TPTP Problem File: ITP147^1.p
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%------------------------------------------------------------------------------
% File : ITP147^1 : 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 : 0.40 v8.2.0, 0.31 v8.1.0, 0.36 v7.5.0
% Syntax : Number of formulae : 395 ( 195 unt; 46 typ; 0 def)
% Number of atoms : 895 ( 396 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 2765 ( 48 ~; 6 |; 31 &;2414 @)
% ( 0 <=>; 266 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 5 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 87 ( 87 >; 0 *; 0 +; 0 <<)
% Number of symbols : 45 ( 42 usr; 14 con; 0-4 aty)
% Number of variables : 663 ( 11 ^; 650 !; 2 ?; 663 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:49:33.538
%------------------------------------------------------------------------------
% Could-be-implicit typings (4)
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (42)
thf(sy_c_Elementary__Metric__Spaces_Oball_001t__Real__Oreal,type,
elemen341675809l_real: real > real > set_real ).
thf(sy_c_Elementary__Metric__Spaces_Oball_001tf__a,type,
elemen154694473ball_a: a > real > set_a ).
thf(sy_c_Flow_Oauto__ll__on__open_Oexistence__ivl0_001tf__a,type,
auto_l612940ivl0_a: ( a > a ) > set_a > a > set_real ).
thf(sy_c_Flow_Oauto__ll__on__open_Oflow0_001tf__a,type,
auto_ll_on_flow0_a: ( a > a ) > set_a > a > real > a ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
abs_abs_real: real > real ).
thf(sy_c_Groups_Oabs__class_Oabs_001tf__a,type,
abs_abs_a: a > a ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Oplus__class_Oplus_001tf__a,type,
plus_plus_a: a > a > a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_Itf__a_Mtf__a_J,type,
uminus_uminus_a_a: ( a > a ) > a > a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001tf__a,type,
uminus_uminus_a: a > a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Groups_Ozero__class_Ozero_001tf__a,type,
zero_zero_a: a ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
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_001tf__a,type,
limit_1865182413oint_a: ( a > a ) > set_a > a > a > $o ).
thf(sy_c_Limit__Set_Oauto__ll__on__open_O_092_060omega_062__limit__point_001tf__a,type,
limit_94460170oint_a: ( a > a ) > set_a > a > a > $o ).
thf(sy_c_Line__Segment_Oopen__segment_001tf__a,type,
line_open_segment_a: a > a > set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
bot_bot_set_real: set_real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001tf__a,type,
ord_less_a: a > a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__a,type,
ord_less_eq_a: a > a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
top_top_set_real: set_real ).
thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Oclosed__orbit_001tf__a,type,
period720806154rbit_a: ( a > a ) > set_a > a > $o ).
thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Operiod_001tf__a,type,
period1305449585riod_a: ( a > a ) > set_a > a > real ).
thf(sy_c_Periodic__Orbit_Oauto__ll__on__open_Operiodic__orbit_001tf__a,type,
period138238489rbit_a: ( a > a ) > set_a > a > $o ).
thf(sy_c_Poincare__Bendixson__Mirabelle__pwkwpzhsyu_Oc1__on__open__R2_Otransversal__segment_001tf__a,type,
poinca522724647ment_a: ( a > a ) > set_a > a > a > $o ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001tf__a_001t__Real__Oreal,type,
topolo1710226732a_real: set_a > ( a > real ) > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001tf__a,type,
member_a: 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 (344)
thf(fact_0_rev_Oopen__segment__trichotomy,axiom,
! [X: a,A: a,B: a,Y: a] :
( ( member_a @ X @ ( line_open_segment_a @ A @ B ) )
=> ( ( member_a @ Y @ ( line_open_segment_a @ A @ B ) )
=> ( ( X = Y )
| ( member_a @ Y @ ( line_open_segment_a @ X @ B ) )
| ( member_a @ Y @ ( line_open_segment_a @ A @ 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] :
( ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ D ) @ T )
=> ( ! [Y2: a] :
( ( member_a @ Y2 @ ( elemen154694473ball_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 @ ( elemen154694473ball_a @ p @ D ) )
=> ( ord_less_real @ ( abs_abs_real @ ( T @ Y2 ) ) @ one_one_real ) )
=> ( ( topolo1710226732a_real @ ( elemen154694473ball_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 )
=> ( ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ Delta ) @ Tau )
=> ( ( ( Tau @ p )
= zero_zero_real )
=> ( ! [Y3: a] :
( ( member_a @ Y3 @ ( elemen154694473ball_a @ p @ Delta ) )
=> ( ord_less_real @ ( abs_abs_real @ ( Tau @ Y3 ) ) @ one_one_real ) )
=> ( ! [Y3: a] :
( ( member_a @ Y3 @ ( elemen154694473ball_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,
poinca522724647ment_a @ f @ x @ a2 @ b ).
% assms(1)
thf(fact_6_assms_I3_J,axiom,
limit_94460170oint_a @ f @ x @ x2 @ p ).
% assms(3)
thf(fact_7_centre__in__ball,axiom,
! [X: real,E: real] :
( ( member_real @ X @ ( elemen341675809l_real @ X @ E ) )
= ( ord_less_real @ zero_zero_real @ E ) ) ).
% centre_in_ball
thf(fact_8_centre__in__ball,axiom,
! [X: a,E: real] :
( ( member_a @ X @ ( elemen154694473ball_a @ X @ E ) )
= ( ord_less_real @ zero_zero_real @ E ) ) ).
% centre_in_ball
thf(fact_9_zero__less__abs__iff,axiom,
! [A: a] :
( ( ord_less_a @ zero_zero_a @ ( abs_abs_a @ A ) )
= ( A != zero_zero_a ) ) ).
% zero_less_abs_iff
thf(fact_10_zero__less__abs__iff,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
= ( A != zero_zero_real ) ) ).
% zero_less_abs_iff
thf(fact_11_cross__time__continuous,axiom,
! [A: a,B: a,X: a,E: real] :
( ( poinca522724647ment_a @ f @ x @ A @ B )
=> ( ( member_a @ X @ ( line_open_segment_a @ A @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ E )
=> ~ ! [D: real] :
( ( ord_less_real @ zero_zero_real @ D )
=> ! [T: a > real] :
( ( topolo1710226732a_real @ ( elemen154694473ball_a @ X @ D ) @ T )
=> ( ! [Y2: a] :
( ( member_a @ Y2 @ ( elemen154694473ball_a @ X @ D ) )
=> ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ Y2 @ ( T @ Y2 ) ) @ ( line_open_segment_a @ A @ B ) ) )
=> ( ! [Y2: a] :
( ( member_a @ Y2 @ ( elemen154694473ball_a @ X @ D ) )
=> ( ord_less_real @ ( abs_abs_real @ ( T @ Y2 ) ) @ E ) )
=> ( ( topolo1710226732a_real @ ( elemen154694473ball_a @ X @ D ) @ T )
=> ( ( T @ X )
!= zero_zero_real ) ) ) ) ) ) ) ) ) ).
% cross_time_continuous
thf(fact_12_abs__1,axiom,
( ( abs_abs_real @ one_one_real )
= one_one_real ) ).
% abs_1
thf(fact_13_abs__0,axiom,
( ( abs_abs_real @ zero_zero_real )
= zero_zero_real ) ).
% abs_0
thf(fact_14_abs__0__eq,axiom,
! [A: real] :
( ( zero_zero_real
= ( abs_abs_real @ A ) )
= ( A = zero_zero_real ) ) ).
