ITP001 Axioms: ITP015_7.ax
%------------------------------------------------------------------------------
% File : ITP015_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 : prim_rec.ax [Gau19]
% : HL4015_7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 153 ( 45 unt; 61 typ; 0 def)
% Number of atoms : 179 ( 84 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 98 ( 11 ~; 6 |; 20 &)
% ( 28 <=>; 33 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 5 avg)
% Maximal term depth : 8 ( 2 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 57 ( 30 >; 27 *; 0 +; 0 <<)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 58 ( 58 usr; 10 con; 0-5 aty)
% Number of variables : 325 ( 270 !; 9 ?; 325 :)
% ( 46 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
tff(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: ( $tType * $tType ) > $tType ).
tff(tyop_2Enum_2Enum,type,
tyop_2Enum_2Enum: $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_2Enum_2E0_2E0,type,
c_2Enum_2E0_2E0: tyop_2Enum_2Enum ).
tff(c_2Eprim__rec_2E_3C_2E0,type,
c_2Eprim__rec_2E_3C_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)) ).
tff(c_2Eprim__rec_2E_3C_2E2,type,
c_2Eprim__rec_2E_3C_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > 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_2Ebool_2E_3F_21_2E0,type,
c_2Ebool_2E_3F_21_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_3F_21_2E1,type,
c_2Ebool_2E_3F_21_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_2Eprim__rec_2EPRE_2E0,type,
c_2Eprim__rec_2EPRE_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).
tff(c_2Eprim__rec_2EPRE_2E1,type,
c_2Eprim__rec_2EPRE_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
tff(c_2Eprim__rec_2EPRIM__REC_2E0,type,
c_2Eprim__rec_2EPRIM__REC_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a))) ).
tff(c_2Eprim__rec_2EPRIM__REC_2E3,type,
c_2Eprim__rec_2EPRIM__REC_2E3:
!>[A_27a: $tType] : ( ( A_27a * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)) * tyop_2Enum_2Enum ) > A_27a ) ).
tff(c_2Eprim__rec_2EPRIM__REC__FUN_2E0,type,
c_2Eprim__rec_2EPRIM__REC__FUN_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)))) ).
tff(c_2Eprim__rec_2EPRIM__REC__FUN_2E2,type,
c_2Eprim__rec_2EPRIM__REC__FUN_2E2:
!>[A_27a: $tType] : ( ( A_27a * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)) ) > tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)) ) ).
tff(c_2Eprim__rec_2EPRIM__REC__FUN_2E4,type,
c_2Eprim__rec_2EPRIM__REC__FUN_2E4:
!>[A_27a: $tType] : ( ( A_27a * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)) * tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > A_27a ) ).
tff(c_2Erelation_2ERTC_2E0,type,
c_2Erelation_2ERTC_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))) ).
tff(c_2Erelation_2ERTC_2E3,type,
c_2Erelation_2ERTC_2E3:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) * A_27a * A_27a ) > tyop_2Emin_2Ebool ) ).
tff(c_2Eprim__rec_2ESIMP__REC_2E0,type,
c_2Eprim__rec_2ESIMP__REC_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a))) ).
tff(c_2Eprim__rec_2ESIMP__REC_2E2,type,
c_2Eprim__rec_2ESIMP__REC_2E2:
!>[A_27a: $tType] : ( ( A_27a * tyop_2Emin_2Efun(A_27a,A_27a) ) > tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a) ) ).
tff(c_2Eprim__rec_2ESIMP__REC_2E3,type,
c_2Eprim__rec_2ESIMP__REC_2E3:
!>[A_27a: $tType] : ( ( A_27a * tyop_2Emin_2Efun(A_27a,A_27a) * tyop_2Enum_2Enum ) > A_27a ) ).
tff(c_2Eprim__rec_2ESIMP__REC__REL_2E0,type,
c_2Eprim__rec_2ESIMP__REC__REL_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)))) ).
tff(c_2Eprim__rec_2ESIMP__REC__REL_2E4,type,
c_2Eprim__rec_2ESIMP__REC__REL_2E4:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a) * A_27a * tyop_2Emin_2Efun(A_27a,A_27a) * tyop_2Enum_2Enum ) > tyop_2Emin_2Ebool ) ).
tff(c_2Enum_2ESUC_2E0,type,
c_2Enum_2ESUC_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).
