ITP001 Axioms: ITP102^5.ax
%------------------------------------------------------------------------------
% File : ITP102^5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : sptree^2.ax [Gau20]
% : HL4102^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 257 ( 15 unt; 44 typ; 0 def)
% Number of atoms : 6863 ( 347 equ; 0 cnn)
% Maximal formula atoms : 273 ( 26 avg)
% Number of connectives : 12892 ( 30 ~; 9 |; 246 &;11908 @)
% ( 41 <=>; 658 =>; 0 <=; 0 <~>)
% Maximal formula depth : 59 ( 12 avg;11908 nst)
% Number of types : 2 ( 1 usr)
% Number of type conns : 48 ( 48 >; 0 *; 0 +; 0 <<)
% Number of symbols : 135 ( 132 usr; 95 con; 0-2 aty)
% Number of variables : 1155 ( 152 ^ 987 !; 16 ?;1155 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_ty_2Esptree_2Espt,type,
ty_2Esptree_2Espt: del > del ).
thf(tp_c_2Esptree_2EBN,type,
c_2Esptree_2EBN: del > $i ).
thf(mem_c_2Esptree_2EBN,axiom,
! [A_27a: del] : ( mem @ ( c_2Esptree_2EBN @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( ty_2Esptree_2Espt @ A_27a ) ) ) ) ).
thf(tp_c_2Esptree_2EBS,type,
c_2Esptree_2EBS: del > $i ).
thf(mem_c_2Esptree_2EBS,axiom,
! [A_27a: del] : ( mem @ ( c_2Esptree_2EBS @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( ty_2Esptree_2Espt @ A_27a ) ) ) ) ) ).
thf(tp_c_2Esptree_2ELN,type,
c_2Esptree_2ELN: del > $i ).
thf(mem_c_2Esptree_2ELN,axiom,
! [A_27a: del] : ( mem @ ( c_2Esptree_2ELN @ A_27a ) @ ( ty_2Esptree_2Espt @ A_27a ) ) ).
thf(tp_c_2Esptree_2ELS,type,
c_2Esptree_2ELS: del > $i ).
thf(mem_c_2Esptree_2ELS,axiom,
! [A_27a: del] : ( mem @ ( c_2Esptree_2ELS @ A_27a ) @ ( arr @ A_27a @ ( ty_2Esptree_2Espt @ A_27a ) ) ) ).
thf(tp_c_2Esptree_2Edelete,type,
c_2Esptree_2Edelete: del > $i ).
thf(mem_c_2Esptree_2Edelete,axiom,
! [A_27a: del] : ( mem @ ( c_2Esptree_2Edelete @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( ty_2Esptree_2Espt @ A_27a ) ) ) ) ).
thf(tp_c_2Esptree_2Edifference,type,
c_2Esptree_2Edifference: del > del > $i ).
thf(mem_c_2Esptree_2Edifference,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Esptree_2Edifference @ A_27a @ A_27b ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27b ) @ ( ty_2Esptree_2Espt @ A_27a ) ) ) ) ).
thf(tp_c_2Esptree_2Edomain,type,
c_2Esptree_2Edomain: del > $i ).
thf(mem_c_2Esptree_2Edomain,axiom,
! [A_27a: del] : ( mem @ ( c_2Esptree_2Edomain @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) ).
thf(tp_c_2Esptree_2Efilter__v,type,
c_2Esptree_2Efilter__v: del > $i ).
thf(mem_c_2Esptree_2Efilter__v,axiom,
! [A_27a: del] : ( mem @ ( c_2Esptree_2Efilter__v @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( ty_2Esptree_2Espt @ A_27a ) ) ) ) ).
thf(tp_c_2Esptree_2Efoldi,type,
c_2Esptree_2Efoldi: del > del > $i ).
thf(mem_c_2Esptree_2Efoldi,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Esptree_2Efoldi @ A_27a @ A_27b ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ ( arr @ A_27a @ A_27a ) ) ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ( arr @ ( ty_2Esptree_2Espt @ A_27b ) @ A_27a ) ) ) ) ) ).
thf(tp_c_2Esptree_2EfromAList,type,
c_2Esptree_2EfromAList: del > $i ).
thf(mem_c_2Esptree_2EfromAList,axiom,
! [A_27a: del] : ( mem @ ( c_2Esptree_2EfromAList @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) @ ( ty_2Esptree_2Espt @ A_27a ) ) ) ).
thf(tp_c_2Esptree_2EfromList,type,
c_2Esptree_2EfromList: del > $i ).
thf(mem_c_2Esptree_2EfromList,axiom,
! [A_27a: del] : ( mem @ ( c_2Esptree_2EfromList @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Esptree_2Espt @ A_27a ) ) ) ).
thf(tp_c_2Esptree_2Einsert,type,
c_2Esptree_2Einsert: del > $i ).
thf(mem_c_2Esptree_2Einsert,axiom,
! [A_27a: del] : ( mem @ ( c_2Esptree_2Einsert @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( ty_2Esptree_2Espt @ A_27a ) ) ) ) ) ).
thf(tp_c_2Esptree_2Einter,type,
c_2Esptree_2Einter: del > del > $i ).
thf(mem_c_2Esptree_2Einter,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Esptree_2Einter @ A_27a @ A_27b ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27b ) @ ( ty_2Esptree_2Espt @ A_27a ) ) ) ) ).
thf(tp_c_2Esptree_2Einter__eq,type,
c_2Esptree_2Einter__eq: del > $i ).
thf(mem_c_2Esptree_2Einter__eq,axiom,
! [A_27a: del] : ( mem @ ( c_2Esptree_2Einter__eq @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( ty_2Esptree_2Espt @ A_27a ) ) ) ) ).
thf(stp_c_ty_2Esptree_2Espt_ty_2Eone_2Eone,type,
tp__c_ty_2Esptree_2Espt_ty_2Eone_2Eone: $tType ).
thf(stp_inj_c_ty_2Esptree_2Espt_ty_2Eone_2Eone,type,
inj__c_ty_2Esptree_2Espt_ty_2Eone_2Eone: tp__c_ty_2Esptree_2Espt_ty_2Eone_2Eone > $i ).
thf(stp_surj_c_ty_2Esptree_2Espt_ty_2Eone_2Eone,type,
surj__c_ty_2Esptree_2Espt_ty_2Eone_2Eone: $i > tp__c_ty_2Esptree_2Espt_ty_2Eone_2Eone ).
thf(stp_inj_surj_c_ty_2Esptree_2Espt_ty_2Eone_2Eone,axiom,
! [X: tp__c_ty_2Esptree_2Espt_ty_2Eone_2Eone] :
( ( surj__c_ty_2Esptree_2Espt_ty_2Eone_2Eone @ ( inj__c_ty_2Esptree_2Espt_ty_2Eone_2Eone @ X ) )
= X ) ).
thf(stp_inj_mem_c_ty_2Esptree_2Espt_ty_2Eone_2Eone,axiom,
! [X: tp__c_ty_2Esptree_2Espt_ty_2Eone_2Eone] : ( mem @ ( inj__c_ty_2Esptree_2Espt_ty_2Eone_2Eone @ X ) @ ( ty_2Esptree_2Espt @ ty_2Eone_2Eone ) ) ).
thf(stp_iso_mem_c_ty_2Esptree_2Espt_ty_2Eone_2Eone,axiom,
! [X: $i] :
( ( mem @ X @ ( ty_2Esptree_2Espt @ ty_2Eone_2Eone ) )
=> ( X
= ( inj__c_ty_2Esptree_2Espt_ty_2Eone_2Eone @ ( surj__c_ty_2Esptree_2Espt_ty_2Eone_2Eone @ X ) ) ) ) ).
thf(tp_c_2Esptree_2Elist__insert,type,
c_2Esptree_2Elist__insert: $i ).
thf(mem_c_2Esptree_2Elist__insert,axiom,
mem @ c_2Esptree_2Elist__insert @ ( arr @ ( ty_2Elist_2Elist @ ty_2Enum_2Enum ) @ ( arr @ ( ty_2Esptree_2Espt @ ty_2Eone_2Eone ) @ ( ty_2Esptree_2Espt @ ty_2Eone_2Eone ) ) ) ).
thf(tp_c_2Esptree_2Elist__to__num__set,type,
c_2Esptree_2Elist__to__num__set: $i ).
thf(mem_c_2Esptree_2Elist__to__num__set,axiom,
mem @ c_2Esptree_2Elist__to__num__set @ ( arr @ ( ty_2Elist_2Elist @ ty_2Enum_2Enum ) @ ( ty_2Esptree_2Espt @ ty_2Eone_2Eone ) ) ).
thf(tp_c_2Esptree_2Elookup,type,
c_2Esptree_2Elookup: del > $i ).
thf(mem_c_2Esptree_2Elookup,axiom,
! [A_27a: del] : ( mem @ ( c_2Esptree_2Elookup @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) ) ).
thf(tp_c_2Esptree_2Elrnext,type,
c_2Esptree_2Elrnext: $i ).
thf(mem_c_2Esptree_2Elrnext,axiom,
mem @ c_2Esptree_2Elrnext @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ).
thf(stp_fo_c_2Esptree_2Elrnext,type,
fo__c_2Esptree_2Elrnext: tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Esptree_2Elrnext,axiom,
! [X0: tp__ty_2Enum_2Enum] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Esptree_2Elrnext @ X0 ) )
= ( ap @ c_2Esptree_2Elrnext @ ( inj__ty_2Enum_2Enum @ X0 ) ) ) ).
thf(tp_c_2Esptree_2Emap,type,
c_2Esptree_2Emap: del > del > $i ).
thf(mem_c_2Esptree_2Emap,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Esptree_2Emap @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27b ) @ ( ty_2Esptree_2Espt @ A_27a ) ) ) ) ).
thf(tp_c_2Esptree_2Emapi,type,
c_2Esptree_2Emapi: del > del > $i ).
thf(mem_c_2Esptree_2Emapi,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Esptree_2Emapi @ A_27a @ A_27b ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ A_27a ) ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27b ) @ ( ty_2Esptree_2Espt @ A_27a ) ) ) ) ).
thf(tp_c_2Esptree_2Emapi0,type,
c_2Esptree_2Emapi0: del > del > $i ).
thf(mem_c_2Esptree_2Emapi0,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Esptree_2Emapi0 @ A_27a @ A_27b ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ A_27a ) ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Esptree_2Espt @ A_27b ) @ ( ty_2Esptree_2Espt @ A_27a ) ) ) ) ) ).
thf(tp_c_2Esptree_2Emk__BN,type,
c_2Esptree_2Emk__BN: del > $i ).
thf(mem_c_2Esptree_2Emk__BN,axiom,
! [A_27a: del] : ( mem @ ( c_2Esptree_2Emk__BN @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( ty_2Esptree_2Espt @ A_27a ) ) ) ) ).
thf(tp_c_2Esptree_2Emk__BS,type,
c_2Esptree_2Emk__BS: del > $i ).
thf(mem_c_2Esptree_2Emk__BS,axiom,
! [A_27a: del] : ( mem @ ( c_2Esptree_2Emk__BS @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( ty_2Esptree_2Espt @ A_27a ) ) ) ) ) ).
thf(tp_c_2Esptree_2Emk__wf,type,
c_2Esptree_2Emk__wf: del > $i ).
thf(mem_c_2Esptree_2Emk__wf,axiom,
! [A_27a: del] : ( mem @ ( c_2Esptree_2Emk__wf @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( ty_2Esptree_2Espt @ A_27a ) ) ) ).
thf(tp_c_2Esptree_2Esize,type,
c_2Esptree_2Esize: del > $i ).
thf(mem_c_2Esptree_2Esize,axiom,
! [A_27a: del] : ( mem @ ( c_2Esptree_2Esize @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ty_2Enum_2Enum ) ) ).
thf(tp_c_2Esptree_2Espt__CASE,type,
c_2Esptree_2Espt__CASE: del > del > $i ).
thf(mem_c_2Esptree_2Espt__CASE,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Esptree_2Espt__CASE @ A_27a @ A_27b ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ A_27b @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ A_27b ) ) @ ( arr @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ A_27b ) ) ) @ A_27b ) ) ) ) ) ) ).
thf(tp_c_2Esptree_2Espt__acc,type,
c_2Esptree_2Espt__acc: $i ).
thf(mem_c_2Esptree_2Espt__acc,axiom,
mem @ c_2Esptree_2Espt__acc @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) ).
thf(stp_fo_c_2Esptree_2Espt__acc,type,
fo__c_2Esptree_2Espt__acc: tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Esptree_2Espt__acc,axiom,
! [X0: tp__ty_2Enum_2Enum,X1: tp__ty_2Enum_2Enum] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Esptree_2Espt__acc @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2Esptree_2Espt__acc @ ( inj__ty_2Enum_2Enum @ X0 ) ) @ ( inj__ty_2Enum_2Enum @ X1 ) ) ) ).
thf(tp_c_2Esptree_2Espt__center,type,
c_2Esptree_2Espt__center: del > $i ).
thf(mem_c_2Esptree_2Espt__center,axiom,
! [A_27a: del] : ( mem @ ( c_2Esptree_2Espt__center @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) ).
thf(tp_c_2Esptree_2Espt__fold,type,
c_2Esptree_2Espt__fold: del > del > $i ).
thf(mem_c_2Esptree_2Espt__fold,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Esptree_2Espt__fold @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ ( ty_2Esptree_2Espt @ A_27b ) @ A_27a ) ) ) ) ).
thf(tp_c_2Esptree_2Espt__left,type,
c_2Esptree_2Espt__left: del > $i ).
thf(mem_c_2Esptree_2Espt__left,axiom,
! [A_27a: del] : ( mem @ ( c_2Esptree_2Espt__left @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( ty_2Esptree_2Espt @ A_27a ) ) ) ).
thf(tp_c_2Esptree_2Espt__right,type,
c_2Esptree_2Espt__right: del > $i ).
thf(mem_c_2Esptree_2Espt__right,axiom,
! [A_27a: del] : ( mem @ ( c_2Esptree_2Espt__right @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( ty_2Esptree_2Espt @ A_27a ) ) ) ).
thf(tp_c_2Esptree_2Espt__size,type,
c_2Esptree_2Espt__size: del > $i ).
thf(mem_c_2Esptree_2Espt__size,axiom,
! [A_27a: del] : ( mem @ ( c_2Esptree_2Espt__size @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ty_2Enum_2Enum ) ) ) ).
thf(tp_c_2Esptree_2Esubspt,type,
c_2Esptree_2Esubspt: del > $i ).
thf(mem_c_2Esptree_2Esubspt,axiom,
! [A_27a: del] : ( mem @ ( c_2Esptree_2Esubspt @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ bool ) ) ) ).
thf(tp_c_2Esptree_2EtoAList,type,
c_2Esptree_2EtoAList: del > $i ).
thf(mem_c_2Esptree_2EtoAList,axiom,
! [A_27a: del] : ( mem @ ( c_2Esptree_2EtoAList @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) ) ) ).
thf(tp_c_2Esptree_2EtoList,type,
c_2Esptree_2EtoList: del > $i ).
thf(mem_c_2Esptree_2EtoList,axiom,
! [A_27a: del] : ( mem @ ( c_2Esptree_2EtoList @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ).
thf(tp_c_2Esptree_2EtoListA,type,
c_2Esptree_2EtoListA: del > $i ).
thf(mem_c_2Esptree_2EtoListA,axiom,
! [A_27a: del] : ( mem @ ( c_2Esptree_2EtoListA @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ).
thf(tp_c_2Esptree_2Eunion,type,
c_2Esptree_2Eunion: del > $i ).
thf(mem_c_2Esptree_2Eunion,axiom,
! [A_27a: del] : ( mem @ ( c_2Esptree_2Eunion @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( ty_2Esptree_2Espt @ A_27a ) ) ) ) ).
thf(tp_c_2Esptree_2Ewf,type,
c_2Esptree_2Ewf: del > $i ).
thf(mem_c_2Esptree_2Ewf,axiom,
! [A_27a: del] : ( mem @ ( c_2Esptree_2Ewf @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ bool ) ) ).
thf(ax_thm_2Esptree_2Espt__TY__DEF,axiom,
! [A_27a: del] :
? [V0rep: $i] :
( ( mem @ V0rep @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( ty_2Eind__type_2Erecspace @ A_27a ) ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Ebool_2ETYPE__DEFINITION @ ( ty_2Eind__type_2Erecspace @ A_27a ) @ ( ty_2Esptree_2Espt @ A_27a ) )
@ ( lam @ ( ty_2Eind__type_2Erecspace @ A_27a )
@ ^ [V1a0_27: $i] :
( ap @ ( c_2Ebool_2E_21 @ ( arr @ ( ty_2Eind__type_2Erecspace @ A_27a ) @ bool ) )
@ ( lam @ ( arr @ ( ty_2Eind__type_2Erecspace @ A_27a ) @ bool )
@ ^ [V2_27spt_27: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap @ ( c_2Ebool_2E_21 @ ( ty_2Eind__type_2Erecspace @ A_27a ) )
@ ( lam @ ( ty_2Eind__type_2Erecspace @ A_27a )
@ ^ [V3a0_27: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap
@ ( ap @ c_2Ebool_2E_5C_2F
@ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eind__type_2Erecspace @ A_27a ) ) @ V3a0_27 )
@ ( ap @ ( ap @ ( ap @ ( c_2Eind__type_2ECONSTR @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( c_2Ebool_2EARB @ A_27a ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( c_2Eind__type_2EBOTTOM @ A_27a ) ) ) ) )
@ ( ap
@ ( ap @ c_2Ebool_2E_5C_2F
@ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V5a: $i] :
( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eind__type_2Erecspace @ A_27a ) ) @ V3a0_27 )
@ ( ap
@ ( lam @ A_27a
@ ^ [V6a: $i] :
( ap @ ( ap @ ( ap @ ( c_2Eind__type_2ECONSTR @ A_27a ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V6a )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V7n: $i] : ( c_2Eind__type_2EBOTTOM @ A_27a ) ) ) )
@ V5a ) ) ) ) )
@ ( ap
@ ( ap @ c_2Ebool_2E_5C_2F
@ ( ap @ ( c_2Ebool_2E_3F @ ( ty_2Eind__type_2Erecspace @ A_27a ) )
@ ( lam @ ( ty_2Eind__type_2Erecspace @ A_27a )
@ ^ [V8a0: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Eind__type_2Erecspace @ A_27a ) )
@ ( lam @ ( ty_2Eind__type_2Erecspace @ A_27a )
@ ^ [V9a1: $i] :
( ap
@ ( ap @ c_2Ebool_2E_2F_5C
@ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eind__type_2Erecspace @ A_27a ) ) @ V3a0_27 )
@ ( ap
@ ( ap
@ ( lam @ ( ty_2Eind__type_2Erecspace @ A_27a )
@ ^ [V10a0: $i] :
( lam @ ( ty_2Eind__type_2Erecspace @ A_27a )
@ ^ [V11a1: $i] :
( ap @ ( ap @ ( ap @ ( c_2Eind__type_2ECONSTR @ A_27a ) @ ( ap @ c_2Enum_2ESUC @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) @ ( c_2Ebool_2EARB @ A_27a ) )
@ ( ap @ ( ap @ ( c_2Eind__type_2EFCONS @ ( ty_2Eind__type_2Erecspace @ A_27a ) ) @ V10a0 )
@ ( ap @ ( ap @ ( c_2Eind__type_2EFCONS @ ( ty_2Eind__type_2Erecspace @ A_27a ) ) @ V11a1 )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V12n: $i] : ( c_2Eind__type_2EBOTTOM @ A_27a ) ) ) ) ) ) )
@ V8a0 )
@ V9a1 ) ) )
@ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V2_27spt_27 @ V8a0 ) ) @ ( ap @ V2_27spt_27 @ V9a1 ) ) ) ) ) ) ) )
@ ( ap @ ( c_2Ebool_2E_3F @ ( ty_2Eind__type_2Erecspace @ A_27a ) )
@ ( lam @ ( ty_2Eind__type_2Erecspace @ A_27a )
@ ^ [V13a0: $i] :
( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V14a1: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Eind__type_2Erecspace @ A_27a ) )
@ ( lam @ ( ty_2Eind__type_2Erecspace @ A_27a )
@ ^ [V15a2: $i] :
( ap
@ ( ap @ c_2Ebool_2E_2F_5C
@ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eind__type_2Erecspace @ A_27a ) ) @ V3a0_27 )
@ ( ap
@ ( ap
@ ( ap
@ ( lam @ ( ty_2Eind__type_2Erecspace @ A_27a )
@ ^ [V16a0: $i] :
( lam @ A_27a
@ ^ [V17a1: $i] :
( lam @ ( ty_2Eind__type_2Erecspace @ A_27a )
@ ^ [V18a2: $i] :
( ap @ ( ap @ ( ap @ ( c_2Eind__type_2ECONSTR @ A_27a ) @ ( ap @ c_2Enum_2ESUC @ ( ap @ c_2Enum_2ESUC @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) @ V17a1 )
@ ( ap @ ( ap @ ( c_2Eind__type_2EFCONS @ ( ty_2Eind__type_2Erecspace @ A_27a ) ) @ V16a0 )
@ ( ap @ ( ap @ ( c_2Eind__type_2EFCONS @ ( ty_2Eind__type_2Erecspace @ A_27a ) ) @ V18a2 )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V19n: $i] : ( c_2Eind__type_2EBOTTOM @ A_27a ) ) ) ) ) ) ) )
@ V13a0 )
@ V14a1 )
@ V15a2 ) ) )
@ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V2_27spt_27 @ V13a0 ) ) @ ( ap @ V2_27spt_27 @ V15a2 ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( ap @ V2_27spt_27 @ V3a0_27 ) ) ) ) )
@ ( ap @ V2_27spt_27 @ V1a0_27 ) ) ) ) ) )
@ V0rep ) ) ) ).
