%------------------------------------------------------------------------------
% File     : ITP083+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    : fmaptree.ax [Gau19]
%          : HL4083+4.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  101 (  66 unt;   0 def)
%            Number of atoms       :  186 ( 102 equ)
%            Maximal formula atoms :    8 (   1 avg)
%            Number of connectives :   88 (   3   ~;   3   |;  20   &)
%                                         (  20 <=>;  42  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   21 (   6 avg)
%            Maximal term depth    :   15 (   3 avg)
%            Number of predicates  :    2 (   1 usr;   0 prp; 1-2 aty)
%            Number of functors    :   90 (  90 usr;  44 con; 0-3 aty)
%            Number of variables   :  445 ( 429   !;  16   ?)
% 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_2Emin_2E_40_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(A_27a,c_2Emin_2E_40_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0))) = s(A_27a,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),A_27a),c_2Emin_2E_40_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0))) ).

fof(arityeq1_2Ec_2Efinite__map_2EFDOM_2E1_2Emono_2EA_27b_20mono_2EA_27c,axiom,
    ! [A_27b,A_27c,X0_2E0] : s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27b,A_27c),X0_2E0))) = s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,A_27c),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),c_2Efinite__map_2EFDOM_2E0),s(tyop_2Efinite__map_2Efmap(A_27b,A_27c),X0_2E0))) ).

fof(arityeq1_2Ec_2Efinite__map_2EFDOM_2E1_2Emono_2EA_27b_20mono_2Etyop_2Efmaptree_2Efmaptree_28A_27b_2CA_27a_29,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),X0_2E0))) = s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),c_2Efinite__map_2EFDOM_2E0),s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),X0_2E0))) ).

fof(arityeq1_2Ec_2Epred__set_2EFINITE_2E1_2Emono_2Etyop_2Elist_2Elist_28A_27a_29,axiom,
    ! [A_27a,X0_2E0] : s(tyop_2Emin_2Ebool,c_2Epred__set_2EFINITE_2E1(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool),X0_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Epred__set_2EFINITE_2E0),s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool),X0_2E0))) ).

fof(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2Etyop_2Elist_2Elist_28A_27a_29_20mono_2Etyop_2Elist_2Elist_28A_27a_29,axiom,
    ! [A_27a,X0_2E0] : s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E1(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Epair_2Eprod(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool)),X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Epair_2Eprod(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool)),c_2Epred__set_2EGSPEC_2E0),s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Epair_2Eprod(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool)),X0_2E0))) ).

fof(arityeq1_2Ec_2Eoption_2ESOME_2E1_2Emono_2Etyop_2Efmaptree_2Efmaptree_28A_27a_2CA_27b_29,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Eoption_2ESOME_2E1(s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),X0_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),c_2Eoption_2ESOME_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),X0_2E0))) ).

fof(arityeq2_2Ec_2Efmaptree_2EFTNode_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),c_2Efmaptree_2EFTNode_2E2(s(A_27b,X0_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),X1_2E0))) = s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),c_2Efmaptree_2EFTNode_2E0),s(A_27b,X0_2E0))),s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),X1_2E0))) ).

fof(arityeq1_2Ec_2Efmaptree_2Emap_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efmaptree_2Emap_2E1(s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),X0_2E0))) = s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),c_2Efmaptree_2Emap_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),X0_2E0))) ).

fof(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2EA_27a_20mono_2Etyop_2Efmaptree_2Efmaptree_28A_27a_2CA_27b_29,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Epair_2Eprod(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Epair_2E_2C_2E2(s(A_27a,X0_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),X1_2E0))) = s(tyop_2Epair_2Eprod(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Epair_2Eprod(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Epair_2Eprod(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)))),c_2Epair_2E_2C_2E0),s(A_27a,X0_2E0))),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),X1_2E0))) ).

fof(arityeq2_2Ec_2Efinite__map_2EFUPDATE_2E2_2Emono_2EA_27a_20mono_2Etyop_2Efmaptree_2Efmaptree_28A_27a_2CA_27b_29,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efinite__map_2EFUPDATE_2E2(s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),X0_2E0),s(tyop_2Epair_2Eprod(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),X1_2E0))) = s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)))),c_2Efinite__map_2EFUPDATE_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),X0_2E0))),s(tyop_2Epair_2Eprod(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),X1_2E0))) ).

fof(arityeq1_2Ec_2Efmaptree_2Eitem_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(A_27b,c_2Efmaptree_2Eitem_2E1(s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),X0_2E0))) = s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),A_27b),c_2Efmaptree_2Eitem_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),X0_2E0))) ).

fof(arityeq2_2Ec_2Efmaptree_2Econstruct_2E2_2Emono_2EA_27b_20mono_2EA_27a,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Eoption_2Eoption(A_27b)),c_2Efmaptree_2Econstruct_2E2(s(A_27b,X0_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Eoption_2Eoption(A_27b))),X1_2E0))) = s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Eoption_2Eoption(A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Eoption_2Eoption(A_27b))),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Eoption_2Eoption(A_27b))),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Eoption_2Eoption(A_27b))),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Eoption_2Eoption(A_27b)))),c_2Efmaptree_2Econstruct_2E0),s(A_27b,X0_2E0))),s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Eoption_2Eoption(A_27b))),X1_2E0))) ).

fof(arityeq1_2Ec_2Efmaptree_2EfromF_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),c_2Efmaptree_2EfromF_2E1(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Eoption_2Eoption(A_27b)),X0_2E0))) = s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Eoption_2Eoption(A_27b)),tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efmaptree_2EfromF_2E0),s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Eoption_2Eoption(A_27b)),X0_2E0))) ).

fof(arityeq2_2Ec_2Efinite__map_2Eo__f_2E2_2Emono_2EA_27a_20mono_2Etyop_2Efmaptree_2Efmaptree_28A_27a_2CA_27b_29_20mono_2Etyop_2Emin_2Efun_28tyop_2Elist_2Elist_28A_27a_29_2Ctyop_2Eoption_2Eoption_28A_27b_29_29,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Eoption_2Eoption(A_27b))),c_2Efinite__map_2Eo__f_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Eoption_2Eoption(A_27b))),X0_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),X1_2E0))) = s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Eoption_2Eoption(A_27b))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),tyop_2Efinite__map_2Efmap(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Eoption_2Eoption(A_27b)))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Eoption_2Eoption(A_27b))),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),tyop_2Efinite__map_2Efmap(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Eoption_2Eoption(A_27b))))),c_2Efinite__map_2Eo__f_2E0),s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Eoption_2Eoption(A_27b))),X0_2E0))),s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),X1_2E0))) ).

