => (apply(B, C) = C))))) <=> ((~(relation(B) & function(B))) | ((B = identity_relation(A)) <=> ((relation_dom(B) = A) & ![C: $i] : ((~in(C, A)) | (apply(B, C) = C)))))))),
% 0.85/1.05 inference(bind,[status(th)],[])).
% 0.85/1.05 tff(37,plain,
% 0.85/1.05 (![A: $i, B: $i] : ((relation(B) & function(B)) => ((B = identity_relation(A)) <=> ((relation_dom(B) = A) & ![C: $i] : (in(C, A) => (apply(B, C) = C))))) <=> ![A: $i, B: $i] : ((~(relation(B) & function(B))) | ((B = identity_relation(A)) <=> ((relation_dom(B) = A) & ![C: $i] : ((~in(C, A)) | (apply(B, C) = C)))))),
% 0.85/1.05 inference(quant_intro,[status(thm)],[36])).
% 0.85/1.05 tff(38,axiom,(![A: $i, B: $i] : ((relation(B) & function(B)) => ((B = identity_relation(A)) <=> ((relation_dom(B) = A) & ![C: $i] : (in(C, A) => (apply(B, C) = C)))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t34_funct_1')).
% 0.85/1.05 tff(39,plain,
% 0.85/1.05 (![A: $i, B: $i] : ((~(relation(B) & function(B))) | ((B = identity_relation(A)) <=> ((relation_dom(B) = A) & ![C: $i] : ((~in(C, A)) | (apply(B, C) = C)))))),
% 0.85/1.05 inference(modus_ponens,[status(thm)],[38, 37])).
% 0.85/1.05 tff(40,plain,
% 0.85/1.05 (![A: $i, B: $i] : ((~(relation(B) & function(B))) | ((B = identity_relation(A)) <=> ((relation_dom(B) = A) & ![C: $i] : ((~in(C, A)) | (apply(B, C) = C)))))),
% 0.85/1.05 inference(modus_ponens,[status(thm)],[39, 35])).
% 0.85/1.05 tff(41,plain,(
% 0.85/1.05 ![A: $i, B: $i] : ((~(relation(B) & function(B))) | (((~(B = identity_relation(A))) | ((relation_dom(B) = A) & ![C: $i] : ((~in(C, A)) | (apply(B, C) = C)))) & ((B = identity_relation(A)) | ((~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A))))))))),
% 0.85/1.05 inference(skolemize,[status(sab)],[40])).
% 0.85/1.05 tff(42,plain,
% 0.85/1.05 (![A: $i, B: $i] : ((~(relation(B) & function(B))) | (((~(B = identity_relation(A))) | ((relation_dom(B) = A) & ![C: $i] : ((~in(C, A)) | (apply(B, C) = C)))) & ((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A)))))))),
% 0.85/1.05 inference(modus_ponens,[status(thm)],[41, 34])).
% 0.85/1.05 tff(43,plain,
% 0.85/1.05 (![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | (~((~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))) | (~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A)))))))))),
% 0.85/1.05 inference(modus_ponens,[status(thm)],[42, 32])).
% 0.85/1.05 tff(44,plain,
% 0.85/1.05 (![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | (~((~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))) | (~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A)))))))))),
% 0.85/1.05 inference(modus_ponens,[status(thm)],[43, 30])).
% 0.85/1.05 tff(45,plain,
% 0.85/1.05 (![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | (~((~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A)))))) | (~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))))))),
% 0.85/1.05 inference(modus_ponens,[status(thm)],[44, 25])).
% 0.85/1.05 tff(46,plain,
% 0.85/1.05 (((~![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | (~((~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A)))))) | (~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))))))) | ((~relation(identity_relation(A!12))) | (~function(identity_relation(A!12))) | (~((~![C: $i] : ((~in(C, A!12)) | (apply(identity_relation(A!12), C) = C))) | (~(relation_dom(identity_relation(A!12)) = A!12)))))) <=> ((~![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | (~((~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A)))))) | (~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))))))) | (~relation(identity_relation(A!12))) | (~function(identity_relation(A!12))) | (~((~![C: $i] : ((~in(C, A!12)) | (apply(identity_relation(A!12), C) = C))) | (~(relation_dom(identity_relation(A!12)) = A!12)))))),
% 0.85/1.05 inference(rewrite,[status(thm)],[])).
% 0.85/1.05 tff(47,plain,
% 0.85/1.05 (((~relation(identity_relation(A!12))) | (~function(identity_relation(A!12))) | (~((~![C: $i] : ((~in(C, A!12)) | (apply(identity_relation(A!12), C) = C))) | (~(relation_dom(identity_relation(A!12)) = A!12))))) <=> ((~relation(identity_relation(A!12))) | (~function(identity_relation(A!12))) | (~((~![C: $i] : ((~in(C, A!12)) | (apply(identity_relation(A!12), C) = C))) | (~(relation_dom(identity_relation(A!12)) = A!12)))))),
% 0.85/1.05 inference(rewrite,[status(thm)],[])).
% 0.85/1.05 tff(48,plain,
% 0.85/1.05 ((~((~((identity_relation(A!12) = identity_relation(A!12)) | (~(relation_dom(identity_relation(A!12)) = A!12)) | (~((~in(tptp_fun_C_9(identity_relation(A!12), A!12), A!12)) | (apply(identity_relation(A!12), tptp_fun_C_9(identity_relation(A!12), A!12)) = tptp_fun_C_9(identity_relation(A!12), A!12)))))) | (~((~(identity_relation(A!12) = identity_relation(A!12))) | (~((~(relation_dom(identity_relation(A!12)) = A!12)) | (~![C: $i] : ((~in(C, A!12)) | (apply(identity_relation(A!12), C) = C))))))))) <=> (~((~![C: $i] : ((~in(C, A!12)) | (apply(identity_relation(A!12), C) = C))) | (~(relation_dom(identity_relation(A!12)) = A!12))))),
% 0.85/1.06 inference(rewrite,[status(thm)],[])).