thf(ax_thm_2Esptree_2Espt__case__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0v: $i] :
( ( mem @ V0v @ A_27b )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2f1: $i] :
( ( mem @ V2f1 @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ A_27b ) ) )
=> ! [V3f2: $i] :
( ( mem @ V3f2 @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ A_27b ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Espt__CASE @ A_27a @ A_27b ) @ ( c_2Esptree_2ELN @ A_27a ) ) @ V0v ) @ V1f ) @ V2f1 ) @ V3f2 )
= V0v ) ) ) ) )
& ! [V4a: $i] :
( ( mem @ V4a @ A_27a )
=> ! [V5v: $i] :
( ( mem @ V5v @ A_27b )
=> ! [V6f: $i] :
( ( mem @ V6f @ ( arr @ A_27a @ A_27b ) )
=> ! [V7f1: $i] :
( ( mem @ V7f1 @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ A_27b ) ) )
=> ! [V8f2: $i] :
( ( mem @ V8f2 @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ A_27b ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Espt__CASE @ A_27a @ A_27b ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V4a ) ) @ V5v ) @ V6f ) @ V7f1 ) @ V8f2 )
= ( ap @ V6f @ V4a ) ) ) ) ) ) )
& ! [V9a0: $i] :
( ( mem @ V9a0 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V10a1: $i] :
( ( mem @ V10a1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V11v: $i] :
( ( mem @ V11v @ A_27b )
=> ! [V12f: $i] :
( ( mem @ V12f @ ( arr @ A_27a @ A_27b ) )
=> ! [V13f1: $i] :
( ( mem @ V13f1 @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ A_27b ) ) )
=> ! [V14f2: $i] :
( ( mem @ V14f2 @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ A_27b ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Espt__CASE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V9a0 ) @ V10a1 ) ) @ V11v ) @ V12f ) @ V13f1 ) @ V14f2 )
= ( ap @ ( ap @ V13f1 @ V9a0 ) @ V10a1 ) ) ) ) ) ) ) )
& ! [V15a0: $i] :
( ( mem @ V15a0 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V16a1: $i] :
( ( mem @ V16a1 @ A_27a )
=> ! [V17a2: $i] :
( ( mem @ V17a2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V18v: $i] :
( ( mem @ V18v @ A_27b )
=> ! [V19f: $i] :
( ( mem @ V19f @ ( arr @ A_27a @ A_27b ) )
=> ! [V20f1: $i] :
( ( mem @ V20f1 @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ A_27b ) ) )
=> ! [V21f2: $i] :
( ( mem @ V21f2 @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ A_27b ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Espt__CASE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V15a0 ) @ V16a1 ) @ V17a2 ) ) @ V18v ) @ V19f ) @ V20f1 ) @ V21f2 )
= ( ap @ ( ap @ ( ap @ V21f2 @ V15a0 ) @ V16a1 ) @ V17a2 ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Esptree_2Espt__size__def,axiom,
! [A_27a: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Esptree_2Espt__size @ A_27a ) @ V0f ) @ ( c_2Esptree_2ELN @ A_27a ) ) )
= fo__c_2Enum_2E0 ) )
& ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Esptree_2Espt__size @ A_27a ) @ V1f ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V2a ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ V1f @ V2a ) ) ) ) ) )
& ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V4a0: $i] :
( ( mem @ V4a0 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V5a1: $i] :
( ( mem @ V5a1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Esptree_2Espt__size @ A_27a ) @ V3f ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V4a0 ) @ V5a1 ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ ( c_2Esptree_2Espt__size @ A_27a ) @ V3f ) @ V4a0 ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Espt__size @ A_27a ) @ V3f ) @ V5a1 ) ) ) ) ) ) ) )
& ! [V6f: $i] :
( ( mem @ V6f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V7a0: $i] :
( ( mem @ V7a0 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V8a1: $i] :
( ( mem @ V8a1 @ A_27a )
=> ! [V9a2: $i] :
( ( mem @ V9a2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Esptree_2Espt__size @ A_27a ) @ V6f ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V7a0 ) @ V8a1 ) @ V9a2 ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ ( c_2Esptree_2Espt__size @ A_27a ) @ V6f ) @ V7a0 ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ V6f @ V8a1 ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Espt__size @ A_27a ) @ V6f ) @ V9a2 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Edatatype__spt,axiom,
! [A_27a: del,V0spt: $i] :
( ( mem @ V0spt @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( ty_2Esptree_2Espt @ A_27a ) ) @ ( arr @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( ty_2Esptree_2Espt @ A_27a ) ) ) @ ( arr @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( ty_2Esptree_2Espt @ A_27a ) ) ) ) @ bool ) ) ) ) )
=> ( p @ ( ap @ ( c_2Ebool_2EDATATYPE @ bool ) @ ( ap @ ( ap @ ( ap @ ( ap @ V0spt @ ( c_2Esptree_2ELN @ A_27a ) ) @ ( c_2Esptree_2ELS @ A_27a ) ) @ ( c_2Esptree_2EBN @ A_27a ) ) @ ( c_2Esptree_2EBS @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Espt__11,axiom,
! [A_27a: del] :
( ! [V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ! [V1a_27: $i] :
( ( mem @ V1a_27 @ A_27a )
=> ( ( ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V0a )
= ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V1a_27 ) )
<=> ( V0a = V1a_27 ) ) ) )
& ! [V2a0: $i] :
( ( mem @ V2a0 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V3a1: $i] :
( ( mem @ V3a1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V4a0_27: $i] :
( ( mem @ V4a0_27 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V5a1_27: $i] :
( ( mem @ V5a1_27 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V2a0 ) @ V3a1 )
= ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V4a0_27 ) @ V5a1_27 ) )
<=> ( ( V2a0 = V4a0_27 )
& ( V3a1 = V5a1_27 ) ) ) ) ) ) )
& ! [V6a0: $i] :
( ( mem @ V6a0 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V7a1: $i] :
( ( mem @ V7a1 @ A_27a )
=> ! [V8a2: $i] :
( ( mem @ V8a2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V9a0_27: $i] :
( ( mem @ V9a0_27 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V10a1_27: $i] :
( ( mem @ V10a1_27 @ A_27a )
=> ! [V11a2_27: $i] :
( ( mem @ V11a2_27 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V6a0 ) @ V7a1 ) @ V8a2 )
= ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V9a0_27 ) @ V10a1_27 ) @ V11a2_27 ) )
<=> ( ( V6a0 = V9a0_27 )
& ( V7a1 = V10a1_27 )
& ( V8a2 = V11a2_27 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Espt__distinct,axiom,
! [A_27a: del] :
( ! [V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ( ( c_2Esptree_2ELN @ A_27a )
!= ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V0a ) ) )
& ! [V1a1: $i] :
( ( mem @ V1a1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V2a0: $i] :
( ( mem @ V2a0 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( c_2Esptree_2ELN @ A_27a )
!= ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V2a0 ) @ V1a1 ) ) ) )
& ! [V3a2: $i] :
( ( mem @ V3a2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V4a1: $i] :
( ( mem @ V4a1 @ A_27a )
=> ! [V5a0: $i] :
( ( mem @ V5a0 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( c_2Esptree_2ELN @ A_27a )
!= ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V5a0 ) @ V4a1 ) @ V3a2 ) ) ) ) )
& ! [V6a1: $i] :
( ( mem @ V6a1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V7a0: $i] :
( ( mem @ V7a0 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V8a: $i] :
( ( mem @ V8a @ A_27a )
=> ( ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V8a )
!= ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V7a0 ) @ V6a1 ) ) ) ) )
& ! [V9a2: $i] :
( ( mem @ V9a2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V10a1: $i] :
( ( mem @ V10a1 @ A_27a )
=> ! [V11a0: $i] :
( ( mem @ V11a0 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V12a: $i] :
( ( mem @ V12a @ A_27a )
=> ( ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V12a )
!= ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V11a0 ) @ V10a1 ) @ V9a2 ) ) ) ) ) )
& ! [V13a2: $i] :
( ( mem @ V13a2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V14a1_27: $i] :
( ( mem @ V14a1_27 @ A_27a )
=> ! [V15a1: $i] :
( ( mem @ V15a1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V16a0_27: $i] :
( ( mem @ V16a0_27 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V17a0: $i] :
( ( mem @ V17a0 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V17a0 ) @ V15a1 )
!= ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V16a0_27 ) @ V14a1_27 ) @ V13a2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Espt__nchotomy,axiom,
! [A_27a: del,V0ss: $i] :
( ( mem @ V0ss @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( V0ss
= ( c_2Esptree_2ELN @ A_27a ) )
| ? [V1a: $i] :
( ( mem @ V1a @ A_27a )
& ( V0ss
= ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V1a ) ) )
| ? [V2s: $i] :
( ( mem @ V2s @ ( ty_2Esptree_2Espt @ A_27a ) )
& ? [V3s0: $i] :
( ( mem @ V3s0 @ ( ty_2Esptree_2Espt @ A_27a ) )
& ( V0ss
= ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V2s ) @ V3s0 ) ) ) )
| ? [V4s: $i] :
( ( mem @ V4s @ ( ty_2Esptree_2Espt @ A_27a ) )
& ? [V5a: $i] :
( ( mem @ V5a @ A_27a )
& ? [V6s0: $i] :
( ( mem @ V6s0 @ ( ty_2Esptree_2Espt @ A_27a ) )
& ( V0ss
= ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V4s ) @ V5a ) @ V6s0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Espt__Axiom,axiom,
! [A_27a: del,A_27b: del,V0f0: $i] :
( ( mem @ V0f0 @ A_27b )
=> ! [V1f1: $i] :
( ( mem @ V1f1 @ ( arr @ A_27a @ A_27b ) )
=> ! [V2f2: $i] :
( ( mem @ V2f2 @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ A_27b @ ( arr @ A_27b @ A_27b ) ) ) ) )
=> ! [V3f3: $i] :
( ( mem @ V3f3 @ ( arr @ A_27a @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ A_27b @ ( arr @ A_27b @ A_27b ) ) ) ) ) )
=> ? [V4fn: $i] :
( ( mem @ V4fn @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ A_27b ) )
& ( ( ap @ V4fn @ ( c_2Esptree_2ELN @ A_27a ) )
= V0f0 )
& ! [V5a: $i] :
( ( mem @ V5a @ A_27a )
=> ( ( ap @ V4fn @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V5a ) )
= ( ap @ V1f1 @ V5a ) ) )
& ! [V6a0: $i] :
( ( mem @ V6a0 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V7a1: $i] :
( ( mem @ V7a1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ V4fn @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V6a0 ) @ V7a1 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ V2f2 @ V6a0 ) @ V7a1 ) @ ( ap @ V4fn @ V6a0 ) ) @ ( ap @ V4fn @ V7a1 ) ) ) ) )
& ! [V8a0: $i] :
( ( mem @ V8a0 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V9a1: $i] :
( ( mem @ V9a1 @ A_27a )
=> ! [V10a2: $i] :
( ( mem @ V10a2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ V4fn @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V8a0 ) @ V9a1 ) @ V10a2 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ V3f3 @ V9a1 ) @ V8a0 ) @ V10a2 ) @ ( ap @ V4fn @ V8a0 ) ) @ ( ap @ V4fn @ V10a2 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Espt__induction,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ bool ) )
=> ( ( ( p @ ( ap @ V0P @ ( c_2Esptree_2ELN @ A_27a ) ) )
& ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ( p @ ( ap @ V0P @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V1a ) ) ) )
& ! [V2s: $i] :
( ( mem @ V2s @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V3s0: $i] :
( ( mem @ V3s0 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ( p @ ( ap @ V0P @ V2s ) )
& ( p @ ( ap @ V0P @ V3s0 ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V2s ) @ V3s0 ) ) ) ) ) )
& ! [V4s: $i] :
( ( mem @ V4s @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V5s0: $i] :
( ( mem @ V5s0 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ( p @ ( ap @ V0P @ V4s ) )
& ( p @ ( ap @ V0P @ V5s0 ) ) )
=> ! [V6a: $i] :
( ( mem @ V6a @ A_27a )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V4s ) @ V6a ) @ V5s0 ) ) ) ) ) ) ) )
=> ! [V7s: $i] :
( ( mem @ V7s @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( p @ ( ap @ V0P @ V7s ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Espt__case__cong,axiom,
! [A_27a: del,A_27b: del,V0v_27: $i] :
( ( mem @ V0v_27 @ A_27b )
=> ! [V1f_27: $i] :
( ( mem @ V1f_27 @ ( arr @ A_27a @ A_27b ) )
=> ! [V2f1_27: $i] :
( ( mem @ V2f1_27 @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ A_27b ) ) )
=> ! [V3f2_27: $i] :
( ( mem @ V3f2_27 @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ A_27b ) ) ) )
=> ! [V4M: $i] :
( ( mem @ V4M @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V5M_27: $i] :
( ( mem @ V5M_27 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V6v: $i] :
( ( mem @ V6v @ A_27b )
=> ! [V7f: $i] :
( ( mem @ V7f @ ( arr @ A_27a @ A_27b ) )
=> ! [V8f1: $i] :
( ( mem @ V8f1 @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ A_27b ) ) )
=> ! [V9f2: $i] :
( ( mem @ V9f2 @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ A_27b ) ) ) )
=> ( ( ( V4M = V5M_27 )
& ( ( V5M_27
= ( c_2Esptree_2ELN @ A_27a ) )
=> ( V6v = V0v_27 ) )
& ! [V10a: $i] :
( ( mem @ V10a @ A_27a )
=> ( ( V5M_27
= ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V10a ) )
=> ( ( ap @ V7f @ V10a )
= ( ap @ V1f_27 @ V10a ) ) ) )
& ! [V11a0: $i] :
( ( mem @ V11a0 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V12a1: $i] :
( ( mem @ V12a1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( V5M_27
= ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V11a0 ) @ V12a1 ) )
=> ( ( ap @ ( ap @ V8f1 @ V11a0 ) @ V12a1 )
= ( ap @ ( ap @ V2f1_27 @ V11a0 ) @ V12a1 ) ) ) ) )
& ! [V13a0: $i] :
( ( mem @ V13a0 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V14a1: $i] :
( ( mem @ V14a1 @ A_27a )
=> ! [V15a2: $i] :
( ( mem @ V15a2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( V5M_27
= ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V13a0 ) @ V14a1 ) @ V15a2 ) )
=> ( ( ap @ ( ap @ ( ap @ V9f2 @ V13a0 ) @ V14a1 ) @ V15a2 )
= ( ap @ ( ap @ ( ap @ V3f2_27 @ V13a0 ) @ V14a1 ) @ V15a2 ) ) ) ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Espt__CASE @ A_27a @ A_27b ) @ V4M ) @ V6v ) @ V7f ) @ V8f1 ) @ V9f2 )
= ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Espt__CASE @ A_27a @ A_27b ) @ V5M_27 ) @ V0v_27 ) @ V1f_27 ) @ V2f1_27 ) @ V3f2_27 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Espt__case__eq,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1v: $i] :
( ( mem @ V1v @ A_27b )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ! [V3f1: $i] :
( ( mem @ V3f1 @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ A_27b ) ) )
=> ! [V4f2: $i] :
( ( mem @ V4f2 @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ A_27b ) ) ) )
=> ! [V5v_27: $i] :
( ( mem @ V5v_27 @ A_27b )
=> ( ( ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Espt__CASE @ A_27a @ A_27b ) @ V0x ) @ V1v ) @ V2f ) @ V3f1 ) @ V4f2 )
= V5v_27 )
<=> ( ( ( V0x
= ( c_2Esptree_2ELN @ A_27a ) )
& ( V1v = V5v_27 ) )
| ? [V6a: $i] :
( ( mem @ V6a @ A_27a )
& ( V0x
= ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V6a ) )
& ( ( ap @ V2f @ V6a )
= V5v_27 ) )
| ? [V7s: $i] :
( ( mem @ V7s @ ( ty_2Esptree_2Espt @ A_27a ) )
& ? [V8s0: $i] :
( ( mem @ V8s0 @ ( ty_2Esptree_2Espt @ A_27a ) )
& ( V0x
= ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V7s ) @ V8s0 ) )
& ( ( ap @ ( ap @ V3f1 @ V7s ) @ V8s0 )
= V5v_27 ) ) )
| ? [V9s: $i] :
( ( mem @ V9s @ ( ty_2Esptree_2Espt @ A_27a ) )
& ? [V10a: $i] :
( ( mem @ V10a @ A_27a )
& ? [V11s0: $i] :
( ( mem @ V11s0 @ ( ty_2Esptree_2Espt @ A_27a ) )
& ( V0x
= ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V9s ) @ V10a ) @ V11s0 ) )
& ( ( ap @ ( ap @ ( ap @ V4f2 @ V9s ) @ V10a ) @ V11s0 )
= V5v_27 ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Esptree_2Ewf__def,axiom,
! [A_27a: del] :
( ( ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ ( c_2Esptree_2ELN @ A_27a ) ) )
<=> $true )
& ! [V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ( ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V0a ) ) )
<=> $true ) )
& ! [V1t1: $i] :
( ( mem @ V1t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V2t2: $i] :
( ( mem @ V2t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V1t1 ) @ V2t2 ) ) )
<=> ( ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ V1t1 ) )
& ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ V2t2 ) )
& ~ ( ( V1t1
= ( c_2Esptree_2ELN @ A_27a ) )
& ( V2t2
= ( c_2Esptree_2ELN @ A_27a ) ) ) ) ) ) )
& ! [V3t1: $i] :
( ( mem @ V3t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V4a: $i] :
( ( mem @ V4a @ A_27a )
=> ! [V5t2: $i] :
( ( mem @ V5t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V3t1 ) @ V4a ) @ V5t2 ) ) )
<=> ( ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ V3t1 ) )
& ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ V5t2 ) )
& ~ ( ( V3t1
= ( c_2Esptree_2ELN @ A_27a ) )
& ( V5t2
= ( c_2Esptree_2ELN @ A_27a ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Elookup__ind,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ bool ) ) )
=> ( ( ! [V1k: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ ( c_2Esptree_2ELN @ A_27a ) ) )
& ! [V2k: tp__ty_2Enum_2Enum,V3a: $i] :
( ( mem @ V3a @ A_27a )
=> ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V3a ) ) ) )
& ! [V4k: tp__ty_2Enum_2Enum,V5t1: $i] :
( ( mem @ V5t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V6t2: $i] :
( ( mem @ V6t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ( V4k != fo__c_2Enum_2E0 )
=> ( p @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V4k ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esptree_2Espt @ A_27a ) ) @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V4k ) ) ) @ V5t1 ) @ V6t2 ) ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V4k ) ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V5t1 ) @ V6t2 ) ) ) ) ) )
& ! [V7k: tp__ty_2Enum_2Enum,V8t1: $i] :
( ( mem @ V8t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V9a: $i] :
( ( mem @ V9a @ A_27a )
=> ! [V10t2: $i] :
( ( mem @ V10t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ( V7k != fo__c_2Enum_2E0 )
=> ( p @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V7k ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esptree_2Espt @ A_27a ) ) @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V7k ) ) ) @ V8t1 ) @ V10t2 ) ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V7k ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V8t1 ) @ V9a ) @ V10t2 ) ) ) ) ) ) ) )
=> ! [V11v: tp__ty_2Enum_2Enum,V12v1: $i] :
( ( mem @ V12v1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V11v ) ) @ V12v1 ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Elookup__def,axiom,
! [A_27a: del] :
( ! [V0k: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Eoption_2ENONE @ A_27a ) )
& ! [V1k: tp__ty_2Enum_2Enum,V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V2a ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V2a ) ) @ ( c_2Eoption_2ENONE @ A_27a ) ) ) )
& ! [V3t2: $i] :
( ( mem @ V3t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V4t1: $i] :
( ( mem @ V4t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V5k: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V5k ) ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V4t1 ) @ V3t2 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V5k ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( c_2Eoption_2ENONE @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V5k ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esptree_2Espt @ A_27a ) ) @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V5k ) ) ) @ V4t1 ) @ V3t2 ) ) ) ) ) )
& ! [V6t2: $i] :
( ( mem @ V6t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V7t1: $i] :
( ( mem @ V7t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V8k: tp__ty_2Enum_2Enum,V9a: $i] :
( ( mem @ V9a @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V8k ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V7t1 ) @ V9a ) @ V6t2 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V8k ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V9a ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V8k ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esptree_2Espt @ A_27a ) ) @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V8k ) ) ) @ V7t1 ) @ V6t2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Einsert__ind,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ bool ) ) ) )
=> ( ( ! [V1k: tp__ty_2Enum_2Enum,V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( ( ( ( ( V1k != fo__c_2Enum_2E0 )
& ( p @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V1k ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V2a ) @ ( c_2Esptree_2ELN @ A_27a ) ) ) )
& ( ( ( V1k != fo__c_2Enum_2E0 )
& ~ ( p @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V1k ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V2a ) @ ( c_2Esptree_2ELN @ A_27a ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ V2a ) @ ( c_2Esptree_2ELN @ A_27a ) ) ) ) )
& ! [V3k: tp__ty_2Enum_2Enum,V4a: $i] :
( ( mem @ V4a @ A_27a )
=> ! [V5a_27: $i] :
( ( mem @ V5a_27 @ A_27a )
=> ( ( ( ( ( V3k != fo__c_2Enum_2E0 )
& ( p @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V3k ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V3k ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V4a ) @ ( c_2Esptree_2ELN @ A_27a ) ) ) )
& ( ( ( V3k != fo__c_2Enum_2E0 )
& ~ ( p @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V3k ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V3k ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V4a ) @ ( c_2Esptree_2ELN @ A_27a ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V3k ) ) @ V4a ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V5a_27 ) ) ) ) ) )
& ! [V6k: tp__ty_2Enum_2Enum,V7a: $i] :
( ( mem @ V7a @ A_27a )
=> ! [V8t1: $i] :
( ( mem @ V8t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V9t2: $i] :
( ( mem @ V9t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ( ( ( V6k != fo__c_2Enum_2E0 )
& ( p @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V6k ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V6k ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V7a ) @ V8t1 ) ) )
