TPTP Problem File: ITP147^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP147^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Poincare_Bendixson problem prob_2118__19618702_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Poincare_Bendixson/prob_2118__19618702_1 [Des21]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.00 v7.5.0
% Syntax : Number of formulae : 386 ( 60 unt; 66 typ; 0 def)
% Number of atoms : 926 ( 300 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 3917 ( 53 ~; 6 |; 28 &;3425 @)
% ( 0 <=>; 405 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 7 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 184 ( 184 >; 0 *; 0 +; 0 <<)
% Number of symbols : 67 ( 64 usr; 8 con; 0-5 aty)
% Number of variables : 723 ( 8 ^; 656 !; 3 ?; 723 :)
% ( 56 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:34:48.401
%------------------------------------------------------------------------------
% Could-be-implicit typings (3)
thf(ty_t_Real_Oreal,type,
real: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (63)
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
thf(sy_cl_Countable_Ocountable,type,
countable:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oabs__if,type,
abs_if:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__abs__sgn,type,
idom_abs_sgn:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__group__add,type,
ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__less__one,type,
zero_less_one:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring,type,
linordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
ordere216010020id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add__abs,type,
ordere142940540dd_abs:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Ometric__space,type,
real_V2090557954_space:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord1659791738miring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add,type,
ordere1818651114id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere223160158up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
ordere516151231imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Executable__Euclidean__Space_Oexecutable__euclidean__space,type,
execut510477386_space:
!>[A: $tType] : $o ).
thf(sy_c_Elementary__Metric__Spaces_Oball,type,
elemen1815086676c_ball:
!>[A: $tType] : ( A > real > ( set @ A ) ) ).
thf(sy_c_Flow_Oauto__ll__on__open_Oexistence__ivl0,type,
auto_l1112008849e_ivl0:
!>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > ( set @ real ) ) ).
thf(sy_c_Flow_Oauto__ll__on__open_Oflow0,type,
auto_ll_on_flow0:
!>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > real > A ) ).
thf(sy_c_Groups_Oabs__class_Oabs,type,
abs_abs:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Initial__Value__Problem_Ointerval,type,
initia826609931terval: ( set @ real ) > $o ).
thf(sy_c_Limit__Set_Oauto__ll__on__open_O_092_060alpha_062__limit__point,type,
limit_1335059408_point:
!>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > A > $o ) ).
thf(sy_c_Limit__Set_Oauto__ll__on__open_O_092_060omega_062__limit__point,type,
limit_1052139091_point:
!>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > A > $o ) ).
thf(sy_c_Line__Segment_Oopen__segment,type,
line_open_segment:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
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_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Oclosed__orbit,type,
period385816147_orbit:
!>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > $o ) ).
thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Operiod,type,
period1153813292period:
!>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > real ) ).
thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Operiodic__orbit,type,
period862636932_orbit:
!>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > $o ) ).
thf(sy_c_Poincare__Bendixson__Mirabelle__helaxgvbop_Oc1__on__open__R2_Otransversal__segment,type,
poinca272511729egment:
!>[A: $tType] : ( ( A > A ) > ( set @ A ) > A > A > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Topological__Spaces_Ocontinuous__on,type,
topolo2071040574ous_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_X,type,
x: set @ a ).
thf(sy_v_a,type,
a2: a ).
thf(sy_v_b,type,
b: a ).
thf(sy_v_f,type,
f: a > a ).
thf(sy_v_p,type,
p: a ).
thf(sy_v_thesisa____,type,
thesisa: $o ).
thf(sy_v_x,type,
x2: a ).
% Relevant facts (254)
thf(fact_0_rev_Oopen__segment__trichotomy,axiom,
! [X: a,A2: a,B2: a,Y: a] :
( ( member @ a @ X @ ( line_open_segment @ a @ A2 @ B2 ) )
=> ( ( member @ a @ Y @ ( line_open_segment @ a @ A2 @ B2 ) )
=> ( ( X = Y )
| ( member @ a @ Y @ ( line_open_segment @ a @ X @ B2 ) )
| ( member @ a @ Y @ ( line_open_segment @ a @ A2 @ X ) ) ) ) ) ).
% rev.open_segment_trichotomy
thf(fact_1__092_060open_062p_A_092_060in_062_AX_092_060close_062,axiom,
member @ a @ p @ x ).
% \<open>p \<in> X\<close>
thf(fact_2_assms_I4_J,axiom,
member @ a @ p @ ( line_open_segment @ a @ a2 @ b ) ).
% assms(4)
thf(fact_3__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062d_At_O_A_092_060lbrakk_0620_A_060_Ad_059_Acontinuous__on_A_Iball_Ap_Ad_J_At_059_A_092_060And_062y_O_Ay_A_092_060in_062_Aball_Ap_Ad_A_092_060Longrightarrow_062_Aflow0_Ay_A_It_Ay_J_A_092_060in_062_A_123a_060_N_N_060b_125_059_A_092_060And_062y_O_Ay_A_092_060in_062_Aball_Ap_Ad_A_092_060Longrightarrow_062_A_092_060bar_062t_Ay_092_060bar_062_A_060_A1_059_Acontinuous__on_A_Iball_Ap_Ad_J_At_059_At_Ap_A_061_A0_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [D: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D )
=> ! [T: a > real] :
( ( topolo2071040574ous_on @ a @ real @ ( elemen1815086676c_ball @ a @ p @ D ) @ T )
=> ( ! [Y2: a] :
( ( member @ a @ Y2 @ ( elemen1815086676c_ball @ a @ p @ D ) )
=> ( member @ a @ ( auto_ll_on_flow0 @ a @ f @ x @ Y2 @ ( T @ Y2 ) ) @ ( line_open_segment @ a @ a2 @ b ) ) )
=> ( ! [Y2: a] :
( ( member @ a @ Y2 @ ( elemen1815086676c_ball @ a @ p @ D ) )
=> ( ord_less @ real @ ( abs_abs @ real @ ( T @ Y2 ) ) @ ( one_one @ real ) ) )
=> ( ( topolo2071040574ous_on @ a @ real @ ( elemen1815086676c_ball @ a @ p @ D ) @ T )
=> ( ( T @ p )
!= ( zero_zero @ real ) ) ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>d t. \<lbrakk>0 < d; continuous_on (ball p d) t; \<And>y. y \<in> ball p d \<Longrightarrow> flow0 y (t y) \<in> {a<--<b}; \<And>y. y \<in> ball p d \<Longrightarrow> \<bar>t y\<bar> < 1; continuous_on (ball p d) t; t p = 0\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_4_that,axiom,
! [Delta: real,Tau: a > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Delta )
=> ( ( topolo2071040574ous_on @ a @ real @ ( elemen1815086676c_ball @ a @ p @ Delta ) @ Tau )
=> ( ( ( Tau @ p )
= ( zero_zero @ real ) )
=> ( ! [Y3: a] :
( ( member @ a @ Y3 @ ( elemen1815086676c_ball @ a @ p @ Delta ) )
=> ( ord_less @ real @ ( abs_abs @ real @ ( Tau @ Y3 ) ) @ ( one_one @ real ) ) )
=> ( ! [Y3: a] :
( ( member @ a @ Y3 @ ( elemen1815086676c_ball @ a @ p @ Delta ) )
=> ( member @ a @ ( auto_ll_on_flow0 @ a @ f @ x @ Y3 @ ( Tau @ Y3 ) ) @ ( line_open_segment @ a @ a2 @ b ) ) )
=> thesisa ) ) ) ) ) ).
