TPTP Problem File: ITP104^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP104^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer ListSlice problem prob_199__5618146_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 : ListSlice/prob_199__5618146_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% Syntax : Number of formulae : 462 ( 122 unt; 112 typ; 0 def)
% Number of atoms : 1046 ( 371 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 3415 ( 124 ~; 18 |; 67 &;2673 @)
% ( 0 <=>; 533 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 18 ( 17 usr)
% Number of type conns : 235 ( 235 >; 0 *; 0 +; 0 <<)
% Number of symbols : 96 ( 95 usr; 4 con; 0-3 aty)
% Number of variables : 1096 ( 94 ^; 941 !; 61 ?;1096 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:32:28.219
%------------------------------------------------------------------------------
% Could-be-implicit typings (17)
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
list_l2071841302list_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
list_list_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
list_list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
set_list_list_a: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
list_set_list_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
list_set_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
list_set_a: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (95)
thf(sy_c_List2_Of__image_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
f_image_list_list_a: list_list_list_a > set_nat > set_list_list_a ).
thf(sy_c_List2_Of__image_001t__List__Olist_It__Nat__Onat_J,type,
f_image_list_nat: list_list_nat > set_nat > set_list_nat ).
thf(sy_c_List2_Of__image_001t__List__Olist_Itf__a_J,type,
f_image_list_a: list_list_a > set_nat > set_list_a ).
thf(sy_c_List2_Of__image_001t__Nat__Onat,type,
f_image_nat: list_nat > set_nat > set_nat ).
thf(sy_c_List2_Of__image_001tf__a,type,
f_image_a: list_a > set_nat > set_a ).
thf(sy_c_List2_Olist__asc_001t__Nat__Onat,type,
list_asc_nat: list_nat > $o ).
thf(sy_c_List2_Olist__asc_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
list_asc_set_list_a: list_set_list_a > $o ).
thf(sy_c_List2_Olist__asc_001t__Set__Oset_It__Nat__Onat_J,type,
list_asc_set_nat: list_set_nat > $o ).
thf(sy_c_List2_Olist__asc_001t__Set__Oset_Itf__a_J,type,
list_asc_set_a: list_set_a > $o ).
thf(sy_c_List2_Olist__desc_001t__Nat__Onat,type,
list_desc_nat: list_nat > $o ).
thf(sy_c_List2_Olist__desc_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
list_desc_set_list_a: list_set_list_a > $o ).
thf(sy_c_List2_Olist__desc_001t__Set__Oset_It__Nat__Onat_J,type,
list_desc_set_nat: list_set_nat > $o ).
thf(sy_c_List2_Olist__desc_001t__Set__Oset_Itf__a_J,type,
list_desc_set_a: list_set_a > $o ).
thf(sy_c_List2_Olist__strict__asc_001t__Nat__Onat,type,
list_strict_asc_nat: list_nat > $o ).
thf(sy_c_List2_Olist__strict__desc_001t__Nat__Onat,type,
list_strict_desc_nat: list_nat > $o ).
thf(sy_c_ListSlice__Mirabelle__csprijwpmv_Olist__slice2_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
listSl769338732list_a: list_list_list_a > nat > list_l2071841302list_a ).
thf(sy_c_ListSlice__Mirabelle__csprijwpmv_Olist__slice2_001t__List__Olist_It__Nat__Onat_J,type,
listSl2099749758st_nat: list_list_nat > nat > list_list_list_nat ).
thf(sy_c_ListSlice__Mirabelle__csprijwpmv_Olist__slice2_001t__List__Olist_Itf__a_J,type,
listSl703198822list_a: list_list_a > nat > list_list_list_a ).
thf(sy_c_ListSlice__Mirabelle__csprijwpmv_Olist__slice2_001t__Nat__Onat,type,
listSl882585454e2_nat: list_nat > nat > list_list_nat ).
thf(sy_c_ListSlice__Mirabelle__csprijwpmv_Olist__slice2_001tf__a,type,
listSl1174287072ice2_a: list_a > nat > list_list_a ).
thf(sy_c_ListSlice__Mirabelle__csprijwpmv_Olist__slice_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
listSl2102238196list_a: list_list_list_a > nat > list_l2071841302list_a ).
thf(sy_c_ListSlice__Mirabelle__csprijwpmv_Olist__slice_001t__List__Olist_It__Nat__Onat_J,type,
listSl729562870st_nat: list_list_nat > nat > list_list_list_nat ).
thf(sy_c_ListSlice__Mirabelle__csprijwpmv_Olist__slice_001t__List__Olist_Itf__a_J,type,
listSl162220270list_a: list_list_a > nat > list_list_list_a ).
thf(sy_c_ListSlice__Mirabelle__csprijwpmv_Olist__slice_001t__Nat__Onat,type,
listSl1630398182ce_nat: list_nat > nat > list_list_nat ).
thf(sy_c_ListSlice__Mirabelle__csprijwpmv_Olist__slice_001tf__a,type,
listSl97544552lice_a: list_a > nat > list_list_a ).
thf(sy_c_ListSlice__Mirabelle__csprijwpmv_Olist__slice__aux_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
listSl902632890list_a: list_list_list_a > nat > nat > list_l2071841302list_a ).
thf(sy_c_ListSlice__Mirabelle__csprijwpmv_Olist__slice__aux_001t__List__Olist_It__Nat__Onat_J,type,
listSl1506485424st_nat: list_list_nat > nat > nat > list_list_list_nat ).
thf(sy_c_ListSlice__Mirabelle__csprijwpmv_Olist__slice__aux_001t__List__Olist_Itf__a_J,type,
listSl856612276list_a: list_list_a > nat > nat > list_list_list_a ).
thf(sy_c_ListSlice__Mirabelle__csprijwpmv_Olist__slice__aux_001t__Nat__Onat,type,
listSl1794011552ux_nat: list_nat > nat > nat > list_list_nat ).
thf(sy_c_ListSlice__Mirabelle__csprijwpmv_Olist__slice__aux_001tf__a,type,
listSl1495374126_aux_a: list_a > nat > nat > list_list_a ).
thf(sy_c_List_Obutlast_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
butlast_list_list_a: list_list_list_a > list_list_list_a ).
thf(sy_c_List_Obutlast_001t__List__Olist_It__Nat__Onat_J,type,
butlast_list_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Obutlast_001t__List__Olist_Itf__a_J,type,
butlast_list_a: list_list_a > list_list_a ).
thf(sy_c_List_Obutlast_001t__Nat__Onat,type,
butlast_nat: list_nat > list_nat ).
thf(sy_c_List_Obutlast_001tf__a,type,
butlast_a: list_a > list_a ).
thf(sy_c_List_Odistinct_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
distinct_list_list_a: list_list_list_a > $o ).
thf(sy_c_List_Odistinct_001t__List__Olist_It__Nat__Onat_J,type,
distinct_list_nat: list_list_nat > $o ).
thf(sy_c_List_Odistinct_001t__List__Olist_Itf__a_J,type,
distinct_list_a: list_list_a > $o ).
thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
distinct_nat: list_nat > $o ).
thf(sy_c_List_Odistinct_001tf__a,type,
distinct_a: list_a > $o ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
set_list_list_a2: list_list_list_a > set_list_list_a ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
set_list_nat2: list_list_nat > set_list_nat ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_Itf__a_J,type,
set_list_a2: list_list_a > set_list_a ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_List_Olist__ex_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
list_ex_list_list_a: ( list_list_a > $o ) > list_list_list_a > $o ).
thf(sy_c_List_Olist__ex_001t__List__Olist_It__Nat__Onat_J,type,
list_ex_list_nat: ( list_nat > $o ) > list_list_nat > $o ).
thf(sy_c_List_Olist__ex_001t__List__Olist_Itf__a_J,type,
list_ex_list_a: ( list_a > $o ) > list_list_a > $o ).
thf(sy_c_List_Olist__ex_001t__Nat__Onat,type,
list_ex_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__ex_001tf__a,type,
list_ex_a: ( a > $o ) > list_a > $o ).
thf(sy_c_List_Olist__update_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
list_u1330012152list_a: list_list_list_a > nat > list_list_a > list_list_list_a ).
thf(sy_c_List_Olist__update_001t__List__Olist_It__Nat__Onat_J,type,
list_update_list_nat: list_list_nat > nat > list_nat > list_list_nat ).
thf(sy_c_List_Olist__update_001t__List__Olist_Itf__a_J,type,
list_update_list_a: list_list_a > nat > list_a > list_list_a ).
thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
list_update_nat: list_nat > nat > nat > list_nat ).
thf(sy_c_List_Olist__update_001tf__a,type,
list_update_a: list_a > nat > a > list_a ).
thf(sy_c_List_Onth_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
nth_list_list_list_a: list_l2071841302list_a > nat > list_list_list_a ).
thf(sy_c_List_Onth_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
nth_list_list_nat: list_list_list_nat > nat > list_list_nat ).
thf(sy_c_List_Onth_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
nth_list_list_a: list_list_list_a > nat > list_list_a ).
thf(sy_c_List_Onth_001t__List__Olist_It__Nat__Onat_J,type,
nth_list_nat: list_list_nat > nat > list_nat ).
thf(sy_c_List_Onth_001t__List__Olist_Itf__a_J,type,
nth_list_a: list_list_a > nat > list_a ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
nth_set_list_a: list_set_list_a > nat > set_list_a ).
thf(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
nth_set_nat: list_set_nat > nat > set_nat ).
thf(sy_c_List_Onth_001t__Set__Oset_Itf__a_J,type,
nth_set_a: list_set_a > nat > set_a ).
thf(sy_c_List_Onth_001tf__a,type,
nth_a: list_a > nat > a ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
size_s1764310658list_a: list_l2071841302list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
size_s1321307156st_nat: list_list_list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
size_s575106428list_a: list_list_list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
size_s1898481668st_nat: list_list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
size_s1427607542list_a: list_list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
size_s1635937238list_a: list_set_list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
size_s577819178et_nat: list_set_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
size_size_list_set_a: list_set_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
ord_le1756736906list_a: set_list_list_a > set_list_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le1747345782st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
ord_le1301786372list_a: set_list_a > set_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
member_list_list_a: list_list_a > set_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_k,type,
k: nat ).
thf(sy_v_m,type,
m: nat ).
thf(sy_v_xs,type,
xs: list_a ).
