ITP001 Axioms: ITP053^5.ax
%------------------------------------------------------------------------------
% File : ITP053^5 : 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 : sorting^2.ax [Gau20]
% : HL4053^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 147 ( 3 unt; 10 typ; 0 def)
% Number of atoms : 3750 ( 82 equ; 0 cnn)
% Maximal formula atoms : 167 ( 25 avg)
% Number of connectives : 6727 ( 3 ~; 2 |; 96 &;6020 @)
% ( 54 <=>; 552 =>; 0 <=; 0 <~>)
% Maximal formula depth : 39 ( 15 avg;6020 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 10 ( 10 >; 0 *; 0 +; 0 <<)
% Number of symbols : 78 ( 76 usr; 67 con; 0-2 aty)
% Number of variables : 621 ( 22 ^ 591 !; 8 ?; 621 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Esorting_2EPART,type,
c_2Esorting_2EPART: del > $i ).
thf(mem_c_2Esorting_2EPART,axiom,
! [A_27a: del] : ( mem @ ( c_2Esorting_2EPART @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ) ) ) ).
thf(tp_c_2Esorting_2EPART3,type,
c_2Esorting_2EPART3: del > $i ).
thf(mem_c_2Esorting_2EPART3,axiom,
! [A_27a: del] : ( mem @ ( c_2Esorting_2EPART3 @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ) ) ) ).
thf(tp_c_2Esorting_2EPARTITION,type,
c_2Esorting_2EPARTITION: del > $i ).
thf(mem_c_2Esorting_2EPARTITION,axiom,
! [A_27a: del] : ( mem @ ( c_2Esorting_2EPARTITION @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ) ).
thf(tp_c_2Esorting_2EPERM,type,
c_2Esorting_2EPERM: del > $i ).
thf(mem_c_2Esorting_2EPERM,axiom,
! [A_27a: del] : ( mem @ ( c_2Esorting_2EPERM @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) ) ) ).
thf(tp_c_2Esorting_2EPERM__SINGLE__SWAP,type,
c_2Esorting_2EPERM__SINGLE__SWAP: del > $i ).
thf(mem_c_2Esorting_2EPERM__SINGLE__SWAP,axiom,
! [A_27a: del] : ( mem @ ( c_2Esorting_2EPERM__SINGLE__SWAP @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) ) ) ).
thf(tp_c_2Esorting_2EQSORT,type,
c_2Esorting_2EQSORT: del > $i ).
thf(mem_c_2Esorting_2EQSORT,axiom,
! [A_27a: del] : ( mem @ ( c_2Esorting_2EQSORT @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ).
thf(tp_c_2Esorting_2EQSORT3,type,
c_2Esorting_2EQSORT3: del > $i ).
thf(mem_c_2Esorting_2EQSORT3,axiom,
! [A_27a: del] : ( mem @ ( c_2Esorting_2EQSORT3 @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ).
thf(tp_c_2Esorting_2ESORTED,type,
c_2Esorting_2ESORTED: del > $i ).
thf(mem_c_2Esorting_2ESORTED,axiom,
! [A_27a: del] : ( mem @ ( c_2Esorting_2ESORTED @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) ) ) ).
thf(tp_c_2Esorting_2ESORTS,type,
c_2Esorting_2ESORTS: del > $i ).
thf(mem_c_2Esorting_2ESORTS,axiom,
! [A_27a: del] : ( mem @ ( c_2Esorting_2ESORTS @ A_27a ) @ ( arr @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ bool ) ) ) ).
thf(tp_c_2Esorting_2ESTABLE,type,
c_2Esorting_2ESTABLE: del > $i ).
thf(mem_c_2Esorting_2ESTABLE,axiom,
! [A_27a: del] : ( mem @ ( c_2Esorting_2ESTABLE @ A_27a ) @ ( arr @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ bool ) ) ) ).
thf(ax_thm_2Esorting_2EPERM__DEF,axiom,
! [A_27a: del,V0L1: $i] :
( ( mem @ V0L1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1L2: $i] :
( ( mem @ V1L2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0L1 ) @ V1L2 ) )
<=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V2x ) ) @ V0L1 )
= ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V2x ) ) @ V1L2 ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__REFL,axiom,
! [A_27a: del,V0L: $i] :
( ( mem @ V0L @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0L ) @ V0L ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__INTRO,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( V0x = V1y )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0x ) @ V1y ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__transitive,axiom,
! [A_27a: del] : ( p @ ( ap @ ( c_2Erelation_2Etransitive @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( c_2Esorting_2EPERM @ A_27a ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__TRANS,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2z: $i] :
( ( mem @ V2z @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0x ) @ V1y ) )
& ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V1y ) @ V2z ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0x ) @ V2z ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__SYM,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0l1 ) @ V1l2 ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V1l2 ) @ V0l1 ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__CONG,axiom,
! [A_27a: del,V0L1: $i] :
( ( mem @ V0L1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1L2: $i] :
( ( mem @ V1L2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2L3: $i] :
( ( mem @ V2L3 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3L4: $i] :
( ( mem @ V3L4 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0L1 ) @ V2L3 ) )
& ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V1L2 ) @ V3L4 ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0L1 ) @ V1L2 ) ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2L3 ) @ V3L4 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__MONO,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0l1 ) @ V1l2 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2x ) @ V0l1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2x ) @ V1l2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__CONS__IFF,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l1: $i] :
( ( mem @ V2l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ V2l1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ V1l2 ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V2l1 ) @ V1l2 ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__NIL,axiom,
! [A_27a: del,V0L: $i] :
( ( mem @ V0L @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0L ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
<=> ( V0L
= ( c_2Elist_2ENIL @ A_27a ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ V0L ) )
<=> ( V0L
= ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__SING,axiom,
! [A_27a: del,V0L: $i] :
( ( mem @ V0L @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0L ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) )
<=> ( V0L
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) @ V0L ) )
<=> ( V0L
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__CONS__EQ__APPEND,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1L: $i] :
( ( mem @ V1L @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h ) @ V0t ) ) @ V1L ) )
<=> ? [V3M: $i] :
( ( mem @ V3M @ ( ty_2Elist_2Elist @ A_27a ) )
& ? [V4N: $i] :
( ( mem @ V4N @ ( ty_2Elist_2Elist @ A_27a ) )
& ( V1L
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V3M ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h ) @ V4N ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0t ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V3M ) @ V4N ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__APPEND,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ V1l2 ) ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1l2 ) @ V0l1 ) ) ) ) ) ).
thf(conj_thm_2Esorting_2ECONS__PERM,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1L: $i] :
( ( mem @ V1L @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2M: $i] :
( ( mem @ V2M @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3N: $i] :
( ( mem @ V3N @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V1L ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2M ) @ V3N ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ V1L ) ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2M ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ V3N ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EAPPEND__PERM__SYM,axiom,
! [A_27a: del,V0A: $i] :
( ( mem @ V0A @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1B: $i] :
( ( mem @ V1B @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2C: $i] :
( ( mem @ V2C @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0A ) @ V1B ) ) @ V2C ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1B ) @ V0A ) ) @ V2C ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__SPLIT__IF,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ! [V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p
@ ( ap
@ ( ap @ ( c_2Elist_2EEVERY @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ bool ) @ ( ap @ V0P @ V3x ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ V1Q @ V3x ) ) ) ) )
@ V2l ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V2l ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V0P ) @ V2l ) ) @ ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V1Q ) @ V2l ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__SPLIT,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V1l ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V0P ) @ V1l ) ) @ ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ bool @ bool ) @ c_2Ebool_2E_7E ) @ V0P ) ) @ V1l ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EFILTER__EQ__REP,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V0x ) ) @ V1l )
= ( ap @ ( ap @ ( c_2Erich__list_2EREPLICATE @ A_27a ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V0x ) ) @ V1l ) ) ) @ V0x ) ) ) ) ).
