ITP001 Axioms: ITP101^5.ax
%------------------------------------------------------------------------------
% File : ITP101^5 : 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 : patricia^2.ax [Gau20]
% : HL4101^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 194 ( 14 unt; 33 typ; 0 def)
% Number of atoms : 4587 ( 152 equ; 0 cnn)
% Maximal formula atoms : 278 ( 23 avg)
% Number of connectives : 8130 ( 26 ~; 12 |; 131 &;7549 @)
% ( 42 <=>; 370 =>; 0 <=; 0 <~>)
% Maximal formula depth : 67 ( 13 avg;7549 nst)
% Number of types : 2 ( 1 usr)
% Number of type conns : 29 ( 29 >; 0 *; 0 +; 0 <<)
% Number of symbols : 123 ( 120 usr; 96 con; 0-2 aty)
% Number of variables : 762 ( 66 ^ 680 !; 16 ?; 762 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_ty_2Epatricia_2Eptree,type,
ty_2Epatricia_2Eptree: del > del ).
thf(tp_c_2Epatricia_2EADD,type,
c_2Epatricia_2EADD: del > $i ).
thf(mem_c_2Epatricia_2EADD,axiom,
! [A_27a: del] : ( mem @ ( c_2Epatricia_2EADD @ A_27a ) @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) ).
thf(tp_c_2Epatricia_2EADD__LIST,type,
c_2Epatricia_2EADD__LIST: del > $i ).
thf(mem_c_2Epatricia_2EADD__LIST,axiom,
! [A_27a: del] : ( mem @ ( c_2Epatricia_2EADD__LIST @ A_27a ) @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) ).
thf(tp_c_2Epatricia_2EBRANCH,type,
c_2Epatricia_2EBRANCH: del > $i ).
thf(mem_c_2Epatricia_2EBRANCH,axiom,
! [A_27a: del] : ( mem @ ( c_2Epatricia_2EBRANCH @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ).
thf(tp_c_2Epatricia_2EBRANCHING__BIT,type,
c_2Epatricia_2EBRANCHING__BIT: $i ).
thf(mem_c_2Epatricia_2EBRANCHING__BIT,axiom,
mem @ c_2Epatricia_2EBRANCHING__BIT @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) ).
thf(stp_fo_c_2Epatricia_2EBRANCHING__BIT,type,
fo__c_2Epatricia_2EBRANCHING__BIT: tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Epatricia_2EBRANCHING__BIT,axiom,
! [X0: tp__ty_2Enum_2Enum,X1: tp__ty_2Enum_2Enum] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Epatricia_2EBRANCHING__BIT @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2Epatricia_2EBRANCHING__BIT @ ( inj__ty_2Enum_2Enum @ X0 ) ) @ ( inj__ty_2Enum_2Enum @ X1 ) ) ) ).
thf(tp_c_2Epatricia_2EBranch,type,
c_2Epatricia_2EBranch: del > $i ).
thf(mem_c_2Epatricia_2EBranch,axiom,
! [A_27a: del] : ( mem @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) ) ) ).
thf(tp_c_2Epatricia_2EDEPTH,type,
c_2Epatricia_2EDEPTH: del > $i ).
thf(mem_c_2Epatricia_2EDEPTH,axiom,
! [A_27a: del] : ( mem @ ( c_2Epatricia_2EDEPTH @ A_27a ) @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ty_2Enum_2Enum ) ) ).
thf(tp_c_2Epatricia_2EEVERY__LEAF,type,
c_2Epatricia_2EEVERY__LEAF: del > $i ).
thf(mem_c_2Epatricia_2EEVERY__LEAF,axiom,
! [A_27a: del] : ( mem @ ( c_2Epatricia_2EEVERY__LEAF @ A_27a ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ bool ) ) ) ).
thf(tp_c_2Epatricia_2EEXISTS__LEAF,type,
c_2Epatricia_2EEXISTS__LEAF: del > $i ).
thf(mem_c_2Epatricia_2EEXISTS__LEAF,axiom,
! [A_27a: del] : ( mem @ ( c_2Epatricia_2EEXISTS__LEAF @ A_27a ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ bool ) ) ) ).
thf(tp_c_2Epatricia_2EEmpty,type,
c_2Epatricia_2EEmpty: del > $i ).
thf(mem_c_2Epatricia_2EEmpty,axiom,
! [A_27a: del] : ( mem @ ( c_2Epatricia_2EEmpty @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ).
thf(tp_c_2Epatricia_2EFIND,type,
c_2Epatricia_2EFIND: del > $i ).
thf(mem_c_2Epatricia_2EFIND,axiom,
! [A_27a: del] : ( mem @ ( c_2Epatricia_2EFIND @ A_27a ) @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ A_27a ) ) ) ).
thf(stp_c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone,type,
tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone: $tType ).
thf(stp_inj_c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone,type,
inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone > $i ).
thf(stp_surj_c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone,type,
surj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone: $i > tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone ).
thf(stp_inj_surj_c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone,axiom,
! [X: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone] :
( ( surj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ X ) )
= X ) ).
thf(stp_inj_mem_c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone,axiom,
! [X: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone] : ( mem @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ X ) @ ( ty_2Epatricia_2Eptree @ ty_2Eone_2Eone ) ) ).
thf(stp_iso_mem_c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone,axiom,
! [X: $i] :
( ( mem @ X @ ( ty_2Epatricia_2Eptree @ ty_2Eone_2Eone ) )
=> ( X
= ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ ( surj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ X ) ) ) ) ).
thf(tp_c_2Epatricia_2EINSERT__PTREE,type,
c_2Epatricia_2EINSERT__PTREE: $i ).
thf(mem_c_2Epatricia_2EINSERT__PTREE,axiom,
mem @ c_2Epatricia_2EINSERT__PTREE @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Epatricia_2Eptree @ ty_2Eone_2Eone ) @ ( ty_2Epatricia_2Eptree @ ty_2Eone_2Eone ) ) ) ).
thf(tp_c_2Epatricia_2EIN__PTREE,type,
c_2Epatricia_2EIN__PTREE: $i ).
thf(mem_c_2Epatricia_2EIN__PTREE,axiom,
mem @ c_2Epatricia_2EIN__PTREE @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Epatricia_2Eptree @ ty_2Eone_2Eone ) @ bool ) ) ).
thf(tp_c_2Epatricia_2EIS__EMPTY,type,
c_2Epatricia_2EIS__EMPTY: del > $i ).
thf(mem_c_2Epatricia_2EIS__EMPTY,axiom,
! [A_27a: del] : ( mem @ ( c_2Epatricia_2EIS__EMPTY @ A_27a ) @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ bool ) ) ).
thf(tp_c_2Epatricia_2EIS__PTREE,type,
c_2Epatricia_2EIS__PTREE: del > $i ).
thf(mem_c_2Epatricia_2EIS__PTREE,axiom,
! [A_27a: del] : ( mem @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ bool ) ) ).
thf(tp_c_2Epatricia_2EJOIN,type,
c_2Epatricia_2EJOIN: del > $i ).
thf(mem_c_2Epatricia_2EJOIN,axiom,
! [A_27a: del] : ( mem @ ( c_2Epatricia_2EJOIN @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ).
thf(tp_c_2Epatricia_2EKEYS,type,
c_2Epatricia_2EKEYS: del > $i ).
thf(mem_c_2Epatricia_2EKEYS,axiom,
! [A_27a: del] : ( mem @ ( c_2Epatricia_2EKEYS @ A_27a ) @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Elist_2Elist @ ty_2Enum_2Enum ) ) ) ).
thf(tp_c_2Epatricia_2ELeaf,type,
c_2Epatricia_2ELeaf: del > $i ).
thf(mem_c_2Epatricia_2ELeaf,axiom,
! [A_27a: del] : ( mem @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) ).
thf(tp_c_2Epatricia_2ENUMSET__OF__PTREE,type,
c_2Epatricia_2ENUMSET__OF__PTREE: $i ).
thf(mem_c_2Epatricia_2ENUMSET__OF__PTREE,axiom,
mem @ c_2Epatricia_2ENUMSET__OF__PTREE @ ( arr @ ( ty_2Epatricia_2Eptree @ ty_2Eone_2Eone ) @ ( arr @ ty_2Enum_2Enum @ bool ) ) ).
thf(tp_c_2Epatricia_2EPEEK,type,
c_2Epatricia_2EPEEK: del > $i ).
thf(mem_c_2Epatricia_2EPEEK,axiom,
! [A_27a: del] : ( mem @ ( c_2Epatricia_2EPEEK @ A_27a ) @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) ) ).
thf(tp_c_2Epatricia_2EPTREE__OF__NUMSET,type,
c_2Epatricia_2EPTREE__OF__NUMSET: $i ).
thf(mem_c_2Epatricia_2EPTREE__OF__NUMSET,axiom,
mem @ c_2Epatricia_2EPTREE__OF__NUMSET @ ( arr @ ( ty_2Epatricia_2Eptree @ ty_2Eone_2Eone ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ bool ) @ ( ty_2Epatricia_2Eptree @ ty_2Eone_2Eone ) ) ) ).
thf(tp_c_2Epatricia_2EREMOVE,type,
c_2Epatricia_2EREMOVE: del > $i ).
thf(mem_c_2Epatricia_2EREMOVE,axiom,
! [A_27a: del] : ( mem @ ( c_2Epatricia_2EREMOVE @ A_27a ) @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) ).
thf(tp_c_2Epatricia_2ESIZE,type,
c_2Epatricia_2ESIZE: del > $i ).
thf(mem_c_2Epatricia_2ESIZE,axiom,
! [A_27a: del] : ( mem @ ( c_2Epatricia_2ESIZE @ A_27a ) @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ty_2Enum_2Enum ) ) ).
thf(tp_c_2Epatricia_2ETRANSFORM,type,
c_2Epatricia_2ETRANSFORM: del > del > $i ).
thf(mem_c_2Epatricia_2ETRANSFORM,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epatricia_2ETRANSFORM @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ A_27a ) @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27b ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) ).
thf(tp_c_2Epatricia_2ETRAVERSE,type,
c_2Epatricia_2ETRAVERSE: del > $i ).
thf(mem_c_2Epatricia_2ETRAVERSE,axiom,
! [A_27a: del] : ( mem @ ( c_2Epatricia_2ETRAVERSE @ A_27a ) @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Elist_2Elist @ ty_2Enum_2Enum ) ) ) ).
thf(tp_c_2Epatricia_2ETRAVERSE__AUX,type,
c_2Epatricia_2ETRAVERSE__AUX: del > $i ).
thf(mem_c_2Epatricia_2ETRAVERSE__AUX,axiom,
! [A_27a: del] : ( mem @ ( c_2Epatricia_2ETRAVERSE__AUX @ A_27a ) @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ ty_2Enum_2Enum ) @ ( ty_2Elist_2Elist @ ty_2Enum_2Enum ) ) ) ) ).
thf(tp_c_2Epatricia_2EUNION__PTREE,type,
c_2Epatricia_2EUNION__PTREE: $i ).
thf(mem_c_2Epatricia_2EUNION__PTREE,axiom,
mem @ c_2Epatricia_2EUNION__PTREE @ ( arr @ ( ty_2Epatricia_2Eptree @ ty_2Eone_2Eone ) @ ( arr @ ( ty_2Epatricia_2Eptree @ ty_2Eone_2Eone ) @ ( ty_2Epatricia_2Eptree @ ty_2Eone_2Eone ) ) ) ).
thf(tp_c_2Epatricia_2Eptree__CASE,type,
c_2Epatricia_2Eptree__CASE: del > del > $i ).
thf(mem_c_2Epatricia_2Eptree__CASE,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epatricia_2Eptree__CASE @ A_27a @ A_27b ) @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( arr @ A_27b @ ( arr @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ A_27b ) ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ A_27b ) ) ) ) @ A_27b ) ) ) ) ) ).
thf(tp_c_2Epatricia_2Eptree__size,type,
c_2Epatricia_2Eptree__size: del > $i ).
thf(mem_c_2Epatricia_2Eptree__size,axiom,
! [A_27a: del] : ( mem @ ( c_2Epatricia_2Eptree__size @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ty_2Enum_2Enum ) ) ) ).
thf(ax_thm_2Epatricia_2Eptree__TY__DEF,axiom,
! [A_27a: del] :
? [V0rep: $i] :
( ( mem @ V0rep @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ A_27a @ ty_2Enum_2Enum ) ) ) ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Ebool_2ETYPE__DEFINITION @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ A_27a @ ty_2Enum_2Enum ) ) ) @ ( ty_2Epatricia_2Eptree @ A_27a ) )
@ ( lam @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ A_27a @ ty_2Enum_2Enum ) ) )
@ ^ [V1a0_27: $i] :
( ap @ ( c_2Ebool_2E_21 @ ( arr @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ A_27a @ ty_2Enum_2Enum ) ) ) @ bool ) )
@ ( lam @ ( arr @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ A_27a @ ty_2Enum_2Enum ) ) ) @ bool )
@ ^ [V2_27ptree_27: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap @ ( c_2Ebool_2E_21 @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ A_27a @ ty_2Enum_2Enum ) ) ) )
@ ( lam @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ A_27a @ ty_2Enum_2Enum ) ) )
@ ^ [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 @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ A_27a @ ty_2Enum_2Enum ) ) ) ) @ V3a0_27 )
@ ( ap @ ( ap @ ( ap @ ( c_2Eind__type_2ECONSTR @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ A_27a @ ty_2Enum_2Enum ) ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ A_27a @ ty_2Enum_2Enum ) ) @ ( c_2Ebool_2EARB @ ty_2Enum_2Enum ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ ty_2Enum_2Enum ) @ ( c_2Ebool_2EARB @ A_27a ) ) @ ( c_2Ebool_2EARB @ ty_2Enum_2Enum ) ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( c_2Eind__type_2EBOTTOM @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ A_27a @ ty_2Enum_2Enum ) ) ) ) ) ) )
@ ( ap
@ ( ap @ c_2Ebool_2E_5C_2F
@ ( ap @ ( c_2Ebool_2E_3F @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5a0: $i] :
( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V6a1: $i] :
( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ A_27a @ ty_2Enum_2Enum ) ) ) ) @ V3a0_27 )
@ ( ap
@ ( ap
@ ( lam @ ty_2Enum_2Enum
@ ^ [V7a0: $i] :
( lam @ A_27a
@ ^ [V8a1: $i] :
( ap @ ( ap @ ( ap @ ( c_2Eind__type_2ECONSTR @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ A_27a @ ty_2Enum_2Enum ) ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ A_27a @ ty_2Enum_2Enum ) ) @ V7a0 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ ty_2Enum_2Enum ) @ V8a1 ) @ ( c_2Ebool_2EARB @ ty_2Enum_2Enum ) ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V9n: $i] : ( c_2Eind__type_2EBOTTOM @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ A_27a @ ty_2Enum_2Enum ) ) ) ) ) ) )
@ V5a0 )
@ V6a1 ) ) ) ) ) ) )
@ ( ap @ ( c_2Ebool_2E_3F @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V10a0: $i] :
( ap @ ( c_2Ebool_2E_3F @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V11a1: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ A_27a @ ty_2Enum_2Enum ) ) ) )
@ ( lam @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ A_27a @ ty_2Enum_2Enum ) ) )
@ ^ [V12a2: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ A_27a @ ty_2Enum_2Enum ) ) ) )
@ ( lam @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ A_27a @ ty_2Enum_2Enum ) ) )
@ ^ [V13a3: $i] :
( ap
@ ( ap @ c_2Ebool_2E_2F_5C
@ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ A_27a @ ty_2Enum_2Enum ) ) ) ) @ V3a0_27 )
@ ( ap
@ ( ap
@ ( ap
@ ( ap
@ ( lam @ ty_2Enum_2Enum
@ ^ [V14a0: $i] :
( lam @ ty_2Enum_2Enum
@ ^ [V15a1: $i] :
( lam @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ A_27a @ ty_2Enum_2Enum ) ) )
@ ^ [V16a2: $i] :
( lam @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ A_27a @ ty_2Enum_2Enum ) ) )
@ ^ [V17a3: $i] :
( ap @ ( ap @ ( ap @ ( c_2Eind__type_2ECONSTR @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ A_27a @ ty_2Enum_2Enum ) ) ) @ ( ap @ c_2Enum_2ESUC @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ A_27a @ ty_2Enum_2Enum ) ) @ V14a0 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ ty_2Enum_2Enum ) @ ( c_2Ebool_2EARB @ A_27a ) ) @ V15a1 ) ) )
@ ( ap @ ( ap @ ( c_2Eind__type_2EFCONS @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ A_27a @ ty_2Enum_2Enum ) ) ) ) @ V16a2 )
@ ( ap @ ( ap @ ( c_2Eind__type_2EFCONS @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ A_27a @ ty_2Enum_2Enum ) ) ) ) @ V17a3 )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V18n: $i] : ( c_2Eind__type_2EBOTTOM @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ A_27a @ ty_2Enum_2Enum ) ) ) ) ) ) ) ) ) ) )
@ V10a0 )
@ V11a1 )
@ V12a2 )
@ V13a3 ) ) )
@ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V2_27ptree_27 @ V12a2 ) ) @ ( ap @ V2_27ptree_27 @ V13a3 ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( ap @ V2_27ptree_27 @ V3a0_27 ) ) ) ) )
@ ( ap @ V2_27ptree_27 @ V1a0_27 ) ) ) ) ) )
@ V0rep ) ) ) ).
