TPTP Problem File: ITP016+1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP016+1 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 syntactic 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+1.p [TPAP]
% Status : Theorem
% Rating : 1.00 v7.5.0
% Syntax : Number of formulae : 102 ( 42 unt; 0 def)
% Number of atoms : 312 ( 73 equ)
% Maximal formula atoms : 15 ( 3 avg)
% Number of connectives : 265 ( 55 ~; 46 |; 48 &)
% ( 67 <=>; 49 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 5 avg)
% Maximal term depth : 11 ( 2 avg)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 59 ( 59 usr; 33 con; 0-2 aty)
% Number of variables : 210 ( 199 !; 11 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
fof(reserved_2Eho_2Eeq__ext,axiom,
! [A_27a,A_27b,V0f_2E0,V1g_2E0] :
( ! [V2x_2E0] : s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V0f_2E0),s(A_27a,V2x_2E0))) = s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0),s(A_27a,V2x_2E0)))
=> s(tyop_2Emin_2Efun(A_27a,A_27b),V0f_2E0) = s(tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0) ) ).
fof(reserved_2Eho_2Eboolext,axiom,
! [V0_2E0,V1_2E0] :
( ( p(s(tyop_2Emin_2Ebool,V0_2E0))
<=> p(s(tyop_2Emin_2Ebool,V1_2E0)) )
=> s(tyop_2Emin_2Ebool,V0_2E0) = s(tyop_2Emin_2Ebool,V1_2E0) ) ).
fof(reserved_2Eho_2Etruth,axiom,
p(s(tyop_2Emin_2Ebool,c_2Ebool_2ET_2E0)) ).
fof(reserved_2Eho_2Enotfalse,axiom,
~ p(s(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0)) ).
fof(reserved_2Eho_2Ebool__cases__ax,axiom,
! [V0t_2E0] :
( s(tyop_2Emin_2Ebool,V0t_2E0) = s(tyop_2Emin_2Ebool,c_2Ebool_2ET_2E0)
| s(tyop_2Emin_2Ebool,V0t_2E0) = s(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0) ) ).
fof(reserved_2Eho_2Ei__thm,axiom,
! [A_27a,V0x_2E0] : s(A_27a,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27a),combin_i_2E0),s(A_27a,V0x_2E0))) = s(A_27a,V0x_2E0) ).
fof(reserved_2Eho_2Ek__thm,axiom,
! [A_27a,A_27b,V0x_2E0,V1y_2E0] : s(A_27a,app_2E2(s(tyop_2Emin_2Efun(A_27b,A_27a),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27a)),combin_k_2E0),s(A_27a,V0x_2E0))),s(A_27b,V1y_2E0))) = s(A_27a,V0x_2E0) ).
fof(reserved_2Eho_2Es__thm,axiom,
! [A_27a,A_27b,A_27c,V0f_2E0,V1g_2E0,V2x_2E0] : s(A_27c,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27c),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c))),combin_s_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),V0f_2E0))),s(tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0))),s(A_27a,V2x_2E0))) = s(A_27c,app_2E2(s(tyop_2Emin_2Efun(A_27b,A_27c),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),V0f_2E0),s(A_27a,V2x_2E0))),s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0),s(A_27a,V2x_2E0))))) ).
fof(reserved_2Elogic_2E_2F_5C,axiom,
! [V0_2E0,V1_2E0] :
( p(s(tyop_2Emin_2Ebool,c_2Ebool_2E_2F_5C_2E2(s(tyop_2Emin_2Ebool,V0_2E0),s(tyop_2Emin_2Ebool,V1_2E0))))
<=> ( p(s(tyop_2Emin_2Ebool,V0_2E0))
& p(s(tyop_2Emin_2Ebool,V1_2E0)) ) ) ).
fof(reserved_2Elogic_2E_5C_2F,axiom,
! [V0_2E0,V1_2E0] :
( p(s(tyop_2Emin_2Ebool,c_2Ebool_2E_5C_2F_2E2(s(tyop_2Emin_2Ebool,V0_2E0),s(tyop_2Emin_2Ebool,V1_2E0))))
<=> ( p(s(tyop_2Emin_2Ebool,V0_2E0))
| p(s(tyop_2Emin_2Ebool,V1_2E0)) ) ) ).
fof(reserved_2Elogic_2E_7E,axiom,
! [V0_2E0] :
( p(s(tyop_2Emin_2Ebool,c_2Ebool_2E_7E_2E1(s(tyop_2Emin_2Ebool,V0_2E0))))
<=> ~ p(s(tyop_2Emin_2Ebool,V0_2E0)) ) ).
fof(reserved_2Elogic_2E_3D_3D_3E,axiom,
! [V0_2E0,V1_2E0] :
( p(s(tyop_2Emin_2Ebool,c_2Emin_2E_3D_3D_3E_2E2(s(tyop_2Emin_2Ebool,V0_2E0),s(tyop_2Emin_2Ebool,V1_2E0))))
<=> ( p(s(tyop_2Emin_2Ebool,V0_2E0))
=> p(s(tyop_2Emin_2Ebool,V1_2E0)) ) ) ).
fof(reserved_2Elogic_2E_3D,axiom,
! [A_27a,V0_2E0,V1_2E0] :
( p(s(tyop_2Emin_2Ebool,c_2Emin_2E_3D_2E2(s(A_27a,V0_2E0),s(A_27a,V1_2E0))))
<=> s(A_27a,V0_2E0) = s(A_27a,V1_2E0) ) ).
fof(reserved_2Equant_2E_21,axiom,
! [A_27a,V0f_2E0] :
( p(s(tyop_2Emin_2Ebool,c_2Ebool_2E_21_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0))))
<=> ! [V1x_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0),s(A_27a,V1x_2E0)))) ) ).
fof(reserved_2Equant_2E_3F,axiom,
! [A_27a,V0f_2E0] :
( p(s(tyop_2Emin_2Ebool,c_2Ebool_2E_3F_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0))))
<=> ? [V1x_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0),s(A_27a,V1x_2E0)))) ) ).
fof(arityeq2_2Ec_2Ebool_2E_2F_5C_2E2,axiom,
! [X0_2E0,X1_2E0] :
( ( p(s(tyop_2Emin_2Ebool,X0_2E0))
& p(s(tyop_2Emin_2Ebool,X1_2E0)) )
<=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)),c_2Ebool_2E_2F_5C_2E0),s(tyop_2Emin_2Ebool,X0_2E0))),s(tyop_2Emin_2Ebool,X1_2E0)))) ) ).
