%------------------------------------------------------------------------------
% File     : ITP068+4 : TPTP v9.0.0. Bugfixed v7.5.0.
% Domain   : Interactive Theorem Proving
% Axioms   : HOL4 syntactic export, chainy mode
% Version  : [BG+19] axioms.
% English  :

% Refs     : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
%          : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source   : [BG+19]
% Names    : path.ax [Gau19]
%          : HL4068+4.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  334 ( 202 unt;   0 def)
%            Number of atoms       :  627 ( 379 equ)
%            Maximal formula atoms :   12 (   1 avg)
%            Number of connectives :  311 (  18   ~;  13   |; 120   &)
%                                         (  56 <=>; 104  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   18 (   6 avg)
%            Maximal term depth    :   21 (   2 avg)
%            Number of predicates  :    2 (   1 usr;   0 prp; 1-2 aty)
%            Number of functors    :  190 ( 190 usr;  95 con; 0-5 aty)
%            Number of variables   : 1582 (1540   !;  42   ?)
% SPC      : FOF_SAT_RFO_SEQ

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
fof(reserved_2Eho_2Eboolext,axiom,
    ! [V0_2E0,V1_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,V0_2E0))
      <=> p(s(tyop_2Emin_2Ebool,V1_2E0)) )
     => s(tyop_2Emin_2Ebool,V0_2E0) = s(tyop_2Emin_2Ebool,V1_2E0) ) ).

fof(reserved_2Eho_2Etruth,axiom,
    p(s(tyop_2Emin_2Ebool,c_2Ebool_2ET_2E0)) ).

fof(reserved_2Eho_2Enotfalse,axiom,
    ~ p(s(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0)) ).

fof(reserved_2Eho_2Ebool__cases__ax,axiom,
    ! [V0t_2E0] :
      ( s(tyop_2Emin_2Ebool,V0t_2E0) = s(tyop_2Emin_2Ebool,c_2Ebool_2ET_2E0)
      | s(tyop_2Emin_2Ebool,V0t_2E0) = s(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0) ) ).

fof(reserved_2Eho_2Ei__thm,axiom,
    ! [A_27a,V0x_2E0] : s(A_27a,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27a),combin_i_2E0),s(A_27a,V0x_2E0))) = s(A_27a,V0x_2E0) ).

fof(reserved_2Eho_2Ek__thm,axiom,
    ! [A_27a,A_27b,V0x_2E0,V1y_2E0] : s(A_27a,app_2E2(s(tyop_2Emin_2Efun(A_27b,A_27a),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27a)),combin_k_2E0),s(A_27a,V0x_2E0))),s(A_27b,V1y_2E0))) = s(A_27a,V0x_2E0) ).

fof(reserved_2Eho_2Es__thm,axiom,
    ! [A_27a,A_27b,A_27c,V0f_2E0,V1g_2E0,V2x_2E0] : s(A_27c,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27c),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c))),combin_s_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),V0f_2E0))),s(tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0))),s(A_27a,V2x_2E0))) = s(A_27c,app_2E2(s(tyop_2Emin_2Efun(A_27b,A_27c),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),V0f_2E0),s(A_27a,V2x_2E0))),s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0),s(A_27a,V2x_2E0))))) ).

fof(reserved_2Elogic_2E_2F_5C,axiom,
    ! [V0_2E0,V1_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2E_2F_5C_2E2(s(tyop_2Emin_2Ebool,V0_2E0),s(tyop_2Emin_2Ebool,V1_2E0))))
    <=> ( p(s(tyop_2Emin_2Ebool,V0_2E0))
        & p(s(tyop_2Emin_2Ebool,V1_2E0)) ) ) ).

fof(reserved_2Elogic_2E_5C_2F,axiom,
    ! [V0_2E0,V1_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2E_5C_2F_2E2(s(tyop_2Emin_2Ebool,V0_2E0),s(tyop_2Emin_2Ebool,V1_2E0))))
    <=> ( p(s(tyop_2Emin_2Ebool,V0_2E0))
        | p(s(tyop_2Emin_2Ebool,V1_2E0)) ) ) ).

fof(reserved_2Elogic_2E_7E,axiom,
    ! [V0_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2E_7E_2E1(s(tyop_2Emin_2Ebool,V0_2E0))))
    <=> ~ p(s(tyop_2Emin_2Ebool,V0_2E0)) ) ).

fof(reserved_2Elogic_2E_3D_3D_3E,axiom,
    ! [V0_2E0,V1_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Emin_2E_3D_3D_3E_2E2(s(tyop_2Emin_2Ebool,V0_2E0),s(tyop_2Emin_2Ebool,V1_2E0))))
    <=> ( p(s(tyop_2Emin_2Ebool,V0_2E0))
       => p(s(tyop_2Emin_2Ebool,V1_2E0)) ) ) ).

fof(reserved_2Elogic_2E_3D,axiom,
    ! [A_27a,V0_2E0,V1_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Emin_2E_3D_2E2(s(A_27a,V0_2E0),s(A_27a,V1_2E0))))
    <=> s(A_27a,V0_2E0) = s(A_27a,V1_2E0) ) ).

fof(reserved_2Equant_2E_21,axiom,
    ! [A_27a,V0f_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2E_21_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0))))
    <=> ! [V1x_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0),s(A_27a,V1x_2E0)))) ) ).

fof(reserved_2Equant_2E_3F,axiom,
    ! [A_27a,V0f_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2E_3F_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0))))
    <=> ? [V1x_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0),s(A_27a,V1x_2E0)))) ) ).

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(arityeq1_2Ec_2Epair_2EFST_2E1_2Emono_2EA_27a_20mono_2Etyop_2Ellist_2Ellist_28tyop_2Epair_2Eprod_28A_27b_2CA_27a_29_29,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(A_27a,c_2Epair_2EFST_2E1(s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),X0_2E0))) = s(A_27a,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),A_27a),c_2Epair_2EFST_2E0),s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),X0_2E0))) ).

fof(arityeq1_2Ec_2Epath_2EfromPath_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),c_2Epath_2EfromPath_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),X0_2E0))) = s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)))),c_2Epath_2EfromPath_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),X0_2E0))) ).

fof(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0] : s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E1(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Epred__set_2EGSPEC_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),X0_2E0))) ).

fof(arityeq1_2Ec_2Ellist_2ELFINITE_2E1_2Emono_2EA_27b,axiom,
    ! [A_27b,X0_2E0] : s(tyop_2Emin_2Ebool,c_2Ellist_2ELFINITE_2E1(s(tyop_2Ellist_2Ellist(A_27b),X0_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(A_27b),tyop_2Emin_2Ebool),c_2Ellist_2ELFINITE_2E0),s(tyop_2Ellist_2Ellist(A_27b),X0_2E0))) ).

fof(arityeq1_2Ec_2Epath_2Elabels_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Ellist_2Ellist(A_27b),c_2Epath_2Elabels_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),X0_2E0))) = s(tyop_2Ellist_2Ellist(A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Ellist_2Ellist(A_27b)),c_2Epath_2Elabels_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),X0_2E0))) ).

fof(arityeq1_2Ec_2Ellist_2ELFINITE_2E1_2Emono_2Etyop_2Epair_2Eprod_28A_27b_2CA_27a_29,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Emin_2Ebool,c_2Ellist_2ELFINITE_2E1(s(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),X0_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Emin_2Ebool),c_2Ellist_2ELFINITE_2E0),s(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),X0_2E0))) ).

fof(arityeq1_2Ec_2Epair_2ESND_2E1_2Emono_2EA_27a_20mono_2Etyop_2Ellist_2Ellist_28tyop_2Epair_2Eprod_28A_27b_2CA_27a_29_29,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),c_2Epair_2ESND_2E1(s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),X0_2E0))) = s(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),c_2Epair_2ESND_2E0),s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),X0_2E0))) ).

fof(arityeq1_2Ec_2Epath_2EPL_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Epath_2EPL_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),X0_2E0))) ).

fof(arityeq1_2Ec_2Epath_2Estopped__at_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,X0_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Epath_2Epath(A_27a,A_27b)),c_2Epath_2Estopped__at_2E0),s(A_27a,X0_2E0))) ).

fof(arityeq2_2Ec_2Epath_2Edrop_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Edrop_2E2(s(tyop_2Enum_2Enum,X0_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),X1_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epath_2Epath(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epath_2Epath(A_27a,A_27b))),c_2Epath_2Edrop_2E0),s(tyop_2Enum_2Enum,X0_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),X1_2E0))) ).

fof(arityeq2_2Ec_2Epath_2Etake_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Etake_2E2(s(tyop_2Enum_2Enum,X0_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),X1_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epath_2Epath(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epath_2Epath(A_27a,A_27b))),c_2Epath_2Etake_2E0),s(tyop_2Enum_2Enum,X0_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),X1_2E0))) ).

fof(arityeq3_2Ec_2Epath_2Epcons_2E3_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,X0_2E0),s(A_27b,X1_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),X2_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epath_2Epath(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epath_2Epath(A_27a,A_27b))),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epath_2Epath(A_27a,A_27b)))),c_2Epath_2Epcons_2E0),s(A_27a,X0_2E0))),s(A_27b,X1_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),X2_2E0))) ).

fof(arityeq3_2Ec_2Epath_2Epmap_2E3_2Emono_2EA_27c_20mono_2EA_27d_20mono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,A_27c,A_27d,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epmap_2E3(s(tyop_2Emin_2Efun(A_27c,A_27a),X0_2E0),s(tyop_2Emin_2Efun(A_27d,A_27b),X1_2E0),s(tyop_2Epath_2Epath(A_27c,A_27d),X2_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27c,A_27d),tyop_2Epath_2Epath(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27c,A_27d),tyop_2Epath_2Epath(A_27a,A_27b))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27c,A_27d),tyop_2Epath_2Epath(A_27a,A_27b)))),c_2Epath_2Epmap_2E0),s(tyop_2Emin_2Efun(A_27c,A_27a),X0_2E0))),s(tyop_2Emin_2Efun(A_27d,A_27b),X1_2E0))),s(tyop_2Epath_2Epath(A_27c,A_27d),X2_2E0))) ).

fof(arityeq3_2Ec_2Epath_2Eseg_2E3_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Eseg_2E3(s(tyop_2Enum_2Enum,X0_2E0),s(tyop_2Enum_2Enum,X1_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),X2_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epath_2Epath(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epath_2Epath(A_27a,A_27b))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epath_2Epath(A_27a,A_27b)))),c_2Epath_2Eseg_2E0),s(tyop_2Enum_2Enum,X0_2E0))),s(tyop_2Enum_2Enum,X1_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),X2_2E0))) ).

fof(arityeq1_2Ec_2Epath_2EPL_2E1_2Emono_2EA_27c_20mono_2EA_27d,axiom,
    ! [A_27c,A_27d,X0_2E0] : s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27c,A_27d),X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27c,A_27d),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Epath_2EPL_2E0),s(tyop_2Epath_2Epath(A_27c,A_27d),X0_2E0))) ).

fof(arityeq1_2Ec_2Epath_2ESN_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Emin_2Ebool,c_2Epath_2ESN_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,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_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),tyop_2Emin_2Ebool),c_2Epath_2ESN_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),X0_2E0))) ).

fof(arityeq1_2Ec_2Eoption_2ESOME_2E1_2Emono_2EA_27b,axiom,
    ! [A_27b,X0_2E0] : s(tyop_2Eoption_2Eoption(A_27b),c_2Eoption_2ESOME_2E1(s(A_27b,X0_2E0))) = s(tyop_2Eoption_2Eoption(A_27b),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(A_27b)),c_2Eoption_2ESOME_2E0),s(A_27b,X0_2E0))) ).

fof(arityeq1_2Ec_2Eoption_2ESOME_2E1_2Emono_2Etyop_2Elist_2Elist_28A_27a_29,axiom,
    ! [A_27a,X0_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Elist_2Elist(A_27a)),c_2Eoption_2ESOME_2E1(s(tyop_2Elist_2Elist(A_27a),X0_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Elist_2Elist(A_27a)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Eoption_2Eoption(tyop_2Elist_2Elist(A_27a))),c_2Eoption_2ESOME_2E0),s(tyop_2Elist_2Elist(A_27a),X0_2E0))) ).

fof(arityeq1_2Ec_2Eoption_2ESOME_2E1_2Emono_2Etyop_2Elist_2Elist_28A_27b_29,axiom,
    ! [A_27b,X0_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Elist_2Elist(A_27b)),c_2Eoption_2ESOME_2E1(s(tyop_2Elist_2Elist(A_27b),X0_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Elist_2Elist(A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(tyop_2Elist_2Elist(A_27b))),c_2Eoption_2ESOME_2E0),s(tyop_2Elist_2Elist(A_27b),X0_2E0))) ).

fof(arityeq1_2Ec_2Eoption_2ESOME_2E1_2Emono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Eoption_2ESOME_2E1(s(tyop_2Enum_2Enum,X0_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)),c_2Eoption_2ESOME_2E0),s(tyop_2Enum_2Enum,X0_2E0))) ).

fof(arityeq1_2Ec_2Earithmetic_2EBIT1_2E1,axiom,
    ! [X0_2E0] : s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,X0_2E0))) = s(tyop_2Enum_2Enum,app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2EBIT1_2E0),s(tyop_2Enum_2Enum,X0_2E0))) ).

fof(arityeq1_2Ec_2Earithmetic_2ENUMERAL_2E1,axiom,
    ! [X0_2E0] : s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,X0_2E0))) = s(tyop_2Enum_2Enum,app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2ENUMERAL_2E0),s(tyop_2Enum_2Enum,X0_2E0))) ).

fof(arityeq2_2Ec_2Earithmetic_2E_2B_2E2,axiom,
    ! [X0_2E0,X1_2E0] : s(tyop_2Enum_2Enum,c_2Earithmetic_2E_2B_2E2(s(tyop_2Enum_2Enum,X0_2E0),s(tyop_2Enum_2Enum,X1_2E0))) = s(tyop_2Enum_2Enum,app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),c_2Earithmetic_2E_2B_2E0),s(tyop_2Enum_2Enum,X0_2E0))),s(tyop_2Enum_2Enum,X1_2E0))) ).

fof(arityeq2_2Ec_2Earithmetic_2E_2D_2E2,axiom,
    ! [X0_2E0,X1_2E0] : s(tyop_2Enum_2Enum,c_2Earithmetic_2E_2D_2E2(s(tyop_2Enum_2Enum,X0_2E0),s(tyop_2Enum_2Enum,X1_2E0))) = s(tyop_2Enum_2Enum,app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),c_2Earithmetic_2E_2D_2E0),s(tyop_2Enum_2Enum,X0_2E0))),s(tyop_2Enum_2Enum,X1_2E0))) ).

fof(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2EA_27a_20mono_2EA_27c,axiom,
    ! [A_27a,A_27c,X0_2E0,X1_2E0] : s(tyop_2Epair_2Eprod(A_27a,A_27c),c_2Epair_2E_2C_2E2(s(A_27a,X0_2E0),s(A_27c,X1_2E0))) = s(tyop_2Epair_2Eprod(A_27a,A_27c),app_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Epair_2Eprod(A_27a,A_27c)),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Epair_2Eprod(A_27a,A_27c))),c_2Epair_2E_2C_2E0),s(A_27a,X0_2E0))),s(A_27c,X1_2E0))) ).

fof(arityeq1_2Ec_2Eoption_2ESOME_2E1_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27c_29,axiom,
    ! [A_27a,A_27c,X0_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c)),c_2Eoption_2ESOME_2E1(s(tyop_2Epair_2Eprod(A_27a,A_27c),X0_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),c_2Eoption_2ESOME_2E0),s(tyop_2Epair_2Eprod(A_27a,A_27c),X0_2E0))) ).

fof(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2EA_27b_20mono_2Etyop_2Epath_2Epath_28A_27a_2CA_27b_29_20mono_2Etyop_2Epair_2Eprod_28tyop_2Epath_2Epath_28A_27a_2CA_27b_29_2Ctyop_2Emin_2Ebool_29,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,tyop_2Epath_2Epath(A_27a,A_27b)),tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),c_2Epair_2EUNCURRY_2E1(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool))),X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,tyop_2Epath_2Epath(A_27a,A_27b)),tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,tyop_2Epath_2Epath(A_27a,A_27b)),tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool))),c_2Epair_2EUNCURRY_2E0),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool))),X0_2E0))) ).

fof(arityeq1_2Ec_2Erelation_2EWF_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(tyop_2Emin_2Ebool,c_2Erelation_2EWF_2E1(s(tyop_2Emin_2Efun(A_27a,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_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool),c_2Erelation_2EWF_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),X0_2E0))) ).

fof(arityeq1_2Ec_2Epath_2Efinite_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),X0_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),c_2Epath_2Efinite_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),X0_2E0))) ).

fof(arityeq2_2Ec_2Epath_2Epgenerate_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epgenerate_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),X0_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27b),X1_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27b),tyop_2Epath_2Epath(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27b),tyop_2Epath_2Epath(A_27a,A_27b))),c_2Epath_2Epgenerate_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),X0_2E0))),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27b),X1_2E0))) ).

fof(arityeq2_2Ec_2Epath_2Eplink_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Eplink_2E2(s(tyop_2Epath_2Epath(A_27a,A_27b),X0_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),X1_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epath_2Epath(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epath_2Epath(A_27a,A_27b))),c_2Epath_2Eplink_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),X0_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),X1_2E0))) ).

fof(arityeq3_2Ec_2Epath_2Epconcat_2E3_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epconcat_2E3(s(tyop_2Epath_2Epath(A_27a,A_27b),X0_2E0),s(A_27b,X1_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),X2_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epath_2Epath(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epath_2Epath(A_27a,A_27b))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epath_2Epath(A_27a,A_27b)))),c_2Epath_2Epconcat_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),X0_2E0))),s(A_27b,X1_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),X2_2E0))) ).

fof(arityeq1_2Ec_2Epath_2Efinite_2E1_2Emono_2EA_27c_20mono_2EA_27d,axiom,
    ! [A_27c,A_27d,X0_2E0] : s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27c,A_27d),X0_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27c,A_27d),tyop_2Emin_2Ebool),c_2Epath_2Efinite_2E0),s(tyop_2Epath_2Epath(A_27c,A_27d),X0_2E0))) ).

fof(arityeq3_2Ec_2Epath_2Epmap_2E3_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27c_20mono_2EA_27d,axiom,
    ! [A_27a,A_27b,A_27c,A_27d,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Epath_2Epath(A_27c,A_27d),c_2Epath_2Epmap_2E3(s(tyop_2Emin_2Efun(A_27a,A_27c),X0_2E0),s(tyop_2Emin_2Efun(A_27b,A_27d),X1_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),X2_2E0))) = s(tyop_2Epath_2Epath(A_27c,A_27d),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epath_2Epath(A_27c,A_27d)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27d),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epath_2Epath(A_27c,A_27d))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27c),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27d),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epath_2Epath(A_27c,A_27d)))),c_2Epath_2Epmap_2E0),s(tyop_2Emin_2Efun(A_27a,A_27c),X0_2E0))),s(tyop_2Emin_2Efun(A_27b,A_27d),X1_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),X2_2E0))) ).

fof(arityeq1_2Ec_2Epath_2Efirst_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(A_27a,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),X0_2E0))) = s(A_27a,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),A_27a),c_2Epath_2Efirst_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),X0_2E0))) ).

fof(arityeq1_2Ec_2Epath_2Efirst_2E1_2Emono_2EA_27b_20mono_2EA_27c,axiom,
    ! [A_27b,A_27c,X0_2E0] : s(A_27b,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27b,A_27c),X0_2E0))) = s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27b,A_27c),A_27b),c_2Epath_2Efirst_2E0),s(tyop_2Epath_2Epath(A_27b,A_27c),X0_2E0))) ).

fof(arityeq1_2Ec_2Epath_2Efirst_2E1_2Emono_2EA_27c_20mono_2EA_27d,axiom,
    ! [A_27c,A_27d,X0_2E0] : s(A_27c,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27c,A_27d),X0_2E0))) = s(A_27c,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27c,A_27d),A_27c),c_2Epath_2Efirst_2E0),s(tyop_2Epath_2Epath(A_27c,A_27d),X0_2E0))) ).

fof(arityeq1_2Ec_2Epath_2Efirst_2E1_2Emono_2Etyop_2Elist_2Elist_28A_27a_29_20mono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(tyop_2Elist_2Elist(A_27a),c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(tyop_2Elist_2Elist(A_27a),A_27a),X0_2E0))) = s(tyop_2Elist_2Elist(A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(tyop_2Elist_2Elist(A_27a),A_27a),tyop_2Elist_2Elist(A_27a)),c_2Epath_2Efirst_2E0),s(tyop_2Epath_2Epath(tyop_2Elist_2Elist(A_27a),A_27a),X0_2E0))) ).

fof(arityeq1_2Ec_2Epath_2Efirst_2E1_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27c_29_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,A_27c,X0_2E0] : s(tyop_2Epair_2Eprod(A_27a,A_27c),c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(tyop_2Epair_2Eprod(A_27a,A_27c),A_27b),X0_2E0))) = s(tyop_2Epair_2Eprod(A_27a,A_27c),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(tyop_2Epair_2Eprod(A_27a,A_27c),A_27b),tyop_2Epair_2Eprod(A_27a,A_27c)),c_2Epath_2Efirst_2E0),s(tyop_2Epath_2Epath(tyop_2Epair_2Eprod(A_27a,A_27c),A_27b),X0_2E0))) ).

fof(arityeq1_2Ec_2Epath_2Efirst__label_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(A_27b,c_2Epath_2Efirst__label_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),X0_2E0))) = s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),A_27b),c_2Epath_2Efirst__label_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),X0_2E0))) ).

fof(arityeq1_2Ec_2Epath_2Efirst__label_2E1_2Emono_2EA_27b_20mono_2EA_27a,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(A_27a,c_2Epath_2Efirst__label_2E1(s(tyop_2Epath_2Epath(A_27b,A_27a),X0_2E0))) = s(A_27a,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27b,A_27a),A_27a),c_2Epath_2Efirst__label_2E0),s(tyop_2Epath_2Epath(A_27b,A_27a),X0_2E0))) ).

fof(arityeq1_2Ec_2Epath_2EtoPath_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2EtoPath_2E1(s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),X0_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),tyop_2Epath_2Epath(A_27a,A_27b)),c_2Epath_2EtoPath_2E0),s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),X0_2E0))) ).

fof(arityeq1_2Ec_2EfixedPoint_2Egfp_2E1_2Emono_2Etyop_2Epath_2Epath_28A_27a_2CA_27b_29,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),c_2EfixedPoint_2Egfp_2E1(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),c_2EfixedPoint_2Egfp_2E0),s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),X0_2E0))) ).

fof(arityeq1_2Ec_2Epath_2Eokpath__f_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),c_2Epath_2Eokpath__f_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool))),c_2Epath_2Eokpath__f_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),X0_2E0))) ).

fof(arityeq1_2Ec_2Epath_2Eis__stopped_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Emin_2Ebool,c_2Epath_2Eis__stopped_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),X0_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),c_2Epath_2Eis__stopped_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),X0_2E0))) ).

fof(arityeq1_2Ec_2Epath_2Eis__stopped_2E1_2Emono_2EA_27c_20mono_2EA_27d,axiom,
    ! [A_27c,A_27d,X0_2E0] : s(tyop_2Emin_2Ebool,c_2Epath_2Eis__stopped_2E1(s(tyop_2Epath_2Epath(A_27c,A_27d),X0_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27c,A_27d),tyop_2Emin_2Ebool),c_2Epath_2Eis__stopped_2E0),s(tyop_2Epath_2Epath(A_27c,A_27d),X0_2E0))) ).