% Relevant facts (348)
thf(fact_0_nth__equalityI,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( ( size_s1898481668st_nat @ Xs )
= ( size_s1898481668st_nat @ Ys ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s1898481668st_nat @ Xs ) )
=> ( ( nth_list_nat @ Xs @ I )
= ( nth_list_nat @ Ys @ I ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_1_nth__equalityI,axiom,
! [Xs: list_list_list_a,Ys: list_list_list_a] :
( ( ( size_s575106428list_a @ Xs )
= ( size_s575106428list_a @ Ys ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s575106428list_a @ Xs ) )
=> ( ( nth_list_list_a @ Xs @ I )
= ( nth_list_list_a @ Ys @ I ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_2_nth__equalityI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I )
= ( nth_nat @ Ys @ I ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_3_nth__equalityI,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( size_s1427607542list_a @ Xs )
= ( size_s1427607542list_a @ Ys ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s1427607542list_a @ Xs ) )
=> ( ( nth_list_a @ Xs @ I )
= ( nth_list_a @ Ys @ I ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_4_nth__equalityI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ Xs @ I )
= ( nth_a @ Ys @ I ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_5_Skolem__list__nth,axiom,
! [K: nat,P: nat > list_nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X: list_nat] : ( P @ I2 @ X ) ) )
= ( ? [Xs2: list_list_nat] :
( ( ( size_s1898481668st_nat @ Xs2 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_list_nat @ Xs2 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_6_Skolem__list__nth,axiom,
! [K: nat,P: nat > list_list_a > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X: list_list_a] : ( P @ I2 @ X ) ) )
= ( ? [Xs2: list_list_list_a] :
( ( ( size_s575106428list_a @ Xs2 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_list_list_a @ Xs2 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_7_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X: nat] : ( P @ I2 @ X ) ) )
= ( ? [Xs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_nat @ Xs2 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_8_Skolem__list__nth,axiom,
! [K: nat,P: nat > list_a > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X: list_a] : ( P @ I2 @ X ) ) )
= ( ? [Xs2: list_list_a] :
( ( ( size_s1427607542list_a @ Xs2 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_list_a @ Xs2 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_9_Skolem__list__nth,axiom,
! [K: nat,P: nat > a > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X: a] : ( P @ I2 @ X ) ) )
= ( ? [Xs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_a @ Xs2 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_10_list__eq__iff__nth__eq,axiom,
( ( ^ [Y: list_list_nat,Z: list_list_nat] : Y = Z )
= ( ^ [Xs2: list_list_nat,Ys2: list_list_nat] :
( ( ( size_s1898481668st_nat @ Xs2 )
= ( size_s1898481668st_nat @ Ys2 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s1898481668st_nat @ Xs2 ) )
=> ( ( nth_list_nat @ Xs2 @ I2 )
= ( nth_list_nat @ Ys2 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_11_list__eq__iff__nth__eq,axiom,
( ( ^ [Y: list_list_list_a,Z: list_list_list_a] : Y = Z )
= ( ^ [Xs2: list_list_list_a,Ys2: list_list_list_a] :
( ( ( size_s575106428list_a @ Xs2 )
= ( size_s575106428list_a @ Ys2 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s575106428list_a @ Xs2 ) )
=> ( ( nth_list_list_a @ Xs2 @ I2 )
= ( nth_list_list_a @ Ys2 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_12_list__eq__iff__nth__eq,axiom,
( ( ^ [Y: list_nat,Z: list_nat] : Y = Z )
= ( ^ [Xs2: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I2 )
= ( nth_nat @ Ys2 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_13_list__eq__iff__nth__eq,axiom,
( ( ^ [Y: list_list_a,Z: list_list_a] : Y = Z )
= ( ^ [Xs2: list_list_a,Ys2: list_list_a] :
( ( ( size_s1427607542list_a @ Xs2 )
= ( size_s1427607542list_a @ Ys2 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s1427607542list_a @ Xs2 ) )
=> ( ( nth_list_a @ Xs2 @ I2 )
= ( nth_list_a @ Ys2 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_14_list__eq__iff__nth__eq,axiom,
( ( ^ [Y: list_a,Z: list_a] : Y = Z )
= ( ^ [Xs2: list_a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs2 ) )
=> ( ( nth_a @ Xs2 @ I2 )
= ( nth_a @ Ys2 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_15_list__slice2__list__slice__nth,axiom,
! [M: nat,Xs: list_list_nat,K: nat] :
( ( ord_less_nat @ M @ ( divide_divide_nat @ ( size_s1898481668st_nat @ Xs ) @ K ) )
=> ( ( nth_list_list_nat @ ( listSl2099749758st_nat @ Xs @ K ) @ M )
= ( nth_list_list_nat @ ( listSl729562870st_nat @ Xs @ K ) @ M ) ) ) ).
% list_slice2_list_slice_nth
thf(fact_16_list__slice2__list__slice__nth,axiom,
! [M: nat,Xs: list_list_list_a,K: nat] :
( ( ord_less_nat @ M @ ( divide_divide_nat @ ( size_s575106428list_a @ Xs ) @ K ) )
=> ( ( nth_list_list_list_a @ ( listSl769338732list_a @ Xs @ K ) @ M )
= ( nth_list_list_list_a @ ( listSl2102238196list_a @ Xs @ K ) @ M ) ) ) ).
% list_slice2_list_slice_nth
thf(fact_17_list__slice2__list__slice__nth,axiom,
! [M: nat,Xs: list_list_a,K: nat] :
( ( ord_less_nat @ M @ ( divide_divide_nat @ ( size_s1427607542list_a @ Xs ) @ K ) )
=> ( ( nth_list_list_a @ ( listSl703198822list_a @ Xs @ K ) @ M )
= ( nth_list_list_a @ ( listSl162220270list_a @ Xs @ K ) @ M ) ) ) ).
% list_slice2_list_slice_nth
thf(fact_18_list__slice2__list__slice__nth,axiom,
! [M: nat,Xs: list_nat,K: nat] :
( ( ord_less_nat @ M @ ( divide_divide_nat @ ( size_size_list_nat @ Xs ) @ K ) )
=> ( ( nth_list_nat @ ( listSl882585454e2_nat @ Xs @ K ) @ M )
= ( nth_list_nat @ ( listSl1630398182ce_nat @ Xs @ K ) @ M ) ) ) ).
% list_slice2_list_slice_nth
thf(fact_19_list__slice2__list__slice__nth,axiom,
! [M: nat,Xs: list_a,K: nat] :
( ( ord_less_nat @ M @ ( divide_divide_nat @ ( size_size_list_a @ Xs ) @ K ) )
=> ( ( nth_list_a @ ( listSl1174287072ice2_a @ Xs @ K ) @ M )
= ( nth_list_a @ ( listSl97544552lice_a @ Xs @ K ) @ M ) ) ) ).
% list_slice2_list_slice_nth
thf(fact_20_length__induct,axiom,
! [P: list_list_nat > $o,Xs: list_list_nat] :
( ! [Xs3: list_list_nat] :
( ! [Ys3: list_list_nat] :
( ( ord_less_nat @ ( size_s1898481668st_nat @ Ys3 ) @ ( size_s1898481668st_nat @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_21_length__induct,axiom,
! [P: list_list_list_a > $o,Xs: list_list_list_a] :
( ! [Xs3: list_list_list_a] :
( ! [Ys3: list_list_list_a] :
( ( ord_less_nat @ ( size_s575106428list_a @ Ys3 ) @ ( size_s575106428list_a @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_22_length__induct,axiom,
! [P: list_list_a > $o,Xs: list_list_a] :
( ! [Xs3: list_list_a] :
( ! [Ys3: list_list_a] :
( ( ord_less_nat @ ( size_s1427607542list_a @ Ys3 ) @ ( size_s1427607542list_a @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_23_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs3: list_nat] :
( ! [Ys3: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys3 ) @ ( size_size_list_nat @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_24_length__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ! [Xs3: list_a] :
( ! [Ys3: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Ys3 ) @ ( size_size_list_a @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_25_list__strict__desc__trans,axiom,
( list_strict_desc_nat
= ( ^ [Xs2: list_nat] :
! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ ( nth_nat @ Xs2 @ J ) @ ( nth_nat @ Xs2 @ I2 ) ) ) ) ) ) ).
% list_strict_desc_trans
thf(fact_26_list__slice__nth__length,axiom,
! [M: nat,Xs: list_list_nat,K: nat] :
( ( ord_less_nat @ M @ ( divide_divide_nat @ ( size_s1898481668st_nat @ Xs ) @ K ) )
=> ( ( size_s1898481668st_nat @ ( nth_list_list_nat @ ( listSl729562870st_nat @ Xs @ K ) @ M ) )
= K ) ) ).
% list_slice_nth_length
thf(fact_27_list__slice__nth__length,axiom,
! [M: nat,Xs: list_list_list_a,K: nat] :
( ( ord_less_nat @ M @ ( divide_divide_nat @ ( size_s575106428list_a @ Xs ) @ K ) )
=> ( ( size_s575106428list_a @ ( nth_list_list_list_a @ ( listSl2102238196list_a @ Xs @ K ) @ M ) )
= K ) ) ).
% list_slice_nth_length
thf(fact_28_list__slice__nth__length,axiom,
! [M: nat,Xs: list_list_a,K: nat] :
( ( ord_less_nat @ M @ ( divide_divide_nat @ ( size_s1427607542list_a @ Xs ) @ K ) )
=> ( ( size_s1427607542list_a @ ( nth_list_list_a @ ( listSl162220270list_a @ Xs @ K ) @ M ) )
= K ) ) ).
% list_slice_nth_length
thf(fact_29_list__slice__nth__length,axiom,
! [M: nat,Xs: list_nat,K: nat] :
( ( ord_less_nat @ M @ ( divide_divide_nat @ ( size_size_list_nat @ Xs ) @ K ) )
=> ( ( size_size_list_nat @ ( nth_list_nat @ ( listSl1630398182ce_nat @ Xs @ K ) @ M ) )
= K ) ) ).
% list_slice_nth_length
thf(fact_30_list__slice__nth__length,axiom,
! [M: nat,Xs: list_a,K: nat] :
( ( ord_less_nat @ M @ ( divide_divide_nat @ ( size_size_list_a @ Xs ) @ K ) )
=> ( ( size_size_list_a @ ( nth_list_a @ ( listSl97544552lice_a @ Xs @ K ) @ M ) )
= K ) ) ).
% list_slice_nth_length
thf(fact_31_list__strict__asc__trans,axiom,
( list_strict_asc_nat
= ( ^ [Xs2: list_nat] :
! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ ( nth_nat @ Xs2 @ I2 ) @ ( nth_nat @ Xs2 @ J ) ) ) ) ) ) ).