% abs_0_eq
thf(fact_15_abs__0__eq,axiom,
! [A: a] :
( ( zero_zero_a
= ( abs_abs_a @ A ) )
= ( A = zero_zero_a ) ) ).
% abs_0_eq
thf(fact_16_abs__eq__0,axiom,
! [A: real] :
( ( ( abs_abs_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% abs_eq_0
thf(fact_17_abs__eq__0,axiom,
! [A: a] :
( ( ( abs_abs_a @ A )
= zero_zero_a )
= ( A = zero_zero_a ) ) ).
% abs_eq_0
thf(fact_18_abs__zero,axiom,
( ( abs_abs_real @ zero_zero_real )
= zero_zero_real ) ).
% abs_zero
thf(fact_19_abs__zero,axiom,
( ( abs_abs_a @ zero_zero_a )
= zero_zero_a ) ).
% abs_zero
thf(fact_20_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_21_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_22_transversal__segment__reverse,axiom,
! [X: a,Y: a] :
( ( poinca522724647ment_a @ f @ x @ X @ Y )
=> ( poinca522724647ment_a @ f @ x @ Y @ X ) ) ).
% transversal_segment_reverse
thf(fact_23_transversal__segment__commute,axiom,
! [X: a,Y: a] :
( ( poinca522724647ment_a @ f @ x @ X @ Y )
= ( poinca522724647ment_a @ f @ x @ Y @ X ) ) ).
% transversal_segment_commute
thf(fact_24_abs__idempotent,axiom,
! [A: real] :
( ( abs_abs_real @ ( abs_abs_real @ A ) )
= ( abs_abs_real @ A ) ) ).
% abs_idempotent
thf(fact_25_abs__idempotent,axiom,
! [A: a] :
( ( abs_abs_a @ ( abs_abs_a @ A ) )
= ( abs_abs_a @ A ) ) ).
% abs_idempotent
thf(fact_26_abs__abs,axiom,
! [A: real] :
( ( abs_abs_real @ ( abs_abs_real @ A ) )
= ( abs_abs_real @ A ) ) ).
% abs_abs
thf(fact_27_transversal__segment__exists,axiom,
! [X: a] :
( ( member_a @ X @ x )
=> ( ( ( f @ X )
!= zero_zero_a )
=> ~ ! [A2: a,B2: a] :
( ( member_a @ X @ ( line_open_segment_a @ A2 @ B2 ) )
=> ~ ( poinca522724647ment_a @ f @ x @ A2 @ B2 ) ) ) ) ).
% transversal_segment_exists
thf(fact_28_c1__on__open__R2_Otransversal__segment_Ocong,axiom,
poinca522724647ment_a = poinca522724647ment_a ).
% c1_on_open_R2.transversal_segment.cong
thf(fact_29_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_30_zero__reorient,axiom,
! [X: a] :
( ( zero_zero_a = X )
= ( X = zero_zero_a ) ) ).
% zero_reorient
thf(fact_31_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_32_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_33_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_34_abs__eq__0__iff,axiom,
! [A: real] :
( ( ( abs_abs_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% abs_eq_0_iff
thf(fact_35_abs__one,axiom,
( ( abs_abs_real @ one_one_real )
= one_one_real ) ).
% abs_one
thf(fact_36_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_37_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_38_abs__not__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).
% abs_not_less_zero
thf(fact_39_abs__not__less__zero,axiom,
! [A: a] :
~ ( ord_less_a @ ( abs_abs_a @ A ) @ zero_zero_a ) ).
% abs_not_less_zero
thf(fact_40_abs__of__pos,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( abs_abs_real @ A )
= A ) ) ).
% abs_of_pos
thf(fact_41_abs__of__pos,axiom,
! [A: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( abs_abs_a @ A )
= A ) ) ).
% abs_of_pos
thf(fact_42_periodic__orbit__imp__flow0__regular,axiom,
! [X: a,T2: real] :
( ( period138238489rbit_a @ f @ x @ X )
=> ( ( f @ ( auto_ll_on_flow0_a @ f @ x @ X @ T2 ) )
!= zero_zero_a ) ) ).
% periodic_orbit_imp_flow0_regular
thf(fact_43_fixed__point__imp__closed__orbit__period__zero_I2_J,axiom,
! [X: a] :
( ( member_a @ X @ x )
=> ( ( ( f @ X )
= zero_zero_a )
=> ( ( period1305449585riod_a @ f @ x @ X )
= zero_zero_real ) ) ) ).
% fixed_point_imp_closed_orbit_period_zero(2)
thf(fact_44_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_45_rev_Otransversal__segment__exists,axiom,
! [X: a] :
( ( member_a @ X @ x )
=> ( ( ( uminus_uminus_a_a @ f @ X )
!= zero_zero_a )
=> ~ ! [A2: a,B2: a] :
( ( member_a @ X @ ( line_open_segment_a @ A2 @ B2 ) )
=> ~ ( poinca522724647ment_a @ ( uminus_uminus_a_a @ f ) @ x @ A2 @ B2 ) ) ) ) ).
% rev.transversal_segment_exists
thf(fact_46_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_47_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_48_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_49_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_50_Collect__mem__eq,axiom,
! [A3: set_real] :
( ( collect_real
@ ^ [X2: real] : ( member_real @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_51_Collect__cong,axiom,
! [P: real > $o,Q: real > $o] :
( ! [X3: real] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_real @ P )
= ( collect_real @ Q ) ) ) ).
% Collect_cong
thf(fact_52_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_53_local_Oflow__undefined0,axiom,
! [T2: real,X: a] :
( ~ ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ X ) )
=> ( ( auto_ll_on_flow0_a @ f @ x @ X @ T2 )
= zero_zero_a ) ) ).
% local.flow_undefined0
thf(fact_54_rev__transversal__segment,axiom,
! [A: a,B: a] :
( ( poinca522724647ment_a @ ( uminus_uminus_a_a @ f ) @ x @ A @ B )
= ( poinca522724647ment_a @ f @ x @ A @ B ) ) ).
% rev_transversal_segment
thf(fact_55_rev_Otransversal__segment__reverse,axiom,
! [X: a,Y: a] :
( ( poinca522724647ment_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ Y )
=> ( poinca522724647ment_a @ ( uminus_uminus_a_a @ f ) @ x @ Y @ X ) ) ).
% rev.transversal_segment_reverse
thf(fact_56_rev_Otransversal__segment__commute,axiom,
! [X: a,Y: a] :
( ( poinca522724647ment_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ Y )
= ( poinca522724647ment_a @ ( uminus_uminus_a_a @ f ) @ x @ Y @ X ) ) ).
% rev.transversal_segment_commute
thf(fact_57_fixed__point__imp__closed__orbit__period__zero_I1_J,axiom,
! [X: a] :
( ( member_a @ X @ x )
=> ( ( ( f @ X )
= zero_zero_a )
=> ( period720806154rbit_a @ f @ x @ X ) ) ) ).
% fixed_point_imp_closed_orbit_period_zero(1)
thf(fact_58_closed__orbit__in__domain,axiom,
! [X: a] :
( ( period720806154rbit_a @ f @ x @ X )
=> ( member_a @ X @ x ) ) ).
