TPTP Problem File: ITP040^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP040^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Coincidence problem prob_718__7224732_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_718__7224732_1 [Des21]
% Status : ContradictoryAxioms
% Rating : 0.33 v8.2.0, 0.00 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 416 ( 108 unt; 83 typ; 0 def)
% Number of atoms : 987 ( 291 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 7700 ( 76 ~; 11 |; 113 &;7111 @)
% ( 0 <=>; 389 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 8 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 246 ( 246 >; 0 *; 0 +; 0 <<)
% Number of symbols : 80 ( 77 usr; 14 con; 0-7 aty)
% Number of variables : 1358 ( 155 ^;1095 !; 10 ?;1358 :)
% ( 98 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_CAX_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:27:45.025
%------------------------------------------------------------------------------
% Could-be-implicit typings (13)
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_Syntax_OODE,type,
ode: $tType > $tType > $tType ).
thf(ty_t_Real_Oreal,type,
real: $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 (70)
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_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_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_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_Euclidean__Algorithm_Oeuclidean__ring__gcd,type,
euclid1678468529ng_gcd:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Algorithm_Onormalization__euclidean__semiring,type,
euclid1155270486miring:
!>[A: $tType] : $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_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_Oconcrete__v,type,
denota1738237502rete_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_Oconcrete__v__rel,type,
denota2028625979_v_rel:
!>[A: $tType,B: $tType,C: $tType] : ( ( product_prod @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( product_prod @ ( ode @ A @ C ) @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( finite_Cartesian_vec @ real @ C ) ) ) ) > ( product_prod @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( product_prod @ ( ode @ A @ C ) @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( finite_Cartesian_vec @ real @ 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_Orepd,type,
denotational_repd:
!>[C: $tType] : ( ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) > C > real > ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) ) ).
thf(sy_c_Denotational__Semantics_Orepv,type,
denotational_repv:
!>[C: $tType] : ( ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) > C > real > ( 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_Finite__Cartesian__Product_Omap__matrix,type,
finite1614961371matrix:
!>[A: $tType,B: $tType,I: $tType,J: $tType] : ( ( A > B ) > ( finite_Cartesian_vec @ ( finite_Cartesian_vec @ A @ I ) @ J ) > ( finite_Cartesian_vec @ ( finite_Cartesian_vec @ B @ I ) @ J ) ) ).
thf(sy_c_Finite__Cartesian__Product_Ovec_Ovec__lambda,type,
finite1990238425lambda:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( finite_Cartesian_vec @ A @ B ) ) ).
thf(sy_c_Finite__Cartesian__Product_Ovec_Ovec__nth,type,
finite1433825200ec_nth:
!>[A: $tType,B: $tType] : ( ( finite_Cartesian_vec @ A @ B ) > B > A ) ).
thf(sy_c_FuncSet_OPi,type,
pi:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( A > B ) ) ) ).
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_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
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_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_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_Oprod_Ofst,type,
product_fst:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
thf(sy_c_Product__Type_Oprod_Osnd,type,
product_snd:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
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_OatLeastAtMost,type,
set_or331188842AtMost:
!>[A: $tType] : ( A > 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_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_Ofsafe,type,
fsafe:
!>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > $o ) ).
thf(sy_c_Syntax_Oosafe,type,
osafe:
!>[A: $tType,C: $tType] : ( ( ode @ A @ C ) > $o ) ).
thf(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
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_ODE____,type,
ode2: ode @ a @ c ).
thf(sy_v_ODEa____,type,
oDEa: ode @ a @ c ).
thf(sy_v_P____,type,
p: formula @ a @ b @ c ).
thf(sy_v__092_060phi_062_H____,type,
phi: formula @ a @ b @ 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_ba____,type,
ba: finite_Cartesian_vec @ real @ c ).
thf(sy_v_bb____,type,
bb: finite_Cartesian_vec @ real @ c ).
thf(sy_v_i____,type,
i2: c ).
thf(sy_v_sol____,type,
sol: real > ( finite_Cartesian_vec @ real @ c ) ).
thf(sy_v_t____,type,
t: real ).
thf(sy_v_x____,type,
x: real ).
% Relevant facts (255)
thf(fact_0_ode__to__fvo,axiom,
! [C: $tType,A: $tType,B: $tType,X: A,I2: denota1663640101rp_ext @ B @ C @ A @ product_unit,ODE: ode @ B @ A] :
( ( member @ A @ X @ ( denota811733865E_vars @ B @ C @ A @ I2 @ ODE ) )
=> ( member @ A @ X @ ( static_FVO @ B @ A @ ODE ) ) ) ).
% ode_to_fvo
thf(fact_1_osafe,axiom,
osafe @ a @ c @ oDEa ).
% osafe
thf(fact_2_eqP,axiom,
p = phi ).
% eqP
thf(fact_3_x,axiom,
ord_less_eq @ real @ ( zero_zero @ real ) @ x ).
% x
thf(fact_4_fsafe,axiom,
fsafe @ a @ b @ c @ phi ).
% fsafe
thf(fact_5__092_060open_062_IInl_Ai_A_092_060in_062_AsemBV_AI_AODE_J_A_061_A_IInr_Ai_A_092_060in_062_AsemBV_AI_AODE_J_092_060close_062,axiom,
( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ i2 ) @ ( denotational_semBV @ a @ b @ c @ i @ oDEa ) )
= ( member @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c @ i2 ) @ ( denotational_semBV @ a @ b @ c @ i @ oDEa ) ) ) ).
% \<open>(Inl i \<in> semBV I ODE) = (Inr i \<in> semBV I ODE)\<close>
thf(fact_6_VA,axiom,
! [X2: real] :
( denotational_Vagree @ c @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ X2 ) )
@ ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb )
@ ( finite1990238425lambda @ c @ real
@ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ X2 ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) )
@ ( static_FVF @ a @ b @ c @ phi ) ) ).
% VA
thf(fact_7_prod_Ocollapse,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_8_bvo__to__fvo,axiom,
! [A: $tType,B: $tType,X: A,ODE: ode @ B @ A] :
( ( member @ ( sum_sum @ A @ A ) @ ( sum_Inl @ A @ A @ X ) @ ( static_BVO @ B @ A @ ODE ) )
=> ( member @ A @ X @ ( static_FVO @ B @ A @ ODE ) ) ) ).
% bvo_to_fvo
thf(fact_9_vec__lambda__eta,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite @ B )
=> ! [G: finite_Cartesian_vec @ A @ B] :
( ( finite1990238425lambda @ B @ A @ ( finite1433825200ec_nth @ A @ B @ G ) )
= G ) ) ).
% vec_lambda_eta
thf(fact_10_Inl__Inr__False,axiom,
! [A: $tType,B: $tType,X: A,Y: B] :
( ( sum_Inl @ A @ B @ X )
!= ( sum_Inr @ B @ A @ Y ) ) ).
% Inl_Inr_False
thf(fact_11_Inr__Inl__False,axiom,
! [B: $tType,A: $tType,X: B,Y: A] :
( ( sum_Inr @ B @ A @ X )
!= ( sum_Inl @ A @ B @ Y ) ) ).
% Inr_Inl_False
thf(fact_12_vec__lambda__beta,axiom,
! [A: $tType,B: $tType] :
( ( finite_finite @ B )
=> ! [G: B > A,I4: B] :
( ( finite1433825200ec_nth @ A @ B @ ( finite1990238425lambda @ B @ A @ G ) @ I4 )
= ( G @ I4 ) ) ) ).
% vec_lambda_beta
thf(fact_13_alt__sem__lemma,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( finite_finite @ C )
& ( finite_finite @ A )
& ( finite_finite @ B ) )
=> ! [ODE: ode @ A @ C,I2: denota1663640101rp_ext @ A @ B @ C @ product_unit,Sol: real > ( finite_Cartesian_vec @ real @ C ),T2: real,Ab: finite_Cartesian_vec @ real @ C] :
( ( osafe @ A @ C @ ODE )
=> ( ( denotational_ODE_sem @ A @ B @ C @ I2 @ ODE @ ( Sol @ T2 ) )
= ( denotational_ODE_sem @ A @ B @ C @ I2 @ ODE
@ ( finite1990238425lambda @ C @ real
@ ^ [I3: C] : ( if @ real @ ( member @ C @ I3 @ ( static_FVO @ A @ C @ ODE ) ) @ ( finite1433825200ec_nth @ real @ C @ ( Sol @ T2 ) @ I3 ) @ ( finite1433825200ec_nth @ real @ C @ Ab @ I3 ) ) ) ) ) ) ) ).
% alt_sem_lemma
thf(fact_14_snd__zero,axiom,
! [B: $tType,A: $tType] :
( ( ( zero @ A )
& ( zero @ B ) )
=> ( ( product_snd @ B @ A @ ( zero_zero @ ( product_prod @ B @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% snd_zero
thf(fact_15_fst__zero,axiom,
! [B: $tType,A: $tType] :
( ( ( zero @ A )
& ( zero @ B ) )
=> ( ( product_fst @ A @ B @ ( zero_zero @ ( product_prod @ A @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% fst_zero
thf(fact_16_zero__index,axiom,
! [B: $tType,A: $tType] :
( ( ( zero @ A )
& ( finite_finite @ B ) )
=> ! [I4: B] :
( ( finite1433825200ec_nth @ A @ B @ ( zero_zero @ ( finite_Cartesian_vec @ A @ B ) ) @ I4 )
= ( zero_zero @ A ) ) ) ).
% zero_index
thf(fact_17_hpsafe__Evolve_OIH,axiom,
coinci1993344360de_fml @ a @ b @ c @ p ).
% hpsafe_Evolve.IH
thf(fact_18_hpsafe__Evolve_Ohyps_I1_J,axiom,
osafe @ a @ c @ ode2 ).
% hpsafe_Evolve.hyps(1)
thf(fact_19_hpsafe__Evolve_Ohyps_I2_J,axiom,
fsafe @ a @ b @ c @ p ).
% hpsafe_Evolve.hyps(2)
thf(fact_20_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_21_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X1 @ X22 )
= ( product_Pair @ A @ B @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_22_t,axiom,
ord_less_eq @ real @ x @ t ).
% t
thf(fact_23_all,axiom,
! [I5: c] :
( ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ I5 ) @ ( static_BVO @ a @ c @ oDEa ) )
=> ( ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I5 )
= ( finite1433825200ec_nth @ real @ c @ ab @ I5 ) ) )
& ( ( member @ ( sum_sum @ c @ d ) @ ( sum_Inl @ c @ d @ I5 ) @ ( image @ c @ ( sum_sum @ c @ d ) @ ( sum_Inl @ c @ d ) @ ( static_FVO @ a @ c @ oDEa ) ) )
=> ( ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I5 )
= ( finite1433825200ec_nth @ real @ c @ ab @ I5 ) ) )
& ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ I5 ) @ ( static_FVF @ a @ b @ c @ phi ) )
=> ( ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I5 )
= ( finite1433825200ec_nth @ real @ c @ ab @ I5 ) ) ) ) ).
% all
thf(fact_24_less__eq__vec__def,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( finite_finite @ B ) )
=> ( ( ord_less_eq @ ( finite_Cartesian_vec @ A @ B ) )
= ( ^ [X3: finite_Cartesian_vec @ A @ B,Y3: finite_Cartesian_vec @ A @ B] :
! [I3: B] : ( ord_less_eq @ A @ ( finite1433825200ec_nth @ A @ B @ X3 @ I3 ) @ ( finite1433825200ec_nth @ A @ B @ Y3 @ I3 ) ) ) ) ) ).
% less_eq_vec_def
thf(fact_25_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A4: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_26_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A4: A,B4: B] :
( Y
!= ( product_Pair @ A @ B @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_27_prod__induct7,axiom,
! [G2: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) ) )] :
( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G3: G2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G2 ) @ E2 @ ( product_Pair @ F @ G2 @ F2 @ G3 ) ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct7
thf(fact_28_prod__induct6,axiom,
! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( 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 ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct6
thf(fact_29_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A4: A,B4: B,C2: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct5
thf(fact_30_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A4: A,B4: B,C2: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_31_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A4: A,B4: B,C2: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_32_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) ) )] :
~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G3: G2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G2 ) @ E2 @ ( product_Pair @ F @ G2 @ F2 @ G3 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_33_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( 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_34_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A4: A,B4: B,C2: C,D2: D,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_35_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A4: A,B4: B,C2: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_36_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A4: A,B4: B,C2: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) ) ).
