ITP001 Axioms: ITP048_7.ax
%------------------------------------------------------------------------------
% File : ITP048_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 : inftree.ax [Gau19]
% : HL4048_7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 133 ( 38 unt; 60 typ; 0 def)
% Number of atoms : 173 ( 87 equ)
% Maximal formula atoms : 10 ( 1 avg)
% Number of connectives : 104 ( 4 ~; 8 |; 24 &)
% ( 27 <=>; 41 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 7 avg)
% Maximal term depth : 10 ( 2 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 52 ( 30 >; 22 *; 0 +; 0 <<)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 58 ( 58 usr; 6 con; 0-8 aty)
% Number of variables : 530 ( 412 !; 20 ?; 530 :)
% ( 98 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tyop_2Einftree_2Einftree,type,
tyop_2Einftree_2Einftree: ( $tType * $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_2Esum_2Esum,type,
tyop_2Esum_2Esum: ( $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_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_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_2Ebool_2EF_2E0,type,
c_2Ebool_2EF_2E0: tyop_2Emin_2Ebool ).
tff(c_2Elist_2EHD_2E0,type,
c_2Elist_2EHD_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),A_27a) ).
tff(c_2Elist_2EHD_2E1,type,
c_2Elist_2EHD_2E1:
!>[A_27a: $tType] : ( tyop_2Elist_2Elist(A_27a) > A_27a ) ).
tff(c_2Esum_2EINL_2E0,type,
c_2Esum_2EINL_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Esum_2Esum(A_27a,A_27b)) ).
tff(c_2Esum_2EINL_2E1,type,
c_2Esum_2EINL_2E1:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > tyop_2Esum_2Esum(A_27a,A_27b) ) ).
tff(c_2Esum_2EINR_2E0,type,
c_2Esum_2EINR_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(A_27b,tyop_2Esum_2Esum(A_27a,A_27b)) ).
tff(c_2Esum_2EINR_2E1,type,
c_2Esum_2EINR_2E1:
!>[A_27a: $tType,A_27b: $tType] : ( A_27b > tyop_2Esum_2Esum(A_27a,A_27b) ) ).
tff(c_2Elist_2ENIL_2E0,type,
c_2Elist_2ENIL_2E0:
!>[A_27a: $tType] : tyop_2Elist_2Elist(A_27a) ).
tff(c_2Ebool_2ET_2E0,type,
c_2Ebool_2ET_2E0: tyop_2Emin_2Ebool ).
tff(c_2Elist_2ETL_2E0,type,
c_2Elist_2ETL_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27a)) ).
tff(c_2Elist_2ETL_2E1,type,
c_2Elist_2ETL_2E1:
!>[A_27a: $tType] : ( tyop_2Elist_2Elist(A_27a) > tyop_2Elist_2Elist(A_27a) ) ).
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_2Einftree_2Efrom__inftree_2E0,type,
c_2Einftree_2Efrom__inftree_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27d: $tType] : tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27b,A_27d),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))) ).
tff(c_2Einftree_2Efrom__inftree_2E1,type,
c_2Einftree_2Efrom__inftree_2E1:
!>[A_27a: $tType,A_27b: $tType,A_27d: $tType] : ( tyop_2Einftree_2Einftree(A_27a,A_27b,A_27d) > tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)) ) ).
tff(c_2Einftree_2Efrom__inftree_2E2,type,
c_2Einftree_2Efrom__inftree_2E2:
!>[A_27a: $tType,A_27b: $tType,A_27d: $tType] : ( ( tyop_2Einftree_2Einftree(A_27a,A_27b,A_27d) * tyop_2Elist_2Elist(A_27d) ) > tyop_2Esum_2Esum(A_27a,A_27b) ) ).
tff(c_2Einftree_2EiLf_2E0,type,
c_2Einftree_2EiLf_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c)) ).
tff(c_2Einftree_2EiLf_2E1,type,
c_2Einftree_2EiLf_2E1:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( A_27a > tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c) ) ).
tff(c_2Einftree_2EiNd_2E0,type,
c_2Einftree_2EiNd_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c)),tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c))) ).
tff(c_2Einftree_2EiNd_2E2,type,
c_2Einftree_2EiNd_2E2:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27b * tyop_2Emin_2Efun(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c)) ) > tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c) ) ).
tff(c_2Einftree_2Einftree__CASE_2E0,type,
c_2Einftree_2Einftree__CASE_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] : tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27d),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c)),A_27d)),A_27d))) ).
tff(c_2Einftree_2Einftree__CASE_2E3,type,
c_2Einftree_2Einftree__CASE_2E3:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] : ( ( tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c) * tyop_2Emin_2Efun(A_27a,A_27d) * tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c)),A_27d)) ) > A_27d ) ).
tff(c_2Einftree_2Einftree__rec_2E0,type,
c_2Einftree_2Einftree__rec_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27a),A_27a)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27b,A_27c,A_27d),A_27a))) ).
tff(c_2Einftree_2Einftree__rec_2E3,type,
c_2Einftree_2Einftree__rec_2E3:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] : ( ( tyop_2Emin_2Efun(A_27b,A_27a) * tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27a),A_27a)) * tyop_2Einftree_2Einftree(A_27b,A_27c,A_27d) ) > A_27a ) ).
tff(c_2Einftree_2Eis__tree_2E0,type,
c_2Einftree_2Eis__tree_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27d: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),tyop_2Emin_2Ebool) ).
tff(c_2Einftree_2Eis__tree_2E1,type,
c_2Einftree_2Eis__tree_2E1:
!>[A_27a: $tType,A_27b: $tType,A_27d: $tType] : ( tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)) > tyop_2Emin_2Ebool ) ).
tff(c_2Ecombin_2Eo_2E0,type,
c_2Ecombin_2Eo_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27c),tyop_2Emin_2Efun(A_27a,A_27b))) ).
tff(c_2Ecombin_2Eo_2E2,type,
c_2Ecombin_2Eo_2E2:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Emin_2Efun(A_27c,A_27b) * tyop_2Emin_2Efun(A_27a,A_27c) ) > tyop_2Emin_2Efun(A_27a,A_27b) ) ).
tff(c_2Einftree_2Erelrec_2E0,type,
c_2Einftree_2Erelrec_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)))) ).
tff(c_2Einftree_2Erelrec_2E4,type,
c_2Einftree_2Erelrec_2E4:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27b) * tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)) * tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d) * A_27b ) > tyop_2Emin_2Ebool ) ).
tff(c_2Einftree_2Eto__inftree_2E0,type,
c_2Einftree_2Eto__inftree_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27d: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),tyop_2Einftree_2Einftree(A_27a,A_27b,A_27d)) ).
tff(c_2Einftree_2Eto__inftree_2E1,type,
c_2Einftree_2Eto__inftree_2E1:
!>[A_27a: $tType,A_27b: $tType,A_27d: $tType] : ( tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)) > tyop_2Einftree_2Einftree(A_27a,A_27b,A_27d) ) ).
