%------------------------------------------------------------------------------
% File     : ITP081+4 : TPTP v8.2.0. 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    : alist.ax [Gau19]
%          : HL4081+4.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  190 ( 129 unt;   0 def)
%            Number of atoms       :  286 ( 210 equ)
%            Maximal formula atoms :    5 (   1 avg)
%            Number of connectives :  102 (   6   ~;   3   |;  18   &)
%                                         (  23 <=>;  52  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   15 (   5 avg)
%            Maximal term depth    :   17 (   3 avg)
%            Number of predicates  :    2 (   1 usr;   0 prp; 1-2 aty)
%            Number of functors    :  131 ( 131 usr;  67 con; 0-3 aty)
%            Number of variables   :  762 ( 760   !;   2   ?)
% 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_2Elist_2EALL__DISTINCT_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(tyop_2Emin_2Ebool,c_2Elist_2EALL__DISTINCT_2E1(s(tyop_2Elist_2Elist(A_27a),X0_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool),c_2Elist_2EALL__DISTINCT_2E0),s(tyop_2Elist_2Elist(A_27a),X0_2E0))) ).

fof(arityeq2_2Ec_2Elist_2EMAP_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29_20mono_2EA_27a,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Elist_2Elist(A_27a),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a),X0_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X1_2E0))) = s(tyop_2Elist_2Elist(A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(A_27a)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(A_27a))),c_2Elist_2EMAP_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a),X0_2E0))),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X1_2E0))) ).

fof(arityeq1_2Ec_2Ealist_2Efmap__to__alist_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Ealist_2Efmap__to__alist_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),X0_2E0))) = s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))),c_2Ealist_2Efmap__to__alist_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),X0_2E0))) ).

fof(arityeq1_2Ec_2Ealist_2EALOOKUP_2E1_2Emono_2EA_27b_20mono_2EA_27a,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27b)),c_2Ealist_2EALOOKUP_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0))) = s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27b))),c_2Ealist_2EALOOKUP_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0))) ).

fof(arityeq2_2Ec_2Elist_2EAPPEND_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Elist_2EAPPEND_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X1_2E0))) = s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)))),c_2Elist_2EAPPEND_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0))),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X1_2E0))) ).

fof(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Elist_2Elist_28A_27a_29_20mono_2Etyop_2Elist_2Elist_28A_27d_29,axiom,
    ! [A_27a,A_27d,X0_2E0,X1_2E0] : s(tyop_2Epair_2Eprod(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27d)),c_2Epair_2E_2C_2E2(s(tyop_2Elist_2Elist(A_27a),X0_2E0),s(tyop_2Elist_2Elist(A_27d),X1_2E0))) = s(tyop_2Epair_2Eprod(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27d)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Epair_2Eprod(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27d))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Epair_2Eprod(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27d)))),c_2Epair_2E_2C_2E0),s(tyop_2Elist_2Elist(A_27a),X0_2E0))),s(tyop_2Elist_2Elist(A_27d),X1_2E0))) ).

fof(arityeq1_2Ec_2Ealist_2EALOOKUP_2E1_2Emono_2EA_27d_20mono_2EA_27a,axiom,
    ! [A_27a,A_27d,X0_2E0] : s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27d)),c_2Ealist_2EALOOKUP_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27d)),X0_2E0))) = s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27d)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27d)),tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27d))),c_2Ealist_2EALOOKUP_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27d)),X0_2E0))) ).

fof(arityeq2_2Ec_2Elist_2EMAP_2E2_2Emono_2EA_27b_20mono_2EA_27d,axiom,
    ! [A_27b,A_27d,X0_2E0,X1_2E0] : s(tyop_2Elist_2Elist(A_27d),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(A_27b,A_27d),X0_2E0),s(tyop_2Elist_2Elist(A_27b),X1_2E0))) = s(tyop_2Elist_2Elist(A_27d),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Elist_2Elist(A_27d)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27d),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Elist_2Elist(A_27d))),c_2Elist_2EMAP_2E0),s(tyop_2Emin_2Efun(A_27b,A_27d),X0_2E0))),s(tyop_2Elist_2Elist(A_27b),X1_2E0))) ).

fof(arityeq1_2Ec_2Elist_2EZIP_2E1_2Emono_2EA_27a_20mono_2EA_27d,axiom,
    ! [A_27a,A_27d,X0_2E0] : s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27d)),c_2Elist_2EZIP_2E1(s(tyop_2Epair_2Eprod(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27d)),X0_2E0))) = s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27d)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27d)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27d))),c_2Elist_2EZIP_2E0),s(tyop_2Epair_2Eprod(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27d)),X0_2E0))) ).

fof(arityeq1_2Ec_2Ealist_2EALOOKUP_2E1_2Emono_2EA_27b_20mono_2EA_27c,axiom,
    ! [A_27b,A_27c,X0_2E0] : s(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(A_27b)),c_2Ealist_2EALOOKUP_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27b)),X0_2E0))) = s(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27b)),tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(A_27b))),c_2Ealist_2EALOOKUP_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27b)),X0_2E0))) ).

fof(arityeq2_2Ec_2Elist_2EMAP_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27c_2CA_27a_29_20mono_2Etyop_2Epair_2Eprod_28A_27c_2CA_27b_29,axiom,
    ! [A_27a,A_27b,A_27c,X0_2E0,X1_2E0] : s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27b)),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27a),tyop_2Epair_2Eprod(A_27c,A_27b)),X0_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27a)),X1_2E0))) = s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27a)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27b))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27a),tyop_2Epair_2Eprod(A_27c,A_27b)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27a)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27b)))),c_2Elist_2EMAP_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27a),tyop_2Epair_2Eprod(A_27c,A_27b)),X0_2E0))),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27a)),X1_2E0))) ).

fof(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2EA_27c_20mono_2EA_27a_20mono_2Etyop_2Epair_2Eprod_28A_27c_2CA_27b_29,axiom,
    ! [A_27a,A_27b,A_27c,X0_2E0] : s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27a),tyop_2Epair_2Eprod(A_27c,A_27b)),c_2Epair_2EUNCURRY_2E1(s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27c,A_27b))),X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27a),tyop_2Epair_2Eprod(A_27c,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27c,A_27b))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27a),tyop_2Epair_2Eprod(A_27c,A_27b))),c_2Epair_2EUNCURRY_2E0),s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27c,A_27b))),X0_2E0))) ).

fof(arityeq1_2Ec_2Ealist_2EALOOKUP_2E1_2Emono_2EA_27d_20mono_2EA_27c,axiom,
    ! [A_27c,A_27d,X0_2E0] : s(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(A_27d)),c_2Ealist_2EALOOKUP_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27d)),X0_2E0))) = s(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(A_27d)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27d)),tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(A_27d))),c_2Ealist_2EALOOKUP_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27d)),X0_2E0))) ).

fof(arityeq1_2Ec_2Ealist_2Efmap__to__alist_2E1_2Emono_2EA_27c_20mono_2EA_27d,axiom,
    ! [A_27c,A_27d,X0_2E0] : s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27d)),c_2Ealist_2Efmap__to__alist_2E1(s(tyop_2Efinite__map_2Efmap(A_27c,A_27d),X0_2E0))) = s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27d)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,A_27d),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27d))),c_2Ealist_2Efmap__to__alist_2E0),s(tyop_2Efinite__map_2Efmap(A_27c,A_27d),X0_2E0))) ).

fof(arityeq1_2Ec_2Epred__set_2ECARD_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(tyop_2Enum_2Enum,c_2Epred__set_2ECARD_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0))) = s(tyop_2Enum_2Enum,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Enum_2Enum),c_2Epred__set_2ECARD_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0))) ).

fof(arityeq1_2Ec_2Efinite__map_2EFDOM_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),X0_2E0))) = s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Efinite__map_2EFDOM_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),X0_2E0))) ).

fof(arityeq1_2Ec_2Ealist_2Ealist__to__fmap_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0))) = s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Efinite__map_2Efmap(A_27a,A_27b)),c_2Ealist_2Ealist__to__fmap_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0))) ).

fof(arityeq1_2Ec_2Efinite__map_2EFDOM_2E1_2Emono_2EA_27a_20mono_2EA_27c,axiom,
    ! [A_27a,A_27c,X0_2E0] : s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,A_27c),X0_2E0))) = s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27c),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Efinite__map_2EFDOM_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,A_27c),X0_2E0))) ).

fof(arityeq1_2Ec_2Efinite__map_2EFLOOKUP_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27b)),c_2Efinite__map_2EFLOOKUP_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),X0_2E0))) = s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27b))),c_2Efinite__map_2EFLOOKUP_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),X0_2E0))) ).

fof(arityeq1_2Ec_2Efinite__map_2EFLOOKUP_2E1_2Emono_2EA_27c_20mono_2EA_27d,axiom,
    ! [A_27c,A_27d,X0_2E0] : s(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(A_27d)),c_2Efinite__map_2EFLOOKUP_2E1(s(tyop_2Efinite__map_2Efmap(A_27c,A_27d),X0_2E0))) = s(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(A_27d)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,A_27d),tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(A_27d))),c_2Efinite__map_2EFLOOKUP_2E0),s(tyop_2Efinite__map_2Efmap(A_27c,A_27d),X0_2E0))) ).

fof(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2EA_27a_20mono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),X0_2E0))) = s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Epred__set_2EGSPEC_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),X0_2E0))) ).

fof(arityeq1_2Ec_2Elist_2ELENGTH_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(tyop_2Enum_2Enum,c_2Elist_2ELENGTH_2E1(s(tyop_2Elist_2Elist(A_27a),X0_2E0))) = s(tyop_2Enum_2Enum,app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Enum_2Enum),c_2Elist_2ELENGTH_2E0),s(tyop_2Elist_2Elist(A_27a),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(arityeq1_2Ec_2Elist_2ELENGTH_2E1_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Enum_2Enum,c_2Elist_2ELENGTH_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0))) = s(tyop_2Enum_2Enum,app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Enum_2Enum),c_2Elist_2ELENGTH_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0))) ).

fof(arityeq1_2Ec_2Elist_2ELIST__TO__SET_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Elist_2ELIST__TO__SET_2E1(s(tyop_2Elist_2Elist(A_27a),X0_2E0))) = s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Elist_2ELIST__TO__SET_2E0),s(tyop_2Elist_2Elist(A_27a),X0_2E0))) ).

fof(arityeq1_2Ec_2Elist_2ELIST__TO__SET_2E1_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),c_2Elist_2ELIST__TO__SET_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),c_2Elist_2ELIST__TO__SET_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0))) ).

fof(arityeq1_2Ec_2Eoption_2ESOME_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(tyop_2Eoption_2Eoption(A_27a),c_2Eoption_2ESOME_2E1(s(A_27a,X0_2E0))) = s(tyop_2Eoption_2Eoption(A_27a),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27a)),c_2Eoption_2ESOME_2E0),s(A_27a,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_2Epair_2ESND_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(A_27b,c_2Epair_2ESND_2E1(s(tyop_2Epair_2Eprod(A_27a,A_27b),X0_2E0))) = s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27b),c_2Epair_2ESND_2E0),s(tyop_2Epair_2Eprod(A_27a,A_27b),X0_2E0))) ).

fof(arityeq2_2Ec_2Elist_2EEL_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Epair_2Eprod(A_27a,A_27b),c_2Elist_2EEL_2E2(s(tyop_2Enum_2Enum,X0_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X1_2E0))) = s(tyop_2Epair_2Eprod(A_27a,A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Epair_2Eprod(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Epair_2Eprod(A_27a,A_27b))),c_2Elist_2EEL_2E0),s(tyop_2Enum_2Enum,X0_2E0))),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X1_2E0))) ).

fof(arityeq1_2Ec_2Ealist_2Ealist__range_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Ealist_2Ealist__range_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),X0_2E0))) = s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Ealist_2Ealist__range_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),X0_2E0))) ).

fof(arityeq1_2Ec_2Elist_2EREVERSE_2E1_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Elist_2EREVERSE_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0))) = s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))),c_2Elist_2EREVERSE_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0))) ).

fof(arityeq1_2Ec_2Ealist_2Ealist__to__fmap_2E1_2Emono_2EA_27b_20mono_2EA_27d,axiom,
    ! [A_27b,A_27d,X0_2E0] : s(tyop_2Efinite__map_2Efmap(A_27b,A_27d),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27d)),X0_2E0))) = s(tyop_2Efinite__map_2Efmap(A_27b,A_27d),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27d)),tyop_2Efinite__map_2Efmap(A_27b,A_27d)),c_2Ealist_2Ealist__to__fmap_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27d)),X0_2E0))) ).

