TPTP Problem File: ITP038^2.p
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%------------------------------------------------------------------------------
% File : ITP038^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Coincidence problem prob_231__7211966_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_231__7211966_1 [Des21]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.50 v7.5.0
% Syntax : Number of formulae : 404 ( 102 unt; 80 typ; 0 def)
% Number of atoms : 1201 ( 241 equ; 0 cnn)
% Maximal formula atoms : 16 ( 3 avg)
% Number of connectives : 7244 ( 57 ~; 2 |; 311 &;6415 @)
% ( 0 <=>; 459 =>; 0 <=; 0 <~>)
% Maximal formula depth : 39 ( 12 avg)
% Number of types : 9 ( 8 usr)
% Number of type conns : 314 ( 314 >; 0 *; 0 +; 0 <<)
% Number of symbols : 75 ( 72 usr; 8 con; 0-13 aty)
% Number of variables : 1935 ( 117 ^;1698 !; 15 ?;1935 :)
% ( 105 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:23:15.294
%------------------------------------------------------------------------------
% Could-be-implicit typings (19)
thf(ty_t_Denotational__Semantics_Ointerp_Ointerp__ext,type,
denota1663640101rp_ext: $tType > $tType > $tType > $tType > $tType ).
thf(ty_t_Frechet__Correctness_Oids_Ogood__interp,type,
frechet_good_interp: $tType > $tType > $tType > $tType ).
thf(ty_t_Bounded__Linear__Function_Oblinfun,type,
bounde2145540817linfun: $tType > $tType > $tType ).
thf(ty_t_Frechet__Correctness_Oids_Ostrm,type,
frechet_strm: $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_Otrm,type,
trm: $tType > $tType > $tType ).
thf(ty_t_Real_Oreal,type,
real: $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_tf_sz,type,
sz: $tType ).
thf(ty_tf_sf,type,
sf: $tType ).
thf(ty_tf_sc,type,
sc: $tType ).
thf(ty_tf_c,type,
c: $tType ).
thf(ty_tf_b,type,
b: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (61)
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector,type,
real_V55928688vector:
!>[A: $tType] : $o ).
thf(sy_cl_Cardinality_OCARD__1,type,
cARD_1:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[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_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Otopological__space,type,
topolo503727757_space:
!>[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_BNF__Def_Oeq__onp,type,
bNF_eq_onp:
!>[A: $tType] : ( ( A > $o ) > A > A > $o ) ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
bNF_Greatest_image2:
!>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Bounded__Linear__Function_Oblinfun_OBlinfun,type,
bounde688532126linfun:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( bounde2145540817linfun @ A @ B ) ) ).
thf(sy_c_Denotational__Semantics_OIagree,type,
denotational_Iagree:
!>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) > $o ) ).
thf(sy_c_Denotational__Semantics_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_Odirectional__derivative,type,
denota2078997598vative:
!>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( trm @ A @ C ) > ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) > real ) ).
thf(sy_c_Denotational__Semantics_Ofrechet,type,
denotational_frechet:
!>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( trm @ A @ C ) > ( finite_Cartesian_vec @ real @ C ) > ( finite_Cartesian_vec @ real @ C ) > real ) ).
thf(sy_c_Denotational__Semantics_Ois__interp,type,
denota2077489681interp:
!>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > $o ) ).
thf(sy_c_Denotational__Semantics_Osterm__sem,type,
denota126604975rm_sem:
!>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( trm @ A @ C ) > ( finite_Cartesian_vec @ real @ C ) > real ) ).
thf(sy_c_Frechet__Correctness_Oids_Oblin__frechet,type,
frechet_blin_frechet:
!>[Sf: $tType,Sc: $tType,Sz: $tType] : ( ( frechet_good_interp @ Sf @ Sc @ Sz ) > ( frechet_strm @ Sf @ Sz ) > ( finite_Cartesian_vec @ real @ Sz ) > ( bounde2145540817linfun @ ( finite_Cartesian_vec @ real @ Sz ) @ real ) ) ).
thf(sy_c_Frechet__Correctness_Oids_Ocr__good__interp,type,
freche457001096interp:
!>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( frechet_good_interp @ A @ B @ C ) > $o ) ).
thf(sy_c_Frechet__Correctness_Oids_Ocr__strm,type,
frechet_cr_strm:
!>[A: $tType,B: $tType] : ( ( trm @ A @ B ) > ( frechet_strm @ A @ B ) > $o ) ).
thf(sy_c_Frechet__Correctness_Oids_Ogood__interp_Ogood__interp,type,
freche227871258interp:
!>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( frechet_good_interp @ A @ B @ C ) ) ).
thf(sy_c_Frechet__Correctness_Oids_Ogood__interp_Oraw__interp,type,
freche229654227interp:
!>[A: $tType,B: $tType,C: $tType] : ( ( frechet_good_interp @ A @ B @ C ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) ) ).
thf(sy_c_Frechet__Correctness_Oids_Ostrm_Oraw__term,type,
frechet_raw_term:
!>[A: $tType,C: $tType] : ( ( frechet_strm @ A @ C ) > ( trm @ A @ C ) ) ).
thf(sy_c_Frechet__Correctness_Oids_Ostrm_Osimple__term,type,
frechet_simple_term:
!>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > ( frechet_strm @ A @ C ) ) ).
thf(sy_c_Ids_Oids,type,
ids:
!>[Sz: $tType,Sf: $tType,Sc: $tType] : ( Sz > Sz > Sz > Sf > Sf > Sf > Sc > Sc > Sc > Sc > $o ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Lifting_OQuotient,type,
quotient:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > ( A > B > $o ) > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
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_Static__Semantics_OFVDiff,type,
static_FVDiff:
!>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > ( set @ ( sum_sum @ C @ C ) ) ) ).
thf(sy_c_Static__Semantics_OFVT,type,
static_FVT:
!>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > ( set @ ( sum_sum @ C @ C ) ) ) ).
thf(sy_c_Static__Semantics_OSIGT,type,
static_SIGT:
!>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > ( set @ A ) ) ).
thf(sy_c_Sum__Type_OInl,type,
sum_Inl:
!>[A: $tType,B: $tType] : ( A > ( sum_sum @ A @ B ) ) ).
thf(sy_c_Sum__Type_Oold_Osum_Orec__sum,type,
sum_rec_sum:
!>[A: $tType,T: $tType,B: $tType] : ( ( A > T ) > ( B > T ) > ( sum_sum @ A @ B ) > T ) ).
thf(sy_c_Syntax_OPredicational,type,
predicational:
!>[B: $tType,A: $tType,C: $tType] : ( B > ( formula @ A @ B @ C ) ) ).
thf(sy_c_Syntax_Odfree,type,
dfree:
!>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > $o ) ).
thf(sy_c_Syntax_Oids_OP,type,
p:
!>[Sc: $tType,Sf: $tType,Sz: $tType] : ( Sc > ( formula @ Sf @ Sc @ Sz ) ) ).
thf(sy_c_Syntax_Oids_Oempty,type,
empty:
!>[B: $tType,A: $tType] : ( B > ( trm @ A @ B ) ) ).
thf(sy_c_Syntax_Oids_Of0,type,
f0:
!>[Sf: $tType,Sz: $tType] : ( Sf > ( trm @ Sf @ Sz ) ) ).
thf(sy_c_Syntax_Otrm_OFunction,type,
function:
!>[A: $tType,C: $tType] : ( A > ( C > ( trm @ A @ C ) ) > ( trm @ A @ C ) ) ).
thf(sy_c_Syntax_Otrm_OPlus,type,
plus:
!>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > ( trm @ A @ C ) > ( trm @ A @ C ) ) ).
thf(sy_c_Syntax_Otrm_OTimes,type,
times:
!>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > ( trm @ A @ C ) > ( trm @ A @ C ) ) ).
thf(sy_c_Topological__Spaces_Ocontinuous__on,type,
topolo2071040574ous_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > $o ) ).
thf(sy_c_Typedef_Otype__definition,type,
type_definition:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( A > B ) > ( set @ 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_J,type,
j: denota1663640101rp_ext @ a @ b @ c @ product_unit ).
thf(sy_v__092_060nu_062,type,
nu: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ).
thf(sy_v__092_060nu_062_H,type,
nu2: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ).
thf(sy_v_t1____,type,
t1: trm @ a @ c ).
thf(sy_v_t2____,type,
t2: trm @ a @ c ).
% Relevant facts (256)
thf(fact_0_agree__plus2,axiom,
! [A: $tType,B: $tType] :
( ( finite_finite @ A )
=> ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),T1: trm @ B @ A,T2: trm @ B @ A] :
( ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ ( plus @ B @ A @ T1 @ T2 ) ) )
=> ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ T2 ) ) ) ) ).
% agree_plus2
thf(fact_1_agree,axiom,
denotational_Vagree @ c @ nu @ nu2 @ ( static_FVDiff @ a @ c @ ( plus @ a @ c @ t1 @ t2 ) ) ).
% agree
thf(fact_2_agree1,axiom,
denotational_Vagree @ c @ nu @ nu2 @ ( static_FVDiff @ a @ c @ t1 ) ).
% agree1
thf(fact_3_dfree2,axiom,
dfree @ a @ c @ t2 ).
% dfree2
thf(fact_4_agree__plus1,axiom,
! [A: $tType,B: $tType] :
( ( finite_finite @ A )
=> ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),T1: trm @ B @ A,T2: trm @ B @ A] :
( ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ ( plus @ B @ A @ T1 @ T2 ) ) )
=> ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ T1 ) ) ) ) ).
% agree_plus1
thf(fact_5_agree__comm,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [A2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),B2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),V: set @ ( sum_sum @ A @ A )] :
( ( denotational_Vagree @ A @ A2 @ B2 @ V )
=> ( denotational_Vagree @ A @ B2 @ A2 @ V ) ) ) ).
% agree_comm
thf(fact_6_agree__refl,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),A2: set @ ( sum_sum @ A @ A )] : ( denotational_Vagree @ A @ Nu @ Nu @ A2 ) ) ).
% agree_refl
thf(fact_7_dfree1,axiom,
dfree @ a @ c @ t1 ).
% dfree1
thf(fact_8_raw__interp__inject,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [X: frechet_good_interp @ A @ B @ C,Y: frechet_good_interp @ A @ B @ C] :
( ( ( freche229654227interp @ A @ B @ C @ X )
= ( freche229654227interp @ A @ B @ C @ Y ) )
= ( X = Y ) ) ) ).
% raw_interp_inject
thf(fact_9_raw__term__inject,axiom,
! [C: $tType,A: $tType,X: frechet_strm @ A @ C,Y: frechet_strm @ A @ C] :
( ( ( frechet_raw_term @ A @ C @ X )
= ( frechet_raw_term @ A @ C @ Y ) )
= ( X = Y ) ) ).
% raw_term_inject
thf(fact_10_agree__times1,axiom,
! [A: $tType,B: $tType] :
( ( finite_finite @ A )
=> ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),T1: trm @ B @ A,T2: trm @ B @ A] :
( ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ ( times @ B @ A @ T1 @ T2 ) ) )
=> ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ T1 ) ) ) ) ).
% agree_times1
thf(fact_11_agree__times2,axiom,
! [A: $tType,B: $tType] :
( ( finite_finite @ A )
=> ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),T1: trm @ B @ A,T2: trm @ B @ A] :
( ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ ( times @ B @ A @ T1 @ T2 ) ) )
=> ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ T2 ) ) ) ) ).
% agree_times2
thf(fact_12_agree__func,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite @ A )
=> ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Var: B,Args: A > ( trm @ B @ A ),I: A] :
( ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ ( function @ B @ A @ Var @ Args ) ) )
=> ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ ( Args @ I ) ) ) ) ) ).
% agree_func
thf(fact_13_P__def,axiom,
( ( p @ sc @ sf @ sz )
= ( predicational @ sc @ sf @ sz ) ) ).
