TPTP Problem File: SEV524^1.p
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%------------------------------------------------------------------------------
% File : SEV524^1 : TPTP v9.0.0. Released v7.5.0.
% Domain : Set Theory
% Problem : ProofGold problem HOSetConstr_20_neg_th0
% Version : Especial.
% English :
% Refs : [Urb20] Urban (2020) Email to Geoff Sutcliffe
% Source : [Urb20]
% Names : HOSetConstr_20_neg_th0.p [Urb20]
% Status : Theorem
% Rating : 0.62 v9.0.0, 0.90 v8.2.0, 0.85 v8.1.0, 0.82 v7.5.0
% Syntax : Number of formulae : 213 ( 99 unt; 104 typ; 0 def)
% Number of atoms : 487 ( 135 equ; 0 cnn)
% Maximal formula atoms : 8 ( 4 avg)
% Number of connectives : 1342 ( 0 ~; 0 |; 0 &;1232 @)
% ( 0 <=>; 110 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 2 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 697 ( 697 >; 0 *; 0 +; 0 <<)
% Number of symbols : 105 ( 104 usr; 3 con; 0-14 aty)
% Number of variables : 504 ( 338 ^; 129 !; 37 ?; 504 :)
% SPC : TH0_THM_EQU_NAR
% Comments : See https://proofgold.org
%------------------------------------------------------------------------------
thf(c_Eps_i_tp,type,
c_Eps_i: ( $i > $o ) > $i ).
thf(c_False_tp,type,
c_False: $o ).
thf(c_True_tp,type,
c_True: $o ).
thf(c_not_tp,type,
c_not: $o > $o ).
thf(c_and_tp,type,
c_and: $o > $o > $o ).
thf(c_or_tp,type,
c_or: $o > $o > $o ).
thf(c_iff_tp,type,
c_iff: $o > $o > $o ).
thf(c_In_tp,type,
c_In: $i > $i > $o ).
thf(c_Subq_tp,type,
c_Subq: $i > $i > $o ).
thf(c_Empty_tp,type,
c_Empty: $i ).
thf(c_Union_tp,type,
c_Union: $i > $i ).
thf(c_Power_tp,type,
c_Power: $i > $i ).
thf(c_Repl_tp,type,
c_Repl: $i > ( $i > $i ) > $i ).
thf(c_TransSet_tp,type,
c_TransSet: $i > $o ).
thf(c_atleast2_tp,type,
c_atleast2: $i > $o ).
thf(c_atleast3_tp,type,
c_atleast3: $i > $o ).
thf(c_atleast4_tp,type,
c_atleast4: $i > $o ).
thf(c_atleast5_tp,type,
c_atleast5: $i > $o ).
thf(c_atleast6_tp,type,
c_atleast6: $i > $o ).
thf(c_exactly2_tp,type,
c_exactly2: $i > $o ).
thf(c_exactly3_tp,type,
c_exactly3: $i > $o ).
thf(c_exactly4_tp,type,
c_exactly4: $i > $o ).
thf(c_exactly5_tp,type,
c_exactly5: $i > $o ).
thf(c_exu_i_tp,type,
c_exu_i: ( $i > $o ) > $o ).
thf(c_reflexive_i_tp,type,
c_reflexive_i: ( $i > $i > $o ) > $o ).
thf(c_irreflexive_i_tp,type,
c_irreflexive_i: ( $i > $i > $o ) > $o ).
thf(c_symmetric_i_tp,type,
c_symmetric_i: ( $i > $i > $o ) > $o ).
thf(c_antisymmetric_i_tp,type,
c_antisymmetric_i: ( $i > $i > $o ) > $o ).
thf(c_transitive_i_tp,type,
c_transitive_i: ( $i > $i > $o ) > $o ).
thf(c_eqreln_i_tp,type,
c_eqreln_i: ( $i > $i > $o ) > $o ).
thf(c_per_i_tp,type,
c_per_i: ( $i > $i > $o ) > $o ).
thf(c_linear_i_tp,type,
c_linear_i: ( $i > $i > $o ) > $o ).
thf(c_trichotomous_or_i_tp,type,
c_trichotomous_or_i: ( $i > $i > $o ) > $o ).
thf(c_partialorder_i_tp,type,
c_partialorder_i: ( $i > $i > $o ) > $o ).
thf(c_totalorder_i_tp,type,
c_totalorder_i: ( $i > $i > $o ) > $o ).
thf(c_strictpartialorder_i_tp,type,
c_strictpartialorder_i: ( $i > $i > $o ) > $o ).
thf(c_stricttotalorder_i_tp,type,
c_stricttotalorder_i: ( $i > $i > $o ) > $o ).
thf(c_If_i_tp,type,
c_If_i: $o > $i > $i > $i ).
thf(c_exactly1of2_tp,type,
c_exactly1of2: $o > $o > $o ).
thf(c_exactly1of3_tp,type,
c_exactly1of3: $o > $o > $o > $o ).
thf(c_nIn_tp,type,
c_nIn: $i > $i > $o ).
thf(c_nSubq_tp,type,
c_nSubq: $i > $i > $o ).
thf(c_UPair_tp,type,
c_UPair: $i > $i > $i ).
thf(c_Sing_tp,type,
c_Sing: $i > $i ).
thf(c_binunion_tp,type,
c_binunion: $i > $i > $i ).
thf(c_SetAdjoin_tp,type,
c_SetAdjoin: $i > $i > $i ).
thf(c_famunion_tp,type,
c_famunion: $i > ( $i > $i ) > $i ).
thf(c_Sep_tp,type,
c_Sep: $i > ( $i > $o ) > $i ).
thf(c_ReplSep_tp,type,
c_ReplSep: $i > ( $i > $o ) > ( $i > $i ) > $i ).
thf(c_binintersect_tp,type,
c_binintersect: $i > $i > $i ).
thf(c_setminus_tp,type,
c_setminus: $i > $i > $i ).
thf(c_inj_tp,type,
c_inj: $i > $i > ( $i > $i ) > $o ).
thf(c_bij_tp,type,
c_bij: $i > $i > ( $i > $i ) > $o ).
thf(c_atleastp_tp,type,
c_atleastp: $i > $i > $o ).
thf(c_equip_tp,type,
c_equip: $i > $i > $o ).
thf(c_In_rec_poly_G_i_tp,type,
c_In_rec_poly_G_i: ( $i > ( $i > $i ) > $i ) > $i > $i > $o ).
thf(c_In_rec_poly_i_tp,type,
c_In_rec_poly_i: ( $i > ( $i > $i ) > $i ) > $i > $i ).
thf(c_ordsucc_tp,type,
c_ordsucc: $i > $i ).
thf(c_nat_p_tp,type,
c_nat_p: $i > $o ).
thf(c_nat_primrec_tp,type,
c_nat_primrec: $i > ( $i > $i > $i ) > $i > $i ).
thf(c_add_nat_tp,type,
c_add_nat: $i > $i > $i ).
thf(c_mul_nat_tp,type,
c_mul_nat: $i > $i > $i ).
thf(c_ordinal_tp,type,
c_ordinal: $i > $o ).
thf(c_V__tp,type,
c_V_: $i > $i ).
thf(c_Inj1_tp,type,
c_Inj1: $i > $i ).
thf(c_Inj0_tp,type,
c_Inj0: $i > $i ).
thf(c_Unj_tp,type,
c_Unj: $i > $i ).
thf(c_combine_funcs_tp,type,
c_combine_funcs: $i > $i > ( $i > $i ) > ( $i > $i ) > $i > $i ).
thf(c_setsum_tp,type,
c_setsum: $i > $i > $i ).
thf(c_proj0_tp,type,
c_proj0: $i > $i ).
thf(c_proj1_tp,type,
c_proj1: $i > $i ).