% closed_orbit_in_domain
thf(fact_59_add_Oinverse__inverse,axiom,
! [A: a] :
( ( uminus_uminus_a @ ( uminus_uminus_a @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_60_add_Oinverse__inverse,axiom,
! [A: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_61_neg__equal__iff__equal,axiom,
! [A: a,B: a] :
( ( ( uminus_uminus_a @ A )
= ( uminus_uminus_a @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_62_neg__equal__iff__equal,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_63_existence__ivl__zero,axiom,
! [X0: a] :
( ( member_a @ X0 @ x )
=> ( member_real @ zero_zero_real @ ( auto_l612940ivl0_a @ f @ x @ X0 ) ) ) ).
% existence_ivl_zero
thf(fact_64_local_Omem__existence__ivl__subset,axiom,
! [T2: real,X0: a] :
( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
=> ( member_real @ T2 @ top_top_set_real ) ) ).
% local.mem_existence_ivl_subset
thf(fact_65_rev_Oclosed__orbit__eq__rev,axiom,
! [X: a] :
( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
= ( period720806154rbit_a @ f @ x @ X ) ) ).
% rev.closed_orbit_eq_rev
thf(fact_66_rev_Oclosed__orbit__in__domain,axiom,
! [X: a] :
( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
=> ( member_a @ X @ x ) ) ).
% rev.closed_orbit_in_domain
thf(fact_67_closed__orbit__flow0,axiom,
! [X: a,T2: real] :
( ( period720806154rbit_a @ f @ x @ X )
=> ( period720806154rbit_a @ f @ x @ ( auto_ll_on_flow0_a @ f @ x @ X @ T2 ) ) ) ).
% closed_orbit_flow0
thf(fact_68_rev_Oexistence__ivl__zero,axiom,
! [X0: a] :
( ( member_a @ X0 @ x )
=> ( member_real @ zero_zero_real @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) ) ) ).
% rev.existence_ivl_zero
thf(fact_69_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_l612940ivl0_a @ f @ x @ X0 ) ) ) ) ).
% local.existence_ivl_initial_time
thf(fact_70_local_Orev_Omem__existence__ivl__subset,axiom,
! [T2: real,X0: a] :
( ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
=> ( member_real @ T2 @ top_top_set_real ) ) ).
% local.rev.mem_existence_ivl_subset
thf(fact_71_fixpoint__sol_I1_J,axiom,
! [X: a] :
( ( member_a @ X @ x )
=> ( ( ( f @ X )
= zero_zero_a )
=> ( ( auto_l612940ivl0_a @ f @ x @ X )
= top_top_set_real ) ) ) ).
% fixpoint_sol(1)
thf(fact_72_rev_Oclosed__orbit__flow0,axiom,
! [X: a,T2: real] :
( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
=> ( period720806154rbit_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_73_closed__orbit__global__existence,axiom,
! [X: a] :
( ( period720806154rbit_a @ f @ x @ X )
=> ( ( auto_l612940ivl0_a @ f @ x @ X )
= top_top_set_real ) ) ).
% closed_orbit_global_existence
thf(fact_74_closed__orbitI,axiom,
! [T2: real,T3: real,Y: a] :
( ( T2 != T3 )
=> ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ Y ) )
=> ( ( member_real @ T3 @ ( auto_l612940ivl0_a @ f @ x @ Y ) )
=> ( ( ( auto_ll_on_flow0_a @ f @ x @ Y @ T2 )
= ( auto_ll_on_flow0_a @ f @ x @ Y @ T3 ) )
=> ( period720806154rbit_a @ f @ x @ Y ) ) ) ) ) ).
% closed_orbitI
thf(fact_75_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 )
=> ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ).
% rev.fixed_point_imp_closed_orbit_period_zero(1)
thf(fact_76_periodic__orbit__period_I2_J,axiom,
! [X: a] :
( ( period138238489rbit_a @ f @ x @ X )
=> ( ( auto_ll_on_flow0_a @ f @ x @ X @ ( period1305449585riod_a @ f @ x @ X ) )
= X ) ) ).
% periodic_orbit_period(2)
thf(fact_77_closed__orbit__periodic,axiom,
! [X: a] :
( ( period720806154rbit_a @ f @ x @ X )
=> ( ( ( f @ X )
!= zero_zero_a )
=> ( period138238489rbit_a @ f @ x @ X ) ) ) ).
% closed_orbit_periodic
thf(fact_78_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_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) ) ) ) ).
% local.rev.existence_ivl_initial_time
thf(fact_79_rev_Ofixpoint__sol_I1_J,axiom,
! [X: a] :
( ( member_a @ X @ x )
=> ( ( ( uminus_uminus_a_a @ f @ X )
= zero_zero_a )
=> ( ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
= top_top_set_real ) ) ) ).
% rev.fixpoint_sol(1)
thf(fact_80_local_Orev_Oflow__undefined0,axiom,
! [T2: real,X: a] :
( ~ ( member_real @ T2 @ ( auto_l612940ivl0_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_81_closed__orbit__def,axiom,
! [X: a] :
( ( period720806154rbit_a @ f @ x @ X )
= ( ? [X2: real] :
( ( member_real @ X2 @ ( auto_l612940ivl0_a @ f @ x @ X ) )
& ( X2 != zero_zero_real )
& ( ( auto_ll_on_flow0_a @ f @ x @ X @ X2 )
= X ) ) ) ) ).
% closed_orbit_def
thf(fact_82_rev_Oclosed__orbit__global__existence,axiom,
! [X: a] :
( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
=> ( ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
= top_top_set_real ) ) ).
% rev.closed_orbit_global_existence
thf(fact_83_rev_Oclosed__orbitI,axiom,
! [T2: real,T3: real,Y: a] :
( ( T2 != T3 )
=> ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ Y ) )
=> ( ( member_real @ T3 @ ( auto_l612940ivl0_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 ) )
=> ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ Y ) ) ) ) ) ).
% rev.closed_orbitI
thf(fact_84_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 )
=> ( ( period1305449585riod_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
= zero_zero_real ) ) ) ).
% rev.fixed_point_imp_closed_orbit_period_zero(2)
thf(fact_85_rev_Operiodic__orbit__imp__flow0__regular,axiom,
! [X: a,T2: real] :
( ( period138238489rbit_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_86_periodic__orbitI,axiom,
! [T2: real,T3: real,Y: a] :
( ( T2 != T3 )
=> ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ Y ) )
=> ( ( member_real @ T3 @ ( auto_l612940ivl0_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 )
=> ( period138238489rbit_a @ f @ x @ Y ) ) ) ) ) ) ).
% periodic_orbitI
thf(fact_87_periodic__orbit__period_I1_J,axiom,
! [X: a] :
( ( period138238489rbit_a @ f @ x @ X )
=> ( ord_less_real @ zero_zero_real @ ( period1305449585riod_a @ f @ x @ X ) ) ) ).
% periodic_orbit_period(1)
thf(fact_88_closed__orbit__period__zero__fixed__point,axiom,
! [X: a] :
( ( period720806154rbit_a @ f @ x @ X )
=> ( ( ( period1305449585riod_a @ f @ x @ X )
= zero_zero_real )
=> ( ( f @ X )
= zero_zero_a ) ) ) ).
% closed_orbit_period_zero_fixed_point
thf(fact_89_rev_Operiodic__orbit__period_I2_J,axiom,
! [X: a] :
( ( period138238489rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
=> ( ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ ( period1305449585riod_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) )
= X ) ) ).
% rev.periodic_orbit_period(2)
thf(fact_90_rev_Oclosed__orbit__periodic,axiom,
! [X: a] :
( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
=> ( ( ( uminus_uminus_a_a @ f @ X )
!= zero_zero_a )
=> ( period138238489rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ).
