TPTP Problem File: SLH0866^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : FOL_Seq_Calc2/0018_EPathHintikka/prob_00578_023761__13269484_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1641 ( 590 unt; 360 typ; 0 def)
% Number of atoms : 3693 (1075 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10273 ( 390 ~; 92 |; 204 &;8048 @)
% ( 0 <=>;1539 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 61 ( 60 usr)
% Number of type conns : 1111 (1111 >; 0 *; 0 +; 0 <<)
% Number of symbols : 303 ( 300 usr; 13 con; 0-4 aty)
% Number of variables : 3571 ( 288 ^;3170 !; 113 ?;3571 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 08:45:47.374
%------------------------------------------------------------------------------
% Could-be-implicit typings (60)
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% Explicit typings (300)
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thf(sy_c_Stream_Osmerge_001t__SeCaV__Ofm,type,
smerge_fm: stream_stream_fm > stream_fm ).
thf(sy_c_Stream_Osmerge_001t__SeCaV__Otm,type,
smerge_tm: stream_stream_tm > stream_tm ).
thf(sy_c_Stream_Osnth_001t__Nat__Onat,type,
snth_nat: stream_nat > nat > nat ).
thf(sy_c_Stream_Osnth_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
snth_P7093566783922538521ist_fm: stream4408948924543953275ist_fm > nat > produc6018962875968178549ist_fm ).
thf(sy_c_Stream_Osnth_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
snth_P6679518042731451922m_rule: stream2709947120125613254m_rule > nat > produc340336539035504054m_rule ).
thf(sy_c_Stream_Osnth_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
snth_P8853764340393315953m_rule: stream6210534828274662995m_rule > nat > produc8828831911945107917m_rule ).
thf(sy_c_Stream_Osnth_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_Mt__Prover__Orule_J,type,
snth_P6395297581347168077e_rule: stream8099677779113257519e_rule > nat > produc9112364199808626345e_rule ).
thf(sy_c_Stream_Osnth_001t__Product____Type__Oprod_It__Prover__Orule_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
snth_P5203088247548055629m_rule: stream8953843411776101167m_rule > nat > produc7920154866009513897m_rule ).
thf(sy_c_Stream_Osnth_001t__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J,type,
snth_P6178434775611311401e_rule: stream4385846686851721995e_rule > nat > produc7694839378271647877e_rule ).
thf(sy_c_Stream_Osnth_001t__Prover__Orule,type,
snth_rule: stream_rule > nat > rule ).
thf(sy_c_Stream_Osnth_001t__Real__Oreal,type,
snth_real: stream_real > nat > real ).
thf(sy_c_Stream_Osnth_001t__SeCaV__Ofm,type,
snth_fm: stream_fm > nat > fm ).
thf(sy_c_Stream_Osnth_001t__SeCaV__Otm,type,
snth_tm: stream_tm > nat > tm ).
thf(sy_c_Stream_Osnth_001t__Stream__Ostream_It__Nat__Onat_J,type,
snth_stream_nat: stream_stream_nat > nat > stream_nat ).
thf(sy_c_Stream_Osnth_001t__Stream__Ostream_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
snth_s6182113952396108578m_rule: stream3752074346242807894m_rule > nat > stream2709947120125613254m_rule ).
thf(sy_c_Stream_Osnth_001t__Stream__Ostream_It__Prover__Orule_J,type,
snth_stream_rule: stream_stream_rule > nat > stream_rule ).
thf(sy_c_Stream_Osnth_001t__Stream__Ostream_It__Real__Oreal_J,type,
snth_stream_real: stream_stream_real > nat > stream_real ).
thf(sy_c_Stream_Osnth_001t__Stream__Ostream_It__SeCaV__Ofm_J,type,
snth_stream_fm: stream_stream_fm > nat > stream_fm ).
thf(sy_c_Stream_Osnth_001t__Stream__Ostream_It__SeCaV__Otm_J,type,
snth_stream_tm: stream_stream_tm > nat > stream_tm ).
thf(sy_c_Stream_Ostream_Oshd_001t__Nat__Onat,type,
shd_nat: stream_nat > nat ).
thf(sy_c_Stream_Ostream_Oshd_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
shd_Pr3211216682057661985ist_fm: stream4408948924543953275ist_fm > produc6018962875968178549ist_fm ).
thf(sy_c_Stream_Ostream_Oshd_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
shd_Pr4562317740776619530m_rule: stream2709947120125613254m_rule > produc340336539035504054m_rule ).
thf(sy_c_Stream_Ostream_Oshd_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
shd_Pr4461660664618831993m_rule: stream6210534828274662995m_rule > produc8828831911945107917m_rule ).
thf(sy_c_Stream_Ostream_Oshd_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_Mt__Prover__Orule_J,type,
shd_Pr2400962586966563157e_rule: stream8099677779113257519e_rule > produc9112364199808626345e_rule ).
thf(sy_c_Stream_Ostream_Oshd_001t__Product____Type__Oprod_It__Prover__Orule_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
shd_Pr1208753253167450709m_rule: stream8953843411776101167m_rule > produc7920154866009513897m_rule ).
thf(sy_c_Stream_Ostream_Oshd_001t__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J,type,
shd_Pr2264621979884435249e_rule: stream4385846686851721995e_rule > produc7694839378271647877e_rule ).
thf(sy_c_Stream_Ostream_Oshd_001t__Prover__Orule,type,
shd_rule: stream_rule > rule ).
thf(sy_c_Stream_Ostream_Oshd_001t__Real__Oreal,type,
shd_real: stream_real > real ).
thf(sy_c_Stream_Ostream_Oshd_001t__SeCaV__Ofm,type,
shd_fm: stream_fm > fm ).
thf(sy_c_Stream_Ostream_Oshd_001t__SeCaV__Otm,type,
shd_tm: stream_tm > tm ).
thf(sy_c_Stream_Ostream_Osset_001t__Nat__Onat,type,
sset_nat: stream_nat > set_nat ).
thf(sy_c_Stream_Ostream_Osset_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
sset_P5379989128061332361ist_fm: stream4408948924543953275ist_fm > set_Pr5202636777678657877ist_fm ).
thf(sy_c_Stream_Ostream_Osset_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
sset_P4484857331586881186m_rule: stream2709947120125613254m_rule > set_Pr1822751329126368876m_rule ).
thf(sy_c_Stream_Ostream_Osset_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
sset_P3768201806174902753m_rule: stream6210534828274662995m_rule > set_Pr4971326047967503661m_rule ).
thf(sy_c_Stream_Ostream_Osset_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_Mt__Prover__Orule_J,type,
sset_P6986346827493259965e_rule: stream8099677779113257519e_rule > set_Pr2701264590931556105e_rule ).
thf(sy_c_Stream_Ostream_Osset_001t__Product____Type__Oprod_It__Prover__Orule_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
sset_P5794137493694147517m_rule: stream8953843411776101167m_rule > set_Pr3555430223594399753m_rule ).
thf(sy_c_Stream_Ostream_Osset_001t__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J,type,
sset_P1894962091411173529e_rule: stream4385846686851721995e_rule > set_Pr7340090144965549541e_rule ).
thf(sy_c_Stream_Ostream_Osset_001t__Prover__Orule,type,
sset_rule: stream_rule > set_rule ).
thf(sy_c_Stream_Ostream_Osset_001t__Real__Oreal,type,
sset_real: stream_real > set_real ).
thf(sy_c_Stream_Ostream_Osset_001t__SeCaV__Ofm,type,
sset_fm: stream_fm > set_fm ).
thf(sy_c_Stream_Ostream_Osset_001t__SeCaV__Otm,type,
sset_tm: stream_tm > set_tm ).
thf(sy_c_Stream_Ostream__all_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
stream2134157564947672044m_rule: ( produc340336539035504054m_rule > $o ) > stream2709947120125613254m_rule > $o ).
thf(sy_c_Stream_Ostream__all_001t__Prover__Orule,type,
stream_all_rule: ( rule > $o ) > stream_rule > $o ).
thf(sy_c_Stream_Oszip_001t__List__Olist_It__SeCaV__Otm_J_001t__List__Olist_It__SeCaV__Ofm_J,type,
szip_list_tm_list_fm: stream_list_tm > stream_list_fm > stream4408948924543953275ist_fm ).
thf(sy_c_Stream_Oszip_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_001t__Prover__Orule,type,
szip_P2924820683901490861m_rule: stream4408948924543953275ist_fm > stream_rule > stream2709947120125613254m_rule ).
thf(sy_c_Stream_Oszip_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
szip_P811719526838699976m_rule: stream2709947120125613254m_rule > stream2709947120125613254m_rule > stream6210534828274662995m_rule ).
thf(sy_c_Stream_Oszip_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_001t__Prover__Orule,type,
szip_P2044787555563963556e_rule: stream2709947120125613254m_rule > stream_rule > stream8099677779113257519e_rule ).
thf(sy_c_Stream_Oszip_001t__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J_001t__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J,type,
szip_P1965874896742159130e_rule: stream4385846686851721995e_rule > stream4385846686851721995e_rule > stream4681790265157167213e_rule ).
thf(sy_c_Stream_Oszip_001t__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J_001t__Prover__Orule,type,
szip_P4230510992641354557e_rule: stream4385846686851721995e_rule > stream_rule > stream128700168137323990e_rule ).
thf(sy_c_Stream_Oszip_001t__Prover__Orule_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
szip_r5451144443050508039ist_fm: stream_rule > stream4408948924543953275ist_fm > stream4490188412977367400ist_fm ).
thf(sy_c_Stream_Oszip_001t__Prover__Orule_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
szip_r6533552703345879972m_rule: stream_rule > stream2709947120125613254m_rule > stream8953843411776101167m_rule ).
thf(sy_c_Stream_Oszip_001t__Prover__Orule_001t__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J,type,
szip_r8329995387344033047e_rule: stream_rule > stream4385846686851721995e_rule > stream6316532330152022520e_rule ).
thf(sy_c_Stream_Oszip_001t__Prover__Orule_001t__Prover__Orule,type,
szip_rule_rule: stream_rule > stream_rule > stream4385846686851721995e_rule ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat3: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
member4699826688122452638ist_fm: produc6018962875968178549ist_fm > set_Pr5202636777678657877ist_fm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
member7231649785386036813m_rule: produc340336539035504054m_rule > set_Pr1822751329126368876m_rule > $o ).
thf(sy_c_member_001t__Prover__Orule,type,
member_rule3: rule > set_rule > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real3: real > set_real > $o ).
thf(sy_c_member_001t__SeCaV__Ofm,type,
member_fm3: fm > set_fm > $o ).
thf(sy_c_member_001t__SeCaV__Otm,type,
member_tm3: tm > set_tm > $o ).
thf(sy_v_n____,type,
n: nat ).
thf(sy_v_p____,type,
p: fm ).
thf(sy_v_steps,type,
steps: stream2709947120125613254m_rule ).
thf(sy_v_t____,type,
t: tm ).
% Relevant facts (1269)
thf(fact_0_t,axiom,
member_tm3 @ t @ ( terms @ ( tree_fms @ steps ) ) ).
% t
thf(fact_1_sdrop__szip,axiom,
! [N: nat,S1: stream_rule,S2: stream_rule] :
( ( sdrop_9113879250048157294e_rule @ N @ ( szip_rule_rule @ S1 @ S2 ) )
= ( szip_rule_rule @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).
% sdrop_szip
thf(fact_2_sdrop__szip,axiom,
! [N: nat,S1: stream4408948924543953275ist_fm,S2: stream_rule] :
( ( sdrop_8169176516188972301m_rule @ N @ ( szip_P2924820683901490861m_rule @ S1 @ S2 ) )
= ( szip_P2924820683901490861m_rule @ ( sdrop_9176333610110415838ist_fm @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).
% sdrop_szip
thf(fact_3_sdrop__szip,axiom,
! [N: nat,S1: stream2709947120125613254m_rule,S2: stream_rule] :
( ( sdrop_1938960342593938834e_rule @ N @ ( szip_P2044787555563963556e_rule @ S1 @ S2 ) )
= ( szip_P2044787555563963556e_rule @ ( sdrop_8169176516188972301m_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).
% sdrop_szip
thf(fact_4_sdrop__szip,axiom,
! [N: nat,S1: stream_rule,S2: stream2709947120125613254m_rule] :
( ( sdrop_746751008794826386m_rule @ N @ ( szip_r6533552703345879972m_rule @ S1 @ S2 ) )
= ( szip_r6533552703345879972m_rule @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_8169176516188972301m_rule @ N @ S2 ) ) ) ).
% sdrop_szip
thf(fact_5_sdrop__szip,axiom,
! [N: nat,S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule] :
( ( sdrop_7192298464603511222m_rule @ N @ ( szip_P811719526838699976m_rule @ S1 @ S2 ) )
= ( szip_P811719526838699976m_rule @ ( sdrop_8169176516188972301m_rule @ N @ S1 ) @ ( sdrop_8169176516188972301m_rule @ N @ S2 ) ) ) ).
% sdrop_szip
thf(fact_6_sdrop__szip,axiom,
! [N: nat,S1: stream_rule,S2: stream4385846686851721995e_rule] :
( ( sdrop_4667533458460693055e_rule @ N @ ( szip_r8329995387344033047e_rule @ S1 @ S2 ) )
= ( szip_r8329995387344033047e_rule @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_9113879250048157294e_rule @ N @ S2 ) ) ) ).
% sdrop_szip
thf(fact_7_sdrop__szip,axiom,
! [N: nat,S1: stream4385846686851721995e_rule,S2: stream_rule] :
( ( sdrop_6431130323856442653e_rule @ N @ ( szip_P4230510992641354557e_rule @ S1 @ S2 ) )
= ( szip_P4230510992641354557e_rule @ ( sdrop_9113879250048157294e_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).
% sdrop_szip
thf(fact_8_sdrop__szip,axiom,
! [N: nat,S1: stream_list_tm,S2: stream_list_fm] :
( ( sdrop_9176333610110415838ist_fm @ N @ ( szip_list_tm_list_fm @ S1 @ S2 ) )
= ( szip_list_tm_list_fm @ ( sdrop_list_tm @ N @ S1 ) @ ( sdrop_list_fm @ N @ S2 ) ) ) ).
% sdrop_szip
thf(fact_9_sdrop__szip,axiom,
! [N: nat,S1: stream_rule,S2: stream4408948924543953275ist_fm] :
( ( sdrop_959413419342957231ist_fm @ N @ ( szip_r5451144443050508039ist_fm @ S1 @ S2 ) )
= ( szip_r5451144443050508039ist_fm @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_9176333610110415838ist_fm @ N @ S2 ) ) ) ).
% sdrop_szip
thf(fact_10_sdrop__szip,axiom,
! [N: nat,S1: stream4385846686851721995e_rule,S2: stream4385846686851721995e_rule] :
( ( sdrop_7704051541046111184e_rule @ N @ ( szip_P1965874896742159130e_rule @ S1 @ S2 ) )
= ( szip_P1965874896742159130e_rule @ ( sdrop_9113879250048157294e_rule @ N @ S1 ) @ ( sdrop_9113879250048157294e_rule @ N @ S2 ) ) ) ).
% sdrop_szip
thf(fact_11_n,axiom,
member_fm3 @ ( neg @ ( uni @ p ) ) @ ( set_fm2 @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ n @ steps ) ) ) ) ).
% n
thf(fact_12__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062n_O_ANeg_A_IUni_Ap_J_A_092_060in_062_Aset_A_Ipseq_A_Ishd_A_Isdrop_An_Asteps_J_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [N2: nat] :
~ ( member_fm3 @ ( neg @ ( uni @ p ) ) @ ( set_fm2 @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N2 @ steps ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>n. Neg (Uni p) \<in> set (pseq (shd (sdrop n steps))) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_13_sdrop__simps_I1_J,axiom,
! [N: nat,S: stream4408948924543953275ist_fm] :
( ( shd_Pr3211216682057661985ist_fm @ ( sdrop_9176333610110415838ist_fm @ N @ S ) )
= ( snth_P7093566783922538521ist_fm @ S @ N ) ) ).
% sdrop_simps(1)
thf(fact_14_sdrop__simps_I1_J,axiom,
! [N: nat,S: stream4385846686851721995e_rule] :
( ( shd_Pr2264621979884435249e_rule @ ( sdrop_9113879250048157294e_rule @ N @ S ) )
= ( snth_P6178434775611311401e_rule @ S @ N ) ) ).
% sdrop_simps(1)
thf(fact_15_sdrop__simps_I1_J,axiom,
! [N: nat,S: stream8953843411776101167m_rule] :
( ( shd_Pr1208753253167450709m_rule @ ( sdrop_746751008794826386m_rule @ N @ S ) )
= ( snth_P5203088247548055629m_rule @ S @ N ) ) ).
% sdrop_simps(1)
thf(fact_16_sdrop__simps_I1_J,axiom,
! [N: nat,S: stream8099677779113257519e_rule] :
( ( shd_Pr2400962586966563157e_rule @ ( sdrop_1938960342593938834e_rule @ N @ S ) )
= ( snth_P6395297581347168077e_rule @ S @ N ) ) ).
% sdrop_simps(1)
thf(fact_17_sdrop__simps_I1_J,axiom,
! [N: nat,S: stream6210534828274662995m_rule] :
( ( shd_Pr4461660664618831993m_rule @ ( sdrop_7192298464603511222m_rule @ N @ S ) )
= ( snth_P8853764340393315953m_rule @ S @ N ) ) ).
% sdrop_simps(1)
thf(fact_18_sdrop__simps_I1_J,axiom,
! [N: nat,S: stream2709947120125613254m_rule] :
( ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N @ S ) )
= ( snth_P6679518042731451922m_rule @ S @ N ) ) ).
% sdrop_simps(1)
thf(fact_19_sdrop__simps_I1_J,axiom,
! [N: nat,S: stream_rule] :
( ( shd_rule @ ( sdrop_rule @ N @ S ) )
= ( snth_rule @ S @ N ) ) ).
% sdrop_simps(1)
thf(fact_20_member,axiom,
( member_rule2
= ( ^ [P: rule,Z: list_rule] : ( member_rule3 @ P @ ( set_rule2 @ Z ) ) ) ) ).
% member
thf(fact_21_member,axiom,
( member_real2
= ( ^ [P: real,Z: list_real] : ( member_real3 @ P @ ( set_real2 @ Z ) ) ) ) ).
% member
thf(fact_22_member,axiom,
( member_nat2
= ( ^ [P: nat,Z: list_nat] : ( member_nat3 @ P @ ( set_nat2 @ Z ) ) ) ) ).
% member
thf(fact_23_member,axiom,
( member_tm2
= ( ^ [P: tm,Z: list_tm] : ( member_tm3 @ P @ ( set_tm2 @ Z ) ) ) ) ).
% member
thf(fact_24_member,axiom,
( member_fm2
= ( ^ [P: fm,Z: list_fm] : ( member_fm3 @ P @ ( set_fm2 @ Z ) ) ) ) ).
% member
thf(fact_25_assms_I2_J,axiom,
abstra6097777249025082867ist_fm @ eff @ rules @ steps ).
% assms(2)
thf(fact_26_in__set__member,axiom,
! [X: rule,Xs: list_rule] :
( ( member_rule3 @ X @ ( set_rule2 @ Xs ) )
= ( member_rule @ Xs @ X ) ) ).
% in_set_member
thf(fact_27_in__set__member,axiom,
! [X: real,Xs: list_real] :
( ( member_real3 @ X @ ( set_real2 @ Xs ) )
= ( member_real @ Xs @ X ) ) ).
% in_set_member
thf(fact_28_in__set__member,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat3 @ X @ ( set_nat2 @ Xs ) )
= ( member_nat @ Xs @ X ) ) ).
% in_set_member
thf(fact_29_in__set__member,axiom,
! [X: tm,Xs: list_tm] :
( ( member_tm3 @ X @ ( set_tm2 @ Xs ) )
= ( member_tm @ Xs @ X ) ) ).
% in_set_member
thf(fact_30_in__set__member,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm3 @ X @ ( set_fm2 @ Xs ) )
= ( member_fm @ Xs @ X ) ) ).
% in_set_member
thf(fact_31__092_060open_062Neg_A_IUni_Ap_J_A_092_060in_062_Atree__fms_Asteps_092_060close_062,axiom,
member_fm3 @ ( neg @ ( uni @ p ) ) @ ( tree_fms @ steps ) ).
% \<open>Neg (Uni p) \<in> tree_fms steps\<close>
thf(fact_32_assms_I1_J,axiom,
abstra6789711989322986974ist_fm @ eff @ rules @ steps ).
% assms(1)
thf(fact_33_sdrop__smap2,axiom,
! [N: nat,F: rule > rule > rule,S1: stream_rule,S2: stream_rule] :
( ( sdrop_rule @ N @ ( smap2_rule_rule_rule @ F @ S1 @ S2 ) )
= ( smap2_rule_rule_rule @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).
% sdrop_smap2
thf(fact_34_sdrop__smap2,axiom,
! [N: nat,F: rule > produc7694839378271647877e_rule > rule,S1: stream_rule,S2: stream4385846686851721995e_rule] :
( ( sdrop_rule @ N @ ( smap2_9025683351474882503e_rule @ F @ S1 @ S2 ) )
= ( smap2_9025683351474882503e_rule @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_9113879250048157294e_rule @ N @ S2 ) ) ) ).
% sdrop_smap2
thf(fact_35_sdrop__smap2,axiom,
! [N: nat,F: produc7694839378271647877e_rule > rule > rule,S1: stream4385846686851721995e_rule,S2: stream_rule] :
( ( sdrop_rule @ N @ ( smap2_3450944986098114541e_rule @ F @ S1 @ S2 ) )
= ( smap2_3450944986098114541e_rule @ F @ ( sdrop_9113879250048157294e_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).
% sdrop_smap2
thf(fact_36_sdrop__smap2,axiom,
! [N: nat,F: rule > rule > produc7694839378271647877e_rule,S1: stream_rule,S2: stream_rule] :
( ( sdrop_9113879250048157294e_rule @ N @ ( smap2_3901795709322785185e_rule @ F @ S1 @ S2 ) )
= ( smap2_3901795709322785185e_rule @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).
% sdrop_smap2
thf(fact_37_sdrop__smap2,axiom,
! [N: nat,F: rule > produc6018962875968178549ist_fm > rule,S1: stream_rule,S2: stream4408948924543953275ist_fm] :
( ( sdrop_rule @ N @ ( smap2_6567943727960754231m_rule @ F @ S1 @ S2 ) )
= ( smap2_6567943727960754231m_rule @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_9176333610110415838ist_fm @ N @ S2 ) ) ) ).
% sdrop_smap2
thf(fact_38_sdrop__smap2,axiom,
! [N: nat,F: produc6018962875968178549ist_fm > rule > rule,S1: stream4408948924543953275ist_fm,S2: stream_rule] :
( ( sdrop_rule @ N @ ( smap2_4866629018036412893e_rule @ F @ S1 @ S2 ) )
= ( smap2_4866629018036412893e_rule @ F @ ( sdrop_9176333610110415838ist_fm @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).
% sdrop_smap2
thf(fact_39_sdrop__smap2,axiom,
! [N: nat,F: produc7694839378271647877e_rule > produc7694839378271647877e_rule > rule,S1: stream4385846686851721995e_rule,S2: stream4385846686851721995e_rule] :
( ( sdrop_rule @ N @ ( smap2_5087128713194948752e_rule @ F @ S1 @ S2 ) )
= ( smap2_5087128713194948752e_rule @ F @ ( sdrop_9113879250048157294e_rule @ N @ S1 ) @ ( sdrop_9113879250048157294e_rule @ N @ S2 ) ) ) ).
% sdrop_smap2
thf(fact_40_sdrop__smap2,axiom,
! [N: nat,F: rule > rule > produc6018962875968178549ist_fm,S1: stream_rule,S2: stream_rule] :
( ( sdrop_9176333610110415838ist_fm @ N @ ( smap2_9094267487109771409ist_fm @ F @ S1 @ S2 ) )
= ( smap2_9094267487109771409ist_fm @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).
% sdrop_smap2
thf(fact_41_sdrop__smap2,axiom,
! [N: nat,F: rule > produc7694839378271647877e_rule > produc7694839378271647877e_rule,S1: stream_rule,S2: stream4385846686851721995e_rule] :
( ( sdrop_9113879250048157294e_rule @ N @ ( smap2_8640882101725621264e_rule @ F @ S1 @ S2 ) )
= ( smap2_8640882101725621264e_rule @ F @ ( sdrop_rule @ N @ S1 ) @ ( sdrop_9113879250048157294e_rule @ N @ S2 ) ) ) ).
% sdrop_smap2
thf(fact_42_sdrop__smap2,axiom,
! [N: nat,F: produc7694839378271647877e_rule > rule > produc7694839378271647877e_rule,S1: stream4385846686851721995e_rule,S2: stream_rule] :
( ( sdrop_9113879250048157294e_rule @ N @ ( smap2_9186613107897627242e_rule @ F @ S1 @ S2 ) )
= ( smap2_9186613107897627242e_rule @ F @ ( sdrop_9113879250048157294e_rule @ N @ S1 ) @ ( sdrop_rule @ N @ S2 ) ) ) ).
% sdrop_smap2
thf(fact_43_in__set__insert,axiom,
! [X: tm,Xs: list_tm] :
( ( member_tm3 @ X @ ( set_tm2 @ Xs ) )
=> ( ( insert_tm @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_44_in__set__insert,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm3 @ X @ ( set_fm2 @ Xs ) )
=> ( ( insert_fm @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_45_in__set__insert,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat3 @ X @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_46_in__set__insert,axiom,
! [X: real,Xs: list_real] :
( ( member_real3 @ X @ ( set_real2 @ Xs ) )
=> ( ( insert_real @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_47_in__set__insert,axiom,
! [X: rule,Xs: list_rule] :
( ( member_rule3 @ X @ ( set_rule2 @ Xs ) )
=> ( ( insert_rule @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_48_list__ex1__iff,axiom,
( list_ex1_tm
= ( ^ [P2: tm > $o,Xs2: list_tm] :
? [X2: tm] :
( ( member_tm3 @ X2 @ ( set_tm2 @ Xs2 ) )
& ( P2 @ X2 )
& ! [Y: tm] :
( ( ( member_tm3 @ Y @ ( set_tm2 @ Xs2 ) )
& ( P2 @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_49_list__ex1__iff,axiom,
( list_ex1_fm
= ( ^ [P2: fm > $o,Xs2: list_fm] :
? [X2: fm] :
( ( member_fm3 @ X2 @ ( set_fm2 @ Xs2 ) )
& ( P2 @ X2 )
& ! [Y: fm] :
( ( ( member_fm3 @ Y @ ( set_fm2 @ Xs2 ) )
& ( P2 @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_50_list__ex1__iff,axiom,
( list_ex1_nat
= ( ^ [P2: nat > $o,Xs2: list_nat] :
? [X2: nat] :
( ( member_nat3 @ X2 @ ( set_nat2 @ Xs2 ) )
& ( P2 @ X2 )
& ! [Y: nat] :
( ( ( member_nat3 @ Y @ ( set_nat2 @ Xs2 ) )
& ( P2 @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_51_list__ex1__iff,axiom,
( list_ex1_real
= ( ^ [P2: real > $o,Xs2: list_real] :
? [X2: real] :
( ( member_real3 @ X2 @ ( set_real2 @ Xs2 ) )
& ( P2 @ X2 )
& ! [Y: real] :
( ( ( member_real3 @ Y @ ( set_real2 @ Xs2 ) )
& ( P2 @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_52_list__ex1__iff,axiom,
( list_ex1_rule
= ( ^ [P2: rule > $o,Xs2: list_rule] :
? [X2: rule] :
( ( member_rule3 @ X2 @ ( set_rule2 @ Xs2 ) )
& ( P2 @ X2 )
& ! [Y: rule] :
( ( ( member_rule3 @ Y @ ( set_rule2 @ Xs2 ) )
& ( P2 @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_53_fm_Oinject_I7_J,axiom,
! [X7: fm,Y7: fm] :
( ( ( neg @ X7 )
= ( neg @ Y7 ) )
= ( X7 = Y7 ) ) ).
% fm.inject(7)
thf(fact_54_fm_Oinject_I6_J,axiom,
! [X6: fm,Y6: fm] :
( ( ( uni @ X6 )
= ( uni @ Y6 ) )
= ( X6 = Y6 ) ) ).
% fm.inject(6)
thf(fact_55_snth__smap2,axiom,
! [F: produc340336539035504054m_rule > produc340336539035504054m_rule > produc340336539035504054m_rule,S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule,N: nat] :
( ( snth_P6679518042731451922m_rule @ ( smap2_6945574070536835298m_rule @ F @ S1 @ S2 ) @ N )
= ( F @ ( snth_P6679518042731451922m_rule @ S1 @ N ) @ ( snth_P6679518042731451922m_rule @ S2 @ N ) ) ) ).
% snth_smap2
thf(fact_56_snth__smap2,axiom,
! [F: produc340336539035504054m_rule > rule > produc340336539035504054m_rule,S1: stream2709947120125613254m_rule,S2: stream_rule,N: nat] :
( ( snth_P6679518042731451922m_rule @ ( smap2_6366920367380388542m_rule @ F @ S1 @ S2 ) @ N )
= ( F @ ( snth_P6679518042731451922m_rule @ S1 @ N ) @ ( snth_rule @ S2 @ N ) ) ) ).
