TPTP Problem File: ITP170^2.p
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%------------------------------------------------------------------------------
% File : ITP170^2 : 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 : 0.00 v7.5.0
% Syntax : Number of formulae : 377 ( 97 unt; 75 typ; 0 def)
% Number of atoms : 820 ( 246 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3738 ( 36 ~; 7 |; 112 &;3252 @)
% ( 0 <=>; 331 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 7 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 213 ( 213 >; 0 *; 0 +; 0 <<)
% Number of symbols : 70 ( 69 usr; 6 con; 0-4 aty)
% Number of variables : 909 ( 25 ^; 772 !; 69 ?; 909 :)
% ( 43 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:23:21.183
%------------------------------------------------------------------------------
% Could-be-implicit typings (8)
thf(ty_t_Views_Oview_Oview__ext,type,
view_ext: $tType > $tType ).
thf(ty_t_Product__Type_Ounit,type,
product_unit: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_RealInt_Oreal__int,type,
real_int: $tType ).
thf(ty_t_Traffic_Otraffic,type,
traffic: $tType ).
thf(ty_t_NatInt_Onat__int,type,
nat_int: $tType ).
thf(ty_t_Real_Oreal,type,
real: $tType ).
thf(ty_t_Cars_Ocars,type,
cars: $tType ).
% Explicit typings (67)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__less__one,type,
zero_less_one:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
ordere216010020id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord1659791738miring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add,type,
ordere1818651114id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
strict797366125id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere223160158up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
strict2144017051up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
unique1598680935umeral:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
ordere516151231imp_le:
!>[A: $tType] : $o ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
unique455577585es_aux:
!>[A: $tType] : ( ( product_prod @ A @ A ) > $o ) ).
thf(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Length_Osensors_Olen,type,
len: ( cars > traffic > cars > real ) > ( view_ext @ product_unit ) > traffic > cars > real_int ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_RealInt_Oleft,type,
left: real_int > real ).
thf(sy_c_RealInt_Oreal__int_OAbs__real__int,type,
real_Abs_real_int: ( product_prod @ real @ 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_ext @ product_unit ) > cars > real_int ).
thf(sy_c_Traffic_Otraffic_OAbs__traffic,type,
abs_traffic: ( product_prod @ ( cars > real ) @ ( product_prod @ ( cars > nat_int ) @ ( product_prod @ ( cars > nat_int ) @ ( product_prod @ ( cars > real > real ) @ ( product_prod @ ( cars > real ) @ ( cars > 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,type,
ext:
!>[Z: $tType] : ( ( view_ext @ Z ) > real_int ) ).
thf(sy_c_Views_Oview_Ohchop,type,
hchop: ( view_ext @ product_unit ) > ( view_ext @ product_unit ) > ( view_ext @ product_unit ) > $o ).
thf(sy_c_Views_Oview_Oown,type,
own:
!>[Z: $tType] : ( ( view_ext @ Z ) > 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: product_prod @ ( cars > real ) @ ( product_prod @ ( cars > nat_int ) @ ( product_prod @ ( cars > nat_int ) @ ( product_prod @ ( cars > real > real ) @ ( product_prod @ ( cars > real ) @ ( cars > real ) ) ) ) ) ).
thf(sy_v_v_H____,type,
v: view_ext @ product_unit ).
thf(sy_v_v____,type,
v2: view_ext @ product_unit ).
% Relevant facts (252)
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_ext @ product_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_ext @ product_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_ext @ product_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_ext @ product_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_ext @ product_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_ext @ product_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_ext @ product_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_ext @ product_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_ext @ product_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_ext @ product_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_ext @ product_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_ext @ product_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_ext @ product_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_ext @ product_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_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_33_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_34__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_35_hmlsl_Olen__sub__int,axiom,
! [V: view_ext @ product_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_36_hmlsl_Olen__empty__subview,axiom,
! [V: view_ext @ product_unit,Ts: traffic,C2: cars,V3: view_ext @ product_unit] :
( ( ( ( real_length @ ( len @ regular_regular @ V @ Ts @ C2 ) )
= ( zero_zero @ real ) )
& ( ord_less_eq @ ( view_ext @ product_unit ) @ V3 @ V ) )
=> ( ( real_length @ ( len @ regular_regular @ V3 @ Ts @ C2 ) )
= ( zero_zero @ real ) ) ) ).
% hmlsl.len_empty_subview
thf(fact_37_lv,axiom,
( ( left @ ( ext @ product_unit @ v2 ) )
= ( zero_zero @ real ) ) ).
% lv
thf(fact_38_rc,axiom,
( ( right @ ( space @ regular_regular @ ts @ v2 @ c ) )
= ( one_one @ real ) ) ).
% rc
thf(fact_39_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X2: A] :
( ( ( one_one @ A )
= X2 )
= ( X2
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_40_sensors_Olen__sub__int,axiom,
! [Sensors: cars > traffic > cars > real,V: view_ext @ product_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_41_sensors_Olen__empty__subview,axiom,
! [Sensors: cars > traffic > cars > real,V: view_ext @ product_unit,Ts: traffic,C2: cars,V3: view_ext @ product_unit] :
( ( sensors @ Sensors )
=> ( ( ( ( real_length @ ( len @ Sensors @ V @ Ts @ C2 ) )
= ( zero_zero @ real ) )
& ( ord_less_eq @ ( view_ext @ product_unit ) @ V3 @ V ) )
=> ( ( real_length @ ( len @ Sensors @ V3 @ Ts @ C2 ) )
= ( zero_zero @ real ) ) ) ) ).
