TPTP Problem File: ITP041^2.p
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%------------------------------------------------------------------------------
% File : ITP041^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Coincidence problem prob_905__7236378_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 : Coincidence/prob_905__7236378_1 [Des21]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 446 ( 118 unt; 105 typ; 0 def)
% Number of atoms : 978 ( 201 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 5664 ( 53 ~; 10 |; 134 &;5084 @)
% ( 0 <=>; 383 =>; 0 <=; 0 <~>)
% Maximal formula depth : 35 ( 9 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 298 ( 298 >; 0 *; 0 +; 0 <<)
% Number of symbols : 100 ( 99 usr; 13 con; 0-13 aty)
% Number of variables : 1428 ( 92 ^;1202 !; 7 ?;1428 :)
% ( 127 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:30:21.538
%------------------------------------------------------------------------------
% Could-be-implicit typings (17)
thf(ty_t_Denotational__Semantics_Ointerp_Ointerp__ext,type,
denota1663640101rp_ext: $tType > $tType > $tType > $tType > $tType ).
thf(ty_t_Finite__Cartesian__Product_Ovec,type,
finite_Cartesian_vec: $tType > $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_Syntax_Oformula,type,
formula: $tType > $tType > $tType > $tType ).
thf(ty_t_Sum__Type_Osum,type,
sum_sum: $tType > $tType > $tType ).
thf(ty_t_Filter_Ofilter,type,
filter: $tType > $tType ).
thf(ty_t_Syntax_Otrm,type,
trm: $tType > $tType > $tType ).
thf(ty_t_Syntax_OODE,type,
ode: $tType > $tType > $tType ).
thf(ty_t_Syntax_Ohp,type,
hp: $tType > $tType > $tType > $tType ).
thf(ty_t_Real_Oreal,type,
real: $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_tf_d,type,
d: $tType ).
thf(ty_tf_c,type,
c: $tType ).
thf(ty_tf_b,type,
b: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (88)
thf(sy_cl_Cardinality_OCARD__1,type,
cARD_1:
!>[A: $tType] : $o ).
thf(sy_cl_Ordered__Euclidean__Space_Oordered__euclidean__space,type,
ordere890947078_space:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ouminus,type,
uminus:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[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_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : $o ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[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_Lattices_Oboolean__algebra,type,
boolean_algebra:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord1659791738miring:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Otopological__space,type,
topolo503727757_space:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Lattice__Algebras_Olattice__ab__group__add,type,
lattic1601792062up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector,type,
real_V55928688vector:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Algorithm_Oeuclidean__ring__gcd,type,
euclid1678468529ng_gcd:
!>[A: $tType] : $o ).
thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
unique1598680935umeral:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Algorithm_Onormalization__euclidean__semiring,type,
euclid1155270486miring:
!>[A: $tType] : $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
bNF_Ca1785829860lChain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
thf(sy_c_Coincidence__Mirabelle__cppqbdunjv_Oids_Ocoincide__fml,type,
coinci1993344360de_fml:
!>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > $o ) ).
thf(sy_c_Coincidence__Mirabelle__cppqbdunjv_Oids_Ocoincide__hp,type,
coinci32832645ide_hp:
!>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > $o ) ).
thf(sy_c_Coincidence__Mirabelle__cppqbdunjv_Oids_Ocoincide__hp_H,type,
coinci554967470ide_hp:
!>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > $o ) ).
thf(sy_c_Coincidence__Mirabelle__cppqbdunjv_Oids_Oode__sem__equiv,type,
coinci1495174342_equiv:
!>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > $o ) ).
thf(sy_c_Denotational__Semantics_OIagree,type,
denotational_Iagree:
!>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) > $o ) ).
thf(sy_c_Denotational__Semantics_OODE__sem,type,
denotational_ODE_sem:
!>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( ode @ A @ C ) > ( finite_Cartesian_vec @ real @ C ) > ( finite_Cartesian_vec @ real @ C ) ) ).
thf(sy_c_Denotational__Semantics_OODE__vars,type,
denota811733865E_vars:
!>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( ode @ A @ C ) > ( set @ C ) ) ).
thf(sy_c_Denotational__Semantics_OVSagree,type,
denotational_VSagree:
!>[C: $tType] : ( ( finite_Cartesian_vec @ real @ C ) > ( finite_Cartesian_vec @ real @ C ) > ( set @ C ) > $o ) ).
thf(sy_c_Denotational__Semantics_OVagree,type,
denotational_Vagree:
!>[C: $tType] : ( ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) > ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) > ( set @ ( sum_sum @ C @ C ) ) > $o ) ).
thf(sy_c_Denotational__Semantics_Ofml__sem,type,
denotational_fml_sem:
!>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( formula @ A @ B @ C ) > ( set @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) ) ) ).
thf(sy_c_Denotational__Semantics_Omk__v,type,
denotational_mk_v:
!>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( ode @ A @ C ) > ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) > ( finite_Cartesian_vec @ real @ C ) > ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) ) ).
thf(sy_c_Denotational__Semantics_Omk__xode,type,
denotational_mk_xode:
!>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( ode @ A @ C ) > ( finite_Cartesian_vec @ real @ C ) > ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) ) ).
thf(sy_c_Denotational__Semantics_OsemBV,type,
denotational_semBV:
!>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( ode @ A @ C ) > ( set @ ( sum_sum @ C @ C ) ) ) ).
thf(sy_c_Deriv_Ohas__vector__derivative,type,
has_ve2132708402vative:
!>[B: $tType] : ( ( real > B ) > B > ( filter @ real ) > $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_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_HOL_OThe,type,
the:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Ids_Oids,type,
ids:
!>[Sz: $tType,Sf: $tType,Sc: $tType] : ( Sz > Sz > Sz > Sf > Sf > Sf > Sc > Sc > Sc > Sc > $o ) ).
thf(sy_c_Initial__Value__Problem_Osolves__ode,type,
initia1685620758es_ode:
!>[A: $tType] : ( ( real > A ) > ( real > A > A ) > ( set @ real ) > ( set @ A ) > $o ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
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_Product__Type_Ointernal__case__prod,type,
produc2004651681e_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
product_rec_set_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T > $o ) ).
thf(sy_c_Product__Type_Oold_Ounit_Orec__set__unit,type,
product_rec_set_unit:
!>[T: $tType] : ( T > product_unit > T > $o ) ).
thf(sy_c_Product__Type_Oold_Ounit_Orec__unit,type,
product_rec_unit:
!>[T: $tType] : ( T > product_unit > T ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeast,type,
set_ord_atLeast:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
set_or331188842AtMost:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatMost,type,
set_ord_atMost:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Static__Semantics_OBVO,type,
static_BVO:
!>[A: $tType,C: $tType] : ( ( ode @ A @ C ) > ( set @ ( sum_sum @ C @ C ) ) ) ).
thf(sy_c_Static__Semantics_OFVF,type,
static_FVF:
!>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > ( set @ ( sum_sum @ C @ C ) ) ) ).
thf(sy_c_Static__Semantics_OFVO,type,
static_FVO:
!>[A: $tType,C: $tType] : ( ( ode @ A @ C ) > ( set @ C ) ) ).
thf(sy_c_Static__Semantics_OSIGF,type,
static_SIGF:
!>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > ( set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ).
thf(sy_c_Sum__Type_OInl,type,
sum_Inl:
!>[A: $tType,B: $tType] : ( A > ( sum_sum @ A @ B ) ) ).
thf(sy_c_Sum__Type_OInr,type,
sum_Inr:
!>[B: $tType,A: $tType] : ( B > ( sum_sum @ A @ B ) ) ).
thf(sy_c_Syntax_OEquiv,type,
equiv:
!>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > ( formula @ A @ B @ C ) > ( formula @ A @ B @ C ) ) ).
thf(sy_c_Syntax_OImplies,type,
implies:
!>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > ( formula @ A @ B @ C ) > ( formula @ A @ B @ C ) ) ).
thf(sy_c_Syntax_OODE__dom,type,
oDE_dom:
!>[A: $tType,C: $tType] : ( ( ode @ A @ C ) > ( set @ C ) ) ).
thf(sy_c_Syntax_Ofsafe,type,
fsafe:
!>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > $o ) ).
thf(sy_c_Syntax_Ohp_OEvolveODE,type,
evolveODE:
!>[A: $tType,C: $tType,B: $tType] : ( ( ode @ A @ C ) > ( formula @ A @ B @ C ) > ( hp @ A @ B @ C ) ) ).
thf(sy_c_Syntax_Ohpsafe,type,
hpsafe:
!>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > $o ) ).
thf(sy_c_Syntax_Oosafe,type,
osafe:
!>[A: $tType,C: $tType] : ( ( ode @ A @ C ) > $o ) ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within,type,
topolo507301023within:
!>[A: $tType] : ( A > ( set @ A ) > ( filter @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_I____,type,
i: denota1663640101rp_ext @ a @ b @ c @ product_unit ).
thf(sy_v_J____,type,
j: denota1663640101rp_ext @ a @ b @ c @ product_unit ).
thf(sy_v_ODE____,type,
ode2: ode @ a @ c ).
thf(sy_v_P____,type,
p: formula @ a @ b @ c ).
thf(sy_v_V____,type,
v: set @ ( sum_sum @ c @ c ) ).
thf(sy_v_a____,type,
a2: finite_Cartesian_vec @ real @ c ).
thf(sy_v_aa____,type,
aa: finite_Cartesian_vec @ real @ c ).
thf(sy_v_ab____,type,
ab: finite_Cartesian_vec @ real @ c ).
thf(sy_v_b____,type,
b2: finite_Cartesian_vec @ real @ c ).
thf(sy_v_ba____,type,
ba: finite_Cartesian_vec @ real @ c ).
thf(sy_v_bb____,type,
bb: finite_Cartesian_vec @ real @ c ).
thf(sy_v_s____,type,
s: real ).
thf(sy_v_sol____,type,
sol: real > ( finite_Cartesian_vec @ real @ c ) ).
thf(sy_v_t____,type,
t: real ).
% Relevant facts (255)
thf(fact_0_osafe,axiom,
osafe @ a @ c @ ode2 ).
% osafe
thf(fact_1__092_060open_062Vagree_A_Imk__v_AJ_AODE_A_Iaa_M_Aba_J_A_Isol_As_J_J_A_Iaa_M_Aba_J_A_I_N_AsemBV_AJ_AODE_J_A_092_060and_062_AVagree_A_Imk__v_AJ_AODE_A_Iaa_M_Aba_J_A_Isol_As_J_J_A_Imk__xode_AJ_AODE_A_Isol_As_J_J_A_IsemBV_AJ_AODE_J_092_060close_062,axiom,
( ( denotational_Vagree @ c @ ( denotational_mk_v @ a @ b @ c @ j @ ode2 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba ) @ ( sol @ s ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba ) @ ( uminus_uminus @ ( set @ ( sum_sum @ c @ c ) ) @ ( denotational_semBV @ a @ b @ c @ j @ ode2 ) ) )
& ( denotational_Vagree @ c @ ( denotational_mk_v @ a @ b @ c @ j @ ode2 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba ) @ ( sol @ s ) ) @ ( denotational_mk_xode @ a @ b @ c @ j @ ode2 @ ( sol @ s ) ) @ ( denotational_semBV @ a @ b @ c @ j @ ode2 ) ) ) ).
