TPTP Problem File: ITP146^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP146^1 : 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 : 0.40 v8.2.0, 0.31 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 380 ( 187 unt; 37 typ; 0 def)
% Number of atoms : 791 ( 396 equ; 0 cnn)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 3066 ( 36 ~; 2 |; 30 &;2756 @)
% ( 0 <=>; 242 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 54 ( 54 >; 0 *; 0 +; 0 <<)
% Number of symbols : 34 ( 33 usr; 13 con; 0-4 aty)
% Number of variables : 731 ( 14 ^; 715 !; 2 ?; 731 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:46:31.992
%------------------------------------------------------------------------------
% 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 (33)
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_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001tf__a,type,
minus_minus_a: a > a > a ).
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_Initial__Value__Problem_Ointerval,type,
initia826609931terval: set_real > $o ).
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_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_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_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_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_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 (342)
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_l612940ivl0_a @ f @ x @ x2 ) ).
% assms(8)
thf(fact_4_tt__ex,axiom,
member_real @ tt @ ( auto_l612940ivl0_a @ f @ x @ xx ) ).
% tt_ex
thf(fact_5_ss__ex,axiom,
member_real @ ss @ ( auto_l612940ivl0_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_l612940ivl0_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_l612940ivl0_a @ f @ x @ X0 ) )
=> ( ( member_real @ T @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
=> ( member_real @ ( minus_minus_real @ T @ T0 ) @ ( auto_l612940ivl0_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_l612940ivl0_a @ f @ x @ X0 ) )
=> ( member_real @ ( minus_minus_real @ T0 @ T ) @ ( auto_l612940ivl0_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_l612940ivl0_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_l612940ivl0_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_general_Omem__existence__ivl__iv__defined_I2_J,axiom,
! [T: real,T0: real,X0: a] :
( ( member_real @ ( minus_minus_real @ T @ T0 ) @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
=> ( member_a @ X0 @ x ) ) ).
% general.mem_existence_ivl_iv_defined(2)
thf(fact_13_closed__orbit__flow0,axiom,
! [X: a,T: real] :
( ( period720806154rbit_a @ f @ x @ X )
=> ( period720806154rbit_a @ f @ x @ ( auto_ll_on_flow0_a @ f @ x @ X @ T ) ) ) ).
% closed_orbit_flow0
thf(fact_14_closed__orbit__in__domain,axiom,
! [X: a] :
( ( period720806154rbit_a @ f @ x @ X )
=> ( member_a @ X @ x ) ) ).
% closed_orbit_in_domain
thf(fact_15_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_16_recurrence__time__flip__sign_I1_J,axiom,
! [T2: real,X: a] :
( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ X ) )
=> ( ( ( auto_ll_on_flow0_a @ f @ x @ X @ T2 )
= X )
=> ( member_real @ ( uminus_uminus_real @ T2 ) @ ( auto_l612940ivl0_a @ f @ x @ X ) ) ) ) ).
% recurrence_time_flip_sign(1)
thf(fact_17_recurrence__time__flip__sign_I2_J,axiom,
! [T2: real,X: a] :
( ( member_real @ T2 @ ( auto_l612940ivl0_a @ f @ x @ X ) )
=> ( ( ( auto_ll_on_flow0_a @ f @ x @ X @ T2 )
= X )
=> ( ( auto_ll_on_flow0_a @ f @ x @ X @ ( uminus_uminus_real @ T2 ) )
= X ) ) ) ).
% recurrence_time_flip_sign(2)
thf(fact_18_local_Oflow__undefined0,axiom,
! [T: real,X: a] :
( ~ ( member_real @ T @ ( auto_l612940ivl0_a @ f @ x @ X ) )
=> ( ( auto_ll_on_flow0_a @ f @ x @ X @ T )
= zero_zero_a ) ) ).
% local.flow_undefined0
thf(fact_19_closed__orbitI,axiom,
! [T: real,T3: real,Y: a] :
( ( T != T3 )
=> ( ( member_real @ T @ ( auto_l612940ivl0_a @ f @ x @ Y ) )
=> ( ( member_real @ T3 @ ( auto_l612940ivl0_a @ f @ x @ Y ) )
=> ( ( ( auto_ll_on_flow0_a @ f @ x @ Y @ T )
= ( auto_ll_on_flow0_a @ f @ x @ Y @ T3 ) )
=> ( period720806154rbit_a @ f @ x @ Y ) ) ) ) ) ).
% closed_orbitI
thf(fact_20_general_Oflow__undefined0,axiom,
! [T: real,T0: real,X: a] :
( ~ ( member_real @ ( minus_minus_real @ T @ T0 ) @ ( auto_l612940ivl0_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_21_local_Omem__existence__ivl__iv__defined_I2_J,axiom,
! [T: real,X0: a] :
( ( member_real @ T @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
=> ( member_a @ X0 @ x ) ) ).