fof(arityeq2_2Ec_2Elist_2EMAP_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27c_29_20mono_2Etyop_2Epair_2Eprod_28A_27b_2CA_27d_29,axiom,
    ! [A_27a,A_27b,A_27c,A_27d,X0_2E0,X1_2E0] : s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27d)),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Epair_2Eprod(A_27b,A_27d)),X0_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27c)),X1_2E0))) = s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27d)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27c)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27d))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Epair_2Eprod(A_27b,A_27d)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27c)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27d)))),c_2Elist_2EMAP_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Epair_2Eprod(A_27b,A_27d)),X0_2E0))),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27c)),X1_2E0))) ).

fof(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2EA_27a_20mono_2EA_27c_20mono_2Etyop_2Epair_2Eprod_28A_27b_2CA_27d_29,axiom,
    ! [A_27a,A_27b,A_27c,A_27d,X0_2E0] : s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Epair_2Eprod(A_27b,A_27d)),c_2Epair_2EUNCURRY_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Epair_2Eprod(A_27b,A_27d))),X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Epair_2Eprod(A_27b,A_27d)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Epair_2Eprod(A_27b,A_27d))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Epair_2Eprod(A_27b,A_27d))),c_2Epair_2EUNCURRY_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Epair_2Eprod(A_27b,A_27d))),X0_2E0))) ).

fof(arityeq1_2Ec_2Ealist_2Ealist__to__fmap_2E1_2Emono_2EA_27c_20mono_2EA_27b,axiom,
    ! [A_27b,A_27c,X0_2E0] : s(tyop_2Efinite__map_2Efmap(A_27c,A_27b),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27b)),X0_2E0))) = s(tyop_2Efinite__map_2Efmap(A_27c,A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27b)),tyop_2Efinite__map_2Efmap(A_27c,A_27b)),c_2Ealist_2Ealist__to__fmap_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27b)),X0_2E0))) ).

fof(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2EA_27c_20mono_2EA_27d,axiom,
    ! [A_27c,A_27d,X0_2E0,X1_2E0] : s(tyop_2Epair_2Eprod(A_27c,A_27d),c_2Epair_2E_2C_2E2(s(A_27c,X0_2E0),s(A_27d,X1_2E0))) = s(tyop_2Epair_2Eprod(A_27c,A_27d),app_2E2(s(tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(A_27c,A_27d)),app_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(A_27c,A_27d))),c_2Epair_2E_2C_2E0),s(A_27c,X0_2E0))),s(A_27d,X1_2E0))) ).

fof(arityeq2_2Ec_2Elist_2ECONS_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27c_2CA_27d_29,axiom,
    ! [A_27c,A_27d,X0_2E0,X1_2E0] : s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27d)),c_2Elist_2ECONS_2E2(s(tyop_2Epair_2Eprod(A_27c,A_27d),X0_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27d)),X1_2E0))) = s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27d)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27d)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27d))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27d),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27d)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27d)))),c_2Elist_2ECONS_2E0),s(tyop_2Epair_2Eprod(A_27c,A_27d),X0_2E0))),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27d)),X1_2E0))) ).

fof(arityeq1_2Ec_2Ealist_2Ealist__to__fmap_2E1_2Emono_2EA_27c_20mono_2EA_27d,axiom,
    ! [A_27c,A_27d,X0_2E0] : s(tyop_2Efinite__map_2Efmap(A_27c,A_27d),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27d)),X0_2E0))) = s(tyop_2Efinite__map_2Efmap(A_27c,A_27d),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27d)),tyop_2Efinite__map_2Efmap(A_27c,A_27d)),c_2Ealist_2Ealist__to__fmap_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27d)),X0_2E0))) ).

fof(arityeq1_2Ec_2Ealist_2Efmap__to__alist_2E1_2Emono_2EA_27c_20mono_2EA_27b,axiom,
    ! [A_27b,A_27c,X0_2E0] : s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27b)),c_2Ealist_2Efmap__to__alist_2E1(s(tyop_2Efinite__map_2Efmap(A_27c,A_27b),X0_2E0))) = s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,A_27b),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27b))),c_2Ealist_2Efmap__to__alist_2E0),s(tyop_2Efinite__map_2Efmap(A_27c,A_27b),X0_2E0))) ).

fof(arityeq2_2Ec_2Efinite__map_2Eo__f_2E2_2Emono_2EA_27c_20mono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,A_27c,X0_2E0,X1_2E0] : s(tyop_2Efinite__map_2Efmap(A_27c,A_27b),c_2Efinite__map_2Eo__f_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),X0_2E0),s(tyop_2Efinite__map_2Efmap(A_27c,A_27a),X1_2E0))) = s(tyop_2Efinite__map_2Efmap(A_27c,A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,A_27a),tyop_2Efinite__map_2Efmap(A_27c,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,A_27a),tyop_2Efinite__map_2Efmap(A_27c,A_27b))),c_2Efinite__map_2Eo__f_2E0),s(tyop_2Emin_2Efun(A_27a,A_27b),X0_2E0))),s(tyop_2Efinite__map_2Efmap(A_27c,A_27a),X1_2E0))) ).

fof(arityeq1_2Ec_2Erelation_2Etotal_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(tyop_2Emin_2Ebool,c_2Erelation_2Etotal_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_2Etotal_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),X0_2E0))) ).

fof(arityeq1_2Ec_2Erelation_2Etransitive_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(tyop_2Emin_2Ebool,c_2Erelation_2Etransitive_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_2Etransitive_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),X0_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_2Elist_2ECONS_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27b_2CA_27a_29,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),c_2Elist_2ECONS_2E2(s(tyop_2Epair_2Eprod(A_27b,A_27a),X0_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),X1_2E0))) = s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)))),c_2Elist_2ECONS_2E0),s(tyop_2Epair_2Eprod(A_27b,A_27a),X0_2E0))),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),X1_2E0))) ).

fof(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epair_2E_2C_2E2(s(A_27a,X0_2E0),s(A_27b,X1_2E0))) = s(tyop_2Epair_2Eprod(A_27a,A_27b),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,A_27b))),c_2Epair_2E_2C_2E0),s(A_27a,X0_2E0))),s(A_27b,X1_2E0))) ).

fof(arityeq2_2Ec_2Ealist_2EALOOKUP_2E2_2Emono_2EA_27b_20mono_2EA_27a,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Eoption_2Eoption(A_27b),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0),s(A_27a,X1_2E0))) = s(tyop_2Eoption_2Eoption(A_27b),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27b))),c_2Ealist_2EALOOKUP_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0))),s(A_27a,X1_2E0))) ).

fof(arityeq1_2Ec_2Eoption_2ETHE_2E1_2Emono_2EA_27b,axiom,
    ! [A_27b,X0_2E0] : s(A_27b,c_2Eoption_2ETHE_2E1(s(tyop_2Eoption_2Eoption(A_27b),X0_2E0))) = s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27b),A_27b),c_2Eoption_2ETHE_2E0),s(tyop_2Eoption_2Eoption(A_27b),X0_2E0))) ).

fof(arityeq2_2Ec_2Efinite__map_2EFAPPLY_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(A_27b,c_2Efinite__map_2EFAPPLY_2E2(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),X0_2E0),s(A_27a,X1_2E0))) = s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b)),c_2Efinite__map_2EFAPPLY_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),X0_2E0))),s(A_27a,X1_2E0))) ).

fof(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2EA_27a_20mono_2Etyop_2Emin_2Ebool,axiom,
    ! [A_27a,X0_2E0,X1_2E0] : s(tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool),c_2Epair_2E_2C_2E2(s(A_27a,X0_2E0),s(tyop_2Emin_2Ebool,X1_2E0))) = s(tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),c_2Epair_2E_2C_2E0),s(A_27a,X0_2E0))),s(tyop_2Emin_2Ebool,X1_2E0))) ).

fof(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2EA_27b_20mono_2EA_27d,axiom,
    ! [A_27b,A_27d,X0_2E0,X1_2E0] : s(tyop_2Epair_2Eprod(A_27b,A_27d),c_2Epair_2E_2C_2E2(s(A_27b,X0_2E0),s(A_27d,X1_2E0))) = s(tyop_2Epair_2Eprod(A_27b,A_27d),app_2E2(s(tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(A_27b,A_27d)),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(A_27b,A_27d))),c_2Epair_2E_2C_2E0),s(A_27b,X0_2E0))),s(A_27d,X1_2E0))) ).

fof(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2EA_27c_20mono_2EA_27b,axiom,
    ! [A_27b,A_27c,X0_2E0,X1_2E0] : s(tyop_2Epair_2Eprod(A_27c,A_27b),c_2Epair_2E_2C_2E2(s(A_27c,X0_2E0),s(A_27b,X1_2E0))) = s(tyop_2Epair_2Eprod(A_27c,A_27b),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27c,A_27b)),app_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27c,A_27b))),c_2Epair_2E_2C_2E0),s(A_27c,X0_2E0))),s(A_27b,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_2Epair_2EFST_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(A_27a,c_2Epair_2EFST_2E1(s(tyop_2Epair_2Eprod(A_27a,A_27b),X0_2E0))) = s(A_27a,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST_2E0),s(tyop_2Epair_2Eprod(A_27a,A_27b),X0_2E0))) ).

fof(arityeq2_2Ec_2Erelation_2Einv__image_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29_20mono_2EA_27a,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),c_2Erelation_2Einv__image_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),X0_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a),X1_2E0))) = s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),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_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)))),c_2Erelation_2Einv__image_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),X0_2E0))),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a),X1_2E0))) ).

fof(arityeq2_2Ec_2Elist_2EFILTER_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Elist_2EFILTER_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),X0_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X1_2E0))) = s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)))),c_2Elist_2EFILTER_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),X0_2E0))),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X1_2E0))) ).

fof(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2EA_27a_20mono_2EA_27b_20mono_2Etyop_2Emin_2Ebool,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),c_2Epair_2EUNCURRY_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),X0_2E0))) = s(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_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),c_2Epair_2EUNCURRY_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),X0_2E0))) ).

fof(arityeq2_2Ec_2Elist_2EMAP_2E2_2Emono_2EA_27a_20mono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0),s(tyop_2Elist_2Elist(A_27a),X1_2E0))) = s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)))),c_2Elist_2EMAP_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0))),s(tyop_2Elist_2Elist(A_27a),X1_2E0))) ).

fof(arityeq2_2Ec_2Ealist_2EALOOKUP_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Eoption_2Eoption(A_27a),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),X0_2E0),s(A_27b,X1_2E0))) = s(tyop_2Eoption_2Eoption(A_27a),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(A_27a)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(A_27a))),c_2Ealist_2EALOOKUP_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),X0_2E0))),s(A_27b,X1_2E0))) ).

fof(arityeq2_2Ec_2Ealist_2EALOOKUP_2E2_2Emono_2EA_27c_20mono_2EA_27b,axiom,
    ! [A_27b,A_27c,X0_2E0,X1_2E0] : s(tyop_2Eoption_2Eoption(A_27c),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27c)),X0_2E0),s(A_27b,X1_2E0))) = s(tyop_2Eoption_2Eoption(A_27c),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(A_27c)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27c)),tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(A_27c))),c_2Ealist_2EALOOKUP_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27c)),X0_2E0))),s(A_27b,X1_2E0))) ).

fof(arityeq2_2Ec_2Elist_2EFILTER_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27b_2CA_27c_29,axiom,
    ! [A_27b,A_27c,X0_2E0,X1_2E0] : s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27c)),c_2Elist_2EFILTER_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27c),tyop_2Emin_2Ebool),X0_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27c)),X1_2E0))) = s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27c)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27c)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27c))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27c),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27c)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27c)))),c_2Elist_2EFILTER_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27c),tyop_2Emin_2Ebool),X0_2E0))),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27c)),X1_2E0))) ).

fof(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2EA_27b_20mono_2EA_27c_20mono_2Etyop_2Emin_2Ebool,axiom,
    ! [A_27b,A_27c,X0_2E0] : s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27c),tyop_2Emin_2Ebool),c_2Epair_2EUNCURRY_2E1(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool)),X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(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_2Ebool)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27c),tyop_2Emin_2Ebool)),c_2Epair_2EUNCURRY_2E0),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool)),X0_2E0))) ).

fof(arityeq2_2Ec_2Elist_2EEL_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0,X1_2E0] : s(A_27a,c_2Elist_2EEL_2E2(s(tyop_2Enum_2Enum,X0_2E0),s(tyop_2Elist_2Elist(A_27a),X1_2E0))) = s(A_27a,app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),A_27a)),c_2Elist_2EEL_2E0),s(tyop_2Enum_2Enum,X0_2E0))),s(tyop_2Elist_2Elist(A_27a),X1_2E0))) ).