& ( ( ( V6k != fo__c_2Enum_2E0 )
& ~ ( p @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V6k ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V6k ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V7a ) @ V9t2 ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V6k ) ) @ V7a ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V8t1 ) @ V9t2 ) ) ) ) ) ) )
& ! [V10k: tp__ty_2Enum_2Enum,V11a: $i] :
( ( mem @ V11a @ A_27a )
=> ! [V12t1: $i] :
( ( mem @ V12t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V13a_27: $i] :
( ( mem @ V13a_27 @ A_27a )
=> ! [V14t2: $i] :
( ( mem @ V14t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ( ( ( V10k != fo__c_2Enum_2E0 )
& ( p @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V10k ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V10k ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V11a ) @ V12t1 ) ) )
& ( ( ( V10k != fo__c_2Enum_2E0 )
& ~ ( p @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V10k ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V10k ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V11a ) @ V14t2 ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V10k ) ) @ V11a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V12t1 ) @ V13a_27 ) @ V14t2 ) ) ) ) ) ) ) ) )
=> ! [V15v: tp__ty_2Enum_2Enum,V16v1: $i] :
( ( mem @ V16v1 @ A_27a )
=> ! [V17v2: $i] :
( ( mem @ V17v2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V15v ) ) @ V16v1 ) @ V17v2 ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Einsert__def,axiom,
! [A_27a: del] :
( ! [V0k: tp__ty_2Enum_2Enum,V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) @ V1a ) @ ( c_2Esptree_2ELN @ A_27a ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esptree_2Espt @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V1a ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esptree_2Espt @ A_27a ) ) @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V0k ) ) ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V0k ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V1a ) @ ( c_2Esptree_2ELN @ A_27a ) ) ) @ ( c_2Esptree_2ELN @ A_27a ) ) ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ ( c_2Esptree_2ELN @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V0k ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V1a ) @ ( c_2Esptree_2ELN @ A_27a ) ) ) ) ) ) )
& ! [V2k: tp__ty_2Enum_2Enum,V3a_27: $i] :
( ( mem @ V3a_27 @ A_27a )
=> ! [V4a: $i] :
( ( mem @ V4a @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ V4a ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V3a_27 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esptree_2Espt @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V4a ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esptree_2Espt @ A_27a ) ) @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V2k ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V4a ) @ ( c_2Esptree_2ELN @ A_27a ) ) ) @ V3a_27 ) @ ( c_2Esptree_2ELN @ A_27a ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ ( c_2Esptree_2ELN @ A_27a ) ) @ V3a_27 ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V4a ) @ ( c_2Esptree_2ELN @ A_27a ) ) ) ) ) ) ) )
& ! [V5t2: $i] :
( ( mem @ V5t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V6t1: $i] :
( ( mem @ V6t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V7k: tp__ty_2Enum_2Enum,V8a: $i] :
( ( mem @ V8a @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V7k ) ) @ V8a ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V6t1 ) @ V5t2 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esptree_2Espt @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V7k ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V6t1 ) @ V8a ) @ V5t2 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esptree_2Espt @ A_27a ) ) @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V7k ) ) ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V7k ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V8a ) @ V6t1 ) ) @ V5t2 ) ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V6t1 ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V7k ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V8a ) @ V5t2 ) ) ) ) ) ) ) )
& ! [V9t2: $i] :
( ( mem @ V9t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V10t1: $i] :
( ( mem @ V10t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V11k: tp__ty_2Enum_2Enum,V12a_27: $i] :
( ( mem @ V12a_27 @ A_27a )
=> ! [V13a: $i] :
( ( mem @ V13a @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V11k ) ) @ V13a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V10t1 ) @ V12a_27 ) @ V9t2 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esptree_2Espt @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V11k ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V10t1 ) @ V13a ) @ V9t2 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esptree_2Espt @ A_27a ) ) @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V11k ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V11k ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V13a ) @ V10t1 ) ) @ V12a_27 ) @ V9t2 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V10t1 ) @ V12a_27 ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V11k ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V13a ) @ V9t2 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Emk__BN__ind,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ bool ) ) )
=> ( ( ( p @ ( ap @ ( ap @ V0P @ ( c_2Esptree_2ELN @ A_27a ) ) @ ( c_2Esptree_2ELN @ A_27a ) ) )
& ! [V1v14: $i] :
( ( mem @ V1v14 @ A_27a )
=> ( p @ ( ap @ ( ap @ V0P @ ( c_2Esptree_2ELN @ A_27a ) ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V1v14 ) ) ) )
& ! [V2v15: $i] :
( ( mem @ V2v15 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V3v16: $i] :
( ( mem @ V3v16 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( c_2Esptree_2ELN @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V2v15 ) @ V3v16 ) ) ) ) )
& ! [V4v17: $i] :
( ( mem @ V4v17 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V5v18: $i] :
( ( mem @ V5v18 @ A_27a )
=> ! [V6v19: $i] :
( ( mem @ V6v19 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( c_2Esptree_2ELN @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V4v17 ) @ V5v18 ) @ V6v19 ) ) ) ) ) )
& ! [V7v2: $i] :
( ( mem @ V7v2 @ A_27a )
=> ! [V8t2: $i] :
( ( mem @ V8t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V7v2 ) ) @ V8t2 ) ) ) )
& ! [V9v3: $i] :
( ( mem @ V9v3 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V10v4: $i] :
( ( mem @ V10v4 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V11t2: $i] :
( ( mem @ V11t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V9v3 ) @ V10v4 ) ) @ V11t2 ) ) ) ) )
& ! [V12v5: $i] :
( ( mem @ V12v5 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V13v6: $i] :
( ( mem @ V13v6 @ A_27a )
=> ! [V14v7: $i] :
( ( mem @ V14v7 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V15t2: $i] :
( ( mem @ V15t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V12v5 ) @ V13v6 ) @ V14v7 ) ) @ V15t2 ) ) ) ) ) ) )
=> ! [V16v: $i] :
( ( mem @ V16v @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V17v1: $i] :
( ( mem @ V17v1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( p @ ( ap @ ( ap @ V0P @ V16v ) @ V17v1 ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Emk__BN__def,axiom,
! [A_27a: del,V0v14: $i] :
( ( mem @ V0v14 @ A_27a )
=> ! [V1v15: $i] :
( ( mem @ V1v15 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V2v16: $i] :
( ( mem @ V2v16 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V3v17: $i] :
( ( mem @ V3v17 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V4v18: $i] :
( ( mem @ V4v18 @ A_27a )
=> ! [V5v19: $i] :
( ( mem @ V5v19 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V6v2: $i] :
( ( mem @ V6v2 @ A_27a )
=> ! [V7t2: $i] :
( ( mem @ V7t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V8v3: $i] :
( ( mem @ V8v3 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V9v4: $i] :
( ( mem @ V9v4 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V10v5: $i] :
( ( mem @ V10v5 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V11v6: $i] :
( ( mem @ V11v6 @ A_27a )
=> ! [V12v7: $i] :
( ( mem @ V12v7 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Esptree_2Emk__BN @ A_27a ) @ ( c_2Esptree_2ELN @ A_27a ) ) @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Esptree_2ELN @ A_27a ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Emk__BN @ A_27a ) @ ( c_2Esptree_2ELN @ A_27a ) ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V0v14 ) )
= ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ ( c_2Esptree_2ELN @ A_27a ) ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V0v14 ) ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Emk__BN @ A_27a ) @ ( c_2Esptree_2ELN @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V1v15 ) @ V2v16 ) )
= ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ ( c_2Esptree_2ELN @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V1v15 ) @ V2v16 ) ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Emk__BN @ A_27a ) @ ( c_2Esptree_2ELN @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V3v17 ) @ V4v18 ) @ V5v19 ) )
= ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ ( c_2Esptree_2ELN @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V3v17 ) @ V4v18 ) @ V5v19 ) ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Emk__BN @ A_27a ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V6v2 ) ) @ V7t2 )
= ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V6v2 ) ) @ V7t2 ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Emk__BN @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V8v3 ) @ V9v4 ) ) @ V7t2 )
= ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V8v3 ) @ V9v4 ) ) @ V7t2 ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Emk__BN @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V10v5 ) @ V11v6 ) @ V12v7 ) ) @ V7t2 )
= ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V10v5 ) @ V11v6 ) @ V12v7 ) ) @ V7t2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Emk__BS__ind,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ bool ) ) ) )
=> ( ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ ( c_2Esptree_2ELN @ A_27a ) ) @ V1x ) @ ( c_2Esptree_2ELN @ A_27a ) ) ) )
& ! [V2v16: $i] :
( ( mem @ V2v16 @ A_27a )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V2v16 ) ) @ V3x ) @ ( c_2Esptree_2ELN @ A_27a ) ) ) ) )
& ! [V4v17: $i] :
( ( mem @ V4v17 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V5v18: $i] :
( ( mem @ V5v18 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V4v17 ) @ V5v18 ) ) @ V6x ) @ ( c_2Esptree_2ELN @ A_27a ) ) ) ) ) )
& ! [V7v19: $i] :
( ( mem @ V7v19 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V8v20: $i] :
( ( mem @ V8v20 @ A_27a )
=> ! [V9v21: $i] :
( ( mem @ V9v21 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V10x: $i] :
( ( mem @ V10x @ A_27a )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V7v19 ) @ V8v20 ) @ V9v21 ) ) @ V10x ) @ ( c_2Esptree_2ELN @ A_27a ) ) ) ) ) ) )
& ! [V11t1: $i] :
( ( mem @ V11t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V12x: $i] :
( ( mem @ V12x @ A_27a )
=> ! [V13v4: $i] :
( ( mem @ V13v4 @ A_27a )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V11t1 ) @ V12x ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V13v4 ) ) ) ) ) )
& ! [V14t1: $i] :
( ( mem @ V14t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V15x: $i] :
( ( mem @ V15x @ A_27a )
=> ! [V16v5: $i] :
( ( mem @ V16v5 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V17v6: $i] :
( ( mem @ V17v6 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V14t1 ) @ V15x ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V16v5 ) @ V17v6 ) ) ) ) ) ) )
& ! [V18t1: $i] :
( ( mem @ V18t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V19x: $i] :
( ( mem @ V19x @ A_27a )
=> ! [V20v7: $i] :
( ( mem @ V20v7 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V21v8: $i] :
( ( mem @ V21v8 @ A_27a )
=> ! [V22v9: $i] :
( ( mem @ V22v9 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V18t1 ) @ V19x ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V20v7 ) @ V21v8 ) @ V22v9 ) ) ) ) ) ) ) ) )
=> ! [V23v: $i] :
( ( mem @ V23v @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V24v1: $i] :
( ( mem @ V24v1 @ A_27a )
=> ! [V25v2: $i] :
( ( mem @ V25v2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V23v ) @ V24v1 ) @ V25v2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Emk__BS__def,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1v16: $i] :
( ( mem @ V1v16 @ A_27a )
=> ! [V2v17: $i] :
( ( mem @ V2v17 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V3v18: $i] :
( ( mem @ V3v18 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V4v19: $i] :
( ( mem @ V4v19 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V5v20: $i] :
( ( mem @ V5v20 @ A_27a )
=> ! [V6v21: $i] :
( ( mem @ V6v21 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V7t1: $i] :
( ( mem @ V7t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V8v4: $i] :
( ( mem @ V8v4 @ A_27a )
=> ! [V9v5: $i] :
( ( mem @ V9v5 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V10v6: $i] :
( ( mem @ V10v6 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V11v7: $i] :
( ( mem @ V11v7 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V12v8: $i] :
( ( mem @ V12v8 @ A_27a )
=> ! [V13v9: $i] :
( ( mem @ V13v9 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Emk__BS @ A_27a ) @ ( c_2Esptree_2ELN @ A_27a ) ) @ V0x ) @ ( c_2Esptree_2ELN @ A_27a ) )
= ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V0x ) )
& ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Emk__BS @ A_27a ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V1v16 ) ) @ V0x ) @ ( c_2Esptree_2ELN @ A_27a ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V1v16 ) ) @ V0x ) @ ( c_2Esptree_2ELN @ A_27a ) ) )
& ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Emk__BS @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V2v17 ) @ V3v18 ) ) @ V0x ) @ ( c_2Esptree_2ELN @ A_27a ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V2v17 ) @ V3v18 ) ) @ V0x ) @ ( c_2Esptree_2ELN @ A_27a ) ) )
& ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Emk__BS @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V4v19 ) @ V5v20 ) @ V6v21 ) ) @ V0x ) @ ( c_2Esptree_2ELN @ A_27a ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V4v19 ) @ V5v20 ) @ V6v21 ) ) @ V0x ) @ ( c_2Esptree_2ELN @ A_27a ) ) )
& ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Emk__BS @ A_27a ) @ V7t1 ) @ V0x ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V8v4 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V7t1 ) @ V0x ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V8v4 ) ) )
& ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Emk__BS @ A_27a ) @ V7t1 ) @ V0x ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V9v5 ) @ V10v6 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V7t1 ) @ V0x ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V9v5 ) @ V10v6 ) ) )
& ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Emk__BS @ A_27a ) @ V7t1 ) @ V0x ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V11v7 ) @ V12v8 ) @ V13v9 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V7t1 ) @ V0x ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V11v7 ) @ V12v8 ) @ V13v9 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Esptree_2Edelete__def,axiom,
! [A_27a: del] :
( ! [V0k: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Esptree_2ELN @ A_27a ) )
& ! [V1k: tp__ty_2Enum_2Enum,V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V2a ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esptree_2Espt @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( c_2Esptree_2ELN @ A_27a ) ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V2a ) ) ) )
& ! [V3k: tp__ty_2Enum_2Enum,V4t1: $i] :
( ( mem @ V4t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V5t2: $i] :
( ( mem @ V5t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V3k ) ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V4t1 ) @ V5t2 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esptree_2Espt @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V3k ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V4t1 ) @ V5t2 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esptree_2Espt @ A_27a ) ) @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V3k ) ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Emk__BN @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V3k ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V4t1 ) ) @ V5t2 ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Emk__BN @ A_27a ) @ V4t1 ) @ ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V3k ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V5t2 ) ) ) ) ) ) )
& ! [V6k: tp__ty_2Enum_2Enum,V7t1: $i] :
( ( mem @ V7t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V8a: $i] :
( ( mem @ V8a @ A_27a )
=> ! [V9t2: $i] :
( ( mem @ V9t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V6k ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V7t1 ) @ V8a ) @ V9t2 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esptree_2Espt @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V6k ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V7t1 ) @ V9t2 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esptree_2Espt @ A_27a ) ) @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V6k ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Emk__BS @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V6k ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V7t1 ) ) @ V8a ) @ V9t2 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Emk__BS @ A_27a ) @ V7t1 ) @ V8a ) @ ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V6k ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V9t2 ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Esptree_2EfromList__def,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Esptree_2EfromList @ A_27a ) @ V0l )
= ( ap @ ( c_2Epair_2ESND @ ty_2Enum_2Enum @ ( ty_2Esptree_2Espt @ A_27a ) )
@ ( ap
@ ( ap
@ ( ap @ ( c_2Elist_2EFOLDL @ A_27a @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Esptree_2Espt @ A_27a ) ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Enum_2Enum @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ A_27a @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Esptree_2Espt @ A_27a ) ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1i: $i] :
( lam @ ( ty_2Esptree_2Espt @ A_27a )
@ ^ [V2t: $i] :
( lam @ A_27a
@ ^ [V3a: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Esptree_2Espt @ A_27a ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V1i ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ V1i ) @ V3a ) @ V2t ) ) ) ) ) ) )
@ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Esptree_2Espt @ A_27a ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( c_2Esptree_2ELN @ A_27a ) ) )
@ V0l ) ) ) ) ).
thf(ax_thm_2Esptree_2Esize__def,axiom,
! [A_27a: del] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Esptree_2Esize @ A_27a ) @ ( c_2Esptree_2ELN @ A_27a ) ) )
= fo__c_2Enum_2E0 )
& ! [V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Esptree_2Esize @ A_27a ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V0a ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
& ! [V1t1: $i] :
( ( mem @ V1t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V2t2: $i] :
( ( mem @ V2t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Esptree_2Esize @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V1t1 ) @ V2t2 ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( c_2Esptree_2Esize @ A_27a ) @ V1t1 ) ) @ ( ap @ ( c_2Esptree_2Esize @ A_27a ) @ V2t2 ) ) ) ) ) )
& ! [V3t1: $i] :
( ( mem @ V3t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V4a: $i] :
( ( mem @ V4a @ A_27a )
=> ! [V5t2: $i] :
( ( mem @ V5t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Esptree_2Esize @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V3t1 ) @ V4a ) @ V5t2 ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( c_2Esptree_2Esize @ A_27a ) @ V3t1 ) ) @ ( ap @ ( c_2Esptree_2Esize @ A_27a ) @ V5t2 ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Einsert__notEmpty,axiom,
! [A_27a: del,V0k: tp__ty_2Enum_2Enum,V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) @ V1a ) @ V2t )
!= ( c_2Esptree_2ELN @ A_27a ) ) ) ) ).
thf(conj_thm_2Esptree_2Ewf__insert,axiom,
! [A_27a: del,V0k: tp__ty_2Enum_2Enum,V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ V2t ) )
=> ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) @ V1a ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Ewf__delete,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1k: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ V0t ) )
=> ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ V0t ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Elookup__insert1,axiom,
! [A_27a: del,V0k: tp__ty_2Enum_2Enum,V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) @ V1a ) @ V2t ) )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V1a ) ) ) ) ).
thf(conj_thm_2Esptree_2Elookup__insert,axiom,
! [A_27a: del,V0k2: tp__ty_2Enum_2Enum,V1v: $i] :
( ( mem @ V1v @ A_27a )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V3k1: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V3k1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0k2 ) ) @ V1v ) @ V2t ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V3k1 ) ) @ ( inj__ty_2Enum_2Enum @ V0k2 ) ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V1v ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V3k1 ) ) @ V2t ) ) ) ) ) ).