% \<open>Vagree (mk_v J ODE (aa, ba) (sol s)) (aa, ba) (- semBV J ODE) \<and> Vagree (mk_v J ODE (aa, ba) (sol s)) (mk_xode J ODE (sol s)) (semBV J ODE)\<close>
thf(fact_2__092_060open_062_092_060And_062sa_O_AVagree_A_Imk__v_AJ_AODE_A_Iaa_M_Aba_J_A_Isol_Asa_J_J_A_Iaa_M_Aba_J_A_I_N_AsemBV_AJ_AODE_J_092_060close_062,axiom,
! [S: real] : ( denotational_Vagree @ c @ ( denotational_mk_v @ a @ b @ c @ j @ ode2 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba ) @ ( sol @ S ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba ) @ ( uminus_uminus @ ( set @ ( sum_sum @ c @ c ) ) @ ( denotational_semBV @ a @ b @ c @ j @ ode2 ) ) ) ).
% \<open>\<And>sa. Vagree (mk_v J ODE (aa, ba) (sol sa)) (aa, ba) (- semBV J ODE)\<close>
thf(fact_3_mk__v__agree,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( finite_finite @ B )
& ( finite_finite @ A )
& ( finite_finite @ C ) )
=> ! [I: denota1663640101rp_ext @ B @ C @ A @ product_unit,ODE: ode @ B @ A,Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Sol: finite_Cartesian_vec @ real @ A] :
( ( denotational_Vagree @ A @ ( denotational_mk_v @ B @ C @ A @ I @ ODE @ Nu @ Sol ) @ Nu @ ( uminus_uminus @ ( set @ ( sum_sum @ A @ A ) ) @ ( denotational_semBV @ B @ C @ A @ I @ ODE ) ) )
& ( denotational_Vagree @ A @ ( denotational_mk_v @ B @ C @ A @ I @ ODE @ Nu @ Sol ) @ ( denotational_mk_xode @ B @ C @ A @ I @ ODE @ Sol ) @ ( denotational_semBV @ B @ C @ A @ I @ ODE ) ) ) ) ).
% mk_v_agree
thf(fact_4_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X1 @ X2 )
= ( product_Pair @ A @ B @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_5_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_6_mk__v__exists,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( finite_finite @ B )
& ( finite_finite @ A )
& ( finite_finite @ C ) )
=> ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),I: denota1663640101rp_ext @ B @ C @ A @ product_unit,ODE: ode @ B @ A,Sol: finite_Cartesian_vec @ real @ A] :
? [Omega: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A )] :
( ( denotational_Vagree @ A @ Omega @ Nu @ ( uminus_uminus @ ( set @ ( sum_sum @ A @ A ) ) @ ( denotational_semBV @ B @ C @ A @ I @ ODE ) ) )
& ( denotational_Vagree @ A @ Omega @ ( denotational_mk_xode @ B @ C @ A @ I @ ODE @ Sol ) @ ( denotational_semBV @ B @ C @ A @ I @ ODE ) ) ) ) ).
% mk_v_exists
thf(fact_7_VA,axiom,
denotational_Vagree @ c @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ a2 @ b2 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba ) @ v ).
% VA
thf(fact_8__092_060open_062_092_060And_062sa_O_AVagree_A_Imk__v_AI_AODE_A_Ia_M_Ab_J_A_Isol_Asa_J_J_A_Imk__xode_AI_AODE_A_Isol_Asa_J_J_A_IsemBV_AI_AODE_J_092_060close_062,axiom,
! [S: real] : ( denotational_Vagree @ c @ ( denotational_mk_v @ a @ b @ c @ i @ ode2 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ a2 @ b2 ) @ ( sol @ S ) ) @ ( denotational_mk_xode @ a @ b @ c @ i @ ode2 @ ( sol @ S ) ) @ ( denotational_semBV @ a @ b @ c @ i @ ode2 ) ) ).
% \<open>\<And>sa. Vagree (mk_v I ODE (a, b) (sol sa)) (mk_xode I ODE (sol sa)) (semBV I ODE)\<close>
thf(fact_9_VAOV,axiom,
denotational_Vagree @ c @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ a2 @ b2 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba ) @ ( static_BVO @ a @ c @ ode2 ) ).
% VAOV
thf(fact_10_agree__comm,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [A4: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),B4: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),V: set @ ( sum_sum @ A @ A )] :
( ( denotational_Vagree @ A @ A4 @ B4 @ V )
=> ( denotational_Vagree @ A @ B4 @ A4 @ V ) ) ) ).
% agree_comm
thf(fact_11_agree__refl,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),A4: set @ ( sum_sum @ A @ A )] : ( denotational_Vagree @ A @ Nu @ Nu @ A4 ) ) ).
% agree_refl
thf(fact_12__092_060open_062_092_060And_062sa_O_AVagree_A_Imk__v_AI_AODE_A_Ia_M_Ab_J_A_Isol_Asa_J_J_A_Ia_M_Ab_J_A_I_N_AsemBV_AI_AODE_J_092_060close_062,axiom,
! [S: real] : ( denotational_Vagree @ c @ ( denotational_mk_v @ a @ b @ c @ i @ ode2 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ a2 @ b2 ) @ ( sol @ S ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ a2 @ b2 ) @ ( uminus_uminus @ ( set @ ( sum_sum @ c @ c ) ) @ ( denotational_semBV @ a @ b @ c @ i @ ode2 ) ) ) ).
% \<open>\<And>sa. Vagree (mk_v I ODE (a, b) (sol sa)) (a, b) (- semBV I ODE)\<close>
thf(fact_13_surj__pair,axiom,
! [A: $tType,B: $tType,P: product_prod @ A @ B] :
? [X: A,Y: B] :
( P
= ( product_Pair @ A @ B @ X @ Y ) ) ).
% surj_pair
thf(fact_14_prod__cases,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
( ! [A5: A,B5: B] : ( P2 @ ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_15_OVsub,axiom,
ord_less_eq @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_BVO @ a @ c @ ode2 ) @ v ).
% OVsub
thf(fact_16_veq,axiom,
( ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb )
= ( denotational_mk_v @ a @ b @ c @ i @ ode2 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ a2 @ b2 ) @ ( sol @ t ) ) ) ).
% veq
thf(fact_17_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A5: A,B5: B] : ( P2 @ ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_18_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
~ ! [A5: A,B5: B] :
( Y3
!= ( product_Pair @ A @ B @ A5 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_19_prod__induct7,axiom,
! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
( ! [A5: A,B5: B,C2: C,D2: D,E2: E,F2: F,G2: G] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct7
thf(fact_20_prod__induct6,axiom,
! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
( ! [A5: A,B5: B,C2: C,D2: D,E2: E,F2: F] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct6
thf(fact_21_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A5: A,B5: B,C2: C,D2: D,E2: E] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct5
thf(fact_22_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A5: A,B5: B,C2: C,D2: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct4
thf(fact_23_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A5: A,B5: B,C2: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C2 ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct3
thf(fact_24_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
~ ! [A5: A,B5: B,C2: C,D2: D,E2: E,F2: F,G2: G] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_25_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
~ ! [A5: A,B5: B,C2: C,D2: D,E2: E,F2: F] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_26_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A5: A,B5: B,C2: C,D2: D,E2: E] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_27_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A5: A,B5: B,C2: C,D2: D] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_28_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A5: A,B5: B,C2: C] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C2 ) ) ) ).
% prod_cases3
thf(fact_29_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_30_OVsub_H_H,axiom,
ord_less_eq @ ( set @ ( sum_sum @ c @ c ) ) @ ( denotational_semBV @ a @ b @ c @ i @ ode2 ) @ v ).
% OVsub''
thf(fact_31_semBVsub,axiom,
ord_less_eq @ ( set @ ( sum_sum @ c @ c ) ) @ ( denotational_semBV @ a @ b @ c @ i @ ode2 ) @ ( static_BVO @ a @ c @ ode2 ) ).
% semBVsub
thf(fact_32_uminus__Pair,axiom,
! [A: $tType,B: $tType] :
( ( ( uminus @ B )
& ( uminus @ A ) )
=> ! [A2: A,B2: B] :
( ( uminus_uminus @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) )
= ( product_Pair @ A @ B @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ B @ B2 ) ) ) ) ).
% uminus_Pair
thf(fact_33_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
= ( F1 @ A2 @ B2 ) ) ).
% old.prod.rec
thf(fact_34_ComplI,axiom,
! [A: $tType,C3: A,A4: set @ A] :
( ~ ( member @ A @ C3 @ A4 )
=> ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A4 ) ) ) ).
% ComplI
thf(fact_35_Compl__iff,axiom,
! [A: $tType,C3: A,A4: set @ A] :
( ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= ( ~ ( member @ A @ C3 @ A4 ) ) ) ).
% Compl_iff
thf(fact_36_Compl__eq__Compl__iff,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ( uminus_uminus @ ( set @ A ) @ A4 )
= ( uminus_uminus @ ( set @ A ) @ B4 ) )
= ( A4 = B4 ) ) ).
% Compl_eq_Compl_iff
thf(fact_37_verit__minus__simplify_I4_J,axiom,
! [B: $tType] :
( ( group_add @ B )
=> ! [B2: B] :
( ( uminus_uminus @ B @ ( uminus_uminus @ B @ B2 ) )
= B2 ) ) ).
% verit_minus_simplify(4)
thf(fact_38_uminus__apply,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A6: A > B,X4: A] : ( uminus_uminus @ B @ ( A6 @ X4 ) ) ) ) ) ).
% uminus_apply
thf(fact_39_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
= A2 ) ) ).
% add.inverse_inverse
thf(fact_40_subset__antisym,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% subset_antisym
thf(fact_41_subsetI,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ! [X: A] :
( ( member @ A @ X @ A4 )
=> ( member @ A @ X @ B4 ) )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% subsetI
thf(fact_42_compl__eq__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X3: A,Y3: A] :
( ( ( uminus_uminus @ A @ X3 )
= ( uminus_uminus @ A @ Y3 ) )
= ( X3 = Y3 ) ) ) ).
% compl_eq_compl_iff
thf(fact_43_double__compl,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X3: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X3 ) )
= X3 ) ) ).
% double_compl
thf(fact_44_neg__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( ( uminus_uminus @ A @ A2 )
= ( uminus_uminus @ A @ B2 ) )
= ( A2 = B2 ) ) ) ).