fof(arityeq2_2Ec_2Efinite__map_2EFLOOKUP_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Eoption_2Eoption(A_27b),c_2Efinite__map_2EFLOOKUP_2E2(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),X0_2E0),s(A_27a,X1_2E0))) = s(tyop_2Eoption_2Eoption(A_27b),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27b))),c_2Efinite__map_2EFLOOKUP_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),X0_2E0))),s(A_27a,X1_2E0))) ).

fof(arityeq2_2Ec_2Efinite__map_2EFUPDATE__LIST_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Efinite__map_2EFUPDATE__LIST_2E2(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),X0_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X1_2E0))) = s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Efinite__map_2Efmap(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Efinite__map_2Efmap(A_27a,A_27b))),c_2Efinite__map_2EFUPDATE__LIST_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),X0_2E0))),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X1_2E0))) ).

fof(arityeq2_2Ec_2Efinite__map_2EFLOOKUP_2E2_2Emono_2EA_27b_20mono_2EA_27a,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Eoption_2Eoption(A_27a),c_2Efinite__map_2EFLOOKUP_2E2(s(tyop_2Efinite__map_2Efmap(A_27b,A_27a),X0_2E0),s(A_27b,X1_2E0))) = s(tyop_2Eoption_2Eoption(A_27a),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(A_27a)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,A_27a),tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(A_27a))),c_2Efinite__map_2EFLOOKUP_2E0),s(tyop_2Efinite__map_2Efmap(A_27b,A_27a),X0_2E0))),s(A_27b,X1_2E0))) ).

fof(arityeq2_2Ec_2Efinite__map_2EFUPDATE__LIST_2E2_2Emono_2EA_27b_20mono_2EA_27a,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Efinite__map_2Efmap(A_27b,A_27a),c_2Efinite__map_2EFUPDATE__LIST_2E2(s(tyop_2Efinite__map_2Efmap(A_27b,A_27a),X0_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),X1_2E0))) = s(tyop_2Efinite__map_2Efmap(A_27b,A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Efinite__map_2Efmap(A_27b,A_27a)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Efinite__map_2Efmap(A_27b,A_27a))),c_2Efinite__map_2EFUPDATE__LIST_2E0),s(tyop_2Efinite__map_2Efmap(A_27b,A_27a),X0_2E0))),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),X1_2E0))) ).

fof(arityeq1_2Ec_2Elist_2EREVERSE_2E1_2Emono_2Etyop_2Epair_2Eprod_28A_27b_2CA_27a_29,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),c_2Elist_2EREVERSE_2E1(s(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_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a))),c_2Elist_2EREVERSE_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),X0_2E0))) ).

fof(arityeq2_2Ec_2Efinite__map_2EFUNION_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Efinite__map_2EFUNION_2E2(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),X0_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),X1_2E0))) = s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Efinite__map_2Efmap(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Efinite__map_2Efmap(A_27a,A_27b))),c_2Efinite__map_2EFUNION_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),X0_2E0))),s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),X1_2E0))) ).

fof(arityeq2_2Ec_2Efinite__map_2EFUPDATE_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Efinite__map_2EFUPDATE_2E2(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),X0_2E0),s(tyop_2Epair_2Eprod(A_27a,A_27b),X1_2E0))) = s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Efinite__map_2Efmap(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Efinite__map_2Efmap(A_27a,A_27b))),c_2Efinite__map_2EFUPDATE_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),X0_2E0))),s(tyop_2Epair_2Eprod(A_27a,A_27b),X1_2E0))) ).

fof(arityeq2_2Ec_2Efinite__map_2EFUPDATE_2E2_2Emono_2EA_27c_20mono_2EA_27d,axiom,
    ! [A_27c,A_27d,X0_2E0,X1_2E0] : s(tyop_2Efinite__map_2Efmap(A_27c,A_27d),c_2Efinite__map_2EFUPDATE_2E2(s(tyop_2Efinite__map_2Efmap(A_27c,A_27d),X0_2E0),s(tyop_2Epair_2Eprod(A_27c,A_27d),X1_2E0))) = s(tyop_2Efinite__map_2Efmap(A_27c,A_27d),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27d),tyop_2Efinite__map_2Efmap(A_27c,A_27d)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,A_27d),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27d),tyop_2Efinite__map_2Efmap(A_27c,A_27d))),c_2Efinite__map_2EFUPDATE_2E0),s(tyop_2Efinite__map_2Efmap(A_27c,A_27d),X0_2E0))),s(tyop_2Epair_2Eprod(A_27c,A_27d),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_2Elist_2EMAP_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27b_2CA_27a_29_20mono_2EA_27a,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Elist_2Elist(A_27a),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),A_27a),X0_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),X1_2E0))) = s(tyop_2Elist_2Elist(A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Elist_2Elist(A_27a)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Elist_2Elist(A_27a))),c_2Elist_2EMAP_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),A_27a),X0_2E0))),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),X1_2E0))) ).

fof(arityeq1_2Ec_2Efinite__map_2EFRANGE_2E1_2Emono_2EA_27b_20mono_2EA_27a,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Efinite__map_2EFRANGE_2E1(s(tyop_2Efinite__map_2Efmap(A_27b,A_27a),X0_2E0))) = s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Efinite__map_2EFRANGE_2E0),s(tyop_2Efinite__map_2Efmap(A_27b,A_27a),X0_2E0))) ).

fof(arityeq1_2Ec_2Ealist_2Ealist__to__fmap_2E1_2Emono_2EA_27b_20mono_2EA_27a,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Efinite__map_2Efmap(A_27b,A_27a),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),X0_2E0))) = s(tyop_2Efinite__map_2Efmap(A_27b,A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Efinite__map_2Efmap(A_27b,A_27a)),c_2Ealist_2Ealist__to__fmap_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),X0_2E0))) ).

fof(arityeq1_2Ec_2Efinite__map_2EFRANGE_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Efinite__map_2EFRANGE_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),X0_2E0))) = s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),c_2Efinite__map_2EFRANGE_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),X0_2E0))) ).

fof(arityeq2_2Ec_2Ebool_2EIN_2E2_2Emono_2EA_27b,axiom,
    ! [A_27b,X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(A_27b,X0_2E0),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),c_2Ebool_2EIN_2E0),s(A_27b,X0_2E0))),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),X1_2E0))) ).

fof(arityeq2_2Ec_2Ebool_2EIN_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Epair_2Eprod(A_27a,A_27b),X0_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),c_2Ebool_2EIN_2E0),s(tyop_2Epair_2Eprod(A_27a,A_27b),X0_2E0))),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),X1_2E0))) ).

fof(arityeq2_2Ec_2Ebool_2EIN_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27b_2CA_27a_29,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Epair_2Eprod(A_27b,A_27a),X0_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),tyop_2Emin_2Ebool),X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),c_2Ebool_2EIN_2E0),s(tyop_2Epair_2Eprod(A_27b,A_27a),X0_2E0))),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),tyop_2Emin_2Ebool),X1_2E0))) ).

fof(arityeq1_2Ec_2Elist_2ELIST__TO__SET_2E1_2Emono_2Etyop_2Epair_2Eprod_28A_27b_2CA_27a_29,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),tyop_2Emin_2Ebool),c_2Elist_2ELIST__TO__SET_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),tyop_2Emin_2Ebool)),c_2Elist_2ELIST__TO__SET_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),X0_2E0))) ).

fof(arityeq1_2Ec_2Elist_2ESET__TO__LIST_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(tyop_2Elist_2Elist(A_27a),c_2Elist_2ESET__TO__LIST_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0))) = s(tyop_2Elist_2Elist(A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Elist_2Elist(A_27a)),c_2Elist_2ESET__TO__LIST_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0))) ).

fof(arityeq2_2Ec_2Elist_2EMAP_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27c_29_20mono_2EA_27a,axiom,
    ! [A_27a,A_27c,X0_2E0,X1_2E0] : s(tyop_2Elist_2Elist(A_27a),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),A_27a),X0_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27c)),X1_2E0))) = s(tyop_2Elist_2Elist(A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27c)),tyop_2Elist_2Elist(A_27a)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27c)),tyop_2Elist_2Elist(A_27a))),c_2Elist_2EMAP_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),A_27a),X0_2E0))),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27c)),X1_2E0))) ).

fof(arityeq1_2Ec_2Ealist_2Efmap__to__alist_2E1_2Emono_2EA_27a_20mono_2EA_27c,axiom,
    ! [A_27a,A_27c,X0_2E0] : s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27c)),c_2Ealist_2Efmap__to__alist_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,A_27c),X0_2E0))) = s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27c)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27c),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27c))),c_2Ealist_2Efmap__to__alist_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,A_27c),X0_2E0))) ).

fof(arityeq1_2Ec_2Ealist_2Efmap__to__alist_2E1_2Emono_2EA_27c_20mono_2EA_27a,axiom,
    ! [A_27a,A_27c,X0_2E0] : s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27a)),c_2Ealist_2Efmap__to__alist_2E1(s(tyop_2Efinite__map_2Efmap(A_27c,A_27a),X0_2E0))) = s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27a)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,A_27a),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27a))),c_2Ealist_2Efmap__to__alist_2E0),s(tyop_2Efinite__map_2Efmap(A_27c,A_27a),X0_2E0))) ).

fof(arityeq2_2Ec_2Efinite__map_2EMAP__KEYS_2E2_2Emono_2EA_27a_20mono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Efinite__map_2EMAP__KEYS_2E2(s(tyop_2Emin_2Efun(A_27a,A_27a),X0_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),X1_2E0))) = s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Efinite__map_2Efmap(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Efinite__map_2Efmap(A_27a,A_27b))),c_2Efinite__map_2EMAP__KEYS_2E0),s(tyop_2Emin_2Efun(A_27a,A_27a),X0_2E0))),s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),X1_2E0))) ).

fof(arityeq2_2Ec_2Efinite__map_2EMAP__KEYS_2E2_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27d,axiom,
    ! [A_27a,A_27b,A_27d,X0_2E0,X1_2E0] : s(tyop_2Efinite__map_2Efmap(A_27b,A_27d),c_2Efinite__map_2EMAP__KEYS_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),X0_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,A_27d),X1_2E0))) = s(tyop_2Efinite__map_2Efmap(A_27b,A_27d),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27d),tyop_2Efinite__map_2Efmap(A_27b,A_27d)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27d),tyop_2Efinite__map_2Efmap(A_27b,A_27d))),c_2Efinite__map_2EMAP__KEYS_2E0),s(tyop_2Emin_2Efun(A_27a,A_27b),X0_2E0))),s(tyop_2Efinite__map_2Efmap(A_27a,A_27d),X1_2E0))) ).

fof(arityeq1_2Ec_2Ealist_2Ealist__to__fmap_2E1_2Emono_2EA_27a_20mono_2EA_27c,axiom,
    ! [A_27a,A_27c,X0_2E0] : s(tyop_2Efinite__map_2Efmap(A_27a,A_27c),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27c)),X0_2E0))) = s(tyop_2Efinite__map_2Efmap(A_27a,A_27c),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27c)),tyop_2Efinite__map_2Efmap(A_27a,A_27c)),c_2Ealist_2Ealist__to__fmap_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27c)),X0_2E0))) ).

fof(arityeq2_2Ec_2Efinite__map_2Eo__f_2E2_2Emono_2EA_27a_20mono_2EA_27c_20mono_2EA_27d,axiom,
    ! [A_27a,A_27c,A_27d,X0_2E0,X1_2E0] : s(tyop_2Efinite__map_2Efmap(A_27a,A_27d),c_2Efinite__map_2Eo__f_2E2(s(tyop_2Emin_2Efun(A_27c,A_27d),X0_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,A_27c),X1_2E0))) = s(tyop_2Efinite__map_2Efmap(A_27a,A_27d),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27c),tyop_2Efinite__map_2Efmap(A_27a,A_27d)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27d),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27c),tyop_2Efinite__map_2Efmap(A_27a,A_27d))),c_2Efinite__map_2Eo__f_2E0),s(tyop_2Emin_2Efun(A_27c,A_27d),X0_2E0))),s(tyop_2Efinite__map_2Efmap(A_27a,A_27c),X1_2E0))) ).

fof(arityeq2_2Ec_2Esorting_2EPERM_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Esorting_2EPERM_2E2(s(tyop_2Elist_2Elist(A_27a),X0_2E0),s(tyop_2Elist_2Elist(A_27a),X1_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(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool)),c_2Esorting_2EPERM_2E0),s(tyop_2Elist_2Elist(A_27a),X0_2E0))),s(tyop_2Elist_2Elist(A_27a),X1_2E0))) ).

fof(arityeq2_2Ec_2Esorting_2EPERM_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Esorting_2EPERM_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Ebool)),c_2Esorting_2EPERM_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0))),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X1_2E0))) ).

