TPTP Problem File: ITP037^2.p
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%------------------------------------------------------------------------------
% File : ITP037^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Coincidence problem prob_129__7211024_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_129__7211024_1 [Des21]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 374 ( 148 unt; 86 typ; 0 def)
% Number of atoms : 791 ( 314 equ; 0 cnn)
% Maximal formula atoms : 15 ( 2 avg)
% Number of connectives : 5363 ( 112 ~; 11 |; 165 &;4764 @)
% ( 0 <=>; 311 =>; 0 <=; 0 <~>)
% Maximal formula depth : 34 ( 10 avg)
% Number of types : 9 ( 8 usr)
% Number of type conns : 253 ( 253 >; 0 *; 0 +; 0 <<)
% Number of symbols : 81 ( 78 usr; 10 con; 0-13 aty)
% Number of variables : 1660 ( 32 ^;1472 !; 35 ?;1660 :)
% ( 121 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:22:14.005
%------------------------------------------------------------------------------
% Could-be-implicit typings (18)
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_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 (68)
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_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : $o ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_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_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_Odterm__sem,type,
denota594965758rm_sem:
!>[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_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_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Ids_Oids,type,
ids:
!>[Sz: $tType,Sf: $tType,Sc: $tType] : ( Sz > Sz > Sz > Sf > Sf > Sf > Sc > Sc > Sc > Sc > $o ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
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_Ointernal__case__prod,type,
produc2004651681e_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Omap__prod,type,
product_map_prod:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
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_Product__Type_Oprod_Oswap,type,
product_swap:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
thf(sy_c_Relation_ODomainp,type,
domainp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > A > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Static__Semantics_OFVDiff,type,
static_FVDiff:
!>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > ( set @ ( sum_sum @ C @ C ) ) ) ).
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_Odsafe,type,
dsafe:
!>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > $o ) ).
thf(sy_c_Syntax_Offree,type,
ffree:
!>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > $o ) ).
thf(sy_c_Syntax_Oformula_OProp,type,
prop:
!>[C: $tType,A: $tType,B: $tType] : ( C > ( C > ( trm @ A @ C ) ) > ( formula @ A @ B @ C ) ) ).
thf(sy_c_Syntax_Oformula_Oset1__formula,type,
set1_formula:
!>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > ( set @ A ) ) ).
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_Oids_Of1,type,
f1:
!>[Sz: $tType,Sf: $tType] : ( Sz > Sf > Sz > ( trm @ Sf @ Sz ) ) ).
thf(sy_c_Syntax_Oids_Op1,type,
p1:
!>[Sz: $tType,Sf: $tType,Sc: $tType] : ( Sz > Sz > Sz > ( formula @ Sf @ Sc @ Sz ) ) ).
thf(sy_c_Syntax_Oids_Osingleton,type,
singleton:
!>[Sz: $tType,A: $tType] : ( Sz > ( trm @ A @ Sz ) > Sz > ( trm @ A @ Sz ) ) ).
thf(sy_c_Syntax_Oids_Osingleton__rel,type,
singleton_rel:
!>[A: $tType,Sz: $tType] : ( ( product_prod @ ( trm @ A @ Sz ) @ Sz ) > ( product_prod @ ( trm @ A @ Sz ) @ Sz ) > $o ) ).
thf(sy_c_Syntax_Otrm_OConst,type,
const:
!>[A: $tType,C: $tType] : ( real > ( trm @ A @ C ) ) ).
thf(sy_c_Syntax_Otrm_ODiffVar,type,
diffVar:
!>[C: $tType,A: $tType] : ( C > ( trm @ A @ C ) ) ).
thf(sy_c_Syntax_Otrm_ODifferential,type,
differential:
!>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > ( trm @ A @ C ) ) ).
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_Syntax_Otrm_OVar,type,
var:
!>[C: $tType,A: $tType] : ( C > ( trm @ A @ C ) ) ).
thf(sy_c_Syntax_Otrm_Oset__trm,type,
set_trm:
!>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > ( set @ A ) ) ).
thf(sy_c_Transfer_Oleft__unique,type,
left_unique:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Transfer_Oright__unique,type,
right_unique:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_I,type,
i: denota1663640101rp_ext @ a @ b @ c @ product_unit ).
thf(sy_v__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_args____,type,
args: c > ( trm @ a @ c ) ).
thf(sy_v_i,type,
i2: c ).
thf(sy_v_var____,type,
var2: a ).
thf(sy_v_vid1,type,
vid1: sz ).
thf(sy_v_vid2,type,
vid2: sz ).
thf(sy_v_vid3,type,
vid3: sz ).
% Relevant facts (255)
thf(fact_0_free,axiom,
! [I: c] : ( dfree @ a @ c @ ( args @ I ) ) ).
% free
thf(fact_1_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_2_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_3_P__def,axiom,
( ( p @ sc @ sf @ sz )
= ( predicational @ sc @ sf @ sz ) ) ).
% P_def
thf(fact_4_strm_Odomain,axiom,
! [C: $tType,A: $tType] :
( ( domainp @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) )
= ( dfree @ A @ C ) ) ).
% strm.domain
thf(fact_5_dfree__DiffVar__simps,axiom,
! [A: $tType,B: $tType,X: B] :
~ ( dfree @ A @ B @ ( diffVar @ B @ A @ X ) ) ).
% dfree_DiffVar_simps
thf(fact_6_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_7_singleton_Ocases,axiom,
! [A: $tType,X: product_prod @ ( trm @ A @ sz ) @ sz] :
~ ! [T2: trm @ A @ sz,I3: sz] :
( X
!= ( product_Pair @ ( trm @ A @ sz ) @ sz @ T2 @ I3 ) ) ).
% singleton.cases
thf(fact_8_dfree__Differential__simps,axiom,
! [B: $tType,A: $tType,X: trm @ A @ B] :
~ ( dfree @ A @ B @ ( differential @ A @ B @ X ) ) ).
% dfree_Differential_simps
thf(fact_9_dfree__Var__simps,axiom,
! [A: $tType,B: $tType,X: B] : ( dfree @ A @ B @ ( var @ B @ A @ X ) ) ).
% dfree_Var_simps
thf(fact_10_dfree__Times__simps,axiom,
! [B: $tType,A: $tType,A2: trm @ A @ B,B2: trm @ A @ B] :
( ( dfree @ A @ B @ ( times @ A @ B @ A2 @ B2 ) )
= ( ( dfree @ A @ B @ A2 )
& ( dfree @ A @ B @ B2 ) ) ) ).
% dfree_Times_simps
thf(fact_11_dfree__Plus__simps,axiom,
! [B: $tType,A: $tType,A2: trm @ A @ B,B2: trm @ A @ B] :
( ( dfree @ A @ B @ ( plus @ A @ B @ A2 @ B2 ) )
= ( ( dfree @ A @ B @ A2 )
& ( dfree @ A @ B @ B2 ) ) ) ).
% dfree_Plus_simps
thf(fact_12_dfree__Fun_Oprems,axiom,
denotational_Vagree @ c @ nu @ nu2 @ ( static_FVDiff @ a @ c @ ( function @ a @ c @ var2 @ args ) ) ).
% dfree_Fun.prems
thf(fact_13_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_14_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_15_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_16_trm_Oinject_I1_J,axiom,
! [A: $tType,C: $tType,X1: C,Y1: C] :
( ( ( var @ C @ A @ X1 )
= ( var @ C @ A @ Y1 ) )
= ( X1 = Y1 ) ) ).
% trm.inject(1)
thf(fact_17_trm_Oinject_I7_J,axiom,
! [C: $tType,A: $tType,X7: trm @ A @ C,Y7: trm @ A @ C] :
( ( ( differential @ A @ C @ X7 )
= ( differential @ A @ C @ Y7 ) )
= ( X7 = Y7 ) ) ).
% trm.inject(7)
thf(fact_18_trm_Oinject_I6_J,axiom,
! [A: $tType,C: $tType,X6: C,Y6: C] :
( ( ( diffVar @ C @ A @ X6 )
= ( diffVar @ C @ A @ Y6 ) )
= ( X6 = Y6 ) ) ).
% trm.inject(6)
thf(fact_19_dfree__Fun__simps,axiom,
! [A: $tType,B: $tType,I4: A,Args: B > ( trm @ A @ B )] :
( ( dfree @ A @ B @ ( function @ A @ B @ I4 @ Args ) )
= ( ! [X2: B] : ( dfree @ A @ B @ ( Args @ X2 ) ) ) ) ).
% dfree_Fun_simps
thf(fact_20_trm_Odistinct_I41_J,axiom,
! [C: $tType,A: $tType,X6: C,X7: trm @ A @ C] :
( ( diffVar @ C @ A @ X6 )
!= ( differential @ A @ C @ X7 ) ) ).
% trm.distinct(41)
thf(fact_21_trm_Odistinct_I39_J,axiom,
! [C: $tType,A: $tType,X51: trm @ A @ C,X52: trm @ A @ C,X7: trm @ A @ C] :
( ( times @ A @ C @ X51 @ X52 )
!= ( differential @ A @ C @ X7 ) ) ).
% trm.distinct(39)
thf(fact_22_trm_Odistinct_I37_J,axiom,
! [A: $tType,C: $tType,X51: trm @ A @ C,X52: trm @ A @ C,X6: C] :
( ( times @ A @ C @ X51 @ X52 )
!= ( diffVar @ C @ A @ X6 ) ) ).
