TPTP Problem File: ITP077^2.p
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%------------------------------------------------------------------------------
% File : ITP077^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Hilbert_Function problem prob_146__11622158_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 : Hilbert_Function/prob_146__11622158_1 [Des21]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.75 v7.5.0
% Syntax : Number of formulae : 336 ( 75 unt; 48 typ; 0 def)
% Number of atoms : 922 ( 240 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 4025 ( 72 ~; 13 |; 46 &;3378 @)
% ( 0 <=>; 516 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 9 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 186 ( 186 >; 0 *; 0 +; 0 <<)
% Number of symbols : 49 ( 46 usr; 8 con; 0-5 aty)
% Number of variables : 1125 ( 57 ^; 982 !; 47 ?;1125 :)
% ( 39 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:22:15.198
%------------------------------------------------------------------------------
% Could-be-implicit typings (5)
thf(ty_t_Multiset_Omultiset,type,
multiset: $tType > $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (43)
thf(sy_cl_Rings_Odioid,type,
dioid:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osize,type,
size:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[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_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : $o ).
thf(sy_c_Fun_Obij__betw,type,
bij_betw:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) > $o ) ).
thf(sy_c_Fun_Oinj__on,type,
inj_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list,type,
groups501227526m_list:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_Hilbert__Choice_Oinv__into,type,
hilbert_inv_into:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
thf(sy_c_Hilbert__Function__Mirabelle__phhewjaonq_Odirect__decomp,type,
hilber580762119decomp:
!>[A: $tType] : ( ( set @ A ) > ( list @ ( set @ A ) ) > $o ) ).
thf(sy_c_List_Olist__ex,type,
list_ex:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olist__update,type,
list_update:
!>[A: $tType] : ( ( list @ A ) > nat > A > ( list @ A ) ) ).
thf(sy_c_List_Olistset,type,
listset:
!>[A: $tType] : ( ( list @ ( set @ A ) ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Onth,type,
nth:
!>[A: $tType] : ( ( list @ A ) > nat > A ) ).
thf(sy_c_List_Orev,type,
rev:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Multiset_Ocomm__monoid__add__class_Osum__mset,type,
comm_m1079429914m_mset:
!>[A: $tType] : ( ( multiset @ A ) > A ) ).
thf(sy_c_Multiset_Omset,type,
mset:
!>[A: $tType] : ( ( list @ A ) > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Oset__mset,type,
set_mset:
!>[A: $tType] : ( ( multiset @ A ) > ( set @ A ) ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Permutation_Operm,type,
perm:
!>[A: $tType] : ( ( list @ A ) > ( list @ 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_Set__Interval_Oord__class_OlessThan,type,
set_ord_lessThan:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_A,type,
a2: set @ a ).
thf(sy_v_a____,type,
a3: a ).
thf(sy_v_f____,type,
f: nat > nat ).
thf(sy_v_g____,type,
g: nat > nat ).
thf(sy_v_qs2____,type,
qs2: list @ a ).
thf(sy_v_qs____,type,
qs: list @ a ).
thf(sy_v_ss1,type,
ss1: list @ ( set @ a ) ).
thf(sy_v_ss2,type,
ss2: list @ ( set @ a ) ).
% Relevant facts (256)
thf(fact_0__092_060open_062mset_Aqs2_A_061_Amset_Aqs_092_060close_062,axiom,
( ( mset @ a @ qs2 )
= ( mset @ a @ qs ) ) ).
% \<open>mset qs2 = mset qs\<close>
thf(fact_1__092_060open_062a_A_092_060in_062_AA_092_060close_062,axiom,
member @ a @ a3 @ a2 ).
% \<open>a \<in> A\<close>
thf(fact_2_a,axiom,
( a3
= ( groups501227526m_list @ a @ qs ) ) ).
% a
thf(fact_3_len__qs2,axiom,
( ( size_size @ ( list @ a ) @ qs2 )
= ( size_size @ ( list @ a ) @ qs ) ) ).
% len_qs2
thf(fact_4__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062qs_O_A_092_060lbrakk_062a_A_061_Asum__list_Aqs_059_Aqs_A_092_060in_062_Alistset_Ass1_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Qs: list @ a] :
( ( a3
= ( groups501227526m_list @ a @ Qs ) )
=> ~ ( member @ ( list @ a ) @ Qs @ ( listset @ a @ ss1 ) ) ) ).
% \<open>\<And>thesis. (\<And>qs. \<lbrakk>a = sum_list qs; qs \<in> listset ss1\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_5_qs2__in,axiom,
member @ ( list @ a ) @ qs2 @ ( listset @ a @ ss2 ) ).
% qs2_in
thf(fact_6_direct__decompD_I1_J,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: set @ A,Ss: list @ ( set @ A ),Qs2: list @ A] :
( ( hilber580762119decomp @ A @ A2 @ Ss )
=> ( ( member @ ( list @ A ) @ Qs2 @ ( listset @ A @ Ss ) )
=> ( member @ A @ ( groups501227526m_list @ A @ Qs2 ) @ A2 ) ) ) ) ).
% direct_decompD(1)
thf(fact_7_direct__decompE,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: set @ A,Ss: list @ ( set @ A ),A3: A] :
( ( hilber580762119decomp @ A @ A2 @ Ss )
=> ( ( member @ A @ A3 @ A2 )
=> ~ ! [Qs: list @ A] :
( ( member @ ( list @ A ) @ Qs @ ( listset @ A @ Ss ) )
=> ( A3
!= ( groups501227526m_list @ A @ Qs ) ) ) ) ) ) ).
% direct_decompE
thf(fact_8_direct__decompI__alt,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [Ss: list @ ( set @ A ),A2: set @ A] :
( ! [Qs: list @ A] :
( ( member @ ( list @ A ) @ Qs @ ( listset @ A @ Ss ) )
=> ( member @ A @ ( groups501227526m_list @ A @ Qs ) @ A2 ) )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A2 )
=> ? [X: list @ A] :
( ( member @ ( list @ A ) @ X @ ( listset @ A @ Ss ) )
& ( A4
= ( groups501227526m_list @ A @ X ) )
& ! [Y: list @ A] :
( ( ( member @ ( list @ A ) @ Y @ ( listset @ A @ Ss ) )
& ( A4
= ( groups501227526m_list @ A @ Y ) ) )
=> ( Y = X ) ) ) )
=> ( hilber580762119decomp @ A @ A2 @ Ss ) ) ) ) ).
% direct_decompI_alt
thf(fact_9_direct__decomp__unique,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: set @ A,Ss: list @ ( set @ A ),Qs2: list @ A,Qs3: list @ A] :
( ( hilber580762119decomp @ A @ A2 @ Ss )
=> ( ( member @ ( list @ A ) @ Qs2 @ ( listset @ A @ Ss ) )
=> ( ( member @ ( list @ A ) @ Qs3 @ ( listset @ A @ Ss ) )
=> ( ( ( groups501227526m_list @ A @ Qs2 )
= ( groups501227526m_list @ A @ Qs3 ) )
=> ( Qs2 = Qs3 ) ) ) ) ) ) ).
% direct_decomp_unique
thf(fact_10_sum__list_Orev,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [Xs: list @ A] :
( ( groups501227526m_list @ A @ ( rev @ A @ Xs ) )
= ( groups501227526m_list @ A @ Xs ) ) ) ).
% sum_list.rev
thf(fact_11__C1_C,axiom,
! [I: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ a ) @ qs ) )
=> ( ( nth @ a @ qs2 @ I )
= ( nth @ a @ qs @ ( g @ I ) ) ) ) ).
% "1"
thf(fact_12_g__bij2,axiom,
bij_betw @ nat @ nat @ g @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ a ) @ qs2 ) ) @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ a ) @ qs ) ) ).
% g_bij2
thf(fact_13_sum__mset__sum__list,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [Xs: list @ A] :
( ( comm_m1079429914m_mset @ A @ ( mset @ A @ Xs ) )
= ( groups501227526m_list @ A @ Xs ) ) ) ).
% sum_mset_sum_list
thf(fact_14_direct__decompD_I3_J,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: set @ A,Ss: list @ ( set @ A )] :
( ( hilber580762119decomp @ A @ A2 @ Ss )
=> ( ( image @ ( list @ A ) @ A @ ( groups501227526m_list @ A ) @ ( listset @ A @ Ss ) )
= A2 ) ) ) ).
% direct_decompD(3)
thf(fact_15_assms_I1_J,axiom,
hilber580762119decomp @ a @ a2 @ ss1 ).
% assms(1)
thf(fact_16_qs__in,axiom,
member @ ( list @ a ) @ qs @ ( listset @ a @ ss1 ) ).
% qs_in
thf(fact_17_mset__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( mset @ A @ ( rev @ A @ Xs ) )
= ( mset @ A @ Xs ) ) ).