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_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(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Esum_2Esum_28A_27a_2CA_27b_29,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Esum_2Esum(A_27a,A_27b),X2_2E0: tyop_2Esum_2Esum(A_27a,A_27b)] : ( c_2Ebool_2ECOND_2E3(tyop_2Esum_2Esum(A_27a,A_27b),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Esum_2Esum(A_27a,A_27b),app_2E2(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Esum_2Esum(A_27a,A_27b)),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Esum_2Esum(A_27a,A_27b))),c_2Ebool_2ECOND_2E0(tyop_2Esum_2Esum(A_27a,A_27b)),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Esum_2Esum_28A_27c_2CA_27a_29,axiom,
! [A_27a: $tType,A_27c: $tType,X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Esum_2Esum(A_27c,A_27a),X2_2E0: tyop_2Esum_2Esum(A_27c,A_27a)] : ( c_2Ebool_2ECOND_2E3(tyop_2Esum_2Esum(A_27c,A_27a),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Esum_2Esum(A_27c,A_27a),tyop_2Esum_2Esum(A_27c,A_27a),app_2E2(tyop_2Esum_2Esum(A_27c,A_27a),tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27c,A_27a),tyop_2Esum_2Esum(A_27c,A_27a)),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27c,A_27a),tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27c,A_27a),tyop_2Esum_2Esum(A_27c,A_27a))),c_2Ebool_2ECOND_2E0(tyop_2Esum_2Esum(A_27c,A_27a)),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq1_2Ec_2Elist_2EHD_2E1_2Emono_2EA_27b,axiom,
! [A_27b: $tType,X0_2E0: tyop_2Elist_2Elist(A_27b)] : ( c_2Elist_2EHD_2E1(A_27b,X0_2E0) = app_2E2(tyop_2Elist_2Elist(A_27b),A_27b,c_2Elist_2EHD_2E0(A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Elist_2EHD_2E1_2Emono_2EA_27c,axiom,
! [A_27c: $tType,X0_2E0: tyop_2Elist_2Elist(A_27c)] : ( c_2Elist_2EHD_2E1(A_27c,X0_2E0) = app_2E2(tyop_2Elist_2Elist(A_27c),A_27c,c_2Elist_2EHD_2E0(A_27c),X0_2E0) ) ).
tff(arityeq1_2Ec_2Elist_2EHD_2E1_2Emono_2EA_27d,axiom,
! [A_27d: $tType,X0_2E0: tyop_2Elist_2Elist(A_27d)] : ( c_2Elist_2EHD_2E1(A_27d,X0_2E0) = app_2E2(tyop_2Elist_2Elist(A_27d),A_27d,c_2Elist_2EHD_2E0(A_27d),X0_2E0) ) ).
tff(arityeq1_2Ec_2Esum_2EINL_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: A_27a] : ( c_2Esum_2EINL_2E1(A_27a,A_27b,X0_2E0) = app_2E2(A_27a,tyop_2Esum_2Esum(A_27a,A_27b),c_2Esum_2EINL_2E0(A_27a,A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Esum_2EINR_2E1_2Emono_2EA_27c_20mono_2EA_27a,axiom,
! [A_27a: $tType,A_27c: $tType,X0_2E0: A_27a] : ( c_2Esum_2EINR_2E1(A_27c,A_27a,X0_2E0) = app_2E2(A_27a,tyop_2Esum_2Esum(A_27c,A_27a),c_2Esum_2EINR_2E0(A_27c,A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Esum_2EINR_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: A_27b] : ( c_2Esum_2EINR_2E1(A_27a,A_27b,X0_2E0) = app_2E2(A_27b,tyop_2Esum_2Esum(A_27a,A_27b),c_2Esum_2EINR_2E0(A_27a,A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Elist_2ETL_2E1_2Emono_2EA_27b,axiom,
! [A_27b: $tType,X0_2E0: tyop_2Elist_2Elist(A_27b)] : ( c_2Elist_2ETL_2E1(A_27b,X0_2E0) = app_2E2(tyop_2Elist_2Elist(A_27b),tyop_2Elist_2Elist(A_27b),c_2Elist_2ETL_2E0(A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Elist_2ETL_2E1_2Emono_2EA_27c,axiom,
! [A_27c: $tType,X0_2E0: tyop_2Elist_2Elist(A_27c)] : ( c_2Elist_2ETL_2E1(A_27c,X0_2E0) = app_2E2(tyop_2Elist_2Elist(A_27c),tyop_2Elist_2Elist(A_27c),c_2Elist_2ETL_2E0(A_27c),X0_2E0) ) ).
tff(arityeq1_2Ec_2Elist_2ETL_2E1_2Emono_2EA_27d,axiom,
! [A_27d: $tType,X0_2E0: tyop_2Elist_2Elist(A_27d)] : ( c_2Elist_2ETL_2E1(A_27d,X0_2E0) = app_2E2(tyop_2Elist_2Elist(A_27d),tyop_2Elist_2Elist(A_27d),c_2Elist_2ETL_2E0(A_27d),X0_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2ETYPE__DEFINITION_2E2_2Emono_2Etyop_2Emin_2Efun_28tyop_2Elist_2Elist_28A_27d_29_2Ctyop_2Esum_2Esum_28A_27a_2CA_27b_29_29_20mono_2Etyop_2Einftree_2Einftree_28A_27a_2CA_27b_2CA_27d_29,axiom,
! [A_27a: $tType,A_27b: $tType,A_27d: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27b,A_27d),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)))] : ( c_2Ebool_2ETYPE__DEFINITION_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),tyop_2Einftree_2Einftree(A_27a,A_27b,A_27d),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27b,A_27d),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27b,A_27d),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))),tyop_2Emin_2Ebool),c_2Ebool_2ETYPE__DEFINITION_2E0(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),tyop_2Einftree_2Einftree(A_27a,A_27b,A_27d)),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_2Einftree_2Efrom__inftree_2E2_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27c,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,X0_2E0: tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c),X1_2E0: tyop_2Elist_2Elist(A_27c)] : ( c_2Einftree_2Efrom__inftree_2E2(A_27a,A_27b,A_27c,X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(A_27c),tyop_2Esum_2Esum(A_27a,A_27b),app_2E2(tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27c),tyop_2Esum_2Esum(A_27a,A_27b)),c_2Einftree_2Efrom__inftree_2E0(A_27a,A_27b,A_27c),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Einftree_2Efrom__inftree_2E1_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27d,axiom,
! [A_27a: $tType,A_27b: $tType,A_27d: $tType,X0_2E0: tyop_2Einftree_2Einftree(A_27a,A_27b,A_27d)] : ( c_2Einftree_2Efrom__inftree_2E1(A_27a,A_27b,A_27d,X0_2E0) = app_2E2(tyop_2Einftree_2Einftree(A_27a,A_27b,A_27d),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),c_2Einftree_2Efrom__inftree_2E0(A_27a,A_27b,A_27d),X0_2E0) ) ).
