TPTP Problem File: ITP016+2.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP016+2 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 set theory export of thm_2Ereal_2ESUP__EPSILON.p, bushy 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 : thm_2Ereal_2ESUP__EPSILON.p [Gau19]
% : HL407501+2.p [TPAP]
% Status : Theorem
% Rating : 1.00 v8.1.0, 0.97 v7.5.0
% Syntax : Number of formulae : 110 ( 31 unt; 0 def)
% Number of atoms : 490 ( 37 equ)
% Maximal formula atoms : 18 ( 4 avg)
% Number of connectives : 434 ( 54 ~; 44 |; 58 &)
% ( 76 <=>; 202 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 5 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 6 ( 3 usr; 2 prp; 0-2 aty)
% Number of functors : 35 ( 35 usr; 28 con; 0-2 aty)
% Number of variables : 172 ( 161 !; 11 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
include('Axioms/ITP001/ITP001+2.ax').
%------------------------------------------------------------------------------
fof(mem_c_2Ebool_2ET,axiom,
mem(c_2Ebool_2ET,bool) ).
fof(ax_true_p,axiom,
p(c_2Ebool_2ET) ).
fof(ne_ty_2Enum_2Enum,axiom,
ne(ty_2Enum_2Enum) ).
fof(mem_c_2Enum_2ESUC,axiom,
mem(c_2Enum_2ESUC,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) ).
fof(mem_c_2Earithmetic_2EZERO,axiom,
mem(c_2Earithmetic_2EZERO,ty_2Enum_2Enum) ).
fof(mem_c_2Earithmetic_2EBIT1,axiom,
mem(c_2Earithmetic_2EBIT1,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) ).
fof(mem_c_2Earithmetic_2ENUMERAL,axiom,
mem(c_2Earithmetic_2ENUMERAL,arr(ty_2Enum_2Enum,ty_2Enum_2Enum)) ).
fof(mem_c_2Earithmetic_2E_3C_3D,axiom,
mem(c_2Earithmetic_2E_3C_3D,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,bool))) ).
fof(mem_c_2Earithmetic_2E_2B,axiom,
mem(c_2Earithmetic_2E_2B,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))) ).
fof(ne_ty_2Erealax_2Ereal,axiom,
ne(ty_2Erealax_2Ereal) ).
fof(mem_c_2Ereal_2E_2F,axiom,
mem(c_2Ereal_2E_2F,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) ).
fof(mem_c_2Ereal_2Ereal__sub,axiom,
mem(c_2Ereal_2Ereal__sub,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) ).
fof(mem_c_2Enum_2E0,axiom,
mem(c_2Enum_2E0,ty_2Enum_2Enum) ).
fof(mem_c_2Ereal_2Esup,axiom,
mem(c_2Ereal_2Esup,arr(arr(ty_2Erealax_2Ereal,bool),ty_2Erealax_2Ereal)) ).
fof(mem_c_2Erealax_2Ereal__neg,axiom,
mem(c_2Erealax_2Ereal__neg,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal)) ).
fof(mem_c_2Ereal_2Ereal__lte,axiom,
mem(c_2Ereal_2Ereal__lte,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,bool))) ).
fof(mem_c_2Erealax_2Ereal__add,axiom,
mem(c_2Erealax_2Ereal__add,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) ).
fof(mem_c_2Erealax_2Ereal__mul,axiom,
mem(c_2Erealax_2Ereal__mul,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal))) ).
fof(mem_c_2Ereal_2Ereal__of__num,axiom,
mem(c_2Ereal_2Ereal__of__num,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal)) ).
fof(mem_c_2Erealax_2Ereal__lt,axiom,
mem(c_2Erealax_2Ereal__lt,arr(ty_2Erealax_2Ereal,arr(ty_2Erealax_2Ereal,bool))) ).
fof(mem_c_2Ebool_2EF,axiom,
mem(c_2Ebool_2EF,bool) ).
fof(ax_false_p,axiom,
~ p(c_2Ebool_2EF) ).
fof(mem_c_2Ebool_2E_5C_2F,axiom,
mem(c_2Ebool_2E_5C_2F,arr(bool,arr(bool,bool))) ).