% prod_cases3
thf(fact_37_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_38_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A4: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_39_surj__pair,axiom,
! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
? [X4: A,Y4: B] :
( P2
= ( product_Pair @ A @ B @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_40_vec__nth__inject,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite @ B )
=> ! [X: finite_Cartesian_vec @ A @ B,Y: finite_Cartesian_vec @ A @ B] :
( ( ( finite1433825200ec_nth @ A @ B @ X )
= ( finite1433825200ec_nth @ A @ B @ Y ) )
= ( X = Y ) ) ) ).
% vec_nth_inject
thf(fact_41_cond__component,axiom,
! [A: $tType,B: $tType] :
( ( finite_finite @ B )
=> ! [B2: $o,X: finite_Cartesian_vec @ A @ B,Y: finite_Cartesian_vec @ A @ B,I4: B] :
( ( B2
=> ( ( finite1433825200ec_nth @ A @ B @ ( if @ ( finite_Cartesian_vec @ A @ B ) @ B2 @ X @ Y ) @ I4 )
= ( finite1433825200ec_nth @ A @ B @ X @ I4 ) ) )
& ( ~ B2
=> ( ( finite1433825200ec_nth @ A @ B @ ( if @ ( finite_Cartesian_vec @ A @ B ) @ B2 @ X @ Y ) @ I4 )
= ( finite1433825200ec_nth @ A @ B @ Y @ I4 ) ) ) ) ) ).
% cond_component
thf(fact_42_vec__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite @ B )
=> ( ( ^ [Y5: finite_Cartesian_vec @ A @ B,Z: finite_Cartesian_vec @ A @ B] : Y5 = Z )
= ( ^ [X3: finite_Cartesian_vec @ A @ B,Y3: finite_Cartesian_vec @ A @ B] :
! [I3: B] :
( ( finite1433825200ec_nth @ A @ B @ X3 @ I3 )
= ( finite1433825200ec_nth @ A @ B @ Y3 @ I3 ) ) ) ) ) ).
% vec_eq_iff
thf(fact_43_fun__cong__unused__0,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( zero @ B )
=> ! [F3: ( A > B ) > C,G: C] :
( ( F3
= ( ^ [X3: A > B] : G ) )
=> ( ( F3
@ ^ [X3: A] : ( zero_zero @ B ) )
= G ) ) ) ).
% fun_cong_unused_0
thf(fact_44_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_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F3: A > B,G: A > B] :
( ! [X4: A] :
( ( F3 @ X4 )
= ( G @ X4 ) )
=> ( F3 = G ) ) ).
% ext
thf(fact_49_fst__conv,axiom,
! [B: $tType,A: $tType,X1: A,X22: B] :
( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_50_fst__eqD,axiom,
! [B: $tType,A: $tType,X: A,Y: B,A2: A] :
( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_51_snd__conv,axiom,
! [Aa: $tType,A: $tType,X1: Aa,X22: A] :
( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_52_snd__eqD,axiom,
! [B: $tType,A: $tType,X: B,Y: A,A2: A] :
( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_53_prod__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( ^ [Y5: product_prod @ A @ B,Z: product_prod @ A @ B] : Y5 = Z )
= ( ^ [S: product_prod @ A @ B,T3: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ S )
= ( product_fst @ A @ B @ T3 ) )
& ( ( product_snd @ A @ B @ S )
= ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_54_prod_Oexpand,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
( ( ( ( product_fst @ A @ B @ Prod )
= ( product_fst @ A @ B @ Prod2 ) )
& ( ( product_snd @ A @ B @ Prod )
= ( product_snd @ A @ B @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_55_prod__eqI,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q2: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ P2 )
= ( product_fst @ A @ B @ Q2 ) )
=> ( ( ( product_snd @ A @ B @ P2 )
= ( product_snd @ A @ B @ Q2 ) )
=> ( P2 = Q2 ) ) ) ).
% prod_eqI
thf(fact_56_vec__nth__inverse,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite @ B )
=> ! [X: finite_Cartesian_vec @ A @ B] :
( ( finite1990238425lambda @ B @ A @ ( finite1433825200ec_nth @ A @ B @ X ) )
= X ) ) ).
% vec_nth_inverse
thf(fact_57_vec__lambda__unique,axiom,
! [A: $tType,B: $tType] :
( ( finite_finite @ A )
=> ! [F3: finite_Cartesian_vec @ B @ A,G: A > B] :
( ( ! [I3: A] :
( ( finite1433825200ec_nth @ B @ A @ F3 @ I3 )
= ( G @ I3 ) ) )
= ( ( finite1990238425lambda @ A @ B @ G )
= F3 ) ) ) ).
% vec_lambda_unique
thf(fact_58_obj__sumE__f,axiom,
! [A: $tType,C: $tType,B: $tType,S2: B,F3: ( sum_sum @ A @ C ) > B,P: $o] :
( ! [X4: A] :
( ( S2
= ( F3 @ ( sum_Inl @ A @ C @ X4 ) ) )
=> P )
=> ( ! [X4: C] :
( ( S2
= ( F3 @ ( sum_Inr @ C @ A @ X4 ) ) )
=> P )
=> ! [X5: sum_sum @ A @ C] :
( ( S2
= ( F3 @ X5 ) )
=> P ) ) ) ).
% obj_sumE_f
thf(fact_59_zero__vec__def,axiom,
! [A: $tType,B: $tType] :
( ( ( finite_finite @ B )
& ( zero @ A ) )
=> ( ( zero_zero @ ( finite_Cartesian_vec @ A @ B ) )
= ( finite1990238425lambda @ B @ A
@ ^ [I3: B] : ( zero_zero @ A ) ) ) ) ).
% zero_vec_def
thf(fact_60_surjective__pairing,axiom,
! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
( T2
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ).
% surjective_pairing
thf(fact_61_prod_Oexhaust__sel,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_62_mk__v__concrete,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ A ) )
=> ( ( denotational_mk_v @ B @ C @ A )
= ( ^ [I6: denota1663640101rp_ext @ B @ C @ A @ product_unit,ODE2: ode @ B @ A,Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Sol2: finite_Cartesian_vec @ real @ A] :
( product_Pair @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A )
@ ( finite1990238425lambda @ A @ real
@ ^ [I3: A] : ( finite1433825200ec_nth @ real @ A @ ( if @ ( finite_Cartesian_vec @ real @ A ) @ ( member @ ( sum_sum @ A @ A ) @ ( sum_Inl @ A @ A @ I3 ) @ ( denotational_semBV @ B @ C @ A @ I6 @ ODE2 ) ) @ Sol2 @ ( product_fst @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) @ Nu ) ) @ I3 ) )
@ ( finite1990238425lambda @ A @ real
@ ^ [I3: A] : ( finite1433825200ec_nth @ real @ A @ ( if @ ( finite_Cartesian_vec @ real @ A ) @ ( member @ ( sum_sum @ A @ A ) @ ( sum_Inr @ A @ A @ I3 ) @ ( denotational_semBV @ B @ C @ A @ I6 @ ODE2 ) ) @ ( denotational_ODE_sem @ B @ C @ A @ I6 @ ODE2 @ Sol2 ) @ ( product_snd @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) @ Nu ) ) @ I3 ) ) ) ) ) ) ).
% mk_v_concrete
thf(fact_63__092_060open_062Vagree_A_Imk__v_AI_AODE_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_A0_A_E_Ai_Aelse_Aab_A_E_Ai_M_Abb_J_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_Ax_A_E_Ai_Aelse_Aab_A_E_Ai_J_J_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_A0_A_E_Ai_Aelse_Aab_A_E_Ai_M_Abb_J_A_I_N_AsemBV_AI_AODE_J_A_092_060and_062_AVagree_A_Imk__v_AI_AODE_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_A0_A_E_Ai_Aelse_Aab_A_E_Ai_M_Abb_J_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_Ax_A_E_Ai_Aelse_Aab_A_E_Ai_J_J_A_Imk__xode_AI_AODE_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_Ax_A_E_Ai_Aelse_Aab_A_E_Ai_J_J_A_IsemBV_AI_AODE_J_092_060close_062,axiom,
( ( denotational_Vagree @ c
@ ( denotational_mk_v @ a @ b @ c @ i @ oDEa
@ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
@ ( finite1990238425lambda @ c @ real
@ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
@ bb )
@ ( finite1990238425lambda @ c @ real
@ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) )
@ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
@ ( finite1990238425lambda @ c @ real
@ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
@ bb )
@ ( uminus_uminus @ ( set @ ( sum_sum @ c @ c ) ) @ ( denotational_semBV @ a @ b @ c @ i @ oDEa ) ) )
& ( denotational_Vagree @ c
@ ( denotational_mk_v @ a @ b @ c @ i @ oDEa
@ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
@ ( finite1990238425lambda @ c @ real
@ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
@ bb )
@ ( finite1990238425lambda @ c @ real
@ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) )
@ ( denotational_mk_xode @ a @ b @ c @ i @ oDEa
@ ( finite1990238425lambda @ c @ real
@ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) )
@ ( denotational_semBV @ a @ b @ c @ i @ oDEa ) ) ) ).
% \<open>Vagree (mk_v I ODE (\<chi>i. if i \<in> FVO ODE then sol 0 $ i else ab $ i, bb) (\<chi>i. if i \<in> FVO ODE then sol x $ i else ab $ i)) (\<chi>i. if i \<in> FVO ODE then sol 0 $ i else ab $ i, bb) (- semBV I ODE) \<and> Vagree (mk_v I ODE (\<chi>i. if i \<in> FVO ODE then sol 0 $ i else ab $ i, bb) (\<chi>i. if i \<in> FVO ODE then sol x $ i else ab $ i)) (mk_xode I ODE (\<chi>i. if i \<in> FVO ODE then sol x $ i else ab $ i)) (semBV I ODE)\<close>
thf(fact_64_Vagree__def,axiom,
! [C: $tType] :
( ( finite_finite @ C )
=> ( ( denotational_Vagree @ C )
= ( ^ [Nu: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),V: set @ ( sum_sum @ C @ C )] :
( ! [I3: C] :
( ( member @ ( sum_sum @ C @ C ) @ ( sum_Inl @ C @ C @ I3 ) @ V )
=> ( ( finite1433825200ec_nth @ real @ C @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Nu ) @ I3 )
= ( finite1433825200ec_nth @ real @ C @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Nu2 ) @ I3 ) ) )
& ! [I3: C] :
( ( member @ ( sum_sum @ C @ C ) @ ( sum_Inr @ C @ C @ I3 ) @ V )
=> ( ( finite1433825200ec_nth @ real @ C @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Nu ) @ I3 )
= ( finite1433825200ec_nth @ real @ C @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Nu2 ) @ I3 ) ) ) ) ) ) ) ).
% Vagree_def
thf(fact_65__092_060open_062Vagree_A_Imk__v_AI_AODE_A_Iab_M_Abb_J_A_Isol_Ax_J_J_A_Iab_M_Abb_J_A_I_N_AsemBV_AI_AODE_J_A_092_060and_062_AVagree_A_Imk__v_AI_AODE_A_Iab_M_Abb_J_A_Isol_Ax_J_J_A_Imk__xode_AI_AODE_A_Isol_Ax_J_J_A_IsemBV_AI_AODE_J_092_060close_062,axiom,
( ( denotational_Vagree @ c @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ x ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( uminus_uminus @ ( set @ ( sum_sum @ c @ c ) ) @ ( denotational_semBV @ a @ b @ c @ i @ oDEa ) ) )
& ( denotational_Vagree @ c @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ x ) ) @ ( denotational_mk_xode @ a @ b @ c @ i @ oDEa @ ( sol @ x ) ) @ ( denotational_semBV @ a @ b @ c @ i @ oDEa ) ) ) ).