thf(c_binrep_tp,type,
c_binrep: $i > $i > $i ).
thf(c_lam_tp,type,
c_lam: $i > ( $i > $i ) > $i ).
thf(c_setprod_tp,type,
c_setprod: $i > $i > $i ).
thf(c_ap_tp,type,
c_ap: $i > $i > $i ).
thf(c_setsum_p_tp,type,
c_setsum_p: $i > $o ).
thf(c_tuple_p_tp,type,
c_tuple_p: $i > $i > $o ).
thf(c_Pi_tp,type,
c_Pi: $i > ( $i > $i ) > $i ).
thf(c_setexp_tp,type,
c_setexp: $i > $i > $i ).
thf(c_Sep2_tp,type,
c_Sep2: $i > ( $i > $i ) > ( $i > $i > $o ) > $i ).
thf(c_set_of_pairs_tp,type,
c_set_of_pairs: $i > $o ).
thf(c_lam2_tp,type,
c_lam2: $i > ( $i > $i ) > ( $i > $i > $i ) > $i ).
thf(c_PNoEq__tp,type,
c_PNoEq_: $i > ( $i > $o ) > ( $i > $o ) > $o ).
thf(c_PNoLt__tp,type,
c_PNoLt_: $i > ( $i > $o ) > ( $i > $o ) > $o ).
thf(c_PNoLt_tp,type,
c_PNoLt: $i > ( $i > $o ) > $i > ( $i > $o ) > $o ).
thf(c_PNoLe_tp,type,
c_PNoLe: $i > ( $i > $o ) > $i > ( $i > $o ) > $o ).
thf(c_PNo_downc_tp,type,
c_PNo_downc: ( $i > ( $i > $o ) > $o ) > $i > ( $i > $o ) > $o ).
thf(c_PNo_upc_tp,type,
c_PNo_upc: ( $i > ( $i > $o ) > $o ) > $i > ( $i > $o ) > $o ).
thf(c_SNoElts__tp,type,
c_SNoElts_: $i > $i ).
thf(c_SNo__tp,type,
c_SNo_: $i > $i > $o ).
thf(c_PSNo_tp,type,
c_PSNo: $i > ( $i > $o ) > $i ).
thf(c_SNo_tp,type,
c_SNo: $i > $o ).
thf(c_SNoLev_tp,type,
c_SNoLev: $i > $i ).
thf(c_SNoEq__tp,type,
c_SNoEq_: $i > $i > $i > $o ).
thf(c_SNoLt_tp,type,
c_SNoLt: $i > $i > $o ).
thf(c_SNoLe_tp,type,
c_SNoLe: $i > $i > $o ).
thf(c_binop_on_tp,type,
c_binop_on: $i > ( $i > $i > $i ) > $o ).
thf(c_Loop_tp,type,
c_Loop: $i > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > $i > $o ).
thf(c_Loop_with_defs_tp,type,
c_Loop_with_defs: $i > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > $i > ( $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > $o ).
thf(c_Loop_with_defs_cex1_tp,type,
c_Loop_with_defs_cex1: $i > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > $i > ( $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > $o ).
thf(c_Loop_with_defs_cex2_tp,type,
c_Loop_with_defs_cex2: $i > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > $i > ( $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > ( $i > $i > $i ) > $o ).
thf(c_combinator_tp,type,
c_combinator: $i > $o ).
thf(c_combinator_equiv_tp,type,
c_combinator_equiv: $i > $i > $o ).
thf(c_equip_mod_tp,type,
c_equip_mod: $i > $i > $i > $o ).
thf(ax1,axiom,
! [X0: $i > $o,X1: $i] :
( ( X0 @ X1 )
=> ( X0 @ ( c_Eps_i @ X0 ) ) ) ).
thf(ax2,axiom,
! [X0: $o] :
( ( c_not @ ( c_not @ X0 ) )
=> X0 ) ).
thf(ax3,axiom,
! [X0: $o,X1: $o] :
( ( c_iff @ X0 @ X1 )
=> ( X0 = X1 ) ) ).
thf(ax4,axiom,
! [X0: $i,X1: $i] :
( ( c_Subq @ X0 @ X1 )
=> ( ( c_Subq @ X1 @ X0 )
=> ( X0 = X1 ) ) ) ).
thf(ax5,axiom,
( c_not
@ ? [X0: $i] : ( c_In @ X0 @ c_Empty ) ) ).
thf(ax6,axiom,
! [X0: $i,X1: $i] :
( c_iff @ ( c_In @ X1 @ ( c_Union @ X0 ) )
@ ? [X2: $i] : ( c_and @ ( c_In @ X1 @ X2 ) @ ( c_In @ X2 @ X0 ) ) ) ).
thf(ax7,axiom,
! [X0: $i,X1: $i] : ( c_iff @ ( c_In @ X1 @ ( c_Power @ X0 ) ) @ ( c_Subq @ X1 @ X0 ) ) ).
thf(ax8,axiom,
! [X0: $i,X1: $i > $i,X2: $i] :
( c_iff
@ ( c_In @ X2
@ ( c_Repl @ X0
@ ^ [X3: $i] : ( X1 @ X3 ) ) )
@ ? [X3: $i] :
( c_and @ ( c_In @ X3 @ X0 )
@ ( X2
= ( X1 @ X3 ) ) ) ) ).
thf(ax9,axiom,
! [X0: $i > $o] :
( ! [X1: $i] :
( ( X0 @ X1 )
=> ! [X2: $i] :
( ( c_In @ X2 @ X1 )
=> ( X0 @ X2 ) ) )
=> ( ( X0 @ c_Empty )
=> ( ! [X1: $i] :
( ( X0 @ X1 )
=> ( X0 @ ( c_Union @ X1 ) ) )
=> ( ! [X1: $i] :
( ( X0 @ X1 )
=> ( X0 @ ( c_Power @ X1 ) ) )
=> ( ! [X1: $i] :
( ( X0 @ X1 )
=> ! [X2: $i > $i] :
( ! [X3: $i] :
( ( c_In @ X3 @ X1 )
=> ( X0 @ ( X2 @ X3 ) ) )
=> ( X0
@ ( c_Repl @ X1
@ ^ [X3: $i] : ( X2 @ X3 ) ) ) ) )
=> ! [X1: $i] : ( X0 @ X1 ) ) ) ) ) ) ).
thf(ax10,axiom,
! [X0: $i > $o] :
( ! [X1: $i] :
( ! [X2: $i] :
( ( c_In @ X2 @ X1 )
=> ( X0 @ X2 ) )
=> ( X0 @ X1 ) )
=> ! [X1: $i] : ( X0 @ X1 ) ) ).
thf(ax11,axiom,
( c_False
= ( ! [X0: $o] : X0 ) ) ).
thf(ax12,axiom,
( c_True
= ( ! [X0: $o] :
( X0
=> X0 ) ) ) ).
thf(ax13,axiom,
( c_not
= ( ^ [X0: $o] :
( X0
=> c_False ) ) ) ).
thf(ax14,axiom,
( c_and
= ( ^ [X0: $o,X1: $o] :
! [X2: $o] :
( ( X0
=> ( X1
=> X2 ) )
=> X2 ) ) ) ).
thf(ax15,axiom,
( c_or
= ( ^ [X0: $o,X1: $o] :
! [X2: $o] :
( ( X0
=> X2 )
=> ( ( X1
=> X2 )
=> X2 ) ) ) ) ).
thf(ax16,axiom,
( c_iff
= ( ^ [X0: $o,X1: $o] :
( c_and
@ ( X0
=> X1 )
@ ( X1
=> X0 ) ) ) ) ).
