TPTP Problem File: DAT152^1.p
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%------------------------------------------------------------------------------
% File : DAT152^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Coinductive stream 429
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [Loc10] Lochbihler (2010), Coinductive
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : coinductive_stream__429.p [Bla16]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.25 v7.5.0, 0.00 v7.1.0
% Syntax : Number of formulae : 323 ( 96 unt; 39 typ; 0 def)
% Number of atoms : 903 ( 179 equ; 0 cnn)
% Maximal formula atoms : 24 ( 3 avg)
% Number of connectives : 3478 ( 119 ~; 22 |; 45 &;2820 @)
% ( 0 <=>; 472 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 8 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 198 ( 198 >; 0 *; 0 +; 0 <<)
% Number of symbols : 39 ( 36 usr; 3 con; 0-4 aty)
% Number of variables : 991 ( 106 ^; 816 !; 38 ?; 991 :)
% ( 31 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 15:14:51.363
%------------------------------------------------------------------------------
%----Could-be-implicit typings (6)
thf(ty_t_Extended__Nat_Oenat,type,
extended_enat: $tType ).
thf(ty_t_Stream_Ostream,type,
stream: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (33)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
condit1656338222tinuum:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit1037483654norder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Oscount,type,
coindu1365464361scount:
!>[S: $tType] : ( ( ( stream @ S ) > $o ) > ( stream @ S ) > extended_enat ) ).
thf(sy_c_Extended__Nat_Oenat,type,
extended_enat2: nat > extended_enat ).
thf(sy_c_Extended__Nat_Oenat_Ocase__enat,type,
extended_case_enat:
!>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
thf(sy_c_Extended__Nat_Oenat_Orec__enat,type,
extended_rec_enat:
!>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity,type,
extend1396239628finity:
!>[A: $tType] : A ).
thf(sy_c_Finite__Set_Ocard,type,
finite_card:
!>[B: $tType] : ( ( set @ B ) > nat ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Linear__Temporal__Logic__on__Streams_Oalw,type,
linear1386806755on_alw:
!>[A: $tType] : ( ( ( stream @ A ) > $o ) > ( stream @ A ) > $o ) ).
thf(sy_c_Linear__Temporal__Logic__on__Streams_Oev,type,
linear505997466_on_ev:
!>[A: $tType] : ( ( ( stream @ A ) > $o ) > ( stream @ A ) > $o ) ).
thf(sy_c_Linear__Temporal__Logic__on__Streams_Onxt,type,
linear1494993505on_nxt:
!>[A: $tType,B: $tType] : ( ( ( stream @ A ) > B ) > ( stream @ A ) > B ) ).
thf(sy_c_Linear__Temporal__Logic__on__Streams_Owait,type,
linear1335279038n_wait:
!>[A: $tType] : ( ( ( stream @ A ) > $o ) > ( stream @ A ) > nat ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Stream_Osdrop,type,
sdrop:
!>[A: $tType] : ( nat > ( stream @ A ) > ( stream @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_P,type,
p: ( stream @ a ) > $o ).
thf(sy_v__092_060omega_062,type,
omega: stream @ a ).
%----Relevant facts (256)
thf(fact_0__092_060open_062ev_A_Ialw_A_I_092_060lambda_062xs_O_A_092_060not_062_AP_Axs_J_J_A_092_060omega_062_A_092_060Longrightarrow_062_Ascount_AP_A_092_060omega_062_A_061_Aenat_A_Icard_A_123i_O_AP_A_Isdrop_Ai_A_092_060omega_062_J_125_J_092_060close_062,axiom,
( ( linear505997466_on_ev @ a
@ ( linear1386806755on_alw @ a
@ ^ [Xs: stream @ a] :
~ ( p @ Xs ) )
@ omega )
=> ( ( coindu1365464361scount @ a @ p @ omega )
= ( extended_enat2
@ ( finite_card @ nat
@ ( collect @ nat
@ ^ [I: nat] : ( p @ ( sdrop @ a @ I @ omega ) ) ) ) ) ) ) ).
% \<open>ev (alw (\<lambda>xs. \<not> P xs)) \<omega> \<Longrightarrow> scount P \<omega> = enat (card {i. P (sdrop i \<omega>)})\<close>
thf(fact_1_not__ev__not,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o] :
( ( ^ [Xs: stream @ A] :
~ ( linear505997466_on_ev @ A
@ ^ [Xt: stream @ A] :
~ ( Phi @ Xt )
@ Xs ) )
= ( linear1386806755on_alw @ A @ Phi ) ) ).
% not_ev_not
thf(fact_2_not__alw__not,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o] :
( ( ^ [Xs: stream @ A] :
~ ( linear1386806755on_alw @ A
@ ^ [Xt: stream @ A] :
~ ( Phi @ Xt )
@ Xs ) )
= ( linear505997466_on_ev @ A @ Phi ) ) ).
% not_alw_not
thf(fact_3_enat__ord__simps_I6_J,axiom,
! [Q: extended_enat] :
~ ( ord_less @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ Q ) ).
% enat_ord_simps(6)
thf(fact_4_enat__ord__simps_I4_J,axiom,
! [Q: extended_enat] :
( ( ord_less @ extended_enat @ Q @ ( extend1396239628finity @ extended_enat ) )
= ( Q
!= ( extend1396239628finity @ extended_enat ) ) ) ).
% enat_ord_simps(4)
thf(fact_5_ev__ev,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o] :
( ( linear505997466_on_ev @ A @ ( linear505997466_on_ev @ A @ Phi ) )
= ( linear505997466_on_ev @ A @ Phi ) ) ).
% ev_ev
thf(fact_6_alw__alw,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o] :
( ( linear1386806755on_alw @ A @ ( linear1386806755on_alw @ A @ Phi ) )
= ( linear1386806755on_alw @ A @ Phi ) ) ).
% alw_alw
thf(fact_7_not__ev,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o] :
( ( ^ [Xs: stream @ A] :
~ ( linear505997466_on_ev @ A @ Phi @ Xs ) )
= ( linear1386806755on_alw @ A
@ ^ [Xs: stream @ A] :
~ ( Phi @ Xs ) ) ) ).
% not_ev
thf(fact_8_not__alw,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o] :
( ( ^ [Xs: stream @ A] :
~ ( linear1386806755on_alw @ A @ Phi @ Xs ) )
= ( linear505997466_on_ev @ A
@ ^ [Xs: stream @ A] :
~ ( Phi @ Xs ) ) ) ).
% not_alw
thf(fact_9_not__ev__iff,axiom,
! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
( ( ~ ( linear505997466_on_ev @ A @ P @ Omega ) )
= ( linear1386806755on_alw @ A
@ ^ [Xs: stream @ A] :
~ ( P @ Xs )
@ Omega ) ) ).
% not_ev_iff
thf(fact_10_ev__alw__aand,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
( ( linear505997466_on_ev @ A @ ( linear1386806755on_alw @ A @ Phi ) @ Xs2 )
=> ( ( linear505997466_on_ev @ A @ ( linear1386806755on_alw @ A @ Psi ) @ Xs2 )
=> ( linear505997466_on_ev @ A
@ ( linear1386806755on_alw @ A
@ ^ [Xs: stream @ A] :
( ( Phi @ Xs )
& ( Psi @ Xs ) ) )
@ Xs2 ) ) ) ).
% ev_alw_aand
thf(fact_11_ev__alw__impl,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
( ( linear505997466_on_ev @ A @ Phi @ Xs2 )
=> ( ( linear1386806755on_alw @ A
@ ^ [Xs: stream @ A] :
( ( Phi @ Xs )
=> ( Psi @ Xs ) )
@ Xs2 )
=> ( linear505997466_on_ev @ A @ Psi @ Xs2 ) ) ) ).
% ev_alw_impl
thf(fact_12_not__alw__iff,axiom,
! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
( ( ~ ( linear1386806755on_alw @ A @ P @ Omega ) )
= ( linear505997466_on_ev @ A
@ ^ [Xs: stream @ A] :
~ ( P @ Xs )
@ Omega ) ) ).