% \<open>Vagree (mk_v I ODE (ab, bb) (sol x)) (ab, bb) (- semBV I ODE) \<and> Vagree (mk_v I ODE (ab, bb) (sol x)) (mk_xode I ODE (sol x)) (semBV I ODE)\<close>
thf(fact_66_concrete__v_Osimps,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ( ( denota1738237502rete_v @ A @ B @ C )
= ( ^ [I6: denota1663640101rp_ext @ A @ B @ C @ product_unit,ODE2: ode @ A @ C,Nu: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Sol2: finite_Cartesian_vec @ real @ C] :
( product_Pair @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )
@ ( finite1990238425lambda @ C @ real
@ ^ [I3: C] : ( finite1433825200ec_nth @ real @ C @ ( if @ ( finite_Cartesian_vec @ real @ C ) @ ( member @ ( sum_sum @ C @ C ) @ ( sum_Inl @ C @ C @ I3 ) @ ( denotational_semBV @ A @ B @ C @ I6 @ ODE2 ) ) @ Sol2 @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Nu ) ) @ I3 ) )
@ ( finite1990238425lambda @ C @ real
@ ^ [I3: C] : ( finite1433825200ec_nth @ real @ C @ ( if @ ( finite_Cartesian_vec @ real @ C ) @ ( member @ ( sum_sum @ C @ C ) @ ( sum_Inr @ C @ C @ I3 ) @ ( denotational_semBV @ A @ B @ C @ I6 @ ODE2 ) ) @ ( denotational_ODE_sem @ A @ B @ C @ I6 @ ODE2 @ Sol2 ) @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Nu ) ) @ I3 ) ) ) ) ) ) ).
% concrete_v.simps
thf(fact_67_concrete__v_Oelims,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( finite_finite @ C )
& ( finite_finite @ A )
& ( finite_finite @ B ) )
=> ! [X: denota1663640101rp_ext @ A @ B @ C @ product_unit,Xa: ode @ A @ C,Xb: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Xc: finite_Cartesian_vec @ real @ C,Y: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] :
( ( ( denota1738237502rete_v @ A @ B @ C @ X @ Xa @ Xb @ Xc )
= Y )
=> ( Y
= ( product_Pair @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )
@ ( finite1990238425lambda @ C @ real
@ ^ [I3: C] : ( finite1433825200ec_nth @ real @ C @ ( if @ ( finite_Cartesian_vec @ real @ C ) @ ( member @ ( sum_sum @ C @ C ) @ ( sum_Inl @ C @ C @ I3 ) @ ( denotational_semBV @ A @ B @ C @ X @ Xa ) ) @ Xc @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Xb ) ) @ I3 ) )
@ ( finite1990238425lambda @ C @ real
@ ^ [I3: C] : ( finite1433825200ec_nth @ real @ C @ ( if @ ( finite_Cartesian_vec @ real @ C ) @ ( member @ ( sum_sum @ C @ C ) @ ( sum_Inr @ C @ C @ I3 ) @ ( denotational_semBV @ A @ B @ C @ X @ Xa ) ) @ ( denotational_ODE_sem @ A @ B @ C @ X @ Xa @ Xc ) @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Xb ) ) @ I3 ) ) ) ) ) ) ).
% concrete_v.elims
thf(fact_68_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_69_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_70_sem,axiom,
! [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 )] :
( ( denotational_Vagree @ c @ Nu3 @ Nu4 @ ( static_FVF @ a @ b @ c @ p ) )
=> ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ Nu3 @ ( denotational_fml_sem @ a @ b @ c @ i @ p ) )
= ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ Nu4 @ ( denotational_fml_sem @ a @ b @ c @ i @ p ) ) ) ) ).
% sem
thf(fact_71_aaba,axiom,
( ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba )
= ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ t ) ) ) ).
% aaba
thf(fact_72_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_73_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
= A2 ) ) ).
% add.inverse_inverse
thf(fact_74_add_Oinverse__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% add.inverse_neutral
thf(fact_75_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_76_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_77_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_78_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_79_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_80_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_81_vector__uminus__component,axiom,
! [A: $tType,B: $tType] :
( ( ( finite_finite @ B )
& ( uminus @ A ) )
=> ! [X: finite_Cartesian_vec @ A @ B,I4: B] :
( ( finite1433825200ec_nth @ A @ B @ ( uminus_uminus @ ( finite_Cartesian_vec @ A @ B ) @ X ) @ I4 )
= ( uminus_uminus @ A @ ( finite1433825200ec_nth @ A @ B @ X @ I4 ) ) ) ) ).
% vector_uminus_component
thf(fact_82_fst__uminus,axiom,
! [B: $tType,A: $tType] :
( ( ( uminus @ A )
& ( uminus @ B ) )
=> ! [X: product_prod @ A @ B] :
( ( product_fst @ A @ B @ ( uminus_uminus @ ( product_prod @ A @ B ) @ X ) )
= ( uminus_uminus @ A @ ( product_fst @ A @ B @ X ) ) ) ) ).
% fst_uminus
thf(fact_83_snd__uminus,axiom,
! [A: $tType,B: $tType] :
( ( ( uminus @ B )
& ( uminus @ A ) )
=> ! [X: product_prod @ B @ A] :
( ( product_snd @ B @ A @ ( uminus_uminus @ ( product_prod @ B @ A ) @ X ) )
= ( uminus_uminus @ A @ ( product_snd @ B @ A @ X ) ) ) ) ).
% snd_uminus
thf(fact_84_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_85_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_86_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_87_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_88_allT,axiom,
! [S3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ S3 )
=> ( ( ord_less_eq @ real @ S3 @ t )
=> ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ S3 ) ) @ ( denotational_fml_sem @ a @ b @ c @ i @ phi ) ) ) ) ).
% allT
thf(fact_89_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_90_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_91_iff__to__impl,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ A ) )
=> ! [Nu5: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),I2: denota1663640101rp_ext @ B @ C @ A @ product_unit,A5: formula @ B @ C @ A,B5: formula @ B @ C @ A] :
( ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu5 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ A5 ) )
= ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu5 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ B5 ) ) )
= ( ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu5 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ A5 ) )
=> ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu5 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ B5 ) ) )
& ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu5 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ B5 ) )
=> ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu5 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ A5 ) ) ) ) ) ) ).
% iff_to_impl
thf(fact_92_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_93_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_94_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_95_agree__supset,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [B5: set @ ( sum_sum @ A @ A ),A5: set @ ( sum_sum @ A @ A ),Nu5: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Nu6: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A )] :
( ( ord_less_eq @ ( set @ ( sum_sum @ A @ A ) ) @ B5 @ A5 )
=> ( ( denotational_Vagree @ A @ Nu5 @ Nu6 @ A5 )
=> ( denotational_Vagree @ A @ Nu5 @ Nu6 @ B5 ) ) ) ) ).
% agree_supset
thf(fact_96_agree__sub,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [A5: set @ ( sum_sum @ A @ A ),B5: set @ ( sum_sum @ A @ A ),Nu5: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Omega: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A )] :
( ( ord_less_eq @ ( set @ ( sum_sum @ A @ A ) ) @ A5 @ B5 )
=> ( ( denotational_Vagree @ A @ Nu5 @ Omega @ B5 )
=> ( denotational_Vagree @ A @ Nu5 @ Omega @ A5 ) ) ) ) ).
% agree_sub
thf(fact_97_mk__v__exists,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( finite_finite @ B )
& ( finite_finite @ A )
& ( finite_finite @ C ) )
=> ! [Nu5: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),I2: denota1663640101rp_ext @ B @ C @ A @ product_unit,ODE: ode @ B @ A,Sol: finite_Cartesian_vec @ real @ A] :
? [Omega2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A )] :
( ( denotational_Vagree @ A @ Omega2 @ Nu5 @ ( uminus_uminus @ ( set @ ( sum_sum @ A @ A ) ) @ ( denotational_semBV @ B @ C @ A @ I2 @ ODE ) ) )
& ( denotational_Vagree @ A @ Omega2 @ ( denotational_mk_xode @ B @ C @ A @ I2 @ ODE @ Sol ) @ ( denotational_semBV @ B @ C @ A @ I2 @ ODE ) ) ) ) ).
% mk_v_exists
thf(fact_98_uminus__vec__def,axiom,
! [B: $tType,A: $tType] :
( ( ( uminus @ A )
& ( finite_finite @ B ) )
=> ( ( uminus_uminus @ ( finite_Cartesian_vec @ A @ B ) )
= ( ^ [X3: finite_Cartesian_vec @ A @ B] :
( finite1990238425lambda @ B @ A
@ ^ [I3: B] : ( uminus_uminus @ A @ ( finite1433825200ec_nth @ A @ B @ X3 @ I3 ) ) ) ) ) ) ).
% uminus_vec_def
thf(fact_99_uminus__prod__def,axiom,
! [B: $tType,A: $tType] :
( ( ( uminus @ A )
& ( uminus @ B ) )
=> ( ( uminus_uminus @ ( product_prod @ A @ B ) )
= ( ^ [X3: product_prod @ A @ B] : ( product_Pair @ A @ B @ ( uminus_uminus @ A @ ( product_fst @ A @ B @ X3 ) ) @ ( uminus_uminus @ B @ ( product_snd @ A @ B @ X3 ) ) ) ) ) ) ).
% uminus_prod_def
thf(fact_100_mk__v__agree,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( finite_finite @ B )
& ( finite_finite @ A )
& ( finite_finite @ C ) )
=> ! [I2: denota1663640101rp_ext @ B @ C @ A @ product_unit,ODE: ode @ B @ A,Nu5: 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 @ I2 @ ODE @ Nu5 @ Sol ) @ Nu5 @ ( uminus_uminus @ ( set @ ( sum_sum @ A @ A ) ) @ ( denotational_semBV @ B @ C @ A @ I2 @ ODE ) ) )
& ( denotational_Vagree @ A @ ( denotational_mk_v @ B @ C @ A @ I2 @ ODE @ Nu5 @ Sol ) @ ( denotational_mk_xode @ B @ C @ A @ I2 @ ODE @ Sol ) @ ( denotational_semBV @ B @ C @ A @ I2 @ ODE ) ) ) ) ).
% mk_v_agree
thf(fact_101_mk__xode_Oelims,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( finite_finite @ C )
& ( finite_finite @ B )
& ( finite_finite @ A ) )
=> ! [X: denota1663640101rp_ext @ A @ B @ C @ product_unit,Xa: ode @ A @ C,Xb: finite_Cartesian_vec @ real @ C,Y: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] :
( ( ( denotational_mk_xode @ A @ B @ C @ X @ Xa @ Xb )
= Y )
=> ( Y
= ( product_Pair @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Xb @ ( denotational_ODE_sem @ A @ B @ C @ X @ Xa @ Xb ) ) ) ) ) ).
% mk_xode.elims
thf(fact_102_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 )
= ( ^ [I6: 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 @ I6 @ ODE2 @ Sol2 ) ) ) ) ) ).
% mk_xode.simps
thf(fact_103_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_104_agree__refl,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [Nu5: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),A5: set @ ( sum_sum @ A @ A )] : ( denotational_Vagree @ A @ Nu5 @ Nu5 @ A5 ) ) ).
% agree_refl
thf(fact_105_agree__comm,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [A5: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),B5: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),V2: set @ ( sum_sum @ A @ A )] :
( ( denotational_Vagree @ A @ A5 @ B5 @ V2 )
=> ( denotational_Vagree @ A @ B5 @ A5 @ V2 ) ) ) ).
% agree_comm
thf(fact_106_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_107_Pair__mono,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [X: A,X6: A,Y: B,Y6: B] :
( ( ord_less_eq @ A @ X @ X6 )
=> ( ( ord_less_eq @ B @ Y @ Y6 )
=> ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( product_Pair @ A @ B @ X6 @ Y6 ) ) ) ) ) ).
% Pair_mono
thf(fact_108_fst__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [X: product_prod @ A @ B,Y: product_prod @ A @ B] :
( ( ord_less_eq @ ( product_prod @ A @ B ) @ X @ Y )
=> ( ord_less_eq @ A @ ( product_fst @ A @ B @ X ) @ ( product_fst @ A @ B @ Y ) ) ) ) ).
% fst_mono
thf(fact_109_snd__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [X: product_prod @ A @ B,Y: product_prod @ A @ B] :
( ( ord_less_eq @ ( product_prod @ A @ B ) @ X @ Y )
=> ( ord_less_eq @ B @ ( product_snd @ A @ B @ X ) @ ( product_snd @ A @ B @ Y ) ) ) ) ).