% trm.distinct(37)
thf(fact_23_trm_Odistinct_I35_J,axiom,
! [C: $tType,A: $tType,X41: trm @ A @ C,X42: trm @ A @ C,X7: trm @ A @ C] :
( ( plus @ A @ C @ X41 @ X42 )
!= ( differential @ A @ C @ X7 ) ) ).
% trm.distinct(35)
thf(fact_24_trm_Odistinct_I33_J,axiom,
! [A: $tType,C: $tType,X41: trm @ A @ C,X42: trm @ A @ C,X6: C] :
( ( plus @ A @ C @ X41 @ X42 )
!= ( diffVar @ C @ A @ X6 ) ) ).
% trm.distinct(33)
thf(fact_25_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_26_trm_Odistinct_I29_J,axiom,
! [C: $tType,A: $tType,X31: A,X32: C > ( trm @ A @ C ),X7: trm @ A @ C] :
( ( function @ A @ C @ X31 @ X32 )
!= ( differential @ A @ C @ X7 ) ) ).
% trm.distinct(29)
thf(fact_27_trm_Odistinct_I27_J,axiom,
! [A: $tType,C: $tType,X31: A,X32: C > ( trm @ A @ C ),X6: C] :
( ( function @ A @ C @ X31 @ X32 )
!= ( diffVar @ C @ A @ X6 ) ) ).
% trm.distinct(27)
thf(fact_28_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_29_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_30_trm_Odistinct_I11_J,axiom,
! [C: $tType,A: $tType,X1: C,X7: trm @ A @ C] :
( ( var @ C @ A @ X1 )
!= ( differential @ A @ C @ X7 ) ) ).
% trm.distinct(11)
thf(fact_31_trm_Odistinct_I9_J,axiom,
! [A: $tType,C: $tType,X1: C,X6: C] :
( ( var @ C @ A @ X1 )
!= ( diffVar @ C @ A @ X6 ) ) ).
% trm.distinct(9)
thf(fact_32_trm_Odistinct_I7_J,axiom,
! [C: $tType,A: $tType,X1: C,X51: trm @ A @ C,X52: trm @ A @ C] :
( ( var @ C @ A @ X1 )
!= ( times @ A @ C @ X51 @ X52 ) ) ).
% trm.distinct(7)
thf(fact_33_trm_Odistinct_I5_J,axiom,
! [C: $tType,A: $tType,X1: C,X41: trm @ A @ C,X42: trm @ A @ C] :
( ( var @ C @ A @ X1 )
!= ( plus @ A @ C @ X41 @ X42 ) ) ).
% trm.distinct(5)
thf(fact_34_trm_Odistinct_I3_J,axiom,
! [A: $tType,C: $tType,X1: C,X31: A,X32: C > ( trm @ A @ C )] :
( ( var @ C @ A @ X1 )
!= ( function @ A @ C @ X31 @ X32 ) ) ).
% trm.distinct(3)
thf(fact_35_dfree_Odfree__Fun,axiom,
! [A: $tType,C: $tType,Args: C > ( trm @ A @ C ),I4: A] :
( ! [I3: C] : ( dfree @ A @ C @ ( Args @ I3 ) )
=> ( dfree @ A @ C @ ( function @ A @ C @ I4 @ Args ) ) ) ).
% dfree.dfree_Fun
thf(fact_36_dfree__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_Plus
thf(fact_37_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_38_dfree__Var,axiom,
! [A: $tType,C: $tType,I4: C] : ( dfree @ A @ C @ ( var @ C @ A @ I4 ) ) ).
% dfree_Var
thf(fact_39_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_40_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_41_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,T22: trm @ B @ A] :
( ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ ( times @ B @ A @ T1 @ T22 ) ) )
=> ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ T1 ) ) ) ) ).
% agree_times1
thf(fact_42_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,T22: trm @ B @ A] :
( ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ ( times @ B @ A @ T1 @ T22 ) ) )
=> ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ T22 ) ) ) ) ).
% agree_times2
thf(fact_43_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,T22: trm @ B @ A] :
( ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ ( plus @ B @ A @ T1 @ T22 ) ) )
=> ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ T1 ) ) ) ) ).
% agree_plus1
thf(fact_44_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,T22: trm @ B @ A] :
( ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ ( plus @ B @ A @ T1 @ T22 ) ) )
=> ( denotational_Vagree @ A @ Nu @ Nu2 @ ( static_FVDiff @ B @ A @ T22 ) ) ) ) ).
% agree_plus2
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
= A3 ) ).
% 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__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 ),I4: 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 @ I4 ) ) ) ) ) ).
% agree_func
thf(fact_50_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X1 @ X22 )
= ( product_Pair @ A @ B @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_51_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A4: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A4 @ B3 ) )
= ( ( A2 = A4 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_52_trm_Oexhaust,axiom,
! [C: $tType,A: $tType,Y: trm @ A @ C] :
( ! [X12: C] :
( Y
!= ( var @ C @ A @ X12 ) )
=> ( ! [X23: real] :
( Y
!= ( const @ A @ C @ X23 ) )
=> ( ! [X312: A,X322: C > ( trm @ A @ C )] :
( Y
!= ( function @ A @ C @ X312 @ X322 ) )
=> ( ! [X412: trm @ A @ C,X422: trm @ A @ C] :
( Y
!= ( plus @ A @ C @ X412 @ X422 ) )
=> ( ! [X512: trm @ A @ C,X522: trm @ A @ C] :
( Y
!= ( times @ A @ C @ X512 @ X522 ) )
=> ( ! [X62: C] :
( Y
!= ( diffVar @ C @ A @ X62 ) )
=> ~ ! [X72: trm @ A @ C] :
( Y
!= ( differential @ A @ C @ X72 ) ) ) ) ) ) ) ) ).
% trm.exhaust
thf(fact_53_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A5: A,B4: B,C2: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) ) ).
% prod_cases3
thf(fact_54_trm_Oinject_I2_J,axiom,
! [C: $tType,A: $tType,X22: real,Y2: real] :
( ( ( const @ A @ C @ X22 )
= ( const @ A @ C @ Y2 ) )
= ( X22 = Y2 ) ) ).
% trm.inject(2)
thf(fact_55_dfree__Const__simps,axiom,
! [A: $tType,B: $tType,R: real] : ( dfree @ A @ B @ ( const @ A @ B @ R ) ) ).
% dfree_Const_simps
thf(fact_56_dfree__Const,axiom,
! [A: $tType,C: $tType,R: real] : ( dfree @ A @ C @ ( const @ A @ C @ R ) ) ).
% dfree_Const
thf(fact_57_trm_Odistinct_I13_J,axiom,
! [A: $tType,C: $tType,X22: real,X31: A,X32: C > ( trm @ A @ C )] :
( ( const @ A @ C @ X22 )
!= ( function @ A @ C @ X31 @ X32 ) ) ).
% trm.distinct(13)
thf(fact_58_trm_Odistinct_I15_J,axiom,
! [C: $tType,A: $tType,X22: real,X41: trm @ A @ C,X42: trm @ A @ C] :
( ( const @ A @ C @ X22 )
!= ( plus @ A @ C @ X41 @ X42 ) ) ).
% trm.distinct(15)
thf(fact_59_trm_Odistinct_I17_J,axiom,
! [C: $tType,A: $tType,X22: real,X51: trm @ A @ C,X52: trm @ A @ C] :
( ( const @ A @ C @ X22 )
!= ( times @ A @ C @ X51 @ X52 ) ) ).
% trm.distinct(17)
thf(fact_60_trm_Odistinct_I1_J,axiom,
! [A: $tType,C: $tType,X1: C,X22: real] :
( ( var @ C @ A @ X1 )
!= ( const @ A @ C @ X22 ) ) ).
% trm.distinct(1)
thf(fact_61_trm_Odistinct_I21_J,axiom,
! [C: $tType,A: $tType,X22: real,X7: trm @ A @ C] :
( ( const @ A @ C @ X22 )
!= ( differential @ A @ C @ X7 ) ) ).
% trm.distinct(21)
thf(fact_62_trm_Odistinct_I19_J,axiom,
! [A: $tType,C: $tType,X22: real,X6: C] :
( ( const @ A @ C @ X22 )
!= ( diffVar @ C @ A @ X6 ) ) ).
% trm.distinct(19)
thf(fact_63_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A5: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_64_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A5: A,B4: B] :
( Y
!= ( product_Pair @ A @ B @ A5 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_65_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A4: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ~ ( ( A2 = A4 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_66_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A5: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_67_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_68_dfree_Oinducts,axiom,
! [C: $tType,A: $tType,X: trm @ A @ C,P: ( trm @ A @ C ) > $o] :
( ( dfree @ A @ C @ X )
=> ( ! [I3: C] : ( P @ ( var @ C @ A @ I3 ) )
=> ( ! [R2: real] : ( P @ ( const @ A @ C @ R2 ) )
=> ( ! [Args2: C > ( trm @ A @ C ),I3: A] :
( ! [Ia: C] : ( dfree @ A @ C @ ( Args2 @ Ia ) )
=> ( ! [Ia: C] : ( P @ ( Args2 @ Ia ) )
=> ( P @ ( function @ A @ C @ I3 @ Args2 ) ) ) )
=> ( ! [Theta_12: trm @ A @ C,Theta_22: trm @ A @ C] :
( ( dfree @ A @ C @ Theta_12 )
=> ( ( P @ Theta_12 )
=> ( ( dfree @ A @ C @ Theta_22 )
=> ( ( P @ Theta_22 )
=> ( P @ ( plus @ A @ C @ Theta_12 @ Theta_22 ) ) ) ) ) )
=> ( ! [Theta_12: trm @ A @ C,Theta_22: trm @ A @ C] :
( ( dfree @ A @ C @ Theta_12 )
=> ( ( P @ Theta_12 )
=> ( ( dfree @ A @ C @ Theta_22 )
=> ( ( P @ Theta_22 )
=> ( P @ ( times @ A @ C @ Theta_12 @ Theta_22 ) ) ) ) ) )
=> ( P @ X ) ) ) ) ) ) ) ).