% P_def
thf(fact_14_IH2,axiom,
( ( denotational_Vagree @ c @ nu @ nu2 @ ( static_FVDiff @ a @ c @ t2 ) )
=> ( ( denotational_Iagree @ a @ b @ c @ i @ j
@ ( collect @ ( sum_sum @ a @ ( sum_sum @ b @ c ) )
@ ^ [Uu: sum_sum @ a @ ( sum_sum @ b @ c )] :
? [X2: a] :
( ( Uu
= ( sum_Inl @ a @ ( sum_sum @ b @ c ) @ X2 ) )
& ( member @ a @ X2 @ ( static_SIGT @ a @ c @ t2 ) ) ) ) )
=> ( ( denotational_frechet @ a @ b @ c @ i @ t2 @ ( product_fst @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu ) @ ( product_snd @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu ) )
= ( denotational_frechet @ a @ b @ c @ j @ t2 @ ( product_fst @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu2 ) @ ( product_snd @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu2 ) ) ) ) ) ).
% IH2
thf(fact_15_seq__sem_Ocases,axiom,
! [X: product_prod @ ( denota1663640101rp_ext @ sf @ sc @ sz @ product_unit ) @ ( product_prod @ ( list @ ( formula @ sf @ sc @ sz ) ) @ ( list @ ( formula @ sf @ sc @ sz ) ) )] :
~ ! [I2: denota1663640101rp_ext @ sf @ sc @ sz @ product_unit,S: product_prod @ ( list @ ( formula @ sf @ sc @ sz ) ) @ ( list @ ( formula @ sf @ sc @ sz ) )] :
( X
!= ( product_Pair @ ( denota1663640101rp_ext @ sf @ sc @ sz @ product_unit ) @ ( product_prod @ ( list @ ( formula @ sf @ sc @ sz ) ) @ ( list @ ( formula @ sf @ sc @ sz ) ) ) @ I2 @ S ) ) ).
% seq_sem.cases
thf(fact_16_raw__term,axiom,
! [C: $tType,A: $tType,X: frechet_strm @ A @ C] : ( member @ ( trm @ A @ C ) @ ( frechet_raw_term @ A @ C @ X ) @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) ) ).
% raw_term
thf(fact_17_raw__term__cases,axiom,
! [C: $tType,A: $tType,Y: trm @ A @ C] :
( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
=> ~ ! [X3: frechet_strm @ A @ C] :
( Y
!= ( frechet_raw_term @ A @ C @ X3 ) ) ) ).
% raw_term_cases
thf(fact_18_raw__term__induct,axiom,
! [C: $tType,A: $tType,Y: trm @ A @ C,P: ( trm @ A @ C ) > $o] :
( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
=> ( ! [X3: frechet_strm @ A @ C] : ( P @ ( frechet_raw_term @ A @ C @ X3 ) )
=> ( P @ Y ) ) ) ).
% raw_term_induct
thf(fact_19_coincidence__frechet,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( finite_finite @ C )
& ( finite_finite @ A )
& ( finite_finite @ B ) )
=> ! [Theta: trm @ A @ 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 ),I3: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( dfree @ A @ C @ Theta )
=> ( ( denotational_Vagree @ C @ Nu @ Nu2 @ ( static_FVDiff @ A @ C @ Theta ) )
=> ( ( denotational_frechet @ A @ B @ C @ I3 @ Theta @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Nu ) @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Nu ) )
= ( denotational_frechet @ A @ B @ C @ I3 @ Theta @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Nu2 ) @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Nu2 ) ) ) ) ) ) ).
% coincidence_frechet
thf(fact_20_IA,axiom,
( denotational_Iagree @ a @ b @ c @ i @ j
@ ( collect @ ( sum_sum @ a @ ( sum_sum @ b @ c ) )
@ ^ [Uu: sum_sum @ a @ ( sum_sum @ b @ c )] :
? [X2: a] :
( ( Uu
= ( sum_Inl @ a @ ( sum_sum @ b @ c ) @ X2 ) )
& ( member @ a @ X2 @ ( static_SIGT @ a @ c @ ( plus @ a @ c @ t1 @ t2 ) ) ) ) ) ) ).
% IA
thf(fact_21_IA1,axiom,
( denotational_Iagree @ a @ b @ c @ i @ j
@ ( collect @ ( sum_sum @ a @ ( sum_sum @ b @ c ) )
@ ^ [Uu: sum_sum @ a @ ( sum_sum @ b @ c )] :
? [X2: a] :
( ( Uu
= ( sum_Inl @ a @ ( sum_sum @ b @ c ) @ X2 ) )
& ( member @ a @ X2 @ ( static_SIGT @ a @ c @ t1 ) ) ) ) ) ).
% IA1
thf(fact_22_IA2,axiom,
( denotational_Iagree @ a @ b @ c @ i @ j
@ ( collect @ ( sum_sum @ a @ ( sum_sum @ b @ c ) )
@ ^ [Uu: sum_sum @ a @ ( sum_sum @ b @ c )] :
? [X2: a] :
( ( Uu
= ( sum_Inl @ a @ ( sum_sum @ b @ c ) @ X2 ) )
& ( member @ a @ X2 @ ( static_SIGT @ a @ c @ t2 ) ) ) ) ) ).
% IA2
thf(fact_23_IH1,axiom,
( ( denotational_Vagree @ c @ nu @ nu2 @ ( static_FVDiff @ a @ c @ t1 ) )
=> ( ( denotational_Iagree @ a @ b @ c @ i @ j
@ ( collect @ ( sum_sum @ a @ ( sum_sum @ b @ c ) )
@ ^ [Uu: sum_sum @ a @ ( sum_sum @ b @ c )] :
? [X2: a] :
( ( Uu
= ( sum_Inl @ a @ ( sum_sum @ b @ c ) @ X2 ) )
& ( member @ a @ X2 @ ( static_SIGT @ a @ c @ t1 ) ) ) ) )
=> ( ( denotational_frechet @ a @ b @ c @ i @ t1 @ ( product_fst @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu ) @ ( product_snd @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu ) )
= ( denotational_frechet @ a @ b @ c @ j @ t1 @ ( product_fst @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu2 ) @ ( product_snd @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu2 ) ) ) ) ) ).
% IH1
thf(fact_24_Iagree__comm,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [A2: denota1663640101rp_ext @ A @ B @ C @ product_unit,B2: denota1663640101rp_ext @ A @ B @ C @ product_unit,V: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) )] :
( ( denotational_Iagree @ A @ B @ C @ A2 @ B2 @ V )
=> ( denotational_Iagree @ A @ B @ C @ B2 @ A2 @ V ) ) ) ).
% Iagree_comm
thf(fact_25_Iagree__refl,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,A2: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) )] : ( denotational_Iagree @ A @ B @ C @ I3 @ I3 @ A2 ) ) ).
% Iagree_refl
thf(fact_26_FunctionFrechet_Ocases,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ C )
& ( finite_finite @ B ) )
=> ! [X: product_prod @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ A] :
~ ! [I2: denota1663640101rp_ext @ A @ B @ C @ product_unit,I4: A] :
( X
!= ( product_Pair @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ A @ I2 @ I4 ) ) ) ).
% FunctionFrechet.cases
thf(fact_27_subs_I2_J,axiom,
! [E: $tType] :
( ord_less_eq @ ( set @ ( sum_sum @ a @ E ) )
@ ( collect @ ( sum_sum @ a @ E )
@ ^ [Uu: sum_sum @ a @ E] :
? [X2: a] :
( ( Uu
= ( sum_Inl @ a @ E @ X2 ) )
& ( member @ a @ X2 @ ( static_SIGT @ a @ c @ t2 ) ) ) )
@ ( collect @ ( sum_sum @ a @ E )
@ ^ [Uu: sum_sum @ a @ E] :
? [X2: a] :
( ( Uu
= ( sum_Inl @ a @ E @ X2 ) )
& ( member @ a @ X2 @ ( static_SIGT @ a @ c @ ( plus @ a @ c @ t1 @ t2 ) ) ) ) ) ) ).
% subs(2)
thf(fact_28_subs_I1_J,axiom,
! [D: $tType] :
( ord_less_eq @ ( set @ ( sum_sum @ a @ D ) )
@ ( collect @ ( sum_sum @ a @ D )
@ ^ [Uu: sum_sum @ a @ D] :
? [X2: a] :
( ( Uu
= ( sum_Inl @ a @ D @ X2 ) )
& ( member @ a @ X2 @ ( static_SIGT @ a @ c @ t1 ) ) ) )
@ ( collect @ ( sum_sum @ a @ D )
@ ^ [Uu: sum_sum @ a @ D] :
? [X2: a] :
( ( Uu
= ( sum_Inl @ a @ D @ X2 ) )
& ( member @ a @ X2 @ ( static_SIGT @ a @ c @ ( plus @ a @ c @ t1 @ t2 ) ) ) ) ) ) ).
% subs(1)
thf(fact_29_simple__term__inverse,axiom,
! [C: $tType,A: $tType,Y: trm @ A @ C] :
( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
=> ( ( frechet_raw_term @ A @ C @ ( frechet_simple_term @ A @ C @ Y ) )
= Y ) ) ).
% simple_term_inverse
thf(fact_30_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_31_cr__good__interp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ( ( freche457001096interp @ A @ B @ C )
= ( ^ [X2: denota1663640101rp_ext @ A @ B @ C @ product_unit,Y2: frechet_good_interp @ A @ B @ C] :
( X2
= ( freche229654227interp @ A @ B @ C @ Y2 ) ) ) ) ) ).
% cr_good_interp_def
thf(fact_32_cr__strm__def,axiom,
! [B: $tType,A: $tType] :
( ( frechet_cr_strm @ A @ B )
= ( ^ [X2: trm @ A @ B,Y2: frechet_strm @ A @ B] :
( X2
= ( frechet_raw_term @ A @ B @ Y2 ) ) ) ) ).
% cr_strm_def
thf(fact_33_dfree__Times__simps,axiom,
! [B: $tType,A: $tType,A3: trm @ A @ B,B3: trm @ A @ B] :
( ( dfree @ A @ B @ ( times @ A @ B @ A3 @ B3 ) )
= ( ( dfree @ A @ B @ A3 )
& ( dfree @ A @ B @ B3 ) ) ) ).
% dfree_Times_simps
thf(fact_34_dfree__Plus__simps,axiom,
! [B: $tType,A: $tType,A3: trm @ A @ B,B3: trm @ A @ B] :
( ( dfree @ A @ B @ ( plus @ A @ B @ A3 @ B3 ) )
= ( ( dfree @ A @ B @ A3 )
& ( dfree @ A @ B @ B3 ) ) ) ).
% dfree_Plus_simps
thf(fact_35_dfree__Fun__simps,axiom,
! [A: $tType,B: $tType,I: A,Args: B > ( trm @ A @ B )] :
( ( dfree @ A @ B @ ( function @ A @ B @ I @ Args ) )
= ( ! [X2: B] : ( dfree @ A @ B @ ( Args @ X2 ) ) ) ) ).
% dfree_Fun_simps
thf(fact_36_simple__term__cases,axiom,
! [C: $tType,A: $tType,X: frechet_strm @ A @ C] :
~ ! [Y3: trm @ A @ C] :
( ( X
= ( frechet_simple_term @ A @ C @ Y3 ) )
=> ~ ( member @ ( trm @ A @ C ) @ Y3 @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) ) ) ).
% simple_term_cases
thf(fact_37_raw__term__inverse,axiom,
! [C: $tType,A: $tType,X: frechet_strm @ A @ C] :
( ( frechet_simple_term @ A @ C @ ( frechet_raw_term @ A @ C @ X ) )
= X ) ).
% raw_term_inverse
thf(fact_38_simple__term__inject,axiom,
! [C: $tType,A: $tType,X: trm @ A @ C,Y: trm @ A @ C] :
( ( member @ ( trm @ A @ C ) @ X @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
=> ( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
=> ( ( ( frechet_simple_term @ A @ C @ X )
= ( frechet_simple_term @ A @ C @ Y ) )
= ( X = Y ) ) ) ) ).
