%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : HAL004+1 : TPTP v9.0.0. Released v2.6.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n002.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 05:08:44 AM UTC 2025
% Result : Theorem 0.20s 0.39s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.08/0.12 % Problem : HAL004+1 : TPTP v9.0.0. Released v2.6.0.
% 0.08/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n002.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Thu Jun 19 17:12:01 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.20/0.39 % SZS status Theorem
% 0.20/0.39 % SZS output start Proof
% 0.20/0.39 tff(apply_type, type, (
% 0.20/0.39 apply: ( $i * $i ) > $i)).
% 0.20/0.39 tff(tptp_fun_ElDom_2_type, type, (
% 0.20/0.39 tptp_fun_ElDom_2: ( $i * $i * $i ) > $i)).
% 0.20/0.39 tff(beta_type, type, (
% 0.20/0.39 beta: $i)).
% 0.20/0.39 tff(b_type, type, (
% 0.20/0.39 b: $i)).
% 0.20/0.39 tff(h_type, type, (
% 0.20/0.39 h: $i)).
% 0.20/0.39 tff(c_type, type, (
% 0.20/0.39 c: $i)).
% 0.20/0.39 tff(tptp_fun_E_8_type, type, (
% 0.20/0.39 tptp_fun_E_8: $i)).
% 0.20/0.39 tff(delta_type, type, (
% 0.20/0.39 delta: $i)).
% 0.20/0.39 tff(g_type, type, (
% 0.20/0.39 g: $i)).
% 0.20/0.39 tff(element_type, type, (
% 0.20/0.39 element: ( $i * $i ) > $o)).
% 0.20/0.39 tff(r_type, type, (
% 0.20/0.39 r: $i)).
% 0.20/0.39 tff(e_type, type, (
% 0.20/0.39 e: $i)).
% 0.20/0.39 tff(zero_type, type, (
% 0.20/0.39 zero: $i > $i)).
% 0.20/0.39 tff(morphism_type, type, (
% 0.20/0.39 morphism: ( $i * $i * $i ) > $o)).
% 0.20/0.39 tff(surjection_type, type, (
% 0.20/0.39 surjection: $i > $o)).
% 0.20/0.39 tff(commute_type, type, (
% 0.20/0.39 commute: ( $i * $i * $i * $i ) > $o)).
% 0.20/0.39 tff(1,plain,
% 0.20/0.39 (morphism(delta, e, r) <=> morphism(delta, e, r)),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(2,axiom,(morphism(delta, e, r)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','delta_morphism')).
% 0.20/0.39 tff(3,plain,
% 0.20/0.39 (morphism(delta, e, r)),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[2, 1])).
% 0.20/0.39 tff(4,plain,
% 0.20/0.39 (^[Morphism: $i, Dom: $i, Cod: $i] : rewrite(((~morphism(Morphism, Dom, Cod)) | (~((~![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod))) | (~(apply(Morphism, zero(Dom)) = zero(Cod)))))) <=> ((~morphism(Morphism, Dom, Cod)) | (~((~(apply(Morphism, zero(Dom)) = zero(Cod))) | (~![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod)))))))),
% 0.20/0.39 inference(bind,[status(th)],[])).
% 0.20/0.39 tff(5,plain,
% 0.20/0.39 (![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~((~![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod))) | (~(apply(Morphism, zero(Dom)) = zero(Cod)))))) <=> ![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~((~(apply(Morphism, zero(Dom)) = zero(Cod))) | (~![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod))))))),
% 0.20/0.39 inference(quant_intro,[status(thm)],[4])).
% 0.20/0.39 tff(6,plain,
% 0.20/0.39 (^[Morphism: $i, Dom: $i, Cod: $i] : refl(((~morphism(Morphism, Dom, Cod)) | (~((~![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod))) | (~(apply(Morphism, zero(Dom)) = zero(Cod)))))) <=> ((~morphism(Morphism, Dom, Cod)) | (~((~![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod))) | (~(apply(Morphism, zero(Dom)) = zero(Cod)))))))),
% 0.20/0.39 inference(bind,[status(th)],[])).
% 0.20/0.39 tff(7,plain,
% 0.20/0.39 (![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~((~![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod))) | (~(apply(Morphism, zero(Dom)) = zero(Cod)))))) <=> ![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~((~![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod))) | (~(apply(Morphism, zero(Dom)) = zero(Cod))))))),
% 0.20/0.39 inference(quant_intro,[status(thm)],[6])).
% 0.20/0.39 tff(8,plain,
% 0.20/0.39 (^[Morphism: $i, Dom: $i, Cod: $i] : rewrite(((~morphism(Morphism, Dom, Cod)) | (~((~![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod))) | (~(apply(Morphism, zero(Dom)) = zero(Cod)))))) <=> ((~morphism(Morphism, Dom, Cod)) | (~((~![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod))) | (~(apply(Morphism, zero(Dom)) = zero(Cod)))))))),
% 0.20/0.39 inference(bind,[status(th)],[])).
% 0.20/0.39 tff(9,plain,
% 0.20/0.39 (![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~((~![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod))) | (~(apply(Morphism, zero(Dom)) = zero(Cod)))))) <=> ![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~((~![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod))) | (~(apply(Morphism, zero(Dom)) = zero(Cod))))))),
% 0.20/0.39 inference(quant_intro,[status(thm)],[8])).
% 0.20/0.39 tff(10,plain,
% 0.20/0.39 (![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~((~![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod))) | (~(apply(Morphism, zero(Dom)) = zero(Cod)))))) <=> ![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~((~![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod))) | (~(apply(Morphism, zero(Dom)) = zero(Cod))))))),
% 0.20/0.39 inference(transitivity,[status(thm)],[9, 7])).
% 0.20/0.39 tff(11,plain,
% 0.20/0.39 (^[Morphism: $i, Dom: $i, Cod: $i] : rewrite(((~morphism(Morphism, Dom, Cod)) | (![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod)) & (apply(Morphism, zero(Dom)) = zero(Cod)))) <=> ((~morphism(Morphism, Dom, Cod)) | (~((~![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod))) | (~(apply(Morphism, zero(Dom)) = zero(Cod)))))))),
% 0.20/0.39 inference(bind,[status(th)],[])).
% 0.20/0.39 tff(12,plain,
% 0.20/0.39 (![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod)) & (apply(Morphism, zero(Dom)) = zero(Cod)))) <=> ![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~((~![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod))) | (~(apply(Morphism, zero(Dom)) = zero(Cod))))))),
% 0.20/0.39 inference(quant_intro,[status(thm)],[11])).
% 0.20/0.39 tff(13,plain,
% 0.20/0.39 (![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod)) & (apply(Morphism, zero(Dom)) = zero(Cod)))) <=> ![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod)) & (apply(Morphism, zero(Dom)) = zero(Cod))))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(14,plain,
% 0.20/0.39 (^[Morphism: $i, Dom: $i, Cod: $i] : trans(monotonicity(rewrite((![El: $i] : (element(El, Dom) => element(apply(Morphism, El), Cod)) & (apply(Morphism, zero(Dom)) = zero(Cod))) <=> (![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod)) & (apply(Morphism, zero(Dom)) = zero(Cod)))), ((morphism(Morphism, Dom, Cod) => (![El: $i] : (element(El, Dom) => element(apply(Morphism, El), Cod)) & (apply(Morphism, zero(Dom)) = zero(Cod)))) <=> (morphism(Morphism, Dom, Cod) => (![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod)) & (apply(Morphism, zero(Dom)) = zero(Cod)))))), rewrite((morphism(Morphism, Dom, Cod) => (![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod)) & (apply(Morphism, zero(Dom)) = zero(Cod)))) <=> ((~morphism(Morphism, Dom, Cod)) | (![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod)) & (apply(Morphism, zero(Dom)) = zero(Cod))))), ((morphism(Morphism, Dom, Cod) => (![El: $i] : (element(El, Dom) => element(apply(Morphism, El), Cod)) & (apply(Morphism, zero(Dom)) = zero(Cod)))) <=> ((~morphism(Morphism, Dom, Cod)) | (![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod)) & (apply(Morphism, zero(Dom)) = zero(Cod))))))),
% 0.20/0.39 inference(bind,[status(th)],[])).
% 0.20/0.39 tff(15,plain,
% 0.20/0.39 (![Morphism: $i, Dom: $i, Cod: $i] : (morphism(Morphism, Dom, Cod) => (![El: $i] : (element(El, Dom) => element(apply(Morphism, El), Cod)) & (apply(Morphism, zero(Dom)) = zero(Cod)))) <=> ![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod)) & (apply(Morphism, zero(Dom)) = zero(Cod))))),
% 0.20/0.39 inference(quant_intro,[status(thm)],[14])).
% 0.20/0.39 tff(16,axiom,(![Morphism: $i, Dom: $i, Cod: $i] : (morphism(Morphism, Dom, Cod) => (![El: $i] : (element(El, Dom) => element(apply(Morphism, El), Cod)) & (apply(Morphism, zero(Dom)) = zero(Cod))))), file('/export/starexec/sandbox2/benchmark/Axioms/HAL001+0.ax','morphism')).
% 0.20/0.39 tff(17,plain,
% 0.20/0.39 (![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod)) & (apply(Morphism, zero(Dom)) = zero(Cod))))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[16, 15])).
% 0.20/0.39 tff(18,plain,
% 0.20/0.39 (![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod)) & (apply(Morphism, zero(Dom)) = zero(Cod))))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[17, 13])).
% 0.20/0.39 tff(19,plain,(
% 0.20/0.39 ![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod)) & (apply(Morphism, zero(Dom)) = zero(Cod))))),
% 0.20/0.40 inference(skolemize,[status(sab)],[18])).
% 0.20/0.40 tff(20,plain,
% 0.20/0.40 (![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~((~![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod))) | (~(apply(Morphism, zero(Dom)) = zero(Cod))))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[19, 12])).
% 0.20/0.40 tff(21,plain,
% 0.20/0.40 (![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~((~![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod))) | (~(apply(Morphism, zero(Dom)) = zero(Cod))))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[20, 10])).
% 0.20/0.40 tff(22,plain,
% 0.20/0.40 (![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~((~(apply(Morphism, zero(Dom)) = zero(Cod))) | (~![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod))))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[21, 5])).
% 0.20/0.40 tff(23,plain,
% 0.20/0.40 (((~![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~((~(apply(Morphism, zero(Dom)) = zero(Cod))) | (~![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod))))))) | ((~morphism(delta, e, r)) | (~((~(apply(delta, zero(e)) = zero(r))) | (~![El: $i] : ((~element(El, e)) | element(apply(delta, El), r))))))) <=> ((~![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~((~(apply(Morphism, zero(Dom)) = zero(Cod))) | (~![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod))))))) | (~morphism(delta, e, r)) | (~((~(apply(delta, zero(e)) = zero(r))) | (~![El: $i] : ((~element(El, e)) | element(apply(delta, El), r))))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(24,plain,
% 0.20/0.40 ((~![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~((~(apply(Morphism, zero(Dom)) = zero(Cod))) | (~![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod))))))) | ((~morphism(delta, e, r)) | (~((~(apply(delta, zero(e)) = zero(r))) | (~![El: $i] : ((~element(El, e)) | element(apply(delta, El), r))))))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(25,plain,
% 0.20/0.40 ((~![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~((~(apply(Morphism, zero(Dom)) = zero(Cod))) | (~![El: $i] : ((~element(El, Dom)) | element(apply(Morphism, El), Cod))))))) | (~morphism(delta, e, r)) | (~((~(apply(delta, zero(e)) = zero(r))) | (~![El: $i] : ((~element(El, e)) | element(apply(delta, El), r)))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[24, 23])).