% dfree.inducts
thf(fact_69_dfree_Osimps,axiom,
! [C: $tType,A: $tType] :
( ( dfree @ A @ C )
= ( ^ [A6: trm @ A @ C] :
( ? [I5: C] :
( A6
= ( var @ C @ A @ I5 ) )
| ? [R3: real] :
( A6
= ( const @ A @ C @ R3 ) )
| ? [Args3: C > ( trm @ A @ C ),I5: A] :
( ( A6
= ( function @ A @ C @ I5 @ Args3 ) )
& ! [X2: C] : ( dfree @ A @ C @ ( Args3 @ X2 ) ) )
| ? [Theta_13: trm @ A @ C,Theta_23: trm @ A @ C] :
( ( A6
= ( plus @ A @ C @ Theta_13 @ Theta_23 ) )
& ( dfree @ A @ C @ Theta_13 )
& ( dfree @ A @ C @ Theta_23 ) )
| ? [Theta_13: trm @ A @ C,Theta_23: trm @ A @ C] :
( ( A6
= ( times @ A @ C @ Theta_13 @ Theta_23 ) )
& ( dfree @ A @ C @ Theta_13 )
& ( dfree @ A @ C @ Theta_23 ) ) ) ) ) ).
% dfree.simps
thf(fact_70_dfree_Ocases,axiom,
! [C: $tType,A: $tType,A2: trm @ A @ C] :
( ( dfree @ A @ C @ A2 )
=> ( ! [I3: C] :
( A2
!= ( var @ C @ A @ I3 ) )
=> ( ! [R2: real] :
( A2
!= ( const @ A @ C @ R2 ) )
=> ( ! [Args2: C > ( trm @ A @ C )] :
( ? [I3: A] :
( A2
= ( function @ A @ C @ I3 @ Args2 ) )
=> ~ ! [I6: C] : ( dfree @ A @ C @ ( Args2 @ I6 ) ) )
=> ( ! [Theta_12: trm @ A @ C,Theta_22: trm @ A @ C] :
( ( A2
= ( plus @ A @ C @ Theta_12 @ Theta_22 ) )
=> ( ( dfree @ A @ C @ Theta_12 )
=> ~ ( dfree @ A @ C @ Theta_22 ) ) )
=> ~ ! [Theta_12: trm @ A @ C,Theta_22: trm @ A @ C] :
( ( A2
= ( times @ A @ C @ Theta_12 @ Theta_22 ) )
=> ( ( dfree @ A @ C @ Theta_12 )
=> ~ ( dfree @ A @ C @ Theta_22 ) ) ) ) ) ) ) ) ).
% dfree.cases
thf(fact_71_prod__induct7,axiom,
! [G2: $tType,F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
( ! [A5: A,B4: B,C2: C,D2: D,E2: E,F3: F2,G3: G2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G2 ) @ E2 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct7
thf(fact_72_prod__induct6,axiom,
! [F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
( ! [A5: A,B4: B,C2: C,D2: D,E2: E,F3: F2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct6
thf(fact_73_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A5: A,B4: B,C2: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct5
thf(fact_74_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A5: A,B4: B,C2: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_75_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,B4: B,C2: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_76_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,G2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
~ ! [A5: A,B4: B,C2: C,D2: D,E2: E,F3: F2,G3: G2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G2 ) @ E2 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_77_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
~ ! [A5: A,B4: B,C2: C,D2: D,E2: E,F3: F2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_78_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A5: A,B4: B,C2: C,D2: D,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_79_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A5: A,B4: B,C2: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_80_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
= ( F1 @ A2 @ B2 ) ) ).
% old.prod.rec
thf(fact_81_f0__def,axiom,
( ( f0 @ sf @ sz )
= ( ^ [F4: sf] : ( function @ sf @ sz @ F4 @ ( empty @ sz @ sf ) ) ) ) ).
% f0_def
thf(fact_82_dsafe_Oinducts,axiom,
! [C: $tType,A: $tType,X: trm @ A @ C,P: ( trm @ A @ C ) > $o] :
( ( dsafe @ A @ C @ X )
=> ( ! [I3: C] : ( P @ ( var @ C @ A @ I3 ) )
=> ( ! [R2: real] : ( P @ ( const @ A @ C @ R2 ) )
=> ( ! [Args2: C > ( trm @ A @ C ),I3: A] :
( ! [Ia: C] : ( dsafe @ A @ C @ ( Args2 @ Ia ) )
=> ( ! [Ia: C] : ( P @ ( Args2 @ Ia ) )
=> ( P @ ( function @ A @ C @ I3 @ Args2 ) ) ) )
=> ( ! [Theta_12: trm @ A @ C,Theta_22: trm @ A @ C] :
( ( dsafe @ A @ C @ Theta_12 )
=> ( ( P @ Theta_12 )
=> ( ( dsafe @ A @ C @ Theta_22 )
=> ( ( P @ Theta_22 )
=> ( P @ ( plus @ A @ C @ Theta_12 @ Theta_22 ) ) ) ) ) )
=> ( ! [Theta_12: trm @ A @ C,Theta_22: trm @ A @ C] :
( ( dsafe @ A @ C @ Theta_12 )
=> ( ( P @ Theta_12 )
=> ( ( dsafe @ A @ C @ Theta_22 )
=> ( ( P @ Theta_22 )
=> ( P @ ( times @ A @ C @ Theta_12 @ Theta_22 ) ) ) ) ) )
=> ( ! [Theta: trm @ A @ C] :
( ( dfree @ A @ C @ Theta )
=> ( P @ ( differential @ A @ C @ Theta ) ) )
=> ( ! [I3: C] : ( P @ ( diffVar @ C @ A @ I3 ) )
=> ( P @ X ) ) ) ) ) ) ) ) ) ).
% dsafe.inducts
thf(fact_83_dsafe_Osimps,axiom,
! [C: $tType,A: $tType] :
( ( dsafe @ A @ C )
= ( ^ [A6: trm @ A @ C] :
( ? [I5: C] :
( A6
= ( var @ C @ A @ I5 ) )
| ? [R3: real] :
( A6
= ( const @ A @ C @ R3 ) )
| ? [Args3: C > ( trm @ A @ C ),I5: A] :
( ( A6
= ( function @ A @ C @ I5 @ Args3 ) )
& ! [X2: C] : ( dsafe @ A @ C @ ( Args3 @ X2 ) ) )
| ? [Theta_13: trm @ A @ C,Theta_23: trm @ A @ C] :
( ( A6
= ( plus @ A @ C @ Theta_13 @ Theta_23 ) )
& ( dsafe @ A @ C @ Theta_13 )
& ( dsafe @ A @ C @ Theta_23 ) )
| ? [Theta_13: trm @ A @ C,Theta_23: trm @ A @ C] :
( ( A6
= ( times @ A @ C @ Theta_13 @ Theta_23 ) )
& ( dsafe @ A @ C @ Theta_13 )
& ( dsafe @ A @ C @ Theta_23 ) )
| ? [Theta2: trm @ A @ C] :
( ( A6
= ( differential @ A @ C @ Theta2 ) )
& ( dfree @ A @ C @ Theta2 ) )
| ? [I5: C] :
( A6
= ( diffVar @ C @ A @ I5 ) ) ) ) ) ).
% dsafe.simps
thf(fact_84_dsafe_Ocases,axiom,
! [A: $tType,C: $tType,A2: trm @ A @ C] :
( ( dsafe @ A @ C @ A2 )
=> ( ! [I3: C] :
( A2
!= ( var @ C @ A @ I3 ) )
=> ( ! [R2: real] :
( A2
!= ( const @ A @ C @ R2 ) )
=> ( ! [Args2: C > ( trm @ A @ C )] :
( ? [I3: A] :
( A2
= ( function @ A @ C @ I3 @ Args2 ) )
=> ~ ! [I6: C] : ( dsafe @ A @ C @ ( Args2 @ I6 ) ) )
=> ( ! [Theta_12: trm @ A @ C,Theta_22: trm @ A @ C] :
( ( A2
= ( plus @ A @ C @ Theta_12 @ Theta_22 ) )
=> ( ( dsafe @ A @ C @ Theta_12 )
=> ~ ( dsafe @ A @ C @ Theta_22 ) ) )
=> ( ! [Theta_12: trm @ A @ C,Theta_22: trm @ A @ C] :
( ( A2
= ( times @ A @ C @ Theta_12 @ Theta_22 ) )
=> ( ( dsafe @ A @ C @ Theta_12 )
=> ~ ( dsafe @ A @ C @ Theta_22 ) ) )
=> ( ! [Theta: trm @ A @ C] :
( ( A2
= ( differential @ A @ C @ Theta ) )
=> ~ ( dfree @ A @ C @ Theta ) )
=> ~ ! [I3: C] :
( A2
!= ( diffVar @ C @ A @ I3 ) ) ) ) ) ) ) ) ) ).