% sensors.len_empty_subview
thf(fact_42_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_43_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X2: A] :
( ( ( zero_zero @ A )
= X2 )
= ( X2
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_44_sensors_Olen_Ocong,axiom,
len = len ).
% sensors.len.cong
thf(fact_45_sensors_Ospace_Ocong,axiom,
space = space ).
% sensors.space.cong
thf(fact_46_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 ) ) ).
% zero_le
thf(fact_47_gr__zeroI,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( N
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% gr_zeroI
thf(fact_48_not__less__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_49_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [M: A,N: A] :
( ( ord_less @ A @ M @ N )
=> ( N
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_50_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ N )
= ( N
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_51_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_52_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_53_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_54_sensors_Oleft__space,axiom,
! [Sensors: cars > traffic > cars > real,Ts: traffic,V: view_ext @ product_unit,C2: cars] :
( ( sensors @ Sensors )
=> ( ( left @ ( space @ Sensors @ Ts @ V @ C2 ) )
= ( pos @ Ts @ C2 ) ) ) ).
% sensors.left_space
thf(fact_55_size,axiom,
! [C: cars] :
( ( physical_size @ ts @ C )
= ( one_one @ real ) ) ).
% size
thf(fact_56_sensors__c,axiom,
( ( regular_regular @ ( own @ product_unit @ v2 ) @ ts @ c )
= ( one_one @ real ) ) ).
% sensors_c
thf(fact_57_hmlsl_Oview__leq__len__leq,axiom,
! [V: view_ext @ product_unit,V3: view_ext @ product_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_58_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_59_hmlsl_Olen__fills__subview,axiom,
! [V: view_ext @ product_unit,Ts: traffic,C2: cars] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_length @ ( len @ regular_regular @ V @ Ts @ C2 ) ) )
=> ? [V1: view_ext @ product_unit,V22: view_ext @ product_unit,V32: view_ext @ product_unit,V4: view_ext @ product_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_60_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_61_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_62_left__leq__right,axiom,
! [R: real_int] : ( ord_less_eq @ real @ ( left @ R ) @ ( right @ R ) ) ).
% left_leq_right
thf(fact_63_length__ge__zero,axiom,
! [R: real_int] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( real_length @ R ) ) ).
% length_ge_zero
thf(fact_64_local_Ohmlsl_Ohchop__assoc,axiom,
! [Phi: traffic > ( view_ext @ product_unit ) > $o,Psi: traffic > ( view_ext @ product_unit ) > $o,Chi: traffic > ( view_ext @ product_unit ) > $o,Ts2: traffic,V2: view_ext @ product_unit] :
( ( ? [W: view_ext @ product_unit,U: view_ext @ product_unit] :
( ( hchop @ V2 @ W @ U )
& ( Phi @ Ts2 @ W )
& ? [X4: view_ext @ product_unit,Y: view_ext @ product_unit] :
( ( hchop @ U @ X4 @ Y )
& ( Psi @ Ts2 @ X4 )
& ( Chi @ Ts2 @ Y ) ) ) )
= ( ? [W: view_ext @ product_unit,U: view_ext @ product_unit] :
( ( hchop @ V2 @ W @ U )
& ? [X4: view_ext @ product_unit,Y: view_ext @ product_unit] :
( ( hchop @ W @ X4 @ Y )
& ( Phi @ Ts2 @ X4 )
& ( Psi @ Ts2 @ Y ) )
& ( Chi @ Ts2 @ U ) ) ) ) ).
% local.hmlsl.hchop_assoc
thf(fact_65_local_Ohmlsl_Ohchop__disj__distr1,axiom,
! [Phi: traffic > ( view_ext @ product_unit ) > $o,Psi: traffic > ( view_ext @ product_unit ) > $o,Chi: traffic > ( view_ext @ product_unit ) > $o,Ts2: traffic,V2: view_ext @ product_unit] :
( ( ? [W: view_ext @ product_unit,U: view_ext @ product_unit] :
( ( hchop @ V2 @ W @ U )
& ( Phi @ Ts2 @ W )
& ( ( Psi @ Ts2 @ U )
| ( Chi @ Ts2 @ U ) ) ) )
= ( ? [W: view_ext @ product_unit,U: view_ext @ product_unit] :
( ( hchop @ V2 @ W @ U )
& ( Phi @ Ts2 @ W )
& ( Psi @ Ts2 @ U ) )
| ? [W: view_ext @ product_unit,U: view_ext @ product_unit] :
( ( hchop @ V2 @ W @ U )
& ( Phi @ Ts2 @ W )
& ( Chi @ Ts2 @ U ) ) ) ) ).
