ITP001 Axioms: ITP012_7.ax
%------------------------------------------------------------------------------
% File : ITP012_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 : sum.ax [Gau19]
% : HL4012_7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 155 ( 53 unt; 60 typ; 0 def)
% Number of atoms : 183 ( 106 equ)
% Maximal formula atoms : 12 ( 1 avg)
% Number of connectives : 102 ( 14 ~; 7 |; 29 &)
% ( 32 <=>; 20 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 6 avg)
% Maximal term depth : 8 ( 2 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 47 ( 29 >; 18 *; 0 +; 0 <<)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 58 ( 58 usr; 6 con; 0-7 aty)
% Number of variables : 507 ( 408 !; 10 ?; 507 :)
% ( 89 !>; 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_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_2Esum_2E_2B_2B_2E0,type,
c_2Esum_2E_2B_2B_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27c),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27d),tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Esum_2Esum(A_27c,A_27d)))) ).
tff(c_2Esum_2E_2B_2B_2E2,type,
c_2Esum_2E_2B_2B_2E2:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27c) * tyop_2Emin_2Efun(A_27b,A_27d) ) > tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Esum_2Esum(A_27c,A_27d)) ) ).
tff(c_2Esum_2E_2B_2B_2E3,type,
c_2Esum_2E_2B_2B_2E3:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27c) * tyop_2Emin_2Efun(A_27b,A_27d) * tyop_2Esum_2Esum(A_27a,A_27b) ) > tyop_2Esum_2Esum(A_27c,A_27d) ) ).
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_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_2Esum_2EABS__sum_2E0,type,
c_2Esum_2EABS__sum_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),tyop_2Esum_2Esum(A_27a,A_27b)) ).
tff(c_2Esum_2EABS__sum_2E1,type,
c_2Esum_2EABS__sum_2E1:
!>[A_27a: $tType,A_27b: $tType] : ( tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))) > tyop_2Esum_2Esum(A_27a,A_27b) ) ).
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_2EDATATYPE_2E0,type,
c_2Ebool_2EDATATYPE_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2EDATATYPE_2E1,type,
c_2Ebool_2EDATATYPE_2E1:
!>[A_27a: $tType] : ( A_27a > tyop_2Emin_2Ebool ) ).
tff(c_2Ebool_2EF_2E0,type,
c_2Ebool_2EF_2E0: tyop_2Emin_2Ebool ).
tff(c_2Ecombin_2EI_2E0,type,
c_2Ecombin_2EI_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(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_2Esum_2EISL_2E0,type,
c_2Esum_2EISL_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Emin_2Ebool) ).
tff(c_2Esum_2EISL_2E1,type,
c_2Esum_2EISL_2E1:
!>[A_27a: $tType,A_27b: $tType] : ( tyop_2Esum_2Esum(A_27a,A_27b) > tyop_2Emin_2Ebool ) ).
tff(c_2Esum_2EISR_2E0,type,
c_2Esum_2EISR_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Emin_2Ebool) ).
tff(c_2Esum_2EISR_2E1,type,
c_2Esum_2EISR_2E1:
!>[A_27a: $tType,A_27b: $tType] : ( tyop_2Esum_2Esum(A_27a,A_27b) > tyop_2Emin_2Ebool ) ).
tff(c_2Esum_2EIS__SUM__REP_2E0,type,
c_2Esum_2EIS__SUM__REP_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),tyop_2Emin_2Ebool) ).
tff(c_2Esum_2EIS__SUM__REP_2E1,type,
c_2Esum_2EIS__SUM__REP_2E1:
!>[A_27a: $tType,A_27b: $tType] : ( tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))) > tyop_2Emin_2Ebool ) ).
tff(c_2Esum_2EOUTL_2E0,type,
c_2Esum_2EOUTL_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),A_27a) ).
tff(c_2Esum_2EOUTL_2E1,type,
c_2Esum_2EOUTL_2E1:
!>[A_27a: $tType,A_27b: $tType] : ( tyop_2Esum_2Esum(A_27a,A_27b) > A_27a ) ).
tff(c_2Esum_2EOUTR_2E0,type,
c_2Esum_2EOUTR_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),A_27b) ).
tff(c_2Esum_2EOUTR_2E1,type,
c_2Esum_2EOUTR_2E1:
!>[A_27a: $tType,A_27b: $tType] : ( tyop_2Esum_2Esum(A_27a,A_27b) > A_27b ) ).
tff(c_2Esum_2EREP__sum_2E0,type,
c_2Esum_2EREP__sum_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)))) ).
tff(c_2Esum_2EREP__sum_2E1,type,
c_2Esum_2EREP__sum_2E1:
!>[A_27a: $tType,A_27b: $tType] : ( tyop_2Esum_2Esum(A_27a,A_27b) > tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))) ) ).
tff(c_2Esum_2ESUM__ALL_2E0,type,
c_2Esum_2ESUM__ALL_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,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Emin_2Ebool))) ).
tff(c_2Esum_2ESUM__ALL_2E3,type,
c_2Esum_2ESUM__ALL_2E3:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool) * tyop_2Esum_2Esum(A_27a,A_27b) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Ebool_2ET_2E0,type,
c_2Ebool_2ET_2E0: tyop_2Emin_2Ebool ).
tff(c_2Ebool_2ETYPE__DEFINITION_2E0,type,
c_2Ebool_2ETYPE__DEFINITION_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Ebool)) ).
tff(c_2Ebool_2ETYPE__DEFINITION_2E2,type,
c_2Ebool_2ETYPE__DEFINITION_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27b,A_27a) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Ebool_2E_5C_2F_2E0,type,
c_2Ebool_2E_5C_2F_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).
tff(c_2Ebool_2E_5C_2F_2E2,type,
c_2Ebool_2E_5C_2F_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).
tff(c_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_2Esum_2Esum__CASE_2E0,type,
c_2Esum_2Esum__CASE_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27c),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27c),A_27c))) ).
tff(c_2Esum_2Esum__CASE_2E3,type,
c_2Esum_2Esum__CASE_2E3:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Esum_2Esum(A_27a,A_27b) * tyop_2Emin_2Efun(A_27a,A_27c) * tyop_2Emin_2Efun(A_27b,A_27c) ) > A_27c ) ).