% dsafe.cases
thf(fact_85_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 ) ) ) )] :
~ ! [S2: A,S3: A,T3: A,T2: A,R4: A,R2: A] :
( X
!= ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) ) @ S2 @ ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) @ S3 @ ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) @ T3 @ ( product_Pair @ A @ ( product_prod @ A @ A ) @ T2 @ ( product_Pair @ A @ A @ R4 @ R2 ) ) ) ) ) ) ) ).
% euclid_ext_aux.cases
thf(fact_86_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B2: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( C3 @ A2 @ B2 ) ) ).
% internal_case_prod_conv
thf(fact_87_strm_Oright__unique,axiom,
! [C: $tType,A: $tType] : ( right_unique @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) ) ).
% strm.right_unique
thf(fact_88_dsafe__Const__simps,axiom,
! [A: $tType,B: $tType,R: real] : ( dsafe @ A @ B @ ( const @ A @ B @ R ) ) ).
% dsafe_Const_simps
thf(fact_89_dsafe__Fun__simps,axiom,
! [A: $tType,B: $tType,I4: A,Args: B > ( trm @ A @ B )] :
( ( dsafe @ A @ B @ ( function @ A @ B @ I4 @ Args ) )
= ( ! [X2: B] : ( dsafe @ A @ B @ ( Args @ X2 ) ) ) ) ).
% dsafe_Fun_simps
thf(fact_90_dsafe__Plus__simps,axiom,
! [B: $tType,A: $tType,A2: trm @ A @ B,B2: trm @ A @ B] :
( ( dsafe @ A @ B @ ( plus @ A @ B @ A2 @ B2 ) )
= ( ( dsafe @ A @ B @ A2 )
& ( dsafe @ A @ B @ B2 ) ) ) ).
% dsafe_Plus_simps
thf(fact_91_dsafe__Times__simps,axiom,
! [B: $tType,A: $tType,A2: trm @ A @ B,B2: trm @ A @ B] :
( ( dsafe @ A @ B @ ( times @ A @ B @ A2 @ B2 ) )
= ( ( dsafe @ A @ B @ A2 )
& ( dsafe @ A @ B @ B2 ) ) ) ).
% dsafe_Times_simps
thf(fact_92_dsafe__Var__simps,axiom,
! [A: $tType,B: $tType,X: B] : ( dsafe @ A @ B @ ( var @ B @ A @ X ) ) ).
% dsafe_Var_simps
thf(fact_93_dsafe__DiffVar__simps,axiom,
! [A: $tType,B: $tType,X: B] : ( dsafe @ A @ B @ ( diffVar @ B @ A @ X ) ) ).
% dsafe_DiffVar_simps
thf(fact_94_dsafe__Diff__simps,axiom,
! [B: $tType,A: $tType,A2: trm @ A @ B] :
( ( dsafe @ A @ B @ ( differential @ A @ B @ A2 ) )
= ( dfree @ A @ B @ A2 ) ) ).
% dsafe_Diff_simps
thf(fact_95_dfree__is__dsafe,axiom,
! [B: $tType,A: $tType,Theta3: trm @ A @ B] :
( ( dfree @ A @ B @ Theta3 )
=> ( dsafe @ A @ B @ Theta3 ) ) ).
% dfree_is_dsafe
thf(fact_96_dsafe__Const,axiom,
! [A: $tType,C: $tType,R: real] : ( dsafe @ A @ C @ ( const @ A @ C @ R ) ) ).
% dsafe_Const
thf(fact_97_dsafe__Fun,axiom,
! [A: $tType,C: $tType,Args: C > ( trm @ A @ C ),I4: A] :
( ! [I3: C] : ( dsafe @ A @ C @ ( Args @ I3 ) )
=> ( dsafe @ A @ C @ ( function @ A @ C @ I4 @ Args ) ) ) ).
% dsafe_Fun
thf(fact_98_dsafe__Plus,axiom,
! [C: $tType,A: $tType,Theta_1: trm @ A @ C,Theta_2: trm @ A @ C] :
( ( dsafe @ A @ C @ Theta_1 )
=> ( ( dsafe @ A @ C @ Theta_2 )
=> ( dsafe @ A @ C @ ( plus @ A @ C @ Theta_1 @ Theta_2 ) ) ) ) ).
% dsafe_Plus
thf(fact_99_dsafe__Times,axiom,
! [C: $tType,A: $tType,Theta_1: trm @ A @ C,Theta_2: trm @ A @ C] :
( ( dsafe @ A @ C @ Theta_1 )
=> ( ( dsafe @ A @ C @ Theta_2 )
=> ( dsafe @ A @ C @ ( times @ A @ C @ Theta_1 @ Theta_2 ) ) ) ) ).
% dsafe_Times
thf(fact_100_dsafe__Var,axiom,
! [A: $tType,C: $tType,I4: C] : ( dsafe @ A @ C @ ( var @ C @ A @ I4 ) ) ).
% dsafe_Var
thf(fact_101_dsafe__DiffVar,axiom,
! [A: $tType,C: $tType,I4: C] : ( dsafe @ A @ C @ ( diffVar @ C @ A @ I4 ) ) ).
% dsafe_DiffVar
thf(fact_102_dsafe__Diff,axiom,
! [C: $tType,A: $tType,Theta3: trm @ A @ C] :
( ( dfree @ A @ C @ Theta3 )
=> ( dsafe @ A @ C @ ( differential @ A @ C @ Theta3 ) ) ) ).
% dsafe_Diff
thf(fact_103_gcd_Ocases,axiom,
! [A: $tType] :
( ( euclid1155270486miring @ A )
=> ! [X: product_prod @ A @ A] :
~ ! [A5: A,B4: A] :
( X
!= ( product_Pair @ A @ A @ A5 @ B4 ) ) ) ).
% gcd.cases
thf(fact_104_local_Oempty__def,axiom,
! [A: $tType,B: $tType] :
( ( empty @ B @ A )
= ( ^ [I5: B] : ( const @ A @ B @ ( zero_zero @ real ) ) ) ) ).
% local.empty_def
thf(fact_105_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_106_dfree__Fun_OIH,axiom,
! [I: c] :
( ( denotational_Vagree @ c @ nu @ nu2 @ ( static_FVDiff @ a @ c @ ( args @ I ) ) )
=> ( ( denotational_frechet @ a @ b @ c @ i @ ( args @ I ) @ ( 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 @ i @ ( args @ I ) @ ( 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 ) ) ) ) ).
% dfree_Fun.IH
thf(fact_107_strm_Oleft__unique,axiom,
! [C: $tType,A: $tType] : ( left_unique @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) ) ).
% strm.left_unique
thf(fact_108_trm_Oinduct,axiom,
! [C: $tType,A: $tType,P: ( trm @ A @ C ) > $o,Trm: trm @ A @ C] :
( ! [X3: C] : ( P @ ( var @ C @ A @ X3 ) )
=> ( ! [X3: real] : ( P @ ( const @ A @ C @ X3 ) )
=> ( ! [X1a: A,X2a: C > ( trm @ A @ C )] :
( ! [X2aa: trm @ A @ C] :
( ( member @ ( trm @ A @ C ) @ X2aa @ ( image @ C @ ( trm @ A @ C ) @ X2a @ ( top_top @ ( set @ C ) ) ) )
=> ( P @ X2aa ) )
=> ( P @ ( function @ A @ C @ X1a @ X2a ) ) )
=> ( ! [X1a: trm @ A @ C,X2a: trm @ A @ C] :
( ( P @ X1a )
=> ( ( P @ X2a )
=> ( P @ ( plus @ A @ C @ X1a @ X2a ) ) ) )
=> ( ! [X1a: trm @ A @ C,X2a: trm @ A @ C] :
( ( P @ X1a )
=> ( ( P @ X2a )
=> ( P @ ( times @ A @ C @ X1a @ X2a ) ) ) )
=> ( ! [X3: C] : ( P @ ( diffVar @ C @ A @ X3 ) )
=> ( ! [X3: trm @ A @ C] :
( ( P @ X3 )
=> ( P @ ( differential @ A @ C @ X3 ) ) )
=> ( P @ Trm ) ) ) ) ) ) ) ) ).
% trm.induct
thf(fact_109_agree__comm,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [A3: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),B5: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),V: set @ ( sum_sum @ A @ A )] :
( ( denotational_Vagree @ A @ A3 @ B5 @ V )
=> ( denotational_Vagree @ A @ B5 @ A3 @ V ) ) ) ).
% agree_comm
thf(fact_110_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_111_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_112_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_113_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_114_Vagree__univ,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [A2: finite_Cartesian_vec @ real @ A,B2: finite_Cartesian_vec @ real @ A,C3: finite_Cartesian_vec @ real @ A,D3: finite_Cartesian_vec @ real @ A] :
( ( denotational_Vagree @ A @ ( product_Pair @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) @ A2 @ B2 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) @ C3 @ D3 ) @ ( top_top @ ( set @ ( sum_sum @ A @ A ) ) ) )
=> ( ( A2 = C3 )
& ( B2 = D3 ) ) ) ) ).
% Vagree_univ
thf(fact_115_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_116_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,A1: Sz] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ! [T2: trm @ A @ Sz,X_1: Sz] : ( P @ T2 @ X_1 )
=> ( P @ A0 @ A1 ) ) ) ) ).
% ids.singleton.induct
thf(fact_117_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,A1: product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) )] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ! [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 @ A1 ) ) ) ) ).
% ids.seq_sem.induct
thf(fact_118_frechet_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( finite_finite @ C )
& ( finite_finite @ A )
& ( finite_finite @ B ) )
=> ! [I7: denota1663640101rp_ext @ A @ B @ C @ product_unit,R: real,V2: finite_Cartesian_vec @ real @ C] :
( ( denotational_frechet @ A @ B @ C @ I7 @ ( const @ A @ C @ R ) @ V2 )
= ( ^ [V3: finite_Cartesian_vec @ real @ C] : ( zero_zero @ real ) ) ) ) ).