thf(ax_thm_2Epatricia_2Eptree__case__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0v: $i] :
( ( mem @ V0v @ A_27b )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ A_27b ) ) )
=> ! [V2f1: $i] :
( ( mem @ V2f1 @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ A_27b ) ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2Eptree__CASE @ A_27a @ A_27b ) @ ( c_2Epatricia_2EEmpty @ A_27a ) ) @ V0v ) @ V1f ) @ V2f1 )
= V0v ) ) ) )
& ! [V3a0: tp__ty_2Enum_2Enum,V4a1: $i] :
( ( mem @ V4a1 @ A_27a )
=> ! [V5v: $i] :
( ( mem @ V5v @ A_27b )
=> ! [V6f: $i] :
( ( mem @ V6f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ A_27b ) ) )
=> ! [V7f1: $i] :
( ( mem @ V7f1 @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ A_27b ) ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2Eptree__CASE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V3a0 ) ) @ V4a1 ) ) @ V5v ) @ V6f ) @ V7f1 )
= ( ap @ ( ap @ V6f @ ( inj__ty_2Enum_2Enum @ V3a0 ) ) @ V4a1 ) ) ) ) ) )
& ! [V8a0: tp__ty_2Enum_2Enum,V9a1: tp__ty_2Enum_2Enum,V10a2: $i] :
( ( mem @ V10a2 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V11a3: $i] :
( ( mem @ V11a3 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V12v: $i] :
( ( mem @ V12v @ A_27b )
=> ! [V13f: $i] :
( ( mem @ V13f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ A_27b ) ) )
=> ! [V14f1: $i] :
( ( mem @ V14f1 @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ A_27b ) ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2Eptree__CASE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V8a0 ) ) @ ( inj__ty_2Enum_2Enum @ V9a1 ) ) @ V10a2 ) @ V11a3 ) ) @ V12v ) @ V13f ) @ V14f1 )
= ( ap @ ( ap @ ( ap @ ( ap @ V14f1 @ ( inj__ty_2Enum_2Enum @ V8a0 ) ) @ ( inj__ty_2Enum_2Enum @ V9a1 ) ) @ V10a2 ) @ V11a3 ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Epatricia_2Eptree__size__def,axiom,
! [A_27a: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epatricia_2Eptree__size @ A_27a ) @ V0f ) @ ( c_2Epatricia_2EEmpty @ A_27a ) ) )
= fo__c_2Enum_2E0 ) )
& ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2a0: tp__ty_2Enum_2Enum,V3a1: $i] :
( ( mem @ V3a1 @ A_27a )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epatricia_2Eptree__size @ A_27a ) @ V1f ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2a0 ) ) @ V3a1 ) ) )
= ( 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 @ ( inj__ty_2Enum_2Enum @ V2a0 ) ) @ ( ap @ V1f @ V3a1 ) ) ) ) ) ) )
& ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V5a0: tp__ty_2Enum_2Enum,V6a1: tp__ty_2Enum_2Enum,V7a2: $i] :
( ( mem @ V7a2 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V8a3: $i] :
( ( mem @ V8a3 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epatricia_2Eptree__size @ A_27a ) @ V4f ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V5a0 ) ) @ ( inj__ty_2Enum_2Enum @ V6a1 ) ) @ V7a2 ) @ V8a3 ) ) )
= ( 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 @ ( inj__ty_2Enum_2Enum @ V5a0 ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V6a1 ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ ( c_2Epatricia_2Eptree__size @ A_27a ) @ V4f ) @ V7a2 ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2Eptree__size @ A_27a ) @ V4f ) @ V8a3 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2Edatatype__ptree,axiom,
! [A_27a: del,V0ptree: $i] :
( ( mem @ V0ptree @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) ) @ bool ) ) ) )
=> ( p @ ( ap @ ( c_2Ebool_2EDATATYPE @ bool ) @ ( ap @ ( ap @ ( ap @ V0ptree @ ( c_2Epatricia_2EEmpty @ A_27a ) ) @ ( c_2Epatricia_2ELeaf @ A_27a ) ) @ ( c_2Epatricia_2EBranch @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2Eptree__11,axiom,
! [A_27a: del] :
( ! [V0a0: tp__ty_2Enum_2Enum,V1a1: $i] :
( ( mem @ V1a1 @ A_27a )
=> ! [V2a0_27: tp__ty_2Enum_2Enum,V3a1_27: $i] :
( ( mem @ V3a1_27 @ A_27a )
=> ( ( ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0a0 ) ) @ V1a1 )
= ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2a0_27 ) ) @ V3a1_27 ) )
<=> ( ( V0a0 = V2a0_27 )
& ( V1a1 = V3a1_27 ) ) ) ) )
& ! [V4a0: tp__ty_2Enum_2Enum,V5a1: tp__ty_2Enum_2Enum,V6a2: $i] :
( ( mem @ V6a2 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V7a3: $i] :
( ( mem @ V7a3 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V8a0_27: tp__ty_2Enum_2Enum,V9a1_27: tp__ty_2Enum_2Enum,V10a2_27: $i] :
( ( mem @ V10a2_27 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V11a3_27: $i] :
( ( mem @ V11a3_27 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V4a0 ) ) @ ( inj__ty_2Enum_2Enum @ V5a1 ) ) @ V6a2 ) @ V7a3 )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V8a0_27 ) ) @ ( inj__ty_2Enum_2Enum @ V9a1_27 ) ) @ V10a2_27 ) @ V11a3_27 ) )
<=> ( ( V4a0 = V8a0_27 )
& ( V5a1 = V9a1_27 )
& ( V6a2 = V10a2_27 )
& ( V7a3 = V11a3_27 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2Eptree__distinct,axiom,
! [A_27a: del] :
( ! [V0a1: $i] :
( ( mem @ V0a1 @ A_27a )
=> ! [V1a0: tp__ty_2Enum_2Enum] :
( ( c_2Epatricia_2EEmpty @ A_27a )
!= ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1a0 ) ) @ V0a1 ) ) )
& ! [V2a3: $i] :
( ( mem @ V2a3 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V3a2: $i] :
( ( mem @ V3a2 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V4a1: tp__ty_2Enum_2Enum,V5a0: tp__ty_2Enum_2Enum] :
( ( c_2Epatricia_2EEmpty @ A_27a )
!= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V5a0 ) ) @ ( inj__ty_2Enum_2Enum @ V4a1 ) ) @ V3a2 ) @ V2a3 ) ) ) )
& ! [V6a3: $i] :
( ( mem @ V6a3 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V7a2: $i] :
( ( mem @ V7a2 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V8a1_27: tp__ty_2Enum_2Enum,V9a1: $i] :
( ( mem @ V9a1 @ A_27a )
=> ! [V10a0_27: tp__ty_2Enum_2Enum,V11a0: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V11a0 ) ) @ V9a1 )
!= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V10a0_27 ) ) @ ( inj__ty_2Enum_2Enum @ V8a1_27 ) ) @ V7a2 ) @ V6a3 ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2Eptree__nchotomy,axiom,
! [A_27a: del,V0pp: $i] :
( ( mem @ V0pp @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( ( V0pp
= ( c_2Epatricia_2EEmpty @ A_27a ) )
| ? [V1n: tp__ty_2Enum_2Enum,V2a: $i] :
( ( mem @ V2a @ A_27a )
& ( V0pp
= ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V2a ) ) )
| ? [V3n0: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum,V5p: $i] :
( ( mem @ V5p @ ( ty_2Epatricia_2Eptree @ A_27a ) )
& ? [V6p0: $i] :
( ( mem @ V6p0 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
& ( V0pp
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V3n0 ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ V5p ) @ V6p0 ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2Eptree__Axiom,axiom,
! [A_27a: del,A_27b: del,V0f0: $i] :
( ( mem @ V0f0 @ A_27b )
=> ! [V1f1: $i] :
( ( mem @ V1f1 @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ A_27b ) ) )
=> ! [V2f2: $i] :
( ( mem @ V2f2 @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( arr @ A_27b @ ( arr @ A_27b @ A_27b ) ) ) ) ) ) )
=> ? [V3fn: $i] :
( ( mem @ V3fn @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ A_27b ) )
& ( ( ap @ V3fn @ ( c_2Epatricia_2EEmpty @ A_27a ) )
= V0f0 )
& ! [V4a0: tp__ty_2Enum_2Enum,V5a1: $i] :
( ( mem @ V5a1 @ A_27a )
=> ( ( ap @ V3fn @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V4a0 ) ) @ V5a1 ) )
= ( ap @ ( ap @ V1f1 @ ( inj__ty_2Enum_2Enum @ V4a0 ) ) @ V5a1 ) ) )
& ! [V6a0: tp__ty_2Enum_2Enum,V7a1: tp__ty_2Enum_2Enum,V8a2: $i] :
( ( mem @ V8a2 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V9a3: $i] :
( ( mem @ V9a3 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( ( ap @ V3fn @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V6a0 ) ) @ ( inj__ty_2Enum_2Enum @ V7a1 ) ) @ V8a2 ) @ V9a3 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ V2f2 @ ( inj__ty_2Enum_2Enum @ V6a0 ) ) @ ( inj__ty_2Enum_2Enum @ V7a1 ) ) @ V8a2 ) @ V9a3 ) @ ( ap @ V3fn @ V8a2 ) ) @ ( ap @ V3fn @ V9a3 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2Eptree__induction,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ bool ) )
=> ( ( ( p @ ( ap @ V0P @ ( c_2Epatricia_2EEmpty @ A_27a ) ) )
& ! [V1n: tp__ty_2Enum_2Enum,V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V2a ) ) ) )
& ! [V3p: $i] :
( ( mem @ V3p @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V4p0: $i] :
( ( mem @ V4p0 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( ( ( p @ ( ap @ V0P @ V3p ) )
& ( p @ ( ap @ V0P @ V4p0 ) ) )
=> ! [V5n: tp__ty_2Enum_2Enum,V6n0: tp__ty_2Enum_2Enum] : ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V6n0 ) ) @ ( inj__ty_2Enum_2Enum @ V5n ) ) @ V3p ) @ V4p0 ) ) ) ) ) ) )
=> ! [V7p: $i] :
( ( mem @ V7p @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( p @ ( ap @ V0P @ V7p ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2Eptree__case__cong,axiom,
! [A_27a: del,A_27b: del,V0v_27: $i] :
( ( mem @ V0v_27 @ A_27b )
=> ! [V1f_27: $i] :
( ( mem @ V1f_27 @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ A_27b ) ) )
=> ! [V2f1_27: $i] :
( ( mem @ V2f1_27 @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ A_27b ) ) ) ) )
=> ! [V3M: $i] :
( ( mem @ V3M @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V4M_27: $i] :
( ( mem @ V4M_27 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V5v: $i] :
( ( mem @ V5v @ A_27b )
=> ! [V6f: $i] :
( ( mem @ V6f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ A_27b ) ) )
=> ! [V7f1: $i] :
( ( mem @ V7f1 @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ A_27b ) ) ) ) )
=> ( ( ( V3M = V4M_27 )
& ( ( V4M_27
= ( c_2Epatricia_2EEmpty @ A_27a ) )
=> ( V5v = V0v_27 ) )
& ! [V8a0: tp__ty_2Enum_2Enum,V9a1: $i] :
( ( mem @ V9a1 @ A_27a )
=> ( ( V4M_27
= ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V8a0 ) ) @ V9a1 ) )
=> ( ( ap @ ( ap @ V6f @ ( inj__ty_2Enum_2Enum @ V8a0 ) ) @ V9a1 )
= ( ap @ ( ap @ V1f_27 @ ( inj__ty_2Enum_2Enum @ V8a0 ) ) @ V9a1 ) ) ) )
& ! [V10a0: tp__ty_2Enum_2Enum,V11a1: tp__ty_2Enum_2Enum,V12a2: $i] :
( ( mem @ V12a2 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V13a3: $i] :
( ( mem @ V13a3 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( ( V4M_27
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V10a0 ) ) @ ( inj__ty_2Enum_2Enum @ V11a1 ) ) @ V12a2 ) @ V13a3 ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ V7f1 @ ( inj__ty_2Enum_2Enum @ V10a0 ) ) @ ( inj__ty_2Enum_2Enum @ V11a1 ) ) @ V12a2 ) @ V13a3 )
= ( ap @ ( ap @ ( ap @ ( ap @ V2f1_27 @ ( inj__ty_2Enum_2Enum @ V10a0 ) ) @ ( inj__ty_2Enum_2Enum @ V11a1 ) ) @ V12a2 ) @ V13a3 ) ) ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2Eptree__CASE @ A_27a @ A_27b ) @ V3M ) @ V5v ) @ V6f ) @ V7f1 )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2Eptree__CASE @ A_27a @ A_27b ) @ V4M_27 ) @ V0v_27 ) @ V1f_27 ) @ V2f1_27 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2Eptree__case__eq,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V1v: $i] :
( ( mem @ V1v @ A_27b )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ A_27b ) ) )
=> ! [V3f1: $i] :
( ( mem @ V3f1 @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ A_27b ) ) ) ) )
=> ! [V4v_27: $i] :
( ( mem @ V4v_27 @ A_27b )
=> ( ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2Eptree__CASE @ A_27a @ A_27b ) @ V0x ) @ V1v ) @ V2f ) @ V3f1 )
= V4v_27 )
<=> ( ( ( V0x
= ( c_2Epatricia_2EEmpty @ A_27a ) )
& ( V1v = V4v_27 ) )
| ? [V5n: tp__ty_2Enum_2Enum,V6a: $i] :
( ( mem @ V6a @ A_27a )
& ( V0x
= ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V5n ) ) @ V6a ) )
& ( ( ap @ ( ap @ V2f @ ( inj__ty_2Enum_2Enum @ V5n ) ) @ V6a )
= V4v_27 ) )
| ? [V7n0: tp__ty_2Enum_2Enum,V8n: tp__ty_2Enum_2Enum,V9p: $i] :
( ( mem @ V9p @ ( ty_2Epatricia_2Eptree @ A_27a ) )
& ? [V10p0: $i] :
( ( mem @ V10p0 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
& ( V0x
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V7n0 ) ) @ ( inj__ty_2Enum_2Enum @ V8n ) ) @ V9p ) @ V10p0 ) )
& ( ( ap @ ( ap @ ( ap @ ( ap @ V3f1 @ ( inj__ty_2Enum_2Enum @ V7n0 ) ) @ ( inj__ty_2Enum_2Enum @ V8n ) ) @ V9p ) @ V10p0 )
= V4v_27 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EBRANCHING__BIT__ind,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) )
=> ( ! [V1p0: tp__ty_2Enum_2Enum,V2p1: tp__ty_2Enum_2Enum] :
( ( ~ ( ( ( p @ ( ap @ c_2Earithmetic_2EODD @ ( inj__ty_2Enum_2Enum @ V1p0 ) ) )
<=> ( p @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V2p1 ) ) ) )
| ( V1p0 = V2p1 ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( ap @ c_2Earithmetic_2EDIV2 @ ( inj__ty_2Enum_2Enum @ V1p0 ) ) ) @ ( ap @ c_2Earithmetic_2EDIV2 @ ( inj__ty_2Enum_2Enum @ V2p1 ) ) ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V1p0 ) ) @ ( inj__ty_2Enum_2Enum @ V2p1 ) ) ) )
