ITP001 Axioms: ITP034^7.ax
%------------------------------------------------------------------------------
% File : ITP034^7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : errorStateMonad.ax [Gau19]
% : HL4034^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 121 ( 49 unt; 63 typ; 0 def)
% Number of atoms : 88 ( 56 equ; 2 cnn)
% Maximal formula atoms : 3 ( 0 avg)
% Number of connectives : 1953 ( 2 ~; 1 |; 7 &;1930 @)
% ( 7 <=>; 6 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 6 avg;1930 nst)
% Number of types : 4 ( 3 usr)
% Number of type conns : 471 ( 471 >; 0 *; 0 +; 0 <<)
% Number of symbols : 62 ( 60 usr; 5 con; 0-9 aty)
% Number of variables : 379 ( 50 ^ 238 !; 1 ?; 379 :)
% ( 90 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Elist_2Elist,type,
tyop_2Elist_2Elist: $tType > $tType ).
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(tyop_2Enum_2Enum,type,
tyop_2Enum_2Enum: $tType ).
thf(tyop_2Eone_2Eone,type,
tyop_2Eone_2Eone: $tType ).
thf(tyop_2Eoption_2Eoption,type,
tyop_2Eoption_2Eoption: $tType > $tType ).
thf(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: $tType > $tType > $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Earithmetic_2E_2B,type,
c_2Earithmetic_2E_2B: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epair_2E_2C,type,
c_2Epair_2E_2C:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).
thf(c_2Earithmetic_2E_2D,type,
c_2Earithmetic_2E_2D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Eprim__rec_2E_3C,type,
c_2Eprim__rec_2E_3C: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Emin_2E_40,type,
c_2Emin_2E_40:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a ) ).
thf(c_2EerrorStateMonad_2EBIND,type,
c_2EerrorStateMonad_2EBIND:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) > ( A_27b > A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27c @ A_27a ) ) ) > A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27c @ A_27a ) ) ) ).
thf(c_2Earithmetic_2EBIT1,type,
c_2Earithmetic_2EBIT1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND:
!>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).
thf(c_2Elist_2ECONS,type,
c_2Elist_2ECONS:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Epair_2ECURRY,type,
c_2Epair_2ECURRY:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27c ) > A_27a > A_27b > A_27c ) ).
thf(c_2EerrorStateMonad_2EES__APPLY,type,
c_2EerrorStateMonad_2EES__APPLY:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ ( A_27c > A_27b ) @ A_27a ) ) ) > ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27c @ A_27a ) ) ) > A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) ).
thf(c_2EerrorStateMonad_2EES__CHOICE,type,
c_2EerrorStateMonad_2EES__CHOICE:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > ( tyop_2Eoption_2Eoption @ A_27a ) ) > ( A_27b > ( tyop_2Eoption_2Eoption @ A_27a ) ) > A_27b > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).
thf(c_2EerrorStateMonad_2EES__FAIL,type,
c_2EerrorStateMonad_2EES__FAIL:
!>[A_27a: $tType,A_27b: $tType] : ( A_27b > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).
thf(c_2EerrorStateMonad_2EES__GUARD,type,
c_2EerrorStateMonad_2EES__GUARD:
!>[A_27a: $tType] : ( $o > A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27a ) ) ) ).
thf(c_2EerrorStateMonad_2EES__LIFT2,type,
c_2EerrorStateMonad_2EES__LIFT2:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] : ( ( A_27c > A_27d > A_27b ) > ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27c @ A_27a ) ) ) > ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27d @ A_27a ) ) ) > A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) ).
thf(c_2EerrorStateMonad_2EEXT,type,
c_2EerrorStateMonad_2EEXT:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27c > A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) > ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27c @ A_27a ) ) ) > A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) ).
thf(c_2Ebool_2EF,type,
c_2Ebool_2EF: $o ).
thf(c_2Elist_2EFOLDR,type,
c_2Elist_2EFOLDR:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > A_27b ) > A_27b > ( tyop_2Elist_2Elist @ A_27a ) > A_27b ) ).
thf(c_2EerrorStateMonad_2EFOR,type,
c_2EerrorStateMonad_2EFOR:
!>[A_27state: $tType] : ( ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Enum_2Enum > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) ) ) > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) ).
thf(c_2EerrorStateMonad_2EFOREACH,type,
c_2EerrorStateMonad_2EFOREACH:
!>[A_27a: $tType,A_27state: $tType] : ( ( tyop_2Epair_2Eprod @ ( tyop_2Elist_2Elist @ A_27a ) @ ( A_27a > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) ) > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) ).
thf(c_2Ecombin_2EI,type,
c_2Ecombin_2EI:
!>[A_27a: $tType] : ( A_27a > A_27a ) ).
thf(c_2EerrorStateMonad_2EIGNORE__BIND,type,
c_2EerrorStateMonad_2EIGNORE__BIND:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27b > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ) > ( A_27b > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27c @ A_27b ) ) ) > A_27b > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27c @ A_27b ) ) ) ).
thf(c_2EerrorStateMonad_2EJOIN,type,
c_2EerrorStateMonad_2EJOIN:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ A_27a ) ) ) > A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) ).
thf(c_2Elist_2EMAP,type,
c_2Elist_2EMAP:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27b ) ) ).
thf(c_2EerrorStateMonad_2EMCOMP,type,
c_2EerrorStateMonad_2EMCOMP:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType] : ( ( A_27d > A_27e > ( tyop_2Eoption_2Eoption @ A_27c ) ) > ( A_27a > A_27b > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27d @ A_27e ) ) ) > A_27a > A_27b > ( tyop_2Eoption_2Eoption @ A_27c ) ) ).
thf(c_2EerrorStateMonad_2EMMAP,type,
c_2EerrorStateMonad_2EMMAP:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27c > A_27b ) > ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27c @ A_27a ) ) ) > A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) ).
thf(c_2EerrorStateMonad_2ENARROW,type,
c_2EerrorStateMonad_2ENARROW:
!>[A_27a: $tType,A_27b: $tType,A_27state: $tType] : ( A_27b > ( ( tyop_2Epair_2Eprod @ A_27b @ A_27state ) > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27state ) ) ) ) > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27a @ A_27state ) ) ) ).
thf(c_2Elist_2ENIL,type,
c_2Elist_2ENIL:
!>[A_27a: $tType] : ( tyop_2Elist_2Elist @ A_27a ) ).
thf(c_2Eoption_2ENONE,type,
c_2Eoption_2ENONE:
!>[A_27a: $tType] : ( tyop_2Eoption_2Eoption @ A_27a ) ).
thf(c_2Earithmetic_2ENUMERAL,type,
c_2Earithmetic_2ENUMERAL: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Eoption_2EOPTION__MCOMP,type,
c_2Eoption_2EOPTION__MCOMP:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27b > ( tyop_2Eoption_2Eoption @ A_27a ) ) > ( A_27c > ( tyop_2Eoption_2Eoption @ A_27b ) ) > A_27c > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).