% 0.20/0.40 tff(26,plain,
% 0.20/0.40 (~((~(apply(delta, zero(e)) = zero(r))) | (~![El: $i] : ((~element(El, e)) | element(apply(delta, El), r))))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[25, 22, 3])).
% 0.20/0.40 tff(27,plain,
% 0.20/0.40 (((~(apply(delta, zero(e)) = zero(r))) | (~![El: $i] : ((~element(El, e)) | element(apply(delta, El), r)))) | ![El: $i] : ((~element(El, e)) | element(apply(delta, El), r))),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(28,plain,
% 0.20/0.40 (![El: $i] : ((~element(El, e)) | element(apply(delta, El), r))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[27, 26])).
% 0.20/0.40 tff(29,plain,
% 0.20/0.40 ((element(E!8, e) & ![B1: $i] : (~(element(apply(delta, apply(g, B1)), r) & (apply(delta, E!8) = apply(delta, apply(g, B1))) & element(B1, b) & (apply(h, apply(beta, B1)) = apply(delta, apply(g, B1)))))) <=> (element(E!8, e) & ![B1: $i] : (~(element(apply(delta, apply(g, B1)), r) & (apply(delta, E!8) = apply(delta, apply(g, B1))) & element(B1, b) & (apply(h, apply(beta, B1)) = apply(delta, apply(g, B1))))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(30,plain,
% 0.20/0.40 ((~![E: $i] : ((~element(E, e)) | ?[B1: $i] : (element(apply(delta, apply(g, B1)), r) & (apply(delta, E) = apply(delta, apply(g, B1))) & element(B1, b) & (apply(h, apply(beta, B1)) = apply(delta, apply(g, B1)))))) <=> (~![E: $i] : ((~element(E, e)) | ?[B1: $i] : (element(apply(delta, apply(g, B1)), r) & (apply(delta, E) = apply(delta, apply(g, B1))) & element(B1, b) & (apply(h, apply(beta, B1)) = apply(delta, apply(g, B1))))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(31,plain,
% 0.20/0.40 ((~![E: $i] : (element(E, e) => ?[R: $i, B1: $i] : ((((element(R, r) & (apply(delta, E) = R)) & element(B1, b)) & (apply(h, apply(beta, B1)) = R)) & (apply(delta, apply(g, B1)) = R)))) <=> (~![E: $i] : ((~element(E, e)) | ?[B1: $i] : (element(apply(delta, apply(g, B1)), r) & (apply(delta, E) = apply(delta, apply(g, B1))) & element(B1, b) & (apply(h, apply(beta, B1)) = apply(delta, apply(g, B1))))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(32,axiom,(~![E: $i] : (element(E, e) => ?[R: $i, B1: $i] : ((((element(R, r) & (apply(delta, E) = R)) & element(B1, b)) & (apply(h, apply(beta, B1)) = R)) & (apply(delta, apply(g, B1)) = R)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','lemma3')).
% 0.20/0.40 tff(33,plain,
% 0.20/0.40 (~![E: $i] : ((~element(E, e)) | ?[B1: $i] : (element(apply(delta, apply(g, B1)), r) & (apply(delta, E) = apply(delta, apply(g, B1))) & element(B1, b) & (apply(h, apply(beta, B1)) = apply(delta, apply(g, B1)))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[32, 31])).
% 0.20/0.40 tff(34,plain,
% 0.20/0.40 (~![E: $i] : ((~element(E, e)) | ?[B1: $i] : (element(apply(delta, apply(g, B1)), r) & (apply(delta, E) = apply(delta, apply(g, B1))) & element(B1, b) & (apply(h, apply(beta, B1)) = apply(delta, apply(g, B1)))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[33, 30])).
% 0.20/0.40 tff(35,plain,
% 0.20/0.40 (~![E: $i] : ((~element(E, e)) | ?[B1: $i] : (element(apply(delta, apply(g, B1)), r) & (apply(delta, E) = apply(delta, apply(g, B1))) & element(B1, b) & (apply(h, apply(beta, B1)) = apply(delta, apply(g, B1)))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[34, 30])).
% 0.20/0.40 tff(36,plain,
% 0.20/0.40 (~![E: $i] : ((~element(E, e)) | ?[B1: $i] : (element(apply(delta, apply(g, B1)), r) & (apply(delta, E) = apply(delta, apply(g, B1))) & element(B1, b) & (apply(h, apply(beta, B1)) = apply(delta, apply(g, B1)))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[35, 30])).
% 0.20/0.40 tff(37,plain,
% 0.20/0.40 (element(E!8, e) & ![B1: $i] : (~(element(apply(delta, apply(g, B1)), r) & (apply(delta, E!8) = apply(delta, apply(g, B1))) & element(B1, b) & (apply(h, apply(beta, B1)) = apply(delta, apply(g, B1)))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[36, 29])).
% 0.20/0.40 tff(38,plain,
% 0.20/0.40 (element(E!8, e)),
% 0.20/0.40 inference(and_elim,[status(thm)],[37])).
% 0.20/0.40 tff(39,plain,
% 0.20/0.40 (((~![El: $i] : ((~element(El, e)) | element(apply(delta, El), r))) | ((~element(E!8, e)) | element(apply(delta, E!8), r))) <=> ((~![El: $i] : ((~element(El, e)) | element(apply(delta, El), r))) | (~element(E!8, e)) | element(apply(delta, E!8), r))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(40,plain,
% 0.20/0.40 ((~![El: $i] : ((~element(El, e)) | element(apply(delta, El), r))) | ((~element(E!8, e)) | element(apply(delta, E!8), r))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(41,plain,
% 0.20/0.40 ((~![El: $i] : ((~element(El, e)) | element(apply(delta, El), r))) | (~element(E!8, e)) | element(apply(delta, E!8), r)),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[40, 39])).
% 0.20/0.40 tff(42,plain,
% 0.20/0.40 (element(apply(delta, E!8), r)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[41, 38, 28])).
% 0.20/0.40 tff(43,plain,
% 0.20/0.40 (surjection(h) <=> surjection(h)),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(44,axiom,(surjection(h)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','h_surjection')).
% 0.20/0.40 tff(45,plain,
% 0.20/0.40 (surjection(h)),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[44, 43])).
% 0.20/0.40 tff(46,plain,
% 0.20/0.40 (morphism(h, c, r) <=> morphism(h, c, r)),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(47,axiom,(morphism(h, c, r)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','h_morphism')).
% 0.20/0.40 tff(48,plain,
% 0.20/0.40 (morphism(h, c, r)),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[47, 46])).
% 0.20/0.40 tff(49,plain,
% 0.20/0.40 (^[Morphism: $i, Dom: $i, Cod: $i] : refl(((~morphism(Morphism, Dom, Cod)) | (~surjection(Morphism)) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (~((~element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom)) | (~(apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod)))))) <=> ((~morphism(Morphism, Dom, Cod)) | (~surjection(Morphism)) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (~((~element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom)) | (~(apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod)))))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(50,plain,
% 0.20/0.40 (![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~surjection(Morphism)) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (~((~element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom)) | (~(apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod)))))) <=> ![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~surjection(Morphism)) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (~((~element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom)) | (~(apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod))))))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[49])).
% 0.20/0.40 tff(51,plain,
% 0.20/0.40 (^[Morphism: $i, Dom: $i, Cod: $i] : rewrite(((~morphism(Morphism, Dom, Cod)) | (~surjection(Morphism)) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (~((~element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom)) | (~(apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod)))))) <=> ((~morphism(Morphism, Dom, Cod)) | (~surjection(Morphism)) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (~((~element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom)) | (~(apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod)))))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(52,plain,
% 0.20/0.40 (![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~surjection(Morphism)) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (~((~element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom)) | (~(apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod)))))) <=> ![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~surjection(Morphism)) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (~((~element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom)) | (~(apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod))))))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[51])).
% 0.20/0.40 tff(53,plain,
% 0.20/0.40 (![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~surjection(Morphism)) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (~((~element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom)) | (~(apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod)))))) <=> ![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~surjection(Morphism)) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (~((~element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom)) | (~(apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod))))))),
% 0.20/0.40 inference(transitivity,[status(thm)],[52, 50])).
% 0.20/0.40 tff(54,plain,
% 0.20/0.40 (^[Morphism: $i, Dom: $i, Cod: $i] : trans(monotonicity(trans(monotonicity(rewrite((surjection(Morphism) & morphism(Morphism, Dom, Cod)) <=> (~((~surjection(Morphism)) | (~morphism(Morphism, Dom, Cod))))), ((~(surjection(Morphism) & morphism(Morphism, Dom, Cod))) <=> (~(~((~surjection(Morphism)) | (~morphism(Morphism, Dom, Cod))))))), rewrite((~(~((~surjection(Morphism)) | (~morphism(Morphism, Dom, Cod))))) <=> ((~surjection(Morphism)) | (~morphism(Morphism, Dom, Cod)))), ((~(surjection(Morphism) & morphism(Morphism, Dom, Cod))) <=> ((~surjection(Morphism)) | (~morphism(Morphism, Dom, Cod))))), quant_intro(proof_bind(^[ElCod: $i] : rewrite(((~element(ElCod, Cod)) | (element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom) & (apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod))) <=> ((~element(ElCod, Cod)) | (~((~element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom)) | (~(apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod))))))), (![ElCod: $i] : ((~element(ElCod, Cod)) | (element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom) & (apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod))) <=> ![ElCod: $i] : ((~element(ElCod, Cod)) | (~((~element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom)) | (~(apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod))))))), (((~(surjection(Morphism) & morphism(Morphism, Dom, Cod))) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom) & (apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod)))) <=> (((~surjection(Morphism)) | (~morphism(Morphism, Dom, Cod))) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (~((~element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom)) | (~(apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod)))))))), rewrite((((~surjection(Morphism)) | (~morphism(Morphism, Dom, Cod))) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (~((~element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom)) | (~(apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod)))))) <=> ((~morphism(Morphism, Dom, Cod)) | (~surjection(Morphism)) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (~((~element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom)) | (~(apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod))))))), (((~(surjection(Morphism) & morphism(Morphism, Dom, Cod))) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom) & (apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod)))) <=> ((~morphism(Morphism, Dom, Cod)) | (~surjection(Morphism)) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (~((~element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom)) | (~(apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod))))))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(55,plain,
% 0.20/0.40 (![Morphism: $i, Dom: $i, Cod: $i] : ((~(surjection(Morphism) & morphism(Morphism, Dom, Cod))) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom) & (apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod)))) <=> ![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~surjection(Morphism)) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (~((~element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom)) | (~(apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod))))))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[54])).