% snd_mono
thf(fact_110_less__eq__prod__def,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ( ( ord_less_eq @ ( product_prod @ A @ B ) )
= ( ^ [X3: product_prod @ A @ B,Y3: product_prod @ A @ B] :
( ( ord_less_eq @ A @ ( product_fst @ A @ B @ X3 ) @ ( product_fst @ A @ B @ Y3 ) )
& ( ord_less_eq @ B @ ( product_snd @ A @ B @ X3 ) @ ( product_snd @ A @ B @ Y3 ) ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_111_ODE__vars__lr,axiom,
! [Sc: $tType,Sz: $tType,Sf: $tType,X: Sz,I2: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,ODE: ode @ Sf @ Sz] :
( ( member @ ( sum_sum @ Sz @ Sz ) @ ( sum_Inl @ Sz @ Sz @ X ) @ ( denotational_semBV @ Sf @ Sc @ Sz @ I2 @ ODE ) )
= ( member @ ( sum_sum @ Sz @ Sz ) @ ( sum_Inr @ Sz @ Sz @ X ) @ ( denotational_semBV @ Sf @ Sc @ Sz @ I2 @ ODE ) ) ) ).
% ODE_vars_lr
thf(fact_112__092_060open_062_092_060lbrakk_062osafe_AODE_059_Afsafe_A_092_060phi_062_059_AODE_____A_061_AODE_059_AP_A_061_A_092_060phi_062_059_A0_A_092_060le_062_At_059_A_Iaa_M_Aba_J_A_061_Amk__v_AI_AODE_A_Iab_M_Abb_J_A_Isol_At_J_059_A_Isol_Asolves__ode_A_I_092_060lambda_062a_O_AODE__sem_AI_AODE_J_J_A_1230_O_Ot_125_A_123x_O_Amk__v_AI_AODE_A_Iab_M_Abb_J_Ax_A_092_060in_062_Afml__sem_AI_A_092_060phi_062_125_059_AVSagree_A_Isol_A0_J_Aab_A_123uu___O_AInl_Auu___A_092_060in_062_ABVO_AODE_A_092_060or_062_AInl_Auu___A_092_060in_062_AInl_A_096_AFVO_AODE_A_092_060or_062_AInl_Auu___A_092_060in_062_AFVF_A_092_060phi_062_125_092_060rbrakk_062_A_092_060Longrightarrow_062_Aab_A_061_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_A0_A_E_Ai_Aelse_Aab_A_E_Ai_J_092_060close_062,axiom,
( ( osafe @ a @ c @ oDEa )
=> ( ( fsafe @ a @ b @ c @ phi )
=> ( ( ode2 = oDEa )
=> ( ( p = phi )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ t )
=> ( ( ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba )
= ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ t ) ) )
=> ( ( initia1685620758es_ode @ ( finite_Cartesian_vec @ real @ c ) @ sol
@ ^ [X7: real] : ( denotational_ODE_sem @ a @ b @ c @ i @ oDEa )
@ ( set_or331188842AtMost @ real @ ( zero_zero @ real ) @ t )
@ ( collect @ ( finite_Cartesian_vec @ real @ c )
@ ^ [X3: 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 @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ X3 ) @ ( denotational_fml_sem @ a @ b @ c @ i @ phi ) ) ) )
=> ( ( denotational_VSagree @ c @ ( sol @ ( zero_zero @ real ) ) @ ab
@ ( collect @ c
@ ^ [Uu: c] :
( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ Uu ) @ ( static_BVO @ a @ c @ oDEa ) )
| ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ Uu ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( static_FVO @ a @ c @ oDEa ) ) )
| ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ Uu ) @ ( static_FVF @ a @ b @ c @ phi ) ) ) ) )
=> ( ab
= ( finite1990238425lambda @ c @ real
@ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% \<open>\<lbrakk>osafe ODE; fsafe \<phi>; ODE__ = ODE; P = \<phi>; 0 \<le> t; (aa, ba) = mk_v I ODE (ab, bb) (sol t); (sol solves_ode (\<lambda>a. ODE_sem I ODE)) {0..t} {x. mk_v I ODE (ab, bb) x \<in> fml_sem I \<phi>}; VSagree (sol 0) ab {uu_. Inl uu_ \<in> BVO ODE \<or> Inl uu_ \<in> Inl ` FVO ODE \<or> Inl uu_ \<in> FVF \<phi>}\<rbrakk> \<Longrightarrow> ab = (\<chi>i. if i \<in> FVO ODE then sol 0 $ i else ab $ i)\<close>
thf(fact_113__092_060open_062_092_060lbrakk_062osafe_AODE_059_Afsafe_A_092_060phi_062_059_AODE_____A_061_AODE_059_AP_A_061_A_092_060phi_062_059_A0_A_092_060le_062_At_059_A_Iaa_M_Aba_J_A_061_Amk__v_AI_AODE_A_Iab_M_Abb_J_A_Isol_At_J_059_A_Isol_Asolves__ode_A_I_092_060lambda_062a_O_AODE__sem_AI_AODE_J_J_A_1230_O_Ot_125_A_123x_O_Amk__v_AI_AODE_A_Iab_M_Abb_J_Ax_A_092_060in_062_Afml__sem_AI_A_092_060phi_062_125_059_AVSagree_A_Isol_A0_J_Aab_A_123uu___O_AInl_Auu___A_092_060in_062_ABVO_AODE_A_092_060or_062_AInl_Auu___A_092_060in_062_AInl_A_096_AFVO_AODE_A_092_060or_062_AInl_Auu___A_092_060in_062_AFVF_A_092_060phi_062_125_092_060rbrakk_062_A_092_060Longrightarrow_062_Amk__v_AI_AODE_A_Iab_M_Abb_J_A_Isol_At_J_A_061_Amk__v_AI_AODE_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_A0_A_E_Ai_Aelse_Aab_A_E_Ai_M_Abb_J_A_I_092_060chi_062i_O_Aif_Ai_A_092_060in_062_AFVO_AODE_Athen_Asol_At_A_E_Ai_Aelse_Aab_A_E_Ai_J_092_060close_062,axiom,
( ( osafe @ a @ c @ oDEa )
=> ( ( fsafe @ a @ b @ c @ phi )
=> ( ( ode2 = oDEa )
=> ( ( p = phi )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ t )
=> ( ( ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba )
= ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ t ) ) )
=> ( ( initia1685620758es_ode @ ( finite_Cartesian_vec @ real @ c ) @ sol
@ ^ [X7: real] : ( denotational_ODE_sem @ a @ b @ c @ i @ oDEa )
@ ( set_or331188842AtMost @ real @ ( zero_zero @ real ) @ t )
@ ( collect @ ( finite_Cartesian_vec @ real @ c )
@ ^ [X3: 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 @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ X3 ) @ ( denotational_fml_sem @ a @ b @ c @ i @ phi ) ) ) )
=> ( ( denotational_VSagree @ c @ ( sol @ ( zero_zero @ real ) ) @ ab
@ ( collect @ c
@ ^ [Uu: c] :
( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ Uu ) @ ( static_BVO @ a @ c @ oDEa ) )
| ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ Uu ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( static_FVO @ a @ c @ oDEa ) ) )
| ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ Uu ) @ ( static_FVF @ a @ b @ c @ phi ) ) ) ) )
=> ( ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ t ) )
= ( denotational_mk_v @ a @ b @ c @ i @ oDEa
@ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
@ ( finite1990238425lambda @ c @ real
@ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
@ bb )
@ ( finite1990238425lambda @ c @ real
@ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ t ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% \<open>\<lbrakk>osafe ODE; fsafe \<phi>; ODE__ = ODE; P = \<phi>; 0 \<le> t; (aa, ba) = mk_v I ODE (ab, bb) (sol t); (sol solves_ode (\<lambda>a. ODE_sem I ODE)) {0..t} {x. mk_v I ODE (ab, bb) x \<in> fml_sem I \<phi>}; VSagree (sol 0) ab {uu_. Inl uu_ \<in> BVO ODE \<or> Inl uu_ \<in> Inl ` FVO ODE \<or> Inl uu_ \<in> FVF \<phi>}\<rbrakk> \<Longrightarrow> mk_v I ODE (ab, bb) (sol t) = mk_v I ODE (\<chi>i. if i \<in> FVO ODE then sol 0 $ i else ab $ i, bb) (\<chi>i. if i \<in> FVO ODE then sol t $ i else ab $ i)\<close>
thf(fact_114__092_060open_062_092_060lbrakk_062osafe_AODE_059_Afsafe_A_092_060phi_062_059_AODE_____A_061_AODE_059_AP_A_061_A_092_060phi_062_059_A0_A_092_060le_062_At_059_A_Iaa_M_Aba_J_A_061_Amk__v_AI_AODE_A_Iab_M_Abb_J_A_Isol_At_J_059_A_Isol_Asolves__ode_A_I_092_060lambda_062a_O_AODE__sem_AI_AODE_J_J_A_1230_O_Ot_125_A_123x_O_Amk__v_AI_AODE_A_Iab_M_Abb_J_Ax_A_092_060in_062_Afml__sem_AI_A_092_060phi_062_125_059_AVSagree_A_Isol_A0_J_Aab_A_123uu___O_AInl_Auu___A_092_060in_062_ABVO_AODE_A_092_060or_062_AInl_Auu___A_092_060in_062_AInl_A_096_AFVO_AODE_A_092_060or_062_AInl_Auu___A_092_060in_062_AFVF_A_092_060phi_062_125_092_060rbrakk_062_A_092_060Longrightarrow_062_A0_A_092_060le_062_At_092_060close_062,axiom,
( ( osafe @ a @ c @ oDEa )
=> ( ( fsafe @ a @ b @ c @ phi )
=> ( ( ode2 = oDEa )
=> ( ( p = phi )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ t )
=> ( ( ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ aa @ ba )
= ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ t ) ) )
=> ( ( initia1685620758es_ode @ ( finite_Cartesian_vec @ real @ c ) @ sol
@ ^ [X7: real] : ( denotational_ODE_sem @ a @ b @ c @ i @ oDEa )
@ ( set_or331188842AtMost @ real @ ( zero_zero @ real ) @ t )
@ ( collect @ ( finite_Cartesian_vec @ real @ c )
@ ^ [X3: 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 @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ X3 ) @ ( denotational_fml_sem @ a @ b @ c @ i @ phi ) ) ) )
=> ( ( denotational_VSagree @ c @ ( sol @ ( zero_zero @ real ) ) @ ab
@ ( collect @ c
@ ^ [Uu: c] :
( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ Uu ) @ ( static_BVO @ a @ c @ oDEa ) )
| ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ Uu ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( static_FVO @ a @ c @ oDEa ) ) )
| ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ Uu ) @ ( static_FVF @ a @ b @ c @ phi ) ) ) ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ t ) ) ) ) ) ) ) ) ) ).
% \<open>\<lbrakk>osafe ODE; fsafe \<phi>; ODE__ = ODE; P = \<phi>; 0 \<le> t; (aa, ba) = mk_v I ODE (ab, bb) (sol t); (sol solves_ode (\<lambda>a. ODE_sem I ODE)) {0..t} {x. mk_v I ODE (ab, bb) x \<in> fml_sem I \<phi>}; VSagree (sol 0) ab {uu_. Inl uu_ \<in> BVO ODE \<or> Inl uu_ \<in> Inl ` FVO ODE \<or> Inl uu_ \<in> FVF \<phi>}\<rbrakk> \<Longrightarrow> 0 \<le> t\<close>
thf(fact_115_mkV,axiom,
( member @ ( real > ( finite_Cartesian_vec @ real @ c ) ) @ sol
@ ( pi @ real @ ( finite_Cartesian_vec @ real @ c ) @ ( set_or331188842AtMost @ real @ ( zero_zero @ real ) @ t )
@ ^ [Uu: real] :
( collect @ ( finite_Cartesian_vec @ real @ c )
@ ^ [X3: 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 @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ X3 ) @ ( denotational_fml_sem @ a @ b @ c @ i @ phi ) ) ) ) ) ).