fof(ax_or_p,axiom,
! [Q] :
( mem(Q,bool)
=> ! [R] :
( mem(R,bool)
=> ( p(ap(ap(c_2Ebool_2E_5C_2F,Q),R))
<=> ( p(Q)
| p(R) ) ) ) ) ).
fof(mem_c_2Emin_2E_3D,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Emin_2E_3D(A_27a),arr(A_27a,arr(A_27a,bool))) ) ).
fof(ax_eq_p,axiom,
! [A] :
( ne(A)
=> ! [X] :
( mem(X,A)
=> ! [Y] :
( mem(Y,A)
=> ( p(ap(ap(c_2Emin_2E_3D(A),X),Y))
<=> X = Y ) ) ) ) ).
fof(mem_c_2Ebool_2E_7E,axiom,
mem(c_2Ebool_2E_7E,arr(bool,bool)) ).
fof(ax_neg_p,axiom,
! [Q] :
( mem(Q,bool)
=> ( p(ap(c_2Ebool_2E_7E,Q))
<=> ~ p(Q) ) ) ).
fof(mem_c_2Eprim__rec_2E_3C,axiom,
mem(c_2Eprim__rec_2E_3C,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,bool))) ).
fof(mem_c_2Ewhile_2ELEAST,axiom,
mem(c_2Ewhile_2ELEAST,arr(arr(ty_2Enum_2Enum,bool),ty_2Enum_2Enum)) ).
fof(mem_c_2Ebool_2E_2F_5C,axiom,
mem(c_2Ebool_2E_2F_5C,arr(bool,arr(bool,bool))) ).
fof(ax_and_p,axiom,
! [Q] :
( mem(Q,bool)
=> ! [R] :
( mem(R,bool)
=> ( p(ap(ap(c_2Ebool_2E_2F_5C,Q),R))
<=> ( p(Q)
& p(R) ) ) ) ) ).
fof(mem_c_2Ebool_2E_3F,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ebool_2E_3F(A_27a),arr(arr(A_27a,bool),bool)) ) ).
fof(ax_ex_p,axiom,
! [A] :
( ne(A)
=> ! [Q] :
( mem(Q,arr(A,bool))
=> ( p(ap(c_2Ebool_2E_3F(A),Q))
<=> ? [X] :
( mem(X,A)
& p(ap(Q,X)) ) ) ) ) ).
fof(mem_c_2Emin_2E_3D_3D_3E,axiom,
mem(c_2Emin_2E_3D_3D_3E,arr(bool,arr(bool,bool))) ).
fof(ax_imp_p,axiom,
! [Q] :
( mem(Q,bool)
=> ! [R] :
( mem(R,bool)
=> ( p(ap(ap(c_2Emin_2E_3D_3D_3E,Q),R))
<=> ( p(Q)
=> p(R) ) ) ) ) ).
fof(mem_c_2Ebool_2E_21,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ebool_2E_21(A_27a),arr(arr(A_27a,bool),bool)) ) ).
fof(ax_all_p,axiom,
! [A] :
( ne(A)
=> ! [Q] :
( mem(Q,arr(A,bool))
=> ( p(ap(c_2Ebool_2E_21(A),Q))
<=> ! [X] :
( mem(X,A)
=> p(ap(Q,X)) ) ) ) ) ).
fof(conj_thm_2Earithmetic_2Enum__CASES,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ( V0m = c_2Enum_2E0
| ? [V1n] :
( mem(V1n,ty_2Enum_2Enum)
& V0m = ap(c_2Enum_2ESUC,V1n) ) ) ) ).
fof(conj_thm_2Earithmetic_2ELESS__EQ__SUC__REFL,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,V0m),ap(c_2Enum_2ESUC,V0m))) ) ).
fof(conj_thm_2Earithmetic_2EADD1,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ap(c_2Enum_2ESUC,V0m) = ap(ap(c_2Earithmetic_2E_2B,V0m),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ) ).
fof(ax_thm_2Ebool_2EBOOL__CASES__AX,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ( p(V0t)
<=> $true )
| ( p(V0t)
<=> $false ) ) ) ).
fof(conj_thm_2Ebool_2ETRUTH,axiom,
$true ).