% local.mem_existence_ivl_iv_defined(2)
thf(fact_22_local_Oflow__in__domain,axiom,
! [T: real,X0: a] :
( ( member_real @ T @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
=> ( member_a @ ( auto_ll_on_flow0_a @ f @ x @ X0 @ T ) @ x ) ) ).
% local.flow_in_domain
thf(fact_23_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_24_mvar_Ointerval__axioms,axiom,
! [X0: a] : ( initia826609931terval @ ( auto_l612940ivl0_a @ f @ x @ X0 ) ) ).
% mvar.interval_axioms
thf(fact_25_periodic__orbitI,axiom,
! [T: real,T3: real,Y: a] :
( ( T != T3 )
=> ( ( member_real @ T @ ( auto_l612940ivl0_a @ f @ x @ Y ) )
=> ( ( member_real @ T3 @ ( auto_l612940ivl0_a @ f @ x @ Y ) )
=> ( ( ( auto_ll_on_flow0_a @ f @ x @ Y @ T )
= ( auto_ll_on_flow0_a @ f @ x @ Y @ T3 ) )
=> ( ( ( f @ Y )
!= zero_zero_a )
=> ( period138238489rbit_a @ f @ x @ Y ) ) ) ) ) ) ).
% periodic_orbitI
thf(fact_26_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_27_periodic__orbit__imp__flow0__regular,axiom,
! [X: a,T: real] :
( ( period138238489rbit_a @ f @ x @ X )
=> ( ( f @ ( auto_ll_on_flow0_a @ f @ x @ X @ T ) )
!= zero_zero_a ) ) ).
% periodic_orbit_imp_flow0_regular
thf(fact_28_diff__0,axiom,
! [A: a] :
( ( minus_minus_a @ zero_zero_a @ A )
= ( uminus_uminus_a @ A ) ) ).
% diff_0
thf(fact_29_diff__0,axiom,
! [A: real] :
( ( minus_minus_real @ zero_zero_real @ A )
= ( uminus_uminus_real @ A ) ) ).
% diff_0
thf(fact_30_verit__minus__simplify_I3_J,axiom,
! [B: a] :
( ( minus_minus_a @ zero_zero_a @ B )
= ( uminus_uminus_a @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_31_verit__minus__simplify_I3_J,axiom,
! [B: real] :
( ( minus_minus_real @ zero_zero_real @ B )
= ( uminus_uminus_real @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_32_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_33_local_Oflows__reverse,axiom,
! [T: real,X0: a] :
( ( member_real @ T @ ( auto_l612940ivl0_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_l612940ivl0_a @ f @ x @ X0 ) )
=> ( member_real @ ( minus_minus_real @ zero_zero_real @ T ) @ ( auto_l612940ivl0_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_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_38_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_39_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_40_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_41_add_Oinverse__inverse,axiom,
! [A: a] :
( ( uminus_uminus_a @ ( uminus_uminus_a @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_42_add_Oinverse__inverse,axiom,
! [A: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_43_verit__minus__simplify_I4_J,axiom,
! [B: a] :
( ( uminus_uminus_a @ ( uminus_uminus_a @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_44_verit__minus__simplify_I4_J,axiom,
! [B: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_45_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_46_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_47_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_48_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_49_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X2: real] : ( member_real @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_50_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_51_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_52_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_53_rev_Ogeneral_Omem__existence__ivl__iv__defined_I2_J,axiom,
! [T: real,T0: real,X0: a] :
( ( member_real @ ( minus_minus_real @ T @ T0 ) @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
=> ( member_a @ X0 @ x ) ) ).
% rev.general.mem_existence_ivl_iv_defined(2)
thf(fact_54_local_Orev_Omem__existence__ivl__subset,axiom,
! [T: real,X0: a] :
( ( member_real @ T @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
=> ( member_real @ T @ top_top_set_real ) ) ).
% local.rev.mem_existence_ivl_subset
thf(fact_55_local_Omem__existence__ivl__subset,axiom,
! [T: real,X0: a] :
( ( member_real @ T @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
=> ( member_real @ T @ top_top_set_real ) ) ).
% local.mem_existence_ivl_subset
thf(fact_56_rev_Orev__eq__flow,axiom,
! [Y: a,T: real] :
( ( auto_ll_on_flow0_a @ f @ x @ Y @ T )
= ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ Y @ ( uminus_uminus_real @ T ) ) ) ).
% rev.rev_eq_flow
thf(fact_57_rev__eq__flow,axiom,
! [Y: a,T: real] :
( ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ Y @ T )
= ( auto_ll_on_flow0_a @ f @ x @ Y @ ( uminus_uminus_real @ T ) ) ) ).
% rev_eq_flow
thf(fact_58_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_59_rev_Oclosed__orbit__flow0,axiom,
! [X: a,T: 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 @ T ) ) ) ).
% rev.closed_orbit_flow0
thf(fact_60_rev_Omvar_Ointerval__axioms,axiom,
! [X0: a] : ( initia826609931terval @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) ) ).
