ITP001 Axioms: ITP083_7.ax
%------------------------------------------------------------------------------
% File : ITP083_7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : fmaptree.ax [Gau19]
% : HL4083_7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 191 ( 66 unt; 90 typ; 0 def)
% Number of atoms : 186 ( 102 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 : 9 ( 2 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 82 ( 46 >; 36 *; 0 +; 0 <<)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 88 ( 88 usr; 6 con; 0-6 aty)
% Number of variables : 574 ( 429 !; 16 ?; 574 :)
% ( 129 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tyop_2Efinite__map_2Efmap,type,
tyop_2Efinite__map_2Efmap: ( $tType * $tType ) > $tType ).
tff(tyop_2Efmaptree_2Efmaptree,type,
tyop_2Efmaptree_2Efmaptree: ( $tType * $tType ) > $tType ).
tff(tyop_2Elist_2Elist,type,
tyop_2Elist_2Elist: $tType > $tType ).
tff(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
tff(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: ( $tType * $tType ) > $tType ).
tff(tyop_2Eoption_2Eoption,type,
tyop_2Eoption_2Eoption: $tType > $tType ).
tff(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: ( $tType * $tType ) > $tType ).
tff(app_2E2,type,
app_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27b) * A_27a ) > A_27b ) ).
tff(p,type,
p: tyop_2Emin_2Ebool > $o ).
tff(combin_i_2E0,type,
combin_i_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,A_27a) ).
tff(combin_k_2E0,type,
combin_k_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27a)) ).
tff(combin_s_2E0,type,
combin_s_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : 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))) ).
tff(c_2Ebool_2E_21_2E0,type,
c_2Ebool_2E_21_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_21_2E1,type,
c_2Ebool_2E_21_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).
tff(c_2Epair_2E_2C_2E0,type,
c_2Epair_2E_2C_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,A_27b))) ).
tff(c_2Epair_2E_2C_2E2,type,
c_2Epair_2E_2C_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a * A_27b ) > tyop_2Epair_2Eprod(A_27a,A_27b) ) ).
tff(c_2Ebool_2E_2F_5C_2E0,type,
c_2Ebool_2E_2F_5C_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).
tff(c_2Ebool_2E_2F_5C_2E2,type,
c_2Ebool_2E_2F_5C_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).
tff(c_2Emin_2E_3D_2E0,type,
c_2Emin_2E_3D_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) ).
tff(c_2Emin_2E_3D_2E2,type,
c_2Emin_2E_3D_2E2:
!>[A_27a: $tType] : ( ( A_27a * A_27a ) > tyop_2Emin_2Ebool ) ).
tff(c_2Emin_2E_3D_3D_3E_2E0,type,
c_2Emin_2E_3D_3D_3E_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).
tff(c_2Emin_2E_3D_3D_3E_2E2,type,
c_2Emin_2E_3D_3D_3E_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).
tff(c_2Ebool_2E_3F_2E0,type,
c_2Ebool_2E_3F_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_3F_2E1,type,
c_2Ebool_2E_3F_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).
tff(c_2Emin_2E_40_2E0,type,
c_2Emin_2E_40_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),A_27a) ).
tff(c_2Emin_2E_40_2E1,type,
c_2Emin_2E_40_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > A_27a ) ).
tff(c_2Elist_2EAPPEND_2E0,type,
c_2Elist_2EAPPEND_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27a))) ).
tff(c_2Elist_2EAPPEND_2E2,type,
c_2Elist_2EAPPEND_2E2:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist(A_27a) * tyop_2Elist_2Elist(A_27a) ) > tyop_2Elist_2Elist(A_27a) ) ).
tff(c_2Ebool_2ECOND_2E0,type,
c_2Ebool_2ECOND_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a))) ).
tff(c_2Ebool_2ECOND_2E3,type,
c_2Ebool_2ECOND_2E3:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Ebool * A_27a * A_27a ) > A_27a ) ).
tff(c_2Elist_2ECONS_2E0,type,
c_2Elist_2ECONS_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27a))) ).
tff(c_2Elist_2ECONS_2E2,type,
c_2Elist_2ECONS_2E2:
!>[A_27a: $tType] : ( ( A_27a * tyop_2Elist_2Elist(A_27a) ) > tyop_2Elist_2Elist(A_27a) ) ).
tff(c_2Ebool_2EF_2E0,type,
c_2Ebool_2EF_2E0: tyop_2Emin_2Ebool ).
tff(c_2Efinite__map_2EFAPPLY_2E0,type,
c_2Efinite__map_2EFAPPLY_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b)) ).
tff(c_2Efinite__map_2EFAPPLY_2E2,type,
c_2Efinite__map_2EFAPPLY_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Efinite__map_2Efmap(A_27a,A_27b) * A_27a ) > A_27b ) ).
tff(c_2Efinite__map_2EFDOM_2E0,type,
c_2Efinite__map_2EFDOM_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) ).
tff(c_2Efinite__map_2EFDOM_2E1,type,
c_2Efinite__map_2EFDOM_2E1:
!>[A_27a: $tType,A_27b: $tType] : ( tyop_2Efinite__map_2Efmap(A_27a,A_27b) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) ).
tff(c_2Epred__set_2EFINITE_2E0,type,
c_2Epred__set_2EFINITE_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Epred__set_2EFINITE_2E1,type,
c_2Epred__set_2EFINITE_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).
tff(c_2Efinite__map_2EFLOOKUP_2E0,type,
c_2Efinite__map_2EFLOOKUP_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27b))) ).
tff(c_2Efinite__map_2EFLOOKUP_2E2,type,
c_2Efinite__map_2EFLOOKUP_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Efinite__map_2Efmap(A_27a,A_27b) * A_27a ) > tyop_2Eoption_2Eoption(A_27b) ) ).
tff(c_2Efmaptree_2EFTNode_2E0,type,
c_2Efmaptree_2EFTNode_2E0:
!>[A_27a: $tType,A_27b: $tType] : 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))) ).
tff(c_2Efmaptree_2EFTNode_2E2,type,
c_2Efmaptree_2EFTNode_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b * tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)) ) > tyop_2Efmaptree_2Efmaptree(A_27a,A_27b) ) ).
tff(c_2Efinite__map_2EFUPDATE_2E0,type,
c_2Efinite__map_2EFUPDATE_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Efinite__map_2Efmap(A_27a,A_27b))) ).
tff(c_2Efinite__map_2EFUPDATE_2E2,type,
c_2Efinite__map_2EFUPDATE_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Efinite__map_2Efmap(A_27a,A_27b) * tyop_2Epair_2Eprod(A_27a,A_27b) ) > tyop_2Efinite__map_2Efmap(A_27a,A_27b) ) ).
tff(c_2Epred__set_2EGSPEC_2E0,type,
c_2Epred__set_2EGSPEC_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) ).
tff(c_2Epred__set_2EGSPEC_2E1,type,
c_2Epred__set_2EGSPEC_2E1:
!>[A_27a: $tType,A_27b: $tType] : ( tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,tyop_2Emin_2Ebool)) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) ).
tff(c_2Ebool_2EIN_2E0,type,
c_2Ebool_2EIN_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)) ).
tff(c_2Ebool_2EIN_2E2,type,
c_2Ebool_2EIN_2E2:
!>[A_27a: $tType] : ( ( A_27a * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Elist_2ENIL_2E0,type,
c_2Elist_2ENIL_2E0:
!>[A_27a: $tType] : tyop_2Elist_2Elist(A_27a) ).
tff(c_2Eoption_2ENONE_2E0,type,
c_2Eoption_2ENONE_2E0:
!>[A_27a: $tType] : tyop_2Eoption_2Eoption(A_27a) ).
tff(c_2Eoption_2ESOME_2E0,type,
c_2Eoption_2ESOME_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Eoption_2Eoption(A_27a)) ).
tff(c_2Eoption_2ESOME_2E1,type,
c_2Eoption_2ESOME_2E1:
!>[A_27a: $tType] : ( A_27a > tyop_2Eoption_2Eoption(A_27a) ) ).
tff(c_2Ebool_2ET_2E0,type,
c_2Ebool_2ET_2E0: tyop_2Emin_2Ebool ).
tff(c_2Ebool_2ETYPE__DEFINITION_2E0,type,
c_2Ebool_2ETYPE__DEFINITION_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Ebool)) ).
tff(c_2Ebool_2ETYPE__DEFINITION_2E2,type,
c_2Ebool_2ETYPE__DEFINITION_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27b,A_27a) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Ebool_2E_5C_2F_2E0,type,
c_2Ebool_2E_5C_2F_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).
tff(c_2Ebool_2E_5C_2F_2E2,type,
c_2Ebool_2E_5C_2F_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).