% frechet.simps(5)
thf(fact_119_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_120_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_121_prod__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( ^ [Y4: product_prod @ A @ B,Z: product_prod @ A @ B] : Y4 = Z )
= ( ^ [S4: product_prod @ A @ B,T5: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ S4 )
= ( product_fst @ A @ B @ T5 ) )
& ( ( product_snd @ A @ B @ S4 )
= ( product_snd @ A @ B @ T5 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_122_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_123_snd__eqD,axiom,
! [B: $tType,A: $tType,X: B,Y: A,A2: A] :
( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_124_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_125_fst__eqD,axiom,
! [B: $tType,A: $tType,X: A,Y: B,A2: A] :
( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_126_ids_Oempty__def,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType,A: $tType,B: $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 )
=> ( ( empty @ B @ A )
= ( ^ [I5: B] : ( const @ A @ B @ ( zero_zero @ real ) ) ) ) ) ) ).
% ids.empty_def
thf(fact_127_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 )
=> ~ ! [T2: trm @ A @ Sz,I3: Sz] :
( X
!= ( product_Pair @ ( trm @ A @ Sz ) @ Sz @ T2 @ I3 ) ) ) ) ).
% ids.singleton.cases
thf(fact_128_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_129_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,I3: A] :
( X
!= ( product_Pair @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ A @ I2 @ I3 ) ) ) ).
% FunctionFrechet.cases
thf(fact_130_agree__refl,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),A3: set @ ( sum_sum @ A @ A )] : ( denotational_Vagree @ A @ Nu @ Nu @ A3 ) ) ).
% agree_refl
thf(fact_131_ids_Ostrm_Odomain,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 )
=> ( ( domainp @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) )
= ( dfree @ A @ C ) ) ) ) ).
% ids.strm.domain
thf(fact_132_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_133_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_134_snd__zero,axiom,
! [B: $tType,A: $tType] :
( ( ( zero @ A )
& ( zero @ B ) )
=> ( ( product_snd @ B @ A @ ( zero_zero @ ( product_prod @ B @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% snd_zero
thf(fact_135_fst__zero,axiom,
! [B: $tType,A: $tType] :
( ( ( zero @ A )
& ( zero @ B ) )
=> ( ( product_fst @ A @ B @ ( zero_zero @ ( product_prod @ A @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% fst_zero
thf(fact_136_zero__prod__def,axiom,
! [A: $tType,B: $tType] :
( ( ( zero @ B )
& ( zero @ A ) )
=> ( ( zero_zero @ ( product_prod @ A @ B ) )
= ( product_Pair @ A @ B @ ( zero_zero @ A ) @ ( zero_zero @ B ) ) ) ) ).
% zero_prod_def
thf(fact_137_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_138_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_139_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_140_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_141_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_142_ids_Ostrm_Oright__unique,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 )
=> ( right_unique @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) ) ) ) ).
% ids.strm.right_unique
thf(fact_143_ids_Ostrm_Oleft__unique,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 )
=> ( left_unique @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) ) ) ) ).
% ids.strm.left_unique
thf(fact_144_directional__derivative__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ( ( denota2078997598vative @ A @ B @ C )
= ( ^ [I8: denota1663640101rp_ext @ A @ B @ C @ product_unit,T5: trm @ A @ C,V4: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] : ( denotational_frechet @ A @ B @ C @ I8 @ T5 @ ( product_fst @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ V4 ) @ ( product_snd @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) @ V4 ) ) ) ) ) ).
% directional_derivative_def
thf(fact_145_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P: A > B > $o,X: A,Y: B,A2: product_prod @ A @ B] :
( ( P @ X @ Y )
=> ( ( A2
= ( product_Pair @ A @ B @ X @ Y ) )
=> ( P @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_146_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_147_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_148_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_149_iso__tuple__UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% iso_tuple_UNIV_I
thf(fact_150_UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_151_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X2: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_152_image__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,X: B,A3: set @ B] :
( ( B2
= ( F @ X ) )
=> ( ( member @ B @ X @ A3 )
=> ( member @ A @ B2 @ ( image @ B @ A @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_153_imageI,axiom,
! [B: $tType,A: $tType,X: A,A3: set @ A,F: A > B] :
( ( member @ A @ X @ A3 )
=> ( member @ B @ ( F @ X ) @ ( image @ A @ B @ F @ A3 ) ) ) ).
% imageI
thf(fact_154_image__iff,axiom,
! [A: $tType,B: $tType,Z2: A,F: B > A,A3: set @ B] :
( ( member @ A @ Z2 @ ( image @ B @ A @ F @ A3 ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_155_bex__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A3: set @ B,P: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ ( image @ B @ A @ F @ A3 ) )
& ( P @ X4 ) )
=> ? [X3: B] :
( ( member @ B @ X3 @ A3 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_156_image__cong,axiom,
! [B: $tType,A: $tType,M: set @ A,N: set @ A,F: A > B,G: A > B] :
( ( M = N )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image @ A @ B @ F @ M )
= ( image @ A @ B @ G @ N ) ) ) ) ).
% image_cong
thf(fact_157_ball__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A3: set @ B,P: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( image @ B @ A @ F @ A3 ) )
=> ( P @ X3 ) )
=> ! [X4: B] :
( ( member @ B @ X4 @ A3 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_158_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X: A,A3: set @ A,B2: B,F: A > B] :
( ( member @ A @ X @ A3 )
=> ( ( B2
= ( F @ X ) )
=> ( member @ B @ B2 @ ( image @ A @ B @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_159_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_160_UNIV__eq__I,axiom,
! [A: $tType,A3: set @ A] :
( ! [X3: A] : ( member @ A @ X3 @ A3 )
=> ( ( top_top @ ( set @ A ) )
= A3 ) ) ).
% UNIV_eq_I
thf(fact_161_UNIV__witness,axiom,
! [A: $tType] :
? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_162_rangeI,axiom,
! [A: $tType,B: $tType,F: B > A,X: B] : ( member @ A @ ( F @ X ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% rangeI
thf(fact_163_range__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,X: B] :
( ( B2
= ( F @ X ) )
=> ( member @ A @ B2 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_eqI
thf(fact_164_sndI,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B,Y: A,Z2: B] :
( ( X
= ( product_Pair @ A @ B @ Y @ Z2 ) )
=> ( ( product_snd @ A @ B @ X )
= Z2 ) ) ).
% sndI
thf(fact_165_eq__snd__iff,axiom,
! [A: $tType,B: $tType,B2: A,P2: product_prod @ B @ A] :
( ( B2
= ( product_snd @ B @ A @ P2 ) )
= ( ? [A6: B] :
( P2
= ( product_Pair @ B @ A @ A6 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_166_top1I,axiom,
! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% top1I
thf(fact_167_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_168_eq__fst__iff,axiom,
! [A: $tType,B: $tType,A2: A,P2: product_prod @ A @ B] :
( ( A2
= ( product_fst @ A @ B @ P2 ) )
= ( ? [B6: B] :
( P2
= ( product_Pair @ A @ B @ A2 @ B6 ) ) ) ) ).
% eq_fst_iff
thf(fact_169_fstI,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,Y: A,Z2: B] :
( ( X
= ( product_Pair @ A @ B @ Y @ Z2 ) )
=> ( ( product_fst @ A @ B @ X )
= Y ) ) ).
% fstI
thf(fact_170_range__snd,axiom,
! [B: $tType,A: $tType] :
( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% range_snd
thf(fact_171_range__fst,axiom,
! [B: $tType,A: $tType] :
( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% range_fst
thf(fact_172_Domain__eq__top,axiom,
! [A: $tType] :
( ( domainp @ A @ A
@ ^ [Y4: A,Z: A] : Y4 = Z )
= ( top_top @ ( A > $o ) ) ) ).
% Domain_eq_top
thf(fact_173_DomainpE,axiom,
! [A: $tType,B: $tType,R: A > B > $o,A2: A] :
( ( domainp @ A @ B @ R @ A2 )
=> ~ ! [B4: B] :
~ ( R @ A2 @ B4 ) ) ).
% DomainpE
thf(fact_174_Domainp__iff,axiom,
! [B: $tType,A: $tType] :
( ( domainp @ A @ B )
= ( ^ [T6: A > B > $o,X2: A] :
? [X5: B] : ( T6 @ X2 @ X5 ) ) ) ).
% Domainp_iff
thf(fact_175_Domainp__refl,axiom,
! [B: $tType,A: $tType] :
( ( domainp @ A @ B )
= ( domainp @ A @ B ) ) ).
% Domainp_refl
thf(fact_176_Domainp_Ocases,axiom,
! [A: $tType,B: $tType,R: A > B > $o,A2: A] :
( ( domainp @ A @ B @ R @ A2 )
=> ~ ! [B4: B] :
~ ( R @ A2 @ B4 ) ) ).
% Domainp.cases
thf(fact_177_Domainp_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( domainp @ A @ B )
= ( ^ [R3: A > B > $o,A6: A] :
? [B6: A,C4: B] :
( ( A6 = B6 )
& ( R3 @ B6 @ C4 ) ) ) ) ).
% Domainp.simps
thf(fact_178_Domainp_ODomainI,axiom,
! [B: $tType,A: $tType,R: A > B > $o,A2: A,B2: B] :
( ( R @ A2 @ B2 )
=> ( domainp @ A @ B @ R @ A2 ) ) ).