% list_strict_asc_trans
thf(fact_32_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_list_nat] :
( ( size_s1898481668st_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_33_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_list_list_a] :
( ( size_s575106428list_a @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_34_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_list_a] :
( ( size_s1427607542list_a @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_35_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_nat] :
( ( size_size_list_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_36_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_a] :
( ( size_size_list_a @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_37_neq__if__length__neq,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( ( size_s1898481668st_nat @ Xs )
!= ( size_s1898481668st_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_38_neq__if__length__neq,axiom,
! [Xs: list_list_list_a,Ys: list_list_list_a] :
( ( ( size_s575106428list_a @ Xs )
!= ( size_s575106428list_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_39_neq__if__length__neq,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( size_s1427607542list_a @ Xs )
!= ( size_s1427607542list_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_40_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_41_neq__if__length__neq,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
!= ( size_size_list_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_42_size__neq__size__imp__neq,axiom,
! [X2: list_list_nat,Y2: list_list_nat] :
( ( ( size_s1898481668st_nat @ X2 )
!= ( size_s1898481668st_nat @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_43_size__neq__size__imp__neq,axiom,
! [X2: list_list_list_a,Y2: list_list_list_a] :
( ( ( size_s575106428list_a @ X2 )
!= ( size_s575106428list_a @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_44_size__neq__size__imp__neq,axiom,
! [X2: list_a,Y2: list_a] :
( ( ( size_size_list_a @ X2 )
!= ( size_size_list_a @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_45_size__neq__size__imp__neq,axiom,
! [X2: list_list_a,Y2: list_list_a] :
( ( ( size_s1427607542list_a @ X2 )
!= ( size_s1427607542list_a @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_46_size__neq__size__imp__neq,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( ( size_size_list_nat @ X2 )
!= ( size_size_list_nat @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_47_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_48_list__slice__length,axiom,
! [Xs: list_list_nat,K: nat] :
( ( size_s1321307156st_nat @ ( listSl729562870st_nat @ Xs @ K ) )
= ( divide_divide_nat @ ( size_s1898481668st_nat @ Xs ) @ K ) ) ).
% list_slice_length
thf(fact_49_list__slice__length,axiom,
! [Xs: list_list_list_a,K: nat] :
( ( size_s1764310658list_a @ ( listSl2102238196list_a @ Xs @ K ) )
= ( divide_divide_nat @ ( size_s575106428list_a @ Xs ) @ K ) ) ).
% list_slice_length
thf(fact_50_list__slice__length,axiom,
! [Xs: list_list_a,K: nat] :
( ( size_s575106428list_a @ ( listSl162220270list_a @ Xs @ K ) )
= ( divide_divide_nat @ ( size_s1427607542list_a @ Xs ) @ K ) ) ).
% list_slice_length
thf(fact_51_list__slice__length,axiom,
! [Xs: list_nat,K: nat] :
( ( size_s1898481668st_nat @ ( listSl1630398182ce_nat @ Xs @ K ) )
= ( divide_divide_nat @ ( size_size_list_nat @ Xs ) @ K ) ) ).
% list_slice_length
thf(fact_52_list__slice__length,axiom,
! [Xs: list_a,K: nat] :
( ( size_s1427607542list_a @ ( listSl97544552lice_a @ Xs @ K ) )
= ( divide_divide_nat @ ( size_size_list_a @ Xs ) @ K ) ) ).
% list_slice_length
thf(fact_53_linorder__neqE__nat,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_54_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_55_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_56_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_57_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_58_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_59_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_60_list__slice__def,axiom,
( listSl729562870st_nat
= ( ^ [Xs2: list_list_nat,K2: nat] : ( listSl1506485424st_nat @ Xs2 @ K2 @ ( divide_divide_nat @ ( size_s1898481668st_nat @ Xs2 ) @ K2 ) ) ) ) ).
% list_slice_def
thf(fact_61_list__slice__def,axiom,
( listSl2102238196list_a
= ( ^ [Xs2: list_list_list_a,K2: nat] : ( listSl902632890list_a @ Xs2 @ K2 @ ( divide_divide_nat @ ( size_s575106428list_a @ Xs2 ) @ K2 ) ) ) ) ).
% list_slice_def
thf(fact_62_list__slice__def,axiom,
( listSl162220270list_a
= ( ^ [Xs2: list_list_a,K2: nat] : ( listSl856612276list_a @ Xs2 @ K2 @ ( divide_divide_nat @ ( size_s1427607542list_a @ Xs2 ) @ K2 ) ) ) ) ).
% list_slice_def
thf(fact_63_list__slice__def,axiom,
( listSl1630398182ce_nat
= ( ^ [Xs2: list_nat,K2: nat] : ( listSl1794011552ux_nat @ Xs2 @ K2 @ ( divide_divide_nat @ ( size_size_list_nat @ Xs2 ) @ K2 ) ) ) ) ).
% list_slice_def
thf(fact_64_list__slice__def,axiom,
( listSl97544552lice_a
= ( ^ [Xs2: list_a,K2: nat] : ( listSl1495374126_aux_a @ Xs2 @ K2 @ ( divide_divide_nat @ ( size_size_list_a @ Xs2 ) @ K2 ) ) ) ) ).
% list_slice_def
thf(fact_65_list__strict__desc__imp__list__desc,axiom,
! [Xs: list_nat] :
( ( list_strict_desc_nat @ Xs )
=> ( list_desc_nat @ Xs ) ) ).
% list_strict_desc_imp_list_desc
thf(fact_66_list__ex__length,axiom,
( list_ex_list_nat
= ( ^ [P2: list_nat > $o,Xs2: list_list_nat] :
? [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_s1898481668st_nat @ Xs2 ) )
& ( P2 @ ( nth_list_nat @ Xs2 @ N3 ) ) ) ) ) ).
% list_ex_length
thf(fact_67_list__ex__length,axiom,
( list_ex_list_list_a
= ( ^ [P2: list_list_a > $o,Xs2: list_list_list_a] :
? [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_s575106428list_a @ Xs2 ) )
& ( P2 @ ( nth_list_list_a @ Xs2 @ N3 ) ) ) ) ) ).
% list_ex_length
thf(fact_68_list__ex__length,axiom,
( list_ex_a
= ( ^ [P2: a > $o,Xs2: list_a] :
? [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs2 ) )
& ( P2 @ ( nth_a @ Xs2 @ N3 ) ) ) ) ) ).
% list_ex_length
thf(fact_69_list__ex__length,axiom,
( list_ex_list_a
= ( ^ [P2: list_a > $o,Xs2: list_list_a] :
? [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_s1427607542list_a @ Xs2 ) )
& ( P2 @ ( nth_list_a @ Xs2 @ N3 ) ) ) ) ) ).
% list_ex_length
thf(fact_70_list__ex__length,axiom,
( list_ex_nat
= ( ^ [P2: nat > $o,Xs2: list_nat] :
? [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs2 ) )
& ( P2 @ ( nth_nat @ Xs2 @ N3 ) ) ) ) ) ).
% list_ex_length
thf(fact_71_f__image__eqI,axiom,
! [X2: list_nat,Xs: list_list_nat,N: nat,A: set_nat] :
( ( X2
= ( nth_list_nat @ Xs @ N ) )
=> ( ( member_nat @ N @ A )
=> ( ( ord_less_nat @ N @ ( size_s1898481668st_nat @ Xs ) )
=> ( member_list_nat @ X2 @ ( f_image_list_nat @ Xs @ A ) ) ) ) ) ).
% f_image_eqI
thf(fact_72_f__image__eqI,axiom,
! [X2: list_list_a,Xs: list_list_list_a,N: nat,A: set_nat] :
( ( X2
= ( nth_list_list_a @ Xs @ N ) )
=> ( ( member_nat @ N @ A )
=> ( ( ord_less_nat @ N @ ( size_s575106428list_a @ Xs ) )
=> ( member_list_list_a @ X2 @ ( f_image_list_list_a @ Xs @ A ) ) ) ) ) ).
% f_image_eqI
thf(fact_73_f__image__eqI,axiom,
! [X2: a,Xs: list_a,N: nat,A: set_nat] :
( ( X2
= ( nth_a @ Xs @ N ) )
=> ( ( member_nat @ N @ A )
=> ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( member_a @ X2 @ ( f_image_a @ Xs @ A ) ) ) ) ) ).
% f_image_eqI
thf(fact_74_f__image__eqI,axiom,
! [X2: list_a,Xs: list_list_a,N: nat,A: set_nat] :
( ( X2
= ( nth_list_a @ Xs @ N ) )
=> ( ( member_nat @ N @ A )
=> ( ( ord_less_nat @ N @ ( size_s1427607542list_a @ Xs ) )
=> ( member_list_a @ X2 @ ( f_image_list_a @ Xs @ A ) ) ) ) ) ).
% f_image_eqI
thf(fact_75_f__image__eqI,axiom,
! [X2: nat,Xs: list_nat,N: nat,A: set_nat] :
( ( X2
= ( nth_nat @ Xs @ N ) )
=> ( ( member_nat @ N @ A )
=> ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ X2 @ ( f_image_nat @ Xs @ A ) ) ) ) ) ).
% f_image_eqI
thf(fact_76_list__strict__asc__imp__list__asc,axiom,
! [Xs: list_nat] :
( ( list_strict_asc_nat @ Xs )
=> ( list_asc_nat @ Xs ) ) ).
% list_strict_asc_imp_list_asc
thf(fact_77_nth__list__update__eq,axiom,
! [I3: nat,Xs: list_list_nat,X2: list_nat] :
( ( ord_less_nat @ I3 @ ( size_s1898481668st_nat @ Xs ) )
=> ( ( nth_list_nat @ ( list_update_list_nat @ Xs @ I3 @ X2 ) @ I3 )
= X2 ) ) ).
% nth_list_update_eq
thf(fact_78_nth__list__update__eq,axiom,
! [I3: nat,Xs: list_list_list_a,X2: list_list_a] :
( ( ord_less_nat @ I3 @ ( size_s575106428list_a @ Xs ) )
=> ( ( nth_list_list_a @ ( list_u1330012152list_a @ Xs @ I3 @ X2 ) @ I3 )
= X2 ) ) ).
% nth_list_update_eq
thf(fact_79_nth__list__update__eq,axiom,
! [I3: nat,Xs: list_a,X2: a] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( list_update_a @ Xs @ I3 @ X2 ) @ I3 )
= X2 ) ) ).
% nth_list_update_eq
thf(fact_80_nth__list__update__eq,axiom,
! [I3: nat,Xs: list_list_a,X2: list_a] :
( ( ord_less_nat @ I3 @ ( size_s1427607542list_a @ Xs ) )
=> ( ( nth_list_a @ ( list_update_list_a @ Xs @ I3 @ X2 ) @ I3 )
= X2 ) ) ).
% nth_list_update_eq
thf(fact_81_nth__list__update__eq,axiom,
! [I3: nat,Xs: list_nat,X2: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ I3 )
= X2 ) ) ).
% nth_list_update_eq
thf(fact_82_rev__f__imageI,axiom,
! [N: nat,A: set_nat,Xs: list_list_nat,X2: list_nat] :
( ( member_nat @ N @ A )
=> ( ( ord_less_nat @ N @ ( size_s1898481668st_nat @ Xs ) )
=> ( ( X2
= ( nth_list_nat @ Xs @ N ) )
=> ( member_list_nat @ X2 @ ( f_image_list_nat @ Xs @ A ) ) ) ) ) ).