% rev.closed_orbit_periodic
thf(fact_91_add_Oinverse__neutral,axiom,
( ( uminus_uminus_a @ zero_zero_a )
= zero_zero_a ) ).
% add.inverse_neutral
thf(fact_92_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_93_neg__0__equal__iff__equal,axiom,
! [A: a] :
( ( zero_zero_a
= ( uminus_uminus_a @ A ) )
= ( zero_zero_a = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_94_neg__0__equal__iff__equal,axiom,
! [A: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A ) )
= ( zero_zero_real = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_95_neg__equal__0__iff__equal,axiom,
! [A: a] :
( ( ( uminus_uminus_a @ A )
= zero_zero_a )
= ( A = zero_zero_a ) ) ).
% neg_equal_0_iff_equal
thf(fact_96_neg__equal__0__iff__equal,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_97_equal__neg__zero,axiom,
! [A: real] :
( ( A
= ( uminus_uminus_real @ A ) )
= ( A = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_98_neg__equal__zero,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= A )
= ( A = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_99_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_100_neg__less__iff__less,axiom,
! [B: a,A: a] :
( ( ord_less_a @ ( uminus_uminus_a @ B ) @ ( uminus_uminus_a @ A ) )
= ( ord_less_a @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_101_neg__less__iff__less,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_102_rev_Oclosed__orbit__def,axiom,
! [X: a] :
( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
= ( ? [X2: real] :
( ( member_real @ X2 @ ( auto_l612940ivl0_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_103_abs__minus,axiom,
! [A: real] :
( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
= ( abs_abs_real @ A ) ) ).
% abs_minus
thf(fact_104_abs__minus__cancel,axiom,
! [A: a] :
( ( abs_abs_a @ ( uminus_uminus_a @ A ) )
= ( abs_abs_a @ A ) ) ).
% abs_minus_cancel
thf(fact_105_abs__minus__cancel,axiom,
! [A: real] :
( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
= ( abs_abs_real @ A ) ) ).
% abs_minus_cancel
thf(fact_106_rev_Operiodic__orbitI,axiom,
! [T2: real,T3: real,Y: a] :
( ( T2 != T3 )
=> ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ Y ) )
=> ( ( member_real @ T3 @ ( auto_l612940ivl0_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 )
=> ( period138238489rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ Y ) ) ) ) ) ) ).
% rev.periodic_orbitI
thf(fact_107_rev_Operiodic__orbit__period_I1_J,axiom,
! [X: a] :
( ( period138238489rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
=> ( ord_less_real @ zero_zero_real @ ( period1305449585riod_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ).
% rev.periodic_orbit_period(1)
thf(fact_108_rev_Oclosed__orbit__period__zero__fixed__point,axiom,
! [X: a] :
( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
=> ( ( ( period1305449585riod_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_109_periodic__orbit__def,axiom,
! [X: a] :
( ( period138238489rbit_a @ f @ x @ X )
= ( ( period720806154rbit_a @ f @ x @ X )
& ( ord_less_real @ zero_zero_real @ ( period1305449585riod_a @ f @ x @ X ) ) ) ) ).
% periodic_orbit_def
thf(fact_110_rev_Operiodic__orbit__def,axiom,
! [X: a] :
( ( period138238489rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
= ( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
& ( ord_less_real @ zero_zero_real @ ( period1305449585riod_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ) ).
% rev.periodic_orbit_def
thf(fact_111_less__neg__neg,axiom,
! [A: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% less_neg_neg
thf(fact_112_neg__less__pos,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_pos
thf(fact_113_neg__0__less__iff__less,axiom,
! [A: a] :
( ( ord_less_a @ zero_zero_a @ ( uminus_uminus_a @ A ) )
= ( ord_less_a @ A @ zero_zero_a ) ) ).
% neg_0_less_iff_less
thf(fact_114_neg__0__less__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% neg_0_less_iff_less
thf(fact_115_neg__less__0__iff__less,axiom,
! [A: a] :
( ( ord_less_a @ ( uminus_uminus_a @ A ) @ zero_zero_a )
= ( ord_less_a @ zero_zero_a @ A ) ) ).
% neg_less_0_iff_less
thf(fact_116_neg__less__0__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_0_iff_less
thf(fact_117_local_Omem__existence__ivl__iv__defined_I2_J,axiom,
! [T2: real,X0: a] :
( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
=> ( member_a @ X0 @ x ) ) ).
% local.mem_existence_ivl_iv_defined(2)
thf(fact_118_local_Orev_Omem__existence__ivl__iv__defined_I2_J,axiom,
! [T2: real,X0: a] :
( ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
=> ( member_a @ X0 @ x ) ) ).
% local.rev.mem_existence_ivl_iv_defined(2)
thf(fact_119_flow0__defined,axiom,
! [Xa: real,X0: a] :
( ( member_real @ Xa @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
=> ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ X0 @ Xa ) @ x ) ) ).
% flow0_defined
thf(fact_120_local_Oflow__in__domain,axiom,
! [T2: real,X0: a] :
( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
=> ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ X0 @ T2 ) @ x ) ) ).
% local.flow_in_domain
thf(fact_121_local_Omem__existence__ivl__iv__defined_I1_J,axiom,
! [T2: real,X0: a] :
( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
=> ( member_real @ zero_zero_real @ top_top_set_real ) ) ).
% local.mem_existence_ivl_iv_defined(1)
thf(fact_122_local_Oexistence__ivl__initial__time__iff,axiom,
! [X0: a] :
( ( member_real @ zero_zero_real @ ( auto_l612940ivl0_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_123_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_124_rev_Oflow0__defined,axiom,
! [Xa: real,X0: a] :
( ( member_real @ Xa @ ( auto_l612940ivl0_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_125_local_Orev_Oflow__in__domain,axiom,
! [T2: real,X0: a] :
( ( member_real @ T2 @ ( auto_l612940ivl0_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_126_local_Orev_Omem__existence__ivl__iv__defined_I1_J,axiom,
! [T2: real,X0: a] :
( ( member_real @ T2 @ ( auto_l612940ivl0_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_127_local_Orev_Oexistence__ivl__initial__time__iff,axiom,
! [X0: a] :
( ( member_real @ zero_zero_real @ ( auto_l612940ivl0_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_128_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_129_equation__minus__iff,axiom,
! [A: a,B: a] :
( ( A
= ( uminus_uminus_a @ B ) )
= ( B
= ( uminus_uminus_a @ A ) ) ) ).
% equation_minus_iff
thf(fact_130_equation__minus__iff,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% equation_minus_iff
thf(fact_131_minus__equation__iff,axiom,
! [A: a,B: a] :
( ( ( uminus_uminus_a @ A )
= B )
= ( ( uminus_uminus_a @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_132_minus__equation__iff,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( uminus_uminus_real @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_133_less__minus__iff,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ ( uminus_uminus_a @ B ) )
= ( ord_less_a @ B @ ( uminus_uminus_a @ A ) ) ) ).
% less_minus_iff
thf(fact_134_less__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% less_minus_iff
thf(fact_135_minus__less__iff,axiom,
! [A: a,B: a] :
( ( ord_less_a @ ( uminus_uminus_a @ A ) @ B )
= ( ord_less_a @ ( uminus_uminus_a @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_136_minus__less__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_137_abs__eq__iff,axiom,
! [X: real,Y: real] :
( ( ( abs_abs_real @ X )
= ( abs_abs_real @ Y ) )
= ( ( X = Y )
| ( X
= ( uminus_uminus_real @ Y ) ) ) ) ).