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_2Esum_2E_2B_2B_2E2_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,A_27a),X1_2E0: tyop_2Emin_2Efun(A_27b,A_27b)] : ( c_2Esum_2E_2B_2B_2E2(A_27a,A_27b,A_27a,A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,A_27b),tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Esum_2Esum(A_27a,A_27b)),app_2E2(tyop_2Emin_2Efun(A_27a,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27b),tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Esum_2Esum(A_27a,A_27b))),c_2Esum_2E_2B_2B_2E0(A_27a,A_27b,A_27a,A_27b),X0_2E0),X1_2E0) ) ).
tff(arityeq3_2Ec_2Esum_2E_2B_2B_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_27a,A_27c),X1_2E0: tyop_2Emin_2Efun(A_27b,A_27d),X2_2E0: tyop_2Esum_2Esum(A_27a,A_27b)] : ( c_2Esum_2E_2B_2B_2E3(A_27a,A_27b,A_27c,A_27d,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Esum_2Esum(A_27c,A_27d),app_2E2(tyop_2Emin_2Efun(A_27b,A_27d),tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Esum_2Esum(A_27c,A_27d)),app_2E2(tyop_2Emin_2Efun(A_27a,A_27c),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27d),tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Esum_2Esum(A_27c,A_27d))),c_2Esum_2E_2B_2B_2E0(A_27a,A_27b,A_27c,A_27d),X0_2E0),X1_2E0),X2_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_2Ebool_2E_3F_21_2E1_2Emono_2Etyop_2Emin_2Efun_28tyop_2Esum_2Esum_28A_27a_2CA_27b_29_2CA_27c_29,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),A_27c),tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_3F_21_2E1(tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),A_27c),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),A_27c),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_3F_21_2E0(tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),A_27c)),X0_2E0) ) ).
tff(arityeq1_2Ec_2Esum_2EABS__sum_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)))] : ( c_2Esum_2EABS__sum_2E1(A_27a,A_27b,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),tyop_2Esum_2Esum(A_27a,A_27b),c_2Esum_2EABS__sum_2E0(A_27a,A_27b),X0_2E0) ) ).
tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Esum_2Esum_28A_27b_2CA_27a_29,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Esum_2Esum(A_27b,A_27a),X2_2E0: tyop_2Esum_2Esum(A_27b,A_27a)] : ( c_2Ebool_2ECOND_2E3(tyop_2Esum_2Esum(A_27b,A_27a),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Esum_2Esum(A_27b,A_27a),tyop_2Esum_2Esum(A_27b,A_27a),app_2E2(tyop_2Esum_2Esum(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27b,A_27a),tyop_2Esum_2Esum(A_27b,A_27a)),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27b,A_27a),tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27b,A_27a),tyop_2Esum_2Esum(A_27b,A_27a))),c_2Ebool_2ECOND_2E0(tyop_2Esum_2Esum(A_27b,A_27a)),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Esum_2Esum_28A_27c_2CA_27d_29,axiom,
! [A_27c: $tType,A_27d: $tType,X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Esum_2Esum(A_27c,A_27d),X2_2E0: tyop_2Esum_2Esum(A_27c,A_27d)] : ( c_2Ebool_2ECOND_2E3(tyop_2Esum_2Esum(A_27c,A_27d),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Esum_2Esum(A_27c,A_27d),tyop_2Esum_2Esum(A_27c,A_27d),app_2E2(tyop_2Esum_2Esum(A_27c,A_27d),tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27c,A_27d),tyop_2Esum_2Esum(A_27c,A_27d)),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27c,A_27d),tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27c,A_27d),tyop_2Esum_2Esum(A_27c,A_27d))),c_2Ebool_2ECOND_2E0(tyop_2Esum_2Esum(A_27c,A_27d)),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Esum_2Esum_28A_27d_2CA_27c_29,axiom,
! [A_27c: $tType,A_27d: $tType,X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Esum_2Esum(A_27d,A_27c),X2_2E0: tyop_2Esum_2Esum(A_27d,A_27c)] : ( c_2Ebool_2ECOND_2E3(tyop_2Esum_2Esum(A_27d,A_27c),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Esum_2Esum(A_27d,A_27c),tyop_2Esum_2Esum(A_27d,A_27c),app_2E2(tyop_2Esum_2Esum(A_27d,A_27c),tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27d,A_27c),tyop_2Esum_2Esum(A_27d,A_27c)),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27d,A_27c),tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27d,A_27c),tyop_2Esum_2Esum(A_27d,A_27c))),c_2Ebool_2ECOND_2E0(tyop_2Esum_2Esum(A_27d,A_27c)),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Esum_2Esum_28A_27e_2CA_27f_29,axiom,
! [A_27e: $tType,A_27f: $tType,X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Esum_2Esum(A_27e,A_27f),X2_2E0: tyop_2Esum_2Esum(A_27e,A_27f)] : ( c_2Ebool_2ECOND_2E3(tyop_2Esum_2Esum(A_27e,A_27f),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Esum_2Esum(A_27e,A_27f),tyop_2Esum_2Esum(A_27e,A_27f),app_2E2(tyop_2Esum_2Esum(A_27e,A_27f),tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27e,A_27f),tyop_2Esum_2Esum(A_27e,A_27f)),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27e,A_27f),tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27e,A_27f),tyop_2Esum_2Esum(A_27e,A_27f))),c_2Ebool_2ECOND_2E0(tyop_2Esum_2Esum(A_27e,A_27f)),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Esum_2Esum_28A_27g_2CA_27h_29,axiom,
! [A_27g: $tType,A_27h: $tType,X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Esum_2Esum(A_27g,A_27h),X2_2E0: tyop_2Esum_2Esum(A_27g,A_27h)] : ( c_2Ebool_2ECOND_2E3(tyop_2Esum_2Esum(A_27g,A_27h),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Esum_2Esum(A_27g,A_27h),tyop_2Esum_2Esum(A_27g,A_27h),app_2E2(tyop_2Esum_2Esum(A_27g,A_27h),tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27g,A_27h),tyop_2Esum_2Esum(A_27g,A_27h)),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27g,A_27h),tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27g,A_27h),tyop_2Esum_2Esum(A_27g,A_27h))),c_2Ebool_2ECOND_2E0(tyop_2Esum_2Esum(A_27g,A_27h)),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq1_2Ec_2Ebool_2EDATATYPE_2E1_2Emono_2EA_27c,axiom,
! [A_27c: $tType,X0_2E0: A_27c] : ( c_2Ebool_2EDATATYPE_2E1(A_27c,X0_2E0) = app_2E2(A_27c,tyop_2Emin_2Ebool,c_2Ebool_2EDATATYPE_2E0(A_27c),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_2EINL_2E1_2Emono_2EA_27b_20mono_2EA_27a,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: A_27b] : ( c_2Esum_2EINL_2E1(A_27b,A_27a,X0_2E0) = app_2E2(A_27b,tyop_2Esum_2Esum(A_27b,A_27a),c_2Esum_2EINL_2E0(A_27b,A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Esum_2EINL_2E1_2Emono_2EA_27c_20mono_2EA_27d,axiom,
! [A_27c: $tType,A_27d: $tType,X0_2E0: A_27c] : ( c_2Esum_2EINL_2E1(A_27c,A_27d,X0_2E0) = app_2E2(A_27c,tyop_2Esum_2Esum(A_27c,A_27d),c_2Esum_2EINL_2E0(A_27c,A_27d),X0_2E0) ) ).