% rev.mvar.interval_axioms
thf(fact_61_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_62_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_63_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_64_rev_Ogeneral_Omem__existence__ivl__iv__defined_I1_J,axiom,
! [T: real,T0: real,X0: a] :
( ( member_real @ ( minus_minus_real @ T @ T0 ) @ ( auto_l612940ivl0_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_65_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_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) ) ) ) ).
% rev.general.existence_ivl_initial_time
thf(fact_66_rev_Omem__existence__ivl__shift__autonomous2,axiom,
! [T: real,S: real,X: a] :
( ( member_real @ ( minus_minus_real @ T @ S ) @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) )
=> ( ( member_a @ X @ x )
=> ( ! [S2: real,T4: real,X3: a] :
( ( member_a @ X3 @ x )
=> ( ( uminus_uminus_a_a @ f @ X3 )
= ( uminus_uminus_a_a @ f @ X3 ) ) )
=> ( ( top_top_set_real = top_top_set_real )
=> ( member_real @ ( minus_minus_real @ T @ S ) @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ) ) ) ).
% rev.mem_existence_ivl_shift_autonomous2
thf(fact_67_rev_Ogeneral_Omem__existence__ivl__subset,axiom,
! [T: real,T0: real,X0: a] :
( ( member_real @ ( minus_minus_real @ T @ T0 ) @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
=> ( member_real @ T @ top_top_set_real ) ) ).
% rev.general.mem_existence_ivl_subset
thf(fact_68_general_Omem__existence__ivl__iv__defined_I1_J,axiom,
! [T: real,T0: real,X0: a] :
( ( member_real @ ( minus_minus_real @ T @ T0 ) @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
=> ( member_real @ T0 @ top_top_set_real ) ) ).
% general.mem_existence_ivl_iv_defined(1)
thf(fact_69_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_l612940ivl0_a @ f @ x @ X0 ) ) ) ) ).
% general.existence_ivl_initial_time
thf(fact_70_mem__existence__ivl__shift__autonomous2,axiom,
! [T: real,S: real,X: a] :
( ( member_real @ ( minus_minus_real @ T @ S ) @ ( auto_l612940ivl0_a @ f @ x @ X ) )
=> ( ( member_a @ X @ x )
=> ( ! [S2: real,T4: real,X3: a] :
( ( member_a @ X3 @ x )
=> ( ( f @ X3 )
= ( f @ X3 ) ) )
=> ( ( top_top_set_real = top_top_set_real )
=> ( member_real @ ( minus_minus_real @ T @ S ) @ ( auto_l612940ivl0_a @ f @ x @ X ) ) ) ) ) ) ).
% mem_existence_ivl_shift_autonomous2
thf(fact_71_general_Omem__existence__ivl__subset,axiom,
! [T: real,T0: real,X0: a] :
( ( member_real @ ( minus_minus_real @ T @ T0 ) @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
=> ( member_real @ T @ top_top_set_real ) ) ).
% general.mem_existence_ivl_subset
thf(fact_72_rev_Odiff__existence__ivl__trans,axiom,
! [T0: real,X0: a,T: real] :
( ( member_real @ T0 @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
=> ( ( member_real @ T @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
=> ( member_real @ ( minus_minus_real @ T @ T0 ) @ ( auto_l612940ivl0_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_73_rev_Ogeneral_Oexistence__ivl__reverse,axiom,
! [T: real,T0: real,X0: a] :
( ( member_real @ ( minus_minus_real @ T @ T0 ) @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
=> ( member_real @ ( minus_minus_real @ T0 @ T ) @ ( auto_l612940ivl0_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_74_rev_Ogeneral_Oflow__in__domain,axiom,
! [T: real,T0: real,X0: a] :
( ( member_real @ ( minus_minus_real @ T @ T0 ) @ ( auto_l612940ivl0_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_75_rev_Ogeneral_Oflows__reverse,axiom,
! [T: real,T0: real,X0: a] :
( ( member_real @ ( minus_minus_real @ T @ T0 ) @ ( auto_l612940ivl0_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_76_rev_Orecurrence__time__flip__sign_I2_J,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 )
= X )
=> ( ( auto_ll_on_flow0_a @ ( uminus_uminus_a_a @ f ) @ x @ X @ ( uminus_uminus_real @ T2 ) )
= X ) ) ) ).
% rev.recurrence_time_flip_sign(2)
thf(fact_77_rev_Orecurrence__time__flip__sign_I1_J,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 )
= X )
=> ( member_real @ ( uminus_uminus_real @ T2 ) @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) ) ) ) ).
% rev.recurrence_time_flip_sign(1)
thf(fact_78_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_79_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_80_local_Orev_Oflow__undefined0,axiom,
! [T: real,X: a] :
( ~ ( member_real @ T @ ( auto_l612940ivl0_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_81_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_82_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_83_rev_Oclosed__orbitI,axiom,
! [T: real,T3: real,Y: a] :
( ( T != T3 )
=> ( ( member_real @ T @ ( 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 @ T )
= ( 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_Operiodic__orbit__imp__flow0__regular,axiom,
! [X: a,T: 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 @ T ) )
!= zero_zero_a ) ) ).