% that
thf(fact_5_assms_I1_J,axiom,
poinca272511729egment @ a @ f @ x @ a2 @ b ).
% assms(1)
thf(fact_6_assms_I3_J,axiom,
limit_1052139091_point @ a @ f @ x @ x2 @ p ).
% assms(3)
thf(fact_7_centre__in__ball,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [X: A,E: real] :
( ( member @ A @ X @ ( elemen1815086676c_ball @ A @ X @ E ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ E ) ) ) ).
% centre_in_ball
thf(fact_8_zero__less__abs__iff,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A2 ) )
= ( A2
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_abs_iff
thf(fact_9_cross__time__continuous,axiom,
! [A2: a,B2: a,X: a,E: real] :
( ( poinca272511729egment @ a @ f @ x @ A2 @ B2 )
=> ( ( member @ a @ X @ ( line_open_segment @ a @ A2 @ B2 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ~ ! [D: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D )
=> ! [T: a > real] :
( ( topolo2071040574ous_on @ a @ real @ ( elemen1815086676c_ball @ a @ X @ D ) @ T )
=> ( ! [Y2: a] :
( ( member @ a @ Y2 @ ( elemen1815086676c_ball @ a @ X @ D ) )
=> ( member @ a @ ( auto_ll_on_flow0 @ a @ f @ x @ Y2 @ ( T @ Y2 ) ) @ ( line_open_segment @ a @ A2 @ B2 ) ) )
=> ( ! [Y2: a] :
( ( member @ a @ Y2 @ ( elemen1815086676c_ball @ a @ X @ D ) )
=> ( ord_less @ real @ ( abs_abs @ real @ ( T @ Y2 ) ) @ E ) )
=> ( ( topolo2071040574ous_on @ a @ real @ ( elemen1815086676c_ball @ a @ X @ D ) @ T )
=> ( ( T @ X )
!= ( zero_zero @ real ) ) ) ) ) ) ) ) ) ) ).
% cross_time_continuous
thf(fact_10_abs__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_1
thf(fact_11_abs__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_0
thf(fact_12_abs__0__eq,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( abs_abs @ A @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% abs_0_eq
thf(fact_13_abs__eq__0,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A] :
( ( ( abs_abs @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% abs_eq_0
thf(fact_14_abs__zero,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_zero
thf(fact_15_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_16_fixpoint__sol_I2_J,axiom,
! [X: a,T2: real] :
( ( member @ a @ X @ x )
=> ( ( ( f @ X )
= ( zero_zero @ a ) )
=> ( ( auto_ll_on_flow0 @ a @ f @ x @ X @ T2 )
= X ) ) ) ).
% fixpoint_sol(2)
thf(fact_17_gt__one__absI,axiom,
! [K: real] :
( ( ord_less @ real @ ( abs_abs @ real @ K ) @ ( one_one @ real ) )
=> ( ord_less @ real @ K @ ( one_one @ real ) ) ) ).
% gt_one_absI
thf(fact_18_transversal__segment__reverse,axiom,
! [X: a,Y: a] :
( ( poinca272511729egment @ a @ f @ x @ X @ Y )
=> ( poinca272511729egment @ a @ f @ x @ Y @ X ) ) ).
% transversal_segment_reverse
thf(fact_19_transversal__segment__commute,axiom,
! [X: a,Y: a] :
( ( poinca272511729egment @ a @ f @ x @ X @ Y )
= ( poinca272511729egment @ a @ f @ x @ Y @ X ) ) ).
% transversal_segment_commute
thf(fact_20_abs__idempotent,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A] :
( ( abs_abs @ A @ ( abs_abs @ A @ A2 ) )
= ( abs_abs @ A @ A2 ) ) ) ).
% abs_idempotent
thf(fact_21_abs__abs,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( abs_abs @ A @ ( abs_abs @ A @ A2 ) )
= ( abs_abs @ A @ A2 ) ) ) ).
% abs_abs
thf(fact_22_transversal__segment__exists,axiom,
! [X: a] :
( ( member @ a @ X @ x )
=> ( ( ( f @ X )
!= ( zero_zero @ a ) )
=> ~ ! [A3: a,B3: a] :
( ( member @ a @ X @ ( line_open_segment @ a @ A3 @ B3 ) )
=> ~ ( poinca272511729egment @ a @ f @ x @ A3 @ B3 ) ) ) ) ).
% transversal_segment_exists
thf(fact_23_c1__on__open__R2_Otransversal__segment_Ocong,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( ( poinca272511729egment @ A )
= ( poinca272511729egment @ A ) ) ) ).
% c1_on_open_R2.transversal_segment.cong
thf(fact_24_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_25_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_26_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X: A] :
( ( ( one_one @ A )
= X )
= ( X
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_27_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ N )
= ( N
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_28_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [M: A,N: A] :
( ( ord_less @ A @ M @ N )
=> ( N
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_29_not__less__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_30_gr__zeroI,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( N
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% gr_zeroI
thf(fact_31_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_32_abs__eq__0__iff,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( ( abs_abs @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% abs_eq_0_iff
thf(fact_33_abs__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_one
thf(fact_34_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_35_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_36_abs__not__less__zero,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ ( abs_abs @ A @ A2 ) @ ( zero_zero @ A ) ) ) ).
% abs_not_less_zero
thf(fact_37_abs__of__pos,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( abs_abs @ A @ A2 )
= A2 ) ) ) ).
% abs_of_pos
thf(fact_38_periodic__orbit__imp__flow0__regular,axiom,
! [X: a,T2: real] :
( ( period862636932_orbit @ a @ f @ x @ X )
=> ( ( f @ ( auto_ll_on_flow0 @ a @ f @ x @ X @ T2 ) )
!= ( zero_zero @ a ) ) ) ).
% periodic_orbit_imp_flow0_regular
thf(fact_39_fixed__point__imp__closed__orbit__period__zero_I2_J,axiom,
! [X: a] :
( ( member @ a @ X @ x )
=> ( ( ( f @ X )
= ( zero_zero @ a ) )
=> ( ( period1153813292period @ a @ f @ x @ X )
= ( zero_zero @ real ) ) ) ) ).
% fixed_point_imp_closed_orbit_period_zero(2)
thf(fact_40_local_Oflow__initial__time__if,axiom,
! [X0: a] :
( ( ( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
& ( member @ a @ X0 @ x ) )
=> ( ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ ( zero_zero @ real ) )
= X0 ) )
& ( ~ ( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
& ( member @ a @ X0 @ x ) )
=> ( ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ ( zero_zero @ real ) )
= ( zero_zero @ a ) ) ) ) ).
% local.flow_initial_time_if
thf(fact_41_rev_Otransversal__segment__exists,axiom,
! [X: a] :
( ( member @ a @ X @ x )
=> ( ( ( uminus_uminus @ ( a > a ) @ f @ X )
!= ( zero_zero @ a ) )
=> ~ ! [A3: a,B3: a] :
( ( member @ a @ X @ ( line_open_segment @ a @ A3 @ B3 ) )
=> ~ ( poinca272511729egment @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ A3 @ B3 ) ) ) ) ).
% rev.transversal_segment_exists
thf(fact_42_rev_Ofixpoint__sol_I2_J,axiom,
! [X: a,T2: real] :
( ( member @ a @ X @ x )
=> ( ( ( uminus_uminus @ ( a > a ) @ f @ X )
= ( zero_zero @ a ) )
=> ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ T2 )
= X ) ) ) ).