tff(arityeq1_2Ec_2Esum_2EINL_2E1_2Emono_2EA_27d_20mono_2EA_27c,axiom,
! [A_27c: $tType,A_27d: $tType,X0_2E0: A_27d] : ( c_2Esum_2EINL_2E1(A_27d,A_27c,X0_2E0) = app_2E2(A_27d,tyop_2Esum_2Esum(A_27d,A_27c),c_2Esum_2EINL_2E0(A_27d,A_27c),X0_2E0) ) ).
tff(arityeq1_2Ec_2Esum_2EINL_2E1_2Emono_2EA_27e_20mono_2EA_27f,axiom,
! [A_27e: $tType,A_27f: $tType,X0_2E0: A_27e] : ( c_2Esum_2EINL_2E1(A_27e,A_27f,X0_2E0) = app_2E2(A_27e,tyop_2Esum_2Esum(A_27e,A_27f),c_2Esum_2EINL_2E0(A_27e,A_27f),X0_2E0) ) ).
tff(arityeq1_2Ec_2Esum_2EINL_2E1_2Emono_2EA_27g_20mono_2EA_27h,axiom,
! [A_27g: $tType,A_27h: $tType,X0_2E0: A_27g] : ( c_2Esum_2EINL_2E1(A_27g,A_27h,X0_2E0) = app_2E2(A_27g,tyop_2Esum_2Esum(A_27g,A_27h),c_2Esum_2EINL_2E0(A_27g,A_27h),X0_2E0) ) ).
tff(arityeq1_2Ec_2Esum_2EINR_2E1_2Emono_2EA_27b_20mono_2EA_27a,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: A_27a] : ( c_2Esum_2EINR_2E1(A_27b,A_27a,X0_2E0) = app_2E2(A_27a,tyop_2Esum_2Esum(A_27b,A_27a),c_2Esum_2EINR_2E0(A_27b,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_2Esum_2EINR_2E1_2Emono_2EA_27d_20mono_2EA_27c,axiom,
! [A_27c: $tType,A_27d: $tType,X0_2E0: A_27c] : ( c_2Esum_2EINR_2E1(A_27d,A_27c,X0_2E0) = app_2E2(A_27c,tyop_2Esum_2Esum(A_27d,A_27c),c_2Esum_2EINR_2E0(A_27d,A_27c),X0_2E0) ) ).
tff(arityeq1_2Ec_2Esum_2EINR_2E1_2Emono_2EA_27c_20mono_2EA_27d,axiom,
! [A_27c: $tType,A_27d: $tType,X0_2E0: A_27d] : ( c_2Esum_2EINR_2E1(A_27c,A_27d,X0_2E0) = app_2E2(A_27d,tyop_2Esum_2Esum(A_27c,A_27d),c_2Esum_2EINR_2E0(A_27c,A_27d),X0_2E0) ) ).
tff(arityeq1_2Ec_2Esum_2EINR_2E1_2Emono_2EA_27e_20mono_2EA_27f,axiom,
! [A_27e: $tType,A_27f: $tType,X0_2E0: A_27f] : ( c_2Esum_2EINR_2E1(A_27e,A_27f,X0_2E0) = app_2E2(A_27f,tyop_2Esum_2Esum(A_27e,A_27f),c_2Esum_2EINR_2E0(A_27e,A_27f),X0_2E0) ) ).
tff(arityeq1_2Ec_2Esum_2EINR_2E1_2Emono_2EA_27g_20mono_2EA_27h,axiom,
! [A_27g: $tType,A_27h: $tType,X0_2E0: A_27h] : ( c_2Esum_2EINR_2E1(A_27g,A_27h,X0_2E0) = app_2E2(A_27h,tyop_2Esum_2Esum(A_27g,A_27h),c_2Esum_2EINR_2E0(A_27g,A_27h),X0_2E0) ) ).
tff(arityeq1_2Ec_2Esum_2EISL_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Esum_2Esum(A_27a,A_27b)] : ( c_2Esum_2EISL_2E1(A_27a,A_27b,X0_2E0) = app_2E2(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Emin_2Ebool,c_2Esum_2EISL_2E0(A_27a,A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Esum_2EISR_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Esum_2Esum(A_27a,A_27b)] : ( c_2Esum_2EISR_2E1(A_27a,A_27b,X0_2E0) = app_2E2(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Emin_2Ebool,c_2Esum_2EISR_2E0(A_27a,A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Esum_2EIS__SUM__REP_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)))] : ( c_2Esum_2EIS__SUM__REP_2E1(A_27a,A_27b,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),tyop_2Emin_2Ebool,c_2Esum_2EIS__SUM__REP_2E0(A_27a,A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Esum_2EOUTL_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Esum_2Esum(A_27a,A_27b)] : ( c_2Esum_2EOUTL_2E1(A_27a,A_27b,X0_2E0) = app_2E2(tyop_2Esum_2Esum(A_27a,A_27b),A_27a,c_2Esum_2EOUTL_2E0(A_27a,A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Esum_2EOUTR_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Esum_2Esum(A_27a,A_27b)] : ( c_2Esum_2EOUTR_2E1(A_27a,A_27b,X0_2E0) = app_2E2(tyop_2Esum_2Esum(A_27a,A_27b),A_27b,c_2Esum_2EOUTR_2E0(A_27a,A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Esum_2EREP__sum_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Esum_2Esum(A_27a,A_27b)] : ( c_2Esum_2EREP__sum_2E1(A_27a,A_27b,X0_2E0) = app_2E2(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),c_2Esum_2EREP__sum_2E0(A_27a,A_27b),X0_2E0) ) ).
