ITP001 Axioms: ITP067^5.ax
%------------------------------------------------------------------------------
% File : ITP067^5 : 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 : mergesort^2.ax [Gau20]
% : HL4067^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 64 ( 0 unt; 11 typ; 0 def)
% Number of atoms : 3132 ( 66 equ; 0 cnn)
% Maximal formula atoms : 308 ( 48 avg)
% Number of connectives : 5172 ( 22 ~; 0 |; 96 &;4687 @)
% ( 8 <=>; 359 =>; 0 <=; 0 <~>)
% Maximal formula depth : 37 ( 16 avg;4687 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 11 ( 11 >; 0 *; 0 +; 0 <<)
% Number of symbols : 55 ( 52 usr; 43 con; 0-2 aty)
% Number of variables : 418 ( 27 ^ 391 !; 0 ?; 418 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Emergesort_2Emerge,type,
c_2Emergesort_2Emerge: del > $i ).
thf(mem_c_2Emergesort_2Emerge,axiom,
! [A_27a: del] : ( mem @ ( c_2Emergesort_2Emerge @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ) ).
thf(tp_c_2Emergesort_2Emerge__tail,type,
c_2Emergesort_2Emerge__tail: del > $i ).
thf(mem_c_2Emergesort_2Emerge__tail,axiom,
! [A_27a: del] : ( mem @ ( c_2Emergesort_2Emerge__tail @ A_27a ) @ ( arr @ bool @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ) ) ) ).
thf(tp_c_2Emergesort_2Emergesort,type,
c_2Emergesort_2Emergesort: del > $i ).
thf(mem_c_2Emergesort_2Emergesort,axiom,
! [A_27a: del] : ( mem @ ( c_2Emergesort_2Emergesort @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ).
thf(tp_c_2Emergesort_2EmergesortN,type,
c_2Emergesort_2EmergesortN: del > $i ).
thf(mem_c_2Emergesort_2EmergesortN,axiom,
! [A_27a: del] : ( mem @ ( c_2Emergesort_2EmergesortN @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ) ).
thf(tp_c_2Emergesort_2EmergesortN__tail,type,
c_2Emergesort_2EmergesortN__tail: del > $i ).
thf(mem_c_2Emergesort_2EmergesortN__tail,axiom,
! [A_27a: del] : ( mem @ ( c_2Emergesort_2EmergesortN__tail @ A_27a ) @ ( arr @ bool @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ) ) ).
thf(tp_c_2Emergesort_2Emergesort__tail,type,
c_2Emergesort_2Emergesort__tail: del > $i ).
thf(mem_c_2Emergesort_2Emergesort__tail,axiom,
! [A_27a: del] : ( mem @ ( c_2Emergesort_2Emergesort__tail @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ).
thf(tp_c_2Emergesort_2Esort2,type,
c_2Emergesort_2Esort2: del > $i ).
thf(mem_c_2Emergesort_2Esort2,axiom,
! [A_27a: del] : ( mem @ ( c_2Emergesort_2Esort2 @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ) ).
thf(tp_c_2Emergesort_2Esort2__tail,type,
c_2Emergesort_2Esort2__tail: del > $i ).
thf(mem_c_2Emergesort_2Esort2__tail,axiom,
! [A_27a: del] : ( mem @ ( c_2Emergesort_2Esort2__tail @ A_27a ) @ ( arr @ bool @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ) ) ).
thf(tp_c_2Emergesort_2Esort3,type,
c_2Emergesort_2Esort3: del > $i ).
thf(mem_c_2Emergesort_2Esort3,axiom,
! [A_27a: del] : ( mem @ ( c_2Emergesort_2Esort3 @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ ( arr @ A_27a @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ) ) ).
thf(tp_c_2Emergesort_2Esort3__tail,type,
c_2Emergesort_2Esort3__tail: del > $i ).
thf(mem_c_2Emergesort_2Esort3__tail,axiom,
! [A_27a: del] : ( mem @ ( c_2Emergesort_2Esort3__tail @ A_27a ) @ ( arr @ bool @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ ( arr @ A_27a @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ) ) ) ).
thf(tp_c_2Emergesort_2Estable,type,
c_2Emergesort_2Estable: del > $i ).
thf(mem_c_2Emergesort_2Estable,axiom,
! [A_27a: del] : ( mem @ ( c_2Emergesort_2Estable @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) ) ) ) ).
thf(ax_thm_2Emergesort_2Estable__def,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 @ ( ap @ ( c_2Emergesort_2Estable @ A_27a ) @ V0R ) @ V1l1 ) @ V2l2 ) )
<=> ! [V3p: $i] :
( ( mem @ V3p @ ( arr @ A_27a @ bool ) )
=> ( ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( ( p @ ( ap @ V3p @ V4x ) )
& ( p @ ( ap @ V3p @ V5y ) ) )
=> ( p @ ( ap @ ( ap @ V0R @ V4x ) @ V5y ) ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V3p ) @ V1l1 )
= ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V3p ) @ V2l2 ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Emergesort_2Esort2__def,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Esort2 @ A_27a ) @ V0R ) @ V1x ) @ V2y )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( ap @ V0R @ V1x ) @ V2y ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2y ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2y ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Emergesort_2Esort3__def,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ! [V3z: $i] :
( ( mem @ V3z @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Esort3 @ A_27a ) @ V0R ) @ V1x ) @ V2y ) @ V3z )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( ap @ V0R @ V1x ) @ V2y ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( ap @ V0R @ V2y ) @ V3z ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2y ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3z ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( ap @ V0R @ V1x ) @ V3z ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3z ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2y ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3z ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2y ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( ap @ V0R @ V2y ) @ V3z ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( ap @ V0R @ V1x ) @ V3z ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2y ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3z ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2y ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3z ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3z ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2y ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emergesort_2Emerge__ind,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) ) ) )
=> ( ( ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V1R ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) )
& ! [V2R: $i] :
( ( mem @ V2R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V3v8: $i] :
( ( mem @ V3v8 @ A_27a )
=> ! [V4v9: $i] :
( ( mem @ V4v9 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V2R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3v8 ) @ V4v9 ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) )
& ! [V5R: $i] :
( ( mem @ V5R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V6v4: $i] :
( ( mem @ V6v4 @ A_27a )
=> ! [V7v5: $i] :
( ( mem @ V7v5 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V5R ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V6v4 ) @ V7v5 ) ) ) ) ) )
& ! [V8R: $i] :
( ( mem @ V8R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V9x: $i] :
( ( mem @ V9x @ A_27a )
=> ! [V10l1: $i] :
( ( mem @ V10l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V11y: $i] :
( ( mem @ V11y @ A_27a )
=> ! [V12l2: $i] :
( ( mem @ V12l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ~ ( p @ ( ap @ ( ap @ V8R @ V9x ) @ V11y ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V8R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V9x ) @ V10l1 ) ) @ V12l2 ) ) )