% neg_equal_iff_equal
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P2: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X: A] :
( ( P2 @ X )
= ( Q @ X ) )
=> ( ( collect @ A @ P2 )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F3: A > B,G3: A > B] :
( ! [X: A] :
( ( F3 @ X )
= ( G3 @ X ) )
=> ( F3 = G3 ) ) ).
% ext
thf(fact_49_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y3 ) )
= ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% compl_le_compl_iff
thf(fact_50_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% neg_le_iff_le
thf(fact_51_Compl__subset__Compl__iff,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ ( uminus_uminus @ ( set @ A ) @ B4 ) )
= ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ).
% Compl_subset_Compl_iff
thf(fact_52_Compl__anti__mono,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B4 ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) ) ) ).
% Compl_anti_mono
thf(fact_53_verit__la__disequality,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( A2 = B2 )
| ~ ( ord_less_eq @ A @ A2 @ B2 )
| ~ ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% verit_la_disequality
thf(fact_54_Collect__mono__iff,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) )
= ( ! [X4: A] :
( ( P2 @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_55_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y4: set @ A,Z: set @ A] : Y4 = Z )
= ( ^ [A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_56_subset__trans,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ C4 ) ) ) ).
% subset_trans
thf(fact_57_Collect__mono,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X: A] :
( ( P2 @ X )
=> ( Q @ X ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_58_subset__refl,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% subset_refl
thf(fact_59_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
! [T2: A] :
( ( member @ A @ T2 @ A6 )
=> ( member @ A @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_60_equalityD2,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( A4 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ).
% equalityD2
thf(fact_61_equalityD1,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( A4 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% equalityD1
thf(fact_62_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( member @ A @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_63_equalityE,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( A4 = B4 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% equalityE
thf(fact_64_subsetD,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( member @ A @ C3 @ A4 )
=> ( member @ A @ C3 @ B4 ) ) ) ).
% subsetD
thf(fact_65_in__mono,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,X3: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( member @ A @ X3 @ A4 )
=> ( member @ A @ X3 @ B4 ) ) ) ).
% in_mono
thf(fact_66_le__imp__neg__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% le_imp_neg_le
thf(fact_67_minus__le__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ).
% minus_le_iff
thf(fact_68_le__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% le_minus_iff
thf(fact_69_compl__le__swap2,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [Y3: A,X3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ X3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ Y3 ) ) ) ).
% compl_le_swap2
thf(fact_70_compl__le__swap1,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [Y3: A,X3: A] :
( ( ord_less_eq @ A @ Y3 @ ( uminus_uminus @ A @ X3 ) )
=> ( ord_less_eq @ A @ X3 @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% compl_le_swap1
thf(fact_71_compl__mono,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X3 ) ) ) ) ).
% compl_mono
thf(fact_72_agree__sub,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [A4: set @ ( sum_sum @ A @ A ),B4: set @ ( sum_sum @ A @ A ),Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Omega2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A )] :
( ( ord_less_eq @ ( set @ ( sum_sum @ A @ A ) ) @ A4 @ B4 )
=> ( ( denotational_Vagree @ A @ Nu @ Omega2 @ B4 )
=> ( denotational_Vagree @ A @ Nu @ Omega2 @ A4 ) ) ) ) ).
% agree_sub
thf(fact_73_agree__supset,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [B4: set @ ( sum_sum @ A @ A ),A4: set @ ( sum_sum @ A @ A ),Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A )] :
( ( ord_less_eq @ ( set @ ( sum_sum @ A @ A ) ) @ B4 @ A4 )
=> ( ( denotational_Vagree @ A @ Nu @ Nu2 @ A4 )
=> ( denotational_Vagree @ A @ Nu @ Nu2 @ B4 ) ) ) ) ).
% agree_supset
thf(fact_74_minus__equation__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( ( uminus_uminus @ A @ A2 )
= B2 )
= ( ( uminus_uminus @ A @ B2 )
= A2 ) ) ) ).
% minus_equation_iff
thf(fact_75_equation__minus__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( uminus_uminus @ A @ B2 ) )
= ( B2
= ( uminus_uminus @ A @ A2 ) ) ) ) ).
% equation_minus_iff
thf(fact_76_fun__Compl__def,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A6: A > B,X4: A] : ( uminus_uminus @ B @ ( A6 @ X4 ) ) ) ) ) ).
% fun_Compl_def
thf(fact_77_verit__negate__coefficient_I3_J,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( A2 = B2 )
=> ( ( uminus_uminus @ A @ A2 )
= ( uminus_uminus @ A @ B2 ) ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_78_double__complement,axiom,
! [A: $tType,A4: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= A4 ) ).
% double_complement
thf(fact_79_ComplD,axiom,
! [A: $tType,C3: A,A4: set @ A] :
( ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
=> ~ ( member @ A @ C3 @ A4 ) ) ).
% ComplD
thf(fact_80_euclid__ext__aux_Ocases,axiom,
! [A: $tType] :
( ( euclid1678468529ng_gcd @ A )
=> ! [X3: product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) )] :
~ ! [S2: A,S3: A,T3: A,T4: A,R: A,R2: A] :
( X3
!= ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) ) @ S2 @ ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) @ S3 @ ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) @ T3 @ ( product_Pair @ A @ ( product_prod @ A @ A ) @ T4 @ ( product_Pair @ A @ A @ R @ R2 ) ) ) ) ) ) ) ).
% euclid_ext_aux.cases
thf(fact_81_Pair__le,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: B,C3: A,D3: B] :
( ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ C3 @ D3 ) )
= ( ( ord_less_eq @ A @ A2 @ C3 )
& ( ord_less_eq @ B @ B2 @ D3 ) ) ) ) ).
% Pair_le
thf(fact_82_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% order_refl
thf(fact_83_Pair__mono,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [X3: A,X5: A,Y3: B,Y5: B] :
( ( ord_less_eq @ A @ X3 @ X5 )
=> ( ( ord_less_eq @ B @ Y3 @ Y5 )
=> ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( product_Pair @ A @ B @ X5 @ Y5 ) ) ) ) ) ).
% Pair_mono
thf(fact_84_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B2: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( C3 @ A2 @ B2 ) ) ).
% internal_case_prod_conv
thf(fact_85_Fsub,axiom,
ord_less_eq @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_FVF @ a @ b @ c @ p ) @ v ).
% Fsub
thf(fact_86_solSem,axiom,
! [X6: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X6 )
=> ( ( ord_less_eq @ real @ X6 @ t )
=> ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( denotational_mk_v @ a @ b @ c @ i @ ode2 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ a2 @ b2 ) @ ( sol @ X6 ) ) @ ( denotational_fml_sem @ a @ b @ c @ i @ p ) ) ) ) ).
% solSem
thf(fact_87_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( bNF_Ca1785829860lChain @ A @ B )
= ( ^ [R3: set @ ( product_prod @ A @ A ),As: A > B] :
! [I2: A,J: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J ) @ R3 )
=> ( ord_less_eq @ B @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ) ).
% relChain_def
thf(fact_88_hpsafe__Evolve_OIH,axiom,
coinci1993344360de_fml @ a @ b @ c @ p ).
% hpsafe_Evolve.IH
thf(fact_89_t,axiom,
ord_less_eq @ real @ ( zero_zero @ real ) @ t ).
% t
thf(fact_90_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_91_add_Oinverse__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% add.inverse_neutral
thf(fact_92_neg__0__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( uminus_uminus @ A @ A2 ) )
= ( ( zero_zero @ A )
= A2 ) ) ) ).
% neg_0_equal_iff_equal
thf(fact_93_neg__equal__0__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( ( uminus_uminus @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_0_iff_equal
thf(fact_94_equal__neg__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( A2
= ( uminus_uminus @ A @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% equal_neg_zero
thf(fact_95_neg__equal__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ( uminus_uminus @ A @ A2 )
= A2 )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_zero
thf(fact_96_IH_H,axiom,
! [A7: finite_Cartesian_vec @ real @ c,B7: finite_Cartesian_vec @ real @ c,Aa: finite_Cartesian_vec @ real @ c,Ba: finite_Cartesian_vec @ real @ c] :
( ( denotational_Vagree @ c @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ A7 @ B7 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ Aa @ Ba ) @ ( static_FVF @ a @ b @ c @ p ) )
=> ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ A7 @ B7 ) @ ( denotational_fml_sem @ a @ b @ c @ i @ p ) )
= ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ Aa @ Ba ) @ ( denotational_fml_sem @ a @ b @ c @ j @ p ) ) ) ) ).
% IH'
thf(fact_97_neg__less__eq__nonneg,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_less_eq_nonneg
thf(fact_98_less__eq__neg__nonpos,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% less_eq_neg_nonpos
thf(fact_99_neg__le__0__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_le_0_iff_le
thf(fact_100_neg__0__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_le_iff_le
thf(fact_101_fsafe,axiom,
fsafe @ a @ b @ c @ p ).
% fsafe
thf(fact_102_zero__prod__def,axiom,
! [A: $tType,B: $tType] :
( ( ( zero @ B )
& ( zero @ A ) )
=> ( ( zero_zero @ ( product_prod @ A @ B ) )
= ( product_Pair @ A @ B @ ( zero_zero @ A ) @ ( zero_zero @ B ) ) ) ) ).
% zero_prod_def
thf(fact_103_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X3: A] :
( ( ( zero_zero @ A )
= X3 )
= ( X3
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_104_iff__to__impl,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ A ) )
=> ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),I: denota1663640101rp_ext @ B @ C @ A @ product_unit,A4: formula @ B @ C @ A,B4: formula @ B @ C @ A] :
( ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ A4 ) )
= ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ B4 ) ) )
= ( ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ A4 ) )
=> ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ B4 ) ) )
& ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ B4 ) )
=> ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ A4 ) ) ) ) ) ) ).
% iff_to_impl
thf(fact_105_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) ) ).
% zero_le
thf(fact_106_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G3: A > B,X3: A] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G3 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) ) ) ).
% le_funD
thf(fact_107_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G3: A > B,X3: A] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G3 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) ) ) ).
% le_funE
thf(fact_108_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G3: A > B] :
( ! [X: A] : ( ord_less_eq @ B @ ( F3 @ X ) @ ( G3 @ X ) )
=> ( ord_less_eq @ ( A > B ) @ F3 @ G3 ) ) ) ).
% le_funI
thf(fact_109_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F4: A > B,G4: A > B] :
! [X4: A] : ( ord_less_eq @ B @ ( F4 @ X4 ) @ ( G4 @ X4 ) ) ) ) ) ).