% 0.85/1.06 tff(49,plain,
% 0.85/1.06 (((~relation(identity_relation(A!12))) | (~function(identity_relation(A!12))) | (~((~((identity_relation(A!12) = identity_relation(A!12)) | (~(relation_dom(identity_relation(A!12)) = A!12)) | (~((~in(tptp_fun_C_9(identity_relation(A!12), A!12), A!12)) | (apply(identity_relation(A!12), tptp_fun_C_9(identity_relation(A!12), A!12)) = tptp_fun_C_9(identity_relation(A!12), A!12)))))) | (~((~(identity_relation(A!12) = identity_relation(A!12))) | (~((~(relation_dom(identity_relation(A!12)) = A!12)) | (~![C: $i] : ((~in(C, A!12)) | (apply(identity_relation(A!12), C) = C)))))))))) <=> ((~relation(identity_relation(A!12))) | (~function(identity_relation(A!12))) | (~((~![C: $i] : ((~in(C, A!12)) | (apply(identity_relation(A!12), C) = C))) | (~(relation_dom(identity_relation(A!12)) = A!12)))))),
% 0.85/1.06 inference(monotonicity,[status(thm)],[48])).
% 0.85/1.06 tff(50,plain,
% 0.85/1.06 (((~relation(identity_relation(A!12))) | (~function(identity_relation(A!12))) | (~((~((identity_relation(A!12) = identity_relation(A!12)) | (~(relation_dom(identity_relation(A!12)) = A!12)) | (~((~in(tptp_fun_C_9(identity_relation(A!12), A!12), A!12)) | (apply(identity_relation(A!12), tptp_fun_C_9(identity_relation(A!12), A!12)) = tptp_fun_C_9(identity_relation(A!12), A!12)))))) | (~((~(identity_relation(A!12) = identity_relation(A!12))) | (~((~(relation_dom(identity_relation(A!12)) = A!12)) | (~![C: $i] : ((~in(C, A!12)) | (apply(identity_relation(A!12), C) = C)))))))))) <=> ((~relation(identity_relation(A!12))) | (~function(identity_relation(A!12))) | (~((~![C: $i] : ((~in(C, A!12)) | (apply(identity_relation(A!12), C) = C))) | (~(relation_dom(identity_relation(A!12)) = A!12)))))),
% 0.85/1.06 inference(transitivity,[status(thm)],[49, 47])).
% 0.85/1.06 tff(51,plain,
% 0.85/1.06 (((~![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | (~((~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A)))))) | (~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))))))) | ((~relation(identity_relation(A!12))) | (~function(identity_relation(A!12))) | (~((~((identity_relation(A!12) = identity_relation(A!12)) | (~(relation_dom(identity_relation(A!12)) = A!12)) | (~((~in(tptp_fun_C_9(identity_relation(A!12), A!12), A!12)) | (apply(identity_relation(A!12), tptp_fun_C_9(identity_relation(A!12), A!12)) = tptp_fun_C_9(identity_relation(A!12), A!12)))))) | (~((~(identity_relation(A!12) = identity_relation(A!12))) | (~((~(relation_dom(identity_relation(A!12)) = A!12)) | (~![C: $i] : ((~in(C, A!12)) | (apply(identity_relation(A!12), C) = C))))))))))) <=> ((~![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | (~((~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A)))))) | (~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))))))) | ((~relation(identity_relation(A!12))) | (~function(identity_relation(A!12))) | (~((~![C: $i] : ((~in(C, A!12)) | (apply(identity_relation(A!12), C) = C))) | (~(relation_dom(identity_relation(A!12)) = A!12))))))),
% 0.85/1.06 inference(monotonicity,[status(thm)],[50])).
% 0.85/1.06 tff(52,plain,
% 0.85/1.06 (((~![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | (~((~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A)))))) | (~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))))))) | ((~relation(identity_relation(A!12))) | (~function(identity_relation(A!12))) | (~((~((identity_relation(A!12) = identity_relation(A!12)) | (~(relation_dom(identity_relation(A!12)) = A!12)) | (~((~in(tptp_fun_C_9(identity_relation(A!12), A!12), A!12)) | (apply(identity_relation(A!12), tptp_fun_C_9(identity_relation(A!12), A!12)) = tptp_fun_C_9(identity_relation(A!12), A!12)))))) | (~((~(identity_relation(A!12) = identity_relation(A!12))) | (~((~(relation_dom(identity_relation(A!12)) = A!12)) | (~![C: $i] : ((~in(C, A!12)) | (apply(identity_relation(A!12), C) = C))))))))))) <=> ((~![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | (~((~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A)))))) | (~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))))))) | (~relation(identity_relation(A!12))) | (~function(identity_relation(A!12))) | (~((~![C: $i] : ((~in(C, A!12)) | (apply(identity_relation(A!12), C) = C))) | (~(relation_dom(identity_relation(A!12)) = A!12)))))),
% 0.85/1.06 inference(transitivity,[status(thm)],[51, 46])).