thf(ax17,axiom,
( c_Subq
= ( ^ [X0: $i,X1: $i] :
! [X2: $i] :
( ( c_In @ X2 @ X0 )
=> ( c_In @ X2 @ X1 ) ) ) ) ).
thf(ax18,axiom,
( c_TransSet
= ( ^ [X0: $i] :
! [X1: $i] :
( ( c_In @ X1 @ X0 )
=> ( c_Subq @ X1 @ X0 ) ) ) ) ).
thf(ax19,axiom,
( c_atleast2
= ( ^ [X0: $i] :
? [X1: $i] : ( c_and @ ( c_In @ X1 @ X0 ) @ ( c_not @ ( c_Subq @ X0 @ ( c_Power @ X1 ) ) ) ) ) ) ).
thf(ax20,axiom,
( c_atleast3
= ( ^ [X0: $i] :
? [X1: $i] : ( c_and @ ( c_Subq @ X1 @ X0 ) @ ( c_and @ ( c_not @ ( c_Subq @ X0 @ X1 ) ) @ ( c_atleast2 @ X1 ) ) ) ) ) ).
thf(ax21,axiom,
( c_atleast4
= ( ^ [X0: $i] :
? [X1: $i] : ( c_and @ ( c_Subq @ X1 @ X0 ) @ ( c_and @ ( c_not @ ( c_Subq @ X0 @ X1 ) ) @ ( c_atleast3 @ X1 ) ) ) ) ) ).
thf(ax22,axiom,
( c_atleast5
= ( ^ [X0: $i] :
? [X1: $i] : ( c_and @ ( c_Subq @ X1 @ X0 ) @ ( c_and @ ( c_not @ ( c_Subq @ X0 @ X1 ) ) @ ( c_atleast4 @ X1 ) ) ) ) ) ).
thf(ax23,axiom,
( c_atleast6
= ( ^ [X0: $i] :
? [X1: $i] : ( c_and @ ( c_Subq @ X1 @ X0 ) @ ( c_and @ ( c_not @ ( c_Subq @ X0 @ X1 ) ) @ ( c_atleast5 @ X1 ) ) ) ) ) ).
thf(ax24,axiom,
( c_exactly2
= ( ^ [X0: $i] : ( c_and @ ( c_atleast2 @ X0 ) @ ( c_not @ ( c_atleast3 @ X0 ) ) ) ) ) ).
thf(ax25,axiom,
( c_exactly3
= ( ^ [X0: $i] : ( c_and @ ( c_atleast3 @ X0 ) @ ( c_not @ ( c_atleast4 @ X0 ) ) ) ) ) ).
thf(ax26,axiom,
( c_exactly4
= ( ^ [X0: $i] : ( c_and @ ( c_atleast4 @ X0 ) @ ( c_not @ ( c_atleast5 @ X0 ) ) ) ) ) ).
thf(ax27,axiom,
( c_exactly5
= ( ^ [X0: $i] : ( c_and @ ( c_atleast5 @ X0 ) @ ( c_not @ ( c_atleast6 @ X0 ) ) ) ) ) ).
thf(ax28,axiom,
( c_exu_i
= ( ^ [X0: $i > $o] :
( c_and
@ ? [X1: $i] : ( X0 @ X1 )
@ ! [X1: $i,X2: $i] :
( ( X0 @ X1 )
=> ( ( X0 @ X2 )
=> ( X1 = X2 ) ) ) ) ) ) ).
thf(ax29,axiom,
( c_reflexive_i
= ( ^ [X0: $i > $i > $o] :
! [X1: $i] : ( X0 @ X1 @ X1 ) ) ) ).
thf(ax30,axiom,
( c_irreflexive_i
= ( ^ [X0: $i > $i > $o] :
! [X1: $i] : ( c_not @ ( X0 @ X1 @ X1 ) ) ) ) ).
thf(ax31,axiom,
( c_symmetric_i
= ( ^ [X0: $i > $i > $o] :
! [X1: $i,X2: $i] :
( ( X0 @ X1 @ X2 )
=> ( X0 @ X2 @ X1 ) ) ) ) ).
thf(ax32,axiom,
( c_antisymmetric_i
= ( ^ [X0: $i > $i > $o] :
! [X1: $i,X2: $i] :
( ( X0 @ X1 @ X2 )
=> ( ( X0 @ X2 @ X1 )
=> ( X1 = X2 ) ) ) ) ) ).
thf(ax33,axiom,
( c_transitive_i
= ( ^ [X0: $i > $i > $o] :
! [X1: $i,X2: $i,X3: $i] :
( ( X0 @ X1 @ X2 )
=> ( ( X0 @ X2 @ X3 )
=> ( X0 @ X1 @ X3 ) ) ) ) ) ).
thf(ax34,axiom,
( c_eqreln_i
= ( ^ [X0: $i > $i > $o] : ( c_and @ ( c_and @ ( c_reflexive_i @ X0 ) @ ( c_symmetric_i @ X0 ) ) @ ( c_transitive_i @ X0 ) ) ) ) ).
thf(ax35,axiom,
( c_per_i
= ( ^ [X0: $i > $i > $o] : ( c_and @ ( c_symmetric_i @ X0 ) @ ( c_transitive_i @ X0 ) ) ) ) ).
thf(ax36,axiom,
( c_linear_i
= ( ^ [X0: $i > $i > $o] :
! [X1: $i,X2: $i] : ( c_or @ ( X0 @ X1 @ X2 ) @ ( X0 @ X2 @ X1 ) ) ) ) ).
thf(ax37,axiom,
( c_trichotomous_or_i
= ( ^ [X0: $i > $i > $o] :
! [X1: $i,X2: $i] : ( c_or @ ( c_or @ ( X0 @ X1 @ X2 ) @ ( X1 = X2 ) ) @ ( X0 @ X2 @ X1 ) ) ) ) ).
thf(ax38,axiom,
( c_partialorder_i
= ( ^ [X0: $i > $i > $o] : ( c_and @ ( c_and @ ( c_reflexive_i @ X0 ) @ ( c_antisymmetric_i @ X0 ) ) @ ( c_transitive_i @ X0 ) ) ) ) ).
thf(ax39,axiom,
( c_totalorder_i
= ( ^ [X0: $i > $i > $o] : ( c_and @ ( c_partialorder_i @ X0 ) @ ( c_linear_i @ X0 ) ) ) ) ).
thf(ax40,axiom,
( c_strictpartialorder_i
= ( ^ [X0: $i > $i > $o] : ( c_and @ ( c_irreflexive_i @ X0 ) @ ( c_transitive_i @ X0 ) ) ) ) ).
thf(ax41,axiom,
( c_stricttotalorder_i
= ( ^ [X0: $i > $i > $o] : ( c_and @ ( c_strictpartialorder_i @ X0 ) @ ( c_trichotomous_or_i @ X0 ) ) ) ) ).
thf(ax42,axiom,
( c_If_i
= ( ^ [X0: $o,X1: $i,X2: $i] :
( c_Eps_i
@ ^ [X3: $i] : ( c_or @ ( c_and @ X0 @ ( X3 = X1 ) ) @ ( c_and @ ( c_not @ X0 ) @ ( X3 = X2 ) ) ) ) ) ) ).
thf(ax43,axiom,
( c_exactly1of2
= ( ^ [X0: $o,X1: $o] : ( c_or @ ( c_and @ X0 @ ( c_not @ X1 ) ) @ ( c_and @ ( c_not @ X0 ) @ X1 ) ) ) ) ).
thf(ax44,axiom,
( c_exactly1of3
= ( ^ [X0: $o,X1: $o,X2: $o] : ( c_or @ ( c_and @ ( c_exactly1of2 @ X0 @ X1 ) @ ( c_not @ X2 ) ) @ ( c_and @ ( c_and @ ( c_not @ X0 ) @ ( c_not @ X1 ) ) @ X2 ) ) ) ) ).