% le_fun_def
thf(fact_110_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F3: B > A,B2: B,C3: B] :
( ( ord_less_eq @ A @ A2 @ ( F3 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C3 )
=> ( ! [X: B,Y: B] :
( ( ord_less_eq @ B @ X @ Y )
=> ( ord_less_eq @ A @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% order_subst1
thf(fact_111_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F3: A > C,C3: C] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C @ ( F3 @ B2 ) @ C3 )
=> ( ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ C @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq @ C @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% order_subst2
thf(fact_112_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F3: B > A,B2: B,C3: B] :
( ( A2
= ( F3 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C3 )
=> ( ! [X: B,Y: B] :
( ( ord_less_eq @ B @ X @ Y )
=> ( ord_less_eq @ A @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_113_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F3: A > B,C3: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F3 @ B2 )
= C3 )
=> ( ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq @ B @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_114_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [X4: A,Y6: A] :
( ( ord_less_eq @ A @ X4 @ Y6 )
& ( ord_less_eq @ A @ Y6 @ X4 ) ) ) ) ) ).
% eq_iff
thf(fact_115_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ) ).
% antisym
thf(fact_116_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
| ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% linear
thf(fact_117_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A] :
( ( X3 = Y3 )
=> ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ).
% eq_refl
thf(fact_118_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ~ ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% le_cases
thf(fact_119_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% order.trans
thf(fact_120_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A,Z2: A] :
( ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X3 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y3 ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ X3 ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_121_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y3: A,X3: A] :
( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ) ).
% antisym_conv
thf(fact_122_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [A8: A,B8: A] :
( ( ord_less_eq @ A @ A8 @ B8 )
& ( ord_less_eq @ A @ B8 @ A8 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_123_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_124_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_125_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_126_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A,Z2: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ Z2 )
=> ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ).
% order_trans
thf(fact_127_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_128_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: A > A > $o,A2: A,B2: A] :
( ! [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
=> ( P2 @ A5 @ B5 ) )
=> ( ! [A5: A,B5: A] :
( ( P2 @ B5 @ A5 )
=> ( P2 @ A5 @ B5 ) )
=> ( P2 @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_129_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C3 @ B2 )
=> ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_130_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [A8: A,B8: A] :
( ( ord_less_eq @ A @ B8 @ A8 )
& ( ord_less_eq @ A @ A8 @ B8 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_131_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_132_gcd_Ocases,axiom,
! [A: $tType] :
( ( euclid1155270486miring @ A )
=> ! [X3: product_prod @ A @ A] :
~ ! [A5: A,B5: A] :
( X3
!= ( product_Pair @ A @ A @ A5 @ B5 ) ) ) ).
% gcd.cases
thf(fact_133_iff__sem,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ A ) )
=> ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),I: denota1663640101rp_ext @ B @ C @ A @ product_unit,A4: formula @ B @ C @ A,B4: formula @ B @ C @ A] :
( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ ( equiv @ B @ C @ A @ A4 @ B4 ) ) )
= ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ A4 ) )
= ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ B4 ) ) ) ) ) ).
% iff_sem
thf(fact_134_IHF,axiom,
! [A7: finite_Cartesian_vec @ real @ c,B7: finite_Cartesian_vec @ real @ c,Aa: finite_Cartesian_vec @ real @ c,Ba: finite_Cartesian_vec @ real @ c] :
( ( denotational_Iagree @ a @ b @ c @ i @ j @ ( static_SIGF @ a @ b @ c @ p ) )
=> ( ( denotational_Vagree @ c @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ A7 @ B7 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ Aa @ Ba ) @ ( static_FVF @ a @ b @ c @ p ) )
=> ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ A7 @ B7 ) @ ( denotational_fml_sem @ a @ b @ c @ i @ p ) )
= ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ Aa @ Ba ) @ ( denotational_fml_sem @ a @ b @ c @ j @ p ) ) ) ) ) ).
% IHF
thf(fact_135_minus__le__self__iff,axiom,
! [A: $tType] :
( ( lattic1601792062up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% minus_le_self_iff
thf(fact_136_le__minus__self__iff,axiom,
! [A: $tType] :
( ( lattic1601792062up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% le_minus_self_iff
thf(fact_137_real__eq__0__iff__le__ge__0,axiom,
! [X3: real] :
( ( X3
= ( zero_zero @ real ) )
= ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( uminus_uminus @ real @ X3 ) ) ) ) ).
% real_eq_0_iff_le_ge_0
thf(fact_138_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_139_coincide__fml__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ( ( coinci1993344360de_fml @ A @ B @ C )
= ( ^ [Phi: formula @ A @ B @ C] :
! [Nu3: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Nu4: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,J2: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( denotational_Iagree @ A @ B @ C @ I3 @ J2 @ ( static_SIGF @ A @ B @ C @ Phi ) )
=> ( ( denotational_Vagree @ C @ Nu3 @ Nu4 @ ( static_FVF @ A @ B @ C @ Phi ) )
=> ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu3 @ ( denotational_fml_sem @ A @ B @ C @ I3 @ Phi ) )
= ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu4 @ ( denotational_fml_sem @ A @ B @ C @ J2 @ Phi ) ) ) ) ) ) ) ) ).
% coincide_fml_def
thf(fact_140_IAP,axiom,
denotational_Iagree @ a @ b @ c @ i @ j @ ( static_SIGF @ a @ b @ c @ p ) ).
% IAP
thf(fact_141_Iagree__sub,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [A4: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ),B4: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ),I: denota1663640101rp_ext @ A @ B @ C @ product_unit,J3: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( ord_less_eq @ ( set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) @ A4 @ B4 )
=> ( ( denotational_Iagree @ A @ B @ C @ I @ J3 @ B4 )
=> ( denotational_Iagree @ A @ B @ C @ I @ J3 @ A4 ) ) ) ) ).
% Iagree_sub
thf(fact_142_Iagree__comm,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [A4: denota1663640101rp_ext @ A @ B @ C @ product_unit,B4: denota1663640101rp_ext @ A @ B @ C @ product_unit,V: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) )] :
( ( denotational_Iagree @ A @ B @ C @ A4 @ B4 @ V )
=> ( denotational_Iagree @ A @ B @ C @ B4 @ A4 @ V ) ) ) ).
% Iagree_comm
thf(fact_143_Iagree__refl,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [I: denota1663640101rp_ext @ A @ B @ C @ product_unit,A4: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) )] : ( denotational_Iagree @ A @ B @ C @ I @ I @ A4 ) ) ).
% Iagree_refl
thf(fact_144_ids_Ocoincide__fml__def,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,B: $tType,A: $tType,C: $tType] :
( ( ( finite_finite @ C )
& ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Phi2: formula @ A @ B @ C] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( coinci1993344360de_fml @ A @ B @ C @ Phi2 )
= ( ! [Nu3: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Nu4: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,J2: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( denotational_Iagree @ A @ B @ C @ I3 @ J2 @ ( static_SIGF @ A @ B @ C @ Phi2 ) )
=> ( ( denotational_Vagree @ C @ Nu3 @ Nu4 @ ( static_FVF @ A @ B @ C @ Phi2 ) )
=> ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu3 @ ( denotational_fml_sem @ A @ B @ C @ I3 @ Phi2 ) )
= ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu4 @ ( denotational_fml_sem @ A @ B @ C @ J2 @ Phi2 ) ) ) ) ) ) ) ) ) ).
% ids.coincide_fml_def
thf(fact_145__092_060open_062ode__sem__equiv_A_IEvolveODE_AODE_AP_J_AJ_092_060close_062,axiom,
coinci1495174342_equiv @ a @ b @ c @ ( evolveODE @ a @ c @ b @ ode2 @ p ) @ j ).
% \<open>ode_sem_equiv (EvolveODE ODE P) J\<close>
thf(fact_146__092_060open_062ode__sem__equiv_A_IEvolveODE_AODE_AP_J_AI_092_060close_062,axiom,
coinci1495174342_equiv @ a @ b @ c @ ( evolveODE @ a @ c @ b @ ode2 @ p ) @ i ).
% \<open>ode_sem_equiv (EvolveODE ODE P) I\<close>
thf(fact_147_impl__sem,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ A ) )
=> ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),I: denota1663640101rp_ext @ B @ C @ A @ product_unit,A4: formula @ B @ C @ A,B4: formula @ B @ C @ A] :
( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ ( implies @ B @ C @ A @ A4 @ B4 ) ) )
= ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ A4 ) )
=> ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu @ ( denotational_fml_sem @ B @ C @ A @ I @ B4 ) ) ) ) ) ).
% impl_sem
thf(fact_148_equiv,axiom,
! [I4: denota1663640101rp_ext @ a @ b @ c @ product_unit] : ( coinci1495174342_equiv @ a @ b @ c @ ( evolveODE @ a @ c @ b @ ode2 @ p ) @ I4 ) ).
% equiv
thf(fact_149_ids_Oseq__sem_Ocases,axiom,
! [Sz: $tType,Sc: $tType,Sf: $tType] :
( ( ( finite_finite @ Sf )
& ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X3: product_prod @ ( denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit ) @ ( product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) )] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ~ ! [I5: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,S4: product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) )] :
( X3
!= ( product_Pair @ ( denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit ) @ ( product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) ) @ I5 @ S4 ) ) ) ) ).
% ids.seq_sem.cases
thf(fact_150_ids_Oseq__sem_Oinduct,axiom,
! [Sz: $tType,Sc: $tType,Sf: $tType] :
( ( ( finite_finite @ Sf )
& ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P2: ( denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit ) > ( product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) ) > $o,A0: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,A1: product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) )] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ! [I5: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,X_1: product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) )] : ( P2 @ I5 @ X_1 )
=> ( P2 @ A0 @ A1 ) ) ) ) ).
% ids.seq_sem.induct
thf(fact_151_coincide__hp_H__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ( ( coinci554967470ide_hp @ A @ B @ C )
= ( ^ [Alpha: hp @ A @ B @ C] :
! [I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,J2: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( coinci32832645ide_hp @ A @ B @ C @ Alpha @ I3 @ J2 )
& ( coinci1495174342_equiv @ A @ B @ C @ Alpha @ I3 ) ) ) ) ) ).
% coincide_hp'_def
thf(fact_152_hp_Oinject_I5_J,axiom,
! [C: $tType,B: $tType,A: $tType,X51: ode @ A @ C,X52: formula @ A @ B @ C,Y51: ode @ A @ C,Y52: formula @ A @ B @ C] :
( ( ( evolveODE @ A @ C @ B @ X51 @ X52 )
= ( evolveODE @ A @ C @ B @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% hp.inject(5)
thf(fact_153_sol,axiom,
( initia1685620758es_ode @ ( finite_Cartesian_vec @ real @ c ) @ sol
@ ^ [A8: real] : ( denotational_ODE_sem @ a @ b @ c @ i @ ode2 )
@ ( set_or331188842AtMost @ real @ ( zero_zero @ real ) @ t )
@ ( collect @ ( finite_Cartesian_vec @ real @ c )
@ ^ [X4: finite_Cartesian_vec @ real @ c] : ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( denotational_mk_v @ a @ b @ c @ i @ ode2 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ a2 @ b2 ) @ X4 ) @ ( denotational_fml_sem @ a @ b @ c @ i @ p ) ) ) ) ).