% rev.fixpoint_sol(2)
thf(fact_43_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_45_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_46_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_47_local_Oflow__undefined0,axiom,
! [T2: real,X: a] :
( ~ ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) )
=> ( ( auto_ll_on_flow0 @ a @ f @ x @ X @ T2 )
= ( zero_zero @ a ) ) ) ).
% local.flow_undefined0
thf(fact_48_rev__transversal__segment,axiom,
! [A2: a,B2: a] :
( ( poinca272511729egment @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ A2 @ B2 )
= ( poinca272511729egment @ a @ f @ x @ A2 @ B2 ) ) ).
% rev_transversal_segment
thf(fact_49_rev_Otransversal__segment__reverse,axiom,
! [X: a,Y: a] :
( ( poinca272511729egment @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ Y )
=> ( poinca272511729egment @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y @ X ) ) ).
% rev.transversal_segment_reverse
thf(fact_50_rev_Otransversal__segment__commute,axiom,
! [X: a,Y: a] :
( ( poinca272511729egment @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ Y )
= ( poinca272511729egment @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y @ X ) ) ).
% rev.transversal_segment_commute
thf(fact_51_fixed__point__imp__closed__orbit__period__zero_I1_J,axiom,
! [X: a] :
( ( member @ a @ X @ x )
=> ( ( ( f @ X )
= ( zero_zero @ a ) )
=> ( period385816147_orbit @ a @ f @ x @ X ) ) ) ).
% fixed_point_imp_closed_orbit_period_zero(1)
thf(fact_52_existence__ivl0__cong,axiom,
! [B: $tType,Y4: set @ a,G: a > a,X0: a] :
( ( x = Y4 )
=> ( ! [X3: a,T: B] :
( ( member @ a @ X3 @ Y4 )
=> ( ( f @ X3 )
= ( G @ X3 ) ) )
=> ( ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 )
= ( auto_l1112008849e_ivl0 @ a @ G @ Y4 @ X0 ) ) ) ) ).
% existence_ivl0_cong
thf(fact_53_closed__orbit__in__domain,axiom,
! [X: a] :
( ( period385816147_orbit @ a @ f @ x @ X )
=> ( member @ a @ X @ x ) ) ).
% closed_orbit_in_domain
thf(fact_54_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
= A2 ) ) ).
% add.inverse_inverse
thf(fact_55_neg__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( ( uminus_uminus @ A @ A2 )
= ( uminus_uminus @ A @ B2 ) )
= ( A2 = B2 ) ) ) ).
% neg_equal_iff_equal
thf(fact_56_existence__ivl__zero,axiom,
! [X0: a] :
( ( member @ a @ X0 @ x )
=> ( member @ real @ ( zero_zero @ real ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) ) ) ).
% existence_ivl_zero
thf(fact_57_rev_Oexistence__ivl0__cong,axiom,
! [B: $tType,Y4: set @ a,G: a > a,X0: a] :
( ( x = Y4 )
=> ( ! [X3: a,T: B] :
( ( member @ a @ X3 @ Y4 )
=> ( ( uminus_uminus @ ( a > a ) @ f @ X3 )
= ( G @ X3 ) ) )
=> ( ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 )
= ( auto_l1112008849e_ivl0 @ a @ G @ Y4 @ X0 ) ) ) ) ).
% rev.existence_ivl0_cong
thf(fact_58_local_Omem__existence__ivl__subset,axiom,
! [T2: real,X0: a] :
( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( member @ real @ T2 @ ( top_top @ ( set @ real ) ) ) ) ).
% local.mem_existence_ivl_subset
thf(fact_59_flow0__cong,axiom,
! [B: $tType,Y4: set @ a,G: a > a,T2: real,X0: a] :
( ( x = Y4 )
=> ( ! [X3: a,T: B] :
( ( member @ a @ X3 @ Y4 )
=> ( ( f @ X3 )
= ( G @ X3 ) ) )
=> ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ T2 )
= ( auto_ll_on_flow0 @ a @ G @ Y4 @ X0 @ T2 ) ) ) ) ) ).
% flow0_cong
thf(fact_60_rev_Oclosed__orbit__eq__rev,axiom,
! [X: a] :
( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
= ( period385816147_orbit @ a @ f @ x @ X ) ) ).
% rev.closed_orbit_eq_rev
thf(fact_61_rev_Oclosed__orbit__in__domain,axiom,
! [X: a] :
( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
=> ( member @ a @ X @ x ) ) ).
% rev.closed_orbit_in_domain
thf(fact_62_closed__orbit__flow0,axiom,
! [X: a,T2: real] :
( ( period385816147_orbit @ a @ f @ x @ X )
=> ( period385816147_orbit @ a @ f @ x @ ( auto_ll_on_flow0 @ a @ f @ x @ X @ T2 ) ) ) ).
% closed_orbit_flow0
thf(fact_63_rev_Oexistence__ivl__zero,axiom,
! [X0: a] :
( ( member @ a @ X0 @ x )
=> ( member @ real @ ( zero_zero @ real ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) ) ) ).
% rev.existence_ivl_zero
thf(fact_64_local_Oexistence__ivl__initial__time,axiom,
! [X0: a] :
( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
=> ( ( member @ a @ X0 @ x )
=> ( member @ real @ ( zero_zero @ real ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) ) ) ) ).
% local.existence_ivl_initial_time
thf(fact_65_local_Orev_Omem__existence__ivl__subset,axiom,
! [T2: real,X0: a] :
( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
=> ( member @ real @ T2 @ ( top_top @ ( set @ real ) ) ) ) ).
% local.rev.mem_existence_ivl_subset
thf(fact_66_rev_Oflow0__cong,axiom,
! [B: $tType,Y4: set @ a,G: a > a,T2: real,X0: a] :
( ( x = Y4 )
=> ( ! [X3: a,T: B] :
( ( member @ a @ X3 @ Y4 )
=> ( ( uminus_uminus @ ( a > a ) @ f @ X3 )
= ( G @ X3 ) ) )
=> ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
=> ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ T2 )
= ( auto_ll_on_flow0 @ a @ G @ Y4 @ X0 @ T2 ) ) ) ) ) ).
% rev.flow0_cong
thf(fact_67_fixpoint__sol_I1_J,axiom,
! [X: a] :
( ( member @ a @ X @ x )
=> ( ( ( f @ X )
= ( zero_zero @ a ) )
=> ( ( auto_l1112008849e_ivl0 @ a @ f @ x @ X )
= ( top_top @ ( set @ real ) ) ) ) ) ).
% fixpoint_sol(1)
thf(fact_68_rev_Oclosed__orbit__flow0,axiom,
! [X: a,T2: real] :
( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
=> ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ T2 ) ) ) ).
% rev.closed_orbit_flow0
thf(fact_69_closed__orbit__global__existence,axiom,
! [X: a] :
( ( period385816147_orbit @ a @ f @ x @ X )
=> ( ( auto_l1112008849e_ivl0 @ a @ f @ x @ X )
= ( top_top @ ( set @ real ) ) ) ) ).
% closed_orbit_global_existence
thf(fact_70_closed__orbitI,axiom,
! [T2: real,T3: real,Y: a] :
( ( T2 != T3 )
=> ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ Y ) )
=> ( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ Y ) )
=> ( ( ( auto_ll_on_flow0 @ a @ f @ x @ Y @ T2 )
= ( auto_ll_on_flow0 @ a @ f @ x @ Y @ T3 ) )
=> ( period385816147_orbit @ a @ f @ x @ Y ) ) ) ) ) ).