fof(arityeq2_2Ec_2Ebool_2E_5C_2F_2E2,axiom,
! [X0_2E0,X1_2E0] :
( ( p(s(tyop_2Emin_2Ebool,X0_2E0))
| p(s(tyop_2Emin_2Ebool,X1_2E0)) )
<=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)),c_2Ebool_2E_5C_2F_2E0),s(tyop_2Emin_2Ebool,X0_2E0))),s(tyop_2Emin_2Ebool,X1_2E0)))) ) ).
fof(arityeq1_2Ec_2Ebool_2E_7E_2E1,axiom,
! [X0_2E0] :
( ~ p(s(tyop_2Emin_2Ebool,X0_2E0))
<=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2E_7E_2E0),s(tyop_2Emin_2Ebool,X0_2E0)))) ) ).
fof(arityeq2_2Ec_2Emin_2E_3D_3D_3E_2E2,axiom,
! [X0_2E0,X1_2E0] :
( ( p(s(tyop_2Emin_2Ebool,X0_2E0))
=> p(s(tyop_2Emin_2Ebool,X1_2E0)) )
<=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)),c_2Emin_2E_3D_3D_3E_2E0),s(tyop_2Emin_2Ebool,X0_2E0))),s(tyop_2Emin_2Ebool,X1_2E0)))) ) ).
fof(arityeq2_2Ec_2Emin_2E_3D_2E2_2Emono_2EA_27a,axiom,
! [A_27a,X0_2E0,X1_2E0] :
( s(A_27a,X0_2E0) = s(A_27a,X1_2E0)
<=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Emin_2E_3D_2E0),s(A_27a,X0_2E0))),s(A_27a,X1_2E0)))) ) ).
fof(arityeq1_2Ec_2Ebool_2E_21_2E1_2Emono_2EA_27a,axiom,
! [A_27a,X0_2E0] : s(tyop_2Emin_2Ebool,c_2Ebool_2E_21_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ebool_2E_21_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0))) ).
fof(arityeq1_2Ec_2Ebool_2E_3F_2E1_2Emono_2EA_27a,axiom,
! [A_27a,X0_2E0] : s(tyop_2Emin_2Ebool,c_2Ebool_2E_3F_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ebool_2E_3F_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0))) ).
fof(arityeq1_2Ec_2Enum_2ESUC_2E1,axiom,
! [X0_2E0] : s(tyop_2Enum_2Enum,c_2Enum_2ESUC_2E1(s(tyop_2Enum_2Enum,X0_2E0))) = s(tyop_2Enum_2Enum,app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Enum_2ESUC_2E0),s(tyop_2Enum_2Enum,X0_2E0))) ).
fof(arityeq1_2Ec_2Erealax_2Ereal__neg_2E1,axiom,
! [X0_2E0] : s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E1(s(tyop_2Erealax_2Ereal,X0_2E0))) = s(tyop_2Erealax_2Ereal,app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Erealax_2Ereal__neg_2E0),s(tyop_2Erealax_2Ereal,X0_2E0))) ).
fof(arityeq2_2Ec_2Erealax_2Ereal__add_2E2,axiom,
! [X0_2E0,X1_2E0] : s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(s(tyop_2Erealax_2Ereal,X0_2E0),s(tyop_2Erealax_2Ereal,X1_2E0))) = s(tyop_2Erealax_2Ereal,app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),c_2Erealax_2Ereal__add_2E0),s(tyop_2Erealax_2Ereal,X0_2E0))),s(tyop_2Erealax_2Ereal,X1_2E0))) ).
fof(arityeq2_2Ec_2Erealax_2Ereal__mul_2E2,axiom,
! [X0_2E0,X1_2E0] : s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(s(tyop_2Erealax_2Ereal,X0_2E0),s(tyop_2Erealax_2Ereal,X1_2E0))) = s(tyop_2Erealax_2Ereal,app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),c_2Erealax_2Ereal__mul_2E0),s(tyop_2Erealax_2Ereal,X0_2E0))),s(tyop_2Erealax_2Ereal,X1_2E0))) ).
fof(arityeq1_2Ec_2Ereal_2Ereal__of__num_2E1,axiom,
! [X0_2E0] : s(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(s(tyop_2Enum_2Enum,X0_2E0))) = s(tyop_2Erealax_2Ereal,app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal),c_2Ereal_2Ereal__of__num_2E0),s(tyop_2Enum_2Enum,X0_2E0))) ).
fof(arityeq2_2Ec_2Earithmetic_2E_2B_2E2,axiom,
! [X0_2E0,X1_2E0] : s(tyop_2Enum_2Enum,c_2Earithmetic_2E_2B_2E2(s(tyop_2Enum_2Enum,X0_2E0),s(tyop_2Enum_2Enum,X1_2E0))) = s(tyop_2Enum_2Enum,app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)),c_2Earithmetic_2E_2B_2E0),s(tyop_2Enum_2Enum,X0_2E0))),s(tyop_2Enum_2Enum,X1_2E0))) ).
fof(arityeq1_2Ec_2Ewhile_2ELEAST_2E1,axiom,
! [X0_2E0] : s(tyop_2Enum_2Enum,c_2Ewhile_2ELEAST_2E1(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X0_2E0))) = s(tyop_2Enum_2Enum,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),tyop_2Enum_2Enum),c_2Ewhile_2ELEAST_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),X0_2E0))) ).
fof(arityeq1_2Ec_2Earithmetic_2EBIT1_2E1,axiom,
! [X0_2E0] : s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,X0_2E0))) = s(tyop_2Enum_2Enum,app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2EBIT1_2E0),s(tyop_2Enum_2Enum,X0_2E0))) ).
fof(arityeq1_2Ec_2Earithmetic_2ENUMERAL_2E1,axiom,
! [X0_2E0] : s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,X0_2E0))) = s(tyop_2Enum_2Enum,app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2ENUMERAL_2E0),s(tyop_2Enum_2Enum,X0_2E0))) ).
fof(arityeq2_2Ec_2Eprim__rec_2E_3C_2E2,axiom,
! [X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Eprim__rec_2E_3C_2E2(s(tyop_2Enum_2Enum,X0_2E0),s(tyop_2Enum_2Enum,X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Eprim__rec_2E_3C_2E0),s(tyop_2Enum_2Enum,X0_2E0))),s(tyop_2Enum_2Enum,X1_2E0))) ).