% local.hmlsl.hchop_disj_distr1
thf(fact_66_local_Ohmlsl_Ohchop__disj__distr2,axiom,
! [Psi: traffic > ( view_ext @ product_unit ) > $o,Chi: traffic > ( view_ext @ product_unit ) > $o,Phi: traffic > ( view_ext @ product_unit ) > $o,Ts2: traffic,V2: view_ext @ product_unit] :
( ( ? [W: view_ext @ product_unit,U: view_ext @ product_unit] :
( ( hchop @ V2 @ W @ U )
& ( ( Psi @ Ts2 @ W )
| ( Chi @ Ts2 @ W ) )
& ( Phi @ Ts2 @ U ) ) )
= ( ? [W: view_ext @ product_unit,U: view_ext @ product_unit] :
( ( hchop @ V2 @ W @ U )
& ( Psi @ Ts2 @ W )
& ( Phi @ Ts2 @ U ) )
| ? [W: view_ext @ product_unit,U: view_ext @ product_unit] :
( ( hchop @ V2 @ W @ U )
& ( Chi @ Ts2 @ W )
& ( Phi @ Ts2 @ U ) ) ) ) ).
% local.hmlsl.hchop_disj_distr2
thf(fact_67_local_Ohmlsl_Ohchop__neg1,axiom,
! [Phi: traffic > ( view_ext @ product_unit ) > $o,Ts2: traffic,V2: view_ext @ product_unit] :
( ~ ? [Va: view_ext @ product_unit] :
( ? [X_1: view_ext @ product_unit] : ( hchop @ V2 @ Va @ X_1 )
& ( Phi @ Ts2 @ Va ) )
=> ? [Va: view_ext @ product_unit] :
( ? [X_1: view_ext @ product_unit] : ( hchop @ V2 @ Va @ X_1 )
& ~ ( Phi @ Ts2 @ Va ) ) ) ).
% local.hmlsl.hchop_neg1
thf(fact_68_local_Ohmlsl_Ohchop__neg2,axiom,
! [Phi: traffic > ( view_ext @ product_unit ) > $o,Ts2: traffic,V2: view_ext @ product_unit] :
( ~ ? [Va: view_ext @ product_unit,U2: view_ext @ product_unit] :
( ( hchop @ V2 @ Va @ U2 )
& ( Phi @ Ts2 @ U2 ) )
=> ? [Va: view_ext @ product_unit,U2: view_ext @ product_unit] :
( ( hchop @ V2 @ Va @ U2 )
& ~ ( Phi @ Ts2 @ U2 ) ) ) ).
% local.hmlsl.hchop_neg2
thf(fact_69_local_Ohmlsl_Ohchop__weaken,axiom,
! [Phi: traffic > ( view_ext @ product_unit ) > $o,Ts2: traffic,V2: view_ext @ product_unit] :
( ( Phi @ Ts2 @ V2 )
=> ? [Va: view_ext @ product_unit,U2: view_ext @ product_unit] :
( ( hchop @ V2 @ Va @ U2 )
& ? [Vb: view_ext @ product_unit] :
( ? [X_1: view_ext @ product_unit] : ( hchop @ U2 @ Vb @ X_1 )
& ( Phi @ Ts2 @ Vb ) ) ) ) ).
% local.hmlsl.hchop_weaken
thf(fact_70_local_Ohmlsl_Ohchop__weaken1,axiom,
! [Phi: traffic > ( view_ext @ product_unit ) > $o,Ts2: traffic,V2: view_ext @ product_unit] :
( ( Phi @ Ts2 @ V2 )
=> ? [Va: view_ext @ product_unit] :
( ? [X_1: view_ext @ product_unit] : ( hchop @ V2 @ Va @ X_1 )
& ( Phi @ Ts2 @ Va ) ) ) ).
% local.hmlsl.hchop_weaken1
thf(fact_71_local_Ohmlsl_Ohchop__weaken2,axiom,
! [Phi: traffic > ( view_ext @ product_unit ) > $o,Ts2: traffic,V2: view_ext @ product_unit] :
( ( Phi @ Ts2 @ V2 )
=> ? [Va: view_ext @ product_unit,U2: view_ext @ product_unit] :
( ( hchop @ V2 @ Va @ U2 )
& ( Phi @ Ts2 @ U2 ) ) ) ).