fof(arityeq2_2Ec_2Esorting_2ESTABLE_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Esorting_2ESTABLE_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)))),X0_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool)),c_2Esorting_2ESTABLE_2E0),s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)))),X0_2E0))),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),X1_2E0))) ).

fof(arityeq2_2Ec_2Epred__set_2EIMAGE_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27b_2CA_27a_29_20mono_2EA_27a,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Epred__set_2EIMAGE_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),A_27a),X0_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,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(tyop_2Epair_2Eprod(A_27b,A_27a),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),c_2Epred__set_2EIMAGE_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),A_27a),X0_2E0))),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),tyop_2Emin_2Ebool),X1_2E0))) ).

fof(arityeq2_2Ec_2Epred__set_2ESUBSET_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Epred__set_2ESUBSET_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),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(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),c_2Epred__set_2ESUBSET_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0))),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0))) ).

fof(arityeq1_2Ec_2Ealist_2EALOOKUP_2E1_2Emono_2EA_27a_20mono_2EA_27c,axiom,
    ! [A_27a,A_27c,X0_2E0] : s(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(A_27a)),c_2Ealist_2EALOOKUP_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27a)),X0_2E0))) = s(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(A_27a)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27a)),tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(A_27a))),c_2Ealist_2EALOOKUP_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27a)),X0_2E0))) ).

fof(arityeq1_2Ec_2Eoption_2EOPTION__MAP_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Eoption_2Eoption(A_27b)),c_2Eoption_2EOPTION__MAP_2E1(s(tyop_2Emin_2Efun(A_27a,A_27b),X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Eoption_2Eoption(A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Eoption_2Eoption(A_27b))),c_2Eoption_2EOPTION__MAP_2E0),s(tyop_2Emin_2Efun(A_27a,A_27b),X0_2E0))) ).

fof(arityeq2_2Ec_2Ecombin_2Eo_2E2_2Emono_2EA_27c_20mono_2Etyop_2Eoption_2Eoption_28A_27b_29_20mono_2Etyop_2Eoption_2Eoption_28A_27a_29,axiom,
    ! [A_27a,A_27b,A_27c,X0_2E0,X1_2E0] : s(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(A_27b)),c_2Ecombin_2Eo_2E2(s(tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Eoption_2Eoption(A_27b)),X0_2E0),s(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(A_27a)),X1_2E0))) = s(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(A_27b))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Eoption_2Eoption(A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(A_27b)))),c_2Ecombin_2Eo_2E0),s(tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Eoption_2Eoption(A_27b)),X0_2E0))),s(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(A_27a)),X1_2E0))) ).

fof(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Elist_2Elist_28A_27a_29_20mono_2Etyop_2Elist_2Elist_28A_27b_29,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Epair_2Eprod(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27b)),c_2Epair_2E_2C_2E2(s(tyop_2Elist_2Elist(A_27a),X0_2E0),s(tyop_2Elist_2Elist(A_27b),X1_2E0))) = s(tyop_2Epair_2Eprod(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Epair_2Eprod(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27b))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Epair_2Eprod(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27b)))),c_2Epair_2E_2C_2E0),s(tyop_2Elist_2Elist(A_27a),X0_2E0))),s(tyop_2Elist_2Elist(A_27b),X1_2E0))) ).

fof(arityeq1_2Ec_2Eoption_2EOPTION__MAP_2E1_2Emono_2EA_27b_20mono_2EA_27d,axiom,
    ! [A_27b,A_27d,X0_2E0] : s(tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27b),tyop_2Eoption_2Eoption(A_27d)),c_2Eoption_2EOPTION__MAP_2E1(s(tyop_2Emin_2Efun(A_27b,A_27d),X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27b),tyop_2Eoption_2Eoption(A_27d)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27d),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27b),tyop_2Eoption_2Eoption(A_27d))),c_2Eoption_2EOPTION__MAP_2E0),s(tyop_2Emin_2Efun(A_27b,A_27d),X0_2E0))) ).

fof(arityeq1_2Ec_2Elist_2EZIP_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Elist_2EZIP_2E1(s(tyop_2Epair_2Eprod(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27b)),X0_2E0))) = s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))),c_2Elist_2EZIP_2E0),s(tyop_2Epair_2Eprod(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27b)),X0_2E0))) ).

fof(arityeq2_2Ec_2Ecombin_2Eo_2E2_2Emono_2EA_27a_20mono_2Etyop_2Eoption_2Eoption_28A_27d_29_20mono_2Etyop_2Eoption_2Eoption_28A_27b_29,axiom,
    ! [A_27a,A_27b,A_27d,X0_2E0,X1_2E0] : s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27d)),c_2Ecombin_2Eo_2E2(s(tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27b),tyop_2Eoption_2Eoption(A_27d)),X0_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27b)),X1_2E0))) = s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27d)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27b)),tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27d))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27b),tyop_2Eoption_2Eoption(A_27d)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27b)),tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27d)))),c_2Ecombin_2Eo_2E0),s(tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27b),tyop_2Eoption_2Eoption(A_27d)),X0_2E0))),s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27b)),X1_2E0))) ).

fof(arityeq1_2Ec_2Ealist_2Ealist__to__fmap_2E1_2Emono_2EA_27c_20mono_2EA_27a,axiom,
    ! [A_27a,A_27c,X0_2E0] : s(tyop_2Efinite__map_2Efmap(A_27c,A_27a),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27a)),X0_2E0))) = s(tyop_2Efinite__map_2Efmap(A_27c,A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27a)),tyop_2Efinite__map_2Efmap(A_27c,A_27a)),c_2Ealist_2Ealist__to__fmap_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27a)),X0_2E0))) ).

fof(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Eoption_2Eoption_28A_27a_29,axiom,
    ! [A_27a,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Eoption_2Eoption(A_27a),c_2Ebool_2ECOND_2E3(s(tyop_2Emin_2Ebool,X0_2E0),s(tyop_2Eoption_2Eoption(A_27a),X1_2E0),s(tyop_2Eoption_2Eoption(A_27a),X2_2E0))) = s(tyop_2Eoption_2Eoption(A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Eoption_2Eoption(A_27a)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Eoption_2Eoption(A_27a))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Eoption_2Eoption(A_27a)))),c_2Ebool_2ECOND_2E0),s(tyop_2Emin_2Ebool,X0_2E0))),s(tyop_2Eoption_2Eoption(A_27a),X1_2E0))),s(tyop_2Eoption_2Eoption(A_27a),X2_2E0))) ).

fof(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Eoption_2Eoption_28A_27b_29,axiom,
    ! [A_27b,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Eoption_2Eoption(A_27b),c_2Ebool_2ECOND_2E3(s(tyop_2Emin_2Ebool,X0_2E0),s(tyop_2Eoption_2Eoption(A_27b),X1_2E0),s(tyop_2Eoption_2Eoption(A_27b),X2_2E0))) = s(tyop_2Eoption_2Eoption(A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27b),tyop_2Eoption_2Eoption(A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27b),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27b),tyop_2Eoption_2Eoption(A_27b))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27b),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27b),tyop_2Eoption_2Eoption(A_27b)))),c_2Ebool_2ECOND_2E0),s(tyop_2Emin_2Ebool,X0_2E0))),s(tyop_2Eoption_2Eoption(A_27b),X1_2E0))),s(tyop_2Eoption_2Eoption(A_27b),X2_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(arityeq1_2Ec_2Ewhile_2ELEAST_2E1,axiom,
    ! [X0_2E0] : s(tyop_2Enum_2Enum,c_2Ewhile_2ELEAST_2E1(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X0_2E0))) = s(tyop_2Enum_2Enum,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Enum_2Enum),c_2Ewhile_2ELEAST_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X0_2E0))) ).

fof(arityeq2_2Ec_2Elist_2EMAP_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Elist_2Elist(A_27b),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27b),X0_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X1_2E0))) = s(tyop_2Elist_2Elist(A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27b),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(A_27b))),c_2Elist_2EMAP_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27b),X0_2E0))),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X1_2E0))) ).

fof(arityeq3_2Ec_2Elist_2EFOLDR_2E3_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29_20mono_2Etyop_2Efinite__map_2Efmap_28A_27a_2CA_27b_29,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Elist_2EFOLDR_2E3(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Efinite__map_2Efmap(A_27a,A_27b))),X0_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),X1_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X2_2E0))) = s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Efinite__map_2Efmap(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Efinite__map_2Efmap(A_27a,A_27b))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Efinite__map_2Efmap(A_27a,A_27b))),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Efinite__map_2Efmap(A_27a,A_27b)))),c_2Elist_2EFOLDR_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Efinite__map_2Efmap(A_27a,A_27b))),X0_2E0))),s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),X1_2E0))),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),X2_2E0))) ).

fof(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2EA_27a_20mono_2EA_27b_20mono_2Etyop_2Emin_2Efun_28tyop_2Efinite__map_2Efmap_28A_27a_2CA_27b_29_2Ctyop_2Efinite__map_2Efmap_28A_27a_2CA_27b_29_29,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Efinite__map_2Efmap(A_27a,A_27b))),c_2Epair_2EUNCURRY_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Efinite__map_2Efmap(A_27a,A_27b)))),X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Efinite__map_2Efmap(A_27a,A_27b))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Efinite__map_2Efmap(A_27a,A_27b)))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Efinite__map_2Efmap(A_27a,A_27b)))),c_2Epair_2EUNCURRY_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Efinite__map_2Efmap(A_27a,A_27b)))),X0_2E0))) ).

fof(arityeq3_2Ec_2Epred__set_2EINJ_2E3_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Emin_2Ebool,c_2Epred__set_2EINJ_2E3(s(tyop_2Emin_2Efun(A_27a,A_27b),X0_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),X2_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))),c_2Epred__set_2EINJ_2E0),s(tyop_2Emin_2Efun(A_27a,A_27b),X0_2E0))),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0))),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),X2_2E0))) ).

fof(arityeq3_2Ec_2Eoption_2Eoption__CASE_2E3_2Emono_2EA_27b_20mono_2Etyop_2Eoption_2Eoption_28A_27b_29,axiom,
    ! [A_27b,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Eoption_2Eoption(A_27b),c_2Eoption_2Eoption__CASE_2E3(s(tyop_2Eoption_2Eoption(A_27b),X0_2E0),s(tyop_2Eoption_2Eoption(A_27b),X1_2E0),s(tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(A_27b)),X2_2E0))) = s(tyop_2Eoption_2Eoption(A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(A_27b)),tyop_2Eoption_2Eoption(A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(A_27b)),tyop_2Eoption_2Eoption(A_27b))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27b),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(A_27b)),tyop_2Eoption_2Eoption(A_27b)))),c_2Eoption_2Eoption__CASE_2E0),s(tyop_2Eoption_2Eoption(A_27b),X0_2E0))),s(tyop_2Eoption_2Eoption(A_27b),X1_2E0))),s(tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(A_27b)),X2_2E0))) ).

fof(thm_2Ealist_2Efmap__to__alist__def,axiom,
    ! [A_27a,A_27b,F0_2E0] :
      ( ! [V0s_2E0,V1k_2E0] : s(tyop_2Epair_2Eprod(A_27a,A_27b),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,A_27b))),F0_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V0s_2E0))),s(A_27a,V1k_2E0))) = s(tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epair_2E_2C_2E2(s(A_27a,V1k_2E0),s(A_27b,c_2Efinite__map_2EFAPPLY_2E2(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V0s_2E0),s(A_27a,V1k_2E0)))))
     => ! [V0s_2E0] : s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Ealist_2Efmap__to__alist_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V0s_2E0))) = s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,A_27b))),F0_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V0s_2E0))),s(tyop_2Elist_2Elist(A_27a),c_2Elist_2ESET__TO__LIST_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V0s_2E0))))))) ) ).

fof(thm_2Ealist_2Ealist__to__fmap__def,axiom,
    ! [A_27a,A_27b,F0_2E0] :
      ( ! [V1k_2E0,V2v_2E0,V3f_2E0] : s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Efinite__map_2Efmap(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Efinite__map_2Efmap(A_27a,A_27b))),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Efinite__map_2Efmap(A_27a,A_27b)))),F0_2E0),s(A_27a,V1k_2E0))),s(A_27b,V2v_2E0))),s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V3f_2E0))) = s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Efinite__map_2EFUPDATE_2E2(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V3f_2E0),s(tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epair_2E_2C_2E2(s(A_27a,V1k_2E0),s(A_27b,V2v_2E0)))))
     => ! [V0s_2E0] : s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0s_2E0))) = s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Elist_2EFOLDR_2E3(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Efinite__map_2Efmap(A_27a,A_27b))),c_2Epair_2EUNCURRY_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Efinite__map_2Efmap(A_27a,A_27b)))),F0_2E0))),s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Efinite__map_2EFEMPTY_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0s_2E0))) ) ).