% 0.85/1.06 tff(53,plain,
% 0.85/1.06 ((~![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | (~((~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A)))))) | (~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))))))) | ((~relation(identity_relation(A!12))) | (~function(identity_relation(A!12))) | (~((~((identity_relation(A!12) = identity_relation(A!12)) | (~(relation_dom(identity_relation(A!12)) = A!12)) | (~((~in(tptp_fun_C_9(identity_relation(A!12), A!12), A!12)) | (apply(identity_relation(A!12), tptp_fun_C_9(identity_relation(A!12), A!12)) = tptp_fun_C_9(identity_relation(A!12), A!12)))))) | (~((~(identity_relation(A!12) = identity_relation(A!12))) | (~((~(relation_dom(identity_relation(A!12)) = A!12)) | (~![C: $i] : ((~in(C, A!12)) | (apply(identity_relation(A!12), C) = C))))))))))),
% 0.85/1.06 inference(quant_inst,[status(thm)],[])).
% 0.85/1.06 tff(54,plain,
% 0.85/1.06 ((~![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | (~((~((B = identity_relation(A)) | (~(relation_dom(B) = A)) | (~((~in(tptp_fun_C_9(B, A), A)) | (apply(B, tptp_fun_C_9(B, A)) = tptp_fun_C_9(B, A)))))) | (~((~(B = identity_relation(A))) | (~((~(relation_dom(B) = A)) | (~![C: $i] : ((~in(C, A)) | (apply(B, C) = C))))))))))) | (~relation(identity_relation(A!12))) | (~function(identity_relation(A!12))) | (~((~![C: $i] : ((~in(C, A!12)) | (apply(identity_relation(A!12), C) = C))) | (~(relation_dom(identity_relation(A!12)) = A!12))))),
% 0.85/1.06 inference(modus_ponens,[status(thm)],[53, 52])).
% 0.85/1.06 tff(55,plain,
% 0.85/1.06 (~((~![C: $i] : ((~in(C, A!12)) | (apply(identity_relation(A!12), C) = C))) | (~(relation_dom(identity_relation(A!12)) = A!12)))),
% 0.85/1.06 inference(unit_resolution,[status(thm)],[54, 45, 23, 14])).
% 0.85/1.06 tff(56,plain,
% 0.85/1.06 (((~![C: $i] : ((~in(C, A!12)) | (apply(identity_relation(A!12), C) = C))) | (~(relation_dom(identity_relation(A!12)) = A!12))) | (relation_dom(identity_relation(A!12)) = A!12)),
% 0.85/1.06 inference(tautology,[status(thm)],[])).
% 0.85/1.06 tff(57,plain,
% 0.85/1.06 (relation_dom(identity_relation(A!12)) = A!12),
% 0.85/1.06 inference(unit_resolution,[status(thm)],[56, 55])).
% 0.85/1.06 tff(58,plain,
% 0.85/1.06 (in(apply(C!10, B!11), relation_dom(identity_relation(A!12))) <=> in(apply(C!10, B!11), A!12)),
% 0.85/1.06 inference(monotonicity,[status(thm)],[57])).
% 0.85/1.06 tff(59,plain,
% 0.85/1.06 (in(apply(C!10, B!11), A!12) <=> in(apply(C!10, B!11), relation_dom(identity_relation(A!12)))),
% 0.85/1.06 inference(symmetry,[status(thm)],[58])).
% 0.85/1.06 tff(60,plain,
% 0.85/1.06 ((~in(apply(C!10, B!11), A!12)) <=> (~in(apply(C!10, B!11), relation_dom(identity_relation(A!12))))),
% 0.85/1.06 inference(monotonicity,[status(thm)],[59])).
% 0.85/1.06 tff(61,assumption,(~in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12))))), introduced(assumption)).
% 0.85/1.06 tff(62,plain,
% 0.85/1.06 ((~(~(((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), A!12))) <=> in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12))))))) <=> (((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), A!12))) <=> in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12)))))),
% 0.85/1.06 inference(rewrite,[status(thm)],[])).
% 0.85/1.06 tff(63,plain,
% 0.85/1.06 ((~(((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), A!12))) <=> in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12)))))) <=> (~(((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), A!12))) <=> in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12))))))),
% 0.85/1.06 inference(rewrite,[status(thm)],[])).
% 0.85/1.06 tff(64,plain,
% 0.85/1.06 ((in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12)))) <=> (~((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), A!12))))) <=> (~(((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), A!12))) <=> in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12))))))),
% 0.85/1.06 inference(rewrite,[status(thm)],[])).
% 0.85/1.06 tff(65,plain,
% 0.85/1.06 ((in(B!11, relation_dom(C!10)) & in(apply(C!10, B!11), A!12)) <=> (~((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), A!12))))),
% 0.85/1.06 inference(rewrite,[status(thm)],[])).
% 0.85/1.06 tff(66,plain,
% 0.85/1.06 ((in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12)))) <=> (in(B!11, relation_dom(C!10)) & in(apply(C!10, B!11), A!12))) <=> (in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12)))) <=> (~((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), A!12)))))),
% 0.85/1.06 inference(monotonicity,[status(thm)],[65])).
% 0.85/1.06 tff(67,plain,
% 0.85/1.06 ((in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12)))) <=> (in(B!11, relation_dom(C!10)) & in(apply(C!10, B!11), A!12))) <=> (~(((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), A!12))) <=> in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12))))))),
% 0.85/1.06 inference(transitivity,[status(thm)],[66, 64])).
% 0.85/1.06 tff(68,plain,
% 0.85/1.06 ((in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12)))) <=> (in(B!11, relation_dom(C!10)) & in(apply(C!10, B!11), A!12))) <=> (~(((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), A!12))) <=> in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12))))))),
% 0.85/1.06 inference(transitivity,[status(thm)],[67, 63])).
% 0.85/1.06 tff(69,plain,
% 0.85/1.06 ((~(in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12)))) <=> (in(B!11, relation_dom(C!10)) & in(apply(C!10, B!11), A!12)))) <=> (~(~(((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), A!12))) <=> in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12)))))))),
% 0.85/1.06 inference(monotonicity,[status(thm)],[68])).