% sol
thf(fact_154_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A6 )
@ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_155_Collect__subset,axiom,
! [A: $tType,A4: set @ A,P2: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A4 )
& ( P2 @ X4 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_156_Compl__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A6: set @ A] :
( collect @ A
@ ^ [X4: A] :
~ ( member @ A @ X4 @ A6 ) ) ) ) ).
% Compl_eq
thf(fact_157_Collect__neg__eq,axiom,
! [A: $tType,P2: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
~ ( P2 @ X4 ) )
= ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P2 ) ) ) ).
% Collect_neg_eq
thf(fact_158_uminus__set__def,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A6: set @ A] :
( collect @ A
@ ( uminus_uminus @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A6 ) ) ) ) ) ).
% uminus_set_def
thf(fact_159_mk__xode_Oelims,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( finite_finite @ C )
& ( finite_finite @ B )
& ( finite_finite @ A ) )
=> ! [X3: denota1663640101rp_ext @ A @ B @ C @ product_unit,Xa: ode @ A @ C,Xb: finite_Cartesian_vec @ real @ C,Y3: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] :
( ( ( denotational_mk_xode @ A @ B @ C @ X3 @ Xa @ Xb )
= Y3 )
=> ( Y3
= ( product_Pair @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Xb @ ( denotational_ODE_sem @ A @ B @ C @ X3 @ Xa @ Xb ) ) ) ) ) ).
% mk_xode.elims
thf(fact_160_mk__xode_Osimps,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ( ( denotational_mk_xode @ A @ B @ C )
= ( ^ [I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,ODE2: ode @ A @ C,Sol2: finite_Cartesian_vec @ real @ C] : ( product_Pair @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Sol2 @ ( denotational_ODE_sem @ A @ B @ C @ I3 @ ODE2 @ Sol2 ) ) ) ) ) ).
% mk_xode.simps
thf(fact_161_ids_Osingleton_Oinduct,axiom,
! [Sf: $tType,Sc: $tType,A: $tType,Sz: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sc )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P2: ( trm @ A @ Sz ) > Sz > $o,A0: trm @ A @ Sz,A1: Sz] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ! [T4: trm @ A @ Sz,X_1: Sz] : ( P2 @ T4 @ X_1 )
=> ( P2 @ A0 @ A1 ) ) ) ) ).
% ids.singleton.induct
thf(fact_162_ids_Osingleton_Ocases,axiom,
! [Sc: $tType,Sf: $tType,A: $tType,Sz: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X3: product_prod @ ( trm @ A @ Sz ) @ Sz] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ~ ! [T4: trm @ A @ Sz,I6: Sz] :
( X3
!= ( product_Pair @ ( trm @ A @ Sz ) @ Sz @ T4 @ I6 ) ) ) ) ).
% ids.singleton.cases
thf(fact_163_ids_Ocoincide__hp_H__def,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Alpha2: hp @ A @ B @ C] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( coinci554967470ide_hp @ A @ B @ C @ Alpha2 )
= ( ! [I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,J2: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( coinci32832645ide_hp @ A @ B @ C @ Alpha2 @ I3 @ J2 )
& ( coinci1495174342_equiv @ A @ B @ C @ Alpha2 @ I3 ) ) ) ) ) ) ).
% ids.coincide_hp'_def
thf(fact_164_atLeastatMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or331188842AtMost @ A @ A2 @ B2 ) @ ( set_or331188842AtMost @ A @ C3 @ D3 ) )
= ( ~ ( ord_less_eq @ A @ A2 @ B2 )
| ( ( ord_less_eq @ A @ C3 @ A2 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_165_atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I7: A,L: A,U: A] :
( ( member @ A @ I7 @ ( set_or331188842AtMost @ A @ L @ U ) )
= ( ( ord_less_eq @ A @ L @ I7 )
& ( ord_less_eq @ A @ I7 @ U ) ) ) ) ).
% atLeastAtMost_iff
thf(fact_166_Icc__eq__Icc,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,H: A,L2: A,H2: A] :
( ( ( set_or331188842AtMost @ A @ L @ H )
= ( set_or331188842AtMost @ A @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq @ A @ L @ H )
& ~ ( ord_less_eq @ A @ L2 @ H2 ) ) ) ) ) ).
% Icc_eq_Icc
thf(fact_167_predicate1I,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X: A] :
( ( P2 @ X )
=> ( Q @ X ) )
=> ( ord_less_eq @ ( A > $o ) @ P2 @ Q ) ) ).
% predicate1I
thf(fact_168_predicate1D,axiom,
! [A: $tType,P2: A > $o,Q: A > $o,X3: A] :
( ( ord_less_eq @ ( A > $o ) @ P2 @ Q )
=> ( ( P2 @ X3 )
=> ( Q @ X3 ) ) ) ).
% predicate1D
thf(fact_169_rev__predicate1D,axiom,
! [A: $tType,P2: A > $o,X3: A,Q: A > $o] :
( ( P2 @ X3 )
=> ( ( ord_less_eq @ ( A > $o ) @ P2 @ Q )
=> ( Q @ X3 ) ) ) ).
% rev_predicate1D
thf(fact_170_pred__subset__eq,axiom,
! [A: $tType,R4: set @ A,S5: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ R4 )
@ ^ [X4: A] : ( member @ A @ X4 @ S5 ) )
= ( ord_less_eq @ ( set @ A ) @ R4 @ S5 ) ) ).
% pred_subset_eq
thf(fact_171_solDeriv,axiom,
! [X6: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X6 )
=> ( ( ord_less_eq @ real @ X6 @ t )
=> ( has_ve2132708402vative @ ( finite_Cartesian_vec @ real @ c ) @ sol @ ( denotational_ODE_sem @ a @ b @ c @ i @ ode2 @ ( sol @ X6 ) ) @ ( topolo507301023within @ real @ X6 @ ( set_or331188842AtMost @ real @ ( zero_zero @ real ) @ t ) ) ) ) ) ).
% solDeriv
thf(fact_172_mk__v__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ( ( denotational_mk_v @ A @ B @ C )
= ( ^ [I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,ODE2: ode @ A @ C,Nu3: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Sol2: finite_Cartesian_vec @ real @ C] :
( the @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) )
@ ^ [Omega3: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] :
( ( denotational_Vagree @ C @ Omega3 @ Nu3 @ ( uminus_uminus @ ( set @ ( sum_sum @ C @ C ) ) @ ( denotational_semBV @ A @ B @ C @ I3 @ ODE2 ) ) )
& ( denotational_Vagree @ C @ Omega3 @ ( denotational_mk_xode @ A @ B @ C @ I3 @ ODE2 @ Sol2 ) @ ( denotational_semBV @ A @ B @ C @ I3 @ ODE2 ) ) ) ) ) ) ) ).
% mk_v_def
thf(fact_173_subrelI,axiom,
! [B: $tType,A: $tType,R5: set @ ( product_prod @ A @ B ),S6: set @ ( product_prod @ A @ B )] :
( ! [X: A,Y: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ R5 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ S6 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R5 @ S6 ) ) ).
% subrelI
thf(fact_174_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S5: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( A > B > $o )
@ ^ [X4: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y6 ) @ R4 )
@ ^ [X4: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y6 ) @ S5 ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R4 @ S5 ) ) ).
% pred_subset_eq2
thf(fact_175_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S5: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X4: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y6 ) @ R4 ) )
= ( ^ [X4: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y6 ) @ S5 ) ) )
= ( R4 = S5 ) ) ).
% pred_equals_eq2
thf(fact_176_old_Orec__prod__def,axiom,
! [T: $tType,B: $tType,A: $tType] :
( ( product_rec_prod @ A @ B @ T )
= ( ^ [F12: A > B > T,X4: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F12 @ X4 ) ) ) ) ).
% old.rec_prod_def
thf(fact_177_has__vector__derivative__Pair,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V55928688vector @ A )
& ( real_V55928688vector @ B ) )
=> ! [F3: real > A,F5: A,X3: real,S6: set @ real,G3: real > B,G5: B] :
( ( has_ve2132708402vative @ A @ F3 @ F5 @ ( topolo507301023within @ real @ X3 @ S6 ) )
=> ( ( has_ve2132708402vative @ B @ G3 @ G5 @ ( topolo507301023within @ real @ X3 @ S6 ) )
=> ( has_ve2132708402vative @ ( product_prod @ A @ B )
@ ^ [X4: real] : ( product_Pair @ A @ B @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( product_Pair @ A @ B @ F5 @ G5 )
@ ( topolo507301023within @ real @ X3 @ S6 ) ) ) ) ) ).
% has_vector_derivative_Pair
thf(fact_178_has__vector__derivative__minus,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ! [F3: real > A,F5: A,Net: filter @ real] :
( ( has_ve2132708402vative @ A @ F3 @ F5 @ Net )
=> ( has_ve2132708402vative @ A
@ ^ [X4: real] : ( uminus_uminus @ A @ ( F3 @ X4 ) )
@ ( uminus_uminus @ A @ F5 )
@ Net ) ) ) ).
% has_vector_derivative_minus
thf(fact_179_predicate2I,axiom,
! [B: $tType,A: $tType,P2: A > B > $o,Q: A > B > $o] :
( ! [X: A,Y: B] :
( ( P2 @ X @ Y )
=> ( Q @ X @ Y ) )
=> ( ord_less_eq @ ( A > B > $o ) @ P2 @ Q ) ) ).
% predicate2I
thf(fact_180_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,X3: A,Y3: B,Q: A > B > $o] :
( ( P2 @ X3 @ Y3 )
=> ( ( ord_less_eq @ ( A > B > $o ) @ P2 @ Q )
=> ( Q @ X3 @ Y3 ) ) ) ).
% rev_predicate2D
thf(fact_181_predicate2D,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,Q: A > B > $o,X3: A,Y3: B] :
( ( ord_less_eq @ ( A > B > $o ) @ P2 @ Q )
=> ( ( P2 @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) ) ) ).
% predicate2D
thf(fact_182_has__vector__derivative__const,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ! [C3: A,Net: filter @ real] :
( has_ve2132708402vative @ A
@ ^ [X4: real] : C3
@ ( zero_zero @ A )
@ Net ) ) ).
% has_vector_derivative_const
thf(fact_183_has__vector__derivative__within__subset,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ! [F3: real > A,F5: A,X3: real,S5: set @ real,T5: set @ real] :
( ( has_ve2132708402vative @ A @ F3 @ F5 @ ( topolo507301023within @ real @ X3 @ S5 ) )
=> ( ( ord_less_eq @ ( set @ real ) @ T5 @ S5 )
=> ( has_ve2132708402vative @ A @ F3 @ F5 @ ( topolo507301023within @ real @ X3 @ T5 ) ) ) ) ) ).