& ( ( p @ ( ap @ ( ap @ V8R @ V9x ) @ V11y ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V8R ) @ V10l1 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V11y ) @ V12l2 ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V8R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V9x ) @ V10l1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V11y ) @ V12l2 ) ) ) ) ) ) ) ) ) )
=> ! [V13v: $i] :
( ( mem @ V13v @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V14v1: $i] :
( ( mem @ V14v1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V15v2: $i] :
( ( mem @ V15v2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V13v ) @ V14v1 ) @ V15v2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Emergesort_2Emerge__def,axiom,
! [A_27a: del] :
( ! [V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Emerge @ A_27a ) @ V0R ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Elist_2ENIL @ A_27a ) ) )
& ! [V1v9: $i] :
( ( mem @ V1v9 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2v8: $i] :
( ( mem @ V2v8 @ A_27a )
=> ! [V3R: $i] :
( ( mem @ V3R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Emerge @ A_27a ) @ V3R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2v8 ) @ V1v9 ) ) @ ( c_2Elist_2ENIL @ A_27a ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2v8 ) @ V1v9 ) ) ) ) )
& ! [V4v5: $i] :
( ( mem @ V4v5 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V5v4: $i] :
( ( mem @ V5v4 @ A_27a )
=> ! [V6R: $i] :
( ( mem @ V6R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Emerge @ A_27a ) @ V6R ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V5v4 ) @ V4v5 ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V5v4 ) @ V4v5 ) ) ) ) )
& ! [V7y: $i] :
( ( mem @ V7y @ A_27a )
=> ! [V8x: $i] :
( ( mem @ V8x @ A_27a )
=> ! [V9l2: $i] :
( ( mem @ V9l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V10l1: $i] :
( ( mem @ V10l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V11R: $i] :
( ( mem @ V11R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Emerge @ A_27a ) @ V11R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V8x ) @ V10l1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V7y ) @ V9l2 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( ap @ V11R @ V8x ) @ V7y ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V8x ) @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Emerge @ A_27a ) @ V11R ) @ V10l1 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V7y ) @ V9l2 ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V7y ) @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Emerge @ A_27a ) @ V11R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V8x ) @ V10l1 ) ) @ V9l2 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emergesort_2EmergesortN__ind,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) ) ) )
=> ( ( ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V1R ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V2l ) ) ) )
& ! [V3R: $i] :
( ( mem @ V3R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5l: $i] :
( ( mem @ V5l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V3R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V4x ) @ V5l ) ) ) ) ) )
& ! [V6R: $i] :
( ( mem @ V6R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V6R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) )
& ! [V7R: $i] :
( ( mem @ V7R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V8x: $i] :
( ( mem @ V8x @ A_27a )
=> ! [V9y: $i] :
( ( mem @ V9y @ A_27a )
=> ! [V10l: $i] :
( ( mem @ V10l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V7R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V8x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V9y ) @ V10l ) ) ) ) ) ) ) )
& ! [V11R: $i] :
( ( mem @ V11R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V12x: $i] :
( ( mem @ V12x @ A_27a )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V11R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V12x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) )
& ! [V13R: $i] :
( ( mem @ V13R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V13R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) )
& ! [V14R: $i] :
( ( mem @ V14R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V15x: $i] :
( ( mem @ V15x @ A_27a )
=> ! [V16y: $i] :
( ( mem @ V16y @ A_27a )
=> ! [V17z: $i] :
( ( mem @ V17z @ A_27a )
=> ! [V18l: $i] :
( ( mem @ V18l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V14R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V15x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V16y ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V17z ) @ V18l ) ) ) ) ) ) ) ) ) )
& ! [V19R: $i] :
( ( mem @ V19R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V20x: $i] :
( ( mem @ V20x @ A_27a )
=> ! [V21y: $i] :
( ( mem @ V21y @ A_27a )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V19R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V20x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V21y ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ) ) )
& ! [V22R: $i] :
( ( mem @ V22R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V23x: $i] :
( ( mem @ V23x @ A_27a )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V22R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V23x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) )
& ! [V24R: $i] :
( ( mem @ V24R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V24R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) )
& ! [V25R: $i] :
( ( mem @ V25R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V26v4: tp__ty_2Enum_2Enum,V27l: $i] :
( ( mem @ V27l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ! [V28len1: tp__ty_2Enum_2Enum] :
( ( ( V26v4 != fo__c_2Enum_2E0 )
& ( V26v4
!= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
& ( V26v4
!= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
& ( V26v4
!= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
& ( V28len1
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EDIV2 @ ( inj__ty_2Enum_2Enum @ V26v4 ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V25R ) @ ( ap @ c_2Earithmetic_2EDIV2 @ ( inj__ty_2Enum_2Enum @ V26v4 ) ) ) @ V27l ) ) )
& ! [V29len1: tp__ty_2Enum_2Enum] :
( ( ( V26v4 != fo__c_2Enum_2E0 )
& ( V26v4
!= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
& ( V26v4
!= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
& ( V26v4
!= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
& ( V29len1
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EDIV2 @ ( inj__ty_2Enum_2Enum @ V26v4 ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V25R ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V26v4 ) ) @ ( inj__ty_2Enum_2Enum @ V29len1 ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2EDROP @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V29len1 ) ) @ V27l ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V25R ) @ ( inj__ty_2Enum_2Enum @ V26v4 ) ) @ V27l ) ) ) ) ) )
=> ! [V30v: $i] :
( ( mem @ V30v @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V31v1: tp__ty_2Enum_2Enum,V32v2: $i] :
( ( mem @ V32v2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V30v ) @ ( inj__ty_2Enum_2Enum @ V31v1 ) ) @ V32v2 ) ) ) ) ) ) ).