tff(arityeq2_2Ec_2Einftree_2Efrom__inftree_2E2_2Emono_2EA_27c_20mono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,X0_2E0: tyop_2Einftree_2Einftree(A_27c,A_27a,A_27b),X1_2E0: tyop_2Elist_2Elist(A_27b)] : ( c_2Einftree_2Efrom__inftree_2E2(A_27c,A_27a,A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(A_27b),tyop_2Esum_2Esum(A_27c,A_27a),app_2E2(tyop_2Einftree_2Einftree(A_27c,A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Esum_2Esum(A_27c,A_27a)),c_2Einftree_2Efrom__inftree_2E0(A_27c,A_27a,A_27b),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Einftree_2EiLf_2E1_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27c,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,X0_2E0: A_27a] : ( c_2Einftree_2EiLf_2E1(A_27a,A_27b,A_27c,X0_2E0) = app_2E2(A_27a,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c),c_2Einftree_2EiLf_2E0(A_27a,A_27b,A_27c),X0_2E0) ) ).
tff(arityeq1_2Ec_2Einftree_2EiLf_2E1_2Emono_2EA_27a_20mono_2EA_27c_20mono_2EA_27d,axiom,
! [A_27a: $tType,A_27c: $tType,A_27d: $tType,X0_2E0: A_27a] : ( c_2Einftree_2EiLf_2E1(A_27a,A_27c,A_27d,X0_2E0) = app_2E2(A_27a,tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),c_2Einftree_2EiLf_2E0(A_27a,A_27c,A_27d),X0_2E0) ) ).
tff(arityeq2_2Ec_2Einftree_2EiNd_2E2_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27c,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,X0_2E0: A_27b,X1_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c))] : ( c_2Einftree_2EiNd_2E2(A_27a,A_27b,A_27c,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c)),tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c)),tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c)),c_2Einftree_2EiNd_2E0(A_27a,A_27b,A_27c),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Einftree_2EiNd_2E2_2Emono_2EA_27a_20mono_2EA_27c_20mono_2EA_27d,axiom,
! [A_27a: $tType,A_27c: $tType,A_27d: $tType,X0_2E0: A_27c,X1_2E0: tyop_2Emin_2Efun(A_27d,tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d))] : ( c_2Einftree_2EiNd_2E2(A_27a,A_27c,A_27d,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27d,tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d)),tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),app_2E2(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d)),tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d)),c_2Einftree_2EiNd_2E0(A_27a,A_27c,A_27d),X0_2E0),X1_2E0) ) ).
tff(arityeq3_2Ec_2Einftree_2Einftree__CASE_2E3_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27c_20mono_2EA_27d,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,X0_2E0: tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c),X1_2E0: tyop_2Emin_2Efun(A_27a,A_27d),X2_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c)),A_27d))] : ( c_2Einftree_2Einftree__CASE_2E3(A_27a,A_27b,A_27c,A_27d,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c)),A_27d)),A_27d,app_2E2(tyop_2Emin_2Efun(A_27a,A_27d),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c)),A_27d)),A_27d),app_2E2(tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27d),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c)),A_27d)),A_27d)),c_2Einftree_2Einftree__CASE_2E0(A_27a,A_27b,A_27c,A_27d),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Einftree_2Einftree__rec_2E3_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27c_20mono_2EA_27d,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,A_27a),X1_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27a),A_27a)),X2_2E0: tyop_2Einftree_2Einftree(A_27b,A_27c,A_27d)] : ( c_2Einftree_2Einftree__rec_2E3(A_27a,A_27b,A_27c,A_27d,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Einftree_2Einftree(A_27b,A_27c,A_27d),A_27a,app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27a),A_27a)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27b,A_27c,A_27d),A_27a),app_2E2(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27a),A_27a)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27b,A_27c,A_27d),A_27a)),c_2Einftree_2Einftree__rec_2E0(A_27a,A_27b,A_27c,A_27d),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq1_2Ec_2Einftree_2Eis__tree_2E1_2Emono_2EA_27c_20mono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Esum_2Esum(A_27c,A_27a))] : ( c_2Einftree_2Eis__tree_2E1(A_27c,A_27a,A_27b,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Esum_2Esum(A_27c,A_27a)),tyop_2Emin_2Ebool,c_2Einftree_2Eis__tree_2E0(A_27c,A_27a,A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Einftree_2Eis__tree_2E1_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27d,axiom,
! [A_27a: $tType,A_27b: $tType,A_27d: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))] : ( c_2Einftree_2Eis__tree_2E1(A_27a,A_27b,A_27d,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),tyop_2Emin_2Ebool,c_2Einftree_2Eis__tree_2E0(A_27a,A_27b,A_27d),X0_2E0) ) ).
tff(arityeq2_2Ec_2Ecombin_2Eo_2E2_2Emono_2EA_27c_20mono_2EA_27d_20mono_2Etyop_2Einftree_2Einftree_28A_27a_2CA_27b_2CA_27c_29,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c),A_27d),X1_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c))] : ( c_2Ecombin_2Eo_2E2(A_27c,A_27d,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c)),tyop_2Emin_2Efun(A_27c,A_27d),app_2E2(tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c),A_27d),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c)),tyop_2Emin_2Efun(A_27c,A_27d)),c_2Ecombin_2Eo_2E0(A_27c,A_27d,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c)),X0_2E0),X1_2E0) ) ).
tff(arityeq4_2Ec_2Einftree_2Erelrec_2E4_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27c_20mono_2EA_27d,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,A_27b),X1_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),X2_2E0: tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),X3_2E0: A_27b] : ( c_2Einftree_2Erelrec_2E4(A_27a,A_27b,A_27c,A_27d,X0_2E0,X1_2E0,X2_2E0,X3_2E0) = app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),c_2Einftree_2Erelrec_2E0(A_27a,A_27b,A_27c,A_27d),X0_2E0),X1_2E0),X2_2E0),X3_2E0) ) ).
tff(arityeq4_2Ec_2Einftree_2Erelrec_2E4_2Emono_2EA_27b_20mono_2EA_27a_20mono_2EA_27c_20mono_2EA_27d,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,A_27a),X1_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27a),A_27a)),X2_2E0: tyop_2Einftree_2Einftree(A_27b,A_27c,A_27d),X3_2E0: A_27a] : ( c_2Einftree_2Erelrec_2E4(A_27b,A_27a,A_27c,A_27d,X0_2E0,X1_2E0,X2_2E0,X3_2E0) = app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Einftree_2Einftree(A_27b,A_27c,A_27d),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27a),A_27a)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27b,A_27c,A_27d),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27a),A_27a)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27b,A_27c,A_27d),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),c_2Einftree_2Erelrec_2E0(A_27b,A_27a,A_27c,A_27d),X0_2E0),X1_2E0),X2_2E0),X3_2E0) ) ).
tff(arityeq1_2Ec_2Einftree_2Eto__inftree_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(tyop_2Elist_2Elist(A_27c),tyop_2Esum_2Esum(A_27a,A_27b))] : ( c_2Einftree_2Eto__inftree_2E1(A_27a,A_27b,A_27c,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27c),tyop_2Esum_2Esum(A_27a,A_27b)),tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c),c_2Einftree_2Eto__inftree_2E0(A_27a,A_27b,A_27c),X0_2E0) ) ).