tff(c_2Enum_2ESUC_2E1,type,
c_2Enum_2ESUC_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
tff(c_2Ebool_2ET_2E0,type,
c_2Ebool_2ET_2E0: tyop_2Emin_2Ebool ).
tff(c_2Erelation_2ETC_2E0,type,
c_2Erelation_2ETC_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))) ).
tff(c_2Erelation_2ETC_2E1,type,
c_2Erelation_2ETC_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) ) ).
tff(c_2Erelation_2ETC_2E3,type,
c_2Erelation_2ETC_2E3:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) * A_27a * A_27a ) > tyop_2Emin_2Ebool ) ).
tff(c_2Erelation_2EWF_2E0,type,
c_2Erelation_2EWF_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool) ).
tff(c_2Erelation_2EWF_2E1,type,
c_2Erelation_2EWF_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) > 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_2Erelation_2Einv__image_2E0,type,
c_2Erelation_2Einv__image_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))) ).
tff(c_2Erelation_2Einv__image_2E1,type,
c_2Erelation_2Einv__image_2E1:
!>[A_27a: $tType,A_27b: $tType] : ( tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)) > tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))) ) ).
tff(c_2Eprim__rec_2Emeasure_2E0,type,
c_2Eprim__rec_2Emeasure_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))) ).
tff(c_2Eprim__rec_2Emeasure_2E1,type,
c_2Eprim__rec_2Emeasure_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) ) ).
tff(c_2Eprim__rec_2Emeasure_2E3,type,
c_2Eprim__rec_2Emeasure_2E3:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum) * A_27a * A_27a ) > tyop_2Emin_2Ebool ) ).
tff(c_2Eprim__rec_2Ewellfounded_2E0,type,
c_2Eprim__rec_2Ewellfounded_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool) ).
tff(c_2Eprim__rec_2Ewellfounded_2E1,type,
c_2Eprim__rec_2Ewellfounded_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) > 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_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_2Eprim__rec_2E_3C_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),c_2Eprim__rec_2E_3C_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_2Ebool_2E_3F_21_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_3F_21_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_3F_21_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Ebool_2E_3F_21_2E1_2Emono_2Etyop_2Emin_2Efun_28tyop_2Enum_2Enum_2CA_27a_29,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_3F_21_2E1(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_3F_21_2E0(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),X0_2E0) ) ).
tff(arityeq1_2Ec_2Emin_2E_40_2E1_2Emono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] : ( c_2Emin_2E_40_2E1(tyop_2Enum_2Enum,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Enum_2Enum,c_2Emin_2E_40_2E0(tyop_2Enum_2Enum),X0_2E0) ) ).
tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Enum_2Enum,X2_2E0: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),c_2Ebool_2ECOND_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq1_2Ec_2Eprim__rec_2EPRE_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2EPRE_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Eprim__rec_2EPRE_2E0,X0_2E0) ) ).
tff(arityeq3_2Ec_2Eprim__rec_2EPRIM__REC_2E3_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),X2_2E0: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2EPRIM__REC_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Enum_2Enum,A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),c_2Eprim__rec_2EPRIM__REC_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq2_2Ec_2Eprim__rec_2EPRIM__REC__FUN_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a))] : ( c_2Eprim__rec_2EPRIM__REC__FUN_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a))),c_2Eprim__rec_2EPRIM__REC__FUN_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq4_2Ec_2Eprim__rec_2EPRIM__REC__FUN_2E4_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),X2_2E0: tyop_2Enum_2Enum,X3_2E0: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2EPRIM__REC__FUN_2E4(A_27a,X0_2E0,X1_2E0,X2_2E0,X3_2E0) = app_2E2(tyop_2Enum_2Enum,A_27a,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a))),c_2Eprim__rec_2EPRIM__REC__FUN_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0),X3_2E0) ) ).