% 0.85/1.06 tff(70,plain,
% 0.85/1.06 ((~(in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12)))) <=> (in(B!11, relation_dom(C!10)) & in(apply(C!10, B!11), A!12)))) <=> (((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), A!12))) <=> in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12)))))),
% 0.85/1.06 inference(transitivity,[status(thm)],[69, 62])).
% 0.85/1.06 tff(71,plain,
% 0.85/1.06 ((~![A: $i, B: $i, C: $i] : ((~(relation(C) & function(C))) | (in(B, relation_dom(relation_composition(C, identity_relation(A)))) <=> (in(B, relation_dom(C)) & in(apply(C, B), A))))) <=> (~![A: $i, B: $i, C: $i] : ((~(relation(C) & function(C))) | (in(B, relation_dom(relation_composition(C, identity_relation(A)))) <=> (in(B, relation_dom(C)) & in(apply(C, B), A)))))),
% 0.85/1.06 inference(rewrite,[status(thm)],[])).
% 0.85/1.06 tff(72,plain,
% 0.85/1.06 ((~![A: $i, B: $i, C: $i] : ((relation(C) & function(C)) => (in(B, relation_dom(relation_composition(C, identity_relation(A)))) <=> (in(B, relation_dom(C)) & in(apply(C, B), A))))) <=> (~![A: $i, B: $i, C: $i] : ((~(relation(C) & function(C))) | (in(B, relation_dom(relation_composition(C, identity_relation(A)))) <=> (in(B, relation_dom(C)) & in(apply(C, B), A)))))),
% 0.85/1.06 inference(rewrite,[status(thm)],[])).
% 0.85/1.06 tff(73,axiom,(~![A: $i, B: $i, C: $i] : ((relation(C) & function(C)) => (in(B, relation_dom(relation_composition(C, identity_relation(A)))) <=> (in(B, relation_dom(C)) & in(apply(C, B), A))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t40_funct_1')).
% 0.85/1.06 tff(74,plain,
% 0.85/1.06 (~![A: $i, B: $i, C: $i] : ((~(relation(C) & function(C))) | (in(B, relation_dom(relation_composition(C, identity_relation(A)))) <=> (in(B, relation_dom(C)) & in(apply(C, B), A))))),
% 0.85/1.06 inference(modus_ponens,[status(thm)],[73, 72])).
% 0.85/1.06 tff(75,plain,
% 0.85/1.06 (~![A: $i, B: $i, C: $i] : ((~(relation(C) & function(C))) | (in(B, relation_dom(relation_composition(C, identity_relation(A)))) <=> (in(B, relation_dom(C)) & in(apply(C, B), A))))),
% 0.85/1.06 inference(modus_ponens,[status(thm)],[74, 71])).
% 0.85/1.06 tff(76,plain,
% 0.85/1.06 (~![A: $i, B: $i, C: $i] : ((~(relation(C) & function(C))) | (in(B, relation_dom(relation_composition(C, identity_relation(A)))) <=> (in(B, relation_dom(C)) & in(apply(C, B), A))))),
% 0.85/1.06 inference(modus_ponens,[status(thm)],[75, 71])).
% 0.85/1.06 tff(77,plain,
% 0.85/1.06 (~![A: $i, B: $i, C: $i] : ((~(relation(C) & function(C))) | (in(B, relation_dom(relation_composition(C, identity_relation(A)))) <=> (in(B, relation_dom(C)) & in(apply(C, B), A))))),
% 0.85/1.06 inference(modus_ponens,[status(thm)],[76, 71])).
% 0.85/1.06 tff(78,plain,(
% 0.85/1.06 ~((~(relation(C!10) & function(C!10))) | (in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12)))) <=> (in(B!11, relation_dom(C!10)) & in(apply(C!10, B!11), A!12))))),
% 0.85/1.06 inference(skolemize,[status(sab)],[77])).
% 0.85/1.06 tff(79,plain,
% 0.85/1.06 (~(in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12)))) <=> (in(B!11, relation_dom(C!10)) & in(apply(C!10, B!11), A!12)))),
% 0.85/1.06 inference(or_elim,[status(thm)],[78])).
% 0.85/1.06 tff(80,plain,
% 0.85/1.06 (((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), A!12))) <=> in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12))))),
% 0.85/1.06 inference(modus_ponens,[status(thm)],[79, 70])).
% 0.85/1.06 tff(81,plain,
% 0.85/1.06 ((~((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), A!12)))) | in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12)))) | (~(((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), A!12))) <=> in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12))))))),
% 0.85/1.06 inference(tautology,[status(thm)],[])).
% 0.85/1.06 tff(82,plain,
% 0.85/1.06 ((~((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), A!12)))) | in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12))))),
% 0.85/1.06 inference(unit_resolution,[status(thm)],[81, 80])).
% 0.85/1.06 tff(83,plain,
% 0.85/1.06 (~((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), A!12)))),
% 0.85/1.06 inference(unit_resolution,[status(thm)],[82, 61])).
% 0.85/1.06 tff(84,plain,
% 0.85/1.06 (((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), A!12))) | in(apply(C!10, B!11), A!12)),
% 0.85/1.06 inference(tautology,[status(thm)],[])).
% 0.85/1.06 tff(85,plain,
% 0.85/1.06 (in(apply(C!10, B!11), A!12)),
% 0.85/1.06 inference(unit_resolution,[status(thm)],[84, 83])).
% 0.85/1.06 tff(86,plain,
% 0.85/1.06 (in(apply(C!10, B!11), relation_dom(identity_relation(A!12)))),
% 0.85/1.06 inference(modus_ponens,[status(thm)],[85, 59])).
