TPTP Problem File: ITP170^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP170^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Safety_Regular problem prob_415__3354378_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 : Safety_Regular/prob_415__3354378_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% Syntax : Number of formulae : 322 ( 122 unt; 54 typ; 0 def)
% Number of atoms : 683 ( 248 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 2476 ( 28 ~; 7 |; 116 &;2101 @)
% ( 0 <=>; 224 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Number of types : 13 ( 12 usr)
% Number of type conns : 189 ( 189 >; 0 *; 0 +; 0 <<)
% Number of symbols : 43 ( 42 usr; 9 con; 0-4 aty)
% Number of variables : 775 ( 46 ^; 661 !; 68 ?; 775 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:38:16.446
%------------------------------------------------------------------------------
% Could-be-implicit typings (12)
thf(ty_n_t__Product____Type__Oprod_I_062_It__Cars__Ocars_Mt__Real__Oreal_J_Mt__Product____Type__Oprod_I_062_It__Cars__Ocars_Mt__NatInt__Onat____int_J_Mt__Product____Type__Oprod_I_062_It__Cars__Ocars_Mt__NatInt__Onat____int_J_Mt__Product____Type__Oprod_I_062_It__Cars__Ocars_M_062_It__Real__Oreal_Mt__Real__Oreal_J_J_Mt__Product____Type__Oprod_I_062_It__Cars__Ocars_Mt__Real__Oreal_J_M_062_It__Cars__Ocars_Mt__Real__Oreal_J_J_J_J_J_J,type,
produc458204144s_real: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Cars__Ocars_Mt__NatInt__Onat____int_J_Mt__Product____Type__Oprod_I_062_It__Cars__Ocars_Mt__NatInt__Onat____int_J_Mt__Product____Type__Oprod_I_062_It__Cars__Ocars_M_062_It__Real__Oreal_Mt__Real__Oreal_J_J_Mt__Product____Type__Oprod_I_062_It__Cars__Ocars_Mt__Real__Oreal_J_M_062_It__Cars__Ocars_Mt__Real__Oreal_J_J_J_J_J,type,
produc1603581272s_real: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Cars__Ocars_Mt__NatInt__Onat____int_J_Mt__Product____Type__Oprod_I_062_It__Cars__Ocars_M_062_It__Real__Oreal_Mt__Real__Oreal_J_J_Mt__Product____Type__Oprod_I_062_It__Cars__Ocars_Mt__Real__Oreal_J_M_062_It__Cars__Ocars_Mt__Real__Oreal_J_J_J_J,type,
produc28608506s_real: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Cars__Ocars_M_062_It__Real__Oreal_Mt__Real__Oreal_J_J_Mt__Product____Type__Oprod_I_062_It__Cars__Ocars_Mt__Real__Oreal_J_M_062_It__Cars__Ocars_Mt__Real__Oreal_J_J_J,type,
produc1622321602s_real: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Cars__Ocars_Mt__Real__Oreal_J_M_062_It__Cars__Ocars_Mt__Real__Oreal_J_J,type,
produc427086159s_real: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
produc957004601l_real: $tType ).
thf(ty_n_t__Views__Oview__Oview____ext_It__Product____Type__Ounit_J,type,
view_e774982825t_unit: $tType ).
thf(ty_n_t__RealInt__Oreal____int,type,
real_int: $tType ).
thf(ty_n_t__NatInt__Onat____int,type,
nat_int: $tType ).
thf(ty_n_t__Traffic__Otraffic,type,
traffic: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Cars__Ocars,type,
cars: $tType ).
% Explicit typings (42)
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Length_Osensors_Olen,type,
len: ( cars > traffic > cars > real ) > view_e774982825t_unit > traffic > cars > real_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__NatInt__Onat____int,type,
bot_bot_nat_int: nat_int ).
thf(sy_c_Orderings_Oord__class_Oless_001t__RealInt__Oreal____int,type,
ord_less_real_int: real_int > real_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__RealInt__Oreal____int,type,
ord_less_eq_real_int: real_int > real_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Views__Oview__Oview____ext_It__Product____Type__Ounit_J,type,
ord_le461438217t_unit: view_e774982825t_unit > view_e774982825t_unit > $o ).
thf(sy_c_Product__Type_OPair_001_062_It__Cars__Ocars_M_062_It__Real__Oreal_Mt__Real__Oreal_J_J_001t__Product____Type__Oprod_I_062_It__Cars__Ocars_Mt__Real__Oreal_J_M_062_It__Cars__Ocars_Mt__Real__Oreal_J_J,type,
produc795767668s_real: ( cars > real > real ) > produc427086159s_real > produc1622321602s_real ).
thf(sy_c_Product__Type_OPair_001_062_It__Cars__Ocars_Mt__NatInt__Onat____int_J_001t__Product____Type__Oprod_I_062_It__Cars__Ocars_M_062_It__Real__Oreal_Mt__Real__Oreal_J_J_Mt__Product____Type__Oprod_I_062_It__Cars__Ocars_Mt__Real__Oreal_J_M_062_It__Cars__Ocars_Mt__Real__Oreal_J_J_J,type,
produc1348924850s_real: ( cars > nat_int ) > produc1622321602s_real > produc28608506s_real ).
thf(sy_c_Product__Type_OPair_001_062_It__Cars__Ocars_Mt__NatInt__Onat____int_J_001t__Product____Type__Oprod_I_062_It__Cars__Ocars_Mt__NatInt__Onat____int_J_Mt__Product____Type__Oprod_I_062_It__Cars__Ocars_M_062_It__Real__Oreal_Mt__Real__Oreal_J_J_Mt__Product____Type__Oprod_I_062_It__Cars__Ocars_Mt__Real__Oreal_J_M_062_It__Cars__Ocars_Mt__Real__Oreal_J_J_J_J,type,
produc1230243850s_real: ( cars > nat_int ) > produc28608506s_real > produc1603581272s_real ).
thf(sy_c_Product__Type_OPair_001_062_It__Cars__Ocars_Mt__Real__Oreal_J_001_062_It__Cars__Ocars_Mt__Real__Oreal_J,type,
produc1439932679s_real: ( cars > real ) > ( cars > real ) > produc427086159s_real ).
thf(sy_c_Product__Type_OPair_001_062_It__Cars__Ocars_Mt__Real__Oreal_J_001t__Product____Type__Oprod_I_062_It__Cars__Ocars_Mt__NatInt__Onat____int_J_Mt__Product____Type__Oprod_I_062_It__Cars__Ocars_Mt__NatInt__Onat____int_J_Mt__Product____Type__Oprod_I_062_It__Cars__Ocars_M_062_It__Real__Oreal_Mt__Real__Oreal_J_J_Mt__Product____Type__Oprod_I_062_It__Cars__Ocars_Mt__Real__Oreal_J_M_062_It__Cars__Ocars_Mt__Real__Oreal_J_J_J_J_J,type,
produc1876576040s_real: ( cars > real ) > produc1603581272s_real > produc458204144s_real ).
thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Real__Oreal,type,
produc705216881l_real: real > real > produc957004601l_real ).
thf(sy_c_RealInt_Oleft,type,
left: real_int > real ).
thf(sy_c_RealInt_Oreal__int_OAbs__real__int,type,
real_Abs_real_int: produc957004601l_real > real_int ).
thf(sy_c_RealInt_Oreal__int_Olength,type,
real_length: real_int > real ).
thf(sy_c_RealInt_Oreal__int_Oshift,type,
real_shift: real_int > real > real_int ).
thf(sy_c_RealInt_Oright,type,
right: real_int > real ).
thf(sy_c_Regular__Sensors_Oregular,type,
regular_regular: cars > traffic > cars > real ).
thf(sy_c_Sensors_Osensors,type,
sensors: ( cars > traffic > cars > real ) > $o ).
thf(sy_c_Sensors_Osensors_Ospace,type,
space: ( cars > traffic > cars > real ) > traffic > view_e774982825t_unit > cars > real_int ).
thf(sy_c_Traffic_Otraffic_OAbs__traffic,type,
abs_traffic: produc458204144s_real > traffic ).
thf(sy_c_Traffic_Otraffic_Ophysical__size,type,
physical_size: traffic > cars > real ).
thf(sy_c_Traffic_Otraffic_Opos,type,
pos: traffic > cars > real ).
thf(sy_c_Views_Oview_Oext_001t__Product____Type__Ounit,type,
ext_Product_unit: view_e774982825t_unit > real_int ).
thf(sy_c_Views_Oview_Ohchop,type,
hchop: view_e774982825t_unit > view_e774982825t_unit > view_e774982825t_unit > $o ).
thf(sy_c_Views_Oview_Oown_001t__Product____Type__Ounit,type,
own_Product_unit: view_e774982825t_unit > cars ).
thf(sy_v_c____,type,
c: cars ).
thf(sy_v_cl____,type,
cl: cars > nat_int ).
thf(sy_v_d____,type,
d: cars ).
thf(sy_v_dy____,type,
dy: cars > real > real ).
thf(sy_v_po____,type,
po: cars > real ).
thf(sy_v_ps____,type,
ps: cars > real ).
thf(sy_v_re____,type,
re: cars > nat_int ).
thf(sy_v_sd____,type,
sd: cars > real ).
thf(sy_v_ts____,type,
ts: traffic ).
thf(sy_v_ts__rep____,type,
ts_rep: produc458204144s_real ).
thf(sy_v_v_H____,type,
v: view_e774982825t_unit ).
thf(sy_v_v____,type,
v2: view_e774982825t_unit ).
% Relevant facts (267)
thf(fact_0__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062dy_O_A_092_060forall_062c_Ax_O_Ady_Ac_Ax_A_061_A0_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Dy: cars > real > real] :
~ ! [C: cars,X: real] :
( ( Dy @ C @ X )
= zero_zero_real ) ).
% \<open>\<And>thesis. (\<And>dy. \<forall>c x. dy c x = 0 \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_1_dyn__def,axiom,
! [C: cars,X: real] :
( ( dy @ C @ X )
= zero_zero_real ) ).