% 0.20/0.40 tff(56,plain,
% 0.20/0.40 (![Morphism: $i, Dom: $i, Cod: $i] : ((~(surjection(Morphism) & morphism(Morphism, Dom, Cod))) | ![ElCod: $i] : ((~element(ElCod, Cod)) | ?[ElDom: $i] : (element(ElDom, Dom) & (apply(Morphism, ElDom) = ElCod)))) <=> ![Morphism: $i, Dom: $i, Cod: $i] : ((~(surjection(Morphism) & morphism(Morphism, Dom, Cod))) | ![ElCod: $i] : ((~element(ElCod, Cod)) | ?[ElDom: $i] : (element(ElDom, Dom) & (apply(Morphism, ElDom) = ElCod))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(57,plain,
% 0.20/0.40 (^[Morphism: $i, Dom: $i, Cod: $i] : trans(monotonicity(quant_intro(proof_bind(^[ElCod: $i] : rewrite((element(ElCod, Cod) => ?[ElDom: $i] : (element(ElDom, Dom) & (apply(Morphism, ElDom) = ElCod))) <=> ((~element(ElCod, Cod)) | ?[ElDom: $i] : (element(ElDom, Dom) & (apply(Morphism, ElDom) = ElCod))))), (![ElCod: $i] : (element(ElCod, Cod) => ?[ElDom: $i] : (element(ElDom, Dom) & (apply(Morphism, ElDom) = ElCod))) <=> ![ElCod: $i] : ((~element(ElCod, Cod)) | ?[ElDom: $i] : (element(ElDom, Dom) & (apply(Morphism, ElDom) = ElCod))))), (((surjection(Morphism) & morphism(Morphism, Dom, Cod)) => ![ElCod: $i] : (element(ElCod, Cod) => ?[ElDom: $i] : (element(ElDom, Dom) & (apply(Morphism, ElDom) = ElCod)))) <=> ((surjection(Morphism) & morphism(Morphism, Dom, Cod)) => ![ElCod: $i] : ((~element(ElCod, Cod)) | ?[ElDom: $i] : (element(ElDom, Dom) & (apply(Morphism, ElDom) = ElCod)))))), rewrite(((surjection(Morphism) & morphism(Morphism, Dom, Cod)) => ![ElCod: $i] : ((~element(ElCod, Cod)) | ?[ElDom: $i] : (element(ElDom, Dom) & (apply(Morphism, ElDom) = ElCod)))) <=> ((~(surjection(Morphism) & morphism(Morphism, Dom, Cod))) | ![ElCod: $i] : ((~element(ElCod, Cod)) | ?[ElDom: $i] : (element(ElDom, Dom) & (apply(Morphism, ElDom) = ElCod))))), (((surjection(Morphism) & morphism(Morphism, Dom, Cod)) => ![ElCod: $i] : (element(ElCod, Cod) => ?[ElDom: $i] : (element(ElDom, Dom) & (apply(Morphism, ElDom) = ElCod)))) <=> ((~(surjection(Morphism) & morphism(Morphism, Dom, Cod))) | ![ElCod: $i] : ((~element(ElCod, Cod)) | ?[ElDom: $i] : (element(ElDom, Dom) & (apply(Morphism, ElDom) = ElCod))))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(58,plain,
% 0.20/0.40 (![Morphism: $i, Dom: $i, Cod: $i] : ((surjection(Morphism) & morphism(Morphism, Dom, Cod)) => ![ElCod: $i] : (element(ElCod, Cod) => ?[ElDom: $i] : (element(ElDom, Dom) & (apply(Morphism, ElDom) = ElCod)))) <=> ![Morphism: $i, Dom: $i, Cod: $i] : ((~(surjection(Morphism) & morphism(Morphism, Dom, Cod))) | ![ElCod: $i] : ((~element(ElCod, Cod)) | ?[ElDom: $i] : (element(ElDom, Dom) & (apply(Morphism, ElDom) = ElCod))))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[57])).
% 0.20/0.41 tff(59,axiom,(![Morphism: $i, Dom: $i, Cod: $i] : ((surjection(Morphism) & morphism(Morphism, Dom, Cod)) => ![ElCod: $i] : (element(ElCod, Cod) => ?[ElDom: $i] : (element(ElDom, Dom) & (apply(Morphism, ElDom) = ElCod))))), file('/export/starexec/sandbox2/benchmark/Axioms/HAL001+0.ax','surjection_properties')).
% 0.20/0.41 tff(60,plain,
% 0.20/0.41 (![Morphism: $i, Dom: $i, Cod: $i] : ((~(surjection(Morphism) & morphism(Morphism, Dom, Cod))) | ![ElCod: $i] : ((~element(ElCod, Cod)) | ?[ElDom: $i] : (element(ElDom, Dom) & (apply(Morphism, ElDom) = ElCod))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[59, 58])).
% 0.20/0.41 tff(61,plain,
% 0.20/0.41 (![Morphism: $i, Dom: $i, Cod: $i] : ((~(surjection(Morphism) & morphism(Morphism, Dom, Cod))) | ![ElCod: $i] : ((~element(ElCod, Cod)) | ?[ElDom: $i] : (element(ElDom, Dom) & (apply(Morphism, ElDom) = ElCod))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[60, 56])).
% 0.20/0.41 tff(62,plain,(
% 0.20/0.41 ![Morphism: $i, Dom: $i, Cod: $i] : ((~(surjection(Morphism) & morphism(Morphism, Dom, Cod))) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom) & (apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod))))),
% 0.20/0.41 inference(skolemize,[status(sab)],[61])).
% 0.20/0.41 tff(63,plain,
% 0.20/0.41 (![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~surjection(Morphism)) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (~((~element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom)) | (~(apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod))))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[62, 55])).
% 0.20/0.41 tff(64,plain,
% 0.20/0.41 (![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~surjection(Morphism)) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (~((~element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom)) | (~(apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod))))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[63, 53])).
% 0.20/0.41 tff(65,plain,
% 0.20/0.41 (((~![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~surjection(Morphism)) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (~((~element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom)) | (~(apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod))))))) | ((~morphism(h, c, r)) | ![ElCod: $i] : ((~element(ElCod, r)) | (~((~element(tptp_fun_ElDom_2(ElCod, c, h), c)) | (~(apply(h, tptp_fun_ElDom_2(ElCod, c, h)) = ElCod))))) | (~surjection(h)))) <=> ((~![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~surjection(Morphism)) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (~((~element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom)) | (~(apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod))))))) | (~morphism(h, c, r)) | ![ElCod: $i] : ((~element(ElCod, r)) | (~((~element(tptp_fun_ElDom_2(ElCod, c, h), c)) | (~(apply(h, tptp_fun_ElDom_2(ElCod, c, h)) = ElCod))))) | (~surjection(h)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(66,plain,
% 0.20/0.41 (((~morphism(h, c, r)) | (~surjection(h)) | ![ElCod: $i] : ((~element(ElCod, r)) | (~((~element(tptp_fun_ElDom_2(ElCod, c, h), c)) | (~(apply(h, tptp_fun_ElDom_2(ElCod, c, h)) = ElCod)))))) <=> ((~morphism(h, c, r)) | ![ElCod: $i] : ((~element(ElCod, r)) | (~((~element(tptp_fun_ElDom_2(ElCod, c, h), c)) | (~(apply(h, tptp_fun_ElDom_2(ElCod, c, h)) = ElCod))))) | (~surjection(h)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(67,plain,
% 0.20/0.41 (((~![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~surjection(Morphism)) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (~((~element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom)) | (~(apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod))))))) | ((~morphism(h, c, r)) | (~surjection(h)) | ![ElCod: $i] : ((~element(ElCod, r)) | (~((~element(tptp_fun_ElDom_2(ElCod, c, h), c)) | (~(apply(h, tptp_fun_ElDom_2(ElCod, c, h)) = ElCod))))))) <=> ((~![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~surjection(Morphism)) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (~((~element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom)) | (~(apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod))))))) | ((~morphism(h, c, r)) | ![ElCod: $i] : ((~element(ElCod, r)) | (~((~element(tptp_fun_ElDom_2(ElCod, c, h), c)) | (~(apply(h, tptp_fun_ElDom_2(ElCod, c, h)) = ElCod))))) | (~surjection(h))))),
% 0.20/0.41 inference(monotonicity,[status(thm)],[66])).
% 0.20/0.41 tff(68,plain,
% 0.20/0.41 (((~![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~surjection(Morphism)) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (~((~element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom)) | (~(apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod))))))) | ((~morphism(h, c, r)) | (~surjection(h)) | ![ElCod: $i] : ((~element(ElCod, r)) | (~((~element(tptp_fun_ElDom_2(ElCod, c, h), c)) | (~(apply(h, tptp_fun_ElDom_2(ElCod, c, h)) = ElCod))))))) <=> ((~![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~surjection(Morphism)) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (~((~element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom)) | (~(apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod))))))) | (~morphism(h, c, r)) | ![ElCod: $i] : ((~element(ElCod, r)) | (~((~element(tptp_fun_ElDom_2(ElCod, c, h), c)) | (~(apply(h, tptp_fun_ElDom_2(ElCod, c, h)) = ElCod))))) | (~surjection(h)))),
% 0.20/0.41 inference(transitivity,[status(thm)],[67, 65])).
% 0.20/0.41 tff(69,plain,
% 0.20/0.41 ((~![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~surjection(Morphism)) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (~((~element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom)) | (~(apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod))))))) | ((~morphism(h, c, r)) | (~surjection(h)) | ![ElCod: $i] : ((~element(ElCod, r)) | (~((~element(tptp_fun_ElDom_2(ElCod, c, h), c)) | (~(apply(h, tptp_fun_ElDom_2(ElCod, c, h)) = ElCod))))))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(70,plain,
% 0.20/0.41 ((~![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~surjection(Morphism)) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (~((~element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom)) | (~(apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod))))))) | (~morphism(h, c, r)) | ![ElCod: $i] : ((~element(ElCod, r)) | (~((~element(tptp_fun_ElDom_2(ElCod, c, h), c)) | (~(apply(h, tptp_fun_ElDom_2(ElCod, c, h)) = ElCod))))) | (~surjection(h))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[69, 68])).
% 0.20/0.41 tff(71,plain,
% 0.20/0.41 (![ElCod: $i] : ((~element(ElCod, r)) | (~((~element(tptp_fun_ElDom_2(ElCod, c, h), c)) | (~(apply(h, tptp_fun_ElDom_2(ElCod, c, h)) = ElCod)))))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[70, 64, 48, 45])).
% 0.20/0.41 tff(72,plain,
% 0.20/0.41 (((~![ElCod: $i] : ((~element(ElCod, r)) | (~((~element(tptp_fun_ElDom_2(ElCod, c, h), c)) | (~(apply(h, tptp_fun_ElDom_2(ElCod, c, h)) = ElCod)))))) | ((~element(apply(delta, E!8), r)) | (~((~element(tptp_fun_ElDom_2(apply(delta, E!8), c, h), c)) | (~(apply(h, tptp_fun_ElDom_2(apply(delta, E!8), c, h)) = apply(delta, E!8))))))) <=> ((~![ElCod: $i] : ((~element(ElCod, r)) | (~((~element(tptp_fun_ElDom_2(ElCod, c, h), c)) | (~(apply(h, tptp_fun_ElDom_2(ElCod, c, h)) = ElCod)))))) | (~element(apply(delta, E!8), r)) | (~((~element(tptp_fun_ElDom_2(apply(delta, E!8), c, h), c)) | (~(apply(h, tptp_fun_ElDom_2(apply(delta, E!8), c, h)) = apply(delta, E!8))))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(73,plain,
% 0.20/0.41 ((~![ElCod: $i] : ((~element(ElCod, r)) | (~((~element(tptp_fun_ElDom_2(ElCod, c, h), c)) | (~(apply(h, tptp_fun_ElDom_2(ElCod, c, h)) = ElCod)))))) | ((~element(apply(delta, E!8), r)) | (~((~element(tptp_fun_ElDom_2(apply(delta, E!8), c, h), c)) | (~(apply(h, tptp_fun_ElDom_2(apply(delta, E!8), c, h)) = apply(delta, E!8))))))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(74,plain,
% 0.20/0.41 ((~![ElCod: $i] : ((~element(ElCod, r)) | (~((~element(tptp_fun_ElDom_2(ElCod, c, h), c)) | (~(apply(h, tptp_fun_ElDom_2(ElCod, c, h)) = ElCod)))))) | (~element(apply(delta, E!8), r)) | (~((~element(tptp_fun_ElDom_2(apply(delta, E!8), c, h), c)) | (~(apply(h, tptp_fun_ElDom_2(apply(delta, E!8), c, h)) = apply(delta, E!8)))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[73, 72])).