% simple_term_inject
thf(fact_39_simple__term__induct,axiom,
! [C: $tType,A: $tType,P: ( frechet_strm @ A @ C ) > $o,X: frechet_strm @ A @ C] :
( ! [Y3: trm @ A @ C] :
( ( member @ ( trm @ A @ C ) @ Y3 @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
=> ( P @ ( frechet_simple_term @ A @ C @ Y3 ) ) )
=> ( P @ X ) ) ).
% simple_term_induct
thf(fact_40_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B] :
( ( ( product_Pair @ A @ B @ A3 @ B3 )
= ( product_Pair @ A @ B @ A4 @ B4 ) )
= ( ( A3 = A4 )
& ( B3 = B4 ) ) ) ).
% old.prod.inject
thf(fact_41_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
( ( ( product_Pair @ A @ B @ X1 @ X22 )
= ( product_Pair @ A @ B @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_42_trm_Oinject_I3_J,axiom,
! [A: $tType,C: $tType,X31: A,X32: C > ( trm @ A @ C ),Y31: A,Y32: C > ( trm @ A @ C )] :
( ( ( function @ A @ C @ X31 @ X32 )
= ( function @ A @ C @ Y31 @ Y32 ) )
= ( ( X31 = Y31 )
& ( X32 = Y32 ) ) ) ).
% trm.inject(3)
thf(fact_43_trm_Oinject_I4_J,axiom,
! [C: $tType,A: $tType,X41: trm @ A @ C,X42: trm @ A @ C,Y41: trm @ A @ C,Y42: trm @ A @ C] :
( ( ( plus @ A @ C @ X41 @ X42 )
= ( plus @ A @ C @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% trm.inject(4)
thf(fact_44_trm_Oinject_I5_J,axiom,
! [C: $tType,A: $tType,X51: trm @ A @ C,X52: trm @ A @ C,Y51: trm @ A @ C,Y52: trm @ A @ C] :
( ( ( times @ A @ C @ X51 @ X52 )
= ( times @ A @ C @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% trm.inject(5)
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_agree__sub,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [A2: set @ ( sum_sum @ A @ A ),B2: set @ ( sum_sum @ A @ A ),Nu: 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 ) ) @ A2 @ B2 )
=> ( ( denotational_Vagree @ A @ Nu @ Omega @ B2 )
=> ( denotational_Vagree @ A @ Nu @ Omega @ A2 ) ) ) ) ).
% agree_sub
thf(fact_50_agree__supset,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [B2: set @ ( sum_sum @ A @ A ),A2: set @ ( sum_sum @ A @ A ),Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A )] :
( ( ord_less_eq @ ( set @ ( sum_sum @ A @ A ) ) @ B2 @ A2 )
=> ( ( denotational_Vagree @ A @ Nu @ Nu2 @ A2 )
=> ( denotational_Vagree @ A @ Nu @ Nu2 @ B2 ) ) ) ) ).
% agree_supset
thf(fact_51_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A5: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_52_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A5: A,B5: B] :
( Y
!= ( product_Pair @ A @ B @ A5 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_53_Pair__inject,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B] :
( ( ( product_Pair @ A @ B @ A3 @ B3 )
= ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ~ ( ( A3 = A4 )
=> ( B3 != B4 ) ) ) ).
% Pair_inject
thf(fact_54_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A5: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_55_surj__pair,axiom,
! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
? [X3: A,Y3: B] :
( P2
= ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_56_prod__induct7,axiom,
! [G2: $tType,F2: $tType,E2: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F2 @ G2 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
( ! [A5: A,B5: B,C2: C,D3: D2,E3: E2,F3: F2,G3: G2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F2 @ G2 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F2 @ G2 ) ) ) @ C2 @ ( product_Pair @ D2 @ ( product_prod @ E2 @ ( product_prod @ F2 @ G2 ) ) @ D3 @ ( product_Pair @ E2 @ ( product_prod @ F2 @ G2 ) @ E3 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct7
thf(fact_57_prod__induct6,axiom,
! [F2: $tType,E2: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F2 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F2 ) ) ) )] :
( ! [A5: A,B5: B,C2: C,D3: D2,E3: E2,F3: F2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F2 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F2 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F2 ) ) @ C2 @ ( product_Pair @ D2 @ ( product_prod @ E2 @ F2 ) @ D3 @ ( product_Pair @ E2 @ F2 @ E3 @ F3 ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct6
thf(fact_58_prod__induct5,axiom,
! [E2: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) )] :
( ! [A5: A,B5: B,C2: C,D3: D2,E3: E2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D2 @ E2 ) @ C2 @ ( product_Pair @ D2 @ E2 @ D3 @ E3 ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct5
thf(fact_59_prod__induct4,axiom,
! [D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
( ! [A5: A,B5: B,C2: C,D3: D2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B5 @ ( product_Pair @ C @ D2 @ C2 @ D3 ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_60_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 )] :
( ! [A5: A,B5: B,C2: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C2 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_61_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D2: $tType,E2: $tType,F2: $tType,G2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
~ ! [A5: A,B5: B,C2: C,D3: D2,E3: E2,F3: F2,G3: G2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F2 @ G2 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F2 @ G2 ) ) ) @ C2 @ ( product_Pair @ D2 @ ( product_prod @ E2 @ ( product_prod @ F2 @ G2 ) ) @ D3 @ ( product_Pair @ E2 @ ( product_prod @ F2 @ G2 ) @ E3 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_62_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D2: $tType,E2: $tType,F2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F2 ) ) ) )] :
~ ! [A5: A,B5: B,C2: C,D3: D2,E3: E2,F3: F2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F2 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F2 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F2 ) ) @ C2 @ ( product_Pair @ D2 @ ( product_prod @ E2 @ F2 ) @ D3 @ ( product_Pair @ E2 @ F2 @ E3 @ F3 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_63_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D2: $tType,E2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) )] :
~ ! [A5: A,B5: B,C2: C,D3: D2,E3: E2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D2 @ E2 ) @ C2 @ ( product_Pair @ D2 @ E2 @ D3 @ E3 ) ) ) ) ) ).
% prod_cases5
thf(fact_64_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
~ ! [A5: A,B5: B,C2: C,D3: D2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B5 @ ( product_Pair @ C @ D2 @ C2 @ D3 ) ) ) ) ).
% prod_cases4
thf(fact_65_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A5: A,B5: B,C2: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C2 ) ) ) ).
% prod_cases3
thf(fact_66_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_67_fst__eqD,axiom,
! [B: $tType,A: $tType,X: A,Y: B,A3: A] :
( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
= A3 )
=> ( X = A3 ) ) ).
% fst_eqD
thf(fact_68_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_69_snd__eqD,axiom,
! [B: $tType,A: $tType,X: B,Y: A,A3: A] :
( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
= A3 )
=> ( Y = A3 ) ) ).
% snd_eqD
thf(fact_70_prod__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( ^ [Y4: product_prod @ A @ B,Z: product_prod @ A @ B] : Y4 = Z )
= ( ^ [S2: product_prod @ A @ B,T3: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ S2 )
= ( product_fst @ A @ B @ T3 ) )
& ( ( product_snd @ A @ B @ S2 )
= ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_71_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_72_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_73_dfree__Fun,axiom,
! [A: $tType,C: $tType,Args: C > ( trm @ A @ C ),I: A] :
( ! [I4: C] : ( dfree @ A @ C @ ( Args @ I4 ) )
=> ( dfree @ A @ C @ ( function @ A @ C @ I @ Args ) ) ) ).
% dfree_Fun
thf(fact_74_dfree_Odfree__Plus,axiom,
! [C: $tType,A: $tType,Theta_1: trm @ A @ C,Theta_2: trm @ A @ C] :
( ( dfree @ A @ C @ Theta_1 )
=> ( ( dfree @ A @ C @ Theta_2 )
=> ( dfree @ A @ C @ ( plus @ A @ C @ Theta_1 @ Theta_2 ) ) ) ) ).
% dfree.dfree_Plus
thf(fact_75_dfree__Times,axiom,
! [C: $tType,A: $tType,Theta_1: trm @ A @ C,Theta_2: trm @ A @ C] :
( ( dfree @ A @ C @ Theta_1 )
=> ( ( dfree @ A @ C @ Theta_2 )
=> ( dfree @ A @ C @ ( times @ A @ C @ Theta_1 @ Theta_2 ) ) ) ) ).
% dfree_Times
thf(fact_76_trm_Odistinct_I23_J,axiom,
! [C: $tType,A: $tType,X31: A,X32: C > ( trm @ A @ C ),X41: trm @ A @ C,X42: trm @ A @ C] :
( ( function @ A @ C @ X31 @ X32 )
!= ( plus @ A @ C @ X41 @ X42 ) ) ).
% trm.distinct(23)
thf(fact_77_trm_Odistinct_I25_J,axiom,
! [C: $tType,A: $tType,X31: A,X32: C > ( trm @ A @ C ),X51: trm @ A @ C,X52: trm @ A @ C] :
( ( function @ A @ C @ X31 @ X32 )
!= ( times @ A @ C @ X51 @ X52 ) ) ).
% trm.distinct(25)
thf(fact_78_trm_Odistinct_I31_J,axiom,
! [C: $tType,A: $tType,X41: trm @ A @ C,X42: trm @ A @ C,X51: trm @ A @ C,X52: trm @ A @ C] :
( ( plus @ A @ C @ X41 @ X42 )
!= ( times @ A @ C @ X51 @ X52 ) ) ).
% trm.distinct(31)
thf(fact_79_surjective__pairing,axiom,
! [B: $tType,A: $tType,T4: product_prod @ A @ B] :
( T4
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T4 ) @ ( product_snd @ A @ B @ T4 ) ) ) ).
% surjective_pairing
thf(fact_80_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_81_Pair__le,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B3: B,C3: A,D4: B] :
( ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Pair @ A @ B @ C3 @ D4 ) )
= ( ( ord_less_eq @ A @ A3 @ C3 )
& ( ord_less_eq @ B @ B3 @ D4 ) ) ) ) ).
% Pair_le
thf(fact_82_directional__derivative__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ( ( denota2078997598vative @ A @ B @ C )
= ( ^ [I5: denota1663640101rp_ext @ A @ B @ C @ product_unit,T3: trm @ A @ C,V2: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] : ( denotational_frechet @ A @ B @ C @ I5 @ T3 @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ V2 ) @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ V2 ) ) ) ) ) ).
% directional_derivative_def
thf(fact_83_raw__interp__inverse,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [X: frechet_good_interp @ A @ B @ C] :
( ( freche227871258interp @ A @ B @ C @ ( freche229654227interp @ A @ B @ C @ X ) )
= X ) ) ).
% raw_interp_inverse
thf(fact_84_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B3: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
= ( F1 @ A3 @ B3 ) ) ).
% old.prod.rec
thf(fact_85_type__definition__strm,axiom,
! [C: $tType,A: $tType] : ( type_definition @ ( frechet_strm @ A @ C ) @ ( trm @ A @ C ) @ ( frechet_raw_term @ A @ C ) @ ( frechet_simple_term @ A @ C ) @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) ) ).
% type_definition_strm
thf(fact_86_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P: A > B > $o,X: A,Y: B,A3: product_prod @ A @ B] :
( ( P @ X @ Y )
=> ( ( A3
= ( product_Pair @ A @ B @ X @ Y ) )
=> ( P @ ( product_fst @ A @ B @ A3 ) @ ( product_snd @ A @ B @ A3 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_87_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_88_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_89_less__eq__prod__def,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ( ( ord_less_eq @ ( product_prod @ A @ B ) )
= ( ^ [X2: product_prod @ A @ B,Y2: product_prod @ A @ B] :
( ( ord_less_eq @ A @ ( product_fst @ A @ B @ X2 ) @ ( product_fst @ A @ B @ Y2 ) )
& ( ord_less_eq @ B @ ( product_snd @ A @ B @ X2 ) @ ( product_snd @ A @ B @ Y2 ) ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_90_Iagree__sub,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [A2: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ),B2: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ),I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,J: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( ord_less_eq @ ( set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) @ A2 @ B2 )
=> ( ( denotational_Iagree @ A @ B @ C @ I3 @ J @ B2 )
=> ( denotational_Iagree @ A @ B @ C @ I3 @ J @ A2 ) ) ) ) ).