% closed_orbitI
thf(fact_71_rev_Ofixed__point__imp__closed__orbit__period__zero_I1_J,axiom,
! [X: a] :
( ( member @ a @ X @ x )
=> ( ( ( uminus_uminus @ ( a > a ) @ f @ X )
= ( zero_zero @ a ) )
=> ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ).
% rev.fixed_point_imp_closed_orbit_period_zero(1)
thf(fact_72_periodic__orbit__period_I2_J,axiom,
! [X: a] :
( ( period862636932_orbit @ a @ f @ x @ X )
=> ( ( auto_ll_on_flow0 @ a @ f @ x @ X @ ( period1153813292period @ a @ f @ x @ X ) )
= X ) ) ).
% periodic_orbit_period(2)
thf(fact_73_closed__orbit__periodic,axiom,
! [X: a] :
( ( period385816147_orbit @ a @ f @ x @ X )
=> ( ( ( f @ X )
!= ( zero_zero @ a ) )
=> ( period862636932_orbit @ a @ f @ x @ X ) ) ) ).
% closed_orbit_periodic
thf(fact_74_local_Orev_Oexistence__ivl__initial__time,axiom,
! [X0: a] :
( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
=> ( ( member @ a @ X0 @ x )
=> ( member @ real @ ( zero_zero @ real ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) ) ) ) ).
% local.rev.existence_ivl_initial_time
thf(fact_75_rev_Ofixpoint__sol_I1_J,axiom,
! [X: a] :
( ( member @ a @ X @ x )
=> ( ( ( uminus_uminus @ ( a > a ) @ f @ X )
= ( zero_zero @ a ) )
=> ( ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
= ( top_top @ ( set @ real ) ) ) ) ) ).
% rev.fixpoint_sol(1)
thf(fact_76_local_Orev_Oflow__undefined0,axiom,
! [T2: real,X: a] :
( ~ ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) )
=> ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ T2 )
= ( zero_zero @ a ) ) ) ).
% local.rev.flow_undefined0
thf(fact_77_closed__orbit__def,axiom,
! [X: a] :
( ( period385816147_orbit @ a @ f @ x @ X )
= ( ? [X2: real] :
( ( member @ real @ X2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) )
& ( X2
!= ( zero_zero @ real ) )
& ( ( auto_ll_on_flow0 @ a @ f @ x @ X @ X2 )
= X ) ) ) ) ).
% closed_orbit_def
thf(fact_78_rev_Oclosed__orbit__global__existence,axiom,
! [X: a] :
( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
=> ( ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
= ( top_top @ ( set @ real ) ) ) ) ).
% rev.closed_orbit_global_existence
thf(fact_79_rev_Oclosed__orbitI,axiom,
! [T2: real,T3: real,Y: a] :
( ( T2 != T3 )
=> ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y ) )
=> ( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y ) )
=> ( ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y @ T2 )
= ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y @ T3 ) )
=> ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y ) ) ) ) ) ).
% rev.closed_orbitI
thf(fact_80_rev_Ofixed__point__imp__closed__orbit__period__zero_I2_J,axiom,
! [X: a] :
( ( member @ a @ X @ x )
=> ( ( ( uminus_uminus @ ( a > a ) @ f @ X )
= ( zero_zero @ a ) )
=> ( ( period1153813292period @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
= ( zero_zero @ real ) ) ) ) ).
% rev.fixed_point_imp_closed_orbit_period_zero(2)
thf(fact_81_rev_Operiodic__orbit__imp__flow0__regular,axiom,
! [X: a,T2: real] :
( ( period862636932_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
=> ( ( uminus_uminus @ ( a > a ) @ f @ ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ T2 ) )
!= ( zero_zero @ a ) ) ) ).
% rev.periodic_orbit_imp_flow0_regular
thf(fact_82_periodic__orbitI,axiom,
! [T2: real,T3: real,Y: a] :
( ( T2 != T3 )
=> ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ Y ) )
=> ( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ Y ) )
=> ( ( ( auto_ll_on_flow0 @ a @ f @ x @ Y @ T2 )
= ( auto_ll_on_flow0 @ a @ f @ x @ Y @ T3 ) )
=> ( ( ( f @ Y )
!= ( zero_zero @ a ) )
=> ( period862636932_orbit @ a @ f @ x @ Y ) ) ) ) ) ) ).
% periodic_orbitI
thf(fact_83_periodic__orbit__period_I1_J,axiom,
! [X: a] :
( ( period862636932_orbit @ a @ f @ x @ X )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ f @ x @ X ) ) ) ).
% periodic_orbit_period(1)
thf(fact_84_closed__orbit__period__zero__fixed__point,axiom,
! [X: a] :
( ( period385816147_orbit @ a @ f @ x @ X )
=> ( ( ( period1153813292period @ a @ f @ x @ X )
= ( zero_zero @ real ) )
=> ( ( f @ X )
= ( zero_zero @ a ) ) ) ) ).
% closed_orbit_period_zero_fixed_point
thf(fact_85_rev_Operiodic__orbit__period_I2_J,axiom,
! [X: a] :
( ( period862636932_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
=> ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ ( period1153813292period @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) )
= X ) ) ).
% rev.periodic_orbit_period(2)
thf(fact_86_rev_Oclosed__orbit__periodic,axiom,
! [X: a] :
( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
=> ( ( ( uminus_uminus @ ( a > a ) @ f @ X )
!= ( zero_zero @ a ) )
=> ( period862636932_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ).
% rev.closed_orbit_periodic
thf(fact_87_add_Oinverse__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% add.inverse_neutral
thf(fact_88_neg__0__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( uminus_uminus @ A @ A2 ) )
= ( ( zero_zero @ A )
= A2 ) ) ) ).
% neg_0_equal_iff_equal
thf(fact_89_neg__equal__0__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( ( uminus_uminus @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_0_iff_equal
thf(fact_90_equal__neg__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( A2
= ( uminus_uminus @ A @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% equal_neg_zero
thf(fact_91_neg__equal__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ( uminus_uminus @ A @ A2 )
= A2 )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_zero
thf(fact_92_local_Orev_Oflow__initial__time__if,axiom,
! [X0: a] :
( ( ( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
& ( member @ a @ X0 @ x ) )
=> ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ ( zero_zero @ real ) )
= X0 ) )
& ( ~ ( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
& ( member @ a @ X0 @ x ) )
=> ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ ( zero_zero @ real ) )
= ( zero_zero @ a ) ) ) ) ).
% local.rev.flow_initial_time_if
thf(fact_93_neg__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% neg_less_iff_less
thf(fact_94_rev_Oclosed__orbit__def,axiom,
! [X: a] :
( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
= ( ? [X2: real] :
( ( member @ real @ X2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) )
& ( X2
!= ( zero_zero @ real ) )
& ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ X2 )
= X ) ) ) ) ).
% rev.closed_orbit_def
thf(fact_95_abs__minus,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( abs_abs @ A @ ( uminus_uminus @ A @ A2 ) )
= ( abs_abs @ A @ A2 ) ) ) ).
% abs_minus
thf(fact_96_abs__minus__cancel,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A] :
( ( abs_abs @ A @ ( uminus_uminus @ A @ A2 ) )
= ( abs_abs @ A @ A2 ) ) ) ).