fof(arityeq2_2Ec_2Earithmetic_2E_3C_3D_2E2,axiom,
! [X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Earithmetic_2E_3C_3D_2E2(s(tyop_2Enum_2Enum,X0_2E0),s(tyop_2Enum_2Enum,X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool)),c_2Earithmetic_2E_3C_3D_2E0),s(tyop_2Enum_2Enum,X0_2E0))),s(tyop_2Enum_2Enum,X1_2E0))) ).
fof(arityeq2_2Ec_2Erealax_2Ereal__lt_2E2,axiom,
! [X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Erealax_2Ereal__lt_2E2(s(tyop_2Erealax_2Ereal,X0_2E0),s(tyop_2Erealax_2Ereal,X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),c_2Erealax_2Ereal__lt_2E0),s(tyop_2Erealax_2Ereal,X0_2E0))),s(tyop_2Erealax_2Ereal,X1_2E0))) ).
fof(arityeq1_2Ec_2Ereal_2Esup_2E1,axiom,
! [X0_2E0] : s(tyop_2Erealax_2Ereal,c_2Ereal_2Esup_2E1(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X0_2E0))) = s(tyop_2Erealax_2Ereal,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal),c_2Ereal_2Esup_2E0),s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),X0_2E0))) ).
fof(arityeq2_2Ec_2Ereal_2Ereal__lte_2E2,axiom,
! [X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Ereal_2Ereal__lte_2E2(s(tyop_2Erealax_2Ereal,X0_2E0),s(tyop_2Erealax_2Ereal,X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),c_2Ereal_2Ereal__lte_2E0),s(tyop_2Erealax_2Ereal,X0_2E0))),s(tyop_2Erealax_2Ereal,X1_2E0))) ).
fof(arityeq2_2Ec_2Ereal_2Ereal__sub_2E2,axiom,
! [X0_2E0,X1_2E0] : s(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__sub_2E2(s(tyop_2Erealax_2Ereal,X0_2E0),s(tyop_2Erealax_2Ereal,X1_2E0))) = s(tyop_2Erealax_2Ereal,app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),c_2Ereal_2Ereal__sub_2E0),s(tyop_2Erealax_2Ereal,X0_2E0))),s(tyop_2Erealax_2Ereal,X1_2E0))) ).
fof(arityeq2_2Ec_2Ereal_2E_2F_2E2,axiom,
! [X0_2E0,X1_2E0] : s(tyop_2Erealax_2Ereal,c_2Ereal_2E_2F_2E2(s(tyop_2Erealax_2Ereal,X0_2E0),s(tyop_2Erealax_2Ereal,X1_2E0))) = s(tyop_2Erealax_2Ereal,app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),c_2Ereal_2E_2F_2E0),s(tyop_2Erealax_2Ereal,X0_2E0))),s(tyop_2Erealax_2Ereal,X1_2E0))) ).
fof(thm_2Earithmetic_2Enum__CASES,axiom,
! [V0m_2E0] :
( s(tyop_2Enum_2Enum,V0m_2E0) = s(tyop_2Enum_2Enum,c_2Enum_2E0_2E0)
| ? [V1n_2E0] : s(tyop_2Enum_2Enum,V0m_2E0) = s(tyop_2Enum_2Enum,c_2Enum_2ESUC_2E1(s(tyop_2Enum_2Enum,V1n_2E0))) ) ).
fof(thm_2Earithmetic_2ELESS__EQ__SUC__REFL,axiom,
! [V0m_2E0] : p(s(tyop_2Emin_2Ebool,c_2Earithmetic_2E_3C_3D_2E2(s(tyop_2Enum_2Enum,V0m_2E0),s(tyop_2Enum_2Enum,c_2Enum_2ESUC_2E1(s(tyop_2Enum_2Enum,V0m_2E0)))))) ).
fof(thm_2Earithmetic_2EADD1,axiom,
! [V0m_2E0] : s(tyop_2Enum_2Enum,c_2Enum_2ESUC_2E1(s(tyop_2Enum_2Enum,V0m_2E0))) = s(tyop_2Enum_2Enum,c_2Earithmetic_2E_2B_2E2(s(tyop_2Enum_2Enum,V0m_2E0),s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EZERO_2E0))))))) ).
fof(thm_2Ebool_2EBOOL__CASES__AX,axiom,
! [V0t_2E0] :
( s(tyop_2Emin_2Ebool,V0t_2E0) = s(tyop_2Emin_2Ebool,c_2Ebool_2ET_2E0)
| s(tyop_2Emin_2Ebool,V0t_2E0) = s(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0) ) ).
fof(thm_2Ebool_2ETRUTH,axiom,
p(s(tyop_2Emin_2Ebool,c_2Ebool_2ET_2E0)) ).
fof(thm_2Ebool_2EIMP__ANTISYM__AX,axiom,
! [V0t1_2E0,V1t2_2E0] :
( ( p(s(tyop_2Emin_2Ebool,V0t1_2E0))
=> p(s(tyop_2Emin_2Ebool,V1t2_2E0)) )
=> ( ( p(s(tyop_2Emin_2Ebool,V1t2_2E0))
=> p(s(tyop_2Emin_2Ebool,V0t1_2E0)) )
=> s(tyop_2Emin_2Ebool,V0t1_2E0) = s(tyop_2Emin_2Ebool,V1t2_2E0) ) ) ).
fof(thm_2Ebool_2EFALSITY,axiom,
! [V0t_2E0] :
( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0))
=> p(s(tyop_2Emin_2Ebool,V0t_2E0)) ) ).
fof(thm_2Ebool_2EEXCLUDED__MIDDLE,axiom,
! [V0t_2E0] :
( p(s(tyop_2Emin_2Ebool,V0t_2E0))
| ~ p(s(tyop_2Emin_2Ebool,V0t_2E0)) ) ).
fof(thm_2Ebool_2EIMP__F,axiom,
! [V0t_2E0] :
( ( p(s(tyop_2Emin_2Ebool,V0t_2E0))
=> p(s(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0)) )
=> ~ p(s(tyop_2Emin_2Ebool,V0t_2E0)) ) ).
fof(thm_2Ebool_2EF__IMP,axiom,
! [V0t_2E0] :
( ~ p(s(tyop_2Emin_2Ebool,V0t_2E0))
=> ( p(s(tyop_2Emin_2Ebool,V0t_2E0))
=> p(s(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0)) ) ) ).