% Iagree_sub
thf(fact_91_subset__Collect__iff,axiom,
! [A: $tType,B2: set @ A,A2: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A2 )
& ( P @ X2 ) ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ B2 )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_92_subset__CollectI,axiom,
! [A: $tType,B2: set @ A,A2: set @ A,Q: A > $o,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B2 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ B2 )
& ( Q @ X2 ) ) )
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A2 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_93_Pair__mono,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [X: A,X4: A,Y: B,Y5: B] :
( ( ord_less_eq @ A @ X @ X4 )
=> ( ( ord_less_eq @ B @ Y @ Y5 )
=> ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) ) ) ) ) ).
% Pair_mono
thf(fact_94_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_95_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_96_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 ) )
=> ~ ! [X3: B,Y3: A] :
~ ( P @ Y3 @ X3 ) ) ).
% exE_realizer'
thf(fact_97_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 ) ) ) )] :
~ ! [S3: A,S4: A,T5: A,T6: A,R: A,R2: A] :
( X
!= ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) ) @ S3 @ ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) @ S4 @ ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) @ T5 @ ( product_Pair @ A @ ( product_prod @ A @ A ) @ T6 @ ( product_Pair @ A @ A @ R @ R2 ) ) ) ) ) ) ) ).
% euclid_ext_aux.cases
thf(fact_98_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_99_old_Osum_Oinject_I1_J,axiom,
! [B: $tType,A: $tType,A3: A,A4: A] :
( ( ( sum_Inl @ A @ B @ A3 )
= ( sum_Inl @ A @ B @ A4 ) )
= ( A3 = A4 ) ) ).
% old.sum.inject(1)
thf(fact_100_good__interp__inverse,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [Y: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
=> ( ( freche229654227interp @ A @ B @ C @ ( freche227871258interp @ A @ B @ C @ Y ) )
= Y ) ) ) ).
% good_interp_inverse
thf(fact_101_f0__def,axiom,
( ( f0 @ sf @ sz )
= ( ^ [F4: sf] : ( function @ sf @ sz @ F4 @ ( empty @ sz @ sf ) ) ) ) ).
% f0_def
thf(fact_102_subsetI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( member @ A @ X3 @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% subsetI
thf(fact_103_raw__interp__induct,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [Y: denota1663640101rp_ext @ A @ B @ C @ product_unit,P: ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > $o] :
( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
=> ( ! [X3: frechet_good_interp @ A @ B @ C] : ( P @ ( freche229654227interp @ A @ B @ C @ X3 ) )
=> ( P @ Y ) ) ) ) ).
% raw_interp_induct
thf(fact_104_raw__interp__cases,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [Y: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
=> ~ ! [X3: frechet_good_interp @ A @ B @ C] :
( Y
!= ( freche229654227interp @ A @ B @ C @ X3 ) ) ) ) ).
% raw_interp_cases
thf(fact_105_raw__interp,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [X: frechet_good_interp @ A @ B @ C] : ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( freche229654227interp @ A @ B @ C @ X ) @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) ) ) ).
% raw_interp
thf(fact_106_good__interp__cases,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [X: frechet_good_interp @ A @ B @ C] :
~ ! [Y3: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( X
= ( freche227871258interp @ A @ B @ C @ Y3 ) )
=> ~ ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y3 @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) ) ) ) ).
% good_interp_cases
thf(fact_107_good__interp__induct,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [P: ( frechet_good_interp @ A @ B @ C ) > $o,X: frechet_good_interp @ A @ B @ C] :
( ! [Y3: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y3 @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
=> ( P @ ( freche227871258interp @ A @ B @ C @ Y3 ) ) )
=> ( P @ X ) ) ) ).
% good_interp_induct
thf(fact_108_good__interp__inject,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ! [X: denota1663640101rp_ext @ A @ B @ C @ product_unit,Y: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ X @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
=> ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
=> ( ( ( freche227871258interp @ A @ B @ C @ X )
= ( freche227871258interp @ A @ B @ C @ Y ) )
= ( X = Y ) ) ) ) ) ).
% good_interp_inject
thf(fact_109_subset__antisym,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_110_gcd_Ocases,axiom,
! [A: $tType] :
( ( euclid1155270486miring @ A )
=> ! [X: product_prod @ A @ A] :
~ ! [A5: A,B5: A] :
( X
!= ( product_Pair @ A @ A @ A5 @ B5 ) ) ) ).
% gcd.cases
thf(fact_111_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_112_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y4: set @ A,Z: set @ A] : Y4 = Z )
= ( ^ [A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_113_subset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ C4 ) ) ) ).
% subset_trans
thf(fact_114_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_115_subset__refl,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% subset_refl
thf(fact_116_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_117_equalityD2,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_118_equalityD1,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_119_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ( member @ A @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_120_equalityE,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_121_subsetD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ C3 @ A2 )
=> ( member @ A @ C3 @ B2 ) ) ) ).
% subsetD
thf(fact_122_in__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ X @ A2 )
=> ( member @ A @ X @ B2 ) ) ) ).
% in_mono
thf(fact_123_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_124_Collect__subset,axiom,
! [A: $tType,A2: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_125_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A6 )
@ ^ [X2: A] : ( member @ A @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_126_sterm__determines__frechet,axiom,
! [A1: $tType,B1: $tType,B22: $tType,A22: $tType,C: $tType] :
( ( ( finite_finite @ C )
& ( finite_finite @ A22 )
& ( finite_finite @ B22 )
& ( finite_finite @ B1 )
& ( finite_finite @ A1 ) )
=> ! [I3: denota1663640101rp_ext @ A1 @ B1 @ C @ product_unit,J: denota1663640101rp_ext @ A22 @ B22 @ C @ product_unit,Theta_12: trm @ A1 @ C,Theta_22: trm @ A22 @ C,Nu: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] :
( ( denota2077489681interp @ A1 @ B1 @ C @ I3 )
=> ( ( denota2077489681interp @ A22 @ B22 @ C @ J )
=> ( ( dfree @ A1 @ C @ Theta_12 )
=> ( ( dfree @ A22 @ C @ Theta_22 )
=> ( ( ( denota126604975rm_sem @ A1 @ B1 @ C @ I3 @ Theta_12 )
= ( denota126604975rm_sem @ A22 @ B22 @ C @ J @ Theta_22 ) )
=> ( ( denotational_frechet @ A1 @ B1 @ C @ I3 @ Theta_12 @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Nu ) @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Nu ) )
= ( denotational_frechet @ A22 @ B22 @ C @ J @ Theta_22 @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Nu ) @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Nu ) ) ) ) ) ) ) ) ) ).
% sterm_determines_frechet
thf(fact_127_type__definition__good__interp,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ( type_definition @ ( frechet_good_interp @ A @ B @ C ) @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( freche229654227interp @ A @ B @ C ) @ ( freche227871258interp @ A @ B @ C ) @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) ) ) ).
% type_definition_good_interp
thf(fact_128_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_129_old_Osum_Osimps_I7_J,axiom,
! [B: $tType,T: $tType,A: $tType,F1: A > T,F22: B > T,A3: A] :
( ( sum_rec_sum @ A @ T @ B @ F1 @ F22 @ ( sum_Inl @ A @ B @ A3 ) )
= ( F1 @ A3 ) ) ).
% old.sum.simps(7)
thf(fact_130_image2__def,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( bNF_Greatest_image2 @ C @ A @ B )
= ( ^ [A6: set @ C,F4: C > A,G4: C > B] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu: product_prod @ A @ B] :
? [A7: C] :
( ( Uu
= ( product_Pair @ A @ B @ ( F4 @ A7 ) @ ( G4 @ A7 ) ) )
& ( member @ C @ A7 @ A6 ) ) ) ) ) ).
% image2_def
thf(fact_131_predicate1I,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% predicate1I
thf(fact_132_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_133_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_134_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_135_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_136_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_137_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F4: A > B,G4: A > B] :
! [X2: A] : ( ord_less_eq @ B @ ( F4 @ X2 ) @ ( G4 @ X2 ) ) ) ) ) ).
% le_fun_def
thf(fact_138_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F: B > A,B3: B,C3: B] :
( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C3 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% order_subst1
thf(fact_139_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B3: A,F: A > C,C3: C] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ C @ ( F @ B3 ) @ C3 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ C @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% order_subst2
thf(fact_140_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F: B > A,B3: B,C3: B] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C3 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_141_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B3: A,F: A > B,C3: B] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C3 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ B @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_142_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
& ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ) ) ).
% eq_iff
thf(fact_143_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_144_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_145_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% eq_refl
thf(fact_146_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_147_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% order.trans
thf(fact_148_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_149_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_150_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [A7: A,B7: A] :
( ( ord_less_eq @ A @ A7 @ B7 )
& ( ord_less_eq @ A @ B7 @ A7 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_151_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( A3 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_152_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( B3 = C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_153_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ) ).
% order_class.order.antisym
thf(fact_154_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_155_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_156_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B3: A] :
( ! [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: A,B5: A] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_157_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ B3 )
=> ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_158_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [A7: A,B7: A] :
( ( ord_less_eq @ A @ B7 @ A7 )
& ( ord_less_eq @ A @ A7 @ B7 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_159_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_160_image2__eqI,axiom,
! [A: $tType,C: $tType,B: $tType,B3: A,F: B > A,X: B,C3: C,G: B > C,A2: set @ B] :
( ( B3
= ( F @ X ) )
=> ( ( C3
= ( G @ X ) )
=> ( ( member @ B @ X @ A2 )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B3 @ C3 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A2 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_161_coincidence__sterm_H,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ A ) )
=> ! [Theta: trm @ A @ B,Nu: product_prod @ ( finite_Cartesian_vec @ real @ B ) @ ( finite_Cartesian_vec @ real @ B ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ B ) @ ( finite_Cartesian_vec @ real @ B ),I3: denota1663640101rp_ext @ A @ C @ B @ product_unit,J: denota1663640101rp_ext @ A @ C @ B @ product_unit] :
( ( dfree @ A @ B @ Theta )
=> ( ( denotational_Vagree @ B @ Nu @ Nu2 @ ( static_FVT @ A @ B @ Theta ) )
=> ( ( denotational_Iagree @ A @ C @ B @ I3 @ J
@ ( collect @ ( sum_sum @ A @ ( sum_sum @ C @ B ) )
@ ^ [Uu: sum_sum @ A @ ( sum_sum @ C @ B )] :
? [X2: A] :
( ( Uu
= ( sum_Inl @ A @ ( sum_sum @ C @ B ) @ X2 ) )
& ( member @ A @ X2 @ ( static_SIGT @ A @ B @ Theta ) ) ) ) )
=> ( ( denota126604975rm_sem @ A @ C @ B @ I3 @ Theta @ ( product_fst @ ( finite_Cartesian_vec @ real @ B ) @ ( finite_Cartesian_vec @ real @ B ) @ Nu ) )
= ( denota126604975rm_sem @ A @ C @ B @ J @ Theta @ ( product_fst @ ( finite_Cartesian_vec @ real @ B ) @ ( finite_Cartesian_vec @ real @ B ) @ Nu2 ) ) ) ) ) ) ) ).
% coincidence_sterm'
thf(fact_162_coincidence__sterm,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ C )
& ( finite_finite @ B ) )
=> ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Theta: trm @ B @ A,I3: denota1663640101rp_ext @ B @ C @ A @ product_unit] :
( ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVT @ B @ A @ Theta ) )
=> ( ( denota126604975rm_sem @ B @ C @ A @ I3 @ Theta @ ( product_fst @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) @ Nu ) )
= ( denota126604975rm_sem @ B @ C @ A @ I3 @ Theta @ ( product_fst @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) @ Nu2 ) ) ) ) ) ).
% coincidence_sterm
thf(fact_163_pred__subset__eq,axiom,
! [A: $tType,R3: set @ A,S5: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ R3 )
@ ^ [X2: A] : ( member @ A @ X2 @ S5 ) )
= ( ord_less_eq @ ( set @ A ) @ R3 @ S5 ) ) ).