tff(arityeq3_2Ec_2Esum_2ESUM__ALL_2E3_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),X2_2E0: tyop_2Esum_2Esum(A_27a,A_27b)] : ( c_2Esum_2ESUM__ALL_2E3(A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Emin_2Ebool)),c_2Esum_2ESUM__ALL_2E0(A_27a,A_27b),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2ETYPE__DEFINITION_2E2_2Emono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Ebool_2Ctyop_2Emin_2Efun_28A_27a_2Ctyop_2Emin_2Efun_28A_27b_2Ctyop_2Emin_2Ebool_29_29_29_20mono_2Etyop_2Esum_2Esum_28A_27a_2CA_27b_29,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))))] : ( c_2Ebool_2ETYPE__DEFINITION_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),tyop_2Esum_2Esum(A_27a,A_27b),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)))),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)))),tyop_2Emin_2Ebool),c_2Ebool_2ETYPE__DEFINITION_2E0(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),tyop_2Esum_2Esum(A_27a,A_27b)),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_2Ecombin_2Eo_2E2_2Emono_2EA_27a_20mono_2Etyop_2Esum_2Esum_28A_27c_2CA_27d_29_20mono_2EA_27c,axiom,
! [A_27a: $tType,A_27c: $tType,A_27d: $tType,X0_2E0: tyop_2Emin_2Efun(A_27c,tyop_2Esum_2Esum(A_27c,A_27d)),X1_2E0: tyop_2Emin_2Efun(A_27a,A_27c)] : ( c_2Ecombin_2Eo_2E2(A_27a,tyop_2Esum_2Esum(A_27c,A_27d),A_27c,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,A_27c),tyop_2Emin_2Efun(A_27a,tyop_2Esum_2Esum(A_27c,A_27d)),app_2E2(tyop_2Emin_2Efun(A_27c,tyop_2Esum_2Esum(A_27c,A_27d)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27c),tyop_2Emin_2Efun(A_27a,tyop_2Esum_2Esum(A_27c,A_27d))),c_2Ecombin_2Eo_2E0(A_27a,tyop_2Esum_2Esum(A_27c,A_27d),A_27c),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ecombin_2Eo_2E2_2Emono_2EA_27b_20mono_2Etyop_2Esum_2Esum_28A_27c_2CA_27d_29_20mono_2EA_27d,axiom,
! [A_27b: $tType,A_27c: $tType,A_27d: $tType,X0_2E0: tyop_2Emin_2Efun(A_27d,tyop_2Esum_2Esum(A_27c,A_27d)),X1_2E0: tyop_2Emin_2Efun(A_27b,A_27d)] : ( c_2Ecombin_2Eo_2E2(A_27b,tyop_2Esum_2Esum(A_27c,A_27d),A_27d,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Esum_2Esum(A_27c,A_27d)),app_2E2(tyop_2Emin_2Efun(A_27d,tyop_2Esum_2Esum(A_27c,A_27d)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27d),tyop_2Emin_2Efun(A_27b,tyop_2Esum_2Esum(A_27c,A_27d))),c_2Ecombin_2Eo_2E0(A_27b,tyop_2Esum_2Esum(A_27c,A_27d),A_27d),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ecombin_2Eo_2E2_2Emono_2EA_27a_20mono_2EA_27c_20mono_2Etyop_2Esum_2Esum_28A_27a_2CA_27b_29,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),A_27c),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Esum_2Esum(A_27a,A_27b))] : ( c_2Ecombin_2Eo_2E2(A_27a,A_27c,tyop_2Esum_2Esum(A_27a,A_27b),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Esum_2Esum(A_27a,A_27b)),tyop_2Emin_2Efun(A_27a,A_27c),app_2E2(tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),A_27c),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Esum_2Esum(A_27a,A_27b)),tyop_2Emin_2Efun(A_27a,A_27c)),c_2Ecombin_2Eo_2E0(A_27a,A_27c,tyop_2Esum_2Esum(A_27a,A_27b)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ecombin_2Eo_2E2_2Emono_2EA_27b_20mono_2EA_27c_20mono_2Etyop_2Esum_2Esum_28A_27a_2CA_27b_29,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),A_27c),X1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Esum_2Esum(A_27a,A_27b))] : ( c_2Ecombin_2Eo_2E2(A_27b,A_27c,tyop_2Esum_2Esum(A_27a,A_27b),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Esum_2Esum(A_27a,A_27b)),tyop_2Emin_2Efun(A_27b,A_27c),app_2E2(tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),A_27c),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Esum_2Esum(A_27a,A_27b)),tyop_2Emin_2Efun(A_27b,A_27c)),c_2Ecombin_2Eo_2E0(A_27b,A_27c,tyop_2Esum_2Esum(A_27a,A_27b)),X0_2E0),X1_2E0) ) ).
tff(arityeq3_2Ec_2Esum_2Esum__CASE_2E3_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27c,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,X0_2E0: tyop_2Esum_2Esum(A_27a,A_27b),X1_2E0: tyop_2Emin_2Efun(A_27a,A_27c),X2_2E0: tyop_2Emin_2Efun(A_27b,A_27c)] : ( c_2Esum_2Esum__CASE_2E3(A_27a,A_27b,A_27c,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,A_27c),A_27c,app_2E2(tyop_2Emin_2Efun(A_27a,A_27c),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27c),A_27c),app_2E2(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27c),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27c),A_27c)),c_2Esum_2Esum__CASE_2E0(A_27a,A_27b,A_27c),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Esum_2Esum__CASE_2E3_2Emono_2EA_27a_20mono_2EA_27b_20mono_2Etyop_2Esum_2Esum_28A_27c_2CA_27d_29,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,X0_2E0: tyop_2Esum_2Esum(A_27a,A_27b),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Esum_2Esum(A_27c,A_27d)),X2_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Esum_2Esum(A_27c,A_27d))] : ( c_2Esum_2Esum__CASE_2E3(A_27a,A_27b,tyop_2Esum_2Esum(A_27c,A_27d),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Esum_2Esum(A_27c,A_27d)),tyop_2Esum_2Esum(A_27c,A_27d),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Esum_2Esum(A_27c,A_27d)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Esum_2Esum(A_27c,A_27d)),tyop_2Esum_2Esum(A_27c,A_27d)),app_2E2(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Esum_2Esum(A_27c,A_27d)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Esum_2Esum(A_27c,A_27d)),tyop_2Esum_2Esum(A_27c,A_27d))),c_2Esum_2Esum__CASE_2E0(A_27a,A_27b,tyop_2Esum_2Esum(A_27c,A_27d)),X0_2E0),X1_2E0),X2_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_2Esum_2EIS__SUM__REP,axiom,
! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)))] :
( p(c_2Esum_2EIS__SUM__REP_2E1(A_27a,A_27b,V0f_2E0))
<=> ? [V1v1_2E0: A_27a,V2v2_2E0: A_27b] :
( ! [V3x_2E0: tyop_2Emin_2Ebool,V4x_27_2E0: A_27a,V5x_27_27_2E0: A_27b] :
( p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V0f_2E0,V3x_2E0),V4x_27_2E0),V5x_27_27_2E0))
<=> ( ( V4x_27_2E0 = V1v1_2E0 )
& p(V3x_2E0) ) )
| ! [V6x_2E0: tyop_2Emin_2Ebool,V7x_27_2E0: A_27a,V8x_27_27_2E0: A_27b] :
( p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V0f_2E0,V6x_2E0),V7x_27_2E0),V8x_27_27_2E0))
<=> ( ( V8x_27_27_2E0 = V2v2_2E0 )
& ~ p(V6x_2E0) ) ) ) ) ).
tff(thm_2Esum_2Esum__TY__DEF,axiom,
! [A_27a: $tType,A_27b: $tType] :
? [V0rep_2E0: tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))))] : p(c_2Ebool_2ETYPE__DEFINITION_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),tyop_2Esum_2Esum(A_27a,A_27b),c_2Esum_2EIS__SUM__REP_2E0(A_27a,A_27b),V0rep_2E0)) ).