% 0.85/1.06 tff(87,plain,
% 0.85/1.06 (relation(C!10) & function(C!10)),
% 0.85/1.06 inference(or_elim,[status(thm)],[78])).
% 0.85/1.06 tff(88,plain,
% 0.85/1.06 (function(C!10)),
% 0.85/1.06 inference(and_elim,[status(thm)],[87])).
% 0.85/1.06 tff(89,plain,
% 0.85/1.06 (relation(C!10)),
% 0.85/1.06 inference(and_elim,[status(thm)],[87])).
% 0.85/1.06 tff(90,plain,
% 0.85/1.06 (^[A: $i, B: $i, C: $i] : trans(monotonicity(trans(monotonicity(trans(monotonicity(rewrite((((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B)))) <=> (((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))), ((~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))) <=> (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))))), rewrite((~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))) <=> (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B)))))), ((~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))) <=> (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))))), (((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B)))))) <=> ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B)))))))), rewrite(((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B)))))) <=> ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))))), (((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B)))))) <=> ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B)))))))), (((~relation(B)) | (~function(B)) | ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))))) <=> ((~relation(B)) | (~function(B)) | ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))))))), rewrite(((~relation(B)) | (~function(B)) | ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))))) <=> ((~relation(B)) | (~relation(C)) | (~function(B)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))))), (((~relation(B)) | (~function(B)) | ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))))) <=> ((~relation(B)) | (~relation(C)) | (~function(B)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))))))),
% 0.85/1.06 inference(bind,[status(th)],[])).
% 0.85/1.06 tff(91,plain,
% 0.85/1.06 (![A: $i, B: $i, C: $i] : ((~relation(B)) | (~function(B)) | ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))))) <=> ![A: $i, B: $i, C: $i] : ((~relation(B)) | (~relation(C)) | (~function(B)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))))),
% 0.85/1.06 inference(quant_intro,[status(thm)],[90])).
% 0.85/1.06 tff(92,plain,
% 0.85/1.06 (^[A: $i, B: $i, C: $i] : refl(((~relation(B)) | (~function(B)) | ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))))) <=> ((~relation(B)) | (~function(B)) | ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))))))),
% 0.85/1.06 inference(bind,[status(th)],[])).
% 0.85/1.06 tff(93,plain,
% 0.85/1.06 (![A: $i, B: $i, C: $i] : ((~relation(B)) | (~function(B)) | ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))))) <=> ![A: $i, B: $i, C: $i] : ((~relation(B)) | (~function(B)) | ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B)))))))),
% 0.85/1.06 inference(quant_intro,[status(thm)],[92])).
% 0.85/1.06 tff(94,plain,
% 0.85/1.06 (![A: $i, B: $i] : ![C: $i] : ((~relation(B)) | (~function(B)) | ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))))) <=> ![A: $i, B: $i, C: $i] : ((~relation(B)) | (~function(B)) | ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B)))))))),
% 0.85/1.06 inference(pull_quant,[status(thm)],[])).
% 0.85/1.06 tff(95,plain,
% 0.85/1.06 (^[A: $i, B: $i] : pull_quant(((~relation(B)) | (~function(B)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))))) <=> ![C: $i] : ((~relation(B)) | (~function(B)) | ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))))))),
% 0.85/1.06 inference(bind,[status(th)],[])).
% 0.85/1.06 tff(96,plain,
% 0.85/1.06 (![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))))) <=> ![A: $i, B: $i] : ![C: $i] : ((~relation(B)) | (~function(B)) | ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B)))))))),
% 0.85/1.06 inference(quant_intro,[status(thm)],[95])).
% 0.85/1.06 tff(97,plain,
% 0.85/1.06 (![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))))) <=> ![A: $i, B: $i, C: $i] : ((~relation(B)) | (~function(B)) | ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B)))))))),
% 0.85/1.06 inference(transitivity,[status(thm)],[96, 94])).
% 0.85/1.06 tff(98,plain,
% 0.85/1.06 (![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))))) <=> ![A: $i, B: $i, C: $i] : ((~relation(B)) | (~function(B)) | ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B)))))))),
% 0.85/1.06 inference(transitivity,[status(thm)],[97, 93])).
% 0.85/1.06 tff(99,plain,
% 0.85/1.06 (^[A: $i, B: $i] : rewrite(((~relation(B)) | (~function(B)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))))) <=> ((~relation(B)) | (~function(B)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))))))),
% 0.85/1.06 inference(bind,[status(th)],[])).
% 0.85/1.06 tff(100,plain,
% 0.85/1.06 (![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))))) <=> ![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B)))))))),
% 0.85/1.06 inference(quant_intro,[status(thm)],[99])).
% 0.85/1.06 tff(101,plain,
% 0.85/1.06 (![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))))) <=> ![A: $i, B: $i, C: $i] : ((~relation(B)) | (~function(B)) | ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B)))))))),
% 0.85/1.06 inference(transitivity,[status(thm)],[100, 98])).