% snth_smap2
thf(fact_57_snth__smap2,axiom,
! [F: rule > produc340336539035504054m_rule > produc340336539035504054m_rule,S1: stream_rule,S2: stream2709947120125613254m_rule,N: nat] :
( ( snth_P6679518042731451922m_rule @ ( smap2_4579943528986984894m_rule @ F @ S1 @ S2 ) @ N )
= ( F @ ( snth_rule @ S1 @ N ) @ ( snth_P6679518042731451922m_rule @ S2 @ N ) ) ) ).
% snth_smap2
thf(fact_58_snth__smap2,axiom,
! [F: rule > rule > produc340336539035504054m_rule,S1: stream_rule,S2: stream_rule,N: nat] :
( ( snth_P6679518042731451922m_rule @ ( smap2_4319354692778897306m_rule @ F @ S1 @ S2 ) @ N )
= ( F @ ( snth_rule @ S1 @ N ) @ ( snth_rule @ S2 @ N ) ) ) ).
% snth_smap2
thf(fact_59_snth__smap2,axiom,
! [F: produc340336539035504054m_rule > produc340336539035504054m_rule > rule,S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule,N: nat] :
( ( snth_rule @ ( smap2_1878155219598472126e_rule @ F @ S1 @ S2 ) @ N )
= ( F @ ( snth_P6679518042731451922m_rule @ S1 @ N ) @ ( snth_P6679518042731451922m_rule @ S2 @ N ) ) ) ).
% snth_smap2
thf(fact_60_snth__smap2,axiom,
! [F: produc340336539035504054m_rule > rule > rule,S1: stream2709947120125613254m_rule,S2: stream_rule,N: nat] :
( ( snth_rule @ ( smap2_8971878701134242202e_rule @ F @ S1 @ S2 ) @ N )
= ( F @ ( snth_P6679518042731451922m_rule @ S1 @ N ) @ ( snth_rule @ S2 @ N ) ) ) ).
% snth_smap2
thf(fact_61_snth__smap2,axiom,
! [F: rule > produc340336539035504054m_rule > rule,S1: stream_rule,S2: stream2709947120125613254m_rule,N: nat] :
( ( snth_rule @ ( smap2_9053961581851756698e_rule @ F @ S1 @ S2 ) @ N )
= ( F @ ( snth_rule @ S1 @ N ) @ ( snth_P6679518042731451922m_rule @ S2 @ N ) ) ) ).
% snth_smap2
thf(fact_62_snth__smap2,axiom,
! [F: rule > rule > rule,S1: stream_rule,S2: stream_rule,N: nat] :
( ( snth_rule @ ( smap2_rule_rule_rule @ F @ S1 @ S2 ) @ N )
= ( F @ ( snth_rule @ S1 @ N ) @ ( snth_rule @ S2 @ N ) ) ) ).
% snth_smap2
thf(fact_63_fm_Odistinct_I41_J,axiom,
! [X6: fm,X7: fm] :
( ( uni @ X6 )
!= ( neg @ X7 ) ) ).
% fm.distinct(41)
thf(fact_64_smap2__alt,axiom,
! [F: produc340336539035504054m_rule > produc340336539035504054m_rule > produc340336539035504054m_rule,S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule,S: stream2709947120125613254m_rule] :
( ( ( smap2_6945574070536835298m_rule @ F @ S1 @ S2 )
= S )
= ( ! [N3: nat] :
( ( F @ ( snth_P6679518042731451922m_rule @ S1 @ N3 ) @ ( snth_P6679518042731451922m_rule @ S2 @ N3 ) )
= ( snth_P6679518042731451922m_rule @ S @ N3 ) ) ) ) ).
% smap2_alt
thf(fact_65_smap2__alt,axiom,
! [F: produc340336539035504054m_rule > produc340336539035504054m_rule > rule,S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule,S: stream_rule] :
( ( ( smap2_1878155219598472126e_rule @ F @ S1 @ S2 )
= S )
= ( ! [N3: nat] :
( ( F @ ( snth_P6679518042731451922m_rule @ S1 @ N3 ) @ ( snth_P6679518042731451922m_rule @ S2 @ N3 ) )
= ( snth_rule @ S @ N3 ) ) ) ) ).
% smap2_alt
thf(fact_66_smap2__alt,axiom,
! [F: produc340336539035504054m_rule > rule > produc340336539035504054m_rule,S1: stream2709947120125613254m_rule,S2: stream_rule,S: stream2709947120125613254m_rule] :
( ( ( smap2_6366920367380388542m_rule @ F @ S1 @ S2 )
= S )
= ( ! [N3: nat] :
( ( F @ ( snth_P6679518042731451922m_rule @ S1 @ N3 ) @ ( snth_rule @ S2 @ N3 ) )
= ( snth_P6679518042731451922m_rule @ S @ N3 ) ) ) ) ).
% smap2_alt
thf(fact_67_smap2__alt,axiom,
! [F: produc340336539035504054m_rule > rule > rule,S1: stream2709947120125613254m_rule,S2: stream_rule,S: stream_rule] :
( ( ( smap2_8971878701134242202e_rule @ F @ S1 @ S2 )
= S )
= ( ! [N3: nat] :
( ( F @ ( snth_P6679518042731451922m_rule @ S1 @ N3 ) @ ( snth_rule @ S2 @ N3 ) )
= ( snth_rule @ S @ N3 ) ) ) ) ).
% smap2_alt
thf(fact_68_smap2__alt,axiom,
! [F: rule > produc340336539035504054m_rule > produc340336539035504054m_rule,S1: stream_rule,S2: stream2709947120125613254m_rule,S: stream2709947120125613254m_rule] :
( ( ( smap2_4579943528986984894m_rule @ F @ S1 @ S2 )
= S )
= ( ! [N3: nat] :
( ( F @ ( snth_rule @ S1 @ N3 ) @ ( snth_P6679518042731451922m_rule @ S2 @ N3 ) )
= ( snth_P6679518042731451922m_rule @ S @ N3 ) ) ) ) ).
% smap2_alt
thf(fact_69_smap2__alt,axiom,
! [F: rule > produc340336539035504054m_rule > rule,S1: stream_rule,S2: stream2709947120125613254m_rule,S: stream_rule] :
( ( ( smap2_9053961581851756698e_rule @ F @ S1 @ S2 )
= S )
= ( ! [N3: nat] :
( ( F @ ( snth_rule @ S1 @ N3 ) @ ( snth_P6679518042731451922m_rule @ S2 @ N3 ) )
= ( snth_rule @ S @ N3 ) ) ) ) ).
% smap2_alt
thf(fact_70_smap2__alt,axiom,
! [F: rule > rule > produc340336539035504054m_rule,S1: stream_rule,S2: stream_rule,S: stream2709947120125613254m_rule] :
( ( ( smap2_4319354692778897306m_rule @ F @ S1 @ S2 )
= S )
= ( ! [N3: nat] :
( ( F @ ( snth_rule @ S1 @ N3 ) @ ( snth_rule @ S2 @ N3 ) )
= ( snth_P6679518042731451922m_rule @ S @ N3 ) ) ) ) ).
% smap2_alt
thf(fact_71_smap2__alt,axiom,
! [F: rule > rule > rule,S1: stream_rule,S2: stream_rule,S: stream_rule] :
( ( ( smap2_rule_rule_rule @ F @ S1 @ S2 )
= S )
= ( ! [N3: nat] :
( ( F @ ( snth_rule @ S1 @ N3 ) @ ( snth_rule @ S2 @ N3 ) )
= ( snth_rule @ S @ N3 ) ) ) ) ).
% smap2_alt
thf(fact_72_epath__sdrop,axiom,
! [Steps: stream2709947120125613254m_rule,N: nat] :
( ( abstra6789711989322986974ist_fm @ eff @ rules @ Steps )
=> ( abstra6789711989322986974ist_fm @ eff @ rules @ ( sdrop_8169176516188972301m_rule @ N @ Steps ) ) ) ).
% epath_sdrop
thf(fact_73_Saturated__sdrop,axiom,
! [Steps: stream2709947120125613254m_rule,N: nat] :
( ( abstra6097777249025082867ist_fm @ eff @ rules @ Steps )
=> ( abstra6097777249025082867ist_fm @ eff @ rules @ ( sdrop_8169176516188972301m_rule @ N @ Steps ) ) ) ).
% Saturated_sdrop
thf(fact_74_tree__fms__in__pseq,axiom,
! [P3: fm,Steps: stream2709947120125613254m_rule] :
( ( member_fm3 @ P3 @ ( tree_fms @ Steps ) )
=> ? [N2: nat] : ( member_fm3 @ P3 @ ( set_fm2 @ ( pseq @ ( snth_P6679518042731451922m_rule @ Steps @ N2 ) ) ) ) ) ).
% tree_fms_in_pseq
thf(fact_75_smap2_Osimps_I1_J,axiom,
! [F: produc340336539035504054m_rule > produc340336539035504054m_rule > produc340336539035504054m_rule,S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule] :
( ( shd_Pr4562317740776619530m_rule @ ( smap2_6945574070536835298m_rule @ F @ S1 @ S2 ) )
= ( F @ ( shd_Pr4562317740776619530m_rule @ S1 ) @ ( shd_Pr4562317740776619530m_rule @ S2 ) ) ) ).
% smap2.simps(1)
thf(fact_76_smap2_Osimps_I1_J,axiom,
! [F: produc340336539035504054m_rule > rule > produc340336539035504054m_rule,S1: stream2709947120125613254m_rule,S2: stream_rule] :
( ( shd_Pr4562317740776619530m_rule @ ( smap2_6366920367380388542m_rule @ F @ S1 @ S2 ) )
= ( F @ ( shd_Pr4562317740776619530m_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).
% smap2.simps(1)
thf(fact_77_smap2_Osimps_I1_J,axiom,
! [F: rule > produc340336539035504054m_rule > produc340336539035504054m_rule,S1: stream_rule,S2: stream2709947120125613254m_rule] :
( ( shd_Pr4562317740776619530m_rule @ ( smap2_4579943528986984894m_rule @ F @ S1 @ S2 ) )
= ( F @ ( shd_rule @ S1 ) @ ( shd_Pr4562317740776619530m_rule @ S2 ) ) ) ).
% smap2.simps(1)
thf(fact_78_smap2_Osimps_I1_J,axiom,
! [F: rule > rule > produc340336539035504054m_rule,S1: stream_rule,S2: stream_rule] :
( ( shd_Pr4562317740776619530m_rule @ ( smap2_4319354692778897306m_rule @ F @ S1 @ S2 ) )
= ( F @ ( shd_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).
% smap2.simps(1)
thf(fact_79_smap2_Osimps_I1_J,axiom,
! [F: produc340336539035504054m_rule > produc340336539035504054m_rule > rule,S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule] :
( ( shd_rule @ ( smap2_1878155219598472126e_rule @ F @ S1 @ S2 ) )
= ( F @ ( shd_Pr4562317740776619530m_rule @ S1 ) @ ( shd_Pr4562317740776619530m_rule @ S2 ) ) ) ).
% smap2.simps(1)
thf(fact_80_smap2_Osimps_I1_J,axiom,
! [F: produc340336539035504054m_rule > rule > rule,S1: stream2709947120125613254m_rule,S2: stream_rule] :
( ( shd_rule @ ( smap2_8971878701134242202e_rule @ F @ S1 @ S2 ) )
= ( F @ ( shd_Pr4562317740776619530m_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).
% smap2.simps(1)
thf(fact_81_smap2_Osimps_I1_J,axiom,
! [F: rule > produc340336539035504054m_rule > rule,S1: stream_rule,S2: stream2709947120125613254m_rule] :
( ( shd_rule @ ( smap2_9053961581851756698e_rule @ F @ S1 @ S2 ) )
= ( F @ ( shd_rule @ S1 ) @ ( shd_Pr4562317740776619530m_rule @ S2 ) ) ) ).
% smap2.simps(1)
thf(fact_82_smap2_Osimps_I1_J,axiom,
! [F: rule > rule > rule,S1: stream_rule,S2: stream_rule] :
( ( shd_rule @ ( smap2_rule_rule_rule @ F @ S1 @ S2 ) )
= ( F @ ( shd_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).
% smap2.simps(1)
thf(fact_83_can__select__set__list__ex1,axiom,
! [P4: tm > $o,A: list_tm] :
( ( can_select_tm @ P4 @ ( set_tm2 @ A ) )
= ( list_ex1_tm @ P4 @ A ) ) ).
% can_select_set_list_ex1
thf(fact_84_can__select__set__list__ex1,axiom,
! [P4: fm > $o,A: list_fm] :
( ( can_select_fm @ P4 @ ( set_fm2 @ A ) )
= ( list_ex1_fm @ P4 @ A ) ) ).
% can_select_set_list_ex1
thf(fact_85_can__select__set__list__ex1,axiom,
! [P4: nat > $o,A: list_nat] :
( ( can_select_nat @ P4 @ ( set_nat2 @ A ) )
= ( list_ex1_nat @ P4 @ A ) ) ).
% can_select_set_list_ex1
thf(fact_86_can__select__set__list__ex1,axiom,
! [P4: real > $o,A: list_real] :
( ( can_select_real @ P4 @ ( set_real2 @ A ) )
= ( list_ex1_real @ P4 @ A ) ) ).
% can_select_set_list_ex1
thf(fact_87_can__select__set__list__ex1,axiom,
! [P4: rule > $o,A: list_rule] :
( ( can_select_rule @ P4 @ ( set_rule2 @ A ) )
= ( list_ex1_rule @ P4 @ A ) ) ).
% can_select_set_list_ex1
thf(fact_88_epath__sdrop__ptms,axiom,
! [Steps: stream2709947120125613254m_rule,N: nat] :
( ( abstra6789711989322986974ist_fm @ eff @ rules @ Steps )
=> ( ord_less_eq_set_tm @ ( set_tm2 @ ( ptms @ ( shd_Pr4562317740776619530m_rule @ Steps ) ) ) @ ( set_tm2 @ ( ptms @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N @ Steps ) ) ) ) ) ) ).
% epath_sdrop_ptms
thf(fact_89_RuleSystem__Defs_OSaturated_Ocong,axiom,
abstra6097777249025082867ist_fm = abstra6097777249025082867ist_fm ).
% RuleSystem_Defs.Saturated.cong
thf(fact_90_RuleSystem__Defs_Oepath_Ocong,axiom,
abstra6789711989322986974ist_fm = abstra6789711989322986974ist_fm ).
% RuleSystem_Defs.epath.cong
thf(fact_91_pseq__in__tree__fms,axiom,
! [X: produc340336539035504054m_rule,Steps: stream2709947120125613254m_rule,P3: fm] :
( ( member7231649785386036813m_rule @ X @ ( sset_P4484857331586881186m_rule @ Steps ) )
=> ( ( member_fm3 @ P3 @ ( set_fm2 @ ( pseq @ X ) ) )
=> ( member_fm3 @ P3 @ ( tree_fms @ Steps ) ) ) ) ).
% pseq_in_tree_fms
thf(fact_92_rules__repeat__sdrop,axiom,
! [K: nat,R: rule] :
? [N2: nat] :
( ( snth_rule @ ( sdrop_rule @ K @ rules ) @ N2 )
= R ) ).
% rules_repeat_sdrop
thf(fact_93_szip_Osimps_I1_J,axiom,
! [S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule] :
( ( shd_Pr4461660664618831993m_rule @ ( szip_P811719526838699976m_rule @ S1 @ S2 ) )
= ( produc6261311607089640965m_rule @ ( shd_Pr4562317740776619530m_rule @ S1 ) @ ( shd_Pr4562317740776619530m_rule @ S2 ) ) ) ).
% szip.simps(1)
thf(fact_94_szip_Osimps_I1_J,axiom,
! [S1: stream2709947120125613254m_rule,S2: stream_rule] :
( ( shd_Pr2400962586966563157e_rule @ ( szip_P2044787555563963556e_rule @ S1 @ S2 ) )
= ( produc4831648765031708129e_rule @ ( shd_Pr4562317740776619530m_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).
% szip.simps(1)
thf(fact_95_szip_Osimps_I1_J,axiom,
! [S1: stream_rule,S2: stream2709947120125613254m_rule] :
( ( shd_Pr1208753253167450709m_rule @ ( szip_r6533552703345879972m_rule @ S1 @ S2 ) )
= ( produc97041875958848737m_rule @ ( shd_rule @ S1 ) @ ( shd_Pr4562317740776619530m_rule @ S2 ) ) ) ).
% szip.simps(1)
thf(fact_96_szip_Osimps_I1_J,axiom,
! [S1: stream_rule,S2: stream_rule] :
( ( shd_Pr2264621979884435249e_rule @ ( szip_rule_rule @ S1 @ S2 ) )
= ( produc5849431337705160893e_rule @ ( shd_rule @ S1 ) @ ( shd_rule @ S2 ) ) ) ).
% szip.simps(1)
thf(fact_97_szip_Osimps_I1_J,axiom,
! [S1: stream4408948924543953275ist_fm,S2: stream_rule] :
( ( shd_Pr4562317740776619530m_rule @ ( szip_P2924820683901490861m_rule @ S1 @ S2 ) )
= ( produc1733806532565653680m_rule @ ( shd_Pr3211216682057661985ist_fm @ S1 ) @ ( shd_rule @ S2 ) ) ) ).
% szip.simps(1)
thf(fact_98_snth__szip,axiom,
! [S1: stream4408948924543953275ist_fm,S2: stream_rule,N: nat] :
( ( snth_P6679518042731451922m_rule @ ( szip_P2924820683901490861m_rule @ S1 @ S2 ) @ N )
= ( produc1733806532565653680m_rule @ ( snth_P7093566783922538521ist_fm @ S1 @ N ) @ ( snth_rule @ S2 @ N ) ) ) ).
% snth_szip
thf(fact_99_snth__szip,axiom,
! [S1: stream_rule,S2: stream_rule,N: nat] :
( ( snth_P6178434775611311401e_rule @ ( szip_rule_rule @ S1 @ S2 ) @ N )
= ( produc5849431337705160893e_rule @ ( snth_rule @ S1 @ N ) @ ( snth_rule @ S2 @ N ) ) ) ).
% snth_szip
thf(fact_100_snth__szip,axiom,
! [S1: stream_rule,S2: stream2709947120125613254m_rule,N: nat] :
( ( snth_P5203088247548055629m_rule @ ( szip_r6533552703345879972m_rule @ S1 @ S2 ) @ N )
= ( produc97041875958848737m_rule @ ( snth_rule @ S1 @ N ) @ ( snth_P6679518042731451922m_rule @ S2 @ N ) ) ) ).
% snth_szip
thf(fact_101_snth__szip,axiom,
! [S1: stream2709947120125613254m_rule,S2: stream_rule,N: nat] :
( ( snth_P6395297581347168077e_rule @ ( szip_P2044787555563963556e_rule @ S1 @ S2 ) @ N )
= ( produc4831648765031708129e_rule @ ( snth_P6679518042731451922m_rule @ S1 @ N ) @ ( snth_rule @ S2 @ N ) ) ) ).
% snth_szip
thf(fact_102_snth__szip,axiom,
! [S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule,N: nat] :
( ( snth_P8853764340393315953m_rule @ ( szip_P811719526838699976m_rule @ S1 @ S2 ) @ N )
= ( produc6261311607089640965m_rule @ ( snth_P6679518042731451922m_rule @ S1 @ N ) @ ( snth_P6679518042731451922m_rule @ S2 @ N ) ) ) ).
% snth_szip
thf(fact_103_not__in__set__insert,axiom,
! [X: fm,Xs: list_fm] :
( ~ ( member_fm3 @ X @ ( set_fm2 @ Xs ) )
=> ( ( insert_fm @ X @ Xs )
= ( cons_fm @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_104_not__in__set__insert,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat3 @ X @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X @ Xs )
= ( cons_nat @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_105_not__in__set__insert,axiom,
! [X: real,Xs: list_real] :
( ~ ( member_real3 @ X @ ( set_real2 @ Xs ) )
=> ( ( insert_real @ X @ Xs )
= ( cons_real @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_106_not__in__set__insert,axiom,
! [X: rule,Xs: list_rule] :
( ~ ( member_rule3 @ X @ ( set_rule2 @ Xs ) )
=> ( ( insert_rule @ X @ Xs )
= ( cons_rule @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_107_not__in__set__insert,axiom,
! [X: tm,Xs: list_tm] :
( ~ ( member_tm3 @ X @ ( set_tm2 @ Xs ) )
=> ( ( insert_tm @ X @ Xs )
= ( cons_tm @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_108_List_Oset__insert,axiom,
! [X: tm,Xs: list_tm] :
( ( set_tm2 @ ( insert_tm @ X @ Xs ) )
= ( insert_tm2 @ X @ ( set_tm2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_109_List_Oset__insert,axiom,
! [X: fm,Xs: list_fm] :
( ( set_fm2 @ ( insert_fm @ X @ Xs ) )
= ( insert_fm2 @ X @ ( set_fm2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_110_List_Oset__insert,axiom,
! [X: nat,Xs: list_nat] :
( ( set_nat2 @ ( insert_nat @ X @ Xs ) )
= ( insert_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_111_List_Oset__insert,axiom,
! [X: real,Xs: list_real] :
( ( set_real2 @ ( insert_real @ X @ Xs ) )
= ( insert_real2 @ X @ ( set_real2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_112_List_Oset__insert,axiom,
! [X: rule,Xs: list_rule] :
( ( set_rule2 @ ( insert_rule @ X @ Xs ) )
= ( insert_rule2 @ X @ ( set_rule2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_113_stream__all__def,axiom,
( stream2134157564947672044m_rule
= ( ^ [P2: produc340336539035504054m_rule > $o,S3: stream2709947120125613254m_rule] :
! [P: nat] : ( P2 @ ( snth_P6679518042731451922m_rule @ S3 @ P ) ) ) ) ).
% stream_all_def
thf(fact_114_stream__all__def,axiom,
( stream_all_rule
= ( ^ [P2: rule > $o,S3: stream_rule] :
! [P: nat] : ( P2 @ ( snth_rule @ S3 @ P ) ) ) ) ).
% stream_all_def
thf(fact_115_snth_Osimps_I1_J,axiom,
! [S: stream2709947120125613254m_rule] :
( ( snth_P6679518042731451922m_rule @ S @ zero_zero_nat )
= ( shd_Pr4562317740776619530m_rule @ S ) ) ).
% snth.simps(1)
thf(fact_116_snth_Osimps_I1_J,axiom,
! [S: stream_rule] :
( ( snth_rule @ S @ zero_zero_nat )
= ( shd_rule @ S ) ) ).
% snth.simps(1)
thf(fact_117_list_Oinject,axiom,
! [X21: tm,X22: list_tm,Y21: tm,Y22: list_tm] :
( ( ( cons_tm @ X21 @ X22 )
= ( cons_tm @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_118_list_Osimps_I15_J,axiom,
! [X21: fm,X22: list_fm] :
( ( set_fm2 @ ( cons_fm @ X21 @ X22 ) )
= ( insert_fm2 @ X21 @ ( set_fm2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_119_list_Osimps_I15_J,axiom,
! [X21: nat,X22: list_nat] :
( ( set_nat2 @ ( cons_nat @ X21 @ X22 ) )
= ( insert_nat2 @ X21 @ ( set_nat2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_120_list_Osimps_I15_J,axiom,
! [X21: real,X22: list_real] :
( ( set_real2 @ ( cons_real @ X21 @ X22 ) )
= ( insert_real2 @ X21 @ ( set_real2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_121_list_Osimps_I15_J,axiom,
! [X21: rule,X22: list_rule] :
( ( set_rule2 @ ( cons_rule @ X21 @ X22 ) )
= ( insert_rule2 @ X21 @ ( set_rule2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_122_list_Osimps_I15_J,axiom,
! [X21: tm,X22: list_tm] :
( ( set_tm2 @ ( cons_tm @ X21 @ X22 ) )
= ( insert_tm2 @ X21 @ ( set_tm2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_123_not__Cons__self2,axiom,
! [X: tm,Xs: list_tm] :
( ( cons_tm @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_124_set__subset__Cons,axiom,
! [Xs: list_fm,X: fm] : ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ ( set_fm2 @ ( cons_fm @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_125_set__subset__Cons,axiom,
! [Xs: list_real,X: real] : ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ ( set_real2 @ ( cons_real @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_126_set__subset__Cons,axiom,
! [Xs: list_rule,X: rule] : ( ord_less_eq_set_rule @ ( set_rule2 @ Xs ) @ ( set_rule2 @ ( cons_rule @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_127_set__subset__Cons,axiom,
! [Xs: list_tm,X: tm] : ( ord_less_eq_set_tm @ ( set_tm2 @ Xs ) @ ( set_tm2 @ ( cons_tm @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_128_set__subset__Cons,axiom,
! [Xs: list_nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_129_sset__sdrop,axiom,
! [N: nat,S: stream4408948924543953275ist_fm] : ( ord_le1771420097867575541ist_fm @ ( sset_P5379989128061332361ist_fm @ ( sdrop_9176333610110415838ist_fm @ N @ S ) ) @ ( sset_P5379989128061332361ist_fm @ S ) ) ).
% sset_sdrop
thf(fact_130_sset__sdrop,axiom,
! [N: nat,S: stream4385846686851721995e_rule] : ( ord_le5312242768264209797e_rule @ ( sset_P1894962091411173529e_rule @ ( sdrop_9113879250048157294e_rule @ N @ S ) ) @ ( sset_P1894962091411173529e_rule @ S ) ) ).
% sset_sdrop
thf(fact_131_sset__sdrop,axiom,
! [N: nat,S: stream8953843411776101167m_rule] : ( ord_le1038668239487504297m_rule @ ( sset_P5794137493694147517m_rule @ ( sdrop_746751008794826386m_rule @ N @ S ) ) @ ( sset_P5794137493694147517m_rule @ S ) ) ).
% sset_sdrop
thf(fact_132_sset__sdrop,axiom,
! [N: nat,S: stream8099677779113257519e_rule] : ( ord_le184502606824660649e_rule @ ( sset_P6986346827493259965e_rule @ ( sdrop_1938960342593938834e_rule @ N @ S ) ) @ ( sset_P6986346827493259965e_rule @ S ) ) ).
% sset_sdrop
thf(fact_133_sset__sdrop,axiom,
! [N: nat,S: stream6210534828274662995m_rule] : ( ord_le8955734628938902733m_rule @ ( sset_P3768201806174902753m_rule @ ( sdrop_7192298464603511222m_rule @ N @ S ) ) @ ( sset_P3768201806174902753m_rule @ S ) ) ).
% sset_sdrop
thf(fact_134_sset__sdrop,axiom,
! [N: nat,S: stream2709947120125613254m_rule] : ( ord_le6390412330253371084m_rule @ ( sset_P4484857331586881186m_rule @ ( sdrop_8169176516188972301m_rule @ N @ S ) ) @ ( sset_P4484857331586881186m_rule @ S ) ) ).
% sset_sdrop
thf(fact_135_sset__sdrop,axiom,
! [N: nat,S: stream_rule] : ( ord_less_eq_set_rule @ ( sset_rule @ ( sdrop_rule @ N @ S ) ) @ ( sset_rule @ S ) ) ).
% sset_sdrop
thf(fact_136_sset__sdrop,axiom,
! [N: nat,S: stream_tm] : ( ord_less_eq_set_tm @ ( sset_tm @ ( sdrop_tm @ N @ S ) ) @ ( sset_tm @ S ) ) ).
% sset_sdrop
thf(fact_137_sset__sdrop,axiom,
! [N: nat,S: stream_nat] : ( ord_less_eq_set_nat @ ( sset_nat @ ( sdrop_nat @ N @ S ) ) @ ( sset_nat @ S ) ) ).
% sset_sdrop
thf(fact_138_shd__sset,axiom,
! [A2: stream_real] : ( member_real3 @ ( shd_real @ A2 ) @ ( sset_real @ A2 ) ) ).
% shd_sset
thf(fact_139_shd__sset,axiom,
! [A2: stream_tm] : ( member_tm3 @ ( shd_tm @ A2 ) @ ( sset_tm @ A2 ) ) ).
% shd_sset
thf(fact_140_shd__sset,axiom,
! [A2: stream_fm] : ( member_fm3 @ ( shd_fm @ A2 ) @ ( sset_fm @ A2 ) ) ).
% shd_sset
thf(fact_141_shd__sset,axiom,
! [A2: stream_nat] : ( member_nat3 @ ( shd_nat @ A2 ) @ ( sset_nat @ A2 ) ) ).
% shd_sset
thf(fact_142_shd__sset,axiom,
! [A2: stream2709947120125613254m_rule] : ( member7231649785386036813m_rule @ ( shd_Pr4562317740776619530m_rule @ A2 ) @ ( sset_P4484857331586881186m_rule @ A2 ) ) ).
% shd_sset
thf(fact_143_shd__sset,axiom,
! [A2: stream_rule] : ( member_rule3 @ ( shd_rule @ A2 ) @ ( sset_rule @ A2 ) ) ).