thf(conj_thm_2Emergesort_2EmergesortN__def,axiom,
! [A_27a: del] :
( ! [V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2EmergesortN @ A_27a ) @ V1R ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V0l )
= ( c_2Elist_2ENIL @ A_27a ) ) ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3l: $i] :
( ( mem @ V3l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V4R: $i] :
( ( mem @ V4R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2EmergesortN @ A_27a ) @ V4R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2x ) @ V3l ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) )
& ! [V5R: $i] :
( ( mem @ V5R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2EmergesortN @ A_27a ) @ V5R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Elist_2ENIL @ A_27a ) ) )
& ! [V6y: $i] :
( ( mem @ V6y @ A_27a )
=> ! [V7x: $i] :
( ( mem @ V7x @ A_27a )
=> ! [V8l: $i] :
( ( mem @ V8l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V9R: $i] :
( ( mem @ V9R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2EmergesortN @ A_27a ) @ V9R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V7x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V6y ) @ V8l ) ) )
= ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Esort2 @ A_27a ) @ V9R ) @ V7x ) @ V6y ) ) ) ) ) )
& ! [V10x: $i] :
( ( mem @ V10x @ A_27a )
=> ! [V11R: $i] :
( ( mem @ V11R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2EmergesortN @ A_27a ) @ V11R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V10x ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V10x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) )
& ! [V12R: $i] :
( ( mem @ V12R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2EmergesortN @ A_27a ) @ V12R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Elist_2ENIL @ A_27a ) ) )
& ! [V13z: $i] :
( ( mem @ V13z @ A_27a )
=> ! [V14y: $i] :
( ( mem @ V14y @ A_27a )
=> ! [V15x: $i] :
( ( mem @ V15x @ A_27a )
=> ! [V16l: $i] :
( ( mem @ V16l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V17R: $i] :
( ( mem @ V17R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2EmergesortN @ A_27a ) @ V17R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V15x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V14y ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V13z ) @ V16l ) ) ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Esort3 @ A_27a ) @ V17R ) @ V15x ) @ V14y ) @ V13z ) ) ) ) ) ) )
& ! [V18y: $i] :
( ( mem @ V18y @ A_27a )
=> ! [V19x: $i] :
( ( mem @ V19x @ A_27a )
=> ! [V20R: $i] :
( ( mem @ V20R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2EmergesortN @ A_27a ) @ V20R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V19x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V18y ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) )
= ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Esort2 @ A_27a ) @ V20R ) @ V19x ) @ V18y ) ) ) ) )
& ! [V21x: $i] :
( ( mem @ V21x @ A_27a )
=> ! [V22R: $i] :
( ( mem @ V22R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2EmergesortN @ A_27a ) @ V22R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V21x ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V21x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) )
& ! [V23R: $i] :
( ( mem @ V23R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2EmergesortN @ A_27a ) @ V23R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Elist_2ENIL @ A_27a ) ) )
& ! [V24v4: tp__ty_2Enum_2Enum,V25l: $i] :
( ( mem @ V25l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V26R: $i] :
( ( mem @ V26R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2EmergesortN @ A_27a ) @ V26R ) @ ( inj__ty_2Enum_2Enum @ V24v4 ) ) @ V25l )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V24v4 ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( c_2Elist_2ENIL @ A_27a ) )
@ ( ap
@ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V24v4 ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
@ ( ap @ ( ap @ ( ap @ ( c_2Elist_2Elist__CASE @ A_27a @ ( ty_2Elist_2Elist @ A_27a ) ) @ V25l ) @ ( c_2Elist_2ENIL @ A_27a ) )
@ ( lam @ A_27a
@ ^ [V27x: $i] :
( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V28l_27: $i] : ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V27x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) )
@ ( ap
@ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V24v4 ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
@ ( ap @ ( ap @ ( ap @ ( c_2Elist_2Elist__CASE @ A_27a @ ( ty_2Elist_2Elist @ A_27a ) ) @ V25l ) @ ( c_2Elist_2ENIL @ A_27a ) )
@ ( lam @ A_27a
@ ^ [V29x_27: $i] :
( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V30v17: $i] :
( ap @ ( ap @ ( ap @ ( c_2Elist_2Elist__CASE @ A_27a @ ( ty_2Elist_2Elist @ A_27a ) ) @ V30v17 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V29x_27 ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
@ ( lam @ A_27a
@ ^ [V31y: $i] :
( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V32l_27_27: $i] : ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Esort2 @ A_27a ) @ V26R ) @ V29x_27 ) @ V31y ) ) ) ) ) ) ) )
@ ( ap
@ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V24v4 ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
@ ( ap @ ( ap @ ( ap @ ( c_2Elist_2Elist__CASE @ A_27a @ ( ty_2Elist_2Elist @ A_27a ) ) @ V25l ) @ ( c_2Elist_2ENIL @ A_27a ) )
@ ( lam @ A_27a
@ ^ [V33x_27_27: $i] :
( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V34v25: $i] :
( ap @ ( ap @ ( ap @ ( c_2Elist_2Elist__CASE @ A_27a @ ( ty_2Elist_2Elist @ A_27a ) ) @ V34v25 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V33x_27_27 ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
@ ( lam @ A_27a
@ ^ [V35y_27: $i] :
( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V36v29: $i] :
( ap @ ( ap @ ( ap @ ( c_2Elist_2Elist__CASE @ A_27a @ ( ty_2Elist_2Elist @ A_27a ) ) @ V36v29 ) @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Esort2 @ A_27a ) @ V26R ) @ V33x_27_27 ) @ V35y_27 ) )
@ ( lam @ A_27a
@ ^ [V37z: $i] :
( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V38l_27_27_27: $i] : ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Esort3 @ A_27a ) @ V26R ) @ V33x_27_27 ) @ V35y_27 ) @ V37z ) ) ) ) ) ) ) ) ) ) )
@ ( ap
@ ( ap @ ( c_2Ebool_2ELET @ ty_2Enum_2Enum @ ( ty_2Elist_2Elist @ A_27a ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V39len1: $i] : ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Emerge @ A_27a ) @ V26R ) @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2EmergesortN @ A_27a ) @ V26R ) @ ( ap @ c_2Earithmetic_2EDIV2 @ ( inj__ty_2Enum_2Enum @ V24v4 ) ) ) @ V25l ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2EmergesortN @ A_27a ) @ V26R ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V24v4 ) ) @ V39len1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2EDROP @ A_27a ) @ V39len1 ) @ V25l ) ) ) ) )
@ ( ap @ c_2Earithmetic_2EDIV2 @ ( inj__ty_2Enum_2Enum @ V24v4 ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Emergesort_2Emergesort__def,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Emergesort_2Emergesort @ A_27a ) @ V0R ) @ V1l )
= ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2EmergesortN @ A_27a ) @ V0R ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l ) ) @ V1l ) ) ) ) ).