thf(ax45,axiom,
( c_nIn
= ( ^ [X0: $i,X1: $i] : ( c_not @ ( c_In @ X0 @ X1 ) ) ) ) ).
thf(ax46,axiom,
( c_nSubq
= ( ^ [X0: $i,X1: $i] : ( c_not @ ( c_Subq @ X0 @ X1 ) ) ) ) ).
thf(ax47,axiom,
( c_UPair
= ( ^ [X0: $i,X1: $i] :
( c_Repl @ ( c_Power @ ( c_Power @ c_Empty ) )
@ ^ [X2: $i] : ( c_If_i @ ( c_In @ c_Empty @ X2 ) @ X0 @ X1 ) ) ) ) ).
thf(ax48,axiom,
( c_Sing
= ( ^ [X0: $i] : ( c_UPair @ X0 @ X0 ) ) ) ).
thf(ax49,axiom,
( c_binunion
= ( ^ [X0: $i,X1: $i] : ( c_Union @ ( c_UPair @ X0 @ X1 ) ) ) ) ).
thf(ax50,axiom,
( c_SetAdjoin
= ( ^ [X0: $i,X1: $i] : ( c_binunion @ X0 @ ( c_Sing @ X1 ) ) ) ) ).
thf(ax51,axiom,
( c_famunion
= ( ^ [X0: $i,X1: $i > $i] :
( c_Union
@ ( c_Repl @ X0
@ ^ [X2: $i] : ( X1 @ X2 ) ) ) ) ) ).
thf(ax52,axiom,
( c_Sep
= ( ^ [X0: $i,X1: $i > $o] :
( c_If_i
@ ? [X2: $i] : ( c_and @ ( c_In @ X2 @ X0 ) @ ( X1 @ X2 ) )
@ ( c_Repl @ X0
@ ^ [X2: $i] :
( ^ [X3: $i] :
( c_If_i @ ( X1 @ X3 ) @ X3
@ ( c_Eps_i
@ ^ [X4: $i] : ( c_and @ ( c_In @ X4 @ X0 ) @ ( X1 @ X4 ) ) ) )
@ X2 ) )
@ c_Empty ) ) ) ).
thf(ax53,axiom,
( c_ReplSep
= ( ^ [X0: $i,X1: $i > $o,X2: $i > $i] :
( c_Repl
@ ( c_Sep @ X0
@ ^ [X3: $i] : ( X1 @ X3 ) )
@ ^ [X3: $i] : ( X2 @ X3 ) ) ) ) ).
thf(ax54,axiom,
( c_binintersect
= ( ^ [X0: $i,X1: $i] :
( c_Sep @ X0
@ ^ [X2: $i] : ( c_In @ X2 @ X1 ) ) ) ) ).
thf(ax55,axiom,
( c_setminus
= ( ^ [X0: $i,X1: $i] :
( c_Sep @ X0
@ ^ [X2: $i] : ( c_nIn @ X2 @ X1 ) ) ) ) ).
thf(ax56,axiom,
( c_inj
= ( ^ [X0: $i,X1: $i,X2: $i > $i] :
( c_and
@ ! [X3: $i] :
( ( c_In @ X3 @ X0 )
=> ( c_In @ ( X2 @ X3 ) @ X1 ) )
@ ! [X3: $i] :
( ( c_In @ X3 @ X0 )
=> ! [X4: $i] :
( ( c_In @ X4 @ X0 )
=> ( ( ( X2 @ X3 )
= ( X2 @ X4 ) )
=> ( X3 = X4 ) ) ) ) ) ) ) ).
thf(ax57,axiom,
( c_bij
= ( ^ [X0: $i,X1: $i,X2: $i > $i] :
( c_and @ ( c_inj @ X0 @ X1 @ X2 )
@ ! [X3: $i] :
( ( c_In @ X3 @ X1 )
=> ? [X4: $i] :
( c_and @ ( c_In @ X4 @ X0 )
@ ( ( X2 @ X4 )
= X3 ) ) ) ) ) ) ).
thf(ax58,axiom,
( c_atleastp
= ( ^ [X0: $i,X1: $i] :
? [X2: $i > $i] : ( c_inj @ X0 @ X1 @ X2 ) ) ) ).
thf(ax59,axiom,
( c_equip
= ( ^ [X0: $i,X1: $i] :
? [X2: $i > $i] : ( c_bij @ X0 @ X1 @ X2 ) ) ) ).
thf(ax60,axiom,
( c_In_rec_poly_G_i
= ( ^ [X0: $i > ( $i > $i ) > $i,X1: $i,X2: $i] :
! [X3: $i > $i > $o] :
( ! [X4: $i,X5: $i > $i] :
( ! [X6: $i] :
( ( c_In @ X6 @ X4 )
=> ( X3 @ X6 @ ( X5 @ X6 ) ) )
=> ( X3 @ X4 @ ( X0 @ X4 @ X5 ) ) )
=> ( X3 @ X1 @ X2 ) ) ) ) ).
thf(ax61,axiom,
( c_In_rec_poly_i
= ( ^ [X0: $i > ( $i > $i ) > $i,X1: $i] :
( c_Eps_i
@ ^ [X2: $i] : ( c_In_rec_poly_G_i @ X0 @ X1 @ X2 ) ) ) ) ).
thf(ax62,axiom,
( c_ordsucc
= ( ^ [X0: $i] : ( c_binunion @ X0 @ ( c_Sing @ X0 ) ) ) ) ).
thf(ax63,axiom,
( c_nat_p
= ( ^ [X0: $i] :
! [X1: $i > $o] :
( ( X1 @ c_Empty )
=> ( ! [X2: $i] :
( ( X1 @ X2 )
=> ( X1 @ ( c_ordsucc @ X2 ) ) )
=> ( X1 @ X0 ) ) ) ) ) ).
thf(ax64,axiom,
( c_nat_primrec
= ( ^ [X0: $i,X1: $i > $i > $i] :
( c_In_rec_poly_i
@ ^ [X2: $i,X3: $i > $i] : ( c_If_i @ ( c_In @ ( c_Union @ X2 ) @ X2 ) @ ( X1 @ ( c_Union @ X2 ) @ ( X3 @ ( c_Union @ X2 ) ) ) @ X0 ) ) ) ) ).
thf(ax65,axiom,
( c_add_nat
= ( ^ [X0: $i,X1: $i] :
( c_nat_primrec @ X0
@ ^ [X2: $i,X3: $i] : ( c_ordsucc @ X3 )
@ X1 ) ) ) ).
thf(ax66,axiom,
( c_mul_nat
= ( ^ [X0: $i,X1: $i] :
( c_nat_primrec @ c_Empty
@ ^ [X2: $i,X3: $i] : ( c_add_nat @ X0 @ X3 )
@ X1 ) ) ) ).
thf(ax67,axiom,
( c_ordinal
= ( ^ [X0: $i] :
( c_and @ ( c_TransSet @ X0 )
@ ! [X1: $i] :
( ( c_In @ X1 @ X0 )
=> ( c_TransSet @ X1 ) ) ) ) ) ).
thf(ax68,axiom,
( c_V_
= ( c_In_rec_poly_i
@ ^ [X0: $i,X1: $i > $i] :
( c_famunion @ X0
@ ^ [X2: $i] : ( c_Power @ ( X1 @ X2 ) ) ) ) ) ).
thf(ax69,axiom,
( c_Inj1
= ( c_In_rec_poly_i
@ ^ [X0: $i,X1: $i > $i] :
( c_binunion @ ( c_Sing @ c_Empty )
@ ( c_Repl @ X0
@ ^ [X2: $i] : ( X1 @ X2 ) ) ) ) ) ).