% shd_sset
thf(fact_144_list_Oset__intros_I2_J,axiom,
! [Y2: fm,X22: list_fm,X21: fm] :
( ( member_fm3 @ Y2 @ ( set_fm2 @ X22 ) )
=> ( member_fm3 @ Y2 @ ( set_fm2 @ ( cons_fm @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_145_list_Oset__intros_I2_J,axiom,
! [Y2: nat,X22: list_nat,X21: nat] :
( ( member_nat3 @ Y2 @ ( set_nat2 @ X22 ) )
=> ( member_nat3 @ Y2 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_146_list_Oset__intros_I2_J,axiom,
! [Y2: real,X22: list_real,X21: real] :
( ( member_real3 @ Y2 @ ( set_real2 @ X22 ) )
=> ( member_real3 @ Y2 @ ( set_real2 @ ( cons_real @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_147_list_Oset__intros_I2_J,axiom,
! [Y2: rule,X22: list_rule,X21: rule] :
( ( member_rule3 @ Y2 @ ( set_rule2 @ X22 ) )
=> ( member_rule3 @ Y2 @ ( set_rule2 @ ( cons_rule @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_148_list_Oset__intros_I2_J,axiom,
! [Y2: tm,X22: list_tm,X21: tm] :
( ( member_tm3 @ Y2 @ ( set_tm2 @ X22 ) )
=> ( member_tm3 @ Y2 @ ( set_tm2 @ ( cons_tm @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_149_list_Oset__intros_I1_J,axiom,
! [X21: fm,X22: list_fm] : ( member_fm3 @ X21 @ ( set_fm2 @ ( cons_fm @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_150_list_Oset__intros_I1_J,axiom,
! [X21: nat,X22: list_nat] : ( member_nat3 @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_151_list_Oset__intros_I1_J,axiom,
! [X21: real,X22: list_real] : ( member_real3 @ X21 @ ( set_real2 @ ( cons_real @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_152_list_Oset__intros_I1_J,axiom,
! [X21: rule,X22: list_rule] : ( member_rule3 @ X21 @ ( set_rule2 @ ( cons_rule @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_153_list_Oset__intros_I1_J,axiom,
! [X21: tm,X22: list_tm] : ( member_tm3 @ X21 @ ( set_tm2 @ ( cons_tm @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_154_list_Oset__cases,axiom,
! [E: fm,A2: list_fm] :
( ( member_fm3 @ E @ ( set_fm2 @ A2 ) )
=> ( ! [Z2: list_fm] :
( A2
!= ( cons_fm @ E @ Z2 ) )
=> ~ ! [Z1: fm,Z2: list_fm] :
( ( A2
= ( cons_fm @ Z1 @ Z2 ) )
=> ~ ( member_fm3 @ E @ ( set_fm2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_155_list_Oset__cases,axiom,
! [E: nat,A2: list_nat] :
( ( member_nat3 @ E @ ( set_nat2 @ A2 ) )
=> ( ! [Z2: list_nat] :
( A2
!= ( cons_nat @ E @ Z2 ) )
=> ~ ! [Z1: nat,Z2: list_nat] :
( ( A2
= ( cons_nat @ Z1 @ Z2 ) )
=> ~ ( member_nat3 @ E @ ( set_nat2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_156_list_Oset__cases,axiom,
! [E: real,A2: list_real] :
( ( member_real3 @ E @ ( set_real2 @ A2 ) )
=> ( ! [Z2: list_real] :
( A2
!= ( cons_real @ E @ Z2 ) )
=> ~ ! [Z1: real,Z2: list_real] :
( ( A2
= ( cons_real @ Z1 @ Z2 ) )
=> ~ ( member_real3 @ E @ ( set_real2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_157_list_Oset__cases,axiom,
! [E: rule,A2: list_rule] :
( ( member_rule3 @ E @ ( set_rule2 @ A2 ) )
=> ( ! [Z2: list_rule] :
( A2
!= ( cons_rule @ E @ Z2 ) )
=> ~ ! [Z1: rule,Z2: list_rule] :
( ( A2
= ( cons_rule @ Z1 @ Z2 ) )
=> ~ ( member_rule3 @ E @ ( set_rule2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_158_list_Oset__cases,axiom,
! [E: tm,A2: list_tm] :
( ( member_tm3 @ E @ ( set_tm2 @ A2 ) )
=> ( ! [Z2: list_tm] :
( A2
!= ( cons_tm @ E @ Z2 ) )
=> ~ ! [Z1: tm,Z2: list_tm] :
( ( A2
= ( cons_tm @ Z1 @ Z2 ) )
=> ~ ( member_tm3 @ E @ ( set_tm2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_159_set__ConsD,axiom,
! [Y2: fm,X: fm,Xs: list_fm] :
( ( member_fm3 @ Y2 @ ( set_fm2 @ ( cons_fm @ X @ Xs ) ) )
=> ( ( Y2 = X )
| ( member_fm3 @ Y2 @ ( set_fm2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_160_set__ConsD,axiom,
! [Y2: nat,X: nat,Xs: list_nat] :
( ( member_nat3 @ Y2 @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) )
=> ( ( Y2 = X )
| ( member_nat3 @ Y2 @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_161_set__ConsD,axiom,
! [Y2: real,X: real,Xs: list_real] :
( ( member_real3 @ Y2 @ ( set_real2 @ ( cons_real @ X @ Xs ) ) )
=> ( ( Y2 = X )
| ( member_real3 @ Y2 @ ( set_real2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_162_set__ConsD,axiom,
! [Y2: rule,X: rule,Xs: list_rule] :
( ( member_rule3 @ Y2 @ ( set_rule2 @ ( cons_rule @ X @ Xs ) ) )
=> ( ( Y2 = X )
| ( member_rule3 @ Y2 @ ( set_rule2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_163_set__ConsD,axiom,
! [Y2: tm,X: tm,Xs: list_tm] :
( ( member_tm3 @ Y2 @ ( set_tm2 @ ( cons_tm @ X @ Xs ) ) )
=> ( ( Y2 = X )
| ( member_tm3 @ Y2 @ ( set_tm2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_164_can__select__def,axiom,
( can_select_rule
= ( ^ [P2: rule > $o,A3: set_rule] :
? [X2: rule] :
( ( member_rule3 @ X2 @ A3 )
& ( P2 @ X2 )
& ! [Y: rule] :
( ( ( member_rule3 @ Y @ A3 )
& ( P2 @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_165_can__select__def,axiom,
( can_select_real
= ( ^ [P2: real > $o,A3: set_real] :
? [X2: real] :
( ( member_real3 @ X2 @ A3 )
& ( P2 @ X2 )
& ! [Y: real] :
( ( ( member_real3 @ Y @ A3 )
& ( P2 @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_166_can__select__def,axiom,
( can_select_tm
= ( ^ [P2: tm > $o,A3: set_tm] :
? [X2: tm] :
( ( member_tm3 @ X2 @ A3 )
& ( P2 @ X2 )
& ! [Y: tm] :
( ( ( member_tm3 @ Y @ A3 )
& ( P2 @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_167_can__select__def,axiom,
( can_select_fm
= ( ^ [P2: fm > $o,A3: set_fm] :
? [X2: fm] :
( ( member_fm3 @ X2 @ A3 )
& ( P2 @ X2 )
& ! [Y: fm] :
( ( ( member_fm3 @ Y @ A3 )
& ( P2 @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_168_can__select__def,axiom,
( can_select_nat
= ( ^ [P2: nat > $o,A3: set_nat] :
? [X2: nat] :
( ( member_nat3 @ X2 @ A3 )
& ( P2 @ X2 )
& ! [Y: nat] :
( ( ( member_nat3 @ Y @ A3 )
& ( P2 @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_169_mem__Collect__eq,axiom,
! [A2: rule,P4: rule > $o] :
( ( member_rule3 @ A2 @ ( collect_rule @ P4 ) )
= ( P4 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_170_mem__Collect__eq,axiom,
! [A2: real,P4: real > $o] :
( ( member_real3 @ A2 @ ( collect_real @ P4 ) )
= ( P4 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_171_mem__Collect__eq,axiom,
! [A2: tm,P4: tm > $o] :
( ( member_tm3 @ A2 @ ( collect_tm @ P4 ) )
= ( P4 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_172_mem__Collect__eq,axiom,
! [A2: fm,P4: fm > $o] :
( ( member_fm3 @ A2 @ ( collect_fm @ P4 ) )
= ( P4 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_173_mem__Collect__eq,axiom,
! [A2: nat,P4: nat > $o] :
( ( member_nat3 @ A2 @ ( collect_nat @ P4 ) )
= ( P4 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_174_Collect__mem__eq,axiom,
! [A: set_rule] :
( ( collect_rule
@ ^ [X2: rule] : ( member_rule3 @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_175_Collect__mem__eq,axiom,
! [A: set_real] :
( ( collect_real
@ ^ [X2: real] : ( member_real3 @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_176_Collect__mem__eq,axiom,
! [A: set_tm] :
( ( collect_tm
@ ^ [X2: tm] : ( member_tm3 @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_177_Collect__mem__eq,axiom,
! [A: set_fm] :
( ( collect_fm
@ ^ [X2: fm] : ( member_fm3 @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_178_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat3 @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_179_snth__sset,axiom,
! [S: stream_real,N: nat] : ( member_real3 @ ( snth_real @ S @ N ) @ ( sset_real @ S ) ) ).
% snth_sset
thf(fact_180_snth__sset,axiom,
! [S: stream_tm,N: nat] : ( member_tm3 @ ( snth_tm @ S @ N ) @ ( sset_tm @ S ) ) ).
% snth_sset
thf(fact_181_snth__sset,axiom,
! [S: stream_fm,N: nat] : ( member_fm3 @ ( snth_fm @ S @ N ) @ ( sset_fm @ S ) ) ).
% snth_sset
thf(fact_182_snth__sset,axiom,
! [S: stream_nat,N: nat] : ( member_nat3 @ ( snth_nat @ S @ N ) @ ( sset_nat @ S ) ) ).
% snth_sset
thf(fact_183_snth__sset,axiom,
! [S: stream2709947120125613254m_rule,N: nat] : ( member7231649785386036813m_rule @ ( snth_P6679518042731451922m_rule @ S @ N ) @ ( sset_P4484857331586881186m_rule @ S ) ) ).
% snth_sset
thf(fact_184_snth__sset,axiom,
! [S: stream_rule,N: nat] : ( member_rule3 @ ( snth_rule @ S @ N ) @ ( sset_rule @ S ) ) ).
% snth_sset
thf(fact_185_subset__code_I1_J,axiom,
! [Xs: list_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ B )
= ( ! [X2: fm] :
( ( member_fm3 @ X2 @ ( set_fm2 @ Xs ) )
=> ( member_fm3 @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_186_subset__code_I1_J,axiom,
! [Xs: list_real,B: set_real] :
( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ B )
= ( ! [X2: real] :
( ( member_real3 @ X2 @ ( set_real2 @ Xs ) )
=> ( member_real3 @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_187_subset__code_I1_J,axiom,
! [Xs: list_rule,B: set_rule] :
( ( ord_less_eq_set_rule @ ( set_rule2 @ Xs ) @ B )
= ( ! [X2: rule] :
( ( member_rule3 @ X2 @ ( set_rule2 @ Xs ) )
=> ( member_rule3 @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_188_subset__code_I1_J,axiom,
! [Xs: list_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ Xs ) @ B )
= ( ! [X2: tm] :
( ( member_tm3 @ X2 @ ( set_tm2 @ Xs ) )
=> ( member_tm3 @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_189_subset__code_I1_J,axiom,
! [Xs: list_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B )
= ( ! [X2: nat] :
( ( member_nat3 @ X2 @ ( set_nat2 @ Xs ) )
=> ( member_nat3 @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_190_sdrop_Osimps_I1_J,axiom,
! [S: stream2709947120125613254m_rule] :
( ( sdrop_8169176516188972301m_rule @ zero_zero_nat @ S )
= S ) ).
% sdrop.simps(1)
thf(fact_191_sdrop_Osimps_I1_J,axiom,
! [S: stream_rule] :
( ( sdrop_rule @ zero_zero_nat @ S )
= S ) ).
% sdrop.simps(1)
thf(fact_192_sdrop_Osimps_I1_J,axiom,
! [S: stream4408948924543953275ist_fm] :
( ( sdrop_9176333610110415838ist_fm @ zero_zero_nat @ S )
= S ) ).
% sdrop.simps(1)
thf(fact_193_sdrop_Osimps_I1_J,axiom,
! [S: stream4385846686851721995e_rule] :
( ( sdrop_9113879250048157294e_rule @ zero_zero_nat @ S )
= S ) ).
% sdrop.simps(1)
thf(fact_194_sdrop_Osimps_I1_J,axiom,
! [S: stream8953843411776101167m_rule] :
( ( sdrop_746751008794826386m_rule @ zero_zero_nat @ S )
= S ) ).
% sdrop.simps(1)
thf(fact_195_sdrop_Osimps_I1_J,axiom,
! [S: stream8099677779113257519e_rule] :
( ( sdrop_1938960342593938834e_rule @ zero_zero_nat @ S )
= S ) ).
% sdrop.simps(1)
thf(fact_196_sdrop_Osimps_I1_J,axiom,
! [S: stream6210534828274662995m_rule] :
( ( sdrop_7192298464603511222m_rule @ zero_zero_nat @ S )
= S ) ).
% sdrop.simps(1)
thf(fact_197_SeCaV_Omember_Osimps_I2_J,axiom,
! [P3: fm,Q: fm,Z3: list_fm] :
( ( member_fm2 @ P3 @ ( cons_fm @ Q @ Z3 ) )
= ( ( P3 != Q )
=> ( member_fm2 @ P3 @ Z3 ) ) ) ).
% SeCaV.member.simps(2)
thf(fact_198_SeCaV_Omember_Osimps_I2_J,axiom,
! [P3: nat,Q: nat,Z3: list_nat] :
( ( member_nat2 @ P3 @ ( cons_nat @ Q @ Z3 ) )
= ( ( P3 != Q )
=> ( member_nat2 @ P3 @ Z3 ) ) ) ).
% SeCaV.member.simps(2)
thf(fact_199_SeCaV_Omember_Osimps_I2_J,axiom,
! [P3: real,Q: real,Z3: list_real] :
( ( member_real2 @ P3 @ ( cons_real @ Q @ Z3 ) )
= ( ( P3 != Q )
=> ( member_real2 @ P3 @ Z3 ) ) ) ).
% SeCaV.member.simps(2)
thf(fact_200_SeCaV_Omember_Osimps_I2_J,axiom,
! [P3: rule,Q: rule,Z3: list_rule] :
( ( member_rule2 @ P3 @ ( cons_rule @ Q @ Z3 ) )
= ( ( P3 != Q )
=> ( member_rule2 @ P3 @ Z3 ) ) ) ).
% SeCaV.member.simps(2)
thf(fact_201_SeCaV_Omember_Osimps_I2_J,axiom,
! [P3: tm,Q: tm,Z3: list_tm] :
( ( member_tm2 @ P3 @ ( cons_tm @ Q @ Z3 ) )
= ( ( P3 != Q )
=> ( member_tm2 @ P3 @ Z3 ) ) ) ).
% SeCaV.member.simps(2)
thf(fact_202_member__rec_I1_J,axiom,
! [X: fm,Xs: list_fm,Y2: fm] :
( ( member_fm @ ( cons_fm @ X @ Xs ) @ Y2 )
= ( ( X = Y2 )
| ( member_fm @ Xs @ Y2 ) ) ) ).
% member_rec(1)
thf(fact_203_member__rec_I1_J,axiom,
! [X: nat,Xs: list_nat,Y2: nat] :
( ( member_nat @ ( cons_nat @ X @ Xs ) @ Y2 )
= ( ( X = Y2 )
| ( member_nat @ Xs @ Y2 ) ) ) ).
% member_rec(1)
thf(fact_204_member__rec_I1_J,axiom,
! [X: real,Xs: list_real,Y2: real] :
( ( member_real @ ( cons_real @ X @ Xs ) @ Y2 )
= ( ( X = Y2 )
| ( member_real @ Xs @ Y2 ) ) ) ).
% member_rec(1)
thf(fact_205_member__rec_I1_J,axiom,
! [X: rule,Xs: list_rule,Y2: rule] :
( ( member_rule @ ( cons_rule @ X @ Xs ) @ Y2 )
= ( ( X = Y2 )
| ( member_rule @ Xs @ Y2 ) ) ) ).
% member_rec(1)
thf(fact_206_member__rec_I1_J,axiom,
! [X: tm,Xs: list_tm,Y2: tm] :
( ( member_tm @ ( cons_tm @ X @ Xs ) @ Y2 )
= ( ( X = Y2 )
| ( member_tm @ Xs @ Y2 ) ) ) ).
% member_rec(1)
thf(fact_207_List_Oinsert__def,axiom,
( insert_fm
= ( ^ [X2: fm,Xs2: list_fm] : ( if_list_fm @ ( member_fm3 @ X2 @ ( set_fm2 @ Xs2 ) ) @ Xs2 @ ( cons_fm @ X2 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_208_List_Oinsert__def,axiom,
( insert_nat
= ( ^ [X2: nat,Xs2: list_nat] : ( if_list_nat @ ( member_nat3 @ X2 @ ( set_nat2 @ Xs2 ) ) @ Xs2 @ ( cons_nat @ X2 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_209_List_Oinsert__def,axiom,
( insert_real
= ( ^ [X2: real,Xs2: list_real] : ( if_list_real @ ( member_real3 @ X2 @ ( set_real2 @ Xs2 ) ) @ Xs2 @ ( cons_real @ X2 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_210_List_Oinsert__def,axiom,
( insert_rule
= ( ^ [X2: rule,Xs2: list_rule] : ( if_list_rule @ ( member_rule3 @ X2 @ ( set_rule2 @ Xs2 ) ) @ Xs2 @ ( cons_rule @ X2 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_211_List_Oinsert__def,axiom,
( insert_tm
= ( ^ [X2: tm,Xs2: list_tm] : ( if_list_tm @ ( member_tm3 @ X2 @ ( set_tm2 @ Xs2 ) ) @ Xs2 @ ( cons_tm @ X2 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_212_insert__subset,axiom,
! [X: rule,A: set_rule,B: set_rule] :
( ( ord_less_eq_set_rule @ ( insert_rule2 @ X @ A ) @ B )
= ( ( member_rule3 @ X @ B )
& ( ord_less_eq_set_rule @ A @ B ) ) ) ).
% insert_subset
thf(fact_213_insert__subset,axiom,
! [X: real,A: set_real,B: set_real] :
( ( ord_less_eq_set_real @ ( insert_real2 @ X @ A ) @ B )
= ( ( member_real3 @ X @ B )
& ( ord_less_eq_set_real @ A @ B ) ) ) ).
% insert_subset
thf(fact_214_insert__subset,axiom,
! [X: fm,A: set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ ( insert_fm2 @ X @ A ) @ B )
= ( ( member_fm3 @ X @ B )
& ( ord_less_eq_set_fm @ A @ B ) ) ) ).
% insert_subset
thf(fact_215_insert__subset,axiom,
! [X: tm,A: set_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ ( insert_tm2 @ X @ A ) @ B )
= ( ( member_tm3 @ X @ B )
& ( ord_less_eq_set_tm @ A @ B ) ) ) ).
% insert_subset
thf(fact_216_insert__subset,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat2 @ X @ A ) @ B )
= ( ( member_nat3 @ X @ B )
& ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_217_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_218_SeCaV_Oext,axiom,
( ext_fm
= ( ^ [Y: list_fm,Z: list_fm] : ( ord_less_eq_set_fm @ ( set_fm2 @ Z ) @ ( set_fm2 @ Y ) ) ) ) ).
% SeCaV.ext
thf(fact_219_SeCaV_Oext,axiom,
( ext_real
= ( ^ [Y: list_real,Z: list_real] : ( ord_less_eq_set_real @ ( set_real2 @ Z ) @ ( set_real2 @ Y ) ) ) ) ).
% SeCaV.ext
thf(fact_220_SeCaV_Oext,axiom,
( ext_rule
= ( ^ [Y: list_rule,Z: list_rule] : ( ord_less_eq_set_rule @ ( set_rule2 @ Z ) @ ( set_rule2 @ Y ) ) ) ) ).
% SeCaV.ext
thf(fact_221_SeCaV_Oext,axiom,
( ext_tm
= ( ^ [Y: list_tm,Z: list_tm] : ( ord_less_eq_set_tm @ ( set_tm2 @ Z ) @ ( set_tm2 @ Y ) ) ) ) ).
% SeCaV.ext
thf(fact_222_SeCaV_Oext,axiom,
( ext_nat
= ( ^ [Y: list_nat,Z: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Z ) @ ( set_nat2 @ Y ) ) ) ) ).
% SeCaV.ext
thf(fact_223_insert__iff,axiom,
! [A2: rule,B2: rule,A: set_rule] :
( ( member_rule3 @ A2 @ ( insert_rule2 @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_rule3 @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_224_insert__iff,axiom,
! [A2: real,B2: real,A: set_real] :
( ( member_real3 @ A2 @ ( insert_real2 @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_real3 @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_225_insert__iff,axiom,
! [A2: tm,B2: tm,A: set_tm] :
( ( member_tm3 @ A2 @ ( insert_tm2 @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_tm3 @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_226_insert__iff,axiom,
! [A2: fm,B2: fm,A: set_fm] :
( ( member_fm3 @ A2 @ ( insert_fm2 @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_fm3 @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_227_insert__iff,axiom,
! [A2: nat,B2: nat,A: set_nat] :
( ( member_nat3 @ A2 @ ( insert_nat2 @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_nat3 @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_228_insertCI,axiom,
! [A2: rule,B: set_rule,B2: rule] :
( ( ~ ( member_rule3 @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_rule3 @ A2 @ ( insert_rule2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_229_insertCI,axiom,
! [A2: real,B: set_real,B2: real] :
( ( ~ ( member_real3 @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_real3 @ A2 @ ( insert_real2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_230_insertCI,axiom,
! [A2: tm,B: set_tm,B2: tm] :
( ( ~ ( member_tm3 @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_tm3 @ A2 @ ( insert_tm2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_231_insertCI,axiom,
! [A2: fm,B: set_fm,B2: fm] :
( ( ~ ( member_fm3 @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_fm3 @ A2 @ ( insert_fm2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_232_insertCI,axiom,
! [A2: nat,B: set_nat,B2: nat] :
( ( ~ ( member_nat3 @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_nat3 @ A2 @ ( insert_nat2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_233_subset__antisym,axiom,
! [A: set_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( ord_less_eq_set_tm @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_234_subset__antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_235_subsetI,axiom,
! [A: set_rule,B: set_rule] :
( ! [X3: rule] :
( ( member_rule3 @ X3 @ A )
=> ( member_rule3 @ X3 @ B ) )
=> ( ord_less_eq_set_rule @ A @ B ) ) ).
% subsetI
thf(fact_236_subsetI,axiom,
! [A: set_real,B: set_real] :
( ! [X3: real] :
( ( member_real3 @ X3 @ A )
=> ( member_real3 @ X3 @ B ) )
=> ( ord_less_eq_set_real @ A @ B ) ) ).
% subsetI
thf(fact_237_subsetI,axiom,
! [A: set_fm,B: set_fm] :
( ! [X3: fm] :
( ( member_fm3 @ X3 @ A )
=> ( member_fm3 @ X3 @ B ) )
=> ( ord_less_eq_set_fm @ A @ B ) ) ).
% subsetI
thf(fact_238_subsetI,axiom,
! [A: set_tm,B: set_tm] :
( ! [X3: tm] :
( ( member_tm3 @ X3 @ A )
=> ( member_tm3 @ X3 @ B ) )
=> ( ord_less_eq_set_tm @ A @ B ) ) ).
% subsetI
thf(fact_239_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat3 @ X3 @ A )
=> ( member_nat3 @ X3 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_240_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_241_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_242_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_243_in__mono,axiom,
! [A: set_rule,B: set_rule,X: rule] :
( ( ord_less_eq_set_rule @ A @ B )
=> ( ( member_rule3 @ X @ A )
=> ( member_rule3 @ X @ B ) ) ) ).
% in_mono
thf(fact_244_in__mono,axiom,
! [A: set_real,B: set_real,X: real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( member_real3 @ X @ A )
=> ( member_real3 @ X @ B ) ) ) ).
% in_mono
thf(fact_245_in__mono,axiom,
! [A: set_fm,B: set_fm,X: fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ( member_fm3 @ X @ A )
=> ( member_fm3 @ X @ B ) ) ) ).
% in_mono
thf(fact_246_in__mono,axiom,
! [A: set_tm,B: set_tm,X: tm] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( member_tm3 @ X @ A )
=> ( member_tm3 @ X @ B ) ) ) ).
% in_mono
thf(fact_247_in__mono,axiom,
! [A: set_nat,B: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat3 @ X @ A )
=> ( member_nat3 @ X @ B ) ) ) ).
% in_mono
thf(fact_248_subsetD,axiom,
! [A: set_rule,B: set_rule,C: rule] :
( ( ord_less_eq_set_rule @ A @ B )
=> ( ( member_rule3 @ C @ A )
=> ( member_rule3 @ C @ B ) ) ) ).
% subsetD
thf(fact_249_subsetD,axiom,
! [A: set_real,B: set_real,C: real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( member_real3 @ C @ A )
=> ( member_real3 @ C @ B ) ) ) ).
% subsetD
thf(fact_250_subsetD,axiom,
! [A: set_fm,B: set_fm,C: fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ( member_fm3 @ C @ A )
=> ( member_fm3 @ C @ B ) ) ) ).
% subsetD
thf(fact_251_subsetD,axiom,
! [A: set_tm,B: set_tm,C: tm] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( member_tm3 @ C @ A )
=> ( member_tm3 @ C @ B ) ) ) ).
% subsetD
thf(fact_252_subsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat3 @ C @ A )
=> ( member_nat3 @ C @ B ) ) ) ).
% subsetD
thf(fact_253_equalityE,axiom,
! [A: set_tm,B: set_tm] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_tm @ A @ B )
=> ~ ( ord_less_eq_set_tm @ B @ A ) ) ) ).
% equalityE
thf(fact_254_equalityE,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_255_subset__eq,axiom,
( ord_less_eq_set_rule
= ( ^ [A3: set_rule,B3: set_rule] :
! [X2: rule] :
( ( member_rule3 @ X2 @ A3 )
=> ( member_rule3 @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_256_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A3: set_real,B3: set_real] :
! [X2: real] :
( ( member_real3 @ X2 @ A3 )
=> ( member_real3 @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_257_subset__eq,axiom,
( ord_less_eq_set_fm
= ( ^ [A3: set_fm,B3: set_fm] :
! [X2: fm] :
( ( member_fm3 @ X2 @ A3 )
=> ( member_fm3 @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_258_subset__eq,axiom,
( ord_less_eq_set_tm
= ( ^ [A3: set_tm,B3: set_tm] :
! [X2: tm] :
( ( member_tm3 @ X2 @ A3 )
=> ( member_tm3 @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_259_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [X2: nat] :
( ( member_nat3 @ X2 @ A3 )
=> ( member_nat3 @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_260_equalityD1,axiom,
! [A: set_tm,B: set_tm] :
( ( A = B )
=> ( ord_less_eq_set_tm @ A @ B ) ) ).
% equalityD1
thf(fact_261_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_262_equalityD2,axiom,
! [A: set_tm,B: set_tm] :
( ( A = B )
=> ( ord_less_eq_set_tm @ B @ A ) ) ).
% equalityD2
thf(fact_263_equalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% equalityD2
thf(fact_264_subset__iff,axiom,
( ord_less_eq_set_rule
= ( ^ [A3: set_rule,B3: set_rule] :
! [T: rule] :
( ( member_rule3 @ T @ A3 )
=> ( member_rule3 @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_265_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A3: set_real,B3: set_real] :
! [T: real] :
( ( member_real3 @ T @ A3 )
=> ( member_real3 @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_266_subset__iff,axiom,
( ord_less_eq_set_fm
= ( ^ [A3: set_fm,B3: set_fm] :
! [T: fm] :
( ( member_fm3 @ T @ A3 )
=> ( member_fm3 @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_267_subset__iff,axiom,
( ord_less_eq_set_tm
= ( ^ [A3: set_tm,B3: set_tm] :
! [T: tm] :
( ( member_tm3 @ T @ A3 )
=> ( member_tm3 @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_268_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [T: nat] :
( ( member_nat3 @ T @ A3 )
=> ( member_nat3 @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_269_subset__refl,axiom,
! [A: set_tm] : ( ord_less_eq_set_tm @ A @ A ) ).
% subset_refl
thf(fact_270_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_271_Collect__mono,axiom,
! [P4: tm > $o,Q2: tm > $o] :
( ! [X3: tm] :
( ( P4 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_less_eq_set_tm @ ( collect_tm @ P4 ) @ ( collect_tm @ Q2 ) ) ) ).
% Collect_mono
thf(fact_272_Collect__mono,axiom,
! [P4: nat > $o,Q2: nat > $o] :
( ! [X3: nat] :
( ( P4 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P4 ) @ ( collect_nat @ Q2 ) ) ) ).