% 0.85/1.06 tff(102,plain,
% 0.85/1.06 (^[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))))), trans(trans(monotonicity(rewrite((in(A, relation_dom(C)) & in(apply(C, A), relation_dom(B))) <=> (~((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))))), ((in(A, relation_dom(relation_composition(C, B))) <=> (in(A, relation_dom(C)) & in(apply(C, A), relation_dom(B)))) <=> (in(A, relation_dom(relation_composition(C, B))) <=> (~((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))))))), rewrite((in(A, relation_dom(relation_composition(C, B))) <=> (~((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))))) <=> (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B)))))), ((in(A, relation_dom(relation_composition(C, B))) <=> (in(A, relation_dom(C)) & in(apply(C, A), relation_dom(B)))) <=> (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))))), rewrite((~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))) <=> (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B)))))), ((in(A, relation_dom(relation_composition(C, B))) <=> (in(A, relation_dom(C)) & in(apply(C, A), relation_dom(B)))) <=> (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))))), (((~(relation(C) & function(C))) | (in(A, relation_dom(relation_composition(C, B))) <=> (in(A, relation_dom(C)) & in(apply(C, A), relation_dom(B))))) <=> (((~relation(C)) | (~function(C))) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B)))))))), rewrite((((~relation(C)) | (~function(C))) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B)))))) <=> ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))))), (((~(relation(C) & function(C))) | (in(A, relation_dom(relation_composition(C, B))) <=> (in(A, relation_dom(C)) & in(apply(C, A), relation_dom(B))))) <=> ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))))))), (![C: $i] : ((~(relation(C) & function(C))) | (in(A, relation_dom(relation_composition(C, B))) <=> (in(A, relation_dom(C)) & in(apply(C, A), relation_dom(B))))) <=> ![C: $i] : ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B)))))))), (((~(relation(B) & function(B))) | ![C: $i] : ((~(relation(C) & function(C))) | (in(A, relation_dom(relation_composition(C, B))) <=> (in(A, relation_dom(C)) & in(apply(C, A), relation_dom(B)))))) <=> (((~relation(B)) | (~function(B))) | ![C: $i] : ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))))))), rewrite((((~relation(B)) | (~function(B))) | ![C: $i] : ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))))) <=> ((~relation(B)) | (~function(B)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B)))))))), (((~(relation(B) & function(B))) | ![C: $i] : ((~(relation(C) & function(C))) | (in(A, relation_dom(relation_composition(C, B))) <=> (in(A, relation_dom(C)) & in(apply(C, A), relation_dom(B)))))) <=> ((~relation(B)) | (~function(B)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B)))))))))),
% 0.85/1.06 inference(bind,[status(th)],[])).
% 0.85/1.06 tff(103,plain,
% 0.85/1.06 (![A: $i, B: $i] : ((~(relation(B) & function(B))) | ![C: $i] : ((~(relation(C) & function(C))) | (in(A, relation_dom(relation_composition(C, B))) <=> (in(A, relation_dom(C)) & in(apply(C, A), relation_dom(B)))))) <=> ![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B)))))))),
% 0.85/1.06 inference(quant_intro,[status(thm)],[102])).
% 0.85/1.06 tff(104,plain,
% 0.85/1.06 (![A: $i, B: $i] : ((~(relation(B) & function(B))) | ![C: $i] : ((~(relation(C) & function(C))) | (in(A, relation_dom(relation_composition(C, B))) <=> (in(A, relation_dom(C)) & in(apply(C, A), relation_dom(B)))))) <=> ![A: $i, B: $i] : ((~(relation(B) & function(B))) | ![C: $i] : ((~(relation(C) & function(C))) | (in(A, relation_dom(relation_composition(C, B))) <=> (in(A, relation_dom(C)) & in(apply(C, A), relation_dom(B))))))),
% 0.85/1.06 inference(rewrite,[status(thm)],[])).
% 0.85/1.06 tff(105,plain,
% 0.85/1.06 (^[A: $i, B: $i] : trans(monotonicity(quant_intro(proof_bind(^[C: $i] : rewrite(((relation(C) & function(C)) => (in(A, relation_dom(relation_composition(C, B))) <=> (in(A, relation_dom(C)) & in(apply(C, A), relation_dom(B))))) <=> ((~(relation(C) & function(C))) | (in(A, relation_dom(relation_composition(C, B))) <=> (in(A, relation_dom(C)) & in(apply(C, A), relation_dom(B))))))), (![C: $i] : ((relation(C) & function(C)) => (in(A, relation_dom(relation_composition(C, B))) <=> (in(A, relation_dom(C)) & in(apply(C, A), relation_dom(B))))) <=> ![C: $i] : ((~(relation(C) & function(C))) | (in(A, relation_dom(relation_composition(C, B))) <=> (in(A, relation_dom(C)) & in(apply(C, A), relation_dom(B))))))), (((relation(B) & function(B)) => ![C: $i] : ((relation(C) & function(C)) => (in(A, relation_dom(relation_composition(C, B))) <=> (in(A, relation_dom(C)) & in(apply(C, A), relation_dom(B)))))) <=> ((relation(B) & function(B)) => ![C: $i] : ((~(relation(C) & function(C))) | (in(A, relation_dom(relation_composition(C, B))) <=> (in(A, relation_dom(C)) & in(apply(C, A), relation_dom(B)))))))), rewrite(((relation(B) & function(B)) => ![C: $i] : ((~(relation(C) & function(C))) | (in(A, relation_dom(relation_composition(C, B))) <=> (in(A, relation_dom(C)) & in(apply(C, A), relation_dom(B)))))) <=> ((~(relation(B) & function(B))) | ![C: $i] : ((~(relation(C) & function(C))) | (in(A, relation_dom(relation_composition(C, B))) <=> (in(A, relation_dom(C)) & in(apply(C, A), relation_dom(B))))))), (((relation(B) & function(B)) => ![C: $i] : ((relation(C) & function(C)) => (in(A, relation_dom(relation_composition(C, B))) <=> (in(A, relation_dom(C)) & in(apply(C, A), relation_dom(B)))))) <=> ((~(relation(B) & function(B))) | ![C: $i] : ((~(relation(C) & function(C))) | (in(A, relation_dom(relation_composition(C, B))) <=> (in(A, relation_dom(C)) & in(apply(C, A), relation_dom(B))))))))),
% 0.85/1.06 inference(bind,[status(th)],[])).
