TPTP Problem File: ITP146^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP146^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Poincare_Bendixson problem prob_1288__19593336_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_1288__19593336_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 357 ( 54 unt; 52 typ; 0 def)
% Number of atoms : 821 ( 291 equ; 0 cnn)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 4287 ( 41 ~; 2 |; 28 &;3844 @)
% ( 0 <=>; 372 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 192 ( 192 >; 0 *; 0 +; 0 <<)
% Number of symbols : 51 ( 50 usr; 9 con; 0-5 aty)
% Number of variables : 727 ( 7 ^; 680 !; 2 ?; 727 :)
% ( 38 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:31:42.052
%------------------------------------------------------------------------------
% 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 (49)
thf(sy_cl_Executable__Euclidean__Space_Oexecutable__euclidean__space,type,
execut510477386_space:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[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_Groups_Oab__group__add,type,
ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[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_Ocomm__monoid__diff,type,
comm_monoid_diff:
!>[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_Ocancel__ab__semigroup__add,type,
cancel146912293up_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__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
strict797366125id_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_Ostrict__ordered__ab__semigroup__add,type,
strict2144017051up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
ordere516151231imp_le:
!>[A: $tType] : $o ).
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_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > 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_Initial__Value__Problem_Ointerval,type,
initia826609931terval: ( set @ real ) > $o ).
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_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_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_X,type,
x: set @ a ).
thf(sy_v_d____,type,
d: real ).
thf(sy_v_f,type,
f: a > a ).
thf(sy_v_i1____,type,
i1: real ).
thf(sy_v_i2____,type,
i2: real ).
thf(sy_v_ss____,type,
ss: real ).
thf(sy_v_t,type,
t: real ).
thf(sy_v_tt____,type,
tt: real ).
thf(sy_v_x,type,
x2: a ).
thf(sy_v_xx____,type,
xx: a ).
% Relevant facts (255)
thf(fact_0__092_060open_062xx_A_092_060in_062_AX_092_060close_062,axiom,
member @ a @ xx @ x ).
% \<open>xx \<in> X\<close>
thf(fact_1__092_060open_062flow0_A_Iflow0_Axx_Att_J_A_I_N_Att_J_A_061_Aflow0_A_Iflow0_Ax_Ass_J_A_I_N_Att_J_092_060close_062,axiom,
( ( auto_ll_on_flow0 @ a @ f @ x @ ( auto_ll_on_flow0 @ a @ f @ x @ xx @ tt ) @ ( uminus_uminus @ real @ tt ) )
= ( auto_ll_on_flow0 @ a @ f @ x @ ( auto_ll_on_flow0 @ a @ f @ x @ x2 @ ss ) @ ( uminus_uminus @ real @ tt ) ) ) ).
% \<open>flow0 (flow0 xx tt) (- tt) = flow0 (flow0 x ss) (- tt)\<close>
thf(fact_2_eq,axiom,
( ( auto_ll_on_flow0 @ a @ f @ x @ xx @ tt )
= ( auto_ll_on_flow0 @ a @ f @ x @ x2 @ ss ) ) ).
% eq
thf(fact_3_assms_I8_J,axiom,
member @ real @ t @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ x2 ) ).
% assms(8)
thf(fact_4_tt__ex,axiom,
member @ real @ tt @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ xx ) ).
% tt_ex
thf(fact_5_ss__ex,axiom,
member @ real @ ss @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ x2 ) ).
% ss_ex
thf(fact_6_fixpoint__sol_I2_J,axiom,
! [X: a,T: real] :
( ( member @ a @ X @ x )
=> ( ( ( f @ X )
= ( zero_zero @ a ) )
=> ( ( auto_ll_on_flow0 @ a @ f @ x @ X @ T )
= X ) ) ) ).
% fixpoint_sol(2)
thf(fact_7_neg__tt__ex,axiom,
member @ real @ ( uminus_uminus @ real @ tt ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ ( auto_ll_on_flow0 @ a @ f @ x @ xx @ tt ) ) ).
% neg_tt_ex
thf(fact_8_diff__existence__ivl__trans,axiom,
! [T0: real,X0: a,T: real] :
( ( member @ real @ T0 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( ( member @ real @ T @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( member @ real @ ( minus_minus @ real @ T @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ T0 ) ) ) ) ) ).
% diff_existence_ivl_trans
thf(fact_9_general_Oexistence__ivl__reverse,axiom,
! [T: real,T0: real,X0: a] :
( ( member @ real @ ( minus_minus @ real @ T @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( member @ real @ ( minus_minus @ real @ T0 @ T ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ ( minus_minus @ real @ T @ T0 ) ) ) ) ) ).
% general.existence_ivl_reverse
thf(fact_10_general_Oflow__in__domain,axiom,
! [T: real,T0: real,X0: a] :
( ( member @ real @ ( minus_minus @ real @ T @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( member @ a @ ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ ( minus_minus @ real @ T @ T0 ) ) @ x ) ) ).
% general.flow_in_domain
thf(fact_11_general_Oflows__reverse,axiom,
! [T: real,T0: real,X0: a] :
( ( member @ real @ ( minus_minus @ real @ T @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( ( auto_ll_on_flow0 @ a @ f @ x @ ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ ( minus_minus @ real @ T @ T0 ) ) @ ( minus_minus @ real @ T0 @ T ) )
= X0 ) ) ).
% general.flows_reverse
thf(fact_12_flow0__cong,axiom,
! [B: $tType,Y: set @ a,G: a > a,T: real,X0: a] :
( ( x = Y )
=> ( ! [X2: a,T2: B] :
( ( member @ a @ X2 @ Y )
=> ( ( f @ X2 )
= ( G @ X2 ) ) )
=> ( ( member @ real @ T @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ T )
= ( auto_ll_on_flow0 @ a @ G @ Y @ X0 @ T ) ) ) ) ) ).