=> ! [V3v: tp__ty_2Enum_2Enum,V4v1: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V3v ) ) @ ( inj__ty_2Enum_2Enum @ V4v1 ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EBRANCHING__BIT__def,axiom,
! [V0p1: tp__ty_2Enum_2Enum,V1p0: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Epatricia_2EBRANCHING__BIT @ ( inj__ty_2Enum_2Enum @ V1p0 ) ) @ ( inj__ty_2Enum_2Enum @ V0p1 ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ bool ) @ ( ap @ c_2Earithmetic_2EODD @ ( inj__ty_2Enum_2Enum @ V1p0 ) ) ) @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V0p1 ) ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1p0 ) ) @ ( inj__ty_2Enum_2Enum @ V0p1 ) ) ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ c_2Enum_2ESUC @ ( ap @ ( ap @ c_2Epatricia_2EBRANCHING__BIT @ ( ap @ c_2Earithmetic_2EDIV2 @ ( inj__ty_2Enum_2Enum @ V1p0 ) ) ) @ ( ap @ c_2Earithmetic_2EDIV2 @ ( inj__ty_2Enum_2Enum @ V0p1 ) ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EPEEK__ind,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ bool ) ) )
=> ( ( ! [V1k: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0P @ ( c_2Epatricia_2EEmpty @ A_27a ) ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) )
& ! [V2j: tp__ty_2Enum_2Enum,V3d: $i] :
( ( mem @ V3d @ A_27a )
=> ! [V4k: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2j ) ) @ V3d ) ) @ ( inj__ty_2Enum_2Enum @ V4k ) ) ) )
& ! [V5p: tp__ty_2Enum_2Enum,V6m: tp__ty_2Enum_2Enum,V7l: $i] :
( ( mem @ V7l @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V8r: $i] :
( ( mem @ V8r @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V9k: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( ap @ ( ap @ c_2Ebit_2EBIT @ ( inj__ty_2Enum_2Enum @ V6m ) ) @ ( inj__ty_2Enum_2Enum @ V9k ) ) ) @ V7l ) @ V8r ) ) @ ( inj__ty_2Enum_2Enum @ V9k ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V5p ) ) @ ( inj__ty_2Enum_2Enum @ V6m ) ) @ V7l ) @ V8r ) ) @ ( inj__ty_2Enum_2Enum @ V9k ) ) ) ) ) ) )
=> ! [V10v: $i] :
( ( mem @ V10v @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V11v1: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0P @ V10v ) @ ( inj__ty_2Enum_2Enum @ V11v1 ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EPEEK__def,axiom,
! [A_27a: del] :
( ! [V0k: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Epatricia_2EPEEK @ A_27a ) @ ( c_2Epatricia_2EEmpty @ A_27a ) ) @ ( inj__ty_2Enum_2Enum @ V0k ) )
= ( c_2Eoption_2ENONE @ A_27a ) )
& ! [V1k: tp__ty_2Enum_2Enum,V2j: tp__ty_2Enum_2Enum,V3d: $i] :
( ( mem @ V3d @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Epatricia_2EPEEK @ A_27a ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2j ) ) @ V3d ) ) @ ( inj__ty_2Enum_2Enum @ V1k ) )
= ( 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 @ V2j ) ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V3d ) ) @ ( c_2Eoption_2ENONE @ A_27a ) ) ) )
& ! [V4r: $i] :
( ( mem @ V4r @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V5p: tp__ty_2Enum_2Enum,V6m: tp__ty_2Enum_2Enum,V7l: $i] :
( ( mem @ V7l @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V8k: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Epatricia_2EPEEK @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V5p ) ) @ ( inj__ty_2Enum_2Enum @ V6m ) ) @ V7l ) @ V4r ) ) @ ( inj__ty_2Enum_2Enum @ V8k ) )
= ( ap @ ( ap @ ( c_2Epatricia_2EPEEK @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( ap @ ( ap @ c_2Ebit_2EBIT @ ( inj__ty_2Enum_2Enum @ V6m ) ) @ ( inj__ty_2Enum_2Enum @ V8k ) ) ) @ V7l ) @ V4r ) ) @ ( inj__ty_2Enum_2Enum @ V8k ) ) ) ) ) ) ).
thf(ax_thm_2Epatricia_2EJOIN__def,axiom,
! [A_27a: del,V0p0: tp__ty_2Enum_2Enum,V1t0: $i] :
( ( mem @ V1t0 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V2p1: tp__ty_2Enum_2Enum,V3t1: $i] :
( ( mem @ V3t1 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( ( ap @ ( c_2Epatricia_2EJOIN @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0p0 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) @ V1t0 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( inj__ty_2Enum_2Enum @ V2p1 ) ) @ V3t1 ) ) ) )
= ( ap
@ ( ap @ ( c_2Ebool_2ELET @ ty_2Enum_2Enum @ ( ty_2Epatricia_2Eptree @ A_27a ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4m: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( ap @ ( ap @ c_2Ebit_2EBIT @ V4m ) @ ( inj__ty_2Enum_2Enum @ V0p0 ) ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( ap @ ( ap @ c_2Ebit_2EMOD__2EXP @ V4m ) @ ( inj__ty_2Enum_2Enum @ V0p0 ) ) ) @ V4m ) @ V1t0 ) @ V3t1 ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( ap @ ( ap @ c_2Ebit_2EMOD__2EXP @ V4m ) @ ( inj__ty_2Enum_2Enum @ V0p0 ) ) ) @ V4m ) @ V3t1 ) @ V1t0 ) ) ) )
@ ( ap @ ( ap @ c_2Epatricia_2EBRANCHING__BIT @ ( inj__ty_2Enum_2Enum @ V0p0 ) ) @ ( inj__ty_2Enum_2Enum @ V2p1 ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EADD__ind,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) @ bool ) ) )
=> ( ( ! [V1k: tp__ty_2Enum_2Enum,V2e: $i] :
( ( mem @ V2e @ A_27a )
=> ( p @ ( ap @ ( ap @ V0P @ ( c_2Epatricia_2EEmpty @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ V2e ) ) ) )
& ! [V3j: tp__ty_2Enum_2Enum,V4d: $i] :
( ( mem @ V4d @ A_27a )
=> ! [V5k: tp__ty_2Enum_2Enum,V6e: $i] :
( ( mem @ V6e @ A_27a )
=> ( p @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V3j ) ) @ V4d ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V5k ) ) @ V6e ) ) ) ) )
& ! [V7p: tp__ty_2Enum_2Enum,V8m: tp__ty_2Enum_2Enum,V9l: $i] :
( ( mem @ V9l @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V10r: $i] :
( ( mem @ V10r @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V11k: tp__ty_2Enum_2Enum,V12e: $i] :
( ( mem @ V12e @ A_27a )
=> ( ( ( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Ebit_2EMOD__2EXP__EQ @ ( inj__ty_2Enum_2Enum @ V8m ) ) @ ( inj__ty_2Enum_2Enum @ V11k ) ) @ ( inj__ty_2Enum_2Enum @ V7p ) ) )
& ~ ( p @ ( ap @ ( ap @ c_2Ebit_2EBIT @ ( inj__ty_2Enum_2Enum @ V8m ) ) @ ( inj__ty_2Enum_2Enum @ V11k ) ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ V10r ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V11k ) ) @ V12e ) ) ) )
& ( ( ( p @ ( ap @ ( ap @ ( ap @ c_2Ebit_2EMOD__2EXP__EQ @ ( inj__ty_2Enum_2Enum @ V8m ) ) @ ( inj__ty_2Enum_2Enum @ V11k ) ) @ ( inj__ty_2Enum_2Enum @ V7p ) ) )
& ( p @ ( ap @ ( ap @ c_2Ebit_2EBIT @ ( inj__ty_2Enum_2Enum @ V8m ) ) @ ( inj__ty_2Enum_2Enum @ V11k ) ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ V9l ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V11k ) ) @ V12e ) ) ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V7p ) ) @ ( inj__ty_2Enum_2Enum @ V8m ) ) @ V9l ) @ V10r ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V11k ) ) @ V12e ) ) ) ) ) ) ) )
=> ! [V13v: $i] :
( ( mem @ V13v @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V14v1: tp__ty_2Enum_2Enum,V15v2: $i] :
( ( mem @ V15v2 @ A_27a )
=> ( p @ ( ap @ ( ap @ V0P @ V13v ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V14v1 ) ) @ V15v2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EADD__def,axiom,
! [A_27a: del] :
( ! [V0k: tp__ty_2Enum_2Enum,V1e: $i] :
( ( mem @ V1e @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Epatricia_2EADD @ A_27a ) @ ( c_2Epatricia_2EEmpty @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) @ V1e ) )
= ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) @ V1e ) ) )
& ! [V2k: tp__ty_2Enum_2Enum,V3j: tp__ty_2Enum_2Enum,V4e: $i] :
( ( mem @ V4e @ A_27a )
=> ! [V5d: $i] :
( ( mem @ V5d @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Epatricia_2EADD @ A_27a ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V3j ) ) @ V5d ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ V4e ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V3j ) ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ V4e ) ) @ ( ap @ ( c_2Epatricia_2EJOIN @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ V4e ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( inj__ty_2Enum_2Enum @ V3j ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V3j ) ) @ V5d ) ) ) ) ) ) ) ) )
& ! [V6r: $i] :
( ( mem @ V6r @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V7p: tp__ty_2Enum_2Enum,V8m: tp__ty_2Enum_2Enum,V9l: $i] :
( ( mem @ V9l @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V10k: tp__ty_2Enum_2Enum,V11e: $i] :
( ( mem @ V11e @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Epatricia_2EADD @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V7p ) ) @ ( inj__ty_2Enum_2Enum @ V8m ) ) @ V9l ) @ V6r ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V10k ) ) @ V11e ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ c_2Ebit_2EMOD__2EXP__EQ @ ( inj__ty_2Enum_2Enum @ V8m ) ) @ ( inj__ty_2Enum_2Enum @ V10k ) ) @ ( inj__ty_2Enum_2Enum @ V7p ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( ap @ ( ap @ c_2Ebit_2EBIT @ ( inj__ty_2Enum_2Enum @ V8m ) ) @ ( inj__ty_2Enum_2Enum @ V10k ) ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V7p ) ) @ ( inj__ty_2Enum_2Enum @ V8m ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2EADD @ A_27a ) @ V9l ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V10k ) ) @ V11e ) ) ) @ V6r ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V7p ) ) @ ( inj__ty_2Enum_2Enum @ V8m ) ) @ V9l ) @ ( ap @ ( ap @ ( c_2Epatricia_2EADD @ A_27a ) @ V6r ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V10k ) ) @ V11e ) ) ) ) ) @ ( ap @ ( c_2Epatricia_2EJOIN @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V10k ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V10k ) ) @ V11e ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( inj__ty_2Enum_2Enum @ V7p ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V7p ) ) @ ( inj__ty_2Enum_2Enum @ V8m ) ) @ V9l ) @ V6r ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Epatricia_2EBRANCH__primitive__def,axiom,
! [A_27a: del] :
( ( c_2Epatricia_2EBRANCH @ A_27a )
= ( ap
@ ( ap @ ( c_2Erelation_2EWFREC @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) @ ( ty_2Epatricia_2Eptree @ A_27a ) )
@ ( ap @ ( c_2Emin_2E_40 @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) @ bool ) ) )
@ ( lam @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) @ bool ) )
@ ^ [V0R: $i] : ( ap @ ( c_2Erelation_2EWF @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) ) @ V0R ) ) ) )
@ ( lam @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) @ ( ty_2Epatricia_2Eptree @ A_27a ) )
@ ^ [V1BRANCH: $i] :
( lam @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) )
@ ^ [V2a: $i] :
( ap @ ( ap @ ( c_2Epair_2Epair__CASE @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) @ V2a )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3p: $i] :
( lam @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) )
@ ^ [V4v1: $i] :
( ap @ ( ap @ ( c_2Epair_2Epair__CASE @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) @ V4v1 )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5m: $i] :
( lam @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) )
@ ^ [V6v3: $i] :
( ap @ ( ap @ ( c_2Epair_2Epair__CASE @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ V6v3 )
@ ( lam @ ( ty_2Epatricia_2Eptree @ A_27a )
@ ^ [V7v4: $i] :
( lam @ ( ty_2Epatricia_2Eptree @ A_27a )
@ ^ [V8t: $i] :
( ap
@ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2Eptree__CASE @ A_27a @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ V7v4 ) @ ( ap @ ( c_2Ecombin_2EI @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ V8t ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V9v18: $i] :
( lam @ A_27a
@ ^ [V10v19: $i] :
( ap
@ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2Eptree__CASE @ A_27a @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ V8t ) @ ( ap @ ( c_2Ecombin_2EI @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ V9v18 ) @ V10v19 ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V11v30: $i] :
( lam @ A_27a
@ ^ [V12v31: $i] : ( ap @ ( c_2Ecombin_2EI @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ V3p ) @ V5m ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ V9v18 ) @ V10v19 ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ V11v30 ) @ V12v31 ) ) ) ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V13v32: $i] :
( lam @ ty_2Enum_2Enum
@ ^ [V14v33: $i] :
( lam @ ( ty_2Epatricia_2Eptree @ A_27a )
@ ^ [V15v34: $i] :
( lam @ ( ty_2Epatricia_2Eptree @ A_27a )
@ ^ [V16v35: $i] : ( ap @ ( c_2Ecombin_2EI @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ V3p ) @ V5m ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ V9v18 ) @ V10v19 ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ V13v32 ) @ V14v33 ) @ V15v34 ) @ V16v35 ) ) ) ) ) ) ) ) ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V17v20: $i] :