fof(thm_2Ealist_2Ealist__range__def,axiom,
    ! [A_27a,A_27b,F1_2E0,F0_2E0] :
      ( ! [V0m_2E0,V1v_2E0] :
          ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F1_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),V0m_2E0))),s(A_27a,V1v_2E0))))
        <=> ? [V2k_2E0] : s(tyop_2Eoption_2Eoption(A_27a),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),V0m_2E0),s(A_27b,V2k_2E0))) = s(tyop_2Eoption_2Eoption(A_27a),c_2Eoption_2ESOME_2E1(s(A_27a,V1v_2E0))) )
     => ( ! [V0m_2E0,V1v_2E0] : s(tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F0_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),V0m_2E0))),s(A_27a,V1v_2E0))) = s(tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool),c_2Epair_2E_2C_2E2(s(A_27a,V1v_2E0),s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F1_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),V0m_2E0))),s(A_27a,V1v_2E0)))))
       => ! [V0m_2E0] : s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Ealist_2Ealist__range_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),V0m_2E0))) = s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool))),F0_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),V0m_2E0))))) ) ) ).

fof(thm_2Ealist_2Efmap__to__alist__FEMPTY,axiom,
    ! [A_27a,A_27b] : s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Ealist_2Efmap__to__alist_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Efinite__map_2EFEMPTY_2E0))) = s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Elist_2ENIL_2E0) ).

fof(thm_2Ealist_2Ealist__to__fmap__thm,axiom,
    ! [A_27a,A_27b,A_27c,A_27d,V0v_2E0,V1t_2E0,V2k_2E0] :
      ( s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Elist_2ENIL_2E0))) = s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Efinite__map_2EFEMPTY_2E0)
      & s(tyop_2Efinite__map_2Efmap(A_27c,A_27d),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27d)),c_2Elist_2ECONS_2E2(s(tyop_2Epair_2Eprod(A_27c,A_27d),c_2Epair_2E_2C_2E2(s(A_27c,V2k_2E0),s(A_27d,V0v_2E0))),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27d)),V1t_2E0))))) = s(tyop_2Efinite__map_2Efmap(A_27c,A_27d),c_2Efinite__map_2EFUPDATE_2E2(s(tyop_2Efinite__map_2Efmap(A_27c,A_27d),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27d)),V1t_2E0))),s(tyop_2Epair_2Eprod(A_27c,A_27d),c_2Epair_2E_2C_2E2(s(A_27c,V2k_2E0),s(A_27d,V0v_2E0))))) ) ).

fof(thm_2Ealist_2EALOOKUP__ind,axiom,
    ! [A_27a,A_27b,V0P_2E0] :
      ( ( ! [V1q_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V0P_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),c_2Elist_2ENIL_2E0))),s(A_27b,V1q_2E0))))
        & ! [V2x_2E0,V3y_2E0,V4t_2E0,V5q_2E0] :
            ( ( s(A_27b,V2x_2E0) != s(A_27b,V5q_2E0)
             => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V0P_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),V4t_2E0))),s(A_27b,V5q_2E0)))) )
           => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V0P_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),c_2Elist_2ECONS_2E2(s(tyop_2Epair_2Eprod(A_27b,A_27a),c_2Epair_2E_2C_2E2(s(A_27b,V2x_2E0),s(A_27a,V3y_2E0))),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),V4t_2E0))))),s(A_27b,V5q_2E0)))) ) )
     => ! [V6v_2E0,V7v1_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V0P_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),V6v_2E0))),s(A_27b,V7v1_2E0)))) ) ).

fof(thm_2Ealist_2EALOOKUP__def,axiom,
    ! [A_27a,A_27b,F0_2E0] :
      ( ! [V2x_2E0,V4q_2E0] :
          ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),F0_2E0),s(A_27b,V2x_2E0))),s(A_27b,V4q_2E0))))
        <=> s(A_27b,V2x_2E0) = s(A_27b,V4q_2E0) )
     => ( ! [V0q_2E0] : s(tyop_2Eoption_2Eoption(A_27a),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),c_2Elist_2ENIL_2E0),s(A_27b,V0q_2E0))) = s(tyop_2Eoption_2Eoption(A_27a),c_2Eoption_2ENONE_2E0)
        & ! [V1y_2E0,V2x_2E0,V3t_2E0,V4q_2E0] : s(tyop_2Eoption_2Eoption(A_27a),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),c_2Elist_2ECONS_2E2(s(tyop_2Epair_2Eprod(A_27b,A_27a),c_2Epair_2E_2C_2E2(s(A_27b,V2x_2E0),s(A_27a,V1y_2E0))),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),V3t_2E0))),s(A_27b,V4q_2E0))) = s(tyop_2Eoption_2Eoption(A_27a),c_2Ebool_2ECOND_2E3(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),F0_2E0),s(A_27b,V2x_2E0))),s(A_27b,V4q_2E0))),s(tyop_2Eoption_2Eoption(A_27a),c_2Eoption_2ESOME_2E1(s(A_27a,V1y_2E0))),s(tyop_2Eoption_2Eoption(A_27a),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),V3t_2E0),s(A_27b,V4q_2E0))))) ) ) ).

fof(thm_2Ealist_2EALOOKUP__FAILS,axiom,
    ! [A_27a,A_27b,V0x_2E0,V1l_2E0] :
      ( s(tyop_2Eoption_2Eoption(A_27a),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),V1l_2E0),s(A_27b,V0x_2E0))) = s(tyop_2Eoption_2Eoption(A_27a),c_2Eoption_2ENONE_2E0)
    <=> ! [V2k_2E0,V3v_2E0] :
          ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Epair_2Eprod(A_27b,A_27a),c_2Epair_2E_2C_2E2(s(A_27b,V2k_2E0),s(A_27a,V3v_2E0))),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),tyop_2Emin_2Ebool),c_2Elist_2ELIST__TO__SET_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),V1l_2E0))))))
         => s(A_27b,V2k_2E0) != s(A_27b,V0x_2E0) ) ) ).

fof(thm_2Ealist_2EALOOKUP__NONE,axiom,
    ! [A_27a,A_27b,V0l_2E0,V1x_2E0] :
      ( s(tyop_2Eoption_2Eoption(A_27b),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0l_2E0),s(A_27a,V1x_2E0))) = s(tyop_2Eoption_2Eoption(A_27b),c_2Eoption_2ENONE_2E0)
    <=> ~ p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(A_27a,V1x_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Elist_2ELIST__TO__SET_2E1(s(tyop_2Elist_2Elist(A_27a),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0l_2E0)))))))) ) ).

fof(thm_2Ealist_2EALOOKUP__TABULATE,axiom,
    ! [A_27a,A_27b,F0_2E0] :
      ( ! [V2f_2E0,V3k_2E0] : s(tyop_2Epair_2Eprod(A_27a,A_27b),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,A_27b))),F0_2E0),s(tyop_2Emin_2Efun(A_27a,A_27b),V2f_2E0))),s(A_27a,V3k_2E0))) = s(tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epair_2E_2C_2E2(s(A_27a,V3k_2E0),s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V2f_2E0),s(A_27a,V3k_2E0)))))
     => ! [V0x_2E0,V1l_2E0,V2f_2E0] :
          ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(A_27a,V0x_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Elist_2ELIST__TO__SET_2E1(s(tyop_2Elist_2Elist(A_27a),V1l_2E0))))))
         => s(tyop_2Eoption_2Eoption(A_27b),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,A_27b))),F0_2E0),s(tyop_2Emin_2Efun(A_27a,A_27b),V2f_2E0))),s(tyop_2Elist_2Elist(A_27a),V1l_2E0))),s(A_27a,V0x_2E0))) = s(tyop_2Eoption_2Eoption(A_27b),c_2Eoption_2ESOME_2E1(s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V2f_2E0),s(A_27a,V0x_2E0))))) ) ) ).

fof(thm_2Ealist_2EALOOKUP__EQ__FLOOKUP,axiom,
    ! [A_27a,A_27b,A_27c,A_27d,V0fm_2E0,V1al_2E0] :
      ( s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27b)),c_2Efinite__map_2EFLOOKUP_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V1al_2E0))))) = s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27b)),c_2Ealist_2EALOOKUP_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V1al_2E0)))
      & s(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(A_27d)),c_2Ealist_2EALOOKUP_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27d)),c_2Ealist_2Efmap__to__alist_2E1(s(tyop_2Efinite__map_2Efmap(A_27c,A_27d),V0fm_2E0))))) = s(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(A_27d)),c_2Efinite__map_2EFLOOKUP_2E1(s(tyop_2Efinite__map_2Efmap(A_27c,A_27d),V0fm_2E0))) ) ).

fof(thm_2Ealist_2EMEM__fmap__to__alist,axiom,
    ! [A_27a,A_27b,V0y_2E0,V1x_2E0,V2fm_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epair_2E_2C_2E2(s(A_27a,V1x_2E0),s(A_27b,V0y_2E0))),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),c_2Elist_2ELIST__TO__SET_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Ealist_2Efmap__to__alist_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V2fm_2E0))))))))
    <=> ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(A_27a,V1x_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V2fm_2E0))))))
        & s(A_27b,c_2Efinite__map_2EFAPPLY_2E2(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V2fm_2E0),s(A_27a,V1x_2E0))) = s(A_27b,V0y_2E0) ) ) ).

fof(thm_2Ealist_2EMEM__fmap__to__alist__FLOOKUP,axiom,
    ! [A_27a,A_27b,V0p_2E0,V1fm_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Epair_2Eprod(A_27a,A_27b),V0p_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),c_2Elist_2ELIST__TO__SET_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Ealist_2Efmap__to__alist_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V1fm_2E0))))))))
    <=> s(tyop_2Eoption_2Eoption(A_27b),c_2Efinite__map_2EFLOOKUP_2E2(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V1fm_2E0),s(A_27a,c_2Epair_2EFST_2E1(s(tyop_2Epair_2Eprod(A_27a,A_27b),V0p_2E0))))) = s(tyop_2Eoption_2Eoption(A_27b),c_2Eoption_2ESOME_2E1(s(A_27b,c_2Epair_2ESND_2E1(s(tyop_2Epair_2Eprod(A_27a,A_27b),V0p_2E0))))) ) ).

fof(thm_2Ealist_2EMEM__pair__fmap__to__alist__FLOOKUP,axiom,
    ! [A_27a,A_27b,V0x_2E0,V1y_2E0,V2fm_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epair_2E_2C_2E2(s(A_27a,V0x_2E0),s(A_27b,V1y_2E0))),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),c_2Elist_2ELIST__TO__SET_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Ealist_2Efmap__to__alist_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V2fm_2E0))))))))
    <=> s(tyop_2Eoption_2Eoption(A_27b),c_2Efinite__map_2EFLOOKUP_2E2(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V2fm_2E0),s(A_27a,V0x_2E0))) = s(tyop_2Eoption_2Eoption(A_27b),c_2Eoption_2ESOME_2E1(s(A_27b,V1y_2E0))) ) ).

fof(thm_2Ealist_2ELENGTH__fmap__to__alist,axiom,
    ! [A_27a,A_27b,V0fm_2E0] : s(tyop_2Enum_2Enum,c_2Elist_2ELENGTH_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Ealist_2Efmap__to__alist_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V0fm_2E0))))) = s(tyop_2Enum_2Enum,c_2Epred__set_2ECARD_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V0fm_2E0))))) ).

fof(thm_2Ealist_2Efmap__to__alist__to__fmap,axiom,
    ! [A_27a,A_27b,V0fm_2E0] : s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Ealist_2Efmap__to__alist_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V0fm_2E0))))) = s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V0fm_2E0) ).

fof(thm_2Ealist_2EALOOKUP__MEM,axiom,
    ! [A_27a,A_27b,V0al_2E0,V1k_2E0,V2v_2E0] :
      ( s(tyop_2Eoption_2Eoption(A_27b),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0al_2E0),s(A_27a,V1k_2E0))) = s(tyop_2Eoption_2Eoption(A_27b),c_2Eoption_2ESOME_2E1(s(A_27b,V2v_2E0)))
     => p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epair_2E_2C_2E2(s(A_27a,V1k_2E0),s(A_27b,V2v_2E0))),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),c_2Elist_2ELIST__TO__SET_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0al_2E0)))))) ) ).