% rev.periodic_orbit_imp_flow0_regular
thf(fact_85_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_86_local_Orev_Oexistence__ivl__reverse,axiom,
! [T: real,X0: a] :
( ( member_real @ T @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
=> ( member_real @ ( minus_minus_real @ zero_zero_real @ T ) @ ( auto_l612940ivl0_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_87_local_Orev_Oflows__reverse,axiom,
! [T: real,X0: a] :
( ( member_real @ T @ ( auto_l612940ivl0_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_88_rev_Oflow__shift__autonomous2,axiom,
! [T: real,S: real,X: a] :
( ( member_real @ ( minus_minus_real @ T @ S ) @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X ) )
=> ( ( member_a @ X @ x )
=> ( ! [S2: real,T4: real,X3: a] :
( ( member_a @ X3 @ x )
=> ( ( uminus_uminus_a_a @ f @ X3 )
= ( uminus_uminus_a_a @ f @ X3 ) ) )
=> ( ( 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_89_flow__shift__autonomous2,axiom,
! [T: real,S: real,X: a] :
( ( member_real @ ( minus_minus_real @ T @ S ) @ ( auto_l612940ivl0_a @ f @ x @ X ) )
=> ( ( member_a @ X @ x )
=> ( ! [S2: real,T4: real,X3: a] :
( ( member_a @ X3 @ x )
=> ( ( f @ X3 )
= ( f @ X3 ) ) )
=> ( ( 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_90_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_91_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_92_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_93_rev_Ogeneral_Oflow__undefined0,axiom,
! [T: real,T0: real,X: a] :
( ~ ( member_real @ ( minus_minus_real @ T @ T0 ) @ ( auto_l612940ivl0_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_94_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_95_rev_Operiodic__orbitI,axiom,
! [T: real,T3: real,Y: a] :
( ( T != T3 )
=> ( ( member_real @ T @ ( 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 @ T )
= ( 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_96_neg__equal__zero,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= A )
= ( A = zero_zero_real ) ) ).
% neg_equal_zero
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__0__iff__equal,axiom,
! [A: a] :
( ( ( uminus_uminus_a @ A )
= zero_zero_a )
= ( A = zero_zero_a ) ) ).
% neg_equal_0_iff_equal
thf(fact_99_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_100_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_101_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_102_add_Oinverse__neutral,axiom,
( ( uminus_uminus_a @ zero_zero_a )
= zero_zero_a ) ).
% add.inverse_neutral
thf(fact_103_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_104_diff__self,axiom,
! [A: a] :
( ( minus_minus_a @ A @ A )
= zero_zero_a ) ).
% diff_self
thf(fact_105_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_106_diff__0__right,axiom,
! [A: a] :
( ( minus_minus_a @ A @ zero_zero_a )
= A ) ).
% diff_0_right
thf(fact_107_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_108_diff__zero,axiom,
! [A: a] :
( ( minus_minus_a @ A @ zero_zero_a )
= A ) ).
% diff_zero
thf(fact_109_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_110_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: a] :
( ( minus_minus_a @ A @ A )
= zero_zero_a ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_111_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_112_minus__diff__eq,axiom,
! [A: a,B: a] :
( ( uminus_uminus_a @ ( minus_minus_a @ A @ B ) )
= ( minus_minus_a @ B @ A ) ) ).
% minus_diff_eq
thf(fact_113_minus__diff__eq,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
= ( minus_minus_real @ B @ A ) ) ).
% minus_diff_eq
thf(fact_114_tt_I1_J,axiom,
ord_less_real @ zero_zero_real @ tt ).
% tt(1)
thf(fact_115_local_Orev_Omem__existence__ivl__iv__defined_I2_J,axiom,
! [T: real,X0: a] :
( ( member_real @ T @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
=> ( member_a @ X0 @ x ) ) ).
% local.rev.mem_existence_ivl_iv_defined(2)
thf(fact_116_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_117_local_Orev_Oflow__in__domain,axiom,
! [T: real,X0: a] :
( ( member_real @ T @ ( auto_l612940ivl0_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_118_local_Orev_Omem__existence__ivl__iv__defined_I1_J,axiom,
! [T: real,X0: a] :
( ( member_real @ T @ ( 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_119_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_120_local_Omem__existence__ivl__iv__defined_I1_J,axiom,
! [T: real,X0: a] :
( ( member_real @ T @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
=> ( member_real @ zero_zero_real @ top_top_set_real ) ) ).
% local.mem_existence_ivl_iv_defined(1)
thf(fact_121_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_122_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_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_Ogeneral_Oexistence__ivl__initial__time__iff,axiom,
! [T0: real,X0: a] :
( ( member_real @ ( minus_minus_real @ T0 @ T0 ) @ ( auto_l612940ivl0_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_125_general_Oexistence__ivl__initial__time__iff,axiom,
! [T0: real,X0: a] :
( ( member_real @ ( minus_minus_real @ T0 @ T0 ) @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
= ( ( member_real @ T0 @ top_top_set_real )
& ( member_a @ X0 @ x ) ) ) ).