fof(arityeq1_2Ec_2Efmaptree_2EfromF_2E1_2Emono_2EA_27key_20mono_2EA_27value,axiom,
    ! [A_27key,A_27value,X0_2E0] : s(tyop_2Efmaptree_2Efmaptree(A_27key,A_27value),c_2Efmaptree_2EfromF_2E1(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value)),X0_2E0))) = s(tyop_2Efmaptree_2Efmaptree(A_27key,A_27value),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value)),tyop_2Efmaptree_2Efmaptree(A_27key,A_27value)),c_2Efmaptree_2EfromF_2E0),s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value)),X0_2E0))) ).

fof(arityeq1_2Ec_2Efmaptree_2EtoF_2E1_2Emono_2EA_27key_20mono_2EA_27value,axiom,
    ! [A_27key,A_27value,X0_2E0] : s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value)),c_2Efmaptree_2EtoF_2E1(s(tyop_2Efmaptree_2Efmaptree(A_27key,A_27value),X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27key,A_27value),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value))),c_2Efmaptree_2EtoF_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27key,A_27value),X0_2E0))) ).

fof(arityeq2_2Ec_2Efmaptree_2EFTNode_2E2_2Emono_2EA_27b_20mono_2EA_27a,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),c_2Efmaptree_2EFTNode_2E2(s(A_27a,X0_2E0),s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),X1_2E0))) = s(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),tyop_2Efmaptree_2Efmaptree(A_27b,A_27a))),c_2Efmaptree_2EFTNode_2E0),s(A_27a,X0_2E0))),s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),X1_2E0))) ).

fof(arityeq1_2Ec_2Efmaptree_2Eitem_2E1_2Emono_2EA_27b_20mono_2EA_27a,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(A_27a,c_2Efmaptree_2Eitem_2E1(s(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),X0_2E0))) = s(A_27a,app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),A_27a),c_2Efmaptree_2Eitem_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),X0_2E0))) ).

fof(arityeq1_2Ec_2Efmaptree_2Emap_2E1_2Emono_2EA_27b_20mono_2EA_27a,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),c_2Efmaptree_2Emap_2E1(s(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),X0_2E0))) = s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a))),c_2Efmaptree_2Emap_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),X0_2E0))) ).

fof(arityeq1_2Ec_2Efmaptree_2Ewf_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Emin_2Ebool,c_2Efmaptree_2Ewf_2E1(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),X0_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Ebool),c_2Efmaptree_2Ewf_2E0),s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),X0_2E0))) ).

fof(arityeq2_2Ec_2Efinite__map_2EFAPPLY_2E2_2Emono_2EA_27b_20mono_2Etyop_2Emin_2Efun_28tyop_2Elist_2Elist_28A_27b_29_2Ctyop_2Eoption_2Eoption_28A_27a_29_29,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),c_2Efinite__map_2EFAPPLY_2E2(s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),X0_2E0),s(A_27b,X1_2E0))) = s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)))),c_2Efinite__map_2EFAPPLY_2E0),s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),X0_2E0))),s(A_27b,X1_2E0))) ).

fof(arityeq2_2Ec_2Efmaptree_2Econstruct_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),c_2Efmaptree_2Econstruct_2E2(s(A_27a,X0_2E0),s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),X1_2E0))) = s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)))),c_2Efmaptree_2Econstruct_2E0),s(A_27a,X0_2E0))),s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),X1_2E0))) ).

fof(arityeq1_2Ec_2Efmaptree_2Ewf_2E1_2Emono_2EA_27value_20mono_2EA_27key,axiom,
    ! [A_27key,A_27value,X0_2E0] : s(tyop_2Emin_2Ebool,c_2Efmaptree_2Ewf_2E1(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value)),X0_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value)),tyop_2Emin_2Ebool),c_2Efmaptree_2Ewf_2E0),s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value)),X0_2E0))) ).

fof(arityeq2_2Ec_2Efinite__map_2EFAPPLY_2E2_2Emono_2EA_27b_20mono_2EA_27c,axiom,
    ! [A_27b,A_27c,X0_2E0,X1_2E0] : s(A_27c,c_2Efinite__map_2EFAPPLY_2E2(s(tyop_2Efinite__map_2Efmap(A_27b,A_27c),X0_2E0),s(A_27b,X1_2E0))) = s(A_27c,app_2E2(s(tyop_2Emin_2Efun(A_27b,A_27c),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,A_27c),tyop_2Emin_2Efun(A_27b,A_27c)),c_2Efinite__map_2EFAPPLY_2E0),s(tyop_2Efinite__map_2Efmap(A_27b,A_27c),X0_2E0))),s(A_27b,X1_2E0))) ).

fof(arityeq2_2Ec_2Efinite__map_2EFAPPLY_2E2_2Emono_2EA_27b_20mono_2Etyop_2Efmaptree_2Efmaptree_28A_27b_2CA_27a_29,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),c_2Efinite__map_2EFAPPLY_2E2(s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),X0_2E0),s(A_27b,X1_2E0))) = s(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),tyop_2Emin_2Efun(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a))),c_2Efinite__map_2EFAPPLY_2E0),s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),X0_2E0))),s(A_27b,X1_2E0))) ).

fof(arityeq2_2Ec_2Efinite__map_2Eo__f_2E2_2Emono_2EA_27b_20mono_2Etyop_2Efmaptree_2Efmaptree_28A_27b_2CA_27a_29_20mono_2EA_27c,axiom,
    ! [A_27a,A_27b,A_27c,X0_2E0,X1_2E0] : s(tyop_2Efinite__map_2Efmap(A_27b,A_27c),c_2Efinite__map_2Eo__f_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),A_27c),X0_2E0),s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),X1_2E0))) = s(tyop_2Efinite__map_2Efmap(A_27b,A_27c),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),tyop_2Efinite__map_2Efmap(A_27b,A_27c)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),A_27c),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),tyop_2Efinite__map_2Efmap(A_27b,A_27c))),c_2Efinite__map_2Eo__f_2E0),s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),A_27c),X0_2E0))),s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),X1_2E0))) ).

fof(arityeq1_2Ec_2Efmaptree_2Efmtreerec_2E1_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27c,axiom,
    ! [A_27a,A_27b,A_27c,X0_2E0] : s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),A_27a),c_2Efmaptree_2Efmtreerec_2E1(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,A_27a),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)),A_27a))),X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,A_27a),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)),A_27a))),tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),A_27a)),c_2Efmaptree_2Efmtreerec_2E0),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,A_27a),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)),A_27a))),X0_2E0))) ).

fof(arityeq2_2Ec_2Efinite__map_2Eo__f_2E2_2Emono_2EA_27c_20mono_2Etyop_2Efmaptree_2Efmaptree_28A_27c_2CA_27b_29_20mono_2EA_27a,axiom,
    ! [A_27a,A_27b,A_27c,X0_2E0,X1_2E0] : s(tyop_2Efinite__map_2Efmap(A_27c,A_27a),c_2Efinite__map_2Eo__f_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),A_27a),X0_2E0),s(tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)),X1_2E0))) = s(tyop_2Efinite__map_2Efmap(A_27c,A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)),tyop_2Efinite__map_2Efmap(A_27c,A_27a)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),A_27a),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)),tyop_2Efinite__map_2Efmap(A_27c,A_27a))),c_2Efinite__map_2Eo__f_2E0),s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),A_27a),X0_2E0))),s(tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)),X1_2E0))) ).