% not_alw_iff
thf(fact_13_enat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( extended_enat2 @ Nat )
= ( extended_enat2 @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% enat.inject
thf(fact_14_not__infinity__eq,axiom,
! [X: extended_enat] :
( ( X
!= ( extend1396239628finity @ extended_enat ) )
= ( ? [I: nat] :
( X
= ( extended_enat2 @ I ) ) ) ) ).
% not_infinity_eq
thf(fact_15_not__enat__eq,axiom,
! [X: extended_enat] :
( ( ! [Y: nat] :
( X
!= ( extended_enat2 @ Y ) ) )
= ( X
= ( extend1396239628finity @ extended_enat ) ) ) ).
% not_enat_eq
thf(fact_16_enat__ex__split,axiom,
( ( ^ [P2: extended_enat > $o] :
? [X2: extended_enat] : ( P2 @ X2 ) )
= ( ^ [P3: extended_enat > $o] :
( ( P3 @ ( extend1396239628finity @ extended_enat ) )
| ? [X3: nat] : ( P3 @ ( extended_enat2 @ X3 ) ) ) ) ) ).
% enat_ex_split
thf(fact_17_enat_Oinducts,axiom,
! [P: extended_enat > $o,Enat: extended_enat] :
( ! [Nat3: nat] : ( P @ ( extended_enat2 @ Nat3 ) )
=> ( ( P @ ( extend1396239628finity @ extended_enat ) )
=> ( P @ Enat ) ) ) ).
% enat.inducts
thf(fact_18_enat_Oexhaust,axiom,
! [Y2: extended_enat] :
( ! [Nat3: nat] :
( Y2
!= ( extended_enat2 @ Nat3 ) )
=> ( Y2
= ( extend1396239628finity @ extended_enat ) ) ) ).
% enat.exhaust
thf(fact_19_enat3__cases,axiom,
! [Y2: extended_enat,Ya: extended_enat,Yb: extended_enat] :
( ( ? [Nat3: nat] :
( Y2
= ( extended_enat2 @ Nat3 ) )
=> ( ? [Nata: nat] :
( Ya
= ( extended_enat2 @ Nata ) )
=> ! [Natb: nat] :
( Yb
!= ( extended_enat2 @ Natb ) ) ) )
=> ( ( ? [Nat3: nat] :
( Y2
= ( extended_enat2 @ Nat3 ) )
=> ( ? [Nata: nat] :
( Ya
= ( extended_enat2 @ Nata ) )
=> ( Yb
!= ( extend1396239628finity @ extended_enat ) ) ) )
=> ( ( ? [Nat3: nat] :
( Y2
= ( extended_enat2 @ Nat3 ) )
=> ( ( Ya
= ( extend1396239628finity @ extended_enat ) )
=> ! [Nata: nat] :
( Yb
!= ( extended_enat2 @ Nata ) ) ) )
=> ( ( ? [Nat3: nat] :
( Y2
= ( extended_enat2 @ Nat3 ) )
=> ( ( Ya
= ( extend1396239628finity @ extended_enat ) )
=> ( Yb
!= ( extend1396239628finity @ extended_enat ) ) ) )
=> ( ( ( Y2
= ( extend1396239628finity @ extended_enat ) )
=> ( ? [Nat3: nat] :
( Ya
= ( extended_enat2 @ Nat3 ) )
=> ! [Nata: nat] :
( Yb
!= ( extended_enat2 @ Nata ) ) ) )
=> ( ( ( Y2
= ( extend1396239628finity @ extended_enat ) )
=> ( ? [Nat3: nat] :
( Ya
= ( extended_enat2 @ Nat3 ) )
=> ( Yb
!= ( extend1396239628finity @ extended_enat ) ) ) )
=> ( ( ( Y2
= ( extend1396239628finity @ extended_enat ) )
=> ( ( Ya
= ( extend1396239628finity @ extended_enat ) )
=> ! [Nat3: nat] :
( Yb
!= ( extended_enat2 @ Nat3 ) ) ) )
=> ~ ( ( Y2
= ( extend1396239628finity @ extended_enat ) )
=> ( ( Ya
= ( extend1396239628finity @ extended_enat ) )
=> ( Yb
!= ( extend1396239628finity @ extended_enat ) ) ) ) ) ) ) ) ) ) ) ).
% enat3_cases
thf(fact_20_enat2__cases,axiom,
! [Y2: extended_enat,Ya: extended_enat] :
( ( ? [Nat3: nat] :
( Y2
= ( extended_enat2 @ Nat3 ) )
=> ! [Nata: nat] :
( Ya
!= ( extended_enat2 @ Nata ) ) )
=> ( ( ? [Nat3: nat] :
( Y2
= ( extended_enat2 @ Nat3 ) )
=> ( Ya
!= ( extend1396239628finity @ extended_enat ) ) )
=> ( ( ( Y2
= ( extend1396239628finity @ extended_enat ) )
=> ! [Nat3: nat] :
( Ya
!= ( extended_enat2 @ Nat3 ) ) )
=> ~ ( ( Y2
= ( extend1396239628finity @ extended_enat ) )
=> ( Ya
!= ( extend1396239628finity @ extended_enat ) ) ) ) ) ) ).
% enat2_cases
thf(fact_21_enat_Odistinct_I1_J,axiom,
! [Nat: nat] :
( ( extended_enat2 @ Nat )
!= ( extend1396239628finity @ extended_enat ) ) ).
% enat.distinct(1)
thf(fact_22_enat_Odistinct_I2_J,axiom,
! [Nat4: nat] :
( ( extend1396239628finity @ extended_enat )
!= ( extended_enat2 @ Nat4 ) ) ).
% enat.distinct(2)
thf(fact_23_enat__iless,axiom,
! [N: extended_enat,M: nat] :
( ( ord_less @ extended_enat @ N @ ( extended_enat2 @ M ) )
=> ? [K: nat] :
( N
= ( extended_enat2 @ K ) ) ) ).
% enat_iless
thf(fact_24_chain__incr,axiom,
! [A: $tType,Y3: A > extended_enat,K2: nat] :
( ! [I2: A] :
? [J: A] : ( ord_less @ extended_enat @ ( Y3 @ I2 ) @ ( Y3 @ J ) )
=> ? [J2: A] : ( ord_less @ extended_enat @ ( extended_enat2 @ K2 ) @ ( Y3 @ J2 ) ) ) ).
% chain_incr
thf(fact_25_alw__iff__sdrop,axiom,
! [A: $tType] :
( ( linear1386806755on_alw @ A )
= ( ^ [P3: ( stream @ A ) > $o,Omega2: stream @ A] :
! [M2: nat] : ( P3 @ ( sdrop @ A @ M2 @ Omega2 ) ) ) ) ).
% alw_iff_sdrop
thf(fact_26_ev__iff__sdrop,axiom,
! [A: $tType] :
( ( linear505997466_on_ev @ A )
= ( ^ [P3: ( stream @ A ) > $o,Omega2: stream @ A] :
? [M2: nat] : ( P3 @ ( sdrop @ A @ M2 @ Omega2 ) ) ) ) ).
% ev_iff_sdrop
thf(fact_27_alw__sdrop,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,N: nat] :
( ( linear1386806755on_alw @ A @ Phi @ Xs2 )
=> ( linear1386806755on_alw @ A @ Phi @ ( sdrop @ A @ N @ Xs2 ) ) ) ).
% alw_sdrop
thf(fact_28_all__imp__alw,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A] :
( ! [X1: stream @ A] : ( Phi @ X1 )
=> ( linear1386806755on_alw @ A @ Phi @ Xs2 ) ) ).
% all_imp_alw
thf(fact_29_alw__mono,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
( ( linear1386806755on_alw @ A @ Phi @ Xs2 )
=> ( ! [Xs3: stream @ A] :
( ( Phi @ Xs3 )
=> ( Psi @ Xs3 ) )
=> ( linear1386806755on_alw @ A @ Psi @ Xs2 ) ) ) ).