% general.existence_ivl_initial_time_iff
thf(fact_126_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_127_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_128_zero__reorient,axiom,
! [X: a] :
( ( zero_zero_a = X )
= ( X = zero_zero_a ) ) ).
% zero_reorient
thf(fact_129_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_130_minus__equation__iff,axiom,
! [A: a,B: a] :
( ( ( uminus_uminus_a @ A )
= B )
= ( ( uminus_uminus_a @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_131_minus__equation__iff,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( uminus_uminus_real @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_132_equation__minus__iff,axiom,
! [A: a,B: a] :
( ( A
= ( uminus_uminus_a @ B ) )
= ( B
= ( uminus_uminus_a @ A ) ) ) ).
% equation_minus_iff
thf(fact_133_equation__minus__iff,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% equation_minus_iff
thf(fact_134_verit__negate__coefficient_I3_J,axiom,
! [A: a,B: a] :
( ( A = B )
=> ( ( uminus_uminus_a @ A )
= ( uminus_uminus_a @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_135_verit__negate__coefficient_I3_J,axiom,
! [A: real,B: real] :
( ( A = B )
=> ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_136_diff__eq__diff__eq,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( minus_minus_a @ A @ B )
= ( minus_minus_a @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_137_diff__eq__diff__eq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_138_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: a,C: a,B: a] :
( ( minus_minus_a @ ( minus_minus_a @ A @ C ) @ B )
= ( minus_minus_a @ ( minus_minus_a @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_139_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_140_eq__iff__diff__eq__0,axiom,
( ( ^ [Y2: a,Z: a] : Y2 = Z )
= ( ^ [A3: a,B2: a] :
( ( minus_minus_a @ A3 @ B2 )
= zero_zero_a ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_141_eq__iff__diff__eq__0,axiom,
( ( ^ [Y2: real,Z: real] : Y2 = Z )
= ( ^ [A3: real,B2: real] :
( ( minus_minus_real @ A3 @ B2 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_142_minus__diff__commute,axiom,
! [B: a,A: a] :
( ( minus_minus_a @ ( uminus_uminus_a @ B ) @ A )
= ( minus_minus_a @ ( uminus_uminus_a @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_143_minus__diff__commute,axiom,
! [B: real,A: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
= ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_144_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_145_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_146_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_147_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_148_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_149_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_150_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_151_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_152_rev_Ogeneral_Oexistence__ivl__trans,axiom,
! [S: real,T0: real,X0: a,T: real] :
( ( member_real @ ( minus_minus_real @ S @ T0 ) @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
=> ( ( member_real @ T @ ( auto_l612940ivl0_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_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) ) ) ) ).
% rev.general.existence_ivl_trans
thf(fact_153_i2_I2_J,axiom,
ord_less_real @ zero_zero_real @ i1 ).
% i2(2)
thf(fact_154_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_155_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_156_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_157_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_158_d,axiom,
ord_less_real @ zero_zero_real @ d ).
% d
thf(fact_159_local_Oflow__trans,axiom,
! [S: real,X0: a,T: real] :
( ( member_real @ S @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
=> ( ( member_real @ T @ ( 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 @ T ) )
= ( auto_ll_on_flow0_a @ f @ x @ ( auto_ll_on_flow0_a @ f @ x @ X0 @ S ) @ T ) ) ) ) ).
% local.flow_trans
thf(fact_160_local_Oexistence__ivl__trans_H,axiom,
! [T: real,S: real,X0: a] :
( ( member_real @ ( plus_plus_real @ T @ S ) @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
=> ( ( member_real @ T @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
=> ( member_real @ S @ ( auto_l612940ivl0_a @ f @ x @ ( auto_ll_on_flow0_a @ f @ x @ X0 @ T ) ) ) ) ) ).
% local.existence_ivl_trans'
thf(fact_161_local_Oexistence__ivl__trans,axiom,
! [S: real,X0: a,T: real] :
( ( member_real @ S @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
=> ( ( member_real @ T @ ( auto_l612940ivl0_a @ f @ x @ ( auto_ll_on_flow0_a @ f @ x @ X0 @ S ) ) )
=> ( member_real @ ( plus_plus_real @ S @ T ) @ ( auto_l612940ivl0_a @ f @ x @ X0 ) ) ) ) ).
% local.existence_ivl_trans
thf(fact_162_i2_I3_J,axiom,
ord_less_real @ i1 @ i2 ).
% i2(3)
thf(fact_163_local_Orev_Oflow__trans,axiom,
! [S: real,X0: a,T: real] :
( ( member_real @ S @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
=> ( ( member_real @ T @ ( 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 @ 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_164_local_Orev_Oexistence__ivl__trans_H,axiom,
! [T: real,S: real,X0: a] :
( ( member_real @ ( plus_plus_real @ T @ S ) @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
=> ( ( member_real @ T @ ( 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 @ T ) ) ) ) ) ).