% has_vector_derivative_within_subset
thf(fact_184_has__vector__derivative__weaken,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ! [F3: real > A,D4: A,X3: real,T5: set @ real,S5: set @ real,G3: real > A] :
( ( has_ve2132708402vative @ A @ F3 @ D4 @ ( topolo507301023within @ real @ X3 @ T5 ) )
=> ( ( member @ real @ X3 @ S5 )
=> ( ( ord_less_eq @ ( set @ real ) @ S5 @ T5 )
=> ( ! [X: real] :
( ( member @ real @ X @ S5 )
=> ( ( F3 @ X )
= ( G3 @ X ) ) )
=> ( has_ve2132708402vative @ A @ G3 @ D4 @ ( topolo507301023within @ real @ X3 @ S5 ) ) ) ) ) ) ) ).
% has_vector_derivative_weaken
thf(fact_185_solves__ode__supset__range,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ! [X3: real > A,F3: real > A > A,T5: set @ real,X7: set @ A,Y7: set @ A] :
( ( initia1685620758es_ode @ A @ X3 @ F3 @ T5 @ X7 )
=> ( ( ord_less_eq @ ( set @ A ) @ X7 @ Y7 )
=> ( initia1685620758es_ode @ A @ X3 @ F3 @ T5 @ Y7 ) ) ) ) ).
% solves_ode_supset_range
thf(fact_186_solves__ode__subset,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ! [X3: real > A,F3: real > A > A,T5: set @ real,X7: set @ A,S5: set @ real] :
( ( initia1685620758es_ode @ A @ X3 @ F3 @ T5 @ X7 )
=> ( ( ord_less_eq @ ( set @ real ) @ S5 @ T5 )
=> ( initia1685620758es_ode @ A @ X3 @ F3 @ S5 @ X7 ) ) ) ) ).
% solves_ode_subset
thf(fact_187_solves__ode__on__subset,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ! [X3: real > A,F3: real > A > A,S5: set @ real,Y7: set @ A,T5: set @ real,X7: set @ A] :
( ( initia1685620758es_ode @ A @ X3 @ F3 @ S5 @ Y7 )
=> ( ( ord_less_eq @ ( set @ real ) @ T5 @ S5 )
=> ( ( ord_less_eq @ ( set @ A ) @ Y7 @ X7 )
=> ( initia1685620758es_ode @ A @ X3 @ F3 @ T5 @ X7 ) ) ) ) ) ).
% solves_ode_on_subset
thf(fact_188_at__le,axiom,
! [A: $tType] :
( ( topolo503727757_space @ A )
=> ! [S6: set @ A,T6: set @ A,X3: A] :
( ( ord_less_eq @ ( set @ A ) @ S6 @ T6 )
=> ( ord_less_eq @ ( filter @ A ) @ ( topolo507301023within @ A @ X3 @ S6 ) @ ( topolo507301023within @ A @ X3 @ T6 ) ) ) ) ).
% at_le
thf(fact_189_intervalE,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,X3: A,B2: A] :
( ( ( ord_less_eq @ A @ A2 @ X3 )
& ( ord_less_eq @ A @ X3 @ B2 ) )
=> ( member @ A @ X3 @ ( set_or331188842AtMost @ A @ A2 @ B2 ) ) ) ) ).
% intervalE
thf(fact_190_eq__subset,axiom,
! [A: $tType,P2: A > A > $o] :
( ord_less_eq @ ( A > A > $o )
@ ^ [Y4: A,Z: A] : Y4 = Z
@ ^ [A8: A,B8: A] :
( ( P2 @ A8 @ B8 )
| ( A8 = B8 ) ) ) ).
% eq_subset
thf(fact_191_conj__subset__def,axiom,
! [A: $tType,A4: set @ A,P2: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A4
@ ( collect @ A
@ ^ [X4: A] :
( ( P2 @ X4 )
& ( Q @ X4 ) ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( collect @ A @ P2 ) )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( collect @ A @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_192_predicate2D__conj,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,Q: A > B > $o,R4: $o,X3: A,Y3: B] :
( ( ( ord_less_eq @ ( A > B > $o ) @ P2 @ Q )
& R4 )
=> ( R4
& ( ( P2 @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) ) ) ) ).
% predicate2D_conj
thf(fact_193_fun__cong__unused__0,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( zero @ B )
=> ! [F3: ( A > B ) > C,G3: C] :
( ( F3
= ( ^ [X4: A > B] : G3 ) )
=> ( ( F3
@ ^ [X4: A] : ( zero_zero @ B ) )
= G3 ) ) ) ).
% fun_cong_unused_0
thf(fact_194_subset__Collect__iff,axiom,
! [A: $tType,B4: set @ A,A4: set @ A,P2: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A4 )
& ( P2 @ X4 ) ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ B4 )
=> ( P2 @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_195_subset__CollectI,axiom,
! [A: $tType,B4: set @ A,A4: set @ A,Q: A > $o,P2: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
=> ( ! [X: A] :
( ( member @ A @ X @ B4 )
=> ( ( Q @ X )
=> ( P2 @ X ) ) )
=> ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ B4 )
& ( Q @ X4 ) ) )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A4 )
& ( P2 @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_196_prop__restrict,axiom,
! [A: $tType,X3: A,Z3: set @ A,X7: set @ A,P2: A > $o] :
( ( member @ A @ X3 @ Z3 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z3
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ X7 )
& ( P2 @ X4 ) ) ) )
=> ( P2 @ X3 ) ) ) ).
% prop_restrict
thf(fact_197_Collect__restrict,axiom,
! [A: $tType,X7: set @ A,P2: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ X7 )
& ( P2 @ X4 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_198_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R5: A,S6: B,R4: set @ ( product_prod @ A @ B ),S7: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R5 @ S6 ) @ R4 )
=> ( ( S7 = S6 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R5 @ S7 ) @ R4 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_199_hpsafe__ODE__simps,axiom,
! [C: $tType,B: $tType,A: $tType,ODE: ode @ A @ C,P: formula @ A @ B @ C] :
( ( hpsafe @ A @ B @ C @ ( evolveODE @ A @ C @ B @ ODE @ P ) )
= ( ( osafe @ A @ C @ ODE )
& ( fsafe @ A @ B @ C @ P ) ) ) ).
% hpsafe_ODE_simps
thf(fact_200_old_Orec__unit__def,axiom,
! [T: $tType] :
( ( product_rec_unit @ T )
= ( ^ [F12: T,X4: product_unit] : ( the @ T @ ( product_rec_set_unit @ T @ F12 @ X4 ) ) ) ) ).
% old.rec_unit_def
thf(fact_201_hpsafe__fsafe_Ohpsafe__Evolve,axiom,
! [C: $tType,B: $tType,A: $tType,ODE: ode @ A @ C,P2: formula @ A @ B @ C] :
( ( osafe @ A @ C @ ODE )
=> ( ( fsafe @ A @ B @ C @ P2 )
=> ( hpsafe @ A @ B @ C @ ( evolveODE @ A @ C @ B @ ODE @ P2 ) ) ) ) ).
% hpsafe_fsafe.hpsafe_Evolve
thf(fact_202_divides__aux__eq,axiom,
! [A: $tType] :
( ( unique1598680935umeral @ A )
=> ! [Q2: A,R5: A] :
( ( unique455577585es_aux @ A @ ( product_Pair @ A @ A @ Q2 @ R5 ) )
= ( R5
= ( zero_zero @ A ) ) ) ) ).
% divides_aux_eq
thf(fact_203_refl__ge__eq,axiom,
! [A: $tType,R4: A > A > $o] :
( ! [X: A] : ( R4 @ X @ X )
=> ( ord_less_eq @ ( A > A > $o )
@ ^ [Y4: A,Z: A] : Y4 = Z
@ R4 ) ) ).
% refl_ge_eq
thf(fact_204_ge__eq__refl,axiom,
! [A: $tType,R4: A > A > $o,X3: A] :
( ( ord_less_eq @ ( A > A > $o )
@ ^ [Y4: A,Z: A] : Y4 = Z
@ R4 )
=> ( R4 @ X3 @ X3 ) ) ).
% ge_eq_refl
thf(fact_205_Icc__subset__Ici__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [L: A,H: A,L2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or331188842AtMost @ A @ L @ H ) @ ( set_ord_atLeast @ A @ L2 ) )
= ( ~ ( ord_less_eq @ A @ L @ H )
| ( ord_less_eq @ A @ L2 @ L ) ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_206_Icc__subset__Iic__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [L: A,H: A,H2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or331188842AtMost @ A @ L @ H ) @ ( set_ord_atMost @ A @ H2 ) )
= ( ~ ( ord_less_eq @ A @ L @ H )
| ( ord_less_eq @ A @ H @ H2 ) ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_207_atMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I7: A,K: A] :
( ( member @ A @ I7 @ ( set_ord_atMost @ A @ K ) )
= ( ord_less_eq @ A @ I7 @ K ) ) ) ).
% atMost_iff
thf(fact_208_atLeast__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I7: A,K: A] :
( ( member @ A @ I7 @ ( set_ord_atLeast @ A @ K ) )
= ( ord_less_eq @ A @ K @ I7 ) ) ) ).
% atLeast_iff
thf(fact_209_atMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ X3 ) @ ( set_ord_atMost @ A @ Y3 ) )
= ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ).
% atMost_subset_iff
thf(fact_210_atLeast__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ X3 ) @ ( set_ord_atLeast @ A @ Y3 ) )
= ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% atLeast_subset_iff
thf(fact_211_not__Ici__le__Icc,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L: A,L2: A,H2: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L ) @ ( set_or331188842AtMost @ A @ L2 @ H2 ) ) ) ).
% not_Ici_le_Icc
thf(fact_212_not__Iic__le__Icc,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [H: A,L2: A,H2: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H ) @ ( set_or331188842AtMost @ A @ L2 @ H2 ) ) ) ).
% not_Iic_le_Icc
thf(fact_213_atLeast__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atLeast @ A )
= ( ^ [L3: A] : ( collect @ A @ ( ord_less_eq @ A @ L3 ) ) ) ) ) ).
% atLeast_def
thf(fact_214_atMost__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atMost @ A )
= ( ^ [U2: A] :
( collect @ A
@ ^ [X4: A] : ( ord_less_eq @ A @ X4 @ U2 ) ) ) ) ) ).
% atMost_def
thf(fact_215_not__Iic__le__Ici,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [H: A,L2: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H ) @ ( set_ord_atLeast @ A @ L2 ) ) ) ).
% not_Iic_le_Ici
thf(fact_216_not__Ici__le__Iic,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L: A,H2: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L ) @ ( set_ord_atMost @ A @ H2 ) ) ) ).
% not_Ici_le_Iic
thf(fact_217_image__uminus__atMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X3: A] :
( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_atMost @ A @ X3 ) )
= ( set_ord_atLeast @ A @ ( uminus_uminus @ A @ X3 ) ) ) ) ).