tff(arityeq3_2Ec_2Erelation_2ERTC_2E3_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),X1_2E0: A_27a,X2_2E0: A_27a] : ( c_2Erelation_2ERTC_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Erelation_2ERTC_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Erelation_2ERTC_2E3_2Emono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),X1_2E0: tyop_2Enum_2Enum,X2_2E0: tyop_2Enum_2Enum] : ( c_2Erelation_2ERTC_2E3(tyop_2Enum_2Enum,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Erelation_2ERTC_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Eprim__rec_2ESIMP__REC_2E3_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: tyop_2Emin_2Efun(A_27a,A_27a),X2_2E0: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2ESIMP__REC_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Enum_2Enum,A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),c_2Eprim__rec_2ESIMP__REC_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq2_2Ec_2Eprim__rec_2ESIMP__REC_2E2_2Emono_2Etyop_2Emin_2Efun_28tyop_2Enum_2Enum_2CA_27a_29,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),X1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a))] : ( c_2Eprim__rec_2ESIMP__REC_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a))),c_2Eprim__rec_2ESIMP__REC_2E0(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),X0_2E0),X1_2E0) ) ).
tff(arityeq4_2Ec_2Eprim__rec_2ESIMP__REC__REL_2E4_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),X1_2E0: A_27a,X2_2E0: tyop_2Emin_2Efun(A_27a,A_27a),X3_2E0: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2ESIMP__REC__REL_2E4(A_27a,X0_2E0,X1_2E0,X2_2E0,X3_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))),c_2Eprim__rec_2ESIMP__REC__REL_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0),X3_2E0) ) ).
tff(arityeq1_2Ec_2Enum_2ESUC_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Enum_2ESUC_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2ESUC_2E0,X0_2E0) ) ).
tff(arityeq3_2Ec_2Erelation_2ETC_2E3_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),X1_2E0: A_27a,X2_2E0: A_27a] : ( c_2Erelation_2ETC_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Erelation_2ETC_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq1_2Ec_2Erelation_2ETC_2E1_2Emono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] : ( c_2Erelation_2ETC_2E1(tyop_2Enum_2Enum,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Erelation_2ETC_2E0(tyop_2Enum_2Enum),X0_2E0) ) ).
tff(arityeq3_2Ec_2Erelation_2ETC_2E3_2Emono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),X1_2E0: tyop_2Enum_2Enum,X2_2E0: tyop_2Enum_2Enum] : ( c_2Erelation_2ETC_2E3(tyop_2Enum_2Enum,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Erelation_2ETC_2E0(tyop_2Enum_2Enum),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq1_2Ec_2Erelation_2EWF_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))] : ( c_2Erelation_2EWF_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,c_2Erelation_2EWF_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Erelation_2EWF_2E1_2Emono_2Etyop_2Enum_2Enum,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] : ( c_2Erelation_2EWF_2E1(tyop_2Enum_2Enum,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,c_2Erelation_2EWF_2E0(tyop_2Enum_2Enum),X0_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(arityeq1_2Ec_2Erelation_2Einv__image_2E1_2Emono_2EA_27a_20mono_2Etyop_2Enum_2Enum,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] : ( c_2Erelation_2Einv__image_2E1(A_27a,tyop_2Enum_2Enum,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),c_2Erelation_2Einv__image_2E0(A_27a,tyop_2Enum_2Enum),X0_2E0) ) ).
tff(arityeq1_2Ec_2Eprim__rec_2Emeasure_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)] : ( c_2Eprim__rec_2Emeasure_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Eprim__rec_2Emeasure_2E0(A_27a),X0_2E0) ) ).
tff(arityeq3_2Ec_2Eprim__rec_2Emeasure_2E3_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),X1_2E0: A_27a,X2_2E0: A_27a] : ( c_2Eprim__rec_2Emeasure_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Eprim__rec_2Emeasure_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq1_2Ec_2Eprim__rec_2Ewellfounded_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))] : ( c_2Eprim__rec_2Ewellfounded_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,c_2Eprim__rec_2Ewellfounded_2E0(A_27a),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_2Eprim__rec_2ELESS__DEF,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0))
<=> ? [V2P_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)] :
( ! [V3n_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V2P_2E0,c_2Enum_2ESUC_2E1(V3n_2E0)))
=> p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V2P_2E0,V3n_2E0)) )
& p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V2P_2E0,V0m_2E0))
& ~ p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,V2P_2E0,V1n_2E0)) ) ) ).
tff(thm_2Eprim__rec_2EPRE__DEF,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
( ! [V0m_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F0_2E0,V0m_2E0))
<=> ( V0m_2E0 = c_2Enum_2E0_2E0 ) )
=> ( ! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F1_2E0,V0m_2E0),V1n_2E0))
<=> ( V0m_2E0 = c_2Enum_2ESUC_2E1(V1n_2E0) ) )
=> ! [V0m_2E0: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2EPRE_2E1(V0m_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,F0_2E0,V0m_2E0),c_2Enum_2E0_2E0,c_2Emin_2E_40_2E1(tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F1_2E0,V0m_2E0))) ) ) ) ).
