%------------------------------------------------------------------------------
% File     : ITP083_4 : TPTP v9.0.0. Bugfixed v7.5.0.
% Domain   : Interactive Theorem Proving
% Axioms   : HOL4 syntactic export, chainy mode
% Version  : [BG+19] axioms.
% English  :

% Refs     : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
%          : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source   : [BG+19]
% Names    : fmaptree.ax [Gau19]
%          : HL4083_4.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  224 (  74 unt; 115 typ;   0 def)
%            Number of atoms       :  194 ( 110 equ)
%            Maximal formula atoms :    8 (   0 avg)
%            Number of connectives :   88 (   3   ~;   3   |;  20   &)
%                                         (  20 <=>;  42  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   21 (   6 avg)
%            Maximal term depth    :   16 (   3 avg)
%            Number of types       :    7 (   6 usr)
%            Number of type conns  :  101 (  59   >;  42   *;   0   +;   0  <<)
%            Number of predicates  :    2 (   1 usr;   0 prp; 1-2 aty)
%            Number of functors    :  108 ( 108 usr;  50 con; 0-3 aty)
%            Number of variables   :  455 ( 439   !;  16   ?; 455   :)
% SPC      : TF0_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(u,type,
    u: $tType ).

tff(d,type,
    d: $tType ).

tff(du,type,
    du: $tType ).

tff(mono_2Etyop_2Emin_2Ebool,type,
    mono_2Etyop_2Emin_2Ebool: $tType ).

tff(mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29,type,
    mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29: $tType ).

tff(mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29_29,type,
    mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29_29: $tType ).

tff(tyop_2Efinite__map_2Efmap,type,
    tyop_2Efinite__map_2Efmap: ( d * d ) > d ).

tff(tyop_2Efmaptree_2Efmaptree,type,
    tyop_2Efmaptree_2Efmaptree: ( d * d ) > d ).

tff(tyop_2Elist_2Elist,type,
    tyop_2Elist_2Elist: d > d ).

tff(tyop_2Emin_2Ebool,type,
    tyop_2Emin_2Ebool: d ).

tff(tyop_2Emin_2Efun,type,
    tyop_2Emin_2Efun: ( d * d ) > d ).

tff(tyop_2Eoption_2Eoption,type,
    tyop_2Eoption_2Eoption: d > d ).

tff(tyop_2Epair_2Eprod,type,
    tyop_2Epair_2Eprod: ( d * d ) > d ).

tff(s,type,
    s: ( d * u ) > du ).

tff(app_2E2,type,
    app_2E2: ( du * du ) > u ).

tff(p,type,
    p: mono_2Etyop_2Emin_2Ebool > $o ).

tff(combin_i_2E0,type,
    combin_i_2E0: u ).

tff(combin_k_2E0,type,
    combin_k_2E0: u ).

tff(combin_s_2E0,type,
    combin_s_2E0: u ).

tff(c_2Ebool_2E_21_2E0,type,
    c_2Ebool_2E_21_2E0: u ).

tff(c_2Ebool_2E_21_2E1,type,
    c_2Ebool_2E_21_2E1: du > u ).

tff(c_2Epair_2E_2C_2E0,type,
    c_2Epair_2E_2C_2E0: u ).

tff(c_2Epair_2E_2C_2E2,type,
    c_2Epair_2E_2C_2E2: ( du * du ) > u ).

tff(c_2Ebool_2E_2F_5C_2E0,type,
    c_2Ebool_2E_2F_5C_2E0: u ).

tff(c_2Ebool_2E_2F_5C_2E2,type,
    c_2Ebool_2E_2F_5C_2E2: ( du * du ) > u ).

tff(c_2Emin_2E_3D_2E0,type,
    c_2Emin_2E_3D_2E0: u ).

tff(c_2Emin_2E_3D_2E2,type,
    c_2Emin_2E_3D_2E2: ( du * du ) > u ).

tff(c_2Emin_2E_3D_3D_3E_2E0,type,
    c_2Emin_2E_3D_3D_3E_2E0: u ).

tff(c_2Emin_2E_3D_3D_3E_2E2,type,
    c_2Emin_2E_3D_3D_3E_2E2: ( du * du ) > u ).

tff(c_2Ebool_2E_3F_2E0,type,
    c_2Ebool_2E_3F_2E0: u ).

tff(c_2Ebool_2E_3F_2E1,type,
    c_2Ebool_2E_3F_2E1: du > u ).

tff(c_2Emin_2E_40_2E0,type,
    c_2Emin_2E_40_2E0: u ).

tff(c_2Emin_2E_40_2E1,type,
    c_2Emin_2E_40_2E1: du > u ).

tff(c_2Elist_2EAPPEND_2E0,type,
    c_2Elist_2EAPPEND_2E0: u ).

tff(c_2Elist_2EAPPEND_2E2,type,
    c_2Elist_2EAPPEND_2E2: ( du * du ) > u ).

tff(c_2Ebool_2ECOND_2E0,type,
    c_2Ebool_2ECOND_2E0: u ).

tff(c_2Ebool_2ECOND_2E3,type,
    c_2Ebool_2ECOND_2E3: ( du * du * du ) > u ).

tff(c_2Elist_2ECONS_2E0,type,
    c_2Elist_2ECONS_2E0: u ).

tff(c_2Elist_2ECONS_2E2,type,
    c_2Elist_2ECONS_2E2: ( du * du ) > u ).

tff(c_2Ebool_2EF_2E0,type,
    c_2Ebool_2EF_2E0: u ).

tff(c_2Efinite__map_2EFAPPLY_2E0,type,
    c_2Efinite__map_2EFAPPLY_2E0: u ).

tff(c_2Efinite__map_2EFAPPLY_2E2,type,
    c_2Efinite__map_2EFAPPLY_2E2: ( du * du ) > u ).

tff(c_2Efinite__map_2EFDOM_2E0,type,
    c_2Efinite__map_2EFDOM_2E0: u ).

tff(c_2Efinite__map_2EFDOM_2E1,type,
    c_2Efinite__map_2EFDOM_2E1: du > u ).

tff(c_2Epred__set_2EFINITE_2E0,type,
    c_2Epred__set_2EFINITE_2E0: u ).

tff(c_2Epred__set_2EFINITE_2E1,type,
    c_2Epred__set_2EFINITE_2E1: du > u ).

tff(c_2Efinite__map_2EFLOOKUP_2E0,type,
    c_2Efinite__map_2EFLOOKUP_2E0: u ).

tff(c_2Efinite__map_2EFLOOKUP_2E2,type,
    c_2Efinite__map_2EFLOOKUP_2E2: ( du * du ) > u ).