% abs_eq_iff
thf(fact_138_abs__less__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( abs_abs_real @ A ) @ B )
= ( ( ord_less_real @ A @ B )
& ( ord_less_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% abs_less_iff
thf(fact_139_abs__if,axiom,
( abs_abs_real
= ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).
% abs_if
thf(fact_140_abs__of__neg,axiom,
! [A: a] :
( ( ord_less_a @ A @ zero_zero_a )
=> ( ( abs_abs_a @ A )
= ( uminus_uminus_a @ A ) ) ) ).
% abs_of_neg
thf(fact_141_abs__of__neg,axiom,
! [A: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( abs_abs_real @ A )
= ( uminus_uminus_real @ A ) ) ) ).
% abs_of_neg
thf(fact_142__092_060alpha_062__limit__point__eq__rev,axiom,
! [X: a,P2: a] :
( ( limit_1865182413oint_a @ f @ x @ X @ P2 )
= ( limit_94460170oint_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ P2 ) ) ).
% \<alpha>_limit_point_eq_rev
thf(fact_143_rev_O_092_060alpha_062__limit__point__eq__rev,axiom,
! [X: a,P2: a] :
( ( limit_1865182413oint_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ P2 )
= ( limit_94460170oint_a @ f @ x @ X @ P2 ) ) ).
% rev.\<alpha>_limit_point_eq_rev
thf(fact_144_rev_Omvar_Ointerval__axioms,axiom,
! [X0: a] : ( initia826609931terval @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) ) ).
% rev.mvar.interval_axioms
thf(fact_145_mvar_Ointerval__axioms,axiom,
! [X0: a] : ( initia826609931terval @ ( auto_l612940ivl0_a @ f @ x @ X0 ) ) ).
% mvar.interval_axioms
thf(fact_146_abs__neg__one,axiom,
( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
= one_one_real ) ).
% abs_neg_one
thf(fact_147_rev_Oclosed__orbitE,axiom,
! [X: a] :
( ( period720806154rbit_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_148_local_Orev_Oexistence__ivl__notempty,axiom,
! [X0: a] :
( ( member_real @ zero_zero_real @ top_top_set_real )
=> ( ( member_a @ X0 @ x )
=> ( ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 )
!= bot_bot_set_real ) ) ) ).
% local.rev.existence_ivl_notempty
thf(fact_149_rev_Oclosed__orbit__period__nonneg,axiom,
! [X: a] :
( ( period720806154rbit_a @ ( uminus_uminus_a_a @ f ) @ x @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( period1305449585riod_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ).
% rev.closed_orbit_period_nonneg
thf(fact_150_rev_Ovec__simps_I11_J,axiom,
! [Ae: a,Be: a,Cd: a] :
( ( plus_plus_a @ ( plus_plus_a @ Ae @ Be ) @ Cd )
= ( plus_plus_a @ Ae @ ( plus_plus_a @ Be @ Cd ) ) ) ).
% rev.vec_simps(11)
thf(fact_151_rev_Ovec__simps_I11_J,axiom,
! [Ae: real,Be: real,Cd: real] :
( ( plus_plus_real @ ( plus_plus_real @ Ae @ Be ) @ Cd )
= ( plus_plus_real @ Ae @ ( plus_plus_real @ Be @ Cd ) ) ) ).
% rev.vec_simps(11)
thf(fact_152_rev_Ovec__simps_I12_J,axiom,
( plus_plus_a
= ( ^ [Ag: a,Bg: a] : ( plus_plus_a @ Bg @ Ag ) ) ) ).
% rev.vec_simps(12)
thf(fact_153_rev_Ovec__simps_I12_J,axiom,
( plus_plus_real
= ( ^ [Ag: real,Bg: real] : ( plus_plus_real @ Bg @ Ag ) ) ) ).
% rev.vec_simps(12)
thf(fact_154_rev_Ovec__simps_I13_J,axiom,
! [Bg2: a,Ag2: a,Cf: a] :
( ( plus_plus_a @ Bg2 @ ( plus_plus_a @ Ag2 @ Cf ) )
= ( plus_plus_a @ Ag2 @ ( plus_plus_a @ Bg2 @ Cf ) ) ) ).
% rev.vec_simps(13)
thf(fact_155_rev_Ovec__simps_I13_J,axiom,
! [Bg2: real,Ag2: real,Cf: real] :
( ( plus_plus_real @ Bg2 @ ( plus_plus_real @ Ag2 @ Cf ) )
= ( plus_plus_real @ Ag2 @ ( plus_plus_real @ Bg2 @ Cf ) ) ) ).
% rev.vec_simps(13)
thf(fact_156_interval__axioms,axiom,
initia826609931terval @ top_top_set_real ).
% interval_axioms
thf(fact_157_add__right__cancel,axiom,
! [B: a,A: a,C: a] :
( ( ( plus_plus_a @ B @ A )
= ( plus_plus_a @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_158_add__right__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_159_add__left__cancel,axiom,
! [A: a,B: a,C: a] :
( ( ( plus_plus_a @ A @ B )
= ( plus_plus_a @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_160_add__left__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_161_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_162_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_163_recurrence__time__flip__sign_I1_J,axiom,
! [T6: real,X: a] :
( ( member_real @ T6 @ ( auto_l612940ivl0_a @ f @ x @ X ) )
=> ( ( ( auto_ll_on_flow0_a @ f @ x @ X @ T6 )
= X )
=> ( member_real @ ( uminus_uminus_real @ T6 ) @ ( auto_l612940ivl0_a @ f @ x @ X ) ) ) ) ).
% recurrence_time_flip_sign(1)
thf(fact_164_recurrence__time__flip__sign_I2_J,axiom,
! [T6: real,X: a] :
( ( member_real @ T6 @ ( auto_l612940ivl0_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_165_rev_Orecurrence__time__flip__sign_I1_J,axiom,
! [T6: real,X: a] :
( ( member_real @ T6 @ ( auto_l612940ivl0_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_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ) ).
% rev.recurrence_time_flip_sign(1)
thf(fact_166_rev_Orecurrence__time__flip__sign_I2_J,axiom,
! [T6: real,X: a] :
( ( member_real @ T6 @ ( auto_l612940ivl0_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_167_local_Oflow__trans,axiom,
! [S: real,X0: a,T2: real] :
( ( member_real @ S @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
=> ( ( member_real @ T2 @ ( auto_l612940ivl0_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_168_local_Oexistence__ivl__trans_H,axiom,
! [T2: real,S: real,X0: a] :
( ( member_real @ ( plus_plus_real @ T2 @ S ) @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
=> ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
=> ( member_real @ S @ ( auto_l612940ivl0_a @ f @ x @ ( auto_ll_on_flow0_a @ f @ x @ X0 @ T2 ) ) ) ) ) ).
% local.existence_ivl_trans'
thf(fact_169_local_Oexistence__ivl__trans,axiom,
! [S: real,X0: a,T2: real] :
( ( member_real @ S @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
=> ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ ( auto_ll_on_flow0_a @ f @ x @ X0 @ S ) ) )
=> ( member_real @ ( plus_plus_real @ S @ T2 ) @ ( auto_l612940ivl0_a @ f @ x @ X0 ) ) ) ) ).