thf(ax70,axiom,
( c_Inj0
= ( ^ [X0: $i] :
( c_Repl @ X0
@ ^ [X1: $i] : ( c_Inj1 @ X1 ) ) ) ) ).
thf(ax71,axiom,
( c_Unj
= ( c_In_rec_poly_i
@ ^ [X0: $i,X1: $i > $i] :
( c_Repl @ ( c_setminus @ X0 @ ( c_Sing @ c_Empty ) )
@ ^ [X2: $i] : ( X1 @ X2 ) ) ) ) ).
thf(ax72,axiom,
( c_combine_funcs
= ( ^ [X0: $i,X1: $i,X2: $i > $i,X3: $i > $i,X4: $i] :
( c_If_i
@ ( X4
= ( c_Inj0 @ ( c_Unj @ X4 ) ) )
@ ( X2 @ ( c_Unj @ X4 ) )
@ ( X3 @ ( c_Unj @ X4 ) ) ) ) ) ).
thf(ax73,axiom,
( c_setsum
= ( ^ [X0: $i,X1: $i] :
( c_binunion
@ ( c_Repl @ X0
@ ^ [X2: $i] : ( c_Inj0 @ X2 ) )
@ ( c_Repl @ X1
@ ^ [X2: $i] : ( c_Inj1 @ X2 ) ) ) ) ) ).
thf(ax74,axiom,
( c_proj0
= ( ^ [X0: $i] :
( c_ReplSep @ X0
@ ^ [X1: $i] :
? [X2: $i] :
( ( c_Inj0 @ X2 )
= X1 )
@ ^ [X1: $i] : ( c_Unj @ X1 ) ) ) ) ).
thf(ax75,axiom,
( c_proj1
= ( ^ [X0: $i] :
( c_ReplSep @ X0
@ ^ [X1: $i] :
? [X2: $i] :
( ( c_Inj1 @ X2 )
= X1 )
@ ^ [X1: $i] : ( c_Unj @ X1 ) ) ) ) ).
thf(ax76,axiom,
( c_binrep
= ( ^ [X0: $i,X1: $i] : ( c_setsum @ X0 @ ( c_Power @ X1 ) ) ) ) ).
thf(ax77,axiom,
( c_lam
= ( ^ [X0: $i,X1: $i > $i] :
( c_famunion @ X0
@ ^ [X2: $i] :
( c_Repl @ ( X1 @ X2 )
@ ^ [X3: $i] : ( c_setsum @ X2 @ X3 ) ) ) ) ) ).
thf(ax78,axiom,
( c_setprod
= ( ^ [X0: $i,X1: $i] :
( c_lam @ X0
@ ^ [X2: $i] : X1 ) ) ) ).
thf(ax79,axiom,
( c_ap
= ( ^ [X0: $i,X1: $i] :
( c_ReplSep @ X0
@ ^ [X2: $i] :
? [X3: $i] :
( X2
= ( c_setsum @ X1 @ X3 ) )
@ ^ [X2: $i] : ( c_proj1 @ X2 ) ) ) ) ).
thf(ax80,axiom,
( c_setsum_p
= ( ^ [X0: $i] :
( ( c_setsum @ ( c_ap @ X0 @ c_Empty ) @ ( c_ap @ X0 @ ( c_ordsucc @ c_Empty ) ) )
= X0 ) ) ) ).
thf(ax81,axiom,
( c_tuple_p
= ( ^ [X0: $i,X1: $i] :
! [X2: $i] :
( ( c_In @ X2 @ X1 )
=> ? [X3: $i] :
( c_and @ ( c_In @ X3 @ X0 )
@ ? [X4: $i] :
( X2
= ( c_setsum @ X3 @ X4 ) ) ) ) ) ) ).
thf(ax82,axiom,
( c_Pi
= ( ^ [X0: $i,X1: $i > $i] :
( c_Sep
@ ( c_Power
@ ( c_lam @ X0
@ ^ [X2: $i] : ( c_Union @ ( X1 @ X2 ) ) ) )
@ ^ [X2: $i] :
! [X3: $i] :
( ( c_In @ X3 @ X0 )
=> ( c_In @ ( c_ap @ X2 @ X3 ) @ ( X1 @ X3 ) ) ) ) ) ) ).
thf(ax83,axiom,
( c_setexp
= ( ^ [X0: $i,X1: $i] :
( c_Pi @ X1
@ ^ [X2: $i] : X0 ) ) ) ).
thf(ax84,axiom,
( c_Sep2
= ( ^ [X0: $i,X1: $i > $i,X2: $i > $i > $o] :
( c_Sep
@ ( c_lam @ X0
@ ^ [X3: $i] : ( X1 @ X3 ) )
@ ^ [X3: $i] : ( X2 @ ( c_ap @ X3 @ c_Empty ) @ ( c_ap @ X3 @ ( c_ordsucc @ c_Empty ) ) ) ) ) ) ).
thf(ax85,axiom,
( c_set_of_pairs
= ( ^ [X0: $i] :
! [X1: $i] :
( ( c_In @ X1 @ X0 )
=> ? [X2: $i,X3: $i] :
( X1
= ( c_lam @ ( c_ordsucc @ ( c_ordsucc @ c_Empty ) )
@ ^ [X4: $i] : ( c_If_i @ ( X4 = c_Empty ) @ X2 @ X3 ) ) ) ) ) ) ).
thf(ax86,axiom,
( c_lam2
= ( ^ [X0: $i,X1: $i > $i,X2: $i > $i > $i] :
( c_lam @ X0
@ ^ [X3: $i] :
( c_lam @ ( X1 @ X3 )
@ ^ [X4: $i] : ( X2 @ X3 @ X4 ) ) ) ) ) ).
thf(ax87,axiom,
( c_PNoEq_
= ( ^ [X0: $i,X1: $i > $o,X2: $i > $o] :
! [X3: $i] :
( ( c_In @ X3 @ X0 )
=> ( c_iff @ ( X1 @ X3 ) @ ( X2 @ X3 ) ) ) ) ) ).
thf(ax88,axiom,
( c_PNoLt_
= ( ^ [X0: $i,X1: $i > $o,X2: $i > $o] :
? [X3: $i] : ( c_and @ ( c_In @ X3 @ X0 ) @ ( c_and @ ( c_and @ ( c_PNoEq_ @ X3 @ X1 @ X2 ) @ ( c_not @ ( X1 @ X3 ) ) ) @ ( X2 @ X3 ) ) ) ) ) ).
thf(ax89,axiom,
( c_PNoLt
= ( ^ [X0: $i,X1: $i > $o,X2: $i,X3: $i > $o] : ( c_or @ ( c_or @ ( c_PNoLt_ @ ( c_binintersect @ X0 @ X2 ) @ X1 @ X3 ) @ ( c_and @ ( c_and @ ( c_In @ X0 @ X2 ) @ ( c_PNoEq_ @ X0 @ X1 @ X3 ) ) @ ( X3 @ X0 ) ) ) @ ( c_and @ ( c_and @ ( c_In @ X2 @ X0 ) @ ( c_PNoEq_ @ X2 @ X1 @ X3 ) ) @ ( c_not @ ( X1 @ X2 ) ) ) ) ) ) ).
thf(ax90,axiom,
( c_PNoLe
= ( ^ [X0: $i,X1: $i > $o,X2: $i,X3: $i > $o] : ( c_or @ ( c_PNoLt @ X0 @ X1 @ X2 @ X3 ) @ ( c_and @ ( X0 = X2 ) @ ( c_PNoEq_ @ X0 @ X1 @ X3 ) ) ) ) ) ).