% alw_mono
thf(fact_30_alw__cong,axiom,
! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A,Q1: ( stream @ A ) > $o,Q2: ( stream @ A ) > $o] :
( ( linear1386806755on_alw @ A @ P @ Omega )
=> ( ! [Omega3: stream @ A] :
( ( P @ Omega3 )
=> ( ( Q1 @ Omega3 )
= ( Q2 @ Omega3 ) ) )
=> ( ( linear1386806755on_alw @ A @ Q1 @ Omega )
= ( linear1386806755on_alw @ A @ Q2 @ Omega ) ) ) ) ).
% alw_cong
thf(fact_31_alw__alwD,axiom,
! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
( ( linear1386806755on_alw @ A @ P @ Omega )
=> ( linear1386806755on_alw @ A @ ( linear1386806755on_alw @ A @ P ) @ Omega ) ) ).
% alw_alwD
thf(fact_32_ev__mono,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
( ( linear505997466_on_ev @ A @ Phi @ Xs2 )
=> ( ! [Xs3: stream @ A] :
( ( Phi @ Xs3 )
=> ( Psi @ Xs3 ) )
=> ( linear505997466_on_ev @ A @ Psi @ Xs2 ) ) ) ).
% ev_mono
thf(fact_33_ev_Obase,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A] :
( ( Phi @ Xs2 )
=> ( linear505997466_on_ev @ A @ Phi @ Xs2 ) ) ).
% ev.base
thf(fact_34_alwD,axiom,
! [A: $tType,P: ( stream @ A ) > $o,X: stream @ A] :
( ( linear1386806755on_alw @ A @ P @ X )
=> ( P @ X ) ) ).
% alwD
thf(fact_35_enat__less__induct,axiom,
! [P: extended_enat > $o,N: extended_enat] :
( ! [N2: extended_enat] :
( ! [M3: extended_enat] :
( ( ord_less @ extended_enat @ M3 @ N2 )
=> ( P @ M3 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% enat_less_induct
thf(fact_36_infinity__ilessE,axiom,
! [M: nat] :
~ ( ord_less @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ ( extended_enat2 @ M ) ) ).
% infinity_ilessE
thf(fact_37_less__infinityE,axiom,
! [N: extended_enat] :
( ( ord_less @ extended_enat @ N @ ( extend1396239628finity @ extended_enat ) )
=> ~ ! [K: nat] :
( N
!= ( extended_enat2 @ K ) ) ) ).
% less_infinityE
thf(fact_38_enat__ord__code_I4_J,axiom,
! [M: nat] : ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ ( extend1396239628finity @ extended_enat ) ) ).
% enat_ord_code(4)
thf(fact_39_alw__ev__sdrop,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,M: nat,Xs2: stream @ A] :
( ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ Phi ) @ ( sdrop @ A @ M @ Xs2 ) )
=> ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ Phi ) @ Xs2 ) ) ).
% alw_ev_sdrop
thf(fact_40_alw__False,axiom,
! [A: $tType,Omega: stream @ A] :
~ ( linear1386806755on_alw @ A
@ ^ [X3: stream @ A] : $false
@ Omega ) ).
% alw_False
thf(fact_41_ev__False,axiom,
! [A: $tType,Omega: stream @ A] :
~ ( linear505997466_on_ev @ A
@ ^ [X3: stream @ A] : $false
@ Omega ) ).
% ev_False
thf(fact_42_alw__aand,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Psi: ( stream @ A ) > $o] :
( ( linear1386806755on_alw @ A
@ ^ [Xs: stream @ A] :
( ( Phi @ Xs )
& ( Psi @ Xs ) ) )
= ( ^ [Xs: stream @ A] :
( ( linear1386806755on_alw @ A @ Phi @ Xs )
& ( linear1386806755on_alw @ A @ Psi @ Xs ) ) ) ) ).
% alw_aand
thf(fact_43_alw__mp,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
( ( linear1386806755on_alw @ A @ Phi @ Xs2 )
=> ( ( linear1386806755on_alw @ A
@ ^ [Xs: stream @ A] :
( ( Phi @ Xs )
=> ( Psi @ Xs ) )
@ Xs2 )
=> ( linear1386806755on_alw @ A @ Psi @ Xs2 ) ) ) ).
% alw_mp
thf(fact_44_ev__or,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Psi: ( stream @ A ) > $o] :
( ( linear505997466_on_ev @ A
@ ^ [Xs: stream @ A] :
( ( Phi @ Xs )
| ( Psi @ Xs ) ) )
= ( ^ [Xs: stream @ A] :
( ( linear505997466_on_ev @ A @ Phi @ Xs )
| ( linear505997466_on_ev @ A @ Psi @ Xs ) ) ) ) ).
% ev_or
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q3: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
= ( Q3 @ X4 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q3 ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X4: A] :
( ( F @ X4 )
= ( G @ X4 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_scount__eq__card,axiom,
! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
( ( linear505997466_on_ev @ A
@ ( linear1386806755on_alw @ A
@ ^ [Xs: stream @ A] :
~ ( P @ Xs ) )
@ Omega )
=> ( ( coindu1365464361scount @ A @ P @ Omega )
= ( extended_enat2
@ ( finite_card @ nat
@ ( collect @ nat
@ ^ [I: nat] : ( P @ ( sdrop @ A @ I @ Omega ) ) ) ) ) ) ) ).
% scount_eq_card
thf(fact_50_ev__alw__imp__alw__ev,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A] :
( ( linear505997466_on_ev @ A @ ( linear1386806755on_alw @ A @ Phi ) @ Xs2 )
=> ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ Phi ) @ Xs2 ) ) ).
% ev_alw_imp_alw_ev
thf(fact_51_ev__cong,axiom,
! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A,Q1: ( stream @ A ) > $o,Q2: ( stream @ A ) > $o] :
( ( linear1386806755on_alw @ A @ P @ Omega )
=> ( ! [Omega3: stream @ A] :
( ( P @ Omega3 )
=> ( ( Q1 @ Omega3 )
= ( Q2 @ Omega3 ) ) )
=> ( ( linear505997466_on_ev @ A @ Q1 @ Omega )
= ( linear505997466_on_ev @ A @ Q2 @ Omega ) ) ) ) ).
% ev_cong
thf(fact_52_alw__ev__imp__ev__alw,axiom,
! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
( ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ P ) @ Omega )
=> ( linear505997466_on_ev @ A
@ ^ [Xs: stream @ A] :
( ( P @ Xs )
& ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ P ) @ Xs ) )
@ Omega ) ) ).
% alw_ev_imp_ev_alw
thf(fact_53_ev__alw__alw__impl,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
( ( linear505997466_on_ev @ A @ ( linear1386806755on_alw @ A @ Phi ) @ Xs2 )
=> ( ( linear1386806755on_alw @ A
@ ^ [Xs: stream @ A] :
( ( linear1386806755on_alw @ A @ Phi @ Xs )
=> ( linear505997466_on_ev @ A @ Psi @ Xs ) )
@ Xs2 )
=> ( linear505997466_on_ev @ A @ Psi @ Xs2 ) ) ) ).
% ev_alw_alw_impl
thf(fact_54_alw__impl__ev__alw,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Psi: ( stream @ A ) > $o,Xs2: stream @ A] :
( ( linear1386806755on_alw @ A
@ ^ [Xs: stream @ A] :
( ( Phi @ Xs )
=> ( linear505997466_on_ev @ A @ Psi @ Xs ) )
@ Xs2 )
=> ( linear1386806755on_alw @ A
@ ^ [Xs: stream @ A] :
( ( linear505997466_on_ev @ A @ Phi @ Xs )
=> ( linear505997466_on_ev @ A @ Psi @ Xs ) )
@ Xs2 ) ) ).