% local.existence_ivl_trans
thf(fact_170_local_Orev_Oflow__trans,axiom,
! [S: real,X0: a,T2: real] :
( ( member_real @ S @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
=> ( ( member_real @ T2 @ ( auto_l612940ivl0_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_171_local_Orev_Oexistence__ivl__trans_H,axiom,
! [T2: real,S: real,X0: a] :
( ( member_real @ ( plus_plus_real @ T2 @ S ) @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
=> ( ( member_real @ T2 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
=> ( member_real @ S @ ( auto_l612940ivl0_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_172_local_Orev_Oexistence__ivl__trans,axiom,
! [S: real,X0: a,T2: real] :
( ( member_real @ S @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
=> ( ( member_real @ T2 @ ( auto_l612940ivl0_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_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) ) ) ) ).
% local.rev.existence_ivl_trans
thf(fact_173_local_Oexistence__ivl__notempty,axiom,
! [X0: a] :
( ( member_real @ zero_zero_real @ top_top_set_real )
=> ( ( member_a @ X0 @ x )
=> ( ( auto_l612940ivl0_a @ f @ x @ X0 )
!= bot_bot_set_real ) ) ) ).
% local.existence_ivl_notempty
thf(fact_174_closed__orbit__period__nonneg,axiom,
! [X: a] :
( ( period720806154rbit_a @ f @ x @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( period1305449585riod_a @ f @ x @ X ) ) ) ).
% closed_orbit_period_nonneg
thf(fact_175_neg__le__iff__le,axiom,
! [B: a,A: a] :
( ( ord_less_eq_a @ ( uminus_uminus_a @ B ) @ ( uminus_uminus_a @ A ) )
= ( ord_less_eq_a @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_176_neg__le__iff__le,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_177_add__cancel__right__right,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ A @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_178_add__cancel__right__right,axiom,
! [A: a,B: a] :
( ( A
= ( plus_plus_a @ A @ B ) )
= ( B = zero_zero_a ) ) ).
% add_cancel_right_right
thf(fact_179_add__cancel__right__left,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ B @ A ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_180_add__cancel__right__left,axiom,
! [A: a,B: a] :
( ( A
= ( plus_plus_a @ B @ A ) )
= ( B = zero_zero_a ) ) ).
% add_cancel_right_left
thf(fact_181_add__cancel__left__right,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_182_add__cancel__left__right,axiom,
! [A: a,B: a] :
( ( ( plus_plus_a @ A @ B )
= A )
= ( B = zero_zero_a ) ) ).
% add_cancel_left_right
thf(fact_183_add__cancel__left__left,axiom,
! [B: real,A: real] :
( ( ( plus_plus_real @ B @ A )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_184_add__cancel__left__left,axiom,
! [B: a,A: a] :
( ( ( plus_plus_a @ B @ A )
= A )
= ( B = zero_zero_a ) ) ).
% add_cancel_left_left
thf(fact_185_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_186_linordered__ab__group__add__class_Odouble__zero,axiom,
! [A: real] :
( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% linordered_ab_group_add_class.double_zero
thf(fact_187_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_188_add_Oright__neutral,axiom,
! [A: a] :
( ( plus_plus_a @ A @ zero_zero_a )
= A ) ).
% add.right_neutral
thf(fact_189_add_Oleft__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.left_neutral
thf(fact_190_add_Oleft__neutral,axiom,
! [A: a] :
( ( plus_plus_a @ zero_zero_a @ A )
= A ) ).
% add.left_neutral
thf(fact_191_add__le__cancel__right,axiom,
! [A: a,C: a,B: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) )
= ( ord_less_eq_a @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_192_add__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_193_add__le__cancel__left,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) )
= ( ord_less_eq_a @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_194_add__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_195_add__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_196_add__less__cancel__right,axiom,
! [A: a,C: a,B: a] :
( ( ord_less_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) )
= ( ord_less_a @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_197_add__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_198_add__less__cancel__left,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) )
= ( ord_less_a @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_199_closed__orbitE,axiom,
! [X: a] :
( ( period720806154rbit_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_200_minus__add__distrib,axiom,
! [A: a,B: a] :
( ( uminus_uminus_a @ ( plus_plus_a @ A @ B ) )
= ( plus_plus_a @ ( uminus_uminus_a @ A ) @ ( uminus_uminus_a @ B ) ) ) ).
% minus_add_distrib
thf(fact_201_minus__add__distrib,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% minus_add_distrib
thf(fact_202_minus__add__cancel,axiom,
! [A: a,B: a] :
( ( plus_plus_a @ ( uminus_uminus_a @ A ) @ ( plus_plus_a @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_203_minus__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_204_add__minus__cancel,axiom,
! [A: a,B: a] :
( ( plus_plus_a @ A @ ( plus_plus_a @ ( uminus_uminus_a @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_205_add__minus__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_206_abs__add__abs,axiom,
! [A: a,B: a] :
( ( abs_abs_a @ ( plus_plus_a @ ( abs_abs_a @ A ) @ ( abs_abs_a @ B ) ) )
= ( plus_plus_a @ ( abs_abs_a @ A ) @ ( abs_abs_a @ B ) ) ) ).
% abs_add_abs
thf(fact_207_abs__add__abs,axiom,
! [A: real,B: real] :
( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
= ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% abs_add_abs
thf(fact_208_neg__0__le__iff__le,axiom,
! [A: a] :
( ( ord_less_eq_a @ zero_zero_a @ ( uminus_uminus_a @ A ) )
= ( ord_less_eq_a @ A @ zero_zero_a ) ) ).
% neg_0_le_iff_le
thf(fact_209_neg__0__le__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% neg_0_le_iff_le
thf(fact_210_neg__le__0__iff__le,axiom,
! [A: a] :
( ( ord_less_eq_a @ ( uminus_uminus_a @ A ) @ zero_zero_a )
= ( ord_less_eq_a @ zero_zero_a @ A ) ) ).
% neg_le_0_iff_le
thf(fact_211_neg__le__0__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_le_0_iff_le
thf(fact_212_less__eq__neg__nonpos,axiom,
! [A: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% less_eq_neg_nonpos
thf(fact_213_neg__less__eq__nonneg,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_214_linordered__ab__group__add__class_Ozero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% linordered_ab_group_add_class.zero_le_double_add_iff_zero_le_single_add
thf(fact_215_linordered__ab__group__add__class_Odouble__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% linordered_ab_group_add_class.double_add_le_zero_iff_single_add_le_zero
thf(fact_216_le__add__same__cancel2,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ ( plus_plus_a @ B @ A ) )
= ( ord_less_eq_a @ zero_zero_a @ B ) ) ).
% le_add_same_cancel2
thf(fact_217_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_218_le__add__same__cancel1,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ ( plus_plus_a @ A @ B ) )
= ( ord_less_eq_a @ zero_zero_a @ B ) ) ).
% le_add_same_cancel1
thf(fact_219_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_220_add__le__same__cancel2,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ A @ B ) @ B )
= ( ord_less_eq_a @ A @ zero_zero_a ) ) ).
% add_le_same_cancel2
thf(fact_221_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_222_add__le__same__cancel1,axiom,
! [B: a,A: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ B @ A ) @ B )
= ( ord_less_eq_a @ A @ zero_zero_a ) ) ).
% add_le_same_cancel1
thf(fact_223_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_224_linordered__ab__group__add__class_Ozero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% linordered_ab_group_add_class.zero_less_double_add_iff_zero_less_single_add
thf(fact_225_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_226_less__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_227_less__add__same__cancel2,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ ( plus_plus_a @ B @ A ) )
= ( ord_less_a @ zero_zero_a @ B ) ) ).