thf(c_2EerrorStateMonad_2EREAD,type,
c_2EerrorStateMonad_2EREAD:
!>[A_27a: $tType,A_27state: $tType] : ( ( A_27state > A_27a ) > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27a @ A_27state ) ) ) ).
thf(c_2Epair_2ESND,type,
c_2Epair_2ESND:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27b ) ).
thf(c_2Eoption_2ESOME,type,
c_2Eoption_2ESOME:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).
thf(c_2Ebool_2ET,type,
c_2Ebool_2ET: $o ).
thf(c_2Epair_2EUNCURRY,type,
c_2Epair_2EUNCURRY:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > A_27b > A_27c ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27c ) ).
thf(c_2EerrorStateMonad_2EUNIT,type,
c_2EerrorStateMonad_2EUNIT:
!>[A_27a: $tType,A_27b: $tType] : ( A_27b > A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) ).
thf(c_2Erelation_2EWF,type,
c_2Erelation_2EWF:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Erelation_2EWFREC,type,
c_2Erelation_2EWFREC:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > $o ) > ( ( A_27a > A_27b ) > A_27a > A_27b ) > A_27a > A_27b ) ).
thf(c_2EerrorStateMonad_2EWIDEN,type,
c_2EerrorStateMonad_2EWIDEN:
!>[A_27a: $tType,A_27b: $tType,A_27state: $tType] : ( ( A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27a @ A_27state ) ) ) > ( tyop_2Epair_2Eprod @ A_27b @ A_27state ) > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27state ) ) ) ) ).
thf(c_2EerrorStateMonad_2EWRITE,type,
c_2EerrorStateMonad_2EWRITE:
!>[A_27state: $tType] : ( ( A_27state > A_27state ) > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) ).
thf(c_2Earithmetic_2EZERO,type,
c_2Earithmetic_2EZERO: tyop_2Enum_2Enum ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Elist_2Elist__CASE,type,
c_2Elist_2Elist__CASE:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > A_27b > ( A_27a > ( tyop_2Elist_2Elist @ A_27a ) > A_27b ) > A_27b ) ).
thf(c_2EerrorStateMonad_2EmapM,type,
c_2EerrorStateMonad_2EmapM:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > A_27b > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27c @ A_27b ) ) ) > ( tyop_2Elist_2Elist @ A_27a ) > A_27b > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ ( tyop_2Elist_2Elist @ A_27c ) @ A_27b ) ) ) ).
thf(c_2Ecombin_2Eo,type,
c_2Ecombin_2Eo:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27c > A_27b ) > ( A_27a > A_27c ) > A_27a > A_27b ) ).
thf(c_2Eone_2Eone,type,
c_2Eone_2Eone: tyop_2Eone_2Eone ).
thf(c_2Eoption_2Eoption__CASE,type,
c_2Eoption_2Eoption__CASE:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Eoption_2Eoption @ A_27a ) > A_27b > ( A_27a > A_27b ) > A_27b ) ).
thf(c_2Epair_2Epair__CASE,type,
c_2Epair_2Epair__CASE:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) > ( A_27b > A_27c > A_27a ) > A_27a ) ).
thf(c_2EerrorStateMonad_2Esequence,type,
c_2EerrorStateMonad_2Esequence:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Elist_2Elist @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) ) > A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ ( tyop_2Elist_2Elist @ A_27b ) @ A_27a ) ) ) ).
thf(c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $o > $o ).
thf(logicdef_2E_2F_5C,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
<=> ( V0
& V1 ) ) ).
thf(logicdef_2E_5C_2F,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
<=> ( V0
| V1 ) ) ).
thf(logicdef_2E_7E,axiom,
! [V0: $o] :
( ( c_2Ebool_2E_7E @ V0 )
<=> ( (~) @ V0 ) ) ).
thf(logicdef_2E_3D_3D_3E,axiom,
! [V0: $o,V1: $o] :
( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
<=> ( V0
=> V1 ) ) ).
thf(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0: A_27a,V1: A_27a] :
( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
<=> ( V0 = V1 ) ) ).
thf(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_21 @ A_27a @ V0f )
<=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_3F @ A_27a @ V0f )
<=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2EerrorStateMonad_2EUNIT__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27b] :
( ( c_2EerrorStateMonad_2EUNIT @ A_27a @ A_27b @ V0x )
= ( ^ [V1s: A_27a] : ( c_2Eoption_2ESOME @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ ( c_2Epair_2E_2C @ A_27b @ A_27a @ V0x @ V1s ) ) ) ) ).
thf(thm_2EerrorStateMonad_2EBIND__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0g: A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ),V1f: A_27b > A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27c @ A_27a ) ),V2s0: A_27a] :
( ( c_2EerrorStateMonad_2EBIND @ A_27a @ A_27b @ A_27c @ V0g @ V1f @ V2s0 )
= ( c_2Eoption_2Eoption__CASE @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27c @ A_27a ) ) @ ( V0g @ V2s0 ) @ ( c_2Eoption_2ENONE @ ( tyop_2Epair_2Eprod @ A_27c @ A_27a ) )
@ ^ [V3v: tyop_2Epair_2Eprod @ A_27b @ A_27a] :
( c_2Epair_2Epair__CASE @ ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27c @ A_27a ) ) @ A_27b @ A_27a @ V3v
@ ^ [V4b: A_27b,V5s: A_27a] : ( V1f @ V4b @ V5s ) ) ) ) ).
thf(thm_2EerrorStateMonad_2EIGNORE__BIND__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27b > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ),V1g: A_27b > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27c @ A_27b ) )] :
( ( c_2EerrorStateMonad_2EIGNORE__BIND @ A_27a @ A_27b @ A_27c @ V0f @ V1g )
= ( c_2EerrorStateMonad_2EBIND @ A_27b @ A_27a @ A_27c @ V0f
@ ^ [V2x: A_27a] : V1g ) ) ).
thf(thm_2EerrorStateMonad_2EMMAP__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27c > A_27b,V1m: A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27c @ A_27a ) )] :
( ( c_2EerrorStateMonad_2EMMAP @ A_27a @ A_27b @ A_27c @ V0f @ V1m )
= ( c_2EerrorStateMonad_2EBIND @ A_27a @ A_27c @ A_27b @ V1m @ ( c_2Ecombin_2Eo @ A_27c @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ A_27b @ ( c_2EerrorStateMonad_2EUNIT @ A_27a @ A_27b ) @ V0f ) ) ) ).
thf(thm_2EerrorStateMonad_2EJOIN__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0z: A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ A_27a ) )] :
( ( c_2EerrorStateMonad_2EJOIN @ A_27a @ A_27b @ V0z )
= ( c_2EerrorStateMonad_2EBIND @ A_27a @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ A_27b @ V0z @ ( c_2Ecombin_2EI @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) ) ) ) ).