thf(ax91,axiom,
( c_PNo_downc
= ( ^ [X0: $i > ( $i > $o ) > $o,X1: $i,X2: $i > $o] :
? [X3: $i] :
( c_and @ ( c_ordinal @ X3 )
@ ? [X4: $i > $o] : ( c_and @ ( X0 @ X3 @ X4 ) @ ( c_PNoLe @ X1 @ X2 @ X3 @ X4 ) ) ) ) ) ).
thf(ax92,axiom,
( c_PNo_upc
= ( ^ [X0: $i > ( $i > $o ) > $o,X1: $i,X2: $i > $o] :
? [X3: $i] :
( c_and @ ( c_ordinal @ X3 )
@ ? [X4: $i > $o] : ( c_and @ ( X0 @ X3 @ X4 ) @ ( c_PNoLe @ X3 @ X4 @ X1 @ X2 ) ) ) ) ) ).
thf(ax93,axiom,
( c_SNoElts_
= ( ^ [X0: $i] :
( c_binunion @ X0
@ ( c_Repl @ X0
@ ^ [X1: $i] :
( ^ [X2: $i] : ( c_SetAdjoin @ X2 @ ( c_Sing @ ( c_ordsucc @ c_Empty ) ) )
@ X1 ) ) ) ) ) ).
thf(ax94,axiom,
( c_SNo_
= ( ^ [X0: $i,X1: $i] :
( c_and @ ( c_Subq @ X1 @ ( c_SNoElts_ @ X0 ) )
@ ! [X2: $i] :
( ( c_In @ X2 @ X0 )
=> ( c_exactly1of2
@ ( c_In
@ ( ^ [X3: $i] : ( c_SetAdjoin @ X3 @ ( c_Sing @ ( c_ordsucc @ c_Empty ) ) )
@ X2 )
@ X1 )
@ ( c_In @ X2 @ X1 ) ) ) ) ) ) ).
thf(ax95,axiom,
( c_PSNo
= ( ^ [X0: $i,X1: $i > $o] :
( c_binunion
@ ( c_Sep @ X0
@ ^ [X2: $i] : ( X1 @ X2 ) )
@ ( c_ReplSep @ X0
@ ^ [X2: $i] : ( c_not @ ( X1 @ X2 ) )
@ ^ [X2: $i] :
( ^ [X3: $i] : ( c_SetAdjoin @ X3 @ ( c_Sing @ ( c_ordsucc @ c_Empty ) ) )
@ X2 ) ) ) ) ) ).
thf(ax96,axiom,
( c_SNo
= ( ^ [X0: $i] :
? [X1: $i] : ( c_and @ ( c_ordinal @ X1 ) @ ( c_SNo_ @ X1 @ X0 ) ) ) ) ).
thf(ax97,axiom,
( c_SNoLev
= ( ^ [X0: $i] :
( c_Eps_i
@ ^ [X1: $i] : ( c_and @ ( c_ordinal @ X1 ) @ ( c_SNo_ @ X1 @ X0 ) ) ) ) ) ).
thf(ax98,axiom,
( c_SNoEq_
= ( ^ [X0: $i,X1: $i,X2: $i] :
( c_PNoEq_ @ X0
@ ^ [X3: $i] : ( c_In @ X3 @ X1 )
@ ^ [X3: $i] : ( c_In @ X3 @ X2 ) ) ) ) ).
thf(ax99,axiom,
( c_SNoLt
= ( ^ [X0: $i,X1: $i] :
( c_PNoLt @ ( c_SNoLev @ X0 )
@ ^ [X2: $i] : ( c_In @ X2 @ X0 )
@ ( c_SNoLev @ X1 )
@ ^ [X2: $i] : ( c_In @ X2 @ X1 ) ) ) ) ).
thf(ax100,axiom,
( c_SNoLe
= ( ^ [X0: $i,X1: $i] :
( c_PNoLe @ ( c_SNoLev @ X0 )
@ ^ [X2: $i] : ( c_In @ X2 @ X0 )
@ ( c_SNoLev @ X1 )
@ ^ [X2: $i] : ( c_In @ X2 @ X1 ) ) ) ) ).
thf(ax101,axiom,
( c_binop_on
= ( ^ [X0: $i,X1: $i > $i > $i] :
! [X2: $i] :
( ( c_In @ X2 @ X0 )
=> ! [X3: $i] :
( ( c_In @ X3 @ X0 )
=> ( c_In @ ( X1 @ X2 @ X3 ) @ X0 ) ) ) ) ) ).
thf(ax102,axiom,
( c_Loop
= ( ^ [X0: $i,X1: $i > $i > $i,X2: $i > $i > $i,X3: $i > $i > $i,X4: $i] :
( c_and
@ ( c_and @ ( c_and @ ( c_and @ ( c_binop_on @ X0 @ X1 ) @ ( c_binop_on @ X0 @ X2 ) ) @ ( c_binop_on @ X0 @ X3 ) )
@ ! [X5: $i] :
( ( c_In @ X5 @ X0 )
=> ( c_and
@ ( ( X1 @ X4 @ X5 )
= X5 )
@ ( ( X1 @ X5 @ X4 )
= X5 ) ) ) )
@ ! [X5: $i] :
( ( c_In @ X5 @ X0 )
=> ! [X6: $i] :
( ( c_In @ X6 @ X0 )
=> ( c_and
@ ( c_and
@ ( c_and
@ ( ( X2 @ X5 @ ( X1 @ X5 @ X6 ) )
= X6 )
@ ( ( X1 @ X5 @ ( X2 @ X5 @ X6 ) )
= X6 ) )
@ ( ( X3 @ ( X1 @ X5 @ X6 ) @ X6 )
= X5 ) )
@ ( ( X1 @ ( X3 @ X5 @ X6 ) @ X6 )
= X5 ) ) ) ) ) ) ) ).
thf(ax103,axiom,
( c_Loop_with_defs
= ( ^ [X0: $i,X1: $i > $i > $i,X2: $i > $i > $i,X3: $i > $i > $i,X4: $i,X5: $i > $i > $i,X6: $i > $i > $i > $i,X7: $i > $i > $i,X8: $i > $i > $i > $i,X9: $i > $i > $i > $i,X10: $i > $i > $i,X11: $i > $i > $i,X12: $i > $i > $i,X13: $i > $i > $i] :
( c_and
@ ( c_and
@ ( c_and
@ ( c_and @ ( c_Loop @ X0 @ X1 @ X2 @ X3 @ X4 )
@ ! [X14: $i] :
( ( c_In @ X14 @ X0 )
=> ! [X15: $i] :
( ( c_In @ X15 @ X0 )
=> ( ( X5 @ X14 @ X15 )
= ( X2 @ ( X1 @ X15 @ X14 ) @ ( X1 @ X14 @ X15 ) ) ) ) ) )
@ ! [X14: $i] :
( ( c_In @ X14 @ X0 )
=> ! [X15: $i] :
( ( c_In @ X15 @ X0 )
=> ! [X16: $i] :
( ( c_In @ X16 @ X0 )
=> ( ( X6 @ X14 @ X15 @ X16 )
= ( X2 @ ( X1 @ X14 @ ( X1 @ X15 @ X16 ) ) @ ( X1 @ ( X1 @ X14 @ X15 ) @ X16 ) ) ) ) ) ) )
@ ! [X14: $i] :
( ( c_In @ X14 @ X0 )
=> ! [X15: $i] :
( ( c_In @ X15 @ X0 )
=> ( c_and
@ ( c_and
@ ( c_and
@ ( c_and
@ ( ( X7 @ X14 @ X15 )
= ( X2 @ X14 @ ( X1 @ X15 @ X14 ) ) )
@ ( ( X10 @ X14 @ X15 )
= ( X1 @ X14 @ ( X1 @ X15 @ ( X2 @ X14 @ X4 ) ) ) ) )
@ ( ( X11 @ X14 @ X15 )
= ( X1 @ ( X1 @ ( X3 @ X4 @ X14 ) @ X15 ) @ X14 ) ) )
@ ( ( X12 @ X14 @ X15 )
= ( X1 @ ( X2 @ X14 @ X15 ) @ ( X2 @ ( X2 @ X14 @ X4 ) @ X4 ) ) ) )
@ ( ( X13 @ X14 @ X15 )
= ( X1 @ ( X3 @ X4 @ ( X3 @ X4 @ X14 ) ) @ ( X3 @ X15 @ X14 ) ) ) ) ) ) )
@ ! [X14: $i] :
( ( c_In @ X14 @ X0 )
=> ! [X15: $i] :
( ( c_In @ X15 @ X0 )
=> ! [X16: $i] :
( ( c_In @ X16 @ X0 )
=> ( c_and
@ ( ( X8 @ X14 @ X15 @ X16 )
= ( X2 @ ( X1 @ X15 @ X14 ) @ ( X1 @ X15 @ ( X1 @ X14 @ X16 ) ) ) )
@ ( ( X9 @ X14 @ X15 @ X16 )
= ( X3 @ ( X1 @ ( X1 @ X16 @ X14 ) @ X15 ) @ ( X1 @ X14 @ X15 ) ) ) ) ) ) ) ) ) ) ).