% less_add_same_cancel2
thf(fact_228_less__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_229_less__add__same__cancel1,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ ( plus_plus_a @ A @ B ) )
= ( ord_less_a @ zero_zero_a @ B ) ) ).
% less_add_same_cancel1
thf(fact_230_add__less__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_231_add__less__same__cancel2,axiom,
! [A: a,B: a] :
( ( ord_less_a @ ( plus_plus_a @ A @ B ) @ B )
= ( ord_less_a @ A @ zero_zero_a ) ) ).
% add_less_same_cancel2
thf(fact_232_add__less__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_233_add__less__same__cancel1,axiom,
! [B: a,A: a] :
( ( ord_less_a @ ( plus_plus_a @ B @ A ) @ B )
= ( ord_less_a @ A @ zero_zero_a ) ) ).
% add_less_same_cancel1
thf(fact_234_add_Oleft__inverse,axiom,
! [A: a] :
( ( plus_plus_a @ ( uminus_uminus_a @ A ) @ A )
= zero_zero_a ) ).
% add.left_inverse
thf(fact_235_add_Oleft__inverse,axiom,
! [A: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
= zero_zero_real ) ).
% add.left_inverse
thf(fact_236_add_Oright__inverse,axiom,
! [A: a] :
( ( plus_plus_a @ A @ ( uminus_uminus_a @ A ) )
= zero_zero_a ) ).
% add.right_inverse
thf(fact_237_add_Oright__inverse,axiom,
! [A: real] :
( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
= zero_zero_real ) ).
% add.right_inverse
thf(fact_238_abs__le__zero__iff,axiom,
! [A: a] :
( ( ord_less_eq_a @ ( abs_abs_a @ A ) @ zero_zero_a )
= ( A = zero_zero_a ) ) ).
% abs_le_zero_iff
thf(fact_239_abs__le__zero__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
= ( A = zero_zero_real ) ) ).
% abs_le_zero_iff
thf(fact_240_abs__le__self__iff,axiom,
! [A: a] :
( ( ord_less_eq_a @ ( abs_abs_a @ A ) @ A )
= ( ord_less_eq_a @ zero_zero_a @ A ) ) ).
% abs_le_self_iff
thf(fact_241_abs__le__self__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% abs_le_self_iff
thf(fact_242_abs__of__nonneg,axiom,
! [A: a] :
( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( abs_abs_a @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_243_abs__of__nonneg,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( abs_abs_real @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_244_ball__trivial,axiom,
! [X: a] :
( ( elemen154694473ball_a @ X @ zero_zero_real )
= bot_bot_set_a ) ).
% ball_trivial
thf(fact_245_ball__trivial,axiom,
! [X: real] :
( ( elemen341675809l_real @ X @ zero_zero_real )
= bot_bot_set_real ) ).
% ball_trivial
thf(fact_246_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% add_neg_numeral_special(7)
thf(fact_247_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
= zero_zero_real ) ).
% add_neg_numeral_special(8)
thf(fact_248_abs__of__nonpos,axiom,
! [A: a] :
( ( ord_less_eq_a @ A @ zero_zero_a )
=> ( ( abs_abs_a @ A )
= ( uminus_uminus_a @ A ) ) ) ).
% abs_of_nonpos
thf(fact_249_abs__of__nonpos,axiom,
! [A: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( abs_abs_real @ A )
= ( uminus_uminus_real @ A ) ) ) ).
% abs_of_nonpos
thf(fact_250_ball__eq__empty,axiom,
! [X: a,E: real] :
( ( ( elemen154694473ball_a @ X @ E )
= bot_bot_set_a )
= ( ord_less_eq_real @ E @ zero_zero_real ) ) ).
% ball_eq_empty
thf(fact_251_ball__eq__empty,axiom,
! [X: real,E: real] :
( ( ( elemen341675809l_real @ X @ E )
= bot_bot_set_real )
= ( ord_less_eq_real @ E @ zero_zero_real ) ) ).
% ball_eq_empty
thf(fact_252_local_Oexistence__ivl__undefined,axiom,
! [X0: a] :
( ~ ( member_a @ X0 @ x )
=> ( ( auto_l612940ivl0_a @ f @ x @ X0 )
= bot_bot_set_real ) ) ).
% local.existence_ivl_undefined
thf(fact_253_local_Orev_Oexistence__ivl__undefined,axiom,
! [X0: a] :
( ~ ( member_a @ X0 @ x )
=> ( ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 )
= bot_bot_set_real ) ) ).
% local.rev.existence_ivl_undefined
thf(fact_254_local_Oexistence__ivl__empty__iff,axiom,
! [X0: a] :
( ( ( auto_l612940ivl0_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_255_local_Oexistence__ivl__empty1,axiom,
! [X0: a] :
( ~ ( member_real @ zero_zero_real @ top_top_set_real )
=> ( ( auto_l612940ivl0_a @ f @ x @ X0 )
= bot_bot_set_real ) ) ).
% local.existence_ivl_empty1
thf(fact_256_local_Orev_Oexistence__ivl__empty__iff,axiom,
! [X0: a] :
( ( ( auto_l612940ivl0_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_257_local_Orev_Oexistence__ivl__empty1,axiom,
! [X0: a] :
( ~ ( member_real @ zero_zero_real @ top_top_set_real )
=> ( ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 )
= bot_bot_set_real ) ) ).
% local.rev.existence_ivl_empty1
thf(fact_258_is__num__normalize_I8_J,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_259_abs__triangle__ineq,axiom,
! [A: a,B: a] : ( ord_less_eq_a @ ( abs_abs_a @ ( plus_plus_a @ A @ B ) ) @ ( plus_plus_a @ ( abs_abs_a @ A ) @ ( abs_abs_a @ B ) ) ) ).
% abs_triangle_ineq
thf(fact_260_abs__triangle__ineq,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% abs_triangle_ineq
thf(fact_261_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_262_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_263_is__num__normalize_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_264_add__le__imp__le__right,axiom,
! [A: a,C: a,B: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) )
=> ( ord_less_eq_a @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_265_add__le__imp__le__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_266_add__le__imp__le__left,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_eq_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) )
=> ( ord_less_eq_a @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_267_add__le__imp__le__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_268_add__less__le__mono,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_eq_a @ C @ D2 )
=> ( ord_less_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_269_add__less__le__mono,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D2 )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_270_add__le__less__mono,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_a @ C @ D2 )
=> ( ord_less_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_271_add__le__less__mono,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C @ D2 )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_272_add__nonpos__eq__0__iff,axiom,
! [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_273_add__nonpos__eq__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_274_add__nonneg__eq__0__iff,axiom,
! [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_275_add__nonneg__eq__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_276_add__nonpos__nonpos,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ zero_zero_a )
=> ( ( ord_less_eq_a @ B @ zero_zero_a )
=> ( ord_less_eq_a @ ( plus_plus_a @ A @ B ) @ zero_zero_a ) ) ) ).
% add_nonpos_nonpos
thf(fact_277_add__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_278_add__nonneg__nonneg,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ zero_zero_a @ B )
=> ( ord_less_eq_a @ zero_zero_a @ ( plus_plus_a @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_279_add__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_280_add__increasing2,axiom,
! [C: a,B: a,A: a] :
( ( ord_less_eq_a @ zero_zero_a @ C )
=> ( ( ord_less_eq_a @ B @ A )
=> ( ord_less_eq_a @ B @ ( plus_plus_a @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_281_add__increasing2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_282_add__decreasing2,axiom,
! [C: a,A: a,B: a] :
( ( ord_less_eq_a @ C @ zero_zero_a )
=> ( ( ord_less_eq_a @ A @ B )
=> ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_283_add__decreasing2,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_284_add__right__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) ) ) ).