fof(thm_2Ebool_2EAND__CLAUSES,axiom,
! [V0t_2E0] :
( ( ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2ET_2E0))
& p(s(tyop_2Emin_2Ebool,V0t_2E0)) )
<=> p(s(tyop_2Emin_2Ebool,V0t_2E0)) )
& ( ( p(s(tyop_2Emin_2Ebool,V0t_2E0))
& p(s(tyop_2Emin_2Ebool,c_2Ebool_2ET_2E0)) )
<=> p(s(tyop_2Emin_2Ebool,V0t_2E0)) )
& ( ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0))
& p(s(tyop_2Emin_2Ebool,V0t_2E0)) )
<=> p(s(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0)) )
& ( ( p(s(tyop_2Emin_2Ebool,V0t_2E0))
& p(s(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0)) )
<=> p(s(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0)) )
& ( ( p(s(tyop_2Emin_2Ebool,V0t_2E0))
& p(s(tyop_2Emin_2Ebool,V0t_2E0)) )
<=> p(s(tyop_2Emin_2Ebool,V0t_2E0)) ) ) ).
fof(thm_2Ebool_2EOR__CLAUSES,axiom,
! [V0t_2E0] :
( ( ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2ET_2E0))
| p(s(tyop_2Emin_2Ebool,V0t_2E0)) )
<=> p(s(tyop_2Emin_2Ebool,c_2Ebool_2ET_2E0)) )
& ( ( p(s(tyop_2Emin_2Ebool,V0t_2E0))
| p(s(tyop_2Emin_2Ebool,c_2Ebool_2ET_2E0)) )
<=> p(s(tyop_2Emin_2Ebool,c_2Ebool_2ET_2E0)) )
& ( ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0))
| p(s(tyop_2Emin_2Ebool,V0t_2E0)) )
<=> p(s(tyop_2Emin_2Ebool,V0t_2E0)) )
& ( ( p(s(tyop_2Emin_2Ebool,V0t_2E0))
| p(s(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0)) )
<=> p(s(tyop_2Emin_2Ebool,V0t_2E0)) )
& ( ( p(s(tyop_2Emin_2Ebool,V0t_2E0))
| p(s(tyop_2Emin_2Ebool,V0t_2E0)) )
<=> p(s(tyop_2Emin_2Ebool,V0t_2E0)) ) ) ).
fof(thm_2Ebool_2EIMP__CLAUSES,axiom,
! [V0t_2E0] :
( ( ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2ET_2E0))
=> p(s(tyop_2Emin_2Ebool,V0t_2E0)) )
<=> p(s(tyop_2Emin_2Ebool,V0t_2E0)) )
& ( ( p(s(tyop_2Emin_2Ebool,V0t_2E0))
=> p(s(tyop_2Emin_2Ebool,c_2Ebool_2ET_2E0)) )
<=> p(s(tyop_2Emin_2Ebool,c_2Ebool_2ET_2E0)) )
& ( ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0))
=> p(s(tyop_2Emin_2Ebool,V0t_2E0)) )
<=> p(s(tyop_2Emin_2Ebool,c_2Ebool_2ET_2E0)) )
& ( ( p(s(tyop_2Emin_2Ebool,V0t_2E0))
=> p(s(tyop_2Emin_2Ebool,V0t_2E0)) )
<=> p(s(tyop_2Emin_2Ebool,c_2Ebool_2ET_2E0)) )
& ( ( p(s(tyop_2Emin_2Ebool,V0t_2E0))
=> p(s(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0)) )
<=> ~ p(s(tyop_2Emin_2Ebool,V0t_2E0)) ) ) ).
fof(thm_2Ebool_2ENOT__CLAUSES,axiom,
( ! [V0t_2E0] :
( ~ ~ p(s(tyop_2Emin_2Ebool,V0t_2E0))
<=> p(s(tyop_2Emin_2Ebool,V0t_2E0)) )
& ( ~ p(s(tyop_2Emin_2Ebool,c_2Ebool_2ET_2E0))
<=> p(s(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0)) )
& ( ~ p(s(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0))
<=> p(s(tyop_2Emin_2Ebool,c_2Ebool_2ET_2E0)) ) ) ).
fof(thm_2Ebool_2EREFL__CLAUSE,axiom,
! [A_27a,V0x_2E0] :
( s(A_27a,V0x_2E0) = s(A_27a,V0x_2E0)
<=> p(s(tyop_2Emin_2Ebool,c_2Ebool_2ET_2E0)) ) ).
fof(thm_2Ebool_2EEQ__SYM__EQ,axiom,
! [A_27a,V0x_2E0,V1y_2E0] :
( s(A_27a,V0x_2E0) = s(A_27a,V1y_2E0)
<=> s(A_27a,V1y_2E0) = s(A_27a,V0x_2E0) ) ).
fof(thm_2Ebool_2EEQ__CLAUSES,axiom,
! [V0t_2E0] :
( ( s(tyop_2Emin_2Ebool,c_2Ebool_2ET_2E0) = s(tyop_2Emin_2Ebool,V0t_2E0)
<=> p(s(tyop_2Emin_2Ebool,V0t_2E0)) )
& ( s(tyop_2Emin_2Ebool,V0t_2E0) = s(tyop_2Emin_2Ebool,c_2Ebool_2ET_2E0)
<=> p(s(tyop_2Emin_2Ebool,V0t_2E0)) )
& ( s(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0) = s(tyop_2Emin_2Ebool,V0t_2E0)
<=> ~ p(s(tyop_2Emin_2Ebool,V0t_2E0)) )
& ( s(tyop_2Emin_2Ebool,V0t_2E0) = s(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0)
<=> ~ p(s(tyop_2Emin_2Ebool,V0t_2E0)) ) ) ).
fof(thm_2Ebool_2ENOT__EXISTS__THM,axiom,
! [A_27a,V0P_2E0] :
( ~ ? [V1x_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(A_27a,V1x_2E0))))
<=> ! [V2x_2E0] : ~ p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0P_2E0),s(A_27a,V2x_2E0)))) ) ).
fof(thm_2Ebool_2EDISJ__ASSOC,axiom,
! [V0A_2E0,V1B_2E0,V2C_2E0] :
( ( p(s(tyop_2Emin_2Ebool,V0A_2E0))
| p(s(tyop_2Emin_2Ebool,V1B_2E0))
| p(s(tyop_2Emin_2Ebool,V2C_2E0)) )
<=> ( p(s(tyop_2Emin_2Ebool,V0A_2E0))
| p(s(tyop_2Emin_2Ebool,V1B_2E0))
| p(s(tyop_2Emin_2Ebool,V2C_2E0)) ) ) ).