tff(c_2Efmaptree_2EFTNode_2E0,type,
    c_2Efmaptree_2EFTNode_2E0: u ).

tff(c_2Efmaptree_2EFTNode_2E2,type,
    c_2Efmaptree_2EFTNode_2E2: ( du * du ) > u ).

tff(c_2Efinite__map_2EFUPDATE_2E0,type,
    c_2Efinite__map_2EFUPDATE_2E0: u ).

tff(c_2Efinite__map_2EFUPDATE_2E2,type,
    c_2Efinite__map_2EFUPDATE_2E2: ( du * du ) > u ).

tff(c_2Epred__set_2EGSPEC_2E0,type,
    c_2Epred__set_2EGSPEC_2E0: u ).

tff(c_2Epred__set_2EGSPEC_2E1,type,
    c_2Epred__set_2EGSPEC_2E1: du > u ).

tff(c_2Ebool_2EIN_2E0,type,
    c_2Ebool_2EIN_2E0: u ).

tff(c_2Ebool_2EIN_2E2,type,
    c_2Ebool_2EIN_2E2: ( du * du ) > u ).

tff(c_2Elist_2ENIL_2E0,type,
    c_2Elist_2ENIL_2E0: u ).

tff(c_2Eoption_2ENONE_2E0,type,
    c_2Eoption_2ENONE_2E0: u ).

tff(c_2Eoption_2ESOME_2E0,type,
    c_2Eoption_2ESOME_2E0: u ).

tff(c_2Eoption_2ESOME_2E1,type,
    c_2Eoption_2ESOME_2E1: du > u ).

tff(c_2Ebool_2ET_2E0,type,
    c_2Ebool_2ET_2E0: u ).

tff(c_2Ebool_2ETYPE__DEFINITION_2E0,type,
    c_2Ebool_2ETYPE__DEFINITION_2E0: u ).

tff(c_2Ebool_2ETYPE__DEFINITION_2E2,type,
    c_2Ebool_2ETYPE__DEFINITION_2E2: ( du * du ) > u ).

tff(c_2Ebool_2E_5C_2F_2E0,type,
    c_2Ebool_2E_5C_2F_2E0: u ).

tff(c_2Ebool_2E_5C_2F_2E2,type,
    c_2Ebool_2E_5C_2F_2E2: ( du * du ) > u ).

tff(c_2Efmaptree_2Eapply__path_2E0,type,
    c_2Efmaptree_2Eapply__path_2E0: u ).

tff(c_2Efmaptree_2Eapply__path_2E2,type,
    c_2Efmaptree_2Eapply__path_2E2: ( du * du ) > u ).

tff(c_2Efmaptree_2Econstruct_2E0,type,
    c_2Efmaptree_2Econstruct_2E0: u ).

tff(c_2Efmaptree_2Econstruct_2E2,type,
    c_2Efmaptree_2Econstruct_2E2: ( du * du ) > u ).

tff(c_2Efmaptree_2Econstruct_2E3,type,
    c_2Efmaptree_2Econstruct_2E3: ( du * du * du ) > u ).

tff(c_2Efmaptree_2Efmtreerec_2E0,type,
    c_2Efmaptree_2Efmtreerec_2E0: u ).

tff(c_2Efmaptree_2Efmtreerec_2E1,type,
    c_2Efmaptree_2Efmtreerec_2E1: du > u ).

tff(c_2Efmaptree_2Efmtreerec_2E2,type,
    c_2Efmaptree_2Efmtreerec_2E2: ( du * du ) > u ).

tff(c_2Efmaptree_2EfromF_2E0,type,
    c_2Efmaptree_2EfromF_2E0: u ).

tff(c_2Efmaptree_2EfromF_2E1,type,
    c_2Efmaptree_2EfromF_2E1: du > u ).

tff(c_2Efmaptree_2Efupd__at__path_2E0,type,
    c_2Efmaptree_2Efupd__at__path_2E0: u ).

tff(c_2Efmaptree_2Efupd__at__path_2E3,type,
    c_2Efmaptree_2Efupd__at__path_2E3: ( du * du * du ) > u ).

tff(c_2Efmaptree_2Eitem_2E0,type,
    c_2Efmaptree_2Eitem_2E0: u ).

tff(c_2Efmaptree_2Eitem_2E1,type,
    c_2Efmaptree_2Eitem_2E1: du > u ).

tff(c_2Elist_2Elist__CASE_2E0,type,
    c_2Elist_2Elist__CASE_2E0: u ).

tff(c_2Elist_2Elist__CASE_2E3,type,
    c_2Elist_2Elist__CASE_2E3: ( du * du * du ) > u ).

tff(c_2Efmaptree_2Emap_2E0,type,
    c_2Efmaptree_2Emap_2E0: u ).

tff(c_2Efmaptree_2Emap_2E1,type,
    c_2Efmaptree_2Emap_2E1: du > u ).

tff(c_2Efinite__map_2Eo__f_2E0,type,
    c_2Efinite__map_2Eo__f_2E0: u ).

tff(c_2Efinite__map_2Eo__f_2E2,type,
    c_2Efinite__map_2Eo__f_2E2: ( du * du ) > u ).

tff(c_2Eoption_2Eoption__CASE_2E0,type,
    c_2Eoption_2Eoption__CASE_2E0: u ).

tff(c_2Eoption_2Eoption__CASE_2E3,type,
    c_2Eoption_2Eoption__CASE_2E3: ( du * du * du ) > u ).

tff(c_2Efmaptree_2Erelrec_2E0,type,
    c_2Efmaptree_2Erelrec_2E0: u ).

tff(c_2Efmaptree_2Erelrec_2E3,type,
    c_2Efmaptree_2Erelrec_2E3: ( du * du * du ) > u ).

tff(c_2Efmaptree_2EtoF_2E0,type,
    c_2Efmaptree_2EtoF_2E0: u ).

tff(c_2Efmaptree_2EtoF_2E1,type,
    c_2Efmaptree_2EtoF_2E1: du > u ).

tff(c_2Efmaptree_2Eupdate__at__path_2E0,type,
    c_2Efmaptree_2Eupdate__at__path_2E0: u ).

tff(c_2Efmaptree_2Eupdate__at__path_2E3,type,
    c_2Efmaptree_2Eupdate__at__path_2E3: ( du * du * du ) > u ).

tff(c_2Efmaptree_2Ewf_2E0,type,
    c_2Efmaptree_2Ewf_2E0: u ).

tff(c_2Efmaptree_2Ewf_2E1,type,
    c_2Efmaptree_2Ewf_2E1: du > u ).

tff(c_2Ebool_2E_7E_2E0,type,
    c_2Ebool_2E_7E_2E0: u ).