% local.hmlsl.hchop_weaken2
thf(fact_72_hmlsl_Olen__view__hchop__right,axiom,
! [V: view_ext @ product_unit,Ts: traffic,C2: cars,V12: view_ext @ product_unit,V23: view_ext @ product_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_73_hmlsl_Olen__view__hchop__left,axiom,
! [V: view_ext @ product_unit,Ts: traffic,C2: cars,V12: view_ext @ product_unit,V23: view_ext @ product_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_74_hmlsl_Olen__compose__hchop,axiom,
! [V: view_ext @ product_unit,V12: view_ext @ product_unit,V23: view_ext @ product_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_75_hmlsl_Oext__eq__len__eq,axiom,
! [V: view_ext @ product_unit,V3: view_ext @ product_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_76_local_Ohmlsl_Olength__split,axiom,
! [Ts2: traffic,V2: view_ext @ product_unit] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_length @ ( ext @ product_unit @ V2 ) ) )
=> ? [Va: view_ext @ product_unit,U2: view_ext @ product_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_77_local_Ohmlsl_Olength__meld,axiom,
! [Ts2: traffic,V2: view_ext @ product_unit] :
( ? [Va2: view_ext @ product_unit,U3: view_ext @ product_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_78_local_Ohmlsl_Olength__dense,axiom,
! [Ts2: traffic,V2: view_ext @ product_unit] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_length @ ( ext @ product_unit @ V2 ) ) )
= ( ? [W: view_ext @ product_unit,U: view_ext @ product_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_79_hmlsl_Olen__empty__on__subview2,axiom,
! [V: view_ext @ product_unit,Ts: traffic,C2: cars,V12: view_ext @ product_unit,V23: view_ext @ product_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_80_hmlsl_Olen__empty__on__subview1,axiom,
! [V: view_ext @ product_unit,Ts: traffic,C2: cars,V12: view_ext @ product_unit,V23: view_ext @ product_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_81_hmlsl_Olen__hchop__right__right__border,axiom,
! [V: view_ext @ product_unit,Ts: traffic,C2: cars,V12: view_ext @ product_unit,V23: view_ext @ product_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_82_hmlsl_Olen__hchop__right__left__border,axiom,
! [V: view_ext @ product_unit,Ts: traffic,C2: cars,V12: view_ext @ product_unit,V23: view_ext @ product_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_83_hmlsl_Olen__hchop__left__right__border,axiom,
! [V: view_ext @ product_unit,Ts: traffic,C2: cars,V12: view_ext @ product_unit,V23: view_ext @ product_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_84_hmlsl_Olen__hchop__left__left__border,axiom,
! [V: view_ext @ product_unit,Ts: traffic,C2: cars,V12: view_ext @ product_unit,V23: view_ext @ product_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_85_regular__sensors_Ospace__eq,axiom,
! [V: view_ext @ product_unit,V3: view_ext @ product_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_86_sensors_Olen__compose__hchop,axiom,
! [Sensors: cars > traffic > cars > real,V: view_ext @ product_unit,V12: view_ext @ product_unit,V23: view_ext @ product_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_87_sensors_Olen__view__hchop__left,axiom,
! [Sensors: cars > traffic > cars > real,V: view_ext @ product_unit,Ts: traffic,C2: cars,V12: view_ext @ product_unit,V23: view_ext @ product_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_88_sensors_Olen__view__hchop__right,axiom,
! [Sensors: cars > traffic > cars > real,V: view_ext @ product_unit,Ts: traffic,C2: cars,V12: view_ext @ product_unit,V23: view_ext @ product_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_89_sensors_Oext__eq__len__eq,axiom,
! [Sensors: cars > traffic > cars > real,V: view_ext @ product_unit,V3: view_ext @ product_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_90_sensors_Olen__empty__on__subview1,axiom,
! [Sensors: cars > traffic > cars > real,V: view_ext @ product_unit,Ts: traffic,C2: cars,V12: view_ext @ product_unit,V23: view_ext @ product_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_91_sensors_Olen__empty__on__subview2,axiom,
! [Sensors: cars > traffic > cars > real,V: view_ext @ product_unit,Ts: traffic,C2: cars,V12: view_ext @ product_unit,V23: view_ext @ product_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_92_sensors_Olen__hchop__right__right__border,axiom,
! [Sensors: cars > traffic > cars > real,V: view_ext @ product_unit,Ts: traffic,C2: cars,V12: view_ext @ product_unit,V23: view_ext @ product_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_93_sensors_Olen__hchop__right__left__border,axiom,
! [Sensors: cars > traffic > cars > real,V: view_ext @ product_unit,Ts: traffic,C2: cars,V12: view_ext @ product_unit,V23: view_ext @ product_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_94_sensors_Olen__hchop__left__right__border,axiom,
! [Sensors: cars > traffic > cars > real,V: view_ext @ product_unit,Ts: traffic,C2: cars,V12: view_ext @ product_unit,V23: view_ext @ product_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_95_sensors_Olen__hchop__left__left__border,axiom,
! [Sensors: cars > traffic > cars > real,V: view_ext @ product_unit,Ts: traffic,C2: cars,V12: view_ext @ product_unit,V23: view_ext @ product_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_96_sensors_Olen__fills__subview,axiom,
! [Sensors: cars > traffic > cars > real,V: view_ext @ product_unit,Ts: traffic,C2: cars] :
( ( sensors @ Sensors )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_length @ ( len @ Sensors @ V @ Ts @ C2 ) ) )
=> ? [V1: view_ext @ product_unit,V22: view_ext @ product_unit,V32: view_ext @ product_unit,V4: view_ext @ product_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_97_sensors_Oview__leq__len__leq,axiom,
! [Sensors: cars > traffic > cars > real,V: view_ext @ product_unit,V3: view_ext @ product_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_98_view_Oh__chop__middle2,axiom,
! [V: view_ext @ product_unit,U4: view_ext @ product_unit,W2: view_ext @ product_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_99_view_Oh__chop__middle1,axiom,
! [V: view_ext @ product_unit,U4: view_ext @ product_unit,W2: view_ext @ product_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_100_view_Ohorizontal__chop__non__empty,axiom,
! [V: view_ext @ product_unit] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_length @ ( ext @ product_unit @ V ) ) )
=> ? [U2: view_ext @ product_unit,W3: view_ext @ product_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_101_hmlsl_Oright__space,axiom,
! [Ts: traffic,V: view_ext @ product_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_102_local_Ohmlsl_Olength__add,axiom,
! [X2: real,Y2: real,Ts2: traffic,V2: view_ext @ product_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_ext @ product_unit,U: view_ext @ product_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_103_local_Ohmlsl_Olength__add2,axiom,
! [X2: real,Y2: real,Ts2: traffic,V2: view_ext @ product_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_ext @ product_unit,U2: view_ext @ product_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_104_traffic_OpsGeZero,axiom,
! [Ts: traffic,C: cars] : ( ord_less @ real @ ( zero_zero @ real ) @ ( physical_size @ Ts @ C ) ) ).