thf(ax_thm_2Emergesort_2Esort2__tail__def,axiom,
! [A_27a: del,V0neg: $i] :
( ( mem @ V0neg @ bool )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Esort2__tail @ A_27a ) @ V0neg ) @ V1R ) @ V2x ) @ V3y )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ bool ) @ ( ap @ ( ap @ V1R @ V2x ) @ V3y ) ) @ V0neg ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3y ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3y ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Emergesort_2Esort3__tail__def,axiom,
! [A_27a: del,V0neg: $i] :
( ( mem @ V0neg @ bool )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ! [V4z: $i] :
( ( mem @ V4z @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Esort3__tail @ A_27a ) @ V0neg ) @ V1R ) @ V2x ) @ V3y ) @ V4z )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ bool ) @ ( ap @ ( ap @ V1R @ V2x ) @ V3y ) ) @ V0neg ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ bool ) @ ( ap @ ( ap @ V1R @ V3y ) @ V4z ) ) @ V0neg ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3y ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V4z ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ bool ) @ ( ap @ ( ap @ V1R @ V2x ) @ V4z ) ) @ V0neg ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V4z ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3y ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V4z ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3y ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ bool ) @ ( ap @ ( ap @ V1R @ V3y ) @ V4z ) ) @ V0neg ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ bool ) @ ( ap @ ( ap @ V1R @ V2x ) @ V4z ) ) @ V0neg ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3y ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V4z ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3y ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V4z ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V4z ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3y ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emergesort_2Emerge__tail__ind,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ bool @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) ) ) ) ) )
=> ( ( ! [V1negate: $i] :
( ( mem @ V1negate @ bool )
=> ! [V2R: $i] :
( ( mem @ V2R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V3acc: $i] :
( ( mem @ V3acc @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V1negate ) @ V2R ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ V3acc ) ) ) ) )
& ! [V4negate: $i] :
( ( mem @ V4negate @ bool )
=> ! [V5R: $i] :
( ( mem @ V5R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V6v12: $i] :
( ( mem @ V6v12 @ A_27a )
=> ! [V7v13: $i] :
( ( mem @ V7v13 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V8acc: $i] :
( ( mem @ V8acc @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V4negate ) @ V5R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V6v12 ) @ V7v13 ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ V8acc ) ) ) ) ) ) )
& ! [V9negate: $i] :
( ( mem @ V9negate @ bool )
=> ! [V10R: $i] :
( ( mem @ V10R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V11v8: $i] :
( ( mem @ V11v8 @ A_27a )
=> ! [V12v9: $i] :
( ( mem @ V12v9 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V13acc: $i] :
( ( mem @ V13acc @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V9negate ) @ V10R ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V11v8 ) @ V12v9 ) ) @ V13acc ) ) ) ) ) ) )
& ! [V14negate: $i] :
( ( mem @ V14negate @ bool )
=> ! [V15R: $i] :
( ( mem @ V15R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V16x: $i] :
( ( mem @ V16x @ A_27a )
=> ! [V17l1: $i] :
( ( mem @ V17l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V18y: $i] :
( ( mem @ V18y @ A_27a )
=> ! [V19l2: $i] :
( ( mem @ V19l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V20acc: $i] :
( ( mem @ V20acc @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ~ ~ ( ( p @ ( ap @ ( ap @ V15R @ V16x ) @ V18y ) )
<=> ( p @ V14negate ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V14negate ) @ V15R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V16x ) @ V17l1 ) ) @ V19l2 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V18y ) @ V20acc ) ) ) )
& ( ~ ( ( p @ ( ap @ ( ap @ V15R @ V16x ) @ V18y ) )
<=> ( p @ V14negate ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V14negate ) @ V15R ) @ V17l1 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V18y ) @ V19l2 ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V16x ) @ V20acc ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V14negate ) @ V15R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V16x ) @ V17l1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V18y ) @ V19l2 ) ) @ V20acc ) ) ) ) ) ) ) ) ) ) )
=> ! [V21v: $i] :
( ( mem @ V21v @ bool )
=> ! [V22v1: $i] :
( ( mem @ V22v1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V23v2: $i] :
( ( mem @ V23v2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V24v3: $i] :
( ( mem @ V24v3 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V25v4: $i] :
( ( mem @ V25v4 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V21v ) @ V22v1 ) @ V23v2 ) @ V24v3 ) @ V25v4 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emergesort_2Emerge__tail__def,axiom,
! [A_27a: del] :
( ! [V0negate: $i] :
( ( mem @ V0negate @ bool )
=> ! [V1acc: $i] :
( ( mem @ V1acc @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2R: $i] :
( ( mem @ V2R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Emerge__tail @ A_27a ) @ V0negate ) @ V2R ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ V1acc )
= V1acc ) ) ) )
& ! [V3v13: $i] :
( ( mem @ V3v13 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V4v12: $i] :
( ( mem @ V4v12 @ A_27a )
=> ! [V5negate: $i] :
( ( mem @ V5negate @ bool )
=> ! [V6acc: $i] :
( ( mem @ V6acc @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V7R: $i] :
( ( mem @ V7R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Emerge__tail @ A_27a ) @ V5negate ) @ V7R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V4v12 ) @ V3v13 ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ V6acc )
= ( ap @ ( ap @ ( c_2Elist_2EREV @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V4v12 ) @ V3v13 ) ) @ V6acc ) ) ) ) ) ) )
& ! [V8v9: $i] :
( ( mem @ V8v9 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V9v8: $i] :
( ( mem @ V9v8 @ A_27a )
=> ! [V10negate: $i] :
( ( mem @ V10negate @ bool )
=> ! [V11acc: $i] :
( ( mem @ V11acc @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V12R: $i] :
( ( mem @ V12R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Emerge__tail @ A_27a ) @ V10negate ) @ V12R ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V9v8 ) @ V8v9 ) ) @ V11acc )
= ( ap @ ( ap @ ( c_2Elist_2EREV @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V9v8 ) @ V8v9 ) ) @ V11acc ) ) ) ) ) ) )
& ! [V13y: $i] :
( ( mem @ V13y @ A_27a )
=> ! [V14x: $i] :