tff(c_2Efmaptree_2Eapply__path_2E0,type,
c_2Efmaptree_2Eapply__path_2E0:
!>[A_27a: $tType,A_27b: $tType] : 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)))) ).
tff(c_2Efmaptree_2Eapply__path_2E2,type,
c_2Efmaptree_2Eapply__path_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Elist_2Elist(A_27a) * tyop_2Efmaptree_2Efmaptree(A_27a,A_27b) ) > tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)) ) ).
tff(c_2Efmaptree_2Econstruct_2E0,type,
c_2Efmaptree_2Econstruct_2E0:
!>[A_27a: $tType,A_27b: $tType] : 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)))) ).
tff(c_2Efmaptree_2Econstruct_2E2,type,
c_2Efmaptree_2Econstruct_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a * 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)) ) ).
tff(c_2Efmaptree_2Econstruct_2E3,type,
c_2Efmaptree_2Econstruct_2E3:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a * tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))) * tyop_2Elist_2Elist(A_27b) ) > tyop_2Eoption_2Eoption(A_27a) ) ).
tff(c_2Efmaptree_2Efmtreerec_2E0,type,
c_2Efmaptree_2Efmtreerec_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : 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)) ).
tff(c_2Efmaptree_2Efmtreerec_2E1,type,
c_2Efmaptree_2Efmtreerec_2E1:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( 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) ) ).
tff(c_2Efmaptree_2Efmtreerec_2E2,type,
c_2Efmaptree_2Efmtreerec_2E2:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( 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_2Efmaptree_2Efmaptree(A_27c,A_27b) ) > A_27a ) ).
tff(c_2Efmaptree_2EfromF_2E0,type,
c_2Efmaptree_2EfromF_2E0:
!>[A_27key: $tType,A_27value: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value)),tyop_2Efmaptree_2Efmaptree(A_27key,A_27value)) ).
tff(c_2Efmaptree_2EfromF_2E1,type,
c_2Efmaptree_2EfromF_2E1:
!>[A_27key: $tType,A_27value: $tType] : ( tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value)) > tyop_2Efmaptree_2Efmaptree(A_27key,A_27value) ) ).
tff(c_2Efmaptree_2Efupd__at__path_2E0,type,
c_2Efmaptree_2Efupd__at__path_2E0:
!>[A_27a: $tType,A_27b: $tType] : 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))))) ).
tff(c_2Efmaptree_2Efupd__at__path_2E3,type,
c_2Efmaptree_2Efupd__at__path_2E3:
!>[A_27a: $tType,A_27b: $tType] : ( ( 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))) * tyop_2Efmaptree_2Efmaptree(A_27a,A_27b) ) > tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)) ) ).
tff(c_2Efmaptree_2Eitem_2E0,type,
c_2Efmaptree_2Eitem_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),A_27b) ).
tff(c_2Efmaptree_2Eitem_2E1,type,
c_2Efmaptree_2Eitem_2E1:
!>[A_27a: $tType,A_27b: $tType] : ( tyop_2Efmaptree_2Efmaptree(A_27a,A_27b) > A_27b ) ).
tff(c_2Elist_2Elist__CASE_2E0,type,
c_2Elist_2Elist__CASE_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),A_27b)),A_27b))) ).
tff(c_2Elist_2Elist__CASE_2E3,type,
c_2Elist_2Elist__CASE_2E3:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Elist_2Elist(A_27a) * A_27b * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),A_27b)) ) > A_27b ) ).
tff(c_2Efmaptree_2Emap_2E0,type,
c_2Efmaptree_2Emap_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))) ).
tff(c_2Efmaptree_2Emap_2E1,type,
c_2Efmaptree_2Emap_2E1:
!>[A_27a: $tType,A_27b: $tType] : ( tyop_2Efmaptree_2Efmaptree(A_27a,A_27b) > tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)) ) ).
tff(c_2Efinite__map_2Eo__f_2E0,type,
c_2Efinite__map_2Eo__f_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Efinite__map_2Efmap(A_27a,A_27b),tyop_2Efinite__map_2Efmap(A_27a,A_27c))) ).
tff(c_2Efinite__map_2Eo__f_2E2,type,
c_2Efinite__map_2Eo__f_2E2:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Emin_2Efun(A_27b,A_27c) * tyop_2Efinite__map_2Efmap(A_27a,A_27b) ) > tyop_2Efinite__map_2Efmap(A_27a,A_27c) ) ).
tff(c_2Eoption_2Eoption__CASE_2E0,type,
c_2Eoption_2Eoption__CASE_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),A_27b))) ).
tff(c_2Eoption_2Eoption__CASE_2E3,type,
c_2Eoption_2Eoption__CASE_2E3:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Eoption_2Eoption(A_27a) * A_27b * tyop_2Emin_2Efun(A_27a,A_27b) ) > A_27b ) ).
tff(c_2Efmaptree_2Erelrec_2E0,type,
c_2Efmaptree_2Erelrec_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : 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))) ).
tff(c_2Efmaptree_2Erelrec_2E3,type,
c_2Efmaptree_2Erelrec_2E3:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( 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_2Efmaptree_2Efmaptree(A_27b,A_27a) * A_27c ) > tyop_2Emin_2Ebool ) ).
tff(c_2Efmaptree_2EtoF_2E0,type,
c_2Efmaptree_2EtoF_2E0:
!>[A_27key: $tType,A_27value: $tType] : tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27key,A_27value),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value))) ).
tff(c_2Efmaptree_2EtoF_2E1,type,
c_2Efmaptree_2EtoF_2E1:
!>[A_27key: $tType,A_27value: $tType] : ( tyop_2Efmaptree_2Efmaptree(A_27key,A_27value) > tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value)) ) ).
tff(c_2Efmaptree_2Eupdate__at__path_2E0,type,
c_2Efmaptree_2Eupdate__at__path_2E0:
!>[A_27a: $tType,A_27b: $tType] : 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))))) ).
tff(c_2Efmaptree_2Eupdate__at__path_2E3,type,
c_2Efmaptree_2Eupdate__at__path_2E3:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Elist_2Elist(A_27a) * A_27b * tyop_2Efmaptree_2Efmaptree(A_27a,A_27b) ) > tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)) ) ).
tff(c_2Efmaptree_2Ewf_2E0,type,
c_2Efmaptree_2Ewf_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Ebool) ).
tff(c_2Efmaptree_2Ewf_2E1,type,
c_2Efmaptree_2Ewf_2E1:
!>[A_27a: $tType,A_27b: $tType] : ( tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)) > tyop_2Emin_2Ebool ) ).
tff(c_2Ebool_2E_7E_2E0,type,
c_2Ebool_2E_7E_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_7E_2E1,type,
c_2Ebool_2E_7E_2E1: tyop_2Emin_2Ebool > tyop_2Emin_2Ebool ).
tff(thm_2Eextra_2Dho_2Eeq__ext,axiom,
! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] :
( ! [V2x_2E0: A_27a] : ( app_2E2(A_27a,A_27b,V0f_2E0,V2x_2E0) = app_2E2(A_27a,A_27b,V1g_2E0,V2x_2E0) )
=> ( V0f_2E0 = V1g_2E0 ) ) ).
tff(thm_2Eextra_2Dho_2Eboolext,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( ( p(V0_2E0)
<=> p(V1_2E0) )
=> ( V0_2E0 = V1_2E0 ) ) ).
tff(thm_2Eextra_2Dho_2Etruth,axiom,
p(c_2Ebool_2ET_2E0) ).
tff(thm_2Eextra_2Dho_2Enotfalse,axiom,
~ p(c_2Ebool_2EF_2E0) ).
tff(thm_2Eextra_2Dho_2Ebool__cases__ax,axiom,
! [V0t_2E0: tyop_2Emin_2Ebool] :
( ( V0t_2E0 = c_2Ebool_2ET_2E0 )
| ( V0t_2E0 = c_2Ebool_2EF_2E0 ) ) ).
tff(thm_2Eextra_2Dho_2Ei__thm,axiom,
! [A_27a: $tType,V0x_2E0: A_27a] : ( app_2E2(A_27a,A_27a,combin_i_2E0(A_27a),V0x_2E0) = V0x_2E0 ) ).
tff(thm_2Eextra_2Dho_2Ek__thm,axiom,
! [A_27a: $tType,A_27b: $tType,V0x_2E0: A_27a,V1y_2E0: A_27b] : ( app_2E2(A_27b,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27a),combin_k_2E0(A_27a,A_27b),V0x_2E0),V1y_2E0) = V0x_2E0 ) ).