% abs_minus_cancel
thf(fact_97_rev_Operiodic__orbitI,axiom,
! [T2: real,T3: real,Y: a] :
( ( T2 != T3 )
=> ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y ) )
=> ( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y ) )
=> ( ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y @ T2 )
= ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y @ T3 ) )
=> ( ( ( uminus_uminus @ ( a > a ) @ f @ Y )
!= ( zero_zero @ a ) )
=> ( period862636932_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y ) ) ) ) ) ) ).
% rev.periodic_orbitI
thf(fact_98_rev_Operiodic__orbit__period_I1_J,axiom,
! [X: a] :
( ( period862636932_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ).
% rev.periodic_orbit_period(1)
thf(fact_99_rev_Oclosed__orbit__period__zero__fixed__point,axiom,
! [X: a] :
( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
=> ( ( ( period1153813292period @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
= ( zero_zero @ real ) )
=> ( ( uminus_uminus @ ( a > a ) @ f @ X )
= ( zero_zero @ a ) ) ) ) ).
% rev.closed_orbit_period_zero_fixed_point
thf(fact_100_periodic__orbit__def,axiom,
! [X: a] :
( ( period862636932_orbit @ a @ f @ x @ X )
= ( ( period385816147_orbit @ a @ f @ x @ X )
& ( ord_less @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ f @ x @ X ) ) ) ) ).
% periodic_orbit_def
thf(fact_101_rev_Operiodic__orbit__def,axiom,
! [X: a] :
( ( period862636932_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
= ( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
& ( ord_less @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ) ).
% rev.periodic_orbit_def
thf(fact_102_less__neg__neg,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% less_neg_neg
thf(fact_103_neg__less__pos,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_less_pos
thf(fact_104_neg__0__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_less_iff_less
thf(fact_105_neg__less__0__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_less_0_iff_less
thf(fact_106_local_Omem__existence__ivl__iv__defined_I2_J,axiom,
! [T2: real,X0: a] :
( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( member @ a @ X0 @ x ) ) ).
% local.mem_existence_ivl_iv_defined(2)
thf(fact_107_local_Orev_Omem__existence__ivl__iv__defined_I2_J,axiom,
! [T2: real,X0: a] :
( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
=> ( member @ a @ X0 @ x ) ) ).
% local.rev.mem_existence_ivl_iv_defined(2)
thf(fact_108_flow0__defined,axiom,
! [Xa: real,X0: a] :
( ( member @ real @ Xa @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( member @ a @ ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ Xa ) @ x ) ) ).
% flow0_defined
thf(fact_109_local_Oflow__in__domain,axiom,
! [T2: real,X0: a] :
( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( member @ a @ ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ T2 ) @ x ) ) ).
% local.flow_in_domain
thf(fact_110_local_Omem__existence__ivl__iv__defined_I1_J,axiom,
! [T2: real,X0: a] :
( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) ) ) ).
% local.mem_existence_ivl_iv_defined(1)
thf(fact_111_local_Oexistence__ivl__initial__time__iff,axiom,
! [X0: a] :
( ( member @ real @ ( zero_zero @ real ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
= ( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
& ( member @ a @ X0 @ x ) ) ) ).
% local.existence_ivl_initial_time_iff
thf(fact_112_local_Oflow__initial__time,axiom,
! [X0: a] :
( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
=> ( ( member @ a @ X0 @ x )
=> ( ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ ( zero_zero @ real ) )
= X0 ) ) ) ).
% local.flow_initial_time
thf(fact_113_rev_Oflow0__defined,axiom,
! [Xa: real,X0: a] :
( ( member @ real @ Xa @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
=> ( member @ a @ ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ Xa ) @ x ) ) ).
% rev.flow0_defined
thf(fact_114_local_Orev_Oflow__in__domain,axiom,
! [T2: real,X0: a] :
( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
=> ( member @ a @ ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ T2 ) @ x ) ) ).
% local.rev.flow_in_domain
thf(fact_115_local_Orev_Omem__existence__ivl__iv__defined_I1_J,axiom,
! [T2: real,X0: a] :
( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
=> ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) ) ) ).
% local.rev.mem_existence_ivl_iv_defined(1)
thf(fact_116_local_Orev_Oexistence__ivl__initial__time__iff,axiom,
! [X0: a] :
( ( member @ real @ ( zero_zero @ real ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
= ( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
& ( member @ a @ X0 @ x ) ) ) ).
% local.rev.existence_ivl_initial_time_iff
thf(fact_117_local_Orev_Oflow__initial__time,axiom,
! [X0: a] :
( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
=> ( ( member @ a @ X0 @ x )
=> ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ ( zero_zero @ real ) )
= X0 ) ) ) ).
% local.rev.flow_initial_time
thf(fact_118_equation__minus__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( uminus_uminus @ A @ B2 ) )
= ( B2
= ( uminus_uminus @ A @ A2 ) ) ) ) ).
% equation_minus_iff
thf(fact_119_minus__equation__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( ( uminus_uminus @ A @ A2 )
= B2 )
= ( ( uminus_uminus @ A @ B2 )
= A2 ) ) ) ).
% minus_equation_iff
thf(fact_120_less__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( ord_less @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% less_minus_iff
thf(fact_121_minus__less__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ).
% minus_less_iff
thf(fact_122_abs__eq__iff,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X: A,Y: A] :
( ( ( abs_abs @ A @ X )
= ( abs_abs @ A @ Y ) )
= ( ( X = Y )
| ( X
= ( uminus_uminus @ A @ Y ) ) ) ) ) ).
% abs_eq_iff
thf(fact_123_abs__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A2 ) @ B2 )
= ( ( ord_less @ A @ A2 @ B2 )
& ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ) ).
% abs_less_iff
thf(fact_124_abs__if,axiom,
! [A: $tType] :
( ( abs_if @ A )
=> ( ( abs_abs @ A )
= ( ^ [A5: A] : ( if @ A @ ( ord_less @ A @ A5 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A5 ) @ A5 ) ) ) ) ).
% abs_if
thf(fact_125_abs__of__neg,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ A2 )
= ( uminus_uminus @ A @ A2 ) ) ) ) ).
% abs_of_neg
thf(fact_126__092_060alpha_062__limit__point__eq__rev,axiom,
! [X: a,P2: a] :
( ( limit_1335059408_point @ a @ f @ x @ X @ P2 )
= ( limit_1052139091_point @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ P2 ) ) ).
% \<alpha>_limit_point_eq_rev
thf(fact_127_rev_O_092_060alpha_062__limit__point__eq__rev,axiom,
! [X: a,P2: a] :
( ( limit_1335059408_point @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ P2 )
= ( limit_1052139091_point @ a @ f @ x @ X @ P2 ) ) ).
% rev.\<alpha>_limit_point_eq_rev
thf(fact_128_rev_Omvar_Ointerval__axioms,axiom,
! [X0: a] : ( initia826609931terval @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) ) ).
% rev.mvar.interval_axioms
thf(fact_129_mvar_Ointerval__axioms,axiom,
! [X0: a] : ( initia826609931terval @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) ) ).
% mvar.interval_axioms
thf(fact_130_abs__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( one_one @ A ) ) ) ).
% abs_neg_one
thf(fact_131_rev_Oclosed__orbitE,axiom,
! [X: a] :
( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
=> ~ ! [T4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T4 )
=> ~ ! [T5: real] :
( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ ( plus_plus @ real @ T5 @ T4 ) )
= ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ T5 ) ) ) ) ).