% mkV
thf(fact_116_euclid__ext__aux_Ocases,axiom,
! [A: $tType] :
( ( euclid1678468529ng_gcd @ A )
=> ! [X: product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) )] :
~ ! [S4: A,S5: A,T4: A,T5: A,R: A,R2: A] :
( X
!= ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) ) @ S4 @ ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) @ S5 @ ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) @ T4 @ ( product_Pair @ A @ ( product_prod @ A @ A ) @ T5 @ ( product_Pair @ A @ A @ R @ R2 ) ) ) ) ) ) ) ).
% euclid_ext_aux.cases
thf(fact_117_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% compl_le_compl_iff
thf(fact_118_iff__sem,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ A ) )
=> ! [Nu5: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),I2: denota1663640101rp_ext @ B @ C @ A @ product_unit,A5: formula @ B @ C @ A,B5: formula @ B @ C @ A] :
( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu5 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ ( equiv @ B @ C @ A @ A5 @ B5 ) ) )
= ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu5 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ A5 ) )
= ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu5 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ B5 ) ) ) ) ) ).
% iff_sem
thf(fact_119_uminus__apply,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A6: A > B,X3: A] : ( uminus_uminus @ B @ ( A6 @ X3 ) ) ) ) ) ).
% uminus_apply
thf(fact_120_double__compl,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X ) )
= X ) ) ).
% double_compl
thf(fact_121_compl__eq__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X: A,Y: A] :
( ( ( uminus_uminus @ A @ X )
= ( uminus_uminus @ A @ Y ) )
= ( X = Y ) ) ) ).
% compl_eq_compl_iff
thf(fact_122_VSagree__sub,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [A5: set @ A,B5: set @ A,Nu5: finite_Cartesian_vec @ real @ A,Omega: finite_Cartesian_vec @ real @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( denotational_VSagree @ A @ Nu5 @ Omega @ B5 )
=> ( denotational_VSagree @ A @ Nu5 @ Omega @ A5 ) ) ) ) ).
% VSagree_sub
thf(fact_123_VSagree__supset,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [B5: set @ A,A5: set @ A,Nu5: finite_Cartesian_vec @ real @ A,Nu6: finite_Cartesian_vec @ real @ A] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( ( denotational_VSagree @ A @ Nu5 @ Nu6 @ A5 )
=> ( denotational_VSagree @ A @ Nu5 @ Nu6 @ B5 ) ) ) ) ).
% VSagree_supset
thf(fact_124_VSagree__refl,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [Nu5: finite_Cartesian_vec @ real @ A,A5: set @ A] : ( denotational_VSagree @ A @ Nu5 @ Nu5 @ A5 ) ) ).
% VSagree_refl
thf(fact_125_VSagree__def,axiom,
! [C: $tType] :
( ( finite_finite @ C )
=> ( ( denotational_VSagree @ C )
= ( ^ [Nu: finite_Cartesian_vec @ real @ C,Nu2: finite_Cartesian_vec @ real @ C,V: set @ C] :
! [X3: C] :
( ( member @ C @ X3 @ V )
=> ( ( finite1433825200ec_nth @ real @ C @ Nu @ X3 )
= ( finite1433825200ec_nth @ real @ C @ Nu2 @ X3 ) ) ) ) ) ) ).
% VSagree_def
thf(fact_126_gcd_Ocases,axiom,
! [A: $tType] :
( ( euclid1155270486miring @ A )
=> ! [X: product_prod @ A @ A] :
~ ! [A4: A,B4: A] :
( X
!= ( product_Pair @ A @ A @ A4 @ B4 ) ) ) ).
% gcd.cases
thf(fact_127_fun__Compl__def,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A6: A > B,X3: A] : ( uminus_uminus @ B @ ( A6 @ X3 ) ) ) ) ) ).
% fun_Compl_def
thf(fact_128_compl__mono,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% compl_mono
thf(fact_129_compl__le__swap1,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ ( uminus_uminus @ A @ X ) )
=> ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% compl_le_swap1
thf(fact_130_compl__le__swap2,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ X )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% compl_le_swap2
thf(fact_131_image__uminus__atLeastAtMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A] :
( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_or331188842AtMost @ A @ X @ Y ) )
= ( set_or331188842AtMost @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_atLeastAtMost
thf(fact_132_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_133_Compl__subset__Compl__iff,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) @ ( uminus_uminus @ ( set @ A ) @ B5 ) )
= ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ).
% Compl_subset_Compl_iff
thf(fact_134_Compl__anti__mono,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B5 ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) ) ) ).
% Compl_anti_mono
thf(fact_135_image__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F3: B > A,X: B,A5: set @ B] :
( ( B2
= ( F3 @ X ) )
=> ( ( member @ B @ X @ A5 )
=> ( member @ A @ B2 @ ( image @ B @ A @ F3 @ A5 ) ) ) ) ).
% image_eqI
thf(fact_136_subset__antisym,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( A5 = B5 ) ) ) ).
% subset_antisym
thf(fact_137_subsetI,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( member @ A @ X4 @ B5 ) )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% subsetI
thf(fact_138_Compl__eq__Compl__iff,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ( uminus_uminus @ ( set @ A ) @ A5 )
= ( uminus_uminus @ ( set @ A ) @ B5 ) )
= ( A5 = B5 ) ) ).
% Compl_eq_Compl_iff
thf(fact_139_Compl__iff,axiom,
! [A: $tType,C3: A,A5: set @ A] :
( ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
= ( ~ ( member @ A @ C3 @ A5 ) ) ) ).
% Compl_iff
thf(fact_140_ComplI,axiom,
! [A: $tType,C3: A,A5: set @ A] :
( ~ ( member @ A @ C3 @ A5 )
=> ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A5 ) ) ) ).
% ComplI
thf(fact_141_image__ident,axiom,
! [A: $tType,Y7: set @ A] :
( ( image @ A @ A
@ ^ [X3: A] : X3
@ Y7 )
= Y7 ) ).
% image_ident
thf(fact_142_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_143_atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I4: A,L: A,U: A] :
( ( member @ A @ I4 @ ( set_or331188842AtMost @ A @ L @ U ) )
= ( ( ord_less_eq @ A @ L @ I4 )
& ( ord_less_eq @ A @ I4 @ U ) ) ) ) ).
% atLeastAtMost_iff
thf(fact_144_imageI,axiom,
! [B: $tType,A: $tType,X: A,A5: set @ A,F3: A > B] :
( ( member @ A @ X @ A5 )
=> ( member @ B @ ( F3 @ X ) @ ( image @ A @ B @ F3 @ A5 ) ) ) ).
% imageI
thf(fact_145_image__iff,axiom,
! [A: $tType,B: $tType,Z2: A,F3: B > A,A5: set @ B] :
( ( member @ A @ Z2 @ ( image @ B @ A @ F3 @ A5 ) )
= ( ? [X3: B] :
( ( member @ B @ X3 @ A5 )
& ( Z2
= ( F3 @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_146_bex__imageD,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B,P: A > $o] :
( ? [X5: A] :
( ( member @ A @ X5 @ ( image @ B @ A @ F3 @ A5 ) )
& ( P @ X5 ) )
=> ? [X4: B] :
( ( member @ B @ X4 @ A5 )
& ( P @ ( F3 @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_147_image__cong,axiom,
! [B: $tType,A: $tType,M: set @ A,N2: set @ A,F3: A > B,G: A > B] :
( ( M = N2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ N2 )
=> ( ( F3 @ X4 )
= ( G @ X4 ) ) )
=> ( ( image @ A @ B @ F3 @ M )
= ( image @ A @ B @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_148_ball__imageD,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B,P: A > $o] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( image @ B @ A @ F3 @ A5 ) )
=> ( P @ X4 ) )
=> ! [X5: B] :
( ( member @ B @ X5 @ A5 )
=> ( P @ ( F3 @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_149_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X: A,A5: set @ A,B2: B,F3: A > B] :
( ( member @ A @ X @ A5 )
=> ( ( B2
= ( F3 @ X ) )
=> ( member @ B @ B2 @ ( image @ A @ B @ F3 @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_150_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_151_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y5: set @ A,Z: set @ A] : Y5 = 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_152_subset__trans,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% subset_trans
thf(fact_153_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_154_subset__refl,axiom,
! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ A5 ) ).
% subset_refl
thf(fact_155_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
! [T3: A] :
( ( member @ A @ T3 @ A6 )
=> ( member @ A @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_156_equalityD2,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( A5 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ).
% equalityD2
thf(fact_157_equalityD1,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( A5 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% equalityD1
thf(fact_158_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( member @ A @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_159_equalityE,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( A5 = B5 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% equalityE
thf(fact_160_subsetD,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( member @ A @ C3 @ A5 )
=> ( member @ A @ C3 @ B5 ) ) ) ).
% subsetD
thf(fact_161_in__mono,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( member @ A @ X @ A5 )
=> ( member @ A @ X @ B5 ) ) ) ).
% in_mono
thf(fact_162_double__complement,axiom,
! [A: $tType,A5: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
= A5 ) ).
% double_complement
thf(fact_163_ComplD,axiom,
! [A: $tType,C3: A,A5: set @ A] :
( ( member @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
=> ~ ( member @ A @ C3 @ A5 ) ) ).
% ComplD
thf(fact_164_imageE,axiom,
! [A: $tType,B: $tType,B2: A,F3: B > A,A5: set @ B] :
( ( member @ A @ B2 @ ( image @ B @ A @ F3 @ A5 ) )
=> ~ ! [X4: B] :
( ( B2
= ( F3 @ X4 ) )
=> ~ ( member @ B @ X4 @ A5 ) ) ) ).
% imageE
thf(fact_165_image__image,axiom,
! [A: $tType,B: $tType,C: $tType,F3: B > A,G: C > B,A5: set @ C] :
( ( image @ B @ A @ F3 @ ( image @ C @ B @ G @ A5 ) )
= ( image @ C @ A
@ ^ [X3: C] : ( F3 @ ( G @ X3 ) )
@ A5 ) ) ).
% image_image
thf(fact_166_Compr__image__eq,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B,P: A > $o] :
( ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ ( image @ B @ A @ F3 @ A5 ) )
& ( P @ X3 ) ) )
= ( image @ B @ A @ F3
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ A5 )
& ( P @ ( F3 @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_167_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ A6 )
@ ^ [X3: A] : ( member @ A @ X3 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_168_Collect__subset,axiom,
! [A: $tType,A5: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A5 )
& ( P @ X3 ) ) )
@ A5 ) ).
% Collect_subset
thf(fact_169_Collect__neg__eq,axiom,
! [A: $tType,P: A > $o] :
( ( collect @ A
@ ^ [X3: A] :
~ ( P @ X3 ) )
= ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) ) ).
% Collect_neg_eq
thf(fact_170_Compl__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A6: set @ A] :
( collect @ A
@ ^ [X3: A] :
~ ( member @ A @ X3 @ A6 ) ) ) ) ).
% Compl_eq
thf(fact_171_uminus__set__def,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A6: set @ A] :
( collect @ A
@ ( uminus_uminus @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ A6 ) ) ) ) ) ).
% uminus_set_def
thf(fact_172_image__mono,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: set @ A,F3: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A5 ) @ ( image @ A @ B @ F3 @ B5 ) ) ) ).
% image_mono
thf(fact_173_image__subsetI,axiom,
! [A: $tType,B: $tType,A5: set @ A,F3: A > B,B5: set @ B] :
( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( member @ B @ ( F3 @ X4 ) @ B5 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A5 ) @ B5 ) ) ).
% image_subsetI
thf(fact_174_subset__imageE,axiom,
! [A: $tType,B: $tType,B5: set @ A,F3: B > A,A5: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image @ B @ A @ F3 @ A5 ) )
=> ~ ! [C5: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C5 @ A5 )
=> ( B5
!= ( image @ B @ A @ F3 @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_175_image__subset__iff,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F3 @ A5 ) @ B5 )
= ( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( member @ A @ ( F3 @ X3 ) @ B5 ) ) ) ) ).
% image_subset_iff
thf(fact_176_subset__image__iff,axiom,
! [A: $tType,B: $tType,B5: set @ A,F3: B > A,A5: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image @ B @ A @ F3 @ A5 ) )
= ( ? [AA: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ AA @ A5 )
& ( B5
= ( image @ B @ A @ F3 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_177_repd_Oelims,axiom,
! [C: $tType] :
( ( finite_finite @ C )
=> ! [X: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Xa: C,Xb: real,Y: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] :
( ( ( denotational_repd @ C @ X @ Xa @ Xb )
= Y )
=> ( Y
= ( product_Pair @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ X )
@ ( finite1990238425lambda @ C @ real
@ ^ [Y3: C] : ( if @ real @ ( Xa = Y3 ) @ Xb @ ( finite1433825200ec_nth @ real @ C @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ X ) @ Y3 ) ) ) ) ) ) ) ).