% Domainp.DomainI
thf(fact_179_Domainp_Oinducts,axiom,
! [B: $tType,A: $tType,R: A > B > $o,X: A,P: A > $o] :
( ( domainp @ A @ B @ R @ X )
=> ( ! [A5: A,B4: B] :
( ( R @ A5 @ B4 )
=> ( P @ A5 ) )
=> ( P @ X ) ) ) ).
% Domainp.inducts
thf(fact_180_right__uniqueD,axiom,
! [A: $tType,B: $tType,A3: A > B > $o,X: A,Y: B,Z2: B] :
( ( right_unique @ A @ B @ A3 )
=> ( ( A3 @ X @ Y )
=> ( ( A3 @ X @ Z2 )
=> ( Y = Z2 ) ) ) ) ).
% right_uniqueD
thf(fact_181_right__uniqueI,axiom,
! [B: $tType,A: $tType,A3: A > B > $o] :
( ! [X3: A,Y3: B,Z3: B] :
( ( A3 @ X3 @ Y3 )
=> ( ( A3 @ X3 @ Z3 )
=> ( Y3 = Z3 ) ) )
=> ( right_unique @ A @ B @ A3 ) ) ).
% right_uniqueI
thf(fact_182_right__unique__eq,axiom,
! [A: $tType] :
( right_unique @ A @ A
@ ^ [Y4: A,Z: A] : Y4 = Z ) ).
% right_unique_eq
thf(fact_183_right__unique__def,axiom,
! [B: $tType,A: $tType] :
( ( right_unique @ A @ B )
= ( ^ [R5: A > B > $o] :
! [X2: A,Y5: B,Z4: B] :
( ( R5 @ X2 @ Y5 )
=> ( ( R5 @ X2 @ Z4 )
=> ( Y5 = Z4 ) ) ) ) ) ).
% right_unique_def
thf(fact_184_left__uniqueD,axiom,
! [B: $tType,A: $tType,A3: A > B > $o,X: A,Z2: B,Y: A] :
( ( left_unique @ A @ B @ A3 )
=> ( ( A3 @ X @ Z2 )
=> ( ( A3 @ Y @ Z2 )
=> ( X = Y ) ) ) ) ).
% left_uniqueD
thf(fact_185_left__uniqueI,axiom,
! [B: $tType,A: $tType,A3: A > B > $o] :
( ! [X3: A,Y3: A,Z3: B] :
( ( A3 @ X3 @ Z3 )
=> ( ( A3 @ Y3 @ Z3 )
=> ( X3 = Y3 ) ) )
=> ( left_unique @ A @ B @ A3 ) ) ).
% left_uniqueI
thf(fact_186_left__unique__eq,axiom,
! [A: $tType] :
( left_unique @ A @ A
@ ^ [Y4: A,Z: A] : Y4 = Z ) ).
% left_unique_eq
thf(fact_187_left__unique__def,axiom,
! [B: $tType,A: $tType] :
( ( left_unique @ A @ B )
= ( ^ [R5: A > B > $o] :
! [X2: A,Y5: A,Z4: B] :
( ( R5 @ X2 @ Z4 )
=> ( ( R5 @ Y5 @ Z4 )
=> ( X2 = Y5 ) ) ) ) ) ).
% left_unique_def
thf(fact_188_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_189_trm_Oset__cases,axiom,
! [C: $tType,A: $tType,E3: A,A2: trm @ A @ C] :
( ( member @ A @ E3 @ ( set_trm @ A @ C @ A2 ) )
=> ( ! [Z22: C > ( trm @ A @ C )] :
( A2
!= ( function @ A @ C @ E3 @ Z22 ) )
=> ( ! [Z1: A,Z22: C > ( trm @ A @ C )] :
( ( A2
= ( function @ A @ C @ Z1 @ Z22 ) )
=> ! [X3: trm @ A @ C] :
( ( member @ ( trm @ A @ C ) @ X3 @ ( image @ C @ ( trm @ A @ C ) @ Z22 @ ( top_top @ ( set @ C ) ) ) )
=> ~ ( member @ A @ E3 @ ( set_trm @ A @ C @ X3 ) ) ) )
=> ( ! [Z1: trm @ A @ C] :
( ? [Z22: trm @ A @ C] :
( A2
= ( plus @ A @ C @ Z1 @ Z22 ) )
=> ~ ( member @ A @ E3 @ ( set_trm @ A @ C @ Z1 ) ) )
=> ( ! [Z1: trm @ A @ C,Z22: trm @ A @ C] :
( ( A2
= ( plus @ A @ C @ Z1 @ Z22 ) )
=> ~ ( member @ A @ E3 @ ( set_trm @ A @ C @ Z22 ) ) )
=> ( ! [Z1: trm @ A @ C] :
( ? [Z22: trm @ A @ C] :
( A2
= ( times @ A @ C @ Z1 @ Z22 ) )
=> ~ ( member @ A @ E3 @ ( set_trm @ A @ C @ Z1 ) ) )
=> ( ! [Z1: trm @ A @ C,Z22: trm @ A @ C] :
( ( A2
= ( times @ A @ C @ Z1 @ Z22 ) )
=> ~ ( member @ A @ E3 @ ( set_trm @ A @ C @ Z22 ) ) )
=> ~ ! [Z3: trm @ A @ C] :
( ( A2
= ( differential @ A @ C @ Z3 ) )
=> ~ ( member @ A @ E3 @ ( set_trm @ A @ C @ Z3 ) ) ) ) ) ) ) ) ) ) ).
% trm.set_cases
thf(fact_190_surjD,axiom,
! [A: $tType,B: $tType,F: B > A,Y: A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ? [X3: B] :
( Y
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_191_surjE,axiom,
! [A: $tType,B: $tType,F: B > A,Y: A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ~ ! [X3: B] :
( Y
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_192_trm_Oset__intros_I1_J,axiom,
! [A: $tType,C: $tType,X31: A,X32: C > ( trm @ A @ C )] : ( member @ A @ X31 @ ( set_trm @ A @ C @ ( function @ A @ C @ X31 @ X32 ) ) ) ).
% trm.set_intros(1)
thf(fact_193_trm_Oset__intros_I4_J,axiom,
! [C: $tType,A: $tType,Yc: A,X42: trm @ A @ C,X41: trm @ A @ C] :
( ( member @ A @ Yc @ ( set_trm @ A @ C @ X42 ) )
=> ( member @ A @ Yc @ ( set_trm @ A @ C @ ( plus @ A @ C @ X41 @ X42 ) ) ) ) ).
% trm.set_intros(4)
thf(fact_194_trm_Oset__intros_I3_J,axiom,
! [C: $tType,A: $tType,Yb: A,X41: trm @ A @ C,X42: trm @ A @ C] :
( ( member @ A @ Yb @ ( set_trm @ A @ C @ X41 ) )
=> ( member @ A @ Yb @ ( set_trm @ A @ C @ ( plus @ A @ C @ X41 @ X42 ) ) ) ) ).
% trm.set_intros(3)
thf(fact_195_trm_Oset__intros_I6_J,axiom,
! [C: $tType,A: $tType,Ye: A,X52: trm @ A @ C,X51: trm @ A @ C] :
( ( member @ A @ Ye @ ( set_trm @ A @ C @ X52 ) )
=> ( member @ A @ Ye @ ( set_trm @ A @ C @ ( times @ A @ C @ X51 @ X52 ) ) ) ) ).
% trm.set_intros(6)
thf(fact_196_trm_Oset__intros_I5_J,axiom,
! [C: $tType,A: $tType,Yd: A,X51: trm @ A @ C,X52: trm @ A @ C] :
( ( member @ A @ Yd @ ( set_trm @ A @ C @ X51 ) )
=> ( member @ A @ Yd @ ( set_trm @ A @ C @ ( times @ A @ C @ X51 @ X52 ) ) ) ) ).
% trm.set_intros(5)
thf(fact_197_trm_Osimps_I126_J,axiom,
! [C: $tType,A: $tType,X7: trm @ A @ C] :
( ( set_trm @ A @ C @ ( differential @ A @ C @ X7 ) )
= ( set_trm @ A @ C @ X7 ) ) ).
% trm.simps(126)
thf(fact_198_trm_Oset__intros_I7_J,axiom,
! [C: $tType,A: $tType,Yf: A,X7: trm @ A @ C] :
( ( member @ A @ Yf @ ( set_trm @ A @ C @ X7 ) )
=> ( member @ A @ Yf @ ( set_trm @ A @ C @ ( differential @ A @ C @ X7 ) ) ) ) ).
% trm.set_intros(7)
thf(fact_199_trm_Oset__intros_I2_J,axiom,
! [A: $tType,C: $tType,Y: trm @ A @ C,X32: C > ( trm @ A @ C ),Ya: A,X31: A] :
( ( member @ ( trm @ A @ C ) @ Y @ ( image @ C @ ( trm @ A @ C ) @ X32 @ ( top_top @ ( set @ C ) ) ) )
=> ( ( member @ A @ Ya @ ( set_trm @ A @ C @ Y ) )
=> ( member @ A @ Ya @ ( set_trm @ A @ C @ ( function @ A @ C @ X31 @ X32 ) ) ) ) ) ).