% mset_rev
thf(fact_18__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062qs2_O_A_092_060lbrakk_062qs2_A_092_060in_062_Alistset_Ass2_059_Alength_Aqs2_A_061_Alength_Aqs_059_A_092_060And_062i_O_Ai_A_060_Alength_Aqs_A_092_060Longrightarrow_062_Aqs2_A_B_Ai_A_061_Aqs_A_B_Ag_Ai_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Qs22: list @ a] :
( ( member @ ( list @ a ) @ Qs22 @ ( listset @ a @ ss2 ) )
=> ( ( ( size_size @ ( list @ a ) @ Qs22 )
= ( size_size @ ( list @ a ) @ qs ) )
=> ~ ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ a ) @ qs ) )
=> ( ( nth @ a @ Qs22 @ I2 )
= ( nth @ a @ qs @ ( g @ I2 ) ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>qs2. \<lbrakk>qs2 \<in> listset ss2; length qs2 = length qs; \<And>i. i < length qs \<Longrightarrow> qs2 ! i = qs ! g i\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_19_assms_I2_J,axiom,
perm @ ( set @ a ) @ ss1 @ ss2 ).
% assms(2)
thf(fact_20_ex__mset,axiom,
! [A: $tType,X2: multiset @ A] :
? [Xs2: list @ A] :
( ( mset @ A @ Xs2 )
= X2 ) ).
% ex_mset
thf(fact_21_mset__eq__length,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( mset @ A @ Xs )
= ( mset @ A @ Ys ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) ) ) ).
% mset_eq_length
thf(fact_22_mset__bij,axiom,
! [A: $tType,F: nat > nat,Xs: list @ A,Ys: list @ A] :
( ( bij_betw @ nat @ nat @ F @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ A ) @ Xs ) ) @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ A ) @ Ys ) ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I3 )
= ( nth @ A @ Ys @ ( F @ I3 ) ) ) )
=> ( ( mset @ A @ Xs )
= ( mset @ A @ Ys ) ) ) ) ).
% mset_bij
thf(fact_23_listset__permE,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ ( set @ A ),F: nat > nat,Xs3: list @ ( set @ A )] :
( ( member @ ( list @ A ) @ Ys @ ( listset @ A @ Xs ) )
=> ( ( bij_betw @ nat @ nat @ F @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( set @ A ) ) @ Xs ) ) @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( set @ A ) ) @ Xs3 ) ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ ( set @ A ) ) @ Xs ) )
=> ( ( nth @ ( set @ A ) @ Xs3 @ I3 )
= ( nth @ ( set @ A ) @ Xs @ ( F @ I3 ) ) ) )
=> ~ ! [Ys2: list @ A] :
( ( member @ ( list @ A ) @ Ys2 @ ( listset @ A @ Xs3 ) )
=> ( ( ( size_size @ ( list @ A ) @ Ys2 )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ~ ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( ( nth @ A @ Ys2 @ I2 )
= ( nth @ A @ Ys @ ( F @ I2 ) ) ) ) ) ) ) ) ) ).
% listset_permE
thf(fact_24_g__bij,axiom,
bij_betw @ nat @ nat @ g @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( set @ a ) ) @ ss1 ) ) @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( set @ a ) ) @ ss2 ) ) ).
% g_bij
thf(fact_25_length__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( rev @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_rev
thf(fact_26_lessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_lessThan @ A @ K ) )
= ( ord_less @ A @ I @ K ) ) ) ).
% lessThan_iff
thf(fact_27_listsetI,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ ( set @ A )] :
( ( ( size_size @ ( list @ A ) @ Ys )
= ( size_size @ ( list @ ( set @ A ) ) @ Xs ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ ( set @ A ) ) @ Xs ) )
=> ( member @ A @ ( nth @ A @ Ys @ I3 ) @ ( nth @ ( set @ A ) @ Xs @ I3 ) ) )
=> ( member @ ( list @ A ) @ Ys @ ( listset @ A @ Xs ) ) ) ) ).
% listsetI
thf(fact_28_len__qs,axiom,
( ( size_size @ ( list @ a ) @ qs )
= ( size_size @ ( list @ ( set @ a ) ) @ ss1 ) ) ).
% len_qs
thf(fact_29_f__bij,axiom,
bij_betw @ nat @ nat @ f @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( set @ a ) ) @ ss2 ) ) @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( set @ a ) ) @ ss1 ) ) ).
% f_bij
thf(fact_30_nth__equalityI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I3 )
= ( nth @ A @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_31_Skolem__list__nth,axiom,
! [A: $tType,K: nat,P: nat > A > $o] :
( ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K )
=> ? [X3: A] : ( P @ I4 @ X3 ) ) )
= ( ? [Xs4: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs4 )
= K )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K )
=> ( P @ I4 @ ( nth @ A @ Xs4 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_32_list__eq__iff__nth__eq,axiom,
! [A: $tType] :
( ( ^ [Y2: list @ A,Z: list @ A] : Y2 = Z )
= ( ^ [Xs4: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs4 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs4 ) )
=> ( ( nth @ A @ Xs4 @ I4 )
= ( nth @ A @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_33_lessThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y3: A] :
( ( ( set_ord_lessThan @ A @ X4 )
= ( set_ord_lessThan @ A @ Y3 ) )
= ( X4 = Y3 ) ) ) ).
% lessThan_eq_iff
thf(fact_34_rev__is__rev__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( rev @ A @ Xs )
= ( rev @ A @ Ys ) )
= ( Xs = Ys ) ) ).
% rev_is_rev_conv
thf(fact_35_rev__rev__ident,axiom,
! [A: $tType,Xs: list @ A] :
( ( rev @ A @ ( rev @ A @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_36_len__ss1,axiom,
( ( size_size @ ( list @ ( set @ a ) ) @ ss1 )
= ( size_size @ ( list @ ( set @ a ) ) @ ss2 ) ) ).
% len_ss1
thf(fact_37_g,axiom,
! [I: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( set @ a ) ) @ ss1 ) )
=> ( ( nth @ ( set @ a ) @ ss2 @ I )
= ( nth @ ( set @ a ) @ ss1 @ ( g @ I ) ) ) ) ).
% g
thf(fact_38_f,axiom,
! [I: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( set @ a ) ) @ ss2 ) )
=> ( ( nth @ ( set @ a ) @ ss1 @ I )
= ( nth @ ( set @ a ) @ ss2 @ ( f @ I ) ) ) ) ).
% f
thf(fact_39__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062f_O_A_092_060lbrakk_062bij__betw_Af_A_123_O_O_060length_Ass2_125_A_123_O_O_060length_Ass1_125_059_A_092_060And_062i_O_Ai_A_060_Alength_Ass2_A_092_060Longrightarrow_062_Ass1_A_B_Ai_A_061_Ass2_A_B_Af_Ai_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [F2: nat > nat] :
( ( bij_betw @ nat @ nat @ F2 @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( set @ a ) ) @ ss2 ) ) @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( set @ a ) ) @ ss1 ) ) )
=> ~ ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ ( set @ a ) ) @ ss2 ) )
=> ( ( nth @ ( set @ a ) @ ss1 @ I2 )
= ( nth @ ( set @ a ) @ ss2 @ ( F2 @ I2 ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>f. \<lbrakk>bij_betw f {..<length ss2} {..<length ss1}; \<And>i. i < length ss2 \<Longrightarrow> ss1 ! i = ss2 ! f i\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_40__092_060open_062_092_060exists_062f_O_Abij__betw_Af_A_123_O_O_060length_Ass1_125_A_123_O_O_060length_Ass2_125_A_092_060and_062_A_I_092_060forall_062i_060length_Ass1_O_Ass1_A_B_Ai_A_061_Ass2_A_B_Af_Ai_J_092_060close_062,axiom,
? [F2: nat > nat] :
( ( bij_betw @ nat @ nat @ F2 @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( set @ a ) ) @ ss1 ) ) @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( set @ a ) ) @ ss2 ) ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ ( set @ a ) ) @ ss1 ) )
=> ( ( nth @ ( set @ a ) @ ss1 @ I2 )
= ( nth @ ( set @ a ) @ ss2 @ ( F2 @ I2 ) ) ) ) ) ).
% \<open>\<exists>f. bij_betw f {..<length ss1} {..<length ss2} \<and> (\<forall>i<length ss1. ss1 ! i = ss2 ! f i)\<close>
thf(fact_41_f__g,axiom,
! [I: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( set @ a ) ) @ ss1 ) )
=> ( ( f @ ( g @ I ) )
= I ) ) ).
% f_g
thf(fact_42_g__f,axiom,
! [I: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( set @ a ) ) @ ss2 ) )
=> ( ( g @ ( f @ I ) )
= I ) ) ).
% g_f
thf(fact_43_listsetD_I2_J,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ ( set @ A ),I: nat] :
( ( member @ ( list @ A ) @ Ys @ ( listset @ A @ Xs ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( set @ A ) ) @ Xs ) )
=> ( member @ A @ ( nth @ A @ Ys @ I ) @ ( nth @ ( set @ A ) @ Xs @ I ) ) ) ) ).