% pred_subset_eq
thf(fact_164_sterm__continuous_H,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ C )
& ( finite_finite @ B ) )
=> ! [I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,Theta: trm @ A @ C,S5: set @ ( finite_Cartesian_vec @ real @ C )] :
( ( denota2077489681interp @ A @ B @ C @ I3 )
=> ( ( dfree @ A @ C @ Theta )
=> ( topolo2071040574ous_on @ ( finite_Cartesian_vec @ real @ C ) @ real @ S5 @ ( denota126604975rm_sem @ A @ B @ C @ I3 @ Theta ) ) ) ) ) ).
% sterm_continuous'
thf(fact_165_agree__func__fvt,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite @ A )
=> ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),F: B,Args: A > ( trm @ B @ A ),I: A] :
( ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVT @ B @ A @ ( function @ B @ A @ F @ Args ) ) )
=> ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVT @ B @ A @ ( Args @ I ) ) ) ) ) ).
% agree_func_fvt
thf(fact_166_subrelI,axiom,
! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S6: set @ ( product_prod @ A @ B )] :
( ! [X3: A,Y3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R4 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S6 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R4 @ S6 ) ) ).
% subrelI
thf(fact_167_FVDiff__sub,axiom,
! [A: $tType,B: $tType,F: trm @ B @ A] : ( ord_less_eq @ ( set @ ( sum_sum @ A @ A ) ) @ ( static_FVT @ B @ A @ F ) @ ( static_FVDiff @ B @ A @ F ) ) ).
% FVDiff_sub
thf(fact_168_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S5: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R3 ) )
= ( ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ S5 ) ) )
= ( R3 = S5 ) ) ).
% pred_equals_eq2
thf(fact_169_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S5: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( A > B > $o )
@ ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R3 )
@ ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ S5 ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S5 ) ) ).
% pred_subset_eq2
thf(fact_170_sterm__continuous,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ C )
& ( finite_finite @ B ) )
=> ! [I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,Theta: trm @ A @ C] :
( ( denota2077489681interp @ A @ B @ C @ I3 )
=> ( ( dfree @ A @ C @ Theta )
=> ( topolo2071040574ous_on @ ( finite_Cartesian_vec @ real @ C ) @ real @ ( top_top @ ( set @ ( finite_Cartesian_vec @ real @ C ) ) ) @ ( denota126604975rm_sem @ A @ B @ C @ I3 @ Theta ) ) ) ) ) ).
% sterm_continuous
thf(fact_171_ids_Ocoincidence__sterm_H,axiom,
! [Sz: $tType,Sf: $tType,Sc: $tType,A: $tType,C: $tType,B: $tType] :
( ( ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ A )
& ( finite_finite @ Sc )
& ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Theta: trm @ A @ B,Nu: product_prod @ ( finite_Cartesian_vec @ real @ B ) @ ( finite_Cartesian_vec @ real @ B ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ B ) @ ( finite_Cartesian_vec @ real @ B ),I3: denota1663640101rp_ext @ A @ C @ B @ product_unit,J: denota1663640101rp_ext @ A @ C @ B @ product_unit] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( dfree @ A @ B @ Theta )
=> ( ( denotational_Vagree @ B @ Nu @ Nu2 @ ( static_FVT @ A @ B @ Theta ) )
=> ( ( denotational_Iagree @ A @ C @ B @ I3 @ J
@ ( collect @ ( sum_sum @ A @ ( sum_sum @ C @ B ) )
@ ^ [Uu: sum_sum @ A @ ( sum_sum @ C @ B )] :
? [X2: A] :
( ( Uu
= ( sum_Inl @ A @ ( sum_sum @ C @ B ) @ X2 ) )
& ( member @ A @ X2 @ ( static_SIGT @ A @ B @ Theta ) ) ) ) )
=> ( ( denota126604975rm_sem @ A @ C @ B @ I3 @ Theta @ ( product_fst @ ( finite_Cartesian_vec @ real @ B ) @ ( finite_Cartesian_vec @ real @ B ) @ Nu ) )
= ( denota126604975rm_sem @ A @ C @ B @ J @ Theta @ ( product_fst @ ( finite_Cartesian_vec @ real @ B ) @ ( finite_Cartesian_vec @ real @ B ) @ Nu2 ) ) ) ) ) ) ) ) ).
% ids.coincidence_sterm'
thf(fact_172_continuous__on__cases__le,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo503727757_space @ A )
& ( topolo503727757_space @ B ) )
=> ! [S6: set @ A,H: A > real,A3: real,F: A > B,G: A > B] :
( ( topolo2071040574ous_on @ A @ B
@ ( collect @ A
@ ^ [T3: A] :
( ( member @ A @ T3 @ S6 )
& ( ord_less_eq @ real @ ( H @ T3 ) @ A3 ) ) )
@ F )
=> ( ( topolo2071040574ous_on @ A @ B
@ ( collect @ A
@ ^ [T3: A] :
( ( member @ A @ T3 @ S6 )
& ( ord_less_eq @ real @ A3 @ ( H @ T3 ) ) ) )
@ G )
=> ( ( topolo2071040574ous_on @ A @ real @ S6 @ H )
=> ( ! [T6: A] :
( ( member @ A @ T6 @ S6 )
=> ( ( ( H @ T6 )
= A3 )
=> ( ( F @ T6 )
= ( G @ T6 ) ) ) )
=> ( topolo2071040574ous_on @ A @ B @ S6
@ ^ [T3: A] : ( if @ B @ ( ord_less_eq @ real @ ( H @ T3 ) @ A3 ) @ ( F @ T3 ) @ ( G @ T3 ) ) ) ) ) ) ) ) ).
% continuous_on_cases_le
thf(fact_173_top__apply,axiom,
! [C: $tType,D2: $tType] :
( ( top @ C )
=> ( ( top_top @ ( D2 > C ) )
= ( ^ [X2: D2] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_174_UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_175_predicate2I,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
( ! [X3: A,Y3: B] :
( ( P @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) )
=> ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% predicate2I
thf(fact_176_fst__top,axiom,
! [B: $tType,A: $tType] :
( ( ( top @ A )
& ( top @ B ) )
=> ( ( product_fst @ A @ B @ ( top_top @ ( product_prod @ A @ B ) ) )
= ( top_top @ A ) ) ) ).
% fst_top
thf(fact_177_snd__top,axiom,
! [B: $tType,A: $tType] :
( ( ( top @ A )
& ( top @ B ) )
=> ( ( product_snd @ B @ A @ ( top_top @ ( product_prod @ B @ A ) ) )
= ( top_top @ A ) ) ) ).
% snd_top
thf(fact_178_continuous__on__cases__1,axiom,
! [A: $tType] :
( ( topolo503727757_space @ A )
=> ! [S6: set @ real,A3: real,F: real > A,G: real > A] :
( ( topolo2071040574ous_on @ real @ A
@ ( collect @ real
@ ^ [T3: real] :
( ( member @ real @ T3 @ S6 )
& ( ord_less_eq @ real @ T3 @ A3 ) ) )
@ F )
=> ( ( topolo2071040574ous_on @ real @ A
@ ( collect @ real
@ ^ [T3: real] :
( ( member @ real @ T3 @ S6 )
& ( ord_less_eq @ real @ A3 @ T3 ) ) )
@ G )
=> ( ( ( member @ real @ A3 @ S6 )
=> ( ( F @ A3 )
= ( G @ A3 ) ) )
=> ( topolo2071040574ous_on @ real @ A @ S6
@ ^ [T3: real] : ( if @ A @ ( ord_less_eq @ real @ T3 @ A3 ) @ ( F @ T3 ) @ ( G @ T3 ) ) ) ) ) ) ) ).
% continuous_on_cases_1
thf(fact_179_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
=> ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_180_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
= ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_181_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_182_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X: A,Y: B,Q: A > B > $o] :
( ( P @ X @ Y )
=> ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( Q @ X @ Y ) ) ) ).
% rev_predicate2D
thf(fact_183_predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X: A,Y: B] :
( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( ( P @ X @ Y )
=> ( Q @ X @ Y ) ) ) ).
% predicate2D
thf(fact_184_subset__UNIV,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_185_ids_Oseq__sem_Ocases,axiom,
! [Sz: $tType,Sc: $tType,Sf: $tType] :
( ( ( finite_finite @ Sf )
& ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: product_prod @ ( denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit ) @ ( product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) )] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ~ ! [I2: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,S: product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) )] :
( X
!= ( product_Pair @ ( denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit ) @ ( product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) ) @ I2 @ S ) ) ) ) ).
% ids.seq_sem.cases
thf(fact_186_UNIV__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A
@ ^ [X2: A] : $true ) ) ).
% UNIV_def
thf(fact_187_UNIV__witness,axiom,
! [A: $tType] :
? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_188_UNIV__eq__I,axiom,
! [A: $tType,A2: set @ A] :
( ! [X3: A] : ( member @ A @ X3 @ A2 )
=> ( ( top_top @ ( set @ A ) )
= A2 ) ) ).
% UNIV_eq_I
thf(fact_189_ids_Osingleton_Oinduct,axiom,
! [Sf: $tType,Sc: $tType,A: $tType,Sz: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sc )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P: ( trm @ A @ Sz ) > Sz > $o,A0: trm @ A @ Sz,A12: Sz] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ! [T6: trm @ A @ Sz,X_1: Sz] : ( P @ T6 @ X_1 )
=> ( P @ A0 @ A12 ) ) ) ) ).
% ids.singleton.induct
thf(fact_190_ids_Oseq__sem_Oinduct,axiom,
! [Sz: $tType,Sc: $tType,Sf: $tType] :
( ( ( finite_finite @ Sf )
& ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P: ( denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit ) > ( product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) ) > $o,A0: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,A12: product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) )] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ! [I2: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,X_1: product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) )] : ( P @ I2 @ X_1 )
=> ( P @ A0 @ A12 ) ) ) ) ).
% ids.seq_sem.induct
thf(fact_191_ids_Osingleton_Ocases,axiom,
! [Sc: $tType,Sf: $tType,A: $tType,Sz: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: product_prod @ ( trm @ A @ Sz ) @ Sz] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ~ ! [T6: trm @ A @ Sz,I4: Sz] :
( X
!= ( product_Pair @ ( trm @ A @ Sz ) @ Sz @ T6 @ I4 ) ) ) ) ).
% ids.singleton.cases
thf(fact_192_top__prod__def,axiom,
! [A: $tType,B: $tType] :
( ( ( top @ B )
& ( top @ A ) )
=> ( ( top_top @ ( product_prod @ A @ B ) )
= ( product_Pair @ A @ B @ ( top_top @ A ) @ ( top_top @ B ) ) ) ) ).
% top_prod_def
thf(fact_193_ids_OP__def,axiom,
! [Sf: $tType,Sz: $tType,Sc: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P2: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( p @ Sc @ Sf @ Sz @ P2 )
= ( predicational @ Sc @ Sf @ Sz @ P2 ) ) ) ) ).
% ids.P_def
thf(fact_194_ids_Ocoincidence__sterm,axiom,
! [Sz: $tType,Sf: $tType,Sc: $tType,B: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ C )
& ( finite_finite @ B )
& ( finite_finite @ Sc )
& ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Theta: trm @ B @ A,I3: denota1663640101rp_ext @ B @ C @ A @ product_unit] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVT @ B @ A @ Theta ) )
=> ( ( denota126604975rm_sem @ B @ C @ A @ I3 @ Theta @ ( product_fst @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) @ Nu ) )
= ( denota126604975rm_sem @ B @ C @ A @ I3 @ Theta @ ( product_fst @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) @ Nu2 ) ) ) ) ) ) ).
% ids.coincidence_sterm
thf(fact_195_ids_Of0__def,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,F: Sf] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( f0 @ Sf @ Sz @ F )
= ( function @ Sf @ Sz @ F @ ( empty @ Sz @ Sf ) ) ) ) ) ).