fof(conj_thm_2Ebool_2EIMP__ANTISYM__AX,axiom,
! [V0t1] :
( mem(V0t1,bool)
=> ! [V1t2] :
( mem(V1t2,bool)
=> ( ( p(V0t1)
=> p(V1t2) )
=> ( ( p(V1t2)
=> p(V0t1) )
=> ( p(V0t1)
<=> p(V1t2) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EFALSITY,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( $false
=> p(V0t) ) ) ).
fof(conj_thm_2Ebool_2EEXCLUDED__MIDDLE,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( p(V0t)
| ~ p(V0t) ) ) ).
fof(conj_thm_2Ebool_2EIMP__F,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ( p(V0t)
=> $false )
=> ~ p(V0t) ) ) ).
fof(conj_thm_2Ebool_2EF__IMP,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ~ p(V0t)
=> ( p(V0t)
=> $false ) ) ) ).
fof(conj_thm_2Ebool_2EAND__CLAUSES,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ( ( $true
& p(V0t) )
<=> p(V0t) )
& ( ( p(V0t)
& $true )
<=> p(V0t) )
& ( ( $false
& p(V0t) )
<=> $false )
& ( ( p(V0t)
& $false )
<=> $false )
& ( ( p(V0t)
& p(V0t) )
<=> p(V0t) ) ) ) ).
fof(conj_thm_2Ebool_2EOR__CLAUSES,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ( ( $true
| p(V0t) )
<=> $true )
& ( ( p(V0t)
| $true )
<=> $true )
& ( ( $false
| p(V0t) )
<=> p(V0t) )
& ( ( p(V0t)
| $false )
<=> p(V0t) )
& ( ( p(V0t)
| p(V0t) )
<=> p(V0t) ) ) ) ).
fof(conj_thm_2Ebool_2EIMP__CLAUSES,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ( ( $true
=> p(V0t) )
<=> p(V0t) )
& ( ( p(V0t)
=> $true )
<=> $true )
& ( ( $false
=> p(V0t) )
<=> $true )
& ( ( p(V0t)
=> p(V0t) )
<=> $true )
& ( ( p(V0t)
=> $false )
<=> ~ p(V0t) ) ) ) ).
fof(conj_thm_2Ebool_2ENOT__CLAUSES,axiom,
( ! [V0t] :
( mem(V0t,bool)
=> ( ~ ~ p(V0t)
<=> p(V0t) ) )
& ( ~ $true
<=> $false )
& ( ~ $false
<=> $true ) ) ).
fof(conj_thm_2Ebool_2EREFL__CLAUSE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ( V0x = V0x
<=> $true ) ) ) ).
fof(conj_thm_2Ebool_2EEQ__SYM__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27a)
=> ( V0x = V1y
<=> V1y = V0x ) ) ) ) ).
fof(conj_thm_2Ebool_2EEQ__CLAUSES,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ( ( $true
<=> p(V0t) )
<=> p(V0t) )
& ( ( p(V0t)
<=> $true )
<=> p(V0t) )
& ( ( $false
<=> p(V0t) )
<=> ~ p(V0t) )
& ( ( p(V0t)
<=> $false )
<=> ~ p(V0t) ) ) ) ).
fof(conj_thm_2Ebool_2ENOT__EXISTS__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ( ~ ? [V1x] :
( mem(V1x,A_27a)
& p(ap(V0P,V1x)) )
<=> ! [V2x] :
( mem(V2x,A_27a)
=> ~ p(ap(V0P,V2x)) ) ) ) ) ).
fof(conj_thm_2Ebool_2EDISJ__ASSOC,axiom,
! [V0A] :
( mem(V0A,bool)
=> ! [V1B] :
( mem(V1B,bool)
=> ! [V2C] :
( mem(V2C,bool)
=> ( ( p(V0A)
| p(V1B)
| p(V2C) )
<=> ( p(V0A)
| p(V1B)
| p(V2C) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EDISJ__SYM,axiom,
! [V0A] :
( mem(V0A,bool)
=> ! [V1B] :
( mem(V1B,bool)
=> ( ( p(V0A)
| p(V1B) )
<=> ( p(V1B)
| p(V0A) ) ) ) ) ).