% repd.elims
thf(fact_178_repd_Osimps,axiom,
! [C: $tType] :
( ( finite_finite @ C )
=> ( ( denotational_repd @ C )
= ( ^ [V3: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),X3: C,R3: real] :
( product_Pair @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ V3 )
@ ( finite1990238425lambda @ C @ real
@ ^ [Y3: C] : ( if @ real @ ( X3 = Y3 ) @ R3 @ ( finite1433825200ec_nth @ real @ C @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ V3 ) @ Y3 ) ) ) ) ) ) ) ).
% repd.simps
thf(fact_179_nth__map__matrix,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType] :
( ( ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [F3: D > A,X: finite_Cartesian_vec @ ( finite_Cartesian_vec @ D @ B ) @ C,I4: C,J2: B] :
( ( finite1433825200ec_nth @ A @ B @ ( finite1433825200ec_nth @ ( finite_Cartesian_vec @ A @ B ) @ C @ ( finite1614961371matrix @ D @ A @ B @ C @ F3 @ X ) @ I4 ) @ J2 )
= ( F3 @ ( finite1433825200ec_nth @ D @ B @ ( finite1433825200ec_nth @ ( finite_Cartesian_vec @ D @ B ) @ C @ X @ I4 ) @ J2 ) ) ) ) ).
% nth_map_matrix
thf(fact_180_image__subset__iff__funcset,axiom,
! [A: $tType,B: $tType,F4: B > A,A5: set @ B,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F4 @ A5 ) @ B5 )
= ( member @ ( B > A ) @ F4
@ ( pi @ B @ A @ A5
@ ^ [Uu: B] : B5 ) ) ) ).
% image_subset_iff_funcset
thf(fact_181_Pi__I,axiom,
! [B: $tType,A: $tType,A5: set @ A,F3: A > B,B5: A > ( set @ B )] :
( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( member @ B @ ( F3 @ X4 ) @ ( B5 @ X4 ) ) )
=> ( member @ ( A > B ) @ F3 @ ( pi @ A @ B @ A5 @ B5 ) ) ) ).
% Pi_I
thf(fact_182_PiE,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ A,B5: A > ( set @ B ),X: A] :
( ( member @ ( A > B ) @ F3 @ ( pi @ A @ B @ A5 @ B5 ) )
=> ( ~ ( member @ B @ ( F3 @ X ) @ ( B5 @ X ) )
=> ~ ( member @ A @ X @ A5 ) ) ) ).
% PiE
thf(fact_183_Pi__I_H,axiom,
! [B: $tType,A: $tType,A5: set @ A,F3: A > B,B5: A > ( set @ B )] :
( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( member @ B @ ( F3 @ X4 ) @ ( B5 @ X4 ) ) )
=> ( member @ ( A > B ) @ F3 @ ( pi @ A @ B @ A5 @ B5 ) ) ) ).
% Pi_I'
thf(fact_184_Pi__iff,axiom,
! [B: $tType,A: $tType,F3: A > B,I2: set @ A,X8: A > ( set @ B )] :
( ( member @ ( A > B ) @ F3 @ ( pi @ A @ B @ I2 @ X8 ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ I2 )
=> ( member @ B @ ( F3 @ X3 ) @ ( X8 @ X3 ) ) ) ) ) ).
% Pi_iff
thf(fact_185_Pi__mem,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ A,B5: A > ( set @ B ),X: A] :
( ( member @ ( A > B ) @ F3 @ ( pi @ A @ B @ A5 @ B5 ) )
=> ( ( member @ A @ X @ A5 )
=> ( member @ B @ ( F3 @ X ) @ ( B5 @ X ) ) ) ) ).
% Pi_mem
thf(fact_186_Pi__cong,axiom,
! [B: $tType,A: $tType,A5: set @ A,F3: A > B,G: A > B,B5: A > ( set @ B )] :
( ! [W: A] :
( ( member @ A @ W @ A5 )
=> ( ( F3 @ W )
= ( G @ W ) ) )
=> ( ( member @ ( A > B ) @ F3 @ ( pi @ A @ B @ A5 @ B5 ) )
= ( member @ ( A > B ) @ G @ ( pi @ A @ B @ A5 @ B5 ) ) ) ) ).
% Pi_cong
thf(fact_187_funcsetI,axiom,
! [B: $tType,A: $tType,A5: set @ A,F3: A > B,B5: set @ B] :
( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( member @ B @ ( F3 @ X4 ) @ B5 ) )
=> ( member @ ( A > B ) @ F3
@ ( pi @ A @ B @ A5
@ ^ [Uu: A] : B5 ) ) ) ).
% funcsetI
thf(fact_188_funcset__id,axiom,
! [A: $tType,A5: set @ A] :
( member @ ( A > A )
@ ^ [X3: A] : X3
@ ( pi @ A @ A @ A5
@ ^ [Uu: A] : A5 ) ) ).
% funcset_id
thf(fact_189_funcset__mem,axiom,
! [A: $tType,B: $tType,F3: A > B,A5: set @ A,B5: set @ B,X: A] :
( ( member @ ( A > B ) @ F3
@ ( pi @ A @ B @ A5
@ ^ [Uu: A] : B5 ) )
=> ( ( member @ A @ X @ A5 )
=> ( member @ B @ ( F3 @ X ) @ B5 ) ) ) ).
% funcset_mem
thf(fact_190_Pi__mono,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: A > ( set @ B ),C4: A > ( set @ B )] :
( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ ( set @ B ) @ ( B5 @ X4 ) @ ( C4 @ X4 ) ) )
=> ( ord_less_eq @ ( set @ ( A > B ) ) @ ( pi @ A @ B @ A5 @ B5 ) @ ( pi @ A @ B @ A5 @ C4 ) ) ) ).
% Pi_mono
thf(fact_191_Pi__anti__mono,axiom,
! [B: $tType,A: $tType,A7: set @ A,A5: set @ A,B5: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ A5 )
=> ( ord_less_eq @ ( set @ ( A > B ) ) @ ( pi @ A @ B @ A5 @ B5 ) @ ( pi @ A @ B @ A7 @ B5 ) ) ) ).
% Pi_anti_mono
thf(fact_192_map__matrix__def,axiom,
! [I: $tType,J: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ J )
& ( finite_finite @ I ) )
=> ( ( finite1614961371matrix @ A @ B @ I @ J )
= ( ^ [F5: A > B,X3: finite_Cartesian_vec @ ( finite_Cartesian_vec @ A @ I ) @ J] :
( finite1990238425lambda @ J @ ( finite_Cartesian_vec @ B @ I )
@ ^ [I3: J] :
( finite1990238425lambda @ I @ B
@ ^ [J3: I] : ( F5 @ ( finite1433825200ec_nth @ A @ I @ ( finite1433825200ec_nth @ ( finite_Cartesian_vec @ A @ I ) @ J @ X3 @ I3 ) @ J3 ) ) ) ) ) ) ) ).
% map_matrix_def
thf(fact_193_funcset__image,axiom,
! [A: $tType,B: $tType,F3: A > B,A5: set @ A,B5: set @ B] :
( ( member @ ( A > B ) @ F3
@ ( pi @ A @ B @ A5
@ ^ [Uu: A] : B5 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A5 ) @ B5 ) ) ).
% funcset_image
thf(fact_194_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_195_repv_Osimps,axiom,
! [C: $tType] :
( ( finite_finite @ C )
=> ( ( denotational_repv @ C )
= ( ^ [V3: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),X3: C,R3: real] :
( product_Pair @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )
@ ( finite1990238425lambda @ C @ real
@ ^ [Y3: C] : ( if @ real @ ( X3 = Y3 ) @ R3 @ ( finite1433825200ec_nth @ real @ C @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ V3 ) @ Y3 ) ) )
@ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ V3 ) ) ) ) ) ).
% repv.simps
thf(fact_196_repv_Oelims,axiom,
! [C: $tType] :
( ( finite_finite @ C )
=> ! [X: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Xa: C,Xb: real,Y: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] :
( ( ( denotational_repv @ C @ X @ Xa @ Xb )
= Y )
=> ( Y
= ( product_Pair @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )
@ ( finite1990238425lambda @ C @ real
@ ^ [Y3: C] : ( if @ real @ ( Xa = Y3 ) @ Xb @ ( finite1433825200ec_nth @ real @ C @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ X ) @ Y3 ) ) )
@ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ X ) ) ) ) ) ).
% repv.elims
thf(fact_197_exI__realizer,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Y: A,X: B] :
( ( P @ Y @ X )
=> ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_198_conjI__realizer,axiom,
! [A: $tType,B: $tType,P: A > $o,P2: A,Q: B > $o,Q2: B] :
( ( P @ P2 )
=> ( ( Q @ Q2 )
=> ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) )
& ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_199_exE__realizer_H,axiom,
! [A: $tType,B: $tType,P: A > B > $o,P2: product_prod @ B @ A] :
( ( P @ ( product_snd @ B @ A @ P2 ) @ ( product_fst @ B @ A @ P2 ) )
=> ~ ! [X4: B,Y4: A] :
~ ( P @ Y4 @ X4 ) ) ).
% exE_realizer'
thf(fact_200_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P: A > B > $o,X: A,Y: B,A2: product_prod @ A @ B] :
( ( P @ X @ Y )
=> ( ( A2
= ( product_Pair @ A @ B @ X @ Y ) )
=> ( P @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_201_concrete__v_Opelims,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( finite_finite @ C )
& ( finite_finite @ A )
& ( finite_finite @ B ) )
=> ! [X: denota1663640101rp_ext @ A @ B @ C @ product_unit,Xa: ode @ A @ C,Xb: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Xc: finite_Cartesian_vec @ real @ C,Y: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] :
( ( ( denota1738237502rete_v @ A @ B @ C @ X @ Xa @ Xb @ Xc )
= Y )
=> ( ( accp @ ( product_prod @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( product_prod @ ( ode @ A @ C ) @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( finite_Cartesian_vec @ real @ C ) ) ) ) @ ( denota2028625979_v_rel @ A @ B @ C ) @ ( product_Pair @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( product_prod @ ( ode @ A @ C ) @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( finite_Cartesian_vec @ real @ C ) ) ) @ X @ ( product_Pair @ ( ode @ A @ C ) @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Xa @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( finite_Cartesian_vec @ real @ C ) @ Xb @ Xc ) ) ) )
=> ~ ( ( Y
= ( product_Pair @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )
@ ( finite1990238425lambda @ C @ real
@ ^ [I3: C] : ( finite1433825200ec_nth @ real @ C @ ( if @ ( finite_Cartesian_vec @ real @ C ) @ ( member @ ( sum_sum @ C @ C ) @ ( sum_Inl @ C @ C @ I3 ) @ ( denotational_semBV @ A @ B @ C @ X @ Xa ) ) @ Xc @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Xb ) ) @ I3 ) )
@ ( finite1990238425lambda @ C @ real
@ ^ [I3: C] : ( finite1433825200ec_nth @ real @ C @ ( if @ ( finite_Cartesian_vec @ real @ C ) @ ( member @ ( sum_sum @ C @ C ) @ ( sum_Inr @ C @ C @ I3 ) @ ( denotational_semBV @ A @ B @ C @ X @ Xa ) ) @ ( denotational_ODE_sem @ A @ B @ C @ X @ Xa @ Xc ) @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Xb ) ) @ I3 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( product_prod @ ( ode @ A @ C ) @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( finite_Cartesian_vec @ real @ C ) ) ) ) @ ( denota2028625979_v_rel @ A @ B @ C ) @ ( product_Pair @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( product_prod @ ( ode @ A @ C ) @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( finite_Cartesian_vec @ real @ C ) ) ) @ X @ ( product_Pair @ ( ode @ A @ C ) @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Xa @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( finite_Cartesian_vec @ real @ C ) @ Xb @ Xc ) ) ) ) ) ) ) ) ).