tff(c_2Ebool_2E_7E_2E1,type,
    c_2Ebool_2E_7E_2E1: du > u ).

tff(mono_2Eapp_2E2_2Emono_2Etyop_2Emin_2Ebool_20mono_2Etyop_2Emin_2Ebool,type,
    mono_2Eapp_2E2_2Emono_2Etyop_2Emin_2Ebool_20mono_2Etyop_2Emin_2Ebool: ( mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29 * mono_2Etyop_2Emin_2Ebool ) > mono_2Etyop_2Emin_2Ebool ).

tff(mono_2Eapp_2E2_2Emono_2Etyop_2Emin_2Ebool_20mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29,type,
    mono_2Eapp_2E2_2Emono_2Etyop_2Emin_2Ebool_20mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29: ( mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29_29 * mono_2Etyop_2Emin_2Ebool ) > mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29 ).

tff(mono_2Ec_2Ebool_2E_2F_5C_2E0,type,
    mono_2Ec_2Ebool_2E_2F_5C_2E0: mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29_29 ).

tff(mono_2Ec_2Ebool_2E_2F_5C_2E2,type,
    mono_2Ec_2Ebool_2E_2F_5C_2E2: ( mono_2Etyop_2Emin_2Ebool * mono_2Etyop_2Emin_2Ebool ) > mono_2Etyop_2Emin_2Ebool ).

tff(mono_2Ec_2Emin_2E_3D_3D_3E_2E0,type,
    mono_2Ec_2Emin_2E_3D_3D_3E_2E0: mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29_29 ).

tff(mono_2Ec_2Emin_2E_3D_3D_3E_2E2,type,
    mono_2Ec_2Emin_2E_3D_3D_3E_2E2: ( mono_2Etyop_2Emin_2Ebool * mono_2Etyop_2Emin_2Ebool ) > mono_2Etyop_2Emin_2Ebool ).

tff(mono_2Ec_2Ebool_2EF_2E0,type,
    mono_2Ec_2Ebool_2EF_2E0: mono_2Etyop_2Emin_2Ebool ).

tff(mono_2Ec_2Ebool_2ET_2E0,type,
    mono_2Ec_2Ebool_2ET_2E0: mono_2Etyop_2Emin_2Ebool ).

tff(mono_2Ec_2Ebool_2E_5C_2F_2E0,type,
    mono_2Ec_2Ebool_2E_5C_2F_2E0: mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29_29 ).

tff(mono_2Ec_2Ebool_2E_5C_2F_2E2,type,
    mono_2Ec_2Ebool_2E_5C_2F_2E2: ( mono_2Etyop_2Emin_2Ebool * mono_2Etyop_2Emin_2Ebool ) > mono_2Etyop_2Emin_2Ebool ).

tff(mono_2Ec_2Ebool_2E_7E_2E0,type,
    mono_2Ec_2Ebool_2E_7E_2E0: mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29 ).

tff(mono_2Ec_2Ebool_2E_7E_2E1,type,
    mono_2Ec_2Ebool_2E_7E_2E1: mono_2Etyop_2Emin_2Ebool > mono_2Etyop_2Emin_2Ebool ).

tff(i_mono_2Etyop_2Emin_2Ebool,type,
    i_mono_2Etyop_2Emin_2Ebool: mono_2Etyop_2Emin_2Ebool > u ).

tff(i_mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29,type,
    i_mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29: mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29 > u ).

tff(i_mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29_29,type,
    i_mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29_29: mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29_29 > u ).

tff(j_mono_2Etyop_2Emin_2Ebool,type,
    j_mono_2Etyop_2Emin_2Ebool: du > mono_2Etyop_2Emin_2Ebool ).

tff(j_mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29,type,
    j_mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29: du > mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29 ).

tff(j_mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29_29,type,
    j_mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29_29: du > mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29_29 ).

tff(reserved_2Eho_2Eeq__ext,axiom,
    ! [A_27a: d,A_27b: d,V0f_2E0: u,V1g_2E0: u] :
      ( ! [V2x_2E0: u] : ( 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) ) ) ).

tff(reserved_2Eho_2Eboolext,axiom,
    ! [V0_2E0: mono_2Etyop_2Emin_2Ebool,V1_2E0: mono_2Etyop_2Emin_2Ebool] :
      ( ( p(V0_2E0)
      <=> p(V1_2E0) )
     => ( V0_2E0 = V1_2E0 ) ) ).

tff(reserved_2Eho_2Etruth,axiom,
    p(mono_2Ec_2Ebool_2ET_2E0) ).

tff(reserved_2Eho_2Enotfalse,axiom,
    ~ p(mono_2Ec_2Ebool_2EF_2E0) ).

tff(reserved_2Eho_2Ebool__cases__ax,axiom,
    ! [V0t_2E0: mono_2Etyop_2Emin_2Ebool] :
      ( ( V0t_2E0 = mono_2Ec_2Ebool_2ET_2E0 )
      | ( V0t_2E0 = mono_2Ec_2Ebool_2EF_2E0 ) ) ).

tff(reserved_2Eho_2Ei__thm,axiom,
    ! [A_27a: d,V0x_2E0: u] : ( 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) ) ).

tff(reserved_2Eho_2Ek__thm,axiom,
    ! [A_27a: d,A_27b: d,V0x_2E0: u,V1y_2E0: u] : ( 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) ) ).

tff(reserved_2Eho_2Es__thm,axiom,
    ! [A_27a: d,A_27b: d,A_27c: d,V0f_2E0: u,V1g_2E0: u,V2x_2E0: u] : ( 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))))) ) ).

tff(reserved_2Elogic_2E_2F_5C,axiom,
    ! [V0_2E0: mono_2Etyop_2Emin_2Ebool,V1_2E0: mono_2Etyop_2Emin_2Ebool] :
      ( p(mono_2Ec_2Ebool_2E_2F_5C_2E2(V0_2E0,V1_2E0))
    <=> ( p(V0_2E0)
        & p(V1_2E0) ) ) ).

tff(reserved_2Elogic_2E_5C_2F,axiom,
    ! [V0_2E0: mono_2Etyop_2Emin_2Ebool,V1_2E0: mono_2Etyop_2Emin_2Ebool] :
      ( p(mono_2Ec_2Ebool_2E_5C_2F_2E2(V0_2E0,V1_2E0))
    <=> ( p(V0_2E0)
        | p(V1_2E0) ) ) ).

tff(reserved_2Elogic_2E_7E,axiom,
    ! [V0_2E0: mono_2Etyop_2Emin_2Ebool] :
      ( p(mono_2Ec_2Ebool_2E_7E_2E1(V0_2E0))
    <=> ~ p(V0_2E0) ) ).