fof(arityeq3_2Ec_2Epath_2Epcons_2E3_2Emono_2EA_27c_20mono_2EA_27d,axiom,
    ! [A_27c,A_27d,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Epath_2Epath(A_27c,A_27d),c_2Epath_2Epcons_2E3(s(A_27c,X0_2E0),s(A_27d,X1_2E0),s(tyop_2Epath_2Epath(A_27c,A_27d),X2_2E0))) = s(tyop_2Epath_2Epath(A_27c,A_27d),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27c,A_27d),tyop_2Epath_2Epath(A_27c,A_27d)),app_2E2(s(tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27c,A_27d),tyop_2Epath_2Epath(A_27c,A_27d))),app_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27c,A_27d),tyop_2Epath_2Epath(A_27c,A_27d)))),c_2Epath_2Epcons_2E0),s(A_27c,X0_2E0))),s(A_27d,X1_2E0))),s(tyop_2Epath_2Epath(A_27c,A_27d),X2_2E0))) ).

fof(arityeq1_2Ec_2Epath_2Elabels_2E1_2Emono_2EA_27a_20mono_2EA_27c,axiom,
    ! [A_27a,A_27c,X0_2E0] : s(tyop_2Ellist_2Ellist(A_27c),c_2Epath_2Elabels_2E1(s(tyop_2Epath_2Epath(A_27a,A_27c),X0_2E0))) = s(tyop_2Ellist_2Ellist(A_27c),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27c),tyop_2Ellist_2Ellist(A_27c)),c_2Epath_2Elabels_2E0),s(tyop_2Epath_2Epath(A_27a,A_27c),X0_2E0))) ).

fof(arityeq1_2Ec_2Epath_2Elabels_2E1_2Emono_2EA_27b_20mono_2EA_27c,axiom,
    ! [A_27b,A_27c,X0_2E0] : s(tyop_2Ellist_2Ellist(A_27c),c_2Epath_2Elabels_2E1(s(tyop_2Epath_2Epath(A_27b,A_27c),X0_2E0))) = s(tyop_2Ellist_2Ellist(A_27c),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27b,A_27c),tyop_2Ellist_2Ellist(A_27c)),c_2Epath_2Elabels_2E0),s(tyop_2Epath_2Epath(A_27b,A_27c),X0_2E0))) ).

fof(arityeq3_2Ec_2Epath_2Eunfold_2E3_2Emono_2EA_27b_20mono_2EA_27c_20mono_2EA_27a,axiom,
    ! [A_27a,A_27b,A_27c,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Epath_2Epath(A_27b,A_27c),c_2Epath_2Eunfold_2E3(s(tyop_2Emin_2Efun(A_27a,A_27b),X0_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),X1_2E0),s(A_27a,X2_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(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,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,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),tyop_2Emin_2Efun(A_27a,tyop_2Epath_2Epath(A_27b,A_27c)))),c_2Epath_2Eunfold_2E0),s(tyop_2Emin_2Efun(A_27a,A_27b),X0_2E0))),s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),X1_2E0))),s(A_27a,X2_2E0))) ).

fof(arityeq1_2Ec_2Epath_2Elabels_2E1_2Emono_2EA_27c_20mono_2EA_27b,axiom,
    ! [A_27b,A_27c,X0_2E0] : s(tyop_2Ellist_2Ellist(A_27b),c_2Epath_2Elabels_2E1(s(tyop_2Epath_2Epath(A_27c,A_27b),X0_2E0))) = s(tyop_2Ellist_2Ellist(A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27c,A_27b),tyop_2Ellist_2Ellist(A_27b)),c_2Epath_2Elabels_2E0),s(tyop_2Epath_2Epath(A_27c,A_27b),X0_2E0))) ).

fof(arityeq1_2Ec_2Epath_2Elabels_2E1_2Emono_2Etyop_2Elist_2Elist_28A_27a_29_20mono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(tyop_2Ellist_2Ellist(A_27a),c_2Epath_2Elabels_2E1(s(tyop_2Epath_2Epath(tyop_2Elist_2Elist(A_27a),A_27a),X0_2E0))) = s(tyop_2Ellist_2Ellist(A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(tyop_2Elist_2Elist(A_27a),A_27a),tyop_2Ellist_2Ellist(A_27a)),c_2Epath_2Elabels_2E0),s(tyop_2Epath_2Epath(tyop_2Elist_2Elist(A_27a),A_27a),X0_2E0))) ).

fof(arityeq1_2Ec_2Epath_2Elabels_2E1_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27c_29_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,A_27c,X0_2E0] : s(tyop_2Ellist_2Ellist(A_27b),c_2Epath_2Elabels_2E1(s(tyop_2Epath_2Epath(tyop_2Epair_2Eprod(A_27a,A_27c),A_27b),X0_2E0))) = s(tyop_2Ellist_2Ellist(A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(tyop_2Epair_2Eprod(A_27a,A_27c),A_27b),tyop_2Ellist_2Ellist(A_27b)),c_2Epath_2Elabels_2E0),s(tyop_2Epath_2Epath(tyop_2Epair_2Eprod(A_27a,A_27c),A_27b),X0_2E0))) ).

fof(arityeq1_2Ec_2Epath_2Elast_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(A_27a,c_2Epath_2Elast_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),X0_2E0))) = s(A_27a,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),A_27a),c_2Epath_2Elast_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),X0_2E0))) ).

fof(arityeq1_2Ec_2Epath_2Elast_2E1_2Emono_2EA_27c_20mono_2EA_27d,axiom,
    ! [A_27c,A_27d,X0_2E0] : s(A_27c,c_2Epath_2Elast_2E1(s(tyop_2Epath_2Epath(A_27c,A_27d),X0_2E0))) = s(A_27c,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27c,A_27d),A_27c),c_2Epath_2Elast_2E0),s(tyop_2Epath_2Epath(A_27c,A_27d),X0_2E0))) ).

fof(arityeq1_2Ec_2Epath_2Elength_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Epath_2Elength_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),X0_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)),c_2Epath_2Elength_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),X0_2E0))) ).

fof(arityeq1_2Ec_2Epath_2Elength_2E1_2Emono_2EA_27a_20mono_2EA_27c,axiom,
    ! [A_27a,A_27c,X0_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Epath_2Elength_2E1(s(tyop_2Epath_2Epath(A_27a,A_27c),X0_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27c),tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)),c_2Epath_2Elength_2E0),s(tyop_2Epath_2Epath(A_27a,A_27c),X0_2E0))) ).

fof(arityeq1_2Ec_2Epath_2Elength_2E1_2Emono_2EA_27b_20mono_2EA_27d,axiom,
    ! [A_27b,A_27d,X0_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Epath_2Elength_2E1(s(tyop_2Epath_2Epath(A_27b,A_27d),X0_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27b,A_27d),tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)),c_2Epath_2Elength_2E0),s(tyop_2Epath_2Epath(A_27b,A_27d),X0_2E0))) ).

fof(arityeq3_2Ec_2Epath_2Epmap_2E3_2Emono_2EA_27a_20mono_2EA_27c_20mono_2EA_27b_20mono_2EA_27d,axiom,
    ! [A_27a,A_27b,A_27c,A_27d,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Epath_2Epath(A_27b,A_27d),c_2Epath_2Epmap_2E3(s(tyop_2Emin_2Efun(A_27a,A_27b),X0_2E0),s(tyop_2Emin_2Efun(A_27c,A_27d),X1_2E0),s(tyop_2Epath_2Epath(A_27a,A_27c),X2_2E0))) = s(tyop_2Epath_2Epath(A_27b,A_27d),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27c),tyop_2Epath_2Epath(A_27b,A_27d)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27d),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27c),tyop_2Epath_2Epath(A_27b,A_27d))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27d),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27c),tyop_2Epath_2Epath(A_27b,A_27d)))),c_2Epath_2Epmap_2E0),s(tyop_2Emin_2Efun(A_27a,A_27b),X0_2E0))),s(tyop_2Emin_2Efun(A_27c,A_27d),X1_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27c),X2_2E0))) ).

fof(arityeq1_2Ec_2Epath_2Elength_2E1_2Emono_2EA_27c_20mono_2EA_27d,axiom,
    ! [A_27c,A_27d,X0_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Epath_2Elength_2E1(s(tyop_2Epath_2Epath(A_27c,A_27d),X0_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27c,A_27d),tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)),c_2Epath_2Elength_2E0),s(tyop_2Epath_2Epath(A_27c,A_27d),X0_2E0))) ).

fof(arityeq1_2Ec_2EfixedPoint_2Emonotone_2E1_2Emono_2Etyop_2Epath_2Epath_28A_27a_2CA_27b_29_20mono_2Etyop_2Epath_2Epath_28A_27a_2CA_27b_29,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Emin_2Ebool,c_2EfixedPoint_2Emonotone_2E1(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),X0_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool),c_2EfixedPoint_2Emonotone_2E0),s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),X0_2E0))) ).

fof(arityeq1_2Ec_2Epath_2Eokpath_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),c_2Epath_2Eokpath_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),c_2Epath_2Eokpath_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),X0_2E0))) ).

fof(arityeq2_2Ec_2Epath_2Eel_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(A_27a,c_2Epath_2Eel_2E2(s(tyop_2Enum_2Enum,X0_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),X1_2E0))) = s(A_27a,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),A_27a)),c_2Epath_2Eel_2E0),s(tyop_2Enum_2Enum,X0_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),X1_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(arityeq1_2Ec_2Epath_2Etail_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Etail_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),X0_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epath_2Epath(A_27a,A_27b)),c_2Epath_2Etail_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),X0_2E0))) ).

fof(arityeq1_2Ec_2Ellist_2EtoList_2E1_2Emono_2EA_27b,axiom,
    ! [A_27b,X0_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Elist_2Elist(A_27b)),c_2Ellist_2EtoList_2E1(s(tyop_2Ellist_2Ellist(A_27b),X0_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Elist_2Elist(A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(A_27b),tyop_2Eoption_2Eoption(tyop_2Elist_2Elist(A_27b))),c_2Ellist_2EtoList_2E0),s(tyop_2Ellist_2Ellist(A_27b),X0_2E0))) ).

fof(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2EA_27a_20mono_2Etyop_2Ellist_2Ellist_28tyop_2Epair_2Eprod_28A_27b_2CA_27a_29_29,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),c_2Epair_2E_2C_2E2(s(A_27a,X0_2E0),s(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),X1_2E0))) = s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)))),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))))),c_2Epair_2E_2C_2E0),s(A_27a,X0_2E0))),s(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),X1_2E0))) ).

fof(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2EA_27b_20mono_2EA_27a,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Epair_2Eprod(A_27b,A_27a),c_2Epair_2E_2C_2E2(s(A_27b,X0_2E0),s(A_27a,X1_2E0))) = s(tyop_2Epair_2Eprod(A_27b,A_27a),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,A_27a)),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,A_27a))),c_2Epair_2E_2C_2E0),s(A_27b,X0_2E0))),s(A_27a,X1_2E0))) ).

fof(arityeq2_2Ec_2Ellist_2ELCONS_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27b_2CA_27a_29,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),c_2Ellist_2ELCONS_2E2(s(tyop_2Epair_2Eprod(A_27b,A_27a),X0_2E0),s(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),X1_2E0))) = s(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)))),c_2Ellist_2ELCONS_2E0),s(tyop_2Epair_2Eprod(A_27b,A_27a),X0_2E0))),s(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),X1_2E0))) ).

fof(arityeq2_2Ec_2Ellist_2ELAPPEND_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27b_2CA_27a_29,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),c_2Ellist_2ELAPPEND_2E2(s(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),X0_2E0),s(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),X1_2E0))) = s(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)))),c_2Ellist_2ELAPPEND_2E0),s(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),X0_2E0))),s(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),X1_2E0))) ).

fof(arityeq2_2Ec_2Ellist_2ELUNFOLD_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27b_2CA_27a_29_20mono_2EA_27c,axiom,
    ! [A_27a,A_27b,A_27c,X0_2E0,X1_2E0] : s(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),c_2Ellist_2ELUNFOLD_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a)))),X0_2E0),s(A_27c,X1_2E0))) = s(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),app_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a)))),tyop_2Emin_2Efun(A_27c,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)))),c_2Ellist_2ELUNFOLD_2E0),s(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a)))),X0_2E0))),s(A_27c,X1_2E0))) ).

fof(arityeq3_2Ec_2Epair_2E_23_23_2E3_2Emono_2EA_27a_20mono_2Etyop_2Ellist_2Ellist_28tyop_2Epair_2Eprod_28A_27b_2CA_27a_29_29_20mono_2EA_27c_20mono_2Etyop_2Ellist_2Ellist_28tyop_2Epair_2Eprod_28A_27d_2CA_27c_29_29,axiom,
    ! [A_27a,A_27b,A_27c,A_27d,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Epair_2Eprod(A_27c,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27d,A_27c))),c_2Epair_2E_23_23_2E3(s(tyop_2Emin_2Efun(A_27a,A_27c),X0_2E0),s(tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27d,A_27c))),X1_2E0),s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),X2_2E0))) = s(tyop_2Epair_2Eprod(A_27c,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27d,A_27c))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),tyop_2Epair_2Eprod(A_27c,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27d,A_27c)))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27d,A_27c))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),tyop_2Epair_2Eprod(A_27c,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27d,A_27c))))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27c),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27d,A_27c))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),tyop_2Epair_2Eprod(A_27c,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27d,A_27c)))))),c_2Epair_2E_23_23_2E0),s(tyop_2Emin_2Efun(A_27a,A_27c),X0_2E0))),s(tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27d,A_27c))),X1_2E0))),s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),X2_2E0))) ).

fof(arityeq2_2Ec_2Epair_2E_23_23_2E2_2Emono_2EA_27b_20mono_2EA_27a_20mono_2EA_27d_20mono_2EA_27c,axiom,
    ! [A_27a,A_27b,A_27c,A_27d,X0_2E0,X1_2E0] : s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),tyop_2Epair_2Eprod(A_27d,A_27c)),c_2Epair_2E_23_23_2E2(s(tyop_2Emin_2Efun(A_27b,A_27d),X0_2E0),s(tyop_2Emin_2Efun(A_27a,A_27c),X1_2E0))) = s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),tyop_2Epair_2Eprod(A_27d,A_27c)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27c),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),tyop_2Epair_2Eprod(A_27d,A_27c))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27d),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27c),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),tyop_2Epair_2Eprod(A_27d,A_27c)))),c_2Epair_2E_23_23_2E0),s(tyop_2Emin_2Efun(A_27b,A_27d),X0_2E0))),s(tyop_2Emin_2Efun(A_27a,A_27c),X1_2E0))) ).

fof(arityeq1_2Ec_2Ellist_2ELMAP_2E1_2Emono_2Etyop_2Epair_2Eprod_28A_27b_2CA_27a_29_20mono_2Etyop_2Epair_2Eprod_28A_27d_2CA_27c_29,axiom,
    ! [A_27a,A_27b,A_27c,A_27d,X0_2E0] : s(tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27d,A_27c))),c_2Ellist_2ELMAP_2E1(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),tyop_2Epair_2Eprod(A_27d,A_27c)),X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27d,A_27c))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),tyop_2Epair_2Eprod(A_27d,A_27c)),tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27d,A_27c)))),c_2Ellist_2ELMAP_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),tyop_2Epair_2Eprod(A_27d,A_27c)),X0_2E0))) ).

fof(arityeq1_2Ec_2Epath_2EtoPath_2E1_2Emono_2EA_27c_20mono_2EA_27d,axiom,
    ! [A_27c,A_27d,X0_2E0] : s(tyop_2Epath_2Epath(A_27c,A_27d),c_2Epath_2EtoPath_2E1(s(tyop_2Epair_2Eprod(A_27c,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27d,A_27c))),X0_2E0))) = s(tyop_2Epath_2Epath(A_27c,A_27d),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27d,A_27c))),tyop_2Epath_2Epath(A_27c,A_27d)),c_2Epath_2EtoPath_2E0),s(tyop_2Epair_2Eprod(A_27c,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27d,A_27c))),X0_2E0))) ).

fof(arityeq2_2Ec_2Epath_2Enth__label_2E2_2Emono_2EA_27b_20mono_2EA_27a,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(A_27b,c_2Epath_2Enth__label_2E2(s(tyop_2Enum_2Enum,X0_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),X1_2E0))) = s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),A_27b)),c_2Epath_2Enth__label_2E0),s(tyop_2Enum_2Enum,X0_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),X1_2E0))) ).

fof(arityeq1_2Ec_2Epath_2Efirst_2E1_2Emono_2EA_27a_20mono_2EA_27c,axiom,
    ! [A_27a,A_27c,X0_2E0] : s(A_27a,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27a,A_27c),X0_2E0))) = s(A_27a,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27c),A_27a),c_2Epath_2Efirst_2E0),s(tyop_2Epath_2Epath(A_27a,A_27c),X0_2E0))) ).

fof(arityeq2_2Ec_2Elist_2EAPPEND_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0,X1_2E0] : s(tyop_2Elist_2Elist(A_27a),c_2Elist_2EAPPEND_2E2(s(tyop_2Elist_2Elist(A_27a),X0_2E0),s(tyop_2Elist_2Elist(A_27a),X1_2E0))) = s(tyop_2Elist_2Elist(A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27a)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27a))),c_2Elist_2EAPPEND_2E0),s(tyop_2Elist_2Elist(A_27a),X0_2E0))),s(tyop_2Elist_2Elist(A_27a),X1_2E0))) ).

fof(arityeq2_2Ec_2Elist_2ECONS_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0,X1_2E0] : s(tyop_2Elist_2Elist(A_27a),c_2Elist_2ECONS_2E2(s(A_27a,X0_2E0),s(tyop_2Elist_2Elist(A_27a),X1_2E0))) = s(tyop_2Elist_2Elist(A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27a)),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27a))),c_2Elist_2ECONS_2E0),s(A_27a,X0_2E0))),s(tyop_2Elist_2Elist(A_27a),X1_2E0))) ).

fof(arityeq1_2Ec_2Epath_2Efirst_2E1_2Emono_2EA_27c_20mono_2EA_27b,axiom,
    ! [A_27b,A_27c,X0_2E0] : s(A_27c,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27c,A_27b),X0_2E0))) = s(A_27c,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27c,A_27b),A_27c),c_2Epath_2Efirst_2E0),s(tyop_2Epath_2Epath(A_27c,A_27b),X0_2E0))) ).

fof(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2EA_27c_20mono_2Etyop_2Epair_2Eprod_28A_27b_2CA_27a_29,axiom,
    ! [A_27a,A_27b,A_27c,X0_2E0,X1_2E0] : s(tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a)),c_2Epair_2E_2C_2E2(s(A_27c,X0_2E0),s(tyop_2Epair_2Eprod(A_27b,A_27a),X1_2E0))) = s(tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a))),app_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a)))),c_2Epair_2E_2C_2E0),s(A_27c,X0_2E0))),s(tyop_2Epair_2Eprod(A_27b,A_27a),X1_2E0))) ).

fof(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Emin_2Ebool,axiom,
    ! [X0_2E0,X1_2E0] : s(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epair_2E_2C_2E2(s(tyop_2Enum_2Enum,X0_2E0),s(tyop_2Emin_2Ebool,X1_2E0))) = s(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),c_2Epair_2E_2C_2E0),s(tyop_2Enum_2Enum,X0_2E0))),s(tyop_2Emin_2Ebool,X1_2E0))) ).

fof(arityeq2_2Ec_2Earithmetic_2E_3C_3D_2E2,axiom,
    ! [X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Earithmetic_2E_3C_3D_2E2(s(tyop_2Enum_2Enum,X0_2E0),s(tyop_2Enum_2Enum,X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Earithmetic_2E_3C_3D_2E0),s(tyop_2Enum_2Enum,X0_2E0))),s(tyop_2Enum_2Enum,X1_2E0))) ).

fof(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Epath_2Epath_28A_27a_2CA_27b_29_20mono_2Etyop_2Emin_2Ebool,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),c_2Epair_2E_2C_2E2(s(tyop_2Epath_2Epath(A_27a,A_27b),X0_2E0),s(tyop_2Emin_2Ebool,X1_2E0))) = s(tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool))),c_2Epair_2E_2C_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),X0_2E0))),s(tyop_2Emin_2Ebool,X1_2E0))) ).

fof(arityeq2_2Ec_2Ebool_2EIN_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(A_27a,X0_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),c_2Ebool_2EIN_2E0),s(A_27a,X0_2E0))),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0))) ).

fof(arityeq2_2Ec_2Eprim__rec_2E_3C_2E2,axiom,
    ! [X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Eprim__rec_2E_3C_2E2(s(tyop_2Enum_2Enum,X0_2E0),s(tyop_2Enum_2Enum,X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Eprim__rec_2E_3C_2E0),s(tyop_2Enum_2Enum,X0_2E0))),s(tyop_2Enum_2Enum,X1_2E0))) ).

fof(arityeq1_2Ec_2Eoption_2ETHE_2E1_2Emono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0] : s(tyop_2Enum_2Enum,c_2Eoption_2ETHE_2E1(s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),X0_2E0))) = s(tyop_2Enum_2Enum,app_2E2(s(tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Enum_2Enum),c_2Eoption_2ETHE_2E0),s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),X0_2E0))) ).

fof(arityeq1_2Ec_2Elist_2ELENGTH_2E1_2Emono_2EA_27b,axiom,
    ! [A_27b,X0_2E0] : s(tyop_2Enum_2Enum,c_2Elist_2ELENGTH_2E1(s(tyop_2Elist_2Elist(A_27b),X0_2E0))) = s(tyop_2Enum_2Enum,app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Enum_2Enum),c_2Elist_2ELENGTH_2E0),s(tyop_2Elist_2Elist(A_27b),X0_2E0))) ).

fof(arityeq2_2Ec_2Elist_2EEL_2E2_2Emono_2EA_27b,axiom,
    ! [A_27b,X0_2E0,X1_2E0] : s(A_27b,c_2Elist_2EEL_2E2(s(tyop_2Enum_2Enum,X0_2E0),s(tyop_2Elist_2Elist(A_27b),X1_2E0))) = s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),A_27b)),c_2Elist_2EEL_2E0),s(tyop_2Enum_2Enum,X0_2E0))),s(tyop_2Elist_2Elist(A_27b),X1_2E0))) ).

fof(arityeq2_2Ec_2Epred__set_2EIMAGE_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0,X1_2E0] : s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epred__set_2EIMAGE_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X0_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X1_2E0))) = s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),c_2Epred__set_2EIMAGE_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X0_2E0))),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X1_2E0))) ).

fof(arityeq2_2Ec_2Ebool_2EIN_2E2_2Emono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,X0_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),c_2Ebool_2EIN_2E0),s(tyop_2Enum_2Enum,X0_2E0))),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X1_2E0))) ).

fof(arityeq1_2Ec_2Enum_2ESUC_2E1,axiom,
    ! [X0_2E0] : s(tyop_2Enum_2Enum,c_2Enum_2ESUC_2E1(s(tyop_2Enum_2Enum,X0_2E0))) = s(tyop_2Enum_2Enum,app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Enum_2ESUC_2E0),s(tyop_2Enum_2Enum,X0_2E0))) ).

fof(arityeq2_2Ec_2Ebool_2EIN_2E2_2Emono_2Etyop_2Epath_2Epath_28A_27a_2CA_27b_29,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Epath_2Epath(A_27a,A_27b),X0_2E0),s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),c_2Ebool_2EIN_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),X0_2E0))),s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),X1_2E0))) ).

fof(arityeq2_2Ec_2Epred__set_2EINSERT_2E2_2Emono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0,X1_2E0] : s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(s(tyop_2Enum_2Enum,X0_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X1_2E0))) = s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),c_2Epred__set_2EINSERT_2E0),s(tyop_2Enum_2Enum,X0_2E0))),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X1_2E0))) ).