tff(thm_2Eprim__rec_2ESIMP__REC__REL,axiom,
! [A_27a: $tType,V0fun_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),V1x_2E0: A_27a,V2f_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V3n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2ESIMP__REC__REL_2E4(A_27a,V0fun_2E0,V1x_2E0,V2f_2E0,V3n_2E0))
<=> ( ( app_2E2(tyop_2Enum_2Enum,A_27a,V0fun_2E0,c_2Enum_2E0_2E0) = V1x_2E0 )
& ! [V4m_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V4m_2E0,V3n_2E0))
=> ( app_2E2(tyop_2Enum_2Enum,A_27a,V0fun_2E0,c_2Enum_2ESUC_2E1(V4m_2E0)) = app_2E2(A_27a,A_27a,V2f_2E0,app_2E2(tyop_2Enum_2Enum,A_27a,V0fun_2E0,V4m_2E0)) ) ) ) ) ).
tff(thm_2Eprim__rec_2ESIMP__REC,axiom,
! [A_27a: $tType,V0x_2E0: A_27a,V1f_27_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V2n_2E0: tyop_2Enum_2Enum] :
? [V3g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)] :
( p(c_2Eprim__rec_2ESIMP__REC__REL_2E4(A_27a,V3g_2E0,V0x_2E0,V1f_27_2E0,c_2Enum_2ESUC_2E1(V2n_2E0)))
& ( c_2Eprim__rec_2ESIMP__REC_2E3(A_27a,V0x_2E0,V1f_27_2E0,V2n_2E0) = app_2E2(tyop_2Enum_2Enum,A_27a,V3g_2E0,V2n_2E0) ) ) ).
tff(thm_2Eprim__rec_2EPRIM__REC__FUN,axiom,
! [A_27a: $tType,F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a))),F0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a))] :
( ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),V3fun_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),V4n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,A_27a,app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),F1_2E0,V1f_2E0),V3fun_2E0),V4n_2E0) = app_2E2(tyop_2Enum_2Enum,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),V1f_2E0,app_2E2(tyop_2Enum_2Enum,A_27a,V3fun_2E0,c_2Eprim__rec_2EPRE_2E1(V4n_2E0))),V4n_2E0) )
=> ( ! [V0x_2E0: A_27a,V2n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),F0_2E0,V0x_2E0),V2n_2E0) = V0x_2E0 )
=> ! [V0x_2E0: A_27a,V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a))] : ( c_2Eprim__rec_2EPRIM__REC__FUN_2E2(A_27a,V0x_2E0,V1f_2E0) = c_2Eprim__rec_2ESIMP__REC_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),F0_2E0,V0x_2E0),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),F1_2E0,V1f_2E0)) ) ) ) ).
tff(thm_2Eprim__rec_2EPRIM__REC,axiom,
! [A_27a: $tType,V0x_2E0: A_27a,V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),V2m_2E0: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2EPRIM__REC_2E3(A_27a,V0x_2E0,V1f_2E0,V2m_2E0) = c_2Eprim__rec_2EPRIM__REC__FUN_2E4(A_27a,V0x_2E0,V1f_2E0,V2m_2E0,c_2Eprim__rec_2EPRE_2E1(V2m_2E0)) ) ).
tff(thm_2Eprim__rec_2Ewellfounded__def,axiom,
! [A_27a: $tType,V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))] :
( p(c_2Eprim__rec_2Ewellfounded_2E1(A_27a,V0R_2E0))
<=> ~ ? [V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)] :
! [V2n_2E0: tyop_2Enum_2Enum] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0R_2E0,app_2E2(tyop_2Enum_2Enum,A_27a,V1f_2E0,c_2Enum_2ESUC_2E1(V2n_2E0))),app_2E2(tyop_2Enum_2Enum,A_27a,V1f_2E0,V2n_2E0))) ) ).