fof(conj_thm_2Ebool_2EDE__MORGAN__THM,axiom,
! [V0A] :
( mem(V0A,bool)
=> ! [V1B] :
( mem(V1B,bool)
=> ( ( ~ ( p(V0A)
& p(V1B) )
<=> ( ~ p(V0A)
| ~ p(V1B) ) )
& ( ~ ( p(V0A)
| p(V1B) )
<=> ( ~ p(V0A)
& ~ p(V1B) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EIMP__DISJ__THM,axiom,
! [V0A] :
( mem(V0A,bool)
=> ! [V1B] :
( mem(V1B,bool)
=> ( ( p(V0A)
=> p(V1B) )
<=> ( ~ p(V0A)
| p(V1B) ) ) ) ) ).
fof(conj_thm_2Ebool_2EAND__IMP__INTRO,axiom,
! [V0t1] :
( mem(V0t1,bool)
=> ! [V1t2] :
( mem(V1t2,bool)
=> ! [V2t3] :
( mem(V2t3,bool)
=> ( ( p(V0t1)
=> ( p(V1t2)
=> p(V2t3) ) )
<=> ( ( p(V0t1)
& p(V1t2) )
=> p(V2t3) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EIMP__CONG,axiom,
! [V0x] :
( mem(V0x,bool)
=> ! [V1x_27] :
( mem(V1x_27,bool)
=> ! [V2y] :
( mem(V2y,bool)
=> ! [V3y_27] :
( mem(V3y_27,bool)
=> ( ( ( p(V0x)
<=> p(V1x_27) )
& ( p(V1x_27)
=> ( p(V2y)
<=> p(V3y_27) ) ) )
=> ( ( p(V0x)
=> p(V2y) )
<=> ( p(V1x_27)
=> p(V3y_27) ) ) ) ) ) ) ) ).
fof(conj_thm_2Eprim__rec_2ELESS__SUC__REFL,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Eprim__rec_2E_3C,V0n),ap(c_2Enum_2ESUC,V0n))) ) ).
fof(conj_thm_2Ereal_2EREAL__ADD__SYM,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ap(ap(c_2Erealax_2Ereal__add,V0x),V1y) = ap(ap(c_2Erealax_2Ereal__add,V1y),V0x) ) ) ).
fof(conj_thm_2Ereal_2EREAL__ADD__ASSOC,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ! [V2z] :
( mem(V2z,ty_2Erealax_2Ereal)
=> ap(ap(c_2Erealax_2Ereal__add,V0x),ap(ap(c_2Erealax_2Ereal__add,V1y),V2z)) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Erealax_2Ereal__add,V0x),V1y)),V2z) ) ) ) ).
fof(conj_thm_2Ereal_2EREAL__MUL__LID,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),V0x) = V0x ) ).
fof(ax_thm_2Ereal_2Ereal__sub,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ap(ap(c_2Ereal_2Ereal__sub,V0x),V1y) = ap(ap(c_2Erealax_2Ereal__add,V0x),ap(c_2Erealax_2Ereal__neg,V1y)) ) ) ).
fof(conj_thm_2Ereal_2EREAL__EQ__LADD,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ! [V2z] :
( mem(V2z,ty_2Erealax_2Ereal)
=> ( ap(ap(c_2Erealax_2Ereal__add,V0x),V1y) = ap(ap(c_2Erealax_2Ereal__add,V0x),V2z)
<=> V1y = V2z ) ) ) ) ).
fof(conj_thm_2Ereal_2EREAL__NEG__ADD,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ap(c_2Erealax_2Ereal__neg,ap(ap(c_2Erealax_2Ereal__add,V0x),V1y)) = ap(ap(c_2Erealax_2Ereal__add,ap(c_2Erealax_2Ereal__neg,V0x)),ap(c_2Erealax_2Ereal__neg,V1y)) ) ) ).
fof(conj_thm_2Ereal_2EREAL__MUL__LZERO,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V0x) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ).
fof(conj_thm_2Ereal_2EREAL__NEGNEG,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ap(c_2Erealax_2Ereal__neg,ap(c_2Erealax_2Ereal__neg,V0x)) = V0x ) ).
fof(conj_thm_2Ereal_2EREAL__NOT__LT,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ( ~ p(ap(ap(c_2Erealax_2Ereal__lt,V0x),V1y))
<=> p(ap(ap(c_2Ereal_2Ereal__lte,V1y),V0x)) ) ) ) ).