( ( mem @ V14x @ A_27a )
=> ! [V15negate: $i] :
( ( mem @ V15negate @ bool )
=> ! [V16l2: $i] :
( ( mem @ V16l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V17l1: $i] :
( ( mem @ V17l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V18acc: $i] :
( ( mem @ V18acc @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V19R: $i] :
( ( mem @ V19R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Emerge__tail @ A_27a ) @ V15negate ) @ V19R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V14x ) @ V17l1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V13y ) @ V16l2 ) ) @ V18acc )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ bool ) @ ( ap @ ( ap @ V19R @ V14x ) @ V13y ) ) @ V15negate ) ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Emerge__tail @ A_27a ) @ V15negate ) @ V19R ) @ V17l1 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V13y ) @ V16l2 ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V14x ) @ V18acc ) ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Emerge__tail @ A_27a ) @ V15negate ) @ V19R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V14x ) @ V17l1 ) ) @ V16l2 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V13y ) @ V18acc ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emergesort_2EmergesortN__tail__ind,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ bool @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) ) ) ) )
=> ( ( ! [V1negate: $i] :
( ( mem @ V1negate @ bool )
=> ! [V2R: $i] :
( ( mem @ V2R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V3l: $i] :
( ( mem @ V3l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V1negate ) @ V2R ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V3l ) ) ) ) )
& ! [V4negate: $i] :
( ( mem @ V4negate @ bool )
=> ! [V5R: $i] :
( ( mem @ V5R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ! [V7l: $i] :
( ( mem @ V7l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V4negate ) @ V5R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V6x ) @ V7l ) ) ) ) ) ) )
& ! [V8negate: $i] :
( ( mem @ V8negate @ bool )
=> ! [V9R: $i] :
( ( mem @ V9R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V8negate ) @ V9R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) )
& ! [V10negate: $i] :
( ( mem @ V10negate @ bool )
=> ! [V11R: $i] :
( ( mem @ V11R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V12x: $i] :
( ( mem @ V12x @ A_27a )
=> ! [V13y: $i] :
( ( mem @ V13y @ A_27a )
=> ! [V14l: $i] :
( ( mem @ V14l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V10negate ) @ V11R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V12x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V13y ) @ V14l ) ) ) ) ) ) ) ) )
& ! [V15negate: $i] :
( ( mem @ V15negate @ bool )
=> ! [V16R: $i] :
( ( mem @ V16R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V17x: $i] :
( ( mem @ V17x @ A_27a )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V15negate ) @ V16R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V17x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ) )
& ! [V18negate: $i] :
( ( mem @ V18negate @ bool )
=> ! [V19R: $i] :
( ( mem @ V19R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V18negate ) @ V19R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) )
& ! [V20negate: $i] :
( ( mem @ V20negate @ bool )
=> ! [V21R: $i] :
( ( mem @ V21R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V22x: $i] :
( ( mem @ V22x @ A_27a )
=> ! [V23y: $i] :
( ( mem @ V23y @ A_27a )
=> ! [V24z: $i] :
( ( mem @ V24z @ A_27a )
=> ! [V25l: $i] :
( ( mem @ V25l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V20negate ) @ V21R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V22x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V23y ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V24z ) @ V25l ) ) ) ) ) ) ) ) ) ) )
& ! [V26negate: $i] :
( ( mem @ V26negate @ bool )
=> ! [V27R: $i] :
( ( mem @ V27R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V28x: $i] :
( ( mem @ V28x @ A_27a )
=> ! [V29y: $i] :
( ( mem @ V29y @ A_27a )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V26negate ) @ V27R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V28x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V29y ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ) ) ) )
& ! [V30negate: $i] :
( ( mem @ V30negate @ bool )
=> ! [V31R: $i] :
( ( mem @ V31R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V32x: $i] :
( ( mem @ V32x @ A_27a )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V30negate ) @ V31R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V32x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ) )
& ! [V33negate: $i] :
( ( mem @ V33negate @ bool )
=> ! [V34R: $i] :
( ( mem @ V34R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V33negate ) @ V34R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) )
& ! [V35negate: $i] :
( ( mem @ V35negate @ bool )
=> ! [V36R: $i] :
( ( mem @ V36R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V37v6: tp__ty_2Enum_2Enum,V38l: $i] :
( ( mem @ V38l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ! [V39len1: tp__ty_2Enum_2Enum,V40neg: $i] :
( ( mem @ V40neg @ bool )
=> ( ( ( V37v6 != fo__c_2Enum_2E0 )
& ( V37v6
!= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
& ( V37v6
!= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
& ( V37v6
!= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
& ( V39len1
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EDIV2 @ ( inj__ty_2Enum_2Enum @ V37v6 ) ) ) )
& ( ( p @ V40neg )
<=> ~ ( p @ V35negate ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V40neg ) @ V36R ) @ ( ap @ c_2Earithmetic_2EDIV2 @ ( inj__ty_2Enum_2Enum @ V37v6 ) ) ) @ V38l ) ) ) )
& ! [V41len1: tp__ty_2Enum_2Enum,V42neg: $i] :
( ( mem @ V42neg @ bool )
=> ( ( ( V37v6 != fo__c_2Enum_2E0 )
& ( V37v6
!= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
& ( V37v6
!= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
& ( V37v6
!= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
& ( V41len1
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EDIV2 @ ( inj__ty_2Enum_2Enum @ V37v6 ) ) ) )
& ( ( p @ V42neg )
<=> ~ ( p @ V35negate ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V42neg ) @ V36R ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V37v6 ) ) @ ( inj__ty_2Enum_2Enum @ V41len1 ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2EDROP @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V41len1 ) ) @ V38l ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V35negate ) @ V36R ) @ ( inj__ty_2Enum_2Enum @ V37v6 ) ) @ V38l ) ) ) ) ) ) )
=> ! [V43v: $i] :
( ( mem @ V43v @ bool )
=> ! [V44v1: $i] :
( ( mem @ V44v1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V45v2: tp__ty_2Enum_2Enum,V46v3: $i] :
( ( mem @ V46v3 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V43v ) @ V44v1 ) @ ( inj__ty_2Enum_2Enum @ V45v2 ) ) @ V46v3 ) ) ) ) ) ) ) ).