% Collect_mono
thf(fact_273_subset__trans,axiom,
! [A: set_tm,B: set_tm,C2: set_tm] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( ord_less_eq_set_tm @ B @ C2 )
=> ( ord_less_eq_set_tm @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_274_subset__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_275_set__eq__subset,axiom,
( ( ^ [Y3: set_tm,Z4: set_tm] : ( Y3 = Z4 ) )
= ( ^ [A3: set_tm,B3: set_tm] :
( ( ord_less_eq_set_tm @ A3 @ B3 )
& ( ord_less_eq_set_tm @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_276_set__eq__subset,axiom,
( ( ^ [Y3: set_nat,Z4: set_nat] : ( Y3 = Z4 ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_277_Collect__mono__iff,axiom,
! [P4: tm > $o,Q2: tm > $o] :
( ( ord_less_eq_set_tm @ ( collect_tm @ P4 ) @ ( collect_tm @ Q2 ) )
= ( ! [X2: tm] :
( ( P4 @ X2 )
=> ( Q2 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_278_Collect__mono__iff,axiom,
! [P4: nat > $o,Q2: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P4 ) @ ( collect_nat @ Q2 ) )
= ( ! [X2: nat] :
( ( P4 @ X2 )
=> ( Q2 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_279_insertE,axiom,
! [A2: rule,B2: rule,A: set_rule] :
( ( member_rule3 @ A2 @ ( insert_rule2 @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_rule3 @ A2 @ A ) ) ) ).
% insertE
thf(fact_280_insertE,axiom,
! [A2: real,B2: real,A: set_real] :
( ( member_real3 @ A2 @ ( insert_real2 @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_real3 @ A2 @ A ) ) ) ).
% insertE
thf(fact_281_insertE,axiom,
! [A2: tm,B2: tm,A: set_tm] :
( ( member_tm3 @ A2 @ ( insert_tm2 @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_tm3 @ A2 @ A ) ) ) ).
% insertE
thf(fact_282_insertE,axiom,
! [A2: fm,B2: fm,A: set_fm] :
( ( member_fm3 @ A2 @ ( insert_fm2 @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_fm3 @ A2 @ A ) ) ) ).
% insertE
thf(fact_283_insertE,axiom,
! [A2: nat,B2: nat,A: set_nat] :
( ( member_nat3 @ A2 @ ( insert_nat2 @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_nat3 @ A2 @ A ) ) ) ).
% insertE
thf(fact_284_insertI1,axiom,
! [A2: rule,B: set_rule] : ( member_rule3 @ A2 @ ( insert_rule2 @ A2 @ B ) ) ).
% insertI1
thf(fact_285_insertI1,axiom,
! [A2: real,B: set_real] : ( member_real3 @ A2 @ ( insert_real2 @ A2 @ B ) ) ).
% insertI1
thf(fact_286_insertI1,axiom,
! [A2: tm,B: set_tm] : ( member_tm3 @ A2 @ ( insert_tm2 @ A2 @ B ) ) ).
% insertI1
thf(fact_287_insertI1,axiom,
! [A2: fm,B: set_fm] : ( member_fm3 @ A2 @ ( insert_fm2 @ A2 @ B ) ) ).
% insertI1
thf(fact_288_insertI1,axiom,
! [A2: nat,B: set_nat] : ( member_nat3 @ A2 @ ( insert_nat2 @ A2 @ B ) ) ).
% insertI1
thf(fact_289_insertI2,axiom,
! [A2: rule,B: set_rule,B2: rule] :
( ( member_rule3 @ A2 @ B )
=> ( member_rule3 @ A2 @ ( insert_rule2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_290_insertI2,axiom,
! [A2: real,B: set_real,B2: real] :
( ( member_real3 @ A2 @ B )
=> ( member_real3 @ A2 @ ( insert_real2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_291_insertI2,axiom,
! [A2: tm,B: set_tm,B2: tm] :
( ( member_tm3 @ A2 @ B )
=> ( member_tm3 @ A2 @ ( insert_tm2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_292_insertI2,axiom,
! [A2: fm,B: set_fm,B2: fm] :
( ( member_fm3 @ A2 @ B )
=> ( member_fm3 @ A2 @ ( insert_fm2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_293_insertI2,axiom,
! [A2: nat,B: set_nat,B2: nat] :
( ( member_nat3 @ A2 @ B )
=> ( member_nat3 @ A2 @ ( insert_nat2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_294_Set_Oset__insert,axiom,
! [X: rule,A: set_rule] :
( ( member_rule3 @ X @ A )
=> ~ ! [B4: set_rule] :
( ( A
= ( insert_rule2 @ X @ B4 ) )
=> ( member_rule3 @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_295_Set_Oset__insert,axiom,
! [X: real,A: set_real] :
( ( member_real3 @ X @ A )
=> ~ ! [B4: set_real] :
( ( A
= ( insert_real2 @ X @ B4 ) )
=> ( member_real3 @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_296_Set_Oset__insert,axiom,
! [X: tm,A: set_tm] :
( ( member_tm3 @ X @ A )
=> ~ ! [B4: set_tm] :
( ( A
= ( insert_tm2 @ X @ B4 ) )
=> ( member_tm3 @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_297_Set_Oset__insert,axiom,
! [X: fm,A: set_fm] :
( ( member_fm3 @ X @ A )
=> ~ ! [B4: set_fm] :
( ( A
= ( insert_fm2 @ X @ B4 ) )
=> ( member_fm3 @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_298_Set_Oset__insert,axiom,
! [X: nat,A: set_nat] :
( ( member_nat3 @ X @ A )
=> ~ ! [B4: set_nat] :
( ( A
= ( insert_nat2 @ X @ B4 ) )
=> ( member_nat3 @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_299_insert__ident,axiom,
! [X: rule,A: set_rule,B: set_rule] :
( ~ ( member_rule3 @ X @ A )
=> ( ~ ( member_rule3 @ X @ B )
=> ( ( ( insert_rule2 @ X @ A )
= ( insert_rule2 @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_300_insert__ident,axiom,
! [X: real,A: set_real,B: set_real] :
( ~ ( member_real3 @ X @ A )
=> ( ~ ( member_real3 @ X @ B )
=> ( ( ( insert_real2 @ X @ A )
= ( insert_real2 @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_301_insert__ident,axiom,
! [X: tm,A: set_tm,B: set_tm] :
( ~ ( member_tm3 @ X @ A )
=> ( ~ ( member_tm3 @ X @ B )
=> ( ( ( insert_tm2 @ X @ A )
= ( insert_tm2 @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_302_insert__ident,axiom,
! [X: fm,A: set_fm,B: set_fm] :
( ~ ( member_fm3 @ X @ A )
=> ( ~ ( member_fm3 @ X @ B )
=> ( ( ( insert_fm2 @ X @ A )
= ( insert_fm2 @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_303_insert__ident,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat3 @ X @ A )
=> ( ~ ( member_nat3 @ X @ B )
=> ( ( ( insert_nat2 @ X @ A )
= ( insert_nat2 @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_304_insert__absorb,axiom,
! [A2: rule,A: set_rule] :
( ( member_rule3 @ A2 @ A )
=> ( ( insert_rule2 @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_305_insert__absorb,axiom,
! [A2: real,A: set_real] :
( ( member_real3 @ A2 @ A )
=> ( ( insert_real2 @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_306_insert__absorb,axiom,
! [A2: tm,A: set_tm] :
( ( member_tm3 @ A2 @ A )
=> ( ( insert_tm2 @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_307_insert__absorb,axiom,
! [A2: fm,A: set_fm] :
( ( member_fm3 @ A2 @ A )
=> ( ( insert_fm2 @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_308_insert__absorb,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat3 @ A2 @ A )
=> ( ( insert_nat2 @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_309_insert__eq__iff,axiom,
! [A2: rule,A: set_rule,B2: rule,B: set_rule] :
( ~ ( member_rule3 @ A2 @ A )
=> ( ~ ( member_rule3 @ B2 @ B )
=> ( ( ( insert_rule2 @ A2 @ A )
= ( insert_rule2 @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_rule] :
( ( A
= ( insert_rule2 @ B2 @ C3 ) )
& ~ ( member_rule3 @ B2 @ C3 )
& ( B
= ( insert_rule2 @ A2 @ C3 ) )
& ~ ( member_rule3 @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_310_insert__eq__iff,axiom,
! [A2: real,A: set_real,B2: real,B: set_real] :
( ~ ( member_real3 @ A2 @ A )
=> ( ~ ( member_real3 @ B2 @ B )
=> ( ( ( insert_real2 @ A2 @ A )
= ( insert_real2 @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_real] :
( ( A
= ( insert_real2 @ B2 @ C3 ) )
& ~ ( member_real3 @ B2 @ C3 )
& ( B
= ( insert_real2 @ A2 @ C3 ) )
& ~ ( member_real3 @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_311_insert__eq__iff,axiom,
! [A2: tm,A: set_tm,B2: tm,B: set_tm] :
( ~ ( member_tm3 @ A2 @ A )
=> ( ~ ( member_tm3 @ B2 @ B )
=> ( ( ( insert_tm2 @ A2 @ A )
= ( insert_tm2 @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_tm] :
( ( A
= ( insert_tm2 @ B2 @ C3 ) )
& ~ ( member_tm3 @ B2 @ C3 )
& ( B
= ( insert_tm2 @ A2 @ C3 ) )
& ~ ( member_tm3 @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_312_insert__eq__iff,axiom,
! [A2: fm,A: set_fm,B2: fm,B: set_fm] :
( ~ ( member_fm3 @ A2 @ A )
=> ( ~ ( member_fm3 @ B2 @ B )
=> ( ( ( insert_fm2 @ A2 @ A )
= ( insert_fm2 @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_fm] :
( ( A
= ( insert_fm2 @ B2 @ C3 ) )
& ~ ( member_fm3 @ B2 @ C3 )
& ( B
= ( insert_fm2 @ A2 @ C3 ) )
& ~ ( member_fm3 @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_313_insert__eq__iff,axiom,
! [A2: nat,A: set_nat,B2: nat,B: set_nat] :
( ~ ( member_nat3 @ A2 @ A )
=> ( ~ ( member_nat3 @ B2 @ B )
=> ( ( ( insert_nat2 @ A2 @ A )
= ( insert_nat2 @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_nat] :
( ( A
= ( insert_nat2 @ B2 @ C3 ) )
& ~ ( member_nat3 @ B2 @ C3 )
& ( B
= ( insert_nat2 @ A2 @ C3 ) )
& ~ ( member_nat3 @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_314_mk__disjoint__insert,axiom,
! [A2: rule,A: set_rule] :
( ( member_rule3 @ A2 @ A )
=> ? [B4: set_rule] :
( ( A
= ( insert_rule2 @ A2 @ B4 ) )
& ~ ( member_rule3 @ A2 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_315_mk__disjoint__insert,axiom,
! [A2: real,A: set_real] :
( ( member_real3 @ A2 @ A )
=> ? [B4: set_real] :
( ( A
= ( insert_real2 @ A2 @ B4 ) )
& ~ ( member_real3 @ A2 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_316_mk__disjoint__insert,axiom,
! [A2: tm,A: set_tm] :
( ( member_tm3 @ A2 @ A )
=> ? [B4: set_tm] :
( ( A
= ( insert_tm2 @ A2 @ B4 ) )
& ~ ( member_tm3 @ A2 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_317_mk__disjoint__insert,axiom,
! [A2: fm,A: set_fm] :
( ( member_fm3 @ A2 @ A )
=> ? [B4: set_fm] :
( ( A
= ( insert_fm2 @ A2 @ B4 ) )
& ~ ( member_fm3 @ A2 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_318_mk__disjoint__insert,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat3 @ A2 @ A )
=> ? [B4: set_nat] :
( ( A
= ( insert_nat2 @ A2 @ B4 ) )
& ~ ( member_nat3 @ A2 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_319_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_320_insert__mono,axiom,
! [C2: set_tm,D: set_tm,A2: tm] :
( ( ord_less_eq_set_tm @ C2 @ D )
=> ( ord_less_eq_set_tm @ ( insert_tm2 @ A2 @ C2 ) @ ( insert_tm2 @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_321_insert__mono,axiom,
! [C2: set_nat,D: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ C2 @ D )
=> ( ord_less_eq_set_nat @ ( insert_nat2 @ A2 @ C2 ) @ ( insert_nat2 @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_322_subset__insert,axiom,
! [X: rule,A: set_rule,B: set_rule] :
( ~ ( member_rule3 @ X @ A )
=> ( ( ord_less_eq_set_rule @ A @ ( insert_rule2 @ X @ B ) )
= ( ord_less_eq_set_rule @ A @ B ) ) ) ).
% subset_insert
thf(fact_323_subset__insert,axiom,
! [X: real,A: set_real,B: set_real] :
( ~ ( member_real3 @ X @ A )
=> ( ( ord_less_eq_set_real @ A @ ( insert_real2 @ X @ B ) )
= ( ord_less_eq_set_real @ A @ B ) ) ) ).
% subset_insert
thf(fact_324_subset__insert,axiom,
! [X: fm,A: set_fm,B: set_fm] :
( ~ ( member_fm3 @ X @ A )
=> ( ( ord_less_eq_set_fm @ A @ ( insert_fm2 @ X @ B ) )
= ( ord_less_eq_set_fm @ A @ B ) ) ) ).
% subset_insert
thf(fact_325_subset__insert,axiom,
! [X: tm,A: set_tm,B: set_tm] :
( ~ ( member_tm3 @ X @ A )
=> ( ( ord_less_eq_set_tm @ A @ ( insert_tm2 @ X @ B ) )
= ( ord_less_eq_set_tm @ A @ B ) ) ) ).
% subset_insert
thf(fact_326_subset__insert,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat3 @ X @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ X @ B ) )
= ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_327_subset__insertI,axiom,
! [B: set_tm,A2: tm] : ( ord_less_eq_set_tm @ B @ ( insert_tm2 @ A2 @ B ) ) ).
% subset_insertI
thf(fact_328_subset__insertI,axiom,
! [B: set_nat,A2: nat] : ( ord_less_eq_set_nat @ B @ ( insert_nat2 @ A2 @ B ) ) ).
% subset_insertI
thf(fact_329_subset__insertI2,axiom,
! [A: set_tm,B: set_tm,B2: tm] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ord_less_eq_set_tm @ A @ ( insert_tm2 @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_330_subset__insertI2,axiom,
! [A: set_nat,B: set_nat,B2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_331_ext_Osimps_I2_J,axiom,
! [Y2: list_fm,P3: fm,Z3: list_fm] :
( ( ext_fm @ Y2 @ ( cons_fm @ P3 @ Z3 ) )
= ( ( ( member_fm2 @ P3 @ Y2 )
=> ( ext_fm @ Y2 @ Z3 ) )
& ( member_fm2 @ P3 @ Y2 ) ) ) ).
% ext.simps(2)
thf(fact_332_ext_Osimps_I2_J,axiom,
! [Y2: list_nat,P3: nat,Z3: list_nat] :
( ( ext_nat @ Y2 @ ( cons_nat @ P3 @ Z3 ) )
= ( ( ( member_nat2 @ P3 @ Y2 )
=> ( ext_nat @ Y2 @ Z3 ) )
& ( member_nat2 @ P3 @ Y2 ) ) ) ).
% ext.simps(2)
thf(fact_333_ext_Osimps_I2_J,axiom,
! [Y2: list_real,P3: real,Z3: list_real] :
( ( ext_real @ Y2 @ ( cons_real @ P3 @ Z3 ) )
= ( ( ( member_real2 @ P3 @ Y2 )
=> ( ext_real @ Y2 @ Z3 ) )
& ( member_real2 @ P3 @ Y2 ) ) ) ).
% ext.simps(2)
thf(fact_334_ext_Osimps_I2_J,axiom,
! [Y2: list_rule,P3: rule,Z3: list_rule] :
( ( ext_rule @ Y2 @ ( cons_rule @ P3 @ Z3 ) )
= ( ( ( member_rule2 @ P3 @ Y2 )
=> ( ext_rule @ Y2 @ Z3 ) )
& ( member_rule2 @ P3 @ Y2 ) ) ) ).
% ext.simps(2)
thf(fact_335_ext_Osimps_I2_J,axiom,
! [Y2: list_tm,P3: tm,Z3: list_tm] :
( ( ext_tm @ Y2 @ ( cons_tm @ P3 @ Z3 ) )
= ( ( ( member_tm2 @ P3 @ Y2 )
=> ( ext_tm @ Y2 @ Z3 ) )
& ( member_tm2 @ P3 @ Y2 ) ) ) ).
% ext.simps(2)
thf(fact_336_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_337_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_338_order__refl,axiom,
! [X: set_tm] : ( ord_less_eq_set_tm @ X @ X ) ).
% order_refl
thf(fact_339_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_340_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_341_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_342_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_343_dual__order_Orefl,axiom,
! [A2: set_tm] : ( ord_less_eq_set_tm @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_344_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_345_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_346_dual__order_Orefl,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_347_dual__order_Orefl,axiom,
! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_348_snth__sset__smerge,axiom,
! [Ss: stream_stream_real,N: nat,M: nat] : ( member_real3 @ ( snth_real @ ( snth_stream_real @ Ss @ N ) @ M ) @ ( sset_real @ ( smerge_real @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_349_snth__sset__smerge,axiom,
! [Ss: stream_stream_tm,N: nat,M: nat] : ( member_tm3 @ ( snth_tm @ ( snth_stream_tm @ Ss @ N ) @ M ) @ ( sset_tm @ ( smerge_tm @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_350_snth__sset__smerge,axiom,
! [Ss: stream_stream_fm,N: nat,M: nat] : ( member_fm3 @ ( snth_fm @ ( snth_stream_fm @ Ss @ N ) @ M ) @ ( sset_fm @ ( smerge_fm @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_351_snth__sset__smerge,axiom,
! [Ss: stream_stream_nat,N: nat,M: nat] : ( member_nat3 @ ( snth_nat @ ( snth_stream_nat @ Ss @ N ) @ M ) @ ( sset_nat @ ( smerge_nat @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_352_snth__sset__smerge,axiom,
! [Ss: stream3752074346242807894m_rule,N: nat,M: nat] : ( member7231649785386036813m_rule @ ( snth_P6679518042731451922m_rule @ ( snth_s6182113952396108578m_rule @ Ss @ N ) @ M ) @ ( sset_P4484857331586881186m_rule @ ( smerge193809993764105000m_rule @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_353_snth__sset__smerge,axiom,
! [Ss: stream_stream_rule,N: nat,M: nat] : ( member_rule3 @ ( snth_rule @ ( snth_stream_rule @ Ss @ N ) @ M ) @ ( sset_rule @ ( smerge_rule @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_354_insert__subsetI,axiom,
! [X: rule,A: set_rule,X4: set_rule] :
( ( member_rule3 @ X @ A )
=> ( ( ord_less_eq_set_rule @ X4 @ A )
=> ( ord_less_eq_set_rule @ ( insert_rule2 @ X @ X4 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_355_insert__subsetI,axiom,
! [X: real,A: set_real,X4: set_real] :
( ( member_real3 @ X @ A )
=> ( ( ord_less_eq_set_real @ X4 @ A )
=> ( ord_less_eq_set_real @ ( insert_real2 @ X @ X4 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_356_insert__subsetI,axiom,
! [X: fm,A: set_fm,X4: set_fm] :
( ( member_fm3 @ X @ A )
=> ( ( ord_less_eq_set_fm @ X4 @ A )
=> ( ord_less_eq_set_fm @ ( insert_fm2 @ X @ X4 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_357_insert__subsetI,axiom,
! [X: tm,A: set_tm,X4: set_tm] :
( ( member_tm3 @ X @ A )
=> ( ( ord_less_eq_set_tm @ X4 @ A )
=> ( ord_less_eq_set_tm @ ( insert_tm2 @ X @ X4 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_358_insert__subsetI,axiom,
! [X: nat,A: set_nat,X4: set_nat] :
( ( member_nat3 @ X @ A )
=> ( ( ord_less_eq_set_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ ( insert_nat2 @ X @ X4 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_359_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_360_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_361_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_362_RuleSystem__Defs_OSaturated__def,axiom,
( abstra6097777249025082867ist_fm
= ( ^ [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules: stream_rule,Steps2: stream2709947120125613254m_rule] :
! [X2: rule] :
( ( member_rule3 @ X2 @ ( sset_rule @ Rules ) )
=> ( abstra2533313685312581075ist_fm @ Eff @ X2 @ Steps2 ) ) ) ) ).
% RuleSystem_Defs.Saturated_def
thf(fact_363_Nat_Oex__has__greatest__nat,axiom,
! [P4: nat > $o,K: nat,B2: nat] :
( ( P4 @ K )
=> ( ! [Y4: nat] :
( ( P4 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B2 ) )
=> ? [X3: nat] :
( ( P4 @ X3 )
& ! [Y5: nat] :
( ( P4 @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_364_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_365_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_366_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_367_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_368_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_369_RuleSystem__Defs_Osaturated_Ocong,axiom,
abstra2533313685312581075ist_fm = abstra2533313685312581075ist_fm ).
% RuleSystem_Defs.saturated.cong
thf(fact_370_order__antisym__conv,axiom,
! [Y2: set_tm,X: set_tm] :
( ( ord_less_eq_set_tm @ Y2 @ X )
=> ( ( ord_less_eq_set_tm @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_371_order__antisym__conv,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( ord_less_eq_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_372_order__antisym__conv,axiom,
! [Y2: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X )
=> ( ( ord_less_eq_set_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_373_order__antisym__conv,axiom,
! [Y2: int,X: int] :
( ( ord_less_eq_int @ Y2 @ X )
=> ( ( ord_less_eq_int @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_374_order__antisym__conv,axiom,
! [Y2: real,X: real] :
( ( ord_less_eq_real @ Y2 @ X )
=> ( ( ord_less_eq_real @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_375_linorder__le__cases,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_376_linorder__le__cases,axiom,
! [X: int,Y2: int] :
( ~ ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_377_linorder__le__cases,axiom,
! [X: real,Y2: real] :
( ~ ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_378_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_379_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_380_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_381_ord__le__eq__subst,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_382_ord__le__eq__subst,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_383_ord__le__eq__subst,axiom,
! [A2: int,B2: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_384_ord__le__eq__subst,axiom,
! [A2: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_385_ord__le__eq__subst,axiom,
! [A2: real,B2: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_386_ord__le__eq__subst,axiom,
! [A2: real,B2: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_387_ord__le__eq__subst,axiom,
! [A2: set_tm,B2: set_tm,F: set_tm > nat,C: nat] :
( ( ord_less_eq_set_tm @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_tm,Y4: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_388_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_389_ord__eq__le__subst,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_390_ord__eq__le__subst,axiom,
! [A2: real,F: nat > real,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_391_ord__eq__le__subst,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_392_ord__eq__le__subst,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_393_ord__eq__le__subst,axiom,
! [A2: real,F: int > real,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_394_ord__eq__le__subst,axiom,
! [A2: nat,F: real > nat,B2: real,C: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_395_ord__eq__le__subst,axiom,
! [A2: int,F: real > int,B2: real,C: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_396_ord__eq__le__subst,axiom,
! [A2: real,F: real > real,B2: real,C: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_397_ord__eq__le__subst,axiom,
! [A2: nat,F: set_tm > nat,B2: set_tm,C: set_tm] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_tm @ B2 @ C )
=> ( ! [X3: set_tm,Y4: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_398_linorder__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_399_linorder__linear,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
| ( ord_less_eq_int @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_400_linorder__linear,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
| ( ord_less_eq_real @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_401_order__eq__refl,axiom,
! [X: set_tm,Y2: set_tm] :
( ( X = Y2 )
=> ( ord_less_eq_set_tm @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_402_order__eq__refl,axiom,
! [X: nat,Y2: nat] :
( ( X = Y2 )
=> ( ord_less_eq_nat @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_403_order__eq__refl,axiom,
! [X: set_nat,Y2: set_nat] :
( ( X = Y2 )
=> ( ord_less_eq_set_nat @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_404_order__eq__refl,axiom,
! [X: int,Y2: int] :
( ( X = Y2 )
=> ( ord_less_eq_int @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_405_order__eq__refl,axiom,
! [X: real,Y2: real] :
( ( X = Y2 )
=> ( ord_less_eq_real @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_406_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_407_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_408_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_409_order__subst2,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_410_order__subst2,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_411_order__subst2,axiom,
! [A2: int,B2: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_412_order__subst2,axiom,
! [A2: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_413_order__subst2,axiom,
! [A2: real,B2: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_414_order__subst2,axiom,
! [A2: real,B2: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_415_order__subst2,axiom,
! [A2: set_tm,B2: set_tm,F: set_tm > nat,C: nat] :
( ( ord_less_eq_set_tm @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_tm,Y4: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_416_order__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_417_order__subst1,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_418_order__subst1,axiom,
! [A2: nat,F: real > nat,B2: real,C: real] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_419_order__subst1,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_420_order__subst1,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_421_order__subst1,axiom,
! [A2: int,F: real > int,B2: real,C: real] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_422_order__subst1,axiom,
! [A2: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_423_order__subst1,axiom,
! [A2: real,F: int > real,B2: int,C: int] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_424_order__subst1,axiom,
! [A2: real,F: real > real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_425_order__subst1,axiom,
! [A2: set_tm,F: nat > set_tm,B2: nat,C: nat] :
( ( ord_less_eq_set_tm @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_tm @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_426_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_tm,Z4: set_tm] : ( Y3 = Z4 ) )
= ( ^ [A4: set_tm,B5: set_tm] :
( ( ord_less_eq_set_tm @ A4 @ B5 )
& ( ord_less_eq_set_tm @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_427_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z4: nat] : ( Y3 = Z4 ) )
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ A4 @ B5 )
& ( ord_less_eq_nat @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_428_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z4: set_nat] : ( Y3 = Z4 ) )
= ( ^ [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_429_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z4: int] : ( Y3 = Z4 ) )
= ( ^ [A4: int,B5: int] :
( ( ord_less_eq_int @ A4 @ B5 )
& ( ord_less_eq_int @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_430_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z4: real] : ( Y3 = Z4 ) )
= ( ^ [A4: real,B5: real] :
( ( ord_less_eq_real @ A4 @ B5 )
& ( ord_less_eq_real @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_431_antisym,axiom,
! [A2: set_tm,B2: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ B2 )
=> ( ( ord_less_eq_set_tm @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_432_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_433_antisym,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_434_antisym,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_435_antisym,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_436_dual__order_Otrans,axiom,
! [B2: set_tm,A2: set_tm,C: set_tm] :
( ( ord_less_eq_set_tm @ B2 @ A2 )
=> ( ( ord_less_eq_set_tm @ C @ B2 )
=> ( ord_less_eq_set_tm @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_437_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_438_dual__order_Otrans,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B2 )
=> ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_439_dual__order_Otrans,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_eq_int @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_440_dual__order_Otrans,axiom,
! [B2: real,A2: real,C: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ( ord_less_eq_real @ C @ B2 )
=> ( ord_less_eq_real @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_441_dual__order_Oantisym,axiom,
! [B2: set_tm,A2: set_tm] :
( ( ord_less_eq_set_tm @ B2 @ A2 )
=> ( ( ord_less_eq_set_tm @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_442_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_443_dual__order_Oantisym,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_444_dual__order_Oantisym,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_445_dual__order_Oantisym,axiom,
! [B2: real,A2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ( ord_less_eq_real @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_446_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_tm,Z4: set_tm] : ( Y3 = Z4 ) )
= ( ^ [A4: set_tm,B5: set_tm] :
( ( ord_less_eq_set_tm @ B5 @ A4 )
& ( ord_less_eq_set_tm @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_447_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z4: nat] : ( Y3 = Z4 ) )
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A4 )
& ( ord_less_eq_nat @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_448_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_nat,Z4: set_nat] : ( Y3 = Z4 ) )
= ( ^ [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
& ( ord_less_eq_set_nat @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_449_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: int,Z4: int] : ( Y3 = Z4 ) )
= ( ^ [A4: int,B5: int] :
( ( ord_less_eq_int @ B5 @ A4 )
& ( ord_less_eq_int @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_450_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: real,Z4: real] : ( Y3 = Z4 ) )
= ( ^ [A4: real,B5: real] :
( ( ord_less_eq_real @ B5 @ A4 )
& ( ord_less_eq_real @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_451_linorder__wlog,axiom,
! [P4: nat > nat > $o,A2: nat,B2: nat] :
( ! [A5: nat,B6: nat] :
( ( ord_less_eq_nat @ A5 @ B6 )
=> ( P4 @ A5 @ B6 ) )
=> ( ! [A5: nat,B6: nat] :
( ( P4 @ B6 @ A5 )
=> ( P4 @ A5 @ B6 ) )
=> ( P4 @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_452_linorder__wlog,axiom,
! [P4: int > int > $o,A2: int,B2: int] :
( ! [A5: int,B6: int] :
( ( ord_less_eq_int @ A5 @ B6 )
=> ( P4 @ A5 @ B6 ) )
=> ( ! [A5: int,B6: int] :
( ( P4 @ B6 @ A5 )
=> ( P4 @ A5 @ B6 ) )
=> ( P4 @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_453_linorder__wlog,axiom,
! [P4: real > real > $o,A2: real,B2: real] :
( ! [A5: real,B6: real] :
( ( ord_less_eq_real @ A5 @ B6 )
=> ( P4 @ A5 @ B6 ) )
=> ( ! [A5: real,B6: real] :
( ( P4 @ B6 @ A5 )
=> ( P4 @ A5 @ B6 ) )
=> ( P4 @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_454_order__trans,axiom,
! [X: set_tm,Y2: set_tm,Z3: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y2 )
=> ( ( ord_less_eq_set_tm @ Y2 @ Z3 )
=> ( ord_less_eq_set_tm @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_455_order__trans,axiom,
! [X: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z3 )
=> ( ord_less_eq_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_456_order__trans,axiom,
! [X: set_nat,Y2: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ Z3 )
=> ( ord_less_eq_set_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_457_order__trans,axiom,
! [X: int,Y2: int,Z3: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z3 )
=> ( ord_less_eq_int @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_458_order__trans,axiom,
! [X: real,Y2: real,Z3: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z3 )
=> ( ord_less_eq_real @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_459_order_Otrans,axiom,
! [A2: set_tm,B2: set_tm,C: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ B2 )
=> ( ( ord_less_eq_set_tm @ B2 @ C )
=> ( ord_less_eq_set_tm @ A2 @ C ) ) ) ).