( lam @ ty_2Enum_2Enum
@ ^ [V18v21: $i] :
( lam @ ( ty_2Epatricia_2Eptree @ A_27a )
@ ^ [V19v22: $i] :
( lam @ ( ty_2Epatricia_2Eptree @ A_27a )
@ ^ [V20v23: $i] :
( ap
@ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2Eptree__CASE @ A_27a @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ V8t ) @ ( ap @ ( c_2Ecombin_2EI @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ V17v20 ) @ V18v21 ) @ V19v22 ) @ V20v23 ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V21v42: $i] :
( lam @ A_27a
@ ^ [V22v43: $i] : ( ap @ ( c_2Ecombin_2EI @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ V3p ) @ V5m ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ V17v20 ) @ V18v21 ) @ V19v22 ) @ V20v23 ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ V21v42 ) @ V22v43 ) ) ) ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V23v44: $i] :
( lam @ ty_2Enum_2Enum
@ ^ [V24v45: $i] :
( lam @ ( ty_2Epatricia_2Eptree @ A_27a )
@ ^ [V25v46: $i] :
( lam @ ( ty_2Epatricia_2Eptree @ A_27a )
@ ^ [V26v47: $i] : ( ap @ ( c_2Ecombin_2EI @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ V3p ) @ V5m ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ V17v20 ) @ V18v21 ) @ V19v22 ) @ V20v23 ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ V23v44 ) @ V24v45 ) @ V25v46 ) @ V26v47 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EBRANCH__ind,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) @ bool ) )
=> ( ( ! [V1p: tp__ty_2Enum_2Enum,V2m: tp__ty_2Enum_2Enum,V3t: $i] :
( ( mem @ V3t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1p ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( c_2Epatricia_2EEmpty @ A_27a ) ) @ V3t ) ) ) ) ) )
& ! [V4p: tp__ty_2Enum_2Enum,V5m: tp__ty_2Enum_2Enum,V6v6: tp__ty_2Enum_2Enum,V7v7: $i] :
( ( mem @ V7v7 @ A_27a )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V4p ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) @ ( inj__ty_2Enum_2Enum @ V5m ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V6v6 ) ) @ V7v7 ) ) @ ( c_2Epatricia_2EEmpty @ A_27a ) ) ) ) ) ) )
& ! [V8p: tp__ty_2Enum_2Enum,V9m: tp__ty_2Enum_2Enum,V10v8: tp__ty_2Enum_2Enum,V11v9: tp__ty_2Enum_2Enum,V12v10: $i] :
( ( mem @ V12v10 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V13v11: $i] :
( ( mem @ V13v11 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V8p ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) @ ( inj__ty_2Enum_2Enum @ V9m ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V10v8 ) ) @ ( inj__ty_2Enum_2Enum @ V11v9 ) ) @ V12v10 ) @ V13v11 ) ) @ ( c_2Epatricia_2EEmpty @ A_27a ) ) ) ) ) ) ) )
& ! [V14p: tp__ty_2Enum_2Enum,V15m: tp__ty_2Enum_2Enum,V16v12: tp__ty_2Enum_2Enum,V17v13: $i] :
( ( mem @ V17v13 @ A_27a )
=> ! [V18v24: tp__ty_2Enum_2Enum,V19v25: $i] :
( ( mem @ V19v25 @ A_27a )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V14p ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) @ ( inj__ty_2Enum_2Enum @ V15m ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V16v12 ) ) @ V17v13 ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V18v24 ) ) @ V19v25 ) ) ) ) ) ) ) )
& ! [V20p: tp__ty_2Enum_2Enum,V21m: tp__ty_2Enum_2Enum,V22v12: tp__ty_2Enum_2Enum,V23v13: $i] :
( ( mem @ V23v13 @ A_27a )
=> ! [V24v26: tp__ty_2Enum_2Enum,V25v27: tp__ty_2Enum_2Enum,V26v28: $i] :
( ( mem @ V26v28 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V27v29: $i] :
( ( mem @ V27v29 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V20p ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) @ ( inj__ty_2Enum_2Enum @ V21m ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V22v12 ) ) @ V23v13 ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V24v26 ) ) @ ( inj__ty_2Enum_2Enum @ V25v27 ) ) @ V26v28 ) @ V27v29 ) ) ) ) ) ) ) ) )
& ! [V28p: tp__ty_2Enum_2Enum,V29m: tp__ty_2Enum_2Enum,V30v14: tp__ty_2Enum_2Enum,V31v15: tp__ty_2Enum_2Enum,V32v16: $i] :
( ( mem @ V32v16 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V33v17: $i] :
( ( mem @ V33v17 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V34v36: tp__ty_2Enum_2Enum,V35v37: $i] :
( ( mem @ V35v37 @ A_27a )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V28p ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) @ ( inj__ty_2Enum_2Enum @ V29m ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V30v14 ) ) @ ( inj__ty_2Enum_2Enum @ V31v15 ) ) @ V32v16 ) @ V33v17 ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V34v36 ) ) @ V35v37 ) ) ) ) ) ) ) ) )
& ! [V36p: tp__ty_2Enum_2Enum,V37m: tp__ty_2Enum_2Enum,V38v14: tp__ty_2Enum_2Enum,V39v15: tp__ty_2Enum_2Enum,V40v16: $i] :
( ( mem @ V40v16 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V41v17: $i] :
( ( mem @ V41v17 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V42v38: tp__ty_2Enum_2Enum,V43v39: tp__ty_2Enum_2Enum,V44v40: $i] :
( ( mem @ V44v40 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V45v41: $i] :
( ( mem @ V45v41 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V36p ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) @ ( inj__ty_2Enum_2Enum @ V37m ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V38v14 ) ) @ ( inj__ty_2Enum_2Enum @ V39v15 ) ) @ V40v16 ) @ V41v17 ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V42v38 ) ) @ ( inj__ty_2Enum_2Enum @ V43v39 ) ) @ V44v40 ) @ V45v41 ) ) ) ) ) ) ) ) ) ) )
=> ! [V46v: tp__ty_2Enum_2Enum,V47v1: tp__ty_2Enum_2Enum,V48v2: $i] :
( ( mem @ V48v2 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V49v3: $i] :
( ( mem @ V49v3 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V46v ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) @ ( inj__ty_2Enum_2Enum @ V47v1 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ V48v2 ) @ V49v3 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EBRANCH__def,axiom,
! [A_27a: del,V0p: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum,V2t: $i] :
( ( mem @ V2t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V3v6: tp__ty_2Enum_2Enum,V4v7: $i] :
( ( mem @ V4v7 @ A_27a )
=> ! [V5v8: tp__ty_2Enum_2Enum,V6v9: tp__ty_2Enum_2Enum,V7v10: $i] :
( ( mem @ V7v10 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V8v11: $i] :
( ( mem @ V8v11 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V9v12: tp__ty_2Enum_2Enum,V10v13: $i] :
( ( mem @ V10v13 @ A_27a )
=> ! [V11v24: tp__ty_2Enum_2Enum,V12v25: $i] :
( ( mem @ V12v25 @ A_27a )
=> ! [V13v26: tp__ty_2Enum_2Enum,V14v27: tp__ty_2Enum_2Enum,V15v28: $i] :
( ( mem @ V15v28 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V16v29: $i] :
( ( mem @ V16v29 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V17v14: tp__ty_2Enum_2Enum,V18v15: tp__ty_2Enum_2Enum,V19v16: $i] :
( ( mem @ V19v16 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V20v17: $i] :
( ( mem @ V20v17 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V21v36: tp__ty_2Enum_2Enum,V22v37: $i] :
( ( mem @ V22v37 @ A_27a )
=> ! [V23v38: tp__ty_2Enum_2Enum,V24v39: tp__ty_2Enum_2Enum,V25v40: $i] :
( ( mem @ V25v40 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V26v41: $i] :
( ( mem @ V26v41 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( ( ( ap @ ( c_2Epatricia_2EBRANCH @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0p ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( c_2Epatricia_2EEmpty @ A_27a ) ) @ V2t ) ) ) )
= V2t )
& ( ( ap @ ( c_2Epatricia_2EBRANCH @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0p ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V3v6 ) ) @ V4v7 ) ) @ ( c_2Epatricia_2EEmpty @ A_27a ) ) ) ) )
= ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V3v6 ) ) @ V4v7 ) )
& ( ( ap @ ( c_2Epatricia_2EBRANCH @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0p ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V5v8 ) ) @ ( inj__ty_2Enum_2Enum @ V6v9 ) ) @ V7v10 ) @ V8v11 ) ) @ ( c_2Epatricia_2EEmpty @ A_27a ) ) ) ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V5v8 ) ) @ ( inj__ty_2Enum_2Enum @ V6v9 ) ) @ V7v10 ) @ V8v11 ) )
& ( ( ap @ ( c_2Epatricia_2EBRANCH @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0p ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V9v12 ) ) @ V10v13 ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V11v24 ) ) @ V12v25 ) ) ) ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0p ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V9v12 ) ) @ V10v13 ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V11v24 ) ) @ V12v25 ) ) )
& ( ( ap @ ( c_2Epatricia_2EBRANCH @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0p ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V9v12 ) ) @ V10v13 ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V13v26 ) ) @ ( inj__ty_2Enum_2Enum @ V14v27 ) ) @ V15v28 ) @ V16v29 ) ) ) ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0p ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V9v12 ) ) @ V10v13 ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V13v26 ) ) @ ( inj__ty_2Enum_2Enum @ V14v27 ) ) @ V15v28 ) @ V16v29 ) ) )
& ( ( ap @ ( c_2Epatricia_2EBRANCH @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0p ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V17v14 ) ) @ ( inj__ty_2Enum_2Enum @ V18v15 ) ) @ V19v16 ) @ V20v17 ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V21v36 ) ) @ V22v37 ) ) ) ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0p ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V17v14 ) ) @ ( inj__ty_2Enum_2Enum @ V18v15 ) ) @ V19v16 ) @ V20v17 ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V21v36 ) ) @ V22v37 ) ) )
& ( ( ap @ ( c_2Epatricia_2EBRANCH @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0p ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V17v14 ) ) @ ( inj__ty_2Enum_2Enum @ V18v15 ) ) @ V19v16 ) @ V20v17 ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V23v38 ) ) @ ( inj__ty_2Enum_2Enum @ V24v39 ) ) @ V25v40 ) @ V26v41 ) ) ) ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0p ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V17v14 ) ) @ ( inj__ty_2Enum_2Enum @ V18v15 ) ) @ V19v16 ) @ V20v17 ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V23v38 ) ) @ ( inj__ty_2Enum_2Enum @ V24v39 ) ) @ V25v40 ) @ V26v41 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Epatricia_2EREMOVE__def,axiom,
! [A_27a: del] :
( ! [V0k: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Epatricia_2EREMOVE @ A_27a ) @ ( c_2Epatricia_2EEmpty @ A_27a ) ) @ ( inj__ty_2Enum_2Enum @ V0k ) )
= ( c_2Epatricia_2EEmpty @ A_27a ) )
& ! [V1j: tp__ty_2Enum_2Enum,V2d: $i] :
( ( mem @ V2d @ A_27a )
=> ! [V3k: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Epatricia_2EREMOVE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1j ) ) @ V2d ) ) @ ( inj__ty_2Enum_2Enum @ V3k ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1j ) ) @ ( inj__ty_2Enum_2Enum @ V3k ) ) ) @ ( c_2Epatricia_2EEmpty @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1j ) ) @ V2d ) ) ) )
& ! [V4p: tp__ty_2Enum_2Enum,V5m: tp__ty_2Enum_2Enum,V6l: $i] :
( ( mem @ V6l @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V7r: $i] :
( ( mem @ V7r @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V8k: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Epatricia_2EREMOVE @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V4p ) ) @ ( inj__ty_2Enum_2Enum @ V5m ) ) @ V6l ) @ V7r ) ) @ ( inj__ty_2Enum_2Enum @ V8k ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ c_2Ebit_2EMOD__2EXP__EQ @ ( inj__ty_2Enum_2Enum @ V5m ) ) @ ( inj__ty_2Enum_2Enum @ V8k ) ) @ ( inj__ty_2Enum_2Enum @ V4p ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( ap @ ( ap @ c_2Ebit_2EBIT @ ( inj__ty_2Enum_2Enum @ V5m ) ) @ ( inj__ty_2Enum_2Enum @ V8k ) ) ) @ ( ap @ ( c_2Epatricia_2EBRANCH @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V4p ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) @ ( inj__ty_2Enum_2Enum @ V5m ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2EREMOVE @ A_27a ) @ V6l ) @ ( inj__ty_2Enum_2Enum @ V8k ) ) ) @ V7r ) ) ) ) ) @ ( ap @ ( c_2Epatricia_2EBRANCH @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V4p ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) @ ( inj__ty_2Enum_2Enum @ V5m ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ V6l ) @ ( ap @ ( ap @ ( c_2Epatricia_2EREMOVE @ A_27a ) @ V7r ) @ ( inj__ty_2Enum_2Enum @ V8k ) ) ) ) ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V4p ) ) @ ( inj__ty_2Enum_2Enum @ V5m ) ) @ V6l ) @ V7r ) ) ) ) ) ) ).