thf(conj_thm_2Esorting_2EFILTER__EQ__LENGTHS__EQ,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V0x ) ) @ V1l1 ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V0x ) ) @ V2l2 ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V0x ) ) @ V1l1 )
= ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V0x ) ) @ V2l2 ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__alt,axiom,
! [A_27a: del,V0L1: $i] :
( ( mem @ V0L1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1L2: $i] :
( ( mem @ V1L2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0L1 ) @ V1L2 ) )
<=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V2x ) ) @ V0L1 ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V2x ) ) @ V1L2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__IND,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) ) )
=> ( ( ( p @ ( ap @ ( ap @ V0P @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
& ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2l1: $i] :
( ( mem @ V2l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3l2: $i] :
( ( mem @ V3l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ V0P @ V2l1 ) @ V3l2 ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ V2l1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ V3l2 ) ) ) ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ! [V6l1: $i] :
( ( mem @ V6l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V7l2: $i] :
( ( mem @ V7l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ V0P @ V6l1 ) @ V7l2 ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V4x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V5y ) @ V6l1 ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V5y ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V4x ) @ V7l2 ) ) ) ) ) ) ) ) )
& ! [V8l1: $i] :
( ( mem @ V8l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V9l2: $i] :
( ( mem @ V9l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V10l3: $i] :
( ( mem @ V10l3 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ V0P @ V8l1 ) @ V9l2 ) )
& ( p @ ( ap @ ( ap @ V0P @ V9l2 ) @ V10l3 ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ V8l1 ) @ V10l3 ) ) ) ) ) ) )
=> ! [V11l1: $i] :
( ( mem @ V11l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V12l2: $i] :
( ( mem @ V12l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V11l1 ) @ V12l2 ) )
=> ( p @ ( ap @ ( ap @ V0P @ V11l1 ) @ V12l2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__SWAP__AT__FRONT,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ! [V2l1: $i] :
( ( mem @ V2l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3l2: $i] :
( ( mem @ V3l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1y ) @ V2l1 ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1y ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ V3l2 ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V2l1 ) @ V3l2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__SWAP__L__AT__FRONT,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3y: $i] :
( ( mem @ V3y @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2x ) @ V3y ) ) @ V0l1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V3y ) @ V2x ) ) @ V1l2 ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0l1 ) @ V1l2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__STRONG__IND,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) ) )
=> ( ( ( p @ ( ap @ ( ap @ V0P @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
& ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2l1: $i] :
( ( mem @ V2l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3l2: $i] :
( ( mem @ V3l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V2l1 ) @ V3l2 ) )
& ( p @ ( ap @ ( ap @ V0P @ V2l1 ) @ V3l2 ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ V2l1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ V3l2 ) ) ) ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ! [V6l1: $i] :
( ( mem @ V6l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V7l2: $i] :
( ( mem @ V7l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V6l1 ) @ V7l2 ) )
& ( p @ ( ap @ ( ap @ V0P @ V6l1 ) @ V7l2 ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V4x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V5y ) @ V6l1 ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V5y ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V4x ) @ V7l2 ) ) ) ) ) ) ) ) )
& ! [V8l1: $i] :
( ( mem @ V8l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V9l2: $i] :
( ( mem @ V9l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V10l3: $i] :
( ( mem @ V10l3 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V8l1 ) @ V9l2 ) )
& ( p @ ( ap @ ( ap @ V0P @ V8l1 ) @ V9l2 ) )
& ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V9l2 ) @ V10l3 ) )
& ( p @ ( ap @ ( ap @ V0P @ V9l2 ) @ V10l3 ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ V8l1 ) @ V10l3 ) ) ) ) ) ) )
=> ! [V11l1: $i] :
( ( mem @ V11l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V12l2: $i] :
( ( mem @ V12l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V11l1 ) @ V12l2 ) )
=> ( p @ ( ap @ ( ap @ V0P @ V11l1 ) @ V12l2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__LENGTH,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0l1 ) @ V1l2 ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l1 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l2 ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__MEM__EQ,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0l1 ) @ 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 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__LIST__TO__SET,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0l1 ) @ V1l2 ) )
=> ( ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V0l1 )
= ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l2 ) ) ) ) ) ).
thf(conj_thm_2Esorting_2ESORTED__IND,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) ) )
=> ( ( ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ V1R ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) )
& ! [V2R: $i] :
( ( mem @ V2R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( p @ ( ap @ ( ap @ V0P @ V2R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) )
& ! [V4R: $i] :
( ( mem @ V4R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ! [V6y: $i] :
( ( mem @ V6y @ A_27a )
=> ! [V7rst: $i] :
( ( mem @ V7rst @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ V0P @ V4R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V6y ) @ V7rst ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ V4R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V5x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V6y ) @ V7rst ) ) ) ) ) ) ) ) ) )
=> ! [V8v: $i] :
( ( mem @ V8v @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V9v1: $i] :
( ( mem @ V9v1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ V0P @ V8v ) @ V9v1 ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2ESORTED__DEF,axiom,
! [A_27a: del] :
( ! [V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V0R ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
<=> $true ) )
& ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2R: $i] :
( ( mem @ V2R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V2R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) )
<=> $true ) ) )
& ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5rst: $i] :
( ( mem @ V5rst @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V6R: $i] :
( ( mem @ V6R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V6R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V4x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3y ) @ V5rst ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ V6R @ V4x ) @ V3y ) )
& ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V6R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3y ) @ V5rst ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Esorting_2ESORTS__DEF,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTS @ A_27a ) @ V0f ) @ V1R ) )
<=> ! [V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V2l ) @ ( ap @ ( ap @ V0f @ V1R ) @ V2l ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V1R ) @ ( ap @ ( ap @ V0f @ V1R ) @ V2l ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2ESORTED__EQ,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1L: $i] :
( ( mem @ V1L @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2x ) @ V1L ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V0R ) @ V1L ) )
& ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1L ) ) )
=> ( p @ ( ap @ ( ap @ V0R @ V2x ) @ V3y ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2ESORTED__APPEND,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1L1: $i] :
( ( mem @ V1L1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2L2: $i] :
( ( mem @ V2L2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V0R ) @ V1L1 ) )
& ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V0R ) @ V2L2 ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ 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 ) @ V4y ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V2L2 ) ) ) )
=> ( p @ ( ap @ ( ap @ V0R @ V3x ) @ V4y ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1L1 ) @ V2L2 ) ) ) ) ) ) ) ).