% rev.closed_orbitE
thf(fact_132_local_Orev_Oexistence__ivl__notempty,axiom,
! [X0: a] :
( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
=> ( ( member @ a @ X0 @ x )
=> ( ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 )
!= ( bot_bot @ ( set @ real ) ) ) ) ) ).
% local.rev.existence_ivl_notempty
thf(fact_133_rev_Oclosed__orbit__period__nonneg,axiom,
! [X: a] :
( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ).
% rev.closed_orbit_period_nonneg
thf(fact_134_rev_Ovec__simps_I11_J,axiom,
! [Ag: $tType] :
( ( semigroup_add @ Ag )
=> ! [Ae: Ag,Be: Ag,Cd: Ag] :
( ( plus_plus @ Ag @ ( plus_plus @ Ag @ Ae @ Be ) @ Cd )
= ( plus_plus @ Ag @ Ae @ ( plus_plus @ Ag @ Be @ Cd ) ) ) ) ).
% rev.vec_simps(11)
thf(fact_135_rev_Ovec__simps_I12_J,axiom,
! [Ai: $tType] :
( ( ab_semigroup_add @ Ai )
=> ( ( plus_plus @ Ai )
= ( ^ [Ag2: Ai,Bg: Ai] : ( plus_plus @ Ai @ Bg @ Ag2 ) ) ) ) ).
% rev.vec_simps(12)
thf(fact_136_rev_Ovec__simps_I13_J,axiom,
! [Ai: $tType] :
( ( ab_semigroup_add @ Ai )
=> ! [Bg2: Ai,Ag3: Ai,Cf: Ai] :
( ( plus_plus @ Ai @ Bg2 @ ( plus_plus @ Ai @ Ag3 @ Cf ) )
= ( plus_plus @ Ai @ Ag3 @ ( plus_plus @ Ai @ Bg2 @ Cf ) ) ) ) ).
% rev.vec_simps(13)
thf(fact_137_interval__axioms,axiom,
initia826609931terval @ ( top_top @ ( set @ real ) ) ).
% interval_axioms
thf(fact_138_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
= ( B2 = C ) ) ) ).
% add_right_cancel
thf(fact_139_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C ) )
= ( B2 = C ) ) ) ).
% add_left_cancel
thf(fact_140_rev__eq__flow,axiom,
! [Y: a,T2: real] :
( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y @ T2 )
= ( auto_ll_on_flow0 @ a @ f @ x @ Y @ ( uminus_uminus @ real @ T2 ) ) ) ).
% rev_eq_flow
thf(fact_141_rev_Orev__eq__flow,axiom,
! [Y: a,T2: real] :
( ( auto_ll_on_flow0 @ a @ f @ x @ Y @ T2 )
= ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y @ ( uminus_uminus @ real @ T2 ) ) ) ).
% rev.rev_eq_flow
thf(fact_142_recurrence__time__flip__sign_I1_J,axiom,
! [T6: real,X: a] :
( ( member @ real @ T6 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) )
=> ( ( ( auto_ll_on_flow0 @ a @ f @ x @ X @ T6 )
= X )
=> ( member @ real @ ( uminus_uminus @ real @ T6 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) ) ) ) ).
% recurrence_time_flip_sign(1)
thf(fact_143_recurrence__time__flip__sign_I2_J,axiom,
! [T6: real,X: a] :
( ( member @ real @ T6 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) )
=> ( ( ( auto_ll_on_flow0 @ a @ f @ x @ X @ T6 )
= X )
=> ( ( auto_ll_on_flow0 @ a @ f @ x @ X @ ( uminus_uminus @ real @ T6 ) )
= X ) ) ) ).
% recurrence_time_flip_sign(2)
thf(fact_144_rev_Orecurrence__time__flip__sign_I1_J,axiom,
! [T6: real,X: a] :
( ( member @ real @ T6 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) )
=> ( ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ T6 )
= X )
=> ( member @ real @ ( uminus_uminus @ real @ T6 ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ) ).
% rev.recurrence_time_flip_sign(1)
thf(fact_145_rev_Orecurrence__time__flip__sign_I2_J,axiom,
! [T6: real,X: a] :
( ( member @ real @ T6 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) )
=> ( ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ T6 )
= X )
=> ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ ( uminus_uminus @ real @ T6 ) )
= X ) ) ) ).
% rev.recurrence_time_flip_sign(2)
thf(fact_146_local_Oflow__trans,axiom,
! [S: real,X0: a,T2: real] :
( ( member @ real @ S @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ S ) ) )
=> ( ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ ( plus_plus @ real @ S @ T2 ) )
= ( auto_ll_on_flow0 @ a @ f @ x @ ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ S ) @ T2 ) ) ) ) ).
% local.flow_trans
thf(fact_147_local_Oexistence__ivl__trans_H,axiom,
! [T2: real,S: real,X0: a] :
( ( member @ real @ ( plus_plus @ real @ T2 @ S ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( member @ real @ S @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ T2 ) ) ) ) ) ).
% local.existence_ivl_trans'
thf(fact_148_local_Oexistence__ivl__trans,axiom,
! [S: real,X0: a,T2: real] :
( ( member @ real @ S @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ S ) ) )
=> ( member @ real @ ( plus_plus @ real @ S @ T2 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) ) ) ) ).
% local.existence_ivl_trans
thf(fact_149_local_Orev_Oflow__trans,axiom,
! [S: real,X0: a,T2: real] :
( ( member @ real @ S @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
=> ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ S ) ) )
=> ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ ( plus_plus @ real @ S @ T2 ) )
= ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ S ) @ T2 ) ) ) ) ).
% local.rev.flow_trans
thf(fact_150_local_Orev_Oexistence__ivl__trans_H,axiom,
! [T2: real,S: real,X0: a] :
( ( member @ real @ ( plus_plus @ real @ T2 @ S ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
=> ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
=> ( member @ real @ S @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ T2 ) ) ) ) ) ).
% local.rev.existence_ivl_trans'
thf(fact_151_local_Orev_Oexistence__ivl__trans,axiom,
! [S: real,X0: a,T2: real] :
( ( member @ real @ S @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
=> ( ( member @ real @ T2 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ S ) ) )
=> ( member @ real @ ( plus_plus @ real @ S @ T2 ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) ) ) ) ).
% local.rev.existence_ivl_trans
thf(fact_152_local_Oexistence__ivl__notempty,axiom,
! [X0: a] :
( ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
=> ( ( member @ a @ X0 @ x )
=> ( ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 )
!= ( bot_bot @ ( set @ real ) ) ) ) ) ).
% local.existence_ivl_notempty
thf(fact_153_closed__orbit__period__nonneg,axiom,
! [X: a] :
( ( period385816147_orbit @ a @ f @ x @ X )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( period1153813292period @ a @ f @ x @ X ) ) ) ).
% closed_orbit_period_nonneg
thf(fact_154_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_155_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% neg_le_iff_le
thf(fact_156_zero__eq__add__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X: A,Y: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ X @ Y ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_157_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X: A,Y: A] :
( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_158_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( plus_plus @ A @ A2 @ B2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_159_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( plus_plus @ A @ B2 @ A2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_160_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A,B2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= A2 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_161_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [B2: A,A2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= A2 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_162_double__zero__sym,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ A2 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero_sym
thf(fact_163_linordered__ab__group__add__class_Odouble__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ( plus_plus @ A @ A2 @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% linordered_ab_group_add_class.double_zero
thf(fact_164_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.right_neutral
thf(fact_165_add_Oleft__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.left_neutral
thf(fact_166_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A2: A,C: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_right
thf(fact_167_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_left
thf(fact_168_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A2: A,C: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_cancel_right
thf(fact_169_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C: A,A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_cancel_left
thf(fact_170_closed__orbitE,axiom,
! [X: a] :
( ( period385816147_orbit @ a @ f @ x @ X )
=> ~ ! [T4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T4 )
=> ~ ! [T5: real] :
( ( auto_ll_on_flow0 @ a @ f @ x @ X @ ( plus_plus @ real @ T5 @ T4 ) )
= ( auto_ll_on_flow0 @ a @ f @ x @ X @ T5 ) ) ) ) ).