tff(reserved_2Elogic_2E_3D_3D_3E,axiom,
    ! [V0_2E0: mono_2Etyop_2Emin_2Ebool,V1_2E0: mono_2Etyop_2Emin_2Ebool] :
      ( p(mono_2Ec_2Emin_2E_3D_3D_3E_2E2(V0_2E0,V1_2E0))
    <=> ( p(V0_2E0)
       => p(V1_2E0) ) ) ).

tff(reserved_2Elogic_2E_3D,axiom,
    ! [A_27a: d,V0_2E0: u,V1_2E0: u] :
      ( p(j_mono_2Etyop_2Emin_2Ebool(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) ) ) ).

tff(reserved_2Equant_2E_21,axiom,
    ! [A_27a: d,V0f_2E0: u] :
      ( p(j_mono_2Etyop_2Emin_2Ebool(s(tyop_2Emin_2Ebool,c_2Ebool_2E_21_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0)))))
    <=> ! [V1x_2E0: u] : p(j_mono_2Etyop_2Emin_2Ebool(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0),s(A_27a,V1x_2E0))))) ) ).

tff(reserved_2Equant_2E_3F,axiom,
    ! [A_27a: d,V0f_2E0: u] :
      ( p(j_mono_2Etyop_2Emin_2Ebool(s(tyop_2Emin_2Ebool,c_2Ebool_2E_3F_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0)))))
    <=> ? [V1x_2E0: u] : p(j_mono_2Etyop_2Emin_2Ebool(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0),s(A_27a,V1x_2E0))))) ) ).

tff(ij_2Emono_2Etyop_2Emin_2Ebool,axiom,
    ! [V0_2E0: u] : ( s(tyop_2Emin_2Ebool,i_mono_2Etyop_2Emin_2Ebool(j_mono_2Etyop_2Emin_2Ebool(s(tyop_2Emin_2Ebool,V0_2E0)))) = s(tyop_2Emin_2Ebool,V0_2E0) ) ).

tff(ij_2Emono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29,axiom,
    ! [V0_2E0: u] : ( s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),i_mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29(j_mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),V0_2E0)))) = s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),V0_2E0) ) ).

tff(ij_2Emono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29_29,axiom,
    ! [V0_2E0: u] : ( s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)),i_mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29_29(j_mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29_29(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)),V0_2E0)))) = s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)),V0_2E0) ) ).

tff(ji_2Emono_2Etyop_2Emin_2Ebool,axiom,
    ! [V0_2E0: mono_2Etyop_2Emin_2Ebool] : ( j_mono_2Etyop_2Emin_2Ebool(s(tyop_2Emin_2Ebool,i_mono_2Etyop_2Emin_2Ebool(V0_2E0))) = V0_2E0 ) ).

tff(ji_2Emono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29,axiom,
    ! [V0_2E0: mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29] : ( j_mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),i_mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29(V0_2E0))) = V0_2E0 ) ).

tff(ji_2Emono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29_29,axiom,
    ! [V0_2E0: mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29_29] : ( j_mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29_29(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)),i_mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29_29(V0_2E0))) = V0_2E0 ) ).

tff(arityeq1_2Ec_2Ebool_2E_21_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: d,X0_2E0: u] : ( j_mono_2Etyop_2Emin_2Ebool(s(tyop_2Emin_2Ebool,c_2Ebool_2E_21_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0)))) = j_mono_2Etyop_2Emin_2Ebool(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)))) ) ).

tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2EA_27a_20mono_2Etyop_2Efmaptree_2Efmaptree_28A_27a_2CA_27b_29,axiom,
    ! [A_27a: d,A_27b: d,X0_2E0: u,X1_2E0: u] : ( 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))) ) ).

tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Elist_2Elist_28A_27a_29_20mono_2Etyop_2Emin_2Ebool,axiom,
    ! [A_27a: d,X0_2E0: u,X1_2E0: mono_2Etyop_2Emin_2Ebool] : ( 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,i_mono_2Etyop_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,i_mono_2Etyop_2Emin_2Ebool(X1_2E0)))) ) ).

tff(arityeq2_2Ec_2Ebool_2E_2F_5C_2E2,axiom,
    ! [X0_2E0: mono_2Etyop_2Emin_2Ebool,X1_2E0: mono_2Etyop_2Emin_2Ebool] :
      ( ( p(X0_2E0)
        & p(X1_2E0) )
    <=> p(mono_2Eapp_2E2_2Emono_2Etyop_2Emin_2Ebool_20mono_2Etyop_2Emin_2Ebool(mono_2Eapp_2E2_2Emono_2Etyop_2Emin_2Ebool_20mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29(mono_2Ec_2Ebool_2E_2F_5C_2E0,X0_2E0),X1_2E0)) ) ).

tff(arityeq2_2Ec_2Emin_2E_3D_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: d,X0_2E0: u,X1_2E0: u] :
      ( ( s(A_27a,X0_2E0) = s(A_27a,X1_2E0) )
    <=> p(j_mono_2Etyop_2Emin_2Ebool(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))))) ) ).

tff(arityeq2_2Ec_2Emin_2E_3D_3D_3E_2E2,axiom,
    ! [X0_2E0: mono_2Etyop_2Emin_2Ebool,X1_2E0: mono_2Etyop_2Emin_2Ebool] :
      ( ( p(X0_2E0)
       => p(X1_2E0) )
    <=> p(mono_2Eapp_2E2_2Emono_2Etyop_2Emin_2Ebool_20mono_2Etyop_2Emin_2Ebool(mono_2Eapp_2E2_2Emono_2Etyop_2Emin_2Ebool_20mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29(mono_2Ec_2Emin_2E_3D_3D_3E_2E0,X0_2E0),X1_2E0)) ) ).

tff(arityeq1_2Ec_2Ebool_2E_3F_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: d,X0_2E0: u] : ( j_mono_2Etyop_2Emin_2Ebool(s(tyop_2Emin_2Ebool,c_2Ebool_2E_3F_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0)))) = j_mono_2Etyop_2Emin_2Ebool(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)))) ) ).

tff(arityeq1_2Ec_2Emin_2E_40_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: d,X0_2E0: u] : ( 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))) ) ).

tff(arityeq2_2Ec_2Elist_2EAPPEND_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: d,X0_2E0: u,X1_2E0: u] : ( 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))) ) ).

tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Eoption_2Eoption_28A_27a_29,axiom,
    ! [A_27a: d,X0_2E0: mono_2Etyop_2Emin_2Ebool,X1_2E0: u,X2_2E0: u] : ( s(tyop_2Eoption_2Eoption(A_27a),c_2Ebool_2ECOND_2E3(s(tyop_2Emin_2Ebool,i_mono_2Etyop_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,i_mono_2Etyop_2Emin_2Ebool(X0_2E0)))),s(tyop_2Eoption_2Eoption(A_27a),X1_2E0))),s(tyop_2Eoption_2Eoption(A_27a),X2_2E0))) ) ).

tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Eoption_2Eoption_28tyop_2Efmaptree_2Efmaptree_28A_27a_2CA_27b_29_29,axiom,
    ! [A_27a: d,A_27b: d,X0_2E0: mono_2Etyop_2Emin_2Ebool,X1_2E0: u,X2_2E0: u] : ( s(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Ebool_2ECOND_2E3(s(tyop_2Emin_2Ebool,i_mono_2Etyop_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,i_mono_2Etyop_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))) ) ).

tff(arityeq2_2Ec_2Elist_2ECONS_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: d,X0_2E0: u,X1_2E0: u] : ( 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))) ) ).

tff(arityeq2_2Ec_2Efinite__map_2EFAPPLY_2E2_2Emono_2EA_27a_20mono_2Etyop_2Efmaptree_2Efmaptree_28A_27a_2CA_27b_29,axiom,
    ! [A_27a: d,A_27b: d,X0_2E0: u,X1_2E0: u] : ( 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))) ) ).

tff(arityeq2_2Ec_2Efinite__map_2EFAPPLY_2E2_2Emono_2EA_27b_20mono_2EA_27c,axiom,
    ! [A_27b: d,A_27c: d,X0_2E0: u,X1_2E0: u] : ( 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))) ) ).

tff(arityeq2_2Ec_2Efinite__map_2EFAPPLY_2E2_2Emono_2EA_27b_20mono_2Etyop_2Efmaptree_2Efmaptree_28A_27b_2CA_27a_29,axiom,
    ! [A_27a: d,A_27b: d,X0_2E0: u,X1_2E0: u] : ( 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))) ) ).

tff(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: d,A_27b: d,X0_2E0: u,X1_2E0: u] : ( 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))) ) ).

tff(arityeq1_2Ec_2Efinite__map_2EFDOM_2E1_2Emono_2EA_27a_20mono_2Etyop_2Efmaptree_2Efmaptree_28A_27a_2CA_27b_29,axiom,
    ! [A_27a: d,A_27b: d,X0_2E0: u] : ( 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))) ) ).

tff(arityeq1_2Ec_2Efinite__map_2EFDOM_2E1_2Emono_2EA_27b_20mono_2EA_27c,axiom,
    ! [A_27b: d,A_27c: d,X0_2E0: u] : ( 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))) ) ).

tff(arityeq1_2Ec_2Efinite__map_2EFDOM_2E1_2Emono_2EA_27b_20mono_2Etyop_2Efmaptree_2Efmaptree_28A_27b_2CA_27a_29,axiom,
    ! [A_27a: d,A_27b: d,X0_2E0: u] : ( 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))) ) ).

tff(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: d,A_27b: d,X0_2E0: u] : ( 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))) ) ).

tff(arityeq1_2Ec_2Epred__set_2EFINITE_2E1_2Emono_2Etyop_2Elist_2Elist_28A_27a_29,axiom,
    ! [A_27a: d,X0_2E0: u] : ( j_mono_2Etyop_2Emin_2Ebool(s(tyop_2Emin_2Ebool,c_2Epred__set_2EFINITE_2E1(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool),X0_2E0)))) = j_mono_2Etyop_2Emin_2Ebool(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)))) ) ).

tff(arityeq2_2Ec_2Efinite__map_2EFLOOKUP_2E2_2Emono_2EA_27a_20mono_2Etyop_2Efmaptree_2Efmaptree_28A_27a_2CA_27b_29,axiom,
    ! [A_27a: d,A_27b: d,X0_2E0: u,X1_2E0: u] : ( 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))) ) ).

tff(arityeq2_2Ec_2Efmaptree_2EFTNode_2E2_2Emono_2EA_27b_20mono_2EA_27a,axiom,
    ! [A_27a: d,A_27b: d,X0_2E0: u,X1_2E0: u] : ( 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))) ) ).

tff(arityeq2_2Ec_2Efmaptree_2EFTNode_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a: d,A_27b: d,X0_2E0: u,X1_2E0: u] : ( 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))) ) ).

tff(arityeq2_2Ec_2Efmaptree_2EFTNode_2E2_2Emono_2EA_27c_20mono_2EA_27b,axiom,
    ! [A_27b: d,A_27c: d,X0_2E0: u,X1_2E0: u] : ( 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))) ) ).

tff(arityeq2_2Ec_2Efinite__map_2EFUPDATE_2E2_2Emono_2EA_27a_20mono_2Etyop_2Efmaptree_2Efmaptree_28A_27a_2CA_27b_29,axiom,
    ! [A_27a: d,A_27b: d,X0_2E0: u,X1_2E0: u] : ( 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))) ) ).

tff(arityeq1_2Ec_2Epred__set_2EGSPEC_2E1_2Emono_2Etyop_2Elist_2Elist_28A_27a_29_20mono_2Etyop_2Elist_2Elist_28A_27a_29,axiom,
    ! [A_27a: d,X0_2E0: u] : ( 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))) ) ).

tff(arityeq2_2Ec_2Ebool_2EIN_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: d,X0_2E0: u,X1_2E0: u] : ( j_mono_2Etyop_2Emin_2Ebool(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(A_27a,X0_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0)))) = j_mono_2Etyop_2Emin_2Ebool(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)))) ) ).

tff(arityeq2_2Ec_2Ebool_2EIN_2E2_2Emono_2EA_27b,axiom,
    ! [A_27b: d,X0_2E0: u,X1_2E0: u] : ( j_mono_2Etyop_2Emin_2Ebool(s(tyop_2Emin_2Ebool,c_2Ebool_2EIN_2E2(s(A_27b,X0_2E0),s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),X1_2E0)))) = j_mono_2Etyop_2Emin_2Ebool(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)))) ) ).

tff(arityeq1_2Ec_2Eoption_2ESOME_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: d,X0_2E0: u] : ( 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))) ) ).

tff(arityeq1_2Ec_2Eoption_2ESOME_2E1_2Emono_2Etyop_2Efmaptree_2Efmaptree_28A_27a_2CA_27b_29,axiom,
    ! [A_27a: d,A_27b: d,X0_2E0: u] : ( 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))) ) ).

tff(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: d,A_27value: d,X0_2E0: u,X1_2E0: u] : ( j_mono_2Etyop_2Emin_2Ebool(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)))) = j_mono_2Etyop_2Emin_2Ebool(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)))) ) ).

