fof(arityeq2_2Ec_2Ebool_2E_2F_5C_2E2,axiom,![X0_2E0, X1_2E0]: ((p(s(tyop_2Emin_2Ebool,X0_2E0)) & p(s(tyop_2Emin_2Ebool,X1_2E0))) <=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)),c_2Ebool_2E_2F_5C_2E0),s(tyop_2Emin_2Ebool,X0_2E0))),s(tyop_2Emin_2Ebool,X1_2E0)))))).
fof(arityeq2_2Ec_2Ebool_2E_5C_2F_2E2,axiom,![X0_2E0, X1_2E0]: ((p(s(tyop_2Emin_2Ebool,X0_2E0)) | p(s(tyop_2Emin_2Ebool,X1_2E0))) <=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)),c_2Ebool_2E_5C_2F_2E0),s(tyop_2Emin_2Ebool,X0_2E0))),s(tyop_2Emin_2Ebool,X1_2E0)))))).
fof(arityeq1_2Ec_2Ebool_2E_7E_2E1,axiom,![X0_2E0]: (~ (p(s(tyop_2Emin_2Ebool,X0_2E0))) <=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2E_7E_2E0),s(tyop_2Emin_2Ebool,X0_2E0)))))).
fof(arityeq2_2Ec_2Emin_2E_3D_3D_3E_2E2,axiom,![X0_2E0, X1_2E0]: ((p(s(tyop_2Emin_2Ebool,X0_2E0)) => p(s(tyop_2Emin_2Ebool,X1_2E0))) <=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)),c_2Emin_2E_3D_3D_3E_2E0),s(tyop_2Emin_2Ebool,X0_2E0))),s(tyop_2Emin_2Ebool,X1_2E0)))))).
fof(arityeq2_2Ec_2Emin_2E_3D_2E2_2Emono_2EA_27a,axiom,![A_27a]: ![X0_2E0, X1_2E0]: ((s(A_27a,X0_2E0) = s(A_27a,X1_2E0)) <=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Emin_2E_3D_2E0),s(A_27a,X0_2E0))),s(A_27a,X1_2E0)))))).
fof(arityeq1_2Ec_2Ebool_2E_21_2E1_2Emono_2EA_27a,axiom,![A_27a]: ![X0_2E0]: (s(tyop_2Emin_2Ebool,c_2Ebool_2E_21_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ebool_2E_21_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0))))).
fof(arityeq1_2Ec_2Ebool_2E_3F_2E1_2Emono_2EA_27a,axiom,![A_27a]: ![X0_2E0]: (s(tyop_2Emin_2Ebool,c_2Ebool_2E_3F_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ebool_2E_3F_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0))))).
fof(arityeq2_2Ec_2Epair_2Epair__CASE_2E2_2Emono_2Etyop_2Epath_2Epath_28A_27b_2CA_27c_29_20mono_2EA_27b_20mono_2Etyop_2Eoption_2Eoption_28tyop_2Epair_2Eprod_28A_27c_2CA_27a_29_29,axiom,![A_27a, A_27b, A_27c]: ![X0_2E0, X1_2E0]: (s(tyop_2Epath_2Epath(A_27b,A_27c),c_2Epair_2Epair__CASE_2E2(s(tyop_2Epair_2Eprod(A_27b,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27a))),X0_2E0),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27a)),tyop_2Epath_2Epath(A_27b,A_27c))),X1_2E0))) = s(tyop_2Epath_2Epath(A_27b,A_27c),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27a)),tyop_2Epath_2Epath(A_27b,A_27c))),tyop_2Epath_2Epath(A_27b,A_27c)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27a))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27a)),tyop_2Epath_2Epath(A_27b,A_27c))),tyop_2Epath_2Epath(A_27b,A_27c))),c_2Epair_2Epair__CASE_2E0),s(tyop_2Epair_2Eprod(A_27b,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27a))),X0_2E0))),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27a)),tyop_2Epath_2Epath(A_27b,A_27c))),X1_2E0))))).
