ITP001 Axioms: ITP034^5.ax
%------------------------------------------------------------------------------
% File : ITP034^5 : 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 : errorStateMonad^2.ax [Gau20]
% : HL4034^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 91 ( 9 unt; 20 typ; 0 def)
% Number of atoms : 2032 ( 55 equ; 0 cnn)
% Maximal formula atoms : 135 ( 22 avg)
% Number of connectives : 3568 ( 1 ~; 0 |; 6 &;3476 @)
% ( 0 <=>; 85 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 11 avg;3476 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 50 ( 50 >; 0 *; 0 +; 0 <<)
% Number of symbols : 70 ( 69 usr; 49 con; 0-5 aty)
% Number of variables : 323 ( 50 ^ 273 !; 0 ?; 323 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2EerrorStateMonad_2EBIND,type,
c_2EerrorStateMonad_2EBIND: del > del > del > $i ).
thf(mem_c_2EerrorStateMonad_2EBIND,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2EerrorStateMonad_2EBIND @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ ( arr @ ( arr @ A_27b @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27c @ A_27a ) ) ) ) @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27c @ A_27a ) ) ) ) ) ) ).
thf(tp_c_2EerrorStateMonad_2EES__APPLY,type,
c_2EerrorStateMonad_2EES__APPLY: del > del > del > $i ).
thf(mem_c_2EerrorStateMonad_2EES__APPLY,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2EerrorStateMonad_2EES__APPLY @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ( arr @ A_27c @ A_27b ) @ A_27a ) ) ) @ ( arr @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27c @ A_27a ) ) ) @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) ) ) ) ).
thf(tp_c_2EerrorStateMonad_2EES__CHOICE,type,
c_2EerrorStateMonad_2EES__CHOICE: del > del > $i ).
thf(mem_c_2EerrorStateMonad_2EES__CHOICE,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2EerrorStateMonad_2EES__CHOICE @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( arr @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) ) ) ).
thf(tp_c_2EerrorStateMonad_2EES__FAIL,type,
c_2EerrorStateMonad_2EES__FAIL: del > del > $i ).
thf(mem_c_2EerrorStateMonad_2EES__FAIL,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2EerrorStateMonad_2EES__FAIL @ A_27a @ A_27b ) @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) ).
thf(tp_c_2EerrorStateMonad_2EES__GUARD,type,
c_2EerrorStateMonad_2EES__GUARD: del > $i ).
thf(mem_c_2EerrorStateMonad_2EES__GUARD,axiom,
! [A_27a: del] : ( mem @ ( c_2EerrorStateMonad_2EES__GUARD @ A_27a ) @ ( arr @ bool @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27a ) ) ) ) ) ).
thf(tp_c_2EerrorStateMonad_2EES__LIFT2,type,
c_2EerrorStateMonad_2EES__LIFT2: del > del > del > del > $i ).
thf(mem_c_2EerrorStateMonad_2EES__LIFT2,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] : ( mem @ ( c_2EerrorStateMonad_2EES__LIFT2 @ A_27a @ A_27b @ A_27c @ A_27d ) @ ( arr @ ( arr @ A_27c @ ( arr @ A_27d @ A_27b ) ) @ ( arr @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27c @ A_27a ) ) ) @ ( arr @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27d @ A_27a ) ) ) @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) ) ) ) ) ).
thf(tp_c_2EerrorStateMonad_2EEXT,type,
c_2EerrorStateMonad_2EEXT: del > del > del > $i ).
thf(mem_c_2EerrorStateMonad_2EEXT,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2EerrorStateMonad_2EEXT @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27c @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) ) @ ( arr @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27c @ A_27a ) ) ) @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) ) ) ) ).
thf(tp_c_2EerrorStateMonad_2EFOR,type,
c_2EerrorStateMonad_2EFOR: del > $i ).
thf(mem_c_2EerrorStateMonad_2EFOR,axiom,
! [A_27state: del] : ( mem @ ( c_2EerrorStateMonad_2EFOR @ A_27state ) @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) ) @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) ).
thf(tp_c_2EerrorStateMonad_2EFOREACH,type,
c_2EerrorStateMonad_2EFOREACH: del > del > $i ).
thf(mem_c_2EerrorStateMonad_2EFOREACH,axiom,
! [A_27a: del,A_27state: del] : ( mem @ ( c_2EerrorStateMonad_2EFOREACH @ A_27a @ A_27state ) @ ( arr @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) ).
thf(tp_c_2EerrorStateMonad_2EIGNORE__BIND,type,
c_2EerrorStateMonad_2EIGNORE__BIND: del > del > del > $i ).
thf(mem_c_2EerrorStateMonad_2EIGNORE__BIND,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2EerrorStateMonad_2EIGNORE__BIND @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) ) @ ( arr @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27c @ A_27b ) ) ) @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27c @ A_27b ) ) ) ) ) ) ).
thf(tp_c_2EerrorStateMonad_2EJOIN,type,
c_2EerrorStateMonad_2EJOIN: del > del > $i ).
thf(mem_c_2EerrorStateMonad_2EJOIN,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2EerrorStateMonad_2EJOIN @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ A_27a ) ) ) @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) ) ) ).
thf(tp_c_2EerrorStateMonad_2EMCOMP,type,
c_2EerrorStateMonad_2EMCOMP: del > del > del > del > del > $i ).
thf(mem_c_2EerrorStateMonad_2EMCOMP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del] : ( mem @ ( c_2EerrorStateMonad_2EMCOMP @ A_27a @ A_27b @ A_27c @ A_27d @ A_27e ) @ ( arr @ ( arr @ A_27d @ ( arr @ A_27e @ ( ty_2Eoption_2Eoption @ A_27c ) ) ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27d @ A_27e ) ) ) ) @ ( arr @ A_27a @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ A_27c ) ) ) ) ) ) ).
thf(tp_c_2EerrorStateMonad_2EMMAP,type,
c_2EerrorStateMonad_2EMMAP: del > del > del > $i ).
thf(mem_c_2EerrorStateMonad_2EMMAP,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2EerrorStateMonad_2EMMAP @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27c @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27c @ A_27a ) ) ) @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) ) ) ) ).
thf(tp_c_2EerrorStateMonad_2ENARROW,type,
c_2EerrorStateMonad_2ENARROW: del > del > del > $i ).
thf(mem_c_2EerrorStateMonad_2ENARROW,axiom,
! [A_27a: del,A_27b: del,A_27state: del] : ( mem @ ( c_2EerrorStateMonad_2ENARROW @ A_27a @ A_27b @ A_27state ) @ ( arr @ A_27b @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27b @ A_27state ) @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27state ) ) ) ) @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27a @ A_27state ) ) ) ) ) ) ).