thf(ax_thm_2Esorting_2EPART__DEF,axiom,
! [A_27a: del] :
( ! [V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esorting_2EPART @ A_27a ) @ V0P ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ V1l1 ) @ V2l2 )
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) @ V1l1 ) @ V2l2 ) ) ) ) )
& ! [V3P: $i] :
( ( mem @ V3P @ ( arr @ A_27a @ bool ) )
=> ! [V4h: $i] :
( ( mem @ V4h @ A_27a )
=> ! [V5rst: $i] :
( ( mem @ V5rst @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V6l1: $i] :
( ( mem @ V6l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V7l2: $i] :
( ( mem @ V7l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esorting_2EPART @ A_27a ) @ V3P ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V4h ) @ V5rst ) ) @ V6l1 ) @ V7l2 )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) @ ( ap @ V3P @ V4h ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esorting_2EPART @ A_27a ) @ V3P ) @ V5rst ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V4h ) @ V6l1 ) ) @ V7l2 ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esorting_2EPART @ A_27a ) @ V3P ) @ V5rst ) @ V6l1 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V4h ) @ V7l2 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPART__LENGTH,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1L: $i] :
( ( mem @ V1L @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l1: $i] :
( ( mem @ V2l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3l2: $i] :
( ( mem @ V3l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V4p: $i] :
( ( mem @ V4p @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V5q: $i] :
( ( mem @ V5q @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) @ V4p ) @ V5q )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esorting_2EPART @ A_27a ) @ V0P ) @ V1L ) @ V2l1 ) @ V3l2 ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1L ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V2l1 ) ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V3l2 ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V4p ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V5q ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPART__LENGTH__LEM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1L: $i] :
( ( mem @ V1L @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l1: $i] :
( ( mem @ V2l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3l2: $i] :
( ( mem @ V3l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V4p: $i] :
( ( mem @ V4p @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V5q: $i] :
( ( mem @ V5q @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) @ V4p ) @ V5q )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esorting_2EPART @ A_27a ) @ V0P ) @ V1L ) @ V2l1 ) @ V3l2 ) )
=> ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V4p ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1L ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V2l1 ) ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V3l2 ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V5q ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1L ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V2l1 ) ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V3l2 ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPARTs__HAVE__PROP,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1L: $i] :
( ( mem @ V1L @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2A: $i] :
( ( mem @ V2A @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3B: $i] :
( ( mem @ V3B @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V4l1: $i] :
( ( mem @ V4l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V5l2: $i] :
( ( mem @ V5l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) @ V2A ) @ V3B )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esorting_2EPART @ A_27a ) @ V0P ) @ V1L ) @ V4l1 ) @ V5l2 ) )
& ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V4l1 ) ) )
=> ( p @ ( ap @ V0P @ V6x ) ) ) )
& ! [V7x: $i] :
( ( mem @ V7x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V7x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V5l2 ) ) )
=> ~ ( p @ ( ap @ V0P @ V7x ) ) ) ) )
=> ( ! [V8z: $i] :
( ( mem @ V8z @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V8z ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V2A ) ) )
=> ( p @ ( ap @ V0P @ V8z ) ) ) )
& ! [V9z: $i] :
( ( mem @ V9z @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V9z ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V3B ) ) )
=> ~ ( p @ ( ap @ V0P @ V9z ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPART__MEM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1L: $i] :
( ( mem @ V1L @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2a1: $i] :
( ( mem @ V2a1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3a2: $i] :
( ( mem @ V3a2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V4l1: $i] :
( ( mem @ V4l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V5l2: $i] :
( ( mem @ V5l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) @ V2a1 ) @ V3a2 )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esorting_2EPART @ A_27a ) @ V0P ) @ V1L ) @ V4l1 ) @ V5l2 ) )
=> ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1L ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V4l1 ) @ V5l2 ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2a1 ) @ V3a2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Esorting_2EPARTITION__DEF,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Esorting_2EPARTITION @ A_27a ) @ V0P ) @ V1l )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esorting_2EPART @ A_27a ) @ V0P ) @ V1l ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EQSORT__IND,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) ) )
=> ( ( ! [V1ord: $i] :
( ( mem @ V1ord @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ V1ord ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) )
& ! [V2ord: $i] :
( ( mem @ V2ord @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V3h: $i] :
( ( mem @ V3h @ A_27a )
=> ! [V4t: $i] :
( ( mem @ V4t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ! [V5l1: $i] :
( ( mem @ V5l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V6l2: $i] :
( ( mem @ V6l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) @ V5l1 ) @ V6l2 )
= ( ap
@ ( ap @ ( c_2Esorting_2EPARTITION @ A_27a )
@ ( lam @ A_27a
@ ^ [V7y: $i] : ( ap @ ( ap @ V2ord @ V7y ) @ V3h ) ) )
@ V4t ) )
=> ( p @ ( ap @ ( ap @ V0P @ V2ord ) @ V6l2 ) ) ) ) )
& ! [V8l1: $i] :
( ( mem @ V8l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V9l2: $i] :
( ( mem @ V9l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) @ V8l1 ) @ V9l2 )
= ( ap
@ ( ap @ ( c_2Esorting_2EPARTITION @ A_27a )
@ ( lam @ A_27a
@ ^ [V10y: $i] : ( ap @ ( ap @ V2ord @ V10y ) @ V3h ) ) )
@ V4t ) )
=> ( p @ ( ap @ ( ap @ V0P @ V2ord ) @ V8l1 ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ V2ord ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3h ) @ V4t ) ) ) ) ) ) ) )
=> ! [V11v: $i] :
( ( mem @ V11v @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V12v1: $i] :
( ( mem @ V12v1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ V0P @ V11v ) @ V12v1 ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EQSORT__DEF,axiom,
! [A_27a: del] :
( ! [V0ord: $i] :
( ( mem @ V0ord @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( c_2Esorting_2EQSORT @ A_27a ) @ V0ord ) @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Elist_2ENIL @ A_27a ) ) )
& ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2ord: $i] :
( ( mem @ V2ord @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V3h: $i] :
( ( mem @ V3h @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Esorting_2EQSORT @ A_27a ) @ V2ord ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3h ) @ V1t ) )
= ( ap
@ ( ap @ ( c_2Ebool_2ELET @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ty_2Elist_2Elist @ A_27a ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) )
@ ( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V4l1: $i] :
( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V5l2: $i] : ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Esorting_2EQSORT @ A_27a ) @ V2ord ) @ V4l1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3h ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) @ ( ap @ ( ap @ ( c_2Esorting_2EQSORT @ A_27a ) @ V2ord ) @ V5l2 ) ) ) ) ) )
@ ( ap
@ ( ap @ ( c_2Esorting_2EPARTITION @ A_27a )
@ ( lam @ A_27a
@ ^ [V6y: $i] : ( ap @ ( ap @ V2ord @ V6y ) @ V3h ) ) )
@ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EQSORT__MEM,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1L: $i] :
( ( mem @ V1L @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Esorting_2EQSORT @ A_27a ) @ V0R ) @ V1L ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1L ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EQSORT__PERM,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1L: $i] :
( ( mem @ V1L @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V1L ) @ ( ap @ ( ap @ ( c_2Esorting_2EQSORT @ A_27a ) @ V0R ) @ V1L ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EQSORT__SORTED,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1L: $i] :
( ( mem @ V1L @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Erelation_2Etotal @ A_27a ) @ V0R ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( c_2Esorting_2EQSORT @ A_27a ) @ V0R ) @ V1L ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EQSORT__SORTS,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Erelation_2Etotal @ A_27a ) @ V0R ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTS @ A_27a ) @ ( c_2Esorting_2EQSORT @ A_27a ) ) @ V0R ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__APPEND__IFF,axiom,
! [A_27a: del] :
( ! [V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l ) @ V1l1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l ) @ V2l2 ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V1l1 ) @ V2l2 ) ) ) ) ) )
& ! [V3l: $i] :
( ( mem @ V3l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V4l1: $i] :
( ( mem @ V4l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V5l2: $i] :
( ( mem @ V5l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V4l1 ) @ V3l ) ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V5l2 ) @ V3l ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V4l1 ) @ V5l2 ) ) ) ) ) ) ) ).