thf(conj_thm_2Emergesort_2EmergesortN__tail__def,axiom,
! [A_27a: del] :
( ! [V0negate: $i] :
( ( mem @ V0negate @ bool )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2R: $i] :
( ( mem @ V2R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2EmergesortN__tail @ A_27a ) @ V0negate ) @ V2R ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V1l )
= ( c_2Elist_2ENIL @ A_27a ) ) ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4negate: $i] :
( ( mem @ V4negate @ bool )
=> ! [V5l: $i] :
( ( mem @ V5l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V6R: $i] :
( ( mem @ V6R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2EmergesortN__tail @ A_27a ) @ V4negate ) @ V6R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3x ) @ V5l ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ) )
& ! [V7negate: $i] :
( ( mem @ V7negate @ bool )
=> ! [V8R: $i] :
( ( mem @ V8R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2EmergesortN__tail @ A_27a ) @ V7negate ) @ V8R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Elist_2ENIL @ A_27a ) ) ) )
& ! [V9y: $i] :
( ( mem @ V9y @ A_27a )
=> ! [V10x: $i] :
( ( mem @ V10x @ A_27a )
=> ! [V11negate: $i] :
( ( mem @ V11negate @ bool )
=> ! [V12l: $i] :
( ( mem @ V12l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V13R: $i] :
( ( mem @ V13R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2EmergesortN__tail @ A_27a ) @ V11negate ) @ V13R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V10x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V9y ) @ V12l ) ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Esort2__tail @ A_27a ) @ V11negate ) @ V13R ) @ V10x ) @ V9y ) ) ) ) ) ) )
& ! [V14x: $i] :
( ( mem @ V14x @ A_27a )
=> ! [V15negate: $i] :
( ( mem @ V15negate @ bool )
=> ! [V16R: $i] :
( ( mem @ V16R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2EmergesortN__tail @ A_27a ) @ V15negate ) @ V16R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V14x ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V14x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) )
& ! [V17negate: $i] :
( ( mem @ V17negate @ bool )
=> ! [V18R: $i] :
( ( mem @ V18R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2EmergesortN__tail @ A_27a ) @ V17negate ) @ V18R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Elist_2ENIL @ A_27a ) ) ) )
& ! [V19z: $i] :
( ( mem @ V19z @ A_27a )
=> ! [V20y: $i] :
( ( mem @ V20y @ A_27a )
=> ! [V21x: $i] :
( ( mem @ V21x @ A_27a )
=> ! [V22negate: $i] :
( ( mem @ V22negate @ bool )
=> ! [V23l: $i] :
( ( mem @ V23l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V24R: $i] :
( ( mem @ V24R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2EmergesortN__tail @ A_27a ) @ V22negate ) @ V24R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V21x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V20y ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V19z ) @ V23l ) ) ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Esort3__tail @ A_27a ) @ V22negate ) @ V24R ) @ V21x ) @ V20y ) @ V19z ) ) ) ) ) ) ) )
& ! [V25y: $i] :
( ( mem @ V25y @ A_27a )
=> ! [V26x: $i] :
( ( mem @ V26x @ A_27a )
=> ! [V27negate: $i] :
( ( mem @ V27negate @ bool )
=> ! [V28R: $i] :
( ( mem @ V28R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2EmergesortN__tail @ A_27a ) @ V27negate ) @ V28R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V26x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V25y ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Esort2__tail @ A_27a ) @ V27negate ) @ V28R ) @ V26x ) @ V25y ) ) ) ) ) )
& ! [V29x: $i] :
( ( mem @ V29x @ A_27a )
=> ! [V30negate: $i] :
( ( mem @ V30negate @ bool )
=> ! [V31R: $i] :
( ( mem @ V31R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2EmergesortN__tail @ A_27a ) @ V30negate ) @ V31R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V29x ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V29x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) )
& ! [V32negate: $i] :
( ( mem @ V32negate @ bool )
=> ! [V33R: $i] :
( ( mem @ V33R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2EmergesortN__tail @ A_27a ) @ V32negate ) @ V33R ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Elist_2ENIL @ A_27a ) ) ) )
& ! [V34v6: tp__ty_2Enum_2Enum,V35negate: $i] :
( ( mem @ V35negate @ bool )
=> ! [V36l: $i] :
( ( mem @ V36l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V37R: $i] :
( ( mem @ V37R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2EmergesortN__tail @ A_27a ) @ V35negate ) @ V37R ) @ ( inj__ty_2Enum_2Enum @ V34v6 ) ) @ V36l )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V34v6 ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( c_2Elist_2ENIL @ A_27a ) )
@ ( ap
@ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V34v6 ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
@ ( ap @ ( ap @ ( ap @ ( c_2Elist_2Elist__CASE @ A_27a @ ( ty_2Elist_2Elist @ A_27a ) ) @ V36l ) @ ( c_2Elist_2ENIL @ A_27a ) )
@ ( lam @ A_27a
@ ^ [V38x: $i] :
( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V39l_27: $i] : ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V38x ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) )
@ ( ap
@ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V34v6 ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
@ ( ap @ ( ap @ ( ap @ ( c_2Elist_2Elist__CASE @ A_27a @ ( ty_2Elist_2Elist @ A_27a ) ) @ V36l ) @ ( c_2Elist_2ENIL @ A_27a ) )
@ ( lam @ A_27a
@ ^ [V40x_27: $i] :
( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V41v19: $i] :
( ap @ ( ap @ ( ap @ ( c_2Elist_2Elist__CASE @ A_27a @ ( ty_2Elist_2Elist @ A_27a ) ) @ V41v19 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V40x_27 ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
@ ( lam @ A_27a
@ ^ [V42y: $i] :
( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V43l_27_27: $i] : ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Esort2__tail @ A_27a ) @ V35negate ) @ V37R ) @ V40x_27 ) @ V42y ) ) ) ) ) ) ) )
@ ( ap
@ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V34v6 ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) )