thf(thm_2EerrorStateMonad_2EEXT__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0g: A_27c > A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ),V1m: A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27c @ A_27a ) )] :
( ( c_2EerrorStateMonad_2EEXT @ A_27a @ A_27b @ A_27c @ V0g @ V1m )
= ( c_2EerrorStateMonad_2EBIND @ A_27a @ A_27c @ A_27b @ V1m @ V0g ) ) ).
thf(thm_2EerrorStateMonad_2EMCOMP__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType,V0g: A_27d > A_27e > ( tyop_2Eoption_2Eoption @ A_27c ),V1f: A_27a > A_27b > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27d @ A_27e ) )] :
( ( c_2EerrorStateMonad_2EMCOMP @ A_27a @ A_27b @ A_27c @ A_27d @ A_27e @ V0g @ V1f )
= ( c_2Epair_2ECURRY @ A_27a @ A_27b @ ( tyop_2Eoption_2Eoption @ A_27c ) @ ( c_2Eoption_2EOPTION__MCOMP @ A_27c @ ( tyop_2Epair_2Eprod @ A_27d @ A_27e ) @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2EUNCURRY @ A_27d @ A_27e @ ( tyop_2Eoption_2Eoption @ A_27c ) @ V0g ) @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27d @ A_27e ) ) @ V1f ) ) ) ) ).
thf(thm_2EerrorStateMonad_2EFOR__primitive__def,axiom,
! [A_27state: $tType] :
( ( c_2EerrorStateMonad_2EFOR @ A_27state )
= ( c_2Erelation_2EWFREC @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Enum_2Enum > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) ) ) @ ( A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) )
@ ( c_2Emin_2E_40 @ ( ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Enum_2Enum > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) ) ) > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Enum_2Enum > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) ) ) > $o )
@ ^ [V0R: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Enum_2Enum > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) ) ) > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Enum_2Enum > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) ) ) > $o] :
( c_2Ebool_2E_2F_5C @ ( c_2Erelation_2EWF @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Enum_2Enum > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) ) ) @ V0R )
@ ( c_2Ebool_2E_21 @ ( tyop_2Enum_2Enum > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) )
@ ^ [V1a: tyop_2Enum_2Enum > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) )] :
( c_2Ebool_2E_21 @ tyop_2Enum_2Enum
@ ^ [V2j: tyop_2Enum_2Enum] :
( c_2Ebool_2E_21 @ tyop_2Enum_2Enum
@ ^ [V3i: tyop_2Enum_2Enum] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V3i @ V2j ) ) @ ( V0R @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Enum_2Enum > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) ) @ ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Eprim__rec_2E_3C @ V3i @ V2j ) @ ( c_2Earithmetic_2E_2B @ V3i @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2E_2D @ V3i @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Enum_2Enum > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) @ V2j @ V1a ) ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Enum_2Enum > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) ) @ V3i @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Enum_2Enum > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) @ V2j @ V1a ) ) ) ) ) ) ) ) )
@ ^ [V4FOR: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Enum_2Enum > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) ) ) > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ),V5a_27: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Enum_2Enum > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) )] :
( c_2Epair_2Epair__CASE @ ( A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Enum_2Enum > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) ) @ V5a_27
@ ^ [V6i: tyop_2Enum_2Enum,V7v1: tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Enum_2Enum > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) )] :
( c_2Epair_2Epair__CASE @ ( A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) @ tyop_2Enum_2Enum @ ( tyop_2Enum_2Enum > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) @ V7v1
@ ^ [V8j: tyop_2Enum_2Enum,V9a: tyop_2Enum_2Enum > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) )] :
( c_2Ecombin_2EI @ ( A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) )
@ ( c_2Ebool_2ECOND @ ( A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V6i @ V8j ) @ ( V9a @ V6i )
@ ( c_2EerrorStateMonad_2EBIND @ A_27state @ tyop_2Eone_2Eone @ tyop_2Eone_2Eone @ ( V9a @ V6i )
@ ^ [V10u: tyop_2Eone_2Eone] : ( V4FOR @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Enum_2Enum > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) ) @ ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Eprim__rec_2E_3C @ V6i @ V8j ) @ ( c_2Earithmetic_2E_2B @ V6i @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2E_2D @ V6i @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Enum_2Enum > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) @ V8j @ V9a ) ) ) ) ) ) ) ) ) ) ).
thf(thm_2EerrorStateMonad_2EFOREACH__primitive__def,axiom,
! [A_27a: $tType,A_27state: $tType] :
( ( c_2EerrorStateMonad_2EFOREACH @ A_27a @ A_27state )
= ( c_2Erelation_2EWFREC @ ( tyop_2Epair_2Eprod @ ( tyop_2Elist_2Elist @ A_27a ) @ ( A_27a > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) ) @ ( A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) )
@ ( c_2Emin_2E_40 @ ( ( tyop_2Epair_2Eprod @ ( tyop_2Elist_2Elist @ A_27a ) @ ( A_27a > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) ) > ( tyop_2Epair_2Eprod @ ( tyop_2Elist_2Elist @ A_27a ) @ ( A_27a > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) ) > $o )
@ ^ [V0R: ( tyop_2Epair_2Eprod @ ( tyop_2Elist_2Elist @ A_27a ) @ ( A_27a > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) ) > ( tyop_2Epair_2Eprod @ ( tyop_2Elist_2Elist @ A_27a ) @ ( A_27a > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) ) > $o] :
( c_2Ebool_2E_2F_5C @ ( c_2Erelation_2EWF @ ( tyop_2Epair_2Eprod @ ( tyop_2Elist_2Elist @ A_27a ) @ ( A_27a > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) ) @ V0R )
@ ( c_2Ebool_2E_21 @ A_27a
@ ^ [V1h: A_27a] :
( c_2Ebool_2E_21 @ ( A_27a > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) )
@ ^ [V2a: A_27a > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) )] :
( c_2Ebool_2E_21 @ ( tyop_2Elist_2Elist @ A_27a )
@ ^ [V3t: tyop_2Elist_2Elist @ A_27a] : ( V0R @ ( c_2Epair_2E_2C @ ( tyop_2Elist_2Elist @ A_27a ) @ ( A_27a > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) @ V3t @ V2a ) @ ( c_2Epair_2E_2C @ ( tyop_2Elist_2Elist @ A_27a ) @ ( A_27a > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) @ ( c_2Elist_2ECONS @ A_27a @ V1h @ V3t ) @ V2a ) ) ) ) ) ) )
@ ^ [V4FOREACH: ( tyop_2Epair_2Eprod @ ( tyop_2Elist_2Elist @ A_27a ) @ ( A_27a > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) ) > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ),V5a_27: tyop_2Epair_2Eprod @ ( tyop_2Elist_2Elist @ A_27a ) @ ( A_27a > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) )] :
( c_2Epair_2Epair__CASE @ ( A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) @ ( tyop_2Elist_2Elist @ A_27a ) @ ( A_27a > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) @ V5a_27
@ ^ [V6v: tyop_2Elist_2Elist @ A_27a,V7a: A_27a > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) )] :
( c_2Elist_2Elist__CASE @ A_27a @ ( A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) @ V6v @ ( c_2Ecombin_2EI @ ( A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) @ ( c_2EerrorStateMonad_2EUNIT @ A_27state @ tyop_2Eone_2Eone @ c_2Eone_2Eone ) )
@ ^ [V8h: A_27a,V9t: tyop_2Elist_2Elist @ A_27a] :
( c_2Ecombin_2EI @ ( A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) )
@ ( c_2EerrorStateMonad_2EBIND @ A_27state @ tyop_2Eone_2Eone @ tyop_2Eone_2Eone @ ( V7a @ V8h )
@ ^ [V10u: tyop_2Eone_2Eone] : ( V4FOREACH @ ( c_2Epair_2E_2C @ ( tyop_2Elist_2Elist @ A_27a ) @ ( A_27a > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) @ V9t @ V7a ) ) ) ) ) ) ) ) ).