% trm.set_intros(2)
thf(fact_200_surj__def,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [Y5: A] :
? [X2: B] :
( Y5
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_201_surjI,axiom,
! [B: $tType,A: $tType,G: B > A,F: A > B] :
( ! [X3: A] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surjI
thf(fact_202_ids_Oexpand__singleton,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,T4: trm @ A @ Sz] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( singleton @ Sz @ A @ Vid1 @ T4 )
= ( ^ [I5: Sz] : ( if @ ( trm @ A @ Sz ) @ ( I5 = Vid1 ) @ T4 @ ( const @ A @ Sz @ ( zero_zero @ real ) ) ) ) ) ) ) ).
% ids.expand_singleton
thf(fact_203_ids_Osingleton_Osimps,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,I4: Sz,T4: trm @ A @ Sz] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( ( I4 = Vid1 )
=> ( ( singleton @ Sz @ A @ Vid1 @ T4 @ I4 )
= T4 ) )
& ( ( I4 != Vid1 )
=> ( ( singleton @ Sz @ A @ Vid1 @ T4 @ I4 )
= ( const @ A @ Sz @ ( zero_zero @ real ) ) ) ) ) ) ) ).
% ids.singleton.simps
thf(fact_204_ids_Oproj__sing1,axiom,
! [Sc: $tType,Sf: $tType,Sz: $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,Theta3: trm @ A @ Sz] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( singleton @ Sz @ A @ Vid1 @ Theta3 @ Vid1 )
= Theta3 ) ) ) ).
% ids.proj_sing1
thf(fact_205_ids_Osingleton_Ocong,axiom,
! [A: $tType,Sz: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz ) )
=> ( ( singleton @ Sz @ A )
= ( singleton @ Sz @ A ) ) ) ).
% ids.singleton.cong
thf(fact_206_ids_Oproj__sing2,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,Y: Sz,Theta3: trm @ A @ Sz] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( Vid1 != Y )
=> ( ( singleton @ Sz @ A @ Vid1 @ Theta3 @ Y )
= ( const @ A @ Sz @ ( zero_zero @ real ) ) ) ) ) ) ).
% ids.proj_sing2
thf(fact_207_ids_Osingleton_Oelims,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: trm @ A @ Sz,Xa: Sz,Y: trm @ A @ Sz] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( ( singleton @ Sz @ A @ Vid1 @ X @ Xa )
= Y )
=> ( ( ( Xa = Vid1 )
=> ( Y = X ) )
& ( ( Xa != Vid1 )
=> ( Y
= ( const @ A @ Sz @ ( zero_zero @ real ) ) ) ) ) ) ) ) ).
% ids.singleton.elims
thf(fact_208_ids_Of1__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,X: Sz] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( f1 @ Sz @ Sf @ Vid1 @ F @ X )
= ( function @ Sf @ Sz @ F @ ( singleton @ Sz @ Sf @ Vid1 @ ( var @ Sz @ Sf @ X ) ) ) ) ) ) ).
% ids.f1_def
thf(fact_209_ids_Osingleton_Opelims,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: trm @ A @ Sz,Xa: Sz,Y: trm @ A @ Sz] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( ( singleton @ Sz @ A @ Vid1 @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( trm @ A @ Sz ) @ Sz ) @ ( singleton_rel @ A @ Sz ) @ ( product_Pair @ ( trm @ A @ Sz ) @ Sz @ X @ Xa ) )
=> ~ ( ( ( ( Xa = Vid1 )
=> ( Y = X ) )
& ( ( Xa != Vid1 )
=> ( Y
= ( const @ A @ Sz @ ( zero_zero @ real ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( trm @ A @ Sz ) @ Sz ) @ ( singleton_rel @ A @ Sz ) @ ( product_Pair @ ( trm @ A @ Sz ) @ Sz @ X @ Xa ) ) ) ) ) ) ) ).
% ids.singleton.pelims
thf(fact_210_ids_Of1_Ocong,axiom,
! [Sf: $tType,Sz: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf ) )
=> ( ( f1 @ Sz @ Sf )
= ( f1 @ Sz @ Sf ) ) ) ).
% ids.f1.cong
thf(fact_211_singleton_Opelims,axiom,
! [A: $tType,X: trm @ A @ sz,Xa: sz,Y: trm @ A @ sz] :
( ( ( singleton @ sz @ A @ vid1 @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( trm @ A @ sz ) @ sz ) @ ( singleton_rel @ A @ sz ) @ ( product_Pair @ ( trm @ A @ sz ) @ sz @ X @ Xa ) )
=> ~ ( ( ( ( Xa = vid1 )
=> ( Y = X ) )
& ( ( Xa != vid1 )
=> ( Y
= ( const @ A @ sz @ ( zero_zero @ real ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( trm @ A @ sz ) @ sz ) @ ( singleton_rel @ A @ sz ) @ ( product_Pair @ ( trm @ A @ sz ) @ sz @ X @ Xa ) ) ) ) ) ).
% singleton.pelims
thf(fact_212_f1__def,axiom,
! [F: sf,X: sz] :
( ( f1 @ sz @ sf @ vid1 @ F @ X )
= ( function @ sf @ sz @ F @ ( singleton @ sz @ sf @ vid1 @ ( var @ sz @ sf @ X ) ) ) ) ).
% f1_def
thf(fact_213_proj__sing1,axiom,
! [A: $tType,Theta3: trm @ A @ sz] :
( ( singleton @ sz @ A @ vid1 @ Theta3 @ vid1 )
= Theta3 ) ).
% proj_sing1
thf(fact_214_expand__singleton,axiom,
! [A: $tType,T4: trm @ A @ sz] :
( ( singleton @ sz @ A @ vid1 @ T4 )
= ( ^ [I5: sz] : ( if @ ( trm @ A @ sz ) @ ( I5 = vid1 ) @ T4 @ ( const @ A @ sz @ ( zero_zero @ real ) ) ) ) ) ).
% expand_singleton
thf(fact_215_proj__sing2,axiom,
! [A: $tType,Y: sz,Theta3: trm @ A @ sz] :
( ( vid1 != Y )
=> ( ( singleton @ sz @ A @ vid1 @ Theta3 @ Y )
= ( const @ A @ sz @ ( zero_zero @ real ) ) ) ) ).
% proj_sing2
thf(fact_216_singleton_Oelims,axiom,
! [A: $tType,X: trm @ A @ sz,Xa: sz,Y: trm @ A @ sz] :
( ( ( singleton @ sz @ A @ vid1 @ X @ Xa )
= Y )
=> ( ( ( Xa = vid1 )
=> ( Y = X ) )
& ( ( Xa != vid1 )
=> ( Y
= ( const @ A @ sz @ ( zero_zero @ real ) ) ) ) ) ) ).
% singleton.elims
thf(fact_217_singleton_Osimps,axiom,
! [A: $tType,I4: sz,T4: trm @ A @ sz] :
( ( ( I4 = vid1 )
=> ( ( singleton @ sz @ A @ vid1 @ T4 @ I4 )
= T4 ) )
& ( ( I4 != vid1 )
=> ( ( singleton @ sz @ A @ vid1 @ T4 @ I4 )
= ( const @ A @ sz @ ( zero_zero @ real ) ) ) ) ) ).
% singleton.simps
thf(fact_218_local_Oid__simps_I15_J,axiom,
vid3 != vid1 ).
% local.id_simps(15)
thf(fact_219_local_Oid__simps_I13_J,axiom,
vid2 != vid1 ).
% local.id_simps(13)
thf(fact_220_vne13,axiom,
vid1 != vid3 ).
% vne13
thf(fact_221_vne12,axiom,
vid1 != vid2 ).
% vne12
thf(fact_222_local_Oid__simps_I1_J,axiom,
vid1 != vid2 ).
% local.id_simps(1)
thf(fact_223_local_Oid__simps_I3_J,axiom,
vid1 != vid3 ).
% local.id_simps(3)
thf(fact_224_vne23,axiom,
vid2 != vid3 ).
% vne23
thf(fact_225_local_Oid__simps_I2_J,axiom,
vid2 != vid3 ).
% local.id_simps(2)
thf(fact_226_local_Oid__simps_I14_J,axiom,
vid3 != vid2 ).
% local.id_simps(14)
thf(fact_227_p1__def,axiom,
! [P2: sz,X: sz] :
( ( p1 @ sz @ sf @ sc @ vid1 @ P2 @ X )
= ( prop @ sz @ sf @ sc @ P2 @ ( singleton @ sz @ sf @ vid1 @ ( var @ sz @ sf @ X ) ) ) ) ).
% p1_def
thf(fact_228_prod_Oswap__def,axiom,
! [B: $tType,A: $tType] :
( ( product_swap @ A @ B )
= ( ^ [P3: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P3 ) @ ( product_fst @ A @ B @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_229_formula_Oinject_I2_J,axiom,
! [B: $tType,A: $tType,C: $tType,X21: C,X222: C > ( trm @ A @ C ),Y21: C,Y22: C > ( trm @ A @ C )] :
( ( ( prop @ C @ A @ B @ X21 @ X222 )
= ( prop @ C @ A @ B @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% formula.inject(2)
thf(fact_230_swap__swap,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P2 ) )
= P2 ) ).
% swap_swap
thf(fact_231_pair__in__swap__image,axiom,
! [A: $tType,B: $tType,Y: A,X: B,A3: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ X ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A3 ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A3 ) ) ).
% pair_in_swap_image
thf(fact_232_swap__simp,axiom,
! [A: $tType,B: $tType,X: B,Y: A] :
( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
= ( product_Pair @ A @ B @ Y @ X ) ) ).
% swap_simp
thf(fact_233_surj__swap,axiom,
! [B: $tType,A: $tType] :
( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% surj_swap
thf(fact_234_snd__swap,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B] :
( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X ) )
= ( product_fst @ A @ B @ X ) ) ).