tff(thm_2Eextra_2Dho_2Es__thm,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),V1g_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2x_2E0: A_27a] : ( app_2E2(A_27a,A_27c,app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c),app_2E2(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(A_27a,A_27b,A_27c),V0f_2E0),V1g_2E0),V2x_2E0) = app_2E2(A_27b,A_27c,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27c),V0f_2E0,V2x_2E0),app_2E2(A_27a,A_27b,V1g_2E0,V2x_2E0)) ) ).
tff(logicdef_2E_2F_5C,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( p(c_2Ebool_2E_2F_5C_2E2(V0_2E0,V1_2E0))
<=> ( p(V0_2E0)
& p(V1_2E0) ) ) ).
tff(logicdef_2E_5C_2F,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( p(c_2Ebool_2E_5C_2F_2E2(V0_2E0,V1_2E0))
<=> ( p(V0_2E0)
| p(V1_2E0) ) ) ).
tff(logicdef_2E_7E,axiom,
! [V0_2E0: tyop_2Emin_2Ebool] :
( p(c_2Ebool_2E_7E_2E1(V0_2E0))
<=> ~ p(V0_2E0) ) ).
tff(logicdef_2E_3D_3D_3E,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( p(c_2Emin_2E_3D_3D_3E_2E2(V0_2E0,V1_2E0))
<=> ( p(V0_2E0)
=> p(V1_2E0) ) ) ).
tff(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0_2E0: A_27a,V1_2E0: A_27a] :
( p(c_2Emin_2E_3D_2E2(A_27a,V0_2E0,V1_2E0))
<=> ( V0_2E0 = V1_2E0 ) ) ).
tff(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2E_21_2E1(A_27a,V0f_2E0))
<=> ! [V1x_2E0: A_27a] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0f_2E0,V1x_2E0)) ) ).
tff(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2E_3F_2E1(A_27a,V0f_2E0))
<=> ? [V1x_2E0: A_27a] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0f_2E0,V1x_2E0)) ) ).
tff(arityeq1_2Ec_2Ebool_2E_21_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_21_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_21_2E0(A_27a),X0_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2EA_27a_20mono_2Etyop_2Efmaptree_2Efmaptree_28A_27a_2CA_27b_29,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: A_27a,X1_2E0: tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)] : ( c_2Epair_2E_2C_2E2(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),X0_2E0,X1_2E0) = app_2E2(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Epair_2Eprod(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),app_2E2(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(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Elist_2Elist_28A_27a_29_20mono_2Etyop_2Emin_2Ebool,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Elist_2Elist(A_27a),X1_2E0: tyop_2Emin_2Ebool] : ( c_2Epair_2E_2C_2E2(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Ebool,tyop_2Epair_2Eprod(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool),app_2E2(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(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2E_2F_5C_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
( ( p(X0_2E0)
& p(X1_2E0) )
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2E_2F_5C_2E0,X0_2E0),X1_2E0)) ) ).
tff(arityeq2_2Ec_2Emin_2E_3D_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: A_27a] :
( ( X0_2E0 = X1_2E0 )
<=> p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Emin_2E_3D_2E0(A_27a),X0_2E0),X1_2E0)) ) ).
tff(arityeq2_2Ec_2Emin_2E_3D_3D_3E_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
( ( p(X0_2E0)
=> p(X1_2E0) )
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Emin_2E_3D_3D_3E_2E0,X0_2E0),X1_2E0)) ) ).
tff(arityeq1_2Ec_2Ebool_2E_3F_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_3F_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_3F_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Emin_2E_40_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Emin_2E_40_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),A_27a,c_2Emin_2E_40_2E0(A_27a),X0_2E0) ) ).
tff(arityeq2_2Ec_2Elist_2EAPPEND_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Elist_2Elist(A_27a),X1_2E0: tyop_2Elist_2Elist(A_27a)] : ( c_2Elist_2EAPPEND_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27a),app_2E2(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27a)),c_2Elist_2EAPPEND_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Eoption_2Eoption_28A_27a_29,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Eoption_2Eoption(A_27a),X2_2E0: tyop_2Eoption_2Eoption(A_27a)] : ( c_2Ebool_2ECOND_2E3(tyop_2Eoption_2Eoption(A_27a),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Eoption_2Eoption(A_27a),tyop_2Eoption_2Eoption(A_27a),app_2E2(tyop_2Eoption_2Eoption(A_27a),tyop_2Emin_2Efun(tyop_2Eoption_2Eoption(A_27a),tyop_2Eoption_2Eoption(A_27a)),app_2E2(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(tyop_2Eoption_2Eoption(A_27a)),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Eoption_2Eoption_28tyop_2Efmaptree_2Efmaptree_28A_27a_2CA_27b_29_29,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),X2_2E0: tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))] : ( c_2Ebool_2ECOND_2E3(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),app_2E2(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(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(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq2_2Ec_2Elist_2ECONS_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: tyop_2Elist_2Elist(A_27a)] : ( c_2Elist_2ECONS_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27a),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27a)),c_2Elist_2ECONS_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Efinite__map_2EFAPPLY_2E2_2Emono_2EA_27a_20mono_2Etyop_2Efmaptree_2Efmaptree_28A_27a_2CA_27b_29,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),X1_2E0: A_27a] : ( c_2Efinite__map_2EFAPPLY_2E2(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),X0_2E0,X1_2E0) = app_2E2(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),app_2E2(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(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Efinite__map_2EFAPPLY_2E2_2Emono_2EA_27b_20mono_2EA_27c,axiom,
! [A_27b: $tType,A_27c: $tType,X0_2E0: tyop_2Efinite__map_2Efmap(A_27b,A_27c),X1_2E0: A_27b] : ( c_2Efinite__map_2EFAPPLY_2E2(A_27b,A_27c,X0_2E0,X1_2E0) = app_2E2(A_27b,A_27c,app_2E2(tyop_2Efinite__map_2Efmap(A_27b,A_27c),tyop_2Emin_2Efun(A_27b,A_27c),c_2Efinite__map_2EFAPPLY_2E0(A_27b,A_27c),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Efinite__map_2EFAPPLY_2E2_2Emono_2EA_27b_20mono_2Etyop_2Efmaptree_2Efmaptree_28A_27b_2CA_27a_29,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),X1_2E0: A_27b] : ( c_2Efinite__map_2EFAPPLY_2E2(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),X0_2E0,X1_2E0) = app_2E2(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),app_2E2(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(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),X0_2E0),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: $tType,A_27b: $tType,X0_2E0: tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),X1_2E0: A_27b] : ( c_2Efinite__map_2EFAPPLY_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),X0_2E0,X1_2E0) = app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),app_2E2(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(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Efinite__map_2EFDOM_2E1_2Emono_2EA_27a_20mono_2Etyop_2Efmaptree_2Efmaptree_28A_27a_2CA_27b_29,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))] : ( c_2Efinite__map_2EFDOM_2E1(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),X0_2E0) = app_2E2(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(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: $tType,A_27c: $tType,X0_2E0: tyop_2Efinite__map_2Efmap(A_27b,A_27c)] : ( c_2Efinite__map_2EFDOM_2E1(A_27b,A_27c,X0_2E0) = app_2E2(tyop_2Efinite__map_2Efmap(A_27b,A_27c),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Efinite__map_2EFDOM_2E0(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: $tType,A_27b: $tType,X0_2E0: tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a))] : ( c_2Efinite__map_2EFDOM_2E1(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),X0_2E0) = app_2E2(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(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: $tType,A_27b: $tType,X0_2E0: tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)))] : ( c_2Efinite__map_2EFDOM_2E1(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),X0_2E0) = app_2E2(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(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: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool)] : ( c_2Epred__set_2EFINITE_2E1(tyop_2Elist_2Elist(A_27a),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Epred__set_2EFINITE_2E0(tyop_2Elist_2Elist(A_27a)),X0_2E0) ) ).