fof(arityeq2_2Ec_2Efinite__map_2EFAPPLY_2E2_2Emono_2EA_27a_20mono_2Etyop_2Efmaptree_2Efmaptree_28A_27a_2CA_27b_29,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),c_2Efinite__map_2EFAPPLY_2E2(s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),X0_2E0),s(A_27a,X1_2E0))) = s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),tyop_2Emin_2Efun(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),c_2Efinite__map_2EFAPPLY_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),X0_2E0))),s(A_27a,X1_2E0))) ).

fof(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Elist_2Elist_28A_27a_29_20mono_2Etyop_2Emin_2Ebool,axiom,
    ! [A_27a,X0_2E0,X1_2E0] : s(tyop_2Epair_2Eprod(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool),c_2Epair_2E_2C_2E2(s(tyop_2Elist_2Elist(A_27a),X0_2E0),s(tyop_2Emin_2Ebool,X1_2E0))) = s(tyop_2Epair_2Eprod(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool))),c_2Epair_2E_2C_2E0),s(tyop_2Elist_2Elist(A_27a),X0_2E0))),s(tyop_2Emin_2Ebool,X1_2E0))) ).

fof(arityeq2_2Ec_2Efinite__map_2EFLOOKUP_2E2_2Emono_2EA_27a_20mono_2Etyop_2Efmaptree_2Efmaptree_28A_27a_2CA_27b_29,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efinite__map_2EFLOOKUP_2E2(s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),X0_2E0),s(A_27a,X1_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)))),c_2Efinite__map_2EFLOOKUP_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),X0_2E0))),s(A_27a,X1_2E0))) ).

fof(arityeq1_2Ec_2Efinite__map_2EFDOM_2E1_2Emono_2EA_27a_20mono_2Etyop_2Efmaptree_2Efmaptree_28A_27a_2CA_27b_29,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,tyop_2Efmaptree_2Efmaptree(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,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Efinite__map_2EFDOM_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),X0_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(arityeq1_2Ec_2Efinite__map_2EFDOM_2E1_2Emono_2EA_27b_20mono_2Etyop_2Emin_2Efun_28tyop_2Elist_2Elist_28A_27b_29_2Ctyop_2Eoption_2Eoption_28A_27a_29_29,axiom,
    ! [A_27a,A_27b,X0_2E0] : s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),X0_2E0))) = s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),c_2Efinite__map_2EFDOM_2E0),s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),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_2ETYPE__DEFINITION_2E2_2Emono_2Etyop_2Emin_2Efun_28tyop_2Elist_2Elist_28A_27key_29_2Ctyop_2Eoption_2Eoption_28A_27value_29_29_20mono_2Etyop_2Efmaptree_2Efmaptree_28A_27key_2CA_27value_29,axiom,
    ! [A_27key,A_27value,X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Ebool_2ETYPE__DEFINITION_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value)),tyop_2Emin_2Ebool),X0_2E0),s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27key,A_27value),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value))),X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27key,A_27value),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value))),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value)),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27key,A_27value),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value))),tyop_2Emin_2Ebool)),c_2Ebool_2ETYPE__DEFINITION_2E0),s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value)),tyop_2Emin_2Ebool),X0_2E0))),s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27key,A_27value),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value))),X1_2E0))) ).

fof(arityeq2_2Ec_2Efmaptree_2Eapply__path_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efmaptree_2Eapply__path_2E2(s(tyop_2Elist_2Elist(A_27a),X0_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),X1_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)))),c_2Efmaptree_2Eapply__path_2E0),s(tyop_2Elist_2Elist(A_27a),X0_2E0))),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),X1_2E0))) ).

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

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

fof(arityeq2_2Ec_2Efmaptree_2Efmtreerec_2E2_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27c,axiom,
    ! [A_27a,A_27b,A_27c,X0_2E0,X1_2E0] : s(A_27a,c_2Efmaptree_2Efmtreerec_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,A_27a),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)),A_27a))),X0_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),X1_2E0))) = s(A_27a,app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,A_27a),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)),A_27a))),tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),A_27a)),c_2Efmaptree_2Efmtreerec_2E0),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,A_27a),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)),A_27a))),X0_2E0))),s(tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),X1_2E0))) ).

fof(arityeq2_2Ec_2Efmaptree_2EFTNode_2E2_2Emono_2EA_27c_20mono_2EA_27b,axiom,
    ! [A_27b,A_27c,X0_2E0,X1_2E0] : s(tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),c_2Efmaptree_2EFTNode_2E2(s(A_27b,X0_2E0),s(tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)),X1_2E0))) = s(tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)),tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)),tyop_2Efmaptree_2Efmaptree(A_27c,A_27b))),c_2Efmaptree_2EFTNode_2E0),s(A_27b,X0_2E0))),s(tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)),X1_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_28tyop_2Efmaptree_2Efmaptree_28A_27a_2CA_27b_29_29,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Ebool_2ECOND_2E3(s(tyop_2Emin_2Ebool,X0_2E0),s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),X1_2E0),s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),X2_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))))),c_2Ebool_2ECOND_2E0),s(tyop_2Emin_2Ebool,X0_2E0))),s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),X1_2E0))),s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),X2_2E0))) ).

fof(arityeq3_2Ec_2Efmaptree_2Efupd__at__path_2E3_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efmaptree_2Efupd__at__path_2E3(s(tyop_2Elist_2Elist(A_27a),X0_2E0),s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),X1_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),X2_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))))),c_2Efmaptree_2Efupd__at__path_2E0),s(tyop_2Elist_2Elist(A_27a),X0_2E0))),s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),X1_2E0))),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),X2_2E0))) ).

fof(arityeq3_2Ec_2Eoption_2Eoption__CASE_2E3_2Emono_2Etyop_2Efmaptree_2Efmaptree_28A_27a_2CA_27b_29_20mono_2Etyop_2Eoption_2Eoption_28tyop_2Efmaptree_2Efmaptree_28A_27a_2CA_27b_29_29,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Eoption_2Eoption__CASE_2E3(s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),X0_2E0),s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),X1_2E0),s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),X2_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))))),c_2Eoption_2Eoption__CASE_2E0),s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),X0_2E0))),s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),X1_2E0))),s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),X2_2E0))) ).

fof(arityeq3_2Ec_2Efmaptree_2Eupdate__at__path_2E3_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efmaptree_2Eupdate__at__path_2E3(s(tyop_2Elist_2Elist(A_27a),X0_2E0),s(A_27b,X1_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),X2_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))))),c_2Efmaptree_2Eupdate__at__path_2E0),s(tyop_2Elist_2Elist(A_27a),X0_2E0))),s(A_27b,X1_2E0))),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),X2_2E0))) ).