% dyn_def
thf(fact_2_local_Ohmlsl_Osensors__ge,axiom,
! [E: cars,Ts: traffic,C2: cars] : ( ord_less_real @ zero_zero_real @ ( regular_regular @ E @ Ts @ C2 ) ) ).
% local.hmlsl.sensors_ge
thf(fact_3_hmlsl_Olen__non__empty__inside,axiom,
! [V: view_e774982825t_unit,Ts: traffic,C2: cars] :
( ( ord_less_real @ zero_zero_real @ ( real_length @ ( len @ regular_regular @ V @ Ts @ C2 ) ) )
=> ( ( ord_less_real @ ( left @ ( space @ regular_regular @ Ts @ V @ C2 ) ) @ ( right @ ( ext_Product_unit @ V ) ) )
& ( ord_less_real @ ( left @ ( ext_Product_unit @ V ) ) @ ( right @ ( space @ regular_regular @ Ts @ V @ C2 ) ) ) ) ) ).
% hmlsl.len_non_empty_inside
thf(fact_4_v_H__d,axiom,
ord_less_real @ zero_zero_real @ ( real_length @ ( len @ regular_regular @ v @ ts @ d ) ) ).
% v'_d
thf(fact_5_hmlsl_Ospace__nonempty,axiom,
! [Ts: traffic,V: view_e774982825t_unit,C2: cars] : ( ord_less_real @ ( left @ ( space @ regular_regular @ Ts @ V @ C2 ) ) @ ( right @ ( space @ regular_regular @ Ts @ V @ C2 ) ) ) ).
% hmlsl.space_nonempty
thf(fact_6_ts__def,axiom,
( ts
= ( abs_traffic @ ts_rep ) ) ).
% ts_def
thf(fact_7_assumption,axiom,
d != c ).
% assumption
thf(fact_8_local_Ohmlsl_Olength__geq__zero,axiom,
! [Ts2: traffic,V2: view_e774982825t_unit] :
( ( ( real_length @ ( ext_Product_unit @ V2 ) )
= zero_zero_real )
| ( ord_less_real @ zero_zero_real @ ( real_length @ ( ext_Product_unit @ V2 ) ) ) ) ).
% local.hmlsl.length_geq_zero
thf(fact_9_v_H__rel__c,axiom,
( ( ord_less_real @ ( left @ ( space @ regular_regular @ ts @ v @ c ) ) @ ( right @ ( ext_Product_unit @ v ) ) )
& ( ord_less_real @ ( left @ ( ext_Product_unit @ v ) ) @ ( right @ ( space @ regular_regular @ ts @ v @ c ) ) ) ) ).
% v'_rel_c
thf(fact_10_hmlsl_Oleft__space,axiom,
! [Ts: traffic,V: view_e774982825t_unit,C2: cars] :
( ( left @ ( space @ regular_regular @ Ts @ V @ C2 ) )
= ( pos @ Ts @ C2 ) ) ).
% hmlsl.left_space
thf(fact_11_len__eq,axiom,
( ( len @ regular_regular @ v @ ts @ c )
= ( len @ regular_regular @ v @ ts @ d ) ) ).
% len_eq
thf(fact_12_hmlsl_Olen__space__left,axiom,
! [Ts: traffic,V: view_e774982825t_unit,C2: cars] :
( ( ord_less_eq_real @ ( left @ ( space @ regular_regular @ Ts @ V @ C2 ) ) @ ( right @ ( ext_Product_unit @ V ) ) )
=> ( ord_less_eq_real @ ( left @ ( space @ regular_regular @ Ts @ V @ C2 ) ) @ ( left @ ( len @ regular_regular @ V @ Ts @ C2 ) ) ) ) ).
% hmlsl.len_space_left
thf(fact_13_hmlsl_Olen__space__right,axiom,
! [V: view_e774982825t_unit,Ts: traffic,C2: cars] :
( ( ord_less_eq_real @ ( left @ ( ext_Product_unit @ V ) ) @ ( right @ ( space @ regular_regular @ Ts @ V @ C2 ) ) )
=> ( ord_less_eq_real @ ( right @ ( len @ regular_regular @ V @ Ts @ C2 ) ) @ ( right @ ( space @ regular_regular @ Ts @ V @ C2 ) ) ) ) ).
% hmlsl.len_space_right
thf(fact_14_regular__sensors_Osensors__le,axiom,
! [E: cars,C2: cars,Ts: traffic] :
( ( E != C2 )
=> ( ord_less_real @ ( regular_regular @ E @ Ts @ C2 ) @ ( regular_regular @ C2 @ Ts @ C2 ) ) ) ).
% regular_sensors.sensors_le
thf(fact_15_ps__ge__zero,axiom,
! [C: cars] : ( ord_less_real @ zero_zero_real @ ( ps @ C ) ) ).
% ps_ge_zero
thf(fact_16_sd__ge__zero,axiom,
! [C: cars] : ( ord_less_real @ zero_zero_real @ ( sd @ C ) ) ).
% sd_ge_zero
thf(fact_17_hmlsl_Olen__left,axiom,
! [V: view_e774982825t_unit,Ts: traffic,C2: cars] : ( ord_less_eq_real @ ( left @ ( ext_Product_unit @ V ) ) @ ( left @ ( len @ regular_regular @ V @ Ts @ C2 ) ) ) ).
% hmlsl.len_left
thf(fact_18_hmlsl_Olen__right,axiom,
! [V: view_e774982825t_unit,Ts: traffic,C2: cars] : ( ord_less_eq_real @ ( right @ ( len @ regular_regular @ V @ Ts @ C2 ) ) @ ( right @ ( ext_Product_unit @ V ) ) ) ).
% hmlsl.len_right
thf(fact_19_v_H__c,axiom,
ord_less_real @ zero_zero_real @ ( real_length @ ( len @ regular_regular @ v @ ts @ c ) ) ).
% v'_c
thf(fact_20_hmlsl_Osensors__axioms,axiom,
sensors @ regular_regular ).
% hmlsl.sensors_axioms
thf(fact_21_dyn__geq__zero,axiom,
! [C: cars,X: real] : ( ord_less_eq_real @ zero_zero_real @ ( dy @ C @ X ) ) ).
% dyn_geq_zero
thf(fact_22_ps__def,axiom,
! [C: cars] :
( ( ps @ C )
= one_one_real ) ).
% ps_def
thf(fact_23_regular__sensors_Osensors__leq,axiom,
! [E: cars,Ts: traffic,C2: cars] : ( ord_less_eq_real @ ( regular_regular @ E @ Ts @ C2 ) @ ( regular_regular @ C2 @ Ts @ C2 ) ) ).
% regular_sensors.sensors_leq
thf(fact_24_sensors_Olen__non__empty__inside,axiom,
! [Sensors: cars > traffic > cars > real,V: view_e774982825t_unit,Ts: traffic,C2: cars] :
( ( sensors @ Sensors )
=> ( ( ord_less_real @ zero_zero_real @ ( real_length @ ( len @ Sensors @ V @ Ts @ C2 ) ) )
=> ( ( ord_less_real @ ( left @ ( space @ Sensors @ Ts @ V @ C2 ) ) @ ( right @ ( ext_Product_unit @ V ) ) )
& ( ord_less_real @ ( left @ ( ext_Product_unit @ V ) ) @ ( right @ ( space @ Sensors @ Ts @ V @ C2 ) ) ) ) ) ) ).
% sensors.len_non_empty_inside
thf(fact_25_other__len__zero,axiom,
! [E2: cars] :
( ( ( E2 != c )
& ( E2 != d ) )
=> ( ( real_length @ ( len @ regular_regular @ v2 @ ts @ E2 ) )
= zero_zero_real ) ) ).
% other_len_zero
thf(fact_26_sensors_Olen__space__right,axiom,
! [Sensors: cars > traffic > cars > real,V: view_e774982825t_unit,Ts: traffic,C2: cars] :
( ( sensors @ Sensors )
=> ( ( ord_less_eq_real @ ( left @ ( ext_Product_unit @ V ) ) @ ( right @ ( space @ Sensors @ Ts @ V @ C2 ) ) )
=> ( ord_less_eq_real @ ( right @ ( len @ Sensors @ V @ Ts @ C2 ) ) @ ( right @ ( space @ Sensors @ Ts @ V @ C2 ) ) ) ) ) ).
% sensors.len_space_right
thf(fact_27_sensors_Olen__space__left,axiom,
! [Sensors: cars > traffic > cars > real,Ts: traffic,V: view_e774982825t_unit,C2: cars] :
( ( sensors @ Sensors )
=> ( ( ord_less_eq_real @ ( left @ ( space @ Sensors @ Ts @ V @ C2 ) ) @ ( right @ ( ext_Product_unit @ V ) ) )
=> ( ord_less_eq_real @ ( left @ ( space @ Sensors @ Ts @ V @ C2 ) ) @ ( left @ ( len @ Sensors @ V @ Ts @ C2 ) ) ) ) ) ).
% sensors.len_space_left
thf(fact_28_lc,axiom,
( ( left @ ( space @ regular_regular @ ts @ v2 @ c ) )
= zero_zero_real ) ).
% lc
thf(fact_29_sensors_Ospace__nonempty,axiom,
! [Sensors: cars > traffic > cars > real,Ts: traffic,V: view_e774982825t_unit,C2: cars] :
( ( sensors @ Sensors )
=> ( ord_less_real @ ( left @ ( space @ Sensors @ Ts @ V @ C2 ) ) @ ( right @ ( space @ Sensors @ Ts @ V @ C2 ) ) ) ) ).
% sensors.space_nonempty
thf(fact_30_sensors_Olen__right,axiom,
! [Sensors: cars > traffic > cars > real,V: view_e774982825t_unit,Ts: traffic,C2: cars] :
( ( sensors @ Sensors )
=> ( ord_less_eq_real @ ( right @ ( len @ Sensors @ V @ Ts @ C2 ) ) @ ( right @ ( ext_Product_unit @ V ) ) ) ) ).
% sensors.len_right
thf(fact_31_sensors_Olen__left,axiom,
! [Sensors: cars > traffic > cars > real,V: view_e774982825t_unit,Ts: traffic,C2: cars] :
( ( sensors @ Sensors )
=> ( ord_less_eq_real @ ( left @ ( ext_Product_unit @ V ) ) @ ( left @ ( len @ Sensors @ V @ Ts @ C2 ) ) ) ) ).