% closed_orbitE
thf(fact_171_minus__add__distrib,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% minus_add_distrib
thf(fact_172_minus__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( plus_plus @ A @ A2 @ B2 ) )
= B2 ) ) ).
% minus_add_cancel
thf(fact_173_add__minus__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) )
= B2 ) ) ).
% add_minus_cancel
thf(fact_174_abs__add__abs,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A,B2: A] :
( ( abs_abs @ A @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) )
= ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_add_abs
thf(fact_175_neg__0__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_le_iff_le
thf(fact_176_neg__le__0__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_le_0_iff_le
thf(fact_177_less__eq__neg__nonpos,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% less_eq_neg_nonpos
thf(fact_178_neg__less__eq__nonneg,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_less_eq_nonneg
thf(fact_179_linordered__ab__group__add__class_Ozero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% linordered_ab_group_add_class.zero_le_double_add_iff_zero_le_single_add
thf(fact_180_linordered__ab__group__add__class_Odouble__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% linordered_ab_group_add_class.double_add_le_zero_iff_single_add_le_zero
thf(fact_181_le__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel2
thf(fact_182_le__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel1
thf(fact_183_add__le__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel2
thf(fact_184_add__le__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel1
thf(fact_185_linordered__ab__group__add__class_Ozero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% linordered_ab_group_add_class.zero_less_double_add_iff_zero_less_single_add
thf(fact_186_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_187_less__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel2
thf(fact_188_less__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel1
thf(fact_189_add__less__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel2
thf(fact_190_add__less__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel1
thf(fact_191_add_Oleft__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% add.left_inverse
thf(fact_192_add_Oright__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
= ( zero_zero @ A ) ) ) ).
% add.right_inverse
thf(fact_193_abs__le__zero__iff,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% abs_le_zero_iff
thf(fact_194_abs__le__self__iff,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ A2 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% abs_le_self_iff
thf(fact_195_abs__of__nonneg,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( abs_abs @ A @ A2 )
= A2 ) ) ) ).
% abs_of_nonneg
thf(fact_196_ball__trivial,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [X: A] :
( ( elemen1815086676c_ball @ A @ X @ ( zero_zero @ real ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ball_trivial
thf(fact_197_add__neg__numeral__special_I7_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( plus_plus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% add_neg_numeral_special(7)
thf(fact_198_add__neg__numeral__special_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% add_neg_numeral_special(8)
thf(fact_199_abs__of__nonpos,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ A2 )
= ( uminus_uminus @ A @ A2 ) ) ) ) ).
% abs_of_nonpos
thf(fact_200_ball__eq__empty,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [X: A,E: real] :
( ( ( elemen1815086676c_ball @ A @ X @ E )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ real @ E @ ( zero_zero @ real ) ) ) ) ).
% ball_eq_empty
thf(fact_201_local_Oexistence__ivl__undefined,axiom,
! [X0: a] :
( ~ ( member @ a @ X0 @ x )
=> ( ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 )
= ( bot_bot @ ( set @ real ) ) ) ) ).
% local.existence_ivl_undefined
thf(fact_202_local_Orev_Oexistence__ivl__undefined,axiom,
! [X0: a] :
( ~ ( member @ a @ X0 @ x )
=> ( ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 )
= ( bot_bot @ ( set @ real ) ) ) ) ).
% local.rev.existence_ivl_undefined
thf(fact_203_local_Oexistence__ivl__empty__iff,axiom,
! [X0: a] :
( ( ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 )
= ( bot_bot @ ( set @ real ) ) )
= ( ~ ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
| ~ ( member @ a @ X0 @ x ) ) ) ).
% local.existence_ivl_empty_iff
thf(fact_204_local_Oexistence__ivl__empty1,axiom,
! [X0: a] :
( ~ ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
=> ( ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 )
= ( bot_bot @ ( set @ real ) ) ) ) ).
% local.existence_ivl_empty1
thf(fact_205_local_Orev_Oexistence__ivl__empty__iff,axiom,
! [X0: a] :
( ( ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 )
= ( bot_bot @ ( set @ real ) ) )
= ( ~ ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
| ~ ( member @ a @ X0 @ x ) ) ) ).
% local.rev.existence_ivl_empty_iff
thf(fact_206_local_Orev_Oexistence__ivl__empty1,axiom,
! [X0: a] :
( ~ ( member @ real @ ( zero_zero @ real ) @ ( top_top @ ( set @ real ) ) )
=> ( ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 )
= ( bot_bot @ ( set @ real ) ) ) ) ).
% local.rev.existence_ivl_empty1
thf(fact_207_is__num__normalize_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% is_num_normalize(8)
thf(fact_208_abs__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( plus_plus @ A @ A2 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_triangle_ineq
thf(fact_209_le__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% le_numeral_extra(3)
thf(fact_210_le__numeral__extra_I4_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% le_numeral_extra(4)
thf(fact_211_is__num__normalize_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A2: A,B2: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% is_num_normalize(1)
thf(fact_212_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A2: A,C: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_right
thf(fact_213_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_left
thf(fact_214_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A2: A,B2: A,C: A,D2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% add_less_le_mono
thf(fact_215_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A2: A,B2: A,C: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% add_le_less_mono
thf(fact_216_add__nonpos__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_217_add__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_218_add__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_nonpos
thf(fact_219_add__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_220_le__iff__add,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
? [C2: A] :
( B4
= ( plus_plus @ A @ A5 @ C2 ) ) ) ) ) ).
% le_iff_add
thf(fact_221_add__increasing2,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [C: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% add_increasing2
thf(fact_222_add__decreasing2,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [C: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ C @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ B2 ) ) ) ) ).
% add_decreasing2
thf(fact_223_add__right__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% add_right_mono
thf(fact_224_less__eqE,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ~ ! [C3: A] :
( B2
!= ( plus_plus @ A @ A2 @ C3 ) ) ) ) ).
% less_eqE
thf(fact_225_add__increasing,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% add_increasing
thf(fact_226_add__decreasing,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,C: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ C @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ B2 ) ) ) ) ).
% add_decreasing
thf(fact_227_add__left__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) ) ) ) ).
% add_left_mono
thf(fact_228_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
=> ( B2 = C ) ) ) ).
% add_right_imp_eq
thf(fact_229_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C ) )
=> ( B2 = C ) ) ) ).
% add_left_imp_eq
thf(fact_230_add__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A2: A,B2: A,C: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C @ D2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% add_mono
thf(fact_231_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B2: A,A2: A,C: A] :
( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% add.left_commute
thf(fact_232_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A5: A,B4: A] : ( plus_plus @ A @ B4 @ A5 ) ) ) ) ).
% add.commute
thf(fact_233_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B2: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
= ( B2 = C ) ) ) ).
% add.right_cancel
thf(fact_234_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C ) )
= ( B2 = C ) ) ) ).