% concrete_v.pelims
thf(fact_202_not__arg__cong__Inr,axiom,
! [B: $tType,A: $tType,X: A,Y: A] :
( ( X != Y )
=> ( ( sum_Inr @ A @ B @ X )
!= ( sum_Inr @ A @ B @ Y ) ) ) ).
% not_arg_cong_Inr
thf(fact_203_subset__Collect__iff,axiom,
! [A: $tType,B5: set @ A,A5: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A5 )
& ( P @ X3 ) ) ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ B5 )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_204_subset__CollectI,axiom,
! [A: $tType,B5: set @ A,A5: set @ A,Q: A > $o,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ B5 )
=> ( ( Q @ X4 )
=> ( P @ X4 ) ) )
=> ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ B5 )
& ( Q @ X3 ) ) )
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A5 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_205_obj__sumE,axiom,
! [A: $tType,B: $tType,S2: sum_sum @ A @ B] :
( ! [X4: A] :
( S2
!= ( sum_Inl @ A @ B @ X4 ) )
=> ~ ! [X4: B] :
( S2
!= ( sum_Inr @ B @ A @ X4 ) ) ) ).
% obj_sumE
thf(fact_206_predicate1I,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% predicate1I
thf(fact_207_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_208_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_209_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 )
=> ( ! [X4: B,Y4: B] :
( ( ord_less_eq @ B @ X4 @ Y4 )
=> ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% order_subst1
thf(fact_210_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 )
=> ( ! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ C @ ( F3 @ X4 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq @ C @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% order_subst2
thf(fact_211_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 )
=> ( ! [X4: B,Y4: B] :
( ( ord_less_eq @ B @ X4 @ Y4 )
=> ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_212_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 )
=> ( ! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_213_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z: A] : Y5 = Z )
= ( ^ [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% eq_iff
thf(fact_214_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% antisym
thf(fact_215_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linear
thf(fact_216_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% eq_refl
thf(fact_217_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% le_cases
thf(fact_218_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_219_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_220_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv
thf(fact_221_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z: A] : Y5 = Z )
= ( ^ [A8: A,B7: A] :
( ( ord_less_eq @ A @ A8 @ B7 )
& ( ord_less_eq @ A @ B7 @ A8 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_222_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_223_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_224_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_225_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z2 )
=> ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% order_trans
thf(fact_226_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_227_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: A,B4: A] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_228_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_229_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z: A] : Y5 = Z )
= ( ^ [A8: A,B7: A] :
( ( ord_less_eq @ A @ B7 @ A8 )
& ( ord_less_eq @ A @ A8 @ B7 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_230_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_231_predicate1D,axiom,
! [A: $tType,P: A > $o,Q: A > $o,X: A] :
( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( ( P @ X )
=> ( Q @ X ) ) ) ).
% predicate1D
thf(fact_232_rev__predicate1D,axiom,
! [A: $tType,P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( Q @ X ) ) ) ).
% rev_predicate1D
thf(fact_233_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_234_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_235_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G: A > B] :
( ! [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G @ X4 ) )
=> ( ord_less_eq @ ( A > B ) @ F3 @ G ) ) ) ).
% le_funI
thf(fact_236_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F5: A > B,G4: A > B] :
! [X3: A] : ( ord_less_eq @ B @ ( F5 @ X3 ) @ ( G4 @ X3 ) ) ) ) ) ).
% le_fun_def
thf(fact_237_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_238_old_Osum_Oinject_I1_J,axiom,
! [B: $tType,A: $tType,A2: A,A3: A] :
( ( ( sum_Inl @ A @ B @ A2 )
= ( sum_Inl @ A @ B @ A3 ) )
= ( A2 = A3 ) ) ).
% old.sum.inject(1)
thf(fact_239_sum_Oinject_I1_J,axiom,
! [B: $tType,A: $tType,X1: A,Y1: A] :
( ( ( sum_Inl @ A @ B @ X1 )
= ( sum_Inl @ A @ B @ Y1 ) )
= ( X1 = Y1 ) ) ).
% sum.inject(1)
thf(fact_240_sum_Oinject_I2_J,axiom,
! [A: $tType,B: $tType,X22: B,Y2: B] :
( ( ( sum_Inr @ B @ A @ X22 )
= ( sum_Inr @ B @ A @ Y2 ) )
= ( X22 = Y2 ) ) ).
% sum.inject(2)
thf(fact_241_old_Osum_Oinject_I2_J,axiom,
! [A: $tType,B: $tType,B2: B,B3: B] :
( ( ( sum_Inr @ B @ A @ B2 )
= ( sum_Inr @ B @ A @ B3 ) )
= ( B2 = B3 ) ) ).
% old.sum.inject(2)
thf(fact_242_Inr__inject,axiom,
! [A: $tType,B: $tType,X: B,Y: B] :
( ( ( sum_Inr @ B @ A @ X )
= ( sum_Inr @ B @ A @ Y ) )
=> ( X = Y ) ) ).
% Inr_inject
thf(fact_243_Inl__inject,axiom,
! [B: $tType,A: $tType,X: A,Y: A] :
( ( ( sum_Inl @ A @ B @ X )
= ( sum_Inl @ A @ B @ Y ) )
=> ( X = Y ) ) ).
% Inl_inject
thf(fact_244_sum_Odistinct_I1_J,axiom,
! [A: $tType,B: $tType,X1: A,X22: B] :
( ( sum_Inl @ A @ B @ X1 )
!= ( sum_Inr @ B @ A @ X22 ) ) ).
% sum.distinct(1)
thf(fact_245_old_Osum_Odistinct_I2_J,axiom,
! [B8: $tType,A9: $tType,B9: B8,A10: A9] :
( ( sum_Inr @ B8 @ A9 @ B9 )
!= ( sum_Inl @ A9 @ B8 @ A10 ) ) ).
% old.sum.distinct(2)
thf(fact_246_old_Osum_Odistinct_I1_J,axiom,
! [A: $tType,B: $tType,A2: A,B3: B] :
( ( sum_Inl @ A @ B @ A2 )
!= ( sum_Inr @ B @ A @ B3 ) ) ).
% old.sum.distinct(1)
thf(fact_247_sumE,axiom,
! [A: $tType,B: $tType,S2: sum_sum @ A @ B] :
( ! [X4: A] :
( S2
!= ( sum_Inl @ A @ B @ X4 ) )
=> ~ ! [Y4: B] :
( S2
!= ( sum_Inr @ B @ A @ Y4 ) ) ) ).
% sumE
thf(fact_248_Inr__not__Inl,axiom,
! [B: $tType,A: $tType,B2: B,A2: A] :
( ( sum_Inr @ B @ A @ B2 )
!= ( sum_Inl @ A @ B @ A2 ) ) ).
% Inr_not_Inl
thf(fact_249_split__sum__ex,axiom,
! [B: $tType,A: $tType] :
( ( ^ [P3: ( sum_sum @ A @ B ) > $o] :
? [X9: sum_sum @ A @ B] : ( P3 @ X9 ) )
= ( ^ [P4: ( sum_sum @ A @ B ) > $o] :
( ? [X3: A] : ( P4 @ ( sum_Inl @ A @ B @ X3 ) )
| ? [X3: B] : ( P4 @ ( sum_Inr @ B @ A @ X3 ) ) ) ) ) ).
% split_sum_ex
thf(fact_250_split__sum__all,axiom,
! [B: $tType,A: $tType] :
( ( ^ [P3: ( sum_sum @ A @ B ) > $o] :
! [X9: sum_sum @ A @ B] : ( P3 @ X9 ) )
= ( ^ [P4: ( sum_sum @ A @ B ) > $o] :
( ! [X3: A] : ( P4 @ ( sum_Inl @ A @ B @ X3 ) )
& ! [X3: B] : ( P4 @ ( sum_Inr @ B @ A @ X3 ) ) ) ) ) ).
% split_sum_all
thf(fact_251_old_Osum_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: sum_sum @ A @ B] :
( ! [A4: A] :
( Y
!= ( sum_Inl @ A @ B @ A4 ) )
=> ~ ! [B4: B] :
( Y
!= ( sum_Inr @ B @ A @ B4 ) ) ) ).
% old.sum.exhaust
thf(fact_252_old_Osum_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( sum_sum @ A @ B ) > $o,Sum: sum_sum @ A @ B] :
( ! [A4: A] : ( P @ ( sum_Inl @ A @ B @ A4 ) )
=> ( ! [B4: B] : ( P @ ( sum_Inr @ B @ A @ B4 ) )
=> ( P @ Sum ) ) ) ).
% old.sum.inducts
thf(fact_253_pred__subset__eq,axiom,
! [A: $tType,R4: set @ A,S6: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ R4 )
@ ^ [X3: A] : ( member @ A @ X3 @ S6 ) )
= ( ord_less_eq @ ( set @ A ) @ R4 @ S6 ) ) ).
% pred_subset_eq
thf(fact_254_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)
% Subclasses (1)
thf(subcl_Finite__Set_Ofinite___HOL_Otype,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( type @ A ) ) ).
% Type constructors (59)
thf(tcon_Finite__Cartesian__Product_Ovec___Ordered__Euclidean__Space_Oordered__euclidean__space,axiom,
! [A9: $tType,A11: $tType] :
( ( ( ordere890947078_space @ A9 )
& ( finite_finite @ A11 ) )
=> ( ordere890947078_space @ ( finite_Cartesian_vec @ A9 @ A11 ) ) ) ).
thf(tcon_Product__Type_Oprod___Ordered__Euclidean__Space_Oordered__euclidean__space_1,axiom,
! [A9: $tType,A11: $tType] :
( ( ( ordere890947078_space @ A9 )
& ( ordere890947078_space @ A11 ) )
=> ( ordere890947078_space @ ( product_prod @ A9 @ A11 ) ) ) ).
thf(tcon_Real_Oreal___Ordered__Euclidean__Space_Oordered__euclidean__space_2,axiom,
ordere890947078_space @ real ).
thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
! [A9: $tType,A11: $tType] :
( ( boolean_algebra @ A11 )
=> ( boolean_algebra @ ( A9 > A11 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A9: $tType,A11: $tType] :
( ( preorder @ A11 )
=> ( preorder @ ( A9 > A11 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A9: $tType,A11: $tType] :
( ( ( finite_finite @ A9 )
& ( finite_finite @ A11 ) )
=> ( finite_finite @ ( A9 > A11 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A9: $tType,A11: $tType] :
( ( order @ A11 )
=> ( order @ ( A9 > A11 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A9: $tType,A11: $tType] :
( ( ord @ A11 )
=> ( ord @ ( A9 > A11 ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A9: $tType,A11: $tType] :
( ( uminus @ A11 )
=> ( uminus @ ( A9 > A11 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Oboolean__algebra_3,axiom,
! [A9: $tType] : ( boolean_algebra @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_5,axiom,
! [A9: $tType] :
( ( finite_finite @ A9 )
=> ( finite_finite @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_7,axiom,
! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_8,axiom,
! [A9: $tType] : ( uminus @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Groups_Ozero,axiom,
! [A9: $tType] :
( ( zero @ A9 )
=> ( zero @ ( set @ A9 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_9,axiom,
boolean_algebra @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_10,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_11,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_12,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_13,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_14,axiom,
uminus @ $o ).
thf(tcon_Real_Oreal___Lattice__Algebras_Olattice__ab__group__add,axiom,
lattic1601792062up_add @ 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___Orderings_Opreorder_15,axiom,
preorder @ real ).
thf(tcon_Real_Oreal___Orderings_Olinorder_16,axiom,
linorder @ real ).
thf(tcon_Real_Oreal___Groups_Ogroup__add,axiom,
group_add @ real ).
thf(tcon_Real_Oreal___Orderings_Oorder_17,axiom,
order @ real ).
thf(tcon_Real_Oreal___Orderings_Oord_18,axiom,
ord @ real ).
thf(tcon_Real_Oreal___Groups_Ouminus_19,axiom,
uminus @ real ).
thf(tcon_Real_Oreal___Groups_Ozero_20,axiom,
zero @ real ).
thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_21,axiom,
! [A9: $tType,A11: $tType] :
( ( ( finite_finite @ A9 )
& ( finite_finite @ A11 ) )
=> ( finite_finite @ ( sum_sum @ A9 @ A11 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Oordered__ab__group__add_22,axiom,
! [A9: $tType,A11: $tType] :
( ( ( ordere890947078_space @ A9 )
& ( ordere890947078_space @ A11 ) )
=> ( ordered_ab_group_add @ ( product_prod @ A9 @ A11 ) ) ) ).
thf(tcon_Product__Type_Oprod___Lattices_Oboolean__algebra_23,axiom,
! [A9: $tType,A11: $tType] :
( ( ( boolean_algebra @ A9 )
& ( boolean_algebra @ A11 ) )
=> ( boolean_algebra @ ( product_prod @ A9 @ A11 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Opreorder_24,axiom,
! [A9: $tType,A11: $tType] :
( ( ( preorder @ A9 )
& ( preorder @ A11 ) )
=> ( preorder @ ( product_prod @ A9 @ A11 ) ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_25,axiom,
! [A9: $tType,A11: $tType] :
( ( ( finite_finite @ A9 )
& ( finite_finite @ A11 ) )
=> ( finite_finite @ ( product_prod @ A9 @ A11 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Ogroup__add_26,axiom,
! [A9: $tType,A11: $tType] :
( ( ( group_add @ A9 )
& ( group_add @ A11 ) )
=> ( group_add @ ( product_prod @ A9 @ A11 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Oorder_27,axiom,
! [A9: $tType,A11: $tType] :
( ( ( order @ A9 )
& ( order @ A11 ) )
=> ( order @ ( product_prod @ A9 @ A11 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Oord_28,axiom,
! [A9: $tType,A11: $tType] :
( ( ( ord @ A9 )
& ( ord @ A11 ) )
=> ( ord @ ( product_prod @ A9 @ A11 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Ouminus_29,axiom,
! [A9: $tType,A11: $tType] :
( ( ( uminus @ A9 )
& ( uminus @ A11 ) )
=> ( uminus @ ( product_prod @ A9 @ A11 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Ozero_30,axiom,
! [A9: $tType,A11: $tType] :
( ( ( zero @ A9 )
& ( zero @ A11 ) )
=> ( zero @ ( product_prod @ A9 @ A11 ) ) ) ).
thf(tcon_Product__Type_Ounit___Lattices_Oboolean__algebra_31,axiom,
boolean_algebra @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_32,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_33,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_34,axiom,
finite_finite @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_35,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_36,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ouminus_37,axiom,
uminus @ product_unit ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Oordered__ab__group__add_38,axiom,
! [A9: $tType,A11: $tType] :
( ( ( ordere890947078_space @ A9 )
& ( finite_finite @ A11 ) )
=> ( ordered_ab_group_add @ ( finite_Cartesian_vec @ A9 @ A11 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Opreorder_39,axiom,
! [A9: $tType,A11: $tType] :
( ( ( order @ A9 )
& ( finite_finite @ A11 ) )
=> ( preorder @ ( finite_Cartesian_vec @ A9 @ A11 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Olinorder_40,axiom,
! [A9: $tType,A11: $tType] :
( ( ( linorder @ A9 )
& ( cARD_1 @ A11 ) )
=> ( linorder @ ( finite_Cartesian_vec @ A9 @ A11 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Finite__Set_Ofinite_41,axiom,
! [A9: $tType,A11: $tType] :
( ( ( finite_finite @ A9 )
& ( finite_finite @ A11 ) )
=> ( finite_finite @ ( finite_Cartesian_vec @ A9 @ A11 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ogroup__add_42,axiom,
! [A9: $tType,A11: $tType] :
( ( ( group_add @ A9 )
& ( finite_finite @ A11 ) )
=> ( group_add @ ( finite_Cartesian_vec @ A9 @ A11 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Oorder_43,axiom,
! [A9: $tType,A11: $tType] :
( ( ( order @ A9 )
& ( finite_finite @ A11 ) )
=> ( order @ ( finite_Cartesian_vec @ A9 @ A11 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Oord_44,axiom,
! [A9: $tType,A11: $tType] :
( ( ( ord @ A9 )
& ( finite_finite @ A11 ) )
=> ( ord @ ( finite_Cartesian_vec @ A9 @ A11 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ouminus_45,axiom,
! [A9: $tType,A11: $tType] :
( ( ( uminus @ A9 )
& ( finite_finite @ A11 ) )
=> ( uminus @ ( finite_Cartesian_vec @ A9 @ A11 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ozero_46,axiom,
! [A9: $tType,A11: $tType] :
( ( ( zero @ A9 )
& ( finite_finite @ A11 ) )
=> ( zero @ ( finite_Cartesian_vec @ A9 @ A11 ) ) ) ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
% 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 (12)
thf(conj_0,hypothesis,
member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ i2 ) @ ( static_FVF @ a @ b @ c @ phi ) ).
thf(conj_1,hypothesis,
member @ c @ i2 @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ).
thf(conj_2,hypothesis,
~ ( member @ c @ i2 @ ( static_FVO @ a @ c @ oDEa ) ) ).
thf(conj_3,hypothesis,
( ( ~ ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
& ~ ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) ) )
=> ( ( ( member @ c @ i2 @ ( static_FVO @ a @ c @ oDEa ) )
=> ( ( finite1433825200ec_nth @ real @ c
@ ( product_fst @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
@ ( denotational_mk_v @ a @ b @ c @ i @ oDEa
@ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
@ ( finite1990238425lambda @ c @ real
@ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
@ bb )
@ ( finite1990238425lambda @ c @ real
@ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) ) )
@ i2 )
= ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ i2 ) ) )
& ( ~ ( member @ c @ i2 @ ( static_FVO @ a @ c @ oDEa ) )
=> ( ( finite1433825200ec_nth @ real @ c
@ ( product_fst @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
@ ( denotational_mk_v @ a @ b @ c @ i @ oDEa
@ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
@ ( finite1990238425lambda @ c @ real
@ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
@ bb )
@ ( finite1990238425lambda @ c @ real
@ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) ) )
@ i2 )
= ( finite1433825200ec_nth @ real @ c @ ab @ i2 ) ) ) ) ) ).
thf(conj_4,hypothesis,
( ( ~ ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
& ~ ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) ) )
=> ( ( finite1433825200ec_nth @ real @ c @ ( product_fst @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ x ) ) ) @ i2 )
= ( finite1433825200ec_nth @ real @ c @ ab @ i2 ) ) ) ).
thf(conj_5,hypothesis,
( ( ~ ( member @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
& ~ ( member @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) ) )
=> ( ( finite1433825200ec_nth @ real @ c
@ ( product_snd @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
@ ( denotational_mk_v @ a @ b @ c @ i @ oDEa
@ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
@ ( finite1990238425lambda @ c @ real
@ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
@ bb )
@ ( finite1990238425lambda @ c @ real
@ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) ) )
@ i2 )
= ( finite1433825200ec_nth @ real @ c @ bb @ i2 ) ) ) ).
thf(conj_6,hypothesis,
( ( ~ ( member @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
& ~ ( member @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) ) )
=> ( ( finite1433825200ec_nth @ real @ c @ ( product_snd @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ x ) ) ) @ i2 )
= ( finite1433825200ec_nth @ real @ c @ bb @ i2 ) ) ) ).
thf(conj_7,hypothesis,
( ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
=> ( ( ( member @ c @ i2 @ ( static_FVO @ a @ c @ oDEa ) )
=> ( ( finite1433825200ec_nth @ real @ c
@ ( product_fst @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
@ ( denotational_mk_v @ a @ b @ c @ i @ oDEa
@ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
@ ( finite1990238425lambda @ c @ real
@ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
@ bb )
@ ( finite1990238425lambda @ c @ real
@ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) ) )
@ i2 )
= ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ i2 ) ) )
& ( ~ ( member @ c @ i2 @ ( static_FVO @ a @ c @ oDEa ) )
=> ( ( finite1433825200ec_nth @ real @ c
@ ( product_fst @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
@ ( denotational_mk_v @ a @ b @ c @ i @ oDEa
@ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
@ ( finite1990238425lambda @ c @ real
@ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
@ bb )
@ ( finite1990238425lambda @ c @ real
@ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) ) )
@ i2 )
= ( finite1433825200ec_nth @ real @ c @ ab @ i2 ) ) ) ) )
& ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
=> ( ( ( member @ c @ i2 @ ( static_FVO @ a @ c @ oDEa ) )
=> ( ( finite1433825200ec_nth @ real @ c
@ ( product_fst @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
@ ( denotational_mk_v @ a @ b @ c @ i @ oDEa
@ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
@ ( finite1990238425lambda @ c @ real
@ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
@ bb )
@ ( finite1990238425lambda @ c @ real
@ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) ) )
@ i2 )
= ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ i2 ) ) )
& ( ~ ( member @ c @ i2 @ ( static_FVO @ a @ c @ oDEa ) )
=> ( ( finite1433825200ec_nth @ real @ c
@ ( product_fst @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
@ ( denotational_mk_v @ a @ b @ c @ i @ oDEa
@ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
@ ( finite1990238425lambda @ c @ real
@ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
@ bb )
@ ( finite1990238425lambda @ c @ real
@ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) ) )
@ i2 )
= ( finite1433825200ec_nth @ real @ c @ ab @ i2 ) ) ) ) ) ) ).
thf(conj_8,hypothesis,
( ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
=> ( ( finite1433825200ec_nth @ real @ c
@ ( product_snd @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
@ ( denotational_mk_v @ a @ b @ c @ i @ oDEa
@ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
@ ( finite1990238425lambda @ c @ real
@ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
@ bb )
@ ( finite1990238425lambda @ c @ real
@ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) ) )
@ i2 )
= ( finite1433825200ec_nth @ real @ c
@ ( denotational_ODE_sem @ a @ b @ c @ i @ oDEa
@ ( finite1990238425lambda @ c @ real
@ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) )
@ i2 ) ) )
& ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
=> ( ( finite1433825200ec_nth @ real @ c
@ ( product_snd @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
@ ( denotational_mk_v @ a @ b @ c @ i @ oDEa
@ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )
@ ( finite1990238425lambda @ c @ real
@ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ ( zero_zero @ real ) ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) )
@ bb )
@ ( finite1990238425lambda @ c @ real
@ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) ) )
@ i2 )
= ( finite1433825200ec_nth @ real @ c
@ ( denotational_ODE_sem @ a @ b @ c @ i @ oDEa
@ ( finite1990238425lambda @ c @ real
@ ^ [I3: c] : ( if @ real @ ( member @ c @ I3 @ ( static_FVO @ a @ c @ oDEa ) ) @ ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ I3 ) @ ( finite1433825200ec_nth @ real @ c @ ab @ I3 ) ) ) )
@ i2 ) ) ) ) ).
thf(conj_9,hypothesis,
( ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
=> ( ( finite1433825200ec_nth @ real @ c @ ( product_fst @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ x ) ) ) @ i2 )
= ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ i2 ) ) )
& ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
=> ( ( finite1433825200ec_nth @ real @ c @ ( product_fst @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ x ) ) ) @ i2 )
= ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ i2 ) ) ) ) ).
thf(conj_10,hypothesis,
( ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inl @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
=> ( ( finite1433825200ec_nth @ real @ c @ ( product_snd @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ x ) ) ) @ i2 )
= ( finite1433825200ec_nth @ real @ c @ ( denotational_ODE_sem @ a @ b @ c @ i @ oDEa @ ( sol @ x ) ) @ i2 ) ) )
& ( ( member @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c @ i2 ) @ ( image @ c @ ( sum_sum @ c @ c ) @ ( sum_Inr @ c @ c ) @ ( denota811733865E_vars @ a @ b @ c @ i @ oDEa ) ) )
=> ( ( finite1433825200ec_nth @ real @ c @ ( product_snd @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ( denotational_mk_v @ a @ b @ c @ i @ oDEa @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ ab @ bb ) @ ( sol @ x ) ) ) @ i2 )
= ( finite1433825200ec_nth @ real @ c @ ( denotational_ODE_sem @ a @ b @ c @ i @ oDEa @ ( sol @ x ) ) @ i2 ) ) ) ) ).
thf(conj_11,conjecture,
( ( finite1433825200ec_nth @ real @ c @ ab @ i2 )
= ( finite1433825200ec_nth @ real @ c @ ( sol @ x ) @ i2 ) ) ).
%------------------------------------------------------------------------------