% sensors.len_left
thf(fact_32__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062ps_O_A_092_060forall_062c_O_Aps_Ac_A_061_A1_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Ps: cars > real] :
~ ! [C: cars] :
( ( Ps @ C )
= one_one_real ) ).
% \<open>\<And>thesis. (\<And>ps. \<forall>c. ps c = 1 \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_33_hmlsl_Olen__sub__int,axiom,
! [V: view_e774982825t_unit,Ts: traffic,C2: cars] : ( ord_less_eq_real_int @ ( len @ regular_regular @ V @ Ts @ C2 ) @ ( ext_Product_unit @ V ) ) ).
% hmlsl.len_sub_int
thf(fact_34_hmlsl_Olen__empty__subview,axiom,
! [V: view_e774982825t_unit,Ts: traffic,C2: cars,V3: view_e774982825t_unit] :
( ( ( ( real_length @ ( len @ regular_regular @ V @ Ts @ C2 ) )
= zero_zero_real )
& ( ord_le461438217t_unit @ V3 @ V ) )
=> ( ( real_length @ ( len @ regular_regular @ V3 @ Ts @ C2 ) )
= zero_zero_real ) ) ).
% hmlsl.len_empty_subview
thf(fact_35_lv,axiom,
( ( left @ ( ext_Product_unit @ v2 ) )
= zero_zero_real ) ).
% lv
thf(fact_36_rc,axiom,
( ( right @ ( space @ regular_regular @ ts @ v2 @ c ) )
= one_one_real ) ).
% rc
thf(fact_37_one__reorient,axiom,
! [X2: real] :
( ( one_one_real = X2 )
= ( X2 = one_one_real ) ) ).
% one_reorient
thf(fact_38_sensors_Olen__sub__int,axiom,
! [Sensors: cars > traffic > cars > real,V: view_e774982825t_unit,Ts: traffic,C2: cars] :
( ( sensors @ Sensors )
=> ( ord_less_eq_real_int @ ( len @ Sensors @ V @ Ts @ C2 ) @ ( ext_Product_unit @ V ) ) ) ).
% sensors.len_sub_int
thf(fact_39_sensors_Olen__empty__subview,axiom,
! [Sensors: cars > traffic > cars > real,V: view_e774982825t_unit,Ts: traffic,C2: cars,V3: view_e774982825t_unit] :
( ( sensors @ Sensors )
=> ( ( ( ( real_length @ ( len @ Sensors @ V @ Ts @ C2 ) )
= zero_zero_real )
& ( ord_le461438217t_unit @ V3 @ V ) )
=> ( ( real_length @ ( len @ Sensors @ V3 @ Ts @ C2 ) )
= zero_zero_real ) ) ) ).
% sensors.len_empty_subview
thf(fact_40_zero__reorient,axiom,
! [X2: real] :
( ( zero_zero_real = X2 )
= ( X2 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_41_sensors_Olen_Ocong,axiom,
len = len ).
% sensors.len.cong
thf(fact_42_sensors_Ospace_Ocong,axiom,
space = space ).
% sensors.space.cong
thf(fact_43_sensors__def,axiom,
( sensors
= ( ^ [Sensors2: cars > traffic > cars > real] :
! [E3: cars,Ts3: traffic,C3: cars] : ( ord_less_real @ zero_zero_real @ ( Sensors2 @ E3 @ Ts3 @ C3 ) ) ) ) ).
% sensors_def
thf(fact_44_sensors_Ointro,axiom,
! [Sensors: cars > traffic > cars > real] :
( ! [E4: cars,Ts4: traffic,C4: cars] : ( ord_less_real @ zero_zero_real @ ( Sensors @ E4 @ Ts4 @ C4 ) )
=> ( sensors @ Sensors ) ) ).
% sensors.intro
thf(fact_45_sensors_Osensors__ge,axiom,
! [Sensors: cars > traffic > cars > real,E: cars,Ts: traffic,C2: cars] :
( ( sensors @ Sensors )
=> ( ord_less_real @ zero_zero_real @ ( Sensors @ E @ Ts @ C2 ) ) ) ).
% sensors.sensors_ge
thf(fact_46_sensors_Oleft__space,axiom,
! [Sensors: cars > traffic > cars > real,Ts: traffic,V: view_e774982825t_unit,C2: cars] :
( ( sensors @ Sensors )
=> ( ( left @ ( space @ Sensors @ Ts @ V @ C2 ) )
= ( pos @ Ts @ C2 ) ) ) ).
% sensors.left_space
thf(fact_47_size,axiom,
! [C: cars] :
( ( physical_size @ ts @ C )
= one_one_real ) ).
% size
thf(fact_48_sensors__c,axiom,
( ( regular_regular @ ( own_Product_unit @ v2 ) @ ts @ c )
= one_one_real ) ).
% sensors_c
thf(fact_49_hmlsl_Oview__leq__len__leq,axiom,
! [V: view_e774982825t_unit,V3: view_e774982825t_unit,Ts: traffic,C2: cars] :
( ( ( ord_less_eq_real_int @ ( ext_Product_unit @ V ) @ ( ext_Product_unit @ V3 ) )
& ( ( own_Product_unit @ V )
= ( own_Product_unit @ V3 ) )
& ( ord_less_real @ zero_zero_real @ ( real_length @ ( len @ regular_regular @ V @ Ts @ C2 ) ) ) )
=> ( ord_less_eq_real_int @ ( len @ regular_regular @ V @ Ts @ C2 ) @ ( len @ regular_regular @ V3 @ Ts @ C2 ) ) ) ).
% hmlsl.view_leq_len_leq
thf(fact_50_length__zero__iff__borders__eq,axiom,
! [R: real_int] :
( ( ( real_length @ R )
= zero_zero_real )
= ( ( left @ R )
= ( right @ R ) ) ) ).
% length_zero_iff_borders_eq
thf(fact_51_hmlsl_Olen__fills__subview,axiom,
! [V: view_e774982825t_unit,Ts: traffic,C2: cars] :
( ( ord_less_real @ zero_zero_real @ ( real_length @ ( len @ regular_regular @ V @ Ts @ C2 ) ) )
=> ? [V1: view_e774982825t_unit,V22: view_e774982825t_unit,V32: view_e774982825t_unit,V4: view_e774982825t_unit] :
( ( hchop @ V @ V1 @ V22 )
& ( hchop @ V22 @ V4 @ V32 )
& ( ( len @ regular_regular @ V4 @ Ts @ C2 )
= ( ext_Product_unit @ V4 ) )
& ( ( real_length @ ( len @ regular_regular @ V4 @ Ts @ C2 ) )
= ( real_length @ ( len @ regular_regular @ V @ Ts @ C2 ) ) ) ) ) ).
% hmlsl.len_fills_subview
thf(fact_52_less__real__int__def,axiom,
( ord_less_real_int
= ( ^ [R2: real_int,S: real_int] :
( ( ord_less_eq_real @ ( left @ S ) @ ( left @ R2 ) )
& ( ord_less_eq_real @ ( right @ R2 ) @ ( right @ S ) )
& ~ ( ( ord_less_eq_real @ ( left @ R2 ) @ ( left @ S ) )
& ( ord_less_eq_real @ ( right @ S ) @ ( right @ R2 ) ) ) ) ) ) ).
% less_real_int_def
thf(fact_53_less__eq__real__int__def,axiom,
( ord_less_eq_real_int
= ( ^ [R2: real_int,S: real_int] :
( ( ord_less_eq_real @ ( left @ S ) @ ( left @ R2 ) )
& ( ord_less_eq_real @ ( right @ R2 ) @ ( right @ S ) ) ) ) ) ).
% less_eq_real_int_def
thf(fact_54_left__leq__right,axiom,
! [R: real_int] : ( ord_less_eq_real @ ( left @ R ) @ ( right @ R ) ) ).
% left_leq_right
thf(fact_55_length__ge__zero,axiom,
! [R: real_int] : ( ord_less_eq_real @ zero_zero_real @ ( real_length @ R ) ) ).
% length_ge_zero
thf(fact_56_local_Ohmlsl_Ohchop__assoc,axiom,
! [Phi: traffic > view_e774982825t_unit > $o,Psi: traffic > view_e774982825t_unit > $o,Chi: traffic > view_e774982825t_unit > $o,Ts2: traffic,V2: view_e774982825t_unit] :
( ( ? [W: view_e774982825t_unit,U: view_e774982825t_unit] :
( ( hchop @ V2 @ W @ U )
& ( Phi @ Ts2 @ W )
& ? [X3: view_e774982825t_unit,Y: view_e774982825t_unit] :
( ( hchop @ U @ X3 @ Y )
& ( Psi @ Ts2 @ X3 )
& ( Chi @ Ts2 @ Y ) ) ) )
= ( ? [W: view_e774982825t_unit,U: view_e774982825t_unit] :
( ( hchop @ V2 @ W @ U )
& ? [X3: view_e774982825t_unit,Y: view_e774982825t_unit] :
( ( hchop @ W @ X3 @ Y )
& ( Phi @ Ts2 @ X3 )
& ( Psi @ Ts2 @ Y ) )
& ( Chi @ Ts2 @ U ) ) ) ) ).
% local.hmlsl.hchop_assoc
thf(fact_57_local_Ohmlsl_Ohchop__disj__distr1,axiom,
! [Phi: traffic > view_e774982825t_unit > $o,Psi: traffic > view_e774982825t_unit > $o,Chi: traffic > view_e774982825t_unit > $o,Ts2: traffic,V2: view_e774982825t_unit] :
( ( ? [W: view_e774982825t_unit,U: view_e774982825t_unit] :
( ( hchop @ V2 @ W @ U )
& ( Phi @ Ts2 @ W )
& ( ( Psi @ Ts2 @ U )
| ( Chi @ Ts2 @ U ) ) ) )
= ( ? [W: view_e774982825t_unit,U: view_e774982825t_unit] :
( ( hchop @ V2 @ W @ U )
& ( Phi @ Ts2 @ W )
& ( Psi @ Ts2 @ U ) )
| ? [W: view_e774982825t_unit,U: view_e774982825t_unit] :
( ( hchop @ V2 @ W @ U )
& ( Phi @ Ts2 @ W )
& ( Chi @ Ts2 @ U ) ) ) ) ).