% 0.85/1.06 tff(106,plain,
% 0.85/1.06 (![A: $i, B: $i] : ((relation(B) & function(B)) => ![C: $i] : ((relation(C) & function(C)) => (in(A, relation_dom(relation_composition(C, B))) <=> (in(A, relation_dom(C)) & in(apply(C, A), relation_dom(B)))))) <=> ![A: $i, B: $i] : ((~(relation(B) & function(B))) | ![C: $i] : ((~(relation(C) & function(C))) | (in(A, relation_dom(relation_composition(C, B))) <=> (in(A, relation_dom(C)) & in(apply(C, A), relation_dom(B))))))),
% 0.85/1.06 inference(quant_intro,[status(thm)],[105])).
% 0.85/1.06 tff(107,axiom,(![A: $i, B: $i] : ((relation(B) & function(B)) => ![C: $i] : ((relation(C) & function(C)) => (in(A, relation_dom(relation_composition(C, B))) <=> (in(A, relation_dom(C)) & in(apply(C, A), relation_dom(B))))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t21_funct_1')).
% 0.85/1.06 tff(108,plain,
% 0.85/1.06 (![A: $i, B: $i] : ((~(relation(B) & function(B))) | ![C: $i] : ((~(relation(C) & function(C))) | (in(A, relation_dom(relation_composition(C, B))) <=> (in(A, relation_dom(C)) & in(apply(C, A), relation_dom(B))))))),
% 0.85/1.06 inference(modus_ponens,[status(thm)],[107, 106])).
% 0.85/1.06 tff(109,plain,
% 0.85/1.06 (![A: $i, B: $i] : ((~(relation(B) & function(B))) | ![C: $i] : ((~(relation(C) & function(C))) | (in(A, relation_dom(relation_composition(C, B))) <=> (in(A, relation_dom(C)) & in(apply(C, A), relation_dom(B))))))),
% 0.85/1.06 inference(modus_ponens,[status(thm)],[108, 104])).
% 0.85/1.06 tff(110,plain,(
% 0.85/1.06 ![A: $i, B: $i] : ((~(relation(B) & function(B))) | ![C: $i] : ((~(relation(C) & function(C))) | (in(A, relation_dom(relation_composition(C, B))) <=> (in(A, relation_dom(C)) & in(apply(C, A), relation_dom(B))))))),
% 0.85/1.06 inference(skolemize,[status(sab)],[109])).
% 0.85/1.06 tff(111,plain,
% 0.85/1.06 (![A: $i, B: $i] : ((~relation(B)) | (~function(B)) | ![C: $i] : ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B)))))))),
% 0.85/1.06 inference(modus_ponens,[status(thm)],[110, 103])).
% 0.85/1.06 tff(112,plain,
% 0.85/1.06 (![A: $i, B: $i, C: $i] : ((~relation(B)) | (~function(B)) | ((~relation(C)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B)))))))),
% 0.85/1.06 inference(modus_ponens,[status(thm)],[111, 101])).
% 0.85/1.06 tff(113,plain,
% 0.85/1.06 (![A: $i, B: $i, C: $i] : ((~relation(B)) | (~relation(C)) | (~function(B)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))))),
% 0.85/1.06 inference(modus_ponens,[status(thm)],[112, 91])).
% 0.85/1.06 tff(114,plain,
% 0.85/1.06 (((~![A: $i, B: $i, C: $i] : ((~relation(B)) | (~relation(C)) | (~function(B)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))))) | ((~relation(identity_relation(A!12))) | (~relation(C!10)) | (~function(identity_relation(A!12))) | (~function(C!10)) | (~(((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), relation_dom(identity_relation(A!12))))) <=> in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12)))))))) <=> ((~![A: $i, B: $i, C: $i] : ((~relation(B)) | (~relation(C)) | (~function(B)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))))) | (~relation(identity_relation(A!12))) | (~relation(C!10)) | (~function(identity_relation(A!12))) | (~function(C!10)) | (~(((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), relation_dom(identity_relation(A!12))))) <=> in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12)))))))),
% 0.85/1.06 inference(rewrite,[status(thm)],[])).
% 0.85/1.06 tff(115,plain,
% 0.85/1.06 ((~![A: $i, B: $i, C: $i] : ((~relation(B)) | (~relation(C)) | (~function(B)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))))) | ((~relation(identity_relation(A!12))) | (~relation(C!10)) | (~function(identity_relation(A!12))) | (~function(C!10)) | (~(((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), relation_dom(identity_relation(A!12))))) <=> in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12)))))))),
% 0.85/1.06 inference(quant_inst,[status(thm)],[])).
% 0.85/1.06 tff(116,plain,
% 0.85/1.06 ((~![A: $i, B: $i, C: $i] : ((~relation(B)) | (~relation(C)) | (~function(B)) | (~function(C)) | (~(((~in(A, relation_dom(C))) | (~in(apply(C, A), relation_dom(B)))) <=> in(A, relation_dom(relation_composition(C, B))))))) | (~relation(identity_relation(A!12))) | (~relation(C!10)) | (~function(identity_relation(A!12))) | (~function(C!10)) | (~(((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), relation_dom(identity_relation(A!12))))) <=> in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12))))))),
% 0.85/1.06 inference(modus_ponens,[status(thm)],[115, 114])).
% 0.85/1.06 tff(117,plain,
% 0.85/1.06 ((~relation(identity_relation(A!12))) | (~function(identity_relation(A!12))) | (~(((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), relation_dom(identity_relation(A!12))))) <=> in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12))))))),
% 0.85/1.06 inference(unit_resolution,[status(thm)],[116, 113, 89, 88])).