tff(arityeq2_2Ec_2Efinite__map_2EFLOOKUP_2E2_2Emono_2EA_27a_20mono_2Etyop_2Efmaptree_2Efmaptree_28A_27a_2CA_27b_29,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),X1_2E0: A_27a] : ( c_2Efinite__map_2EFLOOKUP_2E2(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),X0_2E0,X1_2E0) = app_2E2(A_27a,tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),app_2E2(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(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Efmaptree_2EFTNode_2E2_2Emono_2EA_27b_20mono_2EA_27a,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: A_27a,X1_2E0: tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a))] : ( c_2Efmaptree_2EFTNode_2E2(A_27b,A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),app_2E2(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(A_27b,A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Efmaptree_2EFTNode_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: A_27b,X1_2E0: tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))] : ( c_2Efmaptree_2EFTNode_2E2(A_27a,A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),app_2E2(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(A_27a,A_27b),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Efmaptree_2EFTNode_2E2_2Emono_2EA_27c_20mono_2EA_27b,axiom,
! [A_27b: $tType,A_27c: $tType,X0_2E0: A_27b,X1_2E0: tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b))] : ( c_2Efmaptree_2EFTNode_2E2(A_27c,A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)),tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),app_2E2(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(A_27c,A_27b),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Efinite__map_2EFUPDATE_2E2_2Emono_2EA_27a_20mono_2Etyop_2Efmaptree_2Efmaptree_28A_27a_2CA_27b_29,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),X1_2E0: tyop_2Epair_2Eprod(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))] : ( c_2Efinite__map_2EFUPDATE_2E2(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),X0_2E0,X1_2E0) = app_2E2(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(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(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),X0_2E0),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: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Epair_2Eprod(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool))] : ( c_2Epred__set_2EGSPEC_2E1(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27a),X0_2E0) = app_2E2(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(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27a)),X0_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2EIN_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ebool_2EIN_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ebool_2EIN_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2EIN_2E2_2Emono_2EA_27b,axiom,
! [A_27b: $tType,X0_2E0: A_27b,X1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( c_2Ebool_2EIN_2E2(A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ebool_2EIN_2E0(A_27b),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Eoption_2ESOME_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a] : ( c_2Eoption_2ESOME_2E1(A_27a,X0_2E0) = app_2E2(A_27a,tyop_2Eoption_2Eoption(A_27a),c_2Eoption_2ESOME_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Eoption_2ESOME_2E1_2Emono_2Etyop_2Efmaptree_2Efmaptree_28A_27a_2CA_27b_29,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)] : ( c_2Eoption_2ESOME_2E1(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),X0_2E0) = app_2E2(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Eoption_2ESOME_2E0(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: $tType,A_27value: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value)),tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27key,A_27value),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value)))] : ( c_2Ebool_2ETYPE__DEFINITION_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value)),tyop_2Efmaptree_2Efmaptree(A_27key,A_27value),X0_2E0,X1_2E0) = app_2E2(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(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(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value)),tyop_2Efmaptree_2Efmaptree(A_27key,A_27value)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2E_5C_2F_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
( ( p(X0_2E0)
| p(X1_2E0) )
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_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: $tType,A_27b: $tType,X0_2E0: tyop_2Elist_2Elist(A_27a),X1_2E0: tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)] : ( c_2Efmaptree_2Eapply__path_2E2(A_27a,A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),app_2E2(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(A_27a,A_27b),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Efmaptree_2Econstruct_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: A_27a,X1_2E0: tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)))] : ( c_2Efmaptree_2Econstruct_2E2(A_27a,A_27b,X0_2E0,X1_2E0) = app_2E2(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(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(A_27a,A_27b),X0_2E0),X1_2E0) ) ).
tff(arityeq3_2Ec_2Efmaptree_2Econstruct_2E3_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: A_27a,X1_2E0: tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),X2_2E0: tyop_2Elist_2Elist(A_27b)] : ( c_2Efmaptree_2Econstruct_2E3(A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a),app_2E2(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(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(A_27a,A_27b),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq2_2Ec_2Efmaptree_2Econstruct_2E2_2Emono_2EA_27b_20mono_2EA_27a,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: A_27b,X1_2E0: tyop_2Efinite__map_2Efmap(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Eoption_2Eoption(A_27b)))] : ( c_2Efmaptree_2Econstruct_2E2(A_27b,A_27a,X0_2E0,X1_2E0) = app_2E2(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(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(A_27b,A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Efmaptree_2Efmtreerec_2E1_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27c,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,X0_2E0: 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)))] : ( c_2Efmaptree_2Efmtreerec_2E1(A_27a,A_27b,A_27c,X0_2E0) = app_2E2(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(A_27a,A_27b,A_27c),X0_2E0) ) ).
tff(arityeq2_2Ec_2Efmaptree_2Efmtreerec_2E2_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27c,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,X0_2E0: 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))),X1_2E0: tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)] : ( c_2Efmaptree_2Efmtreerec_2E2(A_27a,A_27b,A_27c,X0_2E0,X1_2E0) = app_2E2(tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),A_27a,app_2E2(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(A_27a,A_27b,A_27c),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Efmaptree_2EfromF_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Eoption_2Eoption(A_27b))] : ( c_2Efmaptree_2EfromF_2E1(A_27a,A_27b,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Eoption_2Eoption(A_27b)),tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),c_2Efmaptree_2EfromF_2E0(A_27a,A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Efmaptree_2EfromF_2E1_2Emono_2EA_27key_20mono_2EA_27value,axiom,
! [A_27key: $tType,A_27value: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value))] : ( c_2Efmaptree_2EfromF_2E1(A_27key,A_27value,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value)),tyop_2Efmaptree_2Efmaptree(A_27key,A_27value),c_2Efmaptree_2EfromF_2E0(A_27key,A_27value),X0_2E0) ) ).
tff(arityeq3_2Ec_2Efmaptree_2Efupd__at__path_2E3_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Elist_2Elist(A_27a),X1_2E0: tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),X2_2E0: tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)] : ( c_2Efmaptree_2Efupd__at__path_2E3(A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),app_2E2(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(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(A_27a,A_27b),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq1_2Ec_2Efmaptree_2Eitem_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)] : ( c_2Efmaptree_2Eitem_2E1(A_27a,A_27b,X0_2E0) = app_2E2(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),A_27b,c_2Efmaptree_2Eitem_2E0(A_27a,A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Efmaptree_2Eitem_2E1_2Emono_2EA_27b_20mono_2EA_27a,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)] : ( c_2Efmaptree_2Eitem_2E1(A_27b,A_27a,X0_2E0) = app_2E2(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),A_27a,c_2Efmaptree_2Eitem_2E0(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: $tType,A_27b: $tType,X0_2E0: tyop_2Elist_2Elist(A_27b),X1_2E0: tyop_2Eoption_2Eoption(A_27a),X2_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)))] : ( c_2Elist_2Elist__CASE_2E3(A_27b,tyop_2Eoption_2Eoption(A_27a),X0_2E0,X1_2E0,X2_2E0) = app_2E2(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(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(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(A_27b,tyop_2Eoption_2Eoption(A_27a)),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq1_2Ec_2Efmaptree_2Emap_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)] : ( c_2Efmaptree_2Emap_2E1(A_27a,A_27b,X0_2E0) = app_2E2(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efmaptree_2Emap_2E0(A_27a,A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Efmaptree_2Emap_2E1_2Emono_2EA_27b_20mono_2EA_27a,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)] : ( c_2Efmaptree_2Emap_2E1(A_27b,A_27a,X0_2E0) = app_2E2(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),c_2Efmaptree_2Emap_2E0(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: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Eoption_2Eoption(A_27b))),X1_2E0: tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))] : ( c_2Efinite__map_2Eo__f_2E2(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Eoption_2Eoption(A_27b)),X0_2E0,X1_2E0) = app_2E2(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(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(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Eoption_2Eoption(A_27b))),X0_2E0),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: $tType,A_27b: $tType,A_27c: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),A_27c),X1_2E0: tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a))] : ( c_2Efinite__map_2Eo__f_2E2(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),A_27c,X0_2E0,X1_2E0) = app_2E2(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),tyop_2Efinite__map_2Efmap(A_27b,A_27c),app_2E2(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(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),A_27c),X0_2E0),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: $tType,A_27b: $tType,A_27c: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),A_27a),X1_2E0: tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b))] : ( c_2Efinite__map_2Eo__f_2E2(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)),tyop_2Efinite__map_2Efmap(A_27c,A_27a),app_2E2(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(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),A_27a),X0_2E0),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: $tType,A_27b: $tType,X0_2E0: tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),X1_2E0: tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),X2_2E0: tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)))] : ( c_2Eoption_2Eoption__CASE_2E3(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),X0_2E0,X1_2E0,X2_2E0) = app_2E2(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(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(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(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Efmaptree_2Erelrec_2E3_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27c,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,X0_2E0: 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))),X1_2E0: tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),X2_2E0: A_27c] : ( c_2Efmaptree_2Erelrec_2E3(A_27a,A_27b,A_27c,X0_2E0,X1_2E0,X2_2E0) = app_2E2(A_27c,tyop_2Emin_2Ebool,app_2E2(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),app_2E2(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(A_27a,A_27b,A_27c),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Efmaptree_2Erelrec_2E3_2Emono_2EA_27b_20mono_2EA_27c_20mono_2EA_27a,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,X0_2E0: 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))),X1_2E0: tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),X2_2E0: A_27a] : ( c_2Efmaptree_2Erelrec_2E3(A_27b,A_27c,A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(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(A_27b,A_27c,A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq1_2Ec_2Efmaptree_2EtoF_2E1_2Emono_2EA_27key_20mono_2EA_27value,axiom,
! [A_27key: $tType,A_27value: $tType,X0_2E0: tyop_2Efmaptree_2Efmaptree(A_27key,A_27value)] : ( c_2Efmaptree_2EtoF_2E1(A_27key,A_27value,X0_2E0) = app_2E2(tyop_2Efmaptree_2Efmaptree(A_27key,A_27value),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value)),c_2Efmaptree_2EtoF_2E0(A_27key,A_27value),X0_2E0) ) ).