% local.hmlsl.hchop_disj_distr1
thf(fact_58_local_Ohmlsl_Ohchop__disj__distr2,axiom,
! [Psi: traffic > view_e774982825t_unit > $o,Chi: traffic > view_e774982825t_unit > $o,Phi: traffic > view_e774982825t_unit > $o,Ts2: traffic,V2: view_e774982825t_unit] :
( ( ? [W: view_e774982825t_unit,U: view_e774982825t_unit] :
( ( hchop @ V2 @ W @ U )
& ( ( Psi @ Ts2 @ W )
| ( Chi @ Ts2 @ W ) )
& ( Phi @ Ts2 @ U ) ) )
= ( ? [W: view_e774982825t_unit,U: view_e774982825t_unit] :
( ( hchop @ V2 @ W @ U )
& ( Psi @ Ts2 @ W )
& ( Phi @ Ts2 @ U ) )
| ? [W: view_e774982825t_unit,U: view_e774982825t_unit] :
( ( hchop @ V2 @ W @ U )
& ( Chi @ Ts2 @ W )
& ( Phi @ Ts2 @ U ) ) ) ) ).
% local.hmlsl.hchop_disj_distr2
thf(fact_59_local_Ohmlsl_Ohchop__neg1,axiom,
! [Phi: traffic > view_e774982825t_unit > $o,Ts2: traffic,V2: view_e774982825t_unit] :
( ~ ? [Va: view_e774982825t_unit] :
( ? [X_1: view_e774982825t_unit] : ( hchop @ V2 @ Va @ X_1 )
& ( Phi @ Ts2 @ Va ) )
=> ? [Va: view_e774982825t_unit] :
( ? [X_1: view_e774982825t_unit] : ( hchop @ V2 @ Va @ X_1 )
& ~ ( Phi @ Ts2 @ Va ) ) ) ).
% local.hmlsl.hchop_neg1
thf(fact_60_local_Ohmlsl_Ohchop__neg2,axiom,
! [Phi: traffic > view_e774982825t_unit > $o,Ts2: traffic,V2: view_e774982825t_unit] :
( ~ ? [Va: view_e774982825t_unit,U2: view_e774982825t_unit] :
( ( hchop @ V2 @ Va @ U2 )
& ( Phi @ Ts2 @ U2 ) )
=> ? [Va: view_e774982825t_unit,U2: view_e774982825t_unit] :
( ( hchop @ V2 @ Va @ U2 )
& ~ ( Phi @ Ts2 @ U2 ) ) ) ).
% local.hmlsl.hchop_neg2
thf(fact_61_local_Ohmlsl_Ohchop__weaken,axiom,
! [Phi: traffic > view_e774982825t_unit > $o,Ts2: traffic,V2: view_e774982825t_unit] :
( ( Phi @ Ts2 @ V2 )
=> ? [Va: view_e774982825t_unit,U2: view_e774982825t_unit] :
( ( hchop @ V2 @ Va @ U2 )
& ? [Vb: view_e774982825t_unit] :
( ? [X_1: view_e774982825t_unit] : ( hchop @ U2 @ Vb @ X_1 )
& ( Phi @ Ts2 @ Vb ) ) ) ) ).
% local.hmlsl.hchop_weaken
thf(fact_62_local_Ohmlsl_Ohchop__weaken1,axiom,
! [Phi: traffic > view_e774982825t_unit > $o,Ts2: traffic,V2: view_e774982825t_unit] :
( ( Phi @ Ts2 @ V2 )
=> ? [Va: view_e774982825t_unit] :
( ? [X_1: view_e774982825t_unit] : ( hchop @ V2 @ Va @ X_1 )
& ( Phi @ Ts2 @ Va ) ) ) ).
% local.hmlsl.hchop_weaken1
thf(fact_63_local_Ohmlsl_Ohchop__weaken2,axiom,
! [Phi: traffic > view_e774982825t_unit > $o,Ts2: traffic,V2: view_e774982825t_unit] :
( ( Phi @ Ts2 @ V2 )
=> ? [Va: view_e774982825t_unit,U2: view_e774982825t_unit] :
( ( hchop @ V2 @ Va @ U2 )
& ( Phi @ Ts2 @ U2 ) ) ) ).
% local.hmlsl.hchop_weaken2
thf(fact_64_hmlsl_Olen__view__hchop__right,axiom,
! [V: view_e774982825t_unit,Ts: traffic,C2: cars,V12: view_e774982825t_unit,V23: view_e774982825t_unit] :
( ( ( ( len @ regular_regular @ V @ Ts @ C2 )
= ( ext_Product_unit @ V ) )
& ( hchop @ V @ V12 @ V23 ) )
=> ( ( len @ regular_regular @ V23 @ Ts @ C2 )
= ( ext_Product_unit @ V23 ) ) ) ).
% hmlsl.len_view_hchop_right
thf(fact_65_hmlsl_Olen__view__hchop__left,axiom,
! [V: view_e774982825t_unit,Ts: traffic,C2: cars,V12: view_e774982825t_unit,V23: view_e774982825t_unit] :
( ( ( ( len @ regular_regular @ V @ Ts @ C2 )
= ( ext_Product_unit @ V ) )
& ( hchop @ V @ V12 @ V23 ) )
=> ( ( len @ regular_regular @ V12 @ Ts @ C2 )
= ( ext_Product_unit @ V12 ) ) ) ).
% hmlsl.len_view_hchop_left
thf(fact_66_hmlsl_Olen__compose__hchop,axiom,
! [V: view_e774982825t_unit,V12: view_e774982825t_unit,V23: view_e774982825t_unit,Ts: traffic,C2: cars] :
( ( ( hchop @ V @ V12 @ V23 )
& ( ( len @ regular_regular @ V12 @ Ts @ C2 )
= ( ext_Product_unit @ V12 ) )
& ( ( len @ regular_regular @ V23 @ Ts @ C2 )
= ( ext_Product_unit @ V23 ) ) )
=> ( ( len @ regular_regular @ V @ Ts @ C2 )
= ( ext_Product_unit @ V ) ) ) ).
% hmlsl.len_compose_hchop
thf(fact_67_hmlsl_Oext__eq__len__eq,axiom,
! [V: view_e774982825t_unit,V3: view_e774982825t_unit,Ts: traffic,C2: cars] :
( ( ( ( ext_Product_unit @ V )
= ( ext_Product_unit @ V3 ) )
& ( ( own_Product_unit @ V )
= ( own_Product_unit @ V3 ) ) )
=> ( ( len @ regular_regular @ V @ Ts @ C2 )
= ( len @ regular_regular @ V3 @ Ts @ C2 ) ) ) ).
% hmlsl.ext_eq_len_eq
thf(fact_68_local_Ohmlsl_Olength__split,axiom,
! [Ts2: traffic,V2: view_e774982825t_unit] :
( ( ord_less_real @ zero_zero_real @ ( real_length @ ( ext_Product_unit @ V2 ) ) )
=> ? [Va: view_e774982825t_unit,U2: view_e774982825t_unit] :
( ( hchop @ V2 @ Va @ U2 )
& ( ord_less_real @ zero_zero_real @ ( real_length @ ( ext_Product_unit @ Va ) ) )
& ( ord_less_real @ zero_zero_real @ ( real_length @ ( ext_Product_unit @ U2 ) ) ) ) ) ).
% local.hmlsl.length_split
thf(fact_69_local_Ohmlsl_Olength__meld,axiom,
! [Ts2: traffic,V2: view_e774982825t_unit] :
( ? [Va2: view_e774982825t_unit,U3: view_e774982825t_unit] :
( ( hchop @ V2 @ Va2 @ U3 )
& ( ord_less_real @ zero_zero_real @ ( real_length @ ( ext_Product_unit @ Va2 ) ) )
& ( ord_less_real @ zero_zero_real @ ( real_length @ ( ext_Product_unit @ U3 ) ) ) )
=> ( ord_less_real @ zero_zero_real @ ( real_length @ ( ext_Product_unit @ V2 ) ) ) ) ).
% local.hmlsl.length_meld
thf(fact_70_local_Ohmlsl_Olength__dense,axiom,
! [Ts2: traffic,V2: view_e774982825t_unit] :
( ( ord_less_real @ zero_zero_real @ ( real_length @ ( ext_Product_unit @ V2 ) ) )
= ( ? [W: view_e774982825t_unit,U: view_e774982825t_unit] :
( ( hchop @ V2 @ W @ U )
& ( ord_less_real @ zero_zero_real @ ( real_length @ ( ext_Product_unit @ W ) ) )
& ( ord_less_real @ zero_zero_real @ ( real_length @ ( ext_Product_unit @ U ) ) ) ) ) ) ).
% local.hmlsl.length_dense
thf(fact_71_hmlsl_Olen__empty__on__subview2,axiom,
! [V: view_e774982825t_unit,Ts: traffic,C2: cars,V12: view_e774982825t_unit,V23: view_e774982825t_unit] :
( ( ( ( real_length @ ( len @ regular_regular @ V @ Ts @ C2 ) )
= zero_zero_real )
& ( hchop @ V @ V12 @ V23 ) )
=> ( ( real_length @ ( len @ regular_regular @ V23 @ Ts @ C2 ) )
= zero_zero_real ) ) ).
% hmlsl.len_empty_on_subview2
thf(fact_72_hmlsl_Olen__empty__on__subview1,axiom,
! [V: view_e774982825t_unit,Ts: traffic,C2: cars,V12: view_e774982825t_unit,V23: view_e774982825t_unit] :
( ( ( ( real_length @ ( len @ regular_regular @ V @ Ts @ C2 ) )
= zero_zero_real )
& ( hchop @ V @ V12 @ V23 ) )
=> ( ( real_length @ ( len @ regular_regular @ V12 @ Ts @ C2 ) )
= zero_zero_real ) ) ).