% 0.20/0.41 tff(75,plain,
% 0.20/0.41 (~((~element(tptp_fun_ElDom_2(apply(delta, E!8), c, h), c)) | (~(apply(h, tptp_fun_ElDom_2(apply(delta, E!8), c, h)) = apply(delta, E!8))))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[74, 71, 42])).
% 0.20/0.41 tff(76,plain,
% 0.20/0.41 (((~element(tptp_fun_ElDom_2(apply(delta, E!8), c, h), c)) | (~(apply(h, tptp_fun_ElDom_2(apply(delta, E!8), c, h)) = apply(delta, E!8)))) | element(tptp_fun_ElDom_2(apply(delta, E!8), c, h), c)),
% 0.20/0.41 inference(tautology,[status(thm)],[])).
% 0.20/0.41 tff(77,plain,
% 0.20/0.41 (element(tptp_fun_ElDom_2(apply(delta, E!8), c, h), c)),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[76, 75])).
% 0.20/0.41 tff(78,plain,
% 0.20/0.41 (surjection(beta) <=> surjection(beta)),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(79,axiom,(surjection(beta)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','beta_surjection')).
% 0.20/0.41 tff(80,plain,
% 0.20/0.41 (surjection(beta)),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[79, 78])).
% 0.20/0.41 tff(81,plain,
% 0.20/0.41 (morphism(beta, b, c) <=> morphism(beta, b, c)),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(82,axiom,(morphism(beta, b, c)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','beta_morphism')).
% 0.20/0.41 tff(83,plain,
% 0.20/0.41 (morphism(beta, b, c)),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[82, 81])).
% 0.20/0.41 tff(84,plain,
% 0.20/0.41 (((~![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~surjection(Morphism)) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (~((~element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom)) | (~(apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod))))))) | ((~morphism(beta, b, c)) | ![ElCod: $i] : ((~element(ElCod, c)) | (~((~element(tptp_fun_ElDom_2(ElCod, b, beta), b)) | (~(apply(beta, tptp_fun_ElDom_2(ElCod, b, beta)) = ElCod))))) | (~surjection(beta)))) <=> ((~![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~surjection(Morphism)) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (~((~element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom)) | (~(apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod))))))) | (~morphism(beta, b, c)) | ![ElCod: $i] : ((~element(ElCod, c)) | (~((~element(tptp_fun_ElDom_2(ElCod, b, beta), b)) | (~(apply(beta, tptp_fun_ElDom_2(ElCod, b, beta)) = ElCod))))) | (~surjection(beta)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(85,plain,
% 0.20/0.41 (((~morphism(beta, b, c)) | (~surjection(beta)) | ![ElCod: $i] : ((~element(ElCod, c)) | (~((~element(tptp_fun_ElDom_2(ElCod, b, beta), b)) | (~(apply(beta, tptp_fun_ElDom_2(ElCod, b, beta)) = ElCod)))))) <=> ((~morphism(beta, b, c)) | ![ElCod: $i] : ((~element(ElCod, c)) | (~((~element(tptp_fun_ElDom_2(ElCod, b, beta), b)) | (~(apply(beta, tptp_fun_ElDom_2(ElCod, b, beta)) = ElCod))))) | (~surjection(beta)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(86,plain,
% 0.20/0.41 (((~![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~surjection(Morphism)) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (~((~element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom)) | (~(apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod))))))) | ((~morphism(beta, b, c)) | (~surjection(beta)) | ![ElCod: $i] : ((~element(ElCod, c)) | (~((~element(tptp_fun_ElDom_2(ElCod, b, beta), b)) | (~(apply(beta, tptp_fun_ElDom_2(ElCod, b, beta)) = ElCod))))))) <=> ((~![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~surjection(Morphism)) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (~((~element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom)) | (~(apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod))))))) | ((~morphism(beta, b, c)) | ![ElCod: $i] : ((~element(ElCod, c)) | (~((~element(tptp_fun_ElDom_2(ElCod, b, beta), b)) | (~(apply(beta, tptp_fun_ElDom_2(ElCod, b, beta)) = ElCod))))) | (~surjection(beta))))),
% 0.20/0.41 inference(monotonicity,[status(thm)],[85])).
% 0.20/0.42 tff(87,plain,
% 0.20/0.42 (((~![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~surjection(Morphism)) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (~((~element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom)) | (~(apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod))))))) | ((~morphism(beta, b, c)) | (~surjection(beta)) | ![ElCod: $i] : ((~element(ElCod, c)) | (~((~element(tptp_fun_ElDom_2(ElCod, b, beta), b)) | (~(apply(beta, tptp_fun_ElDom_2(ElCod, b, beta)) = ElCod))))))) <=> ((~![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~surjection(Morphism)) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (~((~element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom)) | (~(apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod))))))) | (~morphism(beta, b, c)) | ![ElCod: $i] : ((~element(ElCod, c)) | (~((~element(tptp_fun_ElDom_2(ElCod, b, beta), b)) | (~(apply(beta, tptp_fun_ElDom_2(ElCod, b, beta)) = ElCod))))) | (~surjection(beta)))),
% 0.20/0.42 inference(transitivity,[status(thm)],[86, 84])).
% 0.20/0.42 tff(88,plain,
% 0.20/0.42 ((~![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~surjection(Morphism)) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (~((~element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom)) | (~(apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod))))))) | ((~morphism(beta, b, c)) | (~surjection(beta)) | ![ElCod: $i] : ((~element(ElCod, c)) | (~((~element(tptp_fun_ElDom_2(ElCod, b, beta), b)) | (~(apply(beta, tptp_fun_ElDom_2(ElCod, b, beta)) = ElCod))))))),
% 0.20/0.42 inference(quant_inst,[status(thm)],[])).
% 0.20/0.42 tff(89,plain,
% 0.20/0.42 ((~![Morphism: $i, Dom: $i, Cod: $i] : ((~morphism(Morphism, Dom, Cod)) | (~surjection(Morphism)) | ![ElCod: $i] : ((~element(ElCod, Cod)) | (~((~element(tptp_fun_ElDom_2(ElCod, Dom, Morphism), Dom)) | (~(apply(Morphism, tptp_fun_ElDom_2(ElCod, Dom, Morphism)) = ElCod))))))) | (~morphism(beta, b, c)) | ![ElCod: $i] : ((~element(ElCod, c)) | (~((~element(tptp_fun_ElDom_2(ElCod, b, beta), b)) | (~(apply(beta, tptp_fun_ElDom_2(ElCod, b, beta)) = ElCod))))) | (~surjection(beta))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[88, 87])).
% 0.20/0.42 tff(90,plain,
% 0.20/0.42 (![ElCod: $i] : ((~element(ElCod, c)) | (~((~element(tptp_fun_ElDom_2(ElCod, b, beta), b)) | (~(apply(beta, tptp_fun_ElDom_2(ElCod, b, beta)) = ElCod)))))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[89, 64, 83, 80])).
% 0.20/0.42 tff(91,plain,
% 0.20/0.42 (((~![ElCod: $i] : ((~element(ElCod, c)) | (~((~element(tptp_fun_ElDom_2(ElCod, b, beta), b)) | (~(apply(beta, tptp_fun_ElDom_2(ElCod, b, beta)) = ElCod)))))) | ((~element(tptp_fun_ElDom_2(apply(delta, E!8), c, h), c)) | (~((~element(tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta), b)) | (~(apply(beta, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta)) = tptp_fun_ElDom_2(apply(delta, E!8), c, h))))))) <=> ((~![ElCod: $i] : ((~element(ElCod, c)) | (~((~element(tptp_fun_ElDom_2(ElCod, b, beta), b)) | (~(apply(beta, tptp_fun_ElDom_2(ElCod, b, beta)) = ElCod)))))) | (~element(tptp_fun_ElDom_2(apply(delta, E!8), c, h), c)) | (~((~element(tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta), b)) | (~(apply(beta, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta)) = tptp_fun_ElDom_2(apply(delta, E!8), c, h))))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(92,plain,
% 0.20/0.42 ((~![ElCod: $i] : ((~element(ElCod, c)) | (~((~element(tptp_fun_ElDom_2(ElCod, b, beta), b)) | (~(apply(beta, tptp_fun_ElDom_2(ElCod, b, beta)) = ElCod)))))) | ((~element(tptp_fun_ElDom_2(apply(delta, E!8), c, h), c)) | (~((~element(tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta), b)) | (~(apply(beta, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta)) = tptp_fun_ElDom_2(apply(delta, E!8), c, h))))))),
% 0.20/0.42 inference(quant_inst,[status(thm)],[])).
% 0.20/0.42 tff(93,plain,
% 0.20/0.42 ((~![ElCod: $i] : ((~element(ElCod, c)) | (~((~element(tptp_fun_ElDom_2(ElCod, b, beta), b)) | (~(apply(beta, tptp_fun_ElDom_2(ElCod, b, beta)) = ElCod)))))) | (~element(tptp_fun_ElDom_2(apply(delta, E!8), c, h), c)) | (~((~element(tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta), b)) | (~(apply(beta, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta)) = tptp_fun_ElDom_2(apply(delta, E!8), c, h)))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[92, 91])).
% 0.20/0.42 tff(94,plain,
% 0.20/0.42 (~((~element(tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta), b)) | (~(apply(beta, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta)) = tptp_fun_ElDom_2(apply(delta, E!8), c, h))))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[93, 90, 77])).
% 0.20/0.42 tff(95,plain,
% 0.20/0.42 (((~element(tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta), b)) | (~(apply(beta, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta)) = tptp_fun_ElDom_2(apply(delta, E!8), c, h)))) | element(tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta), b)),
% 0.20/0.42 inference(tautology,[status(thm)],[])).
% 0.20/0.42 tff(96,plain,
% 0.20/0.42 (element(tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta), b)),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[95, 94])).
% 0.20/0.42 tff(97,plain,
% 0.20/0.42 (commute(beta, h, g, delta) <=> commute(beta, h, g, delta)),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(98,axiom,(commute(beta, h, g, delta)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','beta_h_g_delta_commute')).
% 0.20/0.42 tff(99,plain,
% 0.20/0.42 (commute(beta, h, g, delta)),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[98, 97])).
% 0.20/0.42 tff(100,plain,
% 0.20/0.42 (morphism(g, b, e) <=> morphism(g, b, e)),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(101,axiom,(morphism(g, b, e)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','g_morphism')).
% 0.20/0.42 tff(102,plain,
% 0.20/0.42 (morphism(g, b, e)),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[101, 100])).