% order.trans
thf(fact_460_order_Otrans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_461_order_Otrans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_462_order_Otrans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% order.trans
thf(fact_463_order_Otrans,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% order.trans
thf(fact_464_order__antisym,axiom,
! [X: set_tm,Y2: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y2 )
=> ( ( ord_less_eq_set_tm @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_465_order__antisym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_466_order__antisym,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_467_order__antisym,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_468_order__antisym,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_469_ord__le__eq__trans,axiom,
! [A2: set_tm,B2: set_tm,C: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_tm @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_470_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_471_ord__le__eq__trans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_472_ord__le__eq__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_473_ord__le__eq__trans,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_474_ord__eq__le__trans,axiom,
! [A2: set_tm,B2: set_tm,C: set_tm] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_tm @ B2 @ C )
=> ( ord_less_eq_set_tm @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_475_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_476_ord__eq__le__trans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_477_ord__eq__le__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( A2 = B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_478_ord__eq__le__trans,axiom,
! [A2: real,B2: real,C: real] :
( ( A2 = B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_479_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_tm,Z4: set_tm] : ( Y3 = Z4 ) )
= ( ^ [X2: set_tm,Y: set_tm] :
( ( ord_less_eq_set_tm @ X2 @ Y )
& ( ord_less_eq_set_tm @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_480_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z4: nat] : ( Y3 = Z4 ) )
= ( ^ [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
& ( ord_less_eq_nat @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_481_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z4: set_nat] : ( Y3 = Z4 ) )
= ( ^ [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
& ( ord_less_eq_set_nat @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_482_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z4: int] : ( Y3 = Z4 ) )
= ( ^ [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
& ( ord_less_eq_int @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_483_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z4: real] : ( Y3 = Z4 ) )
= ( ^ [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
& ( ord_less_eq_real @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_484_le__cases3,axiom,
! [X: nat,Y2: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_485_le__cases3,axiom,
! [X: int,Y2: int,Z3: int] :
( ( ( ord_less_eq_int @ X @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ Z3 ) )
=> ( ( ( ord_less_eq_int @ Y2 @ X )
=> ~ ( ord_less_eq_int @ X @ Z3 ) )
=> ( ( ( ord_less_eq_int @ X @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ Y2 ) )
=> ( ( ( ord_less_eq_int @ Z3 @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ X ) )
=> ( ( ( ord_less_eq_int @ Y2 @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z3 @ X )
=> ~ ( ord_less_eq_int @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_486_le__cases3,axiom,
! [X: real,Y2: real,Z3: real] :
( ( ( ord_less_eq_real @ X @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ Z3 ) )
=> ( ( ( ord_less_eq_real @ Y2 @ X )
=> ~ ( ord_less_eq_real @ X @ Z3 ) )
=> ( ( ( ord_less_eq_real @ X @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ Y2 ) )
=> ( ( ( ord_less_eq_real @ Z3 @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ X ) )
=> ( ( ( ord_less_eq_real @ Y2 @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_real @ Z3 @ X )
=> ~ ( ord_less_eq_real @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_487_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_488_nle__le,axiom,
! [A2: int,B2: int] :
( ( ~ ( ord_less_eq_int @ A2 @ B2 ) )
= ( ( ord_less_eq_int @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_489_nle__le,axiom,
! [A2: real,B2: real] :
( ( ~ ( ord_less_eq_real @ A2 @ B2 ) )
= ( ( ord_less_eq_real @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_490_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_491_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_492_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_493_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_494_Saturated__def,axiom,
! [Steps: stream2709947120125613254m_rule] :
( ( abstra6097777249025082867ist_fm @ eff @ rules @ Steps )
= ( ! [X2: rule] :
( ( member_rule3 @ X2 @ ( sset_rule @ rules ) )
=> ( abstra2533313685312581075ist_fm @ eff @ X2 @ Steps ) ) ) ) ).
% Saturated_def
thf(fact_495_subtermTm__le,axiom,
! [T2: tm,S: tm] :
( ( member_tm3 @ T2 @ ( set_tm2 @ ( subtermTm @ S ) ) )
=> ( ord_less_eq_set_tm @ ( set_tm2 @ ( subtermTm @ T2 ) ) @ ( set_tm2 @ ( subtermTm @ S ) ) ) ) ).
% subtermTm_le
thf(fact_496_Stream_Osmember__def,axiom,
( smember_real
= ( ^ [X2: real,S3: stream_real] : ( member_real3 @ X2 @ ( sset_real @ S3 ) ) ) ) ).
% Stream.smember_def
thf(fact_497_Stream_Osmember__def,axiom,
( smember_tm
= ( ^ [X2: tm,S3: stream_tm] : ( member_tm3 @ X2 @ ( sset_tm @ S3 ) ) ) ) ).
% Stream.smember_def
thf(fact_498_Stream_Osmember__def,axiom,
( smember_fm
= ( ^ [X2: fm,S3: stream_fm] : ( member_fm3 @ X2 @ ( sset_fm @ S3 ) ) ) ) ).
% Stream.smember_def
thf(fact_499_Stream_Osmember__def,axiom,
( smember_nat
= ( ^ [X2: nat,S3: stream_nat] : ( member_nat3 @ X2 @ ( sset_nat @ S3 ) ) ) ) ).
% Stream.smember_def
thf(fact_500_Stream_Osmember__def,axiom,
( smembe4439892024482649336m_rule
= ( ^ [X2: produc340336539035504054m_rule,S3: stream2709947120125613254m_rule] : ( member7231649785386036813m_rule @ X2 @ ( sset_P4484857331586881186m_rule @ S3 ) ) ) ) ).
% Stream.smember_def
thf(fact_501_Stream_Osmember__def,axiom,
( smember_rule
= ( ^ [X2: rule,S3: stream_rule] : ( member_rule3 @ X2 @ ( sset_rule @ S3 ) ) ) ) ).
% Stream.smember_def
thf(fact_502_divides__aux__eq,axiom,
! [Q: nat,R: nat] :
( ( unique5332122412489317741ux_nat @ ( product_Pair_nat_nat @ Q @ R ) )
= ( R = zero_zero_nat ) ) ).
% divides_aux_eq
thf(fact_503_divides__aux__eq,axiom,
! [Q: int,R: int] :
( ( unique5329631941980267465ux_int @ ( product_Pair_int_int @ Q @ R ) )
= ( R = zero_zero_int ) ) ).
% divides_aux_eq
thf(fact_504_paramsts__subset,axiom,
! [A: list_tm,B: list_tm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ A ) @ ( set_tm2 @ B ) )
=> ( ord_less_eq_set_nat @ ( paramsts @ A ) @ ( paramsts @ B ) ) ) ).
% paramsts_subset
thf(fact_505_count__notin,axiom,
! [X: tm,Xs: list_tm] :
( ~ ( member_tm3 @ X @ ( set_tm2 @ Xs ) )
=> ( ( count_list_tm @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_506_count__notin,axiom,
! [X: fm,Xs: list_fm] :
( ~ ( member_fm3 @ X @ ( set_fm2 @ Xs ) )
=> ( ( count_list_fm @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_507_count__notin,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat3 @ X @ ( set_nat2 @ Xs ) )
=> ( ( count_list_nat @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_508_count__notin,axiom,
! [X: real,Xs: list_real] :
( ~ ( member_real3 @ X @ ( set_real2 @ Xs ) )
=> ( ( count_list_real @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_509_count__notin,axiom,
! [X: rule,Xs: list_rule] :
( ~ ( member_rule3 @ X @ ( set_rule2 @ Xs ) )
=> ( ( count_list_rule @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_510_i_OSaturated__def,axiom,
! [Eff2: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm,Rules2: stream_rule,Steps: stream2709947120125613254m_rule] :
( ( abstra6097777249025082867ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ Rules2 @ Steps )
= ( ! [X2: rule] :
( ( member_rule3 @ X2 @ ( sset_rule @ Rules2 ) )
=> ( abstra2533313685312581075ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ X2 @ Steps ) ) ) ) ).
% i.Saturated_def
thf(fact_511_subtermTm__refl,axiom,
! [T2: tm] : ( member_tm3 @ T2 @ ( set_tm2 @ ( subtermTm @ T2 ) ) ) ).
% subtermTm_refl
thf(fact_512_count__list__0__iff,axiom,
! [Xs: list_tm,X: tm] :
( ( ( count_list_tm @ Xs @ X )
= zero_zero_nat )
= ( ~ ( member_tm3 @ X @ ( set_tm2 @ Xs ) ) ) ) ).
% count_list_0_iff
thf(fact_513_count__list__0__iff,axiom,
! [Xs: list_fm,X: fm] :
( ( ( count_list_fm @ Xs @ X )
= zero_zero_nat )
= ( ~ ( member_fm3 @ X @ ( set_fm2 @ Xs ) ) ) ) ).
% count_list_0_iff
thf(fact_514_count__list__0__iff,axiom,
! [Xs: list_nat,X: nat] :
( ( ( count_list_nat @ Xs @ X )
= zero_zero_nat )
= ( ~ ( member_nat3 @ X @ ( set_nat2 @ Xs ) ) ) ) ).
% count_list_0_iff
thf(fact_515_count__list__0__iff,axiom,
! [Xs: list_real,X: real] :
( ( ( count_list_real @ Xs @ X )
= zero_zero_nat )
= ( ~ ( member_real3 @ X @ ( set_real2 @ Xs ) ) ) ) ).
% count_list_0_iff
thf(fact_516_count__list__0__iff,axiom,
! [Xs: list_rule,X: rule] :
( ( ( count_list_rule @ Xs @ X )
= zero_zero_nat )
= ( ~ ( member_rule3 @ X @ ( set_rule2 @ Xs ) ) ) ) ).
% count_list_0_iff
thf(fact_517_saturated__effG__uu__def,axiom,
( abstra8695313511658867272ist_fm
= ( ^ [Eff3: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm] : ( abstra2533313685312581075ist_fm @ ( abstra2682625350522704545ist_fm @ Eff3 ) ) ) ) ).
% saturated_effG_uu_def
thf(fact_518_Saturated__effG__uu__uu__def,axiom,
( abstra1395361944017196648ist_fm
= ( ^ [Eff3: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm] : ( abstra6097777249025082867ist_fm @ ( abstra2682625350522704545ist_fm @ Eff3 ) ) ) ) ).
% Saturated_effG_uu_uu_def
thf(fact_519_pos__least,axiom,
! [N: nat,Rs: stream4408948924543953275ist_fm,R: produc6018962875968178549ist_fm] :
( ( ( shd_Pr3211216682057661985ist_fm @ ( sdrop_9176333610110415838ist_fm @ N @ Rs ) )
= R )
=> ( ord_less_eq_nat @ ( abstra2497064141961437305ist_fm @ Rs @ R ) @ N ) ) ).
% pos_least
thf(fact_520_pos__least,axiom,
! [N: nat,Rs: stream4385846686851721995e_rule,R: produc7694839378271647877e_rule] :
( ( ( shd_Pr2264621979884435249e_rule @ ( sdrop_9113879250048157294e_rule @ N @ Rs ) )
= R )
=> ( ord_less_eq_nat @ ( abstra7798328242894269833e_rule @ Rs @ R ) @ N ) ) ).
% pos_least
thf(fact_521_pos__least,axiom,
! [N: nat,Rs: stream8953843411776101167m_rule,R: produc7920154866009513897m_rule] :
( ( ( shd_Pr1208753253167450709m_rule @ ( sdrop_746751008794826386m_rule @ N @ Rs ) )
= R )
=> ( ord_less_eq_nat @ ( abstra2305546423536801965m_rule @ Rs @ R ) @ N ) ) ).
% pos_least
thf(fact_522_pos__least,axiom,
! [N: nat,Rs: stream8099677779113257519e_rule,R: produc9112364199808626345e_rule] :
( ( ( shd_Pr2400962586966563157e_rule @ ( sdrop_1938960342593938834e_rule @ N @ Rs ) )
= R )
=> ( ord_less_eq_nat @ ( abstra3497755757335914413e_rule @ Rs @ R ) @ N ) ) ).
% pos_least
thf(fact_523_pos__least,axiom,
! [N: nat,Rs: stream6210534828274662995m_rule,R: produc8828831911945107917m_rule] :
( ( ( shd_Pr4461660664618831993m_rule @ ( sdrop_7192298464603511222m_rule @ N @ Rs ) )
= R )
=> ( ord_less_eq_nat @ ( abstra1175832030460196561m_rule @ Rs @ R ) @ N ) ) ).
% pos_least
thf(fact_524_pos__least,axiom,
! [N: nat,Rs: stream2709947120125613254m_rule,R: produc340336539035504054m_rule] :
( ( ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N @ Rs ) )
= R )
=> ( ord_less_eq_nat @ ( abstra4499547390127564210m_rule @ Rs @ R ) @ N ) ) ).
% pos_least
thf(fact_525_pos__least,axiom,
! [N: nat,Rs: stream_rule,R: rule] :
( ( ( shd_rule @ ( sdrop_rule @ N @ Rs ) )
= R )
=> ( ord_less_eq_nat @ ( abstract_pos_rule @ Rs @ R ) @ N ) ) ).
% pos_least
thf(fact_526_s1_I2_J,axiom,
( new_list
= ( ^ [C4: nat,L: list_tm] :
~ ( member_nat3 @ C4 @ ( paramsts @ L ) ) ) ) ).
% s1(2)
thf(fact_527_paramst__liftt_I2_J,axiom,
! [Ts: list_tm] :
( ( paramsts @ ( liftts @ Ts ) )
= ( paramsts @ Ts ) ) ).
% paramst_liftt(2)
thf(fact_528_sset__fenum,axiom,
( ( sset_rule @ ( abstra745658567949189203m_rule @ rules ) )
= ( sset_rule @ rules ) ) ).
% sset_fenum
thf(fact_529_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_530_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_531_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_532_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_533_RuleSystem__Defs_Ofenum_Ocong,axiom,
abstra745658567949189203m_rule = abstra745658567949189203m_rule ).
% RuleSystem_Defs.fenum.cong
thf(fact_534_lt__ex,axiom,
! [X: int] :
? [Y4: int] : ( ord_less_int @ Y4 @ X ) ).
% lt_ex
thf(fact_535_lt__ex,axiom,
! [X: real] :
? [Y4: real] : ( ord_less_real @ Y4 @ X ) ).
% lt_ex
thf(fact_536_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_537_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_538_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_539_dense,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ? [Z5: real] :
( ( ord_less_real @ X @ Z5 )
& ( ord_less_real @ Z5 @ Y2 ) ) ) ).
% dense
thf(fact_540_less__imp__neq,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_541_less__imp__neq,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_542_less__imp__neq,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_543_order_Oasym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order.asym
thf(fact_544_order_Oasym,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ~ ( ord_less_int @ B2 @ A2 ) ) ).
% order.asym
thf(fact_545_order_Oasym,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ~ ( ord_less_real @ B2 @ A2 ) ) ).
% order.asym
thf(fact_546_ord__eq__less__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_547_ord__eq__less__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( A2 = B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_548_ord__eq__less__trans,axiom,
! [A2: real,B2: real,C: real] :
( ( A2 = B2 )
=> ( ( ord_less_real @ B2 @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_549_ord__less__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_550_ord__less__eq__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_551_ord__less__eq__trans,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_552_less__induct,axiom,
! [P4: nat > $o,A2: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P4 @ Y5 ) )
=> ( P4 @ X3 ) )
=> ( P4 @ A2 ) ) ).
% less_induct
thf(fact_553_antisym__conv3,axiom,
! [Y2: nat,X: nat] :
( ~ ( ord_less_nat @ Y2 @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv3
thf(fact_554_antisym__conv3,axiom,
! [Y2: int,X: int] :
( ~ ( ord_less_int @ Y2 @ X )
=> ( ( ~ ( ord_less_int @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv3
thf(fact_555_antisym__conv3,axiom,
! [Y2: real,X: real] :
( ~ ( ord_less_real @ Y2 @ X )
=> ( ( ~ ( ord_less_real @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv3
thf(fact_556_linorder__cases,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_cases
thf(fact_557_linorder__cases,axiom,
! [X: int,Y2: int] :
( ~ ( ord_less_int @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less_int @ Y2 @ X ) ) ) ).
% linorder_cases
thf(fact_558_linorder__cases,axiom,
! [X: real,Y2: real] :
( ~ ( ord_less_real @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less_real @ Y2 @ X ) ) ) ).
% linorder_cases
thf(fact_559_dual__order_Oasym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ~ ( ord_less_nat @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_560_dual__order_Oasym,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ~ ( ord_less_int @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_561_dual__order_Oasym,axiom,
! [B2: real,A2: real] :
( ( ord_less_real @ B2 @ A2 )
=> ~ ( ord_less_real @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_562_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_563_dual__order_Oirrefl,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_564_dual__order_Oirrefl,axiom,
! [A2: real] :
~ ( ord_less_real @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_565_exists__least__iff,axiom,
( ( ^ [P5: nat > $o] :
? [X5: nat] : ( P5 @ X5 ) )
= ( ^ [P2: nat > $o] :
? [N3: nat] :
( ( P2 @ N3 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ~ ( P2 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_566_linorder__less__wlog,axiom,
! [P4: nat > nat > $o,A2: nat,B2: nat] :
( ! [A5: nat,B6: nat] :
( ( ord_less_nat @ A5 @ B6 )
=> ( P4 @ A5 @ B6 ) )
=> ( ! [A5: nat] : ( P4 @ A5 @ A5 )
=> ( ! [A5: nat,B6: nat] :
( ( P4 @ B6 @ A5 )
=> ( P4 @ A5 @ B6 ) )
=> ( P4 @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_567_linorder__less__wlog,axiom,
! [P4: int > int > $o,A2: int,B2: int] :
( ! [A5: int,B6: int] :
( ( ord_less_int @ A5 @ B6 )
=> ( P4 @ A5 @ B6 ) )
=> ( ! [A5: int] : ( P4 @ A5 @ A5 )
=> ( ! [A5: int,B6: int] :
( ( P4 @ B6 @ A5 )
=> ( P4 @ A5 @ B6 ) )
=> ( P4 @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_568_linorder__less__wlog,axiom,
! [P4: real > real > $o,A2: real,B2: real] :
( ! [A5: real,B6: real] :
( ( ord_less_real @ A5 @ B6 )
=> ( P4 @ A5 @ B6 ) )
=> ( ! [A5: real] : ( P4 @ A5 @ A5 )
=> ( ! [A5: real,B6: real] :
( ( P4 @ B6 @ A5 )
=> ( P4 @ A5 @ B6 ) )
=> ( P4 @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_569_order_Ostrict__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_570_order_Ostrict__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_571_order_Ostrict__trans,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ B2 @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_572_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( ( ord_less_nat @ Y2 @ X )
| ( X = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_573_not__less__iff__gr__or__eq,axiom,
! [X: int,Y2: int] :
( ( ~ ( ord_less_int @ X @ Y2 ) )
= ( ( ord_less_int @ Y2 @ X )
| ( X = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_574_not__less__iff__gr__or__eq,axiom,
! [X: real,Y2: real] :
( ( ~ ( ord_less_real @ X @ Y2 ) )
= ( ( ord_less_real @ Y2 @ X )
| ( X = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_575_dual__order_Ostrict__trans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_576_dual__order_Ostrict__trans,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_577_dual__order_Ostrict__trans,axiom,
! [B2: real,A2: real,C: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( ( ord_less_real @ C @ B2 )
=> ( ord_less_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_578_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_579_order_Ostrict__implies__not__eq,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_580_order_Ostrict__implies__not__eq,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_581_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_582_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_583_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: real,A2: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_584_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_585_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_586_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_587_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less_nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_588_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_589_nat__less__induct,axiom,
! [P4: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( P4 @ M3 ) )
=> ( P4 @ N2 ) )
=> ( P4 @ N ) ) ).
% nat_less_induct
thf(fact_590_infinite__descent,axiom,
! [P4: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P4 @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P4 @ M3 ) ) )
=> ( P4 @ N ) ) ).
% infinite_descent
thf(fact_591_linorder__neqE__nat,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
=> ( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_592_linorder__neqE,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
=> ( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_neqE
thf(fact_593_linorder__neqE,axiom,
! [X: int,Y2: int] :
( ( X != Y2 )
=> ( ~ ( ord_less_int @ X @ Y2 )
=> ( ord_less_int @ Y2 @ X ) ) ) ).
% linorder_neqE
thf(fact_594_linorder__neqE,axiom,
! [X: real,Y2: real] :
( ( X != Y2 )
=> ( ~ ( ord_less_real @ X @ Y2 )
=> ( ord_less_real @ Y2 @ X ) ) ) ).
% linorder_neqE
thf(fact_595_order__less__asym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X ) ) ).
% order_less_asym
thf(fact_596_order__less__asym,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X ) ) ).
% order_less_asym
thf(fact_597_order__less__asym,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X ) ) ).
% order_less_asym
thf(fact_598_linorder__neq__iff,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
= ( ( ord_less_nat @ X @ Y2 )
| ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_599_linorder__neq__iff,axiom,
! [X: int,Y2: int] :
( ( X != Y2 )
= ( ( ord_less_int @ X @ Y2 )
| ( ord_less_int @ Y2 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_600_linorder__neq__iff,axiom,
! [X: real,Y2: real] :
( ( X != Y2 )
= ( ( ord_less_real @ X @ Y2 )
| ( ord_less_real @ Y2 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_601_order__less__asym_H,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_602_order__less__asym_H,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ~ ( ord_less_int @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_603_order__less__asym_H,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ~ ( ord_less_real @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_604_order__less__trans,axiom,
! [X: nat,Y2: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_605_order__less__trans,axiom,
! [X: int,Y2: int,Z3: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_606_order__less__trans,axiom,
! [X: real,Y2: real,Z3: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_real @ Y2 @ Z3 )
=> ( ord_less_real @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_607_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_608_ord__eq__less__subst,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_609_ord__eq__less__subst,axiom,
! [A2: real,F: nat > real,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_610_ord__eq__less__subst,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_611_ord__eq__less__subst,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_612_ord__eq__less__subst,axiom,
! [A2: real,F: int > real,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_613_ord__eq__less__subst,axiom,
! [A2: nat,F: real > nat,B2: real,C: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_614_ord__eq__less__subst,axiom,
! [A2: int,F: real > int,B2: real,C: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_615_ord__eq__less__subst,axiom,
! [A2: real,F: real > real,B2: real,C: real] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_616_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_617_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_618_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_619_ord__less__eq__subst,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_620_ord__less__eq__subst,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_621_ord__less__eq__subst,axiom,
! [A2: int,B2: int,F: int > real,C: real] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_622_ord__less__eq__subst,axiom,
! [A2: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_623_ord__less__eq__subst,axiom,
! [A2: real,B2: real,F: real > int,C: int] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_624_ord__less__eq__subst,axiom,
! [A2: real,B2: real,F: real > real,C: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_625_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_626_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_627_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_628_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_629_order__less__subst1,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_630_order__less__subst1,axiom,
! [A2: nat,F: real > nat,B2: real,C: real] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_631_order__less__subst1,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_632_order__less__subst1,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_633_order__less__subst1,axiom,
! [A2: int,F: real > int,B2: real,C: real] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_634_order__less__subst1,axiom,
! [A2: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_635_order__less__subst1,axiom,
! [A2: real,F: int > real,B2: int,C: int] :
( ( ord_less_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_636_order__less__subst1,axiom,
! [A2: real,F: real > real,B2: real,C: real] :
( ( ord_less_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_637_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_638_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_639_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_640_order__less__subst2,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_641_order__less__subst2,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_642_order__less__subst2,axiom,
! [A2: int,B2: int,F: int > real,C: real] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_643_order__less__subst2,axiom,
! [A2: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_644_order__less__subst2,axiom,
! [A2: real,B2: real,F: real > int,C: int] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_645_order__less__subst2,axiom,
! [A2: real,B2: real,F: real > real,C: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_646_order__less__not__sym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X ) ) ).
% order_less_not_sym
thf(fact_647_order__less__not__sym,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X ) ) ).
% order_less_not_sym
thf(fact_648_order__less__not__sym,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X ) ) ).
% order_less_not_sym
thf(fact_649_order__less__imp__triv,axiom,
! [X: nat,Y2: nat,P4: $o] :
( ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_nat @ Y2 @ X )
=> P4 ) ) ).
% order_less_imp_triv
thf(fact_650_order__less__imp__triv,axiom,
! [X: int,Y2: int,P4: $o] :
( ( ord_less_int @ X @ Y2 )
=> ( ( ord_less_int @ Y2 @ X )
=> P4 ) ) ).
% order_less_imp_triv
thf(fact_651_order__less__imp__triv,axiom,
! [X: real,Y2: real,P4: $o] :
( ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_real @ Y2 @ X )
=> P4 ) ) ).
% order_less_imp_triv
thf(fact_652_linorder__less__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
| ( X = Y2 )
| ( ord_less_nat @ Y2 @ X ) ) ).
% linorder_less_linear
thf(fact_653_linorder__less__linear,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
| ( X = Y2 )
| ( ord_less_int @ Y2 @ X ) ) ).
% linorder_less_linear
thf(fact_654_linorder__less__linear,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
| ( X = Y2 )
| ( ord_less_real @ Y2 @ X ) ) ).
% linorder_less_linear
thf(fact_655_order__less__imp__not__eq,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( X != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_656_order__less__imp__not__eq,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( X != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_657_order__less__imp__not__eq,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( X != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_658_order__less__imp__not__eq2,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_659_order__less__imp__not__eq2,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_660_order__less__imp__not__eq2,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_661_order__less__imp__not__less,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X ) ) ).
% order_less_imp_not_less
thf(fact_662_order__less__imp__not__less,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X ) ) ).
% order_less_imp_not_less
thf(fact_663_order__less__imp__not__less,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X ) ) ).
% order_less_imp_not_less
thf(fact_664_order__le__imp__less__or__eq,axiom,
! [X: set_tm,Y2: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y2 )
=> ( ( ord_less_set_tm @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_665_order__le__imp__less__or__eq,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_nat @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_666_order__le__imp__less__or__eq,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ( ord_less_set_nat @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_667_order__le__imp__less__or__eq,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_less_int @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_668_order__le__imp__less__or__eq,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_real @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_669_linorder__le__less__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
| ( ord_less_nat @ Y2 @ X ) ) ).
% linorder_le_less_linear
thf(fact_670_linorder__le__less__linear,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
| ( ord_less_int @ Y2 @ X ) ) ).
% linorder_le_less_linear
thf(fact_671_linorder__le__less__linear,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
| ( ord_less_real @ Y2 @ X ) ) ).