fof(conj_thm_2Ereal_2EREAL__LT__LE,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,V0x),V1y))
<=> ( p(ap(ap(c_2Ereal_2Ereal__lte,V0x),V1y))
& V0x != V1y ) ) ) ) ).
fof(conj_thm_2Ereal_2EREAL__LE__TRANS,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ! [V2z] :
( mem(V2z,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Ereal_2Ereal__lte,V0x),V1y))
& p(ap(ap(c_2Ereal_2Ereal__lte,V1y),V2z)) )
=> p(ap(ap(c_2Ereal_2Ereal__lte,V0x),V2z)) ) ) ) ) ).
fof(conj_thm_2Ereal_2EREAL__LE__RADD,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ! [V2z] :
( mem(V2z,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Erealax_2Ereal__add,V0x),V2z)),ap(ap(c_2Erealax_2Ereal__add,V1y),V2z)))
<=> p(ap(ap(c_2Ereal_2Ereal__lte,V0x),V1y)) ) ) ) ) ).
fof(conj_thm_2Ereal_2EREAL__EQ__RMUL,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ! [V2z] :
( mem(V2z,ty_2Erealax_2Ereal)
=> ( ap(ap(c_2Erealax_2Ereal__mul,V0x),V2z) = ap(ap(c_2Erealax_2Ereal__mul,V1y),V2z)
<=> ( V2z = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)
| V0x = V1y ) ) ) ) ) ).
fof(conj_thm_2Ereal_2EREAL__LE,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,V0m)),ap(c_2Ereal_2Ereal__of__num,V1n)))
<=> p(ap(ap(c_2Earithmetic_2E_3C_3D,V0m),V1n)) ) ) ) ).
fof(conj_thm_2Ereal_2EREAL__INJ,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( ap(c_2Ereal_2Ereal__of__num,V0m) = ap(c_2Ereal_2Ereal__of__num,V1n)
<=> V0m = V1n ) ) ) ).
fof(conj_thm_2Ereal_2EREAL__ADD,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ap(ap(c_2Erealax_2Ereal__add,ap(c_2Ereal_2Ereal__of__num,V0m)),ap(c_2Ereal_2Ereal__of__num,V1n)) = ap(c_2Ereal_2Ereal__of__num,ap(ap(c_2Earithmetic_2E_2B,V0m),V1n)) ) ) ).
fof(conj_thm_2Ereal_2EREAL__DIV__RMUL,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ( V1y != ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)
=> ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Ereal_2E_2F,V0x),V1y)),V1y) = V0x ) ) ) ).
fof(conj_thm_2Ereal_2EREAL__LE__SUB__RADD,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ! [V2z] :
( mem(V2z,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Ereal_2Ereal__sub,V0x),V1y)),V2z))
<=> p(ap(ap(c_2Ereal_2Ereal__lte,V0x),ap(ap(c_2Erealax_2Ereal__add,V2z),V1y))) ) ) ) ) ).
fof(conj_thm_2Ereal_2EREAL__SUB__RZERO,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ap(ap(c_2Ereal_2Ereal__sub,V0x),ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)) = V0x ) ).
fof(conj_thm_2Ereal_2EREAL__EQ__SUB__LADD,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ! [V2z] :
( mem(V2z,ty_2Erealax_2Ereal)
=> ( V0x = ap(ap(c_2Ereal_2Ereal__sub,V1y),V2z)
<=> ap(ap(c_2Erealax_2Ereal__add,V0x),V2z) = V1y ) ) ) ) ).
fof(conj_thm_2Ereal_2EREAL__LE__RMUL,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ! [V2z] :
( mem(V2z,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2z))
=> ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Erealax_2Ereal__mul,V0x),V2z)),ap(ap(c_2Erealax_2Ereal__mul,V1y),V2z)))
<=> p(ap(ap(c_2Ereal_2Ereal__lte,V0x),V1y)) ) ) ) ) ) ).
fof(conj_thm_2Ereal_2EREAL__EQ__NEG,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ( ap(c_2Erealax_2Ereal__neg,V0x) = ap(c_2Erealax_2Ereal__neg,V1y)
<=> V0x = V1y ) ) ) ).