fof(arityeq3_2Ec_2Efmaptree_2Econstruct_2E3_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Eoption_2Eoption(A_27a),c_2Efmaptree_2Econstruct_2E3(s(A_27a,X0_2E0),s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),X1_2E0),s(tyop_2Elist_2Elist(A_27b),X2_2E0))) = s(tyop_2Eoption_2Eoption(A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)))),c_2Efmaptree_2Econstruct_2E0),s(A_27a,X0_2E0))),s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),X1_2E0))),s(tyop_2Elist_2Elist(A_27b),X2_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(arityeq3_2Ec_2Elist_2Elist__CASE_2E3_2Emono_2EA_27b_20mono_2Etyop_2Eoption_2Eoption_28A_27a_29,axiom,
    ! [A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Eoption_2Eoption(A_27a),c_2Elist_2Elist__CASE_2E3(s(tyop_2Elist_2Elist(A_27b),X0_2E0),s(tyop_2Eoption_2Eoption(A_27a),X1_2E0),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),X2_2E0))) = s(tyop_2Eoption_2Eoption(A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),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_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),tyop_2Eoption_2Eoption(A_27a))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),tyop_2Eoption_2Eoption(A_27a)))),c_2Elist_2Elist__CASE_2E0),s(tyop_2Elist_2Elist(A_27b),X0_2E0))),s(tyop_2Eoption_2Eoption(A_27a),X1_2E0))),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),X2_2E0))) ).

fof(arityeq3_2Ec_2Efmaptree_2Erelrec_2E3_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27c,axiom,
    ! [A_27a,A_27b,A_27c,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Emin_2Ebool,c_2Efmaptree_2Erelrec_2E3(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),A_27c))),X0_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),X1_2E0),s(A_27c,X2_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),A_27c))),tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool))),c_2Efmaptree_2Erelrec_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),A_27c))),X0_2E0))),s(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),X1_2E0))),s(A_27c,X2_2E0))) ).

fof(arityeq3_2Ec_2Efmaptree_2Erelrec_2E3_2Emono_2EA_27b_20mono_2EA_27c_20mono_2EA_27a,axiom,
    ! [A_27a,A_27b,A_27c,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Emin_2Ebool,c_2Efmaptree_2Erelrec_2E3(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,A_27a),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)),A_27a))),X0_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),X1_2E0),s(A_27a,X2_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,A_27a),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)),A_27a))),tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),c_2Efmaptree_2Erelrec_2E0),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,A_27a),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)),A_27a))),X0_2E0))),s(tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),X1_2E0))),s(A_27a,X2_2E0))) ).

fof(thm_2Efmaptree_2Econstruct__def,axiom,
    ! [A_27a,A_27b,F0_2E0] :
      ( ! [V1kfm_2E0,V3h_2E0,V4t_2E0] : s(tyop_2Eoption_2Eoption(A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)))),F0_2E0),s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),V1kfm_2E0))),s(A_27b,V3h_2E0))),s(tyop_2Elist_2Elist(A_27b),V4t_2E0))) = s(tyop_2Eoption_2Eoption(A_27a),c_2Ebool_2ECOND_2E3(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(A_27b,V3h_2E0),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),V1kfm_2E0))))),s(tyop_2Eoption_2Eoption(A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),c_2Efinite__map_2EFAPPLY_2E2(s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),V1kfm_2E0),s(A_27b,V3h_2E0))),s(tyop_2Elist_2Elist(A_27b),V4t_2E0))),s(tyop_2Eoption_2Eoption(A_27a),c_2Eoption_2ENONE_2E0)))
     => ! [V0a_2E0,V1kfm_2E0,V2kl_2E0] : s(tyop_2Eoption_2Eoption(A_27a),c_2Efmaptree_2Econstruct_2E3(s(A_27a,V0a_2E0),s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),V1kfm_2E0),s(tyop_2Elist_2Elist(A_27b),V2kl_2E0))) = s(tyop_2Eoption_2Eoption(A_27a),c_2Elist_2Elist__CASE_2E3(s(tyop_2Elist_2Elist(A_27b),V2kl_2E0),s(tyop_2Eoption_2Eoption(A_27a),c_2Eoption_2ESOME_2E1(s(A_27a,V0a_2E0))),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)))),F0_2E0),s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),V1kfm_2E0))))) ) ).

fof(thm_2Efmaptree_2Ewf__def,axiom,
    ! [A_27a,A_27b,V0x_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Efmaptree_2Ewf_2E1(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),V0x_2E0))))
    <=> ! [V1wf_27_2E0] :
          ( ! [V2a0_2E0] :
              ( ? [V3a_2E0,V4fm_2E0] :
                  ( s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),V2a0_2E0) = s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),c_2Efmaptree_2Econstruct_2E2(s(A_27a,V3a_2E0),s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),V4fm_2E0)))
                  & ! [V5k_2E0] :
                      ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(A_27b,V5k_2E0),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),V4fm_2E0))))))
                     => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Ebool),V1wf_27_2E0),s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),c_2Efinite__map_2EFAPPLY_2E2(s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),V4fm_2E0),s(A_27b,V5k_2E0)))))) ) )
             => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Ebool),V1wf_27_2E0),s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),V2a0_2E0)))) )
         => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Ebool),V1wf_27_2E0),s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),V0x_2E0)))) ) ) ).

fof(thm_2Efmaptree_2Efmaptree__TY__DEF,axiom,
    ! [A_27key,A_27value] :
    ? [V0rep_2E0] : p(s(tyop_2Emin_2Ebool,c_2Ebool_2ETYPE__DEFINITION_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value)),tyop_2Emin_2Ebool),c_2Efmaptree_2Ewf_2E0),s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27key,A_27value),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value))),V0rep_2E0)))) ).

fof(thm_2Efmaptree_2Efmap__bij__thm,axiom,
    ! [A_27key,A_27value] :
      ( ! [V0a_2E0] : s(tyop_2Efmaptree_2Efmaptree(A_27key,A_27value),c_2Efmaptree_2EfromF_2E1(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value)),c_2Efmaptree_2EtoF_2E1(s(tyop_2Efmaptree_2Efmaptree(A_27key,A_27value),V0a_2E0))))) = s(tyop_2Efmaptree_2Efmaptree(A_27key,A_27value),V0a_2E0)
      & ! [V1r_2E0] :
          ( p(s(tyop_2Emin_2Ebool,c_2Efmaptree_2Ewf_2E1(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value)),V1r_2E0))))
        <=> s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value)),c_2Efmaptree_2EtoF_2E1(s(tyop_2Efmaptree_2Efmaptree(A_27key,A_27value),c_2Efmaptree_2EfromF_2E1(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value)),V1r_2E0))))) = s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value)),V1r_2E0) ) ) ).