fof(arityeq2_2Ec_2Ellist_2ELAPPEND_2E2_2Emono_2EA_27b,axiom,
    ! [A_27b,X0_2E0,X1_2E0] : s(tyop_2Ellist_2Ellist(A_27b),c_2Ellist_2ELAPPEND_2E2(s(tyop_2Ellist_2Ellist(A_27b),X0_2E0),s(tyop_2Ellist_2Ellist(A_27b),X1_2E0))) = s(tyop_2Ellist_2Ellist(A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(A_27b),tyop_2Ellist_2Ellist(A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(A_27b),tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(A_27b),tyop_2Ellist_2Ellist(A_27b))),c_2Ellist_2ELAPPEND_2E0),s(tyop_2Ellist_2Ellist(A_27b),X0_2E0))),s(tyop_2Ellist_2Ellist(A_27b),X1_2E0))) ).

fof(arityeq2_2Ec_2Ellist_2ELCONS_2E2_2Emono_2EA_27b,axiom,
    ! [A_27b,X0_2E0,X1_2E0] : s(tyop_2Ellist_2Ellist(A_27b),c_2Ellist_2ELCONS_2E2(s(A_27b,X0_2E0),s(tyop_2Ellist_2Ellist(A_27b),X1_2E0))) = s(tyop_2Ellist_2Ellist(A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(A_27b),tyop_2Ellist_2Ellist(A_27b)),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(A_27b),tyop_2Ellist_2Ellist(A_27b))),c_2Ellist_2ELCONS_2E0),s(A_27b,X0_2E0))),s(tyop_2Ellist_2Ellist(A_27b),X1_2E0))) ).

fof(arityeq2_2Ec_2Ellist_2ELMAP_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27b_2CA_27a_29_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Ellist_2Ellist(A_27b),c_2Ellist_2ELMAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),A_27b),X0_2E0),s(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),X1_2E0))) = s(tyop_2Ellist_2Ellist(A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Ellist_2Ellist(A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),A_27b),tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Ellist_2Ellist(A_27b))),c_2Ellist_2ELMAP_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),A_27b),X0_2E0))),s(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),X1_2E0))) ).

fof(arityeq2_2Ec_2Ellist_2ELNTH_2E2_2Emono_2EA_27b,axiom,
    ! [A_27b,X0_2E0,X1_2E0] : s(tyop_2Eoption_2Eoption(A_27b),c_2Ellist_2ELNTH_2E2(s(tyop_2Enum_2Enum,X0_2E0),s(tyop_2Ellist_2Ellist(A_27b),X1_2E0))) = s(tyop_2Eoption_2Eoption(A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(A_27b),tyop_2Eoption_2Eoption(A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(A_27b),tyop_2Eoption_2Eoption(A_27b))),c_2Ellist_2ELNTH_2E0),s(tyop_2Enum_2Enum,X0_2E0))),s(tyop_2Ellist_2Ellist(A_27b),X1_2E0))) ).

fof(arityeq2_2Ec_2Ellist_2ELTAKE_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0,X1_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Elist_2Elist(A_27a)),c_2Ellist_2ELTAKE_2E2(s(tyop_2Enum_2Enum,X0_2E0),s(tyop_2Ellist_2Ellist(A_27a),X1_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Elist_2Elist(A_27a)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(A_27a),tyop_2Eoption_2Eoption(tyop_2Elist_2Elist(A_27a))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(A_27a),tyop_2Eoption_2Eoption(tyop_2Elist_2Elist(A_27a)))),c_2Ellist_2ELTAKE_2E0),s(tyop_2Enum_2Enum,X0_2E0))),s(tyop_2Ellist_2Ellist(A_27a),X1_2E0))) ).

fof(arityeq2_2Ec_2Ellist_2ELTAKE_2E2_2Emono_2EA_27b,axiom,
    ! [A_27b,X0_2E0,X1_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Elist_2Elist(A_27b)),c_2Ellist_2ELTAKE_2E2(s(tyop_2Enum_2Enum,X0_2E0),s(tyop_2Ellist_2Ellist(A_27b),X1_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Elist_2Elist(A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(A_27b),tyop_2Eoption_2Eoption(tyop_2Elist_2Elist(A_27b))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(A_27b),tyop_2Eoption_2Eoption(tyop_2Elist_2Elist(A_27b)))),c_2Ellist_2ELTAKE_2E0),s(tyop_2Enum_2Enum,X0_2E0))),s(tyop_2Ellist_2Ellist(A_27b),X1_2E0))) ).

fof(arityeq2_2Ec_2Ellist_2ELUNFOLD_2E2_2Emono_2EA_27c_20mono_2EA_27a,axiom,
    ! [A_27a,A_27c,X0_2E0,X1_2E0] : s(tyop_2Ellist_2Ellist(A_27c),c_2Ellist_2ELUNFOLD_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),X0_2E0),s(A_27a,X1_2E0))) = s(tyop_2Ellist_2Ellist(A_27c),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Ellist_2Ellist(A_27c)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),tyop_2Emin_2Efun(A_27a,tyop_2Ellist_2Ellist(A_27c))),c_2Ellist_2ELUNFOLD_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),X0_2E0))),s(A_27a,X1_2E0))) ).

fof(arityeq2_2Ec_2Eoption_2EOPTION__MAP_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Enum_2Enum,axiom,
    ! [X0_2E0,X1_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Eoption_2EOPTION__MAP_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X0_2E0),s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),X1_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),app_2E2(s(tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Eoption_2Eoption(tyop_2Enum_2Enum))),c_2Eoption_2EOPTION__MAP_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X0_2E0))),s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),X1_2E0))) ).

fof(arityeq2_2Ec_2Eoption_2EOPTION__MAP_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27c_2CA_27b_29_20mono_2Etyop_2Epair_2Eprod_28A_27c_2Ctyop_2Epair_2Eprod_28A_27b_2CA_27a_29_29,axiom,
    ! [A_27a,A_27b,A_27c,X0_2E0,X1_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a))),c_2Eoption_2EOPTION__MAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27b),tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a))),X0_2E0),s(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27b)),X1_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27b)),tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a)))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27b),tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a))),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27b)),tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a))))),c_2Eoption_2EOPTION__MAP_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27b),tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a))),X0_2E0))),s(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27b)),X1_2E0))) ).

fof(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2EA_27c_20mono_2EA_27b_20mono_2Etyop_2Epair_2Eprod_28A_27c_2Ctyop_2Epair_2Eprod_28A_27b_2CA_27a_29_29,axiom,
    ! [A_27a,A_27b,A_27c,X0_2E0] : s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27b),tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a))),c_2Epair_2EUNCURRY_2E1(s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a)))),X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27b),tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a)))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27b),tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a)))),c_2Epair_2EUNCURRY_2E0),s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a)))),X0_2E0))) ).

fof(arityeq2_2Ec_2Ebool_2ETYPE__DEFINITION_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2Ctyop_2Ellist_2Ellist_28tyop_2Epair_2Eprod_28A_27b_2CA_27a_29_29_29_20mono_2Etyop_2Epath_2Epath_28A_27a_2CA_27b_29,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Ebool_2ETYPE__DEFINITION_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),tyop_2Emin_2Ebool),X0_2E0),s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)))),X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)))),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)))),tyop_2Emin_2Ebool)),c_2Ebool_2ETYPE__DEFINITION_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),tyop_2Emin_2Ebool),X0_2E0))),s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)))),X1_2E0))) ).

fof(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2Etyop_2Epath_2Epath_28A_27a_2CA_27b_29_20mono_2EA_27a,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),c_2Epred__set_2EGSPEC_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),X0_2E0))) ).

fof(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2Etyop_2Epath_2Epath_28A_27a_2CA_27b_29_20mono_2Etyop_2Epair_2Eprod_28A_27a_2Ctyop_2Epair_2Eprod_28A_27b_2Ctyop_2Epath_2Epath_28A_27a_2CA_27b_29_29_29,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E1(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Epath_2Epath(A_27a,A_27b))),tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Epath_2Epath(A_27a,A_27b))),tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),c_2Epred__set_2EGSPEC_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Epath_2Epath(A_27a,A_27b))),tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),X0_2E0))) ).

fof(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2EA_27a_20mono_2Etyop_2Epair_2Eprod_28A_27b_2Ctyop_2Epath_2Epath_28A_27a_2CA_27b_29_29_20mono_2Etyop_2Epair_2Eprod_28tyop_2Epath_2Epath_28A_27a_2CA_27b_29_2Ctyop_2Emin_2Ebool_29,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Epath_2Epath(A_27a,A_27b))),tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),c_2Epair_2EUNCURRY_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,tyop_2Epath_2Epath(A_27a,A_27b)),tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool))),X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Epath_2Epath(A_27a,A_27b))),tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,tyop_2Epath_2Epath(A_27a,A_27b)),tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Epath_2Epath(A_27a,A_27b))),tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool))),c_2Epair_2EUNCURRY_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,tyop_2Epath_2Epath(A_27a,A_27b)),tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool))),X0_2E0))) ).

fof(arityeq2_2Ec_2Epred__set_2EUNION_2E2_2Emono_2Etyop_2Epath_2Epath_28A_27a_2CA_27b_29,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),c_2Epred__set_2EUNION_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),X0_2E0),s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),X1_2E0))) = s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool))),c_2Epred__set_2EUNION_2E0),s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),X0_2E0))),s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),X1_2E0))) ).

fof(arityeq1_2Ec_2Earithmetic_2EBIT2_2E1,axiom,
    ! [X0_2E0] : s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT2_2E1(s(tyop_2Enum_2Enum,X0_2E0))) = s(tyop_2Enum_2Enum,app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2EBIT2_2E0),s(tyop_2Enum_2Enum,X0_2E0))) ).

fof(arityeq2_2Ec_2Epath_2Eel_2E2_2Emono_2EA_27c_20mono_2EA_27d,axiom,
    ! [A_27c,A_27d,X0_2E0,X1_2E0] : s(A_27c,c_2Epath_2Eel_2E2(s(tyop_2Enum_2Enum,X0_2E0),s(tyop_2Epath_2Epath(A_27c,A_27d),X1_2E0))) = s(A_27c,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27c,A_27d),A_27c),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27c,A_27d),A_27c)),c_2Epath_2Eel_2E0),s(tyop_2Enum_2Enum,X0_2E0))),s(tyop_2Epath_2Epath(A_27c,A_27d),X1_2E0))) ).

fof(arityeq2_2Ec_2Epath_2Eevery_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Epath_2Eevery_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),c_2Epath_2Eevery_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),X1_2E0))) ).

fof(arityeq2_2Ec_2Epath_2Efilter_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Efilter_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),X1_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epath_2Epath(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epath_2Epath(A_27a,A_27b))),c_2Epath_2Efilter_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),X1_2E0))) ).

fof(arityeq2_2Ec_2Ecombin_2Eo_2E2_2Emono_2EA_27a_20mono_2Etyop_2Emin_2Ebool_20mono_2Etyop_2Emin_2Ebool,axiom,
    ! [A_27a,X0_2E0,X1_2E0] : s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Ecombin_2Eo_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),X0_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0))) = s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),c_2Ecombin_2Eo_2E0),s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),X0_2E0))),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0))) ).

fof(arityeq2_2Ec_2Epath_2Eevery_2E2_2Emono_2EA_27a_20mono_2EA_27c,axiom,
    ! [A_27a,A_27c,X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Epath_2Eevery_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0),s(tyop_2Epath_2Epath(A_27a,A_27c),X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27c),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27c),tyop_2Emin_2Ebool)),c_2Epath_2Eevery_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27c),X1_2E0))) ).

fof(arityeq3_2Ec_2Epath_2Epcons_2E3_2Emono_2EA_27a_20mono_2EA_27c,axiom,
    ! [A_27a,A_27c,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27c),c_2Epath_2Epcons_2E3(s(A_27a,X0_2E0),s(A_27c,X1_2E0),s(tyop_2Epath_2Epath(A_27a,A_27c),X2_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27c),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27c),tyop_2Epath_2Epath(A_27a,A_27c)),app_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27c),tyop_2Epath_2Epath(A_27a,A_27c))),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27c),tyop_2Epath_2Epath(A_27a,A_27c)))),c_2Epath_2Epcons_2E0),s(A_27a,X0_2E0))),s(A_27c,X1_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27c),X2_2E0))) ).

fof(arityeq2_2Ec_2Epath_2Eexists_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Epath_2Eexists_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),c_2Epath_2Eexists_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),X1_2E0))) ).

fof(arityeq2_2Ec_2Epath_2Eexists_2E2_2Emono_2EA_27a_20mono_2EA_27c,axiom,
    ! [A_27a,A_27c,X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Epath_2Eexists_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0),s(tyop_2Epath_2Epath(A_27a,A_27c),X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27c),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27c),tyop_2Emin_2Ebool)),c_2Epath_2Eexists_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27c),X1_2E0))) ).

fof(arityeq2_2Ec_2Epath_2Emem_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Epath_2Emem_2E2(s(A_27a,X0_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),c_2Epath_2Emem_2E0),s(A_27a,X0_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),X1_2E0))) ).

fof(arityeq2_2Ec_2Epath_2Emem_2E2_2Emono_2EA_27c_20mono_2EA_27d,axiom,
    ! [A_27c,A_27d,X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Epath_2Emem_2E2(s(A_27c,X0_2E0),s(tyop_2Epath_2Epath(A_27c,A_27d),X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27c,A_27d),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27c,A_27d),tyop_2Emin_2Ebool)),c_2Epath_2Emem_2E0),s(A_27c,X0_2E0))),s(tyop_2Epath_2Epath(A_27c,A_27d),X1_2E0))) ).

fof(arityeq2_2Ec_2Epath_2Enth__label_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(A_27a,c_2Epath_2Enth__label_2E2(s(tyop_2Enum_2Enum,X0_2E0),s(tyop_2Epath_2Epath(A_27b,A_27a),X1_2E0))) = s(A_27a,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27b,A_27a),A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27b,A_27a),A_27a)),c_2Epath_2Enth__label_2E0),s(tyop_2Enum_2Enum,X0_2E0))),s(tyop_2Epath_2Epath(A_27b,A_27a),X1_2E0))) ).

fof(arityeq1_2Ec_2Epath_2Etail_2E1_2Emono_2EA_27b_20mono_2EA_27a,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Epath_2Epath(A_27b,A_27a),c_2Epath_2Etail_2E1(s(tyop_2Epath_2Epath(A_27b,A_27a),X0_2E0))) = s(tyop_2Epath_2Epath(A_27b,A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27b,A_27a),tyop_2Epath_2Epath(A_27b,A_27a)),c_2Epath_2Etail_2E0),s(tyop_2Epath_2Epath(A_27b,A_27a),X0_2E0))) ).

fof(arityeq2_2Ec_2Epath_2Enth__label_2E2_2Emono_2EA_27c_20mono_2EA_27d,axiom,
    ! [A_27c,A_27d,X0_2E0,X1_2E0] : s(A_27c,c_2Epath_2Enth__label_2E2(s(tyop_2Enum_2Enum,X0_2E0),s(tyop_2Epath_2Epath(A_27d,A_27c),X1_2E0))) = s(A_27c,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27d,A_27c),A_27c),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27d,A_27c),A_27c)),c_2Epath_2Enth__label_2E0),s(tyop_2Enum_2Enum,X0_2E0))),s(tyop_2Epath_2Epath(A_27d,A_27c),X1_2E0))) ).

fof(arityeq3_2Ec_2Epath_2Epmap_2E3_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27d_20mono_2EA_27c,axiom,
    ! [A_27a,A_27b,A_27c,A_27d,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Epath_2Epath(A_27d,A_27c),c_2Epath_2Epmap_2E3(s(tyop_2Emin_2Efun(A_27a,A_27d),X0_2E0),s(tyop_2Emin_2Efun(A_27b,A_27c),X1_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),X2_2E0))) = s(tyop_2Epath_2Epath(A_27d,A_27c),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epath_2Epath(A_27d,A_27c)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epath_2Epath(A_27d,A_27c))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27d),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epath_2Epath(A_27d,A_27c)))),c_2Epath_2Epmap_2E0),s(tyop_2Emin_2Efun(A_27a,A_27d),X0_2E0))),s(tyop_2Emin_2Efun(A_27b,A_27c),X1_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),X2_2E0))) ).

fof(arityeq2_2Ec_2Epath_2Eokpath_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),X0_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),c_2Epath_2Eokpath_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),X0_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),X1_2E0))) ).

fof(arityeq3_2Ec_2Epath_2Epmap_2E3_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epmap_2E3(s(tyop_2Emin_2Efun(A_27a,A_27a),X0_2E0),s(tyop_2Emin_2Efun(A_27b,A_27b),X1_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),X2_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epath_2Epath(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27b),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epath_2Epath(A_27a,A_27b))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27b),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epath_2Epath(A_27a,A_27b)))),c_2Epath_2Epmap_2E0),s(tyop_2Emin_2Efun(A_27a,A_27a),X0_2E0))),s(tyop_2Emin_2Efun(A_27b,A_27b),X1_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),X2_2E0))) ).

fof(arityeq2_2Ec_2Epath_2Eokpath_2E2_2Emono_2EA_27a_20mono_2EA_27c,axiom,
    ! [A_27a,A_27c,X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),X0_2E0),s(tyop_2Epath_2Epath(A_27a,A_27c),X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27c),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27c),tyop_2Emin_2Ebool)),c_2Epath_2Eokpath_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),X0_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27c),X1_2E0))) ).

fof(arityeq3_2Ec_2Epath_2Epmap_2E3_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29_20mono_2EA_27c_20mono_2EA_27a_20mono_2EA_27c,axiom,
    ! [A_27a,A_27b,A_27c,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27c),c_2Epath_2Epmap_2E3(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a),X0_2E0),s(tyop_2Emin_2Efun(A_27c,A_27c),X1_2E0),s(tyop_2Epath_2Epath(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),X2_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27c),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),tyop_2Epath_2Epath(A_27a,A_27c)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27c),tyop_2Emin_2Efun(tyop_2Epath_2Epath(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),tyop_2Epath_2Epath(A_27a,A_27c))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27c),tyop_2Emin_2Efun(tyop_2Epath_2Epath(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),tyop_2Epath_2Epath(A_27a,A_27c)))),c_2Epath_2Epmap_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a),X0_2E0))),s(tyop_2Emin_2Efun(A_27c,A_27c),X1_2E0))),s(tyop_2Epath_2Epath(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),X2_2E0))) ).

fof(arityeq2_2Ec_2Epath_2Eokpath_2E2_2Emono_2EA_27b_20mono_2EA_27c,axiom,
    ! [A_27b,A_27c,X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),X0_2E0),s(tyop_2Epath_2Epath(A_27b,A_27c),X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27b,A_27c),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27b,A_27c),tyop_2Emin_2Ebool)),c_2Epath_2Eokpath_2E0),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),X0_2E0))),s(tyop_2Epath_2Epath(A_27b,A_27c),X1_2E0))) ).

fof(arityeq3_2Ec_2Epath_2Epmap_2E3_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29_20mono_2EA_27c_20mono_2EA_27b_20mono_2EA_27c,axiom,
    ! [A_27a,A_27b,A_27c,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Epath_2Epath(A_27b,A_27c),c_2Epath_2Epmap_2E3(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27b),X0_2E0),s(tyop_2Emin_2Efun(A_27c,A_27c),X1_2E0),s(tyop_2Epath_2Epath(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),X2_2E0))) = s(tyop_2Epath_2Epath(A_27b,A_27c),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),tyop_2Epath_2Epath(A_27b,A_27c)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27c),tyop_2Emin_2Efun(tyop_2Epath_2Epath(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),tyop_2Epath_2Epath(A_27b,A_27c))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27c),tyop_2Emin_2Efun(tyop_2Epath_2Epath(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),tyop_2Epath_2Epath(A_27b,A_27c)))),c_2Epath_2Epmap_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27b),X0_2E0))),s(tyop_2Emin_2Efun(A_27c,A_27c),X1_2E0))),s(tyop_2Epath_2Epath(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),X2_2E0))) ).

fof(arityeq2_2Ec_2Epath_2Eokpath_2E2_2Emono_2EA_27c_20mono_2EA_27b,axiom,
    ! [A_27b,A_27c,X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool))),X0_2E0),s(tyop_2Epath_2Epath(A_27c,A_27b),X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27c,A_27b),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27c,A_27b),tyop_2Emin_2Ebool)),c_2Epath_2Eokpath_2E0),s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool))),X0_2E0))),s(tyop_2Epath_2Epath(A_27c,A_27b),X1_2E0))) ).

fof(arityeq2_2Ec_2Epath_2Eokpath_2E2_2Emono_2Etyop_2Elist_2Elist_28A_27a_29_20mono_2EA_27a,axiom,
    ! [A_27a,X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool))),X0_2E0),s(tyop_2Epath_2Epath(tyop_2Elist_2Elist(A_27a),A_27a),X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(tyop_2Elist_2Elist(A_27a),A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Epath_2Epath(tyop_2Elist_2Elist(A_27a),A_27a),tyop_2Emin_2Ebool)),c_2Epath_2Eokpath_2E0),s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool))),X0_2E0))),s(tyop_2Epath_2Epath(tyop_2Elist_2Elist(A_27a),A_27a),X1_2E0))) ).

fof(arityeq1_2Ec_2Epath_2Etrace__machine_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool))),c_2Epath_2Etrace__machine_2E1(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool),X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool)))),c_2Epath_2Etrace__machine_2E0),s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool),X0_2E0))) ).

fof(arityeq2_2Ec_2Epath_2Eokpath_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29_20mono_2EA_27c,axiom,
    ! [A_27a,A_27b,A_27c,X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool))),X0_2E0),s(tyop_2Epath_2Epath(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Epath_2Epath(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),tyop_2Emin_2Ebool)),c_2Epath_2Eokpath_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool))),X0_2E0))),s(tyop_2Epath_2Epath(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),X1_2E0))) ).

fof(arityeq2_2Ec_2Epath_2Eparallel__comp_2E2_2Emono_2EA_27a_20mono_2EA_27c_20mono_2EA_27a_20mono_2EA_27b_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,A_27c,X0_2E0,X1_2E0] : s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool))),c_2Epath_2Eparallel__comp_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),X0_2E0),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),X1_2E0))) = s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool))))),c_2Epath_2Eparallel__comp_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),X0_2E0))),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),X1_2E0))) ).

fof(arityeq2_2Ec_2Epath_2Eokpath_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27c_29_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,A_27c,X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Emin_2Ebool))),X0_2E0),s(tyop_2Epath_2Epath(tyop_2Epair_2Eprod(A_27a,A_27c),A_27b),X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(tyop_2Epair_2Eprod(A_27a,A_27c),A_27b),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Epath_2Epath(tyop_2Epair_2Eprod(A_27a,A_27c),A_27b),tyop_2Emin_2Ebool)),c_2Epath_2Eokpath_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Emin_2Ebool))),X0_2E0))),s(tyop_2Epath_2Epath(tyop_2Epair_2Eprod(A_27a,A_27c),A_27b),X1_2E0))) ).

fof(arityeq2_2Ec_2Epath_2Eparallel__comp_2E2_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27a_20mono_2EA_27c_20mono_2EA_27c,axiom,
    ! [A_27a,A_27b,A_27c,X0_2E0,X1_2E0] : s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Emin_2Ebool))),c_2Epath_2Eparallel__comp_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),X0_2E0),s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool))),X1_2E0))) = s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Emin_2Ebool))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Emin_2Ebool)))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Emin_2Ebool))))),c_2Epath_2Eparallel__comp_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),X0_2E0))),s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool))),X1_2E0))) ).

fof(arityeq2_2Ec_2Epath_2Eokpath__f_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),c_2Epath_2Eokpath__f_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),X0_2E0),s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),X1_2E0))) = s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool))),c_2Epath_2Eokpath__f_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),X0_2E0))),s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),X1_2E0))) ).