thf(ax_thm_2Esorting_2EPERM__SINGLE__SWAP__DEF,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM__SINGLE__SWAP @ A_27a ) @ V0l1 ) @ V1l2 ) )
<=> ? [V2x1: $i] :
( ( mem @ V2x1 @ ( ty_2Elist_2Elist @ A_27a ) )
& ? [V3x2: $i] :
( ( mem @ V3x2 @ ( ty_2Elist_2Elist @ A_27a ) )
& ? [V4x3: $i] :
( ( mem @ V4x3 @ ( ty_2Elist_2Elist @ A_27a ) )
& ( V0l1
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2x1 ) @ V3x2 ) ) @ V4x3 ) )
& ( V1l2
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2x1 ) @ V4x3 ) ) @ V3x2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__SINGLE__SWAP__SYM,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM__SINGLE__SWAP @ A_27a ) @ V0l1 ) @ V1l2 ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM__SINGLE__SWAP @ A_27a ) @ V1l2 ) @ V0l1 ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__SINGLE__SWAP__I,axiom,
! [A_27a: del,V0x1: $i] :
( ( mem @ V0x1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1x2: $i] :
( ( mem @ V1x2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2x3: $i] :
( ( mem @ V2x3 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM__SINGLE__SWAP @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0x1 ) @ V1x2 ) ) @ V2x3 ) ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0x1 ) @ V2x3 ) ) @ V1x2 ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__SINGLE__SWAP__APPEND,axiom,
! [A_27a: del,V0x2: $i] :
( ( mem @ V0x2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1x3: $i] :
( ( mem @ V1x3 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM__SINGLE__SWAP @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0x2 ) @ V1x3 ) ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1x3 ) @ V0x2 ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__SINGLE__SWAP__REFL,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM__SINGLE__SWAP @ A_27a ) @ V0l ) @ V0l ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__SINGLE__SWAP__CONS,axiom,
! [A_27a: del,V0M: $i] :
( ( mem @ V0M @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1N: $i] :
( ( mem @ V1N @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM__SINGLE__SWAP @ A_27a ) @ V0M ) @ V1N ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM__SINGLE__SWAP @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2x ) @ V0M ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2x ) @ V1N ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__SINGLE__SWAP__TC__CONS,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1M: $i] :
( ( mem @ V1M @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2N: $i] :
( ( mem @ V2N @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( c_2Esorting_2EPERM__SINGLE__SWAP @ A_27a ) ) @ V1M ) @ V2N ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( c_2Esorting_2EPERM__SINGLE__SWAP @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ V1M ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ V2N ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__TC,axiom,
! [A_27a: del] :
( ( c_2Esorting_2EPERM @ A_27a )
= ( ap @ ( c_2Erelation_2ETC @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( c_2Esorting_2EPERM__SINGLE__SWAP @ A_27a ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__RTC,axiom,
! [A_27a: del] :
( ( c_2Esorting_2EPERM @ A_27a )
= ( ap @ ( c_2Erelation_2ERTC @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( c_2Esorting_2EPERM__SINGLE__SWAP @ A_27a ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__EQC,axiom,
! [A_27a: del] :
( ( c_2Esorting_2EPERM @ A_27a )
= ( ap @ ( c_2Erelation_2EEQC @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( c_2Esorting_2EPERM__SINGLE__SWAP @ A_27a ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__lifts__transitive__relations,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ A_27b ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ( ( ! [V2x1: $i] :
( ( mem @ V2x1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3x2: $i] :
( ( mem @ V3x2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V4x3: $i] :
( ( mem @ V4x3 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ V1Q @ ( ap @ V0f @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2x1 ) @ V3x2 ) ) @ V4x3 ) ) ) @ ( ap @ V0f @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2x1 ) @ V4x3 ) ) @ V3x2 ) ) ) ) ) ) )
& ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27b ) @ V1Q ) ) )
=> ! [V5x: $i] :
( ( mem @ V5x @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V6y: $i] :
( ( mem @ V6y @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V5x ) @ V6y ) )
=> ( p @ ( ap @ ( ap @ V1Q @ ( ap @ V0f @ V5x ) ) @ ( ap @ V0f @ V6y ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__lifts__equalities,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ A_27b ) )
=> ( ! [V1x1: $i] :
( ( mem @ V1x1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2x2: $i] :
( ( mem @ V2x2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3x3: $i] :
( ( mem @ V3x3 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ V0f @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1x1 ) @ V2x2 ) ) @ V3x3 ) )
= ( ap @ V0f @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1x1 ) @ V3x3 ) ) @ V2x2 ) ) ) ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V5y: $i] :
( ( mem @ V5y @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V4x ) @ V5y ) )
=> ( ( ap @ V0f @ V4x )
= ( ap @ V0f @ V5y ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__lifts__invariants,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) )
=> ( ! [V1x1: $i] :
( ( mem @ V1x1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2x2: $i] :
( ( mem @ V2x2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3x3: $i] :
( ( mem @ V3x3 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1x1 ) @ V2x2 ) ) @ V3x3 ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1x1 ) @ V3x3 ) ) @ V2x2 ) ) ) ) ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V5y: $i] :
( ( mem @ V5y @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( p @ ( ap @ V0P @ V4x ) )
& ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V4x ) @ V5y ) ) )
=> ( p @ ( ap @ V0P @ V5y ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__lifts__monotonicities,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27b ) ) )
=> ( ! [V1x1: $i] :
( ( mem @ V1x1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2x2: $i] :
( ( mem @ V2x2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3x3: $i] :
( ( mem @ V3x3 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ? [V4x1_27: $i] :
( ( mem @ V4x1_27 @ ( ty_2Elist_2Elist @ A_27b ) )
& ? [V5x2_27: $i] :
( ( mem @ V5x2_27 @ ( ty_2Elist_2Elist @ A_27b ) )
& ? [V6x3_27: $i] :
( ( mem @ V6x3_27 @ ( ty_2Elist_2Elist @ A_27b ) )
& ( ( ap @ V0f @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1x1 ) @ V2x2 ) ) @ V3x3 ) )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27b ) @ V4x1_27 ) @ V5x2_27 ) ) @ V6x3_27 ) )
& ( ( ap @ V0f @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1x1 ) @ V3x3 ) ) @ V2x2 ) )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27b ) @ V4x1_27 ) @ V6x3_27 ) ) @ V5x2_27 ) ) ) ) ) ) ) )
=> ! [V7x: $i] :
( ( mem @ V7x @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V8y: $i] :
( ( mem @ V8y @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V7x ) @ V8y ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27b ) @ ( ap @ V0f @ V7x ) ) @ ( ap @ V0f @ V8y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__EQUIVALENCE,axiom,
! [A_27a: del] : ( p @ ( ap @ ( c_2Erelation_2Eequivalence @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( c_2Esorting_2EPERM @ A_27a ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__EQUIVALENCE__ALT__DEF,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0x ) @ V1y ) )
<=> ( ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0x )
= ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V1y ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EALL__DISTINCT__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 @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0l1 ) @ V1l2 ) )
=> ( ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ V0l1 ) )
<=> ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ V1l2 ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__ALL__DISTINCT,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_2Esorting_2EALL__DISTINCT__PERM__LIST__TO__SET__TO__LIST,axiom,
! [A_27a: del,V0ls: $i] :
( ( mem @ V0ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ V0ls ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0ls ) @ ( ap @ ( c_2Elist_2ESET__TO__LIST @ A_27a ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V0ls ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__MAP,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V1l1 ) @ V2l2 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27b ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V0f ) @ V1l1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V0f ) @ V2l2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__SUM,axiom,
! [V0l1: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum,V1l2: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ ty_2Enum_2Enum ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V0l1 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1l2 ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Elist_2ESUM @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V0l1 ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Elist_2ESUM @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V1l2 ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__FILTER,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V1l1 ) @ V2l2 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V0P ) @ V1l1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V0P ) @ V2l2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__REVERSE,axiom,