fof(thm_2Ealist_2EALOOKUP__SOME__FAPPLY__alist__to__fmap,axiom,
    ! [A_27a,A_27b,V0al_2E0,V1k_2E0,V2v_2E0] :
      ( s(tyop_2Eoption_2Eoption(A_27b),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0al_2E0),s(A_27a,V1k_2E0))) = s(tyop_2Eoption_2Eoption(A_27b),c_2Eoption_2ESOME_2E1(s(A_27b,V2v_2E0)))
     => s(A_27b,c_2Efinite__map_2EFAPPLY_2E2(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0al_2E0))),s(A_27a,V1k_2E0))) = s(A_27b,V2v_2E0) ) ).

fof(thm_2Ealist_2Ealist__to__fmap__FAPPLY__MEM,axiom,
    ! [A_27a,A_27b,V0al_2E0,V1z_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(A_27a,V1z_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0al_2E0))))))))
     => p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epair_2E_2C_2E2(s(A_27a,V1z_2E0),s(A_27b,c_2Efinite__map_2EFAPPLY_2E2(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0al_2E0))),s(A_27a,V1z_2E0))))),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),c_2Elist_2ELIST__TO__SET_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0al_2E0)))))) ) ).

fof(thm_2Ealist_2EALOOKUP__MAP,axiom,
    ! [A_27a,A_27b,A_27c,F0_2E0] :
      ( ! [V0f_2E0,V2x_2E0,V3y_2E0] : s(tyop_2Epair_2Eprod(A_27c,A_27b),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27c,A_27b)),app_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27c,A_27b))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27c,A_27b)))),F0_2E0),s(tyop_2Emin_2Efun(A_27a,A_27b),V0f_2E0))),s(A_27c,V2x_2E0))),s(A_27a,V3y_2E0))) = s(tyop_2Epair_2Eprod(A_27c,A_27b),c_2Epair_2E_2C_2E2(s(A_27c,V2x_2E0),s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V0f_2E0),s(A_27a,V3y_2E0)))))
     => ! [V0f_2E0,V1al_2E0] : s(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(A_27b)),c_2Ealist_2EALOOKUP_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27b)),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27a),tyop_2Epair_2Eprod(A_27c,A_27b)),c_2Epair_2EUNCURRY_2E1(s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27c,A_27b))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27c,A_27b)))),F0_2E0),s(tyop_2Emin_2Efun(A_27a,A_27b),V0f_2E0))))),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27a)),V1al_2E0))))) = s(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(A_27b)),c_2Ecombin_2Eo_2E2(s(tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Eoption_2Eoption(A_27b)),c_2Eoption_2EOPTION__MAP_2E1(s(tyop_2Emin_2Efun(A_27a,A_27b),V0f_2E0))),s(tyop_2Emin_2Efun(A_27c,tyop_2Eoption_2Eoption(A_27a)),c_2Ealist_2EALOOKUP_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27a)),V1al_2E0))))) ) ).

fof(thm_2Ealist_2EFDOM__alist__to__fmap,axiom,
    ! [A_27a,A_27b,V0al_2E0] : s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0al_2E0))))) = s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Elist_2ELIST__TO__SET_2E1(s(tyop_2Elist_2Elist(A_27a),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0al_2E0))))) ).

fof(thm_2Ealist_2Ealist__to__fmap__prefix,axiom,
    ! [A_27a,A_27b,V0ls_2E0,V1l1_2E0,V2l2_2E0] :
      ( s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V1l1_2E0))) = s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V2l2_2E0)))
     => s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Elist_2EAPPEND_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0ls_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V1l1_2E0))))) = s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Elist_2EAPPEND_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0ls_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V2l2_2E0))))) ) ).

fof(thm_2Ealist_2Ealist__to__fmap__APPEND,axiom,
    ! [A_27a,A_27b,V0l1_2E0,V1l2_2E0] : s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Elist_2EAPPEND_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0l1_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V1l2_2E0))))) = s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Efinite__map_2EFUNION_2E2(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0l1_2E0))),s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V1l2_2E0))))) ).

fof(thm_2Ealist_2EALOOKUP__prefix,axiom,
    ! [A_27a,A_27b,V0v_2E0,V1ls_2E0,V2k_2E0,V3ls2_2E0] :
      ( ( s(tyop_2Eoption_2Eoption(A_27b),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V1ls_2E0),s(A_27a,V2k_2E0))) = s(tyop_2Eoption_2Eoption(A_27b),c_2Eoption_2ESOME_2E1(s(A_27b,V0v_2E0)))
       => s(tyop_2Eoption_2Eoption(A_27b),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Elist_2EAPPEND_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V1ls_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V3ls2_2E0))),s(A_27a,V2k_2E0))) = s(tyop_2Eoption_2Eoption(A_27b),c_2Eoption_2ESOME_2E1(s(A_27b,V0v_2E0))) )
      & ( s(tyop_2Eoption_2Eoption(A_27b),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V1ls_2E0),s(A_27a,V2k_2E0))) = s(tyop_2Eoption_2Eoption(A_27b),c_2Eoption_2ENONE_2E0)
       => s(tyop_2Eoption_2Eoption(A_27b),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Elist_2EAPPEND_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V1ls_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V3ls2_2E0))),s(A_27a,V2k_2E0))) = s(tyop_2Eoption_2Eoption(A_27b),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V3ls2_2E0),s(A_27a,V2k_2E0))) ) ) ).

fof(thm_2Ealist_2EALOOKUP__APPEND,axiom,
    ! [A_27a,A_27b,F0_2E0] :
      ( ! [V3v_2E0] : s(tyop_2Eoption_2Eoption(A_27b),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(A_27b)),F0_2E0),s(A_27b,V3v_2E0))) = s(tyop_2Eoption_2Eoption(A_27b),c_2Eoption_2ESOME_2E1(s(A_27b,V3v_2E0)))
     => ! [V0l1_2E0,V1l2_2E0,V2k_2E0] : s(tyop_2Eoption_2Eoption(A_27b),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Elist_2EAPPEND_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0l1_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V1l2_2E0))),s(A_27a,V2k_2E0))) = s(tyop_2Eoption_2Eoption(A_27b),c_2Eoption_2Eoption__CASE_2E3(s(tyop_2Eoption_2Eoption(A_27b),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0l1_2E0),s(A_27a,V2k_2E0))),s(tyop_2Eoption_2Eoption(A_27b),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V1l2_2E0),s(A_27a,V2k_2E0))),s(tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(A_27b)),F0_2E0))) ) ).

fof(thm_2Ealist_2EFUPDATE__LIST__EQ__APPEND__REVERSE,axiom,
    ! [A_27a,A_27b,V0ls_2E0,V1fm_2E0] : s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Efinite__map_2EFUPDATE__LIST_2E2(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V1fm_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0ls_2E0))) = s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Elist_2EAPPEND_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Elist_2EREVERSE_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0ls_2E0))),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Ealist_2Efmap__to__alist_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V1fm_2E0))))))) ).

fof(thm_2Ealist_2EFLOOKUP__FUPDATE__LIST__ALOOKUP__SOME,axiom,
    ! [A_27a,A_27b,V0v_2E0,V1ls_2E0,V2k_2E0,V3fm_2E0] :
      ( s(tyop_2Eoption_2Eoption(A_27a),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),V1ls_2E0),s(A_27b,V2k_2E0))) = s(tyop_2Eoption_2Eoption(A_27a),c_2Eoption_2ESOME_2E1(s(A_27a,V0v_2E0)))
     => s(tyop_2Eoption_2Eoption(A_27a),c_2Efinite__map_2EFLOOKUP_2E2(s(tyop_2Efinite__map_2Efmap(A_27b,A_27a),c_2Efinite__map_2EFUPDATE__LIST_2E2(s(tyop_2Efinite__map_2Efmap(A_27b,A_27a),V3fm_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),c_2Elist_2EREVERSE_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),V1ls_2E0))))),s(A_27b,V2k_2E0))) = s(tyop_2Eoption_2Eoption(A_27a),c_2Eoption_2ESOME_2E1(s(A_27a,V0v_2E0))) ) ).

fof(thm_2Ealist_2EFLOOKUP__FUPDATE__LIST__ALOOKUP__NONE,axiom,
    ! [A_27a,A_27b,V0ls_2E0,V1k_2E0,V2fm_2E0] :
      ( s(tyop_2Eoption_2Eoption(A_27a),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),V0ls_2E0),s(A_27b,V1k_2E0))) = s(tyop_2Eoption_2Eoption(A_27a),c_2Eoption_2ENONE_2E0)
     => s(tyop_2Eoption_2Eoption(A_27a),c_2Efinite__map_2EFLOOKUP_2E2(s(tyop_2Efinite__map_2Efmap(A_27b,A_27a),c_2Efinite__map_2EFUPDATE__LIST_2E2(s(tyop_2Efinite__map_2Efmap(A_27b,A_27a),V2fm_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),c_2Elist_2EREVERSE_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),V0ls_2E0))))),s(A_27b,V1k_2E0))) = s(tyop_2Eoption_2Eoption(A_27a),c_2Efinite__map_2EFLOOKUP_2E2(s(tyop_2Efinite__map_2Efmap(A_27b,A_27a),V2fm_2E0),s(A_27b,V1k_2E0))) ) ).

fof(thm_2Ealist_2EFUNION__alist__to__fmap,axiom,
    ! [A_27a,A_27b,V0ls_2E0,V1fm_2E0] : s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Efinite__map_2EFUNION_2E2(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0ls_2E0))),s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V1fm_2E0))) = s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Efinite__map_2EFUPDATE__LIST_2E2(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V1fm_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Elist_2EREVERSE_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0ls_2E0))))) ).

fof(thm_2Ealist_2Ealist__to__fmap__MAP,axiom,
    ! [A_27a,A_27b,A_27c,A_27d,F0_2E0] :
      ( ! [V0f1_2E0,V1f2_2E0,V3x_2E0,V4y_2E0] : s(tyop_2Epair_2Eprod(A_27b,A_27d),app_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Epair_2Eprod(A_27b,A_27d)),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Epair_2Eprod(A_27b,A_27d))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27d),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Epair_2Eprod(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(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Epair_2Eprod(A_27b,A_27d))))),F0_2E0),s(tyop_2Emin_2Efun(A_27a,A_27b),V0f1_2E0))),s(tyop_2Emin_2Efun(A_27c,A_27d),V1f2_2E0))),s(A_27a,V3x_2E0))),s(A_27c,V4y_2E0))) = s(tyop_2Epair_2Eprod(A_27b,A_27d),c_2Epair_2E_2C_2E2(s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V0f1_2E0),s(A_27a,V3x_2E0))),s(A_27d,app_2E2(s(tyop_2Emin_2Efun(A_27c,A_27d),V1f2_2E0),s(A_27c,V4y_2E0)))))
     => ! [V0f1_2E0,V1f2_2E0,V2al_2E0] :
          ( p(s(tyop_2Emin_2Ebool,c_2Epred__set_2EINJ_2E3(s(tyop_2Emin_2Efun(A_27a,A_27b),V0f1_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Elist_2ELIST__TO__SET_2E1(s(tyop_2Elist_2Elist(A_27a),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),A_27a),c_2Epair_2EFST_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27c)),V2al_2E0))))),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Epred__set_2EUNIV_2E0))))
         => s(tyop_2Efinite__map_2Efmap(A_27b,A_27d),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27d)),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Epair_2Eprod(A_27b,A_27d)),c_2Epair_2EUNCURRY_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Epair_2Eprod(A_27b,A_27d))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27d),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Epair_2Eprod(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(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Epair_2Eprod(A_27b,A_27d))))),F0_2E0),s(tyop_2Emin_2Efun(A_27a,A_27b),V0f1_2E0))),s(tyop_2Emin_2Efun(A_27c,A_27d),V1f2_2E0))))),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27c)),V2al_2E0))))) = s(tyop_2Efinite__map_2Efmap(A_27b,A_27d),c_2Efinite__map_2EMAP__KEYS_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V0f1_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,A_27d),c_2Efinite__map_2Eo__f_2E2(s(tyop_2Emin_2Efun(A_27c,A_27d),V1f2_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,A_27c),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27c)),V2al_2E0))))))) ) ) ).

fof(thm_2Ealist_2Ealist__to__fmap__to__alist,axiom,
    ! [A_27a,A_27b,F0_2E0] :
      ( ! [V0al_2E0,V1k_2E0] : s(tyop_2Epair_2Eprod(A_27a,A_27b),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,A_27b))),F0_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0al_2E0))),s(A_27a,V1k_2E0))) = s(tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epair_2E_2C_2E2(s(A_27a,V1k_2E0),s(A_27b,c_2Eoption_2ETHE_2E1(s(tyop_2Eoption_2Eoption(A_27b),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0al_2E0),s(A_27a,V1k_2E0)))))))
     => ! [V0al_2E0] : s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Ealist_2Efmap__to__alist_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0al_2E0))))) = s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,A_27b))),F0_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0al_2E0))),s(tyop_2Elist_2Elist(A_27a),c_2Elist_2ESET__TO__LIST_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Elist_2ELIST__TO__SET_2E1(s(tyop_2Elist_2Elist(A_27a),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0al_2E0))))))))) ) ).