thf(ax_thm_2Epatricia_2ETRAVERSE__AUX__def,axiom,
! [A_27a: del] :
( ! [V0a: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] :
( ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epatricia_2ETRAVERSE__AUX @ A_27a ) @ ( c_2Epatricia_2EEmpty @ A_27a ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V0a ) ) )
= V0a )
& ! [V1k: tp__ty_2Enum_2Enum,V2d: $i] :
( ( mem @ V2d @ A_27a )
=> ! [V3a: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] :
( ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epatricia_2ETRAVERSE__AUX @ A_27a ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ V2d ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V3a ) ) )
= ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V3a ) ) ) ) )
& ! [V4p: tp__ty_2Enum_2Enum,V5m: tp__ty_2Enum_2Enum,V6l: $i] :
( ( mem @ V6l @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V7r: $i] :
( ( mem @ V7r @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V8a: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] :
( ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epatricia_2ETRAVERSE__AUX @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V4p ) ) @ ( inj__ty_2Enum_2Enum @ V5m ) ) @ V6l ) @ V7r ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V8a ) ) )
= ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epatricia_2ETRAVERSE__AUX @ A_27a ) @ V6l ) @ ( ap @ ( ap @ ( c_2Epatricia_2ETRAVERSE__AUX @ A_27a ) @ V7r ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V8a ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Epatricia_2ETRAVERSE__def,axiom,
! [A_27a: del] :
( ( ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ A_27a ) @ ( c_2Epatricia_2EEmpty @ A_27a ) ) )
= ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( c_2Elist_2ENIL @ ty_2Enum_2Enum ) ) )
& ! [V0j: tp__ty_2Enum_2Enum,V1d: $i] :
( ( mem @ V1d @ A_27a )
=> ( ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0j ) ) @ V1d ) ) )
= ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0j ) ) @ ( c_2Elist_2ENIL @ ty_2Enum_2Enum ) ) ) ) )
& ! [V2p: tp__ty_2Enum_2Enum,V3m: tp__ty_2Enum_2Enum,V4l: $i] :
( ( mem @ V4l @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V5r: $i] :
( ( mem @ V5r @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2p ) ) @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ V4l ) @ V5r ) ) )
= ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ A_27a ) @ V4l ) ) @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ A_27a ) @ V5r ) ) ) ) ) ) ) ).
thf(ax_thm_2Epatricia_2EKEYS__def,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( ap @ ( c_2Epatricia_2EKEYS @ A_27a ) @ V0t ) )
= ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Esorting_2EQSORT @ ty_2Enum_2Enum ) @ c_2Eprim__rec_2E_3C ) @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ A_27a ) @ V0t ) ) ) ) ) ).
thf(ax_thm_2Epatricia_2ETRANSFORM__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Epatricia_2ETRANSFORM @ A_27a @ A_27b ) @ V0f ) @ ( c_2Epatricia_2EEmpty @ A_27b ) )
= ( c_2Epatricia_2EEmpty @ A_27a ) ) )
& ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ A_27a ) )
=> ! [V2j: tp__ty_2Enum_2Enum,V3d: $i] :
( ( mem @ V3d @ A_27b )
=> ( ( ap @ ( ap @ ( c_2Epatricia_2ETRANSFORM @ A_27a @ A_27b ) @ V1f ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V2j ) ) @ V3d ) )
= ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2j ) ) @ ( ap @ V1f @ V3d ) ) ) ) )
& ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ A_27b @ A_27a ) )
=> ! [V5p: tp__ty_2Enum_2Enum,V6m: tp__ty_2Enum_2Enum,V7l: $i] :
( ( mem @ V7l @ ( ty_2Epatricia_2Eptree @ A_27b ) )
=> ! [V8r: $i] :
( ( mem @ V8r @ ( ty_2Epatricia_2Eptree @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Epatricia_2ETRANSFORM @ A_27a @ A_27b ) @ V4f ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V5p ) ) @ ( inj__ty_2Enum_2Enum @ V6m ) ) @ V7l ) @ V8r ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V5p ) ) @ ( inj__ty_2Enum_2Enum @ V6m ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2ETRANSFORM @ A_27a @ A_27b ) @ V4f ) @ V7l ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2ETRANSFORM @ A_27a @ A_27b ) @ V4f ) @ V8r ) ) ) ) ) ) ) ).
thf(ax_thm_2Epatricia_2EEVERY__LEAF__def,axiom,
! [A_27a: del] :
( ! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epatricia_2EEVERY__LEAF @ A_27a ) @ V0P ) @ ( c_2Epatricia_2EEmpty @ A_27a ) ) )
<=> $true ) )
& ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V2j: tp__ty_2Enum_2Enum,V3d: $i] :
( ( mem @ V3d @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epatricia_2EEVERY__LEAF @ A_27a ) @ V1P ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2j ) ) @ V3d ) ) )
<=> ( p @ ( ap @ ( ap @ V1P @ ( inj__ty_2Enum_2Enum @ V2j ) ) @ V3d ) ) ) ) )
& ! [V4P: $i] :
( ( mem @ V4P @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V5p: tp__ty_2Enum_2Enum,V6m: tp__ty_2Enum_2Enum,V7l: $i] :
( ( mem @ V7l @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V8r: $i] :
( ( mem @ V8r @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epatricia_2EEVERY__LEAF @ A_27a ) @ V4P ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V5p ) ) @ ( inj__ty_2Enum_2Enum @ V6m ) ) @ V7l ) @ V8r ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Epatricia_2EEVERY__LEAF @ A_27a ) @ V4P ) @ V7l ) )
& ( p @ ( ap @ ( ap @ ( c_2Epatricia_2EEVERY__LEAF @ A_27a ) @ V4P ) @ V8r ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Epatricia_2EEXISTS__LEAF__def,axiom,
! [A_27a: del] :
( ! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epatricia_2EEXISTS__LEAF @ A_27a ) @ V0P ) @ ( c_2Epatricia_2EEmpty @ A_27a ) ) )
<=> $false ) )
& ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V2j: tp__ty_2Enum_2Enum,V3d: $i] :
( ( mem @ V3d @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epatricia_2EEXISTS__LEAF @ A_27a ) @ V1P ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2j ) ) @ V3d ) ) )
<=> ( p @ ( ap @ ( ap @ V1P @ ( inj__ty_2Enum_2Enum @ V2j ) ) @ V3d ) ) ) ) )
& ! [V4P: $i] :
( ( mem @ V4P @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V5p: tp__ty_2Enum_2Enum,V6m: tp__ty_2Enum_2Enum,V7l: $i] :
( ( mem @ V7l @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V8r: $i] :
( ( mem @ V8r @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epatricia_2EEXISTS__LEAF @ A_27a ) @ V4P ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V5p ) ) @ ( inj__ty_2Enum_2Enum @ V6m ) ) @ V7l ) @ V8r ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Epatricia_2EEXISTS__LEAF @ A_27a ) @ V4P ) @ V7l ) )
| ( p @ ( ap @ ( ap @ ( c_2Epatricia_2EEXISTS__LEAF @ A_27a ) @ V4P ) @ V8r ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Epatricia_2ESIZE__def,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epatricia_2ESIZE @ A_27a ) @ V0t ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ A_27a ) @ V0t ) ) ) ) ) ).
thf(ax_thm_2Epatricia_2EDEPTH__def,axiom,
! [A_27a: del] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epatricia_2EDEPTH @ A_27a ) @ ( c_2Epatricia_2EEmpty @ A_27a ) ) )
= fo__c_2Enum_2E0 )
& ! [V0j: tp__ty_2Enum_2Enum,V1d: $i] :
( ( mem @ V1d @ A_27a )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epatricia_2EDEPTH @ A_27a ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0j ) ) @ V1d ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
& ! [V2p: tp__ty_2Enum_2Enum,V3m: tp__ty_2Enum_2Enum,V4l: $i] :
( ( mem @ V4l @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V5r: $i] :
( ( mem @ V5r @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epatricia_2EDEPTH @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2p ) ) @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ V4l ) @ V5r ) ) )
= ( 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_2EMAX @ ( ap @ ( c_2Epatricia_2EDEPTH @ A_27a ) @ V4l ) ) @ ( ap @ ( c_2Epatricia_2EDEPTH @ A_27a ) @ V5r ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Epatricia_2EIS__PTREE__def,axiom,
! [A_27a: del] :
( ( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ ( c_2Epatricia_2EEmpty @ A_27a ) ) )
<=> $true )
& ! [V0k: tp__ty_2Enum_2Enum,V1d: $i] :
( ( mem @ V1d @ A_27a )
=> ( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) @ V1d ) ) )
<=> $true ) )
& ! [V2p: tp__ty_2Enum_2Enum,V3m: tp__ty_2Enum_2Enum,V4l: $i] :
( ( mem @ V4l @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V5r: $i] :
( ( mem @ V5r @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2p ) ) @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ V4l ) @ V5r ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2p ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) )
& ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ V4l ) )
& ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ V5r ) )
& ( V4l
!= ( c_2Epatricia_2EEmpty @ A_27a ) )
& ( V5r
!= ( c_2Epatricia_2EEmpty @ A_27a ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Epatricia_2EEVERY__LEAF @ A_27a )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V6k: $i] :
( lam @ A_27a
@ ^ [V7d: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( ap @ c_2Ebit_2EMOD__2EXP__EQ @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ V6k ) @ ( inj__ty_2Enum_2Enum @ V2p ) ) ) @ ( ap @ ( ap @ c_2Ebit_2EBIT @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ V6k ) ) ) ) )
@ V4l ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Epatricia_2EEVERY__LEAF @ A_27a )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V8k: $i] :
( lam @ A_27a
@ ^ [V9d: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( ap @ c_2Ebit_2EMOD__2EXP__EQ @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ V8k ) @ ( inj__ty_2Enum_2Enum @ V2p ) ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ c_2Ebit_2EBIT @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ V8k ) ) ) ) ) )
@ V5r ) ) ) ) ) ) ) ).
thf(ax_thm_2Epatricia_2EIN__PTREE__def,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1t: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone] :
( ( p @ ( ap @ ( ap @ c_2Epatricia_2EIN__PTREE @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V1t ) ) )
<=> ( p @ ( ap @ ( c_2Eoption_2EIS__SOME @ ty_2Eone_2Eone ) @ ( ap @ ( ap @ ( c_2Epatricia_2EPEEK @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V1t ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ).
thf(ax_thm_2Epatricia_2EINSERT__PTREE__def,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1t: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone] :
( ( surj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ ( ap @ ( ap @ c_2Epatricia_2EINSERT__PTREE @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V1t ) ) )
= ( surj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ ( ap @ ( ap @ ( c_2Epatricia_2EADD @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V1t ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Eone_2Eone ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Eone_2Eone @ fo__c_2Eone_2Eone ) ) ) ) ) ).
thf(ax_thm_2Epatricia_2EPTREE__OF__NUMSET__def,axiom,
! [V0t: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( surj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ ( ap @ ( ap @ c_2Epatricia_2EPTREE__OF__NUMSET @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) @ V1s ) )
= ( surj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2EFOLDL @ ty_2Enum_2Enum @ ( ty_2Epatricia_2Eptree @ ty_2Eone_2Eone ) ) @ ( ap @ ( c_2Ecombin_2EC @ ty_2Enum_2Enum @ ( ty_2Epatricia_2Eptree @ ty_2Eone_2Eone ) @ ( ty_2Epatricia_2Eptree @ ty_2Eone_2Eone ) ) @ c_2Epatricia_2EINSERT__PTREE ) ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) @ ( ap @ ( c_2Elist_2ESET__TO__LIST @ ty_2Enum_2Enum ) @ V1s ) ) ) ) ) ).
thf(ax_thm_2Epatricia_2ENUMSET__OF__PTREE__def,axiom,
! [V0t: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone] :
( ( ap @ c_2Epatricia_2ENUMSET__OF__PTREE @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) )
= ( ap @ ( c_2Elist_2ELIST__TO__SET @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) ) ) ).
thf(ax_thm_2Epatricia_2EUNION__PTREE__def,axiom,
! [V0t1: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone,V1t2: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone] :
( ( surj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ ( ap @ ( ap @ c_2Epatricia_2EUNION__PTREE @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t1 ) ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V1t2 ) ) )
= ( surj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ ( ap @ ( ap @ c_2Epatricia_2EPTREE__OF__NUMSET @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t1 ) ) @ ( ap @ c_2Epatricia_2ENUMSET__OF__PTREE @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V1t2 ) ) ) ) ) ).
thf(ax_thm_2Epatricia_2EIS__EMPTY__primitive__def,axiom,
! [A_27a: del] :
( ( c_2Epatricia_2EIS__EMPTY @ A_27a )
= ( ap
@ ( ap @ ( c_2Erelation_2EWFREC @ ( ty_2Epatricia_2Eptree @ A_27a ) @ bool )
@ ( ap @ ( c_2Emin_2E_40 @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ bool ) ) )
@ ( lam @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ bool ) )
@ ^ [V0R: $i] : ( ap @ ( c_2Erelation_2EWF @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ V0R ) ) ) )
@ ( lam @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ bool )
@ ^ [V1IS__EMPTY: $i] :
( lam @ ( ty_2Epatricia_2Eptree @ A_27a )
@ ^ [V2a: $i] :
( ap
@ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2Eptree__CASE @ A_27a @ bool ) @ V2a ) @ ( ap @ ( c_2Ecombin_2EI @ bool ) @ c_2Ebool_2ET ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3v6: $i] :
( lam @ A_27a
@ ^ [V4v7: $i] : ( ap @ ( c_2Ecombin_2EI @ bool ) @ c_2Ebool_2EF ) ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5v8: $i] :
( lam @ ty_2Enum_2Enum
@ ^ [V6v9: $i] :
( lam @ ( ty_2Epatricia_2Eptree @ A_27a )
@ ^ [V7v10: $i] :
( lam @ ( ty_2Epatricia_2Eptree @ A_27a )
@ ^ [V8v11: $i] : ( ap @ ( c_2Ecombin_2EI @ bool ) @ c_2Ebool_2EF ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EIS__EMPTY__ind,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Epatricia_2Eptree @ A_27a ) @ bool ) )
=> ( ( ( p @ ( ap @ V0P @ ( c_2Epatricia_2EEmpty @ A_27a ) ) )
& ! [V1v: tp__ty_2Enum_2Enum,V2v1: $i] :
( ( mem @ V2v1 @ A_27a )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1v ) ) @ V2v1 ) ) ) )
& ! [V3v2: tp__ty_2Enum_2Enum,V4v3: tp__ty_2Enum_2Enum,V5v4: $i] :
( ( mem @ V5v4 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V6v5: $i] :
( ( mem @ V6v5 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V3v2 ) ) @ ( inj__ty_2Enum_2Enum @ V4v3 ) ) @ V5v4 ) @ V6v5 ) ) ) ) ) )
=> ! [V7v: $i] :
( ( mem @ V7v @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( p @ ( ap @ V0P @ V7v ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EIS__EMPTY__def,axiom,
! [A_27a: del,V0v: tp__ty_2Enum_2Enum,V1v1: $i] :
( ( mem @ V1v1 @ A_27a )
=> ! [V2v2: tp__ty_2Enum_2Enum,V3v3: tp__ty_2Enum_2Enum,V4v4: $i] :
( ( mem @ V4v4 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V5v5: $i] :
( ( mem @ V5v5 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( ( ( p @ ( ap @ ( c_2Epatricia_2EIS__EMPTY @ A_27a ) @ ( c_2Epatricia_2EEmpty @ A_27a ) ) )
<=> $true )
& ( ( p @ ( ap @ ( c_2Epatricia_2EIS__EMPTY @ A_27a ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0v ) ) @ V1v1 ) ) )
<=> $false )
& ( ( p @ ( ap @ ( c_2Epatricia_2EIS__EMPTY @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2v2 ) ) @ ( inj__ty_2Enum_2Enum @ V3v3 ) ) @ V4v4 ) @ V5v5 ) ) )
<=> $false ) ) ) ) ) ).