@ ( ap @ ( ap @ ( ap @ ( c_2Elist_2Elist__CASE @ A_27a @ ( ty_2Elist_2Elist @ A_27a ) ) @ V36l ) @ ( c_2Elist_2ENIL @ A_27a ) )
@ ( lam @ A_27a
@ ^ [V44x_27_27: $i] :
( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V45v27: $i] :
( ap @ ( ap @ ( ap @ ( c_2Elist_2Elist__CASE @ A_27a @ ( ty_2Elist_2Elist @ A_27a ) ) @ V45v27 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V44x_27_27 ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
@ ( lam @ A_27a
@ ^ [V46y_27: $i] :
( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V47v31: $i] :
( ap @ ( ap @ ( ap @ ( c_2Elist_2Elist__CASE @ A_27a @ ( ty_2Elist_2Elist @ A_27a ) ) @ V47v31 ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Esort2__tail @ A_27a ) @ V35negate ) @ V37R ) @ V44x_27_27 ) @ V46y_27 ) )
@ ( lam @ A_27a
@ ^ [V48z: $i] :
( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V49l_27_27_27: $i] : ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Esort3__tail @ A_27a ) @ V35negate ) @ V37R ) @ V44x_27_27 ) @ V46y_27 ) @ V48z ) ) ) ) ) ) ) ) ) ) )
@ ( ap
@ ( ap @ ( c_2Ebool_2ELET @ ty_2Enum_2Enum @ ( ty_2Elist_2Elist @ A_27a ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V50len1: $i] :
( ap
@ ( ap @ ( c_2Ebool_2ELET @ bool @ ( ty_2Elist_2Elist @ A_27a ) )
@ ( lam @ bool
@ ^ [V51neg: $i] : ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Emerge__tail @ A_27a ) @ V51neg ) @ V37R ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2EmergesortN__tail @ A_27a ) @ V51neg ) @ V37R ) @ ( ap @ c_2Earithmetic_2EDIV2 @ ( inj__ty_2Enum_2Enum @ V34v6 ) ) ) @ V36l ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2EmergesortN__tail @ A_27a ) @ V51neg ) @ V37R ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V34v6 ) ) @ V50len1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2EDROP @ A_27a ) @ V50len1 ) @ V36l ) ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) )
@ ( ap @ c_2Ebool_2E_7E @ V35negate ) ) ) )
@ ( ap @ c_2Earithmetic_2EDIV2 @ ( inj__ty_2Enum_2Enum @ V34v6 ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Emergesort_2Emergesort__tail__def,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Emergesort_2Emergesort__tail @ A_27a ) @ V0R ) @ V1l )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2EmergesortN__tail @ A_27a ) @ c_2Ebool_2EF ) @ V0R ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l ) ) @ V1l ) ) ) ) ).
thf(conj_thm_2Emergesort_2Esort2__perm,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2y ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Esort2 @ A_27a ) @ V0R ) @ V1x ) @ V2y ) ) ) ) ) ) ).
thf(conj_thm_2Emergesort_2Esort3__perm,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ! [V3z: $i] :
( ( mem @ V3z @ A_27a )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2y ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3z ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Esort3 @ A_27a ) @ V0R ) @ V1x ) @ V2y ) @ V3z ) ) ) ) ) ) ) ).
thf(conj_thm_2Emergesort_2Emerge__perm,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_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1l1 ) @ V2l2 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Emerge @ A_27a ) @ V0R ) @ V1l1 ) @ V2l2 ) ) ) ) ) ) ).
thf(conj_thm_2Emergesort_2EmergesortN__perm,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1n: tp__ty_2Enum_2Enum,V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2EPERM @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V2l ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2EmergesortN @ A_27a ) @ V0R ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V2l ) ) ) ) ) ).
thf(conj_thm_2Emergesort_2Emergesort__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_2Emergesort_2Emergesort @ A_27a ) @ V0R ) @ V1l ) ) ) ) ) ).
thf(conj_thm_2Emergesort_2Esort2__sorted,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( c_2Erelation_2Etotal @ A_27a ) @ V0R ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Esort2 @ A_27a ) @ V0R ) @ V1x ) @ V2y ) ) ) ) ) ) ) ).
thf(conj_thm_2Emergesort_2Esort3__sorted,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ! [V3z: $i] :
( ( mem @ V3z @ A_27a )
=> ( ( p @ ( ap @ ( c_2Erelation_2Etotal @ A_27a ) @ V0R ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Esort3 @ A_27a ) @ V0R ) @ V1x ) @ V2y ) @ V3z ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emergesort_2Emerge__sorted,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 @ ( c_2Erelation_2Etotal @ A_27a ) @ V0R ) )
& ( 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_2ESORTED @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Emerge @ A_27a ) @ V0R ) @ V1l1 ) @ V2l2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Emergesort_2EmergesortN__sorted,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1n: tp__ty_2Enum_2Enum,V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2Etotal @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2EmergesortN @ A_27a ) @ V0R ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V2l ) ) ) ) ) ) ).
thf(conj_thm_2Emergesort_2Emergesort__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_2Emergesort_2Emergesort @ A_27a ) @ V0R ) @ V1l ) ) ) ) ) ) ).
thf(conj_thm_2Emergesort_2Estable__cong,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 ) )
=> ! [V3l3: $i] :
( ( mem @ V3l3 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V4l4: $i] :
( ( mem @ V4l4 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Estable @ A_27a ) @ V0R ) @ V1l1 ) @ V2l2 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Estable @ A_27a ) @ V0R ) @ V3l3 ) @ V4l4 ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Estable @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1l1 ) @ V3l3 ) ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2l2 ) @ V4l4 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emergesort_2Estable__trans,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 ) )
=> ! [V3l3: $i] :
( ( mem @ V3l3 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Estable @ A_27a ) @ V0R ) @ V1l1 ) @ V2l2 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Estable @ A_27a ) @ V0R ) @ V2l2 ) @ V3l3 ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Estable @ A_27a ) @ V0R ) @ V1l1 ) @ V3l3 ) ) ) ) ) ) ) ).