% alw_impl_ev_alw
thf(fact_55_alw__alw__impl__ev,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Psi: ( stream @ A ) > $o] :
( ( linear1386806755on_alw @ A
@ ^ [Xs: stream @ A] :
( ( linear1386806755on_alw @ A @ Phi @ Xs )
=> ( linear505997466_on_ev @ A @ Psi @ Xs ) ) )
= ( ^ [Xs: stream @ A] :
( ( linear505997466_on_ev @ A @ ( linear1386806755on_alw @ A @ Phi ) @ Xs )
=> ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ Psi ) @ Xs ) ) ) ) ).
% alw_alw_impl_ev
thf(fact_56_ev__alw__impl__ev,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
( ( linear505997466_on_ev @ A @ Phi @ Xs2 )
=> ( ( linear1386806755on_alw @ A
@ ^ [Xs: stream @ A] :
( ( Phi @ Xs )
=> ( linear505997466_on_ev @ A @ Psi @ Xs ) )
@ Xs2 )
=> ( linear505997466_on_ev @ A @ Psi @ Xs2 ) ) ) ).
% ev_alw_impl_ev
thf(fact_57_enat_Osimps_I7_J,axiom,
! [T: $tType,F1: nat > T,F2: T] :
( ( extended_rec_enat @ T @ F1 @ F2 @ ( extend1396239628finity @ extended_enat ) )
= F2 ) ).
% enat.simps(7)
thf(fact_58_enat_Osimps_I6_J,axiom,
! [T: $tType,F1: nat > T,F2: T,Nat: nat] :
( ( extended_rec_enat @ T @ F1 @ F2 @ ( extended_enat2 @ Nat ) )
= ( F1 @ Nat ) ) ).
% enat.simps(6)
thf(fact_59_sdrop__wait,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A] :
( ( linear505997466_on_ev @ A @ Phi @ Xs2 )
=> ( Phi @ ( sdrop @ A @ ( linear1335279038n_wait @ A @ Phi @ Xs2 ) @ Xs2 ) ) ) ).
% sdrop_wait
thf(fact_60_enat_Osimps_I5_J,axiom,
! [T: $tType,F1: nat > T,F2: T] :
( ( extended_case_enat @ T @ F1 @ F2 @ ( extend1396239628finity @ extended_enat ) )
= F2 ) ).
% enat.simps(5)
thf(fact_61_enat_Osimps_I4_J,axiom,
! [T: $tType,F1: nat > T,F2: T,Nat: nat] :
( ( extended_case_enat @ T @ F1 @ F2 @ ( extended_enat2 @ Nat ) )
= ( F1 @ Nat ) ) ).
% enat.simps(4)
thf(fact_62_infinite__iff__alw__ev,axiom,
! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
( ( ~ ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [M2: nat] : ( P @ ( sdrop @ A @ M2 @ Omega ) ) ) ) )
= ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ P ) @ Omega ) ) ).
% infinite_iff_alw_ev
thf(fact_63_ev__alw__not__HLD__finite,axiom,
! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
( ( linear505997466_on_ev @ A
@ ( linear1386806755on_alw @ A
@ ^ [Xs: stream @ A] :
~ ( P @ Xs ) )
@ Omega )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [I: nat] : ( P @ ( sdrop @ A @ I @ Omega ) ) ) ) ) ).
% ev_alw_not_HLD_finite
thf(fact_64_ev__alw__imp__nxt,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A] :
( ( linear505997466_on_ev @ A @ Phi @ Xs2 )
=> ( ( linear1386806755on_alw @ A
@ ^ [Xs: stream @ A] :
( ( Phi @ Xs )
=> ( linear1494993505on_nxt @ A @ $o @ Phi @ Xs ) )
@ Xs2 )
=> ( linear505997466_on_ev @ A @ ( linear1386806755on_alw @ A @ Phi ) @ Xs2 ) ) ) ).
% ev_alw_imp_nxt
thf(fact_65_variance,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
( ( Phi @ Xs2 )
=> ( ( linear1386806755on_alw @ A
@ ^ [Xs: stream @ A] :
( ( Phi @ Xs )
=> ( ( Psi @ Xs )
| ( linear1494993505on_nxt @ A @ $o @ Phi @ Xs ) ) )
@ Xs2 )
=> ( ( linear1386806755on_alw @ A @ Phi @ Xs2 )
| ( linear505997466_on_ev @ A @ Psi @ Xs2 ) ) ) ) ).
% variance
thf(fact_66_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
=> ! [A2: A] :
? [B2: A] :
( ( ord_less @ A @ A2 @ B2 )
| ( ord_less @ A @ B2 @ A2 ) ) ) ).
% ex_gt_or_lt
thf(fact_67_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( X != Y2 )
=> ( ~ ( ord_less @ A @ X @ Y2 )
=> ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_68_enat__ord__simps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% enat_ord_simps(2)
thf(fact_69_nxt__mono,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
( ( linear1494993505on_nxt @ A @ $o @ Phi @ Xs2 )
=> ( ! [Xs3: stream @ A] :
( ( Phi @ Xs3 )
=> ( Psi @ Xs3 ) )
=> ( linear1494993505on_nxt @ A @ $o @ Psi @ Xs2 ) ) ) ).
% nxt_mono
thf(fact_70_case__enat__def,axiom,
! [T: $tType] :
( ( extended_case_enat @ T )
= ( extended_rec_enat @ T ) ) ).
% case_enat_def
thf(fact_71_less__enat__def,axiom,
( ( ord_less @ extended_enat )
= ( ^ [M2: extended_enat,N3: extended_enat] :
( extended_case_enat @ $o
@ ^ [M1: nat] : ( extended_case_enat @ $o @ ( ord_less @ nat @ M1 ) @ $true @ N3 )
@ $false
@ M2 ) ) ) ).
% less_enat_def
thf(fact_72_ev__nxt,axiom,
! [A: $tType] :
( ( linear505997466_on_ev @ A )
= ( ^ [Phi2: ( stream @ A ) > $o,Xs: stream @ A] :
( ( Phi2 @ Xs )
| ( linear1494993505on_nxt @ A @ $o @ ( linear505997466_on_ev @ A @ Phi2 ) @ Xs ) ) ) ) ).
% ev_nxt
thf(fact_73_alw__nxt,axiom,
! [A: $tType] :
( ( linear1386806755on_alw @ A )
= ( ^ [Phi2: ( stream @ A ) > $o,Xs: stream @ A] :
( ( Phi2 @ Xs )
& ( linear1494993505on_nxt @ A @ $o @ ( linear1386806755on_alw @ A @ Phi2 ) @ Xs ) ) ) ) ).
% alw_nxt
thf(fact_74_alw__invar,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A] :
( ( Phi @ Xs2 )
=> ( ( linear1386806755on_alw @ A
@ ^ [Xs: stream @ A] :
( ( Phi @ Xs )
=> ( linear1494993505on_nxt @ A @ $o @ Phi @ Xs ) )
@ Xs2 )
=> ( linear1386806755on_alw @ A @ Phi @ Xs2 ) ) ) ).
% alw_invar
thf(fact_75_less__enatE,axiom,
! [N: extended_enat,M: nat] :
( ( ord_less @ extended_enat @ N @ ( extended_enat2 @ M ) )
=> ~ ! [K: nat] :
( ( N
= ( extended_enat2 @ K ) )
=> ~ ( ord_less @ nat @ K @ M ) ) ) ).
% less_enatE
thf(fact_76_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card @ nat
@ ( collect @ nat
@ ^ [I: nat] : ( ord_less @ nat @ I @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_77_finite__Collect__less__nat,axiom,
! [K2: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N3: nat] : ( ord_less @ nat @ N3 @ K2 ) ) ) ).
% finite_Collect_less_nat
thf(fact_78_finite__Collect__disjI,axiom,
! [A: $tType,P: A > $o,Q3: A > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] :
( ( P @ X3 )
| ( Q3 @ X3 ) ) ) )
= ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
& ( finite_finite2 @ A @ ( collect @ A @ Q3 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_79_finite__Collect__conjI,axiom,
! [A: $tType,P: A > $o,Q3: A > $o] :
( ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
| ( finite_finite2 @ A @ ( collect @ A @ Q3 ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] :
( ( P @ X3 )
& ( Q3 @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_80_finite__code,axiom,
! [A: $tType] :
( ( finite_finite @ A @ ( type2 @ A ) )
=> ( ( finite_finite2 @ A )
= ( ^ [A4: set @ A] : $true ) ) ) ).