tff(arityeq3_2Ec_2Efmaptree_2Eupdate__at__path_2E3_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Elist_2Elist(A_27a),X1_2E0: A_27b,X2_2E0: tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)] : ( c_2Efmaptree_2Eupdate__at__path_2E3(A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),app_2E2(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(A_27a,A_27b),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq1_2Ec_2Efmaptree_2Ewf_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))] : ( c_2Efmaptree_2Ewf_2E1(A_27a,A_27b,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Ebool,c_2Efmaptree_2Ewf_2E0(A_27a,A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Efmaptree_2Ewf_2E1_2Emono_2EA_27value_20mono_2EA_27key,axiom,
! [A_27key: $tType,A_27value: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value))] : ( c_2Efmaptree_2Ewf_2E1(A_27value,A_27key,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value)),tyop_2Emin_2Ebool,c_2Efmaptree_2Ewf_2E0(A_27value,A_27key),X0_2E0) ) ).
tff(arityeq1_2Ec_2Ebool_2E_7E_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Ebool] :
( ~ p(X0_2E0)
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,c_2Ebool_2E_7E_2E0,X0_2E0)) ) ).
tff(thm_2Efmaptree_2Econstruct__def,axiom,
! [A_27a: $tType,A_27b: $tType,F0_2E0: 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))))] :
( ! [V1kfm_2E0: tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),V3h_2E0: A_27b,V4t_2E0: tyop_2Elist_2Elist(A_27b)] : ( app_2E2(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),app_2E2(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,V1kfm_2E0),V3h_2E0),V4t_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Eoption_2Eoption(A_27a),c_2Ebool_2EIN_2E2(A_27b,V3h_2E0,c_2Efinite__map_2EFDOM_2E1(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),V1kfm_2E0)),app_2E2(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a),c_2Efinite__map_2EFAPPLY_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),V1kfm_2E0,V3h_2E0),V4t_2E0),c_2Eoption_2ENONE_2E0(A_27a)) )
=> ! [V0a_2E0: A_27a,V1kfm_2E0: tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))),V2kl_2E0: tyop_2Elist_2Elist(A_27b)] : ( c_2Efmaptree_2Econstruct_2E3(A_27a,A_27b,V0a_2E0,V1kfm_2E0,V2kl_2E0) = c_2Elist_2Elist__CASE_2E3(A_27b,tyop_2Eoption_2Eoption(A_27a),V2kl_2E0,c_2Eoption_2ESOME_2E1(A_27a,V0a_2E0),app_2E2(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,V1kfm_2E0)) ) ) ).
tff(thm_2Efmaptree_2Ewf__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0x_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))] :
( p(c_2Efmaptree_2Ewf_2E1(A_27a,A_27b,V0x_2E0))
<=> ! [V1wf_27_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Ebool)] :
( ! [V2a0_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))] :
( ? [V3a_2E0: A_27a,V4fm_2E0: tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)))] :
( ( V2a0_2E0 = c_2Efmaptree_2Econstruct_2E2(A_27a,A_27b,V3a_2E0,V4fm_2E0) )
& ! [V5k_2E0: A_27b] :
( p(c_2Ebool_2EIN_2E2(A_27b,V5k_2E0,c_2Efinite__map_2EFDOM_2E1(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),V4fm_2E0)))
=> p(app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Ebool,V1wf_27_2E0,c_2Efinite__map_2EFAPPLY_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),V4fm_2E0,V5k_2E0))) ) )
=> p(app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Ebool,V1wf_27_2E0,V2a0_2E0)) )
=> p(app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Ebool,V1wf_27_2E0,V0x_2E0)) ) ) ).
tff(thm_2Efmaptree_2Efmaptree__TY__DEF,axiom,
! [A_27key: $tType,A_27value: $tType] :
? [V0rep_2E0: tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27key,A_27value),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value)))] : p(c_2Ebool_2ETYPE__DEFINITION_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value)),tyop_2Efmaptree_2Efmaptree(A_27key,A_27value),c_2Efmaptree_2Ewf_2E0(A_27value,A_27key),V0rep_2E0)) ).
tff(thm_2Efmaptree_2Efmap__bij__thm,axiom,
! [A_27key: $tType,A_27value: $tType] :
( ! [V0a_2E0: tyop_2Efmaptree_2Efmaptree(A_27key,A_27value)] : ( c_2Efmaptree_2EfromF_2E1(A_27key,A_27value,c_2Efmaptree_2EtoF_2E1(A_27key,A_27value,V0a_2E0)) = V0a_2E0 )
& ! [V1r_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27key),tyop_2Eoption_2Eoption(A_27value))] :
( p(c_2Efmaptree_2Ewf_2E1(A_27value,A_27key,V1r_2E0))
<=> ( c_2Efmaptree_2EtoF_2E1(A_27key,A_27value,c_2Efmaptree_2EfromF_2E1(A_27key,A_27value,V1r_2E0)) = V1r_2E0 ) ) ) ).
tff(thm_2Efmaptree_2EFTNode__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0i_2E0: A_27b,V1fm_2E0: tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))] : ( c_2Efmaptree_2EFTNode_2E2(A_27a,A_27b,V0i_2E0,V1fm_2E0) = c_2Efmaptree_2EfromF_2E1(A_27a,A_27b,c_2Efmaptree_2Econstruct_2E2(A_27b,A_27a,V0i_2E0,c_2Efinite__map_2Eo__f_2E2(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Eoption_2Eoption(A_27b)),c_2Efmaptree_2EtoF_2E0(A_27a,A_27b),V1fm_2E0))) ) ).
tff(thm_2Efmaptree_2Eitem__map__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0ft_2E0: tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)] : ( V0ft_2E0 = c_2Efmaptree_2EFTNode_2E2(A_27a,A_27b,c_2Efmaptree_2Eitem_2E1(A_27a,A_27b,V0ft_2E0),c_2Efmaptree_2Emap_2E1(A_27a,A_27b,V0ft_2E0)) ) ).
tff(thm_2Efmaptree_2Eapply__path__def,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0ft_2E0: tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)] : ( c_2Efmaptree_2Eapply__path_2E2(A_27a,A_27b,c_2Elist_2ENIL_2E0(A_27a),V0ft_2E0) = c_2Eoption_2ESOME_2E1(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V0ft_2E0) )
& ! [V1h_2E0: A_27a,V2t_2E0: tyop_2Elist_2Elist(A_27a),V3ft_2E0: tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)] : ( c_2Efmaptree_2Eapply__path_2E2(A_27a,A_27b,c_2Elist_2ECONS_2E2(A_27a,V1h_2E0,V2t_2E0),V3ft_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Ebool_2EIN_2E2(A_27a,V1h_2E0,c_2Efinite__map_2EFDOM_2E1(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),c_2Efmaptree_2Emap_2E1(A_27a,A_27b,V3ft_2E0))),c_2Efmaptree_2Eapply__path_2E2(A_27a,A_27b,V2t_2E0,c_2Efinite__map_2EFAPPLY_2E2(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),c_2Efmaptree_2Emap_2E1(A_27a,A_27b,V3ft_2E0),V1h_2E0)),c_2Eoption_2ENONE_2E0(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))) ) ) ).