thf(tp_c_2EerrorStateMonad_2EREAD,type,
c_2EerrorStateMonad_2EREAD: del > del > $i ).
thf(mem_c_2EerrorStateMonad_2EREAD,axiom,
! [A_27a: del,A_27state: del] : ( mem @ ( c_2EerrorStateMonad_2EREAD @ A_27a @ A_27state ) @ ( arr @ ( arr @ A_27state @ A_27a ) @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27a @ A_27state ) ) ) ) ) ).
thf(tp_c_2EerrorStateMonad_2EUNIT,type,
c_2EerrorStateMonad_2EUNIT: del > del > $i ).
thf(mem_c_2EerrorStateMonad_2EUNIT,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2EerrorStateMonad_2EUNIT @ A_27a @ A_27b ) @ ( arr @ A_27b @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) ) ) ).
thf(tp_c_2EerrorStateMonad_2EWIDEN,type,
c_2EerrorStateMonad_2EWIDEN: del > del > del > $i ).
thf(mem_c_2EerrorStateMonad_2EWIDEN,axiom,
! [A_27a: del,A_27b: del,A_27state: del] : ( mem @ ( c_2EerrorStateMonad_2EWIDEN @ A_27a @ A_27b @ A_27state ) @ ( arr @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27a @ A_27state ) ) ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27b @ A_27state ) @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27state ) ) ) ) ) ) ).
thf(tp_c_2EerrorStateMonad_2EWRITE,type,
c_2EerrorStateMonad_2EWRITE: del > $i ).
thf(mem_c_2EerrorStateMonad_2EWRITE,axiom,
! [A_27state: del] : ( mem @ ( c_2EerrorStateMonad_2EWRITE @ A_27state ) @ ( arr @ ( arr @ A_27state @ A_27state ) @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) ).
thf(tp_c_2EerrorStateMonad_2EmapM,type,
c_2EerrorStateMonad_2EmapM: del > del > del > $i ).
thf(mem_c_2EerrorStateMonad_2EmapM,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2EerrorStateMonad_2EmapM @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27c @ A_27b ) ) ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27c ) @ A_27b ) ) ) ) ) ) ).
thf(tp_c_2EerrorStateMonad_2Esequence,type,
c_2EerrorStateMonad_2Esequence: del > del > $i ).
thf(mem_c_2EerrorStateMonad_2Esequence,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2EerrorStateMonad_2Esequence @ A_27a @ A_27b ) @ ( arr @ ( ty_2Elist_2Elist @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) ) @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27b ) @ A_27a ) ) ) ) ) ).
thf(ax_thm_2EerrorStateMonad_2EUNIT__DEF,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27b )
=> ( ( ap @ ( c_2EerrorStateMonad_2EUNIT @ A_27a @ A_27b ) @ V0x )
= ( lam @ A_27a
@ ^ [V1s: $i] : ( ap @ ( c_2Eoption_2ESOME @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27a ) @ V0x ) @ V1s ) ) ) ) ) ).
thf(ax_thm_2EerrorStateMonad_2EBIND__DEF,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27c @ A_27a ) ) ) ) )
=> ! [V2s0: $i] :
( ( mem @ V2s0 @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EerrorStateMonad_2EBIND @ A_27a @ A_27b @ A_27c ) @ V0g ) @ V1f ) @ V2s0 )
= ( ap @ ( ap @ ( ap @ ( c_2Eoption_2Eoption__CASE @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27c @ A_27a ) ) ) @ ( ap @ V0g @ V2s0 ) ) @ ( c_2Eoption_2ENONE @ ( ty_2Epair_2Eprod @ A_27c @ A_27a ) ) )
@ ( lam @ ( ty_2Epair_2Eprod @ A_27b @ A_27a )
@ ^ [V3v: $i] :
( ap @ ( ap @ ( c_2Epair_2Epair__CASE @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27c @ A_27a ) ) @ A_27b @ A_27a ) @ V3v )
@ ( lam @ A_27b
@ ^ [V4b: $i] :
( lam @ A_27a
@ ^ [V5s: $i] : ( ap @ ( ap @ V1f @ V4b ) @ V5s ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2EerrorStateMonad_2EIGNORE__BIND__DEF,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27c @ A_27b ) ) ) )
=> ( ( ap @ ( ap @ ( c_2EerrorStateMonad_2EIGNORE__BIND @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1g )
= ( ap @ ( ap @ ( c_2EerrorStateMonad_2EBIND @ A_27b @ A_27a @ A_27c ) @ V0f )
@ ( lam @ A_27a
@ ^ [V2x: $i] : V1g ) ) ) ) ) ).
thf(ax_thm_2EerrorStateMonad_2EMMAP__DEF,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27c @ A_27b ) )
=> ! [V1m: $i] :
( ( mem @ V1m @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27c @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( c_2EerrorStateMonad_2EMMAP @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1m )
= ( ap @ ( ap @ ( c_2EerrorStateMonad_2EBIND @ A_27a @ A_27c @ A_27b ) @ V1m ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27c @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ A_27b ) @ ( c_2EerrorStateMonad_2EUNIT @ A_27a @ A_27b ) ) @ V0f ) ) ) ) ) ).
thf(ax_thm_2EerrorStateMonad_2EJOIN__DEF,axiom,
! [A_27a: del,A_27b: del,V0z: $i] :
( ( mem @ V0z @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ A_27a ) ) ) )
=> ( ( ap @ ( c_2EerrorStateMonad_2EJOIN @ A_27a @ A_27b ) @ V0z )
= ( ap @ ( ap @ ( c_2EerrorStateMonad_2EBIND @ A_27a @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ A_27b ) @ V0z ) @ ( c_2Ecombin_2EI @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) ) ) ) ) ).
thf(ax_thm_2EerrorStateMonad_2EEXT__DEF,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27c @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) ) )
=> ! [V1m: $i] :
( ( mem @ V1m @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27c @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( c_2EerrorStateMonad_2EEXT @ A_27a @ A_27b @ A_27c ) @ V0g ) @ V1m )
= ( ap @ ( ap @ ( c_2EerrorStateMonad_2EBIND @ A_27a @ A_27c @ A_27b ) @ V1m ) @ V0g ) ) ) ) ).