% hmlsl.len_empty_on_subview1
thf(fact_73_hmlsl_Olen__hchop__right__right__border,axiom,
! [V: view_e774982825t_unit,Ts: traffic,C2: cars,V12: view_e774982825t_unit,V23: view_e774982825t_unit] :
( ( ( ( len @ regular_regular @ V @ Ts @ C2 )
= ( ext_Product_unit @ V ) )
& ( hchop @ V @ V12 @ V23 ) )
=> ( ( right @ ( len @ regular_regular @ V23 @ Ts @ C2 ) )
= ( right @ ( ext_Product_unit @ V23 ) ) ) ) ).
% hmlsl.len_hchop_right_right_border
thf(fact_74_hmlsl_Olen__hchop__right__left__border,axiom,
! [V: view_e774982825t_unit,Ts: traffic,C2: cars,V12: view_e774982825t_unit,V23: view_e774982825t_unit] :
( ( ( ( len @ regular_regular @ V @ Ts @ C2 )
= ( ext_Product_unit @ V ) )
& ( hchop @ V @ V12 @ V23 ) )
=> ( ( left @ ( len @ regular_regular @ V23 @ Ts @ C2 ) )
= ( left @ ( ext_Product_unit @ V23 ) ) ) ) ).
% hmlsl.len_hchop_right_left_border
thf(fact_75_hmlsl_Olen__hchop__left__right__border,axiom,
! [V: view_e774982825t_unit,Ts: traffic,C2: cars,V12: view_e774982825t_unit,V23: view_e774982825t_unit] :
( ( ( ( len @ regular_regular @ V @ Ts @ C2 )
= ( ext_Product_unit @ V ) )
& ( hchop @ V @ V12 @ V23 ) )
=> ( ( right @ ( len @ regular_regular @ V12 @ Ts @ C2 ) )
= ( right @ ( ext_Product_unit @ V12 ) ) ) ) ).
% hmlsl.len_hchop_left_right_border
thf(fact_76_hmlsl_Olen__hchop__left__left__border,axiom,
! [V: view_e774982825t_unit,Ts: traffic,C2: cars,V12: view_e774982825t_unit,V23: view_e774982825t_unit] :
( ( ( ( len @ regular_regular @ V @ Ts @ C2 )
= ( ext_Product_unit @ V ) )
& ( hchop @ V @ V12 @ V23 ) )
=> ( ( left @ ( len @ regular_regular @ V12 @ Ts @ C2 ) )
= ( left @ ( ext_Product_unit @ V12 ) ) ) ) ).
% hmlsl.len_hchop_left_left_border
thf(fact_77_regular__sensors_Ospace__eq,axiom,
! [V: view_e774982825t_unit,V3: view_e774982825t_unit,Ts: traffic,C2: cars] :
( ( ( own_Product_unit @ V )
= ( own_Product_unit @ V3 ) )
=> ( ( space @ regular_regular @ Ts @ V @ C2 )
= ( space @ regular_regular @ Ts @ V3 @ C2 ) ) ) ).
% regular_sensors.space_eq
thf(fact_78_sensors_Olen__compose__hchop,axiom,
! [Sensors: cars > traffic > cars > real,V: view_e774982825t_unit,V12: view_e774982825t_unit,V23: view_e774982825t_unit,Ts: traffic,C2: cars] :
( ( sensors @ Sensors )
=> ( ( ( hchop @ V @ V12 @ V23 )
& ( ( len @ Sensors @ V12 @ Ts @ C2 )
= ( ext_Product_unit @ V12 ) )
& ( ( len @ Sensors @ V23 @ Ts @ C2 )
= ( ext_Product_unit @ V23 ) ) )
=> ( ( len @ Sensors @ V @ Ts @ C2 )
= ( ext_Product_unit @ V ) ) ) ) ).
% sensors.len_compose_hchop
thf(fact_79_sensors_Olen__view__hchop__left,axiom,
! [Sensors: cars > traffic > cars > real,V: view_e774982825t_unit,Ts: traffic,C2: cars,V12: view_e774982825t_unit,V23: view_e774982825t_unit] :
( ( sensors @ Sensors )
=> ( ( ( ( len @ Sensors @ V @ Ts @ C2 )
= ( ext_Product_unit @ V ) )
& ( hchop @ V @ V12 @ V23 ) )
=> ( ( len @ Sensors @ V12 @ Ts @ C2 )
= ( ext_Product_unit @ V12 ) ) ) ) ).
% sensors.len_view_hchop_left
thf(fact_80_sensors_Olen__view__hchop__right,axiom,
! [Sensors: cars > traffic > cars > real,V: view_e774982825t_unit,Ts: traffic,C2: cars,V12: view_e774982825t_unit,V23: view_e774982825t_unit] :
( ( sensors @ Sensors )
=> ( ( ( ( len @ Sensors @ V @ Ts @ C2 )
= ( ext_Product_unit @ V ) )
& ( hchop @ V @ V12 @ V23 ) )
=> ( ( len @ Sensors @ V23 @ Ts @ C2 )
= ( ext_Product_unit @ V23 ) ) ) ) ).
% sensors.len_view_hchop_right
thf(fact_81_sensors_Oext__eq__len__eq,axiom,
! [Sensors: cars > traffic > cars > real,V: view_e774982825t_unit,V3: view_e774982825t_unit,Ts: traffic,C2: cars] :
( ( sensors @ Sensors )
=> ( ( ( ( ext_Product_unit @ V )
= ( ext_Product_unit @ V3 ) )
& ( ( own_Product_unit @ V )
= ( own_Product_unit @ V3 ) ) )
=> ( ( len @ Sensors @ V @ Ts @ C2 )
= ( len @ Sensors @ V3 @ Ts @ C2 ) ) ) ) ).
% sensors.ext_eq_len_eq
thf(fact_82_sensors_Olen__empty__on__subview1,axiom,
! [Sensors: cars > traffic > cars > real,V: view_e774982825t_unit,Ts: traffic,C2: cars,V12: view_e774982825t_unit,V23: view_e774982825t_unit] :
( ( sensors @ Sensors )
=> ( ( ( ( real_length @ ( len @ Sensors @ V @ Ts @ C2 ) )
= zero_zero_real )
& ( hchop @ V @ V12 @ V23 ) )
=> ( ( real_length @ ( len @ Sensors @ V12 @ Ts @ C2 ) )
= zero_zero_real ) ) ) ).
% sensors.len_empty_on_subview1
thf(fact_83_sensors_Olen__empty__on__subview2,axiom,
! [Sensors: cars > traffic > cars > real,V: view_e774982825t_unit,Ts: traffic,C2: cars,V12: view_e774982825t_unit,V23: view_e774982825t_unit] :
( ( sensors @ Sensors )
=> ( ( ( ( real_length @ ( len @ Sensors @ V @ Ts @ C2 ) )
= zero_zero_real )
& ( hchop @ V @ V12 @ V23 ) )
=> ( ( real_length @ ( len @ Sensors @ V23 @ Ts @ C2 ) )
= zero_zero_real ) ) ) ).
% sensors.len_empty_on_subview2
thf(fact_84_sensors_Olen__hchop__right__right__border,axiom,
! [Sensors: cars > traffic > cars > real,V: view_e774982825t_unit,Ts: traffic,C2: cars,V12: view_e774982825t_unit,V23: view_e774982825t_unit] :
( ( sensors @ Sensors )
=> ( ( ( ( len @ Sensors @ V @ Ts @ C2 )
= ( ext_Product_unit @ V ) )
& ( hchop @ V @ V12 @ V23 ) )
=> ( ( right @ ( len @ Sensors @ V23 @ Ts @ C2 ) )
= ( right @ ( ext_Product_unit @ V23 ) ) ) ) ) ).
% sensors.len_hchop_right_right_border
thf(fact_85_sensors_Olen__hchop__right__left__border,axiom,
! [Sensors: cars > traffic > cars > real,V: view_e774982825t_unit,Ts: traffic,C2: cars,V12: view_e774982825t_unit,V23: view_e774982825t_unit] :
( ( sensors @ Sensors )
=> ( ( ( ( len @ Sensors @ V @ Ts @ C2 )
= ( ext_Product_unit @ V ) )
& ( hchop @ V @ V12 @ V23 ) )
=> ( ( left @ ( len @ Sensors @ V23 @ Ts @ C2 ) )
= ( left @ ( ext_Product_unit @ V23 ) ) ) ) ) ).
% sensors.len_hchop_right_left_border
thf(fact_86_sensors_Olen__hchop__left__right__border,axiom,
! [Sensors: cars > traffic > cars > real,V: view_e774982825t_unit,Ts: traffic,C2: cars,V12: view_e774982825t_unit,V23: view_e774982825t_unit] :
( ( sensors @ Sensors )
=> ( ( ( ( len @ Sensors @ V @ Ts @ C2 )
= ( ext_Product_unit @ V ) )
& ( hchop @ V @ V12 @ V23 ) )
=> ( ( right @ ( len @ Sensors @ V12 @ Ts @ C2 ) )
= ( right @ ( ext_Product_unit @ V12 ) ) ) ) ) ).
% sensors.len_hchop_left_right_border
thf(fact_87_sensors_Olen__hchop__left__left__border,axiom,
! [Sensors: cars > traffic > cars > real,V: view_e774982825t_unit,Ts: traffic,C2: cars,V12: view_e774982825t_unit,V23: view_e774982825t_unit] :
( ( sensors @ Sensors )
=> ( ( ( ( len @ Sensors @ V @ Ts @ C2 )
= ( ext_Product_unit @ V ) )
& ( hchop @ V @ V12 @ V23 ) )
=> ( ( left @ ( len @ Sensors @ V12 @ Ts @ C2 ) )
= ( left @ ( ext_Product_unit @ V12 ) ) ) ) ) ).