% local.rev.existence_ivl_trans'
thf(fact_165_local_Orev_Oexistence__ivl__trans,axiom,
! [S: real,X0: a,T: real] :
( ( member_real @ S @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
=> ( ( member_real @ T @ ( 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 @ T ) @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) ) ) ) ).
% local.rev.existence_ivl_trans
thf(fact_166_general_Oflow__trans,axiom,
! [S: real,T0: real,X0: a,T: real] :
( ( member_real @ ( minus_minus_real @ S @ T0 ) @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
=> ( ( member_real @ T @ ( auto_l612940ivl0_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_167_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_l612940ivl0_a @ f @ x @ X0 ) )
=> ( ( member_real @ ( minus_minus_real @ T @ T0 ) @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
=> ( member_real @ S @ ( auto_l612940ivl0_a @ f @ x @ ( auto_ll_on_flow0_a @ f @ x @ X0 @ ( minus_minus_real @ T @ T0 ) ) ) ) ) ) ).
% general.existence_ivl_trans'
thf(fact_168_general_Oexistence__ivl__trans,axiom,
! [S: real,T0: real,X0: a,T: real] :
( ( member_real @ ( minus_minus_real @ S @ T0 ) @ ( auto_l612940ivl0_a @ f @ x @ X0 ) )
=> ( ( member_real @ T @ ( auto_l612940ivl0_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_l612940ivl0_a @ f @ x @ X0 ) ) ) ) ).
% general.existence_ivl_trans
thf(fact_169_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_170_rev_Ogeneral_Oflow__trans,axiom,
! [S: real,T0: real,X0: a,T: real] :
( ( member_real @ ( minus_minus_real @ S @ T0 ) @ ( auto_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
=> ( ( member_real @ T @ ( auto_l612940ivl0_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_171_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_l612940ivl0_a @ ( uminus_uminus_a_a @ f ) @ x @ X0 ) )
=> ( ( member_real @ ( minus_minus_real @ T @ T0 ) @ ( 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 @ ( minus_minus_real @ T @ T0 ) ) ) ) ) ) ).
% rev.general.existence_ivl_trans'
thf(fact_172_add_Oleft__neutral,axiom,
! [A: a] :
( ( plus_plus_a @ zero_zero_a @ A )
= A ) ).
% add.left_neutral
thf(fact_173_add_Oleft__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.left_neutral
thf(fact_174_add_Oright__neutral,axiom,
! [A: a] :
( ( plus_plus_a @ A @ zero_zero_a )
= A ) ).
% add.right_neutral
thf(fact_175_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_176_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_177_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_178_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_179_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_180_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_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__right__left,axiom,
! [A: a,B: a] :
( ( A
= ( plus_plus_a @ B @ A ) )
= ( B = zero_zero_a ) ) ).
% add_cancel_right_left
thf(fact_183_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_184_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_185_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_186_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_187_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_188_closed__orbitE,axiom,
! [X: a] :
( ( period720806154rbit_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_189_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_190_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_191_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_192_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_193_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_194_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_195_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_196_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_197_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_198_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_199_add__diff__cancel__right_H,axiom,
! [A: a,B: a] :
( ( minus_minus_a @ ( plus_plus_a @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_200_add__diff__cancel__right_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_201_add__diff__cancel__right,axiom,
! [A: a,C: a,B: a] :
( ( minus_minus_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) )
= ( minus_minus_a @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_202_add__diff__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_203_add__diff__cancel__left_H,axiom,
! [A: a,B: a] :
( ( minus_minus_a @ ( plus_plus_a @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_204_add__diff__cancel__left_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_205_add__diff__cancel__left,axiom,
! [C: a,A: a,B: a] :
( ( minus_minus_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) )
= ( minus_minus_a @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_206_add__diff__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_207_diff__add__cancel,axiom,
! [A: a,B: a] :
( ( plus_plus_a @ ( minus_minus_a @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_208_diff__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_209_add__diff__cancel,axiom,
! [A: a,B: a] :
( ( minus_minus_a @ ( plus_plus_a @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_210_add__diff__cancel,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_211_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_212_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_213_tt_I2_J,axiom,
ord_less_real @ tt @ d ).
% tt(2)
thf(fact_214_rev_Oclosed__orbitE,axiom,
! [X: a] :
( ( period720806154rbit_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_215_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_216_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_217_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_218_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_219_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_220_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_221_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_222_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_223_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_224_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_225_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_226_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_227_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_228_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_229_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_230_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_231_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_232_diff__gt__0__iff__gt,axiom,
! [A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ ( minus_minus_a @ A @ B ) )
= ( ord_less_a @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_233_diff__gt__0__iff__gt,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_real @ B @ A ) ) ).