tff(thm_2Efmaptree_2Eupdate__at__path__def,axiom,
! [A_27a: $tType,A_27b: $tType,F0_2E0: 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)))))] :
( ! [V5ft_2E0: tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V2h_2E0: A_27a,V6ft_27_2E0: tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)] : ( app_2E2(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),app_2E2(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,V5ft_2E0),V2h_2E0),V6ft_27_2E0) = c_2Eoption_2ESOME_2E1(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),c_2Efmaptree_2EFTNode_2E2(A_27a,A_27b,c_2Efmaptree_2Eitem_2E1(A_27a,A_27b,V5ft_2E0),c_2Efinite__map_2EFUPDATE_2E2(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),c_2Efmaptree_2Emap_2E1(A_27a,A_27b,V5ft_2E0),c_2Epair_2E_2C_2E2(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V2h_2E0,V6ft_27_2E0)))) )
=> ( ! [V0a_2E0: A_27b,V1ft_2E0: tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)] : ( c_2Efmaptree_2Eupdate__at__path_2E3(A_27a,A_27b,c_2Elist_2ENIL_2E0(A_27a),V0a_2E0,V1ft_2E0) = c_2Eoption_2ESOME_2E1(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),c_2Efmaptree_2EFTNode_2E2(A_27a,A_27b,V0a_2E0,c_2Efmaptree_2Emap_2E1(A_27a,A_27b,V1ft_2E0))) )
& ! [V2h_2E0: A_27a,V3t_2E0: tyop_2Elist_2Elist(A_27a),V4a_2E0: A_27b,V5ft_2E0: tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)] : ( c_2Efmaptree_2Eupdate__at__path_2E3(A_27a,A_27b,c_2Elist_2ECONS_2E2(A_27a,V2h_2E0,V3t_2E0),V4a_2E0,V5ft_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Ebool_2EIN_2E2(A_27a,V2h_2E0,c_2Efinite__map_2EFDOM_2E1(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),c_2Efmaptree_2Emap_2E1(A_27a,A_27b,V5ft_2E0))),c_2Eoption_2Eoption__CASE_2E3(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efmaptree_2Eupdate__at__path_2E3(A_27a,A_27b,V3t_2E0,V4a_2E0,c_2Efinite__map_2EFAPPLY_2E2(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),c_2Efmaptree_2Emap_2E1(A_27a,A_27b,V5ft_2E0),V2h_2E0)),c_2Eoption_2ENONE_2E0(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),app_2E2(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,V5ft_2E0),V2h_2E0)),c_2Eoption_2ENONE_2E0(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))) ) ) ) ).
tff(thm_2Efmaptree_2Efupd__at__path__def,axiom,
! [A_27a: $tType,A_27b: $tType,F0_2E0: 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)))))] :
( ! [V5ft_2E0: tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V2h_2E0: A_27a,V6ft_27_2E0: tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)] : ( app_2E2(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),app_2E2(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,V5ft_2E0),V2h_2E0),V6ft_27_2E0) = c_2Eoption_2ESOME_2E1(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),c_2Efmaptree_2EFTNode_2E2(A_27a,A_27b,c_2Efmaptree_2Eitem_2E1(A_27a,A_27b,V5ft_2E0),c_2Efinite__map_2EFUPDATE_2E2(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),c_2Efmaptree_2Emap_2E1(A_27a,A_27b,V5ft_2E0),c_2Epair_2E_2C_2E2(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V2h_2E0,V6ft_27_2E0)))) )
=> ( ! [V0f_2E0: tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),V1ft_2E0: tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)] : ( c_2Efmaptree_2Efupd__at__path_2E3(A_27a,A_27b,c_2Elist_2ENIL_2E0(A_27a),V0f_2E0,V1ft_2E0) = app_2E2(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),V0f_2E0,V1ft_2E0) )
& ! [V2h_2E0: A_27a,V3t_2E0: tyop_2Elist_2Elist(A_27a),V4f_2E0: tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),V5ft_2E0: tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)] : ( c_2Efmaptree_2Efupd__at__path_2E3(A_27a,A_27b,c_2Elist_2ECONS_2E2(A_27a,V2h_2E0,V3t_2E0),V4f_2E0,V5ft_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Ebool_2EIN_2E2(A_27a,V2h_2E0,c_2Efinite__map_2EFDOM_2E1(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),c_2Efmaptree_2Emap_2E1(A_27a,A_27b,V5ft_2E0))),c_2Eoption_2Eoption__CASE_2E3(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efmaptree_2Efupd__at__path_2E3(A_27a,A_27b,V3t_2E0,V4f_2E0,c_2Efinite__map_2EFAPPLY_2E2(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),c_2Efmaptree_2Emap_2E1(A_27a,A_27b,V5ft_2E0),V2h_2E0)),c_2Eoption_2ENONE_2E0(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),app_2E2(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,V5ft_2E0),V2h_2E0)),c_2Eoption_2ENONE_2E0(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))) ) ) ) ).
tff(thm_2Efmaptree_2Erelrec__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0x_2E0: 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))),V1x_27_2E0: tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),V2x_27_27_2E0: A_27c] :
( p(c_2Efmaptree_2Erelrec_2E3(A_27a,A_27b,A_27c,V0x_2E0,V1x_27_2E0,V2x_27_27_2E0))
<=> ! [V3relrec_27_2E0: tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool))] :
( ! [V4a0_2E0: tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),V5a1_2E0: A_27c] :
( ? [V6i_2E0: A_27a,V7fm_2E0: tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V8rfm_2E0: tyop_2Efinite__map_2Efmap(A_27b,A_27c)] :
( ( V4a0_2E0 = c_2Efmaptree_2EFTNode_2E2(A_27b,A_27a,V6i_2E0,V7fm_2E0) )
& ( V5a1_2E0 = app_2E2(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),A_27c,app_2E2(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(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,V6i_2E0),V8rfm_2E0),V7fm_2E0) )
& ( c_2Efinite__map_2EFDOM_2E1(A_27b,A_27c,V8rfm_2E0) = c_2Efinite__map_2EFDOM_2E1(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),V7fm_2E0) )
& ! [V9d_2E0: A_27b] :
( p(c_2Ebool_2EIN_2E2(A_27b,V9d_2E0,c_2Efinite__map_2EFDOM_2E1(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),V7fm_2E0)))
=> p(app_2E2(A_27c,tyop_2Emin_2Ebool,app_2E2(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),V3relrec_27_2E0,c_2Efinite__map_2EFAPPLY_2E2(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),V7fm_2E0,V9d_2E0)),c_2Efinite__map_2EFAPPLY_2E2(A_27b,A_27c,V8rfm_2E0,V9d_2E0))) ) )
=> p(app_2E2(A_27c,tyop_2Emin_2Ebool,app_2E2(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),V3relrec_27_2E0,V4a0_2E0),V5a1_2E0)) )
=> p(app_2E2(A_27c,tyop_2Emin_2Ebool,app_2E2(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),V3relrec_27_2E0,V1x_27_2E0),V2x_27_27_2E0)) ) ) ).
tff(thm_2Efmaptree_2Efmtreerec__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,F0_2E0: 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)))] :
( ! [V0h_2E0: 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))),V1ft_2E0: tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),V2r_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(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,V0h_2E0),V1ft_2E0),V2r_2E0) = c_2Efmaptree_2Erelrec_2E3(A_27b,A_27c,A_27a,V0h_2E0,V1ft_2E0,V2r_2E0) )
=> ! [V0h_2E0: 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))),V1ft_2E0: tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)] : ( c_2Efmaptree_2Efmtreerec_2E2(A_27a,A_27b,A_27c,V0h_2E0,V1ft_2E0) = c_2Emin_2E_40_2E1(A_27a,app_2E2(tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(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,V0h_2E0),V1ft_2E0)) ) ) ).
tff(thm_2Efmaptree_2Ewf__rules,axiom,
! [A_27a: $tType,A_27b: $tType,V0a_2E0: A_27a,V1fm_2E0: tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)))] :
( ! [V2k_2E0: A_27b] :
( p(c_2Ebool_2EIN_2E2(A_27b,V2k_2E0,c_2Efinite__map_2EFDOM_2E1(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),V1fm_2E0)))
=> p(c_2Efmaptree_2Ewf_2E1(A_27a,A_27b,c_2Efinite__map_2EFAPPLY_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),V1fm_2E0,V2k_2E0))) )
=> p(c_2Efmaptree_2Ewf_2E1(A_27a,A_27b,c_2Efmaptree_2Econstruct_2E2(A_27a,A_27b,V0a_2E0,V1fm_2E0))) ) ).
tff(thm_2Efmaptree_2Ewf__ind,axiom,
! [A_27a: $tType,A_27b: $tType,V0wf_27_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Ebool)] :
( ! [V1a_2E0: A_27a,V2fm_2E0: tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)))] :
( ! [V3k_2E0: A_27b] :
( p(c_2Ebool_2EIN_2E2(A_27b,V3k_2E0,c_2Efinite__map_2EFDOM_2E1(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),V2fm_2E0)))
=> p(app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Ebool,V0wf_27_2E0,c_2Efinite__map_2EFAPPLY_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),V2fm_2E0,V3k_2E0))) )
=> p(app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Ebool,V0wf_27_2E0,c_2Efmaptree_2Econstruct_2E2(A_27a,A_27b,V1a_2E0,V2fm_2E0))) )
=> ! [V4a0_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))] :
( p(c_2Efmaptree_2Ewf_2E1(A_27a,A_27b,V4a0_2E0))
=> p(app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Ebool,V0wf_27_2E0,V4a0_2E0)) ) ) ).