fof(thm_2Ealist_2Ealist__to__fmap__to__alist__PERM,axiom,
    ! [A_27a,A_27b,V0al_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Elist_2EALL__DISTINCT_2E1(s(tyop_2Elist_2Elist(A_27a),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0al_2E0))))))
     => p(s(tyop_2Emin_2Ebool,c_2Esorting_2EPERM_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Ealist_2Efmap__to__alist_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0al_2E0))))),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0al_2E0)))) ) ).

fof(thm_2Ealist_2EALOOKUP__LEAST__EL,axiom,
    ! [A_27a,A_27b,F0_2E0] :
      ( ! [V0ls_2E0,V1k_2E0,V2n_2E0] :
          ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),F0_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0ls_2E0))),s(A_27a,V1k_2E0))),s(tyop_2Enum_2Enum,V2n_2E0))))
        <=> s(A_27a,c_2Elist_2EEL_2E2(s(tyop_2Enum_2Enum,V2n_2E0),s(tyop_2Elist_2Elist(A_27a),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0ls_2E0))))) = s(A_27a,V1k_2E0) )
     => ! [V0ls_2E0,V1k_2E0] : s(tyop_2Eoption_2Eoption(A_27b),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0ls_2E0),s(A_27a,V1k_2E0))) = s(tyop_2Eoption_2Eoption(A_27b),c_2Ebool_2ECOND_2E3(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(A_27a,V1k_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Elist_2ELIST__TO__SET_2E1(s(tyop_2Elist_2Elist(A_27a),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0ls_2E0))))))),s(tyop_2Eoption_2Eoption(A_27b),c_2Eoption_2ESOME_2E1(s(A_27b,c_2Elist_2EEL_2E2(s(tyop_2Enum_2Enum,c_2Ewhile_2ELEAST_2E1(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),F0_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0ls_2E0))),s(A_27a,V1k_2E0))))),s(tyop_2Elist_2Elist(A_27b),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27b),c_2Epair_2ESND_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0ls_2E0))))))),s(tyop_2Eoption_2Eoption(A_27b),c_2Eoption_2ENONE_2E0))) ) ).

fof(thm_2Ealist_2EALOOKUP__ALL__DISTINCT__MEM,axiom,
    ! [A_27a,A_27b,V0v_2E0,V1k_2E0,V2al_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,c_2Elist_2EALL__DISTINCT_2E1(s(tyop_2Elist_2Elist(A_27a),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V2al_2E0))))))
        & p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epair_2E_2C_2E2(s(A_27a,V1k_2E0),s(A_27b,V0v_2E0))),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),c_2Elist_2ELIST__TO__SET_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V2al_2E0)))))) )
     => s(tyop_2Eoption_2Eoption(A_27b),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V2al_2E0),s(A_27a,V1k_2E0))) = s(tyop_2Eoption_2Eoption(A_27b),c_2Eoption_2ESOME_2E1(s(A_27b,V0v_2E0))) ) ).

fof(thm_2Ealist_2EALL__DISTINCT__fmap__to__alist__keys,axiom,
    ! [A_27a,A_27b,V0fm_2E0] : p(s(tyop_2Emin_2Ebool,c_2Elist_2EALL__DISTINCT_2E1(s(tyop_2Elist_2Elist(A_27a),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Ealist_2Efmap__to__alist_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V0fm_2E0)))))))) ).

fof(thm_2Ealist_2Efmap__to__alist__inj,axiom,
    ! [A_27a,A_27b,V0f1_2E0,V1f2_2E0] :
      ( s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Ealist_2Efmap__to__alist_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V0f1_2E0))) = s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Ealist_2Efmap__to__alist_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V1f2_2E0)))
     => s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V0f1_2E0) = s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V1f2_2E0) ) ).

fof(thm_2Ealist_2Efmap__to__alist__preserves__FDOM,axiom,
    ! [A_27a,A_27b,A_27c,V0fm1_2E0,V1fm2_2E0] :
      ( s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V0fm1_2E0))) = s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,A_27c),V1fm2_2E0)))
     => s(tyop_2Elist_2Elist(A_27a),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Ealist_2Efmap__to__alist_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V0fm1_2E0))))) = s(tyop_2Elist_2Elist(A_27a),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),A_27a),c_2Epair_2EFST_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27c)),c_2Ealist_2Efmap__to__alist_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,A_27c),V1fm2_2E0))))) ) ).

fof(thm_2Ealist_2EPERM__fmap__to__alist,axiom,
    ! [A_27a,A_27b,V0fm2_2E0,V1fm1_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Esorting_2EPERM_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Ealist_2Efmap__to__alist_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V1fm1_2E0))),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Ealist_2Efmap__to__alist_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V0fm2_2E0))))))
    <=> s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V1fm1_2E0) = s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V0fm2_2E0) ) ).

fof(thm_2Ealist_2Ealist__to__fmap__PERM,axiom,
    ! [A_27a,A_27b,V0l1_2E0,V1l2_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,c_2Esorting_2EPERM_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0l1_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V1l2_2E0))))
        & p(s(tyop_2Emin_2Ebool,c_2Elist_2EALL__DISTINCT_2E1(s(tyop_2Elist_2Elist(A_27a),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0l1_2E0)))))) )
     => s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0l1_2E0))) = s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V1l2_2E0))) ) ).

fof(thm_2Ealist_2EALOOKUP__ALL__DISTINCT__EL,axiom,
    ! [A_27a,A_27b,V0ls_2E0,V1n_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,c_2Eprim__rec_2E_3C_2E2(s(tyop_2Enum_2Enum,V1n_2E0),s(tyop_2Enum_2Enum,c_2Elist_2ELENGTH_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0ls_2E0))))))
        & p(s(tyop_2Emin_2Ebool,c_2Elist_2EALL__DISTINCT_2E1(s(tyop_2Elist_2Elist(A_27a),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0ls_2E0)))))) )
     => s(tyop_2Eoption_2Eoption(A_27b),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0ls_2E0),s(A_27a,c_2Epair_2EFST_2E1(s(tyop_2Epair_2Eprod(A_27a,A_27b),c_2Elist_2EEL_2E2(s(tyop_2Enum_2Enum,V1n_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0ls_2E0))))))) = s(tyop_2Eoption_2Eoption(A_27b),c_2Eoption_2ESOME_2E1(s(A_27b,c_2Epair_2ESND_2E1(s(tyop_2Epair_2Eprod(A_27a,A_27b),c_2Elist_2EEL_2E2(s(tyop_2Enum_2Enum,V1n_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0ls_2E0))))))) ) ).

fof(thm_2Ealist_2EALOOKUP__ZIP__MAP__SND,axiom,
    ! [A_27a,A_27b,A_27c,A_27d,V0l1_2E0,V1l2_2E0,V2k_2E0,V3f_2E0] :
      ( s(tyop_2Enum_2Enum,c_2Elist_2ELENGTH_2E1(s(tyop_2Elist_2Elist(A_27a),V0l1_2E0))) = s(tyop_2Enum_2Enum,c_2Elist_2ELENGTH_2E1(s(tyop_2Elist_2Elist(A_27b),V1l2_2E0)))
     => s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27d)),c_2Ealist_2EALOOKUP_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27d)),c_2Elist_2EZIP_2E1(s(tyop_2Epair_2Eprod(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27d)),c_2Epair_2E_2C_2E2(s(tyop_2Elist_2Elist(A_27a),V0l1_2E0),s(tyop_2Elist_2Elist(A_27d),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(A_27b,A_27d),V3f_2E0),s(tyop_2Elist_2Elist(A_27b),V1l2_2E0))))))))) = s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27d)),c_2Ecombin_2Eo_2E2(s(tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27b),tyop_2Eoption_2Eoption(A_27d)),c_2Eoption_2EOPTION__MAP_2E1(s(tyop_2Emin_2Efun(A_27b,A_27d),V3f_2E0))),s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27b)),c_2Ealist_2EALOOKUP_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Elist_2EZIP_2E1(s(tyop_2Epair_2Eprod(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27b)),c_2Epair_2E_2C_2E2(s(tyop_2Elist_2Elist(A_27a),V0l1_2E0),s(tyop_2Elist_2Elist(A_27b),V1l2_2E0))))))))) ) ).

fof(thm_2Ealist_2EALOOKUP__FILTER,axiom,
    ! [A_27a,A_27b,F0_2E0] :
      ( ! [V0P_2E0,V3k_2E0,V4v_2E0] : 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)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),F0_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0))),s(A_27a,V3k_2E0))),s(A_27b,V4v_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(A_27a,V3k_2E0)))
     => ! [V0P_2E0,V1ls_2E0,V2x_2E0] : s(tyop_2Eoption_2Eoption(A_27b),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Elist_2EFILTER_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),c_2Epair_2EUNCURRY_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),F0_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0))))),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V1ls_2E0))),s(A_27a,V2x_2E0))) = s(tyop_2Eoption_2Eoption(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_2Eoption_2Eoption(A_27b),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V1ls_2E0),s(A_27a,V2x_2E0))),s(tyop_2Eoption_2Eoption(A_27b),c_2Eoption_2ENONE_2E0))) ) ).

fof(thm_2Ealist_2EALOOKUP__APPEND__same,axiom,
    ! [A_27a,A_27b,V0l1_2E0,V1l2_2E0,V2l_2E0] :
      ( s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27b)),c_2Ealist_2EALOOKUP_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0l1_2E0))) = s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27b)),c_2Ealist_2EALOOKUP_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V1l2_2E0)))
     => s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27b)),c_2Ealist_2EALOOKUP_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Elist_2EAPPEND_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0l1_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V2l_2E0))))) = s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27b)),c_2Ealist_2EALOOKUP_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Elist_2EAPPEND_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V1l2_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V2l_2E0))))) ) ).

fof(thm_2Ealist_2EALOOKUP__IN__FRANGE,axiom,
    ! [A_27a,A_27b,V0ls_2E0,V1k_2E0,V2v_2E0] :
      ( s(tyop_2Eoption_2Eoption(A_27b),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0ls_2E0),s(A_27a,V1k_2E0))) = s(tyop_2Eoption_2Eoption(A_27b),c_2Eoption_2ESOME_2E1(s(A_27b,V2v_2E0)))
     => p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(A_27b,V2v_2E0),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Efinite__map_2EFRANGE_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0ls_2E0)))))))) ) ).

fof(thm_2Ealist_2EFRANGE__alist__to__fmap__SUBSET,axiom,
    ! [A_27a,A_27b,V0ls_2E0] : p(s(tyop_2Emin_2Ebool,c_2Epred__set_2ESUBSET_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Efinite__map_2EFRANGE_2E1(s(tyop_2Efinite__map_2Efmap(A_27b,A_27a),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),V0ls_2E0))))),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Epred__set_2EIMAGE_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),A_27a),c_2Epair_2ESND_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),tyop_2Emin_2Ebool),c_2Elist_2ELIST__TO__SET_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),V0ls_2E0)))))))) ).

fof(thm_2Ealist_2EIN__FRANGE__alist__to__fmap__suff,axiom,
    ! [A_27a,A_27b,V0ls_2E0,V1P_2E0] :
      ( ! [V2v_2E0] :
          ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(A_27a,V2v_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Elist_2ELIST__TO__SET_2E1(s(tyop_2Elist_2Elist(A_27a),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),A_27a),c_2Epair_2ESND_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),V0ls_2E0))))))))
         => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1P_2E0),s(A_27a,V2v_2E0)))) )
     => ! [V3v_2E0] :
          ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(A_27a,V3v_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Efinite__map_2EFRANGE_2E1(s(tyop_2Efinite__map_2Efmap(A_27b,A_27a),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27a)),V0ls_2E0))))))))
         => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1P_2E0),s(A_27a,V3v_2E0)))) ) ) ).