% flow0_cong
thf(fact_13_general_Omem__existence__ivl__iv__defined_I2_J,axiom,
! [T: real,T0: real,X0: a] :
( ( member @ real @ ( minus_minus @ real @ T @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( member @ a @ X0 @ x ) ) ).
% general.mem_existence_ivl_iv_defined(2)
thf(fact_14_closed__orbit__flow0,axiom,
! [X: a,T: real] :
( ( period385816147_orbit @ a @ f @ x @ X )
=> ( period385816147_orbit @ a @ f @ x @ ( auto_ll_on_flow0 @ a @ f @ x @ X @ T ) ) ) ).
% closed_orbit_flow0
thf(fact_15_existence__ivl0__cong,axiom,
! [B: $tType,Y: set @ a,G: a > a,X0: a] :
( ( x = Y )
=> ( ! [X2: a,T2: B] :
( ( member @ a @ X2 @ Y )
=> ( ( f @ X2 )
= ( G @ X2 ) ) )
=> ( ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 )
= ( auto_l1112008849e_ivl0 @ a @ G @ Y @ X0 ) ) ) ) ).
% existence_ivl0_cong
thf(fact_16_closed__orbit__in__domain,axiom,
! [X: a] :
( ( period385816147_orbit @ a @ f @ x @ X )
=> ( member @ a @ X @ x ) ) ).
% closed_orbit_in_domain
thf(fact_17_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_18_recurrence__time__flip__sign_I1_J,axiom,
! [T3: real,X: a] :
( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) )
=> ( ( ( auto_ll_on_flow0 @ a @ f @ x @ X @ T3 )
= X )
=> ( member @ real @ ( uminus_uminus @ real @ T3 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) ) ) ) ).
% recurrence_time_flip_sign(1)
thf(fact_19_recurrence__time__flip__sign_I2_J,axiom,
! [T3: real,X: a] :
( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) )
=> ( ( ( auto_ll_on_flow0 @ a @ f @ x @ X @ T3 )
= X )
=> ( ( auto_ll_on_flow0 @ a @ f @ x @ X @ ( uminus_uminus @ real @ T3 ) )
= X ) ) ) ).
% recurrence_time_flip_sign(2)
thf(fact_20_local_Oflow__undefined0,axiom,
! [T: real,X: a] :
( ~ ( member @ real @ T @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) )
=> ( ( auto_ll_on_flow0 @ a @ f @ x @ X @ T )
= ( zero_zero @ a ) ) ) ).
% local.flow_undefined0
thf(fact_21_closed__orbitI,axiom,
! [T: real,T4: real,Y2: a] :
( ( T != T4 )
=> ( ( member @ real @ T @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ Y2 ) )
=> ( ( member @ real @ T4 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ Y2 ) )
=> ( ( ( auto_ll_on_flow0 @ a @ f @ x @ Y2 @ T )
= ( auto_ll_on_flow0 @ a @ f @ x @ Y2 @ T4 ) )
=> ( period385816147_orbit @ a @ f @ x @ Y2 ) ) ) ) ) ).
% closed_orbitI
thf(fact_22_general_Oflow__undefined0,axiom,
! [T: real,T0: real,X: a] :
( ~ ( member @ real @ ( minus_minus @ real @ T @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) )
=> ( ( auto_ll_on_flow0 @ a @ f @ x @ X @ ( minus_minus @ real @ T @ T0 ) )
= ( zero_zero @ a ) ) ) ).
% general.flow_undefined0
thf(fact_23_local_Omem__existence__ivl__iv__defined_I2_J,axiom,
! [T: real,X0: a] :
( ( member @ real @ T @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( member @ a @ X0 @ x ) ) ).
% local.mem_existence_ivl_iv_defined(2)
thf(fact_24_local_Oflow__in__domain,axiom,
! [T: real,X0: a] :
( ( member @ real @ T @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( member @ a @ ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ T ) @ x ) ) ).
% local.flow_in_domain
thf(fact_25_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_26_mvar_Ointerval__axioms,axiom,
! [X0: a] : ( initia826609931terval @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) ) ).
% mvar.interval_axioms
thf(fact_27_periodic__orbitI,axiom,
! [T: real,T4: real,Y2: a] :
( ( T != T4 )
=> ( ( member @ real @ T @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ Y2 ) )
=> ( ( member @ real @ T4 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ Y2 ) )
=> ( ( ( auto_ll_on_flow0 @ a @ f @ x @ Y2 @ T )
= ( auto_ll_on_flow0 @ a @ f @ x @ Y2 @ T4 ) )
=> ( ( ( f @ Y2 )
!= ( zero_zero @ a ) )
=> ( period862636932_orbit @ a @ f @ x @ Y2 ) ) ) ) ) ) ).
% periodic_orbitI
thf(fact_28_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_29_periodic__orbit__imp__flow0__regular,axiom,
! [X: a,T: real] :
( ( period862636932_orbit @ a @ f @ x @ X )
=> ( ( f @ ( auto_ll_on_flow0 @ a @ f @ x @ X @ T ) )
!= ( zero_zero @ a ) ) ) ).
% periodic_orbit_imp_flow0_regular
thf(fact_30_diff__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
= ( uminus_uminus @ A @ A2 ) ) ) ).
% diff_0
thf(fact_31_verit__minus__simplify_I3_J,axiom,
! [B: $tType] :
( ( group_add @ B )
=> ! [B2: B] :
( ( minus_minus @ B @ ( zero_zero @ B ) @ B2 )
= ( uminus_uminus @ B @ B2 ) ) ) ).
% verit_minus_simplify(3)
thf(fact_32_closed__orbit__def,axiom,
! [X: a] :
( ( period385816147_orbit @ a @ f @ x @ X )
= ( ? [X3: real] :
( ( member @ real @ X3 @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) )
& ( X3
!= ( zero_zero @ real ) )
& ( ( auto_ll_on_flow0 @ a @ f @ x @ X @ X3 )
= X ) ) ) ) ).