% image_uminus_atMost
thf(fact_218_image__uminus__atLeast,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X3: A] :
( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_atLeast @ A @ X3 ) )
= ( set_ord_atMost @ A @ ( uminus_uminus @ A @ X3 ) ) ) ) ).
% image_uminus_atLeast
thf(fact_219_image__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F3: B > A,X3: B,A4: set @ B] :
( ( B2
= ( F3 @ X3 ) )
=> ( ( member @ B @ X3 @ A4 )
=> ( member @ A @ B2 @ ( image @ B @ A @ F3 @ A4 ) ) ) ) ).
% image_eqI
thf(fact_220_image__ident,axiom,
! [A: $tType,Y7: set @ A] :
( ( image @ A @ A
@ ^ [X4: A] : X4
@ Y7 )
= Y7 ) ).
% image_ident
thf(fact_221_image__uminus__atLeastAtMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X3: A,Y3: A] :
( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_or331188842AtMost @ A @ X3 @ Y3 ) )
= ( set_or331188842AtMost @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X3 ) ) ) ) ).
% image_uminus_atLeastAtMost
thf(fact_222_subset__image__iff,axiom,
! [A: $tType,B: $tType,B4: set @ A,F3: B > A,A4: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ ( image @ B @ A @ F3 @ A4 ) )
= ( ? [AA: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ AA @ A4 )
& ( B4
= ( image @ B @ A @ F3 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_223_image__subset__iff,axiom,
! [A: $tType,B: $tType,F3: B > A,A4: set @ B,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F3 @ A4 ) @ B4 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A4 )
=> ( member @ A @ ( F3 @ X4 ) @ B4 ) ) ) ) ).
% image_subset_iff
thf(fact_224_subset__imageE,axiom,
! [A: $tType,B: $tType,B4: set @ A,F3: B > A,A4: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ ( image @ B @ A @ F3 @ A4 ) )
=> ~ ! [C5: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C5 @ A4 )
=> ( B4
!= ( image @ B @ A @ F3 @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_225_image__subsetI,axiom,
! [A: $tType,B: $tType,A4: set @ A,F3: A > B,B4: set @ B] :
( ! [X: A] :
( ( member @ A @ X @ A4 )
=> ( member @ B @ ( F3 @ X ) @ B4 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A4 ) @ B4 ) ) ).
% image_subsetI
thf(fact_226_image__mono,axiom,
! [B: $tType,A: $tType,A4: set @ A,B4: set @ A,F3: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A4 ) @ ( image @ A @ B @ F3 @ B4 ) ) ) ).
% image_mono
thf(fact_227_solves__ode__subset__range,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ! [X3: real > A,F3: real > A > A,T5: set @ real,X7: set @ A,Y7: set @ A] :
( ( initia1685620758es_ode @ A @ X3 @ F3 @ T5 @ X7 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ real @ A @ X3 @ T5 ) @ Y7 )
=> ( initia1685620758es_ode @ A @ X3 @ F3 @ T5 @ Y7 ) ) ) ) ).
% solves_ode_subset_range
thf(fact_228_imageI,axiom,
! [B: $tType,A: $tType,X3: A,A4: set @ A,F3: A > B] :
( ( member @ A @ X3 @ A4 )
=> ( member @ B @ ( F3 @ X3 ) @ ( image @ A @ B @ F3 @ A4 ) ) ) ).
% imageI
thf(fact_229_image__iff,axiom,
! [A: $tType,B: $tType,Z2: A,F3: B > A,A4: set @ B] :
( ( member @ A @ Z2 @ ( image @ B @ A @ F3 @ A4 ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A4 )
& ( Z2
= ( F3 @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_230_bex__imageD,axiom,
! [A: $tType,B: $tType,F3: B > A,A4: set @ B,P2: A > $o] :
( ? [X8: A] :
( ( member @ A @ X8 @ ( image @ B @ A @ F3 @ A4 ) )
& ( P2 @ X8 ) )
=> ? [X: B] :
( ( member @ B @ X @ A4 )
& ( P2 @ ( F3 @ X ) ) ) ) ).
% bex_imageD
thf(fact_231_image__cong,axiom,
! [B: $tType,A: $tType,M: set @ A,N2: set @ A,F3: A > B,G3: A > B] :
( ( M = N2 )
=> ( ! [X: A] :
( ( member @ A @ X @ N2 )
=> ( ( F3 @ X )
= ( G3 @ X ) ) )
=> ( ( image @ A @ B @ F3 @ M )
= ( image @ A @ B @ G3 @ N2 ) ) ) ) ).
% image_cong
thf(fact_232_ball__imageD,axiom,
! [A: $tType,B: $tType,F3: B > A,A4: set @ B,P2: A > $o] :
( ! [X: A] :
( ( member @ A @ X @ ( image @ B @ A @ F3 @ A4 ) )
=> ( P2 @ X ) )
=> ! [X8: B] :
( ( member @ B @ X8 @ A4 )
=> ( P2 @ ( F3 @ X8 ) ) ) ) ).
% ball_imageD
thf(fact_233_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X3: A,A4: set @ A,B2: B,F3: A > B] :
( ( member @ A @ X3 @ A4 )
=> ( ( B2
= ( F3 @ X3 ) )
=> ( member @ B @ B2 @ ( image @ A @ B @ F3 @ A4 ) ) ) ) ).
% rev_image_eqI
thf(fact_234_Compr__image__eq,axiom,
! [A: $tType,B: $tType,F3: B > A,A4: set @ B,P2: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ ( image @ B @ A @ F3 @ A4 ) )
& ( P2 @ X4 ) ) )
= ( image @ B @ A @ F3
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ A4 )
& ( P2 @ ( F3 @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_235_image__image,axiom,
! [A: $tType,B: $tType,C: $tType,F3: B > A,G3: C > B,A4: set @ C] :
( ( image @ B @ A @ F3 @ ( image @ C @ B @ G3 @ A4 ) )
= ( image @ C @ A
@ ^ [X4: C] : ( F3 @ ( G3 @ X4 ) )
@ A4 ) ) ).
% image_image
thf(fact_236_imageE,axiom,
! [A: $tType,B: $tType,B2: A,F3: B > A,A4: set @ B] :
( ( member @ A @ B2 @ ( image @ B @ A @ F3 @ A4 ) )
=> ~ ! [X: B] :
( ( B2
= ( F3 @ X ) )
=> ~ ( member @ B @ X @ A4 ) ) ) ).
% imageE
thf(fact_237_image__Collect__subsetI,axiom,
! [A: $tType,B: $tType,P2: A > $o,F3: A > B,B4: set @ B] :
( ! [X: A] :
( ( P2 @ X )
=> ( member @ B @ ( F3 @ X ) @ B4 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ ( collect @ A @ P2 ) ) @ B4 ) ) ).
% image_Collect_subsetI
thf(fact_238_Osub,axiom,
ord_less_eq @ ( set @ ( sum_sum @ c @ c ) ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( static_FVO @ a @ c @ ode2 ) ) @ v ).
% Osub
thf(fact_239_all__subset__image,axiom,
! [A: $tType,B: $tType,F3: B > A,A4: set @ B,P2: ( set @ A ) > $o] :
( ( ! [B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F3 @ A4 ) )
=> ( P2 @ B6 ) ) )
= ( ! [B6: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B6 @ A4 )
=> ( P2 @ ( image @ B @ A @ F3 @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_240_bvo__to__fvo,axiom,
! [A: $tType,B: $tType,X3: A,ODE: ode @ B @ A] :
( ( member @ ( sum_sum @ A @ A ) @ ( sum_Inl @ A @ A @ X3 ) @ ( static_BVO @ B @ A @ ODE ) )
=> ( member @ A @ X3 @ ( static_FVO @ B @ A @ ODE ) ) ) ).
% bvo_to_fvo
thf(fact_241_ids_Obvo__to__fvo,axiom,
! [Sz: $tType,Sf: $tType,Sc: $tType,A: $tType,B: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X3: A,ODE: ode @ B @ A] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( member @ ( sum_sum @ A @ A ) @ ( sum_Inl @ A @ A @ X3 ) @ ( static_BVO @ B @ A @ ODE ) )
=> ( member @ A @ X3 @ ( static_FVO @ B @ A @ ODE ) ) ) ) ) ).
% ids.bvo_to_fvo
thf(fact_242_VSA,axiom,
( denotational_VSagree @ c @ ( sol @ ( zero_zero @ real ) ) @ a2
@ ( collect @ c
@ ^ [Uu: c] :
( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ Uu ) @ ( static_BVO @ a @ c @ ode2 ) )
| ( member @ ( sum_sum @ c @ d ) @ ( sum_Inl @ c @ d @ Uu ) @ ( image @ c @ ( sum_sum @ c @ d ) @ ( sum_Inl @ c @ d ) @ ( static_FVO @ a @ c @ ode2 ) ) )
| ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ Uu ) @ ( static_FVF @ a @ b @ c @ p ) ) ) ) ) ).
% VSA
thf(fact_243_ode__to__fvo,axiom,
! [C: $tType,A: $tType,B: $tType,X3: A,I: denota1663640101rp_ext @ B @ C @ A @ product_unit,ODE: ode @ B @ A] :
( ( member @ A @ X3 @ ( denota811733865E_vars @ B @ C @ A @ I @ ODE ) )
=> ( member @ A @ X3 @ ( static_FVO @ B @ A @ ODE ) ) ) ).
% ode_to_fvo
thf(fact_244_VSagree__sub,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [A4: set @ A,B4: set @ A,Nu: finite_Cartesian_vec @ real @ A,Omega2: finite_Cartesian_vec @ real @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( denotational_VSagree @ A @ Nu @ Omega2 @ B4 )
=> ( denotational_VSagree @ A @ Nu @ Omega2 @ A4 ) ) ) ) ).
% VSagree_sub
thf(fact_245_VSagree__supset,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [B4: set @ A,A4: set @ A,Nu: finite_Cartesian_vec @ real @ A,Nu2: finite_Cartesian_vec @ real @ A] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
=> ( ( denotational_VSagree @ A @ Nu @ Nu2 @ A4 )
=> ( denotational_VSagree @ A @ Nu @ Nu2 @ B4 ) ) ) ) ).
% VSagree_supset
thf(fact_246_VSagree__refl,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [Nu: finite_Cartesian_vec @ real @ A,A4: set @ A] : ( denotational_VSagree @ A @ Nu @ Nu @ A4 ) ) ).
% VSagree_refl
thf(fact_247_ids_Oode__to__fvo,axiom,
! [C: $tType,Sz: $tType,Sf: $tType,Sc: $tType,A: $tType,B: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X3: A,I: denota1663640101rp_ext @ B @ C @ A @ product_unit,ODE: ode @ B @ A] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( member @ A @ X3 @ ( denota811733865E_vars @ B @ C @ A @ I @ ODE ) )
=> ( member @ A @ X3 @ ( static_FVO @ B @ A @ ODE ) ) ) ) ) ).