% finite_code
thf(fact_81_psubset__card__mono,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
=> ( ord_less @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ) ).
% psubset_card_mono
thf(fact_82_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I3: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( P @ K3 )
& ( ord_less @ nat @ K3 @ I3 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_83_finite__set__choice,axiom,
! [B: $tType,A: $tType,A3: set @ A,P: A > B > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ? [X12: B] : ( P @ X4 @ X12 ) )
=> ? [F3: A > B] :
! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( P @ X5 @ ( F3 @ X5 ) ) ) ) ) ).
% finite_set_choice
thf(fact_84_finite,axiom,
! [A: $tType] :
( ( finite_finite @ A @ ( type2 @ A ) )
=> ! [A3: set @ A] : ( finite_finite2 @ A @ A3 ) ) ).
% finite
thf(fact_85_finite__psubset__induct,axiom,
! [A: $tType,A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [B4: set @ A] :
( ( ord_less @ ( set @ A ) @ B4 @ A5 )
=> ( P @ B4 ) )
=> ( P @ A5 ) ) )
=> ( P @ A3 ) ) ) ).
% finite_psubset_induct
thf(fact_86_bounded__nat__set__is__finite,axiom,
! [N4: set @ nat,N: nat] :
( ! [X4: nat] :
( ( member @ nat @ X4 @ N4 )
=> ( ord_less @ nat @ X4 @ N ) )
=> ( finite_finite2 @ nat @ N4 ) ) ).
% bounded_nat_set_is_finite
thf(fact_87_finite__nat__set__iff__bounded,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N5: set @ nat] :
? [M2: nat] :
! [X3: nat] :
( ( member @ nat @ X3 @ N5 )
=> ( ord_less @ nat @ X3 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_88_not__finite__existsD,axiom,
! [A: $tType,P: A > $o] :
( ~ ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ? [X1: A] : ( P @ X1 ) ) ).
% not_finite_existsD
thf(fact_89_pigeonhole__infinite__rel,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: set @ B,R: A > B > $o] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ? [Xa: B] :
( ( member @ B @ Xa @ B3 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: B] :
( ( member @ B @ X4 @ B3 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A6: A] :
( ( member @ A @ A6 @ A3 )
& ( R @ A6 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_90_unbounded__k__infinite,axiom,
! [K2: nat,S2: set @ nat] :
( ! [M4: nat] :
( ( ord_less @ nat @ K2 @ M4 )
=> ? [N6: nat] :
( ( ord_less @ nat @ M4 @ N6 )
& ( member @ nat @ N6 @ S2 ) ) )
=> ~ ( finite_finite2 @ nat @ S2 ) ) ).
% unbounded_k_infinite
thf(fact_91_infinite__nat__iff__unbounded,axiom,
! [S2: set @ nat] :
( ( ~ ( finite_finite2 @ nat @ S2 ) )
= ( ! [M2: nat] :
? [N3: nat] :
( ( ord_less @ nat @ M2 @ N3 )
& ( member @ nat @ N3 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_92_less__set__def,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A4: set @ A,B5: set @ A] :
( ord_less @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ A4 )
@ ^ [X3: A] : ( member @ A @ X3 @ B5 ) ) ) ) ).
% less_set_def
thf(fact_93_card__psubset,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) )
=> ( ord_less @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% card_psubset
thf(fact_94_sdrop__wait__least,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,N: nat] :
( ( linear505997466_on_ev @ A @ Phi @ Xs2 )
=> ( ( Phi @ ( sdrop @ A @ N @ Xs2 ) )
=> ( ord_less_eq @ nat @ ( linear1335279038n_wait @ A @ Phi @ Xs2 ) @ N ) ) ) ).
% sdrop_wait_least
thf(fact_95_psubset__trans,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less @ ( set @ A ) @ B3 @ C )
=> ( ord_less @ ( set @ A ) @ A3 @ C ) ) ) ).
% psubset_trans
thf(fact_96_subset__antisym,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% subset_antisym
thf(fact_97_subsetI,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( member @ A @ X4 @ B3 ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% subsetI
thf(fact_98_psubsetI,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% psubsetI
thf(fact_99_finite__Collect__subsets,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( finite_finite2 @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ B5 @ A3 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_100_finite__Collect__le__nat,axiom,
! [K2: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N3: nat] : ( ord_less_eq @ nat @ N3 @ K2 ) ) ) ).
% finite_Collect_le_nat
thf(fact_101_infinite__nat__iff__unbounded__le,axiom,
! [S2: set @ nat] :
( ( ~ ( finite_finite2 @ nat @ S2 ) )
= ( ! [M2: nat] :
? [N3: nat] :
( ( ord_less_eq @ nat @ M2 @ N3 )
& ( member @ nat @ N3 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_102_subset__iff__psubset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A4: set @ A,B5: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_103_subset__psubset__trans,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less @ ( set @ A ) @ B3 @ C )
=> ( ord_less @ ( set @ A ) @ A3 @ C ) ) ) ).
% subset_psubset_trans
thf(fact_104_subset__not__subset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A4: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
& ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_105_psubset__subset__trans,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C )
=> ( ord_less @ ( set @ A ) @ A3 @ C ) ) ) ).
% psubset_subset_trans
thf(fact_106_psubset__imp__subset,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% psubset_imp_subset
thf(fact_107_psubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A4: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_108_psubsetE,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B3 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ).
% psubsetE
thf(fact_109_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q3: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q3 ) )
= ( ! [X3: A] :
( ( P @ X3 )
=> ( Q3 @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_110_contra__subsetD,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ~ ( member @ A @ C2 @ B3 )
=> ~ ( member @ A @ C2 @ A3 ) ) ) ).
% contra_subsetD
thf(fact_111_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y4: set @ A,Z: set @ A] : Y4 = Z )
= ( ^ [A4: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
& ( ord_less_eq @ ( set @ A ) @ B5 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_112_subset__trans,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ C ) ) ) ).
% subset_trans
thf(fact_113_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q3: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
=> ( Q3 @ X4 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q3 ) ) ) ).
% Collect_mono
thf(fact_114_subset__refl,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% subset_refl
thf(fact_115_rev__subsetD,axiom,
! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% rev_subsetD
thf(fact_116_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A4: set @ A,B5: set @ A] :
! [T2: A] :
( ( member @ A @ T2 @ A4 )
=> ( member @ A @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_117_set__rev__mp,axiom,
! [A: $tType,X: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ X @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( member @ A @ X @ B3 ) ) ) ).
% set_rev_mp
thf(fact_118_equalityD2,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( A3 = B3 )
=> ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ).
% equalityD2
thf(fact_119_equalityD1,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( A3 = B3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% equalityD1
thf(fact_120_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A4: set @ A,B5: set @ A] :
! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( member @ A @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_121_equalityE,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( A3 = B3 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ).
% equalityE
thf(fact_122_subsetCE,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% subsetCE
thf(fact_123_subsetD,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% subsetD
thf(fact_124_in__mono,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( member @ A @ X @ A3 )
=> ( member @ A @ X @ B3 ) ) ) ).
% in_mono
thf(fact_125_set__mp,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( member @ A @ X @ A3 )
=> ( member @ A @ X @ B3 ) ) ) ).
% set_mp
thf(fact_126_card__seteq,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ B3 ) @ ( finite_card @ A @ A3 ) )
=> ( A3 = B3 ) ) ) ) ).
% card_seteq
thf(fact_127_card__mono,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ) ).