% closed_orbit_def
thf(fact_33_local_Oflows__reverse,axiom,
! [T: real,X0: a] :
( ( member @ real @ T @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( ( auto_ll_on_flow0 @ a @ f @ x @ ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ T ) @ ( minus_minus @ real @ ( zero_zero @ real ) @ T ) )
= X0 ) ) ).
% local.flows_reverse
thf(fact_34_local_Oexistence__ivl__reverse,axiom,
! [T: real,X0: a] :
( ( member @ real @ T @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( member @ real @ ( minus_minus @ real @ ( zero_zero @ real ) @ T ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ T ) ) ) ) ).
% local.existence_ivl_reverse
thf(fact_35_general_Oflow__initial__time__if,axiom,
! [T0: real,X0: a] :
( ( ( ( member @ real @ T0 @ ( top_top @ ( set @ real ) ) )
& ( member @ a @ X0 @ x ) )
=> ( ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ ( minus_minus @ real @ T0 @ T0 ) )
= X0 ) )
& ( ~ ( ( member @ real @ T0 @ ( top_top @ ( set @ real ) ) )
& ( member @ a @ X0 @ x ) )
=> ( ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ ( minus_minus @ real @ T0 @ T0 ) )
= ( zero_zero @ a ) ) ) ) ).
% general.flow_initial_time_if
thf(fact_36_interval__axioms,axiom,
initia826609931terval @ ( top_top @ ( set @ real ) ) ).
% interval_axioms
thf(fact_37_rev_Oexistence__ivl0__cong,axiom,
! [B: $tType,Y: set @ a,G: a > a,X0: a] :
( ( x = Y )
=> ( ! [X2: a,T2: B] :
( ( member @ a @ X2 @ Y )
=> ( ( uminus_uminus @ ( a > a ) @ f @ X2 )
= ( G @ X2 ) ) )
=> ( ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 )
= ( auto_l1112008849e_ivl0 @ a @ G @ Y @ X0 ) ) ) ) ).
% rev.existence_ivl0_cong
thf(fact_38_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_39_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_40_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_41_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
= A2 ) ) ).
% add.inverse_inverse
thf(fact_42_verit__minus__simplify_I4_J,axiom,
! [B: $tType] :
( ( group_add @ B )
=> ! [B2: B] :
( ( uminus_uminus @ B @ ( uminus_uminus @ B @ B2 ) )
= B2 ) ) ).
% verit_minus_simplify(4)
thf(fact_43_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_44_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_47_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X2: A] :
( ( F @ X2 )
= ( G @ X2 ) )
=> ( F = G ) ) ).
% ext
thf(fact_48_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_49_rev_Ogeneral_Omem__existence__ivl__iv__defined_I2_J,axiom,
! [T: real,T0: real,X0: a] :
( ( member @ real @ ( minus_minus @ real @ T @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
=> ( member @ a @ X0 @ x ) ) ).
% rev.general.mem_existence_ivl_iv_defined(2)
thf(fact_50_local_Orev_Omem__existence__ivl__subset,axiom,
! [T: real,X0: a] :
( ( member @ real @ T @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
=> ( member @ real @ T @ ( top_top @ ( set @ real ) ) ) ) ).
% local.rev.mem_existence_ivl_subset
thf(fact_51_local_Omem__existence__ivl__subset,axiom,
! [T: real,X0: a] :
( ( member @ real @ T @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( member @ real @ T @ ( top_top @ ( set @ real ) ) ) ) ).
% local.mem_existence_ivl_subset
thf(fact_52_rev_Oflow0__cong,axiom,
! [B: $tType,Y: set @ a,G: a > a,T: real,X0: a] :
( ( x = Y )
=> ( ! [X2: a,T2: B] :
( ( member @ a @ X2 @ Y )
=> ( ( uminus_uminus @ ( a > a ) @ f @ X2 )
= ( G @ X2 ) ) )
=> ( ( member @ real @ T @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
=> ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ T )
= ( auto_ll_on_flow0 @ a @ G @ Y @ X0 @ T ) ) ) ) ) ).
% rev.flow0_cong
thf(fact_53_rev_Orev__eq__flow,axiom,
! [Y2: a,T: real] :
( ( auto_ll_on_flow0 @ a @ f @ x @ Y2 @ T )
= ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y2 @ ( uminus_uminus @ real @ T ) ) ) ).
% rev.rev_eq_flow
thf(fact_54_rev__eq__flow,axiom,
! [Y2: a,T: real] :
( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y2 @ T )
= ( auto_ll_on_flow0 @ a @ f @ x @ Y2 @ ( uminus_uminus @ real @ T ) ) ) ).
% rev_eq_flow
thf(fact_55_rev_Ofixpoint__sol_I2_J,axiom,
! [X: a,T: 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 @ T )
= X ) ) ) ).
% rev.fixpoint_sol(2)
thf(fact_56_rev_Oclosed__orbit__flow0,axiom,
! [X: a,T: 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 @ T ) ) ) ).
% rev.closed_orbit_flow0
thf(fact_57_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_58_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_59_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_60_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_61_rev_Ogeneral_Omem__existence__ivl__iv__defined_I1_J,axiom,
! [T: real,T0: real,X0: a] :
( ( member @ real @ ( minus_minus @ real @ T @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
=> ( member @ real @ T0 @ ( top_top @ ( set @ real ) ) ) ) ).
% rev.general.mem_existence_ivl_iv_defined(1)
thf(fact_62_rev_Ogeneral_Oexistence__ivl__initial__time,axiom,
! [T0: real,X0: a] :
( ( member @ real @ T0 @ ( top_top @ ( set @ real ) ) )
=> ( ( member @ a @ X0 @ x )
=> ( member @ real @ ( minus_minus @ real @ T0 @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) ) ) ) ).