% linorder_le_less_linear
thf(fact_672_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_673_order__less__le__subst2,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_674_order__less__le__subst2,axiom,
! [A2: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_675_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_676_order__less__le__subst2,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_677_order__less__le__subst2,axiom,
! [A2: real,B2: real,F: real > int,C: int] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_678_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_679_order__less__le__subst2,axiom,
! [A2: int,B2: int,F: int > real,C: real] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_680_order__less__le__subst2,axiom,
! [A2: real,B2: real,F: real > real,C: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_681_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_tm,C: set_tm] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_tm @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_set_tm @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_tm @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_682_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_683_order__less__le__subst1,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_684_order__less__le__subst1,axiom,
! [A2: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_685_order__less__le__subst1,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_686_order__less__le__subst1,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_687_order__less__le__subst1,axiom,
! [A2: real,F: int > real,B2: int,C: int] :
( ( ord_less_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_688_order__less__le__subst1,axiom,
! [A2: nat,F: real > nat,B2: real,C: real] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_689_order__less__le__subst1,axiom,
! [A2: int,F: real > int,B2: real,C: real] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_690_order__less__le__subst1,axiom,
! [A2: real,F: real > real,B2: real,C: real] :
( ( ord_less_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_691_order__less__le__subst1,axiom,
! [A2: nat,F: set_tm > nat,B2: set_tm,C: set_tm] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_tm @ B2 @ C )
=> ( ! [X3: set_tm,Y4: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_692_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_693_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_694_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_695_order__le__less__subst2,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_696_order__le__less__subst2,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_697_order__le__less__subst2,axiom,
! [A2: int,B2: int,F: int > real,C: real] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_698_order__le__less__subst2,axiom,
! [A2: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_699_order__le__less__subst2,axiom,
! [A2: real,B2: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_700_order__le__less__subst2,axiom,
! [A2: real,B2: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_701_order__le__less__subst2,axiom,
! [A2: set_tm,B2: set_tm,F: set_tm > nat,C: nat] :
( ( ord_less_eq_set_tm @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_tm,Y4: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_702_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_703_order__le__less__subst1,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_704_order__le__less__subst1,axiom,
! [A2: nat,F: real > nat,B2: real,C: real] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_705_order__le__less__subst1,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_706_order__le__less__subst1,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_707_order__le__less__subst1,axiom,
! [A2: int,F: real > int,B2: real,C: real] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_708_order__le__less__subst1,axiom,
! [A2: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_709_order__le__less__subst1,axiom,
! [A2: real,F: int > real,B2: int,C: int] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_710_order__le__less__subst1,axiom,
! [A2: real,F: real > real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_711_order__le__less__subst1,axiom,
! [A2: set_tm,F: nat > set_tm,B2: nat,C: nat] :
( ( ord_less_eq_set_tm @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_set_tm @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_tm @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_712_order__less__le__trans,axiom,
! [X: set_tm,Y2: set_tm,Z3: set_tm] :
( ( ord_less_set_tm @ X @ Y2 )
=> ( ( ord_less_eq_set_tm @ Y2 @ Z3 )
=> ( ord_less_set_tm @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_713_order__less__le__trans,axiom,
! [X: nat,Y2: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_714_order__less__le__trans,axiom,
! [X: set_nat,Y2: set_nat,Z3: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ Z3 )
=> ( ord_less_set_nat @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_715_order__less__le__trans,axiom,
! [X: int,Y2: int,Z3: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_716_order__less__le__trans,axiom,
! [X: real,Y2: real,Z3: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z3 )
=> ( ord_less_real @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_717_order__le__less__trans,axiom,
! [X: set_tm,Y2: set_tm,Z3: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y2 )
=> ( ( ord_less_set_tm @ Y2 @ Z3 )
=> ( ord_less_set_tm @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_718_order__le__less__trans,axiom,
! [X: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_719_order__le__less__trans,axiom,
! [X: set_nat,Y2: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ( ord_less_set_nat @ Y2 @ Z3 )
=> ( ord_less_set_nat @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_720_order__le__less__trans,axiom,
! [X: int,Y2: int,Z3: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_721_order__le__less__trans,axiom,
! [X: real,Y2: real,Z3: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_real @ Y2 @ Z3 )
=> ( ord_less_real @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_722_order__neq__le__trans,axiom,
! [A2: set_tm,B2: set_tm] :
( ( A2 != B2 )
=> ( ( ord_less_eq_set_tm @ A2 @ B2 )
=> ( ord_less_set_tm @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_723_order__neq__le__trans,axiom,
! [A2: nat,B2: nat] :
( ( A2 != B2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_724_order__neq__le__trans,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 != B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_725_order__neq__le__trans,axiom,
! [A2: int,B2: int] :
( ( A2 != B2 )
=> ( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_int @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_726_order__neq__le__trans,axiom,
! [A2: real,B2: real] :
( ( A2 != B2 )
=> ( ( ord_less_eq_real @ A2 @ B2 )
=> ( ord_less_real @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_727_order__le__neq__trans,axiom,
! [A2: set_tm,B2: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_tm @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_728_order__le__neq__trans,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_729_order__le__neq__trans,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_730_order__le__neq__trans,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_int @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_731_order__le__neq__trans,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_real @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_732_order__less__imp__le,axiom,
! [X: set_tm,Y2: set_tm] :
( ( ord_less_set_tm @ X @ Y2 )
=> ( ord_less_eq_set_tm @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_733_order__less__imp__le,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_734_order__less__imp__le,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_735_order__less__imp__le,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_eq_int @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_736_order__less__imp__le,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ord_less_eq_real @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_737_linorder__not__less,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_not_less
thf(fact_738_linorder__not__less,axiom,
! [X: int,Y2: int] :
( ( ~ ( ord_less_int @ X @ Y2 ) )
= ( ord_less_eq_int @ Y2 @ X ) ) ).
% linorder_not_less
thf(fact_739_linorder__not__less,axiom,
! [X: real,Y2: real] :
( ( ~ ( ord_less_real @ X @ Y2 ) )
= ( ord_less_eq_real @ Y2 @ X ) ) ).
% linorder_not_less
thf(fact_740_linorder__not__le,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y2 ) )
= ( ord_less_nat @ Y2 @ X ) ) ).
% linorder_not_le
thf(fact_741_linorder__not__le,axiom,
! [X: int,Y2: int] :
( ( ~ ( ord_less_eq_int @ X @ Y2 ) )
= ( ord_less_int @ Y2 @ X ) ) ).
% linorder_not_le
thf(fact_742_linorder__not__le,axiom,
! [X: real,Y2: real] :
( ( ~ ( ord_less_eq_real @ X @ Y2 ) )
= ( ord_less_real @ Y2 @ X ) ) ).
% linorder_not_le
thf(fact_743_order__less__le,axiom,
( ord_less_set_tm
= ( ^ [X2: set_tm,Y: set_tm] :
( ( ord_less_eq_set_tm @ X2 @ Y )
& ( X2 != Y ) ) ) ) ).
% order_less_le
thf(fact_744_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
& ( X2 != Y ) ) ) ) ).
% order_less_le
thf(fact_745_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
& ( X2 != Y ) ) ) ) ).
% order_less_le
thf(fact_746_order__less__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
& ( X2 != Y ) ) ) ) ).
% order_less_le
thf(fact_747_order__less__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
& ( X2 != Y ) ) ) ) ).
% order_less_le
thf(fact_748_order__le__less,axiom,
( ord_less_eq_set_tm
= ( ^ [X2: set_tm,Y: set_tm] :
( ( ord_less_set_tm @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% order_le_less
thf(fact_749_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% order_le_less
thf(fact_750_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X2: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% order_le_less
thf(fact_751_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% order_le_less
thf(fact_752_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% order_le_less
thf(fact_753_dual__order_Ostrict__implies__order,axiom,
! [B2: set_tm,A2: set_tm] :
( ( ord_less_set_tm @ B2 @ A2 )
=> ( ord_less_eq_set_tm @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_754_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_755_dual__order_Ostrict__implies__order,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B2 @ A2 )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_756_dual__order_Ostrict__implies__order,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ord_less_eq_int @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_757_dual__order_Ostrict__implies__order,axiom,
! [B2: real,A2: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( ord_less_eq_real @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_758_order_Ostrict__implies__order,axiom,
! [A2: set_tm,B2: set_tm] :
( ( ord_less_set_tm @ A2 @ B2 )
=> ( ord_less_eq_set_tm @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_759_order_Ostrict__implies__order,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_760_order_Ostrict__implies__order,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_761_order_Ostrict__implies__order,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_eq_int @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_762_order_Ostrict__implies__order,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ord_less_eq_real @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_763_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_tm
= ( ^ [B5: set_tm,A4: set_tm] :
( ( ord_less_eq_set_tm @ B5 @ A4 )
& ~ ( ord_less_eq_set_tm @ A4 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_764_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B5: nat,A4: nat] :
( ( ord_less_eq_nat @ B5 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_765_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B5: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
& ~ ( ord_less_eq_set_nat @ A4 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_766_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B5: int,A4: int] :
( ( ord_less_eq_int @ B5 @ A4 )
& ~ ( ord_less_eq_int @ A4 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_767_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B5: real,A4: real] :
( ( ord_less_eq_real @ B5 @ A4 )
& ~ ( ord_less_eq_real @ A4 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_768_dual__order_Ostrict__trans2,axiom,
! [B2: set_tm,A2: set_tm,C: set_tm] :
( ( ord_less_set_tm @ B2 @ A2 )
=> ( ( ord_less_eq_set_tm @ C @ B2 )
=> ( ord_less_set_tm @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_769_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_770_dual__order_Ostrict__trans2,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B2 )
=> ( ord_less_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_771_dual__order_Ostrict__trans2,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_772_dual__order_Ostrict__trans2,axiom,
! [B2: real,A2: real,C: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( ( ord_less_eq_real @ C @ B2 )
=> ( ord_less_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_773_dual__order_Ostrict__trans1,axiom,
! [B2: set_tm,A2: set_tm,C: set_tm] :
( ( ord_less_eq_set_tm @ B2 @ A2 )
=> ( ( ord_less_set_tm @ C @ B2 )
=> ( ord_less_set_tm @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_774_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_775_dual__order_Ostrict__trans1,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_set_nat @ C @ B2 )
=> ( ord_less_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_776_dual__order_Ostrict__trans1,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_777_dual__order_Ostrict__trans1,axiom,
! [B2: real,A2: real,C: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ( ord_less_real @ C @ B2 )
=> ( ord_less_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_778_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_tm
= ( ^ [B5: set_tm,A4: set_tm] :
( ( ord_less_eq_set_tm @ B5 @ A4 )
& ( A4 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_779_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B5: nat,A4: nat] :
( ( ord_less_eq_nat @ B5 @ A4 )
& ( A4 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_780_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B5: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
& ( A4 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_781_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B5: int,A4: int] :
( ( ord_less_eq_int @ B5 @ A4 )
& ( A4 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_782_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B5: real,A4: real] :
( ( ord_less_eq_real @ B5 @ A4 )
& ( A4 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_783_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_tm
= ( ^ [B5: set_tm,A4: set_tm] :
( ( ord_less_set_tm @ B5 @ A4 )
| ( A4 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_784_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A4: nat] :
( ( ord_less_nat @ B5 @ A4 )
| ( A4 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_785_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B5: set_nat,A4: set_nat] :
( ( ord_less_set_nat @ B5 @ A4 )
| ( A4 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_786_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B5: int,A4: int] :
( ( ord_less_int @ B5 @ A4 )
| ( A4 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_787_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B5: real,A4: real] :
( ( ord_less_real @ B5 @ A4 )
| ( A4 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_788_dense__le__bounded,axiom,
! [X: real,Y2: real,Z3: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ! [W: real] :
( ( ord_less_real @ X @ W )
=> ( ( ord_less_real @ W @ Y2 )
=> ( ord_less_eq_real @ W @ Z3 ) ) )
=> ( ord_less_eq_real @ Y2 @ Z3 ) ) ) ).
% dense_le_bounded
thf(fact_789_dense__ge__bounded,axiom,
! [Z3: real,X: real,Y2: real] :
( ( ord_less_real @ Z3 @ X )
=> ( ! [W: real] :
( ( ord_less_real @ Z3 @ W )
=> ( ( ord_less_real @ W @ X )
=> ( ord_less_eq_real @ Y2 @ W ) ) )
=> ( ord_less_eq_real @ Y2 @ Z3 ) ) ) ).
% dense_ge_bounded
thf(fact_790_order_Ostrict__iff__not,axiom,
( ord_less_set_tm
= ( ^ [A4: set_tm,B5: set_tm] :
( ( ord_less_eq_set_tm @ A4 @ B5 )
& ~ ( ord_less_eq_set_tm @ B5 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_791_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ A4 @ B5 )
& ~ ( ord_less_eq_nat @ B5 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_792_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
& ~ ( ord_less_eq_set_nat @ B5 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_793_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A4: int,B5: int] :
( ( ord_less_eq_int @ A4 @ B5 )
& ~ ( ord_less_eq_int @ B5 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_794_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A4: real,B5: real] :
( ( ord_less_eq_real @ A4 @ B5 )
& ~ ( ord_less_eq_real @ B5 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_795_order_Ostrict__trans2,axiom,
! [A2: set_tm,B2: set_tm,C: set_tm] :
( ( ord_less_set_tm @ A2 @ B2 )
=> ( ( ord_less_eq_set_tm @ B2 @ C )
=> ( ord_less_set_tm @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_796_order_Ostrict__trans2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_797_order_Ostrict__trans2,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_798_order_Ostrict__trans2,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_799_order_Ostrict__trans2,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_800_order_Ostrict__trans1,axiom,
! [A2: set_tm,B2: set_tm,C: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ B2 )
=> ( ( ord_less_set_tm @ B2 @ C )
=> ( ord_less_set_tm @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_801_order_Ostrict__trans1,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_802_order_Ostrict__trans1,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_set_nat @ B2 @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_803_order_Ostrict__trans1,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_804_order_Ostrict__trans1,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_real @ B2 @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_805_order_Ostrict__iff__order,axiom,
( ord_less_set_tm
= ( ^ [A4: set_tm,B5: set_tm] :
( ( ord_less_eq_set_tm @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_806_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_807_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_808_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A4: int,B5: int] :
( ( ord_less_eq_int @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_809_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A4: real,B5: real] :
( ( ord_less_eq_real @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_810_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_tm
= ( ^ [A4: set_tm,B5: set_tm] :
( ( ord_less_set_tm @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_811_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_nat @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_812_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
( ( ord_less_set_nat @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_813_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B5: int] :
( ( ord_less_int @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_814_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B5: real] :
( ( ord_less_real @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_815_not__le__imp__less,axiom,
! [Y2: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y2 @ X )
=> ( ord_less_nat @ X @ Y2 ) ) ).
% not_le_imp_less
thf(fact_816_not__le__imp__less,axiom,
! [Y2: int,X: int] :
( ~ ( ord_less_eq_int @ Y2 @ X )
=> ( ord_less_int @ X @ Y2 ) ) ).
% not_le_imp_less
thf(fact_817_not__le__imp__less,axiom,
! [Y2: real,X: real] :
( ~ ( ord_less_eq_real @ Y2 @ X )
=> ( ord_less_real @ X @ Y2 ) ) ).
% not_le_imp_less
thf(fact_818_less__le__not__le,axiom,
( ord_less_set_tm
= ( ^ [X2: set_tm,Y: set_tm] :
( ( ord_less_eq_set_tm @ X2 @ Y )
& ~ ( ord_less_eq_set_tm @ Y @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_819_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
& ~ ( ord_less_eq_nat @ Y @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_820_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
& ~ ( ord_less_eq_set_nat @ Y @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_821_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
& ~ ( ord_less_eq_int @ Y @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_822_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
& ~ ( ord_less_eq_real @ Y @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_823_dense__le,axiom,
! [Y2: real,Z3: real] :
( ! [X3: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ X3 @ Z3 ) )
=> ( ord_less_eq_real @ Y2 @ Z3 ) ) ).
% dense_le
thf(fact_824_dense__ge,axiom,
! [Z3: real,Y2: real] :
( ! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ord_less_eq_real @ Y2 @ X3 ) )
=> ( ord_less_eq_real @ Y2 @ Z3 ) ) ).
% dense_ge
thf(fact_825_antisym__conv2,axiom,
! [X: set_tm,Y2: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y2 )
=> ( ( ~ ( ord_less_set_tm @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_826_antisym__conv2,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_827_antisym__conv2,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ( ~ ( ord_less_set_nat @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_828_antisym__conv2,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ~ ( ord_less_int @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_829_antisym__conv2,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ~ ( ord_less_real @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_830_antisym__conv1,axiom,
! [X: set_tm,Y2: set_tm] :
( ~ ( ord_less_set_tm @ X @ Y2 )
=> ( ( ord_less_eq_set_tm @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_831_antisym__conv1,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_832_antisym__conv1,axiom,
! [X: set_nat,Y2: set_nat] :
( ~ ( ord_less_set_nat @ X @ Y2 )
=> ( ( ord_less_eq_set_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_833_antisym__conv1,axiom,
! [X: int,Y2: int] :
( ~ ( ord_less_int @ X @ Y2 )
=> ( ( ord_less_eq_int @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_834_antisym__conv1,axiom,
! [X: real,Y2: real] :
( ~ ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_835_nless__le,axiom,
! [A2: set_tm,B2: set_tm] :
( ( ~ ( ord_less_set_tm @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_set_tm @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_836_nless__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_837_nless__le,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ~ ( ord_less_set_nat @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_set_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_838_nless__le,axiom,
! [A2: int,B2: int] :
( ( ~ ( ord_less_int @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_int @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_839_nless__le,axiom,
! [A2: real,B2: real] :
( ( ~ ( ord_less_real @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_real @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_840_leI,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) ) ).
% leI
thf(fact_841_leI,axiom,
! [X: int,Y2: int] :
( ~ ( ord_less_int @ X @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X ) ) ).
% leI
thf(fact_842_leI,axiom,
! [X: real,Y2: real] :
( ~ ( ord_less_real @ X @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X ) ) ).
% leI
thf(fact_843_leD,axiom,
! [Y2: set_tm,X: set_tm] :
( ( ord_less_eq_set_tm @ Y2 @ X )
=> ~ ( ord_less_set_tm @ X @ Y2 ) ) ).
% leD
thf(fact_844_leD,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ~ ( ord_less_nat @ X @ Y2 ) ) ).
% leD
thf(fact_845_leD,axiom,
! [Y2: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X )
=> ~ ( ord_less_set_nat @ X @ Y2 ) ) ).
% leD
thf(fact_846_leD,axiom,
! [Y2: int,X: int] :
( ( ord_less_eq_int @ Y2 @ X )
=> ~ ( ord_less_int @ X @ Y2 ) ) ).
% leD
thf(fact_847_leD,axiom,
! [Y2: real,X: real] :
( ( ord_less_eq_real @ Y2 @ X )
=> ~ ( ord_less_real @ X @ Y2 ) ) ).
% leD
thf(fact_848_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_849_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_850_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_851_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_852_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_853_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_854_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_855_infinite__descent0,axiom,
! [P4: nat > $o,N: nat] :
( ( P4 @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P4 @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P4 @ M3 ) ) ) )
=> ( P4 @ N ) ) ) ).
% infinite_descent0
thf(fact_856_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_857_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_858_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_859_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_860_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_861_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_862_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_863_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_864_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_865_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
| ( M2 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_866_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_867_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
& ( M2 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_868_ex__least__nat__le,axiom,
! [P4: nat > $o,N: nat] :
( ( P4 @ N )
=> ( ~ ( P4 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P4 @ I3 ) )
& ( P4 @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_869_RuleSystem__Defs_Osset__fenum,axiom,
! [Rules2: stream2709947120125613254m_rule] :
( ( sset_P4484857331586881186m_rule @ ( abstra4452268713091674743m_rule @ Rules2 ) )
= ( sset_P4484857331586881186m_rule @ Rules2 ) ) ).
% RuleSystem_Defs.sset_fenum
thf(fact_870_RuleSystem__Defs_Osset__fenum,axiom,
! [Rules2: stream_rule] :
( ( sset_rule @ ( abstra745658567949189203m_rule @ Rules2 ) )
= ( sset_rule @ Rules2 ) ) ).
% RuleSystem_Defs.sset_fenum
thf(fact_871_i_Osset__fenum,axiom,
! [Rules2: stream2709947120125613254m_rule] :
( ( sset_P4484857331586881186m_rule @ ( abstra4452268713091674743m_rule @ Rules2 ) )
= ( sset_P4484857331586881186m_rule @ Rules2 ) ) ).
% i.sset_fenum
thf(fact_872_i_Osset__fenum,axiom,
! [Rules2: stream_rule] :
( ( sset_rule @ ( abstra745658567949189203m_rule @ Rules2 ) )
= ( sset_rule @ Rules2 ) ) ).
% i.sset_fenum
thf(fact_873_rules__repeat,axiom,
! [M: nat,R: rule] :
? [N2: nat] :
( ( ord_less_nat @ M @ N2 )
& ( ( snth_rule @ rules @ N2 )
= R ) ) ).
% rules_repeat
thf(fact_874_fenum__uu__def,axiom,
abstra1582897422107675196u_rule = abstra745658567949189203m_rule ).
% fenum_uu_def
thf(fact_875_nat__descend__induct,axiom,
! [N: nat,P4: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P4 @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K2 @ I3 )
=> ( P4 @ I3 ) )
=> ( P4 @ K2 ) ) )
=> ( P4 @ M ) ) ) ).
% nat_descend_induct
thf(fact_876_s4_I2_J,axiom,
inc_list = liftts ).
% s4(2)
thf(fact_877_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
& ( ord_less_real @ E2 @ D1 )
& ( ord_less_real @ E2 @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_878_complete__interval,axiom,
! [A2: nat,B2: nat,P4: nat > $o] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( P4 @ A2 )
=> ( ~ ( P4 @ B2 )
=> ? [C5: nat] :
( ( ord_less_eq_nat @ A2 @ C5 )
& ( ord_less_eq_nat @ C5 @ B2 )
& ! [X8: nat] :
( ( ( ord_less_eq_nat @ A2 @ X8 )
& ( ord_less_nat @ X8 @ C5 ) )
=> ( P4 @ X8 ) )
& ! [D3: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A2 @ X3 )
& ( ord_less_nat @ X3 @ D3 ) )
=> ( P4 @ X3 ) )
=> ( ord_less_eq_nat @ D3 @ C5 ) ) ) ) ) ) ).
% complete_interval
thf(fact_879_complete__interval,axiom,
! [A2: int,B2: int,P4: int > $o] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( P4 @ A2 )
=> ( ~ ( P4 @ B2 )
=> ? [C5: int] :
( ( ord_less_eq_int @ A2 @ C5 )
& ( ord_less_eq_int @ C5 @ B2 )
& ! [X8: int] :
( ( ( ord_less_eq_int @ A2 @ X8 )
& ( ord_less_int @ X8 @ C5 ) )
=> ( P4 @ X8 ) )
& ! [D3: int] :
( ! [X3: int] :
( ( ( ord_less_eq_int @ A2 @ X3 )
& ( ord_less_int @ X3 @ D3 ) )
=> ( P4 @ X3 ) )
=> ( ord_less_eq_int @ D3 @ C5 ) ) ) ) ) ) ).
% complete_interval
thf(fact_880_complete__interval,axiom,
! [A2: real,B2: real,P4: real > $o] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( P4 @ A2 )
=> ( ~ ( P4 @ B2 )
=> ? [C5: real] :
( ( ord_less_eq_real @ A2 @ C5 )
& ( ord_less_eq_real @ C5 @ B2 )
& ! [X8: real] :
( ( ( ord_less_eq_real @ A2 @ X8 )
& ( ord_less_real @ X8 @ C5 ) )
=> ( P4 @ X8 ) )
& ! [D3: real] :
( ! [X3: real] :
( ( ( ord_less_eq_real @ A2 @ X3 )
& ( ord_less_real @ X3 @ D3 ) )
=> ( P4 @ X3 ) )
=> ( ord_less_eq_real @ D3 @ C5 ) ) ) ) ) ) ).
% complete_interval
thf(fact_881_verit__comp__simplify1_I3_J,axiom,
! [B7: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B7 @ A6 ) )
= ( ord_less_nat @ A6 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_882_verit__comp__simplify1_I3_J,axiom,
! [B7: int,A6: int] :
( ( ~ ( ord_less_eq_int @ B7 @ A6 ) )
= ( ord_less_int @ A6 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_883_verit__comp__simplify1_I3_J,axiom,
! [B7: real,A6: real] :
( ( ~ ( ord_less_eq_real @ B7 @ A6 ) )
= ( ord_less_real @ A6 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_884_pinf_I6_J,axiom,
! [T2: nat] :
? [Z5: nat] :
! [X8: nat] :
( ( ord_less_nat @ Z5 @ X8 )
=> ~ ( ord_less_eq_nat @ X8 @ T2 ) ) ).
% pinf(6)
thf(fact_885_pinf_I6_J,axiom,
! [T2: int] :
? [Z5: int] :
! [X8: int] :
( ( ord_less_int @ Z5 @ X8 )
=> ~ ( ord_less_eq_int @ X8 @ T2 ) ) ).
% pinf(6)
thf(fact_886_pinf_I6_J,axiom,
! [T2: real] :
? [Z5: real] :
! [X8: real] :
( ( ord_less_real @ Z5 @ X8 )
=> ~ ( ord_less_eq_real @ X8 @ T2 ) ) ).
% pinf(6)
thf(fact_887_psubsetI,axiom,
! [A: set_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_tm @ A @ B ) ) ) ).
% psubsetI
thf(fact_888_psubsetI,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% psubsetI
thf(fact_889_psubsetE,axiom,
! [A: set_tm,B: set_tm] :
( ( ord_less_set_tm @ A @ B )
=> ~ ( ( ord_less_eq_set_tm @ A @ B )
=> ( ord_less_eq_set_tm @ B @ A ) ) ) ).
% psubsetE
thf(fact_890_psubsetE,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% psubsetE
thf(fact_891_psubset__eq,axiom,
( ord_less_set_tm
= ( ^ [A3: set_tm,B3: set_tm] :
( ( ord_less_eq_set_tm @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_892_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_893_psubset__imp__subset,axiom,
! [A: set_tm,B: set_tm] :
( ( ord_less_set_tm @ A @ B )
=> ( ord_less_eq_set_tm @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_894_psubset__imp__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_895_psubset__subset__trans,axiom,
! [A: set_tm,B: set_tm,C2: set_tm] :
( ( ord_less_set_tm @ A @ B )
=> ( ( ord_less_eq_set_tm @ B @ C2 )
=> ( ord_less_set_tm @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_896_psubset__subset__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_set_nat @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_897_subset__not__subset__eq,axiom,
( ord_less_set_tm
= ( ^ [A3: set_tm,B3: set_tm] :
( ( ord_less_eq_set_tm @ A3 @ B3 )
& ~ ( ord_less_eq_set_tm @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_898_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ~ ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_899_subset__psubset__trans,axiom,
! [A: set_tm,B: set_tm,C2: set_tm] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( ord_less_set_tm @ B @ C2 )
=> ( ord_less_set_tm @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_900_subset__psubset__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C2 )
=> ( ord_less_set_nat @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_901_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_tm
= ( ^ [A3: set_tm,B3: set_tm] :
( ( ord_less_set_tm @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_902_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_903_verit__comp__simplify1_I2_J,axiom,
! [A2: set_tm] : ( ord_less_eq_set_tm @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_904_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_905_verit__comp__simplify1_I2_J,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_906_verit__comp__simplify1_I2_J,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_907_verit__comp__simplify1_I2_J,axiom,
! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_908_verit__la__disequality,axiom,
! [A2: nat,B2: nat] :
( ( A2 = B2 )
| ~ ( ord_less_eq_nat @ A2 @ B2 )
| ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_909_verit__la__disequality,axiom,
! [A2: int,B2: int] :
( ( A2 = B2 )
| ~ ( ord_less_eq_int @ A2 @ B2 )
| ~ ( ord_less_eq_int @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_910_verit__la__disequality,axiom,
! [A2: real,B2: real] :
( ( A2 = B2 )
| ~ ( ord_less_eq_real @ A2 @ B2 )
| ~ ( ord_less_eq_real @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_911_minf_I8_J,axiom,
! [T2: nat] :
? [Z5: nat] :
! [X8: nat] :
( ( ord_less_nat @ X8 @ Z5 )
=> ~ ( ord_less_eq_nat @ T2 @ X8 ) ) ).
% minf(8)
thf(fact_912_minf_I8_J,axiom,
! [T2: int] :
? [Z5: int] :
! [X8: int] :
( ( ord_less_int @ X8 @ Z5 )
=> ~ ( ord_less_eq_int @ T2 @ X8 ) ) ).
% minf(8)
thf(fact_913_minf_I8_J,axiom,
! [T2: real] :
? [Z5: real] :
! [X8: real] :
( ( ord_less_real @ X8 @ Z5 )
=> ~ ( ord_less_eq_real @ T2 @ X8 ) ) ).
% minf(8)
thf(fact_914_minf_I6_J,axiom,
! [T2: nat] :
? [Z5: nat] :
! [X8: nat] :
( ( ord_less_nat @ X8 @ Z5 )
=> ( ord_less_eq_nat @ X8 @ T2 ) ) ).
% minf(6)
thf(fact_915_minf_I6_J,axiom,
! [T2: int] :
? [Z5: int] :
! [X8: int] :
( ( ord_less_int @ X8 @ Z5 )
=> ( ord_less_eq_int @ X8 @ T2 ) ) ).
% minf(6)
thf(fact_916_minf_I6_J,axiom,
! [T2: real] :
? [Z5: real] :
! [X8: real] :
( ( ord_less_real @ X8 @ Z5 )
=> ( ord_less_eq_real @ X8 @ T2 ) ) ).
% minf(6)
thf(fact_917_pinf_I8_J,axiom,
! [T2: nat] :
? [Z5: nat] :
! [X8: nat] :
( ( ord_less_nat @ Z5 @ X8 )
=> ( ord_less_eq_nat @ T2 @ X8 ) ) ).
% pinf(8)
thf(fact_918_pinf_I8_J,axiom,
! [T2: int] :
? [Z5: int] :
! [X8: int] :
( ( ord_less_int @ Z5 @ X8 )
=> ( ord_less_eq_int @ T2 @ X8 ) ) ).
% pinf(8)
thf(fact_919_pinf_I8_J,axiom,
! [T2: real] :
? [Z5: real] :
! [X8: real] :
( ( ord_less_real @ Z5 @ X8 )
=> ( ord_less_eq_real @ T2 @ X8 ) ) ).
% pinf(8)
thf(fact_920_RuleSystem_Opos__least,axiom,
! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,N: nat,Rs: stream4408948924543953275ist_fm,R: produc6018962875968178549ist_fm] :
( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S4 )
=> ( ( ( shd_Pr3211216682057661985ist_fm @ ( sdrop_9176333610110415838ist_fm @ N @ Rs ) )
= R )
=> ( ord_less_eq_nat @ ( abstra2497064141961437305ist_fm @ Rs @ R ) @ N ) ) ) ).
% RuleSystem.pos_least
thf(fact_921_RuleSystem_Opos__least,axiom,
! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,N: nat,Rs: stream4385846686851721995e_rule,R: produc7694839378271647877e_rule] :
( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S4 )
=> ( ( ( shd_Pr2264621979884435249e_rule @ ( sdrop_9113879250048157294e_rule @ N @ Rs ) )
= R )
=> ( ord_less_eq_nat @ ( abstra7798328242894269833e_rule @ Rs @ R ) @ N ) ) ) ).