fof(arityeq2_2Ec_2Epair_2Epair__CASE_2E2_2Emono_2Etyop_2Epath_2Epath_28A_27b_2CA_27c_29_20mono_2EA_27a_20mono_2EA_27c,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_27a,A_27c),X0_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27c,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_27a,tyop_2Emin_2Efun(A_27c,tyop_2Epath_2Epath(A_27b,A_27c))),tyop_2Epath_2Epath(A_27b,A_27c)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Epath_2Epath(A_27b,A_27c))),tyop_2Epath_2Epath(A_27b,A_27c))),c_2Epair_2Epair__CASE_2E0),s(tyop_2Epair_2Eprod(A_27a,A_27c),X0_2E0))),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27c,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_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_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Eoption_2Eoption_28tyop_2Enum_2Enum_29,axiom,
    ! [X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Ebool_2ECOND_2E3(s(tyop_2Emin_2Ebool,X0_2E0),s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),X1_2E0),s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),X2_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),app_2E2(s(tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Eoption_2Eoption(tyop_2Enum_2Enum))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)))),c_2Ebool_2ECOND_2E0),s(tyop_2Emin_2Ebool,X0_2E0))),s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),X1_2E0))),s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),X2_2E0))) ).

fof(arityeq1_2Ec_2Elist_2ELENGTH_2E1_2Emono_2Etyop_2Epair_2Eprod_28A_27b_2CA_27a_29,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Enum_2Enum,c_2Elist_2ELENGTH_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),X0_2E0))) = s(tyop_2Enum_2Enum,app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Enum_2Enum),c_2Elist_2ELENGTH_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),X0_2E0))) ).

fof(arityeq1_2Ec_2Eoption_2ETHE_2E1_2Emono_2Etyop_2Elist_2Elist_28tyop_2Epair_2Eprod_28A_27b_2CA_27a_29_29,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),c_2Eoption_2ETHE_2E1(s(tyop_2Eoption_2Eoption(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a))),X0_2E0))) = s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a))),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a))),c_2Eoption_2ETHE_2E0),s(tyop_2Eoption_2Eoption(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a))),X0_2E0))) ).

fof(arityeq1_2Ec_2Ellist_2EtoList_2E1_2Emono_2Etyop_2Epair_2Eprod_28A_27b_2CA_27a_29,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a))),c_2Ellist_2EtoList_2E1(s(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),X0_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Eoption_2Eoption(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)))),c_2Ellist_2EtoList_2E0),s(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),X0_2E0))) ).

fof(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Epath_2Epath_28A_27a_2CA_27b_29,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Ebool_2ECOND_2E3(s(tyop_2Emin_2Ebool,X0_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),X1_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),X2_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epath_2Epath(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epath_2Epath(A_27a,A_27b))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epath_2Epath(A_27a,A_27b)))),c_2Ebool_2ECOND_2E0),s(tyop_2Emin_2Ebool,X0_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),X1_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),X2_2E0))) ).

fof(arityeq3_2Ec_2Epath_2EfirstP__at_2E3_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Emin_2Ebool,c_2Epath_2EfirstP__at_2E3(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),X1_2E0),s(tyop_2Enum_2Enum,X2_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),c_2Epath_2EfirstP__at_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),X1_2E0))),s(tyop_2Enum_2Enum,X2_2E0))) ).

fof(arityeq3_2Ec_2Eoption_2Eoption__CASE_2E3_2Emono_2Etyop_2Enum_2Enum_20mono_2Etyop_2Eoption_2Eoption_28tyop_2Enum_2Enum_29,axiom,
    ! [X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Eoption_2Eoption__CASE_2E3(s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),X0_2E0),s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),X1_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)),X2_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)),tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)),tyop_2Eoption_2Eoption(tyop_2Enum_2Enum))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)),tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)))),c_2Eoption_2Eoption__CASE_2E0),s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),X0_2E0))),s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),X1_2E0))),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)),X2_2E0))) ).

fof(arityeq3_2Ec_2Eoption_2Eoption__CASE_2E3_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27c_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_27a,A_27c)),X0_2E0),s(tyop_2Epath_2Epath(A_27b,A_27c),X1_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),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_27a,A_27c),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_27a,A_27c),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_27a,A_27c)),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),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_27a,A_27c)),X0_2E0))),s(tyop_2Epath_2Epath(A_27b,A_27c),X1_2E0))),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Epath_2Epath(A_27b,A_27c)),X2_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(arityeq3_2Ec_2Epath_2Epconcat_2E3_2Emono_2EA_27c_20mono_2EA_27d,axiom,
    ! [A_27c,A_27d,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Epath_2Epath(A_27c,A_27d),c_2Epath_2Epconcat_2E3(s(tyop_2Epath_2Epath(A_27c,A_27d),X0_2E0),s(A_27d,X1_2E0),s(tyop_2Epath_2Epath(A_27c,A_27d),X2_2E0))) = s(tyop_2Epath_2Epath(A_27c,A_27d),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27c,A_27d),tyop_2Epath_2Epath(A_27c,A_27d)),app_2E2(s(tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27c,A_27d),tyop_2Epath_2Epath(A_27c,A_27d))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27c,A_27d),tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27c,A_27d),tyop_2Epath_2Epath(A_27c,A_27d)))),c_2Epath_2Epconcat_2E0),s(tyop_2Epath_2Epath(A_27c,A_27d),X0_2E0))),s(A_27d,X1_2E0))),s(tyop_2Epath_2Epath(A_27c,A_27d),X2_2E0))) ).

fof(arityeq2_2Ec_2Ecombin_2Eo_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2EA_27a_20mono_2Etyop_2Enum_2Enum,axiom,
    ! [A_27a,X0_2E0,X1_2E0] : s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),c_2Ecombin_2Eo_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),X0_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X1_2E0))) = s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a))),c_2Ecombin_2Eo_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),X0_2E0))),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X1_2E0))) ).

fof(arityeq2_2Ec_2Ecombin_2Eo_2E2_2Emono_2Etyop_2Enum_2Enum_20mono_2EA_27b_20mono_2Etyop_2Enum_2Enum,axiom,
    ! [A_27b,X0_2E0,X1_2E0] : s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27b),c_2Ecombin_2Eo_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27b),X0_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X1_2E0))) = s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27b))),c_2Ecombin_2Eo_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27b),X0_2E0))),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),X1_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(arityeq3_2Ec_2Epath_2Epmap_2E3_2Emono_2EA_27a_20mono_2EA_27d_20mono_2EA_27b_20mono_2EA_27c,axiom,
    ! [A_27a,A_27b,A_27c,A_27d,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Epath_2Epath(A_27b,A_27c),c_2Epath_2Epmap_2E3(s(tyop_2Emin_2Efun(A_27a,A_27b),X0_2E0),s(tyop_2Emin_2Efun(A_27d,A_27c),X1_2E0),s(tyop_2Epath_2Epath(A_27a,A_27d),X2_2E0))) = s(tyop_2Epath_2Epath(A_27b,A_27c),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27d),tyop_2Epath_2Epath(A_27b,A_27c)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27c),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27d),tyop_2Epath_2Epath(A_27b,A_27c))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27c),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27d),tyop_2Epath_2Epath(A_27b,A_27c)))),c_2Epath_2Epmap_2E0),s(tyop_2Emin_2Efun(A_27a,A_27b),X0_2E0))),s(tyop_2Emin_2Efun(A_27d,A_27c),X1_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27d),X2_2E0))) ).

fof(arityeq1_2Ec_2Epath_2Estopped__at_2E1_2Emono_2EA_27a_20mono_2EA_27d,axiom,
    ! [A_27a,A_27d,X0_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27d),c_2Epath_2Estopped__at_2E1(s(A_27a,X0_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27d),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Epath_2Epath(A_27a,A_27d)),c_2Epath_2Estopped__at_2E0),s(A_27a,X0_2E0))) ).

fof(arityeq3_2Ec_2Epath_2Epcons_2E3_2Emono_2EA_27a_20mono_2EA_27d,axiom,
    ! [A_27a,A_27d,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27d),c_2Epath_2Epcons_2E3(s(A_27a,X0_2E0),s(A_27d,X1_2E0),s(tyop_2Epath_2Epath(A_27a,A_27d),X2_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27d),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27d),tyop_2Epath_2Epath(A_27a,A_27d)),app_2E2(s(tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27d),tyop_2Epath_2Epath(A_27a,A_27d))),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27d),tyop_2Epath_2Epath(A_27a,A_27d)))),c_2Epath_2Epcons_2E0),s(A_27a,X0_2E0))),s(A_27d,X1_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27d),X2_2E0))) ).

fof(arityeq3_2Ec_2Epath_2Eunfold_2E3_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27c,axiom,
    ! [A_27a,A_27b,A_27c,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Eunfold_2E3(s(tyop_2Emin_2Efun(A_27c,A_27a),X0_2E0),s(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27b))),X1_2E0),s(A_27c,X2_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),app_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Epath_2Epath(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27b))),tyop_2Emin_2Efun(A_27c,tyop_2Epath_2Epath(A_27a,A_27b))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27b))),tyop_2Emin_2Efun(A_27c,tyop_2Epath_2Epath(A_27a,A_27b)))),c_2Epath_2Eunfold_2E0),s(tyop_2Emin_2Efun(A_27c,A_27a),X0_2E0))),s(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27b))),X1_2E0))),s(A_27c,X2_2E0))) ).

fof(arityeq4_2Ec_2Epath_2Etrace__machine_2E4_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0,X1_2E0,X2_2E0,X3_2E0] : s(tyop_2Emin_2Ebool,c_2Epath_2Etrace__machine_2E4(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool),X0_2E0),s(tyop_2Elist_2Elist(A_27a),X1_2E0),s(A_27a,X2_2E0),s(tyop_2Elist_2Elist(A_27a),X3_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool)))),c_2Epath_2Etrace__machine_2E0),s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool),X0_2E0))),s(tyop_2Elist_2Elist(A_27a),X1_2E0))),s(A_27a,X2_2E0))),s(tyop_2Elist_2Elist(A_27a),X3_2E0))) ).

fof(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2EA_27a_20mono_2EA_27d,axiom,
    ! [A_27a,A_27d,X0_2E0,X1_2E0] : s(tyop_2Epair_2Eprod(A_27a,A_27d),c_2Epair_2E_2C_2E2(s(A_27a,X0_2E0),s(A_27d,X1_2E0))) = s(tyop_2Epair_2Eprod(A_27a,A_27d),app_2E2(s(tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(A_27a,A_27d)),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(A_27a,A_27d))),c_2Epair_2E_2C_2E0),s(A_27a,X0_2E0))),s(A_27d,X1_2E0))) ).

fof(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2EA_27c_20mono_2EA_27e,axiom,
    ! [A_27c,A_27e,X0_2E0,X1_2E0] : s(tyop_2Epair_2Eprod(A_27c,A_27e),c_2Epair_2E_2C_2E2(s(A_27c,X0_2E0),s(A_27e,X1_2E0))) = s(tyop_2Epair_2Eprod(A_27c,A_27e),app_2E2(s(tyop_2Emin_2Efun(A_27e,tyop_2Epair_2Eprod(A_27c,A_27e)),app_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27e,tyop_2Epair_2Eprod(A_27c,A_27e))),c_2Epair_2E_2C_2E0),s(A_27c,X0_2E0))),s(A_27e,X1_2E0))) ).

fof(arityeq5_2Ec_2Epath_2Eparallel__comp_2E5_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27c_20mono_2EA_27d_20mono_2EA_27e,axiom,
    ! [A_27a,A_27b,A_27c,A_27d,A_27e,X0_2E0,X1_2E0,X2_2E0,X3_2E0,X4_2E0] : s(tyop_2Emin_2Ebool,c_2Epath_2Eparallel__comp_2E5(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool))),X0_2E0),s(tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),X1_2E0),s(tyop_2Epair_2Eprod(A_27a,A_27d),X2_2E0),s(A_27b,X3_2E0),s(tyop_2Epair_2Eprod(A_27c,A_27e),X4_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27e),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27e),tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27e),tyop_2Emin_2Ebool))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27e),tyop_2Emin_2Ebool)))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27e),tyop_2Emin_2Ebool))))),c_2Epath_2Eparallel__comp_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool))),X0_2E0))),s(tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),X1_2E0))),s(tyop_2Epair_2Eprod(A_27a,A_27d),X2_2E0))),s(A_27b,X3_2E0))),s(tyop_2Epair_2Eprod(A_27c,A_27e),X4_2E0))) ).

fof(thm_2Epath_2Epath__TY__DEF,axiom,
    ! [A_27a,A_27b,F0_2E0] :
      ( ! [V1x_2E0] : s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),tyop_2Emin_2Ebool),F0_2E0),s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),V1x_2E0))) = s(tyop_2Emin_2Ebool,c_2Ebool_2ET_2E0)
     => ? [V0rep_2E0] : p(s(tyop_2Emin_2Ebool,c_2Ebool_2ETYPE__DEFINITION_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),tyop_2Emin_2Ebool),F0_2E0),s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)))),V0rep_2E0)))) ) ).

fof(thm_2Epath_2Epath__absrep__bijections,axiom,
    ! [A_27a,A_27b] :
      ( ! [V0a_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2EtoPath_2E1(s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),c_2Epath_2EfromPath_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0a_2E0))))) = s(tyop_2Epath_2Epath(A_27a,A_27b),V0a_2E0)
      & ! [V1r_2E0] :
          ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2ET_2E0))
        <=> s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),c_2Epath_2EfromPath_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2EtoPath_2E1(s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),V1r_2E0))))) = s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),V1r_2E0) ) ) ).

fof(thm_2Epath_2Efirst__def,axiom,
    ! [A_27a,A_27b,V0p_2E0] : s(A_27a,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))) = s(A_27a,c_2Epair_2EFST_2E1(s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),c_2Epath_2EfromPath_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))))) ).

fof(thm_2Epath_2Estopped__at__def,axiom,
    ! [A_27a,A_27b,V0x_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V0x_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2EtoPath_2E1(s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),c_2Epair_2E_2C_2E2(s(A_27a,V0x_2E0),s(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),c_2Ellist_2ELNIL_2E0))))) ).

fof(thm_2Epath_2Epcons__def,axiom,
    ! [A_27a,A_27b,V0x_2E0,V1r_2E0,V2p_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V0x_2E0),s(A_27b,V1r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2EtoPath_2E1(s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),c_2Epair_2E_2C_2E2(s(A_27a,V0x_2E0),s(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),c_2Ellist_2ELCONS_2E2(s(tyop_2Epair_2Eprod(A_27b,A_27a),c_2Epair_2E_2C_2E2(s(A_27b,V1r_2E0),s(A_27a,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))))),s(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),c_2Epair_2ESND_2E1(s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),c_2Epath_2EfromPath_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))))))))))) ).

fof(thm_2Epath_2Efinite__def,axiom,
    ! [A_27a,A_27b,V0sigma_2E0] : s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0sigma_2E0))) = s(tyop_2Emin_2Ebool,c_2Ellist_2ELFINITE_2E1(s(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),c_2Epair_2ESND_2E1(s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),c_2Epath_2EfromPath_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0sigma_2E0))))))) ).

fof(thm_2Epath_2Elast__thm,axiom,
    ! [A_27a,A_27b] :
      ( ! [V0x_2E0] : s(A_27a,c_2Epath_2Elast_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V0x_2E0))))) = s(A_27a,V0x_2E0)
      & ! [V1x_2E0,V2r_2E0,V3p_2E0] : s(A_27a,c_2Epath_2Elast_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V1x_2E0),s(A_27b,V2r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V3p_2E0))))) = s(A_27a,c_2Epath_2Elast_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V3p_2E0))) ) ).

fof(thm_2Epath_2Epmap__def,axiom,
    ! [A_27a,A_27b,A_27c,A_27d,V0f_2E0,V1g_2E0,V2p_2E0] : s(tyop_2Epath_2Epath(A_27c,A_27d),c_2Epath_2Epmap_2E3(s(tyop_2Emin_2Efun(A_27a,A_27c),V0f_2E0),s(tyop_2Emin_2Efun(A_27b,A_27d),V1g_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))) = s(tyop_2Epath_2Epath(A_27c,A_27d),c_2Epath_2EtoPath_2E1(s(tyop_2Epair_2Eprod(A_27c,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27d,A_27c))),c_2Epair_2E_23_23_2E3(s(tyop_2Emin_2Efun(A_27a,A_27c),V0f_2E0),s(tyop_2Emin_2Efun(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27d,A_27c))),c_2Ellist_2ELMAP_2E1(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),tyop_2Epair_2Eprod(A_27d,A_27c)),c_2Epair_2E_23_23_2E2(s(tyop_2Emin_2Efun(A_27b,A_27d),V1g_2E0),s(tyop_2Emin_2Efun(A_27a,A_27c),V0f_2E0))))),s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),c_2Epath_2EfromPath_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))))))) ).

fof(thm_2Epath_2Etail__def,axiom,
    ! [A_27a,A_27b,V0x_2E0,V1r_2E0,V2p_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Etail_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V0x_2E0),s(A_27b,V1r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))))) = s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0) ).

fof(thm_2Epath_2Efirst__label__def,axiom,
    ! [A_27a,A_27b,V0x_2E0,V1r_2E0,V2p_2E0] : s(A_27b,c_2Epath_2Efirst__label_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V0x_2E0),s(A_27b,V1r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))))) = s(A_27b,V1r_2E0) ).

fof(thm_2Epath_2Elength__def,axiom,
    ! [A_27a,A_27b,V0p_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Epath_2Elength_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Ebool_2ECOND_2E3(s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))),s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Eoption_2ESOME_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2E_2B_2E2(s(tyop_2Enum_2Enum,c_2Elist_2ELENGTH_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),c_2Eoption_2ETHE_2E1(s(tyop_2Eoption_2Eoption(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a))),c_2Ellist_2EtoList_2E1(s(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),c_2Epair_2ESND_2E1(s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),c_2Epath_2EfromPath_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))))))))))),s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EZERO_2E0))))))))),s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Eoption_2ENONE_2E0))) ).

fof(thm_2Epath_2Eel__def,axiom,
    ! [A_27a,A_27b] :
      ( ! [V0p_2E0] : s(A_27a,c_2Epath_2Eel_2E2(s(tyop_2Enum_2Enum,c_2Enum_2E0_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))) = s(A_27a,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0)))
      & ! [V1n_2E0,V2p_2E0] : s(A_27a,c_2Epath_2Eel_2E2(s(tyop_2Enum_2Enum,c_2Enum_2ESUC_2E1(s(tyop_2Enum_2Enum,V1n_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))) = s(A_27a,c_2Epath_2Eel_2E2(s(tyop_2Enum_2Enum,V1n_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Etail_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))))) ) ).

fof(thm_2Epath_2Enth__label__def,axiom,
    ! [A_27a,A_27b] :
      ( ! [V0p_2E0] : s(A_27a,c_2Epath_2Enth__label_2E2(s(tyop_2Enum_2Enum,c_2Enum_2E0_2E0),s(tyop_2Epath_2Epath(A_27b,A_27a),V0p_2E0))) = s(A_27a,c_2Epath_2Efirst__label_2E1(s(tyop_2Epath_2Epath(A_27b,A_27a),V0p_2E0)))
      & ! [V1n_2E0,V2p_2E0] : s(A_27a,c_2Epath_2Enth__label_2E2(s(tyop_2Enum_2Enum,c_2Enum_2ESUC_2E1(s(tyop_2Enum_2Enum,V1n_2E0))),s(tyop_2Epath_2Epath(A_27b,A_27a),V2p_2E0))) = s(A_27a,c_2Epath_2Enth__label_2E2(s(tyop_2Enum_2Enum,V1n_2E0),s(tyop_2Epath_2Epath(A_27b,A_27a),c_2Epath_2Etail_2E1(s(tyop_2Epath_2Epath(A_27b,A_27a),V2p_2E0))))) ) ).

fof(thm_2Epath_2Epconcat__def,axiom,
    ! [A_27a,A_27b,V0p1_2E0,V1lab_2E0,V2p2_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epconcat_2E3(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p1_2E0),s(A_27b,V1lab_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p2_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2EtoPath_2E1(s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),c_2Epair_2E_2C_2E2(s(A_27a,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p1_2E0))),s(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),c_2Ellist_2ELAPPEND_2E2(s(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),c_2Epair_2ESND_2E1(s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),c_2Epath_2EfromPath_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p1_2E0))))),s(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),c_2Ellist_2ELCONS_2E2(s(tyop_2Epair_2Eprod(A_27b,A_27a),c_2Epair_2E_2C_2E2(s(A_27b,V1lab_2E0),s(A_27a,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V2p2_2E0))))),s(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),c_2Epair_2ESND_2E1(s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),c_2Epath_2EfromPath_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V2p2_2E0))))))))))))) ).

fof(thm_2Epath_2EPL__def,axiom,
    ! [A_27a,A_27b,F1_2E0,F0_2E0] :
      ( ! [V0p_2E0,V1i_2E0] :
          ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),F1_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))),s(tyop_2Enum_2Enum,V1i_2E0))))
        <=> ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))))
           => p(s(tyop_2Emin_2Ebool,c_2Eprim__rec_2E_3C_2E2(s(tyop_2Enum_2Enum,V1i_2E0),s(tyop_2Enum_2Enum,c_2Eoption_2ETHE_2E1(s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Epath_2Elength_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0)))))))) ) )
     => ( ! [V0p_2E0,V1i_2E0] : s(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),F0_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))),s(tyop_2Enum_2Enum,V1i_2E0))) = s(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epair_2E_2C_2E2(s(tyop_2Enum_2Enum,V1i_2E0),s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),F1_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))),s(tyop_2Enum_2Enum,V1i_2E0)))))
       => ! [V0p_2E0] : s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))) = s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E1(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),F0_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))))) ) ) ).

fof(thm_2Epath_2EfirstP__at__def,axiom,
    ! [A_27a,A_27b,V0P_2E0,V1p_2E0,V2i_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Epath_2EfirstP__at_2E3(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0),s(tyop_2Enum_2Enum,V2i_2E0))))
    <=> ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,V2i_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))))
        & p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(A_27a,c_2Epath_2Eel_2E2(s(tyop_2Enum_2Enum,V2i_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))))
        & ! [V3j_2E0] :
            ( p(s(tyop_2Emin_2Ebool,c_2Eprim__rec_2E_3C_2E2(s(tyop_2Enum_2Enum,V3j_2E0),s(tyop_2Enum_2Enum,V2i_2E0))))
           => ~ p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(A_27a,c_2Epath_2Eel_2E2(s(tyop_2Enum_2Enum,V3j_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0)))))) ) ) ) ).

fof(thm_2Epath_2Eexists__def,axiom,
    ! [A_27a,A_27b,V0P_2E0,V1p_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Eexists_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))
    <=> ? [V2i_2E0] : p(s(tyop_2Emin_2Ebool,c_2Epath_2EfirstP__at_2E3(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0),s(tyop_2Enum_2Enum,V2i_2E0)))) ) ).

fof(thm_2Epath_2Eevery__def,axiom,
    ! [A_27a,A_27b,V0P_2E0,V1p_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Eevery_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))
    <=> ~ p(s(tyop_2Emin_2Ebool,c_2Epath_2Eexists_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Ecombin_2Eo_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2E_7E_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0)))) ) ).

fof(thm_2Epath_2Emem__def,axiom,
    ! [A_27a,A_27b,V0s_2E0,V1p_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Emem_2E2(s(A_27a,V0s_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))
    <=> ? [V2i_2E0] :
          ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,V2i_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))))
          & s(A_27a,V0s_2E0) = s(A_27a,c_2Epath_2Eel_2E2(s(tyop_2Enum_2Enum,V2i_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))) ) ) ).

fof(thm_2Epath_2Edrop__def,axiom,
    ! [A_27a,A_27b] :
      ( ! [V0p_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Edrop_2E2(s(tyop_2Enum_2Enum,c_2Enum_2E0_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0)
      & ! [V1n_2E0,V2p_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Edrop_2E2(s(tyop_2Enum_2Enum,c_2Enum_2ESUC_2E1(s(tyop_2Enum_2Enum,V1n_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Edrop_2E2(s(tyop_2Enum_2Enum,V1n_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Etail_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))))) ) ).