tff(arityeq2_2Ec_2Ebool_2E_5C_2F_2E2,axiom,
    ! [X0_2E0: mono_2Etyop_2Emin_2Ebool,X1_2E0: mono_2Etyop_2Emin_2Ebool] :
      ( ( p(X0_2E0)
        | p(X1_2E0) )
    <=> p(mono_2Eapp_2E2_2Emono_2Etyop_2Emin_2Ebool_20mono_2Etyop_2Emin_2Ebool(mono_2Eapp_2E2_2Emono_2Etyop_2Emin_2Ebool_20mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29(mono_2Ec_2Ebool_2E_5C_2F_2E0,X0_2E0),X1_2E0)) ) ).

tff(arityeq2_2Ec_2Efmaptree_2Eapply__path_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a: d,A_27b: d,X0_2E0: u,X1_2E0: u] : ( 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))) ) ).

tff(arityeq2_2Ec_2Efmaptree_2Econstruct_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a: d,A_27b: d,X0_2E0: u,X1_2E0: u] : ( 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))) ) ).

tff(arityeq3_2Ec_2Efmaptree_2Econstruct_2E3_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a: d,A_27b: d,X0_2E0: u,X1_2E0: u,X2_2E0: u] : ( 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))) ) ).

tff(arityeq2_2Ec_2Efmaptree_2Econstruct_2E2_2Emono_2EA_27b_20mono_2EA_27a,axiom,
    ! [A_27a: d,A_27b: d,X0_2E0: u,X1_2E0: u] : ( 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))) ) ).

tff(arityeq1_2Ec_2Efmaptree_2Efmtreerec_2E1_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27c,axiom,
    ! [A_27a: d,A_27b: d,A_27c: d,X0_2E0: u] : ( 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))) ) ).

tff(arityeq2_2Ec_2Efmaptree_2Efmtreerec_2E2_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27c,axiom,
    ! [A_27a: d,A_27b: d,A_27c: d,X0_2E0: u,X1_2E0: u] : ( 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))) ) ).

tff(arityeq1_2Ec_2Efmaptree_2EfromF_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a: d,A_27b: d,X0_2E0: u] : ( 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))) ) ).

tff(arityeq1_2Ec_2Efmaptree_2EfromF_2E1_2Emono_2EA_27key_20mono_2EA_27value,axiom,
    ! [A_27key: d,A_27value: d,X0_2E0: u] : ( 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))) ) ).

tff(arityeq3_2Ec_2Efmaptree_2Efupd__at__path_2E3_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a: d,A_27b: d,X0_2E0: u,X1_2E0: u,X2_2E0: u] : ( 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))) ) ).

tff(arityeq1_2Ec_2Efmaptree_2Eitem_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a: d,A_27b: d,X0_2E0: u] : ( 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))) ) ).

tff(arityeq1_2Ec_2Efmaptree_2Eitem_2E1_2Emono_2EA_27b_20mono_2EA_27a,axiom,
    ! [A_27a: d,A_27b: d,X0_2E0: u] : ( 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))) ) ).

tff(arityeq3_2Ec_2Elist_2Elist__CASE_2E3_2Emono_2EA_27b_20mono_2Etyop_2Eoption_2Eoption_28A_27a_29,axiom,
    ! [A_27a: d,A_27b: d,X0_2E0: u,X1_2E0: u,X2_2E0: u] : ( 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))) ) ).

tff(arityeq1_2Ec_2Efmaptree_2Emap_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a: d,A_27b: d,X0_2E0: u] : ( 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))) ) ).

tff(arityeq1_2Ec_2Efmaptree_2Emap_2E1_2Emono_2EA_27b_20mono_2EA_27a,axiom,
    ! [A_27a: d,A_27b: d,X0_2E0: u] : ( 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))) ) ).

tff(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: d,A_27b: d,X0_2E0: u,X1_2E0: u] : ( 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))) ) ).

tff(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: d,A_27b: d,A_27c: d,X0_2E0: u,X1_2E0: u] : ( 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))) ) ).

tff(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: d,A_27b: d,A_27c: d,X0_2E0: u,X1_2E0: u] : ( 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))) ) ).

tff(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: d,A_27b: d,X0_2E0: u,X1_2E0: u,X2_2E0: u] : ( 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))) ) ).

tff(arityeq3_2Ec_2Efmaptree_2Erelrec_2E3_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27c,axiom,
    ! [A_27a: d,A_27b: d,A_27c: d,X0_2E0: u,X1_2E0: u,X2_2E0: u] : ( j_mono_2Etyop_2Emin_2Ebool(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)))) = j_mono_2Etyop_2Emin_2Ebool(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)))) ) ).

tff(arityeq3_2Ec_2Efmaptree_2Erelrec_2E3_2Emono_2EA_27b_20mono_2EA_27c_20mono_2EA_27a,axiom,
    ! [A_27a: d,A_27b: d,A_27c: d,X0_2E0: u,X1_2E0: u,X2_2E0: u] : ( j_mono_2Etyop_2Emin_2Ebool(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)))) = j_mono_2Etyop_2Emin_2Ebool(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)))) ) ).

tff(arityeq1_2Ec_2Efmaptree_2EtoF_2E1_2Emono_2EA_27key_20mono_2EA_27value,axiom,
    ! [A_27key: d,A_27value: d,X0_2E0: u] : ( 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))) ) ).

tff(arityeq3_2Ec_2Efmaptree_2Eupdate__at__path_2E3_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a: d,A_27b: d,X0_2E0: u,X1_2E0: u,X2_2E0: u] : ( 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))) ) ).

tff(arityeq1_2Ec_2Efmaptree_2Ewf_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a: d,A_27b: d,X0_2E0: u] : ( j_mono_2Etyop_2Emin_2Ebool(s(tyop_2Emin_2Ebool,c_2Efmaptree_2Ewf_2E1(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),X0_2E0)))) = j_mono_2Etyop_2Emin_2Ebool(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)))) ) ).

tff(arityeq1_2Ec_2Efmaptree_2Ewf_2E1_2Emono_2EA_27value_20mono_2EA_27key,axiom,
    ! [A_27key: d,A_27value: d,X0_2E0: u] : ( j_mono_2Etyop_2Emin_2Ebool(s(tyop_2Emin_2Ebool,c_2Efmaptree_2Ewf_2E1(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value)),X0_2E0)))) = j_mono_2Etyop_2Emin_2Ebool(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)))) ) ).

tff(arityeq1_2Ec_2Ebool_2E_7E_2E1,axiom,
    ! [X0_2E0: mono_2Etyop_2Emin_2Ebool] :
      ( ~ p(X0_2E0)
    <=> p(mono_2Eapp_2E2_2Emono_2Etyop_2Emin_2Ebool_20mono_2Etyop_2Emin_2Ebool(mono_2Ec_2Ebool_2E_7E_2E0,X0_2E0)) ) ).