fof(thm_2Ebool_2EDISJ__SYM,axiom,
! [V0A_2E0,V1B_2E0] :
( ( p(s(tyop_2Emin_2Ebool,V0A_2E0))
| p(s(tyop_2Emin_2Ebool,V1B_2E0)) )
<=> ( p(s(tyop_2Emin_2Ebool,V1B_2E0))
| p(s(tyop_2Emin_2Ebool,V0A_2E0)) ) ) ).
fof(thm_2Ebool_2EDE__MORGAN__THM,axiom,
! [V0A_2E0,V1B_2E0] :
( ( ~ ( p(s(tyop_2Emin_2Ebool,V0A_2E0))
& p(s(tyop_2Emin_2Ebool,V1B_2E0)) )
<=> ( ~ p(s(tyop_2Emin_2Ebool,V0A_2E0))
| ~ p(s(tyop_2Emin_2Ebool,V1B_2E0)) ) )
& ( ~ ( p(s(tyop_2Emin_2Ebool,V0A_2E0))
| p(s(tyop_2Emin_2Ebool,V1B_2E0)) )
<=> ( ~ p(s(tyop_2Emin_2Ebool,V0A_2E0))
& ~ p(s(tyop_2Emin_2Ebool,V1B_2E0)) ) ) ) ).
fof(thm_2Ebool_2EIMP__DISJ__THM,axiom,
! [V0A_2E0,V1B_2E0] :
( ( p(s(tyop_2Emin_2Ebool,V0A_2E0))
=> p(s(tyop_2Emin_2Ebool,V1B_2E0)) )
<=> ( ~ p(s(tyop_2Emin_2Ebool,V0A_2E0))
| p(s(tyop_2Emin_2Ebool,V1B_2E0)) ) ) ).
fof(thm_2Ebool_2EAND__IMP__INTRO,axiom,
! [V0t1_2E0,V1t2_2E0,V2t3_2E0] :
( ( p(s(tyop_2Emin_2Ebool,V0t1_2E0))
=> ( p(s(tyop_2Emin_2Ebool,V1t2_2E0))
=> p(s(tyop_2Emin_2Ebool,V2t3_2E0)) ) )
<=> ( ( p(s(tyop_2Emin_2Ebool,V0t1_2E0))
& p(s(tyop_2Emin_2Ebool,V1t2_2E0)) )
=> p(s(tyop_2Emin_2Ebool,V2t3_2E0)) ) ) ).
fof(thm_2Ebool_2EIMP__CONG,axiom,
! [V0x_2E0,V1x_27_2E0,V2y_2E0,V3y_27_2E0] :
( ( s(tyop_2Emin_2Ebool,V0x_2E0) = s(tyop_2Emin_2Ebool,V1x_27_2E0)
& ( p(s(tyop_2Emin_2Ebool,V1x_27_2E0))
=> s(tyop_2Emin_2Ebool,V2y_2E0) = s(tyop_2Emin_2Ebool,V3y_27_2E0) ) )
=> ( ( p(s(tyop_2Emin_2Ebool,V0x_2E0))
=> p(s(tyop_2Emin_2Ebool,V2y_2E0)) )
<=> ( p(s(tyop_2Emin_2Ebool,V1x_27_2E0))
=> p(s(tyop_2Emin_2Ebool,V3y_27_2E0)) ) ) ) ).
fof(thm_2Eprim__rec_2ELESS__SUC__REFL,axiom,
! [V0n_2E0] : p(s(tyop_2Emin_2Ebool,c_2Eprim__rec_2E_3C_2E2(s(tyop_2Enum_2Enum,V0n_2E0),s(tyop_2Enum_2Enum,c_2Enum_2ESUC_2E1(s(tyop_2Enum_2Enum,V0n_2E0)))))) ).
fof(thm_2Ereal_2EREAL__ADD__SYM,axiom,
! [V0x_2E0,V1y_2E0] : s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,V1y_2E0))) = s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(s(tyop_2Erealax_2Ereal,V1y_2E0),s(tyop_2Erealax_2Ereal,V0x_2E0))) ).
fof(thm_2Ereal_2EREAL__ADD__ASSOC,axiom,
! [V0x_2E0,V1y_2E0,V2z_2E0] : s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(s(tyop_2Erealax_2Ereal,V1y_2E0),s(tyop_2Erealax_2Ereal,V2z_2E0))))) = s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,V1y_2E0))),s(tyop_2Erealax_2Ereal,V2z_2E0))) ).
fof(thm_2Ereal_2EREAL__MUL__LID,axiom,
! [V0x_2E0] : s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(s(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2EZERO_2E0))))))),s(tyop_2Erealax_2Ereal,V0x_2E0))) = s(tyop_2Erealax_2Ereal,V0x_2E0) ).
fof(thm_2Ereal_2Ereal__sub,axiom,
! [V0x_2E0,V1y_2E0] : s(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__sub_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,V1y_2E0))) = s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E1(s(tyop_2Erealax_2Ereal,V1y_2E0))))) ).
fof(thm_2Ereal_2EREAL__EQ__LADD,axiom,
! [V0x_2E0,V1y_2E0,V2z_2E0] :
( s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,V1y_2E0))) = s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,V2z_2E0)))
<=> s(tyop_2Erealax_2Ereal,V1y_2E0) = s(tyop_2Erealax_2Ereal,V2z_2E0) ) ).
fof(thm_2Ereal_2EREAL__NEG__ADD,axiom,
! [V0x_2E0,V1y_2E0] : s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E1(s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,V1y_2E0))))) = s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E1(s(tyop_2Erealax_2Ereal,V0x_2E0))),s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E1(s(tyop_2Erealax_2Ereal,V1y_2E0))))) ).
fof(thm_2Ereal_2EREAL__MUL__LZERO,axiom,
! [V0x_2E0] : s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(s(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(s(tyop_2Enum_2Enum,c_2Enum_2E0_2E0))),s(tyop_2Erealax_2Ereal,V0x_2E0))) = s(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(s(tyop_2Enum_2Enum,c_2Enum_2E0_2E0))) ).
fof(thm_2Ereal_2EREAL__NEGNEG,axiom,
! [V0x_2E0] : s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E1(s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E1(s(tyop_2Erealax_2Ereal,V0x_2E0))))) = s(tyop_2Erealax_2Ereal,V0x_2E0) ).