fof(thm_2Efmaptree_2EFTNode__def,axiom,
    ! [A_27a,A_27b,V0i_2E0,V1fm_2E0] : s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),c_2Efmaptree_2EFTNode_2E2(s(A_27b,V0i_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),V1fm_2E0))) = s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),c_2Efmaptree_2EfromF_2E1(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Eoption_2Eoption(A_27b)),c_2Efmaptree_2Econstruct_2E2(s(A_27b,V0i_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Eoption_2Eoption(A_27b))),c_2Efinite__map_2Eo__f_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Eoption_2Eoption(A_27b))),c_2Efmaptree_2EtoF_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),V1fm_2E0))))))) ).

fof(thm_2Efmaptree_2Eitem__map__def,axiom,
    ! [A_27a,A_27b,V0ft_2E0] : s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V0ft_2E0) = s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),c_2Efmaptree_2EFTNode_2E2(s(A_27b,c_2Efmaptree_2Eitem_2E1(s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V0ft_2E0))),s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efmaptree_2Emap_2E1(s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V0ft_2E0))))) ).

fof(thm_2Efmaptree_2Eapply__path__def,axiom,
    ! [A_27a,A_27b] :
      ( ! [V0ft_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efmaptree_2Eapply__path_2E2(s(tyop_2Elist_2Elist(A_27a),c_2Elist_2ENIL_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V0ft_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Eoption_2ESOME_2E1(s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V0ft_2E0)))
      & ! [V1h_2E0,V2t_2E0,V3ft_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efmaptree_2Eapply__path_2E2(s(tyop_2Elist_2Elist(A_27a),c_2Elist_2ECONS_2E2(s(A_27a,V1h_2E0),s(tyop_2Elist_2Elist(A_27a),V2t_2E0))),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V3ft_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Ebool_2ECOND_2E3(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(A_27a,V1h_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efmaptree_2Emap_2E1(s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V3ft_2E0))))))),s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efmaptree_2Eapply__path_2E2(s(tyop_2Elist_2Elist(A_27a),V2t_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),c_2Efinite__map_2EFAPPLY_2E2(s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efmaptree_2Emap_2E1(s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V3ft_2E0))),s(A_27a,V1h_2E0))))),s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Eoption_2ENONE_2E0))) ) ).

fof(thm_2Efmaptree_2Eupdate__at__path__def,axiom,
    ! [A_27a,A_27b,F0_2E0] :
      ( ! [V5ft_2E0,V2h_2E0,V6ft_27_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))))),F0_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V5ft_2E0))),s(A_27a,V2h_2E0))),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V6ft_27_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Eoption_2ESOME_2E1(s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),c_2Efmaptree_2EFTNode_2E2(s(A_27b,c_2Efmaptree_2Eitem_2E1(s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V5ft_2E0))),s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efinite__map_2EFUPDATE_2E2(s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efmaptree_2Emap_2E1(s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V5ft_2E0))),s(tyop_2Epair_2Eprod(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Epair_2E_2C_2E2(s(A_27a,V2h_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V6ft_27_2E0)))))))))
     => ( ! [V0a_2E0,V1ft_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efmaptree_2Eupdate__at__path_2E3(s(tyop_2Elist_2Elist(A_27a),c_2Elist_2ENIL_2E0),s(A_27b,V0a_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V1ft_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Eoption_2ESOME_2E1(s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),c_2Efmaptree_2EFTNode_2E2(s(A_27b,V0a_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efmaptree_2Emap_2E1(s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V1ft_2E0)))))))
        & ! [V2h_2E0,V3t_2E0,V4a_2E0,V5ft_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efmaptree_2Eupdate__at__path_2E3(s(tyop_2Elist_2Elist(A_27a),c_2Elist_2ECONS_2E2(s(A_27a,V2h_2E0),s(tyop_2Elist_2Elist(A_27a),V3t_2E0))),s(A_27b,V4a_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V5ft_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Ebool_2ECOND_2E3(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(A_27a,V2h_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efmaptree_2Emap_2E1(s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V5ft_2E0))))))),s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Eoption_2Eoption__CASE_2E3(s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efmaptree_2Eupdate__at__path_2E3(s(tyop_2Elist_2Elist(A_27a),V3t_2E0),s(A_27b,V4a_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),c_2Efinite__map_2EFAPPLY_2E2(s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efmaptree_2Emap_2E1(s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V5ft_2E0))),s(A_27a,V2h_2E0))))),s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Eoption_2ENONE_2E0),s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))))),F0_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V5ft_2E0))),s(A_27a,V2h_2E0))))),s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Eoption_2ENONE_2E0))) ) ) ).

fof(thm_2Efmaptree_2Efupd__at__path__def,axiom,
    ! [A_27a,A_27b,F0_2E0] :
      ( ! [V5ft_2E0,V2h_2E0,V6ft_27_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))))),F0_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V5ft_2E0))),s(A_27a,V2h_2E0))),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V6ft_27_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Eoption_2ESOME_2E1(s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),c_2Efmaptree_2EFTNode_2E2(s(A_27b,c_2Efmaptree_2Eitem_2E1(s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V5ft_2E0))),s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efinite__map_2EFUPDATE_2E2(s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efmaptree_2Emap_2E1(s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V5ft_2E0))),s(tyop_2Epair_2Eprod(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Epair_2E_2C_2E2(s(A_27a,V2h_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V6ft_27_2E0)))))))))
     => ( ! [V0f_2E0,V1ft_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efmaptree_2Efupd__at__path_2E3(s(tyop_2Elist_2Elist(A_27a),c_2Elist_2ENIL_2E0),s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),V0f_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V1ft_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),V0f_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V1ft_2E0)))
        & ! [V2h_2E0,V3t_2E0,V4f_2E0,V5ft_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efmaptree_2Efupd__at__path_2E3(s(tyop_2Elist_2Elist(A_27a),c_2Elist_2ECONS_2E2(s(A_27a,V2h_2E0),s(tyop_2Elist_2Elist(A_27a),V3t_2E0))),s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),V4f_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V5ft_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Ebool_2ECOND_2E3(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(A_27a,V2h_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efmaptree_2Emap_2E1(s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V5ft_2E0))))))),s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Eoption_2Eoption__CASE_2E3(s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efmaptree_2Efupd__at__path_2E3(s(tyop_2Elist_2Elist(A_27a),V3t_2E0),s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),V4f_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),c_2Efinite__map_2EFAPPLY_2E2(s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efmaptree_2Emap_2E1(s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V5ft_2E0))),s(A_27a,V2h_2E0))))),s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Eoption_2ENONE_2E0),s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))))),F0_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V5ft_2E0))),s(A_27a,V2h_2E0))))),s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Eoption_2ENONE_2E0))) ) ) ).