thf(ax_thm_2Esptree_2Eunion__def,axiom,
! [A_27a: del] :
( ! [V0t: $i] :
( ( mem @ V0t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Eunion @ A_27a ) @ ( c_2Esptree_2ELN @ A_27a ) ) @ V0t )
= V0t ) )
& ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Eunion @ A_27a ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V1a ) ) @ V2t )
= ( ap
@ ( ap
@ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Espt__CASE @ A_27a @ ( ty_2Esptree_2Espt @ A_27a ) ) @ V2t ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V1a ) )
@ ( lam @ A_27a
@ ^ [V3b: $i] : ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V1a ) ) )
@ ( lam @ ( ty_2Esptree_2Espt @ A_27a )
@ ^ [V4t1: $i] :
( lam @ ( ty_2Esptree_2Espt @ A_27a )
@ ^ [V5t2: $i] : ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V4t1 ) @ V1a ) @ V5t2 ) ) ) )
@ ( lam @ ( ty_2Esptree_2Espt @ A_27a )
@ ^ [V6t1_27: $i] :
( lam @ A_27a
@ ^ [V7v4: $i] :
( lam @ ( ty_2Esptree_2Espt @ A_27a )
@ ^ [V8t2_27: $i] : ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V6t1_27 ) @ V1a ) @ V8t2_27 ) ) ) ) ) ) ) )
& ! [V9t1: $i] :
( ( mem @ V9t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V10t2: $i] :
( ( mem @ V10t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V11t: $i] :
( ( mem @ V11t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Eunion @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V9t1 ) @ V10t2 ) ) @ V11t )
= ( ap
@ ( ap
@ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Espt__CASE @ A_27a @ ( ty_2Esptree_2Espt @ A_27a ) ) @ V11t ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V9t1 ) @ V10t2 ) )
@ ( lam @ A_27a
@ ^ [V12a: $i] : ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V9t1 ) @ V12a ) @ V10t2 ) ) )
@ ( lam @ ( ty_2Esptree_2Espt @ A_27a )
@ ^ [V13t1_27: $i] :
( lam @ ( ty_2Esptree_2Espt @ A_27a )
@ ^ [V14t2_27: $i] : ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Eunion @ A_27a ) @ V9t1 ) @ V13t1_27 ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Eunion @ A_27a ) @ V10t2 ) @ V14t2_27 ) ) ) ) )
@ ( lam @ ( ty_2Esptree_2Espt @ A_27a )
@ ^ [V15t1_27_27: $i] :
( lam @ A_27a
@ ^ [V16a_27_27: $i] :
( lam @ ( ty_2Esptree_2Espt @ A_27a )
@ ^ [V17t2_27_27: $i] : ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Eunion @ A_27a ) @ V9t1 ) @ V15t1_27_27 ) ) @ V16a_27_27 ) @ ( ap @ ( ap @ ( c_2Esptree_2Eunion @ A_27a ) @ V10t2 ) @ V17t2_27_27 ) ) ) ) ) ) ) ) ) )
& ! [V18t1: $i] :
( ( mem @ V18t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V19a: $i] :
( ( mem @ V19a @ A_27a )
=> ! [V20t2: $i] :
( ( mem @ V20t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V21t: $i] :
( ( mem @ V21t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Eunion @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V18t1 ) @ V19a ) @ V20t2 ) ) @ V21t )
= ( ap
@ ( ap
@ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Espt__CASE @ A_27a @ ( ty_2Esptree_2Espt @ A_27a ) ) @ V21t ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V18t1 ) @ V19a ) @ V20t2 ) )
@ ( lam @ A_27a
@ ^ [V22a_27: $i] : ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V18t1 ) @ V19a ) @ V20t2 ) ) )
@ ( lam @ ( ty_2Esptree_2Espt @ A_27a )
@ ^ [V23t1_27: $i] :
( lam @ ( ty_2Esptree_2Espt @ A_27a )
@ ^ [V24t2_27: $i] : ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Eunion @ A_27a ) @ V18t1 ) @ V23t1_27 ) ) @ V19a ) @ ( ap @ ( ap @ ( c_2Esptree_2Eunion @ A_27a ) @ V20t2 ) @ V24t2_27 ) ) ) ) )
@ ( lam @ ( ty_2Esptree_2Espt @ A_27a )
@ ^ [V25t1_27_27: $i] :
( lam @ A_27a
@ ^ [V26a_27_27_27: $i] :
( lam @ ( ty_2Esptree_2Espt @ A_27a )
@ ^ [V27t2_27_27: $i] : ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Eunion @ A_27a ) @ V18t1 ) @ V25t1_27_27 ) ) @ V19a ) @ ( ap @ ( ap @ ( c_2Esptree_2Eunion @ A_27a ) @ V20t2 ) @ V27t2_27_27 ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2EisEmpty__union,axiom,
! [A_27a: del,V0m1: $i] :
( ( mem @ V0m1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1m2: $i] :
( ( mem @ V1m2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Esptree_2Eunion @ A_27a ) @ V0m1 ) @ V1m2 )
= ( c_2Esptree_2ELN @ A_27a ) )
<=> ( ( V0m1
= ( c_2Esptree_2ELN @ A_27a ) )
& ( V1m2
= ( c_2Esptree_2ELN @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Ewf__union,axiom,
! [A_27a: del,V0m1: $i] :
( ( mem @ V0m1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1m2: $i] :
( ( mem @ V1m2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ V0m1 ) )
& ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ V1m2 ) ) )
=> ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Eunion @ A_27a ) @ V0m1 ) @ V1m2 ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Elookup__union,axiom,
! [A_27a: del,V0m1: $i] :
( ( mem @ V0m1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1m2: $i] :
( ( mem @ V1m2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V2k: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Eunion @ A_27a ) @ V0m1 ) @ V1m2 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Eoption_2Eoption__CASE @ A_27a @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ V0m1 ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ V1m2 ) )
@ ( lam @ A_27a
@ ^ [V3v: $i] : ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V3v ) ) ) ) ) ) ).
thf(ax_thm_2Esptree_2Einter__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0t: $i] :
( ( mem @ V0t @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Einter @ A_27a @ A_27b ) @ ( c_2Esptree_2ELN @ A_27a ) ) @ V0t )
= ( c_2Esptree_2ELN @ A_27a ) ) )
& ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Einter @ A_27a @ A_27b ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V1a ) ) @ V2t )
= ( ap
@ ( ap
@ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Espt__CASE @ A_27b @ ( ty_2Esptree_2Espt @ A_27a ) ) @ V2t ) @ ( c_2Esptree_2ELN @ A_27a ) )
@ ( lam @ A_27b
@ ^ [V3b: $i] : ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V1a ) ) )
@ ( lam @ ( ty_2Esptree_2Espt @ A_27b )
@ ^ [V4t1: $i] :
( lam @ ( ty_2Esptree_2Espt @ A_27b )
@ ^ [V5t2: $i] : ( c_2Esptree_2ELN @ A_27a ) ) ) )
@ ( lam @ ( ty_2Esptree_2Espt @ A_27b )
@ ^ [V6t1_27: $i] :
( lam @ A_27b
@ ^ [V7v4: $i] :
( lam @ ( ty_2Esptree_2Espt @ A_27b )
@ ^ [V8t2_27: $i] : ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V1a ) ) ) ) ) ) ) )
& ! [V9t1: $i] :
( ( mem @ V9t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V10t2: $i] :
( ( mem @ V10t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V11t: $i] :
( ( mem @ V11t @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Einter @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V9t1 ) @ V10t2 ) ) @ V11t )
= ( ap
@ ( ap
@ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Espt__CASE @ A_27b @ ( ty_2Esptree_2Espt @ A_27a ) ) @ V11t ) @ ( c_2Esptree_2ELN @ A_27a ) )
@ ( lam @ A_27b
@ ^ [V12a: $i] : ( c_2Esptree_2ELN @ A_27a ) ) )
@ ( lam @ ( ty_2Esptree_2Espt @ A_27b )
@ ^ [V13t1_27: $i] :
( lam @ ( ty_2Esptree_2Espt @ A_27b )
@ ^ [V14t2_27: $i] : ( ap @ ( ap @ ( c_2Esptree_2Emk__BN @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Einter @ A_27a @ A_27b ) @ V9t1 ) @ V13t1_27 ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Einter @ A_27a @ A_27b ) @ V10t2 ) @ V14t2_27 ) ) ) ) )
@ ( lam @ ( ty_2Esptree_2Espt @ A_27b )
@ ^ [V15t1_27_27: $i] :
( lam @ A_27b
@ ^ [V16a_27_27: $i] :
( lam @ ( ty_2Esptree_2Espt @ A_27b )
@ ^ [V17t2_27_27: $i] : ( ap @ ( ap @ ( c_2Esptree_2Emk__BN @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Einter @ A_27a @ A_27b ) @ V9t1 ) @ V15t1_27_27 ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Einter @ A_27a @ A_27b ) @ V10t2 ) @ V17t2_27_27 ) ) ) ) ) ) ) ) ) )
& ! [V18t1: $i] :
( ( mem @ V18t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V19a: $i] :
( ( mem @ V19a @ A_27a )
=> ! [V20t2: $i] :
( ( mem @ V20t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V21t: $i] :
( ( mem @ V21t @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Einter @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V18t1 ) @ V19a ) @ V20t2 ) ) @ V21t )
= ( ap
@ ( ap
@ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Espt__CASE @ A_27b @ ( ty_2Esptree_2Espt @ A_27a ) ) @ V21t ) @ ( c_2Esptree_2ELN @ A_27a ) )
@ ( lam @ A_27b
@ ^ [V22a_27: $i] : ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V19a ) ) )
@ ( lam @ ( ty_2Esptree_2Espt @ A_27b )
@ ^ [V23t1_27: $i] :
( lam @ ( ty_2Esptree_2Espt @ A_27b )
@ ^ [V24t2_27: $i] : ( ap @ ( ap @ ( c_2Esptree_2Emk__BN @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Einter @ A_27a @ A_27b ) @ V18t1 ) @ V23t1_27 ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Einter @ A_27a @ A_27b ) @ V20t2 ) @ V24t2_27 ) ) ) ) )
@ ( lam @ ( ty_2Esptree_2Espt @ A_27b )
@ ^ [V25t1_27_27: $i] :
( lam @ A_27b
@ ^ [V26a_27_27_27: $i] :
( lam @ ( ty_2Esptree_2Espt @ A_27b )
@ ^ [V27t2_27_27: $i] : ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Emk__BS @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Einter @ A_27a @ A_27b ) @ V18t1 ) @ V25t1_27_27 ) ) @ V19a ) @ ( ap @ ( ap @ ( c_2Esptree_2Einter @ A_27a @ A_27b ) @ V20t2 ) @ V27t2_27_27 ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Esptree_2Einter__eq__def,axiom,
! [A_27a: del] :
( ! [V0t: $i] :
( ( mem @ V0t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Einter__eq @ A_27a ) @ ( c_2Esptree_2ELN @ A_27a ) ) @ V0t )
= ( c_2Esptree_2ELN @ A_27a ) ) )
& ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Einter__eq @ A_27a ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V1a ) ) @ V2t )
= ( ap
@ ( ap
@ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Espt__CASE @ A_27a @ ( ty_2Esptree_2Espt @ A_27a ) ) @ V2t ) @ ( c_2Esptree_2ELN @ A_27a ) )
@ ( lam @ A_27a
@ ^ [V3b: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esptree_2Espt @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V1a ) @ V3b ) ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V1a ) ) @ ( c_2Esptree_2ELN @ A_27a ) ) ) )
@ ( lam @ ( ty_2Esptree_2Espt @ A_27a )
@ ^ [V4t1: $i] :
( lam @ ( ty_2Esptree_2Espt @ A_27a )
@ ^ [V5t2: $i] : ( c_2Esptree_2ELN @ A_27a ) ) ) )
@ ( lam @ ( ty_2Esptree_2Espt @ A_27a )
@ ^ [V6t1_27: $i] :
( lam @ A_27a
@ ^ [V7b_27: $i] :
( lam @ ( ty_2Esptree_2Espt @ A_27a )
@ ^ [V8t2_27: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esptree_2Espt @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V1a ) @ V7b_27 ) ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V1a ) ) @ ( c_2Esptree_2ELN @ A_27a ) ) ) ) ) ) ) ) )
& ! [V9t1: $i] :
( ( mem @ V9t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V10t2: $i] :
( ( mem @ V10t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V11t: $i] :
( ( mem @ V11t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Einter__eq @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V9t1 ) @ V10t2 ) ) @ V11t )
= ( ap
@ ( ap
@ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Espt__CASE @ A_27a @ ( ty_2Esptree_2Espt @ A_27a ) ) @ V11t ) @ ( c_2Esptree_2ELN @ A_27a ) )
@ ( lam @ A_27a
@ ^ [V12a: $i] : ( c_2Esptree_2ELN @ A_27a ) ) )
@ ( lam @ ( ty_2Esptree_2Espt @ A_27a )
@ ^ [V13t1_27: $i] :
( lam @ ( ty_2Esptree_2Espt @ A_27a )
@ ^ [V14t2_27: $i] : ( ap @ ( ap @ ( c_2Esptree_2Emk__BN @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Einter__eq @ A_27a ) @ V9t1 ) @ V13t1_27 ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Einter__eq @ A_27a ) @ V10t2 ) @ V14t2_27 ) ) ) ) )
@ ( lam @ ( ty_2Esptree_2Espt @ A_27a )
@ ^ [V15t1_27_27: $i] :
( lam @ A_27a
@ ^ [V16a_27_27: $i] :
( lam @ ( ty_2Esptree_2Espt @ A_27a )
@ ^ [V17t2_27_27: $i] : ( ap @ ( ap @ ( c_2Esptree_2Emk__BN @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Einter__eq @ A_27a ) @ V9t1 ) @ V15t1_27_27 ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Einter__eq @ A_27a ) @ V10t2 ) @ V17t2_27_27 ) ) ) ) ) ) ) ) ) )
& ! [V18t1: $i] :
( ( mem @ V18t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V19a: $i] :
( ( mem @ V19a @ A_27a )
=> ! [V20t2: $i] :
( ( mem @ V20t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V21t: $i] :
( ( mem @ V21t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Einter__eq @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V18t1 ) @ V19a ) @ V20t2 ) ) @ V21t )
= ( ap
@ ( ap
@ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Espt__CASE @ A_27a @ ( ty_2Esptree_2Espt @ A_27a ) ) @ V21t ) @ ( c_2Esptree_2ELN @ A_27a ) )
@ ( lam @ A_27a
@ ^ [V22a_27: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esptree_2Espt @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V22a_27 ) @ V19a ) ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V19a ) ) @ ( c_2Esptree_2ELN @ A_27a ) ) ) )
@ ( lam @ ( ty_2Esptree_2Espt @ A_27a )
@ ^ [V23t1_27: $i] :
( lam @ ( ty_2Esptree_2Espt @ A_27a )
@ ^ [V24t2_27: $i] : ( ap @ ( ap @ ( c_2Esptree_2Emk__BN @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Einter__eq @ A_27a ) @ V18t1 ) @ V23t1_27 ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Einter__eq @ A_27a ) @ V20t2 ) @ V24t2_27 ) ) ) ) )
@ ( lam @ ( ty_2Esptree_2Espt @ A_27a )
@ ^ [V25t1_27_27: $i] :
( lam @ A_27a
@ ^ [V26a_27_27_27: $i] :
( lam @ ( ty_2Esptree_2Espt @ A_27a )
@ ^ [V27t2_27_27: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esptree_2Espt @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V26a_27_27_27 ) @ V19a ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Emk__BS @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Einter__eq @ A_27a ) @ V18t1 ) @ V25t1_27_27 ) ) @ V19a ) @ ( ap @ ( ap @ ( c_2Esptree_2Einter__eq @ A_27a ) @ V20t2 ) @ V27t2_27_27 ) ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Emk__BN @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Einter__eq @ A_27a ) @ V18t1 ) @ V25t1_27_27 ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Einter__eq @ A_27a ) @ V20t2 ) @ V27t2_27_27 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Esptree_2Edifference__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0t: $i] :
( ( mem @ V0t @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Edifference @ A_27a @ A_27b ) @ ( c_2Esptree_2ELN @ A_27a ) ) @ V0t )
= ( c_2Esptree_2ELN @ A_27a ) ) )
& ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Edifference @ A_27a @ A_27b ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V1a ) ) @ V2t )
= ( ap
@ ( ap
@ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Espt__CASE @ A_27b @ ( ty_2Esptree_2Espt @ A_27a ) ) @ V2t ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V1a ) )
@ ( lam @ A_27b
@ ^ [V3b: $i] : ( c_2Esptree_2ELN @ A_27a ) ) )
@ ( lam @ ( ty_2Esptree_2Espt @ A_27b )
@ ^ [V4t1: $i] :
( lam @ ( ty_2Esptree_2Espt @ A_27b )
@ ^ [V5t2: $i] : ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V1a ) ) ) )
@ ( lam @ ( ty_2Esptree_2Espt @ A_27b )
@ ^ [V6t1_27: $i] :
( lam @ A_27b
@ ^ [V7b_27: $i] :
( lam @ ( ty_2Esptree_2Espt @ A_27b )
@ ^ [V8t2_27: $i] : ( c_2Esptree_2ELN @ A_27a ) ) ) ) ) ) ) )
& ! [V9t1: $i] :
( ( mem @ V9t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V10t2: $i] :
( ( mem @ V10t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V11t: $i] :
( ( mem @ V11t @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Edifference @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V9t1 ) @ V10t2 ) ) @ V11t )
= ( ap
@ ( ap
@ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Espt__CASE @ A_27b @ ( ty_2Esptree_2Espt @ A_27a ) ) @ V11t ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V9t1 ) @ V10t2 ) )
@ ( lam @ A_27b
@ ^ [V12a: $i] : ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V9t1 ) @ V10t2 ) ) )
@ ( lam @ ( ty_2Esptree_2Espt @ A_27b )
@ ^ [V13t1_27: $i] :
( lam @ ( ty_2Esptree_2Espt @ A_27b )
@ ^ [V14t2_27: $i] : ( ap @ ( ap @ ( c_2Esptree_2Emk__BN @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Edifference @ A_27a @ A_27b ) @ V9t1 ) @ V13t1_27 ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Edifference @ A_27a @ A_27b ) @ V10t2 ) @ V14t2_27 ) ) ) ) )
@ ( lam @ ( ty_2Esptree_2Espt @ A_27b )
@ ^ [V15t1_27_27: $i] :
( lam @ A_27b
@ ^ [V16a_27_27: $i] :
( lam @ ( ty_2Esptree_2Espt @ A_27b )
@ ^ [V17t2_27_27: $i] : ( ap @ ( ap @ ( c_2Esptree_2Emk__BN @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Edifference @ A_27a @ A_27b ) @ V9t1 ) @ V15t1_27_27 ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Edifference @ A_27a @ A_27b ) @ V10t2 ) @ V17t2_27_27 ) ) ) ) ) ) ) ) ) )
& ! [V18t1: $i] :
( ( mem @ V18t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V19a: $i] :
( ( mem @ V19a @ A_27a )
=> ! [V20t2: $i] :
( ( mem @ V20t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V21t: $i] :
( ( mem @ V21t @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Edifference @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V18t1 ) @ V19a ) @ V20t2 ) ) @ V21t )
= ( ap
@ ( ap
@ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Espt__CASE @ A_27b @ ( ty_2Esptree_2Espt @ A_27a ) ) @ V21t ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V18t1 ) @ V19a ) @ V20t2 ) )
@ ( lam @ A_27b
@ ^ [V22a_27: $i] : ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V18t1 ) @ V20t2 ) ) )
@ ( lam @ ( ty_2Esptree_2Espt @ A_27b )
@ ^ [V23t1_27: $i] :
( lam @ ( ty_2Esptree_2Espt @ A_27b )
@ ^ [V24t2_27: $i] : ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Emk__BS @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Edifference @ A_27a @ A_27b ) @ V18t1 ) @ V23t1_27 ) ) @ V19a ) @ ( ap @ ( ap @ ( c_2Esptree_2Edifference @ A_27a @ A_27b ) @ V20t2 ) @ V24t2_27 ) ) ) ) )
@ ( lam @ ( ty_2Esptree_2Espt @ A_27b )
@ ^ [V25t1_27_27: $i] :
( lam @ A_27b
@ ^ [V26a_27_27_27: $i] :
( lam @ ( ty_2Esptree_2Espt @ A_27b )
@ ^ [V27t2_27_27: $i] : ( ap @ ( ap @ ( c_2Esptree_2Emk__BN @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Edifference @ A_27a @ A_27b ) @ V18t1 ) @ V25t1_27_27 ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Edifference @ A_27a @ A_27b ) @ V20t2 ) @ V27t2_27_27 ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Ewf__inter,axiom,
! [A_27a: del,A_27b: del,V0m1: $i] :
( ( mem @ V0m1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1m2: $i] :
( ( mem @ V1m2 @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Einter @ A_27a @ A_27b ) @ V0m1 ) @ V1m2 ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Elookup__inter,axiom,
! [A_27a: del,A_27b: del,V0m1: $i] :
( ( mem @ V0m1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1m2: $i] :
( ( mem @ V1m2 @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ! [V2k: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Einter @ A_27a @ A_27b ) @ V0m1 ) @ V1m2 ) )
= ( ap @ ( ap @ ( c_2Epair_2Epair__CASE @ ( ty_2Eoption_2Eoption @ A_27a ) @ ( ty_2Eoption_2Eoption @ A_27a ) @ ( ty_2Eoption_2Eoption @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eoption_2Eoption @ A_27a ) @ ( ty_2Eoption_2Eoption @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ V0m1 ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ V1m2 ) ) )
@ ( lam @ ( ty_2Eoption_2Eoption @ A_27a )
@ ^ [V3v3: $i] :
( lam @ ( ty_2Eoption_2Eoption @ A_27b )
@ ^ [V4v4: $i] :
( ap @ ( ap @ ( ap @ ( c_2Eoption_2Eoption__CASE @ A_27a @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ V3v3 ) @ ( c_2Eoption_2ENONE @ A_27a ) )
@ ( lam @ A_27a
@ ^ [V5v: $i] :
( ap @ ( ap @ ( ap @ ( c_2Eoption_2Eoption__CASE @ A_27b @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ V4v4 ) @ ( c_2Eoption_2ENONE @ A_27a ) )
@ ( lam @ A_27b
@ ^ [V6w: $i] : ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V5v ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Elookup__inter__eq,axiom,
! [A_27a: del,V0m1: $i] :
( ( mem @ V0m1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1m2: $i] :
( ( mem @ V1m2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V2k: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Einter__eq @ A_27a ) @ V0m1 ) @ V1m2 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Eoption_2Eoption__CASE @ A_27a @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ V0m1 ) ) @ ( c_2Eoption_2ENONE @ A_27a ) )
@ ( lam @ A_27a
@ ^ [V3v: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ V1m2 ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V3v ) ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V3v ) ) @ ( c_2Eoption_2ENONE @ A_27a ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Elookup__inter__EQ,axiom,
! [A_27a: del,A_27b: del,V0x: tp__ty_2Enum_2Enum,V1t1: $i] :
( ( mem @ V1t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V2t2: $i] :
( ( mem @ V2t2 @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Einter @ A_27a @ A_27b ) @ V1t1 ) @ V2t2 ) )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V3y ) )
<=> ( ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ V1t1 )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V3y ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ V2t2 )
!= ( c_2Eoption_2ENONE @ A_27b ) ) ) )
& ( ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Einter @ A_27a @ A_27b ) @ V1t1 ) @ V2t2 ) )
= ( c_2Eoption_2ENONE @ A_27a ) )
<=> ( ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ V1t1 )
= ( c_2Eoption_2ENONE @ A_27a ) )
| ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ V2t2 )
= ( c_2Eoption_2ENONE @ A_27b ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Elookup__inter__assoc,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0x: tp__ty_2Enum_2Enum,V1t1: $i] :
( ( mem @ V1t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V2t2: $i] :
( ( mem @ V2t2 @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ! [V3t3: $i] :
( ( mem @ V3t3 @ ( ty_2Esptree_2Espt @ A_27c ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Einter @ A_27a @ A_27b ) @ V1t1 ) @ ( ap @ ( ap @ ( c_2Esptree_2Einter @ A_27b @ A_27c ) @ V2t2 ) @ V3t3 ) ) )
= ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Einter @ A_27a @ A_27c ) @ ( ap @ ( ap @ ( c_2Esptree_2Einter @ A_27a @ A_27b ) @ V1t1 ) @ V2t2 ) ) @ V3t3 ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Elookup__difference,axiom,
! [A_27a: del,A_27b: del,V0m1: $i] :
( ( mem @ V0m1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1m2: $i] :
( ( mem @ V1m2 @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ! [V2k: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Edifference @ A_27a @ A_27b ) @ V0m1 ) @ V1m2 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eoption_2Eoption @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ V1m2 ) ) @ ( c_2Eoption_2ENONE @ A_27b ) ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ V0m1 ) ) @ ( c_2Eoption_2ENONE @ A_27a ) ) ) ) ) ).