% listsetD(2)
thf(fact_44_neq__if__length__neq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
!= ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
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
@ ^ [X5: A] : ( member @ A @ X5 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X6: A] :
( ( P @ X6 )
= ( Q @ X6 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X6: A] :
( ( F @ X6 )
= ( G @ X6 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_Ex__list__of__length,axiom,
! [A: $tType,N: nat] :
? [Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_50_rev__swap,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( rev @ A @ Xs )
= Ys )
= ( Xs
= ( rev @ A @ Ys ) ) ) ).
% rev_swap
thf(fact_51_lessThan__strict__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N: A] :
( ( ord_less @ ( set @ A ) @ ( set_ord_lessThan @ A @ M ) @ ( set_ord_lessThan @ A @ N ) )
= ( ord_less @ A @ M @ N ) ) ) ).
% lessThan_strict_subset_iff
thf(fact_52_length__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ! [Xs2: list @ A] :
( ! [Ys4: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys4 ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P @ Ys4 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_53_listsetD_I1_J,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ ( set @ A )] :
( ( member @ ( list @ A ) @ Ys @ ( listset @ A @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ Ys )
= ( size_size @ ( list @ ( set @ A ) ) @ Xs ) ) ) ).
% listsetD(1)
thf(fact_54_permutation__Ex__bij,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( perm @ A @ Xs @ Ys )
=> ? [F2: nat > nat] :
( ( bij_betw @ nat @ nat @ F2 @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ A ) @ Xs ) ) @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ A ) @ Ys ) ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I2 )
= ( nth @ A @ Ys @ ( F2 @ I2 ) ) ) ) ) ) ).
% permutation_Ex_bij
thf(fact_55_g__def,axiom,
( g
= ( hilbert_inv_into @ nat @ nat @ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( set @ a ) ) @ ss2 ) ) @ f ) ) ).
% g_def
thf(fact_56_perm__refl,axiom,
! [A: $tType,L: list @ A] : ( perm @ A @ L @ L ) ).
% perm_refl
thf(fact_57_image__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,X4: B,A2: set @ B] :
( ( B2
= ( F @ X4 ) )
=> ( ( member @ B @ X4 @ A2 )
=> ( member @ A @ B2 @ ( image @ B @ A @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_58_mset__eq__perm,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( mset @ A @ Xs )
= ( mset @ A @ Ys ) )
= ( perm @ A @ Xs @ Ys ) ) ).
% mset_eq_perm
thf(fact_59_perm__rev,axiom,
! [A: $tType,Xs: list @ A] : ( perm @ A @ ( rev @ A @ Xs ) @ Xs ) ).
% perm_rev
thf(fact_60_perm__length,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( perm @ A @ Xs @ Ys )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) ) ) ).
% perm_length
thf(fact_61_bij__betw__imp__surj__on,axiom,
! [A: $tType,B: $tType,F: A > B,A2: set @ A,B3: set @ B] :
( ( bij_betw @ A @ B @ F @ A2 @ B3 )
=> ( ( image @ A @ B @ F @ A2 )
= B3 ) ) ).
% bij_betw_imp_surj_on
thf(fact_62_mbs_Oless__not__eq,axiom,
! [A: $tType] :
( ( size @ A )
=> ! [X4: A,A2: set @ A,Y3: A] :
( ( member @ A @ X4 @ A2 )
=> ( ( ord_less @ nat @ ( size_size @ A @ X4 ) @ ( size_size @ A @ Y3 ) )
=> ( X4 != Y3 ) ) ) ) ).
% mbs.less_not_eq
thf(fact_63_size__mset,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( multiset @ A ) @ ( mset @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% size_mset
thf(fact_64_psubsetD,axiom,
! [A: $tType,A2: set @ A,B3: set @ A,C: A] :
( ( ord_less @ ( set @ A ) @ A2 @ B3 )
=> ( ( member @ A @ C @ A2 )
=> ( member @ A @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_65_psubset__trans,axiom,
! [A: $tType,A2: set @ A,B3: set @ A,C2: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B3 )
=> ( ( ord_less @ ( set @ A ) @ B3 @ C2 )
=> ( ord_less @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% psubset_trans
thf(fact_66_imageI,axiom,
! [B: $tType,A: $tType,X4: A,A2: set @ A,F: A > B] :
( ( member @ A @ X4 @ A2 )
=> ( member @ B @ ( F @ X4 ) @ ( image @ A @ B @ F @ A2 ) ) ) ).
% imageI
thf(fact_67_image__iff,axiom,
! [A: $tType,B: $tType,Z2: A,F: B > A,A2: set @ B] :
( ( member @ A @ Z2 @ ( image @ B @ A @ F @ A2 ) )
= ( ? [X5: B] :
( ( member @ B @ X5 @ A2 )
& ( Z2
= ( F @ X5 ) ) ) ) ) ).
% image_iff
thf(fact_68_bex__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B,P: A > $o] :
( ? [X: A] :
( ( member @ A @ X @ ( image @ B @ A @ F @ A2 ) )
& ( P @ X ) )
=> ? [X6: B] :
( ( member @ B @ X6 @ A2 )
& ( P @ ( F @ X6 ) ) ) ) ).
% bex_imageD
thf(fact_69_image__cong,axiom,
! [B: $tType,A: $tType,M2: set @ A,N2: set @ A,F: A > B,G: A > B] :
( ( M2 = N2 )
=> ( ! [X6: A] :
( ( member @ A @ X6 @ N2 )
=> ( ( F @ X6 )
= ( G @ X6 ) ) )
=> ( ( image @ A @ B @ F @ M2 )
= ( image @ A @ B @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_70_ball__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B,P: A > $o] :
( ! [X6: A] :
( ( member @ A @ X6 @ ( image @ B @ A @ F @ A2 ) )
=> ( P @ X6 ) )
=> ! [X: B] :
( ( member @ B @ X @ A2 )
=> ( P @ ( F @ X ) ) ) ) ).
% ball_imageD
thf(fact_71_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X4: A,A2: set @ A,B2: B,F: A > B] :
( ( member @ A @ X4 @ A2 )
=> ( ( B2
= ( F @ X4 ) )
=> ( member @ B @ B2 @ ( image @ A @ B @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_72_direct__decomp__def,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ( ( hilber580762119decomp @ A )
= ( ^ [A5: set @ A,Ss2: list @ ( set @ A )] : ( bij_betw @ ( list @ A ) @ A @ ( groups501227526m_list @ A ) @ ( listset @ A @ Ss2 ) @ A5 ) ) ) ) ).
% direct_decomp_def
thf(fact_73_bij__betwE,axiom,
! [A: $tType,B: $tType,F: A > B,A2: set @ A,B3: set @ B] :
( ( bij_betw @ A @ B @ F @ A2 @ B3 )
=> ! [X: A] :
( ( member @ A @ X @ A2 )
=> ( member @ B @ ( F @ X ) @ B3 ) ) ) ).
% bij_betwE
thf(fact_74_bij__betw__inv,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,B3: set @ B] :
( ( bij_betw @ A @ B @ F @ A2 @ B3 )
=> ? [G2: B > A] : ( bij_betw @ B @ A @ G2 @ B3 @ A2 ) ) ).
% bij_betw_inv
thf(fact_75_bij__betw__cong,axiom,
! [A: $tType,B: $tType,A2: set @ A,F: A > B,G: A > B,A6: set @ B] :
( ! [A4: A] :
( ( member @ A @ A4 @ A2 )
=> ( ( F @ A4 )
= ( G @ A4 ) ) )
=> ( ( bij_betw @ A @ B @ F @ A2 @ A6 )
= ( bij_betw @ A @ B @ G @ A2 @ A6 ) ) ) ).
% bij_betw_cong
thf(fact_76_bij__betw__apply,axiom,
! [A: $tType,B: $tType,F: A > B,A2: set @ A,B3: set @ B,A3: A] :
( ( bij_betw @ A @ B @ F @ A2 @ B3 )
=> ( ( member @ A @ A3 @ A2 )
=> ( member @ B @ ( F @ A3 ) @ B3 ) ) ) ).
% bij_betw_apply
thf(fact_77_bij__betw__iff__bijections,axiom,
! [B: $tType,A: $tType] :
( ( bij_betw @ A @ B )
= ( ^ [F3: A > B,A5: set @ A,B4: set @ B] :
? [G3: B > A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ( ( member @ B @ ( F3 @ X5 ) @ B4 )
& ( ( G3 @ ( F3 @ X5 ) )
= X5 ) ) )
& ! [X5: B] :
( ( member @ B @ X5 @ B4 )
=> ( ( member @ A @ ( G3 @ X5 ) @ A5 )
& ( ( F3 @ ( G3 @ X5 ) )
= X5 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_78_perm__sym,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( perm @ A @ Xs @ Ys )
=> ( perm @ A @ Ys @ Xs ) ) ).
% perm_sym
thf(fact_79_perm_Otrans,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( perm @ A @ Xs @ Ys )
=> ( ( perm @ A @ Ys @ Zs )
=> ( perm @ A @ Xs @ Zs ) ) ) ).