% ids.f0_def
thf(fact_196_ids_Ocoincidence__frechet,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,B: $tType,A: $tType,C: $tType] :
( ( ( finite_finite @ C )
& ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Theta: trm @ A @ 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 ),I3: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( dfree @ A @ C @ Theta )
=> ( ( denotational_Vagree @ C @ Nu @ Nu2 @ ( static_FVDiff @ A @ C @ Theta ) )
=> ( ( denotational_frechet @ A @ B @ C @ I3 @ Theta @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Nu ) @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Nu ) )
= ( denotational_frechet @ A @ B @ C @ I3 @ Theta @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Nu2 ) @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Nu2 ) ) ) ) ) ) ) ).
% ids.coincidence_frechet
thf(fact_197_frechet__continuous,axiom,
! [I3: denota1663640101rp_ext @ sf @ sc @ sz @ product_unit,Theta: trm @ sf @ sz] :
( ( denota2077489681interp @ sf @ sc @ sz @ I3 )
=> ( ( dfree @ sf @ sz @ Theta )
=> ( topolo2071040574ous_on @ ( finite_Cartesian_vec @ real @ sz ) @ ( bounde2145540817linfun @ ( finite_Cartesian_vec @ real @ sz ) @ real ) @ ( top_top @ ( set @ ( finite_Cartesian_vec @ real @ sz ) ) ) @ ( frechet_blin_frechet @ sf @ sc @ sz @ ( freche227871258interp @ sf @ sc @ sz @ I3 ) @ ( frechet_simple_term @ sf @ sz @ Theta ) ) ) ) ) ).
% frechet_continuous
thf(fact_198_ids_Osterm__determines__frechet,axiom,
! [Sz: $tType,Sf: $tType,Sc: $tType,A1: $tType,B1: $tType,B22: $tType,A22: $tType,C: $tType] :
( ( ( finite_finite @ C )
& ( finite_finite @ A22 )
& ( finite_finite @ B22 )
& ( finite_finite @ B1 )
& ( finite_finite @ A1 )
& ( finite_finite @ Sc )
& ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,I3: denota1663640101rp_ext @ A1 @ B1 @ C @ product_unit,J: denota1663640101rp_ext @ A22 @ B22 @ C @ product_unit,Theta_12: trm @ A1 @ C,Theta_22: trm @ A22 @ C,Nu: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( denota2077489681interp @ A1 @ B1 @ C @ I3 )
=> ( ( denota2077489681interp @ A22 @ B22 @ C @ J )
=> ( ( dfree @ A1 @ C @ Theta_12 )
=> ( ( dfree @ A22 @ C @ Theta_22 )
=> ( ( ( denota126604975rm_sem @ A1 @ B1 @ C @ I3 @ Theta_12 )
= ( denota126604975rm_sem @ A22 @ B22 @ C @ J @ Theta_22 ) )
=> ( ( denotational_frechet @ A1 @ B1 @ C @ I3 @ Theta_12 @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Nu ) @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Nu ) )
= ( denotational_frechet @ A22 @ B22 @ C @ J @ Theta_22 @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Nu ) @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ Nu ) ) ) ) ) ) ) ) ) ) ).
% ids.sterm_determines_frechet
thf(fact_199_ids_Osterm__continuous,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,B: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ C )
& ( finite_finite @ B )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,Theta: trm @ A @ C] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( denota2077489681interp @ A @ B @ C @ I3 )
=> ( ( dfree @ A @ C @ Theta )
=> ( topolo2071040574ous_on @ ( finite_Cartesian_vec @ real @ C ) @ real @ ( top_top @ ( set @ ( finite_Cartesian_vec @ real @ C ) ) ) @ ( denota126604975rm_sem @ A @ B @ C @ I3 @ Theta ) ) ) ) ) ) ).
% ids.sterm_continuous
thf(fact_200_top__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( top_top @ ( A > B > $o ) )
= ( ^ [X2: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% top_empty_eq2
thf(fact_201_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_202_Vagree__univ,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [A3: finite_Cartesian_vec @ real @ A,B3: finite_Cartesian_vec @ real @ A,C3: finite_Cartesian_vec @ real @ A,D4: finite_Cartesian_vec @ real @ A] :
( ( denotational_Vagree @ A @ ( product_Pair @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) @ A3 @ B3 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) @ C3 @ D4 ) @ ( top_top @ ( set @ ( sum_sum @ A @ A ) ) ) )
=> ( ( A3 = C3 )
& ( B3 = D4 ) ) ) ) ).
% Vagree_univ
thf(fact_203_agree__UNIV__eq,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [Nu: 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 )] :
( ( denotational_Vagree @ A @ Nu @ Omega @ ( top_top @ ( set @ ( sum_sum @ A @ A ) ) ) )
=> ( Nu = Omega ) ) ) ).
% agree_UNIV_eq
thf(fact_204_ids_Oraw__term__inject,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: frechet_strm @ A @ C,Y: frechet_strm @ A @ C] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( ( frechet_raw_term @ A @ C @ X )
= ( frechet_raw_term @ A @ C @ Y ) )
= ( X = Y ) ) ) ) ).
% ids.raw_term_inject
thf(fact_205_ids_Oraw__interp__inject,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: frechet_good_interp @ A @ B @ C,Y: frechet_good_interp @ A @ B @ C] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( ( freche229654227interp @ A @ B @ C @ X )
= ( freche229654227interp @ A @ B @ C @ Y ) )
= ( X = Y ) ) ) ) ).
% ids.raw_interp_inject
thf(fact_206_ids_Ofrechet__continuous,axiom,
! [Sc: $tType,Sz: $tType,Sf: $tType] :
( ( ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,I3: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,Theta: trm @ Sf @ Sz] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( denota2077489681interp @ Sf @ Sc @ Sz @ I3 )
=> ( ( dfree @ Sf @ Sz @ Theta )
=> ( topolo2071040574ous_on @ ( finite_Cartesian_vec @ real @ Sz ) @ ( bounde2145540817linfun @ ( finite_Cartesian_vec @ real @ Sz ) @ real ) @ ( top_top @ ( set @ ( finite_Cartesian_vec @ real @ Sz ) ) ) @ ( frechet_blin_frechet @ Sf @ Sc @ Sz @ ( freche227871258interp @ Sf @ Sc @ Sz @ I3 ) @ ( frechet_simple_term @ Sf @ Sz @ Theta ) ) ) ) ) ) ) ).
% ids.frechet_continuous
thf(fact_207_ids_Oraw__interp__inverse,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: frechet_good_interp @ A @ B @ C] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( freche227871258interp @ A @ B @ C @ ( freche229654227interp @ A @ B @ C @ X ) )
= X ) ) ) ).
% ids.raw_interp_inverse
thf(fact_208_ids_Oraw__term__inverse,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: frechet_strm @ A @ C] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( frechet_simple_term @ A @ C @ ( frechet_raw_term @ A @ C @ X ) )
= X ) ) ) ).
% ids.raw_term_inverse
thf(fact_209_ids_Oraw__term,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: frechet_strm @ A @ C] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( member @ ( trm @ A @ C ) @ ( frechet_raw_term @ A @ C @ X ) @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) ) ) ) ).
% ids.raw_term
thf(fact_210_ids_Oraw__term__cases,axiom,
! [Sz: $tType,Sf: $tType,Sc: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Y: trm @ A @ C] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
=> ~ ! [X3: frechet_strm @ A @ C] :
( Y
!= ( frechet_raw_term @ A @ C @ X3 ) ) ) ) ) ).
% ids.raw_term_cases
thf(fact_211_ids_Oraw__term__induct,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Y: trm @ A @ C,P: ( trm @ A @ C ) > $o] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
=> ( ! [X3: frechet_strm @ A @ C] : ( P @ ( frechet_raw_term @ A @ C @ X3 ) )
=> ( P @ Y ) ) ) ) ) ).
% ids.raw_term_induct
thf(fact_212_ids_Osimple__term__cases,axiom,
! [Sz: $tType,Sf: $tType,Sc: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: frechet_strm @ A @ C] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ~ ! [Y3: trm @ A @ C] :
( ( X
= ( frechet_simple_term @ A @ C @ Y3 ) )
=> ~ ( member @ ( trm @ A @ C ) @ Y3 @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) ) ) ) ) ).
% ids.simple_term_cases
thf(fact_213_ids_Osimple__term__induct,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P: ( frechet_strm @ A @ C ) > $o,X: frechet_strm @ A @ C] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ! [Y3: trm @ A @ C] :
( ( member @ ( trm @ A @ C ) @ Y3 @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
=> ( P @ ( frechet_simple_term @ A @ C @ Y3 ) ) )
=> ( P @ X ) ) ) ) ).
% ids.simple_term_induct
thf(fact_214_ids_Osimple__term__inject,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: trm @ A @ C,Y: trm @ A @ C] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( member @ ( trm @ A @ C ) @ X @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
=> ( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
=> ( ( ( frechet_simple_term @ A @ C @ X )
= ( frechet_simple_term @ A @ C @ Y ) )
= ( X = Y ) ) ) ) ) ) ).
% ids.simple_term_inject
thf(fact_215_ids_Oraw__interp__induct,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Y: denota1663640101rp_ext @ A @ B @ C @ product_unit,P: ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > $o] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
=> ( ! [X3: frechet_good_interp @ A @ B @ C] : ( P @ ( freche229654227interp @ A @ B @ C @ X3 ) )
=> ( P @ Y ) ) ) ) ) ).
% ids.raw_interp_induct
thf(fact_216_ids_Oraw__interp__cases,axiom,
! [Sz: $tType,Sf: $tType,Sc: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ Sc )
& ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Y: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
=> ~ ! [X3: frechet_good_interp @ A @ B @ C] :
( Y
!= ( freche229654227interp @ A @ B @ C @ X3 ) ) ) ) ) ).
% ids.raw_interp_cases
thf(fact_217_ids_Oraw__interp,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: frechet_good_interp @ A @ B @ C] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( freche229654227interp @ A @ B @ C @ X ) @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) ) ) ) ).
% ids.raw_interp
thf(fact_218_ids_Ogood__interp__cases,axiom,
! [Sz: $tType,Sf: $tType,Sc: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ Sc )
& ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: frechet_good_interp @ A @ B @ C] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ~ ! [Y3: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( X
= ( freche227871258interp @ A @ B @ C @ Y3 ) )
=> ~ ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y3 @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) ) ) ) ) ).
% ids.good_interp_cases
thf(fact_219_ids_Ogood__interp__induct,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P: ( frechet_good_interp @ A @ B @ C ) > $o,X: frechet_good_interp @ A @ B @ C] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ! [Y3: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y3 @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
=> ( P @ ( freche227871258interp @ A @ B @ C @ Y3 ) ) )
=> ( P @ X ) ) ) ) ).
% ids.good_interp_induct
thf(fact_220_ids_Ogood__interp__inject,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: denota1663640101rp_ext @ A @ B @ C @ product_unit,Y: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ X @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
=> ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
=> ( ( ( freche227871258interp @ A @ B @ C @ X )
= ( freche227871258interp @ A @ B @ C @ Y ) )
= ( X = Y ) ) ) ) ) ) ).
% ids.good_interp_inject
thf(fact_221_ids_Ocr__strm__def,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( frechet_cr_strm @ A @ B )
= ( ^ [X2: trm @ A @ B,Y2: frechet_strm @ A @ B] :
( X2
= ( frechet_raw_term @ A @ B @ Y2 ) ) ) ) ) ) ).
% ids.cr_strm_def
thf(fact_222_ids_Ocr__good__interp__def,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( freche457001096interp @ A @ B @ C )
= ( ^ [X2: denota1663640101rp_ext @ A @ B @ C @ product_unit,Y2: frechet_good_interp @ A @ B @ C] :
( X2
= ( freche229654227interp @ A @ B @ C @ Y2 ) ) ) ) ) ) ).
% ids.cr_good_interp_def
thf(fact_223_ids_Osimple__term__inverse,axiom,
! [Sz: $tType,Sf: $tType,Sc: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ Sc )
& ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Y: trm @ A @ C] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
=> ( ( frechet_raw_term @ A @ C @ ( frechet_simple_term @ A @ C @ Y ) )
= Y ) ) ) ) ).