! [A_27a: del,V0ls: $i] :
( ( mem @ V0ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0ls ) @ ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ V0ls ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__REVERSE__EQ,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ V0l1 ) ) @ V1l2 ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0l1 ) @ V1l2 ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0l1 ) @ ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ V1l2 ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0l1 ) @ V1l2 ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EFOLDR__PERM,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3e: $i] :
( ( mem @ V3e @ A_27a )
=> ( ( ( p @ ( ap @ ( c_2Ecombin_2EASSOC @ A_27a ) @ V0f ) )
& ( p @ ( ap @ ( c_2Ecombin_2ECOMM @ A_27a @ A_27a ) @ V0f ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V1l1 ) @ V2l2 ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2EFOLDR @ A_27a @ A_27a ) @ V0f ) @ V3e ) @ V1l1 )
= ( ap @ ( ap @ ( ap @ ( c_2Elist_2EFOLDR @ A_27a @ A_27a ) @ V0f ) @ V3e ) @ V2l2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__SET__TO__LIST__count__COUNT__LIST,axiom,
! [V0n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Elist_2ESET__TO__LIST @ ty_2Enum_2Enum ) @ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) @ ( ap @ c_2Erich__list_2ECOUNT__LIST @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ).
thf(conj_thm_2Esorting_2ESUM__IMAGE__count__SUM__GENLIST,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ ty_2Enum_2Enum ) @ V0f ) @ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Elist_2ESUM @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ ty_2Enum_2Enum ) @ V0f ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2ESUM__IMAGE__count__MULT,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
=> ! [V3k: tp__ty_2Enum_2Enum] :
( ! [V4m: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V4m ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V4m ) ) )
= ( surj__ty_2Enum_2Enum
@ ( ap
@ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5x: $i] : ( ap @ V2f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V5x ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V3k ) ) @ ( inj__ty_2Enum_2Enum @ V4m ) ) ) ) ) )
@ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V3k ) ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ ty_2Enum_2Enum ) @ V2f ) @ ( ap @ c_2Epred__set_2Ecount @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V3k ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ ty_2Enum_2Enum ) @ V1g ) @ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2Esum__of__sums,axiom,
! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) )
=> ! [V1a: tp__ty_2Enum_2Enum,V2b: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum
@ ( ap
@ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3m: $i] : ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ ty_2Enum_2Enum ) @ ( ap @ V0f @ V3m ) ) @ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V1a ) ) ) ) )
@ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V2b ) ) ) )
= ( surj__ty_2Enum_2Enum
@ ( ap
@ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4m: $i] : ( ap @ ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ V4m ) @ ( inj__ty_2Enum_2Enum @ V1a ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2EMOD @ V4m ) @ ( inj__ty_2Enum_2Enum @ V1a ) ) ) ) )
@ ( ap @ c_2Epred__set_2Ecount @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V1a ) ) @ ( inj__ty_2Enum_2Enum @ V2b ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2ESORTED__NIL,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V0R ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ).
thf(conj_thm_2Esorting_2ESORTED__SING,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2ESORTED__TL,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2xs: $i] :
( ( mem @ V2xs @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ V2xs ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V0R ) @ V2xs ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2ESORTED__EL__SUC,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1ls: $i] :
( ( mem @ V1ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V0R ) @ V1ls ) )
<=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1ls ) ) )
=> ( p @ ( ap @ ( ap @ V0R @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ V1ls ) ) @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V1ls ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2ESORTED__EL__LESS,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) )
=> ! [V1ls: $i] :
( ( mem @ V1ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V0R ) @ V1ls ) )
<=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1ls ) ) ) )
=> ( p @ ( ap @ ( ap @ V0R @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ V1ls ) ) @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ V1ls ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2ESORTED__APPEND__IFF,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1L1: $i] :
( ( mem @ V1L1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2L2: $i] :
( ( mem @ V2L2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1L1 ) @ V2L2 ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V0R ) @ V1L1 ) )
& ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V0R ) @ V2L2 ) )
& ( ( V1L1
= ( c_2Elist_2ENIL @ A_27a ) )
| ( V2L2
= ( c_2Elist_2ENIL @ A_27a ) )
| ( p @ ( ap @ ( ap @ V0R @ ( ap @ ( c_2Elist_2ELAST @ A_27a ) @ V1L1 ) ) @ ( ap @ ( c_2Elist_2EHD @ A_27a ) @ V2L2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EMEM__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 @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0l1 ) @ V1l2 ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2a ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V0l1 ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2a ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2ESORTED__PERM__EQ,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Erelation_2Eantisymmetric @ A_27a ) @ V0R ) ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V0R ) @ V1l1 ) )
& ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V0R ) @ V2l2 ) )
& ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V1l1 ) @ V2l2 ) ) )
=> ( V1l1 = V2l2 ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EQSORT__eq__if__PERM,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2Etotal @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Erelation_2Eantisymmetric @ A_27a ) @ V0R ) ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Esorting_2EQSORT @ A_27a ) @ V0R ) @ V1l1 )
= ( ap @ ( ap @ ( c_2Esorting_2EQSORT @ A_27a ) @ V0R ) @ V2l2 ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V1l1 ) @ V2l2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2ESORTED__FILTER,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1ls: $i] :
( ( mem @ V1ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2P: $i] :
( ( mem @ V2P @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V0R ) @ V1ls ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V2P ) @ V1ls ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EALL__DISTINCT__SORTED__WEAKEN,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1R_27: $i] :
( ( mem @ V1R_27 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2ls: $i] :
( ( mem @ V2ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V2ls ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4y ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V2ls ) ) )
& ( V3x != V4y ) )
=> ( ( p @ ( ap @ ( ap @ V0R @ V3x ) @ V4y ) )
<=> ( p @ ( ap @ ( ap @ V1R_27 @ V3x ) @ V4y ) ) ) ) ) )
& ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ V2ls ) )
& ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V0R ) @ V2ls ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V1R_27 ) @ V2ls ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__FUN__APPEND__C,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l1_27: $i] :
( ( mem @ V1l1_27 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3l2_27: $i] :
( ( mem @ V3l2_27 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0l1 )
= ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V1l1_27 ) )
=> ( ( ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V2l2 )
= ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V3l2_27 ) )
=> ( ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ V2l2 ) )
= ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1l1_27 ) @ V3l2_27 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__FUN__CONS,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l1_27: $i] :
( ( mem @ V2l1_27 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V1l1 )
= ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V2l1_27 ) )
=> ( ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ V1l1 ) )
= ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ V2l1_27 ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__FUN__APPEND__CONS,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1l1 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ V2l2 ) ) )
= ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ V1l1 ) ) @ V2l2 ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__FUN__SWAP__AT__FRONT,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ! [V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1y ) @ V2l ) ) )
= ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1y ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ V2l ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__FUN__CONS__11__SWAP__AT__FRONT,axiom,
! [A_27a: del,V0y: $i] :