thf(ax_thm_2EerrorStateMonad_2EMCOMP__DEF,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27d @ ( arr @ A_27e @ ( ty_2Eoption_2Eoption @ A_27c ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27d @ A_27e ) ) ) ) )
=> ( ( ap @ ( ap @ ( c_2EerrorStateMonad_2EMCOMP @ A_27a @ A_27b @ A_27c @ A_27d @ A_27e ) @ V0g ) @ V1f )
= ( ap @ ( c_2Epair_2ECURRY @ A_27a @ A_27b @ ( ty_2Eoption_2Eoption @ A_27c ) ) @ ( ap @ ( ap @ ( c_2Eoption_2EOPTION__MCOMP @ A_27c @ ( ty_2Epair_2Eprod @ A_27d @ A_27e ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27d @ A_27e @ ( ty_2Eoption_2Eoption @ A_27c ) ) @ V0g ) ) @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27d @ A_27e ) ) ) @ V1f ) ) ) ) ) ) ).
thf(ax_thm_2EerrorStateMonad_2EFOR__primitive__def,axiom,
! [A_27state: del] :
( ( c_2EerrorStateMonad_2EFOR @ A_27state )
= ( ap
@ ( ap @ ( c_2Erelation_2EWFREC @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) ) @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) )
@ ( ap @ ( c_2Emin_2E_40 @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) ) @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) ) @ bool ) ) )
@ ( lam @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) ) @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) ) @ bool ) )
@ ^ [V0R: $i] :
( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( c_2Erelation_2EWF @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) ) ) @ V0R ) )
@ ( ap @ ( c_2Ebool_2E_21 @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) )
@ ( lam @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) )
@ ^ [V1a: $i] :
( ap @ ( c_2Ebool_2E_21 @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2j: $i] :
( ap @ ( c_2Ebool_2E_21 @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3i: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V3i ) @ V2j ) ) ) @ ( ap @ ( ap @ V0R @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ V3i ) @ V2j ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V3i ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V3i ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) @ V2j ) @ V1a ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) ) @ V3i ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) @ V2j ) @ V1a ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( lam @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) ) @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) )
@ ^ [V4FOR: $i] :
( lam @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) )
@ ^ [V5a_27: $i] :
( ap @ ( ap @ ( c_2Epair_2Epair__CASE @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) ) @ V5a_27 )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V6i: $i] :
( lam @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) )
@ ^ [V7v1: $i] :
( ap @ ( ap @ ( c_2Epair_2Epair__CASE @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) @ V7v1 )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V8j: $i] :
( lam @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) )
@ ^ [V9a: $i] :
( ap @ ( c_2Ecombin_2EI @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) )
@ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V6i ) @ V8j ) ) @ ( ap @ V9a @ V6i ) )
@ ( ap @ ( ap @ ( c_2EerrorStateMonad_2EBIND @ A_27state @ ty_2Eone_2Eone @ ty_2Eone_2Eone ) @ ( ap @ V9a @ V6i ) )
@ ( lam @ ty_2Eone_2Eone
@ ^ [V10u: $i] : ( ap @ V4FOR @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ V6i ) @ V8j ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ V6i ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V6i ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) @ V8j ) @ V9a ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EerrorStateMonad_2EFOR__ind,axiom,
! [A_27state: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) ) @ bool ) )
=> ( ! [V1i: tp__ty_2Enum_2Enum,V2j: tp__ty_2Enum_2Enum,V3a: $i] :
( ( mem @ V3a @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) )
=> ( ( ( V1i != V2j )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ ( inj__ty_2Enum_2Enum @ V2j ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V2j ) ) @ V3a ) ) ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V2j ) ) @ V3a ) ) ) ) ) )
=> ! [V4v: tp__ty_2Enum_2Enum,V5v1: tp__ty_2Enum_2Enum,V6v2: $i] :
( ( mem @ V6v2 @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V4v ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V5v1 ) ) @ V6v2 ) ) ) ) ) ) ) ).
thf(conj_thm_2EerrorStateMonad_2EFOR__def,axiom,
! [A_27state: del,V0j: tp__ty_2Enum_2Enum,V1i: tp__ty_2Enum_2Enum,V2a: $i] :
( ( mem @ V2a @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) )
=> ( ( ap @ ( c_2EerrorStateMonad_2EFOR @ A_27state ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0j ) ) @ V2a ) ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ ( inj__ty_2Enum_2Enum @ V0j ) ) ) @ ( ap @ V2a @ ( inj__ty_2Enum_2Enum @ V1i ) ) )
@ ( ap @ ( ap @ ( c_2EerrorStateMonad_2EBIND @ A_27state @ ty_2Eone_2Eone @ ty_2Eone_2Eone ) @ ( ap @ V2a @ ( inj__ty_2Enum_2Enum @ V1i ) ) )
@ ( lam @ ty_2Eone_2Eone
@ ^ [V3u: $i] : ( ap @ ( c_2EerrorStateMonad_2EFOR @ A_27state ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ ( inj__ty_2Enum_2Enum @ V0j ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V0j ) ) @ V2a ) ) ) ) ) ) ) ) ).
thf(ax_thm_2EerrorStateMonad_2EFOREACH__primitive__def,axiom,
! [A_27a: del,A_27state: del] :
( ( c_2EerrorStateMonad_2EFOREACH @ A_27a @ A_27state )
= ( ap
@ ( ap @ ( c_2Erelation_2EWFREC @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) )
@ ( ap @ ( c_2Emin_2E_40 @ ( arr @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) @ ( arr @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) @ bool ) ) )
@ ( lam @ ( arr @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) @ ( arr @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) @ bool ) )
@ ^ [V0R: $i] :
( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( c_2Erelation_2EWF @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) ) @ V0R ) )
@ ( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V1h: $i] :
( ap @ ( c_2Ebool_2E_21 @ ( arr @ A_27a @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) )
@ ( lam @ ( arr @ A_27a @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) )
@ ^ [V2a: $i] :
( ap @ ( c_2Ebool_2E_21 @ ( ty_2Elist_2Elist @ A_27a ) )
@ ( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V3t: $i] : ( ap @ ( ap @ V0R @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) @ V3t ) @ V2a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1h ) @ V3t ) ) @ V2a ) ) ) ) ) ) ) ) ) ) ) )
@ ( lam @ ( arr @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) )
@ ^ [V4FOREACH: $i] :
( lam @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) )
@ ^ [V5a_27: $i] :
( ap @ ( ap @ ( c_2Epair_2Epair__CASE @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) @ V5a_27 )
@ ( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V6v: $i] :
( lam @ ( arr @ A_27a @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) )
@ ^ [V7a: $i] :
( ap @ ( ap @ ( ap @ ( c_2Elist_2Elist__CASE @ A_27a @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) @ V6v ) @ ( ap @ ( c_2Ecombin_2EI @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) @ ( ap @ ( c_2EerrorStateMonad_2EUNIT @ A_27state @ ty_2Eone_2Eone ) @ ( inj__ty_2Eone_2Eone @ fo__c_2Eone_2Eone ) ) ) )
@ ( lam @ A_27a
@ ^ [V8h: $i] :
( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V9t: $i] :
( ap @ ( c_2Ecombin_2EI @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) )
@ ( ap @ ( ap @ ( c_2EerrorStateMonad_2EBIND @ A_27state @ ty_2Eone_2Eone @ ty_2Eone_2Eone ) @ ( ap @ V7a @ V8h ) )
@ ( lam @ ty_2Eone_2Eone
@ ^ [V10u: $i] : ( ap @ V4FOREACH @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) @ V9t ) @ V7a ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EerrorStateMonad_2EFOREACH__ind,axiom,
! [A_27a: del,A_27state: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) @ bool ) )
=> ( ( ! [V1a: $i] :
( ( mem @ V1a @ ( arr @ A_27a @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ V1a ) ) ) )
& ! [V2h: $i] :
( ( mem @ V2h @ A_27a )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V4a: $i] :
( ( mem @ V4a @ ( arr @ A_27a @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) )
=> ( ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) @ V3t ) @ V4a ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h ) @ V3t ) ) @ V4a ) ) ) ) ) ) ) )
=> ! [V5v: $i] :
( ( mem @ V5v @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V6v1: $i] :
( ( mem @ V6v1 @ ( arr @ A_27a @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) @ V5v ) @ V6v1 ) ) ) ) ) ) ) ).