tff(thm_2Efmaptree_2Ewf__strongind,axiom,
! [A_27a: $tType,A_27b: $tType,V0wf_27_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Ebool)] :
( ! [V1a_2E0: A_27a,V2fm_2E0: tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)))] :
( ! [V3k_2E0: A_27b] :
( p(c_2Ebool_2EIN_2E2(A_27b,V3k_2E0,c_2Efinite__map_2EFDOM_2E1(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),V2fm_2E0)))
=> ( p(c_2Efmaptree_2Ewf_2E1(A_27a,A_27b,c_2Efinite__map_2EFAPPLY_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),V2fm_2E0,V3k_2E0)))
& p(app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Ebool,V0wf_27_2E0,c_2Efinite__map_2EFAPPLY_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),V2fm_2E0,V3k_2E0))) ) )
=> p(app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Ebool,V0wf_27_2E0,c_2Efmaptree_2Econstruct_2E2(A_27a,A_27b,V1a_2E0,V2fm_2E0))) )
=> ! [V4a0_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))] :
( p(c_2Efmaptree_2Ewf_2E1(A_27a,A_27b,V4a0_2E0))
=> p(app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),tyop_2Emin_2Ebool,V0wf_27_2E0,V4a0_2E0)) ) ) ).
tff(thm_2Efmaptree_2Ewf__cases,axiom,
! [A_27a: $tType,A_27b: $tType,V0a0_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a))] :
( p(c_2Efmaptree_2Ewf_2E1(A_27a,A_27b,V0a0_2E0))
<=> ? [V1a_2E0: A_27a,V2fm_2E0: tyop_2Efinite__map_2Efmap(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)))] :
( ( V0a0_2E0 = c_2Efmaptree_2Econstruct_2E2(A_27a,A_27b,V1a_2E0,V2fm_2E0) )
& ! [V3k_2E0: A_27b] :
( p(c_2Ebool_2EIN_2E2(A_27b,V3k_2E0,c_2Efinite__map_2EFDOM_2E1(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),V2fm_2E0)))
=> p(c_2Efmaptree_2Ewf_2E1(A_27a,A_27b,c_2Efinite__map_2EFAPPLY_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Eoption_2Eoption(A_27a)),V2fm_2E0,V3k_2E0))) ) ) ) ).
tff(thm_2Efmaptree_2EFTNode__11,axiom,
! [A_27a: $tType,A_27b: $tType,V0i2_2E0: A_27b,V1i1_2E0: A_27b,V2f2_2E0: tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),V3f1_2E0: tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))] :
( ( c_2Efmaptree_2EFTNode_2E2(A_27a,A_27b,V1i1_2E0,V3f1_2E0) = c_2Efmaptree_2EFTNode_2E2(A_27a,A_27b,V0i2_2E0,V2f2_2E0) )
<=> ( ( V1i1_2E0 = V0i2_2E0 )
& ( V3f1_2E0 = V2f2_2E0 ) ) ) ).
tff(thm_2Efmaptree_2Efmaptree__nchotomy,axiom,
! [A_27a: $tType,A_27b: $tType,V0ft_2E0: tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)] :
? [V1i_2E0: A_27b,V2fm_2E0: tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))] : ( V0ft_2E0 = c_2Efmaptree_2EFTNode_2E2(A_27a,A_27b,V1i_2E0,V2fm_2E0) ) ).
tff(thm_2Efmaptree_2Eitem__thm,axiom,
! [A_27a: $tType,A_27b: $tType,V0i_2E0: A_27a,V1fm_2E0: tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a))] : ( c_2Efmaptree_2Eitem_2E1(A_27b,A_27a,c_2Efmaptree_2EFTNode_2E2(A_27b,A_27a,V0i_2E0,V1fm_2E0)) = V0i_2E0 ) ).
tff(thm_2Efmaptree_2Emap__thm,axiom,
! [A_27a: $tType,A_27b: $tType,V0i_2E0: A_27a,V1fm_2E0: tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a))] : ( c_2Efmaptree_2Emap_2E1(A_27b,A_27a,c_2Efmaptree_2EFTNode_2E2(A_27b,A_27a,V0i_2E0,V1fm_2E0)) = V1fm_2E0 ) ).
tff(thm_2Efmaptree_2Eft__ind,axiom,
! [A_27a: $tType,A_27b: $tType,V0P_2E0: tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Emin_2Ebool)] :
( ! [V1a_2E0: A_27b,V2fm_2E0: tyop_2Efinite__map_2Efmap(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))] :
( ! [V3k_2E0: A_27a] :
( p(c_2Ebool_2EIN_2E2(A_27a,V3k_2E0,c_2Efinite__map_2EFDOM_2E1(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V2fm_2E0)))
=> p(app_2E2(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Emin_2Ebool,V0P_2E0,c_2Efinite__map_2EFAPPLY_2E2(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V2fm_2E0,V3k_2E0))) )
=> p(app_2E2(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Emin_2Ebool,V0P_2E0,c_2Efmaptree_2EFTNode_2E2(A_27a,A_27b,V1a_2E0,V2fm_2E0))) )
=> ! [V4ft_2E0: tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)] : p(app_2E2(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Emin_2Ebool,V0P_2E0,V4ft_2E0)) ) ).
tff(thm_2Efmaptree_2Eapplicable__paths__FINITE,axiom,
! [A_27a: $tType,A_27b: $tType,F1_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Emin_2Ebool)),F0_2E0: 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)))] :
( ! [V1p_2E0: tyop_2Elist_2Elist(A_27a),V0ft_2E0: tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)] :
( p(app_2E2(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Emin_2Ebool,app_2E2(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Emin_2Ebool),F1_2E0,V1p_2E0),V0ft_2E0))
<=> ? [V2ft_27_2E0: tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)] : ( c_2Efmaptree_2Eapply__path_2E2(A_27a,A_27b,V1p_2E0,V0ft_2E0) = c_2Eoption_2ESOME_2E1(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V2ft_27_2E0) ) )
=> ( ! [V0ft_2E0: tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V1p_2E0: tyop_2Elist_2Elist(A_27a)] : ( app_2E2(tyop_2Elist_2Elist(A_27a),tyop_2Epair_2Eprod(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool),app_2E2(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,V0ft_2E0),V1p_2E0) = c_2Epair_2E_2C_2E2(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Ebool,V1p_2E0,app_2E2(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Emin_2Ebool,app_2E2(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Emin_2Ebool),F1_2E0,V1p_2E0),V0ft_2E0)) )
=> ! [V0ft_2E0: tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)] : p(c_2Epred__set_2EFINITE_2E1(tyop_2Elist_2Elist(A_27a),c_2Epred__set_2EGSPEC_2E1(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27a),app_2E2(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,V0ft_2E0)))) ) ) ).
tff(thm_2Efmaptree_2Eapply__path__SNOC,axiom,
! [A_27a: $tType,A_27b: $tType,F0_2E0: 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))))] :
( ! [V1x_2E0: A_27a,V3ft_27_2E0: tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)] : ( app_2E2(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),F0_2E0,V1x_2E0),V3ft_27_2E0) = c_2Efinite__map_2EFLOOKUP_2E2(A_27a,tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),c_2Efmaptree_2Emap_2E1(A_27a,A_27b,V3ft_27_2E0),V1x_2E0) )
=> ! [V0ft_2E0: tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),V1x_2E0: A_27a,V2p_2E0: tyop_2Elist_2Elist(A_27a)] : ( c_2Efmaptree_2Eapply__path_2E2(A_27a,A_27b,c_2Elist_2EAPPEND_2E2(A_27a,V2p_2E0,c_2Elist_2ECONS_2E2(A_27a,V1x_2E0,c_2Elist_2ENIL_2E0(A_27a))),V0ft_2E0) = c_2Eoption_2Eoption__CASE_2E3(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),c_2Efmaptree_2Eapply__path_2E2(A_27a,A_27b,V2p_2E0,V0ft_2E0),c_2Eoption_2ENONE_2E0(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b),tyop_2Eoption_2Eoption(tyop_2Efmaptree_2Efmaptree(A_27a,A_27b))),F0_2E0,V1x_2E0)) ) ) ).