( ( mem @ V0y @ A_27a )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3l2: $i] :
( ( mem @ V3l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V1l1 )
= ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2x ) @ V3l2 ) ) )
=> ( ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0y ) @ V1l1 ) )
= ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0y ) @ V3l2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__FUN__CONS__11__APPEND,axiom,
! [A_27a: del,V0y: $i] :
( ( mem @ V0y @ A_27a )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3l3: $i] :
( ( mem @ V3l3 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V1l1 )
= ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2l2 ) @ V3l3 ) ) )
=> ( ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0y ) @ V1l1 ) )
= ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2l2 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0y ) @ V3l3 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__FUN__CONS__APPEND__1,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3l2: $i] :
( ( mem @ V3l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V1l1 )
= ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2x ) @ V3l2 ) ) )
=> ( ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1l1 ) @ V0l ) )
= ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V3l2 ) @ V0l ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__FUN__CONS__APPEND__2,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3l2: $i] :
( ( mem @ V3l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V1l1 )
= ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2x ) @ V3l2 ) ) )
=> ( ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l ) @ V1l1 ) )
= ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l ) @ V3l2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__FUN__APPEND__APPEND__1,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l3: $i] :
( ( mem @ V2l3 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3l4: $i] :
( ( mem @ V3l4 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0l1 )
= ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1l2 ) @ V2l3 ) ) )
=> ( ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ V3l4 ) )
= ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1l2 ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2l3 ) @ V3l4 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__FUN__APPEND__APPEND__2,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l3: $i] :
( ( mem @ V2l3 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3l4: $i] :
( ( mem @ V3l4 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0l1 )
= ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1l2 ) @ V2l3 ) ) )
=> ( ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V3l4 ) @ V0l1 ) )
= ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1l2 ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V3l4 ) @ V2l3 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__FUN__APPEND,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ V1l2 ) )
= ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1l2 ) @ V0l1 ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__FUN__CONS__IFF,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V1l1 )
= ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V2l2 ) )
=> ( ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ V1l1 ) )
= ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ V2l2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__FUN__APPEND__IFF,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V1l1 )
= ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V2l2 ) )
=> ( ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l ) @ V1l1 ) )
= ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l ) @ V2l2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__FUN__CONG,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l1_27: $i] :
( ( mem @ V1l1_27 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3l2_27: $i] :
( ( mem @ V3l2_27 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0l1 )
= ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V1l1_27 ) )
=> ( ( ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V2l2 )
= ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V3l2_27 ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0l1 ) @ V2l2 ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V1l1_27 ) @ V3l2_27 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__CONG__2,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l1_27: $i] :
( ( mem @ V1l1_27 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3l2_27: $i] :
( ( mem @ V3l2_27 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0l1 ) @ V1l1_27 ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V2l2 ) @ V3l2_27 ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0l1 ) @ V2l2 ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V1l1_27 ) @ V3l2_27 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__CONG__APPEND__IFF,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l1_27: $i] :
( ( mem @ V2l1_27 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3l2: $i] :
( ( mem @ V3l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V4l2_27: $i] :
( ( mem @ V4l2_27 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V1l1 ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l ) @ V2l1_27 ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V3l2 ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l ) @ V4l2_27 ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V1l1 ) @ V3l2 ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V2l1_27 ) @ V4l2_27 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__CONG__APPEND__IFF2,axiom,
! [A_27a: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l1_27: $i] :
( ( mem @ V1l1_27 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l1_27_27: $i] :
( ( mem @ V2l1_27_27 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3l2: $i] :
( ( mem @ V3l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V4l2_27: $i] :
( ( mem @ V4l2_27 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V5l2_27_27: $i] :
( ( mem @ V5l2_27_27 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0l1 ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1l1_27 ) @ V2l1_27_27 ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V3l2 ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V4l2_27 ) @ V5l2_27_27 ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V1l1_27 ) @ V4l2_27 ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0l1 ) @ V3l2 ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V2l1_27_27 ) @ V5l2_27_27 ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__FUN__SPLIT,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l1_27: $i] :
( ( mem @ V2l1_27 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3l2: $i] :
( ( mem @ V3l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0l ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1l1 ) @ V3l2 ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V2l1_27 ) @ V1l1 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0l ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2l1_27 ) @ V3l2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__REWR,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l1: $i] :
( ( mem @ V2l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3l2: $i] :
( ( mem @ V3l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0l ) @ V1r ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l ) @ V2l1 ) ) @ V3l2 ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1r ) @ V2l1 ) ) @ V3l2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__TO__APPEND__SIMPS,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2r1: $i] :
( ( mem @ V2r1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3r2: $i] :
( ( mem @ V3r2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V4xs: $i] :
( ( mem @ V4xs @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V5ys: $i] :
( ( mem @ V5ys @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V6zs: $i] :
( ( mem @ V6zs @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V7r: $i] :
( ( mem @ V7r @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ V1l ) ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ V2r1 ) ) @ V3r2 ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V1l ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2r1 ) @ V3r2 ) ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ V1l ) ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2r1 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ V3r2 ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V1l ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2r1 ) @ V3r2 ) ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V4xs ) @ V5ys ) ) @ V6zs ) ) @ V7r ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V4xs ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V5ys ) @ V6zs ) ) ) @ V7r ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ V5ys ) ) @ V6zs ) ) @ V7r ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0x ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V5ys ) @ V6zs ) ) ) @ V7r ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ V1l ) ) @ V7r ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V1l ) @ V7r ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V4xs ) @ V1l ) ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V4xs ) @ V2r1 ) ) @ V3r2 ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V1l ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2r1 ) @ V3r2 ) ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V4xs ) @ V1l ) ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2r1 ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V4xs ) @ V3r2 ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V1l ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2r1 ) @ V3r2 ) ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27b ) @ ( c_2Elist_2ENIL @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27b ) @ ( c_2Elist_2ENIL @ A_27b ) ) @ ( c_2Elist_2ENIL @ A_27b ) ) ) )