tff(arityeq1_2Ec_2Einftree_2Eto__inftree_2E1_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27d,axiom,
! [A_27a: $tType,A_27b: $tType,A_27d: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))] : ( c_2Einftree_2Eto__inftree_2E1(A_27a,A_27b,A_27d,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),tyop_2Einftree_2Einftree(A_27a,A_27b,A_27d),c_2Einftree_2Eto__inftree_2E0(A_27a,A_27b,A_27d),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_2Einftree_2Eis__tree__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27d: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Emin_2Ebool)] :
( ! [V7x_2E0: tyop_2Elist_2Elist(A_27d)] :
( p(app_2E2(tyop_2Elist_2Elist(A_27d),tyop_2Emin_2Ebool,F0_2E0,V7x_2E0))
<=> ( V7x_2E0 = c_2Elist_2ENIL_2E0(A_27d) ) )
=> ! [V0x_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))] :
( p(c_2Einftree_2Eis__tree_2E1(A_27a,A_27b,A_27d,V0x_2E0))
<=> ! [V1is__tree_27_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),tyop_2Emin_2Ebool)] :
( ! [V2a0_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))] :
( ( ? [V3a_2E0: A_27a] :
! [V4x_2E0: tyop_2Elist_2Elist(A_27d)] : ( app_2E2(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b),V2a0_2E0,V4x_2E0) = c_2Esum_2EINL_2E1(A_27a,A_27b,V3a_2E0) )
| ? [V5f_2E0: tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))),V6b_2E0: A_27b] :
( ! [V7x_2E0: tyop_2Elist_2Elist(A_27d)] : ( app_2E2(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b),V2a0_2E0,V7x_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Esum_2Esum(A_27a,A_27b),app_2E2(tyop_2Elist_2Elist(A_27d),tyop_2Emin_2Ebool,F0_2E0,V7x_2E0),c_2Esum_2EINR_2E1(A_27a,A_27b,V6b_2E0),app_2E2(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b),app_2E2(A_27d,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),V5f_2E0,c_2Elist_2EHD_2E1(A_27d,V7x_2E0)),c_2Elist_2ETL_2E1(A_27d,V7x_2E0))) )
& ! [V8d_2E0: A_27d] : p(app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),tyop_2Emin_2Ebool,V1is__tree_27_2E0,app_2E2(A_27d,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),V5f_2E0,V8d_2E0))) ) )
=> p(app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),tyop_2Emin_2Ebool,V1is__tree_27_2E0,V2a0_2E0)) )
=> p(app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),tyop_2Emin_2Ebool,V1is__tree_27_2E0,V0x_2E0)) ) ) ) ).
tff(thm_2Einftree_2Einftree__TY__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,A_27d: $tType] :
? [V0rep_2E0: tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27b,A_27d),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)))] : p(c_2Ebool_2ETYPE__DEFINITION_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),tyop_2Einftree_2Einftree(A_27a,A_27b,A_27d),c_2Einftree_2Eis__tree_2E0(A_27a,A_27b,A_27d),V0rep_2E0)) ).
tff(thm_2Einftree_2Einftree__bijections,axiom,
! [A_27a: $tType,A_27b: $tType,A_27d: $tType] :
( ! [V0a_2E0: tyop_2Einftree_2Einftree(A_27a,A_27b,A_27d)] : ( c_2Einftree_2Eto__inftree_2E1(A_27a,A_27b,A_27d,c_2Einftree_2Efrom__inftree_2E1(A_27a,A_27b,A_27d,V0a_2E0)) = V0a_2E0 )
& ! [V1r_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))] :
( p(c_2Einftree_2Eis__tree_2E1(A_27a,A_27b,A_27d,V1r_2E0))
<=> ( c_2Einftree_2Efrom__inftree_2E1(A_27a,A_27b,A_27d,c_2Einftree_2Eto__inftree_2E1(A_27a,A_27b,A_27d,V1r_2E0)) = V1r_2E0 ) ) ) ).
tff(thm_2Einftree_2EiLf__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,F0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27c),tyop_2Esum_2Esum(A_27a,A_27b)))] :
( ! [V0a_2E0: A_27a,V1p_2E0: tyop_2Elist_2Elist(A_27c)] : ( app_2E2(tyop_2Elist_2Elist(A_27c),tyop_2Esum_2Esum(A_27a,A_27b),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27c),tyop_2Esum_2Esum(A_27a,A_27b)),F0_2E0,V0a_2E0),V1p_2E0) = c_2Esum_2EINL_2E1(A_27a,A_27b,V0a_2E0) )
=> ! [V0a_2E0: A_27a] : ( c_2Einftree_2EiLf_2E1(A_27a,A_27b,A_27c,V0a_2E0) = c_2Einftree_2Eto__inftree_2E1(A_27a,A_27b,A_27c,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27c),tyop_2Esum_2Esum(A_27a,A_27b)),F0_2E0,V0a_2E0)) ) ) ).
tff(thm_2Einftree_2EiNd__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,F1_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27c),tyop_2Emin_2Ebool),F0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27c),tyop_2Esum_2Esum(A_27a,A_27b))))] :
( ! [V2p_2E0: tyop_2Elist_2Elist(A_27c)] :
( p(app_2E2(tyop_2Elist_2Elist(A_27c),tyop_2Emin_2Ebool,F1_2E0,V2p_2E0))
<=> ( V2p_2E0 = c_2Elist_2ENIL_2E0(A_27c) ) )
=> ( ! [V0b_2E0: A_27b,V1f_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c)),V2p_2E0: tyop_2Elist_2Elist(A_27c)] : ( app_2E2(tyop_2Elist_2Elist(A_27c),tyop_2Esum_2Esum(A_27a,A_27b),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27c),tyop_2Esum_2Esum(A_27a,A_27b)),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27c),tyop_2Esum_2Esum(A_27a,A_27b))),F0_2E0,V0b_2E0),V1f_2E0),V2p_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Esum_2Esum(A_27a,A_27b),app_2E2(tyop_2Elist_2Elist(A_27c),tyop_2Emin_2Ebool,F1_2E0,V2p_2E0),c_2Esum_2EINR_2E1(A_27a,A_27b,V0b_2E0),c_2Einftree_2Efrom__inftree_2E2(A_27a,A_27b,A_27c,app_2E2(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c),V1f_2E0,c_2Elist_2EHD_2E1(A_27c,V2p_2E0)),c_2Elist_2ETL_2E1(A_27c,V2p_2E0))) )
=> ! [V0b_2E0: A_27b,V1f_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c))] : ( c_2Einftree_2EiNd_2E2(A_27a,A_27b,A_27c,V0b_2E0,V1f_2E0) = c_2Einftree_2Eto__inftree_2E1(A_27a,A_27b,A_27c,app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27c),tyop_2Esum_2Esum(A_27a,A_27b)),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27c),tyop_2Esum_2Esum(A_27a,A_27b))),F0_2E0,V0b_2E0),V1f_2E0)) ) ) ) ).