fof(thm_2Ereal_2EREAL__NOT__LT,axiom,
! [V0x_2E0,V1y_2E0] :
( ~ p(s(tyop_2Emin_2Ebool,c_2Erealax_2Ereal__lt_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,V1y_2E0))))
<=> p(s(tyop_2Emin_2Ebool,c_2Ereal_2Ereal__lte_2E2(s(tyop_2Erealax_2Ereal,V1y_2E0),s(tyop_2Erealax_2Ereal,V0x_2E0)))) ) ).
fof(thm_2Ereal_2EREAL__LT__LE,axiom,
! [V0x_2E0,V1y_2E0] :
( p(s(tyop_2Emin_2Ebool,c_2Erealax_2Ereal__lt_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,V1y_2E0))))
<=> ( p(s(tyop_2Emin_2Ebool,c_2Ereal_2Ereal__lte_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,V1y_2E0))))
& s(tyop_2Erealax_2Ereal,V0x_2E0) != s(tyop_2Erealax_2Ereal,V1y_2E0) ) ) ).
fof(thm_2Ereal_2EREAL__LE__TRANS,axiom,
! [V0x_2E0,V1y_2E0,V2z_2E0] :
( ( p(s(tyop_2Emin_2Ebool,c_2Ereal_2Ereal__lte_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,V1y_2E0))))
& p(s(tyop_2Emin_2Ebool,c_2Ereal_2Ereal__lte_2E2(s(tyop_2Erealax_2Ereal,V1y_2E0),s(tyop_2Erealax_2Ereal,V2z_2E0)))) )
=> p(s(tyop_2Emin_2Ebool,c_2Ereal_2Ereal__lte_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,V2z_2E0)))) ) ).
fof(thm_2Ereal_2EREAL__LE__RADD,axiom,
! [V0x_2E0,V1y_2E0,V2z_2E0] : s(tyop_2Emin_2Ebool,c_2Ereal_2Ereal__lte_2E2(s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,V2z_2E0))),s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(s(tyop_2Erealax_2Ereal,V1y_2E0),s(tyop_2Erealax_2Ereal,V2z_2E0))))) = s(tyop_2Emin_2Ebool,c_2Ereal_2Ereal__lte_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,V1y_2E0))) ).
fof(thm_2Ereal_2EREAL__EQ__RMUL,axiom,
! [V0x_2E0,V1y_2E0,V2z_2E0] :
( s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,V2z_2E0))) = s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(s(tyop_2Erealax_2Ereal,V1y_2E0),s(tyop_2Erealax_2Ereal,V2z_2E0)))
<=> ( s(tyop_2Erealax_2Ereal,V2z_2E0) = s(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(s(tyop_2Enum_2Enum,c_2Enum_2E0_2E0)))
| s(tyop_2Erealax_2Ereal,V0x_2E0) = s(tyop_2Erealax_2Ereal,V1y_2E0) ) ) ).
fof(thm_2Ereal_2EREAL__LE,axiom,
! [V0m_2E0,V1n_2E0] : s(tyop_2Emin_2Ebool,c_2Ereal_2Ereal__lte_2E2(s(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(s(tyop_2Enum_2Enum,V0m_2E0))),s(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(s(tyop_2Enum_2Enum,V1n_2E0))))) = s(tyop_2Emin_2Ebool,c_2Earithmetic_2E_3C_3D_2E2(s(tyop_2Enum_2Enum,V0m_2E0),s(tyop_2Enum_2Enum,V1n_2E0))) ).
fof(thm_2Ereal_2EREAL__INJ,axiom,
! [V0m_2E0,V1n_2E0] :
( s(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(s(tyop_2Enum_2Enum,V0m_2E0))) = s(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(s(tyop_2Enum_2Enum,V1n_2E0)))
<=> s(tyop_2Enum_2Enum,V0m_2E0) = s(tyop_2Enum_2Enum,V1n_2E0) ) ).
fof(thm_2Ereal_2EREAL__ADD,axiom,
! [V0m_2E0,V1n_2E0] : s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(s(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(s(tyop_2Enum_2Enum,V0m_2E0))),s(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(s(tyop_2Enum_2Enum,V1n_2E0))))) = s(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(s(tyop_2Enum_2Enum,c_2Earithmetic_2E_2B_2E2(s(tyop_2Enum_2Enum,V0m_2E0),s(tyop_2Enum_2Enum,V1n_2E0))))) ).
fof(thm_2Ereal_2EREAL__DIV__RMUL,axiom,
! [V0x_2E0,V1y_2E0] :
( s(tyop_2Erealax_2Ereal,V1y_2E0) != s(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(s(tyop_2Enum_2Enum,c_2Enum_2E0_2E0)))
=> s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(s(tyop_2Erealax_2Ereal,c_2Ereal_2E_2F_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,V1y_2E0))),s(tyop_2Erealax_2Ereal,V1y_2E0))) = s(tyop_2Erealax_2Ereal,V0x_2E0) ) ).
fof(thm_2Ereal_2EREAL__LE__SUB__RADD,axiom,
! [V0x_2E0,V1y_2E0,V2z_2E0] : s(tyop_2Emin_2Ebool,c_2Ereal_2Ereal__lte_2E2(s(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__sub_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,V1y_2E0))),s(tyop_2Erealax_2Ereal,V2z_2E0))) = s(tyop_2Emin_2Ebool,c_2Ereal_2Ereal__lte_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(s(tyop_2Erealax_2Ereal,V2z_2E0),s(tyop_2Erealax_2Ereal,V1y_2E0))))) ).
fof(thm_2Ereal_2EREAL__SUB__RZERO,axiom,
! [V0x_2E0] : s(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__sub_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(s(tyop_2Enum_2Enum,c_2Enum_2E0_2E0))))) = s(tyop_2Erealax_2Ereal,V0x_2E0) ).
fof(thm_2Ereal_2EREAL__EQ__SUB__LADD,axiom,
! [V0x_2E0,V1y_2E0,V2z_2E0] :
( s(tyop_2Erealax_2Ereal,V0x_2E0) = s(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__sub_2E2(s(tyop_2Erealax_2Ereal,V1y_2E0),s(tyop_2Erealax_2Ereal,V2z_2E0)))
<=> s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,V2z_2E0))) = s(tyop_2Erealax_2Ereal,V1y_2E0) ) ).