thf(thm_2EerrorStateMonad_2EREAD__def,axiom,
! [A_27a: $tType,A_27state: $tType,V0f: A_27state > A_27a] :
( ( c_2EerrorStateMonad_2EREAD @ A_27a @ A_27state @ V0f )
= ( ^ [V1s: A_27state] : ( c_2Eoption_2ESOME @ ( tyop_2Epair_2Eprod @ A_27a @ A_27state ) @ ( c_2Epair_2E_2C @ A_27a @ A_27state @ ( V0f @ V1s ) @ V1s ) ) ) ) ).
thf(thm_2EerrorStateMonad_2EWRITE__def,axiom,
! [A_27state: $tType,V0f: A_27state > A_27state] :
( ( c_2EerrorStateMonad_2EWRITE @ A_27state @ V0f )
= ( ^ [V1s: A_27state] : ( c_2Eoption_2ESOME @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) @ ( c_2Epair_2E_2C @ tyop_2Eone_2Eone @ A_27state @ c_2Eone_2Eone @ ( V0f @ V1s ) ) ) ) ) ).
thf(thm_2EerrorStateMonad_2ENARROW__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27state: $tType,V0v: A_27b,V1f: ( tyop_2Epair_2Eprod @ A_27b @ A_27state ) > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27state ) ) )] :
( ( c_2EerrorStateMonad_2ENARROW @ A_27a @ A_27b @ A_27state @ V0v @ V1f )
= ( ^ [V2s: A_27state] :
( c_2Eoption_2Eoption__CASE @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27state ) ) @ ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27a @ A_27state ) ) @ ( V1f @ ( c_2Epair_2E_2C @ A_27b @ A_27state @ V0v @ V2s ) ) @ ( c_2Eoption_2ENONE @ ( tyop_2Epair_2Eprod @ A_27a @ A_27state ) )
@ ^ [V3v: tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27state )] :
( c_2Epair_2Epair__CASE @ ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27a @ A_27state ) ) @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27state ) @ V3v
@ ^ [V4r: A_27a,V5s1: tyop_2Epair_2Eprod @ A_27b @ A_27state] : ( c_2Eoption_2ESOME @ ( tyop_2Epair_2Eprod @ A_27a @ A_27state ) @ ( c_2Epair_2E_2C @ A_27a @ A_27state @ V4r @ ( c_2Epair_2ESND @ A_27b @ A_27state @ V5s1 ) ) ) ) ) ) ) ).
thf(thm_2EerrorStateMonad_2EWIDEN__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27state: $tType,V0f: A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27a @ A_27state ) )] :
( ( c_2EerrorStateMonad_2EWIDEN @ A_27a @ A_27b @ A_27state @ V0f )
= ( c_2Epair_2EUNCURRY @ A_27b @ A_27state @ ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27state ) ) )
@ ^ [V1s1: A_27b,V2s2: A_27state] :
( c_2Eoption_2Eoption__CASE @ ( tyop_2Epair_2Eprod @ A_27a @ A_27state ) @ ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27state ) ) ) @ ( V0f @ V2s2 ) @ ( c_2Eoption_2ENONE @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27state ) ) )
@ ^ [V3v: tyop_2Epair_2Eprod @ A_27a @ A_27state] :
( c_2Epair_2Epair__CASE @ ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27state ) ) ) @ A_27a @ A_27state @ V3v
@ ^ [V4r: A_27a,V5s3: A_27state] : ( c_2Eoption_2ESOME @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27state ) ) @ ( c_2Epair_2E_2C @ A_27a @ ( tyop_2Epair_2Eprod @ A_27b @ A_27state ) @ V4r @ ( c_2Epair_2E_2C @ A_27b @ A_27state @ V1s1 @ V5s3 ) ) ) ) ) ) ) ).
thf(thm_2EerrorStateMonad_2Esequence__def,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2EerrorStateMonad_2Esequence @ A_27a @ A_27b )
= ( c_2Elist_2EFOLDR @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ ( tyop_2Elist_2Elist @ A_27b ) @ A_27a ) ) )
@ ^ [V0m: A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ),V1ms: A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ ( tyop_2Elist_2Elist @ A_27b ) @ A_27a ) )] :
( c_2EerrorStateMonad_2EBIND @ A_27a @ A_27b @ ( tyop_2Elist_2Elist @ A_27b ) @ V0m
@ ^ [V2x: A_27b] :
( c_2EerrorStateMonad_2EBIND @ A_27a @ ( tyop_2Elist_2Elist @ A_27b ) @ ( tyop_2Elist_2Elist @ A_27b ) @ V1ms
@ ^ [V3xs: tyop_2Elist_2Elist @ A_27b] : ( c_2EerrorStateMonad_2EUNIT @ A_27a @ ( tyop_2Elist_2Elist @ A_27b ) @ ( c_2Elist_2ECONS @ A_27b @ V2x @ V3xs ) ) ) )
@ ( c_2EerrorStateMonad_2EUNIT @ A_27a @ ( tyop_2Elist_2Elist @ A_27b ) @ ( c_2Elist_2ENIL @ A_27b ) ) ) ) ).