tff(thm_2Esum_2Esum__ISO__DEF,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0a_2E0: tyop_2Esum_2Esum(A_27a,A_27b)] : ( c_2Esum_2EABS__sum_2E1(A_27a,A_27b,c_2Esum_2EREP__sum_2E1(A_27a,A_27b,V0a_2E0)) = V0a_2E0 )
& ! [V1r_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)))] :
( p(c_2Esum_2EIS__SUM__REP_2E1(A_27a,A_27b,V1r_2E0))
<=> ( c_2Esum_2EREP__sum_2E1(A_27a,A_27b,c_2Esum_2EABS__sum_2E1(A_27a,A_27b,V1r_2E0)) = V1r_2E0 ) ) ) ).
tff(thm_2Esum_2EINL__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,F0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))))] :
( ! [V0e_2E0: A_27a,V1b_2E0: tyop_2Emin_2Ebool,V2x_2E0: A_27a,V3y_2E0: A_27b] :
( p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),F0_2E0,V0e_2E0),V1b_2E0),V2x_2E0),V3y_2E0))
<=> ( ( V2x_2E0 = V0e_2E0 )
& p(V1b_2E0) ) )
=> ! [V0e_2E0: A_27a] : ( c_2Esum_2EINL_2E1(A_27a,A_27b,V0e_2E0) = c_2Esum_2EABS__sum_2E1(A_27a,A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),F0_2E0,V0e_2E0)) ) ) ).
tff(thm_2Esum_2EINR__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,F0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))))] :
( ! [V0e_2E0: A_27b,V1b_2E0: tyop_2Emin_2Ebool,V2x_2E0: A_27a,V3y_2E0: A_27b] :
( p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),F0_2E0,V0e_2E0),V1b_2E0),V2x_2E0),V3y_2E0))
<=> ( ( V3y_2E0 = V0e_2E0 )
& ~ p(V1b_2E0) ) )
=> ! [V0e_2E0: A_27b] : ( c_2Esum_2EINR_2E1(A_27a,A_27b,V0e_2E0) = c_2Esum_2EABS__sum_2E1(A_27a,A_27b,app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),F0_2E0,V0e_2E0)) ) ) ).
tff(thm_2Esum_2EISL,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0x_2E0: A_27a] : p(c_2Esum_2EISL_2E1(A_27a,A_27b,c_2Esum_2EINL_2E1(A_27a,A_27b,V0x_2E0)))
& ! [V1y_2E0: A_27b] : ~ p(c_2Esum_2EISL_2E1(A_27a,A_27b,c_2Esum_2EINR_2E1(A_27a,A_27b,V1y_2E0))) ) ).
tff(thm_2Esum_2EISR,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0x_2E0: A_27b] : p(c_2Esum_2EISR_2E1(A_27a,A_27b,c_2Esum_2EINR_2E1(A_27a,A_27b,V0x_2E0)))
& ! [V1y_2E0: A_27a] : ~ p(c_2Esum_2EISR_2E1(A_27a,A_27b,c_2Esum_2EINL_2E1(A_27a,A_27b,V1y_2E0))) ) ).
tff(thm_2Esum_2EOUTL,axiom,
! [A_27a: $tType,A_27b: $tType,V0x_2E0: A_27a] : ( c_2Esum_2EOUTL_2E1(A_27a,A_27b,c_2Esum_2EINL_2E1(A_27a,A_27b,V0x_2E0)) = V0x_2E0 ) ).
tff(thm_2Esum_2EOUTR,axiom,
! [A_27a: $tType,A_27b: $tType,V0x_2E0: A_27b] : ( c_2Esum_2EOUTR_2E1(A_27a,A_27b,c_2Esum_2EINR_2E1(A_27a,A_27b,V0x_2E0)) = V0x_2E0 ) ).
tff(thm_2Esum_2Esum__case__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType] :
( ! [V0x_2E0: A_27a,V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27c),V2f1_2E0: tyop_2Emin_2Efun(A_27b,A_27c)] : ( c_2Esum_2Esum__CASE_2E3(A_27a,A_27b,A_27c,c_2Esum_2EINL_2E1(A_27a,A_27b,V0x_2E0),V1f_2E0,V2f1_2E0) = app_2E2(A_27a,A_27c,V1f_2E0,V0x_2E0) )
& ! [V3y_2E0: A_27b,V4f_2E0: tyop_2Emin_2Efun(A_27a,A_27c),V5f1_2E0: tyop_2Emin_2Efun(A_27b,A_27c)] : ( c_2Esum_2Esum__CASE_2E3(A_27a,A_27b,A_27c,c_2Esum_2EINR_2E1(A_27a,A_27b,V3y_2E0),V4f_2E0,V5f1_2E0) = app_2E2(A_27b,A_27c,V5f1_2E0,V3y_2E0) ) ) ).
tff(thm_2Esum_2ESUM__MAP__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] :
( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27c),V1g_2E0: tyop_2Emin_2Efun(A_27b,A_27d),V2a_2E0: A_27a] : ( c_2Esum_2E_2B_2B_2E3(A_27a,A_27b,A_27c,A_27d,V0f_2E0,V1g_2E0,c_2Esum_2EINL_2E1(A_27a,A_27b,V2a_2E0)) = c_2Esum_2EINL_2E1(A_27c,A_27d,app_2E2(A_27a,A_27c,V0f_2E0,V2a_2E0)) )
& ! [V3f_2E0: tyop_2Emin_2Efun(A_27a,A_27c),V4g_2E0: tyop_2Emin_2Efun(A_27b,A_27d),V5b_2E0: A_27b] : ( c_2Esum_2E_2B_2B_2E3(A_27a,A_27b,A_27c,A_27d,V3f_2E0,V4g_2E0,c_2Esum_2EINR_2E1(A_27a,A_27b,V5b_2E0)) = c_2Esum_2EINR_2E1(A_27c,A_27d,app_2E2(A_27b,A_27d,V4g_2E0,V5b_2E0)) ) ) ).
tff(thm_2Esum_2ESUM__ALL__def,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1Q_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V2x_2E0: A_27a] : ( c_2Esum_2ESUM__ALL_2E3(A_27a,A_27b,V0P_2E0,V1Q_2E0,c_2Esum_2EINL_2E1(A_27a,A_27b,V2x_2E0)) = app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,V2x_2E0) )
& ! [V3P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V4Q_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V5y_2E0: A_27b] : ( c_2Esum_2ESUM__ALL_2E3(A_27a,A_27b,V3P_2E0,V4Q_2E0,c_2Esum_2EINR_2E1(A_27a,A_27b,V5y_2E0)) = app_2E2(A_27b,tyop_2Emin_2Ebool,V4Q_2E0,V5y_2E0) ) ) ).