tff(thm_2Eprim__rec_2Emeasure__def,axiom,
! [A_27a: $tType] : ( c_2Eprim__rec_2Emeasure_2E0(A_27a) = c_2Erelation_2Einv__image_2E1(A_27a,tyop_2Enum_2Enum,c_2Eprim__rec_2E_3C_2E0) ) ).
tff(thm_2Eprim__rec_2EINV__SUC__EQ,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ( c_2Enum_2ESUC_2E1(V0m_2E0) = c_2Enum_2ESUC_2E1(V1n_2E0) )
<=> ( V0m_2E0 = V1n_2E0 ) ) ).
tff(thm_2Eprim__rec_2EPRE,axiom,
( ( c_2Eprim__rec_2EPRE_2E1(c_2Enum_2E0_2E0) = c_2Enum_2E0_2E0 )
& ! [V0m_2E0: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2EPRE_2E1(c_2Enum_2ESUC_2E1(V0m_2E0)) = V0m_2E0 ) ) ).
tff(thm_2Eprim__rec_2ELESS__REFL,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : ~ p(c_2Eprim__rec_2E_3C_2E2(V0n_2E0,V0n_2E0)) ).
tff(thm_2Eprim__rec_2ESUC__LESS,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2ESUC_2E1(V0m_2E0),V1n_2E0))
=> p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0)) ) ).
tff(thm_2Eprim__rec_2ENOT__LESS__0,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : ~ p(c_2Eprim__rec_2E_3C_2E2(V0n_2E0,c_2Enum_2E0_2E0)) ).
tff(thm_2Eprim__rec_2ELESS__0,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,c_2Enum_2ESUC_2E1(V0n_2E0))) ).
tff(thm_2Eprim__rec_2ELESS__0__0,axiom,
p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2E0_2E0,c_2Enum_2ESUC_2E1(c_2Enum_2E0_2E0))) ).
tff(thm_2Eprim__rec_2ELESS__MONO,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0))
=> p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2ESUC_2E1(V0m_2E0),c_2Enum_2ESUC_2E1(V1n_2E0))) ) ).
tff(thm_2Eprim__rec_2ELESS__MONO__REV,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(c_2Enum_2ESUC_2E1(V0m_2E0),c_2Enum_2ESUC_2E1(V1n_2E0)))
=> p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0)) ) ).
tff(thm_2Eprim__rec_2ELESS__MONO__EQ,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2E_3C_2E2(c_2Enum_2ESUC_2E1(V0m_2E0),c_2Enum_2ESUC_2E1(V1n_2E0)) = c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0) ) ).
tff(thm_2Eprim__rec_2ETC__IM__RTC__SUC,axiom,
! [F1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
( ! [V4x_2E0: tyop_2Enum_2Enum,V5y_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F1_2E0,V4x_2E0),V5y_2E0))
<=> ( V5y_2E0 = c_2Enum_2ESUC_2E1(V4x_2E0) ) )
=> ( ! [V2x_2E0: tyop_2Enum_2Enum,V3y_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V2x_2E0),V3y_2E0))
<=> ( V3y_2E0 = c_2Enum_2ESUC_2E1(V2x_2E0) ) )
=> ! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] : ( c_2Erelation_2ETC_2E3(tyop_2Enum_2Enum,F0_2E0,V0m_2E0,c_2Enum_2ESUC_2E1(V1n_2E0)) = c_2Erelation_2ERTC_2E3(tyop_2Enum_2Enum,F1_2E0,V0m_2E0,V1n_2E0) ) ) ) ).
tff(thm_2Eprim__rec_2ERTC__IM__TC,axiom,
! [A_27a: $tType,F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V5x_2E0: A_27a,V6y_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F1_2E0,V0f_2E0),V5x_2E0),V6y_2E0))
<=> ( V6y_2E0 = app_2E2(A_27a,A_27a,V0f_2E0,V5x_2E0) ) )
=> ( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V3x_2E0: A_27a,V4y_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V0f_2E0),V3x_2E0),V4y_2E0))
<=> ( V4y_2E0 = app_2E2(A_27a,A_27a,V0f_2E0,V3x_2E0) ) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V1m_2E0: A_27a,V2n_2E0: A_27a] : ( c_2Erelation_2ERTC_2E3(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F0_2E0,V0f_2E0),app_2E2(A_27a,A_27a,V0f_2E0,V1m_2E0),V2n_2E0) = c_2Erelation_2ETC_2E3(A_27a,app_2E2(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),F1_2E0,V0f_2E0),V1m_2E0,V2n_2E0) ) ) ) ).