thf(thm_2EerrorStateMonad_2EmapM__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27c @ A_27b ) )] :
( ( c_2EerrorStateMonad_2EmapM @ A_27a @ A_27b @ A_27c @ V0f )
= ( c_2Ecombin_2Eo @ ( tyop_2Elist_2Elist @ A_27a ) @ ( A_27b > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ ( tyop_2Elist_2Elist @ A_27c ) @ A_27b ) ) ) @ ( tyop_2Elist_2Elist @ ( A_27b > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27c @ A_27b ) ) ) ) @ ( c_2EerrorStateMonad_2Esequence @ A_27b @ A_27c ) @ ( c_2Elist_2EMAP @ A_27a @ ( A_27b > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27c @ A_27b ) ) ) @ V0f ) ) ) ).
thf(thm_2EerrorStateMonad_2EES__FAIL__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27b] :
( ( c_2EerrorStateMonad_2EES__FAIL @ A_27a @ A_27b @ V0s )
= ( c_2Eoption_2ENONE @ A_27a ) ) ).
thf(thm_2EerrorStateMonad_2EES__CHOICE__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0xM: A_27b > ( tyop_2Eoption_2Eoption @ A_27a ),V1yM: A_27b > ( tyop_2Eoption_2Eoption @ A_27a ),V2s: A_27b] :
( ( c_2EerrorStateMonad_2EES__CHOICE @ A_27a @ A_27b @ V0xM @ V1yM @ V2s )
= ( c_2Eoption_2Eoption__CASE @ A_27a @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( V0xM @ V2s ) @ ( V1yM @ V2s )
@ ^ [V3v1: A_27a] : ( c_2Eoption_2ESOME @ A_27a @ V3v1 ) ) ) ).
thf(thm_2EerrorStateMonad_2EES__GUARD__DEF,axiom,
! [A_27a: $tType,V0b: $o] :
( ( c_2EerrorStateMonad_2EES__GUARD @ A_27a @ V0b )
= ( c_2Ebool_2ECOND @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27a ) ) ) @ V0b @ ( c_2EerrorStateMonad_2EUNIT @ A_27a @ tyop_2Eone_2Eone @ c_2Eone_2Eone ) @ ( c_2EerrorStateMonad_2EES__FAIL @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27a ) @ A_27a ) ) ) ).
thf(thm_2EerrorStateMonad_2EES__APPLY__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0fM: A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ ( A_27c > A_27b ) @ A_27a ) ),V1xM: A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27c @ A_27a ) )] :
( ( c_2EerrorStateMonad_2EES__APPLY @ A_27a @ A_27b @ A_27c @ V0fM @ V1xM )
= ( c_2EerrorStateMonad_2EBIND @ A_27a @ ( A_27c > A_27b ) @ A_27b @ V0fM
@ ^ [V2f: A_27c > A_27b] :
( c_2EerrorStateMonad_2EBIND @ A_27a @ A_27c @ A_27b @ V1xM
@ ^ [V3x: A_27c] : ( c_2EerrorStateMonad_2EUNIT @ A_27a @ A_27b @ ( V2f @ V3x ) ) ) ) ) ).
thf(thm_2EerrorStateMonad_2EES__LIFT2__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0f: A_27c > A_27d > A_27b,V1xM: A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27c @ A_27a ) ),V2yM: A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27d @ A_27a ) )] :
( ( c_2EerrorStateMonad_2EES__LIFT2 @ A_27a @ A_27b @ A_27c @ A_27d @ V0f @ V1xM @ V2yM )
= ( c_2EerrorStateMonad_2EES__APPLY @ A_27a @ A_27b @ A_27d @ ( c_2EerrorStateMonad_2EMMAP @ A_27a @ ( A_27d > A_27b ) @ A_27c @ V0f @ V1xM ) @ V2yM ) ) ).
thf(thm_2EerrorStateMonad_2EFOR__ind,axiom,
! [A_27state: $tType,V0P: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Enum_2Enum > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) ) ) > $o] :
( ! [V1i: tyop_2Enum_2Enum,V2j: tyop_2Enum_2Enum,V3a: tyop_2Enum_2Enum > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) )] :
( ( ( (~) @ ( V1i = V2j ) )
=> ( V0P @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Enum_2Enum > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) ) @ ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Eprim__rec_2E_3C @ V1i @ V2j ) @ ( c_2Earithmetic_2E_2B @ V1i @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2E_2D @ V1i @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Enum_2Enum > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) @ V2j @ V3a ) ) ) )
=> ( V0P @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Enum_2Enum > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) ) @ V1i @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Enum_2Enum > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) @ V2j @ V3a ) ) ) )
=> ! [V4v: tyop_2Enum_2Enum,V5v1: tyop_2Enum_2Enum,V6v2: tyop_2Enum_2Enum > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) )] : ( V0P @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Enum_2Enum > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) ) @ V4v @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Enum_2Enum > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) @ V5v1 @ V6v2 ) ) ) ) ).
thf(thm_2EerrorStateMonad_2EFOR__def,axiom,
! [A_27state: $tType,V0j: tyop_2Enum_2Enum,V1i: tyop_2Enum_2Enum,V2a: tyop_2Enum_2Enum > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) )] :
( ( c_2EerrorStateMonad_2EFOR @ A_27state @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Enum_2Enum > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) ) @ V1i @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Enum_2Enum > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) @ V0j @ V2a ) ) )
= ( c_2Ebool_2ECOND @ ( A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V1i @ V0j ) @ ( V2a @ V1i )
@ ( c_2EerrorStateMonad_2EBIND @ A_27state @ tyop_2Eone_2Eone @ tyop_2Eone_2Eone @ ( V2a @ V1i )
@ ^ [V3u: tyop_2Eone_2Eone] : ( c_2EerrorStateMonad_2EFOR @ A_27state @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ ( tyop_2Enum_2Enum > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) ) @ ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Eprim__rec_2E_3C @ V1i @ V0j ) @ ( c_2Earithmetic_2E_2B @ V1i @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2E_2D @ V1i @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ ( tyop_2Enum_2Enum > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) @ V0j @ V2a ) ) ) ) ) ) ).
thf(thm_2EerrorStateMonad_2EFOREACH__ind,axiom,
! [A_27a: $tType,A_27state: $tType,V0P: ( tyop_2Epair_2Eprod @ ( tyop_2Elist_2Elist @ A_27a ) @ ( A_27a > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) ) > $o] :
( ( ! [V1a: A_27a > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) )] : ( V0P @ ( c_2Epair_2E_2C @ ( tyop_2Elist_2Elist @ A_27a ) @ ( A_27a > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) @ ( c_2Elist_2ENIL @ A_27a ) @ V1a ) )
& ! [V2h: A_27a,V3t: tyop_2Elist_2Elist @ A_27a,V4a: A_27a > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) )] :
( ( V0P @ ( c_2Epair_2E_2C @ ( tyop_2Elist_2Elist @ A_27a ) @ ( A_27a > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) @ V3t @ V4a ) )
=> ( V0P @ ( c_2Epair_2E_2C @ ( tyop_2Elist_2Elist @ A_27a ) @ ( A_27a > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) @ ( c_2Elist_2ECONS @ A_27a @ V2h @ V3t ) @ V4a ) ) ) )
=> ! [V5v: tyop_2Elist_2Elist @ A_27a,V6v1: A_27a > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) )] : ( V0P @ ( c_2Epair_2E_2C @ ( tyop_2Elist_2Elist @ A_27a ) @ ( A_27a > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) @ V5v @ V6v1 ) ) ) ).