<=> $true )
& ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V4xs ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V4xs ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) )
<=> $true )
& ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V4xs ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V4xs ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) )
<=> $true ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Esorting_2ESTABLE__DEF,axiom,
! [A_27a: del,V0sort: $i] :
( ( mem @ V0sort @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESTABLE @ A_27a ) @ V0sort ) @ V1r ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTS @ A_27a ) @ V0sort ) @ V1r ) )
& ! [V2p: $i] :
( ( mem @ V2p @ ( arr @ A_27a @ bool ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( p @ ( ap @ V2p @ V3x ) )
& ( p @ ( ap @ V2p @ V4y ) ) )
=> ( p @ ( ap @ ( ap @ V1r @ V3x ) @ V4y ) ) ) ) )
=> ! [V5l: $i] :
( ( mem @ V5l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V2p ) @ V5l )
= ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V2p ) @ ( ap @ ( ap @ V0sort @ V1r ) @ V5l ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Esorting_2EPART3__DEF,axiom,
! [A_27a: del] :
( ! [V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1h: $i] :
( ( mem @ V1h @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esorting_2EPART3 @ A_27a ) @ V0R ) @ V1h ) @ ( c_2Elist_2ENIL @ A_27a ) )
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) )
& ! [V2R: $i] :
( ( mem @ V2R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V3h: $i] :
( ( mem @ V3h @ A_27a )
=> ! [V4hd: $i] :
( ( mem @ V4hd @ A_27a )
=> ! [V5tl: $i] :
( ( mem @ V5tl @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esorting_2EPART3 @ A_27a ) @ V2R ) @ V3h ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V4hd ) @ V5tl ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ V2R @ V3h ) @ V4hd ) ) @ ( ap @ ( ap @ V2R @ V4hd ) @ V3h ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Epair_2E_23_23 @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) @ ( c_2Ecombin_2EI @ ( ty_2Elist_2Elist @ A_27a ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_23_23 @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V4hd ) ) @ ( c_2Ecombin_2EI @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esorting_2EPART3 @ A_27a ) @ V2R ) @ V3h ) @ V5tl ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) @ ( ap @ ( ap @ V2R @ V4hd ) @ V3h ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Epair_2E_23_23 @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V4hd ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_23_23 @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( c_2Ecombin_2EI @ ( ty_2Elist_2Elist @ A_27a ) ) ) @ ( c_2Ecombin_2EI @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esorting_2EPART3 @ A_27a ) @ V2R ) @ V3h ) @ V5tl ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Epair_2E_23_23 @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) @ ( c_2Ecombin_2EI @ ( ty_2Elist_2Elist @ A_27a ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_23_23 @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( c_2Ecombin_2EI @ ( ty_2Elist_2Elist @ A_27a ) ) ) @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V4hd ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esorting_2EPART3 @ A_27a ) @ V2R ) @ V3h ) @ V5tl ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPART3__FILTER,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1tl: $i] :
( ( mem @ V1tl @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2hd: $i] :
( ( mem @ V2hd @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esorting_2EPART3 @ A_27a ) @ V0R ) @ V2hd ) @ V1tl )
= ( ap
@ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) )
@ ( ap
@ ( ap @ ( c_2Elist_2EFILTER @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ V0R @ V3x ) @ V2hd ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ V0R @ V2hd ) @ V3x ) ) ) ) )
@ V1tl ) )
@ ( ap
@ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) )
@ ( ap
@ ( ap @ ( c_2Elist_2EFILTER @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ V0R @ V4x ) @ V2hd ) ) @ ( ap @ ( ap @ V0R @ V2hd ) @ V4x ) ) ) )
@ V1tl ) )
@ ( ap
@ ( ap @ ( c_2Elist_2EFILTER @ A_27a )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ V0R @ V5x ) @ V2hd ) ) ) )
@ V1tl ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EQSORT3__IND,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) ) )
=> ( ( ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ V1R ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) )
& ! [V2R: $i] :
( ( mem @ V2R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V3hd: $i] :
( ( mem @ V3hd @ A_27a )
=> ! [V4tl: $i] :
( ( mem @ V4tl @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ! [V5lo: $i] :
( ( mem @ V5lo @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V6eq: $i] :
( ( mem @ V6eq @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V7hi: $i] :
( ( mem @ V7hi @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) @ V5lo ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) @ V6eq ) @ V7hi ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esorting_2EPART3 @ A_27a ) @ V2R ) @ V3hd ) @ V4tl ) )
=> ( p @ ( ap @ ( ap @ V0P @ V2R ) @ V7hi ) ) ) ) ) )
& ! [V8lo: $i] :
( ( mem @ V8lo @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V9eq: $i] :
( ( mem @ V9eq @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V10hi: $i] :
( ( mem @ V10hi @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) @ V8lo ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) @ V9eq ) @ V10hi ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esorting_2EPART3 @ A_27a ) @ V2R ) @ V3hd ) @ V4tl ) )
=> ( p @ ( ap @ ( ap @ V0P @ V2R ) @ V8lo ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ V2R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3hd ) @ V4tl ) ) ) ) ) ) ) )
=> ! [V11v: $i] :
( ( mem @ V11v @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V12v1: $i] :
( ( mem @ V12v1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ V0P @ V11v ) @ V12v1 ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EQSORT3__DEF,axiom,
! [A_27a: del] :
( ! [V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( c_2Esorting_2EQSORT3 @ A_27a ) @ V0R ) @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Elist_2ENIL @ A_27a ) ) )
& ! [V1tl: $i] :
( ( mem @ V1tl @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2hd: $i] :
( ( mem @ V2hd @ A_27a )
=> ! [V3R: $i] :
( ( mem @ V3R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( c_2Esorting_2EQSORT3 @ A_27a ) @ V3R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2hd ) @ V1tl ) )
= ( ap
@ ( ap @ ( c_2Ebool_2ELET @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) @ ( ty_2Elist_2Elist @ A_27a ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ty_2Elist_2Elist @ A_27a ) )
@ ( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V4lo: $i] :
( ap @ ( c_2Epair_2EUNCURRY @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) )
@ ( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V5eq: $i] :
( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V6hi: $i] : ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Esorting_2EQSORT3 @ A_27a ) @ V3R ) @ V4lo ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2hd ) @ V5eq ) ) ) @ ( ap @ ( ap @ ( c_2Esorting_2EQSORT3 @ A_27a ) @ V3R ) @ V6hi ) ) ) ) ) ) ) )
@ ( ap @ ( ap @ ( ap @ ( c_2Esorting_2EPART3 @ A_27a ) @ V3R ) @ V2hd ) @ V1tl ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM3,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2a_27: $i] :
( ( mem @ V2a_27 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3b: $i] :
( ( mem @ V3b @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V4b_27: $i] :
( ( mem @ V4b_27 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V5c: $i] :
( ( mem @ V5c @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V6c_27: $i] :
( ( mem @ V6c_27 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V1a ) @ V2a_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V3b ) @ V4b_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V5c ) @ V6c_27 ) )
& ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0x ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1a ) @ V3b ) ) @ V5c ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0x ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2a_27 ) @ V4b_27 ) ) @ V6c_27 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM3__FILTER,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ A_27a )
=> ( p
@ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V1l )
@ ( ap
@ ( ap @ ( c_2Elist_2EAPPEND @ A_27a )
@ ( ap
@ ( ap @ ( c_2Elist_2EAPPEND @ A_27a )
@ ( ap
@ ( ap @ ( c_2Elist_2EFILTER @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ V0R @ V3x ) @ V2h ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ V0R @ V2h ) @ V3x ) ) ) ) )
@ V1l ) )
@ ( ap
@ ( ap @ ( c_2Elist_2EFILTER @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ V0R @ V4x ) @ V2h ) ) @ ( ap @ ( ap @ V0R @ V2h ) @ V4x ) ) ) )
@ V1l ) ) )
@ ( ap
@ ( ap @ ( c_2Elist_2EFILTER @ A_27a )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ V0R @ V5x ) @ V2h ) ) ) )
@ V1l ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EPERM__QSORT3,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ V0l ) @ ( ap @ ( ap @ ( c_2Esorting_2EQSORT3 @ A_27a ) @ V1R ) @ V0l ) ) ) ) ) ).