% card_mono
thf(fact_128_complete__interval,axiom,
! [A: $tType] :
( ( condit1037483654norder @ A @ ( type2 @ A ) )
=> ! [A2: A,B6: A,P: A > $o] :
( ( ord_less @ A @ A2 @ B6 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B6 )
=> ? [C3: A] :
( ( ord_less_eq @ A @ A2 @ C3 )
& ( ord_less_eq @ A @ C3 @ B6 )
& ! [X5: A] :
( ( ( ord_less_eq @ A @ A2 @ X5 )
& ( ord_less @ A @ X5 @ C3 ) )
=> ( P @ X5 ) )
& ! [D: A] :
( ! [X4: A] :
( ( ( ord_less_eq @ A @ A2 @ X4 )
& ( ord_less @ A @ X4 @ D ) )
=> ( P @ X4 ) )
=> ( ord_less_eq @ A @ D @ C3 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_129_finite__subset,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( finite_finite2 @ A @ B3 )
=> ( finite_finite2 @ A @ A3 ) ) ) ).
% finite_subset
thf(fact_130_infinite__super,axiom,
! [A: $tType,S2: set @ A,T3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S2 @ T3 )
=> ( ~ ( finite_finite2 @ A @ S2 )
=> ~ ( finite_finite2 @ A @ T3 ) ) ) ).
% infinite_super
thf(fact_131_rev__finite__subset,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( finite_finite2 @ A @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_132_finite__nat__set__iff__bounded__le,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N5: set @ nat] :
? [M2: nat] :
! [X3: nat] :
( ( member @ nat @ X3 @ N5 )
=> ( ord_less_eq @ nat @ X3 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_133_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N2: nat] : ( ord_less_eq @ nat @ N2 @ ( F @ N2 ) )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N3: nat] : ( ord_less_eq @ nat @ ( F @ N3 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_134_card__subset__eq,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ( finite_card @ A @ A3 )
= ( finite_card @ A @ B3 ) )
=> ( A3 = B3 ) ) ) ) ).
% card_subset_eq
thf(fact_135_infinite__arbitrarily__large,axiom,
! [A: $tType,A3: set @ A,N: nat] :
( ~ ( finite_finite2 @ A @ A3 )
=> ? [B7: set @ A] :
( ( finite_finite2 @ A @ B7 )
& ( ( finite_card @ A @ B7 )
= N )
& ( ord_less_eq @ ( set @ A ) @ B7 @ A3 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_136_psubsetD,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
( ( ord_less @ ( set @ A ) @ A3 @ B3 )
=> ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% psubsetD
thf(fact_137_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_138_ex__has__greatest__nat,axiom,
! [A: $tType,P: A > $o,K2: A,M: A > nat,B6: nat] :
( ( P @ K2 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less @ nat @ ( M @ Y5 ) @ B6 ) )
=> ? [X4: A] :
( ( P @ X4 )
& ! [Y6: A] :
( ( P @ Y6 )
=> ( ord_less_eq @ nat @ ( M @ Y6 ) @ ( M @ X4 ) ) ) ) ) ) ).
% ex_has_greatest_nat
thf(fact_139_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I3: nat,J3: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq @ nat @ I3 @ J3 )
=> ( ord_less_eq @ nat @ ( F @ I3 ) @ ( F @ J3 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_140_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_141_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_142_enat__ord__simps_I5_J,axiom,
! [Q: extended_enat] :
( ( ord_less_eq @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ Q )
= ( Q
= ( extend1396239628finity @ extended_enat ) ) ) ).
% enat_ord_simps(5)
thf(fact_143_enat__ord__code_I3_J,axiom,
! [Q: extended_enat] : ( ord_less_eq @ extended_enat @ Q @ ( extend1396239628finity @ extended_enat ) ) ).
% enat_ord_code(3)
thf(fact_144_enat__ord__simps_I1_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% enat_ord_simps(1)
thf(fact_145_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F4: A > B,G2: A > B] :
( ( ord_less_eq @ ( A > B ) @ F4 @ G2 )
& ~ ( ord_less_eq @ ( A > B ) @ G2 @ F4 ) ) ) ) ) ).
% less_fun_def
thf(fact_146_finite__enat__bounded,axiom,
! [A3: set @ extended_enat,N: nat] :
( ! [Y5: extended_enat] :
( ( member @ extended_enat @ Y5 @ A3 )
=> ( ord_less_eq @ extended_enat @ Y5 @ ( extended_enat2 @ N ) ) )
=> ( finite_finite2 @ extended_enat @ A3 ) ) ).
% finite_enat_bounded
thf(fact_147_enat__ile,axiom,
! [N: extended_enat,M: nat] :
( ( ord_less_eq @ extended_enat @ N @ ( extended_enat2 @ M ) )
=> ? [K: nat] :
( N
= ( extended_enat2 @ K ) ) ) ).
% enat_ile
thf(fact_148_enat__ord__simps_I3_J,axiom,
! [Q: extended_enat] : ( ord_less_eq @ extended_enat @ Q @ ( extend1396239628finity @ extended_enat ) ) ).
% enat_ord_simps(3)
thf(fact_149_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A4: set @ A,B5: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ A4 )
@ ^ [X3: A] : ( member @ A @ X3 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_150_infinity__ileE,axiom,
! [M: nat] :
~ ( ord_less_eq @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ ( extended_enat2 @ M ) ) ).
% infinity_ileE
thf(fact_151_enat__ord__code_I5_J,axiom,
! [N: nat] :
~ ( ord_less_eq @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ ( extended_enat2 @ N ) ) ).
% enat_ord_code(5)
thf(fact_152_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_153_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_154_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B] :
( ! [X4: A] : ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_155_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F4: A > B,G2: A > B] :
! [X3: A] : ( ord_less_eq @ B @ ( F4 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% le_fun_def
thf(fact_156_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B6: B,C2: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B6 ) )
=> ( ( ord_less_eq @ B @ B6 @ C2 )
=> ( ! [X4: B,Y5: B] :
( ( ord_less_eq @ B @ X4 @ Y5 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_157_order__subst2,axiom,
! [A: $tType,C4: $tType] :
( ( ( order @ C4 @ ( type2 @ C4 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B6: A,F: A > C4,C2: C4] :
( ( ord_less_eq @ A @ A2 @ B6 )
=> ( ( ord_less_eq @ C4 @ ( F @ B6 ) @ C2 )
=> ( ! [X4: A,Y5: A] :
( ( ord_less_eq @ A @ X4 @ Y5 )
=> ( ord_less_eq @ C4 @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq @ C4 @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_158_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B6: B,C2: B] :
( ( A2
= ( F @ B6 ) )
=> ( ( ord_less_eq @ B @ B6 @ C2 )
=> ( ! [X4: B,Y5: B] :
( ( ord_less_eq @ B @ X4 @ Y5 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_159_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B6: A,F: A > B,C2: B] :
( ( ord_less_eq @ A @ A2 @ B6 )
=> ( ( ( F @ B6 )
= C2 )
=> ( ! [X4: A,Y5: A] :
( ( ord_less_eq @ A @ X4 @ Y5 )
=> ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_160_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
& ( ord_less_eq @ A @ Y @ X3 ) ) ) ) ) ).
% eq_iff
thf(fact_161_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ X )
=> ( X = Y2 ) ) ) ) ).
% antisym
thf(fact_162_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
| ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% linear
thf(fact_163_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( X = Y2 )
=> ( ord_less_eq @ A @ X @ Y2 ) ) ) ).
% eq_refl
thf(fact_164_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ~ ( ord_less_eq @ A @ X @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% le_cases
thf(fact_165_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B6: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B6 )
=> ( ( ord_less_eq @ A @ B6 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% order.trans
thf(fact_166_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z2: A] :
( ( ( ord_less_eq @ A @ X @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ X )
=> ~ ( ord_less_eq @ A @ X @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ X ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y2 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_167_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y2: A,X: A] :
( ( ord_less_eq @ A @ Y2 @ X )
=> ( ( ord_less_eq @ A @ X @ Y2 )
= ( X = Y2 ) ) ) ) ).