% sensors.len_hchop_left_left_border
thf(fact_88_sensors_Olen__fills__subview,axiom,
! [Sensors: cars > traffic > cars > real,V: view_e774982825t_unit,Ts: traffic,C2: cars] :
( ( sensors @ Sensors )
=> ( ( ord_less_real @ zero_zero_real @ ( real_length @ ( len @ Sensors @ V @ Ts @ C2 ) ) )
=> ? [V1: view_e774982825t_unit,V22: view_e774982825t_unit,V32: view_e774982825t_unit,V4: view_e774982825t_unit] :
( ( hchop @ V @ V1 @ V22 )
& ( hchop @ V22 @ V4 @ V32 )
& ( ( len @ Sensors @ V4 @ Ts @ C2 )
= ( ext_Product_unit @ V4 ) )
& ( ( real_length @ ( len @ Sensors @ V4 @ Ts @ C2 ) )
= ( real_length @ ( len @ Sensors @ V @ Ts @ C2 ) ) ) ) ) ) ).
% sensors.len_fills_subview
thf(fact_89_sensors_Oview__leq__len__leq,axiom,
! [Sensors: cars > traffic > cars > real,V: view_e774982825t_unit,V3: view_e774982825t_unit,Ts: traffic,C2: cars] :
( ( sensors @ Sensors )
=> ( ( ( ord_less_eq_real_int @ ( ext_Product_unit @ V ) @ ( ext_Product_unit @ V3 ) )
& ( ( own_Product_unit @ V )
= ( own_Product_unit @ V3 ) )
& ( ord_less_real @ zero_zero_real @ ( real_length @ ( len @ Sensors @ V @ Ts @ C2 ) ) ) )
=> ( ord_less_eq_real_int @ ( len @ Sensors @ V @ Ts @ C2 ) @ ( len @ Sensors @ V3 @ Ts @ C2 ) ) ) ) ).
% sensors.view_leq_len_leq
thf(fact_90_view_Oh__chop__middle2,axiom,
! [V: view_e774982825t_unit,U4: view_e774982825t_unit,W2: view_e774982825t_unit] :
( ( hchop @ V @ U4 @ W2 )
=> ( ord_less_eq_real @ ( left @ ( ext_Product_unit @ W2 ) ) @ ( right @ ( ext_Product_unit @ V ) ) ) ) ).
% view.h_chop_middle2
thf(fact_91_view_Oh__chop__middle1,axiom,
! [V: view_e774982825t_unit,U4: view_e774982825t_unit,W2: view_e774982825t_unit] :
( ( hchop @ V @ U4 @ W2 )
=> ( ord_less_eq_real @ ( left @ ( ext_Product_unit @ V ) ) @ ( right @ ( ext_Product_unit @ U4 ) ) ) ) ).
% view.h_chop_middle1
thf(fact_92_view_Ohorizontal__chop__non__empty,axiom,
! [V: view_e774982825t_unit] :
( ( ord_less_real @ zero_zero_real @ ( real_length @ ( ext_Product_unit @ V ) ) )
=> ? [U2: view_e774982825t_unit,W3: view_e774982825t_unit] :
( ( hchop @ V @ U2 @ W3 )
& ( ord_less_real @ zero_zero_real @ ( real_length @ ( ext_Product_unit @ U2 ) ) )
& ( ord_less_real @ zero_zero_real @ ( real_length @ ( ext_Product_unit @ W3 ) ) ) ) ) ).
% view.horizontal_chop_non_empty
thf(fact_93_hmlsl_Oright__space,axiom,
! [Ts: traffic,V: view_e774982825t_unit,C2: cars] :
( ( right @ ( space @ regular_regular @ Ts @ V @ C2 ) )
= ( plus_plus_real @ ( pos @ Ts @ C2 ) @ ( regular_regular @ ( own_Product_unit @ V ) @ Ts @ C2 ) ) ) ).
% hmlsl.right_space
thf(fact_94_local_Ohmlsl_Olength__add,axiom,
! [X2: real,Y2: real,Ts2: traffic,V2: view_e774982825t_unit] :
( ( ( ord_less_eq_real @ zero_zero_real @ X2 )
& ( ord_less_eq_real @ zero_zero_real @ Y2 ) )
=> ( ( ( real_length @ ( ext_Product_unit @ V2 ) )
= ( plus_plus_real @ X2 @ Y2 ) )
= ( ? [W: view_e774982825t_unit,U: view_e774982825t_unit] :
( ( hchop @ V2 @ W @ U )
& ( ( real_length @ ( ext_Product_unit @ W ) )
= X2 )
& ( ( real_length @ ( ext_Product_unit @ U ) )
= Y2 ) ) ) ) ) ).
% local.hmlsl.length_add
thf(fact_95_local_Ohmlsl_Olength__add2,axiom,
! [X2: real,Y2: real,Ts2: traffic,V2: view_e774982825t_unit] :
( ( ( ord_less_eq_real @ zero_zero_real @ X2 )
& ( ord_less_eq_real @ zero_zero_real @ Y2 ) )
=> ( ( ( real_length @ ( ext_Product_unit @ V2 ) )
= ( plus_plus_real @ X2 @ Y2 ) )
=> ? [Va: view_e774982825t_unit,U2: view_e774982825t_unit] :
( ( hchop @ V2 @ Va @ U2 )
& ( ( real_length @ ( ext_Product_unit @ Va ) )
= X2 )
& ( ( real_length @ ( ext_Product_unit @ U2 ) )
= Y2 ) ) ) ) ).
% local.hmlsl.length_add2
thf(fact_96_traffic_OpsGeZero,axiom,
! [Ts: traffic,C: cars] : ( ord_less_real @ zero_zero_real @ ( physical_size @ Ts @ C ) ) ).
% traffic.psGeZero
thf(fact_97_local_Ohmlsl_Olength__add1,axiom,
! [X2: real,Y2: real,Ts2: traffic,V2: view_e774982825t_unit] :
( ? [Va2: view_e774982825t_unit,U3: view_e774982825t_unit] :
( ( hchop @ V2 @ Va2 @ U3 )
& ( ( real_length @ ( ext_Product_unit @ Va2 ) )
= X2 )
& ( ( real_length @ ( ext_Product_unit @ U3 ) )
= Y2 ) )
=> ( ( real_length @ ( ext_Product_unit @ V2 ) )
= ( plus_plus_real @ X2 @ Y2 ) ) ) ).
% local.hmlsl.length_add1
thf(fact_98_add__left__cancel,axiom,
! [A: real,B: real,C2: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_99_add__right__cancel,axiom,
! [B: real,A: real,C2: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_100_hmlsl_Olen__hchop__add,axiom,
! [V: view_e774982825t_unit,V12: view_e774982825t_unit,V23: view_e774982825t_unit,Ts: traffic,C2: cars] :
( ( hchop @ V @ V12 @ V23 )
=> ( ( real_length @ ( len @ regular_regular @ V @ Ts @ C2 ) )
= ( plus_plus_real @ ( real_length @ ( len @ regular_regular @ V12 @ Ts @ C2 ) ) @ ( real_length @ ( len @ regular_regular @ V23 @ Ts @ C2 ) ) ) ) ) ).
% hmlsl.len_hchop_add
thf(fact_101_add__le__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_102_add__le__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_103_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_104_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_105_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_106_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_107_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_108_double__zero,axiom,
! [A: real] :
( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% double_zero
thf(fact_109_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_110_add_Oleft__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.left_neutral
thf(fact_111_add__less__cancel__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_112_add__less__cancel__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_113_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_114_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_115_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_116_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_117_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_118_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_119_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_120_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_121_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_122_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_123_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_124_zero__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 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_125_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_126_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_127_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_128_group__cancel_Oadd2,axiom,
! [B2: real,K: real,B: real,A: real] :
( ( B2
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A @ B2 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_129_add_Oassoc,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_130_add_Oleft__cancel,axiom,
! [A: real,B: real,C2: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C2 ) )
= ( B = C2 ) ) ).
% add.left_cancel
thf(fact_131_add_Oright__cancel,axiom,
! [B: real,A: real,C2: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C2 @ A ) )
= ( B = C2 ) ) ).
% add.right_cancel
thf(fact_132_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A3: real,B3: real] : ( plus_plus_real @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_133_add_Oleft__commute,axiom,
! [B: real,A: real,C2: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C2 ) )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_134_add__left__imp__eq,axiom,
! [A: real,B: real,C2: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_135_add__right__imp__eq,axiom,
! [B: real,A: real,C2: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_136_add__le__imp__le__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_137_add__le__imp__le__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_138_add__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_139_add__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_140_add__mono,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C2 @ D )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_mono
thf(fact_141_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_142_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_143_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_144_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_145_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_146_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_147_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_148_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_149_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_150_add__strict__mono,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C2 @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_151_add__strict__left__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_152_add__strict__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_153_add__less__imp__less__left,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_154_add__less__imp__less__right,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_155_add__nonpos__eq__0__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
=> ( ( ( plus_plus_real @ X2 @ Y2 )
= zero_zero_real )
= ( ( X2 = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_156_add__nonneg__eq__0__iff,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ( plus_plus_real @ X2 @ Y2 )
= zero_zero_real )
= ( ( X2 = zero_zero_real )
& ( Y2 = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_157_add__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_158_add__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_159_add__increasing2,axiom,
! [C2: real,B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ C2 )
=> ( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_160_add__decreasing2,axiom,
! [C2: real,A: real,B: real] :
( ( ord_less_eq_real @ C2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_161_add__increasing,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C2 ) ) ) ) ).
% add_increasing
thf(fact_162_add__decreasing,axiom,
! [A: real,C2: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ C2 @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ B ) ) ) ).
% add_decreasing
thf(fact_163_add__less__le__mono,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C2 @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_164_add__le__less__mono,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C2 @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_165_add__mono__thms__linordered__field_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_166_add__mono__thms__linordered__field_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_167_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_168_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_169_pos__add__strict,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C2 )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_170_add__strict__increasing2,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C2 )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C2 ) ) ) ) ).
% add_strict_increasing2
thf(fact_171_add__strict__increasing,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C2 ) ) ) ) ).
% add_strict_increasing
thf(fact_172_add__pos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_173_add__nonpos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_neg
thf(fact_174_add__nonneg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_175_add__neg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_nonpos
thf(fact_176_consec__add,axiom,
! [R: real_int,S2: real_int,T: real_int] :
( ( ( ( left @ R )
= ( left @ S2 ) )
& ( ( right @ R )
= ( right @ T ) )
& ( ( right @ S2 )
= ( left @ T ) ) )
=> ( ( real_length @ R )
= ( plus_plus_real @ ( real_length @ S2 ) @ ( real_length @ T ) ) ) ) ).