% RuleSystem.pos_least
thf(fact_922_RuleSystem_Opos__least,axiom,
! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,N: nat,Rs: stream8953843411776101167m_rule,R: produc7920154866009513897m_rule] :
( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S4 )
=> ( ( ( shd_Pr1208753253167450709m_rule @ ( sdrop_746751008794826386m_rule @ N @ Rs ) )
= R )
=> ( ord_less_eq_nat @ ( abstra2305546423536801965m_rule @ Rs @ R ) @ N ) ) ) ).
% RuleSystem.pos_least
thf(fact_923_RuleSystem_Opos__least,axiom,
! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,N: nat,Rs: stream8099677779113257519e_rule,R: produc9112364199808626345e_rule] :
( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S4 )
=> ( ( ( shd_Pr2400962586966563157e_rule @ ( sdrop_1938960342593938834e_rule @ N @ Rs ) )
= R )
=> ( ord_less_eq_nat @ ( abstra3497755757335914413e_rule @ Rs @ R ) @ N ) ) ) ).
% RuleSystem.pos_least
thf(fact_924_RuleSystem_Opos__least,axiom,
! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,N: nat,Rs: stream6210534828274662995m_rule,R: produc8828831911945107917m_rule] :
( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S4 )
=> ( ( ( shd_Pr4461660664618831993m_rule @ ( sdrop_7192298464603511222m_rule @ N @ Rs ) )
= R )
=> ( ord_less_eq_nat @ ( abstra1175832030460196561m_rule @ Rs @ R ) @ N ) ) ) ).
% RuleSystem.pos_least
thf(fact_925_RuleSystem_Opos__least,axiom,
! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,N: nat,Rs: stream2709947120125613254m_rule,R: produc340336539035504054m_rule] :
( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S4 )
=> ( ( ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N @ Rs ) )
= R )
=> ( ord_less_eq_nat @ ( abstra4499547390127564210m_rule @ Rs @ R ) @ N ) ) ) ).
% RuleSystem.pos_least
thf(fact_926_RuleSystem_Opos__least,axiom,
! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,N: nat,Rs: stream_rule,R: rule] :
( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S4 )
=> ( ( ( shd_rule @ ( sdrop_rule @ N @ Rs ) )
= R )
=> ( ord_less_eq_nat @ ( abstract_pos_rule @ Rs @ R ) @ N ) ) ) ).
% RuleSystem.pos_least
thf(fact_927_new__list_Osimps_I2_J,axiom,
! [C: nat,T2: tm,L2: list_tm] :
( ( new_list @ C @ ( cons_tm @ T2 @ L2 ) )
= ( ( ( new_term @ C @ T2 )
=> ( new_list @ C @ L2 ) )
& ( new_term @ C @ T2 ) ) ) ).
% new_list.simps(2)
thf(fact_928_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_929_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_930_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_931_pos,axiom,
! [Rs: stream_rule,R: rule] :
( ( abstra3799686578551160190r_rule @ rules @ Rs )
=> ( ( member_rule3 @ R @ ( sset_rule @ rules ) )
=> ( ( shd_rule @ ( sdrop_rule @ ( abstract_pos_rule @ Rs @ R ) @ Rs ) )
= R ) ) ) ).
% pos
thf(fact_932_subterm__Fun__refl,axiom,
! [Ts: list_tm,N: nat] : ( ord_less_eq_set_tm @ ( set_tm2 @ Ts ) @ ( set_tm2 @ ( subtermTm @ ( fun @ N @ Ts ) ) ) ) ).
% subterm_Fun_refl
thf(fact_933_inc__list_Osimps_I2_J,axiom,
! [T2: tm,L2: list_tm] :
( ( inc_list @ ( cons_tm @ T2 @ L2 ) )
= ( cons_tm @ ( inc_term @ T2 ) @ ( inc_list @ L2 ) ) ) ).
% inc_list.simps(2)
thf(fact_934_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_935_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_936_tm_Oinject_I1_J,axiom,
! [X11: nat,X12: list_tm,Y11: nat,Y12: list_tm] :
( ( ( fun @ X11 @ X12 )
= ( fun @ Y11 @ Y12 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 ) ) ) ).
% tm.inject(1)
thf(fact_937_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_938_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_939_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_940_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_941_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_942_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_943_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_944_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_945_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_946_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_947_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_948_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_949_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_950_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_951_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_952_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_953_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_954_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_955_psubsetD,axiom,
! [A: set_rule,B: set_rule,C: rule] :
( ( ord_less_set_rule @ A @ B )
=> ( ( member_rule3 @ C @ A )
=> ( member_rule3 @ C @ B ) ) ) ).
% psubsetD
thf(fact_956_psubsetD,axiom,
! [A: set_real,B: set_real,C: real] :
( ( ord_less_set_real @ A @ B )
=> ( ( member_real3 @ C @ A )
=> ( member_real3 @ C @ B ) ) ) ).
% psubsetD
thf(fact_957_psubsetD,axiom,
! [A: set_tm,B: set_tm,C: tm] :
( ( ord_less_set_tm @ A @ B )
=> ( ( member_tm3 @ C @ A )
=> ( member_tm3 @ C @ B ) ) ) ).
% psubsetD
thf(fact_958_psubsetD,axiom,
! [A: set_fm,B: set_fm,C: fm] :
( ( ord_less_set_fm @ A @ B )
=> ( ( member_fm3 @ C @ A )
=> ( member_fm3 @ C @ B ) ) ) ).
% psubsetD
thf(fact_959_psubsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( member_nat3 @ C @ A )
=> ( member_nat3 @ C @ B ) ) ) ).
% psubsetD
thf(fact_960_RuleSystem__Defs_Ofair_Ocong,axiom,
abstra3799686578551160190r_rule = abstra3799686578551160190r_rule ).
% RuleSystem_Defs.fair.cong
thf(fact_961_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_962_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B5: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_963_inc__term_Osimps_I2_J,axiom,
! [I: nat,L2: list_tm] :
( ( inc_term @ ( fun @ I @ L2 ) )
= ( fun @ I @ ( inc_list @ L2 ) ) ) ).
% inc_term.simps(2)
thf(fact_964_new__term_Osimps_I2_J,axiom,
! [C: nat,I: nat,L2: list_tm] :
( ( new_term @ C @ ( fun @ I @ L2 ) )
= ( ( I != C )
& ( ( I != C )
=> ( new_list @ C @ L2 ) ) ) ) ).
% new_term.simps(2)
thf(fact_965_fair__rules,axiom,
abstra3799686578551160190r_rule @ rules @ rules ).
% fair_rules
thf(fact_966_RuleSystem_Oenabled__R,axiom,
! [Eff4: rule > rule > fset_rule > $o,Rules2: stream_rule,S4: set_rule,S: rule] :
( ( abstra6805424310982398927e_rule @ Eff4 @ Rules2 @ S4 )
=> ( ( member_rule3 @ S @ S4 )
=> ? [X3: rule] :
( ( member_rule3 @ X3 @ ( sset_rule @ Rules2 ) )
& ? [X_1: fset_rule] : ( Eff4 @ X3 @ S @ X_1 ) ) ) ) ).
% RuleSystem.enabled_R
thf(fact_967_RuleSystem_Oenabled__R,axiom,
! [Eff4: rule > real > fset_real > $o,Rules2: stream_rule,S4: set_real,S: real] :
( ( abstra7065561124429741161e_real @ Eff4 @ Rules2 @ S4 )
=> ( ( member_real3 @ S @ S4 )
=> ? [X3: rule] :
( ( member_rule3 @ X3 @ ( sset_rule @ Rules2 ) )
& ? [X_1: fset_real] : ( Eff4 @ X3 @ S @ X_1 ) ) ) ) ).
% RuleSystem.enabled_R
thf(fact_968_RuleSystem_Oenabled__R,axiom,
! [Eff4: rule > tm > fset_tm > $o,Rules2: stream_rule,S4: set_tm,S: tm] :
( ( abstra4909020524820736232ule_tm @ Eff4 @ Rules2 @ S4 )
=> ( ( member_tm3 @ S @ S4 )
=> ? [X3: rule] :
( ( member_rule3 @ X3 @ ( sset_rule @ Rules2 ) )
& ? [X_1: fset_tm] : ( Eff4 @ X3 @ S @ X_1 ) ) ) ) ).
% RuleSystem.enabled_R
thf(fact_969_RuleSystem_Oenabled__R,axiom,
! [Eff4: rule > fm > fset_fm > $o,Rules2: stream_rule,S4: set_fm,S: fm] :
( ( abstra4909020524819817846ule_fm @ Eff4 @ Rules2 @ S4 )
=> ( ( member_fm3 @ S @ S4 )
=> ? [X3: rule] :
( ( member_rule3 @ X3 @ ( sset_rule @ Rules2 ) )
& ? [X_1: fset_fm] : ( Eff4 @ X3 @ S @ X_1 ) ) ) ) ).
% RuleSystem.enabled_R
thf(fact_970_RuleSystem_Oenabled__R,axiom,
! [Eff4: rule > nat > fset_nat > $o,Rules2: stream_rule,S4: set_nat,S: nat] :
( ( abstra3263948797621512205le_nat @ Eff4 @ Rules2 @ S4 )
=> ( ( member_nat3 @ S @ S4 )
=> ? [X3: rule] :
( ( member_rule3 @ X3 @ ( sset_rule @ Rules2 ) )
& ? [X_1: fset_nat] : ( Eff4 @ X3 @ S @ X_1 ) ) ) ) ).
% RuleSystem.enabled_R
thf(fact_971_RuleSystem_Oenabled__R,axiom,
! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,S: produc6018962875968178549ist_fm] :
( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S4 )
=> ( ( member4699826688122452638ist_fm @ S @ S4 )
=> ? [X3: rule] :
( ( member_rule3 @ X3 @ ( sset_rule @ Rules2 ) )
& ? [X_1: fset_P8989946509869081563ist_fm] : ( Eff4 @ X3 @ S @ X_1 ) ) ) ) ).
% RuleSystem.enabled_R
thf(fact_972_RuleSystem_Oenabled__R,axiom,
! [Eff4: produc340336539035504054m_rule > rule > fset_rule > $o,Rules2: stream2709947120125613254m_rule,S4: set_rule,S: rule] :
( ( abstra7792261008698710771e_rule @ Eff4 @ Rules2 @ S4 )
=> ( ( member_rule3 @ S @ S4 )
=> ? [X3: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X3 @ ( sset_P4484857331586881186m_rule @ Rules2 ) )
& ? [X_1: fset_rule] : ( Eff4 @ X3 @ S @ X_1 ) ) ) ) ).
% RuleSystem.enabled_R
thf(fact_973_RuleSystem_Oenabled__R,axiom,
! [Eff4: produc340336539035504054m_rule > real > fset_real > $o,Rules2: stream2709947120125613254m_rule,S4: set_real,S: real] :
( ( abstra9019047391330260365e_real @ Eff4 @ Rules2 @ S4 )
=> ( ( member_real3 @ S @ S4 )
=> ? [X3: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X3 @ ( sset_P4484857331586881186m_rule @ Rules2 ) )
& ? [X_1: fset_real] : ( Eff4 @ X3 @ S @ X_1 ) ) ) ) ).
% RuleSystem.enabled_R
thf(fact_974_RuleSystem_Oenabled__R,axiom,
! [Eff4: produc340336539035504054m_rule > tm > fset_tm > $o,Rules2: stream2709947120125613254m_rule,S4: set_tm,S: tm] :
( ( abstra6746019703890151108ule_tm @ Eff4 @ Rules2 @ S4 )
=> ( ( member_tm3 @ S @ S4 )
=> ? [X3: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X3 @ ( sset_P4484857331586881186m_rule @ Rules2 ) )
& ? [X_1: fset_tm] : ( Eff4 @ X3 @ S @ X_1 ) ) ) ) ).
% RuleSystem.enabled_R
thf(fact_975_RuleSystem_Oenabled__R,axiom,
! [Eff4: produc340336539035504054m_rule > fm > fset_fm > $o,Rules2: stream2709947120125613254m_rule,S4: set_fm,S: fm] :
( ( abstra6746019703889232722ule_fm @ Eff4 @ Rules2 @ S4 )
=> ( ( member_fm3 @ S @ S4 )
=> ? [X3: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X3 @ ( sset_P4484857331586881186m_rule @ Rules2 ) )
& ? [X_1: fset_fm] : ( Eff4 @ X3 @ S @ X_1 ) ) ) ) ).
% RuleSystem.enabled_R
thf(fact_976_i_Osdrop__fair,axiom,
! [Rules2: stream2709947120125613254m_rule,Rs: stream2709947120125613254m_rule,M: nat] :
( ( abstra3665372904625986210m_rule @ Rules2 @ Rs )
=> ( abstra3665372904625986210m_rule @ Rules2 @ ( sdrop_8169176516188972301m_rule @ M @ Rs ) ) ) ).
% i.sdrop_fair
thf(fact_977_i_Osdrop__fair,axiom,
! [Rules2: stream4408948924543953275ist_fm,Rs: stream4408948924543953275ist_fm,M: nat] :
( ( abstra7670258426472572809ist_fm @ Rules2 @ Rs )
=> ( abstra7670258426472572809ist_fm @ Rules2 @ ( sdrop_9176333610110415838ist_fm @ M @ Rs ) ) ) ).
% i.sdrop_fair
thf(fact_978_i_Osdrop__fair,axiom,
! [Rules2: stream4385846686851721995e_rule,Rs: stream4385846686851721995e_rule,M: nat] :
( ( abstra2528290469512735897e_rule @ Rules2 @ Rs )
=> ( abstra2528290469512735897e_rule @ Rules2 @ ( sdrop_9113879250048157294e_rule @ M @ Rs ) ) ) ).
% i.sdrop_fair
thf(fact_979_i_Osdrop__fair,axiom,
! [Rules2: stream8953843411776101167m_rule,Rs: stream8953843411776101167m_rule,M: nat] :
( ( abstra4920968922616850365m_rule @ Rules2 @ Rs )
=> ( abstra4920968922616850365m_rule @ Rules2 @ ( sdrop_746751008794826386m_rule @ M @ Rs ) ) ) ).
% i.sdrop_fair
thf(fact_980_i_Osdrop__fair,axiom,
! [Rules2: stream8099677779113257519e_rule,Rs: stream8099677779113257519e_rule,M: nat] :
( ( abstra6113178256415962813e_rule @ Rules2 @ Rs )
=> ( abstra6113178256415962813e_rule @ Rules2 @ ( sdrop_1938960342593938834e_rule @ M @ Rs ) ) ) ).
% i.sdrop_fair
thf(fact_981_i_Osdrop__fair,axiom,
! [Rules2: stream6210534828274662995m_rule,Rs: stream6210534828274662995m_rule,M: nat] :
( ( abstra1378428917839123937m_rule @ Rules2 @ Rs )
=> ( abstra1378428917839123937m_rule @ Rules2 @ ( sdrop_7192298464603511222m_rule @ M @ Rs ) ) ) ).
% i.sdrop_fair
thf(fact_982_i_Osdrop__fair,axiom,
! [Rules2: stream_rule,Rs: stream_rule,M: nat] :
( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
=> ( abstra3799686578551160190r_rule @ Rules2 @ ( sdrop_rule @ M @ Rs ) ) ) ).
% i.sdrop_fair
thf(fact_983_RuleSystem__Defs_Osdrop__fair,axiom,
! [Rules2: stream2709947120125613254m_rule,Rs: stream2709947120125613254m_rule,M: nat] :
( ( abstra3665372904625986210m_rule @ Rules2 @ Rs )
=> ( abstra3665372904625986210m_rule @ Rules2 @ ( sdrop_8169176516188972301m_rule @ M @ Rs ) ) ) ).
% RuleSystem_Defs.sdrop_fair
thf(fact_984_RuleSystem__Defs_Osdrop__fair,axiom,
! [Rules2: stream4408948924543953275ist_fm,Rs: stream4408948924543953275ist_fm,M: nat] :
( ( abstra7670258426472572809ist_fm @ Rules2 @ Rs )
=> ( abstra7670258426472572809ist_fm @ Rules2 @ ( sdrop_9176333610110415838ist_fm @ M @ Rs ) ) ) ).
% RuleSystem_Defs.sdrop_fair
thf(fact_985_RuleSystem__Defs_Osdrop__fair,axiom,
! [Rules2: stream4385846686851721995e_rule,Rs: stream4385846686851721995e_rule,M: nat] :
( ( abstra2528290469512735897e_rule @ Rules2 @ Rs )
=> ( abstra2528290469512735897e_rule @ Rules2 @ ( sdrop_9113879250048157294e_rule @ M @ Rs ) ) ) ).
% RuleSystem_Defs.sdrop_fair
thf(fact_986_RuleSystem__Defs_Osdrop__fair,axiom,
! [Rules2: stream8953843411776101167m_rule,Rs: stream8953843411776101167m_rule,M: nat] :
( ( abstra4920968922616850365m_rule @ Rules2 @ Rs )
=> ( abstra4920968922616850365m_rule @ Rules2 @ ( sdrop_746751008794826386m_rule @ M @ Rs ) ) ) ).
% RuleSystem_Defs.sdrop_fair
thf(fact_987_RuleSystem__Defs_Osdrop__fair,axiom,
! [Rules2: stream8099677779113257519e_rule,Rs: stream8099677779113257519e_rule,M: nat] :
( ( abstra6113178256415962813e_rule @ Rules2 @ Rs )
=> ( abstra6113178256415962813e_rule @ Rules2 @ ( sdrop_1938960342593938834e_rule @ M @ Rs ) ) ) ).
% RuleSystem_Defs.sdrop_fair
thf(fact_988_RuleSystem__Defs_Osdrop__fair,axiom,
! [Rules2: stream6210534828274662995m_rule,Rs: stream6210534828274662995m_rule,M: nat] :
( ( abstra1378428917839123937m_rule @ Rules2 @ Rs )
=> ( abstra1378428917839123937m_rule @ Rules2 @ ( sdrop_7192298464603511222m_rule @ M @ Rs ) ) ) ).
% RuleSystem_Defs.sdrop_fair
thf(fact_989_RuleSystem__Defs_Osdrop__fair,axiom,
! [Rules2: stream_rule,Rs: stream_rule,M: nat] :
( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
=> ( abstra3799686578551160190r_rule @ Rules2 @ ( sdrop_rule @ M @ Rs ) ) ) ).
% RuleSystem_Defs.sdrop_fair
thf(fact_990_i_Ofair__fenum,axiom,
! [Rules2: stream_rule] : ( abstra3799686578551160190r_rule @ Rules2 @ ( abstra745658567949189203m_rule @ Rules2 ) ) ).
% i.fair_fenum
thf(fact_991_RuleSystem__Defs_Ofair__fenum,axiom,
! [Rules2: stream_rule] : ( abstra3799686578551160190r_rule @ Rules2 @ ( abstra745658567949189203m_rule @ Rules2 ) ) ).
% RuleSystem_Defs.fair_fenum
thf(fact_992_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_993_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_994_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% of_nat_0_le_iff
thf(fact_995_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_996_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_997_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_998_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_999_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_1000_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% of_nat_mono
thf(fact_1001_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_1002_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_1003_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_1004_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_1005_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_1006_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_1007_RuleSystem_Opos,axiom,
! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,Rs: stream_rule,R: rule] :
( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S4 )
=> ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
=> ( ( member_rule3 @ R @ ( sset_rule @ Rules2 ) )
=> ( ( shd_rule @ ( sdrop_rule @ ( abstract_pos_rule @ Rs @ R ) @ Rs ) )
= R ) ) ) ) ).
% RuleSystem.pos
thf(fact_1008_sdrop__fair,axiom,
! [Rs: stream_rule,M: nat] :
( ( abstra3799686578551160190r_rule @ rules @ Rs )
=> ( abstra3799686578551160190r_rule @ rules @ ( sdrop_rule @ M @ Rs ) ) ) ).
% sdrop_fair
thf(fact_1009_fair__fenum,axiom,
abstra3799686578551160190r_rule @ rules @ ( abstra745658567949189203m_rule @ rules ) ).
% fair_fenum
thf(fact_1010_fair__uu__def,axiom,
abstra4598247580560492629u_rule = abstra3799686578551160190r_rule ).
% fair_uu_def
thf(fact_1011_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1012_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_1013_real__arch__simple,axiom,
! [X: real] :
? [N2: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% real_arch_simple
thf(fact_1014_RuleSystem_OminWait__ex,axiom,
! [Eff4: rule > rule > fset_rule > $o,Rules2: stream_rule,S4: set_rule,S: rule,Rs: stream_rule] :
( ( abstra6805424310982398927e_rule @ Eff4 @ Rules2 @ S4 )
=> ( ( member_rule3 @ S @ S4 )
=> ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
=> ? [N2: nat] : ( abstra7234149737463204544e_rule @ Eff4 @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).
% RuleSystem.minWait_ex
thf(fact_1015_RuleSystem_OminWait__ex,axiom,
! [Eff4: rule > real > fset_real > $o,Rules2: stream_rule,S4: set_real,S: real,Rs: stream_rule] :
( ( abstra7065561124429741161e_real @ Eff4 @ Rules2 @ S4 )
=> ( ( member_real3 @ S @ S4 )
=> ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
=> ? [N2: nat] : ( abstra3964006582584171482e_real @ Eff4 @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).
% RuleSystem.minWait_ex
thf(fact_1016_RuleSystem_OminWait__ex,axiom,
! [Eff4: rule > tm > fset_tm > $o,Rules2: stream_rule,S4: set_tm,S: tm,Rs: stream_rule] :
( ( abstra4909020524820736232ule_tm @ Eff4 @ Rules2 @ S4 )
=> ( ( member_tm3 @ S @ S4 )
=> ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
=> ? [N2: nat] : ( abstra8423890812398148919ule_tm @ Eff4 @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).
% RuleSystem.minWait_ex
thf(fact_1017_RuleSystem_OminWait__ex,axiom,
! [Eff4: rule > fm > fset_fm > $o,Rules2: stream_rule,S4: set_fm,S: fm,Rs: stream_rule] :
( ( abstra4909020524819817846ule_fm @ Eff4 @ Rules2 @ S4 )
=> ( ( member_fm3 @ S @ S4 )
=> ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
=> ? [N2: nat] : ( abstra8423890812397230533ule_fm @ Eff4 @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).
% RuleSystem.minWait_ex
thf(fact_1018_RuleSystem_OminWait__ex,axiom,
! [Eff4: rule > nat > fset_nat > $o,Rules2: stream_rule,S4: set_nat,S: nat,Rs: stream_rule] :
( ( abstra3263948797621512205le_nat @ Eff4 @ Rules2 @ S4 )
=> ( ( member_nat3 @ S @ S4 )
=> ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
=> ? [N2: nat] : ( abstra8624044281687814142le_nat @ Eff4 @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).
% RuleSystem.minWait_ex
thf(fact_1019_RuleSystem_OminWait__ex,axiom,
! [Eff4: produc7694839378271647877e_rule > rule > fset_rule > $o,Rules2: stream4385846686851721995e_rule,S4: set_rule,S: rule,Rs: stream4385846686851721995e_rule] :
( ( abstra2905114889036010542e_rule @ Eff4 @ Rules2 @ S4 )
=> ( ( member_rule3 @ S @ S4 )
=> ( ( abstra2528290469512735897e_rule @ Rules2 @ Rs )
=> ? [N2: nat] : ( abstra6715931297792236797e_rule @ Eff4 @ ( shd_Pr2264621979884435249e_rule @ ( sdrop_9113879250048157294e_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).
% RuleSystem.minWait_ex
thf(fact_1020_RuleSystem_OminWait__ex,axiom,
! [Eff4: produc7694839378271647877e_rule > real > fset_real > $o,Rules2: stream4385846686851721995e_rule,S4: set_real,S: real,Rs: stream4385846686851721995e_rule] :
( ( abstra7180780039604567112e_real @ Eff4 @ Rules2 @ S4 )
=> ( ( member_real3 @ S @ S4 )
=> ( ( abstra2528290469512735897e_rule @ Rules2 @ Rs )
=> ? [N2: nat] : ( abstra1743222340184358039e_real @ Eff4 @ ( shd_Pr2264621979884435249e_rule @ ( sdrop_9113879250048157294e_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).
% RuleSystem.minWait_ex
thf(fact_1021_RuleSystem_OminWait__ex,axiom,
! [Eff4: produc7694839378271647877e_rule > tm > fset_tm > $o,Rules2: stream4385846686851721995e_rule,S4: set_tm,S: tm,Rs: stream4385846686851721995e_rule] :
( ( abstra1261657321278996809ule_tm @ Eff4 @ Rules2 @ S4 )
=> ( ( member_tm3 @ S @ S4 )
=> ( ( abstra2528290469512735897e_rule @ Rules2 @ Rs )
=> ? [N2: nat] : ( abstra5903841454926380346ule_tm @ Eff4 @ ( shd_Pr2264621979884435249e_rule @ ( sdrop_9113879250048157294e_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).
% RuleSystem.minWait_ex
thf(fact_1022_RuleSystem_OminWait__ex,axiom,
! [Eff4: produc7694839378271647877e_rule > fm > fset_fm > $o,Rules2: stream4385846686851721995e_rule,S4: set_fm,S: fm,Rs: stream4385846686851721995e_rule] :
( ( abstra1261657321278078423ule_fm @ Eff4 @ Rules2 @ S4 )
=> ( ( member_fm3 @ S @ S4 )
=> ( ( abstra2528290469512735897e_rule @ Rules2 @ Rs )
=> ? [N2: nat] : ( abstra5903841454925461960ule_fm @ Eff4 @ ( shd_Pr2264621979884435249e_rule @ ( sdrop_9113879250048157294e_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).
% RuleSystem.minWait_ex
thf(fact_1023_RuleSystem_OminWait__ex,axiom,
! [Eff4: produc7694839378271647877e_rule > nat > fset_nat > $o,Rules2: stream4385846686851721995e_rule,S4: set_nat,S: nat,Rs: stream4385846686851721995e_rule] :
( ( abstra5123640173530818412le_nat @ Eff4 @ Rules2 @ S4 )
=> ( ( member_nat3 @ S @ S4 )
=> ( ( abstra2528290469512735897e_rule @ Rules2 @ Rs )
=> ? [N2: nat] : ( abstra7127153494424084283le_nat @ Eff4 @ ( shd_Pr2264621979884435249e_rule @ ( sdrop_9113879250048157294e_rule @ N2 @ Rs ) ) @ S ) ) ) ) ).
% RuleSystem.minWait_ex
thf(fact_1024_liftts_Osimps_I2_J,axiom,
! [T2: tm,Ts: list_tm] :
( ( liftts @ ( cons_tm @ T2 @ Ts ) )
= ( cons_tm @ ( liftt @ T2 ) @ ( liftts @ Ts ) ) ) ).
% liftts.simps(2)
thf(fact_1025_s4_I1_J,axiom,
inc_term = liftt ).
% s4(1)
thf(fact_1026_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1027_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( K
!= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_1028_RuleSystem__Defs_Oenabled_Ocong,axiom,
abstra1707737187183579335ist_fm = abstra1707737187183579335ist_fm ).
% RuleSystem_Defs.enabled.cong
thf(fact_1029_RuleSystem__Defs_Oenabled__def,axiom,
( abstra1707737187183579335ist_fm
= ( ^ [Eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,R2: rule,S3: produc6018962875968178549ist_fm] :
? [X9: fset_P8989946509869081563ist_fm] : ( Eff @ R2 @ S3 @ X9 ) ) ) ).
% RuleSystem_Defs.enabled_def
thf(fact_1030_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1031_conj__le__cong,axiom,
! [X: int,X10: int,P4: $o,P6: $o] :
( ( X = X10 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X10 )
=> ( P4 = P6 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P4 )
= ( ( ord_less_eq_int @ zero_zero_int @ X10 )
& P6 ) ) ) ) ).
% conj_le_cong
thf(fact_1032_imp__le__cong,axiom,
! [X: int,X10: int,P4: $o,P6: $o] :
( ( X = X10 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X10 )
=> ( P4 = P6 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P4 )
= ( ( ord_less_eq_int @ zero_zero_int @ X10 )
=> P6 ) ) ) ) ).
% imp_le_cong
thf(fact_1033_verit__la__generic,axiom,
! [A2: int,X: int] :
( ( ord_less_eq_int @ A2 @ X )
| ( A2 = X )
| ( ord_less_eq_int @ X @ A2 ) ) ).
% verit_la_generic
thf(fact_1034_enabled__def,axiom,
! [R: rule,S: produc6018962875968178549ist_fm] :
( ( abstra1707737187183579335ist_fm @ eff @ R @ S )
= ( ? [X9: fset_P8989946509869081563ist_fm] : ( eff @ R @ S @ X9 ) ) ) ).
% enabled_def
thf(fact_1035_i_Oenabled__def,axiom,
! [Eff2: rule > produc6018962875968178549ist_fm > option6967287582980624417ist_fm,R: rule,S: produc6018962875968178549ist_fm] :
( ( abstra1707737187183579335ist_fm @ ( abstra2682625350522704545ist_fm @ Eff2 ) @ R @ S )
= ( ? [X9: fset_P8989946509869081563ist_fm] : ( abstra2682625350522704545ist_fm @ Eff2 @ R @ S @ X9 ) ) ) ).