fof(arityeq2_2Ec_2Epair_2Epair__CASE_2E2_2Emono_2Etyop_2Epath_2Epath_28A_27b_2CA_27c_29_20mono_2EA_27c_20mono_2EA_27a,axiom,![A_27a, A_27b, A_27c]: ![X0_2E0, X1_2E0]: (s(tyop_2Epath_2Epath(A_27b,A_27c),c_2Epair_2Epair__CASE_2E2(s(tyop_2Epair_2Eprod(A_27c,A_27a),X0_2E0),s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Epath_2Epath(A_27b,A_27c))),X1_2E0))) = s(tyop_2Epath_2Epath(A_27b,A_27c),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Epath_2Epath(A_27b,A_27c))),tyop_2Epath_2Epath(A_27b,A_27c)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Epath_2Epath(A_27b,A_27c))),tyop_2Epath_2Epath(A_27b,A_27c))),c_2Epair_2Epair__CASE_2E0),s(tyop_2Epair_2Eprod(A_27c,A_27a),X0_2E0))),s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Epath_2Epath(A_27b,A_27c))),X1_2E0))))).
fof(arityeq3_2Ec_2Eoption_2Eoption__CASE_2E3_2Emono_2Etyop_2Epair_2Eprod_28A_27c_2CA_27a_29_20mono_2Etyop_2Epath_2Epath_28A_27b_2CA_27c_29,axiom,![A_27a, A_27b, A_27c]: ![X0_2E0, X1_2E0, X2_2E0]: (s(tyop_2Epath_2Epath(A_27b,A_27c),c_2Eoption_2Eoption__CASE_2E3(s(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27a)),X0_2E0),s(tyop_2Epath_2Epath(A_27b,A_27c),X1_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27a),tyop_2Epath_2Epath(A_27b,A_27c)),X2_2E0))) = s(tyop_2Epath_2Epath(A_27b,A_27c),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27a),tyop_2Epath_2Epath(A_27b,A_27c)),tyop_2Epath_2Epath(A_27b,A_27c)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27a),tyop_2Epath_2Epath(A_27b,A_27c)),tyop_2Epath_2Epath(A_27b,A_27c))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27a)),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27a),tyop_2Epath_2Epath(A_27b,A_27c)),tyop_2Epath_2Epath(A_27b,A_27c)))),c_2Eoption_2Eoption__CASE_2E0),s(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27a)),X0_2E0))),s(tyop_2Epath_2Epath(A_27b,A_27c),X1_2E0))),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27a),tyop_2Epath_2Epath(A_27b,A_27c)),X2_2E0))))).
fof(arityeq1_2Ec_2Epath_2Estopped__at_2E1_2Emono_2EA_27b_20mono_2EA_27c,axiom,![A_27b, A_27c]: ![X0_2E0]: (s(tyop_2Epath_2Epath(A_27b,A_27c),c_2Epath_2Estopped__at_2E1(s(A_27b,X0_2E0))) = s(tyop_2Epath_2Epath(A_27b,A_27c),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Epath_2Epath(A_27b,A_27c)),c_2Epath_2Estopped__at_2E0),s(A_27b,X0_2E0))))).
fof(arityeq3_2Ec_2Epath_2Epcons_2E3_2Emono_2EA_27b_20mono_2EA_27c,axiom,![A_27b, A_27c]: ![X0_2E0, X1_2E0, X2_2E0]: (s(tyop_2Epath_2Epath(A_27b,A_27c),c_2Epath_2Epcons_2E3(s(A_27b,X0_2E0),s(A_27c,X1_2E0),s(tyop_2Epath_2Epath(A_27b,A_27c),X2_2E0))) = s(tyop_2Epath_2Epath(A_27b,A_27c),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27b,A_27c),tyop_2Epath_2Epath(A_27b,A_27c)),app_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27b,A_27c),tyop_2Epath_2Epath(A_27b,A_27c))),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27b,A_27c),tyop_2Epath_2Epath(A_27b,A_27c)))),c_2Epath_2Epcons_2E0),s(A_27b,X0_2E0))),s(A_27c,X1_2E0))),s(tyop_2Epath_2Epath(A_27b,A_27c),X2_2E0))))).