thf(conj_thm_2EerrorStateMonad_2EFOREACH__def,axiom,
! [A_27a: del,A_27state: del] :
( ! [V0a: $i] :
( ( mem @ V0a @ ( arr @ A_27a @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) )
=> ( ( ap @ ( c_2EerrorStateMonad_2EFOREACH @ A_27a @ A_27state ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ V0a ) )
= ( ap @ ( c_2EerrorStateMonad_2EUNIT @ A_27state @ ty_2Eone_2Eone ) @ ( inj__ty_2Eone_2Eone @ fo__c_2Eone_2Eone ) ) ) )
& ! [V1t: $i] :
( ( mem @ V1t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ A_27a )
=> ! [V3a: $i] :
( ( mem @ V3a @ ( arr @ A_27a @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) )
=> ( ( ap @ ( c_2EerrorStateMonad_2EFOREACH @ A_27a @ A_27state ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h ) @ V1t ) ) @ V3a ) )
= ( ap @ ( ap @ ( c_2EerrorStateMonad_2EBIND @ A_27state @ ty_2Eone_2Eone @ ty_2Eone_2Eone ) @ ( ap @ V3a @ V2h ) )
@ ( lam @ ty_2Eone_2Eone
@ ^ [V4u: $i] : ( ap @ ( c_2EerrorStateMonad_2EFOREACH @ A_27a @ A_27state ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) ) ) ) @ V1t ) @ V3a ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2EerrorStateMonad_2EREAD__def,axiom,
! [A_27a: del,A_27state: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27state @ A_27a ) )
=> ( ( ap @ ( c_2EerrorStateMonad_2EREAD @ A_27a @ A_27state ) @ V0f )
= ( lam @ A_27state
@ ^ [V1s: $i] : ( ap @ ( c_2Eoption_2ESOME @ ( ty_2Epair_2Eprod @ A_27a @ A_27state ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27state ) @ ( ap @ V0f @ V1s ) ) @ V1s ) ) ) ) ) ).
thf(ax_thm_2EerrorStateMonad_2EWRITE__def,axiom,
! [A_27state: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27state @ A_27state ) )
=> ( ( ap @ ( c_2EerrorStateMonad_2EWRITE @ A_27state ) @ V0f )
= ( lam @ A_27state
@ ^ [V1s: $i] : ( ap @ ( c_2Eoption_2ESOME @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27state ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Eone_2Eone @ A_27state ) @ ( inj__ty_2Eone_2Eone @ fo__c_2Eone_2Eone ) ) @ ( ap @ V0f @ V1s ) ) ) ) ) ) ).
thf(ax_thm_2EerrorStateMonad_2ENARROW__def,axiom,
! [A_27a: del,A_27b: del,A_27state: del,V0v: $i] :
( ( mem @ V0v @ A_27b )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ( ty_2Epair_2Eprod @ A_27b @ A_27state ) @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27state ) ) ) ) )
=> ( ( ap @ ( ap @ ( c_2EerrorStateMonad_2ENARROW @ A_27a @ A_27b @ A_27state ) @ V0v ) @ V1f )
= ( lam @ A_27state
@ ^ [V2s: $i] :
( ap @ ( ap @ ( ap @ ( c_2Eoption_2Eoption__CASE @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27state ) ) @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27a @ A_27state ) ) ) @ ( ap @ V1f @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27state ) @ V0v ) @ V2s ) ) ) @ ( c_2Eoption_2ENONE @ ( ty_2Epair_2Eprod @ A_27a @ A_27state ) ) )
@ ( lam @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27state ) )
@ ^ [V3v: $i] :
( ap @ ( ap @ ( c_2Epair_2Epair__CASE @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27a @ A_27state ) ) @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27state ) ) @ V3v )
@ ( lam @ A_27a
@ ^ [V4r: $i] :
( lam @ ( ty_2Epair_2Eprod @ A_27b @ A_27state )
@ ^ [V5s1: $i] : ( ap @ ( c_2Eoption_2ESOME @ ( ty_2Epair_2Eprod @ A_27a @ A_27state ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27state ) @ V4r ) @ ( ap @ ( c_2Epair_2ESND @ A_27b @ A_27state ) @ V5s1 ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2EerrorStateMonad_2EWIDEN__def,axiom,
! [A_27a: del,A_27b: del,A_27state: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27a @ A_27state ) ) ) )
=> ( ( ap @ ( c_2EerrorStateMonad_2EWIDEN @ A_27a @ A_27b @ A_27state ) @ V0f )
= ( ap @ ( c_2Epair_2EUNCURRY @ A_27b @ A_27state @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27state ) ) ) )
@ ( lam @ A_27b
@ ^ [V1s1: $i] :
( lam @ A_27state
@ ^ [V2s2: $i] :
( ap @ ( ap @ ( ap @ ( c_2Eoption_2Eoption__CASE @ ( ty_2Epair_2Eprod @ A_27a @ A_27state ) @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27state ) ) ) ) @ ( ap @ V0f @ V2s2 ) ) @ ( c_2Eoption_2ENONE @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27state ) ) ) )
@ ( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27state )
@ ^ [V3v: $i] :
( ap @ ( ap @ ( c_2Epair_2Epair__CASE @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27state ) ) ) @ A_27a @ A_27state ) @ V3v )
@ ( lam @ A_27a
@ ^ [V4r: $i] :
( lam @ A_27state
@ ^ [V5s3: $i] : ( ap @ ( c_2Eoption_2ESOME @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27state ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27state ) ) @ V4r ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27state ) @ V1s1 ) @ V5s3 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2EerrorStateMonad_2Esequence__def,axiom,
! [A_27a: del,A_27b: del] :
( ( c_2EerrorStateMonad_2Esequence @ A_27a @ A_27b )
= ( ap
@ ( ap @ ( c_2Elist_2EFOLDR @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27b ) @ A_27a ) ) ) )
@ ( lam @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) )
@ ^ [V0m: $i] :
( lam @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27b ) @ A_27a ) ) )
@ ^ [V1ms: $i] :
( ap @ ( ap @ ( c_2EerrorStateMonad_2EBIND @ A_27a @ A_27b @ ( ty_2Elist_2Elist @ A_27b ) ) @ V0m )
@ ( lam @ A_27b
@ ^ [V2x: $i] :
( ap @ ( ap @ ( c_2EerrorStateMonad_2EBIND @ A_27a @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ V1ms )
@ ( lam @ ( ty_2Elist_2Elist @ A_27b )
@ ^ [V3xs: $i] : ( ap @ ( c_2EerrorStateMonad_2EUNIT @ A_27a @ ( ty_2Elist_2Elist @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V2x ) @ V3xs ) ) ) ) ) ) ) ) )
@ ( ap @ ( c_2EerrorStateMonad_2EUNIT @ A_27a @ ( ty_2Elist_2Elist @ A_27b ) ) @ ( c_2Elist_2ENIL @ A_27b ) ) ) ) ).