% 0.85/1.06 tff(118,plain,
% 0.85/1.06 (~(((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), relation_dom(identity_relation(A!12))))) <=> in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12)))))),
% 0.85/1.06 inference(unit_resolution,[status(thm)],[117, 14, 23])).
% 0.85/1.06 tff(119,plain,
% 0.85/1.06 ((((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), relation_dom(identity_relation(A!12))))) <=> in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12))))) | ((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), relation_dom(identity_relation(A!12))))) | in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12))))),
% 0.85/1.06 inference(tautology,[status(thm)],[])).
% 0.85/1.06 tff(120,plain,
% 0.85/1.06 (((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), relation_dom(identity_relation(A!12))))) | in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12))))),
% 0.85/1.06 inference(unit_resolution,[status(thm)],[119, 118])).
% 0.85/1.06 tff(121,plain,
% 0.85/1.06 ((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), relation_dom(identity_relation(A!12))))),
% 0.85/1.06 inference(unit_resolution,[status(thm)],[120, 61])).
% 0.85/1.06 tff(122,plain,
% 0.85/1.06 (((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), A!12))) | in(B!11, relation_dom(C!10))),
% 0.85/1.06 inference(tautology,[status(thm)],[])).
% 0.85/1.06 tff(123,plain,
% 0.85/1.06 (in(B!11, relation_dom(C!10))),
% 0.85/1.06 inference(unit_resolution,[status(thm)],[122, 83])).
% 0.85/1.06 tff(124,plain,
% 0.85/1.06 ((~((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), relation_dom(identity_relation(A!12)))))) | (~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), relation_dom(identity_relation(A!12))))),
% 0.85/1.06 inference(tautology,[status(thm)],[])).
% 0.85/1.06 tff(125,plain,
% 0.85/1.06 (~in(apply(C!10, B!11), relation_dom(identity_relation(A!12)))),
% 0.85/1.06 inference(unit_resolution,[status(thm)],[124, 123, 121])).
% 0.85/1.06 tff(126,plain,
% 0.85/1.06 ($false),
% 0.85/1.06 inference(unit_resolution,[status(thm)],[125, 86])).
% 0.85/1.06 tff(127,plain,(in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12))))), inference(lemma,lemma(discharge,[]))).
% 0.85/1.06 tff(128,plain,
% 0.85/1.06 (((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), A!12))) | (~in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12))))) | (~(((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), A!12))) <=> in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12))))))),
% 0.85/1.06 inference(tautology,[status(thm)],[])).
% 0.85/1.06 tff(129,plain,
% 0.85/1.06 (((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), A!12))) | (~in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12)))))),
% 0.85/1.06 inference(unit_resolution,[status(thm)],[128, 80])).
% 0.85/1.06 tff(130,plain,
% 0.85/1.06 ((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), A!12))),
% 0.85/1.06 inference(unit_resolution,[status(thm)],[129, 127])).
% 0.85/1.06 tff(131,plain,
% 0.85/1.06 ((((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), relation_dom(identity_relation(A!12))))) <=> in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12))))) | (~((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), relation_dom(identity_relation(A!12)))))) | (~in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12)))))),
% 0.85/1.06 inference(tautology,[status(thm)],[])).
% 0.85/1.06 tff(132,plain,
% 0.85/1.06 ((~((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), relation_dom(identity_relation(A!12)))))) | (~in(B!11, relation_dom(relation_composition(C!10, identity_relation(A!12)))))),
% 0.85/1.06 inference(unit_resolution,[status(thm)],[131, 118])).
% 0.85/1.06 tff(133,plain,
% 0.85/1.06 (~((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), relation_dom(identity_relation(A!12)))))),
% 0.85/1.06 inference(unit_resolution,[status(thm)],[132, 127])).
% 0.85/1.06 tff(134,plain,
% 0.85/1.06 (((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), relation_dom(identity_relation(A!12))))) | in(B!11, relation_dom(C!10))),
% 0.85/1.07 inference(tautology,[status(thm)],[])).
% 0.85/1.07 tff(135,plain,
% 0.85/1.07 (in(B!11, relation_dom(C!10))),
% 0.85/1.07 inference(unit_resolution,[status(thm)],[134, 133])).
% 0.85/1.07 tff(136,plain,
% 0.85/1.07 ((~((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), A!12)))) | (~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), A!12))),
% 0.85/1.07 inference(tautology,[status(thm)],[])).
% 0.85/1.07 tff(137,plain,
% 0.85/1.07 (~in(apply(C!10, B!11), A!12)),
% 0.85/1.07 inference(unit_resolution,[status(thm)],[136, 135, 130])).
% 0.85/1.07 tff(138,plain,
% 0.85/1.07 (~in(apply(C!10, B!11), relation_dom(identity_relation(A!12)))),
% 0.85/1.07 inference(modus_ponens,[status(thm)],[137, 60])).
% 0.85/1.07 tff(139,plain,
% 0.85/1.07 (((~in(B!11, relation_dom(C!10))) | (~in(apply(C!10, B!11), relation_dom(identity_relation(A!12))))) | in(apply(C!10, B!11), relation_dom(identity_relation(A!12)))),
% 0.85/1.07 inference(tautology,[status(thm)],[])).
% 0.85/1.07 tff(140,plain,
% 0.85/1.07 (in(apply(C!10, B!11), relation_dom(identity_relation(A!12)))),
% 0.85/1.07 inference(unit_resolution,[status(thm)],[139, 133])).
% 0.85/1.07 tff(141,plain,
% 0.85/1.07 ($false),
% 0.85/1.07 inference(unit_resolution,[status(thm)],[140, 138])).
% 0.85/1.07 % SZS output end Proof
% 0.85/1.07 % E exiting
%------------------------------------------------------------------------------