% rev.general.existence_ivl_initial_time
thf(fact_63_rev_Omem__existence__ivl__shift__autonomous2,axiom,
! [T: real,S: real,X: a] :
( ( member @ real @ ( minus_minus @ real @ T @ S ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) )
=> ( ( member @ a @ X @ x )
=> ( ! [S2: real,T2: real,X2: a] :
( ( member @ a @ X2 @ x )
=> ( ( uminus_uminus @ ( a > a ) @ f @ X2 )
= ( uminus_uminus @ ( a > a ) @ f @ X2 ) ) )
=> ( ( ( top_top @ ( set @ real ) )
= ( top_top @ ( set @ real ) ) )
=> ( member @ real @ ( minus_minus @ real @ T @ S ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ) ) ) ).
% rev.mem_existence_ivl_shift_autonomous2
thf(fact_64_rev_Ogeneral_Omem__existence__ivl__subset,axiom,
! [T: real,T0: real,X0: a] :
( ( member @ real @ ( minus_minus @ real @ T @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
=> ( member @ real @ T @ ( top_top @ ( set @ real ) ) ) ) ).
% rev.general.mem_existence_ivl_subset
thf(fact_65_general_Omem__existence__ivl__iv__defined_I1_J,axiom,
! [T: real,T0: real,X0: a] :
( ( member @ real @ ( minus_minus @ real @ T @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( member @ real @ T0 @ ( top_top @ ( set @ real ) ) ) ) ).
% general.mem_existence_ivl_iv_defined(1)
thf(fact_66_general_Oexistence__ivl__initial__time,axiom,
! [T0: real,X0: a] :
( ( member @ real @ T0 @ ( top_top @ ( set @ real ) ) )
=> ( ( member @ a @ X0 @ x )
=> ( member @ real @ ( minus_minus @ real @ T0 @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) ) ) ) ).
% general.existence_ivl_initial_time
thf(fact_67_mem__existence__ivl__shift__autonomous2,axiom,
! [T: real,S: real,X: a] :
( ( member @ real @ ( minus_minus @ real @ T @ S ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) )
=> ( ( member @ a @ X @ x )
=> ( ! [S2: real,T2: real,X2: a] :
( ( member @ a @ X2 @ x )
=> ( ( f @ X2 )
= ( f @ X2 ) ) )
=> ( ( ( top_top @ ( set @ real ) )
= ( top_top @ ( set @ real ) ) )
=> ( member @ real @ ( minus_minus @ real @ T @ S ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) ) ) ) ) ) ).
% mem_existence_ivl_shift_autonomous2
thf(fact_68_general_Omem__existence__ivl__subset,axiom,
! [T: real,T0: real,X0: a] :
( ( member @ real @ ( minus_minus @ real @ T @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( member @ real @ T @ ( top_top @ ( set @ real ) ) ) ) ).
% general.mem_existence_ivl_subset
thf(fact_69_rev_Odiff__existence__ivl__trans,axiom,
! [T0: real,X0: a,T: real] :
( ( member @ real @ T0 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
=> ( ( member @ real @ T @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
=> ( member @ real @ ( minus_minus @ real @ T @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ T0 ) ) ) ) ) ).
% rev.diff_existence_ivl_trans
thf(fact_70_rev_Ogeneral_Oexistence__ivl__reverse,axiom,
! [T: real,T0: real,X0: a] :
( ( member @ real @ ( minus_minus @ real @ T @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
=> ( member @ real @ ( minus_minus @ real @ T0 @ T ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ ( minus_minus @ real @ T @ T0 ) ) ) ) ) ).
% rev.general.existence_ivl_reverse
thf(fact_71_rev_Ogeneral_Oflow__in__domain,axiom,
! [T: real,T0: real,X0: a] :
( ( member @ real @ ( minus_minus @ real @ T @ T0 ) @ ( 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 @ ( minus_minus @ real @ T @ T0 ) ) @ x ) ) ).
% rev.general.flow_in_domain
thf(fact_72_rev_Ogeneral_Oflows__reverse,axiom,
! [T: real,T0: real,X0: a] :
( ( member @ real @ ( minus_minus @ real @ T @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
=> ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ ( minus_minus @ real @ T @ T0 ) ) @ ( minus_minus @ real @ T0 @ T ) )
= X0 ) ) ).
% rev.general.flows_reverse
thf(fact_73_rev_Orecurrence__time__flip__sign_I2_J,axiom,
! [T3: real,X: a] :
( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) )
=> ( ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ T3 )
= X )
=> ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ ( uminus_uminus @ real @ T3 ) )
= X ) ) ) ).
% rev.recurrence_time_flip_sign(2)
thf(fact_74_rev_Orecurrence__time__flip__sign_I1_J,axiom,
! [T3: real,X: a] :
( ( member @ real @ T3 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) )
=> ( ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ T3 )
= X )
=> ( member @ real @ ( uminus_uminus @ real @ T3 ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) ) ) ) ).
% rev.recurrence_time_flip_sign(1)
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_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_77_local_Orev_Oflow__undefined0,axiom,
! [T: real,X: a] :
( ~ ( member @ real @ T @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) )
=> ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ T )
= ( zero_zero @ a ) ) ) ).
% local.rev.flow_undefined0
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_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_80_rev_Oclosed__orbitI,axiom,
! [T: real,T4: real,Y2: a] :
( ( T != T4 )
=> ( ( member @ real @ T @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y2 ) )
=> ( ( member @ real @ T4 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y2 ) )
=> ( ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y2 @ T )
= ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y2 @ T4 ) )
=> ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y2 ) ) ) ) ) ).
% rev.closed_orbitI
thf(fact_81_rev_Operiodic__orbit__imp__flow0__regular,axiom,
! [X: a,T: 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 @ T ) )
!= ( zero_zero @ a ) ) ) ).
% rev.periodic_orbit_imp_flow0_regular
thf(fact_82_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_83_local_Orev_Oexistence__ivl__reverse,axiom,
! [T: real,X0: a] :
( ( member @ real @ T @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
=> ( member @ real @ ( minus_minus @ real @ ( zero_zero @ real ) @ T ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ T ) ) ) ) ).
