ITP001 Axioms: ITP010+4.ax
%------------------------------------------------------------------------------
% File : ITP010+4 : TPTP v9.0.0. 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 : one.ax [Gau19]
% : HL4010+4.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 36 ( 16 unt; 0 def)
% Number of atoms : 64 ( 25 equ)
% Maximal formula atoms : 3 ( 1 avg)
% Number of connectives : 31 ( 3 ~; 3 |; 2 &)
% ( 15 <=>; 8 =>; 0 <=; 0 <~>)
% Maximal formula depth : 7 ( 4 avg)
% Maximal term depth : 11 ( 2 avg)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 33 ( 33 usr; 19 con; 0-2 aty)
% Number of variables : 84 ( 81 !; 3 ?)
% SPC : FOF_SAT_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
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_2Ebool_2E_3F_21_2E1_2Emono_2Etyop_2Emin_2Efun_28tyop_2Eone_2Eone_2CA_27a_29,axiom,
! [A_27a,X0_2E0] : s(tyop_2Emin_2Ebool,c_2Ebool_2E_3F_21_2E1(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Eone_2Eone,A_27a),tyop_2Emin_2Ebool),X0_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Eone_2Eone,A_27a),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ebool_2E_3F_21_2E0),s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Eone_2Eone,A_27a),tyop_2Emin_2Ebool),X0_2E0))) ).
fof(arityeq1_2Ec_2Emin_2E_40_2E1_2Emono_2Etyop_2Eone_2Eone,axiom,
! [X0_2E0] : s(tyop_2Eone_2Eone,c_2Emin_2E_40_2E1(s(tyop_2Emin_2Efun(tyop_2Eone_2Eone,tyop_2Emin_2Ebool),X0_2E0))) = s(tyop_2Eone_2Eone,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Eone_2Eone,tyop_2Emin_2Ebool),tyop_2Eone_2Eone),c_2Emin_2E_40_2E0),s(tyop_2Emin_2Efun(tyop_2Eone_2Eone,tyop_2Emin_2Ebool),X0_2E0))) ).
fof(arityeq2_2Ec_2Ebool_2ETYPE__DEFINITION_2E2_2Emono_2Etyop_2Emin_2Ebool_20mono_2Etyop_2Eone_2Eone,axiom,
! [X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Ebool_2ETYPE__DEFINITION_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),X0_2E0),s(tyop_2Emin_2Efun(tyop_2Eone_2Eone,tyop_2Emin_2Ebool),X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Eone_2Eone,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Eone_2Eone,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),c_2Ebool_2ETYPE__DEFINITION_2E0),s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),X0_2E0))),s(tyop_2Emin_2Efun(tyop_2Eone_2Eone,tyop_2Emin_2Ebool),X1_2E0))) ).
fof(arityeq2_2Ec_2Eone_2Eone__CASE_2E2_2Emono_2EA_27a,axiom,
! [A_27a,X0_2E0,X1_2E0] : s(A_27a,c_2Eone_2Eone__CASE_2E2(s(tyop_2Eone_2Eone,X0_2E0),s(A_27a,X1_2E0))) = s(A_27a,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Eone_2Eone,tyop_2Emin_2Efun(A_27a,A_27a)),c_2Eone_2Eone__CASE_2E0),s(tyop_2Eone_2Eone,X0_2E0))),s(A_27a,X1_2E0))) ).
fof(thm_2Eone_2Eone__TY__DEF,axiom,
! [F0_2E0] :
( ! [V1b_2E0] : s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),F0_2E0),s(tyop_2Emin_2Ebool,V1b_2E0))) = s(tyop_2Emin_2Ebool,V1b_2E0)
=> ? [V0rep_2E0] : p(s(tyop_2Emin_2Ebool,c_2Ebool_2ETYPE__DEFINITION_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),F0_2E0),s(tyop_2Emin_2Efun(tyop_2Eone_2Eone,tyop_2Emin_2Ebool),V0rep_2E0)))) ) ).
