ITP001 Axioms: ITP055^7.ax
%------------------------------------------------------------------------------
% File : ITP055^7 : 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 : sum_num.ax [Gau19]
% : HL4055^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 54 ( 10 unt; 27 typ; 0 def)
% Number of atoms : 74 ( 34 equ; 5 cnn)
% Maximal formula atoms : 9 ( 1 avg)
% Number of connectives : 463 ( 5 ~; 3 |; 24 &; 403 @)
% ( 13 <=>; 15 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 8 avg; 403 nst)
% Number of types : 3 ( 2 usr)
% Number of type conns : 79 ( 79 >; 0 *; 0 +; 0 <<)
% Number of symbols : 27 ( 25 usr; 3 con; 0-5 aty)
% Number of variables : 109 ( 2 ^ 97 !; 3 ?; 109 :)
% ( 7 !>; 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_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_2Enum_2E0,type,
c_2Enum_2E0: tyop_2Enum_2Enum ).
thf(c_2Eprim__rec_2E_3C,type,
c_2Eprim__rec_2E_3C: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Earithmetic_2E_3C_3D,type,
c_2Earithmetic_2E_3C_3D: 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_2Earithmetic_2EBIT1,type,
c_2Earithmetic_2EBIT1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2EBIT2,type,
c_2Earithmetic_2EBIT2: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Erich__list_2ECOUNT__LIST,type,
c_2Erich__list_2ECOUNT__LIST: tyop_2Enum_2Enum > ( tyop_2Elist_2Elist @ tyop_2Enum_2Enum ) ).
thf(c_2Elist_2EFOLDL,type,
c_2Elist_2EFOLDL:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > A_27a > A_27b ) > A_27b > ( tyop_2Elist_2Elist @ A_27a ) > A_27b ) ).
thf(c_2Esum__num_2EGSUM,type,
c_2Esum__num_2EGSUM: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > ( tyop_2Enum_2Enum > tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2ENUMERAL,type,
c_2Earithmetic_2ENUMERAL: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Enum_2ESUC,type,
c_2Enum_2ESUC: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Esum__num_2ESUM,type,
c_2Esum__num_2ESUM: tyop_2Enum_2Enum > ( tyop_2Enum_2Enum > tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).
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_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_2Esum__num_2ESUM__def,axiom,
( ! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ( c_2Esum__num_2ESUM @ c_2Enum_2E0 @ V0f )
= c_2Enum_2E0 )
& ! [V1m: tyop_2Enum_2Enum,V2f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ( c_2Esum__num_2ESUM @ ( c_2Enum_2ESUC @ V1m ) @ V2f )
= ( c_2Earithmetic_2E_2B @ ( c_2Esum__num_2ESUM @ V1m @ V2f ) @ ( V2f @ V1m ) ) ) ) ).
thf(thm_2Esum__num_2EGSUM__ind,axiom,
! [V0P: ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) > ( tyop_2Enum_2Enum > tyop_2Enum_2Enum ) > $o] :
( ( ! [V1n: tyop_2Enum_2Enum,V2f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] : ( V0P @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V1n @ c_2Enum_2E0 ) @ V2f )
& ! [V3n: tyop_2Enum_2Enum,V4m: tyop_2Enum_2Enum,V5f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ( V0P @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V3n @ V4m ) @ V5f )
=> ( V0P @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V3n @ ( c_2Enum_2ESUC @ V4m ) ) @ V5f ) ) )
=> ! [V6v: tyop_2Enum_2Enum,V7v1: tyop_2Enum_2Enum,V8v2: tyop_2Enum_2Enum > tyop_2Enum_2Enum] : ( V0P @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V6v @ V7v1 ) @ V8v2 ) ) ).
thf(thm_2Esum__num_2EGSUM__def,axiom,
( ! [V0n: tyop_2Enum_2Enum,V1f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ( c_2Esum__num_2EGSUM @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V0n @ c_2Enum_2E0 ) @ V1f )
= c_2Enum_2E0 )
& ! [V2n: tyop_2Enum_2Enum,V3m: tyop_2Enum_2Enum,V4f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ( c_2Esum__num_2EGSUM @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V2n @ ( c_2Enum_2ESUC @ V3m ) ) @ V4f )
= ( c_2Earithmetic_2E_2B @ ( c_2Esum__num_2EGSUM @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V2n @ V3m ) @ V4f ) @ ( V4f @ ( c_2Earithmetic_2E_2B @ V2n @ V3m ) ) ) ) ) ).