thf(ax104,axiom,
( c_Loop_with_defs_cex1
= ( ^ [X0: $i,X1: $i > $i > $i,X2: $i > $i > $i,X3: $i > $i > $i,X4: $i,X5: $i > $i > $i,X6: $i > $i > $i > $i,X7: $i > $i > $i,X8: $i > $i > $i > $i,X9: $i > $i > $i > $i,X10: $i > $i > $i,X11: $i > $i > $i,X12: $i > $i > $i,X13: $i > $i > $i] :
( c_and @ ( c_Loop_with_defs @ X0 @ X1 @ X2 @ X3 @ X4 @ X5 @ X6 @ X7 @ X8 @ X9 @ X10 @ X11 @ X12 @ X13 )
@ ? [X14: $i] :
( c_and @ ( c_In @ X14 @ X0 )
@ ? [X15: $i] :
( c_and @ ( c_In @ X15 @ X0 )
@ ? [X16: $i] :
( c_and @ ( c_In @ X16 @ X0 )
@ ? [X17: $i] :
( c_and @ ( c_In @ X17 @ X0 )
@ ( c_not
@ ( ( X5 @ ( X1 @ ( X2 @ ( X8 @ X15 @ X16 @ X14 ) @ X4 ) @ X14 ) @ X17 )
= X4 ) ) ) ) ) ) ) ) ) ).
thf(ax105,axiom,
( c_Loop_with_defs_cex2
= ( ^ [X0: $i,X1: $i > $i > $i,X2: $i > $i > $i,X3: $i > $i > $i,X4: $i,X5: $i > $i > $i,X6: $i > $i > $i > $i,X7: $i > $i > $i,X8: $i > $i > $i > $i,X9: $i > $i > $i > $i,X10: $i > $i > $i,X11: $i > $i > $i,X12: $i > $i > $i,X13: $i > $i > $i] :
( c_and @ ( c_Loop_with_defs @ X0 @ X1 @ X2 @ X3 @ X4 @ X5 @ X6 @ X7 @ X8 @ X9 @ X10 @ X11 @ X12 @ X13 )
@ ? [X14: $i] :
( c_and @ ( c_In @ X14 @ X0 )
@ ? [X15: $i] :
( c_and @ ( c_In @ X15 @ X0 )
@ ? [X16: $i] :
( c_and @ ( c_In @ X16 @ X0 )
@ ? [X17: $i] :
( c_and @ ( c_In @ X17 @ X0 )
@ ? [X18: $i] :
( c_and @ ( c_In @ X18 @ X0 )
@ ( c_not
@ ( ( X6 @ X18 @ ( X1 @ ( X3 @ X4 @ X14 ) @ ( X9 @ X15 @ X16 @ X14 ) ) @ X17 )
= X4 ) ) ) ) ) ) ) ) ) ) ).
thf(ax106,axiom,
( c_combinator
= ( ^ [X0: $i] :
! [X1: $i > $o] :
( ( X1 @ ( c_Inj0 @ c_Empty ) )
=> ( ( X1 @ ( c_Inj0 @ ( c_Power @ c_Empty ) ) )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X1 @ ( c_Inj1 @ ( c_setsum @ X2 @ X3 ) ) ) ) )
=> ( X1 @ X0 ) ) ) ) ) ) ).
thf(ax107,axiom,
( c_combinator_equiv
= ( ^ [X0: $i,X1: $i] :
! [X2: $i > $i > $o] :
( ^ [X3: $i,X4: $i,X5: $i > $i > $i] :
( ( c_per_i @ X2 )
=> ( ! [X6: $i] :
( ( c_combinator @ X6 )
=> ( X2 @ X6 @ X6 ) )
=> ( ! [X6: $i,X7: $i,X8: $i,X9: $i] :
( ( c_combinator @ X6 )
=> ( ( c_combinator @ X7 )
=> ( ( c_combinator @ X8 )
=> ( ( c_combinator @ X9 )
=> ( ( X2 @ X6 @ X8 )
=> ( ( X2 @ X7 @ X9 )
=> ( X2 @ ( X5 @ X6 @ X7 ) @ ( X5 @ X8 @ X9 ) ) ) ) ) ) ) )
=> ( ! [X6: $i,X7: $i] : ( X2 @ ( X5 @ ( X5 @ X3 @ X6 ) @ X7 ) @ X6 )
=> ( ! [X6: $i,X7: $i,X8: $i] : ( X2 @ ( X5 @ ( X5 @ ( X5 @ X4 @ X6 ) @ X7 ) @ X8 ) @ ( X5 @ ( X5 @ X6 @ X8 ) @ ( X5 @ X7 @ X8 ) ) )
=> ( X2 @ X0 @ X1 ) ) ) ) ) )
@ ( c_Inj0 @ c_Empty )
@ ( c_Inj0 @ ( c_Power @ c_Empty ) )
@ ^ [X3: $i,X4: $i] : ( c_Inj1 @ ( c_setsum @ X3 @ X4 ) ) ) ) ) ).
thf(ax108,axiom,
( c_equip_mod
= ( ^ [X0: $i,X1: $i,X2: $i] :
? [X3: $i,X4: $i] : ( c_or @ ( c_and @ ( c_equip @ ( c_setsum @ X0 @ X3 ) @ X1 ) @ ( c_equip @ ( c_setprod @ X4 @ X3 ) @ X2 ) ) @ ( c_and @ ( c_equip @ ( c_setsum @ X1 @ X3 ) @ X0 ) @ ( c_equip @ ( c_setprod @ X4 @ X3 ) @ X2 ) ) ) ) ) ).