thf(ax_thm_2EerrorStateMonad_2EmapM__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27c @ A_27b ) ) ) ) )
=> ( ( ap @ ( c_2EerrorStateMonad_2EmapM @ A_27a @ A_27b @ A_27c ) @ V0f )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ( ty_2Elist_2Elist @ A_27c ) @ A_27b ) ) ) @ ( ty_2Elist_2Elist @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27c @ A_27b ) ) ) ) ) @ ( c_2EerrorStateMonad_2Esequence @ A_27b @ A_27c ) ) @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27c @ A_27b ) ) ) ) @ V0f ) ) ) ) ).
thf(conj_thm_2EerrorStateMonad_2EBIND__LEFT__UNIT,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0k: $i] :
( ( mem @ V0k @ ( arr @ A_27a @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27c @ A_27b ) ) ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( ap @ ( ap @ ( c_2EerrorStateMonad_2EBIND @ A_27b @ A_27a @ A_27c ) @ ( ap @ ( c_2EerrorStateMonad_2EUNIT @ A_27b @ A_27a ) @ V1x ) ) @ V0k )
= ( ap @ V0k @ V1x ) ) ) ) ).
thf(conj_thm_2EerrorStateMonad_2EMCOMP__THM,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27d @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27c @ A_27b ) ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27d @ A_27b ) ) ) ) )
=> ( ( ap @ ( ap @ ( c_2EerrorStateMonad_2EMCOMP @ A_27a @ A_27b @ ( ty_2Epair_2Eprod @ A_27c @ A_27b ) @ A_27d @ A_27b ) @ V0g ) @ V1f )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27c @ A_27b ) ) ) @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27d @ A_27b ) ) ) ) @ ( ap @ ( c_2EerrorStateMonad_2EEXT @ A_27b @ A_27c @ A_27d ) @ V0g ) ) @ V1f ) ) ) ) ).
thf(conj_thm_2EerrorStateMonad_2EMCOMP__ASSOC,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,A_27f: del,A_27g: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27d @ ( arr @ A_27e @ ( ty_2Eoption_2Eoption @ A_27c ) ) ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27f @ ( arr @ A_27g @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27d @ A_27e ) ) ) ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ ( arr @ A_27a @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27f @ A_27g ) ) ) ) )
=> ( ( ap @ ( ap @ ( c_2EerrorStateMonad_2EMCOMP @ A_27a @ A_27b @ A_27c @ A_27d @ A_27e ) @ V0f ) @ ( ap @ ( ap @ ( c_2EerrorStateMonad_2EMCOMP @ A_27a @ A_27b @ ( ty_2Epair_2Eprod @ A_27d @ A_27e ) @ A_27f @ A_27g ) @ V1g ) @ V2h ) )
= ( ap @ ( ap @ ( c_2EerrorStateMonad_2EMCOMP @ A_27a @ A_27b @ A_27c @ A_27f @ A_27g ) @ ( ap @ ( ap @ ( c_2EerrorStateMonad_2EMCOMP @ A_27f @ A_27g @ A_27c @ A_27d @ A_27e ) @ V0f ) @ V1g ) ) @ V2h ) ) ) ) ) ).
thf(conj_thm_2EerrorStateMonad_2EUNIT__CURRY,axiom,
! [A_27a: del,A_27b: del] :
( ( c_2EerrorStateMonad_2EUNIT @ A_27b @ A_27a )
= ( ap @ ( c_2Epair_2ECURRY @ A_27a @ A_27b @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) ) @ ( c_2Eoption_2ESOME @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) ) ) ).
thf(conj_thm_2EerrorStateMonad_2EMCOMP__ID,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,A_27f: del,A_27g: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27a @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ A_27c ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27d @ ( arr @ A_27e @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27f @ A_27g ) ) ) ) )
=> ( ( ( ap @ ( ap @ ( c_2EerrorStateMonad_2EMCOMP @ A_27a @ A_27b @ A_27c @ A_27a @ A_27b ) @ V0g ) @ ( c_2EerrorStateMonad_2EUNIT @ A_27b @ A_27a ) )
= V0g )
& ( ( ap @ ( ap @ ( c_2EerrorStateMonad_2EMCOMP @ A_27d @ A_27e @ ( ty_2Epair_2Eprod @ A_27f @ A_27g ) @ A_27f @ A_27g ) @ ( c_2EerrorStateMonad_2EUNIT @ A_27g @ A_27f ) ) @ V1f )
= V1f ) ) ) ) ).
thf(conj_thm_2EerrorStateMonad_2EBIND__RIGHT__UNIT,axiom,
! [A_27a: del,A_27b: del,V0k: $i] :
( ( mem @ V0k @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( c_2EerrorStateMonad_2EBIND @ A_27a @ A_27b @ A_27b ) @ V0k ) @ ( c_2EerrorStateMonad_2EUNIT @ A_27a @ A_27b ) )
= V0k ) ) ).