% perm.trans
thf(fact_80_bij__betw__inv__into__right,axiom,
! [A: $tType,B: $tType,F: A > B,A2: set @ A,A6: set @ B,A7: B] :
( ( bij_betw @ A @ B @ F @ A2 @ A6 )
=> ( ( member @ B @ A7 @ A6 )
=> ( ( F @ ( hilbert_inv_into @ A @ B @ A2 @ F @ A7 ) )
= A7 ) ) ) ).
% bij_betw_inv_into_right
thf(fact_81_bij__betw__inv__into__left,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,A6: set @ B,A3: A] :
( ( bij_betw @ A @ B @ F @ A2 @ A6 )
=> ( ( member @ A @ A3 @ A2 )
=> ( ( hilbert_inv_into @ A @ B @ A2 @ F @ ( F @ A3 ) )
= A3 ) ) ) ).
% bij_betw_inv_into_left
thf(fact_82_inv__into__inv__into__eq,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,A6: set @ B,A3: A] :
( ( bij_betw @ A @ B @ F @ A2 @ A6 )
=> ( ( member @ A @ A3 @ A2 )
=> ( ( hilbert_inv_into @ B @ A @ A6 @ ( hilbert_inv_into @ A @ B @ A2 @ F ) @ A3 )
= ( F @ A3 ) ) ) ) ).
% inv_into_inv_into_eq
thf(fact_83_bij__betw__inv__into,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,B3: set @ B] :
( ( bij_betw @ A @ B @ F @ A2 @ B3 )
=> ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ A2 @ F ) @ B3 @ A2 ) ) ).
% bij_betw_inv_into
thf(fact_84_inv__into__injective,axiom,
! [A: $tType,B: $tType,A2: set @ A,F: A > B,X4: B,Y3: B] :
( ( ( hilbert_inv_into @ A @ B @ A2 @ F @ X4 )
= ( hilbert_inv_into @ A @ B @ A2 @ F @ Y3 ) )
=> ( ( member @ B @ X4 @ ( image @ A @ B @ F @ A2 ) )
=> ( ( member @ B @ Y3 @ ( image @ A @ B @ F @ A2 ) )
=> ( X4 = Y3 ) ) ) ) ).
% inv_into_injective
thf(fact_85_inv__into__into,axiom,
! [A: $tType,B: $tType,X4: A,F: B > A,A2: set @ B] :
( ( member @ A @ X4 @ ( image @ B @ A @ F @ A2 ) )
=> ( member @ B @ ( hilbert_inv_into @ B @ A @ A2 @ F @ X4 ) @ A2 ) ) ).
% inv_into_into
thf(fact_86_f__inv__into__f,axiom,
! [B: $tType,A: $tType,Y3: A,F: B > A,A2: set @ B] :
( ( member @ A @ Y3 @ ( image @ B @ A @ F @ A2 ) )
=> ( ( F @ ( hilbert_inv_into @ B @ A @ A2 @ F @ Y3 ) )
= Y3 ) ) ).
% f_inv_into_f
thf(fact_87_list__ex__length,axiom,
! [A: $tType] :
( ( list_ex @ A )
= ( ^ [P2: A > $o,Xs4: list @ A] :
? [N3: nat] :
( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs4 ) )
& ( P2 @ ( nth @ A @ Xs4 @ N3 ) ) ) ) ) ).
% list_ex_length
thf(fact_88_direct__decompI,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [Ss: list @ ( set @ A ),A2: set @ A] :
( ( inj_on @ ( list @ A ) @ A @ ( groups501227526m_list @ A ) @ ( listset @ A @ Ss ) )
=> ( ( ( image @ ( list @ A ) @ A @ ( groups501227526m_list @ A ) @ ( listset @ A @ Ss ) )
= A2 )
=> ( hilber580762119decomp @ A @ A2 @ Ss ) ) ) ) ).
% direct_decompI
thf(fact_89_elem__le__sum__list,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [K: nat,Ns: list @ A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Ns ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Ns @ K ) @ ( groups501227526m_list @ A @ Ns ) ) ) ) ).
% elem_le_sum_list
thf(fact_90_perm__swap,axiom,
! [A: $tType,I: nat,Xs: list @ A,J: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( perm @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ J ) ) @ J @ ( nth @ A @ Xs @ I ) ) @ Xs ) ) ) ).
% perm_swap
thf(fact_91_mset__swap,axiom,
! [A: $tType,I: nat,Ls: list @ A,J: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ls ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Ls ) )
=> ( ( mset @ A @ ( list_update @ A @ ( list_update @ A @ Ls @ J @ ( nth @ A @ Ls @ I ) ) @ I @ ( nth @ A @ Ls @ J ) ) )
= ( mset @ A @ Ls ) ) ) ) ).
% mset_swap
thf(fact_92_nth__mem__mset,axiom,
! [A: $tType,I: nat,Ls: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ls ) )
=> ( member @ A @ ( nth @ A @ Ls @ I ) @ ( set_mset @ A @ ( mset @ A @ Ls ) ) ) ) ).
% nth_mem_mset
thf(fact_93_in__Union__mset__iff,axiom,
! [A: $tType,X4: A,MM: multiset @ ( multiset @ A )] :
( ( member @ A @ X4 @ ( set_mset @ A @ ( comm_m1079429914m_mset @ ( multiset @ A ) @ MM ) ) )
= ( ? [M3: multiset @ A] :
( ( member @ ( multiset @ A ) @ M3 @ ( set_mset @ ( multiset @ A ) @ MM ) )
& ( member @ A @ X4 @ ( set_mset @ A @ M3 ) ) ) ) ) ).
% in_Union_mset_iff
thf(fact_94_psubsetI,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less @ ( set @ A ) @ A2 @ B3 ) ) ) ).
% psubsetI
thf(fact_95_inj__on__rev,axiom,
! [A: $tType,A2: set @ ( list @ A )] : ( inj_on @ ( list @ A ) @ ( list @ A ) @ ( rev @ A ) @ A2 ) ).
% inj_on_rev
thf(fact_96_list__update__overwrite,axiom,
! [A: $tType,Xs: list @ A,I: nat,X4: A,Y3: A] :
( ( list_update @ A @ ( list_update @ A @ Xs @ I @ X4 ) @ I @ Y3 )
= ( list_update @ A @ Xs @ I @ Y3 ) ) ).
% list_update_overwrite
thf(fact_97_lessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_lessThan @ A @ X4 ) @ ( set_ord_lessThan @ A @ Y3 ) )
= ( ord_less_eq @ A @ X4 @ Y3 ) ) ) ).
% lessThan_subset_iff
thf(fact_98_list__update__beyond,axiom,
! [A: $tType,Xs: list @ A,I: nat,X4: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I )
=> ( ( list_update @ A @ Xs @ I @ X4 )
= Xs ) ) ).
% list_update_beyond
thf(fact_99_length__list__update,axiom,
! [A: $tType,Xs: list @ A,I: nat,X4: A] :
( ( size_size @ ( list @ A ) @ ( list_update @ A @ Xs @ I @ X4 ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_list_update
thf(fact_100_inv__into__f__f,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,X4: A] :
( ( inj_on @ A @ B @ F @ A2 )
=> ( ( member @ A @ X4 @ A2 )
=> ( ( hilbert_inv_into @ A @ B @ A2 @ F @ ( F @ X4 ) )
= X4 ) ) ) ).
% inv_into_f_f
thf(fact_101_list__update__id,axiom,
! [A: $tType,Xs: list @ A,I: nat] :
( ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ I ) )
= Xs ) ).
% list_update_id
thf(fact_102_nth__list__update__neq,axiom,
! [A: $tType,I: nat,J: nat,Xs: list @ A,X4: A] :
( ( I != J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X4 ) @ J )
= ( nth @ A @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_103_list__ex__rev,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( list_ex @ A @ P @ ( rev @ A @ Xs ) )
= ( list_ex @ A @ P @ Xs ) ) ).
% list_ex_rev
thf(fact_104_inv__into__image__cancel,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,S: set @ A] :
( ( inj_on @ A @ B @ F @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ S @ A2 )
=> ( ( image @ B @ A @ ( hilbert_inv_into @ A @ B @ A2 @ F ) @ ( image @ A @ B @ F @ S ) )
= S ) ) ) ).
% inv_into_image_cancel
thf(fact_105_nth__list__update__eq,axiom,
! [A: $tType,I: nat,Xs: list @ A,X4: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X4 ) @ I )
= X4 ) ) ).
% nth_list_update_eq
thf(fact_106_subset__image__inj,axiom,
! [A: $tType,B: $tType,S: set @ A,F: B > A,T: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ S @ ( image @ B @ A @ F @ T ) )
= ( ? [U: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ U @ T )
& ( inj_on @ B @ A @ F @ U )
& ( S
= ( image @ B @ A @ F @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_107_inj__on__inv__into,axiom,
! [B: $tType,A: $tType,B3: set @ A,F: B > A,A2: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B3 @ ( image @ B @ A @ F @ A2 ) )
=> ( inj_on @ A @ B @ ( hilbert_inv_into @ B @ A @ A2 @ F ) @ B3 ) ) ).