% rev_f_imageI
thf(fact_83_rev__f__imageI,axiom,
! [N: nat,A: set_nat,Xs: list_list_list_a,X2: list_list_a] :
( ( member_nat @ N @ A )
=> ( ( ord_less_nat @ N @ ( size_s575106428list_a @ Xs ) )
=> ( ( X2
= ( nth_list_list_a @ Xs @ N ) )
=> ( member_list_list_a @ X2 @ ( f_image_list_list_a @ Xs @ A ) ) ) ) ) ).
% rev_f_imageI
thf(fact_84_rev__f__imageI,axiom,
! [N: nat,A: set_nat,Xs: list_a,X2: a] :
( ( member_nat @ N @ A )
=> ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( X2
= ( nth_a @ Xs @ N ) )
=> ( member_a @ X2 @ ( f_image_a @ Xs @ A ) ) ) ) ) ).
% rev_f_imageI
thf(fact_85_rev__f__imageI,axiom,
! [N: nat,A: set_nat,Xs: list_list_a,X2: list_a] :
( ( member_nat @ N @ A )
=> ( ( ord_less_nat @ N @ ( size_s1427607542list_a @ Xs ) )
=> ( ( X2
= ( nth_list_a @ Xs @ N ) )
=> ( member_list_a @ X2 @ ( f_image_list_a @ Xs @ A ) ) ) ) ) ).
% rev_f_imageI
thf(fact_86_rev__f__imageI,axiom,
! [N: nat,A: set_nat,Xs: list_nat,X2: nat] :
( ( member_nat @ N @ A )
=> ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( X2
= ( nth_nat @ Xs @ N ) )
=> ( member_nat @ X2 @ ( f_image_nat @ Xs @ A ) ) ) ) ) ).
% rev_f_imageI
thf(fact_87_f__image__iff,axiom,
! [X2: list_nat,Xs: list_list_nat,A: set_nat] :
( ( member_list_nat @ X2 @ ( f_image_list_nat @ Xs @ A ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ord_less_nat @ X3 @ ( size_s1898481668st_nat @ Xs ) )
& ( X2
= ( nth_list_nat @ Xs @ X3 ) ) ) ) ) ).
% f_image_iff
thf(fact_88_f__image__iff,axiom,
! [X2: list_list_a,Xs: list_list_list_a,A: set_nat] :
( ( member_list_list_a @ X2 @ ( f_image_list_list_a @ Xs @ A ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ord_less_nat @ X3 @ ( size_s575106428list_a @ Xs ) )
& ( X2
= ( nth_list_list_a @ Xs @ X3 ) ) ) ) ) ).
% f_image_iff
thf(fact_89_f__image__iff,axiom,
! [X2: a,Xs: list_a,A: set_nat] :
( ( member_a @ X2 @ ( f_image_a @ Xs @ A ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ord_less_nat @ X3 @ ( size_size_list_a @ Xs ) )
& ( X2
= ( nth_a @ Xs @ X3 ) ) ) ) ) ).
% f_image_iff
thf(fact_90_f__image__iff,axiom,
! [X2: list_a,Xs: list_list_a,A: set_nat] :
( ( member_list_a @ X2 @ ( f_image_list_a @ Xs @ A ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ord_less_nat @ X3 @ ( size_s1427607542list_a @ Xs ) )
& ( X2
= ( nth_list_a @ Xs @ X3 ) ) ) ) ) ).
% f_image_iff
thf(fact_91_f__image__iff,axiom,
! [X2: nat,Xs: list_nat,A: set_nat] :
( ( member_nat @ X2 @ ( f_image_nat @ Xs @ A ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ord_less_nat @ X3 @ ( size_size_list_nat @ Xs ) )
& ( X2
= ( nth_nat @ Xs @ X3 ) ) ) ) ) ).
% f_image_iff
thf(fact_92_nth__butlast,axiom,
! [N: nat,Xs: list_list_nat] :
( ( ord_less_nat @ N @ ( size_s1898481668st_nat @ ( butlast_list_nat @ Xs ) ) )
=> ( ( nth_list_nat @ ( butlast_list_nat @ Xs ) @ N )
= ( nth_list_nat @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_93_nth__butlast,axiom,
! [N: nat,Xs: list_list_list_a] :
( ( ord_less_nat @ N @ ( size_s575106428list_a @ ( butlast_list_list_a @ Xs ) ) )
=> ( ( nth_list_list_a @ ( butlast_list_list_a @ Xs ) @ N )
= ( nth_list_list_a @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_94_nth__butlast,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ ( butlast_a @ Xs ) ) )
=> ( ( nth_a @ ( butlast_a @ Xs ) @ N )
= ( nth_a @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_95_nth__butlast,axiom,
! [N: nat,Xs: list_list_a] :
( ( ord_less_nat @ N @ ( size_s1427607542list_a @ ( butlast_list_a @ Xs ) ) )
=> ( ( nth_list_a @ ( butlast_list_a @ Xs ) @ N )
= ( nth_list_a @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_96_nth__butlast,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ ( butlast_nat @ Xs ) ) )
=> ( ( nth_nat @ ( butlast_nat @ Xs ) @ N )
= ( nth_nat @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_97_f__imageI,axiom,
! [N: nat,A: set_nat,Xs: list_list_nat] :
( ( member_nat @ N @ A )
=> ( ( ord_less_nat @ N @ ( size_s1898481668st_nat @ Xs ) )
=> ( member_list_nat @ ( nth_list_nat @ Xs @ N ) @ ( f_image_list_nat @ Xs @ A ) ) ) ) ).
% f_imageI
thf(fact_98_f__imageI,axiom,
! [N: nat,A: set_nat,Xs: list_list_list_a] :
( ( member_nat @ N @ A )
=> ( ( ord_less_nat @ N @ ( size_s575106428list_a @ Xs ) )
=> ( member_list_list_a @ ( nth_list_list_a @ Xs @ N ) @ ( f_image_list_list_a @ Xs @ A ) ) ) ) ).
% f_imageI
thf(fact_99_f__imageI,axiom,
! [N: nat,A: set_nat,Xs: list_a] :
( ( member_nat @ N @ A )
=> ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( member_a @ ( nth_a @ Xs @ N ) @ ( f_image_a @ Xs @ A ) ) ) ) ).
% f_imageI
thf(fact_100_f__imageI,axiom,
! [N: nat,A: set_nat,Xs: list_list_a] :
( ( member_nat @ N @ A )
=> ( ( ord_less_nat @ N @ ( size_s1427607542list_a @ Xs ) )
=> ( member_list_a @ ( nth_list_a @ Xs @ N ) @ ( f_image_list_a @ Xs @ A ) ) ) ) ).
% f_imageI
thf(fact_101_f__imageI,axiom,
! [N: nat,A: set_nat,Xs: list_nat] :
( ( member_nat @ N @ A )
=> ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ ( nth_nat @ Xs @ N ) @ ( f_image_nat @ Xs @ A ) ) ) ) ).
% f_imageI
thf(fact_102_f__imageE,axiom,
! [X2: list_nat,Xs: list_list_nat,A: set_nat] :
( ( member_list_nat @ X2 @ ( f_image_list_nat @ Xs @ A ) )
=> ~ ! [N2: nat] :
( ( X2
= ( nth_list_nat @ Xs @ N2 ) )
=> ( ( member_nat @ N2 @ A )
=> ~ ( ord_less_nat @ N2 @ ( size_s1898481668st_nat @ Xs ) ) ) ) ) ).
% f_imageE
thf(fact_103_f__imageE,axiom,
! [X2: list_list_a,Xs: list_list_list_a,A: set_nat] :
( ( member_list_list_a @ X2 @ ( f_image_list_list_a @ Xs @ A ) )
=> ~ ! [N2: nat] :
( ( X2
= ( nth_list_list_a @ Xs @ N2 ) )
=> ( ( member_nat @ N2 @ A )
=> ~ ( ord_less_nat @ N2 @ ( size_s575106428list_a @ Xs ) ) ) ) ) ).
% f_imageE
thf(fact_104_f__imageE,axiom,
! [X2: a,Xs: list_a,A: set_nat] :
( ( member_a @ X2 @ ( f_image_a @ Xs @ A ) )
=> ~ ! [N2: nat] :
( ( X2
= ( nth_a @ Xs @ N2 ) )
=> ( ( member_nat @ N2 @ A )
=> ~ ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) ) ) ) ) ).
% f_imageE
thf(fact_105_f__imageE,axiom,
! [X2: list_a,Xs: list_list_a,A: set_nat] :
( ( member_list_a @ X2 @ ( f_image_list_a @ Xs @ A ) )
=> ~ ! [N2: nat] :
( ( X2
= ( nth_list_a @ Xs @ N2 ) )
=> ( ( member_nat @ N2 @ A )
=> ~ ( ord_less_nat @ N2 @ ( size_s1427607542list_a @ Xs ) ) ) ) ) ).
% f_imageE
thf(fact_106_f__imageE,axiom,
! [X2: nat,Xs: list_nat,A: set_nat] :
( ( member_nat @ X2 @ ( f_image_nat @ Xs @ A ) )
=> ~ ! [N2: nat] :
( ( X2
= ( nth_nat @ Xs @ N2 ) )
=> ( ( member_nat @ N2 @ A )
=> ~ ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% f_imageE
thf(fact_107_list__update__overwrite,axiom,
! [Xs: list_nat,I3: nat,X2: nat,Y2: nat] :
( ( list_update_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ I3 @ Y2 )
= ( list_update_nat @ Xs @ I3 @ Y2 ) ) ).
% list_update_overwrite
thf(fact_108_list__update__overwrite,axiom,
! [Xs: list_list_a,I3: nat,X2: list_a,Y2: list_a] :
( ( list_update_list_a @ ( list_update_list_a @ Xs @ I3 @ X2 ) @ I3 @ Y2 )
= ( list_update_list_a @ Xs @ I3 @ Y2 ) ) ).
% list_update_overwrite
thf(fact_109_list__update__overwrite,axiom,
! [Xs: list_a,I3: nat,X2: a,Y2: a] :
( ( list_update_a @ ( list_update_a @ Xs @ I3 @ X2 ) @ I3 @ Y2 )
= ( list_update_a @ Xs @ I3 @ Y2 ) ) ).
% list_update_overwrite
thf(fact_110_length__list__update,axiom,
! [Xs: list_list_nat,I3: nat,X2: list_nat] :
( ( size_s1898481668st_nat @ ( list_update_list_nat @ Xs @ I3 @ X2 ) )
= ( size_s1898481668st_nat @ Xs ) ) ).
% length_list_update
thf(fact_111_length__list__update,axiom,
! [Xs: list_list_list_a,I3: nat,X2: list_list_a] :
( ( size_s575106428list_a @ ( list_u1330012152list_a @ Xs @ I3 @ X2 ) )
= ( size_s575106428list_a @ Xs ) ) ).