tff(thm_2Einftree_2Erelrec__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0x_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1x_27_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),V2x_27_27_2E0: tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),V3x_27_27_27_2E0: A_27b] :
( p(c_2Einftree_2Erelrec_2E4(A_27a,A_27b,A_27c,A_27d,V0x_2E0,V1x_27_2E0,V2x_27_27_2E0,V3x_27_27_27_2E0))
<=> ! [V4relrec_27_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))))] :
( ! [V5a0_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V6a1_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),V7a2_2E0: tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),V8a3_2E0: A_27b] :
( ( ? [V9a_2E0: A_27a] :
( ( V7a2_2E0 = c_2Einftree_2EiLf_2E1(A_27a,A_27c,A_27d,V9a_2E0) )
& ( V8a3_2E0 = app_2E2(A_27a,A_27b,V5a0_2E0,V9a_2E0) ) )
| ? [V10b_2E0: A_27c,V11df_2E0: tyop_2Emin_2Efun(A_27d,tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d)),V12g_2E0: tyop_2Emin_2Efun(A_27d,A_27b)] :
( ( V7a2_2E0 = c_2Einftree_2EiNd_2E2(A_27a,A_27c,A_27d,V10b_2E0,V11df_2E0) )
& ( V8a3_2E0 = app_2E2(tyop_2Emin_2Efun(A_27d,A_27b),A_27b,app_2E2(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b),V6a1_2E0,V10b_2E0),V12g_2E0) )
& ! [V13d_2E0: A_27d] : p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),V4relrec_27_2E0,V5a0_2E0),V6a1_2E0),app_2E2(A_27d,tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),V11df_2E0,V13d_2E0)),app_2E2(A_27d,A_27b,V12g_2E0,V13d_2E0))) ) )
=> p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),V4relrec_27_2E0,V5a0_2E0),V6a1_2E0),V7a2_2E0),V8a3_2E0)) )
=> p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),V4relrec_27_2E0,V0x_2E0),V1x_27_2E0),V2x_27_27_2E0),V3x_27_27_27_2E0)) ) ) ).
tff(thm_2Einftree_2Einftree__rec__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27a),A_27a)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27b,A_27c,A_27d),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))))] :
( ! [V0lf_2E0: tyop_2Emin_2Efun(A_27b,A_27a),V1nd_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27a),A_27a)),V2t_2E0: tyop_2Einftree_2Einftree(A_27b,A_27c,A_27d),V3r_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Einftree_2Einftree(A_27b,A_27c,A_27d),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27a),A_27a)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27b,A_27c,A_27d),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27a),A_27a)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27b,A_27c,A_27d),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V0lf_2E0),V1nd_2E0),V2t_2E0),V3r_2E0) = c_2Einftree_2Erelrec_2E4(A_27b,A_27a,A_27c,A_27d,V0lf_2E0,V1nd_2E0,V2t_2E0,V3r_2E0) )
=> ! [V0lf_2E0: tyop_2Emin_2Efun(A_27b,A_27a),V1nd_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27a),A_27a)),V2t_2E0: tyop_2Einftree_2Einftree(A_27b,A_27c,A_27d)] : ( c_2Einftree_2Einftree__rec_2E3(A_27a,A_27b,A_27c,A_27d,V0lf_2E0,V1nd_2E0,V2t_2E0) = c_2Emin_2E_40_2E1(A_27a,app_2E2(tyop_2Einftree_2Einftree(A_27b,A_27c,A_27d),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27a),A_27a)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27b,A_27c,A_27d),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27a),A_27a)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27b,A_27c,A_27d),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V0lf_2E0),V1nd_2E0),V2t_2E0)) ) ) ).
tff(thm_2Einftree_2Einftree__case__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] :
( ! [V0a_2E0: A_27a,V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27d),V2f1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c)),A_27d))] : ( c_2Einftree_2Einftree__CASE_2E3(A_27a,A_27b,A_27c,A_27d,c_2Einftree_2EiLf_2E1(A_27a,A_27b,A_27c,V0a_2E0),V1f_2E0,V2f1_2E0) = app_2E2(A_27a,A_27d,V1f_2E0,V0a_2E0) )
& ! [V3b_2E0: A_27b,V4d_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c)),V5f_2E0: tyop_2Emin_2Efun(A_27a,A_27d),V6f1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c)),A_27d))] : ( c_2Einftree_2Einftree__CASE_2E3(A_27a,A_27b,A_27c,A_27d,c_2Einftree_2EiNd_2E2(A_27a,A_27b,A_27c,V3b_2E0,V4d_2E0),V5f_2E0,V6f1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c)),A_27d,app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c)),A_27d),V6f1_2E0,V3b_2E0),V4d_2E0) ) ) ).
tff(thm_2Einftree_2Eis__tree__rules,axiom,
! [A_27a: $tType,A_27b: $tType,A_27d: $tType,F2_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Emin_2Ebool),F1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)))),F0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)))] :
( ! [V5p_2E0: tyop_2Elist_2Elist(A_27d)] :
( p(app_2E2(tyop_2Elist_2Elist(A_27d),tyop_2Emin_2Ebool,F2_2E0,V5p_2E0))
<=> ( V5p_2E0 = c_2Elist_2ENIL_2E0(A_27d) ) )
=> ( ! [V3b_2E0: A_27b,V2f_2E0: tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))),V5p_2E0: tyop_2Elist_2Elist(A_27d)] : ( app_2E2(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b),app_2E2(tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))),F1_2E0,V3b_2E0),V2f_2E0),V5p_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Esum_2Esum(A_27a,A_27b),app_2E2(tyop_2Elist_2Elist(A_27d),tyop_2Emin_2Ebool,F2_2E0,V5p_2E0),c_2Esum_2EINR_2E1(A_27a,A_27b,V3b_2E0),app_2E2(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b),app_2E2(A_27d,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),V2f_2E0,c_2Elist_2EHD_2E1(A_27d,V5p_2E0)),c_2Elist_2ETL_2E1(A_27d,V5p_2E0))) )
=> ( ! [V0a_2E0: A_27a,V1p_2E0: tyop_2Elist_2Elist(A_27d)] : ( app_2E2(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),F0_2E0,V0a_2E0),V1p_2E0) = c_2Esum_2EINL_2E1(A_27a,A_27b,V0a_2E0) )
=> ( ! [V0a_2E0: A_27a] : p(c_2Einftree_2Eis__tree_2E1(A_27a,A_27b,A_27d,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),F0_2E0,V0a_2E0)))
& ! [V2f_2E0: tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))),V3b_2E0: A_27b] :
( ! [V4d_2E0: A_27d] : p(c_2Einftree_2Eis__tree_2E1(A_27a,A_27b,A_27d,app_2E2(A_27d,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),V2f_2E0,V4d_2E0)))
=> p(c_2Einftree_2Eis__tree_2E1(A_27a,A_27b,A_27d,app_2E2(tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))),F1_2E0,V3b_2E0),V2f_2E0))) ) ) ) ) ) ).