thf(conj,conjecture,
( c_not
@ ! [X0: ( $i > $i > ( ( $i > $i ) > $i > $i ) > $i ) > $i > $i > $o,X1: ( $i > ( ( $i > $i ) > $i ) > $i ) > $i > $o,X2: ( $i > $i ) > ( ( ( ( $i > $i ) > $i ) > $i ) > ( $i > $i > $i ) > $i > $i > $i ) > $i > $o,X3: ( $i > ( $i > $i > $i ) > $i ) > $i > $i > $o] :
( ! [X4: ( ( $i > $i > $i ) > ( $i > $i ) > $i > $i ) > $i,X5: $i,X6: $i,X7: $i] :
( ( c_In @ ( c_Inj0 @ X6 ) @ ( c_setsum @ ( c_setsum @ c_Empty @ c_Empty ) @ ( c_setsum @ X6 @ ( c_Inj0 @ ( c_setsum @ c_Empty @ c_Empty ) ) ) ) )
=> ( ( X2
@ ^ [X8: $i] : ( c_Inj0 @ ( c_Inj0 @ c_Empty ) )
@ ^ [X8: ( ( $i > $i ) > $i ) > $i,X9: $i > $i > $i,X10: $i,X11: $i] : X10
@ ( c_setsum
@ ( X4
@ ^ [X8: $i > $i > $i,X9: $i > $i,X10: $i] : ( c_Inj0 @ ( c_setsum @ c_Empty @ c_Empty ) ) )
@ c_Empty ) )
=> ( X3
@ ^ [X8: $i,X9: $i > $i > $i] : ( c_Inj0 @ X8 )
@ X5
@ ( c_setsum @ ( c_setsum @ ( c_setsum @ c_Empty @ ( c_setsum @ c_Empty @ c_Empty ) ) @ ( c_Inj0 @ c_Empty ) ) @ c_Empty ) ) ) )
=> ( ! [X4: ( ( ( $i > $i ) > $i > $i ) > $i ) > $i,X5: $i > $i,X6: $i,X7: $i] :
( ( X3
@ ^ [X8: $i,X9: $i > $i > $i] : c_Empty
@ X7
@ ( c_setsum @ ( c_setsum @ ( c_setsum @ ( c_setsum @ c_Empty @ c_Empty ) @ c_Empty ) @ ( c_setsum @ ( c_setsum @ c_Empty @ c_Empty ) @ ( c_Inj0 @ c_Empty ) ) ) @ ( c_setsum @ c_Empty @ ( c_Inj1 @ ( c_setsum @ c_Empty @ c_Empty ) ) ) ) )
=> ( X3
@ ^ [X8: $i,X9: $i > $i > $i] : X7
@ ( c_setsum
@ ( X4
@ ^ [X8: ( $i > $i ) > $i > $i] : X7 )
@ X6 )
@ ( c_Inj0 @ ( c_Inj1 @ ( c_setsum @ ( c_setsum @ c_Empty @ c_Empty ) @ ( c_setsum @ c_Empty @ c_Empty ) ) ) ) ) )
=> ( ! [X4: $i,X5: $i > ( ( $i > $i ) > $i > $i ) > $i,X6: ( ( ( $i > $i ) > $i > $i ) > ( $i > $i ) > $i > $i ) > ( $i > $i > $i ) > $i,X7: $i] :
( ( c_In @ ( c_Inj0 @ c_Empty )
@ ( c_setsum @ c_Empty
@ ( X5 @ ( c_Inj0 @ ( c_setsum @ c_Empty @ c_Empty ) )
@ ^ [X8: $i > $i,X9: $i] : ( c_Inj0 @ c_Empty ) ) ) )
=> ( ( X0
@ ^ [X8: $i,X9: $i,X10: ( $i > $i ) > $i > $i] : X8
@ X7
@ ( X6
@ ^ [X8: ( $i > $i ) > $i > $i,X9: $i > $i,X10: $i] : ( c_Inj1 @ c_Empty )
@ ^ [X8: $i,X9: $i] : c_Empty ) )
=> ( X2
@ ^ [X8: $i] : X8
@ ^ [X8: ( ( $i > $i ) > $i ) > $i,X9: $i > $i > $i,X10: $i,X11: $i] : X10
@ ( c_setsum @ ( c_Inj0 @ c_Empty )
@ ( X6
@ ^ [X8: ( $i > $i ) > $i > $i,X9: $i > $i,X10: $i] : ( c_Inj0 @ c_Empty )
@ ^ [X8: $i,X9: $i] : X9 ) ) ) ) )
=> ( ! [X4: $i > $i > $i,X5: $i > $i,X6: $i,X7: $i > $i] :
( ( c_In @ ( c_Inj0 @ ( c_Inj1 @ c_Empty ) ) @ ( c_setsum @ ( c_setsum @ ( c_Inj0 @ c_Empty ) @ ( c_setsum @ ( c_setsum @ c_Empty @ c_Empty ) @ ( c_setsum @ c_Empty @ c_Empty ) ) ) @ ( c_setsum @ c_Empty @ ( c_setsum @ ( c_Inj0 @ c_Empty ) @ ( X4 @ c_Empty @ c_Empty ) ) ) ) )
=> ( ( X2
@ ^ [X8: $i] : c_Empty
@ ^ [X8: ( ( $i > $i ) > $i ) > $i,X9: $i > $i > $i,X10: $i,X11: $i] : c_Empty
@ ( c_setsum @ ( c_setsum @ c_Empty @ ( c_setsum @ ( c_setsum @ c_Empty @ c_Empty ) @ ( c_Inj0 @ c_Empty ) ) ) @ ( c_Inj1 @ X6 ) ) )
=> ( X1
@ ^ [X8: $i,X9: ( $i > $i ) > $i] : ( c_Inj0 @ c_Empty )
@ ( X4 @ c_Empty @ ( X5 @ ( c_setsum @ ( c_Inj1 @ c_Empty ) @ c_Empty ) ) ) ) ) )
=> ( ! [X4: $i,X5: $i > $i,X6: $i > $i,X7: $i] :
( ( X3
@ ^ [X8: $i,X9: $i > $i > $i] : c_Empty
@ c_Empty
@ c_Empty )
=> ( X1
@ ^ [X8: $i,X9: ( $i > $i ) > $i] :
( X9
@ ^ [X10: $i] : ( c_setsum @ ( c_Inj1 @ ( c_Inj0 @ c_Empty ) ) @ ( c_setsum @ c_Empty @ c_Empty ) ) )
@ c_Empty ) )
=> ( ! [X4: $i,X5: $i,X6: $i,X7: $i] :
( ( c_In @ ( c_Inj0 @ X6 ) @ ( c_setsum @ ( c_setsum @ X4 @ ( c_setsum @ X5 @ ( c_setsum @ c_Empty @ c_Empty ) ) ) @ ( c_Inj1 @ ( c_setsum @ X5 @ c_Empty ) ) ) )
=> ( ( X1
@ ^ [X8: $i,X9: ( $i > $i ) > $i] : X7
@ c_Empty )
=> ( X2
@ ^ [X8: $i] : c_Empty
@ ^ [X8: ( ( $i > $i ) > $i ) > $i,X9: $i > $i > $i,X10: $i,X11: $i] : ( c_setsum @ c_Empty @ c_Empty )
@ X5 ) ) )
=> ( ! [X4: ( ( ( $i > $i ) > $i > $i ) > $i ) > $i,X5: $i,X6: $i,X7: $i] :
( ( c_In @ ( c_Inj1 @ c_Empty ) @ X5 )
=> ( ( X1
@ ^ [X8: $i,X9: ( $i > $i ) > $i] : c_Empty
@ c_Empty )
=> ( X0
@ ^ [X8: $i,X9: $i,X10: ( $i > $i ) > $i > $i] : ( c_Inj1 @ c_Empty )
@ ( c_Inj1
@ ( c_Inj1
@ ( X4
@ ^ [X8: ( $i > $i ) > $i > $i] : c_Empty ) ) )
@ ( c_Inj1 @ ( c_setsum @ ( c_setsum @ ( c_Inj0 @ c_Empty ) @ c_Empty ) @ ( c_Inj1 @ ( c_Inj0 @ c_Empty ) ) ) ) ) ) )
=> ( ! [X4: $i > $i,X5: $i,X6: $i > $i,X7: $i] :
( ( X0
@ ^ [X8: $i,X9: $i,X10: ( $i > $i ) > $i > $i] : c_Empty
@ ( X4 @ c_Empty )
@ X7 )
=> c_False )
=> c_False ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------