% i.enabled_def
thf(fact_1036_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_1037_liftt_Osimps_I2_J,axiom,
! [A2: nat,Ts: list_tm] :
( ( liftt @ ( fun @ A2 @ Ts ) )
= ( fun @ A2 @ ( liftts @ Ts ) ) ) ).
% liftt.simps(2)
thf(fact_1038_RuleSystem_OminWait__le__pos,axiom,
! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,Rs: stream_rule,R: rule,S: produc6018962875968178549ist_fm] :
( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S4 )
=> ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
=> ( ( member_rule3 @ R @ ( sset_rule @ Rules2 ) )
=> ( ( abstra1707737187183579335ist_fm @ Eff4 @ R @ S )
=> ( ord_less_eq_nat @ ( abstra1963046427795717555ist_fm @ Eff4 @ Rs @ S ) @ ( abstract_pos_rule @ Rs @ R ) ) ) ) ) ) ).
% RuleSystem.minWait_le_pos
thf(fact_1039_minWait__le__pos,axiom,
! [Rs: stream_rule,R: rule,S: produc6018962875968178549ist_fm] :
( ( abstra3799686578551160190r_rule @ rules @ Rs )
=> ( ( member_rule3 @ R @ ( sset_rule @ rules ) )
=> ( ( abstra1707737187183579335ist_fm @ eff @ R @ S )
=> ( ord_less_eq_nat @ ( abstra1963046427795717555ist_fm @ eff @ Rs @ S ) @ ( abstract_pos_rule @ Rs @ R ) ) ) ) ) ).
% minWait_le_pos
thf(fact_1040_RuleSystem_OminWait__least,axiom,
! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,N: nat,Rs: stream_rule,S: produc6018962875968178549ist_fm] :
( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S4 )
=> ( ( abstra1707737187183579335ist_fm @ Eff4 @ ( shd_rule @ ( sdrop_rule @ N @ Rs ) ) @ S )
=> ( ord_less_eq_nat @ ( abstra1963046427795717555ist_fm @ Eff4 @ Rs @ S ) @ N ) ) ) ).
% RuleSystem.minWait_least
thf(fact_1041_minWait__least,axiom,
! [N: nat,Rs: stream_rule,S: produc6018962875968178549ist_fm] :
( ( abstra1707737187183579335ist_fm @ eff @ ( shd_rule @ ( sdrop_rule @ N @ Rs ) ) @ S )
=> ( ord_less_eq_nat @ ( abstra1963046427795717555ist_fm @ eff @ Rs @ S ) @ N ) ) ).
% minWait_least
thf(fact_1042_minWait__ex,axiom,
! [S: produc6018962875968178549ist_fm,Rs: stream_rule] :
( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
=> ( ( abstra3799686578551160190r_rule @ rules @ Rs )
=> ? [N2: nat] : ( abstra1707737187183579335ist_fm @ eff @ ( shd_rule @ ( sdrop_rule @ N2 @ Rs ) ) @ S ) ) ) ).
% minWait_ex
thf(fact_1043_RuleSystem_Otrim__enabled,axiom,
! [Eff4: rule > fm > fset_fm > $o,Rules2: stream_rule,S4: set_fm,S: fm,Rs: stream_rule] :
( ( abstra4909020524819817846ule_fm @ Eff4 @ Rules2 @ S4 )
=> ( ( member_fm3 @ S @ S4 )
=> ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
=> ( abstra8423890812397230533ule_fm @ Eff4 @ ( shd_rule @ ( abstra4814164703916120982ule_fm @ Eff4 @ Rs @ S ) ) @ S ) ) ) ) ).
% RuleSystem.trim_enabled
thf(fact_1044_RuleSystem_Otrim__enabled,axiom,
! [Eff4: rule > nat > fset_nat > $o,Rules2: stream_rule,S4: set_nat,S: nat,Rs: stream_rule] :
( ( abstra3263948797621512205le_nat @ Eff4 @ Rules2 @ S4 )
=> ( ( member_nat3 @ S @ S4 )
=> ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
=> ( abstra8624044281687814142le_nat @ Eff4 @ ( shd_rule @ ( abstra8319304012324366829le_nat @ Eff4 @ Rs @ S ) ) @ S ) ) ) ) ).
% RuleSystem.trim_enabled
thf(fact_1045_RuleSystem_Otrim__enabled,axiom,
! [Eff4: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o,Rules2: stream_rule,S4: set_Pr5202636777678657877ist_fm,S: produc6018962875968178549ist_fm,Rs: stream_rule] :
( ( abstra5221733350967904376ist_fm @ Eff4 @ Rules2 @ S4 )
=> ( ( member4699826688122452638ist_fm @ S @ S4 )
=> ( ( abstra3799686578551160190r_rule @ Rules2 @ Rs )
=> ( abstra1707737187183579335ist_fm @ Eff4 @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ Eff4 @ Rs @ S ) ) @ S ) ) ) ) ).
% RuleSystem.trim_enabled
thf(fact_1046_trim__fair,axiom,
! [S: produc6018962875968178549ist_fm,Rs: stream_rule] :
( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
=> ( ( abstra3799686578551160190r_rule @ rules @ Rs )
=> ( abstra3799686578551160190r_rule @ rules @ ( abstra7806236140005899928ist_fm @ eff @ Rs @ S ) ) ) ) ).
% trim_fair
thf(fact_1047_trim__alt,axiom,
! [S: produc6018962875968178549ist_fm,Rs: stream_rule] :
( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
=> ( ( abstra3799686578551160190r_rule @ rules @ Rs )
=> ( ( abstra7806236140005899928ist_fm @ eff @ Rs @ S )
= ( sdrop_rule @ ( abstra1963046427795717555ist_fm @ eff @ Rs @ S ) @ Rs ) ) ) ) ).
% trim_alt
thf(fact_1048_RuleSystem__axioms,axiom,
abstra5221733350967904376ist_fm @ eff @ rules @ top_to1730629564055774885ist_fm ).
% RuleSystem_axioms
thf(fact_1049_trim__enabled,axiom,
! [S: produc6018962875968178549ist_fm,Rs: stream_rule] :
( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
=> ( ( abstra3799686578551160190r_rule @ rules @ Rs )
=> ( abstra1707737187183579335ist_fm @ eff @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs @ S ) ) @ S ) ) ) ).
% trim_enabled
thf(fact_1050_trim__in__R,axiom,
! [S: produc6018962875968178549ist_fm,Rs: stream_rule] :
( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
=> ( ( abstra3799686578551160190r_rule @ rules @ Rs )
=> ( member_rule3 @ ( shd_rule @ ( abstra7806236140005899928ist_fm @ eff @ Rs @ S ) ) @ ( sset_rule @ rules ) ) ) ) ).
% trim_in_R
thf(fact_1051_enabled__R,axiom,
! [S: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ S @ top_to1730629564055774885ist_fm )
=> ? [X3: rule] :
( ( member_rule3 @ X3 @ ( sset_rule @ rules ) )
& ? [X_1: fset_P8989946509869081563ist_fm] : ( eff @ X3 @ S @ X_1 ) ) ) ).
% enabled_R
thf(fact_1052_all__rules__persistent,axiom,
! [R3: rule] :
( ( member_rule3 @ R3 @ ( sset_rule @ rules ) )
=> ( abstra5255361903751151037ist_fm @ eff @ rules @ top_to1730629564055774885ist_fm @ R3 ) ) ).
% all_rules_persistent
thf(fact_1053_per,axiom,
! [R: rule] :
( ( member_rule3 @ R @ ( sset_rule @ rules ) )
=> ( abstra5255361903751151037ist_fm @ eff @ rules @ top_to1730629564055774885ist_fm @ R ) ) ).
% per
thf(fact_1054_PersistentRuleSystem__axioms,axiom,
abstra3967921542344661089ist_fm @ eff @ rules @ top_to1730629564055774885ist_fm ).
% PersistentRuleSystem_axioms
thf(fact_1055_paramst__subtermTm_I1_J,axiom,
! [T2: tm,X8: nat] :
( ( member_nat3 @ X8 @ ( paramst @ T2 ) )
=> ? [L3: list_tm] : ( member_tm3 @ ( fun @ X8 @ L3 ) @ ( set_tm2 @ ( subtermTm @ T2 ) ) ) ) ).
% paramst_subtermTm(1)
thf(fact_1056_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% neg_int_cases
thf(fact_1057_paramst__liftt_I1_J,axiom,
! [T2: tm] :
( ( paramst @ ( liftt @ T2 ) )
= ( paramst @ T2 ) ) ).
% paramst_liftt(1)
thf(fact_1058_s1_I1_J,axiom,
( new_term
= ( ^ [C4: nat,T: tm] :
~ ( member_nat3 @ C4 @ ( paramst @ T ) ) ) ) ).
% s1(1)
thf(fact_1059_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_1060_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zle
thf(fact_1061_int__cases4,axiom,
! [M: int] :
( ! [N2: nat] :
( M
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( M
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% int_cases4
thf(fact_1062_int__zle__neg,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_1063_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N2: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% nonpos_int_cases
thf(fact_1064_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1065_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% int_cases3
thf(fact_1066_p1,axiom,
paramst2 = paramst ).
% p1
thf(fact_1067_nat__less__iff,axiom,
! [W2: int,M: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ M )
= ( ord_less_int @ W2 @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% nat_less_iff
thf(fact_1068_zero__less__nat__eq,axiom,
! [Z3: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z3 ) )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% zero_less_nat_eq
thf(fact_1069_nat__le__0,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ Z3 @ zero_zero_int )
=> ( ( nat2 @ Z3 )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_1070_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= zero_zero_nat )
= ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_1071_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_1072_int__nat__eq,axiom,
! [Z3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= Z3 ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_1073_real__arch__inverse,axiom,
! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
= ( ? [N3: nat] :
( ( N3 != zero_zero_nat )
& ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) @ E ) ) ) ) ).
% real_arch_inverse
thf(fact_1074_forall__pos__mono,axiom,
! [P4: real > $o,E: real] :
( ! [D4: real,E2: real] :
( ( ord_less_real @ D4 @ E2 )
=> ( ( P4 @ D4 )
=> ( P4 @ E2 ) ) )
=> ( ! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( P4 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E )
=> ( P4 @ E ) ) ) ) ).
% forall_pos_mono
thf(fact_1075_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_1076_nat__mono,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y2 ) ) ) ).
% nat_mono
thf(fact_1077_ex__nat,axiom,
( ( ^ [P5: nat > $o] :
? [X5: nat] : ( P5 @ X5 ) )
= ( ^ [P2: nat > $o] :
? [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
& ( P2 @ ( nat2 @ X2 ) ) ) ) ) ).
% ex_nat
thf(fact_1078_all__nat,axiom,
( ( ^ [P5: nat > $o] :
! [X5: nat] : ( P5 @ X5 ) )
= ( ^ [P2: nat > $o] :
! [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( P2 @ ( nat2 @ X2 ) ) ) ) ) ).
% all_nat
thf(fact_1079_eq__nat__nat__iff,axiom,
! [Z3: int,Z6: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
=> ( ( ( nat2 @ Z3 )
= ( nat2 @ Z6 ) )
= ( Z3 = Z6 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_1080_nat__le__iff,axiom,
! [X: int,N: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
= ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% nat_le_iff
thf(fact_1081_int__eq__iff,axiom,
! [M: nat,Z3: int] :
( ( ( semiri1314217659103216013at_int @ M )
= Z3 )
= ( ( M
= ( nat2 @ Z3 ) )
& ( ord_less_eq_int @ zero_zero_int @ Z3 ) ) ) ).
% int_eq_iff
thf(fact_1082_nat__0__le,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= Z3 ) ) ).
% nat_0_le
thf(fact_1083_nat__less__eq__zless,axiom,
! [W2: int,Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ord_less_int @ W2 @ Z3 ) ) ) ).
% nat_less_eq_zless
thf(fact_1084_nat__eq__iff2,axiom,
! [M: nat,W2: int] :
( ( M
= ( nat2 @ W2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_1085_nat__eq__iff,axiom,
! [W2: int,M: nat] :
( ( ( nat2 @ W2 )
= M )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_1086_nat__le__eq__zle,axiom,
! [W2: int,Z3: int] :
( ( ( ord_less_int @ zero_zero_int @ W2 )
| ( ord_less_eq_int @ zero_zero_int @ Z3 ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ord_less_eq_int @ W2 @ Z3 ) ) ) ).
% nat_le_eq_zle
thf(fact_1087_split__nat,axiom,
! [P4: nat > $o,I: int] :
( ( P4 @ ( nat2 @ I ) )
= ( ! [N3: nat] :
( ( I
= ( semiri1314217659103216013at_int @ N3 ) )
=> ( P4 @ N3 ) )
& ( ( ord_less_int @ I @ zero_zero_int )
=> ( P4 @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_1088_le__nat__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% le_nat_iff
thf(fact_1089_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% less_eq_real_def
thf(fact_1090_fun__arguments__subterm,axiom,
! [N: nat,Ts: list_tm,P3: fm] :
( ( member_tm3 @ ( fun @ N @ Ts ) @ ( set_tm2 @ ( subtermFm @ P3 ) ) )
=> ( ord_less_eq_set_tm @ ( set_tm2 @ Ts ) @ ( set_tm2 @ ( subtermFm @ P3 ) ) ) ) ).
% fun_arguments_subterm
thf(fact_1091_complete__real,axiom,
! [S4: set_real] :
( ? [X8: real] : ( member_real3 @ X8 @ S4 )
=> ( ? [Z7: real] :
! [X3: real] :
( ( member_real3 @ X3 @ S4 )
=> ( ord_less_eq_real @ X3 @ Z7 ) )
=> ? [Y4: real] :
( ! [X8: real] :
( ( member_real3 @ X8 @ S4 )
=> ( ord_less_eq_real @ X8 @ Y4 ) )
& ! [Z7: real] :
( ! [X3: real] :
( ( member_real3 @ X3 @ S4 )
=> ( ord_less_eq_real @ X3 @ Z7 ) )
=> ( ord_less_eq_real @ Y4 @ Z7 ) ) ) ) ) ).
% complete_real
thf(fact_1092_subtermFm_Osimps_I7_J,axiom,
! [P3: fm] :
( ( subtermFm @ ( neg @ P3 ) )
= ( subtermFm @ P3 ) ) ).
% subtermFm.simps(7)
thf(fact_1093_subtermFm_Osimps_I6_J,axiom,
! [P3: fm] :
( ( subtermFm @ ( uni @ P3 ) )
= ( subtermFm @ P3 ) ) ).
% subtermFm.simps(6)
thf(fact_1094_subtermFm__subset__params,axiom,
! [P3: fm,A: list_tm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ ( subtermFm @ P3 ) ) @ ( set_tm2 @ A ) )
=> ( ord_less_eq_set_nat @ ( params @ P3 ) @ ( paramsts @ A ) ) ) ).
% subtermFm_subset_params
thf(fact_1095_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ! [M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ M4 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X ) @ C ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1096_params__subtermFm,axiom,
! [P3: fm,X8: nat] :
( ( member_nat3 @ X8 @ ( params @ P3 ) )
=> ? [L3: list_tm] : ( member_tm3 @ ( fun @ X8 @ L3 ) @ ( set_tm2 @ ( subtermFm @ P3 ) ) ) ) ).
% params_subtermFm
thf(fact_1097_nat__ceiling__le__eq,axiom,
! [X: real,A2: nat] :
( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A2 )
= ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A2 ) ) ) ).
% nat_ceiling_le_eq
thf(fact_1098_real__minus__mult__self__le,axiom,
! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X @ X ) ) ).
% real_minus_mult_self_le
thf(fact_1099_params_Osimps_I7_J,axiom,
! [P3: fm] :
( ( params @ ( neg @ P3 ) )
= ( params @ P3 ) ) ).
% params.simps(7)
thf(fact_1100_params_Osimps_I6_J,axiom,
! [P3: fm] :
( ( params @ ( uni @ P3 ) )
= ( params @ P3 ) ) ).
% params.simps(6)
thf(fact_1101_real__nat__ceiling__ge,axiom,
! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% real_nat_ceiling_ge
thf(fact_1102_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1103_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1104_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1105_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1106_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1107_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1108_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1109_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1110_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1111_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L2 ) ) ) ) ).
% mult_le_mono
thf(fact_1112_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1113_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1114_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1115_nat__mult__distrib,axiom,
! [Z3: int,Z6: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( nat2 @ ( times_times_int @ Z3 @ Z6 ) )
= ( times_times_nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z6 ) ) ) ) ).
% nat_mult_distrib
thf(fact_1116_nat__mult__distrib__neg,axiom,
! [Z3: int,Z6: int] :
( ( ord_less_eq_int @ Z3 @ zero_zero_int )
=> ( ( nat2 @ ( times_times_int @ Z3 @ Z6 ) )
= ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z3 ) ) @ ( nat2 @ ( uminus_uminus_int @ Z6 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_1117_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1118_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1119_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1120_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1121_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1122_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1123_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1124_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1125_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1126_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1127_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1128_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1129_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1130_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1131_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1132_int__one__le__iff__zero__less,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ one_one_int @ Z3 )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1133_one__less__nat__eq,axiom,
! [Z3: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z3 ) )
= ( ord_less_int @ one_one_int @ Z3 ) ) ).
% one_less_nat_eq
thf(fact_1134_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1135_nat_Oinject,axiom,
! [X23: nat,Y23: nat] :
( ( ( suc @ X23 )
= ( suc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% nat.inject
thf(fact_1136_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1137_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1138_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_1139_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_1140_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1141_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1142_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1143_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1144_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1145_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_1146_floor__eq4,axiom,
! [N: nat,X: real] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X )
=> ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
= N ) ) ) ).
% floor_eq4
thf(fact_1147_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1148_strict__inc__induct,axiom,
! [I: nat,J: nat,P4: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P4 @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P4 @ ( suc @ I2 ) )
=> ( P4 @ I2 ) ) )
=> ( P4 @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1149_less__Suc__induct,axiom,
! [I: nat,J: nat,P4: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P4 @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P4 @ I2 @ J2 )
=> ( ( P4 @ J2 @ K2 )
=> ( P4 @ I2 @ K2 ) ) ) ) )
=> ( P4 @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1150_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1151_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_1152_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1153_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M5: nat] :
( ( M
= ( suc @ M5 ) )
& ( ord_less_nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1154_All__less__Suc,axiom,
! [N: nat,P4: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P4 @ I4 ) ) )
= ( ( P4 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P4 @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_1155_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1156_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_1157_Ex__less__Suc,axiom,
! [N: nat,P4: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P4 @ I4 ) ) )
= ( ( P4 @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P4 @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1158_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1159_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1160_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1161_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1162_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1163_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1164_exists__least__lemma,axiom,
! [P4: nat > $o] :
( ~ ( P4 @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P4 @ X_12 )
=> ? [N2: nat] :
( ~ ( P4 @ N2 )
& ( P4 @ ( suc @ N2 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1165_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( zero_zero_nat
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_1166_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1167_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1168_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1169_old_Onat_Oexhaust,axiom,
! [Y2: nat] :
( ( Y2 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y2
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1170_nat__induct,axiom,
! [P4: nat > $o,N: nat] :
( ( P4 @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P4 @ N2 )
=> ( P4 @ ( suc @ N2 ) ) )
=> ( P4 @ N ) ) ) ).
% nat_induct
thf(fact_1171_diff__induct,axiom,
! [P4: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P4 @ X3 @ zero_zero_nat )
=> ( ! [Y4: nat] : ( P4 @ zero_zero_nat @ ( suc @ Y4 ) )
=> ( ! [X3: nat,Y4: nat] :
( ( P4 @ X3 @ Y4 )
=> ( P4 @ ( suc @ X3 ) @ ( suc @ Y4 ) ) )
=> ( P4 @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1172_zero__induct,axiom,
! [P4: nat > $o,K: nat] :
( ( P4 @ K )
=> ( ! [N2: nat] :
( ( P4 @ ( suc @ N2 ) )
=> ( P4 @ N2 ) )
=> ( P4 @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1173_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1174_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1175_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1176_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_1177_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1178_Suc__inject,axiom,
! [X: nat,Y2: nat] :
( ( ( suc @ X )
= ( suc @ Y2 ) )
=> ( X = Y2 ) ) ).
% Suc_inject
thf(fact_1179_real__of__int__div4,axiom,
! [N: int,X: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) ) ).
% real_of_int_div4
thf(fact_1180_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1181_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1182_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1183_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M4: nat] :
( M6
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_1184_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1185_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1186_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_1187_full__nat__induct,axiom,
! [P4: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
=> ( P4 @ M3 ) )
=> ( P4 @ N2 ) )
=> ( P4 @ N ) ) ).
% full_nat_induct
thf(fact_1188_nat__induct__at__least,axiom,
! [M: nat,N: nat,P4: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P4 @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P4 @ N2 )
=> ( P4 @ ( suc @ N2 ) ) ) )
=> ( P4 @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1189_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R4: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R4 @ X3 @ X3 )
=> ( ! [X3: nat,Y4: nat,Z5: nat] :
( ( R4 @ X3 @ Y4 )
=> ( ( R4 @ Y4 @ Z5 )
=> ( R4 @ X3 @ Z5 ) ) )
=> ( ! [N2: nat] : ( R4 @ N2 @ ( suc @ N2 ) )
=> ( R4 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1190_floor__divide__real__eq__div,axiom,
! [B2: int,A2: real] :
( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A2 @ ( ring_1_of_int_real @ B2 ) ) )
= ( divide_divide_int @ ( archim6058952711729229775r_real @ A2 ) @ B2 ) ) ) ).
% floor_divide_real_eq_div
thf(fact_1191_real__of__nat__div4,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% real_of_nat_div4
thf(fact_1192_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_1193_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1194_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_1195_All__less__Suc2,axiom,
! [N: nat,P4: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P4 @ I4 ) ) )
= ( ( P4 @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P4 @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1196_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1197_Ex__less__Suc2,axiom,
! [N: nat,P4: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P4 @ I4 ) ) )
= ( ( P4 @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P4 @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1198_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1199_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1200_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_1201_dec__induct,axiom,
! [I: nat,J: nat,P4: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P4 @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P4 @ N2 )
=> ( P4 @ ( suc @ N2 ) ) ) ) )
=> ( P4 @ J ) ) ) ) ).
% dec_induct
thf(fact_1202_inc__induct,axiom,
! [I: nat,J: nat,P4: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P4 @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P4 @ ( suc @ N2 ) )
=> ( P4 @ N2 ) ) ) )
=> ( P4 @ I ) ) ) ) ).
% inc_induct
thf(fact_1203_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_1204_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1205_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_1206_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1207_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1208_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1209_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1210_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1211_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1212_ex__least__nat__less,axiom,
! [P4: nat > $o,N: nat] :
( ( P4 @ N )
=> ( ~ ( P4 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K2 )
=> ~ ( P4 @ I3 ) )
& ( P4 @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1213_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_1214_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1215_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1216_nat__induct__non__zero,axiom,
! [N: nat,P4: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P4 @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P4 @ N2 )
=> ( P4 @ ( suc @ N2 ) ) ) )
=> ( P4 @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1217_nat__floor__neg,axiom,
! [X: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
= zero_zero_nat ) ) ).
% nat_floor_neg
thf(fact_1218_le__nat__floor,axiom,
! [X: nat,A2: real] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ A2 )
=> ( ord_less_eq_nat @ X @ ( nat2 @ ( archim6058952711729229775r_real @ A2 ) ) ) ) ).
% le_nat_floor
thf(fact_1219_verit__less__mono__div__int2,axiom,
! [A: int,B: int,N: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
=> ( ord_less_eq_int @ ( divide_divide_int @ B @ N ) @ ( divide_divide_int @ A @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_1220_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% not_zle_0_negative
thf(fact_1221_div__pos__pos__trivial,axiom,
! [K: int,L2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L2 )
=> ( ( divide_divide_int @ K @ L2 )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_1222_div__neg__neg__trivial,axiom,
! [K: int,L2: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L2 @ K )
=> ( ( divide_divide_int @ K @ L2 )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_1223_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_1224_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
= M ) ).
% div_by_Suc_0
thf(fact_1225_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1226_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
= M ) ) ).
% div_mult_self_is_m
thf(fact_1227_div__le__mono,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_1228_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_1229_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( divide_divide_nat @ M @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1230_Suc__div__le__mono,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% Suc_div_le_mono
thf(fact_1231_times__div__less__eq__dividend,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_1232_div__times__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_1233_div__le__mono2,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_1234_div__greater__zero__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ N @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_1235_div__less__iff__less__mult,axiom,
! [Q: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q ) @ N )
= ( ord_less_nat @ M @ ( times_times_nat @ N @ Q ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1236_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( divide_divide_nat @ M @ N )
= M )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1237_div__less__dividend,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% div_less_dividend
thf(fact_1238_zdiv__mono1,axiom,
! [A2: int,A6: int,B2: int] :
( ( ord_less_eq_int @ A2 @ A6 )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B2 ) @ ( divide_divide_int @ A6 @ B2 ) ) ) ) ).
% zdiv_mono1
thf(fact_1239_zdiv__mono2,axiom,
! [A2: int,B7: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B7 )
=> ( ( ord_less_eq_int @ B7 @ B2 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B2 ) @ ( divide_divide_int @ A2 @ B7 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1240_zdiv__eq__0__iff,axiom,
! [I: int,K: int] :
( ( ( divide_divide_int @ I @ K )
= zero_zero_int )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K @ I ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_1241_zdiv__mono1__neg,axiom,
! [A2: int,A6: int,B2: int] :
( ( ord_less_eq_int @ A2 @ A6 )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A6 @ B2 ) @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1242_zdiv__mono2__neg,axiom,
! [A2: int,B7: int,B2: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B7 )
=> ( ( ord_less_eq_int @ B7 @ B2 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B7 ) @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1243_div__int__pos__iff,axiom,
! [K: int,L2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L2 ) )
= ( ( K = zero_zero_int )
| ( L2 = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_eq_int @ zero_zero_int @ L2 ) )
| ( ( ord_less_int @ K @ zero_zero_int )
& ( ord_less_int @ L2 @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_1244_div__nonneg__neg__le0,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1245_div__nonpos__pos__le0,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1246_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
= ( ord_less_eq_int @ K @ I ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1247_neg__imp__zdiv__nonneg__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B2 ) )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1248_pos__imp__zdiv__nonneg__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B2 ) )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1249_nonneg1__imp__zdiv__pos__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B2 ) )
= ( ( ord_less_eq_int @ B2 @ A2 )
& ( ord_less_int @ zero_zero_int @ B2 ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1250_zdiv__zmult2__eq,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( divide_divide_int @ A2 @ ( times_times_int @ B2 @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A2 @ B2 ) @ C ) ) ) ).
% zdiv_zmult2_eq
thf(fact_1251_nat__div__distrib_H,axiom,
! [Y2: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( nat2 @ ( divide_divide_int @ X @ Y2 ) )
= ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y2 ) ) ) ) ).
% nat_div_distrib'
thf(fact_1252_nat__div__distrib,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( nat2 @ ( divide_divide_int @ X @ Y2 ) )
= ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y2 ) ) ) ) ).
% nat_div_distrib
thf(fact_1253_less__eq__div__iff__mult__less__eq,axiom,
! [Q: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q )
=> ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M @ Q ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1254_div__nat__eqI,axiom,
! [N: nat,Q: nat,M: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q ) @ M )
=> ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q ) ) )
=> ( ( divide_divide_nat @ M @ N )
= Q ) ) ) ).
% div_nat_eqI
thf(fact_1255_split__div_H,axiom,
! [P4: nat > $o,M: nat,N: nat] :
( ( P4 @ ( divide_divide_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
& ( P4 @ zero_zero_nat ) )
| ? [Q3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q3 ) @ M )
& ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q3 ) ) )
& ( P4 @ Q3 ) ) ) ) ).
% split_div'
thf(fact_1256_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1257_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1258_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1259_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1260_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_1261_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1262_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_1263_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1264_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1265_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1266_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1267_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1268_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
% Helper facts (11)
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y2: list_nat] :
( ( if_list_nat @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y2: list_nat] :
( ( if_list_nat @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__SeCaV__Ofm_J_T,axiom,
! [X: list_fm,Y2: list_fm] :
( ( if_list_fm @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__SeCaV__Ofm_J_T,axiom,
! [X: list_fm,Y2: list_fm] :
( ( if_list_fm @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__SeCaV__Otm_J_T,axiom,
! [X: list_tm,Y2: list_tm] :
( ( if_list_tm @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__SeCaV__Otm_J_T,axiom,
! [X: list_tm,Y2: list_tm] :
( ( if_list_tm @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Real__Oreal_J_T,axiom,
! [X: list_real,Y2: list_real] :
( ( if_list_real @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Real__Oreal_J_T,axiom,
! [X: list_real,Y2: list_real] :
( ( if_list_real @ $true @ X @ Y2 )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Prover__Orule_J_T,axiom,
! [P4: $o] :
( ( P4 = $true )
| ( P4 = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Prover__Orule_J_T,axiom,
! [X: list_rule,Y2: list_rule] :
( ( if_list_rule @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Prover__Orule_J_T,axiom,
! [X: list_rule,Y2: list_rule] :
( ( if_list_rule @ $true @ X @ Y2 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
? [M3: nat] : ( member_tm3 @ t @ ( set_tm2 @ ( subterms @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ M3 @ steps ) ) ) ) ) ) ).
%------------------------------------------------------------------------------