% consec_add
thf(fact_177_view_Ohorizontal__chop__empty__right,axiom,
! [V2: view_e774982825t_unit] :
? [X_1: view_e774982825t_unit] : ( hchop @ V2 @ V2 @ X_1 ) ).
% view.horizontal_chop_empty_right
thf(fact_178_view_Ohorizontal__chop__empty__left,axiom,
! [V2: view_e774982825t_unit] :
? [U2: view_e774982825t_unit] : ( hchop @ V2 @ U2 @ V2 ) ).
% view.horizontal_chop_empty_left
thf(fact_179_view_Ohorizontal__chop__assoc2,axiom,
! [V: view_e774982825t_unit,V12: view_e774982825t_unit,V23: view_e774982825t_unit,V33: view_e774982825t_unit,V42: view_e774982825t_unit] :
( ( ( hchop @ V @ V12 @ V23 )
& ( hchop @ V12 @ V33 @ V42 ) )
=> ? [V4: view_e774982825t_unit] :
( ( hchop @ V @ V33 @ V4 )
& ( hchop @ V4 @ V42 @ V23 ) ) ) ).
% view.horizontal_chop_assoc2
thf(fact_180_view_Ohorizontal__chop__assoc1,axiom,
! [V: view_e774982825t_unit,V12: view_e774982825t_unit,V23: view_e774982825t_unit,V33: view_e774982825t_unit,V42: view_e774982825t_unit] :
( ( ( hchop @ V @ V12 @ V23 )
& ( hchop @ V23 @ V33 @ V42 ) )
=> ? [V4: view_e774982825t_unit] :
( ( hchop @ V @ V4 @ V42 )
& ( hchop @ V4 @ V12 @ V33 ) ) ) ).
% view.horizontal_chop_assoc1
thf(fact_181_view_Ohorizontal__chop1,axiom,
! [V: view_e774982825t_unit] :
? [U2: view_e774982825t_unit,X_1: view_e774982825t_unit] : ( hchop @ V @ U2 @ X_1 ) ).
% view.horizontal_chop1
thf(fact_182_view_Ohorizontal__chop__split__add,axiom,
! [X2: real,Y2: real,V: view_e774982825t_unit] :
( ( ( ord_less_eq_real @ zero_zero_real @ X2 )
& ( ord_less_eq_real @ zero_zero_real @ Y2 ) )
=> ( ( ( real_length @ ( ext_Product_unit @ V ) )
= ( plus_plus_real @ X2 @ Y2 ) )
=> ? [U2: view_e774982825t_unit,W3: view_e774982825t_unit] :
( ( hchop @ V @ U2 @ W3 )
& ( ( real_length @ ( ext_Product_unit @ U2 ) )
= X2 )
& ( ( real_length @ ( ext_Product_unit @ W3 ) )
= Y2 ) ) ) ) ).
% view.horizontal_chop_split_add
thf(fact_183_sensors_Olen__hchop__add,axiom,
! [Sensors: cars > traffic > cars > real,V: view_e774982825t_unit,V12: view_e774982825t_unit,V23: view_e774982825t_unit,Ts: traffic,C2: cars] :
( ( sensors @ Sensors )
=> ( ( hchop @ V @ V12 @ V23 )
=> ( ( real_length @ ( len @ Sensors @ V @ Ts @ C2 ) )
= ( plus_plus_real @ ( real_length @ ( len @ Sensors @ V12 @ Ts @ C2 ) ) @ ( real_length @ ( len @ Sensors @ V23 @ Ts @ C2 ) ) ) ) ) ) ).
% sensors.len_hchop_add
thf(fact_184_sensors_Oright__space,axiom,
! [Sensors: cars > traffic > cars > real,Ts: traffic,V: view_e774982825t_unit,C2: cars] :
( ( sensors @ Sensors )
=> ( ( right @ ( space @ Sensors @ Ts @ V @ C2 ) )
= ( plus_plus_real @ ( pos @ Ts @ C2 ) @ ( Sensors @ ( own_Product_unit @ V ) @ Ts @ C2 ) ) ) ) ).
% sensors.right_space
thf(fact_185_view_Ohorizontal__chop__own__trans,axiom,
! [V: view_e774982825t_unit,U4: view_e774982825t_unit,W2: view_e774982825t_unit] :
( ( hchop @ V @ U4 @ W2 )
=> ( ( own_Product_unit @ U4 )
= ( own_Product_unit @ W2 ) ) ) ).
% view.horizontal_chop_own_trans
thf(fact_186_view_Ohorizontal__chop__leq2,axiom,
! [V: view_e774982825t_unit,U4: view_e774982825t_unit,W2: view_e774982825t_unit] :
( ( hchop @ V @ U4 @ W2 )
=> ( ord_le461438217t_unit @ W2 @ V ) ) ).
% view.horizontal_chop_leq2
thf(fact_187_view_Ohorizontal__chop__leq1,axiom,
! [V: view_e774982825t_unit,U4: view_e774982825t_unit,W2: view_e774982825t_unit] :
( ( hchop @ V @ U4 @ W2 )
=> ( ord_le461438217t_unit @ U4 @ V ) ) ).
% view.horizontal_chop_leq1
thf(fact_188_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% zero_less_two
thf(fact_189_field__le__epsilon,axiom,
! [X2: real,Y2: real] :
( ! [E4: real] :
( ( ord_less_real @ zero_zero_real @ E4 )
=> ( ord_less_eq_real @ X2 @ ( plus_plus_real @ Y2 @ E4 ) ) )
=> ( ord_less_eq_real @ X2 @ Y2 ) ) ).
% field_le_epsilon
thf(fact_190_add__mono1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% add_mono1
thf(fact_191_less__add__one,axiom,
! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% less_add_one
thf(fact_192_linorder__neqE__linordered__idom,axiom,
! [X2: real,Y2: real] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_real @ Y2 @ X2 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_193_linordered__field__no__ub,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% linordered_field_no_ub
thf(fact_194_linordered__field__no__lb,axiom,
! [X: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X ) ).
% linordered_field_no_lb
thf(fact_195_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_196_zero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_le_one
thf(fact_197_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_198_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_199_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_200_add__less__zeroD,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ ( plus_plus_real @ X2 @ Y2 ) @ zero_zero_real )
=> ( ( ord_less_real @ X2 @ zero_zero_real )
| ( ord_less_real @ Y2 @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_201_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_202_space__c,axiom,
( ( space @ regular_regular @ ts @ v2 @ c )
= ( real_Abs_real_int @ ( produc705216881l_real @ zero_zero_real @ one_one_real ) ) ) ).
% space_c
thf(fact_203_hmlsl_Ospace__def,axiom,
! [Ts: traffic,V: view_e774982825t_unit,C2: cars] :
( ( space @ regular_regular @ Ts @ V @ C2 )
= ( real_Abs_real_int @ ( produc705216881l_real @ ( pos @ Ts @ C2 ) @ ( plus_plus_real @ ( pos @ Ts @ C2 ) @ ( regular_regular @ ( own_Product_unit @ V ) @ Ts @ C2 ) ) ) ) ) ).
% hmlsl.space_def
thf(fact_204_len__c,axiom,
( ( len @ regular_regular @ v2 @ ts @ c )
= ( real_Abs_real_int @ ( produc705216881l_real @ zero_zero_real @ one_one_real ) ) ) ).
% len_c
thf(fact_205_sensors_Ospace__def,axiom,
! [Sensors: cars > traffic > cars > real,Ts: traffic,V: view_e774982825t_unit,C2: cars] :
( ( sensors @ Sensors )
=> ( ( space @ Sensors @ Ts @ V @ C2 )
= ( real_Abs_real_int @ ( produc705216881l_real @ ( pos @ Ts @ C2 ) @ ( plus_plus_real @ ( pos @ Ts @ C2 ) @ ( Sensors @ ( own_Product_unit @ V ) @ Ts @ C2 ) ) ) ) ) ) ).
% sensors.space_def
thf(fact_206_is__num__normalize_I1_J,axiom,
! [A: real,B: real,C2: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C2 )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).
% is_num_normalize(1)
thf(fact_207_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_208_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_209_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_210_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_211_real__int__class_Oshift__def,axiom,
( real_shift
= ( ^ [R2: real_int,X3: real] : ( real_Abs_real_int @ ( produc705216881l_real @ ( plus_plus_real @ ( left @ R2 ) @ X3 ) @ ( plus_plus_real @ ( right @ R2 ) @ X3 ) ) ) ) ) ).
% real_int_class.shift_def
thf(fact_212_shift__additivity,axiom,
! [R: real_int,X2: real,Y2: real] :
( ( real_shift @ R @ ( plus_plus_real @ X2 @ Y2 ) )
= ( real_shift @ ( real_shift @ R @ X2 ) @ Y2 ) ) ).
% shift_additivity
thf(fact_213_shift__keeps__length,axiom,
! [X2: real] :
( real_length
= ( ^ [R2: real_int] : ( real_length @ ( real_shift @ R2 @ X2 ) ) ) ) ).
% shift_keeps_length
thf(fact_214_shift__zero,axiom,
! [R: real_int] :
( ( real_shift @ R @ zero_zero_real )
= R ) ).
% shift_zero
thf(fact_215_shift__left__eq__right,axiom,
! [R: real_int,X2: real] : ( ord_less_eq_real @ ( left @ ( real_shift @ R @ X2 ) ) @ ( right @ ( real_shift @ R @ X2 ) ) ) ).
% shift_left_eq_right
thf(fact_216_ts__rep__def,axiom,
( ts_rep
= ( produc1876576040s_real @ po @ ( produc1230243850s_real @ re @ ( produc1348924850s_real @ cl @ ( produc795767668s_real @ dy @ ( produc1439932679s_real @ ps @ sd ) ) ) ) ) ) ).
% ts_rep_def
thf(fact_217_order__refl,axiom,
! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% order_refl
thf(fact_218_order__refl,axiom,
! [X2: real_int] : ( ord_less_eq_real_int @ X2 @ X2 ) ).