fof(thm_2Efmaptree_2Erelrec__def,axiom,
    ! [A_27a,A_27b,A_27c,V0x_2E0,V1x_27_2E0,V2x_27_27_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Efmaptree_2Erelrec_2E3(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),A_27c))),V0x_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),V1x_27_2E0),s(A_27c,V2x_27_27_2E0))))
    <=> ! [V3relrec_27_2E0] :
          ( ! [V4a0_2E0,V5a1_2E0] :
              ( ? [V6i_2E0,V7fm_2E0,V8rfm_2E0] :
                  ( s(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),V4a0_2E0) = s(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),c_2Efmaptree_2EFTNode_2E2(s(A_27a,V6i_2E0),s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V7fm_2E0)))
                  & s(A_27c,V5a1_2E0) = s(A_27c,app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),A_27c),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),A_27c)),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),A_27c))),V0x_2E0),s(A_27a,V6i_2E0))),s(tyop_2Efinite__map_2Efmap(A_27b,A_27c),V8rfm_2E0))),s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V7fm_2E0)))
                  & s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27b,A_27c),V8rfm_2E0))) = s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V7fm_2E0)))
                  & ! [V9d_2E0] :
                      ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(A_27b,V9d_2E0),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V7fm_2E0))))))
                     => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool)),V3relrec_27_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),c_2Efinite__map_2EFAPPLY_2E2(s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V7fm_2E0),s(A_27b,V9d_2E0))))),s(A_27c,c_2Efinite__map_2EFAPPLY_2E2(s(tyop_2Efinite__map_2Efmap(A_27b,A_27c),V8rfm_2E0),s(A_27b,V9d_2E0)))))) ) )
             => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool)),V3relrec_27_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),V4a0_2E0))),s(A_27c,V5a1_2E0)))) )
         => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool)),V3relrec_27_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),V1x_27_2E0))),s(A_27c,V2x_27_27_2E0)))) ) ) ).

fof(thm_2Efmaptree_2Efmtreerec__def,axiom,
    ! [A_27a,A_27b,A_27c,F0_2E0] :
      ( ! [V0h_2E0,V1ft_2E0,V2r_2E0] : s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,A_27a),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)),A_27a))),tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F0_2E0),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,A_27a),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)),A_27a))),V0h_2E0))),s(tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),V1ft_2E0))),s(A_27a,V2r_2E0))) = s(tyop_2Emin_2Ebool,c_2Efmaptree_2Erelrec_2E3(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,A_27a),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)),A_27a))),V0h_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),V1ft_2E0),s(A_27a,V2r_2E0)))
     => ! [V0h_2E0,V1ft_2E0] : s(A_27a,c_2Efmaptree_2Efmtreerec_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,A_27a),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)),A_27a))),V0h_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),V1ft_2E0))) = s(A_27a,c_2Emin_2E_40_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,A_27a),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)),A_27a))),tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F0_2E0),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,A_27a),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)),A_27a))),V0h_2E0))),s(tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),V1ft_2E0))))) ) ).

fof(thm_2Efmaptree_2Ewf__rules,axiom,
    ! [A_27a,A_27b,V0a_2E0,V1fm_2E0] :
      ( ! [V2k_2E0] :
          ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(A_27b,V2k_2E0),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),V1fm_2E0))))))
         => p(s(tyop_2Emin_2Ebool,c_2Efmaptree_2Ewf_2E1(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),c_2Efinite__map_2EFAPPLY_2E2(s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),V1fm_2E0),s(A_27b,V2k_2E0)))))) )
     => p(s(tyop_2Emin_2Ebool,c_2Efmaptree_2Ewf_2E1(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),c_2Efmaptree_2Econstruct_2E2(s(A_27a,V0a_2E0),s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),V1fm_2E0)))))) ) ).

fof(thm_2Efmaptree_2Ewf__ind,axiom,
    ! [A_27a,A_27b,V0wf_27_2E0] :
      ( ! [V1a_2E0,V2fm_2E0] :
          ( ! [V3k_2E0] :
              ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(A_27b,V3k_2E0),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),V2fm_2E0))))))
             => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Ebool),V0wf_27_2E0),s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),c_2Efinite__map_2EFAPPLY_2E2(s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),V2fm_2E0),s(A_27b,V3k_2E0)))))) )
         => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Ebool),V0wf_27_2E0),s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),c_2Efmaptree_2Econstruct_2E2(s(A_27a,V1a_2E0),s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),V2fm_2E0)))))) )
     => ! [V4a0_2E0] :
          ( p(s(tyop_2Emin_2Ebool,c_2Efmaptree_2Ewf_2E1(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),V4a0_2E0))))
         => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Ebool),V0wf_27_2E0),s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),V4a0_2E0)))) ) ) ).

fof(thm_2Efmaptree_2Ewf__strongind,axiom,
    ! [A_27a,A_27b,V0wf_27_2E0] :
      ( ! [V1a_2E0,V2fm_2E0] :
          ( ! [V3k_2E0] :
              ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(A_27b,V3k_2E0),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),V2fm_2E0))))))
             => ( p(s(tyop_2Emin_2Ebool,c_2Efmaptree_2Ewf_2E1(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),c_2Efinite__map_2EFAPPLY_2E2(s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),V2fm_2E0),s(A_27b,V3k_2E0))))))
                & p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Ebool),V0wf_27_2E0),s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),c_2Efinite__map_2EFAPPLY_2E2(s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),V2fm_2E0),s(A_27b,V3k_2E0)))))) ) )
         => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Ebool),V0wf_27_2E0),s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),c_2Efmaptree_2Econstruct_2E2(s(A_27a,V1a_2E0),s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),V2fm_2E0)))))) )
     => ! [V4a0_2E0] :
          ( p(s(tyop_2Emin_2Ebool,c_2Efmaptree_2Ewf_2E1(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),V4a0_2E0))))
         => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Ebool),V0wf_27_2E0),s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),V4a0_2E0)))) ) ) ).

fof(thm_2Efmaptree_2Ewf__cases,axiom,
    ! [A_27a,A_27b,V0a0_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Efmaptree_2Ewf_2E1(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),V0a0_2E0))))
    <=> ? [V1a_2E0,V2fm_2E0] :
          ( s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),V0a0_2E0) = s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),c_2Efmaptree_2Econstruct_2E2(s(A_27a,V1a_2E0),s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),V2fm_2E0)))
          & ! [V3k_2E0] :
              ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(A_27b,V3k_2E0),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),V2fm_2E0))))))
             => p(s(tyop_2Emin_2Ebool,c_2Efmaptree_2Ewf_2E1(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),c_2Efinite__map_2EFAPPLY_2E2(s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),V2fm_2E0),s(A_27b,V3k_2E0)))))) ) ) ) ).

fof(thm_2Efmaptree_2EFTNode__11,axiom,
    ! [A_27a,A_27b,V0i2_2E0,V1i1_2E0,V2f2_2E0,V3f1_2E0] :
      ( s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),c_2Efmaptree_2EFTNode_2E2(s(A_27b,V1i1_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),V3f1_2E0))) = s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),c_2Efmaptree_2EFTNode_2E2(s(A_27b,V0i2_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),V2f2_2E0)))
    <=> ( s(A_27b,V1i1_2E0) = s(A_27b,V0i2_2E0)
        & s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),V3f1_2E0) = s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),V2f2_2E0) ) ) ).