tff(thm_2Esum_2EINL__11,axiom,
! [A_27a: $tType,A_27b: $tType,V0y_2E0: A_27a,V1x_2E0: A_27a] :
( ( c_2Esum_2EINL_2E1(A_27a,A_27b,V1x_2E0) = c_2Esum_2EINL_2E1(A_27a,A_27b,V0y_2E0) )
<=> ( V1x_2E0 = V0y_2E0 ) ) ).
tff(thm_2Esum_2EINR__11,axiom,
! [A_27a: $tType,A_27b: $tType,V0y_2E0: A_27b,V1x_2E0: A_27b] :
( ( c_2Esum_2EINR_2E1(A_27a,A_27b,V1x_2E0) = c_2Esum_2EINR_2E1(A_27a,A_27b,V0y_2E0) )
<=> ( V1x_2E0 = V0y_2E0 ) ) ).
tff(thm_2Esum_2EINR__INL__11,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0y_2E0: A_27a,V1x_2E0: A_27a] :
( ( c_2Esum_2EINL_2E1(A_27a,A_27b,V1x_2E0) = c_2Esum_2EINL_2E1(A_27a,A_27b,V0y_2E0) )
<=> ( V1x_2E0 = V0y_2E0 ) )
& ! [V2y_2E0: A_27b,V3x_2E0: A_27b] :
( ( c_2Esum_2EINR_2E1(A_27a,A_27b,V3x_2E0) = c_2Esum_2EINR_2E1(A_27a,A_27b,V2y_2E0) )
<=> ( V3x_2E0 = V2y_2E0 ) ) ) ).
tff(thm_2Esum_2EINR__neq__INL,axiom,
! [A_27a: $tType,A_27b: $tType,V0v1_2E0: A_27a,V1v2_2E0: A_27b] : ( c_2Esum_2EINR_2E1(A_27a,A_27b,V1v2_2E0) != c_2Esum_2EINL_2E1(A_27a,A_27b,V0v1_2E0) ) ).
tff(thm_2Esum_2Esum__axiom,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27c),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),A_27c),tyop_2Emin_2Ebool)))] :
( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27c),V1g_2E0: tyop_2Emin_2Efun(A_27b,A_27c),V2h_2E0: tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),A_27c)] :
( p(app_2E2(tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),A_27c),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),A_27c),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,A_27c),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),A_27c),tyop_2Emin_2Ebool)),F0_2E0,V0f_2E0),V1g_2E0),V2h_2E0))
<=> ( ( c_2Ecombin_2Eo_2E2(A_27a,A_27c,tyop_2Esum_2Esum(A_27a,A_27b),V2h_2E0,c_2Esum_2EINL_2E0(A_27a,A_27b)) = V0f_2E0 )
& ( c_2Ecombin_2Eo_2E2(A_27b,A_27c,tyop_2Esum_2Esum(A_27a,A_27b),V2h_2E0,c_2Esum_2EINR_2E0(A_27a,A_27b)) = V1g_2E0 ) ) )
=> ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27c),V1g_2E0: tyop_2Emin_2Efun(A_27b,A_27c)] : p(c_2Ebool_2E_3F_21_2E1(tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),A_27c),app_2E2(tyop_2Emin_2Efun(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),A_27c),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,A_27c),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),A_27c),tyop_2Emin_2Ebool)),F0_2E0,V0f_2E0),V1g_2E0))) ) ).
tff(thm_2Esum_2Esum__INDUCT,axiom,
! [A_27a: $tType,A_27b: $tType,V0P_2E0: tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Emin_2Ebool)] :
( ( ! [V1x_2E0: A_27a] : p(app_2E2(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Emin_2Ebool,V0P_2E0,c_2Esum_2EINL_2E1(A_27a,A_27b,V1x_2E0)))
& ! [V2y_2E0: A_27b] : p(app_2E2(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Emin_2Ebool,V0P_2E0,c_2Esum_2EINR_2E1(A_27a,A_27b,V2y_2E0))) )
=> ! [V3s_2E0: tyop_2Esum_2Esum(A_27a,A_27b)] : p(app_2E2(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Emin_2Ebool,V0P_2E0,V3s_2E0)) ) ).
tff(thm_2Esum_2EFORALL__SUM,axiom,
! [A_27a: $tType,A_27b: $tType,V0P_2E0: tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Emin_2Ebool)] :
( ! [V1s_2E0: tyop_2Esum_2Esum(A_27a,A_27b)] : p(app_2E2(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Emin_2Ebool,V0P_2E0,V1s_2E0))
<=> ( ! [V2x_2E0: A_27a] : p(app_2E2(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Emin_2Ebool,V0P_2E0,c_2Esum_2EINL_2E1(A_27a,A_27b,V2x_2E0)))
& ! [V3y_2E0: A_27b] : p(app_2E2(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Emin_2Ebool,V0P_2E0,c_2Esum_2EINR_2E1(A_27a,A_27b,V3y_2E0))) ) ) ).
tff(thm_2Esum_2EEXISTS__SUM,axiom,
! [A_27a: $tType,A_27b: $tType,V0P_2E0: tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Emin_2Ebool)] :
( ? [V1s_2E0: tyop_2Esum_2Esum(A_27a,A_27b)] : p(app_2E2(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Emin_2Ebool,V0P_2E0,V1s_2E0))
<=> ( ? [V2x_2E0: A_27a] : p(app_2E2(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Emin_2Ebool,V0P_2E0,c_2Esum_2EINL_2E1(A_27a,A_27b,V2x_2E0)))
| ? [V3y_2E0: A_27b] : p(app_2E2(tyop_2Esum_2Esum(A_27a,A_27b),tyop_2Emin_2Ebool,V0P_2E0,c_2Esum_2EINR_2E1(A_27a,A_27b,V3y_2E0))) ) ) ).
tff(thm_2Esum_2Esum__Axiom,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27c),V1g_2E0: tyop_2Emin_2Efun(A_27b,A_27c)] :
? [V2h_2E0: tyop_2Emin_2Efun(tyop_2Esum_2Esum(A_27a,A_27b),A_27c)] :
( ! [V3x_2E0: A_27a] : ( app_2E2(tyop_2Esum_2Esum(A_27a,A_27b),A_27c,V2h_2E0,c_2Esum_2EINL_2E1(A_27a,A_27b,V3x_2E0)) = app_2E2(A_27a,A_27c,V0f_2E0,V3x_2E0) )
& ! [V4y_2E0: A_27b] : ( app_2E2(tyop_2Esum_2Esum(A_27a,A_27b),A_27c,V2h_2E0,c_2Esum_2EINR_2E1(A_27a,A_27b,V4y_2E0)) = app_2E2(A_27b,A_27c,V1g_2E0,V4y_2E0) ) ) ).