fof(thm_2Epath_2Etake__def,axiom,
    ! [A_27a,A_27b] :
      ( ! [V0p_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Etake_2E2(s(tyop_2Enum_2Enum,c_2Enum_2E0_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0)))))
      & ! [V1n_2E0,V2p_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Etake_2E2(s(tyop_2Enum_2Enum,c_2Enum_2ESUC_2E1(s(tyop_2Enum_2Enum,V1n_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))),s(A_27b,c_2Epath_2Efirst__label_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Etake_2E2(s(tyop_2Enum_2Enum,V1n_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Etail_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))))))) ) ).

fof(thm_2Epath_2Eseg__def,axiom,
    ! [A_27a,A_27b,V0i_2E0,V1j_2E0,V2p_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Eseg_2E3(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Enum_2Enum,V1j_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Etake_2E2(s(tyop_2Enum_2Enum,c_2Earithmetic_2E_2D_2E2(s(tyop_2Enum_2Enum,V1j_2E0),s(tyop_2Enum_2Enum,V0i_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Edrop_2E2(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))))) ).

fof(thm_2Epath_2Elabels__def,axiom,
    ! [A_27a,A_27b] :
      ( ! [V0x_2E0] : s(tyop_2Ellist_2Ellist(A_27b),c_2Epath_2Elabels_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V0x_2E0))))) = s(tyop_2Ellist_2Ellist(A_27b),c_2Ellist_2ELNIL_2E0)
      & ! [V1x_2E0,V2r_2E0,V3p_2E0] : s(tyop_2Ellist_2Ellist(A_27b),c_2Epath_2Elabels_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V1x_2E0),s(A_27b,V2r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V3p_2E0))))) = s(tyop_2Ellist_2Ellist(A_27b),c_2Ellist_2ELCONS_2E2(s(A_27b,V2r_2E0),s(tyop_2Ellist_2Ellist(A_27b),c_2Epath_2Elabels_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V3p_2E0))))) ) ).

fof(thm_2Epath_2Eis__stopped__def,axiom,
    ! [A_27a,A_27b,V0p_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Eis__stopped_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))))
    <=> ? [V1x_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V1x_2E0))) ) ).

fof(thm_2Epath_2Efilter__def,axiom,
    ! [A_27a,A_27b,V0P_2E0] :
      ( ! [V1x_2E0] :
          ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(A_27a,V1x_2E0))))
         => s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Efilter_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V1x_2E0))))) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V1x_2E0))) )
      & ! [V2x_2E0,V3r_2E0,V4p_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Efilter_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V2x_2E0),s(A_27b,V3r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V4p_2E0))))) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Ebool_2ECOND_2E3(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(A_27a,V2x_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Ebool_2ECOND_2E3(s(tyop_2Emin_2Ebool,c_2Epath_2Eexists_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V4p_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V2x_2E0),s(A_27b,V3r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Efilter_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V4p_2E0))))),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V2x_2E0))))),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Efilter_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V4p_2E0))))) ) ).

fof(thm_2Epath_2Epgenerate__def,axiom,
    ! [A_27a,A_27b,V0f_2E0,V1g_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epgenerate_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),V0f_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27b),V1g_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),V0f_2E0),s(tyop_2Enum_2Enum,c_2Enum_2E0_2E0))),s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27b),V1g_2E0),s(tyop_2Enum_2Enum,c_2Enum_2E0_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epgenerate_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),c_2Ecombin_2Eo_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),V0f_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Enum_2ESUC_2E0))),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27b),c_2Ecombin_2Eo_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27b),V1g_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Enum_2ESUC_2E0))))))) ).

fof(thm_2Epath_2Eokpath__f__def,axiom,
    ! [A_27a,A_27b,F3_2E0,F2_2E0,F1_2E0,F0_2E0] :
      ( ! [V0R_2E0,V3x_2E0,V4r_2E0,V5p_2E0,V1X_2E0] :
          ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))))),F3_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0))),s(A_27a,V3x_2E0))),s(A_27b,V4r_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),V5p_2E0))),s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),V1X_2E0))))
        <=> ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0),s(A_27a,V3x_2E0))),s(A_27b,V4r_2E0))),s(A_27a,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V5p_2E0))))))
            & p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Epath_2Epath(A_27a,A_27b),V5p_2E0),s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),V1X_2E0)))) ) )
     => ( ! [V3x_2E0,V0R_2E0,V1X_2E0,V4r_2E0,V5p_2E0] : s(tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool))))),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)))))),F2_2E0),s(A_27a,V3x_2E0))),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0))),s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),V1X_2E0))),s(A_27b,V4r_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),V5p_2E0))) = s(tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),c_2Epair_2E_2C_2E2(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V3x_2E0),s(A_27b,V4r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V5p_2E0))),s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))))),F3_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0))),s(A_27a,V3x_2E0))),s(A_27b,V4r_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),V5p_2E0))),s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),V1X_2E0)))))
       => ( ! [V0R_2E0,V1X_2E0,V3x_2E0] : s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,tyop_2Epath_2Epath(A_27a,A_27b)),tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,tyop_2Epath_2Epath(A_27a,A_27b)),tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,tyop_2Epath_2Epath(A_27a,A_27b)),tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,tyop_2Epath_2Epath(A_27a,A_27b)),tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool))))),F1_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0))),s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),V1X_2E0))),s(A_27a,V3x_2E0))) = s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,tyop_2Epath_2Epath(A_27a,A_27b)),tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),c_2Epair_2EUNCURRY_2E1(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool))))),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)))))),F2_2E0),s(A_27a,V3x_2E0))),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0))),s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),V1X_2E0)))))
         => ( ! [V2x_2E0] : s(tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),F0_2E0),s(A_27a,V2x_2E0))) = s(tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),c_2Epair_2E_2C_2E2(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V2x_2E0))),s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(A_27a,V2x_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Epred__set_2EUNIV_2E0)))))
           => ! [V0R_2E0,V1X_2E0] : s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),c_2Epath_2Eokpath__f_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0),s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),V1X_2E0))) = s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),c_2Epred__set_2EUNION_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),F0_2E0))),s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E1(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Epair_2Eprod(A_27b,tyop_2Epath_2Epath(A_27a,A_27b))),tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),c_2Epair_2EUNCURRY_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,tyop_2Epath_2Epath(A_27a,A_27b)),tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,tyop_2Epath_2Epath(A_27a,A_27b)),tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,tyop_2Epath_2Epath(A_27a,A_27b)),tyop_2Epair_2Eprod(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool))))),F1_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0))),s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),V1X_2E0))))))))) ) ) ) ) ).

fof(thm_2Epath_2Eokpath__def,axiom,
    ! [A_27a,A_27b,V0R_2E0] : s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),c_2Epath_2Eokpath_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0))) = s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),c_2EfixedPoint_2Egfp_2E1(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),c_2Epath_2Eokpath__f_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0))))) ).

fof(thm_2Epath_2Eplink__def,axiom,
    ! [A_27a,A_27b] :
      ( ! [V0x_2E0,V1p_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Eplink_2E2(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V0x_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0)
      & ! [V2x_2E0,V3r_2E0,V4p1_2E0,V5p2_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Eplink_2E2(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V2x_2E0),s(A_27b,V3r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V4p1_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),V5p2_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V2x_2E0),s(A_27b,V3r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Eplink_2E2(s(tyop_2Epath_2Epath(A_27a,A_27b),V4p1_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V5p2_2E0))))) ) ).

fof(thm_2Epath_2ESN__def,axiom,
    ! [A_27a,A_27b,F0_2E0] :
      ( ! [V0R_2E0,V1x_2E0,V2y_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)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F0_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0))),s(A_27a,V1x_2E0))),s(A_27a,V2y_2E0))))
        <=> ? [V3l_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0),s(A_27a,V2y_2E0))),s(A_27b,V3l_2E0))),s(A_27a,V1x_2E0)))) )
     => ! [V0R_2E0] : s(tyop_2Emin_2Ebool,c_2Epath_2ESN_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0))) = s(tyop_2Emin_2Ebool,c_2Erelation_2EWF_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F0_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0))))) ) ).

fof(thm_2Epath_2Eunfold__def,axiom,
    ! [A_27a,A_27b,A_27c,F1_2E0,F0_2E0] :
      ( ! [V0proj_2E0,V4next__s_2E0,V5lbl_2E0] : s(tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a)),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a))),app_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a)))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27a),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a))))),F1_2E0),s(tyop_2Emin_2Efun(A_27c,A_27a),V0proj_2E0))),s(A_27c,V4next__s_2E0))),s(A_27b,V5lbl_2E0))) = s(tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a)),c_2Epair_2E_2C_2E2(s(A_27c,V4next__s_2E0),s(tyop_2Epair_2Eprod(A_27b,A_27a),c_2Epair_2E_2C_2E2(s(A_27b,V5lbl_2E0),s(A_27a,app_2E2(s(tyop_2Emin_2Efun(A_27c,A_27a),V0proj_2E0),s(A_27c,V4next__s_2E0)))))))
     => ( ! [V0proj_2E0,V1f_2E0,V3s_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a))),app_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a)))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27b))),tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a))))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27b))),tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a)))))),F0_2E0),s(tyop_2Emin_2Efun(A_27c,A_27a),V0proj_2E0))),s(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27b))),V1f_2E0))),s(A_27c,V3s_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a))),c_2Eoption_2EOPTION__MAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27b),tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a))),c_2Epair_2EUNCURRY_2E1(s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a)))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27a),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a))))),F1_2E0),s(tyop_2Emin_2Efun(A_27c,A_27a),V0proj_2E0))))),s(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27b)),app_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27b))),V1f_2E0),s(A_27c,V3s_2E0)))))
       => ! [V0proj_2E0,V1f_2E0,V2s_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Eunfold_2E3(s(tyop_2Emin_2Efun(A_27c,A_27a),V0proj_2E0),s(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27b))),V1f_2E0),s(A_27c,V2s_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2EtoPath_2E1(s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),c_2Epair_2E_2C_2E2(s(A_27a,app_2E2(s(tyop_2Emin_2Efun(A_27c,A_27a),V0proj_2E0),s(A_27c,V2s_2E0))),s(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),c_2Ellist_2ELUNFOLD_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a)))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27b))),tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a))))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27b))),tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,tyop_2Epair_2Eprod(A_27b,A_27a)))))),F0_2E0),s(tyop_2Emin_2Efun(A_27c,A_27a),V0proj_2E0))),s(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27c,A_27b))),V1f_2E0))),s(A_27c,V2s_2E0))))))) ) ) ).

fof(thm_2Epath_2Etrace__machine__def,axiom,
    ! [A_27a,V0P_2E0,V1s_2E0,V2l_2E0,V3s_27_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Etrace__machine_2E4(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Elist_2Elist(A_27a),V1s_2E0),s(A_27a,V2l_2E0),s(tyop_2Elist_2Elist(A_27a),V3s_27_2E0))))
    <=> ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Elist_2Elist(A_27a),c_2Elist_2EAPPEND_2E2(s(tyop_2Elist_2Elist(A_27a),V1s_2E0),s(tyop_2Elist_2Elist(A_27a),c_2Elist_2ECONS_2E2(s(A_27a,V2l_2E0),s(tyop_2Elist_2Elist(A_27a),c_2Elist_2ENIL_2E0))))))))
        & s(tyop_2Elist_2Elist(A_27a),V3s_27_2E0) = s(tyop_2Elist_2Elist(A_27a),c_2Elist_2EAPPEND_2E2(s(tyop_2Elist_2Elist(A_27a),V1s_2E0),s(tyop_2Elist_2Elist(A_27a),c_2Elist_2ECONS_2E2(s(A_27a,V2l_2E0),s(tyop_2Elist_2Elist(A_27a),c_2Elist_2ENIL_2E0))))) ) ) ).

fof(thm_2Epath_2Eparallel__comp__def,axiom,
    ! [A_27a,A_27b,A_27c,A_27d,A_27e,V0m1_2E0,V1m2_2E0,V2s1_2E0,V3s2_2E0,V4l_2E0,V5s1_27_2E0,V6s2_27_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Eparallel__comp_2E5(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool))),V0m1_2E0),s(tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),V1m2_2E0),s(tyop_2Epair_2Eprod(A_27a,A_27d),c_2Epair_2E_2C_2E2(s(A_27a,V2s1_2E0),s(A_27d,V3s2_2E0))),s(A_27b,V4l_2E0),s(tyop_2Epair_2Eprod(A_27c,A_27e),c_2Epair_2E_2C_2E2(s(A_27c,V5s1_27_2E0),s(A_27e,V6s2_27_2E0))))))
    <=> ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool))),V0m1_2E0),s(A_27a,V2s1_2E0))),s(A_27b,V4l_2E0))),s(A_27c,V5s1_27_2E0))))
        & p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27e,tyop_2Emin_2Ebool))),V1m2_2E0),s(A_27d,V3s2_2E0))),s(A_27b,V4l_2E0))),s(A_27e,V6s2_27_2E0)))) ) ) ).

fof(thm_2Epath_2Epath__rep__bijections__thm,axiom,
    ! [A_27a,A_27b] :
      ( ! [V0a_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2EtoPath_2E1(s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),c_2Epath_2EfromPath_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0a_2E0))))) = s(tyop_2Epath_2Epath(A_27a,A_27b),V0a_2E0)
      & ! [V1r_2E0] : s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),c_2Epath_2EfromPath_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2EtoPath_2E1(s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),V1r_2E0))))) = s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),V1r_2E0) ) ).

fof(thm_2Epath_2EtoPath__11,axiom,
    ! [A_27a,A_27b,V0r_2E0,V1r_27_2E0] :
      ( s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2EtoPath_2E1(s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),V0r_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2EtoPath_2E1(s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),V1r_27_2E0)))
    <=> s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),V0r_2E0) = s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),V1r_27_2E0) ) ).

fof(thm_2Epath_2EfromPath__11,axiom,
    ! [A_27a,A_27b,V0a_2E0,V1a_27_2E0] :
      ( s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),c_2Epath_2EfromPath_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0a_2E0))) = s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),c_2Epath_2EfromPath_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1a_27_2E0)))
    <=> s(tyop_2Epath_2Epath(A_27a,A_27b),V0a_2E0) = s(tyop_2Epath_2Epath(A_27a,A_27b),V1a_27_2E0) ) ).

fof(thm_2Epath_2EfromPath__onto,axiom,
    ! [A_27a,A_27b,V0r_2E0] :
    ? [V1a_2E0] : s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),V0r_2E0) = s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),c_2Epath_2EfromPath_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1a_2E0))) ).

fof(thm_2Epath_2EtoPath__onto,axiom,
    ! [A_27a,A_27b,V0a_2E0] :
    ? [V1r_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),V0a_2E0) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2EtoPath_2E1(s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),V1r_2E0))) ).

fof(thm_2Epath_2Estopped__at__11,axiom,
    ! [A_27a,A_27b,V0x_2E0,V1y_2E0] :
      ( s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V0x_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V1y_2E0)))
    <=> s(A_27a,V0x_2E0) = s(A_27a,V1y_2E0) ) ).

fof(thm_2Epath_2Epcons__11,axiom,
    ! [A_27a,A_27b,V0x_2E0,V1r_2E0,V2p_2E0,V3y_2E0,V4s_2E0,V5q_2E0] :
      ( s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V0x_2E0),s(A_27b,V1r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V3y_2E0),s(A_27b,V4s_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V5q_2E0)))
    <=> ( s(A_27a,V0x_2E0) = s(A_27a,V3y_2E0)
        & s(A_27b,V1r_2E0) = s(A_27b,V4s_2E0)
        & s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0) = s(tyop_2Epath_2Epath(A_27a,A_27b),V5q_2E0) ) ) ).

fof(thm_2Epath_2Estopped__at__not__pcons,axiom,
    ! [A_27a,A_27b,V0x_2E0,V1y_2E0,V2r_2E0,V3p_2E0] :
      ( s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V0x_2E0))) != s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V1y_2E0),s(A_27b,V2r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V3p_2E0)))
      & s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V1y_2E0),s(A_27b,V2r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V3p_2E0))) != s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V0x_2E0))) ) ).

fof(thm_2Epath_2Epath__cases,axiom,
    ! [A_27a,A_27b,V0p_2E0] :
      ( ? [V1x_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V1x_2E0)))
      | ? [V2x_2E0,V3r_2E0,V4q_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V2x_2E0),s(A_27b,V3r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V4q_2E0))) ) ).

fof(thm_2Epath_2EFORALL__path,axiom,
    ! [A_27a,A_27b,V0P_2E0] :
      ( ! [V1p_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))
    <=> ( ! [V2x_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V2x_2E0))))))
        & ! [V3x_2E0,V4r_2E0,V5p_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V3x_2E0),s(A_27b,V4r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V5p_2E0)))))) ) ) ).

fof(thm_2Epath_2EEXISTS__path,axiom,
    ! [A_27a,A_27b,V0P_2E0] :
      ( ? [V1p_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))
    <=> ( ? [V2x_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V2x_2E0))))))
        | ? [V3x_2E0,V4r_2E0,V5p_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V3x_2E0),s(A_27b,V4r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V5p_2E0)))))) ) ) ).

fof(thm_2Epath_2Efirst__thm,axiom,
    ! [A_27a,A_27b] :
      ( ! [V0x_2E0] : s(A_27a,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V0x_2E0))))) = s(A_27a,V0x_2E0)
      & ! [V1x_2E0,V2r_2E0,V3p_2E0] : s(A_27a,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V1x_2E0),s(A_27b,V2r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V3p_2E0))))) = s(A_27a,V1x_2E0) ) ).

fof(thm_2Epath_2Efinite__thm,axiom,
    ! [A_27a,A_27b] :
      ( ! [V0x_2E0] : s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V0x_2E0))))) = s(tyop_2Emin_2Ebool,c_2Ebool_2ET_2E0)
      & ! [V1x_2E0,V2r_2E0,V3p_2E0] : s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V1x_2E0),s(A_27b,V2r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V3p_2E0))))) = s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V3p_2E0))) ) ).

fof(thm_2Epath_2Epath__bisimulation,axiom,
    ! [A_27a,A_27b,V0p1_2E0,V1p2_2E0] :
      ( s(tyop_2Epath_2Epath(A_27a,A_27b),V0p1_2E0) = s(tyop_2Epath_2Epath(A_27a,A_27b),V1p2_2E0)
    <=> ? [V2R_2E0] :
          ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),V2R_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V0p1_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p2_2E0))))
          & ! [V3q1_2E0,V4q2_2E0] :
              ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),V2R_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V3q1_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),V4q2_2E0))))
             => ( ? [V5x_2E0] :
                    ( s(tyop_2Epath_2Epath(A_27a,A_27b),V3q1_2E0) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V5x_2E0)))
                    & s(tyop_2Epath_2Epath(A_27a,A_27b),V4q2_2E0) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V5x_2E0))) )
                | ? [V6x_2E0,V7r_2E0,V8q1_27_2E0,V9q2_27_2E0] :
                    ( s(tyop_2Epath_2Epath(A_27a,A_27b),V3q1_2E0) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V6x_2E0),s(A_27b,V7r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V8q1_27_2E0)))
                    & s(tyop_2Epath_2Epath(A_27a,A_27b),V4q2_2E0) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V6x_2E0),s(A_27b,V7r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V9q2_27_2E0)))
                    & p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),V2R_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V8q1_27_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),V9q2_27_2E0)))) ) ) ) ) ) ).

fof(thm_2Epath_2Efinite__path__ind,axiom,
    ! [A_27a,A_27b,V0P_2E0] :
      ( ( ! [V1x_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V1x_2E0))))))
        & ! [V2x_2E0,V3r_2E0,V4p_2E0] :
            ( ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V4p_2E0))))
              & p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V4p_2E0)))) )
           => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V2x_2E0),s(A_27b,V3r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V4p_2E0)))))) ) )
     => ! [V5q_2E0] :
          ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V5q_2E0))))
         => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V5q_2E0)))) ) ) ).

fof(thm_2Epath_2Epmap__thm,axiom,
    ! [A_27a,A_27b,A_27c,A_27d,V0g_2E0,V1f_2E0] :
      ( ! [V2x_2E0] : s(tyop_2Epath_2Epath(A_27b,A_27c),c_2Epath_2Epmap_2E3(s(tyop_2Emin_2Efun(A_27a,A_27b),V1f_2E0),s(tyop_2Emin_2Efun(A_27d,A_27c),V0g_2E0),s(tyop_2Epath_2Epath(A_27a,A_27d),c_2Epath_2Estopped__at_2E1(s(A_27a,V2x_2E0))))) = s(tyop_2Epath_2Epath(A_27b,A_27c),c_2Epath_2Estopped__at_2E1(s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V1f_2E0),s(A_27a,V2x_2E0)))))
      & ! [V3x_2E0,V4r_2E0,V5p_2E0] : s(tyop_2Epath_2Epath(A_27b,A_27c),c_2Epath_2Epmap_2E3(s(tyop_2Emin_2Efun(A_27a,A_27b),V1f_2E0),s(tyop_2Emin_2Efun(A_27d,A_27c),V0g_2E0),s(tyop_2Epath_2Epath(A_27a,A_27d),c_2Epath_2Epcons_2E3(s(A_27a,V3x_2E0),s(A_27d,V4r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27d),V5p_2E0))))) = s(tyop_2Epath_2Epath(A_27b,A_27c),c_2Epath_2Epcons_2E3(s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V1f_2E0),s(A_27a,V3x_2E0))),s(A_27c,app_2E2(s(tyop_2Emin_2Efun(A_27d,A_27c),V0g_2E0),s(A_27d,V4r_2E0))),s(tyop_2Epath_2Epath(A_27b,A_27c),c_2Epath_2Epmap_2E3(s(tyop_2Emin_2Efun(A_27a,A_27b),V1f_2E0),s(tyop_2Emin_2Efun(A_27d,A_27c),V0g_2E0),s(tyop_2Epath_2Epath(A_27a,A_27d),V5p_2E0))))) ) ).

fof(thm_2Epath_2Efirst__pmap,axiom,
    ! [A_27a,A_27b,A_27c,A_27d,V0g_2E0,V1f_2E0,V2p_2E0] : s(A_27c,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27c,A_27d),c_2Epath_2Epmap_2E3(s(tyop_2Emin_2Efun(A_27a,A_27c),V1f_2E0),s(tyop_2Emin_2Efun(A_27b,A_27d),V0g_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))))) = s(A_27c,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27c),V1f_2E0),s(A_27a,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))))) ).

fof(thm_2Epath_2Elast__pmap,axiom,
    ! [A_27a,A_27b,A_27c,A_27d,V0g_2E0,V1f_2E0,V2p_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))))
     => s(A_27c,c_2Epath_2Elast_2E1(s(tyop_2Epath_2Epath(A_27c,A_27d),c_2Epath_2Epmap_2E3(s(tyop_2Emin_2Efun(A_27a,A_27c),V1f_2E0),s(tyop_2Emin_2Efun(A_27b,A_27d),V0g_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))))) = s(A_27c,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27c),V1f_2E0),s(A_27a,c_2Epath_2Elast_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))))) ) ).

fof(thm_2Epath_2Efinite__pmap,axiom,
    ! [A_27a,A_27b,A_27c,A_27d,V0f_2E0,V1g_2E0,V2p_2E0] : s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27c,A_27d),c_2Epath_2Epmap_2E3(s(tyop_2Emin_2Efun(A_27a,A_27c),V0f_2E0),s(tyop_2Emin_2Efun(A_27b,A_27d),V1g_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))))) = s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))) ).