% traffic.psGeZero
thf(fact_105_local_Ohmlsl_Olength__add1,axiom,
! [X2: real,Y2: real,Ts2: traffic,V2: view_ext @ product_unit] :
( ? [Va2: view_ext @ product_unit,U3: view_ext @ product_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_106_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% add_left_cancel
thf(fact_107_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ).
% add_right_cancel
thf(fact_108_hmlsl_Olen__hchop__add,axiom,
! [V: view_ext @ product_unit,V12: view_ext @ product_unit,V23: view_ext @ product_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_109_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_right
thf(fact_110_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_left
thf(fact_111_zero__eq__add__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X2: A,Y2: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ X2 @ Y2 ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( Y2
= ( zero_zero @ A ) ) ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_112_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X2: A,Y2: A] :
( ( ( plus_plus @ A @ X2 @ Y2 )
= ( zero_zero @ A ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( Y2
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_113_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( plus_plus @ A @ A2 @ B2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_114_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( plus_plus @ A @ B2 @ A2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_115_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A,B2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= A2 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_116_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [B2: A,A2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= A2 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_117_double__zero__sym,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ A2 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero_sym
thf(fact_118_double__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ( plus_plus @ A @ A2 @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero
thf(fact_119_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.right_neutral
thf(fact_120_add_Oleft__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.left_neutral
thf(fact_121_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_cancel_left
thf(fact_122_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_cancel_right
thf(fact_123_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_124_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_125_le__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel2
thf(fact_126_le__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel1
thf(fact_127_add__le__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel2
thf(fact_128_add__le__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel1
thf(fact_129_add__less__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel1
thf(fact_130_add__less__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel2
thf(fact_131_less__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel1
thf(fact_132_less__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel2
thf(fact_133_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_134_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_135_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_136_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus @ A @ I @ K )
= ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_137_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: A,K: A,A2: A,B2: A] :
( ( A3
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( plus_plus @ A @ A3 @ B2 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_138_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B3: A,K: A,B2: A,A2: A] :
( ( B3
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( plus_plus @ A @ A2 @ B3 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_139_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add.assoc
thf(fact_140_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% add.left_cancel
thf(fact_141_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ).
% add.right_cancel
thf(fact_142_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A4: A,B4: A] : ( plus_plus @ A @ B4 @ A4 ) ) ) ) ).
% add.commute
thf(fact_143_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add.left_commute
thf(fact_144_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
=> ( B2 = C2 ) ) ) ).
% add_left_imp_eq
thf(fact_145_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
=> ( B2 = C2 ) ) ) ).
% add_right_imp_eq
thf(fact_146_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_right
thf(fact_147_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_left
thf(fact_148_le__iff__add,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B4: A] :
? [C3: A] :
( B4
= ( plus_plus @ A @ A4 @ C3 ) ) ) ) ) ).
% le_iff_add
thf(fact_149_add__right__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add_right_mono
thf(fact_150_less__eqE,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ~ ! [C4: A] :
( B2
!= ( plus_plus @ A @ A2 @ C4 ) ) ) ) ).
% less_eqE
thf(fact_151_add__left__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% add_left_mono
thf(fact_152_add__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% add_mono
thf(fact_153_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_154_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_155_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( K = L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_156_add_Ogroup__left__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.group_left_neutral
thf(fact_157_add_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.comm_neutral
thf(fact_158_comm__monoid__add__class_Oadd__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% comm_monoid_add_class.add_0
thf(fact_159_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_160_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_161_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( K = L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_162_add__strict__mono,axiom,
! [A: $tType] :
( ( strict2144017051up_add @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% add_strict_mono
thf(fact_163_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% add_strict_left_mono
thf(fact_164_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add_strict_right_mono
thf(fact_165_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_imp_less_left
thf(fact_166_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_imp_less_right
thf(fact_167_add__nonpos__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y2 @ ( zero_zero @ A ) )
=> ( ( ( plus_plus @ A @ X2 @ Y2 )
= ( zero_zero @ A ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( Y2
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_168_add__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ( ( plus_plus @ A @ X2 @ Y2 )
= ( zero_zero @ A ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( Y2
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_169_add__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_nonpos
thf(fact_170_add__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_171_add__increasing2,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% add_increasing2
thf(fact_172_add__decreasing2,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% add_decreasing2
thf(fact_173_add__increasing,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% add_increasing
thf(fact_174_add__decreasing,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% add_decreasing
thf(fact_175_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% add_less_le_mono
thf(fact_176_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% add_le_less_mono
thf(fact_177_add__mono__thms__linordered__field_I3_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_178_add__mono__thms__linordered__field_I4_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_179_add__neg__neg,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_neg
thf(fact_180_add__pos__pos,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_pos_pos
thf(fact_181_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ! [C4: A] :
( ( B2
= ( plus_plus @ A @ A2 @ C4 ) )
=> ( C4
= ( zero_zero @ A ) ) ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_182_pos__add__strict,axiom,
! [A: $tType] :
( ( strict797366125id_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% pos_add_strict
thf(fact_183_add__strict__increasing2,axiom,
! [A: $tType] :
( ( ordere1818651114id_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% add_strict_increasing2
thf(fact_184_add__strict__increasing,axiom,
! [A: $tType] :
( ( ordere1818651114id_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% add_strict_increasing
thf(fact_185_add__pos__nonneg,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_pos_nonneg
thf(fact_186_add__nonpos__neg,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_neg
thf(fact_187_add__nonneg__pos,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_nonneg_pos
thf(fact_188_add__neg__nonpos,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_nonpos
thf(fact_189_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_190_view_Ohorizontal__chop__empty__right,axiom,
! [V2: view_ext @ product_unit] :
? [X_1: view_ext @ product_unit] : ( hchop @ V2 @ V2 @ X_1 ) ).