fof(thm_2Ereal_2EREAL__LE__RMUL,axiom,
! [V0x_2E0,V1y_2E0,V2z_2E0] :
( p(s(tyop_2Emin_2Ebool,c_2Erealax_2Ereal__lt_2E2(s(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(s(tyop_2Enum_2Enum,c_2Enum_2E0_2E0))),s(tyop_2Erealax_2Ereal,V2z_2E0))))
=> s(tyop_2Emin_2Ebool,c_2Ereal_2Ereal__lte_2E2(s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,V2z_2E0))),s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(s(tyop_2Erealax_2Ereal,V1y_2E0),s(tyop_2Erealax_2Ereal,V2z_2E0))))) = s(tyop_2Emin_2Ebool,c_2Ereal_2Ereal__lte_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,V1y_2E0))) ) ).
fof(thm_2Ereal_2EREAL__EQ__NEG,axiom,
! [V0x_2E0,V1y_2E0] :
( s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E1(s(tyop_2Erealax_2Ereal,V0x_2E0))) = s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E1(s(tyop_2Erealax_2Ereal,V1y_2E0)))
<=> s(tyop_2Erealax_2Ereal,V0x_2E0) = s(tyop_2Erealax_2Ereal,V1y_2E0) ) ).
fof(thm_2Ereal_2EREAL__SUP__LE,axiom,
! [V0P_2E0] :
( ( ? [V1x_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Erealax_2Ereal,V1x_2E0))))
& ? [V2z_2E0] :
! [V3x_2E0] :
( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Erealax_2Ereal,V3x_2E0))))
=> p(s(tyop_2Emin_2Ebool,c_2Ereal_2Ereal__lte_2E2(s(tyop_2Erealax_2Ereal,V3x_2E0),s(tyop_2Erealax_2Ereal,V2z_2E0)))) ) )
=> ! [V4y_2E0] :
( ? [V5x_2E0] :
( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Erealax_2Ereal,V5x_2E0))))
& p(s(tyop_2Emin_2Ebool,c_2Erealax_2Ereal__lt_2E2(s(tyop_2Erealax_2Ereal,V4y_2E0),s(tyop_2Erealax_2Ereal,V5x_2E0)))) )
<=> p(s(tyop_2Emin_2Ebool,c_2Erealax_2Ereal__lt_2E2(s(tyop_2Erealax_2Ereal,V4y_2E0),s(tyop_2Erealax_2Ereal,c_2Ereal_2Esup_2E1(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0)))))) ) ) ).
fof(thm_2Ereal_2EREAL__MUL__LNEG,axiom,
! [V0x_2E0,V1y_2E0] : s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E1(s(tyop_2Erealax_2Ereal,V0x_2E0))),s(tyop_2Erealax_2Ereal,V1y_2E0))) = s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E1(s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,V1y_2E0))))) ).
fof(thm_2Ereal_2Ereal__lt,axiom,
! [V0y_2E0,V1x_2E0] :
( p(s(tyop_2Emin_2Ebool,c_2Erealax_2Ereal__lt_2E2(s(tyop_2Erealax_2Ereal,V1x_2E0),s(tyop_2Erealax_2Ereal,V0y_2E0))))
<=> ~ p(s(tyop_2Emin_2Ebool,c_2Ereal_2Ereal__lte_2E2(s(tyop_2Erealax_2Ereal,V0y_2E0),s(tyop_2Erealax_2Ereal,V1x_2E0)))) ) ).
fof(thm_2Ereal_2EREAL__ADD__RDISTRIB,axiom,
! [V0x_2E0,V1y_2E0,V2z_2E0] : s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,V1y_2E0))),s(tyop_2Erealax_2Ereal,V2z_2E0))) = s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,V2z_2E0))),s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(s(tyop_2Erealax_2Ereal,V1y_2E0),s(tyop_2Erealax_2Ereal,V2z_2E0))))) ).
fof(thm_2Ereal_2EREAL__BIGNUM,axiom,
! [V0r_2E0] :
? [V1n_2E0] : p(s(tyop_2Emin_2Ebool,c_2Erealax_2Ereal__lt_2E2(s(tyop_2Erealax_2Ereal,V0r_2E0),s(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(s(tyop_2Enum_2Enum,V1n_2E0)))))) ).
fof(thm_2Esat_2ENOT__NOT,axiom,
! [V0t_2E0] :
( ~ ~ p(s(tyop_2Emin_2Ebool,V0t_2E0))
<=> p(s(tyop_2Emin_2Ebool,V0t_2E0)) ) ).
fof(thm_2Esat_2EAND__INV__IMP,axiom,
! [V0A_2E0] :
( p(s(tyop_2Emin_2Ebool,V0A_2E0))
=> ( ~ p(s(tyop_2Emin_2Ebool,V0A_2E0))
=> p(s(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0)) ) ) ).
fof(thm_2Esat_2EOR__DUAL2,axiom,
! [V0B_2E0,V1A_2E0] :
( ( ~ ( p(s(tyop_2Emin_2Ebool,V1A_2E0))
| p(s(tyop_2Emin_2Ebool,V0B_2E0)) )
=> p(s(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0)) )
<=> ( ( p(s(tyop_2Emin_2Ebool,V1A_2E0))
=> p(s(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0)) )
=> ( ~ p(s(tyop_2Emin_2Ebool,V0B_2E0))
=> p(s(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0)) ) ) ) ).
fof(thm_2Esat_2EOR__DUAL3,axiom,
! [V0B_2E0,V1A_2E0] :
( ( ~ ( ~ p(s(tyop_2Emin_2Ebool,V1A_2E0))
| p(s(tyop_2Emin_2Ebool,V0B_2E0)) )
=> p(s(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0)) )
<=> ( p(s(tyop_2Emin_2Ebool,V1A_2E0))
=> ( ~ p(s(tyop_2Emin_2Ebool,V0B_2E0))
=> p(s(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0)) ) ) ) ).
fof(thm_2Esat_2EAND__INV2,axiom,
! [V0A_2E0] :
( ( ~ p(s(tyop_2Emin_2Ebool,V0A_2E0))
=> p(s(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0)) )
=> ( ( p(s(tyop_2Emin_2Ebool,V0A_2E0))
=> p(s(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0)) )
=> p(s(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0)) ) ) ).