fof(thm_2Epath_2Elength__thm,axiom,
    ! [A_27a,A_27b,A_27c,A_27d] :
      ( ! [V0x_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Epath_2Elength_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V0x_2E0))))) = s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Eoption_2ESOME_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EZERO_2E0)))))))
      & ! [V1x_2E0,V2r_2E0,V3p_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Epath_2Elength_2E1(s(tyop_2Epath_2Epath(A_27c,A_27d),c_2Epath_2Epcons_2E3(s(A_27c,V1x_2E0),s(A_27d,V2r_2E0),s(tyop_2Epath_2Epath(A_27c,A_27d),V3p_2E0))))) = s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Ebool_2ECOND_2E3(s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27c,A_27d),V3p_2E0))),s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Eoption_2ESOME_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2E_2B_2E2(s(tyop_2Enum_2Enum,c_2Eoption_2ETHE_2E1(s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Epath_2Elength_2E1(s(tyop_2Epath_2Epath(A_27c,A_27d),V3p_2E0))))),s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EZERO_2E0))))))))),s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Eoption_2ENONE_2E0))) ) ).

fof(thm_2Epath_2Ealt__length__thm,axiom,
    ! [A_27a,A_27b,A_27c,A_27d] :
      ( ! [V0x_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Epath_2Elength_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V0x_2E0))))) = s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Eoption_2ESOME_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EZERO_2E0)))))))
      & ! [V1x_2E0,V2r_2E0,V3p_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Epath_2Elength_2E1(s(tyop_2Epath_2Epath(A_27c,A_27d),c_2Epath_2Epcons_2E3(s(A_27c,V1x_2E0),s(A_27d,V2r_2E0),s(tyop_2Epath_2Epath(A_27c,A_27d),V3p_2E0))))) = s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Eoption_2EOPTION__MAP_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Enum_2ESUC_2E0),s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Epath_2Elength_2E1(s(tyop_2Epath_2Epath(A_27c,A_27d),V3p_2E0))))) ) ).

fof(thm_2Epath_2Elength__never__zero,axiom,
    ! [A_27a,A_27b,V0p_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Epath_2Elength_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))) != s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Eoption_2ESOME_2E1(s(tyop_2Enum_2Enum,c_2Enum_2E0_2E0))) ).

fof(thm_2Epath_2Efinite__length,axiom,
    ! [A_27a,A_27b,V0p_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))))
      <=> ? [V1n_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Epath_2Elength_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Eoption_2ESOME_2E1(s(tyop_2Enum_2Enum,V1n_2E0))) )
      & ( ~ p(s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))))
      <=> s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Epath_2Elength_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Eoption_2ENONE_2E0) ) ) ).

fof(thm_2Epath_2Elength__pmap,axiom,
    ! [A_27a,A_27b,A_27c,A_27d,V0f_2E0,V1g_2E0,V2p_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Epath_2Elength_2E1(s(tyop_2Epath_2Epath(A_27b,A_27d),c_2Epath_2Epmap_2E3(s(tyop_2Emin_2Efun(A_27a,A_27b),V0f_2E0),s(tyop_2Emin_2Efun(A_27c,A_27d),V1g_2E0),s(tyop_2Epath_2Epath(A_27a,A_27c),V2p_2E0))))) = s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Epath_2Elength_2E1(s(tyop_2Epath_2Epath(A_27a,A_27c),V2p_2E0))) ).

fof(thm_2Epath_2Eel__def__compute,axiom,
    ! [A_27a,A_27b] :
      ( ! [V0p_2E0] : s(A_27a,c_2Epath_2Eel_2E2(s(tyop_2Enum_2Enum,c_2Enum_2E0_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))) = s(A_27a,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0)))
      & ! [V1n_2E0,V2p_2E0] : s(A_27a,c_2Epath_2Eel_2E2(s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,V1n_2E0))))),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))) = s(A_27a,c_2Epath_2Eel_2E2(s(tyop_2Enum_2Enum,c_2Earithmetic_2E_2D_2E2(s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,V1n_2E0))))),s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EZERO_2E0))))))),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Etail_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0)))))
      & ! [V3n_2E0,V4p_2E0] : s(A_27a,c_2Epath_2Eel_2E2(s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT2_2E1(s(tyop_2Enum_2Enum,V3n_2E0))))),s(tyop_2Epath_2Epath(A_27a,A_27b),V4p_2E0))) = s(A_27a,c_2Epath_2Eel_2E2(s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,V3n_2E0))))),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Etail_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V4p_2E0))))) ) ).

fof(thm_2Epath_2Enth__label__def__compute,axiom,
    ! [A_27a,A_27b] :
      ( ! [V0p_2E0] : s(A_27a,c_2Epath_2Enth__label_2E2(s(tyop_2Enum_2Enum,c_2Enum_2E0_2E0),s(tyop_2Epath_2Epath(A_27b,A_27a),V0p_2E0))) = s(A_27a,c_2Epath_2Efirst__label_2E1(s(tyop_2Epath_2Epath(A_27b,A_27a),V0p_2E0)))
      & ! [V1n_2E0,V2p_2E0] : s(A_27a,c_2Epath_2Enth__label_2E2(s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,V1n_2E0))))),s(tyop_2Epath_2Epath(A_27b,A_27a),V2p_2E0))) = s(A_27a,c_2Epath_2Enth__label_2E2(s(tyop_2Enum_2Enum,c_2Earithmetic_2E_2D_2E2(s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,V1n_2E0))))),s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EZERO_2E0))))))),s(tyop_2Epath_2Epath(A_27b,A_27a),c_2Epath_2Etail_2E1(s(tyop_2Epath_2Epath(A_27b,A_27a),V2p_2E0)))))
      & ! [V3n_2E0,V4p_2E0] : s(A_27a,c_2Epath_2Enth__label_2E2(s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT2_2E1(s(tyop_2Enum_2Enum,V3n_2E0))))),s(tyop_2Epath_2Epath(A_27b,A_27a),V4p_2E0))) = s(A_27a,c_2Epath_2Enth__label_2E2(s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,V3n_2E0))))),s(tyop_2Epath_2Epath(A_27b,A_27a),c_2Epath_2Etail_2E1(s(tyop_2Epath_2Epath(A_27b,A_27a),V4p_2E0))))) ) ).

fof(thm_2Epath_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))))) ) ) ) ).

fof(thm_2Epath_2Epconcat__thm,axiom,
    ! [A_27a,A_27b,A_27c,A_27d] :
      ( ! [V0x_2E0,V1lab_2E0,V2p2_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epconcat_2E3(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V0x_2E0))),s(A_27b,V1lab_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p2_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V0x_2E0),s(A_27b,V1lab_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p2_2E0)))
      & ! [V3x_2E0,V4r_2E0,V5p_2E0,V6lab_2E0,V7p2_2E0] : s(tyop_2Epath_2Epath(A_27c,A_27d),c_2Epath_2Epconcat_2E3(s(tyop_2Epath_2Epath(A_27c,A_27d),c_2Epath_2Epcons_2E3(s(A_27c,V3x_2E0),s(A_27d,V4r_2E0),s(tyop_2Epath_2Epath(A_27c,A_27d),V5p_2E0))),s(A_27d,V6lab_2E0),s(tyop_2Epath_2Epath(A_27c,A_27d),V7p2_2E0))) = s(tyop_2Epath_2Epath(A_27c,A_27d),c_2Epath_2Epcons_2E3(s(A_27c,V3x_2E0),s(A_27d,V4r_2E0),s(tyop_2Epath_2Epath(A_27c,A_27d),c_2Epath_2Epconcat_2E3(s(tyop_2Epath_2Epath(A_27c,A_27d),V5p_2E0),s(A_27d,V6lab_2E0),s(tyop_2Epath_2Epath(A_27c,A_27d),V7p2_2E0))))) ) ).

fof(thm_2Epath_2Epconcat__eq__stopped,axiom,
    ! [A_27a,A_27b,V0p1_2E0,V1lab_2E0,V2p2_2E0,V3x_2E0] :
      ( s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epconcat_2E3(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p1_2E0),s(A_27b,V1lab_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p2_2E0))) != s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V3x_2E0)))
      & s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V3x_2E0))) != s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epconcat_2E3(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p1_2E0),s(A_27b,V1lab_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p2_2E0))) ) ).

fof(thm_2Epath_2Epconcat__eq__pcons,axiom,
    ! [A_27a,A_27b,V0x_2E0,V1r_2E0,V2p_2E0,V3p1_2E0,V4lab_2E0,V5p2_2E0] :
      ( ( s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epconcat_2E3(s(tyop_2Epath_2Epath(A_27a,A_27b),V3p1_2E0),s(A_27b,V4lab_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V5p2_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V0x_2E0),s(A_27b,V1r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0)))
      <=> ( ( s(A_27b,V4lab_2E0) = s(A_27b,V1r_2E0)
            & s(tyop_2Epath_2Epath(A_27a,A_27b),V3p1_2E0) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V0x_2E0)))
            & s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0) = s(tyop_2Epath_2Epath(A_27a,A_27b),V5p2_2E0) )
          | ? [V6p1_27_2E0] :
              ( s(tyop_2Epath_2Epath(A_27a,A_27b),V3p1_2E0) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V0x_2E0),s(A_27b,V1r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V6p1_27_2E0)))
              & s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epconcat_2E3(s(tyop_2Epath_2Epath(A_27a,A_27b),V6p1_27_2E0),s(A_27b,V4lab_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V5p2_2E0))) ) ) )
      & ( s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V0x_2E0),s(A_27b,V1r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epconcat_2E3(s(tyop_2Epath_2Epath(A_27a,A_27b),V3p1_2E0),s(A_27b,V4lab_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V5p2_2E0)))
      <=> ( ( s(A_27b,V4lab_2E0) = s(A_27b,V1r_2E0)
            & s(tyop_2Epath_2Epath(A_27a,A_27b),V3p1_2E0) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V0x_2E0)))
            & s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0) = s(tyop_2Epath_2Epath(A_27a,A_27b),V5p2_2E0) )
          | ? [V7p1_27_2E0] :
              ( s(tyop_2Epath_2Epath(A_27a,A_27b),V3p1_2E0) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V0x_2E0),s(A_27b,V1r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V7p1_27_2E0)))
              & s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epconcat_2E3(s(tyop_2Epath_2Epath(A_27a,A_27b),V7p1_27_2E0),s(A_27b,V4lab_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V5p2_2E0))) ) ) ) ) ).

fof(thm_2Epath_2Efinite__pconcat,axiom,
    ! [A_27a,A_27b,V0p1_2E0,V1lab_2E0,V2p2_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epconcat_2E3(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p1_2E0),s(A_27b,V1lab_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p2_2E0))))))
    <=> ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p1_2E0))))
        & p(s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V2p2_2E0)))) ) ) ).

fof(thm_2Epath_2Einfinite__PL,axiom,
    ! [A_27a,A_27b,V0p_2E0] :
      ( ~ p(s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))))
     => ! [V1i_2E0] : p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,V1i_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0)))))) ) ).

fof(thm_2Epath_2EPL__pcons,axiom,
    ! [A_27a,A_27b,V0x_2E0,V1r_2E0,V2q_2E0] : s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V0x_2E0),s(A_27b,V1r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V2q_2E0))))) = s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(s(tyop_2Enum_2Enum,c_2Enum_2E0_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epred__set_2EIMAGE_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Enum_2ESUC_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V2q_2E0))))))) ).

fof(thm_2Epath_2EPL__stopped__at,axiom,
    ! [A_27a,A_27b,V0x_2E0] : s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V0x_2E0))))) = s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(s(tyop_2Enum_2Enum,c_2Enum_2E0_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epred__set_2EEMPTY_2E0))) ).

fof(thm_2Epath_2EPL__thm,axiom,
    ! [A_27a,A_27b] :
      ( ! [V0x_2E0] : s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V0x_2E0))))) = s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(s(tyop_2Enum_2Enum,c_2Enum_2E0_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epred__set_2EEMPTY_2E0)))
      & ! [V1x_2E0,V2r_2E0,V3q_2E0] : s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V1x_2E0),s(A_27b,V2r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V3q_2E0))))) = s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epred__set_2EINSERT_2E2(s(tyop_2Enum_2Enum,c_2Enum_2E0_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epred__set_2EIMAGE_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Enum_2ESUC_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V3q_2E0))))))) ) ).

fof(thm_2Epath_2EPL__0,axiom,
    ! [A_27a,A_27b,V0p_2E0] : p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,c_2Enum_2E0_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0)))))) ).

fof(thm_2Epath_2EPL__downward__closed,axiom,
    ! [A_27a,A_27b,V0i_2E0,V1p_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))))
     => ! [V2j_2E0] :
          ( p(s(tyop_2Emin_2Ebool,c_2Eprim__rec_2E_3C_2E2(s(tyop_2Enum_2Enum,V2j_2E0),s(tyop_2Enum_2Enum,V0i_2E0))))
         => p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,V2j_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0)))))) ) ) ).

fof(thm_2Epath_2EPL__pmap,axiom,
    ! [A_27a,A_27b,A_27c,A_27d,V0p_2E0,V1g_2E0,V2f_2E0] : s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epmap_2E3(s(tyop_2Emin_2Efun(A_27c,A_27a),V2f_2E0),s(tyop_2Emin_2Efun(A_27d,A_27b),V1g_2E0),s(tyop_2Epath_2Epath(A_27c,A_27d),V0p_2E0))))) = s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27c,A_27d),V0p_2E0))) ).

fof(thm_2Epath_2Eel__pmap,axiom,
    ! [A_27a,A_27b,A_27c,A_27d,V0g_2E0,V1f_2E0,V2i_2E0,V3p_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,V2i_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V3p_2E0))))))
     => s(A_27c,c_2Epath_2Eel_2E2(s(tyop_2Enum_2Enum,V2i_2E0),s(tyop_2Epath_2Epath(A_27c,A_27d),c_2Epath_2Epmap_2E3(s(tyop_2Emin_2Efun(A_27a,A_27c),V1f_2E0),s(tyop_2Emin_2Efun(A_27b,A_27d),V0g_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V3p_2E0))))) = s(A_27c,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27c),V1f_2E0),s(A_27a,c_2Epath_2Eel_2E2(s(tyop_2Enum_2Enum,V2i_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V3p_2E0))))) ) ).

fof(thm_2Epath_2Enth__label__pmap,axiom,
    ! [A_27a,A_27b,A_27c,A_27d,V0g_2E0,V1f_2E0,V2i_2E0,V3p_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,c_2Enum_2ESUC_2E1(s(tyop_2Enum_2Enum,V2i_2E0))),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V3p_2E0))))))
     => s(A_27c,c_2Epath_2Enth__label_2E2(s(tyop_2Enum_2Enum,V2i_2E0),s(tyop_2Epath_2Epath(A_27d,A_27c),c_2Epath_2Epmap_2E3(s(tyop_2Emin_2Efun(A_27a,A_27d),V1f_2E0),s(tyop_2Emin_2Efun(A_27b,A_27c),V0g_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V3p_2E0))))) = s(A_27c,app_2E2(s(tyop_2Emin_2Efun(A_27b,A_27c),V0g_2E0),s(A_27b,c_2Epath_2Enth__label_2E2(s(tyop_2Enum_2Enum,V2i_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V3p_2E0))))) ) ).

fof(thm_2Epath_2EfirstP__at__thm,axiom,
    ! [A_27a,A_27b] :
      ( ! [V0P_2E0,V1x_2E0,V2n_2E0] :
          ( p(s(tyop_2Emin_2Ebool,c_2Epath_2EfirstP__at_2E3(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V1x_2E0))),s(tyop_2Enum_2Enum,V2n_2E0))))
        <=> ( s(tyop_2Enum_2Enum,V2n_2E0) = s(tyop_2Enum_2Enum,c_2Enum_2E0_2E0)
            & p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(A_27a,V1x_2E0)))) ) )
      & ! [V3P_2E0,V4n_2E0,V5x_2E0,V6r_2E0,V7p_2E0] :
          ( p(s(tyop_2Emin_2Ebool,c_2Epath_2EfirstP__at_2E3(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V5x_2E0),s(A_27b,V6r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V7p_2E0))),s(tyop_2Enum_2Enum,V4n_2E0))))
        <=> ( ( s(tyop_2Enum_2Enum,V4n_2E0) = s(tyop_2Enum_2Enum,c_2Enum_2E0_2E0)
              & p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3P_2E0),s(A_27a,V5x_2E0)))) )
            | ( p(s(tyop_2Emin_2Ebool,c_2Eprim__rec_2E_3C_2E2(s(tyop_2Enum_2Enum,c_2Enum_2E0_2E0),s(tyop_2Enum_2Enum,V4n_2E0))))
              & ~ p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3P_2E0),s(A_27a,V5x_2E0))))
              & p(s(tyop_2Emin_2Ebool,c_2Epath_2EfirstP__at_2E3(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V7p_2E0),s(tyop_2Enum_2Enum,c_2Earithmetic_2E_2D_2E2(s(tyop_2Enum_2Enum,V4n_2E0),s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EZERO_2E0)))))))))) ) ) ) ) ).

fof(thm_2Epath_2EfirstP__at__zero,axiom,
    ! [A_27a,A_27b,V0P_2E0,V1p_2E0] : s(tyop_2Emin_2Ebool,c_2Epath_2EfirstP__at_2E3(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0),s(tyop_2Enum_2Enum,c_2Enum_2E0_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(A_27a,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))) ).

fof(thm_2Epath_2Eexists__thm,axiom,
    ! [A_27a,A_27b,A_27c,V0P_2E0] :
      ( ! [V1x_2E0] : s(tyop_2Emin_2Ebool,c_2Epath_2Eexists_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V1x_2E0))))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(A_27a,V1x_2E0)))
      & ! [V2x_2E0,V3r_2E0,V4p_2E0] :
          ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Eexists_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27c),c_2Epath_2Epcons_2E3(s(A_27a,V2x_2E0),s(A_27c,V3r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27c),V4p_2E0))))))
        <=> ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(A_27a,V2x_2E0))))
            | p(s(tyop_2Emin_2Ebool,c_2Epath_2Eexists_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27c),V4p_2E0)))) ) ) ) ).

fof(thm_2Epath_2Eevery__thm,axiom,
    ! [A_27a,A_27b,A_27c,V0P_2E0] :
      ( ! [V1x_2E0] : s(tyop_2Emin_2Ebool,c_2Epath_2Eevery_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V1x_2E0))))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(A_27a,V1x_2E0)))
      & ! [V2x_2E0,V3r_2E0,V4p_2E0] :
          ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Eevery_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27c),c_2Epath_2Epcons_2E3(s(A_27a,V2x_2E0),s(A_27c,V3r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27c),V4p_2E0))))))
        <=> ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(A_27a,V2x_2E0))))
            & p(s(tyop_2Emin_2Ebool,c_2Epath_2Eevery_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27c),V4p_2E0)))) ) ) ) ).

fof(thm_2Epath_2Enot__every,axiom,
    ! [A_27a,A_27b,V0P_2E0,V1p_2E0] :
      ( ~ p(s(tyop_2Emin_2Ebool,c_2Epath_2Eevery_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))
    <=> p(s(tyop_2Emin_2Ebool,c_2Epath_2Eexists_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Ecombin_2Eo_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2E_7E_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0)))) ) ).

fof(thm_2Epath_2Enot__exists,axiom,
    ! [A_27a,A_27b,V0P_2E0,V1p_2E0] :
      ( ~ p(s(tyop_2Emin_2Ebool,c_2Epath_2Eexists_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))
    <=> p(s(tyop_2Emin_2Ebool,c_2Epath_2Eevery_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Ecombin_2Eo_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2E_7E_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0)))) ) ).

fof(thm_2Epath_2Eexists__el,axiom,
    ! [A_27a,A_27b,V0P_2E0,V1p_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Eexists_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))
    <=> ? [V2i_2E0] :
          ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,V2i_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))))
          & p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(A_27a,c_2Epath_2Eel_2E2(s(tyop_2Enum_2Enum,V2i_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0)))))) ) ) ).

fof(thm_2Epath_2Eevery__el,axiom,
    ! [A_27a,A_27b,V0P_2E0,V1p_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Eevery_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))
    <=> ! [V2i_2E0] :
          ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,V2i_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))))
         => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(A_27a,c_2Epath_2Eel_2E2(s(tyop_2Enum_2Enum,V2i_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0)))))) ) ) ).

fof(thm_2Epath_2Eevery__coinduction,axiom,
    ! [A_27a,A_27b,V0P_2E0,V1Q_2E0] :
      ( ( ! [V2x_2E0] :
            ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V2x_2E0))))))
           => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1Q_2E0),s(A_27a,V2x_2E0)))) )
        & ! [V3x_2E0,V4r_2E0,V5p_2E0] :
            ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V3x_2E0),s(A_27b,V4r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V5p_2E0))))))
           => ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1Q_2E0),s(A_27a,V3x_2E0))))
              & p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V5p_2E0)))) ) ) )
     => ! [V6p_2E0] :
          ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V6p_2E0))))
         => p(s(tyop_2Emin_2Ebool,c_2Epath_2Eevery_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1Q_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V6p_2E0)))) ) ) ).

fof(thm_2Epath_2Eexists__induction,axiom,
    ! [A_27a,A_27b,V0Q_2E0,V1P_2E0] :
      ( ( ! [V2x_2E0] :
            ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0Q_2E0),s(A_27a,V2x_2E0))))
           => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),V1P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V2x_2E0)))))) )
        & ! [V3x_2E0,V4r_2E0,V5p_2E0] :
            ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0Q_2E0),s(A_27a,V3x_2E0))))
           => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),V1P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V3x_2E0),s(A_27b,V4r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V5p_2E0)))))) )
        & ! [V6x_2E0,V7r_2E0,V8p_2E0] :
            ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),V1P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V8p_2E0))))
           => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),V1P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V6x_2E0),s(A_27b,V7r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V8p_2E0)))))) ) )
     => ! [V9p_2E0] :
          ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Eexists_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0Q_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V9p_2E0))))
         => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),V1P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V9p_2E0)))) ) ) ).

fof(thm_2Epath_2Emem__thm,axiom,
    ! [A_27a,A_27b,A_27c,A_27d] :
      ( ! [V0x_2E0,V1s_2E0] :
          ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Emem_2E2(s(A_27a,V1s_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V0x_2E0))))))
        <=> s(A_27a,V1s_2E0) = s(A_27a,V0x_2E0) )
      & ! [V2x_2E0,V3r_2E0,V4p_2E0,V5s_2E0] :
          ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Emem_2E2(s(A_27c,V5s_2E0),s(tyop_2Epath_2Epath(A_27c,A_27d),c_2Epath_2Epcons_2E3(s(A_27c,V2x_2E0),s(A_27d,V3r_2E0),s(tyop_2Epath_2Epath(A_27c,A_27d),V4p_2E0))))))
        <=> ( s(A_27c,V5s_2E0) = s(A_27c,V2x_2E0)
            | p(s(tyop_2Emin_2Ebool,c_2Epath_2Emem_2E2(s(A_27c,V5s_2E0),s(tyop_2Epath_2Epath(A_27c,A_27d),V4p_2E0)))) ) ) ) ).

fof(thm_2Epath_2Edrop__def__compute,axiom,
    ! [A_27a,A_27b] :
      ( ! [V0p_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Edrop_2E2(s(tyop_2Enum_2Enum,c_2Enum_2E0_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0)
      & ! [V1n_2E0,V2p_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Edrop_2E2(s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,V1n_2E0))))),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Edrop_2E2(s(tyop_2Enum_2Enum,c_2Earithmetic_2E_2D_2E2(s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,V1n_2E0))))),s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EZERO_2E0))))))),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Etail_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0)))))
      & ! [V3n_2E0,V4p_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Edrop_2E2(s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT2_2E1(s(tyop_2Enum_2Enum,V3n_2E0))))),s(tyop_2Epath_2Epath(A_27a,A_27b),V4p_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Edrop_2E2(s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,V3n_2E0))))),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Etail_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V4p_2E0))))) ) ).

fof(thm_2Epath_2Enumeral__drop,axiom,
    ! [A_27a,A_27b] :
      ( ! [V0n_2E0,V1p_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Edrop_2E2(s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,V0n_2E0))))),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Edrop_2E2(s(tyop_2Enum_2Enum,c_2Earithmetic_2E_2D_2E2(s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,V0n_2E0))))),s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EZERO_2E0))))))),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Etail_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0)))))
      & ! [V2n_2E0,V3p_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Edrop_2E2(s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT2_2E1(s(tyop_2Enum_2Enum,V2n_2E0))))),s(tyop_2Epath_2Epath(A_27a,A_27b),V3p_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Edrop_2E2(s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,V2n_2E0))))),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Etail_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V3p_2E0))))) ) ).