% local.rev.existence_ivl_reverse
thf(fact_84_local_Orev_Oflows__reverse,axiom,
! [T: real,X0: a] :
( ( member @ real @ T @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
=> ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ T ) @ ( minus_minus @ real @ ( zero_zero @ real ) @ T ) )
= X0 ) ) ).
% local.rev.flows_reverse
thf(fact_85_rev_Oflow__shift__autonomous2,axiom,
! [T: real,S: real,X: a] :
( ( member @ real @ ( minus_minus @ real @ T @ S ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) )
=> ( ( member @ a @ X @ x )
=> ( ! [S2: real,T2: real,X2: a] :
( ( member @ a @ X2 @ x )
=> ( ( uminus_uminus @ ( a > a ) @ f @ X2 )
= ( uminus_uminus @ ( a > a ) @ f @ X2 ) ) )
=> ( ( ( top_top @ ( set @ real ) )
= ( top_top @ ( set @ real ) ) )
=> ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ ( minus_minus @ real @ T @ S ) )
= ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ ( minus_minus @ real @ T @ S ) ) ) ) ) ) ) ).
% rev.flow_shift_autonomous2
thf(fact_86_flow__shift__autonomous2,axiom,
! [T: real,S: real,X: a] :
( ( member @ real @ ( minus_minus @ real @ T @ S ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X ) )
=> ( ( member @ a @ X @ x )
=> ( ! [S2: real,T2: real,X2: a] :
( ( member @ a @ X2 @ x )
=> ( ( f @ X2 )
= ( f @ X2 ) ) )
=> ( ( ( top_top @ ( set @ real ) )
= ( top_top @ ( set @ real ) ) )
=> ( ( auto_ll_on_flow0 @ a @ f @ x @ X @ ( minus_minus @ real @ T @ S ) )
= ( auto_ll_on_flow0 @ a @ f @ x @ X @ ( minus_minus @ real @ T @ S ) ) ) ) ) ) ) ).
% flow_shift_autonomous2
thf(fact_87_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_88_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_89_rev_Oclosed__orbit__def,axiom,
! [X: a] :
( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
= ( ? [X3: real] :
( ( member @ real @ X3 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) )
& ( X3
!= ( zero_zero @ real ) )
& ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ X3 )
= X ) ) ) ) ).
% rev.closed_orbit_def
thf(fact_90_rev_Ogeneral_Oflow__undefined0,axiom,
! [T: real,T0: real,X: a] :
( ~ ( member @ real @ ( minus_minus @ real @ T @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X ) )
=> ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ ( minus_minus @ real @ T @ T0 ) )
= ( zero_zero @ a ) ) ) ).
% rev.general.flow_undefined0
thf(fact_91_rev_Ogeneral_Oflow__initial__time__if,axiom,
! [T0: real,X0: a] :
( ( ( ( member @ real @ T0 @ ( top_top @ ( set @ real ) ) )
& ( member @ a @ X0 @ x ) )
=> ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ ( minus_minus @ real @ T0 @ T0 ) )
= X0 ) )
& ( ~ ( ( member @ real @ T0 @ ( top_top @ ( set @ real ) ) )
& ( member @ a @ X0 @ x ) )
=> ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ ( minus_minus @ real @ T0 @ T0 ) )
= ( zero_zero @ a ) ) ) ) ).
% rev.general.flow_initial_time_if
thf(fact_92_rev_Operiodic__orbitI,axiom,
! [T: real,T4: real,Y2: a] :
( ( T != T4 )
=> ( ( member @ real @ T @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y2 ) )
=> ( ( member @ real @ T4 @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y2 ) )
=> ( ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y2 @ T )
= ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y2 @ T4 ) )
=> ( ( ( uminus_uminus @ ( a > a ) @ f @ Y2 )
!= ( zero_zero @ a ) )
=> ( period862636932_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ Y2 ) ) ) ) ) ) ).
% rev.periodic_orbitI
thf(fact_93_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_94_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_95_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_96_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_97_add_Oinverse__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% add.inverse_neutral
thf(fact_98_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_99_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_0_right
thf(fact_100_zero__diff,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% zero_diff
thf(fact_101_diff__zero,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_zero
thf(fact_102_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_103_minus__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( minus_minus @ A @ A2 @ B2 ) )
= ( minus_minus @ A @ B2 @ A2 ) ) ) ).
% minus_diff_eq
thf(fact_104_tt_I1_J,axiom,
ord_less @ real @ ( zero_zero @ real ) @ tt ).
% tt(1)
thf(fact_105_local_Orev_Omem__existence__ivl__iv__defined_I2_J,axiom,
! [T: real,X0: a] :
( ( member @ real @ T @ ( 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_106_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_107_local_Orev_Oflow__in__domain,axiom,
! [T: real,X0: a] :
( ( member @ real @ T @ ( 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 @ T ) @ x ) ) ).
% local.rev.flow_in_domain
thf(fact_108_local_Orev_Omem__existence__ivl__iv__defined_I1_J,axiom,
! [T: real,X0: a] :
( ( member @ real @ T @ ( 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_109_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_110_local_Omem__existence__ivl__iv__defined_I1_J,axiom,
! [T: real,X0: a] :
( ( member @ real @ T @ ( 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_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_113_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_114_rev_Ogeneral_Oexistence__ivl__initial__time__iff,axiom,
! [T0: real,X0: a] :
( ( member @ real @ ( minus_minus @ real @ T0 @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
= ( ( member @ real @ T0 @ ( top_top @ ( set @ real ) ) )
& ( member @ a @ X0 @ x ) ) ) ).