thf(conj_thm_2EerrorStateMonad_2EBIND__ASSOC,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0k: $i] :
( ( mem @ V0k @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) )
=> ! [V1m: $i] :
( ( mem @ V1m @ ( arr @ A_27b @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27c @ A_27a ) ) ) ) )
=> ! [V2n: $i] :
( ( mem @ V2n @ ( arr @ A_27c @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27d @ A_27a ) ) ) ) )
=> ( ( ap @ ( ap @ ( c_2EerrorStateMonad_2EBIND @ A_27a @ A_27b @ A_27d ) @ V0k )
@ ( lam @ A_27b
@ ^ [V3a: $i] : ( ap @ ( ap @ ( c_2EerrorStateMonad_2EBIND @ A_27a @ A_27c @ A_27d ) @ ( ap @ V1m @ V3a ) ) @ V2n ) ) )
= ( ap @ ( ap @ ( c_2EerrorStateMonad_2EBIND @ A_27a @ A_27c @ A_27d ) @ ( ap @ ( ap @ ( c_2EerrorStateMonad_2EBIND @ A_27a @ A_27b @ A_27c ) @ V0k ) @ V1m ) ) @ V2n ) ) ) ) ) ).
thf(conj_thm_2EerrorStateMonad_2EMMAP__ID,axiom,
! [A_27a: del,A_27b: del] :
( ( ap @ ( c_2EerrorStateMonad_2EMMAP @ A_27a @ A_27b @ A_27b ) @ ( c_2Ecombin_2EI @ A_27b ) )
= ( c_2Ecombin_2EI @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) ) ) ).
thf(conj_thm_2EerrorStateMonad_2EMMAP__COMP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27c @ A_27a ) )
=> ( ( ap @ ( c_2EerrorStateMonad_2EMMAP @ A_27d @ A_27b @ A_27c ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27c @ A_27b @ A_27a ) @ V0f ) @ V1g ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( arr @ A_27d @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27c @ A_27d ) ) ) @ ( arr @ A_27d @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27d ) ) ) @ ( arr @ A_27d @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27a @ A_27d ) ) ) ) @ ( ap @ ( c_2EerrorStateMonad_2EMMAP @ A_27d @ A_27b @ A_27a ) @ V0f ) ) @ ( ap @ ( c_2EerrorStateMonad_2EMMAP @ A_27d @ A_27a @ A_27c ) @ V1g ) ) ) ) ) ).
thf(conj_thm_2EerrorStateMonad_2EMMAP__UNIT,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ ( arr @ A_27c @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) ) @ ( arr @ A_27c @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) ) ) ) @ ( ap @ ( c_2EerrorStateMonad_2EMMAP @ A_27c @ A_27b @ A_27a ) @ V0f ) ) @ ( c_2EerrorStateMonad_2EUNIT @ A_27c @ A_27a ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ ( arr @ A_27c @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) ) @ A_27b ) @ ( c_2EerrorStateMonad_2EUNIT @ A_27c @ A_27b ) ) @ V0f ) ) ) ).
thf(conj_thm_2EerrorStateMonad_2EMMAP__JOIN,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( arr @ A_27c @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ( arr @ A_27c @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) ) ) @ A_27c ) ) ) @ ( arr @ A_27c @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) ) @ ( arr @ A_27c @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) ) ) ) @ ( ap @ ( c_2EerrorStateMonad_2EMMAP @ A_27c @ A_27b @ A_27a ) @ V0f ) ) @ ( c_2EerrorStateMonad_2EJOIN @ A_27c @ A_27a ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( arr @ A_27c @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ( arr @ A_27c @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) ) ) @ A_27c ) ) ) @ ( arr @ A_27c @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) ) @ ( arr @ A_27c @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ( arr @ A_27c @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) ) @ A_27c ) ) ) ) @ ( c_2EerrorStateMonad_2EJOIN @ A_27c @ A_27b ) ) @ ( ap @ ( c_2EerrorStateMonad_2EMMAP @ A_27c @ ( arr @ A_27c @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) ) @ ( arr @ A_27c @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) ) ) ) @ ( ap @ ( c_2EerrorStateMonad_2EMMAP @ A_27c @ A_27b @ A_27a ) @ V0f ) ) ) ) ) ).
thf(conj_thm_2EerrorStateMonad_2EJOIN__UNIT,axiom,
! [A_27a: del,A_27b: del] :
( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ A_27a ) ) ) ) @ ( c_2EerrorStateMonad_2EJOIN @ A_27a @ A_27b ) ) @ ( c_2EerrorStateMonad_2EUNIT @ A_27a @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) ) )
= ( c_2Ecombin_2EI @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) ) ) ).
thf(conj_thm_2EerrorStateMonad_2EJOIN__MMAP__UNIT,axiom,
! [A_27a: del,A_27b: del] :
( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ A_27a ) ) ) ) @ ( c_2EerrorStateMonad_2EJOIN @ A_27a @ A_27b ) ) @ ( ap @ ( c_2EerrorStateMonad_2EMMAP @ A_27a @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ A_27b ) @ ( c_2EerrorStateMonad_2EUNIT @ A_27a @ A_27b ) ) )
= ( c_2Ecombin_2EI @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) ) ) ).
thf(conj_thm_2EerrorStateMonad_2EJOIN__MAP__JOIN,axiom,
! [A_27a: del,A_27b: del] :
( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ A_27a ) ) ) @ A_27a ) ) ) @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ A_27a ) ) ) ) @ ( c_2EerrorStateMonad_2EJOIN @ A_27a @ A_27b ) ) @ ( ap @ ( c_2EerrorStateMonad_2EMMAP @ A_27a @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ A_27a ) ) ) ) @ ( c_2EerrorStateMonad_2EJOIN @ A_27a @ A_27b ) ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ A_27a ) ) ) @ A_27a ) ) ) @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ A_27a ) ) ) ) @ ( c_2EerrorStateMonad_2EJOIN @ A_27a @ A_27b ) ) @ ( c_2EerrorStateMonad_2EJOIN @ A_27a @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2EerrorStateMonad_2EJOIN__MAP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0k: $i] :
( ( mem @ V0k @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) )
=> ! [V1m: $i] :
( ( mem @ V1m @ ( arr @ A_27b @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27c @ A_27a ) ) ) ) )
=> ( ( ap @ ( ap @ ( c_2EerrorStateMonad_2EBIND @ A_27a @ A_27b @ A_27c ) @ V0k ) @ V1m )
= ( ap @ ( c_2EerrorStateMonad_2EJOIN @ A_27a @ A_27c ) @ ( ap @ ( ap @ ( c_2EerrorStateMonad_2EMMAP @ A_27a @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27c @ A_27a ) ) ) @ A_27b ) @ V1m ) @ V0k ) ) ) ) ) ).