tff(monoeq_2Emono_2Eapp_2E2_2Emono_2Etyop_2Emin_2Ebool_20mono_2Etyop_2Emin_2Ebool,axiom,
    ! [V0_2E0: mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29,V1_2E0: mono_2Etyop_2Emin_2Ebool] : ( s(tyop_2Emin_2Ebool,i_mono_2Etyop_2Emin_2Ebool(mono_2Eapp_2E2_2Emono_2Etyop_2Emin_2Ebool_20mono_2Etyop_2Emin_2Ebool(V0_2E0,V1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),i_mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29(V0_2E0)),s(tyop_2Emin_2Ebool,i_mono_2Etyop_2Emin_2Ebool(V1_2E0)))) ) ).

tff(monoeq_2Emono_2Eapp_2E2_2Emono_2Etyop_2Emin_2Ebool_20mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29,axiom,
    ! [V0_2E0: mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29_29,V1_2E0: mono_2Etyop_2Emin_2Ebool] : ( s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),i_mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29(mono_2Eapp_2E2_2Emono_2Etyop_2Emin_2Ebool_20mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29(V0_2E0,V1_2E0))) = 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)),i_mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Ebool_29_29(V0_2E0)),s(tyop_2Emin_2Ebool,i_mono_2Etyop_2Emin_2Ebool(V1_2E0)))) ) ).

tff(thm_2Efmaptree_2Econstruct__def,axiom,
    ! [A_27a: d,A_27b: d,F0_2E0: u] :
      ( ! [V1kfm_2E0: u,V3h_2E0: u,V4t_2E0: u] : ( 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: u,V1kfm_2E0: u,V2kl_2E0: u] : ( 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))))) ) ) ).

tff(thm_2Efmaptree_2Ewf__def,axiom,
    ! [A_27a: d,A_27b: d,V0x_2E0: u] :
      ( p(j_mono_2Etyop_2Emin_2Ebool(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: u] :
          ( ! [V2a0_2E0: u] :
              ( ? [V3a_2E0: u,V4fm_2E0: u] :
                  ( ( 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: u] :
                      ( p(j_mono_2Etyop_2Emin_2Ebool(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(j_mono_2Etyop_2Emin_2Ebool(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(j_mono_2Etyop_2Emin_2Ebool(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(j_mono_2Etyop_2Emin_2Ebool(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))))) ) ) ).

tff(thm_2Efmaptree_2Efmaptree__TY__DEF,axiom,
    ! [A_27key: d,A_27value: d] :
    ? [V0rep_2E0: u] : p(j_mono_2Etyop_2Emin_2Ebool(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))))) ).

tff(thm_2Efmaptree_2Efmap__bij__thm,axiom,
    ! [A_27key: d,A_27value: d] :
      ( ! [V0a_2E0: u] : ( 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: u] :
          ( p(j_mono_2Etyop_2Emin_2Ebool(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) ) ) ) ).

tff(thm_2Efmaptree_2EFTNode__def,axiom,
    ! [A_27a: d,A_27b: d,V0i_2E0: u,V1fm_2E0: u] : ( 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))))))) ) ).

tff(thm_2Efmaptree_2Eitem__map__def,axiom,
    ! [A_27a: d,A_27b: d,V0ft_2E0: u] : ( 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))))) ) ).

tff(thm_2Efmaptree_2Eapply__path__def,axiom,
    ! [A_27a: d,A_27b: d] :
      ( ! [V0ft_2E0: u] : ( 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: u,V2t_2E0: u,V3ft_2E0: u] : ( 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))) ) ) ).

tff(thm_2Efmaptree_2Eupdate__at__path__def,axiom,
    ! [A_27a: d,A_27b: d,F0_2E0: u] :
      ( ! [V5ft_2E0: u,V2h_2E0: u,V6ft_27_2E0: u] : ( 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: u,V1ft_2E0: u] : ( 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: u,V3t_2E0: u,V4a_2E0: u,V5ft_2E0: u] : ( 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))) ) ) ) ).

tff(thm_2Efmaptree_2Efupd__at__path__def,axiom,
    ! [A_27a: d,A_27b: d,F0_2E0: u] :
      ( ! [V5ft_2E0: u,V2h_2E0: u,V6ft_27_2E0: u] : ( 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: u,V1ft_2E0: u] : ( 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: u,V3t_2E0: u,V4f_2E0: u,V5ft_2E0: u] : ( 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))) ) ) ) ).

tff(thm_2Efmaptree_2Erelrec__def,axiom,
    ! [A_27a: d,A_27b: d,A_27c: d,V0x_2E0: u,V1x_27_2E0: u,V2x_27_27_2E0: u] :
      ( p(j_mono_2Etyop_2Emin_2Ebool(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: u] :
          ( ! [V4a0_2E0: u,V5a1_2E0: u] :
              ( ? [V6i_2E0: u,V7fm_2E0: u,V8rfm_2E0: u] :
                  ( ( 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: u] :
                      ( p(j_mono_2Etyop_2Emin_2Ebool(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(j_mono_2Etyop_2Emin_2Ebool(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(j_mono_2Etyop_2Emin_2Ebool(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(j_mono_2Etyop_2Emin_2Ebool(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))))) ) ) ).

tff(thm_2Efmaptree_2Efmtreerec__def,axiom,
    ! [A_27a: d,A_27b: d,A_27c: d,F0_2E0: u] :
      ( ! [V0h_2E0: u,V1ft_2E0: u,V2r_2E0: u] : ( j_mono_2Etyop_2Emin_2Ebool(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)))) = j_mono_2Etyop_2Emin_2Ebool(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: u,V1ft_2E0: u] : ( 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))))) ) ) ).

tff(thm_2Efmaptree_2Ewf__rules,axiom,
    ! [A_27a: d,A_27b: d,V0a_2E0: u,V1fm_2E0: u] :
      ( ! [V2k_2E0: u] :
          ( p(j_mono_2Etyop_2Emin_2Ebool(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(j_mono_2Etyop_2Emin_2Ebool(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(j_mono_2Etyop_2Emin_2Ebool(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))))))) ) ).

tff(thm_2Efmaptree_2Ewf__ind,axiom,
    ! [A_27a: d,A_27b: d,V0wf_27_2E0: u] :
      ( ! [V1a_2E0: u,V2fm_2E0: u] :
          ( ! [V3k_2E0: u] :
              ( p(j_mono_2Etyop_2Emin_2Ebool(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(j_mono_2Etyop_2Emin_2Ebool(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(j_mono_2Etyop_2Emin_2Ebool(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: u] :
          ( p(j_mono_2Etyop_2Emin_2Ebool(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(j_mono_2Etyop_2Emin_2Ebool(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))))) ) ) ).