fof(thm_2Efmaptree_2Efmaptree__nchotomy,axiom,
    ! [A_27a,A_27b,V0ft_2E0] :
    ? [V1i_2E0,V2fm_2E0] : s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V0ft_2E0) = s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),c_2Efmaptree_2EFTNode_2E2(s(A_27b,V1i_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),V2fm_2E0))) ).

fof(thm_2Efmaptree_2Eitem__thm,axiom,
    ! [A_27a,A_27b,V0i_2E0,V1fm_2E0] : s(A_27a,c_2Efmaptree_2Eitem_2E1(s(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),c_2Efmaptree_2EFTNode_2E2(s(A_27a,V0i_2E0),s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V1fm_2E0))))) = s(A_27a,V0i_2E0) ).

fof(thm_2Efmaptree_2Emap__thm,axiom,
    ! [A_27a,A_27b,V0i_2E0,V1fm_2E0] : s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),c_2Efmaptree_2Emap_2E1(s(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),c_2Efmaptree_2EFTNode_2E2(s(A_27a,V0i_2E0),s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V1fm_2E0))))) = s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V1fm_2E0) ).

fof(thm_2Efmaptree_2Eft__ind,axiom,
    ! [A_27a,A_27b,V0P_2E0] :
      ( ! [V1a_2E0,V2fm_2E0] :
          ( ! [V3k_2E0] :
              ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(A_27a,V3k_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),V2fm_2E0))))))
             => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),c_2Efinite__map_2EFAPPLY_2E2(s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),V2fm_2E0),s(A_27a,V3k_2E0)))))) )
         => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),c_2Efmaptree_2EFTNode_2E2(s(A_27b,V1a_2E0),s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),V2fm_2E0)))))) )
     => ! [V4ft_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V4ft_2E0)))) ) ).

fof(thm_2Efmaptree_2Eapplicable__paths__FINITE,axiom,
    ! [A_27a,A_27b,F1_2E0,F0_2E0] :
      ( ! [V1p_2E0,V0ft_2E0] :
          ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Emin_2Ebool)),F1_2E0),s(tyop_2Elist_2Elist(A_27a),V1p_2E0))),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V0ft_2E0))))
        <=> ? [V2ft_27_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efmaptree_2Eapply__path_2E2(s(tyop_2Elist_2Elist(A_27a),V1p_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V0ft_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Eoption_2ESOME_2E1(s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V2ft_27_2E0))) )
     => ( ! [V0ft_2E0,V1p_2E0] : s(tyop_2Epair_2Eprod(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Epair_2Eprod(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Epair_2Eprod(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool))),F0_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V0ft_2E0))),s(tyop_2Elist_2Elist(A_27a),V1p_2E0))) = s(tyop_2Epair_2Eprod(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool),c_2Epair_2E_2C_2E2(s(tyop_2Elist_2Elist(A_27a),V1p_2E0),s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Emin_2Ebool)),F1_2E0),s(tyop_2Elist_2Elist(A_27a),V1p_2E0))),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V0ft_2E0)))))
       => ! [V0ft_2E0] : p(s(tyop_2Emin_2Ebool,c_2Epred__set_2EFINITE_2E1(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool),c_2Epred__set_2EGSPEC_2E1(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Epair_2Eprod(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Epair_2Eprod(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool))),F0_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V0ft_2E0)))))))) ) ) ).

fof(thm_2Efmaptree_2Eapply__path__SNOC,axiom,
    ! [A_27a,A_27b,F0_2E0] :
      ( ! [V1x_2E0,V3ft_27_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)))),F0_2E0),s(A_27a,V1x_2E0))),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V3ft_27_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efinite__map_2EFLOOKUP_2E2(s(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efmaptree_2Emap_2E1(s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V3ft_27_2E0))),s(A_27a,V1x_2E0)))
     => ! [V0ft_2E0,V1x_2E0,V2p_2E0] : s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efmaptree_2Eapply__path_2E2(s(tyop_2Elist_2Elist(A_27a),c_2Elist_2EAPPEND_2E2(s(tyop_2Elist_2Elist(A_27a),V2p_2E0),s(tyop_2Elist_2Elist(A_27a),c_2Elist_2ECONS_2E2(s(A_27a,V1x_2E0),s(tyop_2Elist_2Elist(A_27a),c_2Elist_2ENIL_2E0))))),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V0ft_2E0))) = s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Eoption_2Eoption__CASE_2E3(s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efmaptree_2Eapply__path_2E2(s(tyop_2Elist_2Elist(A_27a),V2p_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V0ft_2E0))),s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Eoption_2ENONE_2E0),s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)))),F0_2E0),s(A_27a,V1x_2E0))))) ) ).

fof(thm_2Efmaptree_2Erelrec__rules,axiom,
    ! [A_27a,A_27b,A_27c,V0h_2E0,V1i_2E0,V2fm_2E0,V3rfm_2E0] :
      ( ( s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27b,A_27c),V3rfm_2E0))) = s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V2fm_2E0)))
        & ! [V4d_2E0] :
            ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(A_27b,V4d_2E0),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V2fm_2E0))))))
           => p(s(tyop_2Emin_2Ebool,c_2Efmaptree_2Erelrec_2E3(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),A_27c))),V0h_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),c_2Efinite__map_2EFAPPLY_2E2(s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V2fm_2E0),s(A_27b,V4d_2E0))),s(A_27c,c_2Efinite__map_2EFAPPLY_2E2(s(tyop_2Efinite__map_2Efmap(A_27b,A_27c),V3rfm_2E0),s(A_27b,V4d_2E0)))))) ) )
     => p(s(tyop_2Emin_2Ebool,c_2Efmaptree_2Erelrec_2E3(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),A_27c))),V0h_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),c_2Efmaptree_2EFTNode_2E2(s(A_27a,V1i_2E0),s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V2fm_2E0))),s(A_27c,app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),A_27c),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),A_27c)),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),A_27c))),V0h_2E0),s(A_27a,V1i_2E0))),s(tyop_2Efinite__map_2Efmap(A_27b,A_27c),V3rfm_2E0))),s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V2fm_2E0)))))) ) ).