tff(thm_2Einftree_2Eis__tree__ind,axiom,
! [A_27a: $tType,A_27b: $tType,A_27d: $tType,F2_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Emin_2Ebool),F1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)))),F0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)))] :
( ! [V6p_2E0: tyop_2Elist_2Elist(A_27d)] :
( p(app_2E2(tyop_2Elist_2Elist(A_27d),tyop_2Emin_2Ebool,F2_2E0,V6p_2E0))
<=> ( V6p_2E0 = c_2Elist_2ENIL_2E0(A_27d) ) )
=> ( ! [V4b_2E0: A_27b,V3f_2E0: tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))),V6p_2E0: tyop_2Elist_2Elist(A_27d)] : ( app_2E2(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b),app_2E2(tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))),F1_2E0,V4b_2E0),V3f_2E0),V6p_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Esum_2Esum(A_27a,A_27b),app_2E2(tyop_2Elist_2Elist(A_27d),tyop_2Emin_2Ebool,F2_2E0,V6p_2E0),c_2Esum_2EINR_2E1(A_27a,A_27b,V4b_2E0),app_2E2(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b),app_2E2(A_27d,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),V3f_2E0,c_2Elist_2EHD_2E1(A_27d,V6p_2E0)),c_2Elist_2ETL_2E1(A_27d,V6p_2E0))) )
=> ( ! [V1a_2E0: A_27a,V2p_2E0: tyop_2Elist_2Elist(A_27d)] : ( app_2E2(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),F0_2E0,V1a_2E0),V2p_2E0) = c_2Esum_2EINL_2E1(A_27a,A_27b,V1a_2E0) )
=> ! [V0is__tree_27_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),tyop_2Emin_2Ebool)] :
( ( ! [V1a_2E0: A_27a] : p(app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),tyop_2Emin_2Ebool,V0is__tree_27_2E0,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),F0_2E0,V1a_2E0)))
& ! [V3f_2E0: tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))),V4b_2E0: A_27b] :
( ! [V5d_2E0: A_27d] : p(app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),tyop_2Emin_2Ebool,V0is__tree_27_2E0,app_2E2(A_27d,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),V3f_2E0,V5d_2E0)))
=> p(app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),tyop_2Emin_2Ebool,V0is__tree_27_2E0,app_2E2(tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))),F1_2E0,V4b_2E0),V3f_2E0))) ) )
=> ! [V7a0_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))] :
( p(c_2Einftree_2Eis__tree_2E1(A_27a,A_27b,A_27d,V7a0_2E0))
=> p(app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),tyop_2Emin_2Ebool,V0is__tree_27_2E0,V7a0_2E0)) ) ) ) ) ) ).
tff(thm_2Einftree_2Eis__tree__strongind,axiom,
! [A_27a: $tType,A_27b: $tType,A_27d: $tType,F2_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Emin_2Ebool),F1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)))),F0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)))] :
( ! [V6p_2E0: tyop_2Elist_2Elist(A_27d)] :
( p(app_2E2(tyop_2Elist_2Elist(A_27d),tyop_2Emin_2Ebool,F2_2E0,V6p_2E0))
<=> ( V6p_2E0 = c_2Elist_2ENIL_2E0(A_27d) ) )
=> ( ! [V4b_2E0: A_27b,V3f_2E0: tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))),V6p_2E0: tyop_2Elist_2Elist(A_27d)] : ( app_2E2(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b),app_2E2(tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))),F1_2E0,V4b_2E0),V3f_2E0),V6p_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Esum_2Esum(A_27a,A_27b),app_2E2(tyop_2Elist_2Elist(A_27d),tyop_2Emin_2Ebool,F2_2E0,V6p_2E0),c_2Esum_2EINR_2E1(A_27a,A_27b,V4b_2E0),app_2E2(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b),app_2E2(A_27d,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),V3f_2E0,c_2Elist_2EHD_2E1(A_27d,V6p_2E0)),c_2Elist_2ETL_2E1(A_27d,V6p_2E0))) )
=> ( ! [V1a_2E0: A_27a,V2p_2E0: tyop_2Elist_2Elist(A_27d)] : ( app_2E2(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),F0_2E0,V1a_2E0),V2p_2E0) = c_2Esum_2EINL_2E1(A_27a,A_27b,V1a_2E0) )
=> ! [V0is__tree_27_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),tyop_2Emin_2Ebool)] :
( ( ! [V1a_2E0: A_27a] : p(app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),tyop_2Emin_2Ebool,V0is__tree_27_2E0,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),F0_2E0,V1a_2E0)))
& ! [V3f_2E0: tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))),V4b_2E0: A_27b] :
( ! [V5d_2E0: A_27d] :
( p(c_2Einftree_2Eis__tree_2E1(A_27a,A_27b,A_27d,app_2E2(A_27d,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),V3f_2E0,V5d_2E0)))
& p(app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),tyop_2Emin_2Ebool,V0is__tree_27_2E0,app_2E2(A_27d,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),V3f_2E0,V5d_2E0))) )
=> p(app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),tyop_2Emin_2Ebool,V0is__tree_27_2E0,app_2E2(tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))),F1_2E0,V4b_2E0),V3f_2E0))) ) )
=> ! [V7a0_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))] :
( p(c_2Einftree_2Eis__tree_2E1(A_27a,A_27b,A_27d,V7a0_2E0))
=> p(app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),tyop_2Emin_2Ebool,V0is__tree_27_2E0,V7a0_2E0)) ) ) ) ) ) ).
tff(thm_2Einftree_2Eis__tree__cases,axiom,
! [A_27a: $tType,A_27b: $tType,A_27d: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Emin_2Ebool)] :
( ! [V5x_2E0: tyop_2Elist_2Elist(A_27d)] :
( p(app_2E2(tyop_2Elist_2Elist(A_27d),tyop_2Emin_2Ebool,F0_2E0,V5x_2E0))
<=> ( V5x_2E0 = c_2Elist_2ENIL_2E0(A_27d) ) )
=> ! [V0a0_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))] :
( p(c_2Einftree_2Eis__tree_2E1(A_27a,A_27b,A_27d,V0a0_2E0))
<=> ( ? [V1a_2E0: A_27a] :
! [V2x_2E0: tyop_2Elist_2Elist(A_27d)] : ( app_2E2(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b),V0a0_2E0,V2x_2E0) = c_2Esum_2EINL_2E1(A_27a,A_27b,V1a_2E0) )
| ? [V3f_2E0: tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b))),V4b_2E0: A_27b] :
( ! [V5x_2E0: tyop_2Elist_2Elist(A_27d)] : ( app_2E2(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b),V0a0_2E0,V5x_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Esum_2Esum(A_27a,A_27b),app_2E2(tyop_2Elist_2Elist(A_27d),tyop_2Emin_2Ebool,F0_2E0,V5x_2E0),c_2Esum_2EINR_2E1(A_27a,A_27b,V4b_2E0),app_2E2(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b),app_2E2(A_27d,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),V3f_2E0,c_2Elist_2EHD_2E1(A_27d,V5x_2E0)),c_2Elist_2ETL_2E1(A_27d,V5x_2E0))) )
& ! [V6d_2E0: A_27d] : p(c_2Einftree_2Eis__tree_2E1(A_27a,A_27b,A_27d,app_2E2(A_27d,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27d),tyop_2Esum_2Esum(A_27a,A_27b)),V3f_2E0,V6d_2E0))) ) ) ) ) ).