% antisym_conv
thf(fact_168_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B6: A,C2: A] :
( ( A2 = B6 )
=> ( ( ord_less_eq @ A @ B6 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_169_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B6: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B6 )
=> ( ( B6 = C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_170_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B6: A] :
( ( ord_less_eq @ A @ A2 @ B6 )
=> ( ( ord_less_eq @ A @ B6 @ A2 )
=> ( A2 = B6 ) ) ) ) ).
% order_class.order.antisym
thf(fact_171_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z2 )
=> ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% order_trans
thf(fact_172_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_173_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A2: A,B6: A] :
( ! [A7: A,B2: A] :
( ( ord_less_eq @ A @ A7 @ B2 )
=> ( P @ A7 @ B2 ) )
=> ( ! [A7: A,B2: A] :
( ( P @ B2 @ A7 )
=> ( P @ A7 @ B2 ) )
=> ( P @ A2 @ B6 ) ) ) ) ).
% linorder_wlog
thf(fact_174_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B6: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B6 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B6 )
=> ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_175_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B6: A,A2: A] :
( ( ord_less_eq @ A @ B6 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B6 )
=> ( A2 = B6 ) ) ) ) ).
% dual_order.antisym
thf(fact_176_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B6: A,A2: A] :
( ( ord_less @ A @ B6 @ A2 )
=> ( A2 != B6 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_177_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B6: A] :
( ( ord_less @ A @ A2 @ B6 )
=> ( A2 != B6 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_178_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ~ ( ord_less @ A @ X @ Y2 ) )
= ( ( ord_less @ A @ Y2 @ X )
| ( X = Y2 ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_179_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B6: A,A2: A,C2: A] :
( ( ord_less @ A @ B6 @ A2 )
=> ( ( ord_less @ A @ C2 @ B6 )
=> ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_180_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X ) ) ) ).
% less_imp_not_less
thf(fact_181_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B6: A,C2: A] :
( ( ord_less @ A @ A2 @ B6 )
=> ( ( ord_less @ A @ B6 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% order.strict_trans
thf(fact_182_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_183_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ~ ( ord_less @ A @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% linorder_cases
thf(fact_184_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,P: $o] :
( ( ord_less @ A @ X @ Y2 )
=> ( ( ord_less @ A @ Y2 @ X )
=> P ) ) ) ).
% less_imp_triv
thf(fact_185_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( Y2 != X ) ) ) ).
% less_imp_not_eq2
thf(fact_186_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y2: A,X: A] :
( ~ ( ord_less @ A @ Y2 @ X )
=> ( ( ~ ( ord_less @ A @ X @ Y2 ) )
= ( X = Y2 ) ) ) ) ).
% antisym_conv3
thf(fact_187_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,A2: A] :
( ! [X4: A] :
( ! [Y6: A] :
( ( ord_less @ A @ Y6 @ X4 )
=> ( P @ Y6 ) )
=> ( P @ X4 ) )
=> ( P @ A2 ) ) ) ).
% less_induct
thf(fact_188_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X ) ) ) ).
% less_not_sym
thf(fact_189_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( X != Y2 ) ) ) ).
% less_imp_not_eq
thf(fact_190_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B6: A,A2: A] :
( ( ord_less @ A @ B6 @ A2 )
=> ~ ( ord_less @ A @ A2 @ B6 ) ) ) ).
% dual_order.asym
thf(fact_191_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B6: A,C2: A] :
( ( ord_less @ A @ A2 @ B6 )
=> ( ( B6 = C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_192_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B6: A,C2: A] :
( ( A2 = B6 )
=> ( ( ord_less @ A @ B6 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_193_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% less_irrefl
thf(fact_194_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
| ( X = Y2 )
| ( ord_less @ A @ Y2 @ X ) ) ) ).
% less_linear
thf(fact_195_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ( ord_less @ A @ Y2 @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% less_trans
thf(fact_196_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B6: A] :
( ( ord_less @ A @ A2 @ B6 )
=> ~ ( ord_less @ A @ B6 @ A2 ) ) ) ).
% less_asym'
thf(fact_197_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X ) ) ) ).
% less_asym
thf(fact_198_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( X != Y2 ) ) ) ).
% less_imp_neq
thf(fact_199_dense,axiom,
! [A: $tType] :
( ( dense_order @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ? [Z3: A] :
( ( ord_less @ A @ X @ Z3 )
& ( ord_less @ A @ Z3 @ Y2 ) ) ) ) ).
% dense
thf(fact_200_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B6: A] :
( ( ord_less @ A @ A2 @ B6 )
=> ~ ( ord_less @ A @ B6 @ A2 ) ) ) ).
% order.asym
thf(fact_201_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( X != Y2 )
= ( ( ord_less @ A @ X @ Y2 )
| ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% neq_iff
thf(fact_202_neqE,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( X != Y2 )
=> ( ~ ( ord_less @ A @ X @ Y2 )
=> ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% neqE
thf(fact_203_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% gt_ex
thf(fact_204_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [Y5: A] : ( ord_less @ A @ Y5 @ X ) ) ).
% lt_ex
thf(fact_205_order__less__subst2,axiom,
! [A: $tType,C4: $tType] :
( ( ( order @ C4 @ ( type2 @ C4 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B6: A,F: A > C4,C2: C4] :
( ( ord_less @ A @ A2 @ B6 )
=> ( ( ord_less @ C4 @ ( F @ B6 ) @ C2 )
=> ( ! [X4: A,Y5: A] :
( ( ord_less @ A @ X4 @ Y5 )
=> ( ord_less @ C4 @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less @ C4 @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% order_less_subst2
thf(fact_206_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B6: B,C2: B] :
( ( ord_less @ A @ A2 @ ( F @ B6 ) )
=> ( ( ord_less @ B @ B6 @ C2 )
=> ( ! [X4: B,Y5: B] :
( ( ord_less @ B @ X4 @ Y5 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_207_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B6: A,F: A > B,C2: B] :
( ( ord_less @ A @ A2 @ B6 )
=> ( ( ( F @ B6 )
= C2 )
=> ( ! [X4: A,Y5: A] :
( ( ord_less @ A @ X4 @ Y5 )
=> ( ord_less @ B @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_208_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B6: B,C2: B] :
( ( A2
= ( F @ B6 ) )
=> ( ( ord_less @ B @ B6 @ C2 )
=> ( ! [X4: B,Y5: B] :
( ( ord_less @ B @ X4 @ Y5 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_209_less__eq__enat__def,axiom,
( ( ord_less_eq @ extended_enat )
= ( ^ [M2: extended_enat] :
( extended_case_enat @ $o
@ ^ [N1: nat] :
( extended_case_enat @ $o
@ ^ [M1: nat] : ( ord_less_eq @ nat @ M1 @ N1 )
@ $false
@ M2 )
@ $true ) ) ) ).
% less_eq_enat_def
thf(fact_210_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V: A > nat,X: A] :
( ! [X4: A] :
( ~ ( P @ X4 )
=> ? [Y6: A] :
( ( ord_less @ nat @ ( V @ Y6 ) @ ( V @ X4 ) )
& ~ ( P @ Y6 ) ) )
=> ( P @ X ) ) ).
% infinite_descent_measure
thf(fact_211_measure__induct__rule,axiom,
! [A: $tType,F: A > nat,P: A > $o,A2: A] :
( ! [X4: A] :
( ! [Y6: A] :
( ( ord_less @ nat @ ( F @ Y6 ) @ ( F @ X4 ) )
=> ( P @ Y6 ) )
=> ( P @ X4 ) )
=> ( P @ A2 ) ) ).
% measure_induct_rule
thf(fact_212_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_213_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N2 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_214_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less @ nat @ M3 @ N2 )
=> ( P @ M3 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_215_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_216_measure__induct,axiom,
! [A: $tType,F: A > nat,P: A > $o,A2: A] :
( ! [X4: A] :
( ! [Y6: A] :
( ( ord_less @ nat @ ( F @ Y6 ) @ ( F @ X4 ) )
=> ( P @ Y6 ) )
=> ( P @ X4 ) )
=> ( P @ A2 ) ) ).