thf(thm_2Esum__num_2EGSUM__def__compute,axiom,
( ! [V0n: tyop_2Enum_2Enum,V1f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ( c_2Esum__num_2EGSUM @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V0n @ c_2Enum_2E0 ) @ V1f )
= c_2Enum_2E0 )
& ! [V2n: tyop_2Enum_2Enum,V3m: tyop_2Enum_2Enum,V4f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ( c_2Esum__num_2EGSUM @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V2n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V3m ) ) ) @ V4f )
= ( c_2Earithmetic_2E_2B @ ( c_2Esum__num_2EGSUM @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V2n @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V3m ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V4f ) @ ( V4f @ ( c_2Earithmetic_2E_2B @ V2n @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V3m ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) )
& ! [V5n: tyop_2Enum_2Enum,V6m: tyop_2Enum_2Enum,V7f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ( c_2Esum__num_2EGSUM @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V5n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V6m ) ) ) @ V7f )
= ( c_2Earithmetic_2E_2B @ ( c_2Esum__num_2EGSUM @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V5n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V6m ) ) ) @ V7f ) @ ( V7f @ ( c_2Earithmetic_2E_2B @ V5n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V6m ) ) ) ) ) ) ) ).
thf(thm_2Esum__num_2EGSUM__1,axiom,
! [V0m: tyop_2Enum_2Enum,V1f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ( c_2Esum__num_2EGSUM @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V0m @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V1f )
= ( V1f @ V0m ) ) ).
thf(thm_2Esum__num_2EGSUM__ADD,axiom,
! [V0p: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum,V3f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ( c_2Esum__num_2EGSUM @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V0p @ ( c_2Earithmetic_2E_2B @ V1m @ V2n ) ) @ V3f )
= ( c_2Earithmetic_2E_2B @ ( c_2Esum__num_2EGSUM @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V0p @ V1m ) @ V3f ) @ ( c_2Esum__num_2EGSUM @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ ( c_2Earithmetic_2E_2B @ V0p @ V1m ) @ V2n ) @ V3f ) ) ) ).
thf(thm_2Esum__num_2EGSUM__ZERO,axiom,
! [V0p: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ! [V3m: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V0p @ V3m )
& ( c_2Eprim__rec_2E_3C @ V3m @ ( c_2Earithmetic_2E_2B @ V0p @ V1n ) ) )
=> ( ( V2f @ V3m )
= c_2Enum_2E0 ) )
<=> ( ( c_2Esum__num_2EGSUM @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V0p @ V1n ) @ V2f )
= c_2Enum_2E0 ) ) ).
thf(thm_2Esum__num_2EGSUM__MONO,axiom,
! [V0p: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum,V3f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V1m @ V2n )
& ( (~)
@ ( ( V3f @ ( c_2Earithmetic_2E_2B @ V0p @ V2n ) )
= c_2Enum_2E0 ) ) )
=> ( c_2Eprim__rec_2E_3C @ ( c_2Esum__num_2EGSUM @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V0p @ V1m ) @ V3f ) @ ( c_2Esum__num_2EGSUM @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V0p @ ( c_2Enum_2ESUC @ V2n ) ) @ V3f ) ) ) ).
thf(thm_2Esum__num_2EGSUM__LESS,axiom,
! [V0p: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum,V3f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ? [V4q: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2E_2B @ V1m @ V0p ) @ V4q )
& ( c_2Eprim__rec_2E_3C @ V4q @ ( c_2Earithmetic_2E_2B @ V2n @ V0p ) )
& ( (~)
@ ( ( V3f @ V4q )
= c_2Enum_2E0 ) ) )
<=> ( c_2Eprim__rec_2E_3C @ ( c_2Esum__num_2EGSUM @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V0p @ V1m ) @ V3f ) @ ( c_2Esum__num_2EGSUM @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V0p @ V2n ) @ V3f ) ) ) ).
thf(thm_2Esum__num_2EGSUM__EQUAL,axiom,
! [V0p: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum,V3f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ( ( c_2Esum__num_2EGSUM @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V0p @ V1m ) @ V3f )
= ( c_2Esum__num_2EGSUM @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V0p @ V2n ) @ V3f ) )
<=> ( ( ( c_2Earithmetic_2E_3C_3D @ V1m @ V2n )
& ! [V4q: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2E_2B @ V0p @ V1m ) @ V4q )
& ( c_2Eprim__rec_2E_3C @ V4q @ ( c_2Earithmetic_2E_2B @ V0p @ V2n ) ) )
=> ( ( V3f @ V4q )
= c_2Enum_2E0 ) ) )
| ( ( c_2Eprim__rec_2E_3C @ V2n @ V1m )
& ! [V5q: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2E_2B @ V0p @ V2n ) @ V5q )
& ( c_2Eprim__rec_2E_3C @ V5q @ ( c_2Earithmetic_2E_2B @ V0p @ V1m ) ) )
=> ( ( V3f @ V5q )
= c_2Enum_2E0 ) ) ) ) ) ).
thf(thm_2Esum__num_2EGSUM__FUN__EQUAL,axiom,
! [V0p: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2f: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V3g: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ! [V4x: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V0p @ V4x )
& ( c_2Eprim__rec_2E_3C @ V4x @ ( c_2Earithmetic_2E_2B @ V0p @ V1n ) ) )
=> ( ( V2f @ V4x )
= ( V3g @ V4x ) ) )
=> ( ( c_2Esum__num_2EGSUM @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V0p @ V1n ) @ V2f )
= ( c_2Esum__num_2EGSUM @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ V0p @ V1n ) @ V3g ) ) ) ).