% length_list_update
thf(fact_112_length__list__update,axiom,
! [Xs: list_a,I3: nat,X2: a] :
( ( size_size_list_a @ ( list_update_a @ Xs @ I3 @ X2 ) )
= ( size_size_list_a @ Xs ) ) ).
% length_list_update
thf(fact_113_length__list__update,axiom,
! [Xs: list_list_a,I3: nat,X2: list_a] :
( ( size_s1427607542list_a @ ( list_update_list_a @ Xs @ I3 @ X2 ) )
= ( size_s1427607542list_a @ Xs ) ) ).
% length_list_update
thf(fact_114_length__list__update,axiom,
! [Xs: list_nat,I3: nat,X2: nat] :
( ( size_size_list_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_list_update
thf(fact_115_list__update__id,axiom,
! [Xs: list_list_nat,I3: nat] :
( ( list_update_list_nat @ Xs @ I3 @ ( nth_list_nat @ Xs @ I3 ) )
= Xs ) ).
% list_update_id
thf(fact_116_list__update__id,axiom,
! [Xs: list_list_list_a,I3: nat] :
( ( list_u1330012152list_a @ Xs @ I3 @ ( nth_list_list_a @ Xs @ I3 ) )
= Xs ) ).
% list_update_id
thf(fact_117_list__update__id,axiom,
! [Xs: list_list_a,I3: nat] :
( ( list_update_list_a @ Xs @ I3 @ ( nth_list_a @ Xs @ I3 ) )
= Xs ) ).
% list_update_id
thf(fact_118_list__update__id,axiom,
! [Xs: list_a,I3: nat] :
( ( list_update_a @ Xs @ I3 @ ( nth_a @ Xs @ I3 ) )
= Xs ) ).
% list_update_id
thf(fact_119_list__update__id,axiom,
! [Xs: list_nat,I3: nat] :
( ( list_update_nat @ Xs @ I3 @ ( nth_nat @ Xs @ I3 ) )
= Xs ) ).
% list_update_id
thf(fact_120_nth__list__update__neq,axiom,
! [I3: nat,J2: nat,Xs: list_list_nat,X2: list_nat] :
( ( I3 != J2 )
=> ( ( nth_list_nat @ ( list_update_list_nat @ Xs @ I3 @ X2 ) @ J2 )
= ( nth_list_nat @ Xs @ J2 ) ) ) ).
% nth_list_update_neq
thf(fact_121_nth__list__update__neq,axiom,
! [I3: nat,J2: nat,Xs: list_list_list_a,X2: list_list_a] :
( ( I3 != J2 )
=> ( ( nth_list_list_a @ ( list_u1330012152list_a @ Xs @ I3 @ X2 ) @ J2 )
= ( nth_list_list_a @ Xs @ J2 ) ) ) ).
% nth_list_update_neq
thf(fact_122_nth__list__update__neq,axiom,
! [I3: nat,J2: nat,Xs: list_list_a,X2: list_a] :
( ( I3 != J2 )
=> ( ( nth_list_a @ ( list_update_list_a @ Xs @ I3 @ X2 ) @ J2 )
= ( nth_list_a @ Xs @ J2 ) ) ) ).
% nth_list_update_neq
thf(fact_123_nth__list__update__neq,axiom,
! [I3: nat,J2: nat,Xs: list_a,X2: a] :
( ( I3 != J2 )
=> ( ( nth_a @ ( list_update_a @ Xs @ I3 @ X2 ) @ J2 )
= ( nth_a @ Xs @ J2 ) ) ) ).
% nth_list_update_neq
thf(fact_124_nth__list__update__neq,axiom,
! [I3: nat,J2: nat,Xs: list_nat,X2: nat] :
( ( I3 != J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ J2 )
= ( nth_nat @ Xs @ J2 ) ) ) ).
% nth_list_update_neq
thf(fact_125_list__update__swap,axiom,
! [I3: nat,I4: nat,Xs: list_nat,X2: nat,X4: nat] :
( ( I3 != I4 )
=> ( ( list_update_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ I4 @ X4 )
= ( list_update_nat @ ( list_update_nat @ Xs @ I4 @ X4 ) @ I3 @ X2 ) ) ) ).
% list_update_swap
thf(fact_126_list__update__swap,axiom,
! [I3: nat,I4: nat,Xs: list_list_a,X2: list_a,X4: list_a] :
( ( I3 != I4 )
=> ( ( list_update_list_a @ ( list_update_list_a @ Xs @ I3 @ X2 ) @ I4 @ X4 )
= ( list_update_list_a @ ( list_update_list_a @ Xs @ I4 @ X4 ) @ I3 @ X2 ) ) ) ).
% list_update_swap
thf(fact_127_list__update__swap,axiom,
! [I3: nat,I4: nat,Xs: list_a,X2: a,X4: a] :
( ( I3 != I4 )
=> ( ( list_update_a @ ( list_update_a @ Xs @ I3 @ X2 ) @ I4 @ X4 )
= ( list_update_a @ ( list_update_a @ Xs @ I4 @ X4 ) @ I3 @ X2 ) ) ) ).
% list_update_swap
thf(fact_128_list__slice__aux__length,axiom,
! [Xs: list_nat,K: nat,N: nat] :
( ( size_s1898481668st_nat @ ( listSl1794011552ux_nat @ Xs @ K @ N ) )
= N ) ).
% list_slice_aux_length
thf(fact_129_list__slice__aux__length,axiom,
! [Xs: list_list_a,K: nat,N: nat] :
( ( size_s575106428list_a @ ( listSl856612276list_a @ Xs @ K @ N ) )
= N ) ).
% list_slice_aux_length
thf(fact_130_list__slice__aux__length,axiom,
! [Xs: list_a,K: nat,N: nat] :
( ( size_s1427607542list_a @ ( listSl1495374126_aux_a @ Xs @ K @ N ) )
= N ) ).
% list_slice_aux_length
thf(fact_131_nth__list__update,axiom,
! [I3: nat,Xs: list_list_nat,J2: nat,X2: list_nat] :
( ( ord_less_nat @ I3 @ ( size_s1898481668st_nat @ Xs ) )
=> ( ( ( I3 = J2 )
=> ( ( nth_list_nat @ ( list_update_list_nat @ Xs @ I3 @ X2 ) @ J2 )
= X2 ) )
& ( ( I3 != J2 )
=> ( ( nth_list_nat @ ( list_update_list_nat @ Xs @ I3 @ X2 ) @ J2 )
= ( nth_list_nat @ Xs @ J2 ) ) ) ) ) ).
% nth_list_update
thf(fact_132_nth__list__update,axiom,
! [I3: nat,Xs: list_list_list_a,J2: nat,X2: list_list_a] :
( ( ord_less_nat @ I3 @ ( size_s575106428list_a @ Xs ) )
=> ( ( ( I3 = J2 )
=> ( ( nth_list_list_a @ ( list_u1330012152list_a @ Xs @ I3 @ X2 ) @ J2 )
= X2 ) )
& ( ( I3 != J2 )
=> ( ( nth_list_list_a @ ( list_u1330012152list_a @ Xs @ I3 @ X2 ) @ J2 )
= ( nth_list_list_a @ Xs @ J2 ) ) ) ) ) ).
% nth_list_update
thf(fact_133_nth__list__update,axiom,
! [I3: nat,Xs: list_a,J2: nat,X2: a] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
=> ( ( ( I3 = J2 )
=> ( ( nth_a @ ( list_update_a @ Xs @ I3 @ X2 ) @ J2 )
= X2 ) )
& ( ( I3 != J2 )
=> ( ( nth_a @ ( list_update_a @ Xs @ I3 @ X2 ) @ J2 )
= ( nth_a @ Xs @ J2 ) ) ) ) ) ).
% nth_list_update
thf(fact_134_nth__list__update,axiom,
! [I3: nat,Xs: list_list_a,J2: nat,X2: list_a] :
( ( ord_less_nat @ I3 @ ( size_s1427607542list_a @ Xs ) )
=> ( ( ( I3 = J2 )
=> ( ( nth_list_a @ ( list_update_list_a @ Xs @ I3 @ X2 ) @ J2 )
= X2 ) )
& ( ( I3 != J2 )
=> ( ( nth_list_a @ ( list_update_list_a @ Xs @ I3 @ X2 ) @ J2 )
= ( nth_list_a @ Xs @ J2 ) ) ) ) ) ).
% nth_list_update
thf(fact_135_nth__list__update,axiom,
! [I3: nat,Xs: list_nat,J2: nat,X2: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( I3 = J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ J2 )
= X2 ) )
& ( ( I3 != J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X2 ) @ J2 )
= ( nth_nat @ Xs @ J2 ) ) ) ) ) ).
% nth_list_update
thf(fact_136_list__update__same__conv,axiom,
! [I3: nat,Xs: list_list_nat,X2: list_nat] :
( ( ord_less_nat @ I3 @ ( size_s1898481668st_nat @ Xs ) )
=> ( ( ( list_update_list_nat @ Xs @ I3 @ X2 )
= Xs )
= ( ( nth_list_nat @ Xs @ I3 )
= X2 ) ) ) ).
% list_update_same_conv
thf(fact_137_list__update__same__conv,axiom,
! [I3: nat,Xs: list_list_list_a,X2: list_list_a] :
( ( ord_less_nat @ I3 @ ( size_s575106428list_a @ Xs ) )
=> ( ( ( list_u1330012152list_a @ Xs @ I3 @ X2 )
= Xs )
= ( ( nth_list_list_a @ Xs @ I3 )
= X2 ) ) ) ).
% list_update_same_conv
thf(fact_138_list__update__same__conv,axiom,
! [I3: nat,Xs: list_a,X2: a] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
=> ( ( ( list_update_a @ Xs @ I3 @ X2 )
= Xs )
= ( ( nth_a @ Xs @ I3 )
= X2 ) ) ) ).
% list_update_same_conv
thf(fact_139_list__update__same__conv,axiom,
! [I3: nat,Xs: list_list_a,X2: list_a] :
( ( ord_less_nat @ I3 @ ( size_s1427607542list_a @ Xs ) )
=> ( ( ( list_update_list_a @ Xs @ I3 @ X2 )
= Xs )
= ( ( nth_list_a @ Xs @ I3 )
= X2 ) ) ) ).
% list_update_same_conv
thf(fact_140_list__update__same__conv,axiom,
! [I3: nat,Xs: list_nat,X2: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( list_update_nat @ Xs @ I3 @ X2 )
= Xs )
= ( ( nth_nat @ Xs @ I3 )
= X2 ) ) ) ).