tff(thm_2Efmaptree_2Erelrec__rules,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0h_2E0: 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))),V1i_2E0: A_27a,V2fm_2E0: tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V3rfm_2E0: tyop_2Efinite__map_2Efmap(A_27b,A_27c)] :
( ( ( c_2Efinite__map_2EFDOM_2E1(A_27b,A_27c,V3rfm_2E0) = c_2Efinite__map_2EFDOM_2E1(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),V2fm_2E0) )
& ! [V4d_2E0: A_27b] :
( p(c_2Ebool_2EIN_2E2(A_27b,V4d_2E0,c_2Efinite__map_2EFDOM_2E1(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),V2fm_2E0)))
=> p(c_2Efmaptree_2Erelrec_2E3(A_27a,A_27b,A_27c,V0h_2E0,c_2Efinite__map_2EFAPPLY_2E2(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),V2fm_2E0,V4d_2E0),c_2Efinite__map_2EFAPPLY_2E2(A_27b,A_27c,V3rfm_2E0,V4d_2E0))) ) )
=> p(c_2Efmaptree_2Erelrec_2E3(A_27a,A_27b,A_27c,V0h_2E0,c_2Efmaptree_2EFTNode_2E2(A_27b,A_27a,V1i_2E0,V2fm_2E0),app_2E2(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),A_27c,app_2E2(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(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,V1i_2E0),V3rfm_2E0),V2fm_2E0))) ) ).
tff(thm_2Efmaptree_2Erelrec__ind,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0h_2E0: 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))),V1relrec_27_2E0: tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool))] :
( ! [V2i_2E0: A_27a,V3fm_2E0: tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V4rfm_2E0: tyop_2Efinite__map_2Efmap(A_27b,A_27c)] :
( ( ( c_2Efinite__map_2EFDOM_2E1(A_27b,A_27c,V4rfm_2E0) = c_2Efinite__map_2EFDOM_2E1(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),V3fm_2E0) )
& ! [V5d_2E0: A_27b] :
( p(c_2Ebool_2EIN_2E2(A_27b,V5d_2E0,c_2Efinite__map_2EFDOM_2E1(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),V3fm_2E0)))
=> p(app_2E2(A_27c,tyop_2Emin_2Ebool,app_2E2(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),V1relrec_27_2E0,c_2Efinite__map_2EFAPPLY_2E2(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),V3fm_2E0,V5d_2E0)),c_2Efinite__map_2EFAPPLY_2E2(A_27b,A_27c,V4rfm_2E0,V5d_2E0))) ) )
=> p(app_2E2(A_27c,tyop_2Emin_2Ebool,app_2E2(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),V1relrec_27_2E0,c_2Efmaptree_2EFTNode_2E2(A_27b,A_27a,V2i_2E0,V3fm_2E0)),app_2E2(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),A_27c,app_2E2(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(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,V2i_2E0),V4rfm_2E0),V3fm_2E0))) )
=> ! [V6a0_2E0: tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),V7a1_2E0: A_27c] :
( p(c_2Efmaptree_2Erelrec_2E3(A_27a,A_27b,A_27c,V0h_2E0,V6a0_2E0,V7a1_2E0))
=> p(app_2E2(A_27c,tyop_2Emin_2Ebool,app_2E2(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),V1relrec_27_2E0,V6a0_2E0),V7a1_2E0)) ) ) ).
tff(thm_2Efmaptree_2Erelrec__strongind,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0h_2E0: 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))),V1relrec_27_2E0: tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool))] :
( ! [V2i_2E0: A_27a,V3fm_2E0: tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V4rfm_2E0: tyop_2Efinite__map_2Efmap(A_27b,A_27c)] :
( ( ( c_2Efinite__map_2EFDOM_2E1(A_27b,A_27c,V4rfm_2E0) = c_2Efinite__map_2EFDOM_2E1(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),V3fm_2E0) )
& ! [V5d_2E0: A_27b] :
( p(c_2Ebool_2EIN_2E2(A_27b,V5d_2E0,c_2Efinite__map_2EFDOM_2E1(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),V3fm_2E0)))
=> ( p(c_2Efmaptree_2Erelrec_2E3(A_27a,A_27b,A_27c,V0h_2E0,c_2Efinite__map_2EFAPPLY_2E2(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),V3fm_2E0,V5d_2E0),c_2Efinite__map_2EFAPPLY_2E2(A_27b,A_27c,V4rfm_2E0,V5d_2E0)))
& p(app_2E2(A_27c,tyop_2Emin_2Ebool,app_2E2(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),V1relrec_27_2E0,c_2Efinite__map_2EFAPPLY_2E2(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),V3fm_2E0,V5d_2E0)),c_2Efinite__map_2EFAPPLY_2E2(A_27b,A_27c,V4rfm_2E0,V5d_2E0))) ) ) )
=> p(app_2E2(A_27c,tyop_2Emin_2Ebool,app_2E2(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),V1relrec_27_2E0,c_2Efmaptree_2EFTNode_2E2(A_27b,A_27a,V2i_2E0,V3fm_2E0)),app_2E2(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),A_27c,app_2E2(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(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,V2i_2E0),V4rfm_2E0),V3fm_2E0))) )
=> ! [V6a0_2E0: tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),V7a1_2E0: A_27c] :
( p(c_2Efmaptree_2Erelrec_2E3(A_27a,A_27b,A_27c,V0h_2E0,V6a0_2E0,V7a1_2E0))
=> p(app_2E2(A_27c,tyop_2Emin_2Ebool,app_2E2(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Ebool),V1relrec_27_2E0,V6a0_2E0),V7a1_2E0)) ) ) ).
tff(thm_2Efmaptree_2Erelrec__cases,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0h_2E0: 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))),V1a0_2E0: tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),V2a1_2E0: A_27c] :
( p(c_2Efmaptree_2Erelrec_2E3(A_27a,A_27b,A_27c,V0h_2E0,V1a0_2E0,V2a1_2E0))
<=> ? [V3i_2E0: A_27a,V4fm_2E0: tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),V5rfm_2E0: tyop_2Efinite__map_2Efmap(A_27b,A_27c)] :
( ( V1a0_2E0 = c_2Efmaptree_2EFTNode_2E2(A_27b,A_27a,V3i_2E0,V4fm_2E0) )
& ( V2a1_2E0 = app_2E2(tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a)),A_27c,app_2E2(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(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,V3i_2E0),V5rfm_2E0),V4fm_2E0) )
& ( c_2Efinite__map_2EFDOM_2E1(A_27b,A_27c,V5rfm_2E0) = c_2Efinite__map_2EFDOM_2E1(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),V4fm_2E0) )
& ! [V6d_2E0: A_27b] :
( p(c_2Ebool_2EIN_2E2(A_27b,V6d_2E0,c_2Efinite__map_2EFDOM_2E1(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),V4fm_2E0)))
=> p(c_2Efmaptree_2Erelrec_2E3(A_27a,A_27b,A_27c,V0h_2E0,c_2Efinite__map_2EFAPPLY_2E2(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),V4fm_2E0,V6d_2E0),c_2Efinite__map_2EFAPPLY_2E2(A_27b,A_27c,V5rfm_2E0,V6d_2E0))) ) ) ) ).
tff(thm_2Efmaptree_2Efmtreerec__thm,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0i_2E0: A_27b,V1h_2E0: 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))),V2fm_2E0: tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b))] : ( c_2Efmaptree_2Efmtreerec_2E2(A_27a,A_27b,A_27c,V1h_2E0,c_2Efmaptree_2EFTNode_2E2(A_27c,A_27b,V0i_2E0,V2fm_2E0)) = app_2E2(tyop_2Efinite__map_2Efmap(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b)),A_27a,app_2E2(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(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,V0i_2E0),c_2Efinite__map_2Eo__f_2E2(A_27c,tyop_2Efmaptree_2Efmaptree(A_27c,A_27b),A_27a,c_2Efmaptree_2Efmtreerec_2E1(A_27a,A_27b,A_27c,V1h_2E0),V2fm_2E0)),V2fm_2E0) ) ).
tff(thm_2Efmaptree_2Efmtree__Axiom,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0h_2E0: 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)))] :
? [V1f_2E0: tyop_2Emin_2Efun(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),A_27c)] :
! [V2i_2E0: A_27a,V3fm_2E0: tyop_2Efinite__map_2Efmap(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a))] : ( app_2E2(tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),A_27c,V1f_2E0,c_2Efmaptree_2EFTNode_2E2(A_27b,A_27a,V2i_2E0,V3fm_2E0)) = app_2E2(tyop_2Efinite__map_2Efmap(A_27b,A_27c),A_27c,app_2E2(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(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,V2i_2E0),V3fm_2E0),c_2Efinite__map_2Eo__f_2E2(A_27b,tyop_2Efmaptree_2Efmaptree(A_27b,A_27a),A_27c,V1f_2E0,V3fm_2E0)) ) ).
%------------------------------------------------------------------------------