fof(pathTheory_2Epath__Axiom,axiom,![A_27a, A_27b, A_27c]: ![F2_2E0, F1_2E0, F0_2E0]: (![V3y_2E0, V1g_2E0, V6l_2E0, V7v_2E0]: (s(tyop_2Epath_2Epath(A_27b,A_27c),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Epath_2Epath(A_27b,A_27c)),app_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Epath_2Epath(A_27b,A_27c))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Epath_2Epath(A_27b,A_27c)),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Epath_2Epath(A_27b,A_27c)))),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Epath_2Epath(A_27b,A_27c)),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Epath_2Epath(A_27b,A_27c))))),F2_2E0),s(A_27b,V3y_2E0))),s(tyop_2Emin_2Efun(A_27a,tyop_2Epath_2Epath(A_27b,A_27c)),V1g_2E0))),s(A_27c,V6l_2E0))),s(A_27a,V7v_2E0))) = s(tyop_2Epath_2Epath(A_27b,A_27c),c_2Epath_2Epcons_2E3(s(A_27b,V3y_2E0),s(A_27c,V6l_2E0),s(tyop_2Epath_2Epath(A_27b,A_27c),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Epath_2Epath(A_27b,A_27c)),V1g_2E0),s(A_27a,V7v_2E0)))))) => (![V3y_2E0, V1g_2E0, V5v3_2E0]: (s(tyop_2Epath_2Epath(A_27b,A_27c),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27a),tyop_2Epath_2Epath(A_27b,A_27c)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Epath_2Epath(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27a),tyop_2Epath_2Epath(A_27b,A_27c))),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Epath_2Epath(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27a),tyop_2Epath_2Epath(A_27b,A_27c)))),F1_2E0),s(A_27b,V3y_2E0))),s(tyop_2Emin_2Efun(A_27a,tyop_2Epath_2Epath(A_27b,A_27c)),V1g_2E0))),s(tyop_2Epair_2Eprod(A_27c,A_27a),V5v3_2E0))) = s(tyop_2Epath_2Epath(A_27b,A_27c),c_2Epair_2Epair__CASE_2E2(s(tyop_2Epair_2Eprod(A_27c,A_27a),V5v3_2E0),s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Epath_2Epath(A_27b,A_27c))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Epath_2Epath(A_27b,A_27c)),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Epath_2Epath(A_27b,A_27c)))),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Epath_2Epath(A_27b,A_27c)),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Epath_2Epath(A_27b,A_27c))))),F2_2E0),s(A_27b,V3y_2E0))),s(tyop_2Emin_2Efun(A_27a,tyop_2Epath_2Epath(A_27b,A_27c)),V1g_2E0)))))) => (![V1g_2E0, V3y_2E0, V4v2_2E0]: (s(tyop_2Epath_2Epath(A_27b,A_27c),app_2E2(s(tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27a)),tyop_2Epath_2Epath(A_27b,A_27c)),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27a)),tyop_2Epath_2Epath(A_27b,A_27c))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Epath_2Epath(A_27b,A_27c)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27a)),tyop_2Epath_2Epath(A_27b,A_27c)))),F0_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Epath_2Epath(A_27b,A_27c)),V1g_2E0))),s(A_27b,V3y_2E0))),s(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27a)),V4v2_2E0))) = s(tyop_2Epath_2Epath(A_27b,A_27c),c_2Eoption_2Eoption__CASE_2E3(s(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27a)),V4v2_2E0),s(tyop_2Epath_2Epath(A_27b,A_27c),c_2Epath_2Estopped__at_2E1(s(A_27b,V3y_2E0))),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27a),tyop_2Epath_2Epath(A_27b,A_27c)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Epath_2Epath(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27a),tyop_2Epath_2Epath(A_27b,A_27c))),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Epath_2Epath(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27a),tyop_2Epath_2Epath(A_27b,A_27c)))),F1_2E0),s(A_27b,V3y_2E0))),s(tyop_2Emin_2Efun(A_27a,tyop_2Epath_2Epath(A_27b,A_27c)),V1g_2E0)))))) => ![V0f_2E0]: ?[V1g_2E0]: ![V2x_2E0]: (s(tyop_2Epath_2Epath(A_27b,A_27c),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Epath_2Epath(A_27b,A_27c)),V1g_2E0),s(A_27a,V2x_2E0))) = s(tyop_2Epath_2Epath(A_27b,A_27c),c_2Epair_2Epair__CASE_2E2(s(tyop_2Epair_2Eprod(A_27b,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27a))),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27a)))),V0f_2E0),s(A_27a,V2x_2E0))),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27a)),tyop_2Epath_2Epath(A_27b,A_27c))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Epath_2Epath(A_27b,A_27c)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27a)),tyop_2Epath_2Epath(A_27b,A_27c)))),F0_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Epath_2Epath(A_27b,A_27c)),V1g_2E0)))))))))).