% rev.general.existence_ivl_initial_time_iff
thf(fact_115_general_Oexistence__ivl__initial__time__iff,axiom,
! [T0: real,X0: a] :
( ( member @ real @ ( minus_minus @ real @ T0 @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
= ( ( member @ real @ T0 @ ( top_top @ ( set @ real ) ) )
& ( member @ a @ X0 @ x ) ) ) ).
% general.existence_ivl_initial_time_iff
thf(fact_116_rev_Ogeneral_Oflow__initial__time,axiom,
! [T0: real,X0: a] :
( ( member @ real @ T0 @ ( top_top @ ( set @ real ) ) )
=> ( ( member @ a @ X0 @ x )
=> ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ ( minus_minus @ real @ T0 @ T0 ) )
= X0 ) ) ) ).
% rev.general.flow_initial_time
thf(fact_117_general_Oflow__initial__time,axiom,
! [T0: real,X0: a] :
( ( member @ real @ T0 @ ( top_top @ ( set @ real ) ) )
=> ( ( member @ a @ X0 @ x )
=> ( ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ ( minus_minus @ real @ T0 @ T0 ) )
= X0 ) ) ) ).
% general.flow_initial_time
thf(fact_118_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
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_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_121_verit__negate__coefficient_I3_J,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( A2 = B2 )
=> ( ( uminus_uminus @ A @ A2 )
= ( uminus_uminus @ A @ B2 ) ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_122_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C: A,D: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C @ D ) )
=> ( ( A2 = B2 )
= ( C = D ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_123_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A2: A,C: A,B2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C ) @ B2 )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C ) ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_124_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( ^ [Y3: A,Z: A] : Y3 = Z )
= ( ^ [A4: A,B3: A] :
( ( minus_minus @ A @ A4 @ B3 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_125_minus__diff__commute,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B2: A,A2: A] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ B2 ) @ A2 )
= ( minus_minus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).
% minus_diff_commute
thf(fact_126_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_127_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_128_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_129_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_130_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_131_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_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_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_134_rev_Ogeneral_Oexistence__ivl__trans,axiom,
! [S: real,T0: real,X0: a,T: real] :
( ( member @ real @ ( minus_minus @ real @ S @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
=> ( ( member @ real @ T @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ ( minus_minus @ real @ S @ T0 ) ) ) )
=> ( member @ real @ ( minus_minus @ real @ ( plus_plus @ real @ S @ T ) @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) ) ) ) ).
% rev.general.existence_ivl_trans
thf(fact_135_i2_I2_J,axiom,
ord_less @ real @ ( zero_zero @ real ) @ i1 ).
% i2(2)
thf(fact_136_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_137_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_138_d,axiom,
ord_less @ real @ ( zero_zero @ real ) @ d ).
% d
thf(fact_139_local_Oflow__trans,axiom,
! [S: real,X0: a,T: real] :
( ( member @ real @ S @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( ( member @ real @ T @ ( 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 @ T ) )
= ( auto_ll_on_flow0 @ a @ f @ x @ ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ S ) @ T ) ) ) ) ).
% local.flow_trans
thf(fact_140_local_Oexistence__ivl__trans_H,axiom,
! [T: real,S: real,X0: a] :
( ( member @ real @ ( plus_plus @ real @ T @ S ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( ( member @ real @ T @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( member @ real @ S @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ T ) ) ) ) ) ).
% local.existence_ivl_trans'
thf(fact_141_local_Oexistence__ivl__trans,axiom,
! [S: real,X0: a,T: real] :
( ( member @ real @ S @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( ( member @ real @ T @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ S ) ) )
=> ( member @ real @ ( plus_plus @ real @ S @ T ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) ) ) ) ).
% local.existence_ivl_trans
thf(fact_142_i2_I3_J,axiom,
ord_less @ real @ i1 @ i2 ).
% i2(3)
thf(fact_143_local_Orev_Oflow__trans,axiom,
! [S: real,X0: a,T: real] :
( ( member @ real @ S @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
=> ( ( member @ real @ T @ ( 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 @ T ) )
= ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ S ) @ T ) ) ) ) ).
% local.rev.flow_trans
thf(fact_144_local_Orev_Oexistence__ivl__trans_H,axiom,
! [T: real,S: real,X0: a] :
( ( member @ real @ ( plus_plus @ real @ T @ S ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
=> ( ( member @ real @ T @ ( 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 @ T ) ) ) ) ) ).
% local.rev.existence_ivl_trans'
thf(fact_145_local_Orev_Oexistence__ivl__trans,axiom,
! [S: real,X0: a,T: real] :
( ( member @ real @ S @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
=> ( ( member @ real @ T @ ( 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 @ T ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) ) ) ) ).
% local.rev.existence_ivl_trans
thf(fact_146_general_Oflow__trans,axiom,
! [S: real,T0: real,X0: a,T: real] :
( ( member @ real @ ( minus_minus @ real @ S @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( ( member @ real @ T @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ ( minus_minus @ real @ S @ T0 ) ) ) )
=> ( ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ ( minus_minus @ real @ ( plus_plus @ real @ S @ T ) @ T0 ) )
= ( auto_ll_on_flow0 @ a @ f @ x @ ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ ( minus_minus @ real @ S @ T0 ) ) @ T ) ) ) ) ).
% general.flow_trans
thf(fact_147_general_Oexistence__ivl__trans_H,axiom,
! [T: real,S: real,T0: real,X0: a] :
( ( member @ real @ ( minus_minus @ real @ ( plus_plus @ real @ T @ S ) @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( ( member @ real @ ( minus_minus @ real @ T @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( member @ real @ S @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ ( minus_minus @ real @ T @ T0 ) ) ) ) ) ) ).
% general.existence_ivl_trans'
thf(fact_148_general_Oexistence__ivl__trans,axiom,
! [S: real,T0: real,X0: a,T: real] :
( ( member @ real @ ( minus_minus @ real @ S @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) )
=> ( ( member @ real @ T @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ ( auto_ll_on_flow0 @ a @ f @ x @ X0 @ ( minus_minus @ real @ S @ T0 ) ) ) )
=> ( member @ real @ ( minus_minus @ real @ ( plus_plus @ real @ S @ T ) @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ f @ x @ X0 ) ) ) ) ).