% add_right_mono
thf(fact_285_add__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_right_mono
thf(fact_286_add__increasing,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_eq_a @ B @ ( plus_plus_a @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_287_add__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_288_add__decreasing,axiom,
! [A: a,C: a,B: a] :
( ( ord_less_eq_a @ A @ zero_zero_a )
=> ( ( ord_less_eq_a @ C @ B )
=> ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_289_add__decreasing,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_290_add__left__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ord_less_eq_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) ) ) ).
% add_left_mono
thf(fact_291_add__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_left_mono
thf(fact_292_add__right__imp__eq,axiom,
! [B: a,A: a,C: a] :
( ( ( plus_plus_a @ B @ A )
= ( plus_plus_a @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_293_add__right__imp__eq,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_294_add__left__imp__eq,axiom,
! [A: a,B: a,C: a] :
( ( ( plus_plus_a @ A @ B )
= ( plus_plus_a @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_295_add__left__imp__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_296_add__mono,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ C @ D2 )
=> ( ord_less_eq_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_297_add__mono,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_298_add_Oleft__commute,axiom,
! [B: a,A: a,C: a] :
( ( plus_plus_a @ B @ ( plus_plus_a @ A @ C ) )
= ( plus_plus_a @ A @ ( plus_plus_a @ B @ C ) ) ) ).
% add.left_commute
thf(fact_299_add_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.left_commute
thf(fact_300_add_Ocommute,axiom,
( plus_plus_a
= ( ^ [A4: a,B3: a] : ( plus_plus_a @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_301_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A4: real,B3: real] : ( plus_plus_real @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_302_add_Oright__cancel,axiom,
! [B: a,A: a,C: a] :
( ( ( plus_plus_a @ B @ A )
= ( plus_plus_a @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_303_add_Oright__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_304_add_Oleft__cancel,axiom,
! [A: a,B: a,C: a] :
( ( ( plus_plus_a @ A @ B )
= ( plus_plus_a @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_305_add_Oleft__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_306_add_Oassoc,axiom,
! [A: a,B: a,C: a] :
( ( plus_plus_a @ ( plus_plus_a @ A @ B ) @ C )
= ( plus_plus_a @ A @ ( plus_plus_a @ B @ C ) ) ) ).
% add.assoc
thf(fact_307_add_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.assoc
thf(fact_308_group__cancel_Oadd2,axiom,
! [B4: a,K: a,B: a,A: a] :
( ( B4
= ( plus_plus_a @ K @ B ) )
=> ( ( plus_plus_a @ A @ B4 )
= ( plus_plus_a @ K @ ( plus_plus_a @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_309_group__cancel_Oadd2,axiom,
! [B4: real,K: real,B: real,A: real] :
( ( B4
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A @ B4 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_310_group__cancel_Oadd1,axiom,
! [A3: a,K: a,A: a,B: a] :
( ( A3
= ( plus_plus_a @ K @ A ) )
=> ( ( plus_plus_a @ A3 @ B )
= ( plus_plus_a @ K @ ( plus_plus_a @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_311_group__cancel_Oadd1,axiom,
! [A3: real,K: real,A: real,B: real] :
( ( A3
= ( plus_plus_real @ K @ A ) )
=> ( ( plus_plus_real @ A3 @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_312_add__mono__thms__linordered__semiring_I1_J,axiom,
! [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_313_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_314_add__mono__thms__linordered__semiring_I2_J,axiom,
! [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_315_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_316_add__mono__thms__linordered__semiring_I3_J,axiom,
! [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_317_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_318_add__mono__thms__linordered__semiring_I4_J,axiom,
! [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_319_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_real @ I @ K )
= ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_320_add__mono__thms__linordered__field_I3_J,axiom,
! [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_321_add__mono__thms__linordered__field_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_322_add__mono__thms__linordered__field_I4_J,axiom,
! [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_323_add__mono__thms__linordered__field_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_324_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: a,B: a,C: a] :
( ( plus_plus_a @ ( plus_plus_a @ A @ B ) @ C )
= ( plus_plus_a @ A @ ( plus_plus_a @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_325_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_326_add__strict__increasing2,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ B @ C )
=> ( ord_less_a @ B @ ( plus_plus_a @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_327_add__strict__increasing2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_328_add__strict__increasing,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_a @ B @ ( plus_plus_a @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_329_add__strict__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_330_add__pos__nonneg,axiom,
! [A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_eq_a @ zero_zero_a @ B )
=> ( ord_less_a @ zero_zero_a @ ( plus_plus_a @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_331_add__pos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_332_add__nonpos__neg,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ zero_zero_a )
=> ( ( ord_less_a @ B @ zero_zero_a )
=> ( ord_less_a @ ( plus_plus_a @ A @ B ) @ zero_zero_a ) ) ) ).
% add_nonpos_neg
thf(fact_333_add__nonpos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_neg
thf(fact_334_add__nonneg__pos,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ zero_zero_a @ B )
=> ( ord_less_a @ zero_zero_a @ ( plus_plus_a @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_335_add__nonneg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_336_add__neg__nonpos,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ zero_zero_a )
=> ( ( ord_less_eq_a @ B @ zero_zero_a )
=> ( ord_less_a @ ( plus_plus_a @ A @ B ) @ zero_zero_a ) ) ) ).
% add_neg_nonpos
thf(fact_337_add__neg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_nonpos
thf(fact_338_Elementary__Metric__Spaces_Oball__empty,axiom,
! [E: real,X: a] :
( ( ord_less_eq_real @ E @ zero_zero_real )
=> ( ( elemen154694473ball_a @ X @ E )
= bot_bot_set_a ) ) ).
% Elementary_Metric_Spaces.ball_empty
thf(fact_339_Elementary__Metric__Spaces_Oball__empty,axiom,
! [E: real,X: real] :
( ( ord_less_eq_real @ E @ zero_zero_real )
=> ( ( elemen341675809l_real @ X @ E )
= bot_bot_set_real ) ) ).
% Elementary_Metric_Spaces.ball_empty
thf(fact_340_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_341_add_Ogroup__left__neutral,axiom,
! [A: a] :
( ( plus_plus_a @ zero_zero_a @ A )
= A ) ).
% add.group_left_neutral
thf(fact_342_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_343_add_Ocomm__neutral,axiom,
! [A: a] :
( ( plus_plus_a @ A @ zero_zero_a )
= A ) ).
% add.comm_neutral
% Helper facts (3)
thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $true @ X @ Y )
= X ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
! [Delta2: real,Tau2: a > real] :
( ( ord_less_real @ zero_zero_real @ Delta2 )
=> ( ( topolo1710226732a_real @ ( elemen154694473ball_a @ p @ Delta2 ) @ Tau2 )
=> ( ( ( Tau2 @ p )
= zero_zero_real )
=> ( ! [Y3: a] :
( ( member_a @ Y3 @ ( elemen154694473ball_a @ p @ Delta2 ) )
=> ( ord_less_real @ ( abs_abs_real @ ( Tau2 @ Y3 ) ) @ one_one_real ) )
=> ( ! [Y3: a] :
( ( member_a @ Y3 @ ( elemen154694473ball_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 ).
%------------------------------------------------------------------------------