% list_update_same_conv
thf(fact_141_set__swap,axiom,
! [I3: nat,Xs: list_list_nat,J2: nat] :
( ( ord_less_nat @ I3 @ ( size_s1898481668st_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_s1898481668st_nat @ Xs ) )
=> ( ( set_list_nat2 @ ( list_update_list_nat @ ( list_update_list_nat @ Xs @ I3 @ ( nth_list_nat @ Xs @ J2 ) ) @ J2 @ ( nth_list_nat @ Xs @ I3 ) ) )
= ( set_list_nat2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_142_set__swap,axiom,
! [I3: nat,Xs: list_list_list_a,J2: nat] :
( ( ord_less_nat @ I3 @ ( size_s575106428list_a @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_s575106428list_a @ Xs ) )
=> ( ( set_list_list_a2 @ ( list_u1330012152list_a @ ( list_u1330012152list_a @ Xs @ I3 @ ( nth_list_list_a @ Xs @ J2 ) ) @ J2 @ ( nth_list_list_a @ Xs @ I3 ) ) )
= ( set_list_list_a2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_143_set__swap,axiom,
! [I3: nat,Xs: list_a,J2: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_a @ Xs ) )
=> ( ( set_a2 @ ( list_update_a @ ( list_update_a @ Xs @ I3 @ ( nth_a @ Xs @ J2 ) ) @ J2 @ ( nth_a @ Xs @ I3 ) ) )
= ( set_a2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_144_set__swap,axiom,
! [I3: nat,Xs: list_list_a,J2: nat] :
( ( ord_less_nat @ I3 @ ( size_s1427607542list_a @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_s1427607542list_a @ Xs ) )
=> ( ( set_list_a2 @ ( list_update_list_a @ ( list_update_list_a @ Xs @ I3 @ ( nth_list_a @ Xs @ J2 ) ) @ J2 @ ( nth_list_a @ Xs @ I3 ) ) )
= ( set_list_a2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_145_set__swap,axiom,
! [I3: nat,Xs: list_nat,J2: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I3 @ ( nth_nat @ Xs @ J2 ) ) @ J2 @ ( nth_nat @ Xs @ I3 ) ) )
= ( set_nat2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_146_list__desc__trans,axiom,
( list_desc_set_nat
= ( ^ [Xs2: list_set_nat] :
! [J: nat] :
( ( ord_less_nat @ J @ ( size_s577819178et_nat @ Xs2 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_eq_set_nat @ ( nth_set_nat @ Xs2 @ J ) @ ( nth_set_nat @ Xs2 @ I2 ) ) ) ) ) ) ).
% list_desc_trans
thf(fact_147_list__desc__trans,axiom,
( list_desc_set_list_a
= ( ^ [Xs2: list_set_list_a] :
! [J: nat] :
( ( ord_less_nat @ J @ ( size_s1635937238list_a @ Xs2 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_le1301786372list_a @ ( nth_set_list_a @ Xs2 @ J ) @ ( nth_set_list_a @ Xs2 @ I2 ) ) ) ) ) ) ).
% list_desc_trans
thf(fact_148_list__desc__trans,axiom,
( list_desc_set_a
= ( ^ [Xs2: list_set_a] :
! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_set_a @ Xs2 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_eq_set_a @ ( nth_set_a @ Xs2 @ J ) @ ( nth_set_a @ Xs2 @ I2 ) ) ) ) ) ) ).
% list_desc_trans
thf(fact_149_list__desc__trans,axiom,
( list_desc_nat
= ( ^ [Xs2: list_nat] :
! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ J ) @ ( nth_nat @ Xs2 @ I2 ) ) ) ) ) ) ).
% list_desc_trans
thf(fact_150_list__asc__trans,axiom,
( list_asc_set_nat
= ( ^ [Xs2: list_set_nat] :
! [J: nat] :
( ( ord_less_nat @ J @ ( size_s577819178et_nat @ Xs2 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_eq_set_nat @ ( nth_set_nat @ Xs2 @ I2 ) @ ( nth_set_nat @ Xs2 @ J ) ) ) ) ) ) ).
% list_asc_trans
thf(fact_151_list__asc__trans,axiom,
( list_asc_set_list_a
= ( ^ [Xs2: list_set_list_a] :
! [J: nat] :
( ( ord_less_nat @ J @ ( size_s1635937238list_a @ Xs2 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_le1301786372list_a @ ( nth_set_list_a @ Xs2 @ I2 ) @ ( nth_set_list_a @ Xs2 @ J ) ) ) ) ) ) ).
% list_asc_trans
thf(fact_152_list__asc__trans,axiom,
( list_asc_set_a
= ( ^ [Xs2: list_set_a] :
! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_set_a @ Xs2 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_eq_set_a @ ( nth_set_a @ Xs2 @ I2 ) @ ( nth_set_a @ Xs2 @ J ) ) ) ) ) ) ).
% list_asc_trans
thf(fact_153_list__asc__trans,axiom,
( list_asc_nat
= ( ^ [Xs2: list_nat] :
! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ I2 ) @ ( nth_nat @ Xs2 @ J ) ) ) ) ) ) ).
% list_asc_trans
thf(fact_154_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_155_mem__Collect__eq,axiom,
! [A2: list_a,P: list_a > $o] :
( ( member_list_a @ A2 @ ( collect_list_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_156_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_157_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_158_Collect__mem__eq,axiom,
! [A: set_list_a] :
( ( collect_list_a
@ ^ [X3: list_a] : ( member_list_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_159_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_160_distinct__swap,axiom,
! [I3: nat,Xs: list_list_nat,J2: nat] :
( ( ord_less_nat @ I3 @ ( size_s1898481668st_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_s1898481668st_nat @ Xs ) )
=> ( ( distinct_list_nat @ ( list_update_list_nat @ ( list_update_list_nat @ Xs @ I3 @ ( nth_list_nat @ Xs @ J2 ) ) @ J2 @ ( nth_list_nat @ Xs @ I3 ) ) )
= ( distinct_list_nat @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_161_distinct__swap,axiom,
! [I3: nat,Xs: list_list_list_a,J2: nat] :
( ( ord_less_nat @ I3 @ ( size_s575106428list_a @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_s575106428list_a @ Xs ) )
=> ( ( distinct_list_list_a @ ( list_u1330012152list_a @ ( list_u1330012152list_a @ Xs @ I3 @ ( nth_list_list_a @ Xs @ J2 ) ) @ J2 @ ( nth_list_list_a @ Xs @ I3 ) ) )
= ( distinct_list_list_a @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_162_distinct__swap,axiom,
! [I3: nat,Xs: list_a,J2: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_a @ Xs ) )
=> ( ( distinct_a @ ( list_update_a @ ( list_update_a @ Xs @ I3 @ ( nth_a @ Xs @ J2 ) ) @ J2 @ ( nth_a @ Xs @ I3 ) ) )
= ( distinct_a @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_163_distinct__swap,axiom,
! [I3: nat,Xs: list_list_a,J2: nat] :
( ( ord_less_nat @ I3 @ ( size_s1427607542list_a @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_s1427607542list_a @ Xs ) )
=> ( ( distinct_list_a @ ( list_update_list_a @ ( list_update_list_a @ Xs @ I3 @ ( nth_list_a @ Xs @ J2 ) ) @ J2 @ ( nth_list_a @ Xs @ I3 ) ) )
= ( distinct_list_a @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_164_distinct__swap,axiom,
! [I3: nat,Xs: list_nat,J2: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ( distinct_nat @ ( list_update_nat @ ( list_update_nat @ Xs @ I3 @ ( nth_nat @ Xs @ J2 ) ) @ J2 @ ( nth_nat @ Xs @ I3 ) ) )
= ( distinct_nat @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_165_f__rangeE,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X2 @ ( f_image_list_nat @ Xs @ top_top_set_nat ) )
=> ~ ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_s1898481668st_nat @ Xs ) )
=> ( X2
!= ( nth_list_nat @ Xs @ N2 ) ) ) ) ).
% f_rangeE
thf(fact_166_f__rangeE,axiom,
! [X2: list_list_a,Xs: list_list_list_a] :
( ( member_list_list_a @ X2 @ ( f_image_list_list_a @ Xs @ top_top_set_nat ) )
=> ~ ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_s575106428list_a @ Xs ) )
=> ( X2
!= ( nth_list_list_a @ Xs @ N2 ) ) ) ) ).
% f_rangeE
thf(fact_167_f__rangeE,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( f_image_a @ Xs @ top_top_set_nat ) )
=> ~ ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( X2
!= ( nth_a @ Xs @ N2 ) ) ) ) ).
% f_rangeE
thf(fact_168_f__rangeE,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( member_list_a @ X2 @ ( f_image_list_a @ Xs @ top_top_set_nat ) )
=> ~ ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_s1427607542list_a @ Xs ) )
=> ( X2
!= ( nth_list_a @ Xs @ N2 ) ) ) ) ).
% f_rangeE
thf(fact_169_f__rangeE,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( f_image_nat @ Xs @ top_top_set_nat ) )
=> ~ ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( X2
!= ( nth_nat @ Xs @ N2 ) ) ) ) ).
% f_rangeE
thf(fact_170_f__rangeI,axiom,
! [N: nat,Xs: list_list_nat] :
( ( ord_less_nat @ N @ ( size_s1898481668st_nat @ Xs ) )
=> ( member_list_nat @ ( nth_list_nat @ Xs @ N ) @ ( f_image_list_nat @ Xs @ top_top_set_nat ) ) ) ).
% f_rangeI
thf(fact_171_f__rangeI,axiom,
! [N: nat,Xs: list_list_list_a] :
( ( ord_less_nat @ N @ ( size_s575106428list_a @ Xs ) )
=> ( member_list_list_a @ ( nth_list_list_a @ Xs @ N ) @ ( f_image_list_list_a @ Xs @ top_top_set_nat ) ) ) ).
% f_rangeI
thf(fact_172_f__rangeI,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( member_a @ ( nth_a @ Xs @ N ) @ ( f_image_a @ Xs @ top_top_set_nat ) ) ) ).
% f_rangeI
thf(fact_173_f__rangeI,axiom,
! [N: nat,Xs: list_list_a] :
( ( ord_less_nat @ N @ ( size_s1427607542list_a @ Xs ) )
=> ( member_list_a @ ( nth_list_a @ Xs @ N ) @ ( f_image_list_a @ Xs @ top_top_set_nat ) ) ) ).
% f_rangeI
thf(fact_174_f__rangeI,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ ( nth_nat @ Xs @ N ) @ ( f_image_nat @ Xs @ top_top_set_nat ) ) ) ).
% f_rangeI
thf(fact_175_f__range__eqI,axiom,
! [X2: list_nat,Xs: list_list_nat,N: nat] :
( ( X2
= ( nth_list_nat @ Xs @ N ) )
=> ( ( ord_less_nat @ N @ ( size_s1898481668st_nat @ Xs ) )
=> ( member_list_nat @ X2 @ ( f_image_list_nat @ Xs @ top_top_set_nat ) ) ) ) ).
% f_range_eqI
thf(fact_176_f__range__eqI,axiom,
! [X2: list_list_a,Xs: list_list_list_a,N: nat] :
( ( X2
= ( nth_list_list_a @ Xs @ N ) )
=> ( ( ord_less_nat @ N @ ( size_s575106428list_a @ Xs ) )
=> ( member_list_list_a @ X2 @ ( f_image_list_list_a @ Xs @ top_top_set_nat ) ) ) ) ).