fof(conj_thm_2Ereal_2EREAL__SUP__LE,axiom,
! [V0P] :
( mem(V0P,arr(ty_2Erealax_2Ereal,bool))
=> ( ( ? [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
& p(ap(V0P,V1x)) )
& ? [V2z] :
( mem(V2z,ty_2Erealax_2Ereal)
& ! [V3x] :
( mem(V3x,ty_2Erealax_2Ereal)
=> ( p(ap(V0P,V3x))
=> p(ap(ap(c_2Ereal_2Ereal__lte,V3x),V2z)) ) ) ) )
=> ! [V4y] :
( mem(V4y,ty_2Erealax_2Ereal)
=> ( ? [V5x] :
( mem(V5x,ty_2Erealax_2Ereal)
& p(ap(V0P,V5x))
& p(ap(ap(c_2Erealax_2Ereal__lt,V4y),V5x)) )
<=> p(ap(ap(c_2Erealax_2Ereal__lt,V4y),ap(c_2Ereal_2Esup,V0P))) ) ) ) ) ).
fof(conj_thm_2Ereal_2EREAL__MUL__LNEG,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ap(ap(c_2Erealax_2Ereal__mul,ap(c_2Erealax_2Ereal__neg,V0x)),V1y) = ap(c_2Erealax_2Ereal__neg,ap(ap(c_2Erealax_2Ereal__mul,V0x),V1y)) ) ) ).
fof(conj_thm_2Ereal_2Ereal__lt,axiom,
! [V0y] :
( mem(V0y,ty_2Erealax_2Ereal)
=> ! [V1x] :
( mem(V1x,ty_2Erealax_2Ereal)
=> ( p(ap(ap(c_2Erealax_2Ereal__lt,V1x),V0y))
<=> ~ p(ap(ap(c_2Ereal_2Ereal__lte,V0y),V1x)) ) ) ) ).
fof(conj_thm_2Ereal_2EREAL__ADD__RDISTRIB,axiom,
! [V0x] :
( mem(V0x,ty_2Erealax_2Ereal)
=> ! [V1y] :
( mem(V1y,ty_2Erealax_2Ereal)
=> ! [V2z] :
( mem(V2z,ty_2Erealax_2Ereal)
=> ap(ap(c_2Erealax_2Ereal__mul,ap(ap(c_2Erealax_2Ereal__add,V0x),V1y)),V2z) = ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Erealax_2Ereal__mul,V0x),V2z)),ap(ap(c_2Erealax_2Ereal__mul,V1y),V2z)) ) ) ) ).
fof(conj_thm_2Ereal_2EREAL__BIGNUM,axiom,
! [V0r] :
( mem(V0r,ty_2Erealax_2Ereal)
=> ? [V1n] :
( mem(V1n,ty_2Enum_2Enum)
& p(ap(ap(c_2Erealax_2Ereal__lt,V0r),ap(c_2Ereal_2Ereal__of__num,V1n))) ) ) ).
fof(conj_thm_2Esat_2ENOT__NOT,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ~ ~ p(V0t)
<=> p(V0t) ) ) ).
fof(conj_thm_2Esat_2EAND__INV__IMP,axiom,
! [V0A] :
( mem(V0A,bool)
=> ( p(V0A)
=> ( ~ p(V0A)
=> $false ) ) ) ).
fof(conj_thm_2Esat_2EOR__DUAL2,axiom,
! [V0A] :
( mem(V0A,bool)
=> ! [V1B] :
( mem(V1B,bool)
=> ( ( ~ ( p(V0A)
| p(V1B) )
=> $false )
<=> ( ( p(V0A)
=> $false )
=> ( ~ p(V1B)
=> $false ) ) ) ) ) ).
fof(conj_thm_2Esat_2EOR__DUAL3,axiom,
! [V0A] :
( mem(V0A,bool)
=> ! [V1B] :
( mem(V1B,bool)
=> ( ( ~ ( ~ p(V0A)
| p(V1B) )
=> $false )
<=> ( p(V0A)
=> ( ~ p(V1B)
=> $false ) ) ) ) ) ).
fof(conj_thm_2Esat_2EAND__INV2,axiom,
! [V0A] :
( mem(V0A,bool)
=> ( ( ~ p(V0A)
=> $false )
=> ( ( p(V0A)
=> $false )
=> $false ) ) ) ).