tff(thm_2Einftree_2EiNd__is__tree,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,F1_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Emin_2Ebool),F0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Einftree_2Einftree(A_27c,A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Esum_2Esum(A_27c,A_27a))))] :
( ! [V2p_2E0: tyop_2Elist_2Elist(A_27b)] :
( p(app_2E2(tyop_2Elist_2Elist(A_27b),tyop_2Emin_2Ebool,F1_2E0,V2p_2E0))
<=> ( V2p_2E0 = c_2Elist_2ENIL_2E0(A_27b) ) )
=> ( ! [V0b_2E0: A_27a,V1f_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Einftree_2Einftree(A_27c,A_27a,A_27b)),V2p_2E0: tyop_2Elist_2Elist(A_27b)] : ( app_2E2(tyop_2Elist_2Elist(A_27b),tyop_2Esum_2Esum(A_27c,A_27a),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Einftree_2Einftree(A_27c,A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Esum_2Esum(A_27c,A_27a)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Einftree_2Einftree(A_27c,A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Esum_2Esum(A_27c,A_27a))),F0_2E0,V0b_2E0),V1f_2E0),V2p_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Esum_2Esum(A_27c,A_27a),app_2E2(tyop_2Elist_2Elist(A_27b),tyop_2Emin_2Ebool,F1_2E0,V2p_2E0),c_2Esum_2EINR_2E1(A_27c,A_27a,V0b_2E0),c_2Einftree_2Efrom__inftree_2E2(A_27c,A_27a,A_27b,app_2E2(A_27b,tyop_2Einftree_2Einftree(A_27c,A_27a,A_27b),V1f_2E0,c_2Elist_2EHD_2E1(A_27b,V2p_2E0)),c_2Elist_2ETL_2E1(A_27b,V2p_2E0))) )
=> ! [V0b_2E0: A_27a,V1f_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Einftree_2Einftree(A_27c,A_27a,A_27b))] : p(c_2Einftree_2Eis__tree_2E1(A_27c,A_27a,A_27b,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Einftree_2Einftree(A_27c,A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Esum_2Esum(A_27c,A_27a)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Einftree_2Einftree(A_27c,A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2Esum_2Esum(A_27c,A_27a))),F0_2E0,V0b_2E0),V1f_2E0))) ) ) ).
tff(thm_2Einftree_2Einftree__11,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f2_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c)),V1f1_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c)),V2b2_2E0: A_27b,V3b1_2E0: A_27b,V4a2_2E0: A_27a,V5a1_2E0: A_27a] :
( ( ( c_2Einftree_2EiLf_2E1(A_27a,A_27b,A_27c,V5a1_2E0) = c_2Einftree_2EiLf_2E1(A_27a,A_27b,A_27c,V4a2_2E0) )
<=> ( V5a1_2E0 = V4a2_2E0 ) )
& ( ( c_2Einftree_2EiNd_2E2(A_27a,A_27b,A_27c,V3b1_2E0,V1f1_2E0) = c_2Einftree_2EiNd_2E2(A_27a,A_27b,A_27c,V2b2_2E0,V0f2_2E0) )
<=> ( ( V3b1_2E0 = V2b2_2E0 )
& ( V1f1_2E0 = V0f2_2E0 ) ) ) ) ).
tff(thm_2Einftree_2Einftree__distinct,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c)),V1b_2E0: A_27b,V2a_2E0: A_27a] : ( c_2Einftree_2EiLf_2E1(A_27a,A_27b,A_27c,V2a_2E0) != c_2Einftree_2EiNd_2E2(A_27a,A_27b,A_27c,V1b_2E0,V0f_2E0) ) ).
tff(thm_2Einftree_2Einftree__ind,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0P_2E0: tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c),tyop_2Emin_2Ebool)] :
( ( ! [V1a_2E0: A_27a] : p(app_2E2(tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c),tyop_2Emin_2Ebool,V0P_2E0,c_2Einftree_2EiLf_2E1(A_27a,A_27b,A_27c,V1a_2E0)))
& ! [V2b_2E0: A_27b,V3f_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c))] :
( ! [V4d_2E0: A_27c] : p(app_2E2(tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c),tyop_2Emin_2Ebool,V0P_2E0,app_2E2(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c),V3f_2E0,V4d_2E0)))
=> p(app_2E2(tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c),tyop_2Emin_2Ebool,V0P_2E0,c_2Einftree_2EiNd_2E2(A_27a,A_27b,A_27c,V2b_2E0,V3f_2E0))) ) )
=> ! [V5t_2E0: tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c)] : p(app_2E2(tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c),tyop_2Emin_2Ebool,V0P_2E0,V5t_2E0)) ) ).
tff(thm_2Einftree_2Erelrec__rules,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] :
( ! [V0lf_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1nd_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),V2a_2E0: A_27a] : p(c_2Einftree_2Erelrec_2E4(A_27a,A_27b,A_27c,A_27d,V0lf_2E0,V1nd_2E0,c_2Einftree_2EiLf_2E1(A_27a,A_27c,A_27d,V2a_2E0),app_2E2(A_27a,A_27b,V0lf_2E0,V2a_2E0)))
& ! [V3lf_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V4nd_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),V5b_2E0: A_27c,V6df_2E0: tyop_2Emin_2Efun(A_27d,tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d)),V7g_2E0: tyop_2Emin_2Efun(A_27d,A_27b)] :
( ! [V8d_2E0: A_27d] : p(c_2Einftree_2Erelrec_2E4(A_27a,A_27b,A_27c,A_27d,V3lf_2E0,V4nd_2E0,app_2E2(A_27d,tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),V6df_2E0,V8d_2E0),app_2E2(A_27d,A_27b,V7g_2E0,V8d_2E0)))
=> p(c_2Einftree_2Erelrec_2E4(A_27a,A_27b,A_27c,A_27d,V3lf_2E0,V4nd_2E0,c_2Einftree_2EiNd_2E2(A_27a,A_27c,A_27d,V5b_2E0,V6df_2E0),app_2E2(tyop_2Emin_2Efun(A_27d,A_27b),A_27b,app_2E2(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b),V4nd_2E0,V5b_2E0),V7g_2E0))) ) ) ).