% ids.simple_term_inverse
thf(fact_224_ids_Ogood__interp__inverse,axiom,
! [Sz: $tType,Sf: $tType,Sc: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ Sc )
& ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Y: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
=> ( ( freche229654227interp @ A @ B @ C @ ( freche227871258interp @ A @ B @ C @ Y ) )
= Y ) ) ) ) ).
% ids.good_interp_inverse
thf(fact_225_ids_Osterm__continuous_H,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,B: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ C )
& ( finite_finite @ B )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,Theta: trm @ A @ C,S5: set @ ( finite_Cartesian_vec @ real @ C )] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( denota2077489681interp @ A @ B @ C @ I3 )
=> ( ( dfree @ A @ C @ Theta )
=> ( topolo2071040574ous_on @ ( finite_Cartesian_vec @ real @ C ) @ real @ S5 @ ( denota126604975rm_sem @ A @ B @ C @ I3 @ Theta ) ) ) ) ) ) ).
% ids.sterm_continuous'
thf(fact_226_ids_Otype__definition__strm,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( type_definition @ ( frechet_strm @ A @ C ) @ ( trm @ A @ C ) @ ( frechet_raw_term @ A @ C ) @ ( frechet_simple_term @ A @ C ) @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) ) ) ) ).
% ids.type_definition_strm
thf(fact_227_ids_Otype__definition__good__interp,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( type_definition @ ( frechet_good_interp @ A @ B @ C ) @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( freche229654227interp @ A @ B @ C ) @ ( freche227871258interp @ A @ B @ C ) @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) ) ) ) ).
% ids.type_definition_good_interp
thf(fact_228_frechet__blin,axiom,
! [I3: denota1663640101rp_ext @ sf @ sc @ sz @ product_unit,Theta: trm @ sf @ sz] :
( ( denota2077489681interp @ sf @ sc @ sz @ I3 )
=> ( ( dfree @ sf @ sz @ Theta )
=> ( ( ^ [V2: finite_Cartesian_vec @ real @ sz] : ( bounde688532126linfun @ ( finite_Cartesian_vec @ real @ sz ) @ real @ ( denotational_frechet @ sf @ sc @ sz @ I3 @ Theta @ V2 ) ) )
= ( frechet_blin_frechet @ sf @ sc @ sz @ ( freche227871258interp @ sf @ sc @ sz @ I3 ) @ ( frechet_simple_term @ sf @ sz @ Theta ) ) ) ) ) ).
% frechet_blin
thf(fact_229_continuous__on__snd,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo503727757_space @ A )
& ( topolo503727757_space @ B )
& ( topolo503727757_space @ C ) )
=> ! [S6: set @ A,F: A > ( product_prod @ B @ C )] :
( ( topolo2071040574ous_on @ A @ ( product_prod @ B @ C ) @ S6 @ F )
=> ( topolo2071040574ous_on @ A @ C @ S6
@ ^ [X2: A] : ( product_snd @ B @ C @ ( F @ X2 ) ) ) ) ) ).
% continuous_on_snd
thf(fact_230_top1I,axiom,
! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% top1I
thf(fact_231_top2I,axiom,
! [A: $tType,B: $tType,X: A,Y: B] : ( top_top @ ( A > B > $o ) @ X @ Y ) ).
% top2I
thf(fact_232_continuous__on__const,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo503727757_space @ A )
& ( topolo503727757_space @ B ) )
=> ! [S6: set @ A,C3: B] :
( topolo2071040574ous_on @ A @ B @ S6
@ ^ [X2: A] : C3 ) ) ).
% continuous_on_const
thf(fact_233_continuous__on__id,axiom,
! [A: $tType] :
( ( topolo503727757_space @ A )
=> ! [S6: set @ A] :
( topolo2071040574ous_on @ A @ A @ S6
@ ^ [X2: A] : X2 ) ) ).
% continuous_on_id
thf(fact_234_continuous__on__subset,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo503727757_space @ A )
& ( topolo503727757_space @ B ) )
=> ! [S6: set @ A,F: A > B,T4: set @ A] :
( ( topolo2071040574ous_on @ A @ B @ S6 @ F )
=> ( ( ord_less_eq @ ( set @ A ) @ T4 @ S6 )
=> ( topolo2071040574ous_on @ A @ B @ T4 @ F ) ) ) ) ).
% continuous_on_subset
thf(fact_235_continuous__on__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo503727757_space @ A )
& ( topolo503727757_space @ B )
& ( topolo503727757_space @ C ) )
=> ! [S6: set @ A,F: A > B,G: A > C] :
( ( topolo2071040574ous_on @ A @ B @ S6 @ F )
=> ( ( topolo2071040574ous_on @ A @ C @ S6 @ G )
=> ( topolo2071040574ous_on @ A @ ( product_prod @ B @ C ) @ S6
@ ^ [X2: A] : ( product_Pair @ B @ C @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% continuous_on_Pair
thf(fact_236_ids_Ofrechet__blin,axiom,
! [Sc: $tType,Sz: $tType,Sf: $tType] :
( ( ( finite_finite @ Sf )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,I3: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,Theta: trm @ Sf @ Sz] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( denota2077489681interp @ Sf @ Sc @ Sz @ I3 )
=> ( ( dfree @ Sf @ Sz @ Theta )
=> ( ( ^ [V2: finite_Cartesian_vec @ real @ Sz] : ( bounde688532126linfun @ ( finite_Cartesian_vec @ real @ Sz ) @ real @ ( denotational_frechet @ Sf @ Sc @ Sz @ I3 @ Theta @ V2 ) ) )
= ( frechet_blin_frechet @ Sf @ Sc @ Sz @ ( freche227871258interp @ Sf @ Sc @ Sz @ I3 ) @ ( frechet_simple_term @ Sf @ Sz @ Theta ) ) ) ) ) ) ) ).
% ids.frechet_blin
thf(fact_237_continuous__on__fst,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo503727757_space @ A )
& ( topolo503727757_space @ B )
& ( topolo503727757_space @ C ) )
=> ! [S6: set @ A,F: A > ( product_prod @ B @ C )] :
( ( topolo2071040574ous_on @ A @ ( product_prod @ B @ C ) @ S6 @ F )
=> ( topolo2071040574ous_on @ A @ B @ S6
@ ^ [X2: A] : ( product_fst @ B @ C @ ( F @ X2 ) ) ) ) ) ).
% continuous_on_fst
thf(fact_238_continuous__on__product__then__coordinatewise__UNIV,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( topolo503727757_space @ A )
& ( topolo503727757_space @ C ) )
=> ! [F: A > B > C,I: B] :
( ( topolo2071040574ous_on @ A @ ( B > C ) @ ( top_top @ ( set @ A ) ) @ F )
=> ( topolo2071040574ous_on @ A @ C @ ( top_top @ ( set @ A ) )
@ ^ [X2: A] : ( F @ X2 @ I ) ) ) ) ).
% continuous_on_product_then_coordinatewise_UNIV
thf(fact_239_blin__frechet_Oabs__eq,axiom,
! [Xb: denota1663640101rp_ext @ sf @ sc @ sz @ product_unit,Xa: trm @ sf @ sz,X: finite_Cartesian_vec @ real @ sz] :
( ( bNF_eq_onp @ ( denota1663640101rp_ext @ sf @ sc @ sz @ product_unit ) @ ( denota2077489681interp @ sf @ sc @ sz ) @ Xb @ Xb )
=> ( ( bNF_eq_onp @ ( trm @ sf @ sz ) @ ( dfree @ sf @ sz ) @ Xa @ Xa )
=> ( ( frechet_blin_frechet @ sf @ sc @ sz @ ( freche227871258interp @ sf @ sc @ sz @ Xb ) @ ( frechet_simple_term @ sf @ sz @ Xa ) @ X )
= ( bounde688532126linfun @ ( finite_Cartesian_vec @ real @ sz ) @ real @ ( denotational_frechet @ sf @ sc @ sz @ Xb @ Xa @ X ) ) ) ) ) ).
% blin_frechet.abs_eq
thf(fact_240_ids_Oblin__frechet_Oabs__eq,axiom,
! [Sf: $tType,Sc: $tType,Sz: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sc )
& ( finite_finite @ Sf ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Xb: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,Xa: trm @ Sf @ Sz,X: finite_Cartesian_vec @ real @ Sz] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( bNF_eq_onp @ ( denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit ) @ ( denota2077489681interp @ Sf @ Sc @ Sz ) @ Xb @ Xb )
=> ( ( bNF_eq_onp @ ( trm @ Sf @ Sz ) @ ( dfree @ Sf @ Sz ) @ Xa @ Xa )
=> ( ( frechet_blin_frechet @ Sf @ Sc @ Sz @ ( freche227871258interp @ Sf @ Sc @ Sz @ Xb ) @ ( frechet_simple_term @ Sf @ Sz @ Xa ) @ X )
= ( bounde688532126linfun @ ( finite_Cartesian_vec @ real @ Sz ) @ real @ ( denotational_frechet @ Sf @ Sc @ Sz @ Xb @ Xa @ X ) ) ) ) ) ) ) ).
% ids.blin_frechet.abs_eq
thf(fact_241_continuous__on__coordinatewise__then__product,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo503727757_space @ A )
& ( topolo503727757_space @ C ) )
=> ! [S5: set @ A,F: A > B > C] :
( ! [I4: B] :
( topolo2071040574ous_on @ A @ C @ S5
@ ^ [X2: A] : ( F @ X2 @ I4 ) )
=> ( topolo2071040574ous_on @ A @ ( B > C ) @ S5 @ F ) ) ) ).
% continuous_on_coordinatewise_then_product
thf(fact_242_continuous__on__product__then__coordinatewise,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( topolo503727757_space @ A )
& ( topolo503727757_space @ C ) )
=> ! [S5: set @ A,F: A > B > C,I: B] :
( ( topolo2071040574ous_on @ A @ ( B > C ) @ S5 @ F )
=> ( topolo2071040574ous_on @ A @ C @ S5
@ ^ [X2: A] : ( F @ X2 @ I ) ) ) ) ).
% continuous_on_product_then_coordinatewise
thf(fact_243_Quotient__good__interp,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ( quotient @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( bNF_eq_onp @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) @ ( freche227871258interp @ A @ B @ C ) @ ( freche229654227interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) ) ) ).
% Quotient_good_interp
thf(fact_244_Quotient__strm,axiom,
! [C: $tType,A: $tType] : ( quotient @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( bNF_eq_onp @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) @ ( frechet_simple_term @ A @ C ) @ ( frechet_raw_term @ A @ C ) @ ( frechet_cr_strm @ A @ C ) ) ).
% Quotient_strm
thf(fact_245_ids_OQuotient__strm,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( quotient @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( bNF_eq_onp @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) @ ( frechet_simple_term @ A @ C ) @ ( frechet_raw_term @ A @ C ) @ ( frechet_cr_strm @ A @ C ) ) ) ) ).
% ids.Quotient_strm
thf(fact_246_ids_OQuotient__good__interp,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C )
& ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( quotient @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( bNF_eq_onp @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) @ ( freche227871258interp @ A @ B @ C ) @ ( freche229654227interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) ) ) ) ).
% ids.Quotient_good_interp
thf(fact_247_typedef__to__Quotient,axiom,
! [A: $tType,B: $tType,Rep: A > B,Abs: B > A,S5: set @ B,T7: B > A > $o] :
( ( type_definition @ A @ B @ Rep @ Abs @ S5 )
=> ( ( T7
= ( ^ [X2: B,Y2: A] :
( X2
= ( Rep @ Y2 ) ) ) )
=> ( quotient @ B @ A
@ ( bNF_eq_onp @ B
@ ^ [X2: B] : ( member @ B @ X2 @ S5 ) )
@ Abs
@ Rep
@ T7 ) ) ) ).