fof(thm_2Epath_2Efinite__drop,axiom,
    ! [A_27a,A_27b,V0p_2E0,V1n_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,V1n_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))))))
     => s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Edrop_2E2(s(tyop_2Enum_2Enum,V1n_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))))) = s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))) ) ).

fof(thm_2Epath_2Elength__drop,axiom,
    ! [A_27a,A_27b,F0_2E0] :
      ( ! [V1n_2E0,V2m_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eoption_2Eoption(tyop_2Enum_2Enum))),F0_2E0),s(tyop_2Enum_2Enum,V1n_2E0))),s(tyop_2Enum_2Enum,V2m_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Eoption_2ESOME_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2E_2D_2E2(s(tyop_2Enum_2Enum,V2m_2E0),s(tyop_2Enum_2Enum,V1n_2E0)))))
     => ! [V0p_2E0,V1n_2E0] :
          ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,V1n_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))))))
         => s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Epath_2Elength_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Edrop_2E2(s(tyop_2Enum_2Enum,V1n_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))))) = s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Eoption_2Eoption__CASE_2E3(s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Epath_2Elength_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))),s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Eoption_2ENONE_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eoption_2Eoption(tyop_2Enum_2Enum)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eoption_2Eoption(tyop_2Enum_2Enum))),F0_2E0),s(tyop_2Enum_2Enum,V1n_2E0))))) ) ) ).

fof(thm_2Epath_2EPL__drop,axiom,
    ! [A_27a,A_27b,F0_2E0] :
      ( ! [V1i_2E0,V2n_2E0] : s(tyop_2Enum_2Enum,app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),F0_2E0),s(tyop_2Enum_2Enum,V1i_2E0))),s(tyop_2Enum_2Enum,V2n_2E0))) = s(tyop_2Enum_2Enum,c_2Earithmetic_2E_2D_2E2(s(tyop_2Enum_2Enum,V2n_2E0),s(tyop_2Enum_2Enum,V1i_2E0)))
     => ! [V0p_2E0,V1i_2E0] :
          ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,V1i_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))))))
         => s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Edrop_2E2(s(tyop_2Enum_2Enum,V1i_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))))) = s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epred__set_2EIMAGE_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),F0_2E0),s(tyop_2Enum_2Enum,V1i_2E0))),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))))) ) ) ).

fof(thm_2Epath_2EIN__PL__drop,axiom,
    ! [A_27a,A_27b,V0i_2E0,V1j_2E0,V2p_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))))))
     => s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,V1j_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Edrop_2E2(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))))))) = s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,c_2Earithmetic_2E_2B_2E2(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Enum_2Enum,V1j_2E0))),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))))) ) ).

fof(thm_2Epath_2Efirst__drop,axiom,
    ! [A_27a,A_27b,V0i_2E0,V1p_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))))
     => s(A_27a,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Edrop_2E2(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))) = s(A_27a,c_2Epath_2Eel_2E2(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))) ) ).

fof(thm_2Epath_2Efirst__label__drop,axiom,
    ! [A_27a,A_27b,V0i_2E0,V1p_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))))
     => s(A_27b,c_2Epath_2Efirst__label_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Edrop_2E2(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))) = s(A_27b,c_2Epath_2Enth__label_2E2(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))) ) ).

fof(thm_2Epath_2Etail__drop,axiom,
    ! [A_27a,A_27b,V0i_2E0,V1p_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,c_2Earithmetic_2E_2B_2E2(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EZERO_2E0))))))),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))))
     => s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Etail_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Edrop_2E2(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Edrop_2E2(s(tyop_2Enum_2Enum,c_2Earithmetic_2E_2B_2E2(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EZERO_2E0))))))),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))) ) ).

fof(thm_2Epath_2Eel__drop,axiom,
    ! [A_27a,A_27b,V0i_2E0,V1j_2E0,V2p_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,c_2Earithmetic_2E_2B_2E2(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Enum_2Enum,V1j_2E0))),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))))))
     => s(A_27a,c_2Epath_2Eel_2E2(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Edrop_2E2(s(tyop_2Enum_2Enum,V1j_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))))) = s(A_27a,c_2Epath_2Eel_2E2(s(tyop_2Enum_2Enum,c_2Earithmetic_2E_2B_2E2(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Enum_2Enum,V1j_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))) ) ).

fof(thm_2Epath_2Enth__label__drop,axiom,
    ! [A_27a,A_27b,V0i_2E0,V1j_2E0,V2p_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,c_2Enum_2ESUC_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2E_2B_2E2(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Enum_2Enum,V1j_2E0))))),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))))))
     => s(A_27b,c_2Epath_2Enth__label_2E2(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Edrop_2E2(s(tyop_2Enum_2Enum,V1j_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))))) = s(A_27b,c_2Epath_2Enth__label_2E2(s(tyop_2Enum_2Enum,c_2Earithmetic_2E_2B_2E2(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Enum_2Enum,V1j_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))) ) ).

fof(thm_2Epath_2Etake__def__compute,axiom,
    ! [A_27a,A_27b] :
      ( ! [V0p_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Etake_2E2(s(tyop_2Enum_2Enum,c_2Enum_2E0_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0)))))
      & ! [V1n_2E0,V2p_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Etake_2E2(s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,V1n_2E0))))),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))),s(A_27b,c_2Epath_2Efirst__label_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Etake_2E2(s(tyop_2Enum_2Enum,c_2Earithmetic_2E_2D_2E2(s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,V1n_2E0))))),s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EZERO_2E0))))))),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Etail_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0)))))))
      & ! [V3n_2E0,V4p_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Etake_2E2(s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT2_2E1(s(tyop_2Enum_2Enum,V3n_2E0))))),s(tyop_2Epath_2Epath(A_27a,A_27b),V4p_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V4p_2E0))),s(A_27b,c_2Epath_2Efirst__label_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V4p_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Etake_2E2(s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,V3n_2E0))))),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Etail_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V4p_2E0))))))) ) ).

fof(thm_2Epath_2Efirst__take,axiom,
    ! [A_27a,A_27b,V0p_2E0,V1i_2E0] : s(A_27a,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Etake_2E2(s(tyop_2Enum_2Enum,V1i_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))))) = s(A_27a,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))) ).

fof(thm_2Epath_2Efinite__take,axiom,
    ! [A_27a,A_27b,V0p_2E0,V1i_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,V1i_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))))))
     => p(s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Etake_2E2(s(tyop_2Enum_2Enum,V1i_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0)))))) ) ).

fof(thm_2Epath_2Elength__take,axiom,
    ! [A_27a,A_27b,V0p_2E0,V1i_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,V1i_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))))))
     => s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Epath_2Elength_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Etake_2E2(s(tyop_2Enum_2Enum,V1i_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))))) = s(tyop_2Eoption_2Eoption(tyop_2Enum_2Enum),c_2Eoption_2ESOME_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2E_2B_2E2(s(tyop_2Enum_2Enum,V1i_2E0),s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EZERO_2E0))))))))) ) ).

fof(thm_2Epath_2EPL__take,axiom,
    ! [A_27a,A_27b,F0_2E0] :
      ( ! [V1i_2E0,V2n_2E0] : s(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),F0_2E0),s(tyop_2Enum_2Enum,V1i_2E0))),s(tyop_2Enum_2Enum,V2n_2E0))) = s(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epair_2E_2C_2E2(s(tyop_2Enum_2Enum,V2n_2E0),s(tyop_2Emin_2Ebool,c_2Earithmetic_2E_3C_3D_2E2(s(tyop_2Enum_2Enum,V2n_2E0),s(tyop_2Enum_2Enum,V1i_2E0)))))
     => ! [V0p_2E0,V1i_2E0] :
          ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,V1i_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))))))
         => s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Etake_2E2(s(tyop_2Enum_2Enum,V1i_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))))) = s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E1(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),F0_2E0),s(tyop_2Enum_2Enum,V1i_2E0))))) ) ) ).

fof(thm_2Epath_2Elast__take,axiom,
    ! [A_27a,A_27b,V0i_2E0,V1p_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))))
     => s(A_27a,c_2Epath_2Elast_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Etake_2E2(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))) = s(A_27a,c_2Epath_2Eel_2E2(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))) ) ).

fof(thm_2Epath_2Enth__label__take,axiom,
    ! [A_27a,A_27b,V0n_2E0,V1p_2E0,V2i_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,c_2Eprim__rec_2E_3C_2E2(s(tyop_2Enum_2Enum,V2i_2E0),s(tyop_2Enum_2Enum,V0n_2E0))))
        & p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,V0n_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0)))))) )
     => s(A_27b,c_2Epath_2Enth__label_2E2(s(tyop_2Enum_2Enum,V2i_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Etake_2E2(s(tyop_2Enum_2Enum,V0n_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))) = s(A_27b,c_2Epath_2Enth__label_2E2(s(tyop_2Enum_2Enum,V2i_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))) ) ).

fof(thm_2Epath_2Esingleton__seg,axiom,
    ! [A_27a,A_27b,V0i_2E0,V1p_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))))
     => s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Eseg_2E3(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,c_2Epath_2Eel_2E2(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))) ) ).

fof(thm_2Epath_2Erecursive__seg,axiom,
    ! [A_27a,A_27b,V0i_2E0,V1j_2E0,V2p_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,c_2Eprim__rec_2E_3C_2E2(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Enum_2Enum,V1j_2E0))))
        & p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,V1j_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0)))))) )
     => s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Eseg_2E3(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Enum_2Enum,V1j_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,c_2Epath_2Eel_2E2(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))),s(A_27b,c_2Epath_2Enth__label_2E2(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Eseg_2E3(s(tyop_2Enum_2Enum,c_2Earithmetic_2E_2B_2E2(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EZERO_2E0))))))),s(tyop_2Enum_2Enum,V1j_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))))) ) ).

fof(thm_2Epath_2EPL__seg,axiom,
    ! [A_27a,A_27b,F0_2E0] :
      ( ! [V1j_2E0,V0i_2E0,V3n_2E0] : s(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)))),F0_2E0),s(tyop_2Enum_2Enum,V1j_2E0))),s(tyop_2Enum_2Enum,V0i_2E0))),s(tyop_2Enum_2Enum,V3n_2E0))) = s(tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epair_2E_2C_2E2(s(tyop_2Enum_2Enum,V3n_2E0),s(tyop_2Emin_2Ebool,c_2Earithmetic_2E_3C_3D_2E2(s(tyop_2Enum_2Enum,V3n_2E0),s(tyop_2Enum_2Enum,c_2Earithmetic_2E_2D_2E2(s(tyop_2Enum_2Enum,V1j_2E0),s(tyop_2Enum_2Enum,V0i_2E0)))))))
     => ! [V0i_2E0,V1j_2E0,V2p_2E0] :
          ( ( p(s(tyop_2Emin_2Ebool,c_2Earithmetic_2E_3C_3D_2E2(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Enum_2Enum,V1j_2E0))))
            & p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,V1j_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0)))))) )
         => s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Eseg_2E3(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Enum_2Enum,V1j_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))))) = s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E1(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Epair_2Eprod(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)))),F0_2E0),s(tyop_2Enum_2Enum,V1j_2E0))),s(tyop_2Enum_2Enum,V0i_2E0))))) ) ) ).

fof(thm_2Epath_2Efinite__seg,axiom,
    ! [A_27a,A_27b,V0p_2E0,V1i_2E0,V2j_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,c_2Earithmetic_2E_3C_3D_2E2(s(tyop_2Enum_2Enum,V1i_2E0),s(tyop_2Enum_2Enum,V2j_2E0))))
        & p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,V2j_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0)))))) )
     => p(s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Eseg_2E3(s(tyop_2Enum_2Enum,V1i_2E0),s(tyop_2Enum_2Enum,V2j_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0)))))) ) ).

fof(thm_2Epath_2Efirst__seg,axiom,
    ! [A_27a,A_27b,V0i_2E0,V1j_2E0,V2p_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,c_2Earithmetic_2E_3C_3D_2E2(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Enum_2Enum,V1j_2E0))))
        & p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,V1j_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0)))))) )
     => s(A_27a,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Eseg_2E3(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Enum_2Enum,V1j_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))))) = s(A_27a,c_2Epath_2Eel_2E2(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))) ) ).

fof(thm_2Epath_2Elast__seg,axiom,
    ! [A_27a,A_27b,V0i_2E0,V1j_2E0,V2p_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,c_2Earithmetic_2E_3C_3D_2E2(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Enum_2Enum,V1j_2E0))))
        & p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,V1j_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0)))))) )
     => s(A_27a,c_2Epath_2Elast_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Eseg_2E3(s(tyop_2Enum_2Enum,V0i_2E0),s(tyop_2Enum_2Enum,V1j_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))))) = s(A_27a,c_2Epath_2Eel_2E2(s(tyop_2Enum_2Enum,V1j_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_2E0))) ) ).

fof(thm_2Epath_2EfirstP__at__unique,axiom,
    ! [A_27a,A_27b,V0P_2E0,V1p_2E0,V2n_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Epath_2EfirstP__at_2E3(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0),s(tyop_2Enum_2Enum,V2n_2E0))))
     => ! [V3m_2E0] :
          ( p(s(tyop_2Emin_2Ebool,c_2Epath_2EfirstP__at_2E3(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0),s(tyop_2Enum_2Enum,V3m_2E0))))
        <=> s(tyop_2Enum_2Enum,V3m_2E0) = s(tyop_2Enum_2Enum,V2n_2E0) ) ) ).

fof(thm_2Epath_2Eis__stopped__thm,axiom,
    ! [A_27a,A_27b,A_27c,A_27d] :
      ( ! [V0x_2E0] : s(tyop_2Emin_2Ebool,c_2Epath_2Eis__stopped_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V0x_2E0))))) = s(tyop_2Emin_2Ebool,c_2Ebool_2ET_2E0)
      & ! [V1x_2E0,V2r_2E0,V3p_2E0] : s(tyop_2Emin_2Ebool,c_2Epath_2Eis__stopped_2E1(s(tyop_2Epath_2Epath(A_27c,A_27d),c_2Epath_2Epcons_2E3(s(A_27c,V1x_2E0),s(A_27d,V2r_2E0),s(tyop_2Epath_2Epath(A_27c,A_27d),V3p_2E0))))) = s(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0) ) ).

fof(thm_2Epath_2Efilter__every,axiom,
    ! [A_27a,A_27b,V0P_2E0,V1p_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Eexists_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))
     => p(s(tyop_2Emin_2Ebool,c_2Epath_2Eevery_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Efilter_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0)))))) ) ).

fof(thm_2Epath_2Epgenerate__infinite,axiom,
    ! [A_27a,A_27b,V0f_2E0,V1g_2E0] : ~ p(s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epgenerate_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),V0f_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27b),V1g_2E0)))))) ).

fof(thm_2Epath_2Epgenerate__not__stopped,axiom,
    ! [A_27a,A_27b,V0f_2E0,V1g_2E0,V2x_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V2x_2E0))) != s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epgenerate_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),V0f_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27b),V1g_2E0))) ).

fof(thm_2Epath_2Eel__pgenerate,axiom,
    ! [A_27a,A_27b,V0n_2E0,V1f_2E0,V2g_2E0] : s(A_27a,c_2Epath_2Eel_2E2(s(tyop_2Enum_2Enum,V0n_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epgenerate_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),V1f_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27b),V2g_2E0))))) = s(A_27a,app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),V1f_2E0),s(tyop_2Enum_2Enum,V0n_2E0))) ).

fof(thm_2Epath_2Enth__label__pgenerate,axiom,
    ! [A_27a,A_27b,V0n_2E0,V1f_2E0,V2g_2E0] : s(A_27b,c_2Epath_2Enth__label_2E2(s(tyop_2Enum_2Enum,V0n_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epgenerate_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),V1f_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27b),V2g_2E0))))) = s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27b),V2g_2E0),s(tyop_2Enum_2Enum,V0n_2E0))) ).

fof(thm_2Epath_2Epgenerate__11,axiom,
    ! [A_27a,A_27b,V0f1_2E0,V1g1_2E0,V2f2_2E0,V3g2_2E0] :
      ( s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epgenerate_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),V0f1_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27b),V1g1_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epgenerate_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),V2f2_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27b),V3g2_2E0)))
    <=> ( s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),V0f1_2E0) = s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),V2f2_2E0)
        & s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27b),V1g1_2E0) = s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27b),V3g2_2E0) ) ) ).

fof(thm_2Epath_2Epgenerate__onto,axiom,
    ! [A_27a,A_27b,V0p_2E0] :
      ( ~ p(s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))))
     => ? [V1f_2E0,V2g_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epgenerate_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),V1f_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27b),V2g_2E0))) ) ).

fof(thm_2Epath_2Eokpath__monotone,axiom,
    ! [A_27a,A_27b,V0R_2E0] : p(s(tyop_2Emin_2Ebool,c_2EfixedPoint_2Emonotone_2E1(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool)),c_2Epath_2Eokpath__f_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0)))))) ).

fof(thm_2Epath_2Eokpath__co__ind,axiom,
    ! [A_27a,A_27b,V0R_2E0,V1P_2E0] :
      ( ! [V2x_2E0,V3r_2E0,V4p_2E0] :
          ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),V1P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V2x_2E0),s(A_27b,V3r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V4p_2E0))))))
         => ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0),s(A_27a,V2x_2E0))),s(A_27b,V3r_2E0))),s(A_27a,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V4p_2E0))))))
            & p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),V1P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V4p_2E0)))) ) )
     => ! [V5p_2E0] :
          ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),V1P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V5p_2E0))))
         => p(s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V5p_2E0)))) ) ) ).

fof(thm_2Epath_2Eokpath__cases,axiom,
    ! [A_27a,A_27b,V0R_2E0,V1x_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1x_2E0))))
    <=> ( ? [V2x_27_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),V1x_2E0) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V2x_27_2E0)))
        | ? [V3x_27_2E0,V4r_2E0,V5p_2E0] :
            ( s(tyop_2Epath_2Epath(A_27a,A_27b),V1x_2E0) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V3x_27_2E0),s(A_27b,V4r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V5p_2E0)))
            & p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0),s(A_27a,V3x_27_2E0))),s(A_27b,V4r_2E0))),s(A_27a,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V5p_2E0))))))
            & p(s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V5p_2E0)))) ) ) ) ).

fof(thm_2Epath_2Eokpath__thm,axiom,
    ! [A_27a,A_27b,V0R_2E0] :
      ( ! [V1x_2E0] : p(s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V1x_2E0))))))
      & ! [V2x_2E0,V3r_2E0,V4p_2E0] :
          ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V2x_2E0),s(A_27b,V3r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V4p_2E0))))))
        <=> ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0),s(A_27a,V2x_2E0))),s(A_27b,V3r_2E0))),s(A_27a,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V4p_2E0))))))
            & p(s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V4p_2E0)))) ) ) ) ).

fof(thm_2Epath_2Efinite__okpath__ind,axiom,
    ! [A_27a,A_27b,V0P_2E0,V1R_2E0] :
      ( ( ! [V2x_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V2x_2E0))))))
        & ! [V3x_2E0,V4r_2E0,V5p_2E0] :
            ( ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V1R_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V5p_2E0))))
              & p(s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V5p_2E0))))
              & p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V1R_2E0),s(A_27a,V3x_2E0))),s(A_27b,V4r_2E0))),s(A_27a,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V5p_2E0))))))
              & p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V5p_2E0)))) )
           => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V3x_2E0),s(A_27b,V4r_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V5p_2E0)))))) ) )
     => ! [V6sigma_2E0] :
          ( ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V1R_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V6sigma_2E0))))
            & p(s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V6sigma_2E0)))) )
         => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epath_2Epath(A_27a,A_27b),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V6sigma_2E0)))) ) ) ).

fof(thm_2Epath_2Eokpath__pmap,axiom,
    ! [A_27a,A_27b,V0R_2E0,V1f_2E0,V2g_2E0,V3p_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V3p_2E0))))
        & ! [V4x_2E0,V5r_2E0,V6y_2E0] :
            ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0),s(A_27a,V4x_2E0))),s(A_27b,V5r_2E0))),s(A_27a,V6y_2E0))))
           => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0),s(A_27a,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27a),V1f_2E0),s(A_27a,V4x_2E0))))),s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27b,A_27b),V2g_2E0),s(A_27b,V5r_2E0))))),s(A_27a,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27a),V1f_2E0),s(A_27a,V6y_2E0)))))) ) )
     => p(s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epmap_2E3(s(tyop_2Emin_2Efun(A_27a,A_27a),V1f_2E0),s(tyop_2Emin_2Efun(A_27b,A_27b),V2g_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V3p_2E0)))))) ) ).

fof(thm_2Epath_2Efinite__plink,axiom,
    ! [A_27a,A_27b,V0p1_2E0,V1p2_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Eplink_2E2(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p1_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p2_2E0))))))
    <=> ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p1_2E0))))
        & p(s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p2_2E0)))) ) ) ).

fof(thm_2Epath_2Efirst__plink,axiom,
    ! [A_27a,A_27b,V0p1_2E0,V1p2_2E0] :
      ( s(A_27a,c_2Epath_2Elast_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p1_2E0))) = s(A_27a,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p2_2E0)))
     => s(A_27a,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Eplink_2E2(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p1_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p2_2E0))))) = s(A_27a,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p1_2E0))) ) ).

fof(thm_2Epath_2Elast__plink,axiom,
    ! [A_27a,A_27b,V0p1_2E0,V1p2_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p1_2E0))))
        & p(s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p2_2E0))))
        & s(A_27a,c_2Epath_2Elast_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p1_2E0))) = s(A_27a,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p2_2E0))) )
     => s(A_27a,c_2Epath_2Elast_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Eplink_2E2(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p1_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p2_2E0))))) = s(A_27a,c_2Epath_2Elast_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p2_2E0))) ) ).

fof(thm_2Epath_2Eokpath__plink,axiom,
    ! [A_27a,A_27b,V0R_2E0,V1p1_2E0,V2p2_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p1_2E0))))
        & s(A_27a,c_2Epath_2Elast_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p1_2E0))) = s(A_27a,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V2p2_2E0))) )
     => ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Eplink_2E2(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p1_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p2_2E0))))))
      <=> ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p1_2E0))))
          & p(s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V2p2_2E0)))) ) ) ) ).

fof(thm_2Epath_2Eokpath__take,axiom,
    ! [A_27a,A_27b,V0R_2E0,V1p_2E0,V2i_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,V2i_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))))
        & p(s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0)))) )
     => p(s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Etake_2E2(s(tyop_2Enum_2Enum,V2i_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0)))))) ) ).

fof(thm_2Epath_2Eokpath__drop,axiom,
    ! [A_27a,A_27b,V0R_2E0,V1p_2E0,V2i_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,V2i_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))))
        & p(s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0)))) )
     => p(s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Edrop_2E2(s(tyop_2Enum_2Enum,V2i_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0)))))) ) ).

fof(thm_2Epath_2Eokpath__seg,axiom,
    ! [A_27a,A_27b,V0R_2E0,V1p_2E0,V2i_2E0,V3j_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,c_2Earithmetic_2E_3C_3D_2E2(s(tyop_2Enum_2Enum,V2i_2E0),s(tyop_2Enum_2Enum,V3j_2E0))))
        & p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,V3j_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))))
        & p(s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0)))) )
     => p(s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Eseg_2E3(s(tyop_2Enum_2Enum,V2i_2E0),s(tyop_2Enum_2Enum,V3j_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0)))))) ) ).

fof(thm_2Epath_2ESN__finite__paths,axiom,
    ! [A_27a,A_27b,V0R_2E0,V1p_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,c_2Epath_2ESN_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0))))
        & p(s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0)))) )
     => p(s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0)))) ) ).