% f_range_eqI
thf(fact_177_f__range__eqI,axiom,
! [X2: a,Xs: list_a,N: nat] :
( ( X2
= ( nth_a @ Xs @ N ) )
=> ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( member_a @ X2 @ ( f_image_a @ Xs @ top_top_set_nat ) ) ) ) ).
% f_range_eqI
thf(fact_178_f__range__eqI,axiom,
! [X2: list_a,Xs: list_list_a,N: nat] :
( ( X2
= ( nth_list_a @ Xs @ N ) )
=> ( ( ord_less_nat @ N @ ( size_s1427607542list_a @ Xs ) )
=> ( member_list_a @ X2 @ ( f_image_list_a @ Xs @ top_top_set_nat ) ) ) ) ).
% f_range_eqI
thf(fact_179_f__range__eqI,axiom,
! [X2: nat,Xs: list_nat,N: nat] :
( ( X2
= ( nth_nat @ Xs @ N ) )
=> ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ X2 @ ( f_image_nat @ Xs @ top_top_set_nat ) ) ) ) ).
% f_range_eqI
thf(fact_180_f__image__subsetI,axiom,
! [A: set_nat,Xs: list_list_nat,B: set_list_nat] :
( ! [N2: nat] :
( ( ( member_nat @ N2 @ A )
& ( ord_less_nat @ N2 @ ( size_s1898481668st_nat @ Xs ) ) )
=> ( member_list_nat @ ( nth_list_nat @ Xs @ N2 ) @ B ) )
=> ( ord_le1747345782st_nat @ ( f_image_list_nat @ Xs @ A ) @ B ) ) ).
% f_image_subsetI
thf(fact_181_f__image__subsetI,axiom,
! [A: set_nat,Xs: list_list_list_a,B: set_list_list_a] :
( ! [N2: nat] :
( ( ( member_nat @ N2 @ A )
& ( ord_less_nat @ N2 @ ( size_s575106428list_a @ Xs ) ) )
=> ( member_list_list_a @ ( nth_list_list_a @ Xs @ N2 ) @ B ) )
=> ( ord_le1756736906list_a @ ( f_image_list_list_a @ Xs @ A ) @ B ) ) ).
% f_image_subsetI
thf(fact_182_f__image__subsetI,axiom,
! [A: set_nat,Xs: list_a,B: set_a] :
( ! [N2: nat] :
( ( ( member_nat @ N2 @ A )
& ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) ) )
=> ( member_a @ ( nth_a @ Xs @ N2 ) @ B ) )
=> ( ord_less_eq_set_a @ ( f_image_a @ Xs @ A ) @ B ) ) ).
% f_image_subsetI
thf(fact_183_f__image__subsetI,axiom,
! [A: set_nat,Xs: list_list_a,B: set_list_a] :
( ! [N2: nat] :
( ( ( member_nat @ N2 @ A )
& ( ord_less_nat @ N2 @ ( size_s1427607542list_a @ Xs ) ) )
=> ( member_list_a @ ( nth_list_a @ Xs @ N2 ) @ B ) )
=> ( ord_le1301786372list_a @ ( f_image_list_a @ Xs @ A ) @ B ) ) ).
% f_image_subsetI
thf(fact_184_f__image__subsetI,axiom,
! [A: set_nat,Xs: list_nat,B: set_nat] :
( ! [N2: nat] :
( ( ( member_nat @ N2 @ A )
& ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) ) )
=> ( member_nat @ ( nth_nat @ Xs @ N2 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( f_image_nat @ Xs @ A ) @ B ) ) ).
% f_image_subsetI
thf(fact_185_f__image__subset__iff,axiom,
! [Xs: list_a,A: set_nat,B: set_a] :
( ( ord_less_eq_set_a @ ( f_image_a @ Xs @ A ) @ B )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( ord_less_nat @ X3 @ ( size_size_list_a @ Xs ) )
=> ( member_a @ ( nth_a @ Xs @ X3 ) @ B ) ) ) ) ) ).
% f_image_subset_iff
thf(fact_186_f__image__subset__iff,axiom,
! [Xs: list_list_a,A: set_nat,B: set_list_a] :
( ( ord_le1301786372list_a @ ( f_image_list_a @ Xs @ A ) @ B )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( ord_less_nat @ X3 @ ( size_s1427607542list_a @ Xs ) )
=> ( member_list_a @ ( nth_list_a @ Xs @ X3 ) @ B ) ) ) ) ) ).
% f_image_subset_iff
thf(fact_187_f__image__subset__iff,axiom,
! [Xs: list_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( f_image_nat @ Xs @ A ) @ B )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( ord_less_nat @ X3 @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ ( nth_nat @ Xs @ X3 ) @ B ) ) ) ) ) ).
% f_image_subset_iff
thf(fact_188_list__update__beyond,axiom,
! [Xs: list_a,I3: nat,X2: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ I3 )
=> ( ( list_update_a @ Xs @ I3 @ X2 )
= Xs ) ) ).
% list_update_beyond
thf(fact_189_list__update__beyond,axiom,
! [Xs: list_list_a,I3: nat,X2: list_a] :
( ( ord_less_eq_nat @ ( size_s1427607542list_a @ Xs ) @ I3 )
=> ( ( list_update_list_a @ Xs @ I3 @ X2 )
= Xs ) ) ).
% list_update_beyond
thf(fact_190_list__update__beyond,axiom,
! [Xs: list_nat,I3: nat,X2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I3 )
=> ( ( list_update_nat @ Xs @ I3 @ X2 )
= Xs ) ) ).
% list_update_beyond
thf(fact_191_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I3: nat,J2: nat] :
( ! [I: nat,J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( ord_less_nat @ ( F @ I ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_192_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_193_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_194_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_195_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_196_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_197_distinct__Ex1,axiom,
! [Xs: list_a,X2: a] :
( ( distinct_a @ Xs )
=> ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ? [X5: nat] :
( ( ord_less_nat @ X5 @ ( size_size_list_a @ Xs ) )
& ( ( nth_a @ Xs @ X5 )
= X2 )
& ! [Y3: nat] :
( ( ( ord_less_nat @ Y3 @ ( size_size_list_a @ Xs ) )
& ( ( nth_a @ Xs @ Y3 )
= X2 ) )
=> ( Y3 = X5 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_198_distinct__Ex1,axiom,
! [Xs: list_list_a,X2: list_a] :
( ( distinct_list_a @ Xs )
=> ( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ? [X5: nat] :
( ( ord_less_nat @ X5 @ ( size_s1427607542list_a @ Xs ) )
& ( ( nth_list_a @ Xs @ X5 )
= X2 )
& ! [Y3: nat] :
( ( ( ord_less_nat @ Y3 @ ( size_s1427607542list_a @ Xs ) )
& ( ( nth_list_a @ Xs @ Y3 )
= X2 ) )
=> ( Y3 = X5 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_199_distinct__Ex1,axiom,
! [Xs: list_nat,X2: nat] :
( ( distinct_nat @ Xs )
=> ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ? [X5: nat] :
( ( ord_less_nat @ X5 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ X5 )
= X2 )
& ! [Y3: nat] :
( ( ( ord_less_nat @ Y3 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ Y3 )
= X2 ) )
=> ( Y3 = X5 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_200_list__strict__asc__distinct,axiom,
! [Xs: list_nat] :
( ( list_strict_asc_nat @ Xs )
=> ( distinct_nat @ Xs ) ) ).
% list_strict_asc_distinct
thf(fact_201_list__strict__desc__distinct,axiom,
! [Xs: list_nat] :
( ( list_strict_desc_nat @ Xs )
=> ( distinct_nat @ Xs ) ) ).
% list_strict_desc_distinct
thf(fact_202_list__strict__asc__trans__le,axiom,
! [Xs: list_nat] :
( ( list_strict_asc_nat @ Xs )
=> ! [J4: nat] :
( ( ord_less_nat @ J4 @ ( size_size_list_nat @ Xs ) )
=> ! [I5: nat] :
( ( ord_less_eq_nat @ I5 @ J4 )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I5 ) @ ( nth_nat @ Xs @ J4 ) ) ) ) ) ).
% list_strict_asc_trans_le
thf(fact_203_list__asc__trans__le,axiom,
( list_asc_nat
= ( ^ [Xs2: list_nat] :
! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ! [I2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ I2 ) @ ( nth_nat @ Xs2 @ J ) ) ) ) ) ) ).
% list_asc_trans_le
thf(fact_204_list__desc__trans__le,axiom,
( list_desc_nat
= ( ^ [Xs2: list_nat] :
! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ! [I2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ J ) @ ( nth_nat @ Xs2 @ I2 ) ) ) ) ) ) ).