% snd_swap
thf(fact_235_fst__swap,axiom,
! [A: $tType,B: $tType,X: product_prod @ B @ A] :
( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X ) )
= ( product_snd @ B @ A @ X ) ) ).
% fst_swap
thf(fact_236_ids_Op1_Ocong,axiom,
! [Sc: $tType,Sf: $tType,Sz: $tType] :
( ( ( finite_finite @ Sz )
& ( linorder @ Sz )
& ( finite_finite @ Sf )
& ( finite_finite @ Sc ) )
=> ( ( p1 @ Sz @ Sf @ Sc )
= ( p1 @ Sz @ Sf @ Sc ) ) ) ).
% ids.p1.cong
thf(fact_237_ids_Op1__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,P2: Sz,X: Sz] :
( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( p1 @ Sz @ Sf @ Sc @ Vid1 @ P2 @ X )
= ( prop @ Sz @ Sf @ Sc @ P2 @ ( singleton @ Sz @ Sf @ Vid1 @ ( var @ Sz @ Sf @ X ) ) ) ) ) ) ).
% ids.p1_def
thf(fact_238_formula_Oset__intros_I3_J,axiom,
! [B: $tType,A: $tType,C: $tType,Yb: trm @ A @ C,X222: C > ( trm @ A @ C ),Yc: A,X21: C] :
( ( member @ ( trm @ A @ C ) @ Yb @ ( image @ C @ ( trm @ A @ C ) @ X222 @ ( top_top @ ( set @ C ) ) ) )
=> ( ( member @ A @ Yc @ ( set_trm @ A @ C @ Yb ) )
=> ( member @ A @ Yc @ ( set1_formula @ A @ B @ C @ ( prop @ C @ A @ B @ X21 @ X222 ) ) ) ) ) ).
% formula.set_intros(3)
thf(fact_239_hpfree__ffree_Ointros_I10_J,axiom,
! [B: $tType,A: $tType,C: $tType,Args: C > ( trm @ A @ C ),P2: C] :
( ! [Arg: trm @ A @ C] :
( ( member @ ( trm @ A @ C ) @ Arg @ ( image @ C @ ( trm @ A @ C ) @ Args @ ( top_top @ ( set @ C ) ) ) )
=> ( dfree @ A @ C @ Arg ) )
=> ( ffree @ A @ B @ C @ ( prop @ C @ A @ B @ P2 @ Args ) ) ) ).
% hpfree_ffree.intros(10)
thf(fact_240_hpfree__ffree_Ointros_I15_J,axiom,
! [A: $tType,C: $tType,B: $tType,P: B] : ( ffree @ A @ B @ C @ ( predicational @ B @ A @ C @ P ) ) ).
% hpfree_ffree.intros(15)
thf(fact_241_map__prod__surj,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: A > B,G: C > D] :
( ( ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( ( ( image @ C @ D @ G @ ( top_top @ ( set @ C ) ) )
= ( top_top @ ( set @ D ) ) )
=> ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F @ G ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
= ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% map_prod_surj
thf(fact_242_good__interp_Oright__unique,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ( right_unique @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) ) ) ).
% good_interp.right_unique
thf(fact_243_good__interp_Oleft__unique,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ( left_unique @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) ) ) ).
% good_interp.left_unique
thf(fact_244_map__prod__simp,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G: D > B,A2: C,B2: D] :
( ( product_map_prod @ C @ A @ D @ B @ F @ G @ ( product_Pair @ C @ D @ A2 @ B2 ) )
= ( product_Pair @ A @ B @ ( F @ A2 ) @ ( G @ B2 ) ) ) ).
% map_prod_simp
thf(fact_245_fst__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > A,G: D > B,X: product_prod @ C @ D] :
( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F @ G @ X ) )
= ( F @ ( product_fst @ C @ D @ X ) ) ) ).
% fst_map_prod
thf(fact_246_snd__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > B,G: D > A,X: product_prod @ C @ D] :
( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F @ G @ X ) )
= ( G @ ( product_snd @ C @ D @ X ) ) ) ).
% snd_map_prod
thf(fact_247_map__prod__imageI,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,A2: A,B2: B,R6: set @ ( product_prod @ A @ B ),F: A > C,G: B > D] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R6 )
=> ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F @ A2 ) @ ( G @ B2 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F @ G ) @ R6 ) ) ) ).
% map_prod_imageI
thf(fact_248_prod__fun__imageE,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,C3: product_prod @ A @ B,F: C > A,G: D > B,R6: set @ ( product_prod @ C @ D )] :
( ( member @ ( product_prod @ A @ B ) @ C3 @ ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F @ G ) @ R6 ) )
=> ~ ! [X3: C,Y3: D] :
( ( C3
= ( product_Pair @ A @ B @ ( F @ X3 ) @ ( G @ Y3 ) ) )
=> ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X3 @ Y3 ) @ R6 ) ) ) ).
% prod_fun_imageE
thf(fact_249_ids_Ogood__interp_Oright__unique,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 )
=> ( right_unique @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) ) ) ) ).
% ids.good_interp.right_unique
thf(fact_250_ids_Ogood__interp_Oleft__unique,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 )
=> ( left_unique @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) ) ) ) ).
% ids.good_interp.left_unique
thf(fact_251_good__interp_Odomain,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ B )
& ( finite_finite @ C ) )
=> ( ( domainp @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) )
= ( denota2077489681interp @ A @ B @ C ) ) ) ).
% good_interp.domain
thf(fact_252_dterm__sem_Osimps_I6_J,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( finite_finite @ A )
& ( finite_finite @ C )
& ( finite_finite @ B ) )
=> ! [I7: denota1663640101rp_ext @ A @ B @ C @ product_unit,T4: trm @ A @ C] :
( ( denota594965758rm_sem @ A @ B @ C @ I7 @ ( differential @ A @ C @ T4 ) )
= ( denota2078997598vative @ A @ B @ C @ I7 @ T4 ) ) ) ).
% dterm_sem.simps(6)
thf(fact_253_ids_Ogood__interp_Odomain,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 )
=> ( ( domainp @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) )
= ( denota2077489681interp @ A @ B @ C ) ) ) ) ).
% ids.good_interp.domain
thf(fact_254_dterm__sem_Osimps_I7_J,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( finite_finite @ C )
& ( finite_finite @ B )
& ( finite_finite @ A ) )
=> ! [I7: denota1663640101rp_ext @ A @ B @ C @ product_unit,C3: real] :
( ( denota594965758rm_sem @ A @ B @ C @ I7 @ ( const @ A @ C @ C3 ) )
= ( ^ [V4: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] : C3 ) ) ) ).
% dterm_sem.simps(7)
% Subclasses (2)
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 ) ) ).
% Type constructors (20)
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A7: $tType,A8: $tType] :
( ( ( finite_finite @ A7 )
& ( finite_finite @ A8 ) )
=> ( finite_finite @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A7: $tType,A8: $tType] :
( ( top @ A8 )
=> ( top @ ( A7 > A8 ) ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_1,axiom,
! [A7: $tType] :
( ( finite_finite @ A7 )
=> ( finite_finite @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_2,axiom,
! [A7: $tType] : ( top @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Groups_Ozero,axiom,
! [A7: $tType] :
( ( zero @ A7 )
=> ( zero @ ( set @ A7 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_3,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_4,axiom,
top @ $o ).
thf(tcon_Real_Oreal___Orderings_Olinorder_5,axiom,
linorder @ real ).
thf(tcon_Real_Oreal___Groups_Ozero_6,axiom,
zero @ real ).
thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_7,axiom,
! [A7: $tType,A8: $tType] :
( ( ( finite_finite @ A7 )
& ( finite_finite @ A8 ) )
=> ( finite_finite @ ( sum_sum @ A7 @ A8 ) ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_8,axiom,
! [A7: $tType,A8: $tType] :
( ( ( finite_finite @ A7 )
& ( finite_finite @ A8 ) )
=> ( finite_finite @ ( product_prod @ A7 @ A8 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Otop_9,axiom,
! [A7: $tType,A8: $tType] :
( ( ( top @ A7 )
& ( top @ A8 ) )
=> ( top @ ( product_prod @ A7 @ A8 ) ) ) ).
thf(tcon_Product__Type_Oprod___Groups_Ozero_10,axiom,
! [A7: $tType,A8: $tType] :
( ( ( zero @ A7 )
& ( zero @ A8 ) )
=> ( zero @ ( product_prod @ A7 @ A8 ) ) ) ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_11,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_12,axiom,
finite_finite @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Otop_13,axiom,
top @ product_unit ).
thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Olinorder_14,axiom,
! [A7: $tType,A8: $tType] :
( ( ( linorder @ A7 )
& ( cARD_1 @ A8 ) )
=> ( linorder @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Finite__Set_Ofinite_15,axiom,
! [A7: $tType,A8: $tType] :
( ( ( finite_finite @ A7 )
& ( finite_finite @ A8 ) )
=> ( finite_finite @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
thf(tcon_Finite__Cartesian__Product_Ovec___Groups_Ozero_16,axiom,
! [A7: $tType,A8: $tType] :
( ( ( zero @ A7 )
& ( finite_finite @ A8 ) )
=> ( zero @ ( finite_Cartesian_vec @ A7 @ A8 ) ) ) ).
% 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 @ a ).
thf(tfree_1,hypothesis,
finite_finite @ c ).
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,
dfree @ a @ c @ ( args @ i2 ) ).
%------------------------------------------------------------------------------