tff(thm_2Eprim__rec_2ELESS__ALT,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
( ! [V0x_2E0: tyop_2Enum_2Enum,V1y_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V0x_2E0),V1y_2E0))
<=> ( V1y_2E0 = c_2Enum_2ESUC_2E1(V0x_2E0) ) )
=> ( c_2Eprim__rec_2E_3C_2E0 = c_2Erelation_2ETC_2E1(tyop_2Enum_2Enum,F0_2E0) ) ) ).
tff(thm_2Eprim__rec_2ELESS__SUC__REFL,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : p(c_2Eprim__rec_2E_3C_2E2(V0n_2E0,c_2Enum_2ESUC_2E1(V0n_2E0))) ).
tff(thm_2Eprim__rec_2ELESS__SUC,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0))
=> p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,c_2Enum_2ESUC_2E1(V1n_2E0))) ) ).
tff(thm_2Eprim__rec_2ELESS__LEMMA1,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,c_2Enum_2ESUC_2E1(V1n_2E0)))
=> ( ( V0m_2E0 = V1n_2E0 )
| p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0)) ) ) ).
tff(thm_2Eprim__rec_2ELESS__LEMMA2,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ( ( V0m_2E0 = V1n_2E0 )
| p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0)) )
=> p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,c_2Enum_2ESUC_2E1(V1n_2E0))) ) ).
tff(thm_2Eprim__rec_2ELESS__THM,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,c_2Enum_2ESUC_2E1(V1n_2E0)))
<=> ( ( V0m_2E0 = V1n_2E0 )
| p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0)) ) ) ).
tff(thm_2Eprim__rec_2ELESS__SUC__IMP,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,c_2Enum_2ESUC_2E1(V1n_2E0)))
=> ( ( V0m_2E0 != V1n_2E0 )
=> p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0)) ) ) ).
tff(thm_2Eprim__rec_2EEQ__LESS,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ( c_2Enum_2ESUC_2E1(V0m_2E0) = V1n_2E0 )
=> p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0)) ) ).
tff(thm_2Eprim__rec_2ESUC__ID,axiom,
! [V0n_2E0: tyop_2Enum_2Enum] : ( c_2Enum_2ESUC_2E1(V0n_2E0) != V0n_2E0 ) ).
tff(thm_2Eprim__rec_2ENOT__LESS__EQ,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( ( V0m_2E0 = V1n_2E0 )
=> ~ p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0)) ) ).
tff(thm_2Eprim__rec_2ELESS__NOT__EQ,axiom,
! [V0m_2E0: tyop_2Enum_2Enum,V1n_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,V1n_2E0))
=> ( V0m_2E0 != V1n_2E0 ) ) ).
tff(thm_2Eprim__rec_2ESIMP__REC__EXISTS,axiom,
! [A_27a: $tType,V0x_2E0: A_27a,V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V2n_2E0: tyop_2Enum_2Enum] :
? [V3fun_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)] : p(c_2Eprim__rec_2ESIMP__REC__REL_2E4(A_27a,V3fun_2E0,V0x_2E0,V1f_2E0,V2n_2E0)) ).
tff(thm_2Eprim__rec_2ESIMP__REC__REL__UNIQUE,axiom,
! [A_27a: $tType,V0x_2E0: A_27a,V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V2g1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),V3g2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),V4m1_2E0: tyop_2Enum_2Enum,V5m2_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Eprim__rec_2ESIMP__REC__REL_2E4(A_27a,V2g1_2E0,V0x_2E0,V1f_2E0,V4m1_2E0))
& p(c_2Eprim__rec_2ESIMP__REC__REL_2E4(A_27a,V3g2_2E0,V0x_2E0,V1f_2E0,V5m2_2E0)) )
=> ! [V6n_2E0: tyop_2Enum_2Enum] :
( ( p(c_2Eprim__rec_2E_3C_2E2(V6n_2E0,V4m1_2E0))
& p(c_2Eprim__rec_2E_3C_2E2(V6n_2E0,V5m2_2E0)) )
=> ( app_2E2(tyop_2Enum_2Enum,A_27a,V2g1_2E0,V6n_2E0) = app_2E2(tyop_2Enum_2Enum,A_27a,V3g2_2E0,V6n_2E0) ) ) ) ).