% typedef_to_Quotient
thf(fact_248_open__typedef__to__Quotient,axiom,
! [A: $tType,B: $tType,Rep: A > B,Abs: B > A,P: B > $o,T7: B > A > $o] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( collect @ B @ P ) )
=> ( ( T7
= ( ^ [X2: B,Y2: A] :
( X2
= ( Rep @ Y2 ) ) ) )
=> ( quotient @ B @ A @ ( bNF_eq_onp @ B @ P ) @ Abs @ Rep @ T7 ) ) ) ).
% open_typedef_to_Quotient
thf(fact_249_QuotientI,axiom,
! [A: $tType,B: $tType,Abs: B > A,Rep: A > B,R3: B > B > $o,T7: B > A > $o] :
( ! [A5: A] :
( ( Abs @ ( Rep @ A5 ) )
= A5 )
=> ( ! [A5: A] : ( R3 @ ( Rep @ A5 ) @ ( Rep @ A5 ) )
=> ( ! [R2: B,S4: B] :
( ( R3 @ R2 @ S4 )
= ( ( R3 @ R2 @ R2 )
& ( R3 @ S4 @ S4 )
& ( ( Abs @ R2 )
= ( Abs @ S4 ) ) ) )
=> ( ( T7
= ( ^ [X2: B,Y2: A] :
( ( R3 @ X2 @ X2 )
& ( ( Abs @ X2 )
= Y2 ) ) ) )
=> ( quotient @ B @ A @ R3 @ Abs @ Rep @ T7 ) ) ) ) ) ).
% QuotientI
thf(fact_250_Quotient__def,axiom,
! [B: $tType,A: $tType] :
( ( quotient @ A @ B )
= ( ^ [R5: A > A > $o,Abs2: A > B,Rep2: B > A,T8: A > B > $o] :
( ! [A7: B] :
( ( Abs2 @ ( Rep2 @ A7 ) )
= A7 )
& ! [A7: B] : ( R5 @ ( Rep2 @ A7 ) @ ( Rep2 @ A7 ) )
& ! [R6: A,S2: A] :
( ( R5 @ R6 @ S2 )
= ( ( R5 @ R6 @ R6 )
& ( R5 @ S2 @ S2 )
& ( ( Abs2 @ R6 )
= ( Abs2 @ S2 ) ) ) )
& ( T8
= ( ^ [X2: A,Y2: B] :
( ( R5 @ X2 @ X2 )
& ( ( Abs2 @ X2 )
= Y2 ) ) ) ) ) ) ) ).
% Quotient_def
thf(fact_251_Quotient__cr__rel,axiom,
! [B: $tType,A: $tType,R3: A > A > $o,Abs: A > B,Rep: B > A,T7: A > B > $o] :
( ( quotient @ A @ B @ R3 @ Abs @ Rep @ T7 )
=> ( T7
= ( ^ [X2: A,Y2: B] :
( ( R3 @ X2 @ X2 )
& ( ( Abs @ X2 )
= Y2 ) ) ) ) ) ).
% Quotient_cr_rel
thf(fact_252_eq__onp__le__eq,axiom,
! [A: $tType,P: A > $o] :
( ord_less_eq @ ( A > A > $o ) @ ( bNF_eq_onp @ A @ P )
@ ^ [Y4: A,Z: A] : Y4 = Z ) ).
% eq_onp_le_eq
thf(fact_253_Quotient__rep__abs__eq,axiom,
! [B: $tType,A: $tType,R3: A > A > $o,Abs: A > B,Rep: B > A,T7: A > B > $o,T4: A] :
( ( quotient @ A @ B @ R3 @ Abs @ Rep @ T7 )
=> ( ( R3 @ T4 @ T4 )
=> ( ( ord_less_eq @ ( A > A > $o ) @ R3
@ ^ [Y4: A,Z: A] : Y4 = Z )
=> ( ( Rep @ ( Abs @ T4 ) )
= T4 ) ) ) ) ).
% Quotient_rep_abs_eq
thf(fact_254_UNIV__typedef__to__Quotient,axiom,
! [A: $tType,B: $tType,Rep: A > B,Abs: B > A,T7: B > A > $o] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( T7
= ( ^ [X2: B,Y2: A] :
( X2
= ( Rep @ Y2 ) ) ) )
=> ( quotient @ B @ A
@ ^ [Y4: B,Z: B] : Y4 = Z
@ Abs
@ Rep
@ T7 ) ) ) ).
% UNIV_typedef_to_Quotient
thf(fact_255_Quotient__eq__onp__typedef,axiom,
! [B: $tType,A: $tType,P: A > $o,Abs: A > B,Rep: B > A,Cr: A > B > $o] :
( ( quotient @ A @ B @ ( bNF_eq_onp @ A @ P ) @ Abs @ Rep @ Cr )
=> ( type_definition @ B @ A @ Rep @ Abs @ ( collect @ A @ P ) ) ) ).
% Quotient_eq_onp_typedef
% Subclasses (5)
thf(subcl_Finite__Set_Ofinite___HOL_Otype,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( type @ A ) ) ).
thf(subcl_Orderings_Olinorder___HOL_Otype,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( type @ A ) ) ).
thf(subcl_Orderings_Olinorder___Orderings_Oord,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ord @ A ) ) ).
thf(subcl_Orderings_Olinorder___Orderings_Oorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( order @ A ) ) ).
thf(subcl_Orderings_Olinorder___Orderings_Opreorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( preorder @ A ) ) ).
% Type constructors (52)
thf(tcon_Bounded__Linear__Function_Oblinfun___Real__Vector__Spaces_Oreal__normed__vector,axiom,
! [A8: $tType,A9: $tType] :
( ( ( real_V55928688vector @ A8 )
& ( real_V55928688vector @ A9 ) )
=> ( real_V55928688vector @ ( bounde2145540817linfun @ A8 @ A9 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Real__Vector__Spaces_Oreal__normed__vector_1,axiom,
! [A8: $tType,A9: $tType] :
( ( ( real_V55928688vector @ A8 )
& ( finite_finite @ A9 ) )
=> ( real_V55928688vector @ ( finite_Cartesian_vec @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Real__Vector__Spaces_Oreal__normed__vector_2,axiom,
! [A8: $tType,A9: $tType] :
( ( ( real_V55928688vector @ A8 )
& ( real_V55928688vector @ A9 ) )
=> ( real_V55928688vector @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector_3,axiom,
real_V55928688vector @ real ).
thf(tcon_fun___Topological__Spaces_Otopological__space,axiom,
! [A8: $tType,A9: $tType] :
( ( topolo503727757_space @ A9 )
=> ( topolo503727757_space @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A8: $tType,A9: $tType] :
( ( order_top @ A9 )
=> ( order_top @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A9: $tType] :
( ( preorder @ A9 )
=> ( preorder @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A8: $tType,A9: $tType] :
( ( ( finite_finite @ A8 )
& ( finite_finite @ A9 ) )
=> ( finite_finite @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A9: $tType] :
( ( order @ A9 )
=> ( order @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A8: $tType,A9: $tType] :
( ( top @ A9 )
=> ( top @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 )
=> ( ord @ ( A8 > A9 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_4,axiom,
! [A8: $tType] : ( order_top @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_5,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_6,axiom,
! [A8: $tType] :
( ( finite_finite @ A8 )
=> ( finite_finite @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_7,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_8,axiom,
! [A8: $tType] : ( top @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_9,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_10,axiom,
topolo503727757_space @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_11,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_12,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_13,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_14,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_15,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_16,axiom,
ord @ $o ).
thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_17,axiom,
topolo503727757_space @ real ).
thf(tcon_Real_Oreal___Orderings_Opreorder_18,axiom,
preorder @ real ).
thf(tcon_Real_Oreal___Orderings_Olinorder_19,axiom,
linorder @ real ).
thf(tcon_Real_Oreal___Orderings_Oorder_20,axiom,
order @ real ).
thf(tcon_Real_Oreal___Orderings_Oord_21,axiom,
ord @ real ).
thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_22,axiom,
! [A8: $tType,A9: $tType] :
( ( ( finite_finite @ A8 )
& ( finite_finite @ A9 ) )
=> ( finite_finite @ ( sum_sum @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_23,axiom,
! [A8: $tType,A9: $tType] :
( ( ( topolo503727757_space @ A8 )
& ( topolo503727757_space @ A9 ) )
=> ( topolo503727757_space @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Oorder__top_24,axiom,
! [A8: $tType,A9: $tType] :
( ( ( order_top @ A8 )
& ( order_top @ A9 ) )
=> ( order_top @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Opreorder_25,axiom,
! [A8: $tType,A9: $tType] :
( ( ( preorder @ A8 )
& ( preorder @ A9 ) )
=> ( preorder @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_26,axiom,
! [A8: $tType,A9: $tType] :
( ( ( finite_finite @ A8 )
& ( finite_finite @ A9 ) )
=> ( finite_finite @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Oorder_27,axiom,
! [A8: $tType,A9: $tType] :
( ( ( order @ A8 )
& ( order @ A9 ) )
=> ( order @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Otop_28,axiom,
! [A8: $tType,A9: $tType] :
( ( ( top @ A8 )
& ( top @ A9 ) )
=> ( top @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Oord_29,axiom,
! [A8: $tType,A9: $tType] :
( ( ( ord @ A8 )
& ( ord @ A9 ) )
=> ( ord @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_30,axiom,
order_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_31,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_32,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_33,axiom,
finite_finite @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_34,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Otop_35,axiom,
top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_36,axiom,
ord @ product_unit ).
thf(tcon_Finite__Cartesian__Product_Ovec___Topological__Spaces_Otopological__space_37,axiom,
! [A8: $tType,A9: $tType] :
( ( ( topolo503727757_space @ A8 )
& ( finite_finite @ A9 ) )
=> ( topolo503727757_space @ ( finite_Cartesian_vec @ A8 @ A9 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Opreorder_38,axiom,
! [A8: $tType,A9: $tType] :
( ( ( order @ A8 )
& ( finite_finite @ A9 ) )
=> ( preorder @ ( finite_Cartesian_vec @ A8 @ A9 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Olinorder_39,axiom,
! [A8: $tType,A9: $tType] :
( ( ( linorder @ A8 )
& ( cARD_1 @ A9 ) )
=> ( linorder @ ( finite_Cartesian_vec @ A8 @ A9 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Finite__Set_Ofinite_40,axiom,
! [A8: $tType,A9: $tType] :
( ( ( finite_finite @ A8 )
& ( finite_finite @ A9 ) )
=> ( finite_finite @ ( finite_Cartesian_vec @ A8 @ A9 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Oorder_41,axiom,
! [A8: $tType,A9: $tType] :
( ( ( order @ A8 )
& ( finite_finite @ A9 ) )
=> ( order @ ( finite_Cartesian_vec @ A8 @ A9 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Oord_42,axiom,
! [A8: $tType,A9: $tType] :
( ( ( ord @ A8 )
& ( finite_finite @ A9 ) )
=> ( ord @ ( finite_Cartesian_vec @ A8 @ A9 ) ) ) ).
thf(tcon_Bounded__Linear__Function_Oblinfun___Topological__Spaces_Otopological__space_43,axiom,
! [A8: $tType,A9: $tType] :
( ( ( real_V55928688vector @ A8 )
& ( real_V55928688vector @ A9 ) )
=> ( topolo503727757_space @ ( bounde2145540817linfun @ A8 @ A9 ) ) ) ).
% 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 (7)
thf(tfree_0,hypothesis,
finite_finite @ c ).
thf(tfree_1,hypothesis,
finite_finite @ a ).
thf(tfree_2,hypothesis,
finite_finite @ sz ).
thf(tfree_3,hypothesis,
linorder @ sz ).
thf(tfree_4,hypothesis,
finite_finite @ sf ).
thf(tfree_5,hypothesis,
finite_finite @ sc ).
thf(tfree_6,hypothesis,
finite_finite @ b ).
% Conjectures (1)
thf(conj_0,conjecture,
denotational_Vagree @ c @ nu @ nu2 @ ( static_FVDiff @ a @ c @ t2 ) ).
%------------------------------------------------------------------------------