% inj_on_inv_into
thf(fact_108_list__update__swap,axiom,
! [A: $tType,I: nat,I5: nat,Xs: list @ A,X4: A,X7: A] :
( ( I != I5 )
=> ( ( list_update @ A @ ( list_update @ A @ Xs @ I @ X4 ) @ I5 @ X7 )
= ( list_update @ A @ ( list_update @ A @ Xs @ I5 @ X7 ) @ I @ X4 ) ) ) ).
% list_update_swap
thf(fact_109_inj__on__image__eq__iff,axiom,
! [B: $tType,A: $tType,F: A > B,C2: set @ A,A2: set @ A,B3: set @ A] :
( ( inj_on @ A @ B @ F @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C2 )
=> ( ( ( image @ A @ B @ F @ A2 )
= ( image @ A @ B @ F @ B3 ) )
= ( A2 = B3 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_110_inj__on__image__mem__iff,axiom,
! [B: $tType,A: $tType,F: A > B,B3: set @ A,A3: A,A2: set @ A] :
( ( inj_on @ A @ B @ F @ B3 )
=> ( ( member @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ( member @ B @ ( F @ A3 ) @ ( image @ A @ B @ F @ A2 ) )
= ( member @ A @ A3 @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_111_inj__on__inverseI,axiom,
! [B: $tType,A: $tType,A2: set @ A,G: B > A,F: A > B] :
( ! [X6: A] :
( ( member @ A @ X6 @ A2 )
=> ( ( G @ ( F @ X6 ) )
= X6 ) )
=> ( inj_on @ A @ B @ F @ A2 ) ) ).
% inj_on_inverseI
thf(fact_112_inj__on__contraD,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,X4: A,Y3: A] :
( ( inj_on @ A @ B @ F @ A2 )
=> ( ( X4 != Y3 )
=> ( ( member @ A @ X4 @ A2 )
=> ( ( member @ A @ Y3 @ A2 )
=> ( ( F @ X4 )
!= ( F @ Y3 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_113_subset__inj__on,axiom,
! [B: $tType,A: $tType,F: A > B,B3: set @ A,A2: set @ A] :
( ( inj_on @ A @ B @ F @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( inj_on @ A @ B @ F @ A2 ) ) ) ).
% subset_inj_on
thf(fact_114_inj__on__subset,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,B3: set @ A] :
( ( inj_on @ A @ B @ F @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A2 )
=> ( inj_on @ A @ B @ F @ B3 ) ) ) ).
% inj_on_subset
thf(fact_115_inj__on__eq__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,X4: A,Y3: A] :
( ( inj_on @ A @ B @ F @ A2 )
=> ( ( member @ A @ X4 @ A2 )
=> ( ( member @ A @ Y3 @ A2 )
=> ( ( ( F @ X4 )
= ( F @ Y3 ) )
= ( X4 = Y3 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_116_inj__on__cong,axiom,
! [B: $tType,A: $tType,A2: set @ A,F: A > B,G: A > B] :
( ! [A4: A] :
( ( member @ A @ A4 @ A2 )
=> ( ( F @ A4 )
= ( G @ A4 ) ) )
=> ( ( inj_on @ A @ B @ F @ A2 )
= ( inj_on @ A @ B @ G @ A2 ) ) ) ).
% inj_on_cong
thf(fact_117_inj__on__def,axiom,
! [B: $tType,A: $tType] :
( ( inj_on @ A @ B )
= ( ^ [F3: A > B,A5: set @ A] :
! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ A5 )
=> ( ( ( F3 @ X5 )
= ( F3 @ Y4 ) )
=> ( X5 = Y4 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_118_inj__onI,axiom,
! [B: $tType,A: $tType,A2: set @ A,F: A > B] :
( ! [X6: A,Y: A] :
( ( member @ A @ X6 @ A2 )
=> ( ( member @ A @ Y @ A2 )
=> ( ( ( F @ X6 )
= ( F @ Y ) )
=> ( X6 = Y ) ) ) )
=> ( inj_on @ A @ B @ F @ A2 ) ) ).
% inj_onI
thf(fact_119_inj__onD,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,X4: A,Y3: A] :
( ( inj_on @ A @ B @ F @ A2 )
=> ( ( ( F @ X4 )
= ( F @ Y3 ) )
=> ( ( member @ A @ X4 @ A2 )
=> ( ( member @ A @ Y3 @ A2 )
=> ( X4 = Y3 ) ) ) ) ) ).
% inj_onD
thf(fact_120_linorder__inj__onI,axiom,
! [B: $tType,A: $tType] :
( ( order @ A )
=> ! [A2: set @ A,F: A > B] :
( ! [X6: A,Y: A] :
( ( ord_less @ A @ X6 @ Y )
=> ( ( member @ A @ X6 @ A2 )
=> ( ( member @ A @ Y @ A2 )
=> ( ( F @ X6 )
!= ( F @ Y ) ) ) ) )
=> ( ! [X6: A,Y: A] :
( ( member @ A @ X6 @ A2 )
=> ( ( member @ A @ Y @ A2 )
=> ( ( ord_less_eq @ A @ X6 @ Y )
| ( ord_less_eq @ A @ Y @ X6 ) ) ) )
=> ( inj_on @ A @ B @ F @ A2 ) ) ) ) ).
% linorder_inj_onI
thf(fact_121_inj__on__image__iff,axiom,
! [B: $tType,A: $tType,A2: set @ A,G: A > B,F: A > A] :
( ! [X6: A] :
( ( member @ A @ X6 @ A2 )
=> ! [Xa: A] :
( ( member @ A @ Xa @ A2 )
=> ( ( ( G @ ( F @ X6 ) )
= ( G @ ( F @ Xa ) ) )
= ( ( G @ X6 )
= ( G @ Xa ) ) ) ) )
=> ( ( inj_on @ A @ A @ F @ A2 )
=> ( ( inj_on @ A @ B @ G @ ( image @ A @ A @ F @ A2 ) )
= ( inj_on @ A @ B @ G @ A2 ) ) ) ) ).
% inj_on_image_iff
thf(fact_122_bij__betw__imp__inj__on,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,B3: set @ B] :
( ( bij_betw @ A @ B @ F @ A2 @ B3 )
=> ( inj_on @ A @ B @ F @ A2 ) ) ).
% bij_betw_imp_inj_on
thf(fact_123_inv__into__f__eq,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,X4: A,Y3: B] :
( ( inj_on @ A @ B @ F @ A2 )
=> ( ( member @ A @ X4 @ A2 )
=> ( ( ( F @ X4 )
= Y3 )
=> ( ( hilbert_inv_into @ A @ B @ A2 @ F @ Y3 )
= X4 ) ) ) ) ).
% inv_into_f_eq
thf(fact_124_subset__image__iff,axiom,
! [A: $tType,B: $tType,B3: set @ A,F: B > A,A2: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B3 @ ( image @ B @ A @ F @ A2 ) )
= ( ? [AA: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ AA @ A2 )
& ( B3
= ( image @ B @ A @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_125_image__subset__iff,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ B3 )
= ( ! [X5: B] :
( ( member @ B @ X5 @ A2 )
=> ( member @ A @ ( F @ X5 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_126_subset__imageE,axiom,
! [A: $tType,B: $tType,B3: set @ A,F: B > A,A2: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B3 @ ( image @ B @ A @ F @ A2 ) )
=> ~ ! [C3: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C3 @ A2 )
=> ( B3
!= ( image @ B @ A @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_127_image__subsetI,axiom,
! [A: $tType,B: $tType,A2: set @ A,F: A > B,B3: set @ B] :
( ! [X6: A] :
( ( member @ A @ X6 @ A2 )
=> ( member @ B @ ( F @ X6 ) @ B3 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_128_image__mono,axiom,
! [B: $tType,A: $tType,A2: set @ A,B3: set @ A,F: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ ( image @ A @ B @ F @ B3 ) ) ) ).
% image_mono
thf(fact_129_psubsetE,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B3 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ord_less_eq @ ( set @ A ) @ B3 @ A2 ) ) ) ).
% psubsetE
thf(fact_130_psubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% psubset_eq
thf(fact_131_psubset__imp__subset,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B3 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B3 ) ) ).
% psubset_imp_subset
thf(fact_132_psubset__subset__trans,axiom,
! [A: $tType,A2: set @ A,B3: set @ A,C2: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C2 )
=> ( ord_less @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_133_subset__not__subset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
& ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_134_subset__psubset__trans,axiom,
! [A: $tType,A2: set @ A,B3: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ( ord_less @ ( set @ A ) @ B3 @ C2 )
=> ( ord_less @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_135_subset__iff__psubset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B4: set @ A] :
( ( ord_less @ ( set @ A ) @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_136_inj__on__strict__subset,axiom,
! [B: $tType,A: $tType,F: A > B,B3: set @ A,A2: set @ A] :
( ( inj_on @ A @ B @ F @ B3 )
=> ( ( ord_less @ ( set @ A ) @ A2 @ B3 )
=> ( ord_less @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ ( image @ A @ B @ F @ B3 ) ) ) ) ).