thf(conj_thm_2EerrorStateMonad_2Esequence__nil,axiom,
! [A_27a: del,A_27b: del] :
( ( ap @ ( c_2EerrorStateMonad_2Esequence @ A_27a @ A_27b ) @ ( c_2Elist_2ENIL @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) ) )
= ( ap @ ( c_2EerrorStateMonad_2EUNIT @ A_27a @ ( ty_2Elist_2Elist @ A_27b ) ) @ ( c_2Elist_2ENIL @ A_27b ) ) ) ).
thf(conj_thm_2EerrorStateMonad_2EmapM__nil,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27c @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) ) )
=> ( ( ap @ ( ap @ ( c_2EerrorStateMonad_2EmapM @ A_27c @ A_27a @ A_27b ) @ V0f ) @ ( c_2Elist_2ENIL @ A_27c ) )
= ( ap @ ( c_2EerrorStateMonad_2EUNIT @ A_27a @ ( ty_2Elist_2Elist @ A_27b ) ) @ ( c_2Elist_2ENIL @ A_27b ) ) ) ) ).
thf(conj_thm_2EerrorStateMonad_2EmapM__cons,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27c @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27c )
=> ! [V2xs: $i] :
( ( mem @ V2xs @ ( ty_2Elist_2Elist @ A_27c ) )
=> ( ( ap @ ( ap @ ( c_2EerrorStateMonad_2EmapM @ A_27c @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27c ) @ V1x ) @ V2xs ) )
= ( ap @ ( ap @ ( c_2EerrorStateMonad_2EBIND @ A_27a @ A_27b @ ( ty_2Elist_2Elist @ A_27b ) ) @ ( ap @ V0f @ V1x ) )
@ ( lam @ A_27b
@ ^ [V3y: $i] :
( ap @ ( ap @ ( c_2EerrorStateMonad_2EBIND @ A_27a @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Elist_2Elist @ A_27b ) ) @ ( ap @ ( ap @ ( c_2EerrorStateMonad_2EmapM @ A_27c @ A_27a @ A_27b ) @ V0f ) @ V2xs ) )
@ ( lam @ ( ty_2Elist_2Elist @ A_27b )
@ ^ [V4ys: $i] : ( ap @ ( c_2EerrorStateMonad_2EUNIT @ A_27a @ ( ty_2Elist_2Elist @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V3y ) @ V4ys ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2EerrorStateMonad_2EES__FAIL__DEF,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ A_27b )
=> ( ( ap @ ( c_2EerrorStateMonad_2EES__FAIL @ A_27a @ A_27b ) @ V0s )
= ( c_2Eoption_2ENONE @ A_27a ) ) ) ).
thf(ax_thm_2EerrorStateMonad_2EES__CHOICE__DEF,axiom,
! [A_27a: del,A_27b: del,V0xM: $i] :
( ( mem @ V0xM @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ A_27a ) ) )
=> ! [V1yM: $i] :
( ( mem @ V1yM @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ A_27a ) ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ A_27b )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EerrorStateMonad_2EES__CHOICE @ A_27a @ A_27b ) @ V0xM ) @ V1yM ) @ V2s )
= ( ap @ ( ap @ ( ap @ ( c_2Eoption_2Eoption__CASE @ A_27a @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ V0xM @ V2s ) ) @ ( ap @ V1yM @ V2s ) )
@ ( lam @ A_27a
@ ^ [V3v1: $i] : ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V3v1 ) ) ) ) ) ) ) ).
thf(ax_thm_2EerrorStateMonad_2EES__GUARD__DEF,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ bool )
=> ( ( ap @ ( c_2EerrorStateMonad_2EES__GUARD @ A_27a ) @ V0b )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27a ) ) ) ) @ V0b ) @ ( ap @ ( c_2EerrorStateMonad_2EUNIT @ A_27a @ ty_2Eone_2Eone ) @ ( inj__ty_2Eone_2Eone @ fo__c_2Eone_2Eone ) ) ) @ ( c_2EerrorStateMonad_2EES__FAIL @ ( ty_2Epair_2Eprod @ ty_2Eone_2Eone @ A_27a ) @ A_27a ) ) ) ) ).
thf(conj_thm_2EerrorStateMonad_2EES__CHOICE__ASSOC,axiom,
! [A_27a: del,A_27b: del,V0xM: $i] :
( ( mem @ V0xM @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ A_27b ) ) )
=> ! [V1yM: $i] :
( ( mem @ V1yM @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ A_27b ) ) )
=> ! [V2zM: $i] :
( ( mem @ V2zM @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ A_27b ) ) )
=> ( ( ap @ ( ap @ ( c_2EerrorStateMonad_2EES__CHOICE @ A_27b @ A_27a ) @ V0xM ) @ ( ap @ ( ap @ ( c_2EerrorStateMonad_2EES__CHOICE @ A_27b @ A_27a ) @ V1yM ) @ V2zM ) )
= ( ap @ ( ap @ ( c_2EerrorStateMonad_2EES__CHOICE @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( c_2EerrorStateMonad_2EES__CHOICE @ A_27b @ A_27a ) @ V0xM ) @ V1yM ) ) @ V2zM ) ) ) ) ) ).
thf(conj_thm_2EerrorStateMonad_2EES__CHOICE__FAIL__LID,axiom,
! [A_27a: del,A_27b: del,V0xM: $i] :
( ( mem @ V0xM @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ A_27b ) ) )
=> ( ( ap @ ( ap @ ( c_2EerrorStateMonad_2EES__CHOICE @ A_27b @ A_27a ) @ ( c_2EerrorStateMonad_2EES__FAIL @ A_27b @ A_27a ) ) @ V0xM )
= V0xM ) ) ).
thf(conj_thm_2EerrorStateMonad_2EES__CHOICE__FAIL__RID,axiom,
! [A_27a: del,A_27b: del,V0xM: $i] :
( ( mem @ V0xM @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ A_27b ) ) )
=> ( ( ap @ ( ap @ ( c_2EerrorStateMonad_2EES__CHOICE @ A_27b @ A_27a ) @ V0xM ) @ ( c_2EerrorStateMonad_2EES__FAIL @ A_27b @ A_27a ) )
= V0xM ) ) ).
thf(conj_thm_2EerrorStateMonad_2EBIND__FAIL__L,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0fM: $i] :
( ( mem @ V0fM @ ( arr @ A_27c @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) ) )
=> ( ( ap @ ( ap @ ( c_2EerrorStateMonad_2EBIND @ A_27a @ A_27c @ A_27b ) @ ( c_2EerrorStateMonad_2EES__FAIL @ ( ty_2Epair_2Eprod @ A_27c @ A_27a ) @ A_27a ) ) @ V0fM )
= ( c_2EerrorStateMonad_2EES__FAIL @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) @ A_27a ) ) ) ).