thf(ax_thm_2Esptree_2Elrnext__primitive__def,axiom,
( c_2Esptree_2Elrnext
= ( ap
@ ( ap @ ( c_2Erelation_2EWFREC @ ty_2Enum_2Enum @ ty_2Enum_2Enum )
@ ( ap @ ( c_2Emin_2E_40 @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) )
@ ( lam @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) )
@ ^ [V0R: $i] :
( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( c_2Erelation_2EWF @ ty_2Enum_2Enum ) @ V0R ) )
@ ( ap @ ( c_2Ebool_2E_21 @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V1n ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) @ ( ap @ ( ap @ V0R @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V1n ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V1n ) ) ) ) ) ) ) )
@ ( lam @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum )
@ ^ [V2lrnext: $i] :
( lam @ ty_2Enum_2Enum
@ ^ [V3a: $i] : ( ap @ ( c_2Ecombin_2EI @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V3a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ V2lrnext @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V3a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Elrnext__ind,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ! [V1n: tp__ty_2Enum_2Enum] :
( ( ( V1n != fo__c_2Enum_2E0 )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
=> ( p @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
=> ! [V2v: tp__ty_2Enum_2Enum] : ( p @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V2v ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Elrnext__def,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Esptree_2Elrnext @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ c_2Esptree_2Elrnext @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Elrnext__thm,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Esptree_2Elrnext @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
& ! [V1n: tp__ty_2Enum_2Enum,V2a: $i] :
( ( mem @ V2a @ A_27b )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Esptree_2Elrnext @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Esptree_2Elrnext @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Esptree_2Elrnext @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
& ! [V3n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Esptree_2Elrnext @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ c_2Esptree_2Elrnext @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) )
& ! [V4n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Esptree_2Elrnext @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ c_2Esptree_2Elrnext @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) ) ) ).
thf(ax_thm_2Esptree_2Edomain__def,axiom,
! [A_27a: del] :
( ( ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) )
& ! [V0v0: $i] :
( ( mem @ V0v0 @ A_27a )
=> ( ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V0v0 ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) ) ) )
& ! [V1t1: $i] :
( ( mem @ V1t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V2t2: $i] :
( ( mem @ V2t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V1t1 ) @ V2t2 ) )
= ( ap
@ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Enum_2Enum )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3n: $i] : ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ V3n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
@ ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ V1t1 ) ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ V4n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
@ ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ V2t2 ) ) ) ) ) )
& ! [V5t1: $i] :
( ( mem @ V5t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V6v1: $i] :
( ( mem @ V6v1 @ A_27a )
=> ! [V7t2: $i] :
( ( mem @ V7t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V5t1 ) @ V6v1 ) @ V7t2 ) )
= ( ap
@ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Enum_2Enum )
@ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V8n: $i] : ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ V8n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
@ ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ V5t1 ) ) ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V9n: $i] : ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ V9n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
@ ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ V7t2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2EFINITE__domain,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ V0t ) ) ) ) ).
thf(conj_thm_2Esptree_2Esize__insert,axiom,
! [A_27a: del,V0k: tp__ty_2Enum_2Enum,V1v: $i] :
( ( mem @ V1v @ A_27a )
=> ! [V2m: $i] :
( ( mem @ V2m @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Esptree_2Esize @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) @ V1v ) @ V2m ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ V2m ) ) ) @ ( ap @ ( c_2Esptree_2Esize @ A_27a ) @ V2m ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( c_2Esptree_2Esize @ A_27a ) @ V2m ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Elookup__fromList,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ ( c_2Esptree_2EfromList @ A_27a ) @ V1l ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l ) ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V1l ) ) ) @ ( c_2Eoption_2ENONE @ A_27a ) ) ) ) ).
thf(conj_thm_2Esptree_2Edomain__lookup,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1k: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ V0t ) ) )
<=> ? [V2v: $i] :
( ( mem @ V2v @ A_27a )
& ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ V0t )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V2v ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Elookup__inter__alt,axiom,
! [A_27a: del,A_27b: del,V0x: tp__ty_2Enum_2Enum,V1t1: $i] :
( ( mem @ V1t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V2t2: $i] :
( ( mem @ V2t2 @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Einter @ A_27a @ A_27b ) @ V1t1 ) @ V2t2 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27b ) @ V2t2 ) ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ V1t1 ) ) @ ( c_2Eoption_2ENONE @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Elookup__NONE__domain,axiom,
! [A_27a: del,V0k: tp__ty_2Enum_2Enum,V1t: $i] :
( ( mem @ V1t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) @ V1t )
= ( c_2Eoption_2ENONE @ A_27a ) )
<=> ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Edomain__union,axiom,
! [A_27a: del,V0t1: $i] :
( ( mem @ V0t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Eunion @ A_27a ) @ V0t1 ) @ V1t2 ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ V0t1 ) ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ V1t2 ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Edomain__inter,axiom,
! [A_27a: del,A_27b: del,V0t1: $i] :
( ( mem @ V0t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ( ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Einter @ A_27a @ A_27b ) @ V0t1 ) @ V1t2 ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ V0t1 ) ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27b ) @ V1t2 ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Edomain__insert,axiom,
! [A_27a: del,V0k: tp__ty_2Enum_2Enum,V1v: $i] :
( ( mem @ V1v @ A_27a )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) @ V1v ) @ V2t ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ V2t ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Edomain__difference,axiom,
! [A_27a: del,A_27b: del,V0t1: $i] :
( ( mem @ V0t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ( ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Edifference @ A_27a @ A_27b ) @ V0t1 ) @ V1t2 ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ V0t1 ) ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27b ) @ V1t2 ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Edomain__sing,axiom,
! [A_27a: del,V0k: tp__ty_2Enum_2Enum,V1v: $i] :
( ( mem @ V1v @ A_27a )
=> ( ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) @ V1v ) @ ( c_2Esptree_2ELN @ A_27a ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) ) ) ) ).
thf(conj_thm_2Esptree_2Edomain__fromList,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ ( ap @ ( c_2Esptree_2EfromList @ A_27a ) @ V0l ) )
= ( ap @ c_2Epred__set_2Ecount @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l ) ) ) ) ).
thf(conj_thm_2Esptree_2Elookup__delete,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1k1: tp__ty_2Enum_2Enum,V2k2: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1k1 ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2k2 ) ) @ V0t ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1k1 ) ) @ ( inj__ty_2Enum_2Enum @ V2k2 ) ) ) @ ( c_2Eoption_2ENONE @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1k1 ) ) @ V0t ) ) ) ) ).
thf(conj_thm_2Esptree_2Edomain__delete,axiom,
! [A_27a: del,V0k: tp__ty_2Enum_2Enum,V1t: $i] :
( ( mem @ V1t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) @ V1t ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ V1t ) ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) ) ) ).
thf(ax_thm_2Esptree_2Efoldi__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ ( arr @ A_27a @ A_27a ) ) ) )
=> ! [V1i: tp__ty_2Enum_2Enum,V2acc: $i] :
( ( mem @ V2acc @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Efoldi @ A_27a @ A_27b ) @ V0f ) @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ V2acc ) @ ( c_2Esptree_2ELN @ A_27b ) )
= V2acc ) ) )
& ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ ( arr @ A_27a @ A_27a ) ) ) )
=> ! [V4i: tp__ty_2Enum_2Enum,V5acc: $i] :
( ( mem @ V5acc @ A_27a )
=> ! [V6a: $i] :
( ( mem @ V6a @ A_27b )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Efoldi @ A_27a @ A_27b ) @ V3f ) @ ( inj__ty_2Enum_2Enum @ V4i ) ) @ V5acc ) @ ( ap @ ( c_2Esptree_2ELS @ A_27b ) @ V6a ) )
= ( ap @ ( ap @ ( ap @ V3f @ ( inj__ty_2Enum_2Enum @ V4i ) ) @ V6a ) @ V5acc ) ) ) ) )
& ! [V7f: $i] :
( ( mem @ V7f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ ( arr @ A_27a @ A_27a ) ) ) )
=> ! [V8i: tp__ty_2Enum_2Enum,V9acc: $i] :
( ( mem @ V9acc @ A_27a )
=> ! [V10t1: $i] :
( ( mem @ V10t1 @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ! [V11t2: $i] :
( ( mem @ V11t2 @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Efoldi @ A_27a @ A_27b ) @ V7f ) @ ( inj__ty_2Enum_2Enum @ V8i ) ) @ V9acc ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27b ) @ V10t1 ) @ V11t2 ) )
= ( ap
@ ( ap @ ( c_2Ebool_2ELET @ ty_2Enum_2Enum @ A_27a )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V12inc: $i] : ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Efoldi @ A_27a @ A_27b ) @ V7f ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V8i ) ) @ V12inc ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Efoldi @ A_27a @ A_27b ) @ V7f ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V8i ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ V12inc ) ) ) @ V9acc ) @ V10t1 ) ) @ V11t2 ) ) )
@ ( ap @ c_2Esptree_2Elrnext @ ( inj__ty_2Enum_2Enum @ V8i ) ) ) ) ) ) ) )
& ! [V13f: $i] :
( ( mem @ V13f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ ( arr @ A_27a @ A_27a ) ) ) )
=> ! [V14i: tp__ty_2Enum_2Enum,V15acc: $i] :
( ( mem @ V15acc @ A_27a )
=> ! [V16t1: $i] :
( ( mem @ V16t1 @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ! [V17a: $i] :
( ( mem @ V17a @ A_27b )
=> ! [V18t2: $i] :
( ( mem @ V18t2 @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Efoldi @ A_27a @ A_27b ) @ V13f ) @ ( inj__ty_2Enum_2Enum @ V14i ) ) @ V15acc ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27b ) @ V16t1 ) @ V17a ) @ V18t2 ) )
= ( ap
@ ( ap @ ( c_2Ebool_2ELET @ ty_2Enum_2Enum @ A_27a )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V19inc: $i] : ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Efoldi @ A_27a @ A_27b ) @ V13f ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V14i ) ) @ V19inc ) ) @ ( ap @ ( ap @ ( ap @ V13f @ ( inj__ty_2Enum_2Enum @ V14i ) ) @ V17a ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Efoldi @ A_27a @ A_27b ) @ V13f ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V14i ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ V19inc ) ) ) @ V15acc ) @ V16t1 ) ) ) @ V18t2 ) ) )
@ ( ap @ c_2Esptree_2Elrnext @ ( inj__ty_2Enum_2Enum @ V14i ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Espt__acc__ind,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) )
=> ( ( ! [V1i: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
& ! [V2i: tp__ty_2Enum_2Enum,V3k: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2EEVEN @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3k ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ c_2Esptree_2Elrnext @ ( inj__ty_2Enum_2Enum @ V2i ) ) ) ) @ ( ap @ c_2Esptree_2Elrnext @ ( inj__ty_2Enum_2Enum @ V2i ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( inj__ty_2Enum_2Enum @ V3k ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3k ) ) ) ) ) )
=> ! [V4v: tp__ty_2Enum_2Enum,V5v1: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V4v ) ) @ ( inj__ty_2Enum_2Enum @ V5v1 ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Espt__acc__def,axiom,
( ! [V0i: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Esptree_2Espt__acc @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= V0i )
& ! [V1k: tp__ty_2Enum_2Enum,V2i: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Esptree_2Espt__acc @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1k ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Esptree_2Espt__acc @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2EEVEN @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1k ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ c_2Esptree_2Elrnext @ ( inj__ty_2Enum_2Enum @ V2i ) ) ) ) @ ( ap @ c_2Esptree_2Elrnext @ ( inj__ty_2Enum_2Enum @ V2i ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Espt__acc__def__compute,axiom,
( ! [V0i: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Esptree_2Espt__acc @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
= V0i )
& ! [V1k: tp__ty_2Enum_2Enum,V2i: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Esptree_2Espt__acc @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1k ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Esptree_2Espt__acc @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2EEVEN @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1k ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ c_2Esptree_2Elrnext @ ( inj__ty_2Enum_2Enum @ V2i ) ) ) ) @ ( ap @ c_2Esptree_2Elrnext @ ( inj__ty_2Enum_2Enum @ V2i ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1k ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
& ! [V3k: tp__ty_2Enum_2Enum,V4i: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Esptree_2Espt__acc @ ( inj__ty_2Enum_2Enum @ V4i ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V3k ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Esptree_2Espt__acc @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V4i ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2EEVEN @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V3k ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ c_2Esptree_2Elrnext @ ( inj__ty_2Enum_2Enum @ V4i ) ) ) ) @ ( ap @ c_2Esptree_2Elrnext @ ( inj__ty_2Enum_2Enum @ V4i ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V3k ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Espt__acc__thm,axiom,
! [V0i: tp__ty_2Enum_2Enum,V1k: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Esptree_2Espt__acc @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( ap @ ( ap @ c_2Esptree_2Espt__acc @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V1k ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ c_2Esptree_2Elrnext @ ( inj__ty_2Enum_2Enum @ V0i ) ) ) ) @ ( ap @ c_2Esptree_2Elrnext @ ( inj__ty_2Enum_2Enum @ V0i ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Espt__acc__eqn,axiom,
! [V0k: tp__ty_2Enum_2Enum,V1i: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Esptree_2Espt__acc @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Esptree_2Elrnext @ ( inj__ty_2Enum_2Enum @ V1i ) ) ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) ) @ ( inj__ty_2Enum_2Enum @ V1i ) ) ) ) ).
thf(conj_thm_2Esptree_2Espt__acc__0,axiom,
! [V0k: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Esptree_2Espt__acc @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) )
= V0k ) ).
thf(conj_thm_2Esptree_2Eset__foldi__keys,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V2i: tp__ty_2Enum_2Enum] :
( ( ap
@ ( ap
@ ( ap
@ ( ap @ ( c_2Esptree_2Efoldi @ ( arr @ ty_2Enum_2Enum @ bool ) @ A_27a )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3k: $i] :
( lam @ A_27a
@ ^ [V4v: $i] :
( lam @ ( arr @ ty_2Enum_2Enum @ bool )
@ ^ [V5a: $i] : ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Enum_2Enum ) @ V3k ) @ V5a ) ) ) ) )
@ ( inj__ty_2Enum_2Enum @ V2i ) )
@ V1a )
@ V0t )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Enum_2Enum ) @ V1a )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V6n: $i] : ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Esptree_2Elrnext @ ( inj__ty_2Enum_2Enum @ V2i ) ) ) @ V6n ) ) ) )
@ ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ V0t ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Edomain__foldi,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ V0t )
= ( ap
@ ( ap
@ ( ap
@ ( ap @ ( c_2Esptree_2Efoldi @ ( arr @ ty_2Enum_2Enum @ bool ) @ A_27a )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1k: $i] :
( lam @ A_27a
@ ^ [V2v: $i] :
( lam @ ( arr @ ty_2Enum_2Enum @ bool )
@ ^ [V3a: $i] : ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Enum_2Enum ) @ V1k ) @ V3a ) ) ) ) )
@ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
@ ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) )
@ V0t ) ) ) ).
thf(ax_thm_2Esptree_2Emapi0__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ A_27a ) ) )
=> ! [V1i: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Emapi0 @ A_27a @ A_27b ) @ V0f ) @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ ( c_2Esptree_2ELN @ A_27b ) )
= ( c_2Esptree_2ELN @ A_27a ) ) )
& ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ A_27a ) ) )
=> ! [V3i: tp__ty_2Enum_2Enum,V4a: $i] :
( ( mem @ V4a @ A_27b )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Emapi0 @ A_27a @ A_27b ) @ V2f ) @ ( inj__ty_2Enum_2Enum @ V3i ) ) @ ( ap @ ( c_2Esptree_2ELS @ A_27b ) @ V4a ) )
= ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ ( ap @ ( ap @ V2f @ ( inj__ty_2Enum_2Enum @ V3i ) ) @ V4a ) ) ) ) )
& ! [V5f: $i] :
( ( mem @ V5f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ A_27a ) ) )
=> ! [V6i: tp__ty_2Enum_2Enum,V7t1: $i] :
( ( mem @ V7t1 @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ! [V8t2: $i] :
( ( mem @ V8t2 @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Emapi0 @ A_27a @ A_27b ) @ V5f ) @ ( inj__ty_2Enum_2Enum @ V6i ) ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27b ) @ V7t1 ) @ V8t2 ) )
= ( ap
@ ( ap @ ( c_2Ebool_2ELET @ ty_2Enum_2Enum @ ( ty_2Esptree_2Espt @ A_27a ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V9inc: $i] : ( ap @ ( ap @ ( c_2Esptree_2Emk__BN @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Emapi0 @ A_27a @ A_27b ) @ V5f ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V6i ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ V9inc ) ) ) @ V7t1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Emapi0 @ A_27a @ A_27b ) @ V5f ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V6i ) ) @ V9inc ) ) @ V8t2 ) ) ) )
@ ( ap @ c_2Esptree_2Elrnext @ ( inj__ty_2Enum_2Enum @ V6i ) ) ) ) ) ) )
& ! [V10f: $i] :
( ( mem @ V10f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ A_27a ) ) )
=> ! [V11i: tp__ty_2Enum_2Enum,V12t1: $i] :
( ( mem @ V12t1 @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ! [V13a: $i] :
( ( mem @ V13a @ A_27b )
=> ! [V14t2: $i] :
( ( mem @ V14t2 @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Emapi0 @ A_27a @ A_27b ) @ V10f ) @ ( inj__ty_2Enum_2Enum @ V11i ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27b ) @ V12t1 ) @ V13a ) @ V14t2 ) )
= ( ap
@ ( ap @ ( c_2Ebool_2ELET @ ty_2Enum_2Enum @ ( ty_2Esptree_2Espt @ A_27a ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V15inc: $i] : ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Emk__BS @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Emapi0 @ A_27a @ A_27b ) @ V10f ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V11i ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ V15inc ) ) ) @ V12t1 ) ) @ ( ap @ ( ap @ V10f @ ( inj__ty_2Enum_2Enum @ V11i ) ) @ V13a ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Emapi0 @ A_27a @ A_27b ) @ V10f ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V11i ) ) @ V15inc ) ) @ V14t2 ) ) ) )
@ ( ap @ c_2Esptree_2Elrnext @ ( inj__ty_2Enum_2Enum @ V11i ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Esptree_2Emapi__def,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ A_27a ) ) )
=> ! [V1pt: $i] :
( ( mem @ V1pt @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Emapi @ A_27a @ A_27b ) @ V0f ) @ V1pt )
= ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Emapi0 @ A_27a @ A_27b ) @ V0f ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V1pt ) ) ) ) ).