% 0.20/0.42 tff(103,plain,
% 0.20/0.42 (^[M1: $i, M2: $i, M3: $i, M4: $i, Dom: $i, DomCod1: $i, DomCod2: $i, Cod: $i] : rewrite((![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))) | (~morphism(M1, Dom, DomCod1)) | (~morphism(M2, DomCod1, Cod)) | (~morphism(M3, Dom, DomCod2)) | (~morphism(M4, DomCod2, Cod)) | (~commute(M1, M2, M3, M4))) <=> ((~morphism(M1, Dom, DomCod1)) | (~morphism(M2, DomCod1, Cod)) | (~morphism(M3, Dom, DomCod2)) | (~morphism(M4, DomCod2, Cod)) | (~commute(M1, M2, M3, M4)) | ![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom))))))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(104,plain,
% 0.20/0.42 (![M1: $i, M2: $i, M3: $i, M4: $i, Dom: $i, DomCod1: $i, DomCod2: $i, Cod: $i] : (![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))) | (~morphism(M1, Dom, DomCod1)) | (~morphism(M2, DomCod1, Cod)) | (~morphism(M3, Dom, DomCod2)) | (~morphism(M4, DomCod2, Cod)) | (~commute(M1, M2, M3, M4))) <=> ![M1: $i, M2: $i, M3: $i, M4: $i, Dom: $i, DomCod1: $i, DomCod2: $i, Cod: $i] : ((~morphism(M1, Dom, DomCod1)) | (~morphism(M2, DomCod1, Cod)) | (~morphism(M3, Dom, DomCod2)) | (~morphism(M4, DomCod2, Cod)) | (~commute(M1, M2, M3, M4)) | ![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))))),
% 0.20/0.42 inference(quant_intro,[status(thm)],[103])).
% 0.20/0.42 tff(105,plain,
% 0.20/0.42 (^[M1: $i, M2: $i, M3: $i, M4: $i, Dom: $i, DomCod1: $i, DomCod2: $i, Cod: $i] : refl((![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))) | (~morphism(M1, Dom, DomCod1)) | (~morphism(M2, DomCod1, Cod)) | (~morphism(M3, Dom, DomCod2)) | (~morphism(M4, DomCod2, Cod)) | (~commute(M1, M2, M3, M4))) <=> (![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))) | (~morphism(M1, Dom, DomCod1)) | (~morphism(M2, DomCod1, Cod)) | (~morphism(M3, Dom, DomCod2)) | (~morphism(M4, DomCod2, Cod)) | (~commute(M1, M2, M3, M4))))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(106,plain,
% 0.20/0.42 (![M1: $i, M2: $i, M3: $i, M4: $i, Dom: $i, DomCod1: $i, DomCod2: $i, Cod: $i] : (![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))) | (~morphism(M1, Dom, DomCod1)) | (~morphism(M2, DomCod1, Cod)) | (~morphism(M3, Dom, DomCod2)) | (~morphism(M4, DomCod2, Cod)) | (~commute(M1, M2, M3, M4))) <=> ![M1: $i, M2: $i, M3: $i, M4: $i, Dom: $i, DomCod1: $i, DomCod2: $i, Cod: $i] : (![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))) | (~morphism(M1, Dom, DomCod1)) | (~morphism(M2, DomCod1, Cod)) | (~morphism(M3, Dom, DomCod2)) | (~morphism(M4, DomCod2, Cod)) | (~commute(M1, M2, M3, M4)))),
% 0.20/0.43 inference(quant_intro,[status(thm)],[105])).
% 0.20/0.43 tff(107,plain,
% 0.20/0.43 (^[M1: $i, M2: $i, M3: $i, M4: $i, Dom: $i, DomCod1: $i, DomCod2: $i, Cod: $i] : rewrite((![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))) | (~morphism(M1, Dom, DomCod1)) | (~morphism(M2, DomCod1, Cod)) | (~morphism(M3, Dom, DomCod2)) | (~morphism(M4, DomCod2, Cod)) | (~commute(M1, M2, M3, M4))) <=> (![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))) | (~morphism(M1, Dom, DomCod1)) | (~morphism(M2, DomCod1, Cod)) | (~morphism(M3, Dom, DomCod2)) | (~morphism(M4, DomCod2, Cod)) | (~commute(M1, M2, M3, M4))))),
% 0.20/0.43 inference(bind,[status(th)],[])).
% 0.20/0.43 tff(108,plain,
% 0.20/0.43 (![M1: $i, M2: $i, M3: $i, M4: $i, Dom: $i, DomCod1: $i, DomCod2: $i, Cod: $i] : (![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))) | (~morphism(M1, Dom, DomCod1)) | (~morphism(M2, DomCod1, Cod)) | (~morphism(M3, Dom, DomCod2)) | (~morphism(M4, DomCod2, Cod)) | (~commute(M1, M2, M3, M4))) <=> ![M1: $i, M2: $i, M3: $i, M4: $i, Dom: $i, DomCod1: $i, DomCod2: $i, Cod: $i] : (![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))) | (~morphism(M1, Dom, DomCod1)) | (~morphism(M2, DomCod1, Cod)) | (~morphism(M3, Dom, DomCod2)) | (~morphism(M4, DomCod2, Cod)) | (~commute(M1, M2, M3, M4)))),
% 0.20/0.43 inference(quant_intro,[status(thm)],[107])).
% 0.20/0.43 tff(109,plain,
% 0.20/0.43 (![M1: $i, M2: $i, M3: $i, M4: $i, Dom: $i, DomCod1: $i, DomCod2: $i, Cod: $i] : (![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))) | (~morphism(M1, Dom, DomCod1)) | (~morphism(M2, DomCod1, Cod)) | (~morphism(M3, Dom, DomCod2)) | (~morphism(M4, DomCod2, Cod)) | (~commute(M1, M2, M3, M4))) <=> ![M1: $i, M2: $i, M3: $i, M4: $i, Dom: $i, DomCod1: $i, DomCod2: $i, Cod: $i] : (![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))) | (~morphism(M1, Dom, DomCod1)) | (~morphism(M2, DomCod1, Cod)) | (~morphism(M3, Dom, DomCod2)) | (~morphism(M4, DomCod2, Cod)) | (~commute(M1, M2, M3, M4)))),
% 0.20/0.43 inference(transitivity,[status(thm)],[108, 106])).
% 0.20/0.43 tff(110,plain,
% 0.20/0.43 (^[M1: $i, M2: $i, M3: $i, M4: $i, Dom: $i, DomCod1: $i, DomCod2: $i, Cod: $i] : trans(monotonicity(trans(monotonicity(rewrite((commute(M1, M2, M3, M4) & morphism(M1, Dom, DomCod1) & morphism(M2, DomCod1, Cod) & morphism(M3, Dom, DomCod2) & morphism(M4, DomCod2, Cod)) <=> (~((~morphism(M1, Dom, DomCod1)) | (~morphism(M2, DomCod1, Cod)) | (~morphism(M3, Dom, DomCod2)) | (~morphism(M4, DomCod2, Cod)) | (~commute(M1, M2, M3, M4))))), ((~(commute(M1, M2, M3, M4) & morphism(M1, Dom, DomCod1) & morphism(M2, DomCod1, Cod) & morphism(M3, Dom, DomCod2) & morphism(M4, DomCod2, Cod))) <=> (~(~((~morphism(M1, Dom, DomCod1)) | (~morphism(M2, DomCod1, Cod)) | (~morphism(M3, Dom, DomCod2)) | (~morphism(M4, DomCod2, Cod)) | (~commute(M1, M2, M3, M4))))))), rewrite((~(~((~morphism(M1, Dom, DomCod1)) | (~morphism(M2, DomCod1, Cod)) | (~morphism(M3, Dom, DomCod2)) | (~morphism(M4, DomCod2, Cod)) | (~commute(M1, M2, M3, M4))))) <=> ((~morphism(M1, Dom, DomCod1)) | (~morphism(M2, DomCod1, Cod)) | (~morphism(M3, Dom, DomCod2)) | (~morphism(M4, DomCod2, Cod)) | (~commute(M1, M2, M3, M4)))), ((~(commute(M1, M2, M3, M4) & morphism(M1, Dom, DomCod1) & morphism(M2, DomCod1, Cod) & morphism(M3, Dom, DomCod2) & morphism(M4, DomCod2, Cod))) <=> ((~morphism(M1, Dom, DomCod1)) | (~morphism(M2, DomCod1, Cod)) | (~morphism(M3, Dom, DomCod2)) | (~morphism(M4, DomCod2, Cod)) | (~commute(M1, M2, M3, M4))))), (((~(commute(M1, M2, M3, M4) & morphism(M1, Dom, DomCod1) & morphism(M2, DomCod1, Cod) & morphism(M3, Dom, DomCod2) & morphism(M4, DomCod2, Cod))) | ![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom))))) <=> (((~morphism(M1, Dom, DomCod1)) | (~morphism(M2, DomCod1, Cod)) | (~morphism(M3, Dom, DomCod2)) | (~morphism(M4, DomCod2, Cod)) | (~commute(M1, M2, M3, M4))) | ![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom))))))), rewrite((((~morphism(M1, Dom, DomCod1)) | (~morphism(M2, DomCod1, Cod)) | (~morphism(M3, Dom, DomCod2)) | (~morphism(M4, DomCod2, Cod)) | (~commute(M1, M2, M3, M4))) | ![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom))))) <=> (![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))) | (~morphism(M1, Dom, DomCod1)) | (~morphism(M2, DomCod1, Cod)) | (~morphism(M3, Dom, DomCod2)) | (~morphism(M4, DomCod2, Cod)) | (~commute(M1, M2, M3, M4)))), (((~(commute(M1, M2, M3, M4) & morphism(M1, Dom, DomCod1) & morphism(M2, DomCod1, Cod) & morphism(M3, Dom, DomCod2) & morphism(M4, DomCod2, Cod))) | ![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom))))) <=> (![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))) | (~morphism(M1, Dom, DomCod1)) | (~morphism(M2, DomCod1, Cod)) | (~morphism(M3, Dom, DomCod2)) | (~morphism(M4, DomCod2, Cod)) | (~commute(M1, M2, M3, M4)))))),
% 0.20/0.43 inference(bind,[status(th)],[])).
% 0.20/0.43 tff(111,plain,
% 0.20/0.43 (![M1: $i, M2: $i, M3: $i, M4: $i, Dom: $i, DomCod1: $i, DomCod2: $i, Cod: $i] : ((~(commute(M1, M2, M3, M4) & morphism(M1, Dom, DomCod1) & morphism(M2, DomCod1, Cod) & morphism(M3, Dom, DomCod2) & morphism(M4, DomCod2, Cod))) | ![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom))))) <=> ![M1: $i, M2: $i, M3: $i, M4: $i, Dom: $i, DomCod1: $i, DomCod2: $i, Cod: $i] : (![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))) | (~morphism(M1, Dom, DomCod1)) | (~morphism(M2, DomCod1, Cod)) | (~morphism(M3, Dom, DomCod2)) | (~morphism(M4, DomCod2, Cod)) | (~commute(M1, M2, M3, M4)))),
% 0.20/0.43 inference(quant_intro,[status(thm)],[110])).