thf(ax_thm_2Epatricia_2EFIND__def,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V1k: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Epatricia_2EFIND @ A_27a ) @ V0t ) @ ( inj__ty_2Enum_2Enum @ V1k ) )
= ( ap @ ( c_2Eoption_2ETHE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epatricia_2EPEEK @ A_27a ) @ V0t ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) ) ) ) ).
thf(ax_thm_2Epatricia_2EADD__LIST__def,axiom,
! [A_27a: del] :
( ( c_2Epatricia_2EADD__LIST @ A_27a )
= ( ap @ ( c_2Elist_2EFOLDL @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( c_2Epatricia_2EADD @ A_27a ) ) ) ).
thf(conj_thm_2Epatricia_2EBRANCHING__BIT,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum] :
( ( V0a != V1b )
=> ~ ( ( p @ ( ap @ ( ap @ c_2Ebit_2EBIT @ ( ap @ ( ap @ c_2Epatricia_2EBRANCHING__BIT @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ebit_2EBIT @ ( ap @ ( ap @ c_2Epatricia_2EBRANCHING__BIT @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EBRANCHING__BIT__ZERO,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Epatricia_2EBRANCHING__BIT @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
= fo__c_2Enum_2E0 )
<=> ( ( ( p @ ( ap @ c_2Earithmetic_2EODD @ ( inj__ty_2Enum_2Enum @ V0a ) ) )
<=> ( p @ ( ap @ c_2Earithmetic_2EEVEN @ ( inj__ty_2Enum_2Enum @ V1b ) ) ) )
| ( V0a = V1b ) ) ) ).
thf(conj_thm_2Epatricia_2EBRANCHING__BIT__SYM,axiom,
! [V0a: tp__ty_2Enum_2Enum,V1b: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Epatricia_2EBRANCHING__BIT @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ ( inj__ty_2Enum_2Enum @ V1b ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Epatricia_2EBRANCHING__BIT @ ( inj__ty_2Enum_2Enum @ V1b ) ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) ) ) ).
thf(conj_thm_2Epatricia_2EEVERY__LEAF__ADD,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V2k: tp__ty_2Enum_2Enum,V3d: $i] :
( ( mem @ V3d @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ V3d ) )
& ( p @ ( ap @ ( ap @ ( c_2Epatricia_2EEVERY__LEAF @ A_27a ) @ V0P ) @ V1t ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epatricia_2EEVERY__LEAF @ A_27a ) @ V0P ) @ ( ap @ ( ap @ ( c_2Epatricia_2EADD @ A_27a ) @ V1t ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ V3d ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EMONO__EVERY__LEAF,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( ( ! [V3k: tp__ty_2Enum_2Enum,V4d: $i] :
( ( mem @ V4d @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V3k ) ) @ V4d ) )
=> ( p @ ( ap @ ( ap @ V1Q @ ( inj__ty_2Enum_2Enum @ V3k ) ) @ V4d ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epatricia_2EEVERY__LEAF @ A_27a ) @ V0P ) @ V2t ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epatricia_2EEVERY__LEAF @ A_27a ) @ V1Q ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2ENOT__ADD__EMPTY,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V1k: tp__ty_2Enum_2Enum,V2d: $i] :
( ( mem @ V2d @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Epatricia_2EADD @ A_27a ) @ V0t ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ V2d ) )
!= ( c_2Epatricia_2EEmpty @ A_27a ) ) ) ) ).
thf(conj_thm_2Epatricia_2EEMPTY__IS__PTREE,axiom,
! [A_27a: del] : ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ ( c_2Epatricia_2EEmpty @ A_27a ) ) ) ).
thf(conj_thm_2Epatricia_2EADD__IS__PTREE,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ V0t ) )
=> ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epatricia_2EADD @ A_27a ) @ V0t ) @ V1x ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EEVERY__LEAF__BRANCH,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V1p: tp__ty_2Enum_2Enum,V2m: tp__ty_2Enum_2Enum,V3l: $i] :
( ( mem @ V3l @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V4r: $i] :
( ( mem @ V4r @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epatricia_2EEVERY__LEAF @ A_27a ) @ V0P ) @ ( ap @ ( c_2Epatricia_2EBRANCH @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1p ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ V3l ) @ V4r ) ) ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Epatricia_2EEVERY__LEAF @ A_27a ) @ V0P ) @ V3l ) )
& ( p @ ( ap @ ( ap @ ( c_2Epatricia_2EEVERY__LEAF @ A_27a ) @ V0P ) @ V4r ) ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EEVERY__LEAF__REMOVE,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V2k: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Epatricia_2EEVERY__LEAF @ A_27a ) @ V0P ) @ V1t ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epatricia_2EEVERY__LEAF @ A_27a ) @ V0P ) @ ( ap @ ( ap @ ( c_2Epatricia_2EREMOVE @ A_27a ) @ V1t ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EIS__PTREE__BRANCH,axiom,
! [A_27a: del,V0p: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum,V2l: $i] :
( ( mem @ V2l @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V3r: $i] :
( ( mem @ V3r @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0p ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) )
& ~ ( ( V2l
= ( c_2Epatricia_2EEmpty @ A_27a ) )
& ( V3r
= ( c_2Epatricia_2EEmpty @ A_27a ) ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Epatricia_2EEVERY__LEAF @ A_27a )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4k: $i] :
( lam @ A_27a
@ ^ [V5d: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( ap @ c_2Ebit_2EMOD__2EXP__EQ @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ V4k ) @ ( inj__ty_2Enum_2Enum @ V0p ) ) ) @ ( ap @ ( ap @ c_2Ebit_2EBIT @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ V4k ) ) ) ) )
@ V2l ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Epatricia_2EEVERY__LEAF @ A_27a )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V6k: $i] :
( lam @ A_27a
@ ^ [V7d: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( ap @ c_2Ebit_2EMOD__2EXP__EQ @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ V6k ) @ ( inj__ty_2Enum_2Enum @ V0p ) ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ c_2Ebit_2EBIT @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ V6k ) ) ) ) ) )
@ V3r ) )
& ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ V2l ) )
& ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ V3r ) ) )
=> ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ ( ap @ ( c_2Epatricia_2EBRANCH @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0p ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ V2l ) @ V3r ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EREMOVE__IS__PTREE,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V1k: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ V0t ) )
=> ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epatricia_2EREMOVE @ A_27a ) @ V0t ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EPEEK__NONE,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V2k: tp__ty_2Enum_2Enum] :
( ( ! [V3d: $i] :
( ( mem @ V3d @ A_27a )
=> ~ ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ V3d ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epatricia_2EEVERY__LEAF @ A_27a ) @ V0P ) @ V1t ) ) )
=> ( ( ap @ ( ap @ ( c_2Epatricia_2EPEEK @ A_27a ) @ V1t ) @ ( inj__ty_2Enum_2Enum @ V2k ) )
= ( c_2Eoption_2ENONE @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EPEEK__ADD,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V1k: tp__ty_2Enum_2Enum,V2d: $i] :
( ( mem @ V2d @ A_27a )
=> ! [V3j: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ V0t ) )
=> ( ( ap @ ( ap @ ( c_2Epatricia_2EPEEK @ A_27a ) @ ( ap @ ( ap @ ( c_2Epatricia_2EADD @ A_27a ) @ V0t ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ V2d ) ) ) @ ( inj__ty_2Enum_2Enum @ V3j ) )
= ( 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 @ V3j ) ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V2d ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2EPEEK @ A_27a ) @ V0t ) @ ( inj__ty_2Enum_2Enum @ V3j ) ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EBRANCH,axiom,
! [A_27a: del,V0p: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum,V2l: $i] :
( ( mem @ V2l @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V3r: $i] :
( ( mem @ V3r @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( ( ap @ ( c_2Epatricia_2EBRANCH @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0p ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ V2l ) @ V3r ) ) ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ V2l ) @ ( c_2Epatricia_2EEmpty @ A_27a ) ) ) @ V3r ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ V3r ) @ ( c_2Epatricia_2EEmpty @ A_27a ) ) ) @ V2l ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0p ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ V2l ) @ V3r ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EPEEK__REMOVE,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V1k: tp__ty_2Enum_2Enum,V2j: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ V0t ) )
=> ( ( ap @ ( ap @ ( c_2Epatricia_2EPEEK @ A_27a ) @ ( ap @ ( ap @ ( c_2Epatricia_2EREMOVE @ A_27a ) @ V0t ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) ) @ ( inj__ty_2Enum_2Enum @ V2j ) )
= ( 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 @ V2j ) ) ) @ ( c_2Eoption_2ENONE @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2EPEEK @ A_27a ) @ V0t ) @ ( inj__ty_2Enum_2Enum @ V2j ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EEVERY__LEAF__TRANSFORM,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ bool ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( ( ! [V4k: tp__ty_2Enum_2Enum,V5d: $i] :
( ( mem @ V5d @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V4k ) ) @ V5d ) )
=> ( p @ ( ap @ ( ap @ V1Q @ ( inj__ty_2Enum_2Enum @ V4k ) ) @ ( ap @ V2f @ V5d ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epatricia_2EEVERY__LEAF @ A_27a ) @ V0P ) @ V3t ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epatricia_2EEVERY__LEAF @ A_27b ) @ V1Q ) @ ( ap @ ( ap @ ( c_2Epatricia_2ETRANSFORM @ A_27b @ A_27a ) @ V2f ) @ V3t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2ETRANSFORM__EMPTY,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Epatricia_2ETRANSFORM @ A_27b @ A_27a ) @ V0f ) @ V1t )
= ( c_2Epatricia_2EEmpty @ A_27b ) )
<=> ( V1t
= ( c_2Epatricia_2EEmpty @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2ETRANSFORM__IS__PTREE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ V1t ) )
=> ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27b ) @ ( ap @ ( ap @ ( c_2Epatricia_2ETRANSFORM @ A_27b @ A_27a ) @ V0f ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EPEEK__TRANSFORM,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V2k: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Epatricia_2EPEEK @ A_27b ) @ ( ap @ ( ap @ ( c_2Epatricia_2ETRANSFORM @ A_27b @ A_27a ) @ V0f ) @ V1t ) ) @ ( inj__ty_2Enum_2Enum @ V2k ) )
= ( ap @ ( ap @ ( ap @ ( c_2Eoption_2Eoption__CASE @ A_27a @ ( ty_2Eoption_2Eoption @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2EPEEK @ A_27a ) @ V1t ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) ) @ ( c_2Eoption_2ENONE @ A_27b ) )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( c_2Eoption_2ESOME @ A_27b ) @ ( ap @ V0f @ V3x ) ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EADD__TRANSFORM,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V2k: tp__ty_2Enum_2Enum,V3d: $i] :
( ( mem @ V3d @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Epatricia_2ETRANSFORM @ A_27b @ A_27a ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epatricia_2EADD @ A_27a ) @ V1t ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ V3d ) ) )
= ( ap @ ( ap @ ( c_2Epatricia_2EADD @ A_27b ) @ ( ap @ ( ap @ ( c_2Epatricia_2ETRANSFORM @ A_27b @ A_27a ) @ V0f ) @ V1t ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ ( ap @ V0f @ V3d ) ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2ETRANSFORM__BRANCH,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1p: tp__ty_2Enum_2Enum,V2m: tp__ty_2Enum_2Enum,V3l: $i] :
( ( mem @ V3l @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V4r: $i] :
( ( mem @ V4r @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Epatricia_2ETRANSFORM @ A_27b @ A_27a ) @ V0f ) @ ( ap @ ( c_2Epatricia_2EBRANCH @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1p ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epatricia_2Eptree @ A_27a ) @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ V3l ) @ V4r ) ) ) ) )
= ( ap @ ( c_2Epatricia_2EBRANCH @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27b ) @ ( ty_2Epatricia_2Eptree @ A_27b ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1p ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epatricia_2Eptree @ A_27b ) @ ( ty_2Epatricia_2Eptree @ A_27b ) ) ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epatricia_2Eptree @ A_27b ) @ ( ty_2Epatricia_2Eptree @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2ETRANSFORM @ A_27b @ A_27a ) @ V0f ) @ V3l ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2ETRANSFORM @ A_27b @ A_27a ) @ V0f ) @ V4r ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EREMOVE__TRANSFORM,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V2k: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Epatricia_2ETRANSFORM @ A_27b @ A_27a ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epatricia_2EREMOVE @ A_27a ) @ V1t ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) )
= ( ap @ ( ap @ ( c_2Epatricia_2EREMOVE @ A_27b ) @ ( ap @ ( ap @ ( c_2Epatricia_2ETRANSFORM @ A_27b @ A_27a ) @ V0f ) @ V1t ) ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EREMOVE__ADD__EQ,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V1k: tp__ty_2Enum_2Enum,V2d: $i] :
( ( mem @ V2d @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Epatricia_2EREMOVE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epatricia_2EADD @ A_27a ) @ V0t ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ V2d ) ) ) @ ( inj__ty_2Enum_2Enum @ V1k ) )
= ( ap @ ( ap @ ( c_2Epatricia_2EREMOVE @ A_27a ) @ V0t ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EADD__ADD,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V1k: tp__ty_2Enum_2Enum,V2d: $i] :
( ( mem @ V2d @ A_27a )
=> ! [V3e: $i] :
( ( mem @ V3e @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Epatricia_2EADD @ A_27a ) @ ( ap @ ( ap @ ( c_2Epatricia_2EADD @ A_27a ) @ V0t ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ V2d ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ V3e ) )
= ( ap @ ( ap @ ( c_2Epatricia_2EADD @ A_27a ) @ V0t ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ V3e ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EEVERY__LEAF__PEEK,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V2k: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ ( c_2Epatricia_2EEVERY__LEAF @ A_27a ) @ V0P ) @ V1t ) )
& ( p @ ( ap @ ( c_2Eoption_2EIS__SOME @ A_27a ) @ ( ap @ ( ap @ ( c_2Epatricia_2EPEEK @ A_27a ) @ V1t ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ ( ap @ ( c_2Eoption_2ETHE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epatricia_2EPEEK @ A_27a ) @ V1t ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EIS__PTREE__PEEK,axiom,
! [A_27a: del,A_27b: del,A_27c: del] :
( ! [V0k: tp__ty_2Enum_2Enum] :
~ ( p @ ( ap @ ( c_2Eoption_2EIS__SOME @ A_27a ) @ ( ap @ ( ap @ ( c_2Epatricia_2EPEEK @ A_27a ) @ ( c_2Epatricia_2EEmpty @ A_27a ) ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) ) )