thf(conj_thm_2Emergesort_2Esort2__stable,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Estable @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2y ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Esort2 @ A_27a ) @ V0R ) @ V1x ) @ V2y ) ) ) ) ) ) ).
thf(conj_thm_2Emergesort_2Esort3__stable,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ! [V3z: $i] :
( ( mem @ V3z @ A_27a )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2Etotal @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Estable @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2y ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3z ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Esort3 @ A_27a ) @ V0R ) @ V1x ) @ V2y ) @ V3z ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emergesort_2Efilter__merge,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2l1: $i] :
( ( mem @ V2l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3l2: $i] :
( ( mem @ V3l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V1R ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( ( p @ ( ap @ V0P @ V4x ) )
& ( p @ ( ap @ V0P @ V5y ) ) )
=> ( p @ ( ap @ ( ap @ V1R @ V4x ) @ V5y ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V1R ) @ V2l1 ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V0P ) @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Emerge @ A_27a ) @ V1R ) @ V2l1 ) @ V3l2 ) )
= ( ap @ ( ap @ ( c_2Elist_2EFILTER @ A_27a ) @ V0P ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V2l1 ) @ V3l2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emergesort_2Emerge__stable,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 @ ( ap @ ( c_2Emergesort_2Estable @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1l1 ) @ V2l2 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Emerge @ A_27a ) @ V0R ) @ V1l1 ) @ V2l2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Emergesort_2EmergesortN__stable,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1n: tp__ty_2Enum_2Enum,V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2Etotal @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Estable @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( c_2Elist_2ETAKE @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V2l ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2EmergesortN @ A_27a ) @ V0R ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V2l ) ) ) ) ) ) ).
thf(conj_thm_2Emergesort_2Emergesort__stable,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 @ ( ap @ ( c_2Emergesort_2Estable @ A_27a ) @ V0R ) @ V1l ) @ ( ap @ ( ap @ ( c_2Emergesort_2Emergesort @ A_27a ) @ V0R ) @ V1l ) ) ) ) ) ) ).
thf(conj_thm_2Emergesort_2Emergesort__STABLE__SORT,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_2Emergesort_2Emergesort @ A_27a ) ) @ V0R ) ) ) ) ).
thf(conj_thm_2Emergesort_2Emergesort__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_2Emergesort_2Emergesort @ 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_2Emergesort_2Esort2__tail__correct,axiom,
! [A_27a: del,V0neg: $i] :
( ( mem @ V0neg @ bool )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Esort2__tail @ A_27a ) @ V0neg ) @ V1R ) @ V2x ) @ V3y )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ V0neg ) @ ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Esort2 @ A_27a ) @ V1R ) @ V2x ) @ V3y ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Esort2 @ A_27a ) @ V1R ) @ V2x ) @ V3y ) ) ) ) ) ) ) ).
thf(conj_thm_2Emergesort_2Esort3__tail__correct,axiom,
! [A_27a: del,V0neg: $i] :
( ( mem @ V0neg @ bool )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ! [V4z: $i] :
( ( mem @ V4z @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Esort3__tail @ A_27a ) @ V0neg ) @ V1R ) @ V2x ) @ V3y ) @ V4z )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ V0neg ) @ ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Esort3 @ A_27a ) @ V1R ) @ V2x ) @ V3y ) @ V4z ) ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Esort3 @ A_27a ) @ V1R ) @ V2x ) @ V3y ) @ V4z ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emergesort_2Emerge__tail__correct1,axiom,
! [A_27a: del,V0neg: $i] :
( ( mem @ V0neg @ bool )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2l1: $i] :
( ( mem @ V2l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3l2: $i] :
( ( mem @ V3l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V4acc: $i] :
( ( mem @ V4acc @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( p @ V0neg )
<=> $false )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Emerge__tail @ A_27a ) @ V0neg ) @ V1R ) @ V2l1 ) @ V3l2 ) @ V4acc )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Emerge @ A_27a ) @ V1R ) @ V2l1 ) @ V3l2 ) ) ) @ V4acc ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emergesort_2Emerge__empty,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2acc: $i] :
( ( mem @ V2acc @ A_27b )
=> ( ( ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Emerge @ A_27a ) @ V0R ) @ V1l ) @ ( c_2Elist_2ENIL @ A_27a ) )
= V1l )
& ( ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Emerge @ A_27a ) @ V0R ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ V1l )
= V1l ) ) ) ) ) ).
thf(conj_thm_2Emergesort_2Emerge__tail__correct2,axiom,
! [A_27a: del,V0neg: $i] :
( ( mem @ V0neg @ bool )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2l1: $i] :
( ( mem @ V2l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3l2: $i] :
( ( mem @ V3l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V4acc: $i] :
( ( mem @ V4acc @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ( p @ V0neg )
<=> $true )
& ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V1R ) )
& ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V1R ) @ ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ V2l1 ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Esorting_2ESORTED @ A_27a ) @ V1R ) @ ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ V3l2 ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Emerge__tail @ A_27a ) @ V0neg ) @ V1R ) @ V2l1 ) @ V3l2 ) @ V4acc )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2Emerge @ A_27a ) @ V1R ) @ ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ V2l1 ) ) @ ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ V3l2 ) ) ) @ V4acc ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emergesort_2EmergesortN__correct,axiom,
! [A_27a: del,V0negate: $i] :
( ( mem @ V0negate @ bool )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2n: tp__ty_2Enum_2Enum,V3l: $i] :
( ( mem @ V3l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2Etotal @ A_27a ) @ V1R ) )
& ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V1R ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2EmergesortN__tail @ A_27a ) @ V0negate ) @ V1R ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ V3l )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ V0negate ) @ ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2EmergesortN @ A_27a ) @ V1R ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ V3l ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Emergesort_2EmergesortN @ A_27a ) @ V1R ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ V3l ) ) ) ) ) ) ) ).
thf(conj_thm_2Emergesort_2Emergesort__tail__correct,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_2Etotal @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) ) )
=> ( ( ap @ ( ap @ ( c_2Emergesort_2Emergesort__tail @ A_27a ) @ V0R ) @ V1l )
= ( ap @ ( ap @ ( c_2Emergesort_2Emergesort @ A_27a ) @ V0R ) @ V1l ) ) ) ) ) ).
%------------------------------------------------------------------------------