% view.horizontal_chop_empty_right
thf(fact_191_view_Ohorizontal__chop__empty__left,axiom,
! [V2: view_ext @ product_unit] :
? [U2: view_ext @ product_unit] : ( hchop @ V2 @ U2 @ V2 ) ).
% view.horizontal_chop_empty_left
thf(fact_192_view_Ohorizontal__chop__assoc2,axiom,
! [V: view_ext @ product_unit,V12: view_ext @ product_unit,V23: view_ext @ product_unit,V33: view_ext @ product_unit,V42: view_ext @ product_unit] :
( ( ( hchop @ V @ V12 @ V23 )
& ( hchop @ V12 @ V33 @ V42 ) )
=> ? [V4: view_ext @ product_unit] :
( ( hchop @ V @ V33 @ V4 )
& ( hchop @ V4 @ V42 @ V23 ) ) ) ).
% view.horizontal_chop_assoc2
thf(fact_193_view_Ohorizontal__chop__assoc1,axiom,
! [V: view_ext @ product_unit,V12: view_ext @ product_unit,V23: view_ext @ product_unit,V33: view_ext @ product_unit,V42: view_ext @ product_unit] :
( ( ( hchop @ V @ V12 @ V23 )
& ( hchop @ V23 @ V33 @ V42 ) )
=> ? [V4: view_ext @ product_unit] :
( ( hchop @ V @ V4 @ V42 )
& ( hchop @ V4 @ V12 @ V33 ) ) ) ).
% view.horizontal_chop_assoc1
thf(fact_194_view_Ohorizontal__chop1,axiom,
! [V: view_ext @ product_unit] :
? [U2: view_ext @ product_unit,X_1: view_ext @ product_unit] : ( hchop @ V @ U2 @ X_1 ) ).
% view.horizontal_chop1
thf(fact_195_view_Ohorizontal__chop__split__add,axiom,
! [X2: real,Y2: real,V: view_ext @ product_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_ext @ product_unit,W3: view_ext @ product_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_196_sensors_Olen__hchop__add,axiom,
! [Sensors: cars > traffic > cars > real,V: view_ext @ product_unit,V12: view_ext @ product_unit,V23: view_ext @ product_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_197_sensors_Oright__space,axiom,
! [Sensors: cars > traffic > cars > real,Ts: traffic,V: view_ext @ product_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_198_view_Ohorizontal__chop__own__trans,axiom,
! [V: view_ext @ product_unit,U4: view_ext @ product_unit,W2: view_ext @ product_unit] :
( ( hchop @ V @ U4 @ W2 )
=> ( ( own @ product_unit @ U4 )
= ( own @ product_unit @ W2 ) ) ) ).
% view.horizontal_chop_own_trans
thf(fact_199_view_Ohorizontal__chop__leq2,axiom,
! [V: view_ext @ product_unit,U4: view_ext @ product_unit,W2: view_ext @ product_unit] :
( ( hchop @ V @ U4 @ W2 )
=> ( ord_less_eq @ ( view_ext @ product_unit ) @ W2 @ V ) ) ).
% view.horizontal_chop_leq2
thf(fact_200_view_Ohorizontal__chop__leq1,axiom,
! [V: view_ext @ product_unit,U4: view_ext @ product_unit,W2: view_ext @ product_unit] :
( ( hchop @ V @ U4 @ W2 )
=> ( ord_less_eq @ ( view_ext @ product_unit ) @ U4 @ V ) ) ).
% view.horizontal_chop_leq1
thf(fact_201_zero__less__two,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ).
% zero_less_two
thf(fact_202_discrete,axiom,
! [A: $tType] :
( ( unique1598680935umeral @ A )
=> ( ( ord_less @ A )
= ( ^ [A4: A] : ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ ( one_one @ A ) ) ) ) ) ) ).
% discrete
thf(fact_203_field__le__epsilon,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y2: A] :
( ! [E4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E4 )
=> ( ord_less_eq @ A @ X2 @ ( plus_plus @ A @ Y2 @ E4 ) ) )
=> ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).
% field_le_epsilon
thf(fact_204_add__mono1,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( plus_plus @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% add_mono1
thf(fact_205_less__add__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A] : ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) ) ) ).