% diff_gt_0_iff_gt
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_uminus__add__conv__diff,axiom,
! [A: a,B: a] :
( ( plus_plus_a @ ( uminus_uminus_a @ A ) @ B )
= ( minus_minus_a @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_239_uminus__add__conv__diff,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
= ( minus_minus_real @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_240_diff__minus__eq__add,axiom,
! [A: a,B: a] :
( ( minus_minus_a @ A @ ( uminus_uminus_a @ B ) )
= ( plus_plus_a @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_241_diff__minus__eq__add,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
= ( plus_plus_real @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_242_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_243_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_244_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_245_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_246_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_247_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_248_add__neg__neg,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ zero_zero_a )
=> ( ( ord_less_a @ B @ zero_zero_a )
=> ( ord_less_a @ ( plus_plus_a @ A @ B ) @ zero_zero_a ) ) ) ).
% add_neg_neg
thf(fact_249_add__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_neg
thf(fact_250_add__pos__pos,axiom,
! [A: a,B: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ zero_zero_a @ B )
=> ( ord_less_a @ zero_zero_a @ ( plus_plus_a @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_251_add__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_252_pos__add__strict,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ zero_zero_a @ A )
=> ( ( ord_less_a @ B @ C )
=> ( ord_less_a @ B @ ( plus_plus_a @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_253_pos__add__strict,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_254_add__less__imp__less__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_imp_less_right
thf(fact_255_add__less__imp__less__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_imp_less_right
thf(fact_256_add__less__imp__less__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_imp_less_left
thf(fact_257_add__less__imp__less__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_imp_less_left
thf(fact_258_add__strict__right__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ord_less_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_259_add__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_260_add__strict__left__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ord_less_a @ ( plus_plus_a @ C @ A ) @ ( plus_plus_a @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_261_add__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_262_add__strict__mono,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ C @ D )
=> ( ord_less_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_263_add__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_264_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_265_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_266_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_267_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_268_less__diff__eq,axiom,
! [A: a,C: a,B: a] :
( ( ord_less_a @ A @ ( minus_minus_a @ C @ B ) )
= ( ord_less_a @ ( plus_plus_a @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_269_less__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
= ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_270_diff__less__eq,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ ( minus_minus_a @ A @ B ) @ C )
= ( ord_less_a @ A @ ( plus_plus_a @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_271_diff__less__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
= ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_272_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_273_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_274_add_Ocommute,axiom,
( plus_plus_a
= ( ^ [A3: a,B2: a] : ( plus_plus_a @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_275_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A3: real,B2: real] : ( plus_plus_real @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_276_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_277_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_278_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_279_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_280_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_281_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_282_group__cancel_Oadd2,axiom,
! [B3: a,K: a,B: a,A: a] :
( ( B3
= ( plus_plus_a @ K @ B ) )
=> ( ( plus_plus_a @ A @ B3 )
= ( plus_plus_a @ K @ ( plus_plus_a @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_283_group__cancel_Oadd2,axiom,
! [B3: real,K: real,B: real,A: real] :
( ( B3
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A @ B3 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_284_group__cancel_Oadd1,axiom,
! [A2: a,K: a,A: a,B: a] :
( ( A2
= ( plus_plus_a @ K @ A ) )
=> ( ( plus_plus_a @ A2 @ B )
= ( plus_plus_a @ K @ ( plus_plus_a @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_285_group__cancel_Oadd1,axiom,
! [A2: real,K: real,A: real,B: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( plus_plus_real @ A2 @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_286_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_287_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_288_add__mono__thms__linordered__field_I1_J,axiom,
! [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_289_add__mono__thms__linordered__field_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_290_add__mono__thms__linordered__field_I2_J,axiom,
! [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_291_add__mono__thms__linordered__field_I2_J,axiom,
! [I: real,J: real,K: real,L: 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(2)
thf(fact_292_add__mono__thms__linordered__field_I5_J,axiom,
! [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_293_add__mono__thms__linordered__field_I5_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_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(5)
thf(fact_294_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_295_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_296_verit__comp__simplify1_I1_J,axiom,
! [A: a] :
~ ( ord_less_a @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_297_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_298_verit__sum__simplify,axiom,
! [A: a] :
( ( plus_plus_a @ A @ zero_zero_a )
= A ) ).
% verit_sum_simplify
thf(fact_299_verit__sum__simplify,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% verit_sum_simplify
thf(fact_300_comm__monoid__add__class_Oadd__0,axiom,
! [A: a] :
( ( plus_plus_a @ zero_zero_a @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_301_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_302_add_Ocomm__neutral,axiom,
! [A: a] :
( ( plus_plus_a @ A @ zero_zero_a )
= A ) ).
% add.comm_neutral
thf(fact_303_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_304_add_Ogroup__left__neutral,axiom,
! [A: a] :
( ( plus_plus_a @ zero_zero_a @ A )
= A ) ).