% 0.20/0.43 tff(112,plain,
% 0.20/0.43 (![M1: $i, M2: $i, M3: $i, M4: $i, Dom: $i, DomCod1: $i, DomCod2: $i, Cod: $i] : ((~(commute(M1, M2, M3, M4) & morphism(M1, Dom, DomCod1) & morphism(M2, DomCod1, Cod) & morphism(M3, Dom, DomCod2) & morphism(M4, DomCod2, Cod))) | ![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom))))) <=> ![M1: $i, M2: $i, M3: $i, M4: $i, Dom: $i, DomCod1: $i, DomCod2: $i, Cod: $i] : ((~(commute(M1, M2, M3, M4) & morphism(M1, Dom, DomCod1) & morphism(M2, DomCod1, Cod) & morphism(M3, Dom, DomCod2) & morphism(M4, DomCod2, Cod))) | ![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(113,plain,
% 0.20/0.43 (^[M1: $i, M2: $i, M3: $i, M4: $i, Dom: $i, DomCod1: $i, DomCod2: $i, Cod: $i] : trans(monotonicity(trans(monotonicity(trans(monotonicity(rewrite(((commute(M1, M2, M3, M4) & morphism(M1, Dom, DomCod1)) & morphism(M2, DomCod1, Cod)) <=> (commute(M1, M2, M3, M4) & morphism(M1, Dom, DomCod1) & morphism(M2, DomCod1, Cod))), ((((commute(M1, M2, M3, M4) & morphism(M1, Dom, DomCod1)) & morphism(M2, DomCod1, Cod)) & morphism(M3, Dom, DomCod2)) <=> ((commute(M1, M2, M3, M4) & morphism(M1, Dom, DomCod1) & morphism(M2, DomCod1, Cod)) & morphism(M3, Dom, DomCod2)))), rewrite(((commute(M1, M2, M3, M4) & morphism(M1, Dom, DomCod1) & morphism(M2, DomCod1, Cod)) & morphism(M3, Dom, DomCod2)) <=> (commute(M1, M2, M3, M4) & morphism(M1, Dom, DomCod1) & morphism(M2, DomCod1, Cod) & morphism(M3, Dom, DomCod2))), ((((commute(M1, M2, M3, M4) & morphism(M1, Dom, DomCod1)) & morphism(M2, DomCod1, Cod)) & morphism(M3, Dom, DomCod2)) <=> (commute(M1, M2, M3, M4) & morphism(M1, Dom, DomCod1) & morphism(M2, DomCod1, Cod) & morphism(M3, Dom, DomCod2)))), (((((commute(M1, M2, M3, M4) & morphism(M1, Dom, DomCod1)) & morphism(M2, DomCod1, Cod)) & morphism(M3, Dom, DomCod2)) & morphism(M4, DomCod2, Cod)) <=> ((commute(M1, M2, M3, M4) & morphism(M1, Dom, DomCod1) & morphism(M2, DomCod1, Cod) & morphism(M3, Dom, DomCod2)) & morphism(M4, DomCod2, Cod)))), rewrite(((commute(M1, M2, M3, M4) & morphism(M1, Dom, DomCod1) & morphism(M2, DomCod1, Cod) & morphism(M3, Dom, DomCod2)) & morphism(M4, DomCod2, Cod)) <=> (commute(M1, M2, M3, M4) & morphism(M1, Dom, DomCod1) & morphism(M2, DomCod1, Cod) & morphism(M3, Dom, DomCod2) & morphism(M4, DomCod2, Cod))), (((((commute(M1, M2, M3, M4) & morphism(M1, Dom, DomCod1)) & morphism(M2, DomCod1, Cod)) & morphism(M3, Dom, DomCod2)) & morphism(M4, DomCod2, Cod)) <=> (commute(M1, M2, M3, M4) & morphism(M1, Dom, DomCod1) & morphism(M2, DomCod1, Cod) & morphism(M3, Dom, DomCod2) & morphism(M4, DomCod2, Cod)))), quant_intro(proof_bind(^[ElDom: $i] : rewrite((element(ElDom, Dom) => (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))) <=> ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))))), (![ElDom: $i] : (element(ElDom, Dom) => (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))) <=> ![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))))), ((((((commute(M1, M2, M3, M4) & morphism(M1, Dom, DomCod1)) & morphism(M2, DomCod1, Cod)) & morphism(M3, Dom, DomCod2)) & morphism(M4, DomCod2, Cod)) => ![ElDom: $i] : (element(ElDom, Dom) => (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom))))) <=> ((commute(M1, M2, M3, M4) & morphism(M1, Dom, DomCod1) & morphism(M2, DomCod1, Cod) & morphism(M3, Dom, DomCod2) & morphism(M4, DomCod2, Cod)) => ![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom))))))), rewrite(((commute(M1, M2, M3, M4) & morphism(M1, Dom, DomCod1) & morphism(M2, DomCod1, Cod) & morphism(M3, Dom, DomCod2) & morphism(M4, DomCod2, Cod)) => ![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom))))) <=> ((~(commute(M1, M2, M3, M4) & morphism(M1, Dom, DomCod1) & morphism(M2, DomCod1, Cod) & morphism(M3, Dom, DomCod2) & morphism(M4, DomCod2, Cod))) | ![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))))), ((((((commute(M1, M2, M3, M4) & morphism(M1, Dom, DomCod1)) & morphism(M2, DomCod1, Cod)) & morphism(M3, Dom, DomCod2)) & morphism(M4, DomCod2, Cod)) => ![ElDom: $i] : (element(ElDom, Dom) => (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom))))) <=> ((~(commute(M1, M2, M3, M4) & morphism(M1, Dom, DomCod1) & morphism(M2, DomCod1, Cod) & morphism(M3, Dom, DomCod2) & morphism(M4, DomCod2, Cod))) | ![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))))))),
% 0.20/0.43 inference(bind,[status(th)],[])).
% 0.20/0.43 tff(114,plain,
% 0.20/0.43 (![M1: $i, M2: $i, M3: $i, M4: $i, Dom: $i, DomCod1: $i, DomCod2: $i, Cod: $i] : (((((commute(M1, M2, M3, M4) & morphism(M1, Dom, DomCod1)) & morphism(M2, DomCod1, Cod)) & morphism(M3, Dom, DomCod2)) & morphism(M4, DomCod2, Cod)) => ![ElDom: $i] : (element(ElDom, Dom) => (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom))))) <=> ![M1: $i, M2: $i, M3: $i, M4: $i, Dom: $i, DomCod1: $i, DomCod2: $i, Cod: $i] : ((~(commute(M1, M2, M3, M4) & morphism(M1, Dom, DomCod1) & morphism(M2, DomCod1, Cod) & morphism(M3, Dom, DomCod2) & morphism(M4, DomCod2, Cod))) | ![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))))),
% 0.20/0.43 inference(quant_intro,[status(thm)],[113])).
% 0.20/0.43 tff(115,axiom,(![M1: $i, M2: $i, M3: $i, M4: $i, Dom: $i, DomCod1: $i, DomCod2: $i, Cod: $i] : (((((commute(M1, M2, M3, M4) & morphism(M1, Dom, DomCod1)) & morphism(M2, DomCod1, Cod)) & morphism(M3, Dom, DomCod2)) & morphism(M4, DomCod2, Cod)) => ![ElDom: $i] : (element(ElDom, Dom) => (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))))), file('/export/starexec/sandbox2/benchmark/Axioms/HAL001+0.ax','commute_properties')).
% 0.20/0.44 tff(116,plain,
% 0.20/0.44 (![M1: $i, M2: $i, M3: $i, M4: $i, Dom: $i, DomCod1: $i, DomCod2: $i, Cod: $i] : ((~(commute(M1, M2, M3, M4) & morphism(M1, Dom, DomCod1) & morphism(M2, DomCod1, Cod) & morphism(M3, Dom, DomCod2) & morphism(M4, DomCod2, Cod))) | ![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[115, 114])).
% 0.20/0.44 tff(117,plain,
% 0.20/0.44 (![M1: $i, M2: $i, M3: $i, M4: $i, Dom: $i, DomCod1: $i, DomCod2: $i, Cod: $i] : ((~(commute(M1, M2, M3, M4) & morphism(M1, Dom, DomCod1) & morphism(M2, DomCod1, Cod) & morphism(M3, Dom, DomCod2) & morphism(M4, DomCod2, Cod))) | ![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[116, 112])).
% 0.20/0.44 tff(118,plain,(
% 0.20/0.44 ![M1: $i, M2: $i, M3: $i, M4: $i, Dom: $i, DomCod1: $i, DomCod2: $i, Cod: $i] : ((~(commute(M1, M2, M3, M4) & morphism(M1, Dom, DomCod1) & morphism(M2, DomCod1, Cod) & morphism(M3, Dom, DomCod2) & morphism(M4, DomCod2, Cod))) | ![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))))),
% 0.20/0.44 inference(skolemize,[status(sab)],[117])).
% 0.20/0.44 tff(119,plain,
% 0.20/0.44 (![M1: $i, M2: $i, M3: $i, M4: $i, Dom: $i, DomCod1: $i, DomCod2: $i, Cod: $i] : (![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))) | (~morphism(M1, Dom, DomCod1)) | (~morphism(M2, DomCod1, Cod)) | (~morphism(M3, Dom, DomCod2)) | (~morphism(M4, DomCod2, Cod)) | (~commute(M1, M2, M3, M4)))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[118, 111])).
% 0.20/0.44 tff(120,plain,
% 0.20/0.44 (![M1: $i, M2: $i, M3: $i, M4: $i, Dom: $i, DomCod1: $i, DomCod2: $i, Cod: $i] : (![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))) | (~morphism(M1, Dom, DomCod1)) | (~morphism(M2, DomCod1, Cod)) | (~morphism(M3, Dom, DomCod2)) | (~morphism(M4, DomCod2, Cod)) | (~commute(M1, M2, M3, M4)))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[119, 109])).
% 0.20/0.44 tff(121,plain,
% 0.20/0.44 (![M1: $i, M2: $i, M3: $i, M4: $i, Dom: $i, DomCod1: $i, DomCod2: $i, Cod: $i] : ((~morphism(M1, Dom, DomCod1)) | (~morphism(M2, DomCod1, Cod)) | (~morphism(M3, Dom, DomCod2)) | (~morphism(M4, DomCod2, Cod)) | (~commute(M1, M2, M3, M4)) | ![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[120, 104])).
% 0.20/0.44 tff(122,plain,
% 0.20/0.44 (((~![M1: $i, M2: $i, M3: $i, M4: $i, Dom: $i, DomCod1: $i, DomCod2: $i, Cod: $i] : ((~morphism(M1, Dom, DomCod1)) | (~morphism(M2, DomCod1, Cod)) | (~morphism(M3, Dom, DomCod2)) | (~morphism(M4, DomCod2, Cod)) | (~commute(M1, M2, M3, M4)) | ![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))))) | ((~morphism(g, b, e)) | (~morphism(beta, b, c)) | (~morphism(h, c, r)) | (~morphism(delta, e, r)) | ![ElDom: $i] : ((~element(ElDom, b)) | (apply(h, apply(beta, ElDom)) = apply(delta, apply(g, ElDom)))) | (~commute(beta, h, g, delta)))) <=> ((~![M1: $i, M2: $i, M3: $i, M4: $i, Dom: $i, DomCod1: $i, DomCod2: $i, Cod: $i] : ((~morphism(M1, Dom, DomCod1)) | (~morphism(M2, DomCod1, Cod)) | (~morphism(M3, Dom, DomCod2)) | (~morphism(M4, DomCod2, Cod)) | (~commute(M1, M2, M3, M4)) | ![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))))) | (~morphism(g, b, e)) | (~morphism(beta, b, c)) | (~morphism(h, c, r)) | (~morphism(delta, e, r)) | ![ElDom: $i] : ((~element(ElDom, b)) | (apply(h, apply(beta, ElDom)) = apply(delta, apply(g, ElDom)))) | (~commute(beta, h, g, delta)))),
% 0.20/0.44 inference(rewrite,[status(thm)],[])).