% less_add_one
thf(fact_206_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y2: A] :
( ( X2 != Y2 )
=> ( ~ ( ord_less @ A @ X2 @ Y2 )
=> ( ord_less @ A @ Y2 @ X2 ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_207_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
? [X_1: A] : ( ord_less @ A @ X @ X_1 ) ) ).
% linordered_field_no_ub
thf(fact_208_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X ) ) ).
% linordered_field_no_lb
thf(fact_209_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_210_zero__le__one,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_le_one
thf(fact_211_not__one__le__zero,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_le_zero
thf(fact_212_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_213_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_214_add__less__zeroD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ X2 @ Y2 ) @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ X2 @ ( zero_zero @ A ) )
| ( ord_less @ A @ Y2 @ ( zero_zero @ A ) ) ) ) ) ).
% add_less_zeroD
thf(fact_215_less__numeral__extra_I1_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% less_numeral_extra(1)
thf(fact_216_space__c,axiom,
( ( space @ regular_regular @ ts @ v2 @ c )
= ( real_Abs_real_int @ ( product_Pair @ real @ real @ ( zero_zero @ real ) @ ( one_one @ real ) ) ) ) ).
% space_c
thf(fact_217_hmlsl_Ospace__def,axiom,
! [Ts: traffic,V: view_ext @ product_unit,C2: cars] :
( ( space @ regular_regular @ Ts @ V @ C2 )
= ( real_Abs_real_int @ ( product_Pair @ real @ real @ ( pos @ Ts @ C2 ) @ ( plus_plus @ real @ ( pos @ Ts @ C2 ) @ ( regular_regular @ ( own @ product_unit @ V ) @ Ts @ C2 ) ) ) ) ) ).
% hmlsl.space_def
thf(fact_218_len__c,axiom,
( ( len @ regular_regular @ v2 @ ts @ c )
= ( real_Abs_real_int @ ( product_Pair @ real @ real @ ( zero_zero @ real ) @ ( one_one @ real ) ) ) ) ).
% len_c
thf(fact_219_sensors_Ospace__def,axiom,
! [Sensors: cars > traffic > cars > real,Ts: traffic,V: view_ext @ product_unit,C2: cars] :
( ( sensors @ Sensors )
=> ( ( space @ Sensors @ Ts @ V @ C2 )
= ( real_Abs_real_int @ ( product_Pair @ real @ real @ ( pos @ Ts @ C2 ) @ ( plus_plus @ real @ ( pos @ Ts @ C2 ) @ ( Sensors @ ( own @ product_unit @ V ) @ Ts @ C2 ) ) ) ) ) ) ).
% sensors.space_def
thf(fact_220_is__num__normalize_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% is_num_normalize(1)
thf(fact_221_le__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% le_numeral_extra(3)
thf(fact_222_less__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% less_numeral_extra(3)
thf(fact_223_le__numeral__extra_I4_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% le_numeral_extra(4)
thf(fact_224_less__numeral__extra_I4_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% less_numeral_extra(4)
thf(fact_225_divides__aux__eq,axiom,
! [A: $tType] :
( ( unique1598680935umeral @ A )
=> ! [Q: A,R: A] :
( ( unique455577585es_aux @ A @ ( product_Pair @ A @ A @ Q @ R ) )
= ( R
= ( zero_zero @ A ) ) ) ) ).
% divides_aux_eq
thf(fact_226_real__int__class_Oshift__def,axiom,
( real_shift
= ( ^ [R2: real_int,X4: real] : ( real_Abs_real_int @ ( product_Pair @ real @ real @ ( plus_plus @ real @ ( left @ R2 ) @ X4 ) @ ( plus_plus @ real @ ( right @ R2 ) @ X4 ) ) ) ) ) ).
% real_int_class.shift_def
thf(fact_227_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_228_shift__keeps__length,axiom,
! [X2: real] :
( real_length
= ( ^ [R2: real_int] : ( real_length @ ( real_shift @ R2 @ X2 ) ) ) ) ).
% shift_keeps_length
thf(fact_229_shift__zero,axiom,
! [R: real_int] :
( ( real_shift @ R @ ( zero_zero @ real ) )
= R ) ).
% shift_zero
thf(fact_230_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_231_ts__rep__def,axiom,
( ts_rep
= ( product_Pair @ ( cars > real ) @ ( product_prod @ ( cars > nat_int ) @ ( product_prod @ ( cars > nat_int ) @ ( product_prod @ ( cars > real > real ) @ ( product_prod @ ( cars > real ) @ ( cars > real ) ) ) ) ) @ po @ ( product_Pair @ ( cars > nat_int ) @ ( product_prod @ ( cars > nat_int ) @ ( product_prod @ ( cars > real > real ) @ ( product_prod @ ( cars > real ) @ ( cars > real ) ) ) ) @ re @ ( product_Pair @ ( cars > nat_int ) @ ( product_prod @ ( cars > real > real ) @ ( product_prod @ ( cars > real ) @ ( cars > real ) ) ) @ cl @ ( product_Pair @ ( cars > real > real ) @ ( product_prod @ ( cars > real ) @ ( cars > real ) ) @ dy @ ( product_Pair @ ( cars > real ) @ ( cars > real ) @ ps @ sd ) ) ) ) ) ) ).