% general.existence_ivl_trans
thf(fact_149_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_150_rev_Ogeneral_Oflow__trans,axiom,
! [S: real,T0: real,X0: a,T: real] :
( ( member @ real @ ( minus_minus @ real @ S @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
=> ( ( member @ real @ T @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ ( minus_minus @ real @ S @ T0 ) ) ) )
=> ( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ ( minus_minus @ real @ ( plus_plus @ real @ S @ T ) @ T0 ) )
= ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 @ ( minus_minus @ real @ S @ T0 ) ) @ T ) ) ) ) ).
% rev.general.flow_trans
thf(fact_151_rev_Ogeneral_Oexistence__ivl__trans_H,axiom,
! [T: real,S: real,T0: real,X0: a] :
( ( member @ real @ ( minus_minus @ real @ ( plus_plus @ real @ T @ S ) @ T0 ) @ ( auto_l1112008849e_ivl0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X0 ) )
=> ( ( member @ real @ ( minus_minus @ real @ T @ T0 ) @ ( 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 @ ( minus_minus @ real @ T @ T0 ) ) ) ) ) ) ).
% rev.general.existence_ivl_trans'
thf(fact_152_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_153_add_Oleft__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.left_neutral
thf(fact_154_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.right_neutral
thf(fact_155_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_156_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_157_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_158_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_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__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_161_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X: A,Y2: A] :
( ( ( plus_plus @ A @ X @ Y2 )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y2
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_162_zero__eq__add__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X: A,Y2: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ X @ Y2 ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y2
= ( zero_zero @ A ) ) ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_163_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_164_closed__orbitE,axiom,
! [X: a] :
( ( period385816147_orbit @ a @ f @ x @ X )
=> ~ ! [T5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T5 )
=> ~ ! [T6: real] :
( ( auto_ll_on_flow0 @ a @ f @ x @ X @ ( plus_plus @ real @ T6 @ T5 ) )
= ( auto_ll_on_flow0 @ a @ f @ x @ X @ T6 ) ) ) ) ).
% closed_orbitE
thf(fact_165_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_166_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_167_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_168_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_169_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_170_add__diff__cancel__right_H,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% add_diff_cancel_right'
thf(fact_171_add__diff__cancel__right,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A2: A,C: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
= ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% add_diff_cancel_right
thf(fact_172_add__diff__cancel__left_H,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ A2 )
= B2 ) ) ).
% add_diff_cancel_left'
thf(fact_173_add__diff__cancel__left,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [C: A,A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
= ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% add_diff_cancel_left
thf(fact_174_diff__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% diff_add_cancel
thf(fact_175_add__diff__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% add_diff_cancel
thf(fact_176_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_177_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_178_tt_I2_J,axiom,
ord_less @ real @ tt @ d ).
% tt(2)
thf(fact_179_rev_Oclosed__orbitE,axiom,
! [X: a] :
( ( period385816147_orbit @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X )
=> ~ ! [T5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T5 )
=> ~ ! [T6: real] :
( ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ ( plus_plus @ real @ T6 @ T5 ) )
= ( auto_ll_on_flow0 @ a @ ( uminus_uminus @ ( a > a ) @ f ) @ x @ X @ T6 ) ) ) ) ).
% rev.closed_orbitE
thf(fact_180_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_181_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_182_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_183_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_184_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_185_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_186_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_187_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_188_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_189_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_190_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_191_diff__gt__0__iff__gt,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ).
% diff_gt_0_iff_gt
thf(fact_192_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_193_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_194_diff__add__zero,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( zero_zero @ A ) ) ) ).
% diff_add_zero
thf(fact_195_uminus__add__conv__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( minus_minus @ A @ B2 @ A2 ) ) ) ).
% uminus_add_conv_diff
thf(fact_196_diff__minus__eq__add,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( plus_plus @ A @ A2 @ B2 ) ) ) ).
% diff_minus_eq_add
thf(fact_197_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_198_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_199_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_200_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_201_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_202_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_203_add__neg__neg,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_neg
thf(fact_204_add__pos__pos,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_pos_pos
thf(fact_205_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ! [C2: A] :
( ( B2
= ( plus_plus @ A @ A2 @ C2 ) )
=> ( C2
= ( zero_zero @ A ) ) ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_206_pos__add__strict,axiom,
! [A: $tType] :
( ( strict797366125id_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B2 @ C )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% pos_add_strict
thf(fact_207_add__less__imp__less__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_imp_less_right
thf(fact_208_add__less__imp__less__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_imp_less_left
thf(fact_209_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% add_strict_right_mono
thf(fact_210_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) ) ) ) ).
% add_strict_left_mono
thf(fact_211_add__strict__mono,axiom,
! [A: $tType] :
( ( strict2144017051up_add @ A )
=> ! [A2: A,B2: A,C: A,D: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C @ D )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% add_strict_mono
thf(fact_212_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_213_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_214_less__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,C: A,B2: A] :
( ( ord_less @ A @ A2 @ ( minus_minus @ A @ C @ B2 ) )
= ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C ) ) ) ).
% less_diff_eq
thf(fact_215_diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C )
= ( ord_less @ A @ A2 @ ( plus_plus @ A @ C @ B2 ) ) ) ) ).
% diff_less_eq
thf(fact_216_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_217_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A4: A,B3: A] : ( plus_plus @ A @ B3 @ A4 ) ) ) ) ).