% inj_on_strict_subset
thf(fact_137_bij__betw__def,axiom,
! [B: $tType,A: $tType] :
( ( bij_betw @ A @ B )
= ( ^ [F3: A > B,A5: set @ A,B4: set @ B] :
( ( inj_on @ A @ B @ F3 @ A5 )
& ( ( image @ A @ B @ F3 @ A5 )
= B4 ) ) ) ) ).
% bij_betw_def
thf(fact_138_bij__betw__imageI,axiom,
! [A: $tType,B: $tType,F: A > B,A2: set @ A,B3: set @ B] :
( ( inj_on @ A @ B @ F @ A2 )
=> ( ( ( image @ A @ B @ F @ A2 )
= B3 )
=> ( bij_betw @ A @ B @ F @ A2 @ B3 ) ) ) ).
% bij_betw_imageI
thf(fact_139_inj__on__imp__bij__betw,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A] :
( ( inj_on @ A @ B @ F @ A2 )
=> ( bij_betw @ A @ B @ F @ A2 @ ( image @ A @ B @ F @ A2 ) ) ) ).
% inj_on_imp_bij_betw
thf(fact_140_mbs_Ominimal,axiom,
! [A: $tType] :
( ( size @ A )
=> ! [X4: A,A2: set @ A,P: A > $o] :
( ( member @ A @ X4 @ A2 )
=> ( ( P @ X4 )
=> ? [X6: A] :
( ( member @ A @ X6 @ A2 )
& ( ord_less_eq @ nat @ ( size_size @ A @ X6 ) @ ( size_size @ A @ X4 ) )
& ( P @ X6 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A2 )
=> ( ( ord_less @ nat @ ( size_size @ A @ Xa2 ) @ ( size_size @ A @ X6 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ) ) ).
% mbs.minimal
thf(fact_141_bij__betw__subset,axiom,
! [A: $tType,B: $tType,F: A > B,A2: set @ A,A6: set @ B,B3: set @ A,B5: set @ B] :
( ( bij_betw @ A @ B @ F @ A2 @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A2 )
=> ( ( ( image @ A @ B @ F @ B3 )
= B5 )
=> ( bij_betw @ A @ B @ F @ B3 @ B5 ) ) ) ) ).
% bij_betw_subset
thf(fact_142_bij__betw__byWitness,axiom,
! [A: $tType,B: $tType,A2: set @ A,F4: B > A,F: A > B,A6: set @ B] :
( ! [X6: A] :
( ( member @ A @ X6 @ A2 )
=> ( ( F4 @ ( F @ X6 ) )
= X6 ) )
=> ( ! [X6: B] :
( ( member @ B @ X6 @ A6 )
=> ( ( F @ ( F4 @ X6 ) )
= X6 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F4 @ A6 ) @ A2 )
=> ( bij_betw @ A @ B @ F @ A2 @ A6 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_143_image__inv__into__cancel,axiom,
! [B: $tType,A: $tType,F: B > A,A2: set @ B,A6: set @ A,B5: set @ A] :
( ( ( image @ B @ A @ F @ A2 )
= A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A6 )
=> ( ( image @ B @ A @ F @ ( image @ A @ B @ ( hilbert_inv_into @ B @ A @ A2 @ F ) @ B5 ) )
= B5 ) ) ) ).
% image_inv_into_cancel
thf(fact_144_listset__mono,axiom,
! [A: $tType,Xs: list @ ( set @ A ),Ys: list @ ( set @ A )] :
( ( ( size_size @ ( list @ ( set @ A ) ) @ Xs )
= ( size_size @ ( list @ ( set @ A ) ) @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ ( set @ A ) ) @ Ys ) )
=> ( ord_less_eq @ ( set @ A ) @ ( nth @ ( set @ A ) @ Xs @ I3 ) @ ( nth @ ( set @ A ) @ Ys @ I3 ) ) )
=> ( ord_less_eq @ ( set @ ( list @ A ) ) @ ( listset @ A @ Xs ) @ ( listset @ A @ Ys ) ) ) ) ).
% listset_mono
thf(fact_145_nth__list__update,axiom,
! [A: $tType,I: nat,Xs: list @ A,J: nat,X4: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X4 ) @ J )
= X4 ) )
& ( ( I != J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X4 ) @ J )
= ( nth @ A @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_146_list__update__same__conv,axiom,
! [A: $tType,I: nat,Xs: list @ A,X4: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( list_update @ A @ Xs @ I @ X4 )
= Xs )
= ( ( nth @ A @ Xs @ I )
= X4 ) ) ) ).
% list_update_same_conv
thf(fact_147_bij__betw__inv__into__subset,axiom,
! [B: $tType,A: $tType,F: A > B,A2: set @ A,A6: set @ B,B3: set @ A,B5: set @ B] :
( ( bij_betw @ A @ B @ F @ A2 @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A2 )
=> ( ( ( image @ A @ B @ F @ B3 )
= B5 )
=> ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ A2 @ F ) @ B5 @ B3 ) ) ) ) ).
% bij_betw_inv_into_subset
thf(fact_148_direct__decompD_I2_J,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: set @ A,Ss: list @ ( set @ A )] :
( ( hilber580762119decomp @ A @ A2 @ Ss )
=> ( inj_on @ ( list @ A ) @ A @ ( groups501227526m_list @ A ) @ ( listset @ A @ Ss ) ) ) ) ).
% direct_decompD(2)
thf(fact_149_Schroeder__Bernstein,axiom,
! [A: $tType,B: $tType,F: A > B,A2: set @ A,B3: set @ B,G: B > A] :
( ( inj_on @ A @ B @ F @ A2 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ B3 )
=> ( ( inj_on @ B @ A @ G @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ G @ B3 ) @ A2 )
=> ? [H: A > B] : ( bij_betw @ A @ B @ H @ A2 @ B3 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_150_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A] : ( ord_less_eq @ A @ X4 @ X4 ) ) ).
% order_refl
thf(fact_151_subset__imageE__inj,axiom,
! [A: $tType,B: $tType,B3: set @ A,F: B > A,A2: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B3 @ ( image @ B @ A @ F @ A2 ) )
=> ~ ! [C3: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C3 @ A2 )
=> ( ( B3
= ( image @ B @ A @ F @ C3 ) )
=> ~ ( inj_on @ B @ A @ F @ C3 ) ) ) ) ).
% subset_imageE_inj
thf(fact_152_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z3 )
=> ~ ( ord_less_eq @ A @ T2 @ X ) ) ) ).
% minf(8)
thf(fact_153_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z3 )
=> ( ord_less_eq @ A @ X @ T2 ) ) ) ).
% minf(6)
thf(fact_154_subset__antisym,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_155_subsetI,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ! [X6: A] :
( ( member @ A @ X6 @ A2 )
=> ( member @ A @ X6 @ B3 ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B3 ) ) ).
% subsetI
thf(fact_156_bounded__Max__nat,axiom,
! [P: nat > $o,X4: nat,M2: nat] :
( ( P @ X4 )
=> ( ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq @ nat @ X6 @ M2 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X: nat] :
( ( P @ X )
=> ( ord_less_eq @ nat @ X @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_157_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X5: A] :
( ( P @ X5 )
=> ( Q @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_158_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y2: set @ A,Z: set @ A] : Y2 = Z )
= ( ^ [A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
& ( ord_less_eq @ ( set @ A ) @ B4 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_159_subset__trans,axiom,
! [A: $tType,A2: set @ A,B3: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_160_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X6: A] :
( ( P @ X6 )
=> ( Q @ X6 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_161_subset__refl,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% subset_refl
thf(fact_162_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B4: set @ A] :
! [T3: A] :
( ( member @ A @ T3 @ A5 )
=> ( member @ A @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_163_equalityD2,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ( A2 = B3 )
=> ( ord_less_eq @ ( set @ A ) @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_164_equalityD1,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ( A2 = B3 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_165_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B4: set @ A] :
! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ( member @ A @ X5 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_166_equalityE,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ( A2 = B3 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_167_subsetD,axiom,
! [A: $tType,A2: set @ A,B3: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ( member @ A @ C @ A2 )
=> ( member @ A @ C @ B3 ) ) ) ).
% subsetD
thf(fact_168_in__mono,axiom,
! [A: $tType,A2: set @ A,B3: set @ A,X4: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
=> ( ( member @ A @ X4 @ A2 )
=> ( member @ A @ X4 @ B3 ) ) ) ).