thf(thm_2EerrorStateMonad_2EFOREACH__def,axiom,
! [A_27a: $tType,A_27state: $tType] :
( ! [V0a: A_27a > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) )] :
( ( c_2EerrorStateMonad_2EFOREACH @ A_27a @ A_27state @ ( c_2Epair_2E_2C @ ( tyop_2Elist_2Elist @ A_27a ) @ ( A_27a > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) @ ( c_2Elist_2ENIL @ A_27a ) @ V0a ) )
= ( c_2EerrorStateMonad_2EUNIT @ A_27state @ tyop_2Eone_2Eone @ c_2Eone_2Eone ) )
& ! [V1t: tyop_2Elist_2Elist @ A_27a,V2h: A_27a,V3a: A_27a > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) )] :
( ( c_2EerrorStateMonad_2EFOREACH @ A_27a @ A_27state @ ( c_2Epair_2E_2C @ ( tyop_2Elist_2Elist @ A_27a ) @ ( A_27a > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) @ ( c_2Elist_2ECONS @ A_27a @ V2h @ V1t ) @ V3a ) )
= ( c_2EerrorStateMonad_2EBIND @ A_27state @ tyop_2Eone_2Eone @ tyop_2Eone_2Eone @ ( V3a @ V2h )
@ ^ [V4u: tyop_2Eone_2Eone] : ( c_2EerrorStateMonad_2EFOREACH @ A_27a @ A_27state @ ( c_2Epair_2E_2C @ ( tyop_2Elist_2Elist @ A_27a ) @ ( A_27a > A_27state > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ tyop_2Eone_2Eone @ A_27state ) ) ) @ V1t @ V3a ) ) ) ) ) ).
thf(thm_2EerrorStateMonad_2EBIND__LEFT__UNIT,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0k: A_27a > A_27b > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27c @ A_27b ) ),V1x: A_27a] :
( ( c_2EerrorStateMonad_2EBIND @ A_27b @ A_27a @ A_27c @ ( c_2EerrorStateMonad_2EUNIT @ A_27b @ A_27a @ V1x ) @ V0k )
= ( V0k @ V1x ) ) ).
thf(thm_2EerrorStateMonad_2EMCOMP__THM,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0g: A_27d > A_27b > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27c @ A_27b ) ),V1f: A_27a > A_27b > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27d @ A_27b ) )] :
( ( c_2EerrorStateMonad_2EMCOMP @ A_27a @ A_27b @ ( tyop_2Epair_2Eprod @ A_27c @ A_27b ) @ A_27d @ A_27b @ V0g @ V1f )
= ( c_2Ecombin_2Eo @ A_27a @ ( A_27b > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27c @ A_27b ) ) ) @ ( A_27b > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27d @ A_27b ) ) ) @ ( c_2EerrorStateMonad_2EEXT @ A_27b @ A_27c @ A_27d @ V0g ) @ V1f ) ) ).
thf(thm_2EerrorStateMonad_2EMCOMP__ASSOC,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType,A_27f: $tType,A_27g: $tType,V0h: A_27a > A_27b > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27f @ A_27g ) ),V1g: A_27f > A_27g > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27d @ A_27e ) ),V2f: A_27d > A_27e > ( tyop_2Eoption_2Eoption @ A_27c )] :
( ( c_2EerrorStateMonad_2EMCOMP @ A_27a @ A_27b @ A_27c @ A_27d @ A_27e @ V2f @ ( c_2EerrorStateMonad_2EMCOMP @ A_27a @ A_27b @ ( tyop_2Epair_2Eprod @ A_27d @ A_27e ) @ A_27f @ A_27g @ V1g @ V0h ) )
= ( c_2EerrorStateMonad_2EMCOMP @ A_27a @ A_27b @ A_27c @ A_27f @ A_27g @ ( c_2EerrorStateMonad_2EMCOMP @ A_27f @ A_27g @ A_27c @ A_27d @ A_27e @ V2f @ V1g ) @ V0h ) ) ).
thf(thm_2EerrorStateMonad_2EUNIT__CURRY,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2EerrorStateMonad_2EUNIT @ A_27b @ A_27a )
= ( c_2Epair_2ECURRY @ A_27a @ A_27b @ ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( c_2Eoption_2ESOME @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ) ) ).
thf(thm_2EerrorStateMonad_2EMCOMP__ID,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType,A_27f: $tType,A_27g: $tType,V0g: A_27a > A_27b > ( tyop_2Eoption_2Eoption @ A_27c ),V1f: A_27d > A_27e > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27f @ A_27g ) )] :
( ( ( c_2EerrorStateMonad_2EMCOMP @ A_27a @ A_27b @ A_27c @ A_27a @ A_27b @ V0g @ ( c_2EerrorStateMonad_2EUNIT @ A_27b @ A_27a ) )
= V0g )
& ( ( c_2EerrorStateMonad_2EMCOMP @ A_27d @ A_27e @ ( tyop_2Epair_2Eprod @ A_27f @ A_27g ) @ A_27f @ A_27g @ ( c_2EerrorStateMonad_2EUNIT @ A_27g @ A_27f ) @ V1f )
= V1f ) ) ).
thf(thm_2EerrorStateMonad_2EBIND__RIGHT__UNIT,axiom,
! [A_27a: $tType,A_27b: $tType,V0k: A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) )] :
( ( c_2EerrorStateMonad_2EBIND @ A_27a @ A_27b @ A_27b @ V0k @ ( c_2EerrorStateMonad_2EUNIT @ A_27a @ A_27b ) )
= V0k ) ).
thf(thm_2EerrorStateMonad_2EBIND__ASSOC,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0k: A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ),V1m: A_27b > A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27c @ A_27a ) ),V2n: A_27c > A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27d @ A_27a ) )] :
( ( c_2EerrorStateMonad_2EBIND @ A_27a @ A_27b @ A_27d @ V0k
@ ^ [V3a: A_27b] : ( c_2EerrorStateMonad_2EBIND @ A_27a @ A_27c @ A_27d @ ( V1m @ V3a ) @ V2n ) )
= ( c_2EerrorStateMonad_2EBIND @ A_27a @ A_27c @ A_27d @ ( c_2EerrorStateMonad_2EBIND @ A_27a @ A_27b @ A_27c @ V0k @ V1m ) @ V2n ) ) ).