% 0.20/0.44 tff(123,plain,
% 0.20/0.44 (((~morphism(beta, b, c)) | (~morphism(h, c, r)) | (~morphism(g, b, e)) | (~morphism(delta, e, r)) | (~commute(beta, h, g, delta)) | ![ElDom: $i] : ((~element(ElDom, b)) | (apply(h, apply(beta, ElDom)) = apply(delta, apply(g, ElDom))))) <=> ((~morphism(g, b, e)) | (~morphism(beta, b, c)) | (~morphism(h, c, r)) | (~morphism(delta, e, r)) | ![ElDom: $i] : ((~element(ElDom, b)) | (apply(h, apply(beta, ElDom)) = apply(delta, apply(g, ElDom)))) | (~commute(beta, h, g, delta)))),
% 0.20/0.44 inference(rewrite,[status(thm)],[])).
% 0.20/0.44 tff(124,plain,
% 0.20/0.44 (((~![M1: $i, M2: $i, M3: $i, M4: $i, Dom: $i, DomCod1: $i, DomCod2: $i, Cod: $i] : ((~morphism(M1, Dom, DomCod1)) | (~morphism(M2, DomCod1, Cod)) | (~morphism(M3, Dom, DomCod2)) | (~morphism(M4, DomCod2, Cod)) | (~commute(M1, M2, M3, M4)) | ![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))))) | ((~morphism(beta, b, c)) | (~morphism(h, c, r)) | (~morphism(g, b, e)) | (~morphism(delta, e, r)) | (~commute(beta, h, g, delta)) | ![ElDom: $i] : ((~element(ElDom, b)) | (apply(h, apply(beta, ElDom)) = apply(delta, apply(g, ElDom)))))) <=> ((~![M1: $i, M2: $i, M3: $i, M4: $i, Dom: $i, DomCod1: $i, DomCod2: $i, Cod: $i] : ((~morphism(M1, Dom, DomCod1)) | (~morphism(M2, DomCod1, Cod)) | (~morphism(M3, Dom, DomCod2)) | (~morphism(M4, DomCod2, Cod)) | (~commute(M1, M2, M3, M4)) | ![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))))) | ((~morphism(g, b, e)) | (~morphism(beta, b, c)) | (~morphism(h, c, r)) | (~morphism(delta, e, r)) | ![ElDom: $i] : ((~element(ElDom, b)) | (apply(h, apply(beta, ElDom)) = apply(delta, apply(g, ElDom)))) | (~commute(beta, h, g, delta))))),
% 0.20/0.44 inference(monotonicity,[status(thm)],[123])).
% 0.20/0.44 tff(125,plain,
% 0.20/0.44 (((~![M1: $i, M2: $i, M3: $i, M4: $i, Dom: $i, DomCod1: $i, DomCod2: $i, Cod: $i] : ((~morphism(M1, Dom, DomCod1)) | (~morphism(M2, DomCod1, Cod)) | (~morphism(M3, Dom, DomCod2)) | (~morphism(M4, DomCod2, Cod)) | (~commute(M1, M2, M3, M4)) | ![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))))) | ((~morphism(beta, b, c)) | (~morphism(h, c, r)) | (~morphism(g, b, e)) | (~morphism(delta, e, r)) | (~commute(beta, h, g, delta)) | ![ElDom: $i] : ((~element(ElDom, b)) | (apply(h, apply(beta, ElDom)) = apply(delta, apply(g, ElDom)))))) <=> ((~![M1: $i, M2: $i, M3: $i, M4: $i, Dom: $i, DomCod1: $i, DomCod2: $i, Cod: $i] : ((~morphism(M1, Dom, DomCod1)) | (~morphism(M2, DomCod1, Cod)) | (~morphism(M3, Dom, DomCod2)) | (~morphism(M4, DomCod2, Cod)) | (~commute(M1, M2, M3, M4)) | ![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))))) | (~morphism(g, b, e)) | (~morphism(beta, b, c)) | (~morphism(h, c, r)) | (~morphism(delta, e, r)) | ![ElDom: $i] : ((~element(ElDom, b)) | (apply(h, apply(beta, ElDom)) = apply(delta, apply(g, ElDom)))) | (~commute(beta, h, g, delta)))),
% 0.20/0.44 inference(transitivity,[status(thm)],[124, 122])).
% 0.20/0.44 tff(126,plain,
% 0.20/0.44 ((~![M1: $i, M2: $i, M3: $i, M4: $i, Dom: $i, DomCod1: $i, DomCod2: $i, Cod: $i] : ((~morphism(M1, Dom, DomCod1)) | (~morphism(M2, DomCod1, Cod)) | (~morphism(M3, Dom, DomCod2)) | (~morphism(M4, DomCod2, Cod)) | (~commute(M1, M2, M3, M4)) | ![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))))) | ((~morphism(beta, b, c)) | (~morphism(h, c, r)) | (~morphism(g, b, e)) | (~morphism(delta, e, r)) | (~commute(beta, h, g, delta)) | ![ElDom: $i] : ((~element(ElDom, b)) | (apply(h, apply(beta, ElDom)) = apply(delta, apply(g, ElDom)))))),
% 0.20/0.44 inference(quant_inst,[status(thm)],[])).
% 0.20/0.44 tff(127,plain,
% 0.20/0.44 ((~![M1: $i, M2: $i, M3: $i, M4: $i, Dom: $i, DomCod1: $i, DomCod2: $i, Cod: $i] : ((~morphism(M1, Dom, DomCod1)) | (~morphism(M2, DomCod1, Cod)) | (~morphism(M3, Dom, DomCod2)) | (~morphism(M4, DomCod2, Cod)) | (~commute(M1, M2, M3, M4)) | ![ElDom: $i] : ((~element(ElDom, Dom)) | (apply(M2, apply(M1, ElDom)) = apply(M4, apply(M3, ElDom)))))) | (~morphism(g, b, e)) | (~morphism(beta, b, c)) | (~morphism(h, c, r)) | (~morphism(delta, e, r)) | ![ElDom: $i] : ((~element(ElDom, b)) | (apply(h, apply(beta, ElDom)) = apply(delta, apply(g, ElDom)))) | (~commute(beta, h, g, delta))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[126, 125])).
% 0.20/0.44 tff(128,plain,
% 0.20/0.44 (![ElDom: $i] : ((~element(ElDom, b)) | (apply(h, apply(beta, ElDom)) = apply(delta, apply(g, ElDom))))),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[127, 121, 83, 3, 102, 48, 99])).
% 0.20/0.45 tff(129,plain,
% 0.20/0.45 (((~![ElDom: $i] : ((~element(ElDom, b)) | (apply(h, apply(beta, ElDom)) = apply(delta, apply(g, ElDom))))) | ((~element(tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta), b)) | (apply(h, apply(beta, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))) = apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta)))))) <=> ((~![ElDom: $i] : ((~element(ElDom, b)) | (apply(h, apply(beta, ElDom)) = apply(delta, apply(g, ElDom))))) | (~element(tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta), b)) | (apply(h, apply(beta, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))) = apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta)))))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(130,plain,
% 0.20/0.45 ((~![ElDom: $i] : ((~element(ElDom, b)) | (apply(h, apply(beta, ElDom)) = apply(delta, apply(g, ElDom))))) | ((~element(tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta), b)) | (apply(h, apply(beta, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))) = apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta)))))),
% 0.20/0.45 inference(quant_inst,[status(thm)],[])).
% 0.20/0.45 tff(131,plain,
% 0.20/0.45 ((~![ElDom: $i] : ((~element(ElDom, b)) | (apply(h, apply(beta, ElDom)) = apply(delta, apply(g, ElDom))))) | (~element(tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta), b)) | (apply(h, apply(beta, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))) = apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[130, 129])).
% 0.20/0.45 tff(132,plain,
% 0.20/0.45 (apply(h, apply(beta, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))) = apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta)))),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[131, 128, 96])).
% 0.20/0.45 tff(133,plain,
% 0.20/0.45 (((~element(tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta), b)) | (~(apply(beta, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta)) = tptp_fun_ElDom_2(apply(delta, E!8), c, h)))) | (apply(beta, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta)) = tptp_fun_ElDom_2(apply(delta, E!8), c, h))),
% 0.20/0.45 inference(tautology,[status(thm)],[])).
% 0.20/0.45 tff(134,plain,
% 0.20/0.45 (apply(beta, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta)) = tptp_fun_ElDom_2(apply(delta, E!8), c, h)),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[133, 94])).
% 0.20/0.45 tff(135,plain,
% 0.20/0.45 (apply(h, apply(beta, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))) = apply(h, tptp_fun_ElDom_2(apply(delta, E!8), c, h))),
% 0.20/0.45 inference(monotonicity,[status(thm)],[134])).
% 0.20/0.45 tff(136,plain,
% 0.20/0.45 (apply(h, tptp_fun_ElDom_2(apply(delta, E!8), c, h)) = apply(h, apply(beta, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta)))),
% 0.20/0.45 inference(symmetry,[status(thm)],[135])).
% 0.20/0.45 tff(137,plain,
% 0.20/0.45 (((~element(tptp_fun_ElDom_2(apply(delta, E!8), c, h), c)) | (~(apply(h, tptp_fun_ElDom_2(apply(delta, E!8), c, h)) = apply(delta, E!8)))) | (apply(h, tptp_fun_ElDom_2(apply(delta, E!8), c, h)) = apply(delta, E!8))),
% 0.20/0.45 inference(tautology,[status(thm)],[])).
% 0.20/0.45 tff(138,plain,
% 0.20/0.45 (apply(h, tptp_fun_ElDom_2(apply(delta, E!8), c, h)) = apply(delta, E!8)),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[137, 75])).
% 0.20/0.45 tff(139,plain,
% 0.20/0.45 (apply(delta, E!8) = apply(h, tptp_fun_ElDom_2(apply(delta, E!8), c, h))),
% 0.20/0.45 inference(symmetry,[status(thm)],[138])).
% 0.20/0.45 tff(140,plain,
% 0.20/0.45 (apply(delta, E!8) = apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta)))),
% 0.20/0.45 inference(transitivity,[status(thm)],[139, 136, 132])).
% 0.20/0.45 tff(141,plain,
% 0.20/0.45 (apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))) = apply(h, apply(beta, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta)))),
% 0.20/0.45 inference(symmetry,[status(thm)],[132])).
% 0.20/0.45 tff(142,plain,
% 0.20/0.45 (apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))) = apply(delta, E!8)),
% 0.20/0.45 inference(transitivity,[status(thm)],[141, 135, 138])).
% 0.20/0.45 tff(143,plain,
% 0.20/0.45 (element(apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))), r) <=> element(apply(delta, E!8), r)),
% 0.20/0.45 inference(monotonicity,[status(thm)],[142])).
% 0.20/0.45 tff(144,plain,
% 0.20/0.45 (element(apply(delta, E!8), r) <=> element(apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))), r)),
% 0.20/0.45 inference(symmetry,[status(thm)],[143])).