% order_refl
thf(fact_219_order__refl,axiom,
! [X2: view_e774982825t_unit] : ( ord_le461438217t_unit @ X2 @ X2 ) ).
% order_refl
thf(fact_220_clm__def,axiom,
! [C: cars] :
( ( cl @ C )
= bot_bot_nat_int ) ).
% clm_def
thf(fact_221_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_222_dual__order_Oantisym,axiom,
! [B: real_int,A: real_int] :
( ( ord_less_eq_real_int @ B @ A )
=> ( ( ord_less_eq_real_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_223_dual__order_Oantisym,axiom,
! [B: view_e774982825t_unit,A: view_e774982825t_unit] :
( ( ord_le461438217t_unit @ B @ A )
=> ( ( ord_le461438217t_unit @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_224_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: real,Z: real] : Y4 = Z )
= ( ^ [A3: real,B3: real] :
( ( ord_less_eq_real @ B3 @ A3 )
& ( ord_less_eq_real @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_225_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: real_int,Z: real_int] : Y4 = Z )
= ( ^ [A3: real_int,B3: real_int] :
( ( ord_less_eq_real_int @ B3 @ A3 )
& ( ord_less_eq_real_int @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_226_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: view_e774982825t_unit,Z: view_e774982825t_unit] : Y4 = Z )
= ( ^ [A3: view_e774982825t_unit,B3: view_e774982825t_unit] :
( ( ord_le461438217t_unit @ B3 @ A3 )
& ( ord_le461438217t_unit @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_227_dual__order_Otrans,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C2 @ B )
=> ( ord_less_eq_real @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_228_dual__order_Otrans,axiom,
! [B: real_int,A: real_int,C2: real_int] :
( ( ord_less_eq_real_int @ B @ A )
=> ( ( ord_less_eq_real_int @ C2 @ B )
=> ( ord_less_eq_real_int @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_229_dual__order_Otrans,axiom,
! [B: view_e774982825t_unit,A: view_e774982825t_unit,C2: view_e774982825t_unit] :
( ( ord_le461438217t_unit @ B @ A )
=> ( ( ord_le461438217t_unit @ C2 @ B )
=> ( ord_le461438217t_unit @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_230_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: real,B4: real] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_231_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_232_dual__order_Orefl,axiom,
! [A: real_int] : ( ord_less_eq_real_int @ A @ A ) ).
% dual_order.refl
thf(fact_233_dual__order_Orefl,axiom,
! [A: view_e774982825t_unit] : ( ord_le461438217t_unit @ A @ A ) ).
% dual_order.refl
thf(fact_234_order__trans,axiom,
! [X2: real,Y2: real,Z2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z2 )
=> ( ord_less_eq_real @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_235_order__trans,axiom,
! [X2: real_int,Y2: real_int,Z2: real_int] :
( ( ord_less_eq_real_int @ X2 @ Y2 )
=> ( ( ord_less_eq_real_int @ Y2 @ Z2 )
=> ( ord_less_eq_real_int @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_236_order__trans,axiom,
! [X2: view_e774982825t_unit,Y2: view_e774982825t_unit,Z2: view_e774982825t_unit] :
( ( ord_le461438217t_unit @ X2 @ Y2 )
=> ( ( ord_le461438217t_unit @ Y2 @ Z2 )
=> ( ord_le461438217t_unit @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_237_order__class_Oorder_Oantisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_238_order__class_Oorder_Oantisym,axiom,
! [A: real_int,B: real_int] :
( ( ord_less_eq_real_int @ A @ B )
=> ( ( ord_less_eq_real_int @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_239_order__class_Oorder_Oantisym,axiom,
! [A: view_e774982825t_unit,B: view_e774982825t_unit] :
( ( ord_le461438217t_unit @ A @ B )
=> ( ( ord_le461438217t_unit @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_240_ord__le__eq__trans,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_real @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_241_ord__le__eq__trans,axiom,
! [A: real_int,B: real_int,C2: real_int] :
( ( ord_less_eq_real_int @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_real_int @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_242_ord__le__eq__trans,axiom,
! [A: view_e774982825t_unit,B: view_e774982825t_unit,C2: view_e774982825t_unit] :
( ( ord_le461438217t_unit @ A @ B )
=> ( ( B = C2 )
=> ( ord_le461438217t_unit @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_243_ord__eq__le__trans,axiom,
! [A: real,B: real,C2: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ord_less_eq_real @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_244_ord__eq__le__trans,axiom,
! [A: real_int,B: real_int,C2: real_int] :
( ( A = B )
=> ( ( ord_less_eq_real_int @ B @ C2 )
=> ( ord_less_eq_real_int @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_245_ord__eq__le__trans,axiom,
! [A: view_e774982825t_unit,B: view_e774982825t_unit,C2: view_e774982825t_unit] :
( ( A = B )
=> ( ( ord_le461438217t_unit @ B @ C2 )
=> ( ord_le461438217t_unit @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_246_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: real,Z: real] : Y4 = Z )
= ( ^ [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
& ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_247_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: real_int,Z: real_int] : Y4 = Z )
= ( ^ [A3: real_int,B3: real_int] :
( ( ord_less_eq_real_int @ A3 @ B3 )
& ( ord_less_eq_real_int @ B3 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_248_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: view_e774982825t_unit,Z: view_e774982825t_unit] : Y4 = Z )
= ( ^ [A3: view_e774982825t_unit,B3: view_e774982825t_unit] :
( ( ord_le461438217t_unit @ A3 @ B3 )
& ( ord_le461438217t_unit @ B3 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_249_antisym__conv,axiom,
! [Y2: real,X2: real] :
( ( ord_less_eq_real @ Y2 @ X2 )
=> ( ( ord_less_eq_real @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv
thf(fact_250_antisym__conv,axiom,
! [Y2: real_int,X2: real_int] :
( ( ord_less_eq_real_int @ Y2 @ X2 )
=> ( ( ord_less_eq_real_int @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv
thf(fact_251_antisym__conv,axiom,
! [Y2: view_e774982825t_unit,X2: view_e774982825t_unit] :
( ( ord_le461438217t_unit @ Y2 @ X2 )
=> ( ( ord_le461438217t_unit @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv
thf(fact_252_le__cases3,axiom,
! [X2: real,Y2: real,Z2: real] :
( ( ( ord_less_eq_real @ X2 @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ Y2 @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ X2 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_real @ Z2 @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_real @ Y2 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_real @ Z2 @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_253_order_Otrans,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C2 )
=> ( ord_less_eq_real @ A @ C2 ) ) ) ).
% order.trans
thf(fact_254_order_Otrans,axiom,
! [A: real_int,B: real_int,C2: real_int] :
( ( ord_less_eq_real_int @ A @ B )
=> ( ( ord_less_eq_real_int @ B @ C2 )
=> ( ord_less_eq_real_int @ A @ C2 ) ) ) ).
% order.trans
thf(fact_255_order_Otrans,axiom,
! [A: view_e774982825t_unit,B: view_e774982825t_unit,C2: view_e774982825t_unit] :
( ( ord_le461438217t_unit @ A @ B )
=> ( ( ord_le461438217t_unit @ B @ C2 )
=> ( ord_le461438217t_unit @ A @ C2 ) ) ) ).
% order.trans
thf(fact_256_le__cases,axiom,
! [X2: real,Y2: real] :
( ~ ( ord_less_eq_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X2 ) ) ).
% le_cases
thf(fact_257_eq__refl,axiom,
! [X2: real,Y2: real] :
( ( X2 = Y2 )
=> ( ord_less_eq_real @ X2 @ Y2 ) ) ).
% eq_refl
thf(fact_258_eq__refl,axiom,
! [X2: real_int,Y2: real_int] :
( ( X2 = Y2 )
=> ( ord_less_eq_real_int @ X2 @ Y2 ) ) ).
% eq_refl
thf(fact_259_eq__refl,axiom,
! [X2: view_e774982825t_unit,Y2: view_e774982825t_unit] :
( ( X2 = Y2 )
=> ( ord_le461438217t_unit @ X2 @ Y2 ) ) ).
% eq_refl
thf(fact_260_linear,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
| ( ord_less_eq_real @ Y2 @ X2 ) ) ).
% linear
thf(fact_261_antisym,axiom,
! [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% antisym
thf(fact_262_antisym,axiom,
! [X2: real_int,Y2: real_int] :
( ( ord_less_eq_real_int @ X2 @ Y2 )
=> ( ( ord_less_eq_real_int @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% antisym
thf(fact_263_antisym,axiom,
! [X2: view_e774982825t_unit,Y2: view_e774982825t_unit] :
( ( ord_le461438217t_unit @ X2 @ Y2 )
=> ( ( ord_le461438217t_unit @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% antisym
thf(fact_264_eq__iff,axiom,
( ( ^ [Y4: real,Z: real] : Y4 = Z )
= ( ^ [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ Y )
& ( ord_less_eq_real @ Y @ X3 ) ) ) ) ).
% eq_iff
thf(fact_265_eq__iff,axiom,
( ( ^ [Y4: real_int,Z: real_int] : Y4 = Z )
= ( ^ [X3: real_int,Y: real_int] :
( ( ord_less_eq_real_int @ X3 @ Y )
& ( ord_less_eq_real_int @ Y @ X3 ) ) ) ) ).
% eq_iff
thf(fact_266_eq__iff,axiom,
( ( ^ [Y4: view_e774982825t_unit,Z: view_e774982825t_unit] : Y4 = Z )
= ( ^ [X3: view_e774982825t_unit,Y: view_e774982825t_unit] :
( ( ord_le461438217t_unit @ X3 @ Y )
& ( ord_le461438217t_unit @ Y @ X3 ) ) ) ) ).
% eq_iff
% Conjectures (1)
thf(conj_0,conjecture,
( ( ord_less_real @ ( left @ ( space @ regular_regular @ ts @ v @ d ) ) @ ( right @ ( ext_Product_unit @ v ) ) )
& ( ord_less_real @ ( left @ ( ext_Product_unit @ v ) ) @ ( right @ ( space @ regular_regular @ ts @ v @ d ) ) ) ) ).
%------------------------------------------------------------------------------