thf(thm_2EerrorStateMonad_2EMMAP__ID,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2EerrorStateMonad_2EMMAP @ A_27a @ A_27b @ A_27b @ ( c_2Ecombin_2EI @ A_27b ) )
= ( c_2Ecombin_2EI @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) ) ) ).
thf(thm_2EerrorStateMonad_2EMMAP__COMP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0f: A_27a > A_27b,V1g: A_27c > A_27a] :
( ( c_2EerrorStateMonad_2EMMAP @ A_27d @ A_27b @ A_27c @ ( c_2Ecombin_2Eo @ A_27c @ A_27b @ A_27a @ V0f @ V1g ) )
= ( c_2Ecombin_2Eo @ ( A_27d > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27c @ A_27d ) ) ) @ ( A_27d > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27d ) ) ) @ ( A_27d > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27a @ A_27d ) ) ) @ ( c_2EerrorStateMonad_2EMMAP @ A_27d @ A_27b @ A_27a @ V0f ) @ ( c_2EerrorStateMonad_2EMMAP @ A_27d @ A_27a @ A_27c @ V1g ) ) ) ).
thf(thm_2EerrorStateMonad_2EMMAP__UNIT,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b] :
( ( c_2Ecombin_2Eo @ A_27a @ ( A_27c > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) ) @ ( A_27c > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) ) ) @ ( c_2EerrorStateMonad_2EMMAP @ A_27c @ A_27b @ A_27a @ V0f ) @ ( c_2EerrorStateMonad_2EUNIT @ A_27c @ A_27a ) )
= ( c_2Ecombin_2Eo @ A_27a @ ( A_27c > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) ) @ A_27b @ ( c_2EerrorStateMonad_2EUNIT @ A_27c @ A_27b ) @ V0f ) ) ).
thf(thm_2EerrorStateMonad_2EMMAP__JOIN,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b] :
( ( c_2Ecombin_2Eo @ ( A_27c > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ ( A_27c > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) ) ) @ A_27c ) ) ) @ ( A_27c > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) ) @ ( A_27c > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) ) ) @ ( c_2EerrorStateMonad_2EMMAP @ A_27c @ A_27b @ A_27a @ V0f ) @ ( c_2EerrorStateMonad_2EJOIN @ A_27c @ A_27a ) )
= ( c_2Ecombin_2Eo @ ( A_27c > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ ( A_27c > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) ) ) @ A_27c ) ) ) @ ( A_27c > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) ) @ ( A_27c > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ ( A_27c > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) ) @ A_27c ) ) ) @ ( c_2EerrorStateMonad_2EJOIN @ A_27c @ A_27b ) @ ( c_2EerrorStateMonad_2EMMAP @ A_27c @ ( A_27c > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) ) @ ( A_27c > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) ) ) @ ( c_2EerrorStateMonad_2EMMAP @ A_27c @ A_27b @ A_27a @ V0f ) ) ) ) ).
thf(thm_2EerrorStateMonad_2EJOIN__UNIT,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Ecombin_2Eo @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ A_27a ) ) ) @ ( c_2EerrorStateMonad_2EJOIN @ A_27a @ A_27b ) @ ( c_2EerrorStateMonad_2EUNIT @ A_27a @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) ) )
= ( c_2Ecombin_2EI @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) ) ) ).
thf(thm_2EerrorStateMonad_2EJOIN__MMAP__UNIT,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Ecombin_2Eo @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ A_27a ) ) ) @ ( c_2EerrorStateMonad_2EJOIN @ A_27a @ A_27b ) @ ( c_2EerrorStateMonad_2EMMAP @ A_27a @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ A_27b @ ( c_2EerrorStateMonad_2EUNIT @ A_27a @ A_27b ) ) )
= ( c_2Ecombin_2EI @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) ) ) ).
thf(thm_2EerrorStateMonad_2EJOIN__MAP__JOIN,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Ecombin_2Eo @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ A_27a ) ) ) @ A_27a ) ) ) @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ A_27a ) ) ) @ ( c_2EerrorStateMonad_2EJOIN @ A_27a @ A_27b ) @ ( c_2EerrorStateMonad_2EMMAP @ A_27a @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ A_27a ) ) ) @ ( c_2EerrorStateMonad_2EJOIN @ A_27a @ A_27b ) ) )
= ( c_2Ecombin_2Eo @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ A_27a ) ) ) @ A_27a ) ) ) @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) @ A_27a ) ) ) @ ( c_2EerrorStateMonad_2EJOIN @ A_27a @ A_27b ) @ ( c_2EerrorStateMonad_2EJOIN @ A_27a @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) ) ) ) ).
thf(thm_2EerrorStateMonad_2EJOIN__MAP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0k: A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ),V1m: A_27b > A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27c @ A_27a ) )] :
( ( c_2EerrorStateMonad_2EBIND @ A_27a @ A_27b @ A_27c @ V0k @ V1m )
= ( c_2EerrorStateMonad_2EJOIN @ A_27a @ A_27c @ ( c_2EerrorStateMonad_2EMMAP @ A_27a @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27c @ A_27a ) ) ) @ A_27b @ V1m @ V0k ) ) ) ).
thf(thm_2EerrorStateMonad_2Esequence__nil,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2EerrorStateMonad_2Esequence @ A_27a @ A_27b @ ( c_2Elist_2ENIL @ ( A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) ) ) ) )
= ( c_2EerrorStateMonad_2EUNIT @ A_27a @ ( tyop_2Elist_2Elist @ A_27b ) @ ( c_2Elist_2ENIL @ A_27b ) ) ) ).
thf(thm_2EerrorStateMonad_2EmapM__nil,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27c > A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) )] :
( ( c_2EerrorStateMonad_2EmapM @ A_27c @ A_27a @ A_27b @ V0f @ ( c_2Elist_2ENIL @ A_27c ) )
= ( c_2EerrorStateMonad_2EUNIT @ A_27a @ ( tyop_2Elist_2Elist @ A_27b ) @ ( c_2Elist_2ENIL @ A_27b ) ) ) ).
thf(thm_2EerrorStateMonad_2EmapM__cons,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0xs: tyop_2Elist_2Elist @ A_27c,V1x: A_27c,V2f: A_27c > A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) )] :
( ( c_2EerrorStateMonad_2EmapM @ A_27c @ A_27a @ A_27b @ V2f @ ( c_2Elist_2ECONS @ A_27c @ V1x @ V0xs ) )
= ( c_2EerrorStateMonad_2EBIND @ A_27a @ A_27b @ ( tyop_2Elist_2Elist @ A_27b ) @ ( V2f @ V1x )
@ ^ [V3y: A_27b] :
( c_2EerrorStateMonad_2EBIND @ A_27a @ ( tyop_2Elist_2Elist @ A_27b ) @ ( tyop_2Elist_2Elist @ A_27b ) @ ( c_2EerrorStateMonad_2EmapM @ A_27c @ A_27a @ A_27b @ V2f @ V0xs )
@ ^ [V4ys: tyop_2Elist_2Elist @ A_27b] : ( c_2EerrorStateMonad_2EUNIT @ A_27a @ ( tyop_2Elist_2Elist @ A_27b ) @ ( c_2Elist_2ECONS @ A_27b @ V3y @ V4ys ) ) ) ) ) ).