fof(thm_2Eone_2Eone__DEF,axiom,
! [F0_2E0] :
( ! [V0x_2E0] : s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Eone_2Eone,tyop_2Emin_2Ebool),F0_2E0),s(tyop_2Eone_2Eone,V0x_2E0))) = s(tyop_2Emin_2Ebool,c_2Ebool_2ET_2E0)
=> s(tyop_2Eone_2Eone,c_2Eone_2Eone_2E0) = s(tyop_2Eone_2Eone,c_2Emin_2E_40_2E1(s(tyop_2Emin_2Efun(tyop_2Eone_2Eone,tyop_2Emin_2Ebool),F0_2E0))) ) ).
fof(thm_2Eone_2Eone__case__def,axiom,
! [A_27a,V0u_2E0,V1x_2E0] : s(A_27a,c_2Eone_2Eone__CASE_2E2(s(tyop_2Eone_2Eone,V0u_2E0),s(A_27a,V1x_2E0))) = s(A_27a,V1x_2E0) ).
fof(thm_2Eone_2Eone__axiom,axiom,
! [A_27a,V0f_2E0,V1g_2E0] : s(tyop_2Emin_2Efun(A_27a,tyop_2Eone_2Eone),V0f_2E0) = s(tyop_2Emin_2Efun(A_27a,tyop_2Eone_2Eone),V1g_2E0) ).
fof(thm_2Eone_2Eone,axiom,
! [V0v_2E0] : s(tyop_2Eone_2Eone,V0v_2E0) = s(tyop_2Eone_2Eone,c_2Eone_2Eone_2E0) ).
fof(thm_2Eone_2Eone__Axiom,axiom,
! [A_27a,F0_2E0] :
( ! [V0e_2E0,V1fn_2E0] :
( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Eone_2Eone,A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Eone_2Eone,A_27a),tyop_2Emin_2Ebool)),F0_2E0),s(A_27a,V0e_2E0))),s(tyop_2Emin_2Efun(tyop_2Eone_2Eone,A_27a),V1fn_2E0))))
<=> s(A_27a,app_2E2(s(tyop_2Emin_2Efun(tyop_2Eone_2Eone,A_27a),V1fn_2E0),s(tyop_2Eone_2Eone,c_2Eone_2Eone_2E0))) = s(A_27a,V0e_2E0) )
=> ! [V0e_2E0] : p(s(tyop_2Emin_2Ebool,c_2Ebool_2E_3F_21_2E1(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Eone_2Eone,A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Eone_2Eone,A_27a),tyop_2Emin_2Ebool)),F0_2E0),s(A_27a,V0e_2E0)))))) ) ).
fof(thm_2Eone_2Eone__prim__rec,axiom,
! [A_27a,V0e_2E0] :
? [V1fn_2E0] : s(A_27a,app_2E2(s(tyop_2Emin_2Efun(tyop_2Eone_2Eone,A_27a),V1fn_2E0),s(tyop_2Eone_2Eone,c_2Eone_2Eone_2E0))) = s(A_27a,V0e_2E0) ).
fof(thm_2Eone_2Eone__induction,axiom,
! [V0P_2E0] :
( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Eone_2Eone,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Eone_2Eone,c_2Eone_2Eone_2E0))))
=> ! [V1x_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Eone_2Eone,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Eone_2Eone,V1x_2E0)))) ) ).
fof(thm_2Eone_2EFORALL__ONE,axiom,
! [V0P_2E0] :
( ! [V1x_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Eone_2Eone,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Eone_2Eone,V1x_2E0))))
<=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Eone_2Eone,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Eone_2Eone,c_2Eone_2Eone_2E0)))) ) ).
fof(thm_2Eone_2Eone__case__thm,axiom,
! [A_27a,V0x_2E0] : s(A_27a,c_2Eone_2Eone__CASE_2E2(s(tyop_2Eone_2Eone,c_2Eone_2Eone_2E0),s(A_27a,V0x_2E0))) = s(A_27a,V0x_2E0) ).
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) ) ).
%------------------------------------------------------------------------------