thf(thm_2Esum__num_2ESUM__def__compute,axiom,
( ! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ( c_2Esum__num_2ESUM @ c_2Enum_2E0 @ V0f )
= c_2Enum_2E0 )
& ! [V1m: tyop_2Enum_2Enum,V2f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ( c_2Esum__num_2ESUM @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V1m ) ) @ V2f )
= ( c_2Earithmetic_2E_2B @ ( c_2Esum__num_2ESUM @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V1m ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V2f ) @ ( V2f @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V1m ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
& ! [V3m: tyop_2Enum_2Enum,V4f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ( c_2Esum__num_2ESUM @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V3m ) ) @ V4f )
= ( c_2Earithmetic_2E_2B @ ( c_2Esum__num_2ESUM @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V3m ) ) @ V4f ) @ ( V4f @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V3m ) ) ) ) ) ) ).
thf(thm_2Esum__num_2ESUM,axiom,
! [V0m: tyop_2Enum_2Enum,V1f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ( c_2Esum__num_2ESUM @ V0m @ V1f )
= ( c_2Esum__num_2EGSUM @ ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ V0m ) @ V1f ) ) ).
thf(thm_2Esum__num_2ESUM__1,axiom,
! [V0f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ( c_2Esum__num_2ESUM @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ V0f )
= ( V0f @ c_2Enum_2E0 ) ) ).
thf(thm_2Esum__num_2ESUM__MONO,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V0m @ V1n )
& ( (~)
@ ( ( V2f @ V1n )
= c_2Enum_2E0 ) ) )
=> ( c_2Eprim__rec_2E_3C @ ( c_2Esum__num_2ESUM @ V0m @ V2f ) @ ( c_2Esum__num_2ESUM @ ( c_2Enum_2ESUC @ V1n ) @ V2f ) ) ) ).
thf(thm_2Esum__num_2ESUM__LESS,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ? [V3q: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V0m @ V3q )
& ( c_2Eprim__rec_2E_3C @ V3q @ V1n )
& ( (~)
@ ( ( V2f @ V3q )
= c_2Enum_2E0 ) ) )
<=> ( c_2Eprim__rec_2E_3C @ ( c_2Esum__num_2ESUM @ V0m @ V2f ) @ ( c_2Esum__num_2ESUM @ V1n @ V2f ) ) ) ).
thf(thm_2Esum__num_2ESUM__EQUAL,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ( ( c_2Esum__num_2ESUM @ V0m @ V2f )
= ( c_2Esum__num_2ESUM @ V1n @ V2f ) )
<=> ( ( ( c_2Earithmetic_2E_3C_3D @ V0m @ V1n )
& ! [V3q: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V0m @ V3q )
& ( c_2Eprim__rec_2E_3C @ V3q @ V1n ) )
=> ( ( V2f @ V3q )
= c_2Enum_2E0 ) ) )
| ( ( c_2Eprim__rec_2E_3C @ V1n @ V0m )
& ! [V4q: tyop_2Enum_2Enum] :
( ( ( c_2Earithmetic_2E_3C_3D @ V1n @ V4q )
& ( c_2Eprim__rec_2E_3C @ V4q @ V0m ) )
=> ( ( V2f @ V4q )
= c_2Enum_2E0 ) ) ) ) ) ).
thf(thm_2Esum__num_2ESUM__FUN__EQUAL,axiom,
! [V0n: tyop_2Enum_2Enum,V1f: tyop_2Enum_2Enum > tyop_2Enum_2Enum,V2g: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ! [V3x: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V3x @ V0n )
=> ( ( V1f @ V3x )
= ( V2g @ V3x ) ) )
=> ( ( c_2Esum__num_2ESUM @ V0n @ V1f )
= ( c_2Esum__num_2ESUM @ V0n @ V2g ) ) ) ).
thf(thm_2Esum__num_2ESUM__ZERO,axiom,
! [V0n: tyop_2Enum_2Enum,V1f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ! [V2m: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V2m @ V0n )
=> ( ( V1f @ V2m )
= c_2Enum_2E0 ) )
<=> ( ( c_2Esum__num_2ESUM @ V0n @ V1f )
= c_2Enum_2E0 ) ) ).
thf(thm_2Esum__num_2ESUM__FOLDL,axiom,
! [V0n: tyop_2Enum_2Enum,V1f: tyop_2Enum_2Enum > tyop_2Enum_2Enum] :
( ( c_2Esum__num_2ESUM @ V0n @ V1f )
= ( c_2Elist_2EFOLDL @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum
@ ^ [V2x: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2B @ ( V1f @ V3n ) @ V2x )
@ c_2Enum_2E0
@ ( c_2Erich__list_2ECOUNT__LIST @ V0n ) ) ) ).
%------------------------------------------------------------------------------