thf(conj_thm_2EerrorStateMonad_2EBIND__ESGUARD,axiom,
! [A_27a: del,A_27b: del,V0fM: $i] :
( ( mem @ V0fM @ ( arr @ ty_2Eone_2Eone @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) ) )
=> ( ( ( ap @ ( ap @ ( c_2EerrorStateMonad_2EBIND @ A_27a @ ty_2Eone_2Eone @ A_27b ) @ ( ap @ ( c_2EerrorStateMonad_2EES__GUARD @ A_27a ) @ c_2Ebool_2EF ) ) @ V0fM )
= ( c_2EerrorStateMonad_2EES__FAIL @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) @ A_27a ) )
& ( ( ap @ ( ap @ ( c_2EerrorStateMonad_2EBIND @ A_27a @ ty_2Eone_2Eone @ A_27b ) @ ( ap @ ( c_2EerrorStateMonad_2EES__GUARD @ A_27a ) @ c_2Ebool_2ET ) ) @ V0fM )
= ( ap @ V0fM @ ( inj__ty_2Eone_2Eone @ fo__c_2Eone_2Eone ) ) ) ) ) ).
thf(conj_thm_2EerrorStateMonad_2EIGNORE__BIND__ESGUARD,axiom,
! [A_27a: del,A_27b: del,V0xM: $i] :
( ( mem @ V0xM @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) )
=> ( ( ( ap @ ( ap @ ( c_2EerrorStateMonad_2EIGNORE__BIND @ ty_2Eone_2Eone @ A_27a @ A_27b ) @ ( ap @ ( c_2EerrorStateMonad_2EES__GUARD @ A_27a ) @ c_2Ebool_2EF ) ) @ V0xM )
= ( c_2EerrorStateMonad_2EES__FAIL @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) @ A_27a ) )
& ( ( ap @ ( ap @ ( c_2EerrorStateMonad_2EIGNORE__BIND @ ty_2Eone_2Eone @ A_27a @ A_27b ) @ ( ap @ ( c_2EerrorStateMonad_2EES__GUARD @ A_27a ) @ c_2Ebool_2ET ) ) @ V0xM )
= V0xM ) ) ) ).
thf(conj_thm_2EerrorStateMonad_2EIGNORE__BIND__FAIL,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0xM: $i] :
( ( mem @ V0xM @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) ) )
=> ( ( ( ap @ ( ap @ ( c_2EerrorStateMonad_2EIGNORE__BIND @ A_27c @ A_27a @ A_27b ) @ ( c_2EerrorStateMonad_2EES__FAIL @ ( ty_2Epair_2Eprod @ A_27c @ A_27a ) @ A_27a ) ) @ V0xM )
= ( c_2EerrorStateMonad_2EES__FAIL @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) @ A_27a ) )
& ( ( ap @ ( ap @ ( c_2EerrorStateMonad_2EIGNORE__BIND @ A_27b @ A_27a @ A_27d ) @ V0xM ) @ ( c_2EerrorStateMonad_2EES__FAIL @ ( ty_2Epair_2Eprod @ A_27d @ A_27a ) @ A_27a ) )
= ( c_2EerrorStateMonad_2EES__FAIL @ ( ty_2Epair_2Eprod @ A_27d @ A_27a ) @ A_27a ) ) ) ) ).
thf(ax_thm_2EerrorStateMonad_2EES__APPLY__DEF,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0fM: $i] :
( ( mem @ V0fM @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ ( arr @ A_27c @ A_27b ) @ A_27a ) ) ) )
=> ! [V1xM: $i] :
( ( mem @ V1xM @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27c @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( c_2EerrorStateMonad_2EES__APPLY @ A_27a @ A_27b @ A_27c ) @ V0fM ) @ V1xM )
= ( ap @ ( ap @ ( c_2EerrorStateMonad_2EBIND @ A_27a @ ( arr @ A_27c @ A_27b ) @ A_27b ) @ V0fM )
@ ( lam @ ( arr @ A_27c @ A_27b )
@ ^ [V2f: $i] :
( ap @ ( ap @ ( c_2EerrorStateMonad_2EBIND @ A_27a @ A_27c @ A_27b ) @ V1xM )
@ ( lam @ A_27c
@ ^ [V3x: $i] : ( ap @ ( c_2EerrorStateMonad_2EUNIT @ A_27a @ A_27b ) @ ( ap @ V2f @ V3x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EerrorStateMonad_2EAPPLY__UNIT,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27c @ A_27b ) )
=> ! [V1xM: $i] :
( ( mem @ V1xM @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27c @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( c_2EerrorStateMonad_2EES__APPLY @ A_27a @ A_27b @ A_27c ) @ ( ap @ ( c_2EerrorStateMonad_2EUNIT @ A_27a @ ( arr @ A_27c @ A_27b ) ) @ V0f ) ) @ V1xM )
= ( ap @ ( ap @ ( c_2EerrorStateMonad_2EMMAP @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1xM ) ) ) ) ).
thf(conj_thm_2EerrorStateMonad_2EAPPLY__UNIT__UNIT,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27c @ A_27b ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27c )
=> ( ( ap @ ( ap @ ( c_2EerrorStateMonad_2EES__APPLY @ A_27a @ A_27b @ A_27c ) @ ( ap @ ( c_2EerrorStateMonad_2EUNIT @ A_27a @ ( arr @ A_27c @ A_27b ) ) @ V0f ) ) @ ( ap @ ( c_2EerrorStateMonad_2EUNIT @ A_27a @ A_27c ) @ V1x ) )
= ( ap @ ( c_2EerrorStateMonad_2EUNIT @ A_27a @ A_27b ) @ ( ap @ V0f @ V1x ) ) ) ) ) ).
thf(ax_thm_2EerrorStateMonad_2EES__LIFT2__DEF,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27c @ ( arr @ A_27d @ A_27b ) ) )
=> ! [V1xM: $i] :
( ( mem @ V1xM @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27c @ A_27a ) ) ) )
=> ! [V2yM: $i] :
( ( mem @ V2yM @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ ( ty_2Epair_2Eprod @ A_27d @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EerrorStateMonad_2EES__LIFT2 @ A_27a @ A_27b @ A_27c @ A_27d ) @ V0f ) @ V1xM ) @ V2yM )
= ( ap @ ( ap @ ( c_2EerrorStateMonad_2EES__APPLY @ A_27a @ A_27b @ A_27d ) @ ( ap @ ( ap @ ( c_2EerrorStateMonad_2EMMAP @ A_27a @ ( arr @ A_27d @ A_27b ) @ A_27c ) @ V0f ) @ V1xM ) ) @ V2yM ) ) ) ) ) ).
%------------------------------------------------------------------------------