& ! [V1k: tp__ty_2Enum_2Enum,V2j: tp__ty_2Enum_2Enum,V3b: $i] :
( ( mem @ V3b @ A_27b )
=> ( ( p @ ( ap @ ( c_2Eoption_2EIS__SOME @ A_27b ) @ ( ap @ ( ap @ ( c_2Epatricia_2EPEEK @ A_27b ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V2j ) ) @ V3b ) ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) ) )
<=> ( V2j = V1k ) ) )
& ! [V4p: tp__ty_2Enum_2Enum,V5m: tp__ty_2Enum_2Enum,V6l: $i] :
( ( mem @ V6l @ ( ty_2Epatricia_2Eptree @ A_27c ) )
=> ! [V7r: $i] :
( ( mem @ V7r @ ( ty_2Epatricia_2Eptree @ A_27c ) )
=> ( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27c ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27c ) @ ( inj__ty_2Enum_2Enum @ V4p ) ) @ ( inj__ty_2Enum_2Enum @ V5m ) ) @ V6l ) @ V7r ) ) )
=> ( ? [V8k: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Ebit_2EBIT @ ( inj__ty_2Enum_2Enum @ V5m ) ) @ ( inj__ty_2Enum_2Enum @ V8k ) ) )
& ( p @ ( ap @ ( c_2Eoption_2EIS__SOME @ A_27c ) @ ( ap @ ( ap @ ( c_2Epatricia_2EPEEK @ A_27c ) @ V6l ) @ ( inj__ty_2Enum_2Enum @ V8k ) ) ) ) )
& ? [V9k: tp__ty_2Enum_2Enum] :
( ~ ( p @ ( ap @ ( ap @ c_2Ebit_2EBIT @ ( inj__ty_2Enum_2Enum @ V5m ) ) @ ( inj__ty_2Enum_2Enum @ V9k ) ) )
& ( p @ ( ap @ ( c_2Eoption_2EIS__SOME @ A_27c ) @ ( ap @ ( ap @ ( c_2Epatricia_2EPEEK @ A_27c ) @ V7r ) @ ( inj__ty_2Enum_2Enum @ V9k ) ) ) ) )
& ! [V10k: tp__ty_2Enum_2Enum,V11n: tp__ty_2Enum_2Enum] :
( ( ~ ( p @ ( ap @ ( ap @ ( ap @ c_2Ebit_2EMOD__2EXP__EQ @ ( inj__ty_2Enum_2Enum @ V5m ) ) @ ( inj__ty_2Enum_2Enum @ V10k ) ) @ ( inj__ty_2Enum_2Enum @ V4p ) ) )
| ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V11n ) ) @ ( inj__ty_2Enum_2Enum @ V5m ) ) )
& ~ ( ( p @ ( ap @ ( ap @ c_2Ebit_2EBIT @ ( inj__ty_2Enum_2Enum @ V11n ) ) @ ( inj__ty_2Enum_2Enum @ V4p ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Ebit_2EBIT @ ( inj__ty_2Enum_2Enum @ V11n ) ) @ ( inj__ty_2Enum_2Enum @ V10k ) ) ) ) ) )
=> ( ~ ( p @ ( ap @ ( c_2Eoption_2EIS__SOME @ A_27c ) @ ( ap @ ( ap @ ( c_2Epatricia_2EPEEK @ A_27c ) @ V6l ) @ ( inj__ty_2Enum_2Enum @ V10k ) ) ) )
& ~ ( p @ ( ap @ ( c_2Eoption_2EIS__SOME @ A_27c ) @ ( ap @ ( ap @ ( c_2Epatricia_2EPEEK @ A_27c ) @ V7r ) @ ( inj__ty_2Enum_2Enum @ V10k ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EPTREE__EQ,axiom,
! [A_27a: del,V0t1: $i] :
( ( mem @ V0t1 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( ( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ V0t1 ) )
& ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ V1t2 ) ) )
=> ( ! [V2k: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Epatricia_2EPEEK @ A_27a ) @ V0t1 ) @ ( inj__ty_2Enum_2Enum @ V2k ) )
= ( ap @ ( ap @ ( c_2Epatricia_2EPEEK @ A_27a ) @ V1t2 ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) )
<=> ( V0t1 = V1t2 ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EREMOVE__REMOVE,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V1k: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ V0t ) )
=> ( ( ap @ ( ap @ ( c_2Epatricia_2EREMOVE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epatricia_2EREMOVE @ A_27a ) @ V0t ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) ) @ ( inj__ty_2Enum_2Enum @ V1k ) )
= ( ap @ ( ap @ ( c_2Epatricia_2EREMOVE @ A_27a ) @ V0t ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EREMOVE__ADD,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V1k: tp__ty_2Enum_2Enum,V2d: $i] :
( ( mem @ V2d @ A_27a )
=> ! [V3j: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ V0t ) )
=> ( ( ap @ ( ap @ ( c_2Epatricia_2EREMOVE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epatricia_2EADD @ A_27a ) @ V0t ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ V2d ) ) ) @ ( inj__ty_2Enum_2Enum @ V3j ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Epatricia_2Eptree @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ ( inj__ty_2Enum_2Enum @ V3j ) ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2EREMOVE @ A_27a ) @ V0t ) @ ( inj__ty_2Enum_2Enum @ V3j ) ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2EADD @ A_27a ) @ ( ap @ ( ap @ ( c_2Epatricia_2EREMOVE @ A_27a ) @ V0t ) @ ( inj__ty_2Enum_2Enum @ V3j ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ V2d ) ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EADD__ADD__SYM,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V1k: tp__ty_2Enum_2Enum,V2j: tp__ty_2Enum_2Enum,V3d: $i] :
( ( mem @ V3d @ A_27a )
=> ! [V4e: $i] :
( ( mem @ V4e @ A_27a )
=> ( ( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ V0t ) )
& ( V1k != V2j ) )
=> ( ( ap @ ( ap @ ( c_2Epatricia_2EADD @ A_27a ) @ ( ap @ ( ap @ ( c_2Epatricia_2EADD @ A_27a ) @ V0t ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ V3d ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2j ) ) @ V4e ) )
= ( ap @ ( ap @ ( c_2Epatricia_2EADD @ A_27a ) @ ( ap @ ( ap @ ( c_2Epatricia_2EADD @ A_27a ) @ V0t ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2j ) ) @ V4e ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ V3d ) ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EFILTER__ALL,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ! [V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l ) ) )
=> ~ ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ V1l ) ) ) )
<=> ( ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V0P ) @ V1l )
= ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2ETRAVERSE__TRANSFORM,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ A_27b ) @ ( ap @ ( ap @ ( c_2Epatricia_2ETRANSFORM @ A_27b @ A_27a ) @ V0f ) @ V1t ) ) )
= ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ A_27a ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EMEM__TRAVERSE__PEEK,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V1k: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ V0t ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ A_27a ) @ V0t ) ) ) )
<=> ( p @ ( ap @ ( c_2Eoption_2EIS__SOME @ A_27a ) @ ( ap @ ( ap @ ( c_2Epatricia_2EPEEK @ A_27a ) @ V0t ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EIN__NUMSET__OF__PTREE,axiom,
! [V0t: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ c_2Epatricia_2ENUMSET__OF__PTREE @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Epatricia_2EIN__PTREE @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EADD__LIST__IS__PTREE,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) )
=> ( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ V0t ) )
=> ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epatricia_2EADD__LIST @ A_27a ) @ V0t ) @ V1l ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EADD__LIST__TO__EMPTY__IS__PTREE,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ A_27a ) ) )
=> ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epatricia_2EADD__LIST @ A_27a ) @ ( c_2Epatricia_2EEmpty @ A_27a ) ) @ V0l ) ) ) ) ).
thf(conj_thm_2Epatricia_2EPTREE__OF__NUMSET__IS__PTREE,axiom,
! [V0t: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) )
=> ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ ty_2Eone_2Eone ) @ ( ap @ ( ap @ c_2Epatricia_2EPTREE__OF__NUMSET @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EPTREE__OF__NUMSET__IS__PTREE__EMPTY,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ ty_2Eone_2Eone ) @ ( ap @ ( ap @ c_2Epatricia_2EPTREE__OF__NUMSET @ ( c_2Epatricia_2EEmpty @ ty_2Eone_2Eone ) ) @ V0s ) ) ) ) ).
thf(conj_thm_2Epatricia_2ENOT__KEY__LEFT__AND__RIGHT,axiom,
! [A_27a: del,V0p: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum,V2l: $i] :
( ( mem @ V2l @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V3r: $i] :
( ( mem @ V3r @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V4k: tp__ty_2Enum_2Enum,V5j: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0p ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ V2l ) @ V3r ) ) )
& ( p @ ( ap @ ( c_2Eoption_2EIS__SOME @ A_27a ) @ ( ap @ ( ap @ ( c_2Epatricia_2EPEEK @ A_27a ) @ V2l ) @ ( inj__ty_2Enum_2Enum @ V4k ) ) ) )
& ( p @ ( ap @ ( c_2Eoption_2EIS__SOME @ A_27a ) @ ( ap @ ( ap @ ( c_2Epatricia_2EPEEK @ A_27a ) @ V3r ) @ ( inj__ty_2Enum_2Enum @ V5j ) ) ) ) )
=> ( V4k != V5j ) ) ) ) ).
thf(conj_thm_2Epatricia_2EALL__DISTINCT__TRAVERSE,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ V0t ) )
=> ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ A_27a ) @ V0t ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EMEM__ALL__DISTINCT__IMP__PERM,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ V0l1 ) )
& ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ V1l2 ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V0l1 ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l2 ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0l1 ) @ V1l2 ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EMEM__TRAVERSE,axiom,
! [V0t: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone,V1k: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ ( ap @ c_2Epatricia_2ENUMSET__OF__PTREE @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EINSERT__PTREE__IS__PTREE,axiom,
! [V0t: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone,V1x: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) )
=> ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ ty_2Eone_2Eone ) @ ( ap @ ( ap @ c_2Epatricia_2EINSERT__PTREE @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EFINITE__NUMSET__OF__PTREE,axiom,
! [V0t: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone] : ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ ( ap @ c_2Epatricia_2ENUMSET__OF__PTREE @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) ) ) ).
thf(conj_thm_2Epatricia_2EADD__INSERT,axiom,
! [V0v: tp__ty_2Eone_2Eone,V1t: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone,V2n: tp__ty_2Enum_2Enum] :
( ( surj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ ( ap @ ( ap @ ( c_2Epatricia_2EADD @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V1t ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Eone_2Eone ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Eone_2Eone @ V0v ) ) ) )
= ( surj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ ( ap @ ( ap @ c_2Epatricia_2EINSERT__PTREE @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V1t ) ) ) ) ).
thf(conj_thm_2Epatricia_2EPEEK__INSERT__PTREE,axiom,
! [V0t: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone,V1k: tp__ty_2Enum_2Enum,V2j: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) )
=> ( ( surj__c_ty_2Eoption_2Eoption_ty_2Eone_2Eone @ ( ap @ ( ap @ ( c_2Epatricia_2EPEEK @ ty_2Eone_2Eone ) @ ( ap @ ( ap @ c_2Epatricia_2EINSERT__PTREE @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) ) @ ( inj__ty_2Enum_2Enum @ V2j ) ) )
= ( surj__c_ty_2Eoption_2Eoption_ty_2Eone_2Eone @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ ty_2Eone_2Eone ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ ( inj__ty_2Enum_2Enum @ V2j ) ) ) @ ( ap @ ( c_2Eoption_2ESOME @ ty_2Eone_2Eone ) @ ( inj__ty_2Eone_2Eone @ fo__c_2Eone_2Eone ) ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2EPEEK @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) @ ( inj__ty_2Enum_2Enum @ V2j ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EMEM__TRAVERSE__INSERT__PTREE,axiom,
! [V0t: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone,V1x: tp__ty_2Enum_2Enum,V2h: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ ty_2Eone_2Eone ) @ ( ap @ ( ap @ c_2Epatricia_2EINSERT__PTREE @ ( inj__ty_2Enum_2Enum @ V2h ) ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) ) ) ) )
<=> ( ( V1x = V2h )
| ( ( V1x != V2h )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EPERM__INSERT__PTREE,axiom,
! [V0t: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V1s ) )
=> ( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ ty_2Eone_2Eone ) @ ( ap @ ( ap @ ( ap @ ( c_2Elist_2EFOLDL @ ty_2Enum_2Enum @ ( ty_2Epatricia_2Eptree @ ty_2Eone_2Eone ) ) @ ( ap @ ( c_2Ecombin_2EC @ ty_2Enum_2Enum @ ( ty_2Epatricia_2Eptree @ ty_2Eone_2Eone ) @ ( ty_2Epatricia_2Eptree @ ty_2Eone_2Eone ) ) @ c_2Epatricia_2EINSERT__PTREE ) ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) @ ( ap @ ( c_2Elist_2ESET__TO__LIST @ ty_2Enum_2Enum ) @ V1s ) ) ) ) @ ( ap @ ( c_2Elist_2ESET__TO__LIST @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Enum_2Enum ) @ ( ap @ c_2Epatricia_2ENUMSET__OF__PTREE @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EIN__PTREE__OF__NUMSET,axiom,
! [V0t: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V1s ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Epatricia_2EIN__PTREE @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ ( ap @ c_2Epatricia_2EPTREE__OF__NUMSET @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) @ V1s ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Epatricia_2EIN__PTREE @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) )
| ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EIN__PTREE__EMPTY,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
~ ( p @ ( ap @ ( ap @ c_2Epatricia_2EIN__PTREE @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( c_2Epatricia_2EEmpty @ ty_2Eone_2Eone ) ) ) ).
thf(conj_thm_2Epatricia_2EIN__PTREE__OF__NUMSET__EMPTY,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V0s ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V0s ) )
<=> ( p @ ( ap @ ( ap @ c_2Epatricia_2EIN__PTREE @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( ap @ c_2Epatricia_2EPTREE__OF__NUMSET @ ( c_2Epatricia_2EEmpty @ ty_2Eone_2Eone ) ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EPTREE__EXTENSION,axiom,
! [V0t1: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone,V1t2: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone] :
( ( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t1 ) ) )
& ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V1t2 ) ) ) )
=> ( ( V0t1 = V1t2 )
<=> ! [V2x: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Epatricia_2EIN__PTREE @ ( inj__ty_2Enum_2Enum @ V2x ) ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t1 ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Epatricia_2EIN__PTREE @ ( inj__ty_2Enum_2Enum @ V2x ) ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V1t2 ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EPTREE__OF__NUMSET__NUMSET__OF__PTREE,axiom,
! [V0t: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V1s ) ) )
=> ( ( surj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ ( ap @ ( ap @ c_2Epatricia_2EPTREE__OF__NUMSET @ ( c_2Epatricia_2EEmpty @ ty_2Eone_2Eone ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Enum_2Enum ) @ ( ap @ c_2Epatricia_2ENUMSET__OF__PTREE @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) ) @ V1s ) ) )
= ( surj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ ( ap @ ( ap @ c_2Epatricia_2EPTREE__OF__NUMSET @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2ENUMSET__OF__PTREE__PTREE__OF__NUMSET,axiom,
! [V0t: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V1s ) ) )
=> ( ( ap @ c_2Epatricia_2ENUMSET__OF__PTREE @ ( ap @ ( ap @ c_2Epatricia_2EPTREE__OF__NUMSET @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) @ V1s ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Enum_2Enum ) @ ( ap @ c_2Epatricia_2ENUMSET__OF__PTREE @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) ) @ V1s ) ) ) ) ).