% 0.20/0.45 tff(145,plain,
% 0.20/0.45 (element(apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))), r)),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[42, 144])).
% 0.20/0.45 tff(146,plain,
% 0.20/0.45 (^[B1: $i] : refl(((~element(apply(delta, apply(g, B1)), r)) | (~(apply(delta, E!8) = apply(delta, apply(g, B1)))) | (~element(B1, b)) | (~(apply(h, apply(beta, B1)) = apply(delta, apply(g, B1))))) <=> ((~element(apply(delta, apply(g, B1)), r)) | (~(apply(delta, E!8) = apply(delta, apply(g, B1)))) | (~element(B1, b)) | (~(apply(h, apply(beta, B1)) = apply(delta, apply(g, B1))))))),
% 0.20/0.45 inference(bind,[status(th)],[])).
% 0.20/0.45 tff(147,plain,
% 0.20/0.45 (![B1: $i] : ((~element(apply(delta, apply(g, B1)), r)) | (~(apply(delta, E!8) = apply(delta, apply(g, B1)))) | (~element(B1, b)) | (~(apply(h, apply(beta, B1)) = apply(delta, apply(g, B1))))) <=> ![B1: $i] : ((~element(apply(delta, apply(g, B1)), r)) | (~(apply(delta, E!8) = apply(delta, apply(g, B1)))) | (~element(B1, b)) | (~(apply(h, apply(beta, B1)) = apply(delta, apply(g, B1)))))),
% 0.20/0.45 inference(quant_intro,[status(thm)],[146])).
% 0.20/0.45 tff(148,plain,
% 0.20/0.45 (^[B1: $i] : trans(monotonicity(rewrite((element(apply(delta, apply(g, B1)), r) & (apply(delta, E!8) = apply(delta, apply(g, B1))) & element(B1, b) & (apply(h, apply(beta, B1)) = apply(delta, apply(g, B1)))) <=> (~((~element(apply(delta, apply(g, B1)), r)) | (~(apply(delta, E!8) = apply(delta, apply(g, B1)))) | (~element(B1, b)) | (~(apply(h, apply(beta, B1)) = apply(delta, apply(g, B1))))))), ((~(element(apply(delta, apply(g, B1)), r) & (apply(delta, E!8) = apply(delta, apply(g, B1))) & element(B1, b) & (apply(h, apply(beta, B1)) = apply(delta, apply(g, B1))))) <=> (~(~((~element(apply(delta, apply(g, B1)), r)) | (~(apply(delta, E!8) = apply(delta, apply(g, B1)))) | (~element(B1, b)) | (~(apply(h, apply(beta, B1)) = apply(delta, apply(g, B1))))))))), rewrite((~(~((~element(apply(delta, apply(g, B1)), r)) | (~(apply(delta, E!8) = apply(delta, apply(g, B1)))) | (~element(B1, b)) | (~(apply(h, apply(beta, B1)) = apply(delta, apply(g, B1))))))) <=> ((~element(apply(delta, apply(g, B1)), r)) | (~(apply(delta, E!8) = apply(delta, apply(g, B1)))) | (~element(B1, b)) | (~(apply(h, apply(beta, B1)) = apply(delta, apply(g, B1)))))), ((~(element(apply(delta, apply(g, B1)), r) & (apply(delta, E!8) = apply(delta, apply(g, B1))) & element(B1, b) & (apply(h, apply(beta, B1)) = apply(delta, apply(g, B1))))) <=> ((~element(apply(delta, apply(g, B1)), r)) | (~(apply(delta, E!8) = apply(delta, apply(g, B1)))) | (~element(B1, b)) | (~(apply(h, apply(beta, B1)) = apply(delta, apply(g, B1)))))))),
% 0.20/0.45 inference(bind,[status(th)],[])).
% 0.20/0.45 tff(149,plain,
% 0.20/0.45 (![B1: $i] : (~(element(apply(delta, apply(g, B1)), r) & (apply(delta, E!8) = apply(delta, apply(g, B1))) & element(B1, b) & (apply(h, apply(beta, B1)) = apply(delta, apply(g, B1))))) <=> ![B1: $i] : ((~element(apply(delta, apply(g, B1)), r)) | (~(apply(delta, E!8) = apply(delta, apply(g, B1)))) | (~element(B1, b)) | (~(apply(h, apply(beta, B1)) = apply(delta, apply(g, B1)))))),
% 0.20/0.45 inference(quant_intro,[status(thm)],[148])).
% 0.20/0.45 tff(150,plain,
% 0.20/0.45 (![B1: $i] : (~(element(apply(delta, apply(g, B1)), r) & (apply(delta, E!8) = apply(delta, apply(g, B1))) & element(B1, b) & (apply(h, apply(beta, B1)) = apply(delta, apply(g, B1)))))),
% 0.20/0.45 inference(and_elim,[status(thm)],[37])).
% 0.20/0.46 tff(151,plain,
% 0.20/0.46 (![B1: $i] : ((~element(apply(delta, apply(g, B1)), r)) | (~(apply(delta, E!8) = apply(delta, apply(g, B1)))) | (~element(B1, b)) | (~(apply(h, apply(beta, B1)) = apply(delta, apply(g, B1)))))),
% 0.20/0.46 inference(modus_ponens,[status(thm)],[150, 149])).
% 0.20/0.46 tff(152,plain,
% 0.20/0.46 (![B1: $i] : ((~element(apply(delta, apply(g, B1)), r)) | (~(apply(delta, E!8) = apply(delta, apply(g, B1)))) | (~element(B1, b)) | (~(apply(h, apply(beta, B1)) = apply(delta, apply(g, B1)))))),
% 0.20/0.46 inference(modus_ponens,[status(thm)],[151, 147])).
% 0.20/0.46 tff(153,plain,
% 0.20/0.46 (((~![B1: $i] : ((~element(apply(delta, apply(g, B1)), r)) | (~(apply(delta, E!8) = apply(delta, apply(g, B1)))) | (~element(B1, b)) | (~(apply(h, apply(beta, B1)) = apply(delta, apply(g, B1)))))) | ((~element(tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta), b)) | (~element(apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))), r)) | (~(apply(delta, E!8) = apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))))) | (~(apply(h, apply(beta, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))) = apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))))))) <=> ((~![B1: $i] : ((~element(apply(delta, apply(g, B1)), r)) | (~(apply(delta, E!8) = apply(delta, apply(g, B1)))) | (~element(B1, b)) | (~(apply(h, apply(beta, B1)) = apply(delta, apply(g, B1)))))) | (~element(tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta), b)) | (~element(apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))), r)) | (~(apply(delta, E!8) = apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))))) | (~(apply(h, apply(beta, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))) = apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))))))),
% 0.20/0.46 inference(rewrite,[status(thm)],[])).
% 0.20/0.46 tff(154,plain,
% 0.20/0.46 (((~element(apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))), r)) | (~(apply(delta, E!8) = apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))))) | (~element(tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta), b)) | (~(apply(h, apply(beta, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))) = apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta)))))) <=> ((~element(tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta), b)) | (~element(apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))), r)) | (~(apply(delta, E!8) = apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))))) | (~(apply(h, apply(beta, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))) = apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))))))),
% 0.20/0.46 inference(rewrite,[status(thm)],[])).
% 0.20/0.46 tff(155,plain,
% 0.20/0.46 (((~![B1: $i] : ((~element(apply(delta, apply(g, B1)), r)) | (~(apply(delta, E!8) = apply(delta, apply(g, B1)))) | (~element(B1, b)) | (~(apply(h, apply(beta, B1)) = apply(delta, apply(g, B1)))))) | ((~element(apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))), r)) | (~(apply(delta, E!8) = apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))))) | (~element(tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta), b)) | (~(apply(h, apply(beta, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))) = apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))))))) <=> ((~![B1: $i] : ((~element(apply(delta, apply(g, B1)), r)) | (~(apply(delta, E!8) = apply(delta, apply(g, B1)))) | (~element(B1, b)) | (~(apply(h, apply(beta, B1)) = apply(delta, apply(g, B1)))))) | ((~element(tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta), b)) | (~element(apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))), r)) | (~(apply(delta, E!8) = apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))))) | (~(apply(h, apply(beta, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))) = apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta)))))))),
% 0.20/0.46 inference(monotonicity,[status(thm)],[154])).
% 0.20/0.46 tff(156,plain,
% 0.20/0.46 (((~![B1: $i] : ((~element(apply(delta, apply(g, B1)), r)) | (~(apply(delta, E!8) = apply(delta, apply(g, B1)))) | (~element(B1, b)) | (~(apply(h, apply(beta, B1)) = apply(delta, apply(g, B1)))))) | ((~element(apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))), r)) | (~(apply(delta, E!8) = apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))))) | (~element(tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta), b)) | (~(apply(h, apply(beta, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))) = apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))))))) <=> ((~![B1: $i] : ((~element(apply(delta, apply(g, B1)), r)) | (~(apply(delta, E!8) = apply(delta, apply(g, B1)))) | (~element(B1, b)) | (~(apply(h, apply(beta, B1)) = apply(delta, apply(g, B1)))))) | (~element(tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta), b)) | (~element(apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))), r)) | (~(apply(delta, E!8) = apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))))) | (~(apply(h, apply(beta, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))) = apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))))))),
% 0.20/0.46 inference(transitivity,[status(thm)],[155, 153])).
% 0.20/0.46 tff(157,plain,
% 0.20/0.46 ((~![B1: $i] : ((~element(apply(delta, apply(g, B1)), r)) | (~(apply(delta, E!8) = apply(delta, apply(g, B1)))) | (~element(B1, b)) | (~(apply(h, apply(beta, B1)) = apply(delta, apply(g, B1)))))) | ((~element(apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))), r)) | (~(apply(delta, E!8) = apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))))) | (~element(tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta), b)) | (~(apply(h, apply(beta, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))) = apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))))))),
% 0.20/0.46 inference(quant_inst,[status(thm)],[])).
% 0.20/0.46 tff(158,plain,
% 0.20/0.46 ((~![B1: $i] : ((~element(apply(delta, apply(g, B1)), r)) | (~(apply(delta, E!8) = apply(delta, apply(g, B1)))) | (~element(B1, b)) | (~(apply(h, apply(beta, B1)) = apply(delta, apply(g, B1)))))) | (~element(tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta), b)) | (~element(apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))), r)) | (~(apply(delta, E!8) = apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))))) | (~(apply(h, apply(beta, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))) = apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta)))))),
% 0.20/0.46 inference(modus_ponens,[status(thm)],[157, 156])).
% 0.20/0.46 tff(159,plain,
% 0.20/0.46 ((~element(apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta))), r)) | (~(apply(delta, E!8) = apply(delta, apply(g, tptp_fun_ElDom_2(tptp_fun_ElDom_2(apply(delta, E!8), c, h), b, beta)))))),
% 0.20/0.46 inference(unit_resolution,[status(thm)],[158, 152, 96, 132])).
% 0.20/0.46 tff(160,plain,
% 0.20/0.46 ($false),
% 0.20/0.46 inference(unit_resolution,[status(thm)],[159, 145, 140])).
% 0.20/0.46 % SZS output end Proof
% 0.20/0.46 % E exiting
%------------------------------------------------------------------------------