tff(thm_2Eprim__rec_2ESIMP__REC__REL__UNIQUE__RESULT,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))))] :
( ! [V0x_2E0: A_27a,V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V2n_2E0: tyop_2Enum_2Enum,V3y_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V0x_2E0),V1f_2E0),V2n_2E0),V3y_2E0))
<=> ? [V4g_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)] :
( p(c_2Eprim__rec_2ESIMP__REC__REL_2E4(A_27a,V4g_2E0,V0x_2E0,V1f_2E0,c_2Enum_2ESUC_2E1(V2n_2E0)))
& ( V3y_2E0 = app_2E2(tyop_2Enum_2Enum,A_27a,V4g_2E0,V2n_2E0) ) ) )
=> ! [V0x_2E0: A_27a,V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27a),V2n_2E0: tyop_2Enum_2Enum] : p(c_2Ebool_2E_3F_21_2E1(A_27a,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))),F0_2E0,V0x_2E0),V1f_2E0),V2n_2E0))) ) ).
tff(thm_2Eprim__rec_2ELESS__SUC__SUC,axiom,
! [V0m_2E0: tyop_2Enum_2Enum] :
( p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,c_2Enum_2ESUC_2E1(V0m_2E0)))
& p(c_2Eprim__rec_2E_3C_2E2(V0m_2E0,c_2Enum_2ESUC_2E1(c_2Enum_2ESUC_2E1(V0m_2E0)))) ) ).
tff(thm_2Eprim__rec_2ESIMP__REC__THM,axiom,
! [A_27a: $tType,V0x_2E0: A_27a,V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27a)] :
( ( c_2Eprim__rec_2ESIMP__REC_2E3(A_27a,V0x_2E0,V1f_2E0,c_2Enum_2E0_2E0) = V0x_2E0 )
& ! [V2m_2E0: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2ESIMP__REC_2E3(A_27a,V0x_2E0,V1f_2E0,c_2Enum_2ESUC_2E1(V2m_2E0)) = app_2E2(A_27a,A_27a,V1f_2E0,c_2Eprim__rec_2ESIMP__REC_2E3(A_27a,V0x_2E0,V1f_2E0,V2m_2E0)) ) ) ).
tff(thm_2Eprim__rec_2EPRIM__REC__EQN,axiom,
! [A_27a: $tType,V0x_2E0: A_27a,V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a))] :
( ! [V2n_2E0: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2EPRIM__REC__FUN_2E4(A_27a,V0x_2E0,V1f_2E0,c_2Enum_2E0_2E0,V2n_2E0) = V0x_2E0 )
& ! [V3m_2E0: tyop_2Enum_2Enum,V4n_2E0: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2EPRIM__REC__FUN_2E4(A_27a,V0x_2E0,V1f_2E0,c_2Enum_2ESUC_2E1(V3m_2E0),V4n_2E0) = app_2E2(tyop_2Enum_2Enum,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),V1f_2E0,c_2Eprim__rec_2EPRIM__REC__FUN_2E4(A_27a,V0x_2E0,V1f_2E0,V3m_2E0,c_2Eprim__rec_2EPRE_2E1(V4n_2E0))),V4n_2E0) ) ) ).
tff(thm_2Eprim__rec_2EPRIM__REC__THM,axiom,
! [A_27a: $tType,V0x_2E0: A_27a,V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a))] :
( ( c_2Eprim__rec_2EPRIM__REC_2E3(A_27a,V0x_2E0,V1f_2E0,c_2Enum_2E0_2E0) = V0x_2E0 )
& ! [V2m_2E0: tyop_2Enum_2Enum] : ( c_2Eprim__rec_2EPRIM__REC_2E3(A_27a,V0x_2E0,V1f_2E0,c_2Enum_2ESUC_2E1(V2m_2E0)) = app_2E2(tyop_2Enum_2Enum,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),V1f_2E0,c_2Eprim__rec_2EPRIM__REC_2E3(A_27a,V0x_2E0,V1f_2E0,V2m_2E0)),V2m_2E0) ) ) ).