fof(thm_2Efmaptree_2Erelrec__ind,axiom,
    ! [A_27a,A_27b,A_27c,V0h_2E0,V1relrec_27_2E0] :
      ( ! [V2i_2E0,V3fm_2E0,V4rfm_2E0] :
          ( ( s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27b,A_27c),V4rfm_2E0))) = s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V3fm_2E0)))
            & ! [V5d_2E0] :
                ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(A_27b,V5d_2E0),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V3fm_2E0))))))
               => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool)),V1relrec_27_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),c_2Efinite__map_2EFAPPLY_2E2(s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V3fm_2E0),s(A_27b,V5d_2E0))))),s(A_27c,c_2Efinite__map_2EFAPPLY_2E2(s(tyop_2Efinite__map_2Efmap(A_27b,A_27c),V4rfm_2E0),s(A_27b,V5d_2E0)))))) ) )
         => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool)),V1relrec_27_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),c_2Efmaptree_2EFTNode_2E2(s(A_27a,V2i_2E0),s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V3fm_2E0))))),s(A_27c,app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),A_27c),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),A_27c)),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),A_27c))),V0h_2E0),s(A_27a,V2i_2E0))),s(tyop_2Efinite__map_2Efmap(A_27b,A_27c),V4rfm_2E0))),s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V3fm_2E0)))))) )
     => ! [V6a0_2E0,V7a1_2E0] :
          ( p(s(tyop_2Emin_2Ebool,c_2Efmaptree_2Erelrec_2E3(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),A_27c))),V0h_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),V6a0_2E0),s(A_27c,V7a1_2E0))))
         => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool)),V1relrec_27_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),V6a0_2E0))),s(A_27c,V7a1_2E0)))) ) ) ).

fof(thm_2Efmaptree_2Erelrec__strongind,axiom,
    ! [A_27a,A_27b,A_27c,V0h_2E0,V1relrec_27_2E0] :
      ( ! [V2i_2E0,V3fm_2E0,V4rfm_2E0] :
          ( ( s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27b,A_27c),V4rfm_2E0))) = s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V3fm_2E0)))
            & ! [V5d_2E0] :
                ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(A_27b,V5d_2E0),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V3fm_2E0))))))
               => ( p(s(tyop_2Emin_2Ebool,c_2Efmaptree_2Erelrec_2E3(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),A_27c))),V0h_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),c_2Efinite__map_2EFAPPLY_2E2(s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V3fm_2E0),s(A_27b,V5d_2E0))),s(A_27c,c_2Efinite__map_2EFAPPLY_2E2(s(tyop_2Efinite__map_2Efmap(A_27b,A_27c),V4rfm_2E0),s(A_27b,V5d_2E0))))))
                  & p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool)),V1relrec_27_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),c_2Efinite__map_2EFAPPLY_2E2(s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V3fm_2E0),s(A_27b,V5d_2E0))))),s(A_27c,c_2Efinite__map_2EFAPPLY_2E2(s(tyop_2Efinite__map_2Efmap(A_27b,A_27c),V4rfm_2E0),s(A_27b,V5d_2E0)))))) ) ) )
         => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool)),V1relrec_27_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),c_2Efmaptree_2EFTNode_2E2(s(A_27a,V2i_2E0),s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V3fm_2E0))))),s(A_27c,app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),A_27c),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),A_27c)),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),A_27c))),V0h_2E0),s(A_27a,V2i_2E0))),s(tyop_2Efinite__map_2Efmap(A_27b,A_27c),V4rfm_2E0))),s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V3fm_2E0)))))) )
     => ! [V6a0_2E0,V7a1_2E0] :
          ( p(s(tyop_2Emin_2Ebool,c_2Efmaptree_2Erelrec_2E3(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),A_27c))),V0h_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),V6a0_2E0),s(A_27c,V7a1_2E0))))
         => p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool)),V1relrec_27_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),V6a0_2E0))),s(A_27c,V7a1_2E0)))) ) ) ).

fof(thm_2Efmaptree_2Erelrec__cases,axiom,
    ! [A_27a,A_27b,A_27c,V0h_2E0,V1a0_2E0,V2a1_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Efmaptree_2Erelrec_2E3(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),A_27c))),V0h_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),V1a0_2E0),s(A_27c,V2a1_2E0))))
    <=> ? [V3i_2E0,V4fm_2E0,V5rfm_2E0] :
          ( s(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),V1a0_2E0) = s(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),c_2Efmaptree_2EFTNode_2E2(s(A_27a,V3i_2E0),s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V4fm_2E0)))
          & s(A_27c,V2a1_2E0) = s(A_27c,app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),A_27c),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),A_27c)),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),A_27c))),V0h_2E0),s(A_27a,V3i_2E0))),s(tyop_2Efinite__map_2Efmap(A_27b,A_27c),V5rfm_2E0))),s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V4fm_2E0)))
          & s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27b,A_27c),V5rfm_2E0))) = s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V4fm_2E0)))
          & ! [V6d_2E0] :
              ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(A_27b,V6d_2E0),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E1(s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V4fm_2E0))))))
             => p(s(tyop_2Emin_2Ebool,c_2Efmaptree_2Erelrec_2E3(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),A_27c))),V0h_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),c_2Efinite__map_2EFAPPLY_2E2(s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V4fm_2E0),s(A_27b,V6d_2E0))),s(A_27c,c_2Efinite__map_2EFAPPLY_2E2(s(tyop_2Efinite__map_2Efmap(A_27b,A_27c),V5rfm_2E0),s(A_27b,V6d_2E0)))))) ) ) ) ).

fof(thm_2Efmaptree_2Efmtreerec__thm,axiom,
    ! [A_27a,A_27b,A_27c,V0i_2E0,V1h_2E0,V2fm_2E0] : s(A_27a,c_2Efmaptree_2Efmtreerec_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,A_27a),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)),A_27a))),V1h_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),c_2Efmaptree_2EFTNode_2E2(s(A_27b,V0i_2E0),s(tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)),V2fm_2E0))))) = s(A_27a,app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)),A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,A_27a),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)),A_27a)),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,A_27a),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)),A_27a))),V1h_2E0),s(A_27b,V0i_2E0))),s(tyop_2Efinite__map_2Efmap(A_27c,A_27a),c_2Efinite__map_2Eo__f_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),A_27a),c_2Efmaptree_2Efmtreerec_2E1(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,A_27a),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)),A_27a))),V1h_2E0))),s(tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)),V2fm_2E0))))),s(tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)),V2fm_2E0))) ).

fof(thm_2Efmaptree_2Efmtree__Axiom,axiom,
    ! [A_27a,A_27b,A_27c,V0h_2E0] :
    ? [V1f_2E0] :
    ! [V2i_2E0,V3fm_2E0] : s(A_27c,app_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),A_27c),V1f_2E0),s(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),c_2Efmaptree_2EFTNode_2E2(s(A_27a,V2i_2E0),s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V3fm_2E0))))) = s(A_27c,app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,A_27c),A_27c),app_2E2(s(tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,A_27c),A_27c)),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27b,A_27c),A_27c))),V0h_2E0),s(A_27a,V2i_2E0))),s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V3fm_2E0))),s(tyop_2Efinite__map_2Efmap(A_27b,A_27c),c_2Efinite__map_2Eo__f_2E2(s(tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),A_27c),V1f_2E0),s(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V3fm_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) ) ).

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