tff(thm_2Einftree_2Erelrec__ind,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0relrec_27_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))))] :
( ( ! [V1lf_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2nd_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),V3a_2E0: A_27a] : p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),V0relrec_27_2E0,V1lf_2E0),V2nd_2E0),c_2Einftree_2EiLf_2E1(A_27a,A_27c,A_27d,V3a_2E0)),app_2E2(A_27a,A_27b,V1lf_2E0,V3a_2E0)))
& ! [V4lf_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V5nd_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),V6b_2E0: A_27c,V7df_2E0: tyop_2Emin_2Efun(A_27d,tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d)),V8g_2E0: tyop_2Emin_2Efun(A_27d,A_27b)] :
( ! [V9d_2E0: A_27d] : p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),V0relrec_27_2E0,V4lf_2E0),V5nd_2E0),app_2E2(A_27d,tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),V7df_2E0,V9d_2E0)),app_2E2(A_27d,A_27b,V8g_2E0,V9d_2E0)))
=> p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),V0relrec_27_2E0,V4lf_2E0),V5nd_2E0),c_2Einftree_2EiNd_2E2(A_27a,A_27c,A_27d,V6b_2E0,V7df_2E0)),app_2E2(tyop_2Emin_2Efun(A_27d,A_27b),A_27b,app_2E2(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b),V5nd_2E0,V6b_2E0),V8g_2E0))) ) )
=> ! [V10a0_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V11a1_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),V12a2_2E0: tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),V13a3_2E0: A_27b] :
( p(c_2Einftree_2Erelrec_2E4(A_27a,A_27b,A_27c,A_27d,V10a0_2E0,V11a1_2E0,V12a2_2E0,V13a3_2E0))
=> p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),V0relrec_27_2E0,V10a0_2E0),V11a1_2E0),V12a2_2E0),V13a3_2E0)) ) ) ).
tff(thm_2Einftree_2Erelrec__strongind,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0relrec_27_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))))] :
( ( ! [V1lf_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2nd_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),V3a_2E0: A_27a] : p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),V0relrec_27_2E0,V1lf_2E0),V2nd_2E0),c_2Einftree_2EiLf_2E1(A_27a,A_27c,A_27d,V3a_2E0)),app_2E2(A_27a,A_27b,V1lf_2E0,V3a_2E0)))
& ! [V4lf_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V5nd_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),V6b_2E0: A_27c,V7df_2E0: tyop_2Emin_2Efun(A_27d,tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d)),V8g_2E0: tyop_2Emin_2Efun(A_27d,A_27b)] :
( ! [V9d_2E0: A_27d] :
( p(c_2Einftree_2Erelrec_2E4(A_27a,A_27b,A_27c,A_27d,V4lf_2E0,V5nd_2E0,app_2E2(A_27d,tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),V7df_2E0,V9d_2E0),app_2E2(A_27d,A_27b,V8g_2E0,V9d_2E0)))
& p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),V0relrec_27_2E0,V4lf_2E0),V5nd_2E0),app_2E2(A_27d,tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),V7df_2E0,V9d_2E0)),app_2E2(A_27d,A_27b,V8g_2E0,V9d_2E0))) )
=> p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),V0relrec_27_2E0,V4lf_2E0),V5nd_2E0),c_2Einftree_2EiNd_2E2(A_27a,A_27c,A_27d,V6b_2E0,V7df_2E0)),app_2E2(tyop_2Emin_2Efun(A_27d,A_27b),A_27b,app_2E2(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b),V5nd_2E0,V6b_2E0),V8g_2E0))) ) )
=> ! [V10a0_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V11a1_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),V12a2_2E0: tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),V13a3_2E0: A_27b] :
( p(c_2Einftree_2Erelrec_2E4(A_27a,A_27b,A_27c,A_27d,V10a0_2E0,V11a1_2E0,V12a2_2E0,V13a3_2E0))
=> p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),V0relrec_27_2E0,V10a0_2E0),V11a1_2E0),V12a2_2E0),V13a3_2E0)) ) ) ).
tff(thm_2Einftree_2Erelrec__cases,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0a0_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1a1_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b)),V2a2_2E0: tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),V3a3_2E0: A_27b] :
( p(c_2Einftree_2Erelrec_2E4(A_27a,A_27b,A_27c,A_27d,V0a0_2E0,V1a1_2E0,V2a2_2E0,V3a3_2E0))
<=> ( ? [V4a_2E0: A_27a] :
( ( V2a2_2E0 = c_2Einftree_2EiLf_2E1(A_27a,A_27c,A_27d,V4a_2E0) )
& ( V3a3_2E0 = app_2E2(A_27a,A_27b,V0a0_2E0,V4a_2E0) ) )
| ? [V5b_2E0: A_27c,V6df_2E0: tyop_2Emin_2Efun(A_27d,tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d)),V7g_2E0: tyop_2Emin_2Efun(A_27d,A_27b)] :
( ( V2a2_2E0 = c_2Einftree_2EiNd_2E2(A_27a,A_27c,A_27d,V5b_2E0,V6df_2E0) )
& ( V3a3_2E0 = app_2E2(tyop_2Emin_2Efun(A_27d,A_27b),A_27b,app_2E2(A_27c,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27d,A_27b),A_27b),V1a1_2E0,V5b_2E0),V7g_2E0) )
& ! [V8d_2E0: A_27d] : p(c_2Einftree_2Erelrec_2E4(A_27a,A_27b,A_27c,A_27d,V0a0_2E0,V1a1_2E0,app_2E2(A_27d,tyop_2Einftree_2Einftree(A_27a,A_27c,A_27d),V6df_2E0,V8d_2E0),app_2E2(A_27d,A_27b,V7g_2E0,V8d_2E0))) ) ) ) ).
tff(thm_2Einftree_2Einftree__Axiom,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0lf_2E0: tyop_2Emin_2Efun(A_27a,A_27d),V1nd_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27d),A_27d)))] :
? [V2f_2E0: tyop_2Emin_2Efun(tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c),A_27d)] :
( ! [V3a_2E0: A_27a] : ( app_2E2(tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c),A_27d,V2f_2E0,c_2Einftree_2EiLf_2E1(A_27a,A_27b,A_27c,V3a_2E0)) = app_2E2(A_27a,A_27d,V0lf_2E0,V3a_2E0) )
& ! [V4b_2E0: A_27b,V5d_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c))] : ( app_2E2(tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c),A_27d,V2f_2E0,c_2Einftree_2EiNd_2E2(A_27a,A_27b,A_27c,V4b_2E0,V5d_2E0)) = app_2E2(tyop_2Emin_2Efun(A_27c,A_27d),A_27d,app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27d),A_27d),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27d),A_27d)),V1nd_2E0,V4b_2E0),V5d_2E0),c_2Ecombin_2Eo_2E2(A_27c,A_27d,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c),V2f_2E0,V5d_2E0)) ) ) ).
tff(thm_2Einftree_2Einftree__nchotomy,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0t_2E0: tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c)] :
( ? [V1a_2E0: A_27a] : ( V0t_2E0 = c_2Einftree_2EiLf_2E1(A_27a,A_27b,A_27c,V1a_2E0) )
| ? [V2b_2E0: A_27b,V3d_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Einftree_2Einftree(A_27a,A_27b,A_27c))] : ( V0t_2E0 = c_2Einftree_2EiNd_2E2(A_27a,A_27b,A_27c,V2b_2E0,V3d_2E0) ) ) ).
%------------------------------------------------------------------------------