tff(thm_2Esum_2Esum__CASES,axiom,
! [A_27a: $tType,A_27b: $tType,V0ss_2E0: tyop_2Esum_2Esum(A_27a,A_27b)] :
( ? [V1x_2E0: A_27a] : ( V0ss_2E0 = c_2Esum_2EINL_2E1(A_27a,A_27b,V1x_2E0) )
| ? [V2y_2E0: A_27b] : ( V0ss_2E0 = c_2Esum_2EINR_2E1(A_27a,A_27b,V2y_2E0) ) ) ).
tff(thm_2Esum_2Esum__distinct,axiom,
! [A_27a: $tType,A_27b: $tType,V0x_2E0: A_27a,V1y_2E0: A_27b] : ( c_2Esum_2EINL_2E1(A_27a,A_27b,V0x_2E0) != c_2Esum_2EINR_2E1(A_27a,A_27b,V1y_2E0) ) ).
tff(thm_2Esum_2Esum__distinct1,axiom,
! [A_27a: $tType,A_27b: $tType,V0v1_2E0: A_27a,V1v2_2E0: A_27b] : ( c_2Esum_2EINR_2E1(A_27a,A_27b,V1v2_2E0) != c_2Esum_2EINL_2E1(A_27a,A_27b,V0v1_2E0) ) ).
tff(thm_2Esum_2EISL__OR__ISR,axiom,
! [A_27a: $tType,A_27b: $tType,V0x_2E0: tyop_2Esum_2Esum(A_27a,A_27b)] :
( p(c_2Esum_2EISL_2E1(A_27a,A_27b,V0x_2E0))
| p(c_2Esum_2EISR_2E1(A_27a,A_27b,V0x_2E0)) ) ).
tff(thm_2Esum_2EINL,axiom,
! [A_27a: $tType,A_27b: $tType,V0x_2E0: tyop_2Esum_2Esum(A_27a,A_27b)] :
( p(c_2Esum_2EISL_2E1(A_27a,A_27b,V0x_2E0))
=> ( c_2Esum_2EINL_2E1(A_27a,A_27b,c_2Esum_2EOUTL_2E1(A_27a,A_27b,V0x_2E0)) = V0x_2E0 ) ) ).
tff(thm_2Esum_2EINR,axiom,
! [A_27a: $tType,A_27b: $tType,V0x_2E0: tyop_2Esum_2Esum(A_27a,A_27b)] :
( p(c_2Esum_2EISR_2E1(A_27a,A_27b,V0x_2E0))
=> ( c_2Esum_2EINR_2E1(A_27a,A_27b,c_2Esum_2EOUTR_2E1(A_27a,A_27b,V0x_2E0)) = V0x_2E0 ) ) ).
tff(thm_2Esum_2Esum__case__cong,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f1_27_2E0: tyop_2Emin_2Efun(A_27b,A_27c),V1f_27_2E0: tyop_2Emin_2Efun(A_27a,A_27c),V2M_2E0: tyop_2Esum_2Esum(A_27a,A_27b),V3M_27_2E0: tyop_2Esum_2Esum(A_27a,A_27b),V4f_2E0: tyop_2Emin_2Efun(A_27a,A_27c),V5f1_2E0: tyop_2Emin_2Efun(A_27b,A_27c)] :
( ( ( V2M_2E0 = V3M_27_2E0 )
& ! [V6x_2E0: A_27a] :
( ( V3M_27_2E0 = c_2Esum_2EINL_2E1(A_27a,A_27b,V6x_2E0) )
=> ( app_2E2(A_27a,A_27c,V4f_2E0,V6x_2E0) = app_2E2(A_27a,A_27c,V1f_27_2E0,V6x_2E0) ) )
& ! [V7y_2E0: A_27b] :
( ( V3M_27_2E0 = c_2Esum_2EINR_2E1(A_27a,A_27b,V7y_2E0) )
=> ( app_2E2(A_27b,A_27c,V5f1_2E0,V7y_2E0) = app_2E2(A_27b,A_27c,V0f1_27_2E0,V7y_2E0) ) ) )
=> ( c_2Esum_2Esum__CASE_2E3(A_27a,A_27b,A_27c,V2M_2E0,V4f_2E0,V5f1_2E0) = c_2Esum_2Esum__CASE_2E3(A_27a,A_27b,A_27c,V3M_27_2E0,V1f_27_2E0,V0f1_27_2E0) ) ) ).
tff(thm_2Esum_2ESUM__MAP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27c),V1g_2E0: tyop_2Emin_2Efun(A_27b,A_27d),V2z_2E0: tyop_2Esum_2Esum(A_27a,A_27b)] : ( c_2Esum_2E_2B_2B_2E3(A_27a,A_27b,A_27c,A_27d,V0f_2E0,V1g_2E0,V2z_2E0) = c_2Ebool_2ECOND_2E3(tyop_2Esum_2Esum(A_27c,A_27d),c_2Esum_2EISL_2E1(A_27a,A_27b,V2z_2E0),c_2Esum_2EINL_2E1(A_27c,A_27d,app_2E2(A_27a,A_27c,V0f_2E0,c_2Esum_2EOUTL_2E1(A_27a,A_27b,V2z_2E0))),c_2Esum_2EINR_2E1(A_27c,A_27d,app_2E2(A_27b,A_27d,V1g_2E0,c_2Esum_2EOUTR_2E1(A_27a,A_27b,V2z_2E0)))) ) ).
tff(thm_2Esum_2ESUM__MAP__CASE,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27c),V1g_2E0: tyop_2Emin_2Efun(A_27b,A_27d),V2z_2E0: tyop_2Esum_2Esum(A_27a,A_27b)] : ( c_2Esum_2E_2B_2B_2E3(A_27a,A_27b,A_27c,A_27d,V0f_2E0,V1g_2E0,V2z_2E0) = c_2Esum_2Esum__CASE_2E3(A_27a,A_27b,tyop_2Esum_2Esum(A_27c,A_27d),V2z_2E0,c_2Ecombin_2Eo_2E2(A_27a,tyop_2Esum_2Esum(A_27c,A_27d),A_27c,c_2Esum_2EINL_2E0(A_27c,A_27d),V0f_2E0),c_2Ecombin_2Eo_2E2(A_27b,tyop_2Esum_2Esum(A_27c,A_27d),A_27d,c_2Esum_2EINR_2E0(A_27c,A_27d),V1g_2E0)) ) ).
tff(thm_2Esum_2ESUM__MAP__I,axiom,
! [A_27a: $tType,A_27b: $tType] : ( c_2Esum_2E_2B_2B_2E2(A_27a,A_27b,A_27a,A_27b,c_2Ecombin_2EI_2E0(A_27a),c_2Ecombin_2EI_2E0(A_27b)) = c_2Ecombin_2EI_2E0(tyop_2Esum_2Esum(A_27a,A_27b)) ) ).