thf(conj_thm_2Esorting_2ESORTED__EQ__PART,axiom,
! [A_27a: del,V0hd: $i] :
( ( mem @ V0hd @ A_27a )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2R: $i] :
( ( mem @ V2R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V2R ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V2R )
@ ( ap
@ ( ap @ ( c_2Elist_2EFILTER @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ V2R @ V3x ) @ V0hd ) ) @ ( ap @ ( ap @ V2R @ V0hd ) @ V3x ) ) ) )
@ V1l ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EQSORT3__SORTS,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Erelation_2Etotal @ A_27a ) @ V0R ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTS @ A_27a ) @ ( c_2Esorting_2EQSORT3 @ A_27a ) ) @ V0R ) ) ) ) ).
thf(conj_thm_2Esorting_2EQSORT3__SPLIT,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Erelation_2Etotal @ A_27a ) @ V0R ) ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2e: $i] :
( ( mem @ V2e @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Esorting_2EQSORT3 @ A_27a ) @ V0R ) @ V1l )
= ( ap
@ ( ap @ ( c_2Elist_2EAPPEND @ A_27a )
@ ( ap
@ ( ap @ ( c_2Elist_2EAPPEND @ A_27a )
@ ( ap @ ( ap @ ( c_2Esorting_2EQSORT3 @ A_27a ) @ V0R )
@ ( ap
@ ( ap @ ( c_2Elist_2EFILTER @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ V0R @ V3x ) @ V2e ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ V0R @ V2e ) @ V3x ) ) ) ) )
@ V1l ) ) )
@ ( ap
@ ( ap @ ( c_2Elist_2EFILTER @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ V0R @ V4x ) @ V2e ) ) @ ( ap @ ( ap @ V0R @ V2e ) @ V4x ) ) ) )
@ V1l ) ) )
@ ( ap @ ( ap @ ( c_2Esorting_2EQSORT3 @ A_27a ) @ V0R )
@ ( ap
@ ( ap @ ( c_2Elist_2EFILTER @ A_27a )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ V0R @ V5x ) @ V2e ) ) ) )
@ V1l ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EQSORT3__STABLE,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Erelation_2Etotal @ A_27a ) @ V0R ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESTABLE @ A_27a ) @ ( c_2Esorting_2EQSORT3 @ A_27a ) ) @ V0R ) ) ) ) ).
thf(conj_thm_2Esorting_2EQSORT3__MEM,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1L: $i] :
( ( mem @ V1L @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2Esorting_2EQSORT3 @ A_27a ) @ V0R ) @ V1L ) ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1L ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2EQSORT3__SORTED,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1L: $i] :
( ( mem @ V1L @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Erelation_2Etotal @ A_27a ) @ V0R ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( c_2Esorting_2EQSORT3 @ A_27a ) @ V0R ) @ V1L ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2Esorted__count__list,axiom,
! [V0n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ ty_2Enum_2Enum ) @ c_2Earithmetic_2E_3C_3D ) @ ( ap @ c_2Erich__list_2ECOUNT__LIST @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ).
thf(conj_thm_2Esorting_2Esorted__map,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ A_27a ) )
=> ! [V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27b @ A_27a ) @ V1f ) @ V2l ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27b ) @ ( ap @ ( ap @ ( c_2Erelation_2Einv__image @ A_27b @ A_27a ) @ V0R ) @ V1f ) ) @ V2l ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2Esorted__perm__count__list,axiom,
! [A_27a: del,A_27b: del,V0y: $i] :
( ( mem @ V0y @ A_27a )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ ty_2Enum_2Enum ) )
=> ! [V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V3n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27b ) @ ( ap @ ( ap @ ( c_2Erelation_2Einv__image @ A_27b @ ty_2Enum_2Enum ) @ c_2Earithmetic_2E_3C_3D ) @ V1f ) ) @ V2l ) )
& ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27b @ ty_2Enum_2Enum ) @ V1f ) @ V2l ) ) @ ( ap @ c_2Erich__list_2ECOUNT__LIST @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) )
=> ( ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27b @ ty_2Enum_2Enum ) @ V1f ) @ V2l ) )
= ( surj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ ( ap @ c_2Erich__list_2ECOUNT__LIST @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2ESORTED__weaken,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1R_27: $i] :
( ( mem @ V1R_27 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2ls: $i] :
( ( mem @ V2ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V0R ) @ V2ls ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V2ls ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4y ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V2ls ) ) )
& ( p @ ( ap @ ( ap @ V0R @ V3x ) @ V4y ) ) )
=> ( p @ ( ap @ ( ap @ V1R_27 @ V3x ) @ V4y ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V1R_27 ) @ V2ls ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2Eless__sorted__eq,axiom,
! [V0L: tp__c_ty_2Elist_2Elist_ty_2Enum_2Enum,V1x: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ ty_2Enum_2Enum ) @ c_2Eprim__rec_2E_3C ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V0L ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ ty_2Enum_2Enum ) @ c_2Eprim__rec_2E_3C ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V0L ) ) )
& ! [V2y: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2y ) ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ ty_2Enum_2Enum ) @ ( inj__c_ty_2Elist_2Elist_ty_2Enum_2Enum @ V0L ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ ( inj__ty_2Enum_2Enum @ V2y ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2ESORTED__GENLIST__PLUS,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1k: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ ty_2Enum_2Enum ) @ c_2Eprim__rec_2E_3C ) @ ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1k ) ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ).
thf(conj_thm_2Esorting_2ESORTED__ALL__DISTINCT,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2Eirreflexive @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) ) )
=> ! [V1ls: $i] :
( ( mem @ V1ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V0R ) @ V1ls ) )
=> ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ V1ls ) ) ) ) ) ) ).
thf(conj_thm_2Esorting_2Esorted__filter,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2ls: $i] :
( ( mem @ V2ls @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V1R ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V1R ) @ V2ls ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V1R ) @ ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V0P ) @ V2ls ) ) ) ) ) ) ) ) ).
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