tff(thm_2Efmaptree_2Ewf__strongind,axiom,
    ! [A_27a: d,A_27b: d,V0wf_27_2E0: u] :
      ( ! [V1a_2E0: u,V2fm_2E0: u] :
          ( ! [V3k_2E0: u] :
              ( p(j_mono_2Etyop_2Emin_2Ebool(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(j_mono_2Etyop_2Emin_2Ebool(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(j_mono_2Etyop_2Emin_2Ebool(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(j_mono_2Etyop_2Emin_2Ebool(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: u] :
          ( p(j_mono_2Etyop_2Emin_2Ebool(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(j_mono_2Etyop_2Emin_2Ebool(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))))) ) ) ).

tff(thm_2Efmaptree_2Ewf__cases,axiom,
    ! [A_27a: d,A_27b: d,V0a0_2E0: u] :
      ( p(j_mono_2Etyop_2Emin_2Ebool(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: u,V2fm_2E0: u] :
          ( ( 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: u] :
              ( p(j_mono_2Etyop_2Emin_2Ebool(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(j_mono_2Etyop_2Emin_2Ebool(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))))))) ) ) ) ).

tff(thm_2Efmaptree_2EFTNode__11,axiom,
    ! [A_27a: d,A_27b: d,V0i2_2E0: u,V1i1_2E0: u,V2f2_2E0: u,V3f1_2E0: u] :
      ( ( 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) ) ) ) ).

tff(thm_2Efmaptree_2Efmaptree__nchotomy,axiom,
    ! [A_27a: d,A_27b: d,V0ft_2E0: u] :
    ? [V1i_2E0: u,V2fm_2E0: u] : ( 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))) ) ).

tff(thm_2Efmaptree_2Eitem__thm,axiom,
    ! [A_27a: d,A_27b: d,V0i_2E0: u,V1fm_2E0: u] : ( 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) ) ).

tff(thm_2Efmaptree_2Emap__thm,axiom,
    ! [A_27a: d,A_27b: d,V0i_2E0: u,V1fm_2E0: u] : ( 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) ) ).

tff(thm_2Efmaptree_2Eft__ind,axiom,
    ! [A_27a: d,A_27b: d,V0P_2E0: u] :
      ( ! [V1a_2E0: u,V2fm_2E0: u] :
          ( ! [V3k_2E0: u] :
              ( p(j_mono_2Etyop_2Emin_2Ebool(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(j_mono_2Etyop_2Emin_2Ebool(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(j_mono_2Etyop_2Emin_2Ebool(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: u] : p(j_mono_2Etyop_2Emin_2Ebool(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))))) ) ).

tff(thm_2Efmaptree_2Eapplicable__paths__FINITE,axiom,
    ! [A_27a: d,A_27b: d,F1_2E0: u,F0_2E0: u] :
      ( ! [V1p_2E0: u,V0ft_2E0: u] :
          ( p(j_mono_2Etyop_2Emin_2Ebool(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: u] : ( 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: u,V1p_2E0: u] : ( 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: u] : p(j_mono_2Etyop_2Emin_2Ebool(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))))))))) ) ) ).

tff(thm_2Efmaptree_2Eapply__path__SNOC,axiom,
    ! [A_27a: d,A_27b: d,F0_2E0: u] :
      ( ! [V1x_2E0: u,V3ft_27_2E0: u] : ( 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: u,V1x_2E0: u,V2p_2E0: u] : ( 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))))) ) ) ).

tff(thm_2Efmaptree_2Erelrec__rules,axiom,
    ! [A_27a: d,A_27b: d,A_27c: d,V0h_2E0: u,V1i_2E0: u,V2fm_2E0: u,V3rfm_2E0: u] :
      ( ( ( 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: u] :
            ( p(j_mono_2Etyop_2Emin_2Ebool(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(j_mono_2Etyop_2Emin_2Ebool(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(j_mono_2Etyop_2Emin_2Ebool(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))))))) ) ).

tff(thm_2Efmaptree_2Erelrec__ind,axiom,
    ! [A_27a: d,A_27b: d,A_27c: d,V0h_2E0: u,V1relrec_27_2E0: u] :
      ( ! [V2i_2E0: u,V3fm_2E0: u,V4rfm_2E0: u] :
          ( ( ( 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: u] :
                ( p(j_mono_2Etyop_2Emin_2Ebool(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(j_mono_2Etyop_2Emin_2Ebool(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(j_mono_2Etyop_2Emin_2Ebool(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: u,V7a1_2E0: u] :
          ( p(j_mono_2Etyop_2Emin_2Ebool(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(j_mono_2Etyop_2Emin_2Ebool(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))))) ) ) ).

tff(thm_2Efmaptree_2Erelrec__strongind,axiom,
    ! [A_27a: d,A_27b: d,A_27c: d,V0h_2E0: u,V1relrec_27_2E0: u] :
      ( ! [V2i_2E0: u,V3fm_2E0: u,V4rfm_2E0: u] :
          ( ( ( 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: u] :
                ( p(j_mono_2Etyop_2Emin_2Ebool(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(j_mono_2Etyop_2Emin_2Ebool(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(j_mono_2Etyop_2Emin_2Ebool(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(j_mono_2Etyop_2Emin_2Ebool(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: u,V7a1_2E0: u] :
          ( p(j_mono_2Etyop_2Emin_2Ebool(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(j_mono_2Etyop_2Emin_2Ebool(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))))) ) ) ).

tff(thm_2Efmaptree_2Erelrec__cases,axiom,
    ! [A_27a: d,A_27b: d,A_27c: d,V0h_2E0: u,V1a0_2E0: u,V2a1_2E0: u] :
      ( p(j_mono_2Etyop_2Emin_2Ebool(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: u,V4fm_2E0: u,V5rfm_2E0: u] :
          ( ( 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: u] :
              ( p(j_mono_2Etyop_2Emin_2Ebool(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(j_mono_2Etyop_2Emin_2Ebool(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))))))) ) ) ) ).

tff(thm_2Efmaptree_2Efmtreerec__thm,axiom,
    ! [A_27a: d,A_27b: d,A_27c: d,V0i_2E0: u,V1h_2E0: u,V2fm_2E0: u] : ( 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))) ) ).

tff(thm_2Efmaptree_2Efmtree__Axiom,axiom,
    ! [A_27a: d,A_27b: d,A_27c: d,V0h_2E0: u] :
    ? [V1f_2E0: u] :
    ! [V2i_2E0: u,V3fm_2E0: u] : ( 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))))) ) ).

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