% add.left_cancel
thf(fact_235_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% add.assoc
thf(fact_236_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B5: A,K: A,B2: A,A2: A] :
( ( B5
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( plus_plus @ A @ A2 @ B5 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_237_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: A,K: A,A2: A,B2: A] :
( ( A4
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( plus_plus @ A @ A4 @ B2 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_238_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_239_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_240_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( K = L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_241_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus @ A @ I @ K )
= ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_242_add__mono__thms__linordered__field_I3_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_243_add__mono__thms__linordered__field_I4_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_244_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_245_add__strict__increasing2,axiom,
! [A: $tType] :
( ( ordere1818651114id_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B2 @ C )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% add_strict_increasing2
thf(fact_246_add__strict__increasing,axiom,
! [A: $tType] :
( ( ordere1818651114id_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% add_strict_increasing
thf(fact_247_add__pos__nonneg,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_pos_nonneg
thf(fact_248_add__nonpos__neg,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_neg
thf(fact_249_add__nonneg__pos,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_nonneg_pos
thf(fact_250_add__neg__nonpos,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_nonpos
thf(fact_251_Elementary__Metric__Spaces_Oball__empty,axiom,
! [A: $tType] :
( ( real_V2090557954_space @ A )
=> ! [E: real,X: A] :
( ( ord_less_eq @ real @ E @ ( zero_zero @ real ) )
=> ( ( elemen1815086676c_ball @ A @ X @ E )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Elementary_Metric_Spaces.ball_empty
thf(fact_252_add_Ogroup__left__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.group_left_neutral
thf(fact_253_add_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.comm_neutral
% Subclasses (18)
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ozero,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( zero @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ogroup__add,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( group_add @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Omonoid__add,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( monoid_add @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oab__group__add,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( ab_group_add @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Osemigroup__add,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( semigroup_add @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ocomm__monoid__add,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( comm_monoid_add @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oab__semigroup__add,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( ab_semigroup_add @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ocancel__semigroup__add,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( cancel_semigroup_add @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__ab__group__add,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( ordered_ab_group_add @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ocancel__comm__monoid__add,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( cancel1352612707id_add @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__comm__monoid__add,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( ordere216010020id_add @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__ab__group__add__abs,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( ordere142940540dd_abs @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__ab__semigroup__add,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( ordere779506340up_add @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Real__Vector__Spaces_Ometric__space,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( real_V2090557954_space @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__cancel__comm__monoid__add,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( ordere1818651114id_add @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( ordere236663937imp_le @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__cancel__ab__semigroup__add,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( ordere223160158up_add @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( ordere516151231imp_le @ A ) ) ).
% Type constructors (42)
thf(tcon_HOL_Obool___Countable_Ocountable,axiom,
countable @ $o ).
thf(tcon_Set_Oset___Countable_Ocountable_1,axiom,
! [A6: $tType] :
( ( finite_finite @ A6 )
=> ( countable @ ( set @ A6 ) ) ) ).
thf(tcon_fun___Countable_Ocountable_2,axiom,
! [A6: $tType,A7: $tType] :
( ( ( finite_finite @ A6 )
& ( countable @ A7 ) )
=> ( countable @ ( A6 > A7 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A6: $tType,A7: $tType] :
( ( ( finite_finite @ A6 )
& ( finite_finite @ A7 ) )
=> ( finite_finite @ ( A6 > A7 ) ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_3,axiom,
! [A6: $tType] :
( ( finite_finite @ A6 )
=> ( finite_finite @ ( set @ A6 ) ) ) ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_4,axiom,
finite_finite @ $o ).
thf(tcon_fun___Real__Vector__Spaces_Ometric__space,axiom,
! [A6: $tType,A7: $tType] :
( ( ( countable @ A6 )
& ( real_V2090557954_space @ A7 ) )
=> ( real_V2090557954_space @ ( A6 > A7 ) ) ) ).
thf(tcon_Set_Oset___Groups_Oab__semigroup__add,axiom,
! [A6: $tType] :
( ( ab_semigroup_add @ A6 )
=> ( ab_semigroup_add @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Groups_Ocomm__monoid__add,axiom,
! [A6: $tType] :
( ( comm_monoid_add @ A6 )
=> ( comm_monoid_add @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Groups_Osemigroup__add,axiom,
! [A6: $tType] :
( ( semigroup_add @ A6 )
=> ( semigroup_add @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Groups_Omonoid__add,axiom,
! [A6: $tType] :
( ( monoid_add @ A6 )
=> ( monoid_add @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Groups_Ozero,axiom,
! [A6: $tType] :
( ( zero @ A6 )
=> ( zero @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Groups_Oone,axiom,
! [A6: $tType] :
( ( one @ A6 )
=> ( one @ ( set @ A6 ) ) ) ).
thf(tcon_Real_Oreal___Executable__Euclidean__Space_Oexecutable__euclidean__space,axiom,
execut510477386_space @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere516151231imp_le @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add,axiom,
ordere1818651114id_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring,axiom,
linord1659791738miring @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space_5,axiom,
real_V2090557954_space @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs,axiom,
ordere142940540dd_abs @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__comm__monoid__add,axiom,
ordere216010020id_add @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_6,axiom,
ab_semigroup_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_7,axiom,
comm_monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring,axiom,
linordered_ring @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__idom,axiom,
linordered_idom @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__add_8,axiom,
semigroup_add @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__less__one,axiom,
zero_less_one @ real ).
thf(tcon_Real_Oreal___Groups_Oab__group__add,axiom,
ab_group_add @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__neq__one,axiom,
zero_neq_one @ real ).
thf(tcon_Real_Oreal___Rings_Oidom__abs__sgn,axiom,
idom_abs_sgn @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__add_9,axiom,
monoid_add @ real ).
thf(tcon_Real_Oreal___Groups_Ogroup__add,axiom,
group_add @ real ).
thf(tcon_Real_Oreal___Num_Oneg__numeral,axiom,
neg_numeral @ real ).
thf(tcon_Real_Oreal___Rings_Oabs__if,axiom,
abs_if @ real ).
thf(tcon_Real_Oreal___Groups_Ozero_10,axiom,
zero @ real ).
thf(tcon_Real_Oreal___Groups_Oone_11,axiom,
one @ real ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
% Free types (1)
thf(tfree_0,hypothesis,
execut510477386_space @ a ).
% Conjectures (2)
thf(conj_0,hypothesis,
! [Delta2: real,Tau2: a > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Delta2 )
=> ( ( topolo2071040574ous_on @ a @ real @ ( elemen1815086676c_ball @ a @ p @ Delta2 ) @ Tau2 )
=> ( ( ( Tau2 @ p )
= ( zero_zero @ real ) )
=> ( ! [Y3: a] :
( ( member @ a @ Y3 @ ( elemen1815086676c_ball @ a @ p @ Delta2 ) )
=> ( ord_less @ real @ ( abs_abs @ real @ ( Tau2 @ Y3 ) ) @ ( one_one @ real ) ) )
=> ( ! [Y3: a] :
( ( member @ a @ Y3 @ ( elemen1815086676c_ball @ a @ p @ Delta2 ) )
=> ( member @ a @ ( auto_ll_on_flow0 @ a @ f @ x @ Y3 @ ( Tau2 @ Y3 ) ) @ ( line_open_segment @ a @ a2 @ b ) ) )
=> thesisa ) ) ) ) ) ).
thf(conj_1,conjecture,
thesisa ).
%------------------------------------------------------------------------------