% ids.ode_to_fvo
thf(fact_248_OVsub_H,axiom,
ord_less_eq @ ( set @ ( sum_sum @ c @ c ) ) @ ( sup_sup @ ( set @ ( sum_sum @ c @ c ) ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( oDE_dom @ a @ c @ ode2 ) ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c ) @ ( oDE_dom @ a @ c @ ode2 ) ) ) @ v ).
% OVsub'
thf(fact_249_ids_Ovne23,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Vid2 != Vid3 ) ) ) ).
% ids.vne23
thf(fact_250_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ B2 )
= ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.right_idem
thf(fact_251_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,Y3: A] :
( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ X3 @ Y3 ) )
= ( sup_sup @ A @ X3 @ Y3 ) ) ) ).
% sup_left_idem
thf(fact_252_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] :
( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) )
= ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.left_idem
thf(fact_253_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A] :
( ( sup_sup @ A @ X3 @ X3 )
= X3 ) ) ).
% sup_idem
thf(fact_254_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A] :
( ( sup_sup @ A @ A2 @ A2 )
= A2 ) ) ).
% sup.idem
% Subclasses (1)
thf(subcl_Finite__Set_Ofinite___HOL_Otype,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( type @ A ) ) ).
% Type constructors (81)
thf(tcon_Finite__Cartesian__Product_Ovec___Ordered__Euclidean__Space_Oordered__euclidean__space,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ordere890947078_space @ A9 )
& ( finite_finite @ A10 ) )
=> ( ordere890947078_space @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Ordered__Euclidean__Space_Oordered__euclidean__space_1,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ordere890947078_space @ A9 )
& ( ordere890947078_space @ A10 ) )
=> ( ordere890947078_space @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Real_Oreal___Ordered__Euclidean__Space_Oordered__euclidean__space_2,axiom,
ordere890947078_space @ real ).
thf(tcon_fun___Topological__Spaces_Otopological__space,axiom,
! [A9: $tType,A10: $tType] :
( ( topolo503727757_space @ A10 )
=> ( topolo503727757_space @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A9: $tType,A10: $tType] :
( ( semilattice_sup @ A10 )
=> ( semilattice_sup @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
! [A9: $tType,A10: $tType] :
( ( boolean_algebra @ A10 )
=> ( boolean_algebra @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A9: $tType,A10: $tType] :
( ( preorder @ A10 )
=> ( preorder @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A9: $tType,A10: $tType] :
( ( ( finite_finite @ A9 )
& ( finite_finite @ A10 ) )
=> ( finite_finite @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A9: $tType,A10: $tType] :
( ( order @ A10 )
=> ( order @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A9: $tType,A10: $tType] :
( ( ord @ A10 )
=> ( ord @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A9: $tType,A10: $tType] :
( ( uminus @ A10 )
=> ( uminus @ ( A9 > A10 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_3,axiom,
! [A9: $tType] : ( semilattice_sup @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Lattices_Oboolean__algebra_4,axiom,
! [A9: $tType] : ( boolean_algebra @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_5,axiom,
! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_6,axiom,
! [A9: $tType] :
( ( finite_finite @ A9 )
=> ( finite_finite @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_7,axiom,
! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_8,axiom,
! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_9,axiom,
! [A9: $tType] : ( uminus @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Groups_Ozero,axiom,
! [A9: $tType] :
( ( zero @ A9 )
=> ( zero @ ( set @ A9 ) ) ) ).
thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_10,axiom,
topolo503727757_space @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_11,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_12,axiom,
boolean_algebra @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_13,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_14,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_15,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_16,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_17,axiom,
uminus @ $o ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector,axiom,
real_V55928688vector @ real ).
thf(tcon_Real_Oreal___Lattice__Algebras_Olattice__ab__group__add,axiom,
lattic1601792062up_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_18,axiom,
topolo503727757_space @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring,axiom,
linord1659791738miring @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add @ real ).
thf(tcon_Real_Oreal___Lattices_Osemilattice__sup_19,axiom,
semilattice_sup @ real ).
thf(tcon_Real_Oreal___Orderings_Opreorder_20,axiom,
preorder @ real ).
thf(tcon_Real_Oreal___Orderings_Olinorder_21,axiom,
linorder @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__top,axiom,
no_top @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__bot,axiom,
no_bot @ real ).
thf(tcon_Real_Oreal___Groups_Ogroup__add,axiom,
group_add @ real ).
thf(tcon_Real_Oreal___Orderings_Oorder_22,axiom,
order @ real ).
thf(tcon_Real_Oreal___Orderings_Oord_23,axiom,
ord @ real ).
thf(tcon_Real_Oreal___Groups_Ouminus_24,axiom,
uminus @ real ).
thf(tcon_Real_Oreal___Groups_Ozero_25,axiom,
zero @ real ).
thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_26,axiom,
! [A9: $tType,A10: $tType] :
( ( ( finite_finite @ A9 )
& ( finite_finite @ A10 ) )
=> ( finite_finite @ ( sum_sum @ A9 @ A10 ) ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_27,axiom,
! [A9: $tType] : ( semilattice_sup @ ( filter @ A9 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Opreorder_28,axiom,
! [A9: $tType] : ( preorder @ ( filter @ A9 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder_29,axiom,
! [A9: $tType] : ( order @ ( filter @ A9 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oord_30,axiom,
! [A9: $tType] : ( ord @ ( filter @ A9 ) ) ).
thf(tcon_Product__Type_Oprod___Real__Vector__Spaces_Oreal__normed__vector_31,axiom,
! [A9: $tType,A10: $tType] :
( ( ( real_V55928688vector @ A9 )
& ( real_V55928688vector @ A10 ) )
=> ( real_V55928688vector @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_32,axiom,
! [A9: $tType,A10: $tType] :
( ( ( topolo503727757_space @ A9 )
& ( topolo503727757_space @ A10 ) )
=> ( topolo503727757_space @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Oordered__ab__group__add_33,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ordere890947078_space @ A9 )
& ( ordere890947078_space @ A10 ) )
=> ( ordered_ab_group_add @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Lattices_Osemilattice__sup_34,axiom,
! [A9: $tType,A10: $tType] :
( ( ( semilattice_sup @ A9 )
& ( semilattice_sup @ A10 ) )
=> ( semilattice_sup @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Lattices_Oboolean__algebra_35,axiom,
! [A9: $tType,A10: $tType] :
( ( ( boolean_algebra @ A9 )
& ( boolean_algebra @ A10 ) )
=> ( boolean_algebra @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Opreorder_36,axiom,
! [A9: $tType,A10: $tType] :
( ( ( preorder @ A9 )
& ( preorder @ A10 ) )
=> ( preorder @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_37,axiom,
! [A9: $tType,A10: $tType] :
( ( ( finite_finite @ A9 )
& ( finite_finite @ A10 ) )
=> ( finite_finite @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Ogroup__add_38,axiom,
! [A9: $tType,A10: $tType] :
( ( ( group_add @ A9 )
& ( group_add @ A10 ) )
=> ( group_add @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Oorder_39,axiom,
! [A9: $tType,A10: $tType] :
( ( ( order @ A9 )
& ( order @ A10 ) )
=> ( order @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Oord_40,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ord @ A9 )
& ( ord @ A10 ) )
=> ( ord @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Ouminus_41,axiom,
! [A9: $tType,A10: $tType] :
( ( ( uminus @ A9 )
& ( uminus @ A10 ) )
=> ( uminus @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Ozero_42,axiom,
! [A9: $tType,A10: $tType] :
( ( ( zero @ A9 )
& ( zero @ A10 ) )
=> ( zero @ ( product_prod @ A9 @ A10 ) ) ) ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_43,axiom,
semilattice_sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Oboolean__algebra_44,axiom,
boolean_algebra @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_45,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_46,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_47,axiom,
finite_finite @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_48,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_49,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ouminus_50,axiom,
uminus @ product_unit ).
thf(tcon_Finite__Cartesian__Product_Ovec___Real__Vector__Spaces_Oreal__normed__vector_51,axiom,
! [A9: $tType,A10: $tType] :
( ( ( real_V55928688vector @ A9 )
& ( finite_finite @ A10 ) )
=> ( real_V55928688vector @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Topological__Spaces_Otopological__space_52,axiom,
! [A9: $tType,A10: $tType] :
( ( ( topolo503727757_space @ A9 )
& ( finite_finite @ A10 ) )
=> ( topolo503727757_space @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oordered__ab__group__add_53,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ordere890947078_space @ A9 )
& ( finite_finite @ A10 ) )
=> ( ordered_ab_group_add @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Lattices_Osemilattice__sup_54,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ordere890947078_space @ A9 )
& ( finite_finite @ A10 ) )
=> ( semilattice_sup @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Opreorder_55,axiom,
! [A9: $tType,A10: $tType] :
( ( ( order @ A9 )
& ( finite_finite @ A10 ) )
=> ( preorder @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Olinorder_56,axiom,
! [A9: $tType,A10: $tType] :
( ( ( linorder @ A9 )
& ( cARD_1 @ A10 ) )
=> ( linorder @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Finite__Set_Ofinite_57,axiom,
! [A9: $tType,A10: $tType] :
( ( ( finite_finite @ A9 )
& ( finite_finite @ A10 ) )
=> ( finite_finite @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ogroup__add_58,axiom,
! [A9: $tType,A10: $tType] :
( ( ( group_add @ A9 )
& ( finite_finite @ A10 ) )
=> ( group_add @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Oorder_59,axiom,
! [A9: $tType,A10: $tType] :
( ( ( order @ A9 )
& ( finite_finite @ A10 ) )
=> ( order @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Oord_60,axiom,
! [A9: $tType,A10: $tType] :
( ( ( ord @ A9 )
& ( finite_finite @ A10 ) )
=> ( ord @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ouminus_61,axiom,
! [A9: $tType,A10: $tType] :
( ( ( uminus @ A9 )
& ( finite_finite @ A10 ) )
=> ( uminus @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ozero_62,axiom,
! [A9: $tType,A10: $tType] :
( ( ( zero @ A9 )
& ( finite_finite @ A10 ) )
=> ( zero @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% Free types (3)
thf(tfree_0,hypothesis,
finite_finite @ c ).
thf(tfree_1,hypothesis,
finite_finite @ a ).
thf(tfree_2,hypothesis,
finite_finite @ b ).
% Conjectures (1)
thf(conj_0,conjecture,
denotational_Vagree @ c @ ( denotational_mk_v @ a @ b @ c @ j @ ode2 @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba ) @ ( sol @ s ) ) @ ( denotational_mk_xode @ a @ b @ c @ j @ ode2 @ ( sol @ s ) ) @ ( denotational_semBV @ a @ b @ c @ j @ ode2 ) ).
%------------------------------------------------------------------------------