fof(thm_2Ealist_2Ealist__to__fmap__MAP__matchable,axiom,
    ! [A_27a,A_27b,A_27c,A_27d,F0_2E0] :
      ( ! [V0f1_2E0,V1f2_2E0,V5x_2E0,V6y_2E0] : s(tyop_2Epair_2Eprod(A_27b,A_27d),app_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Epair_2Eprod(A_27b,A_27d)),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Epair_2Eprod(A_27b,A_27d))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27d),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Epair_2Eprod(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(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Epair_2Eprod(A_27b,A_27d))))),F0_2E0),s(tyop_2Emin_2Efun(A_27a,A_27b),V0f1_2E0))),s(tyop_2Emin_2Efun(A_27c,A_27d),V1f2_2E0))),s(A_27a,V5x_2E0))),s(A_27c,V6y_2E0))) = s(tyop_2Epair_2Eprod(A_27b,A_27d),c_2Epair_2E_2C_2E2(s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V0f1_2E0),s(A_27a,V5x_2E0))),s(A_27d,app_2E2(s(tyop_2Emin_2Efun(A_27c,A_27d),V1f2_2E0),s(A_27c,V6y_2E0)))))
     => ! [V0f1_2E0,V1f2_2E0,V2al_2E0,V3mal_2E0,V4v_2E0] :
          ( ( p(s(tyop_2Emin_2Ebool,c_2Epred__set_2EINJ_2E3(s(tyop_2Emin_2Efun(A_27a,A_27b),V0f1_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Elist_2ELIST__TO__SET_2E1(s(tyop_2Elist_2Elist(A_27a),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),A_27a),c_2Epair_2EFST_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27c)),V2al_2E0))))),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Epred__set_2EUNIV_2E0))))
            & s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27d)),V3mal_2E0) = s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27d)),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Epair_2Eprod(A_27b,A_27d)),c_2Epair_2EUNCURRY_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Epair_2Eprod(A_27b,A_27d))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27d),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Epair_2Eprod(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(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Epair_2Eprod(A_27b,A_27d))))),F0_2E0),s(tyop_2Emin_2Efun(A_27a,A_27b),V0f1_2E0))),s(tyop_2Emin_2Efun(A_27c,A_27d),V1f2_2E0))))),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27c)),V2al_2E0)))
            & s(tyop_2Efinite__map_2Efmap(A_27b,A_27d),V4v_2E0) = s(tyop_2Efinite__map_2Efmap(A_27b,A_27d),c_2Efinite__map_2EMAP__KEYS_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V0f1_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,A_27d),c_2Efinite__map_2Eo__f_2E2(s(tyop_2Emin_2Efun(A_27c,A_27d),V1f2_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,A_27c),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27c)),V2al_2E0))))))) )
         => s(tyop_2Efinite__map_2Efmap(A_27b,A_27d),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27d)),V3mal_2E0))) = s(tyop_2Efinite__map_2Efmap(A_27b,A_27d),V4v_2E0) ) ) ).

fof(thm_2Ealist_2EMAP__values__fmap__to__alist,axiom,
    ! [A_27a,A_27b,A_27c,F0_2E0] :
      ( ! [V0f_2E0,V2k_2E0,V3v_2E0] : s(tyop_2Epair_2Eprod(A_27c,A_27b),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27c,A_27b)),app_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27c,A_27b))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27c,A_27b)))),F0_2E0),s(tyop_2Emin_2Efun(A_27a,A_27b),V0f_2E0))),s(A_27c,V2k_2E0))),s(A_27a,V3v_2E0))) = s(tyop_2Epair_2Eprod(A_27c,A_27b),c_2Epair_2E_2C_2E2(s(A_27c,V2k_2E0),s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V0f_2E0),s(A_27a,V3v_2E0)))))
     => ! [V0f_2E0,V1fm_2E0] : s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27b)),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27a),tyop_2Epair_2Eprod(A_27c,A_27b)),c_2Epair_2EUNCURRY_2E1(s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27c,A_27b))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27c,A_27b)))),F0_2E0),s(tyop_2Emin_2Efun(A_27a,A_27b),V0f_2E0))))),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27a)),c_2Ealist_2Efmap__to__alist_2E1(s(tyop_2Efinite__map_2Efmap(A_27c,A_27a),V1fm_2E0))))) = s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27b)),c_2Ealist_2Efmap__to__alist_2E1(s(tyop_2Efinite__map_2Efmap(A_27c,A_27b),c_2Efinite__map_2Eo__f_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V0f_2E0),s(tyop_2Efinite__map_2Efmap(A_27c,A_27a),V1fm_2E0))))) ) ).

fof(thm_2Ealist_2EMAP__KEYS__I,axiom,
    ! [A_27a,A_27b,V0fm_2E0] : s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Efinite__map_2EMAP__KEYS_2E2(s(tyop_2Emin_2Efun(A_27a,A_27a),c_2Ecombin_2EI_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V0fm_2E0))) = s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V0fm_2E0) ).

fof(thm_2Ealist_2Ealist__to__fmap__MAP__values,axiom,
    ! [A_27a,A_27b,A_27c,F0_2E0] :
      ( ! [V0f_2E0,V2k_2E0,V3v_2E0] : s(tyop_2Epair_2Eprod(A_27c,A_27b),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27c,A_27b)),app_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27c,A_27b))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27c,A_27b)))),F0_2E0),s(tyop_2Emin_2Efun(A_27a,A_27b),V0f_2E0))),s(A_27c,V2k_2E0))),s(A_27a,V3v_2E0))) = s(tyop_2Epair_2Eprod(A_27c,A_27b),c_2Epair_2E_2C_2E2(s(A_27c,V2k_2E0),s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V0f_2E0),s(A_27a,V3v_2E0)))))
     => ! [V0f_2E0,V1al_2E0] : s(tyop_2Efinite__map_2Efmap(A_27c,A_27b),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27b)),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27c,A_27a),tyop_2Epair_2Eprod(A_27c,A_27b)),c_2Epair_2EUNCURRY_2E1(s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27c,A_27b))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27c,A_27b)))),F0_2E0),s(tyop_2Emin_2Efun(A_27a,A_27b),V0f_2E0))))),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27a)),V1al_2E0))))) = s(tyop_2Efinite__map_2Efmap(A_27c,A_27b),c_2Efinite__map_2Eo__f_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V0f_2E0),s(tyop_2Efinite__map_2Efmap(A_27c,A_27a),c_2Ealist_2Ealist__to__fmap_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27c,A_27a)),V1al_2E0))))) ) ).

fof(thm_2Ealist_2Eset__MAP__FST__fmap__to__alist,axiom,
    ! [A_27a,A_27b,V0fm_2E0] : s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Elist_2ELIST__TO__SET_2E1(s(tyop_2Elist_2Elist(A_27a),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Ealist_2Efmap__to__alist_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V0fm_2E0))))))) = s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V0fm_2E0))) ).

fof(thm_2Ealist_2Ealookup__distinct__reverse,axiom,
    ! [A_27a,A_27b,V0l_2E0,V1k_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Elist_2EALL__DISTINCT_2E1(s(tyop_2Elist_2Elist(A_27a),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0l_2E0))))))
     => s(tyop_2Eoption_2Eoption(A_27b),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Elist_2EREVERSE_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0l_2E0))),s(A_27a,V1k_2E0))) = s(tyop_2Eoption_2Eoption(A_27b),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0l_2E0),s(A_27a,V1k_2E0))) ) ).

fof(thm_2Ealist_2Eflookup__fupdate__list,axiom,
    ! [A_27a,A_27b,F0_2E0] :
      ( ! [V3v_2E0] : s(tyop_2Eoption_2Eoption(A_27b),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(A_27b)),F0_2E0),s(A_27b,V3v_2E0))) = s(tyop_2Eoption_2Eoption(A_27b),c_2Eoption_2ESOME_2E1(s(A_27b,V3v_2E0)))
     => ! [V0l_2E0,V1k_2E0,V2m_2E0] : s(tyop_2Eoption_2Eoption(A_27b),c_2Efinite__map_2EFLOOKUP_2E2(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Efinite__map_2EFUPDATE__LIST_2E2(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V2m_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0l_2E0))),s(A_27a,V1k_2E0))) = s(tyop_2Eoption_2Eoption(A_27b),c_2Eoption_2Eoption__CASE_2E3(s(tyop_2Eoption_2Eoption(A_27b),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Elist_2EREVERSE_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0l_2E0))),s(A_27a,V1k_2E0))),s(tyop_2Eoption_2Eoption(A_27b),c_2Efinite__map_2EFLOOKUP_2E2(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V2m_2E0),s(A_27a,V1k_2E0))),s(tyop_2Emin_2Efun(A_27b,tyop_2Eoption_2Eoption(A_27b)),F0_2E0))) ) ).

fof(thm_2Ealist_2Efupdate__list__funion,axiom,
    ! [A_27a,A_27b,V0m_2E0,V1l_2E0] : s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Efinite__map_2EFUPDATE__LIST_2E2(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V0m_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V1l_2E0))) = s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Efinite__map_2EFUNION_2E2(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Efinite__map_2EFUPDATE__LIST_2E2(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Efinite__map_2EFEMPTY_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V1l_2E0))),s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),V0m_2E0))) ).

fof(thm_2Ealist_2Emem__to__flookup,axiom,
    ! [A_27a,A_27b,V0x_2E0,V1y_2E0,V2l_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,c_2Elist_2EALL__DISTINCT_2E1(s(tyop_2Elist_2Elist(A_27a),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V2l_2E0))))))
        & p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epair_2E_2C_2E2(s(A_27a,V0x_2E0),s(A_27b,V1y_2E0))),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),c_2Elist_2ELIST__TO__SET_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V2l_2E0)))))) )
     => s(tyop_2Eoption_2Eoption(A_27b),c_2Efinite__map_2EFLOOKUP_2E2(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Efinite__map_2EFUPDATE__LIST_2E2(s(tyop_2Efinite__map_2Efmap(A_27a,A_27b),c_2Efinite__map_2EFEMPTY_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V2l_2E0))),s(A_27a,V0x_2E0))) = s(tyop_2Eoption_2Eoption(A_27b),c_2Eoption_2ESOME_2E1(s(A_27b,V1y_2E0))) ) ).

fof(thm_2Ealist_2Ealookup__filter,axiom,
    ! [A_27a,A_27b,A_27c,F0_2E0] :
      ( ! [V2x_2E0,V3x_27_2E0,V4y_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_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool))),F0_2E0),s(A_27b,V2x_2E0))),s(A_27b,V3x_27_2E0))),s(A_27c,V4y_2E0))))
        <=> s(A_27b,V2x_2E0) = s(A_27b,V3x_27_2E0) )
     => ! [V0f_2E0,V1l_2E0,V2x_2E0] : s(tyop_2Eoption_2Eoption(A_27c),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27c)),V1l_2E0),s(A_27b,V2x_2E0))) = s(tyop_2Eoption_2Eoption(A_27c),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27c)),c_2Elist_2EFILTER_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27c),tyop_2Emin_2Ebool),c_2Epair_2EUNCURRY_2E1(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool))),F0_2E0),s(A_27b,V2x_2E0))))),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27b,A_27c)),V1l_2E0))),s(A_27b,V2x_2E0))) ) ).

fof(thm_2Ealist_2Ealookup__stable__sorted,axiom,
    ! [A_27a,A_27b,V0R_2E0,V1sort_2E0,V2x_2E0,V3l_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,c_2Erelation_2Etransitive_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V0R_2E0))))
        & p(s(tyop_2Emin_2Ebool,c_2Erelation_2Etotal_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V0R_2E0))))
        & p(s(tyop_2Emin_2Ebool,c_2Esorting_2ESTABLE_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)))),V1sort_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),c_2Erelation_2Einv__image_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V0R_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST_2E0)))))) )
     => s(tyop_2Eoption_2Eoption(A_27b),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)))),V1sort_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),c_2Erelation_2Einv__image_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V0R_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST_2E0))))),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V3l_2E0))),s(A_27a,V2x_2E0))) = s(tyop_2Eoption_2Eoption(A_27b),c_2Ealist_2EALOOKUP_2E2(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V3l_2E0),s(A_27a,V2x_2E0))) ) ).

fof(thm_2Ealist_2EALOOKUP__ALL__DISTINCT__PERM__same,axiom,
    ! [A_27a,A_27b,V0l1_2E0,V1l2_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,c_2Elist_2EALL__DISTINCT_2E1(s(tyop_2Elist_2Elist(A_27a),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0l1_2E0))))))
        & p(s(tyop_2Emin_2Ebool,c_2Esorting_2EPERM_2E2(s(tyop_2Elist_2Elist(A_27a),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0l1_2E0))),s(tyop_2Elist_2Elist(A_27a),c_2Elist_2EMAP_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a),c_2Epair_2EFST_2E0),s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V1l2_2E0))))))
        & s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),c_2Elist_2ELIST__TO__SET_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0l1_2E0))) = s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),c_2Elist_2ELIST__TO__SET_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V1l2_2E0))) )
     => s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27b)),c_2Ealist_2EALOOKUP_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V0l1_2E0))) = s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27b)),c_2Ealist_2EALOOKUP_2E1(s(tyop_2Elist_2Elist(tyop_2Epair_2Eprod(A_27a,A_27b)),V1l2_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) ) ).

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