tff(thm_2Esum_2Econd__sum__expand,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType,A_27f: $tType,A_27g: $tType,A_27h: $tType,V0P_2E0: tyop_2Emin_2Ebool] :
( ! [V1x_2E0: A_27a,V2y_2E0: A_27b,V3z_2E0: A_27a] :
( ( c_2Ebool_2ECOND_2E3(tyop_2Esum_2Esum(A_27b,A_27a),V0P_2E0,c_2Esum_2EINR_2E1(A_27b,A_27a,V1x_2E0),c_2Esum_2EINL_2E1(A_27b,A_27a,V2y_2E0)) = c_2Esum_2EINR_2E1(A_27b,A_27a,V3z_2E0) )
<=> ( p(V0P_2E0)
& ( V3z_2E0 = V1x_2E0 ) ) )
& ! [V4x_2E0: A_27c,V5y_2E0: A_27d,V6z_2E0: A_27d] :
( ( c_2Ebool_2ECOND_2E3(tyop_2Esum_2Esum(A_27d,A_27c),V0P_2E0,c_2Esum_2EINR_2E1(A_27d,A_27c,V4x_2E0),c_2Esum_2EINL_2E1(A_27d,A_27c,V5y_2E0)) = c_2Esum_2EINL_2E1(A_27d,A_27c,V6z_2E0) )
<=> ( ~ p(V0P_2E0)
& ( V6z_2E0 = V5y_2E0 ) ) )
& ! [V7x_2E0: A_27e,V8y_2E0: A_27f,V9z_2E0: A_27e] :
( ( c_2Ebool_2ECOND_2E3(tyop_2Esum_2Esum(A_27e,A_27f),V0P_2E0,c_2Esum_2EINL_2E1(A_27e,A_27f,V7x_2E0),c_2Esum_2EINR_2E1(A_27e,A_27f,V8y_2E0)) = c_2Esum_2EINL_2E1(A_27e,A_27f,V9z_2E0) )
<=> ( p(V0P_2E0)
& ( V9z_2E0 = V7x_2E0 ) ) )
& ! [V10x_2E0: A_27g,V11y_2E0: A_27h,V12z_2E0: A_27h] :
( ( c_2Ebool_2ECOND_2E3(tyop_2Esum_2Esum(A_27g,A_27h),V0P_2E0,c_2Esum_2EINL_2E1(A_27g,A_27h,V10x_2E0),c_2Esum_2EINR_2E1(A_27g,A_27h,V11y_2E0)) = c_2Esum_2EINR_2E1(A_27g,A_27h,V12z_2E0) )
<=> ( ~ p(V0P_2E0)
& ( V12z_2E0 = V11y_2E0 ) ) ) ) ).
tff(thm_2Esum_2ENOT__ISL__ISR,axiom,
! [A_27a: $tType,A_27b: $tType,V0x_2E0: tyop_2Esum_2Esum(A_27a,A_27b)] :
( ~ p(c_2Esum_2EISL_2E1(A_27a,A_27b,V0x_2E0))
<=> p(c_2Esum_2EISR_2E1(A_27a,A_27b,V0x_2E0)) ) ).
tff(thm_2Esum_2ENOT__ISR__ISL,axiom,
! [A_27a: $tType,A_27b: $tType,V0x_2E0: tyop_2Esum_2Esum(A_27a,A_27b)] :
( ~ p(c_2Esum_2EISR_2E1(A_27a,A_27b,V0x_2E0))
<=> p(c_2Esum_2EISL_2E1(A_27a,A_27b,V0x_2E0)) ) ).
tff(thm_2Esum_2ESUM__ALL__MONO,axiom,
! [A_27a: $tType,A_27b: $tType,V0s_2E0: tyop_2Esum_2Esum(A_27a,A_27b),V1Q_27_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V2Q_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V3P_27_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V4P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( ! [V5x_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V4P_2E0,V5x_2E0))
=> p(app_2E2(A_27a,tyop_2Emin_2Ebool,V3P_27_2E0,V5x_2E0)) )
& ! [V6y_2E0: A_27b] :
( p(app_2E2(A_27b,tyop_2Emin_2Ebool,V2Q_2E0,V6y_2E0))
=> p(app_2E2(A_27b,tyop_2Emin_2Ebool,V1Q_27_2E0,V6y_2E0)) ) )
=> ( p(c_2Esum_2ESUM__ALL_2E3(A_27a,A_27b,V4P_2E0,V2Q_2E0,V0s_2E0))
=> p(c_2Esum_2ESUM__ALL_2E3(A_27a,A_27b,V3P_27_2E0,V1Q_27_2E0,V0s_2E0)) ) ) ).
tff(thm_2Esum_2ESUM__ALL__CONG,axiom,
! [A_27a: $tType,A_27b: $tType,V0s_2E0: tyop_2Esum_2Esum(A_27a,A_27b),V1s_27_2E0: tyop_2Esum_2Esum(A_27a,A_27b),V2P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3P_27_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V4Q_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V5Q_27_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
( ( ( V0s_2E0 = V1s_27_2E0 )
& ! [V6a_2E0: A_27a] :
( ( V1s_27_2E0 = c_2Esum_2EINL_2E1(A_27a,A_27b,V6a_2E0) )
=> ( app_2E2(A_27a,tyop_2Emin_2Ebool,V2P_2E0,V6a_2E0) = app_2E2(A_27a,tyop_2Emin_2Ebool,V3P_27_2E0,V6a_2E0) ) )
& ! [V7b_2E0: A_27b] :
( ( V1s_27_2E0 = c_2Esum_2EINR_2E1(A_27a,A_27b,V7b_2E0) )
=> ( app_2E2(A_27b,tyop_2Emin_2Ebool,V4Q_2E0,V7b_2E0) = app_2E2(A_27b,tyop_2Emin_2Ebool,V5Q_27_2E0,V7b_2E0) ) ) )
=> ( c_2Esum_2ESUM__ALL_2E3(A_27a,A_27b,V2P_2E0,V4Q_2E0,V0s_2E0) = c_2Esum_2ESUM__ALL_2E3(A_27a,A_27b,V3P_27_2E0,V5Q_27_2E0,V1s_27_2E0) ) ) ).
tff(thm_2Esum_2Edatatype__sum,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0sum_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Esum_2Esum(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Esum_2Esum(A_27a,A_27b)),A_27c))] : p(c_2Ebool_2EDATATYPE_2E1(A_27c,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Esum_2Esum(A_27a,A_27b)),A_27c,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Esum_2Esum(A_27a,A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Esum_2Esum(A_27a,A_27b)),A_27c),V0sum_2E0,c_2Esum_2EINL_2E0(A_27a,A_27b)),c_2Esum_2EINR_2E0(A_27a,A_27b)))) ).
%------------------------------------------------------------------------------