% ts_rep_def
thf(fact_232_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% order_refl
thf(fact_233_clm__def,axiom,
! [C: cars] :
( ( cl @ C )
= ( bot_bot @ nat_int ) ) ).
% clm_def
thf(fact_234_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_235_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z2: A] : Y4 = Z2 )
= ( ^ [A4: A,B4: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
& ( ord_less_eq @ A @ A4 @ B4 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_236_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_237_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: A,B5: A] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_238_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_239_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y2: A,Z3: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z3 )
=> ( ord_less_eq @ A @ X2 @ Z3 ) ) ) ) ).
% order_trans
thf(fact_240_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% order_class.order.antisym
thf(fact_241_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_242_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_243_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z2: A] : Y4 = Z2 )
= ( ^ [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
& ( ord_less_eq @ A @ B4 @ A4 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_244_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y2: A,X2: A] :
( ( ord_less_eq @ A @ Y2 @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ) ).
% antisym_conv
thf(fact_245_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A,Z3: A] :
( ( ( ord_less_eq @ A @ X2 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ X2 @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ Y2 ) )
=> ( ( ( ord_less_eq @ A @ Z3 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq @ A @ Z3 @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_246_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% order.trans
thf(fact_247_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A] :
( ~ ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% le_cases
thf(fact_248_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y2: A] :
( ( X2 = Y2 )
=> ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).
% eq_refl
thf(fact_249_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
| ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% linear
thf(fact_250_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ) ).
% antisym
thf(fact_251_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z2: A] : Y4 = Z2 )
= ( ^ [X4: A,Y: A] :
( ( ord_less_eq @ A @ X4 @ Y )
& ( ord_less_eq @ A @ Y @ X4 ) ) ) ) ) ).
% eq_iff
% Type constructors (49)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A6: $tType,A7: $tType] :
( ( preorder @ A7 )
=> ( preorder @ ( A6 > A7 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A6: $tType,A7: $tType] :
( ( order @ A7 )
=> ( order @ ( A6 > A7 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A6: $tType,A7: $tType] :
( ( ord @ A7 )
=> ( ord @ ( A6 > A7 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_1,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_2,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_3,axiom,
ord @ $o ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere516151231imp_le @ real ).
thf(tcon_Real_Oreal___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict2144017051up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le @ real ).
thf(tcon_Real_Oreal___Groups_Ostrict__ordered__comm__monoid__add,axiom,
strict797366125id_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add,axiom,
ordere1818651114id_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring,axiom,
linord1659791738miring @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__comm__monoid__add,axiom,
ordere216010020id_add @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semidom,axiom,
linordered_semidom @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ real ).
thf(tcon_Real_Oreal___Fields_Olinordered__field,axiom,
linordered_field @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__idom,axiom,
linordered_idom @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__add,axiom,
semigroup_add @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__less__one,axiom,
zero_less_one @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__neq__one,axiom,
zero_neq_one @ real ).
thf(tcon_Real_Oreal___Orderings_Opreorder_4,axiom,
preorder @ real ).
thf(tcon_Real_Oreal___Orderings_Olinorder_5,axiom,
linorder @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__add,axiom,
monoid_add @ real ).
thf(tcon_Real_Oreal___Groups_Ogroup__add,axiom,
group_add @ real ).
thf(tcon_Real_Oreal___Orderings_Oorder_6,axiom,
order @ real ).
thf(tcon_Real_Oreal___Num_Oneg__numeral,axiom,
neg_numeral @ real ).
thf(tcon_Real_Oreal___Orderings_Oord_7,axiom,
ord @ real ).
thf(tcon_Real_Oreal___Groups_Ozero,axiom,
zero @ real ).
thf(tcon_Real_Oreal___Groups_Oone,axiom,
one @ real ).
thf(tcon_NatInt_Onat__int___Orderings_Opreorder_8,axiom,
preorder @ nat_int ).
thf(tcon_NatInt_Onat__int___Orderings_Oorder_9,axiom,
order @ nat_int ).
thf(tcon_NatInt_Onat__int___Orderings_Oord_10,axiom,
ord @ nat_int ).
thf(tcon_RealInt_Oreal__int___Orderings_Opreorder_11,axiom,
preorder @ real_int ).
thf(tcon_RealInt_Oreal__int___Orderings_Oorder_12,axiom,
order @ real_int ).
thf(tcon_RealInt_Oreal__int___Orderings_Oord_13,axiom,
ord @ real_int ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_14,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_15,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_16,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_17,axiom,
ord @ product_unit ).
thf(tcon_Views_Oview_Oview__ext___Orderings_Opreorder_18,axiom,
! [A6: $tType] :
( ( order @ A6 )
=> ( preorder @ ( view_ext @ A6 ) ) ) ).
thf(tcon_Views_Oview_Oview__ext___Orderings_Oorder_19,axiom,
! [A6: $tType] :
( ( order @ A6 )
=> ( order @ ( view_ext @ A6 ) ) ) ).
thf(tcon_Views_Oview_Oview__ext___Orderings_Oord_20,axiom,
! [A6: $tType] :
( ( order @ A6 )
=> ( ord @ ( view_ext @ A6 ) ) ) ).
% 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 ) ) ) ) ).
%------------------------------------------------------------------------------