tff(thm_2Eprim__rec_2EDC,axiom,
! [A_27a: $tType,V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V2a_2E0: A_27a] :
( ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V2a_2E0))
& ! [V3x_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V3x_2E0))
=> ? [V4y_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V4y_2E0))
& p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1R_2E0,V3x_2E0),V4y_2E0)) ) ) )
=> ? [V5f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)] :
( ( app_2E2(tyop_2Enum_2Enum,A_27a,V5f_2E0,c_2Enum_2E0_2E0) = V2a_2E0 )
& ! [V6n_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,app_2E2(tyop_2Enum_2Enum,A_27a,V5f_2E0,V6n_2E0)))
& p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1R_2E0,app_2E2(tyop_2Enum_2Enum,A_27a,V5f_2E0,V6n_2E0)),app_2E2(tyop_2Enum_2Enum,A_27a,V5f_2E0,c_2Enum_2ESUC_2E1(V6n_2E0)))) ) ) ) ).
tff(thm_2Eprim__rec_2Enum__Axiom__old,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Ebool)))] :
( ! [V0e_2E0: A_27a,V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),V2fn1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Ebool)),F0_2E0,V0e_2E0),V1f_2E0),V2fn1_2E0))
<=> ( ( app_2E2(tyop_2Enum_2Enum,A_27a,V2fn1_2E0,c_2Enum_2E0_2E0) = V0e_2E0 )
& ! [V3n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,A_27a,V2fn1_2E0,c_2Enum_2ESUC_2E1(V3n_2E0)) = app_2E2(tyop_2Enum_2Enum,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),V1f_2E0,app_2E2(tyop_2Enum_2Enum,A_27a,V2fn1_2E0,V3n_2E0)),V3n_2E0) ) ) )
=> ! [V0e_2E0: A_27a,V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a))] : p(c_2Ebool_2E_3F_21_2E1(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Ebool)),F0_2E0,V0e_2E0),V1f_2E0))) ) ).
tff(thm_2Eprim__rec_2Enum__Axiom,axiom,
! [A_27a: $tType,V0e_2E0: A_27a,V1f_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,A_27a))] :
? [V2fn_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a)] :
( ( app_2E2(tyop_2Enum_2Enum,A_27a,V2fn_2E0,c_2Enum_2E0_2E0) = V0e_2E0 )
& ! [V3n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,A_27a,V2fn_2E0,c_2Enum_2ESUC_2E1(V3n_2E0)) = app_2E2(A_27a,A_27a,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,A_27a),V1f_2E0,V3n_2E0),app_2E2(tyop_2Enum_2Enum,A_27a,V2fn_2E0,V3n_2E0)) ) ) ).
tff(thm_2Eprim__rec_2EWF__IFF__WELLFOUNDED,axiom,
! [A_27a: $tType,V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))] : ( c_2Erelation_2EWF_2E1(A_27a,V0R_2E0) = c_2Eprim__rec_2Ewellfounded_2E1(A_27a,V0R_2E0) ) ).
tff(thm_2Eprim__rec_2EWF__PRED,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool))] :
( ! [V0x_2E0: tyop_2Enum_2Enum,V1y_2E0: tyop_2Enum_2Enum] :
( p(app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Ebool,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),F0_2E0,V0x_2E0),V1y_2E0))
<=> ( V1y_2E0 = c_2Enum_2ESUC_2E1(V0x_2E0) ) )
=> p(c_2Erelation_2EWF_2E1(tyop_2Enum_2Enum,F0_2E0)) ) ).
tff(thm_2Eprim__rec_2EWF__LESS,axiom,
p(c_2Erelation_2EWF_2E1(tyop_2Enum_2Enum,c_2Eprim__rec_2E_3C_2E0)) ).
tff(thm_2Eprim__rec_2EWF__measure,axiom,
! [A_27a: $tType,V0m_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)] : p(c_2Erelation_2EWF_2E1(A_27a,c_2Eprim__rec_2Emeasure_2E1(A_27a,V0m_2E0))) ).
tff(thm_2Eprim__rec_2Emeasure__thm,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1x_2E0: A_27a,V2y_2E0: A_27a] : ( c_2Eprim__rec_2Emeasure_2E3(A_27a,V0f_2E0,V1x_2E0,V2y_2E0) = c_2Eprim__rec_2E_3C_2E2(app_2E2(A_27a,tyop_2Enum_2Enum,V0f_2E0,V1x_2E0),app_2E2(A_27a,tyop_2Enum_2Enum,V0f_2E0,V2y_2E0)) ) ).
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