% list_desc_trans_le
thf(fact_205_nth__eq__iff__index__eq,axiom,
! [Xs: list_a,I3: nat,J2: nat] :
( ( distinct_a @ Xs )
=> ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_a @ Xs ) )
=> ( ( ( nth_a @ Xs @ I3 )
= ( nth_a @ Xs @ J2 ) )
= ( I3 = J2 ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_206_nth__eq__iff__index__eq,axiom,
! [Xs: list_list_a,I3: nat,J2: nat] :
( ( distinct_list_a @ Xs )
=> ( ( ord_less_nat @ I3 @ ( size_s1427607542list_a @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_s1427607542list_a @ Xs ) )
=> ( ( ( nth_list_a @ Xs @ I3 )
= ( nth_list_a @ Xs @ J2 ) )
= ( I3 = J2 ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_207_nth__eq__iff__index__eq,axiom,
! [Xs: list_nat,I3: nat,J2: nat] :
( ( distinct_nat @ Xs )
=> ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ Xs @ I3 )
= ( nth_nat @ Xs @ J2 ) )
= ( I3 = J2 ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_208_distinct__conv__nth,axiom,
( distinct_a
= ( ^ [Xs2: list_a] :
! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs2 ) )
=> ! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_a @ Xs2 ) )
=> ( ( I2 != J )
=> ( ( nth_a @ Xs2 @ I2 )
!= ( nth_a @ Xs2 @ J ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_209_distinct__conv__nth,axiom,
( distinct_list_a
= ( ^ [Xs2: list_list_a] :
! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s1427607542list_a @ Xs2 ) )
=> ! [J: nat] :
( ( ord_less_nat @ J @ ( size_s1427607542list_a @ Xs2 ) )
=> ( ( I2 != J )
=> ( ( nth_list_a @ Xs2 @ I2 )
!= ( nth_list_a @ Xs2 @ J ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_210_distinct__conv__nth,axiom,
( distinct_nat
= ( ^ [Xs2: list_nat] :
! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
=> ! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ( ( I2 != J )
=> ( ( nth_nat @ Xs2 @ I2 )
!= ( nth_nat @ Xs2 @ J ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_211_all__set__conv__all__nth,axiom,
! [Xs: list_a,P: a > $o] :
( ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( P @ X3 ) ) )
= ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( P @ ( nth_a @ Xs @ I2 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_212_all__set__conv__all__nth,axiom,
! [Xs: list_list_a,P: list_a > $o] :
( ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
=> ( P @ X3 ) ) )
= ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s1427607542list_a @ Xs ) )
=> ( P @ ( nth_list_a @ Xs @ I2 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_213_all__set__conv__all__nth,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( P @ X3 ) ) )
= ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I2 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_214_all__nth__imp__all__set,axiom,
! [Xs: list_a,P: a > $o,X2: a] :
( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( P @ ( nth_a @ Xs @ I ) ) )
=> ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_215_all__nth__imp__all__set,axiom,
! [Xs: list_list_a,P: list_a > $o,X2: list_a] :
( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s1427607542list_a @ Xs ) )
=> ( P @ ( nth_list_a @ Xs @ I ) ) )
=> ( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_216_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P: nat > $o,X2: nat] :
( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I ) ) )
=> ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_217_in__set__conv__nth,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
& ( ( nth_a @ Xs @ I2 )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_218_in__set__conv__nth,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s1427607542list_a @ Xs ) )
& ( ( nth_list_a @ Xs @ I2 )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_219_in__set__conv__nth,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I2 )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_220_list__ball__nth,axiom,
! [N: nat,Xs: list_a,P: a > $o] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ! [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
=> ( P @ X5 ) )
=> ( P @ ( nth_a @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_221_list__ball__nth,axiom,
! [N: nat,Xs: list_list_a,P: list_a > $o] :
( ( ord_less_nat @ N @ ( size_s1427607542list_a @ Xs ) )
=> ( ! [X5: list_a] :
( ( member_list_a @ X5 @ ( set_list_a2 @ Xs ) )
=> ( P @ X5 ) )
=> ( P @ ( nth_list_a @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_222_list__ball__nth,axiom,
! [N: nat,Xs: list_nat,P: nat > $o] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
=> ( P @ X5 ) )
=> ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_223_nth__mem,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( member_a @ ( nth_a @ Xs @ N ) @ ( set_a2 @ Xs ) ) ) ).
% nth_mem
thf(fact_224_nth__mem,axiom,
! [N: nat,Xs: list_list_a] :
( ( ord_less_nat @ N @ ( size_s1427607542list_a @ Xs ) )
=> ( member_list_a @ ( nth_list_a @ Xs @ N ) @ ( set_list_a2 @ Xs ) ) ) ).
% nth_mem
thf(fact_225_nth__mem,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_226_set__update__memI,axiom,
! [N: nat,Xs: list_a,X2: a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( member_a @ X2 @ ( set_a2 @ ( list_update_a @ Xs @ N @ X2 ) ) ) ) ).
% set_update_memI
thf(fact_227_set__update__memI,axiom,
! [N: nat,Xs: list_list_a,X2: list_a] :
( ( ord_less_nat @ N @ ( size_s1427607542list_a @ Xs ) )
=> ( member_list_a @ X2 @ ( set_list_a2 @ ( list_update_list_a @ Xs @ N @ X2 ) ) ) ) ).
% set_update_memI
thf(fact_228_set__update__memI,axiom,
! [N: nat,Xs: list_nat,X2: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ X2 @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X2 ) ) ) ) ).
% set_update_memI
thf(fact_229_f__image__eq__set,axiom,
! [Xs: list_a,A: set_nat] :
( ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( member_nat @ N2 @ A ) )
=> ( ( f_image_a @ Xs @ A )
= ( set_a2 @ Xs ) ) ) ).
% f_image_eq_set
thf(fact_230_f__image__eq__set,axiom,
! [Xs: list_list_a,A: set_nat] :
( ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_s1427607542list_a @ Xs ) )
=> ( member_nat @ N2 @ A ) )
=> ( ( f_image_list_a @ Xs @ A )
= ( set_list_a2 @ Xs ) ) ) ).
% f_image_eq_set
thf(fact_231_f__image__eq__set,axiom,
! [Xs: list_nat,A: set_nat] :
( ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ N2 @ A ) )
=> ( ( f_image_nat @ Xs @ A )
= ( set_nat2 @ Xs ) ) ) ).
% f_image_eq_set
thf(fact_232_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_233_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B2 ) )
=> ? [X5: nat] :
( ( P @ X5 )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X5 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_234_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_235_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_236_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_237_le__trans,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K )
=> ( ord_less_eq_nat @ I3 @ K ) ) ) ).
% le_trans
thf(fact_238_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_239_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_240_dual__order_Oeq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : Y = Z )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_241_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_242_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_243_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_244_order__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_245_order__class_Oorder_Oantisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% order_class.order.antisym
thf(fact_246_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_247_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_248_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : Y = Z )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_249_antisym__conv,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv
thf(fact_250_le__cases3,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_251_order_Otrans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_252_le__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% le_cases
thf(fact_253_eq__refl,axiom,
! [X2: nat,Y2: nat] :
( ( X2 = Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% eq_refl
thf(fact_254_linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linear
thf(fact_255_antisym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% antisym
thf(fact_256_eq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : Y = Z )
= ( ^ [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_257_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_258_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_259_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_260_order__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_261_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_262_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_263_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_264_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_265_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_266_neqE,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% neqE
thf(fact_267_neq__iff,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
= ( ( ord_less_nat @ X2 @ Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% neq_iff
thf(fact_268_order_Oasym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order.asym
thf(fact_269_less__imp__neq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_270_less__asym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% less_asym
thf(fact_271_less__asym_H,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% less_asym'
thf(fact_272_less__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% less_trans
thf(fact_273_less__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ).
% less_linear
thf(fact_274_less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% less_irrefl
thf(fact_275_ord__eq__less__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_276_ord__less__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_277_dual__order_Oasym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ~ ( ord_less_nat @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_278_less__imp__not__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_not_eq
thf(fact_279_less__not__sym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% less_not_sym
thf(fact_280_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X5: nat] :
( ! [Y3: nat] :
( ( ord_less_nat @ Y3 @ X5 )
=> ( P @ Y3 ) )
=> ( P @ X5 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_281_antisym__conv3,axiom,
! [Y2: nat,X2: nat] :
( ~ ( ord_less_nat @ Y2 @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_282_less__imp__not__eq2,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% less_imp_not_eq2
thf(fact_283_less__imp__triv,axiom,
! [X2: nat,Y2: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ X2 )
=> P ) ) ).
% less_imp_triv
thf(fact_284_linorder__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_285_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_286_order_Ostrict__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_287_less__imp__not__less,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% less_imp_not_less
thf(fact_288_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X: nat] : ( P3 @ X ) )
= ( ^ [P2: nat > $o] :
? [N3: nat] :
( ( P2 @ N3 )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ~ ( P2 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_289_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_290_dual__order_Ostrict__trans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_291_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( ( ord_less_nat @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_292_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_293_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_294_order_Onot__eq__order__implies__strict,axiom,
! [A2: nat,B2: nat] :
( ( A2 != B2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_295_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_296_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_297_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_298_order_Ostrict__implies__order,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_299_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_300_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_301_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_302_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_303_order_Ostrict__trans2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_304_order_Ostrict__trans1,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_305_not__le__imp__less,axiom,
! [Y2: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ord_less_nat @ X2 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_306_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_307_le__imp__less__or__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% le_imp_less_or_eq
thf(fact_308_le__less__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ).
% le_less_linear
thf(fact_309_less__le__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% less_le_trans
thf(fact_310_le__less__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% le_less_trans
thf(fact_311_less__imp__le,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% less_imp_le
thf(fact_312_antisym__conv2,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_313_antisym__conv1,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_314_le__neq__trans,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% le_neq_trans
thf(fact_315_not__less,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% not_less
thf(fact_316_not__le,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y2 ) )
= ( ord_less_nat @ Y2 @ X2 ) ) ).
% not_le
thf(fact_317_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_318_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_319_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_320_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_321_less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
& ( X3 != Y5 ) ) ) ) ).
% less_le
thf(fact_322_le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
| ( X3 = Y5 ) ) ) ) ).
% le_less
thf(fact_323_leI,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% leI
thf(fact_324_leD,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y2 ) ) ).
% leD
thf(fact_325_div__gr__imp__gr__divisor,axiom,
! [X2: nat,N: nat,M: nat] :
( ( ord_less_nat @ X2 @ ( divide_divide_nat @ N @ M ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% div_gr_imp_gr_divisor
thf(fact_326_div__le__mono,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_327_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_328_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
=> ( ! [I5: nat] :
( ( ord_less_nat @ K3 @ I5 )
=> ( P @ I5 ) )
=> ( P @ K3 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_329_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X6 ) ) ).
% minf(8)
thf(fact_330_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ~ ( ord_less_nat @ T @ X6 ) ) ).
% minf(7)
thf(fact_331_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ord_less_nat @ X6 @ T ) ) ).
% minf(5)
thf(fact_332_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_333_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_334_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_335_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_336_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ord_less_nat @ T @ X6 ) ) ).
% pinf(7)
thf(fact_337_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ~ ( ord_less_nat @ X6 @ T ) ) ).
% pinf(5)
thf(fact_338_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_339_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_340_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_341_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_342_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T ) ) ).
% pinf(6)
thf(fact_343_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ord_less_eq_nat @ T @ X6 ) ) ).
% pinf(8)
thf(fact_344_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ord_less_eq_nat @ X6 @ T ) ) ).
% minf(6)
thf(fact_345_complete__interval,axiom,
! [A2: nat,B2: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
& ( ord_less_eq_nat @ C2 @ B2 )
& ! [X6: nat] :
( ( ( ord_less_eq_nat @ A2 @ X6 )
& ( ord_less_nat @ X6 @ C2 ) )
=> ( P @ X6 ) )
& ! [D: nat] :
( ! [X5: nat] :
( ( ( ord_less_eq_nat @ A2 @ X5 )
& ( ord_less_nat @ X5 @ D ) )
=> ( P @ X5 ) )
=> ( ord_less_eq_nat @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_346_le__greater__neq__conv,axiom,
( ord_less_eq_nat
= ( ^ [N3: nat,A3: nat] :
! [X3: nat] :
( ( ord_less_nat @ A3 @ X3 )
=> ( N3 != X3 ) ) ) ) ).
% le_greater_neq_conv
thf(fact_347_ge__less__neq__conv,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,N3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ A3 )
=> ( N3 != X3 ) ) ) ) ).
% ge_less_neq_conv
% Conjectures (2)
thf(conj_0,hypothesis,
ord_less_nat @ m @ ( divide_divide_nat @ ( size_size_list_a @ xs ) @ k ) ).
thf(conj_1,conjecture,
( ( size_size_list_a @ ( nth_list_a @ ( listSl1174287072ice2_a @ xs @ k ) @ m ) )
= k ) ).
%------------------------------------------------------------------------------