% in_mono
thf(fact_169_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( A3 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_170_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y2: A,Z: A] : Y2 = Z )
= ( ^ [A8: A,B6: A] :
( ( ord_less_eq @ A @ B6 @ A8 )
& ( ord_less_eq @ A @ A8 @ B6 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_171_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C @ B2 )
=> ( ord_less_eq @ A @ C @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_172_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B2: A] :
( ! [A4: A,B7: A] :
( ( ord_less_eq @ A @ A4 @ B7 )
=> ( P @ A4 @ B7 ) )
=> ( ! [A4: A,B7: A] :
( ( P @ B7 @ A4 )
=> ( P @ A4 @ B7 ) )
=> ( P @ A3 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_173_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_174_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A,Y3: A,Z2: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ Z2 )
=> ( ord_less_eq @ A @ X4 @ Z2 ) ) ) ) ).
% order_trans
thf(fact_175_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ) ).
% order_class.order.antisym
thf(fact_176_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% ord_le_eq_trans
thf(fact_177_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C: A] :
( ( A3 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% ord_eq_le_trans
thf(fact_178_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y2: A,Z: A] : Y2 = Z )
= ( ^ [A8: A,B6: A] :
( ( ord_less_eq @ A @ A8 @ B6 )
& ( ord_less_eq @ A @ B6 @ A8 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_179_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y3: A,X4: A] :
( ( ord_less_eq @ A @ Y3 @ X4 )
=> ( ( ord_less_eq @ A @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ) ).
% antisym_conv
thf(fact_180_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y3: A,Z2: A] :
( ( ( ord_less_eq @ A @ X4 @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ X4 )
=> ~ ( ord_less_eq @ A @ X4 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X4 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y3 ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ X4 ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X4 ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X4 )
=> ~ ( ord_less_eq @ A @ X4 @ Y3 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_181_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% order.trans
thf(fact_182_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y3: A] :
( ~ ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ).
% le_cases
thf(fact_183_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A,Y3: A] :
( ( X4 = Y3 )
=> ( ord_less_eq @ A @ X4 @ Y3 ) ) ) ).
% eq_refl
thf(fact_184_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
| ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ).
% linear
thf(fact_185_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ) ).
% antisym
thf(fact_186_le__rel__bool__arg__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_less_eq @ ( $o > A ) )
= ( ^ [X8: $o > A,Y5: $o > A] :
( ( ord_less_eq @ A @ ( X8 @ $false ) @ ( Y5 @ $false ) )
& ( ord_less_eq @ A @ ( X8 @ $true ) @ ( Y5 @ $true ) ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_187_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y2: A,Z: A] : Y2 = Z )
= ( ^ [X5: A,Y4: A] :
( ( ord_less_eq @ A @ X5 @ Y4 )
& ( ord_less_eq @ A @ Y4 @ X5 ) ) ) ) ) ).
% eq_iff
thf(fact_188_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B2: A,F: A > B,C: B] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X6: A,Y: A] :
( ( ord_less_eq @ A @ X6 @ Y )
=> ( ord_less_eq @ B @ ( F @ X6 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ B @ ( F @ A3 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_189_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F: B > A,B2: B,C: B] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X6: B,Y: B] :
( ( ord_less_eq @ B @ X6 @ Y )
=> ( ord_less_eq @ A @ ( F @ X6 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_190_order__subst2,axiom,
! [A: $tType,C4: $tType] :
( ( ( order @ C4 )
& ( order @ A ) )
=> ! [A3: A,B2: A,F: A > C4,C: C4] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ C4 @ ( F @ B2 ) @ C )
=> ( ! [X6: A,Y: A] :
( ( ord_less_eq @ A @ X6 @ Y )
=> ( ord_less_eq @ C4 @ ( F @ X6 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ C4 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_191_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F: B > A,B2: B,C: B] :
( ( ord_less_eq @ A @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X6: B,Y: B] :
( ( ord_less_eq @ B @ X6 @ Y )
=> ( ord_less_eq @ A @ ( F @ X6 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_192_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F3: A > B,G3: A > B] :
! [X5: A] : ( ord_less_eq @ B @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) ) ) ) ).
% le_fun_def
thf(fact_193_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B] :
( ! [X6: A] : ( ord_less_eq @ B @ ( F @ X6 ) @ ( G @ X6 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_194_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X4: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) ) ) ) ).
% le_funE
thf(fact_195_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X4: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) ) ) ) ).
% le_funD
thf(fact_196_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( A3 != B2 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_197_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( A3 != B2 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_198_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y3: A] :
( ( ~ ( ord_less @ A @ X4 @ Y3 ) )
= ( ( ord_less @ A @ Y3 @ X4 )
| ( X4 = Y3 ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_199_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A,C: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C @ B2 )
=> ( ord_less @ A @ C @ A3 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_200_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B2: A] :
( ! [A4: A,B7: A] :
( ( ord_less @ A @ A4 @ B7 )
=> ( P @ A4 @ B7 ) )
=> ( ! [A4: A] : ( P @ A4 @ A4 )
=> ( ! [A4: A,B7: A] :
( ( P @ B7 @ A4 )
=> ( P @ A4 @ B7 ) )
=> ( P @ A3 @ B2 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_201_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P3: A > $o] :
? [X3: A] : ( P3 @ X3 ) )
= ( ^ [P2: A > $o] :
? [N3: A] :
( ( P2 @ N3 )
& ! [M5: A] :
( ( ord_less @ A @ M5 @ N3 )
=> ~ ( P2 @ M5 ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_202_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ~ ( ord_less @ A @ Y3 @ X4 ) ) ) ).
% less_imp_not_less
thf(fact_203_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A,C: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ B2 @ C )
=> ( ord_less @ A @ A3 @ C ) ) ) ) ).
% order.strict_trans
thf(fact_204_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 ) ) ).
% dual_order.irrefl
thf(fact_205_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y3: A] :
( ~ ( ord_less @ A @ X4 @ Y3 )
=> ( ( X4 != Y3 )
=> ( ord_less @ A @ Y3 @ X4 ) ) ) ) ).
% linorder_cases
thf(fact_206_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A,Y3: A,P: $o] :
( ( ord_less @ A @ X4 @ Y3 )
=> ( ( ord_less @ A @ Y3 @ X4 )
=> P ) ) ) ).
% less_imp_triv
thf(fact_207_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ( Y3 != X4 ) ) ) ).
% less_imp_not_eq2
thf(fact_208_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y3: A,X4: A] :
( ~ ( ord_less @ A @ Y3 @ X4 )
=> ( ( ~ ( ord_less @ A @ X4 @ Y3 ) )
= ( X4 = Y3 ) ) ) ) ).
% antisym_conv3
thf(fact_209_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A3: A] :
( ! [X6: A] :
( ! [Y6: A] :
( ( ord_less @ A @ Y6 @ X6 )
=> ( P @ Y6 ) )
=> ( P @ X6 ) )
=> ( P @ A3 ) ) ) ).
% less_induct
thf(fact_210_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ~ ( ord_less @ A @ Y3 @ X4 ) ) ) ).
% less_not_sym
thf(fact_211_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ( X4 != Y3 ) ) ) ).
% less_imp_not_eq
thf(fact_212_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ~ ( ord_less @ A @ A3 @ B2 ) ) ) ).
% dual_order.asym
thf(fact_213_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( B2 = C )
=> ( ord_less @ A @ A3 @ C ) ) ) ) ).
% ord_less_eq_trans
thf(fact_214_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C: A] :
( ( A3 = B2 )
=> ( ( ord_less @ A @ B2 @ C )
=> ( ord_less @ A @ A3 @ C ) ) ) ) ).
% ord_eq_less_trans
thf(fact_215_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A] :
~ ( ord_less @ A @ X4 @ X4 ) ) ).
% less_irrefl
thf(fact_216_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
| ( X4 = Y3 )
| ( ord_less @ A @ Y3 @ X4 ) ) ) ).
% less_linear
thf(fact_217_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A,Y3: A,Z2: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ( ( ord_less @ A @ Y3 @ Z2 )
=> ( ord_less @ A @ X4 @ Z2 ) ) ) ) ).
% less_trans
thf(fact_218_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A3 ) ) ) ).
% less_asym'
thf(fact_219_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ~ ( ord_less @ A @ Y3 @ X4 ) ) ) ).
% less_asym
thf(fact_220_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ( X4 != Y3 ) ) ) ).
% less_imp_neq
thf(fact_221_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ? [Z3: A] :
( ( ord_less @ A @ X4 @ Z3 )
& ( ord_less @ A @ Z3 @ Y3 ) ) ) ) ).
% dense
thf(fact_222_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A3 ) ) ) ).
% order.asym
thf(fact_223_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y3: A] :
( ( X4 != Y3 )
= ( ( ord_less @ A @ X4 @ Y3 )
| ( ord_less @ A @ Y3 @ X4 ) ) ) ) ).
% neq_iff
thf(fact_224_neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y3: A] :
( ( X4 != Y3 )
=> ( ~ ( ord_less @ A @ X4 @ Y3 )
=> ( ord_less @ A @ Y3 @ X4 ) ) ) ) ).
% neqE
thf(fact_225_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X4: A] :
? [X_1: A] : ( ord_less @ A @ X4 @ X_1 ) ) ).
% gt_ex
thf(fact_226_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X4: A] :
? [Y: A] : ( ord_less @ A @ Y @ X4 ) ) ).