% add.group_left_neutral
thf(fact_305_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_306_verit__negate__coefficient_I2_J,axiom,
! [A: a,B: a] :
( ( ord_less_a @ A @ B )
=> ( ord_less_a @ ( uminus_uminus_a @ B ) @ ( uminus_uminus_a @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_307_verit__negate__coefficient_I2_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_308_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_309_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_310_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_311_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_312_diff__strict__right__mono,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ord_less_a @ ( minus_minus_a @ A @ C ) @ ( minus_minus_a @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_313_diff__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_314_diff__strict__left__mono,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_a @ B @ A )
=> ( ord_less_a @ ( minus_minus_a @ C @ A ) @ ( minus_minus_a @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_315_diff__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_316_diff__eq__diff__less,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( minus_minus_a @ A @ B )
= ( minus_minus_a @ C @ D ) )
=> ( ( ord_less_a @ A @ B )
= ( ord_less_a @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_317_diff__eq__diff__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_real @ A @ B )
= ( ord_less_real @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_318_diff__strict__mono,axiom,
! [A: a,B: a,D: a,C: a] :
( ( ord_less_a @ A @ B )
=> ( ( ord_less_a @ D @ C )
=> ( ord_less_a @ ( minus_minus_a @ A @ C ) @ ( minus_minus_a @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_319_diff__strict__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ D @ C )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_320_group__cancel_Oneg1,axiom,
! [A2: a,K: a,A: a] :
( ( A2
= ( plus_plus_a @ K @ A ) )
=> ( ( uminus_uminus_a @ A2 )
= ( plus_plus_a @ ( uminus_uminus_a @ K ) @ ( uminus_uminus_a @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_321_group__cancel_Oneg1,axiom,
! [A2: real,K: real,A: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( uminus_uminus_real @ A2 )
= ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_322_add_Oinverse__distrib__swap,axiom,
! [A: a,B: a] :
( ( uminus_uminus_a @ ( plus_plus_a @ A @ B ) )
= ( plus_plus_a @ ( uminus_uminus_a @ B ) @ ( uminus_uminus_a @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_323_add_Oinverse__distrib__swap,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_324_add__implies__diff,axiom,
! [C: a,B: a,A: a] :
( ( ( plus_plus_a @ C @ B )
= A )
=> ( C
= ( minus_minus_a @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_325_add__implies__diff,axiom,
! [C: real,B: real,A: real] :
( ( ( plus_plus_real @ C @ B )
= A )
=> ( C
= ( minus_minus_real @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_326_diff__diff__add,axiom,
! [A: a,B: a,C: a] :
( ( minus_minus_a @ ( minus_minus_a @ A @ B ) @ C )
= ( minus_minus_a @ A @ ( plus_plus_a @ B @ C ) ) ) ).
% diff_diff_add
thf(fact_327_diff__diff__add,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% diff_diff_add
thf(fact_328_diff__add__eq__diff__diff__swap,axiom,
! [A: a,B: a,C: a] :
( ( minus_minus_a @ A @ ( plus_plus_a @ B @ C ) )
= ( minus_minus_a @ ( minus_minus_a @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_329_diff__add__eq__diff__diff__swap,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_330_diff__add__eq,axiom,
! [A: a,B: a,C: a] :
( ( plus_plus_a @ ( minus_minus_a @ A @ B ) @ C )
= ( minus_minus_a @ ( plus_plus_a @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_331_diff__add__eq,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_332_diff__diff__eq2,axiom,
! [A: a,B: a,C: a] :
( ( minus_minus_a @ A @ ( minus_minus_a @ B @ C ) )
= ( minus_minus_a @ ( plus_plus_a @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_333_diff__diff__eq2,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_334_add__diff__eq,axiom,
! [A: a,B: a,C: a] :
( ( plus_plus_a @ A @ ( minus_minus_a @ B @ C ) )
= ( minus_minus_a @ ( plus_plus_a @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_335_add__diff__eq,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_336_eq__diff__eq,axiom,
! [A: a,C: a,B: a] :
( ( A
= ( minus_minus_a @ C @ B ) )
= ( ( plus_plus_a @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_337_eq__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( A
= ( minus_minus_real @ C @ B ) )
= ( ( plus_plus_real @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_338_diff__eq__eq,axiom,
! [A: a,B: a,C: a] :
( ( ( minus_minus_a @ A @ B )
= C )
= ( A
= ( plus_plus_a @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_339_diff__eq__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( minus_minus_real @ A @ B )
= C )
= ( A
= ( plus_plus_real @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_340_group__cancel_Osub1,axiom,
! [A2: a,K: a,A: a,B: a] :
( ( A2
= ( plus_plus_a @ K @ A ) )
=> ( ( minus_minus_a @ A2 @ B )
= ( plus_plus_a @ K @ ( minus_minus_a @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_341_group__cancel_Osub1,axiom,
! [A2: real,K: real,A: real,B: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( minus_minus_real @ A2 @ B )
= ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% group_cancel.sub1
% Conjectures (1)
thf(conj_0,conjecture,
( xx
= ( auto_ll_on_flow0_a @ f @ x @ x2 @ ( minus_minus_real @ ss @ tt ) ) ) ).
%------------------------------------------------------------------------------