fof(conj_thm_2Esat_2Edc__eq,axiom,
! [V0p] :
( mem(V0p,bool)
=> ! [V1q] :
( mem(V1q,bool)
=> ! [V2r] :
( mem(V2r,bool)
=> ( ( p(V0p)
<=> ( p(V1q)
<=> p(V2r) ) )
<=> ( ( p(V0p)
| p(V1q)
| p(V2r) )
& ( p(V0p)
| ~ p(V2r)
| ~ p(V1q) )
& ( p(V1q)
| ~ p(V2r)
| ~ p(V0p) )
& ( p(V2r)
| ~ p(V1q)
| ~ p(V0p) ) ) ) ) ) ) ).
fof(conj_thm_2Esat_2Edc__conj,axiom,
! [V0p] :
( mem(V0p,bool)
=> ! [V1q] :
( mem(V1q,bool)
=> ! [V2r] :
( mem(V2r,bool)
=> ( ( p(V0p)
<=> ( p(V1q)
& p(V2r) ) )
<=> ( ( p(V0p)
| ~ p(V1q)
| ~ p(V2r) )
& ( p(V1q)
| ~ p(V0p) )
& ( p(V2r)
| ~ p(V0p) ) ) ) ) ) ) ).
fof(conj_thm_2Esat_2Edc__disj,axiom,
! [V0p] :
( mem(V0p,bool)
=> ! [V1q] :
( mem(V1q,bool)
=> ! [V2r] :
( mem(V2r,bool)
=> ( ( p(V0p)
<=> ( p(V1q)
| p(V2r) ) )
<=> ( ( p(V0p)
| ~ p(V1q) )
& ( p(V0p)
| ~ p(V2r) )
& ( p(V1q)
| p(V2r)
| ~ p(V0p) ) ) ) ) ) ) ).
fof(conj_thm_2Esat_2Edc__imp,axiom,
! [V0p] :
( mem(V0p,bool)
=> ! [V1q] :
( mem(V1q,bool)
=> ! [V2r] :
( mem(V2r,bool)
=> ( ( p(V0p)
<=> ( p(V1q)
=> p(V2r) ) )
<=> ( ( p(V0p)
| p(V1q) )
& ( p(V0p)
| ~ p(V2r) )
& ( ~ p(V1q)
| p(V2r)
| ~ p(V0p) ) ) ) ) ) ) ).
fof(conj_thm_2Esat_2Edc__neg,axiom,
! [V0p] :
( mem(V0p,bool)
=> ! [V1q] :
( mem(V1q,bool)
=> ( ( p(V0p)
<=> ~ p(V1q) )
<=> ( ( p(V0p)
| p(V1q) )
& ( ~ p(V1q)
| ~ p(V0p) ) ) ) ) ) ).
fof(conj_thm_2Ewhile_2ELEAST__EXISTS__IMP,axiom,
! [V0p] :
( mem(V0p,arr(ty_2Enum_2Enum,bool))
=> ( ? [V1n] :
( mem(V1n,ty_2Enum_2Enum)
& p(ap(V0p,V1n)) )
=> ( p(ap(V0p,ap(c_2Ewhile_2ELEAST,V0p)))
& ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Eprim__rec_2E_3C,V2n),ap(c_2Ewhile_2ELEAST,V0p)))
=> ~ p(ap(V0p,V2n)) ) ) ) ) ) ).
fof(conj_thm_2Ereal_2ESUP__EPSILON,conjecture,
! [V0p] :
( mem(V0p,arr(ty_2Erealax_2Ereal,bool))
=> ! [V1e] :
( mem(V1e,ty_2Erealax_2Ereal)
=> ( ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1e))
& ? [V2x] :
( mem(V2x,ty_2Erealax_2Ereal)
& p(ap(V0p,V2x)) )
& ? [V3z] :
( mem(V3z,ty_2Erealax_2Ereal)
& ! [V4x] :
( mem(V4x,ty_2Erealax_2Ereal)
=> ( p(ap(V0p,V4x))
=> p(ap(ap(c_2Ereal_2Ereal__lte,V4x),V3z)) ) ) ) )
=> ? [V5x] :
( mem(V5x,ty_2Erealax_2Ereal)
& p(ap(V0p,V5x))
& p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Esup,V0p)),ap(ap(c_2Erealax_2Ereal__add,V5x),V1e))) ) ) ) ) ).
%------------------------------------------------------------------------------