% measure_induct
thf(fact_217_less__not__refl3,axiom,
! [S3: nat,T4: nat] :
( ( ord_less @ nat @ S3 @ T4 )
=> ( S3 != T4 ) ) ).
% less_not_refl3
thf(fact_218_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_219_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_220_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_221_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_222_le__trans,axiom,
! [I3: nat,J3: nat,K2: nat] :
( ( ord_less_eq @ nat @ I3 @ J3 )
=> ( ( ord_less_eq @ nat @ J3 @ K2 )
=> ( ord_less_eq @ nat @ I3 @ K2 ) ) ) ).
% le_trans
thf(fact_223_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_224_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_225_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_226_ex__has__least__nat,axiom,
! [A: $tType,P: A > $o,K2: A,M: A > nat] :
( ( P @ K2 )
=> ? [X4: A] :
( ( P @ X4 )
& ! [Y6: A] :
( ( P @ Y6 )
=> ( ord_less_eq @ nat @ ( M @ X4 ) @ ( M @ Y6 ) ) ) ) ) ).
% ex_has_least_nat
thf(fact_227_order_Onot__eq__order__implies__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B6: A] :
( ( A2 != B6 )
=> ( ( ord_less_eq @ A @ A2 @ B6 )
=> ( ord_less @ A @ A2 @ B6 ) ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_228_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B6: A,A2: A] :
( ( ord_less @ A @ B6 @ A2 )
=> ( ord_less_eq @ A @ B6 @ A2 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_229_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [B8: A,A6: A] :
( ( ord_less_eq @ A @ B8 @ A6 )
& ( A6 != B8 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_230_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [B8: A,A6: A] :
( ( ord_less @ A @ B8 @ A6 )
| ( A6 = B8 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_231_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B6: A] :
( ( ord_less @ A @ A2 @ B6 )
=> ( ord_less_eq @ A @ A2 @ B6 ) ) ) ).
% order.strict_implies_order
thf(fact_232_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ! [W: A] :
( ( ord_less @ A @ X @ W )
=> ( ( ord_less @ A @ W @ Y2 )
=> ( ord_less_eq @ A @ W @ Z2 ) ) )
=> ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% dense_le_bounded
thf(fact_233_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z2: A,X: A,Y2: A] :
( ( ord_less @ A @ Z2 @ X )
=> ( ! [W: A] :
( ( ord_less @ A @ Z2 @ W )
=> ( ( ord_less @ A @ W @ X )
=> ( ord_less_eq @ A @ Y2 @ W ) ) )
=> ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% dense_ge_bounded
thf(fact_234_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B6: A,A2: A,C2: A] :
( ( ord_less @ A @ B6 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B6 )
=> ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_235_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B6: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B6 @ A2 )
=> ( ( ord_less @ A @ C2 @ B6 )
=> ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_236_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [A6: A,B8: A] :
( ( ord_less_eq @ A @ A6 @ B8 )
& ( A6 != B8 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_237_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B8: A] :
( ( ord_less @ A @ A6 @ B8 )
| ( A6 = B8 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_238_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B6: A,C2: A] :
( ( ord_less @ A @ A2 @ B6 )
=> ( ( ord_less_eq @ A @ B6 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% order.strict_trans2
thf(fact_239_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B6: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B6 )
=> ( ( ord_less @ A @ B6 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% order.strict_trans1
thf(fact_240_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y2: A,X: A] :
( ~ ( ord_less_eq @ A @ Y2 @ X )
=> ( ord_less @ A @ X @ Y2 ) ) ) ).
% not_le_imp_less
thf(fact_241_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
& ~ ( ord_less_eq @ A @ Y @ X3 ) ) ) ) ) ).
% less_le_not_le
thf(fact_242_le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less @ A @ X @ Y2 )
| ( X = Y2 ) ) ) ) ).
% le_imp_less_or_eq
thf(fact_243_le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
| ( ord_less @ A @ Y2 @ X ) ) ) ).
% le_less_linear
thf(fact_244_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Y2: A,Z2: A] :
( ! [X4: A] :
( ( ord_less @ A @ X4 @ Y2 )
=> ( ord_less_eq @ A @ X4 @ Z2 ) )
=> ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% dense_le
thf(fact_245_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z2: A,Y2: A] :
( ! [X4: A] :
( ( ord_less @ A @ Z2 @ X4 )
=> ( ord_less_eq @ A @ Y2 @ X4 ) )
=> ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% dense_ge
thf(fact_246_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% less_le_trans
thf(fact_247_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less @ A @ Y2 @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% le_less_trans
thf(fact_248_antisym__conv2,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ~ ( ord_less @ A @ X @ Y2 ) )
= ( X = Y2 ) ) ) ) ).
% antisym_conv2
thf(fact_249_antisym__conv1,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ~ ( ord_less @ A @ X @ Y2 )
=> ( ( ord_less_eq @ A @ X @ Y2 )
= ( X = Y2 ) ) ) ) ).
% antisym_conv1
thf(fact_250_less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ord_less_eq @ A @ X @ Y2 ) ) ) ).
% less_imp_le
thf(fact_251_le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B6: A] :
( ( ord_less_eq @ A @ A2 @ B6 )
=> ( ( A2 != B6 )
=> ( ord_less @ A @ A2 @ B6 ) ) ) ) ).
% le_neq_trans
thf(fact_252_not__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ~ ( ord_less @ A @ X @ Y2 ) )
= ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% not_less
thf(fact_253_not__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ~ ( ord_less_eq @ A @ X @ Y2 ) )
= ( ord_less @ A @ Y2 @ X ) ) ) ).
% not_le
thf(fact_254_order__less__le__subst2,axiom,
! [A: $tType,C4: $tType] :
( ( ( order @ C4 @ ( type2 @ C4 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B6: A,F: A > C4,C2: C4] :
( ( ord_less @ A @ A2 @ B6 )
=> ( ( ord_less_eq @ C4 @ ( F @ B6 ) @ C2 )
=> ( ! [X4: A,Y5: A] :
( ( ord_less @ A @ X4 @ Y5 )
=> ( ord_less @ C4 @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less @ C4 @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_255_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B6: B,C2: B] :
( ( ord_less @ A @ A2 @ ( F @ B6 ) )
=> ( ( ord_less_eq @ B @ B6 @ C2 )
=> ( ! [X4: B,Y5: B] :
( ( ord_less_eq @ B @ X4 @ Y5 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y5 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% order_less_le_subst1
%----Type constructors (26)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A9: $tType] :
( ( preorder @ A9 @ ( type2 @ A9 ) )
=> ( preorder @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A8: $tType,A9: $tType] :
( ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
& ( finite_finite @ A9 @ ( type2 @ A9 ) ) )
=> ( finite_finite @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A9: $tType] :
( ( order @ A9 @ ( type2 @ A9 ) )
=> ( order @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 @ ( type2 @ A9 ) )
=> ( ord @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit1037483654norder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
preorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
no_top @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
ord @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_5,axiom,
! [A8: $tType] :
( ( finite_finite @ A8 @ ( type2 @ A8 ) )
=> ( finite_finite @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_7,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_8,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_9,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_10,axiom,
finite_finite @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_11,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_12,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_13,axiom,
condit1037483654norder @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Owellorder_14,axiom,
wellorder @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_15,axiom,
preorder @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_16,axiom,
linorder @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_17,axiom,
order @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oord_18,axiom,
ord @ extended_enat @ ( type2 @ extended_enat ) ).
%----Conjectures (2)
thf(conj_0,hypothesis,
( linear505997466_on_ev @ a
@ ( linear1386806755on_alw @ a
@ ^ [Xs: stream @ a] :
~ ( p @ Xs ) )
@ omega ) ).
thf(conj_1,conjecture,
ord_less @ extended_enat @ ( coindu1365464361scount @ a @ p @ omega ) @ ( extend1396239628finity @ extended_enat ) ).
%------------------------------------------------------------------------------