% add.commute
thf(fact_218_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_219_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_220_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_221_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B4: A,K: A,B2: A,A2: A] :
( ( B4
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( plus_plus @ A @ A2 @ B4 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_222_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: A,K: A,A2: A,B2: A] :
( ( A3
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( plus_plus @ A @ A3 @ B2 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_223_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_224_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( K = L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_225_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I: A,J: A,K: A,L: 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(2)
thf(fact_226_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ 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(5)
thf(fact_227_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_228_verit__comp__simplify1_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% verit_comp_simplify1(1)
thf(fact_229_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_230_not__less__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_231_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_232_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_233_verit__sum__simplify,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% verit_sum_simplify
thf(fact_234_comm__monoid__add__class_Oadd__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% comm_monoid_add_class.add_0
thf(fact_235_add_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.comm_neutral
thf(fact_236_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_237_verit__negate__coefficient_I2_J,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_238_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_239_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_240_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B2 @ C ) ) ) ) ).
% diff_strict_right_mono
thf(fact_241_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ord_less @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B2 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_242_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C: A,D: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C @ D ) )
=> ( ( ord_less @ A @ A2 @ B2 )
= ( ord_less @ A @ C @ D ) ) ) ) ).
% diff_eq_diff_less
thf(fact_243_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,D: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ D @ C )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B2 @ D ) ) ) ) ) ).
% diff_strict_mono
thf(fact_244_group__cancel_Oneg1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,K: A,A2: A] :
( ( A3
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( uminus_uminus @ A @ A3 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ) ).
% group_cancel.neg1
thf(fact_245_add_Oinverse__distrib__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% add.inverse_distrib_swap
thf(fact_246_add__implies__diff,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [C: A,B2: A,A2: A] :
( ( ( plus_plus @ A @ C @ B2 )
= A2 )
=> ( C
= ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% add_implies_diff
thf(fact_247_diff__diff__add,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C )
= ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% diff_diff_add
thf(fact_248_diff__add__eq__diff__diff__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C ) @ B2 ) ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_249_diff__add__eq,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C ) @ B2 ) ) ) ).
% diff_add_eq
thf(fact_250_diff__diff__eq2,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( minus_minus @ A @ A2 @ ( minus_minus @ A @ B2 @ C ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C ) @ B2 ) ) ) ).
% diff_diff_eq2
thf(fact_251_add__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ C ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C ) ) ) ).
% add_diff_eq
thf(fact_252_eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,C: A,B2: A] :
( ( A2
= ( minus_minus @ A @ C @ B2 ) )
= ( ( plus_plus @ A @ A2 @ B2 )
= C ) ) ) ).
% eq_diff_eq
thf(fact_253_diff__eq__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= C )
= ( A2
= ( plus_plus @ A @ C @ B2 ) ) ) ) ).
% diff_eq_eq
thf(fact_254_group__cancel_Osub1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,K: A,A2: A,B2: A] :
( ( A3
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( minus_minus @ A @ A3 @ B2 )
= ( plus_plus @ A @ K @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.sub1
% Subclasses (20)
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___HOL_Otype,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( type @ A ) ) ).
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___Orderings_Oorder,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( order @ 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_Ocancel__ab__semigroup__add,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( cancel146912293up_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__semigroup__add,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( ordere779506340up_add @ A ) ) ).
thf(subcl_Executable__Euclidean__Space_Oexecutable__euclidean__space___Groups_Ostrict__ordered__comm__monoid__add,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( strict797366125id_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_Ostrict__ordered__ab__semigroup__add,axiom,
! [A: $tType] :
( ( execut510477386_space @ A )
=> ( strict2144017051up_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 (28)
thf(tcon_fun___Orderings_Oorder,axiom,
! [A5: $tType,A6: $tType] :
( ( order @ A6 )
=> ( order @ ( A5 > A6 ) ) ) ).
thf(tcon_Set_Oset___Groups_Oab__semigroup__add,axiom,
! [A5: $tType] :
( ( ab_semigroup_add @ A5 )
=> ( ab_semigroup_add @ ( set @ A5 ) ) ) ).
thf(tcon_Set_Oset___Groups_Ocomm__monoid__add,axiom,
! [A5: $tType] :
( ( comm_monoid_add @ A5 )
=> ( comm_monoid_add @ ( set @ A5 ) ) ) ).
thf(tcon_Set_Oset___Groups_Osemigroup__add,axiom,
! [A5: $tType] :
( ( semigroup_add @ A5 )
=> ( semigroup_add @ ( set @ A5 ) ) ) ).
thf(tcon_Set_Oset___Groups_Omonoid__add,axiom,
! [A5: $tType] :
( ( monoid_add @ A5 )
=> ( monoid_add @ ( set @ A5 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_1,axiom,
! [A5: $tType] : ( order @ ( set @ A5 ) ) ).
thf(tcon_Set_Oset___Groups_Ozero,axiom,
! [A5: $tType] :
( ( zero @ A5 )
=> ( zero @ ( set @ A5 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_2,axiom,
order @ $o ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere516151231imp_le @ real ).
thf(tcon_Real_Oreal___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict2144017051up_add @ 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_Ostrict__ordered__comm__monoid__add,axiom,
strict797366125id_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add @ 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__ab__semigroup__add,axiom,
cancel146912293up_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_3,axiom,
ab_semigroup_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_4,axiom,
comm_monoid_add @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__add_5,axiom,
semigroup_add @ real ).
thf(tcon_Real_Oreal___Groups_Oab__group__add,axiom,
ab_group_add @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__add_6,axiom,
monoid_add @ real ).
thf(tcon_Real_Oreal___Groups_Ogroup__add,axiom,
group_add @ real ).
thf(tcon_Real_Oreal___Orderings_Oorder_7,axiom,
order @ real ).
thf(tcon_Real_Oreal___Groups_Ozero_8,axiom,
zero @ real ).
% Free types (1)
thf(tfree_0,hypothesis,
execut510477386_space @ a ).
% Conjectures (1)
thf(conj_0,conjecture,
( xx
= ( auto_ll_on_flow0 @ a @ f @ x @ x2 @ ( minus_minus @ real @ ss @ tt ) ) ) ).
%------------------------------------------------------------------------------