thf(conj_thm_2Esptree_2Elookup__mapi0,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ A_27b ) ) )
=> ! [V1pt: $i] :
( ( mem @ V1pt @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V2i: tp__ty_2Enum_2Enum,V3k: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V3k ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Emapi0 @ A_27b @ A_27a ) @ V0f ) @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ V1pt ) )
= ( ap @ ( ap @ ( ap @ ( c_2Eoption_2Eoption__CASE @ A_27a @ ( ty_2Eoption_2Eoption @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V3k ) ) @ V1pt ) ) @ ( c_2Eoption_2ENONE @ A_27b ) )
@ ( lam @ A_27a
@ ^ [V4v: $i] : ( ap @ ( c_2Eoption_2ESOME @ A_27b ) @ ( ap @ ( ap @ V0f @ ( ap @ ( ap @ c_2Esptree_2Espt__acc @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ ( inj__ty_2Enum_2Enum @ V3k ) ) ) @ V4v ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Elookup__mapi,axiom,
! [A_27a: del,A_27b: del,V0k: tp__ty_2Enum_2Enum,V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ A_27a ) ) )
=> ! [V2pt: $i] :
( ( mem @ V2pt @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Emapi @ A_27a @ A_27b ) @ V1f ) @ V2pt ) )
= ( ap @ ( ap @ ( c_2Eoption_2EOPTION__MAP @ A_27b @ A_27a ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V0k ) ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) @ V2pt ) ) ) ) ) ).
thf(ax_thm_2Esptree_2EtoAList__def,axiom,
! [A_27a: del] :
( ( c_2Esptree_2EtoAList @ A_27a )
= ( ap
@ ( ap
@ ( ap @ ( c_2Esptree_2Efoldi @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) @ A_27a )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V0k: $i] :
( lam @ A_27a
@ ^ [V1v: $i] :
( lam @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) )
@ ^ [V2a: $i] : ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ V0k ) @ V1v ) ) @ V2a ) ) ) ) )
@ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
@ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) ) ) ).
thf(conj_thm_2Esptree_2EMEM__toAList,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1k: tp__ty_2Enum_2Enum,V2v: $i] :
( ( mem @ V2v @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ V2v ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) @ ( ap @ ( c_2Esptree_2EtoAList @ A_27a ) @ V0t ) ) ) )
<=> ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ V0t )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V2v ) ) ) ) ) ).
thf(conj_thm_2Esptree_2EALOOKUP__toAList,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1x: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Ealist_2EALOOKUP @ A_27a @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Esptree_2EtoAList @ A_27a ) @ V0t ) ) @ ( inj__ty_2Enum_2Enum @ V1x ) )
= ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ V0t ) ) ) ).
thf(conj_thm_2Esptree_2Einsert__union,axiom,
! [A_27a: del,V0k: tp__ty_2Enum_2Enum,V1v: $i] :
( ( mem @ V1v @ A_27a )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) @ V1v ) @ V2s )
= ( ap @ ( ap @ ( c_2Esptree_2Eunion @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) @ V1v ) @ ( c_2Esptree_2ELN @ A_27a ) ) ) @ V2s ) ) ) ) ).
thf(conj_thm_2Esptree_2Edomain__empty,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ V0t ) )
=> ( ( V0t
= ( c_2Esptree_2ELN @ A_27a ) )
<=> ( ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ V0t )
= ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) ) ) ) ) ).
thf(conj_thm_2Esptree_2EALL__DISTINCT__MAP__FST__toAList,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) @ ty_2Enum_2Enum ) @ ( c_2Epair_2EFST @ ty_2Enum_2Enum @ A_27a ) ) @ ( ap @ ( c_2Esptree_2EtoAList @ A_27a ) @ V0t ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Efoldi__FOLDR__toAList,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27b )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Efoldi @ A_27b @ A_27a ) @ V0f ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V1a ) @ V2t )
= ( ap @ ( ap @ ( ap @ ( c_2Elist_2EFOLDR @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) @ A_27b ) @ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Enum_2Enum @ A_27a @ ( arr @ A_27b @ A_27b ) ) @ V0f ) ) @ V1a ) @ ( ap @ ( c_2Esptree_2EtoAList @ A_27a ) @ V2t ) ) ) ) ) ) ).
thf(ax_thm_2Esptree_2EtoListA__def,axiom,
! [A_27a: del] :
( ! [V0acc: $i] :
( ( mem @ V0acc @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2EtoListA @ A_27a ) @ V0acc ) @ ( c_2Esptree_2ELN @ A_27a ) )
= V0acc ) )
& ! [V1acc: $i] :
( ( mem @ V1acc @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Esptree_2EtoListA @ A_27a ) @ V1acc ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V2a ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2a ) @ V1acc ) ) ) )
& ! [V3acc: $i] :
( ( mem @ V3acc @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V4t1: $i] :
( ( mem @ V4t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V5t2: $i] :
( ( mem @ V5t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2EtoListA @ A_27a ) @ V3acc ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V4t1 ) @ V5t2 ) )
= ( ap @ ( ap @ ( c_2Esptree_2EtoListA @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2EtoListA @ A_27a ) @ V3acc ) @ V5t2 ) ) @ V4t1 ) ) ) ) )
& ! [V6acc: $i] :
( ( mem @ V6acc @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V7t1: $i] :
( ( mem @ V7t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V8a: $i] :
( ( mem @ V8a @ A_27a )
=> ! [V9t2: $i] :
( ( mem @ V9t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2EtoListA @ A_27a ) @ V6acc ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V7t1 ) @ V8a ) @ V9t2 ) )
= ( ap @ ( ap @ ( c_2Esptree_2EtoListA @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V8a ) @ ( ap @ ( ap @ ( c_2Esptree_2EtoListA @ A_27a ) @ V6acc ) @ V9t2 ) ) ) @ V7t1 ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2EtoListA__append,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1acc: $i] :
( ( mem @ V1acc @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2EtoListA @ A_27a ) @ V1acc ) @ V0t )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2EtoListA @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ V0t ) ) @ V1acc ) ) ) ) ).
thf(conj_thm_2Esptree_2EisEmpty__toListA,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1acc: $i] :
( ( mem @ V1acc @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ V0t ) )
=> ( ( V0t
= ( c_2Esptree_2ELN @ A_27a ) )
<=> ( ( ap @ ( ap @ ( c_2Esptree_2EtoListA @ A_27a ) @ V1acc ) @ V0t )
= V1acc ) ) ) ) ) ).
thf(ax_thm_2Esptree_2EtoList__def,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( c_2Esptree_2EtoList @ A_27a ) @ V0m )
= ( ap @ ( ap @ ( c_2Esptree_2EtoListA @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ V0m ) ) ) ).
thf(conj_thm_2Esptree_2EisEmpty__toList,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ V0t ) )
=> ( ( V0t
= ( c_2Esptree_2ELN @ A_27a ) )
<=> ( ( ap @ ( c_2Esptree_2EtoList @ A_27a ) @ V0t )
= ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Esptree_2EMEM__toList,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( c_2Esptree_2EtoList @ A_27a ) @ V1t ) ) ) )
<=> ? [V2k: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ V1t )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V0x ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Espt__eq__thm,axiom,
! [A_27a: del,V0t1: $i] :
( ( mem @ V0t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ V0t1 ) )
& ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ V1t2 ) ) )
=> ( ( V0t1 = V1t2 )
<=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ V0t1 )
= ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ V1t2 ) ) ) ) ) ) ).
thf(ax_thm_2Esptree_2Emk__wf__def,axiom,
! [A_27a: del] :
( ( ( ap @ ( c_2Esptree_2Emk__wf @ A_27a ) @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Esptree_2ELN @ A_27a ) )
& ! [V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( ap @ ( c_2Esptree_2Emk__wf @ A_27a ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V0x ) )
= ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V0x ) ) )
& ! [V1t1: $i] :
( ( mem @ V1t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V2t2: $i] :
( ( mem @ V2t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( c_2Esptree_2Emk__wf @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V1t1 ) @ V2t2 ) )
= ( ap @ ( ap @ ( c_2Esptree_2Emk__BN @ A_27a ) @ ( ap @ ( c_2Esptree_2Emk__wf @ A_27a ) @ V1t1 ) ) @ ( ap @ ( c_2Esptree_2Emk__wf @ A_27a ) @ V2t2 ) ) ) ) )
& ! [V3t1: $i] :
( ( mem @ V3t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5t2: $i] :
( ( mem @ V5t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( c_2Esptree_2Emk__wf @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V3t1 ) @ V4x ) @ V5t2 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Emk__BS @ A_27a ) @ ( ap @ ( c_2Esptree_2Emk__wf @ A_27a ) @ V3t1 ) ) @ V4x ) @ ( ap @ ( c_2Esptree_2Emk__wf @ A_27a ) @ V5t2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Ewf__mk__wf,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ ( ap @ ( c_2Esptree_2Emk__wf @ A_27a ) @ V0t ) ) ) ) ).
thf(conj_thm_2Esptree_2Ewf__mk__id,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ V0t ) )
=> ( ( ap @ ( c_2Esptree_2Emk__wf @ A_27a ) @ V0t )
= V0t ) ) ) ).
thf(conj_thm_2Esptree_2Elookup__mk__wf,axiom,
! [A_27a: del,V0x: tp__ty_2Enum_2Enum,V1t: $i] :
( ( mem @ V1t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( ap @ ( c_2Esptree_2Emk__wf @ A_27a ) @ V1t ) )
= ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ V1t ) ) ) ).
thf(conj_thm_2Esptree_2Edomain__mk__wf,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ ( ap @ ( c_2Esptree_2Emk__wf @ A_27a ) @ V0t ) )
= ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ V0t ) ) ) ).
thf(conj_thm_2Esptree_2Emk__wf__eq,axiom,
! [A_27a: del,V0t1: $i] :
( ( mem @ V0t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ( ap @ ( c_2Esptree_2Emk__wf @ A_27a ) @ V0t1 )
= ( ap @ ( c_2Esptree_2Emk__wf @ A_27a ) @ V1t2 ) )
<=> ! [V2x: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2x ) ) @ V0t1 )
= ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2x ) ) @ V1t2 ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Einter__eq,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0t1: $i] :
( ( mem @ V0t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ! [V2t3: $i] :
( ( mem @ V2t3 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V3t4: $i] :
( ( mem @ V3t4 @ ( ty_2Esptree_2Espt @ A_27c ) )
=> ( ( ( ap @ ( ap @ ( c_2Esptree_2Einter @ A_27a @ A_27b ) @ V0t1 ) @ V1t2 )
= ( ap @ ( ap @ ( c_2Esptree_2Einter @ A_27a @ A_27c ) @ V2t3 ) @ V3t4 ) )
<=> ! [V4x: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V4x ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Einter @ A_27a @ A_27b ) @ V0t1 ) @ V1t2 ) )
= ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V4x ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Einter @ A_27a @ A_27c ) @ V2t3 ) @ V3t4 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Eunion__mk__wf,axiom,
! [A_27a: del,V0t1: $i] :
( ( mem @ V0t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Eunion @ A_27a ) @ ( ap @ ( c_2Esptree_2Emk__wf @ A_27a ) @ V0t1 ) ) @ ( ap @ ( c_2Esptree_2Emk__wf @ A_27a ) @ V1t2 ) )
= ( ap @ ( c_2Esptree_2Emk__wf @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Eunion @ A_27a ) @ V0t1 ) @ V1t2 ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Einter__mk__wf,axiom,
! [A_27a: del,A_27b: del,V0t1: $i] :
( ( mem @ V0t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Einter @ A_27a @ A_27b ) @ ( ap @ ( c_2Esptree_2Emk__wf @ A_27a ) @ V0t1 ) ) @ ( ap @ ( c_2Esptree_2Emk__wf @ A_27b ) @ V1t2 ) )
= ( ap @ ( c_2Esptree_2Emk__wf @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Einter @ A_27a @ A_27b ) @ V0t1 ) @ V1t2 ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Einsert__mk__wf,axiom,
! [A_27a: del,V0x: tp__ty_2Enum_2Enum,V1v: $i] :
( ( mem @ V1v @ A_27a )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ V1v ) @ ( ap @ ( c_2Esptree_2Emk__wf @ A_27a ) @ V2t ) )
= ( ap @ ( c_2Esptree_2Emk__wf @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ V1v ) @ V2t ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Edelete__mk__wf,axiom,
! [A_27a: del,V0x: tp__ty_2Enum_2Enum,V1t: $i] :
( ( mem @ V1t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( ap @ ( c_2Esptree_2Emk__wf @ A_27a ) @ V1t ) )
= ( ap @ ( c_2Esptree_2Emk__wf @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ V1t ) ) ) ) ).
thf(conj_thm_2Esptree_2Eunion__LN,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Esptree_2Eunion @ A_27a ) @ V0t ) @ ( c_2Esptree_2ELN @ A_27a ) )
= V0t )
& ( ( ap @ ( ap @ ( c_2Esptree_2Eunion @ A_27a ) @ ( c_2Esptree_2ELN @ A_27a ) ) @ V0t )
= V0t ) ) ) ).
thf(conj_thm_2Esptree_2Einter__LN,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Esptree_2Einter @ A_27a @ A_27b ) @ V0t ) @ ( c_2Esptree_2ELN @ A_27b ) )
= ( c_2Esptree_2ELN @ A_27a ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Einter @ A_27c @ A_27a ) @ ( c_2Esptree_2ELN @ A_27c ) ) @ V0t )
= ( c_2Esptree_2ELN @ A_27c ) ) ) ) ).
thf(conj_thm_2Esptree_2Eunion__assoc,axiom,
! [A_27a: del,V0t1: $i] :
( ( mem @ V0t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V2t3: $i] :
( ( mem @ V2t3 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Eunion @ A_27a ) @ V0t1 ) @ ( ap @ ( ap @ ( c_2Esptree_2Eunion @ A_27a ) @ V1t2 ) @ V2t3 ) )
= ( ap @ ( ap @ ( c_2Esptree_2Eunion @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Eunion @ A_27a ) @ V0t1 ) @ V1t2 ) ) @ V2t3 ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Einter__assoc,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0t1: $i] :
( ( mem @ V0t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ! [V2t3: $i] :
( ( mem @ V2t3 @ ( ty_2Esptree_2Espt @ A_27c ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Einter @ A_27a @ A_27b ) @ V0t1 ) @ ( ap @ ( ap @ ( c_2Esptree_2Einter @ A_27b @ A_27c ) @ V1t2 ) @ V2t3 ) )
= ( ap @ ( ap @ ( c_2Esptree_2Einter @ A_27a @ A_27c ) @ ( ap @ ( ap @ ( c_2Esptree_2Einter @ A_27a @ A_27b ) @ V0t1 ) @ V1t2 ) ) @ V2t3 ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Elookup__compute,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1t: $i] :
( ( mem @ V1t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ! [V3t1: $i] :
( ( mem @ V3t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V4t2: $i] :
( ( mem @ V4t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ V1t )
= ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V1t ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Eoption_2ENONE @ A_27a ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V2a ) )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V2a ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V3t1 ) @ V4t2 ) )
= ( c_2Eoption_2ENONE @ A_27a ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V3t1 ) @ V2a ) @ V4t2 ) )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V2a ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Eoption_2ENONE @ A_27a ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V2a ) )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V2a ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V3t1 ) @ V4t2 ) )
= ( c_2Eoption_2ENONE @ A_27a ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V3t1 ) @ V2a ) @ V4t2 ) )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V2a ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Eoption_2ENONE @ A_27a ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V2a ) )
= ( c_2Eoption_2ENONE @ A_27a ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V3t1 ) @ V4t2 ) )
= ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V4t2 ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V3t1 ) @ V2a ) @ V4t2 ) )
= ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V4t2 ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Eoption_2ENONE @ A_27a ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V2a ) )
= ( c_2Eoption_2ENONE @ A_27a ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V3t1 ) @ V4t2 ) )
= ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V3t1 ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V3t1 ) @ V2a ) @ V4t2 ) )
= ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V3t1 ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Einsert__compute,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V3a_27: $i] :
( ( mem @ V3a_27 @ A_27a )
=> ! [V4t1: $i] :
( ( mem @ V4t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V5t2: $i] :
( ( mem @ V5t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ V1a ) @ V2t )
= ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V1a ) @ V2t ) )
& ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V1a ) @ ( c_2Esptree_2ELN @ A_27a ) )
= ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V1a ) )
& ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V1a ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V3a_27 ) )
= ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V1a ) )
& ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V1a ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V4t1 ) @ V5t2 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V4t1 ) @ V1a ) @ V5t2 ) )
& ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V1a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V4t1 ) @ V3a_27 ) @ V5t2 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V4t1 ) @ V1a ) @ V5t2 ) )
& ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ V1a ) @ ( c_2Esptree_2ELN @ A_27a ) )
= ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V1a ) )
& ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ V1a ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V3a_27 ) )
= ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V1a ) )
& ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ V1a ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V4t1 ) @ V5t2 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V4t1 ) @ V1a ) @ V5t2 ) )
& ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ V1a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V4t1 ) @ V3a_27 ) @ V5t2 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V4t1 ) @ V1a ) @ V5t2 ) )
& ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ V1a ) @ ( c_2Esptree_2ELN @ A_27a ) )
= ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ ( c_2Esptree_2ELN @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V1a ) @ ( c_2Esptree_2ELN @ A_27a ) ) ) )
& ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ V1a ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V3a_27 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ ( c_2Esptree_2ELN @ A_27a ) ) @ V3a_27 ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V1a ) @ ( c_2Esptree_2ELN @ A_27a ) ) ) )
& ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ V1a ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V4t1 ) @ V5t2 ) )
= ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V4t1 ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V1a ) @ V5t2 ) ) )
& ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ V1a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V4t1 ) @ V3a_27 ) @ V5t2 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V4t1 ) @ V3a_27 ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V1a ) @ V5t2 ) ) )
& ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ V1a ) @ ( c_2Esptree_2ELN @ A_27a ) )
= ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V1a ) @ ( c_2Esptree_2ELN @ A_27a ) ) ) @ ( c_2Esptree_2ELN @ A_27a ) ) )
& ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ V1a ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V3a_27 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V1a ) @ ( c_2Esptree_2ELN @ A_27a ) ) ) @ V3a_27 ) @ ( c_2Esptree_2ELN @ A_27a ) ) )
& ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ V1a ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V4t1 ) @ V5t2 ) )
= ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V1a ) @ V4t1 ) ) @ V5t2 ) )
& ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ V1a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V4t1 ) @ V3a_27 ) @ V5t2 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V1a ) @ V4t1 ) ) @ V3a_27 ) @ V5t2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Edelete__compute,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1t: $i] :
( ( mem @ V1t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ! [V3t1: $i] :
( ( mem @ V3t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V4t2: $i] :
( ( mem @ V4t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ V1t )
= ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V1t ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Esptree_2ELN @ A_27a ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V2a ) )
= ( c_2Esptree_2ELN @ A_27a ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V3t1 ) @ V4t2 ) )
= ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V3t1 ) @ V4t2 ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V3t1 ) @ V2a ) @ V4t2 ) )
= ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V3t1 ) @ V4t2 ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Esptree_2ELN @ A_27a ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V2a ) )
= ( c_2Esptree_2ELN @ A_27a ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V3t1 ) @ V4t2 ) )
= ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V3t1 ) @ V4t2 ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V3t1 ) @ V2a ) @ V4t2 ) )
= ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V3t1 ) @ V4t2 ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Esptree_2ELN @ A_27a ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V2a ) )
= ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V2a ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V3t1 ) @ V4t2 ) )
= ( ap @ ( ap @ ( c_2Esptree_2Emk__BN @ A_27a ) @ V3t1 ) @ ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V4t2 ) ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V3t1 ) @ V2a ) @ V4t2 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Emk__BS @ A_27a ) @ V3t1 ) @ V2a ) @ ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V4t2 ) ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Esptree_2ELN @ A_27a ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V2a ) )
= ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V2a ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V3t1 ) @ V4t2 ) )
= ( ap @ ( ap @ ( c_2Esptree_2Emk__BN @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V3t1 ) ) @ V4t2 ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V3t1 ) @ V2a ) @ V4t2 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Emk__BS @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V3t1 ) ) @ V2a ) @ V4t2 ) ) ) ) ) ) ) ).