thf(conj_thm_2Epatricia_2ENUMSET__OF__PTREE__EMPTY,axiom,
( ( ap @ c_2Epatricia_2ENUMSET__OF__PTREE @ ( c_2Epatricia_2EEmpty @ ty_2Eone_2Eone ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) ) ).
thf(conj_thm_2Epatricia_2EPTREE__OF__NUMSET__EMPTY,axiom,
! [V0t: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone] :
( ( surj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ ( ap @ ( ap @ c_2Epatricia_2EPTREE__OF__NUMSET @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) ) )
= V0t ) ).
thf(conj_thm_2Epatricia_2ENUMSET__OF__PTREE__PTREE__OF__NUMSET__EMPTY,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V0s ) )
=> ( ( ap @ c_2Epatricia_2ENUMSET__OF__PTREE @ ( ap @ ( ap @ c_2Epatricia_2EPTREE__OF__NUMSET @ ( c_2Epatricia_2EEmpty @ ty_2Eone_2Eone ) ) @ V0s ) )
= V0s ) ) ) ).
thf(conj_thm_2Epatricia_2EIN__PTREE__INSERT__PTREE,axiom,
! [V0t: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone,V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Epatricia_2EIN__PTREE @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ ( ap @ c_2Epatricia_2EINSERT__PTREE @ ( inj__ty_2Enum_2Enum @ V1m ) ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) ) )
<=> ( ( V1m = V2n )
| ( p @ ( ap @ ( ap @ c_2Epatricia_2EIN__PTREE @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EIN__PTREE__REMOVE,axiom,
! [V0t: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone,V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Epatricia_2EIN__PTREE @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ ( ap @ ( c_2Epatricia_2EREMOVE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) )
<=> ( ( V2n != V1m )
& ( p @ ( ap @ ( ap @ c_2Epatricia_2EIN__PTREE @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EIN__PTREE__UNION__PTREE,axiom,
! [V0t1: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone,V1t2: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone,V2n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t1 ) ) )
& ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V1t2 ) ) ) )
=> ( ( p @ ( ap @ ( ap @ c_2Epatricia_2EIN__PTREE @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ ( ap @ c_2Epatricia_2EUNION__PTREE @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t1 ) ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V1t2 ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ c_2Epatricia_2EIN__PTREE @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t1 ) ) )
| ( p @ ( ap @ ( ap @ c_2Epatricia_2EIN__PTREE @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V1t2 ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EUNION__PTREE__IS__PTREE,axiom,
! [V0t1: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone,V1t2: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone] :
( ( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t1 ) ) )
& ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V1t2 ) ) ) )
=> ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ ty_2Eone_2Eone ) @ ( ap @ ( ap @ c_2Epatricia_2EUNION__PTREE @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t1 ) ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V1t2 ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EUNION__PTREE__COMM,axiom,
! [V0t1: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone,V1t2: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone] :
( ( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t1 ) ) )
& ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V1t2 ) ) ) )
=> ( ( surj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ ( ap @ ( ap @ c_2Epatricia_2EUNION__PTREE @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t1 ) ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V1t2 ) ) )
= ( surj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ ( ap @ ( ap @ c_2Epatricia_2EUNION__PTREE @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V1t2 ) ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t1 ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EUNION__PTREE__COMM__EMPTY,axiom,
! [V0t: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone] :
( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) )
=> ( ( surj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ ( ap @ ( ap @ c_2Epatricia_2EUNION__PTREE @ ( c_2Epatricia_2EEmpty @ ty_2Eone_2Eone ) ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) )
= ( surj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ ( ap @ ( ap @ c_2Epatricia_2EUNION__PTREE @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) @ ( c_2Epatricia_2EEmpty @ ty_2Eone_2Eone ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EUNION__PTREE__EMPTY,axiom,
( ! [V0t: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone] :
( ( surj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ ( ap @ ( ap @ c_2Epatricia_2EUNION__PTREE @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) @ ( c_2Epatricia_2EEmpty @ ty_2Eone_2Eone ) ) )
= V0t )
& ! [V1t: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone] :
( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V1t ) ) )
=> ( ( surj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ ( ap @ ( ap @ c_2Epatricia_2EUNION__PTREE @ ( c_2Epatricia_2EEmpty @ ty_2Eone_2Eone ) ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V1t ) ) )
= V1t ) ) ) ).
thf(conj_thm_2Epatricia_2EUNION__PTREE__ASSOC,axiom,
! [V0t1: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone,V1t2: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone,V2t3: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone] :
( ( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t1 ) ) )
& ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V1t2 ) ) )
& ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V2t3 ) ) ) )
=> ( ( surj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ ( ap @ ( ap @ c_2Epatricia_2EUNION__PTREE @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t1 ) ) @ ( ap @ ( ap @ c_2Epatricia_2EUNION__PTREE @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V1t2 ) ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V2t3 ) ) ) )
= ( surj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ ( ap @ ( ap @ c_2Epatricia_2EUNION__PTREE @ ( ap @ ( ap @ c_2Epatricia_2EUNION__PTREE @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t1 ) ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V1t2 ) ) ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V2t3 ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EPTREE__OF__NUMSET__UNION,axiom,
! [V0t: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone,V1s1: $i] :
( ( mem @ V1s1 @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V2s2: $i] :
( ( mem @ V2s2 @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V1s1 ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V2s2 ) ) )
=> ( ( surj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ ( ap @ ( ap @ c_2Epatricia_2EPTREE__OF__NUMSET @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Enum_2Enum ) @ V1s1 ) @ V2s2 ) ) )
= ( surj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ ( ap @ ( ap @ c_2Epatricia_2EPTREE__OF__NUMSET @ ( ap @ ( ap @ c_2Epatricia_2EPTREE__OF__NUMSET @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) @ V1s1 ) ) @ V2s2 ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EPTREE__OF__NUMSET__INSERT,axiom,
! [V0t: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V2x: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V1s ) ) )
=> ( ( surj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ ( ap @ ( ap @ c_2Epatricia_2EPTREE__OF__NUMSET @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2x ) ) @ V1s ) ) )
= ( surj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ ( ap @ ( ap @ c_2Epatricia_2EINSERT__PTREE @ ( inj__ty_2Enum_2Enum @ V2x ) ) @ ( ap @ ( ap @ c_2Epatricia_2EPTREE__OF__NUMSET @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EPTREE__OF__NUMSET__INSERT__EMPTY,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V1x: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V0s ) )
=> ( ( surj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ ( ap @ ( ap @ c_2Epatricia_2EPTREE__OF__NUMSET @ ( c_2Epatricia_2EEmpty @ ty_2Eone_2Eone ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ V0s ) ) )
= ( surj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ ( ap @ ( ap @ c_2Epatricia_2EINSERT__PTREE @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ ( ap @ ( ap @ c_2Epatricia_2EPTREE__OF__NUMSET @ ( c_2Epatricia_2EEmpty @ ty_2Eone_2Eone ) ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EPTREE__OF__NUMSET__DELETE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V1x: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V0s ) )
=> ( ( surj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ ( ap @ ( ap @ c_2Epatricia_2EPTREE__OF__NUMSET @ ( c_2Epatricia_2EEmpty @ ty_2Eone_2Eone ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ ty_2Enum_2Enum ) @ V0s ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) )
= ( surj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ ( ap @ ( ap @ ( c_2Epatricia_2EREMOVE @ ty_2Eone_2Eone ) @ ( ap @ ( ap @ c_2Epatricia_2EPTREE__OF__NUMSET @ ( c_2Epatricia_2EEmpty @ ty_2Eone_2Eone ) ) @ V0s ) ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2ETRAVERSE__AUX,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ A_27a ) @ V0t ) )
= ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epatricia_2ETRAVERSE__AUX @ A_27a ) @ V0t ) @ ( c_2Elist_2ENIL @ ty_2Enum_2Enum ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EPTREE__TRAVERSE__EQ,axiom,
! [A_27a: del,A_27b: del,V0t1: $i] :
( ( mem @ V0t1 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ ( ty_2Epatricia_2Eptree @ A_27b ) )
=> ( ( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ V0t1 ) )
& ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27b ) @ V1t2 ) ) )
=> ( ! [V2k: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ A_27a ) @ V0t1 ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2k ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ A_27b ) @ V1t2 ) ) ) ) )
<=> ( ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ A_27a ) @ V0t1 ) )
= ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ A_27b ) @ V1t2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EQSORT__MEM__EQ,axiom,
! [A_27a: del,V0l2: $i] :
( ( mem @ V0l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2R: $i] :
( ( mem @ V2R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( ap @ ( ap @ ( c_2Esorting_2EQSORT @ A_27a ) @ V2R ) @ V1l1 )
= ( ap @ ( ap @ ( c_2Esorting_2EQSORT @ A_27a ) @ V2R ) @ V0l2 ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l1 ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V0l2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EKEYS__PEEK,axiom,
! [A_27a: del,A_27b: del,V0t1: $i] :
( ( mem @ V0t1 @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ ( ty_2Epatricia_2Eptree @ A_27b ) )
=> ( ( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ V0t1 ) )
& ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27b ) @ V1t2 ) ) )
=> ( ( ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( ap @ ( c_2Epatricia_2EKEYS @ A_27a ) @ V0t1 ) )
= ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( ap @ ( c_2Epatricia_2EKEYS @ A_27b ) @ V1t2 ) ) )
<=> ( ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ A_27a ) @ V0t1 ) )
= ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ A_27b ) @ V1t2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EPERM__ADD,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V1k: tp__ty_2Enum_2Enum,V2d: $i] :
( ( mem @ V2d @ A_27a )
=> ( ( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ V0t ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ A_27a ) @ V0t ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epatricia_2EADD @ A_27a ) @ V0t ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ V2d ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ A_27a ) @ V0t ) ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EPERM__NOT__ADD,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V1k: tp__ty_2Enum_2Enum,V2d: $i] :
( ( mem @ V2d @ A_27a )
=> ( ( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ V0t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ A_27a ) @ V0t ) ) ) ) )
=> ( ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epatricia_2EADD @ A_27a ) @ V0t ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ V2d ) ) ) )
= ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ A_27a ) @ V0t ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EPERM__NOT__REMOVE,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V1k: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ V0t ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ A_27a ) @ V0t ) ) ) ) )
=> ( ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epatricia_2EREMOVE @ A_27a ) @ V0t ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) ) )
= ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ A_27a ) @ V0t ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EPERM__DELETE__PTREE,axiom,
! [V0t: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone,V1k: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ ty_2Eone_2Eone ) @ ( ap @ ( ap @ ( c_2Epatricia_2EREMOVE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) ) )
@ ( ap
@ ( ap @ ( c_2Elist_2EFILTER @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2x: $i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V2x ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) ) ) )
@ ( ap @ ( c_2Epatricia_2ETRAVERSE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EFILTER__NONE,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ! [V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ V1l ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V0P ) @ V1l )
= V1l ) ) ) ) ).
thf(conj_thm_2Epatricia_2EPERM__REMOVE,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V1k: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ V0t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ A_27a ) @ V0t ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epatricia_2EREMOVE @ A_27a ) @ V0t ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) ) )
@ ( ap
@ ( ap @ ( c_2Elist_2EFILTER @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2x: $i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V2x ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) ) ) )
@ ( ap @ ( c_2Epatricia_2ETRAVERSE @ A_27a ) @ V0t ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2ESIZE__ADD,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V1k: tp__ty_2Enum_2Enum,V2d: $i] :
( ( mem @ V2d @ A_27a )
=> ( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ V0t ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epatricia_2ESIZE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epatricia_2EADD @ A_27a ) @ V0t ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) @ V2d ) ) ) )
= ( 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 @ V1k ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ A_27a ) @ V0t ) ) ) ) @ ( ap @ ( c_2Epatricia_2ESIZE @ A_27a ) @ V0t ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( c_2Epatricia_2ESIZE @ A_27a ) @ V0t ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2ESIZE__REMOVE,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V1k: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ A_27a ) @ V0t ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epatricia_2ESIZE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epatricia_2EREMOVE @ A_27a ) @ V0t ) @ ( inj__ty_2Enum_2Enum @ V1k ) ) ) )
= ( 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 @ V1k ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ A_27a ) @ V0t ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( c_2Epatricia_2ESIZE @ A_27a ) @ V0t ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( c_2Epatricia_2ESIZE @ A_27a ) @ V0t ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2ESIZE,axiom,
! [A_27a: del] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epatricia_2ESIZE @ A_27a ) @ ( c_2Epatricia_2EEmpty @ A_27a ) ) )
= fo__c_2Enum_2E0 )
& ! [V0k: tp__ty_2Enum_2Enum,V1d: $i] :
( ( mem @ V1d @ A_27a )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epatricia_2ESIZE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epatricia_2ELeaf @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0k ) ) @ V1d ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
& ! [V2p: tp__ty_2Enum_2Enum,V3m: tp__ty_2Enum_2Enum,V4l: $i] :
( ( mem @ V4l @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ! [V5r: $i] :
( ( mem @ V5r @ ( ty_2Epatricia_2Eptree @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epatricia_2ESIZE @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epatricia_2EBranch @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2p ) ) @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ V4l ) @ V5r ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( c_2Epatricia_2ESIZE @ A_27a ) @ V4l ) ) @ ( ap @ ( c_2Epatricia_2ESIZE @ A_27a ) @ V5r ) ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2ESIZE__PTREE__OF__NUMSET,axiom,
! [V0t: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V1s ) )
=> ( ( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) )
& ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Epatricia_2ETRAVERSE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) ) @ ( ap @ ( c_2Elist_2ESET__TO__LIST @ ty_2Enum_2Enum ) @ V1s ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epatricia_2ESIZE @ ty_2Eone_2Eone ) @ ( ap @ ( ap @ c_2Epatricia_2EPTREE__OF__NUMSET @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) @ V1s ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( c_2Epatricia_2ESIZE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ ty_2Enum_2Enum ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2ESIZE__PTREE__OF__NUMSET__EMPTY,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V0s ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epatricia_2ESIZE @ ty_2Eone_2Eone ) @ ( ap @ ( ap @ c_2Epatricia_2EPTREE__OF__NUMSET @ ( c_2Epatricia_2EEmpty @ ty_2Eone_2Eone ) ) @ V0s ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ ty_2Enum_2Enum ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2ECARD__LIST__TO__SET,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ V0l ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V0l ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2ECARD__NUMSET__OF__PTREE,axiom,
! [V0t: tp__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone] :
( ( p @ ( ap @ ( c_2Epatricia_2EIS__PTREE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ ty_2Enum_2Enum ) @ ( ap @ c_2Epatricia_2ENUMSET__OF__PTREE @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epatricia_2ESIZE @ ty_2Eone_2Eone ) @ ( inj__c_ty_2Epatricia_2Eptree_ty_2Eone_2Eone @ V0t ) ) ) ) ) ).
thf(conj_thm_2Epatricia_2EDELETE__UNION,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s1: $i] :
( ( mem @ V1s1 @ ( arr @ A_27a @ bool ) )
=> ! [V2s2: $i] :
( ( mem @ V2s2 @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s1 ) @ V2s2 ) ) @ V0x )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V1s1 ) @ V0x ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V2s2 ) @ V0x ) ) ) ) ) ) ).
%------------------------------------------------------------------------------