% lt_ex
thf(fact_227_order__less__subst2,axiom,
! [A: $tType,C4: $tType] :
( ( ( order @ C4 )
& ( order @ A ) )
=> ! [A3: A,B2: A,F: A > C4,C: C4] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ C4 @ ( F @ B2 ) @ C )
=> ( ! [X6: A,Y: A] :
( ( ord_less @ A @ X6 @ Y )
=> ( ord_less @ C4 @ ( F @ X6 ) @ ( F @ Y ) ) )
=> ( ord_less @ C4 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% order_less_subst2
thf(fact_228_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F: B > A,B2: B,C: B] :
( ( ord_less @ A @ A3 @ ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C )
=> ( ! [X6: B,Y: B] :
( ( ord_less @ B @ X6 @ Y )
=> ( ord_less @ A @ ( F @ X6 ) @ ( F @ Y ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_229_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B2: A,F: A > B,C: B] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X6: A,Y: A] :
( ( ord_less @ A @ X6 @ Y )
=> ( ord_less @ B @ ( F @ X6 ) @ ( F @ Y ) ) )
=> ( ord_less @ B @ ( F @ A3 ) @ C ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_230_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F: B > A,B2: B,C: B] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C )
=> ( ! [X6: B,Y: B] :
( ( ord_less @ B @ X6 @ Y )
=> ( ord_less @ A @ ( F @ X6 ) @ ( F @ Y ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_231_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X6: A] :
( ( ord_less @ A @ Z4 @ X6 )
=> ( ( P @ X6 )
= ( P4 @ X6 ) ) )
=> ( ? [Z4: A] :
! [X6: A] :
( ( ord_less @ A @ Z4 @ X6 )
=> ( ( Q @ X6 )
= ( Q2 @ X6 ) ) )
=> ? [Z3: A] :
! [X: A] :
( ( ord_less @ A @ Z3 @ X )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P4 @ X )
& ( Q2 @ X ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_232_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X6: A] :
( ( ord_less @ A @ Z4 @ X6 )
=> ( ( P @ X6 )
= ( P4 @ X6 ) ) )
=> ( ? [Z4: A] :
! [X6: A] :
( ( ord_less @ A @ Z4 @ X6 )
=> ( ( Q @ X6 )
= ( Q2 @ X6 ) ) )
=> ? [Z3: A] :
! [X: A] :
( ( ord_less @ A @ Z3 @ X )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P4 @ X )
| ( Q2 @ X ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_233_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X: A] :
( ( ord_less @ A @ Z3 @ X )
=> ( X != T2 ) ) ) ).
% pinf(3)
thf(fact_234_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X: A] :
( ( ord_less @ A @ Z3 @ X )
=> ( X != T2 ) ) ) ).
% pinf(4)
thf(fact_235_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X: A] :
( ( ord_less @ A @ Z3 @ X )
=> ~ ( ord_less @ A @ X @ T2 ) ) ) ).
% pinf(5)
thf(fact_236_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X: A] :
( ( ord_less @ A @ Z3 @ X )
=> ( ord_less @ A @ T2 @ X ) ) ) ).
% pinf(7)
thf(fact_237_pinf_I11_J,axiom,
! [C4: $tType,D: $tType] :
( ( ord @ C4 )
=> ! [F5: D] :
? [Z3: C4] :
! [X: C4] :
( ( ord_less @ C4 @ Z3 @ X )
=> ( F5 = F5 ) ) ) ).
% pinf(11)
thf(fact_238_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z4 )
=> ( ( P @ X6 )
= ( P4 @ X6 ) ) )
=> ( ? [Z4: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z4 )
=> ( ( Q @ X6 )
= ( Q2 @ X6 ) ) )
=> ? [Z3: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z3 )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P4 @ X )
& ( Q2 @ X ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_239_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z4 )
=> ( ( P @ X6 )
= ( P4 @ X6 ) ) )
=> ( ? [Z4: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z4 )
=> ( ( Q @ X6 )
= ( Q2 @ X6 ) ) )
=> ? [Z3: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z3 )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P4 @ X )
| ( Q2 @ X ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_240_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z3 )
=> ( X != T2 ) ) ) ).
% minf(3)
thf(fact_241_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z3 )
=> ( X != T2 ) ) ) ).
% minf(4)
thf(fact_242_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z3 )
=> ( ord_less @ A @ X @ T2 ) ) ) ).
% minf(5)
thf(fact_243_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z3 )
=> ~ ( ord_less @ A @ T2 @ X ) ) ) ).
% minf(7)
thf(fact_244_minf_I11_J,axiom,
! [C4: $tType,D: $tType] :
( ( ord @ C4 )
=> ! [F5: D] :
? [Z3: C4] :
! [X: C4] :
( ( ord_less @ C4 @ X @ Z3 )
=> ( F5 = F5 ) ) ) ).
% minf(11)
thf(fact_245_order_Onot__eq__order__implies__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( A3 != B2 )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_246_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_247_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B6: A,A8: A] :
( ( ord_less_eq @ A @ B6 @ A8 )
& ( A8 != B6 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_248_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B6: A,A8: A] :
( ( ord_less @ A @ B6 @ A8 )
| ( A8 = B6 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_249_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% order.strict_implies_order
thf(fact_250_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X4: A,Y3: A,Z2: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ( ! [W: A] :
( ( ord_less @ A @ X4 @ W )
=> ( ( ord_less @ A @ W @ Y3 )
=> ( ord_less_eq @ A @ W @ Z2 ) ) )
=> ( ord_less_eq @ A @ Y3 @ Z2 ) ) ) ) ).
% dense_le_bounded
thf(fact_251_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z2: A,X4: A,Y3: A] :
( ( ord_less @ A @ Z2 @ X4 )
=> ( ! [W: A] :
( ( ord_less @ A @ Z2 @ W )
=> ( ( ord_less @ A @ W @ X4 )
=> ( ord_less_eq @ A @ Y3 @ W ) ) )
=> ( ord_less_eq @ A @ Y3 @ Z2 ) ) ) ) ).
% dense_ge_bounded
thf(fact_252_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A,C: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C @ B2 )
=> ( ord_less @ A @ C @ A3 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_253_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C @ B2 )
=> ( ord_less @ A @ C @ A3 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_254_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A8: A,B6: A] :
( ( ord_less_eq @ A @ A8 @ B6 )
& ( A8 != B6 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_255_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A8: A,B6: A] :
( ( ord_less @ A @ A8 @ B6 )
| ( A8 = B6 ) ) ) ) ) ).
% order.order_iff_strict
% Subclasses (1)
thf(subcl_Groups_Ocomm__monoid__add___HOL_Otype,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ( type @ A ) ) ).
% Type constructors (29)
thf(tcon_Nat_Onat___Rings_Odioid,axiom,
dioid @ nat ).
thf(tcon_fun___Rings_Odioid_1,axiom,
! [A9: $tType,A10: $tType] :
( ( dioid @ A10 )
=> ( dioid @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Groups_Ocanonically__ordered__monoid__add,axiom,
! [A9: $tType,A10: $tType] :
( ( dioid @ A10 )
=> ( canoni770627133id_add @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Groups_Ocomm__monoid__add,axiom,
! [A9: $tType,A10: $tType] :
( ( comm_monoid_add @ A10 )
=> ( comm_monoid_add @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A9: $tType,A10: $tType] :
( ( preorder @ A10 )
=> ( preorder @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A9: $tType,A10: $tType] :
( ( order @ A10 )
=> ( order @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A9: $tType,A10: $tType] :
( ( ord @ A10 )
=> ( ord @ ( A9 > A10 ) ) ) ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add_2,axiom,
canoni770627133id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_3,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_4,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
no_top @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_5,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_6,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Nat_Osize,axiom,
size @ nat ).
thf(tcon_Set_Oset___Orderings_Opreorder_7,axiom,
! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_8,axiom,
! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_9,axiom,
! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_10,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_11,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_12,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_13,axiom,
ord @ $o ).
thf(tcon_List_Olist___Nat_Osize_14,axiom,
! [A9: $tType] : ( size @ ( list @ A9 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_15,axiom,
! [A9: $tType] : ( comm_monoid_add @ ( multiset @ A9 ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Opreorder_16,axiom,
! [A9: $tType] :
( ( preorder @ A9 )
=> ( preorder @ ( multiset @ A9 ) ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oorder_17,axiom,
! [A9: $tType] :
( ( preorder @ A9 )
=> ( order @ ( multiset @ A9 ) ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oord_18,axiom,
! [A9: $tType] :
( ( preorder @ A9 )
=> ( ord @ ( multiset @ A9 ) ) ) ).
thf(tcon_Multiset_Omultiset___Nat_Osize_19,axiom,
! [A9: $tType] : ( size @ ( multiset @ A9 ) ) ).
% Free types (1)
thf(tfree_0,hypothesis,
comm_monoid_add @ a ).
% Conjectures (1)
thf(conj_0,conjecture,
( a3
= ( groups501227526m_list @ a @ qs2 ) ) ).
%------------------------------------------------------------------------------