thf(ax_thm_2Esptree_2EfromAList__primitive__def,axiom,
! [A_27a: del] :
( ( c_2Esptree_2EfromAList @ A_27a )
= ( ap
@ ( ap @ ( c_2Erelation_2EWFREC @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) @ ( ty_2Esptree_2Espt @ A_27a ) )
@ ( ap @ ( c_2Emin_2E_40 @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) @ bool ) ) )
@ ( lam @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) @ bool ) )
@ ^ [V0R: $i] :
( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( c_2Erelation_2EWF @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) ) @ V0R ) )
@ ( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V1y: $i] :
( ap @ ( c_2Ebool_2E_21 @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2x: $i] :
( ap @ ( c_2Ebool_2E_21 @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) )
@ ( lam @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) )
@ ^ [V3xs: $i] : ( ap @ ( ap @ V0R @ V3xs ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ V2x ) @ V1y ) ) @ V3xs ) ) ) ) ) ) ) ) ) ) ) )
@ ( lam @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) @ ( ty_2Esptree_2Espt @ A_27a ) )
@ ^ [V4fromAList: $i] :
( lam @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) )
@ ^ [V5a: $i] :
( ap @ ( ap @ ( ap @ ( c_2Elist_2Elist__CASE @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) @ ( ty_2Esptree_2Espt @ A_27a ) ) @ V5a ) @ ( ap @ ( c_2Ecombin_2EI @ ( ty_2Esptree_2Espt @ A_27a ) ) @ ( c_2Esptree_2ELN @ A_27a ) ) )
@ ( lam @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a )
@ ^ [V6v: $i] :
( lam @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) )
@ ^ [V7xs: $i] :
( ap @ ( ap @ ( c_2Epair_2Epair__CASE @ ( ty_2Esptree_2Espt @ A_27a ) @ ty_2Enum_2Enum @ A_27a ) @ V6v )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V8x: $i] :
( lam @ A_27a
@ ^ [V9y: $i] : ( ap @ ( c_2Ecombin_2EI @ ( ty_2Esptree_2Espt @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ V8x ) @ V9y ) @ ( ap @ V4fromAList @ V7xs ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2EfromAList__ind,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) @ bool ) )
=> ( ( ( p @ ( ap @ V0P @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) ) )
& ! [V1x: tp__ty_2Enum_2Enum,V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ! [V3xs: $i] :
( ( mem @ V3xs @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) )
=> ( ( p @ ( ap @ V0P @ V3xs ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ V2y ) ) @ V3xs ) ) ) ) ) ) )
=> ! [V4v: $i] :
( ( mem @ V4v @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) )
=> ( p @ ( ap @ V0P @ V4v ) ) ) ) ) ).
thf(conj_thm_2Esptree_2EfromAList__def,axiom,
! [A_27a: del] :
( ( ( ap @ ( c_2Esptree_2EfromAList @ A_27a ) @ ( c_2Elist_2ENIL @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) )
= ( c_2Esptree_2ELN @ A_27a ) )
& ! [V0y: $i] :
( ( mem @ V0y @ A_27a )
=> ! [V1xs: $i] :
( ( mem @ V1xs @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) )
=> ! [V2x: tp__ty_2Enum_2Enum] :
( ( ap @ ( c_2Esptree_2EfromAList @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2x ) ) @ V0y ) ) @ V1xs ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2x ) ) @ V0y ) @ ( ap @ ( c_2Esptree_2EfromAList @ A_27a ) @ V1xs ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Elookup__fromAList,axiom,
! [A_27a: del,V0ls: $i] :
( ( mem @ V0ls @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) )
=> ! [V1x: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ ( ap @ ( c_2Esptree_2EfromAList @ A_27a ) @ V0ls ) )
= ( ap @ ( ap @ ( c_2Ealist_2EALOOKUP @ A_27a @ ty_2Enum_2Enum ) @ V0ls ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) ) ).
thf(conj_thm_2Esptree_2Edomain__fromAList,axiom,
! [A_27a: del,V0ls: $i] :
( ( mem @ V0ls @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) )
=> ( ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ ( ap @ ( c_2Esptree_2EfromAList @ A_27a ) @ V0ls ) )
= ( ap @ ( c_2Elist_2ELIST__TO__SET @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) @ ty_2Enum_2Enum ) @ ( c_2Epair_2EFST @ ty_2Enum_2Enum @ A_27a ) ) @ V0ls ) ) ) ) ).
thf(conj_thm_2Esptree_2Elookup__fromAList__toAList,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1x: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ ( ap @ ( c_2Esptree_2EfromAList @ A_27a ) @ ( ap @ ( c_2Esptree_2EtoAList @ A_27a ) @ V0t ) ) )
= ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ V0t ) ) ) ).
thf(conj_thm_2Esptree_2Ewf__fromAList,axiom,
! [A_27a: del,V0ls: $i] :
( ( mem @ V0ls @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) )
=> ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ ( ap @ ( c_2Esptree_2EfromAList @ A_27a ) @ V0ls ) ) ) ) ).
thf(conj_thm_2Esptree_2EfromAList__toAList,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ V0t ) )
=> ( ( ap @ ( c_2Esptree_2EfromAList @ A_27a ) @ ( ap @ ( c_2Esptree_2EtoAList @ A_27a ) @ V0t ) )
= V0t ) ) ) ).
thf(conj_thm_2Esptree_2Eunion__insert__LN,axiom,
! [A_27a: del,V0x: tp__ty_2Enum_2Enum,V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ! [V2t2: $i] :
( ( mem @ V2t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Eunion @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ V1y ) @ ( c_2Esptree_2ELN @ A_27a ) ) ) @ V2t2 )
= ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ V1y ) @ V2t2 ) ) ) ) ).
thf(conj_thm_2Esptree_2EfromAList__append,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) )
=> ( ( ap @ ( c_2Esptree_2EfromAList @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) @ V0l1 ) @ V1l2 ) )
= ( ap @ ( ap @ ( c_2Esptree_2Eunion @ A_27a ) @ ( ap @ ( c_2Esptree_2EfromAList @ A_27a ) @ V0l1 ) ) @ ( ap @ ( c_2Esptree_2EfromAList @ A_27a ) @ V1l2 ) ) ) ) ) ).
thf(ax_thm_2Esptree_2Emap__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Emap @ A_27a @ A_27b ) @ V0f ) @ ( c_2Esptree_2ELN @ A_27b ) )
= ( c_2Esptree_2ELN @ A_27a ) ) )
& ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ A_27a ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27b )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Emap @ A_27a @ A_27b ) @ V1f ) @ ( ap @ ( c_2Esptree_2ELS @ A_27b ) @ V2a ) )
= ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ ( ap @ V1f @ V2a ) ) ) ) )
& ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27b @ A_27a ) )
=> ! [V4t1: $i] :
( ( mem @ V4t1 @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ! [V5t2: $i] :
( ( mem @ V5t2 @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Emap @ A_27a @ A_27b ) @ V3f ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27b ) @ V4t1 ) @ V5t2 ) )
= ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Emap @ A_27a @ A_27b ) @ V3f ) @ V4t1 ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Emap @ A_27a @ A_27b ) @ V3f ) @ V5t2 ) ) ) ) ) )
& ! [V6f: $i] :
( ( mem @ V6f @ ( arr @ A_27b @ A_27a ) )
=> ! [V7t1: $i] :
( ( mem @ V7t1 @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ! [V8a: $i] :
( ( mem @ V8a @ A_27b )
=> ! [V9t2: $i] :
( ( mem @ V9t2 @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Emap @ A_27a @ A_27b ) @ V6f ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27b ) @ V7t1 ) @ V8a ) @ V9t2 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Emap @ A_27a @ A_27b ) @ V6f ) @ V7t1 ) ) @ ( ap @ V6f @ V8a ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Emap @ A_27a @ A_27b ) @ V6f ) @ V9t2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2EtoList__map,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( c_2Esptree_2EtoList @ A_27b ) @ ( ap @ ( ap @ ( c_2Esptree_2Emap @ A_27b @ A_27a ) @ V0f ) @ V1s ) )
= ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( c_2Esptree_2EtoList @ A_27a ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Edomain__map,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( c_2Esptree_2Edomain @ A_27b ) @ ( ap @ ( ap @ ( c_2Esptree_2Emap @ A_27b @ A_27a ) @ V0f ) @ V1s ) )
= ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ V1s ) ) ) ) ).
thf(conj_thm_2Esptree_2Elookup__map,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V2x: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V2x ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Emap @ A_27b @ A_27a ) @ V0f ) @ V1s ) )
= ( ap @ ( ap @ ( c_2Eoption_2EOPTION__MAP @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2x ) ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Emap__LN,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Esptree_2Emap @ A_27b @ A_27a ) @ V0f ) @ V1t )
= ( c_2Esptree_2ELN @ A_27b ) )
<=> ( V1t
= ( c_2Esptree_2ELN @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Ewf__map,axiom,
! [A_27a: del,A_27b: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27b ) @ ( ap @ ( ap @ ( c_2Esptree_2Emap @ A_27b @ A_27a ) @ V1f ) @ V0t ) ) )
<=> ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ V0t ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Emap__map__o,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ A_27c ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Emap @ A_27c @ A_27b ) @ V1f ) @ ( ap @ ( ap @ ( c_2Esptree_2Emap @ A_27b @ A_27a ) @ V2g ) @ V0t ) )
= ( ap @ ( ap @ ( c_2Esptree_2Emap @ A_27c @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ A_27b ) @ V1f ) @ V2g ) ) @ V0t ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Emap__insert,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1x: tp__ty_2Enum_2Enum,V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ! [V3z: $i] :
( ( mem @ V3z @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Emap @ A_27b @ A_27a ) @ V0f ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ V2y ) @ V3z ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ ( ap @ V0f @ V2y ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Emap @ A_27b @ A_27a ) @ V0f ) @ V3z ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Emap__fromAList,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ A_27a ) )
=> ! [V1ls: $i] :
( ( mem @ V1ls @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27b ) ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Emap @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( c_2Esptree_2EfromAList @ A_27b ) @ V1ls ) )
= ( ap @ ( c_2Esptree_2EfromAList @ A_27a )
@ ( ap
@ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27b ) @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Enum_2Enum @ A_27b @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2k: $i] :
( lam @ A_27b
@ ^ [V3v: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ V2k ) @ ( ap @ V0f @ V3v ) ) ) ) ) )
@ V1ls ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Einsert__insert,axiom,
! [A_27a: del,V0x1: tp__ty_2Enum_2Enum,V1x2: tp__ty_2Enum_2Enum,V2v1: $i] :
( ( mem @ V2v1 @ A_27a )
=> ! [V3v2: $i] :
( ( mem @ V3v2 @ A_27a )
=> ! [V4t: $i] :
( ( mem @ V4t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0x1 ) ) @ V2v1 ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1x2 ) ) @ V3v2 ) @ V4t ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esptree_2Espt @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0x1 ) ) @ ( inj__ty_2Enum_2Enum @ V1x2 ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0x1 ) ) @ V2v1 ) @ V4t ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1x2 ) ) @ V3v2 ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0x1 ) ) @ V2v1 ) @ V4t ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Einsert__shadow,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1a: tp__ty_2Enum_2Enum,V2b: $i] :
( ( mem @ V2b @ A_27a )
=> ! [V3c: $i] :
( ( mem @ V3c @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1a ) ) @ V2b ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1a ) ) @ V3c ) @ V0t ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1a ) ) @ V2b ) @ V0t ) ) ) ) ) ).
thf(ax_thm_2Esptree_2Espt__left__def,axiom,
! [A_27a: del] :
( ( ( ap @ ( c_2Esptree_2Espt__left @ A_27a ) @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Esptree_2ELN @ A_27a ) )
& ! [V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( ap @ ( c_2Esptree_2Espt__left @ A_27a ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V0x ) )
= ( c_2Esptree_2ELN @ A_27a ) ) )
& ! [V1t1: $i] :
( ( mem @ V1t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V2t2: $i] :
( ( mem @ V2t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( c_2Esptree_2Espt__left @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V1t1 ) @ V2t2 ) )
= V1t1 ) ) )
& ! [V3t1: $i] :
( ( mem @ V3t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5t2: $i] :
( ( mem @ V5t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( c_2Esptree_2Espt__left @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V3t1 ) @ V4x ) @ V5t2 ) )
= V3t1 ) ) ) ) ) ).
thf(ax_thm_2Esptree_2Espt__right__def,axiom,
! [A_27a: del] :
( ( ( ap @ ( c_2Esptree_2Espt__right @ A_27a ) @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Esptree_2ELN @ A_27a ) )
& ! [V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( ap @ ( c_2Esptree_2Espt__right @ A_27a ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V0x ) )
= ( c_2Esptree_2ELN @ A_27a ) ) )
& ! [V1t1: $i] :
( ( mem @ V1t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V2t2: $i] :
( ( mem @ V2t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( c_2Esptree_2Espt__right @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V1t1 ) @ V2t2 ) )
= V2t2 ) ) )
& ! [V3t1: $i] :
( ( mem @ V3t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5t2: $i] :
( ( mem @ V5t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( c_2Esptree_2Espt__right @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V3t1 ) @ V4x ) @ V5t2 ) )
= V5t2 ) ) ) ) ) ).
thf(ax_thm_2Esptree_2Espt__center__primitive__def,axiom,
! [A_27a: del] :
( ( c_2Esptree_2Espt__center @ A_27a )
= ( ap
@ ( ap @ ( c_2Erelation_2EWFREC @ ( ty_2Esptree_2Espt @ A_27a ) @ ( ty_2Eoption_2Eoption @ A_27a ) )
@ ( ap @ ( c_2Emin_2E_40 @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ bool ) ) )
@ ( lam @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ bool ) )
@ ^ [V0R: $i] : ( ap @ ( c_2Erelation_2EWF @ ( ty_2Esptree_2Espt @ A_27a ) ) @ V0R ) ) ) )
@ ( lam @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ ( ty_2Eoption_2Eoption @ A_27a ) )
@ ^ [V1spt__center: $i] :
( lam @ ( ty_2Esptree_2Espt @ A_27a )
@ ^ [V2a: $i] :
( ap
@ ( ap
@ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Espt__CASE @ A_27a @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ V2a ) @ ( ap @ ( c_2Ecombin_2EI @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( c_2Eoption_2ENONE @ A_27a ) ) )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( c_2Ecombin_2EI @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V3x ) ) ) )
@ ( lam @ ( ty_2Esptree_2Espt @ A_27a )
@ ^ [V4v7: $i] :
( lam @ ( ty_2Esptree_2Espt @ A_27a )
@ ^ [V5v8: $i] : ( ap @ ( c_2Ecombin_2EI @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( c_2Eoption_2ENONE @ A_27a ) ) ) ) )
@ ( lam @ ( ty_2Esptree_2Espt @ A_27a )
@ ^ [V6t1: $i] :
( lam @ A_27a
@ ^ [V7x_27: $i] :
( lam @ ( ty_2Esptree_2Espt @ A_27a )
@ ^ [V8t2: $i] : ( ap @ ( c_2Ecombin_2EI @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V7x_27 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Espt__center__ind,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Esptree_2Espt @ A_27a ) @ bool ) )
=> ( ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( p @ ( ap @ V0P @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V1x ) ) ) )
& ! [V2t1: $i] :
( ( mem @ V2t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4t2: $i] :
( ( mem @ V4t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V2t1 ) @ V3x ) @ V4t2 ) ) ) ) ) )
& ( p @ ( ap @ V0P @ ( c_2Esptree_2ELN @ A_27a ) ) )
& ! [V5v1: $i] :
( ( mem @ V5v1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V6v2: $i] :
( ( mem @ V6v2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V5v1 ) @ V6v2 ) ) ) ) ) )
=> ! [V7v: $i] :
( ( mem @ V7v @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( p @ ( ap @ V0P @ V7v ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Espt__center__def,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1t1: $i] :
( ( mem @ V1t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V2t2: $i] :
( ( mem @ V2t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V3v1: $i] :
( ( mem @ V3v1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V4v2: $i] :
( ( mem @ V4v2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ( ap @ ( c_2Esptree_2Espt__center @ A_27a ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V0x ) )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V0x ) )
& ( ( ap @ ( c_2Esptree_2Espt__center @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V1t1 ) @ V0x ) @ V2t2 ) )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V0x ) )
& ( ( ap @ ( c_2Esptree_2Espt__center @ A_27a ) @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Eoption_2ENONE @ A_27a ) )
& ( ( ap @ ( c_2Esptree_2Espt__center @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V3v1 ) @ V4v2 ) )
= ( c_2Eoption_2ENONE @ A_27a ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Esubspt__eq,axiom,
! [A_27a: del] :
( ! [V0t: $i] :
( ( mem @ V0t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esptree_2Esubspt @ A_27a ) @ ( c_2Esptree_2ELN @ A_27a ) ) @ V0t ) )
<=> $true ) )
& ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esptree_2Esubspt @ A_27a ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V1x ) ) @ V2t ) )
<=> ( ( ap @ ( c_2Esptree_2Espt__center @ A_27a ) @ V2t )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V1x ) ) ) ) )
& ! [V3t1: $i] :
( ( mem @ V3t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V4t2: $i] :
( ( mem @ V4t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V5t: $i] :
( ( mem @ V5t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esptree_2Esubspt @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V3t1 ) @ V4t2 ) ) @ V5t ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Esptree_2Esubspt @ A_27a ) @ V3t1 ) @ ( ap @ ( c_2Esptree_2Espt__left @ A_27a ) @ V5t ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Esptree_2Esubspt @ A_27a ) @ V4t2 ) @ ( ap @ ( c_2Esptree_2Espt__right @ A_27a ) @ V5t ) ) ) ) ) ) ) )
& ! [V6t1: $i] :
( ( mem @ V6t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V7x: $i] :
( ( mem @ V7x @ A_27a )
=> ! [V8t2: $i] :
( ( mem @ V8t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V9t: $i] :
( ( mem @ V9t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esptree_2Esubspt @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V6t1 ) @ V7x ) @ V8t2 ) ) @ V9t ) )
<=> ( ( ( ap @ ( c_2Esptree_2Espt__center @ A_27a ) @ V9t )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V7x ) )
& ( p @ ( ap @ ( ap @ ( c_2Esptree_2Esubspt @ A_27a ) @ V6t1 ) @ ( ap @ ( c_2Esptree_2Espt__left @ A_27a ) @ V9t ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Esptree_2Esubspt @ A_27a ) @ V8t2 ) @ ( ap @ ( c_2Esptree_2Espt__right @ A_27a ) @ V9t ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Esubspt__lookup,axiom,
! [A_27a: del,V0t1: $i] :
( ( mem @ V0t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esptree_2Esubspt @ A_27a ) @ V0t1 ) @ V1t2 ) )
<=> ! [V2x: tp__ty_2Enum_2Enum,V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2x ) ) @ V0t1 )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V3y ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2x ) ) @ V1t2 )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V3y ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Esubspt__domain,axiom,
! [V0t1: tp__c_ty_2Esptree_2Espt_ty_2Eone_2Eone,V1t2: tp__c_ty_2Esptree_2Espt_ty_2Eone_2Eone] :
( ( p @ ( ap @ ( ap @ ( c_2Esptree_2Esubspt @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Esptree_2Espt_ty_2Eone_2Eone @ V0t1 ) ) @ ( inj__c_ty_2Esptree_2Espt_ty_2Eone_2Eone @ V1t2 ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Esptree_2Edomain @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Esptree_2Espt_ty_2Eone_2Eone @ V0t1 ) ) ) @ ( ap @ ( c_2Esptree_2Edomain @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Esptree_2Espt_ty_2Eone_2Eone @ V1t2 ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Esubspt__def,axiom,
! [A_27a: del,V0sp1: $i] :
( ( mem @ V0sp1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1sp2: $i] :
( ( mem @ V1sp2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esptree_2Esubspt @ A_27a ) @ V0sp1 ) @ V1sp2 ) )
<=> ! [V2k: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ V0sp1 ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ V1sp2 ) ) )
& ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ V1sp2 )
= ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ V0sp1 ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Esubspt__refl,axiom,
! [A_27a: del,V0sp: $i] :
( ( mem @ V0sp @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esptree_2Esubspt @ A_27a ) @ V0sp ) @ V0sp ) ) ) ).
thf(conj_thm_2Esptree_2Esubspt__trans,axiom,
! [A_27a: del,V0sp1: $i] :
( ( mem @ V0sp1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1sp2: $i] :
( ( mem @ V1sp2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V2sp3: $i] :
( ( mem @ V2sp3 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Esptree_2Esubspt @ A_27a ) @ V0sp1 ) @ V1sp2 ) )
& ( p @ ( ap @ ( ap @ ( c_2Esptree_2Esubspt @ A_27a ) @ V1sp2 ) @ V2sp3 ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esptree_2Esubspt @ A_27a ) @ V0sp1 ) @ V2sp3 ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Esubspt__LN,axiom,
! [A_27a: del,V0sp: $i] :
( ( mem @ V0sp @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Esptree_2Esubspt @ A_27a ) @ ( c_2Esptree_2ELN @ A_27a ) ) @ V0sp ) )
<=> $true )
& ( ( p @ ( ap @ ( ap @ ( c_2Esptree_2Esubspt @ A_27a ) @ V0sp ) @ ( c_2Esptree_2ELN @ A_27a ) ) )
<=> ( ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ V0sp )
= ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) ) ) ) ) ).