fof(thm_2Epath_2Efinite__paths__SN,axiom,
    ! [A_27a,A_27b,V0R_2E0] :
      ( ! [V1p_2E0] :
          ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))
         => p(s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0)))) )
     => p(s(tyop_2Emin_2Ebool,c_2Epath_2ESN_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0)))) ) ).

fof(thm_2Epath_2ESN__finite__paths__EQ,axiom,
    ! [A_27a,A_27b,V0R_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Epath_2ESN_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0))))
    <=> ! [V1p_2E0] :
          ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0R_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))
         => p(s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0)))) ) ) ).

fof(thm_2Epath_2Elabels__LMAP,axiom,
    ! [A_27a,A_27b,V0p_2E0] : s(tyop_2Ellist_2Ellist(A_27b),c_2Epath_2Elabels_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))) = s(tyop_2Ellist_2Ellist(A_27b),c_2Ellist_2ELMAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),A_27b),c_2Epair_2EFST_2E0),s(tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a)),c_2Epair_2ESND_2E1(s(tyop_2Epair_2Eprod(A_27a,tyop_2Ellist_2Ellist(tyop_2Epair_2Eprod(A_27b,A_27a))),c_2Epath_2EfromPath_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))))))) ).

fof(thm_2Epath_2Elabels__plink,axiom,
    ! [A_27a,A_27b,V0p1_2E0,V1p2_2E0] : s(tyop_2Ellist_2Ellist(A_27b),c_2Epath_2Elabels_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Eplink_2E2(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p1_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p2_2E0))))) = s(tyop_2Ellist_2Ellist(A_27b),c_2Ellist_2ELAPPEND_2E2(s(tyop_2Ellist_2Ellist(A_27b),c_2Epath_2Elabels_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p1_2E0))),s(tyop_2Ellist_2Ellist(A_27b),c_2Epath_2Elabels_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p2_2E0))))) ).

fof(thm_2Epath_2Efinite__labels,axiom,
    ! [A_27a,A_27b,V0p_2E0] : s(tyop_2Emin_2Ebool,c_2Ellist_2ELFINITE_2E1(s(tyop_2Ellist_2Ellist(A_27b),c_2Epath_2Elabels_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))))) = s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))) ).

fof(thm_2Epath_2Eunfold__thm,axiom,
    ! [A_27a,A_27b,A_27c,F1_2E0,F0_2E0] :
      ( ! [V0proj_2E0,V2s_2E0,V1f_2E0,V4s_27_2E0,V5l_2E0] : s(tyop_2Epath_2Epath(A_27b,A_27c),app_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Epath_2Epath(A_27b,A_27c)),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Epath_2Epath(A_27b,A_27c))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Epath_2Epath(A_27b,A_27c)))),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Epath_2Epath(A_27b,A_27c))))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Epath_2Epath(A_27b,A_27c)))))),F1_2E0),s(tyop_2Emin_2Efun(A_27a,A_27b),V0proj_2E0))),s(A_27a,V2s_2E0))),s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),V1f_2E0))),s(A_27a,V4s_27_2E0))),s(A_27c,V5l_2E0))) = s(tyop_2Epath_2Epath(A_27b,A_27c),c_2Epath_2Epcons_2E3(s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V0proj_2E0),s(A_27a,V2s_2E0))),s(A_27c,V5l_2E0),s(tyop_2Epath_2Epath(A_27b,A_27c),c_2Epath_2Eunfold_2E3(s(tyop_2Emin_2Efun(A_27a,A_27b),V0proj_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),V1f_2E0),s(A_27a,V4s_27_2E0)))))
     => ( ! [V0proj_2E0,V2s_2E0,V1f_2E0,V3v_2E0] : s(tyop_2Epath_2Epath(A_27b,A_27c),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Epath_2Epath(A_27b,A_27c)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Epath_2Epath(A_27b,A_27c))),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Epath_2Epath(A_27b,A_27c)))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Epath_2Epath(A_27b,A_27c))))),F0_2E0),s(tyop_2Emin_2Efun(A_27a,A_27b),V0proj_2E0))),s(A_27a,V2s_2E0))),s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),V1f_2E0))),s(tyop_2Epair_2Eprod(A_27a,A_27c),V3v_2E0))) = s(tyop_2Epath_2Epath(A_27b,A_27c),c_2Epair_2Epair__CASE_2E2(s(tyop_2Epair_2Eprod(A_27a,A_27c),V3v_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Epath_2Epath(A_27b,A_27c))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Epath_2Epath(A_27b,A_27c)))),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Epath_2Epath(A_27b,A_27c))))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Epath_2Epath(A_27b,A_27c)))))),F1_2E0),s(tyop_2Emin_2Efun(A_27a,A_27b),V0proj_2E0))),s(A_27a,V2s_2E0))),s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),V1f_2E0)))))
       => ! [V0proj_2E0,V1f_2E0,V2s_2E0] : s(tyop_2Epath_2Epath(A_27b,A_27c),c_2Epath_2Eunfold_2E3(s(tyop_2Emin_2Efun(A_27a,A_27b),V0proj_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),V1f_2E0),s(A_27a,V2s_2E0))) = s(tyop_2Epath_2Epath(A_27b,A_27c),c_2Eoption_2Eoption__CASE_2E3(s(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c)),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),V1f_2E0),s(A_27a,V2s_2E0))),s(tyop_2Epath_2Epath(A_27b,A_27c),c_2Epath_2Estopped__at_2E1(s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V0proj_2E0),s(A_27a,V2s_2E0))))),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Epath_2Epath(A_27b,A_27c)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Epath_2Epath(A_27b,A_27c))),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Epath_2Epath(A_27b,A_27c)))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Epath_2Epath(A_27b,A_27c))))),F0_2E0),s(tyop_2Emin_2Efun(A_27a,A_27b),V0proj_2E0))),s(A_27a,V2s_2E0))),s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),V1f_2E0))))) ) ) ).

fof(thm_2Epath_2Eunfold__thm2,axiom,
    ! [A_27a,A_27b,A_27c,V0proj_2E0,V1f_2E0,V2x_2E0,V3v1_2E0,V4v2_2E0] :
      ( ( s(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c)),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),V1f_2E0),s(A_27a,V2x_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c)),c_2Eoption_2ENONE_2E0)
       => s(tyop_2Epath_2Epath(A_27b,A_27c),c_2Epath_2Eunfold_2E3(s(tyop_2Emin_2Efun(A_27a,A_27b),V0proj_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),V1f_2E0),s(A_27a,V2x_2E0))) = s(tyop_2Epath_2Epath(A_27b,A_27c),c_2Epath_2Estopped__at_2E1(s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V0proj_2E0),s(A_27a,V2x_2E0))))) )
      & ( s(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c)),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),V1f_2E0),s(A_27a,V2x_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c)),c_2Eoption_2ESOME_2E1(s(tyop_2Epair_2Eprod(A_27a,A_27c),c_2Epair_2E_2C_2E2(s(A_27a,V3v1_2E0),s(A_27c,V4v2_2E0)))))
       => s(tyop_2Epath_2Epath(A_27b,A_27c),c_2Epath_2Eunfold_2E3(s(tyop_2Emin_2Efun(A_27a,A_27b),V0proj_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),V1f_2E0),s(A_27a,V2x_2E0))) = s(tyop_2Epath_2Epath(A_27b,A_27c),c_2Epath_2Epcons_2E3(s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V0proj_2E0),s(A_27a,V2x_2E0))),s(A_27c,V4v2_2E0),s(tyop_2Epath_2Epath(A_27b,A_27c),c_2Epath_2Eunfold_2E3(s(tyop_2Emin_2Efun(A_27a,A_27b),V0proj_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),V1f_2E0),s(A_27a,V3v1_2E0))))) ) ) ).

fof(thm_2Epath_2Elabels__unfold,axiom,
    ! [A_27a,A_27b,A_27c,V0proj_2E0,V1f_2E0,V2s_2E0] : s(tyop_2Ellist_2Ellist(A_27c),c_2Epath_2Elabels_2E1(s(tyop_2Epath_2Epath(A_27b,A_27c),c_2Epath_2Eunfold_2E3(s(tyop_2Emin_2Efun(A_27a,A_27b),V0proj_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),V1f_2E0),s(A_27a,V2s_2E0))))) = s(tyop_2Ellist_2Ellist(A_27c),c_2Ellist_2ELUNFOLD_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),V1f_2E0),s(A_27a,V2s_2E0))) ).

fof(thm_2Epath_2Eokpath__unfold,axiom,
    ! [A_27a,A_27b,A_27c,V0P_2E0,V1m_2E0,V2proj_2E0,V3f_2E0,V4s_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(A_27a,V4s_2E0))))
        & ! [V5s_2E0,V6s_27_2E0,V7l_2E0] :
            ( ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(A_27a,V5s_2E0))))
              & s(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c)),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),V3f_2E0),s(A_27a,V5s_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c)),c_2Eoption_2ESOME_2E1(s(tyop_2Epair_2Eprod(A_27a,A_27c),c_2Epair_2E_2C_2E2(s(A_27a,V6s_27_2E0),s(A_27c,V7l_2E0))))) )
           => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(A_27a,V6s_27_2E0)))) )
        & ! [V8s_2E0,V9s_27_2E0,V10l_2E0] :
            ( ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(A_27a,V8s_2E0))))
              & s(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c)),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),V3f_2E0),s(A_27a,V8s_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c)),c_2Eoption_2ESOME_2E1(s(tyop_2Epair_2Eprod(A_27a,A_27c),c_2Epair_2E_2C_2E2(s(A_27a,V9s_27_2E0),s(A_27c,V10l_2E0))))) )
           => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),V1m_2E0),s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V2proj_2E0),s(A_27a,V8s_2E0))))),s(A_27c,V10l_2E0))),s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V2proj_2E0),s(A_27a,V9s_27_2E0)))))) ) )
     => p(s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),V1m_2E0),s(tyop_2Epath_2Epath(A_27b,A_27c),c_2Epath_2Eunfold_2E3(s(tyop_2Emin_2Efun(A_27a,A_27b),V2proj_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Epair_2Eprod(A_27a,A_27c))),V3f_2E0),s(A_27a,V4s_2E0)))))) ) ).

fof(thm_2Epath_2Etrace__machine__thm,axiom,
    ! [A_27a,V0P_2E0,V1tr_2E0] :
      ( ! [V2n_2E0,V3l_2E0] :
          ( s(tyop_2Eoption_2Eoption(tyop_2Elist_2Elist(A_27a)),c_2Ellist_2ELTAKE_2E2(s(tyop_2Enum_2Enum,V2n_2E0),s(tyop_2Ellist_2Ellist(A_27a),V1tr_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Elist_2Elist(A_27a)),c_2Eoption_2ESOME_2E1(s(tyop_2Elist_2Elist(A_27a),V3l_2E0)))
         => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Elist_2Elist(A_27a),V3l_2E0)))) )
     => ? [V4p_2E0] :
          ( s(tyop_2Ellist_2Ellist(A_27a),V1tr_2E0) = s(tyop_2Ellist_2Ellist(A_27a),c_2Epath_2Elabels_2E1(s(tyop_2Epath_2Epath(tyop_2Elist_2Elist(A_27a),A_27a),V4p_2E0)))
          & p(s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool))),c_2Epath_2Etrace__machine_2E1(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool),V0P_2E0))),s(tyop_2Epath_2Epath(tyop_2Elist_2Elist(A_27a),A_27a),V4p_2E0))))
          & s(tyop_2Elist_2Elist(A_27a),c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(tyop_2Elist_2Elist(A_27a),A_27a),V4p_2E0))) = s(tyop_2Elist_2Elist(A_27a),c_2Elist_2ENIL_2E0) ) ) ).

fof(thm_2Epath_2Etrace__machine__thm2,axiom,
    ! [A_27a,A_27b,V0n_2E0,V1l_2E0,V2P_2E0,V3p_2E0,V4init_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool))),c_2Epath_2Etrace__machine_2E1(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool),V2P_2E0))),s(tyop_2Epath_2Epath(tyop_2Elist_2Elist(A_27a),A_27a),V3p_2E0))))
        & p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool),V2P_2E0),s(tyop_2Elist_2Elist(A_27a),c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(tyop_2Elist_2Elist(A_27a),A_27a),V3p_2E0)))))) )
     => ( s(tyop_2Eoption_2Eoption(tyop_2Elist_2Elist(A_27a)),c_2Ellist_2ELTAKE_2E2(s(tyop_2Enum_2Enum,V0n_2E0),s(tyop_2Ellist_2Ellist(A_27a),c_2Epath_2Elabels_2E1(s(tyop_2Epath_2Epath(tyop_2Elist_2Elist(A_27a),A_27a),V3p_2E0))))) = s(tyop_2Eoption_2Eoption(tyop_2Elist_2Elist(A_27a)),c_2Eoption_2ESOME_2E1(s(tyop_2Elist_2Elist(A_27a),V1l_2E0)))
       => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool),V2P_2E0),s(tyop_2Elist_2Elist(A_27a),c_2Elist_2EAPPEND_2E2(s(tyop_2Elist_2Elist(A_27a),c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(tyop_2Elist_2Elist(A_27a),A_27a),V3p_2E0))),s(tyop_2Elist_2Elist(A_27a),V1l_2E0)))))) ) ) ).

fof(thm_2Epath_2ELTAKE__labels,axiom,
    ! [A_27a,A_27b,V0n_2E0,V1p_2E0,V2l_2E0] :
      ( s(tyop_2Eoption_2Eoption(tyop_2Elist_2Elist(A_27b)),c_2Ellist_2ELTAKE_2E2(s(tyop_2Enum_2Enum,V0n_2E0),s(tyop_2Ellist_2Ellist(A_27b),c_2Epath_2Elabels_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))) = s(tyop_2Eoption_2Eoption(tyop_2Elist_2Elist(A_27b)),c_2Eoption_2ESOME_2E1(s(tyop_2Elist_2Elist(A_27b),V2l_2E0)))
    <=> ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,V0n_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))))
        & s(tyop_2Eoption_2Eoption(tyop_2Elist_2Elist(A_27b)),c_2Ellist_2EtoList_2E1(s(tyop_2Ellist_2Ellist(A_27b),c_2Epath_2Elabels_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Etake_2E2(s(tyop_2Enum_2Enum,V0n_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))))) = s(tyop_2Eoption_2Eoption(tyop_2Elist_2Elist(A_27b)),c_2Eoption_2ESOME_2E1(s(tyop_2Elist_2Elist(A_27b),V2l_2E0))) ) ) ).

fof(thm_2Epath_2Edrop__eq__pcons,axiom,
    ! [A_27a,A_27b,V0n_2E0,V1p_2E0,V2h_2E0,V3l_2E0,V4t_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,V0n_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))))
        & s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Edrop_2E2(s(tyop_2Enum_2Enum,V0n_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,V2h_2E0),s(A_27b,V3l_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V4t_2E0))) )
     => p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,c_2Earithmetic_2E_2B_2E2(s(tyop_2Enum_2Enum,V0n_2E0),s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EZERO_2E0))))))),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0)))))) ) ).

fof(thm_2Epath_2Eokpath__parallel__comp,axiom,
    ! [A_27a,A_27b,A_27c,F1_2E0,F0_2E0] :
      ( ! [V4x_2E0] : s(A_27c,app_2E2(s(tyop_2Emin_2Efun(A_27c,A_27c),F1_2E0),s(A_27c,V4x_2E0))) = s(A_27c,V4x_2E0)
     => ( ! [V3x_2E0] : s(A_27c,app_2E2(s(tyop_2Emin_2Efun(A_27c,A_27c),F0_2E0),s(A_27c,V3x_2E0))) = s(A_27c,V3x_2E0)
       => ! [V0p_2E0,V1m1_2E0,V2m2_2E0] :
            ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool))),c_2Epath_2Eparallel__comp_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V1m1_2E0),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),V2m2_2E0))),s(tyop_2Epath_2Epath(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),V0p_2E0))))
          <=> ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V1m1_2E0),s(tyop_2Epath_2Epath(A_27a,A_27c),c_2Epath_2Epmap_2E3(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST_2E0),s(tyop_2Emin_2Efun(A_27c,A_27c),F0_2E0),s(tyop_2Epath_2Epath(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),V0p_2E0))))))
              & p(s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),V2m2_2E0),s(tyop_2Epath_2Epath(A_27b,A_27c),c_2Epath_2Epmap_2E3(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27b),c_2Epair_2ESND_2E0),s(tyop_2Emin_2Efun(A_27c,A_27c),F1_2E0),s(tyop_2Epath_2Epath(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),V0p_2E0)))))) ) ) ) ) ).

fof(thm_2Epath_2Ebuild__pcomp__trace,axiom,
    ! [A_27a,A_27b,A_27c,V0m1_2E0,V1p1_2E0,V2m2_2E0,V3p2_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0m1_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p1_2E0))))
        & p(s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool))),V2m2_2E0),s(tyop_2Epath_2Epath(A_27c,A_27b),V3p2_2E0))))
        & s(tyop_2Ellist_2Ellist(A_27b),c_2Epath_2Elabels_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p1_2E0))) = s(tyop_2Ellist_2Ellist(A_27b),c_2Epath_2Elabels_2E1(s(tyop_2Epath_2Epath(A_27c,A_27b),V3p2_2E0))) )
     => ? [V4p_2E0] :
          ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Emin_2Ebool))),c_2Epath_2Eparallel__comp_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V0m1_2E0),s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool))),V2m2_2E0))),s(tyop_2Epath_2Epath(tyop_2Epair_2Eprod(A_27a,A_27c),A_27b),V4p_2E0))))
          & s(tyop_2Ellist_2Ellist(A_27b),c_2Epath_2Elabels_2E1(s(tyop_2Epath_2Epath(tyop_2Epair_2Eprod(A_27a,A_27c),A_27b),V4p_2E0))) = s(tyop_2Ellist_2Ellist(A_27b),c_2Epath_2Elabels_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p1_2E0)))
          & s(tyop_2Epair_2Eprod(A_27a,A_27c),c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(tyop_2Epair_2Eprod(A_27a,A_27c),A_27b),V4p_2E0))) = s(tyop_2Epair_2Eprod(A_27a,A_27c),c_2Epair_2E_2C_2E2(s(A_27a,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p1_2E0))),s(A_27c,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27c,A_27b),V3p2_2E0))))) ) ) ).

fof(thm_2Epath_2Enth__label__LNTH,axiom,
    ! [A_27a,A_27b,V0n_2E0,V1p_2E0,V2x_2E0] :
      ( s(tyop_2Eoption_2Eoption(A_27b),c_2Ellist_2ELNTH_2E2(s(tyop_2Enum_2Enum,V0n_2E0),s(tyop_2Ellist_2Ellist(A_27b),c_2Epath_2Elabels_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))) = s(tyop_2Eoption_2Eoption(A_27b),c_2Eoption_2ESOME_2E1(s(A_27b,V2x_2E0)))
    <=> ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Enum_2Enum,c_2Earithmetic_2E_2B_2E2(s(tyop_2Enum_2Enum,V0n_2E0),s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EZERO_2E0))))))),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Epath_2EPL_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))))
        & s(A_27b,c_2Epath_2Enth__label_2E2(s(tyop_2Enum_2Enum,V0n_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))) = s(A_27b,V2x_2E0) ) ) ).

fof(thm_2Epath_2Enth__label__LTAKE,axiom,
    ! [A_27a,A_27b,A_27c,V0n_2E0,V1p_2E0,V2l_2E0,V3i_2E0,V4v_2E0] :
      ( ( s(tyop_2Eoption_2Eoption(tyop_2Elist_2Elist(A_27b)),c_2Ellist_2ELTAKE_2E2(s(tyop_2Enum_2Enum,V0n_2E0),s(tyop_2Ellist_2Ellist(A_27b),c_2Epath_2Elabels_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))))) = s(tyop_2Eoption_2Eoption(tyop_2Elist_2Elist(A_27b)),c_2Eoption_2ESOME_2E1(s(tyop_2Elist_2Elist(A_27b),V2l_2E0)))
        & p(s(tyop_2Emin_2Ebool,c_2Eprim__rec_2E_3C_2E2(s(tyop_2Enum_2Enum,V3i_2E0),s(tyop_2Enum_2Enum,c_2Elist_2ELENGTH_2E1(s(tyop_2Elist_2Elist(A_27b),V2l_2E0)))))) )
     => s(A_27b,c_2Epath_2Enth__label_2E2(s(tyop_2Enum_2Enum,V3i_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),V1p_2E0))) = s(A_27b,c_2Elist_2EEL_2E2(s(tyop_2Enum_2Enum,V3i_2E0),s(tyop_2Elist_2Elist(A_27b),V2l_2E0))) ) ).

fof(thm_2Epath_2Efinite__path__end__cases,axiom,
    ! [A_27a,A_27b,V0p_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Efinite_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0))))
     => ( ? [V1x_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V1x_2E0)))
        | ? [V2p_27_2E0,V3l_2E0,V4s_2E0] : s(tyop_2Epath_2Epath(A_27a,A_27b),V0p_2E0) = s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Eplink_2E2(s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_27_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Epcons_2E3(s(A_27a,c_2Epath_2Elast_2E1(s(tyop_2Epath_2Epath(A_27a,A_27b),V2p_27_2E0))),s(A_27b,V3l_2E0),s(tyop_2Epath_2Epath(A_27a,A_27b),c_2Epath_2Estopped__at_2E1(s(A_27a,V4s_2E0))))))) ) ) ).

fof(thm_2Epath_2Esimulation__trace__inclusion,axiom,
    ! [A_27a,A_27b,A_27c,V0R_2E0,V1M1_2E0,V2M2_2E0,V3p_2E0,V4t__init_2E0] :
      ( ( ! [V5s1_2E0,V6l_2E0,V7s2_2E0,V8t1_2E0] :
            ( ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V0R_2E0),s(A_27a,V5s1_2E0))),s(A_27b,V8t1_2E0))))
              & p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V1M1_2E0),s(A_27a,V5s1_2E0))),s(A_27c,V6l_2E0))),s(A_27a,V7s2_2E0)))) )
           => ? [V9t2_2E0] :
                ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V0R_2E0),s(A_27a,V7s2_2E0))),s(A_27b,V9t2_2E0))))
                & p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),V2M2_2E0),s(A_27b,V8t1_2E0))),s(A_27c,V6l_2E0))),s(A_27b,V9t2_2E0)))) ) )
        & p(s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),V1M1_2E0),s(tyop_2Epath_2Epath(A_27a,A_27c),V3p_2E0))))
        & p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V0R_2E0),s(A_27a,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27a,A_27c),V3p_2E0))))),s(A_27b,V4t__init_2E0)))) )
     => ? [V10q_2E0] :
          ( p(s(tyop_2Emin_2Ebool,c_2Epath_2Eokpath_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),V2M2_2E0),s(tyop_2Epath_2Epath(A_27b,A_27c),V10q_2E0))))
          & s(tyop_2Ellist_2Ellist(A_27c),c_2Epath_2Elabels_2E1(s(tyop_2Epath_2Epath(A_27a,A_27c),V3p_2E0))) = s(tyop_2Ellist_2Ellist(A_27c),c_2Epath_2Elabels_2E1(s(tyop_2Epath_2Epath(A_27b,A_27c),V10q_2E0)))
          & s(A_27b,c_2Epath_2Efirst_2E1(s(tyop_2Epath_2Epath(A_27b,A_27c),V10q_2E0))) = s(A_27b,V4t__init_2E0) ) ) ).

fof(reserved_2Eho_2Eeq__ext,axiom,
    ! [A_27a,A_27b,V0f_2E0,V1g_2E0] :
      ( ! [V2x_2E0] : s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V0f_2E0),s(A_27a,V2x_2E0))) = s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0),s(A_27a,V2x_2E0)))
     => s(tyop_2Emin_2Efun(A_27a,A_27b),V0f_2E0) = s(tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0) ) ).

%------------------------------------------------------------------------------