fof(thm_2Esat_2Edc__eq,axiom,
! [V0r_2E0,V1q_2E0,V2p_2E0] :
( ( p(s(tyop_2Emin_2Ebool,V2p_2E0))
<=> s(tyop_2Emin_2Ebool,V1q_2E0) = s(tyop_2Emin_2Ebool,V0r_2E0) )
<=> ( ( p(s(tyop_2Emin_2Ebool,V2p_2E0))
| p(s(tyop_2Emin_2Ebool,V1q_2E0))
| p(s(tyop_2Emin_2Ebool,V0r_2E0)) )
& ( p(s(tyop_2Emin_2Ebool,V2p_2E0))
| ~ p(s(tyop_2Emin_2Ebool,V0r_2E0))
| ~ p(s(tyop_2Emin_2Ebool,V1q_2E0)) )
& ( p(s(tyop_2Emin_2Ebool,V1q_2E0))
| ~ p(s(tyop_2Emin_2Ebool,V0r_2E0))
| ~ p(s(tyop_2Emin_2Ebool,V2p_2E0)) )
& ( p(s(tyop_2Emin_2Ebool,V0r_2E0))
| ~ p(s(tyop_2Emin_2Ebool,V1q_2E0))
| ~ p(s(tyop_2Emin_2Ebool,V2p_2E0)) ) ) ) ).
fof(thm_2Esat_2Edc__conj,axiom,
! [V0r_2E0,V1q_2E0,V2p_2E0] :
( ( p(s(tyop_2Emin_2Ebool,V2p_2E0))
<=> ( p(s(tyop_2Emin_2Ebool,V1q_2E0))
& p(s(tyop_2Emin_2Ebool,V0r_2E0)) ) )
<=> ( ( p(s(tyop_2Emin_2Ebool,V2p_2E0))
| ~ p(s(tyop_2Emin_2Ebool,V1q_2E0))
| ~ p(s(tyop_2Emin_2Ebool,V0r_2E0)) )
& ( p(s(tyop_2Emin_2Ebool,V1q_2E0))
| ~ p(s(tyop_2Emin_2Ebool,V2p_2E0)) )
& ( p(s(tyop_2Emin_2Ebool,V0r_2E0))
| ~ p(s(tyop_2Emin_2Ebool,V2p_2E0)) ) ) ) ).
fof(thm_2Esat_2Edc__disj,axiom,
! [V0r_2E0,V1q_2E0,V2p_2E0] :
( ( p(s(tyop_2Emin_2Ebool,V2p_2E0))
<=> ( p(s(tyop_2Emin_2Ebool,V1q_2E0))
| p(s(tyop_2Emin_2Ebool,V0r_2E0)) ) )
<=> ( ( p(s(tyop_2Emin_2Ebool,V2p_2E0))
| ~ p(s(tyop_2Emin_2Ebool,V1q_2E0)) )
& ( p(s(tyop_2Emin_2Ebool,V2p_2E0))
| ~ p(s(tyop_2Emin_2Ebool,V0r_2E0)) )
& ( p(s(tyop_2Emin_2Ebool,V1q_2E0))
| p(s(tyop_2Emin_2Ebool,V0r_2E0))
| ~ p(s(tyop_2Emin_2Ebool,V2p_2E0)) ) ) ) ).
fof(thm_2Esat_2Edc__imp,axiom,
! [V0r_2E0,V1q_2E0,V2p_2E0] :
( ( p(s(tyop_2Emin_2Ebool,V2p_2E0))
<=> ( p(s(tyop_2Emin_2Ebool,V1q_2E0))
=> p(s(tyop_2Emin_2Ebool,V0r_2E0)) ) )
<=> ( ( p(s(tyop_2Emin_2Ebool,V2p_2E0))
| p(s(tyop_2Emin_2Ebool,V1q_2E0)) )
& ( p(s(tyop_2Emin_2Ebool,V2p_2E0))
| ~ p(s(tyop_2Emin_2Ebool,V0r_2E0)) )
& ( ~ p(s(tyop_2Emin_2Ebool,V1q_2E0))
| p(s(tyop_2Emin_2Ebool,V0r_2E0))
| ~ p(s(tyop_2Emin_2Ebool,V2p_2E0)) ) ) ) ).
fof(thm_2Esat_2Edc__neg,axiom,
! [V0q_2E0,V1p_2E0] :
( ( p(s(tyop_2Emin_2Ebool,V1p_2E0))
<=> ~ p(s(tyop_2Emin_2Ebool,V0q_2E0)) )
<=> ( ( p(s(tyop_2Emin_2Ebool,V1p_2E0))
| p(s(tyop_2Emin_2Ebool,V0q_2E0)) )
& ( ~ p(s(tyop_2Emin_2Ebool,V0q_2E0))
| ~ p(s(tyop_2Emin_2Ebool,V1p_2E0)) ) ) ) ).
fof(thm_2Ewhile_2ELEAST__EXISTS__IMP,axiom,
! [V0p_2E0] :
( ? [V1n_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0p_2E0),s(tyop_2Enum_2Enum,V1n_2E0))))
=> ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0p_2E0),s(tyop_2Enum_2Enum,c_2Ewhile_2ELEAST_2E1(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0p_2E0))))))
& ! [V2n_2E0] :
( p(s(tyop_2Emin_2Ebool,c_2Eprim__rec_2E_3C_2E2(s(tyop_2Enum_2Enum,V2n_2E0),s(tyop_2Enum_2Enum,c_2Ewhile_2ELEAST_2E1(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0p_2E0))))))
=> ~ p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Ebool),V0p_2E0),s(tyop_2Enum_2Enum,V2n_2E0)))) ) ) ) ).
fof(thm_2Ereal_2ESUP__EPSILON,conjecture,
! [V0p_2E0,V1e_2E0] :
( ( p(s(tyop_2Emin_2Ebool,c_2Erealax_2Ereal__lt_2E2(s(tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E1(s(tyop_2Enum_2Enum,c_2Enum_2E0_2E0))),s(tyop_2Erealax_2Ereal,V1e_2E0))))
& ? [V2x_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0p_2E0),s(tyop_2Erealax_2Ereal,V2x_2E0))))
& ? [V3z_2E0] :
! [V4x_2E0] :
( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0p_2E0),s(tyop_2Erealax_2Ereal,V4x_2E0))))
=> p(s(tyop_2Emin_2Ebool,c_2Ereal_2Ereal__lte_2E2(s(tyop_2Erealax_2Ereal,V4x_2E0),s(tyop_2Erealax_2Ereal,V3z_2E0)))) ) )
=> ? [V5x_2E0] :
( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0p_2E0),s(tyop_2Erealax_2Ereal,V5x_2E0))))
& p(s(tyop_2Emin_2Ebool,c_2Ereal_2Ereal__lte_2E2(s(tyop_2Erealax_2Ereal,c_2Ereal_2Esup_2E1(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0p_2E0))),s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(s(tyop_2Erealax_2Ereal,V5x_2E0),s(tyop_2Erealax_2Ereal,V1e_2E0)))))) ) ) ).
%------------------------------------------------------------------------------