thf(ax_thm_2Esptree_2Efilter__v__def,axiom,
! [A_27a: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Efilter__v @ A_27a ) @ V0f ) @ ( c_2Esptree_2ELN @ A_27a ) )
= ( c_2Esptree_2ELN @ A_27a ) ) )
& ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Efilter__v @ A_27a ) @ V1f ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V2x ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esptree_2Espt @ A_27a ) ) @ ( ap @ V1f @ V2x ) ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V2x ) ) @ ( c_2Esptree_2ELN @ A_27a ) ) ) ) )
& ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27a @ bool ) )
=> ! [V4l: $i] :
( ( mem @ V4l @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V5r: $i] :
( ( mem @ V5r @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Efilter__v @ A_27a ) @ V3f ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V4l ) @ V5r ) )
= ( ap @ ( ap @ ( c_2Esptree_2Emk__BN @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Efilter__v @ A_27a ) @ V3f ) @ V4l ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Efilter__v @ A_27a ) @ V3f ) @ V5r ) ) ) ) ) )
& ! [V6f: $i] :
( ( mem @ V6f @ ( arr @ A_27a @ bool ) )
=> ! [V7l: $i] :
( ( mem @ V7l @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V8x: $i] :
( ( mem @ V8x @ A_27a )
=> ! [V9r: $i] :
( ( mem @ V9r @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Efilter__v @ A_27a ) @ V6f ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V7l ) @ V8x ) @ V9r ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esptree_2Espt @ A_27a ) ) @ ( ap @ V6f @ V8x ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Emk__BS @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Efilter__v @ A_27a ) @ V6f ) @ V7l ) ) @ V8x ) @ ( ap @ ( ap @ ( c_2Esptree_2Efilter__v @ A_27a ) @ V6f ) @ V9r ) ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Emk__BN @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Efilter__v @ A_27a ) @ V6f ) @ V7l ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Efilter__v @ A_27a ) @ V6f ) @ V9r ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Elookup__filter__v,axiom,
! [A_27a: del,V0k: tp__ty_2Enum_2Enum,V1t: $i] :
( ( mem @ V1t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Efilter__v @ A_27a ) @ V2f ) @ V1t ) )
= ( ap @ ( ap @ ( ap @ ( c_2Eoption_2Eoption__CASE @ A_27a @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) @ V1t ) ) @ ( c_2Eoption_2ENONE @ A_27a ) )
@ ( lam @ A_27a
@ ^ [V3v: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ V2f @ V3v ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V3v ) ) @ ( c_2Eoption_2ENONE @ A_27a ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Ewf__filter__v,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ V0t ) )
=> ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Efilter__v @ A_27a ) @ V1f ) @ V0t ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Ewf__mk__BN,axiom,
! [A_27a: del,V0t1: $i] :
( ( mem @ V0t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ V0t1 ) )
& ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ V1t2 ) ) )
=> ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Emk__BN @ A_27a ) @ V0t1 ) @ V1t2 ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Ewf__mk__BS,axiom,
! [A_27a: del,V0t1: $i] :
( ( mem @ V0t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( ( ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ V0t1 ) )
& ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ V1t2 ) ) )
=> ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Emk__BS @ A_27a ) @ V0t1 ) @ V2a ) @ V1t2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Ewf__mapi,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ A_27a ) ) )
=> ! [V1pt: $i] :
( ( mem @ V1pt @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Emapi @ A_27a @ A_27b ) @ V0f ) @ V1pt ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Elookup__mk__BN,axiom,
! [A_27a: del,V0i: tp__ty_2Enum_2Enum,V1t1: $i] :
( ( mem @ V1t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V2t2: $i] :
( ( mem @ V2t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Emk__BN @ A_27a ) @ V1t1 ) @ V2t2 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( c_2Eoption_2ENONE @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Esptree_2Espt @ A_27a ) ) @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V0i ) ) ) @ V1t1 ) @ V2t2 ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2EMAP__foldi,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) @ A_27b ) )
=> ! [V1pt: $i] :
( ( mem @ V1pt @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V2n: tp__ty_2Enum_2Enum,V3acc: $i] :
( ( mem @ V3acc @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) @ A_27b ) @ V0f )
@ ( ap
@ ( ap
@ ( ap
@ ( ap @ ( c_2Esptree_2Efoldi @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) @ A_27a )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4k: $i] :
( lam @ A_27a
@ ^ [V5v: $i] :
( lam @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) )
@ ^ [V6a: $i] : ( ap @ ( ap @ ( c_2Elist_2ECONS @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ V4k ) @ V5v ) ) @ V6a ) ) ) ) )
@ ( inj__ty_2Enum_2Enum @ V2n ) )
@ V3acc )
@ V1pt ) )
= ( ap
@ ( ap
@ ( ap
@ ( ap @ ( c_2Esptree_2Efoldi @ ( ty_2Elist_2Elist @ A_27b ) @ A_27a )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V7k: $i] :
( lam @ A_27a
@ ^ [V8v: $i] :
( lam @ ( ty_2Elist_2Elist @ A_27b )
@ ^ [V9a: $i] : ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ ( ap @ V0f @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ V7k ) @ V8v ) ) ) @ V9a ) ) ) ) )
@ ( inj__ty_2Enum_2Enum @ V2n ) )
@ ( ap @ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) @ A_27b ) @ V0f ) @ V3acc ) )
@ V1pt ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Emapi__Alist,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ A_27a ) ) )
=> ! [V1pt: $i] :
( ( mem @ V1pt @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Emapi @ A_27a @ A_27b ) @ V0f ) @ V1pt )
= ( ap @ ( c_2Esptree_2EfromAList @ A_27a )
@ ( ap
@ ( ap @ ( c_2Elist_2EMAP @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27b ) @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) )
@ ( lam @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27b )
@ ^ [V2kv: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ ( ap @ ( c_2Epair_2EFST @ ty_2Enum_2Enum @ A_27b ) @ V2kv ) ) @ ( ap @ ( ap @ V0f @ ( ap @ ( c_2Epair_2EFST @ ty_2Enum_2Enum @ A_27b ) @ V2kv ) ) @ ( ap @ ( c_2Epair_2ESND @ ty_2Enum_2Enum @ A_27b ) @ V2kv ) ) ) ) )
@ ( ap @ ( c_2Esptree_2EtoAList @ A_27b ) @ V1pt ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Edomain__mapi,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ A_27a ) ) )
=> ! [V1pt: $i] :
( ( mem @ V1pt @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ( ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Emapi @ A_27a @ A_27b ) @ V0f ) @ V1pt ) )
= ( ap @ ( c_2Esptree_2Edomain @ A_27b ) @ V1pt ) ) ) ) ).
thf(conj_thm_2Esptree_2Esize__domain,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Esptree_2Esize @ A_27a ) @ V0t ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ V0t ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Enum__set__domain__eq,axiom,
! [V0t1: tp__c_ty_2Esptree_2Espt_ty_2Eone_2Eone,V1t2: tp__c_ty_2Esptree_2Espt_ty_2Eone_2Eone] :
( ( ( p @ ( ap @ ( c_2Esptree_2Ewf @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Esptree_2Espt_ty_2Eone_2Eone @ V0t1 ) ) )
& ( p @ ( ap @ ( c_2Esptree_2Ewf @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Esptree_2Espt_ty_2Eone_2Eone @ V1t2 ) ) ) )
=> ( ( ( ap @ ( c_2Esptree_2Edomain @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Esptree_2Espt_ty_2Eone_2Eone @ V0t1 ) )
= ( ap @ ( c_2Esptree_2Edomain @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Esptree_2Espt_ty_2Eone_2Eone @ V1t2 ) ) )
<=> ( V0t1 = V1t2 ) ) ) ).
thf(conj_thm_2Esptree_2Eunion__num__set__sym,axiom,
! [V0t1: tp__c_ty_2Esptree_2Espt_ty_2Eone_2Eone,V1t2: tp__c_ty_2Esptree_2Espt_ty_2Eone_2Eone] :
( ( surj__c_ty_2Esptree_2Espt_ty_2Eone_2Eone @ ( ap @ ( ap @ ( c_2Esptree_2Eunion @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Esptree_2Espt_ty_2Eone_2Eone @ V0t1 ) ) @ ( inj__c_ty_2Esptree_2Espt_ty_2Eone_2Eone @ V1t2 ) ) )
= ( surj__c_ty_2Esptree_2Espt_ty_2Eone_2Eone @ ( ap @ ( ap @ ( c_2Esptree_2Eunion @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Esptree_2Espt_ty_2Eone_2Eone @ V1t2 ) ) @ ( inj__c_ty_2Esptree_2Espt_ty_2Eone_2Eone @ V0t1 ) ) ) ) ).
thf(conj_thm_2Esptree_2Edifference__sub,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ( ( ( ap @ ( ap @ ( c_2Esptree_2Edifference @ A_27a @ A_27b ) @ V0a ) @ V1b )
= ( c_2Esptree_2ELN @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ V0a ) ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27b ) @ V1b ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Ewf__difference,axiom,
! [A_27a: del,A_27b: del,V0t1: $i] :
( ( mem @ V0t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ( ( ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ V0t1 ) )
& ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27b ) @ V1t2 ) ) )
=> ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Edifference @ A_27a @ A_27b ) @ V0t1 ) @ V1t2 ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Edelete__fail,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1t: $i] :
( ( mem @ V1t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esptree_2Ewf @ A_27a ) @ V1t ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ V1t ) ) )
<=> ( ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V1t )
= V1t ) ) ) ) ).
thf(conj_thm_2Esptree_2Esize__delete,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1t: $i] :
( ( mem @ V1t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Esptree_2Esize @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Edelete @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V1t ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V1t ) ) @ ( c_2Eoption_2ENONE @ A_27a ) ) ) @ ( ap @ ( c_2Esptree_2Esize @ A_27a ) @ V1t ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( c_2Esptree_2Esize @ A_27a ) @ V1t ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Elookup__fromList__outside,axiom,
! [A_27a: del,V0args: $i] :
( ( mem @ V0args @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1k: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0args ) ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ ( ap @ ( c_2Esptree_2EfromList @ A_27a ) @ V0args ) )
= ( c_2Eoption_2ENONE @ A_27a ) ) ) ) ).
thf(conj_thm_2Esptree_2EIN__domain,axiom,
! [A_27a: del,A_27b: del,V0n: tp__ty_2Enum_2Enum,V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2t1: $i] :
( ( mem @ V2t1 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V3t2: $i] :
( ( mem @ V3t2 @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27b ) @ ( c_2Esptree_2ELN @ A_27b ) ) ) )
<=> $false )
& ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ ( ap @ ( c_2Esptree_2ELS @ A_27a ) @ V1x ) ) ) )
<=> ( V0n = fo__c_2Enum_2E0 ) )
& ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27a ) @ V2t1 ) @ V3t2 ) ) ) )
<=> ( ( V0n != fo__c_2Enum_2E0 )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ V2t1 ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ V3t2 ) ) ) ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27a ) @ V2t1 ) @ V1x ) @ V3t2 ) ) ) )
<=> ( ( V0n = fo__c_2Enum_2E0 )
| ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ V2t1 ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ V3t2 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Emap__map__K,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0a: $i] :
( ( mem @ V0a @ A_27b )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27c ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Esptree_2Emap @ A_27b @ A_27c ) @ ( ap @ ( c_2Ecombin_2EK @ A_27b @ A_27c ) @ V0a ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Emap @ A_27c @ A_27a ) @ V1f ) @ V2t ) )
= ( ap @ ( ap @ ( c_2Esptree_2Emap @ A_27b @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ A_27b @ A_27a ) @ V0a ) ) @ V2t ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Elookup__map__K,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27b )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Esptree_2Elookup @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ ( ap @ ( c_2Esptree_2Emap @ A_27b @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ A_27b @ A_27a ) @ V0x ) ) @ V1t ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ V1t ) ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27b ) @ V0x ) ) @ ( c_2Eoption_2ENONE @ A_27b ) ) ) ) ) ).
thf(ax_thm_2Esptree_2Espt__fold__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V1acc: $i] :
( ( mem @ V1acc @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Espt__fold @ A_27a @ A_27b ) @ V0f ) @ V1acc ) @ ( c_2Esptree_2ELN @ A_27b ) )
= V1acc ) ) )
& ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27b @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V3acc: $i] :
( ( mem @ V3acc @ A_27a )
=> ! [V4a: $i] :
( ( mem @ V4a @ A_27b )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Espt__fold @ A_27a @ A_27b ) @ V2f ) @ V3acc ) @ ( ap @ ( c_2Esptree_2ELS @ A_27b ) @ V4a ) )
= ( ap @ ( ap @ V2f @ V4a ) @ V3acc ) ) ) ) )
& ! [V5f: $i] :
( ( mem @ V5f @ ( arr @ A_27b @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V6acc: $i] :
( ( mem @ V6acc @ A_27a )
=> ! [V7t1: $i] :
( ( mem @ V7t1 @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ! [V8t2: $i] :
( ( mem @ V8t2 @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Espt__fold @ A_27a @ A_27b ) @ V5f ) @ V6acc ) @ ( ap @ ( ap @ ( c_2Esptree_2EBN @ A_27b ) @ V7t1 ) @ V8t2 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Espt__fold @ A_27a @ A_27b ) @ V5f ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Espt__fold @ A_27a @ A_27b ) @ V5f ) @ V6acc ) @ V7t1 ) ) @ V8t2 ) ) ) ) ) )
& ! [V9f: $i] :
( ( mem @ V9f @ ( arr @ A_27b @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V10acc: $i] :
( ( mem @ V10acc @ A_27a )
=> ! [V11t1: $i] :
( ( mem @ V11t1 @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ! [V12a: $i] :
( ( mem @ V12a @ A_27b )
=> ! [V13t2: $i] :
( ( mem @ V13t2 @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Espt__fold @ A_27a @ A_27b ) @ V9f ) @ V10acc ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2EBS @ A_27b ) @ V11t1 ) @ V12a ) @ V13t2 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Espt__fold @ A_27a @ A_27b ) @ V9f ) @ ( ap @ ( ap @ V9f @ V12a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Espt__fold @ A_27a @ A_27b ) @ V9f ) @ V10acc ) @ V11t1 ) ) ) @ V13t2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2EIMP__size__LESS__size,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Esptree_2Esubspt @ A_27a ) @ V0x ) @ V1y ) )
& ( ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ V0x )
!= ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ V1y ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ ( c_2Esptree_2Esize @ A_27a ) @ V0x ) ) @ ( ap @ ( c_2Esptree_2Esize @ A_27a ) @ V1y ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Esize__diff__less,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ! [V2z: $i] :
( ( mem @ V2z @ ( ty_2Esptree_2Espt @ A_27c ) )
=> ! [V3t: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27c ) @ V2z ) ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27b ) @ V1y ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V3t ) ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27b ) @ V1y ) ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V3t ) ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27c ) @ V2z ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V3t ) ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ V0x ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ ( c_2Esptree_2Esize @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Edifference @ A_27a @ A_27b ) @ V0x ) @ V1y ) ) ) @ ( ap @ ( c_2Esptree_2Esize @ A_27a ) @ ( ap @ ( ap @ ( c_2Esptree_2Edifference @ A_27a @ A_27c ) @ V0x ) @ V2z ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Einter__eq__LN,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( ty_2Esptree_2Espt @ A_27b ) )
=> ( ( ( ap @ ( ap @ ( c_2Esptree_2Einter @ A_27a @ A_27b ) @ V0x ) @ V1y )
= ( c_2Esptree_2ELN @ A_27a ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ V0x ) ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27b ) @ V1y ) ) ) ) ) ) ).
thf(ax_thm_2Esptree_2Elist__to__num__set__def,axiom,
( ( ( surj__c_ty_2Esptree_2Espt_ty_2Eone_2Eone @ ( ap @ c_2Esptree_2Elist__to__num__set @ ( c_2Elist_2ENIL @ ty_2Enum_2Enum ) ) )
= ( surj__c_ty_2Esptree_2Espt_ty_2Eone_2Eone @ ( c_2Esptree_2ELN @ ty_2Eone_2Eone ) ) )
& ! [V0n: tp__ty_2Enum_2Enum,V1ns: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] :
( ( surj__c_ty_2Esptree_2Espt_ty_2Eone_2Eone @ ( ap @ c_2Esptree_2Elist__to__num__set @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1ns ) ) ) )
= ( surj__c_ty_2Esptree_2Espt_ty_2Eone_2Eone @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ ty_2Eone_2Eone ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Eone_2Eone @ fo__c_2Eone_2Eone ) ) @ ( ap @ c_2Esptree_2Elist__to__num__set @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1ns ) ) ) ) ) ) ).
thf(ax_thm_2Esptree_2Elist__insert__def,axiom,
( ! [V0t: tp__c_ty_2Esptree_2Espt_ty_2Eone_2Eone] :
( ( surj__c_ty_2Esptree_2Espt_ty_2Eone_2Eone @ ( ap @ ( ap @ c_2Esptree_2Elist__insert @ ( c_2Elist_2ENIL @ ty_2Enum_2Enum ) ) @ ( inj__c_ty_2Esptree_2Espt_ty_2Eone_2Eone @ V0t ) ) )
= V0t )
& ! [V1n: tp__ty_2Enum_2Enum,V2ns: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum,V3t: tp__c_ty_2Esptree_2Espt_ty_2Eone_2Eone] :
( ( surj__c_ty_2Esptree_2Espt_ty_2Eone_2Eone @ ( ap @ ( ap @ c_2Esptree_2Elist__insert @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V2ns ) ) ) @ ( inj__c_ty_2Esptree_2Espt_ty_2Eone_2Eone @ V3t ) ) )
= ( surj__c_ty_2Esptree_2Espt_ty_2Eone_2Eone @ ( ap @ ( ap @ c_2Esptree_2Elist__insert @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V2ns ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esptree_2Einsert @ ty_2Eone_2Eone ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Eone_2Eone @ fo__c_2Eone_2Eone ) ) @ ( inj__c_ty_2Esptree_2Espt_ty_2Eone_2Eone @ V3t ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Edomain__list__to__num__set,axiom,
! [V0x: tp__ty_2Enum_2Enum,V1xs: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( ap @ ( c_2Esptree_2Edomain @ ty_2Eone_2Eone ) @ ( ap @ c_2Esptree_2Elist__to__num__set @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1xs ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ty_2Enum_2Enum ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1xs ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Edomain__list__insert,axiom,
! [V0xs: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum,V1x: tp__ty_2Enum_2Enum,V2t: tp__c_ty_2Esptree_2Espt_ty_2Eone_2Eone] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ ( ap @ ( c_2Esptree_2Edomain @ ty_2Eone_2Eone ) @ ( ap @ ( ap @ c_2Esptree_2Elist__insert @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V0xs ) ) @ ( inj__c_ty_2Esptree_2Espt_ty_2Eone_2Eone @ V2t ) ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ty_2Enum_2Enum ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V0xs ) ) ) )
| ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ ( ap @ ( c_2Esptree_2Edomain @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Esptree_2Espt_ty_2Eone_2Eone @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Edomain__FOLDR__delete,axiom,
! [A_27a: del,V0ls: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum,V1live: $i] :
( ( mem @ V1live @ ( ty_2Esptree_2Espt @ A_27a ) )
=> ( ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2EFOLDR @ ty_2Enum_2Enum @ ( ty_2Esptree_2Espt @ A_27a ) ) @ ( c_2Esptree_2Edelete @ A_27a ) ) @ V1live ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V0ls ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Esptree_2Edomain @ A_27a ) @ V1live ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ty_2Enum_2Enum ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V0ls ) ) ) ) ) ).
thf(conj_thm_2Esptree_2Elookup__list__to__num__set,axiom,
! [V0x: tp__ty_2Enum_2Enum,V1xs: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] :
( ( surj__c_ty_2Eoption_2Eoption_ty_2Eone_2Eone @ ( ap @ ( ap @ ( c_2Esptree_2Elookup @ ty_2Eone_2Eone ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( ap @ c_2Esptree_2Elist__to__num__set @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1xs ) ) ) )
= ( surj__c_ty_2Eoption_2Eoption_ty_2Eone_2Eone @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ ty_2Eone_2Eone ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ty_2Enum_2Enum ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1xs ) ) ) ) @ ( ap @ ( c_2Eoption_2ESOME @ ty_2Eone_2Eone ) @ ( inj__ty_2Eone_2Eone @ fo__c_2Eone_2Eone ) ) ) @ ( c_2Eoption_2ENONE @ ty_2Eone_2Eone ) ) ) ) ).
thf(conj_thm_2Esptree_2Elist__to__num__set__append,axiom,
! [V0l1: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum,V1l2: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] :
( ( surj__c_ty_2Esptree_2Espt_ty_2Eone_2Eone @ ( ap @ c_2Esptree_2Elist__to__num__set @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ ty_2Enum_2Enum ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V0l1 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1l2 ) ) ) )
= ( surj__c_ty_2Esptree_2Espt_ty_2Eone_2Eone @ ( ap @ ( ap @ ( c_2Esptree_2Eunion @ ty_2Eone_2Eone ) @ ( ap @ c_2Esptree_2Elist__to__num__set @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V0l1 ) ) ) @ ( ap @ c_2Esptree_2Elist__to__num__set @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1l2 ) ) ) ) ) ).
%------------------------------------------------------------------------------