thf(thm_2EerrorStateMonad_2EES__CHOICE__ASSOC,axiom,
! [A_27a: $tType,A_27b: $tType,V0zM: A_27a > ( tyop_2Eoption_2Eoption @ A_27b ),V1yM: A_27a > ( tyop_2Eoption_2Eoption @ A_27b ),V2xM: A_27a > ( tyop_2Eoption_2Eoption @ A_27b )] :
( ( c_2EerrorStateMonad_2EES__CHOICE @ A_27b @ A_27a @ V2xM @ ( c_2EerrorStateMonad_2EES__CHOICE @ A_27b @ A_27a @ V1yM @ V0zM ) )
= ( c_2EerrorStateMonad_2EES__CHOICE @ A_27b @ A_27a @ ( c_2EerrorStateMonad_2EES__CHOICE @ A_27b @ A_27a @ V2xM @ V1yM ) @ V0zM ) ) ).
thf(thm_2EerrorStateMonad_2EES__CHOICE__FAIL__LID,axiom,
! [A_27a: $tType,A_27b: $tType,V0xM: A_27a > ( tyop_2Eoption_2Eoption @ A_27b )] :
( ( c_2EerrorStateMonad_2EES__CHOICE @ A_27b @ A_27a @ ( c_2EerrorStateMonad_2EES__FAIL @ A_27b @ A_27a ) @ V0xM )
= V0xM ) ).
thf(thm_2EerrorStateMonad_2EES__CHOICE__FAIL__RID,axiom,
! [A_27a: $tType,A_27b: $tType,V0xM: A_27a > ( tyop_2Eoption_2Eoption @ A_27b )] :
( ( c_2EerrorStateMonad_2EES__CHOICE @ A_27b @ A_27a @ V0xM @ ( c_2EerrorStateMonad_2EES__FAIL @ A_27b @ A_27a ) )
= V0xM ) ).
thf(thm_2EerrorStateMonad_2EBIND__FAIL__L,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0fM: A_27c > A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) )] :
( ( c_2EerrorStateMonad_2EBIND @ A_27a @ A_27c @ A_27b @ ( c_2EerrorStateMonad_2EES__FAIL @ ( tyop_2Epair_2Eprod @ A_27c @ A_27a ) @ A_27a ) @ V0fM )
= ( c_2EerrorStateMonad_2EES__FAIL @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ A_27a ) ) ).
thf(thm_2EerrorStateMonad_2EBIND__ESGUARD,axiom,
! [A_27a: $tType,A_27b: $tType,V0fM: tyop_2Eone_2Eone > A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) )] :
( ( ( c_2EerrorStateMonad_2EBIND @ A_27a @ tyop_2Eone_2Eone @ A_27b @ ( c_2EerrorStateMonad_2EES__GUARD @ A_27a @ c_2Ebool_2EF ) @ V0fM )
= ( c_2EerrorStateMonad_2EES__FAIL @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ A_27a ) )
& ( ( c_2EerrorStateMonad_2EBIND @ A_27a @ tyop_2Eone_2Eone @ A_27b @ ( c_2EerrorStateMonad_2EES__GUARD @ A_27a @ c_2Ebool_2ET ) @ V0fM )
= ( V0fM @ c_2Eone_2Eone ) ) ) ).
thf(thm_2EerrorStateMonad_2EIGNORE__BIND__ESGUARD,axiom,
! [A_27a: $tType,A_27b: $tType,V0xM: A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) )] :
( ( ( c_2EerrorStateMonad_2EIGNORE__BIND @ tyop_2Eone_2Eone @ A_27a @ A_27b @ ( c_2EerrorStateMonad_2EES__GUARD @ A_27a @ c_2Ebool_2EF ) @ V0xM )
= ( c_2EerrorStateMonad_2EES__FAIL @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ A_27a ) )
& ( ( c_2EerrorStateMonad_2EIGNORE__BIND @ tyop_2Eone_2Eone @ A_27a @ A_27b @ ( c_2EerrorStateMonad_2EES__GUARD @ A_27a @ c_2Ebool_2ET ) @ V0xM )
= V0xM ) ) ).
thf(thm_2EerrorStateMonad_2EIGNORE__BIND__FAIL,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0xM: A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) )] :
( ( ( c_2EerrorStateMonad_2EIGNORE__BIND @ A_27c @ A_27a @ A_27b @ ( c_2EerrorStateMonad_2EES__FAIL @ ( tyop_2Epair_2Eprod @ A_27c @ A_27a ) @ A_27a ) @ V0xM )
= ( c_2EerrorStateMonad_2EES__FAIL @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ A_27a ) )
& ( ( c_2EerrorStateMonad_2EIGNORE__BIND @ A_27b @ A_27a @ A_27d @ V0xM @ ( c_2EerrorStateMonad_2EES__FAIL @ ( tyop_2Epair_2Eprod @ A_27d @ A_27a ) @ A_27a ) )
= ( c_2EerrorStateMonad_2EES__FAIL @ ( tyop_2Epair_2Eprod @ A_27d @ A_27a ) @ A_27a ) ) ) ).
thf(thm_2EerrorStateMonad_2EAPPLY__UNIT,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0xM: A_27a > ( tyop_2Eoption_2Eoption @ ( tyop_2Epair_2Eprod @ A_27c @ A_27a ) ),V1f: A_27c > A_27b] :
( ( c_2EerrorStateMonad_2EES__APPLY @ A_27a @ A_27b @ A_27c @ ( c_2EerrorStateMonad_2EUNIT @ A_27a @ ( A_27c > A_27b ) @ V1f ) @ V0xM )
= ( c_2EerrorStateMonad_2EMMAP @ A_27a @ A_27b @ A_27c @ V1f @ V0xM ) ) ).
thf(thm_2EerrorStateMonad_2EAPPLY__UNIT__UNIT,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0x: A_27c,V1f: A_27c > A_27b] :
( ( c_2EerrorStateMonad_2EES__APPLY @ A_27a @ A_27b @ A_27c @ ( c_2EerrorStateMonad_2EUNIT @ A_27a @ ( A_27c > A_27b ) @ V1f ) @ ( c_2EerrorStateMonad_2EUNIT @ A_27a @ A_27c @ V0x ) )
= ( c_2EerrorStateMonad_2EUNIT @ A_27a @ A_27b @ ( V1f @ V0x ) ) ) ).
%------------------------------------------------------------------------------